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+{"text":"\\section{Introduction}\nThe interaction of protostellar outflows with the ambient molecular cloud occurs through radiative\nshocks that compress and heat the gas, which in turn cools down through line emission at\ndifferent wavelengths. In the dense medium where the\nstill very embedded protostars (the so called class 0 sources) are located, shocks are primarily non-dissociative, and hence the cooling is mainly through emission from \nabundant molecules. Molecular hydrogen is by far the most abundant species in these environments, \nand although H$_2$\\, emits only through quadrupole transitions with low radiative rates,\nit represents the main gas coolant in  flows from young protostars. H$_2$\\, shocked emission in outflows has been widely studied in the past mainly through its\nro-vibrational emission in the near-IR (e.g. Eisl{\\\"o}ffel et al. 2000; Giannini et al. 2004; Caratti o Garatti et al. 2006) that traces the dense\ngas at T$\\sim$2000-4000 K. Most of the thermal energy associated with the shocks is however radiated away through\nthe emission of H$_2$\\ rotational transitions of the ground state vibrational level at $\\lambda \\le 28\\mu$m\n(e.g. Kaufman \\& Neufeld 1996). \nMid-IR H$_2$\\, lines are easily excited at low densities and temperatures between 300 and 1500 K: therefore they\nare very good tracers of the molecular shocks associated with the acceleration of ambient gas by\nmatter ejection from the protostar. Given the low excitation temperature, \nthey can also probe regions where H$_2$\\, has not yet reached the ortho-to-para equilibrium, thus \ngiving information on the thermal history of the shocked gas (Neufeld et al. 1998; Wilgenbus et al. 2000).\nIn addition, given the different excitation temperature and critical densities of the v=0--0 and v$\\ge$ 1 H$_2$\\ lines,\nthe combination of mid-IR with near-IR observations is a very powerful tool to constrain \nthe global physical structure and the shock conditions giving rise to the observed emission.\n\nThe study of the 0--0 rotational emission in outflows started in some detail with the \n\\emph{Infrared Space Observatory}. Thanks to the observations performed with the SWS and ISOCAM instruments,\nthe shock conditions, the ortho-para ratio and the global H$_2$\\, cooling  have been derived in a handful of \nflows (e.g. Neufeld et al. 1998; Nisini et al. 2000; Molinari et al. 2000;  Lefloch et al. 2003). More recently, the \\emph{Infrared Spectrometer} (Houck et al. 2004) on board\n\\emph{Spitzer}, with its enhanced spatial resolution and sensitivity with respect to the ISO spectrometers, \nhas been used to obtain detailed images of the H$_2$\\, rotational emission, from S(0) to S(7), of several\noutflows, from which maps of  important physical parameters, such as temperature, column density and o\/p\nratio, have been constructed (Neufeld et al. 2006; Maret et al. 2009;  Dionatos et al. 2010). \nIn this framework, Neufeld et al. (2009, hereafter N09)  have recently presented IRS spectroscopic maps observations of five young \nprotostellar outflows at wavelengths between 5.2 and 37$\\mu$m, and discussed their averaged physical properties \nand overall energetics. In all the flows, the H$_2$\\, S(0)-S(7) emission has been detected and contributes to more than 95\\%  of the \ntotal line luminosity in the 5.2-37$\\mu$m\\, range, while atomic emission, in the form  of FeII and SI fine structure lines, accounts for only the remaining $\\sim$5\\%. \n\nIn the present paper, we will analyse the H$_2$\\, line maps obtained by N09 towards the L1157 outflow,\nwith the aim of deriving the main physical conditions pertaining to the molecular gas and their variations \nwithin the flow. This in turn will give information on the thermal history of the flow and on how \nenergy is progressively transferred from the primary ejection event to the slow moving ambient gas. \n\nFor this first detailed analysis, L1157 has been chosen among the sources observed by N09 given its uniqueness as a very active and well studied flow at different wavelengths.   \nMore than 20 different chemical species have been indeed \ndetected in the shocked spots of this object (Bachiller \\& Perez Gutierrez 1997; Benedettini et al. 2007), some of them for the first time in outflows (e.g. HNCO, Rodr{\\'{\\i}}guez-Fern{\\'a}ndez et al. 2010, and complex organic molecules, Arce et al. 2008, Codella et al. 2009)\n, testifying for a rich shock induced chemistry. Warm H$_2$\\ shocked emission in L1157\nis also evidenced through near-IR maps (e.g. Davis \\& Eisl\\\"offel, 1995) and Spitzer-IRAC\nimages (Looney et al. 2007). The L1157\noutflow has been also recently investigated with the \\textit{Herschel Space Observatory}, showing\nto be very strong also at far-IR wavelengths (Codella et al. 2010, Lefloch et al. 2010,\nNisini et al. 2010).\n\nThe L1157 outflow extends about 0.7 pc in length. Its distance is uncertain and has been estimated between 250 and 440 pc. Here we will adopt D=440 pc for an easier comparison with other works.\nThe outflow is driven by a highly embedded, low mass class 0 source (L1157-mm or IRAS20386+6751) having $L_{bol} \\sim $ 8.3 $L_\\odot$ (Froebrich 2005). \nIt is a very nice example of an outflow driven by a precessing and pulsed jet, possessing an S-shaped structure and different cavities, whose morphology has been reproduced assuming that the outflow\nis inclined by $\\sim$80$^\\circ$ to the line of sight and the axis of the underlying jet precesses\non a cone of 6$^\\circ$ opening angle (Gueth et al. 1996). The episodic mass ejection events are\nevidenced by the presence, along the flow, of individual clumps that are symmetrically displaced \nwith respect to the central source.\nIt is therefore a very interesting target for a  study of the physical conditions pertaining to these active regions through an H$_2$\\ excitation analysis.\n\nThe paper is organized as follow: the observations and the main results are summarized in \\S 2. In \\S 3 \nwe describe the analysis performed on the H$_2$\\ images to derive maps of temperature, column density and \northo-to-para ratio. A more detailed NLTE analysis on individual emission peaks is also presented here, where the \nSpitzer data are combined with near-IR data to further constrain the excitation conditions.  The implications of these results for the shock conditions along the L1157  flow are discussed in \\S 4, together with an analysis of the global\nenergy budget in the flow.  A brief summary follows in $\\S 5$.\n\n\\section{Observations and results \\label{analysis}}\n\nObservations of the L1157 outflow were obtained in November 2007 with the IRS instrument, during Cycle 4 of the Spitzer mission . \nThe full IRS spectral range (5.2-36.5$\\mu$m) was observed with the Long-High (LH), Short-High (SH) (R $\\sim$ 600) \nand Short-Low (SL) (R between 64 and 128) modules. \nThe L1157 outflow region was covered through 5  individual IRS maps of $\\sim$ 1\\arcmin x1\\arcmin\\,  of size each, arranged along the outflow axis. Each map was obtained by stepping the IRS slit by half of its width in the direction perpendicular\nto the slit length. For the SH and LH modules the slit was stepped also parallel to its axis by 4\/5 (SL) and 1\/5 (LH)  of its length. \nDetails on the data reduction that generated the individual line maps from the IRS scans are given in N09. The final maps have been resampled to a grid of 2\\arcsec\\, spacing allowing a pixel by pixel comparison of maps obtained with the different IRS modules.\nMaps of the brightest detected lines as well as the full spectrum in a representative position are shown in  Fig.\\,7 and Fig.\\,12 of N09. As regards to H$_2$, all the pure rotational lines of the first vibrational levels, from S(0) to S(7),\n are detected at various intensity along the flow. Here we report, in Tab. \\ref{fluxes},  the H$_2$\\, brightness measured in a 20\\arcsec\\, FWHM Gaussian aperture towards different positions.\n\n Fig\\,1  shows the L1157 maps of the S(1) and S(2) lines while Fig.\\,2 displays the S(5) line with superimposed contours of the CO 2--1 emission from Bachiller et al. 2001.\n In the same figure, a map of the 2.12 $\\mu$m 1--0 S(1) line is also presented. The morphology of the 0--0 S(5) and 1--0 S(1) is very similar, with  peaks of mid-IR emission \nlocated at the near-IR knots from A to D, as identified by Davis \\& Eisl\\\"offel (1995). \n When compared with the CO map, the mid-IR H$_2$\\, emission appears to follow the curved chain of clumps (labelled as B0-B1-B2 and R0-R1-R, for the blue-shifted and red-shifted lobes, respectively) that also correspond to peaks of SiO emission, as resolved in interferometric observation by Gueth et al. (1998) and Zhang et al. (2000). \n The L1157 outflow  morphology has been suggested to delineate a precessing flow (Gueth et al. 1996), where the H$_2$\\, and SiO peak emission knots follow  the location of the actual working surface of the precessing jet and are thus associated with the youngest ejection episodes. \nDiffuse H$_2$\\ emission is also detected in the S(1)-S(2) maps, that delineates the wall of a cavity that connects the central source with both\nthe R0 and B0 clumps. Such a cavity has been recognized in the CO 1-0 interferometric maps and it is likely created by the propagation of large bow-shocks.\nThe S(1)-S(2) maps of Fig.\\,1 show  extended emission of H$_2$\\,  also in the SE direction (i.e. where the B2 clump is located) and in the eastern edge of the northern lobe, that also follow  quite closely the CO morphology: these regions at lower excitation might trace additional cavities created by an older ejection episode of the precessing jet.\n\n\\section{ H$_2$ Analysis }\n\\subsection{LTE 2D analysis of the rotational lines: maps of averaged parameters }\n\\label{sec:maps}\nWe have used the  H$_2$\\, line maps to obtain the 2D distribution of basic H$_2$\\, physical parameters, through the analysis \nof the rotational diagrams in each individual pixel. As described in N09, who analysed the global \nH$_2$\\, excitation conditions in L1157, the distribution of upper level column densities of the S(0)-S(7) lines as a function of their \nexcitation energy, does not follow a straight line, indicating that a  temperature stratification in the \nobserved medium exists. The exact form of this temperature stratification depends on the type of shock\nthe H$_2$\\, lines are tracing, as they probe the post-shock regions where the gas cools from $\\sim  $ 1000 K \nto $ \\sim $ 200 K. \n\nThe simplest way to parametrize the post-shock temperature stratification\nis to assume a power-law distribution: this approach was applied by \nNeufeld \\& Yuan (2008), which also show that this type of distribution is expected  \nin gas ahead of unresolved bow-shocks. On this basis, and following also N09, we fit the observations \nassuming a slab of gas where the H$_2$\\, column density in each layer at a given $T$ varies as \n$ dN \\propto T^{-\\beta}dT$.\n\nThis law is integrated, to find the total column density, between a minimum ($T_{min}  $) and a maximum ($T_{max}$) temperatures . For our calculations \nwe have kept $T_{max}$ fixed at 4000 K, since gas at temperatures larger than this value is not expected to contribute to the emission of the observed pure rotational lines. $T_{min} $ was instead assumed to be equal, in each position, to the minimum temperature probed by the observed lines. This $T_{min} $ is taken as the excitation temperature giving\nrise to the observed ratio of the S(0) and S(1) column densities, assuming a Boltzman\ndistribution. \nThe $T_{min}  $ value ranges between $\\sim$150 and 400 K.\n\nWe found that the approach of a variable  $T_{min}$\nproduces always fits with a better $\\chi^2$ than assuming a fixed low value in all positions.\nWe also assume the gas is in LTE conditions. Critical densities of rotational lines from S(0) to S(7)\nrange between 4.9 cm$^{-3}$\\, (S(0)) and 4.4$\\times$10$^5$cm$^{-3}$(S(7)) at T=1000 K assuming only H$_2$\\, collisions\n(Le Bourlot et~al. 1999): critical densities decrease if collisions with H and He are not negligible. \nDeviations from LTE can be therefore expected only for the high-$J$ S(6) and S(7) transitions: the S\/N of these \ntransitions in the individual pixels is however not high enough to disentangle, in the rotational diagrams, NLTE \neffects from the effects caused by the variations of the other considered parameters. In particular, as also\ndiscussed in  N09, there is a certain degree of degeneracy in the density and the $\\beta$ parameter \nof the temperature power law in a NLTE treatment that we are not able to remove in the analysis of the\nindividual pixels. This issue will be further discussed in \\S \\ref{sec:NLTE}. \nAn additional parameter of our fit is the ortho-to-para ratio (OPR) value. It is indeed recognized that the 0--0 H$_2$\\, lines are\noften far from being in ortho-to-para equilibrium, an effect that in a rotational diagram is evidenced by\na characteristic zigzag behavior in which column densities of lines with even-$J$ lie systematically above those of odd-$J$ lines. In order not to introduce too many parameters, we assume a single OPR value as a free parameter for the\nfit. In reality, the OPR value is temperature dependent (e.g. N09 and  \\S \\ref{sec:NLTE}), and therefore\nthe high-$J$ lines  might present an OPR value closer to equilibrium then the low-$J$ transitions. \nOur fit gives therefore only a value averaged over the temperature range probed by the lines considered (i.e. $\\sim$200-1500 K). \n\nIn summary, we have varied only three parameters, namely the total H$_2$\\, column density N(H$_2$), the OPR, and the temperature power law index $\\beta$,  in order to obtain the best model fit through a $\\chi^2$ minimization procedure and assuming a 20$\\%$ flux uncertainty for all the lines. \nThe fit was performed only in those pixels where at least four lines with an S\/N larger\nthan 3 have been detected.  \nBefore performing the fit, the line column densities were corrected for extinction assuming $A_v$ = 2 (Caratti o Garatti et al. 2006) and \nadopting the Rieke \\& Lebofsky (1985) extinction law. At the considered wavelengths,\nvariations of $A_v$ of the order of 1-2mag do not affect any of the derived results.\n\nFigure \\ref{fig:ncol} shows the derived map of the H$_2$\\, column density, while in Fig. \\ref{fig:b_tmin} and \\ref{fig:op_tcold}  \nmaps of the OPR and  $\\beta$ are displayed together with temperature maps relative to the ''cold'' and ''warm'' components ($T_{cold}$ and $T_{warm}$),\ni.e. the temperature derived from linearly fitting the S(0)-S(1)-S(2) and S(5)-S(6)-S(7) lines, once corrected for the derived OPR value. \nIn Fig. 6 we also show the individual excitation diagrams for selected positions along the flow, obtained from intensities\nmeasured in a 20$\"$ FWHM Gaussian aperture centered towards emission peaks (Tab. 1). Values of the\nfitted parameters in these positions are reported in Tab. \\ref{param}. In addition to $T_{cold}$ and $T_{warm}$,\nwe give in this table also the values of the average temperature in each knot, derived through a\nlinear fit through all the H$_2$\\ lines ($T_{med}$).\n\nThe maps show significant variations in the inferred parameters along the outflow. \nThe H$_2$\\ column density ranges between 5$\\times$10$ ^{19} $ and \n3$\\times$10$ ^{20} $cm$^{-3}$. The region at the highest column density is located towards the B1 molecular bullet (see Fig. \\ref{fig:h2co})\\footnote{Fig. \\ref{fig:h2co} shows that the molecular clumps B1, R0 and\nR coincide in position with the NIR H$_2$\\ knots A, C and D. In the paper, both nomenclature\nwill be used specifying if we refer to the NIR or mm condensations}. \nThis is consistent with the higher column density of CO found in B1 with respect \nto other positions in the blue lobe (Bachiller \\& Perez Gutierrez 1997), and might suggest that this is a zone where the outflowing gas is compressed due \nto the impact with a region of higher density (Nisini et al. 2007). Towards the NW, red-shifted \noutflow, the column density has a more uniform distribution, with a plateau at $ \\sim 10 ^{20} $ cm$^{-2}$\\ that follows the H$_2$\\ intensity distribution. \nThe N(H$_2$) decreases at the apex of the red-shifted outflow, with a value slightly below $ 10 ^{20} $ cm$^{-2}$\\ at the position\nof the D near-IR knot.\n\n$T_{cold}$ ranges between $ \\sim $ 250 and 550 K. The highest values are found at the tip of the northern outflow lobe, while local maxima corresponds to the positions of line intensity peaks.\n$T_{warm}$ ranges between  $ \\sim $ 1000 and 1500 K. In this case the highest values are in the southern lobe, at the position of the A NIR knot. \nAs a general trend, the $T_{warm}$ value decreases going from the southern to the northern peaks of emission, with\nthe minimum value at the position of the D NIR knot. \n\n$\\beta$ values range between $ \\sim $4-4.5 in the blue-shifted lobe while it is larger in the red-shifted\nlobe, with maximum values of $ \\sim $ 5.5 at the tip of the flow. \nDue to the degeneracy between $\\beta$ and density discussed in the previous section, these\nvalues can be considered as upper limits because of our assumption of LTE conditions. \nNeufeld \\& Yuan (2008) have discussed the $\\beta$ index expectations in bow-shock excitation.\nA $\\beta$ index of $ \\sim $ 3.8 is expected in paraboloid bow shocks having a velocity at the bow apex\nhigh enough to dissociate H$_2$,  in which case the temperature distribution\nextends to the maximum allowed temperature. Slower shocks that are not able to attain the maximum\ntemperature, produce steeper temperature distributions, i.e. with values of $ \\beta $ greater than 3.8.\nThis is consistent with our findings: low values of $ \\beta $ (of the order of 4) are found in the blue-shifted\nlobe: here, evidence of H$_2$\\ dissociation is given by the detection\nof atomic lines (i.e. [SiII], [FeII] and [SI]) in the IRS spectrum. In the red-shifted lobe, where values of $\\beta$ larger than 4 are derived in the LTE\nassumption, no atomic emission is detected and the $T_{warm}$ values\nare lower than those measured in the blue-shifted lobe, indicating a maximum temperature lower than in the blue flow .\n\nThe OPR varies significantly along the outflow, spanning from $ \\sim $ 0.6 to 2.8. Hence it is always below the equilibrium value of 3.  Although a one-to-one correlation between temperature and OPR cannot be discerned, some trends can be\ninferred from inspection of Fig. \\ref{fig:op_tcold}. In general the OPR minima are observed in plateau regions between two consecutive\nintensity peaks, where also the cold temperature has its lowest values. At variance with this trend, the emission\nfilaments delineating the outflow cavity within $\\pm$ 20$ \\arcsec  $ from the mm source, where the cold \ntemperature reaches a minimum value of $ \\sim $ 250 K,   show rather high\nvalues of the OPR, $ \\sim $2.4-2.8. This might suggests that this region has experienced an older \nshock event that has raised the OPR, though not to the equilibrium value, and where the gas\nhad time to cool at a temperature close to the pre-shock gas temperature. \nOn the other hand, at the apex of both  the blue- and red-shifted lobes, where the cold temperatures \nare relatively high (i.e. 500-550 K), the OPR is rather low, $ 1.5-2.0 $. Evidence of regions of low OPR and high \ntemperatures at the outflow tips has been already given in other flows (Neufeld et al. 2006; Maret et al. 2009). It has been suggested that these represent zones subject to recent shocks where the OPR has not had time yet to reach the\nequilibrium value. \n\n\n\\subsection{ NLTE analysis: constraints on H and H$_2$ particle density}\n\\label{sec:NLTE}\n\nAdditional constraints on the physical conditions responsible for the H$_2$\\, excitation, are provided by combining\nthe emission of the mid-IR H$_2$\\ pure-rotational lines from the ground vibrational level with the emission from near-IR ro-vibrational lines.\nIt can be seen from Fig. \\ref{fig:h2co} that the 2.12$\\mu$m\\, emission follows quite closely the\nemission of the 0--0 lines at higher $J$. In addition to the 2.12$\\mu$m\\, data presented in Fig. \\ref{fig:h2co}, we have also considered\nthe NIR long-slit spectra obtained on the A and C NIR knots by Caratti o Garatti (2006). These knots, at the spatial\nresolution of the NIR observations (i.e. $\\sim$0.8\\arcsec), are separated in several different sub-structures that have been individually investigated with the long-slit spectroscopic observations. For our analysis we have considered the\ndata obtained on the brightest of the sub-structures,  that coincide, in position, \nwith peaks of the 1--0 S(1)  line.\n\nIn order to inter-calibrate in flux the Spitzer data with these NIR long-slit data, obtained  with a slit-width of 0.5 arcsec, \nwe have proceeded as follows: we first convolved the \n2.12$\\mu$m\\, image at the resolution of the Spitzer images and than performed photometry on the\nA and C peaks positions with a 20\\arcsec\\, diameter Gaussian aperture, i.e. with the same\naperture adopted for the brightness given in Tab. \\ref{fluxes}. We have then scaled the fluxes of the individual lines given in \nCaratti o Garatti (2006) in order to match the 2.12$\\mu$m\\ flux gathered in the slit with that measured by the\nimage photometry. In doing this, we assumed that the average excitation conditions within the 20$\\arcsec$ aperture are\nnot very different from those of the A-C peaks. This assumption is \nobservationally supported by the fact that the ratios of different H$_2$\\ NIR lines \ndo not change significantly (i.e. less than 20\\%) in the A-C knot substructures separately\ninvestigated in Caratti o Garatti (2006).\nWe have considered only those lines detected with S\/N larger \nthan 5; in practice this means considering lines from the first four and three vibrational levels for knots A and C, respectively. \nThe excitation diagrams obtained by combining the Spitzer and NIR data for these two knots are displayed in Fig.\\ref{fig:ACfit}.\n\nIn order to model together the 0--0 lines and the near-IR ro-vibrational lines, we have implemented\n two  modifications to the approach adopted previously. First of all, the NIR lines probe gas\nat temperatures higher than the pure rotational lines, of few thousands of K, at which values it is expected\nthat the OPR has already reached equilibrium. Thus, \nthe ortho-para conversion time as a function of temperature needs to be included in the\nfitting procedure, since lines excited at different temperatures have different OPRs.\nWe  have here adopted the approach of N09 and used an analytical expression for the OPR as a function of the temperature, considering a gas that had an initial value of the ortho-to-para ratio OPR$_0  $ and has been heated to a temperature T for a time $  \\tau$. Assuming that the para-to-ortho conversion occurs through reactive collisions with atomic hydrogen, we have:\n\n\\begin{equation}\n\\frac{OPR(\\tau)}{1+OPR(\\tau)}  = \\frac{OPR_0}{1+OPR_0}\\,e^{-n(H)k\\tau} + {\\frac{OPR_{LTE}}{1+OPR_{LTE}}}\\,\\left(  1 - e^{-n(H)k\\tau}\\right) \n\\end{equation}\n\nIn this expression, n(H) is the number density of atomic hydrogen and OPR$_{LTE}$ is the ortho-to-para ratio equlibrium value.\nThe parameter $ k $ is given by the sum of the rates coefficients for para-to-ortho conversion ($ k_{po} $), estimated\nas 8$ \\times $10$ ^{-11} $exp(-3900\/T) cm$ ^{3} $\\,s$ ^{-1} $, and for ortho-to-para conversion, $ k_{op} \\sim k_{po}\/3 $\n(Schofield et al. 1967). Thus the dependence of the OPR on the temperature is implicitly given by the dependence on T of the\n$ k $ coefficient. The inclusion of a function of OPR on $T$, introduces one additional parameter to our fit: while we have previously \nconsidered only the average OPR of the 0--0 lines, we will fit here the initial OPR$_0  $ value and the coefficient $ K = n(H)\\tau $.\n\nThe second important change that we have introduced with respect to the previous fitting procedure, \nis to include a NLTE treatment of the H$_2$\\, level column densities. In fact, the critical densities\nof the NIR lines are much higher than those of the pure rotational lines (see Le Bourlot et~al. 1999). For example, the \n$n _{crit} $ of the 1-0 S(1) 2.12$\\mu$m\\, line is 10$^7$cm$^{-3}$\\, assuming only collisions with H$_2$\\, and T=2000 K. \nTherefore the previously adopted LTE approximation might not be valid when combining lines from different \nvibrational levels.\nThis is illustrated in Fig.\\ref{fig:plot_dens}, where we plot the results obtained by varying the H$_2$\\ density between \n10$ ^{3} $ and 10$ ^{7} $cm$^{-3}$\\, while keeping the other model parameters fixed . The observed column densities in the A\nposition are displayed for comparison. \nFor the NLTE statistical equilibrium computation we have adopted the H$_2$\\, collisional rate coefficients given by Le Bourlot et~al. (1999) \\footnote{The  rate coefficients for collisions with ortho- and para-H$_2$, HI and He, computed in Le Bourlot et~al. (1999) \nare available at the web-site: http:\/\/ccp7.dur.ac.uk\/cooling$\\_$by$\\_$h2\/} .\nThis figure demonstrates the sensitivity of the relative ratios  between 0--0 and 1--0 transitions to density variations.   For example,  in this particular case, the ratio N(H$_2$)$_{0-0 S(7)}$\/N(H$_2$)$_{1-0 S(1)}$ is 64.6 at n(H$_2$)=10$ ^{4} $ cm$^{-3}$\\, and 1.9 at n(H$_2$) $ \\gtrsim $ 10$ ^{7} $ cm$^{-3}$. The figure also shows that the observational points display only a small misalignment in column densities between the 0--0 lines and the 1--0 lines, already indicating that the ro-vibrational lines are close to\nLTE conditions at high density.\n\nIn Fig.\\ref{fig:ACfit}, we show the final best-fit models for the combined mid- and near-IR column densities \nin the A and C positions. \nAs anticipated, the derived n(H$_2$) densities are large, of the order of 10$ ^{7} $ and 6$\\times$10$ ^{6}$ cm$^{-3}$,  for the A and C positions, respectively. The two positions indeed show very similar excitation conditions: only the column density is a factor of 3 smaller in knot C. \nHence, we conclude that the lack of detection of rotational lines with v$ > $ 3 in knot C, in contrast to knot A (Caratti o Garatti et al. 2006), is due\npurely to a smaller number of emitting molecules along the line of sight and not to different excitation conditions.\n\nThe derived H$_2$\\, densities are much higher than  previous estimates based on other tracers. Nisini et al. (2007)\nderive a density of 4$\\times$10$ ^{5} $ cm$^{-3}$\\, at the position of knot A from multi-line SiO observations, thus more than an order of magnitude smaller than those inferred from our analysis. The high-J CO lines observed along the blue-shifted lobe of L1157 by \nHirano et al. (2001), indicate a density even smaller, of the order of 4$\\times$10$ ^{4} $ cm$^{-3}$. \nSiO is synthesized and excited in a post-shock cooling zone where the maximum compression is reached, therefore\nit should trace post-shock regions at densities higher than H$_2$ (e.g. Gusdorf et al. 2008). \nOne possibility at the origin of the discrepancy is our assumption of collisions with only H$_2$\\ molecules, and thus of a \nnegligible abundance of H. This can be considered roughly true in the case of non-dissociative C-shocks, where H atoms are \nproduced primarely in the chemical reactions that form H$_2$O from O and H$_2$, with an abundance n(H)\/n(H$_2$+H) $ \\sim  $ 10$^{-3}$\n(e.g. Kaufmann \\& Neufeld 1996). However, if the shock is partially dissociative, the abundance of H can increase considerably and \ncollisions with atomic hydrogen cannot be neglected, in view of its  large efficiency in the H$_2$\\, excitation.\nThis situation cannot be excluded at least for the knot A, where atomic emission from [FeII] and [SI] has been \ndetected in our Spitzer observations.\nSince we cannot introduce the n(H) as an additional independent parameter of our fit, we have fixed n(H$_2$) at the value\nderived from SiO observations (4$\\times$10$ ^{5} $ cm$^{-3}$, Nisini et al. 2007) and varied the H\/H$_2$\\, abundance ratio. \nThe best fit is in this case obtained with a ratio H\/H$_2$=0.3: this indicates that our observational data are\nconsistent with previous H$_2$\\ density determinations only if a large fraction of the gas is in atomic form.\n\nTurning back to the inferred OPR variations with temperature, our fit implies that the OPR in the cold gas component at T=300 K\nis significantly below the equilibrium value, while the value of  \nOPR=3 is reached in the hot gas at T=2000 K traced by the NIR lines. The parameter  $K=n(H)\\tau$\nis constrained to be $\\sim$10$^6$ and 10$^7$ yr\\,cm$^{-3}$\\, for knots A and C, respectively. \nWe can also estimate the time needed for the gas to reach this distribution of OPR, from the limits on the\natomic hydrogen abundance previously discussed.\nOur data implies a high value of the n(H) density: a minimum value of n(H) $\\sim$0.6-1$\\times$10$^4$cm$^{-3}$, (for knots C and A, respectively) \n is given if we assume H\/H$_2$$\\sim$ 10$^{-3}$ (and thus the n(H$_2$) $\\sim$ 10$ ^{7} $cm$^{-3}$, given\n by our fit), while a maximum value of $\\sim$10$^5$cm$^{-3}$,\nis derived from the fit where n(H$_2$) is kept equal to 4$\\times$10$ ^{5} $ cm$^{-3}$.\nThe high abundance of atomic H ensures that conversion of para- to\northo-H$_2$\\, proceeds very rapidly: the fitted values of the K parameter indeed imply that the observed range of\nOPR as function of temperature have been attained in a timescale between 100 and 1000 yrs for both the knots.\n\nFinally, given the column density and particle density discussed above, we can estimate the H$_2$\\,\ncooling length ($L \\sim$ N(H$_2$)\/n(H$_2$)). If we consider the case of n(H$_2$) $\\sim$ 10$ ^{7} $cm$^{-3}$,\nand negligible n(H), we have $L \\sim$ 10$ ^{13} $ cm while a length of $\\sim$ 10$ ^{15} $ cm is inferred\nin the case of n(H$_2$)$\\sim$ 4$\\times$10$^5$cm$^{-3}$. \nAll the parameters derived from the above analysis are summarised in Tab. \\ref{shock} and they will be discussed\nin the next section in the framework of different shock models.\n\n\\section{Discussion}\n\n\\subsection{Shock conditions giving rise to the H$_2$\\, emission}\n\nThe copious H$_2$\\, emission at low excitation observed along the L1157 outflow \nindicates that the interaction of the flow with the ambient medium occurs\nprevalently through non-dissociative shocks. \nBoth the Spitzer IRS maps of N09, and the NIR narrow band images of Caratti o Garatti et al. 2006, show that significant gas dissociation in L1157 occurs only at the A peak, where both mid-IR \nlines from [FeII], [SII] and [SI] and weak [FeII] at 1.64$\\mu$m\\ have been detected. \nWeaker [SI]\\,25$\\mu$m\\ and [FeII]\\,26$\\mu$m\\ emission have been also detected on the C spot, but\noverall the atomic transitions give a negligible contribution to the total gas cooling,\nas pointed out in N09.\nThese considerations suggest that most of the shocks along the outflow occur at speeds\n below $\\sim$ 40km\\,s$^{-1}$,  as this is the velocity limit above which H$_2$\\ is expected to be dissociated.\nThe knot A is the only one showing a clear bow-shock structure. Here the velocity at\nthe bow apex is probably high, causing H$_2$\\ dissociation and atomic line excitation,\nconsistent with the fitted temperature power law $\\beta$ of $\\sim$ 4, as discussed in \\S 3.1,\nwhile the bulk of the H$_2$\\, emission comes from shocks at lower velocities originating in\nthe bow wings.\n\nConstraints on the shock velocity that gives rise to the molecular emission in L1157\n have been already given in previous works. The sub-mm SiO emission and\nabundances, measured in different outflow spots, suggest shock velocities of the \norder of 20-30km\\,s$^{-1}$\\ (Nisini et al. 2007). The comparison of SiO and H$_2$\\ \nemission against detailed shock models performed by Gusdorf et al. (2008) confirm a similar\nrange of velocities in the NIR-A knot, although the authors could not find a unique \nshock model that well represents both the emissions. \n\nCabrit et al. (1999) found that the column density of the mid-IR H$_2$\\, emission lines, \nfrom S(2) to S(8), observed by ISO-ISOCAM was consistent either with C-shocks having\nvelocities of $\\sim$\\,25km\\,s$^{-1}$\\, or with J-shocks at lower velocity, of the order\nof $\\sim$\\,10km\\,s$^{-1}$. Gusdorf et al. (2008), however, conclude\nthat stationary shock models, either of C- or J-type, are not able to reproduce the observed\nrotational diagram on the NIR-A position, constructed combining ISOCAM data and \nNIR vibrational lines emission. A better fit was obtained \nby these authors considering non-stationary shock models, which have developed a magnetic precursor but which retain a J-type discontinuity (the so-called CJ shocks, Flower et al. 2003). \nSimilar conclusions, but on a different outflow, have been reached by Giannini et al. 2006\nwho studied the H$_2$\\ mid- and near-IR emission in HH54: in general, stady-state C- and J-type \nshocks fail to reproduce simultaneously the column densities of both the ro-vibrational \nand the v=0, pure-rotational H$_2$\\ levels.  \n\nA different way to look at the issue of the prevailing shock conditions in the observed\nregions, is to compare the set of physical parameters that we have inferred from our analysis \nto those expected from different shocks. With this aim, we summarize in   \nTab. \\ref{shock} the physical properties derived on the A and C H$_2$\\, knots. \nIn addition to the parameters derived from the NLTE analysis \n reported in Section \\ref{analysis}, namely H$_2$ post-shock density, H\/H$_2$\\ fraction, \n initial OPR, cooling length and time, the table reports also the average values of \nOPR and rotational temperature, as they are measured from a simple linear fit\n of the rotational diagrams presented in Fig. \\ref{fig:fit_nir}.\n\n\nAs mentioned in \\S 3.2, the high fraction of atomic hydrogen inferred by our analysis rule out excitation in a pure\n C-shock. In fact, dissociation in C-shocks is always too low to have a \nH\/H$_2$ ratio higher than 5$\\times$10$^{-3}$, irrespective from the shock velocity and magnetic field strength \n(Kaufman \\& Neufeld 1996; Wilgenbus et al. 2000).\nC-shocks are not consistent with the derived parameters even if we consider the model fit with\nthe high H$_2$\\ post-shock density of the order of 10$^{7}$ cm$^{-3}$ and negligible atomic hydrogen: in this case we derive an emission length of 10$^{13}$ cm,\nwhich is much lower than the cooling length expected in C-shocks, which, although \ndecreasing with the pre-shock density, is never less than 10$^{15}$ cm (Neufeld et al. 2006) .\n\nStationary J-shock models better reproduce some of our derived parameters.\nFor example, in J-type shocks the fraction of hydrogen in the post-shocked gas\ncan reach the values of 0.1-0.3 we have inferred, provided that the shock velocity is larger\nthan $\\sim$ 20 km\\,s$^{-1}$. In general, a reasonable agreement with the inferred post-shock density and \nH\/H$_2$\\ ratio is achieved with models having $v_s$=20-25 km\\,s$^{-1}$ and pre-shock densities of 10$^3$cm$^{-3}$\n(Wilgenbus et al. 2000). Such models predict a shock flow time of the order of 100 yr or less, which\nis also in agreement with the value estimated in our analysis at least in knot A.\n In such models, however, the cooling length is an order of magnitude smaller than the \ninferred value of $ \\sim $10$ ^{15} $ cm. In addition, the gas temperature remains high for most of the post-shocked region: the average rotational temperature of \nthe v=0 vibrational level is predicted to be, according to the Wilgenbus et al. (2000)\ngrid of models, always about 1600 K or larger, as compared with the value of about 800-900 K inferred from observations. \nThe consequence of the above inconsistencies is that J-type shocks tend to underestimate \nthe column densities of the lowest H$_2$\\, rotational levels in L1157, an effect already pointed \nout by Gusdorf et al. (2008). \n\nAs mentioned before, Gusdorf et al. (2008) conclude that the H$_2$\\ pure rotational emission in L1157 \nis better fitted with a non-stationary C+J shock model with either $v_s$ between 20 and 25 km\\,s$^{-1}$\\ and pre-shock densities $n_H = 10^4$ cm$^{-3}$, or with $v_s \\sim 15$ km\\,s$^{-1}$\\, and\nhigher pre-shock densities of $n_H = 10^5$ cm$^{-3}$. Such models, however, still underestimate\nthe column densities of the near-IR transitions: the post-shocked H$_2$\\ gas density  \nremains lower than the NIR transitions critical density and the atomic hydrogen\nproduced from H$_2$\\ dissociation is not high enough to populate the vibrational\nlevels to equilibrium conditions.\n\nThe difficulty of finding a suitable single model that reproduce the derived physical \nconditions is likely related to possible geometrical effects and to the fact that multiple shocks with different velocities might be present along the line of sight. It would be indeed  interesting to explore whether bow-shock models might be able to predict the averaged physical characteristics \nalong the line of sight that we infer from our analysis.\n\n\\subsection{Flow energetics}\n\nH$_2$\\ emission represents one of the main contributor to the energy radiated\naway in shocks along outflows from very young stars. Kaufman \\& Neufeld (1996) predicted that\nbetween 40 and 70\\% of the total shock luminosity is emitted in H$_2$\\, lines\nfor shocks with pre-shock density lower than 10$^5$ cm$^{-3}$\\ and shock velocities larger\nthan 20 km\\,s$^{-1}$, the other main contributions being in CO and H$_2$O rotational emission. \nThis has been also observationally tested by Giannini et al. (2001) who measured the \nrelative contribution of the different species to the outflow cooling in a sample of class 0 \nobjects observed with ISO-LWS. \n\nWe will discuss here the role of the H$_2$\\, cooling in the global radiated energy of the L1157 outflow.\nFrom the best fit model obtained for the knots A and C, we have derived the total, extinction corrected,\nH$_2$\\ luminosity by integrating over all the ro-vibrational transitions considered by our\nmodel. $L_{\\rm H_2}$ is found\nto be 8.4$\\times$10$^{-2}$ and 3.7$\\times$10$^{-2}$ L$_\\odot$ for the A and C knots, respectively. Out of this total\nluminosity, the contribution of only the rotational lines is 5.6$\\times$10$^{-2}$(A) and 2.7$\\times$10$^{-2}$(C) L$_\\odot$,\nwhich means that in both cases they represent about 70\\% of the total H$_2$\\, luminosity.\n\nN09 have found that the total luminosity of the H$_2$\\ rotational lines from S(0) to\nS(7), integrated over the entire L1157 outflow, \namount to 0.15 L$_\\odot$. If we take into account an additional 30\\% of contribution from the v$>$0 \nvibrational levels, we estimate a total H$_2$\\ luminosity of 0.21 L$_\\odot$. This is a 30\\% larger\nthan the total H$_2$\\ luminosity estimated by Caratti o Garatti (2006) in this outflow, \nassuming a single component gas at temperature between 2000 and 3000 K that fit the NIR H$_2$\\ lines. \n\nIf we separately compute the H$_2$\\ luminosity\nin the two outflow lobes, we derive $L_{\\rm H_2}$ = 8.5$\\times$10$^{-2}$ L$_\\odot$ in the blue lobe and 1.3$\\times$10$^{-1}$ L$_\\odot$ in the red\nlobe. Comparing these numbers with those derived in the individual A and C knots, \nwe note that the A knot alone contributes to most of the H$_2$\\ luminosity in the blue lobe. By contrast,\nthe H$_2$\\ luminosity of the red lobe is distributed among several peaks of similar values. \nThis might suggest that most of the energy carried out by the blue-shifted jet is\nreleased when the leading bow-shock encounters a density enhancement at the position of the A knot. \nOn the other hand, the red-shifted gas flows more freely without large density discontinuities, and \nthe corresponding shocks are internal bow-shocks, all with similar luminosities.\n\nThe integrated luminosity radiated by CO, H$_2$O and OI in L1157 has been estimated, through ISO and\nrecent Herschel observations, as $\\sim$0.2 L$_\\odot$ (Giannini et al. 2001, Nisini et al. 2010), \nwhich means that H$_2$\\ alone contributes about 50\\% of the\ntotal luminosity radiated by the outflow. \nIncluding all contributions, the total shock cooling along the L1157 outflow amounts to about 0.4 L$_\\odot$, i.e. $L_{cool}\/L_{bol}$ $\\sim$ 5$\\times$10$^{-2}$, assuming $L_{bol}$=8.4 L$_\\odot$ \nfor L1157-mm (Froebrich 2005). This ratio is consistent with the range of values derived from other class 0 sources \nfrom ISO observations (Nisini et al. 2002).\n\nThe total kinetic energy of the L1157 molecular outflow \nestimated by Bachiller et al. (2001) amounts to 0.2 L$_\\odot$ without any correction for the\noutflow inclination angle, or to 1.2 L$_\\odot$ if an inclination angle of 80 degrees \nis assumed. Considering that the derivation of the L$_{kin}$ value has normally\nan uncertainty of a factor of five (Downes \\& Cabrit 2007), we conclude that the mechanical energy flux \ninto the shock, estimated as $L_{cool}$,  is comparable to the kinetic energy of the swept-out \noutflow and thus that the shocks giving rise to the H$_2$ emission have \nenough power to accelerate the molecular outflow.\n\nThe total shock cooling derived above can be also used to infer the momentum flux \nthrough the shock, i.e. $\\dot{P}$ = 2$L_{cool}$\/V$_s$, where V$_s$ is the shock velocity that\nwe can assume, on the basis of the discussion in the previous section, to be of the order of 20 km\\,s$^{-1}$.\nComputing the momentum flux separately for the blue and red outflow lobe, we derive \n$\\dot{P}_{red} \\sim$ 1.7$\\times$10$^{-4}$ and $\\dot{P}_{blue} \\sim$ 1.1$\\times$10$^{-4} $M$_\\odot$ yr$^{-1}$ km\\,s$^{-1}$. \nIn this calculation, we have assumed that the contribution from cooling species\n different from H$_2$, as estimated by ISO and Herschel, is distributed among the two lobes \n in proportion to the H$_2$\\ luminosity. If we assume that the molecular \n outflow is accelerated at the shock front through momentum conservation, then the\n above derived momentum flux should results comparable to the thrust of the outflow,\n derived from the mass, velocity and age measured through CO observations.\nThe momentum flux measured in this way by Bachiller et al. (2001) is 1.1$\\times$10$^{-4}$and 2$\\times$10$^{-4}$ M$_\\odot$ yr$^{-1}$ km\\,s$^{-1}$\nin the blue and red lobes, respectively, i.e. comparable to our derived values. It is interesting to note that \nthe $\\dot{P}$ determination from the shock luminosity confirms the asymmetry between\nthe momentum fluxes derived in two lobes. As shown by Bachiller et al. (2001), the L1157 red lobe \nhas a 30\\% smaller mass with respect to the blue lobe, but a higher momentum flux due to the larger flow velocity. The northern red lobe is in fact more extended than the southern lobe:\nhowever, given the higher velocity of the red-shifted gas, the mean kinematical ages of the two lobes is very\nsimilar. \n\n\n\\section{Conclusions \\label{conclusions}}\n\nWe have analysed the H$_2$\\ pure rotational line emission, from S(0) to S(7),\nalong the outflow driven by the L1157-mm protostar, mapped with the Spitzer - IRS \ninstrument. The data have been analysed assuming a gas temperature stratification where the H$_2$\\ column \ndensity varies as $T^{-\\beta}$ and 2D maps of the H$_2$\\ column density,\northo-to-para ratio (OPR) and temperature spectral index $\\beta$\nhave been constructed. \nFurther constraints on the physical conditions of the shocked gas have been derived \nin two bright emission knots by combining the Spitzer observations with near-IR \ndata of H$_2$\\ ro-vibrational emission. Finally, the global H$_2$\\ radiated energy of the\noutflow has been discussed in comparison with the energy budget of the associated\nCO outflow.\n\nThe main conclusions derived by our analysis are the following:\n\\begin{itemize}\n\\item H$_2$\\ transitions with $J_{lower} \\le$ 2 follows the morphology of the CO molecular\noutflow, with peaks correlated with individual CO clumps and more diffuse\nemission that delineates the CO cavities created by the precessing jet. \nLines with higher $J$ are localized on the shocked peaks, presenting a morphology\nsimilar to that of the H$_2$\\, 2.12$\\mu$m\\, ro-vibrational emission.\n\\item  Significant variations of the derived parameters are observed along the flow. \nThe H$_2$\\ column density ranges between 5$\\times$10$^{19} $ and 3$\\times$10$^{20} $cm$^{-2}$: \nthe highest values are found in the blue-shifted lobe, suggesting that here the outflowing\ngas is compressed due to the impact with a high density region.\nGas components in a wide range  of temperature values, from $ \\sim $ 250 to $ \\sim $ 1500 K\ncontribute to the H$_2$\\ emission along individual lines of sight. The largest range\nof temperature variations is derived towards the intensity peaks closer to the \ndriving source, while a more uniform temperature distribution, with $ T $\nbetween $ 400 $ and  $ 1000$ K, is found at the tip of the northern outflow lobe.\n\\item The OPR is in general lower than the equilibrium value at high temperatures and spans a range from $\\sim$0.6 to 2.8, with the lowest values\nfound in low temperature plateau regions between consecutive intensity peaks. \nAs in previous studies, we also found the presence of regions at low OPR (1.5-1.8) \nbut with relatively high temperatures. These might represent zones subject to recent shocks \nwhere the OPR has not had time yet to reach the equilibrium value.\n\n\\item Additional shock parameters have been derived in the two bright near-IR \nknots A and C,  located \nin the blue- and red-shifted outflow lobes, where the mid- and near-IR H$_2$\\ \ndata have been combined. \nThe ratio between mid- and near-IR lines is very sensitive to the molecular  plus atomic hydrogen particle density. A high \nabundance of atomic hydrogen (H\/H$_2$ $\\sim$ 0.1-0.3) is implied by the \nthe observed H$_2$\\ column densities if we assume n(H$_2$) values as derived by independent \nmm observations. With this assumption, the cooling lengths  of the shock result \nof the order of 7$\\times$10$ ^{14} $  and 10$ ^{15} $ cm for the A and C knot, respectively.\nThe distribution of OPR values as a function of temperature  and the \nderived abundance of atomic hydrogen, implies that the shock passing time is of the\norder of 100 yr for knot A and 1000 yr  for  knot C, given the assumption that the para-to-ortho\nconversion occurs through reactive collisions with atomic hydrogen.\nWe find that planar shock models, either of C- or J-type, are\nnot able to consistently reproduce all the physical parameters derived from our analysis \nof the H$_2$\\ emission. \n\\item Globally, H$_2$\\ emission contributes to about 50\\% of the total shock radiated energy in the L1157 outflow. We find that the momentum flux through the shocks derived from the radiated luminosity is\ncomparable to the thrust of the associated molecular outflow, supporting a scenario\nwhere the working surface of the shocks drives the molecular outflow. \n\\end{itemize}\n\n\\acknowledgments\n\nThis work is based on observations made with the Spitzer Space Telescope, which is operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with NASA. Financial support from contract ASI I\/016\/07\/0 is acknowledged. \n\n\\bibliographystyle{plainnat}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFrames are overcomplete (or redundant) sets of vectors that serve to faithfully represent signals. They were introduced in $1952$ by Duffin and Schaeffer \\cite{duscha52}, and reemerged with the advent of wavelets \\cite{Christensen:2003ab,  Dau92,Ehler:2007aa, Ehler:2008ab, hw89}. \nThough the overcompleteness of frames precludes signals from having unique representation in the frame expansions, it is, in fact, the driving force behind the use of frames in signal processing \\cite{Casazza:2003aa, koche1, koche2}.\n\nIn the finite dimensional setting, frames are exactly spanning sets. However, many applications require ``custom-built'' frames that possess additional properties which are dictated by these applications. As a result, the construction of frames with prescribed structures  has been actively pursued. For instance, a special class called {\\it finite unit norm tight frames } (FUNTFs) that provide a Parseval-type representation very similar to orthonormal bases, has been customized to model data transmissions \\cite{Casazza:2003aa, Goyal:2001aa}. Since then the characterization and construction of FUNTFs and some of their generalizations have received a lot of attention \\cite{Casazza:2003aa, koche1, koche2}. Beyond their use in applications, FUNTFs are also related to some deep open problems in pure mathematics such as the Kadison-Singer conjecture \\cite{cftw}. FUNTFs appear also in statistics where, for instance,  Tyler used them to construct $M$-estimators of multivariate scatter \\cite{Tyler:1987}. We elaborate more on the connection between the $M$-estimators and FUNTFs in Remark~\\ref{remark:M estimator FUNTF}. These $M$-estimators were subsequently used to construct maximum likelihood estimators for the the wrapped Cauchy distribution on the circle in \\cite{KentTaylor1994} and for the angular central Gaussian distribution on the sphere in \\cite{Tyler:1988}. \n\nFUNTFs are exactly the minimizers of a functional called the frame potential \\cite{Benedetto:2003aa}. This was extended to characterize all finite tight frames in \\cite{Waldron:2003aa}. Furthermore, in \\cite{fjko, jbok}, finite tight frames with a convolutional structure, which can be used to model filter banks, have been characterized as minimizers of an appropriate potential. All these potentials are connected to other functionals whose extremals have long been investigated in various settings. We refer to \\cite{cokum06, Delsarte:1977aa, Seidel:2001aa, Venkov:2001aa, Welch:1974aa} for details and related results. \n\nIn the present paper, we study objects beyond both FUNTFs and the frame potential. In fact, we consider a family of functionals, the {\\it $p$-frame potentials}, which are defined on  sets $\\{x_{i}\\}_{i=1}^{N}$ of unit vectors in $\\R^d$; see Section~\\ref{section:pfp}. These potentials have been studied in the context of spherical $t$-designs for even integers $p$, cf.~Seidel in \\cite{Seidel:2001aa}, and their minimizers are not just FUNTFs but FUNTFs that inherit additional properties and structure. Common FUNTFs are recovered only for $p=2$. In the process, we extend Seidel's results on spherical $t$-designs in \\cite{Seidel:2001aa} to the entire range of positive real $p$. \n\nIn Section~\\ref{section:estimates}, we give lower estimates on the $p$-frame potentials, and prove that in certain cases their minimizers are FUNTFs, which possess additional properties and structure. In particular, if $0<p \\leq 2$, we completely characterize the minimizers of the $p$-frame potentials when $N=kd$ for some positive integer $k$. Moreover, when $N=d+1$ and $0<p\\leq 2$, we characterize the minimizers of the $p$-frame potentials, under a technical condition, which, we have only been able to establish when $d=2$. We conjecture that this technical condition holds when $d>2$. Finally in Section \\ref{section:intro prob}, we introduce {\\it probabilistic $p$-frames} that generalize the concepts of frames and $p$-frames. We characterize the minimizers of {\\it probabilistic $p$-frame potentials} in terms of probabilistic $p$-frames. The latter problem is solved completely for $0<p\\leq 2$, and for all even integers $p$. In particular, these last results generalize \\cite{Seidel:2001aa} as well as the recently introduced notion of the probabilistic frame potential in \\cite{Ehler:2010aa}\n\n\\smallskip \n\n\\textbf{Further relations to statistics:}\nBesides the results on FUNTFs used in \\cite{KentTaylor1994,Tyler:1987, Tyler:1988}, and mentioned above, frame theory has essentially evolved independently of statistical fields such as statistical shape analysis \\cite{Dryden:1998aa} and directional statistics \\cite{Mardia:2008aa}. Nevertheless, there still exist several overlaps, and to the best of our knowledge, these overlaps have not yet been fully explored. Recently, frame theory has been used in directional statistics \\cite{Ehler:2010ac}, where FUNTFs are utilized to investigate on statistical tests for directional uniformity and to model and analyze patterns found in granular rod experiments. We must point out that similar results were obtained earlier by Tyler in  \\cite{Tyler:1988}. \n\nProbabilistic tight frames are multivariate probability distributions whose second moments' matrix is a multiple of the identity, and they are used in \\cite{Ehler:2010aa} to obtain approximate FUNTFs. The latter approximation procedure is connected to a classical problem in multivariate statistics, namely estimating the population covariance from a sample, which is closely related to the $M$-estimators addressed in \\cite{KentTaylor1994,Tyler:1987,Tyler:1988}. The $p$-frame potentials and their probabilistic counterparts that we consider in the sequel, are linked to the notion of shape measure, shape space, and mean shape used in statistical shape analysis. In Section \\ref{label:section p frame potential}, we establish a precise connection between the full Procrustes estimate of mean shape \\cite[Definition 3.3]{Kent:1994aa} which is the eigenvector corresponding to the largest eigenvalue of the frame operator. Moreover, this eigenvalue coincides with the upper frame redundancy as introduced in \\cite{Bodmann:2010aa}. The full Procrustes estimate of mean shape also saturates the upper frame inequality. Moreover, the $p$-frame potentials form size measures as required in statistical shape analysis, and their minimizers among all collections of $N$ points on the sphere define a shape space modulo rotations. \n\nWe hope that the present paper will renew interests in more investigation on the role of frames and the $p$-th frame potential in directional statistics and statistical shape analysis. \n\n\n\n \n \\section{The $p$-frame potential}\\label{section:pfp}\n\n\\subsection{Background on frames and the frame potential}\nTo introduce frames and their elementary properties, we follow the textbook \\cite{Christensen:2003aa}.\n\\begin{definition}\\label{framedef}\nA collection of points $\\{x_i\\}_{i=1}^N\\subset\\R^d$ is called a \\emph{finite frame for $\\R^d$} if there are two constants $0<A\\leq B$ such that\n\\begin{equation}\\label{frameineq}\nA\\|x\\|^2 \\leq \\sum_{i=1}^N |\\langle x,x_i\\rangle|^2 \\leq B\\|x\\|^2,\\quad\\text{for all $x\\in\\R^d$.}\n\\end{equation}\nIf the frame bounds $A$ and $B$ are equal, the frame $\\{x_i\\}_{i=1}^N\\subset\\R^d$ is called \\emph{a finite tight frame for $\\R^d$}. In this case, \n\\begin{equation}\\label{deftightf}\nA\\|x\\|^2 = \\sum_{i=1}^N |\\langle x,x_i\\rangle|^2,\\quad\\text{for all $x\\in\\R^d$.}\n\\end{equation}\nA finite tight frame $\\{x_i\\}_{i=1}^N\\subset \\R^d$  consisting of unit norm vectors is  called a \\emph{finite unit norm tight frame (FUNTF)} for $\\R^d$. In this case, the frame bound is $A=N\/d$. \n\nA collection of unit vectors $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ is called \\emph{equiangular} if there exists a nonnegative constant $C$ such that  $|\\langle x_i,x_j\\rangle|=C$, for $i\\neq j$. \n\\end{definition}\n\n\n\nGiven a collection of $N$ points $\\{x_i\\}_{i=1}^N$ in $\\R^d$, the \\emph{analysis operator} is the mapping \n \\begin{equation*}\n F: \\R^d \\rightarrow \\R^N, \\quad x\\mapsto \\big(\\langle x,x_i \\rangle\\big)_{i=1}^N.\n \\end{equation*} \n Its adjoint operator is called the \\emph{synthesis operator} and given by \n \\begin{equation*}\n  F^*: \\R^N \\rightarrow\\R^d , \\quad (c_i)_{i=1}^N\\mapsto \\sum_{i=1}^N c_i x_i.\n \\end{equation*}\nUsing these operators, it is easy to see that $\\{x_i\\}_{i=1}^N$ is a frame if and only if the \\emph{frame operator} defined by\n\\begin{equation*}\nS:= F^* F : \\R^d \\rightarrow \\R^d,\\quad x\\mapsto \\sum_{i=1}^N \\langle x,x_i\\rangle x_i \n\\end{equation*}\nis positive, self-adjoint, and invertible. In this case, the following reconstruction formula holds\n\\begin{equation}\\label{eq:reconstruction for frames}\nx = \\sum_{j=1}^N  \\langle S^{-1} x_i,x\\rangle x_i,\\text{ for all $x\\in\\R^d$,}\n\\end{equation}\nand $\\{S^{-1} x_i\\}_{i=1}^N$, in fact, is a frame too, called the \\emph{canonical dual frame}. If $\\{x_i\\}_{i=1}^N$ is a frame, then $\\{S^{-1\/2}x_i\\}_{i=1}^N$ is a finite tight frame. Moreover, note that $\\{x_i\\}_{i=1}^N$ is a FUNTF if and only if its frame operator $S$ is $\\frac{N}{d}$ times the identity. \n\n\nAs mentioned in the introduction, the question of the existence and characterization of FUNTFs was settled in \\cite{Benedetto:2003aa}, where the \\emph{frame potential}, defined by \n\\begin{equation}\\label{eq:frame potential}\n\\FP(\\{x_i\\}_{i=1}^N) = \\sum_{i=1}^N\\sum_{j=1}^N |\\langle x_i,x_j\\rangle |^2,\n\\end{equation} was introduced and used to give a characterization of its minimizers in terms of FUNTFs. More specifically, they prove the following result: \n\n\n\\begin{theorem}\\cite[Theorem 7.1]{Benedetto:2003aa}\\label{theorem:Benedetto Fickus}\nLet $N$ be fixed and consider the minimization of the frame potential among all collections of $N$ points on the sphere $S^{d-1}$.\n\\begin{itemize}\n\\item[a)] If $N\\leq d$, then the minimum of the frame potential is $N$. The minimizers are exactly the orthonormal systems for $\\R^d$ with $N$ elements.\n\\item[b)] If $N\\geq d$, then the minimum of the frame potential is $\\frac{N^2}{d}$. The minimizers are exactly the FUNTFs for $\\R^d$ with $N$ elements.\n\\end{itemize}\n\\end{theorem}\n\nWe shall prove in the sequel that the frame potential is just an example in a family of functionals defined on points on the sphere, and whose minimizers have approximation properties similar to those of the frame potential. But first, we briefly comment on the relation between FUNTFs and $M$-estimators of multivariate scatter:\n\\begin{remark}\\label{remark:M estimator FUNTF}\nThe concept of FUNTFs is used in signal processing to represent a signal $x\\in\\R^d$ by means of $x=\\frac{d}{N}\\sum_{i=1}^N\\langle x,x_i\\rangle x_i$, which is similar to the well-known expansion in an orthonormal basis. FUNTFs have also been used in statistics to derive $M$-estimators of multivariate scatter: For a sample $\\{x_i\\}_{i=1}^N\\subset \\R^d$, where $x_i\\neq 0$, for $i=1\\ldots,N$, Tyler aims to find a symmetric positive definite matrix $\\Gamma$ such that \n\\begin{equation*\nM(\\Gamma) = \\frac{d}{N}\\sum_{i=1}^N \\frac{\\Gamma^{1\/2}x_i x_i' \\Gamma^{1\/2}}{x_i'\\Gamma x_i}\n\\end{equation*}\nis the identity matrix. Whenever this is possible, the estimate $V$ of the population scatter matrix is then given by $V=\\Gamma^{-1}$. We refer to  \\cite{Tyler:1987} for details. Note that $M(\\Gamma)=I_d$ implies that \n\\begin{equation*}\n\\bigg\\{\\frac{\\Gamma^{1\/2}x_i}{\\sqrt{x_i'\\Gamma x_i}}\\bigg\\}_{i=1}^{N}=\\bigg\\{\\frac{\\Gamma^{1\/2}x_i}{\\|\\Gamma^{1\/2}x_{i}\\|}\\bigg\\}_{i=1}^{N} \\subset S^{d-1}\n\\end{equation*}\n forms a FUNTF. Moreover, $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ is a FUNTF if and only if $M(I_d)=I_d$.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n\\subsection{Definition of the $p$-frame potential}\\label{label:section p frame potential}\n\n\\begin{definition}\\label{pframepotential}\nLet $N$ be a positive integer, and $0<p< \\infty$. Given a collection of unit vectors $\\{x_i\\}_{i=1}^{N}\\subset S^{d-1}$, the \\emph{$p$-frame potential} is the functional\n  \\begin{equation}\\label{eq:pth frame potential}\n  \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})=\\sum_{i, j=1}^{N}|\\langle x_i,x_j\\rangle |^p. \n \\end{equation}\nWhen, $p=\\infty$, the definition reduces to\n\n\\begin{equation*}\n\\FP_{\\infty, N}(\\{x_{i}\\}_{i=1}^{N})=\\sup_{i\\neq j} |\\langle x_i,x_j\\rangle |.\n\\end{equation*}\n\\end{definition}\n\n\nIt is clear that the $p$-frame potential generalizes the frame potential in \\eqref{eq:frame potential}. Finite frames and the $p$-frame potential also extend to complex $\\{z_i\\}_{i=1}^N\\subset S_{\\C}^{d-1}=\\{z\\in\\C^d : \\|z\\|=1\\}$ and are related to statistical shape analysis, a tool to quantitatively track the physical deformation of objects. We refer to \\cite[Chapters 2, 3 $\\&$ 4]{Dryden:1998aa} for more details on shape analysis, but we briefly indicate here its link to the $p$-frame potential.  The shape of an object is specified by landmark points that altogether form the shape space. Often, a suitable transformation is applied first in order to study shape independently on the object's size. To remove the feature of size, we must specify a size measure $g$ (\\cite[Definition 2.2]{Dryden:1998aa}), which is a positive function defined on  $S_{\\C}^{d-1}$ that satisfies  \n\\begin{equation*}\ng(s\\{z_i\\}_{i=1}^N) = s g(\\{z_i\\}_{i=1}^N),\\quad\\text{for all $s\\in\\R_{+}$ and $\\{z_i\\}_{i=1}^N\\subset S_{\\C}^{d-1}$}.\n\\end{equation*}\nIt is immediate that the following family of functionals can be seen as size measures: \n\\begin{equation*}\ng_p(\\{z_i\\}_{i=1}^N) := \\big(\\sum_{i, j=1}^{N}|\\langle z_i,z_j\\rangle |^{p}\\big)^{\\frac{1}{p}} =\\bigg(\\FP_{N,p}(\\{z_i\\}_{i=1}^N)\\bigg)^{1\/p}.\n\\end{equation*}\nIn particular, $g_1$ represents the centroid size (when the shape is centered at the origin), which is one of the most common size measures in statistical shape analysis. Given two complex configurations $z_1,z_2\\in\\C^d$ derived from landmarks that code two-dimensional shape, the full Procrustes distance (\\cite[Definition 3.2]{Dryden:1998aa})  is \n\\begin{equation*}\nd_F(z_1,z_2)= \\inf_{\\beta,\\theta,a,b} \\bigg{\\|}\\frac{z_1}{\\|z_1\\|} - \\frac{z_2}{\\|z_2\\|}\\beta e^{i\\theta} -a -ib \\bigg{\\|}.\n\\end{equation*}\n Given $N$ configurations $\\{z_i\\}_{i=1}^N\\subset \\C^d$, the full Procrustes estimate of mean shape is defined by \n\\begin{equation*}\n\\bar{z}_P = \\arg\\inf_{\\|z\\|=1} \\big( \\sum_{i=1}^N d_F^2(z_i,z) \\big),\n\\end{equation*}\ncf.~\\cite[Definition 3.3]{Dryden:1998aa}. The average axis from the complex Bingham maximum likelihood estimator is the same as the full Procrustes estimate of mean shape for two-dimensional shapes, and the same holds for the complex Watson distribution, cf.~\\cite[Sections 6.2 $\\&$ 6.3]{Dryden:1998aa}. If we assume that $\\{z_i\\}_{i=1}^N$ are centered around zero, then $\\bar{z}_P$ is given by the eigenvector corresponding to the largest eigenvalue $\\lambda$ of the frame operator of the normalized collection $\\{\\frac{z_i}{\\|z_i\\|}\\}_{i=1}^N$. Furthermore, one observes that this eigenvalue $\\lambda$ is the upper frame redundancy of $\\{z_i\\}_{i=1}^N$ as introduced in \\cite{Bodmann:2010aa}, which also coincides with the optimal upper frame bound $B$ in \\eqref{frameineq}. Therefore, the full Procrustes estimate of mean shape satisfies the upper frame inequality with equality. Moreover, we have observed that the root mean square of the full Procrustes estimate of mean shape is $1-\\frac{\\lambda}{N}$.\n\n\n\n\n\nBefore stating the next elementary result, we recall some basic definitions in physics: A {\\it conservative force} $F$, is a vector field defined on $\\R^d$, such that $-F$ is the gradient of some {\\it potential} $P$ that is then induced by the conservative force. Lemma~\\ref{conservative} below was proved for the frame potential in \\cite{Benedetto:2003aa}, and we extend it to all $p$-frame potentials with $1<p<\\infty$:\n\\begin{lemma}\\label{conservative}\nFor each $p\\in (1, \\infty)$, the $p$-frame potential $\\FP_{p, N}$ is induced by the conservative force $F_p= f_p(\\|a-b\\|)(a-b)$, for $a,b\\in S^{d-1}$, where\n\\begin{equation*}\n f_p(x):=\\begin{cases}\n p(1-\\frac{x^2}{2})^{p-1},&\\text{for } |x|\\leq \\sqrt{2},\\\\\n -p(\\frac{x^2}{2}-1)^{p-1},&\\text{otherwise.}\n \\end{cases}\n \\end{equation*}\n\\end{lemma}\n$F_p$ is a central force between the `particles' $a$ and $b$ that we call the \\emph{$p$-frame force}.\n\\begin{proof}   The function  \n \\begin{equation*}\n \\mathbf{p}_p(x):=|1-\\frac{x^2}{2}|^p=\\begin{cases}\n (1-\\frac{x^2}{2})^p,&\\text{for } |x|\\leq \\sqrt{2},\\\\\n (\\frac{x^2}{2}-1)^p,&\\text{otherwise,}\n \\end{cases}\n \\end{equation*}\nis differentiable and satisfies  $ \\mathbf{p}_p'(x)=-xf_p(x)$. This is sufficient to verify that the potential $P_p(a,b):=\\mathbf{p}_p(\\|a-b\\|)$, defined for $a,b\\in S^{d-1}$, satisfies $\\nabla_a P_p(a,b) = -F(a,b)$, where $b$ is held fixed. Thus, $F_p$ is a conservative vector field. The physical meaningful potential $P_p(a,b)$ is in fact given by \n$$ P_p(a,b)  =\\mathbf{p}_p(\\|a-b\\|) = \\big |1-\\frac{1}{2}\\|a-b\\|^2 \\big |^p = \\big|\\langle a,b\\rangle\\big|^p,$$\n where we used that $\\|a\\|=\\|b\\|=1$. \n Consequently, the $p$-frame potential is induced by the conservative central force $F_p$. \n \\end{proof}\n \n As a consequence of the above lemma, $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ are in \\emph{equilibrium} under the $p$-frame force if they minimize the $p$-frame potential among all collections of $N$ points on the sphere. Note that such a collection of equilibria modulo rotations form a shape space. \n\n\\begin{remark}\nWe will use repeatedly the fact that for a fixed $\\{x_{i}\\}_{i=1}^{N} \\subset S^{d-1}$, the $p$-frame potential $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N})$ is a decreasing and continuous  function of $p \\in (0, \\infty)$. \n\n\\end{remark}\n\n\n \n\\section{Lower estimates for the $p$-frame potential}\\label{section:estimates}\nWe start this section with a few elementary results about the minimizers of the $p$-frame potential as well as their connection to $t$-designs. In fact, potentials on the sphere, and $t$-designs have been well investigated \\cite{Bachoc:2005aa, cokum06, Delsarte:1977aa, Seidel:2001aa, Venkov:2001aa}. However, one of the key differences between $t$-designs and our $p$-frame potential is that the former is considered only for positive integers $t$ while the latter is investigated for $p\\in (0, \\infty)$.  \n\n\\subsection{The Welch bound revisited}\\label{subsection:Welch}\nIf $p=2k$ is an even integer, one can use Welch's results~\\cite{Welch:1974aa} to conclude that, for $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$,\n\\begin{equation}\\label{eq:Welch}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\geq \\frac{N^2}{\\binom{d+k-1}{k}}.\n\\end{equation}\nWe shall verify that this estimate is not optimal for small $N$, by proving an  estimate for $\\FP_{p, N}$ when $2<p<\\infty$. The following Proposition first appeared in \\cite{Oktay:2007}: \n\\begin{proposition}\\label{prop:potential for p>2}\nLet $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, $N\\geq d$, and $2<p<\\infty$, then\n\\begin{equation}\\label{eq:potential for p>2}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N(N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^{p\/2}+N,\n\\end{equation}\n and equality holds if and only if  $\\{x_i\\}_{i=1}^N$ is an equiangular FUNTF.\n\\end{proposition}\n\n\n\\begin{proof}\nFor $\\frac{1}{2}=\\frac{1}{p}+\\frac{1}{r}$, H\\\"older's inequality yields\n\\begin{equation}\\label{eq:hoelder numer 1}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_2} \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}(N(N-1))^{1\/r}.\n\\end{equation}\nRaising to the $p$-th power and applying $\\frac{1}{r}=\\frac{1}{2}-\\frac{1}{p}$ leads to \n\\begin{equation}\\label{eq:raising to the power}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_2}^p \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}^p(N(N-1))^{p\/2-1}.\n\\end{equation}\nTherefore, \n$$\n \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p  \\geq  \\big(\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2\\big)^{p\/2} (N(N-1))^{1-p\/2}.$$\nUsing the fact that $\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2\\geq \\frac{N^2}{d}-N$ (see Theorem~\\ref{theorem:Benedetto Fickus}) implies that\n$$ \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p \\geq \\big(N(\\frac{N}{d}-1)\\big)^{p\/2} (N(N-1))^{1-p\/2} = N(N-1)\\bigg(\\frac{N-d}{d(N-1)}\\bigg)^{p\/2},$$\nwhich proves~\\eqref{eq:potential for p>2}. \n\nTo establish the last part of the Proposition, we recall that an equiangular FUNTF $\\{x_{k}\\}_{k=1}^{N} \\subset \\R^d$ satisfies \n\\begin{equation}\\label{eq:equi}\n|\\langle x_i,x_j\\rangle | = \\sqrt{\\frac{N-d}{d(N-1)}},\\quad \\text{ for all }i\\neq j\n\\end{equation} \nsee, \\cite{Casazza:2008ab,Sustik:2007aa}, for details. Consequently, if $\\{x_{k}\\}_{k=1}^{N}$ is an equiangular FUNTF, then~\\eqref{eq:potential for p>2} holds with equality. \n\nOn the other hand, if equality holds in \\eqref{eq:potential for p>2}, then $\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^2 = \\frac{N^2}{d}-N$ and $\\{x_i\\}_{i=1}^N$ is a FUNTF due to Theorem \\ref{theorem:Benedetto Fickus}. Moreover, the H\\\"older estimate \\eqref{eq:hoelder numer 1} must have been an equality which means that $|\\langle x_i,x_j\\rangle|=C$ for $i\\neq j$, and some constant $C\\geq 0$. Thus, the FUNTF must be equiangular.\n\\end{proof}\n\n\n By comparing \\eqref{eq:Welch} with \\eqref{eq:potential for p>2}, it is easily seen that the Welch bound is not optimal for small $N$: \n \n \n \\begin{proposition}\\label{prop:second one}\n Let $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ and $p=2k>2$ be an even integer. If $d<N \\leq  \\binom{d+k-1}{k}$, then \n  \\begin{equation}\\label{eq:abc}\n \\FP_{p,N}(\\{x_{k}\\}_{k=1}^{N}) \\geq N(N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k +N >\\frac{N^2}{\\binom{d+k-1}{k}}.\n \\end{equation} \n \\end{proposition}\n \n \n \\begin{proof} \n The condition on $N$ implies $1 \\geq   \\frac{N}{\\binom{d+k-1}{k}}$, and adding $ (N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k>0$ to the right hand side leads to\n \\begin{equation*}\n  (N-1)\\big(\\frac{N-d}{d(N-1)}\\big)^k +1 >  \\frac{N}{\\binom{d+k-1}{k}}.\n \\end{equation*}\n Multiplication by $N$ and Proposition \\ref{prop:potential for p>2} then yield \\eqref{eq:abc}.\n \\end{proof}\n\n\\begin{remark} The estimate in Proposition \\ref{prop:potential for p>2} is sharp if and only if an equiangular FUNTF exists. In \\cite[Sections 4 $\\&$ 6]{Sustik:2007aa}, construction (and hence existence) of equiangular FUNTFs was established when $d+2 \\leq N \\leq 100$. For general $d$ and $N$, a necessary condition for existence of equiangular FUNTFs is given, and it is conjectured that the conditions are sufficient as well. The authors essentially provide on upper bound on $N$ that depends on the dimension $d$. \nTherefore, Proposition~\\ref{prop:potential for p>2} might not be optimal when the redundancy $N\/d$ is much larger than $1$. \n\\end{remark}\n\n\n\\subsection{Relations to spherical $t$-designs}\\label{subsection:t design}\nA \\emph{spherical $t$-design} is a finite subset $\\{x_i\\}_{i=1}^N$ of the unit sphere $S^{d-1}$ in $\\R^d$,\nsuch that,\n\\begin{equation*}\n\\frac{1}{N}\\sum_{i=1}^N h(x_i) = \\int_{S^{d-1}} h(x)d\\sigma(x),\n\\end{equation*}\nfor all homogeneous polynomials $h$ of total degree equals or less than $t$ in $d$ variables and where $\\sigma$ denotes the uniform surface measure on $S^{d-1}$ normalized to have mass one. The following result is due to \\cite[Theorem 8.1]{Venkov:2001aa} (see \\cite{Seidel:2001aa}, \\cite{Delsarte:1977aa} for similar results). \n\n \n \\begin{theorem}\\label{theorem:p even integer discrete}\\cite[Theorem 8.1]{Venkov:2001aa}\nLet $p=2k$ be an even integer and $\\{x_i\\}_{i=1}^N=\\{-x_i\\}_{i=1}^N\\subset S^{d-1}$, then \n  \\begin{equation*}\n\\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})  \\geq  \\frac{1\\cdot 3\\cdot 5\\cdots(p-1)}{d(d+2)\\cdots (d+p-2) }N^2,\n \\end{equation*}\nand equality holds if and only if $\\{x_i\\}_{i=1}^N$ is a spherical $p$-design. \n \\end{theorem} \n \n \n \n\\subsection{Optimal configurations for  the $p$-frame potential}  \nWe first use Theorem~\\ref{theorem:Benedetto Fickus} to characterize the minimizers of the $p$-frame potential for $0<p<2$ provided that the number of points $N$ is a multiple of the dimension $d$:\n\n\n\\begin{theorem}\\label{prop:k multiples and small p}\nLet $0<p<2$ and assume that $N=kd$ for some positive integer $k$. Then the minimizers of the $p$-frame potential are exactly the $k$ copies of any orthonormal basis modulo multiplications by $\\pm 1$. The minimum of~\\eqref{eq:pth frame potential}, over all sets of $N=kd$ unit norm vectors, is $k^{2}d$.\n\\end{theorem}\n\\begin{proof}\nIf we fix a collection of vectors $\\{x_i\\}_{i=1}^N$, then the frame potential is a decreasing function in $p\\in(0, 2)$. Therefore, $$N^{2}\/d=\\FP_{2,N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}).$$\n\nNow suppose that $\\{y_{i}\\}_{i=1}^{N}$ consists of $k$ copies of an  orthonormal basis $\\{e_{i}\\}_{i=1}^{d}$for $\\R^d$. Then,  $$\\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\FP_{p, N}(\\{y_{i}\\}_{i=1}^{N})=\\FP(\\{y_{i}\\}_{i=1}^{N})=N^{2}\/d.$$ \nConsequently, $$\\min \\FP_{p, N}(\\{x_{k}\\}_{k=1}^{N})=\\min \\FP(\\{x_{k}\\}_{k=1}^{N})=N^{2}\/d.$$ \nThus, $\\FP_{p,N}$ has the same minimum as $\\FP$. Clearly $k$ copies of an orthonormal basis of $\\R^d$ form a  FUNTF, and hence minimize the $\\FP$ due to Theorem \\ref{theorem:Benedetto Fickus}. Consequently, the $k$ copies of an orthonormal basis also minimize the $p$-frame potential  for $0<p<2$. On the other hand, the inner products must be $0$ or $1$ to obtain a minimizer. The smallest $p$-frame potential then have the $k$ copies of an orthonormal basis.\n\\end{proof}\n\nWe now consider the $p$-frame potential for $N=d+1$. The case $p=2$ is covered by Theorem \\ref{theorem:Benedetto Fickus}. Note that any FUNTF with $N=d+1$ vectors is equiangular~\\cite{Goyal:2001aa,Strohmer:2003aa}. Hence, the case $2<p<\\infty$ is settled by Proposition~\\ref{prop:potential for p>2},  so we focus on $p\\in (0,2)$. \n\nOne easily verifies that, for $p_0=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}$, an orthonormal basis plus one repeated vector and an equiangular FUNTF have the same $p_0$-frame potential $\\FP_{p_{0}, d+1}$. Under the assumption that those two systems are exactly the minimizers of $\\FP_{p_{0}, d+1}$, the next result will give a complete characterization of the minimizers of $\\FP_{p, d+1}$, for $0<p<2$. However, we have only been able to establish the validity of this assumption when $d=2$, cf.~Corollary \\ref{theorem:3 points}.\n\\begin{theorem}\\label{theorem:d+1}\nLet $N=d+1$ and set $p_{0}=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}= \\frac{\\log(\\frac{N(N-1)}{2})}{\\log(N-1)}$. Let $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, and assume that $\\FP_{p_{0},N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N+2,$ with equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF. Then,\n\\begin{itemize}\n\\item[\\textnormal{(1)}] for $0<p<p_{0} $, then for any $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, we have  $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}) \\geq N+2,$ and equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector,\n\\item[\\textnormal{(2)}] for $p_{0}<p<2$, then for any $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$, we have $\\FP_{p,N}(\\{x_{i}\\}_{i=1}^{N}) \\geq 2^{\\frac{p}{p_{0}}}\\, (Nd)^{1-\\frac{p}{p_{0}}} + N,$ \nand equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an equiangular FUNTF.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nUnder the assumptions of the Theorem, let $0<p<p_0$, then \n\\begin{equation*}\n\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}) \\leq \\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N}),\n\\end{equation*}\nand using $\\{y_{i}\\}_{i=1}^{N}$ consisting of an orthonormal basis of $\\R^d$ with one repeated vector, yields that \n\\begin{equation*}\n\\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})\\leq \\FP_{p, N}(\\{y_{i}\\}_{i=1}^{N})=N+2=\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}). \n\\end{equation*}\nConsequently, $\\min \\FP_{p_{0}, N}(\\{x_{i}\\}_{i=1}^{N}) = \\min \\FP_{p, N}(\\{x_{i}\\}_{i=1}^{N})=N+2$. Since an orthonormal basis plus one repeated vector minimizes the $p$-frame potential for $p=p_{0}$, it must also minimize $\\FP_{p, N}$ for  $0<p<p_{0}$, which proves $(1)$.\n\nAssume now that $p_0 < p < 2$. Choose $r$ such that $\\frac{1}{p_{0}}=\\frac{1}{p}+\\frac{1}{r}$, H\\\"older's  inequality yields\n\\begin{equation*}\n\\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_{p_{0}}}^p \\leq \\|(\\langle x_i,x_j\\rangle)_{i\\neq j}\\|_{\\ell_p}^p(N^{2}-N)^{p\/r}.\n\\end{equation*}\nSince we assume that $(2)$ holds, we have \n$ \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^{p_{0}} \\geq  2 $, which leads to \n\\begin{align*}\n \\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^p & \\geq  \\bigg(\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle |^{p_{0}}\\bigg){\\frac{p}{p_{0}}} (N^{2}-N)^{1-\\frac{p}{p_{0}}}\\\\\n &\\geq 2^{\\frac{p}{p_{0}}}\\, (Nd)^{1-\\tfrac{p}{p_{0}}}.\n\\end{align*}\nThis concludes the proof of $(2)$. \nBy applying \\eqref{eq:equi}, one then checks that an equiangular FUNTF satisfies $(2)$ with equality.\n\n\nThe ``only'' part comes from the fact that the H\\\"older inequality becomes an equality  only if the sequences are linearly dependent. This means that the $\\{x_i\\}_{i=1}^N$ are equiangular. They must then satisfy $|\\langle x_i,x_j\\rangle | = \\frac{1}{d}$. Thus, by~\\eqref{eq:equi}  they form an equiangular FUNTF \\cite{Casazza:2008ab,Sustik:2007aa}.\n\\end{proof}\n\nWhen $d=2$ we can in fact verify the main hypothesis of Theorem~\\ref{theorem:d+1}, which leads to the following result: \n\n\n\\begin{corollary}\\label{theorem:3 points}\nLet $\\{x_i\\}_{i=1}^3\\subset S^{1}$, and set $p_{0}=\\frac{\\log(3)}{\\log(2)}$. Then, $$\\FP_{p_{0},3}(\\{x_{i}\\}_{i=1}^{3}) \\geq 5,$$ and equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF. \n\nConsequently,\n\\begin{itemize}\n\\item[\\textnormal{(1)}] for $0<p<p_{0} $, then for any $\\{x_i\\}_{i=1}^3\\subset S^{1}$, we have $\\FP_{p,3}(\\{x_{i}\\}_{i=1}^{3}) \\geq 5,$ and equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an orthonormal basis plus one repeated vector,\n\\item[\\textnormal{(2)}] for $p_{0}<p<\\infty$, then for any $\\{x_i\\}_{i=1}^3\\subset S^{1}$, we have $\\FP_{p,3}(\\{x_{i}\\}_{i=1}^{3}) \\geq \\frac{6}{2^{p}} +3,$ \nand equality holds if and only if $\\{x_i\\}_{i=1}^3$ is an equiangular FUNTF.\n\\end{itemize}\n\\end{corollary}\nThe minimum of $\\FP_{p,3}$, for $0<p<\\infty$ is plotted in Figure \\ref{figure:min of 2d 3 vectors}. \n \\begin{figure}\n \\centering\n  \\includegraphics[width=.35\\textwidth]{p_plot_2_b.eps}\\hspace{1ex}\n \\caption{Minimum of $\\FP_{p,3}$ in Corollary \\ref{theorem:3 points}, for $0<p<10$.}\\label{figure:min of 2d 3 vectors}\n \\end{figure}\n\\begin{proof}\nClearly  $(1)$ and $(2)$ follows from Theorem~\\ref{theorem:d+1} once the minimizers of $\\FP_{p_{0}, N}$ are characterized.  \n\nWithout loss of generality, let $\\beta$ be the smallest angle between $x_1$, $x_2$, and $x_3$ and let $\\alpha$ be the second smallest angle between them. This yields, of course, $0\\leq \\beta\\leq \\alpha$.\n\nCase 1: For $0\\leq  \\alpha+\\beta\\leq\\frac{\\pi}{2}$, we have \n\\begin{equation*}\n\\frac{1}{2}\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle|^p = \\cos^p(\\alpha) +\\cos^p(\\beta)+\\cos^p(\\alpha+\\beta)=F(\\alpha, \\beta).\n\\end{equation*} \nSince $1<p_{0}$, the $p$-frame potential is differentiable in $\\alpha$ and $\\beta$, and its critical points are \n\\begin{align*}\n0 & =-p\\cos^{p-1}(\\alpha)\\sin(\\alpha)-p\\cos^{p-1}(\\alpha+\\beta)\\sin(\\alpha+\\beta)\\\\\n0 & = -p\\cos^{p-1}(\\beta)\\sin(\\beta)-p\\cos^{p-1}(\\alpha+\\beta)\\sin(\\alpha+\\beta).\n\\end{align*}\nThis implies that either $\\alpha=\\beta=0$ or $\\beta=0$ and $\\alpha=\\frac{\\pi}{2}$. In the first case, we have a maximum since it implies $x_1=x_2=x_3$. The latter case means that two points are identical and the third one is perpendicular which is a potential minimum of the $p$-frame potential. \n\nCase 2: We can assume that $\\frac{\\pi}{4}\\leq \\alpha$ (otherwise we are in Case 1). We can further assume that $\\frac{\\pi}{4}\\leq \\alpha \\leq \\frac{\\pi}{2}\\leq \\alpha+\\beta\\leq \\frac{2\\pi}{3}$ (if $\\frac{2\\pi}{3}<\\alpha+\\beta\\leq\\pi$. Otherwise, substitute $x_i$ with $-x_i$.\nWe now have \n\\begin{equation*}\n\\frac{1}{2}\\sum_{i\\neq j}|\\langle x_i,x_j\\rangle|^p =\\cos^p(\\alpha) +\\cos^p(\\beta)+(-\\cos(\\alpha+\\beta))^p=G(\\alpha, \\beta).\n\\end{equation*} \nThe critical points of $G$ are given by\n\\begin{align*}\n0 & =-p\\cos^{p-1}(\\alpha)\\sin(\\alpha)+p(-\\cos(\\alpha+\\beta))^{p-1}\\sin(\\alpha+\\beta)\\\\\n0 & = -p\\cos^{p-1}(\\beta)\\sin(\\beta)+p(-\\cos(\\alpha+\\beta))^{p-1}\\sin(\\alpha+\\beta).\n\\end{align*}\nBy subtracting one equation from the other and raising to the second power, we obtain\n \\begin{equation*}\n z_1^{p-1}(1-z_1) = z_2^{p-1}(1-z_2),\n \\end{equation*}\n where $z_1=\\cos(\\alpha)^2$ and $z_2=\\cos^2(\\alpha+\\beta)$. Since $\\sin^2(x)-1\/2\\geq \\cos^2(x)$ for all $\\pi\/2\\leq x\\leq 2\\pi\/3$, we have $0\\leq z_1\\leq 1\/2$ and $0\\leq z_2\\leq 1\/4$ because $z_2\\leq 1-z_2-1\/2$.\n \n We consider the function $F(z)=z^{p-1}(1-z)$ on $0\\leq z\\leq 1\/2$. $F$ achieves its maximum at $z=\\frac{p-1}{p}\\approx 0.3691$ and is convex, cf.~Figure \\ref{figure:1}.%\n \\begin{figure}\n \\centering\n \\subfigure[$F(z)=z^{p-1}(z-1)$, $0\\leq z\\leq \\frac{1}{2}$]{\\hspace{1ex}\n \\includegraphics[width=.35\\textwidth]{function_1.eps}\\hspace{1ex}}\n \\hspace{2ex}\n \\subfigure[$f(\\alpha) =  2\\cos^p(\\alpha) + (-\\cos(2\\alpha))^p$, $\\frac{\\pi}{3}\\leq \\alpha\\leq \\frac{\\pi}{2}$]{\\hspace{7ex}\n \\includegraphics[width=.35\\textwidth]{function_3.eps}\\hspace{7ex}}\n \\caption{$F$ and $f$ in the proof of Theorem \\ref{theorem:3 points}}\\label{figure:1}\n \\end{figure}\n Therefore, for $0\\leq z_1\\leq 1\/2$ and $0\\leq z_2\\leq 1\/4$, we have $F(z_1)=F(z_2)$ if and only if $z_1=z_2$ or $z_1=1\/2$ and $z_2=1\/4$. For $z_1=\\cos^2(\\alpha)=1\/2$, we have $\\alpha=\\pi\/3$ and $z_2=\\cos^2(\\pi\/3+\\beta)=1\/4$ yields $\\beta=\\pi\/3$. The case $z_1=z_2$ leads to $\\cos^2(\\alpha)=\\cos^2(\\alpha+\\beta)$ which implies $\\alpha+\\beta-\\pi\/2 = \\pi\/2-\\alpha$. This is equivalent to $\\beta=\\pi-2\\alpha$. Since we assume $\\beta\\leq \\alpha$, we obtain $\\pi\/3\\leq \\alpha \\leq \\pi\/2$. \n\nWe now check the minima of the function\n \\begin{align*\nf(\\alpha)  & =  \\cos^p(\\alpha)+\\cos^p(\\pi-2\\alpha)+(-\\cos(\\pi-\\alpha))^p\\\\\n& = 2\\cos^p(\\alpha) + (-\\cos(2\\alpha))^p,\n \\end{align*}\n on $\\pi\/3\\leq\\alpha\\leq \\pi\/2$, cf.~Figure \\ref{figure:1}. Its derivative \n \\begin{equation*}\n \\frac{\\partial f}{\\partial \\alpha} (\\alpha) = -2p\\cos^{p-1}(\\alpha)\\sin(\\alpha) + 2p(-\\cos(2\\alpha))^{p-1}\\sin(2\\alpha)\n \\end{equation*}\nvanishes if and only if \n \\begin{equation*}\n \\cos^{p-1}(\\alpha)\\sin(\\alpha) = (\\sin^2(\\alpha)-\\cos^2(\\alpha))^{p-1}2\\cos(\\alpha)\\sin(\\alpha)\n \\end{equation*}\n For $\\alpha\\neq \\pi\/2$, this yields\n  \\begin{equation*}\n \\cos^{p-1}(\\alpha) = (1-2\\cos^2(\\alpha))^{p-1}2\\cos(\\alpha).\n \\end{equation*}\n By substituting $x=\\cos(\\alpha)$, we obtain, for $0<x\\leq 1\/2$\n  \\begin{equation*}\n x^{p-1} = (1-2x^2)^{p-1}2x,\n \\end{equation*}\nwhich is equivalent to \n  \\begin{equation*}\n 1\/2 = (1\/x-2x)^{p-1}x \\Leftrightarrow\n 0 = (1\/x-2x)x^{1\/(p-1)} -(1\/2)^{1\/(p-1)}\n \\end{equation*}\nand define the new function \n   \\begin{equation*}\n g(x) = -2x^{q+1}+x^{q-1} -(1\/2)^{q},\n \\end{equation*}\n where $q=1\/(p-1)$. To show that $g$ has only one extremal point on $0<x\\leq 1\/2$, we differentiate\n   \\begin{equation*}\n  \\frac{\\partial g}{\\partial x}(x) = -2(q+1)x^q+(q-1)x^{q-2}.\n \\end{equation*}\n The term $ \\frac{\\partial g}{\\partial x}(x)$ vanishes if and only if \n \\begin{equation*}\n (q-1)x^{q-2} = 2(q+1)x^q,\n \\end{equation*}\nwhich is equivalent to\n \\begin{equation*}\n x^2 = \\frac{q-1}{2(q+1)} =\\frac{2-\\log_2(3)}{2\\log_2(3)}.\n \\end{equation*}\n Hence, $x\\approx 0.3618$ and $ \\frac{\\partial g}{\\partial x}$ does not have any other zeros on $0<x\\leq 1\/2$. This means that $g$ has only one extremal point and can then only have two zeros on $0<x\\leq 1\/2$. The zero at $x=1\/2$ corresponds to a minimum. This means that the zero of $ \\frac{\\partial g}{\\partial x}$ at $x\\approx 0.3618$ is a maximum of $g$. Hence the other zero of g is between $0$ and $\\approx 0.3618$. However, this other zero corresponds to a maximum of $f$. The minimum of $f$ can thus be at $x=0$ or $x=1\/2$. This implies $\\alpha=\\pi\/3$ or $\\alpha=\\pi\/2$. It is easy to verify that $\\alpha=\\pi\/3$ would lead to $\\beta=\\pi\/3$, and $\\alpha=\\pi\/2$ yields $\\beta=0$. Thus, the minimum of the $p$-frame potential corresponds to either an orthonormal basis plus one repeated element ($\\alpha=\\pi\/2$, $\\beta=0$) or an equiangular FUNTF ($\\alpha=\\beta=\\pi\/3$). One easily checks that both situations lead to the same global minimum.\n \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\nIn view of Theorem \\ref{theorem:d+1} and Corollary \\ref{theorem:3 points}, we have the following conjecture: \n\\begin{conjecture}\\label{conjecture:p_0}\nLet $N=d+1$  and $p_0=\\frac{\\log(\\frac{d(d+1)}{2})}{\\log(d)}$. Then $$\\FP_{p_{0},N}(\\{x_{k}\\}_{k=1}^{N}) \\geq N+2,$$ and equality holds if and only if $\\{x_i\\}_{i=1}^N$ is an orthonormal basis plus one repeated vector or an equiangular FUNTF.\n\\end{conjecture}\n\nOne can check that $1<p_0<2$, for $d>1$. According to Proposition \\ref{prop:potential for p>2}, the minimizers of the $p$-frame potential for $2<p<\\infty$ are exactly the equiangular FUNTFs. Thus, our conjecture essentially addresses the range $0<p<2$. \n\n\\begin{remark}\nAlthough we are primarily interested in real unit norm vectors $\\{x_i\\}_{i=1}^N\\subset \\R^d$, we should mention that Theorem \\ref{theorem:Benedetto Fickus}, Theorem~\\ref{prop:k multiples and small p}, Equation \\eqref{eq:Welch}, and Propositions \\ref{prop:potential for p>2} and \\ref{prop:second one} still hold for complex vectors $\\{z_i\\}_{i=1}^N\\subset \\C^d$ that have unit norm. The constraints on $N$ and $d$ that allow for the existence of a complex FUNTF are slightly weaker than in the real case~\\cite{Sustik:2007aa}.\n\\end{remark}\n\n\n\n\n\n\n\n\n\n \n\\section{The probabilistic $p$-frame potential}\\label{section:intro prob}\n\nThe present section is dedicated to introducing a probabilistic version of the previous section. We shall consider probability distributions on the sphere rather than finite point sets. Let $\\mathcal{M}(S^{d-1},\\mathcal{B})$ denote the collection of probability distributions on the sphere with respect to the Borel sigma algebra $\\mathcal{B}$. \n\nWe begin by introducing the probabilistic $p$-frame which generalizes the notion of probabilistic frames introduced in~\\cite{Ehler:2010aa}. \n\n\\begin{definition}\\label{probpframe}\nFor $0<p<\\infty$, we call $\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})$ a \\emph{probabilistic $p$-frame} for $\\R^d$ if and only if there are constants $A,B>0$ such that\n \\begin{equation}\\label{ppframeineq}\nA\\|y\\|^p\\leq  \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) \\leq B\\|y\\|^p, \\quad\\forall y\\in\\R^d.\n \\end{equation}\n We call $\\mu$ a \\emph{tight probabilistic $p$-frame} if and only if we can choose $A=B$.\n \\end{definition}\n Due to Cauchy-Schwartz, the upper bound $B$ always exists. \nConsequently, in order to check that $\\mu$ is a probabilistic $p$-frame one only needs to focus on the lower bound $A$. \n\n Since the uniform surface measure $\\sigma$ on $S^{d-1}$ is invariant under orthogonal transformations, one can easily check that it constitutes a tight probabilistic $p$-frame, for any $0<p<\\infty$. \n\n \n\n\nGiven a probability measure $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, we call \n\\begin{equation*}\nF : \\R^d \\rightarrow L_p(S^{d-1},\\mu),\\quad x\\mapsto \\langle x,\\cdot\\rangle \n\\end{equation*}\nthe \\emph{analysis operator}. It is trivially seen that $$\\|F(y)\\|_{L_{p}(S^{d-1}, \\mu)}=\\|\\langle x,y\\rangle\\|_{L_{p}(S^{d-1}, \\mu)} \\leq \\|y\\|$$ for all $0<p\\leq \\infty$. The dual of $F$ is called \\emph{synthesis operator} and is given by\n\\begin{equation*}\nF^* : L_q(S^{d-1},\\mu) \\rightarrow \\R^d,\\quad f\\mapsto \\int_{S^{d-1}} f(x)xd\\mu(x),\n\\end{equation*}\nwhere $1=\\frac{1}{p}+\\frac{1}{q}$, and $1\\leq p \\leq \\infty$. In fact, $F^*$ is well-defined and bounded operator on all $L_{r}(S^{d-1}, \\mu)$ where \n$1\\leq r \\leq \\infty$. Indeed, for  $f \\in L_{r}(S^{d-1}, \\mu)$ we have \n $$\n\\|F^{*}(f)\\|\\leq \\int_{S^{d-1}}|f(x)|\\|x\\|d\\mu(x)= \\int_{S^{d-1}}|f(x)|d\\mu(x)\\leq \\|f\\|_{L_{r}}.$$ \nGiven $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, the second moments matrix of $\\mu$ is the $d\\times d$ matrix defined by \n\\begin{equation}\\label{ppfoperator}\nS=F^* F = \\bigg(\\int_{S^{d-1}} x_i x_j d\\mu(x) \\bigg)_{i,j}\n\\end{equation}\nwith respect to the canonical basis for $\\R^d$. As we show next, the second moments matrix $S$ plays a key role in determining if $\\mu$ is a probabilistic $p$-frame. We refer to \\cite{Christensen:2003ab}, where a  result was proved for similar discrete $p$-frames. \n\n\n\\begin{proposition}\\label{pfonto}\nIf $1\\leq p <\\infty$, then $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame if and only if $F^*$ is onto. \n\\end{proposition}\n\n\\begin{proof}\nAs mentioned earlier the upper bound in the probabilistic $p$-frame definition always holds. So we only need to show the equivalence between the lower bound and the surjectivity of $F^{*}$. \n\nAssume that $F^{*}$ is surjective. Since $F$ is injective, then $S$ is invertible and for each $y \\in \\R^d$ we have\n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L^{p}\\to L^{p'}}|,$$ which can be estimated as follows:\n$$\\|y\\|^{2} \\leq \\|F(y)\\|_{L^{p}}\\|F(S^{-1} y)\\|_{L^{p'}}\\leq C \\|F(y)\\|_{L^{p}}\\|S^{-1} (y)\\|_{\\R^{d}}\\leq C \\|F(y)\\|_{L^{p}}\\|y\\|_{\\R^{d}}.$$\nTherefore, for all $y\\neq 0 \\in \\R^d$ we have $1\/C \\|y\\| \\leq \\|F(y)\\|_{L^{p}}$ that is $$ 1\/C \\|y\\|^{p}\\leq \\int_{S^{d-1}}|\\langle x,y\\rangle|^p d\\mu(x).$$ Thus $\\mu$ is a probabilistic $p$-frame.\n\nNow assume that $\\mu$ is a probabilistic $p$-frame, but that $F^{*}$ is not surjective. Then, there exists $z \\neq 0 \\in \\R^d$ such that $\\langle z, F^{*}f\\rangle=0$ for all $f \\in L_{p'}(S^{d-1}, \\mu)$. Consequently $$\\langle z, F^{*}f\\rangle_{\\R^{d}}=\\langle F(z), f\\rangle_{L_{p}\\to L_{p'}}=\\int_{S^{d-1}}f(x)\\langle x , z\\rangle, d\\mu(x)=0$$ for all $f \\in L_{p'}$. A contradiction argument leads to $\\langle x, z\\rangle =0$ for all $x \\in S^{d-1}$ which implies that $z=0$. Thus $F^{*}$ is surjective.\n\\end{proof}\n \nThe second moments matrix can also be  used to show that a probabilistic $p$-frame gives rise to a reconstruction formula that extends the finite frame expansion in \\eqref{eq:reconstruction for frames}. In addition, the next result generalizes the reconstruction formula for tight probabilistic frames obtained in \\cite[Lemma 3.7]{Ehler:2010aa}.\n\n\n\\begin{proposition}\\label{reconppf} Let $0<p<\\infty$ and assume that $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame for $\\R^d$. Set $\\tilde{\\mu}=\\mu \\circ S$. Then for each $y \\in \\R^d$, we have\n\\begin{equation}\\label{reconsppf}\ny= \\int_{S^{-1}(S^{d-1})}Sz \\, \\langle z, y\\rangle \\, d\\tilde{\\mu}(z)= \\int_{S^{-1}(S^{d-1})}z \\, \\langle Sz, y\\rangle \\, d\\tilde{\\mu}(z).\n\\end{equation} \n\\end{proposition}\n\n\\begin{proof}\nThe result follows by noticing that $y=SS^{-1}y=S^{-1}Sy$.\n\\end{proof}\n\n\nThe above result motivates the following definition:\n\n\\begin{definition}\\label{candualppf} Let $0< p < \\infty$. If $\\mu \\in \\mathcal{M}(S^{d-1},\\mathcal{B})$ is a probabilistic $p$-frame with frame operator $S$, then $\\tilde{\\mu}=\\mu \\circ S \\in \\mathcal{M}(S^{-1}(S^{d-1}),S^{-1}\\mathcal{B})$ is called the \\emph{probabilistic canonical dual frame} of $\\mu$\n\\end{definition}\n\nNote that if $\\mu$ in Proposition \\ref{reconppf} is the counting measure corresponding to a FUNTF $\\{x_i\\}_{i=1}^N$, then $\\tilde{\\mu}$ is the counting measure associated to the canonical dual frame of $\\{x_i\\}_{i=1}^N$. \n\n\n\n\n\n\n\n\\begin{lemma}\\label{equivappf} a) If $\\mu$ is probabilistic frame, then it is a probabilistic $p$-frame for all $1\\leq p< \\infty$. Conversely, if $\\mu$ is a probabilistic $p$-frame for some $1\\leq p < \\infty$, then it is a probabilistic frame.\n\n\\noindent b) Let $1\\leq p < \\infty$. If $\\mu$ is a probabilistic $p$-frame, then so is the canonical dual $\\tilde{\\mu}$.\n\\end{lemma}\n\n\n\\begin{proof}\na) \nAssume that $\\mu$ is a probabilistic frame and let $1\\leq p < \\infty$. Then, we only need to check that the lower inequality of~\\eqref{ppframeineq} holds, since the corresponding upper bound is trivial. \nBy Proposition~\\ref{pfonto} (applied to $p=2$), $S=F^{*}F$ is invertible and for each $y \\in \\R^d$ we have\n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L^{p}\\to L^{p'}}|,$$ which can be estimated as follows:\n$$\\|y\\|^{2} \\leq \\|F(y)\\|_{L_{p}}\\|F(S^{-1} y)\\|_{L_{p'}}\\leq C \\|F(y)\\|_{L_{p}}\\|S^{-1} (y)\\|_{\\R^{d}}\\leq C \\|F(y)\\|_{L_{p}}\\|y\\|_{\\R^{d}}.$$ Therefore, for all $y\\neq 0 \\in \\R^d$ we have $1\/C \\|y\\| \\leq \\|F(y)\\|_{L_{p}}$ that is $$ 1\/C \\|y\\|^{p}\\leq \\int_{S^{d-1}}|\\langle x,y\\rangle|^p d\\mu(x).$$ \nFor the converse, assume that $\\mu$ is a probabilistic $p$-frame for some $p>2$. Then, for all $y\\neq 0 \\in \\R^d$, \n\\begin{align*}\nA\\|y\\|^p&\\leq  \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)\\\\\n&=\\int_{S^{d-1}} |\\langle x,y\\rangle|^2\\, |\\langle x,y\\rangle|^{p-2}\\, d\\mu(x)\\\\\n&  \\leq \\int_{S^{d-1}} \\|x\\|^{p-2}\\, \\|y\\|^{p-2}\\, |\\langle x,y\\rangle|^2 d\\mu(x) \\\\\n&= \\|y\\|^{p-2}\\,  \\int_{S^{d-1}} |\\langle x,y\\rangle|^2 d\\mu(x),\n\\end{align*}from which it follows that $$A\\|y\\|^2 \\leq \\int_{S^{d-1}} |\\langle x,y\\rangle|^2 d\\mu(x).$$\n\nIf $\\mu$ is a probabilistic $p$-frame for some $p<2$. Then, for all $y\\neq 0 \\in \\R^d$, \n$$\\|y\\|^{2}=|\\langle Sy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle F^{*}Fy,S^{-1}y\\rangle_{\\R^{d}}|=|\\langle Fy,FS^{-1}y\\rangle_{L_{p}\\to L_{p'}}|,$$ which can be estimated by  \n$$\\|y\\|^{2} \\leq \\|Fy\\|_{L_{p}}\\|FS^{-1}y\\|_{L_{p'}}\\leq C \\|Fy\\|_{L_{2}}\\|y\\|,$$ where we have used the fact that for $p<2$, $L_{2}(S^{d-1}, \\mu) \\subset L_{p}(S^{d-1}, \\mu)$. This conclude the proof of a). \n\n\nb) If $\\mu$ is a probabilistic $p$-frame for some $1\\leq p < \\infty,$ then by a) $\\mu$ is a probabilistic frame. In this case, $\\tilde{\\mu}$ is known to be a probabilistic frame, cf.~\\cite{Ehler:2010aa}, and thus a probabilistic $p$-frame. \n\\end{proof}\n\n\n \n\n\n\nWe are particularly interested in tight probabilistic $p$-frame potentials, which we seek to characterize in terms of minimizers of appropriate potentials. This motivates the following definition: \n\n \\begin{definition}\\label{profframpot}\nFor $0<p<\\infty$ and $\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})$, the \\emph{probabilistic $p$-frame potential} is defined by  \n  \\begin{equation}\\label{eq:prob p pot}\n \\PFP(\\mu,p) = \\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y).\n \\end{equation}\n\\end{definition}\n\nFrom the weak-star-compactness of the collection of all probability distributions on the sphere, we can deduce that $\\PFP(\\mu, p)$ admits a minimizer which satisfies\n\\begin{equation}\\label{minppfp}\n\\PFP(p) = \\min_{\\mu\\in\\mathcal{M}(S^{d-1},\\mathcal{B})} \\PFP(\\mu,p).\n\\end{equation}\n\n \n\n \nWe now turn to the minimizers of the probabilistic frame potential $\\PFP(\\mu)$. In the process, we extend some ideas developed  in \\cite{Bjoerck:1955aa} to the probabilistic frame potential. \n\\begin{proposition}\\label{prop:1}\nLet $0<p<\\infty$ and let $\\mu$ be a minimizer of \\eqref{eq:prob p pot}, then \n\\begin{itemize}\n\\item[\\textnormal{(1)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) = \\PFP(p)$, for all $y\\in\\supp(\\mu)$,\n\\item[\\textnormal{(2)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)\\geq \\PFP(p)$, for all $y\\in S^{d-1}$.\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nThe proof will use the following observation. Let  $\\mu$ be a probability measure on $S^{d-1}$ and choose a measure  $\\nu$, such that $\\nu(S^{d-1})=0$ and $\\mu+\\varepsilon \\nu\\geq 0$, for all $0\\leq \\varepsilon\\leq 1$. Let us also introduce the notation $$ \\PFP(\\mu,\\nu,p):=\\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\nu(y).$$ We then obtain\n\\begin{align*}\n\\PFP(\\mu) & \\leq \\PFP(\\mu+\\varepsilon \\nu,p)\\\\\n& = \\PFP(\\mu,p)+\\varepsilon^2\\PFP(\\nu,p) + 2\\varepsilon \\PFP(\\mu,\\nu,p)\\\\\n& = \\PFP(\\mu)+\\varepsilon^2\\PFP(\\nu,p) + 2\\varepsilon \\PFP(\\mu,\\nu,p).\n\\end{align*} \nWe thus have $0\\leq \\varepsilon \\PFP(\\nu,p)+2\\PFP(\\mu,\\nu,p)$, for all $0\\leq \\varepsilon\\leq 1$, which implies $\\PFP(\\mu,\\nu,p)\\geq 0$. \n\nWe now prove $(1)$ using a contradiction argument. In particular, assume that $(1)$ does not hold. This implies that there are $y_1, y_2\\in \\supp(\\mu)$ such that \n\\begin{equation*}\na := \\int_{S^{d-1}} |\\langle x,y_2\\rangle|^p d\\mu(x) < \\int_{S^{d-1}} |\\langle x,y_1\\rangle|^p d\\mu(x) =: b.\n\\end{equation*}\nSet $P_\\mu(y)=\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x)$. \nLet $K$ be an open ball around $y_1$ in $S^{d-1}$ and so small that $y_2\\not\\in K$ and that the oscillation of $P_\\mu(y)$ on $K$ is smaller than $\\frac{b-a}{2}$. Let $m=\\mu(K)>0$. One can check that the measure $\\nu$ defined by \n\\begin{equation*}\n\\nu(E) := m\\delta_{y_2}(E)-\\mu(E\\cap K),\\quad E\\in\\mathcal{B},\n\\end{equation*}\nsatisfies $\\nu(S^{d-1})=0$, and $\\mu +\\epsilon \\nu \\geq 0$. Hence, $\\PFP(\\mu,\\nu,p)\\geq 0$. On the other hand, we can estimate\n$$\n\\PFP(\\mu,\\nu,p)  = \\int_{S^{d-1}} P_\\mu(y)d\\nu(y)= P_\\mu(y_2)m - \\int_K P_\\mu(y)d\\mu(y)= am - \\int_K P_\\mu(y)d\\mu(y)$$ and so\n\n$$\\PFP(\\mu,\\nu,p)  \\leq am - (b-\\frac{b-a}{2})m  = - \\frac{b-a}{2}m <0.$$\n\nThis is a contradiction to $\\PFP(\\mu,\\nu,p)\\geq 0$ and implies that there is a constant $C$ such that $P_\\mu(y)=C$, for all $y\\in\\supp(\\mu)$.  \nWe still have to verify that the constant $C$ is in fact $\\PFP(p)$: \n\\begin{align*}\n\\PFP(p)  = \\PFP(\\mu,p) & =  \\int_{S^{d-1}} P_\\mu(y)d\\mu(y) \\\\\n& =  \\int_{\\supp(\\mu)} P_\\mu(y)d\\mu(y)\\\\\n& =  \\int_{\\supp(\\mu)} C d\\mu(y) = C.\n\\end{align*}\n\nThe proof of $(2)$ is similar to the one above, and so we omit it. \n\\end{proof}\nThe following result is an immediate consequence of Proposition~\\ref{prop:1}. \n\n\\begin{corollary}\\label{theorem:tight p frame is necessary}\nLet $0<p<\\infty$ and let $\\mu$ be a minimizer of \\eqref{eq:prob p pot}, then\n\\begin{itemize}\n\\item[\\textnormal{(1)}] $\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) = \\PFP(p)\\|y\\|^p$, for all $\\frac{y}{\\|y\\|}\\in\\supp(\\mu)$,\n\\item[\\textnormal{(2)}] $\\supp(\\mu)$ is a complete subset of $\\R^d$. \n\\end{itemize}\n\\end{corollary}\n\\begin{proof}\n$ (1)$ directly follows from $(1)$ in Proposition \\ref{prop:1}.\n\n$(2)$ If $\\supp(\\mu)$ is not complete in $\\R^d$, then there is an element $y\\in S^{d-1}$ that is in the orthogonal complement. However, this contradicts (2) in Proposition \\ref{prop:1}. \n\\end{proof}\n\nWe can now characterize the minimizers of the probabilistic $p$-frame potential when  $0<p<2$. In fact, we shall show that these minimizers are discrete probability measures, and the following theorem is the analogue of Proposition \\ref{prop:k multiples and small p}:\n \\begin{theorem}\\label{theorem:p less than 2}\n Let $0<p<2$, then the minimizers of  \\eqref{eq:prob p pot} are exactly those probability distributions $\\mu$ that satisfy both, \n \\begin{itemize}\n \\item[(i)] there is an orthonormal basis $\\{x_1,\\ldots,x_d\\}$ for $\\R^d$ such that \n \\begin{equation*}\n \\{x_1,\\ldots,x_d\\} \\subset \\supp(\\mu) \\subset \\{\\pm x_1,\\ldots,\\pm x_d\\},\n \\end{equation*}\n \\item[(ii)] there is $f:S^{d-1}\\rightarrow \\R$ such that $\\mu(x) = f(x) \\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$ and\n \\begin{equation*}\n f(x_i) +f(-x_i) = \\frac{1}{d}.\n \\end{equation*} \n \\end{itemize} \n \\end{theorem}\n The measure $\\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$ in Theorem \\ref{theorem:p less than 2} denotes the counting measure of the set $\\{\\pm x_i : i=1,\\ldots,d\\}$.\n \n \n \\begin{proof}\n Since $0\\leq |\\langle x,y\\rangle|\\leq 1$, for $x,y\\in S^{d-1}$, we have $\\PFP(\\mu,p)\\geq \\PFP(\\mu,2)$. In ~\\cite[Theorem 3.10]{Ehler:2010aa} it was shown that the normalized counting measure $\\frac{1}{d}\\nu_{x_1,\\ldots,x_d}$ of an orthonormal basis minimizes $\\PFP(\\cdot,2)$. Due to $\\PFP(\\frac{1}{d}\\nu_{x_1,\\ldots,x_d},2) = \\PFP(\\frac{1}{d}\\nu_{x_1,\\ldots,x_d},p)$, we obtain that $\\frac{1}{d}\\nu_{x_1,\\ldots,x_d}$ also minimizes $\\PFP(\\cdot,p)$ and hence $\\PFP(p)=\\PFP(2)$. \n \n In the following, we prove that all minimizers of $\\PFP(\\cdot,p)$ are essentially induced by an orthonormal basis. Let $\\mu$ be a minimizer and let $v,w\\in \\supp(\\mu)$. We first show that $|\\langle v,w\\rangle |\\in \\{0,1\\}$. The implications $\\langle v,w\\rangle =1$ if and only if $v=w$ and $\\langle v,w\\rangle =-1$ if and only if  $v=-w$ are trivial. \n \n Suppose now that $v\\neq \\pm w$ and $\\langle v,w\\rangle \\neq 0$, then there exist  $\\varepsilon>0$ and $\\delta_\\varepsilon>0$ such that \n \\begin{itemize}\n \\item[(a)] $B_\\varepsilon(v)\\cap B_\\varepsilon(w)=\\emptyset$ and $\\mu(B_\\varepsilon(v)), \\mu(B_\\varepsilon(w)) \\geq \\delta_\\varepsilon$. \n \\item[(b)] for all $x\\in B_\\varepsilon(v)$ and $y\\in B_\\varepsilon(w)$, $|\\langle x,y\\rangle |^p\\geq |\\langle x,y\\rangle |^2+\\varepsilon$.\n \\end{itemize}\nBy using $B=B_\\varepsilon(v)\\times B_\\varepsilon(w)$, this implies\n\\begin{align*}\n\\PFP(\\mu,p) & = \\int_{B} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) + \\int_{S^{d-1}\\times S^{d-1}\\setminus B } |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y)\\\\\n& \\geq \\int_{B} (|\\langle x,y\\rangle|^2+\\varepsilon) d\\mu(x) d\\mu(y) + \\int_{S^{d-1}\\times S^{d-1}\\setminus B } |\\langle x,y\\rangle|^2 d\\mu(x) d\\mu(y)\\\\\n& = \\PFP(\\mu,2) + \\varepsilon \\mu(B_\\varepsilon(v)) \\mu(B_\\varepsilon(w))\\\\\n&\\geq \\PFP(\\mu,2) +\\varepsilon \\delta_\\varepsilon^2 > \\PFP(\\mu,2),\n\\end{align*}\n which is a contradiction. Thus, we have verified that $|\\langle x,y\\rangle|\\in \\{0,1\\}$, for all $x,y\\in \\supp(\\mu)$. Distinct elements in $\\supp(\\mu)$ are then either orthogonal to each other or antipodes. According to Corollary \\ref{theorem:tight p frame is necessary}, $\\supp(\\mu)$ is complete in $\\R^d$. Thus, there must be an orthonormal basis $\\{x_i\\}_{i=1}^d$ such that\n \\begin{equation*}\n  \\{x_1,\\ldots,x_d\\} \\subset \\supp(\\mu) \\subset \\{\\pm x_1,\\ldots,\\pm x_d\\}.\n \\end{equation*} \nConsequently, there is a density $f:S^{d-1}\\rightarrow\\R$ that vanishes on $S^{d-1}\\setminus \\supp(\\mu)$ such that $\\mu(x)=f(x)\\nu_{\\pm x_1,\\ldots,\\pm x_d}(x)$.  \n \nTo verify that $f$ satisfies (ii), let us define $\\tilde{f}:S^{d-1}\\rightarrow \\R$ by \n\\begin{equation*}\n\\tilde{f}(x)=\\begin{cases} f(x)+f(-x),& x\\in\\{x_1,\\ldots,x_d\\}\\\\\n0,& \\text{ otherwise. } \n\\end{cases}\n\\end{equation*}\nThis implies that $\\tilde{\\mu}(x)=\\tilde{f}(x)\\nu_{x_1,\\ldots,x_d}(x)$ is also a minimizer of $\\PFP(\\cdot,2)$. But the minimizers of the  probabilistic frame potential for $p=2$ have been investigated in~\\cite[Section 3]{Ehler:2010aa}. We can follow the arguments given there to obtain $\\tilde{f}(x_i)=\\frac{1}{d}$, for all $i=1,\\ldots,d$. \n\\end{proof}\n\n\n \n For even integers $p$, we can  give the minimum of $\\PFP(\\mu, p)$ and characterize its minimizers. The following theorem generalizes Theorem \\ref{theorem:p even integer discrete}. Moreover, note that the bounds are now sharp, i.e., for any even integer $p$, there is a probabilistic tight $p$-frame: \n \n \\begin{theorem}\\label{theorem:p even integer}\n Let $p$ be an even integer. For any probability distribution $\\mu$  on $S^{d-1}$,  \n  \\begin{equation*}\n \\PFP(\\mu, p)=\\int_{S^{d-1}}\\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) \\geq  \\frac{1\\cdot 3\\cdot 5\\cdots(p-1)}{d(d+2)\\cdots (d+p-2) },\n \\end{equation*}\nand equality holds if and only if $\\mu$ is a probabilistic tight $p$-frame. \n \\end{theorem}\n \n\n\n\n \\begin{proof}\n Let $\\alpha=\\frac{d}{2}-1$ and consider the Gegenbauer polynomials $\\{C_{n}^{\\alpha}\\}_{n\\geq 0}$ defined by \n \\begin{equation*}\n C_0^\\alpha(x)  = 1, \\qquad C_1^\\alpha(x) = 2 \\alpha x,\n \\end{equation*}\n \\begin{align*}\nC_{n}^\\alpha(x) &= \\frac{1}{n}[2x(n+\\alpha-1)C_{n-1}^\\alpha(x) - (n+2\\alpha-2)C_{n-2}^\\alpha(x)]\\\\\n&= C_n^{(\\alpha)}(z)=\\sum_{k=0}^{\\lfloor n\/2\\rfloor} (-1)^k\\frac{\\Gamma(n-k+\\alpha)}{\\Gamma(\\alpha)k!(n-2k)!}(2z)^{n-2k}.\n\\end{align*}\n$\\{C_{n}^{(\\alpha)}\\}_{n=1}^s$ is an orthogonal basis for the collection of polynomials of degree less or equal to $s$ on the interval $[-1,1]$ with respect to the weight\n\\begin{equation*}\nw(z) = \\left(1-z^2\\right)^{\\alpha-\\frac{1}{2}},\n\\end{equation*} \ni.e., for $m\\neq n$,\n \\begin{equation*}\n \\int_{-1}^1 C_n^{(\\alpha)}(x)C_m^{(\\alpha)}(x)w(x)\\,dx = 0.\n \\end{equation*}\n They are normalized by\n \\begin{equation*}\n \\int_{-1}^1 \\left[C_n^{(\\alpha)}(x)\\right]^2(1-x^2)^{\\alpha-\\frac{1}{2}}\\,dx = \\frac{\\pi 2^{1-2\\alpha}\\Gamma(n+2\\alpha)}{n!(n+\\alpha)[\\Gamma(\\alpha)]^2}.\n \\end{equation*}\nThe polynomials $t^p$, $p$ an even integer, can be represented by means of\n\\begin{equation*}\nt^p=\\sum_{k=0}^p \\lambda_k C^{\\alpha}_k(t).\n\\end{equation*}\nIt is known (see, e.g.,~\\cite{Bachoc:2005aa,Delsarte:1977aa}) that $\\lambda_i> 0$, $i=0,\\ldots,p$, and $\\lambda_0$ is given by\n \\begin{equation*}\n\\lambda_0= \\frac{1}{c}\\int_{-1}^1 t^p w(t) dt,\n \\end{equation*}\n where \n \\begin{equation*}\n c = \\frac{\\pi 2^{d+3}\\Gamma(d-2) }{(\\frac{d}{2}-1)\\Gamma(\\frac{d}{2}-1)^2}.\n \\end{equation*}\n Moreover,  $C^\\alpha_k$ induces a positive kernel, i.e., for $\\{x_i\\}_{i=1}^N\\subset S^{d-1}$ and $\\{u_i\\}_{i=1}^N\\subset \\R$,\n \\begin{equation*\n \\sum_{i,j=1}^{N} u_iC^{\\alpha}_k (\\langle x_i,x_j\\rangle )u_j \\geq  0, \\quad \\forall k=0,1,2,...\n \\end{equation*}\n see~\\cite{Bachoc:2005aa,Delsarte:1977aa}. Note that the probability measures with finite support are weak star dense in $\\mathcal{M}(S^{d-1},\\mathcal{B})$. Since $C^{\\alpha}_k$ is continuous, we obtain, for all $\\mu\\in \\mathcal{M}(S^{d-1},\\mathcal{B})$, \n \\begin{equation*} \n \\int_{S^{d-1}} \\int_{S^{d-1}} C^{\\alpha}_k (\\langle x,y\\rangle )d\\mu(x) d\\mu(y) \\geq  0, \\quad \\forall k=0,1,2,...\n \\end{equation*}\nWe can then estimate\n\\begin{align*}\n\\int_{S^{d-1}} \\int_{S^{d-1}} |\\langle x,y\\rangle|^p d\\mu(x) d\\mu(y) & = \\int_{S^{d-1}} \\int_{S^{d-1}}\\sum_{k=0}^p \\lambda_k C^{\\alpha}_k (\\langle x,y\\rangle )d\\mu(x) d\\mu(y)\\\\\n& = \\sum_{k=0}^p \\lambda_k \\int_{S^{d-1}} \\int_{S^{d-1}}C^{\\alpha}_k (\\langle x,y\\rangle ) d\\mu(x) d\\mu(y) \\geq \\lambda_0.\n\\end{align*}\nFrom the results in \\cite{Seidel:2001aa}, one can deduce that \n \\begin{equation*}\n\\lambda_0=  \\frac{1\\cdot 3\\cdot 5\\cdots(2t-1)}{d(d+2)\\cdots (d+2t-2) },\n \\end{equation*}\nwhich provides the desired estimate.\n\nWe still have to address the ``if and only if'' part. Equality holds if and only if $\\mu$ satisfies \n \\begin{equation*}\n  \\int_{S^{d-1}}\\int_{S^{d-1}} C^{\\alpha}_k (\\langle x,y\\rangle ) d\\mu(x) d\\mu(y) = 0, \\quad \\forall k=1,\\ldots, p. \n \\end{equation*}\nWe shall follow the approach outlined in \\cite{Venkov:2001aa} in which the analog  of Theorem~\\ref{theorem:p even integer discrete} was addressed  for finite symmetric collections of points. In this case, the finite symmetric sets of points lead to finite sums rather than integrals as above. The key ideas that we need in order to use the approach presented in \\cite{Venkov:2001aa} are: First, $\\tilde{\\mu}(E):=\\frac{1}{2}(\\mu(E)+\\mu(-E))$, for $E\\in\\mathcal{B}$, satisfies $\\PFP(\\tilde{\\mu},p) = \\PFP(\\mu,p)$. Thus, we can assume that $\\mu$ is symmetric. Secondly and more critically, the map \n\\begin{equation*}\ny\\mapsto \\int_{S^{d-1}} |\\langle x,y\\rangle |^p d\\mu(x)\n\\end{equation*}\nis a polynomial in $y$. In fact, the integral resolves in the polynomial's coefficients. These two observations enable us to follow the lines in \\cite{Venkov:2001aa}, and we can conclude the proof.\n\\end{proof}\n \n \\begin{remark}\nOne may speculate that Theorem \\ref{theorem:p even integer} could be extended to $p\\geq 2$ that are not even integers. This is not true in general. For $d=2$ and $p=3$, for instance, the equiangular FUNTF with $3$ elements induces a smaller potential than the uniform distribution. The uniform distribution is a probabilistic tight $3$-frame, but the equiangular FUNTF is not.\n\\end{remark}\n\n\n\n\n\\section*{Acknowledgements}\nThe authors would like to thank C.~Bachoc, W.~Czaja, C.~Wickman, and W.~S.~Yu for discussions leading to some of the results presented here. M.~Ehler was supported by the Intramural Research Program of the National Institute of Child Health and Human Development and by NIH\/DFG Research Career Transition Awards Program (EH 405\/1-1\/575910).  K.~A.~Okoudjou was partially supported by ONR grant N000140910324, by RASA from the Graduate School of UMCP, and by the Alexander von Humboldt foundation. \n\n\n\n\n\\bibliographystyle{plain}\n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\thispagestyle{empty}\n\nOne equivalent characterization of the amenability of an infinite group $G$, called the \\textit{F{\\o}lner condition}, is that the isoperimetric constant (also known as Cheeger constant) of its Cayley graph should be $0$.\nThat constant is defined as the infimum of $\\frac{|\\partial F|}{|F|}$ over all finite sets $F\\subset G$ with $|F|\\leq\\frac{1}{2}|G|$.\nAs the quotient cannot reach $0$, amenability of infinite groups is therefore characterized by the existence of a sequence of sets $F_n$ such that $\\frac{|\\partial  F_n|}{|F_n|}$ converges towards $0$, also known as a \\textit{F{\\o}lner sequence}.\nOne natural direction for studying the possible F{\\o}lner sequences on a given group is to ask how small the sets can be.\nWe consider the F{\\o}lner function.\nIt has classically been defined using the inner boundary:\n\\begin{equation}\\label{defdin}\n\t\\partial_{in}F=\\left\\{g\\in F:\\exists s\\in S\\bigcup S^{-1}:gs\\notin F\\right\\}.\n\\end{equation}\n\\begin{defi}\\label{foldef}\n\tThe \\textit{F{\\o}lner function} $\\Fol$ (or $\\Fol_S$; or $\\Fol_{G,S}$) of a group $G$ with a given finite generating set $S$ is defined on $\\N$ by\n\t$$\\Fol(n)=\\min\\left(|F|:F\\subset G,\\frac{|\\partial_{in}F|}{|F|}\\leq\\frac{1}{n}\\right).$$\n\\end{defi}\n\nRemark that $\\Fol(1)=1$ and that the values of the function are finite if and only if $G$ is amenable.\nIts values clearly depend on the choice of a generating set, but the functions arising from different generating sets (and more generally, functions arising from quasi-isometric spaces) are asymptotically equivalent.\nTwo functions are asymptotically equivalent if there are constants $A$ and $B$ such that $f(x\/A)\/B<g(X)<f(xA)B$.\nVarious articles have classified F{\\o}lner functions up to asymptotic equivalence, see for example~\\cite{BrieusselZheng,Erschler2003,Erschler2017,Gromov2009,Osserman1978,Pittet1995,Pittet2003,saloffcostezheng}.\nIn this paper, we will determine its exact values for one of the most basic examples of groups with exponential growth (we state our results in Section~\\ref{statesect}).\n\nThe classical isoperimetric theorem states that among domains of given volume in $\\R^n$, the minimal surface area is obtained on a ball (see survey by Osserman~\\cite[Section~2]{Osserman1978}).\nThe fact that if a minimum exists, it is realized only on the ball is obtained (in $\\R^2$) by Steiner in the XIX\\textsuperscript{th} century, using what is now called Steiner symmetrization (see Hehl~\\cite{hehl13}, Hopf~\\cite{hopf40}, Froehlich~\\cite{froehlich09}).\nThe existence of a minimum is obtained, in $\\R^3$, by Schwarz~\\cite{Schwarz1884}.\nAs $\\Z^n$ is quasi-isometric to $\\R^n$, this is also a first isoperimetric result for discrete groups.\nVaropoulos~\\cite{Varopoulis1985} shows more generally an isoperimetric inequality for direct products.\nPansu~\\cite{pansu83} (see also~\\cite{pansu82}) obtains one for the Heisenberg group $H_3$.\nOne central result is the Coulhon and Saloff-Coste inequality~\\cite{Coulhon1993}:\n\n\\begin{thm}[Coulhon and Saloff-Coste inequality]\\label{csc}\nConsider a group $G$ generated by a finite set $S$ and let $\\phi(\\lambda)=\\min(n:V(n)>\\lambda)$.\nThen for all finite sets $F$:\n\n$$\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{1}{8|S|\\phi(2|F|)}.$$\n\\end{thm}\nThe multiplicative constants can improved (see G\u00e1bor Pete~\\cite[Theorem~5.11]{pete}, Bruno Luiz Santos Correia~\\cite{csc2020}):\n\\begin{equation}\\label{csceq}\n\t\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{1}{2\\phi(2|F|)}.\n\\end{equation}\nThe result of Santos Correia is also announced for finite groups for $|F|\\leq\\frac{1}{2}|G|$.\nSantos Correia and Troyanov~\\cite{csc2021} show:\n\\begin{equation}\\label{csceqlam}\n\t\\frac{|\\partial_{in}F|}{|F|}\\geq\\left(1-\\frac{1}{\\lambda}\\right)\\frac{1}{\\phi(\\lambda|F|)}\n\\end{equation}\nfor $1<\\lambda\\leq\\frac{|G|}{|S|}$ (in particular, for arbitrarily large $\\lambda$ if $G$ is infinite).\nThe result is replicated in~\\cite{csccollab} without an upper bound on $\\lambda$.\n\nThe Coulhon and Saloff-Coste inequality (Theorem~\\ref{csc}) implies in particular that for a group with exponential growth, the F{\\o}lner function must also grow at least exponentially.\nIt can be obtained then that it is exactly exponential if one can describe F{\\o}lner sets with exponential growth.\nOne simple example is the lamplighter group $\\Z\\wr\\Z\/2\\Z$ with the standard generating set (\\ref{defst}).\nSimilarly, it is known that the F{\\o}lner functions of groups with polynomial growth are polynomial (see for example~\\cite[Section~I.4.C]{woess2000random}).\nAnother inequality on group isoperimetry is given by \u017buk~\\cite{Zuk2000}.\nVershik~\\cite{vershik1973countable} asks if F{\\o}lner function can be super-exponential, initiating the study of F{\\o}lner functions.\nHe suggests studying the wreath product $\\Z\\wr\\Z$ as a possible example.\nPittet~\\cite{Pittet1995} shows that the F{\\o}lner functions of polycyclic groups are at most exponential (and are therefore exponential for polycyclic groups with exponential growth).\nThis is true more generally for solvable groups of finite Pr\u00fcfer rank, see~\\cite{Pittet2003} and~\\cite{Kropholler2020}.\nThe first example of a group with super-exponential F{\\o}lner function is obtained by Pittet and Saloff-Coste~\\cite{PittetSaloffCoste} for $\\Z^d\\wr\\Z\/2\\Z$ with $d\\geq3$.\nLater the F{\\o}lner functions of wreath products with certain regularity conditions are described by Erschler~\\cite{Erschler2003} up to asymptotic equivalence.\nSpecifically, say that a function $f$ verifies property $(*)$ if for all $C>0$ there is $k>0$ such that $f(kn)>Cf(n)$.\nThe result of \\cite{Erschler2003} than states that if the F{\\o}lner function of a group $A$ verifies property $(*)$ (for some fixed generating set), then for any non-trivial group $B$, the F{\\o}lner function of $A\\wr B$ is $\\Fol_{A\\wr B}(n)=\\Fol_B(n)^{\\Fol_A(n)}$.\n\nOther examples with know F{\\o}lner functions have been presented by Gromov~\\cite[Section~8.2,Remark~(b)]{Gromov2009} for all functions with sufficiently fast growing derivatives.\nSaloff-Coste and Zheng~\\cite{saloffcostezheng} provide upper and lower bounds for it on, among others, \"bubble\" groups and cyclic Neumann-Segal groups, and those two bounds are asymptotically equivalent under certain conditions.\nRecently, Brieussel and Zheng~\\cite{BrieusselZheng} show that for any function $g$ that can be written as the inverse function of $x\/f(x)$ for some non-decreasing $f$ with $f(1)=1$ and $x\/f(x)$ also non-decreasing, there is a group the F{\\o}lner function of which is asymptotically equivalent to $\\exp(g(n))$.\nErschler and Zheng~\\cite{Erschler2017} obtain examples for a class of super-exponential functions under $\\exp(n^2)$ with weaker regularity conditions.\nSpecifically, for any $d$ and any non-decreasing $\\tau$ such that $\\tau(n)\\leq n^d$, there is a group $G$ and a constant $C$ such that\n\\begin{equation}\\label{annatyani}\n\tCn\\exp(n+\\tau(n))\\geq\\Fol_G(n)\\geq\\exp(\\frac{1}{C}(n+\\tau(n\/C))).\n\\end{equation}\nThe left-hand side of this inequality is always asymptotically equivalent to $\\exp(n+\\tau(n))$, and it suffices therefore that the right-hand side be asymptotically equivalent to that function to have a description of the F{\\o}lner function of $G$.\nNotice in particular that if $\\tau$ verifies condition $(*)$, this is verified.\nRemark that the conditions we mentioned only consider functions at least as large as $\\exp(n)$; it is an open question whether a F{\\o}lner function can have intermediate growth (see Grigorchuk~\\cite[Conjecture~5(ii)]{grigsurvey}).\nA negative answer would imply the Growth Gap Conjecture~\\cite[Conjecture~2]{grigsurvey}, which conjectures that the volume growth function must be either polynomial or at least as fast as $\\exp(\\sqrt{n})$.\nThose conjectures also have weak versions, which are equivalent to each other (see discussion after Conjecture~6 in~\\cite{grigsurvey}).\n\n\\section{Statement of results}\\label{statesect}\n\nIn this paper, we obtain the exact values of the F{\\o}lner function for the standard generating set $S=\\{t,\\delta\\}$ (see~(\\ref{defst})) on the lamplighter group $\\Z\\wr\\Z_2$ (see Definition~\\ref{deflamp}) where by $\\Z_2$ we denote $\\Z\/2\\Z$.\n\n\\begin{thm}\\label{thmmain}\nFor $n\\geq2$, the F{\\o}lner function of the lamplighter group $\\Z\\wr\\Z_2$ is, for the standard generating set: \n$$\\Fol(n)=2n2^{2(n-1)}.$$\n\\end{thm}\n\nWe also describe the sets that give rise to this function.\nSpecifically, we obtain that the standard F{\\o}lner sets $F_n=\\{(k,f):k\\in[\\![1,n]\\!],\\supp(f)\\subset[\\![1,n]\\!]\\}$ are optimal (see Definition~\\ref{optimal}) for the outer and edge boundary (see Section~\\ref{prel1}).\nWe then show that by Lemma~\\ref{equiv}, $F_n\\bigcup\\partial_{out}F_n$ is optimal for the inner boundary, from which Theorem~\\ref{thmmain} follows:\n\n\\begin{thm}\\label{thmlamp}\nConsider the lamplighter group $\\Z\\wr\\Z_2$ with the standard generating set.\n\\begin{enumerate}\n\t\\item For any $n\\geq2$ and any $F\\subset\\Z\\wr\\Z_2$ such that $|F|\\leq|F_n|$, we have\n\t$$\\frac{|\\partial_{edge}F|}{|F|}\\geq\\frac{|\\partial_{out}F|}{|F|}\\geq\\frac{|\\partial_{out}F_n|}{|F_n|}=\\frac{|\\partial_{edge}F_n|}{|F_n|},$$\n\tand if $|F|<|F_n|$, the inequality $\\frac{|\\partial_{out}F|}{|F|}>\\frac{|\\partial_{out}F_n|}{|F_n|}$ is strict,\n\t\\item From point $(1)$ it follows that for any $n\\geq2$ and any $F\\subset\\Z\\wr\\Z_2$ such that $|F|\\leq|F_n\\bigcup\\partial_{out}F_n|$, we have\n\t$$\\frac{|\\partial_{in}F|}{|F|}\\geq\\frac{|\\partial_{in}(F_n\\bigcup\\partial_{out}F_n)|}{|F_n\\bigcup\\partial_{out}F_n|},$$\n\tand if $|F|<|F_n\\bigcup\\partial_{out}F_n|$, the inequality is strict.\n\\end{enumerate}\nFurthermore, the sets that give equality are unique up to translation.\n\\end{thm}\n\nWe can substitute those values in the Coulhon and Saloff-Coste inequality in order to study the multiplicative constant.\nAs in~\\cite{csccollab}, we define\n\\begin{defi}\\label{constq}\nFor a group $G$ and a generating set $S$, denote\n$$C_{G,S}=\\sup\\left\\{c\\geq0:\\exists\\alpha\\geq0\\mbox{ such that }\\forall F\\subset G,\\frac{|\\partial_{in}F|}{|F|}\\geq c\\frac{1}{\\phi((1+\\alpha)|F|)}\\right\\},$$\nwhere $F$ is assumed to be finite and non-empty.\n\\end{defi}\nThe original inequality obtains that for all $G,S$, $C_{G,S}\\geq\\frac{1}{8|S|}$.\nThe results of \\cite[Theorem~5.11]{pete} and \\cite{csc2020} that we cited as Equation~\\ref{csceq} give a lower bound of $\\frac{1}{2}$.\nEquation~\\ref{csceqlam} from~\\cite{csc2021} further implies that $C_{G,S}\\geq1$ for all $G,S$.\n\nIn \\cite{csccollab}, it is shown that for groups of exponential growth:\n$$C_{G,S}=\\frac{\\liminf\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}.$$\n\\begin{prop}\\label{const}\nThe lamplighter group verifies\n\n$$C_{\\Z\\wr\\Z_2,S}=\\frac{\\lim\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}=\\frac{\\ln4}{\\ln(\\frac{1}{2}(1+\\sqrt{5}))}\\approx2,88$$\nfor the standard generating set. \n\\end{prop}\n\n\\begin{remark}\\label{const2}\nFor the switch-walk-switch generating set $S'=\\{t,\\delta,t\\delta,\\delta t,\\delta t\\delta\\}$, we have\n\n$$C_{\\Z\\wr\\Z_2,S'}=\\frac{\\liminf\\frac{\\ln\\Fol_{sws}(n)}{n}}{\\lim\\frac{\\ln V_{sws}(n)}{n}}\\leq2.$$\n\\end{remark}\n\nAnother direction that can be considered once one has exact evaluations of a F{\\o}lner function is studying the power series $\\sum_n\\Fol(n)x^n$.\nThe equivalent series have been studied for volume growth (see Grigorchuk-de la Harpe~\\cite[Section~(4)]{Grigorchuk1997}).\nOne central question that a lot of authors have considered is the rationality of those series as a function.\nFor the example shown here, the power series of the F{\\o}lner function is a rational function:\n$$\\sum_{n\\in\\N}\\Fol(n)x^n=\\frac{2x}{(4x-1)^2}.$$\n\nWe also obtain results for the Baumslag-Solitar group $BS(1,2)$ (see Definition~\\ref{defbs}), however only in respect to the edge boundary.\nTaking the notation from the definition, its standard sets are defined the same way as in the lamplighter group.\n\\begin{thm}\\label{bsthm}\nConsider the Baumslag-Solitar group $BS(1,2)$ with the standard generating set.\nThen for any $n\\geq2$ and any $F\\subset BS(1,2)$ such that $|F|\\leq|F_n|$, we have $\\frac{|\\partial_{edge}F|}{|F|}\\geq\\frac{|\\partial_{edge}F_n|}{|F_n|}$ (where $F_n$ are the standard F{\\o}lner sets), and if $|F|<|F_n|$, the inequality is strict.\n\\end{thm}\nThis result is not always true for $B(1,p)$ for larger $p$, and we provide a counter example in Example~\\ref{exbsp}.\nHowever this counter example uses that $p$ is significant when compared to the length of the interval defining the standard set, and it is possible that for $B(1,p)$ as well, standard sets are optimal above a certain size.\n\nWe present more detailed definitions in the next section.\nIn Section~\\ref{prelim2}, we present associated graphs, which are the main tool of the proof, and prove some general results.\nIn particular, we show Lemma~\\ref{equiv}, which will be used to obtain that part $(2)$ of Theorem~\\ref{thmlamp} follows from part $(1)$.\nIn Section~\\ref{mainsection}, we prove Theorem~\\ref{thmlamp}.\nIn Section~\\ref{csc-const-sect}, we prove Proposition~\\ref{const} and Remark~\\ref{const2}.\nFinally, in Section~\\ref{bssect}, we prove Theorem~\\ref{bsthm} and Example~\\ref{exbsp}.\n\n\\section{Preliminaries}\\label{prel1}\n\nThe concept of amenability finds its origins in a 1924 result by Banach and Tarski~\\cite{Banach-Tarski-original}, where they decompose a solid ball in $\\R^3$ into five pieces, and reassemble them into two balls using rotations.\nThat is now called the Banach-Tarski paradox.\nThe proof makes use of the fact that the group of rotations of $\\R^3$ admits a free subgroup.\nVon Neumann~\\cite{Neumann1929} considers it as a group property and introduces the concept of amenable groups.\nNowadays, there are multiple different characterizations of amenability; see books by Greenleaf~\\cite{greenleaf} and Wagon~\\cite{Banach-Tarski}, or an article by Ceccherini-Silberstein-Grigorchuk-la~Harpe~\\cite{MR1721355}, or a recent survey by Bartholdi~\\cite{bartholdi}.\n\n\\begin{defi}[F{\\o}lner criterion]\\label{folamdef}\n\tA group $G$ is amenable if and only if for every finite set $S\\subset G$ and every $\\varepsilon>0$ there exists a set $F$ such that\n\t\n\t$$|F\\Delta F.S|\\leq\\varepsilon|F|.$$\n\\end{defi}\n\nIf $G$ is finitely generated, it suffices to consider a single generating set $S$ instead of all finite sets.\nWe can also apply Definition~\\ref{folamdef} for $S\\bigcup S^{-1}\\bigcup\\{\\Id\\}$.\nThen $|F\\Delta(S\\bigcup S^{-1}\\bigcup\\{\\Id\\}).F|$ is the set of vertices in the Cayley graph of $G$ that are at a distance exactly $1$ from $F$.\nWe denote that the outer boundary $\\partial_{out}F$.\nThen the condition can be written as $\\frac{|\\partial_{out}F_n|}{|F_n|}\\leq\\varepsilon$, or in other words that the  infimum of those quotients should be $0$.\nRecall the definition (\\ref{defdin}) of the inner boundary.\nFinally, we consider $\\partial_{edge}F$ to be the set of edges between $F$ and its complement.\nRemark that while those values can differ, whether the infimum of $\\frac{|\\partial F|}{|F|}$ is $0$ or not does not depend on which boundary we consider.\n\nFor groups of subexponential growth, for every $\\varepsilon$, there is some $n$ such that the ball around the identity of radius $n$ is a corresponding F{\\o}lner set.\nNote that to obtain a F{\\o}lner sequence from this, one needs to consider a subsequence of the sequence of balls of radius $n$.\nIt is an open question whether in every group of subexponential growth, all balls form a F{\\o}lner sequence.\nFor groups of exponential growth, it is generally not sufficient to consider balls, and it is an open question whether there exists any group of exponential growth where some subsequence of balls forms a F{\\o}lner sequence (see for example Tessera~\\cite[Question~15]{Tessera2007}).\n\nFor two groups $A$ and $B$ and a function $f\\in B^A$, denote\n$$\\supp(f)=\\{a\\in A:f(a)\\neq\\Id_B\\}.$$\nLet $B^{(A)}$ be the set of functions from $A$ onto $B$ with finite support.\n\\begin{defi}\\label{deflamp}\n\tThe (restricted) wreath product $A\\wr B$ is the semidirect product $A\\ltimes B^{(A)}$ where $A$ acts on $B^{(A)}$ by translation.\n\\end{defi}\nWe can write the elements as $(a,f)$ with $a\\in A$ and $f\\in B^{(A)}$.\nThe group law is then $(a,f)(a',f')=(aa',x\\mapsto f(x)f'(a^{-1}x))$.\n\nGiven generating sets $S$ and $S'$ on $A$ and $B$ respectively, we can define a standard generating set on $A\\wr B$.\nIt consists of the elements of the form $(s,\\mathbb{Id_B})$ for $s\\in S$ (where $\\mathbb{Id_B}=\\Id_B$ for all $x\\in A$), as well as $(\\Id_A,\\delta_{\\Id_A}^{s'})$ for $s'\\in S'$ where $\\delta_{\\Id_A}^{s'}(\\Id_A)=s'$ and $\\delta_{\\Id_A}^{s'}(x)=\\Id_B$ for all other $x$.\nOne can verify that $(a,f)(s,\\mathbb{Id_B})=(as,f)$, and $(a,f)(\\Id_A,\\delta_{\\Id_A}^{s'})=(a,f+\\delta_a^{s'})$, or in other words the value of $f$ at the point $a$ is changed by $s'$.\n\nSimilarly, given F{\\o}lner sets $F_A$ and $F_B$ on $A$ and $B$ respectively, one obtains standard F{\\o}lner sets on $A\\wr B$: \n$$F=\\{(a,f):a\\in F_A,\\supp(f)\\subset F_A,\\forall x:f(x)\\in F_B\\}.$$\nTheir outer boundary is\n\\begin{equation*}\n\\begin{split}\n\\partial_{out}F&=\\{(a,f):a\\in\\partial_{out}F_A,\\supp(f)\\subset F_A,\\forall x:f(x)\\in F_B\\}\\\\\n&\\cup\\{(a,f):a\\in F_A,\\supp(f)\\subset F_A,f(a)\\in\\partial_{out}F_B,\\forall x\\neq a:f(x)\\in F_B\\}.\n\\end{split}\n\\end{equation*}\nAs $|F|=|F_A||F_B|^{|F_A|}$ and $|\\partial_{out}F|=|\\partial_{out}F_A||F_B|^{|F_A|}+|F_A||F_B|^{|F_A|-1}|\\partial_{out}F_B|$, we have\n$$\\frac{|\\partial_{out}F|}{|F|}=\\frac{|\\partial_{out}F_A|}{|F_A|}+\\frac{|\\partial_{out}F_B|}{|F_B|}.$$\n\nWe will focus on the lamplighter group $\\Z\\wr\\Z_2$.\nAs both of those groups have standard generating sets, this gives us a standard generating set on the lamplighter group:\n\n\\begin{equation}\\label{defst}\nS=\\{t,\\delta\\}\\mbox{ where }t=(1,\\mathbbold{0})\\mbox{ and }\\delta=(0,\\delta^1_0).\n\\end{equation}\n\nThe Baumslag-Solitar groups are defined as follows:\n\\begin{defi}\\label{defbs}\n\tThe Baumslag-Solitar group $BS(m,n)$ is the two-generator group given by the presentation $\\langle a,b:a^{-1}b^ma=b^n\\rangle$.\n\\end{defi}\nThe standard generating set is $\\{a,b\\}$.\n\nWe will focus on the groups $BS(1,p)$.\nThat group is isomorphic to the group generated by $x\\mapsto px$ and $x\\mapsto x+1$ (by mapping $a^{-1}$ and $b$ to them respectively).\nBy abuse of notation, we will also denote the images of $a$ and $b$ with the same letters.\nIn that group, any element can be written as $x\\mapsto p^nx+f$ with $n\\in\\Z$ and $f\\in\\Z[\\frac{1}{p}]$.\nWe then have $(x\\mapsto p^nx+f)a=x\\mapsto p^{n-1}x+f$ and $(x\\mapsto p^nx+f)b=x\\mapsto p^nx+(f+p^n)$.\n\nRemark that the subgroup $N=\\{x\\mapsto x+f:f\\in\\Z[\\frac{1}{p}]\\}$ is normal.\nIndeed, we have $(x\\mapsto p^{-n}(x-f))\\circ(x\\mapsto p^nx+f)=\\Id$ and\n$$(x\\mapsto p^{-n}(x-f'))\\circ(x\\mapsto x+f)\\circ(x\\mapsto p^nx+f')=x\\mapsto x+p^nf.$$\nThus $BS(1,p)$ is isomorphic to the semidirect product $\\Z\\ltimes\\Z[\\frac{1}{p}]$ defined by the action $n.f=p^nf$.\nWe therefore write the element $x\\mapsto p^nx+f$ as $(n,f)$.\nThe standard F{\\o}lner sets are then expressed in the same way as for wreath products.\nIn other words:\n$$F_n=\\{(k,f):k\\in[\\![0,n-1]\\!],f\\in\\Z,0\\leq f<p^n\\}.$$\n\n\\section{Main concepts of the proof}\\label{prelim2}\n\n\\begin{defi}\\label{optimal}\n\tWe will call a set $F$ in a group $G$ \\textit{optimal} with respect to the inner (respectively outer, edge) boundary if for any $F'$ with $|F'|\\leq|F|$, it is true that $\\frac{|\\partial_{in}F'|}{|F'|}\\geq\\frac{|\\partial_{in}F|}{|F|}$ (respectively $\\frac{|\\partial_{out}F'|}{|F'|}\\geq\\frac{|\\partial_{out}F|}{|F|}$, $\\frac{|\\partial_{edge}F'|}{|F'|}\\geq\\frac{|\\partial_{edge}F|}{|F|}$), and if $|F'|<|F|$, the inequality is strict.\n\\end{defi}\n\n\\begin{lemma}\\label{equiv}\nIf $F$ is optimal for the outer boundary, up to replacing $F$ with another optimal set of the same size, $F\\bigcup\\partial_{out}F$ is optimal for the inner boundary.\nFurthermore, this describes all optimal sets of that size.\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the large inequality.\nThe case for the strict inequality is equivalent.\n\nLet $F$ be optimal for the outer boundary and consider $F'$ such that $|F'|\\leq|F\\bigcup\\partial_{out}F|$ and the quotient of the inner boundary is smaller.\nWithout loss of generality, we can assume that $F'$ is optimal for the inner boundary.\n\nLet $F''=F'\\setminus\\partial_{in}F'$.\nObserve that $\\partial_{out}F''\\subset\\partial_{in}F'$.\nWe first claim that $F'$ being optimal implies that $\\partial_{out}F''=\\partial_{in}F'$.\nIndeed, $|F''\\bigcup\\partial_{out}F''|\\leq|F'|$, and\n$$\\frac{|\\partial_{in}(F''\\bigcup\\partial_{out}F'')|}{|F''\\bigcup\\partial_{out}F''|}\\leq\\frac{|\\partial_{out}F''|}{|F''\\bigcup\\partial_{out}F''|}=\\frac{\\frac{|\\partial_{out}F''|}{|F''|}}{1+\\frac{|\\partial_{out}F''|}{|F''|}},$$\nwhile\n$$\\frac{|\\partial_{in}F'|}{|F'|}=\\frac{\\frac{|\\partial_{in}F'|}{|F''|}}{1+\\frac{|\\partial_{in}F'|}{|F''|}}.$$\nAs $\\partial_{out}F''\\subset\\partial_{in}F'$ and $\\frac{x}{x+1}$ is an increasing function in $\\R_+$, the first quantity is smaller then the second, and by $F'$ being optimal, we have an equality.\n\nWe now consider cases for the size of $F''$.\nIf $|F''|<|F|$, then we can apply the assumption that $F$ is optimal and we get\n$$\\frac{|\\partial_{out}F|}{|F|}\\leq\\frac{|\\partial_{out}F''|}{|F''|}=\\frac{|\\partial_{in}F'|}{|F'\\setminus\\partial_{in}F'|}=\\frac{\\frac{|\\partial_{in}F'|}{|F'|}}{1-\\frac{|\\partial_{in}F'|}{|F'|}}.$$\nHowever, applying the initial assumption by which we chose $F'$ gives us the inverse inequality, and strict.\n\nWe are left with the case where $|F''|\\geq|F|$.\nLet $k=|F''|-|F|$ and remove any $k$ points from $F''$ to obtain $F'''$.\nWe obtain a set that is the same size as $F$ and has an outer boundary no larger than that of $F$.\nIt is therefore another optimal set of the same size, and by the optimality of $F'$ for the inner boundary, $F'''\\bigcup\\partial_{out}F'''=F'$, which concludes the proof.\n\\end{proof}\n\nThe central idea of this paper is to work on an associated graph structure which we define for semidirect products.\n\\begin{defi}\\label{assgr}\nConsider a semidirect product $G=H\\ltimes N$.\nConsider a generating set $S$ of $G$.\nWe define the \\textit{associated graph} as the directed labeled graph $\\Gamma=\\Gamma_S$ with vertex set $V(\\Gamma)=N$ and edge set \n\t\n$$\\overrightarrow{E}(\\Gamma)=\\left\\{\\overrightarrow{(f_1,f_2)}:\\exists s\\in S,h_1,h_2\\in H\\mbox{ such that }h_1f_1s=h_2f_2\\right\\}.$$\nWith these notations, the edge $\\overrightarrow{(f_1,f_2)}$ is labeled $s$.\n\\end{defi}\nAs mentioned, the two examples we will consider here are the lamplighter group $\\Z\\wr\\Z_2$ and the Baumslag-Solitar group $BS(1,p)$.\nIn both examples we have $H=\\Z$.\nIn the case of the lamplighter group we have $N=\\Z_2^{(\\Z)}$, and for $BS(1,p)$, $N$ is the set of $p$-adic numbers.\n\nWe define an associating function $\\phi:G\\rightarrow\\mathrm{P}(\\Gamma)$ by \n$$\\phi(hf)=\\left\\{\\overrightarrow{(f,f')}:hfs=h'f'\\mbox{ for some }h'\\in H,s\\in S\\right\\}.$$\n\\begin{defi}\\label{asssubgr}\nConsider a semidirect product $G=H\\ltimes N$.\nLet $F$ be a finite subset of $G$.\nThe \\textit{associated subgraph} of $F$ is the subgraph of the associated graph $\\Gamma$ made of the edges\n$$\\bigcup_{x\\in F}\\phi(x)$$\nand all adjacent vertices.\n\\end{defi}\nWe will provide a bound for the boundary of a set based on a formula on the associated subgraph, and maximize the value of that formula over all subgraphs of $\\Gamma$ no larger (in terms of number of edges) than the associated subgraph.\n\n\\section{The lamplighter group}\\label{mainsection}\n\nIn this section we provide the proof of Theorem~\\ref{thmlamp}, the larger part being a proof of Theorem~\\ref{thmlamp}$(1)$.\nIn other words, we show that for the lamplighter group with the standard generating set, the standard sets are optimal with respect to the outer boundary, and uniquely so up to translation.\nWe first prove an inequality relying the isoperimetry of a subset with values on the associated graph.\n\n\\begin{lemma}\\label{main}\nLet $F$ be a finite set in $\\Z\\wr\\Z_2$ with $\\frac{|\\partial_{out}F|}{|F|}\\leq1$.\nLet $\\Gamma$ be the associated graph (see Definition~\\ref{assgr}).\nThen\n\n$$\\frac{|\\partial_{out}F|}{|F|}\\geq\\min\\left(\\frac{2|V(G)|}{|E(G)|}\\mbox{ for }G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(G)|\\leq|F|\\right).$$\n\\end{lemma}\nWe will obtain that by estimating the number of points in the boundary that are reachable from the set by multiplication by $t$ or $t^{-1}$.\n\n\\begin{proof}\nAs mentioned below the definition of associated graph, the vertex set of $\\Gamma$ is $\\Z_2^{(\\Z)}$.\nThe edge set is $\\left\\{\\overrightarrow{(f,g)}:\\exists x_0:f(x)=g(x)\\iff x\\neq x_0\\right\\}$, and all of them have the label $\\delta$.\nAs all edges have the same label, we will think of $\\Gamma$ as unlabeled.\t\n\nConsider a finite set $F$ of elements of $\\Z\\wr\\Z_2$.\nLet $\\widetilde{F}$ be the associated subgraph (see Definition~\\ref{asssubgr}).\nA \\textbf{leaf} we call a vertex which is included in exactly one edge of the subgraph and is at the tail of that edge.\nThe set of leaves we denote by $L(\\widetilde{F})$.\n\nWe would like to estimate, for each vertex $f$ in $\\widetilde{F}$, the number of points of the form $(n,f)$ in $\\partial_{out}F$.\nWe will show that if $f$ is a leaf, there is at least one such point, and if $f$ is not a leaf, there are at least two.\nThe first case follows directly from the definition of a leaf.\n\nAssume that $f$ is not a leaf.\nThen either there are at least two edges ending in $f$, or there is at least one edge starting at $f$.\nOnce again, the first case follows directly from the definition.\nIn the second case, there is an element $(k,f)\\in F$.\nLet $a=\\max\\{k:(k,f)\\in F\\}$ and $b=\\min\\{k:(k,f)\\in F\\}$.\nThen $(a+1,f)$ and $(b-1,f)$ are in the outer boundary of $F$.\n\nWe obtain:\n\n\\begin{equation}\\label{ineq}\n|\\partial_{out}F|\\geq2(|V(\\widetilde{F})|-|L(\\widetilde{F})|)+|L(\\widetilde{F})|=2|V(\\widetilde{F})|-|L(\\widetilde{F})|.\n\\end{equation}\n\nRemark that if there are no leaves, the desired result follows.\n\n\\begin{lemma}\\label{leaf}\nFix $n$.\nAssume that\n$$\\min\\left(\\frac{2|V(G)|-|L(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|L(G)|\\neq0\\mbox{ and }|E(G)|\\leq n\\right)\\leq1.$$\nThen\n\\begin{equation*}\n\\begin{split}\n&\\min\\left(\\frac{2|V(G)|-|L(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|L(G)|\\neq0\\mbox{ and }|E(G)|\\leq n\\right)\\\\\n&\\geq\\min\\left(\\frac{2|V(G)|}{|E(G)|}:G\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(G)|\\leq n-1\\right).\n\\end{split}\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nWe will prove the lemma by induction on $n$.\nFor small enough $n$, the assumed inequality is impossible, which gives us the base of the induction.\n\nAssume that the statement is true for $n-1$ and consider $G$ with $n$ edges such that\n\n$$\\frac{2|V(G)|-|L(G)|}{|E(G)|}=\\min\\left(\\frac{2|V(H)|-|L(H)|}{|E(H)|}:|L(H)|\\neq0\\mbox{ and }|E(H)|\\leq n\\right)\\leq1.$$\n\nIf $G$ does not contain a leaf, the statement is trivial.\nWe now assume that $G$ contains a leaf.\nLet $G'$ be the subgraph obtained by removing that leaf and the edge leading to it.\nThen $|V(G')|=|V(G)|-1$ and $|E(G')|=n-1$.\n\nNotice also that while new leaves may have appeared, at most one was removed, and thus $|L(G')|\\geq|L(G)|-1$.\nTherefore, as $\\frac{2|V(G)|-|L(G)|}{|E(G)|}\\leq1$, we have:\n$$\\frac{2|V(G')|-|L(G')|}{|E(G')|}\\leq\\frac{2|V(G)|-|L(G)|-1}{|E(G)|-1}\\leq\\frac{2|V(G)|-|L(G)|}{|E(G)|}.$$\n\nIt is in particular less than $1$, and we obtain:\n$$\\frac{2|V(G')|-|L(G')|}{|E(G')|}\\geq\\min\\left(\\frac{2|V(H)|}{|E(H)|}:H\\mbox{ subgraph of }\\Gamma\\mbox{ with }|E(H)|\\leq n-1\\right),$$\neither from the induction hypothesis or directly, depending on whether $G'$ has a leaf or not.\n\\end{proof}\nThe result of Lemma~\\ref{main} then follows from equation (\\ref{ineq}) and applying Lemma~\\ref{leaf} for $n=|E(\\widetilde{F})|$.\n\\end{proof}\n\nHaving proven Lemma~\\ref{main}, we now need to show that the standard sets $F_n$ minimize $\\frac{2|V(G)|}{|E(G)|}$ over subgraphs of $\\Gamma$ with a fixed amount of edges.\nThe associated subgraph of $F_n$ (see Definition~\\ref{asssubgr}) is the subgraph of $\\Gamma$ with vertices $\\{f:\\supp(f)\\subset[\\![1,n]\\!]\\}$ and all edges of $\\Gamma$ between two of those vertices.\nIt has $2^n$ vertices and $2n2^n$ edges.\n\n\\begin{lemma}\nLet $G$ be a subgraph of $\\Gamma$ with at most $2n2^n$ edges.\nThen $\\frac{2|V(G)|}{|E(G)|}\\geq\\frac{1}{n}$, and if they are equal, $G$ is a hypercube (with double edges) with $2^n$ vertices.\n\\end{lemma}\n\n\\begin{proof}\nWe will first see that we can assume that any directed edge is present simultaneously with its inverse.\nIndeed, consider the set of directed edges $\\overrightarrow{(f,g)}$ in $G$ such that $\\overrightarrow{(g,f)}$ is not in $G$.\nIf there are at least two such edges, removing one and adding the inverse of the other changes neither $|V(G)|$ nor $|E(G)|$.\nIf there is exactly one such edge $\\overrightarrow{(f,g)}$, then $|E(G)|$ is odd, and so it is at most $2n2^n-1$.\nIn that case by adding the edge $\\overrightarrow{(g,f)}$ we obtain another subgraph with at most $2n2^n$ edges, and such that the quotient we consider is smaller.\n\nWe can therefore assume that any directed edge is present simultaneously with its inverse.\nWe replace every couple of edges $\\overrightarrow{(f,g)}$ and $\\overrightarrow{(g,f)}$ with one undirected edge $(f,g)$.\nWe will call the graphs we obtain $\\overline{\\Gamma}$ and $\\overline{G}$.\nWe need to prove that $\\frac{|V(\\overline{G})|}{|E(\\overline{G})|}\\geq\\frac{1}{n}$, and if they are equal, $\\overline{G}$ is a hypercube with $2^n$ vertices.\n\nRemark that $\\overline{\\Gamma}$ is the bi-infinite hypercube.\nAs $\\overline{G}$ is finite, it is contained in some finite hypercube.\nIn that case, the question of maximizing the number of edges on a fixed number of vertices has already been answered in literature.\nOne proof is presented in Harper's book~\\cite[Section~1.2.3]{Harper2004}.\nTaking notations from the book, consider a subset $C\\subset\\Z$ of cardinal $c$, and a vertex $f$ of $\\overline{\\Gamma}$ with $\\supp(f)\\bigcap C=\\emptyset$.\nThe set\n$$\\left\\{g\\in V(\\overline{\\Gamma}):f(x)=g(x)\\forall x\\notin C\\right\\}$$\nis called a \\textbf{$c$-subcube}.\nA vertex set $S$ of cardinal $k=\\sum_{i=1}^K2^{c_i}$ with $0\\leq c_i<c_j$ for $i<j$ is \\textbf{cubal} if it is a disjoint union of $c_i$-subcubes with the $c_i$-subcube being contained in the neighborhood of the $c_j$-subcube for all $i<j$.\nHere, neighborhood means the set of points at distance $1$ in graph distance.\nRemark that two cubal sets of the same cardinality are isomorphic.\n\nBy abuse of notation, for a set $S$ of vertices in $\\Gamma$, we will denote by $E(S)$ the set of all edges between two vertices of $S$.\nThen Theorem~$1.1$ from the cited section states\n\\begin{thm}[{\\cite[Section~1.2.3~-~Theorem~1.1]{Harper2004}}]\\label{thmharper}\n\t$S$ maximizes $|E(S)|$ over sets with cardinal $|S|$ if and only if $S$ is cubal.\n\\end{thm}\n\nAs $|E(S)|$ increases strictly with $|S|$, and equality can be achieved for $|E(S)|=n2^n$, it therefore suffices to consider the case where $\\overline{G}$ is cubal.\nLet $|V(\\overline{G})|=\\sum_{i=1}^K2^{c_i}$.\nIf $|V(\\overline{G})|>2^n$, the result follows immediately.\nAssume that $|V(\\overline{G})|\\leq2^n$, and thus $c_K\\leq n$, and $c_i\\leq n-K+i$.\nThen\n$$|E(\\overline{G})|=\\sum_{i=1}^Kc_i2^{c_i}+\\sum_{i<j}2^{c_i}=\\sum_{i=1}^{K}(c_i+K-i)2^{c_i}\\leq n\\sum_{i=1}^K2^{c_i},$$\nwhich concludes the proof of the inequality.\nFurthermore, if there is equality, then $c_K=n$, in which case $|V(\\overline{G})|=n2^n$ and $K=1$.\n\\end{proof}\n\nTheorem~\\ref{thmlamp}$(1)$ follows from this lemma and Lemma~\\ref{main}.\nBy Lemma~\\ref{equiv}, Theorem~\\ref{thmlamp}$(2)$ follows from Theorem~\\ref{thmlamp}$(1)$ (and the unicity of the optimal sets).\n\n\\section{Bounds for the Coulhon and Saloff-Coste inequality for the lamplighter group}\\label{csc-const-sect}\n\nWe will now prove Proposition~\\ref{const}.\nRecall its statement:\n\\begin{prop*}\nThe lamplighter group verifies\n\n$$C_{\\Z\\wr\\Z_2,S}=\\frac{\\lim\\frac{\\ln\\Fol(n)}{n}}{\\lim\\frac{\\ln V(n)}{n}}=\\frac{\\ln4}{\\ln(\\frac{1}{2}(1+\\sqrt{5}))}\\approx2,88$$\nfor the standard generating set.\n\n\\end{prop*}\n\n\\begin{proof}\nWe have obtained in Theorem~\\ref{thmmain} that $\\lim\\sqrt[n]{\\Fol(n)}=4$.\nIt is therefore sufficient to calculate the exponent of its volume growth.\n\nConsider first the standard generating set.\nConsider an element $g$ of length $n$ in the group and let its support be $[m,p]$.\nWe can write $g=t^mAt^i$ where $A$ is a non-reducible word on $t$ and $\\delta$ (without their inverses).\nIt is not hard to see that $A$ contains the letter $t$ exactly $p-m$ times.\nRemark that any representation of $g$ in the standard generators must contain $t$ or $t^{-1}$ at least $p-m$ times.\nAdditionally, it contains $\\delta$ at least as many times as $A$ does.\nTherefore $|A|\\leq|g|=n$.\n\nNotice also that $|m|\\leq n$ and $|i|\\leq2n$.\nTherefore, if we denote by $V'(n)$ the amount of non-reducible words on $t$ and $\\delta$ of length at most $n$, we obtain\n$$V'(n)8n^3\\geq V(n)\\geq V'(n).$$\n\nWe now seek to calculate $V'(n)$.\nNotice that the only condition on those words in the lamplighter group is to not have two consecutive $\\delta$-s.\nTherefore $V'(n)$ is same as the amount of subsets of $[1,n]$ without two consecutive elements.\nA simple induction shows that those are (up to translation) the Fibonacci numbers.\nIndeed, such a subset either has $n$ in it, in which case the rest of the subset is contained in $[1,n-2]$, or it does not have $n$ and the rest of the subset is contained in $[1,n-1]$.\nTherefore\n$$\\lim\\sqrt[n]{V(n)}=\\lim\\sqrt[n]{V'(n)}=\\frac{1+\\sqrt{5}}{2}.$$\n\\end{proof}\n\nIt is worth noting that the exact value of the volume growth power series $\\sum_nV(n)x^n$ for the standard generating set has been described by Parry~\\cite{Parry1992a}.\n\nWe then prove Remark~\\ref{const2}.\nIt states:\n\\begin{remark*}\nFor the switch-walk-switch generating set $S'=\\{t,\\delta,t\\delta,\\delta t,\\delta t\\delta\\}$, we have\n\n$$C_{\\Z\\wr\\Z_2,S'}=\\frac{\\liminf\\frac{\\ln\\Fol_{sws}(n)}{n}}{\\lim\\frac{\\ln V_{sws}(n)}{n}}\\leq2.$$\n\\end{remark*}\n\\begin{proof}\nThe standard F{\\o}lner sets verify that $\\frac{|\\partial'_{out}F_n|}{|F_n|}=\\frac{2}{n}$ where by $\\partial'_{out}$ we denote the outer boundary with regards to $S'$.\nTherefore $\\lim\\sqrt[n]{\\Fol_{sws}(n)}\\leq4$.\nSimilarly, we now need to calculate the exponent of its volume growth.\n\nThe switch-walk-switch volume is also controlled by the size of the set of possible configurations given a certain interval.\nHowever, in its case all configurations with support in that interval are found on an element in the ball.\nThus $8n^32^n\\geq V_{sws}(n)\\geq2^n$ and $\\lim\\sqrt[n]{V_{sws}(n)}=2$.\n\\end{proof}\n\n\\section{The Baumslag-Solitar group $BS(1,2)$}\\label{bssect}\n\nWe now prove Theorem~\\ref{bsthm}.\nRemark that for the edge boundary, the contributions by each element of the generating set are always disjoint.\nRecall that the vertex set of the associated graph $\\Gamma'$ is the dyadic numbers.\nFor the standard generating set, the edges are the couples $\\overrightarrow{(f,g)}$ such that $|f-g|$ is a power of $2$, and are labeled $b$.\nFor any finite subgraph of $\\Gamma'$, up to translation we can assume that its vertices are all integers, in which case edges will be defined by positive powers of $2$.\n\nWe introduce the following notations.\nConsider a subset $F\\subset BS(1,2)$ and let $\\widetilde{F}$ be the associated subgraph.\nThe set of edges $\\overrightarrow{(f,g)}\\in E(\\widetilde{F})$ such that $\\overrightarrow{(g,f)}\\notin E(\\widetilde{F})$ we will denote $L(\\widetilde{F})$ (remark that unlike Section~\\ref{mainsection}, this is a set of edges rather than vertices).\nLet $\\overline{\\Gamma'}$ be the graph obtained by replacing every couple of edges $\\overrightarrow{(f,g)}$ and $\\overrightarrow{(g,f)}$ in $\\Gamma'$ with one undirected edge.\nLet $\\overline{\\widetilde{F}}$ be the subgraph of $\\overline{\\Gamma'}$ obtained by taking the images of $E(\\widetilde{F})\\setminus L(\\widetilde{F})$ and all adjacent vertices.\n\nFor any $i\\in\\Z$, consider the transformation $x\\mapsto x+2^i$ which is well defined on the vertices of $\\overline{\\Gamma'}$.\nDenote by $o(\\overline{\\widetilde{F}})$ the total amount of distinct non-trivial orbits in $V(\\overline{\\widetilde{F}})$ over all $i$.\nRemark that this quantity is well defined for any subgraph $\\overline{\\widetilde{F}}$ of $\\overline{\\Gamma'}$.\n\n\\begin{lemma}\\label{comparebs}\nConsider a subset $F\\subset BS(1,2)$ with $\\frac{|\\partial_{edge}F|}{|F|}\\leq1$.\nThen\n\n$$|F|\\geq|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})$$\nand\n$$\\frac{|\\partial_{edge}F|}{|F|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}{|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}.$$\n\\end{lemma}\n\n\\begin{proof}\nWe start by estimating $|F|$.\nBy definition of associated subgraph, we have\n$$2|F|=|E(\\widetilde{F})|.$$\nBy definition of $L(\\widetilde{F})$ and $\\overline{\\widetilde{F}}$ we have\n$$|E(\\widetilde{F})=|L(\\widetilde{F})|+2|E(\\overline{\\widetilde{F}})|.$$\nWe now need to prove that $|L(\\widetilde{F})|\\geq2o(\\overline{\\widetilde{F}})$.\nIndeed, consider any non-trivial orbit and let $f$ and $g$ be respectively its smallest and largest elements.\nIf the step of the orbit is $2^i$, we have that $(i,f)$ and $(i,g)$ are elements of $F$.\nThen $\\overrightarrow{(f-2^i,f)}$ and $\\overrightarrow{(g,g+2^i)}$ must both be edges in $\\widetilde{F}$.\nAs $f-2^i$ and $g+2^i$ are not in $\\overline{\\widetilde{F}}$, those two edges are then in $L(\\widetilde{F})$.\n\nWe now estimate the boundary.\nWe will denote by $\\partial_{edge}^aF$ and $\\partial_{edge}^bF$ the edges in $\\partial_{edge}F$ labeled $a$ and $b$ respectively.\nIt follows directly from the definition of associated graph and $L(\\widetilde{F})$ that\n$$|\\partial_{edge}^bF|=|L(\\widetilde{F})|.$$\n\nConsider a vertex $f\\in V(\\overline{\\widetilde{F}})$.\nBy definition of $\\overline{\\widetilde{F}}$, there is $g$ such that $\\overrightarrow{(f,g)}\\in E(\\widetilde{F})$.\nIt follows that there is $x$ such that $(x,f)\\in F$.\nThere are thus at least two elements of $\\partial_{edge}^aF$ such that their corresponding configuration is $f$.\nWe obtain:\n$$|\\partial_{edge}^aF|\\geq2|V(\\overline{\\widetilde{F}})|.$$\nThen:\n$$\\frac{|\\partial_{edge}F|}{|F|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+\\frac{1}{2}|L(\\widetilde{F})|}{|E(\\overline{\\widetilde{F}})|+\\frac{1}{2}|L(\\widetilde{F})|}\\geq2\\frac{|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}{|E(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})}.$$\n\\end{proof}\n\nWe now seek to understand subgraphs of $\\overline{\\Gamma'}$ that optimize this quotient.\nWe claim that the subgraphs with vertices $[\\![1,n]\\!]$ do so.\nWe will denote $e(n)$ (respectively $o(n)$) the amount of edges (respectively orbits) in the subgraph with vertices $[\\![1,n]\\!]$ and all induced edges.\nWe prove results similar to Theorem~\\ref{thmharper}, which we cited from Haprer's book~\\cite{Harper2004}.\n\n\\begin{lemma}\\label{baums}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ with $n$ vertices.\nThen\n\n$$|E(G)|\\leq e(n).$$\n\\end{lemma}\n\n\\begin{proof}\nWe start by estimating $e(n)$.\nFor $n\\in\\N$ with $2^{k-1}<n\\leq2^k$ we have that $e(n)$ is the sum over $i$ of the amount of couples of elements of $[\\![1,n]\\!]$ with difference $2^i$.\nIn other words,\n$$e(n)=\\sum_{i=0}^{k-1}(n-2^i)=kn-2^k+1.$$\n\nWe will now prove the result of the lemma by induction on $n$.\nThe base $n=1$ is trivial.\nFix a subgraph $G$ with $n$ vertices.\nUp to translation by an integer we can assume that the vertices of $G$ are integer and the smallest is $1$.\n\nAssume first that all vertices of $G$ are odd integers.\nLet $i$ be the largest integer such that $2^i$ divides all elements of $\\{f-1:f\\in V(G)\\}$.\nDefine $\\psi(f)=\\frac{f-1}{2^i}+1$.\nThen the set $\\psi(V(G))$ is a set of integers with the smallest element being $1$.\nFurthermore, for any $f$ and $f'$ divisible by $2^i$, $|f-f'|$ is a power of $2$ if and only if $|\\psi(f)-\\psi(f')|$ is.\nWithout loss of generality we can replace $G$ by the subgraph of $\\overline{\\Gamma'}$ with vertices $\\psi(V(G))$.\n\nWe can therefore assume $G$ has both even and odd elements.\nLet $G_1$ be the subgraph induced by vertices of $G$ that are odd integers, and $G_2$ the subgraph induced by even integers.\nDenote the cardinal of their vertex sets by $n_1$ and $n_2$ respectively.\nNotice that $n=n_1+n_2$.\n\nAs the difference (in terms of integer values) between vertices of $G_1$ and $G_2$ is always odd, there can be an edge if and only if the difference is $1$.\nThen by induction hypothesis the number of edges in $F$ is not greater than\n$$e(n_1)+e(n_2)+2\\min(n_1,n_2)-\\varepsilon$$\nwhere $\\varepsilon=1$ if $n_1$ and $n_2$ are equal, and $0$ otherwise.\n\nLet us denote $2^{k_1-1}<n_1\\leq2^{k_1}$, $2^{k_2-1}<n_2\\leq2^{k_2}$.\nWithout loss of generality, assume $n_1\\leq n_2$.\nThen $k-1\\leq k_2\\leq k$.\nLet $\\delta=k-k_2$.\nWe calculate\n\n\\begin{align*}\n& |E(G)|-e(n)\\leq e(n_1)+e(n_2)+2n_1-\\varepsilon-e(n) \\\\\n& =k_1n_1-2^{k_1}+1+k_2n_2-2^{k_2}+1+2n_1-\\varepsilon-k(n_1+n_2)+2^k-1 \\\\\n& =(k_1-k+2)n_1-\\delta n_2+1+2^k-2^{k_1}-2^{k_2}-\\varepsilon=A.\n\\end{align*}\n\nWe seek to prove that $A\\leq0$.\nWe have $\\delta=0$ or $1$.\nFirst, assume $\\delta=0$.\nThen $2^k=2^{k_2}$ and\n$$A=(k_1+2-k)n_1+1-2^{k_1}-\\varepsilon.$$\nAs $k_1\\leq k-1$, we have $A\\leq n_1+1-2^{k_1}-\\varepsilon\\leq 0$.\n\nAssume now $\\delta=1$.\nThen\n$$A=(k_1+2-k)n_1-n_2+1+2^{k-1}-2^{k_1}-\\varepsilon.$$\n\nAssume first that $k_1\\leq k-2$.\nThen $A\\leq 2^{k_2}-n_2+1-\\varepsilon\\leq0$.\n\nAs $k_1\\leq k_2=k-1$, the only case left is $k_1=k-1$.\nThen\n$$A=n_1-n_2+1-\\varepsilon.$$\nIf $n_1-n_2=0$, then $\\varepsilon=1$ and $A=0$.\nIf $n_1-n_2\\leq-1$, then $A\\leq-\\varepsilon\\leq0$.\n\\end{proof}\n\n\\begin{lemma}\\label{baums2}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ such that $|V(G)|+o(G)=2n-1$ or $2n$.\nThen\n\n$$|E(G)|\\leq e(n).$$\n\\end{lemma}\n\nRemark that $o(n)=n-1$.\nIndeed, for each non-trivial orbit, consider the second smallest (in terms of the natural order on the integers) element of the orbit.\nIf it has step $2^i$, that element is between $1+2^i$ and $2^{i+1}$.\nTherefore these elements are disjoint, and cover every element in $[\\![1,n]\\!]$ except the element $1$.\n\n\\begin{proof}\nWe will now prove the result of the lemma by induction on $n$.\nThe base $n=1$ is trivial.\nFix a subgraph $G$ such that $|V(G)|+o(F)=2n-1$ or $2n$.\nUp to translation by an integer we can assume that the vertices of $G$ are integer and the smallest is $1$.\nAs in Lemma~\\ref{baums}, we can assume that $G$ has even vertices, and split it into $G_1$ and $G_2$ based on the parity of the vertices.\n\nLet $n_1=\\lceil\\frac{|V(G_1)|+o(G_1)}{2}\\rceil$ and $n_2=\\lceil\\frac{|V(G_2)|+o(G_2)}{2}\\rceil$.\nLet $2^{k_1-1}<n_1\\leq2^k_1$ and $2^{k_2-1}<n_2\\leq2^k_2$.\nWithout loss of generality, let $n_1\\leq n_2$.\n\nConsider first the case where there is at least one edge between $G_1$ and $G_2$.\nThen $o(G)\\geq o(G_1)+o(G_2)+1$ and $2n-1\\geq|V(G)|+o(G)-1\\geq|V(G_1)|+o(G_1)+|V(G_2)|+o(G_2)$.\nThis implies that $n_1+n_2\\leq n$.\nThe result then follows from the proof of Lemma~\\ref{baums}.\n\nConsider now the case where there is no edge between $G_1$ and $G_2$.\nThen $o(G)\\geq o(G_1)+o(G_2)$ and $2n\\geq|V(G)|+o(G)\\geq|V(G_1)|+o(G_1)+|V(G_2)|+o(G_2)$.\nThis implies that $n_1+n_2\\leq n+1$.\nBy induction hypothesis we have\n$$|E(G)|=|E(G_1)|+|E(G_2)|\\leq e(n_1)+e(n_2).$$\nWe calculate\n$$e(n_1)+e(n_2)-e(n)=k_1n_1+k_2n_2-kn+2^k-2^{k_1}-2^{k_2}+1=A.$$\nAssume first that $k_2=k$.\nThen\n$$A\\leq(k_1-k)n_1-2^{k_1}+1+k=(k_1-k)(n_1-1)-2^{k_1}+k_1+1\\leq0$$\nas $n_1\\geq1$.\n\nAssume now that $k_2\\leq k-2$.\nHowever, in that case $n_1+n_2\\leq2^{k-1}<n$, and the result follows from the proof of Lemma~\\ref{baums}.\n\nAssume that $k_2=k-1$.\nThen\n\\begin{equation*}\n\\begin{split}\nA&=(k_1+1-k)n_1-n+2^{k-1}-2^{k_1}+k\\\\\n&=(k_1+1-k)(n_1-1)-(n-2^{k-1})-(2^{k_1}-k_1-1)\\leq0,\n\\end{split}\n\\end{equation*}\nas $n_1\\geq1$ and $k_1\\leq k_2=k-1$.\n\\end{proof}\n\nCombining those two lemma, we obtain:\n\n\\begin{cor}\nLet $G$ be a subgraph of $\\overline{\\Gamma'}$ such that $|V(G)|+o(G)\\leq2n$ and $\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\leq1$.\nThen\n\n$$\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\geq\\frac{n+o(n)}{e(n)+o(n)}.$$\n\\end{cor}\n\nRemark that $n+o(n)=2n-1$.\n\n\\begin{proof}\nIf $o(G)\\leq o(n)$, the result follows trivially from Lemma~\\ref{baums2}.\nAssume then that $o(G)>o(n)=n-1$.\n\nThus $|V(G)|=n'\\leq n$.\nFrom Lemma~\\ref{baums} we obtain $|E(G)|\\leq e(n')$.\nWe have:\n\n$$\\frac{|V(G)|+o(G)}{|E(G)|+o(G)}\\geq\\frac{n'+o(n')+(o(G)-o(n'))}{e(n')+o(n')+(o(G)-o(n'))}\\geq\\frac{n'+o(n')}{e(n')+o(n')}\\geq\\frac{n+o(n)}{e(n)+o(n)}.$$\n\\end{proof}\n\nThe large inequality of Theorem~\\ref{bsthm} follows directly from this corollary and Lemma~\\ref{comparebs}.\nThe strict inequality is obtained by noticing that if $|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})<n+o(n)$, then $|V(\\overline{\\widetilde{F}})|+o(\\overline{\\widetilde{F}})\\leq2n-2$ and we can apply the corollary for $n-1$.\nThis concludes the proof of Theorem~\\ref{bsthm}.\n\nFinally, we consider $BS(1,p)$ with the standard generating set.\n\\begin{ex}\\label{exbsp}\nThere exist natural numbers $p$ and $n$ and a set $F\\subset BS(1,p)$ such that $|F|\\leq|F_n|$ (where $F_n=\\{(k,f):k\\in[\\![0,n-1]\\!],f\\in\\Z,0\\leq f<p^n\\}$) and\n\n$$\\frac{|\\partial_{edge}F|}{|F|}<\\frac{|\\partial_{edge}F_n|}{|F_n|}\\leq1.$$\nIn particular, this is true for $n=3$ and $p$ divisible by $3$ with $36\\leq p\\leq60$.\n\\end{ex}\n\n\\begin{proof}\nThe standard F{\\o}lner set $F_n$ has $np^n$ elements.\nWe calculate $|\\partial_{edge}^aF_n|=2p^n$ and $|\\partial_{edge}^bF_n|=2\\frac{p^n-1}{p-1}$.\nThen\n$$|\\partial_{edge}F_n|=2p^n\\frac{p-\\frac{1}{p^n}}{p-1}\\leq2p^n\\frac{p}{p-1},$$\nand $\\frac{|\\partial_{edge}F_n|}{|F_n|}\\leq\\frac{2}{n}\\frac{p}{p-1}$.\nTherefore for $n=3$ and $p\\geq3$ we have $\\frac{|\\partial_{edge}F_3|}{|F_3|}\\leq1$.\nAssume that $p$ is divisible by $3$ and let\n$$F=\\{(k,f):k\\in[\\![0,3]\\!],f=\\sum_{i=0}^3\\varepsilon_ip^i\\mbox{ with }0\\leq\\varepsilon_i<\\frac{p}{3}\\}.$$\nThen $|F|=4(\\frac{p}{3})^4=3p^3\\frac{4p}{243}$ and for $p\\leq60$ we obtain $|F|\\leq|F_3|$.\nFurthermore, $|\\partial_{edge}^aF|=2(\\frac{p}{3})^4$ and $|\\partial_{edge}^bF|=8(\\frac{p}{3})^3$.\nThus\n$$|\\partial_{edge}F|=4\\left(\\frac{p}{3}\\right)^4\\left(\\frac{1}{2}+\\frac{6}{p}\\right),$$\nand for $p\\geq36$ we have:\n$$\\frac{|\\partial_{edge}F|}{|F|}=\\frac{1}{2}+\\frac{6}{p}\\leq\\frac{2}{3}<\\frac{2}{3}\\frac{p-\\frac{1}{p^3}}{p-1}=\\frac{|\\partial_{edge}F_3|}{|F_3|}.$$\n\\end{proof}\nTherefore the result of Theorem~\\ref{bsthm} is not true for $BS(1,p)$ for all $p$.\n\n\\bibliographystyle{plainurl}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nThe reionization of the universe is the process where neutral hydrogen (H$_{I}$) becomes \nionized and determines the transition from an opaque state to the transparent intergalactic \nmedium (IGM) we observe today. Although the duration of this phase seems to be well \nestablished \\citep{Fan06,Bec15,Planck}, the source population causing this effect \nis still elusive. The  debate is still ongoing as to whether the main contributors \nare faint, high-redshift, star-forming galaxies (SFGs) or active galactic nuclei (AGNs).\nFor both populations there are critical issues. The assumption for the galaxies being \nmain contributors is that all faint SFGs should present an escape fraction of \nionizing radiation {\\it $f_{esc}$} of 10-20\\% \\citep{Fin15,Bow16}, which has not \nbeen observed so far, apart from in a handful of sources \\citep{Sha16,Van16,Bian17,Ste18}.\nRecent results \\citep{Fle18,Jon18,Nai18,Tan18} do not provide a clear evidence, but \nindicate that it is difficult for the global SFG population to reach the 10-20\\% level\nof escape fraction required to drive the reionization process. The main objection against \nAGNs being the main contributors is that the number of bright quasars at $z>$4 is not \nenough \\citep{Fan06,Cow09,HM12} and the number of faint AGNs at high redshifts is \nstill not well constrained. Based on X-ray samples, at low luminosities in this redshift \nrange, the space density of obscured AGNs is at least two times higher than the unobscured \npopulation \\citep{Mar16b}, indicating that optically selected luminosity functions (LFs) \ncould only be a lower limit.     \n\nIn order to answer the question of whether faint AGNs can contribute to the ionizing \nultraviolet background (UVB), three aspects need to be quantified: (i) the exact level \nof the UVB; (ii) the fraction of ionizing radiation escaping these sources \n({\\it $f_{esc}$}); and (iii) the faint slope of the AGN LF. \n\nObservations of the ionizing UVB intensity in the redshift range 2$<z<$5 and the global \nemissivity of ionizing photons indicate a relatively flat hydrogen photoionization \nrate \\citep{BeB13}, but not spatially uniform \\citep{Bos18}. Such large opacity \nfluctuations cannot be easily explained by low clustering populations like ultra-faint \ngalaxies and could require the existence of rare bright sources at high redshift \n\\citep{Bec15,Bec18,Cha15,Cha17}. As models start including larger AGN contributions, \npredicted temperatures are in agreement with observational constraints at $z\\sim$4-6 \n\\citep{Kea18} (but see \\citet{Puc18} for a different interpretation). \n\nThus, a sizable population of faint (L$\\sim$L$^{*}$) AGNs could contribute significantly \nto the UVB, as long as enough H$_{I}$ ionizing photons manage to escape the host galaxy \n\\citep{MH15,Kha16}. In this respect, a recent study by \\citet{Gra18}, using deep optical\/UV \nspectroscopy, found that faint AGNs (L$\\sim$L$^{*}$) at 3.6$<z<$4.2 present high escape \nfractions of ionizing radiation, with a mean value of 74\\%. This is in agreement with \nsimilar studies of bright quasars (M$_{1450}\\leq$ -26) at the same redshift range \n\\citep{Cri16}, and there is no indication of dependence on absolute luminosities. This \nmeans that, if such results are extrapolated to higher redshift (5$<z<$7), the AGN \ncontribution to the cosmic reionization process can become significant. At this point, \nknowledge of the exact number of faint AGNs at redshifts $z>4$ becomes crucial in \naccurately determining the level of this contribution.\n\nCurrently the consensus is that the LF for bright AGNs is well constrained, showing a \npeak at $z\\sim$3 and then rapidly declining \\citep{Bon07,Cro09}. However, for $z>$3 the \ndebate is still open, with various studies presenting contradicting results. Works \npresented by \\citet{Ike11} and \\citet{Gli11} suggest that the number of faint AGNs \nat $z>$3 is higher than expected, producing a steeper slope at the faint end of the LF. \nBut although the faint-end slopes are similar, the normalization factor $\\Phi^{*}$ derived \nby \\citet{Gli11} is three times higher than what calculated by \\citet{Ike11} and \nsubsequently reproduced by other studies \\citep[i.e][]{Mas12,Aki18}. These latter \nstudies report a strong decline in AGN numbers going from z=3 to z=4. In other words, \nthere is still wide disagreement on the actual shape and normalization of the \nLF at $z\\sim$4.\n\nWork by \\citet{Gia15}, including photometric and spectroscopic redshifts of X-ray-selected \nAGN candidates in the CANDELS GOODS-South region, has shown that at $z>$4 the probed AGN \npopulation could produce the necessary ionization rate to keep the IGM highly ionized \n\\citep{MH15}. This result is still controversial, with recent works claiming the opposite \n\\citep[i.e.][]{DAl17,Ric17,Aki18,Has18,Par18}. In fact, so far, the optical LFs at this \nredshift range and luminosities are based on a handful of spectroscopically confirmed \nsources (e.g., eight for \\citet{Ike11} and five for \\citet{Gia15}). Since the bulk of \nionizing photons come from AGNs close to L$^{*}$, it is mandatory to measure their LF \nat $z>$4 in this luminosity range. For this reason, we started a pilot study in the \nCOSMOS field, ideal for this kind of analysis thanks to its multi-wavelength catalog, \nX-ray, and radio coverage, which allows us to robustly select our AGN candidates. Here \nwe present the bright part of our spectroscopically confirmed sample of \nintermediate-\/low-luminosity AGNs, reaching an absolute magnitude of M$_{1450}$=-23 \nand discuss a robust determination of the space density at $z\\sim$4.\n\nThroughout the paper we adopt the $\\Lambda$ cold dark matter ($\\Lambda$CMD) concordance \ncosmological model (H$_{0}$ = 70 km s$^{-1}$ Mpc$^{-1}$, $\\Omega_{M}$ = 0.3, and \n$\\Omega_{\\Lambda}$ = 0.7). All magnitudes are in the AB system.\n\n\\section{AGN Candidate Selection} \nThe selection of our sample is based on: (i) photometric redshifts, (ii) color-color \nselection, and (iii) X-ray emission. \n\n\\begin{deluxetable*}{ccccccccc}\n\\tablecaption{Color-Color Candidates \\label{tab:colorsel}}\n\\tablehead{\n\\colhead{ID} & \\colhead{R.A.} &\n\\colhead{Decl.} & \\colhead{i$_{AB}$} & \\colhead{$z_{phot}$} &\n\\colhead{$z_{spec}$} & \\colhead{(B$_{J}$-V$_{J}$)} &\n\\colhead{(r-i)} & \\colhead{X-ray}\\\\}\n\\startdata\n658294\\tablenotemark{*} & 149.467350 &1.855592 &21.056&-1.000 & 4.174 &1.40&0.25 & no \\\\\n1856470\\tablenotemark{*}& 150.475680 &2.798362 &21.282&0.000 & 4.110 &1.42&0.32 & yes \\\\\n1581239& 150.746170 &2.674495 &21.556&0.293 & -1.000&1.77&0.48 & no \\\\\n507779& 150.485630 &1.871927 &22.034&0.605 & 4.450 &4.94&0.55 & yes \\\\\n38736\\tablenotemark{*}& 150.732540 &1.516127 &22.088&-1.000 & 4.183 &1.69&0.64 & no \\\\\n1226535& 150.100980 &2.419435 &22.325&0.480 & 4.637 &1.68&0.43 & yes \\\\\n422327 & 149.701500 &1.638375 &22.409&0.343 & 3.201 &1.54&0.14 & no \\\\\n664641\\tablenotemark{*} & 149.533720 &1.809260 &22.436&0.338 & 3.986 &1.69&0.30 & no \\\\\n1163086\\tablenotemark{*}& 150.703770 &2.370019 &22.444&-1.000 & 3.748 &1.44&0.25 & yes \\\\\n330806\\tablenotemark{*} & 150.107380 &1.759201 &22.555&3.848 & 4.140 &1.48&0.30 & yes \\\\\n344777 & 150.188180 &1.664540 &22.634&0.392 & -1.000&1.89&-0.44 & no \\\\\n1450499& 150.115830 &2.563627 &22.685&0.280 & 3.355 &1.94&0.63 & no \\\\\n1687778& 150.006940 &2.779943 &22.715&0.437 & -1.000&1.96&0.44 & no \\\\\n96886  & 150.289380 &1.559480 &22.765&3.860 & -1.000&1.77&0.27 & no \\\\\n1573716& 150.729200 &2.739130 &22.783&0.376 & -1.000&1.35&0.48 & no \\\\\n346317 & 150.205950 &1.654837 &22.800&0.352 & -1.000&1.450&-0.21 & no \\\\\n1257518& 150.025190 &2.371214 &22.810&0.241 & -1.000&1.60&0.34 & no \\\\\n1322738& 149.444050 &2.424602 &22.839&0.428 & -1.000&1.92&0.71 & no \\\\\n1663056& 150.185000 &2.779340 &22.862&3.658 & -1.000&2.29&0.52 & no \\\\\n1719143& 149.755390 &2.738555 &22.873&-1.000 & 3.535 &1.76&0.23 & yes \\\\\n125420 & 150.222680 &1.510574 &22.898&0.181 & -1.000&1.83&0.53 & no \\\\\n867305 & 149.446230 &2.115336 &22.950&0.651 & -1.000&2.11&0.71 & no \\\\\n612661 & 149.838500 &1.829048 &23.011&4.229 & 4.351 &1.93&0.60  & no \\\\\n\\enddata\n\\tablenotetext{*}{Used for the LF}\n\\end{deluxetable*}\n\nWe use the photometric catalog and redshifts presented by \\citet{Ilb09}. This is a 30-band \ncatalog, spanning from NUV photometry to IRAC data, with calculated $z_{phot}$ in a \nregion covering 1.73 deg$^{2}$ in COSMOS. The reported $z_{phot}$ dispersion \nis $\\sigma_{(\\Delta z)\/(z_{s}+1)}$=0.007 at i$_{AB}<$22.5 and increases to \n$\\sigma_{(\\Delta z)\/(z_{s}+1)}$=0.012 at i$_{AB}<$24. As discussed in \\citet{Ilb09}, \ntheir $z_{phot}$ determination is mostly based on galaxy templates. However, as showed \nby \\citet{Gia15}, for z$>$4 the accuracy on the photometric redshift estimate is weakly \ndependent on the adopted spectral libraries but it is mainly driven by the Lyman break \nfeature at rest frame wavelength (912\\AA). To take into account possible larger errors \non photometric redshifts for the AGN population, we extended the redshift interval. Thus, \nwe obtained a list of 42 candidates that have a photometric redshift estimate in the \ninterval $3.0\\leq z_{phot}\\leq 5.0$ and a magnitude i$_{AB}<$23.0.\n\nTo increase our selection efficiency and mitigate shortcomings of the $z_{phot}$ technique, \nwe include a color criterion. Since we have a wide number of bands available, initially \nwe explored various combinations of color-color selections, i.e., (B$_{J}$-V$_{J}$) versus \n(r-i), (B$_{J}$-r) versus (r-i), (g-r) versus (r-i), (u$_{*}$-B$_{J}$) versus (r-i), and \n(u$_{*}$-g) versus (r-i). Cross-correlating those candidates with known AGNs from the \nliterature, and after exploratory spectroscopy with \nLDSS-3\\footnote{http:\/\/www.lco.cl\/telescopes-information\/magellan\/instruments\/ldss-3}, \nfor this pilot study we narrowed down our selection to the most promising criterion, \ni.e. (B$_{J}$-V$_{J}$) versus (r-i). In the (B$_{J}$-V$_{J}$) versus (r-i) color-color \ndiagram we consider as high-redshift AGN candidates the sources found in the locus \ndelimited by: \\\\\n\\\\\n(B$_{J}$-V$_{J}$)$>$1.3 \\\\\nand \\\\\n(r-i)$\\leq$0.60$\\times$(B$_{J}$-V$_{J}$) - 0.30.\\\\\n\\\\\nWith this criterion we obtained 23 candidates down to i$_{AB}$=23.0, summarized in \nTable \\ref{tab:colorsel}. We decided not to put any constraints on the morphology, \nsince the population of low-luminosity AGNs (M$_{1450}\\sim$-23) includes Seyferts, \nwhere the host galaxy could be visible. \n\n\\begin{figure}\n\\label{fig:spec}\n\\plotone{fig1.pdf}\n\\caption{The spectra of the six AGNs with $3.6\\leq z\\leq 4.2$ and i$_{AB}\\leq $23.0 \ndiscovered during our spectroscopic campaign with IMACS and LDSS-3. The red line corresponds \nto zero flux F$_{\\lambda}$, in arbitrary units.} \n\\end{figure}\n\nThere is a relatively small overlap between the candidates selected by the various methods. \nMore specifically, only 7\\% of the candidates selected by photometric redshifts are also \nincluded in the color-selected sample (three out of 42 objects), which is useful to \nincrease our completeness. This is a clear advantage with respect to the works of \n\\citet{Gli11} and \\citet{Ike11} which only used four bands for their selections.\n\nThe final criterion for the creation of our sample was X-ray emission. In practice, we \nselected 38 sources detected in X-rays by deep Chandra observations in the COSMOS field \n\\citep{Civ16} with $z_{phot}\\geq$3 and a limiting magnitude i$_{AB}<$23. These photometric \nredshifts were provided by \\citet{Mar16a} based on AGNs, galaxies or hybrid templates, \nas described in \\citet{Sal11}. This sample consists both of type-1 and type-2 AGNs, and \nrepresents an unbiased census of the faint AGN population at this redshift. Only eight of \nthe sources selected with the first two criteria present also emission in X-rays, while six \ncandidates have been selected both by X-ray and color criteria. \n\nOur final sample consists of 92 AGN candidates with magnitudes i$_{AB}<$23, that have been \nselected by at least one of the methods mentioned above. Thanks to extensive spectroscopic \ncampaigns carried out in the COSMOS field \\citep[e.g.][]{Bru09,Ike11,Civ12,Mar16a,Hasin18}, \n22 of our 92 candidates have secure spectroscopic redshifts. To establish the nature of \nthe remaining 70 sources (five of which have uncertain spectroscopic redshifts), we started \nan exploratory spectroscopic campaign at the Magellan Telescopes.  \n\n\\section{Spectroscopic Follow-up}  \n\nWe were awarded 2.5 nights with the wide-field Inamori-Magellan Areal Camera and \nSpectrograph \\citep[IMACS,][]{Dre11} on the 6.5m Magellan-Baade telescope at Las \nCampanas Observatory to obtain spectra for our AGN candidates. We observed a total of five \nmulti-slit masks with the IMACS f\/2 camera (27$\\arcmin$ diameter field of view) \nwith total exposure times ranging from 3hr to 6hr, during dark time in 2018 February and \nMarch. The width of the slits was 1$\\arcsec$.0 and the detector was used without binning \n(0$\\arcsec$.2\/pixel in the spatial direction). \n\nFor the three 6hr masks we used the 300 line mm$^{-1}$ red-blazed grism (300\\_26.7) with \nspectral sampling of 1.25{\\AA} pixel$^{-1}$, while for the two 3hr masks we used the 200 \nline mm$^{-1}$ grism that has a slightly lower resolution, sampling 2.04{\\AA} pixel$^{-1}$. \nIt is worth noting that the space density of our AGN candidates is such that only around \nthree objects typically fall in an IMACS field of view at this magnitude limit.\n\nWe observed a total of 16 AGN candidates with magnitudes ranging from i$_{AB}=$20 to 23.0, \nand for 14 of them we obtained robust redshift determination at $z>3$, resulting in an \nefficiency of $\\sim 88\\%$, and two uncertain redshifts at $z>3$. Out of the sub-sample with \nsecure redshifts, we found six AGNs in the redshift range $3.6\\leq z_{spec}\\leq 4.2$, and \neight AGNs with a measured redshift of either $3.1<z_{spec}<3.6$ or $4.2<z_{spec}<4.7$.\nThe masks were completed with less reliable AGN candidates at $z\\sim$4 obtained by relaxing \nthe criteria mentioned above and exploring different color selections with respect to the \n(B$_{J}$-V$_{J}$) versus (r-i) one. We also added, as fillers, fainter candidates, \ni$_{AB}\\leq$ 24.0, in order to explore the fainter regime of the AGN LF. \n\nAfter the spectroscopic campaign, 36 sources of the parent sample of 92 candidates had \nsecure spectroscopic redshifts. In addition, seven sources had uncertain spectroscopic \nclassification, either because of low signal-to-noise ratio or because only one line was \nvisible. This left 49 candidates with no redshift information. Figure 1 shows the spectra \nof the six AGNs with $3.6\\leq z_{spec}\\leq 4.2$ and i$_{AB}\\leq $23.0 discovered during \nour spectroscopic campaign with IMACS and LDSS-3. Most sources show characteristic AGN \nlines like Si$_{IV}$ and C$_{IV}$. The two sources that don not show Si$_{IV}$ and C$_{IV}$, \nhave been included for other strong high-ionization lines that are also characteristic of \nAGNs. More specifically, COSMOS664641 has strong O$_{VI}$ and N$_{V}$ lines, while the \nC$_{IV}$ line falls in a region of telluric absorption and this could be the reason it is \nnot observed. In the case of COSMOS247934 we also detect strong O$_{VI}$\/Ly$\\beta$, \nas well as He$_{II}$ lines, and it is relatively bright with $M_{1450}\\sim -23.2$. The fact \nthat these sources lack X-ray emission does not preclude their AGN nature, since there is \na number of examples of AGNs not detected by deep X-ray surveys \\citep{Ste02}. Moreover, \ntheir luminosities ($M_{1450}\\le -23.5$) are another indication of their nuclear activity. \nIn the subsequent analysis we only consider sources with secure spectroscopic redshifts \neither from our campaign or from the literature.\n\n\\begin{figure}\n\\label{fig:col}\n\\plotone{fig2.pdf}\n\\caption{Color-color diagram. Blue triangles show candidates selected by color. \nPentagons show candidates selected by photometric redshifts or X-ray emission and \nconfirmed by spectroscopy as AGNs. Green pentagons correspond to AGNs at 3.6$<z_{spec}<$4.2 \nand magenta pentagons show confirmed AGNs that have either z$_{spec}<$3.6 or z$_{spec}>$4.2. \nNotice that half of the confirmed AGNs are found outside the color selection locus and that \nhalf of the color selected candidates still need to be observed.} \n\\end{figure}\n\nThe distribution of our candidates in the color-color space can be seen in Figure 2. \nHere we plot the entire color-selected sample and indicate which sources were confirmed as \nAGNs, in the redshift range of interest, either after our spectroscopic campaign or from \nthe literature. We also indicate sources that lie in the color locus but their spectroscopic \nredshifts are either $z_{spec}<$3.6 or $z_{spec}>$4.2. A detailed presentation of the \nfull spectroscopic sample and a comprehensive description of the different color criteria \nare not the main aims of the present paper and will be discussed in a future work.\n\n\\begin{deluxetable*}{ccccccccc}\n\\tablecaption{Confirmed AGNs Used for Determining Space Density \\label{tab:spaced}}\n\\tablehead{\n\\colhead{ID} & \\colhead{R.A.} &\n\\colhead{Decl.} & \\colhead{$i_{AB}$} & \n\\colhead{$z_{spec}$} & \\colhead{$r_{AB}$} &\\colhead{M$_{1450}$} & References \\\\}\n\\startdata\n38736 & 150.732540 & 1.516127 & 22.088 & 4.183 & 22.897 & -23.341   & our spectroscopy \\\\\n247934 & 150.801300 & 1.657550 & 22.334 & 3.772 & 22.817 & -23.182  & our spectroscopy \\\\\n330806 & 150.107380 & 1.759201 & 22.555 & 4.140 & 23.105 & -23.110  & \\citet{Ike11} \\\\\n658294 & 149.467350 & 1.855592 & 21.056 & 4.174 & 21.603 & -24.630  & \\citet{Tru09} \\\\\n664641 & 149.533720 & 1.809260 & 22.436 & 3.986 & 22.946 & -23.182  & our spectroscopy \\\\\n899256 & 150.782210 & 2.285049 & 21.927 & 3.626 & 22.363 & -23.545  & our spectroscopy \\\\\n1054048\\tablenotemark{*} & 149.879200 & 2.225839 & 22.697 & 3.650 & 23.200 & -22.722  & \\citet{Mar16a} \\\\\n1159815 & 150.638440 & 2.391350 & 22.157 & 3.650 & 22.539 & -23.383  & \\citet{Ike11} \\\\\n1163086 & 150.703770 & 2.370019 & 22.444 & 3.748 & 22.863 & -23.122  & \\citet{Mar16a} \\\\\n1208399 & 150.259540 & 2.376141 & 21.424 & 3.717 & 21.488 & -24.478  & \\citet{Mar16a} \\\\\n1224733 & 150.208990 & 2.438466 & 21.147 & 3.715 & 21.485 & -24.480  & \\citet{Mar16a} \\\\\n1273346\\tablenotemark{*} & 149.776910 & 2.444306 & 22.779 & 4.170 & 23.274 & -22.952  & \\citet{Mar16a} \\\\\n1730531\\tablenotemark{*} & 149.843220 & 2.659095 & 22.900 & 3.748 & 23.439 & -22.545  & our spectroscopy \\\\\n1856470 & 150.475680 & 2.798362 & 21.282 & 4.110 & 21.753 & -24.445  & \\citet{Mar16a} \\\\\n1938843 & 149.845860 & 2.860459 & 22.160 & 3.630 & 22.619 & -23.290  & our spectroscopy \\\\\n1971812 & 149.472870 & 2.793400 & 21.887 & 3.610 & 22.179 & -23.717  & \\citet{Mar16a} \\\\\n\\enddata\n\\tablenotetext{*}{Not included in the space density bins because M$_{1450}>$-23.0}\n\\end{deluxetable*}\n\n\\section{Space Density Determination}\n\nThe advantage of doing this study in the COSMOS field is that it already contains extensive \nspectroscopic follow-up and extensive multi-wavelength data from radio to X-rays. Thus, \ncombining the confirmed candidates presented above, with known AGNs from the literature, \nwe obtain a sample of 16 spectroscopically confirmed AGNs with 3.6$<z<$4.2 and i$_{AB}<$23, \npresented in Table \\ref{tab:spaced}, along with the relative r$_{AB}$ for each source. The \ni$_{AB}$ magnitudes are from HST F814W band, while the r$_{AB}$ magnitudes are from the \nSubaru telescope and they are described in \\citet{Ilb09}.\n\nThe absolute magnitude at 1450 {\\AA} rest frame (M$_{1450}$) for each source has been \nderived from the $r_{AB}$ magnitude applying a K-correction according to the following \nformula:\n\\begin{equation}\nM_{1450} = r_{AB} - 2.5log(1+z_{spec})+K_{corr} \n\\end{equation}\nwhere\n\\begin{equation}\nK_{corr}=2.5\\alpha_\\nu log_{10}(\\lambda_{obs}\/(1+z_{spec})\/\\lambda_{rest})\n\\end{equation}\nThe AGN intrinsic slope $\\alpha_\\nu$ is fixed to -0.7, while $\\lambda_{obs}=6284$ {\\AA} \nis the central wavelength of the $r_{AB}$ filter and $\\lambda_{rest}=1450$ {\\AA}. The reason \nwe chose the $r_{AB}$ band is because it is not affected by strong quasar emission lines, \nlike C$_{IV}$ that falls in the $i_{AB}$ band for this redshift range. The K-correction \nis redshift dependent and ranges from 0.05mag at z=3.6 to 0.14mag at z=4.2. To check the \nrobustness of our absolute magnitude determination, we have used various methods to \ncalculate it (using the i and r band, with and without K-correction). The point at \nM$_{1450}$=-24.5 remains basically unaltered, and we only see small changes in it at \nM$_{1450}$=-23.5, with the current determination resulting in the faintest absolute \nmagnitudes, thus being the most conservative.\n\nWe find four sources for -25$<M_{1450}<$-24 and 9 for -24$<M_{1450}<$-23. Based on these \nnumbers, we calculate the space density of AGNs at this redshift range in the two magnitude \nbins. This space density, summarized in Table \\ref{tab:spacedn}, has been derived by \ndividing the actual number of the spectroscopically confirmed AGNs with the comoving \nvolume between $3.6\\leq z\\leq 4.2$, without any correction for incompleteness. Thus it \nrepresents a robust lower limit to the real space density of $z\\sim 4$ AGNs. As can be \nseen in Figure 3, without any corrections (blue filled squares), our measurements agree \nwell with the \\citet{Gli11} analysis and put more stringent constraints on the knee \nof the LF.\n\nConsidering completeness and contamination, we can estimate rough corrections. \nIn the color selection, we have six AGNs with 3.6$<z_{spec}<$4.2 out of 12 AGNs with \nknown $z_{spec}$. Thus, out of the 11 candidates without $z_{spec}$, we expect five or six \n($\\sim$50\\%) to have 3.6$<z_{spec}<$4.2, resulting to 11 or 12 potential AGNs with the \n(B$_{J}$-V$_{J}$) versus (r-i) selection. The known AGNs with 3.6$<z_{spec}<$4.2 and \ni$_{AB}\\leq$23.0 are 16, of which we recover 37.5\\% with the (B$_{J}$-V$_{J}$) versus \n(r-i) criterion (six out of 16). This means that the total number of AGNs expected in \nCOSMOS at 3.6$<z_{spec}<$4.2 and i$_{AB}<$23.0 could be N$_{tot}$=11$\\times$16\/6=29.3 or \nN$_{tot}$=12$\\times$16\/6=32. The space density determination using only the 16 AGNs known \nat 3.6$<z_{spec}<$4.2, i$_{AB}\\leq$23.0 and -26.0$<M_{1450}<$-23.0 is incomplete by a factor \nof 0.50-0.55 at least. If we correct for this incompleteness factor, we will go to a level \nhigher than \\cite{Gli11} (green open squares in Figure 3). Adopting a slightly different \ncolor criterion, with ($B_{J}-V_{J})>1.1$ instead of 1.3 as threshold, the expected total \nnumber of AGNs is 34 and the completeness corrections remain at the $\\sim 50\\%$ level. \nThis indicates that the green squares (corrected space density) in Figure 3 are quite \nrobust with respect to the details of the adopted color criterion.\n\n\\begin{deluxetable*}{cccccc}\n\\tablecaption{AGN Space Density ($<z>$=3.9)\\label{tab:spacedn}}\n\\tablehead{\n\\colhead{M$_{1450}$} & \\colhead{$\\Phi$} &\n\\colhead{$\\sigma_\\Phi^{up}$} & \\colhead{$\\sigma_\\Phi^{low}$} & \n\\colhead{N$_{AGN}$} & \\colhead{$\\Phi_{corr}$}\\\\ & $Mpc^{-3} Mag^{-1}$ & & & & \\\\}\n\\startdata\n-24.5 & 3.509e-07& 2.789e-07& 1.699e-07& 4 &7.018e-07\\\\\n-23.5 &7.895e-07& 3.616e-07& 2.595e-07& 9 & 1.579e-06 \\\\\n\\enddata\n\\end{deluxetable*}\n\nIn Figure 3 we also present the LFs calculated by \\citet{Aki18}, \\citet{Par18}, and \n\\citet{Mas12} for comparison. The sample created by \\citet{Aki18} is limited to g-band \ndropout (i.e. $3.5<z<4.0$) point-like sources, for which they have derived photometric \nredshifts based on five filters (g,r,i,z,y). The faint-end slope presented in \ntheir work is too shallow to be reconciled with our measurements, which correspond to \nspectroscopically confirmed sources. The LF by \\citet{Ike11} is also lower than our space \ndensity determination. On the other hand \\citet{Par18}, after performing an independent \nphotometric redshift estimate of the X-ray-selected sample presented by \\citet{Gia15}, \ndiscarded 10 of the 22 sources that were supposed to lie at $z>$4. Even though the faint-end \nslope in \\citet{Par18} is steeper than that found by \\citet{Gli11}, their space density in \nabsolute magnitudes M$_{1450}<$-23 is marginally in agreement with our estimates. We also \nshow the space density derived by \\citet{Mar16b}, based on X-ray data, after being converted \nto UV \\citep{Ric17}. Although these points are higher than most optical LFs, they are slightly \nlower than our estimate. In Table \\ref{tab:spacedn} we present the estimate of the AGN space \ndensity $\\Phi$, based on our analysis, in the two absolute magnitude bins. Even excluding \nthe COSMOS247934, which is the least certain among our sources, the uncorrected space density \nat M$_{1450}$=-23.5 becomes 7.018e-07 $Mpc^{-3}Mag^{-1}$, which is still higher than all LFs \npresented in Figure 3, except for \\citet{Par18}. Considering the space density corrected for\nincompleteness, also the \\citet{Par18} LF also turns out to be underestimated.\n\nAn important aspect, made clear by our sample, is that selections based on color criteria \ncan be highly incomplete, since out of the 16 spectroscopically confirmed AGNs only six \nhave been selected by color. So far, the majority of studies on the AGN LF at this redshift \nrange is based on color-selected samples and this could be the reason why faint AGN number \ndensities have been underestimated. When a first attempt was made by \\citet{Gia15} to create \nan AGN sample based on non-traditional criteria, a different picture emerged. Given that AGNs, \neven at faint magnitudes, have a large escape fraction as shown by \\citet{Gra18}, an increase \nof the estimate of their population can have significant implications on the contribution \nof AGNs to the H$_{I}$ ionizing background.  \n\n\\section{Discussion and Conclusions}\n\nOur estimates of the space density in the range $-24.5<M_{1450}<-23.5$, shown by filled \nblue squares in Figure 3, are not corrected for any incompleteness factor, thus they \nrepresent firm lower limits, assuming that the density fluctuations due to cosmic variance \nare not important.\n\nTo explore this possibility, we checked how the volume density of AGNs at $z\\sim$4 in COSMOS \ncompares to the average derived from the SDSS survey. In the COSMOS area there is no known \nquasi stellar object (QSO) brighter than i$_{AB}$=21.0 in the redshift interval $3.6<z<4.2$. \nConsidering the SDSS DR14 catalog \\citep{Pas18} in areas of different sizes, ranging from \n10-100 deg$^{2}$, centered on the COSMOS field, we find a mean density of $0.59\\pm 0.10~deg^{-2}$ \ncompared to the mean SDSS density of $0.61\\pm 0.01~deg^{-2}$ (5683 QSOs in 9376 deg$^{2}$).\nThis indicates that the COSMOS field is not particularly overdense or underdense at $z\\sim 4$. \nFor this reason, the completion of a spectroscopic AGN survey in this field, such as the \none presented here, is fundamental to address the role of AGNs in the reionization epoch. \n\nIndeed, the AGN space density derived by our study is a conservative estimate and might \nincrease in the future. As can be seen in Table \\ref{tab:colorsel}, six confirmed AGNs at \n$z_{spec}\\geq 3$ have $z_{phot}<$1 in \\cite{Ilb09}. In two cases, where both the spectroscopic \nand the photometric redshifts are larger than 3, the $z_{phot}$ tends to systematically \nunderestimate the actual $z_{spec}$. This is an indication that our selection based on \n$z_{phot}$ could still be biased against $z\\sim 4$ AGNs, and that different $z_{phot}$ recipes \ncould increase the number of AGN candidates selected by photometric redshifts. As an example, \n\\citet{Sal11} provided photometric redshifts only for two of our sources in \nTable \\ref{tab:colorsel}, i.e., source id=330806 with $z_{phot}=3.949$ and id=1226535 \nwith $z_{phot}=4.545$. Their estimates are in reasonable agreement with the spectroscopic \nredshifts of these AGNs.\n\n\\begin{figure*}\n\\label{fig:spDen}\n\\plotone{fig3.pdf}\n\\caption{Luminosity function of quasi stellar objects\/AGNs at $z\\sim$4. Black triangles \nshow the bright end determined by the SDSS \\citep{Aki18}, green triangles are the points \npresented by \\citet{Sch18}, red circles show the bins calculated by \\citet{Gli11}, \nblue stars show the bins by \\citet{Gia15}, filled blue squares represent the two magnitude \nbins from this work (no corrections), and the open green squares correspond to the space \ndensity derived in this work after applying the corrections discussed in Section 4. Notice \nthat our data actually compensate for the apparent dip present in the work by \\citet{Gli11}. \nThe results by \\citet{Mar16b}, converted into UV, are presented as orange asterisks. For \ncomparison we also present the LFs derived by \\citet{Ike11}, \\citet{Mas12}, \\citet{Ric17}, \n\\citet{Aki18}, and \\citet{Par18}. All LFs have been evolved to z=3.9 following the density \nevolution law by \\citet{Fan01}.}\n\\end{figure*}\n\n\\citet{Stev18}, in their analysis of the UV LF of a mixed sample including both SFGs and \ngalaxies dominated by AGNs at $z\\sim$4 in the SHELA survey, found a space density of \n$10^{-6} Mpc^{-3}$ at $M_{1450}=-23.5$. Our measurements in the same redshift and absolute \nmagnitude is $\\sim1.6\\times10^{-6} Mpc^{-3}$ (corrected), and this is an indication that \ntheir LF is dominated by AGNs at $M_{1450}<-23.5$. Based on their discussion, a high AGN \nspace density would mean that AGNs could be largely responsible for the H$_{I}$ ionizing \nUVB at $z=4$.\n\nWe are confident that on the net of all random and systematic effects, the corrected \nestimate of the LF presented in this work represents a robust determination of the space \ndensity of L$^{*}$ AGNs at $z\\sim 4$. The agreement of our measurement with the \\citet{Gli11} \nresults favors at $M_{1450}<-23.5$ a steeper slope of the LF for the COSMOS field than that \ndetermined by \\citet{Ike11}, \\citet{Mas12}, \\citet{Ric17}, \\citet{Aki18}, and \\citet{Par18}. \n\nThis study poses an open challenge that should be addressed in the future with major \nobservational effort: to measure with great accuracy the space density of AGNs at L=L$^{*}$ \nand $z>4$. In fact, in a subsequent work, once our spectroscopic sample is complete, we will \npresent the global shape of the LF at $z\\sim$4 and the associated emissivity. This can have \ndeep implications on the extrapolation of the number of QSOs expected at high-z in wide and \ndeep large area surveys, either ground based, e.g. LSST, or from space, e.g., e-Rosita, \nEuclid, WFIRST. An upward revision of the number density of L=L$^{*}$ AGNs would certainly \nimply a reconsideration of the expected QSO and AGN numbers at $z>4$ in these future missions. \n\n\\acknowledgments\nWe would like to thank the anonymous referee for useful suggestions and constructive \ncomments that helped us improve this paper. This paper includes data gathered with the \n6.5 meter Magellan Telescopes located at Las Campanas Observatory (LCO), Chile.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nWe consider the equation\n\\begin{equation}\n  \\label{eq:1}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + |\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n\\end{equation}\nand both semi-classical ($\\eps\\to 0$) and large time ($t\\to \\pm\n\\infty$) limits. Of course these limits must not be expected to\ncommute, and one of the goals of this paper is to analyze this lack of\ncommutation on specific asymptotic data, under the form of coherent\nstates as described below. Even though our main  result\n(Theorem~\\ref{theo:cv}) is proven specifically for the above case of a\ncubic three-dimensional equation, two important intermediate results\n(Theorems~\\ref{theo:scatt-quant} and \\ref{theo:scatt-class}) are\nestablished in a more general setting. Unless specified otherwise, we\nshall from now on consider $\\psi^\\eps:\\R_t \\times \\R^d_x\\to \\C$, $d\\ge\n1$. \n\n\n\\subsection{Propagation of initial coherent states}\n\\label{sec:prop-init-coher}\n\nIn this subsection, we consider the initial value problem, as opposed\nto the scattering problem treated throughout this paper. More\nprecisely, we assume here that the wave function is, at time $t=0$,\ngiven by the coherent state\n\\begin{equation}\n  \\label{eq:ci}\n  \\psi^\\eps(0,x) = \\frac{1}{\\eps^{d\/4}}a\\(\\frac{x-q_0}{\\sqrt\\eps}\\)\n  e^{ip_0\\cdot (x-q_0)\/\\eps},\n\\end{equation}\nwhere $q_0,p_0\\in \\R^d$ denote the initial position and velocity,\nrespectively. The function $a$ belongs to the Schwartz class,\ntypically. In the case where $a$ is a (complex) Gaussian, many\nexplicit computations are available in the linear case (see\n\\cite{Hag80}). Note that the $L^2$-norm of $\\psi^\\eps$ is independent\nof $\\eps$, $\\|\\psi^\\eps(t,\\cdot)\\|_{L^2(\\R^d)} =\\|a\\|_{L^2(\\R^d)}$. \n\n Throughout this subsection, we assume that the external\npotential $V$ is smooth and real-valued, $V\\in C^\\infty(\\R^d;\\R)$, and\nat most quadratic, in the sense that \n\\begin{equation*}\n  \\d^\\alpha V\\in L^\\infty(\\R^d),\\quad \\forall |\\alpha|\\ge 2.\n\\end{equation*}\nThis assumption will be strengthened when large time behavior is\nanalyzed. \n\\subsubsection{Linear case}\n\\label{sec:linear-case}\n Resume \\eqref{eq:1} in the absence of nonlinear term:\n\\begin{equation}\n  \\label{eq:lin}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps,\\quad x\\in \\R^d,\n\\end{equation}\nassociated with the initial datum \\eqref{eq:ci}. To derive an\napproximate solution, and to describe the propagation of the initial\nwave packet, introduce the Hamiltonian flow\n\\begin{equation}\n  \\label{eq:hamil}\n  \\dot q(t)= p(t),\\quad \\dot p(t)=-\\nabla V\\(q(t)\\),\n\\end{equation}\nand prescribe the initial data $q(0)=q_0$, $p(0)=p_0$. Since the\npotential $V$ is smooth and at most quadratic, the solution\n$(q(t),p(t))$ is smooth, defined for all time, and grows at most\nexponentially.\nThe classical action is given by\n\\begin{equation}\\label{eq:action}\nS(t)=\\int_0^t \\left( \\frac{1}{2} |p(s)|^2-V(q(s))\\right)\\,ds.\n\\end{equation}\nWe observe that if we change the unknown function $\\psi^\\eps$ to\n$u^\\eps$ by\n\\begin{equation}\n  \\label{eq:chginc}\n  \\psi^\\eps(t,x)=\\eps^{-d\/4} u^\\eps \n\\left(t,\\frac{x-q(t)}{\\sqrt\\eps}\\right)e^{i\\left(S(t)+p(t)\\cdot\n    (x-q(t))\\right)\/\\eps},\n\\end{equation}\nthen, in terms of $u^\\eps=u^\\eps(t,y)$, the Cauchy problem\n\\eqref{eq:lin}--\\eqref{eq:ci} is equivalent to \n\\begin{equation}\\label{eq:ueps0}\ni\\d_t\nu^\\eps+\\frac{1}{2}\\Delta u^\\eps=V^\\eps(t,y)\nu^\\eps\\quad ;\\quad u^\\eps(0,y) = a(y),\n\\end{equation}\nwhere the external time-dependent potential $V^\\eps$ is given by\n\\begin{equation}\n  \\label{eq:Veps}\n  V^\\eps(t,y)= \\frac{1}{\\eps}\\(V(x(t)+\n\\sqrt{\\eps}y)-V(x(t))-\\sqrt{\\eps}\\<\\nabla V(x(t)),y\\>\\).\n\\end{equation}\nThis potential corresponds to the first term of a Taylor expansion of\n$V$ about the point $q(t)$, and we naturally introduce \n$u=u(t,y)$ solution to \n\\begin{equation}\\label{eq:ulin}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u\\quad\n;\\quad u(0,y)=a(y),\n\\end{equation}\nwhere\n\\begin{equation*}\n  Q(t):= \\nabla^2 V\\(q(t)\\), \\quad \\text{so that } \\frac{1}{2}\\<\n  Q(t)y,y\\>  = \\lim_{\\eps \\to 0} V^\\eps(t,y). \n\\end{equation*}\nThe obvious candidate to approximate the initial wave function\n$\\psi^\\eps$ is then:\n\\begin{equation}\n  \\label{eq:phi}\n  \\varphi^\\eps(t,x)=\\eps^{-d\/4} u\n\\left(t,\\frac{x-q(t)}{\\sqrt\\eps}\\right)e^{i\\left(S(t)+p(t)\\cdot\n    (x-q(t))\\right)\/\\eps}.\n\\end{equation}\nIndeed, it can be proven (see\ne.g. \\cite{BGP99,BR02,CoRoBook,Hag80,HaJo00,HaJo01}) that there\nexists \n$C>0$ independent of $\\eps$ such that\n\\begin{equation*}\n  \\|\\psi^\\eps(t,\\cdot)-\\varphi^\\eps (t,\\cdot)\\|_{L^2(\\R^d)}\\le\n  C\\sqrt\\eps e^{Ct}. \n\\end{equation*}\nTherefore, $\\varphi^\\eps$ is a good approximation of $\\psi^\\eps$ at least up to time\nof order $c\\ln\\frac{1}{\\eps}$ (Ehrenfest time). \n\n\\subsubsection{Nonlinear case}\n\\label{sec:nonlinear}\n When adding a nonlinear term to \\eqref{eq:lin}, one has to be\n cautious about the size of the solution, which rules the importance\n of the nonlinear term. To simplify the discussions, we restrict our\n analysis to the case of a gauge invariant, defocusing, power nonlinearity,\n $|\\psi^\\eps|^{2\\si}\\psi^\\eps$. We choose to measure the importance of\n nonlinear effects not directly through the size of the initial data,\n but through an $\\eps$-dependent coupling factor: we keep the initial\n datum \\eqref{eq:ci} (with an $L^2$-norm independent of $\\eps$), and\n consider\n \\begin{equation*}\n   i\\eps\\d_t \\psi^\\eps + \\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n   V(x)\\psi^\\eps + \\eps^\\alpha|\\psi^\\eps|^{2\\si}\\psi^\\eps.\n \\end{equation*}\nSince the nonlinearity is homogeneous, this approach is equivalent to\nconsidering $\\alpha=0$, up to multiplying the initial datum by\n$\\eps^{\\alpha\/(2\\si)}$. \nWe assume $\\si>0$, with $\\si<2\/(d-2)$ if $d\\ge 3$: for $a\\in \\Sigma$,\ndefined by\n\\begin{equation*}\n  \\Sigma = \\{f\\in H^1(\\R^d),\\quad x\\mapsto \\<x\\> f(x)\\in\n  L^2(\\R^d)\\},\\quad \\<x\\>=\\(1+|x|^2\\)^{1\/2},\n\\end{equation*}\nwe have, for fixed $\\eps>0$, $\\psi^\\eps_{\\mid t=0}\\in \\Sigma$, and the\nCauchy problem is globally well-posed, $\\psi^\\eps\\in C(\\R_t;\\Sigma)$\n(see e.g. \\cite{Ca11}). It was established in \n\\cite{CaFe11} that the value \n\\begin{equation*}\n  \\alpha_c = 1+\\frac{d\\si}{2}\n\\end{equation*}\nis critical in terms of the effect of the nonlinearity in the\nsemi-classical limit $\\eps\\to 0$. If $\\alpha>\\alpha_c$, then \n$\\varphi_{\\rm lin}^\\eps$, given by \\eqref{eq:ulin}-\\eqref{eq:phi},  is\nstill a good approximation of $\\psi^\\eps$ at least up to time\nof order $c\\ln\\frac{1}{\\eps}$. On the other hand, if\n$\\alpha=\\alpha_c$, nonlinear effects alter the behavior of $\\psi^\\eps$\nat leading order, through its envelope only. Replacing \\eqref{eq:ulin}\nby \n\\begin{equation}\\label{eq:u}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u+|u|^{2\\si}u,\n\\end{equation}\nand keeping the relation \\eqref{eq:phi}, $\\varphi^\\eps$ is now a good\napproximation of $\\psi^\\eps$. In \\cite{CaFe11} though, the time of\nvalidity of the approximation is not always proven to be of order at\nleast $c\\ln\\frac{1}{\\eps}$, sometimes shorter time scales (of the\norder $c\\ln\\ln\\frac{1}{\\eps}$) have to be considered, most likely for\ntechnical reasons only. Some of these restrictions have been removed in\n\\cite{Ha13}, by considering decaying external\npotentials $V$. \n\n\n\\subsection{Linear scattering theory and coherent states}\n\\label{sec:line-scatt-theory}\n\nWe now consider the aspect of large time, and instead of prescribing\n$\\psi^\\eps$ at $t=0$ (or more generally at some finite time), we\nimpose its behavior at $t=-\\infty$.\n In the linear case \\eqref{eq:lin},\nthere are several results addressing the question mentioned above,\nconsidering different forms of asymptotic states at $t=-\\infty$.\nBefore describing them, we recall important facts concerning quantum\nand classical scattering. \n\n\\subsubsection{Quantum scattering}\n\\label{sec:quantum-scattering}\n\nThroughout this paper, we assume that the external potential is\nshort-range, and satisfies the following properties:\n\\begin{hyp}\\label{hyp:V}\n  We suppose that $V$ is smooth and real-valued, $V\\in\n  C^\\infty(\\R^d;\\R)$. In addition, it is short range in the following\n  sense: there exists $\\mu>1$ such that\n  \\begin{equation}\n    \\label{eq:short}\n    |\\d^\\alpha V(x)|\\le \\frac{C_\\alpha}{(1+|x|)^{\\mu+|\\alpha|}},\\quad\n   \\forall \\alpha\\in \\N^d. \n  \\end{equation}\n\\end{hyp}\nOur final result is established under the stronger condition\n$\\mu>2$ (a condition which is needed in several steps of the proof), but\nsome results are established under the mere assumption\n$\\mu>1$. Essentially, the analysis of the approximate solution is valid\nfor $\\mu>1$\n(see Section~\\ref{sec:class}), while the rest of the analysis requires $\\mu>2$. \n\\smallbreak\n\nDenote by \n\\begin{equation*}\nH_0^\\eps= -\\frac{\\eps^2}{2}\\Delta\\quad \\text{and}\\quad\nH^\\eps=-\\frac{\\eps^2}{2}\\Delta+V(x) \n\\end{equation*}\nthe underlying Hamiltonians. For fixed $\\eps>0$, the (linear) wave\noperators are given by\n\\begin{equation*}\n  W_\\pm^\\eps = \\lim_{t\\to \\pm \\infty}e^{i\\frac{t}{\\eps}H^\\eps}e^{-i\\frac{t}{\\eps}H^\\eps_0},\n\\end{equation*}\nand the (quantum) scattering operator is defined by\n\\begin{equation*}\n  S^\\eps_{\\rm lin} = \\(W_+^\\eps\\)^* W_-^\\eps. \n\\end{equation*}\nSee for instance \\cite{DG}.\n\n\\subsubsection{Classical scattering}\n\\label{sec:classical-scattering}\n\nLet $V$ satisfying Assumption~\\ref{hyp:V}. \nFor $(q^-,p^-)\\in \\R^d\\times \\R^d$, we consider the classical\ntrajectories  $(q(t),p(t))$ defined by \\eqref{eq:hamil}, \nalong with the prescribed asymptotic behavior as $t\\to -\\infty$:\n\\begin{equation}\n  \\label{eq:CI-hamilton}\n  \\lim_{t\\to -\\infty}\\left| q(t)-p^- t -q^-\\right| = \\lim_{t\\to\n    -\\infty} |p(t)-p^-|=0. \n\\end{equation}\nThe existence and uniqueness of such a trajectory can be found in\ne.g. \\cite{DG,ReedSimon3}, provided that $p^-\\not =0$. Moreover, there\nexists a closed set $\\mathcal N_0$ of Lebesgue measure zero in\n$\\R^{2d}$ such that for all $(q^-,p^-)\\in \\R^{2d}\\setminus \\mathcal\nN_0$, there exists $(q^+,p^+)\\in \\R^d\\times\n  \\(\\R^d\\setminus\\{0\\}\\)$ such that\n\\begin{equation*}\n  \\lim_{t\\to +\\infty}\\left| q(t)-p^+ t -q^+\\right| = \\lim_{t\\to\n    +\\infty} |p(t)-p^+|=0. \n\\end{equation*}\nThe classical scattering operator is $S^{\\rm cl}:(q^-,p^-)\\mapsto\n(q^+,p^+)$. Choosing  $(q^-,p^-)\\in \\R^{2d}\\setminus \\mathcal\nN_0$ implies that the following assumption is satisfied:\n\\begin{assumption}\\label{hyp:flot}\n  The asymptotic center in phase space, $(q^-,p^-)\\in \\R^d\\times\n  \\(\\R^d\\setminus\\{0\\}\\)$ is such that the classical scattering\n  operator is well-defined, \n  \\begin{equation*}\n    S^{\\rm cl}(q^-,p^-)=\n(q^+,p^+),\\quad p^+\\not =0,\n  \\end{equation*}\nand the classical action has  limits as $t\\to \\pm\\infty$:\n\\begin{equation*}\n  \\lim_{t\\to -\\infty}\\left|S(t)-t\\frac{|p^-|^2}{2}\\right| =\n  \\lim_{t\\to +\\infty}\\left|S(t)-t\\frac{|p^+|^2}{2}-S_+\\right|  =0,\n\\end{equation*}\nfor some $S_+\\in \\R$. \n\\end{assumption}\n \n\\subsubsection{Some previous results}\n\\label{sec:some-prev-results}\n\n\nIt seems that the first mathematical result involving both the\nsemi-classical and large time limits appears in\n\\cite{GV79mean}, where the classical field limit of non-relativistic\nmany-boson theories is studied in space dimension $d\\ge 3$. \n\nIn\n\\cite{Yajima79}, the\ncase of a short range potential (Assumption~\\ref{hyp:V}) is\nconsidered, with asymptotic states \nunder the form of semi-classically concentrated functions,\n\\begin{equation*}\n  e^{-i\\frac{\\eps t}{2}\\Delta}\\psi^\\eps(t)_{\\mid t =-\\infty}\n  =\\frac{1}{\\eps^{d\/2}}\\widehat f\\(\\frac{x-q^-}{\\eps}\\),\\quad f\\in L^2(\\R^d),\n\\end{equation*}\nwhere $\\widehat f$ denotes the standard Fourier transform (whose\ndefinition is independent of $\\eps$). The main result from\n\\cite{Yajima79} shows that the semi-classical limit for $S^\\eps_{\\rm lin}$\ncan be expressed in terms of the classical scattering operator, of the\nclassical action, and of\nthe Maslov index associated to each classical trajectory. We refer to\n\\cite{Yajima79} for a precise statement, and to \\cite{Yaj81} for the\ncase of long range potentials, requiring modifications of the\ndynamics. \n\\smallbreak\n\nIn \\cite{Hag81,HaJo00}, coherent  states are considered,\n\\begin{equation}\n  \\label{eq:asym-state}\n e^{-i\\frac{\\eps t}{2}\\Delta}\\psi^\\eps(t)_{\\mid t =-\\infty}=\n  \\frac{1}{\\eps^{d\/4}}u_-\\(\\frac{x-q^-}{\\sqrt\\eps}\\) \n  e^{ip^-\\cdot (x-q^-)\/\\eps+iq^-\\cdot p^-\/(2\\eps)}=:\\psi_-^\\eps(x).\n\\end{equation}\nMore precisely, in \\cite{Hag81,HaJo00}, the asymptotic state $u_-$ is assumed\nto be a complex Gaussian function. Introduce the notation\n\\begin{equation*}\n  \\delta(t) = S(t)-\\frac{q(t)\\cdot p(t) - q^-\\cdot p^-}{2}.\n\\end{equation*}\nThen Assumption~\\ref{hyp:flot} implies that there exists $\\delta^+\\in\n\\R$ such that\n\\begin{equation*}\n  \\delta(t)\\Tend t {-\\infty}0\\quad \\text{and}\\quad \\delta(t)\\Tend t\n  {+\\infty} \\delta^+. \n\\end{equation*}\n In \n\\cite{CoRoBook,HaJo00}, we find the following general result  (an asymptotic\nexpansion in powers of $\\sqrt\\eps$ is actually given, but we stick to\nthe first term to ease the presentation):\n\\begin{theorem}\\label{theo:version-lineaire}\n  Let Assumptions~\\ref{hyp:V} and \\ref{hyp:flot} be satisfied, and let\n  \\begin{equation*}\n    u_-(y) = a_- \\exp \\(\\frac{i}{2}\\<\\Gamma_-y,y\\>\\),\n  \\end{equation*}\nwhere $a_-\\in \\C$ and $\\Gamma_-$ is a complex symmetric $d\\times d$\nmatrix whose \nimaginary part is positive and non-degenerate. Consider $\\psi^\\eps$\nsolution to \\eqref{eq:lin}, with \\eqref{eq:asym-state}. Then the\nfollowing asymptotic expansion holds in $L^2(\\R^d)$:\n\\begin{equation*}\n  S^\\eps_{\\rm lin} \\psi_-^\\eps = \\frac{1}{\\eps^{d\/4}}e^{i\\delta^+\/\\eps} e^{ip^+\\cdot\n    (x-q^+)\/\\eps+iq^+\\cdot p^+\/(2\\eps)} \\hat R(G_+)\n  u_-\\(\\frac{x-q^+}{\\sqrt\\eps}\\) +\\O(\\sqrt\\eps),\n\\end{equation*}\nwhere $\\hat R(G_+)$ is the metaplectic transformation associated to\n$G_+ = \\frac{\\d (q^+,p^+)}{\\d(q^-,p^-)}$. \n\\end{theorem}\nAs a corollary, our main result yields another interpretation of the above\nstatement. It turns out that a complete scattering theory is available\nfor \\eqref{eq:ulin}. As a particular case of\nTheorem~\\ref{theo:scatt-class} (which addresses the nonlinear case), given $u_-\\in\n\\Sigma$, there exist a unique $u\\in C(\\R;\\Sigma)$ solution to\n\\eqref{eq:ulin} and a unique $u_+\\in\n\\Sigma$ such that  \n\\begin{equation*}\n  \\|e^{-i\\frac{t}{2}\\Delta}u(t)-u_\\pm \\|_\\Sigma \\Tend t {\\pm \\infty}\n  0. \n\\end{equation*}\nThen in the above theorem (where $u_-$ is restricted to be a Gaussian), we have\n\\begin{equation*}\n  u_+ = \\hat R(G_+)\n  u_-.\n\\end{equation*}\nFinally, we mention in passing the paper \\cite{NierENS}, where similar\nissues and results are obtained for\n\\begin{equation*}\n  i\\eps\\d_t \\psi^\\eps + \\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V\\(\\frac{x}{\\eps}\\) \\psi^\\eps + U(x)\\psi^\\eps,\n\\end{equation*}\nfor $V$ a short-range potential, and $U$ is bounded as well as its\nderivatives. The special scaling in $V$ implies that initially\nconcentrated waves (at scaled $\\eps$) first undergo the effects of $V$,\nthen exit a time layer of order $\\eps$, through which the main action of $V$\ncorresponds to the above quantum scattering operator (but with $\\eps=1$\ndue to the new scaling in the equation). Then, the action of $V$\nbecomes negligible, and the propagation of the wave is dictated by\nthe classical dynamics associated to $U$. \n\n\n\n\n\n\\subsection{Main results}\n\\label{sec:main}\n We now consider the nonlinear equation\n\\begin{equation}\n  \\label{eq:psi-eps}\n  i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps = V(x)\\psi^\\eps +\n  \\eps^\\alpha|\\psi^\\eps|^{2\\si}\\psi^\\eps,\n\\end{equation}\nalong with asymptotic data \\eqref{eq:asym-state}. We first prove that\nfor fixed $\\eps>0$, a scattering theory is available for\n\\eqref{eq:psi-eps}: at this stage, the value of $\\alpha$ is naturally\nirrelevant, as well as the form \\eqref{eq:asym-state}.\nTo establish a large data scattering theory for \\eqref{eq:psi}, we\nassume that the attractive part of the potential,\n\\begin{equation*}\n  (\\d_r V(x))_+=\n  \\(\\frac{x}{|x|}\\cdot \\nabla V(x)\\)_+\n\\end{equation*}\n is not too large, where $f_+=\\max (0,f)$ for any real number $f$.\n\\begin{theorem}\\label{theo:scatt-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}<\\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)$ such\n  that if the attractive part of the potential $(\\d_r V)_+$ satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+\\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\none can define a\n  scattering operator for \\eqref{eq:psi} in $H^1(\\R^d)$: for\n  all $\\psi_-^\\eps\\in H^1(\\R^d)$, there exist a unique $\\psi^\\eps\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} and a unique $\\psi_+^\\eps\\in\n  H^1(\\R^d)$ such that\n  \\begin{equation*}\n    \\|\\psi^\\eps(t)-e^{i\\frac{\\eps t}{2}\\Delta}\\psi_\\pm^\\eps\\|_{H^1(\\R^d)}\\Tend t {\\pm\n      \\infty} 0.\n  \\end{equation*}\nThe (quantum)  scattering operator is the map\n$S^\\eps:\\psi_-^\\eps\\mapsto \\psi_+^\\eps$.\n\\end{theorem}\nWe emphasize the fact that several recent results address the same\nissue, under various assumptions on the external potential $V$:\n\\cite{ZhZh14} treats the case where $V$ is an inverse square (a\nframework which is ruled out in our contribution), while in\n\\cite{CaDa-p}, the potential is more general than merely inverse\nsquare. In \\cite{CaDa-p}, a magnetic field is also included, and the\nLaplacian is perturbed with variable coefficients. We make more\ncomparisons with \\cite{CaDa-p} in Section~\\ref{sec:quant}. \n\\smallbreak\n\nThe second result of this paper concerns the scattering theory for the\nenvelope equation:\n\n\\begin{theorem}\\label{theo:scatt-class}\n   Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. One can define a\n  scattering operator for \\eqref{eq:u} in $\\Sigma$: for\n  all $u_-\\in \\Sigma$, there exist a unique $u\\in\n  C(\\R;\\Sigma)$ solution to \\eqref{eq:u} and a unique $u_+\\in\n  \\Sigma$ such that\n  \\begin{equation*}\n    \\|e^{-i\\frac{t}{2}\\Delta}u(t)-u_\\pm\\|_{\\Sigma}\\Tend t {\\pm\n      \\infty} 0.\n  \\end{equation*}\n\\end{theorem}\nAs mentioned above, the proof includes the construction of a linear\nscattering operator, comparing the dynamics associated to\n\\eqref{eq:ulin} to the free dynamics $e^{i\\frac{t}{2}\\Delta}$. In the\nabove formula, we have incorporated the information that\n$e^{i\\frac{t}{2}\\Delta}$ is unitary on $H^1(\\R^d)$, but \\emph{not on\n}$\\Sigma$ (see e.g. \\cite{CazCourant}). \n\\smallbreak\n\nWe can now state the nonlinear analogue to\nTheorem~\\ref{theo:version-lineaire}. Since\nTheorem~\\ref{theo:scatt-quant} requires $d\\ge 3$, we naturally have to\nmake this assumption. On the other hand, we will need the\napproximate envelope $u$ to be rather smooth, which requires a smooth\nnonlinearity, $\\si\\in \\N$. Intersecting this property with the\nassumptions of Theorem~\\ref{theo:scatt-quant} leaves only one case:\n$d=3$ and $\\si=1$, that is \\eqref{eq:1}, up to the scaling. We will\nsee in Section~\\ref{sec:cv} that considering $d=3$ is also crucial,\nsince the argument uses dispersive estimates which are known only in\nthe three-dimensional case for $V$ satisfying Assumption~\\ref{hyp:V}\nwith $\\mu>2$ (larger values for $\\mu$ could be considered in higher\ndimensions, though).  Introduce\nthe notation\n\\begin{equation*}\n  \\Sigma^k=\\{ f\\in H^k(\\R^d),\\quad x\\mapsto |x|^k f(x)\\in\n  L^2(\\R^d)\\}. \n\\end{equation*}\n\n\\begin{theorem}\\label{theo:cv}\n Let Assumptions~\\ref{hyp:V} and \\ref{hyp:flot} be satisfied, with\n $\\mu>2$ and $V$ as in Theorem~\\ref{theo:scatt-quant}. Consider \n $\\psi^\\eps$ solution to \n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{5\/2}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nand such that \\eqref{eq:asym-state} holds, with $u_-\\in \\Sigma^7$.\nThen the\nfollowing asymptotic expansion holds in $L^2(\\R^3)$:\n\\begin{equation}\\label{eq:asym-finale}\n  S^\\eps \\psi_-^\\eps = \\frac{1}{\\eps^{3\/4}}e^{i\\delta^+\/\\eps} e^{ip^+\\cdot\n    (x-q^+)\/\\eps+iq^+\\cdot p^+\/(2\\eps)} u_+\\(\\frac{x-q^+}{\\sqrt\\eps}\\)\n  +\\O(\\sqrt\\eps), \n\\end{equation}\nwhere $S^\\eps$ is given by Theorem~\\ref{theo:scatt-quant} and $u_+$\nstems from Theorem~\\ref{theo:scatt-class}. \n\\end{theorem}\n\\begin{remark}\n  In the subcritical case, that is if we consider\n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{\\alpha}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nalong with \\eqref{eq:asym-state}, for some $\\alpha>5\/2$, the argument\nof the proof shows that \\eqref{eq:asym-finale} remains true, but with $u_+$ given\nby the scattering operator associated to \\eqref{eq:ulin} (as opposed\nto \\eqref{eq:u}), that is, the same conclusion as in\nTheorem~\\ref{theo:version-lineaire} when $u_-$ is a Gaussian. \n\\end{remark}\nAs a corollary of the proof of the above result, and of the analysis\nfrom \\cite{CaFe11}, we infer:\n\\begin{corollary}[Asymptotic decoupling]\\label{cor:decoupling}\n  Let Assumption~\\ref{hyp:V} be satisfied, with\n $\\mu>2$ and $V$ as in Theorem~\\ref{theo:scatt-quant}. Consider \n $\\psi^\\eps$ solution to \n \\begin{equation*}\n    i\\eps\\d_t \\psi^\\eps +\\frac{\\eps^2}{2}\\Delta \\psi^\\eps =\n  V(x)\\psi^\\eps + \\eps^{5\/2}|\\psi^\\eps|^2 \\psi^\\eps,\\quad (t,x)\\in \\R\\times \\R^3,\n \\end{equation*}\nwith initial datum\n\\begin{equation*}\n  \\psi^\\eps(0,x) = \\sum_{j=1}^N\\frac{1}{\\eps^{3\/4}}a_j\\(\\frac{x-q_{0j}}{\\sqrt\\eps}\\)\n  e^{ip_{0j}\\cdot (x-q_{0j})\/\\eps}=:\\psi_0^\\eps(x),\n\\end{equation*}\nwhere $N\\ge 2$, $q_{0j},p_{0j}\\in \\R^3$, $p_{0j}\\not =0$ so that scattering   is\navailable as\n$t\\to +\\infty$ for  $(q_j(t),p_j(t))$, in the sense of\nAssumption~\\ref{hyp:flot}, and $a_j\\in \\Sch(\\R^3)$. We suppose\n$(q_{0j},p_{0j})\\not =(q_{0k},p_{0k})$ for $j\\not =k$. Then we have\nthe uniform estimate: \n\\begin{equation*}\n  \\sup_{t\\in \\R}\\left\\| \\psi^\\eps(t) - \\sum_{j=1}^N\n    \\varphi_j^\\eps(t)\\right\\|_{L^2(\\R^3)} \\Tend \\eps\n  0 0 ,\n\\end{equation*}\nwhere $\\varphi_j^\\eps$ is the approximate solution with the $j$-th\nwave packet as an initial datum. As a consequence, the \n asymptotic expansion holds in $L^2(\\R^3)$, as $\\eps \\to 0$:\n\\begin{equation*}\n  \\(W^\\eps_\\pm\\)^{-1} \\psi_0^\\eps =\\sum_{j=1}^N\n  \\frac{1}{\\eps^{3\/4}}e^{i\\delta_{j}^\\pm\/\\eps} e^{ip_{j}^\\pm\\cdot \n    (x-q_{j}^\\pm)\/\\eps+iq_{j}^\\pm\\cdot p_{j}^\\pm\/(2\\eps)}\n  u_{j\\pm}\\(\\frac{x-q_{j}^\\pm}{\\sqrt\\eps}\\) \n  +o(1), \n\\end{equation*}\nwhere the inverse wave operators $\\(W^\\eps_\\pm\\)^{-1} $ stem from\nTheorem~\\ref{theo:scatt-quant}, the $u_{j\\pm}$'s \nare the asymptotic states emanating from $a_j$, and \n\\begin{equation*}\n  \\delta_{j}^\\pm = \\lim_{t\\to \\pm\\infty}\\(S_j(t) - \\frac{q_j(t)\\cdot\n    p_j(t)-q_{0j}\\cdot p_{0j}}{2}\\)\\in \\R. \n\\end{equation*}\n\\end{corollary}\n\\begin{remark}\n  In the case $V=0$, the approximation by wave packets is actually\n  exact, since then $Q(t)\\equiv 0$, hence $u^\\eps=u$. For one wave\n  packet, Theorem~\\ref{theo:cv} \n  becomes empty, since it is merely a rescaling. On the other hand,\n  for two initial wave packets, even in the case $V=0$,\n  Corollary~\\ref{cor:decoupling} brings some information, reminiscent\n  of profile decomposition. More precisely, define $u^\\eps$ by\n  \\eqref{eq:chginc}, and choose (arbitrarily) to privilege the trajectory\n  $(q_1,p_1)$. The Cauchy problem is then equivalent to \n  \\begin{equation*}\n\\left\\{\n\\begin{aligned}\n    &i\\d_t u^\\eps+\\frac{1}{2}\\Delta u^\\eps = |u^\\eps|^2 u^\\eps,\\\\\n&    u^\\eps(0,y) = a_1(y) + a_2\\( y +\\frac{q_{01}-q_{02}}{\\sqrt\\eps}\\)\n    e^{ip_{02}\\cdot \\delta q_0\/\\eps -i\\delta p_0\\cdot y\/\\sqrt\\eps},\n  \\end{aligned}\n\\right.\n\\end{equation*}\nwhere we have set $\\delta p_0 = p_{01}-p_{02}$ and $\\delta q_0\n=q_{01}-q_{02}$. \nNote however that the initial datum is uniformly bounded in\n$L^2(\\R^3)$, but in no $H^s(\\R^3)$ for $s>0$ (if $p_{01}\\not =\np_{02}$), while the equation is \n$\\dot H^{1\/2}$-critical, Therefore, even in the case\n$V=0$, \nCorollary~\\ref{cor:decoupling} does not seem to be a consequence of\nprofile decompositions like in\ne.g. \\cite{DuHoRo08,Keraani01,MerleVega98}. In view of\n\\eqref{eq:hamil}, the approximation provided by\nCorollary~\\ref{cor:decoupling} reads, in that case:\n\\begin{equation*}\n  u^\\eps(t,y) = u_1(t,y) + u_2\\(t,y\n  +\\frac{t \\delta p_0+\\delta q_0}{\\sqrt\\eps}\\)\n  e^{i\\phi_2^\\eps(t,y)}+o(1)\\quad \\text{in }L^\\infty(\\R;L^2(\\R^3)),\n\\end{equation*}\nwhere the phase shift is given by\n\\begin{align*}\n  \\phi^\\eps_2(t,y) &= \\frac{1}{\\eps}p_{02}\\cdot \\( t\\delta p_0+\\delta\n  q_0\\) -\\frac{1}{\\sqrt\\eps}\\delta p_0\\cdot y +\\frac{t}{2\\eps}\n  \\( |p_{02}|^2-|p_{01}|^2\\) \\\\\n&= \\frac{1}{\\eps}p_{02}\\cdot \\delta\n  q_0 -\\frac{1}{\\sqrt\\eps}\\delta p_0\\cdot y -\\frac{t}{2\\eps}|\\delta\n  p_0|^2. \n\\end{align*}\n\\end{remark}\n\n\n\n\\noindent {\\bf Notation.} We write $a^\\eps(t)\\lesssim b^\\eps(t)$\nwhenever there exists $C$ independent of $\\eps\\in (0,1]$ and $t$ such\nthat $a^\\eps(t)\\le C b^\\eps(t)$. \n \n\n\n\n\n\n\n\n\n\n\\section{Spectral properties and consequences}\n\\label{sec:spectral}\n\nIn this section, we derive some useful properties for the Hamiltonian\n\\begin{equation*}\n  H=-\\frac{1}{2}\\Delta +V.\n\\end{equation*}\nSince the dependence upon $\\eps$ is not addressed in this\nsection, we assume $\\eps=1$.\n\\smallbreak\n\nFirst, it follows for instance from \\cite{Mourre} that\nAssumption~\\ref{hyp:V} implies that $H$ has no\nsingular spectrum. Based on Morawetz estimates, we show that $H$ has\nno eigenvalue, provided that the  attractive part of $V$ is\nsufficiently small. Therefore, the spectrum of $H$ is\npurely absolutely continuous. \nFinally, again if  the  attractive part of $V$ is\nsufficiently small, zero is not a resonance of $H$, so Strichartz\nestimates are available for $e^{-itH}$. \n\n\n\n\n\\subsection{Morawetz estimates and a first consequence}\n\\label{sec:morawetz}\n\nIn this section, we want to treat both linear and nonlinear equations,\nso we consider\n\\begin{equation}\n  \\label{eq:psi-gen}\n  i\\d_t \\psi +\\frac{1}{2}\\Delta \\psi = V\\psi + \\lambda\n  |\\psi|^{2\\si}\\psi,\\quad \\l \\in \\R.\n\\end{equation}\nMorawetz estimate in the linear case $\\l=0$ will show the absence of\neigenvalues. In the nonlinear case $\\l>0$, these estimates will be a\ncrucial tool for prove scattering in the quantum case. \n The following lemma and its proof are essentially a rewriting of the\n presentation from \\cite{BaRuVe06}.  \n\\begin{proposition}[Morawetz inequality]\\label{prop:Morawetz}\n   Let $d\\ge 3$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)>0$ such\n  that if the attractive part of the potential satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+ \\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\nthen any solution $\\psi\\in L^\\infty(\\R;H^1(\\R^d))$ to \\eqref{eq:psi-gen}\nsatisfies\n\\begin{equation}\\label{eq:morawetz}\n \\l  \\iint_{\\R\\times \\R^d}\\frac{|\\psi(t,x)|^{2\\si+2}}{|x|}dtdx +\n \\iint_{\\R\\times \\R^d}\\frac{|\\psi(t,x)|^{2}}{(1+|x|)^{\\mu+1}}dtdx\\lesssim \n  \\|\\psi\\|_{L^\\infty(\\R;H^1)}^2. \n\\end{equation}\n\\end{proposition}\nIn other words, the main obstruction to global dispersion for $V$\ncomes from $(\\d_r V)_+$, which is the attractive contribution of $V$\nin classical trajectories, while $(\\d_r V)_-$ is the repulsive part,\nwhich does not ruin the dispersion associated to $-\\Delta$ (it may\n reinforce it, see e.g. \\cite{CaDCDS}, but repulsive\npotentials do not necessarily improve the dispersion, see \\cite{GoVeVi06}). \n\\begin{proof}\n  The proof follows standard arguments, based on virial identities\n  with a suitable weight. We resume the main steps of the\n  computations, and give more details on the choice of the weight in\n  our context. For a real-valued function $h(x)$, we compute, for $\\psi$ solution\n  to \\eqref{eq:psi},\n  \\begin{equation*}\n    \\frac{d}{dt}\\int h(x)|\\psi(t,x)|^2dx = \\IM \\int \\bar \\psi(t,x) \\nabla\n    h(x)\\cdot \\nabla \\psi(t,x)dx,\n  \\end{equation*}\n  \\begin{equation}\n    \\label{eq:viriel}\n    \\begin{aligned}\n        \\frac{d}{dt}\\IM \\int \\bar \\psi(t,x) \\nabla\n    h(x)\\cdot \\nabla \\psi(t,x)dx &= \\int \\nabla \\bar \\psi(t,x)\\cdot\n    \\nabla^2h(x)\\nabla \\psi(t,x)dx \\\\\n-\\frac{1}{4}\\int |\\psi(t,x)|^2 &\\Delta^2\n    h(x)dx -\\int |\\psi(t,x)|^2\\nabla V\\cdot \\nabla h(x)dx\\\\\n &   +\\frac{\\l\\si}{\\si+1}\\int |\\psi(t,x)|^{2\\si+2}\\Delta h(x) dx. \n \\end{aligned}\n   \\end{equation}\nIn the case $V=0$, the standard choice is $h(x)=|x|$, for which\n\\begin{equation*}\n  \\nabla h=\\frac{x}{|x|},\\quad \\nabla^2_{jk}h =\n  \\frac{1}{|x|}\\(\\delta_{jk}-\\frac{x_jx_k}{|x|^2}\\),\\quad \\Delta h\\ge\n  \\frac{d-1}{h},\\quad \\text{and }\\Delta^2 h\\le 0\\text{ for }d\\ge 3. \n\\end{equation*}\nThis readily yields Proposition~\\ref{prop:Morawetz} in the repulsive case\n$\\d_r V\\le 0$, since $\\nabla h\\in L^\\infty$. \n\\smallbreak\n\nIn the same spirit as in \\cite{BaRuVe06}, we proceed by perturbation to\nconstruct a suitable weight when the attractive part of the potential\nis not too large. We seek a priori a radial weight, $h=h(|x|)\\ge 0$, so we\nhave \n\\begin{align*}\n & \\Delta h = h'' +\\frac{d-1}{r} h',\\\\\n& \\Delta^2 h =  h^{(4)}\n  +2\\frac{d-1}{r} h^{(3)} +\\frac{(d-1)(d-3)}{r^2}h'' -\n  \\frac{(d-1)(d-3)}{r^3}h',\\\\\n&\\nabla^2_{jk} h = \\frac{1}{r}\\(\\delta_{jk}-\\frac{x_jx_k}{r^2}\\) h'\n+\\frac{x_jx_k}{r^2}h''. \n\\end{align*}\nWe construct a function $h$ such that $h',h''\\ge 0$, so the condition\n$\\nabla^2 h\\ge 0$ will remain. The goal is then to construct a radial\nfunction $h$ such that the second line in \\eqref{eq:viriel} is\nnon-negative, along with $\\Delta h \\ge \\eta\/|x|$ for some $\\eta>0$.\n\\smallbreak\n\n\\noindent {\\bf Case $d=3$.} In this case, the expression for $\\Delta^2\nh$ is simpler, and the above conditions read\n\\begin{align*}\n &\\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} + \\nabla V(x)\\cdot \\nabla h\\le 0,\\\\\n& h''+\\frac{2}{r}h'\\ge \\frac{\\eta}{r},\\quad h',h''\\ge 0. \n\\end{align*}\nSince we do not suppose a priori that $V$ is a radial potential, the\nfirst condition is not rigorous. We actually use the fact that for\n$h'\\ge 0$, Assumption~\\ref{hyp:V} implies\n\\begin{equation*}\n  \\nabla V(x)\\cdot \\nabla h \\le \\(\\d_r V(x)\\)_+ h'(r)\\le\n  \\frac{M}{(1+r)^{\\mu+1}}h'(r). \n\\end{equation*}\nTo achieve our goal, it is therefore sufficient to require:\n\\begin{align}\n \\label{eq:h1}&\\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} +  \\frac{M}{(1+r)^{\\mu+1}}h' \\le 0,\\\\\n\\label{eq:h2}& h''+\\frac{2}{r}h'\\ge \\frac{\\eta}{r},\\quad h'\\in L^\\infty(\\R_+), \\\nh',h''\\ge 0. \n\\end{align}\nIn view of \\eqref{eq:h2}, we seek\n\\begin{equation*}\n  h'(r) = \\eta +\\int_0^r h''(\\rho)d\\rho.\n\\end{equation*}\nTherefore, if $h''\\ge 0$ with $h''\\in L^1(\\R_+)$, \\eqref{eq:h2} will\nbe automatically fulfilled. We now turn to \\eqref{eq:h1}. Since we\nwant $h'\\in L^\\infty$, we may even replace $h'$ by a constant in\n\\eqref{eq:h1}, and solve, for $C>0$, the ODE\n\\begin{equation*}\n  \\frac{1}{4} h^{(4)}\n  +\\frac{1}{r} h^{(3)} +  \\frac{C}{(1+r)^{\\mu+1}}=0.\n\\end{equation*}\nWe readily have\n\\begin{equation*}\n  h^{(3)}(r) = -\\frac{4C}{r^4}\\int_0^r\\frac{\\rho^4}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation*}\nalong with the properties $h^{(3)}(0)=0$, \n\\begin{equation*}\n  h^{(3)}(r)\\Eq r \\infty -\\frac{k}{r^{\\min (\\mu,4)}},\\quad \\text{for some }k>0.\n\\end{equation*}\nIt is now natural to set\n\\begin{equation*}\n  h''(r) = -\\int_r^\\infty h^{(3)}(\\rho)d\\rho,\n\\end{equation*}\nso we have $h''\\in C([0,\\infty);\\R_+)$ and\n\\begin{equation*}\n  h''(r) \\Eq r \\infty \\frac{\\kappa}{r^{\\min (\\mu-1,3)}},\\quad \\text{for some }\\kappa>0.\n\\end{equation*}\nThis function is indeed in $L^1$ if and only if $\\mu>2$. We \ndefine $h$ by  $h(r)= \\int_0^r h'(\\rho)d\\rho$,\n\\begin{equation}\\label{eq:h3}\n   h^{(3)}(r) = -\\frac{K}{r^4}\\int_0^r\\frac{\\rho^4}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation}\nfor some $K>0$, $h''$ and $h'$ being given by the above relations:\n\\eqref{eq:h2} is satisfied for any value of $K>0$, and \\eqref{eq:h1}\nboils down to an inequality of the form\n\\begin{equation}\\label{eq:M}\n  -\\frac{K}{4} +M\\(\\eta +C(\\mu)K\\)\\le 0,\n\\end{equation}\nwhere $C(\\mu)$ is proportional to \n\\begin{equation*}\n \\frac{1}{K} \\|h'\\|_{L^\\infty} = \\int_0^\\infty \\int_r^\\infty\n \\frac{1}{\\rho^4}\\int_0^\\rho \\frac{s^4}{(1+s)^{\\mu+1}}dsd\\rho dr.\n\\end{equation*}\nWe infer that \\eqref{eq:h3} is satisfied for $K\\gg \\eta$, provided\nthat $M<\\frac{1}{4C(\\mu)}$. Note then that by construction, we may also\nrequire\n\\begin{equation*}\n  \\frac{1}{4}\\Delta^2 h +\\nabla V\\cdot \\nabla h\\le \\frac{-c_0}{(1+|x|)^{\\mu+1}},\n\\end{equation*}\nfor $c_0>0$ morally very small. \n\\smallbreak\n\n\\noindent {\\bf Case $d\\ge 4$.} Resume the above reductions, pretending\nthat the last two terms in $\\Delta^2 h$ are not present: \\eqref{eq:h3}\njust becomes\n\\begin{equation*}\n   h^{(3)}(r) = -\\frac{K}{r^{2d-2}}\\int_0^r\\frac{\\rho^{2d-2}}{(1+\\rho)^{\\mu+1}}d\\rho,\n\\end{equation*}\nand we see that with $h''$ and $h'$ defined like before, we have\n\\begin{equation*}\n  rh''-h'= -\\eta- \\int_0^r h'' +rh''.\n\\end{equation*}\nSince this term is negative at $r=0$ and has a non-positive\nderivative, we have $rh''-h'\\le 0$, so finally $\\Delta^2 h\\le 0$. \n\\end{proof}\n\nWe infer that $H$ has no eigenvalue. Indeed, if there were an $L^2$ solution\n$\\psi=\\psi(x)$ \nto $H\\psi =E\\psi$, $E\\in \\R$, then $\\psi\\in\nH^2(\\R^d)$, and $\\psi(x)e^{-iEt}$ would be an $H^1$\nsolution to \\eqref{eq:psi-gen} for $\\l=0$. This is contradiction with\nthe global integrability in time from \\eqref{eq:morawetz}, so\n$\\si_{\\rm pp}(H)=\\emptyset$. \n\n\\subsection{Strichartz estimates}\n\\label{sec:strichartz}\n\nIn\n\\cite[Proposition~3.1]{BaRuVe06}, it is proved that zero is\nnot a resonance of $H$, but with a definition of resonance which is\nnot quite the definition in \\cite{RodnianskiSchlag}, which contains a\nresult that we want to use. So we shall resume the argument.\n\nBy definition (as in \\cite{RodnianskiSchlag}), zero is a resonance of\n$H$, if there is a distributional solution \n$\\psi\\not\\in L^2$, such that  $\\<x\\>^{-s}\\psi\\in L^2(\\R^d)$ for all\n$s>\\frac{1}{2}$, to\n$H\\psi=0$.  \n\\begin{corollary}\n  Under the assumptions of Proposition~\\ref{prop:Morawetz}, zero is not a\n  resonance of $H$.\n\\end{corollary}\n\\begin{proof}\n  Suppose that zero is a resonance of $H$. Then by definition, we\n  obtain a stationary  distributional solution of \\eqref{eq:psi-gen} (case\n  $\\l=0$), $\\psi= \\psi(x)$, and we may assume that it is\n  real-valued. Since $\\Delta \\psi = \n  2V\\psi$, Assumption~\\ref{hyp:V} implies\n  \\begin{equation*}\n    \\<x\\>^{\\mu-s}\\Delta \\psi\\in L^2(\\R^d),\\quad \\forall s>\\frac{1}{2}. \n  \\end{equation*}\nThis implies that $\\nabla \\psi\\in L^2$, by taking for instance $s=1$ in\n\\begin{equation*}\n  \\int|\\nabla \\psi|^2 = -\\int \\<x\\>^{-s}\\psi \\<x\\>^s\\Delta \\psi.\n\\end{equation*}\nBy definition, for all test function $\\varphi$,\n\\begin{equation}\\label{eq:variat}\n  \\frac{1}{2}\\int_{\\R^d}\\nabla \\varphi(x) \\cdot \\nabla \\psi(x)dx\n  +\\int_{\\R^d}V(x)\\varphi(x)\\psi(x)dx =0.\n\\end{equation}\nLet $h$ be the weight constructed in the proof of\nProposition~\\ref{prop:Morawetz}, and consider\n\\begin{equation*}\n  \\varphi = \\psi\\Delta h +2\\nabla \\psi\\cdot \\nabla h. \n\\end{equation*}\nSince $\\nabla h\\in L^\\infty$, $\\nabla^2 h(x)=\\O(\\<x\\>^{-1})$, and\n$\\nabla^3 h(x)= \\O(\\<x\\>^{-2})$, we see\nthat $\\varphi \\in H^1$, and that this choice is allowed in\n\\eqref{eq:variat}. Integration by parts then yields \\eqref{eq:viriel}\n(where the left hand side is now zero):\n\\begin{equation*}\n  0=\\int \\nabla \\psi\\cdot \\nabla^2h \\nabla \\psi -\\frac{1}{4}\\int\n  \\psi^2\\Delta^2 h -\\int \\psi^2 \\nabla V\\cdot \\nabla h.\n\\end{equation*}\nBy construction of $h$, this implies\n\\begin{equation*}\n  \\int_{\\R^d}\\frac{\\psi(x)^2}{(1+|x|)^{\\mu+1}}dx\\le 0,\n\\end{equation*}\nhence $\\psi\\equiv 0$. \n\\end{proof}\nTherefore,\n\\cite[Theorem~1.4]{RodnianskiSchlag} implies non-endpoint global in\ntime Strichartz estimates.  In the case $d=3$, we know from\n\\cite{Go06} that (in view of the above spectral properties)\n\\begin{equation*}\n  \\|e^{-itH}\\|_{L^1\\to L^\\infty}\\le C |t|^{-d\/2},\\quad \\forall t\\not\n  =0,\n\\end{equation*}\na property which is stronger than Strichartz estimates, and yields the\nendpoint Strichartz estimate missing in \\cite{RodnianskiSchlag}, from\n\\cite{KT}. On the other \nhand, this dispersive estimate does not seem to be known under\nAssumption~\\ref{hyp:V} with $\\mu>2$ when $d\\ge 4$: stronger assumptions\nare always present so far (see e.g. \\cite{CaCuVo09,ErGr10}). However,\nendpoint Strichartz estimates for $d\\ge 4$ are a consequence of\n\\cite[Theorem~1.1]{AFVV10}, under the assumptions of\nProposition~\\ref{prop:Morawetz}. \n\n\\begin{proposition}\\label{prop:StrichartzRS}\nLet $d\\ge 3$. Under the assumptions of\nProposition~\\ref{prop:Morawetz}, for all $(q,r)$ such that \n  \\begin{equation}\\label{eq:adm}\n    \\frac{2}{q}=d\\(\\frac{1}{2}-\\frac{1}{r}\\),\\quad 2<q\\le \\infty,\n  \\end{equation}\nthere exists $C=C(q,d)$ such that \n\\begin{equation*}\n  \\|e^{-itH} f\\|_{L^q(\\R;L^r(\\R^d))}\\le C \\|f\\|_{L^2(\\R^d)},\\quad\n  \\forall f\\in L^2(\\R^d). \n\\end{equation*}\n\\end{proposition}\nIt is classical that this homogeneous Strichartz estimate, a duality\nargument and Christ-Kiselev's Theorem imply the inhomogeneous\ncounterpart. For two admissible pairs $(q_1,r_1)$ and $(q_2,r_2)$\n(that is, satisfying \\eqref{eq:adm}), there exists $C_{q_1,q_2}$\nindependent of the time interval $I$ such\nthat if we denote by\n\\begin{equation*}\n  R(F)(t,x) = \\int_{I\\cap \\{s\\le t\\}} e^{-i(t-s)H}F(s,x)ds,\n\\end{equation*}\nwe have\n\\begin{equation*}\n  \\|R(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^d))}\\le\n  C_{q_1,q_2}\\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^d))},\\quad \\forall F\\in\n  L^{q_2'}(I;L^{r_2'}(\\R^d)). \n\\end{equation*}\n\\smallbreak\n\nNote that the assumption $\\mu>2$ seems essentially sharp in order to have\nglobal in time Strichartz estimates. The result remains true for $\\mu\n=2$ (\\cite{BPST03,BPST04}), but in \\cite{GoVeVi06}, the authors\nprove that for repulsive potentials which are homogeneous of degree\nsmaller than $2$, global Strichartz estimates fail to exist.\n\n\n\n\n\n\\section{Quantum scattering}\n\\label{sec:quant}\n\nIn this section, we prove Theorem~\\ref{theo:scatt-quant}. Since the\ndependence upon $\\eps$ is not measured in\nTheorem~\\ref{theo:scatt-quant}, we shall \nconsider the case $\\eps=1$, corresponding to \n\\begin{equation}\n  \\label{eq:psi}\n  i\\d_t \\psi +\\frac{1}{2}\\Delta \\psi = V\\psi + |\\psi|^{2\\si}\\psi.\n\\end{equation}\nWe split the proof of Theorem~\\ref{theo:scatt-quant} into two\nsteps. First, we solve the Cauchy problem with data prescribed at\n$t=-\\infty$, that is, we show the existence of wave operators. Then,\ngiven an initial datum at $t=0$, we show that the (global) solution to\n\\eqref{eq:psi} behaves asymptotically like a free solution, which\ncorresponds to asymptotic completeness. \n\\smallbreak\n\nFor each of these two steps, we first show that the nonlinearity is\nnegligible for large time, and then recall that the potential is\nnegligible for large time (linear scattering). This means that for any $\\tilde \\psi_-\\in\nH^1(\\R^d)$, there exists  a unique $\\psi\\in \n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi}  such that\n  \\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0,\n  \\end{equation*}\nand for any $\\varphi\\in H^1(\\R^d)$, there exist a unique  $\\psi\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} and a unique $\\tilde\\psi_+\\in\n  H^1(\\R^d)$ such that\n\\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_+\\|_{H^1(\\R^d)}\\Tend t {+\n      \\infty} 0.\n  \\end{equation*}\nThen, we recall that the potential $V$ is negligible for large\ntime. We will adopt the following notations for the propagators,\n\\begin{equation*}\n  U(t)=e^{i\\frac{t}{2}\\Delta},\\quad U_V(t)= e^{-itH}. \n\\end{equation*}\n\n\n\nIn order to construct wave operators which show that the nonlinearity\ncan be neglected for large time, we shall work with an $H^1$\nregularity, on the Duhamel's formula associated to \\eqref{eq:psi} in\nterms of $U_V$, with a prescribed asymptotic behavior as $t\\to\n-\\infty$:\n\\begin{equation}\n  \\label{eq:duhamel-}\n  \\psi(t) = U_V(t)\\tilde \\psi_- -i\\int_{-\\infty}^t\n  U_V(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds. \n\\end{equation}\nApplying the gradient to this formulation brings up the problem of\nnon-commutativity with $U_V$. The worst term is actually the linear\none, $U_V(t)\\tilde \\psi_-$, since\n\\begin{equation*}\n  \\nabla \\(U_V(t)\\tilde \\psi_-\\) = U_V(t)\\nabla \\tilde \\psi_-\n  -i\\int_0^t U_V(t-s)\\((U_V(s)\\tilde \\psi_-)\\nabla V\\)ds.\n\\end{equation*}\nSince the construction of wave operators relies on the use of\nStrichartz estimates, it would be necessary to have an estimate of\n\\begin{equation*}\n \\left\\|\\nabla \\(U_V(t)\\tilde \\psi_-\\)\\right\\|_{L^qL^r}\n\\end{equation*}\nin terms of $\\psi_-$, for admissible pairs\n$(q,r)$. Proposition~\\ref{prop:StrichartzRS} yields\n\\begin{equation*}\n  \\left\\|\\nabla \\(U_V(t)\\tilde \\psi_-\\)\\right\\|_{L^qL^r} \\lesssim \\|\\nabla \\tilde\n  \\psi_-\\|_{L^2} + \\|(U_V(t)\\tilde \\psi_-)\\nabla V\\|_{L^{\\tilde\n      q'}L^{\\tilde r'}},\n\\end{equation*}\nfor any admissible pair $(\\tilde q,\\tilde r)$. In the last factor,\ntime is present only in the term $U_V(t)\\tilde \\psi_-$, so to be able\nto use Strichartz estimates again, we need to consider $\\tilde\nq=2$, in which case $\\tilde r=2^*:=\\frac{2d}{d-2}$:\n\\begin{equation*}\n  \\|(U_V(t)\\tilde \\psi_-)\\nabla V\\|_{L^2L^{{2^*}'}}\\le \\|U_V(t)\\tilde\n  \\psi_-\\|_{L^2L^{2^*}}\\|\\nabla V\\|_{L^{d\/2}},\n\\end{equation*}\nwhere Assumption~\\ref{hyp:V} implies $\\nabla V\\in L^{d\/2}(\\R^d)$ as\nsoon as $\\mu>1$. Using the endpoint Strichartz estimate from\nProposition~\\ref{prop:StrichartzRS}, we have\n\\begin{equation*}\n  \\|U_V(t)\\tilde\n  \\psi_-\\|_{L^2L^{2^*}} \\lesssim \\|\\tilde \\psi_-\\|_{L^2}, \n\\end{equation*}\nand we have:\n\\begin{lemma}\\label{lem:stri2}\n  Let $d\\ge 3$. Under the assumptions of\n  Proposition~\\ref{prop:Morawetz}, for all admissible pair $(q,r)$, \n  \\begin{equation*}\n    \\|e^{-itH}f\\|_{L^q(\\R;W^{1,r}(\\R^d))}\\lesssim \\|f\\|_{H^1(\\R^d)}. \n  \\end{equation*}\n\\end{lemma}\nWe shall rather use a vector-field, for we believe this approach may be\ninteresting in other contexts.\n\n\n\\subsection{Vector-field}\n\\label{sec:vector-field}\n\n We\nintroduce a vector-field which naturally commutes with $U_V$, and\nis comparable with the gradient. \n\\smallbreak\n\nFrom Assumption~\\ref{hyp:V}, $V$ is bounded, so there exists $c_0\\ge\n0$ such that $V+c_0\\ge 0$. We shall consider the operator\n\\begin{equation*}\n  A = \\sqrt{H+c_0}=\\sqrt{-\\frac{1}{2}\\Delta +V+c_0}.\n\\end{equation*}\n\\begin{lemma}\\label{lem:A}\n Let $d\\ge 3$, and $V$ satisfying Assumption~\\ref{hyp:V} with\n $V+c_0\\ge 0$. For every $1<r<\\infty$, there exists $C_r,K_r$  such\n that for all $f\\in    W^{1,r}(\\R^d)$,\n  \\begin{equation}\n    \\label{eq:A}\n    \\|A f\\|_{L^r}\\le C_r\n    \\(\\|f\\|_{L^r}+\\|\\nabla f\\|_{L^r}\\)\\le K_r \\(\\|f\\|_{L^r}+\\|A f\\|_{L^r}\\) .\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\n  The first inequality is very close to \\cite[Theorem~1.2]{AFVV10}, and the\n  proof can readily be adapted. On the other hand, the second\n  inequality would require the restriction $4\/3<r<4$ if we followed \n  the same approach, based on  Stein's interpolation theorem (a\n  similar approach for followed in e.g. \\cite{KiViZh09}). We\n  actually take  advantage of the smoothness of the potential $V$ to\n  rather apply Calder\\'on--Zygmund result on the action of\n  pseudo-differential operators. \n\\smallbreak\n\nWe readily check that the two functions\n\\begin{equation*}\n  a(x,\\xi)=\n  \\sqrt{\\frac{\\frac{|\\xi|^2}{2}+V(x)+c_0}{1+|\\xi|^2}},\\quad\n  b(x,\\xi) = \\sqrt{\\frac{|\\xi|^2}{\\frac{|\\xi|^2}{2}+V(x)+c_0+1}} ,\n\\end{equation*}\nare symbols of order zero, in the sense\nthat they satisfy\n\\begin{equation*}\n  |\\d_x^\\alpha\\d_\\xi^\\beta a(x,\\xi)| +  |\\d_x^\\alpha\\d_\\xi^\\beta\n  b(x,\\xi)| \\le C_{\\alpha,\\beta}\\<\\xi\\>^{-|\\beta|},\n\\end{equation*}\nfor all $\\alpha,\\beta\\in \\N^d$. This implies that the\npseudo-differential operators of symbol \n$a$ and $b$, respectively, are bounded on $L^r(\\R^d)$, for\nall $1<r<\\infty$; see e.g. \\cite[Theorem~5.2]{Taylor3}. In the case of\n$a$, this yields the first inequality in \\eqref{eq:A}, and in the case of $b$, this\nyields the second inequality. \n\\end{proof}\n\n\\subsection{Wave operators}\n\\label{sec:wave-op}\n\nWith the tools presented in the previous section, we can prove the\nfollowing result by adapting the standard proof of the case $V=0$, as\nestablished in \\cite{GV85}.\n\\begin{proposition}\\label{prop:waveop-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}\\le \\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. For\n  all $\\tilde \\psi_-\\in H^1(\\R^d)$, there exists a unique \n$$\\psi\\in\n  C((-\\infty,0];H^1(\\R^d))\\cap\n  L^{\\frac{4\\si+4}{d\\si}}((-\\infty,0);L^{2\\si+2}(\\R^d))$$\n solution to    \\eqref{eq:psi}  such that \n  \\begin{equation*} \n    \\|\\psi(t)-e^{-it H}\\tilde\\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0.\n  \\end{equation*} \n\\end{proposition}\n\\begin{proof}\n  The main part of the proof is to prove that \\eqref{eq:duhamel-} has\n  a fixed point. Let\n  \\begin{equation*}\n   q=\\frac{4\\si+4}{d\\si}.\n  \\end{equation*}\nThe pair $(q,2\\si+2)$ is admissible, in the sense that it satisfies\n\\eqref{eq:adm}.  \nWith the notation $L^\\beta_TY=L^\\beta(]-\\infty,-T];Y)$, we\n  introduce:  \n  \\begin{align*}\n    X_T:=\\Big\\{ \\psi\\in C(]-\\infty,-T];H^1)\\ ;\\ &\\left\\|\n    \\psi\\right\\|_{L^q_TL^{2\\si+2}} \\le  K\\|\\tilde\n                                                  \\psi_-\\|_{L^2},\\\\\n\\left\\|\n    \\nabla \\psi\\right\\|_{L^q_TL^{,2\\si+2}} \\le  K\\|\\tilde\n                                                  \\psi_-\\|_{H^1},\\quad &\n \\left\\|  \\psi\\right\\|_{L^\\infty_TL^2} \\le 2 \\|\\tilde \\psi_-\\|_{L^2}\\,\n    ,\\\\\n \\left\\| \\nabla  \\psi\\right\\|_{L^\\infty_TL^2} \\le K \\|\\tilde \\psi_-\\|_{H^1}\n,\\quad \n&\\left\\|  \\psi\\right\\|_{L^q_T L^{2\\si+2}} \\le 2 \\left\\|\n    U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_T L^{2\\si+2}}\\Big\\},\n  \\end{align*}\nwhere $K$ will be chosen sufficiently large in terms of the\nconstants present in Strichartz estimates presented in\nProposition~\\ref{prop:StrichartzRS}. Set\n$r=s=2\\si +2$: we have\n\\begin{equation*}\n  \\frac{1}{r'}= \\frac{1}{r}+\\frac{2\\si}{s},\\quad\n\\frac{1}{q'}= \\frac{1}{q}+\\frac{2\\si}{k},\n\\end{equation*}\nwhere $q\\le k<\\infty$ since $2\/d\\le\n\\si<2\/(d-2)$. Denote by $\\Phi(\\psi)$ the right hand side of\n\\eqref{eq:duhamel-}. For $\\psi\\in X_T$, Strichartz estimates and H\\\"older\ninequality yield, for all admissible pairs $(q_1,r_1)$:\n\\begin{align*}\n  \\left\\| \\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} &\\le C_{q_1}\\|\\tilde    \n\\psi_-\\|_{L^2} +  C\\left\\| |\\psi|^{2\\si}\\psi\\right\\|_{L^{q'}_TL^{r'}}  \n  \\\\\n&\\le C_{q_1}\\|\\tilde \\psi_-\\|_{L^2}  +\nC\\|\\psi\\|_{L^k_TL^s}^{2\\si}\\|\\psi\\|_{L^q_T L^r}\\\\\n&\\le C_{q_1}\\|\\tilde \\psi_-\\|_{L^2} + C\\|\\psi\\|_{L^q_TL^r}^{2\\si\\theta\n  }\\|\\psi\\|_{L^\\infty_TL^r}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r} ,\n\\end{align*}\nfor some $0<\\theta\\le 1$, where we have used the property\n$r=s=2\\si+2$. Sobolev embedding and the definition of $X_T$ then imply:\n\\begin{align*}\n \\left\\| \\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} \\le C_{q_1}\\|\\tilde\n  \\psi_-\\|_{L^2} + C\\left\\|U_V(\\cdot) \\tilde \n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r} .\n\\end{align*}\nWe now apply the operator $A$. Since $A$ commutes with $H$, we have\n\\begin{equation*}\n  \\left\\| A\\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} \\lesssim\n  \\|A\\tilde \\psi_-\\|_{L^2} + \\left\\|\n    A\\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}}. \n\\end{equation*}\nIn view of Lemma~\\ref{lem:A}, we have successively,\n\\begin{align*}\n  \\|A\\tilde \\psi_-\\|_{L^2} &\\lesssim \\|\\tilde \\psi_-\\|_{H^1},\\\\\n \\left\\|\n    A\\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}}&\\lesssim   \\left\\|\n    |\\psi|^{2\\si}\\psi\\right\\|_{L^{q'}_TL^{r'}} + \\left\\|\n  \\nabla   \\(|\\psi|^{2\\si}\\psi\\)\\right\\|_{L^{q'}_TL^{r'}} \\\\\n&\\lesssim \\|\\psi\\|_{L^k_TL^s}^{2\\si}\\(\\|\\psi\\|_{L^q_T L^r} +\n  \\|\\nabla\\psi\\|_{L^q_T L^r} \\)\\\\\n&\\lesssim \\|\\psi\\|_{L^k_TL^s}^{2\\si}\\(\\|\\psi\\|_{L^q_T L^r} +\n  \\|A\\psi\\|_{L^q_T L^r} \\).\n\\end{align*}\nWe infer along the same lines as above,\n\\begin{align*}\n \\left\\| \\nabla\\Phi(\\psi)\\right\\|_{L^{q_1}_T L^{r_1}} &\\lesssim\n  \\|\\tilde \\psi_-\\|_{H^1} +\\left\\|U_V(\\cdot) \\tilde\n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\( \n\\| \\psi\\|_{L^q_T L^r} + \\|A\\psi\\|_{L^q_T L^r}\\) .\n \\end{align*}\nWe have also\n\\begin{align*}\n\\left\\| \\Phi(\\psi)\\right\\|_{L^{q}_T L^{r}}& \\le\n  \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^{q}_TL^{r}} \n+ C\\left\\|U_V(\\cdot) \\tilde\n  \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta \n  }\\|\\psi\\|_{L^\\infty_TH^1}^{2\\si(1-\\theta) } \\|\\psi\\|_{L^q_T L^r}.\n\\end{align*}\nFrom Strichartz estimates, $U_V(\\cdot)\\tilde \\psi_- \\in L^{q}(\\R;L^{r})$, so \n\\begin{equation*}\n  \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_TL^r}  \\to 0\\quad \\text{as }T\\to +\\infty.\n\\end{equation*}\nSince $\\theta>0$, we infer that $\\Phi$ sends $X_T$ to itself, for\n$T$ sufficiently large. \n\\smallbreak\n\nWe have also, for $\\psi_2,\\psi_1\\in X_T$:\n\\begin{align*}\n  \\left\\| \\Phi(\\psi_2)-\\Phi(\\psi_1)\\right\\|_{L^q_T L^r}&\\lesssim\n  \\max_{j=1,2}\\| \\psi_j\\|_{L^k_TL^s}^{2\\si} \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r}\\\\\n&\\lesssim \\left\\|U_V(\\cdot)\\tilde \\psi_-\\right\\|_{L^q_TL^r}^{2\\si\\theta\n  }\\|\\tilde \\psi_-\\|_{H^1}^{2\\si(1-\\theta) }\\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r}.\n\\end{align*}\nUp to choosing $T$ larger, $\\Phi$ is a contraction on $X_T$, equipped\nwith the distance\n\\begin{equation*}\n  d(\\psi_2,\\psi_1) = \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^q_T L^r} + \\left\\|\n  \\psi_2-\\psi_1\\right\\|_{L^\\infty_T L^2},\n\\end{equation*}\nwhich makes it a Banach space (see \\cite{CazCourant}). \nTherefore, $\\Phi$\nhas a unique fixed point in $X_T$, solution to\n\\eqref{eq:duhamel-}. It follows from \\eqref{eq:A} that this solution\nhas indeed an $H^1$ regularity with \n\\begin{equation*}\n   \\|\\psi(t)-e^{-it H}\\tilde\\psi_-\\|_{H^1(\\R^d)}\\Tend t {-\n      \\infty} 0.\n\\end{equation*}\n In view\nof the global well-posedness results for the Cauchy problem associated\nto \\eqref{eq:psi} (see e.g. \\cite{CazCourant}),\nthe proposition follows.\n\\end{proof}\n\\subsection{Asymptotic completeness}\n\\label{sec:AC-quant}\n\nThere are mainly three approaches to prove asymptotic completeness for\nnonlinear Schr\\\"odinger equations (without potential). The initial\napproach (\\cite{GV79Scatt}) consists in working with a\n$\\Sigma$ regularity. This  \nmakes it possible to use the operator $x+it\\nabla$, which enjoys\nseveral nice properties, and to which an important evolution law (the\npseudo-conformal conservation law) is associated; see\nSection~\\ref{sec:class} for more details. This law provides\nimportant a priori estimates, from which asymptotic completeness\nfollows very easily the the case $\\si\\ge 2\/d$, and less easily for\nsome range of $\\si$ below $2\/d$; see e.g. \\cite{CazCourant}. \n\\smallbreak\n\nThe second historical approach relaxes the localization assumption,\nand allows \nto work in $H^1(\\R^d)$, provided that $\\si>2\/d$. It is based on\nMorawetz inequalities: asymptotic completeness is then established in\n\\cite{LiSt78,GV85} for the case $d\\ge 3$, and in \\cite{NakanishiJFA} for the low\ndimension cases $d=1,2$, by introducing more intricate Morawetz\nestimates. Note that the case $d\\le 2$ is already left out in our case, since we\nhave assumed $d\\ge 3$ to prove Proposition~\\ref{prop:waveop-quant}.  \n\\smallbreak\n\nThe most recent approach to prove asymptotic completeness in $H^1$\nrelies on the introduction of interaction Morawetz estimates in \\cite{CKSTTCPAM},\nan approach which has been revisited since, in particular in\n\\cite{PlVe09} and \\cite{GiVe10}. See also \\cite{Vi09} for a very nice\nalternative approach of the use of interaction Morawetz estimates.  In\nthe presence of an external potential, this approach was used in\n\\cite{CaDa-p}, by working with Morrey-Campanato type norms. \n\\smallbreak\n\nAn analogue for the pseudo-conformal evolution law is available (see\ne.g. \\cite {CazCourant}), but it seems that in the presence of $V$\nsatisfying Assumption~\\ref{hyp:V}, it cannot be exploited to get\nsatisfactory estimates. We shall rather consider\nMorawetz estimates as in \\cite{GV85}, and thus give an alternative\nproof of the corresponding result from \\cite{CaDa-p}: note that for $\\l=1$,\nthe first part of \\eqref{eq:morawetz} provides exactly the same a\npriori estimate as in \\cite{GV85}. \n\\begin{proposition}\\label{prop:AC-quant}\n  Let $d\\ge 3$, $\\frac{2}{d}<\\si<\\frac{2}{d-2}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>2$. There exists $M=M(\\mu,d)$ such\n  that if the attractive part of the potential satisfies\n  \\begin{equation*}\n    (\\d_r V(x))_+\\le \\frac{M}{(1+|x|)^{\\mu+1}},\\quad \\forall x\\in\n    \\R^d,\n  \\end{equation*}\nthen  for\n  all $\\varphi\\in H^1(\\R^d)$, there exist a unique $\\psi\\in\n  C(\\R;H^1(\\R^d))$ solution to \\eqref{eq:psi} with $\\psi_{\\mid\n    t=0}=\\varphi$, and a unique $\\tilde\\psi_+\\in\n  H^1(\\R^d)$ such that\n  \\begin{equation*}\n    \\|\\psi(t)-e^{-itH}\\tilde \\psi_+\\|_{H^1(\\R^d)}\\Tend t {+\n      \\infty} 0.\n  \\end{equation*}\nIn addition, $\\psi,\\nabla \\psi\\in L^q(\\R_+,L^r(\\R^d))$ for all\nadmissible pairs $(q,r)$. \n\\end{proposition}\n\\begin{proof}\n  The proof follows that argument presented in \\cite{GV85} (and\n  resumed in \\cite{GinibreDEA}), so we shall only described the main\n  steps and the modifications needed in the present context.  The key\n  property in the proof consists in showing that there exists\n  $2<r<\\frac{2d}{d-2}$ such that \n  \\begin{equation}\\label{eq:Lp0}\n    \\|\\psi(t)\\|_{L^r}\\Tend t {+\\infty} 0.\n  \\end{equation}\nSince $\\psi\\in L^\\infty(\\R;H^1)$ (see e.g. \\cite{CazCourant}), we\ninfer that the above property is true for \\emph{all}\n$2<r<\\frac{2d}{d-2}$. This aspect is the only one that requires some\nadaptation in our case. Indeed, once this property is at hand, the end\nof the proof relies on Strichartz estimates applied to Duhamel's\nformula. In our framework, since we first want to get rid of the\nnonlinearity only (and not the potential $V$ yet), we consider\n\\begin{equation*}\n  \\psi(t) = U_V(t)\\varphi-i\\int_0^t U_V(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds,\n\\end{equation*}\nand thanks to Proposition~\\ref{prop:StrichartzRS}, it is possible to\nfollow exactly the same lines as in \\cite{GV85} (see also \\cite{TzVi-p})\nin order to infer Proposition~\\ref{prop:AC-quant}. \n\\smallbreak\n\nTherefore, the only delicate point is to show that \\eqref{eq:Lp0}\nholds for some $2<r<\\frac{2d}{d-2}$. This corresponds to Corollary~5.1\nin \\cite{GV85} (Lemme~12.6 in \\cite{GinibreDEA}). The main technical\nremark is that once Morawetz estimate is available (the one given in\nProposition~\\ref{prop:Morawetz}, whose final conclusion does not depend on\n$V$), one uses dispersive properties of the group $U(t)$. As mentioned\nabove, we do not want to use dispersive properties of $U_V(t)$, since\nthey are known only in the case $d=3$ (on the other hand, this means\nthat the result is straightforward in the case $d=3$, from \\cite{GV85}\nand \\cite{Go06}). So instead, we consider\nDuhamel's formula for \\eqref{eq:psi} in terms of $U(t)$, which reads\n\\begin{equation}\n  \\label{eq:DuhamelU}\n  \\psi(t)= U(t)\\varphi-i\\int_0^t\n  U(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds-i\\int_0^t\n  U(t-s)\\(V\\psi(s)\\)ds. \n\\end{equation}\nThe new term compared to \\cite{GV85} is of course the last term in\n\\eqref{eq:DuhamelU}, and so the nonlinearity is now\n\\begin{equation*}\n  f(\\psi) = |\\psi|^{2\\si}\\psi +V\\psi. \n\\end{equation*}\nFollowing the argument from \\cite{GV85} (or \\cite{GinibreDEA}), it\nsuffices to prove the following two properties: \\\\\n\n\\noindent 1. There exist\n$r_1>2^*=\\frac{2d}{d-2}$ and $\\alpha>0$ \nsuch that \n\\begin{equation}\n  \\left\\|\\int_{t_0}^{t-\\ell}\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{r_1}(\\R^d)}\\le C\n\\ell^{-\\alpha}\\|\\psi\\|_{L^\\infty(\\R;H^1)}, \n\\end{equation}\nConsider a Lebesgue index $r_1$\nslightly larger than \n$2^*$, \n\\begin{equation*}\n  \\frac{1}{r_1} = \\frac{1}{2^*} -\\eta,\\quad 0<\\eta\\ll 1. \n\\end{equation*}\nLet $\\ell>0$, and consider\n\\begin{equation*}\n  I_1(t) = \\left\\|\\int_{t_0}^{t-\\ell}\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{r_1}(\\R^d)}.\n\\end{equation*}\nStandard dispersive estimates for $U$ yield\n\\begin{equation*}\n  I_1(t) \\lesssim \\int_{t_0}^{t-\\ell} (t-s)^{-\\delta_1} \\|V\\psi(s)\\|_{L^{r'_1}}ds,\n\\end{equation*}\nwhere $\\delta_1$ is given by\n\\begin{equation*}\n  \\delta_1 = d\\(\\frac{1}{2}-\\frac{1}{r_1}\\) = 1+\\eta d.\n\\end{equation*}\nNow we apply H\\\"older inequality in space, in view of the identity\n\\begin{equation*}\n  \\frac{1}{r'_1} = \\frac{1}{2}+\\frac{1}{d}-\\eta =\n  \\underbrace{\\frac{1}{2}-\\frac{1}{d} +\\eta}_{1\/k} +\n  \\underbrace{\\frac{2}{d} -2\\eta}_{1\/q}. \n\\end{equation*}\nFor $\\eta>0$ sufficiently small, $V\\in L^q(\\R^d)$ since $\\mu>2$, and so\n\\begin{equation*}\n  \\|V\\psi(s)\\|_{L^{r'_1}} \\le \\|V\\|_{L^{q}}\\|\\psi(s)\\|_{L^k}\\lesssim \\|\\psi\\|_{L^\\infty(\\R;H^1)},\n\\end{equation*}\nwhere we have used Sobolev embedding, since $2<k<2^*$. We infer\n\\begin{align*}\n  I_1(t) &\\lesssim  \\int_{t_0}^{t-\\ell} (t-s)^{-\\delta_1} ds\n  \\|\\psi\\|_{L^\\infty(\\R;H^1)} \\lesssim  \\int_{\\ell}^{\\infty}\n  s^{-\\delta_1} ds\\|\\psi\\|_{L^\\infty(\\R;H^1)}\\\\\n& \\lesssim\n  \\ell^{1-\\delta_1}\\|\\psi\\|_{L^\\infty(\\R;H^1)}=\\ell^{-\\eta d}\\|\\psi\\|_{L^\\infty(\\R;H^1)}. \n\\end{align*}\n\n\\noindent 2. Now for fixed $\\ell>0$, let \n\\begin{equation*}\n  I_2(t) = \\left\\|\\int_{t-\\ell}^t\n  U(t-s)\\(V\\psi(s)\\)ds\\right\\|_{L^{2\\si+2}(\\R^d)}.\n\\end{equation*}\nWe show that for any  $\\ell>0$, $I_2(t)\\to 0$ as $t\\to \\infty$. \nDispersive estimates for $U(t)$ yield\n\\begin{equation*}\n  I_2(t) \\lesssim \\int_{t-\\ell}^t\n  (t-s)^{-\\delta}\\|V\\psi(s)\\|_{L^{\\frac{2\\si+2}{2\\si+1}}}ds,\\quad\n  \\delta = d\\(\\frac{1}{2}-\\frac{1}{2\\si+2}\\) = \\frac{d\\si}{2\\si+2}<1. \n\\end{equation*}\nFor (a small) $\\alpha$ to be fixed later, H\\\"older inequality yields\n\\begin{equation*}\n  \\|V\\psi(s)\\|_{L^{\\frac{2\\si+2}{2\\si+1}}} =\\left\\| |x|^{\\alpha}V\n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{\\frac{2\\si+2}{2\\si+1}}}\n  \\le \\left\\| |x|^{\\alpha}V \\right\\|_{L^{\\frac{\\si+1}{\\si}}}\n\\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}}.\n\\end{equation*}\nNote that for $0<\\alpha\\ll 1$, $\\left\\| |x|^{\\alpha}V\n\\right\\|_{L^{\\frac{\\si+1}{\\si}}}$ is finite, since\n$\\frac{\\si+1}{\\si}>\\frac{d}{2}$ and $\\mu>2$. For $0<\\theta<1$, write \n\\begin{align*}\n  \\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}}=  \\left\\| \n    \\frac{|\\psi(s)|^{\\theta}}{|x|^\\alpha}\n      |\\psi(s)|^{1-\\theta}\\right\\|_{L^{2\\si+2}}&\\le \\left\\| \n    \\frac{\\psi(s)}{|x|^{\\alpha\/\\theta}}\\right\\|_{L^{2\\si+2}}^\\theta\n      \\left\\|\\psi(s)\\right\\|_{L^{2\\si+2}}^{1-\\theta}\\\\\n   &   \\lesssim  \\left\\| \n    \\frac{\\psi(s)}{|x|^{\\alpha\/\\theta}}\\right\\|_{L^{2\\si+2}}^\\theta\n      \\left\\|\\psi\\right\\|_{L^\\infty(\\R;H^1)}^{1-\\theta}.\n\\end{align*}\nTo use Morawetz estimate, we impose $\\alpha\/\\theta= 1\/(2\\si+2)$, so\nthat we have\n\\begin{equation*}\n   \\left\\| \n    \\frac{\\psi(s)}{|x|^\\alpha} \\right\\|_{L^{2\\si+2}} \\lesssim\n\\(  \\int_{\\R^d}\\frac{|\\psi(s,x)|^{2\\si+2}}{|x|}dx\\)^{\\theta\/(2\\si+2)}\n\\left\\|\\psi\\right\\|_{L^\\infty(\\R;H^1)}^{1-\\theta}. \n\\end{equation*}\nWe conclude by applying H\\\"older inequality in time: since $\\delta<1$,\nthe map $s\\mapsto (t-s)^{-\\delta}$ belongs to $L^q_{\\rm loc}$ for $1\\le\n  q\\le 1+\\gamma$ and $\\gamma>0$ sufficiently small. Let $q=1+\\gamma$\n  with $0<\\gamma\\ll 1$ so that $s\\mapsto (t-s)^{-\\delta}\\in L^q_{\\rm\n    loc}$: we have $q'<\\infty$, and we can choose  $0<\\theta\\ll 1$ (or\n  equivalently $0<\\eta\\ll 1$) so\n  that \n  \\begin{equation*}\n    \\theta q'=2\\si+2. \n  \\end{equation*}\nWe end up with \n\\begin{equation*}\n I_2(t) \\lesssim \\ell^\\beta \\(\\iint_{[t-\\ell,t]\\times\\R^d}\n   \\frac{|\\psi(s,x)|^{2\\si+2}}{|x|}dsdx\\)^{1\/(2\\si+2)q'},\n\\end{equation*}\nfor some $\\beta>0$. The last factor goes to zero as $t\\to \\infty$ from\nProposition~\\ref{prop:Morawetz}. \n\\end{proof}\n\n\n\\subsection{Scattering}\n\\label{sec:conclusion}\n\nUnder Assumption~\\ref{hyp:V}, a linear scattering theory is available,\nprovided that $\\mu>1$; see e.g. \\cite[Section~4.6]{DG}. This means that\nthe following strong limits exist in $L^2(\\R^d)$,\n\\begin{equation*}\n  \\lim_{t\\to -\\infty} U_V(-t)U(t),\\quad\\text{and}\\quad  \\lim_{t\\to +\\infty} U(-t)U_V(t),\n\\end{equation*}\nwhere the second limit usually requires to project on the continuous\nspectrum. Recall that this projection is the identity in our\nframework. \n\\begin{lemma}\\label{lem:Cook-quant}\n  Let $d\\ge 3$, $V$ satisfying Assumption~\\ref{hyp:V} with $p>1$. Then \nthe strong limit\n \\begin{equation*}\n  \\lim_{t\\to -\\infty} U_V(-t)U(t)\n\\end{equation*}\nexists in $H^1(\\R^d)$. \n\\end{lemma}\n\\begin{proof}\n  Following Cook's method (\\cite[Theorem~XI.4]{ReedSimon3}), it\n  suffices to prove that for all $\\varphi \\in \\Sch(\\R^d)$,\n  \\begin{equation*}\n    t\\mapsto \\left\\| U_V(-t) VU(t)\\varphi\\right\\|_{H^1}\\in\n    L^1((-\\infty,-1]). \n  \\end{equation*}\nFor the $L^2$ norm, we have\n\\begin{equation*}\n  \\left\\| U_V(-t) VU(t)\\varphi\\right\\|_{L^2} = \\left\\|\n    VU(t)\\varphi\\right\\|_{L^2} .\n\\end{equation*}\nAssumption~\\ref{hyp:V} implies that $V\\in L^q(\\R^d)$ for all\n$q>d\/\\mu$. For $\\mu>1$, let $q$ be given by \n\\begin{equation*}\n  \\frac{1}{q} = \\frac{1}{d}+\\eta,\\text{ with } \\eta>0\\text{ and }q>\\frac{d}{\\mu}. \n\\end{equation*}\nWe apply H\\\"older inequality with the identity\n\\begin{equation*}\n  \\frac{1}{2} = \\frac{1}{q} +\\underbrace{\\frac{1}{2}-\\frac{1}{d}-\\eta}_{1\/r}.\n\\end{equation*}\nUsing dispersive estimates for $U(t)$, we have\n\\begin{equation*}\n  \\left\\|\n    VU(t)\\varphi\\right\\|_{L^2} \\lesssim \\|U(t)\\varphi\\|_{L^r}\\lesssim\n  |t|^{-d\\(\\frac{1}{2}-\\frac{1}{r}\\)}\\|\\varphi\\|_{L^{r'}}= |t|^{-1-d\\eta}\\|\\varphi\\|_{L^{r'}},\n\\end{equation*}\nhence the existence of the strong limit in $L^2$. \n\\smallbreak\n\nFor the $H^1$ limit, recall that from Lemma~\\ref{lem:A}, \n\\begin{equation*}\n  \\left\\| \\nabla U_V(-t) VU(t)\\varphi\\right\\|_{L^2}\\lesssim \\left\\| A\n    U_V(-t) VU(t)\\varphi\\right\\|_{L^2} \n\\end{equation*}\nSince $A$ commutes with $U_V$ which is unitary on $L^2$, the right\nhand side is equal to \n\\begin{equation*}\n\\left\\| A\n    VU(t)\\varphi\\right\\|_{L^2}\\lesssim \\|VU(t)\\varphi\\|_{H^1},\n\\end{equation*}\nwhere we have used Lemma~\\ref{lem:A} again. Now\n\\begin{equation*}\n  \\|VU(t)\\varphi\\|_{H^1} \\le \\|VU(t)\\varphi\\|_{L^2}+ \\|\\nabla V\\times\n  U(t)\\varphi\\|_{L^2} + \\|VU(t)\\nabla \\varphi\\|_{L^2},\n\\end{equation*}\nand each term is integrable, like for the $L^2$ limit, from\nAssumption~\\ref{hyp:V}. \n\\end{proof}\n\n  In the case $d=3$, the dispersive estimates established by Goldberg\n  \\cite{Go06} make it possible to prove asymptotic completeness in\n  $H^1$ by Cook's method as well: for all $\\varphi\\in\n  \\Sch(\\R^d)$,\n  \\begin{equation*}\n    t\\mapsto \\|U(-t)VU_V(t)\\varphi\\|_{H^1}\\in L^1(\\R),\n  \\end{equation*}\na property which can be proven by the same computations as above, up\nto changing the order of the arguments. To complete the proof of\nTheorem~\\ref{theo:scatt-quant}, it therefore remains to prove that for\n$d\\ge 4$, $\\psi_+\\in H^1(\\R^d)$ and\n  \\begin{equation}\\label{eq:cvH1}\n    \\|\\psi(t)-U(t)\\psi_+\\|_{H^1(\\R^d)}\\Tend t \\infty 0. \n  \\end{equation}\nIt follows from the above results that\n\\begin{equation*}\n  \\psi(t) = U(t) \\psi_+ +i\\int_t^{+\\infty}\n  U(t-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds\n  +i\\int_t^{+\\infty}U(t-s)\\(V(\\psi(s)\\)ds,\n\\end{equation*}\nand that $\\psi,\\nabla \\psi \\in L^q(\\R;L^r(\\R^d))$ for all admissible\npairs $(q,r)$. Since we have\n\\begin{equation*}\n  \\psi_+= U(-t)\\psi(t) - i\\int_t^{+\\infty}\n  U(-s)\\(|\\psi|^{2\\si}\\psi(s)\\)ds\n -i\\int_t^{+\\infty}U(-s)\\(V(\\psi(s)\\)ds,\n\\end{equation*}\nthe previous estimates show that $\\psi_+\\in H^1(\\R^d)$, along with\n\\eqref{eq:cvH1}. \n\n\n\n\\section{Scattering for the asymptotic envelope}\n\\label{sec:class}\n\n\nIn this section, we prove Theorem~\\ref{theo:scatt-class}. The general\nargument is similar to the quantum case: we first prove that the\nnonlinear term can be neglected to large time, and then rely on\nprevious results to neglect the potential. \nRecall that in view of Assumption~\\ref{hyp:V}, the time dependent\nharmonic potential $\\frac{1}{2}\\<Q(t)y,y\\>$ satisfies\n \\begin{equation}\\label{eq:decayQ}\n   \\left\\|\\frac{d^\\alpha}{dt^\\alpha}Q(t)\\right\\|\\lesssim\n   \\<t\\>^{-\\mu-2-\\alpha},\\quad \\alpha \\in \\N,\n \\end{equation}\nwhere $\\|\\cdot\\|$ denotes any matricial norm. \nWe denote by \n\\begin{equation*}\n  H_Q = -\\frac{1}{2}\\Delta + \\frac{1}{2}\\<Q(t)y,y\\>\n\\end{equation*}\nthe time-dependent Hamiltonian present in \\eqref{eq:u}. Like in the\nquantum case, we show that the nonlinearity is negligible for large\ntime by working on Duhamel's formula associated to \\eqref{eq:u} in\nterms of $H_Q$. Since $H_Q$ depends on time, we recall that the\npropagator $U_Q(t,s)$ is the operator which maps $u_0$ to $u_{\\rm lin}(t)$,\nwhere $u_{\\rm lin}$ solves\n\\begin{equation*}\n  i\\d_t u_{\\rm lin} +\\frac{1}{2}\\Delta u_{\\rm lin} =\n  \\frac{1}{2}\\<Q(t)y,y\\>u_{\\rm lin};\\quad u_{{\\rm lin}}(s,y)=u_0(y). \n\\end{equation*}\nIt is a unitary dynamics, in the sense that $U_Q(s,s)=1$, and\n$U_Q(t,\\tau)U_Q(\\tau,s)=U_Q(t,s)$;\nsee e.g. \\cite{DG}. Then to prove the existence of wave operators, we consider the\nintegral formulation\n\\begin{equation}\n  \\label{eq:duhamel-wave-class}\n  u(t) = U_Q(t,0)\\tilde u_--i\\int_{-\\infty}^t U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds.\n\\end{equation}\nA convenient tool is given by Strichartz estimates associated to\n$U_Q$. Local in time Strichartz estimates follow from general results\ngiven in \\cite{Fujiwara}, where local dispersive estimates are\nproven for more general potential. To address large time,  we take\nadvantage of the fact that the \npotential is exactly quadratic with respect to the space variable, so\nan explicit formula is available for $U_Q$, entering the general\nfamily of Mehler's formulas (see e.g. \\cite{Feyn,HormanderQuad}). \n\\subsection{Mehler's formula}\n\\label{sec:mehler}\n\nConsider, for $t_0\\ll -1$,\n\\begin{equation*}\ni\\d_tu+\\frac{1}{2}\\Delta u=\\frac{1}{2}\\< Q(t)y,y\\> u\\quad\n;\\quad u(t_0,y)=u_0(y).\n\\end{equation*}\nWe seek a solution of the form\n\\begin{equation}\n  \\label{eq:mehler}\n  u(t,y) = \\frac{1}{h(t)}\\int_{\\R^d}\n  e^{\\frac{i}{2}\\(\\<M_1(t)y,y\\>+\\<M_2(t)z,z\\>+2\\<P(t)y,z\\>\\)}u_0(z)dz, \n\\end{equation}\nwith symmetric matrices $M_1, M_2,P\\in \\mathcal S_d(\\R)$. \nExperience shows that no linear term is needed in this formula, since\nthe potential is exactly quadratic (see\ne.g. \\cite{CLSS08}). \n\\smallbreak\n\nWe compute:\n\\begin{align*}\n  i\\d_t u & = -i\\frac{\\dot h}{h}u -\\frac{1}{2}\\<\\dot M_1(t)y,y\\>u\\\\\n&\\quad \n  +\\frac{1}{h}\\int e^{\\frac{i}{2}\\(\\dots\\)} \\(-\\frac{1}{2}\\<\\dot\n  M_2(t)z,z\\>-\\<\\dot P(t)y,z\\>\\)u_0(z)dz,\n\\end{align*}\n\\begin{align*}\n  \\d_{j}^2 u &= \\frac{1}{h}\\int e^{\\frac{i}{2}\\(\\dots\\)}\n  \\(-\\(\\(M_1(t)y\\)_j + \\(P(t)z\\)_j\\)^2 -i\\(M_1\\)_{jj}\\)u_0(z)dz,\n\\end{align*}\nhence\n\\begin{align*}\n & i\\d_tu+\\frac{1}{2}\\Delta u =  -i\\frac{\\dot h}{h}u\n  +\\frac{i}{2}\\operatorname{tr} M_1 - \\frac{1}{2}\\<\\dot M_1(t)y,y\\>u\\\\\n&+ \\frac{1}{2h}\\int\n  e^{\\frac{i}{2}\\(\\<M_1(t)y,y\\>+\\<M_2(t)z,z\\>+2\\<P(t)y,z\\>\\)}u_0(z)\\times\\\\\n&\\times\\( \n-\\<\\dot  M_2(t)z,z\\>-2\\<\\dot\nP(t)y,z\\>-|M_1(t)y|^2 -|P(t)z|^2 -2\n\\<M_1(t)y,P(t)z\\>\\)dz. \n\\end{align*}\nIdentifying the quadratic forms (recall that the matrices $M_j$ and\n$P$ are symmetric), we find:\n\\begin{align*}\n&\\frac{\\dot h}{h}=  \\frac{1}{2}\\operatorname{tr} M_1,\\\\\n& \\dot M_1+M_1^2+Q=0,\\\\\n& \\dot M_2 +P^2=0,\\\\\n& \\dot P + PM_1=0.\n\\end{align*}\nDispersion is given by\n\\begin{equation*}\n  h(t)  = h(t_1)\\exp\\(\\frac{1}{2}\\int_{t_1}^t \\operatorname{tr} M_1(s)ds\\),\n\\end{equation*}\nwhere $M_1$ solves the matrix Riccati equation\n\\begin{equation}\n  \\label{eq:riccati}\n   \\dot M_1 + M_1^2 + Q=0;\\quad M_1(t_0)=\\frac{1}{t_0}{\\rm I}_d.\n\\end{equation}\nNote that in general, solutions to Riccati equations develop\nsingularities in finite time. What saves the day here is that\n\\eqref{eq:riccati} is not translation invariant, and can be\nconsidered, for $t\\le t_0\\ll -1$, \nas a perturbation of the Cauchy problem\n\\begin{equation*}\n  \\dot M + M^2 =0;\\quad M(t_0)=\\frac{1}{t_0}{\\rm I}_d,\n\\end{equation*}\nwhose solution is given by \n\\begin{equation*}\n  M(t) = \\frac{1}{t}{\\rm I}_d. \n\\end{equation*}\n\\begin{lemma}\\label{lem:riccati}\n  Let $Q$ be a symmetric matrix satisfying \\eqref{eq:decayQ} for $\\mu>1$. There\n  exists $t_0<0$ such that \\eqref{eq:riccati} has a unique solution\n  $M_1\\in C((-\\infty,t_0];\\mathcal S_d(\\R))$. In addition, it\n  satisfies\n  \\begin{equation*}\n    M_1(t)= \\frac{1}{t}{\\rm I}_d +\\O\\(\\frac{1}{t^2}\\)\\quad \\text{as\n    }t\\to -\\infty. \n  \\end{equation*}\n\\end{lemma}\n\\begin{proof}\nSeek a solution of the form $M_1(t) =   \\frac{1}{t}{\\rm I}_d +R(t)$,\nwhere $R$ is s symmetric matrix solution of\n\\begin{equation*}\n  \\dot R + \\frac{2}{t}R+R^2 +Q= 0;\\quad R(t_0)=0. \n\\end{equation*}\nEquivalently, the new unknown $\\tilde R = t^2 R$ must satisfy\n\\begin{equation}\\label{eq:Rmatrix}\n   \\dot {\\tilde R} + \\frac{1}{t^2}\\tilde R^2 +t^2Q= 0;\\quad \\tilde R(t_0)=0. \n\\end{equation}\nCauchy-Lipschitz Theorem yields a local solution: we show that it is\ndefined on $(-\\infty,t_0]$, along with the announced decay. \nIntegrating between $t_0$ and $t$, we find\n\\begin{equation*}\n  \\tilde R(t) = -\\int_{t_0}^t \\frac{1}{s^2}\\tilde R(s)^2ds -\n  \\int_{t_0}^ts^2 Q(s)ds.\n\\end{equation*}\nNote that $s\\mapsto s^2 Q$ is integrable as $s\\to -\\infty$ from \\eqref{eq:decayQ}\n(we assume $\\mu>1$). Setting \n\\begin{equation*}\n  \\rho(t) =\\sup_{t\\le s\\le t_0}\\|\\tilde R(s)\\|,\n\\end{equation*}\nwhere $\\|\\cdot\\|$ denotes any matricial norm, we have\n\\begin{equation*}\n  \\rho(t) \\le \\frac{C}{t_0}\\rho(t)^2 + \\frac{C}{t_0^{\\mu-1}},\n\\end{equation*}\nfor some constant $C$. Choosing $t_0\\ll -1$, global existence follows\nfrom the following bootstrap argument (see \\cite{BG3}):\nLet $f=f(t)$ be a nonnegative continuous function on $[0,T]$ such\nthat, for every $t\\in [0,T]$, \n\\begin{equation*}\n  f(t)\\le \\eps_1 + \\eps_2 f(t)^\\theta,\n\\end{equation*}\nwhere $\\eps_1,\\eps_2>0$ and $\\theta >1$ are constants such that\n\\begin{equation*}\n  \\eps_1 <\\left(1-\\frac{1}{\\theta} \\right)\\frac{1}{(\\theta \\eps_2)^{1\/(\\theta\n-1)}}\\ ,\\ \\ \\ f(0)\\le  \\frac{1}{(\\theta \\eps_2)^{1\/(\\theta-1)}}.\n\\end{equation*}\nThen, for every $t\\in [0,T]$, we have\n\\begin{equation*}\n  f(t)\\le \\frac{\\theta}{\\theta -1}\\ \\eps_1.\n\\end{equation*}\nThis shows that for $|t_0|$ sufficiently large, the matrix $R$ (hence\n$M_1$) is defined on $(-\\infty,t_0]$. Moreover, since $\\tilde R$ is\nbounded, $R(t)=\\O(t^{-2})$ as $t\\to -\\infty$, hence the result. \n\\end{proof}\nWe infer\n\\begin{equation*}\n  h(t)\\Eq t {-\\infty} c|t|^{d\/2},\n\\end{equation*}\nwhich is the same dispersion as in the case without\npotential. Putting this result together with local dispersive estimates from\n\\cite{Fujiwara}, we have:\n\\begin{lemma}\\label{lem:strichartz-quad}\n  Let $Q$ be a symmetric matrix satisfying \\eqref{eq:decayQ} for\n  $\\mu>1$. Then for all admissible pairs $(q,r)$, \nthere exists $C=C(q,d)$ such that for all $s\\in \\R$,\n\\begin{equation*}\n  \\|U_Q(\\cdot,s)f\\|_{L^q(\\R;L^r(\\R^d))}\\le C \\|f\\|_{L^2(\\R^d)},\\quad\n  \\forall f\\in L^2(\\R^d). \n\\end{equation*}\nFor two admissible pairs $(q_1,r_1)$ and $(q_2,r_2)$, there exists $C_{q_1,q_2}$ such\nthat for all time interval $I$, if we denote by\n\\begin{equation*}\n  R(F)(t,y) = \\int_{I\\cap \\{s\\le t\\}} U_Q(t,s)F(s,y)ds,\n\\end{equation*}\nwe have\n\\begin{equation*}\n  \\|R(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^d))}\\le\n  C_{q_1,q_2}\\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^d))},\\quad \\forall F\\in\n  L^{q_2'}(I;L^{r_2'}(\\R^d)). \n\\end{equation*}\n\\end{lemma}\n\\begin{remark}\n  Since we have dispersive estimates, end-point Strichartz estimates\n  ($q=2$ when $d\\ge 3$)\n  are also available from \\cite{KT}. \n\\end{remark}\n\\subsection{Wave operators}\n\\label{sec:wave-class}\n\nIn this section, we prove:\n\\begin{proposition}\\label{prop:wave-class}\n   Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. For\n  all $\\tilde u_-\\in \\Sigma$, there exists a unique $u\\in\n  C(\\R;\\Sigma)$ solution to \\eqref{eq:u}  such that\n  \\begin{equation*}\n    \\|U_Q(0,t)u(t)-\\tilde u_-\\|_{\\Sigma}\\Tend t {-\n      \\infty} 0.\n  \\end{equation*}\n\\end{proposition}\n\\begin{remark}\n  The assumption $\\si\\ge \\frac{2}{d}$ could easily be relaxed,\n  following the classical argument (see e.g. \\cite{CazCourant}). We do\n  not present the argument, since Theorem~\\ref{theo:scatt-quant} is\n  proven only for $\\si>\\frac{2}{d}$. \n\\end{remark}\n\n\\begin{proof}\n  The proof follows closely the approach without potential\n  ($Q=0$). From this perspective, a key tool is the vector field\n  \\begin{equation*}\n    J(t)=y+it\\nabla.\n  \\end{equation*}\nIt satisfies three important properties:\n\\begin{itemize}\n\\item It commutes with the free Schr\\\"odinger dynamics,\n  \\begin{equation*}\n   \\left[ i\\d_t +\\frac{1}{2}\\Delta,J\\right]=0.  \n  \\end{equation*}\n\\item It acts like a derivative on gauge invariant nonlinearities. If\n  $F(z)$ is of the form $F(z)=G(|z|^2)z$, then \n  \\begin{equation*}\n    J(t)\\(F(u)\\) = \\d_z F(u)J(t)u -\\d_{\\bar z}F(u)\\overline{J(t)u}.\n  \\end{equation*}\n\\item It provides weighted Gagliardo-Nirenberg inequalities:\n  \\begin{align*}\n    \\|f\\|_{L^r}\\lesssim &\n    \\frac{1}{|t|^{\\delta(r)}}\\|f\\|_{L^2}^{1-\\delta(r)}\\|J(t)f\\|_{L^2}^{\\delta(r)},\n    \\quad \\delta(r)=d\\(\\frac{1}{2}-\\frac{1}{r}\\), \\\\\n&\\text{with }\n\\left\\{\n  \\begin{aligned}\n    2\\le r\\le \\infty &\\text{ if }d=1,\\\\\n2\\le r<\\infty &\\text{ if }d=2,\\\\\n2\\le r\\le \\frac{2d}{d-2}&\\text{ if }d\\ge 3. \n  \\end{aligned}\n\\right.\n  \\end{align*}\n\\end{itemize}\nThe last two properties stem from the factorization $J(t)f =\nit e^{i\\frac{|y|^2}{2t}}\\nabla \\(e^{-i\\frac{|y|^2}{2t}}f\\)$. Note that\nthe commutation property does not incorporate the quadratic potential:\n\\begin{align*}\n  \\left[ i\\d_t -H_Q,J\\right]= itQ(t)y=itQ(t)J(t) +t^2 Q(t)\\nabla. \n\\end{align*}\nNow the important remark is that $t\\mapsto t^2Q(t)$ is integrable,\nfrom \\eqref{eq:decayQ} since $\\mu>1$.  \n\\smallbreak\n\nTo prove Proposition~\\ref{prop:wave-class}, we apply a fixed point argument \nto the Duhamel's formula \\eqref{eq:duhamel-wave-class}. As in the case\nof the quantum scattering operator, we have to deal with the fact that\nthe gradient does not commute with $U_Q$, leading to the problem\ndescribed in Section~\\ref{sec:vector-field}. Above, we have sketched\nhow to deal with the inhomogeneous term in\n\\eqref{eq:duhamel-wave-class}, while in\nSection~\\ref{sec:vector-field}, we had underscored the difficulty\nrelated to the homogeneous term. We therefore start by showing that\nfor any admissible pair $(q_1,r_1)$, there exists $K_{q_1}$ such that\n\\begin{equation}\\label{eq:Qhomo}\n  \\|\\nabla U_Q(t,0)f\\|_{L^{q_1}(\\R;L^{r_1})} +\n  \\|J(t)U_Q(t,0)f\\|_{L^{q_1}(\\R;L^{r_1})} \\le K_{q_1} \\|f\\|_{\\Sigma}. \n\\end{equation}\nTo prove this, denote \n\\begin{equation*}\n  v_0(t)=U_Q(t,0)f,\\quad v_1(t)= \\nabla U_Q(t,0)f, \\quad v_2(t)=\nJ(t)U_Q(t,0)f.\n\\end{equation*}\nSince $yv_0 =v_2-it v_1$, we have:\n\\begin{align*}\n  &i\\d_t v_1=H_Q v_1 +Q(t)yv_0 = Hv_1 +Q(t)v_2-it Q(t)v_1;\\quad\n    v_1(0,y)=\\nabla f(y),\\\\\n& i\\d_t v_2 = H_Qv_2 +itQ(t)v_2+t^2Q(t)v_1;\\quad v_2(0,y)=yf(y). \n\\end{align*}\nLemma~\\ref{lem:strichartz-quad} yields\n\\begin{align*}\n  \\|v_1\\|_{L^{q_1}(\\R;L^{r_1})} + \\|v_2\\|_{L^{q_1}(\\R;L^{r_1})}\n  &\\lesssim \\|f\\|_\\Sigma + \\int_{-\\infty}^\\infty \\|\\<t\\>Q(t)v_2(t)\\|_{L^2}dt \\\\\n&\\quad +\n  \\int_{-\\infty}^\\infty \\|\\<t\\>^2Q(t)v_1(t)\\|_{L^2}dt ,\n\\end{align*}\nwhere we have chosen $(q_2,r_2)=(\\infty,2)$. The fact that $U_Q$ is\nunitary on $L^2$ and \\eqref{eq:decayQ} imply\n\\begin{equation*}\n  \\|\\<t\\>Q(t)v_2(t)\\|_{L^2}\\lesssim \\<t\\>^{-\\mu-1}\\|yf\\|_{L^2},\\quad\n  \\|\\<t\\>^2Q(t)v_1(t)\\|_{L^2}\\lesssim \\<t\\>^{-\\mu}\\|\\nabla f\\|_{L^2}, \n\\end{equation*}\nhence \\eqref{eq:Qhomo}. \nWe then apply a fixed point\nargument in\n\\begin{align*}\n  X(T) =&\\Big\\{ u\\in L^\\infty((-\\infty,-T];H^1),  \\\\\n& \\quad\\sum_{B\\in \\{{\\rm Id}, \\nabla, J\\}}\n\\( \\|B   u\\|_{L^\\infty((-\\infty,-T];L^2)}+\n\\|B  u\\|_{L^q((-\\infty,-T];L^r)}\\)\\le {\\mathbf K}\\|\\tilde u_-\\|_{\\Sigma}\\Big\\},\n\\end{align*}\nwhere the admissible pair $(q,r)$ is given by\n\\begin{equation*}\n  (q,r) = \\(\\frac{4\\si+4}{d\\si},2\\si+2\\),\n\\end{equation*}\n and the constant $\\mathbf K$ is related to the constants $C_q$  from\n Strichartz inequalities \n(Lemma~\\ref{lem:strichartz-quad}), and $K_q$  from\n\\eqref{eq:Qhomo}, whose value we do not try to optimize. The fixed\npoint argument is applied \nto the Duhamel's formula \\eqref{eq:duhamel-wave-class}: we denote by\n$\\Phi(u)$ the left hand side, and let $u\\in X(T)$. We have \n\\begin{equation*}\n  \\|\\Phi(u)\\|_{L^\\infty((-\\infty,-T];L^2)}\\le \\|\\tilde u_-\\|_{L^2} + C\n    \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}},\n\\end{equation*}\nwhere $L^a_T$ stands for $L^a((-\\infty,-T])$. H\\\"older inequality\nyields\n\\begin{equation*}\n  \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}} \\le\n  \\|u\\|_{L^k_TL^r}^{2\\si} \\|u\\|_{L^q_TL^r},\n\\end{equation*}\nwhere $k$ is given by\n\\begin{equation*}\n  \\frac{1}{q'}=\\frac{1}{q}+\\frac{2\\si}{k},\\text{ that is }k =\n  \\frac{4\\si(\\si+1)}{2-(d-2)\\si}. \n\\end{equation*}\nWeighted Gagliardo-Nirenberg inequality and the definition of $X(T)$ yield\n\\begin{equation*}\n  \\|u(t)\\|_{L^r}\\lesssim \\frac{1}{|t|^{\\frac{d\\si}{2\\si+2}}}\\|u_-\\|_{\\Sigma}.\n\\end{equation*}\nWe check that for $\\si\\ge \\frac{2}{d}$, \n\\begin{equation*}\n  k\\times \\frac{d\\si}{2\\si+2} = \\frac{2d\\si^2}{2-(d-2)\\si}\\ge 2,\n\\end{equation*}\nand so\n\\begin{equation*}\n  \\|u\\|_{L^k_TL^r}^k =\\O\\(\\frac{1}{T}\\) \\text{ as }T\\to \\infty. \n\\end{equation*}\nBy using Strichartz estimates again,\n\\begin{equation*}\n  \\|\\Phi(u)\\|_{L^q_TL^r}\\le C_q\\|\\tilde u_-\\|_{L^2} + C\n    \\left\\| |u|^{2\\si}u\\right\\|_{L^{q'}_TL^{r'}},\n\\end{equation*}\nwhich shows, like above, that if $T$ is sufficiently large,\n$\\|\\Phi(u)\\|_{L^q_TL^r}\\le 2C_q\\|\\tilde u_-\\|_{L^2}$. \n\\smallbreak\n\nWe now apply $\\nabla$ and $J(t)$ to $\\Phi$, and get a closed system of\nestimates:\n\\begin{align*}\n  \\nabla \\Phi(u) &= \\nabla U_Q(t,0)\\tilde u_- - i \\int_{-\\infty}^t\n  U_Q(t,s)\\nabla \\(|u|^{2\\si}u(s)\\)ds \\\\\n& -i\\int_{-\\infty}^t U_Q(t,s)\\(Q(s)J(s)\\Phi(u)\\)ds - \\int_{-\\infty}^t\n  U_Q(t,s)\\(sQ(s)\\nabla\\Phi(u)\\)ds, \\\\\n J(t) \\Phi(u) &= J(t) U_Q(t,0)\\tilde u_- - i \\int_{-\\infty}^t\n  U_Q(t,s)J(s) \\(|u|^{2\\si}u(s)\\)ds \\\\\n& +\\int_{-\\infty}^t U_Q(t,s)\\(sQ(s)J(s)\\Phi(u)\\)ds - i\\int_{-\\infty}^t\n  U_Q(t,s)\\(s^2Q(s)\\nabla\\Phi(u)\\)ds,\n\\end{align*}\nwhere we have used the same algebraic properties as in the proof of\n\\eqref{eq:Qhomo}. Set \n\\begin{equation*}\n  M(T) = \\sum_{B\\in \\{\\nabla, J\\}} \\( \\|B(t)\\Phi(u)\\|_{L^\\infty_TL^2}\n  + \\|B(t)\\Phi(u)\\|_{L^q_TL^r}\\).\n\\end{equation*}\nLemma~\\ref{lem:strichartz-quad} and\n\\eqref{eq:Qhomo} yield\n\\begin{align*}\n  M(T)&\\lesssim \\|\\tilde u_-\\|_\\Sigma + \\sum_{B\\in \\{\\nabla, J\\}}\\left\\||u|^{2\\si}B\n        u\\right\\|_{L^{q'}_TL^{r'}} \\\\\n&\\quad+\\|\\<t\\>Q(t)J(t)\\Phi(u)\\|_{L^1_TL^2} +\n        \\|\\<t\\>^2Q(t)\\nabla\\Phi(u)\\|_{L^1_TL^2}, \n\\end{align*}\nwhere we have also used the fact that $J(t)$ acts like a derivative on\ngauge invariant nonlinearities. The same H\\\"older inequalities as\nabove yield\n\\begin{equation*}\n  \\left\\||u|^{2\\si}B\n        u\\right\\|_{L^{q'}_TL^{r'}} \\le\n      \\|u\\|_{L^k_TL^r}^{2\\si}\\|Bu\\|_{L^q_TL^r}\\lesssim \\frac{1}{T^{2\\si\/k}}\\|Bu\\|_{L^q_TL^r}.\n\\end{equation*}\nOn the other hand, from \\eqref{eq:decayQ},\n\\begin{equation*}\n  \\|\\<t\\>Q(t)J(t)\\Phi(u)\\|_{L^1_TL^2} +\n        \\|\\<t\\>^2Q(t)\\nabla\\Phi(u)\\|_{L^1_TL^2}\\lesssim \\frac{1}{T^{\\mu-1}}M(T),\n\\end{equation*}\nand so\n\\begin{equation*}\n  M(T) \\lesssim \\|\\tilde u_-\\|_\\Sigma  +\n  \\frac{1}{T^{2\\si\/k}}\\sum_{B\\in \\{\\nabla, J\\}} \\|Bu\\|_{L^q_TL^r} + \\frac{1}{T^{\\mu-1}}M(T).\n\\end{equation*}\nBy choosing $T$ sufficiently large, we infer\n\\begin{equation*}\n  M(T) \\lesssim \\|\\tilde u_-\\|_\\Sigma  +\n  \\frac{1}{T^{2\\si\/k}}\\sum_{B\\in \\{\\nabla, J\\}} \\|Bu\\|_{L^q_TL^r},\n\\end{equation*}\nand we conclude that $\\Phi$ maps $X(T)$ to $X(T)$ for $T$\nsufficiently large. Up to choosing $T$ even larger, $\\Phi$ is a\ncontraction on $X(T)$ with respect to the weaker norm $L^q_TL^r$,\nsince for $u,v\\in X(T)$, we have \n\\begin{align*}\n  \\|\\Phi(u)-\\Phi(v)\\|_{L^q_TL^r}&\\lesssim \\left\\| |u|^{2\\si}u\n    -|v|^{2\\si}v\\right\\|_{L^{q'}_TL^{r'}}\\lesssim \\(\n  \\|u\\|_{L^k_TL^r}^{2\\si}+\\|v\\|_{L^k_TL^r}^{2\\si}\\)\\|u-v\\|_{L^q_TL^r}\\\\\n&\\lesssim \\frac{1}{T^{2\\si\/k}}\\|u-v\\|_{L^q_TL^r},\n\\end{align*}\nwhere we have used the previous estimate. Therefore, there exists\n$T>0$ such that $\\Phi$ has a unique fixed point in $X(T)$. This\nsolution actually belongs to $C(\\R;\\Sigma)$ from \\cite{CaSi15}.  \nUnconditional uniqueness (in $\\Sigma$, without referring to mixed\nspace-time norms) stems from the approach in \\cite{TzVi-p}. \n\\end{proof}\n\n\\subsection{Vector field}\n\\label{sec:vector-field-Q}\n\n  It is possible to construct a vector field adapted to the presence\n  of $Q$, even though it is not needed to prove\n  Proposition~\\ref{prop:wave-class}. Such a vector field will be useful\n  in Section~\\ref{sec:cv}, and since its construction is very much in\n  the continuity of Section~\\ref{sec:mehler}, we present it now. Set,\n  for a scalar function $f$, \n\\begin{equation*}\n  {\\mathcal A} f= i W(t) e^{i\\phi(t,y)}\\nabla \\(\n    e^{-i\\phi(t,y)}f\\)= W(t) \\(f\\nabla \\phi +i\\nabla f\\),\n\\end{equation*}\nwhere $W$ is a matrix and the phase $\\phi$ solves the eikonal equation\n\\begin{equation*}\n \\d_t \\phi +\\frac{1}{2}|\\nabla \\phi|^2 + \\frac{1}{2}\\< Q(t)y,y\\>=0. \n\\end{equation*}\nSince the underlying Hamiltonian is quadratic, $\\phi$ has the form\n\\begin{equation*}\n  \\phi(t,y) = \\frac{1}{2}\\<K(t)y,y\\>,\n\\end{equation*}\nwhere $K(t)$ is a symmetric matrix. For $ {\\mathcal A}$ to commute\nwith $i\\d_t -H_Q$, we come up with the conditions\n\\begin{equation*}\n   \\dot K + K^2 + Q=0,\\quad  \\dot W = W \\nabla^2\\phi= WK .\n\\end{equation*}\nWe see that we can take $K=M_1$ as in the proof of\nLemma~\\ref{lem:riccati}, and $ {\\mathcal A}$ will then satisfy the\nsame three properties as $J$, up to the fact that the commutation\nproperty now includes the quadratic potential. \n\\smallbreak\n\nSince the construction of this vector field\nboils down to solving a matricial Riccati equation with initial data\nprescribed at large time (see \\eqref{eq:riccati}), we naturally\nconstruct two vector fields $\\mathcal A_\\pm$, associated to $t\\to \\pm\n\\infty$. In view of Lemma~\\ref{lem:riccati}, $\\mathcal A_-$ is defined\non $(-\\infty,-T]$, while $\\mathcal A_+$ is defined\non $[T,\\infty)$, for a common $T\\gg 1$, with\n\\begin{equation*}\n  \\mathcal A_\\pm  = W_\\pm(t)\\(\\nabla \\phi_\\pm + i\\nabla\\), \\quad\n  \\phi_\\pm (t,y) = \\frac{1}{2}\\<K_\\pm (t)y,y\\>,\n\\end{equation*}\nwhere $K_\\pm$ and $W_\\pm$ satisfy\n\\begin{equation*}\n  \\dot K_\\pm +K_\\pm^2+Q=0,\\quad \\dot W_\\pm =W_\\pm K_\\pm,\n\\end{equation*}\nso that Lemma~\\ref{lem:riccati} also yields\n\\begin{equation}\\label{eq:vector-asym}\n  K_\\pm(t)\\sim  \\frac{1}{t}{\\rm I}_d,\\quad  W_\\pm (t)\\sim t {\\rm\n    I}_d\\quad \\text{as }t\\to \\pm \\infty.  \n\\end{equation} \nWe construct commuting vector fields for large time only, essentially\nbecause on finite time intervals, the absence of commutation is not a\nproblem, so we can use $\\nabla$, $y$ or $J$. \n\n\n\n\\subsection{Asymptotic completeness}\n\\label{sec:ac-class}\n\nIn this section we prove:\n\\begin{proposition}\\label{prop:AC-class}\n  Let $d\\ge 1$, $\\frac{2}{d}\\le \\si<\\frac{2}{(d-2)_+}$, and $V$\n  satisfying Assumption~\\ref{hyp:V} for some $\\mu>1$. For all $u_0\\in\n  \\Sigma$, there exists a unique $\\tilde u_+\\in \\Sigma$ such that the\n  solution $u\\in C(\\R;\\Sigma)$ to \\eqref{eq:u} with $u_{\\mid t=0}=u_0$\n  satisfies\n  \\begin{equation*}\n   \\sum_{\\Gamma\\in \\{{\\rm Id},\\nabla, J\\}} \\|\\Gamma(t) u(t)-\n   \\Gamma(t)U_Q(t,0)\\tilde u_+\\|_{L^2} \\Tend t {+\\infty} 0. \n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  In the case $Q=0$, such a result is a rather direct consequence of\n  the \\emph{pseudo-conformal conservation law}, established in\n  \\cite{GV79Scatt}. Recalling that $J(t)=y+it\\nabla$, this law reads\n\\begin{equation*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  =\\frac{t}{\\si+1}(2-d\\si)\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}. \n\\end{equation*}\nA way to derive this relation is to apply $J$ to \\eqref{eq:u}. The operator $J$\ncommutes with the linear part ($Q=0$), and the standard $L^2$\nestimate, which consists in multiplying the outcome by\n$\\overline{Ju}$, integrating in space, and taking the imaginary part, yields:\n\\begin{equation*}\n\\frac{1}{2} \\frac{d}{dt}\\|J(t)u\\|_{L^2}^2  = \\IM \\int \\overline {Ju}J\\(|u|^{2\\si}u\\).\n\\end{equation*}\nSince we have $J= i t  e^{i\\frac{|y|^2}{2t}} \\nabla\\( \\cdot\ne^{-i\\frac{|y|^2}{2t}}\\)$, \n\\begin{equation*}\n  J\\(|u|^{2\\si}u\\) = (\\si+1)|u|^{2\\si}Ju + \\si u^{\\si+1}\\bar\n  u^{\\si-1}\\overline{Ju}.\n\\end{equation*}\nThe first term is real, and the rest of the computation consists in\nexpanding the remaining term. \n\\smallbreak\n\nIn the case where $Q\\not =0$, we resume the above approach: the new\ncontribution is due to the fact that $J$ does not commute with the\nexternal potential, so we find:\n\\begin{align*}\n  \\frac{1}{2} \\frac{d}{dt}\\|J(t)u\\|_{L^2}^2 & =\\text{like before} + \\RE \\int\n  t Q(t)xu\\cdot \\overline {Ju}\\\\\n&=\\text{like before} + \n  t\\RE\\int_{\\R^d} \\<Q(t) J(t)u,J(t)u\\> +t^2 \\IM \\int_{\\R^d} \\<Q(t)\\nabla u,Ju\\>.\n\\end{align*}\nOn the other hand, we still have\n\\begin{align*}\n  \\frac{d}{dt}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}& =2 (\\si+1)\\int\n  |u|^{2\\si}\\RE \\(\\bar u\\d_tu\\) = 2 (\\si+1)\\int\n  |u|^{2\\si}\\RE \\(\\bar u \\times\\frac{i}{2}\\Delta u\\) ,\n\\end{align*}\nand so,\n\\begin{align*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  &=\\frac{t}{\\si+1}(2-d\\si)\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\\\\n+ \n t\\RE\\int_{\\R^d} \\<Q(t) J(t)u,J(t)u\\>& +t^2 \\IM \\int_{\\R^d} \\<Q(t)\\nabla u,Ju\\> . \n\\end{align*}\nThus for $t\\ge 0$ and $\\si\\ge\\frac{2}{d}$, \\eqref{eq:decayQ} implies\n\\begin{equation*}\n  \\frac{d}{dt}\\(\\frac{1}{2}\\|J(t)u\\|_{L^2}^2\n  +\\frac{t^2}{\\si+1}\\|u(t)\\|_{L^{2\\si+2}}^{2\\si+2}\\)\n  \\lesssim  \n  \\<t\\>^{-\\mu-1}\\|J(t)u\\|_{L^2}^2 +\\<t\\>^{-\\mu}\\| \\nabla u\\|_{L^2}  \\|Ju\\|_{L^2}. \n\\end{equation*}\nEven though there is no conservation of the energy for \\eqref{eq:u}\nsince the potential depends on time, we know from \\cite{Ha13} that $u\\in\nL^\\infty(\\R;H^1(\\R^d))$. As a matter of fact, the proof given in\n\\cite[Section~4]{Ha13} concerns the case $\\si=1$ in $d=2$ or $3$, but\nthe argument, based on energy estimates,  remains valid for $d\\ge 1$,\n$\\si<\\frac{2}{(d-2)_+}$, since we then know that $u\\in\nC(\\R;\\Sigma)$. Since $\\mu>1$, we infer\n\\begin{equation}\\label{eq:Jborne}\n  Ju\\in L^\\infty(\\R_+;L^2). \n\\end{equation}\nWriting Duhamel's formula for \\eqref{eq:u} with initial datum $u_0$,\nin terms of $U_Q$, we have\n\\begin{equation*}\n  u(t) = U_Q(t,0)u_0-i\\int_0^t U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds.\n\\end{equation*}\nResuming the computations presented in the proof of\nProposition~\\ref{prop:wave-class}, \\eqref{eq:Jborne} and (weighted)\nGagliardo-Nirenberg inequalities make it possible to prove that\n\\begin{equation*}\n  Bu \\in L^{q_1}(\\R_+;L^{r_1}),\\ \\forall (q_1,r_1)\\text{ admissible},\n  \\ \\forall B\\in \\{{\\rm Id},\\nabla, J\\}. \n\\end{equation*}\nDuhamel's formula then yields, for $0<t_1<t_2$,\n\\begin{equation*}\n  U_Q(0,t_2)u(t_2)-U_Q(0,t_1)u(t_1) =\n  -i\\int_{t_1}^{t_2}U_Q(0,s)\\(|u|^{2\\si}u(s)\\)ds. \n\\end{equation*}\nFrom Strichartz estimates,\n\\begin{equation*}\n  \\| U_Q(0,t_2)u(t_2)-U_Q(0,t_1)u(t_1) \\|_{L^2}\\lesssim \\left\\|\n    |u|^{2\\si}u\\right\\|_{L^{q'}([t_1,t_2]:L^{r'})}, \n\\end{equation*}\nand the right hand side goes to zero as $t_1,t_2\\to\n+\\infty$. Therefore, there exists (a unique) $\\tilde u_+\\in L^2$ such that\n\\begin{equation*}\n   \\| U_Q(0,t)u(t)-\\tilde u_+ \\|_{L^2}\\Tend t {+\\infty} 0 ,\n\\end{equation*}\nand we have\n\\begin{equation*}\n  u(t) = U_Q(t,0)\\tilde u_+ +i\\int_t^\\infty\n  U_Q(t,s)\\(|u|^{2\\si}u(s)\\)ds. \n\\end{equation*}\nUsing the same estimates as in the proof of\nProposition~\\ref{prop:wave-class}, we infer\n\\begin{align*}\n  \\|\\nabla u(t) - \\nabla  U_Q(t,0)\\tilde u_+\\|_{L^2} &+  \\|J(t) u(t) -\n  J(t)  U_Q(t,0)\\tilde u_+\\|_{L^2} \\\\\n& \\lesssim \\left\\| |u|^{2\\si}\\nabla\n  u\\right\\|_{L^{q'}(t,\\infty;L^{r'})} +  \\left\\| |u|^{2\\si} J\n  u\\right\\|_{L^{q'}(t,\\infty;L^{r'})}  \\\\\n&\\quad+\n  \\|\\<s\\>^{-\\mu-1}J(s)u\\|_{L^1(t,\\infty;L^2)} +  \\|\\<s\\>^{-\\mu}\\nabla u\\|_{L^1(t,\\infty;L^2)}. \n\\end{align*}\nThe right hand side goes to zero as $t\\to \\infty$, hence the\nproposition. \n\\end{proof}\n\n\\begin{remark}\n  As pointed out in the previous section, it would be possible to\n  prove the existence of wave operators by using an adapted vector\n  field $\\mathcal A$. On the other hand, if $Q(t)$ is not proportional\n  to the identity matrix, it seems that no (exploitable) analogue of\n  the pseudo-conformal conservation law is available in terms of\n  $\\mathcal A$ rather than in terms of $J$. \n\\end{remark}\n\\subsection{Conclusion}\n\\label{sec:concl-class}\n\nLike in the case of quantum scattering, we use a stronger version of\nthe linear scattering theory:\n\\begin{proposition}\\label{prop:Cook-class}\n  Let $d\\ge 1$, $V$ satisfying Assumption~\\ref{hyp:V} with $\\mu>1$. Then \nthe strong limits\n \\begin{equation*}\n  \\lim_{t\\to \\pm \\infty} U_Q(0,t)U(t) \\quad \\text{and}\\quad  \\lim_{t\\to\n    \\pm\\infty} U(-t) U_Q(t,0) \\quad \\text{and}\\quad  \n\\end{equation*}\nexist in $\\Sigma$. \n\\end{proposition}\n\\begin{proof}\n  For the first limit (existence of wave operators), again in view of\n  Cook's method, we prove that for all $\\varphi\\in \n  \\Sch(\\R^d)$, \n\\begin{equation*}\n    t\\mapsto \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{\\Sigma}\\in\n    L^1(\\R). \n  \\end{equation*}\nFor the $L^2$ norm, we have, in view of \\eqref{eq:decayQ},\n\\begin{equation*}\n  \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{L^2} \\lesssim\n  \\<t\\>^{-\\mu-2}\\sum_{j=1}^d\\| y_j^2 U(t)\\varphi\\|_{L^2}.\n\\end{equation*}\nWrite\n\\begin{equation*}\n  y_j^2 = (y_j+it\\d_j)^2 +t^2\\d_j^2 -2ity_j\\d_j =  (y_j+it\\d_j)^2\n  -t^2\\d_j^2 -2it(y_j+it\\d_j)\\d_j,\n\\end{equation*}\nto take advantage of the commutation\n\\begin{equation*}\n  (y_j+it\\d_j)U(t) = U(t)y_j,\n\\end{equation*}\nand infer\n\\begin{equation*}\n  \\left\\| U_Q(0,t) \\<Q(t)y,y\\>U(t)\\varphi\\right\\|_{L^2} \\lesssim\n  \\<t\\>^{-\\mu-2}\\(\\||y|^2\\varphi\\|_{L^2} +t^2\\|\\Delta \\varphi\\|_{L^2}\n  \\)\\lesssim \\<t\\>^{-\\mu}.\n\\end{equation*}\nThe right hand side is integrable since $\\mu>1$, so the strong limits\n\\begin{equation*}\n   \\lim_{t\\to \\pm\\infty} U_Q(0,t)U(t)\n\\end{equation*}\nexist in $L^2$. \nTo infer that these strong limits actually exist in $\\Sigma$, we\nsimply invoke \\eqref{eq:Qhomo} in the case $(q_1,r_1)=(\\infty,2)$, so\nthe above computation are easily adapted. \n\\smallbreak\n\nFor asymptotic completeness, we can adopt the same strategy. Indeed,\nit suffices to prove that  for all $\\varphi\\in \n  \\Sch(\\R^d)$, \n\\begin{equation*}\n    t\\mapsto \\left\\| U(-t) \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{\\Sigma}\\in\n    L^1(\\R). \n  \\end{equation*}\nFor the $L^2$ norm, we have\n\\begin{align*}\n  \\left\\| U(-t) \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{L^2}&= \\left\\|\n   \\<Q(t)y,y\\>U_Q(t,0)\\varphi\\right\\|_{L^2}\\\\\n& \\lesssim\n \\<t\\>^{-\\mu-2}\\sum_{j=1}^d  \\left\\|\n   y_j^2U_Q(t,0)\\varphi\\right\\|_{L^2}.\n\\end{align*}\nWe first proceed like above, and write\n\\begin{equation*}\n  y_j^2 =  (y_j+it\\d_j)^2\n  -t^2\\d_j^2 -2it(y_j+it\\d_j)\\d_j.\n\\end{equation*}\nThe operator $J$ does not commute with $U_Q$, but this lack of\ncommutation is harmless for our present goal, from\n\\eqref{eq:Qhomo}. By considering the system satisfied by\n$$(y_j+it\\d_j)^2U_Q(t,0)\\varphi, \\d_j^2 U_Q(t,0)\\varphi,\n\\d_j(y_j+it\\d_j)U_Q(t,0)\\varphi,$$ \nwe obtain \n\\begin{align*}\n \\sum_{j=1}^d&\\( \\| (y_j+it\\d_j)^2U_Q(t,0)\\varphi\\|_{L^2} + \\|\\d_j^2\n U_Q(t,0)\\varphi\\|_{L^2} +\n \\|\\d_j(y_j+it\\d_j)U_Q(t,0)\\varphi\\|_{L^2}\\)\\\\\n&\\le C \\|\\varphi\\|_{\\Sigma^2},\n\\end{align*}\nwhere $\\Sigma^k$ is the space of $H^k$ functions with $k$ momenta in\n$L^2$, and $C$ does not depend on time. Finally, we also have a\nsimilar estimate by considering one more derivative or momentum. The\nkey remark in the computation is that the external\npotential $\\<Q(t)y,y\\>$ is exactly quadratic in space, and so\ndifferentiating it three times with any space variables yields zero. \n\\end{proof}\n\n\n\n\\section{Proof of Theorem~\\ref{theo:cv}}\n\\label{sec:cv}\n\nThe main result of this section is:\n\\begin{theorem}\\label{theo:cv-unif}\n  Let $d=3$, $\\si=1$, $V$ as in Theorem~\\ref{theo:scatt-quant}, and\n  $u_-\\in \\Sigma^7$. Suppose that Assumption~\\ref{hyp:flot} \n  is satisfied. Let $\\psi^\\eps$ be given by\n  Theorem~\\ref{theo:scatt-quant}, $u$ be given by\n  Theorem~\\ref{theo:scatt-class}, $\\varphi^\\eps$ defined by\n  \\eqref{eq:phi}.  We have\n  the uniform error \n  estimate:\n  \\begin{equation*}\n   \\sup_{t\\in \\R}\\|\\psi^\\eps(t)-\\varphi^\\eps(t)\\|_{L^2(\\R^3)}  =\n    \\O\\(\\sqrt\\eps\\). \n  \\end{equation*}\n\\end{theorem}\nTheorem~\\ref{theo:cv} is a direct consequence of the above\nresult, whose proof is the core of\nSection~\\ref{sec:cv}. From now on, we assume $d=3$ and $\\si=1$. \n\\subsection{Extra properties for the approximate solution}\n\\label{sec:extra-u}\n\nFurther regularity and localization properties on $u$ will be\nneeded. \n\\begin{proposition}\\label{prop:extra-u}\n  Let $\\si=1$, $1\\le d\\le 3$, $k\\ge 2$ and $V$ satisfying\n  Assumption~\\ref{hyp:V} for some $\\mu>1$. If $u_-\\in \\Sigma^k$, then\n  the solution $u\\in C(\\R;\\Sigma)$ provided by Theorem~\\ref{theo:scatt-class}\n  satisfies $u\\in C(\\R;\\Sigma^k)$.  The momenta\n  of $u$ satisfy\n  \\begin{equation*}\n    \\lVert \\lvert y\\rvert^\\ell u(t,y)\\|_{L^2(\\R^d)}\\le C_\\ell\n    \\<t\\>^\\ell,\\quad 0\\le \\ell\\le k,\n  \\end{equation*}\nwhere $C_\\ell$ is independent of $t\\in \\R$.\n \\end{proposition}\n\\begin{proof}\n  We know from the proof of Theorem~\\ref{theo:scatt-class} that since\n  $u_-\\in \\Sigma$,\n\\begin{equation*}\n  u,\\nabla u, Ju \\in L^\\infty(\\R;L^2(\\R^d)).\n  \\end{equation*}\nThe natural approach is  then to proceed by induction on $k$,\nto prove that \n\\begin{align*}\n  \\nabla^k u,J^k u\\in L^\\infty(\\R;L^2(\\R^d)). \n\\end{align*}\n We have, as we have seen in the proof of\nProposition~\\ref{prop:wave-class},\n\\begin{align*}\n   i\\d_t \\nabla u &= H_Q \\nabla u + Q(t)y u +\\nabla\n                    \\(|u|^2 u\\)\\\\\n&+ H_Q \\nabla u + Q(t)J(t)u -it Q(t)\\nabla u +\\nabla\n                    \\(|u|^2 u\\),\\\\\ni\\d_t Ju  & = H_Q Ju   +it Q(t)y u +J\n                    \\(|u|^2 u\\)\\\\\n& = H_Q J u + itQ(t)J(t)u +t^2Q(t)\\nabla u +J\n                    \\(|u|^2 u\\).\n\\end{align*}\nApplying the operators $\\nabla$ and $J$ again, we find\n\\begin{align*}\n   i\\d_t \\nabla^2 u &= H_Q \\nabla^2 u + 2Q(t)y \\nabla u +Q(t) u +\\nabla^2\n                    \\(|u|^2 u\\)\\\\\n&+ H_Q \\nabla u + 2Q(t)J(t)\\nabla u -2it Q(t)\\nabla^2 u+Q(t)u +\\nabla^2\n                    \\(|u|^2 u\\),\\\\\ni\\d_t J^2u  & = H_Q J^2u   -2t^2 Q(t)y \\nabla u -t^2Q(t)u+J^2\n                    \\(|u|^2 u\\)\\\\\n& = H_Q J^2 u - 2t^2Q(t)J\\nabla u +2it^3Q(t) J^2 u +itQ(t)u+J^2\n                    \\(|u|^2 u\\).\n\\end{align*}\nIn view of \\eqref{eq:decayQ}, we see that $t\\mapsto t^3 Q(t)$ need not be\nintegrable (unless we make stronger and stronger assumptions of $\\mu$,\nas $k$ increases), so the commutator seems to be fatal to this approach. To\novercome this issue, we use the vector field mentioned in\nSection~\\ref{sec:vector-field-Q}. \nFor bounded time $t\\in\n[-T,T]$, the above mentioned lack of commutation is not a problem, and\nwe can use the operator $J$, which is defined for all time. \nWe note that either of the operators $\\mathcal A_\\pm$ or \n$J$ satisfies more generally the pointwise identity\n\\begin{equation*}\n  B\\(u_1\\overline u_2 u_3\\) =\\( B u_1\\) \\overline u_2 u_3 +\n  u_1\\(\\overline{B u_2}\\) u_3 + u_1\\overline u_2\\( Bu_3\\),\n\\end{equation*}\nfor all differentiable functions $u_1,u_2,u_3$. \n\nNow we have all the tools to proceed by induction, and mimic the\nproof from \\cite[Appendix]{Ca11}. The main idea is that the proof is\nsimilar to the propagation of higher regularity for energy-subcritical\nproblems, with the difference that large time is handled thanks to\nvector fields. We leave out the details, which are not difficult but\nrather cumbersome: considering\n\\begin{equation*}\n  B(t) =\n\\left\\{\n  \\begin{aligned}\n    \\mathcal A_-(t)&\\text{ for }t\\le -T,\\\\\nJ(t)&\\text{ for }t\\in [-T,T],\\\\\n \\mathcal A_+(t)&\\text{ for }t\\ge T,\n  \\end{aligned}\n\\right.\n\\end{equation*}\n we can then prove that \n \\begin{equation*}\n   \\nabla^k u,B^k u\\in L^\\infty(\\R;L^2(\\R^d)).  \n \\end{equation*}\nBack to the definition of $\\mathcal A_\\pm$, \n\\begin{equation*}\n  \\mathcal A_\\pm (t) = W_\\pm (t)K_\\pm (t)y +iW_\\pm (t)\\nabla,\n\\end{equation*}\n\\eqref{eq:vector-asym}\nthen yields the result. \n\\end{proof}\n\n\\subsection{Strichartz estimates}\n\\label{sec:strichartz-raff}\n\nIntroduce the following notations, taking the dependence upon $\\eps$\ninto account:\n\\begin{equation*}\n  H^\\eps=-\\frac{\\eps^2}{2}\\Delta+V(x),\\quad U_V^\\eps(t) =\n  e^{-i\\frac{t}{\\eps} H^\\eps}. \n\\end{equation*}\nSince we now work only in space dimension $d=3$, we can use the result\nfrom \\cite{Go06}. Resuming the proof from \\cite{Go06} (a mere scaling\nargument is not sufficient), we have, along with the preliminary\nanalysis from Section~\\ref{sec:spectral}, the global dispersive estimate\n\\begin{equation}\n  \\label{eq:disp-semi-glob}\n  \\|U^\\eps_V(t)\\|_{L^1(\\R^3)\\to L^\\infty(\\R^3)}\\lesssim\n  \\frac{1}{(\\eps|t|)^{3\/2}},\\quad t\\not =0.\n\\end{equation}\nFor $|t|\\le \\delta$, $\\delta>0$ independent of $\\eps$, the above\nrelation stems initially from \\cite{Fujiwara}. As a consequence, we\ncan measure the dependence upon $\\eps$ in Strichartz estimates. We\nrecall the definition of admissible pairs related to Sobolev\nregularity.\n\\begin{definition}\n  Let $d=3$ and $s\\in \\R$. A pair $(q,r)$ is called $\\dot H^s$-admissible if \n  \\begin{equation*}\n    \\frac{2}{q}+\\frac{3}{r} = \\frac{3}{2}-s. \n  \\end{equation*}\n\\end{definition}\nFor $t_0\\in \\R\\cup \\{-\\infty\\}$, we denote by\n\\begin{equation*}\n  R^\\eps_{t_0}(F)(t) = \\int_{t_0}^t U_V^\\eps(t-s)F(s)ds\n\\end{equation*}\nthe retarded term related to Duhamel's formula. Since the dispersive\nestimate \\eqref{eq:disp-semi-glob} is the same as the one for\n$e^{i\\eps t\\Delta}$, we get the same scaled Strichartz estimates as\nfor this operator, which can in turn be obtained by scaling\narguments from the case $\\eps=1$. \n\\begin{lemma}[Scaled $L^2$-Strichartz estimates]\\label{lem:stri-eps}\n  Let $t_0\\in \\R\\cup\\{-\\infty\\}$, and let $(q_1,r_1)$ and $(q_2,r_2)$\n  be $L^2$-admissible pairs, $2\\le  r_j\\le 6$. We have\n  \\begin{equation*}\n   \\eps^{\\frac{1}{q_1}} \\|U_V^\\eps(\\cdot) f\\|_{L^{q_1}(\\R;L^{r_1}(\\R^3))}\\lesssim\n    \\|f\\|_{L^2(\\R^3)}, \n  \\end{equation*}\n  \\begin{equation*}\n  \\eps^{\\frac{1}{q_1}+\\frac{1}{q_2}}\n  \\|R^\\eps_{t_0}(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^3))}\\le C_{q_1,q_2} \\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^3))},\n  \\end{equation*}\nwhere $C_{q_1,q_2}$ is independent of $\\eps$, $t_0$, and of $I$ such that\n$t_0\\in \\bar I$. \n\\end{lemma}\nWe will also use Strichartz estimates for non-admissible pairs, as\nestablished in \\cite{Kat94} (see also \\cite{CW92,FoschiStri}).\n\\begin{lemma}[Scaled inhomogeneous Strichartz\n  estimates]\\label{lem:stri-inhom-eps} \n   Let $t_0\\in \\R\\cup\\{-\\infty\\}$, and let $(q_1,r_1)$ be an $\\dot\n   H^{1\/2}$-admissible pair, and $(q_2,r_2)$\n  be an $\\dot H^{-1\/2}$-admissible pair, with \n  \\begin{equation*}\n    3\\le r_1,r_2<6.\n  \\end{equation*}\nWe have\n\\begin{equation*}\n  \\eps^{\\frac{1}{q_1}+\\frac{1}{q_2}}\n  \\|R^\\eps_{t_0}(F)\\|_{L^{q_1}(I;L^{r_1}(\\R^3))}\\le C_{q_1,q_2} \\|F\\|_{L^{q_2'}(I;L^{r_2'}(\\R^3))},\n  \\end{equation*}\nwhere $C_{q_1,q_2}$ is independent of $\\eps$, $t_0$, and of $I$ such that\n$t_0\\in \\bar I$. \n\\end{lemma}\n\\subsection{Preparing the proof}\n\\label{sec:preparing-proof}\n\n\n\n\n\n Subtracting the equations satisfied by\n$\\psi^\\eps$ and $\\varphi^\\eps$, respectively, we obtain as in\n\\cite{CaFe11}: $w^\\eps=\\psi^\\eps-\\varphi^\\eps$ satisfies\n\\begin{equation}\\label{eq:restecrit}\n  i\\eps\\d_t w^\\eps +\\frac{\\eps^2}{2}\\Delta w^\\eps =V w^\\eps -\\mathcal L^\\eps\n  + \n  \\eps^{5\/2}\\(|\\psi^\\eps|^{2}\\psi^\\eps\n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\),\n\\end{equation}\nalong with the initial condition\n\\begin{equation*}\n    e^{-i\\frac{\\eps\n      t}{2}\\Delta}w^\\eps_{\\mid t=-\\infty}=0,  \n\\end{equation*}\nwhere the source term is given by \n\\begin{equation*}\n  {\\mathcal L}^\\eps(t,x) = \\(V(x) - V\\(q(t)\\)\n  -\\sqrt\\eps \\<\\nabla V\\(q(t)\\),y\\>\n  -\\frac{\\eps}{2}\\<Q(t)y,y\\>\\)\\Big|_{y=\\frac{x-q(t)}{\\sqrt\\eps}}\n  \\varphi^\\eps(t,x). \n\\end{equation*}\nDuhamel's\nformula for $w^\\eps$ reads\n\\begin{align*}\n  w^\\eps(t) &= -i\\eps^{3\/2}\\int_{-\\infty}^t U^\\eps_V(t-s)\\(|\\psi^\\eps|^{2}\\psi^\\eps\n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\)(s)ds\\\\\n&\\quad  +i\\eps^{-1}\\int_{-\\infty}^t U^\\eps_V(t-s) \\mathcal L^\\eps(s)ds. \n\\end{align*}\nDenoting $L^a(]-\\infty,t];L^b(\\R^3))$ by $L^a_tL^b$, Strichartz\nestimates yield, for any $L^2$-admissible pair $(q_1,r_1)$,\n\\begin{equation}\\label{eq:stri-weps}\n  \\eps^{1\/q_1}\\|w^\\eps\\|_{L^{q_1}_t L^{r_1}} \\lesssim\n  \\eps^{3\/2-1\/q}\\left\\||\\psi^\\eps|^{2}\\psi^\\eps \n  -|\\varphi^\\eps|^{2}\\varphi^\\eps\\right\\|_{L^{q'}_tL^{r'}} +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2},\n\\end{equation}\nwhere $(q,r)$ is the admissible pair chosen in the proof of\nProposition~\\ref{prop:waveop-quant}, that is $r=2\\si+2$. Since we now\nhave  $d=3$ and $\\si=1$, this means:\n\\begin{equation*}\n  q=\\frac{8}{3},\\quad k=8,\n\\end{equation*}\nand \\eqref{eq:stri-weps} yields\n\\begin{equation}\\label{eq:w-presque}\n  \\eps^{1\/q_1}\\|w^\\eps\\|_{L^{q_1}_t L^{r_1}} \\lesssim\n  \\eps^{9\/8}\\( \\|w^\\eps\\|^2_{L^8_t L^4}+ \\|\\varphi^\\eps\\|^2_{L^8_t\n    L^4}\\)\\|w^\\eps\\|_{L^{8\/3}_tL^4}  +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2}.\n\\end{equation}\nThe strategy is then to first\nobtain an a priori estimate for $w^\\eps$ in $L^8_tL^4$, and then to\nuse it in the above estimate. In order to do so, we begin by\nestimating the source term $\\mathcal L^\\eps$, in the next subsection. \n\\subsection{Estimating the source term}\n\\label{sec:estim-source-term}\n\n\\begin{proposition}\\label{prop:est-source}\n  Let $d= 3$, $\\si=1$, $V$ satisfying Assumption~\\ref{hyp:V}\n  with $\\mu>2$, and $u_-\\in \\Sigma^k$ for some $k\\ge\n  7$. Suppose that Assumption~\\ref{hyp:flot} is satisfied.\nLet $u\\in C(\\R;\\Sigma^k)$ given by\n  Theorem~\\ref{theo:scatt-class} and \n  Proposition~\\ref{prop:extra-u}. The source term $\\mathcal L^\\eps$ satisfies\n  \\begin{equation*}\n   \\frac{1}{\\eps} \\|\\mathcal L^\\eps(t)\\|_{L^2(\\R^3)}\\lesssim \\frac{\\sqrt\n     \\eps}{\\<t\\>^{3\/2} }\\quad\\text{and}\\quad  \\frac{1}{\\eps}\n   \\|\\mathcal L^\\eps(t)\\|_{L^{3\/2}(\\R^3)}\\lesssim \\frac{\n     \\eps^{3\/4}}{\\<t\\>^{3\/2} },\n\\quad \\forall\n    t\\in \\R.\n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\nTo ease notation, we note that\n\\begin{equation*}\n \\frac{1}{\\eps} \\mathcal L^\\eps(t,x) = \\frac{1}{\\eps^{3\/4}} {\\mathcal\n    S}^\\eps(t,y)\\Big|_{y=\\frac{x-q(t)}{\\sqrt\\eps}}\n  e^{i(S(t)+ip(t)\\cdot (x-q(t)))\/\\eps}, \n\\end{equation*}\nwhere\n\\begin{equation*}\n  {\\mathcal S}^\\eps(t,y) = \\frac{1}{\\eps}\\( V\\(q(t)+y\\sqrt \\eps\\) -V\\(q(t)\\)\n  -\\sqrt\\eps \\<\\nabla V\\(q(t)\\),y\\> -\\frac{\\eps}{2}\\<Q(t)y,y\\>\\)u(t,y).\n\\end{equation*}\nIn particular,\n\\begin{equation*}\n  \\frac{1}{\\eps}\\| \\mathcal L^\\eps(t)\\|_{L^2(\\R^3)} = \\|{\\mathcal\n    S}^\\eps(t)\\|_{L^2(\\R^3)} ,\\quad  \\frac{1}{\\eps}\\| \\mathcal\n  L^\\eps(t)\\|_{L^{3\/2}(\\R^3)} = \\eps^{1\/4}\\|{\\mathcal \n    S}^\\eps(t)\\|_{L^{3\/2}(\\R^3)}. \n\\end{equation*}\n  Taylor's formula and Assumption~\\ref{hyp:V} yield the pointwise estimate\n  \\begin{equation*}\n    | {\\mathcal S}^\\eps(t,y) | \\lesssim \\sqrt\\eps |y|^3 \\int_0^1\n    \\frac{1}{\\<q(t)+\\theta y\\sqrt \\eps\\>^{\\mu+3}}d\\theta |u(t,y)|. \n  \\end{equation*}\nTo simplify notations, we consider only positive times. Recall that\nfrom Assumption~\\ref{hyp:flot}, $p^+\\not =0$. Introduce, for\n$0<\\eta< |p^+|\/2$,\n\\begin{equation*}\n  \\Omega  = \\left\\{y\\in \\R^3,\\quad |y|\\ge \\eta\\frac{t}{\\sqrt\\eps}\\right\\}.\n\\end{equation*}\nSince $q(t)\\sim p^+ t$ as\n$t\\to \\infty$, on the complement of $\\Omega$, we can use the decay of $V$,\n\\eqref{eq:short}, to infer the pointwise estimate\n\\begin{equation}\\label{eq:Spoint}\n   | {\\mathcal S}^\\eps(t,y) | \\lesssim \\sqrt\\eps |y|^3\n    \\frac{1}{\\<t\\>^{\\mu+3}} |u(t,y)| \\quad \\text{on }\\Omega^c.\n\\end{equation}\nTaking the $L^2$-norm, we have\n\\begin{equation*}\n  \\|\\mathcal S^\\eps(t)\\|_{L^2(\\Omega^c)}\\le \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^2(\\R^3)}\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu}},\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:extra-u}. On $\\Omega$\nhowever, the argument of the potential in Taylor's formula is not\nnecessarily going to infinity, so the decay of the potential is\napparently useless. Back to the definition of $\\mathcal L^\\eps$, that is leaving\nout Taylor's formula, we see that all the terms but the first one can\nbe easily estimated on $\\Omega$. Indeed, the definition of $\\Omega$ implies\n\\begin{equation*}\n  |V(q(t))u(t,y)| \\lesssim \\frac{1}{\\<t\\>^\\mu}|u(t,y)|\\lesssim\n  \\frac{1}{\\<t\\>^\\mu} \\left| \\frac{y\\sqrt \\eps}{t}\\right|^k |u(t,y)|,\n\\end{equation*}\nwhere $k$ will be chosen shortly. Taking the $L^2$ norm, we find\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V(q(t))u(t)\\|_{L^2(\\Omega)} \\lesssim\n  \\frac{\\eps^{k\/2-1}}{\\<t\\>^{\\mu+k}}\\||y|^k u(t,y)\\|_{L^2(\\R^3)}\n  \\lesssim \\frac{\\eps^{k\/2-1}}{\\<t\\>^{\\mu}},\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:extra-u} again. Choosing\n$k=3$ yields the expected estimate. The last two terms in $\\mathcal\nL^\\eps$ can be estimated accordingly. For the first term in $\\mathcal\nL^\\eps$ however, we face the same problem as above: the argument of\n$V$ has to be considered as bounded. A heuristic argument goes as\nfollows. In view of Theorem~\\ref{theo:scatt-class},\n\\begin{equation*}\n  u(t,y) \\Eq t \\infty e^{i\\frac{t}{2}\\Delta}u_+ \\Eq t \\infty\n  \\frac{1}{t^{3\/2}}\\widehat u_+\\(\\frac{y}{t}\\)e^{i|y|^2\/(2t)},\n\\end{equation*}\nwhere the last behavior stems from standard analysis of the\nSchr\\\"odinger group (see e.g. \\cite{Rauch91}). In view of\nthe definition of $\\Omega$, we have, formally for $y\\in \\Omega$,  \n\\begin{equation*}\n  |u(t,y)|\\lesssim\n  \\frac{1}{t^{3\/2}}\\sup_{ |z|\\ge \\eta}\\left |\\widehat\n    u_+\\(\\frac{z}{\\sqrt\\eps}\\)\\right|. \n\\end{equation*}\nThen the idea is to keep the linear dispersion measured by the factor\n$t^{-3\/2}$ (which is integrable since $d=3$), and use decay properties\nfor $\\widehat u_+$ to gain powers of $\\eps$. To make this argument\nrigorous, we keep the idea \nthat $u$ must be assessed in $L^\\infty$ rather than in $L^2$, and write\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^2(\\Omega)} \\le\n  \\frac{1}{\\eps}\\|u(t)\\|_{L^\\infty(\\Omega)} \\|V\\(q(t)+y\\sqrt \\eps\\)\\|_{L^2(\\Omega)} .\n\\end{equation*}\nFor the last factor, we have\n\\begin{equation*}\n  \\|V\\(q(t)+y\\sqrt\n  \\eps\\)\\|_{L^2(\\Omega)}\\le \\eps^{-3\/4}\\|V\\|_{L^2(\\R^3)},\n\\end{equation*}\nwhere the last norm is finite since $\\mu>2$. For the $L^\\infty$ norm of\n$u$, we use Gagliardo-Nirenberg inequality and the previous\nvector-fields. To take advantage of the localization in space,\nintroduce a non-negative cut-off function $\\chi\\in C^\\infty(\\R^3)$, such that:\n\\begin{equation*}\n  \\chi(z)=\\left\\{\n    \\begin{aligned}\n      1& \\text{ if }|z|\\ge \\eta,\\\\\n0 & \\text{ if }|z|\\le\\frac{\\eta}{2}.\n    \\end{aligned}\n\\right.\n\\end{equation*}\nIn view of the definition of $\\Omega$, \n\\begin{equation*}\n   \\|u(t)\\|_{L^\\infty(\\Omega)}  \\le \\left\\|\n     \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)u(t,y)\\right\\|_{L^\\infty(\\R^3)}. \n\\end{equation*}\nNow with $B$ as defined in the proof of\nProposition~\\ref{prop:extra-u}, Gagliardo-Nirenberg inequality yields,\nfor any smooth function $f$\n (recall that $y\\in \\R^3$),\n \\begin{equation*}\n   \\|f\\|_{L^\\infty(\\R^3)}\\lesssim\n   \\frac{1}{t^{3\/2}}\\|f\\|_{L^2(\\R^3)}^{1\/4}\\|B^2(t)f\\|_{L^2(\\R^3)}^{3\/4}.\n \\end{equation*}\nWe use this inequality with\n\\begin{equation*}\n  f(t,y) = \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)u(t,y),\n\\end{equation*}\nand note that\n\\begin{equation*}\n  B(t)f (t,y)= \\chi\\(\\frac{y\\sqrt\\eps}{t}\\)B(t)u(t,y) + i\\frac{\\sqrt\n    \\eps}{t}W(t) \\nabla \\chi \\(\\frac{y\\sqrt\\eps}{t}\\) \\times u(t,y),\n\\end{equation*}\nwhere $W(t)$ stands for $W_\\pm$ or $t$. Recall that $t\\mapsto W(t)\/t$\nis bounded, so the last term is actually ``nice''. \nProceeding in the same way as above, we obtain\n\\begin{equation*}\n  \\|u(t)\\|_{L^2(\\Omega)}\\lesssim\n  \\left\\| \\left| \\frac{y\\sqrt \\eps}{t}\\right|^k\n    u(t,y)\\right\\|_{L^2(\\Omega)}\\lesssim \\eps^{k\/2},\n\\end{equation*}\nprovided that $u_-\\in \\Sigma^k$. Similarly,\n\\begin{equation*}\n  \\|B^2(t)u\\|_{L^2(\\Omega)} \\lesssim \\eps^{k\/2-1},\n\\end{equation*}\nand so \n\\begin{equation*}\n   \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^2(\\Omega)}\\lesssim \n   \\frac{1}{t^{3\/2}}\\eps^{-7\/4 +k\/8+3(k\/2-1)\/4}= \\frac{\\eps^{k\/2-5\/2}}{t^{3\/2}}.\n\\end{equation*}\nTherefore, the $L^2$ estimate follows as soon as $k \\ge 6$. For the\n$L^{3\/2}$-estimate, we resume the same computations, and use the extra\nestimate: for all $s>1\/2$, \n\\begin{equation}\\label{eq:localizing}\n  \\|f\\|_{L^{3\/2}(\\R^3)}\\lesssim \\|f\\|_{L^2(\\R^3)}^{1-1\/2s}\\||x|^sf\\|_{L^2(\\R^3)}^{1\/2s}.\n\\end{equation}\nThis estimate can easily be proven by writing\n\\begin{equation*}\n   \\|f\\|_{L^{3\/2}(\\R^3)} \\le \\|f\\|_{L^{3\/2}(|y|<R)} + \\left\\|\n     \\frac{1}{|x|^s} |x|^s f\\right\\|_{L^{3\/2}(|x|>R)},\n\\end{equation*}\nso H\\\"older inequality yields, provided that $s>1\/2$ (so that\n$y\\mapsto |y|^{-s}\\in L^6(|y|>R)$)\n\\begin{equation*}\n   \\|f\\|_{L^{3\/2}(\\R^3)} \\le \\sqrt R \\|f\\|_{L^2} + \\frac{1}{R^{s-1\/2}}\\||x|^s f\\|_{L^2},\n\\end{equation*}\nand by optimizing in $R$. Now from \\eqref{eq:Spoint}, we have\n\\begin{align*}\n  \\|\\mathcal S^\\eps(t)\\|_{L^{3\/2}(\\Omega^c)}&\\le \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^{3\/2}(\\R^d)}\\\\\n&\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu+3}}\\||y|^3u(t,y)\\|_{L^2(\\R^d)}^{1\/2}\\||y|^4u(t,y)\\|_{L^2(\\R^d)}^{1\/2}\\\\\n&\n\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{\\mu-1\/2}}\\lesssim \\frac{\\sqrt \\eps\n  }{\\<t\\>^{3\/2}}\n\\end{align*}\nwhere we have used \\eqref{eq:localizing} with $s=1$, \nProposition~\\ref{prop:extra-u}, and the fact that $\\mu>2$. \n\nOn $\\Omega$, we can repeat the computations from the $L^2$-estimate\n(up to incorporating \\eqref{eq:localizing}): for the last term, we\nnote that\n\\begin{equation*}\n  \\frac{1}{\\eps}\\|V\\(q(t)+y\\sqrt \\eps\\)u(t,y)\\|_{L^{3\/2}(\\Omega)} \\le\n  \\frac{1}{\\eps}\\|u(t)\\|_{L^\\infty(\\Omega)} \\|V\\(q(t)+y\\sqrt \\eps\\)\\|_{L^{3\/2}(\\Omega)} ,\n\\end{equation*}\nand that\n\\begin{equation*}\n  \\|V\\(q(t)+y\\sqrt\n  \\eps\\)\\|_{L^{3\/2}(\\Omega)}\\le \\eps^{-1}\\|V\\|_{L^{3\/2}(\\R^3)},\n\\end{equation*}\nwhere the last norm is finite since $\\mu>2$. Up to taking $u$ in\n$\\Sigma^7$, we conclude\n\\begin{equation*}\n   \\|\\mathcal S^\\eps(t)\\|_{L^{3\/2}(\\R^3)}\\lesssim \\frac{\\sqrt\\eps}{\\<t\\>^{3\/2}},\n\\end{equation*}\nand the proposition follows. \n\\end{proof}\n\n\\subsection{A priori estimate for the error in the critical norm}\n\\label{sec:priori-estim-error}\n\nIn this subsection, we prove:\n\\begin{proposition}\\label{prop:w-crit}\n  Under the assumptions of Theorem~\\ref{theo:cv-unif}, the error\n  $w^\\eps=\\psi^\\eps-\\varphi^\\eps$  satisfies the a priori estimate,\n  for any $\\dot H^{1\/2}$-admissible pair \n  $(q,r)$, \n  \\begin{equation*}\n    \\eps^{\\frac{1}{q}}\\|w^\\eps\\|_{L^q(\\R;L^r(\\R^3))}\\lesssim \\eps^{1\/4}. \n  \\end{equation*}\n\\end{proposition}\n\\begin{proof}\n  The reason for considering $\\dot H^{1\/2}$-admissible pairs is that\n  the cubic three-dimensional Schr\\\"odinger equation is $\\dot\n  H^{1\/2}$-critical; see e.g. \\cite{CW90}. The proof of\n  Proposition~\\ref{prop:w-crit} is then very similar to the proof of\n  \\cite[Proposition~2.3]{HoRo08}. \n\nAn important tool is the known estimate for the approximate solution\n$\\varphi^\\eps$: we have, in view of the fact\nthat $u,Bu\\in L^\\infty L^2$,\n\\begin{equation}\\label{eq:est-a-priori-phi}\n  \\|\\varphi^\\eps(t)\\|_{L^r(\\R^3)}\\lesssim\n  \\(\\frac{1}{\\<t\\>\\sqrt\\eps}\\)^{3\\(\\frac{1}{2}-\\frac{1}{r}\\)},\\quad\n  2\\le r\\le 6.\n\\end{equation}\nNote that for an $\\dot H^{1\/2}$ admissible pair, we infer\n\\begin{equation*}\n  \\|\\varphi^\\eps(t)\\|_{L^q(\\R;L^r(\\R^3))}\\lesssim\n  \\eps^{-\\frac{3}{2}\\(\\frac{1}{2}-\\frac{1}{r}\\)} = \\eps^{-\\frac{1}{q}-\\frac{1}{4}},\n\\end{equation*}\nso Proposition~\\ref{prop:w-crit} shows a $\\sqrt\\eps$ gain\nfor $w^\\eps$ compared to $\\varphi^\\eps$, which is the order of\nmagnitude we eventually prove in $L^\\infty L^2$, and stated in\nTheorem~\\ref{theo:cv-unif}.  \nLet $0<\\eta\\ll 1$, and set\n\\begin{equation*}\n  \\|w^\\eps\\|_{\\mathcal N^\\eps(I)} :=\\sup_{{(q,r)\\ \\dot\n      H^{1\/2}-\\text{admissible}}\\atop 3\\le r\\le\n    6-\\eta}\\eps^{\\frac{1}{q}}\\|w^\\eps\\|_{L^q(I;L^r(\\R^3)}. \n\\end{equation*}\nDuhamel's formula for \\eqref{eq:restecrit} reads, given $w^\\eps_{\\mid\n  t=-\\infty}=0$,\n\\begin{equation*}\n  w^\\eps(t) =-i\\eps^{3\/2} \\int_{-\\infty}^t\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\n+i\\eps^{-1} \\int_{-\\infty}^t  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds. \n\\end{equation*}\nSince we have the point-wise estimate\n\\begin{equation*}\n  \\left|\n    |\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\right|\\lesssim\n  \\(|w^\\eps|^2+|\\varphi^\\eps|^2\\)|w^\\eps|, \n\\end{equation*}\nLemma~\\ref{lem:stri-inhom-eps} yields, with\n$(q_2,r_2)=(\\frac{10}{7},5)$ for the first term of the right hand\nside, and with $(q_2,r_2)=(2,3)$ for the second term,\n\\begin{align*}\n  \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}&\\lesssim \\eps^{3\/2-7\/10}\\left\\|\n    \\(|w^\\eps|^2+|\\varphi^\\eps|^2\\)w^\\eps\\right\\|_{L^{10\/3}_tL^{5\/4}}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}\\\\\n&\\lesssim \\eps^{4\/5}\\( \\|w^\\eps\\|_{L^{20}_tL^{10\/3}}^2 +\n\\|\\varphi^\\eps\\|_{L^{20}_tL^{10\/3}}^2 \\)\n\\|w^\\eps\\|_{L^{5}_tL^{5}}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}},\n\\end{align*}\nwhere we have used H\\\"older inequality. Note that the pairs\n$(20,\\frac{10}{3})$ and $(5,5)$ are $\\dot H^{1\/2}$-admissible. Denote\nby \n\\begin{equation*}\n  \\omega(t) =\\frac{1}{\\<t\\>^{3\/5}}. \n\\end{equation*}\nThis function obviously belongs to $L^{20}(\\R)$. \nThe estimate \\eqref{eq:est-a-priori-phi} and the definition of the\nnorm $\\mathcal N^\\eps$ yield\n\\begin{equation*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3 + \n\\|\\omega\\|_{L^{20}(-\\infty,t)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}.\n\\end{equation*}\nTaking $t\\ll -1$, we infer\n\\begin{equation*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3 \n  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2_t L^{3\/2}}\\lesssim \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t)}^3  + \\eps^{1\/4},\n\\end{equation*}\nwhere we have use Proposition~\\ref{prop:est-source}. We can now use a\nstandard bootstrap argument, as recalled in Section~\\ref{sec:class}.\nWe infer that for $t_1\\ll -1$,\n\\begin{equation*}\n    \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t_1)}\\lesssim \\eps^{1\/4}.\n\\end{equation*}\nUsing Duhamel's formula again, we have\n\\begin{align*}\n    U_V^\\eps(t-t_1)w^\\eps(t_1) &=-i\\eps^{3\/2} \\int_{-\\infty}^{t_1}\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\\\\\n&+i\\eps^{-1} \\int_{-\\infty}^{t_1}  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds,\n\\end{align*}\nso we infer\n\\begin{align*}\n    \\| U_V^\\eps(t-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(\\R)}& \\lesssim  \\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(-\\infty,t_1)}^3 + \n\\|\\omega\\|_{L^{20}(-\\infty,t_1)}^2  \\|w^\\eps\\|_{\\mathcal\n                                                           N^\\eps(-\\infty,t_1)}\\\\\n& \\quad  + \\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2((-\\infty,t_1]; L^{3\/2})}\\\\\n&\\le\nC_0\\eps^{1\/4}.\n\\end{align*}\nWe now rewrite Duhamel's formula with some initial time $t_j$:\n\\begin{align*}\n  w^\\eps(t) &= U_V^\\eps(t-t_j)w^\\eps(t_j)-i\\eps^{3\/2} \\int_{t_j}^t\n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\\\\\n&\\quad\n+i\\eps^{-1} \\int_{t_j}^t  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds. \n\\end{align*}\nFor $t\\ge t_j$ and $I=[t_j,t]$, the same estimates as above yield\n\\begin{align*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}&\\le  \\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I)}\n  + C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}^3 + \nC\\|\\omega\\|_{L^{20}(I)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(I)}\\\\\n&  \\quad+ C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I; L^{3\/2})},\n\\end{align*}\nwhere the above constant $C$ is independent of $\\eps, t_j$ and $t$. We\nsplit $\\R_t$ into finitely many intervals\n\\begin{equation*}\n  \\R = (-\\infty,t_1]\\cup \\bigcup_{j=1}^N [t_j,t_{j+1}]\\cup\n[t_N,\\infty)=:\\bigcup_{j=0}^{N+1} I_j, \n\\end{equation*}\n on which \n\\begin{equation*}\n  C\\|\\omega\\|_{L^{20}(I_j)}^2 \\le\\frac{1}{2},\n\\end{equation*}\nso that we have\n\\begin{align*}\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}&\\le   2\\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I_j)}\n  +2 C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3  +\n  2 C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I_j; L^{3\/2})}\\\\\n&\\le  2\\|U_V^\\eps(\\cdot-t_j)w^\\eps(t_j)\\|_{\\mathcal N^\\eps(I_j)}\n  +2 C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3  + \\tilde C\n  \\eps^{1\/4}\\left\\|\\<t\\>^{-3\/2}\\right\\|_{L^2(I_j)}, \n\\end{align*}\nwhere we have used Proposition~\\ref{prop:est-source} again. Since we\nhave\n\\begin{equation*}\n   \\| U_V^\\eps(t-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(\\R)}\\le C_0\\eps^{1\/4},\n\\end{equation*}\nthe bootstrap argument shows that at least for $\\eps\\le \\eps_1$\n($\\eps_1>0$),\n\\begin{equation*}\n    \\|w^\\eps\\|_{\\mathcal N^\\eps(I_1)}\\le 3\n    \\|U_V^\\eps(\\cdot-t_1)w^\\eps(t_1)\\|_{\\mathcal N^\\eps(I_1)} + \\frac{3}{2} \\tilde C\n  \\eps^{1\/4}\\left\\|\\<t\\>^{-3\/2}\\right\\|_{L^2(I_1)}.\n\\end{equation*}\nOn the other hand, Duhamel's formula implies\n\\begin{align*}\n  U_V^\\eps(t-t_{j+1})w^\\eps(t_{j+1}) &=\n                                       U_V^\\eps(t-t_j)w^\\eps(t_j)\n+i\\eps^{-1} \\int_{t_j}^{t_{j+1}}  U_V^\\eps(t-s)\\mathcal L^\\eps(s)ds\\\\\n&\\quad -i\\eps^{3\/2}\n                                       \\int_{t_j}^{t_{j+1}} \n  U_V^\\eps(t-s)\\(|\\psi^\\eps|^2\\psi^2-|\\varphi^\\eps|^2\\varphi^\\eps\\)(s)ds\n. \n\\end{align*}\nTherefore, we infer\n\\begin{align*}\n    \\| U_V^\\eps(t-t_{j+1})w^\\eps(t_{j+1})\\|_{\\mathcal N^\\eps(\\R)}&\\le\n    \\|U_V^\\eps(t-t_{j})w^\\eps(t_{j})\\|_{\\mathcal N^\\eps(\\R)}+ \n  + C\\sqrt\\eps\n   \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}^3 \\\\\n&  \\quad + \nC\\|\\omega\\|_{L^{20}(I_j)}^2  \\|w^\\eps\\|_{\\mathcal N^\\eps(I_j)}\n+ C\\eps^{-3\/2}\\|\\mathcal L^\\eps\\|_{L^2(I_j; L^{3\/2})}.\n\\end{align*}\nBy induction (carrying over finitely many steps), we conclude\n\\begin{equation*}\n   \\|U_V^\\eps(t-t_{j})w^\\eps(t_{j})\\|_{\\mathcal N^\\eps(\\R)}\n   =\\O\\(\\eps^{1\/4}\\),\\quad 0\\le j\\le N+1,\n\\end{equation*}\nand $\\|w^\\eps\\|_{\\mathcal N^\\eps(\\R)}=\\O\\(\\eps^{1\/4}\\)$ as announced. \n\\end{proof}\n\n\\subsection{End of the argument}\n\\label{sec:end-argument}\n\nResume the estimate \\eqref{eq:w-presque} with the $L^2$-admissible\npair $(q_1,r_1)= (\\frac{8}{3},4)$:\n\\begin{equation*}\n   \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}_t L^{4}} \\lesssim\n  \\eps^{3\/4}\\( \\|w^\\eps\\|^2_{L^8_t L^4}+ \\|\\varphi^\\eps\\|^2_{L^8_t\n    L^4}\\) \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}_tL^4}  +\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1_tL^2}.\n\\end{equation*}\nFrom Proposition~\\ref{prop:w-crit} (the pair $(8,4)$ is $\\dot H^{1\/2}$-admissible),\n\\begin{equation*}\n  \\|w^\\eps\\|_{L^8(\\R; L^4)}\\lesssim \\eps^{1\/8},\n\\end{equation*}\nand we have seen in the course of the proof that\n\\begin{equation*}\n  \\|\\varphi^\\eps\\|_{L^8(\\R; L^4)}\\lesssim \\eps^{-3\/8}.\n\\end{equation*}\nTherefore, we can split $\\R_t$ into finitely many intervals, in a way\nwhich is independent of $\\eps $, so that \n\\begin{equation*}\n   \\eps^{3\/4}\\( \\|w^\\eps\\|^2_{L^8(I; L^4)}+ \\|\\varphi^\\eps\\|^2_{L^8(I;\n    L^4)}\\)\\le \\eta \n\\end{equation*}\non each of these intervals, with $\\eta$ so small that we infer\n\\begin{equation*}\n   \\eps^{3\/8}\\|w^\\eps\\|_{L^{8\/3}(\\R; L^{4})} \\lesssim\n  \\frac{1}{\\eps}\\|\\mathcal L^\\eps\\|_{L^1(\\R;L^2)}\\lesssim \\sqrt\\eps,\n\\end{equation*}\nwhere we have used Proposition~\\ref{prop:est-source}. Plugging this\nestimate into  \\eqref{eq:w-presque} and now taking $(q_1,r_1)$,\nTheorem~\\ref{theo:cv-unif} follows.\n\n\n\n\n\n\\section{Superposition}\n\\label{sec:superp}\n\nIn this section, we sketch the proof of\nCorollary~\\ref{cor:decoupling}. This result heavily relies on the\n(finite time) superposition \n  principle established in \\cite{CaFe11}, in the case of two initial\n  coherent states with different centers in phase space. We present\n  the argument in the case of two initial wave packets, and explain\n  why it can be generalized to any finite number of initial coherent\n  states. \n\\smallbreak\n\nFollowing the proof of\n\\cite[Proposition~1.14]{CaFe11}, we introduce the approximate\nevolution of each individual initial wave packet:\n\\begin{equation*}\n   \\varphi_j^\\eps(t,x)=\\eps^{-3\/4} u_j\n\\left(t,\\frac{x-q_j(t)}{\\sqrt\\eps}\\right)e^{i\\left(S_j(t)+p_j(t)\\cdot\n    (x-q_j(t))\\right)\/\\eps},\n\\end{equation*}\nwhere $u_j$ solves \\eqref{eq:u} with initial datum $a_j$. In the proof\nof \\cite[Proposition~1.14]{CaFe11}, the main remark is that all that\nis needed is the control of a new source term, corresponding to the\ninteractions of the approximate solutions. Set \n\\begin{equation*}\n  w^\\eps = \\psi^\\eps -\\varphi_1^\\eps-\\varphi_2^\\eps.\n\\end{equation*}\nIt solves \n\\begin{equation*}\n   i\\eps\\d_t w^\\eps +\\frac{\\eps^2}{2}\\Delta w^\\eps = Vw^\\eps -\\mathcal\n  L^\\eps+\\mathcal N_I^\\eps+ \\mathcal N_s^\\eps\\quad ;\\quad w^\\eps_{\\mid t=0}=0,\n\\end{equation*}\nwhere the linear source term is the same as in Section~\\ref{sec:cv}\n(except than now we consider the sums of two such terms), $\\mathcal\nN_s^\\eps$ is the semilinear term\n\\begin{equation*}\n  \\mathcal N_s^\\eps =\\eps^{5\/2}\\( |w^\\eps+\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (w^\\eps+\n  \\varphi_1^\\eps+\\varphi_2^\\eps) - |\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (\n  \\varphi_1^\\eps+\\varphi_2^\\eps)\\),\n\\end{equation*}\nand $\\mathcal N_I^\\eps$ is precisely the new interaction term,\n\\begin{equation*}\n   \\mathcal N_I^\\eps =\\eps^{5\/2}\\( |\n  \\varphi_1^\\eps+\\varphi_2^\\eps|^2 (\n  \\varphi_1^\\eps+\\varphi_2^\\eps) - |\n  \\varphi_1^\\eps|^2 \\varphi_1^\\eps-|\n  \\varphi_2^\\eps|^2  \\varphi_2^\\eps\\).\n\\end{equation*}\nIn \\cite{CaFe11}, it is proven that if $(q_{01},p_{01})\\not\n=(q_{02},p_{02})$, then the possible interactions between\n$\\varphi_1^\\eps$ and $\\varphi_2^\\eps$ are negligible on every finite\ntime interval, in the sense that\n\\begin{equation*}\n  \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps\\|_{L^1(0,T;L^2)}\\le C(T,\\gamma) \\eps^\\gamma,\n\\end{equation*}\nfor every $\\gamma<1\/2$. We infer that\n$\\|w^\\eps\\|_{L^\\infty(0,T;L^2)}=\\O(\\eps^\\gamma)$ for every $T>0$. For\n$t\\ge T$, we have\n\\begin{align*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^2}& \\lesssim\n  \\sum_{\\ell_1,\\ell_2\\ge 1,\\ \\ell_1+\\ell_2 =3}\\left\\|\n  u_1^{\\ell_1}\\(t,y-\\frac{q_1(t)-q_2(t)}{\\sqrt\\eps } \\)\n  u_2^{\\ell_2}(t,y)\\right\\|_{L^2}\\\\\n&\\lesssim\n  \\sum_{\\ell_1,\\ell_2\\ge 1,\\ \\ell_1+\\ell_2 =3}\\|\n  u_1(t)\\|_{L^\\infty}^{\\ell_1} \n\\| u_2(t)\\|_{L^\\infty}^{\\ell_2-1} \n\\|  u_2(t)\\|_{L^2}\\lesssim \\frac{1}{t^3}.\n\\end{align*}\nSimilarly, resuming the same estimates as in the proof of\nProposition~\\ref{prop:est-source}, \n\\begin{equation*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^{3\/2}}\\lesssim\n   \\frac{\\eps^{1\/4}}{t^{5\/2}}. \n\\end{equation*}\nBy resuming the proof of Theorem~\\ref{theo:cv-unif} on the time\ninterval $[T,\\infty)$, we infer\n\\begin{equation*}\n  \\|w^\\eps\\|_{L^\\infty(0,\\infty;L^2)}\\le C(T,\\gamma)\\eps^\\gamma + \\frac{C}{T^2}.\n\\end{equation*}\nTherefore, \n\\begin{equation*}\n  \\limsup_{\\eps\\to 0}  \\|w^\\eps\\|_{L^\\infty(0,\\infty;L^2)}\\lesssim \\frac{1}{T^2},\n\\end{equation*}\nfor all $T>0$, hence the result by letting $T\\to \\infty$. \n\\smallbreak\n\nIn the case of more than two initial coherent states, the idea is that\nthe nonlinear interaction term, $\\mathcal N_I^\\eps$, always contains\nthe product of two approximate solutions corresponding to different\ntrajectories in phase space. This is enough for the proof of\n\\cite[Proposition~1.14]{CaFe11} to go through: we always have\n\\begin{align*}\n   \\frac{1}{\\eps} \\| \\mathcal N_I^\\eps(t)\\|_{L^2}& \\\\\n\\lesssim\n  \\sum_{{j\\not =k, \\ \\ell_j,\\ell_k\\ge 1}\\atop {\\ell_j+\\ell_k+\\ell_m =3}}&\\left\\|\n  u_j^{\\ell_j}\\(t,y-\\frac{q_j(t)-q_k(t)}{\\sqrt\\eps } \\)\n  u_k^{\\ell_k}(t,y)u_m^{\\ell_m}\\(t,y-\\frac{q_m(t)-q_k(t)}{\\sqrt\\eps } \\)\\right\\|_{L^2}\\\\\n\\lesssim\n  \\sum_{{j\\not =k,\\  \\ell_j,\\ell_k\\ge 1}\\atop {\\ell_j+\\ell_k+\\ell_m\n  =3}}&\\|u_m(t)\\|_{L^\\infty}^{\\ell_m}\\left\\| \n  u_j^{\\ell_j}\\(t,y-\\frac{q_j(t)-q_k(t)}{\\sqrt\\eps } \\)\n  u_k^{\\ell_k}(t,y)\\right\\|_{L^2},\n\\end{align*}\nso the last factor is exactly the one considered in \\cite{CaFe11} and\n  above. \n\\subsection*{Acknowledgements}\nThe author is grateful to Jean-Fran\\c cois Bony, Clotilde Fermanian,\nIsabelle Gallagher and Fabricio Maci\\`a for fruitful discussions about\nthis work. \n\n\\bibliographystyle{siam}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{Intro}Introduction}\n\nOver four decades passed since it was first shown that plasmas and\nbeam-plasma systems immersed in an external magnetic field can\nsupport travelling electromagnetic waves with specific features.\nThese waves propagate parallel to the applied magnetic field being\ncircularly polarized in a plane transverse to the direction of\npropagation. It has become conventional in the physics of\nmagnetized plasmas to call such structures waves in the whistler\nmode.\n\nAlthough the linear stability properties of the electromagnetic\nwaves in the whistler mode are relatively well studied\n\\cite{Weibel,Neufeld,Bell,Sudan}, there is a serious gap in the\nunderstanding of their nonlinear behaviour. Chen et al.\n\\cite{Wurtele} have shown that electromagnetic whistler waves can\nbe considered as complementary to the nonlinear travelling\nelectrostatic waves, known as the Bernstein-Greene-Kruskal (BGK)\nmodes \\cite{BGK}. While the BGK modes are longitudinal, the\nwhistler modes are transverse, in other words, the components of\nthe electric and magnetic field of the whistler wave parallel to\nthe external magnetic field are both zero. The study of the\nnonlinear behaviour of whistler waves has been initiated by\nTaniuti and Washimi \\cite{Taniuti}, who obtained a nonlinear\nSchr\\\"{o}dinger equation for the slowly varying amplitude (see\nalso Reference \\cite{Shukla}).\n\nThe present paper is aimed at filling the gap in the understanding\nof the nonlinear evolution of whistler waves. The method adopted\nhere is the renormalization group (RG) method \\cite{Oono,Tzenov}.\nThe basic feature of this approach is that it provides a\nconvenient and straightforward tool to obtain an adequate\ndescription of the physically essential properties of\nself-organization and formation of patterns in complex systems.\nCoherent structures which result from the nonlinear interaction\nbetween plane waves evolve on time and\/or spatial scales\ncomparatively large compared to those the fast oscillations occur.\nThe RG method can be considered as a powerful systematic procedure\nto separate the relatively slow dynamics from the fast one, which\nis of no considerable physical relevance.  In a context similar to\nthat of the present paper, it has been successfully applied by one\nof the authors \\cite{Tzenov,Tzenov1} to study collective effects\nin intense charged-particle beams.\n\nThe paper is organized as follows. In the next section, we state\nthe basic equations which will be the subject of the\nrenormalization group reduction in section III. Starting from a\nsingle equation [see equation (\\ref{Waveeqallord})] for the\nelectromagnetic vector potential, we obtain a formal perturbation\nexpansion of its solution to second order. As expected, it\ncontains secular terms proportional to powers of the time variable\nwhich is the only renormalization parameter adopted in our\napproach. In section IV, the arbitrary constant amplitudes of the\nperturbation expansion are renormalized such as to eliminate the\nsecular terms. As a result, a set of equations for the\nrenormalized slowly varying amplitudes is obtained, known as the\nrenormalization group equations (RGEs). These equations comprise\nan infinite system of coupled nonlinear Schr\\\"{o}dinger equations.\nIn section V, the latter are analyzed in the simplest case.\nFinally, section VI is dedicated to discussion and conclusions.\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\setcounter{equation}{0}\n\n\\section{Formulation of the Problem and Basic Equations}\n\nPlasmas and beam-plasma systems considered in the present paper\nare assumed to be weakly collisional. Therefore, the dynamics of\nplasma species is well described by the hydrodynamic equations\ncoupled with the equations for the electromagnetic self-fields. We\nstart with the equations for plasma in an external constant\nmagnetic field ${\\bf B}_0$, which can be written as follows\n\\begin{equation}\n{\\frac {\\partial n_a} {\\partial t}} + \\nabla \\cdot {\\left( n_a\n{\\bf V}_a \\right)} = 0, \\label{Continuity}\n\\end{equation}\n\\begin{equation}\n{\\frac {{\\rm D}_a {\\bf V}_a} {{\\rm D} t}} = - {\\frac {k_B T_a}\n{m_a n_a}} \\nabla n_a + {\\frac {e q_a} {m_a}} {\\left[ {\\bf E} +\n{\\bf V}_a \\times {\\left( {\\bf B}_0 + {\\bf B} \\right)} \\right]},\n\\label{Mombalance}\n\\end{equation}\nwhere $n_a$ and ${\\bf V}_a$ are the density and the current\nvelocity of the species $a$. Furthermore, $m_a$, $q_a$ and $T_a$\nare the mass, the relative charge and the temperature,\nrespectively, while $k_B$ is the Boltzmann constant. The\nsubstantional derivative on the left-hand-side of equation\n(\\ref{Mombalance}) is defined as\n\\begin{equation}\n{\\frac {{\\rm D}_a} {{\\rm D} t}} = {\\frac {\\partial} {\\partial t}}\n+ {\\bf V}_a \\cdot \\nabla. \\label{Substant}\n\\end{equation}\nThe electromagnetic self-fields ${\\bf E}$ and ${\\bf B}$ can be\nobtained in terms of the electromagnetic vector ${\\bf A}$ and\nscalar $\\varphi$ potentials according to the well-known relations\n\\begin{equation}\n{\\bf E} = - \\nabla \\varphi - {\\frac {\\partial {\\bf A}} {\\partial\nt}}, \\qquad {\\bf B} = \\nabla \\times {\\bf A}. \\label{Elmagfield}\n\\end{equation}\nThe latter satisfy the wave equations\n\\begin{equation}\n\\Box {\\bf A} = - \\mu_0 e \\sum \\limits_a n_a q_a {\\bf V}_a, \\qquad\n\\Box \\varphi = - {\\frac {e} {\\epsilon_0}} \\sum \\limits_a n_a q_a,\n\\label{Waveequatvec}\n\\end{equation}\nin the Lorentz gauge\n\\begin{equation}\n{\\frac {1} {c^2}} {\\frac {\\partial \\varphi} {\\partial t}} + \\nabla\n\\cdot {\\bf A} = 0. \\label{Lorgauge}\n\\end{equation}\nHere $\\Box$ denotes the well-known d'Alembert operator. In what\nfollows, we consider the case of a quasineutral plasma\n\\begin{equation}\n\\sum \\limits_a n_a q_a = 0, \\label{Quasineutr}\n\\end{equation}\nin a constant external magnetic field along the $x$-axis ${\\bf\nB}_0 = {\\left( B_0, 0, 0 \\right)}$. Then, equations\n(\\ref{Continuity})--(\\ref{Lorgauge}) possess a stationary solution\n\\begin{equation}\nn_a = n_{a0} = {\\rm const}, \\quad {\\bf V}_a = 0, \\quad {\\bf A} =\n0, \\quad \\varphi = 0. \\label{Statsol}\n\\end{equation}\nThe frequency of the wave will be taken as much higher than the\nion-cyclotron frequency. Therefore, we can further neglect the ion\nmotion and scale the hydrodynamic and field variables as\n\\begin{equation}\nn_e = n_0 + \\epsilon N, \\quad {\\bf V}_e = \\epsilon {\\bf V}, \\quad\n{\\bf A} \\longrightarrow \\epsilon {\\bf A}, \\quad \\varphi\n\\longrightarrow \\epsilon \\varphi, \\label{Scale}\n\\end{equation}\nwhere $\\epsilon$ is a formal small parameter introduced for\nconvenience, which will be set equal to one at the end of the\ncalculations. Thus, the basic equations to be used for the\nsubsequent analysis can be written in the form\n\\begin{equation}\n{\\frac {\\partial N} {\\partial t}} + n_0 \\nabla \\cdot {\\bf V} +\n\\epsilon \\nabla \\cdot {\\left( N {\\bf V} \\right)} = 0,\n\\label{Continuit}\n\\end{equation}\n\\begin{equation}\n{\\frac {\\partial {\\bf V}} {\\partial t}} + \\epsilon {\\bf V} \\cdot\n\\nabla {\\bf V} = - {\\frac {k_B T} {m {\\left( n_0 + \\epsilon N\n\\right)}}} \\nabla N \\nonumber\n\\end{equation}\n\\begin{equation}\n- {\\frac {e} {m}} {\\left[ {\\bf E} + {\\bf V} \\times {\\left( {\\bf\nB}_0 + \\epsilon {\\bf B} \\right)} \\right]}, \\label{Mombalanc}\n\\end{equation}\n\\begin{equation}\n\\Box {\\bf A} = \\mu_0 e {\\left( n_0 + \\epsilon N \\right)} {\\bf V},\n\\qquad {\\frac {1} {c^2}} {\\frac {\\partial \\varphi} {\\partial t}} +\n\\nabla \\cdot {\\bf A} = 0. \\label{Waveequat}\n\\end{equation}\nBefore we continue with the renormalization group reduction of the\nsystem of equations (\\ref{Continuit})--(\\ref{Waveequat}) in the\nnext section, let us assume that the actual dependence of the\nquantities $N$, ${\\bf V}$, ${\\bf A}$ and $\\varphi$ on the spatial\nvariables is represented by the expression\n\\begin{equation}\n{\\widehat{\\Psi}} = {\\widehat{\\Psi}} {\\left( {\\bf x}, {\\bf X}; t\n\\right)}, \\qquad {\\widehat{\\Psi}} = {\\left( N, {\\bf V}, {\\bf A},\n\\varphi \\right)}, \\label{Actualdep}\n\\end{equation}\nwhere ${\\bf X} = \\epsilon {\\bf x}$ is a slow spatial variable.\nThus, the only renormalization parameter left at our disposal is\nthe time $t$ which will prove extremely convenient and simplify\ntedious algebra in the sequel.\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\setcounter{equation}{0}\n\n\\section{Renormalization Group Reduction of the Magnetohydrodynamic\nEquations}\n\nFollowing the standard procedure of the renormalization group\nmethod, we represent ${\\widehat{\\Psi}}$ as a perturbation\nexpansion\n\\begin{equation}\n{\\widehat{\\Psi}} = \\sum \\limits_{n=0}^{\\infty} \\epsilon^n\n{\\widehat{\\Psi}}_n, \\label{Perturbexp}\n\\end{equation}\nin the formal small parameter $\\epsilon$. The next step consists\nin expanding the system of hydrodynamic and field equations\n(\\ref{Continuit})-(\\ref{Waveequat}) in the small parameter\n$\\epsilon$, and obtaining their naive perturbation solution order\nby order. Note that in all orders the perturbation equations\nacquire the general form\n\\begin{equation}\n{\\frac {\\partial N_n} {\\partial t}} + n_0 \\nabla \\cdot {\\bf V}_n =\n\\alpha_n, \\label{Continuitn}\n\\end{equation}\n\\begin{equation}\n{\\frac {\\partial {\\bf V}_n} {\\partial t}} = - {\\frac {v_T^2}\n{n_0}} \\nabla N_n - {\\frac {e} {m}} {\\bf E}_n - \\omega_c {\\bf V}_n\n\\times {\\bf e}_x + {\\bf W}_n, \\label{Mombalancn}\n\\end{equation}\n\\begin{equation}\n\\Box {\\bf A}_n = \\mu_0 e n_0 {\\bf V}_n + {\\bf U}_n, \\qquad {\\frac\n{1} {c^2}} {\\frac {\\partial \\varphi_n} {\\partial t}} + \\nabla\n\\cdot {\\bf A}_n = \\beta_n, \\label{Waveequatn}\n\\end{equation}\nwhere $\\alpha_n$, $\\beta_n$, ${\\bf U}_n$ and ${\\bf W}_n$ are\nquantities, that have been already determined from previous\norders. Here\n\\begin{equation}\nv_T^2 = {\\frac {k_B T} {m}}, \\qquad \\omega_c = {\\frac {e B_0} {m}}\n\\label{Parameters}\n\\end{equation}\nare the thermal velocity of electrons and the electron-cyclotron\nfrequency, respectively and ${\\bf e}_x = {\\left( 1, 0, 0 \\right)}$\nis the unit vector in the $x$-direction. Manipulating in an\nobvious manner equations (\\ref{Continuitn})--(\\ref{Waveequatn}),\nit is possible to obtain a single equation for ${\\bf A}_n$. The\nlatter reads as\n\\begin{equation}\n\\Box {\\frac {\\partial^2 {\\bf A}_n} {\\partial t^2}} - v_T^2 \\Box\n\\nabla {\\left( \\nabla \\cdot {\\bf A}_n \\right)} + \\omega_c \\Box\n{\\frac {\\partial {\\bf A}_n} {\\partial t}} \\times {\\bf e}_x\n\\nonumber\n\\end{equation}\n\\begin{equation}\n- {\\frac {\\omega_p^2} {c^2}} {\\frac {\\partial^2 {\\bf A}_n}\n{\\partial t^2}} + \\omega_p^2 \\nabla {\\left( \\nabla \\cdot {\\bf A}_n\n\\right)} = \\mu_0 e n_0 {\\frac {\\partial {\\bf W}_n} {\\partial t}} +\n{\\frac {\\partial^2 {\\bf U}_n} {\\partial t^2}} \\nonumber\n\\end{equation}\n\\begin{equation}\n- \\mu_0 e v_T^2 \\nabla \\alpha_n - v_T^2 \\nabla {\\left( \\nabla\n\\cdot {\\bf U}_n \\right)} + \\omega_c {\\frac {\\partial {\\bf U}_n}\n{\\partial t}} \\times {\\bf e}_x + \\omega_p^2 \\nabla \\beta_n,\n\\label{Waveeqallord}\n\\end{equation}\nwhere\n\\begin{equation}\n\\omega_p^2 = {\\frac {e^2 n_0} {\\epsilon_0 m}}, \\label{Plasmafreq}\n\\end{equation}\nis the electron plasma frequency. Note that the thermal velocity\n$v_T$ as defined by equation (\\ref{Parameters}) can be\nalternatively expressed according to the expression\n\\begin{equation}\nv_T = \\omega_p r_D, \\qquad r_D^2 = {\\frac {\\epsilon_0 k_B T} {e^2\nn_0}}, \\label{Thermvel}\n\\end{equation}\nwhere $r_D$ is the electron Debye radius. Equation\n(\\ref{Waveeqallord}) represents the starting point for the\nrenormalization group reduction, the final goal of which is to\nobtain a description of the relatively slow dynamics leading to\nformation of patterns and coherent structures.\n\nLet us proceed order by order. We assume that the dependence on\nthe fast spatial variables ${\\bf x} = {\\left( x, y, z \\right)}$ is\nthrough the longitudinal (parallel to the external magnetic field\n${\\bf B}_0$) $x$-coordinate only. The solution to the zero-order\nperturbation equations (\\ref{Waveeqallord}) can be written as\n\\begin{equation}\n{\\bf A}_0 = \\sum \\limits_{k} {\\bf A}_{k}^{(0)} {\\cal A}_{k} {\\rm\ne}^{i \\psi_{k}}, \\label{Zeroordera}\n\\end{equation}\nwhere\n\\begin{equation}\n\\psi_{k} {\\left( x; t \\right)} = k x - \\omega_{k} t, \\label{Phase}\n\\end{equation}\nand ${\\cal A}_{k}$ is an infinite set of constant complex\namplitudes, which will be the subject of the renormalization\nprocedure in the sequel. Here \"constant\" means that the amplitudes\n${\\cal A}_{k}$ do not depend on the fast spatial variable $x$ and\non the time $t$, however, it can depend on the slow spatial\nvariables ${\\bf X}$. The summation sign in equation\n(\\ref{Zeroordera}) and throughout the paper implies summation over\nthe wave number $k$ in the case where it takes discrete values, or\nintegration in the continuous case. From the dispersion equation\n\\begin{equation}\n{\\cal D} {\\left( k; \\omega_{k} \\right)} = \\omega_k^2 {\\left[\n\\omega_k^2 {\\left( \\Box_k - {\\frac {\\omega_p^2} {c^2}} \\right)}^2\n- \\omega_c^2 \\Box_k^2 \\right]} = 0, \\label{Disperequat}\n\\end{equation}\nit follows that the wave frequency $\\omega_{k}$ can be expressed\nin terms of the wave number $k$, where the Fourier-image $\\Box_k$\nof the d'Alembert operator can be written according to\n\\begin{equation}\n\\Box_{k} = {\\frac {\\omega_{k}^2} {c^2}} - k^2. \\label{Dalembert}\n\\end{equation}\nMoreover, it can be verified in a straightforward manner that the\nconstant vector ${\\bf A}_{k}^{(0)}$ can be expressed as\n\\begin{equation}\n{\\bf A}_{k}^{(0)} = {\\left( 0, 1, -i {\\rm sgn} (k) \\right)},\n\\label{Constvect}\n\\end{equation}\nwhere ${\\rm sgn} (k)$ is the well-known sign-function. Details\nconcerning the derivation of the dispersion law\n(\\ref{Disperequat}) and equation (\\ref{Constvect}) can be found in\nthe Appendix. Note that equation (\\ref{Constvect}) is an\nalternative representation of the solvability condition\n(\\ref{Appsolcond}). It is important to emphasize that\n\\begin{equation}\n\\omega_{-k} = - \\omega_k, \\qquad {\\cal A}_{-k} = {\\cal\nA}_{k}^{\\ast}, \\label{Importnote}\n\\end{equation}\nwhere the asterisk denotes complex conjugation. The latter assures\nthat the vector potential as defined by equation\n(\\ref{Zeroordera}) is a real quantity. The zero-order current\nvelocity ${\\bf V}_0$ obtained directly from the first equation\n(\\ref{Waveequatn}) can be written as\n\\begin{equation}\n{\\bf V}_0 = \\sum \\limits_{k} {\\bf V}_{k}^{(0)} {\\cal A}_{k} {\\rm\ne}^{i \\psi_{k}}, \\qquad {\\bf V}_{k}^{(0)} = {\\frac {\\Box_{k}}\n{\\mu_0 e n_0}} {\\bf A}_{k}^{(0)}. \\label{Zeroorderv}\n\\end{equation}\nIn addition, the zero-order density, scalar potential and magnetic\nfield are represented by the expressions\n\\begin{equation}\nN_0 \\equiv 0, \\qquad \\varphi_0 \\equiv 0, \\qquad {\\bf B}_0 = \\sum\n\\limits_{k} {\\bf B}_{k}^{(0)} {\\cal A}_{k} {\\rm e}^{i \\psi_{k}},\n\\label{Zeroordern}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bf B}_{k}^{(0)} = -k {\\bf A}_{k}^{(0)} {\\rm sgn} (k) = {\\left(\n0, -k {\\rm sgn} (k), ik \\right)}. \\label{Zeroorderb}\n\\end{equation}\n\nIt has been mentioned that the first-order \"source terms\" on the\nright-hand-side of equation (\\ref{Waveeqallord}) can be expressed\nvia quantities already known from zero order. Thus, we have\n\\begin{equation}\n\\alpha_1 = - n_0 {\\widehat{\\nabla}} \\cdot {\\bf V}_0, \\qquad\n\\beta_1 = - {\\widehat{\\nabla}} \\cdot {\\bf A}_0,\n\\label{Firstordalp}\n\\end{equation}\n\\begin{equation}\n{\\bf U}_1 = - 2 \\nabla \\cdot {\\widehat{\\nabla}} {\\bf A}_0, \\qquad\n{\\bf W}_1 = - {\\frac {e} {m}} {\\bf V}_0 \\times {\\bf B}_0,\n\\label{Firstordu}\n\\end{equation}\nwhere the shorthand notation\n\\begin{equation}\n{\\widehat{\\nabla}} = {\\frac {\\partial} {\\partial {\\bf X}}}\n\\label{Firstorddef}\n\\end{equation}\nhas been introduced. Note that the vector ${\\bf W}_1$ representing\nthe zero-order Lorentz force has the only nonzero component along\nthe external magnetic field, that is\n\\begin{equation}\n{\\bf W}_1 = {\\bf e}_x \\sum \\limits_{k,l} \\alpha_{kl} {\\cal A}_k\n{\\cal A}_l {\\rm e}^{i {\\left( \\psi_k + \\psi_l \\right)}},\n\\label{Zeroordlf}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha_{kl} = - {\\frac {i} {2 \\mu_0 n_0 m}} {\\left( k \\Box_l + l\n\\Box_k \\right)} {\\left[ 1 - {\\rm sgn} (k) {\\rm sgn} (l) \\right]}.\n\\label{Firstordalph}\n\\end{equation}\n\nEquation (\\ref{Waveeqallord}) has now two types of solutions. The\nfirst is a secular solution linearly dependent on the time\nvariable in the first-order approximation. As a rule, the highest\npower in the renormalization parameter of the secular terms\ncontained in the standard perturbation expansion is equal to the\ncorresponding order in the small perturbation parameter. The\nsecond solution of equation (\\ref{Waveeqallord}) arising from the\nnonlinear interaction between waves in the first order, is\nregular. Omitting tedious but standard algebra, we present here\nonly the result\n\\begin{equation}\n{\\bf A}_{1} = \\sum \\limits_{k} {\\widehat{\\bf A}}_{k}^{(1)} {\\cal\nA}_{k} {\\rm e}^{i \\psi_{k}} + {\\bf e}_x \\sum \\limits_{k,l}\nA_{kl}^{(1)} {\\cal A}_{k} {\\cal A}_{l} {\\rm e}^{i {\\left( \\psi_k +\n\\psi_l \\right)}} , \\label{Firstordvps}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\widehat{\\bf A}}_{k}^{(1)} = {\\left( {\\widehat{A}}_{kx}^{(1)}, t\n{\\widehat{A}}_{ky}^{(1)}, -i t {\\widehat{A}}_{ky}^{(1)} {\\rm sgn}\n(k) \\right)}, \\label{Firstordvec}\n\\end{equation}\nSome of the details of the calculations are presented in the\nAppendix. In explicit form, the components of the vector operator\n${\\widehat{\\bf A}}_{\\bf k}^{(1)}$ and those of the infinite matrix\n$A_{kl}^{(1)}$ are given by the expressions\n\\begin{equation}\n{\\widehat{A}}_{kx}^{(1)} = - {\\frac {i k \\beta_k} {\\gamma_k\n\\Box_k}} {\\widehat{\\nabla}}_k, \\qquad {\\widehat{\\nabla}}_k = {\\bf\nA}_{k}^{(0)} \\cdot {\\widehat{\\nabla}}, \\label{Firstordakx}\n\\end{equation}\n\\begin{equation}\n{\\widehat{A}}_{ky}^{(1)} = - {\\frac {{\\widehat{F}}_{k}} {2\n\\omega_k \\alpha_k {\\rm sgn} (k) + \\omega_c \\chi_k}},\n\\label{Firstordaky}\n\\end{equation}\n\\begin{equation}\nA_{kl}^{(1)} = {\\frac {e} {2m v_T^2}} {\\frac {\\omega_k + \\omega_l}\n{\\Box_{kl} {\\cal D}_{kl}}} {\\left( k \\Box_l + l \\Box_k \\right)}\n{\\left[ 1 - {\\rm sgn} (k) {\\rm sgn} (l) \\right]},\n\\label{Firstordaklx}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\widehat{F}}_{k} = 2k \\omega_k {\\left[ \\omega_k {\\rm sgn} (k) +\n\\omega_c \\right]} {\\widehat{\\nabla}}_{X}, \\label{Firstordoper}\n\\end{equation}\n\\begin{equation}\n\\Box_{kl} = {\\frac {{\\left( \\omega_k + \\omega_l \\right)}^2} {c^2}}\n- (k+l)^2, \\label{Firstordconst}\n\\end{equation}\n\\begin{equation}\n{\\cal D}_{kl} = {\\frac {{\\left( \\omega_k + \\omega_l \\right)}^2}\n{v_T^2}} - (k+l)^2 - {\\frac {1} {r_D^2}}. \\label{Firstordconsta}\n\\end{equation}\nIn addition, the constants $\\alpha_k$, $\\beta_k$, $\\gamma_k$ and\n$\\chi_k$ entering the expressions above are given by\n\\begin{equation}\n\\alpha_k = \\Box_k + {\\frac {\\omega_k^2 - \\omega_p^2} {c^2}},\n\\qquad \\beta_k = \\Box_k - {\\frac {1} {r_D^2}}, \\label{Constantsfo}\n\\end{equation}\n\\begin{equation}\n\\gamma_k = {\\frac {\\omega_k^2} {v_T^2}} - k^2 - {\\frac {1}\n{r_D^2}}, \\qquad \\chi_k = \\Box_k + {\\frac {2 \\omega_k^2} {c^2}}.\n\\label{Constantsfor}\n\\end{equation}\nFurthermore, the first-order  current velocity can be expressed as\n\\begin{equation}\n{\\bf V}_{1} = \\sum \\limits_{k} {\\widehat{\\bf V}}_{k}^{(1)} {\\cal\nA}_{k} {\\rm e}^{i \\psi_{k}} + {\\bf e}_x \\sum \\limits_{k,l}\nV_{kl}^{(1)} {\\cal A}_{k} {\\cal A}_{l} {\\rm e}^{i {\\left( \\psi_k +\n\\psi_l \\right)}}, \\label{Firstordcvel}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\widehat{\\bf V}}_{k}^{(1)} = {\\left( {\\widehat{V}}_{kx}^{(1)},\n{\\widehat{V}}_{ky}^{(1)}, -i {\\widehat{V}}_{ky}^{(1)} {\\rm sgn}\n(k) \\right)}. \\label{Firstordvcvel}\n\\end{equation}\nThe corresponding operators and matrix coefficients can be written\nexplicitly according to the expressions\n\\begin{equation}\n{\\widehat{V}}_{kx}^{(1)} = {\\frac {\\Box_k} {\\mu_0 e n_0}}\n{\\widehat{A}}_{kx}^{(1)}, \\qquad V_{kl}^{(1)} = {\\frac {\\Box_{kl}}\n{\\mu_0 e n_0}} A_{kl}^{(1)}, \\label{Currentvelx}\n\\end{equation}\n\\begin{equation}\n{\\widehat{V}}_{ky}^{(1)} = {\\frac {1} {\\mu_0 e n_0}} {\\left[ t\n\\Box_k {\\widehat{A}}_{ky}^{(1)} + 2i {\\left( {\\frac {\\omega_k}\n{c^2}} {\\widehat{A}}_{ky}^{(1)} + k {\\widehat{\\nabla}}_X \\right)}\n\\right]}, \\label{Currentvely}\n\\end{equation}\nCalculating the first-order density $N_1$ from equation\n(\\ref{Continuitn}), we obtain\n\\begin{equation}\nN_{1} = \\sum \\limits_{k} {\\widehat{N}}_{k}^{(1)} {\\cal A}_{k} {\\rm\ne}^{i \\psi_{k}} + \\sum \\limits_{k,l} N_{kl}^{(1)} {\\cal A}_{k}\n{\\cal A}_{l} {\\rm e}^{i {\\left( \\psi_k + \\psi_l \\right)}},\n\\label{Firstordden}\n\\end{equation}\n\\begin{equation}\n{\\widehat{N}}_{k}^{(1)} = {\\frac {\\Box_k} {\\mu_0 e \\omega_k}}\n{\\left( k {\\widehat{A}}_{kx}^{(1)} - i {\\widehat{\\nabla}}_k\n\\right)}, \\label{Firstorddenc}\n\\end{equation}\n\\begin{equation}\nN_{kl}^{(1)} = {\\frac {k + l} {2 \\mu_0 m v_T^2 {\\cal D}_{kl}}}\n{\\left( k \\Box_l + l \\Box_k \\right)} {\\left[ 1 - {\\rm sgn} (k)\n{\\rm sgn} (l) \\right]}. \\label{Firstorddenco}\n\\end{equation}\nAnalogously, for the first-order scalar potential $\\varphi_1$, we\nfind\n\\begin{equation}\n\\varphi_1 = \\sum \\limits_{k} {\\widehat{\\varphi}}_{k}^{(1)} {\\cal\nA}_{k} {\\rm e}^{i \\psi_{k}} + \\sum \\limits_{k,l}\n\\varphi_{kl}^{(1)} {\\cal A}_{k} {\\cal A}_{l} {\\rm e}^{i {\\left(\n\\psi_k + \\psi_l \\right)}}, \\label{Firstordscp}\n\\end{equation}\n\\begin{equation}\n{\\widehat{\\varphi}}_{k}^{(1)} = {\\frac {e} {\\epsilon_0 \\Box_k}}\n{\\widehat{N}}_{k}^{(1)} = {\\frac {c^2} {\\omega_k}} {\\left( k\n{\\widehat{A}}_{kx}^{(1)} - i {\\widehat{\\nabla}}_k \\right)},\n\\label{Firstordscpc}\n\\end{equation}\n\\begin{equation}\n\\varphi_{kl}^{(1)} = {\\frac {e c^2 (k+l)} {2 m v_T^2 \\Box_{kl}\n{\\cal D}_{kl}}} {\\left( k \\Box_l + l \\Box_k \\right)} {\\left[ 1 -\n{\\rm sgn} (k) {\\rm sgn} (l) \\right]}. \\label{Firstordscpco}\n\\end{equation}\nFinally, the first-order magnetic field is calculated to be\n\\begin{equation}\n{\\bf B}_{1} = \\sum \\limits_{k} {\\widehat{\\bf B}}_{k}^{(1)} {\\cal\nA}_{k} {\\rm e}^{i \\psi_{k}}, \\label{Firstordmagf}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\widehat{\\bf B}}_{k}^{(1)} = {\\left( - i {\\rm sgn} (k)\n{\\widehat{\\nabla}}_k, {\\widehat{B}}_{ky}^{(1)}, -i\n{\\widehat{B}}_{ky}^{(1)} {\\rm sgn} (k) \\right)},\n\\label{Firstordmagfi}\n\\end{equation}\n\\begin{equation}\n{\\widehat{B}}_{ky}^{(1)} = - {\\rm sgn} (k) {\\left( t k\n{\\widehat{A}}_{ky}^{(1)} - i {\\widehat{\\nabla}}_X \\right)}.\n\\label{Firstordmafi}\n\\end{equation}\n\nA couple of interesting features of the zero and first-order\nperturbation solution are noteworthy to be commented at this\npoint. First of all, the zero-order density $N_0$ vanishes which\nmeans that no density waves are induced by the whistler\neigenmodes. The second terms in the expressions for the\nfirst-order density $N_1$ and current velocity ${\\bf V}_1$ [see\nequations (\\ref{Firstordcvel}) and (\\ref{Firstordden})] imply\ncontribution from nonlinear interaction between waves according to\nthe nonlinear Lorentz force. It will be shown in the remainder\nthat these terms give rise to nonlinear terms in the\nrenormalization group equation and describe solitary wave\nbehaviour of the whistler mode.\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\setcounter{equation}{0}\n\n\\section{The Renormalization Group Equation}\n\nPassing over to the final stage of our renormalization group\nprocedure, we note that in second order the quantities ${\\bf U}_2$\nand ${\\bf W}_2$ entering the right-hand-side of equation\n(\\ref{Waveeqallord}) can be written as\n\\begin{equation}\n{\\bf U}_2 = - 2 \\nabla \\cdot {\\widehat{\\nabla}} {\\bf A}_1 -\n{\\widehat{\\nabla}}^2 {\\bf A}_0 + \\mu_0 e N_1 {\\bf V}_0,\n\\label{Secondordu}\n\\end{equation}\n\\begin{equation}\n{\\bf W}_2 = {\\frac {e} {m}} {\\widehat{\\nabla}} \\varphi_1 - {\\frac\n{v_T^2} {n_0}} {\\widehat{\\nabla}} N_1 - {\\bf V}_1 \\cdot \\nabla\n{\\bf V}_0 - {\\frac {e} {m}} {\\bf V}_1 \\times {\\bf B}_0,\n\\label{Secondordw}\n\\end{equation}\nSince we are interested only in the secular terms in second order,\nappearing in the expressions for the $y$ and $z$ components of the\nelectromagnetic vector potential ${\\bf A}_2$, contributions in the\nsource vectors ${\\bf U}_2$ and ${\\bf W}_2$ leading to such terms\nare sufficient for completing the renormalization group procedure.\nThus, we can write\n\\begin{equation}\n{\\bf A}_2 = \\sum \\limits_{k} {\\left( t {\\widehat{\\bf A}}_k^{(2)} +\nt^2 {\\widehat{\\bf C}}_k \\right)} {\\cal A}_{k} {\\rm e}^{i \\psi_{k}}\n\\nonumber\n\\end{equation}\n\\begin{equation}\n+ t \\sum \\limits_{k} {\\widehat{\\bf D}}_k^{(2)} {\\cal A}_{k} {\\rm\ne}^{i \\psi_{k}} + t \\sum \\limits_{k,l} {\\mathbf \\Gamma}_{kl}\n{\\left| {\\cal A}_{l} \\right|}^2 {\\cal A}_{k} {\\rm e}^{i \\psi_k}.\n\\label{Secondordvps}\n\\end{equation}\nAn important remark is in order at this point. From the\nsolvability condition (\\ref{Appsolcond}) it follows that the\ncomplex amplitude ${\\cal A}_k$ must satisfy the complex Poisson\nequation\n\\begin{equation}\n{\\widehat{\\nabla}}_k^2 {\\cal A}_k = 0. \\label{Secondordscon}\n\\end{equation}\nThe latter imposes additional restrictions on the dependence of\nthe wave amplitudes ${\\cal A}_k$ on the slow transverse\nindependent variables $Y$ and $Z$. Straightforward calculations\nyield (see the Appendix for details)\n\\begin{equation}\n{\\widehat{A}}_{ky}^{(2)} = - {\\frac {i {\\rm sgn} (k)} {2 \\omega_k\n\\alpha_k {\\rm sgn} (k) + \\omega_c \\chi_k}} {\\left( \\beta_k^{(2)}\n{\\widehat{A}}_{ky}^{(1) {\\bf 2}} - {\\widehat{G}}_k \\right)},\n\\label{Secondordaky}\n\\end{equation}\n\\begin{equation}\n{\\widehat{D}}_{ky}^{(2)} = {\\frac {i v_T^2 \\beta_k {\\rm sgn} (k)}\n{2 \\omega_k \\alpha_k {\\rm sgn} (k) + \\omega_c \\chi_k}} {\\left( 1 +\n{\\frac {k^2 \\beta_k} {\\gamma_k \\Box_k}} \\right)}\n{\\widehat{\\nabla}}_Y {\\widehat{\\nabla}}_k, \\label{Secondorddky}\n\\end{equation}\n\\begin{equation}\n{\\widehat{C}}_{ky} = {\\frac {1} {2}} {\\widehat{A}}_{ky}^{(1) {\\bf\n2}}, \\label{Secondordcky}\n\\end{equation}\nwhere\n\\begin{equation}\n\\beta_k^{(2)} = \\alpha_k + {\\frac {4 \\omega_k^2} {c^2}} + {\\frac\n{3 \\omega_c \\omega_k} {c^2}} {\\rm sgn} (k), \\label{Secondordcon}\n\\end{equation}\n\\begin{equation}\n{\\widehat{G}}_k = \\omega_k {\\rm sgn} (k) {\\left[ \\omega_k {\\rm\nsgn} (k) + \\omega_c \\right]} {\\widehat{\\nabla}}^2.\n\\label{Secondordoper}\n\\end{equation}\nThe matrix coefficient $\\Gamma_{kly}$ determining the nonlinear\ncontribution represented by the second term in equation\n(\\ref{Secondordvps}) reads explicitly as\n\\begin{equation}\n\\Gamma_{kly} = - {\\frac {1 - {\\rm sgn} (k) {\\rm sgn} (l)} {\\mu_0\nn_0 m v_T^2 \\omega_l {\\cal D}_{kl}}} {\\frac {i \\omega_k \\Box_l\n{\\left( k \\Box_l + l \\Box_k \\right)} {\\rm sgn} (k)} {2 \\omega_k\n\\alpha_k {\\rm sgn} (k) + \\omega_c \\chi_k}} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\times {\\left[ \\omega_c {\\left( l \\omega_k - k \\omega_l \\right)}\n{\\rm sgn} (l) + (k+l) \\omega_k \\omega_l \\right]}.\n\\label{Secondordgam}\n\\end{equation}\nFollowing the standard procedure \\cite{Tzenov} of the RG method,\nwe finally obtain the desired RG equation\n\\begin{equation}\n{\\frac {\\partial {\\widetilde{\\cal A}}_k} {\\partial t}} - \\epsilon\n{\\widehat{A}}_{ky}^{(1)} {\\widetilde{\\cal A}}_k \\nonumber\n\\end{equation}\n\\begin{equation}\n= \\epsilon^2 {\\left( {\\widehat{A}}_{ky}^{(2)} +\n{\\widehat{D}}_{ky}^{(2)} \\right)} {\\widetilde{\\cal A}}_k +\n\\epsilon^2 \\sum \\limits_l \\Gamma_{kly} {\\left| {\\widetilde{\\cal\nA}}_l \\right|}^2 {\\widetilde{\\cal A}}_k, \\label{Rgroupeq}\n\\end{equation}\nwhere now ${\\widetilde{\\cal A}}_k$ is the renormalized complex\namplitude \\cite{Tzenov}. Thus, the renormalized solution for the\nelectromagnetic vector potential acquires the form\n\\begin{equation}\n{\\bf A} = \\sum \\limits_{k} {\\bf A}_{k}^{(0)} {\\widetilde{\\cal\nA}}_{k} {\\rm e}^{i \\psi_{k}}. \\label{Renormsolut}\n\\end{equation}\nAnalogously, for the electric and magnetic field of the whistler\nwave, one can obtain in a straightforward manner the following\nexpressions\n\\begin{equation}\n{\\bf B} = \\sum \\limits_{k} {\\bf B}_{k}^{(0)} {\\widetilde{\\cal\nA}}_{k} {\\rm e}^{i \\psi_{k}}, \\qquad {\\bf E} = i \\sum \\limits_{k}\n\\omega_k {\\bf A}_{k}^{(0)} {\\widetilde{\\cal A}}_{k} {\\rm e}^{i\n\\psi_{k}}. \\label{Renormsolb}\n\\end{equation}\n\nIt is important to mention that the plasma density remains\nunchanged ($N = 0$) contrary to the case of electrostatic waves,\nwhere the evolution of the induced electrostatic waves follows the\nevolution of the density waves.\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\setcounter{equation}{0}\n\n\\section{\\label{Essent}System of Coupled Nonlinear Schr\\\"{o}dinger\nEquations}\n\nThe simplest case of the validity of the solvability condition\n(\\ref{Secondordscon}) consists in the assumption that the slow\nwave amplitudes ${\\cal A}_k$ do not depend on the transverse\ncoordinates. Setting $\\epsilon = 1$ in equation (\\ref{Rgroupeq}),\nwe obtain the following system of coupled nonlinear\nSchr\\\"{o}dinger equations\n\\begin{equation}\ni {\\rm sgn} (k) {\\frac {\\partial {\\cal A}_k} {\\partial t}} + i\n\\nu_k {\\rm sgn} (k) {\\frac {\\partial {\\cal A}_k} {\\partial x}} =\n\\lambda_k {\\frac {\\partial^2 {\\cal A}_k} {\\partial x^2}} + \\sum\n\\limits_l \\mu_{kl} {\\left| {\\cal A}_l \\right|}^2 {\\cal A}_k,\n\\label{Couplednse}\n\\end{equation}\nwhere for simplicity the tilde-sign over the renormalized\namplitude has been dropped. Moreover, the coefficients $\\nu_k$,\n$\\lambda_k$ and $\\mu_{kl}$ are given by the expressions\n\\begin{equation}\n\\nu_k = {\\frac {2k \\omega_k {\\left[ \\omega_k {\\rm sgn} (k) +\n\\omega_c \\right]}} {2 \\omega_k \\alpha_k {\\rm sgn} (k) + \\omega_c\n\\chi_k}}, \\label{Coefficientnu}\n\\end{equation}\n\\begin{equation}\n\\lambda_k = {\\frac {\\omega_k {\\left[ \\omega_k {\\rm sgn} (k) +\n\\omega_c \\right]}} {2 \\omega_k \\alpha_k {\\rm sgn} (k) + \\omega_c\n\\chi_k}} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\times {\\left\\{ {\\frac {4k^2 \\omega_k \\beta_k^{(2)} {\\left[\n\\omega_k {\\rm sgn} (k) + \\omega_c \\right]}} {{\\left[ 2 \\omega_k\n\\alpha_k {\\rm sgn} (k) + \\omega_c \\chi_k \\right]}^2}} - {\\rm sgn}\n(k) \\right\\}}, \\label{Coefficientla}\n\\end{equation}\n\\begin{equation}\n\\mu_{kl} = {\\frac {1 - {\\rm sgn} (k) {\\rm sgn} (l)} {\\mu_0 n_0 m\nv_T^2 \\omega_l {\\cal D}_{kl}}} {\\frac {\\omega_k \\Box_l {\\left( k\n\\Box_l + l \\Box_k \\right)}} {2 \\omega_k \\alpha_k {\\rm sgn} (k) +\n\\omega_c \\chi_k}} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\times {\\left[ \\omega_c {\\left( l \\omega_k - k \\omega_l \\right)}\n{\\rm sgn} (l) + (k+l) \\omega_k \\omega_l \\right]}.\n\\label{Coefficientmu}\n\\end{equation}\nInterestingly enough, the infinite matrix of coupling coefficients\n$\\mu_{kl}$ represents a sort of selection rules. Clearly,\n\\begin{equation}\n\\mu_{kk} = 0, \\qquad \\mu_{k, -k} = 0, \\label{Selectrule1}\n\\end{equation}\nand\n\\begin{equation}\n\\mu_{kl} = 0, \\qquad {\\rm for} \\quad {\\rm sgn} (k) {\\rm sgn} (l) =\n1. \\label{Selectrule2}\n\\end{equation}\nThis means that a generic mode with a wave number $k$ cannot\ncouple with itself, neither can it couple with another mode with a\nwave number of the same sign. Note that this feature is a\nconsequence of the vector character of the nonlinear coupling\nbetween modes and is due to the nonlinear Lorentz force.\nTherefore, for a given mode $k$ the simplest nontrivial reduction\nof the infinite system of coupled nonlinear Schr\\\"{o}dinger\nequations consists of minimum two coupled equations.\n\nWithout loss of generality, we can assume in what follows that the\nsign of an arbitrary mode $k$ under consideration is positive ($k\n> 0$). Suppose that for a particular whistler mode with a\npositive wave number $k$ there exist a mode with wave number $-l$\nfor which the coupling coefficient $\\mu_{k, -l}$ is maximum.\nNeglecting all other modes but the modes $k$ and $-l$, we can\nwrite\n\\begin{equation}\ni {\\frac {\\partial {\\cal A}_k} {\\partial t}} + i \\nu_k {\\frac\n{\\partial {\\cal A}_k} {\\partial x}} = \\lambda_k {\\frac {\\partial^2\n{\\cal A}_k} {\\partial x^2}} + \\mu_1 {\\left| {\\cal A}_l \\right|}^2\n{\\cal A}_k, \\label{Couplednsek}\n\\end{equation}\n\\begin{equation}\ni {\\frac {\\partial {\\cal A}_l} {\\partial t}} + i \\nu_l {\\frac\n{\\partial {\\cal A}_l} {\\partial x}} = \\lambda_l {\\frac {\\partial^2\n{\\cal A}_l} {\\partial x^2}} + \\mu_2 {\\left| {\\cal A}_k \\right|}^2\n{\\cal A}_l, \\label{Couplednsel}\n\\end{equation}\nwhere\n\\begin{equation}\n\\mu_1 = {\\frac {2} {\\mu_0 n_0 m v_T^2 \\omega_l {\\cal D}_{k, -l}}}\n{\\frac {\\omega_k \\Box_l {\\left( k \\Box_l - l \\Box_k \\right)}} {2\n\\omega_k \\alpha_k + \\omega_c \\chi_k}} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\times {\\left[ \\omega_c {\\left( k \\omega_l - l \\omega_k \\right)} +\n(k-l) \\omega_k \\omega_l \\right]}. \\label{Coefficientmu1}\n\\end{equation}\n\\begin{equation}\n\\mu_2 = {\\frac {2} {\\mu_0 n_0 m v_T^2 \\omega_k {\\cal D}_{k, -l}}}\n{\\frac {\\omega_l \\Box_k {\\left( k \\Box_l - l \\Box_k \\right)}} {2\n\\omega_l \\alpha_l + \\omega_c \\chi_l}} \\nonumber\n\\end{equation}\n\\begin{equation}\n\\times {\\left[ \\omega_c {\\left( k \\omega_l - l \\omega_k \\right)} +\n(k-l) \\omega_k \\omega_l \\right]}. \\label{Coefficientmu2}\n\\end{equation}\n\nThe system of coupled nonlinear Schr\\\"{o}dinger equations\n(\\ref{Couplednsek}) and (\\ref{Couplednsel}) is non integrable in\ngeneral \\cite{Manakov}. It represents an important starting point\nfor further investigations on the nonlinear dynamics and evolution\nof whistler waves in magnetized plasmas.\n\n\\renewcommand{\\theequation}{\\thesection.\\arabic{equation}}\n\n\\setcounter{equation}{0}\n\n\\section{Discussion and conclusions}\n\nWe studied the nonlinear dynamics of whistler waves in magnetized\nplasmas. Since plasmas and beam-plasma systems considered here are\nassumed to be weakly collisional, the point of reference for the\nanalysis performed in the present paper is the system of\nhydrodynamic and field equations. We apply the renormalization\ngroup method to obtain dynamical equations for the slowly varying\namplitudes of whistler waves. As a result of the investigation\nperformed, it has been shown that the amplitudes of eigenmodes\nsatisfy an infinite system of coupled nonlinear Schr\\\"{o}dinger\nequations.  In this sense, the whistler eigenmodes form a sort of\na gas of interacting quasiparticles, while the slowly varying\namplitudes can be considered as dynamical variables heralding the\nrelevant information about the system.\n\nAn important feature of our description is that whistler waves do\nnot perturb the initial uniform density of plasma electrons. The\nplasma response to the induced whistler waves consists in velocity\nredistribution which follows exactly the behaviour of the\nwhistlers. Another interesting peculiarity are the selection rules\ngoverning the nonlinear mode coupling. According to these rules\nmodes with the same sign do not couple, which is a direct\nconsequence of the vector character of the interaction. Careful\ninspection shows that the initial source of the nonlinear\ninteraction between waves in the whistler mode is the zero-order\nLorentz force [see equation (\\ref{Zeroordlf})]. Since the quantity\n${\\bf W}_1$ is proportional to ${\\bf A}_k^{(0)} \\times {\\bf\nA}_l^{(0)}$, the above mentioned selection rules follow directly,\nprovided the only case in which the cross product does not vanish\nis the case, where modes $k$ and $l$ have different sign.\n\nWe believe that the results obtained in the present paper might\nhave a wide class of possible applications ranging from laboratory\nexperiments to observations of a variety of effects relevant to\nspace plasmas.\n\n\\begin{acknowledgments}\nIt is a pleasure to thank B. Baizakov for many interesting and\nuseful discussions concerning the subject of the present paper.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe  detection of  primordial gravitational waves (GWs) via B-mode\npolarization of Cosmic Microwave Background Radiation (CMBR) by\nBICEP2~\\cite{Ade:2014xna} has shown that the cosmic  inflation\noccurred at a high scale of $10^{16}$ GeV is the most plausible\nsource of generating primordial GWs.  However, more data are\nrequired to confirm  the above situation.  The primordial GWs can be\nimprinted in the anisotropies and polarization spectrum of CMBR\n by making  the photon redshifts. The B-mode signal\nobserved by BICEP2 might  contain contributions from other sources\nof vector modes and  cosmic strings,  in addition to tensor\nmodes~\\cite{Moss:2014cra}.\n\nThe conformal gravity\n$C^{\\mu\\nu\\rho\\sigma}C_{\\mu\\nu\\rho\\sigma}(=C^2)$ of being invariant\nunder the conformal transformation of $g_{\\mu\\nu}\\to\n\\Omega^2(x)g_{\\mu\\nu}$ has its own interests in gravity and\ncosmology. On the gravity side, it gives us  a promising combination\nof $R_{\\mu\\nu}^2-R^2\/3$ up to the Gauss-Bonnet term which kills\nmassive scalar GWs  when it couples to Einstein gravity (known to be\nthe Einstein-Weyl gravity)~\\cite{Lu:2011zk}.\nStelle~\\cite{Stelle:1976gc} has first introduced the quadratic\ncurvature gravity  of $a(R_{\\mu\\nu}^2-R^2\/3)+b R^2$ to improve  the\nperturbatively renormalizable property of Einstein gravity. In case\nof $ab\\not=0$, the renormalizability was achieved but the unitarity\nwas violated unless $a=0$, showing that the renormalizability and\nunitarity exclude to each other.  Although the $a$-term of providing\nmassive GWs improves the ultraviolet divergence, it induces\nsimultaneously  ghost excitations which spoil the unitarity.  The\nprice one has to pay for making the theory renormalizable is the\nloss of unitarity. This issue is not resolved completely until now.\n\nHowever, the conformal gravity itself is renormalizable. Also, it\nprovides the AdS black hole solution~\\cite{Riegert:1984zz} and its\nthermodynamic properties and holography were discussed extensively\nin the  literature~\\cite{Klemm:1998kf,Lu:2012xu,Grumiller:2013mxa}.\nThe authors have investigated the AdS black hole thermodynamics and\nstability in the Einstein-Weyl gravity and in the limit of  the\nconformal gravity~\\cite{Myung:2013uka}.\n\nOn the cosmology side of the conformal gravity, it provides surely a\nmassive vector propagation generated during de Sitter inflation in\naddition to massive tensor ghosts when it couples to  Einstein\ngravity~\\cite{Clunan:2009er,Deruelle:2010kf,Deruelle:2012xv}.\nRecently, the authors have shown that in the limit of $m^2 \\to 0$\n(keeping the conformal gravity only), the vector and tensor power\nspectra disappear. It implies that their power spectra are not\ngravitationally produced because the vector and tensor perturbations\nare decoupled from the expanding de Sitter background. This occurs\ndue to conformal invariance as a transversely  massive vector has\nbeen shown in the $m^2\\to 0$ limit of the massive Maxwell theory\n($-F^2\/4+m^2A^2\/2$)~\\cite{Dimopoulos:2006ms}. We note here that\n$F^2$ is conformally invariant  like $C^2$ under the transformation\nof  $g_{\\mu\\nu}\\to \\Omega^2(x)g_{\\mu\\nu}$~\\cite{Myung:2014aia}.\n The\nconformal gravity implication to cosmological perturbation was first\nstudied in~\\cite{Mannheim:2011is} which might indicate  that there\nexists a difference between conformal and Einstein gravities in\ntheir perturbed equations around de Sitter background. Even though\nhe has obtained a ``degenerate fourth-order equation\" for the metric\nperturbation tensor $h_{\\mu\\nu}$ from the conformal gravity,  any\nrelevant quantity was not found because he did not split\n$h_{\\mu\\nu}$ according to the SO(3) decomposition for cosmological\nperturbations. As far as we know, there is no definite computation\nof an observable like the power spectrum in the conformal gravity.\n\nIn this Letter, we will study the conformal  gravity  as a\nhigher-order gravity theory to compute the vector and tensor power\nspectra generated from de Sitter inflation. Considering the\nconformal invariant of the conformal gravity seriously, we expect to\nobtain the constant power spectra for vector and tensor\nperturbations.\n\n\n\\section{Conformal  gravity }\n\n\nLet us first consider the conformal  gravity whose action is given\nby\n\\begin{equation} \\label{ECSW}\nS_{\\rm CG}=\\frac{1}{4\\kappa m^2}\\int d^4x\n\\sqrt{-g}\\Big[C^{\\mu\\nu\\rho\\sigma}C_{\\mu\\nu\\rho\\sigma}\\Big],\n\\end{equation}\nwhere the Weyl-squared term is given by\n\\begin{eqnarray}\nC^{\\mu\\nu\\rho\\sigma}C_{\\mu\\nu\\rho\\sigma}&=&2\\Big(R^{\\mu\\nu}R_{\\mu\\nu}-\\frac{1}{3}R^2\\Big)+\n(R^{\\mu\\nu\\rho\\sigma}R_{\\mu\\nu\\rho\\sigma}-4R^{\\mu\\nu}R_{\\mu\\nu}+R^2)\n\\end{eqnarray}\nwith the Weyl tensor\n\\begin{equation}\nC_{\\mu\\nu\\rho\\sigma}=R_{\\mu\\nu\\rho\\sigma}-\\frac{1}{2}\\Big(\ng_{\\mu\\rho}R_{\\nu\\sigma}-g_{\\mu\\sigma}R_{\\nu\\rho}-g_{\\nu\\rho}R_{\\mu\\sigma}+g_{\\nu\\sigma}R_{\\mu\\rho}\\Big)+\\frac{1}{6}R(g_{\\mu\\rho}g_{\\nu\\sigma}-g_{\\mu\\sigma}g_{\\nu\\rho}).\n\\end{equation}\n Here we have\n$\\kappa=8\\pi G=1\/M^2_{\\rm P}$, $M_{\\rm P}$ being the reduced Planck\nmass and a mass-squared $m^2$ is introduced to make the action\ndimensionless. Greek indices run from 0 to 3 with conventions\n$(-+++)$, while Latin indices run from 1 to 3. Further, we note that\nthe Weyl-squared term is invariant under the conformal\ntransformation of $g_{\\mu\\nu} \\to \\Omega^2(x) g_{\\mu\\nu}$.\n\nIts  equation takes the form\n\\begin{equation} \\label{ein-eq}\n2 \\nabla^\\rho\\nabla^\\sigma\nC_{\\mu\\rho\\nu\\sigma}+G^{\\rho\\sigma}C_{\\mu\\rho\\nu\\sigma}=0\n\\end{equation} with  the Einstein tensor $G_{\\mu\\nu}$.\nThe solution is de Sitter space whose  curvature quantities are\ngiven by\n\\begin{equation}\n\\bar{R}_{\\mu\\nu\\rho\\sigma}=H^2(\\bar{g}_{\\mu\\rho}\\bar{g}_{\\nu\\sigma}-\\bar{g}_{\\mu\\sigma}\\bar{g}_{\\nu\\rho}),~~\\bar{R}_{\\mu\\nu}=3H^2\\bar{g}_{\\mu\\nu},~~\\bar{R}=12H^2\n\\end{equation}\nwith $H$=constant. We choose  de Sitter  background explicitly  by\nchoosing a conformal time $\\eta$\n\\begin{eqnarray} \\label{frw}\nds^2_{\\rm dS}=\\bar{g}_{\\mu\\nu}dx^\\mu\ndx^\\nu=a(\\eta)^2\\Big[-d\\eta^2+\\delta_{ij}dx^idx^j\\Big],\n\\end{eqnarray}\nwhere the conformal  scale factor is\n\\begin{eqnarray}\na(\\eta)=-\\frac{1}{H\\eta}\\to a(t)=e^{Ht}.\n\\end{eqnarray}\nHere the latter denotes  the scale factor with respect to cosmic\ntime $t$.\n\nWe   choose  the Newtonian gauge of $B=E=0 $ and $\\bar{E}_i=0$ which\nleads to $10-4=6$ degrees of freedom (DOF). In this case, the\ncosmologically perturbed metric can be simplified to be\n\\begin{eqnarray} \\label{so3-met}\nds^2=a(\\eta)^2\\Big[-(1+2\\Psi)d\\eta^2+2\\Psi_i d\\eta\ndx^{i}+\\Big\\{(1+2\\Phi)\\delta_{ij}+h_{ij}\\Big\\}dx^idx^j\\Big]\n\\end{eqnarray}\nwith the transverse vector $\\partial_i\\Psi^i=0$ and\ntransverse-traceless tensor $\\partial_ih^{ij}=h=0$. We emphasize\nthat choosing the SO(3)-perturbed metric (\\ref{so3-met}) contrasts\nsharply with the covariant approach to the  cosmological conformal\ngravity~\\cite{Mannheim:2011is}.\n\nIn order to get  the cosmological perturbed equations,  one  is\nfirst to obtain the bilinear action and then, varying it to yield\nthe perturbed equations.  We expand the conformal gravity action\n(\\ref{ECSW}) up to quadratic order in the perturbations of\n$\\Psi,\\Phi,\\Psi_i,$ and $h_{ij}$ around  the de Sitter\nbackground~\\cite{Deruelle:2010kf}.  Then,  the bilinear  actions for\nscalar, vector and tensor perturbations can be found as\n\\begin{eqnarray}\n&&\\hspace*{-2.3em}S_{\\rm CG}^{({\\rm S})}=\\frac{1}{3\\kappa m^2}\\int\nd^4x\\Big[\\nabla^2 (\\Psi-\\Phi)\\Big]^2,\n\\label{scalar}\\\\\n&&\\nonumber\\\\\n &&\\hspace*{-2.3em}S_{\\rm CG}^{({\\rm V})}=\\frac{1}{4\\kappa m^2}\\int\nd^4x\\Big(\\partial_i\\Psi'_{j}\\partial^{i}\\Psi'^{j}\n-\\nabla^2\\Psi_i\\nabla^2\\Psi^i\\Big),\\label{vpeq}\\\\\n&&\\nonumber\\\\\n&&\\hspace*{-2.3em} S_{\\rm CG}^{({\\rm T})}=\\frac{1}{8\\kappa m^2}\\int\nd^4x\\Big(h''_{ij}h''^{ij} -2\\partial_kh'_{ij}\\partial^{k}h'^{ij}\n+\\nabla^2h_{ij}\\nabla^2h^{ij}\\Big)\\label{hpeq}.\n\\end{eqnarray}\nVarying the actions (\\ref{vpeq}) and (\\ref{hpeq}) with respect to\n$\\Psi^{i}$ and $h^{ij}$  leads to the equations of motion for vector\nand tensor perturbations\n\\begin{eqnarray}\n&&\\Box\\Psi_i=0,\\label{veq}\\\\\n &&\\Box^2h_{ij}=0,\\label{heq}\n\\end{eqnarray}\nwhere $\\Box=d^2\/d\\eta^2-\\nabla^2$ with $\\nabla^2$ the Laplacian\noperator.\n It is worth noting  that  Eqs.(\\ref{veq}) and (\\ref{heq})  are independent of\nthe expanding de Sitter background in the conformal  gravity.\n\nFinally, we would like to mention two scalars $\\Phi$ and $\\Phi$. Two\nscalar equations\n are given by $\\nabla^2\\Psi=\\nabla^2\\Phi=0$,  which implies\nthat they are obviously\n non-propagating modes  in the de Sitter background. This means that the cosmological conformal gravity describes 4 DOF of vector and tensor modes. Hereafter,\n thus, we will not consider the scalar sector.\n\n\\section{Primordial power spectra}\nThe power spectrum is  given by the two-point correlation function\nwhich could be  computed when one chooses   the vacuum state\n$|0\\rangle$. It is defined by\n\\begin{equation}\n\\langle0|{\\cal F}(\\eta,\\bold{x}){\\cal\nF}(\\eta,\\bold{x}')|0\\rangle=\\int d^3\\bold{k} \\frac{{\\cal P}_{\\cal\nF}}{4\\pi k^3}e^{-i \\bold{k}\\cdot (\\bold{x}-\\bold{x}')},\n\\end{equation}\nwhere ${\\cal F}$ denotes vector and tensor and $k=|\\bold{k}|$ is the\nwave number. In general, fluctuations are created on all length\nscales with wave number $k$. Cosmologically relevant fluctuations\nstart their lives inside the Hubble radius which defines the\nsubhorizon: $k~\\gg aH~(z=-k\\eta\\gg 1)$.  On the other hand, the\ncomoving Hubble radius $(aH)^{-1}$ shrinks during inflation while\nthe comoving wavenumber $k$ is constant. Therefore, eventually all\nfluctuations exit the comoving  Hubble radius which defines the\nsuperhorizon: $k~\\ll aH~(z=-k\\eta\\ll 1)$.\n\nOne may compute the two-point function by taking the Bunch-Davies\nvacuum $|0\\rangle$.\n In the de Sitter inflation, we choose the subhorizon limit\nof  $z\\to \\infty$  to define the Bunch-Davies vacuum, while we\nchoose the superhorizon limit of $z\\to 0$ to get a definite form of\nthe power spectrum which stays alive after decaying.  For example,\nfluctuations of scalar and tensor originate on subhorizon scales and\nthey propagate for a long time on superhorizon scales. This can be\nchecked by computing their power spectra which are scale-invariant.\nAccordingly, it would be interesting  to check what happens when one\ncomputes the power spectra for vector and tensor perturbations\ngenerated from de Sitter inflation in the frame work of conformal\ngravity.\n\n\n\\subsection{Vector power spectrum}\n\n\nLet us  consider Eq.(\\ref{veq}) for vector perturbation and then,\nexpand  $\\Psi_i$ in plane waves with the linearly polarized states\n\\begin{eqnarray}\\label{psim}\n\\Psi_i(\\eta,{\\bf x})=\\frac{1}{(2\\pi)^{\\frac{3}{2}}}\\int d^3{\\bf\nk}\\sum_{s=1,2}p_i^{s}({\\bf k})\\Psi_{\\bf k}^{s}(\\eta)e^{i{\\bf\nk}\\cdot{\\bf x}},\n\\end{eqnarray}\nwhere  $p^{1\/2}_{i}$ are linear polarization   vectors with\n $p^{1\/2}_i\np^{1\/2, i}=1$.  Also, $\\Psi_{\\bf k}^{s}$ denote linearly polarized\nvector modes. Plugging (\\ref{psim}) into the equation (\\ref{veq}),\none finds the equation\n\\begin{eqnarray}\\label{v0eq}\n\\Bigg[\\frac{d^2}{d\\eta^2}+k^2\\Bigg]\\Psi_{\\bf k}^s(\\eta)=0.\n\\end{eqnarray}\nIntroducing  $z=-k\\eta$, Eq.(\\ref{v0eq}) takes a simple form\n\\begin{equation}\\label{v1eq}\n\\Big[\\frac{d^2}{dz^2}+1\\Big]\\Psi_{\\bf k}^{s}(z)=0\\end{equation}\nwhose positive frequency solution is given by \\begin{equation}\n\\Psi_{\\bf k}^{s}(z)\\sim e^{iz}\\end{equation} up to the\nnormalization.\n\n We are willing to  calculate\nvector power spectrum. For this purpose, we define a commutation\nrelation for the vector. In the bilinear action (\\ref{vpeq}), the\nconjugate momentum for the field $\\Psi_j$ is found to be\n\\begin{eqnarray}\\label{vconj}\n\\pi_{\\Psi}^{j}=-\\frac{1}{2\\kappa m^2}\\nabla^2\\Psi'^{j},\n\\end{eqnarray}\nwhere one observes an unusual factor $\\nabla^2$ which reflects that\nthe vector $\\Psi_i$ is not a canonically defined vector, but it is\nfrom the cosmological conformal gravity.  The canonical quantization\nis implemented by imposing the commutation relation\n\\begin{eqnarray}\\label{vcomm}\n[\\hat{\\Psi}_{j}(\\eta,{\\bf x}),\\hat{\\pi}_{\\Psi}^{j}(\\eta,{\\bf\nx}^{\\prime})]=2i\\delta({\\bf x}-{\\bf x}^{\\prime})\n\\end{eqnarray}\nwith $\\hbar=1$.\n\n\nNow, the operator $\\hat{\\Psi}_{j}$ can be expanded in Fourier modes\nas\n\\begin{eqnarray}\\label{vex}\n\\hat{\\Psi}_{j}(\\eta,{\\bf x})=\\frac{1}{(2\\pi)^{\\frac{3}{2}}}\\int\nd^3{\\bf k}\\sum_{s=1,2}\\Big(p_{j}^{s}({\\bf k})\\hat{a}_{\\bf\nk}^{s}\\Psi_{\\bf k}^{s}(\\eta)e^{i{\\bf k}\\cdot{\\bf x}}+{\\rm h.c.}\\Big)\n\\end{eqnarray}\nand the operator $\\hat{\\pi}_{\\Psi}^{j}=\\frac{k^2}{2\\kappa\nm^2}\\hat{\\Psi}'^{j}$ can be easily obtained from (\\ref{vex}).\nPlugging (\\ref{vex}) and $\\hat{\\pi}_{\\Psi}^{j}$ into (\\ref{vcomm}),\nwe find the commutation relation and Wronskian condition as\n\\begin{eqnarray}\n&&\\hspace*{-2em}[\\hat{a}_{\\bf k}^{s},\\hat{a}_{\\bf k^{\\prime}}^{\ns^{\\prime}\\dag}]=\\delta^{ss^{\\prime}}\\delta^3({\\bf k}-{\\bf\nk}^{\\prime}),\\label{comm0}\\\\\n&&\\hspace*{-2em}\\Psi_{\\bf k}^{s}\\Big(\\frac{k^2}{2\\kappa\nm^2}\\Big)(\\Psi_{\\bf k}^{*s})^{\\prime}-{\\rm c.c.}=i \\to \\Psi_{\\bf\nk}^{s}\\frac{d\\Psi_{\\bf k}^{*s}}{dz}-{\\rm c.c.}=-\\frac{2i\\kappa\nm^2}{k^3}. \\label{vwcon}\n\\end{eqnarray}\n  We\nchoose the  positive frequency mode for a Bunch-Davies vacuum\n$|0\\rangle$ normalized by the Wronskian condition\n\\begin{eqnarray} \\label{vecsol}\n\\Psi_{\\bf k}^{s}(z) =\\sqrt{\\frac{\\kappa m^2}{k^3}} e^{iz}\n\\end{eqnarray}\nas the solution to (\\ref{v1eq}).\n On the other hand, the\nvector power spectrum is defined by\n\\begin{eqnarray}\\label{powerv}\n\\langle0|\\hat{\\Psi}_{j}(\\eta,{\\bf x})\\hat{\\Psi}^{j}(\\eta,{\\bf\nx}')|0\\rangle=\\int d^3{\\bf k}\\frac{{\\cal P}_{\\Psi}}{4\\pi\nk^3}e^{i{\\bf k}\\cdot({\\bf x}-{\\bf x^{\\prime}})},\n\\end{eqnarray}\nwhere we take  the Bunch-Davies vacuum state $|0\\rangle$  by\nimposing $\\hat{a}_{\\bf k}^{s}|0\\rangle=0$. The vector  power\nspectrum ${\\cal P}_{\\Psi}$ takes the form  \\begin{equation}\n\\label{vecpt}{\\cal\nP}_{\\Psi}\\equiv\\sum_{s=1,2}\\frac{k^3}{2\\pi^2}\\Big|\\Psi_{\\bf\nk}^{s}\\Big|^2.\\end{equation}\n Plugging (\\ref{vecsol}) into\n(\\ref{vecpt}), we find a constant power  spectrum for a vector\nperturbation\n\\begin{eqnarray} \\label{vec-pow}\n{\\cal P}_{\\Psi}=\\frac{m^2}{\\pi^2 M^2_{\\rm P}}.\n\\end{eqnarray}\n\n\\subsection{Tensor power spectrum}\n\n\nWe take   Eq.(\\ref{heq}) to compute  tensor power spectrum. In this\ncase, the metric tensor $h_{ij}$ can be expanded in Fourier modes\n\\begin{eqnarray}\\label{hijm}\nh_{ij}(\\eta,{\\bf x})=\\frac{1}{(2\\pi)^{\\frac{3}{2}}}\\int d^3{\\bf\nk}\\sum_{s={\\rm +,\\times}}p_{ij}^{s}({\\bf k})h_{\\bf\nk}^{s}(\\eta)e^{i{\\bf k}\\cdot{\\bf x}},\n\\end{eqnarray}\nwhere  $p^{s}_{ij}$ linear polarization tensors with $p^{s}_{ij}\np^{s,ij}=1$. Also, $h_{\\bf k}^{s}(\\eta)$ represent linearly\npolarized tensor modes. Plugging (\\ref{hijm}) into (\\ref{heq}) leads\nto the fourth-order differential equation\n\\begin{eqnarray}\n&&(h_{\\bf k}^{s})^{''''}+2k^2(h_{\\bf k}^{s})^{''}+k^4h_{\\bf k}^{s}\n=0,\\label{heq2}\n\\end{eqnarray}\nwhich is further rewritten as a factorized form\n\\begin{eqnarray}\n&&\\Bigg[\\frac{d^2}{d\\eta^2}+k^2\\Bigg]^2h_{\\bf k}^{s}(\\eta)=0.\n\\label{hee}\n\\end{eqnarray}\nIntroducing  $z=-k\\eta$, Eq.(\\ref{hee}) takes the form of a\ndegenerate fourth-order equation\n\\begin{equation}\\label{hc0}\n\\Big[\\frac{d^2}{dz^2}+1\\Big]^2h_{\\bf k}^{s}(z)=0.\\end{equation} This\nis  the same equation for a degenerate Pais-Uhlenbeck (PU)\noscillator and its solution is given by\n\\begin{equation} \\label{desol}\nh_{\\bf k}^{s}(z)=\\frac{N}{2k^2}\\Big[(a_2^s+a_1^s z)e^{iz}+c.c.\\Big]\n\\end{equation}\n with $N$ the normalization constant. After quantization, $a^s_2$\n and $a^s_1$ are promoted to operators $\\hat{a}^s_2({\\bf k})$ and $\\hat{a}^s_1({\\bf\n k})$.   The presence of $z$\nin $(\\cdots)$ reflects clearly that $ h_{\\bf k}^{s}(z)$ is a\nsolution to the degenerate\n equation (\\ref{hc0}).\nHowever, it is difficult to quantize $h_{ij}$ in the subhorizon\nregion directly  because it satisfies the degenerate fourth-order\nequation (\\ref{heq}). In order to quantize $h_{ij}$, we have to\nconsider (\\ref{heq}) as a final equation obtained by making use of\nan auxiliary tensor $\\beta_{ij}$.\n\nFor this purpose,  one rewrites the fourth-order action (\\ref{hpeq})\nas a second-order action\n\\begin{equation}\nS_{\\rm AC}^{({\\rm T})}=-\\frac{1}{4 \\kappa m^2}\\int\nd^4x\\Big(\\eta^{\\mu\\nu}\\partial_\\mu \\beta_{ij}\\partial_\\nu h^{ij}\n+\\frac{1}{2} \\beta_{ij}\\beta^{ij}\\Big).\\label{ahpeq}\n\\end{equation}\nTheir equations are given by\n\\begin{equation}\n\\Box h_{ij}=\\beta_{ij},~~\\Box \\beta_{ij}=0,\n\\end{equation}\nwhich are combined to give the fourth-order tensor equation\n(\\ref{heq}). Explicitly, acting $\\Box$ on the first equation leads\nto (\\ref{heq}) when one uses the second one. Actually, this is an\nextension of the singleton  action describing a dipole ghost pair as\nthe fourth-order scalar\ntheory~\\cite{Myung:1999nd,Rivelles:2003jd,Kim:2013waf,Kim:2013mfa}.\nThis is related to not a non-degenerate PU oscillator and its\nquantization, but a degenerate PU and\nquantization~\\cite{Mannheim:2004qz,Smilga:2008pr}. The canonical\nconjugate momenta are given by\n\\begin{equation}\n\\pi_h^{ij}=\\frac{1}{4\\kappa\nm^2}\\beta'^{ij},~~\\pi_\\beta^{ij}=\\frac{1}{4\\kappa m^2}h'^{ij}.\n\\end{equation}\nAfter expanding $\\hat{h}_{ij}$ and $\\hat{\\beta}_{ij}$ in their\nFourier modes, their amplitudes at each mode are given by\n\\begin{eqnarray}\n\\label{sol1}\\hat{\\beta}^s_{\\bf k}(z)&=&iN\\Big(\\hat{a}^s_1({\\bf k})e^{iz}-\\hat{a}^{s\\dagger}_1({\\bf k})e^{-iz}\\Big),\\\\\n \\label{sol2}\\hat{h}^s_{\\bf k}(z)&=&\\frac{N}{2k^2}\\Big[\\left(\\hat{a}^s_2({\\bf k})+\\hat{a}^s_1({\\bf\n k})z\\right)e^{iz}+{\\rm h.c.}\\Big].\n \\end{eqnarray}\nNow, the canonical quantization is accomplished by imposing\nequal-time commutation relations:\n\\begin{eqnarray}\\label{comm}\n[\\hat{h}_{ij}(\\eta,{\\bf x}),\\hat{\\pi}_{h}^{ij}(\\eta,{\\bf\nx}^{\\prime})]=2i\\delta^3({\\bf x}-{\\bf\nx}^{\\prime}),~~[\\hat{\\beta}_{ij}(\\eta,{\\bf\nx}),\\hat{\\pi}_{\\beta}^{ij}(\\eta,{\\bf x}^{\\prime})]=2i\\delta^3({\\bf\nx}-{\\bf x}^{\\prime}),\n\\end{eqnarray}\nwhere the factor 2 is coming  from the fact that $h_{ij}$ and\n$\\beta_{ij}$ represent 2 DOF, respectively. Taking (\\ref{sol1}) and\n(\\ref{sol2}) into account, the two operators $\\hat{\\beta}_{ij}$ and\n$\\hat{h}_{ij}$ are given by\n\\begin{eqnarray}\\label{hex1}\n\\hat{\\beta}_{ij}(z,{\\bf x})&=&\\frac{ N}{(2\\pi)^{\\frac{3}{2}}}\\int\nd^3{\\bf k}\\Bigg[\\sum_{s=+,\\times}\\Big(ip_{ij}^{s}({\\bf\nk})\\hat{a}_1^s({\\bf k})e^{iz}e^{i{\\bf k}\\cdot{\\bf\nx}}\\Big)+{\\rm h.c.}\\Bigg], \\\\\n\\label{hex2} \\hat{h}_{ij}(z,{\\bf\nx})&=&\\frac{1}{(2\\pi)^{\\frac{3}{2}}}\\int d^3{\\bf\nk}\\frac{N}{2k^2}\\Bigg[\\sum_{s=+,\\times}\\Big\\{p_{ij}^{s}({\\bf\nk})\\Big(\\hat{a}_2^s({\\bf k})+\\hat{a}_1^s({\\bf k})z\n\\Big)e^{iz}e^{i{\\bf k}\\cdot{\\bf x}}\\Big\\}+{\\rm h.c.}\\Bigg].\n\\end{eqnarray}\nPlugging (\\ref{hex1}) and (\\ref{hex2}) into (\\ref{comm}) determines\nthe normalization constant $N=\\sqrt{2\\kappa m^2}$ and commutation\nrelations between $\\hat{a}_i^s({\\bf k})$ and\n$\\hat{a}^{s\\dagger}_j({\\bf k}')$ as\n \\begin{equation} \\label{scft}\n [\\hat{a}_i^s({\\bf k}), \\hat{a}^{s^{\\prime}\\dagger}_j({\\bf k}')]= 2k \\delta^{ss'}\n \\left(\n  \\begin{array}{cc}\n   0 & -i  \\\\\n    i & 1 \\\\\n  \\end{array}\n \\right)\\delta^3({\\bf k}-{\\bf k}').\n \\end{equation}\nHere the commutation  relation of $ [\\hat{a}_2^s({\\bf k}),\n\\hat{a}^{s^{\\prime}\\dagger}_2({\\bf k}')]$ is determined by the\ncondition of  \\begin{equation} [\\hat{h}_{ij}(\\eta,{\\bf\nx}),\\hat{\\pi}_{\\beta}^{ij}(\\eta,{\\bf x}^{\\prime})]=0.\n\\end{equation}\nWe are ready to compute  the power spectrum of the\ngravitational waves defined by\n\\begin{eqnarray}\\label{power}\n\\langle0|\\hat{h}_{ij}(\\eta,{\\bf x})\\hat{h}^{ij}(\\eta,{\\bf\nx^{\\prime}})|0\\rangle=\\int d^3k\\frac{{\\cal P}_{\\rm h}}{4\\pi\nk^3}e^{i{\\bf k}\\cdot({\\bf x}-{\\bf x^{\\prime}})}.\n\\end{eqnarray}\nHere we choose the Bunch-Davies vacuum $|0\\rangle$ by imposing\n$\\hat{a}_i^s({\\bf k})|0\\rangle=0$.  The tensor power spectrum ${\\cal\nP}_{\\rm h}$ in (\\ref{power}) denotes ${\\cal P}_{\\rm\nh}\\equiv\\sum_{s={+,\\times}}{\\cal P}^s_{\\rm h}$ where $ {\\cal\nP}^s_{\\rm h}$ is given as\n\\begin{eqnarray}\n{\\cal P}^s_{\\rm h}=\\frac{k^3}{2\\pi^2}|h_{\\bf\nk}^{s}\\Big|^2=\\frac{m^2}{2\\pi^2M^2_{\\rm P}}.\n\\end{eqnarray}\nFinally, we obtain the tensor power spectrum\n\\begin{equation}\n{\\cal P}_{\\rm h}=\\frac{m^2}{\\pi^2M^2_{\\rm P}}\n\\end{equation}\nwhich corresponds to a constant power spectrum. This is the same\nform as for the vector power spectrum (\\ref{vec-pow}).\n\nOn the other hand, the power spectrum of auxiliary tensor\n$\\beta_{ij}$ is  defined  by\n\\begin{equation}\n\\langle0|\\hat{\\beta}_{ij}(\\eta,{\\bf x})\\hat{\\beta}^{ij}(\\eta,{\\bf\nx^{\\prime}})|0\\rangle=\\int d^3k\\frac{{\\cal P}_{\\rm \\beta}}{4\\pi\nk^3}e^{i{\\bf k}\\cdot({\\bf x}-{\\bf x^{\\prime}})}. \\end{equation} Here\nwe obtain the zero power spectrum as\n\\begin{equation}\n{\\cal P}_{\\rm \\beta}=0\n\\end{equation}\nwhen one used  the commutation relation $[\\hat{a}_1^s({\\bf k}),\n\\hat{a}^{s^{\\prime}\\dagger}_1({\\bf k}')]=0$. This is clear because\n$\\beta_{ij}$ is an auxiliary tensor to lower the fourth-order action\nto the second-order action. However, it is not  understand why\n$\\hat{h}_{ij}$ could be expanded  by $\\hat{a}_2^s({\\bf k})$ and\n$\\hat{a}_1^s({\\bf k})$ without introducing  $\\beta_{ij}$, because\n$\\beta_{ij}$ was expanded by $\\hat{a}_1^s({\\bf k})$ solely.\n\n\\section{Discussions}\n\nWe have found the constant vector and tensor power spectra generated\nduring de Sitter inflation from conformal gravity. These constant\npower spectra could be understood because the conformal gravity is\ninvariant under conformal (Weyl) transformation. This means that\ntheir power spectra are constant with respect to $z=-k\\eta$ since\nvector and tensor perturbations are decoupled from the expanding de\nSitter inflation. In other words, this is so because the bilinear\nactions (\\ref{vpeq}) and (\\ref{hpeq}) are independent of the\nconformal scale factor $a(\\eta)$ as a result of conformal\ninvariance. On the contrary of Ref.\\cite{Mannheim:2011is},  it is\nless interesting for the conformal gravity to further investigate\nits cosmological implications.\n\n Hence, our\nanalysis implies that the Einstein-Weyl gravity is more promising\nthan the conformal gravity to obtain the physical  tensor power\nspectrum because the Einstein-Hilbert term provides the coupling of\nscale factor $a$ like as\n$a^2(h'_{ij}h'^{ij}-\\partial_lh_{ij}\\partial^lh^{ij})$. Also, the\nsingleton Lagrangian of ${\\cal\nL}_s=-\\sqrt{-\\bar{g}}(\\frac{1}{2}\\bar{g}^{\\mu\\nu}\\partial_\\mu\\phi_1\\partial_\\nu\\phi_2+\\frac{1}{2}\\phi_1^2)$\nis quite interesting because it provides two scalar equations $(\\Box\n+2aH\\partial_\\eta)\\phi_2=\\phi_1$ and $(\\Box +2aH\\partial_\\eta)\n\\phi_1=0$ which are combined to yield the degenerate fourth-order\nequation $(\\Box +2aH\\partial_\\eta)^2 \\phi_2=0$. Here we observe the\npresence of the scale factor $a$  in the perturbed equation of the\nsingleton.\n\n Consequently, the\nconformal invariance of the Lagrangian  like  $\\sqrt{-g}C^2$ or\n$\\sqrt{-g}F^2$ has no responsibility for generating the observed\nfluctuations during inflation.\n\n\\vspace{0.25cm}\n\n {\\bf Acknowledgement}\n\n\\vspace{0.25cm}\n This work was supported by the National\nResearch Foundation of Korea (NRF) grant funded by the Korea\ngovernment (MEST) (No.2012-R1A1A2A10040499).\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nDeep neural networks (DNNs) are a popular machine learning technique and have shown superior performance in many scientific problems. Despite of their high prediction accuracy, DNNs are often criticized for a lack of interpretation of how changes of the input variables influence the output. Indeed, for applications in many scientific fields such as biology and medicine, understanding the statistical models described by the networks can be as important as, if not more important than, the prediction accuracy. In a DNN, because of its nonlinearity and inherent complexity, generally one should not expect a concise relationship between each input variable and the output, such as the conditional monotonicity in linear regression or logistic regression. A more realistic approach for interpreting the DNN model can be selecting a subset of variables, among all input variables, that have significant predictive power on the output, which is known as ``variable selection''. This paper considers the variable selection problem in DNNs.\n\nDuring the past decades, many methods have been proposed for this task. The variable selection methods for neural networks, similar to the ones for other machine learning techniques, can be broadly classified into three categories: filters, wrappers and embedded methods \\cite{may_review_2011-1,guyon_introduction_2003,chandrashekar_survey_2014}. Filters select variables by information theoretic criteria such as mutual information \\cite{battiti_using_1994} and partial mutual information \\cite{may_non-linear_2008}, and the selection procedure does not involve network training. In contrast, both wrappers and embedded methods are based on the training of neural networks. Wrappers wrap the training phase with a search strategy, which searches through the set, or a subset, of all possible combinations of input variables and selects the combination whose corresponding network gives the highest prediction accuracy. A number of sequential \\cite{sung1998ranking,maier1998use} and heuristic search strategies \\cite{brill_fast_1992,tong_genetic_2010,sivagaminathan_hybrid_2007} have been used. Embedded methods, unlike wrappers, select variables during the training of the network of interest. This can be done by gradually removing\/pruning weights or variables according to their importance measured in various ways (a detailed review is given in the Methods section) or by incorporating a regularization term into the loss function of the neural network to impose sparsity on the weights \\cite{grandvalet_outcomes_1999,chapados_input_2001,simila_combined_2009,scardapane_group_2017}. For a more exhaustive review of variable selection methods in neural networks, see \\cite{may_review_2011-1,zhang_neural_2000}.\n\nWhile a lot of variable selection methods have been developed for neural networks, there are still challenges that hinder them from being widely used. First and foremost, these methods lack a control on the quality of selected variables. When selecting from a large number of variables, a standard way of quality control is to calculate false discovery rate (FDR) \\cite{benjamini1995controlling} and control it at a certain level, particularly in biological and medical studies. In the context of variable selection, FDR is the (expected) proportion of false positives among all variables called significant; for example, if 20 variables are selected (called significant), and two of them are actually null, then the FDR is $2\/20=0.1$. However, no variable selection methods for neural networks so far have tried to estimate FDR or keep FDR under control. Second, among these methods, many were developed for specific types of networks, especially very shallow networks, and they do not work, or work inefficiently, for deeper networks. Third, many of the methods are not applicable to large datasets, on which their computational loads can be prohibitively high.\n\nIn this paper, we develop a method called SurvNet for variable selection in neural networks that overcomes these limitations. It is an embedded method that gradually removes least relevant variables until the FDR of remaining variables reaches a desired threshold. Figure \\ref{fig:1} is the flowchart of SurvNet. It starts by adding a set of simulated input variables called ``surrogate variables'' that will help estimate FDR and training a network with all variables, including both original and surrogate variables. Then it calculates the importance of each variable (original or surrogate) and eliminates the variables that are least important. When eliminating a variable, its corresponding input neuron and all outgoing connections of this neuron are removed from the network. After this, SurvNet estimates the FDR of the original variables that remain in the model. If the estimated FDR is greater than the pre-set threshold, SurvNet will go back to the step of training the (updated) network; otherwise, the elimination stops, and all remaining surrogate variables are removed before the final model is trained. Note that each updated network is trained using the values of weights in the last trained network as initial values for a ``warm start''.\n\nThere are three major novelties in this backward elimination procedure of SurvNet. First, it proposes a new measure\/score of variable importance, which works regardless of the type of problems (classification or regression), the number of output neurons (one or multiple), and the number of hidden layers (one or multiple) in neural networks. In fact, this score can be readily computed for networks with arbitrary depths and activation functions. Second, SurvNet proposes an easy and quick way of estimating FDRs. Statistical estimation of FDRs requires obtaining the null distribution of the importance scores, that is, the distribution of the scores of irrelevant variables \\cite{storey2003statistical}. This is often done by permuting the output values of samples and training multiple independent models in parallel, each of which corresponds to a permuted dataset, but the computational cost is typically unaffordable for neural networks. SurvNet proposes a distinct way. It generates a set of null variables which serve as surrogates of the (unknown) null original variables to obtain the null distribution. With the introduction of surrogate variables, an estimate of FDR can be given by a simple mathematical formula without training a large number of networks at each step. Third, instead of eliminating one variable or any pre-specified number of variables at each step, SurvNet is able to adaptively determine an appropriate number of variables to eliminate by itself. This number, expressed in a concise mathematical formula, makes the elimination highly efficient while having the FDR well controlled on the desired level. The formula includes a parameter called ``elimination rate'', which is a constant between 0 and 1 and controls the ``aggressiveness'' of elimination. When this parameter is chosen to be 1, the elimination is the most aggressive.\n\nPut together, SurvNet is a computationally efficient mechanism for variable selection in neural networks that needs little manual intervention. After setting the initial network structure, an FDR cutoff $\\eta ^*$ (0.1 is the most commonly used value), and an elimination rate $\\varepsilon$ (1 is often an acceptable choice), the elimination procedure will automatically determine how many and which variables to eliminate at each step and stop when the estimated FDR is no greater than $\\eta ^*$.\n\n\\section*{Data and results}\nWe applied SurvNet to digits 4's and 9's in the MNIST database (Dataset 5), a single-cell RNA-Seq dataset (Dataset 6), as well as four simulation datasets (Datasets 1 $\\sim$ 4).\n\nMNIST \\cite{lecun1998gradient} contains 60,000 training images (including 5,000 validation images) and 10,000 testing images of ten handwritten digits from 0 to 9. Each image contains $28\\times28 = 784$ pixels, which are treated as 784 input variables.\n\nSingle-cell RNA-Seq \\cite{kolodziejczyk2015technology} is a biological technique for measuring gene expression in cells. Along with other single-cell techniques, it was recognized as the ``2018 Breakthrough of the Year'' by the \\textit{Science} magazine on account of its important applications in biomedical and genomic research. In single-cell RNA-Seq data, the samples are the cells, the inputs are expression levels of individual genes, and the output is the cell type. Biologically, it is often believed that the cell type is determined by a small set of genes, and thus single-cell RNA-Seq data can be a good choice to study variable selection. \n\nThe classification accuracy of SurvNet for these real data was evaluated by several criteria, including initial test loss, initial test error, final test loss and final test error. Here ``test loss'' and ``test error'' refer to the cross-entropy loss and the misclassification rate on the test data, respectively; and their ``initial'' and ``final'' values were derived by using the network with all original variables and with selected variables only, respectively. See Supplementary Materials for details about how they were calculated.\n\nFor these real datasets (Datasets 5 $\\sim$ 6), however, it is unknown which variables are truly significant. Hence we relied on simulated data to quantitatively assess the accuracy of selected variables, the most important aspect of SurvNet. Four datasets were simulated under different schemes. Datasets 1 $\\sim$ 3 were for classification and Dataset 4 was for regression.\n\nExcept for the MNIST data, each dataset was divided into a training set and a test set, with 80\\% of the samples in the training set and 20\\% in the test set, and 30\\% of training samples were further separated for validation (used to decide when to stop training, see Supplementary Materials).\n\nSurvNet was implemented on TensorFlow 1.8 \\cite{tensorflow2015-whitepaper}. For each dataset, we used a common and simple network structure with two hidden layers, which consisted of 40 and 20 nodes respectively. The ReLU activation function was used, together with a batch size of 50 and a learning rate of 0.05 (0.01 for the regression problem).\n\nIn our experiments, Datasets 1 $\\sim$ 4 were simulated 25 times, and the results of variable selection using SurvNet are averaged over these 25 simulations. For Dataset 1, we demonstrate how SurvNet works step by step to look into its behavior, and we also study the influence of the elimination rate by setting $\\varepsilon$ to different values. On other simulation datasets, results are similar and thus are not given.\n\n\\subsection*{Dataset 1: simulated data with independent variables}\nWe simulated a $10,000\\times784$ matrix $\\bm{X}$, with $x_{ij} \\sim {\\rm i.i.d.}\\ U(0,1)$ for $1 \\le i \\le 10,000$, $1 \\le j \\le 784$, where $U$ means uniform distribution, and treated its rows and columns as samples and variables respectively. The samples were randomly assigned into two classes $C_1$ and $C_2$ of equal size. Then $p^\\prime = 64$ variables were chosen at random and their values in one class were shifted: for each of these variables, we generated a shift value $\\delta_j \\sim U(0.1,0.3)$, with its direction having equal probability of being positive and negative. More precisely, $x_{ij} \\leftarrow x_{ij} + (2\\alpha_j-1) \\cdot \\delta_j$ for $i \\in C_1$, $j \\in \\Omega_{p^\\prime}$, where $\\alpha_j \\sim {\\rm Bernoulli}(\\frac{1}{2})$ and $\\Omega_{p^\\prime}$ was the set of $p^\\prime$ randomly chosen variables. In this way, the 784 variables were independent from each other, and the 64 variables were significant because each of them had different mean values in the two classes. This ``independent-variable differential-mean'' scheme is a very widely used simulation scheme for studying variable selection.\n\nWe ran SurvNet on this dataset with an FDR cutoff $\\eta^* = 0.1$ and an elimination rate $\\varepsilon = 1$. To demonstrate how SurvNet works step by step, Figure \\ref{fig:2}a shows, in one instance of simulation, the number of original variables and surrogate variables left at each step of a selection process as well as the corresponding estimated FDR. The number of variables to be eliminated in the subsequent step is also displayed, and notice that our algorithm was efficient: it eliminated a large number of variables at the beginning and gradually slowed down the elimination as the number of remaining variables decreased and the estimated FDR got closer to the desired value. When the estimated FDR became less than 0.1, the selection process stopped, and the final model turned out to contain all the 64 truly significant variables. On the same data, we studied the influence of elimination rate, and the results of using $\\varepsilon = 1$ and $\\varepsilon = 0.5$ are shown in Figure \\ref{fig:2}b and \\ref{fig:2}c. It is found that while a larger elimination rate led to a faster selection process with fewer steps, the number of variables left at the end of the selection was almost the same (Figure \\ref{fig:2}b). Moreover, regardless of elimination rate, our method gave an accurate estimate of FDR, and the true value of FDR was well controlled throughout the selection process (Figure \\ref{fig:2}c).\n\nThe overall performance of SurvNet under $\\eta^* = 0.1$ and $\\varepsilon = 1$ was summarized in Table \\ref{tab:1}. The test loss and test error on the model with selected variables were both less than those on the model that contains all original variables, indicating enhanced predictive power of the network. More importantly, SurvNet accurately selected the significant variables: it kept 61.92 of the 64 significant variables, along with 7.42 false positives, and the selected variables had an FDR of 0.105, which was very close to the cutoff value 0.1. The estimated FDR, 0.093, was also close to the actual FDR.\n\nThe results under different elimination rates ($\\varepsilon = 1$ and $\\varepsilon = 0.5$), different FDR cutoffs ($\\eta^* = 0.1$ and $\\eta^* = 0.05$), and different numbers of significant variables ($p^\\prime = 64$ and $p^\\prime = 32$) are shown in Table S1.\n\n\\subsection*{Dataset 2: simulated data with correlated variables}\nWe considered correlated variables in this simulation dataset. It is well known that variable dependence often makes FDR estimation difficult \\cite{benjamini2001control, heesen2015inequalities}, and we wondered whether SurvNet was still valid in this case. Images are perfect examples of data with correlated variables, as the value of a pixel usually highly depends on the value of its surrounding pixels. Here we used all images of digit 0 in the MNIST data and randomly assigned them into two classes, and all variables were supposed to be non-significant for classification at this time. Then we picked $p^\\prime = 64$ variables and shifted their mean values in one class in the same way we did in Dataset 1.\n\nTable \\ref{tab:1} shows the performance of SurvNet under $\\eta^* = 0.1$ and $\\varepsilon = 1$. Similar to that on Dataset 1, the test loss decreased after variable selection. The test error before and after variable selection were both zero, possibly due to the positive correlation between pixels, which reduced the difficulty of the classification problem. Although SurvNet identified slightly fewer significant variables (59.36 of the 64 significant variables) than it did in Dataset 1, the FDR 0.107 was still very close to the desired cutoff, and its estimated value 0.094 was accurate as well. For results under different sets of parameter values, see Table S2.\n\n\\subsection*{Dataset 3: simulated data with variance-inflated variables}\nThe third simulation scheme is very challenging. Unlike in the previous two datasets, the significant variables did not differ in the mean values of the two classes; instead, they differed only in the variances. Same as in Dataset 1, we simulated a $10,000\\times784$ matrix $\\bm{X}$ whose element $x_{ij} \\sim {\\rm i.i.d.}\\ U(0,1)$ and divided the samples into two equal-size classes $C_1$ and $C_2$. But then, to make $p^\\prime=64$ randomly chosen variables significant, we let $x_{ij} \\leftarrow x_{ij} + (2\\alpha_{ij}-1) \\cdot \\delta_{ij}$ for $i \\in C_1$, $j \\in \\Omega_{p^\\prime}$, where $\\alpha_{ij} \\sim {\\rm Bernoulli}(\\frac{1}{2})$, and $\\delta_{ij} \\sim U(0.8,1)$. Note that different from the first two simulation schemes, here $\\delta$ and $\\alpha$ depend on both $i$ and $j$. Thus the means of these variables remained unchanged, but their standard deviations inflated from 0.29 to 0.95 (see Supplementary Materials for calculations). In other words, the only difference between the two classes was that the values of 64 out of 784 pixels were ``noisier''. In this case, classifiers and tests based on discrepancies in the mean values would fail. For example, t-test merely identified 0.20 (averaged over 25 instances) of the 64 significant variables.\n\nThe results of applying SurvNet with $\\eta^* = 0.1$ and $\\varepsilon = 1$ were shown in Table \\ref{tab:1}, and the first thing to notice is the dramatic improvement of classification accuracy on the test set. While the test error given by the network with all 784 vriables was 49.42\\%, it dropped to 0.47\\% after variable selection by SurvNet; that is, from an almost random guess to an almost perfect classification. This implies that the variable selection gives back to the DNN the ability of utilizing all types of information useful for classification, which was masked by the overwhelming irrelevant variables. \nAmong the selected variables, 23.00 were truly significant variables, and 3.40 were false positives. Although only 36\\% of the significant variables were successfully identified, the FDR of the remaining variables, 0.114, was close to the cutoff, and the estimated FDR was acceptably accurate.\n\nWe then scrutinized the selection process of SurvNet on this dataset, and found that the reason only a proportion of significant variables were retained was that the initial network that made almost random guesses could not accurately determine the importance of variables and thus many significant variables were removed. As the selection proceeded, the network gained higher classification accuracy and also stronger ability to distinguish the significant variables. When we used a smaller elimination rate, say $\\varepsilon = 0.5$, SurvNet was able to keep a larger proportion of significant variables (see Table S3 for details).\n\n\n\\subsection*{Dataset 4: simulated regression data}\nSuppose the data matrix is $\\bm{X}=(x_{ij})_{10,000 \\times 784}$, and each $x_{ij} \\sim U(-1,1)$. Of the 784 variables, 64 were randomly chosen as significant variables (denoted by $x_{k_j}, j=1,\\ldots,64$), and $y$ was set to be the linear combination of $x_{k_j}$ or its nonlinear functions, plus a few interaction terms and a random error term:\n\\[\n\\begin{split}\ny_i=\\sum_{j=1}^{16} \\beta_j x_{ik_j} + \\sum_{j=17}^{32} \\beta_j \\sin x_{ik_j} + \\sum_{j=33}^{48} \\beta_j e^{x_{ik_j}} + \\sum_{j=49}^{64} \\beta_j \\max (0,x_{ik_j}) \\\\\n+ \\beta_1^\\prime x_{ik_{15}} x_{ik_{16}} + \\beta_2^\\prime x_{ik_{31}} x_{ik_{32}} + \\beta_3^\\prime x_{ik_{47}} x_{ik_{48}} + \\beta_4^\\prime x_{ik_{63}} x_{ik_{64}} + \\varepsilon_i,\n\\end{split}\n\\]\nwhere $\\beta_j=(2\\alpha_j-1) \\cdot b_j$, $\\alpha_j \\sim {\\rm Bernoulli}(\\frac{1}{2})$, $b_j \\sim U(1,3)$, $\\varepsilon_i \\sim N(0,1)$ for $i=1,\\ldots,10,000$, $j=1,\\ldots,64$, and $\\beta_1^\\prime$, $\\beta_2^\\prime$, $\\beta_3^\\prime$, $\\beta_4^\\prime$ have the same distribution as $\\beta_j$.\n\nWe ran SurvNet with $\\eta^*=0.1$ and $\\varepsilon=1$ on 25 instances of simulation and the results are reported in the format of mean $\\pm$ standard deviation. After variable selection, the test loss was reduced greatly, from $33.013\\pm27.059$ to $8.901\\pm1.988$. The number of remaining original variables was $71.16\\pm5.02$ on average, and $63.96\\pm0.20$ of the 64 significant variables were kept. The actual FDR of the selected variables was $0.097\\pm0.061$, close to the desired value 0.1, and the estimated FDR, $0.094\\pm0.004$, was accurate. The results suggest that SurvNet is highly effective for this regression dataset.\n\n\\subsection*{Dataset 5: digits 4 and 9 in MNIST}\nAfter four simulation datasets, we applied SurvNet to the MNIST data. Here we only used the images of two digits that look alike (4 and 9), as they are similar in most pixels and are only different in pixels in certain regions. In Figure \\ref{fig:3}a, we show two representative 4's that differ in the width of top opening and two representative 9's that differ in the presence of a bottom hook. The four regions circled in red are likely to be most significant in differentiating 4's and 9's, especially the region in the upper middle denoting whether the top is closed or open, and the region in the lower middle denoting whether there is a hook at the bottom.\n\nFrom left to right, Figure \\ref{fig:3}b shows the pixels that were selected by SurvNet under four combinations of FDR cutoffs ($\\eta^* = 0.1$ or 0.01) and elimination rates ($\\varepsilon = 1$ or 0.5). The colors display the relative importance, defined by equation \\ref{eq:Sj_L2} (see Methods), of the selected pixels, and a darker color means greater importance. We found that different parameter settings gave quite consistent results, and they all picked out the four regions that were speculated to be significant.\n\n\\subsection*{Dataset 6: single-cell RNA-Seq data}\nChen \\textit{et al.} performed single-cell RNA-Seq analysis of the adult mouse hypothalamus and identified 45 cell types based on clustering analysis \\cite{chen_single-cell_2017}. We used 5,282 cells in two non-neuronal clusters, oligodendrocyte precursor cell (OPC) and myelinating oligodendrocyte (MO), which reflected two distinct stages of oligodendrocyte maturation. Following a standard pre-processing protocol of single-cell RNA-Seq data \\cite{hwang2018single}, we filtered out the genes whose expression could not be detected in more than 30\\% of these cells, which left 1,046 genes for further analysis, and used $\\log({\\rm TPM}+1)$ for measuring gene expression levels, where TPM standed for ``transcripts per million''.\n\nWith $\\eta^*=0.01$ and $\\varepsilon=1$, SurvNet selected 145 genes in one realization. Figure \\ref{fig:4} shows the heatmap of the expression values of these genes, in which rows are genes and columns are cells. The top banner shows the class labels for the samples. For gene expression data, the set of significant genes are typically identified by ``differential expression'' analysis, which finds differences in the mean expression levels of genes between classes. Indeed, as the heatmap shows, most genes have evidently different mean expression levels in the OPCs and MOs. However, among the 145 significant genes identified by SurvNet, 16 have log-fold-changes (logFCs) less than 1, meaning that their mean expression levels are not very different in the OPCs and MOs. In Figure \\ref{fig:4}, these genes are marked in purple on the left banner, in contrast to green for the other genes. Actually, Bartlett's test, which tests the difference in variance, claimed that 14 of these 16 genes had unequal variances in the two groups of cells (p-value < 0.05); thus, they are instances of variance-inflated variables selected by SurvNet, in addition to the ones in Dataset 3. Again, SurvNet demonstrates its ability to identify various types of significant variables, not just variables with different means.\n\nFurther, the functional interpretations of the selected genes match the biological characteristics of OPCs and MOs. We conducted Gene Ontology (GO) analysis using DAVID 6.8 program \\cite{huang2009systematic,da2009bioinformatics}, and found that these genes were likely to play an important role in a number of biological processes, for example, substantia nigra development (with p-value $8.8\\times10^{-9}$, fold enrichment 29.1), nervous system development ($1.8\\times10^{-5}$, 4.8), positive regulation of dendritic spine development ($1.2\\times10^{-3}$, 19.0) and astrocyte differentiation ($3.2\\times10^{-3}$, 34.5). In particular, oligodendrocyte differentiation ($1.8\\times10^{-3}$, 16.2) defines the transition from OPCs to their mature form (MOs) \\cite{rubio2004vitro,barateiro2014temporal}, and myelination ($7.1\\times10^{-5}$, 13.8), which is the process of generating myelin and is a kind of axon ensheathment ($3.5\\times10^{-2}$, 55.2), is unique to MOs \\cite{menn2006origin,barateiro2014temporal}. Corresponding to these processes, the selected genes were also enriched for cellular components such as myelin sheath ($2.4\\times10^{-19}$, 16.2), axon ($1.2\\times10^{-5}$, 5.0) as well as internode region of axon ($2.9\\times10^{-4}$, 106.1), and molecular functions like structural constituent of myelin sheath ($3.1\\times10^{-6}$, 115.3). Besides, among the 16 selected genes whose expression levels had no obvious differences in the OPCs and MOs, \\textit{Cd9} was involved in oligodendrocyte development \\cite{terada2002tetraspanin}, and \\textit{Ckb}, \\textit{Actb}, \\textit{Tuba1a} as well as \\textit{Gpm6b} were related to myelin sheath or myelin proteolipid protein PLP \\cite{jahn2009myelin,werner2013critical}.\n\nAfter variable selection, the test loss was reduced from $4.230\\times10^{-3}$ to $3.460\\times10^{-3}$, and the test error dropped from 0.083\\% to 0.076\\% (averaged over 25 realizations).\n\n\\section*{Conclusions and discussion}\nWe have presented a largely automatic procedure for variable selection in neural networks (SurvNet). It is based on a new measure of variable importance that applies to a variety of networks, deep or shallow, for regression or classification, and with one or multiple output units. More importantly, SurvNet is the first method that estimates and controls the FDR of selected variables, which is essential for applications where the trustworthiness of variable selection is pivotal. By introducing surrogate variables, it avoids training multiple networks in parallel. SurvNet also adjusts the number of variables to eliminate at each step, and the ``warm start'' nature of backward elimination facilitates the training of networks. On multiple simulation datasets and real datasets, SurvNet has effectively identified the significant variables and given a dependable estimate of FDR.\n\nSurvNet takes advantages of modern developments of DNNs. The importance scores of input variables that are based on derivatives with respect to the inputs can be efficiently computed by functions in deep-learning packages such as TensorFlow, PyTorch, and Theano. Moreover, advances in optimization techniques and computation platforms have made the training of DNNs highly scalable. In particular, DNNs can accommodate a large number of input variables, which enables the introduction of surrogate variables.\n\nGiven a dataset, SurvNet may select different sets of significant variables at different runs owing to the randomness originated from the generation of surrogate variables and the training of networks (e.g., the random initial values of weights). While the former is unique to SurvNet, the latter is ubiquitous to any applications of neural networks. The randomness caused by generating surrogate variables may be lowered by, for example, using a larger number of surrogate variables or assembling results from multiple runs, but this randomness should not be a major concern if it is not much larger than the inevitable randomness coming from network training. To study this, we take Dataset 5 as an example. Using $\\eta^*=0.1$ and $\\varepsilon=1$, we ran SurvNet 25 times, and found that SurvNet selected $114.16\\pm11.36$ variables; and the overlapped proportion of the selected variables in each pair of realizations was approximately 0.77. These results reflected both sources of randomness. Then we fixed the surrogate variables in each realization, and SurvNet selected $118.32\\pm7.94$ variables, with the overlapped proportion of the selected variables of each pair of realizations around 0.79. This indicates that for this dataset, the randomness brought by surrogate variables was much less than that by the training of networks. And (hopefully) as peace of mind, some other well-known techniques for statistical tests and variable selection, such as permutation tests and bootstrap tests (and especially, parametric bootstrap tests), also have extra randomness caused by permutations or random number generations, but they are still very widely used.\n\nNext we discuss how many surrogate variables should be generated. In all experiments in this paper, we simply set the number of surrogate variables ($q$) to be the same as the number of original variables ($p$). A larger $q$ may lower the randomness brought by the surrogate variables and thus give a more stable selection of variables and a more accurate estimate of FDR. These improvements can be noticeable and worth pursuing when the number of original variables is small. On the other hand, a larger number of surrogate variables may increase the computational load. As a rule of thumb, we recommend using $q=p$ for datasets with moderate to large sample size, and $q$ can be a few times larger than $p$ if $p$ is small and be smaller than $p$ if $p$ is very large.\n\nAlthough variable selection is critical to many real applications and is often considered one of the most fundamental problems in machine learning \\cite{trevor2009elements, tibshirani2015statistical}, it is worth noting that this task does not apply to certain problems or certain types of DNNs. As an example, for some image datasets like ImageNet \\cite{deng2009imagenet}, deep convolutional neural networks are often a good choice due to their translation invariance characteristics, as the object of interest, such as a dog, may appear at any position in an image and thus theoretically every pixel should be relevant. Also, in the area of natural language processing, where recurrent neural networks are often used, the number of input variables (i.e. the length of input sequence) is not fixed and variable selection makes little sense.\n\nThe main aim of variable selection is to identify significant variables, which may, for example, shed light on the mechanisms of biological processes or guide further experimental validation. Apart from that, an additional aim may be to improve the classification accuracy. Although we did observe an improvement of generalization accuracy on all our simulated and real datasets, such an improvement is not guaranteed even if the variable selection procedure works perfectly. In some datasets, except for a set of significant variables, all other variables are almost completely irrelevant to the outcome, and variable selection may give extra power in prediction. However, in some other datasets, the relevances of variables are not polarized; there are many variables each having a very small influence on the output, but their accumulative contribution is non-negligible. For these datasets, such variables are likely to be ruled out during selection since it is hard to confidently determine their individual significance, but ignoring all of them could cause a loss of prediction power.\n\n\\section*{Methods}\n\n\\subsection*{Measures of variable importance}\n\n\\subsubsection*{Notation}\nWe use a tuple ($\\bm{x}$,$\\bm{y}$) to represent the input and the output of the network, with $\\bm{y}$ being either one-dimensional or multi-dimensional. $x_j$ denotes the $j^\\mathrm{th}$ component of $\\bm{x}$, namely the $j^\\mathrm{th}$ variable, and ($\\bm{x}^{(i)}$,$\\bm{y}^{(i)}$) ($i=1,\\ldots,n$) is the $i^\\mathrm{th}$ sample, where $n$ is the total number of samples (in the training set). Given a proper form of the loss $L(\\cdot,\\cdot)$, the loss function $L^*=\\sum_{i=1}^n L(\\bm{y}^{(i)},f(\\bm{x}^{(i)}))$, where $f$ denotes the output function of the network. The most popular choices for $L(\\cdot,\\cdot)$ are the squared error loss for regression problems and the cross-entropy loss for classification problems.\n\n\\subsubsection*{Existing measures}\nMany statistics have been proposed to measure the importance of variables in neural networks, and they generally fall into two categories \\cite{tetko_neural_1996, steppe_feature_1997}.\n\nOne category of methods estimate the importance of $x_j$, denoted by $S_j$, based on the magnitudes of the connection weights in the network \\cite{sen_predicting_1995, yacoub_hvs:_1997, garson_interpreting_1991, nath_determining_1997, gevrey_review_2003}. A simple example is the sum of absolute values of input weights \\cite{sen_predicting_1995}, but larger values of weights in the input layer do not mean greater importance if connections in hidden layers have small weights, and a better alternative is to replace the input weights with the products of the weights on each path from this input to the output \\cite{yacoub_hvs:_1997}. These measures were developed for networks with only one hidden layer, and they are unlikely to work well for deeper networks as the outgoing weights of a neuron does not reflect its importance once the neuron is inactive (e.g., when the input of a sigmoid neuron is far from zero or the input of a ReLU neuron is negative).\n\nThe other category of methods estimate $S_j$ by the sum of influences of the input weights on the loss function, i.e. $S_j=\\sum_{k \\in \\Omega_j} \\delta L^*_k$, where $\\Omega_j$ is the set of outgoing weights from the $j^\\mathrm{th}$ input neuron, and $\\delta L^*_k$ is the increment of the loss function caused by the removal of weight $w_k$ \\cite{tetko_neural_1996}. $\\delta L^*_k$ can be approximated by a Taylor series of the loss function using first-order terms \\cite{mozer_skeletonization:_1989, karnin_simple_1990} or second-order terms \\cite{lecun_optimal_1990, cibas_variable_1994, hassibi_second_1993}. However, it is unclear why $S_j$ equals the (unweighted) sum of $\\delta L^*_k$'s.\n\nApart from these two major categories of measures, it was also proposed to use $S_j=\\frac{\\partial f}{\\partial x_j}$, i.e. $S_j=\\frac{\\partial y}{\\partial x_j}$, when the output $y$ is one-dimensional \\cite{dimopoulos_use_1995,dimopoulos_neural_1999}. But it is unclear how $S_j$ should be defined when there are multiple output units. Let $y_1,\\ldots,y_K$ be the output values of $K$ output units, and one definition of $S_j$ was given by $S_j=\\sum_{k=1}^{K} |\\frac{\\partial y_k}{\\partial x_j}|$ \\cite{ruck_feature_1990}. However, using this summation seems problematic in some cases, especially when $y_1,\\ldots,y_K$ are the outputs of softmax functions.\n\n\\subsubsection*{Our new measure}\nWe propose a simple and direct measure of the importance of variable $j$ based on $\\frac{\\partial L}{\\partial x_j}$, which describes how the loss changes with $x_j$. There are a few advantages of using $\\frac{\\partial L}{\\partial x_j}$. First, regardless of the structure of the network and whether the output(s) is\/are continuous or categorical, $L$ is always well defined since it is the target for the optimization\/training of the network. Thus the proposed measure is applicable to a wide variety of networks. Second, no matter how many output units there are, $L$ is always a scalar and hence $\\frac{\\partial L}{\\partial x_j}$ is always a scalar. There is no trouble in how to combine effects from multiple output units. Third, $\\frac{\\partial L}{\\partial x_j}$ is easily computable with the backpropogation method, and popular frameworks\/libraries for DNN computations (e.g., TensorFlow, PyTorch and Theano) all use differentiators that efficiently compute partial derivatives (gradients) of arbitrary forms.\n\nNote that $\\frac{\\partial L}{\\partial x_j}$ is a function of the tuple ($\\bm{x}$,$\\bm{y}$), and hence it is natural to estimate it by its mean over all observations in the training set. To avoid cancellation of positive and negative values, we measure the importance of $x_j$ by the mean of absolute values\n\\begin{equation}\n\tS_j=\\frac{1}{n} \\sum_{i=1}^n |\\frac{\\partial L}{\\partial x_j}(\\bm{y}^{(i)},f(\\bm{x}^{(i)}))|,\n\t\\label{eq:Sj_L1}\n\\end{equation}\nor the mean of squares\n\\begin{equation}\n\tS_j=\\frac{1}{n} \\sum_{i=1}^n \\frac{\\partial L}{\\partial x_j}(\\bm{y}^{(i)},f(\\bm{x}^{(i)}))^2,\n\t\\label{eq:Sj_L2}\n\\end{equation}\nwhere $\\frac{\\partial L}{\\partial x_j}(\\bm{y}^{(i)},f(\\bm{x}^{(i)}))$ is the value of $\\frac{\\partial L}{\\partial x_j}$ at the $i$'th training sample.\n\nThe importance scores given by equation \\ref{eq:Sj_L1} and equation \\ref{eq:Sj_L2} implicitly assume that all the input values have similar range, which is typically the case for DNNs, since it is common practice to standardize\/scale the variables before supplying them to the network for the sake of faster and more stable training of the network \\cite{bishop1995neural, lecun2012efficient}. If this is not the case, we suggest the score in equation \\ref{eq:Sj_L1} be multiplied by the (sample) standard deviation of $x_j$ and the score in equation \\ref{eq:Sj_L2} be multiplied by the (sample) variance of $x_j$.\n\nNote that in the case of multiple linear regression, $L = \\frac{1}{2} (y-\\hat{y})^2 = \\frac{1}{2} (y-\\sum_j \\beta_j x_j)^2$, where $y$ is a scalar response and $\\beta_j$ is the $j^\\mathrm{th}$ regression coefficient, then $\\frac{\\partial L}{\\partial x_j}=-(y-\\hat{y})\\beta_j$. Thus, $S_j$ is defined as $|\\beta_j| \\cdot \\frac{1}{n} \\sum_{i=1}^n |e_i|$ or $\\beta_j^2 \\cdot \\frac{1}{n} \\sum_{i=1}^n e_i^2 $ by (1) and (2) respectively, where $e_i=y^{(i)}-\\hat{y}^{(i)}$. Note that $S_j$ is proportional to $|\\beta_j|$ or $\\beta_j^2$ as $\\frac{1}{n} \\sum_{i=1}^n |e_i|$ and $\\frac{1}{n} \\sum_{i=1}^n e_i^2$ are constants. Therefore, both of them are reasonable measures of the contribution of the $j^\\mathrm{th}$ variable, and they are actually equivalent in this case. The meaning of $S_j$ in some other special cases, such as linear regression with multiple outputs and logistic regression with one or multiple outputs, is elaborated in Supplementary Materials.\n\nAll results in the main text were obtained using equation \\ref{eq:Sj_L2}. Results obtained using equation \\ref{eq:Sj_L1} (given in Supplementary Materials) are not significantly different.\n\n\\subsection*{Elimination procedure with FDR control}\nIn this section, we first introduce how we estimate FDR and then talk about how we use this estimate to determine the number of variables to eliminate at each step.\n\n\\subsubsection*{Introduction of surrogate variables}\nThe key of estimating FDR \\cite{storey2003statistical} is to estimate\/generate the null distribution of the test statistic. In our case, it is to obtain the distribution of the importance score $S_j$ defined by equation \\ref{eq:Sj_L2} or equation \\ref{eq:Sj_L1} for variables that are not significant. Since the network is a complicated and highly nonlinear model, a theoretical distribution that applies to various network structure and various types of data may not exist. This null distribution needs to be obtained for the network and the data in hand.\n\nHowever, it is usually unknown which variables are truly null. If we construct the null distribution by permuting the output values of the data, it seems inevitable to train multiple networks from scratch in parallel. For this reason, we propose to introduce\/add a number of variables that are known\/generated to be null. We call these variables ``surrogate null variables'' (or ``surrogate variables'' for short). These variables will be concatenated with the original variables to form a larger data matrix.\n\nTo be precise, suppose there are $p$ original variables and $n$ training samples (including validation samples). Then after we add $q$ surrogate variables, the new data matrix will be of size $n\\times(p+q)$, which binds the original $n\\times p$ data matrix $\\bm{X}$ with a $n\\times q$ data matrix for surrogate variables $\\bm{X}_s$. It is assumed that the original variables are distributed in similar ranges or have been standardized, which is a suggested pre-processing step as it benefits the training of the network, and the elements in $\\bm{X}_s$ are sampled with replacement (or without replacement when $q \\le p$) from the elements in $\\bm{X}$. As a result, the $q$ surrogate variables are null, and their importance scores give the null distribution.\n\nWe recommend $q$ to be on the same scale as $p$ (see Conclusions and Discussion for a more detailed discussion about the choice of $q$). For convenience, $q$ takes the same value as $p$ in all experiments in this paper. In this case, the elements in $\\bm{X}_s$ can be generated by permuting the elements in $\\bm{X}$.\n\nThe selection procedure of SurvNet starts with using all $p+q$ variables as inputs. Then at each step, it eliminates a number of least important variables, including both original variables and surrogate variables. The remaining variables are used to continue training the network, and the elimination stops once the FDR falls below the cutoff.\n\n\\subsubsection*{FDR estimation}\nThen we consider how to estimate FDR at any given time of the selection process. Suppose $r$ variables are retained in the network, among which there are $r_0$ surrogate variables, then $r_0\/q$ proportion of surrogate (null) variables have not been eliminated yet. Accordingly, one would expect that roughly the same proportion of null original variables still exist at this time, that is, approximately $\\frac{r_0}{q} \\cdot p_0$ variables among the remaining original variables are falsely called significant, where $p_0$ is the number of null variables in the original dataset. Thus, an estimate of the FDR of the $r-r_0$ original variables is given by\n\\begin{equation}\n\t\\tilde{\\eta}=\\frac{\\frac{r_0}{q} \\cdot p_0}{r-r_0}\n\\end{equation}\nIn practice, however, $p_0$ is unknown, and a common strategy is to replace it with its upper bound $p$ \\cite{storey2003statistical}. Hence we have the following estimated FDR,\n\\begin{equation}\n\t\\hat{\\eta}=\\frac{\\frac{r_0}{q} \\cdot p}{r-r_0}=\\frac{r_0}{r-r_0} \\cdot \\frac{p}{q}\n\\label{eq:est_fdr}\n\\end{equation}\nApparently, when $\\hat{\\eta}$ is controlled to be no greater than a pre-specified threshold $\\eta^*$, $\\tilde{\\eta}$ is guaranteed to be no greater than $\\eta^*$ as well. When $q=p$, $\\hat{\\eta}$ can be simplified as $\\frac{r_0}{r-r_0}$.\n\n\n\\subsubsection*{Determination of the number of variables to eliminate}\nIf the estimated FDR $\\hat{\\eta}$ (given by equation \\ref{eq:est_fdr}) is less than or equal to the FDR cutoff $\\eta^*$, the variable selection procedure stops. Otherwise, the procedure proceeds, and we want to decide how many variables to eliminate among the $r$ variables that are still in the model. Let this number be $m$, and the determination of $m$ is based on the following considerations. On one hand, we expect that the elimination process is time-saving and reaches the FDR threshold quickly; on the other hand, we want to avoid eliminating too many variables at each step, in which case the FDR may fall much lower than the threshold. We have\n\\begin{thm}\n\tIf $m$ variables are further eliminated from the current model, the smallest possible estimated FDR after this step of elimination is\n\t\\begin{equation}\n\t\\min \\hat{\\eta}^{\\rm new}= (1-\\frac{m}{r_0})\\cdot \\hat{\\eta},\n\t\\label{eq:rm_new}\n\t\\end{equation}\n\twhere $r_0$ is the number of surrogate variables that are in the model before this step of elimination.\n\\end{thm}\n\\begin{proof}\n\tSuppose there are $m_0$ surrogate variables among the $m$ variables to be eliminated, $0 \\le m_0 \\le m$, then according to equation \\ref{eq:est_fdr}, $\\hat{\\eta}$ will be updated to\n\t\\begin{equation}\n\t\\hat{\\eta}^{\\rm new}=\\frac{r_0-m_0}{r-r_0-(m-m_0)} \\cdot \\frac{p}{q}.\n\t\\end{equation}\n\tNote that $\\hat{\\eta}^{\\rm new}$ is monotonically decreasing with respect to $m_0$ for any fixed  $m$, we have\n\t\\begin{equation}\n\t\\min \\hat{\\eta}^{\\rm new}=\\hat{\\eta}^{\\rm new}|_{m_0=m}=\\frac{r_0-m}{r-r_0} \\cdot \\frac{p}{q}.\n\t\\label{eq:rm_new_int}\n\t\\end{equation}\n\tEquation \\ref{eq:est_fdr} indicates that $\\frac{1}{r - r_0} \\cdot \\frac{p}{q}=\\frac{\\hat{\\eta}}{r_0}$. Plugging it into \\ref{eq:rm_new_int}, we have\n\t\\[\n\t\\min \\hat{\\eta}^{\\rm new}=(r_0-m) \\cdot \\frac{\\hat{\\eta}}{r_0} = (1-\\frac{m}{r_0})\\cdot \\hat{\\eta}.\n\t\\]\n\\end{proof}\n\nIt follows from equation \\ref{eq:rm_new} that $\\min \\hat{\\eta}^{\\rm new} = \\eta^*$ when $m=(1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$. Also, note that $\\min \\hat{\\eta}^{\\rm new}$ is a monotonically decreasing function of $m$. Therefore, when $m < (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$, $\\min \\hat{\\eta}^{\\rm new} > \\eta^*$ and thus $\\hat{\\eta}^{\\rm new} > \\eta^*$. That is,\n\\begin{cor}\n\tWhen $m < (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$, the estimated FDR after this step of elimination $\\hat{\\eta}^{\\rm new}$ is guaranteed to be still greater than the FDR cutoff $\\eta ^*$.\n\\end{cor}\nOn the other hand, when $m \\ge (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$, $\\min \\hat{\\eta}^{\\rm new} \\le \\eta^*$. That is,\n\\begin{cor}\n\tWhen $m \\ge (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$, the estimated FDR after this step of elimination $\\hat{\\eta}^{\\rm new}$ may reach the FDR cutoff $\\eta ^*$.\n\\end{cor}\n\nCorollary 1 says that $m$ values less than $(1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$ are ``safe'' but the elimination will not stop after this step. Corollary 2 says that $m$ values much larger than $(1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0$ may not be ``safe'' anymore. Taking both into consideration, we choose the step size to be\n\\begin{equation}\n    m=\\lceil (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0 \\rceil,\n    \\label{eq:m}\n\\end{equation}\nwhere $\\lceil \\cdot \\rceil$ denotes ``ceiling'', i.e. the smallest integer that is no less than $\\cdot$. Notice that when $\\hat{\\eta} > \\eta^*$, which is the premise of continuing to eliminate variables, $1-\\frac{\\eta^*}{\\hat{\\eta}}>0$, and $r_0>0$ as well since $\\hat{\\eta}$ is positive. Thus $m$ is ensured to be no less than 1 at each step of variable elimination.\n\nThis form of $m$ seems to be quite reasonable for the following reasons. First, if there still remain a great number of surrogate variables in the network, clearly more of them should be taken out. As $r_0$ decreases, $m$ will be smaller, and this makes sense since one should be more careful in further elimination. Second, when $\\hat{\\eta}$ is much higher than $\\eta^*$, one will naturally expect a larger $m$ so that the updated estimated FDR will approach this cutoff.\n\nUsing the $m$ determined by equation \\ref{eq:m}, there is a chance that the estimated FDR will get to the cutoff in only one step. Many times such a fast pace is not preferred as removing too many inputs at a time may make our warm start of the training not warm any more. Hence we may introduce an ``elimination rate'' $\\varepsilon$, which is a constant between 0 and 1, and take\n\\begin{equation}\n    m=\\lceil \\varepsilon \\cdot (1-\\frac{\\eta^*}{\\hat{\\eta}}) \\cdot r_0 \\rceil.\n\\end{equation}\n\n\n\\iffalse\n\n\\begin{itemize}\n\\item Bullet point one\n\\end{itemize}\n\n\\begin{enumerate}\n\\item Numbered list item one\n\\end{enumerate}\n\n\\fi\n\n\n\\section*{Author Contributions}\nJ.L. conceived the study, J.L. and Z.S. proposed the methods, Z.S. implemented the methods and constructed the data analysis, Z.S. drafted the manuscript, J.L. substantively revised it.\n\n\\section*{Competing Interests statement}\nThe authors declare no competing interests.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nOne of the fundamental results in commutative algebra is the irreducible decomposition theorem \\cite[Satz II and Satz IV]{N21} proved by Emmy Noether in 1921. In this paper she had showed that any ideal $I$ of a Noetherian ring $R$ can be expressed as a finite intersection of irreducible ideals, and the number of irreducible ideals in such an irredundant irreducible decomposition is independent of the choice of the decomposition. This number is then called the index of reducibility of $I$ and denoted by $\\mathrm{ir}_R(I)$. Although irreducible ideals belong to basic objects of commutative algebra, there are not so much papers on the study of irreducible ideals and the index of reducibility. Maybe the first important paper on irreducible ideals after Noether's work is of W. Gr\\\"{o}bner \\cite{G35} (1935). Since then there are interesting works on the index of reducibility of parameter ideals on local rings by D.G. Northcott \\cite{No57} (1957), S. Endo and M. Narita \\cite{EN64} (1964) or S. Goto and N. Suzuki \\cite{GS84} (1984). Especially, W. Heinzer, L.J. Ratliff and K. Shah propounded in a series of papers \\cite{HRS94}, \\cite{HRS95-1}, \\cite{HRS95-2}, \\cite{HRS95-3} a theory of maximal embedded components which is useful for the study of irreducible ideals. It is clear that the concepts of irreducible ideals, the index of reducibility and maximal embedded components can be extended for finitely generated modules.  Then the purpose of this paper is to investigate the index of reducibility of submodules of a finitely generated $R$-module $M$  concerning its maximal embedded components as well as the behaviour of the function ir$_M(I^nM)$, where $I$ is an ideal of $R$,  and to present  applications of the index of reducibility for studying the structure of the module $M$. The paper is divided into 5 sections. Let $M$ be a finitely generated module over a Noetherian ring and $N$ a submodule of $M$. We present in the next section a formula to compute the index of reducibility $\\mathrm{ir}_M(N)$ by using the socle dimension of the module $(M\/N)_{\\frak p}$ for all $\\frak p \\in \\mathrm{Ass}_R(M\/N)$ (see Lemma 2.3). This formula is a generalization of a well-known result which says that $\\mathrm{ir}_M(N) = \\dim_{R\/\\frak m} \\mathrm{Soc}(M\/N)$ provided $(R, \\frak m)$ is a local ring and $\\lambda_R(M\/N) < \\infty$. Section 3 is devoted to answer the following question: When is the index of reducibility of a submodule $N$  equal to the sum of the indices of reducibility of their primary components in a given irredundant primary decomposition of $N$? It turns out here that the notion of maximal embedded components of $N$ introduced by Heinzer, Ratliff and Shah is the key for answering this question (see Theorem \\ref{T3.2}). In Section 4, we consider the index of reducibility $\\mathrm{ir}_M(I^nM)$ of powers of an ideal $I$ as a function in $n$ and show that this function is in fact a polynomial for sufficiently large $n$. Moreover, we can prove that $\\mathrm{bight}_M(I)-1$ is a lower bound and the $\\ell_M(I)-1$ is an upper bound for the degree of this polynomial (see Theorem \\ref{T4.1}), where $\\mathrm{bight}_M(I)$ is the big height and \n $\\ell_M(I)$ is the analytic spread of $M$ with respect to the ideal $I$. However, the degree of this polynomial is still mysterious to us. We can only give examples to show that these bounds are optimal. In the last section, we involve in working out some applications of the index of reducibility. A classical result of Northcott  \\cite{No57} says that the index of reducibility of a parameter ideal in a Cohen-Macaulay local ring is dependent only on the ring and not on the choice of parameter ideals. We will generalize Northcott's result in this section and get a characterization for the Cohen-Macaulayness of a Noetherian module in terms of the index of reducibility of parameter ideals (see Theorem \\ref{T4.7}).\n\n\\section{Index of reducibility of submodules}\nThroughout this paper $R$ is a Noetherian ring and $M$ is a finitely generated $R$-module. For an $R$-module $L$, $\\lambda_R(L)$ denotes the length of $L$.\n\n\\begin{definition}\\rm A submodule $N$ of $M$ is called an {\\it irreducible submodule} if $N$ can not be written as an intersection of two properly larger submodules of $M$. The number of irreducible components of an irredundant irreducible decomposition of $N$, which is independent of the choice of the decomposition  by Noether \\cite{N21}, is called the {\\it index of reducibility} of $N$ and denoted by $\\mathrm{ir}_M(N)$.\n\\end{definition}\n\n\\begin{remark}\\rm We denoted by $\\mathrm{Soc}(M)$ the sum of all simple submodules of $M$. $\\mathrm{Soc}(M)$ is called the socle of $M$. If $R$ is a local ring with the unique maximal ideal $\\frak m$ and $\\frak k =  R\/\\frak m$ its residue field, then it is well-known that $\\mathrm{Soc}(M) = 0:_M\\frak m$ is a $\\frak k$-vector space of finite dimension. Let $N$ be a submodule of $M$ with $\\lambda_R(M\/N) < \\infty$. Then it is easy to check that $\\mathrm{ir}_M(N) = \\lambda_R((N:\\frak m)\/N) = \\dim_{\\frak k} \\mathrm{Soc}(M\/N).$\n\\end{remark}\n\nThe following lemma presents a formula for computing the index of reducibility $\\mathrm{ir}_M(N)$  without the requirement that $R$ is local and $\\lambda_R(M\/N) < \\infty$. It should be mentioned here that the first conclusion of the lemma would be known to experts. But, we cannot find its proof anywhere. So for the completeness, we give a short proof for it. Moreover, from this proof we obtain immediately a second conclusion which is useful for proofs of  further results in this paper. For a prime ideal $\\frak p$, we use $k(\\frak p)$ to denote the residue field $ R_{\\frak p}\/\\frak pR_{\\frak p}$ of the local ring $R_{\\frak p}$.\n\n\\begin{lemma}\\label{L2.3} Let $N$ be a submodule of $M$. Then\n  $$\\mathrm{ir}_M(N) = \\sum_{\\frak p \\in \\mathrm{Ass}_R(M\/N)} \\dim_{k(\\frak p)} \\mathrm{Soc}(M\/N)_{\\frak p} .$$\nMoreover, for any $\\frak p \\in \\mathrm{Ass}_R(M\/N)$, there is a $\\frak p$-primary submodule $N(\\frak p)$ of $M$ with $\\mathrm{ir}_M(N(\\frak p)) = \\dim_{k(\\frak p)} \\mathrm{Soc}(M\/N)_{\\frak p} $ such that\n$$N = \\bigcap_{\\frak p \\in \\mathrm{Ass}_R(M\/N)}N(\\frak p)$$\nis an irredundant primary decomposition of $N$.\n\\end{lemma}\n\\begin{proof}\n  Passing to the quotient $M\/N$ we may assume without any loss of generality that $N=0$. Let $\\mathrm{Ass}_R(M) = \\{ \\frak p_1,..., \\frak p_n\\}$. We set $t_i = \\dim_{k(\\frak p_i)}\\mathrm{Soc}(M_{\\frak p_i})$ and $t = t_1 + \\cdots + t_n$. Let $\\mathcal{F} = \\{\\frak p_{11},..., \\frak p_{1t_1}, \\frak p_{21},..., \\frak p_{2t_2}, ..., \\frak p_{n1},..., \\frak p_{nt_n}\\}$ be a family of prime ideals of $R$ such that $\\frak p_{i1} = \\cdots = \\frak p_{it_i} = \\frak p_i$ for all $i = 1,..., n$. Denote $E(M)$ the injective envelop of $M$. Then we can write\n  $$E(M) = \\bigoplus_{i=1}^n E(R\/\\frak p_i)^{t_i} = \\bigoplus_{\\frak p_{ij} \\in \\mathcal{F}}E(R\/\\frak p_{ij}).$$\n  Let $$ \\pi_i: \\oplus_{i=1}^n E(R\/\\frak p_i)^{t_i} \\to E(R\/\\frak p_i)^{t_i} \\ \\ \\mathrm{and} \\ \\  \\pi_{ij}: \\oplus_{\\frak p_{ij} \\in \\mathcal{F}}E(R\/\\frak p_{ij}) \\to E(R\/\\frak p_{ij})$$ be the canonical projections for all $i = 1,..., n$ and $j = 1,..., t_i$, and set $N(\\frak p_i) = M \\cap \\ker \\pi_i$, $N_{ij} = M \\cap \\ker \\pi_{ij}$. Since $E(R\/\\frak p_{ij})$ are indecomposible, $N_{ij}$ are irreducible submodules of $M$. Then it is easy to check that $N(\\frak p_i)$ is a $\\frak p_i$-primary submodule of $M$ having an irreducible decomposition  $N(\\frak p_i) = N_{i1} \\cap \\cdots \\cap N_{it_i}$ for all $i=1,\\ldots , n$. \n Moreover, because of the minimality of $E(M)$ among injective modules containing $M$, the finite intersection $$0 = N_{11} \\cap \\cdots \\cap N_{1t_1} \\cap \\cdots \\cap N_{n1} \\cap \\cdots \\cap N_{nt_n}$$ \n    is an irredundant irreducible decomposition of $0$. Therefore $0 = N(\\frak p_1) \\cap \\cdots \\cap N(\\frak p_n)$ is an irredundant primary decomposition of $0$ with\n  $\\mathrm{ir}_M(N(\\frak p_i)) = \\dim_{k(\\frak p_i)} \\mathrm{Soc}(M\/N)_{\\frak p_i} $ and $\\mathrm{ir}_M(0) = \\sum_{\\frak p \\in \\mathrm{Ass}(M)} \\dim_{k(\\frak p)} \\mathrm{Soc}(M)_{\\frak p}$\n as required.\n \\end{proof}\n\n\\section{Index of reducibility of maximal embedded components}\n\nLet $N$ be a submodule of $M$ and $\\frak p \\in \\mathrm{Ass}_R(M\/N)$. We use $\\bigwedge_{\\frak p}(N)$ to denote the set of all $\\frak p$-primary submodules of $M$ which appear in an irredundant primary decomposition of $N$. We say that a $\\frak p$-primary submodule $Q$ of $M$ is a $\\frak p$-primary component of $N$ if $Q \\in \\bigwedge_{\\frak p}(N)$, and $Q$ is said to be a {\\it maximal embedded component} (or more precisely, $\\frak p$-maximal embedded component) of $N$ if $Q$ is a maximal element in the set $\\bigwedge_{\\frak p}(N)$. It should be mentioned that the notion of maximal embedded components was first introduced for commutative rings  by Heinzer, Ratliff and Shah. They proved in the papers \\cite{HRS94}, \\cite{HRS95-1}, \\cite{HRS95-2}, \\cite{HRS95-3} many interesting properties of maximal embedded components as well as they showed that this notion is an important tool for  studying irreducible ideals.\\\\\n\nWe recall now a result of Y. Yao \\cite{Y02} which is often used in the proof of the next theorem.\n\\begin{theorem}[Yao \\cite{Y02}, Theorem 1.1] \\label{T3.1} Let $N$ be a submodule of $M$, $\\mathrm{Ass}_R(M\/N) = \\{\\frak p_1,..., \\frak p_n\\}$ and $Q_i \\in \\bigwedge_{\\frak p_i}(N)$, $i = 1,..., n$. Then $N = Q_1 \\cap \\cdots \\cap Q_n$ is an irredundant primary decomposition of $N$.\n\\end{theorem}\nThe following theorem is the main result of this section.\n\\begin{theorem}\\label{T3.2} Let $N$ be a submodule of $M$ and $\\mathrm{Ass}_R(M\/N) = \\{\\frak p_1,..., \\frak p_n\\}$. Let $N = Q_1 \\cap \\cdots \\cap Q_n$ be an irredundant primary decomposition of $N$, where $Q_i$ is $\\frak p_i$-primary for all $i= 1, \\ldots , n$. Then $\\mathrm{ir}_M(N) = \\mathrm{ir}_M(Q_1) + \\cdots + \\mathrm{ir}_M(Q_n)$ if and only if $Q_i$ is a $\\frak p_i$-maximal embedded component of $N$ for all embedded associated prime ideals $\\frak p_i$ of $N$.\n  \\end{theorem}\n  \\begin{proof} As in the proof of Lemma \\ref{L2.3}, we may assume that $N = 0$.\\\\\n  {\\it Sufficient condition}: Let $0 = Q_1 \\cap \\cdots \\cap Q_n$ be an irredundant primary decomposition of the zero submodule $0$, where $Q_i$ is maximal in $\\bigwedge_{\\frak p_i}(0)$, $i = 1,..., n$. Setting $\\mathrm{ir}_M(Q_i) = t_i$, and let $Q_i = Q_{i1} \\cap \\cdots \\cap Q_{it_i}$ be an irredundant irreducible decomposition of $Q_i$. Suppose that\n  $$t_1 + \\cdots + t_n = \\mathrm{ir}_M(Q_1) + \\cdots + \\mathrm{ir}_M(Q_n) > \\mathrm{ir}_M(0).$$\n  Then there exist an $i \\in \\{1,..., n \\}$ and a $j \\in \\{1,..., t_i\\}$ such that\n  $$Q_1 \\cap \\cdots \\cap Q_{i-1} \\cap Q_i' \\cap Q_{i+1} \\cap \\cdots \\cap Q_n \\subseteq Q_{ij},$$\n  where $Q_i' = Q_{i1} \\cap \\cdots \\cap Q_{i(j-1)} \\cap Q_{i(j+1)} \\cap \\cdots \\cap Q_{it_i} \\supsetneqq Q_i$. Therefore\n  $$Q_i' \\bigcap (\\cap_{k \\neq i} Q_k) = Q_i \\bigcap (\\cap_{k \\neq i} Q_k) = 0$$\n  is also an irredundant primary decomposition of $0$. Hence $Q_i' \\in \\bigwedge_{\\frak p_i}(0)$ which contradicts the maximality of $Q_i$ in $\\bigwedge_{\\frak p_i}(0)$. Thus $\\mathrm{ir}_R(0) = \\mathrm{ir}_R(Q_1) + \\cdots + \\mathrm{ir}_R(Q_n)$ as required.\\\\\n  {\\it Necessary condition}: Assume that $0 = Q_1 \\cap \\cdots \\cap Q_n$ is an irredundant primary decomposition of $0$ such that $\\mathrm{ir}_M(0) = \\mathrm{ir}_M(Q_1) + \\cdots + \\mathrm{ir}_M(Q_n)$. We have to prove that $Q_i$ are maximal in $\\bigwedge_{\\frak p_i}(0)$ for all $i = 1,..., n$. Indeed, let $N_1=N(\\frak p_1),..., N_n=N(\\frak p_n)$ be primary submodules of $M$ as in Lemma \\ref{L2.3}, that is $N_i \\in \\bigwedge_{\\frak p_i}(0)$, $0 = N_1 \\cap \\cdots \\cap N_n$ and $\\mathrm{ir}_M (0)  = \\sum_{i=1}^n \\mathrm{ir}_M(N_i) = \\sum_{i=1}^n \\dim_{k(\\frak p_i)} \\mathrm{Soc}(M_{\\frak p_i})$. Then by Theorem \\ref{T3.1} we see for any $0 \\leq i \\leq n$ that\n  $$0 = N_1 \\cap \\cdots \\cap N_{i-1} \\cap Q_i \\cap N_{i+1} \\cap \\cdots \\cap N_n = N_1 \\cap \\cdots \\cap N_n$$\nare two irredundant primary decompositions of $0$. Therefore\n$$\\mathrm{ir}_M(Q_i) + \\sum_{j \\neq i} \\mathrm{ir}_M(N_j) \\geq \\mathrm{ir}_M(0) = \\sum_{j =1}^n \\mathrm{ir}_M(N_j),$$\nand so $\\mathrm{ir}_M(Q_i) \\geq \\mathrm{ir}_M(N_i) = \\dim_{k(\\frak p_i)} \\mathrm{Soc}(M_{\\frak p_i})$ by Lemma \\ref{L2.3}.\\\\\nSimilarly, it follows from the two irredundant primary decompositions\n$$0 = Q_1 \\cap \\cdots \\cap Q_{i-1} \\cap N_i \\cap Q_{i+1} \\cap \\cdots \\cap Q_n = Q_1 \\cap \\cdots \\cap Q_n$$\nand the hypothesis that $\\mathrm{ir}_M(N_i) \\geq \\mathrm{ir}_M(Q_i)$. Thus we get\n$$\\mathrm{ir}_M(Q_i) = \\mathrm{ir}_M(N_i) = \\dim_{k(\\frak p_i)} \\mathrm{Soc}(M_{\\frak p_i})$$\nfor all $i = 1, ..., n$. Now, let $Q_i'$ be a maximal element of  $\\bigwedge_{\\frak p_i}(0)$ and $Q_i \\subseteq Q_i'$. It remains to prove that $Q_i = Q_i'$. By localization at $\\frak p_i$, we may assume that $R$ is a local ring with the unique maximal ideal $\\frak m = \\frak p_i$. Then, since $Q_i$ is an $\\frak m$-primary submodule and by the equality above we have\n$$\\lambda_R((Q_i:\\frak m)\/Q_i) = \\mathrm{ir}_M(Q_i) = \\dim_{\\frak k}\\mathrm{Soc}(M) = \\lambda_R(0:_M \\frak m) = \\lambda_R \\big( (Q_i + 0:_M \\frak m)\/Q_i\\big).$$\nIt follows that $Q_i: \\frak m = Q_i + 0:_M \\frak m$. If $Q_i \\subsetneqq Q_i'$, there is an element $x \\in Q_i' \\setminus Q_i$. Then we can find a positive integer $l$ such that $\\frak m^l x \\subseteq Q_i$ but $\\frak m^{l-1} x \\nsubseteq Q_i$. Choose $y \\in \\frak m^{l-1} x \\setminus Q_i$. We see that\n$$y \\in Q_i' \\cap (Q_i : \\frak m) = Q_i' \\cap (Q_i + 0:_M \\frak m) = Q_i + (Q_i' \\cap 0:_M \\frak m).$$\nSince $0:_M \\frak m \\subseteq \\cap_{j \\neq i} Q_j$ and $Q_i' \\cap (\\cap_{j \\neq i} Q_j) = 0$ by Theorem \\ref{T3.1}, $Q_i' \\cap (0:_M \\frak m) = 0$. Therefore $y \\in Q_i$ which is a contradiction with the choice of $y$. Thus $Q_i = Q_i'$ and the proof is complete.\n  \\end{proof}\n\nThe following characterization of maximal embedded components of $N$ in terms of the index of reducibility follows immediately from the proof of Theorem \\ref{T3.2}.\n\\begin{corollary}\\label{T3.3}\nLet $N$ be a submodule of $M$ and $\\frak p$ an embedded associated prime ideal of $N$. Then an element $Q \\in \\bigwedge_{\\frak p}(N)$ is a maximal embedded component of $N$ if and only if $\\mathrm{ir}_M(Q) = \\dim_{k(\\frak p)} \\mathrm{Soc}(M\/N)_{\\frak p}$.\n\\end{corollary}\n\nAs consequences of Theorem \\ref{T3.2}, we can obtain again several results on maximal embedded components proved by Heinzer, Ratliff and Shah. The following corollary is one of that results stated for modules. For a submodule $L$ of $M$ and $\\frak p$ a prime ideal, we denote by $\\mathrm{IC}_{\\frak p}(L)$ the set of all irreducible $\\frak p$-primary submodules of $M$ that appear in an irredundant irreducible decomposition of $L$, and denote by $\\mathrm{ir}_{\\frak p}(L)$ the number of irreducible $\\frak p$-primary components in an irredundant irreducible decomposition of $L$ (this number is well defined by Noether \\cite[Satz VII]{N21}).\n\n\\begin{corollary}[see \\cite{HRS95-3}, Theorems 2.3 and 2.7] Let $N$ be a submodule of $M$ and $\\frak p$ an embedded associated prime ideal of $N$. Then\n\\begin{enumerate}[{(i)}]\\rm\n\\item {\\it  $\\mathrm{ir}_{\\frak p}(N) = \\mathrm{ir}_{\\frak p}(Q) = \\dim_{k(\\frak p)} \\mathrm{Soc}(M\/N)_{\\frak p}$ for any $\\frak p$-maximal embedded component $Q$ of $N$.}\n\\item {\\it $\\mathrm{IC}_{\\frak p}(N) = \\bigcup_Q \\mathrm{IC}_{\\frak p}(Q)$, where the submodule $Q$ in the union runs over all $\\frak p$-maximal embedded components of $N$.}\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\n(i) follows immediately from the proof of Theorem \\ref{T3.2} and Corollary \\ref{T3.3}.\\\\\n(ii) Let $Q_1 \\in \\mathrm{IC}_{\\frak p}(N)$ and $t_1 = \\dim_{k(\\frak p)} \\mathrm{Soc}(M\/N)_{\\frak p}$. By the hypothesis and (i) there exists an irredundant irreducible decomposition $N= Q_{11}\\cap  \\ldots  \\cap Q_{1 t_1} \\cap Q_2 \\cap \\ldots \\cap Q_l$ such that $Q_{11} = Q_1 , \\  Q_{12},  \\ldots , Q_{1 t_1}$ are all $\\frak p$-primary submodules in this decomposition. Therefore $Q=Q_{11}\\cap  \\ldots  \\cap Q_{1 t_1}$ is a maximal embedded component of $N$ by Corollary \\ref{T3.3}, and so $Q_1\\in \\mathrm{IC}_{\\frak p}(Q)$. The converse inclusion can be easily proved by applying Theorems \\ref{T3.1} and \\ref{T3.2}.\n\\end{proof}\n\n\\section{Index of reducibility of powers of an ideal}\nLet $I$ be an ideal of $R$. It is well known by \\cite{B79} that the $\\mathrm{Ass}_R(M\/I^nM)$ is stable for sufficiently large  $n$ ($n\\gg 0$ for short). We will denote this stable set by $\\text{A}_M(I)$. The big height, $\\mathrm{bight}_M(I)$, of $I$ on $M$ is defined by\n$$\\mathrm{bight}_M(I)=\\max\\{\\dim_{R_{\\frak p}} M_{\\frak p}\\mid \\text{ for all minimal prime ideals } {\\frak p}\\in \\mathrm{Ass}_R(M\/IM)\\}.$$\nLet $G(I)=\\bigoplus\\limits_{n\\geq 0} I^n\/I^{n+1}$ be the associated graded ring of $R$ with respect to $I$ and $G_M(I)=\\bigoplus\\limits_{n\\geq 0} I^nM\/I^{n+1}M$ the associated graded $G(I)$-module of $M$ with respect to $I$. If $R$ is a local ring with the unique maximal ideal $\\frak m$, then the analytic spread $\\ell_M(I)$ of $I$ on $M$ is defined by\n$$\\ell_M(I)=\\dim_{G(I)}(G_M(I)\/{\\frak m} G_M(I)).$$\nIf $R$ is not local, the analytic spread $\\ell_M(I)$ is also defined by\n$$\n\\begin{aligned}\n\\ell_M(I)=\\max\\{ \\ell_{M_{\\frak m}}(IR_{\\frak m})\\mid {\\frak m} &\\text{ is a maximal ideal and }\n\\\\&\n\\text{ there is a prime ideal }{\\frak p} \\in A_M(I) \\text{ such that } {\\frak p}\\subseteq {\\frak m}\\}.\n\\end{aligned}$$\nWe use $\\ell(I)$ to denote the analytic spread of the ideal $I$ on $R$.\nThe following theorem is the main result of this section.\n\n\\begin{theorem} \\label{T4.1} Let $I$ be an ideal of $R$. Then there exists a polynomial $\\mathrm{Ir}_{M,I}(n)$ with rational coefficients such that $\\mathrm{Ir}_{M,I}(n)=\\mathrm{ir}_M(I^nM)$ for sufficiently large  $n$. Moreover, we have\n$$\\mathrm{bight}_M(I)-1\\le \\deg(\\mathrm{Ir}_{M,I}(n))\\le \\ell_M(I)-1.$$\n\\end{theorem}\n\nTo prove Theorem \\ref{T4.1}, we need the following lemma.\n\n\\begin{lemma}\\label{L4.2} Suppose that $R$ is a local ring with the unique maximal ideal $\\frak m$ and $I$ an ideal of $R$. Then\n\\begin{enumerate}[{(i)}]\\rm\n    \\item {\\it $\\dim_{\\frak k}\\mathrm{Soc}(M\/I^nM)=\\lambda_R(I^nM:{\\frak m}\/I^nM)$ is a polynomial of degree $\\le \\ell_M(I)-1$ for  $n\\gg 0$.}\n    \\item {\\it Assume  that $I$ is an ${\\frak m}$-primary ideal. Then $\\mathrm{ir}_M(I^nM)=\\lambda_R(I^nM:{\\frak m}\/I^nM)$ is a polynomial of degree $\\dim_RM-1$ for  $n\\gg 0$.}\n  \\end{enumerate}\n  \\end{lemma}\n\n  \\begin{proof}\n\n(i) Consider the homogeneous submodule $0:_{G_M(I)}{\\frak m}G(I)$. Then \n$$\\lambda_R(0:_{G_M(I)}{\\frak m}G(I))_n=\\lambda_R(((I^{n+1}M:{\\frak m})\\cap I^nM)\/I^{n+1}M)$$ is a polynomial for $n\\gg 0$. Using a result proved by P. Schenzel \\cite[Proposition 2.1]{S98}, proved  for rings but  easily extendible to  modules, we find a positive integer $l$ such that for all $n\\ge l$, $0:_M{\\frak m}\\cap I^nM=0$ and\n$$I^{n+1}M:{\\frak m}= I^{n+1-l}(I^l M:{\\frak m})+ 0:_M{\\frak m}.$$\nTherefore \n$$\n\\begin{aligned}\n(I^{n+1}M:{\\frak m})\\cap I^nM&=I^{n+1-l}(I^l M:{\\frak m})+0:_M{\\frak m}\\cap I^nM\\\\\n&=I^{n+1-l}(I^l M:{\\frak m}).\n\\end{aligned}\n$$\nHence, $\\lambda_R(I^{n+1-l}(I^l M:{\\frak m})\/I^{n+1}M)=\\lambda_R(((I^{n+1}M:{\\frak m})\\cap I^nM)\/I^{n+1}M)$ is a polynomial for $n\\gg 0$. It follows that\n$$\\dim_{\\frak k}\\mathrm{Soc}(M\/I^nM) =\\lambda_R((I^n M:{\\frak m})\/I^{n}M)=\\lambda_R(I^{n-l}(I^l M:{\\frak m})\/I^{n}M)+\\lambda_R(0:_M{\\frak m})$$\nis a polynomial for $n\\gg 0$, and the degree of this polynomial is just equal to\n$$\\dim_{G(I)}(0:_{G_M(I)}{\\frak m}G(I))-1\\le \\dim_{G(I)}({G_M(I)}\/{\\frak m}G_M(I))-1=\\ell_M(I)-1.$$\n(ii) The second statement follows from the first one and the fact that\n$$\n\\begin{aligned}\n\\lambda_R(I^nM\/I^{n+1}M) &=\\lambda_R(\\mathrm{Hom}_R(R\/I,I^nM\/I^{n+1}M))\\\\\n &\\le \\lambda_R(R\/I)\\lambda_R(\\mathrm{Hom}_R(R\/{\\frak m},I^nM\/I^{n+1}M)) \\le \\lambda_R(R\/I)\\mathrm{ir}_M(I^{n+1}M).\n\\end{aligned}\n$$\n\\end{proof}\n\nWe are now able to prove Theorem \\ref{T4.1}.\n\n\\begin{proof}[Proof of Theorem \\ref{T4.1}] Let $\\text{A}_M(I)$ denote  the stable set  $\\mathrm{Ass}_R(M\/I^nM)$ for $n\\gg 0$.   Then, by Lemma \\ref{L2.3} we get that\n$$\\mathrm{ir}_M(I^nM)=\\sum\\limits_{\\frak p\\in A_M(I)}\\dim_{k({\\frak p})}\\mathrm{Soc}(M\/I^nM)_{\\frak p}$$ for all $n\\gg 0$.\nFrom Lemma \\ref{L4.2}, (i), $\\dim_{k({\\frak p})}\\mathrm{Soc}(M\/I^nM)_{\\frak p}$ is a polynomial of degree $\\le \\ell_{M_{\\frak p}}(IR_{\\frak p})-1$ for  $n\\gg0$. Therefore there exists a polynomial $\\mathrm{Ir}_{M,I}(n)$ of  such that $\\mathrm{Ir}_{M,I}(n)=\\mathrm{ir}_M(I^nM)$ for $n\\gg 0$ and \n$$\\mathrm{deg}(\\mathrm{Ir}_{M,I}(n))\\le \\max \\{\\ell_{M_{\\frak p}}(IR_{\\frak p})-1\\mid \\frak p \\in A_M(I)\\}\\le \\ell_M(I)-1.$$\nLet $\\mathrm{Min}(M\/IM)=\\{{\\frak p_1},\\ldots,{\\frak p_m}\\}$ be the set of all minimal associated prime ideals of $IM$. It is clear that ${\\frak p_i}$ is also minimal in $A_M(I)$. Hence $\\Lambda_{\\frak p_i}(I^nM)$ has only one element, says $Q_{in}$. It is easy to check that \n$$\\mathrm{ir}_M(Q_{in})={\\rm ir}_{M_{\\frak p_i}}(Q_{in})_{\\frak p_i}=\\mathrm{ir}_{M_{\\frak p_i}}(I^nM_{\\frak p_i})$$ for $i=1, \\ldots , m$.  This implies by Theorem \\ref{T3.2} that\n$\\mathrm{ir}_M(I^nM)\\ge \\sum\\limits_{i=1}^m \\mathrm{ir}_{M_{\\frak p_i}}(I^nM_{\\frak p_i})$. It follows from Lemma \\ref{L4.2}, (ii) for  $n\\gg 0$ that\n$$\\mathrm{deg}(\\mathrm{Ir}_{M,I}(n))\\ge \\max\\{\\dim_{R_{\\frak p_i} }M_{\\frak p_i}-1\\mid i=1, \\ldots ,m\\}=\\mathrm{bight}_M(I)-1.$$\n\\end{proof}\n\n The following corollaries are immediate consequences of Theorem \\ref{T4.1}.\n An ideal $I$ of a local ring $R$ is called an equimultiple ideal if  $\\ell(I)=\\mathrm{ht}(I)$, and therefore $\\mathrm{bight}_R(I)=\\mathrm{ht}(I)$.\n\n\\begin{corollary} \\label{C4.3} Let $I$ be an ideal of $R$ satisfying $\\ell_M(I)= \\mathrm{bight}_M(I)$. Then $$\\deg(\\mathrm{Ir}_{M,I}(n))=\\ell_M(I)-1.$$\n\\end{corollary}\n\n\\begin{corollary} \\label{C4.4} Let $I$ be an equimultiple ideal of a local ring $R$ with the unique maximal ideal $\\frak m$. Then  $$\\mathrm{deg}(\\mathrm{Ir}_{R,I}(n))=\\mathrm{ht}(I)-1$$. \n\\end{corollary}\n\nExcepting the corollaries above, the authors of the paper do not know how to compute exactly the degree of the polynomial of index of reducibility $\\mathrm{Ir}_{M,I}(n)$. Therefore it is maybe interesting to find a formula for this degree in terms of known invariants associated to $I$ and $M$. Below we give examples to show that although these bounds are sharp, neither  $\\mathrm{bight}_M(I)-1$ nor  $\\ell_M(I)-1$ equal to $\\mathrm{deg}(\\mathrm{Ir}_{M,I}(n))$ in general.\n\n\\begin{example} \\label{E4.5}{\\rm\n(1) Let $R=K[X,Y]$ be the polynomial ring of two variables $X$, $Y$ over a field $K$ and $I=(X^2,XY)=X(X,Y)$ an ideal of $R$. Then we have\n$$\\mathrm{bight}_R(I)=\\mathrm{ht}(I)=1, \\text{  }\\ell(I)=2,$$ and by Lemma \\ref{L2.3}\n$$\\mathrm{ir}_R(I^n)=\\mathrm{ir}_R(X^n(X,Y)^n)=\\mathrm{ir}_R((X,Y)^n) +1=n+1.$$\nTherefore \n$$\\mathrm{bight}_R(I)-1=0 <1= \\mathrm{deg}(\\mathrm{Ir}_{R,I}(n))=\\ell (I)-1.$$\n\n(2) Let $T=K[X_1,X_2,X_3,X_4,X_5,X_6]$ be the polynomial ring in six variables over a field $K$ and $R=T_{(X_1,\\ldots,X_6)}$ the localization of $T$ at the homogeneous maximal ideal $(X_1,\\ldots,X_6)$. Consider the monomial ideal\n$$\n\\begin{aligned}\nI&=(X_1X_2,X_2X_3,X_3X_4,X_4X_5,X_5X_6,X_6X_1)= (X_1,X_3,X_5)\\cap (X_2,X_4,X_6)\\cap \\\\\n&\\cap (X_1,X_2,X_4,X_5) \\cap (X_2,X_3,X_5,X_6)\\cap (X_3,X_4,X_6,X_1).\n\\end{aligned}\n$$\nSince the associated graph to this monomial ideal is a bipartite graph, it follows from \\cite[Theorem 5.9]{SVV94} that\n$\\mathrm{Ass}(R\/I^n)=\\mathrm{Ass}(R\/I)=\\mathrm{Min}(R\/I)$ for all $n\\geq 1$. Therefore $\\mathrm{deg}(\\mathrm{Ir}_{R,I}(n))= \\mathrm{bight}(I)-1= 3$ by Theorem \\ref{T3.2} and Lemma \\ref{L4.2} (ii). On the other hand, by \\cite[Exercise 8.21]{HS06} $\\ell(I)=5$, so\n$$\\mathrm{deg}(\\mathrm{Ir}_{R,I}(n))=3< 4=\\ell(I)-1.$$\n}\n\\end{example}\n\nLet $I$ be an ideal of $R$ and $n$ a positive integer. The $n$th symbolic power $I^{(n)}$ of $I$ is defined by\n$$I^{(n)} = \\bigcap_{\\frak p \\in \\mathrm{Min}(I)}(I^nR_{\\frak p}\\cap R),$$ where ${\\rm Min} (I)$ is the set of all minimal associated prime ideals in Ass$(R\/I)$. Contrary to the function ${\\rm ir}(I^n)$, the behaviour of the function ${\\rm ir}(I^{(n)})$ seems to be better.\n\\begin{proposition} \\label{C4.2}\nLet $I$ be an ideal of $R$. Then there exists a polynomial $p_I(n)$ with rational coefficients that such\n$p_I(n)={\\rm ir}_R (I^{(n)}) $ for  sufficiently large $n$ and $${\\rm deg}(p_I(n))={\\rm bight}(I)-1.$$\n\\end{proposition} \\label{C4.2}\n\\begin{proof} It should be mentioned that Ass$(R\/I^{(n)})={\\rm Min} (I)$ for all positive integer $n$. Thus, by virtue of Theorem \\ref{T3.2}, we can show as in the proof of Theorem \\ref{T4.1} that \n$${\\rm ir} _R(I^{(n)})=\\sum\\limits_{\\frak p\\in {\\rm Min}(I)}\\mathrm{ir}_{R_{\\frak p}}(I^nR_{\\frak p})$$ for all $n$. So the proposition follows from Lemma \\ref{L4.2}, (ii).\n\\end{proof}\n\n\\section{Index of reducibility in Cohen-Macaulay modules}\nIn this section, we assume in addition that $R$ is a local ring with the unique maximal ideal $\\frak m$, and $\\frak k = R\/\\frak m$ is the residue field. Let $\\frak q=(x_1,\\ldots, x_d)$ be a parameter ideal of $M$ ($d=\\dim M$). Let $H^i({\\frak q}, M)$ be the $i$-th Koszul cohomology module of $M$ with respect to $\\frak q$ and $H^i_{\\frak m}(M)$ the $i$-th local cohomology module of $M$ with respect to the maximal ideal $\\frak m$. In order to state the next theorem, we need the following result of Goto and Sakurai \\cite[Lemma 3.12]{GS03}.\n\n\\begin{lemma}\\label{L4.6}\nThere exists a positive integer $l$ such that for all parameter ideals $\\frak q$ of $M$ contained in $\\frak m^l$, the canonical homomorphisms on socles\n$$\\mathrm{Soc}(H^i({\\frak q}, M))\\to \\mathrm{Soc}(H^i_{\\frak m}(M))$$\nare surjective for all $i$.\n \\end{lemma}\n\n\\begin{theorem} \\label{T4.7} Let $M$ be a finitely generated $R$-module of $\\dim M=d$. Then the following conditions are equivalent:\n\\begin{enumerate}[{(i)}]\\rm\n    \\item {\\it $M$ is a Cohen-Macaulay module.}\n    \\item {\\it $\\mathrm{ir}_M({\\frak q}^{n+1}M)=\\dim_{\\frak k}\\mathrm{Soc}(H^d_{\\frak m}(M)) \\binom{n+d-1}{d-1}$ for all parameter ideals $\\frak q$ of $M$ and all $n \\geq 0$.}\n    \\item {\\it  $\\mathrm{ir}_M({\\frak q}M)=\\dim_{\\frak k} \\mathrm{Soc}(H^d_{\\frak m}(M))$ for all parameter ideals $\\frak q$ of $M$.}\n    \\item {\\it There exists a parameter ideal $\\frak q$ of $M$ contained in $\\frak m^l$, where $l$ is a positive integer as in Lemma \\ref{L4.6}, such that $\\mathrm{ir}_M({\\frak q}M)=\\dim_{\\frak k}\\mathrm{Soc}(H^d_{\\frak m}(M))$.}\n  \\end{enumerate}\n\\end{theorem}\n\\begin{proof}\n(i) $\\Rightarrow$ (ii) Let $\\frak q$ be a parameter ideal of $M$. Since $M$ is Cohen-Macaulay, we have a natural isomorphism of graded modules\n\n$$G_M(\\frak q)=\\bigoplus\\limits_{n\\ge 0}\\frak q^nM\/\\frak q^{n+1}M\\to M\/\\frak q M[T_1,\\ldots,T_d],$$ where $T_1,\\ldots,T_d$ are indeterminates.\nThis deduces $R$-isomomorphisms on graded parts\n$$\\frak q^nM\/\\frak q^{n+1}M\\to\\big( M\/\\frak q M[T_1,\\ldots,T_d]\\big)_n\\cong M\/\\frak q M^{\\binom{n+d-1}{d-1}}$$ for all $n\\geq 0$.\nOn the other hand, since $\\frak q$ is a parameter ideal of a Cohen-Macaulay module, $\\frak q^{n+1}M:\\frak m\\subseteq \\frak q^{n+1}M:\\frak q=\\frak q^{n}M$. It follows that\n$$\n\\begin{aligned}\n\\mathrm{ir}_M({\\frak q}^{n+1}M)&=\\lambda_R(\\frak q^{n+1}M:\\frak m\/\\frak q^{n+1}M)=\\lambda_R(0:_{\\frak q^{n}M\/\\frak q^{n+1}M}\\frak m)\\\\\n&=\\lambda_R(0:_{M\/\\frak q M}\\frak m) \\binom{n+d-1}{d-1}=\\dim_{\\frak k}(\\mathrm{Soc}(M\/\\frak q M))\\binom{n+d-1}{d-1}.\n\\end{aligned}\n$$\nSo the conclusion is proved, if we show that $\\dim_{\\frak k} \\mathrm{Soc}(M\/\\frak q M)=\\dim_{\\frak k} \\mathrm{Soc}(H^d_{\\frak m}(M))$. Indeed, let $\\frak q=(x_1,\\ldots, x_d)$ and $\\overline{M}= M\/x_1M$. Then, it is easy to show by induction on $d$ that\n$$\n\\begin{aligned}\n\\dim_{\\frak k} \\mathrm{Soc}(M\/\\frak q M)&=\\dim_{\\frak k} \\mathrm{Soc}(\\overline{M}\/\\frak q \\overline{M})\\\\\n&=\\dim_{\\frak k} \\mathrm{Soc}(H^{d-1}_{\\frak m}(\\overline{M})) = \\dim_{\\frak k} \\mathrm{Soc}(H^d_{\\frak m}(M)).\n\\end{aligned}\n$$\n\n(ii) $\\Rightarrow$ (iii) and (iii) $\\Rightarrow$ (iv) are trivial.\n\n(iv) $\\Rightarrow$ (i) Let $\\frak q =(x_1, \\ldots , x_d)$ be a parameter ideal of $M$ such that $\\frak q\\subseteq \\frak m^l$, where $l$ is a positive integer as in Lemma 5.1 such that the canonical homomorphism on socles\n$$\\mathrm{Soc}(M\/\\frak q M)=\\mathrm{Soc}(H^d({\\frak q}, M))\\to \\mathrm{Soc}(H^d_{\\frak m}(M))$$\nis surjective. Consider the submodule  $(\\underline{x})_M^{\\mathrm{lim}}=\\bigcup\\limits_{t\\ge 0}(x_1^{t+1},\\ldots,x_d^{t+1}):(x_1\\ldots x_d)^{t}$ of $M$. This submodule is called the limit closure of the sequence $x_1,\\ldots,x_d$. Then \n$(\\underline{x})_M^{\\mathrm{lim}}\/\\frak q M$ is just the kernel of the canonical homomorphism $M\/\\frak q M\\to H^d_{\\frak m}(M)$ (see \\cite{CHL99}, \\cite{CQ10}). Moreover, it was proved in \\cite[Corollary 2.4]{CHL99} that the module $M$ is Cohen-Macaulay if and only if $(\\underline{x})_M^{\\mathrm{lim}}=\\frak q M$. Now we assume that $\\mathrm{ir}_M({\\frak q}M)=\\dim_{\\frak k} \\mathrm{Soc}(H^d_{\\frak m}(M))$, therefore $\\dim_{\\frak k} \\mathrm{Soc}(H^d_{\\frak m}(M)) =\\dim_{\\frak k} \\mathrm{Soc}(M\/\\frak q M)$. Then it follows from the exact sequence\n$$0\\to (\\underline{x})_M^{\\mathrm{lim}}\/\\frak q M \\to M\/\\frak q M\\to H^d_{\\frak m}(M) $$\n and the choice of $l$ that the sequence\n$$0\\to \\mathrm{Soc}((\\underline{x})_M^{\\mathrm{lim}}\/\\frak q M) \\to \\mathrm{Soc}( M\/\\frak q M)\\to\\mathrm{Soc}( H^d_{\\frak m}(M))\\to 0 $$\n is a short exact sequence. Hence $\\dim_{\\frak k} \\mathrm{Soc}((\\underline{x})_M^{\\mathrm{lim}}\/\\frak q M)=0 $ by the hypothesis. So $(\\underline{x})_M^{\\mathrm{lim}}=\\frak q M$, and therefore $M$ is a Cohen-Macaulay module.\n  \\end{proof}\n  \\rm It should be mentioned here that the proof of implication (iv) $\\Rightarrow$ (i) of Theorem \\ref{T4.7} is essentially following the proof  of \\cite [Theorem 2.7] {MRS08}.  It is well-known that a Noetherian local ring $R$ with $\\dim R =d$ is Gorenstein if and only if $R$ is Cohen-Macaulay with the Cohen-Macaulay \ntype r$(R)=\\dim_{\\frak k}\\mathrm{Ext}^d(\\frak k,M))= 1$. Therefore the following result, which is the main result of \\cite[Theorem]{MRS08}, is an immediate consequence of Theorem \\ref{T4.7}.\n \n \\begin{corollary}\\label{5.3}\n Let $(R, \\frak m)$ be a Noetherian local ring of dimension $d$. Then $R$ is Gorenstein if and only if there exists an irreducible parameter ideal $\\frak q$ contained in $\\frak m^l$, where $l$ is a positive integer as in Lemma \\ref{L4.6}. Moreover, if $R$ is Gorenstein, then for any parameter ideal $\\frak q$ it holds ${\\rm ir}_R(\\frak q^{n+1}) = \\binom{n+d-1}{d-1}$ for all $n\\geq 0$. \n \\end{corollary}\n \\begin{proof}\nLet $\\frak q=(x_1,\\ldots , x_d)$ be an irreducible parameter ideal contained in $\\frak m^l$ such that the map\n $$\\mathrm{Soc}( M\/\\frak q M)\\to\\mathrm{Soc}( H^d_{\\frak m}(M))$$ is surjective.\n Since $\\dim_{\\frak k}\\mathrm{Soc}( H^d_{\\frak m}(M))\\not= 0$ and $\\dim_{\\frak k} \\mathrm{Soc}(M\/\\frak q M)=1$ by the hypothesis, \n $\\dim_{\\frak k} \\mathrm{Soc}( H^d_{\\frak m}(M))=1. $ This imples by Theorem \\ref{T4.7} that  $M$ is a Cohen-Macaulay module with \n $${\\rm r}(R)=\\dim_{\\frak k}\\mathrm{Ext}^d(\\frak k,M)=\\dim_{\\frak k} \\mathrm{Soc}(M\/\\frak q M)=  1,$$ and so $R$ is Gorenstein. The last conclusion follows from Theorem \\ref{T4.7}.\n \\end{proof}\n \\begin{remark} \\rm  Recently, it was shown by many works that the index of reducibility of parameter ideals can be used to deduce a lot of information on the structure of some classes of modules such as Buchsbaum modules \\cite{GS03}, generalized Cohen-Macaulay modules \\cite{CT08}, \n\\cite{Q12} and sequentially Cohen-Macaulay modules \\cite{T13}.\n It follows from Theorem \\ref{T4.7} that $M$ is a Cohen-Macaulay module if and only if there exists a positive integer $l$ such that\n    $\\mathrm{ir}_M({\\frak q}M)=\\dim_{\\frak k}\\mathrm{Soc}(H^d_{\\frak m}(M))$ for all parameter ideals $\\frak q$ of $M$ contained in $\\frak m^l$. The necessary condition of this result can be extended for a large class of modules called generalized Cohen-Macaulay modules. An $R$-module $M$ of dimension $d$ is said to be a generalized Cohen-Macaulay module (see \\cite{CST78}) if $H^i_{\\frak m}(M)$ is of finite length for all $i=0,\\ldots , d-1$. We proved in \\cite [Theorem 1.1]{CT08} (see also \\cite [Corollary 4.4]{CQ11}) that if $M$ is a generalized Cohen-Macaulay module, then there exists an integer $l$ such that \n  $${\\rm ir}_M(\\frak qM) = \\sum_{i=0}^d \\binom{d}{i}\\dim_{\\frak k}\\mathrm{Soc}(H^i_{\\frak m}(M)).$$\n  for all parameter ideals $\\frak q \\subseteq \\frak m^l$. Therefore, we close this paper with the following two open questions, which are suggested during the work in this paper, on the characterization of the Cohen-Macaulayness and of the generalized Cohen-Macaulayness in terms of the index of reducibility of parameter ideals as follows.\n  \\vskip0.1cm\n  \\noindent\n  {\\bf Open questions 5.5.} Let $M$ be a finitely generated module of dimension $d$ over a local ring $R$. Then our questions are \\\\\n  1. Is $M$  a Cohen-Macaulay module if and only if there exists a parameter ideal $\\frak q$ of $M$ such that $$\\mathrm{ir}_M({\\frak qM}^{n+1}M)=\\dim_{\\frak k}\\mathrm{Soc}(H^d_{\\frak m}(M)) \\binom{n+d-1}{d-1}$$ for all $n\\geq 0$?\n \n  \\noindent\n  2. Is $M$ a generalized Cohen-Macaulay module if and only if  there exists  a positive integer $l$ such that  $${\\rm ir}_M(\\frak qM)= \\sum_{i=0}^d \\binom{d}{i}\\dim_{\\frak k}\\mathrm{Soc}(H^i_{\\frak m}(M))$$ for all parameter ideals $\\frak q \\subseteq \\frak m^l$?\n \\end{remark}\n \n\\bigskip\n\\noindent{\\bf Acknowledgments.} The authors would like to thank the anonymous referee for helpful comments on the earlier version. This paper was finished  during the  authors' visit at the Vietnam\nInstitute for Advanced Study in Mathematics (VIASM).\nThey would like to thank VIASM for their support and hospitality.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Motivation}\n\nIn 1998,  Ellis \\cite{el98} extended the theory of the Schur multiplier for a pair of groups. By a pair of groups $(G,N)$ we mean a group $G$ with a normal subgroup $N$ of $G$. The Schur multiplier of a pair $(G,N)$ of groups is a\nfunctorial abelian group $M(G,N)$ whose principal feature is a\nnatural exact sequence\n\\begin{eqnarray*}\n H_3(G) &\\rightarrow& H_3(G\/N) \\rightarrow M(G,N) \\rightarrow M(G)\n\\rightarrow M(G\/N)\\\\\n&\\rightarrow& N\/[N,G]\\rightarrow (G)^{ab} \\rightarrow (G\/N)^{ab} \\rightarrow 0\n\\end{eqnarray*}\nin which  $H_3(G)$ is the third homology group of $G$ with integer coefficients.\nIn particular, if $N=G$, then $M(G,G)$ is the usual Schur multiplier $M(G)$.\n\nIt has been a considerable question that when $\\exp(M(G))$ divides $\\exp(G)$, in which $\\exp(G)$ denotes the exponent of $G$.\nMacdonald and Wamsely (see \\cite{bay})\nconstructed an example of a group of exponent 4, whereas its Schur\nmultiplier has exponent 8, hence the conjecture is not true in\ngeneral. In 1973, Jones \\cite{jons} proved\nthat the exponent of the Schur multiplier of a finite $p$-group of\nclass $c \\geq 2$ and exponent $p^{e}$ is at most $p^{e(c-1)}$ and hence $\\exp(M(G))$ divides $\\exp(G)$ when $G$ is a $p$-group of class 2. In 1987, Lubotzky and Mann \\cite{lub} proved that $\\exp(M(G))$ divides $\\exp(G)$ when $G$ is a powerful $p$-group. A\nresult of Ellis \\cite{el2001} shows that if $G$ is a $p$-group of class $k\n\\geq 2$ and exponent $p^{e}$, then $\\exp(M(G))\\leq p^{e\\lceil k\/2\\rceil}$, where $\\lceil k\/2\\rceil$ denotes the smallest integer\n$n$ such that $n \\geq k\/2$. Moravec \\cite{mor} showed that\n$\\lceil k\/2\\rceil$ can be replaced by $2\\lfloor \\log_2{k}\\rfloor$\nwhich is an improvement if $k\\geq 11$. He \\cite{mor} also proved that if $G$ is\na metabelian group of exponent $p$, then $\\exp(M(G))$ divides $p$.\nKayvanfar and  Sanati \\cite{kay} proved that if $G$ is a $p$-group, then $\\exp(M(G))$ divides\n$\\exp(G)$ when $G$ is a finite $p$-group of class 3, 4 or 5 with some conditions.  The authors \\cite{mhm}  extended the result and proved that $\\exp(M(G))$ divides $\\exp(G)$ when $G$ is a finite $p$-group of class at most $p-1$.\\\n\nOn the other hand, Ellis \\cite{el98} proved  that $\\exp(M(G,N))$ divides $|N|$ for any pair $(G,N)$ of finite groups, in which $|N|$ denotes  the order of $N$. Now a question that can naturally arise, is whether $\\exp(M(G,N))$ divides $\\exp(N)$ when $N$ is a proper normal subgroup of $G$. In this paper, first we present an example to give\n a negative answer to the question. Second, we give some conditions\n under which the exponent of $M(G,N)$ divides the exponent of $N$.\n\n In Section 2, we give an upper bound for $\\exp(M(G,N))$  in terms of $\\exp(N)$, when $(G,N)$ is a pair of finite $p$-groups such that $N$ admits a complement in $G$, and apply it to  prove that if $(G,N)$ is a pair of finite $p$-groups of class at most $p-1$ (i.e $[N, \\ _{p-1}G]=1$), then $\\exp(M(G,N))$ divides $\\exp(N)$.\nFinally in Section 3, we show that if $(G,N)$ is a pair of finite $p$-groups and $N$ is powerfully embedded in $G$, then $\\exp(M(G,N))$ divides $\\exp(N)$.\n\n\\section{ Nilpotent pairs of $p$-groups}\n\nI. D. Macdonald and J.W. Wamsley \\cite{bay} gave an example which shows that $\\exp(M(G,G))$ dose not divide $\\exp(G)$, in general. The following example shows that $\\exp(M(G,N))$ dose not divide $\\exp(N)$ when $N$ is a proper normal subgroup of $G$. \\\\\n\n\\begin{ex}\nLet $D=A >\\!\\!\\!\\! \\lhd <x_1>$, where $A=<x_2>\\times <x_3>\\times<x_4>\\times<x_5>\\cong {\\mathbf Z}_{{4}}\\times{\\mathbf Z}_{{4}}\\times{\\mathbf Z}_{{4}}\\times{\\mathbf Z}_{{2}}$ and $x_1$ is an automorphism of order 2 of $A$ acting in the following way:\n$$[x_2,x_1]=x_2^2,\\ \\ [x_3,x_1]=x_3^2,\\ \\ [x_4,x_1]=x_4^2,\\ \\ [x_5,x_1]=1.$$\nThere exists an automorphism $a$ of $D$ of order 4 acting on $D$ as follows:\n$$[x_1,a]=x_3,\\ \\ [x_2,a]=x_2^2 x_3^2 x_4^3\\ \\ [x_3,a]=x_5,\\ \\ [x_4,a]=x_2^2,\\ \\ [x_5,a]=x_3^2.$$\nForm $N=D >\\!\\!\\! \\lhd <a>$ and put $G=N >\\!\\!\\! \\lhd <b>$, where $b^2=1$ and\n$[x_1,b]=x_2,\\  [x_2,b]=x_2^2x_4^3x_5,\\  [x_3,b]=x_4,\\  [x_4,b]=x_3^2x_4^2,\\ [x_5,b]=x_2^2x_3^2x_4^2,\\  [a,b]=x_1.$\n Moravec \\cite{mor} showed that the group $G$ is a nilpotent group of class 6 and exponent 4 and $M(G)\\cong{\\mathbf Z}_{{2}}\\times{\\mathbf Z}_{{4}}\\times{\\mathbf Z}_{{8}}$.\n Ellis \\cite{el98} proved that if $G=K >\\!\\!\\!\\! \\lhd Q$, then $M(G)\\cong M(G,K)\\oplus M(Q)$. This implies that $M(G,N)\\cong M(G)$. Therefore $\\exp(M(G,N))=8$ dose not divide $\\exp(N)=4$.\n \\end{ex}\n\nHere we first give an upper bound for the exponent of $M(G,N)$ in terms of the exponent of $N$, when $(G,N)$ is a pair of finite $p$-groups such that $N$ admits a complement in $G$. Since our proof relies on commutator calculations, we need to state the following lemmas.\n\n\\begin{lem} (\\cite{st}). Let $x_1, x_2,\n\\ldots , \\ x_r$ be any elements of a group and $\\alpha$ be a nonnegative integer. Then\n$$(x_1 x_2 ... x_r)^{\\alpha}=x_{i_1}^{\\alpha}x_{i_2}^{\\alpha} ...x_{i_r}^{\\alpha} \\upsilon_1^\n{f_1(\\alpha)}\\upsilon_2^{f_2(\\alpha)}\\cdots \\ ,$$\nwhere $\\{i_1, \\ i_2, \\ldots,\\ i_r \\} = \\{1, 2, \\ldots, r \\}$ and $\\upsilon_1,\\upsilon_2, \\cdots$  are commutators of weight at least two in the letters $x_i^,$ in ascending order and\n\\begin{equation}\nf_i(\\alpha)=a_1{\\alpha \\choose 1}+ a_2{\\alpha \\choose 2}+\\cdots+a_{w_i}{\\alpha \\choose w_i},\n\\end{equation}\nwith $a_1, \\ldots, a_{wi} \\in \\mathbf{Z} $ and $w_i$ is the\nweight of $\\upsilon_i$ in elements $x_1, \\ldots, x_r$.\n\\end{lem}\n\n\\begin{lem} (\\cite{st}). Let $\\alpha$\nbe a fixed integer and $G$ be a nilpotent group of class at most\n$k$. If $b_1, \\ldots, b_r \\in G$ and $r<k$, then\n$$[b_1,\\ldots,b_{i-1},b_i^{\\alpha},b_{i+1},\\ldots,b_r]=[b_1,...,b_r]^{\\alpha}\n\\upsilon_1^{f_1(\\alpha)}\\upsilon_2^{f_2(\\alpha)}\\cdots,$$ where\n$\\upsilon_1, \\upsilon_2, \\ldots$ are commutators in $b_1, \\ldots,b_r$ of weight strictly\ngreater than $r$, and every $b_j$, $1\\leq j \\leq r$, appears in\neach commutator $\\upsilon_i$. The $f_i(\\alpha)$ are of the form (2.1), with $a_1, \\ldots, a_{w_i} \\in \\mathbf{Z}$ and $w_i$ is the weight of $\\upsilon_i$ (in $b_1, \\ldots, b_r$)\nminus $(r-1)$.\n\\end{lem}\n\nIt is noted by Struik \\cite{st} that the above lemma can be proved similarly if $[b_1, \\dots ,b_{i-1},b_i^{\\alpha},b_{i+1}, \\dots ,b_r]$ and $[b_1, \\dots ,b_r]$ are replaced by arbitrary commutators (that is monimial commutators with parentheses arranged arbitrarily).\n\nTo prove the main results we require the following notions.\n\\begin{defn}\n A relative central\nextension of a pair $(G,N)$ of groups consists of a group homomorphism\n$\\sigma: M \\rightarrow G$ together with an action of $G$ on $M$\nsuch that \\\\\ni) $\\sigma (M)=N$;\\\\\nii) $\\sigma (m^g)=g^{-1} \\sigma (m)g$, for all $m \\in M, g \\in G$;\\\\\niii) $m^{\\sigma(m_1)}=m_1^{-1} m m_1$, for all $m \\in M, g \\in G$;\\\\\niv) $G$ acts trivially on $ker \\sigma$.\n\\end{defn}\n\nLet $(G,N)$ be a pair of groups and $\\sigma: M \\rightarrow G$ be a relative central\nextension of $(G,N)$. The $G$-commutator subgroup of $M$ is defined\nthe subgroup $[M,G]$ generated by all the $G$-commutators $[m,g]=m^{-1}m^g ,$\nwhere $m^{g}$ is the action of $g$ on $m$, for all $g \\in G, m \\in M$. Also for all positive integer $n$, we define\n$$Z_n(M,G)= \\{ m \\in M | \\ [m,g_1,g_2,\\ldots,g_n]=1 , \\ for \\ all\\ g_1, g_2, \\ldots , g_n \\in G \\},$$ in which $ [m,g_1,g_2, \\ldots,g_n]$ denotes $[\\cdots[[m,g_1],g_2], \\ldots,g_n]$. It is easy to see that $Z_n(M,G)\\leq Z_n(M)$. The subgroup $Z_1(M,G)$ is well known as the $G$-center of $M$ which is denoted by $Z(M,G)$.\n\nLet $(G,N)$ be a pair of groups and $k$ be a positive integer. We define\n$\\gamma_{k+1}(N,G)= [N, \\ _kG]$ in which $[N, \\ _kG]=[ \\cdots [[N,\\underbrace{G],G], \\ldots ,G]}_{k-times}$.\nA pair $(G,N)$ of groups is called nilpotent of class $k$ if $\\gamma_{k+1}(N,G)=1$ and $\\gamma_{k}(N,G)\\neq 1$. It is clear that any pair of finite $p$-groups is nilpotent.\n\n\\begin{defn}\n A relative central\nextension $\\sigma: N^*\\rightarrow G$ of a pair $(G,N)$ is\ncalled a covering pair if there exists a subgroup $A$ of $N^*$\nsuch that \\\\\ni) $A \\leq Z(N^*,G)\\cap [N^*,G]$;\\\\\nii) $A\\cong M(G,N)$;\\\\\niii) $N\\cong N^*\/A$.\\\n\nEllis proved that any pair of finite groups has at least one covering pair [2, Theorem 5.4].\n\n\\end{defn}\n\nHereafter in this section, we suppose that $(G,N)$ is a pair of finite groups and $K$ is the complement of $N$ in $G$. Also, suppose that\n$\\sigma: N^*\\rightarrow G$ is a covering pair of $(G,N)$ with a subgroup of $A$ of $N^*$ such that $A \\leq Z(N^*,G)\\cap [N^*,G]$, $A\\cong M(G,N)$ and $N\\cong N^*\/A$. Then for all $k \\in K$, the homomorphism $\\psi_k:N^* \\rightarrow N^*$ defined by $ n^* \\mapsto {n^*}^k$ is an automorphism of  $N^*$ in which ${n^*}^k$ is induced by the\naction of $G$ on $N^*$. Considering the homomorphism $\\psi:K \\rightarrow Aut(N^*)$ given by $\\psi(k)=\\psi_k$ for all $k \\in K$, we form the semidirect product of $N^*$\nby $K$ and denote it by $G^*=N^*K$. Then it is easy to check\nthat the subgroups $[N^*,G]$ and $Z(N^*,G)$ are contained in\n$[N^*,G^*]$ and $Z(N^*, G^*)$, respectively. If $\\delta : G^*\n\\rightarrow G$ is the map given by\n$\\delta(n^*k)=\\sigma(n^*)k,$ for all $n^* \\in N^*$ and $k\\in K$,\nthen $\\delta$ is an epimorphism with $\\ker \\delta=\\ker \\sigma$.\n\n\\begin{lem}\n By the above notation, let  $(G,N)$ be\na nilpotent pair of finite groups of class $k$ and $\\exp(N)=p^e$.\n Then every commutator of weight $w$ ($w \\geq 2$) in $[N^*,\\ _{w-1}G^*]$ has an order dividing $p^{e+m(k+1-w)}$, where $m= \\lfloor \\log_p k \\rfloor $.\n \\end{lem}\n\n\\begin{proof}\nWe use reverse induction on $w$ to prove the lemma.\nSince $(G,N)$ is nilpotent of class $k$ and $N\\cong N^*\/A$, $G\\cong G^*\/A$ and $A\\leq Z(N^*,G^*)$, we have $[N^*,\\ _{k+1}G^*]=1$. On the other hand, $\\exp(N)=p^e$ implies that $[{N^*}^{p^e},G^*]=1$. Hence the result follows for $w \\geq k+1$ by Lemma 2.2.\n Now assume that $l<k+1$ and the result is true for all $w>l$. We will prove the result for $l$. Put\n$\\alpha=p^{e+m(k+1-l)}$ with $m= \\lfloor \\log_p k \\rfloor $ and let $u=[n, x_2, \\ldots , x_l]$ be a commutator of weight $l$, where $n\\in N^*$ and $x_2, \\ldots , x_l \\in G^*$. Then by Lemma 2.2, we have\n$$ [n^{\\alpha}, x_2, \\ldots , x_l]=[n, x_2, \\ldots , x_l]^{\\alpha}\n\\upsilon_1^{f_1(\\alpha)} \\upsilon_2^{f_2(\\alpha)} \\cdots,$$\nwhere $\\upsilon_i$ is a commutator on $n, x_2, \\ldots, x_l$ of weight $w_i$ such that $l < w_i \\leq k+1 $, and  $f_i(\\alpha)=a_1{\\alpha \\choose 1}+\na_2{\\alpha \\choose 2}+ \\dots +a_{k_i}{\\alpha \\choose k_i}$, where\n$k_i=w_i - l + 1 \\leq k$, for all $ i \\geq 1$.\nOne can easily check that $p^t$ divides $p^{t+m} \\choose s$ with $m=\\lfloor \\log_p k \\rfloor$, for any prime $p$ and any positive integers $t,s$ with $s \\leq k$.\nThis implies that $p^{e+m(k-l)}$ divides the $f_i(\\alpha)$'s and so by induction hypothesis\n$\\upsilon_i^{f_i(\\alpha)} =1$, for all $ i \\geq 1$.\nOn the other hand, it is clear that $[n^{\\alpha},x_2, \\ldots, x_l] =1$. Therefore $u^{\\alpha}=1$ and this\ncompletes the proof.\n\\end{proof}\n\n\\begin{thm}\nIf  $(G,N)$ is\na nilpotent pair of finite groups of class $k$ and $N$ is a $p$-group of exponent\n$p^e$, then $\\exp([N^*,G^*])$ divides $p^{e+m(k-1)}$,\nwhere $m= \\lfloor \\log_p k \\rfloor $.\n\\end{thm}\n\n\\begin{proof}\nEvery element $g \\in [N^*,G^*]$ can be expressed as $g=y_1 y_2 \\cdots\ny_n$, where $y_i=[n_i,g_i]$ for $ n_i \\in N^*, g_i \\in G^*$. Put $\\alpha = p^{e+m(k-1)} $.\nBy Lemma 2.1, we have\n$$g^{\\alpha} = y_{i_1}^{\\alpha} y_{i_2}^{\\alpha} \\cdots y_{i_n}^{\\alpha}\n\\upsilon_1^{f_1(\\alpha)} \\upsilon_2^{f_2(\\alpha)} \\cdots,$$\nwhere $\\{i_1, i_2, \\ldots,i_n \\}=\\{1, 2, \\ldots, n \\}$ and $\\upsilon_i$ is a basic commutator of weight $w_i$ in $y_1, y_2, \\ldots, y_n$, with $2\\leq w_i \\leq k$, for all $i \\geq 1$, and also $f_i(\\alpha)$ is of the form (2.1). Hence by an argument similar to the  proof of Lemma 2.5 $p^{e+m(k-2)}$\ndivides $f_i(\\alpha)$. Then applying Lemma 2.5, we have $\\upsilon_i^{f_i(\\alpha)} =1$,\nfor all $i \\geq 1$, and $y_j^{\\alpha} =1$, for all $j$, $1\\leq j \\leq n$.\nWe therefore have $g^{\\alpha}=1$ and the desired result follows.\n\\end{proof}\n\nAn upper bound  for the exponent of the Schur multiplier of some pairs of finite groups is given in the following theorem.\n\\begin{thm}\nLet $(G,N)$ be a nilpotent pair of finite groups of class $k$ such that $\\exp(N)=p^e$. Then $\\exp(M(G,N))$ is a divisor of $p^{e+m(k-1)}$, where $m= \\lfloor \\log_p k \\rfloor $.\n\\end{thm}\n\n\\begin{proof}\n The result follows by Theorem 2.6 and the fact that $M(G,N)\\cong A \\leq [N^*,G]\\leq [N^*,G^*]$.\n\\end{proof}\n\nThe following corollary gives a condition under which the exponent of the Schur multiplier of a pair $(G,N)$ divides the exponent of $N$.\n\\begin{cor}\nLet $(G,N)$ be a pair of finite $p$-groups of class at most $p-1$. Then $\\exp(M(G,N))$ divides $\\exp(N)$.\n\\end{cor}\n\n\\begin{rem}\nLet $G$ be a finite $p$-group of class $k$ with $exp(G)=p^e$ . Since $M(G,G)=M(G)$, Theorem 2.7 implies that $exp(M(G))$ divides $p^{e+[\\log_pk](k-1)}$. It is easy to see that this bound improves the bound $p^{(2e\\lfloor \\log_2{k}\\rfloor)}$ given by Moravec \\cite{mor}. For example for any $p$-group $G$ of class $k$, $2\\leq k \\leq p-1$ with $exp(G)=p^e$, we have $p^{e+[\\log_pk](k-1)} \\leq p^{(2e\\lfloor \\log_2{k}\\rfloor)}$.\n\\end{rem}\n\n\\begin{rem}\nLet $(G,N)$ be a pair of finite nilpotent groups of class at most $k$. Let $S_1,S_2, \\ldots, S_n$ be all the Sylow subgroups of $G$.\nBy [2, Corollary 1.2], we have $$ M(G,N)=M(S_{1} , S_{1} \\cap N) \\times \\dots\n\\times M(S_{n} , S_{n} \\cap N).$$ Put  $m_i= \\lfloor \\log_{p_i} k \\rfloor $, for all $i$,\n $1\\leq i \\leq n$. Then by Theorem 2.7, we have\n$$\\exp( M(G,N) \\mid \\prod^n_{i=1}p_i^{e_i+m_i(k-1)},$$ where\n$p_i^{e_i} = \\exp(S_{i})$.\n\\end{rem}\n\n\\section{  Pairs of powerful $p$-groups}\n\nIn 1987, A. Lubotzky and A. Mann \\cite{lub} defined powerful $p$-groups which are used for studying $p$-groups.\nThey gave some bounds for the order, the exponent and the number of\ngenerators of the Schur multiplier of a powerful $p$-group. Also, they showed that $\\exp(M(G))$ divides $\\exp(G)$ when $G$ is a powerful $p$-group. The purpose of this section is to show that\nif $(G,N)$ is a pair of finite $p$-groups and $N$ is powerfully embedded in $G$, then the exponent of $M(G,N)$ divides the exponent of $N$.\n Throughout this section $\\mho_i(G)$ denotes the subgroup of\n$G$ generated by all $p^{i}$th powers of elements of $G$. It is easy to see that $\\mho_{i+j}(G) \\subseteq \\mho_i(\\mho_j(G))$, for all positive integers $i,j$.\n\\begin{defn}\n$(i)$ A $p$-group $G$ is\ncalled powerful if $p$ is odd and $G' \\leq \\mho_1(G)$, or\n$p=2$ and $G' \\leq \\mho_2(G)$.\\\\\n$(ii)$ Let $G$  is a $p$-group and $N\\leq G$. Then $N$ is   powerfully\nembedded in $G$ if $p$ is odd and $[N,G]\\leq \\mho_1(N)$, or $p=2$ and $[N,G]\\leq \\mho_2(N)$.\n\\end{defn}\n\nAny powerfully embedded subgroup is itself a powerful\n$p$-group and must be normal in the whole group. Also a $p$-group is\npowerful exactly when it is powerfully embedded in itself. While it\nis obvious that factor groups and direct products of powerful\n$p$-groups are powerful, this property is not subgroup-inherited \\cite{lub}.\nThe following lemma gives some properties of powerful $p$-groups.\\\\\n\n\\begin{lem}\n (\\cite{lub}). The following statements\nhold for a powerful $p$-group\n$G$.\\\\\n$(i)$ $\\gamma_i(G), G^{i}, \\mho_i(G), \\Phi(G)$ are powerfully\nembedded in $G$.\\\\\n$(ii)$ $\\mho_i(\\mho_j(G))=\\mho_{i+j}(G)$.\\\\\n$(iii)$ Each element of $\\mho_i(G)$ can be written as $a^{p^{i}},$\nfor\nsome $a\\in G$ and hence $\\mho_i(G)=\\{g^{p^{i}}: g\\in G\\}$.\\\\\n$(iv)$ If $G=\\langle a_1,a_2,...,a_d\\rangle$, then $\\mho_i(G)=\\langle\na_1^{p^{i}},a_2^{p^{i}},...,a_d^{p^{i}}\\rangle$.\n\\end{lem}\n\n\\begin{lem}\n (\\cite{lub}). Let $N$ be powerfully embedded in $G$. Then $\\mho_i(N)$ is  powerfully embedded in $G$.\n\\end{lem}\n\nThe proof of the following lemma is straightforward.\n\\begin{lem}\nLet $M$ and $G$ be two groups with an action of $G$ on $M$. Then for all $m,n \\in M$, $g,h \\in G$, and any integer $k$ we have the following equalities. \\\\\n$(i)$ $[mn,g]=[m,g]^n[n,g]$;\\\\\n$(ii)$ $[m,gh]=[m,h][m,g]^h$;\\\\\n$(iii)$ $[m^{-1},g]^{-1}=[m,g]^{m^{-1}}$;\\\\\n$(iv)$ $[m,g^{-1}]^{-1}=[m,g]^{g^{-1}}$;\\\\\n$(v)$ $[m,g^{-1},h]^g[m,[g,h^{-1}]]^h[[m^{-1},h]^{-1},g]^m=1$;\\\\\n$(vi)$ $[m^k,g]=[m,g]^k [m,g,m]^{k(k-1)\/2}  \\pmod{[M, \\ _3G]}$.\n\\end{lem}\n\n\n\\begin{lem}\nLet $(G,N)$ be a pair of finite $p$-groups and $\\sigma : N^* \\rightarrow G$ be a relative central extension of $(G,N)$. Suppose that $M$ and $K$ are two normal subgroups of $N^*$. Then $M \\leq K$ if  $M \\leq K[M,G]$.\n \\end{lem}\n\n\\begin{proof}\n Applying Lemma 3.4 we have\n$$ M \\leq K[M,G]\\leq K[K[M,G],G]\\leq K[K,G][M,G,G]\\leq \\dots \\leq K[M, \\ _iG], $$\nfor all $i\\geq 1$. On the other hand, since $G$ is a finite $p$-group, there exists an integer $l$ such that  $[N, \\ _lG]=1 $. Hence $[N^*, \\ _{l+1}\\ G]=1$ and the result follows.\n\\end{proof}\n\n\\begin{lem}\n Let $(G,N)$ be a pair of finite $p$-groups and $\\sigma : N^* \\rightarrow G$ be a relative central extension of $(G,N)$. Let $M$ be a normal\nsubgroup of $H$. Then the following statements hold.\\\\\n$(i)$ If $p>2$, then $[\\mho_1(M),G]\\subseteq \\mho_1([M,G] )[M, \\ _3G]$.\\\\\n$(ii)$ If $p=2$, then $[\\mho_2(M),G]\\subseteq \\mho_2([M,G] ) \\mho_1([M, \\ _2G])[M, \\ _3G]$.\n\\end{lem}\n\\begin{proof} $(i)$ It is enough to show that $[m^p,g]\\in \\mho_1([M,G] )[M, \\ _3G]$, for all $m \\in M, g \\in G$.\n By Lemma 3.4  $[m^p,g]={[m,g]}^p {[m,g,m]}^{p(p-1)\/2}\\ \\ \\ \\pmod {[M, \\ _3G]}.$ Since $p$ is odd and $p \\mid \\frac{p(p-1)}{2}$ we have ${[m,g]}^p {[m,g,m]}^{p(p-1)\/2} \\in \\mho_1([M,G] )$. Now the result holds.\\\\\n $(ii)$ The proof is similar to $(i)$.\n\\end{proof}\n\n\\begin{lem}\nLet $(G,N)$ be a pair of finite $p$-groups and $\\sigma : N^* \\rightarrow G$ be a relative central extension of $(G,N)$.  Suppose that $K \\leq N^*$. Then the following statements hold.\\\\\n$(i)$ If $p>2$, then $[K,G] \\leq \\mho_1 (K)$ if and only if $[K\/[K,\\  _2G], G]\\leq \\mho_1( K\/[K,\\  _2G])$.\\\\\n$(ii)$ If $p=2$, then $[K,G] \\leq \\mho_2 (K)$ if and only if $[K\/[K,\\  _2G], G]\\leq \\mho_2( K\/[K,\\  _2G])$.\\\\\n$(iii)$ If $p=2$, then $[K,G] \\leq \\mho_2 (K)$ if and only if $[K\/\\mho_1([K, G]), G]\\leq \\mho_2( K\/\\mho_1([K, G]))$.\n\\end{lem}\n\\begin{proof} $(i)$ Let $[K,G] \\leq \\mho_1 (K)$ and put $H=[K, \\ _2G]$. Then\n$$ [\\frac{K}{H},G]=\\frac{[K,G]H}{H} \\leq \\frac{\\mho_1(K)H}{H}=\\mho_1(\\frac{K}{H}),$$ as desired. Sufficiency follows by Lemma 3.5.\\\\\n$(ii)$ The proof is similar to $(i)$.\\\\\n$(iii)$  Necessity follows as for (i). Let $[K\/\\mho_1([K, G]), G]\\leq \\mho_2( K\/\\mho_1([K, G]))$. Then $[K,G]\\leq \\mho_2(K) \\mho_1([K,G])$. On the other hand, $\\mho_1([K\/\\mho_1([K, G]), G])$. Thus $[K\/\\mho_1([K, G]), G]$ is abelian and so $\\Phi([K\/\\mho_1([K, G]), G])=1$. This implies that $\\Phi([K,G])=\\mho_1([K,G])$. Therefore $[K,G]\\leq \\mho_2(K)$.\n\\end{proof}\n\nThe following useful remark is a consequence of Lemma 3.7.\n\\begin{rem}\nLet $(G,N)$ be a pair of finite $p$-groups and $\\sigma : N^* \\rightarrow G$ be a relative central extension of $(G,N)$. Let $K \\leq N^*$. Then to prove that $[K,G] \\leq \\mho_1 (K)$ ( $[K,G] \\leq \\mho_2 (K)$ for $p=2$)  we can assume that\\\\\n$(i)$ $[K, \\ _2G]=1$;\\\\\n$(ii)$ $\\mho_1(K)=1\\ (  \\mho_2(K)=1$ for $p=2$ ) and try to show that $[K,G]=1$;\\\\\n$(iii)$ $\\mho_1([K,G])=1$ whenever $p=2$.\n\\end{rem}\n\\begin{lem}\nLet $(G,N)$ be a pair of finite $p$-groups and $\\sigma : N^* \\rightarrow G$ be a covering pair of $(G,N)$. Let $N$ be powerfully embedded in $G$.  \\\\\n$(i)$ If $p>2$, then $[\\mho_n([N^*,G]),G]\\leq \\mho_1(\\mho_n([N^*,G]))$ .\\\\\n$(ii)$ If $p=2$, then $[\\mho_n([N^*,G]),G]\\leq \\mho_2(\\mho_n([N^*,G]))$.\n\\end{lem}\n\\begin{proof}\n $N^*$ has a subgroup $A$ such that $A \\leq Z(N^*,G)\\cap [N^*,G]$, $A\\cong M(G,N)$ and $N\\cong N^*\/A$.\\\\\n$(i)$ Let $p>2$. We use induction on $n$. If $n=0$, then by Remark 3.8 we may assume that $[[N^*,G],G,G]=1$, $\\mho_1([N^*,G])=1$ and\n we should show that $[[N^*,G],G]=1$.\n  Since $N$ is powerfully embedded in $G$, we have  $[N,G]\\leq  \\mho_1(N)$, and therefore  $[N^*,G]\\leq  \\mho_1(N^*)A$. Now we claim that  $\\mho_1(N^*)\\leq Z(N^*,G)$.\n  To prove the claim, let $ a \\in N^*$ and $b\\in G$. Since $\\gamma_3(\\langle a, [N^*,G]\\rangle)=1$, we have  $cl(<a,a^b>) \\leq cl(<a, [N^*,G]>)\\leq 2$  ($cl(H)$ denotes the nilpotency class of $H$). On the other hand, Lemma 3.4 implies that $$\n  (a^p)^b=a^p[a^p,b]\\equiv  a^p[a,b]^p[a,b,a]^{p(p-1)\/2} \\pmod{[<a>, \\ _3G]}.$$  Therefore $(a^p)^b=a^p$ since $[[N^*,G],G,G]=1$ and $\\mho_1([N^*,G])=1$.\n  Hence  $\\mho_1(N^*)\\leq Z(N^*,G)$ as desired. Thus $[N^*,G] \\leq  \\mho_1(N^*)A\\leq Z(N^*,G)$ and the result follows for $n=0$.\n  \n  Now suppose that the induction hypothesis is true for $n=k$. The first step of induction implies that $[N^*,G]$ is powerful.\n  Using Lemmas 3.5 and 3.6, one can see that if $H$ is a subgroup of $N^*$ and $[H,G] \\leq  \\mho_1(H)$, then $[\\mho_1(H),G] \\leq \\mho_1(\\mho_1(H))$.  Hence by Lemma 3.2 and induction hypothesis we have\n$$[\\mho_{k+1}([N^*,G]),G]= [\\mho_1(\\mho_{k}([N^*,G])),G]\\leq \\mho_1(\\mho_1(\\mho_{k}([N^*,G])))$$ $$=\\mho_1(\\mho_{k+1}([N^*,G]))$$ which completes the proof.\\\\\n$(ii)$ Let $p=2$. The proof is similar to (i), but we need to prove that if $H$ is a subgroup of $N^*$ and $[H,G]\\leq \\mho_2(H)$, then $[ \\mho_1(H),G] \\leq \\mho_2(\\mho_1(H))$. By  Remark 3.8,\n  for $a \\in H, b \\in G$ we have $[a^4,b]=[a^2,b]^2=1$. So $a^4 \\in Z(H,G)$ and $\\mho_2(H)\\leq Z(H,G)$. Then $[H,G]\\leq \\mho_2(H)\\leq Z(H,G)$. Therefore $[a^2,b]=[a,b]^2$ and\n  \\begin{equation}\n  [\\mho_1( H),G]=\\mho_1([H,G]).\n \\end{equation}\n On the other hand, since $\\mho_2(H)\\leq Z(H,G)$, we have\n $$\\mho_1(\\mho_2(H))=<({a_1}^4 \\dots {a_k}^4)^2| a_i \\in H>=<{a_1}^8 \\dots {a_k}^8>=\\mho_3(H)\\leq \\mho_2(\\mho_1(H)). $$\n Hence (3.1) implies that $[\\mho_1(H),G] \\leq \\mho_2(\\mho_1(H))$ which completes the proof of the above claim.\n\\end{proof}\n\\begin{lem}\nLet $H$ and $G$ be two arbitrary groups with an action of $G$ on $N$. If $x \\in H$ and $g\\in G$, then $$[x^n,g]=[x,g]^n c, $$ where $M=\\langle x,[x,g] \\rangle$ and $ c \\in \\gamma_2(M)$.\n\\end{lem}\n\\begin{proof} Applying Lemma 2.1, we have\n$$  [x^n,g]=(x^n)^{-1}(x^n)^{g}=(x)^{-n}(x^g)^n=(x)^{-n}(x[x,g])^n=[x,g]^n c, $$\nwhere $M=\\langle x,[x,g] \\rangle, c \\in \\gamma_2(M)$.\n\\end{proof}\n\nNow we can state the main result of this section.\n\\begin{thm}\n  Let $(G,N)$ be a pair of finite $p$-groups in which $N$ is powerfully embedded in $G$. Then\n  $\\exp(M(G,N))$ divides $ \\exp(N)$.\n  \\end{thm}\n\\begin{proof} Let $p>2$ and $\\sigma : N^* \\rightarrow G$ be a covering pair of $(G,N)$ with a subgroup $A$ such that $A \\leq Z(N^*,G)\\cap [N^*,G]$, $A\\cong M(G,N)$ and $N\\cong N^*\/A$. It is enough to show that $\\exp([N^*,G])=\\exp(N^*\/Z(N^*,G))$. For this we use induction on $k$ and show that\n\\begin{equation}\n\\mho_k([N^*,G])=[\\mho_k(N^*),G].\n\\end{equation}\nIf $k=0$, then (3.2) holds. Now assume that (3.2) holds, for  $k=n$.\nWorking in powerful $p$-group  $N^*\/A$ we get  $\\mho_{n+1}(N^*\/A)=\\mho_1 (\\mho_n(N^*\/A))$ by Lemma 3.2. Hence\n\\begin{equation}\n \\frac{\\mho_{n+1}(N^*)A}{A}=\\mho_1(\\frac{\n\\mho_n(N^*)A}{A})=\\frac{\\mho_1(\\mho_n(N^*)A)A}{A}.\n\\end{equation}\nThen Lemmas 3.6 and 3.9 and induction hypothesis imply that\n\\begin{eqnarray*}\n [\\mho_{n+1}(N^*),G]=\n[\\mho_1 (\\mho_n(N^*)A)A,G] & \\leq &\n\\mho_1 ([\\mho_n(N^*)A,G])[\\mho_n(N^*)A, \\ _3G] \\\\& \\leq & \\mho_1 ([\\mho_n(N^*),G])[\\mho_n(N^*)A, \\ _2G] \\\\&\\leq& \\mho_1 (\\mho_n([N^*,G])[\\mho_n([N^*,G]),G] \\\\&\\leq& \\mho_1 (\\mho_n([N^*,G])=\\mho_{n+1}([N^*,G]).\n\\end{eqnarray*}\nFor the reverse inclusion, we show that\n$$\\mho_{n+1}([N^*,G]) \\equiv 1 \\pmod{[\\mho_{n+1}(N^*),G]}.$$\nSince by $(3.4)$, $[\\mho_{n+1}(N^*),G])=[\\mho_1 (\\mho_n(N^*)A)A,G]=[\\mho_1 (\\mho_n(N^*)A),G]$, it follows that\n$\\mho_1 (\\mho_n(N^*)A)A \\leq Z(N^*,G)  \\pmod{[\\mho_{n+1}(N^*),G]}$.\\\\\nOn the other hand, since $N$ is powerfully embedded in $G$, we have\n\\begin{eqnarray*}\n [\\mho_{n}(N^*),G]=[\\mho_{n}(N^*)A,G] &\\leq& \\mho_1 (\\mho_n(N^*)A)A \\\\&\\leq& Z(N^*,G) \\pmod{[\\mho_{n+1}(N^*),G]}.\n \\end{eqnarray*}\nTherefore $[\\mho_{n}(N^*),G,G]\\equiv 1 \\pmod{[\\mho_{n+1}(N^*),G]}.$\\\\\n Moreover, by Lemma 3.10 we have\n $$[\\mho_1 (\\mho_n(N^*)A),G][\\mho_{n}(N^*),G,G]=\\mho_1( [ \\mho_n(N^*),G])[\\mho_{n}(N^*),G,G].$$\n It follows that   $\\mho_1 [(\\mho_n(N^*)),G]\\equiv 1 \\pmod{[\\mho_{n+1}(N^*),G]}$. Then by induction hypothesis, we have\n $$\\mho_{n+1}([N^*,G])= \\mho_1(\\mho_{n}[N^*,G])=\\mho_1 ([\\mho_n(N^*),G])\\equiv 1 \\pmod{[\\mho_{n+1}(N^*),G]}.$$\n This completes the proof for odd primes $p$. The proof for the case $p=2$ is similar.\n\\end{proof}\n\\section*{Acknowledgement}\nThe authors would like to thank the referee for the valuable comments and useful suggestions to improve the present paper.\n\nThis research was supported by a grant from Ferdowsi University of Mashhad; (No. MP90259MSH).\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n$Q$-balls represent stationary localized solutions \nof a complex scalar field theory with\na suitable self-interaction in flat space\n\\cite{Friedberg:1976me,Coleman:1985ki}.\nThe global phase invariance of the scalar field theory\nis associated with a conserved charge $Q$ \\cite{Friedberg:1976me},\nwhich represents the electromagnetic charge,\nonce the theory is promoted to a gauge theory.\n\nThe simplest type of $Q$-balls is spherically symmetric.\nThese possess a finite mass and charge, but carry no angular momentum.\nConsidering their mass as a function of their charge,\nthere are two branches of $Q$-balls, merging and ending at a cusp,\nwhere mass and charge assume their minimal values \\cite{Friedberg:1976me}.\n\nRecently, a new type of scalar potential for $Q$-balls was considered,\nleading to the signum-Gordan equation for the scalar field\n\\cite{Arodz:2008jk,Arodz:2008nm}.\nThis potential gives rise to spatially compact $Q$-balls, \nwhere the scalar field vanishes identically outside a critical\nradius $r_o$ \\cite{Arodz:2008jk}.\nWhen coupled to electromagnetism, a new type of solution appears,\n$Q$-shells \\cite{Arodz:2008nm}.\nIn $Q$-shells the scalar field vanishes identically both inside\na critical radius $r_i$ and outside a critical radius $r_o$,\nthus forming a finite shell $r_i < r < r_o$ of charged matter.\n\nWhen gravity is coupled to $Q$-balls, boson stars arise,\nrepresenting globally regular self-gravitating solutions\n\\cite{Lee:1991ax,Jetzer:1991jr,Mielke:2000mh,Schunck:2003kk}.\nThe presence of gravity has a crucial influence\non the domain of existence of the classical solutions.\nInstead of only two branches of solutions joint at a single cusp, \nthe boson stars exhibit an intricate cusp structure,\nwhere mass and charge oscillate endlessly.\nFor black holes with scalar fields, on the other hand, \na number of theorems exist, which exclude their existence\nunder a large variety of conditions \n\\cite{Bekenstein:1971hc,Bekenstein:1995un,Mayo:1996mv}.\n\nHere we consider the effect of gravity on the $Q$-balls and\n$Q$-shells of the signum-Gordon model coupled to a Maxwell field.\nWe construct these charged boson stars and gravitating $Q$-shells\nand analyze their properties and their domains of existence.\nWe observe that at certain critical values of the mass and charge,\nthe space-times form a throat at the (outer) radius $r_o$, \nrendering the respective exterior space-time\nan exterior extremal Reissner-Nordstr\\\"om space-time.\n\nMoreover, we show that in this model\nthe black hole theorems can be elluded,\nthat forbid black holes with scalar hair.\nIndeed, the gravitating $Q$-shells can be endowed with a horizon $r_H$\nin the interior region $0< r_H < r_i$,\nwhere the scalar field vanishes and the gauge potential is constant.\nThis Schwarzschild-type black hole in the interior\nis surrounded by a shell of charged matter, $r_i<r<r_o$,\nleading to an exterior space-time $r_o<r<\\infty$ of\nReissner-Nordstr\\\"om type. \nWe analyze the global and horizon properties of these black holes \nwithin $Q$-shells, and present a mass relation.\nWhen a throat develops at the outer radius $r_o$,\nwhich renders the respective exterior space-time\nan exterior extremal Reissner-Nordstr\\\"om space-time,\nthe temperature of the horizon $r_H$ tends to zero.\n\nIn section 2 we recall the action and ans\\\"atze for the fields.\nWe present the gravitating $Q$-balls and $Q$-shells in section 3 and 4, \nrespectively, and the black holes within $Q$-shells in section 5.\nWe end with a conclusion and an outlook.\n\n\\section{Action}\n\nWe consider the action of a self-interacting complex scalar field\n$\\Phi$ coupled to a U(1) gauge field and to Einstein gravity\n\\begin{equation}\nS=\\int \\left[ \\frac{R}{16\\pi G}\n   - \\frac{1}{4} F^{\\mu\\nu} F_{\\mu\\nu}\n   -  \\left( D_\\mu \\Phi \\right)^* \\left( D^\\mu \\Phi \\right)\n - U( \\left| \\Phi \\right|) \n \\right] \\sqrt{-g} d^4x\n , \\label{action}\n\\end{equation}\nwith field strength tensor\n\\begin{equation}\nF_{\\mu\\nu} = \\partial_\\mu A_\\nu - \\partial_\\nu A_\\mu \n , \\label{action2}\n\\end{equation}\ncovariant derivative\n\\begin{equation}\nD_\\mu \\Phi = \\partial_\\mu \\Phi + i e A_\\mu \\Phi\n , \\label{action3}\n\\end{equation}\ncurvature scalar $R$, Newton's constant $G$, \ngauge coupling constant $e$,\nand the asterisk denotes complex conjugation.\nThe scalar potential $U$ is chosen as\n\\begin{equation}\nU(|\\Phi|) =  \\lambda  |\\Phi| \n . \\label{U} \\end{equation} \n\nVariation of the action with respect to the metric and the matter fields\nleads, respectively, to the Einstein equations\n\\begin{equation}\nG_{\\mu\\nu}= R_{\\mu\\nu}-\\frac{1}{2}g_{\\mu\\nu}R = 8\\pi G T_{\\mu\\nu}\n\\  \\label{ee} \\end{equation}\nwith stress-energy tensor\n\\begin{eqnarray}\nT_{\\mu\\nu} &=& g_{\\mu\\nu}{L}_M\n-2 \\frac{\\partial {L}_M}{\\partial g^{\\mu\\nu}}\n=\n    ( F_{\\mu\\alpha} F_{\\nu\\beta} g^{\\alpha\\beta}\n   -\\frac{1}{4} g_{\\mu\\nu} F_{\\alpha\\beta} F^{\\alpha\\beta})\n\\nonumber\\\\\n&-& \n   \\frac{1}{2} g_{\\mu\\nu} \\left(     (D_\\alpha \\Phi)^* (D_\\beta \\Phi)\n  + (D_\\beta \\Phi)^* (D_\\alpha \\Phi)    \\right) g^{\\alpha\\beta}\n  + (D_\\mu \\Phi)^* (D_\\nu \\Phi) + (D_\\nu \\Phi)^* (D_\\mu \\Phi)\n\\nonumber\\\\\n&-& \n    {\\lambda} g_{\\mu\\nu}  |\\Phi| \n , \\label{tmunu}\n\\end{eqnarray}\nand the matter field equations,\n\\begin{eqnarray}\n& & \\partial_\\mu \\left ( \\sqrt{-g} F^{\\mu\\nu} \\right) =\n   \\sqrt{-g} e \\Phi^* D^\\nu \\Phi \n\\label{feqA} \\end{eqnarray}\n\\begin{eqnarray}\n& &D_\\mu D^\\mu \\Phi = - \\frac{\\lambda}{2} \\frac{\\Phi}{|\\Phi|}\n . \\label{feqH} \\end{eqnarray}\n\nTo construct static spherically symmetric solutions\nwe employ Schwarz\\-schild-like coordinates and adopt\nthe spherically symmetric metric\n\\begin{equation}\nds^2=g_{\\mu\\nu}dx^\\mu dx^\\nu=\n  -A^2N dt^2 + N^{-1} dr^2 + r^2 (d\\theta^2 + \\sin^2\\theta d\\phi^2)\n , \\end{equation}\nwith\n\\begin{equation}\nN=1-\\frac{2m(r)}{r}\n . \\end{equation}\n\nFor solutions with vanishing magnetic field\nthe Ansatz for the matter fields has the form\n\\begin{equation}\n \\Phi = \\phi(r) e^{i \\omega t}\n , \\label{phi} \\end{equation}\n\\begin{equation}\n A_\\mu d x^\\mu = A_0(r) dt\n . \\label{A_0} \\end{equation}\n \nFor notational simplicity, we \nintroduce new coupling constants \\cite{Arodz:2008nm}\n\\begin{equation}\n\\alpha^2 = a = 4\\pi G \\frac{\\beta^{1\/3}}{e^2},\n\\ \\ \\\n\\beta = \\frac{\\lambda e}{\\sqrt{2}} \n , \\label{constants} \\end{equation}\nand redefine the matter field functions,\n\\begin{equation}\nh(r) = \\sqrt{2}e \\phi(r) ,\n\\ \\ \\\nb(r) = \\omega + e A_0(r)\n . \\label{functions} \\end{equation}\nThe latter corresponds to\nperforming a gauge transformation to make the scalar field real\nand absorbing the frequency $\\omega$ of the scalar field into the gauge\ntransformed vector potential.\nNote, that the parameter $\\beta$ can be removed by rescaling \nand will therefore be set to one\n\\cite{Arodz:2008nm}.\nThus the only parameter left is the gravitational coupling $\\alpha$.\n\nLet us now specify the boundary conditions for the \nmetric and matter functions.\nFor the metric function $A$ we adopt\n\\begin{equation}\nA(r_o)=1\n\\ , \\end{equation}\nwhere $r_o$ is the outer radius,\nthus fixing the time coordinate. \nFor the mass function $m(r)$ we require for globally regular ball-like\nboson star solutions\n\\begin{equation}\nm(0)=0 \n, \\end{equation}\nfor globally regular shell-like solutions \n\\begin{equation}\nm(r_i)=0 \n, \\end{equation}\nwhere $r_i$ is the inner radius of the shell,\nand for black hole solutions\n\\begin{equation}\nm(r_H)= \\frac{r_H}{2}\n  \\end{equation}\nwhere $r_H < r_i$ denotes the horizon.\n\nTo be able to specify more than four boundary conditions \nfor the matter functions we introduce one or more\nauxiliary variables.\nFor globally regular boson star solutions we require\nat the origin and at the outer radius $r_o$\nthe conditions\n\\begin{equation}\nb'(0)=0 , \\ \\ \\ h'(0)=0 , \\ \\ \\ h(r_o)=0 , \\ \\ \\ h'(r_o)=0\n, \\end{equation}\nwhere the prime denotes differentiation with respect to $r$.\nIn order to choose also the value of $b(0)$ as a boundary condition,\nwe make the outer radius $r_o$ an auxiliary (constant) variable,\nand thus add the differential equation $r_o\\,'=0$,\nwithout imposing a boundary condition.\n\nFor globally regular shell solutions as well as for black holes\nwe require at the inner radius $r_i$ and at the outer radius $r_o$\nthe conditions\n\\begin{equation}\nb'(r_i)=0 , \\ \\ \\ h(r_i)=0 , \\ \\ \\ h'(r_i)=0 , \\ \\ \\ h(r_o)=0 , \\ \\ \\ h'(r_o)=0\n . \\end{equation}\nIn order to choose also the value of $b(r_i)$ as a boundary condition,\nwe now also make the ratio of inner and outer radius $r_i\/r_o$\nan auxiliary (constant) variable.\nAlternatively to demanding a certain value for $b(r_i)$,\nwe may also specify the value of the electric charge $Q$.\n\nThe charge $Q$ of the solutions is associated with the conserved current\n\\begin{eqnarray}\nj^{\\mu} & = &  - i \\left( \\Phi^* D^{\\mu} \\Phi\n - \\Phi D^{\\mu}\\Phi ^* \\right) \\ , \\ \\ \\\nj^{\\mu} _{\\ ; \\, \\mu}  =  0 \\ ,\n\\end{eqnarray}\ni.e., the charge is obtained as the spatial integral\n\\begin{equation}\nQ = -i \\int j^0 \\sqrt{-g} dr d\\theta d\\varphi\n. \\end{equation}\n\nThe mass $M$ of the stationary asymptotically flat space-times\nis obtained from the corresponding Komar expression.\nFor globally regular space-times like boson stars\nand shells of boson matter the mass is given by\n\\begin{equation}\nM = \n \\frac{1}{{4\\pi G}} \\int_{\\Sigma}\n R_{\\mu\\nu}n^\\mu\\xi^\\nu dV\n , \\label{komarM1}\n\\end{equation}\nwhere $\\Sigma$ denotes an asymptotically flat spacelike hypersurface,\n$n^\\mu$ is normal to $\\Sigma$ with $n_\\mu n^\\mu = -1$,\n$dV$ is the natural volume element on $\\Sigma$,\nand $\\xi$ denotes an asymptotically timelike Killing vector field\n\\cite{wald}.\nReplacing the Ricci tensor via the Einstein equations by the\nstress-energy tensor yields\n\\begin{equation}\nM = \n  \\, 2 \\int_{\\Sigma} \\left(  T_{\\mu \\nu}\n-\\frac{1}{2} \\, g_{\\mu\\nu} \\, T_{\\gamma}^{\\ \\gamma}\n \\right) n^{\\mu }\\xi^{\\nu} dV\n . \\label{komarM2}\n\\end{equation}\n\nFor black hole space-times the corresponding Komar expression is given by\n\\begin{equation}\nM = M_H +\n  \\, 2 \\int_{\\Sigma} \\left(  T_{\\mu \\nu}\n-\\frac{1}{2} \\, g_{\\mu\\nu} \\, T_{\\gamma}^{\\ \\gamma}\n \\right) n^{\\mu }\\xi^{\\nu} dV \n , \\label{komarM3}\n\\end{equation}\nwhere $M_H$ is the horizon mass of the black hole.\nThe mass of all (gravitating) solutions\ncan be directly obtained from the asymptotic form of their metric.\nIn the units employed, we find\n\\begin{equation}\nM= \\frac{1}{\\alpha^2}\\, \\lim_{r \\rightarrow \\infty} m(r)\n . \\label{mass} \\end{equation}\n\n\\section{Boson Stars}\n\n\\begin{figure}[p]\n\\begin{center}\n\\vspace{-1.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1a.eps}\n\\label{phasediag}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1b.eps}\n\\label{QB_r0_vs_b0}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1c.eps}\n\\label{QB_M_vs_b0}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1d.eps}\n\\label{QB_Q_vs_b0}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-1.5cm}\n\\subfigure[][]{\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1e.eps}\n\\label{QB_om_vs_b0}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig1f.eps}\n\\label{QB_Mvs_Q}\n}\n}\n\\end{center}\n\\caption{Properties of the boson star solutions shown versus $b(0)$,\nthe value of the gauge field function $b(r)$ at the origin:\n(a) $h(0)$, the value of the scalar field function $h(r)$ at the origin,\nforms with $b(0)$ the phase diagram of the solutions;\n(b) $r_o$ indicates the size of the solutions;\n(c) and (d) exhibit logarithmically\nthe mass $M$ and the charge $Q$, respectively;\nand (e) shows the value $b(\\infty)$ of the gauge field function\n$b(r)$ at infinity (which corresponds to the value\nof the scalar field frequency $\\omega$).\nIn (f) the ratio of mass and charge $M\/Q$ is exhibited versus \nthe charge $Q$.\nNote that $a=\\alpha^2$, and the asterisks mark the transition\npoints from boson stars ($Q$-balls) to $Q$-shells.\n\\label{bosonstar}\n}\n\\end{figure}\n\nWe first address the set of globally regular \nball-like boson star solutions,\nobtained by coupling the $Q$-ball solutions \\cite{Arodz:2008nm} to gravity.\nWe illustrate the physical properties of these solutions\nin Fig.~\\ref{bosonstar}.\n\nFig.~\\ref{phasediag} represents the phase diagram for the boson star\nsolutions. Here the sets of solutions for a sequence of values of the\ngravitational coupling $\\alpha$ are exhibited in terms of\nthe values of the matter field functions at the center of the solutions,\ni.e., the value of the scalar field function $h(0)$ \nand the value of the gauge field function $b(0)$.\nThe figure consists of four distinct regions,\nwhich we have labelled I, Ia, II and IIa.\nFor $\\alpha=0$, the $Q$-ball solutions form a continuous set,\nrepresented by a single curve in the lower part of the figure,\ndenoted as region I.\nThese non-gravitating solutions are\nbounded by some maximal value of $h(0)$, by some minimal value of $b(0)$,\nand by the bifurcation point with the shell-like solutions,\nwhere $h(0)$ reaches zero.\n\nAs the gravitational coupling constant $\\alpha$ is increased from zero, \nthe resulting sets of solutions smoothly deform and fill region I.\nThe maximal value of $h(0)$ \nof these sets of solutions (for each value of $\\alpha$ given by \na single continuous curve) then increases,\nthe corresponding minimal value of $b(0)$ decreases, \nand the bifurcation point with the shell-like solutions decreases as well.\nThis decrease of the bifurcation point continues until \nzero is reached at the critical value of \nthe gravitational coupling constant, $\\alpha_{sh} \\approx 0.1861$.\nBeyond $\\alpha_{sh}$ no shell-like solutions exist.\nHowever,\nthe set of solutions continues to deform smoothly with increasing $\\alpha$.\n\nThis smooth evolution continues, until a second \ncritical value of the gravitational coupling constant $\\alpha$ is reached,\n$\\alpha_{cr} \\approx 0.1989$.\nHere a bifurcation with a second set of solutions is encountered,\nwhich constitute the boundary of region Ia.\nThis second type of solutions is present for each finite value of\n$\\alpha \\le \\alpha_{cr}$. \nFor this type of solutions $h(0)$ has a minimal value for fixed $\\alpha$,\nwhich decreases with increasing $\\alpha$,\nuntil at $\\alpha_{cr}$ the bifurcation is reached.\n\nAt $\\alpha_{cr}$ the set I and set Ia\nsolutions touch and bifurcate. \nFor $\\alpha > \\alpha_{cr}$ \nthey then split into a right and left set of solutions,\nforming regions II and IIa, respectively.\nThe solutions in region II correspond to the larger values of $b(0)$,\nwhile the solutions in region IIa are restricted to the\nsmaller values of $b(0)$.\nWith increasing $\\alpha$, the sets of solutions in region IIa\nmove towards smaller values of $b(0)$,\npossibly disappearing at some critical value of the gravitational coupling,\nwhereas the sets of solutions in region II\nmove towards larger values of $b(0)$.\n\nFig.~\\ref{QB_r0_vs_b0} shows the outer radius $r_o$ for these sets of solutions,\nand thus the size of the corresponding boson stars.\nClearly, the biggest size for a given $\\alpha \\le \\alpha_{cr}$\nis always reached in region I at the bifurcation point\nwith the shell-like solutions.\nThe oscillations of the gauge field value $b(0)$ \nwith increasing scalar field value $h(0)$ seen in region II in\nFig.~\\ref{phasediag}\nare reflected in the spirals\nformed by the outer radius $r_o$ in region II in Fig.~\\ref{QB_r0_vs_b0}.\nThey are also present in regions IIa and Ia, whenever\nthe gauge field value $b(0)$ exhibits oscillations.\n\nThe mass $M$ and the charge $Q$ of these sets of boson star solutions\nare exhibited in Figs.~\\ref{QB_M_vs_b0} and \\ref{QB_Q_vs_b0}.\nBoth show a very similar pattern.\nAgain, the biggest mass and charge for a given $\\alpha \\le \\alpha_{cr}$\nare reached in region I at the respective bifurcation point\nwith the shell-like solutions,\nwhile the oscillations of $b(0)$ seen in regions Ia, II and IIa\nlead to spiral patterns for the mass and charge.\n\nThe corresponding family of curves for the asymptotic\nvalue $b(\\infty)$ of the gauge field function $b(r)$ at infinity\n(which can be indentified with the value of the\nscalar field frequency $\\omega$ in the gauge,\nwhere the gauge field vanishes at infinity)\nis exhibited in Fig.~\\ref{QB_om_vs_b0}.\nHere the overall pattern is different, but spirals occur as well.\nFinally, in Fig.~\\ref{QB_Mvs_Q} we exhibit\nthe ratio of mass and charge $M\/Q$ versus $Q$.\nWe observe a linear increase of $M\/Q$ with $Q$ for the larger values\nof $Q$ in regions I and IIa, where\nthe slope decreases with increasing $\\alpha$,\nmaking $M\/Q$ almost constant for larger values of \n$\\alpha$ (e.g.,~$\\alpha=0.22$).\n\nWhile the occurrence of spirals is a typical feature of\nboson star solutions \\cite{Friedberg:1976me}, \nthe present sets of solutions exhibit a for boson stars new phenomenon,\nnamely the formation of throats.\nAs a throat is formed,\nthe minimum of the metric function $N(r)$ tends to zero,\nand the zero is reached precisely at the outer radius $r_o$.\nAt the same time the metric function $A(r)$ tends to a step\nfunction, that vanishes inside $r_o$, and assumes the asymptotic value\n$A(r)=1$ outside $r_o$.\n(In Fig.~\\ref{fun} the functions close to throat formation\nare exhibited in the case of black holes.)\n\nThe space-time for $r \\ge r_o$ then corresponds to the exterior\nspace-time of an extremal Reissner-Nordstr\\\"om (RN) black hole.\nIndeed, there the metric function $N(r)$ can be expressed as\n\\begin{equation}\nN(r) = 1 - \\frac{2 \\alpha^2 M}{r} + \\frac{\\alpha^2 Q^2}{r^2} \n = \\left( 1 - \\frac{\\alpha Q}{r} \\right)^2 \n , \\label{RN} \\end{equation}\ni.e., $r_H = r_o= \\alpha^2 M = \\alpha Q$ for the extremal RN solution\n(in the units employed).\nAs seen in Fig.~\\ref{QB_Mvs_Q},\nthis relation is precisely satisfied, when $b(0) \\rightarrow 0$.\nThus a throat is formed, when in a set of solutions the value $b(0)$ \nof the gauge field function tends to zero.\nIn fact, the function $b(r)$ then tends to zero in the whole region\n$r< r_o$, and its derivative $b'(r)$ does so as well.\nHowever, at $r_o$ the derivative $b'(r)$ jumps to a finite\nvalue, necessary for the \nCoulomb fall-off of a solution with charge $Q$.\n\nFinally we note, that the sets of boson star solutions\nwith fixed gravitational coupling constant $\\alpha$ \nsatisfy a mass relation.\nThis relation is based on the observation, that\n\\begin{equation}\nd M = b(\\infty) d Q \n , \\label{Mreg2} \\end{equation}\nshown to hold for the regular solutions in flat space \\cite{Arodz:2008nm}.\nSince (\\ref{Mreg2}) continues to hold for the gravitating solutions,\nintegration \nyields the mass relation\n\\begin{equation}\nM_2 =  M_1 + M_Q =\n M_1 + \\int_{Q_1}^{Q_2} b(\\infty) d Q\n , \\label{Mreg} \\end{equation}\nwhere the mass $M_2$ of a regular solution with charge $Q_2$\nis obtained \nby integrating from any regular solution $M_1$ with charge $Q_1$\nalong the curve of intermediate solutions of the set.\n\n\n\n\\section{Gravitating $Q$-Shells}\n \n\\begin{figure}[t!]\n\\begin{center}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig2a.eps}\n\\label{fun1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig2b.eps}\n\\label{fun2}\n}\n}\n\\end{center}\n\\caption{Functions of the gravitating $Q$-shell solutions \nshown versus the radial coordinate $r$ for $Q=10$ and $\\alpha^2=0.1$:\n(a) metric functions $A(r)$ and $N(r)$;\n(b) matter functions $h(r)$ and $b(r)$.\nAlso shown are the corresponding functions for\nblack hole solutions with several horizon radii $r_H$.\nThe largest $r_H$ is close to the critical\nvalue, where the throat is formed.\n\\label{fun}\n}\n\\end{figure}\n\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3a.eps}\n\\label{phaseSdiag2}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3b.eps}\n\\label{QS_r0_vs_b1}\n}\n}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3c.eps}\n\\label{QBS_M_vs_b1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig3d.eps}\n\\label{QS_MQ_vs_b1}\n}\n}\n\\end{center}\n\\caption{Properties of the gravitating $Q$-shell solutions shown versus \n$b(r_i)$,\nthe value of the gauge field function $b(r)$ at the inner \nshell radius $r_i$:\n(a) $r_i\/r_o$, the ratio of inner and outer shell radii;\n(b) the outer shell radius $r_o$;\n(c) the mass $M$ of shell-like solutions and boson stars \n(resp.~$Q$-balls for $\\alpha=0$);\n(d) the scaled mass $\\alpha^2 M$ and the scaled charge $\\alpha Q$.\nNote that $a=\\alpha^2$, and the asterisks mark the transition\npoints from boson stars ($Q$-balls) to $Q$-shells.\n\\label{Q_shell}\n}\n\\end{figure}\n\nLet us next consider the gravitating shell-like solutions.\nHere the space-time consists of 3 parts.\nIn the inner part $0  \\le r < r_i$ \nthe gauge potential is constant and the scalar field vanishes.\nConsequently, it is Minkowski-like,\nwith $N(r)=1$ and $A(r)={\\rm const}<1$. \nThe middle region $r_i < r < r_o$ then represents the shell of\ncharged bosonic matter, while the outer region\n$r_o < r < \\infty$ corresponds to part of a Reissner-Nordstr\\\"om\nspace-time, where the gauge field exhibits the standard Coulomb fall-off,\nwhile the scalar field vanishes identically.\nThis behaviour of the functions is demonstrated in Fig.~\\ref{fun} for\nthe shell-like solution with charge $Q=10$ and \ngravitational coupling constant $\\alpha^2=0.1$.\n\nWe exhibit in Fig.~\\ref{Q_shell} the properties of\nshell-like solutions.\nFig.~\\ref{phaseSdiag2} shows the ratio of the\ninner radius $r_i$ to the outer radius $r_o$ for these solutions.\nFor a given finite value of the gravitational coupling,\nthe branch of gravitating shells emerges\nat the corresponding boson star solution and ends,\nwhen a throat is formed at the outer radius $r_o$.\nAs this happens, the value of\n$b(r_i)$ reaches zero (or equivalently $b(0)\\rightarrow 0$,\nsince $b(r)$ is constant in the interior, $0 \\le r \\le r_i$).\nThe exterior space-time $r > r_o$ then corresponds to the exterior of an\nextremal RN space-time.\n\nThus in contrast to $Q$-shells in flat space, which grow rapidly\nin size, mass and charge as the ratio $r_i\/r_o \\rightarrow 1$,\nthe growth of gravitating $Q$-shells is limited by gravity,\nand the restriction in size, mass and charge is the stronger, \nthe greater the value of the gravitational coupling constant $\\alpha$.\nThis is demonstrated in Figs.~\\ref{QS_r0_vs_b1}, \\ref{QBS_M_vs_b1}\nand \\ref{QS_MQ_vs_b1}, \nwhere the outer radius $r_o$, the mass $M$\nand the charge $Q$ are exhibited\nfor a sequence of values of the gravitational coupling constant.\n\nIn Fig.~\\ref{QBS_M_vs_b1} for comparison also the mass of the\ncorresponding boson star solutions (resp.~$Q$-ball solutions\nfor vanishing gravitational coupling constant) are exhibited.\nThe transitions from the ball-like to the shell-like solutions\nare indicated in the figure by the small asterisks.\nWith increasing $\\alpha$ the sets of shell-like solutions\ndecrease rapidly, until at the critical value\n$\\alpha_{sh}$ (see Fig.~\\ref{phasediag})\nthey cease to exist.\n\nFig.~\\ref{QS_MQ_vs_b1} exhibits the scaled mass $\\alpha^2 M$\nand the scaled charge $\\alpha Q$ for several sets of\ngravitating $Q$-shells. Together with Fig.~\\ref{QS_r0_vs_b1}\nthe figure demonstrates,\nthat the condition \nfor extremal RN solutions,\n$r_o= \\alpha^2 M = \\alpha Q$,\nis satisfied\nfor the shell-like solutions,\nas the throat forms at the outer radius $r_o$.\n\nFinally we note, that the shell-like solutions satisfy the mass relation\n(\\ref{Mreg}) as well.\nConsequently the mass relation holds for any two globally regular solutions\nof a set with given gravitational coupling constant,\nthus relating also ball-like and shell-like solutions.\n\n\\section{Black Holes}\n\n\\begin{figure}[p!]\n\\begin{center}\n\\vspace{-1.0cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4a.eps}\n\\label{BHQ10_b1_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4b.eps}\n\\label{BHQ100_b1_vs_rh}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4c.eps}\n\\label{BHQ10_M_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4d.eps}\n\\label{BHQ100_M_vs_rh}\n}\n}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4e.eps}\n\\label{BHQ10_T_vs_rh}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig4f.eps}\n\\label{BHQ100_T_vs_rh}\n}\n}\n\\end{center}\n\\caption{Properties of the black hole solutions \nwith Schwarzschild-type interior shown versus \nthe horizon radius $r_H$:\nThe left column exhibits for solutions with fixed charge $Q=10$\n(a) \n$b(r_i)$, the value of the gauge\nfield function $b(r)$ at the inner shell radius $r_i$;\n(c) the mass $M$;\n(e) the ratio of the temperature $T$ at the black hole horizon $r_H$ to the\ncorresponding Schwarzschild black hole temperature $T_S$.\nThe right column ((b), (d), (f)) exhibits the\nsame for solutions with fixed charge $Q=100$.\nNote that $a=\\alpha^2$. \n\\label{BH1}\n}\n\\end{figure}\n\nLet us finally address black holes in this model.\nThe simplest type of black holes is obtained,\nwhen the Minkowski-like inner part of the space-time,\n$0 \\le r \\le r_i$, of gravitating $Q$-shell solutions\nis replaced by the inner part of a curved Schwarzschild-like space-time.\nThe metric in the interior region $0 \\le r \\le r_i$\nis then determined by the function $N(r)= 1 - (r_H\/r)$ \nand a constant function $A(r)$.\nThus the event horizon resides at $r_H < r_i$.\nBut the presence of the $Q$-shell outside the event horizon,\nmakes the properties of the black hole differ from those\nof a pure Schwarzschild black hole.\n\nSince with the event horizon size a further variable appears,\nwhich is an important physical quantity,\nwe discuss the black hole properties with respect to the\nhorizon radius $r_H$ in the following.\nThe metric and matter field functions \nfor black holes with charge $Q=10$ \nat gravitational coupling $\\alpha^2=0.1$\nare exhibited in Fig.~\\ref{fun} \nfor several values of the horizon radius $r_H$.\n\nTo illustrate the domain of existence of such black hole solutions,\nwe again choose a sequence\nof values for the gravitational coupling constant,\nbut we now keep the charge $Q$ fixed, as we vary \nthe horizon radius, starting from the corresponding\nglobally regular $Q$-shell solution.\nA respective set of solutions is shown in Fig.~\\ref{BH1}\nfor $Q=$10 and 100.\n\nFirst of all we note, that the horizon radius is always limited\nin size, where the maximal size grows with the charge $Q$.\nFor small $Q$, e.g.~$Q=$10, we observe two distinct patterns\nfor the black hole solutions.\nThe first pattern arises when the\nfixed gravitational coupling constant has a value\nbelow a certain critical value.\nHere a maximal horizon size is reached,\nwhen the horizon radius $r_H$ gets close to the inner radius \nof the shell $r_i$.\nThere a bifurcation occurs and a second branch emerges,\nwhich ends at a second bifurcation, where a third branch emerges, etc.\nThis results in a spiralling pattern,\nwhere the mass $M$ and the temperature $T$ of the solutions tend towards\nfinite limiting values.\n(The first few branches are apparent in Fig.~\\ref{BHQ10_b1_vs_rh},\nand enlarged in the inlet for a representative value\nof the gravitational coupling constant, $\\alpha^2=0.01$,\nwhile the higher branches are too small to be resolved there.)\n\nThe second pattern is present above that critical value\nof the coupling constant. Here the set of black hole solutions\nfor fixed gravitational coupling ends, when a throat is formed\nat the outer shell radius $r_o$.\nThere the condition for extremal RN solutions,\n$r_o= \\alpha^2 M = \\alpha Q$, is satisfied again,\nas seen in Figs.~\\ref{BHQ10_M_vs_rh} and \\ref{BHQ100_M_vs_rh}.\nAs the throat forms,\nthe temperature $T$ at the event horizon of the Schwarzschild-like\nblack hole $r_H < r_i$ tends to zero,\nas seen in Figs.~\\ref{BHQ10_T_vs_rh} and \\ref{BHQ100_T_vs_rh}.\n\nWhile appearing at first unexpected,\nthe reason for the vanishing of the temperature $T$\nis the behaviour of the metric function $A(r)$\nin $g_{tt}$, since $A(r)$ tends to zero in the interior,\nwhen the throat is formed,\nas seen in Fig.~\\ref{fun1}.\nWe recall, that the ratio of the temperature $T$ of\nthe black hole within the $Q$-shell \nto the temperature \n$T_{\\rm S} = (4 \\pi r_{\\rm H})^{-1}$ \nof the Schwarzschild black hole is given by\n$\\displaystyle T \/ T_{\\rm S}\n  =  \\left. r A N' \\right|_{r_{\\rm H}} = A(r_H)$.\n\nFor larger (fixed) values of the charge we always observe\nthis second pattern,\nalthough the throat may either be reached directly after a monotonic\nincrease of the horizon radius $r_H$ to its maximum value,\nor along a second branch, where the\nhorizon radius is decreasing again (having passed a bifurcation),\nas seen in Fig.~\\ref{BHQ100_b1_vs_rh}.\n\nAs seen in the figure,\nwhenever bifurcations occur, there are two (or more) black hole\nspace-times with the same value of the charge $Q$ and the\nsame horizon radius $r_H$ (within a certain range of values),\nbut different values of the total mass $M$\nas measured at infinity.\nSurprisingly, however, \nthere are also two (or more) black hole space-times\nwith the same value of the charge $Q$ and the same \nvalue of the total mass $M$ (within a certain range of values).\nThese black holes thus have the same set of global charges\nbut are otherwise distinct solutions of the Einstein-matter equations.\nConsequently black hole uniqueness does not hold in this model\nof scalar electrodynamics.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{-0.5cm}\n\\mbox{\\hspace{-0.5cm}\n\\subfigure[][]{\\hspace{-1.0cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig5a.eps}\n\\label{figiso1}\n}\n\\subfigure[][]{\\hspace{-0.5cm}\n\\includegraphics[height=.27\\textheight, angle =0]{Fig5b.eps}\n\\label{figiso2}\n}\n}\n\\end{center}\n\\caption{Mass formulae for black hole solutions:\n(a) the mass $M$ of solutions with fixed charge $Q=10$ and fixed\ngravitational coupling constant $\\alpha$,\nobtained from the asymptotic metric (\\ref{mass}) and the\nmass formula (\\ref{IHmu});\n(b) the mass $M$ of solutions with variable charge $Q$ but\nfixed ratio of inner and outer shell radii $r_i\/r_o$ for several fixed\nvalues of the gravitational coupling constant $\\alpha$,\nobtained from the asymptotic metric (\\ref{mass}) and the\nmass formula (\\ref{IHmuDQ2}).\nNote that $a=\\alpha^2$.\n\\label{BH2}\n}\n\\end{figure}\n\nLet us finally consider some mass relations\nfor these black holes space-times possessing $Q$-shells.\nWe begin by recalling an interesting result\nobtained in the isolated horizon framework\n\\cite{Ashtekar:2004cn}.\nIt states that the mass $M$ \nof a black hole space-time with horizon radius $r_H$\nand the mass $M_{\\rm reg}$\nof the corresponding globally regular space-time\nobtained in the limit $r_H \\rightarrow 0$ are related via\n\\cite{Corichi:1999nw,Ashtekar:2000nx,Ashtekar:2004cn}\n\\begin{equation}\nM = M_{\\rm reg} + M_\\Delta  ,\n\\label{IHmu} \\end{equation}\nwhere the mass contribution $M_\\Delta$ is defined by\n\\begin{equation}\nM_\\Delta = \\frac{1}{\\alpha^2} \\, \\int_0^{r_H} \\kappa(r'_H)r'_H d r'_H  .\n\\label{IHmuD}\n\\end{equation}\nHere $\\kappa(r_H)$ represents the surface gravity \nof the black hole with horizon radius $r_H$,\n$\\kappa = 2 \\pi T$.\nAccordingly, the mass $M$ \nof a space-time with a black hole\nwith horizon radius $r_H$ within a $Q$-shell with total charge $Q$\nshould be obtained as the sum of the globally regular gravitating $Q$-shell\nwith charge $Q$ and the integral $M_\\Delta$ along the set of black hole\nspace-times, obtained by increasing the horizon radius for fixed charge\nfrom zero to $r_H$.\n\nThis relation is demonstrated in Fig.~\\ref{figiso1}\nfor the set of solutions with charge $Q=10$ and gravitational coupling\nconstant $\\alpha^2=0.01$.\nThe values for the mass $M$\nobtained from the relation (\\ref{IHmuD}) are seen to agree with\nthe values for the black hole mass $M$ obtained from the\nasymptotics (\\ref{mass}).\nThe set of solutions exhibited has spiralling character,\ni.e., it has besides the main first branch\na second branch, also exhibited, and further branches, \nnot resolved in the figure.\n\nWhen the charge is allowed to vary, too, one expects\na change of the above relation in accordance with (\\ref{Mreg2})\nand the first law (in the units empoyed), i.e.,\n\\begin{equation}\ndM = \\frac{\\kappa}{8 \\pi \\alpha^2} d{\\cal A} + b(\\infty) dQ\n , \\label{firstlaw} \\end{equation}\nwhere ${\\cal A} = 4 \\pi r_H^2$ denotes the area of the horizon\nand $b(\\infty)$ represents the electrostatic potential at infinity.\nThus we generalize the above relation (\\ref{IHmu}) to read\n\\begin{equation}\nM = M_{\\rm reg} + M_\\Delta + M_Q =\n M_{\\rm reg} + M_\\Delta \n  +  \\int_{Q_{\\rm reg}}^{Q} b(\\infty) d Q' .\n\\label{IHmuDQ2}\n\\end{equation}\nThis relation is demonstrated in Fig.~\\ref{figiso2},\nwhere for several values of the gravitational coupling constant\nand for fixed ratio\nof inner and outer shell radii $r_i\/r_o$,\nthe values for the mass $M$\nobtained from the relation (\\ref{IHmuDQ2}) are shown together with\nthe values for the mass $M$ obtained from the\nasymptotics (\\ref{mass}).\n\n\n\\section{Conclusion and Outlook}\n\nWe have considered boson stars, gravitating $Q$-shells and\nblack holes within $Q$-shells in scalar electrodynamics\nwith a $V$-shaped scalar potential,\nwhere the scalar field is finite only in compact ball-like or\nshell-like regions.\n\nThe gravitating $Q$-shells surround a flat Minkoswki-like interior region,\nwhile their exterior represents part of an exterior\nRN space-time.\nWhen the flat interior is replaced by a Schwarzschild-like\ninterior, black hole space-times result, where\na Schwarzschild-like black hole is surrounded by a compact shell of\ncharged matter, whose exterior again represents part of an exterior\nRN space-time.\n\nThese black hole space-times violate black hole uniqueness,\nin certain regions of parameter space.\nHere for the same values of the mass $M$ and the charge $Q$\ntwo or more distinct solutions of the Einstein-matter equations\nexist.\n\nThe solutions satisfy certain relations of the type obtained first in the\nisolated horizon formalism, which connect the mass $M$ of \na black hole solution with the mass $M_{\\rm reg}$ of the\nassociated globally regular solution.\nThe masses of two regular solutions are related in an analogous \n(simpler) manner.\nThis formalism further suggests to interpret the\nblack hole space-times as bound states of\nSchwarzschild-type black holes and gravitating $Q$-shells\n\\cite{Ashtekar:2000nx}.\n\nWhile we have restricted our discussion here to\nSchwarzschild-type black holes in the interior,\nthere are also black hole space-times with charged, i.e.,\nReissner-Nordstr\\\"om-type interior solutions.\nThese more general black hole space-times \nwill be discussed elsewhere.\n\nThe inclusion of rotation presents another interesting generalization\nof the solution considered here, \nsince rotating boson stars are well-known \n\\cite{Mielke:2000mh,Schunck:2003kk,Yoshida:1997qf,Kleihaus:2005me,Kleihaus:2007vk}.\nThe construction of the corresponding rotating shells\nand their black hole generalizations, however, still poses a challenge.\n\n\\vspace{0.5cm}\n{\\bf Acknowledgement}\n\n\\noindent\nBK gratefully acknowledges support by the DFG,\nCL and ML by the DLR.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nHurwitz theory is the study of maps of algebraic curves, viewed as ramified covers of orientable surfaces. At the intersection of geometry, representation theory and combinatorics, it is an area that naturally lends itself to making bridges and connections. In this paper we study the combinatorial structure of certain Hurwitz spaces via a parallel investigation in tropical geometry. \n\n\n\\subsection{Summary}\n\nThe study of the connections between classical and tropical Hurwitz theory was initiated in \\cite{CJM10} and \\cite{CJMwc};  the tropical point of view provided a combinatorial interpretation of double Hurwitz numbers which was very well tuned to describing the polynomial aspects and the wall crossing phenomena occurring in the theory. We  continue this exploration by studying rational double Hurwitz loci inside spaces of (relative\/tropical) stable maps, generically parameterizing covers of $\\mathbb{P}^1$ with two prescribed ramification profiles and a part of the branch divisor fixed, and their pushforwards to the moduli space of curves, which we call double Hurwitz cycles. Besides the genus, the discrete invariants we fix are the total length $n$ of the special ramification profile, and the dimension of the loci we want to study. Then we study families of Hurwitz loci parameterized by integral points in an $(n-1)$ dimensional vector space. \n\nOn the classical side,  we realize Hurwitz loci as the pullback via a natural branch morphism of appropriate strata in spaces of pointed  chains of projective lines (Losev-Manin spaces, Section \\ref{sec:mscm}). This gives a boundary expression for Hurwitz cycles where the coefficients are piecewise polynomials in the entries of the special ramification data (Theorem \\ref{thm:poly}).  \n\nOn the tropical side, we realize the Hurwitz loci as tropical Gromov-Witten cycles. Our tropical Hurwitz cycles are balanced polyhedral complexes; their topology is constant in the chambers of polynomiality of (classical) Hurwitz cycles, whereas their geometry (affine integral structure, weights and coordinates of vertices) varies in a polynomial way in terms of the special ramification profiles. \n\nNaturally, we also study the connection between classical and tropical Hurwitz cycles (Section \\ref{sec:tcc}) and observe a natural combinatorial duality between tropical and classical strata. To be more precise, the stratification on the tropical side is the polyhedral structure inherited from the moduli space of tropical curves. The stratification on the classical side is the usual stratification in boundary classes. For Hurwitz cycles of dimension $d$, $k$-dimensional classical strata correspond to $d-k$-dimensional tropical strata, and the combinatorial type of the tropical stratum can be encoded in terms of the dual graph of the classical stratum.\n\n\nWe conclude the paper by studying how Hurwitz cycles vary across the walls of the chambers of polynomiality, both on the classical and on the tropical side. In a similar fashion to Hurwitz numbers, the wall crossing  formulae have an inductive  form:  the cycles in the formula can be obtained as pushforwards via appropriate gluing morphism  of pairs of boundary strata  coming from Hurwitz cycles where the profile data is split according to the equation of the wall, and the dimensions are split in all possible ways adding up to the correct one. \nEven though the final form of the tropical and classical wall-crossing formulae is essentially the same, there are some subtleties involved in even making sense of what a tropical wall crossing formula may be: that's why we treat the two cases separately. The classical wall crossing formula is Theorem \\ref{thm:wc}, the (cleanest form of  the) tropical one Corollary  \\ref{cor:twc}.\n\nTo make our exposition easier to follow, throughout the paper we use the one-dimensional case ((tropical) Hurwitz curves) as a running example.\n\n\\subsection{Context, Motivation and further directions of our research}\n\nBecause of the many and diverse categories equivalent to curves and their maps,  Hurwitz theory is by nature ``interdisciplinary'': exploring the dictionary between the tropical and classical approaches to Hurwitz theory is a natural thing to do. It has already been fruitful and hopefully will bear even more applications in the future. \n\nBefore Hurwitz theory made its appearance in tropical geometry, the area of tropical enumerative geometry was pioneered by Mikhalkin's celebrated Correspondence Theorem which relates classical numbers of plane curves to their tropical counterparts \\cite{Mi03}. Nowadays, numbers of tropical curves can (at least in the rational case) be understood as intersection products in an appropriate moduli space of tropical curves, just as in the classical world. For higher genus, understanding the appropriate moduli space of tropical curves and its connection to the moduli space of algebraic curves is an active area of research (see e.g.\\ \\cites{CMV12,Cap12a}). Also in the rational case, the connection between the intersection theory of the moduli space of algebraic curves and the moduli space of tropical curves is not yet completely understood. Combinatorial dualities such as the one we observe in Section \\ref{sec:tcc} are present in many situations, but in our opinion they do not fully explain the success of tropical methods in enumerative geometry. We expect deeper connections to be discovered. Our paper contributes some interesting and geometrically meaningful families of cycles in the intersection ring of tropical $\\mathcal{M}_{0,n}$, and makes important steps in understanding the correspondence of these cycles to their classical counterparts. We hope to extent the study to higher genus, and to contribute to the understanding of the deeper connections between moduli spaces of algebraic and tropical curves.\n\n\nClassically double Hurwitz loci were a key ingredient in the proof of  Theorem $\\star$, the main result of  \\cite{gv:rvl}: tautological classes in the moduli space of curves of degree greater than $g-1$ (say $g+k$ with $k$ a non-negative integer) must admit an expression supported on the boundary, and more specifically on strata parameterizing curves with at least $k$ rational components. Then again they were applied to the study of tautological classes in \\cite{gjv:last}, even though in neither of these cases they were viewed as families of loci with any sort of polynomial structure. This makes its appearence for the case of $0$-dimensional cycles, or more mundanely double Hurwitz numbers, in \\cite{gjv:ttgodhn}. After \\cites{ssv:cbodhn,CJM10,CJMwc} the algebro-combinatorial aspects of the piecewise polynomiality\nof double Hurwitz numbers are well understood, leading the way to some really interesting geometric questions: \n\\begin{description}\n\\item[ELSV-type formula for double Hurwitz numbers] can double Hurwitz numbers be obtained as intersections of tautological classes on some family of birational moduli spaces --- constant in the polynomiality chambers ---  in a way that naturally explains the structure of the polynomials? Can the wall crossings be somehow seen as coming from the birational transformations occurring when crossing the walls?\n\\item[Higher dimensional Hurwitz loci] How well does the piecewise polynomial structure carry over to higher dimensional loci? In particular can we understand full Hurwitz spaces (or rather their compactifications such as Admissible Covers or Relative Stable Maps) as tautological classes in the moduli space of curves?\n\\end{description}\nThis paper provides an exhaustive answer for the second question in genus $0$: here the full moduli space is birational to $\\overline{M}_{0,n}$, and we show that every time we  increase the codimension by one by fixing a simple branch point in the branch divisor we obtain a polynomial class of one degree higher. In genus $0$ an ELSV formula  is trivially showed to hold in the one part double Hurwitz number case (\\cite{gjv:ttgodhn}), and we are hopeful that a geometric understanding of the wall crossings will be complete soon. In higher genus, the situation is both more complicated and more interesting: here the Hurwitz moduli space represents a codimension $g$ tautological class, which has recently been at the center of attention because of its connections with symplectic field theory (\\cite{eliashberg}).  Recently Hain (\\cite{H11}) has produced  tautological  classes on $\\overline{M}_{g,n}$,  which agree with double Hurwitz loci when restricted to the partial compactification of curves of compact type (however simple intersection computations show that Hain's class does not agree with either the Admissible Cover nor the Relative Stable Map compactification of the Hurwitz space already in genus one). Interestingly, Hain's class is homogeneous polynomial of degree $2g$. Understanding how Hain's class compares with Admissible Covers or Relative Stable Maps, besides being a very interesting question on its own, is likely a useful ingredient in the quest for an ELSV formula for double Hurwitz numbers. Our work is aiming in that direction in a couple different ways. On the one hand we interpolate a family of classes  starting from the zero dimensional loci which we understand very well. On the other hand we make a connection with tropical geometry, which is usually well tuned to give information about the  deeper boundary strata of the classical moduli spaces of curves.\n\n\n\n\\subsection{Acknowledgements}\nThis work is the result of a two week long \\textit{Research in Pairs} at the Oberwolfach Institute for Mathematics, which the authors thank for its hospitality. The second author was at the intersection of supports by a Simons Collaboration Grant and NSF grant  DMS110549. The third author was partially supported by DFG-grant MA 4797\/1-2.\n \n\n\n\\section{Background and Notation}\nIn this section we recall the basic definitions and constructions that are needed for the set-up of the theory. \n\\subsection{Moduli Spaces of Curves and  Maps}\n\\label{sec:mscm}\n\nWe assume that the reader is familiar with $\\overline{M}_{0,n}$, the moduli space of rational pointed stable curves, a smooth projective variety of dimension $n-3$. The Chow ring of $\\overline{M}_{0,n}$ is generated by irreducible boundary divisors, with the only relations (besides the obvious ones given by empty intersections) generated by the WDVV relations (\\cite{sk:m0n}). Irreducible boundary strata are identified by their dual graph: given a graph $\\Gamma$, we denote the corresponding stratum by $\\Delta_\\Gamma$.\n\nIn \\textit{weighted stable curves} one tweaks the stability of a rational pointed curve $(X= \\cup_j X_j, p_1, \\ldots, p_n)$ by assigning weights $a_i$ to the marked points and requiring the restriction to each $X_j$ of  $\\omega_X +\\sum a_ip_i$ to be ample (this amounts to the combinatorial condition that  $\\sum_{p_i\\in X_j} a_i + n_j>2$, where $n_j$ is the number of shadows of nodes on the $j$-th component of the normalization of $X$). \n\nWhen two points are given weight $1$ and all other points  very small weight, the space $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$ is classically known as the \\textit{Losev-Manin} space {\\cite{lm:lms}}: it parameterizes chains of $\\mathbb{P}^1$'s with the heavy points on the two terminal components and light points (possibly overlapping amongst themselves) in the smooth locus of the chain.\n\nLet  $\\mathbf{x}$ be an $n$-tuple  of integers adding to $0$, and denote $\\mathbf{x^+}$ (resp. $\\mathbf{x^-}$) the sub-tuple of positive (resp. negative) parts. We consider moduli space of relative stable maps $\\overline{M}_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty)$ and their ``rubber'' variant  $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty)$ (see \\cites{gv:rvl,mp:tvgwt}).  An important technical detail is we mark the preimages of the relative points. In order to mark some of the simple ramification points on the source curve, we introduce a space of relative weighted stable maps, where in addition to $0$ and $\\infty$ there are $j$ moving simple transposition points that are marked and given weight $\\varepsilon$.  These spaces  are typically denoted $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty, \\varepsilon t_1, \\ldots, \\varepsilon t_j)$. \n\\begin{note}\nIn all our spaces of maps we make the notation lighter by  forgetting the target (always $\\mathbb{P}^1$),  noting that $\\mathbf{x}$ gives sufficient information to   determine the  relative points fixed at $0 $ and $\\infty$ and that the additional transposition points are understood to be ``light''. For example we write $\\overline{M}^\\sim_0(\\mathbf{x}, t_1, \\ldots, t_j)$ for $\\overline{M}^\\sim_0(\\mathbb{P}^1; \\mathbf{x^+}0, \\mathbf{x^-}\\infty, \\varepsilon t_1, \\ldots, \\varepsilon t_j)$. \n\\end{note}\n\n\nThere is a natural stabilization map $\\st$ to $\\overline{M}_{0,n}$ that forgets the map and remembers the (marked) points over $0$ and $\\infty$, and a branch map to an appropriate quotient of a Losev-Manin space, recording the position of the $r+2=n$ branch points. \nSince a degree $0$ divisor on $\\mathbb{P}^1$ determines a rational function up to a multiplicative constant, the map $\\st: \\overline{M}^\\sim_0(\\mathbf{x}) \\to \\overline{M}_{0,n}$ is birational. Marking $j$ simple transpositions makes $\\st$ into a degree ${{r}\\choose{j}}$ cover. The branch map is a cover of the Losev-Manin space of degree the double Hurwitz number $H_0(\\mathbf{x})$. \n\n\\subsubsection{Multiplicities of boundary strata.}\nBoundary strata in moduli spaces of relative stable maps corresponding to breaking the target are naturally described in terms of products of other moduli spaces of relative stable maps. It is important to keep careful track of various multiplicities coming both from combinatorics of the gluing and infinitesimal automorphisms (see \\cite{gv:rvl}*{Theorem 4.5}). \nLet $S$ be a boundary stratum in $\\overline{M}^\\sim_0(\\mathbf{x})$, parameterizing maps to a chain $T^N$  of $N$ projective lines. $S$ can be seen as the image of:\n$$\n\\gl: \\prod_{i=1}^N \\mathcal{M}^\\bullet_i \\to S \\subset \\overline{M}^\\sim_0(\\mathbf{x}),\n$$\n\nwhere the $\\mathcal{M}^\\bullet_i$ are moduli spaces of possibly disconnected relative stable maps, where the relative condition imposed at the point $\\infty$ in the $i$-th line matches the condition at $0$ in the $(i+1)$-th line. We denote by $\\mathbf{z_i}=(z_i^1, \\ldots, z_i^{r_i})$ such relative condition and by abuse of notation we say it is the relative condition at the $i$-th node of $T^N$. Then,\n\\begin{equation}\n\\label{bound:multi}\n[S] = \\prod_{i=1}^{N-1}\\frac{\\prod_{j=1}^{r_i} z_i^j}{|\\Aut(\\mathbf{z_i})|} \\left[\\gl_\\ast\\left( \\prod_{i=1}^N \\mathcal{M}^\\bullet_i \\right)\\right].\n\\end{equation}\n\nEquation \\eqref{bound:multi} seems horrendous, but it amounts to the following recipe: the general element in $S$ is represented by a map from a nodal curve  $X$ to $T^N$, with matching ramification on each side of each node of $X$. The multiplicity $m(S)$ is the product of ramification orders for each node of $X$ divided by the (product of the order of the group of) automorphisms of each partition of the degree prescribing the ramification profile over each node of $T^N$.\n\n\n\n\n\n\\subsection{Tropical Geometry}\n\\label{sec:tg}\nWe assume that the reader is familiar with tropical varieties and tropical cycles in a vector space with a lattice, i.e.\\ with weighted polyhedral complexes (possibly with negative weights in the case of cycles) satisfying the balancing condition around each cell of codimension one.\nThe exact polyhedral complex structure is not important. We do not distinguish between equivalent tropical cycles, i.e.\\ cycles which allow a common refinement respecting the weights. \n\n A rational function\n  on a tropical cycle is a continuous function that is\n  affine on each cell, and whose linear part is\n  integer. To a tropical cycle $X$ and a rational function $\\varphi$, we can associate the divisor $\\varphi\\cdot X$, a tropical subcycle of codimension one supported on\n  the subset of $X$ where $\\varphi$ is not linear \\cite{AR07}*{Construction 3.3}.\nWe can also form multiple intersection products $\\varphi_1 \\cdot \\ldots \\cdot\n  \\varphi_m \\cdot X$. They are commutative by \\cite{AR07}*{Proposition 3.7}. The weights of cells of  intersection products can be computed locally as follows.\n  \n\\begin{rem}\\label{rem-intersect}\n  Let $h_1, \\ldots, h_m$ be linearly independent integer linear\n  functions on $\\mathbb{R}^n$. By $H : \\mathbb{Z}^n \\rightarrow \\mathbb{Z}^m $ we\n  denote the linear map given by $x \\mapsto (h_1(x), \\ldots, h_m(x))$. Consider\n  the rational functions $ \\varphi_i = \\max\\{h_i,p_i\\}$ on $\\mathbb{R}^n$, where $p_i$ are fixed constants in $\\mathbb{R}$. These rational functions give rise\n  to an intersection product, which obviously consists of only one cone with \n weight equal to the order of the torsion part of $ \\mathbb{Z}^m \/ \\text{Im}(\n  H)$, i.e. the greatest common divisor of\n  the absolute values of the maximal minors of $H$ (see e.g.\\ \\cite{MR08}*{Lemma 5.1}). \n  \\end{rem}\n\nA morphism between tropical cycles\nis a locally affine linear map, with\nthe linear part induced by a map between the underlying lattices.\n A rational function $\\varphi$ on a tropical cycle $Y$ can be pulled back along a morphism $f:X\\rightarrow Y$\n  to the rational function $f^*(\\varphi) = \\varphi \\circ f$ on $X$. Also, we can push forward\n  subcycles $Z$ of $X$ to subcycles $f_*(Z)$ of $Y$ \\cite{AR07}*{Proposition\n  4.6 and Corollary 7.4}. \n\\vspace{0.3cm}\n\nWe refer the  reader  to \\cite{GKM07,Mi07,CJM10} for comprehensive background on moduli spaces of tropical curves and maps. A (marked, rational, abstract) tropical curve is a metric tree $\n\\Gamma $ without 2-valent vertices. Edges leading to 1-valent vertices have infinite length, are marked by the numbers $1,\\ldots,N$ and are called ends. The space of all marked tropical curves with\n$N$ ends is denoted $ \\mathcal{M}_{0,N} $. It can be embedded into $\\mathbb{R}^{\\binom{N}{2}-N}$ using the distance map. It follows from \\cite{SS04a}*{Theorem 3.4},\n\\cite{Mi07}*{Section 2}, or \\cite{GKM07}*{Theorem 3.7} that $\\mathcal{M}_{0,N}$ is a\ntropical variety which is even a fan. All top-dimensional cones have weight one. \nFor a subset $I\\subset\\left[N\\right]$ of cardinality $1<|I|<N-1$ define $v_I$\nto be the image under the distance map of a tree $\\Gamma$ with exactly one bounded\nedge of length one, the marked ends $i\\in I$ on one side and the\nends $i$ for $i\\notin I$ on the other. By abuse of notation, we often do not distinguish between a tree and its image under the distance map. We consider $\\mathbb{R}^{\\binom{N}{2}-N}$ not with its usual lattice, but with the lattice generated by the $v_I$.\nThe set of curves of a given combinatorial type is the interior of a\n$k$-dimensional cone of $\\mathcal{M}_{0,N}$, where $k$ denotes the number of bounded\nedges. The vectors $v_I$ generate the rays of $\\mathcal{M}_{0,N}$.\n\nWe denote by $\\TP= \\mathbb{R}\\cup {\\pm \\infty}$ the simplest model of tropical $\\mathbb{P}^1$.\nA tropical cover is a map $h:\\Gamma \\rightarrow \\TP$ where $\\Gamma$ is a tropical curve and $h$ is a continuous map which is integral affine on each edge. For each bounded edge, the stretching factor with which it is mapped to $\\mathbb{R}\\subset\\TP$ is called its weight.  \nEdges which are contracted have weight $0$. Two tropical covers are called isomorphic if they differ by a translation of the base $\\TP$.\n\nWe sometimes fix a reference orientation for the edges of $\\Gamma$. We then use the convention that (nonzero) weights are positive whenever the image of the tail in $\\mathbb{R}\\cup {\\pm \\infty}$ is smaller than the image of the head, and negative otherwise. At each vertex, a tropical cover satisfies the balancing condition, i.e.\\ the sum of the weights of incoming edges equals the sum of the weights of outgoing edges.\nThe degree $\\mathbf{x}$ of a tropical cover is the multiset of weights of its non-contracted ends, where we make the convention that all ends point away from their 1-valent vertex.\n\nWe fix  a degree $\\mathbf{x}$ and set $n=\\ell(\\mathbf{x})$. The space of isomorphism classes of tropical covers of degree $\\mathbf{x}$ and with additional $r$ marked ends which are contracted to points is called $ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $.\n\n\n\\begin{conv}\n\\label{conv}\nFor practical purposes in computations, it is useful to pick a distinguished representative for each equivalence class of tropical covers. By\nconvention, we introduce a contracted end labeled  $n+1$ and require that\nit is mapped to $0\\in \\TP$ by the map $h$. We say the end $n+1$ fixes a parameterization of $\\TP$.\n\\end{conv}\n\nBy \\cite{GKM07}*{Proposition 4.7}, $ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $ is a\n  tropical variety. It can be identified with $\\mathcal{M}_{0,N}$, where $N=n+r$, via the\n  map that forgets the map $h$.\nIf we later want to count tropical covers using the space \n$ \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) $ we overcount in the situation where we have multiple ends of the same weight, since we mark all the ends here and thus make them artificially distinguishable even if they have the same weight. For counting purposes, we thus have to divide by $|\\Aut(\\mathbf{x})|$. Since we are mostly interested in the situation where the degree $\\mathbf{x}$ is variable, this does not play an important role.\n\nTo compute intersection products in cells of\n  tropical moduli spaces, we work in the local coordinates of the appropriate cone of $\\mathcal{M}_{0,N}$, i.e.\\ in coordinates given by the lengths of each bounded edge.\n\nFor a subset $I\\subset\\left[N\\right]$, there\nis a forgetful map $\\ft_I:\\mathcal{M}_{0,N}\\longrightarrow \\mathcal{M}_{0,N-|I|}$ which\nmaps a tropical curve to the curve where we remove all ends with labels in $I$ (and possibly straighten $2$-valent vertices). Forgetful maps are\nmorphisms by \\cite{GKM07}*{Proposition 3.9}.\nThe forgetful map $\\ft:  \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) = \\mathcal{M}_{0,N} \\rightarrow \\mathcal{M}_{0,n}$ which forgets all contracted ends is most important.\n\n For each contracted end with label $i$  the evaluation map\n    $ \\ev_i : \\mathcal{M}_{0,r}(\\TP,\\mathbf{x}) \\rightarrow \\mathbb{R}\\subset\\TP $\n  assigns to a tropical curve the point in $\\mathbb{R}\\subset \\TP$ to which the end is contracted by $h$. It is shown in \\cite{GKM07}*{Proposition\n  4.8} that these maps are morphisms.\n\nFor each marked end $i$  the tropical Psi-class $\\Psi_i$, as a divisor, consists of all closed codimension one cones of $\\mathcal{M}_{0,N}$ parameterizing graphs  where the end $i$ is adjacent to a 4-valent vertex \\cite{Mi07}. All these cones come with weight one. It is also known how to intersect tropical Psi-classes \\cite{KM07}: every top-dimensional cone appearing in the intersection $\\Psi_1^{k_1}\\cdot \\ldots \\cdot \\Psi_N^{k_N}$ corresponds to a combinatorial type satisfying the following: if the ends $i_1,\\ldots,i_s$ are adjacent to a vertex $V$ then this vertex has valency $k_{i_1}+\\cdots+k_{i_s}+3$. We associate the weight $(k_{i_1}+\\cdots+k_{i_s})! \\cdot (k_{i_1}!\\ldots k_{i_s}!)^{-1}$ to the vertex $V$. The weight of a cone equals the product of these vertex weights for the corresponding combinatorial type.\n\nFor a subset $I\\subset \\{1,\\ldots,N\\}$ with $N-2 \\geq |I|\\geq 2$ we define the tropical boundary divisor $D_{I}$ to be the divisor cut out by the rational function $\\varphi_{I}$ that sends $v_{I}$ to one and all other vectors $v_{I'}\\neq v_I$ to zero \\cite{Rau08}*{Definition 2.4}.\n\n\\subsection{Combinatorial Set-Up}\n\\label{sec:csu}\n\nIn this section we introduce some basic combinatorial objects that play a role in the development of Hurwitz cycles. We think of Hurwitz loci as functions of the special ramification profiles. We define the domain of definition of such functions and outline the linear structure it has. \n\n\\begin{defn}\nFix a positive integer $n$. The lattice\n$$\n\\mathcal{H}=\\mathbb{Z}^n \\cap \\left\\{ \\sum_{i=1}^n x_i = 0 \\right\\}\n$$\nis the parameter space for double Hurwitz cycles. \nFor any proper subset $I\\subset \\{1, \\ldots, n\\}$, the hyperplane\n$$\nW_I= \\left\\{ \\sum_{i\\in I} x_i = 0 \\right\\}\n$$\nis called a \\textbf{wall}.  Any connected component of $\\mathcal{H}\\otimes \\mathbb{R} \\smallsetminus \\cup_I W_I$ is called a \\textbf{chamber}  (of polynomiality) and denoted by $\\mathfrak{c}$.\n\\end{defn}\n\nNext we introduce some notation for sets of decorated trees.\n\\begin{defn}\nLet $\\mathbf{x}\\in \\mathcal{H}$. We denote by $\\mathcal{T}_{r-k}(\\mathbf{x})$ the set of (non-metric) trees with $n$ ends labelled by the $x_i$'s and $r-k$ internal vertices of valence at least $3$.\nWe denote by $\\mathcal{T}^{\\mathfrak{c}}_{r-k}(\\mathbf{x})$ the set of tropical covers whose underlying tree is in $\\mathcal{T}_{r-k}(\\mathbf{x})$. Combinatorially, trees in $\\mathcal{T}^{\\mathfrak{c}}_{r-k}(\\mathbf{x})$ are further decorated with the following data:\n\\begin{enumerate}\n\t\\item A direction for the edges.\n\t\\item Positive weights for each edge in such a way that the balancing condition is satisfied at every vertex.\n\t\\item A total ordering of the vertices compatible with the direction of the edges.\n\\end{enumerate}\n\\end{defn}\n\nDecorations (1) and (2) can in fact be deduced uniquely from imposing the balancing condition, and depend on the chamber $\\mathfrak{c}$ that $\\mathbf{x}$ belongs to.\n\n\\begin{lemma}\n\\label{lem:flat}\nAny $\\Gamma$ in $\\mathcal{T}_{r-k}(\\mathbf{x})$ can be  ``lifted'' to a tropical cover as in Section \\ref{sec:tg}.\n\\end{lemma}\n\\begin{proof}\nFor most combinatorially versed people Lemma \\ref{lem:flat} is obvious. We include a brief sketch for the combinatorially impaired such as the majority of the authors of this paper (see \\cite{GKM07}*{Lemma 4.6} for a more honest discussion).\nAny tree with leaves labelled by an $n$-tuple of integers adding to $0$ can be given positive weights and orientations to all internal edges in a unique way by imposing the balancing condition:  give an arbitrary orientation to all edges in the tree. Cut any internal edge to obtain two smaller connected graphs. The cut edge can be assigned weight by requesting that each the sum of labels on each of the two connected graphs is $0$. Now the statement follows by induction. Wherever  one obtained a negative weight at an internal edge, switch the orientation of the edge with respect to the arbitrary one chosen earlier and switch the sign of the weight. Having given an orientation to any internal edge immediately allows to flatten the graph, concluding the proof. \n\\end{proof}\n\n\n\n\\begin{defn}\n\\label{mult}\nThe number of ways that a graph $\\Gamma$ can be flattened is  the number of vertex orderings for the graph that are compatible with the direction of the edges of the graph.  We call such number $m_\\mathfrak{c}(\\Gamma)$. If the chamber $\\mathfrak{c}$ is clear from the context, we also write $m(\\Gamma)$. \\end{defn}\n\nIn other words, the forgetful morphism $f:\\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})\\to \\mathcal{T}_{r-k}(\\mathbf{x})$ is surjective and $m_\\mathfrak{c}(\\Gamma)= |f^{-1}(\\Gamma)|$.\n\n\\begin{defn}\n\\label{phi}\nFor $\\Gamma \\in \\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x}) $ (or in $\\mathcal{T}_{r-k}(\\mathbf{x})$ ) we define $\\varphi(\\Gamma)$ to be the product of the weights of all internal edges of $\\Gamma$.\n\\end{defn}\n\n\nBecause in each chamber $\\mathfrak{c}$ all edge weights are linear homogeneous polynomials in the $x_i$'s, the functions   $\\varphi(\\Gamma)$ are homogeneous polynomials in each chamber $\\mathfrak{c}$ of degree equal the number of internal edges of $\\Gamma$.\n\n\n\n\n\n\\section{Hurwitz Loci and Classes}\n\\label{sec:hl}\n\n\nIn this section we define Hurwitz classes and study their (piecewise) polynomial properties. We say that a family of Chow cycles $\\alpha(\\mathbf{x})$  in the moduli space of rational pointed stable curves is polynomial of degree $d$ and dimension $k$ if  $\\alpha(\\mathbf{x}) \\in Z_k(\\overline{M}_{0,n})[x_1, \\ldots, x_n]_d$. This is equivalent to \n$\\alpha$  having an expression as a combination of dimension $k$ boundary strata with coefficients polynomials in the $x_i$'s of degree $d$. \n\n\\begin{figure}[tb]\n\\input{T.pstex_t}\n\\caption{The union of strata we denote by $T^{r-k}\\subset \\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$ paramereterizes  chains of $r-k$ rational curves, where the $i$-th component hosts the $i$-th (light) marked point, and the remaining $k$ points are distributed in all possible ways among the various twigs.}\n\\label{fig:T}\n\\end{figure}\n\n\n\\begin{defn}\nLet $\\mathbf{x}\\in \\mathcal{H}$. Consider the union  of boundary strata $$T^{r-k}\\subset \\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$$ parameterizing  chains of $r-k$ projective lines, where the $i$-th component hosts the $i$-th (light) marked point, as illustrated in Figure \\ref{fig:T}. Referring to Figure \\ref{fig:natmorph} for the names of the natural morphisms, we define the $k$-dimensional Hurwitz cycle:\n\\begin{equation}\n\\mathbb{H}_k(\\mathbf{x}):=\\st_\\ast( \\br^\\ast(T^{r-k})) \\in Z_k(\\overline{M}_{0,n}).\n\\end{equation}\nSometimes we want to look at the $k$-dimensional Hurwitz locus in the appropriate space of maps. We make the definition:\n\\begin{equation}\n\\tilde\\mathbb{H}_k(\\mathbf{x}):= \\br^{-1}(T^{r-k}) \\subseteq \\overline{M}^\\sim_{0}(\\mathbf{x},t_1,\\ldots, t_{r-k}).\n\\end{equation}\n\\end{defn}\n\n\\begin{figure}[tb]\n\\centerline{\n\\xymatrix{\n\\tilde{\\mathbb{H}}_k(\\mathbf{x})   \\ar[d]  \\ar[r]   &  \\overline{M}^\\sim_0(\\mathbf{x}, t_1, \\ldots, t_j)  \\ar[d]^{\\br}\\ar[r]^{\\st}& \\overline{M}_{0,n}\\\\\n [T^{r-k}] \\ar[r]&  [\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})\/S_k] & \\\\\n}}\n\\caption{The $k$-dimensional Hurwitz locus is the inverse image via the branch map of the stratum $T^{r-k}$.}\n\\label{fig:natmorph}\n\\end{figure}\n\n\\begin{rem}\nHurwitz loci were introduced in \\cite{gv:rvl}*{Section 4.4}. The only difference  in our definition is that we ``mark'' the branch points that we are fixing. The use of the more refined branch morphism to the Losev-Manin space gives us a more convenient expression of the locus in terms of the pull-back of a boundary stratum in $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$. The next Lemma establishes the equivalence with Graber-Vakil's definition and is parallel to the definition we make on the tropical side.\n\\end{rem}\n\n\n\\begin{lemma}\nConsider the space of relative stable maps to a parameterized $\\mathbb{P}^1$, with   the natural stabilization morphism $\\st: \\overline{M}_{0,r-k}(\\mathbf{x})\\to \\overline{M}_{0,n}$.  Let $\\psi_i$ denote the first Chern class of the $i$-th cotangent line bundle, and $\\ev_i$ the $i$-the evaluation morphism. Then\n\\begin{equation}\n\\mathbb{H}_k(\\mathbf{x})=\\st_\\ast\\left(\\prod_{i=1}^{r-k} \\psi_i \\ev^\\ast_i([pt])\\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nInformally, the result is a combination of the three following facts: putting a $\\psi$ class is equivalent to requesting the mark to be a point of simple ramification for the map (\\cite{op:cc}); fixing one of the branch points gives the equivalence with the space of rubber-maps (\\cite{gv:rvl}*{Lemma 4.6}); via degeneration formula, any additional branch point that is fixed amounts to breaking the target into a chain with one more projective line. We expand on this third fact for the benefit of a reader who is not familiar with degeneration techniques.\n\nWe assume by induction that the class $\\prod_{i=1}^j \\ev_i^\\ast{[pt]}$ can be represented by the cycle of maps to a chain $T^j=\\cup_{i=1}^jT_i$ of $j$ projective lines with the $i$-th mark on the $i$-th line. Consider the trivial family $X=\\mathbb{C}\\times T^j$ and note that it comes with two horizontal sections: $\\mathbf{0_1}$ on $T_1$ and $\\mathbf{\\infty_j}$ on $T_j$. On $T_j$ consider the section $s(t)=1\/t$ that meets the section  $\\mathbf{\\infty_j}$ at $t=0$. Now let $X'$ be the blow up the point $(0,\\infty_j)$ in $X$ and consider the locus $\\ev_{j+1}^{\\ast}([s(t)])$ in the space of relative stable maps to $X'$ of degree $dF$ (where $F$ is the class of a fiber of the family and $d= \\sum x_i^+$ ), relative to the divisor $\\mathbf{x^+0_1}+\\mathbf{x^-\\infty_j}$. The degeneration formula (\\cites{lr:df, l:df})  asserts that it is equivalent to represent $F$ by a general fiber or by the central fiber of the family. In the first case we have $\\prod_{i=1}^{j+1} \\ev_i^\\ast{[pt]}$, in the second case we obtain the cycle of maps to the chain $T^{j+1}$ with the $i$-th mark on the $i$-th line.  \n\\end{proof}\n\n\nThe description of Hurwitz loci in terms of boundary strata in spaces of maps, together with an analysis of the multiplicities of the push-forwards to $\\overline{M}_{0,n}$ naturally leads to discover the (piecewise) polynomiality of the Hurwitz cycles.\n\n\\begin{thm}\n\\label{thm:poly}\nFor $\\mathbf{x}\\in \\mathfrak{c}$, $\\mathbb{H}_k(\\mathbf{x})$ is a homogeneous polynomial cycle of degree $n-3-k$.\n\\end{thm}\n\\begin{proof}\nTheorem \\ref{thm:poly} is a consequence of Lemmas \\ref{spec}, \\ref{flatten} and \\ref{coeff}, where we also give an explicit description of the (polynomial) multiplicity of each boundary stratum.\n\\end{proof}\n\nA simple observation that gives us a surprising amount of mileage, is that many boundary strata in $\\tilde\\bbH_k(\\mathbf{x})$ don't survive being pushed forward to $\\overline{M}_{0,n}$.\n\n\\begin{lemma}\n\\label{spec}\nLet  $\\mathbf{x}\\in \\mathfrak{c}$ and $\\tilde\\Delta$ an irreducible component of $\\tilde\\mathbb{H}_k(\\mathbf{x})$, corresponding to a stratum of maps to the rational chain $T=T_1\\cup\\ldots\\cup T_{r-k}$. Then $\\pi_\\ast(\\tilde\\Delta)\\not=0$ if and only if for every $i=1, \\ldots, r-k$ there is exactly one connected rational curve mapping non-trivially to $T_i$. \n\\end{lemma}\n\\begin{proof}\nLet $f:X\\to T$ correspond to a general point in $\\tilde\\Delta$.\nOver each $T_i$ there must be at least one connected component mapping non-trivially. By a trivial dimension count the push forward of $\\tilde\\Delta$ does not vanish precisely when the stabilization of $X$ is a rational curve with $r-k$ components and therefore there cannot be more than one non-trivial component over each $T_i$. \n\\end{proof}\n\nNext we observe that each boundary stratum of dimension $k$ in $\\overline{M}_{0,n}$ appears as the push forward of some stratum in $\\tilde\\bbH_k(\\mathbf{x})$.\n\n\\begin{lemma}\n\\label{flatten}\nLet   $\\mathbf{x} \\in \\mathfrak{c}$,  $\\Gamma\\in \\mathcal{T}_{r-k}(\\mathbf{x})$. Then there exists  an irreducible component $\\tilde\\Delta$ in $\\tilde\\mathbb{H}_k(\\mathbf{x})$ such that the stabilization of the source curve of the general element of $\\tilde\\Delta$ has dual graph $\\Gamma$.\n\\end{lemma}\n\\begin{proof}\nThis statement becomes  transparent after noting that boundary strata in $\\overline{M}^\\sim_{0}(\\mathbf{x},t_1,\\ldots, t_{r-k})$ are in bijective correspondence with ``tropical dual graphs''. Given a boundary stratum  $[S]$ whose general element is given by a map $f: X\\to T^{r-k}$: \n\\begin{itemize}\n\\item the chain $T^{r-k}$ is replaced by the interval $[0, r-k+1]$ with the $i$-th projective line corresponds to the point $i$ and $i$-th node corresponding to the segment $(i, i+1)$; the points $0$ and $r-k+1$ correspond to the relative points.\n\\item over point $i$ draw one vertex for each connected component of $f^{-1}(T_i)$; \n\\item over segment $(i,i+1)$ draw one edge for each node  of $X$ above the $i$-th node of $T^{r-k}$; the edge connects the appropriate vertices and is weighted with the ramification order at the node;\n\\item over $[0,1)$ and $(r-k,r-k+1]$ draw edges corresponding to the relative condition $\\mathbf{x}$.\n\\end{itemize}\nWe call the graph thus obtained the \\textit{pre-stable} tropical dual of $[S]$.  We stabilize the graph by forgetting all two-valent vertices to obtain what we call the tropical dual of $[S]$. \nThis procedure is illustrated in Figure \\ref{fig:flatten}  and clearly it can be inverted to identify a boundary stratum of maps given a tropical cover. With this translation Lemma \\ref{flatten} is equivalent to the purely combinatorial statement of Lemma \\ref{lem:flat}.\n\\end{proof}\n\n \n\n\\begin{figure}[tb]\n\\input{dual.pstex_t}\n\n\\caption{The construction of the tropical dual graph associated to a boundary stratum of relative maps. The $y_i^j$ are the degrees of the trivial covers of the $i$-th twig, and the $z_i^j$ the ramification orders over the $i$-th node. Note that the $z$'s on either side of a trivial cover are equal to the corresponding $y$ (e.g $x_1=y_1^1=z_1^1=y_2^1=z_2^1$). This makes it possible to erase the two valent vertices and have the $z_i^j$ become the weights of the edges of the tropical dual graph.}\n\\label{fig:flatten}\n\\end{figure}\n\n\\begin{lemma}\n\\label{coeff}\nLet   $\\mathbf{x} \\in \\mathfrak{c}$,  $\\Gamma\\in \\mathcal{T}_{r-k}(\\mathbf{x})$ and $\\Delta_{\\Gamma}$ the corresponding boundary stratum in $\\overline{M}_{0,n}$.  Let $m_\\mathfrak{c}(\\Gamma)$  and $\\varphi(\\Gamma)$ as in Definitions \\ref{mult}, \\ref{phi} and $f:\\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})\\to \\mathcal{T}_{r-k}(\\mathbf{x})$ be the natural forgetful morphism. Then\n\\begin{align}\n\\label{eq:pomult}\n\\mathbb{H}_k(\\mathbf{x}) & =    \\sum_{\\tilde\\Gamma \\in \\mathcal{T}^\\mathfrak{c}_{r-k}(\\mathbf{x})} \\varphi(\\tilde\\Gamma)\\prod_{v}(\\val(v)-2)\\ \\Delta_{f(\\tilde\\Gamma)} \\nonumber\\\\\n & =\\sum_{\\Gamma \\in  \\mathcal{T}_{r-k}(\\mathbf{x})}  m_\\mathfrak{c}(\\Gamma)\\varphi(\\Gamma) \\prod_{v}(\\val(v)-2)\\Delta_{\\Gamma}.\n\\end{align}\n\\end{lemma}\n\\begin{proof}\nWe observed that  for each  boundary stratum  $\\Delta_{\\Gamma}$ in $\\overline{M}_{0,n}$ there are precisely $ m_\\mathfrak{c}(\\Gamma)$ boundary strata in $\\tilde\\bbH_k(\\mathbf{x})$ pushing forward to it. It remains to show that each such stratum $[S]$ pushes forward with multiplicity $ \\varphi(\\Gamma)\\prod_{v}(\\val(v)-2)$.  We obtain this multiplicity by adapting formula \\eqref{bound:multi} to the simple shape of the boundary strata we are observing. Over each $T_i$ in the chain there is only one connected component  of $X$ that maps non-trivially. Therefore we can replace the moduli space of possibly disconnected covers $\\mathcal{M}^\\bullet_i$ of \\eqref{bound:multi} by the unique moduli space of non-trivial connected covers to $T_i$, call it $\\mathcal{M}_i$, times an automorphism factor of $1\/y_i^j$ for every connected component of degree $y_i^j$ mapping  trivially to $T_i$; we denote by $\\triv_i$ the number of  trivial covers of the $i$-th component. Also the factors of $1\/|\\Aut(\\mathbf{z_i})|$ need to be replaced by the automorphisms of each sub-partition of $\\mathbf{z_i}$ corresponding to nodes on the same pair of curves on the two sides of the $i$-th node of $T^{r-k}$. But $X$ is a rational curve and hence all such sub-partitions have length one and trivially no automorphisms.  Hence \\eqref{bound:multi} becomes:\n\n\\begin{equation}\n\\label{eq:popo}\n[S]= \\frac{ \\prod_{i=1}^{r-k-1}\\prod_{j=1}^{r_i} z_i^j\n}{ \\prod_{i=1}^{r-k}\\prod_{j=1}^{{\\triv}_i} y_i^j}\n \\left[\\gl_\\ast\\left( \\prod_{i=1}^{r-k} \\mathcal{M}_i \\right)\\right].\n\\end{equation}\n\n\n\nEquation \\eqref{eq:pomult} is deduced from \\eqref{eq:popo} via the following observations:\n\\begin{enumerate}\n\\item There is a factor of  $y_i^j$ for each $2$-valent vertex  of the pre-stable tropical dual of $[S]$, and $y_i^j$ is equal to the weight of (either) edge on each side of the vertex. Therefore all $y_i^j$'s cancel with some of the $z$'s. The remaining multiplicity from the first part of \\eqref{eq:popo} is therefore the product of weights of all internal edges of the tropical dual graph: by definition this is   $\\varphi(\\Gamma)$.\n\\item Each $\\mathcal{M}_i$ is a moduli space of connected, rubber relative stable maps, with one simple transposition marked. By our discussion in Section \\ref{sec:mscm}, we know  $\\st$ maps  $\\mathcal{M}_i$ onto $\\overline{M}_{0,n}$ with degree $r={{r}\\choose{1}}$. If we call $v_i$ the vertex of the tropical dual graph corresponding to $\\mathcal{M}_i$, then in this case $n= \\val (v_i)$ and thus $r=\\val (v_i)-2$.\n\\end{enumerate}\n\\end{proof}\n\nTo obtain Theorem \\ref{thm:poly} it is now sufficient to remark that the weights of the edges of the tropical dual graph are linear homogeneous polynomials in the $x_i$'s and there are precisely $n-3-k$ internal edges in the tropical dual graph of any stratum  in $\\tilde\\bbH_k(\\mathbf{x})$ that pushes forward non-trivially to $\\overline{M}_{0,n}$.\n\nWe conclude this section by noting that our arguments do not apply exclusively to Hurwitz cycles, but to any cycle obtained by pull-pushing a boundary stratum from Losev-Manin to $\\overline{M}_{0,n}$. We thus obtain the following:\n\n\\begin{cor}\nLet $\\mathbf{x} \\in \\mathfrak{c}$  and $\\overline\\Delta$ be a union of boundary strata of dimension $k$ in $\\overline{M}_{0,2+r}(1^2,\\varepsilon^{r})$. Then $\\st_\\ast \\br^\\ast(\\overline\\Delta)$ is a homogeneous polynomial class of degree $n-3-k$ in $\\mathfrak{c}$.\n\\end{cor}\n\n\n\n\n\n\n\\section{Tropical Hurwitz Loci}\n\n\\label{sec:thl}\n\n\\begin{defn}Let $\\mathbf{x}\\in \\mathcal{H}$ and $r=n-2$. We define the $k$-dimensional tropical Hurwitz cycle as\n$$\\mathbb{H}^{\\trop}_k(\\mathbf{x}):= \\ft_{\\ast}\\Big(\\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})\\Big) \\subset \\mathcal{M}_{0,n},$$\nwhere the $p_i$ are fixed arbitrary points in $\\mathbb{R}\\subset \\TP$. (The notions of tropical moduli spaces, intersection theory and Psi-classes we use here are as recalled in Section \\ref{sec:tg}.) We also define the Hurwitz locus:\n$$\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x}):= \\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x}).$$\n\\end{defn}\n\n\n\nWe describe the (general) tropical curves parameterized by $\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\subset \\mathcal{M}_{0,n}$: if the $p_i$'s and  $0$ are distinct then the evaluation pullbacks force the contracted ends $n+1,\\ldots,n+r-k$ to be adjacent to distinct vertices. Each Psi-class makes the corresponding contracted end adjacent to a $4$-valent vertex.  All contracted ends are forgotten in the push-forward. Thus the top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ consist of  tropical curves that admit a cover of degree $\\mathbf{x}$ to $\\TP$ such that $r-k$ of its $r$ $3$-valent vertices are mapped to the points $p_i$ and $0$.\n\n\\begin{rem}\nIf $k=0$, all $r$ $3$-valent vertices are fixed. \nSince the evaluation maps at the contracted ends in the intersection of Psi-classes coincides with the tropical branch map (see \\cite{CJM10}*{Definition 5.14}),\neach point in the $0$-dimensional cycle $\\mathbb{H}^{\\trop}_0(\\mathbf{x})$ comes with a multiplicity which equals the multiplicity of the tropical branch map. The degree of the $0$-dimensional cycle $\\mathbb{H}^{\\trop}_0(\\mathbf{x})$ thus equals $|\\Aut(\\mathbf{x})|$ times the tropical Hurwitz number $H^0(\\mathbf{x})$ as  defined in \\cite{CJM10}. \n\\end{rem}\n\n\\begin{rem}\\label{rem:equivalent}\nThe cycle $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ depends on the choice of the $p_i$.\n By \\cite{AR07} however, different choices of $p_i$ yield rationally equivalent cycles. \n\\end{rem}\n\n\\begin{lemma}\\label{lem-evproduct}\n Let $\\tilde{\\Gamma}$ be a combinatorial type of a cover in  $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ with $r-k$ vertices of valence at least $4$ and with each contracted end $n+1,\\ldots,n+r-k$ adjacent to one of the $r-k$ vertices. Then the evaluation map $\\ev_{n+2}\\times \\ldots \\times \\ev_{n+r-k}$ in local coordinates of the cone of $\\tilde{\\Gamma}$ is a square matrix such that the absolute value of its determinant equals the product of the weights of the bounded edges of $\\tilde{\\Gamma}$.\n\\end{lemma}\n\\begin{proof}\n Inductively we can order the vertices which are not adjacent to $n+1$ and the bounded edges of $\\tilde{\\Gamma}$ so that going from end $n+1$ to another contracted end\n only bounded edges with a smaller index are traversed\n\n (see also \\cite{CJM10}*{Lemma 5.26}). Then the evaluation map in local coordinates is a square triangular matrix such that the diagonal entries are the weights of the bounded edges of $\\tilde{\\Gamma}$.\n\\end{proof}\n\n\n\n\n\\begin{defn}\\label{def:weights} Let $\\mathbf{x}\\in \\mathcal{H}$, and\n $\\Gamma$  a combinatorial type of a marked tropical curve in $\\mathcal{M}_{0,n}$. Associate to each end $i$ of $\\Gamma$ the weight $x_i$. The weights of the other edges are determined by the balancing condition (see Lemma \\ref{lem:flat}). Orient the edges of $\\Gamma$ so that all weights are positive. The orientation of the edges induces a partial ordering of the vertices of $\\Gamma$.\nFor any choice of $k$ vertices $V_1,\\ldots,V_k$ of $\\Gamma$ we denote by $m_{V_1,\\ldots,V_k}(\\Gamma)$ the number of ways to totally order the vertices different from $V_1,\\ldots,V_k$ of $\\Gamma$ respecting the partial order induced by the orientation (see also \\cite{CJMwc}*{Definition 2.1 and 2.15}). (If $k=0$, this agrees with $m(\\Gamma)$ from Definition \\ref{mult}.) We call the vertices $V_1,\\ldots,V_k$ which are not ordered the \\emph{moving vertices}.\n\nFor a fixed choice of $k$ moving vertices, let $\\Gamma'$ be a graph obtained from $\\Gamma$ by shrinking edges in such a way that all moving vertices are identified with a non-moving vertex, and no pair of non-moving vertices is identified. We denote by $w_{V_1,\\ldots,V_k}(\\Gamma)$ the greatest common divisor, over all possibilities to choose $\\Gamma'$, of the products of the weights of the bounded edges of $\\Gamma'$.\n\\end{defn}\n\n\\begin{defn}\n For a combinatorial type $\\tilde{\\Gamma}$ of tropical covers and a vertex $V$ of $\\Gamma$, we define the \\emph{moving vector} of $V$ to be the vector in $\\mathcal{M}_{0,N}$ that we have to add to a tropical cover of type $\\Gamma$ in order to move $h(V)$ to the right by an interval of legths $\\prod w(e)$ where the product goes over all bounded edges $e$ adjacent to $V$ and $w(e)$ denotes their weight.\nThe moving vector  corresponds to a tree with nonzero lengths for the bounded edges of $\\Gamma$ adjacent to $V$: each edge $e$ pointing out of $V$ has  length $-\\prod_{e'\\neq e}w(e')$, each edge pointing into $V$ has length $\\prod_{e'\\neq e}w(e')$; in both cases the product is over bounded edges $e'$ adjacent to $V$ and not equal to $e$ (see \\cite{GMO}*{Definition 5.5}).\nWe also call the image of a moving vector in $\\mathcal{M}_{0,n}$ under the forgetful map $\\ft$ a moving vector.\n\\end{defn}\n\n\n\n\\begin{lemma}\\label{lem:cellsweights} Let $\\Gamma$ be a combinatorial type of a top-dimensional cone of $\\mathcal{M}_{0,n}$.\nIf the points $p_i$ are chosen generically (i.e.\\ pairwise distinct and different from $0$), there are $\\sum m_{V_1,\\ldots,V_k}(\\Gamma)$ top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ contained in the cone of $\\Gamma$, where the sum goes over all $\\binom{r}{k}$ choices of $k$ moving vertices $V_1,\\ldots,V_k$ of $\\Gamma$. For a fixed choice of $k$ moving vertices, each of the $m_{V_1,\\ldots,V_k}(\\Gamma)$ cells corresponding to this choice has weight $w_{V_1,\\ldots,V_k}(\\Gamma)$. Moreover, all cells corresponding to a fixed choice of $k$ moving vertices are parallel, i.e.\\ their linear part is the same.\n\\end{lemma}\n\\begin{proof}\n A cell of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ maps to the cone of $\\Gamma$ in $\\mathcal{M}_{0,n}$ if we obtain the combinatorial type $\\Gamma$ after forgetting the ends $n+1,\\ldots,n+r-k$. \nThus to recover such a cell from the graph $\\Gamma$ one must to pick $r-k$ vertices to which we attach the marked ends $n+1,\\ldots,n+r-k$ in such a way that we get a cover mapping the end $i$ to the point $p_i$ for $i=n+2,\\ldots,n+r-k$ and end $n+1$ to $0$. One can do this if and only if the orientation of the edges \nrespects the order of the image points $p_i$ of the $r-k$ vertices to which the mark ends are attached. \n\nThe graphs $\\Gamma'$ in Definition \\ref{def:weights}\n correspond to faces of the boundary of the cone of $\\Gamma$ that each cell corresponding to $V_1,\\ldots,V_k$ intersects in a point. \nThus two cells of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ corresponding to different choices of moving vertices are pushed forward to different cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ by $\\ft$ --- their images under $\\ft_\\ast$ intersect different boundary cones of $\\Gamma$ in points. But also two cells corresponding to the same choice of moving vertices are pushed forward to different cells: they intersect the same boundary cones of $\\Gamma$, but they do so in different points as the coordinates of the intersection points are given by the lengths of the bounded edges of each $\\Gamma'$, and these lengths depend on the choice of which of the $r-k$ non-moving vertices is mapped to which of the $p_i$.\n\nIt follows that there are $\\sum m_{V_1,\\ldots,V_k}(\\Gamma)$ top-dimensional cells of  $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ contained in the cone of $\\Gamma$.\n\nWe now compute the weight of one such cell. We start by computing the weight of a cell in $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$. Each cell of $\\prod_{i=n+1}^{n+r-k}\\Psi_i$ corresponds to a combinatorial type where the ends $n+1,\\ldots,n+r-k$ are adjacent to different $4$-valent vertices. The weight of such a cone in the intersection $\\prod_{i=n+1}^{n+r-k}\\Psi_i$ equals one by \\cite{KM07}. \nBy Remark \\ref{rem-intersect} the intersection multiplicity from the evaluation pullbacks is the gcd of the absolute values of the maximal minors of the matrix of the evaluation map in local coordinates. \nWe evaluate each of the $r-k-1$ non-moving vertices different from $n+1$ (which is sent to $0$ by convention). A maximal minor corresponds to a choice of $r-k-1$ bounded edges: we contract the remaining bounded edges and get a graph $\\Gamma'$ with $r-k$ vertices.\n\n If some of the non-moving vertices  in $\\Gamma'$  are identified, then the minor has some identical rows (resp.\\ a zero row) and is thus zero. \n\n All the $r-k$ non-moving vertices must thus remain separate; each of the moving vertices is identified with some non-moving vertex, as in the definition of $w_{V_1,\\ldots,V_k}(\\Gamma)$. \nFinally Lemma \\ref{lem-evproduct} implies that the absolute value of the maximal minor corresponding to a possible choice of $\\Gamma'$ equals the product of the weights of the bounded edges of $\\Gamma'$. \n\n The weight of a cell of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ equals $w_{V_1,\\ldots,V_k}(\\Gamma)$. The push forward maps cells of this intersection to cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ one by one, and the forgetful map has index one since it is just a projection. Thus the claim about the weight of cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ follows.\n\nFor any cell  $\\tilde{\\Gamma}\\in \\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ where $V_1,\\ldots,V_k$ are the moving vertices, the linear part of the affine hull of the cell $\\alpha$ is  spanned by the moving vectors of the vertices $V_1,\\ldots,V_k$ in $\\mathcal{M}_{0,N}$. Thus all cells of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ in the cone of $\\Gamma$ corresponding to a fixed choice of $k$ moving vertices are parallel.\n\\end{proof}\n\nLemma \\ref{lem:cellsweights} implies that a (top-dimensional) cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ is specified by a (3-valent) graph $\\Gamma$, a choice of $k$ moving vertices and an ordering of the remaining vertices that respects the edge orientations given by $\\mathbf{x}$.\nWe call this data  the \\emph{combinatorial type} of a cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$.\n\n\n\n\\begin{rem}\n $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ is not necessarily a simplicial polyhedral complex. Figure \\ref{fig:notsimplic} shows a combinatorial type corresponding to a two-dimensional cell. The moving vertices are labelled with an $M$. This cell has four vertices, since the moving vertices can merge with 1 and 3, or 1 and 2, or 2 and 3, or both with 2.\n\n\\begin{figure}[tb]\n\\input{notsimplic.pstex_t}\n\\caption{The combinatorial type of a twodimensional cell with four vertices.}\n\\label{fig:notsimplic}\n\\end{figure}\n\n\n\\end{rem}\n\n\n\nSince we define $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ as a tropical intersection product, it is of course a tropical cycle itself and thus in particular balanced around each cell of codimension one. We find it useful to have a concrete local description of the balancing of tropical Hurwitz cycles. We conclude this section by discussing this. \n\n\n\n\n\nThe Hurwitz cycle $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ has the structure of a weighted marked polyhedral complex (see \\cite{GMO}*{Definition 5.1}): let $\\sigma$ denote a top-dimensional cell of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ and $\\tilde{\\sigma}$ its unique preimage in $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$. Let $\\tilde{\\Gamma}$ be the combinatorial type of the cell $\\tilde{\\sigma}$. \nWe denote by $\\Gamma$ the image of $\\tilde{\\Gamma}$ under the forgetful map. A bounded edge of $\\Gamma$ is called \\emph{a fixed edge}, if the two end vertices of its preimage in $\\tilde{\\Gamma}$ are adjacent to two contracted ends (i.e.\\ their image is fixed by the evaluation pullback). We call a bounded edge \\emph{moveable}, if neither of the two end vertices of its preimage is adjacent to a contracted end.\nWe define the weight factor $\\omega'(\\sigma)$ to be the product of the weights of all fixed edges divided by the product of the weights of moveable edges.\nThe $k$ moving vectors of the moving vertices are the marked vectors of $\\sigma$. \n\nWe equipped the polyhedral complex underlying $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ with two a priori different weight functions for its top-dimensional cells: one weight function is defined by the tropical intersection product, the other by the structure of a marked polyhedral complex. It can be shown that these two structures agree. Assuming this, we now check the balancing condition (as in \\cite{GMO}*{Lemma 5.2}) locally around a codimension one face.\n\n\n\n\nA cell $\\tau$ of codimension one of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ \nparameterizes trees with exactly one $4$-valent vertex. There are two types of cells of codimension one: those where two moving vertices have merged into a $4$-valent vertex, and those where a moving vertex has merged with a non-moving vertex.\n\nWhen two moving vertices have merged, there are three top-dimensional neighbors of $\\tau$, corresponding to the three resolutions of the $4$-valent vertex.\n\n\nThe three neighboring cones  share $k-2$ marked vectors: the remaining two marked vectors  (for each cone) correspond to the moving vectors of the  new $3$-valent vertices obtained resolving the $4$-valent vertex. \nIn all three cases we can replace one of the two new moving vectors by the moving vector of the $4$-valent vertex of $\\tau$. Now the hypothesis of Proposition 5.2 in \\cite{GMO} apply, and we can show just as in Proposition 5.8 of \\cite{GMO} that the balancing condition at $\\tau$ is equivalent to  the appropriate weighted sum of the three other new moving vectors equal to the moving vector of the $4$-valent vertex. \n\nAssume now that a moving vertex is merged into a non-moving vertex to form the $4$-valent vertex $V$ of $\\tau$.\nWithout restriction, assume that one bounded edge of weight $a$ points into $V$ and three bounded edges of weights $b$, $c$ and $d$ point out of $V$.\n Then there are six top-dimensional neighbors of $\\tau$, illustrated in Figure \\ref{fig:neighbours}.\n\n\nAll but one of the marked vectors coincide for the six top-dimensional neighbors, they only differ by the moving vector of the new vertex. The moving vertex is indicated with an $M$.\nThe weight factors of the six neighboring cells differ only by two of the four bounded edges adjacent to $V$ which are fixed edges according to where the new moving vertex is.\n\n\\begin{figure}[tb]\n\\input{neighbours.pstex_t}\n\\caption{The six neighbors of a cell of codimension one where a moving vertex is merged with a non-moving vertex.}\n\\label{fig:neighbours}\n\\end{figure}\n\n\nWe apply Proposition 5.2 of \\cite{GMO} and deduce that the balancing condition around $\\tau$ is equivalent to the equation: \n\\begin{align*}\n&ab\\cdot (cd\\cdot v_{ab}- (a-b)d \\cdot v_c - (a-b) c \\cdot v_d)\\\\\n&-  cd\\cdot (-ab\\cdot v_{ab}+ (a-b)b \\cdot v_a - (a-b) a \\cdot v_b)\\\\\n&+ac\\cdot (bd\\cdot v_{ac}- (a-c)d \\cdot v_b - (a-c) b \\cdot v_d)\\\\\n&-bd\\cdot (-ac\\cdot v_{ac}+ (a-c)c \\cdot v_a - (a-c) a \\cdot v_c)\\\\\n&+ad\\cdot (bc\\cdot v_{ad}- (a-d)c \\cdot v_b - (a-d) b \\cdot v_c)\\\\\n&-bc\\cdot (-ad\\cdot v_{ad}+ (a-d)d \\cdot v_a - (a-d) a \\cdot v_d)\n\\\\&= 2abcd\\cdot (v_{ab}+v_{ac}+v_{bc}-(v_a+v_b+v_c+v_{abc}))=0,\n\\end{align*}\nwhere we use the equation $a=b+c+d$ and relations among the vectors $v_I$ as in \\cite{KM07}*{Lemma 2.6} (the vector $v_a$ here stands for the tree with all ends that we can reach from $V$ via $a$ on one side).\n\n\\subsection{Extended Example: Hurwitz Curves}\nWe study the tropical structure of the Hurwitz curve $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$. Top dimensional cells (segments) parameterize graphs with one moving vertex.\nEach codimension one cell --- i.e.\\ vertex --- of the Hurwitz curve\ncorresponds to curves where the moving vertex is merged with another (fixed) vertex, and has six top-dimensional cells incident to it. For any $\\mathbf{x}$, the Hurwitz curve is a six-valent tropical curve in  $\\mathcal{M}_{0,n}$. We  distinguish several types of edges and compute their weights $w_{V_1}(\\Gamma)$. \n\n\n\n\n\\begin{figure}[tb]\n\\input{easyend.pstex_t}\n\\caption{The combinatorial type of a constant end of the Hurwitz curve.}\n\\label{fig:easyend}\n\\end{figure}\n\n\\emph{Constant Ends} (Figure \\ref{fig:easyend}) of the Hurwitz curve are edges corresponding to a combinatorial type where the moving vertex is adjacent to two ends $i$ and $j$. The moving vector equals $\\pm v_{ij}$. \nA constant end is incident to a unique vertex, corresponding to shrinking the only bounded edge adjacent to the moving vertex. \nThe weight of a constant end equals the product of all bounded edge weights except for the bounded edge adjacent to the moving vertex. \n\n\n\\begin{figure}[tb]\n\\input{complicatedend.pstex_t}\n\\caption{The combinatorial type of a linear end of the Hurwitz curve.}\n\\label{fig:complicatedend}\n\\end{figure}\n\n\\emph{Linear Ends} (Figure \\ref{fig:complicatedend}):\n the moving vertex is adjacent to two bounded edges $e_1$ and $e_2$ of the same direction and one end $i$, necessarily of opposite direction. The moving vector of a linear end equals $\\pm( w(e_2) v_{I}+w(e_1) v_{I\\cup \\{i\\}})$, where $I$ denotes the set of ends separated from $i$ via $e_1$.\n A linear end is incident to only one $0$-dimensional cell, corresponding to the graph where the moving vertex coincides with the vertex of the shorter of the two edges (the end vertex of $e_1$ in the picture). To obtain possible graphs $\\Gamma'$, we can shrink either $e_1$ or $e_2$. Thus the weight of a linear end equals $\\prod_{e\\neq e_1,e_2} w(e)\\cdot \\mbox{gcd}(w(e_1),w(e_2))$ where the product goes over all bounded edges $e$ except $e_1$ and $e_2$.\n\n\n\\begin{figure}[tb]\n\\input{easybounded.pstex_t}\n\\caption{The combinatorial type of a linear edge of the Hurwitz curve.}\n\\label{fig:easybounded}\n\\end{figure}\n\n\n\\emph{Linear Edges} (Figure \\ref{fig:easybounded}) are edges corresponding to a combinatorial type where the moving vertex is adjacent to two bounded edges $e_1$ and $e_2$ of opposite direction, and to an end $i$. The moving vector of an linear edge equals $\\pm( w(e_2) v_{I}- w(e_1) v_{I\\cup \\{i\\}})$ where $I$ denotes the set of ends separated from $i$ via $e_1$. We reach the two vertices of a linear edge by shrinking either $e_1$ or $e_2$. The weight of a linear edge is exactly as for a linear end: $\\prod_{e\\neq e_1,e_2} w(e)\\cdot \\mbox{gcd}(w(e_1),w(e_2))$.\n\n\\begin{figure}[tb]\n\\input{complicatedbounded.pstex_t}\n\\caption{The combinatorial type of a quadratic edge of the Hurwitz curve.}\n\\label{fig:complicatedbounded}\n\\end{figure}\n\n\n\\emph{Quadratic Edges} (Figure \\ref{fig:complicatedbounded}) are edges corresponding to a combinatorial type where the moving vertex is adjacent to three bounded edges $e_1$, $e_2$ and $e_3$. Without restriction we can assume that $e_1$ and $e_2$ are of the same direction, and $e_3$ is of opposite direction. We can also assume that the non-moving vertices are ordered such that the other end vertex of $e_1$ is between the end vertex of $e_2$ and the moving vertex. A quadratic edge connects the two vertices where we shrink $e_3$ resp.\\ $e_1$.\n The moving vector of a quadratic edge equals $\\pm( w(e_2)\\cdot w(e_3) v_{I}+ w(e_1)w(e_3) v_{J}- w(e_1)w(e_2) v_{(I\\cup J)^c})$ where $I$ denotes the set of ends that we can reach from the moving vertex via $e_1$ and $J$ denotes the set of ends that we can reach via $e_2$. \nWe can shrink each of the three adjacent edges to obtain a possible graph $\\Gamma'$ to compute the weight. Thus the weight of a complicated bounded edge equals $\\prod_{e\\neq e_1,e_2,e_3} w(e)\\cdot \\mbox{gcd}(w(e_1)w(e_2),w(e_1)w(e_3),w(e_2)w(e_3))$.\n\n\\begin{rem}\nThe weight of a one-dimensional cell $H$ in the Hurwitz curve is a function of the weights of the bounded edges of a curve $\\tilde{\\Gamma}$ parameterized by the cell. If we artificially define the weight of ends of  $\\tilde{\\Gamma}$ to be $1$ (these are not the right tropical weights!) then we can describe the weight of $H$ in a uniform way as the product of weights of all edges of  $\\tilde{\\Gamma}$ that are not incident to the moving vertex times the $gcd$ of the three products of  weights of pairs of edges incident to the moving vertex. Given that  weights of edges  of  $\\tilde{\\Gamma}$ are either $1$ or linear polynomials, the degree of the above $gcd$ can range from $0$ to $2$, thus explaining the names we chose above.\\end{rem}\n\n\n\n\n\\begin{example}\\label{ex:m05hurwitzcurve}\n We compute the Hurwitz curve $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ for $\\mathbf{x}=(x_1,\n\\ldots,x_5)$ with $x_1, x_2>0$, $x_3,x_4,x_5<0$, $x_1>|x_i+x_j|$ for all $i\\not=j\\in\\{3,4,5\\}$ and $x_2<-x_i$ for all $i=3,\\ldots,5$ in $\\mathcal{M}_{0,5}$.\nWe start with the cells in the cone spanned by $v_{12}$ and $v_{34}$ in $\\mathcal{M}_{0,5}$. To fix a combinatorial type of covers such that the corresponding cell of $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ lives in this cone, we first have to choose a moving vertex among the three vertices of the tree with ends $1$ and $2$ coming together and ends $3$ and $4$ coming together. For each of these three choices, there is one ordering of the remaining vertices which is compatible with the orientation of the edges (see Figure \\ref{fig:m05}). We thus have three cells of $\\mathbb{H}^{\\trop}_1(\\mathbf{x})$ in this cone, two of these cells are constant ends, the other one is an linear edge. Figure \\ref{fig:m052} shows the cone and the three cells.\nBy symmetry, the cones spanned by $v_{12}$ and $v_{35}$ resp.\\ by $v_{12}$ and $v_{45}$ look analogous. Similar arguments also show that all remaining cones except the cones spanned by $v_{23}$ and $v_{45}$, resp.\\  $v_{24}$ and $v_{35}$, resp.\\  $v_{25}$ and $v_{35}$, look alike.\n\n\\end{example}\n\\begin{figure}[tb]\n\\input{m05.pstex_t}\n\\caption{The combinatorial types of the Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{12}$ and $v_{34}$.}\n\\label{fig:m05}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\input{m052.pstex_t}\n\\caption{The Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{12}$ and $v_{34}$.  We require the vertex $6$ to be mapped to $0$ as usual and $7$ to $1$.}\n\\label{fig:m052}\n\\end{figure}\n\nIn the cone spanned by $v_{23}$ and $v_{45}$, we have two constant ends when the moving vertex is adjacent to two ends. If the third vertex is moving, there are two orderings of the remaining vertices compatible with the orientation of the edges (see Figure \\ref{fig:m053}). Altogether, we get two constant and two linear ends as depicted in Figure \\ref{fig:m054}.\n\n\\begin{figure}[tb]\n\\input{m053.pstex_t}\n\\caption{The combinatorial types of the Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{23}$ and $v_{45}$.}\n\\label{fig:m053}\n\\end{figure}\n\n\\begin{figure}[tb]\n\\input{m054.pstex_t}\n\\caption{The Hurwitz curve in the cone of $\\mathcal{M}_{0,5}$ spanned by $v_{23}$ and $v_{45}$.}\n\\label{fig:m054}\n\\end{figure}\n\n\n\n\n\n\\section{Tropical-Classical Correspondence}\n\\label{sec:tcc}\n\nOur work in Sections \\ref{sec:hl} and \\ref{sec:thl} has highlighted a combinatorial correspondence between  classical and tropical Hurwitz cycles. In this section we make a precise statement of such a correspondence, and illustrate it in the example of one dimensional cycles. A satisfactory correspondence should also encode the polynomial multiplicities of strata. In Section \\ref{sec:int} we interpret the (classical) multiplicity of a stratum in the Hurwitz cycle as an intersection multiplicity of the corresponding tropical face with the  $k$-dimensional skeleton of $\\mathcal{M}_{0,n}$.\n\\begin{co}\n\\label{co:tropclas}\nThere is a natural bijection between $i$-dimensional faces of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ and connected components in  $\\tilde\\mathbb{H}_k(\\mathbf{x})$ of the inverse image via $\\st$ of irreducible  strata in $\\mathbb{H}_k(\\mathbf{x})$ of dimension $k-i$. Further, incidence of faces on the tropical side corresponds to intersection of strata on the classical side.  \n\\end{co}\n\nThe key ingredient here is the correspondence between tropical graphs and boundary strata of moduli spaces of relative stable maps outlined in Lemma \\ref{flatten}. The subtlety to observe is that a cell of the tropical Hurwitz cycle is sensitive to the type of the directed graph parameterized and to the ordering of the fixed vertices, but not to the ordering of moving vertices amongst themselves or with respect to the fixed ones. So two general points  $P_1,P_2$ in the same $i$-dimensional tropical cell may correspond to graphs where two adjacent vertices (at least one of which is moving) have switched order, and hence to different $(k-i)$-dimensional boundary strata $\\tilde{\\Delta}_1, \\tilde{\\Delta}_2$ of relative stable maps. The segment joining $P_1$ and $P_2$ contains a point $P$ where the two incriminated vertices map to the same image point: this corresponds to a $(k-i+1)$-dimensional boundary stratum $\\Delta$ that contains $\\tilde{\\Delta}_1$ and $\\tilde{\\Delta}_2$ as specializations. The stratum $\\Delta$ does not belong to $\\tilde{\\mathbb{H}}_k(\\mathbf{x})$, but it does belong to the inverse image via $\\st$ of the Hurwitz cycle.  We feel that describing this phenomenon in full generality would only mire us in notational confusion, so we choose  to illustrate it in one very specific example.\n\n\n\\subsection{Hurwitz Curves Continued}\n\nConsider the one dimensional cell labelled $I$ in Figure \\ref{fig:m054}, and the corresponding graphs parameterized by points in such cell (these are illustrated in Figure \\ref{fig:m053}). Let $\\ell$ be a coordinate for this cell, corresponding to the length of the segment joining vertices $6$ and $M$. For $\\ell < -\\frac{1}{x_4+x_5}$ (resp. $\\ell > -\\frac{1}{x_4+x_5}$) any point of $I$ is the tropical dual graph to the stratum $\\tilde{\\Delta}_1$ (resp. $\\tilde{\\Delta}_2$) as depicted in Figure \\ref{fig:contract}. The point $\\ell = -\\frac{1}{x_4+x_5}$ correspond to the stratum $\\tilde{\\Delta}$, which is a ${\\mathbb{P}^1}$ connecting $\\tilde{\\Delta}_1$ and  $\\tilde{\\Delta}_2$ in $\\st^{-1}({\\mathbb{H}_1}(\\mathbf{x}))$.\n\n\\begin{figure}[tb]\n\\input{strata.pstex_t}\n\\caption{Strata of relative stable maps in $\\st^{-1}({\\mathbb{H}_1}(\\mathbf{x}))$.}\n\\label{fig:contract}\n\\end{figure}\n\n\\subsection{The Hurwitz cycle \nintersecting the codimension $k$-skeleton of $\\mathcal{M}_{0,n}$}\n\n\\label{sec:int}\n\n\n\nIn this section we realise the (classical) multiplicities of the strata in the Hurwitz cycle as intersection numbers of the tropical Hurwitz cycle with the codimension  $k$-skeleton of $\\mathcal{M}_{0,n}$. To do so we must  view each codimension $k$ cell as a part of an intersection product of divisors. \nWe recall the boundary divisors $D_I$  that ``play well'' in the intersection theory of  $ \\mathcal{M}_{0,n}$ are defined as divisors of appropriate rational functions \\cite{Rau08}*{Definition 2.4}. Each $D_I$ is a linear combination of codimension one cells of $ \\mathcal{M}_{0,n}$ with appropriate weights. In the following lemma we describe some intersection products of these boundary divisors in term of the tropical curves they parameterize.\n\\begin{lemma}\\label{lem:divintersect}\nThe intersection of the tropical boundary divisors $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$ for some $j\\geq 2$ in $\\mathcal{M}_{0,m}$ consists of all cones corresponding to a type with the ends $1,\\ldots,j$ at\n\\begin{itemize}\n \\item  a $j+2$-valent vertex and only $3$-valent vertices otherwise with weight one,\n\\item  a $j+1$-valent vertex adjacent to a $4$-valent vertex and only $3$-valent vertices otherwise with weight $-1$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\nWe show this by induction. The induction beginning is \\cite{Rau08}*{Lemma 2.5}. For the induction step, assume that the statement is true for $j-1$. \nThe codimension $j$ cones in  $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j-1}$ then consist of all cones corresponding to a type with the ends $1,\\ldots,j-1$ at\n\\begin{itemize}\n \\item a $j+2$-valent vertex and only $3$-valent vertices otherwise,\n\\item  a $j+1$-valent vertex and one $4$-valent vertex,\n\\item  a $j$-valent vertex adjacent to a $5$-valent vertex,\n\\item  a $j$-valent vertex adjacent to a $4$-valent vertex and one other $4$-valent vertex.\n\\end{itemize}\nTo obtain the coefficients of such cones in the product   $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$ we must compute the intersection with $\\varphi_{1...j}$ around each of these cones.\n\nIn the first case, the neighbors of  $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j-1}$ are the resolutions of the $j+2$-valent vertex in a $j+1$-valent vertex with $1,\\ldots,j-1$ adjacent. Each such neighbor is spanned by a vector corresponding to this resolution. Since only the vector $v_{1...j}$ is mapped to one by $\\varphi_{1...j}$, we get a contribution of one for the weight of the codimension one cone only if also $j$ is adjacent to the $j+2$-valent vertex. The weight then equals one.\n\nIn the second case, we can resolve the $j+1$-valent vertex, but none of the resolutions is spanned by the vector $v_{1..j}$. We can also resolve the $4$-valent vertex. Again, none of the resolutions is spanned by the vector $v_{1...j}$, however, this vector is contained in the codimension one cone if also $j$ is adjacent to the $j+1$-valent vertex. If in addition the $4$-valent vertex is adjacent to the $j+1$-valent vertex, the sum of the vectors spanning its three resolutions contains $v_{1...j}$ as a summand: if we denote the four edges adjacent to the $4$-valent vertex by $e_1,\\ldots,e_4$ and assume that the subset of ends that can be reached via $e_i$ from the $4$-valent vertex is $A_i$, then the three resolutions are spanned by $v_{A_1\\cup A_2}$, $v_{A_1\\cup A_3}$ and $v_{A_2\\cup A_3}$ and their sum satisfies $v_{A_1\\cup A_2}+v_{A_1\\cup A_3}+v_{A_2\\cup A_3}=v_{A_1}+v_{A_2}+v_{A_3}+v_{A_1\\cup A_2 \\cup A_3}= v_{A_1}+v_{A_2}+v_{A_3}+v_{A_4}$ by \\cite{KM07}*{Lemma 2.6}. This yields a contribution of minus one for the weight of this cone.\n\nIn the third and fourth case, neither the codimension one cone itself nor any of its neighbors contains the vector $v_{1...j}$. Therefore it gets weight zero. \nThe claim follows.\n\\end{proof}\n\n\\begin{rem}\\label{rem:coneintersect}\nThe important consequence of this lemma is the statement that a cone with a $j+2$-valent vertex adjacent to the ends $1,\\ldots,j$ appears with weight one in the intersection $ D_{12}\\cdot D_{123}\\cdot  \\ldots \\cdot D_{1...j}$. It is straight-forward to generalize this statement to a cone $C$ with an arbitrary $j+2$-valent vertex $V$, adjacent to the edges $e_1,\\ldots,e_{j+2}$.\nWe denote by $A_i$ the subset of ends that can be reached from $V$ via $e_i$. Then $C$ appears with weight one in the intersection $ D_{A_1\\cup A_2}\\cdot D_{A_1\\cup A_2 \\cup A_3}\\cdot  \\ldots \\cdot D_{A_1\\cup \\ldots\\cup A_{j-1}}$.\nAlso, to cut out a cone  with several higher-valent vertices, we can combine several such intersection products.\n\\end{rem}\n\n\n\n\\begin{lemma}\\label{lem:pullbackintersect}\n The intersection $\\Psi_\\alpha\\cdot \\ft_\\alpha^\\ast( D_{12})\\cdot\\ft_\\alpha^\\ast( D_{123})\\cdot   \\ldots \\cdot \\ft_\\alpha^\\ast(D_{1...j})$ for some $j\\geq 2$ in $\\mathcal{M}_{0,m+1}$ consists of all cones corresponding to a type with the ends $1,\\ldots,j$ \n\\begin{itemize}\n \\item and $\\alpha$ at a $j+3$-valent vertex with weight $j$, \n\\item at a $j+2$-valent vertex, and $\\alpha$ adjacent to some other vertex with weight $1$,\n\\item and $\\alpha$ at a $j+2$-valent vertex adjacent to a $4$-valent vertex with weight $-(j-1)$,\n\\item at a $j+1$-valent vertex adjacent to a $5$-valent vertex with $\\alpha$ with weight $-2$,\n\\item at a $j+1$-valent vertex adjacent to a $4$-valent vertex and $\\alpha$ adjacent to some other vertex with weight $-1$.\n\\end{itemize}\n\\end{lemma}\n\\begin{proof}\n The proof is again by induction. For $j=2$, we intersect $\\Psi_\\alpha$ with $\\ft_\\alpha^\\ast(\\varphi_{12})=\\varphi_{12}\\circ \\ft_\\alpha$. This map sends the vector $v_{12}$ and the vector $v_{12\\alpha}$ to one and all other $v_i$ to zero. Codimension one cones of $\\Psi_\\alpha$ either have a $5$-valent vertex with $\\alpha$ or a $4$-valent vertex with $\\alpha$ and another $4$-valent vertex. In the first case, if $1$ and $2$ are also adjacent to the $5$-valent vertex, two of the six neighbors are spanned by vectors mapping to one, so we get weight $2$. In the second case, the vector $v_{12\\alpha}$ is contained in the codimension one cone itself and appears in the sum of the vectors spanning the neighbors if $1$ and $2$ are adjacent to the vertex with $\\alpha$ and the other $4$-valent vertex is adjacent. Such a cone then comes with weight minus one.\n\nFor the induction step, assume the statement is true for $j-1$. The codimension one cones of $\\Psi_\\alpha\\cdot \\ft_\\alpha^\\ast( D_{12})\\cdot\\ft_\\alpha^\\ast( D_{123})\\cdot   \\ldots \\cdot \\ft_\\alpha^\\ast(D_{1...j})$ then consist of all cones corresponding to a type with the ends $1,\\ldots,j-1$ \n\\begin{enumerate}\n \\item and $\\alpha$ at a $j+3$-valent vertex, \n\\item and $\\alpha$ at a $j+2$-valent vertex and one $4$-valent vertex,\n\\item at a $j+2$-valent vertex, $\\alpha$ somewhere else ,\n\\item at a $j+1$-valent vertex and a $5$-valent vertex with $\\alpha$,\n\\item at a $j+1$-valent vertex and a $4$-valent vertex, $\\alpha$ somewhere else,\n\\item and $\\alpha$ at a $j+1$-valent vertex, next to a $5$-valent vertex,\n\\item and $\\alpha$ at a $j+1$-valent vertex next to a $4$-valent vertex and another $4$-valent vertex,\n\\item at a $j$-valent vertex next to a $6$-valent vertex with $\\alpha$,\n\\item at a $j$-valent vertex next to a $5$-valent vertex with $\\alpha$ and another $4$-valent vertex,\n\\item at a $j$-valent vertex next to a $5$-valent vertex and $\\alpha$ somewhere else,\n\\item at a $j$-valent vertex next to a $4$-valent vertex, and another $4$-valent vertex and $\\alpha$ somewhere else. \n\\end{enumerate}\nFor the cases (6)-(11) it is easy to see that neither the codimension one cone itself nor any of its neighbors contains the vectors $v_{1..j}$ or $v_{1..j\\alpha}$ which are mapped to one by $\\ft_\\alpha^\\ast(\\varphi_{1..j})=\\varphi_{1..j}\\circ \\ft_\\alpha$. Thus any of these cones is taken with weight zero.\n\nIn the first case, if $j$ is also adjacent to the $j+3$-valent vertex, we have a neighbor spanned by $v_{1...j\\alpha}$ with weight $j-1$, and a neighbor spanned by $v_{1...j}$  with weight one. Altogether the weight  is $j$.\n\nIn the second case, if $j$ is also adjacent to the $j+2$-valent vertex the vector $v_{1...j\\alpha}$ is contained in the codimension one cone itself. If in addition the $4$-valent vertex is adjacent to the $j+2$-valent vertex, the vector $v_{1...j\\alpha}$ appears in the sum of the three neighbors corresponding to the resolutions of the $4$-valent vertex. Since any such resolution comes with weight $j-1$, this codimension one cone has weight $-(j-1)$.\n\nIn the third case, if $j$ is also adjacent to the $j+2$-valent vertex, we have one neighbor spanned by $v_{1...j}$  with weight one, so we also get weight one for this cone.\n\nIn the fourth case, none of the resolutions of the $j+1$-valent vertex contains the vectors $v_{1...j}$ or $v_{1...j\\alpha}$.\n The vector $v_{1...j}$ is contained in the cone itself however if $j$ is also adjacent to the $j+1$-valent vertex. We can also resolve the $5$-valent with $\\alpha$ in such a way that $\\alpha$ is still at a $4$-valent vertex. All these six neighbors have weight one. Denote the four edges not equal to the end $\\alpha$ but adjacent to the $5$-valent vertex by $e_1,\\ldots,e_4$ and denote by $A_i$ the subset of ends that can be reached from the $5$-valent vertex via $e_i$. Then the six neighbors are spanned by the vectors $v_{A_1\\cup A_2}$, $v_{A_1\\cup A_3}$, $v_{A_2\\cup A_3}$, $v_{A_1\\cup A_2\\cup\\{\\alpha\\}}$, $v_{A_1\\cup A_3\\cup\\{\\alpha\\}}$ and $v_{A_2\\cup A_3\\cup\\{\\alpha\\}}$ whose sum equals $2v_{A_1}+2v_{A_2}+2v_{A_3}+2\\cdot v_{A_1\\cup A_2\\cup A_3\\cup\\{\\alpha\\}}=2v_{A_1}+2v_{A_2}+2v_{A_3}+2\\cdot v_{A_4}$ by \\cite{KM07}{Lemma 2.6}. Thus we get weight $-2$ if and only if $A_i=\\{1,\\ldots,j\\}$ for $i=1,2,3$ or $4$ which is the case if and only if the $5$-valent vertex with $\\alpha$ is adjacent to the $j+1$-valent vertex with $1,\\ldots,j$.\n\nThe fifth case is analogous to the second case of Lemma \\ref{lem:divintersect}. All neighbors have weight one. Therefore we get weight minus one if the $4$-valent vertex is adjacent to the $j+1$-valent vertex and $j$ is adjacent to the $j+1$-valent vertex. The claim follows.\n\\end{proof}\n\n\n Now  we intersect $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ with the codimension $k$-skeleton of $\\mathcal{M}_{0,n}$. Let $K$ denote a cone of the codimension $k$-skeleton. It corresponds to a combinatorial type of a tree $\\Gamma$ with $r-k$ vertices $V_1,\\ldots,V_{r-k}$ of valence $\\val(V_i)=k_i$ with $\\sum (k_i-3)=k$. \nRemark \\ref{rem:coneintersect} tells us how to pick functions $\\varphi_1,\\ldots,\\varphi_k$ that cut out $K$ with weight one. Thus we want to compute \n$\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k $ locally around $K$. We have\n\\begin{align*}\n& \\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k = \\\\\n& \\ft_{\\ast}\\Big(\\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\Big) \\cdot\\varphi_1\\cdot\\ldots\\cdot\\varphi_k =\\\\\n& \\ft_{\\ast}\\Big(  \\Psi_{n+1}\\cdot \\prod_{i=n+2}^{n+r-k} \\big(\\Psi_i \\cdot \\ev_i^{\\ast}(p_i) \\big) \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k)    \\Big) =\\\\\n& \\ft_{\\ast}\\Big(  \\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k)  \\cdot \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i)  \\Big)\n\\end{align*}\nwhere the second equality holds by the projection formula \\cite{AR07}*{Proposition 4.8}.\n\nTo get a nonzero intersection of $\\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k) $ with the cycle $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $, the ends $n+1,\\ldots, n+r-k$ must be adjacent to different vertices. The type $\\Gamma$ corresponding to $K$ has $r-k$ vertices, and so we can attach one new end to each vertex of $\\Gamma$. There are $m(\\Gamma)$ ways to do this, where we use the notation from Definition \\ref{def:weights} (we do not need to pick moving vertices here). Hence $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ intersects $K$ in $m(\\Gamma)$ points with a nonzero weight.\nEach such point is the push-forward of an intersection point of  $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $ with a cone $\\tilde{K}$ of $\\prod_{i=n+1}^{n+r-k}\\Psi_{i}$ where all ends are adjacent to different vertices. To compute the weight of $\\tilde{K}$ in $\\prod_{i=n+1}^{n+r-k}\\Psi_{i} \\cdot\\ft^\\ast(\\varphi_1)\\cdot\\ldots\\cdot\\ft^\\ast(\\varphi_k) $, we use a generalization of Lemma \\ref{lem:pullbackintersect} analogous to Remark \\ref{rem:coneintersect}: in $\\tilde{K}$, we have $r-k$ vertices $V_i$ of valence $k_i+1$ each of which is adjacent to an end with a Psi-class condition. We thus get  weight $\\prod (k_i-2)$. From Lemma \\ref{lem-evproduct}, the further intersection with $ \\prod_{i=n+2}^{n+r-k}  \\ev_i^{\\ast}(p_i) $ yields a factor equal to the product of weights of all bounded edges.\nWe have thus proved the following statement that again illustrates the analogy between classical and tropical Hurwitz loci (compare with Lemma \\ref{coeff}):\n\n\\begin{prop}\n Let $K$ be a cone of $\\mathcal{M}_{0,N}$ of codimension $k$, corresponding to the type $\\Gamma$.\nThen the intersection $\\mathbb{H}^{\\trop}_k(\\mathbf{x})\\cdot K$ consists of $m(\\Gamma)$ points, each with weight $\\prod_v (\\val(v)-2)\\cdot \\varphi(\\Gamma)$\nwhere the product goes over all vertices $v$ of $\\Gamma$.\n\\end{prop}\n\n\n\n\\section{Wall Crossings}\n\n\n\nWe have seen that Hurwitz cycles are polynomials in each chamber $\\mathfrak{c}$. In this section we investigate wall-crossings, i.e. how the cycles change from chamber to chamber. \n\\begin{defn}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\sum_{i\\in I}x_i =0\\} $. Let $\\mathfrak{c}^+$  and $\\mathfrak{c}^-$ be two adjacent chambers: $\\sum_{i\\in I}x_i >0$ in $\\mathfrak{c}^+$,  $\\sum_{i\\in I}x_i < 0$ in $\\mathfrak{c}^-$ and for every $J\\not=I \\subseteq  \\{1, \\ldots, n\\}$ the sign of $ \\sum_{i\\in J}x_i$ is the same in both chambers. Let $\\mathbb{H}_k^+(\\mathbf{x})$ (resp. $\\mathbb{H}_k^-(\\mathbf{x})$) be the polynomial class giving the Hurwitz cycle in $\\mathfrak{c}^+$(resp. $\\mathfrak{c}^-$). By {wall crossing formula} at the wall $I$ we mean the formal difference of cycles:\n\\begin{equation}\nWC_{I,k}(\\mathbf{x}):= \\mathbb{H}_k^+(\\mathbf{x}) -\\mathbb{H}_k^-(\\mathbf{x}) \\in Z_k(\\overline{M}_{0,n}).\n\\end{equation}\n\n\\end{defn}\nNaturally the difference of two polynomial cycles is a polynomial cycle: the upshot is that such a cycle can be expressed inductively in terms of Hurwitz cycles.\n\n\\begin{note} Let $\\mathbf{x}$ and $\\mathbf{y}$ be two tuples of integers such that $\\sum{x_i}=-\\sum{y_j}=\\epsilon\\not=0$. Consider the Hurwitz cycles $\\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\in \\overline{M}_{0,n_1+1}$ and $\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon)\\in \\overline{M}_{0,n_2+1}$ and the gluing morphism \n$\\gl:\\overline{M}_{0,n_1+1}\\times\\overline{M}_{0,n_2+1}\\to \\overline{M}_{0,n_1+n_2}$. We denote:\n$$\n\\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\boxtimes\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon):= \\gl_\\ast\\left( \\mathbb{H}_{k_1}(\\mathbf{x},-\\epsilon)\\times\\mathbb{H}_{k_2}(\\mathbf{y},\\epsilon)\\right) \\in Z_{k_1+k_2}( \\overline{M}_{0,n_1+n_2}).\n$$\n\\end{note}\n\nWith this notation in place we are ready to state the wall crossing formulas.\n\n\\begin{thm}[Classical Wall Crossing]\n\\label{thm:wc}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\\begin{equation}\n\\label{eq:wc}\nWC_{I,k}(\\mathbf{x})= \\epsilon \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  {{r-k}\\choose{|I|-1-j}}\\mathbb{H}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\n\\end{equation}\n\\end{thm}\n\\begin{proof}\nThe proof of Theorem \\ref{thm:wc} is parallel to \\cite{CJM10}*{Theorem 6.10}.\nWe first remark that the bounds of the summation are simply  recording the fact that $j$ (resp. $k-j$) must be less than or equal that the dimension of $\\overline{M}_0^{\\sim}(\\mathbf{x}_I,-\\epsilon)$ (resp. $\\overline{M}_0^{\\sim}(\\mathbf{x}_{I^c},\\epsilon)$). One may make the summation simply from $0$ to $k$ by noting that the Hurwitz loci of dimension greater than the corresponding moduli spaces of maps are empty.\n\nHurwitz cycles are completely described by the tropical dual graphs of the boundary strata in the moduli spaces of maps. In order for a tropical dual graph to contribute to the wall crossing, it must have an edge with weight equal to the equation of the wall. For a given tropical dual graph $\\Gamma$, if such an edge exists, then it is unique and we call it the special edge. \n\nCutting the special edge separates the graph into two subtrees  $\\Gamma_I$ and $\\Gamma_{I^c}$. \nThe ends  of $\\Gamma_I$ (resp. $\\Gamma_{I^c}$) are labelled by $x_i\\in I$ and $-\\epsilon$ (resp. $x_i\\in I^c$ and $\\epsilon$). We note immediately that $\\Gamma_I,\\Gamma_{I^c}$ are a pair of graphs identifying a boundary stratum appearing in the product of  Hurwitz cycles on the right hand side of formula \\eqref{eq:wc}. We make this connection more precise in order to extract quantitative information.\n\nFor $0\\leq j\\leq k$, let\n$\nR_j=\\left\\{ \\left(\\Gamma_1, \\Gamma_2, \\mathfrak{m} \\right)\\right\\},\n$\nwhere:\n\\begin{itemize}\n\\item $\\Gamma_1$ is the tropical dual graph of a stratum in  $\\tilde{\\mathbb{H}}_j(\\mathbf{x}_I,-\\epsilon)$ (pushing forward non-trivially to $\\overline{M}_{0,n}$).\n\\item $\\Gamma_2$ is the tropical dual graph of a stratum in  $\\tilde{\\mathbb{H}}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)$ (pushing forward non-trivially to $\\overline{M}_{0,n}$).\n\\item $\\mathfrak{m}$ is a total ordering of the vertices of $\\Gamma_1 \\cup \\Gamma_2$, compatible with the total ordering of the vertices of  $\\Gamma_1$ and $\\Gamma_2$.\n\\end{itemize}\n\nCutting the special edge gives a function $\\Cut$ from the set of graphs contributing to the wall crossing formula to the union $R= \\cup_{j=0}^k R_j$. We claim that $\\Cut$ is a bijection, and it will come as little  surprise that the inverse function  $\\Glue$ consists in gluing  the two graphs along the special edge labelled $\\pm \\epsilon$. The total ordering $\\mathfrak{m}$ is precisely the information needed to make such gluing well defined. We note in particular that $\\mathfrak{m}$ determines in which direction the special edge is pointing once it is glued.\n\nGiven a graph  $\\Gamma$ contributing to the wall crossing, we note that the multiplicity it contributes to the wall crossing formulas is $\\epsilon$ times the product of all weights of all non-special internal edges (this is obvious if $\\Gamma$ comes from $\\mathbb{H}_k^+(\\mathbf{x})$. If  $\\Gamma$ comes from $\\mathbb{H}_k^-(\\mathbf{x})$, then the weight of the special edge is $-\\epsilon$, and there is another minus sign coming from the wall crossing formula). On the other hand the pair of graphs $\\Gamma_1$ and $\\Gamma_2$ in $\\Cut(\\Gamma)$ have multiplicity in $\\mathbb{H}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)$ equal to the product of all non-special internal edges of $\\Gamma$. Therefore $\\Cut$ is a bijection that preserves the multiplicities on both sides of formula \\eqref{eq:wc}.\n\n The proof is then concluded by remarking that if $\\Gamma_1, \\Gamma_2$ appear in $R_j$, there are ${{r-k}\\choose{|I|-1-j}}$ possible ways of giving a total ordering of the vertices of $\\Gamma_1 \\cup \\Gamma_2$, compatible with the total ordering of the vertices of  $\\Gamma_1$ and $\\Gamma_2$. \n\\end{proof}\n\nThe wall crossing formula on the tropical side is similar: the only apparent difference is the lack of the multiplicative factor $\\epsilon$, reflecting the fact that polynomiality does not appear in the generic representative of a tropical Hurwitz cycle.\nDifferently from the classical side however, it is not only the weights that depend on $\\mathbf{x}$ but the cycles themselves, making even the statement of a wall crossing formula more subtle.\n\nFix a wall $W_I$ and two adjacent chambers $\\mathfrak{c}^+$  and $\\mathfrak{c}^-$ with $\\epsilon:=\\sum_{i\\in I}x_i >0$ in $\\mathfrak{c}^+$,  $\\epsilon < 0$ in $\\mathfrak{c}^-$. We denote by $\\mathbb{H}^{\\trop,+}_k(\\mathbf{x})$ resp.\\ $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$ the Hurwitz cycles in the two chambers.\nWe would like to evaluate  both $\\mathbb{H}^{\\trop,+}_k(\\mathbf{x})$  and $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$\n at $\\mathbf{x}\\in \\mathfrak{c}^+$\n\n and then consider the difference. However the fact that $\\epsilon$ changes sign when crossing the wall requires some interpretation, since the edge lengths of tropical covers\nare required to be positive. \nConsider a point in $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ corresponding to a tropical cover with an edge of weight $-\\epsilon$ connecting two vertices that are mapped to two points $p<q$ in $\\TP$. The length of the special edge of this tropical cover then equals $\\frac{q-p}{-\\epsilon}$, which is positive in $\\mathfrak{c}^-$, but negative in $\\mathfrak{c}^+$. In the latter case, $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ does not live in  $\\mathcal{M}_{0,n}$ but in the real vector space $\\mathbb{R}\\mathcal{M}_{0,n}$ surrounding it.\nTo define $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})\\subset \\mathcal{M}_{0,n}$ for $\\mathbf{x}\\in \\mathfrak{c}^+$, we consider  the linear map $c_\\epsilon$ that maps $v_I$ to itself for all $I\\neq \\epsilon$ and $v_\\epsilon$ to $-v_\\epsilon$, and redefine $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ to be the image of $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})\\subset \\mathbb{R}\\mathcal{M}_{0,n}$  under $c_\\epsilon$. \n\n\nAlso the weights of the cycle $\\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ require some interpretation: in Lemma \\ref{lem:cellsweights} we have seen that the weight of a cell equals the gcd of products of weights of trees where we shrink edges in such a way that all moving vertices merge with a non-moving vertex. When computing $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$, we use the convention to use the negative gcd whenever $-\\epsilon$ appears in a product.\n\n\\begin{defn}\nWith the notation from the above paragraph, we now define the \\emph{tropical Wall Crossing} as\n$$WC^{\\trop}_{I,k}(\\mathbf{x}):= \\mathbb{H}^{\\trop,+}_k(\\mathbf{x}) - \\mathbb{H}^{\\trop,-}_k(\\mathbf{x}).$$\n\\end{defn}\n\n\n\\begin{example}\\label{ex:wcm05}\n In Example \\ref{ex:m05hurwitzcurve}, we  consider the Hurwitz curve for  $\\mathbf{x}=(x_1,\n\\ldots,x_5)$ in $\\mathfrak{c}^+$ defined by $x_1, x_2>0$, $x_3,x_4,x_5<0$, $x_1>|x_i+x_j|$ for  $i,j \\in\\{3,4,5\\}$ and $x_2<-x_i$ for  $i=3,4,5$ in $\\mathcal{M}_{0,5}$. We now cross the wall $\\epsilon:=x_1+x_4+x_5=0$. \n\n$\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ and $ \\mathbb{H}^{\\trop,-}_1(\\mathbf{x})$ only differ in cones whose corresponding type contains an edge with weight $\\pm\\epsilon$, i.e.\\ in any cone containing the vector $v_{23}$. There are three such cones. For two of these cones, the Hurwitz curve $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ looks as depicted in Figure \\ref{fig:m052} and for the third cone as in Figure \\ref{fig:m054}. In $ \\mathbb{H}^{\\trop,-}_1(\\mathbf{x})$, any edge with weight $-\\epsilon$ changes direction. In the cone spanned by $v_{23}$ and $v_{14}$, if we choose the vertex adjacent to end $5$ to be the moving vertex, we now have two ways to order the remaining vertices, both corresponding to linear ends. Thus one linear edge is replaced by two linear ends in this cone. The same is true for the cone spanned by $v_{23}$ and $v_{15}$ by symmetry. In the cone spanned by $v_{23}$ and $v_{45}$ contrarily, if the moving vertex is adjacent to end $1$, we have only one way to order the remaining vertices corresponding to a linear edge instead of the two linear ends (see Figures \\ref{fig:m053} and \\ref{fig:m054}). \nNote also that the constant ends of direction $v_{23}$ do not have a factor of $\\epsilon$ in their weight, thus they appear with the same sign both in $\\mathbb{H}^{\\trop,+}(_1\\mathbf{x}) $ and $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ and cancel in the difference. The other constant ends have weight $\\epsilon$ in $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ but weight $-\\epsilon$ in $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$, so they do not cancel but add up to a constant end with weight $2\\epsilon$. Figure \\ref{fig:wc} depicts the tropical wall crossing curve for these two chambers. Blue edges are edges of $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $, red edges are edges of $ \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$. The green constant ends appear in both and add up to the weight $2\\epsilon$. The picture only shows the three cones of $\\mathcal{M}_{0,5}$ in which the difference $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) - \\mathbb{H}^{\\trop,-}_k(\\mathbf{x})$ is nonzero.\n\n\n\\begin{figure}[tb]\n\\input{wc.pstex_t}\n\\caption{The wall crossing curve in $\\mathcal{M}_{0,5}$ for $\\mathfrak{c}^+$ and $\\mathfrak{c}^-$.}\n\\label{fig:wc}\n\\end{figure}\n\n\\end{example}\n\n\nAs in the classical world, we want to describe a tropical wall crossing in terms of cutting and regluing. \n Any cell contributing to the wall crossing parameterizes graphs with a special edge that we may cut to obtain two subgraphs each of which is a tropical cover of the projective line.  Remembering the length of the edge that gets cut accounts for the fact that we want to mod out by translations only once. We make this process precise in the following paragraph.\n\n\n\\begin{construction}\n \nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. \nAssume that $|I|=n_1$ and $|I^c|=n_2$ and denote $r_i=n_i-1$ for $i=1,2$.\nLet $\\Gamma$ be a tropical cover in a $k$-dimensional cell that contributes to the wall crossing, and hence it contains an edge $e$ with weight $\\pm\\epsilon$. Cutting $e$ we obtain two subgraphs $\\Gamma_1$ and $\\Gamma_2$ that are themselves tropical covers of the projective line. We assume that $\\Gamma_1$ contains the ends in $I$ and $\\Gamma_2$ contains the ends in $I^c$. Both $\\Gamma_1$ and $\\Gamma_2$ have an extra end that we denote by $E_1$ (resp.\\ $E_2$). According to our conventions ends are oriented inward, and hence the balancing condition gives $E_1$ weight $-\\epsilon$ and $E_2$ weight $\\epsilon$. \nAssuming that $r_1-j$ fixed vertices are in $\\Gamma_1$,  we can interpret $\\Gamma_1$ as an element in $\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))$ and $\\Gamma_2$ as an element in $\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))$ (adjusting the labeling of the ends). We wish to remember the length of the edge we cut and whether $\\Gamma$ belonged to $\\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) $ or $\\mathbb{H}^{\\trop,-}_1(\\mathbf{x}) $, so we define:\n$\n\\Cut (\\Gamma) \\in \\left(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\right)\\times \\left(\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}\\right)\n$\nby \n\n\\begin{equation} \n\\label{eq:cut}\n\\Cut(\\Gamma) =\\begin{cases} \\left((\\Gamma_1, 0) , (\\Gamma_2, l(e))\\right) & \\Gamma \\in  \\mathbb{H}^{\\trop,+}_1(\\mathbf{x}) \\\\  \\left((\\Gamma_1, 0) , (\\Gamma_2, -l(e))\\right) & \\Gamma \\in  \\mathbb{H}^{\\trop,-}_1(\\mathbf{x}). \\end{cases}\n\\end{equation}\n\n\n\n\n\\begin{rem} \nOur choice to have two ${\\mathbb R}$ coordinates and assigning one of them to be $0$ seems, and in fact is, somewhat arbitrary. However, it will be handy when comparing weights of the same cells appearing on opposite sides of the wall crossing formula.\n\\end{rem}\n\\begin{rem}\nIn order for  equation \\eqref{eq:cut} to make sense we need to drop Convention \\ref{conv} (see page \\pageref{conv}) and remember tropical covers are equivalent up to translation. We therefore use one of the vertices in each of the subgraphs to fix a parameterization of $\\TP$.\n\\end{rem}\n\nWe reverse this operation to glue two graphs in  $\\left(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\right) \\times\\left(\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}\\right)$. Denote by $l_i$ the ${\\mathbb R}$ coordinate function for the $i$-th factor in the product.\nIf $V_i$ is the interior vertex of $\\Gamma_i$ adjacent to $E_i$, then  define $\\ev_{E_i}: \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}\\to {\\mathbb R}$ by $$\\ev_{E_i}(\\Gamma_i, l_i)=\\ev_{V_i}+(-1)^il_i\\cdot \\epsilon. $$\n\n\n\nWe can glue any two pieces in $(\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot (\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}) \\times (\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}) $ to one tropical cover in $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$. To make this operation the inverse to $\\Cut$ defined above, we further impose $l_1=0$. \n\n\nThe above discussion shows that \n\\begin{align*}\\mathcal{G} := (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0) \\cdot  \\Big( &(\\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon))\\times {\\mathbb R}) \\times \\\\  &  (\\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon))\\times {\\mathbb R}) \\Big)\\end{align*}\n is in bijection to the set of covers in $\\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ contributing to the wall crossing \n(see \\cite{GMO} for a similar gluing construction for moduli spaces). \n\nWe now define the folding map $\\fold:\\mathcal{G} \\rightarrow \\mathcal{M}_{0,r-k}(\\TP,\\mathbf{x})$ that maps $l_2$ to its absolute value.\nThe folding map is not globally a tropical morphism, it is only locally a morphism away from $\\mathcal{G}\\cdot l_2^\\ast(0)$.\nConsequently, while $\\mathcal{G}$\nis a tropical variety, its image under $\\fold$ is not. Since the wall crossing curve is also not a tropical variety however, this is not disturbing.\n\nTo make the image of $\\fold$ a weighted polyhedral complex, we give each cell the sum of the weights of its preimages (cells are subdivided by \n$\\mathcal{G}\\cdot l_2^\\ast(0))$. This coincides with the weight of the push forward of $\\fold$ locally where it is a morphism.\n\\end{construction}\n\n\n\nWe  now state the first version of the tropical wall crossing. \n \\begin{prop}[Tropical Wall Crossing, first version]\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\n\n\n\n\n\n\n\n\\begin{equation}\\hspace{-1cm}\n\\begin{array}{cl}\nWC^{\\trop}_{I,k}(\\mathbf{x}) &=  \\ft_{\\ast} \\bigg(  \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}} \\;\\;\\;\\;  \\sum_{n+1 \\leq i_1<\\ldots<i_{r_1-j}\\leq n+r-k}  \n \\fold   \\Big(  \\\\ & \\\\\n & \\left.\\left.  \\hspace{-1cm}\n\\Big( \\Psi_{n_1+2}\\cdot \\prod_{l=2}^{r_1-j} \\big(\\Psi_{l+n_1+1} \\cdot \\ev_{l+n_1+1}^{\\ast}(p_{i_{l}}) \\big)      \n\\cdot \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon)) \\Big) \\times {\\mathbb R}  \n\\times   \\right.\\right.\\\\ & \\\\\n& \\left.\\left.\\hspace{-1cm}\n\\Big( \\Psi_{n_2+2}\\cdot \\prod_{l=2}^{r_2-(k-j)} \\big(\\Psi_{l+n_2+1} \\cdot \\ev_{l+n_2+1}^{\\ast}(p_{j_{l}}) \\big)   \\cdot \\mathcal{M}_{0,r_2-(k-j)}(\\TP,(\\mathbf{x}_{I^c},\\epsilon)) \\Big) \\times {\\mathbb R} \\times \\right.\\right.\\\\ & \\\\\n&\\hspace{-1cm}\n (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\n\\Big)\\bigg), \n\\end{array}\n\\label{twc-one}\n\\end{equation}\nwhere $j_1<\\ldots<j_{r_2-(k-j)}$ are defined by $$\\{n+1,\\ldots,n+r-k\\}=\\{i_1,\\ldots,i_{r_1-j}, j_1,\\ldots, j_{r_2-(k-j)}\\}.$$ \n\n\\end{prop}\n\n\\begin{proof}\n We first show that graphs where the special edge separates two pieces $\\Gamma_1$ and $\\Gamma_2$ such that there is no fixed (i.e. $4$-valent) vertex in $\\Gamma_1$ do not contribute to the wall crossing. \n Analogously the same is true for types with no $4$-valent vertex in $\\Gamma_2$. \n This justifies the bounds of the first sum: $j \\geq 1+k-r_2$ is equivalent to $r_2-(k-j)\\geq 1$ which means that there is at least one $4$-valent vertex in $\\Gamma_2$, the upper bound $j\\leq r_1-1$ guarantees that there is at least one $4$-valent vertex in $\\Gamma_1$. \n\nAssume $\\Gamma$ is a type with no $4$-valent vertex in $\\Gamma_1$. Then there are only moving vertices in $\\Gamma_1$. The type $\\Gamma$ appears both in $\\mathbb{H}_k^{\\trop,+}(\\mathbf{x}) $ and in $\\mathbb{H}_k^{\\trop,-}(\\mathbf{x}) $ : the special edge can point in any direction if none of the vertices past  the special edge  are free to move. The corresponding cells are identical set-theoretically.\n To compute their weight, we have to consider types $\\Gamma'$ where all moving vertices merge with some fixed vertices. In any such type, the special edge must be shrunk since there are only moving vertices on one side of it. Hence the weight which equals the gcd of products of weights of possible $\\Gamma'$ is identical for both cells, which thus cancel in the difference $\\mathbb{H}_k^{\\trop,+}(\\mathbf{x}) - \\mathbb{H}_k^{\\trop,-}(\\mathbf{x})$.\n\nThe above construction shows that  formula \\eqref{twc-one} holds set-theoretically. \nIt remains to show that also the weights agree. Since the cells of $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ are in weight-preserving bijection with the cells of the push forward $\\mathbb{H}^{\\trop}_k(\\mathbf{x}) \\subset \\mathcal{M}_{0,n}$, we can show the equality of the weights for the cells before pushing forward with the forgetful map. Also, we can separate the contributions from the two sides of the wall, i.e. ``unfold'' on the cut-and-glue side of the equation.\nLet $K$ be a cell \ncontributing to the wall crossing\n\n  corresponding to the combinatorial type $\\Gamma$.\n \n  Following the construction above we produce $\\Gamma_1$ and $\\Gamma_2$ which  identify a unique cell $K'$ on the right hand side of equation \\eqref{twc-one}.  By Remark \\ref{rem-intersect}  the weight of $K$ is the gcd of the absolute values of the maximal minors of the matrix of the evaluation maps evaluating the fixed vertices of $\\Gamma$ in local coordinates. Analogously, the weight of the corresponding cut-and-glue cell $K'$ is the gcd of the absolute values of the maximal minors of the matrix of the evaluation maps evaluating the fixed vertices of $\\Gamma_1$, the fixed vertices of $\\Gamma_2$, of the map $\\ev_{E_1}-\\ev_{E_2}$\n  and of the map $l_1$.\nLocal coordinates of the cell $K$ are given by the lengths of all bounded edges of $\\Gamma$,\nwhereas for the cell $K'$ they are given by the bounded edges of $\\Gamma$ except the special edge, plus the lengths $l_1$ and $l_2$. The map $l_1$ produces a row for the matrix with a one in the $l_1$-column and only zeros in the other columns. Any nonzero maximal minor must contain this column and equals the determinant of the submatrix where we erase the $l_1$-column and the row for the map $l_1$. Therefore we can erase this column and row without changing any minor in the matrix for $K'$. We show that the matrix for $K$ and the reduced matrix for $K'$ coincide up to row operations. Since row operations do not change the absolute value of any maximal minor, the claim follows. In the matrix for $K$, columns are the lengths in $\\Gamma_1$, the length $l$ of the special edge and the lengths in $\\Gamma_2$. As rows we have the evaluations of the contracted ends $n+2,\\ldots,n+r-k$ which for each end $i$ are given by following the unique path in $\\Gamma$ from the reference end $n+1$ (that we require to map to $0$) to $i$.\nIn the reduced matrix for $K'$, we have the analogous columns. Assume without restriction that $i_1=n+1$, i.e.\\ the end mapping to zero is contained in $\\Gamma_1$. We evaluate all other contracted ends of $\\Gamma_1$ with respect to this reference end, i.e.\\ we follow the path from this end to any other in $\\Gamma_1$. This gives exactly the same rows for the evaluation of ends of $\\Gamma_1$. In the matrix of $K$, we have a row evaluating the end that becomes the reference end in $\\Gamma_2$ (that we fix to be at the point $p_{j_1}$). This row is exactly equal to the row evaluating $\\ev_{E_1}-\\ev_{E_2}$ --- the unique path in $\\Gamma$ connecting the two reference ends. The remaining rows evaluate ends in $\\Gamma_2$. For an end $i$ in $\\Gamma_2$, we evaluate it in $K$ by following the path from the reference end in $\\Gamma_1$ to $i$. We evaluate it in $K'$ by following the path from the reference end in $\\Gamma_2$ to $i$. These two rows just differ by the path connecting the two reference ends which by the above is also a row of both matrices. Hence we can perform row operations\n to produce the matrix for $K$ from the matrix for $K'$.\n\\end{proof}\n\n\\begin{example}\nWe establish the wall crossing formula \\eqref{twc-one} for the wall crossing curve in $\\mathcal{M}_{0,5}$ from Example \\ref{ex:wcm05}. \nThe wall $\\epsilon=x_1+x_4+x_5=0$ separates any cover $\\Gamma$ containing an edge $\\epsilon$ into $\\Gamma_1$ and $\\Gamma_2$, where $\\Gamma_1$ has the ends $x_1,x_4,x_5$ and $-\\epsilon$ and $\\Gamma_2$ has the ends $x_2$, $x_3$ and $\\epsilon$. We have two contracted marked ends. Since both pieces must contain at least one to have at least one $4$-valent vertex, both contain exactly one. \nWe first assume that the end $6$  (required to map to zero) is in $\\Gamma_1$. For $\\Gamma_2$, there is exactly one type, where all four ends (including the contracted end $7$) are incident to one $4$-valent vertex. We have  a copy of ${\\mathbb R}$ parametrizing all possibilities with coordinate $l_2$.\n\nFor $\\Gamma_1$, there are six possible types, since there are three types in $\\mathcal{M}_{0,4}$ each of which has two $3$-valent vertices, and we can decide which of the two vertices should be equipped with the contracted end $6$ and become $4$-valent. \nWe neglect the length of the extra end $l_1$ that is required to be zero.\nWe  have a ray for each of the six types, the local coordinate is the length of the bounded edge that we denote by $l$. \nFigure \\ref{fig:6types} shows the six types together with the part to which they are glued.\n\n\\begin{figure}[tb]\n\\input{6types.pstex_t}\n\\caption{The possible types for $\\Gamma_1$ and $\\Gamma_2$ after cutting, assuming $6$ is in $\\Gamma_1$.}\n\\label{fig:6types}\n\\end{figure}\n\n\n\nFor each type, we  write down the gluing equation $\\ev_{E_1}-\\ev_{E_2}=0$ in local coordinates. Remember we require $6$ to be mapped to zero and $7$ to one.\nThe equations are:\n\n\\begin{align*}\n \\mathrm{I}:\\qquad &  0+l (x_1+x_4)+\\epsilon l_2=1, \\\\\n \\mathrm{II, IV,V}:\\qquad &  0+l_2\\epsilon=1,\\\\\n \\mathrm{III}:\\qquad &0+l (x_1+x_5)+\\epsilon l_2=1, \\\\\n \\mathrm{VI}:\\qquad & 0-l (-x_4-x_5)+l_2\\epsilon=1.\n\\end{align*}\n\n\n\nThese gluing equations cut out the tropical variety depicted in Figure \\ref{fig:gluing} from $\\Psi_6\\cdot \\mathcal{M}_{0,1}(\\TP,x_1,x_4,x_5,-\\epsilon)\\times \\Psi_7\\cdot \\mathcal{M}_{0,1}(\\TP,x_2,x_3,\\epsilon)\\times {\\mathbb R}$ which consists of six rays \nWe also draw the image under the folding map. The rays that are mapped from the orthant where $l_2$ is negative via the folding map are drawn with dotted lines.\n\n\\begin{figure}[tb]\n\\input{gluing.pstex_t}\n\\caption{The variety obtained from gluing and its image under folding for the first summand.}\n\\label{fig:gluing}\n\\end{figure}\n\n\n\nAnalogously, if end $7$ is in $\\Gamma_1$ and $6$ is in $\\Gamma_2$, we have the types depicted in Figure \\ref{fig:6types} but with $6$ and $7$ exchanged.\nThe gluing equations in local coordinates are as above, but with the role of $0$ and $1$ exchanged.\nFigure \\ref{fig:gluing2} shows the tropical variety cut out by the gluing equation from $\\Psi_7\\cdot \\mathcal{M}_{0,1}(\\TP,x_1,x_4,x_5,-\\epsilon)\\times \\Psi_6\\cdot \\mathcal{M}_{0,1}(\\TP,x_2,x_3,\\epsilon)\\times {\\mathbb R}$ and its image under folding. As before, rays that appear in the positive orthant after folding are drawn as dotted lines.\n\n\\begin{figure}[tb]\n\\input{gluing2.pstex_t}\n\\caption{The variety obtained from gluing and its image under folding for the second summand.}\n\\label{fig:gluing2}\n\\end{figure}\n\n\nUnder the forgetful map, the four orthants labeled I and II go to the cone spanned by $v_{23}$ and $v_{14}$, the four orthants labeled III and IV go to the cone spanned by $v_{23}$ and $v_{15}$ and the remaining four to the cone spanned by $v_{23}$ and $v_{45}$. We get the picture  seen in  Example \\ref{ex:wcm05}.\n\n\n\\end{example}\n\n\n\n\nWe now group summands in the cut-and-glue side of the equation to give a nicer interpretation of formula \\eqref{twc-one}.\nIf we denote by $\\tilde{\\mathbb{H}}^{\\trop}_k(\\mathbf{x})$ the preimage of $\\mathbb{H}^{\\trop}_k(\\mathbf{x})$ under the forgetful map, then as in Remark \\ref{rem:equivalent} the rational equivalence class of the loci \n\\begin{align*}& \\Big( \\Psi_{n_1+2}\\cdot \\prod_{l=2}^{r_1-j} \\big(\\Psi_{l+n_1+1} \\cdot \\ev_{l+n_1+1}^{\\ast}(p_{i_{l}}) \\big)      \n\\cdot \\mathcal{M}_{0,r_1-j}(\\TP,(\\mathbf{x}_I,-\\epsilon)) \\Big) \\times {\\mathbb R}  \\\\&\\sim \\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R}\\end{align*}\n does not depend on the exact choice of $i_1<\\ldots<i_{r_1-j}$, and analogously for the other loci in the sum. Up to rational equivalence we can thus group all summands of the second sum into one just counting how many such summands there are: $\\binom{r-k}{r_1-j}=\\binom{r-k}{|I|-1-j}$. Then \\eqref{twc-one} becomes\n\\begin{align*}\n &WC^{\\trop}_{I,k}(\\mathbf{x})\\sim\\\\ & \\ft_{\\ast} \\Bigg(  \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  \n\\binom{r-k}{r_1-j}  \\fold   \\bigg( \\Big(\\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R} \\times \\tilde{\\mathbb{H}}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\\times {\\mathbb R} \\Big)\n\\\\&\n\\times \n (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\\bigg)\\Bigg).\n\\end{align*}\n\nIf we furthermore denote\n\\begin{align*}\n &\\mathbb{H}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon) :=\\\\&\n\\fold   \\left( \\Big(\\tilde{\\mathbb{H}}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon) \\times {\\mathbb R} \\times \\tilde{\\mathbb{H}}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon)\\times {\\mathbb R} \\Big) (\\ev_{E_1}-\\ev_{E_2})^\\ast(0)\\cdot l_1^\\ast(0)\\right)\n\\end{align*}\nwe obtain a formula for the tropical wall crossing that looks very similar to the classical wall crossing formula  \\eqref{eq:wc}.\n\n\n\\begin{cor}[Tropical Wall Crossing, reloaded]\n\\label{cor:twc}\nLet $I\\subseteq \\{1, \\ldots, n\\}$ and consider the wall $W_I=\\{\\epsilon:=\\sum_{i\\in I}x_i =0\\} $. Then:\n\\begin{align*}\n &WC^{\\trop}_{I,k}(\\mathbf{x})\\sim\\\\ & \\ft_{\\ast} \\left( \\sum_{j=\\max\\{0,1+k-r_2\\}}^{\\min\\{k,r_1-1\\}}  \\binom{r-k}{|I|-1-j}          \\mathbb{H}^{\\trop}_j(\\mathbf{x}_I,-\\epsilon)\\boxtimes\\mathbb{H}^{\\trop}_{k-j}(\\mathbf{x}_{I^c},\\epsilon) \\right).\n\\end{align*}\n\n\\end{cor}\n\n\n\n\n\n\\subsection{Hurwitz Curves: Take three}\n\nA few features of Example \\ref{ex:wcm05} can be generalized to the general one-dimensional wall crossing  case immediatly. Constant ends of direction $v_I$ cancel in the wall crossing. Constant ends of other directions can appear on both sides of the wall but will show up in the wall crossing with a positive sign. Let $\\Gamma^+$ be a graph in $\\tilde{\\mathbb{H}}^{\\trop,+}_k(\\mathbf{x})$ with a special edge and $V$ a choice of a moving vertex corresponding to a linear end. Then we have $m_V(\\Gamma^+)$ linear ends in $\\tilde{\\mathbb{H}}^{\\trop,+}_k(\\mathbf{x})$, and $m_V(\\Gamma^-)$ linear edges in $\\tilde{\\mathbb{H}}^{\\trop,-}_k(\\mathbf{x})$ (where $\\Gamma^-$ denotes the graph with the special edge reversed). All such edges come with the same weight in the wall crossing.  Analogously, $m_V(\\Gamma^+)$ quadratic edges are replaced by $m_V(\\Gamma^-)$ quadratic edges when crossing the wall, also coming with the same weight.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhen two liquid drops meet, a dramatic topological transformation takes place. \nA microscopic liquid neck forms between the drops, which rapidly expands due to surface tension forces (see Fig.\\ \\ref{sequenceWater}). \nThis coalescence is ubiquitous in nature on a vast range of scales, from raindrops merging in clouds \\cite{Pruppacher2010} to the coalescence of gas clouds during star formation \\cite{Montmerle2006}. \nIn industry, coalescence occurs in a multitude of situations, including viscous sintering \\cite{Djohari2009}, oil desalting \\cite{Eow2002}, dense spray systems \\cite{Ashgriz1990}, and the processing and shelf-life of emulsions \\cite{Evans1994}. \n\n\\begin{figure}[b]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.4in]{sequenceWater.pdf} \n\\end{center}\n\\caption{\nInitial moments of coalescence for two water drops of radius $A=2$ mm in air. \nFrames are 120 $\\mu$s apart. The central white spot is from the light source located behind the drops. \nA connecting neck of radius $r_{\\text{min}}$ grows from infinitesimal size until it reaches the macroscopic size of the drops. \n}\n\\label{sequenceWater}\n\\end{figure}\n\nThe singularity inherent in drop coalescence has received significant attention. \nIn the idealized limit where the drops form their initial contact over an infinitesimal region, the interfacial curvature, and therefore the pressure, diverge at the moment of contact, $t_0$. \nAlthough in reality this singularity is cut off (at least by the finite size of the molecules), it nonetheless controls the dynamics of the liquid neck over many orders of magnitude. \nThese dynamics have recently been investigated by theoretical work \\cite{Hopper1984,Hopper1990,Hopper1993a,Hopper1993b,Eggers1999,Eggers2003,Thompson2012_2}, numerical simulations \\cite{Herrera1995,MenchacaRocha2001,Eggers2003,Lee2006,Thompson2012_2,Sprittles2012}, high-speed imaging experiments \\cite{MenchacaRocha2001,Wu2004,Yao2005,Thoroddsen2005,Bonn2005,Burton2007,Yokota2011}, ultrafast x-ray phase-contrast imaging \\cite{Fezzaa2008}, and an ultrafast electrical method \\cite{Case2008,Case2009}. \nThese studies sought to understand the dynamics governing the growth of the neck radius, $r_{\\text{min}}$, as a function of time. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{phaseDiagram.pdf}\n\\end{center}\n\\caption{\n(Color online) \nPrevious and revised phase diagram for drop coalescence in vacuum or air. \nAxes: non-dimensional neck radius, $r_{\\text{min}}\/A$, and $Oh=\\mu\/\\sqrt{\\rho \\gamma A}$ (non-dimensional viscosity). \n(a) Old picture: there are two regimes, one dominated by viscous forces where inertia can be completely ignored (Stokes regime) and one dominated by inertia. \nSolid line: $r_{\\text{min}}\/A=Oh^2$. \n(b) Revised understanding: the ``inertially limited viscous\" (ILV) regime intervenes at early times for finite-viscosity drops. \nThe Stokes and inertial regimes do not share a phase boundary. \nDashed line: Eq.\\ (\\ref{phase_boundary_3D}) with a proportionality constant of $1.4$. \nSolid line: Eq.\\ (\\ref{rcformula}). \n}\n\\label{phaseDiagram}\n\\end{figure}\n\nThe consensus from this work had been that there are just two regimes when the drops of radius $A$ are brought together in vacuum or air: (i) a highly viscous one dominated by macroscopic flows pulling the two drops together, and (ii) an inertial one described by local deformations near the growing neck. \nAt early times, drops obey purely viscous behavior, and low-viscosity drops transition into the inertial regime later on. \nThus, a coalescence phase diagram had been constructed \\cite{Eggers1999}, which is shown in Fig.\\ \\ref{phaseDiagram}(a) in terms of the dimensionless neck radius, $r_{\\text{min}}\/A$, and the Ohnesorge number, $Oh=\\mu\/\\sqrt{\\rho \\gamma A}$, which is a dimensionless viscosity (where $\\mu$ is the dynamic viscosity, $\\rho$ is the liquid density, and $\\gamma$ is the interfacial tension). \n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} demonstrated that a third regime---one that intervenes at early times for drops of any finite viscosity---had been missed. \nUsing high-speed imaging experiments, an ultrafast electrical method, and high-accuracy computation, they showed that at small neck radius, the surface tension force pulling the drops together is too weak to overcome the inertia of the drops. \nTherefore, coalescence \\textit{cannot} be in the Stokes-flow limit at early times, and an ``inertially limited viscous\" regime occurs. \nIn this regime, surface tension, viscous forces, and inertia all balance. \nAt late times, viscous drops ($Oh>1$) \\textit{can} reach the Stokes-flow limit, so there is an inertially limited viscous to Stokes crossover that is traversed as the neck radius grows. \n\nThe final piece of the coalescence phase diagram is the viscous-to-inertial crossover time, $\\tau_c$ (or the crossover radius, $r_c$), where the dynamics switch from the inertially limited viscous regime to a regime where only inertia is important. \nFor many fluid flows, this crossover is easily identified by computing the dimensionless Reynolds number, $Re=\\rho U L\/\\mu$, where $U$ and $L$ are characteristic velocity- and length-scales in the flows, respectively. \nCrossover behavior is expected when $Re\\approx 1$. \nFor coalescence, it was always assumed that $L=r_{\\text{min}}$ \\cite{Eggers1999,Eggers2003,Wu2004,Yao2005,Thoroddsen2005,Bonn2005,Burton2007,Case2008,Case2009,Thompson2012_2}. \n\nTo observe the viscous-to-inertial crossover, Paulsen \\textit{et al.}\\ \\cite{Paulsen2011} used an ultrafast electrical method (following refs.\\ \\cite{Case2008,Case2009}), which measures the neck radius down to tens of nanoseconds after the drops touch. \nFor salt-water drops, Paulsen \\textit{et al.}\\ observed viscous behavior more than $3$ decades later than the prediction using the accepted Reynolds number for coalescence. \nTo explain this discrepancy, they proposed that the dominant length-scale for the flows is instead given by the neck height, $L=r_{\\text{min}}^2\/A$. \nThus, a revised phase diagram for coalescence was constructed, which is pictured in Fig.\\ \\ref{phaseDiagram}(b). \n\n\nThis paper provides a more detailed experimental description and presents additional evidence for the picture developed in refs.\\ \\cite{Paulsen2012, Paulsen2011}. \nFirst, section \\ref{ExperDesc} describes the electrical method, the fluids used, and the high-speed imaging technique, and section \\ref{StokesInertialTheory} outlines several theoretical predictions for the purely viscous (Stokes) regime and the inertial regime. \n\nSection \\ref{ILVregime} provides measurements and analysis of coalescence in the Stokes regime and the inertially limited viscous regime. \nWhereas Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} identified the inertially limited viscous to Stokes crossover by the motion of the back of the drops, I show that the same motions occur in the center-of-mass of the drops. \nThe neck shapes in the inertially limited viscous and Stokes regimes are consistent with two distinct similarity solutions, and the interfacial curvature at the neck minimum can be used to distinguish between the regimes. \nThe phase diagram is robust to different boundary conditions.\n\nSection \\ref{VIcrossover} provides measurements and analysis of the viscous-to-inertial crossover. \nI collapse the electrical data with a different analysis from ref.\\ \\cite{Paulsen2011}, to demonstrate that the results are not sensitive to the details of the collapse protocol. \nI argue for a new Reynolds number for coalescence, as was done in ref.\\ \\cite{Paulsen2011}, now coming from the viscous side of the transition. \nI present high-speed imaging data where the surface tension is varied, which follows the crossover scaling calculated with the new Reynolds number. \n\nSection \\ref{approach} studies the drops during their approach.\nUsing optical, electrical resistance, and capacitance measurements, I show that at low approach-speed, the drops coalesce as undeformed spheres at finite separation. \nThe data suggest that at low voltage, Van der Waals forces form the initial liquid neck (instead of forces due to the applied voltage).\nThe measurements provide an upper bound on the initial neck radius, $r_0$, which is smaller than previous estimates \\cite{Thoroddsen2005,Fezzaa2008}.\n\n\nAppendix \\ref{elecSystematic} reports checks on the electrical method, which show that the applied voltages and resulting electric fields do not affect the coalescence dynamics. Appendix \\ref{prevCross} addresses previous measurements of the viscous-to-inertial crossover in the literature. \n\nThis work gives a consistent picture wherein the inertially limited viscous regime is the asymptotic regime of liquid drop coalescence in vacuum or air. \nViscous drops ($Oh>1$) transition into the Stokes regime later on, and low-viscosity drops ($Oh<1$) crossover into the inertial regime. \nIn the inertially limited viscous regime and the inertial regime, the dominant flow gradients are on the scale of the neck height, $r_{\\text{min}}^2\/A$. \n\n\n\n\n\n\\section{Experimental Description}\n\\label{ExperDesc}\n\n\\subsection{Ultrafast electrical method}\n\nIn the experiment, two drops are formed on vertically aligned teflon nozzles of radius $A=2$ mm, which are separated by a distance $2A$. \nThe pendant drop is fixed while the sessile drop is slowly grown with a syringe pump until the drops coalesce. \nExcept for section \\ref{approach}, the experiments are at sufficiently low approach-speed ($U_{\\text{app}}< 9 \\times 10^{-5}$ m\/s) where the drops do not deform before contact.\n\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.0in]{Schematic_Signals.pdf}\n\\end{center}\n\\caption{\n(Color online) \nElectrical method. \n(a) Coalescence cell and measurement circuit. \nLiquid hemispheres are formed on nozzles. \nOne drop is grown slowly with a syringe pump (Razel Scientific, R-99) to initiate coalescence, while an AC voltage, $V_{\\text{in}}$ (Hewlett-Packard, HP3325A), is applied across the drops and known circuit elements ($R_k$, $C_k$). \nVoltages $V_1$ and $V_2$ are recorded with a high-speed digitizer (NI PCI-5105, National Instruments) and converted to the time-varying complex impedance of the coalescence cell. \n$Z_{\\text{CR}}$: impedance of the coalescence region (dashed box). \n$Z_t$, $Z_b$: impedances of the fluid-filled nozzles. \n$C_p$: stray capacitance between the nozzles. \n(b-d) Signals for a single saturated aqueous NaCl coalescence versus $\\tau\\equiv t-t_0$. \n(b) The phase angle, $\\Delta\\phi$, between $V_1$ and $V_2$ decreases sharply when the drops touch, which is used to measure $t_0$. \n(c) Capacitance of the coalescence region, $C_{\\text{CR}}$, is roughly constant before and after contact. \n(d) Resistance of the coalescence region, $R_{\\text{CR}}$, after contact.\n}\n\\label{Schematic}\n\\end{figure}\n\nFollowing the AC electrical method developed by Case \\textit{et al.}\\ \\cite{Case2008,Case2009} and used in refs.\\ \\cite{Paulsen2012, Paulsen2011}, I measure the time-varying complex impedance, $Z_{\\text{CR}}$, of two liquid hemispheres while they are coalescing (see Fig.\\ \\ref{Schematic}). \nSalt (NaCl) is added to the drops to make them electrically conductive. \nA high-frequency ($0.6 \\leq f \\leq 10$ MHz), low-amplitude ($V_{\\text{in}} \\leq 2$ V) AC voltage is applied across the drops by gold electrodes that are submerged in the fluid. \nBy simultaneously sampling the voltage below the coalescence cell and the voltage below known passive circuit elements, the impedance of the coalescence cell is determined. \nTwo backgrounds are subtracted: one is measured by bringing the nozzles together, and the second is a small parallel capacitance, $C_p=0.61\\pm 0.12$ pF, that is measured before forming drops on the nozzles. \nThis isolates the impedance of the coalescing drops, $Z_{\\text{CR}}$, which is modeled as a time-varying resistor, $R_{\\text{CR}}$, and capacitor, $C_{\\text{CR}}$, in parallel. \nAt the instant the drops touch, there is a sharp decrease in the phase difference, $\\Delta\\phi$, between the two measured voltages, which indicates the moment of contact, $t_0$, to within 1\/$f$. \n\nExamples of these measured quantities are shown in Fig.\\ \\ref{Schematic}(b-d) as a function of $\\tau \\equiv t-t_0$, which measures time elapsed since the moment of contact, $t_0$. \nMore than $10^4$ points are sampled, thereby capturing a large dynamic range from a single coalescence event. \n\nTo ensure that the applied voltage and the resulting electric fields between the drops do not alter the coalescence dynamics, a variety of checks were performed on the electrical method (see Appendix \\ref{elecSystematic}). \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.15in]{EStat.pdf} \n\\end{center}\n\\caption{\n(Color online) \nConversion between electrical resistance, $R_{\\text{CR}}$, and neck radius, $r_{\\text{min}}$. \n(a) Three axisymmetric models of the coalescence region. \nTop to bottom: two truncated hemispheres are joined with a cylindrical neck of radius $r_{\\text{min}}$, with planar equipotentials (dashed lines) sandwiching the neck; the same geometry without the equipotentials; two hemispheres joined smoothly with a circular arc. \n(b) Electrical resistance versus $r_{\\text{min}}$ calculated numerically for $\\sigma=1$ $\\Omega^{-1}$m$^{-1}$ and $A=2$ mm. \nThe data from all three models are well described by $R_{\\text{CR}}=2\/(\\xi \\sigma r_{\\text{min}})+1\/(\\sigma \\pi A)$ (solid line: Eq.\\ (\\ref{conversion})), where $\\xi=3.62\\pm 0.05$ is a fitting parameter. \nFor small $r_{\\text{min}}$, the data follow $R_{\\text{CR}}=2\/(\\xi \\sigma r_{\\text{min}})$ (dashed line). \n}\n\\label{EStat}\n\\end{figure}\n\nThe conversion between $R_{\\text{CR}}$ and $r_{\\text{min}}$ is geometrical, and was determined numerically using the electrostatics calculation package EStat (FieldCo). \nTo assess the dependance of the conversion on the choice of the model, this conversion was calculated in three different ways, pictured in Fig.\\ \\ref{EStat}(a). \nFirst, the conversion by Case \\textit{et al.}\\ \\cite{Case2008,Case2009} was repeated, in which equipotentials are fixed on two planes that sandwich a cylindrical neck of radius $r_{\\text{min}}$ and height $r_{\\text{min}}^2\/A$, so that the drops and their connecting neck are treated as series contributions to the total resistance. \nThis calculation was compared with a second model with the same geometry but no such restriction on the field lines. \nIn the third model, the shape of the interface is given by a circular arc connecting two hemispheres smoothly. \nAs shown in Fig.\\ \\ref{EStat}(b), the three conversions agree within error bars, and the data are well described by:\n\\begin{equation}\nR_{\\text{CR}} = \\frac{2}{\\xi \\sigma r_{\\text{min}}} + \\frac{1}{\\sigma\\pi A},\n\\label{conversion}\n\\end{equation}\n\n\\noindent where $\\sigma$ is the electrical conductivity of the fluid and the dimensionless constant $\\xi=3.62\\pm 0.05$ is determined empirically. \n\nThe first term in the conversion, $2\/(\\xi \\sigma r_{\\text{min}})$, is twice the resistance of a hemisphere with an opening of radius $r_{\\text{min}}$. \nThe constant term in the conversion, $1\/(\\sigma\\pi A)$, can be understood as coming from the fluid neck itself. \n(This expression is the electrical resistance of a cylinder with radius $r_{\\text{min}}$ and height $r_{\\text{min}}^2\/A$, with equipotentials on its flat faces.) \nOther neck geometries (e.g., an overturned neck shape, which is predicted to occur in the inertial regime \\cite{Eggers2003, Fezzaa2008}) are expected to give the same conversion when $r_{\\text{min}}$ is small, since the dominant term in the resistance comes from the general feature of a conducting hemisphere with an opening of radius $r_{\\text{min}}$. \n\n\n\n\n\n\\subsection{Varying the liquid viscosity}\n\nFor the electrical measurements, the drops were mixtures of glycerol and water, with salt (NaCl) added to make the fluids electrically conductive. \nDe-ionized water was saturated with NaCl at room temperature and mixed with glycerol.\nEach mixture was characterized by measuring its density, surface tension, viscosity, and electrical conductivity. \nDensity was measured by weighing a known volume of fluid. \nSurface tension was measured by matching numerical solutions of the Young-Laplace equation to an image of a pendant drop. \nViscosity was measured with glass capillary viscometers (Cannon-Fenske). \nElectrical conductivity was determined by measuring the electrical impedance of a thin cylindrical channel filled with fluid, using the coalescence cell and measurement circuit. \n\nThe measured fluid parameters are shown in Fig.\\ \\ref{fluidParams}. \nBy changing the volume fraction of glycerol, the liquid viscosity was varied over two decades (from $1.9$ mPa s to $230$ mPa s) while the density and surface tension remained nearly constant, changing by factors of only $1.04$ and $1.6$, respectively. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{FluidParams.pdf} \n\\end{center}\n\\caption{\nFluid parameters for glycerol-water-NaCl mixtures used for electrical measurements. \n(a) Mass density, $\\rho$, is approximately constant over the range of mixtures used. \n(b) Surface tension, $\\gamma$, is approximately constant. \n(Aqueous NaCl has $\\gamma=88.5 \\pm 2$ mN\/m, which is higher than for pure water.) \n(c) Viscosity, $\\mu$, varies over a large range. \n(d) AC electrical conductivity, $\\sigma$ (at 1 to 10 MHz), decreases with increasing glycerol concentration. \n(e) AC electrical conductivity as a function of viscosity decreases slightly faster than $\\mu^{-1}$. \nThe low electrical conductivity at high viscosity sets the upper viscosity limit for the electrical method. \n}\n\\label{fluidParams}\n\\end{figure}\n\nAs shown in Fig.\\ \\ref{fluidParams}(e), the electrical conductivity decreases with increasing viscosity, which limits the experimental range of the viscosity of these mixtures with the electrical method. \nFor a fixed, dilute concentration of NaCl, the relationship would obey: $\\sigma\\propto \\mu^{-1}$. \nThis expression comes from combining the Nernst-Einstein law (which relates conductivity to the ionic diffusion coefficients, $D$, at low ionic concentration: $\\sigma\\propto D$) with the Stokes-Einstein equation ($D\\propto \\mu^{-1}$). \nThe conductivity falls off slightly faster than $\\mu^{-1}$, which is consistent with the lower concentration of NaCl in the mixtures as the glycerol fraction is increased.\n(There is another, smaller correction because the mixtures are not at low concentration, which has the opposite effect on the scaling.) \n\n\n\n\\subsection{High-speed imaging}\n\nA high-speed camera (Phantom v12, Vision Research) was used to observe other aspects of the coalescence dynamics, and to measure the neck radius versus time for silicone oils, which are non-conductive. \nThe drops were precisely aligned with respect to the line-of-sight of the camera.\nNeck radii were measured using an edge-locating analysis on the images.\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.3in]{VideoElectric_rmin.pdf} \n\\end{center}\n\\caption{\nNeck radius versus time for coalescing aqueous NaCl drops ($\\mu=1.88$ mPa s, $\\gamma=88.5$ mN\/m, $\\rho=1180$ kg\/m$^3$). \n(a) Data from the electrical method ($\\bullet$) and high-speed imaging ($\\circ$), where $t_0$ is determined from a simultaneous electrical measurement.\nThe two methods are in good agreement. \nThe electrical data extends to far earlier times. \n(b) Data from the same experiments, on linear-linear axes, showing every camera frame. \nBefore contact, the drop geometry and finite spatial resolution create an apparent neck of radius $110$ $\\mu$m.\nThe earliest imaging point that corresponds to the actual fluid neck is the third frame after $t_0$ ($\\tau=27.0$ $\\mu$s). \n}\n\\label{VideoElectric_rmin}\n\\end{figure}\n\nTo compare electrical measurements with high-speed imaging data, $r_{\\text{min}}$ was measured both electrically and optically for saturated aqueous NaCl drops. \nFor the optical data used in this comparison, $t_0$ was determined from a simultaneous electrical measurement, which was converted to the camera's time-base with a precision of $0.1$ $\\mu$s. \nAs shown in Fig.\\ \\ref{VideoElectric_rmin}(a), the two methods are in good agreement. \nThe comparison serves as a quantitative check on the electrical method, and additionally illustrates the dynamic range gained by the electrical method versus high-speed imaging. \n\nIn the current configuration, the dynamic range of high-speed imaging is determined by spatial resolution, as opposed to timing resolution.\nTo see this, observe that imaging a neck of radius $r_{\\text{min}}$ requires resolving a much smaller feature: the vertical gap between the drops, $r_{\\text{min}}^2\/A$. \nThus, the minimum observable neck radius is set by the condition that $r_{\\text{min}}^2\/A$ is approximately equal to the spatial resolution of the optical setup (i.e., the neck height limits measurements of the neck width). \nFor the experiment in Fig.\\ \\ref{VideoElectric_rmin}, the spatial resolution is 5.3 $\\mu$m\/pixel, so this estimate predicts that $r_{\\text{min}}$ can be seen down to $100$ $\\mu$m, which is consistent with the data. \n(To avoid this optical limitation, one can alternatively image \\textit{through} the neck, as was done in recent drop spreading experiments \\cite{Eddi2013_1}.) \n\nWhen comparing electrical and optical signals, a recent high-speed imaging study of coalescence reported a short delay ($20$ to $60$ $\\mu$s) between the moment of electrical contact (from an electrical trigger for their ultrafast camera) and the first visible motion of the neck \\cite{Thoroddsen2005}. \nThe apparent delay between the electrical signal and visualized motion is now easily accounted for by the period of time when the neck height is smaller than the optical resolution.\nThis explanation is also consistent with those authors' observation that the delay is shorter for smaller drops. \nTo illustrate this point, Fig. \\ref{VideoElectric_rmin}(b) compares electrical and optical measurements of the apparent neck size, $r_{\\text{min}}$. \nIndeed, the early-time optical data give a constant value of $110$ $\\mu$m, corresponding to the radius of the darkened region where the gap between the drops is smaller than the optical resolution. \n\n\n\n\n\n\\section{Purely viscous (Stokes) and inertial regimes}\n\\label{StokesInertialTheory}\n\nFor purely viscous Stokes flow in two dimensions (2D), an exact analytic solution of coalescence was given by Hopper \\cite{Hopper1984,Hopper1990,Hopper1993a,Hopper1993b}. \nThe shape of the fluid interface at any instant during coalescence is an inverse ellipse, given parametrically by: \n\\begin{subequations}\n \\label{Hopper_contour}\n \\begin{align}\n  r(\\theta) & = \\sqrt{2}A\\frac{(1-m^2)(1+m)\\cos \\theta}{\\sqrt{1+m^2}(1+2m \\cos 2\\theta +m^2)}, \\label{Hopper_contour_r} \\\\\n  z(\\theta) & = \\sqrt{2}A\\frac{(1-m^2)(1-m)\\sin \\theta}{\\sqrt{1+m^2}(1+2m \\cos 2\\theta +m^2)}, \\label{Hopper_contour_z}\n \\end{align}\n\\end{subequations}\n\n\\noindent where $0\\leq \\theta <2\\pi$, and the parameter $m$ is mapped to a neck radius by: \n\\begin{equation}\nr_{\\text{min}}=A\\sqrt{2}(1-m)\/\\sqrt{1+m^2}. \n\\label{rmin_vs_m}\n\\end{equation}\n\n\\noindent This family of curves interpolates between two kissing circles ($m=1$) and a single circle ($m=0$). \nFor small neck radius, these shapes limit to: \n\\begin{equation}\n(r^2+z^2)^2=4 A^2 z^2+r_{\\text{min}}^2 r^2.\n\\label{inverse_ellipse}\n\\end{equation}\n\nIn the solution, the neck radius is given as a function of time by:\n\\begin{equation}\n\\frac{\\gamma\\tau}{\\mu A} = \\frac{\\pi \\sqrt{2}}{4} \\int_{m^2}^1 \\frac{ds}{s (1+s)^{1\/2} K(s)},\n\\label{Hopper_exact}\n\\end{equation}\n\n\\noindent where $K(s)$ is the complete elliptic integral of the first kind, and $m$ is related to $r_{\\text{min}}$ by Eq.\\ (\\ref{rmin_vs_m}). \nThe asymptotic behavior of Eq.\\ (\\ref{Hopper_exact}) (in the limit that $r_{\\text{min}}\/A \\rightarrow 0$) is given by the simple expression: \n\\begin{equation}\n\\frac{\\gamma\\tau}{\\mu A} = \\frac{\\pi r_{\\text{min}}}{A} \\left| \\ln\\left(\\frac{r_{\\text{min}}}{8 A}\\right) \\right|^{-1}.\n\\label{Hopper_approx}\n\\end{equation}\n\nThe early-time asymptotic form of 2D Stokes coalescence was extended to three dimensions (3D) by Eggers \\textit{et al.}\\ \\cite{Eggers1999}. \nFor asymptotically small neck radius,\n\\begin{equation}\nr_{\\text{min}} = \\frac{\\gamma\\tau}{\\pi\\mu} \\left|\\ln\\left({\\frac{\\gamma \\tau}{\\mu A}}\\right)\\right|,\n\\label{viscScaling}\n\\end{equation}\n\n\\noindent which they report is a reasonable approximation for $r_{\\text{min}}\\lesssim 0.03 A$. \n\n\n\nFor inertially-dominated flows where the fluid viscosity is negligible, a scaling argument \\cite{Eggers1999} predicted that in this regime,\n\\begin{equation}\nr_{\\text{min}} = D_0 \\left( \\frac{\\gamma A}{\\rho} \\right)^{1\/4} \\tau^{1\/2},\n\\label{invScaling}\n\\end{equation}\n\n\\noindent where $D_0$ is a dimensionless prefactor.\nThis scaling was seen in numerical simulations, which report $D_0=1.62$ \\cite{Eggers2003}. \nHigh-speed imaging experiments \\cite{MenchacaRocha2001, Wu2004, Thoroddsen2005, Bonn2005, Fezzaa2008} and other numerical simulations \\cite{MenchacaRocha2001, Lee2006, Baroudi2014} have also observed this scaling regime and all report $D_0\\approx 1$. \n\n\n\n\n\n\\section{The inertially limited viscous (ILV) regime}\n\\label{ILVregime}\n\n\n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2012} showed that there is a third regime of liquid drop coalescence, which had been missed by previous experiments and was unanticipated by theory. \nThe regime arises because the analytical Stokes solution cannot apply at early times, because it violates a simple force-balance when the neck is small. \nNamely, the macroscopic motion of the drops inherent in the Stokes solution requires a larger force than the vanishingly small neck can provide. \nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} used simulation and experiment to show that at later times when the neck is larger, the Stokes regime is entered. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} called this regime the ``inertially limited viscous\" (ILV) regime, because the inertia of the drops prevents the Stokes solution from applying. \nIn the ILV regime, the neck radius is empirically found to follow: \n\\begin{equation}\nr_{\\text{min}} = C_0 \\frac{\\gamma}{\\mu}\\tau.\n\\label{ILV_Scaling}\n\\end{equation}\n\\noindent Previous experiments had observed this linear growth, but incorrectly assumed the drops to be in the Stokes regime \\cite{Bonn2005,Thoroddsen2005,Burton2007,Paulsen2011,Yokota2011}. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} used a force-balance argument to predict that for 3D drops, the Stokes regime is entered when: \n\\begin{equation}\nOh \\propto \\left| \\ln\\left(\\frac{1}{8}\\frac{r_{\\text{min}}}{A}\\right)\\right| \\left(\\frac{r_{\\text{min}}}{A}\\right)^{-1\/2}.\n\\label{phase_boundary_3D}\n\\end{equation} \nFig.\\ \\ref{phaseDiagram}(b) shows the phase diagram for liquid drop coalescence in 3D. \nThe ILV regime occupies an increasingly larger portion of the phase-space as $r_{\\text{min}}\/A\\rightarrow 0$.\nThus, surface tension, inertia, and viscosity combine to form the true asymptotic regime of liquid drop coalescence. \n(However, if the drop viscosity is extremely large or small, the range where the ILV regime occurs may be below atomic scales, and so coalescence will start in the Stokes or the inertial regime.) \n\nIn this section, I provide additional measurements and analysis of the ILV and Stokes regimes. \nThese measurements support the new picture of coalescence developed by Paulsen \\textit{et al.}\\ \\cite{Paulsen2012}. \n\n\n\n\n\n\n\n\n\n\\subsection{Change in velocity scaling}\n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} observed that the transition from the ILV regime to the Stokes regime would be accompanied by a change in the macroscopic velocity scaling of the drops. \nTo observe this macroscopic motion, they used a geometry where two pendant drops hang from nozzles and are translated horizontally to initiate contact on their equators. \nPaulsen \\textit{et al.}\\ \\cite{Paulsen2012} measured the velocity of a point on the back of one drop, $v_{\\text{b.o.d.}}$, as a probe of the global motion of the drops, thus identifying the phase boundary between the ILV and Stokes regimes. \nHere, I measure the center-of-mass velocity of each drop, $v_{\\text{c.o.m.}}$, and show that it gives consistent results. \n\nIn the ILV regime, a force balance argument \\cite{Paulsen2012} gives: \n\\begin{equation}\nv_{\\text{c.o.m.}} \\approx \\frac{3\\mu}{4A^3 \\rho} r_{\\text{min}}^2,\n\\label{vcom_ILV}\n\\end{equation}\n\n\\noindent In the Stokes regime, the 2D Stokes solution gives the asymptotic relationship: \n\\begin{equation}\nv_{\\text{c.o.m.}} \\approx \\frac{\\gamma}{2\\pi\\mu} \\left(\\frac{r_{\\text{min}}}{A}\\right) \\left|\\ln\\left(\\frac{1}{8}\\frac{r_{\\text{min}}}{A}\\right)\\right|,\n\\label{vcom_Stokes}\n\\end{equation}\n\\noindent which should apply for 3D drops as well \\cite{Eggers1999}.\n\nHigh-speed movies of the coalescing drops are analyzed to give the position of the center-of-mass of one drop, which is numerically differentiated to give $v_{\\text{c.o.m.}}$, and averaged to suppress noise. \n(Because the movies only give the planar drop contour, I calculate the center-of-mass of the shape that is given by revolving the contour of the bottom half of one drop around the axis passing through the center of both drops.) \nThe neck radius is measured directly from the same movie. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=2.8in]{Vcom_vs_rmin.pdf}\n\\end{center}\n\\caption{\n(Color online) \nILV-to-Stokes crossover. \n(a) Rescaled center-of-mass velocity of the drops, $v_{\\text{c.o.m.}}\\mu\/\\gamma$, versus $r_{\\text{min}}\/A$ at several viscosities. \nSolid line: asymptotic result from 2D Stokes theory, Eq.\\ (\\ref{vcom_Stokes}). \nThe data show super-linear growth at early times and merge onto the Stokes solution at late times. \nHigher-viscosity drops enter the Stokes regime at smaller neck radius. \n(b) The center-of-mass motion follows the motion of the back of one drop, here shown for $Oh=3.1$. \n}\n\\label{Vcom_vs_rmin}\n\\end{figure}\n\nFigure \\ref{Vcom_vs_rmin}(a) shows $v_{\\text{c.o.m.}}$ rescaled by the viscous-capillary velocity, $\\gamma\/\\mu$, versus the non-dimensional neck radius, $r_{\\text{min}}\/A$, for several viscosities. \nThe data capture both an early dynamics where the global drop velocity is growing approximately as $r_{\\text{min}}^2$ (as predicted for the ILV regime by Eq.\\ (\\ref{vcom_ILV})), and a late dynamics, where the data merge onto a master curve that is consistent with the Stokes theory, Eq.\\ (\\ref{vcom_Stokes}). \nThe higher the fluid viscosity, the earlier the transition into the Stokes regime. \nIn Fig.\\ \\ref{Vcom_vs_rmin}(b), $v_{\\text{c.o.m.}}$ is shown for one of the viscosities along with $v_{\\text{b.o.d.}}$ obtained from the same movie. \nThe two measurements are in good agreement. \nThis crossover in global drop motion marks the phase boundary between the ILV regime and the Stokes regime, which was reported in ref.\\ \\cite{Paulsen2012}, and is plotted in Fig.\\ \\ref{phaseDiagram}(b). \n\n\n\n\\subsection{Neck shapes}\n\nThe shape of the fluid neck connecting the coalescing drops offers another means of identifying the Stokes regime from the ILV regime. \nThe neck shapes were compared by Paulsen \\textit{et al.}\\ \\cite{Paulsen2012}, and a more detailed comparison is provided here. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=3.3in]{DropProfileCurvature.pdf}\n\\end{center}\n\\caption{\n(Color online) \nNeck shape versus $r_{\\text{min}}\/A$ and viscosity. \n(a-c) Neck shape for high viscosity (a) ($\\mu=58600$ mPa s, $Oh=370$) and intermediate viscosity (b,c) ($\\mu=96.6$ mPa s, $Oh=0.62$). \nAt high viscosity (a), the data agree with the exact Stokes-flow shapes (solid lines: Eq.\\ (\\ref{Hopper_contour})). \nAt intermediate viscosity (b), the neck is much broader, and the Stokes theory is a poor fit to the data. \nInstead, the data is well described by two spheres joined smoothly with a parabolic neck (c). \n(d) Neck shape similarity solutions versus rescaled coordinates, $\\tilde{r} \\equiv r\/r_{\\text{min}}$, $\\tilde{z} \\equiv z\/(r_{\\text{min}}^2\/A)$. \nDotted line: parabolic neck connected to spherical drops, Eq.\\ (\\ref{simsol_Parabola}). \nSolid line: Stokes solution, Eq.\\ (\\ref{simsol_Hopper}). \n(e) Dimensionless radius of curvature at neck minimum, $1\/\\kappa A$, versus $r_{\\text{min}}\/A$. \nHigh-viscosity data ($\\bullet$ $Oh=370$) agree with the Stokes theory (dashed line), which is approximated by Eq.\\ (\\ref{neck_curve}) with $C=1\/4$ at early times (solid line). \nIntermediate-viscosity data ($\\circ$ $Oh=0.62$) follow the result for a parabolic neck (dotted line: Eq.\\ (\\ref{neck_curve}) with $C=32\/27$). \n(f) Curvature scaling prefactor, $C$, measured at fixed radius ($0.1<r_{\\text{min}}\/A<0.2$), versus $Oh$.\nAs viscosity increases, the data transition from the value for a parabolic neck (dotted line: $32\/27$) to the Stokes theory (solid line: $1\/4$). \n}\n\\label{DropProfileFig}\n\\end{figure}\n\nIn Fig.\\ \\ref{DropProfileFig}(a) and (b), the exact shapes, Eq.\\ (\\ref{Hopper_contour}), are compared with experiments at $Oh=370$ and $Oh=0.62$.\nAt each viscosity, the neck shapes are shown at three different times. \nTo demonstrate that the regime can be identified by shape alone (without a knowledge of the time-dependance), I plot the shape that matches the minimum neck radius, $r_{\\text{min}}$, of the experimental data. \nThe Stokes solution agrees well with the high-viscosity data, where the drops have transitioned into the Stokes regime. \nHowever, it clearly fails to describe the shapes at $Oh=0.62$, where the drops are in the ILV regime, as shown in Fig.\\ \\ref{DropProfileFig}(b). \nIn this regime, the neck is broader, perhaps because the drops have not translated towards each other appreciably. \nInstead, the neck shapes in this regime can be described by two kissing spheres joined smoothly by a parabolic neck, as shown in Fig.\\ \\ref{DropProfileFig}(c). \n(The parabola is uniquely determined by specifying $r_{\\text{min}}$ and requiring that the parabolic region joins the two spheres continuously and with a continuous first derivative.)\n\nOn intermediate scales that are larger than the neck but smaller than the drops, the spherical drops are well-approximated by parabolas: $r=\\sqrt{2Az}$ for $r_{\\text{min}}\\ll r \\ll A$. \nA change of variables is made by rescaling the radial direction by the neck radius, $\\tilde{r} \\equiv r\/r_{\\text{min}}$, and the axial direction by the neck height, $\\tilde{z} \\equiv z\/(r_{\\text{min}}^2\/A)$. \nThis transformation collapses the profile onto itself: \n\\begin{equation}\n\\tilde{r}(\\tilde{z}) =\n\\begin{cases} \n1+\\frac{27}{64}\\tilde{z}^2, & \\tilde{z}< \\frac{8}{9}; \\\\\n\\sqrt{2 \\tilde{z}}, & \\tilde{z}\\geq \\frac{8}{9}. \\\\\n\\end{cases}\n\\label{simsol_Parabola}\n\\end{equation}\n\n\\noindent Thus, the parabolic-neck geometry is a similarity solution with radial scale $r_{\\text{min}}$ and axial scale $r_{\\text{min}}^2\/A$. \nThe solution is plotted in Fig.\\ \\ref{DropProfileFig}(d), with a point at $(\\tilde{z},\\tilde{r})=(8\/9,4\/3)$ marking where the neck merges onto the drops.\n(The same rescaled coordinates, $r\/r_{\\text{min}}$ and $z\/(r_{\\text{min}}^2\/A)$, were found to collapse the neck shapes in the inertial regime, for the case of neighboring spherical drops deposited on a substrate with a 90$^{\\circ}$ contact angle \\cite{Eddi2013_2}.) \n\nThe drop shapes in the Stokes regime also form a similarity solution near the neck with the same scaling. \nAs noted in ref.\\ \\cite{Eggers1999}, expansion of the drop contour near the neck for $z\\ll r_{\\text{min}}\\ll A$ (using Eq.\\ (\\ref{Hopper_contour}) or Eq.\\ (\\ref{inverse_ellipse})) gives $r(z)=(\\frac{1}{2}r_{\\text{min}}^2+\\sqrt{(\\frac{1}{2}r_{\\text{min}}^2)^2+4A^2z^2})^{1\/2}$. \nRescaling this shape yields:\n\\begin{equation}\n\\tilde{r}(\\tilde{z}) = \\left(\\frac{1}{2}+\\sqrt{\\frac{1}{4}+4\\tilde{z}^2}\\right)^{1\/2},\n\\label{simsol_Hopper}\n\\end{equation}\n\n\\noindent which is plotted in Fig.\\ \\ref{DropProfileFig}(d). \n\nAlthough I have only identified the self-similarity of the drop \\textit{shapes} during coalescence, it is possible that the flows share this self-similarity. \nIf so, then radial flows will scale with the neck radius, $r_{\\text{min}}$, and axial flows will scale with the neck height, $r_{\\text{min}}^2\/A$, in both the ILV and Stokes regimes. \nThis further solidifies a claim argued in ref.\\ \\cite{Paulsen2011} and in section \\ref{VIcrossover} that the flow scale in the axial direction is given by the neck height, $r_{\\text{min}}^2\/A$. \n\nThe interfacial curvature in the $(z,r)$ plane at the neck minimum, $\\kappa$, is a scalar quantity that distinguishes between the asymptotic neck shapes in the ILV and Stokes regimes. \nIn both regimes, the similarity solutions obey: \n\\begin{equation}\n\\frac{1}{\\kappa A}=C \\left(\\frac{r_{\\text{min}}}{A}\\right)^3,\n\\label{neck_curve}\n\\end{equation}\n \n\\noindent where $C=\\frac{32}{27}\\approx 1.19$ in the ILV regime, and $C=\\frac{1}{4}$ in the Stokes regime. \nThese predictions are in good agreement with the data shown in Fig.\\ \\ref{DropProfileFig}(e) for the two regimes. \n\nFigure \\ref{DropProfileFig}(f) shows $C$ at fixed radius ($0.1<r_{\\text{min}}\/A<0.2$) as a function of dimensionless viscosity, $Oh$. \nNear $Oh=1.5$, there is a transition between the values predicted for the Stokes and ILV regimes, marking the phase boundary at that radius. \n\n\n\n\\subsection{Effect of drop boundary condition}\n\nIn ref.\\ \\cite{Paulsen2012} and in this work, it has been shown that the Stokes and ILV regimes have different macroscopic drop motion. \nOne natural question is whether the boundary condition on the drops can influence this motion and determine the coalescence regime (e.g., is the Stokes regime precluded if the drops are fixed to rigid objects?). \nIn terms of the coalescence singularity, the crucial question is whether the boundary condition can affect the dynamics in the limit that $r_{\\text{min}}\/A\\rightarrow 0$. \n\n\\begin{figure}[tb]\n\\begin{center}\n\\includegraphics[width=3.2in]{NozzleGeomCompare.pdf}\n\\end{center}\n\\caption{\n(Color online) Effect of drop boundary conditions.\n(a) High-viscosity coalescence (silicone oil with $\\mu=58600$ mPa s, $Oh=370$) for vertically-aligned hemispherical drops formed on nozzles ($\\circ$) and for hanging pendant drops ($\\triangle$). \nAt early times, the neck growth does not depend on the boundary condition. \nThe data follow the Stokes solution (solid line: Eq.\\ (\\ref{Hopper_exact})). \n(b) Intermediate-viscosity coalescence in these two drop geometries (glycerol-water-NaCl with $\\mu=230$ mPa s ($\\circ$) and $\\mu=215$ mPa s ($\\triangle$); $Oh\\approx 1$).\nThe neck growth does not depend on the boundary condition. \nThe data follow the ILV scaling (dashed line: Eq.\\ (\\ref{ILV_Scaling})). \n}\n\\label{NozzleGeomCompare}\n\\end{figure}\n\nTo address this issue, I compare measurements of $r_{\\text{min}}$ under two contrasting boundary conditions: (i) the drops are hemispheres attached to fixed nozzles, and (ii) the drops are hanging and are coalesced at their equators, so that they can freely translate towards each other as they coalesce. \nI measure $r_{\\text{min}}$ with both boundary conditions at high viscosity and intermediate viscosity. \n\nFigure \\ref{NozzleGeomCompare}(a) compares high viscosity data with the 2D Stokes theory, Eq.\\ (\\ref{Hopper_exact}). \n(While the 2D and 3D theories match at early times, the 3D case does not make a prediction for $r_{\\text{min}}> 0.03 A$, where all of the data lie. \nTherefore, following ref.\\ \\cite{Paulsen2012}, the data are compared with the 2D exact analytic solution.) \nThe data follow the theory for small $r_{\\text{min}}\/A$; only at later times do the curves begin to depart from each other. \nFigure \\ref{NozzleGeomCompare}(b) shows that for intermediate-viscosity drops, the boundary condition has a negligible effect on the dynamics, and the data matches the neck scaling for the ILV regime, Eq.\\ (\\ref{ILV_Scaling}). \nThus, the phase diagram shown in Fig.\\ \\ref{phaseDiagram}(b) applies to coalescing drops that are fixed or free. \n\n\n\n\n\n\\section{Viscous-to-inertial crossover}\n\\label{VIcrossover}\n\nThus far, I have reported coalescence measurements in the Stokes and the ILV regimes, and I have observed the ILV-to-Stokes crossover by measuring the macroscopic motion of the drops. \nThe remaining component of the coalescence phase diagram is the viscous-to-inertial crossover (from the ILV regime to the inertial regime). \n\nRecently, Paulsen \\textit{et al.}\\ \\cite{Paulsen2011} used an ultrafast electrical method to measure this crossover for salt-water drops, and reported a major discrepancy with the theory \\cite{Eggers1999,Eggers2003}. \nWhereas the theory predicts a crossover time between these regimes of $t_c \\approx 0.7$ ns, the experiments show $t_c \\approx 2$ $\\mu$s. \nIn terms of the neck size, the crossover radius was predicted to be $r_c \\approx 30$ nm, whereas experiment showed $r_c \\approx 20$ $\\mu$m.\n\nTo investigate this discrepancy, experiments were carried out where the liquid viscosity was varied over a large range \\cite{Paulsen2011}. \nThe data was found to be consistent with a newly proposed Reynolds number for coalescence, which is based on a smaller length scale for the dominant flow gradients given by the neck height, $L=r_{\\text{min}}^2\/A$. \nHere, I provide additional measurements and analysis of the viscous-to-inertial crossover, which support the conclusions of ref.\\ \\cite{Paulsen2011}. \n\n\n\n\n\\subsection{Collapse of electrical data}\n\\label{PrelimComp}\n\n\n\n\n\nThe inset to Fig.\\ \\ref{muCollapse} shows $r_{\\text{min}}$ versus time for 4 viscosities, ranging from $1.9$ to $82$ mPa s, which were measured electrically. \nIn ref.\\ \\cite{Paulsen2011}, these data were rescaled to fall onto a master plot by rescaling the vertical and horizontal axes with free parameters, $r_c$ and $\\tau_c$, at each viscosity to produce the best collapse. \nI perform a different analysis here, in order to demonstrate that the results are not sensitive to the particular way in which the data collapse is obtained. \n\nIn ref.\\ \\cite{Paulsen2011}, after collapsing the data, it was noted that all of the data followed the simple interpolation:\n\\begin{equation}\n\\frac{r_{\\text{min}}}{r_c}=2 \\left(\\frac{1}{\\tau\/\\tau_c} + \\frac{1}{\\sqrt{\\tau\/\\tau_c}}\\right)^{-1}.\n\\label{interpolate}\n\\end{equation}\n\n\\noindent Here, I start with Eq.\\ (\\ref{interpolate}) and use it to collapse the data. \nFor each viscosity, Eq.\\ (\\ref{interpolate}) is fit to the data, where $r_c$ and $\\tau_c$ are fitting parameters. \nThe data is then rescaled by the $r_c$ and $\\tau_c$.\n\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{muCollapse.pdf}\n\\end{center}\n\\caption{\n(Color online) \n\\textit{Inset}: Neck radius versus time for glycerol-water-NaCl mixtures of different viscosities, from $1.9$ to $82$ mPa s. \nAt each viscosity, data from 5 or more coalescence events are logarithmically binned and averaged. \n\\textit{Main}: The data is collapsed by rescaling the x- and y-axes. \nRescaling parameters $\\tau_c$ and $r_c$ are obtained for each viscosity by fitting the data to Eq.\\ (\\ref{interpolate}) (solid line).\nThe collapsed data and the fit exhibit asymptotic behavior of $2\\tau\/\\tau_c$ (dotted line) at early times and $2\\sqrt{\\tau\/\\tau_c}$ (dashed line) at late times. \n} \n\\label{muCollapse}\n\\end{figure}\n\nFigure \\ref{muCollapse} shows the collapsed data, which fall cleanly onto a single curve given by Eq.\\ (\\ref{interpolate}). \nThe scaling parameters, $r_c$ and $\\tau_c$, determine the coefficients for the early- and late-time scaling laws, $C_0$ and $D_0$ (defined by eqns.\\ \\ref{ILV_Scaling} and \\ref{invScaling}, respectively). \nFigure \\ref{rcvsOh}(a) and (b) shows these coefficients as a function of dimensionless viscosity, $Oh$. \nThe ILV scaling prefactor, $C_0$, is of order 1 across the entire range of viscosity (although there is a slight increase in $C_0$ as the viscosity is increased). \nThe inertial scaling prefactor, $D_0$, is in good agreement with the value from numerical simulations \\cite{Eggers2003}, $D_0=1.62$. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center}\n\\includegraphics[width=3.4in]{rcOh.pdf} \n\\end{center}\n\\caption{\n(a,b) Measured dimensionless scaling-law prefactors, $C_0$ and $D_0$, versus $Oh$. \nIn (a), the dashed line is $C_0=1$. \nIn (b), the dashed line is the value from simulation \\cite{Eggers2003}: $D_0=1.62$. \n(c) Rescaled viscous-to-inertial crossover time versus $Oh$. \nThe dashed line shows $\\tau_c\/\\tau_{\\text{v}} = Oh^2$ (where $\\tau_{\\text{v}}= \\mu A\/\\gamma$ is the viscous timescale), as predicted in the literature \\cite{Eggers1999, Eggers2003}. \nClearly this is a poor description of the data. \nThe crossover radius proposed by ref.\\ \\cite{Paulsen2011} (with $\\tau_c\/\\tau_{\\text{v}} \\propto Oh$) is consistent with the data (solid line: Eq.\\ (\\ref{ourtc})). \n(d) Rescaled viscous-to-inertial crossover radius versus $Oh$.\nThe dashed line shows $r_c\/A = Oh^2$, which was proposed in the literature \\cite{Eggers1999, Eggers2003}.\nThis fails to capture the data. \nThe crossover radius proposed in this work describes the data well, with $r_c\/A \\propto Oh$ (solid line: Eq.\\ (\\ref{rcformula})). \nIn (a-d), the error bars are determined by the fits to Eq.\\ (\\ref{interpolate}). \n} \n\\label{rcvsOh}\n\\end{figure}\n\nFigure \\ref{rcvsOh}(c) shows the dimensionless crossover time, $\\tau_c\/\\tau_{\\text{v}}$, as a function of $Oh$ (where $\\tau_{\\text{v}}= \\mu A\/\\gamma$ is the viscous timescale). \nClearly, the accepted formula for the crossover time, $\\tau_c\/\\tau_{\\text{v}} \\approx Oh^2$, does not agree with the data. \nThe measurements are better described by a linear dependence on $Oh$. \n\nThe discrepancy between theory and experiment is also evident in the dimensionless crossover radius, $r_c\/A$, versus $Oh$, shown in Fig.\\ \\ref{rcvsOh}(d). \nThe predicted crossover radius is $r_c\/A \\approx Oh^2$, whereas the data follow $r_c\/A \\approx Oh$. \nThis suggests that the conventional Reynolds number for coalescence, $Re = \\rho \\gamma r_{\\text{min}}\/\\mu^2$, is wrong. \n\n\n\n\\subsection{Reynolds number for coalescence}\n\nThe viscous-to-inertial crossover can be estimated by the condition that the dimensionless Reynolds number for the flows, $Re=\\rho U L\/\\mu$, is of order unity (where $U$ and $L$ are characteristic velocity- and length-scales in the flows, respectively). \nAs was argued in ref.\\ \\cite{Paulsen2011}, the dominant flows in the viscous-to-inertial crossover correspond to a different Reynolds number than the one used in the literature \\cite{Eggers1999, Eggers2003, Wu2004, Yao2005, Thoroddsen2005, Bonn2005}. \nInstead of the conventionally used length scale given by the neck radius, $L=r_{\\text{min}}$, a much smaller length scale---the neck height, $r_{\\text{min}}^2\/A$---describes the size of the flow gradients. \n\nPaulsen \\textit{et al.}\\ \\cite{Paulsen2011} gave an estimate for the Reynolds number coming from the inertial side of the crossover. \nThey found that the crossover time, $\\tau_c$, is given by: \n\\begin{equation}\n\\frac{\\tau_c}{\\tau_{\\text{v}}} \\approx \\frac{64}{D_0^6} \\left(\\frac{\\mu}{\\sqrt{\\rho \\gamma A}}\\right) = \\frac{64}{D_0^6} \\mbox{ } Oh,\n\\label{ourtc}\n\\end{equation}\n\\noindent which is written here using the viscous timescale, $\\tau_{\\text{v}}$, and the Ohnesorge number. \nFigure \\ref{rcvsOh}(c) shows that this prediction is consistent with the crossover times measured in this work. \n\n\n\n\n\n\n\nA similar argument can be made coming from the viscous side of the crossover, which is presented here. \nIn the early (ILV) regime, the characteristic speed of the flows is $U=\\gamma\/\\mu$, and the characteristic length-scale is $L=r_{\\text{min}}^2\/(2A)$, since liquid from each drop moves in to advance the neck. \nUsing these scales, the Reynolds number is: $Re=\\rho\\gamma r_{\\text{min}}^2\/(2 A \\mu^2)$. \n\\noindent The dimensionless crossover radius, $r_c\/A$, is obtained by setting $Re=1$: \n\\begin{equation}\n\\frac{r_c}{A}\\approx \\sqrt{2}\\left(\\frac{\\mu}{\\sqrt{\\rho \\gamma A}}\\right)=\\sqrt{2}\\mbox{ }Oh.\n\\label{rcformula}\n\\end{equation}\n\n\\noindent Figure \\ref{rcvsOh}(d) shows that this prediction gives excellent agreement with the data\n\nIn Appendix \\ref{prevCross}, I compare the calculated crossover time, Eq.\\ (\\ref{ourtc}), with previous measurements of the viscous-to-inertial crossover in the literature. \n\n\n\n\\subsection{High-speed imaging collapse}\n\nHigh-speed videos of coalescence show that these results also capture the dependance of the crossover on surface tension. \nI coalesce glycerol-water-NaCl mixtures with viscosities ranging from 1.9 to 230 mPa s, and silicone oils with viscosities ranging from 0.82 to 97 mPa s. \n(The silicone oils are electrically insulating, and therefore cannot be measured with the electrical method.) \nUsing these liquids, the surface tension is varied by a factor of 5.\nThe liquids have $Oh<1$ so that the behavior should be described by the ILV regime and the inertial regime, but not the Stokes regime. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.1in]{videoCollapse.pdf} \n\\end{center}\n\\caption{\n(Color online)\nHigh-speed imaging of coalescence. \n\\textit{Inset:} Neck radius versus time for glycerol-water-NaCl mixtures ($\\mu=1.9$, 30, and 230 mPa s) and silicone oils ($\\mu=0.82$ and 97 mPa s). \nOther parameters are listed in the legend. \n\\textit{Main:} The data is collapsed by rescaling the axes with the crossover radius, $r_c$ (calculated with Eq.\\ (\\ref{rcformula})), and the crossover time, $\\tau_c$ (calculated with Eq.\\ (\\ref{ourtc})). \nThe rescaled data are consistent with Eq.\\ (\\ref{interpolate}) (solid line). \n} \n\\label{videoCollapse}\n\\end{figure}\n\nThe inset of Fig.\\ \\ref{videoCollapse} shows $r_{\\text{min}}$ versus $\\tau$ for these liquids. \nWhen the axes are rescaled with $r_c$ (given by Eq.\\ (\\ref{rcformula})) and $\\tau_c$ (given by Eq.\\ (\\ref{ourtc})), the data collapse to a master curve, shown in Fig.\\ \\ref{videoCollapse}. \nThe collapsed data follow Eq.\\ (\\ref{interpolate}), and therefore fall on the electrical data collapse, Fig.\\ \\ref{muCollapse}. \nThese experiments further solidify the new phase diagram for coalescence, shown in Fig.\\ \\ref{phaseDiagram}(b). \n\n\n\n\n\n\\section{Dynamics of drops during approach}\n\\label{approach}\n\n\\subsection{Drop deformation}\n\nThe experiments in this work were performed at ambient air pressure. \nBecause the drops approach at finite speed, they can be deformed by the viscous stresses in the air layer between them \\cite{Neitzel2002}. \nThis deformation could affect the subsequent coalescence dynamics. \nPrevious experiments \\cite{Case2008,Case2009} using the same electrical method suggest that deformation may be present for approach speeds as low as $10^{-4}$ m\/s. \n\nHere, aqueous NaCl drops are coalesced in air at an approach speed that is varied over $7$ orders of magnitude down to $17$ nm\/s, to examine the effects of the ambient gas during approach. \nTo achieve constant approach-speeds lower than $10^{-3}$ m\/s, a variable-speed syringe pump was used with a wide range of syringe sizes.\nThe approach speed, $U_{\\text{app}}$, was calculated based on the geometry and the flow rate. \nThe coalescence cell was fixed to a vibration-isolation table to suppress disturbances on the drops. \nFor high approach-speeds, a gravity-fed system was used: the bottom drop was fed by a reservoir held at a variable height above the coalescence cell, so that hydrostatic pressure caused the bottom drop to grow and impact the top drop. \nFor the gravity-fed system, $U_{\\text{app}}$ was measured directly with a high-speed camera. \n\nFigure \\ref{RCRva}(a) shows an image taken within one frame of $t_0$ for $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s. \nThe drops appear to be undeformed at the moment of contact. \nAt much higher approach-speed, the drops visibly deform before they merge, as shown in Fig.\\ \\ref{RCRva}(b), for $U_{\\text{app}}=3.3\\times 10^{-2}$ m\/s. \nThis transient non-coalescence is due to the pressure provided by the lubricating air layer between the drops. \nAlthough the drops appear undeformed in the low approach-speed case shown in Fig.\\ \\ref{RCRva}(a), the image does not rule out the possibility of a small flattened region at the drop tips. \n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{dropsRCRva.pdf} \n\\end{center}\n\\caption{\nOptical and electrical measurements of coalescence at low and high approach-speed. \n(a,b) Visual indications of drop deformation. \nDrops are shown within one frame of $t_0$ at low approach-speed: (a) $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s, and high approach-speed: (b) $U_{\\text{app}}=3.3\\times 10^{-2}$ m\/s. \nAt low approach-speed, the drops appear to coalesce as undeformed spheres, whereas the high approach-speed drops are flattened. \n(c,d) Electrical measurements corresponding to the experiments shown in (a,b). \n$R_{\\text{CR}}-R_0$ versus $\\tau$, where $R_0=1\/(\\sigma \\pi A)$. \nAt low approach-speed (c), the resistance follows $\\tau^{-1}$ (ILV scaling, dashed line) at early times and $\\tau^{-1\/2}$ (inertial scaling, dotted line) at late times. \nAt high approach-speed (d), the resistance follows $\\tau^{-0.72}$ at early times (solid line). \n}\n\\label{RCRva}\n\\end{figure}\n\nThe electrical method was used to access these small scales. \nFigure \\ref{RCRva}(c) shows electrical measurements of the coalescing drops for the low approach-speed case. \nThe data follow the behavior shown in earlier sections of this work (e.g., Fig.\\ \\ref{Schematic}(d)). \nHowever, for the high approach-speed case, the electrical measurements are qualitatively different, as shown in Fig.\\ \\ref{RCRva}(d). \nAt early times, the resistance appears to follow an approximate power-law with a scaling exponent of $-0.72$.\nAt late times, there is an abrupt crossover out of this scaling.\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=2.7in]{tauc_Cfinal_Uapp.pdf} \n\\end{center}\n\\caption{\n(a) Crossover time between the early and late electrical-resistance scalings versus approach speed. \nThe crossover time is constant for $U_{\\text{app}}<U_{\\text{app}}^*$, where $U_{\\text{app}}^*=(3 \\pm 1)\\times 10^{-4}$ m\/s. \nWhen $U_{\\text{app}}>U_{\\text{app}}^*$ (shaded region), $\\tau_c$ depends on the approach speed, and is delayed as $U_{\\text{app}}$ increases. \n(b) $C_{\\text{final}}$ versus approach speed. \n$C_{\\text{final}}$ is constant for $U_{\\text{app}}<U_{\\text{app}}^*$. \nFor $U_{\\text{app}}>U_{\\text{app}}^*$ (shaded region), $C_{\\text{final}}$ increases with $U_{\\text{app}}$, consistent with an increase in the flattening of the drops before they touch. \nThe data are averaged over 800 samples within the final 100 $\\mu$s before $t_0$, and $V_{\\text{in}} \\leq 275$ mV.  \n} \n\\label{Uapp}\n\\end{figure}\n\nA crossover time, $\\tau_c$, is measured by fitting the early- and late-time data to separate power-laws and determining the point of intersection of the fits. \n(This criteria is equivalent to fitting to Eq.\\ (\\ref{interpolate}) if the two scalings are the ILV and inertial scalings.) \nFigure \\ref{Uapp}(a) shows $\\tau_c$ versus approach speed. \nThe crossover time is insensitive to the drop approach-speed for $U_{\\text{app}}<3 \\times 10^{-4}$ m\/s. \nFor $U_{\\text{app}}>3 \\times 10^{-4}$ m\/s, the crossover time increases approximately linearly with $U_{\\text{app}}$, which is correlated with an increase in flattening in the high-speed videos. \nA threshold approach-speed, $U_{\\text{app}}^*=(3 \\pm 1)\\times 10^{-4}$ m\/s, separates the two behaviors. \n\nThe capacitance of the drops at the moment of contact, $C_{\\text{final}}\\equiv C_{\\text{CR}}(\\tau=0)$, should be sensitive to the amount of drop deformation as well.\nIn particular, $C_{\\text{final}}$ should grow with the area of the deformed region.\nFig.\\ \\ref{Uapp}(b) shows $C_{\\text{final}}$ versus approach speed. \nThe capacitance shows two behaviors, which fall on either side of the threshold approach-speed, $U_{\\text{app}}^*$. \nAt high approach-speed ($U_{\\text{app}}>U_{\\text{app}}^*$), $C_{\\text{final}}$ increases with $U_{\\text{app}}$ as the drops are increasingly deformed. \nAt low approach-speed ($U_{\\text{app}}<U_{\\text{app}}^*$), $C_{\\text{final}}$ is independent of $U_{\\text{app}}$, which is consistent with a picture where the drops are undeformed. \n\nThe crossover time and capacitance measurements suggest that drop deformation is absent at low approach-speed, since in the flattening scenario, one expects the amount of flattening to change with $U_{\\text{app}}$ \\cite{Law2007}.\nIn general, $U_{\\text{app}}^*$ may depend on the drop size, surface tension, the ambient fluid viscosity, and the drop viscosity. \n\n\n\\subsection{Inception of coalescence}\n\nTo probe the dynamics up to the moment of contact, I measure the capacitance of the coalescence region, $C_{\\text{CR}}$, during approach at low speed ($U_{\\text{app}}<U_{\\text{app}}^*$). \nThe separation of the drop surfaces is denoted by $z$ (see Fig.\\ \\ref{CfinalVa}(c)). \nThe mutual capacitance of the two hemispherical drops at small separation ($z\\ll A$) should be comparable to the capacitance of two conducting spheres, since the surfaces in close proximity contribute the most charge. \nThe latter arrangement can be solved exactly as a series expansion. \nFor small $z\/A$, the capacitance is given to a good approximation \\cite{Love1979} by: \n\\begin{equation}\nC_{\\text{spheres}}(z)\\approx \\pi\\epsilon_0 A \\left[\\ln\\left(\\frac{A}{z}\\right)+2.54\\right].\n\\label{Ceq}\n\\end{equation}\n\n\\begin{figure}[bt]\n\\centering \n\\begin{center} \n\\includegraphics[width=3.2in]{CversusMinusTau.pdf} \n\\end{center}\n\\caption{\n(Color online)\nDrop capacitance before coalescence for drops approaching at speed $U_{\\text{app}}=8.8\\times 10^{-5}$ m\/s $<U_{\\text{app}}^*$. \n(a) Capacitance of the coalescence region, $C_{\\text{CR}}$, versus time remaining until coalescence, $-\\tau$, with $V_{\\text{in}}=275$ mV and $f=10$ MHz. \nThe drops initiate contact at finite separation, shown by the excellent fit to Eq.\\ (\\ref{Ceq}) with $z=-U_{\\text{app}} \\tau+z_0$ using $z_0=160$ nm (solid line). \nFor comparison, the same equation is plotted with $z_0=0$ (dashed line). \nThe data is averaged and logarithmically binned over 12 experiments.\n(b) $C_{\\text{final}}$ versus applied voltage, $V_{\\text{in}}$. \nAt high voltage, the data is consistent with coalescence initiating when the peak electric field between the drops reaches a threshold value (dotted line: Eq.\\ (\\ref{Ceq}) with $z_0=V_{\\text{in}}\/E_{\\text{thresh}}$ using $E_{\\text{thresh}}=1.2$ MV\/m). \nAt lower voltage, $C_{\\text{final}}$ is consistent with a constant value that is independent of $V_{\\text{in}}$ (solid line: $C_{\\text{final}}=0.63 \\pm 0.05$ pF). \nEach data point is averaged over 800 samples within the final 100 $\\mu$s before $t_0$. \nThree measurements were made at each voltage to show the run-to-run variation. \n(c) Approach geometry: two drops of radius $A$ separated by distance $z$. \n} \n\\label{CfinalVa}\n\\end{figure}\n\nFigure \\ref{CfinalVa}(a) shows the capacitance of the coalescence region, $C_{\\text{CR}}$, measured as a function of the time remaining until coalescence, $-\\tau$, for aqueous NaCl drops approaching at low speed. \nThe capacitance grows logarithmically with $-\\tau$ at first but does not diverge at $\\tau=0$. \nThis behavior is consistent with the drops initiating contact at a finite separation, $z_0$. \nPlugging $z=-U_{\\text{app}} \\tau+z_0$ into Eq.\\ (\\ref{Ceq}) and using $z_0=160$ nm gives excellent agreement with the data. \n\n\nDespite the low applied voltage, the electric field between the drops can be very strong when the drop separation is small. \nThis is apparent when estimating the peak magnitude of the electric field between the drops during one AC cycle as $E_{\\text{max}}\\approx V_{\\text{in}}\/z$. \nOne expects the large electric field to attract the drops towards each other when they are very close, thus promoting coalescence and distorting the drop shapes before contact. \n\nTo test the effect of the applied voltage on the initiation of coalescence, $C_{\\text{final}}$ was measured as a function of applied voltage, $V_{\\text{in}}$. \nFigure \\ref{CfinalVa}(b) shows the results for $V_{\\text{in}}$ ranging from 2.5 mV to 5 V.\nFor $V_{\\text{in}}>0.3$ V, the data are consistent with the drops forming a connecting neck when the intervening electric field exceeds a threshold value, $E_{\\text{thresh}}$. \nThe data are fit well by $E_{\\text{thresh}}=1.2 \\pm 0.2$ MV\/m, using Eq.\\ (\\ref{Ceq}) for the capacitance of the drops as a function of the final separation, $z_0$, and substituting $z_0\\approx V_{\\text{in}}\/E_{\\text{thresh}}$. \nWhile this value is only slightly smaller than the approximate dielectric strength of air at large distances (3 MV\/m), the dielectric strength of air at these short distances is much greater \\cite{Townsend1915}.\n\nHaving argued that dielectric breakdown does not occur, I now address whether the applied voltage deforms the drops. \nA recent study measured the deformation of two nearby drops with an applied DC electric potential difference \\cite{Bird2009}. \nTheir experiments showed that the drops sharpen into cones, and they measured a cone angle of roughly $20^{\\circ}$ for a potential difference of $500$ V (where $0^{\\circ}$ corresponds to no deformation). \nTheir measurements of the cone angle are approximately linear for electric potentials between $0$ and $500$ V, suggesting that the cone angle would be less than $0.08^{\\circ}$ for the applied voltages used in this work ($V_{\\text{in}}\\leq 2$ V). \n(The angle is likely diminished even further since the measurements in this work are AC instead of DC.) \nThus, any deformation of the drops is expected to be on a small scale, although it could contribute to forming the initial microscopic neck for $V_{\\text{in}}>0.3$ V. \n\nFigure \\ref{CfinalVa}(b) shows that at lower voltages, $V_{\\text{in}}<0.3$ V, $C_{\\text{final}}$ is roughly constant within error, and the description invoking a threshold electric field is a poor fit. \nInstead, the data are consistent with a picture where Van der Waals forces initiate coalescence at finite separation when $V_{\\text{in}}$ is small.\nFor the data at low voltages, I measure $C_{\\text{final}}=0.63 \\pm 0.05$ pF, giving a best fit of $z_0=280$ nm. \nBecause the capacitance is logarithmic in drop separation, the experimental error on $z_0$ is large; the data are consistent with $z_0$ ranging from $120$ nm to $650$ nm. \n\n\n\n\n\\subsection{Initial neck size}\n\nFinally, I address the finite length and width of the liquid neck that is formed at the inception of coalescence. \nDue to the finite separation of the drops at the moment of contact, the separation between the drop interfaces at radius $r$ will be given by $r^2\/A+z_0$, instead of simply $r^2\/A$ as was assumed in previous sections. \nHowever, this correction becomes relatively smaller as $r_{\\text{min}}$ grows. \nNumerical simulations \\cite{Baroudi2013} where low-viscosity drops initiate contact by forming a small fluid neck at finite separation show that after a short delay, the dynamics converge onto the predicted scaling (i.e. Eq.\\ (\\ref{invScaling})). \n\nPrevious high-speed imaging studies have reported values for the initial finite radius of the fluid neck (referred to as $r_0$) at the inception of liquid drop coalescence in air. \nValues reported were $r_0=50$ $\\mu$m for $U_{\\text{app}} \\lesssim 0.1$ mm\/s (ref.\\ \\cite{Thoroddsen2005}) and $r_0=43.8 \\pm 4.3$ $\\mu$m for $U_{\\text{app}}=6.6$ mm\/s (ref.\\ \\cite{Fezzaa2008}). \nIn contrast, I measure $r_{\\text{min}}$ down to $0.7$ $\\mu$m at $\\tau=50$ ns for aqueous NaCl drops at low approach-speed.\nThis is a significantly smaller upper bound for the initial size of the neck for aqueous NaCl drops at low approach-speed.\n\n\n\n\n\n\\section{Conclusion}\n\nIn summary, I have presented supporting evidence for the new phase diagram for liquid drop coalescence in vacuum or air, developed in refs.\\ \\cite{Paulsen2012, Paulsen2011}. \nThe theoretically unanticipated inertially limited viscous regime was found to be the true asymptotic regime of coalescence for drops of any finite viscosity. \nIn this regime, surface-tension, viscosity, and inertia all balance. \nViscous drops ($Oh>1$) transition into the Stokes regime once the neck is sufficiently large to pull the drops towards each other. \nLow-viscosity drops ($Oh<1$) transition into the inertial regime at late times. \n\nIn the inertially limited viscous regime and the Stokes regime, the center-of-mass motion of the drops was found to track with the motion of the backs of the drops, further solidifying the force balance argument that identified the inertially limited viscous regime in ref.\\ \\cite{Paulsen2012}. \nThis work provides similarity solutions for the neck shapes in these two regimes, and the new phase diagram for coalescence was shown to apply for different boundary conditions. \n\nAdditional evidence was provided for the surprisingly late viscous-to-inertial crossover (from the inertially limited viscous regime to the inertial regime), including an alternative method of data-collapse, a Reynolds-number argument coming from the viscous side, and high-speed imaging experiments where the surface tension was varied. \nThe agreement of the new coalescence Reynolds number with the data supports the new picture for the flows, which must have a significant gradient on a small axial length scale set by the neck height, $r_{\\text{min}}^2\/A$. \n\nMany of the results are based on electrical measurements, which were shown to have an insignificant effect on the coalescence dynamics reported here. \nAt low approach-speed and low applied-voltage, the drops coalesce at finite separation as undeformed spheres. \n\nWhereas this work has established the behavior of liquid drop coalescence in vacuum or air, further work is needed to determine how an outer fluid with significant density or viscosity alters the coalescence phase diagram. \n\n\n\n\n\n\n\\begin{acknowledgments} \nI thank Sidney Nagel and Justin Burton for their guidance, support, and keen insight throughout this work. \nI am also particularly grateful to Santosh Appathurai, Osman Basaran, Sarah Case, Thomas Rosenbaum, Savdeep Sethi, and Wendy Zhang. \nI thank Michelle Driscoll, Efi Efrati, Nathan Keim, and Tom Witten for their assistance and for many illuminating discussions. \nThis work was supported by NSF Grant DMR-1105145 and by NSF MRSEC DMR-0820054. \n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:Int}\n\nExoplanets -- planets orbiting stars other than the Sun -- are most often identified through indirect means. We observe a star with periodic behaviour that would otherwise be unexpected and conclude that the best explanation is the presence of one (or more) planets. We direct the interested reader to \\cite{2018Perryman} for a summary of the various exoplanet detection techniques. By piecing together the observations of the unexpected behaviour it is possible to constrain, to some degree, the orbit and physical nature of the planets in question. Such inference is, however, not perfect -- particularly when the planets in question have been detected through observations of the ``wobble'' of their host star, as is the case for planets found using the radial velocity technique \\citep[e.g. ][]{51peg,47UMa,HarpsM,114613,3167c}, or candidate planets claimed on the basis of binary star eclipse timing variability \\citep[e.g. ][]{hwvir,nnser,silly1,uzfor}.\n\nThe accurate determination of the (minimum) masses and orbits of newly discovered exoplanets provides the key data by which we can understand the variety of outcomes of the planet formation process. As such, it behooves us to ensure that exoplanet catalogues contain information which is as accurate and realistic as possible. Such accurate solutions do not just enable us to properly ascertain the distribution of planets at the current epoch -- they also provide an important window into the history of the planetary systems we discover \\citep[e.g. ][]{2014Ford,2015PuWu,2017Fulton,2019Wu}, and allow us to predict and plan follow up observations through population synthesis models \\citep[e.g. ][]{2013Hasegawa,2018Mordasini,2020Dulz}. For example, the migration and mutual gravitational interaction of planets have been identified as being of critical importance to both the observed architectures and predicted long-term stability of the menagerie of known multi-planet systems heretofore identified through radial velocity and transit surveys \\citep[e.g.][]{prototrap,2012cWittenmyer,chain,trappist,2017Mustill,2017Hamers,2019Childs}. \n\nHowever, the accuracy of orbital parameters of the planetary companions presented in discovery works is frequently limited by the time period covered by the observations that led to the discovery, which are often enough to claim detection and little more \\citep{JupiterAnalogues,CoolJupiters}. Long term follow-up of known planet host systems is therefore desirable to refine the orbital parameters for known companions, to infer the presence of additional companions at lower masses and\/or larger semi-major axes \\citep[e.g.][]{2017BeckerAdams,EightSingle,RVDItech,181433,2019Denham,Rick19}, and to disentangle the complex signals produced by planets on resonant \\citep[e.g.][]{Ang10,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer} or eccentric orbits \\citep[e.g.][]{2013Wittenmyer,2017cWittenmyer}. Equally, due to the relative paucity of data on which planet discoveries are often based, it is possible for those initial solutions to change markedly as more data are acquired. The ultimate extension of this is that, on occasion, the process by which planetary solutions are fit to observational data can yield false solutions -- essentially finding local minima in the phase space of all possible orbital solutions that represent a good theoretical fit to the data whilst being unphysical. It is therefore important to check the dynamical feasibility of multi-planet solutions that appear to present a good fit to observational data -- particularly in those cases where such solutions invoke planets on orbits that offer the potential for close encounters between the candidate planets.\n\nFollowing this logic, we have in the past tested the stability of multi-planet systems in a variety of environments including around main sequence \\citep{2010Marshall,2012bWittenmyer,2015Wittenmyer,2017bWittenmyer}, evolved \\citep{2017aWittenmyer,2017cWittenmyer,2019Marshall}, and post-main sequence stars \\citep{2011Horner,2012Horner,QSVir,2012aWittenmyer,2013Mustill}. In some cases, our results confirmed that the proposed systems were dynamically feasible as presented in the discovery work, whilst in others, our analysis demonstrated that alternative explanations must be sought for the observed behaviour of the claimed ``planet-host'' star \\citep[e.g.][]{2011Horner,QSVir}. To ensure that our own work remains robust, we have incorporated such analysis as a standard part of our own exoplanet discovery papers. We test all published multi-planet solutions for dynamical stability before placing too great a confidence in a particular outcome. As an extension to this approach, we presented a revised Bayesian method to the previously adopted frequentist stability analysis in \\cite{2019Marshall}, and demonstrated the consistency between these approaches. \n\nRather than using direct dynamical simulations, the stability of a planetary system can also be inferred from a criterion derived from the planetary masses, semi-major axes, and conservation of the angular momentum deficit \\citep[AMD,][]{2000Laskar,2017Laskar}. AMD can be interpreted as measuring the degree of excitation of planetary orbits, with less excited orbits implying greater stability. The definition of AMD stability has been revised to account for the effect of mean motion resonances and close encounters on orbital stability \\citep{2017Petit,2018Petit}. Of the systems examined in this work, HD~110014 has been identified as being weakly stable, whilst HD~67087 and HD~133131A are both considered unstable according to AMD \\citep[see Figs. 6 and 7,][]{2017Laskar}. In our previous dynamical studies, we find good agreement between the stability inferred from AMD and our dynamical simulations with 13 systems in common between them, of which nine were classified unstable and four stable, one marginally so \\citep{2012Horner,2012Robertson,2012aWittenmyer,2012bWittenmyer,2012cWittenmyer,2014aWittenmyer,2014bWittenmyer,2015Wittenmyer,2016Wittenmyer,2016Endl}. In this paper we examine the dynamical stability of the three multi-planet systems, HD~67087, HD~110014, and HD~133131A, as a critical examination of their stability and a further test of the reliability of AMD for the identification of instability in exoplanet systems. \n\nHD~67087, observed as part of the Japanese Okayama Planet Search programme \\citep{2005Sato}, was discovered to host a pair of exoplanets by \\cite{2015Harakawa}. The candidate planets are super-Jupiters, with $m~\\sin i$ of 3.1\\,$M_{\\rm Jup}$\\ and 4.9\\,$M_{\\rm Jup}$, respectively.  They move on orbits with (${a,e}$) of ({1.08, 0.17}) and ({3.86, 0.76}), respectively, which would place the outer planet amongst the most eccentric Jovian planets identified thus far. The authors noted that the orbit and mass of the outer planet are poorly constrained.   \n\nHD\\,110014 was found to host a planet by \\citet{2009deMedeiros}; the second companion was identified through re-analysis of archival spectra taken by the FEROS instrument \\citep{1998KauferPasquini} looking to derive an updated orbit for planet b \\citep{2015Soto}.  The two candidate planets have super-Jupiter masses, and \\citet{2015Soto} cautioned that the proposed second planet was worryingly close in period to the typical rotation period of K giant stars. However, their analysis of the stellar photometry was inconclusive in identifying its activity as the root cause for the secondary signal. \n\nHD~133131A's planetary companions were reported in \\citet{2016Teske}, based on precise radial velocities primarily from the Magellan Planet Finder Spectrograph \\citep{2006Crane,2008Crane,2010Crane}. Their data supported the presence of two planets, where the outer planet is poorly constrained due to its long period. \\citet{2016Teske} ran a single dynamical stability simulation on the adopted solution and found it to remain stable for the full $10^5$ yr duration. The authors presented both a low- and high- eccentricity solution, reasoning that in a formal sense, the two solutions were essentially indistinguishable. They favoured the low-eccentricity model ($e_2=0.2$) for dynamical stability reasons. There is precedent in the literature for this choice, since it does happen that the formal best fit can be dynamically unfeasible whilst a slightly worse fit pushes the system into a region of stability \\citep[e.g.][]{2013Mustill, 2014Trifonov, 2017bWittenmyer}.  \n\nThe remainder of the paper is laid out as follows. We present a brief summary of the radial velocity observations and other data (e.g. stellar parameters) used for our reanalysis in Sect. \\ref{sec:Obs} along with an explanation of our modelling approach. The results of the reanalyses for each target are shown in Sect. \\ref{sec:Res}. A brief discussion of our findings in comparison to previous work on these systems is presented in Sect. \\ref{sec:Dis}. Finally, we present our conclusions in Sect. \\ref{sec:Con}.\n\n\\section{Observations and methods}\n\\label{sec:Obs}\n\n\\subsection{Radial Velocity Data}\n\nWe compiled radial velocity values from the literature for the three systems examined in this work; the origin of these data are summarised in Table \\ref{tab:rvs}. \n\n\\begin{table}\n\\centering\n\\caption{Table of references for the radial velocities data used in this work. \\label{tab:rvs}}\n\\begin{tabular}{ll}\n    \\hline\n    Target & References \\\\\n    \\hline\\hline\n    HD~67087   & \\cite{2015Harakawa} \\\\\n    HD~110114  & \\cite{2009deMedeiros} \\\\\n    HD~133131A & \\cite{2016Teske} \\\\\n    \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table*}\n\\centering\n\\caption{Planetary orbital parameters based on \\textit{Systemic} fits to radial velocity data. Semi-major axes were calculated using measured orbital periods and stellar masses taken from the NASA Exoplanet Archive. \\label{tab:systemic}}\n\\begin{tabular}{lcccccc}\n    \\hline\n    & HD\\,67087 b & HD\\,67087 c & HD\\,110014 b & HD\\,110014 c & HD\\,133131A b & HD\\,133131A c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] &  74.0~$\\pm$~3.0 & 54.0~$\\pm$~4.0 & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 352.3$\\pm$1.7 & 2380$^{+167}_{-141}$ & 877.5$\\pm$5.2 & 130.125$\\pm$0.096 & 648.$\\pm$3 & 5342$^{+7783}_{-2009}$ \\\\\n    Mean anomaly [deg] & 35$^{+20}_{-16}$ & 94$^{+28}_{-35}$ & 155$\\pm$4 & 231$\\pm$3 & 265$\\pm$11 & 188$^{+104}_{-123}$ \\\\\n    Longitude [deg] & 281$^{+18}_{-15}$ & 256 (fixed) & 41$\\pm$3 & 302$\\pm$4 & 16.6$^{+4.7}_{-4.5}$ & 110$^{+25}_{-43}$ \\\\\n    Eccentricity & 0.18$^{+0.07}_{-0.06}$ & 0.51 (fixed) & 0.259$\\pm$0.017 & 0.410$\\pm$0.022 & 0.340$\\pm$0.032 & 0.63$^{+0.25}_{-0.20}$ \\\\\n    $M$ sin $i$ [$M_{\\rm Jup}$] & 3.10$^{+0.15}_{-0.14}$ & 3.73$^{+0.47}_{-0.45}$ & 10.61$\\pm$0.25 & 3.228$\\pm$0.098 & 1.418$\\pm$0.036 & 0.52$^{+0.45}_{-0.17}$ \\\\\n    Semi-major Axis [au] & 1.08 & 3.87 & 2.32 & 0.65 & 1.44 & 5.88 \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Results from \\textsc{Astroemperor} exploration of parameter space around \\textit{Systemic} nominal best fit values for planetary companions to HD~133131A and HD~110014. \\label{tab:mcmc}}\n\\begin{tabular}{lcccc}\n    \\hline\n    & HD\\,133131A b & HD\\,133131A c & HD\\,110014 b & HD\\,110014 c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 647.816$\\pm$1.575 & 3205.648$\\pm$948.063 & 865.206$\\pm$6.170 & 132.431$\\pm$0.279 \\\\\n    Phase [deg] & 261.620$\\pm$4.850 & 31.734$\\pm$98.433 & 43.753$\\pm$72.179 & 341.373$\\pm$64.346 \\\\\n    Longitude [deg] & 18.550$\\pm$2.165 & 113.777$\\pm$81.302 & 146.633$\\pm$17.229 & 236.903$\\pm$16.452 \\\\\n    Eccentricity & 0.341$\\pm$0.021 & 0.263$\\pm$0.145 & 0.011$\\pm$0.015 & 0.294$\\pm$0.076 \\\\\n    M sin $i$ [$M_{\\rm Jup}$]& 1.428$\\pm$0.099 & 0.388$\\pm$0.124 & 10.622$\\pm$0.757 & 2.581$\\pm$0.247 \\\\\n    Semimajor Axis [au] & 1.435$\\pm$0.046 & 4.153$\\pm$0.800 & 2.350$\\pm$0.075 & 0.668$\\pm$0.023 \\\\\n    \\hline\n    Jitter [\\mbox{m\\,s$^{-1}$}] & 3.557$\\pm$1.254 & 0.466$\\pm$0.419 & 6.060$\\pm$1.856 & 13.350$\\pm$1.492 \\\\\n    Offset [\\mbox{m\\,s$^{-1}$}] & -9.333$\\pm$4.787 & 12.321$\\pm$7.543 & 52.575$\\pm$4.737 & 72.198$\\pm$4.541 \\\\\n    MA coefficient & 0.714$\\pm$0.531 & 0.466$\\pm$0.419 & 0.697$\\pm$0.214 & 13.350$\\pm$1.492 \\\\\n    MA Timescale [days] & 4.158$\\pm$2.815 & 12.321$\\pm$7.543 & 9.793$\\pm$2.488 & 72.198$\\pm$4.541 \\\\\n    Acceleration [\\mbox{m\\,s$^{-1}$}\/yr] & -1.435 & & -21.620 & \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Modelling}\n\nTo test the dynamical stability of these proposed planetary systems, we follow the updated dynamical methodology outlined in our previous work \\citep[][]{2017bWittenmyer,2019Marshall}.\n\nIn brief, we perform a fit to the published velocity data using the \\textit{Systemic Console} \\citep{2009Meschiari}. We then use the MCMC tool within \\textit{Systemic} to explore the parameter space about the best fit. The MCMC chain runs for $10^7$ steps, discarding the first 10,000, and we then draw the trial solutions for our dynamical stability simulations from these posteriors. Using this data, we populate three ``annuli'' in $\\chi^2$ space corresponding to the ranges $0-1\\sigma$, $1-2\\sigma$, and $2-3\\sigma$ from the best fit.  Each annulus contains 42,025 unique realisations drawn from the MCMC chain. The innermost annulus was drawn from the lowest 68.3 per cent of all $\\chi^2$ values, the middle annulus contained the next best 27.2 per cent of values, and the outer annulus contained the worst 4.5 per cent of solutions (i.e. those falling $2-3\\sigma$ away from the best fit). The result is a set of ``clones'' which fall within $3\\sigma$ of the best-fit solution, thus representing a reasonable region of parameter space within which we explore the dynamical stability of the proposed planetary system, using the constraints afforded by the existing observational data.\n\nWe then proceed to perform lengthy dynamical simulations of each of the 126,075 solutions generated by this method. We used the Hybrid integrator within the $n$-body dynamics package \\textit{Mercury} \\citep{1999Chambers} to integrate the solutions forwards in time for a period of 100 Myr. The simulations are brought to a premature end if either of the planets being simulated is ejected from the system, is flung in to the central star, or if the two planets collide with one another. When such events occur, the time at which the collision or ejection occurred is recorded, giving us the lifetime for that particular run. As such, our suite of simulations yield 126,075 tests of the candidate planetary system, allowing us to study how its stability varies as a function of the particular details of the solution chosen to explain the observational data.\n\nWe determine the best-fit parameters and uncertainties for each system using the code Exoplanet Mcmc Parallel tEmpering Radial velOcity fitteR\\footnote{\\href{https:\/\/github.com\/ReddTea\/astroEMPEROR}{https:\/\/github.com\/ReddTea\/astroEMPEROR}} ({\\sc astroEMPEROR}), which uses thermodynamic methods combined with MCMC. Our approach has previously been established and described in \\cite{2019Marshall} and \\cite{EightSingle}. We summarise the input values and constraints used in the fitting presented in this work for the sake of reproducibility. Given that our goal was to test the feasibility of the exoplanetary systems as presented in the literature, we restricted {\\sc astroEMPEROR} to consider zero, one, or two planetary signals in the radial velocity data; dynamical configurations with additional planetary companions in orbits that could mimic a single planetary companion, e.g. two resonant planets looking like a single eccentric planet (for a total of three planetary companions), were not considered in this analysis. The planetary fitting parameters were the orbital period ($P$), line-of-sight mass ($M\\sin i$), orbital eccentricity ($e$), longitude of periastron ($\\omega$), and mean anomaly ($M$). We also include an additional jitter term when fitting the data. We initialised the locations of the walkers in the MCMC fitting at their best-fit values from the \\textit{Systemic} console fit, plus a small random scatter. The priors on each parameter were flat and unbounded i.e. with uniform probability between $\\pm\\infty$, except for the orbital eccentricities which had folded Gaussian priors, and the jitter term, which was a Jeffries function (but still unbound between $\\pm\\infty$). The parameter space was surveyed by 150 walkers at five temperatures over 15\\,000 steps, with the first 5\\,000 steps being discarded as the burn-in phase.\n\n\\section{Results}\n\\label{sec:Res}\n\n\\subsection{HD~67087}\n\nThe HD~67087 system is catastrophically unstable, as illustrated by the results of our stability analysis in Fig. \\ref{fig:HD67087_stability}. In this plot it is clear that the most stable solutions cluster toward the largest ratios of semi-major axes, and the smallest eccentricities. Even in this limit, the longest-lived solutions that plausibly represent the observations are still only stable for 10$^{6}$ yrs, out of a total integration time of 10$^{8}$ yrs. This leads us to the interpretation that the HD~67087 system, as inferred from the available radial velocity data, is dynamically infeasible. Given this high degree of instability, we do not attempt to determine a global best-fit solution for the system parameters.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD67087_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD67087_MRatio_MaxEcc_200_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~67087 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~67087b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~67087b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find no stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets, with the only stable solutions lying at the extreme edges of the parameter space toward low eccentricities, large separations and low mass ratios. \\label{fig:HD67087_stability}}\n\\end{figure*} \n\n\\subsection{HD~110014}\n\nThe HD~110014 system is found to be dynamically stable, with a broad swathe of parameter space centred on the nominal solution producing system architectures that last for the full 10$^{8}$ yrs of our dynamical integrations. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD110014_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD110014_confusogram}.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD110014_MaxEcc_ARatio_1800_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD110014_MRatio_MaxEcc_1800_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~110014 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~110014b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~110014b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets. \\label{fig:HD110014_stability}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\t\\includegraphics[width=\\textwidth]{HD110014Triangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~110014 b and HD~110014 c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD110014_confusogram}\n\\end{figure*}\n\n\\subsection{HD~133131A}\n\nThe HD~133131A system shows a very complex parameter space in the stability plots. As one would expect, the stability of the system generally increases towards lower orbital eccentricities and lower mass ratios between the two planetary components. The overall stability appears to be insensitive to the ratio of the semi-major axes for the planets, with long-lived solutions possible across the full range of values probed for this parameter. Interestingly, we demonstrate that stable architectures for the planetary system exist in both the high and low orbital eccentricity scenarios for the system. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD133131A_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD133131A_confusogram}. Further observations to refine the planet properties of this system will be required to definitively characterise its dynamical stability.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MRatio_MaxEcc_200_45.png}\\\\\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MRatio_MaxEcc_200_45.png}\n    \\caption{Plots of the dynamical stability of the HD~133131A planetary system for both the high eccentricity (top) and low eccentricity (bottom) orbital solutions. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~133131Ab and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~133131Ab and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. The stability revealed by our dynamical simulations is complex, with regions of both extreme stability (log(lifetime) $\\sim$ 100 Myrs) and instability (log(lifetime) $\\sim$ 100 yrs) lying within the 3-$\\sigma$ reach of the nominal best-fit orbital parameters.  \\label{fig:HD133131A_stability}}\n\\end{figure*} \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{HD133131ATriangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~133131A b and HD~133131A c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD133131A_confusogram}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{sec:Dis}\n\nThe results of our dynamical modelling for the three systems considered in this work, HD~67087, HD~110014 and HD~133131A show three distinctly different outcomes. For the first system tested, HD~67087, we find no orbital solutions that exhibit long-term dynamical stability. As a result, we are forced to conclude that, if the planets proposed to orbit that star are real, they must move on orbits significantly different from those proposed in the discovery work, and sampled in our simulations. It seems likely that new radial velocity observations of HD~67087, extending the temporal baseline over which the star has been observed, will yield fresh insights to the system -- either significantly constraining and altering the proposed orbit for the outermost planet, or even revealing that that eccentric solution is in fact the result of multiple unresolved planets at large orbital radii. Such an outcome is far from unusual -- and indeed, it is often the case that, with more data, a single eccentric planet seen in RV data is resolved to actually be two planets moving on near circular orbits \\citep[e.g.][]{2013Wittenmyer,EightSingle}. For now, however, we can do no more than to call the existence of HD~67087~c into question, pending the acquisition of such additional data.\n\nIn contrast to the instability of HD~67087, our simulations of the HD~110014 system reveal that the best-fit solution for that two-planet system lies in a broad region of strong dynamical stability. In this case, our simulations simply reveal that the system, as proposed in the discovery work, is dynamically feasible -- and in a sense, the simulations add little beyond that.\n\nThe case of HD~133131A is somewhat more interesting. Here, our simulations reveal that solutions that fit the observational data can exhibit both strong dynamical stability, and extreme instability (with dynamical lifetimes of just a few years). Both the high- and low-eccentricity solutions considered in \\citet{2016Teske} can produce scenarios that are stable for the full 100 Myr of our simulations. In both the high- and low-eccentricity cases, the stable solutions cluster around the least eccentric available scenarios. The more widely separated the two planets, the more eccentric their orbits can be before instability occurs -- a natural result of the stability being driven by the minimum separation between the planets, rather than their orbital semi-major axes. The more widely the semi-major axes of the orbits are spaced, the more eccentric they must be to bring the planets into close proximity. These results show once again the benefits inherent to such dynamical analysis -- reminding us how studying the dynamical evolution of a given system can help to provide stronger constraints on the orbits of the planets contained therein than is possible by studying the observational data on their own.\n\nA comparison of our results to the analysis of the AMD stability criterion presented in \\cite{2017Laskar} shows agreement between the two different techniques for the dynamical stability of the three systems. Whilst HD~67087 and HD~110014 are respectively very clear cut cases of an unstable and a stable system, HD~133131A exhibits a more complex behaviour. HD~133131A may be dynamically stable, but the inferred lifetime for the planetary system as proposed is sensitive to the chosen initial conditions; this system therefore represents an edge case of stability where limitations of available data and the respective analyses provide no clear answer to the veracity of the previously inferred planetary system.\n\nCombining these new results with our previous dynamical analyses, as summarised in the introduction, we may consider that the AMD criterion is a reliable estimator of stability for planetary systems. There are 13 systems (out of 131 considered in that work) from \\cite{2017Laskar} that have had dynamical modelling of their stability. In \\cite{2017Laskar}, a planetary system is considered strongly stable if all planet-pairs have $\\beta$ values less than 1, such that collisions are impossible whilst weakly stable planetary systems are those in which the inner-most planet might collide with the star without disrupting the remainder of the planetary system. In five systems, both the AMD criterion and dynamical modelling agree on their dynamical stability (HD~142, HD~159868, NN Ser (AB), GJ~832, and HD~110014); the planets in each of these systems are dynamically well separated and therefore not strongly interacting \\citep[][this work]{2011Horner,2012bWittenmyer,2014bWittenmyer}. Six systems are unstable according to the AMD criterion with values of $\\beta$ in the range 1 to 5 for the planet pair (HD~155358, 24 Sex, HD~200964, HD~73526, HD~33844, HD~47366), but all are in mean motion resonances and have been demonstrated to be dynamically stable through $n$-body simulations \\citep{2012Robertson,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer,2019Marshall}. The remaining two systems (HD~67087, HD~133131A) are dynamically unstable in both the AMD and dynamical analysis (this work). However, dynamical analysis of the HD~133131A system reveals regions of dynamical stability consistent with the observed radial velocities, prompting the need for further investigation of this system and its architecture. Neither of these two unstable planetary systems have $\\beta$ values radically different from those of the planetary systems in resonance, or each other, such that determining their stability can only be carried out using dynamical simulations. The existence of such systems in the known planet population as demonstrated in our analysis therefore showcases the necessity of performing long duration dynamical analyses of proposed planetary system architectures to reveal the complex dynamical interplay between high mass planets, the evolution of their orbital elements, and determine what constraints this places on the available parameter space for the endurance of the proposed planetary system over its lifetime.\n\n\\section{Conclusions}\n\\label{sec:Con}\n\nWe re-analysed the dynamical stability of the exoplanet systems around HD~67087, HD~110014, and HD~133131A, using available radial velocity data. These three planetary systems have poorly constrained orbital parameters, and had previously been identified as being potentially unstable. We combine a determination of the best-fit orbital parameters from least-squares fitting to the data with $n$-body simulations to determine the global best-fit solution for the planetary system architectures, and thereafter determine the probability distribution of the orbital solutions through Bayesian inference. \n\nOur dynamical analysis confirms that the published planetary system parameters for HD~67087bc are dynamically unstable on very short timescales, and we must conclude that the system, as published, is dynamically unfeasible. As more data are collected for the HD~67087 system, it seems likely that the true nature of the candidate planets therein will be revealed, and that future planetary solutions for that system will veer towards dynamical stability as the planetary orbits become better constrained. \n\nIn the case of HD~110014 bc we demonstrate that the system parameters can be dynamically stable for the full duration of our 100 Myr integrations. The third system, HD~133131A , exhibits much more complex behaviour, with HD~133131A bc being strongly unstable over much of the parameter space exhibited in this work including the region encompassing the nominal best-fit to the orbital parameters. In agreement with previous analysis of this system, we strongly disfavour a high eccentricity orbital solution for planet c. Additional observations of this system will be required to more precisely determine the planetary properties for HD~133131A bc and thereby categorically rule on the plausibility of the proposed planetary system.\n\nThese results demonstrate the complementarity of various techniques to deduce the stability of planetary systems, with good agreement between the results of our various works, and that of the AMD approach. We highlight the appropriateness of dynamical simulations for determination the long-term stability of planetary systems in the presence of strongly interacting planets, which although costly in a computing sense capture the full essence of planetary interaction in such systems which is not possible with other techniques. We finally assert that the orbital parameters for these three systems which have been determined in this work (as summarised in Table 3) should be the accepted values adopted by exoplanet archives or elsewhere. This work is thus one additional thread in the tapestry of cross-checking of published results through various means that ensures the reliability of archival information on planetary properties and the architectures of planetary systems which are essential to inform models of the formation and evolution of the exoplanet population \\citep[e.g. ][]{2019Childs,2019Denham,2020He,2020VolkMalhotra}.\n\n\\section*{Acknowledgements}\n\nThis is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS  following peer review. The version of record Marshall et al., 2020, MNRAS, 494, 2, 2280--2288 is available online \\href{https:\/\/academic.oup.com\/mnras\/article\/494\/2\/2280\/5819459}{here}.\n\nWe thank the anonymous referee for their comments which helped to improve the article.\n\nThis research has made use of NASA's Astrophysics Data System and the SIMBAD database, operated at CDS, Strasbourg, France.\n\nJPM acknowledges research support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and Academia Sinica under grant AS-IA-106-M03.\n\n\\textit{Software}: This research has made use of the following Python packages: \\textsc{matplotlib} \\citep{2007Hunter}; \\textsc{numpy} \\citep{2006Oliphant}; \\textsc{pygtc} \\citep{2016Bocquet}; \\textsc{emcee} \\citep{2013ForemanMackey}; \\textsc{corner} \\citep{2016ForemanMackey}; \\textsc{mercury} \\citep{1999Chambers}.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n\n\\section{Introduction}\n\\label{sec:Int}\n\nExoplanets -- planets orbiting stars other than the Sun -- are most often identified through indirect means. We observe a star with periodic behaviour that would otherwise be unexpected and conclude that the best explanation is the presence of one (or more) planets. We direct the interested reader to \\cite{2018Perryman} for a summary of the various exoplanet detection techniques. By piecing together the observations of the unexpected behaviour it is possible to constrain, to some degree, the orbit and physical nature of the planets in question. Such inference is, however, not perfect -- particularly when the planets in question have been detected through observations of the ``wobble'' of their host star, as is the case for planets found using the radial velocity technique \\citep[e.g. ][]{51peg,47UMa,HarpsM,114613,3167c}, or candidate planets claimed on the basis of binary star eclipse timing variability \\citep[e.g. ][]{hwvir,nnser,silly1,uzfor}.\n\nThe accurate determination of the (minimum) masses and orbits of newly discovered exoplanets provides the key data by which we can understand the variety of outcomes of the planet formation process. As such, it behooves us to ensure that exoplanet catalogues contain information which is as accurate and realistic as possible. Such accurate solutions do not just enable us to properly ascertain the distribution of planets at the current epoch -- they also provide an important window into the history of the planetary systems we discover \\citep[e.g. ][]{2014Ford,2015PuWu,2017Fulton,2019Wu}, and allow us to predict and plan follow up observations through population synthesis models \\citep[e.g. ][]{2013Hasegawa,2018Mordasini,2020Dulz}. For example, the migration and mutual gravitational interaction of planets have been identified as being of critical importance to both the observed architectures and predicted long-term stability of the menagerie of known multi-planet systems heretofore identified through radial velocity and transit surveys \\citep[e.g.][]{prototrap,2012cWittenmyer,chain,trappist,2017Mustill,2017Hamers,2019Childs}. \n\nHowever, the accuracy of orbital parameters of the planetary companions presented in discovery works is frequently limited by the time period covered by the observations that led to the discovery, which are often enough to claim detection and little more \\citep{JupiterAnalogues,CoolJupiters}. Long term follow-up of known planet host systems is therefore desirable to refine the orbital parameters for known companions, to infer the presence of additional companions at lower masses and\/or larger semi-major axes \\citep[e.g.][]{2017BeckerAdams,EightSingle,RVDItech,181433,2019Denham,Rick19}, and to disentangle the complex signals produced by planets on resonant \\citep[e.g.][]{Ang10,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer} or eccentric orbits \\citep[e.g.][]{2013Wittenmyer,2017cWittenmyer}. Equally, due to the relative paucity of data on which planet discoveries are often based, it is possible for those initial solutions to change markedly as more data are acquired. The ultimate extension of this is that, on occasion, the process by which planetary solutions are fit to observational data can yield false solutions -- essentially finding local minima in the phase space of all possible orbital solutions that represent a good theoretical fit to the data whilst being unphysical. It is therefore important to check the dynamical feasibility of multi-planet solutions that appear to present a good fit to observational data -- particularly in those cases where such solutions invoke planets on orbits that offer the potential for close encounters between the candidate planets.\n\nFollowing this logic, we have in the past tested the stability of multi-planet systems in a variety of environments including around main sequence \\citep{2010Marshall,2012bWittenmyer,2015Wittenmyer,2017bWittenmyer}, evolved \\citep{2017aWittenmyer,2017cWittenmyer,2019Marshall}, and post-main sequence stars \\citep{2011Horner,2012Horner,QSVir,2012aWittenmyer,2013Mustill}. In some cases, our results confirmed that the proposed systems were dynamically feasible as presented in the discovery work, whilst in others, our analysis demonstrated that alternative explanations must be sought for the observed behaviour of the claimed ``planet-host'' star \\citep[e.g.][]{2011Horner,QSVir}. To ensure that our own work remains robust, we have incorporated such analysis as a standard part of our own exoplanet discovery papers. We test all published multi-planet solutions for dynamical stability before placing too great a confidence in a particular outcome. As an extension to this approach, we presented a revised Bayesian method to the previously adopted frequentist stability analysis in \\cite{2019Marshall}, and demonstrated the consistency between these approaches. \n\nRather than using direct dynamical simulations, the stability of a planetary system can also be inferred from a criterion derived from the planetary masses, semi-major axes, and conservation of the angular momentum deficit \\citep[AMD,][]{2000Laskar,2017Laskar}. AMD can be interpreted as measuring the degree of excitation of planetary orbits, with less excited orbits implying greater stability. The definition of AMD stability has been revised to account for the effect of mean motion resonances and close encounters on orbital stability \\citep{2017Petit,2018Petit}. Of the systems examined in this work, HD~110014 has been identified as being weakly stable, whilst HD~67087 and HD~133131A are both considered unstable according to AMD \\citep[see Figs. 6 and 7,][]{2017Laskar}. In our previous dynamical studies, we find good agreement between the stability inferred from AMD and our dynamical simulations with 13 systems in common between them, of which nine were classified unstable and four stable, one marginally so \\citep{2012Horner,2012Robertson,2012aWittenmyer,2012bWittenmyer,2012cWittenmyer,2014aWittenmyer,2014bWittenmyer,2015Wittenmyer,2016Wittenmyer,2016Endl}. In this paper we examine the dynamical stability of the three multi-planet systems, HD~67087, HD~110014, and HD~133131A, as a critical examination of their stability and a further test of the reliability of AMD for the identification of instability in exoplanet systems. \n\nHD~67087, observed as part of the Japanese Okayama Planet Search programme \\citep{2005Sato}, was discovered to host a pair of exoplanets by \\cite{2015Harakawa}. The candidate planets are super-Jupiters, with $m~\\sin i$ of 3.1\\,$M_{\\rm Jup}$\\ and 4.9\\,$M_{\\rm Jup}$, respectively.  They move on orbits with (${a,e}$) of ({1.08, 0.17}) and ({3.86, 0.76}), respectively, which would place the outer planet amongst the most eccentric Jovian planets identified thus far. The authors noted that the orbit and mass of the outer planet are poorly constrained.   \n\nHD\\,110014 was found to host a planet by \\citet{2009deMedeiros}; the second companion was identified through re-analysis of archival spectra taken by the FEROS instrument \\citep{1998KauferPasquini} looking to derive an updated orbit for planet b \\citep{2015Soto}.  The two candidate planets have super-Jupiter masses, and \\citet{2015Soto} cautioned that the proposed second planet was worryingly close in period to the typical rotation period of K giant stars. However, their analysis of the stellar photometry was inconclusive in identifying its activity as the root cause for the secondary signal. \n\nHD~133131A's planetary companions were reported in \\citet{2016Teske}, based on precise radial velocities primarily from the Magellan Planet Finder Spectrograph \\citep{2006Crane,2008Crane,2010Crane}. Their data supported the presence of two planets, where the outer planet is poorly constrained due to its long period. \\citet{2016Teske} ran a single dynamical stability simulation on the adopted solution and found it to remain stable for the full $10^5$ yr duration. The authors presented both a low- and high- eccentricity solution, reasoning that in a formal sense, the two solutions were essentially indistinguishable. They favoured the low-eccentricity model ($e_2=0.2$) for dynamical stability reasons. There is precedent in the literature for this choice, since it does happen that the formal best fit can be dynamically unfeasible whilst a slightly worse fit pushes the system into a region of stability \\citep[e.g.][]{2013Mustill, 2014Trifonov, 2017bWittenmyer}.  \n\nThe remainder of the paper is laid out as follows. We present a brief summary of the radial velocity observations and other data (e.g. stellar parameters) used for our reanalysis in Sect. \\ref{sec:Obs} along with an explanation of our modelling approach. The results of the reanalyses for each target are shown in Sect. \\ref{sec:Res}. A brief discussion of our findings in comparison to previous work on these systems is presented in Sect. \\ref{sec:Dis}. Finally, we present our conclusions in Sect. \\ref{sec:Con}.\n\n\\section{Observations and methods}\n\\label{sec:Obs}\n\n\\subsection{Radial Velocity Data}\n\nWe compiled radial velocity values from the literature for the three systems examined in this work; the origin of these data are summarised in Table \\ref{tab:rvs}. \n\n\\begin{table}\n\\centering\n\\caption{Table of references for the radial velocities data used in this work. \\label{tab:rvs}}\n\\begin{tabular}{ll}\n    \\hline\n    Target & References \\\\\n    \\hline\\hline\n    HD~67087   & \\cite{2015Harakawa} \\\\\n    HD~110114  & \\cite{2009deMedeiros} \\\\\n    HD~133131A & \\cite{2016Teske} \\\\\n    \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table*}\n\\centering\n\\caption{Planetary orbital parameters based on \\textit{Systemic} fits to radial velocity data. Semi-major axes were calculated using measured orbital periods and stellar masses taken from the NASA Exoplanet Archive. \\label{tab:systemic}}\n\\begin{tabular}{lcccccc}\n    \\hline\n    & HD\\,67087 b & HD\\,67087 c & HD\\,110014 b & HD\\,110014 c & HD\\,133131A b & HD\\,133131A c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] &  74.0~$\\pm$~3.0 & 54.0~$\\pm$~4.0 & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 352.3$\\pm$1.7 & 2380$^{+167}_{-141}$ & 877.5$\\pm$5.2 & 130.125$\\pm$0.096 & 648.$\\pm$3 & 5342$^{+7783}_{-2009}$ \\\\\n    Mean anomaly [deg] & 35$^{+20}_{-16}$ & 94$^{+28}_{-35}$ & 155$\\pm$4 & 231$\\pm$3 & 265$\\pm$11 & 188$^{+104}_{-123}$ \\\\\n    Longitude [deg] & 281$^{+18}_{-15}$ & 256 (fixed) & 41$\\pm$3 & 302$\\pm$4 & 16.6$^{+4.7}_{-4.5}$ & 110$^{+25}_{-43}$ \\\\\n    Eccentricity & 0.18$^{+0.07}_{-0.06}$ & 0.51 (fixed) & 0.259$\\pm$0.017 & 0.410$\\pm$0.022 & 0.340$\\pm$0.032 & 0.63$^{+0.25}_{-0.20}$ \\\\\n    $M$ sin $i$ [$M_{\\rm Jup}$] & 3.10$^{+0.15}_{-0.14}$ & 3.73$^{+0.47}_{-0.45}$ & 10.61$\\pm$0.25 & 3.228$\\pm$0.098 & 1.418$\\pm$0.036 & 0.52$^{+0.45}_{-0.17}$ \\\\\n    Semi-major Axis [au] & 1.08 & 3.87 & 2.32 & 0.65 & 1.44 & 5.88 \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\begin{table*}\n\\centering\n\\caption{Results from \\textsc{Astroemperor} exploration of parameter space around \\textit{Systemic} nominal best fit values for planetary companions to HD~133131A and HD~110014. \\label{tab:mcmc}}\n\\begin{tabular}{lcccc}\n    \\hline\n    & HD\\,133131A b & HD\\,133131A c & HD\\,110014 b & HD\\,110014 c \\\\\n    \\hline \\hline\n    Amplitude [\\mbox{m\\,s$^{-1}$}] & 36.949$\\pm$0.750 & 5.956$\\pm$1.617 & 135.315$\\pm$3.640 & 64.660$\\pm$3.966 \\\\\n    Period [days] & 647.816$\\pm$1.575 & 3205.648$\\pm$948.063 & 865.206$\\pm$6.170 & 132.431$\\pm$0.279 \\\\\n    Phase [deg] & 261.620$\\pm$4.850 & 31.734$\\pm$98.433 & 43.753$\\pm$72.179 & 341.373$\\pm$64.346 \\\\\n    Longitude [deg] & 18.550$\\pm$2.165 & 113.777$\\pm$81.302 & 146.633$\\pm$17.229 & 236.903$\\pm$16.452 \\\\\n    Eccentricity & 0.341$\\pm$0.021 & 0.263$\\pm$0.145 & 0.011$\\pm$0.015 & 0.294$\\pm$0.076 \\\\\n    M sin $i$ [$M_{\\rm Jup}$]& 1.428$\\pm$0.099 & 0.388$\\pm$0.124 & 10.622$\\pm$0.757 & 2.581$\\pm$0.247 \\\\\n    Semimajor Axis [au] & 1.435$\\pm$0.046 & 4.153$\\pm$0.800 & 2.350$\\pm$0.075 & 0.668$\\pm$0.023 \\\\\n    \\hline\n    Jitter [\\mbox{m\\,s$^{-1}$}] & 3.557$\\pm$1.254 & 0.466$\\pm$0.419 & 6.060$\\pm$1.856 & 13.350$\\pm$1.492 \\\\\n    Offset [\\mbox{m\\,s$^{-1}$}] & -9.333$\\pm$4.787 & 12.321$\\pm$7.543 & 52.575$\\pm$4.737 & 72.198$\\pm$4.541 \\\\\n    MA coefficient & 0.714$\\pm$0.531 & 0.466$\\pm$0.419 & 0.697$\\pm$0.214 & 13.350$\\pm$1.492 \\\\\n    MA Timescale [days] & 4.158$\\pm$2.815 & 12.321$\\pm$7.543 & 9.793$\\pm$2.488 & 72.198$\\pm$4.541 \\\\\n    Acceleration [\\mbox{m\\,s$^{-1}$}\/yr] & -1.435 & & -21.620 & \\\\\n    \\hline\n\\end{tabular}\n\\end{table*}\n\n\n\\subsection{Modelling}\n\nTo test the dynamical stability of these proposed planetary systems, we follow the updated dynamical methodology outlined in our previous work \\citep[][]{2017bWittenmyer,2019Marshall}.\n\nIn brief, we perform a fit to the published velocity data using the \\textit{Systemic Console} \\citep{2009Meschiari}. We then use the MCMC tool within \\textit{Systemic} to explore the parameter space about the best fit. The MCMC chain runs for $10^7$ steps, discarding the first 10,000, and we then draw the trial solutions for our dynamical stability simulations from these posteriors. Using this data, we populate three ``annuli'' in $\\chi^2$ space corresponding to the ranges $0-1\\sigma$, $1-2\\sigma$, and $2-3\\sigma$ from the best fit.  Each annulus contains 42,025 unique realisations drawn from the MCMC chain. The innermost annulus was drawn from the lowest 68.3 per cent of all $\\chi^2$ values, the middle annulus contained the next best 27.2 per cent of values, and the outer annulus contained the worst 4.5 per cent of solutions (i.e. those falling $2-3\\sigma$ away from the best fit). The result is a set of ``clones'' which fall within $3\\sigma$ of the best-fit solution, thus representing a reasonable region of parameter space within which we explore the dynamical stability of the proposed planetary system, using the constraints afforded by the existing observational data.\n\nWe then proceed to perform lengthy dynamical simulations of each of the 126,075 solutions generated by this method. We used the Hybrid integrator within the $n$-body dynamics package \\textit{Mercury} \\citep{1999Chambers} to integrate the solutions forwards in time for a period of 100 Myr. The simulations are brought to a premature end if either of the planets being simulated is ejected from the system, is flung in to the central star, or if the two planets collide with one another. When such events occur, the time at which the collision or ejection occurred is recorded, giving us the lifetime for that particular run. As such, our suite of simulations yield 126,075 tests of the candidate planetary system, allowing us to study how its stability varies as a function of the particular details of the solution chosen to explain the observational data.\n\nWe determine the best-fit parameters and uncertainties for each system using the code Exoplanet Mcmc Parallel tEmpering Radial velOcity fitteR\\footnote{\\href{https:\/\/github.com\/ReddTea\/astroEMPEROR}{https:\/\/github.com\/ReddTea\/astroEMPEROR}} ({\\sc astroEMPEROR}), which uses thermodynamic methods combined with MCMC. Our approach has previously been established and described in \\cite{2019Marshall} and \\cite{EightSingle}. We summarise the input values and constraints used in the fitting presented in this work for the sake of reproducibility. Given that our goal was to test the feasibility of the exoplanetary systems as presented in the literature, we restricted {\\sc astroEMPEROR} to consider zero, one, or two planetary signals in the radial velocity data; dynamical configurations with additional planetary companions in orbits that could mimic a single planetary companion, e.g. two resonant planets looking like a single eccentric planet (for a total of three planetary companions), were not considered in this analysis. The planetary fitting parameters were the orbital period ($P$), line-of-sight mass ($M\\sin i$), orbital eccentricity ($e$), longitude of periastron ($\\omega$), and mean anomaly ($M$). We also include an additional jitter term when fitting the data. We initialised the locations of the walkers in the MCMC fitting at their best-fit values from the \\textit{Systemic} console fit, plus a small random scatter. The priors on each parameter were flat and unbounded i.e. with uniform probability between $\\pm\\infty$, except for the orbital eccentricities which had folded Gaussian priors, and the jitter term, which was a Jeffries function (but still unbound between $\\pm\\infty$). The parameter space was surveyed by 150 walkers at five temperatures over 15\\,000 steps, with the first 5\\,000 steps being discarded as the burn-in phase.\n\n\\section{Results}\n\\label{sec:Res}\n\n\\subsection{HD~67087}\n\nThe HD~67087 system is catastrophically unstable, as illustrated by the results of our stability analysis in Fig. \\ref{fig:HD67087_stability}. In this plot it is clear that the most stable solutions cluster toward the largest ratios of semi-major axes, and the smallest eccentricities. Even in this limit, the longest-lived solutions that plausibly represent the observations are still only stable for 10$^{6}$ yrs, out of a total integration time of 10$^{8}$ yrs. This leads us to the interpretation that the HD~67087 system, as inferred from the available radial velocity data, is dynamically infeasible. Given this high degree of instability, we do not attempt to determine a global best-fit solution for the system parameters.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD67087_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD67087_MRatio_MaxEcc_200_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~67087 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~67087b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~67087b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find no stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets, with the only stable solutions lying at the extreme edges of the parameter space toward low eccentricities, large separations and low mass ratios. \\label{fig:HD67087_stability}}\n\\end{figure*} \n\n\\subsection{HD~110014}\n\nThe HD~110014 system is found to be dynamically stable, with a broad swathe of parameter space centred on the nominal solution producing system architectures that last for the full 10$^{8}$ yrs of our dynamical integrations. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD110014_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD110014_confusogram}.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD110014_MaxEcc_ARatio_1800_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD110014_MRatio_MaxEcc_1800_45.png}\n    \\caption{Visualisation of the dynamical stability of the HD~110014 planetary system. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~110014b and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~110014b and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. We find stable solutions that last the full 100 Myr duration of the dynamical simulations close to the nominal best-fit orbital solution for the planets. \\label{fig:HD110014_stability}}\n\\end{figure*}\n\n\\begin{figure*}[h]\n\t\\includegraphics[width=\\textwidth]{HD110014Triangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~110014 b and HD~110014 c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD110014_confusogram}\n\\end{figure*}\n\n\\subsection{HD~133131A}\n\nThe HD~133131A system shows a very complex parameter space in the stability plots. As one would expect, the stability of the system generally increases towards lower orbital eccentricities and lower mass ratios between the two planetary components. The overall stability appears to be insensitive to the ratio of the semi-major axes for the planets, with long-lived solutions possible across the full range of values probed for this parameter. Interestingly, we demonstrate that stable architectures for the planetary system exist in both the high and low orbital eccentricity scenarios for the system. We show the results of the stability analysis, sampling the 3-$\\sigma$ parameter space around the nominal orbital solution determined from the radial velocities in Fig. \\ref{fig:HD133131A_stability}. The results of the Bayesian analysis, showing what we infer to be the global best-fit parameters for the system, are presented in Fig. \\ref{fig:HD133131A_confusogram}. Further observations to refine the planet properties of this system will be required to definitively characterise its dynamical stability.\n\n\\begin{figure*}\n    \\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AH_MRatio_MaxEcc_200_45.png}\\\\\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MaxEcc_ARatio_200_45.png}\n\t\\includegraphics[width=0.45\\textwidth]{plot_HD133131AL_MRatio_MaxEcc_200_45.png}\n    \\caption{Plots of the dynamical stability of the HD~133131A planetary system for both the high eccentricity (top) and low eccentricity (bottom) orbital solutions. On the left we show the log(lifetime) as a function of the largest initial eccentricity fit to HD~133131Ab and c and the ratio of their orbital semi-major axes, whilst on the right we show the log(lifetime) as a function of of the largest initial eccentricity fit and the mass ratio between HD~133131Ab and c. The colour bar shows the goodness of fit ($\\chi^{2}$) of each solution tested. The stability revealed by our dynamical simulations is complex, with regions of both extreme stability (log(lifetime) $\\sim$ 100 Myrs) and instability (log(lifetime) $\\sim$ 100 yrs) lying within the 3-$\\sigma$ reach of the nominal best-fit orbital parameters.  \\label{fig:HD133131A_stability}}\n\\end{figure*} \n\n\\begin{figure*}\n\t\\includegraphics[width=\\textwidth]{HD133131ATriangle.pdf}\n\t\\caption{Bayesian posterior distributions of HD~133131A b and HD~133131A c's orbital parameters derived from {\\sc astroemperor}. From left to right (top to bottom), the parameters are $K_{b}$, $P_{b}$, $\\omega_{b}$, $\\phi_{b}$, $e_{b}$, $K_{c}$, $P_{c}$, $\\omega_{c}$, $\\phi_{c}$ and $e_{c}$. Credible intervals are denoted by the solid contours with increments of 1-$\\sigma$.}\n    \\label{fig:HD133131A_confusogram}\n\\end{figure*}\n\n\\section{Discussion}\n\\label{sec:Dis}\n\nThe results of our dynamical modelling for the three systems considered in this work, HD~67087, HD~110014 and HD~133131A show three distinctly different outcomes. For the first system tested, HD~67087, we find no orbital solutions that exhibit long-term dynamical stability. As a result, we are forced to conclude that, if the planets proposed to orbit that star are real, they must move on orbits significantly different from those proposed in the discovery work, and sampled in our simulations. It seems likely that new radial velocity observations of HD~67087, extending the temporal baseline over which the star has been observed, will yield fresh insights to the system -- either significantly constraining and altering the proposed orbit for the outermost planet, or even revealing that that eccentric solution is in fact the result of multiple unresolved planets at large orbital radii. Such an outcome is far from unusual -- and indeed, it is often the case that, with more data, a single eccentric planet seen in RV data is resolved to actually be two planets moving on near circular orbits \\citep[e.g.][]{2013Wittenmyer,EightSingle}. For now, however, we can do no more than to call the existence of HD~67087~c into question, pending the acquisition of such additional data.\n\nIn contrast to the instability of HD~67087, our simulations of the HD~110014 system reveal that the best-fit solution for that two-planet system lies in a broad region of strong dynamical stability. In this case, our simulations simply reveal that the system, as proposed in the discovery work, is dynamically feasible -- and in a sense, the simulations add little beyond that.\n\nThe case of HD~133131A is somewhat more interesting. Here, our simulations reveal that solutions that fit the observational data can exhibit both strong dynamical stability, and extreme instability (with dynamical lifetimes of just a few years). Both the high- and low-eccentricity solutions considered in \\citet{2016Teske} can produce scenarios that are stable for the full 100 Myr of our simulations. In both the high- and low-eccentricity cases, the stable solutions cluster around the least eccentric available scenarios. The more widely separated the two planets, the more eccentric their orbits can be before instability occurs -- a natural result of the stability being driven by the minimum separation between the planets, rather than their orbital semi-major axes. The more widely the semi-major axes of the orbits are spaced, the more eccentric they must be to bring the planets into close proximity. These results show once again the benefits inherent to such dynamical analysis -- reminding us how studying the dynamical evolution of a given system can help to provide stronger constraints on the orbits of the planets contained therein than is possible by studying the observational data on their own.\n\nA comparison of our results to the analysis of the AMD stability criterion presented in \\cite{2017Laskar} shows agreement between the two different techniques for the dynamical stability of the three systems. Whilst HD~67087 and HD~110014 are respectively very clear cut cases of an unstable and a stable system, HD~133131A exhibits a more complex behaviour. HD~133131A may be dynamically stable, but the inferred lifetime for the planetary system as proposed is sensitive to the chosen initial conditions; this system therefore represents an edge case of stability where limitations of available data and the respective analyses provide no clear answer to the veracity of the previously inferred planetary system.\n\nCombining these new results with our previous dynamical analyses, as summarised in the introduction, we may consider that the AMD criterion is a reliable estimator of stability for planetary systems. There are 13 systems (out of 131 considered in that work) from \\cite{2017Laskar} that have had dynamical modelling of their stability. In \\cite{2017Laskar}, a planetary system is considered strongly stable if all planet-pairs have $\\beta$ values less than 1, such that collisions are impossible whilst weakly stable planetary systems are those in which the inner-most planet might collide with the star without disrupting the remainder of the planetary system. In five systems, both the AMD criterion and dynamical modelling agree on their dynamical stability (HD~142, HD~159868, NN Ser (AB), GJ~832, and HD~110014); the planets in each of these systems are dynamically well separated and therefore not strongly interacting \\citep[][this work]{2011Horner,2012bWittenmyer,2014bWittenmyer}. Six systems are unstable according to the AMD criterion with values of $\\beta$ in the range 1 to 5 for the planet pair (HD~155358, 24 Sex, HD~200964, HD~73526, HD~33844, HD~47366), but all are in mean motion resonances and have been demonstrated to be dynamically stable through $n$-body simulations \\citep{2012Robertson,2012cWittenmyer,2014aWittenmyer,2016Wittenmyer,2019Marshall}. The remaining two systems (HD~67087, HD~133131A) are dynamically unstable in both the AMD and dynamical analysis (this work). However, dynamical analysis of the HD~133131A system reveals regions of dynamical stability consistent with the observed radial velocities, prompting the need for further investigation of this system and its architecture. Neither of these two unstable planetary systems have $\\beta$ values radically different from those of the planetary systems in resonance, or each other, such that determining their stability can only be carried out using dynamical simulations. The existence of such systems in the known planet population as demonstrated in our analysis therefore showcases the necessity of performing long duration dynamical analyses of proposed planetary system architectures to reveal the complex dynamical interplay between high mass planets, the evolution of their orbital elements, and determine what constraints this places on the available parameter space for the endurance of the proposed planetary system over its lifetime.\n\n\\section{Conclusions}\n\\label{sec:Con}\n\nWe re-analysed the dynamical stability of the exoplanet systems around HD~67087, HD~110014, and HD~133131A, using available radial velocity data. These three planetary systems have poorly constrained orbital parameters, and had previously been identified as being potentially unstable. We combine a determination of the best-fit orbital parameters from least-squares fitting to the data with $n$-body simulations to determine the global best-fit solution for the planetary system architectures, and thereafter determine the probability distribution of the orbital solutions through Bayesian inference. \n\nOur dynamical analysis confirms that the published planetary system parameters for HD~67087bc are dynamically unstable on very short timescales, and we must conclude that the system, as published, is dynamically unfeasible. As more data are collected for the HD~67087 system, it seems likely that the true nature of the candidate planets therein will be revealed, and that future planetary solutions for that system will veer towards dynamical stability as the planetary orbits become better constrained. \n\nIn the case of HD~110014 bc we demonstrate that the system parameters can be dynamically stable for the full duration of our 100 Myr integrations. The third system, HD~133131A , exhibits much more complex behaviour, with HD~133131A bc being strongly unstable over much of the parameter space exhibited in this work including the region encompassing the nominal best-fit to the orbital parameters. In agreement with previous analysis of this system, we strongly disfavour a high eccentricity orbital solution for planet c. Additional observations of this system will be required to more precisely determine the planetary properties for HD~133131A bc and thereby categorically rule on the plausibility of the proposed planetary system.\n\nThese results demonstrate the complementarity of various techniques to deduce the stability of planetary systems, with good agreement between the results of our various works, and that of the AMD approach. We highlight the appropriateness of dynamical simulations for determination the long-term stability of planetary systems in the presence of strongly interacting planets, which although costly in a computing sense capture the full essence of planetary interaction in such systems which is not possible with other techniques. We finally assert that the orbital parameters for these three systems which have been determined in this work (as summarised in Table 3) should be the accepted values adopted by exoplanet archives or elsewhere. This work is thus one additional thread in the tapestry of cross-checking of published results through various means that ensures the reliability of archival information on planetary properties and the architectures of planetary systems which are essential to inform models of the formation and evolution of the exoplanet population \\citep[e.g. ][]{2019Childs,2019Denham,2020He,2020VolkMalhotra}.\n\n\\section*{Acknowledgements}\n\nThis is a pre-copyedited, author-produced PDF of an article accepted for publication in MNRAS  following peer review. The version of record Marshall et al., 2020, MNRAS, 494, 2, 2280--2288 is available online \\href{https:\/\/academic.oup.com\/mnras\/article\/494\/2\/2280\/5819459}{here}.\n\nWe thank the anonymous referee for their comments which helped to improve the article.\n\nThis research has made use of NASA's Astrophysics Data System and the SIMBAD database, operated at CDS, Strasbourg, France.\n\nJPM acknowledges research support by the Ministry of Science and Technology of Taiwan under grants MOST104-2628-M-001-004-MY3 and MOST107-2119-M-001-031-MY3, and Academia Sinica under grant AS-IA-106-M03.\n\n\\textit{Software}: This research has made use of the following Python packages: \\textsc{matplotlib} \\citep{2007Hunter}; \\textsc{numpy} \\citep{2006Oliphant}; \\textsc{pygtc} \\citep{2016Bocquet}; \\textsc{emcee} \\citep{2013ForemanMackey}; \\textsc{corner} \\citep{2016ForemanMackey}; \\textsc{mercury} \\citep{1999Chambers}.\n\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\subsection{Motivation}\n\\IEEEPARstart{T}{he} emerging of many dazzling innovative technologies such as ultra-massive MIMO, large-scale intelligent reflective surfaces (IRS) and wireless artificial intelligence (AI) \\cite{Saad,Yang}, etc., boosts the rapid development of broadband wireless communications. On the other hand, in the foreseeable future, many revolutionary and challenging application scenarios such as smart city \\cite{Zanella}, autonomous driving \\cite{Toutouh} and unmanned aerial vehicle (UAV) positioning \\cite{Bor}, etc, may call for not only broadband connections but also accurate environment information which include but are not limited to, the locations, shapes, status and electromagnetic characteristic of the stationary or moving objects and\/or the background scatterers, within that environment.\n\nSuch kind of environment sensing has long been accomplished by traditional Radar technology which has been regarded as a related but separate field from communication. However, the channel state information (CSI) obtained during the communication process usually contains certain knowledge of the environment \\cite{Sen, Rao}, and likewise, the environment sensing result also helps improve the accuracy of channel estimation and enhance the performance of communication \\cite{Jiaoicc,Jiaotwc}. As the wireless network operates in higher frequency with wider bandwidth and deploys denser base stations with a larger amount of antennas, its longly-neglected sensing capability inherited from the intrinsic nature of electromagnetic wave propagation becomes even more explorable. The signal processing principles employed in these two fields also tend to converge. As visioned in \\cite{Wild}, this may give rise to a promising new technology of JCAS or sensing-communication integration.\n\nOne major challenge in JCAS lies in the potentially large amount of unknown variables brought by the environment, even many more than those contained in the statistical channel models, which may make the problem rank-deficient and eventually insolvable. As a result, certain sparsity should be exploited. Fortunately, the targeted environment itself usually does possess certain sparsity, which in turn makes the communication channel sparse. For example, in a cellular communication network, buildings are sparsely located within the wireless network coverage, and in the indoor scenario, furniture and other items are sparsely distributed in the entire room. \n\nAs a matter of fact, either in conventional communication or in conventional sensing, sparsity has been fully exploited to achieve better and lower-complexity solutions under the compressed sensing (CS) framework. As in wireless communications, the CS-based channel estimation approaches exploiting the channel sparsity usually exhibit superiority in signaling overhead and computational complexity \\cite{Rao} compared to the conventional channel estimation approaches \\textit{et al.}\\cite{Sen}. Likewise, in the broad area of Radar sensing or computational imaging, the utilization of the intrinsic sparsity of objects or scatterers within an environment is key to their effective detection. Such problems are often modeled as sparse signal recovery problems based on pixel division according to the CS theory \\cite{Donoho, Candes}, and solved by the widely used methods like Sparse Bayesian Learning (SBL) \\cite{Zhang}, orthogonal matching pursuit (OMP) \\cite{Cai}, and Generalized Approximate Message Passing (GAMP) \\cite{Rangan}, and so on. This has been showcased by recent works in \\cite{taoy, yaojj}, where innovative microwave computational imaging methods with the aid of intelligent reflecting surfaces (IRS) are proposed based on a fast block sparse Bayesian learning (BSBL) algorithm. However, how to explore and exploit the sparsity in the JCAS scenario still lacks adequate study.\n\n\\subsection{Related Works}\nSo far, there have been different sorts of attempts to implement joint environment sensing and communication. \n\nTo list a few, in the Radar-Communication Coexistence (RCC) sort of approaches \\cite{wang2008, Saruthirathanaworakun}, effective interference cancellation and management mechanism are designed to achieve flexible coexistence between the radar and the communication systems. In contrast to the RCC system, the second sort of approach, i.e., Dual-Functional Radar-Communication (DFRC), aim to achieve an integration of radar and communication through sharing a common hardware platform, with improved sensing and communication performance through collaborative operation \\cite{Paul, Blunt}. In such systems, separate sensing and communication operations with shared radio resources have also been extensively studied. For instance, a multi-beam scheme is proposed in \\cite{ZhangAndrew}, which uses an analog array to generate multiple beams for simultaneous communication and radar scanning. \n\nThe third sort of works mainly concentrates on the purpose of environment sensing by pure usage of the conventional communication signals or by proper joint design of the radio signals, which can be found in \\cite{AndrewEnabling} and the references therein. As in one of such works on environment sensing with the aid of real deployed communication systems, \\cite{Tan}  identified the behavior of a human body by extracting the Doppler frequency shift from the CSI conveyed by the WIFI communication signals in the indoor scenario. In \\cite{Daniels}, the authors used the orthogonal frequency division multiplexing communication waveform as a radar signal to achieve joint communication and environment sensing between vehicles. As an illustration of joint sensing and communication signaling, the authors in \\cite{ChenPerformance} designed a cooperative sensing unmanned aerial vehicle network (CSUN) with joint sensing and communication beams based on a common transceiver device. Considering the non-ideal factors of the channel, \\cite{Shahi} analyzed the communication channel capacity under the joint effect of Gaussian random noise and non-Gaussian radar sensing interference. In \\cite{Ahmadipour}, the authors theoretically analyzed the performance of the JCAS system under the condition of the memoryless broadcast channel. The related systematic design rules and methodologies for signaling and processing in such a JCAS system have raised increasing research effort recently, resulting in some interesting JCAS implementations based on machine learning \\cite{Aoudia}, joint data sensing and fusion \\cite{Schmitt}, and time-frequency-space full dimension utilization \\cite{Gaudio}, etc. \n\n\\subsection{Main Ideas and Contributions}\n\nIn this paper, we exploit the sparsity of both the structured multi-user signals and the unstructured environment to design a low-complexity joint multi-user communication and environment sensing scheme based on microwave computational imaging.\nDifferent from the above-mentioned application scenarios, we aim at designing a system with integrated sensing and communication capability based on existing wireless communication systems, which is capable of accomplishing the environment sensing (imaging) using the multi-user transmission signals and in turn, assisting the multi-user information detection with the channel information derived from the sensing results. To the best of our knowledge, there still lacks sufficient study on the design of such a JCAS system in the literature.\n\nIn order to achieve these goals, we employ the Sparse Code Multiple Access (SCMA) protocol \\cite{Nikopour,Taherzadeh} for multi-user uplink access, and employ an IRS \\cite{Garcia} to assist signal propagation and collection. SCMA is an elegant code-domain non-orthogonal multiple access (C-NOMA) method, which has extracted extensive research effort due to its superior performance and low detection complexity. In the SCMA scheme, the codebook for users to send data is sparse, and each user occupies a few but not all subcarriers. The sparsity of the user codebook effectively enhances the decoding performance of the received data. IRS is a promising technology to manipulate the electromagnetic environment with low-cost passive reflective elements by adjusting the phase of incident signals, which has been extensively used in wireless communications \\cite{Huang, Garcia, Wu, Chenw}.  Based on the idea of computational imaging, IRS can cause known and diverse changes to electromagnetic signals, and such changes are beneficial to environment sensing. Therefore, IRS also exhibited great potential in environment sensing as recently described in \\cite{yaojj, taoy}. But in these cases, the base station is only used as an environment sensing device instead of a communication signal transmitter. It is noteworthy that, in this work, the environment sensing is just accomplished by making use of the IRS' reflection characteristics rather than by actively manipulating it. Our method not only enables environment sensing to obtain the help of IRS but also retains the ability of IRS to assist communication.\n\nOur design is depicted as follows. First, in the multiple access part, the user uses the SCMA protocol to communicate with the wireless AP. The signals are reflected by the IRS and then arrive at the AP. With the limited channel information obtained by an initial pilot sequence, we use the proposed SCMA-IRS-MPA algorithm (see Section. \\ref{mainalgm} for details) to conduct multi-user detection based on the sparse codebook of the transmitted signals. Then a sliding-window-based environment sensing algorithm is proposed to accomplish the environment sensing (imaging) with the received signal and recovered users' data, again based on the CS principle. Note that the proposed multiple user detection algorithm requires the environment (channel) knowledge, and the proposed environment sensing algorithm also needs to know the decoded data, so the two processes, i.e., multiple access and environment sensing, rely on each other. Therefore, finally, we propose an iterative and incremental algorithm to jointly recover the users' data and accomplish environment sensing at the same time with significantly reduced pilot overhead.\n\nThe main contributions of this paper are summarized as follows:\n\n\\begin{itemize}\n    \\item We design a joint communication and environment sensing scheme, which exploits the sparsity of both the structured multi-user signals and the unstructured environment to achieve the integration of multiple access and environment sensing. \n    \n    \\item We develop a low-complexity iterative algorithm based on CS and generalized message passing theory to conduct the multi-user information detection and environment sensing (imaging). It is sliding-window based and runs alternately between the states of sensing with the decoded multi-user data and data decoding with the sensing results. This way, the overall system performance can be incrementally improved.   \n    \n    \\item We analyze the decoding error and sensing accuracy performances as well as the computational complexity of the proposed algorithm, and investigate the impact of access user number on system performances, base on which, we approximate the optimal operating point. Extensive simulation results verify the convergence and effectiveness of the proposed algorithm.\n        \n\\end{itemize}\n\nThe rest of this paper is organized as follows. Section \\uppercase\\expandafter{\\romannumeral2} presents the environment setting and system model in the uplink communication scenario. Section \\uppercase\\expandafter{\\romannumeral3} proposes the multiple access method based on the SCMA scheme. Section \\uppercase\\expandafter{\\romannumeral4} proposes the environment sensing method base on the CS theory. Section \\uppercase\\expandafter{\\romannumeral5} proposes the iterative and incremental algorithm based on low-density pilots jointly recovers the users' data and achieves environment sensing. In section \\uppercase\\expandafter{\\romannumeral6}, we discuss the trade-off relationship between the number of access users and system performance. Finally, section \\uppercase\\expandafter{\\romannumeral7} presents the numerical results, and section \\uppercase\\expandafter{\\romannumeral8} concludes the paper.\n\n\\textit{Notation}: Fonts $a$ and $\\mathbf{A}$ represent scalars and matrices, respectively.\n$\\mathbf{A}^{\\rm{T}}$ and $\\|\\mathbf{A}\\|_F$ denote transpose and Frobenius norm of $ \\mathbf{A} $, respectively.\n$[\\mathbf{A}](i,j)$ represents $\\mathbf{A}$'s $(i,j)$-th element.\n$|\\cdot|$ and $[\\cdot]$ denote the modulus and the catenation of the matrix, respectively.\n$\\odot $ represents the Hadamard product between two matrices.\nFinally, notation ${\\rm diag}(\\mathbf{a})$ represents a diagonal matrix with the entries of $\\mathbf{a}$ on its main diagonal, and $\\delta(\\cdot)$ is the Dirac delta function.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=3in]{fig1.eps}\n  \\caption{The uplink multi-user communication scenario.}\n  \\label{figsetting}\n  \\end{figure}\n\n\\section{Environment Setting and System Model}\nAs shown in Fig. \\ref{figsetting}, a millimeter-wave multi-antenna AP and multiple single-antenna user equipments (UEs) are deployed in an indoor scenario. An IRS is deployed near the AP to assist the users' communication, and there are some target objects (serve as scatterers) in the scenario. We consider that in the uplink regime, i.e., multiple users simultaneously send data to the AP via a shared channel. Our goal is to accomplish the environment sensing while reliably obtaining the communication data, that is, to sense the distribution position and scattering rate of the scatterers in the environment and obtain the communication data of all the users at the same time.\n\n\\subsection{Environment Setting}\nLet the number of users in the environment be $N_{\\rm{u}}$ and the number of AP receiving antennas be $N_{\\rm{R}}$. In the uplink communication scenario, the channel from the user to the AP has mainly composed of three parts: The first part is the line-of-sight (LOS) path from the user directly to the AP. The second part is the path from the user to the IRS and then reflected to the AP. The third part is the multipath propagation path which the user signal is scattered by the scatterers and reflected to the AP by the IRS. They are denoted as ${{\\bf{H}}^{{\\rm{LOS}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$,  ${{\\bf{H}}^{{\\rm{IRS}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$, and ${{\\bf{H}}^{{\\rm{s}}}} \\in \\mathbb{C}^{{N_{\\rm{u}}} \\times {N_{\\rm{R}}}}$ respectively. Let the number of reflective elements of the IRS be $N_{\\rm{I}}$, and each element can set amplitude reflection coefficient and phase shift independently, thereby controlling the relationship between the reflected signal and the incident signal. The reflection characteristic matrix of IRS is expressed as\n\\begin{equation}\n{\\bf{\\Theta }} = {\\rm{diag}}\\left( {{\\theta _1}, \\cdots, {\\theta _{{N_{\\rm{I}}}}}} \\right) \\in \\mathbb{C}^{{N_{\\rm{I}}} \\times {N_{\\rm{I}}}}, \\label{eq1}\n\\end{equation}\nwhere ${\\theta _{{n_{\\rm{I}}}}} = {\\rho _{{n_{\\rm{I}}}}}{e^{j{\\varphi _{{n_{\\rm{I}}}}}}}$ represents the reflection characteristic of the $n_{\\rm{I}}$ element of the IRS, ${\\rho _{{n_{\\rm{I}}}}} \\in \\left[ {0,1} \\right]$ and ${\\varphi _{{n_{\\rm{I}}}}} \\in \\left[ {{\\rm{0}},2\\pi } \\right]$ represents the amplitude reflection coefficient and phase shift of the $n_{\\rm{I}}$ element respectively, and the diag function represents the construction of a diagonal matrix.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=2in]{fig14.eps}\n  \\caption{The discretized targeted environment object.}\n  \\label{pixel}\n  \\end{figure}\n\nWe discretize the room environment information and regard the environment information of the entire room as a point cloud. Each point in the point cloud represents the environment information of small cubes with sizes $l_{\\rm{s}}$, $w_{\\rm{s}}$, and $h_{\\rm{s}}$ around this point. These small cubes are called pixels. Assuming that the length, width, and height of the room are $L_{\\rm{s}}$, $W_{\\rm{s}}$, and $H_{\\rm{s}}$ respectively, the number of point clouds in the space is ${N_{\\rm{s}}} = {{{L_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{L_{\\rm{s}}}} {{l_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{l_{\\rm{s}}}}} \\times {{{W_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{W_{\\rm{s}}}} {{w_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{w_{\\rm{s}}}}} \\times {{{H_{\\rm{s}}}} \\mathord{\\left\/\n{\\vphantom {{{H_{\\rm{s}}}} {{h_{\\rm{s}}}}}} \\right.\n\\kern-\\nulldelimiterspace} {{h_{\\rm{s}}}}}$. The inside of each pixel may be empty, or there may be scatterers. We use a scattering coefficient ${x_{{n_{\\rm{s}}}}}$ to represent the scattering coefficient of the pixel where the $n_{\\rm{s}}$-th point cloud point is located. If the inside of the small cube is empty, then ${x_{{n_{\\rm{s}}}}} = 0$. As shown in the Fig. \\ref{pixel}, the target scatterer in Fig. \\ref{figsetting} is discretized. Therefore, the environmental information of the entire room can be expressed as\n\\begin{equation}\n{\\bf{x}} = {\\left[ {{x_1},{x_2}, \\cdots ,{x_{{N_{\\rm{s}}}}}} \\right]^{\\rm{T}}}. \\label{eq2}\n\\end{equation}\n\n\\subsection{System Model}\nMultiple users in the space share the same time-frequency resources. The frequency resources of communication are divided into $R$ orthogonal resource elements (OREs), and $N_u$ users send data on all OREs. Therefore, on the $r$-th ORE, the channel from the user to the AP can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {{\\bf{H}}_r} = {\\bf{H}}_r^{{\\rm{LOS}}} + {\\bf{H}}_r^{{\\rm{IRS}}} + {\\bf{H}}_r^{\\rm{s}}\\\\\n  \\quad  = {\\bf{H}}_r^{{\\rm{LOS}}} + {\\bf{H}}_r^{{\\rm{IRS1}}}{\\bf{\\Theta H}}_r^{\\rm{s1}}\\\\\n  \\quad  + \\left[ {\\begin{array}{*{20}{c}}\n  {{\\bf{H}}_r^{\\rm{s}}\\left(1\\right)} & \\cdots & {{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right)}& \\cdots &{{\\bf{H}}_r^{\\rm{s}}\\left(N_{\\rm{u}}\\right)}\n  \\end{array}} \\right]\n  \\end{array}, \\label{eq3}\n\\end{equation}\nwhere\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{\\rm{s3}}}\\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{\\rm{s2}}{\\bf{\\Theta H}}_r^{\\rm{s1}} \\in \\mathbb{C}^{{\\rm{1}} \\times {N_{\\rm{R}}}} \\label{eq4}\n\\end{equation}\nrepresents the channel coefficient from the $n_u$-th user to the AP after being scattered by the scatterers, and\n${\\bf{H}}_r^{{\\rm{LOS}}} = {\\bm{\\alpha }}_r^{{\\rm{LOS}}}\\odot   {e^{j{\\bm{\\varphi }}_r^{{\\rm{LOS}}}}} \\in \\mathbb{C}^{{{N_{\\rm{u}}}} \\times {N_{\\rm{R}}}}$ represents the LOS channel coefficient from the user directly to the AP, with ${\\bm{\\alpha }}_r^{{\\rm{LOS}}}$ denoting the amplitude of the channel and ${e^{j{\\bm{\\varphi }}_r^{{\\rm{LOS}}}}}$ its phase shift. Similarly, ${\\bf{H}}_r^{{\\rm{IRS1}}} = {\\bm{\\alpha }}_r^{{\\rm{IRS1}}}\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{IRS1}}}}} \\in \\mathbb{C}^{{{N_{\\rm{u}}}} \\times {N_{\\rm{I}}}}$, ${{\\bf{H}}_r^{{\\rm{s1}}}} = {\\bm{\\alpha }}_r^{\\rm{s1}}\\odot {e^{j{\\bm{\\varphi }}_r^{\\rm{s1}}}} \\in \\mathbb{C}^{{{N_{\\rm{I}}}} \\times {N_{\\rm{R}}}}$, ${\\bf{H}}_r^{\\rm{s3}}\\left(n_{\\rm{u}}\\right) = {\\bm{\\alpha }}_r^{\\rm{s3}}\\left(n_{\\rm{u}}\\right)\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{s3}}}\\left(n_{\\rm{u}}\\right)}} \\in \\mathbb{C}^{{\\rm{1}} \\times {N_{\\rm{s}}}}$, ${\\bf{H}}_r^{{\\rm{s2}}} = {\\bm{\\alpha }}_r^{{\\rm{s2}}}\\odot {e^{j{\\bm{\\varphi }}_r^{{\\rm{s2}}}}} \\in \\mathbb{C}^{{{N_{\\rm{s}}}} \\times {N_{\\rm{I}}}}$ represent the LOS channel coefficient from the user to the IRS, from the IRS to the AP, from the user to the spatial point cloud location, and from the spatial point cloud location to the IRS, respectively.\n\nLet the $n_{\\rm{u}}$-th user's transmitted symbols on $R$ OREs be ${{\\bf{s}}_{{n_{\\rm{u}}}}} \\in \\mathbb{C}^{{{R}} \\times {1}}$, then at the $n_{\\rm{R}}$-th AP receiving antenna, the received data on all OREs can be expressed as\n\\begin{equation}\n{{\\bf{y}}_{{n_{\\rm{R}}}}} = \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {{\\rm{diag}}\\left\\{ {{{\\bf{H}}\\left({n_{\\rm{u}}},{n_{\\rm{R}}}\\right)}} \\right\\}{{\\bf{s}}_{{n_{\\rm{u}}}}}}  + {\\bf{w}}, \\label{eq5}\n\\end{equation}\nwhere ${{\\bf{H}}\\left({n_{\\rm{u}}},{n_{\\rm{R}}}\\right)} = \\left[ {\\begin{array}{*{20}{c}}\n  {{{\\bf{H}}_1}\\left( {{n_u},{n_R}} \\right)}& \\cdots &{{{\\bf{H}}_R}\\left( {{n_u},{n_R}} \\right)}\n  \\end{array}} \\right]$ , ${\\bf{H}}_r\\left(n_{\\rm{u}},n_{\\rm{R}}\\right) $ represents the $n_{\\rm{R}}$ column and $n_{\\rm{u}}$ row of ${{\\bf{H}}_r}$ calculated in (\\ref{eq3}), and $\\bf{w}$ the Gaussian white noise.\n\n\\section{The Multiple Access Scheme}\nIn order to recover multi-user communication data accurately, the wireless access algorithm can be used to achieve the separation and detection of multi-user transmission symbols under the premise of environmental prior information.\n\n\\subsection{Sparse Code Multiple Access}\nSCMA is an efficient code-domain non-orthogonal multiple access technology. It is based on low-density spectrum spreading. That means, a single user does not completely occupy all OREs, but only a few of them, which greatly reduces the difficulty of signal decoding. In the uplink multi-user SCMA communication scenario considered in this article, the $N_{\\rm{u}}$ users use the SCMA protocol to send their data to the AP and $N_{\\rm{u}}$ users share $R$ OREs. Each user has a total of $M$ input possibilities, and each user accesses the channel by using a unique sparse codebook ${{\\bf{C}}_{{n_{\\rm{u}}}}} \\in \\mathbb{C}^{{{R}} \\times {M}}$. Therefore, each user's codebook contains $M$ codewords. Let ${{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)} \\in \\mathbb{C}^{{{R}} \\times {1}}$ represent the $m$-th codeword of user $n_{\\rm{u}}$. Then the (\\ref{eq5}) can be expressed as\n\\begin{equation}\n{{\\bf{y}}_{{n_{\\rm{R}}}}} = \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {{\\rm{diag}}\\left\\{ {{{\\bf{H}}\\left({{n_{\\rm{u}}},{n_{\\rm{R}}}}\\right)}} \\right\\}{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)}}  + {\\bf{w}}, \\label{eq6}\n\\end{equation}\nwhere ${{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m\\right)}$ represents the sending symbols, it contains $d_{\\rm{v}}$ non-zero elements, that is, each user will only transmit on the OREs represented by $d_{\\rm{v}}$ non-zero elements. $N_{\\rm{u}}$ users perform overload transmission on all OREs, and the number of users $d_{\\rm{f}}$ transmitted on each ORE is constant. Since the codewords $\\bf{C}$ is sparse, not all users' codewords will collide on a single ORE. Fig. \\ref{figSCMA} shows an example of an uplink SCMA system, in which 6 users transmit on 4 OREs, thus $N_{\\rm{u}}$ = 6, $R$ = 4. Each user has its own codebook, and the codebook determines the OREs occupied by the user. In Fig. \\ref{figSCMA}, user 1 transmits on ORE 1 and 2, and user 2 transmits on ORE 3 and 4. The user's bitstream is mapped to the codewords by SCMA encoder after channel-coded, then transmitted to the receiver through the channel, and finally separated and decoded by the SCMA detector.\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=6in]{fig2.eps}\n  \\caption{The uplink SCMA system ($N_{\\rm{u}} = 6$, $R = 4$).}\n  \\label{figSCMA}\n  \\end{figure*}\n\nCodebook design is an important physical layer technology in the SCMA system. The sparsity of the codebook makes it possible for the SCMA receiver to use message passing algorithm (MPA) to decode. As mentioned above, the process of SCMA encoding is the process of mapping the binary bit stream to the complex domain. The codebook of each user is an ${R}\\times{M}$-dimensional matrix. Therefore, the SCMA encoder can be defined as: $f:\\mathbb{B}^{{{\\log }_2}M} \\to {\\cal X}$, where. ${\\cal X} \\subset \\mathbb{C} ^R,\\left| {\\cal X} \\right| = M$. Let $\\bf{b}$ represent the user's input bits, the corresponding codeword output can be expressed as ${\\cal X} = f\\left( {\\bf{b}} \\right)$, the codeword ${\\cal X}$ is an $R$-dimensional sparse complex vector, and the vector contains $N_{\\rm{c}} < R$ non-zero elements. Since SCMA encoding process combines bit-to-constellation mapping and spreading spectrum, the bit-to-constellation mapping can be expressed as: $g:\\mathbb{B} ^{{{\\log }_2}M} \\to {\\cal C},\\;\\;{\\cal C} \\subset \\mathbb{C} ^{N_{\\rm{c}}}$, where ${\\cal C}$ represents the constellation point of the $N_{\\rm{c}}$-dimensional complex constellation, so the SCMA encoder can also be expressed as : $f = {\\bf{V}}g$, where ${\\cal C} = f\\left( {\\bf{b}} \\right)$ and ${\\bf{V}} \\in \\mathbb{B} ^{R \\times N}$ is a binary mapping matrix, and the mapping matrix can map $N_{\\rm{c}}$-dimensional constellation points to $R$-dimensional SCMA codewords. Meanwhile, the mapping matrix of each user is different, and contains $R-N_{\\rm{c}} $ all-zero rows.\n\nDefine the codebook structure of SCMA as ${\\cal S}\\left( {{\\cal V},{\\cal G};{N_{\\rm{u}}},M,N_{\\rm{c}},R} \\right)$, where ${\\cal V}: = \\left[ {{{\\bf{V}}_{{n_{\\rm{u}}}}}} \\right]_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}}$, ${\\cal G}: = \\left[ {{g_{{n_{\\rm{u}}}}}} \\right]_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}}$. Therefore, the SCMA codebook design problem can be expressed as\n\\begin{equation}\n{{\\cal V}^*},{{\\cal G}^*} = \\arg \\mathop {\\max }\\limits_{{\\cal V},{\\cal G}} {\\cal M}\\left( {{\\cal S}\\left( {{\\cal V},{\\cal G};{N_{\\rm{u}}},M,N_{\\rm{c}},R} \\right)} \\right), \\label{eq7}\n\\end{equation}\nwhere ${\\cal M}$ is a codebook design standard. Since there is no unified design standard at present, there are many methods for SCMA codebook design problems, such as rearranging the real and imaginary parts of the constellation points and designing codebooks based on theories such as constellation interleaving and rotation. These methods can achieve a suboptimal solution to the SCMA codebook design problem.\n\n\\subsection{SCMA-IRS-MPA Decoder}\\label{mainalgm}\nIn the above-mentioned SCMA-IRS uplink transmission scheme, there are a total of ${M^{{N_{\\rm{u}}}}}$ combinations of user codewords. The Maximum Likelihood (ML) decoder can provide the theoretically optimal symbol error rate (SER) performance by performing a traversal search on all codewords combinations. The estimated transmit codewords of all users by the ML decoder can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {{{\\bf{\\hat C}}}_{{\\rm{ML}}}} = \\arg \\mathop {\\min }\\limits_{j \\in {M^{{n_{\\rm{u}}}}}} \\\\\n  \\quad \\quad {\\left\\| {{{\\bf{y}}_{{n_{\\rm{R}}}}} - \\sum\\limits_{{n_{\\rm{u}}} = 1}^{{N_{\\rm{u}}}} {\\left( {{\\rm{diag}}\\left( {{\\bf{H}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right)} \\right){{\\bf{C}}_{{n_{\\rm{u}}}}}\\left( {{\\bf{m}}\\left( j \\right)} \\right)} \\right)} } \\right\\|^2}\n  \\end{array}, \\label{eq8}\n\\end{equation}\nwhere ${{\\bf{\\hat C}}_{\\rm{ML}}} = \\left[ {{{{\\bf{\\hat c}}}_{1}}, \\cdots ,{{{\\bf{\\hat c}}}_{{N_{\\rm{u}}}}}} \\right] \\in \\mathbb{C} ^{R \\times {N_{\\rm{u}}}}$, ${\\bf{m}}(j)$ represents the value of the $j$-th combination among ${M^{{N_{\\rm{u}}}}}$ user codewords combinations. Although the ML decoder can provide the theoretical optimal value, it uses an exhaustive method to search for the optimal solution, which is impractical in the actual implementation process. MPA decoder is an iterative decoder, which can nearly achieve the performance of an ML decoder while requiring an achievable computational complexity. And the MPA decoder obtains the corresponding user codewords by calculating the maximum joint message probability.\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7cm]{fig3.eps}\n  \\caption{The MPA decoder factor graph ($N_{\\rm{u}} = 6$, $R = 4$).}\n  \\label{figFG}\n  \\end{figure}\n\nMPA is a belief propagation algorithm that uses a factor graph model to solve probabilistic reasoning problems. The proposed SCMA-IRS-MPA uses the factor graph method shown in the Fig. \\ref{figFG}, where the function nodes (FNs) represent OREs, the variable nodes (VNs) represent the users, and the connection between the FN and VN represents the user transmitting data on the corresponding ORE. The MPA decoder achieves decoding by iteratively updating the message probability between FNs and VNs, and let the MPA decoder stop after ${K_{\\rm{it}}}$ iterations. In order to estimate the transmission codewords in the SCMA-IRS-MPA scheme, we modified the traditional SCMA-MPA. In our method, multiple antennas at the AP perform independent and parallel decoding. For the $n_{\\rm{R}}$-th receiving antenna, we use $p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ to denote the probability of transmitting a message from the $n_{\\rm{u}}$-th VN to the $r$-th FN, and use $p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ to denote the probability of transmitting a message from the $r$-th FN to the $n_{\\rm{u}}$-th VN. The above all represent the probability in the ${K_{\\rm{it}}}$ round iteration, ${k_{\\rm{it}}} = 1,2, \\cdots ,{K_{\\rm{it}}}$. Assuming that at the beginning, in the first iteration, all messages sent from VN to FN have the same probability,\n\\begin{equation}\np_{{v_u} \\to {f_r}}^{\\left( 0 \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right) = \\frac{1}{M},\\left( {\\forall {n_{\\rm{u}}},\\forall r,\\forall m} \\right). \\label{eq9}\n\\end{equation}\n\nTherefore, $p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ can be expressed as\n\\begin{equation}\n\\begin{array}{l}\n  p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right){\\rm{ = }}\\\\\n  \\quad \\quad \\sum\\limits_{\\psi \\left( i \\right),i \\in {{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}} {\\left\\{ {p\\left( {{\\bf{y}}|\\psi \\left( i \\right),\\psi \\left( u \\right) = {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}}} \\right)} \\right.} \\\\\n  \\quad \\quad \\left. { \\times \\prod\\limits_{i \\in {{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}} {p_{{v_i} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}} \\left( {\\psi \\left( i \\right)} \\right)} \\right\\},\\left( {\\forall m,\\forall r,{n_{\\rm{u}}} \\in {{\\bf{\\Lambda }}_r}} \\right)\n  \\end{array}, \\label{eq10}\n\\end{equation}\nwhere ${{\\bf{\\Lambda }}_r}$ represents a set of user indexes sharing the $r$-th ORE, ${{\\bf{\\Lambda }}_r}\\backslash {n_{\\rm{u}}}$ represents ${{\\bf{\\Lambda }}_r}$ except for the $n_{\\rm{u}}$-th user, and\n\\begin{equation}\n\\begin{array}{l}\n  p\\left( {{\\bf{y}}|{{\\bf{\\Psi }}_r}} \\right) = \\frac{1}{{\\sqrt {2\\pi } \\sigma }}\\exp \\left( { - \\left| {{{\\bf{y}}_r} - \\sum\\nolimits_{{n_{\\rm{u}}} \\in {{\\bf{\\Lambda }}_r}} {\\left( {{\\bf{H}}_r^{{\\rm{LOS}}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right)} \\right.} } \\right.} \\right.\\\\\n   \\quad \\left. {{{{{\\left. {\\left. { + {\\bf{H}}_r^{{\\rm{IRS}}}\\left( {{n_{\\rm{u}}},{n_{\\rm{R}}}} \\right) + {\\bf{H}}_r^{\\rm{s}}\\left( {{n_{\\rm{u}},n_{\\rm{R}}}} \\right)} \\right){{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right|}^2}} \/ {\\left( {2{\\sigma ^2}} \\right)}}} \\right)\n  \\end{array}, \\label{eq11}\n\\end{equation}\nwhere ${{\\bf{\\Psi }}_r}$ represents the possible codewords of all users sharing the $r$-th ORE, then $p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)$ is updated to,\n\\begin{equation}\n\\begin{array}{l}\n  p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right) = \\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\\\\n  \\quad \\quad \\quad  \\times \\prod\\limits_{j \\in {\\Omega _u}\\backslash r} {p_{{f_r} \\to {v_u}}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} ,\\forall m,\\forall {n_{\\rm{u}}},r \\in {{\\bf{\\Omega }}_u}\n  \\end{array}, \\label{eq12}\n\\end{equation}\nwhere ${{\\bf{\\Omega }}_u}$ represents the ORE index corresponding to the $d_{\\rm{v}}$ non-zero element positions of the codeword of the $n_{\\rm{u}}$-th user, ${{\\bf{\\Omega }}_u}\\backslash r$ represents ${{\\bf{\\Omega }}_u}$ except for the $r$-th ORE, and $\\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}$ can be expressed as\n\\begin{equation}\n\\gamma _{{v_u},r}^{\\left( {{k_{\\rm{it}}} + 1} \\right)}{\\rm{ = }}{\\left( {\\sum\\limits_{m = 1}^M {p_{{v_u} \\to {f_r}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} } \\right)^{ - 1}}. \\label{eq13}\n\\end{equation}\n\nAfter $K_{\\rm{it}}$ iterations, the estimated transmission codewords of the $n_{\\rm{u}}$-th user can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat C}}}_{{n_{\\rm{u}}}} ^{\\left( {{k_{it}}} \\right)}} = \\arg \\mathop {\\max }\\limits_{m = 1, \\cdots M} \\prod\\limits_{j \\in {\\Omega _u}} {p_{{f_j} \\to {v_u}}^{\\left( {{k_{\\rm{it}}}} \\right)}\\left( {{{\\bf{C}}_{{n_{\\rm{u}}}}\\left(m,r\\right)}} \\right)} ,\\forall n_{\\rm{u}}. \\label{eq14}\n\\end{equation}\n\nThe set of all user transmission codewords obtained by using the SCMA-IRS-MPA decoder is\n\\begin{equation}\n{{\\bf{\\hat C}}_{{\\rm{MPA}}}} = \\left\\{ {{{ {{{{\\bf{\\hat c}}}_1}} }^{\\left( {{k_{\\rm{it}}}} \\right)}}, \\cdots ,{{ {{{{\\bf{\\hat c}}}_{{N_{\\rm{u}}}}}} }^{\\left( {{k_{\\rm{it}}}} \\right)}}} \\right\\}. \\label{eq15}\n\\end{equation}\n\nThe above is the MPA decoder of the SCMA-IRS scheme. We express the decoding computational complexity of the MPA decoder according to the number of addition operations and the number of multiplication operations. Therefore, the number of additions and multiplications required by the MPA detector are $R{d_{\\rm{f}}}\\left( {{M^{{d_{\\rm{f}}}}}\\left( {4{d_{\\rm{f}}} + {K_{\\rm{it}}} + 1} \\right) + N - {K_{\\rm{it}}}} \\right) + 1$ and $R{d_{\\rm{f}}}\\left( {{M^{{d_{\\rm{f}}}}}\\left( {4{d_{\\rm{f}}} + {K_{\\rm{it}}}{d_{\\rm{f}}} + 3} \\right) + N + M{K_{\\rm{it}}}\\left( {{d_{\\rm{v}}} - 1} \\right)} \\right) + {N_{\\rm{u}}}M\\left( {{d_{\\rm{v}}} - 1} \\right)$ respectively.\n\n\\section{Environment Sensing}\nContrary to the multiple access process, the algorithm proposed in this section can sense the environmental information with the data sent by the user has been decoded correctly. Since the distribution of scatterers in the environment is sparse, sensing environmental information is essential to solve the CS reconstruction problem. As shown in (\\ref{eq5}), on the $r$-th ORE, the solution of environmental information can be expressed as\n\\begin{equation}\n{\\bf{\\hat x}} = \\arg \\mathop {\\min }\\limits_{\\bf{x}} {\\left\\| {\\bf{x}} \\right\\|_1}\\quad \\quad {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\| {{{\\bf{y}}_r} - {{\\bf{s}}_r}{{\\bf{H}}_r}} \\right\\|_2} \\le {\\varepsilon _{\\rm{x}}}, \\label{eq16}\n\\end{equation}\nwhere $\\varepsilon _{\\rm{x}}$ is the slack variable, ${{\\bf{y}}_r} \\in \\mathbb{C}^{{N_{\\rm{T}}} \\times {N_{\\rm{R}}}}$ is the symbol sequence received by the AP receiving antennas, $N_{\\rm{T}}$ is the time sequence length, and ${{\\bf{s}}_r} \\in \\mathbb{C} ^{{N_{\\rm{T}}} \\times {N_{\\rm{u}}}}$ is the transmitted symbol sequence of $N_{\\rm{u}}$ users. In the received signal model, when both the transmitted data ${\\bf{s}}_r$ and the received data ${\\bf{y}}_r$ are known, as shown in (\\ref{eq6}), the channel coefficients ${{\\bf{H}}\\left({{n_{\\rm{u}}},{n_R}}\\right)}$ can be obtained by simply solving the linear equations. After performing the same analysis on all the receiving antennas of the AP, the channel coefficient ${\\bf{H}}_r$ on the $r$-th ORE is solved.\n\nAs shown in (\\ref{eq3}), the ${\\bf{H}}_r^{{\\rm{LOS}}}$ and ${\\bf{H}}_r^{{\\rm{IRS}}}$ in the channel coefficient ${\\bf{H}}_r$ are composed of LOS channels. Meanwhile, the reflection characteristic matrix $\\bf{\\Theta} $ of the IRS used for assist communication is given, and only ${\\bf{H}}_r^{\\rm{s}}$ contains unknown environmental information. The $n_{\\rm{u}}$-th row of ${\\bf{H}}_r^{\\rm{s}}$ is expressed as\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}}\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{{\\rm{s3}}}} \\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{{\\rm{s2}}}{\\bf{\\Theta H}}_r^{{\\rm{s1}}}, \\label{eq17}\n\\end{equation}\n\\begin{equation}\n{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right)\\right)^{\\rm{T}}} = {{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right){\\bf{x}}, \\label{eq18}\n\\end{equation}\nwhere ${{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}}\\right) \\in \\mathbb{C}^{{N_{\\rm{R}}} \\times {N_{\\rm{s}}}}$ is the known channel coefficient, which is also called the measurement matrix in the CS problem. For $N_{\\rm{u}}$ users, the (\\ref{eq18}) is expressed as the matrix form of the CS problem,\n\\begin{equation}\n{\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(1\\right) \\right)}^{\\rm{T}}}}\\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(2\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(N_{\\rm{u}}\\right) \\right)}^{\\rm{T}}}}\n  \\end{array}} \\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}} \\times 1}} = {\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(1\\right)\\\\\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(2\\right)\\\\\n   \\vdots \\\\\n  {{{\\bf{A}}_r^{\\rm{s}}}}\\left(N_{\\rm{u}}\\right)\n  \\end{array}} \\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}} \\times {N_{\\rm{s}}}}}{\\left[ {\\bf{x}} \\right]_{{N_{\\rm{s}}} \\times 1}}\n \\end{equation} \n\\begin{equation}\n\\Rightarrow  {{\\bf{\\tilde{H}}}_r^{\\rm{s}}}  = {\\bf{\\tilde{A}}}_r^{\\rm{s}}{\\bf{x}}. \\label{eq19}\n\\end{equation}\n\n\\subsection{Generalized Approximate Message Passing}\nThe GAMP Algorithm\\cite{Rangan} solves the problem of CS sparse reconstruction by iterative decomposition. The above problem formula (\\ref{eq19}) is abbreviated as ${\\bf{y}} = {\\bf{\\Phi x}} + {\\bf{w}}$, where ${\\bf{\\Phi }} \\in \\mathbb{C} ^{{M_\\phi } \\times {N_\\phi }}$ is the CS measurement matrix and ${\\bf{w}} \\sim {\\cal C}{\\cal N}\\left( {0,{\\sigma ^{\\rm{w}}}} \\right)$ represents noise. In this article, we assume that the distribution of environmental scatterers information as a Bernoulli-Gaussian distribution in a limited interval which probability density function is expressed as\n\\begin{equation}\n\\begin{array}{l}\n  {p_{X{\\rm{|}}{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right) = \\left( {1 - \\lambda  + \\alpha } \\right)\\delta \\left( x \\right)\\\\\n  \\quad \\quad \\quad  + \\lambda {\\cal N}\\left( {x|\\theta ,{\\sigma ^{\\rm{x}}}} \\right)\\left[ {u\\left( x \\right) - u\\left( {x - 1} \\right)} \\right]\n  \\end{array}, \\label{eq20}\n\\end{equation}\nwhere all parameters be expressed as ${\\bf{q}} \\buildrel \\Delta \\over = \\left[ {\\lambda ,\\alpha ,\\theta ,{\\sigma ^{\\rm{x}}}} \\right]$, $\\delta \\left(  \\cdot  \\right)$ is the Dirac function, $\\lambda $ is the sparsity coefficient, $\\alpha {\\rm{ = }}\\int_{x \\in \\left( { - \\infty ,0} \\right] \\cup \\left[ {1, + \\infty } \\right)} {\\lambda {\\cal N}\\left( {x|\\theta ,{\\sigma ^{\\rm{x}}}} \\right)} dx$. $\\theta  \\in \\left[ {{\\rm{0}},{\\rm{1}}} \\right]$ and ${\\sigma ^{\\rm{x}}}$ represent the mean and variance of the environmental scatterers information distribution respectively.\n\nThe GAMP algorithm has defined two parameterized functions ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$ and the specific algorithm is shown in Algorithm 1. At this point, we will show how to specify the parameterized functions ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$, ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$, ${g'_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g'_{\\rm{out}}}\\left(  \\cdot  \\right)$, based on the maximum posterior probability (MAP) estimation, the input function can be written as\n\\begin{equation}\n{g_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\arg \\mathop {\\max }\\limits_x {F_{\\rm{in}}}\\left( {x,\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right), \\label{eq22}\n\\end{equation}\n\\begin{equation}\n{F_{\\rm{in}}}\\left( {x,\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\log {p_{X|{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right) - \\frac{1}{{2{\\sigma ^{\\rm{v}}}}}{\\left( {\\hat v - x} \\right)^2}, \\label{eq23}\n\\end{equation}\n\\begin{equation}\n{g'_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right) = \\frac{\\partial {g_{\\rm{in}}}\\left( {\\hat v,{\\sigma ^{\\rm{v}}},{\\bf{q}}} \\right)}{\\partial \\hat v} = \\frac{1}{{1 - {\\sigma ^{\\rm{v}}}\\frac{\\partial ^2}{\\partial  x^2} {\\rm log}\\left[{p_{X|{\\bf{Q}}}}\\left( {x|{\\bf{q}}} \\right)\\right]}}, \\label{eq24}\n\\end{equation}\nthe output function can be expressed as\n\\begin{equation}\n{g_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right) = \\frac{{y - \\hat p}}{{{\\sigma ^{\\rm{w}}}} + {\\sigma ^{\\rm{z}}}}, \\label{eq25}\n\\end{equation}\n\\begin{equation}\n{g'_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right) = \\frac{\\partial {g'_{\\rm{out}}}\\left( {y,\\hat p,{\\sigma ^{\\rm{z}}}} \\right)}{\\partial y} =  - \\frac{1}{{{\\sigma ^{\\rm{w}}}} + {\\sigma ^{\\rm{z}}}}. \\label{eq26}\n\\end{equation}\n\n\\begin{algorithm}[htb]\n  \\caption{The GAMP Algorithm\\cite{Rangan}}\n \n  \\begin{algorithmic}[1]\n  \\REQUIRE\n  Given measurement matrix ${\\bf{\\Phi }} \\in {\\mathbb{C}^{{M_\\phi } \\times {N_\\phi }}}$ and sequence of measurement value ${\\bf{y}} \\in \\mathbb{C} ^{{M_\\phi } \\times 1}$.\n  \\STATE\n  \\textbf{Initialization}: Set environment prior parameter $\\bf{q}$. Defined ${g_{\\rm{in}}}\\left(  \\cdot  \\right)$ and ${g_{\\rm{out}}}\\left(  \\cdot  \\right)$ from (\\ref{eq22}), (\\ref{eq24}). Set $t_i = 0$, ${\\bf{\\hat s}}\\left( { - 1} \\right) = 0$, ${\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right) > 0$, $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right) > 0$.\n\n  \\WHILE {$\\sum\\limits_{{m_\\phi }} {\\left( {{y_{{m_\\phi }}} - {{\\hat z}_{{m_\\phi }}}\\left( {{t_i}} \\right)} \\right)}  > \\varepsilon_{\\rm{t}} $, where $\\varepsilon_{\\rm{t}} $ is a given error tolerance value}\n  \\STATE\n  For each $m_\\phi $:\n\n  $\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right) = \\sum\\limits_{{n_\\phi }} {\\Phi _{{m_\\phi },{n_\\phi }}^2} \\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right),$\n\n  ${\\hat p_{m_\\phi }}\\left( {t_i} \\right) = \\sum\\limits_{n_\\phi } {\\Phi _{{m_\\phi },{n_\\phi }}}{{\\hat x}_{n_\\phi }}\\left( {t_i} \\right) - \\sigma_{m_\\phi }^{\\rm{z}}\\left( t \\right) {\\hat s_{{m_\\phi }}}\\left( {{t_i} - 1} \\right),$\n\n  ${\\hat z_{{m_\\phi }}}\\left( {{t_i}} \\right){\\rm{ = }}\\sum\\limits_{{n_\\phi }} {{\\Phi _{{m_\\phi },{n_\\phi }}}} {\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right).$\n  \\STATE\n  For each $m_\\phi $:\n\n  ${\\hat s_{{m_\\phi }}}\\left( {{t_i}} \\right) = {g_{\\rm{out}}}\\left( {{t_i},{y_{{m_\\phi }}},{{\\hat p}_{{m_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right)} \\right),$\n\n  $\\sigma _{{m_\\phi }}^{\\rm{s}}\\left( {{t_i}} \\right) =  - {g'_{\\rm{out}}}\\left( {{t_i},{y_{{m_\\phi }}},{{\\hat p}_{{m_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{m_\\phi }}^{\\rm{z}}\\left( {{t_i}} \\right)} \\right).$\n  \\STATE\n  For each $n_\\phi $:\n\n  $\\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right) = {\\left[ {\\sum\\limits_{{n_\\phi }} {\\Phi _{{m_\\phi },{n_\\phi }}^2\\sigma _{{n_\\phi }}^{\\rm{s}}\\left( {{t_i}} \\right)} } \\right]^{ - 1}},$\n\n  ${\\hat v_{{n_\\phi }}}\\left( {{t_i}} \\right) = {\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right) + \\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right)\\sum\\limits_{{m_\\phi }} {{\\Phi _{{m_\\phi },{n_\\phi }}}{{\\hat s}_{{m_\\phi }}}\\left( {{t_i}} \\right)}.$\n  \\STATE\n  For each $n_\\phi $:\n\n  ${\\hat x_{{n_\\phi }}}\\left( {{t_i}{\\rm{ + 1}}} \\right) = {g_{\\rm{in}}}\\left( {{t_i},{{\\hat v}_{{n_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right),{\\bf{q}}} \\right),$\n\n  $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}{\\rm{ + 1}}} \\right) = \\sigma _{{n_\\phi }}^{\\rm{v}}\\left( {{t_i}} \\right){g'_{\\rm{in}}}\\left( {{t_i},{{\\hat v}_{{n_\\phi }}}\\left( {{t_i}} \\right),\\sigma _{{n_\\phi }}^r\\left( {{t_i}} \\right),{\\bf{q}}} \\right).$\n  \\STATE\n  ${t_i} = {t_i} + 1.$\n  \\ENDWHILE\n  \\ENSURE\n  Estimated sparse vector ${\\hat x_{{n_\\phi }}}\\left( {{t_i}} \\right)$ and $\\sigma _{{n_\\phi }}^{\\rm{x}}\\left( {{t_i}} \\right)$.\n  \\end{algorithmic}\n  \\end{algorithm}\n\n\\subsection{Proposed Environment Sensing Algorithm}\nIn the process of solving the CS sparse reconstruction problem, the product $N_{\\rm{u}}N_{\\rm{R}}$ of the number of users and the number of receiving antennas, and the number of spatial pixels $N_{\\rm{s}}$ are orders of magnitude different, that is ${N_{\\rm{u}}}{N_{\\rm{R}}} \\ll {N_{\\rm{s}}}$, so that the number of columns in the CS measurement matrix ${\\bf{\\tilde{A}}}_r^{\\rm{s}}$ is much larger than the number of rows, and the compression ratio is too high. Therefore, the environmental information $\\bf{x}$ cannot be recovered accurately. We improve the above algorithm to adapt to the continuous data stream sent from users in the proposed scenario. We use multiple data packets to recover the environmental information after multiple observations. The CS problem is redefined as\n\\begin{equation}\n{\\bf{H}}_r^{\\rm{s}}\\left(n_{\\rm{u}},k\\right) = {{\\bf{x}}^{\\rm{T}}}{\\rm{diag}}\\left( {{\\bf{H}}_r^{{\\rm{s3}}}}\\left(n_{\\rm{u}}\\right) \\right){\\bf{H}}_r^{{\\rm{s2}}}{{\\bf{\\Theta }}\\left(k\\right)}{\\bf{H}}_r^{{\\rm{s1}}}, \\label{eq27}\n\\end{equation}\n\\begin{equation}\n{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right) \\right)^{\\rm{T}}} = {\\bf{A}}_r^{\\rm{s}}\\left(n_{\\rm{u}},k\\right){\\bf{x}}, \\label{eq28}\n\\end{equation}\n\\begin{equation}\n{\\left[ {\\begin{array}{*{20}{c}}\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(1,1\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right) \\right)}^{\\rm{T}}}}\\\\\n   \\vdots \\\\\n  {{{\\left( {{\\bf{H}}_r^{\\rm{s}}}\\left(N_{\\rm{u}},K\\right) \\right)}^{\\rm{T}}}}\n  \\end{array}} \\right]} = {\\left[ {\\begin{array}{*{20}{c}}\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(1,1\\right)\\\\\n   \\vdots \\\\\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(n_{\\rm{u}},k\\right)\\\\\n   \\vdots \\\\\n  {{\\bf{A}}_r^{\\rm{s}}}\\left(N_{\\rm{u}},K\\right)\n  \\end{array}} \\right]} \\left[{\\bf{x}}\\right] , \\label{eq29}\n\\end{equation}\n\\begin{equation}\n\\Rightarrow  \\left[{{\\bf{\\tilde{H}}}_r^{\\rm{s}}}\\left(K\\right)\\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}}K \\times 1}  = \\left[{\\bf{\\tilde{A}}}_r^{\\rm{s}}\\left(K\\right)\\right]_{{N_{\\rm{u}}}{N_{\\rm{R}}}K \\times {N_{\\rm{s}}}}\\left[{\\bf{x}}\\right]_{{N_{\\rm{s}}}\\times 1}, \\label{eq30}\n\\end{equation}\nwhere ${\\bf{\\Theta }}\\left(k\\right)$ represents the IRS reflection characteristic matrix when the $k$-th data packet is received, and $K$ is the number of data packets. After multiple observations, the difference between the number of rows of the observation matrix $N_{\\rm{u}}N_{\\rm{R}}K$ and the number of columns $N_{\\rm{s}}$ is relatively small, and environmental information can be sensed more accurately.\n\nSince data packets are continuously transmitted during the communication process, and the amount of data is very large, it is impossible to store all the data packets $K$ sent at all times. We set a time sliding-window with a length of $n_{\\rm{f}}$, store the received data packet $k$ at the current moment to the $n_{\\rm{f}}$ data packets previously received, and use the data in the sliding-window for environment sensing.\n\nCompared with the communication data that exists all the time, the data in the sliding-window is very limited. Therefore, a large amount of communication data transmitted earlier will be wasted in the process of environment sensing. To solve this problem, we propose a ``momentum-mode'', which combines the sensing results calculated at the previous moments to calculate the current environment sensing results and makes the current sensing result contain part of the information outside the sliding-window. According to (\\ref{eq16}), (\\ref{eq18}), the ``momentum-mode'' can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat x}}}_k}{\\rm{ = }}\\arg \\mathop {\\min }\\limits_{\\bf{x}} \\left( {{{\\left\\| {\\bf{x}} \\right\\|}_1}} \\right){\\rm{ + }}\\mu {{{\\bf{\\hat x}}}_{k{\\rm{ - 1}}}}, \\label{eq31}\n\\end{equation}\n\\begin{equation}\n{\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\|{ {{\\bf{\\tilde{H}}}_r^{\\rm{s}}}\\left(k\\right) - {\\bf{\\tilde{A}}}_r^{\\rm{s}}\\left(k\\right)}{\\bf{x}} \\right\\|_2} \\le {\\varepsilon _{\\rm{x}}}, \\label{eq32}\n\\end{equation}\nwhere $\\mu $ is the momentum coefficient. The larger the momentum coefficient $\\mu $, the more previous data information the current sensing result depends on. Therefore, the setting of the momentum coefficient $\\mu $ should be set according to the actual system and will be further analyzed in section \\uppercase\\expandafter{\\romannumeral7}.\n\n\\section{Joint Multi-User Detection and Environment Sensing Algorithm}\nAs mentioned in Section \\uppercase\\expandafter{\\romannumeral3} and Section \\uppercase\\expandafter{\\romannumeral4}, the accurate decoding of user communication data requires the knowledge of the environment, and if the data sent by the user is not recovered, the environmental information cannot be sensed accurately. Although a sufficient number of pilots can be used to implement the proposed environment sensing algorithm, an excessive number of pilots will cause a decrease in communication efficiency. To tackle this issue, we propose an iterative and incremental algorithm based on low-density pilots to jointly recover users' communication data and environmental information. Let ${{\\bf{s}}_k}\\left( {0 < k \\le K} \\right)$ denote the $k$-th data packets of all users. We insert a pilot $\\bf{P}$ before each $K$ data packets, and the AP can obtain the received data ${\\bf{y}}_{\\rm{p}}$. According to the previous section, based on the pilot $\\bf{P}$, the environmental information ${{\\bf{\\hat x}}_{\\rm{p}}}$ can be roughly estimated. Meanwhile, we need to use subsequent communication data packets to further improve the environment sensing results. Therefore, we use the pilot $\\bf{P}$, the received data ${\\bf{y}}_{\\rm{p}}$ and the estimated environmental information ${{\\bf{\\hat x}}_{\\rm{p}}}$ as initial terms to start the iterative algorithm.\n\n\\subsection{The Proposed Iterative Algorithm}\n\n\\begin{figure*}\n  \\centering\n  \\includegraphics[width=6in]{fig4.eps}\n  \\caption{The proposed iterative algorithm.}\n  \\label{figIT}\n  \\end{figure*}\n\nAs shown in Fig. \\ref{figIT}, after receiving the $k$-th packet data ${\\bf{y}}_k$, the proposed iterative algorithm is divided into three parts:\n\n1. Forward propagation: First, use the estimated environment information ${{\\bf{\\hat x}}_{k - 1}}$ after receiving the $(k-1)$-th data packet to estimate the current channel ${{\\bf{\\hat H}}_k}$, and then the decoder decodes the received data ${\\bf{y}}_k$ to send data ${{\\bf{\\hat s}}_k}$ based on the estimation of current channel ${{\\bf{\\hat H}}_k}$. Finally, based on current ${\\bf{y}}_k$, ${{\\bf{\\hat s}}_k}$, the received data ${{\\bf{y}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{y}}_{k - 1}}$ and the decoded data ${{\\bf{\\hat s}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ in the previous $n_{\\rm{f}}$ data packets, the current environmental information ${{\\bf{\\hat x}}_k}$ can be estimated more accurately. In the initial stage of the proposed algorithm, the results have not converged. Therefore, a certain number of iterations are required to make the system performance converge to a relatively accurate estimation of the transmitted data and environmental information, when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert _2 < {\\varepsilon _{\\rm{k}}}$, the ``momentum-mode'' is enabled.\n\n2. Self-iteration: First, estimate the current environment ${{\\bf{\\hat x}}_k}$ by using the channel estimated ${{\\bf{\\hat H}}_k}$ in the forward propagation part. Then, the decoder decodes the currently transmitted data ${{\\bf{\\hat s}}_k}$ again and estimates the current environment information ${{\\bf{\\hat x}}_k}$ again. \nFinally, iterate $K_{\\rm{s}}$ times to obtain more accurate transmission data and environmental information. $K_s$ can be set to a fixed value, or it can be gradually reduced as the algorithm converges, especially when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$, $K_s$ should be set to a small value. Enable the ``momentum-mode'' when $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$.\n\n3. Feedback: feedback the estimated environmental information ${{\\bf{\\hat x}}_k}$ in the $k$-th data packet to the previous $n_{\\rm{b}}$ data packets, and based on the more accurate environmental information ${{\\bf{\\hat x}}_k}$ estimated by $k$-th data packet, estimate the received signal ${{\\bf{\\hat s}}_{k - {n_{\\rm{b}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ again to improve the accuracy of decoding. Stop feedback when $\\left\\lVert {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - n_{\\rm{b}} - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$.\n\n\\begin{figure*}[htb]\n  \\centering\n  \\includegraphics[width=6in]{fig15.eps}\n  \\caption{The Factor graph representation of the proposed iterative algorithm. }\n  \\label{figITGH}\n  \\end{figure*}\n\n\\begin{algorithm}[t]\n  \\caption{The proposed iterative algorithm}\n  \\label{alg2}\n  \\begin{algorithmic}[1] \n  \\REQUIRE\n  Calculated LOS channel matrix ${\\bf{H}}_r^{{\\rm{LOS}}}$, ${\\bf{H}}_r^{{\\rm{IRS1}}}$, ${\\bf{H}}_r^{{\\rm{s1}}}$, ${\\bf{H}}_r^{{\\rm{s2}}}$and ${\\bf{H}}_r^{{\\rm{s3}}}$. Given IRS reflection characteristic control matrix $\\bf{\\Theta}$.\n  \\STATE\n  \\textbf{Initialization}: Set pilot $\\bf{P}$. Defined sliding-window length $n_{\\rm{f}}$, $n_{\\rm{b}}$. Set ${\\varepsilon _{\\rm{k}}} > 0$, $0 < \\mu  < 1$, $K_{\\rm{s}} > 0$.\n  \\STATE\n  Estimate ${{\\bf{\\hat x}}_{\\rm{p}}}$ from $\\bf{P}$ and ${\\bf{y}}_{\\rm{p}}$ by GAMP. Let ${{\\bf{\\hat x}}_{\\rm{0}}} = {{\\bf{\\hat x}}_{\\rm{p}}}$, ${\\bf{y}}_0 = {\\bf{y}}_{\\rm{p}}$, ${{\\bf{\\hat s}}_0} = {\\bf{P}}$.\n  \\FOR {$k = 1, 2, \\cdots ,K$}\n\n \n  \\STATE\n  Estimate ${{\\bf{\\hat H}}_k}$ from ${{\\bf{\\hat x}}_{k - 1}}$ according to (\\ref{eq3}).\n  \\STATE\n  Estimate ${{\\bf{\\hat s}}_k}$ from ${\\bf{y}}_k$ and ${{\\bf{\\hat H}}_k}$ based on SCMA-IRS-MPA decoder.\n  \\STATE\n  Estimate ${{\\bf{\\hat x}}_k}$ from ${{\\bf{y}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{y}}_{k - 1}}$ and ${{\\bf{\\hat s}}_{k - {n_{\\rm{f}}} - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - 1}}$ according to formula (\\ref{eq31}).\n  \\STATE\n  Replace ${{\\bf{\\hat x}}_{k - 1}}$ with ${{\\bf{\\hat x}}_k}$, Repeat steps 3 to 5 $K_{\\rm{s}}$ times.\n \n  \\STATE\n  Estimate ${{\\bf{\\hat H}}_{k - 1}}, \\cdots ,{{\\bf{\\hat H}}_{k - {n_{\\rm{b}}} - 1}}$ from ${{\\bf{\\hat x}}_k}$ according to (\\ref{eq3}).\n  \\STATE\n  Estimate ${{\\bf{\\hat s}}_{k - 1}}, \\cdots ,{{\\bf{\\hat s}}_{k - {n_{\\rm{b}}} - 1}}$ from  ${{\\bf{y}}_{k - 1}}, \\cdots ,{{\\bf{y}}_{k - {n_{\\rm{b}}} - 1}}$ and ${{\\bf{\\hat H}}_{k - 1}}, \\cdots ,{{\\bf{\\hat H}}_{k - {n_{\\rm{b}}} - 1}}$ according to (\\ref{eq31}).\n  \\STATE\n  If $\\left\\lVert  {{{{\\bf{\\hat x}}}_k} - {{{\\bf{\\hat x}}}_{k - 1}}} \\right\\rVert_2 < {\\varepsilon _{\\rm{k}}}$, start ``momentum-mode'', else set $\\mu  = 0$.\n  \\ENDFOR\n  \\ENSURE\n  Estimated environment information ${{\\bf{\\hat x}}_k}$ and data packet ${{\\bf{\\hat s}}_1}, \\cdots ,{{\\bf{\\hat s}}_K}$.\n  \\end{algorithmic}\n  \\end{algorithm}\n\nWe summarize the proposed iterative algorithm in Algorithm \\ref{alg2}. During the execution process of the algorithm, the forward propagation process can be executed every time a new data packet is received. The self-iteration process can be executed as many times as necessary at any time, and the feedback process should be executed after the self-iteration process. Since the cached data cannot be too much, the forward propagation sliding-window size $n_{\\rm{f}}$ and the feedback window size $n_{\\rm{b}}$ need to be adjusted according to the actual system ability.\n\nThe effectiveness of the proposed algorithm can be explained by the message passing theory. Fig. \\ref{figITGH} shows the factor graph of the proposed iterative algorithm. The decoding algorithm and the sensing algorithm use each other's solution results as side information to achieve their performance. As the number of iterations increases, the environmental information and data information contained in the received data are separated and recovered. The components of the proposed iterative algorithm also reflect this idea: forward propagation passes the previously sensed environmental information and received data information to the next time slot so that the algorithm can incrementally optimize performance based on the continuously received data. The self-iteration process is executed repeatedly and iteratively based on the existing data, and the environmental information in the received data is fully obtained. The feedback process passes more accurate environmental information to the previous time slot and reduces the error caused by inaccurate decoding and sensing in the initial stage of the proposed algorithm.\n\n\\subsection{Computational Complexity Analysis}\nThe computational complexity of the proposed algorithm is mainly composed of two parts: \n\\begin{itemize}\n    \\item [(1)] \n    From the perspective of data decoding, we use the MPA decoder, whose computational complexity is $\\mathcal{O} \\left(Rd_{\\rm{f}}M^{d_{\\rm{f}}}+N_{\\rm{u}}M\\right)$. Compared with the ML decoder whose computational complexity is $\\mathcal{O}\\left(RC^{N_{\\rm{u}}}\\right)$, the computational complexity of the MPA decoder is lower. For example, when there are more users, the computational complexity of the MPA-based decoder just increases linearly. \n    \\item [(2)] \n    From the perspective of environment sensing, we use the GAMP algorithm, whose computational complexity is $\\mathcal{O}\\left(N_{\\rm{u}}N_{\\rm{R}}KN_{\\rm{s}}\\right)$. Compared with the OMP algorithm, whose computational complexity is $\\mathcal{O}(N_{\\rm{u}}N_{\\rm{R}}KN_{\\rm{s}} + (\\lambda N_{\\rm{s}})^3 )$, it can be seen that the GAMP algorithm is a relatively low-complexity CS reconstruction algorithm.\n   \n   \n\\end{itemize}\n\nIn summary, during the execution of the algorithm, replace $K$ with the sliding window sizes $n_{\\rm{f}}$ and $n_{\\rm{b}}$.\nThe computational complexity of the proposed iterative algorithm is $\\mathcal{O}( {R{d_{\\rm{f}}}{M^{{d_{\\rm{f}}}}}} $ $+ {N_{\\rm{u}}}M + {N_{\\rm{u}}}{N_{\\rm{R}}}{n_{\\rm{f}}}{n_{\\rm{b}}}{N_{\\rm{s}}} )$, where $n_{\\rm{f}}$ and $n_{\\rm{b}}$ can be controlled according to the convergence of the algorithm to save computing resources.\nIt can be seen that the computational complexity of the proposed algorithm is mainly determined by the number of users $N_{\\rm{u}}$ and the SCMA codebook parameter $d_{\\rm{f}}$. In contrast, if the OMP algorithm and the ML decoder are used to design the iterative algorithm, then its computational complexity will be $\\mathcal{O}( R{C^{{N_{\\rm{u}}}}} + {N_{\\rm{u}}}{N_{\\rm{R}}}{n_{\\rm{f}}}{n_{\\rm{b}}}{N_{\\rm{s}}} + (\\lambda N_{\\rm{s}})^3  )$, which is much higher than our algorithm.\n\nIn addition, the low complexity of the proposed iterative algorithm is also reflected in the use of low-density pilots, which effectively reduces the time-frequency resources and computing resources consumed by the pilots.\n\n\\section{System Performance Analysis}\nIn this section, we analyze the influence of the number of users on the decoding results of communication data and the accuracy of environment sensing. After receiving the $k$-th data packet, we calculate the mean square error (MSE) between the estimated environmental information ${{\\bf{\\hat x}}_k}$ and the actual environmental information $\\bf{x}$ to evaluate the accuracy of the environment sensing,\n\\begin{equation}\n{\\rm{MSE}} = \\frac{1}{{{N_{\\rm{s}}}}}\\left\\| {{{{\\bf{\\hat x}}}_k} - {\\bf{x}}} \\right\\|_2^2, \\label{eq33}\n\\end{equation}\nwhere $N_{\\rm{s}}$ is the total number of point clouds in the environment, and the SER between the decoding results ${{\\bf{\\hat s}}_k}$ and the original transmission data ${{\\bf{s}}_k}$ is calculated to evaluate the accuracy of data decoding.\n\nBased on the received data ${{\\bf{y}}_k}$ and decoded data ${{\\bf{\\hat s}}_k}$, the essence of estimating the environmental information ${{\\bf{\\hat x}}_k}$ is to solve the CS sparse reconstruction problem, and (16) can be expressed as\n\\begin{equation}\n{{{\\bf{\\hat x}}}_k} = \\arg \\mathop {\\min }\\limits_{\\bf{x}} {\\left\\| {\\bf{x}} \\right\\|_1}, \\label{eq34}\n\\end{equation}\n\\begin{equation}\n\\begin{array}{l}\n  {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {\\left\\| {{{\\bf{y}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}{{\\bf{x}}^{\\rm{T}}} {{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)} } \\right\\|_2}\\\\\n  \\quad  {\\left\\| { {\\left( {{{\\bf{s}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}} \\right){{\\bf{x}}^{\\rm{T}}}{{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}} } \\right\\|_2} + {\\varepsilon _{\\rm{x}}}\n  \\end{array}, \\label{eq35}\n\\end{equation}\nwhere ${{\\bf{y}}_{k-{n_{\\rm{f}}},k}}$, ${{\\bf{\\hat s}}_{k-{n_{\\rm{f}}},k}}$, and ${{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}$ represent the received packets, the decoding results, and the measurement matrix in the forward propagation window of size $n_{\\rm{f}}$ respectively, ${\\rm{SER}} \\propto {\\left\\| { {\\left( {{{\\bf{s}}_{k-{n_{\\rm{f}}},k}} - {{{\\bf{\\hat s}}}_{k-{n_{\\rm{f}}},k}}} \\right){{\\bf{x}}^{\\rm{T}}}{{\\bf{\\tilde{A}}}^{\\rm{s}}\\left(\\left[k-n_{\\rm{f}},k\\right]\\right)}} } \\right\\|_2}$. Therefore, when the decoding error rate (SER) increases, the constraint conditions of the sparse reconstruction problem become more slack, and the estimated environmental information error (MSE) increased. According to the theory of CS \\cite{Donoho}, the theoretical upper bound of environment sensing accuracy is,\n\\begin{equation}\n{\\left\\| {{\\bf{x}} - {{{\\bf{\\hat x}}}_k}} \\right\\|_2} \\le c \\cdot {R_p} \\cdot {\\left( {\\frac{{{N_{\\rm{u}}} \\cdot {N_{\\rm{R}}} \\cdot {n_{\\rm{f}}}}}{{\\log {N_{\\rm{s}}}}}} \\right)^{{1 \\mathord{\\left\/\n {\\vphantom {1 2}} \\right.\n \\kern-\\nulldelimiterspace} 2} - {1 \\mathord{\\left\/\n {\\vphantom {1 p}} \\right.\n \\kern-\\nulldelimiterspace} p}}}, \\label{eq36}\n\\end{equation}\nwhere $c > 0$ is a constant, ${\\left\\| {\\bf{x}} \\right\\|_p} \\le {R_p},\\left( {0 < p < 2} \\right)$ is the sparsity condition, ${\\rm{MSE}} \\propto {\\left\\| {{\\bf{x}} - {{{\\bf{\\hat x}}}_k}} \\right\\|_2}$. Therefore, when the number of users $N_{\\rm{u}}$ increases, the rank of the measurement matrix in the sparse reconstruction problem increases, and the environment sensing error (MSE) decreases. In addition, when the number of non-zero elements in the environment increases, $R_p$ increases, and the environment sensing error (MSE) increases.\n\nWhen using MPA decoder to decode the received signal based on the ML theory, for a single ORE, let $C = N_{\\rm{u}} \\times M$ be the number of symbols in the codebook, then $N_{\\rm{u}}$ users at each moment have ${C^{{N_{\\rm{u}}}}}$ ways to send symbols. The theoretical upper bound of the average decoding error rate (SER) is,\n\\begin{equation}\n{\\rm{SER}} \\le \\frac{1}{{{C^{{N_{\\rm{u}}}}}}}\\sum\\limits_{{{\\bf{s}}_{\\rm{a}}}} {\\sum\\limits_{{{\\bf{s}}_{\\rm{b}}},{{\\bf{s}}_{\\rm{a}}} \\ne {{\\bf{s}}_{\\rm{b}}}} {P\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right)} }, \\label{eq37}\n\\end{equation}\nwhere $P\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right)$ represents the pairwise error probability (PEP) that the symbol ${\\bf{s}}_{\\rm{a}}$ is incorrectly decoded to ${\\bf{s}}_{\\rm{b}}$. In (\\ref{eq37}), there are a total of $\\left( {C - 1} \\right){C^{2{N_{\\rm{u}}} - 1}}$ items for summation. Therefore, when other conditions are the same, SER increases as the number of users $N_{\\rm{u}}$ increases. Due to the random distribution of scatterers in the environment, the channels from the users to the AP can be expressed as Rayleigh fading channels. According to the ML theory, the decoding process in (\\ref{eq8}) can be expressed as\n\\begin{equation}\n  {{{\\bf{\\hat s}}}_k} = \\arg \\mathop {\\min }\\limits_{j \\in {C^{{n_{\\rm{u}}}}}}{\\left\\| {{{\\bf{y}}_k} - {{{\\bf{\\hat H}}_k}{{\\bf{s}}_k}\\left( j \\right)} } \\right\\|^2}, \\label{eq38}\n\\end{equation}\n\\begin{equation}\n  {\\rm{s}}{\\rm{.t}}{\\rm{.}}\\quad {{\\bf{y}}_k} = {{{\\bf{\\hat H}}}_k}{{\\bf{s}}_k} + \\left( {{{\\bf{H}}_k} - {{{\\bf{\\hat H}}}_k}} \\right){{\\bf{s}}_k} + {\\bf{w}}, \\label{eq39}\n\\end{equation}\nwhere $\\bf{w}$ represents Gaussian white noise with variance $N_0$. Due to the random distribution of scatterers in the environment, the interference caused by channel estimation errors is also Gaussian. Then the PEP in (\\ref{eq37}) can be expressed as\n\\begin{equation}\nP\\left( {{{\\bf{s}}_{\\rm{a}}} \\to {{\\bf{s}}_{\\rm{b}}}} \\right) = {{\\mathbb{E}}_{{{{\\bf{\\hat H}}}_k}}}\\left[ {Q\\left( {\\sqrt {\\frac{{{{\\left\\| {{{{\\bf{\\hat H}}}_k}\\left( {{{\\bf{s}}_{\\rm{a}}} - {{\\bf{s}}_{\\rm{b}}}} \\right)} \\right\\|}^2}}}{{{N_0} + \\mathbb{D} \\left( {\\left( {{{\\bf{H}}_k} - {{{\\bf{\\hat H}}}_k}} \\right){{\\bf{s}}_k}} \\right)}}} } \\right)} \\right], \\label{eq40}\n\\end{equation}\nWhere $Q$ is the error function. Therefore, the channel estimation error caused by inaccurate environmental information estimation will lead to an increase in the decoding symbol error rate (SER).\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=7.5cm]{fig5.eps}\n  \\caption{The trade-off relationship between the number of users and system performance.}\n  \\label{figtrade}\n  \\end{figure}\n\nAs shown in Fig. \\ref{figtrade}, the solid line indicates that the increase in the number of users $N_{\\rm{u}}$ promotes the accuracy of environment sensing, and the decrease of MSE reduces the channel prior information error $\\Delta {{\\bf{\\hat H}}_k}$ for decoding, so the SER also decreases and the decoding becomes more accurate. On the other hand, the dotted line indicates that an increase in the number of users $N_{\\rm{u}}$ increases the error of decoding (SER), and the decoded data error $\\Delta {{\\bf{\\hat s}}_k}$ also increases, which increases the environment sensing error (MSE). Therefore, firstly, when the number of users decreases, the environment sensing error (MSE) increases. Secondly, when the number of users increases, the SCMA decoding error (SER) increases. Finally, because the proposed iterative algorithm repeatedly executes environment sensing and data decoding, their performance affects each other, resulting in the same trend of SER and MSE, the number of users should be a compromise choice. The number of users $N_{\\rm{u}}$ has a trade-off relationship with system performance (MSE\/SER) because the number of users $N_{\\rm{u}}$ is traded for system performance. Finally, the optimal operating point could be estimated as\n\\begin{equation}\n{\\tilde N_{\\rm{u}}} = \\arg \\mathop {\\min }\\limits_{{N_{\\rm{u}}}} \\quad {a_1} \\cdot {\\rm{MSE}} + {a_2} \\cdot {\\rm{SER}}. \\label{eq41}\n\\end{equation}\n\nWhen the number of users is ${\\tilde N_{\\rm{u}}}$, the system performance MSE and SER reach the best. In practical applications, we need to adjust the coefficients $a_1$ and $a_2$ to suit the system's requirements for communication and environment sensing performance. And select the appropriate SCMA codebook and system parameters according to the number of users in the scene to ensure that the number of actual users is within the optimal working range of the system.\n\n\\section{Numerical Results}\nIn this section, we simulated the performance of the algorithm and all simulations are conducted in MATLAB 2017b on a computing server with a Xeon E5-2697 v3 processer and 128GB memory. The simulation scenario is set in a room with a size of $4\\rm{m} \\times 4m \\times 4m$, and the point cloud with a size of $8 \\times 8 \\times 8$ is used to represent the environmental information. The transmission signal frequency is set to 28 - 30GHz, and the bandwidth is 2GHz. A $20 \\times 20$ IRS is used for assist communication. In order to meet the actual system design, we set the IRS amplitude reflection coefficient $\\rho_{{n_{\\rm{I}}}}  = 1$, the phase shift $\\varphi_{{n_{\\rm{I}}}}   = 0$ or $\\pi $. The position of the scatterers distributed in the space is random, and the scattering coefficient ${\\bf{x}}_{{n_{\\rm{s}}}} \\in \\left[ {0,1} \\right]$. As shown in Fig. \\ref{figanswer}, small cubes are used to indicate the position distribution and scattering coefficient of the point cloud in space. The lower the transparency of the small cube, the larger the scattering coefficient of the point.\n\\begin{figure*}[t]\n  \\centering\n  \\subfigure[The original environment scatterer distribution.]{\n  \\includegraphics[width=0.4\\textwidth]{fig6.eps}}\n  \\subfigure[The sensing result when $\\rm{E_b\/N_0}=0dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig7.eps}}\n \\subfigure[The sensing result when $\\rm{E_b\/N_0}=5dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig8.eps}}\n  \\subfigure[The sensing result when $\\rm{E_b\/N_0}=10dB$.]{\n  \\includegraphics[width=0.4\\textwidth]{fig9.eps}}\n  \\caption{The original environment scatterer distribution and the sensing results under different SNR conditions.}\n  \\label{figanswer}\n  \\end{figure*}\n\nThe distribution of the environment scatterer is shown in Fig. \\ref{figanswer}(a), and the system parameters are set to the number of users $N_{\\rm{u}}$ = 6, the number of OREs $R$ = 4, and $d_{\\rm{v}}$ = 2. According to the convergence of the algorithm, we set $K_{\\rm{s}} = 5$ in this section. After the proposed algorithm is iterated to convergence, the intuitive sensing results are shown in Fig. \\ref{figanswer}(b), Fig. \\ref{figanswer}(c), and Fig. \\ref{figanswer}(d), when the signal-to-noise ratio (SNR, $\\rm{E_b\/N_0}$) are 0dB, 5dB, and 10dB respectively. It can be seen that the sensing result is very blurred when $\\rm{E_b\/N_0}=0dB$, and the shape of the target object can only be barely distinguished. As the SNR condition becomes better, the number of misidentified scatterers gradually decreases until $\\rm{E_b\/N_0}=10dB$ can clearly distinguish the shape of the target. This is because the SNR conditions affect the accuracy of communication data decoding, and therefore affect the accuracy of the sensing results.\n\nAs shown in Fig. \\ref{IT-MSE} and Fig. \\ref{IT-SER}, the system parameters are set to the number of users $N_{\\rm{u}}$ = 6, the number of OREs $R$ = 4, $d_{\\rm{v}}$ = 2, and the sparsity of randomly generated environmental scatterers is 1.5\\%. Set the forward propagation window size $n_{\\rm{f}}$ = 10 and the feedback window size $n_{\\rm{b}}$ = 1. We use MSE to evaluate the environmental sensing accuracy, when the number of data packets increases, the iterative algorithm converged, and the environment sensing result gradually becomes accurate. We use SER to evaluate the accuracy of transmission data decoding. As the number of data packets increases, the iterative algorithm converged, and the transmission data decoding results become more accurate. \nAt the same time, due to the existence of feedback, as the number of data packets increases, the data packets at the previous time are decoded based on the more accurate environmental information at the later time. The feedback process improves the decoding accuracy during convergence.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig17.eps}\n  \\caption{The relationship between the data packet number (number of iterations) and MSE.}\n  \\label{IT-MSE}\n  \\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig16.eps}\n  \\caption{The relationship between the data packet number (number of iterations) and SER.}\n  \\label{IT-SER}\n  \\end{figure}\n\nAfter the proposed algorithm iterates to convergence, the relationship between the number of users and system performance is shown in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER}. We set a tough condition and the system parameters are set to the number of OREs $R$ = 7, $d_{\\rm{v}}$ = 2, $\\rm{E_b\/N_0}=10dB$, and the sparsity of randomly generated environmental scatterers is 3\\%. \nAs analyzed in the section \\uppercase\\expandafter{\\romannumeral6}, there is a trade-off relationship between the number of users and the system performance indicators SER and MSE. \nAs the number of users changes, the environment sensing performance and the multi-user communication performance cannot reach the best at the same time.\nAs shown in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER}, \nwhen the number of users is small, as the number of users increases, the sensing accuracy is significantly improved (as analyzed in \\eqref{eq36}), and therefore the decoding accuracy is improved, until the optimal operating point ${\\tilde N_{\\rm{u}}} = 12$ is reached under simulation conditions. When the sensing of the environment is accurate sufficiently, a further increase in the number of users causes a decrease in the accuracy of decoding (as analyzed in \\eqref{eq37} and \\eqref{eq40}), and therefore the accuracy of environment sensing also decreases slightly.\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig12.eps}\n  \\caption{The trade-off relationship between the number of users and MSE.}\n  \\label{UE-MSE}\n  \\end{figure}\n\n\\begin{figure}\n  \\centering\n  \\includegraphics[width=8cm]{fig13.eps}\n  \\caption{The trade-off relationship between the number of users and SER.}\n  \\label{UE-SER}\n  \\end{figure}\n\nFig. \\ref{UE-MSE} and Fig. \\ref{UE-SER} also show the impact of ``momentum-mode'' on system performance. The ``momentum-mode'' uses data information outside the sliding-window for environmental sensing. As analyzed in Section \\uppercase\\expandafter{\\romannumeral6}, the environment sensing accuracy is low when the number of users is small, so the ``momentum-mode'' accumulates errors and cannot improve system performance. However, ``momentum-mode'' can improve system performance when the number of users is higher than the optimal operating point, Since. It can be seen from the simulation results that when $\\mu $ = 0.9, the momentum coefficient is large, the MSE and SER are greatly improved when there is a large number of users, and there is a negative impact on the SER in the case of few users. When $\\mu $ = 0.5, the momentum coefficient is moderate, the MSE and SER indicators are slightly improved when there is a large number of users, and there is no significant impact on the system performance in the case of few users. When $\\mu $ = 0.1, the momentum coefficient is small, there is no significant impact on the system performance. We recommend using a larger momentum coefficient when there is a large number of users, and using a moderate or small momentum coefficient in the case of few users.\n\nFinally, in Table \\ref{tab1}, we provide the run time of using the $k$-th received data package for data decoding and environment sensing (steps 4 to 10 in Algorithm \\ref{alg2}). The simulation parameter settings are the same as those in Fig. \\ref{UE-MSE} and Fig. \\ref{UE-SER} and the momentum mode is not enabled. As shown in Table \\ref{tab1}, we verify that the computational complexity of the algorithm increases with the increase in the number of users.\n\\begin{table}[h]\n  \\centering\n  \\caption{The run time of proposed algorithm.}\n  \\begin{tabular}{|p{1in}|p{0.3in}|p{0.3in}|p{0.3in}|p{0.3in}|}\n      \\arrayrulecolor{black}\n      \\hline  \n      \\makecell[l]{Number of users $N_{\\rm{u}}$} & \\makecell[c]{5} & \\makecell[c]{10} & \\makecell[c]{15} & \\makecell[c]{20}\\\\\n      \\hline \n      \\makecell[l]{Run time (s)} & \\makecell[c]{4.95} & \\makecell[c]{5.11} & \\makecell[c]{5.33} & \\makecell[c]{5.64}\\\\\n      \\hline \n      \\end{tabular}\n      \\label{tab1}\n  \\end{table}\n\n\\section{Conclusion}\nIn the diverse wireless communication application scenarios in the future, environment sensing is an important component of the wireless communication system. In the scenario of IRS-assisted indoor uplink communication, we design a multiple access method and an environment sensing method. The multiple access method is based on SCMA. With the assistance of the IRS, based on the sparse codebook of the transmitted signals, the SCMA-IRS-MPA decoder is used. The environment sensing algorithm is based on the CS theory, including time sliding-window and ``momentum-mode'' which keep on sensing the environment while continuously receiving the data stream sent by the user. In this paper, the proposed multiple access algorithm and the proposed environment sensing algorithm rely on each other. Therefore, we propose a novel iterative algorithm based on low-density pilots to jointly solve the multiple access and environment sensing problems and achieve the integration of environment sensing and communication. Finally, numerical simulation has verified the convergence and effectiveness of the iterative and incremental algorithm and analyzed the trade-off relationship between the number of users and system performance. We also give a system parameter configuration method. The sensing-communication integration ideas and algorithms proposed in this paper will provide references for the development of new wireless communication technologies in the future.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nCan the holographic dictionary of AdS\/CFT be generalized to gravitational theories defined on a finite patch of spacetime? This question has recently attracted renewed attention due to the discovery of a new class of solvable irrelevant deformations of two-dimensional conformal field theory, known as the $T\\bar{T}$ deformation \\cite{Smirnov:2016lqw, Zamolodchikov:2004ce, Cavaglia:2016oda}. It was conjectured in \\cite{McGough:2016lol} (see also \\cite{Kraus:2018xrn})  that the $T\\bar{T}$ deformed CFT can be interpreted as the holographic dual of a finite patch of asymptotically AdS$_3$ spacetime. A key piece of evidence in support of this conjectured duality is that the conformal Ward identify of the CFT gets deformed into a second order functional differential equation that formally matches with the Wheeler-DeWitt equation of AdS${}_3$ gravity.  This relationship is akin to the familiar duality between Chern-Simons field theory and Wess-Zumimo-Witten conformal field theory \\cite{Witten:1988hf}, and points to the possible identification between the wave functionals of gravitational theories in $D+1$-dimensions and partition functions of a special class of $D$-dimensional QFTs.\\footnote{For studies of higher-dimensional generalisations of the $T\\bar{T}$ deformation and their potential holographic interpretation, see \\cite{Hartman:2018tkw, Taylor:2018xcy}.} If such a precise relationship could indeed be established, it would be an important generalization of the standard holographic dictionary that would open up a new avenue for studying gravitational physics in the bulk space-time.\n\nHowever, while the new duality AdS${}_3$ gravity and the $T\\bar{T}$ deformed CFTs passes several non-trivial checks,  the precise status of the correspondence is still unclear.   In particular, it was found that for large enough energies and fixed deformation parameter, the energy spectrum of the boundary QFT complexifies, indicating a possible breakdown of unitary. This apparent breakdown has a natural interpretation from the point of view of the bulk: it corresponds to a physical cut-off on the spectrum of black hole states, that removes all black holes with Schwarzschild radii that would extend beyond the finite radial cutoff. Nonetheless, the existence of this cut-off in the energy spectrum raises several conceptual questions, that require better understanding of the UV properties of the boundary QFT.\n\nTo circumvent the complications of field theory, in this paper we turn to analyzing the problem in one dimension lower. In particular we will consider the finite cutoff version of two-dimensional Jackiw-Teitelboim (JT) gravity with a negative cosmological constant and its dual formulation in terms of a deformed version of Schwarzian quantum mechanics proposed in \\cite{Gross:2019ach, Gross:2019uxi}.  Traditionally, the JT path integral is computed with Dirichlet boundary conditions in the limit where the proper boundary length and boundary value for the dilaton become very large \\cite{Maldacena:2016upp}. In that limit, JT gravity reduces to a soluble 1D quantum theory, the Schwarzian theory \\cite{Bagrets:2016cdf, Stanford:2017thb, Mertens:2017mtv}.\\footnote{For further investigations of Schwarzian  quantum mechanics and JT gravity, see \\cite{kitaevTalks, Maldacena:2016hyu, Jensen:2016pah, Engelsoy:2016xyb, Bagrets:2016cdf, Mertens:2017mtv, Lam:2018pvp, Goel:2018ubv, Kitaev:2018wpr, Yang:2018gdb, Saad:2019lba, Mertens:2019tcm, Iliesiu:2019xuh, Suh:2019uec}.}   At finite cut-off, however, the boundary theory is expected to become highly non-linear and the computation of the JT partition function in this regime has thus far been an open problem. In the following, we will discuss two different ways to compute it, relying on either canonical quantization or path integral quantization. \n\nIn the canonical approach one foliates the spacetime with a certain, usually time-like, coordinate and parametrizes the metric in an ADM decomposition \\cite{PhysRev.160.1113}. As a result of diffeomorphism invariance, the quantum mechanical wave functionals satisfy a set of local Wheeler-DeWitt constraints,  that uniquely determine their dependence on local data defined on the chosen foliation. In general these constraints are difficult to solve, except possibly in the so-called mini-superspace approximation. Luckily, for two-dimensional dilaton gravity theories, the constraints reduce to two first order functional differential equations that can be solved exactly \\cite{Henneaux:1985nw} \\cite{LouisMartinez:1993eh} in the form of explicit diffeomorphism invariant wavefunctionals of the  boundary metric and dilaton profile.  The WDW wavefunctionals relevant for our analysis are defined through radial quantisation in Euclidean signature. This provides a way to compute the path integral at finite cutoff.\n\nIn the second approach, we compute the Euclidean path integral directly. Again, the analysis at finite proper boundary length becomes more intricate as some of the gravitational modes, that were frozen in the large volume limit, now become dynamical. After integrating out the dilaton, the JT path integral localizes to one over a boundary action given by the extrinsic curvature $K$ of the boundary. By using constraints from the $SL(2,\\mathbb{R})$ isometry of AdS$_2$, we manage to express $K$ in an expansion containing solely powers of the Schwarzian derivative and its derivatives. This greatly facilitates our computations and allows us to express the partition function as the expectation value of an operator in the Schwarzian theory. Using integrability properties of the Schwarzian theory, we manage to exactly compute the partition function to all orders in a perturbative expansion in the cutoff. \n\nBoth the canonical and path integral approach use widely different techniques to compute the finite cutoff partition function, yet, as expected from the equivalence between the two quantisation procedures, the results agree. Moreover, we find a perfect agreement between the result obtained via two approaches with a proposed deformation of the Schwarzian partition function, analogous to the $T\\bar T$ deformation for $2D$ CFTs \\cite{Gross:2019ach, Gross:2019uxi}. Before diving in the computations, lwe first review (for completeness and later reference) this one-dimensional analog of $T\\bar{T}$ and then present a more detailed summary of our results.\n\n\\subsection{Review $1d$ $T\\bar{T}$}\n\nIn previous work \\cite{Gross:2019ach}, a particular deformation of the Schwarzian quantum mechanics\nwas shown to be classically equivalent to JT gravity with Dirichlet boundary conditions for the metric and dilaton. The deformation on the Schwarzian theory follows from a dimensional reduction of the $T\\bar T$ deformation in $2D$ CFTs \\footnote{This reduction is valid in the classical limit and should be seen as a motivation for the proposed deformation. It would be interesting to extend it to a precise statement using the methods of \\cite{Ghosh:2019rcj}.}.   Explicitly the deformation involves a flow of the action $S$ of the quantum mechanical theory, \n\\begin{eqnarray}\\label{1dTTbar}\n\\partial_{\\l} S = \\int_0^1 d\\theta \\frac{T^2}{1\/2 - 2\\l T} \n\\end{eqnarray}\nwhere $T$ is the trace of the stress-`scalar' of the quantum mechanical theory and $\\l$ is the deformation parameter. By going from the Lagrangian to the Hamiltonian formulation, we can write an equivalent flow for the Hamiltonian instead of $S$ and find the flow of the energy eigenvalues,\\footnote{Since the deformation is a function of the Hamiltonian, the eigenfunctions do not change under the flow.}\n\\begin{eqnarray}\\label{deformedE}\n\\partial_{\\l}H = \\frac{H^2}{1\/2-2\\l H}\\quad \\Rightarrow\n\\quad \\mathcal{E}_{\\pm}(\\l) = \\frac{1}{4\\l}\\left(1 \\mp \\sqrt{1- 8 \\l E}\\right).\n\\end{eqnarray}\nHere $E$ are the energy levels of the undeformed theory and matching onto the original spectrum as $\\l \\to 0$ results in picking the minus sign for the branch of the root in \\eqref{deformedE}. In section \\ref{sec:nonpert} we will see that the other branch of the root will also make its appearance. In the case of the Schwarzian theory, which has a partition function that can be exactly computed \\cite{Stanford:2017thb},\\footnote{In the gravitational theory $C$ is equal to $\\phi_r$, the renormalised boundary value of the dilaton. Furthermore, here we picked a convenient normalisation of the partition function.}\n\\begin{eqnarray}\\label{SchZ}\nZ(\\b) = \\int_0^{\\infty} dE\\; \\frac{\\sinh(2\\pi \\sqrt{2 C E})}{\\sqrt{2C\\pi^3}} e^{-\\b E} = \\frac{e^{2C\\pi^2\/\\b}}{\\b^{3\/2}} ,\n\\end{eqnarray}\nthe deformed partition function is,\n\\begin{eqnarray}\\label{deformedZ}\nZ_{\\l}(\\b) = \\int_0^{\\infty} dE\\; \\frac{\\sinh(2\\pi \\sqrt{2 C E})}{\\sqrt{2C\\pi^3}} e^{-\\b \\mathcal{E}_+(\\l)}.\n\\end{eqnarray}\nLet us make two observations. First, the integral over $E$ runs over the full positive real axis and therefore will also include complex energies $\\mathcal{E}_{+}(\\l)$ when $\\l > 0$, i.e. for $E > 1\/8\\l$ the deformed spectrum complexifies. This violates unitarity and needs to be dealt with. We will come back to this issue in section \\ref{sec:nonpert}. Second, given that there is a closed from expression of the original Schwarzian partition function, one can wonder whether this is also the case for the deformed partition function. This turns out to be the case. For the moment let us assume $\\l < 0$ so that there are no complex energies, then it was shown in \\cite{Gross:2019ach} that the deformed partition function is given by an integral transform of the original one, analogous to the result of \\cite{Dubovsky_2018} in $2D$. The integral transform reads,\n\\begin{eqnarray}\\label{intTransform}\nZ_{\\l}(\\b) = \\frac{\\b}{\\sqrt{-8\\pi \\l}} \\int_0^{\\infty} \\frac{d\\b'}{\\b'^{3\/2}} e^{\\frac{(\\b-\\b')^2}{8\\l \\b'}} Z(\\b'),\n\\end{eqnarray}\nPlugging \\eqref{SchZ} into this expression and performing the integral over $\\b'$ yields,\n\\begin{eqnarray}\\label{K2}\nZ_{\\l}(\\b) = \\frac{ \\b e^{-\\frac{\\b}{4\\l}}}{\\sqrt{-2 \\pi \\l}(\\b^2 + 16 C \\pi^2 \\l)}K_2\\left( -\\frac{1}{4\\l}\\sqrt{\\b^2 + 16 C \\pi^2 \\l} \\right).\n\\end{eqnarray}\nwith the associated density of states given by\n\\begin{eqnarray}\\label{deformedrho}\n\\rho_\\l(E) = \\frac{1-4\\l E}{\\sqrt{2\\pi^3 C}} \\sinh\\left(2\\pi \\sqrt{2CE(1-2\\l E)} \\right)\n\\end{eqnarray}\nAlthough we have derived this formula assuming that $\\l < 0$, we will simply analytically continue to $\\l > 0$ to obtain the partition function of the deformed Schwarzian theory that describes JT gravity at finite cutoff. One might be worried that this would not yield the same as \\eqref{deformedZ} and indeed there are a few subtleties involved in doing that analytic continuation as discussed in the end of section \\ref{sec:JT-gravity-review} and in section \\ref{sec:nonpert}.\n\n\\subsection{Summary of results and outline}\n\nThe purpose of this paper is give two independent bulk computation that reproduce the partition function \\eqref{deformedZ}. In section \\ref{sec:WdW-wavefunctional-main} we present a derivation of the partition function of JT gravity (with negative cosmological constant) at finite cutoff by computing the radial Wheeler-de Witt (WdW) wavefunctional. Due to Henneaux it is known since the 80's that the contraints of $2D$ dilaton gravity can be solved exactly in the full quantum theory \\cite{Henneaux:1985nw}. We will review this computation and fix the solution by imposing Hartle-Hawking boundary conditions. In particular we find that\n\\begin{equation}\\label{eq:AdSfinitecutoffJTintro}\n\\Psi_{\\rm HH}[\\phi_b(u),L] = \\int_0^\\infty dM\\; \\sinh(2\\pi \\sqrt{M}) ~e^{ \\int_0^L du \\left[ \\sqrt{\\phi^2_b-M-( \\partial_u \\phi_b)^2} - \\partial_u \\phi \\tan^{-1} \\left(\\sqrt{\\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}-1}\\right)  \\right]}.\n\\end{equation}\nThis wavefunction is computed in a basis of fixed dilaton $\\phi_b(u)$, where $u$ corresponds to the proper length along the boundary, and $L$ the total proper length of the boundary. The above results obtained through the WdW constraint are non-perturbative in both $L$ and $\\phi_b(u)$.\n\n When considering a constant dilaton profile $\\phi_b(u) = \\phi_b$, the wavefunction \\eqref{eq:AdSfinitecutoffJTintro} reproduces the $T\\bar T$ partition function in \\eqref{deformedZ}, with the identification \n\\begin{equation}\nM \\to 2 C E,\\quad \\phi^2_b \\to \\frac{C}{4\\l},\\quad L \\to \\frac{\\b}{\\sqrt{4C\\l}}, \n\\end{equation}\nIn terms of these variables, \\eqref{eq:AdSfinitecutoffJTintro} matches with $T\\bar{T}$ up to a shift in the ground state energy, which can be accounted for by a boundary counterterm $e^{-I_{\\rm ct}} = e^{- \\phi_b L}$ added to the gravitational theory. An important aspect that this analysis emphasizes if the fact that, for JT gravity, studying boundary conditions with a constant dilaton is enough. As we explain in section \\ref{minisuper}, if the wavefunction for a constant dilaton is known, the general answer \\eqref{eq:AdSfinitecutoffJTintro} is fixed by the constraints and does not constrain any further dynamical information \\cite{MTY}.\n\n\n\n\nThe partition function  \\eqref{deformedZ} is also directly computed from the  path integral in JT gravity at finite cutoff in section \\ref{sec:JT-gravity-review}. We will impose dilaton and metric Dirichlet boundary conditions, in terms of $\\phi_b$ and the total proper length $L$. For the reasons explain in the previous paragraph, it is enough to focus on the case of a constant dilaton. It is convenient to parametrize these quantities in the following way \n\\begin{eqnarray}\n\\label{eq:Dirichlet-bdy-conditions}\n \\phi_b = \\frac{\\phi_{r}}{\\e},~~~~~L = \\frac{\\beta}{\\e}, \n\\end{eqnarray}\nin terms of a renormalized length $\\beta$ and dilaton $\\phi_r$. We will refer to $\\e$ as the cutoff parameter \\footnote{In Poincar\\'e coordinates $\\e$ corresponds semiclassically to the bulk coordinate of the cutoff surface.}. When comparing with the $T\\bar{T}$ approach this parameter is $\\e = \\sqrt{2\\lambda}$ (in units for which we set $\\phi_r \\to 1\/2 $). In order to compare to the asymptotically $AdS_2$ case previously studied in the literature \\cite{Maldacena:2016upp, Saad:2019lba}, we need to take $\\phi_b, L \\to \\infty$ with a fixed renormalized length $L\/\\phi_b$. In terms of the cutoff parameter, this limit corresponds to $\\e \\to 0$, keeping $\\phi_r$ and $\\beta$ fixed. \n\n\nWe will solve this path integral perturbatively in the cutoff $\\e$, to all orders. We integrate out the dilaton and reduce the path integral to a boundary action comprised of the extrinsic curvature $K$ and possible counter-terms. We find an explicit form of the extrinsic curvature valid to all orders in perturbation theory in $\\e$. A key observation in obtaining this result is the realisation of a (local) $SL(2,\\mathbb{R})$ invariance of $K$ in terms of lightcone coordinates $z = \\tau - i x$, $\\bar{z} = \\tau + i x$:\n\\begin{eqnarray}\nK[z,\\bar{z}] = K\\left[ \\frac{a z + b}{c z + d}, \\frac{a \\bar{z} + b}{c \\bar{z} + d} \\right].\n\\end{eqnarray}\nSolving the Dirichlet boundary condition for the metric allows us to write $K$ as a functional of the Schwarzian derivative of the coordinate $z$.\\footnote{This generalizes the computation of \\cite{Maldacena:2016upp} which found the relation between the extrinsic curvature and the Schwarzian derivative in the infinite cutoff limit. } As we will explain in detail, the remaining path integral can be computed exactly using integrability properties in the Schwarzian theory to all orders in $\\e^2$.\n\nThus, by the end of section \\ref{sec:JT-gravity-review}, we find agreement between the WdW wavefunctional, the Euclidean partition function and the  $T\\bar T$ partition function from \\eqref{deformedZ}:\n\\begin{eqnarray}\n  e^{-I_{\\rm ct}}\\Psi_{\\rm HH}[\\phi_b,\\, L] \\overset{{\\rm non-pert.}}{=} Z_\\l(\\b) \\overset{{\\rm pert.}}{=} Z_{\\rm JT}[\\phi_b,\\, L] \\,.\n\\end{eqnarray}\nHere we emphasize again that we show that the first equality is true non-perturbatively in $\\e$ (respectively in $\\l$), whereas we prove the second equality to all orders in perturbation theory.\n\nIn section \\ref{sec:nonpert} we discuss various extensions of the deformed partition function including further corrections. In particular we discuss two types of corrections in the path integral and in the integral over energies in \\eqref{deformedZ}: first, we analyze non-perturbative terms in $\\e$ coming from contributions that cannot be written as a path integral on the disk (the contracting branch of the wavefunction) and second, we speculate about non-perturbative corrections coming from the genus expansion. Related to the first kind of ambiguity, given the exact results we obtained for the wavefunctional and partition function, we explore how the complexification of the energy levels (that we mentioned above) can be cured. In particular, we propose that it requires the inclusion of the other branch of the root in \\eqref{deformedE}, but still results in a negative density of states. The structure of the negative density of states suggests that the (unitary) partition function is not an ordinary one, but one with a chemical potential turned on. Related to the second type, we compute the partition function of the finite cutoff ``trumpet'' which is a necessary ingredient when constructing higher genus hyperbolic surfaces. Finally, we speculate about the range of the remaining Weil-Petersson integral which is needed in order to compute the  finite cutoff partition function when including the contribution of surfaces with arbitrary topology. \n\nSection \\ref{sec:desitter} applies the computation from section \\ref{sec:WdW-wavefunctional-main}  to the case of JT gravity with a positive cosmological constant and finds the wavefunctional on a de Sitter time-slice at finite time. This wavefunctional has some interesting behaviour, similar to the Hagedorn divergence present in \\eqref{K2}. We finish with a discussion of our results and future directions in section \\ref{sec:discussion}.\n\\vspace{0.25cm}\n\n\\textbf{Note:} While this work was in progress, we became aware of a closely related project by D. Stanford and Z. Yang \\cite{SY}. They analyze finite cutoff JT gravity from yet a different perspective, finding different results. We leave understanding how these approaches are related for future work, but we believe they correspond to different ways to regularize (and therefore define) the theory. \n\n\n\\section{Wheeler-DeWitt wavefunction}\n\\label{sec:WdW-wavefunctional-main}\n\nIn this section, we will start by reviewing the canonical quantization of $2D$ dilaton-gravity following the approach of \\cite{Henneaux:1985nw, LouisMartinez:1993eh}. In these references, the authors find the space of exact solutions for both the momentum and Wheeler-DeWitt constraints. Later, in subsections \\ref{sec:WdW-finite-cutoff-JT} and \\ref{sec:HH-boundary-conditions-JT-wavefunctional}, we will focus on JT gravity, and we will explain how to impose the Hartle-Hawking condition appropriately to pick a solution corresponding to finite cutoff AdS$_2$. \n\nLet us consider the more general two dimensional dilaton gravity in Lorentzian signature,\n\\begin{equation}\n\\label{eq:action-dilaton-general}\nI = \\frac{1}{2}\\int_{M} d^2 x \\sqrt{g} [\\phi R - U(\\phi)] + \\int_{\\partial M} du \\sqrt{\\g_{uu}}\\;\\phi K,\n\\end{equation}\nwith an arbitrary potential $U(\\phi)$. $g$ is the two-dimensional space-time metric on $M$ and $\\g$ the induced metric on its boundary $\\partial M$. The boundary term in \\eqref{eq:action-dilaton-general} is necessary in order for the variational principle to be satisfied when imposing Dirichlet boundary conditions for the metric and dilaton. In  \\eqref{eq:action-dilaton-general} we could also add the topological term $ \\frac{1}{2}\\int_{M} d^2 x \\sqrt{g} \\,\\phi_0 R  + \\int_{\\partial M} du \\sqrt{\\g_{uu}}\\;\\phi_0 K = 2\\pi \\phi_0$ which will be relevant in section \\ref{sec:nptopo}.\n\nIt will be useful to define also the prepotential $W(\\phi)$ by the relation $\\partial_\\phi W(\\phi)=U(\\phi)$. In the case of JT gravity with negative (or positive) cosmological constant we will pick $U(\\phi) = -2 \\phi$ (or $U(\\phi)=2\\phi$) and $W(\\phi)=-\\phi^2$ ($W(\\phi)=\\phi^2$), which has as a metric solution AdS$_2$ (dS$_2$) space with unit radius.\n\nWe will assume the topology of space to be a closed circle, and will use the following ADM decomposition of the metric \n\\begin{equation}\\label{ADMdecomp}\nds^2 = - N^2 dt^2 + h (dx + N_{\\perp} dt)^2,~~~~h=e^{2\\sigma}\n\\end{equation}\nwhere $N$ is the lapse, $N_{\\perp}$ the shift, $h$ the boundary metric (which in this simple case is an arbitrary function of $x$) and we identify $x\\sim x+1$. After integrating by parts and using the boundary terms, the action can then be written as \n\\begin{eqnarray}\nI &=& \\int d^2x ~e^{\\sigma} \\Big[ \\frac{\\dot{\\phi}}{N} \\left( N_{\\perp} \\partial_x\\sigma+\\partial_x N_{\\perp} -\\dot{\\sigma}\\right) \\nonumber\\\\\n&&~~~~+ \\frac{\\partial_x\\phi}{N}\\left(\\frac{N\\partial_xN}{e^{2\\sigma}} - N_{\\perp} \\partial_xN_{\\perp}+N_{\\perp} \\dot{\\sigma} - N_{\\perp}^2 \\partial_x \\sigma \\right) - \\frac{1}{2}N U(\\phi) \\Big]\n\\end{eqnarray}\nwhere the dots correspond to derivatives with respect to $t$. As usual the action does not involve time derivatives of fields $N$ and $N_{\\perp}$ and therefore \n\\begin{equation}\n\\Pi_{N} = \\Pi_{N_{\\perp}} =0,\n\\end{equation}\nwhich act as primary constraints. The momenta conjugate to the dilaton and scale factor are \n\\begin{equation}\n\\Pi_\\phi = \\frac{e^\\sigma}{N}(N_{\\perp} \\partial_x \\sigma+\\partial_x N_{\\perp} -\\dot{\\sigma}),~~~~\\Pi_\\sigma = \\frac{e^\\sigma}{N}(N_{\\perp}\\partial_x\\phi-\\dot{\\phi}) .\n\\end{equation}\nWith these equations we can identify the momentum conjugate to the dilaton with the extrinsic curvature $\\Pi_\\phi \\sim K$, and the momentum of $\\sigma$ with the normal derivative of the dilaton $\\Pi_\\sigma \\sim \\partial_n \\phi$. The classical Hamiltonian then becomes \n\\begin{equation}\nH = \\int dx \\left[ N_{\\perp} \\mathcal{P} + e^{-\\sigma}N \\mathcal{H}_{\\text{\\tiny{WdW}}} \\right]\n\\end{equation}\nwhere \n\\begin{eqnarray}\\label{constraintP}\n\\mathcal{P} &\\equiv & \\Pi_\\sigma \\partial_x  \\sigma +  \\Pi_\\phi \\partial_x  \\phi -\\partial_x \\Pi_\\sigma,  \\\\[2mm] \\label{constraintH}\n\\mathcal{H}_{\\text{\\tiny{WdW}}} &\\equiv & - \\Pi_\\phi \\Pi_\\sigma + \\frac{1}{2}e^{2\\sigma}U(\\phi) + \\partial_x^2\\phi- \\partial_x\\phi  \\partial_x\\sigma,\n\\end{eqnarray}\nand classically the momentum and Wheeler-DeWitt constraints are respectively $\\mathcal{P}=0$ and $\\mathcal{H}_{\\text{\\tiny{WdW}}}=0$. \n\nSo far the discussion has been classical. Now we turn to quantum mechanics by promoting field to operators. We will be interested in wavefunctions obtained from path integrals over the metric and dilaton, and we will write them in configuration space. The state will be described by a wave functional $\\Psi[\\phi,\\sigma]$ and the momentum operators are replaced by \n\\begin{equation}\n\\hat{\\Pi}_\\sigma = - i \\frac{\\delta}{\\delta \\sigma(x)},~~~\\hat{\\Pi}_\\phi = - i \\frac{\\delta}{\\delta \\phi(x)},~~~\n\\end{equation}\nThe physical wavefunctions will only depend on the boundary dilaton profile and metric. \n\nUsually, when quantizing a theory, one needs to be careful with the measure and whether it can contribute Liouville terms to the action. Such terms only appear when in conformal gauge, which is not what we are working in presently. Actually, the ADM decomposition \\eqref{ADMdecomp} captures a general metric and is merely a parametrization of all $2D$ metrics and so we have not fixed any gauge. The quantum theory is thus defined through the quantum mechanical version of the classical constraints \\eqref{constraintP} and \\eqref{constraintH} \\footnote{From the path integral perspective, we are assuming an infinite range of integration over the lapse. Different choices for the contour of integration can drastically modify the constraints after quantization. We thank S. Giddings for discussions on this point.}. As a result, we do not need to include any Liouville term in our action in the case of pure gravity. If matter would have been present, there could be Liouville terms coming from integrating out the matter, but that is beyond the scope of this paper.\n\n\n\\subsection{Solution} \n\nIn references \\cite{Henneaux:1985nw, LouisMartinez:1993eh}, the physical wavefunctions that solve the dilaton gravity constraints are constructed as follows. The key step is to notice that the constraints $\\mathcal{P}$ and $\\mathcal{H}_{\\rm WdW}$ are simple enough that we can solve for $\\Pi_{\\sigma}$ and $\\Pi_{\\phi}$ separately. For instance, by combing $\\Pi_{\\sigma}\\mathcal{P}$ with the WdW constraint, we get \n\\begin{eqnarray}\n\\label{solPirho}\n\\partial_x (e^{-2\\sigma}\\Pi_{\\sigma}^2) = \\partial_x (e^{-2\\sigma}(\\partial_x \\phi)^2 + W(\\phi)) ~\\Rightarrow~\\Pi_{\\sigma} = \\pm \\sqrt{(\\partial_x \\phi)^2 + e^{2\\sigma}[M+W(\\phi)]},\n\\end{eqnarray}\nwith $M$ an integration constant that is proportional to the ADM mass of the system as we will see momentarily. It is then straightforward to plug this into the WdW constraint to find an expression for $\\Pi_{\\phi}$. Quantum mechanically, we want the physical wavefunction to satisfy, \n\\begin{equation} \\label{pis}\n\\hat{\\Pi}_\\sigma \\Psi_{\\rm phys} = \\pm Q[M;\\phi,\\sigma]   \\Psi_{\\rm phys},~~~~\\hat{\\Pi}_\\phi \\Psi_{\\rm phys} =  \\pm \\frac{g[\\phi,\\sigma]}{Q[M;\\phi,\\sigma]}\\Psi_{\\rm phys},\n\\end{equation}\nwhere we defined the functions \n\\begin{equation}\\label{eq:Qdef}\nQ[E;\\phi,\\sigma] \\equiv  \\sqrt{(\\partial_x \\phi)^2 + e^{2\\sigma}[M+W(\\phi)]},~~~~g[\\phi,\\sigma]\\equiv \\frac{1}{2}e^{2\\sigma}U(\\phi) + \\partial_x^2\\phi- \\partial_x\\phi  \\partial_x\\sigma.\n\\end{equation}\nWavefunctions that solve these constraints also solve the momentum and Wheeler-DeWitt constraints as explained in \\cite{Henneaux:1985nw, LouisMartinez:1993eh}. In particular they solve the following WdW equation with factor ordering,\\footnote{Here we think of $\\hat{Q}$ as well as $\\hat{M}$ as operators. The physical wavefunctions can be written as linear combinations of eigenfunctions of the operator $\\hat{M}$ with eigenvalue $M$.}\n\\begin{eqnarray}\n\\left(g - \\hat{Q} \\hat{\\Pi}_{\\phi} \\hat{Q}^{-1} \\hat{\\Pi}_{\\sigma} \\right) \\Psi_{\\rm phys} = 0 \n\\end{eqnarray}\nThe most general solution can be written as \n\\begin{equation}\n\\Psi =\\Psi_+ + \\Psi_- ,~~~\\Psi_{\\pm}= \\int dM \\rho_\\pm(M) \\Psi_\\pm(M),\n\\end{equation}\nwhere we will distinguish the two contributions\n\\begin{equation}\\label{eq:psinoninv}\n \\Psi_\\pm(M) = \\exp{\\left[ \\pm i \\int dx \\left( Q[M;\\phi,\\sigma] -  \\partial_x \\phi \\tanh^{-1} \\left(\\frac{Q[M;\\phi,\\sigma]}{2\\partial_x\\phi} \\right)\\right)\\right]},\n\\end{equation}\nwith the function $Q$ defined in \\eqref{eq:Qdef} which depends on the particular dilaton potential. We will refer in general to $\\Psi_+$ ($\\Psi_-$) as the expanding (contracting) branch.\n\nThis makes explicit the fact that solutions to the physical constraints reduce the naive Hilbert space from infinite dimensional to two dimensional with coordinate $M$ (and its conjugate). The most general solution of the Wheeler-DeWitt equation can then be expanded in the base $\\Psi_\\pm(M)$ with coefficients $\\rho_\\pm (M)$. The new ingredient in this paper will be to specify appropriate boundary conditions to pick $\\rho_\\pm(M)$ and extract the full Hartle-Hawking wavefunction. We will see this is only possible for JT gravity for reasons that should will be clear in the next section. \n\n\nIt will be useful to write the physical wavefunction in terms of diffeomorphism invariant quantities. This is possible thanks to the fact that we are satisfying the momentum constraints. In order to do this we will define the proper length $u$ of the spacelike circle as \n\\begin{equation}\ndu = e^{\\sigma} dx,~~~~L\\equiv \\int_0^1 e^{\\sigma}dx,\n\\end{equation}\nwhere $L$ denotes the total length. The only gauge invariant data that the wavefunction can depend on is then $L$ and $\\phi(u)$, a dilaton profile specified as a function of proper length along the boundary. The wavefunction \\eqref{eq:psinoninv} can be rewritten as \n\\begin{equation}\n\\Psi_\\pm(M) = e^{\\pm i \\int_0^L du \\left[ \\sqrt{W(\\phi)+M+( \\partial_u \\phi)^2} - \\partial_u \\phi\\, \\tanh^{-1} \\left(\\sqrt{1+ \\frac{W(\\phi)+M}{(\\partial_u \\phi)^2}}\\right)  \\right]},\n\\end{equation}\nwhich is then manifestly diffeomorphism invariant.\n\nThe results of this section indicate the space of physical states that solve the gravitational constraints is one dimensional, labeled by $M$. In the context of radial quantization of $AdS_2$ that we will analyze in the next section, this parameter corresponds to the ADM mass of the state, while in the case of $dS_2$, it corresponds to the generator of rotations in the spatial circle. Phase space is even-dimensional, and the conjugate variable to $E$ is given by \n\\begin{equation}\n\\Pi_M = - \\int dx \\frac{e^{2\\sigma} \\Pi_\\rho}{\\Pi_\\rho^2 - 2 (\\partial_x \\phi)^2} \n\\end{equation}\nsuch that $[M , \\Pi_M] =i$.\\footnote{The simplicity of the phase space of dilaton gravity theories was also noted in \\cite{Thiemann:1992jj}.}\n\n\n\\subsection{Phase space reduction}\n\\label{minisuper}\n\nHaving the full solution to the WdW equation, we now study the minisuperspace limit. In this limit, the dilaton $\\phi$ and boundary metric $e^{2\\s}$ are taken to be constants. In a general theory of gravity, minisuperspace is an approximation. In JT gravity, as we saw above, the physical phase space is finite-dimensional (two dimensional to be precise). Therefore giving the wavefunction in the minisuperspace regime encodes all the dynamical information of the theory, while the generalization to varying dilaton is fixed purely by the constraints. In this section, we will directly extract the equation satisfied by the wavefunction as a function of constant dilaton and metric, from the more general case considered in the previous section. \n\nIf we start with the WdW equation and fix the dilaton and metric to be constant, the functional derivatives then become ordinary derivatives and the equation reduces to \n\\begin{eqnarray}\n\\label{WdWmini}\n\\left(\\frac{1}{2}e^{2\\s} U(\\phi) - \\hat{Q}\\partial_\\phi \\hat{Q}^{-1}\\partial_\\s \\right) \\Psi(\\phi,\\sigma) = 0.\n\\end{eqnarray}\nwith $\\hat{Q} = (\\hat{M} + W(\\phi))^{1\/2}$. Due to the factor ordering, this differential equation still depends on the operator $\\hat{M}$, which is a bit unsatisfactory. Fortunately, we know that a $\\s$ derivative acting on $\\Psi$ is the same as acting with $Q^2\/g \\partial_\\phi$. In the minisuperspace limit, we can therefore write \\eqref{WdWmini} as \n\\begin{eqnarray}\n\\label{minisupereq}\n(L U(\\phi) - 2L \\partial_L(L^{-1} \\partial_\\phi)  ) \\Psi(\\phi,L) = 0,\n\\end{eqnarray}\nwhere $L$ is the total boundary length. This equation is the exact constraint that wavefunctions with a constant dilaton should satisfy even though it was derived in a limit. We can explicitly check this by using \\eqref{eq:psinoninv} and noticing that any physical wavefunction, evaluated in the minisuperspace limit, will satisfy precisely this equation. \n\nThis equation differs from the one obtained in \\cite{MTY} by $\\Psi_{\\rm here} = L \\Psi_{\\rm there}$ and, therefore, changes the asymptotics of the wavefunctions, something we will analyze more closely in the next subsection. \n\n\n\n\\subsection{Wheeler-DeWitt in JT gravity: radial quantization}\n\\label{sec:WdW-finite-cutoff-JT} \n\nIn this section, we will specialize the previous discussion to JT gravity with a negative cosmological constant. We fix units such that $U(\\phi)=-2\\phi$. We will analytically continue the results of the previous section to Euclidean space and interpret them in the context of radial quantization, such that the wavefunction is identified with the path integral in a finite cutoff surface.  Then, we will explain how to implement Hartle-Hawking boundary conditions, obtaining a proposal for the exact finite cutoff JT gravity path integral that can be compared with results for the analog of the $T\\bar{T}$ deformation in $1d$ \\cite{Gross:2019ach, Gross:2019uxi}.   \n\nLets begin by recalling some small changes that appear when going from Lorenzian to Euclidean radial quantization. The action we will work with is \n\\begin{equation}\\label{IJT}\nI_{\\rm JT}=- \\frac{1}{2}\\int_{M} \\sqrt{g} \\phi( R+2) - \\int_{\\partial M} \\sqrt{\\g}\\phi K,\n\\end{equation}\nand the ADM decomposition of the metric we will use is \n\\begin{equation}\nds^2 = N^2 dr^2 + h (d\\theta + N_{\\perp} dr)^2,~~~~h=e^{2\\sigma}\\,,\n\\end{equation}\nwhere $r$ is the radial direction while $\\theta \\sim \\theta+1$ corresponds to the angular direction that we will interpret as Euclidean time. We show these coordinates in figure \\ref{fig:ads2rq}. In terms of holography we will eventually interpret $\\theta$ as related to the Euclidean time of a boundary quantum mechanical theory. \n\\begin{figure}[t!]\n\\centering\n\\begin{tikzpicture}[scale=0.9]\n\\pgftext{\\includegraphics[scale=0.5]{ads2.pdf}} at (0,0);\n\\draw (2.6,0) node  { $\\theta$ };\n\\draw (1,0.5) node  { $r$ };\n\\draw (0,-3.5) node  { $(a)$ };\n\\end{tikzpicture}\n\\hspace{2cm}\n\\begin{tikzpicture}[scale=0.9]\n\\pgftext{\\includegraphics[scale=0.5]{ads22.pdf}} at (0,0);\n\\draw (3.5,0) node  { $r$ };\n\\draw (-0.4,-3) node  { $(b)$ };\n\\end{tikzpicture}\n\\caption{\\small (a) We show the slicing we use for Euclidean JT gravity in asymptotically $AdS_2$, which has disk topology (but not necessarily rigid hyperbolic metric). (b) Frame where the geometry is rigid $EAdS_2$ with $r$ increasing upwards and a wiggly boundary denoted by the blue curve. \\label{fig:ads2rq}}\n\\end{figure}\n\nAs shown in figure \\ref{fig:ads2rq}, and as we will explicitly show in section \\ref{sec:JT-gravity-review}, the radial quantization wavefunction is identified with the gravitational path integral at a finite cutoff (inside the black circle) with Dirichlet boundary conditions\n\\begin{equation}\n\\Psi[\\phi_b(u),\\sigma(u)] = \\int \\mathcal{D}g \\mathcal{D} \\phi ~e^{- I_{\\rm JT}[\\phi,g]},\\qquad {\\rm with} \\qquad \\phi|_{\\partial} = \\phi_b(u),\\qquad g|_\\partial = \\gamma_{uu}=e^{2\\sigma(u)}.\n\\end{equation}\n The geometry inside the disk in figure \\ref{fig:ads2rq} is asymptotically $EAdS_2$. From this path integral we can derive the WDW and momentum constraints and therefore solving the latter with the appropriate choice of state should be equivalent to doing the path integral directly.  \n\nThe result of previous section implies that this path integral is given by a linear combination of\n\\begin{eqnarray}\n\\label{expanding}\n\\hspace{-1cm}&&\\text{Expanding branch:}~~~\\hspace{0.05cm}\\Psi_+(M) = e^{ \\int_0^L du \\left[ \\sqrt{\\phi_b^2-M-( \\partial_u \\phi_b)^2} - \\partial_u \\phi_b \\tan^{-1} \\left(\\sqrt{\\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}-1}\\right)  \\right]},\\\\\n\\label{contracting} \\hspace{-1cm}&&\\text{Contracting branch:}~~\\Psi_-(M) = e^{- \\int_0^L du \\left[ \\sqrt{\\phi_b^2-M-( \\partial_u \\phi_b)^2} - \\partial_u \\phi_b \\tan^{-1} \\left(\\sqrt{\\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}-1}\\right)  \\right]}.\n\\end{eqnarray}\n\nWe will focus on the purely expanding branch of the solution \\eqref{expanding}, as proposed in \\cite{Freidel:2008sh} and \\cite{MTY} to correspond to the path integral in the disk and therefore set $\\rho_-(M)=0$. We will go back to possible effects coming from turning on this term later. Thus, we will study the solutions\n\\begin{equation}\n\\Psi_{\\rm disk}[\\phi_b(u),\\sigma(u)] = \\int dM \\rho(M) ~e^{ \\int_0^L du \\left[ \\sqrt{\\phi_b^2-M-( \\partial_u \\phi_b)^2} - \\partial_u \\phi_b \\tan^{-1} \\left(\\sqrt{\\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}-1}\\right)  \\right]}\\,.\n\\end{equation}\nTo make a choice of boundary conditions that fix the boundary curve very close to the boundary of the disk we will eventually take the limit of large $L$ and $\\phi_b$.\n\n\\subsection{Hartle-Hawking boundary conditions and the JT wavefunctional} \n\\label{sec:HH-boundary-conditions-JT-wavefunctional}\n\nTo determine the unknown function $\\rho(M)$, we will need to impose a condition that picks the Hartle-Hawking state. For this, one usually analyses the limit $L\\to0$ \\cite{PhysRevD.28.2960}. Such a regime is useful semiclassically but not in general. From the no-boundary condition, $L\\to0$ should reproduce the path integral over JT gravity inside tiny patches deep inside the hyperbolic disk; performing such a calculation is difficult. Instead, it will be simpler to impose the Hartle-Hawking condition at large $L\\to \\infty$. In this case, we know how to do the path integral directly using the Schwarzian theory. The derivation of the Schwarzian action from \\cite{Maldacena:2016upp} explicitly uses the no-boundary condition, so we will take this limit instead, which will be enough to identify a preferred solution of the WdW equation.\n\nTo match the wavefunction with the partition function of the Schwarzian theory, it is enough to consider the case of constant dilaton and metric. Then, the wavefunction simplifies to\\footnote{It is interesting to note that this partition function first appeared in \\cite{Kunstatter:1997my}.}\n\\begin{equation}\n\\Psi[\\phi_b,\\,\\sigma] = \\int dM \\rho(M) ~e^{ \\int_0^1 d\\theta e^{\\sigma}  \\sqrt{\\phi_b^2-M}} =\\int dM \\rho(M) ~e^{ \\int_0^L du \\sqrt{\\phi_b^2-M}}\n\\end{equation}\nwith $\\phi_b$ and $\\sigma$ constants. Expanding the root at large $\\phi_b$ and large $L=e^{\\sigma}$ gives,\n\\begin{equation}\\label{eq:schlimwdw}\n\\Psi[\\phi_b,\\sigma] =e^{L\\phi_b} \\int dM \\rho(M) ~e^{ -L \\frac{M}{2\\phi_b}+\\ldots}\n\\end{equation}\nWe find the usual divergence for large $L$ and $\\phi$, which can be removed by adding to \\eqref{IJT} the counter term, $I_{\\rm ct} = \\int_0^L du \\,\\phi_b$.  In fact, we will identify the JT path integral with this counter term as computing the thermal partition function at a temperature specified by the boundary conditions. At large $L$ and $\\phi_b$ we know that the gravity partition function is given by the Schwarzian theory:\n\\begin{equation}\n\\int \\mathcal{D}g \\mathcal{D} \\phi ~e^{- I_{\\rm JT}[\\phi,g]} \\to e^{L\\phi_b} \\int \\frac{\\mathcal{D}f}{SL(2, \\mR)}~e^{ \\phi_{b} \\int_0^L du  \\hspace{0.05cm}{\\rm Sch}(\\tan \\frac{\\pi}{L} f,u) },\n\\end{equation}\nwhere ${\\rm Sch}(F(u),u) \\equiv \\frac{F'''}{F'} - \\frac{3}{2} \\left( \\frac{F''}{F'} \\right)^2 $. By rescaling time we can see the path integral only depends on $L\/\\phi_b$ which we will sometimes refer to as renormalized length. This result can be derived by first integrating over the dilaton over an imaginary contour, localizing the geometry to rigid $AdS_2$. Then the remaining degree of freedom is the shape of the boundary curve, from which the Schwarzian theory arises.\n\nThe Schwarzian partition function can be computed exactly and gives \n\\begin{equation}\nZ_{\\rm Sch} (\\ell) \\equiv  \\int \\frac{\\mathcal{D}f}{SL(2, \\mR)}e^{\\int_0^\\ell du\\hspace{0.05cm} {\\rm Sch}(\\tan \\frac{\\pi}{\\ell} f,u) } = \\left(\\frac{\\pi}{\\ell}\\right)^{3\/2} e^{\\frac{2\\pi^2}{\\ell}}= \\int dk^2 \\sinh(2\\pi k) e^{- \\ell k^2\/2}\n\\end{equation}\nApplying this result to the JT gravity path integral with the replacement $\\ell \\to L\/\\phi_b$ gives the partition function directly in the form of equation \\eqref{eq:schlimwdw} where we can straightforward identify the Schwarzian density of states with the function of $M$ as\n\\begin{equation}\n\\rho_{\\rm HH}(M) = \\sinh(2 \\pi \\sqrt{M}),\n\\end{equation}\nwhere the subscript indicates that we picked the Hartle-Hawking state. It is important that we are able to compute the path integral of JT gravity for $\\phi_b,\\, L\\to \\infty$ but fixed $L\/\\phi_b$. This involves an exact treatment of the Schwarzian mode since otherwise we would only obtain $\\rho_{\\rm HH}(M)$ in some limits. This ingredient was missing in \\cite{Henneaux:1985nw, LouisMartinez:1993eh} making them unable to identify the HH state from the full space of physical states. \n\nTo summarize, the solution of the gravitational constraints gives the finite cutoff JT gravity path integral as \n\\begin{equation}\\label{eq:AdSfinitecutoffJT}\n\\Psi_{\\rm HH}[\\phi_b(u),L] =\\hspace{-0.1cm} \\int_0^\\infty \\hspace{-0.3cm}dM\\; \\hspace{-0.1cm} \\sinh(2\\pi \\sqrt{M}) ~e^{ \\int_0^L du \\left[ \\sqrt{\\phi^2_b-M-( \\partial_u \\phi_b)^2} - \\partial_u \\phi \\tan^{-1} \\left(\\sqrt{\\frac{\\phi^2-M}{(\\partial_u \\phi)^2}-1}\\right)  \\right]}.\n\\end{equation}\nBy construction, this matches the Schwarzian limit when $\\phi_b$ and $\\sigma$ are constant. \n\nWhen the dilaton is constant but $\\sigma(u)$ is not, it is clear that we can simply go to coordinates $d\\tilde{\\theta} = e^{\\s}d\\theta$ in both the bulk path integral and the WdW wavefunction and see that they give the same result. Since we can always choose time-slices with a constant value for the dilaton, this situation will suffice for comparing our result to the analog of the $T\\bar T$ deformation in the next subsection. \n\nThe more non-trivial case is for non-constant dilaton profiles. We provide a further check of our result in appendix \\ref{sec:vardil}, where we compare the wavefunctional \\eqref{eq:AdSfinitecutoffJT} to the partition function of JT gravity with a non-constant dilaton profile when the cutoff is taken to infinity.\n\n\\subsection{Comparison to $T\\bar{T}$}\n\nLet us now compare the wavefunctional \\eqref{eq:AdSfinitecutoffJT} to the partition function obtained from the $1D$ analog of the $T\\bar T$ deformation \\eqref{deformedZ}. First of all, let us consider configurations of constant $\\phi_b$, so $\\partial_u \\phi_b = 0$. This will simplify $\\Psi_{\\rm HH}$ to \n\\begin{eqnarray}\n\\Psi_{\\rm HH}[\\phi_b,\\,L] = \\int_0^{\\infty} dM\\; \\sinh(2\\pi \\sqrt{M})e^{\\phi_b L \\sqrt{1- M\/\\phi^2_b}}.\n\\end{eqnarray}\nThe partition function is then obtained by multiplying this wavefunction by $e^{-I_{\\rm ct}} = e^{- L \\phi_b}$. The resulting partition function agrees with \\eqref{deformedZ} with identifications:\n\\begin{eqnarray}\nM \\to 2 C E,\\quad \\phi^2_b \\to \\frac{C}{4\\l},\\quad L \\to \\frac{\\b}{\\sqrt{4C\\l}},\n\\end{eqnarray}\nup to an unimportant normalization. In fact, we can say a little more than just mapping solution onto each other. In section \\ref{minisuper} we showed that in the minisuperspace approximation the wave\\emph{functions} satisfy \\eqref{minisupereq}. With the identifications made above and the inclusion of the counter term, the partition function $Z_{\\l}(\\b)$ satisfies \n\\begin{eqnarray}\\label{diffeqZ}\n\\left[4 \\l \\partial_{\\l} \\partial_{\\b} + 2 \\b \\partial_\\b^2 - \\left(\\frac{4\\l}{\\b} - 1\\right) \\partial_\\l \\right] Z_{\\l}(\\b) = 0.\n\\end{eqnarray}\nThis is now purely written in terms of field theory variables and is precisely the flow equation as expected from \\eqref{1dTTbar}, i.e. solutions to this differential equation have the deformed spectrum \\eqref{deformedE}. This is also the flow of the partition function found in two dimensions in \\cite{Aharony:2018bad}, specialised to purely imaginary modular parameter of the torus. We will analyze the associated non-perturbative ambiguities associated to this flow in section \\ref{sec:nonpert}.\n\nLet us summarise. We have seen that the partition function of the deformed Schwarzian theory is mapped to the exact dilaton gravity wavefunctions for constant $\\phi_b$ and $\\g_{uu}$. In fact, \\emph{any} quantum mechanics theory that is deformed according to \\eqref{1dTTbar} will obey the quantum WdW equation (for constant $\\phi_b$ and $\\s$). This principle can be thought of as the two-dimensional version of \\cite{Freidel:2008sh}.  It is only the boundary condition at $\\l \\to 0$ (or large $\\phi_b L$), where we know the bulk JT path integral gives the Schwarzian theory, that tells us that the density of states is $\\sinh(2\\pi \\sqrt{M})$. Next, we will show that the wavefunction for constant  $\\phi_b$ and $\\gamma_{uu}$ can be reproduced by explicitly computing the Euclidean path integral in the bulk, at finite cutoff. \n\n\n\n\\section{The Euclidean path integral}\n\\label{sec:JT-gravity-review}\n\nWe will once again consider the JT gravity action, \\eqref{IJT}, and impose Dirichlet boundary conditions for the dilaton field $\\phi|_{\\partial M_2} \\equiv \\phi_b  \\equiv \\phi_{r}\/\\e$, boundary metric $\\g_{uu}$, and proper length $L \\equiv \\b\/\\e$ and with the addition the counter-term, \n\\begin{eqnarray}\nI_{\\rm ct} = \\int du \\sqrt{\\g} \\phi\\,,\n\\end{eqnarray}\nwhose addition leads to an easy comparison between our results and the infinite cutoff results in JT gravity. As in the previous section we will once again focus on disk topologies. \n\nAs discussed in section \\ref{sec:HH-boundary-conditions-JT-wavefunctional}, the path integral over the dilaton $\\phi$ yields a constrain on the curvature of the space, with $R = -2$. Therefore, in the path integral we are simply summing over different patches of $AdS_2$, which we parametrize in Euclidean signature using Poincar\\'e coordinates as $ds^2 = (d\\tau^2 + dx^2)\/x^2$. To describe the properties which we require of the boundary of this patch we choose a proper boundary time $u$, with a fixed boundary metric $\\g_{uu}= 1\/\\e^2$ (related to the fix proper length $L = \\int_0^\\b du \\sqrt{\\g_{uu}})$. Fixing the intrinsic boundary metric to a constant, requires:   \n\\begin{eqnarray}\n\\label{eq:Poincare-bdy-metric}\n\\frac{\\tau'^2 + x'^2 }{x^2} = \\frac{1}{\\e^2 }\\,, \\qquad \\frac{-t'^2 + x'^2 }{x^2} = \\frac{1}{\\e^2 } \\,, \\qquad \\tau = -i t\\,.\n\\end{eqnarray}\nIf choosing some constant $\\e \\in \\mR$ then we require that the boundary has the following properties: \n\\begin{itemize}\n\\item If working in Euclidean signature, the boundary should never self-intersect. Consequently if working on manifolds with the topology of a disk this implies that the Euler number $\\chi(M_2) = 1$. \n\\item If working in Lorentzian signature, the boundary should always remain time-like since \\eqref{eq:Poincare-bdy-metric} implies that $-(t')^2 + (x')^2 = (x'-t')(t'+x')>0$.\\footnote{While fixing the metric $\\g_{uu}$ to be a constant is not diffeomorphism invariant, the notion of the boundary being time-like  ($\\text{sgn}\\, \\g_{uu}$) is in fact diffeomorphism invariant.} From now on we will assume without loss of generality that $t' >0$. \n\\end{itemize}\n Both conditions are  important constraints which we should impose at the level of the path integral. Such conditions are not typical if considering the boundary of the gravitational theory as the worldline of a particle moving on $H^2$ or $AdS_2$: in Euclidean signature, the worldline could self-intersect, while in Lorentzian signature the worldline could still self-intersect but could also become space-like. These are the two deficiencies that \\cite{Kitaev:2018wpr, Yang:2018gdb} encountered in their analysis, when viewing the path integral of JT gravity as that of a particle moving in an imaginary magnetic field on $H^2$.  \n\nFor the purposes of this paper it will also prove convenient to introduce the light-cone coordinates (with $z =- ix+\\tau\\,, \\bar z = i x+  \\tau$), for  which fixing the intrinsic boundary metric implies: \n\\begin{eqnarray}\n\\label{eq:lightcone-bdy-metric}\n-\\frac{4z' \\bar z'}{(z-\\bar z)^2 } =\\frac{1}{\\e^2} \\,.\n\\end{eqnarray}\nIn Euclidean signature $z = \\bar z^*$, while in Lorentzian signature $z$, $\\bar z \\in i\\mR$. The constraint that the boundary is time-like implies that $i z'>0$ and $i \\bar z'<0$ (alternatively, if assuming $t' <0$, $i z'<0$ and $i \\bar z'>0$). In order to solve the path integral for the remaining boundary fluctuations in the 1D system it will prove convenient to use light-cone coordinates and require that the path integral obeys the two properties described above. \n\n\\subsection{Light-cone coordinates and $SL(2, \\mR)$ isometries in $AdS_2$}\n\\label{sec:light-cone-coordinates-SL(2,R)}\n\nAs is well known, $AdS_2$, even at finite cutoff, exhibits an $SL(2, \\mR)$ isometry. This isometry becomes manifest when considering the coordinate transformations:\n\\begin{eqnarray}\n\\label{eq:SL2-isometry}\n&\\text{E \\& L} :\\qquad  z \\to \\frac{a z+ b}{c z+d}\\,, \\hspace{3.9cm} \\bar z \\to \\frac{a \\bar z+ b}{c \\bar z+d}\\,,\\nonumber \\\\[2mm] &\\text{E}:\\qquad x+i  \\tau \\to  \\frac{a (x+i \\tau)+ b}{c (x+i \\tau )+d}\\,, \\quad \\hspace{0.5cm} \\text{L}:\\quad t+ x \\to  \\frac{a (t+ x)+ b}{c (t+ x)+d}\\,,\n\\end{eqnarray}\nIt is straightforward to check that under such transformations the boundary metrics, \\eqref{eq:Poincare-bdy-metric} and \\eqref{eq:lightcone-bdy-metric}, both remain invariant. The same is true of the extrinsic curvature, which is the light-cone parametrization of the boundary degrees of freedom can be expressed as\n\\begin{eqnarray}\n\\label{eq:K-lightcone}\nK[z(u), \\bar z(u)] = \\frac{2 z'^2 \\bar z' + ( \\bar z-z )\\bar z' z'' + z'(2\\bar z'^2 + (z- \\bar z)\\bar z'')}{4(z' \\bar z')^{3\/2}} \\,.\n\\end{eqnarray}\nConsequently, invariance under $SL(2, \\mR)$ transformations gives: \n\\begin{eqnarray}\nK[z, \\bar z] =  K\\left[\\frac{a z+ b}{c z + d}, \\frac{a \\bar z+ b}{c \\bar z + d}\\right]\\,,\n\\end{eqnarray}\nTherefore, upon solving for $\\bar z[ z(u)]$ (as a functional of $z(u)$) we will find that \n\\begin{eqnarray}\n\\bar z[z(u)] \\qquad  \\Rightarrow \\qquad K[z] =  K\\left[\\frac{a z+ b}{c z + d}\\right]\n\\end{eqnarray}\nAs we will see in the next subsection, such a simple invariance under $SL(2, \\mR)$ transformations will be crucial to being able to relate the path integral of the boundary fluctuations to that of some deformation of the Schwarzian theory. An important related point is that when solving for $\\tau[x(u)]$ as a functional of $x(u)$, the resulting extrinsic curvature is not invariant under the $SL(2, \\mR)$ transformations, $\\tau \\to \\frac{a \\tau + b}{c\\tau+d}$. Rather this is only a valid symmetry in the $\\e \\to 0$ limit, for which $x \\to 0$, while $\\tau$ is kept finite. It is only in the asymptotically $AdS_2$ limit that the transformation in the second line of \\eqref{eq:SL2-isometry} can be identified with $\\tau \\to \\frac{a \\tau + b}{c\\tau+d}$. If keeping track of higher orders in $\\e$, the transformation on $\\tau$ would involve a growing number of derivatives on the $\\tau$ field which should be proportional to the order of the $\\e$-expansion.\n\n\\subsection{Restricting the extrinsic curvature}\n\\label{sec:restricting-extrinsic-curvature}\n\nNext, we discuss the expansion of the extrinsic curvature $K[z]$ to all orders in perturbation in $\\e$: \n\\begin{eqnarray}\nK[z] = \\sum_{n=0}^\\infty \\e^n K_n[z]\\,, \\qquad K_n[z] = K_n\\left[\\frac{a z+b}{c z + d}\\right]\\,,\n\\end{eqnarray}\nWe could in principle explicitly solve for $\\bar z[z(u)]$ to first few orders in perturbation theory in $\\e$ and then plug the result into \\eqref{eq:K-in-terms-of-z}. The first few orders in the expansion can be solved explicitly and yield:\n\\begin{eqnarray} \n\\label{eq:K3-K4}\nK_0[z] &= 1, \\qquad K_1[z]=0, \\qquad K_2[z]= \\Sch(z, u), \\nonumber\\\\ K_3[z] &= -i\\, \\partial_u \\Sch(z, u) \\,, \\qquad K_4[z]= -\\frac{1}2\\Sch(z, u)^2+\\partial_{u}^2 \\,\\Sch(z, u) \\,.\n\\end{eqnarray}\nThe fact that all orders in $K_n[z(u)]$ solely depend on the Schwarzian and its derivatives is not a coincidence. In fact, one generally finds that:  \n\\begin{eqnarray}\nK_n[z]  = \\cK_n[\\Sch(z, u),\\, \\partial_u ]\\,.\n\\end{eqnarray}\n\nThe reason for this is as follows. $K_n[z]$ is a local function of $z(u)$ since solving for $\\bar z[z(u)]$ involves only derivatives of $z(u)$.   The Schwarzian can be written as the Casimir of the $\\mathfrak{sl}(2, \\mR)$ transformation, $z \\to \\frac{a\\,z+b}{c\\,z+d}$ \\cite{Maldacena:2016upp}. Because the rank of the $\\mathfrak{sl}(2, \\mR)$ algebra is $1$, higher-order Casimirs of $\\mathfrak{sl}(2, \\mR)$ can all be expressed as a polynomial (or derivatives of powers)  of the quadratic Casimir. Since local functions in $u$ that are $SL(2, \\mR)$ invariant, can also only be written in terms of the Casimirs of $\\mathfrak{sl}(2, \\mR)$ this implies that they should also be linear combinations of powers (or derivatives of powers) of the quadratic Casimir, which is itself the Schwarzian. \n\nAlternatively, we can prove that  $K_n[z(u)]$ is a functional of the Schwarzian by once again noting that $K_n[z(u)]$ only contains derivatives of $z(u)$ up to some finite order. Then we can check explicitly how each infintesimal $SL(2,\\mR)$ transformation constrains  $K_n[z(u)]$. For instance, translation transformations $z \\to z + b$ imply that $K_n$ solely depends on derivatives of  $z(u)$. The transformation $z(u) \\to a z(u)$ implies that $K_n[z(u)]$ depends solely on ratios of derivatives with a matching order in $z$ between the numerator and denominator of each ratio, of the type ${(\\prod_k z^{(k_i)})}\/{(\\prod_k z^{(\\tilde k_i)})}$. Finally considering all possible linear combinations between ratios of derivatives of the type ${(\\prod_k z^{(k_i)})}\/{(\\prod_k z^{(\\tilde k_i)})}$ and requiring invariance under the transformation $z(u) \\to 1\/z(u)$, fixes the coefficients of the linear combination to those encountered in arbitrary products of Schwarzians and of its derivatives. \n\nOnce again, we emphasize that this does not happen when using the standard Poincar\\'e parametrization \\eqref{eq:Poincare-bdy-metric}  in $\\tau$ and $x$. When solving for $\\tau[x]$ and plugging into $K[\\tau(u)]$, since we have that $K[\\tau(u)] \\neq K[a\\tau(u)+b\/(c \\tau(u)+d)]$ and consequently $K[\\tau(u)]$ is not a functional of the Schwarzian; it is only a functional of the Schwarzian at second-order in $\\e$. This can be observed by going to fourth order in the $\\e$-expansion, where\n\\begin{eqnarray}\nK_4[\\tau(u)] =  \\frac{\\tau^{(3)}(u)^2}{\\tau'(u)^2}+\\frac{27 \\tau''(u)^4}{8 \\tau'(u)^4}+\\frac{\\tau^{(4)}(u)\n   \\tau''(u)}{\\tau'(u)^2}-\\frac{11 \\tau^{(3)}(u) \\tau''(u)^2}{2 \\tau'(u)^3}\\,,\n\\end{eqnarray}\nwhich cannot be written in terms of $\\Sch(\\tau(u), u)$ and of its derivatives.\n\n\n\\subsection{Finding the extrinsic curvature: perturbative terms in $K[z(u)]$ }\n\\label{sec:finding-extrinsic-curvature}\n\n\nThe previous subsection identified the abstract dependence of the extrinsic curvature as a function of the Schwarzian. To quantize the theory, we need to find the explicit dependence of $K_n$ on the Schwarzian.  To do this, we employ the following trick. Consider the specific configuration for $z(u)$:\\footnote{While \\eqref{eq:specific-configuration-z(u)} is, in fact, a solution to the equation of motion for the Schwarzian theory it is not necessarily a solution to the equation of motion in the theory with finite cutoff. }\n\\begin{eqnarray}\n\\label{eq:specific-configuration-z(u)}\nz(u) = \\exp( a u)\\,, \\qquad  \\qquad \\Sch(z, u) = -\\frac{ a^2}{2}\\,.\n\\end{eqnarray}\n\nSince $K[z(u)]$ is a functional of the $\\Sch(z, u)$ and of its derivatives to all orders in perturbation theory in $\\e$, then $K_n[z(u) = \\exp(au)] = \\cK_n[\\Sch(z, u),\\, \\partial_u ] = \\cK_n[a]$. On the other hand, when using a specific configuration for $z(u)$ we can go back to the boundary metric constraint \\eqref{eq:lightcone-bdy-metric} and explicitly solve for $\\bar z(u)$. Plugging-in this solution together with \\eqref{eq:specific-configuration-z(u)} into the formula for the extrinsic curvature $K[z(u), \\bar z(u)]$ \\eqref{eq:K-lightcone}, we can find  $\\cK_n[a]$ and, consequently, find the powers of the Schwarzian in $\\cK_n[\\Sch(z, u), \\, \\partial_u]$. \n\nThe metric constraint involves solving the first order differential equation\n\\begin{eqnarray}\n\\label{eq:metric-a-barZ-config}\n - \\frac{4 \\,a\\, e^{a u}\\bar z'}{\\left(e^{a u}  -  \\,\\bar z\\right)^2} = \\frac{1}{\\e^2}\\,,\n\\end{eqnarray}\nwhose solution, to all orders in perturbation theory in $\\e$, is given by \n\\begin{eqnarray} \n\\label{eq:bar-z-diff-eq-sol}\n\\bar z(u) = e^{a u }\\left(1-2a^2 \\e^2 -2a \\e \\sqrt{-1+a^2 \\e^2}\\right)\\,.\n\\end{eqnarray}\n\nWe can plug this solution for $\\bar z(u)$ together with the configuration $z(u) = \\exp(a u)$ to find that \n\\begin{eqnarray}\n\\label{eq:K-in-terms-of-a}\nK\\left[z(u) = \\exp(a u)\\right] &= \\sqrt{1- \\e^2 a^2}\\,.\n\\end{eqnarray}\nDepending on the choice of branch one can reverse the sign of \\eqref{eq:K-in-terms-of-a} to find that $K\\left[z(u) = \\exp(a u)\\right] = -\\sqrt{1- \\e^2 a^2}$ which corresponds to the considering the exterior of an $AdS_2$ patch as our surface (instead of a regular $AdS_2$ patch). This is analogous to the contracting branch in of the WDW functional in \\eqref{contracting}. \n\nConsequently, it follows that in a perturbative series in $\\e$ we find:\\footnote{The terms containing derivatives of the Schwarzian are not necessarily total derivatives and thus we need to explain why they do not contribute to the path integral.  }\n\\begin{eqnarray}\n\\label{eq:K-in-terms-of-z}\nK_\\pm[z(u)] &= \\pm \\left(\\sqrt{1+ 2{\\e^2}\\,\\Sch(z, u)} \\,\\,\\,+ \\,\\,\\,  \\text{derivatives of Sch.}\\right) \\,,\n\\end{eqnarray}\nwhere we find that the quadratic term in $\\e$ for the $+$ branch of \\eqref{eq:K-in-terms-of-z} agrees with the expansion of $K$ in terms of $\\e$ in JT gravity in asymptotic $AdS_2$ \\cite{Maldacena:2016upp} (which found that $K[z(u)] = 1+\\e^2 \\Sch(z, u) + \\dots$). The $+$ branch in \\eqref{eq:K-in-terms-of-z} corresponds to compact patches of $AdS_2$ for which the normal vector points outwards; the $-$ branch corresponds to non-compact surfaces (the complement of the aforementioned $AdS_2$ patches) for which the normal vector is pointing inwards. While the $+$ branch has a convergent path integral for real values of $\\phi_r$, for a normal choice of countour for $z(u)$, the path integral of the $-$ branch will be divergent. Even for a potential contour choice for which the path integral were convergent, the $-$ branch is non-perturbatively suppressed by $O(e^{-\\int_0^\\b  du\\, \\phi_b\/\\e}) = O(e^{-1\/\\e^2})$.   Therefore, for now, we will ignore the effect of this different branch ($-$) and set $K[z(u)] \\equiv K_+[z(u)]$; we will revisit this problem in section \\ref{sec:nonpert} when studying non-perturbative corrections in $\\e$.\n\n \n\nIn principle, one can also solve for the derivative of the Schwarzian in \\eqref{eq:K-in-terms-of-z} following a similar strategy to that outlined above. Namely, it is straightforward to find that when $\\Sch(z, u)= a u^n$, for some $n \\in \\mathbb Z$, then $z(u)$ is related to a Bessel function. Following the steps above, and using the fact that $\\partial^{n+1} \\Sch(z, u) = 0$ for such configurations, one can then determine all possible terms appearing in the extrinsic curvature. However, since we are interested in quantizing the theory in a constant dilaton configuration, we will shortly see that we can avoid this more laborious process. \n\nTherefore, the JT action that we are interested in quantizing is given by:\n\\begin{eqnarray}\n\\label{eq:JT-simplified-action}\nI_{JT} = -\\int_0^\\b \\frac{du}{\\e^2} \\phi_{ r} \\bigg(\\sqrt{1+ 2\\e^2\\,\\Sch(z, u)} - 1 + \\text{derivatives of Sch.}  \\bigg)   \\,,\n\\end{eqnarray}\nwhere we have added the correct counter-term needed in order to cancel the $1\/\\e^2$ divergence in the $\\e \\to 0$ limit.\n\nWhile we have found $K[z(u)]$ and $I_{JT}$ to all orders in perturbation theory in $\\e$, we have not yet studied other non-perturbative pieces in $\\e$ (that do not come from the $-$ branch in \\eqref{eq:K-in-terms-of-z}). Such corrections  could contain non-local terms in $u$ since all terms containing a finite number of derivatives in $u$ are captured by the $\\e$-perturbative expansion. The full solution of \\eqref{eq:metric-a-barZ-config} provides clues that such non-perturbative corrections could exist and are, indeed, non-local (as they will not be a functional of the Schwarzian). The full solution to \\eqref{eq:metric-a-barZ-config} is\n\\begin{eqnarray}\n\\label{eq:bar-z-diff-eq-sol}\n\\bar z(u) = e^{a u }\\left(1-2a^2 \\e^2 +2a \\e \\left(\\sqrt{-1+a^2 \\e^2}-\\frac{2\\e}{\\frac{\\e}{\\sqrt{-1+a^2 \\e^2}}+\\cC_1 e^{\\frac{u}{\\e}\\sqrt{-1+a^2 \\e^2}}} \\right)\\right)\\,,\n\\end{eqnarray} \nfor some integration constant $\\cC_1$. When $\\cC_1 \\neq 0$, note that the correction to $\\bar z(u)$ in \\eqref{eq:bar-z-diff-eq-sol} are exponentially suppressed in $1\/\\e$ and do not contribute to the series expansion $\\cK_n$. However, when taking $\\cC_1 \\neq 0$, \\eqref{eq:bar-z-diff-eq-sol} there is no way of making $\\bar z(u)$ periodic (while it is possible to make $z(u)$ periodic). While we cannot make sure that every solution has the feature that non-perturbative corrections are inconsistent with the thermal boundary conditions, for the remainder of this section we will only focus on the perturbative expansion of $K[z(u)]$ with the branch choice for the square root given by \\eqref{eq:K-in-terms-of-z}. We will make further comments about the nature of non-perturbative corrections in section \\ref{sec:nonpert}. \n\n\n\n\\subsection{Path integral measure}\n\\label{sec:path-integral-measure}\n\nBefore we proceed by solving the path integral of \\eqref{eq:JT-simplified-action}, it is important to discuss the integration measure and integration contour for $z(u)$. Initially, before imposing the constraint \\eqref{eq:lightcone-bdy-metric} on the boundary metric, we can integrate over both $z(u)$ and $\\bar z(u)$, with the two variables being complex conjugates in Euclidean signature. However, once we integrate out $\\bar z(u)$ we are free to choose an integration contour consistent with the constraint \\eqref{eq:lightcone-bdy-metric} and with the topological requirements discussed at the beginning of this section. Thus, for instance if we choose $z(u) \\in \\mR$ then the constraint \\eqref{eq:lightcone-bdy-metric} would imply that $z'(u) >0$ (or $z'(u) <0$); this, in turn, implies that we solely need to integrate over strictly monotonic functions $z(u)$. The boundary conditions for $z(u)$ should nevertheless be independent of the choice of contour; therefore we will impose that $z(u)$ is periodic, $z(0) = z(\\b)$. Of course, this implies that $z(u)$ has a divergence. In order to impose that the boundary is never self-intersecting we will impose that this divergence occurs solely once.\\footnote{All this is also the case in the Schwarzian theory whose classical solution is $\\tau(u) = \\tan(\\pi u\/\\b)$. \\cite{Maldacena:2016upp} has found that if considering  solutions where $\\tau(u)$ diverges multiple times ($\\tau(u) = \\tan(n\\pi u\/\\b)$ with $n \\in \\mathbb Z$) then the fluctuations around such solutions are unbounded, and the path integral is divergent (one can still make sense of this theory though, as explained in \\cite{Mertens:2019tcm}).    }  \nSuch a choice of contour therefore satisfies the following two criteria: \n\\begin{itemize}\n\\item That the boundary is not self-intersecting.\n\\item The boundary is time-like when going to Lorentzian signature.  This is because redefining $z(u) \\to z^\\text{Lor.}(u) =  -i z(u) \\in \\mR $ leaves the action invariant and describes the boundary of a Lorentzian manifold. Since $i  (z^\\text{Lor.})' > 0$, it then follows that the boundary would be time-like.  \n\\end{itemize} \n\n\nFurthermore, while we have chosen a specific diffeomorphism gauge which fixes $\\g_{uu}=1\/\\e^2$, the path integral measure (as opposed to the action) should be unaffected by this choice of gauge and should rather be diffeomorphism invariant. The only possible local diffeomorphism invariant path integral measure is that encountered in the Schwarzian theory \\cite{Alekseev:1988ce, Bagrets:2016cdf, Stanford:2017thb} and, in JT gravity at infinite cutoff \\cite{Saad:2019lba}: \n\\begin{eqnarray} \n\\label{eq:path-integral-measure}\nD\\mu[z] = \\prod_{z \\in [0, \\b)} \\frac{dz(u)}{z'(u)}\\,.\n\\end{eqnarray}   \nIn principle, one should also be able to derive \\eqref{eq:path-integral-measure} by considering the symplectic form for JT gravity obtained from an equivalent $\\mathfrak{sl}(2, \\mR)$ BF-theory. In \\cite{Saad:2019lba} this symplectic form (which in turn yields the path integral measure \\eqref{eq:path-integral-measure}) was derived in the limit $\\e \\to 0$. It would however be interesting to rederive the result of \\cite{Saad:2019lba} at finite $\\e$ in order to find a more concrete  derivation of \\eqref{eq:path-integral-measure}.\n\nTo summarize, we have therefore argued that both the path integration measure, as well as the integration contour, in the finite-$\\e$ theory, can be taken to be the same as those in the pure Schwarzian theory.  \n\n\n\n\\subsection{Finite cutoff partition function as a correlator in the Schwarzian theory} \n\\label{sec:correlators-and-Schwarzian}\n\nThe path integral which we have to compute is given by \n\\begin{eqnarray}\n\\label{eq:JT-path-integral-what-to-solve}\nZ_{JT}[\\phi_b, L] = \\int_{z'(u)>0} D\\mu[z]\\exp\\bigg[\\int_0^\\b \\frac{du}{\\e^2} \\phi_{ r} &\\bigg(\\sqrt{1+ 2{\\e^2} \\Sch(z, u)}  - 1 + \\nonumber \\\\ &+  \\,\\,\\,\\, \\text{derivatives of Sch.}  \\,\\bigg)\\bigg]   \\,,\n\\end{eqnarray}\nOf course, due to the agreement of integration contour and measure, we can view \\eqref{eq:JT-path-integral-what-to-solve} as the expectation value of the operator in the pure Schwarzian theory with coupling $\\phi_{r}$:\n\\begin{eqnarray}\n\\label{eq:expectation-val-op}\n&Z_{JT}[\\phi_b, L] = \\< \\cO_\\defo\\> \\equiv  \\\\ \n &\\equiv \\bigg\\<\\exp\\bigg[\\int_0^\\b \\frac{du}{\\e^2} \\phi_{r}\\,\\bigg(\\sqrt{1+ 2{\\e^2} \\Sch(z, u)}  - 1 - \\e^2 \\,\\Sch(z, u) + \\text{derivatives of Sch.} \\bigg) \\,\\bigg]  \\bigg\\>\\,.\\nonumber\n\\end{eqnarray}\nA naive analysis (whose downsides will be mention shortly) would conclude that, since in the pure Schwarzian theory, the Schwarzian can be identified with the Hamiltonian of the theory  ($-\\frac{H}{2\\phi_{r}^2} = \\Sch(z, u)$), then computing \\eqref{eq:expectation-val-op} amounts to computing the expectation value for some function of the Hamiltonian and of its derivatives. In the naive analysis, one can use that the Hamiltonian is conserved and therefore all derivatives of the Schwarzian in \\eqref{eq:expectation-val-op} can be neglected. The conservation of the Hamiltonian would also imply that the remaining terms in the integral in the exponent \\eqref{eq:expectation-val-op} are constant. Therefore, the partition function simplifies to\n\\begin{eqnarray} \nZ_{JT}[\\phi_b, L] \\, =_\\text{naive} \\bigg\\<\\exp\\bigg[  \\frac{\\b\\phi_{r}}{\\e^2} &\\bigg(\\sqrt{1- \\frac{\\e^2}{\\phi_{r}^2} H}  - 1 + \\frac{\\e^2}{2\\phi_{r}^2} \\,H\\bigg)\\bigg]  \\bigg\\>\\,.\n\\end{eqnarray} \nwhich can be conveniently rewritten in terms of the actual boundary value of the dilaton $\\phi_b = \\phi_{r}\/\\e$ and the proper length $L = \\b\/\\e$ as\n\\begin{eqnarray} \nZ_{JT}[\\phi_b, L]\\,  =_\\text{naive} \\bigg\\<\\exp\\bigg[ L \\phi_b &\\bigg(\\sqrt{1- \\frac{H}{\\phi_b^2}}  - 1 + \\frac{H}{\\phi_b}\\,\\bigg]  \\bigg\\>\\,.\n\\end{eqnarray} \nThe result for this expectation value in the Schwarzian path integral is given by \n\\begin{eqnarray}\n\\label{eq:final-naive-result}\nZ_{JT}[\\phi_b, L] \\, = _\\text{naive} \\int ds \\,s \\sinh(2\\pi s)e^{L \\phi_b \\left(\\sqrt{1-\\frac{s^2}{\\phi^2_b}} - 1 \\right)}\n\\end{eqnarray}\nwhere we have identified the energy of the Schwarzian theory in terms of the $\\mathfrak{sl}(2, \\mR)$ Casimir for which (for the principal series) $E = C_2(\\l = i s+\\frac{1}2) + \\frac{1}4 = s^2$ (see \\cite{Kitaev:2017hnr, Kitaev:2018wpr, Yang:2018gdb, Iliesiu:2019xuh}). The result \\eqref{eq:final-naive-result} agrees with both the result for the WDW wavefunctional presented in section \\ref{sec:WdW-wavefunctional-main} (up to an overall counter-term)  and with the results of \\cite{Gross:2019ach,Gross:2019uxi} (reviewed in the introduction), obtained by studying an analogue of the $T\\bar T$ deformation in $1d$.\\footnote{We identify the deformation parameter $\\l = \\frac{\\e^2}{4\\phi_{ r}}$ in \\cite{Gross:2019ach,Gross:2019uxi}.} \n\nAs previously hinted, the argument presented above is incomplete. Namely, the problem appears because correlation functions of the $\\Sch(z, u)$ are not precisely the same as those of a quantum mechanical Hamiltonian. While at separated points correlation functions of the Schwarzian are constant (just like those of $1d$ Hamiltonians), the problem appears at identical points where contact-terms are present. Therefore, the rest of this section will be focused on a technical analysis of the contribution of these contact-terms, and we will show that the final result \\eqref{eq:final-naive-result} is indeed correct even when including such terms.\n\n\\subsubsection*{The generating functional}\n\nTo organize the calculation we will first present a generating functional for the Schwarzian operator in the undeformed theory. This generating functional is defined by \n\\begin{equation}\nZ_{\\text{Sch}}[j(u)] \\equiv \\int \\frac{D\\mu[z]}{SL(2, \\mR)}\\,  e^{ \\int_0^\\b du j(u) \\Sch(z(u), u)} ,\n\\end{equation}\nfor an arbitrary function $j(u)$ which acts as a source for Schwarzian insertions. This path integral can be computed repeating the procedure in \\cite{Stanford:2017thb}, which we also review in appendix \\ref{sec:vardil}. The final answer is given by \n\\begin{eqnarray}\n\\label{eq:generating-functional}\nZ_\\text{Sch}[j(u)] \\sim e^{\\int_0^\\b {du}\\frac{j'(u) ^2}{2j(u)}} \\int ds \\,{s} \\sinh(2\\pi s) e^{-\\frac{s^2}2 \\int_0^\\b \\frac{du}{j(u)} }\\,.\n\\end{eqnarray}\nWe will use \\eqref{eq:generating-functional} to evaluate the integrated correlator \\eqref{eq:expectation-val-op}, by rewriting it as \n\\begin{eqnarray} \n\\label{eq:correlator-to-functional-derivative}\n\\<\\cO_\\defo \\> =\\bigg[\\exp\\left(\\int_0^\\b \\frac{du}{\\e^2} \\phi_{r} : \\left(\\sqrt{1+ 2{\\e^2} \\frac{\\delta}{\\delta j(u)}}  - 1  + \\cK\\left[\\partial_u \\frac{\\delta}{\\delta j(u)}\\right] \\right):\\,\\right) \\nonumber \\\\   \\times \\,Z_\\text{Sch}[j(u)]\\bigg] \\bigg|_{j(u) = 0}\\,,\n\\end{eqnarray}\nwhere  $\\cK\\left[\\partial_u \\frac{\\delta}{\\delta j(u)}\\right]$ is a placeholder for terms containing derivative terms of the Schwarzian and, equivalently, for terms of the from $\\dots \\partial_u \\frac{\\delta}{\\delta j(u)} \\dots $. Finally, $:\\cO:$ is a point-splitting operation whose role we will clarify shortly. \n\n\n\n\\subsubsection*{Computing the full path integral}\n\nTo understand the point splitting procedure necessary in \\eqref{eq:new-expectation-value}, we start by analyzing the  structure of correlators when taking functional derivatives  of $Z_{JT}[j(u)]$. Schematically, we have that\n\\begin{eqnarray}\n\\label{eq:example-generating-functional}\n \\bigg(\\frac{\\delta}{\\delta j(u_1)} \\dots \\frac{\\delta}{\\delta j(u_n)} Z_\\text{Sch}[j(u)]\\bigg)\\bigg|_{j(u) = \\phi_{r}} = a_1 + a_2[\\delta(u_{ij})] + a_3[\\partial_u \\delta(u_{ij})] + \\dots\\,,\n\\end{eqnarray}\nwhere $a_1$ is a constant determined by the value of the coupling constant $\\phi_{r}$ and $a_2[\\delta(u_{ij})]]$ captures terms which have $\\delta$-functions in the distances $u_{ij} = u_i - u_j$, while  $a_3[\\partial_u \\delta(u_{ij})]$ contains terms with at least one derivative of  the same $\\delta$-functions for each term.\\footnote{For example, when $n=2$ the exact structure of \\eqref{eq:example-generating-functional} is computed in \\cite{Stanford:2017thb} and is reviewed in appendix \\ref{app:Kexplicit}. } The $\\dots$ in \\eqref{eq:example-generating-functional} capture potential higher-derivative contact-terms.\n\nIf in the expansion of the square root in the exponent of \\eqref{eq:correlator-to-functional-derivative} one takes the functional derivative $\\delta\/\\delta j(u)$ at identical points then the contact terms in \\eqref{eq:example-generating-functional} become divergent (containing $\\delta(0)$, $\\delta'(0)$, $\\dots$). An explicit example about such divergences is given in appendix \\ref{app:Kexplicit} when evaluating the contribution of $K_4[z]$ in the perturbative series. In order to eliminate such divergences we define the point-splitting procedure\n\\begin{eqnarray}\n\\label{eq:ordering-prescription}\n:\\frac{\\delta^n}{\\delta j(u)^n}  : \\,\\equiv \\, \\lim_{(u_1,\\, \\dots,\\, u_n) \\to u}\\, \\frac{\\delta}{\\delta j(u_1)} \\dots \\frac{\\delta}{\\delta j(u_n)} \\,.\n\\end{eqnarray}\nSuch a procedure eliminates the terms containing $\\delta(0)$ or its derivatives since we first evaluate the functional derivatives in the expansion of \\eqref{eq:new-expectation-value} at separated points. \n\nThe structure of the generating functional also suggests that when integrating the correlator \\eqref{eq:example-generating-functional} the contribution of the derivatives of $\\delta(u_{ij})$ vanish after integration by parts since we will be evaluating \\eqref{eq:new-expectation-value} for constant dilaton values. As we explain in more detail in appendix \\ref{app:Kexplicit}, the origin of the derivatives of $\\delta(u_{ij})$ is two-fold: they either come by taking functional derivatives $\\delta\/\\delta j(u)$ of the term $\\exp\\left({\\int_0^\\b {du}\\frac{j'(u) ^2}{2j(u)}}\\right)$  in $Z_{Sch}[j(u)]$, or they come from the contribution of  the derivative terms $\\cK\\left[\\partial_u \\frac{\\delta}{\\delta j(u)}\\right]$. In either case, both sources only contribute terms containing  derivatives of $\\delta$-functions (no constant terms or regular $\\delta$-functions). Thus, since such terms vanish after integration by parts, neither  $\\cK\\left[\\partial_u \\frac{\\delta}{\\delta j(u)}\\right]$ nor $\\exp\\left({\\int_0^\\b {du}\\frac{j'(u) ^2}{2j(u)}}\\right)$ contribute to the partition function. Consequently, we have to evaluate\n\\begin{eqnarray} \n\\label{eq:new-expectation-value}\n&\\<\\cO_\\defo\\>\\nonumber \\\\ &=\\bigg(\\int ds\\, s \\sinh(2\\pi s)\\,\\exp\\bigg[\\int_0^\\b \\frac{du}{\\e^2} \\phi_{ r} \\bigg(:\\sqrt{1+ 2{\\e^2}\\frac{\\delta}{\\delta j(u)}}:  - 1 \\bigg)\\bigg] \\,  e^{-\\frac{s^2}2 \\int_0^\\b du \\frac{1}{j(u)}}\\bigg)\\bigg|_{j(u) = 0}\\,.\n\\end{eqnarray}\nTo avoid having to deal with the divergences eliminated by the point-splitting discussed in the continuum limit, we proceed by discretizing the thermal circle into $\\b\/\\delta$ units of length $\\delta$ (and will ultimately consider the limit $\\delta \\to 0$).\\footnote{Sums and products of the type $\\sum_{u \\in [0, \\b)}$ and $\\prod_{u \\in [0, \\b)}$ will iterate over all $\\b\/\\delta $ intervals.   }  Divergent terms containing $\\delta$ in the final result correspond to terms that contain $\\delta(0)$ in the continuum limit and thus should be eliminated by through the point-splitting procedure \\eqref{eq:ordering-prescription}. Therefore, once we obtain the final form of \\eqref{eq:new-expectation-value}, we will select the universal diffeomorphism invariant $\\delta$-independent term. \n\nTo start, we can use that \n  \\begin{eqnarray} \n  \\label{eq:Laplce-transf-1}\ne^{-\\frac{s^2\\delta}{2 j(u)}} = \\frac{1}{2\\pi i} \\int_{-c - i\\oo}^{-c + i \\oo} d\\a_u \\left[-\\frac{\\pi Y_1(2\\sqrt{\\a_u})}{\\sqrt{\\a_u}}\\right] e^{-\\frac{2\\a_u j_u}{s^2 \\delta}}\n\\end{eqnarray}\nwhere we have introduced a Lagrange multiplier $\\a_u$ for each segment in the thermal circle.  The integration contours for all $\\a_u$ are chosen along the imaginary axis for some real constant $c$. The next step is to apply the differential operator in the exponent in \\eqref{eq:new-expectation-value} to \\eqref{eq:Laplce-transf-1},\n\\begin{eqnarray} \n\\label{eq:action-of-diff-ops-on-exp}\n&\\,\\,\\,\\,\\,\\,\\,\\left(e^{\\int_0^{\\b} du  \\frac{\\phi_{ r}}{\\e^2}\\,\\left(:\\sqrt{1+2\\e^2\\frac{\\delta}{\\delta j(u)}}:-1\\right)}  \\right)\\prod_{u \\in [0, \\b) }e^{-\\frac{2\\a_u j_u}{ s^2 \\delta}} \\bigg|_{j_u=0} =  \\nonumber \\\\ \n&= \\left(e^{\\int_0^{\\b} du  \\frac{\\phi_{ r}}{\\e^2}\\,\\left(:\\sqrt{1+2\\e^2\\frac{\\delta}{\\delta j(u)}}:-1\\right)}\\right)  e^{- \\int_0^\\b du  \\,\\frac{2 \\a_u j_u}{s^2 \\delta^2}} \\bigg|_{j_u=0}\\nonumber \\\\ &= \\,:  \\exp\\left[\\sum_{u \\in [0, \\b)} \\frac{\\delta\\phi_{r}}{\\e^2} \\left(\\sqrt{1-\\frac{4\\a_u\\e^2}{s^2 \\delta^2}}-1\\right)\\right]:\\,,\n\\end{eqnarray}\nwhere $:\\dots:$ indicates that we will be extracting the part  independent of the UV cutoff, $\n\\delta$, when taking the limit $\\delta \\to 0$. Thus, we now need to compute \n\\begin{eqnarray} \nZ_{JT}\\left[\\phi_b,\\, L\\right]=\\frac{1}{2\\pi i} :\\int_0^\\infty ds \\,{s} \\sinh(2\\pi s)&\\int_{-c - i\\oo}^{-c + i \\oo} \\left(\\prod \\,d\\a_u \\right)\\left[-\\frac{\\pi Y_1(2\\sqrt{\\a_u})}{\\sqrt{\\a_u}}\\right] \\nonumber \\\\ &\\times e^{\\sum_{u \\in [0, \\b)} \\frac{\\delta\\phi_{r}}{\\e^2} \\left(\\sqrt{1-\\frac{4\\a_u\\e^2}{s^2 \\delta^2}}-1\\right)}\\,:.\n\\end{eqnarray}\nIn order to do these integrals we introduce an additional field $\\s_u$, such that\n\\begin{eqnarray}\n\\label{eq:Laplce-transf-2}\ne^{ \\frac{\\delta\\phi_{r}}{\\e^2}  \\left(\\sqrt{1-\\frac{4\\a_u\\e^2}{s^2 \\delta^2}}-1\\right)} = \\int_0^\\infty \\frac{d \\sigma_u}{\\sigma_u^{3\/2}} \\,\\sqrt{-\\frac{\\delta \\phi_{r}}{2\\pi \\e^2}}\\, e^{- \\frac{2\\sigma_u \\a_u \\phi_{r}}{s^2 \\delta} + \\frac{\\delta \\phi_{r}}{2\\sigma_u \\e^2}(1-\\sigma_u)^2}\\,,\n\\end{eqnarray}\nwhere in order for the integral \\eqref{eq:Laplce-transf-2} to be convergent, we can analytically continue $\\phi_{r}$ to complex values. \nWe can now perform the integral over $\\a_u$ using \\eqref{eq:Laplce-transf-1}, since $\\a_u$ now appears once again in the numerator of the exponent:\n\\begin{eqnarray}\nZ_{JT}\\left[\\phi_b,\\,L\\right]=&\\,:\\int_0^\\infty  ds \\,{s} \\sinh(2\\pi s) \\nonumber \\\\ & \\times \\int_0^\\infty \\left( \\prod_{u\\in[0, \\b) }\\frac{d \\sigma_u}{\\sigma_u^{3\/2}} \\,\\sqrt{-\\frac{\\delta \\phi_{r}}{2\\pi \\e^2}} \\right)\\, e^{\\sum_{u\\in[0, \\b) }\\left[- \\frac{s^2 \\delta}{2\\sigma_u \\phi_{r}}+ \\frac{\\delta \\phi_{r}}{2\\sigma_u \\e^2}(1-\\sigma_u)^2\\right]}:\\,.\n\\end{eqnarray}\nWe now change variable in the equation above from $\\sigma_u \\to 1\/\\tilde \\sigma_u$ and perform the Laplace transform, once again using \\eqref{eq:Laplce-transf-2}. We finally find that (when keeping the finite terms in $\\delta$) the partition function is given by:\\footnote{Once again to integrate over $\\tilde \\sigma_u$ we have to analytically continue $\\phi_{r}$ to complex values. Finally, to perform the integral over $s$ in \\eqref{eq:JT-analytic-final-result} we analytically continue back to real values of $\\phi_{r}$ and, equivalently, $\\phi_b$. } \n\\begin{eqnarray}\n\\label{eq:JT-analytic-final-result}\n Z_{JT}\\left[\\phi_b, \\,L\\right] &\\sim\\int_0^\\infty  ds \\,{s}  \\sinh(2\\pi s)\\, e^{\\frac{\\beta \\phi_{r}}{\\e^2}\\left(\\sqrt{1-\\frac{s^2\\e^2}{\\phi_{r}^2}}-1 \\right)}\n\\nonumber \\\\ \n&\\sim\\int_0^\\infty  ds\\,{s}  \\sinh(2\\pi s)\\, e^{\\frac{\\b}{4\\l}\\left(\\sqrt{1-4\\l s^2\/\\phi_{r}}-1 \\right)}\\,,\n\\end{eqnarray}\nwhere we defined $\\l = \\e^2\/(4\\phi_{r})$. This partition function agrees with the naive result \\eqref{eq:final-naive-result} obtained by replacing the Schwarzian with the Hamiltonian of the pure theory. Consequently, we arrive to the previously mentioned matching between the Euclidean partition function, the WDW wavefunctional and the partition function of the $T\\bar T$ deformed Schwarzian theory,\n\\begin{eqnarray}\ne^{-I_{\\rm ct}}\\Psi_{HH}[\\phi_b,\\, L] = Z_{\\l=\\e^2\/(4\\phi_{r})}(\\b) = Z_{JT}[\\phi_b,\\, L] \\,.\n\\end{eqnarray}\n\nAs a final comment,  the Euclidean path integral approach hides two ambiguities. First, as we briefly commented in section \\ref{sec:finding-extrinsic-curvature}, the finite cutoff expansion of the extrinsic curvature might involve terms that are non-perturbatively suppressed in $\\varepsilon$. As we have mentioned before, such terms can either come from considering non-local terms in the extrinsic curvature $K[z(u)]$ or by considering the contribution of the negative branch in \\eqref{eq:K-in-terms-of-z}.  Second, even if these terms would vanish, the perturbative series is only asymptotic. Performing the integral \\eqref{eq:JT-analytic-final-result} over energies explicitly gives a finite cutoff partition function \n\\begin{equation}\\label{eqn:k2fullanswer}\nZ_{JT}[\\phi_b, L] =  \\frac{L \\phi^2_b e^{-L \\phi_b} }{L^2 +4\\pi^2} K_2\\Big( -\\sqrt{\\phi^2_b(L^2 + 4\\pi^2)} \\Big).\n\\end{equation} \nThis formal result is not well defined since the Bessel function is evaluated at a branch cut \\footnote{This can be tracked to the fact that we are sitting at a Stokes line. It is curious that this explicit answer gives a complex function even though the perturbative terms we found from the path integral are all real (this phenomenon also happens in more familiar setups like WKB \\cite{WKB}).}. The ambiguity related to the presence of this branch cut can be regulated by analytic continuation; for example, in $L \\to L e^{i \\epsilon}$, and the $\\epsilon \\to 0$ limit we find different answers depending on the sign of $\\epsilon$.  The ambiguity given by the choice of analytic continuation can be quantified by the discontinuity of the partition function ${\\rm Disc}~Z$ for real $\\phi$ and $L$.   \n\nA similar effect is reproduced by the contracting branch of the wavefunction from the canonical approach, there are two orthogonal solutions to the gravitational constraint $\\Psi_\\pm$, defined by their small cutoff behavior $\\Psi_\\pm \\sim e^{\\pm \\phi L}Z_\\pm$, where $Z_\\pm$ is finite. In the language of the Euclidean path integral, the different choice of wavefunctionals correspond to different choices for the square root in the extrinsic curvature \\eqref{eq:K-in-terms-of-z}.  Imposing Hartle-Hawking boundary conditions fixes $\\Psi_+$, which matches the perturbative expansion of the Euclidean path integral. The corrections to the partition function from the other branch are exponentially suppressed $\\Psi_- \/ \\Psi_+ \\sim e^{-\\frac{1}{\\varepsilon^2}}$. \n\nAs previously hinted, contributions from turning on $\\Psi_-$ are not only related to the choice of branch for $K[z(u)]$, but is the same as the branch-cut ambiguity mentioned above for \\eqref{eqn:k2fullanswer}. To see this, we can notice that ${\\rm Disc}~Z$ is a difference of two functions that separately satisfy the WDW equation and goes to zero at small cutoff. Therefore it has to be of the same form as the $\\Psi_-$ branch given in \\eqref{contracting}.\n\n\\section{The contracting branch and other topologies}\n\\label{sec:nonpert}\nIn this section, we will analyze two different kinds of non-perturbative corrections to the partition function. First we will study corrections that are non-perturbative in the cutoff parameter $\\e$ in sections \\ref{sec:npco} and \\ref{sec:npco2}, which come from turning on the contracting branch of the wavefunction.\nThen, we will comment on non-perturbative corrections coming from non-trivial topologies in section \\ref{sec:nptopo}.\n\n\\subsection{Unitarity at finite cutoff}\n\\label{sec:npco}\n\nGiven the exact form of the wavefunction for general cutoff surfaces, we can study some of the more detailed questions about $T\\bar{T}$ in $AdS_2$. One such question is whether the theory can be corrected to become unitarity. As can be seen from the expression for the dressed energy levels \\eqref{deformedE}, the energies go complex whenever $\\l > 1\/(8E)$. This is unsatisfactory if we want to interpret the finite cutoff JT gravity partition function as being described by a $0+1$ dimensional theory, just like the Schwarzian theory describes the full $AdS_2$ bulk of JT gravity. There are a few ways in which one can go around this complexification. \n\nFirstly, we can truncate the spectrum of the initial theory so that $E$ is smaller than some $E_{\\rm max}$. This is totally acceptable, but if we want to have an initial theory that describes the full $AdS_2$ geometry, we cannot do that without making the flow irreversible. In other words, the truncated Schwarzian partition function is not enough to describe the entire JT bulk. The second option is to accept there are complex energies along the flow but truncate the spectrum to real energies after one has flowed in the bulk. In $1D$ this was emphasized in \\cite{Gross:2019ach} (and in \\cite{McGough:2016lol, Smirnov:2016lqw} for $2D$ CFTs). The projection operator that achieves such a truncation will then depend on $\\l$ and, in general, will not solve the flow equation \\eqref{diffeqZ} of the partition function. A third option is that we use the other branch of the deformed energy levels $\\mathcal{E}_{-}$ (see \\eqref{deformedE}) to make the partition function real. In doing so, we will be guaranteed a solution to the Wheeler-de-Witt equation. Let us pursue option three in more detail and show that we can write down a real partition function $Z_{\\l}(\\b)$ with the correct (Schwarzian) boundary condition at $\\l \\to 0$. \n\nThe solution to the $T\\bar{T}$ flow equation \\eqref{diffeqZ} that takes the form of a partition function is, \n\\begin{eqnarray}\\label{solZ}\n{Z}_{\\l}^{\\rm non-pert.}(\\b) =  \\int_{0}^{\\infty} dE \\rho_+(E) e^{-\\b \\mathcal{E}_+(E,\\l)} + \\int_{-\\infty}^{\\infty} dE \\rho_-(E) e^{-\\b \\mathcal{E}_-(E,\\l)}.\n\\end{eqnarray}\nHere, we took the ranges of $E$ to be such that $\\mathcal{E}_{\\pm}$ are bounded from below. As $\\l \\to 0$, we see that the first term goes to some constant (as we already saw previously), but the second term goes to zero non-perturbatively in $\\l$ as $e^{-\\b\/(2\\l)}$. From the boundary condition $\\l \\to 0$ we can therefore not fix the general solution, but only $\\rho_+(E) = \\sinh(2\\pi \\sqrt{2CE})$. If we demand the partition function to be real, then both integrals over $E$ in \\eqref{solZ} should be cutoff at $E = 1\/(8\\l)$ and it will therefore not be a solution to \\eqref{diffeqZ} anymore, because the derivatives with respect to $\\l$ can then act on the integration limit. However, by picking\n\\begin{eqnarray}\\label{rho}\n\\rho_- = \\left\\{\\begin{array}{cc}\n-\\sinh(2\\pi \\sqrt{2CE}) & 0 < E < \\frac{1}{8\\l}\\\\\n\\hat{\\rho}(E) & E < 0\n\\end{array}\\right.,\n\\end{eqnarray}\nwith $\\hat{\\rho}(E)$ an arbitrary function of $E$, the boundary terms cancel and we obtain a valid solution to \\eqref{diffeqZ} and the associated wavefunction $\\Psi = e^{L\\phi_b}Z$ will solve the WDW equation \\eqref{minisupereq}. The final partition function is then given by (see appendix \\ref{app:flowsol} for details),\n\\begin{eqnarray}\n\\label{Znp}\n{Z}^{\\rm non-pert.}_{\\l}(\\b) = \\frac{\\pi \\b e^{-\\frac{\\b}{4\\l}}}{\\sqrt{2 \\l}(\\b^2 + 16 C \\pi^2 \\l)}I_2\\left( \\frac{1}{4\\l}\\sqrt{\\b^2 + 16 C \\pi^2 \\l} \\right) + \\int_{-\\infty}^0 dE \\hat{\\rho}(E) e^{-\\b \\mathcal{E}_-(E,\\l)}.\n\\end{eqnarray}\nNotice that when we redefine $E$ such that we have the canonical Boltzman weight in the second term of \\eqref{Znp}, the support of $\\hat{\\rho}$ is for $E > \\frac{1}{2\\l}$, because for this redefined energy $E = 0$ maps to $\\frac{1}{2\\l}$. Let us comment on this partition function. First, because of the sign in \\eqref{rho}, the first part of \\eqref{Znp} has a negative density of states and turns out the be equal to \\eqref{deformedrho} with support between $0 \\leq E \\leq \\frac{1}{2\\l}$, see Fig. \\ref{fig:DOS}.\n\\begin{figure}\n\\centering\n\\includegraphics[scale=1]{PlotDOS.pdf}\n\\caption{\\label{fig:DOS} In orange, we show the undeformed density of states $\\sinh(2\\pi \\sqrt{E})$ of JT gravity at infinite cutoff. In dashed black, we show the density of states of the theory with just the branch of the root, $\\mathcal{E}_-$, that connects to the undeformed energies, until the energy complexifies. In blue, we show the density of states $\\rho_{\\l}(E)$ of the deformed partition function \\eqref{Znp} which includes non-perturbative corrections in $\\l$. Above we have set $\\hat{\\rho}(E)=0$ and $\\l = 1\/4$ and $C = 1\/2$, the black line therefore ends at $E = \\frac{1}{4\\l} = 1$. The vertical dashed line indicates the energy beyond which $\\hat{\\rho}$ has support. }\n\\end{figure}\nSecond, there is a whole function worth of non-perturbative ambiguities coming from the second term in \\eqref{Znp} that cannot be fixed by the Schwarzian boundary condition. From the Euclidean path integral approach, assuming that the extrinsic curvature does not receive non-perturbative corrections, we could fix $\\hat{\\rho}(E)=0$ by choosing an appropriate analytic continuation on $L$ when defining the partition function.\n \n\n\\subsection{Relation to $3D$ gravity}\\label{sec:npco2}\n\nThe analysis in the previous section can be repeated in the context of $3D$ gravity and $T\\bar{T}$ deformations of $2D$ CFTs on a torus of parameters $\\tau$ and $\\bar{\\tau}$. The deformed partition function satisfies an equation similar to \\eqref{diffeqZ} derived in \\cite{Aharony:2018bad}. This is given by\n\\begin{eqnarray}\n-\\partial_{\\l} Z_{\\l} = \\left[8 \\t_2 \\partial_{\\t}\\partial_{\\bar{\\t}} + 4\\left(i (\\partial_{\\t} - \\partial_{\\bar{\\t}}) - \\frac{1}{\\t_2} \\right) \\l \\partial_{\\l} \\right]Z_{\\l}\n\\end{eqnarray}\nThe solutions of this equation, written in a form of a deformed partition function, can be written as\n\\begin{eqnarray}\nZ(\\t,\\bar{\\t},\\l) = \\sum_{\\pm, \\,k} \\int_{E_0}^{\\infty} dE \\rho_\\pm(E) e^{-\\t_2 \\mathcal{E}_\\pm(E,k) + 2\\pi i k \\t_1}\n\\end{eqnarray}\n where $\\tau = \\t_1+i \\t_2$ and $\\bar \\tau = \\t_1 - i \\t_2$. \nHere we have set the radius to one and\n\\begin{eqnarray}\\label{deformedE3d}\n\\mathcal{E}_\\pm(E,k) = \\frac{1}{4\\l}\\left(1 \\mp \\sqrt{ 1 - 8 \\l E + 64 \\pi^2 k^2 \\l^2 }\\right).\n\\end{eqnarray}\nAs usual we pick the minus sign of the root as that connects to the undeformed energy levels at $\\l = 0$. The energy levels of the deformed partition function complexify when $E_c = \\frac{1}{8\\l} + 8 k^2 \\pi^2 \\l^2$. So we would like to cutoff the integral there. Similarly, a hard cutoff in the energy will not solve the above differential flow equation anymore. We can resolve this by subtracting the same partition function but with the other sign of the root in \\eqref{deformedE3d}. This is again a solution, but (again) with negative density of states. \n\n\\subsection{Comments about other topologies}\\label{sec:nptopo}\n\nFinally, we discuss the contribution to the path integral of manifolds with different topologies. The contribution of such surfaces is non-perturbatively suppressed by $e^{-\\phi_0 \\chi(M)}$, where $\\chi(M)$ is the Euler characteristic of the manifold. \n\nWe start with surfaces with two boundaries of zero genus, where one boundary has the Dirichlet boundary conditions \\eqref{eq:Dirichlet-bdy-conditions} and the other ends on a closed geodesic with proper length $b$. The contribution of such surfaces to the partition function, referred to as ``trumpets'', has been computed in the infinite cutoff limit in \\cite{Saad:2019lba}. We can repeat the method of section \\ref{sec:HH-boundary-conditions-JT-wavefunctional} to a spacetime with the geodesic hole of length $b$ by applying the WDW constraints to the boundary on which we have imposed the Dirichlet boundary conditions. This constraint gives the trumpet finite cutoff partition function\n\\begin{equation}\n\\label{eq:trumpet-part-function}\nZ_{\\rm trumpet} [\\phi_b, L, b] = \\frac{\\phi L}{\\sqrt{L^2 - b^2}} K_1\\Big(-\\sqrt{\\phi_b^2(L^2-b^2)} \\Big)\\,.\n\\end{equation}\nThe partition function diverges as $L \\to b$, indicating the fact that the boundary with Dirichlet boundary conditions overlaps with the geodesic boundary. \n\nIn order to construct higher genus surfaces or surfaces with more Dirichlet boundaries one can naviely glue the trumpet to either a higher genus Riemann bordered surface or to another trumpet.  In order to recover the contribution to the partition function of such configurations we have to integrate over the closed geodesic length $b$ using the Weil-Petersson measure, $d\\mu[b] = db \\,b$. However, if integrating over $b$ in the range  from $0$ to $\\infty$ for a fixed value of $L$ we encounter the divergence at $L=b$. \n\n One way to resolve the appearance of this divergence is to once again consider the non-perturbative corrections in $\\e$ discussed in section \\ref{sec:npco} for the trumpet partition function \\eqref{eq:trumpet-part-function}. We can repeat the same procedure as in \\ref{sec:npco} by accounting for the other WDW branch thus making the density of states of the ``trumpet'' real.  Accounting for the other branch we find that\n\\begin{eqnarray} \n\\label{eq:trumpet-part-function-1}\nZ_{\\rm trumpet}^{\\rm non-pert.} [\\phi_b, L, b] = \\frac{2\\pi\\phi L}{\\sqrt{L^2 - b^2}} I_1\\Big(\\sqrt{\\phi_b^2(L^2-b^2)} \\Big)  \\,,\n\\end{eqnarray}\nwhere we set the density of states for negative energies for the contracting branch to $0$. Interestingly, the partition function \\eqref{eq:trumpet-part-function-1} no longer has a divergence at $L=b$ which was present in \\eqref{eq:trumpet-part-function} and precluded us previously from performing the integral over $b$. We could now integrate \\footnote{Alternatively, one might hope to directly use WDW together with the results of \\cite{Saad:2019lba} for arbitrary genus to directly compute the partition function at finite cutoff. However, as pointed out in \\cite{Giddings:1988wv}, the WDW framework is insufficient for such a computation; instead, computing the full partition function requires a third-quantized framework which greatly complicates the computation.  } \n\\begin{eqnarray} \n\\label{eq:cylinder-partition-function}\nZ_{\\rm cyl.}^{\\rm non-pert.}[\\phi_{b_1}, L_1, \\phi_{b_2}, L_2] =_\\text{naive} \\int_0^\\infty db\\, b\\,Z_{\\rm trumpet}^{\\rm non-pert.} [\\phi_{b_1}, L_1, b] Z_{\\rm trumpet}^{\\rm non-pert.} [\\phi_{b_2}, L_2, b]\\,,\n\\end{eqnarray}\n to obtain a potential partition function for the cylinder.\\footnote{ While unfortunately we cannot compute the integral over $b$ exactly it would be interesting to check whether the partition function for the cylinder can be reproduced by a matrix integral whose leading density of states is given by the one found from the disk contribution. \n} \n\nBesides the ambiguity related to the non-perturbative corrections, there is another issue with the formula for the cylinder partition function \\eqref{eq:cylinder-partition-function}. Specifically, for any value of the proper length $L_1$ and $L_2$ and for a closed geodesic length $b$ (with $b<L_1$ and $b<L_2$) there exist cylinders for which the Dirichlet boundaries intersect with the closed geodesic of length $b$. Such surfaces cannot be obtained by gluing two trumpets along a closed geodesic as  \\eqref{eq:cylinder-partition-function} suggests when using the result \\eqref{eq:trumpet-part-function}. Given that the partition function \\eqref{eq:trumpet-part-function-1} does not have a clear geometric interpretation when including the contributions from the contracting branch, it is unclear if  \\eqref{eq:cylinder-partition-function} accounts for such geometries. Given these difficulties, we hope to revisit the problem of summing over arbitrary topologies in the near future.\n\nAs another example of non-trivial topology, one can study the finite cutoff path integral in a disk with a conical defect in the center. Such defects were previously studied at infinite cutoff in \\cite{Mertens:2019tcm}.  The answer from the canonical approach is given by\n\\begin{equation}\nZ_{\\rm defect} [\\phi_b, L] = \\frac{\\phi L}{\\sqrt{L^2 + 4\\pi^2 \\alpha^2}} K_1\\Big(-\\sqrt{\\phi_b^2(L^2+4\\pi^2 \\alpha^2)} \\Big),\n\\end{equation}\nwhere $\\alpha$ is the opening angle, and $\\alpha =1$ gives back the smooth disk wavefunction. This function is finite for all $L$. \n \n\n\n\\section{de Sitter: Hartle-Hawking wavefunction}\\label{sec:desitter}\n\nAs a final application of the results in this paper, we will study JT gravity with positive cosmological constant, in two-dimensional nearly $dS$ spaces. We will focus on the computation of the Hartle-Hawking wavefunction, see \\cite{MTY}, and \\cite{Cotler:2019nbi}. The results in these references focus on wavefunctions at late times, with an accurate Schwarzian description. Using the methods in this paper, we will be able to compute the exact wavefunction at arbitrary times. \n\n The Lorenzian action for positive cosmological constant JT gravity is given by\n\\begin{equation}\\label{IJTds}\nI_{\\rm JT}=\\frac{1}{2}\\int_{M} \\sqrt{g} \\phi( R-2) - \\int_{\\partial M} \\sqrt{\\g}\\phi K.\n\\end{equation}\nFollowing section \\ref{sec:WdW-finite-cutoff-JT}, we use the ADM decomposition of the metric  \n\\begin{equation}\nds^2 =- N^2 dt^2 + h (d\\theta + N_{\\perp} dt)^2,~~~~h=e^{2\\sigma}\n\\end{equation}\nwhere now $t$ is Lorenzian time and $\\theta$ the spatial direction. We will compute the wavefunction of the universe $\\Psi[L,\\phi_b(u)]$ as a function of the total proper length of the universe $L$ and the dilaton profile $\\phi_b(u)$ along a spatial slice. The proper spatial length along the boundary is defined by $du= e^{\\sigma} d\\theta$.\n\\begin{figure}[t]\n\\centering\n\\begin{tikzpicture}[scale=0.9]\n\\pgftext{\\includegraphics[scale=0.7]{ds2.pdf}} at (0,0);\n\\draw (3,0.5) node  { $t$ };\n\\end{tikzpicture}\n\\caption{\\small Frame in which geometry is rigid $dS_2$. Time runs upwards. We show the wiggly curve where we compute the wavefunction in blue (defined by its length and dilaton profile). \\label{fig:ds2q}}\n\\end{figure}\nThe solution satisfying the gravitational constraints is given by \n\\begin{equation}\\label{ds2ansatz}\n\\Psi_+[\\phi_b(u), L] =\\int dM \\rho(M) e^{-i \\int_0^L du \\left[ \\sqrt{\\phi_b^2 - M+( \\partial_u \\phi_b)^2} - \\partial_u \\phi_b \\tanh^{-1} \\left(\\sqrt{1+ \\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}}\\right)  \\right]},\n\\end{equation}\nwhere the index $+$ indicates we will focus on the expanding branch of the wavefunction. This is defined by its behavior $\\Psi_+ \\sim e^{- i \\int_0^L du~ \\phi_b(u)} $ in the limit of large universe (large $L$). \n\nTo get the wavefunction of the universe, we need to impose the Hartle-Hawking boundary condition. We will look again to the limit of large $L$, and for simplicity, we can evaluate it for a constant dilaton setting $\\partial_u \\phi_b=0$ (this is enough to fix the expanding branch of the wavefunction completely). \n\nAs explained in \\cite{MTY}, one can independently compute the path integral with Hartle-Hawking boundary conditions in this limit by integrating out the dilaton first. This fixes the geometry to be rigid $dS_2$, up to the choice of embedding of the boundary curve inside rigid $dS_2$, see figure \\ref{fig:ds2q}. Then the result reduces to a Schwarzian path integral parametrizing boundary curves, just like in $AdS_2$. The final result for a constant dilaton and total length $L$ is given by \n\\begin{equation}\n\\Psi_+[ \\phi_b, L] \\sim e^{- i \\phi_b L} \\int dM \\sinh{(2\\pi\\sqrt{M})} e^{i L \\frac{M}{2\\phi_b}},~~~~L\\phi_b \\to \\infty,~~~\\phi_b\/L~{\\rm fixed.}\n\\end{equation}\nThis boundary condition fixes the function $\\rho(M)$ in \\eqref{ds2ansatz}, analogously to the procedure in section \\ref{sec:HH-boundary-conditions-JT-wavefunctional}. \n\n\nThen the final answer for the expanding branch of the Hartle-Hawking wavefunction of JT gravity is \n\\begin{equation}\\label{dshhfinal}\n\\Psi_+[\\phi_b, L] =\\int_0^\\infty dM \\sinh(2\\pi \\sqrt{M}) e^{-i \\int_0^L du \\left[ \\sqrt{\\phi_b^2 - M+( \\partial_u \\phi_b)^2} - \\partial_u \\phi_b \\tanh^{-1} \\left(\\sqrt{1+ \\frac{\\phi_b^2-M}{(\\partial_u \\phi_b)^2}}\\right)  \\right]}.\n\\end{equation}\nThe same result can be reproduced for constant values of $\\phi_b(u) = \\phi_b$ by following the procedure in section \\ref{sec:JT-gravity-review}, writing the extrinsic curvature along the spatial slice as a functional of the Schwarzian derivative. Following the same steps as in section \\ref{sec:correlators-and-Schwarzian}, one could then recover the wavefunction \\eqref{dshhfinal} by computing the Lorentzian path integral exactly, to all orders in cutoff parameter $\\e$. \n\nThe procedure outlined so far parallels the original method of Hartle and Hawking \\cite{PhysRevD.28.2960}. First, we solve the WDW equation, which for this simple theory can be done exactly. Then, we impose the constraints from the no-boundary condition. The only subtlety is that, while Hartle and Hawking impose their boundary conditions in the past, we are forced to impose the boundary condition at late times. This is a technical issue since the limit $L\\to0$ is strongly coupled. Nevertheless, we could, in principle, do it at early times if we would know the correct boundary condition in that regime.  \n\nA different procedure was proposed by Maldacena \\cite{Maldacena:2002vr}. The idea is to compute the no-boundary wavefunction by analytic continuation, where one fills the geometry with `$-AdS$' instead of $dS$.\\footnote{For a review in the context of JT gravity see section 2.3 of \\cite{MTY}.} We can check now in this simple model that both prescriptions give the same result. For simplicity, after fixing the dilaton profile to be constant, one can easily check that the result \\eqref{dshhfinal} found following Hartle and Hawking matches with the analytic continuation of the finite cutoff Euclidean path integral in AdS computed in section \\ref{sec:JT-gravity-review}. \n\nFor a constant dilaton profile, we can perform the integral to compute the wavefunction\n\\begin{equation}\\label{K2wavefunction}\n\\Psi_+[\\phi_b, L] =  \\frac{L \\phi^2_b }{L^2 -4\\pi^2-i\\epsilon} K_2\\Big( i \\sqrt{\\phi^2_b(L^2 - 4\\pi^2-i\\epsilon)} \\Big),\n\\end{equation}\nwhere the $i\\epsilon$ prescription is needed to make the final answer well defined (see also section \\ref{sec:nonpert}). This wavefunction satisfies the reduced WDW equation \\footnote{This differs from the wavefunction written in \\cite{MTY} since we found a modification in the WDW equation. The solutions are related by $\\Psi_{\\rm here} = L \\Psi_{\\rm there}$. The Klein-Gordon inner product defined in \\cite{MTY} should also be modified accordingly.} \n\\begin{equation}\n(L \\phi - L \\partial_L(L^{-1} \\partial_\\phi)  ) \\Psi[L,\\phi] = 0.\n\\end{equation}\nOne interesting feature of this formula is the fact that it also satisfies the naive no-boundary condition since $\\Psi_+[L\\to 0,\\phi_b] \\to 0$.\nNevertheless, even though it behaves as expected for small lengths, it has a divergence at $L_{\\rm div} = 2\\pi$ (the Bessel function blows up near the origin). Semiclassically, the geometry that dominates the path integral when $L=2\\pi$ is the lower hemisphere of the Euclidean $S^2$ (dashed line in figure \\ref{fig:ds2q}). This is reasonable from the perspective of the JT gravity path integral since this boundary is also a geodesic, but it would be nice to understand whether this divergence is unique to JT gravity, or would it also be present in theories of gravity in higher dimensions.  \n\nWe can also comment on the $T\\bar{T}$ interpretation of dS gravity. For large $L$ it was argued in \\cite{MTY} that a possible observable in a dual QM theory computing the wavefunction can be $\\Psi_+ [L]\\sim {\\rm Tr}[e^{i L H}]$, with an example provided after summing over non-trivial topologies by a matrix integral (giving a dS version of the AdS story in \\cite{Saad:2019lba}). We can extend this (before summing over topologies) to a calculation of the wavefunction at finite $L$ by $T\\bar{T}$ deforming the same QM system. This is basically an analytic continuation of the discussion for $AdS_2$ given in previous sections. \n\nSo far we focused on the expanding branch of the wavefunction following \\cite{MTY}. We can also find a real wavefunction analogous to the one originally computed by Hartle and Hawking \\cite{PhysRevD.28.2960}, which we will call $\\Psi_{\\rm HH,real}$. This is easy to do in the context of JT gravity and the answer is \n\\begin{equation}\n\\Psi_{\\rm HH,real} [L,\\phi_b] =  \\frac{\\pi L \\phi^2_b }{L^2 -4\\pi^2} I_2\\Big( i \\sqrt{\\phi^2_b(L^2 - 4\\pi^2)} \\Big)\n\\end{equation}\nThis wavefunction is real, smooth at $L=2\\pi$ and also satisfies $\\Psi_{\\rm HH,real} [L\\to 0,\\phi_b]\\to 0$. We plotted the wavefunction in Fig. \\ref{fig:dSPsi}. For large universes this state has an expanding and contracting branch with equal weight.  \n\\begin{figure}\n\\centering\n\\includegraphics[scale=1]{PlotI2dS.pdf}\n\\caption{\\label{fig:dSPsi} Plot of the wavefuntion $\\Psi_{\\rm{HH, real}}$ for $\\phi_b = 1\/4$. The vertical dashed line indicates the location, $L = 2\\pi$, where the expanding branch $\\Psi_+$ of the wavefunction \\eqref{K2wavefunction} diverges, but $\\Psi_{\\rm{HH, real}}$ remains finite.}\n\\end{figure}\n\nFinally, the results of this section can be extended to pure $3D$ gravity with positive cosmological constant $\\Lambda = 2\/\\ell^2$. Using Freidel reconstruction kernel, the wavefunction $\\Psi[e^\\pm]$ satisfying WDW, as a function of the boundary frame fields $e^\\pm$, is given by \n\\begin{equation}\n\\Psi_+ [e^\\pm] = e^{i\\frac{\\ell}{16\\pi G_N}\\int e} \\int \\mathcal{D}E~e^{-i \\frac{\\ell}{8\\pi G_N}\\int E^+\\wedge E^-} Z(E+e).\n\\end{equation}\nThis is the most general, purely expanding, solution of WDW up to an arbitrary function of the boundary metric $Z(E)$. We can fix $\\Psi_+$ uniquely by looking at the late time limit, or more accurately, boundary metrics with large volume. In this limit Freidel formula gives $\\Psi_+ [ T e^\\pm] \\sim e^{i S_{\\rm c.t.}(T,e)} Z(e)$ for large $T$. The first term is rapidly oscillating with the volume $T$ at late times and we see the finite piece is precisely the boundary condition we need $Z(e)$. The path integral calculation of the finite piece $Z(e)$ was done in \\cite{Cotler:2019nbi} for the case of a boundary torus (see their equation 4.121 and also \\cite{Castro:2012gc}) and gives a sum over $SL(2,\\mathbb{Z})$ images of a Virasoro vacuum character. We leave the study of the properties of this wavefunction for future work.\n  \n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n \n JT gravity serves as an essential toolbox to probe some universal features of quantum gravity. In the context of this paper, we have shown that the WDW wavefunctional at finite cutoff and dilaton value in $AdS_2$ agrees with an explicit computation of the Euclidean path integral; this, in turn, matches the partition function of the Schwarzian theory deformed by a $1D$ analog of the $T\\bar T$ deformation. Consequently, our computation serves as a check for the conjectured holographic duality between a theory deformed by $T\\bar T$ and gravity, in $AdS$, at a finite radial distance.    \n\n\\begin{center}\n\\textit{Finite cutoff unitarity}\n\\end{center}\n\nBeyond providing a check, our computations indicate paths to resolve several open problems related to this conjectured duality. One such issue is that of complex energies that were present when deforming by $T \\bar T$ (both in $1$ and $2D$), and were also present in the WDW wavefunctional when solely accounting for the expanding branch. However, from the WDW perspective, one could also consider the contribution of the contracting branch, and, equivalently, in the Euclidean path integral, one could also account for the contribution of non-compact geometries. In both cases, such corrections are non-perturbative in the cutoff parameter $\\e$ or, in the context of $T\\bar T$, in the coupling of the deformation $\\l$.  Nevertheless, we have shown that there exists a linear combination between the two wavefunctional branches that leads to a density of states which is real for all energies. Thus, this suggests that a natural resolution to the problem of complex energy levels is the addition of the other branch, instead of the proposed artificial cutoff for the spectrum once the energies complexify \\cite{McGough:2016lol}.  While the problem of complex energy levels is resolved with the addition of the contracting branch, a new issue appears: the partition function now has a negative density of states. This new density of states implies that, even with such a resolution, the partition function is not that of a single unitary quantum system. In three bulk dimensions, one has a similar state of affairs. The energy levels again complexify, and the other branch of the solution space can cure this, with the caveat that the density of states will become negative. A possible resolution consistent with unitarity would be that the finite cutoff path integral is not computing a boundary partition function but something like an index, where certain states are weighted with a negative sign. \n\nA related issue that leads to the ambiguity in the choice of branches is that the non-perturbative piece of the partition function that cannot be fixed by the $\\l \\to 0$ boundary condition. This ambiguity can be cured by putting additional conditions on the partition function. Fixing the $\\l$-derivative of ${Z}_{\\l}^{\\rm non-pert.}(\\b)$ does not work, but for instance ${Z}_{\\l}^{\\rm non-pert.}(\\b) \\to 0$ as $\\b \\to 0$ would be enough to fix the partition function completely. One other possibility, motivated by the bulk, is to fix the extrinsic curvature $K$ at $\\e \\to 0$. This will eliminate one of the two branches and, therefore, also $\\hat{\\rho}$ in \\eqref{Znp}.\\footnote{However, such a resolution appears to bring back the complex energies.} One can also try to foliate the spacetime with different slices, for instance, by taking constant extrinsic curvature slices.\\footnote{Appendix \\ref{sec:Neumann-bc}, in fact, provides a non-trivial check of the form of the extrinsic curvature $K[z(u)]$ by considering boundary conditions with fixed extrinsic curvature slices. We will provide further comments about such boundary conditions in \\cite{GIY}. } In $3D$, this was done explicitly in \\cite{Hosoya2dTorus} for a toroidal boundary and in \\cite{Hosoya:1989yj} for more general Rieman surfaces. In particular, for the toroidal boundary, it was found that the wavefunction in the mini-superspace approximation inherits a particular modular invariance, and it would be interesting to compare that analysis to the one done in \\cite{Aharony:2018bad}.\n\nIn the AdS$_3$\/CFT$_2$ context, it would also be interesting to understand the non-perturbative corrections to the partition function purely from the field theory. As the $T\\bar{T}$ deformation is a particular irrelevant coupling, it is not unreasonable to suspect that such corrections are due to instanton effects contributing at $O(e^{-1\/\\l})$. The fate of such instantons can be studied using, for example, the kernel methods \\cite{Dubovsky_2018, Mazenc:2019cfg} or the various string interpretation of $T\\bar{T}$ \\cite{Callebaut:2019omt, Giveon:2017nie}; through such an analysis, one could hope to shed some light on the complexification of the energy levels.  \n\n\\begin{center}\n\\textit{Application: Wavefunction of the universe}\n\\end{center}\n\nThe techniques presented in this paper also apply to geometries with constant positive curvature. We do this calculation in two ways. On one hand we solve the WDW constraint that this wavefunction satisfies, imposing the Hartle-Hawking boundary condition. On the other hand, we compute the wavefunction as an analytic continuation from the Euclidean path integral on `-AdS'. As expected, we find that both results match. We also analyze two possible choices to define the wavefunction. The first solely includes the contribution of the expanding branch and has a pole when the size of the universe coincides with the dS radius. The second is a real wavefunctional, which includes the non-perturbative contribution of the contracting branch and is now smooth at the gluing location. It would be interesting to identify whether this divergence is present in higher dimensions or if it is special to JT gravity. We also leave for future work a better understanding of the appropriate definition of an inner product between these states.\\footnote{ In the limit of large universes, some progress in this direction was made in \\cite{MTY}.} Finally, we outlined how a similar analysis can be used to find the no-boundary wavefunction for pure $3D$ gravity with a positive cosmological constant, the simplest example corresponding to a toroidal universe.\n\n\\begin{center}\n\\textit{Sum over topologies}\n\\end{center}\n\nAn important open question that remains unanswered is the computation of the JT gravity partition function when including the contribution of manifolds with arbitrary topology. While we have determined the partition function of finite cutoff trumpets using the WDW constraint, this type of surface is insufficient for performing the gluing necessary to obtain any higher genus manifold with a fixed proper boundary length. It would be interesting to understand whether the contribution of such manifolds to the path integral can be accounted for by using an alternative gluing procedure that would work for any higher genus manifold. \n\nFor the cylinder, we can actually avoid the gluing. From a third quantisation point of view, one way to think about the cylinder partition function, or double trumpet, is as the propagator associated to the WDW equation in mini-superspace, \n\\begin{eqnarray}\\label{WdWprop}\n\\left[-L\\phi + L \\partial_L(L^{-1} \\partial_\\phi)  \\right] \\Psi_{\\rm cylinder}(\\phi,\\phi',L,L') = \\d(L-L')\\d(\\phi - \\phi').\n\\end{eqnarray}\nThis avoids the integral over $b$ and since the WDW equation \\eqref{WdWprop} is just the propagator of a massive particle in a constant electric field\\footnote{In the coordinates $u = \\phi^2$ and $v = L^2$, \\eqref{minisupereq} reduces to $\\left(\\partial_u \\partial_v - \\frac{1}{4} - \\frac{1}{2v} \\partial_u\\right)\\Psi = 0$. This is the KG equation for $m^2 = 1\/2$ and external gauge field $A = \\frac{i}{2v}dv$. Notice that the mini-superspace is Lorentzian, whereas the geometries $\\Psi$ describes are Euclidean.}, we can solve it with standard methods. The resulting propagator is proportional to a Hankel function of the geodesic distance on mini-superspace, but does not have the same form as the double trumpet computed in \\cite{Saad:2019lba} once $L,L', \\phi$ and $\\phi'$ are taken large. In fact, it vanishes in that limit. Furthermore, there is a logarithmic divergence when the geodesic distance in mini-superspace vanishes, i.e. when $L = L'$ and\/or $\\phi = \\phi'$. There are several reasons for this discrepancy. The obvious one would be that the cylinder is not the propagator in third quantisation language, but this then raises the question, what is this propagator? Does it have a geometric interpretation? It would be interesting to understand this discrepancy better and what the role of the third quantised picture is.  \n\n\\begin{center}\n\\textit{Coupling to matter $\\&$ generalizations}\n\\end{center}\n\nFinally, it would be interesting to understand the coupling of the bulk theory to matter. When adding gauge degrees of freedom to a $3D$ bulk and imposing mixed boundary conditions between the graviton and the gauge field, the theory is dual to a $2D$ CFT deformed by the $J\\bar T$ deformation  \\cite{Bzowski:2018pcy}.\\footnote{Here, $J \\bar T$ is a composite operator containing $J$, a chiral $U(1)$ current, and $\\bar T$, a component of the stress tensor.} In $2D$, the partition function of the theory coupled to gauge degrees of freedom can be computed exactly even at finite cutoff; this can be done by combining the techniques presented in this paper with those in \\cite{Iliesiu:2019lfc} \\footnote{Another possible direction could be to understand the result for $2D$ gravity as a limit of $3D$ (either for near extremal states \\cite{Ghosh:2019rcj} or in relation to SYK-like models \\cite{Turiaci:2017zwd}).} . It would be interesting to explore the possibility of a $1D$ deformation, analogous to the  $J\\bar T$ deformation in $2D$, which would lead to the correct boundary dual for the gravitational gauge theory. Since gauge fields do not have any propagating degrees of freedom in $2D$, it would also be interesting to explore the coupling of JT gravity to other forms of matter.\\footnote{One intriguing possibility is to couple JT gravity to a $2D$ CFT. The effect of the CFT on the partition function has been studied in \\cite{Yang:2018gdb, Iliesiu:2020qvm} through the contribution of the Weyl anomaly in the infinite cutoff limit.  It would be interesting to see whether the effect of the Weyl anomaly can be determined at finite cutoff solely in terms of the light-cone coordinate $z(u)$. }   In the usual finite cutoff $AdS_3$\/$T\\bar T$ deformed CFT correspondence,  adding matter results in the dual gravitational theory having mixed boundary conditions for the non-dynamical graviton \\cite{Guica:2019nzm}. Only when matter fields are turned off are these mixed boundary conditions equivalent to the typical finite radius Dirichlet boundary conditions. In $2D$ this was done for the matterless case in \\cite{Gross:2019uxi} and it would be interesting to generalise this to include matter. \n\n\n\\section*{Acknowledgements}\nWe thank Alexandre Belin, Steve Giddings, Henry Lin, Juan Maldacena, Don Marolf, Mark Mezei, Onkar Parrikar, Eric Perlmutter, Silviu Pufu, Ronak Soni, Eva Silverstein, Douglas Stanford, Edward Witten and Zhenbin Yang for valuable discussions. Special thanks go to Edgar Shaghoulian for comments on a draft. JK is supported by the Simons Foundation. LVI is supported in part by the US NSF under Grant No. PHY-1820651 and by the Simons Foundation Grant No. 488653. GJT is supported by a Fundamental Physics Fellowship. The research of HV is supported by NSF grant PHY-1620059. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through through the ministry of Research and Innovation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe approximation of finite time reachable sets of linear control systems \nhas been studied by a number of mathematicians, engineers and computer scientists,\nusing a variety of approaches, so the following list is by no means exhaustive. \nIn \\cite{Baier}, a number of optimal control problems is solved to obtain \na discretization of the support function of the reachable set in fixed directions,\nwhile \\cite{Graettinger} and \\cite{Shao} essentially apply Benson's algorithm\nto construct the reachable set adaptively.\nInner and outer approximations by zonotopes have been explored in \\cite{Girard},\nand inner and outer approximations by displacements of homothetic bodies\nhave been investigated in \\cite{Fiacchini}.\nApproximations by polytopes with fixed facet normals have been \ndiscussed in \\cite{BenSassi}. \n\n\\medskip \n\nFor a strictly stable system with compact control input set, the limit of the finite\ntime reachable sets (as time tends to infinity) is a well-defined object, \nsee \\cite{Kolmanovsky}. \nWe refer to it as the infinite time reachable set.\nIt is a compact subset of the state space, which is invariant under\nthe control dynamics and attracts any trajectory of the system.\nThe contracting dynamics allow the design of a priori and a posteriori estimates\nfor the Hausdorff error between an iterated compact set and the limit set,\nsee \\cite{Artstein} and \\cite{Rakovic05}.\nAs a consequence, all methods for the approximation of finite-time reachable sets\nmentioned above can in principle be used to approximate the limit by forward\niteration.\n\n\\medskip\n\nIn the present paper, we use some of the above ideas, but pursue a completely\ndifferent approach to the approximation of the limit set, which is in a vague sense\nconceptually similar with the papers \\cite{Dorea}, \\cite{Gaitsgory} \nand \\cite{Korda} on weakly invariant sets of control systems.\nIn a first step, we analyze the induced dynamics on the space of the\nnonempty compact sets, extending some of the results in \\cite{Artstein}.\nThen we use this insight and an approximation theorem from \\cite{Rieger:17} \nto prove that an outer approximation by a polytope with fixed facet normals\ncan be computed as the solution to a disjunctive optimization problem,\navoiding the need for forward iteration.\n\n\\medskip\n\nAs the solution of a disjunctive program is difficult to compute, \nwe construct a dual problem, which is a linear program and possesses \nthe same global optimizer as the original disjunctive program.\nIn order to ensure uniqueness of the dual optimum, we inflate \nthe system slightly to push the optimum away from some critical constraints\nto create a situation in which this is relatively simple.\nThen we use a stability result from \\cite{Robinson} to conclude \nthat the unperturbed dual problem has a unique solution, which coincides \nwith the unique primal minimizer we wish to compute.\n\n\n\n\n\n\n\n\n\n\n\\section{Setting and Notation} \\label{SN}\n\nWe fix a matrix $C\\in\\mathbbm{R}^{d\\times d}$ with spectral radius $\\rho(C)<1$, \na matrix $D\\in\\mathbbm{R}^{d\\times m}$ and a nonempty convex and compact control set \n$U\\subset\\mathbbm{R}^m$, and we analyze the behavior of the control system\n\\begin{equation} \\label{cs}\nx_{k+1}=Cx_k+Du_k,\\quad u_k\\in U,\n\\end{equation}\non the unbounded time interval.\nIn particular, we will approximate its infinite time reachable set and\nits all time reachable set by polytopes.\n\nThroughout this paper, we fix a number $\\ell\\in(0,\\rho(C))$.\nBy Lemma 5.6.10 in \\cite{Horn:13}, there exists a norm \n$\\|\\cdot\\|_\\ell:\\mathbbm{R}^d\\to\\mathbbm{R}_+$ such that the induced matrix norm \nsatisfies $\\|C\\|_\\ell\\le\\ell$.\nSince $\\mathbbm{R}^d$ is finite-dimensional, there exist $c_{2,\\ell},c_{\\ell,2}>0$ with\n\\begin{equation}\\label{equivalence}\nc_{2,\\ell}^{-1}\\|x\\|_\\ell\\le\\|x\\|\n\\le c_{\\ell,2}\\|x\\|_\\ell\\quad\\forall\\,x\\in\\mathbbm{R}^d,\n\\end{equation}\nwhere $\\|\\cdot\\|:\\mathbbm{R}^d\\to\\mathbbm{R}_+$ denotes the Euclidean norm.\n\nWe denote the space of all nonempty compact subsets of $\\mathbbm{R}^d$ by $\\mathcal{K}(\\mathbbm{R}^d)$\nand the space of all nonempty compact and convex subsets of $\\mathbbm{R}^d$ \nby $\\mathcal{K}_c(\\mathbbm{R}^d)$.\nThe Hausdorff semi-distance $\\dist:\\mathcal{K}(\\mathbbm{R}^d)\\times\\mathcal{K}(\\mathbbm{R}^d)\\to\\mathbbm{R}_+$ \nand the corresponding symmetric Hausdorff distance \n$\\dist_H:\\mathcal{K}(\\mathbbm{R}^d)\\times\\mathcal{K}(\\mathbbm{R}^d)\\to\\mathbbm{R}_+$ \nare defined by\n\\begin{align*}\n&\\dist(X,X')=\\sup_{x\\in X}\\inf_{x'\\in X'}\\|x-x'\\|,\\\\\n&\\dist_H(X,X')=\\max\\{\\dist(X,X'),\\dist(X',X)\\}.\n\\end{align*}\nFor any $X\\in\\mathcal{K}(\\mathbbm{R}^d)$ and $R>0$, we write\n\\[B_R(X):=\\{x\\in\\mathbbm{R}^d:\\dist(x,X)\\le R\\}\\quad\\text{and}\\quad \n\\|X\\|:=\\sup_{x\\in X}\\|x\\|.\\]\nIdentical notation with a subscript or superscript $\\ell$ will be used\nwhen the underlying norm is $\\|\\cdot\\|_\\ell:\\mathbbm{R}^d\\to\\mathbbm{R}_+$.\nNote that for any $X\\in\\mathcal{K}_c(\\mathbbm{R}^d)$ and $R>0$, the property\n$B_R^\\ell(X)\\in\\mathcal{K}_c(\\mathbbm{R}^d)$ still holds in this non-Euclidean\ngeometry by triangle inequality.\n\nThe support function of a set $X\\in\\mathcal{K}_c(\\mathbbm{R}^d)$ is a mapping\n\\[\\sigma_X:\\mathbbm{R}^d\\to\\mathbbm{R},\\quad\\sigma_X(p):=\\max_{x\\in X}p^Tx.\\]\nFor a set-valued map $F:\\mathbbm{R}^d\\to\\mathcal{K}(\\mathbbm{R}^d)$ and $X\\in\\mathcal{K}(\\mathbbm{R}^d)$,\nwe denote the image and the preimage of $X$ by\n\\[F(X):=\\cup_{x\\in X}F(x)\\quad\\text{and}\\quad\nF^{-1}(X):=\\{x\\in\\mathbbm{R}^d:F(x)\\cap X\\neq\\emptyset\\}.\\]\nThe vector $\\mathbbm{1}\\in\\mathbbm{R}^N$ is the vector with the number $1$ \nin all $N$ components, and the vector $e_i$ is the $i$-th unit vector.\nFor any convex set $X\\subset\\mathbbm{R}^d$, the set of extreme points of $X$\nis denoted $\\ext(X)$, and its interior is denoted $\\interior(X)$.\n\n\n\n\n\\section{Preliminaries and auxiliary results} \\label{PAR}\n\nThe fact that all norms on $\\mathbbm{R}^d$ are equivalent is reflected by a similar\nstatement for Hausdorff semi-distances and Hausdorff distances on $\\mathcal{K}(\\mathbbm{R}^d)$.\n\n\\begin{lemma} \\label{equivalent:Hausdorff}\nThe Hausdorff semi-distances $\\dist$ and $\\dist^\\ell$ as well as the \nHausdorff distances $\\dist_H$ and $\\dist_H^\\ell$ are equivalent.\n\\end{lemma}\n\n\\begin{proof}\nFor any $X,X'\\in\\mathcal{K}(\\mathbbm{R}^d)$, we compute\n\\begin{align*}\n&c_{2,\\ell}^{-1}\\dist^\\ell(X,X')\n=c_{2,\\ell}^{-1}\\sup_{x\\in X}\\inf_{x'\\in X'}\\|x-x'\\|_\\ell\n\\le \\sup_{x\\in X}\\inf_{x'\\in X'}\\|x-x'\\|\n=\\dist(X,X'),\\\\\n&\\dist(X,X')=\\sup_{x\\in X}\\inf_{x'\\in X'}\\|x-x'\\|\n\\le c_{\\ell,2}\\sup_{x\\in X}\\inf_{x'\\in X'}\\|x-x'\\|_\\ell\n=c_{\\ell,2}\\dist^\\ell(X,X'),\n\\end{align*}\nand hence\n\\[c_{2,\\ell}^{-1}\\dist_H^\\ell(X,X')\\le\\dist_H(X,X')\\le c_{\\ell,2}\\dist_H^\\ell(X,X').\\]\n\\end{proof}\n\nGiven a matrix $A\\in\\mathbbm{R}^{N\\times d}$, we define a space of polyhedra \nby setting \n\\[\\mathcal{G}_A:=\\{Q_{A,b}:b\\in\\mathbbm{R}^N\\}\\setminus\\{\\emptyset\\},\\quad\nQ_{A,b}:=\\{x\\in\\mathbbm{R}^d: Ax\\le b\\}.\\]\nThis space has been explored in depth in the paper \\cite{Rieger:17}.\nWe recapitulate the relevant facts as briefly as possible\nand refer to \\cite{Rieger:17} for technical details.\n\nThroughout the rest of the paper, we require the following assumptions.\n\n\\begin{assumption} \\label{A:ass}\nThe matrix $A\\in\\mathbbm{R}^{N\\times d}$ has the following properties.\n\\begin{itemize}\n\\item [a)] It consists of pairwise distinct rows $a_1^T,\\ldots,a_N^T$ \nsatisfying $a_i\\in\\mathbbm{R}^d$ and $\\|a_i\\|_2=1$ for $i=1,\\ldots,N$.\n\\item [b)] We have $Q_{A,0}=\\{0\\}$.\n\\end{itemize}\n\\end{assumption}\n\nAssumption \\ref{A:ass}b) holds whenever the rows of $A$ \nare reasonably dense in the sphere, see Theorem 16 in \\cite{Rieger:17},\nand by Corollary 17 in \\cite{Rieger:17}, it\nguarantees that the space $\\mathcal{G}_A$ consists of (bounded) polytopes. \nBy Theorem 13 in \\cite{Rieger:17}, the mapping $b\\mapsto Q_{A,b}$\nis bi-Lipschitz w.r.t.\\ Hausdorff distance.\n\n\\medskip\n\nIntersections of polytopes can be expressed as the componentwise infimum\nof their representations.\n\n\\begin{lemma} \\label{intersections}\nLet $\\mathcal{B}\\subset\\mathbbm{R}^N$ be a subset with $\\cap_{b\\in\\mathcal{B}}Q_{A,b}\\neq\\emptyset$,\nand let $b^*\\in\\mathbbm{R}^N$ be given by $b^*_i:=\\inf_{b\\in\\mathcal{B}}b_i$.\nThen $Q_{A,b*}=\\cap_{b\\in\\mathcal{B}}Q_{A,b}$.\n\\end{lemma}\n\n\\begin{proof}\nIf $x\\in Q_{A,b^*}$, then $a_i^Tx\\le b^*_i\\le b_i$ for all $b\\in\\mathcal{B}$\nand $i\\in\\{1,\\ldots,N\\}$, so $x\\in\\cap_{b\\in\\mathcal{B}}Q_{A,b}$.\nIf, on the other hand, we have $x\\notin Q_{A,b^*}$, then there exists\n$i\\in\\{1,\\ldots,N\\}$ with $b_i^*<a_i^Tx$.\nBy definition of the infimum, there exists $b'\\in\\mathcal{B}$ \nwith $b_i^*\\le b'_i<a_i^Tx$, and hence we have \n$x\\notin Q_{A,b'}\\supset(\\cap_{b\\in\\mathcal{B}}Q_{A,b})$.\n\\end{proof}\n\nThe quantity \n\\[\\kappa_A:=\\sup_{\\|c\\|_2=1}\\,\n\\inf\\Big\\{\\sum_{k=1}^Np_k\\|a_k-\\frac{1}{\\|p\\|_1}c\\|_2:\np\\in\\ext(\\{q\\in\\mathbbm{R}^N:A^Tq=c,\\ q\\ge 0\\})\\Big\\}\\]\nfrom Proposition 46 in \\cite{Rieger:17} measures, roughly speaking, how easily\npoints on the unit sphere can be positively combined from the rows of $A$. \nIt controls the approximation properties of the space $\\mathcal{G}_A$\nas a subspace of $\\mathcal{K}_c(\\mathbbm{R}^d)$ in the following sense, \nsee Theorem 47 in \\cite{Rieger:17}.\n\\begin{theorem} \\label{projector}\nThe mapping\n\\begin{align*}\n&\\pi_{\\mathcal{G}_A}:\\mathcal{K}_c(\\mathbbm{R}^d)\\to\\mathcal{G}_A,\\quad \n\\pi_{\\mathcal{G}_A}(X):=Q_{A,b_X}\\\\\n&(b_X)_i:=\\max_{x\\in X}a_i^Tx\\quad\\text{for}\\quad i=1,\\ldots,N,\n\\end{align*}\nis a projector from $\\mathcal{K}_c(\\mathbbm{R}^d)$ onto $\\mathcal{G}_A$, it is\nLipschitz w.r.t.\\ $\\dist_H$, it is monotone w.r.t.\\ inclusion, and it satisfies \n\\[X\\subset\\pi_{\\mathcal{G}_A}(X)\\subset B(X,\\kappa_A\\|X\\|)\\quad\n\\forall\\,X\\in\\mathcal{K}_c(\\mathbbm{R}^d).\\]\n\\end{theorem}\n\nThroughout this paper, we assume the following relation between the \ncontraction rate $\\ell$ of the matrix $C$ and the quality $\\kappa_A$\nof the approximation of $\\mathcal{K}_c(\\mathbbm{R}^d)$ by $\\mathcal{G_A}$.\nIt can be achieved by choosing $A$ such that its rows are sufficiently dense \nin the sphere, see the context of Proposition 46 in \\cite{Rieger:17}.\n\n\\begin{assumption} \\label{kappaineqass}\nThe matrix $A$ satisfies\n\\begin{equation*} \n\\kappa_A<\\frac{1-\\ell}{c_{2,\\ell}c_{\\ell,2}\\ell}.\n\\end{equation*}\n\\end{assumption}\n\nWe exploit this relation in the following lemma.\n\n\\begin{lemma} \\label{not:worse:than:ell}\nLet $X\\in\\mathcal{K}(\\mathbbm{R}^d)$, and define\n\\begin{equation*}\nR_A^X:=\\frac{c_{2,\\ell}c_{\\ell,2}\\ell\\kappa_A}\n{1-\\ell-c_{2,\\ell}c_{\\ell,2}\\ell\\kappa_A}\\|X\\|_\\ell.\n\\end{equation*}\nFor every $R\\ge R_A^X$, we obtain\nthe inclusion\n\\[\\pi_{\\mathcal{G}_A}(B_R^\\ell(X))\n\\subset B_{\\ell^{-1}R}^\\ell(X).\\]\n\\end{lemma}\n\n\\begin{proof}\nSince\n\\begin{align*}\nc_{\\ell,2}c_{2,\\ell}\\kappa_A\\|X\\|_\\ell\n=((1-\\ell)\\ell^{-1}-c_{\\ell,2}c_{2,\\ell}\\kappa_A)R_A^X\n\\le((1-\\ell)\\ell^{-1}-c_{\\ell,2}c_{2,\\ell}\\kappa_A)R,\n\\end{align*}\nwe can use Theorem \\ref{projector} to compute\n\\begin{align*}\n&\\dist^\\ell(\\pi_{\\mathcal{G}_A}(B_R^\\ell(X)),B_R^\\ell(X))\n\\le c_{\\ell,2}\\dist(\\pi_{\\mathcal{G}_A}(B_R^\\ell(X)),B_R^\\ell(X))\\\\\n&\\le c_{\\ell,2}\\kappa_A\\|B_R^\\ell(X)\\|\n\\le c_{\\ell,2}c_{2,\\ell}\\kappa_A\\|B_R^\\ell(X)\\|_\\ell\\\\\n&\\le c_{\\ell,2}c_{2,\\ell}\\kappa_A(\\|X\\|_\\ell+R)\n\\le(1-\\ell)\\ell^{-1}R,\n\\end{align*}\nand we conclude that\n\\begin{align*}\n&\\dist^\\ell(\\pi_{\\mathcal{G}_A}(B_R^\\ell(X)),X)\\\\\n&\\le\\dist^\\ell(\\pi_{\\mathcal{G}_A}(B_R^\\ell(X)),B_R^\\ell(X))\n+\\dist^\\ell(B_R^\\ell(X),X)\\\\\n&\\le(1-\\ell)\\ell^{-1}R+R=\\ell^{-1}R,\n\\end{align*}\nwhich implies the desired inclusion.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{Properties of the infinite time reachable set} \\label{ISP}\n\nLet $V\\in\\mathcal{K}_c(\\mathbbm{R}^d)$, and consider the mapping\n\\[F:\\mathbbm{R}^d\\to\\mathcal{K}_c(\\mathbbm{R}^d),\\quad F(x):=Cx+V.\\]\nThis setting includes system \\eqref{cs} for $V=DU$, and it allows us \nto treat $\\varepsilon$-inflations of $F$ with minimal notational complication.\n\nAs we need precise statements in the norms we work with, we prove a few \nfacts that may in principle be well-known. \nPart c) of the following proposition is, e.g.,\nsimilar to Proposition 4.3 in \\cite{Artstein}.\n\n\\begin{proposition} \\label{inf:reach}\nThe following statements hold.\n\\begin{itemize}\n\\item [a)] The map $F:\\mathbbm{R}^d\\to\\mathcal{K}_c(\\mathbbm{R}^d)$ is $\\ell$-Lipschitz w.r.t.\\\n$\\|\\cdot\\|_\\ell:\\mathbbm{R}^d\\to\\mathbbm{R}_+$, i.e.\n\\[\\dist_H^\\ell(F(x),F(x'))\\le\\ell\\|x-x'\\|_\\ell\\quad\\forall\\,x,x'\\in\\mathbbm{R}^d.\\]\n\\item [b)] The mapping \n$\\mathcal{F}:\\mathcal{K}(\\mathbbm{R}^d)\\to\\mathcal{K}(\\mathbbm{R}^d)$ given by $\\mathcal{F}(X):=F(X)$\nmaps $\\mathcal{K}_c(\\mathbbm{R}^d)$ into itself and satisfies\n\\[\\dist_H^\\ell(\\mathcal{F}(X),\\mathcal{F}(X'))\n\\le\\ell\\dist_H^\\ell(X,X')\\quad\\forall\\,X,X'\\in\\mathcal{K}(\\mathbbm{R}^d).\\]\n\\item [c)] There exists a unique set $X^*\\in\\mathcal{K}(\\mathbbm{R}^d)$ with $X^*=F(X^*)$,\nand for any $X_0\\in\\mathcal{K}(\\mathbbm{R}^d)$, the sequence \n$\\{X_k\\}_{k\\in\\mathbbm{N}}\\subset\\mathcal{K}(\\mathbbm{R}^d)$ given by\n$X_{k+1}=F(X_k)$ for all $k\\in\\mathbbm{N}$ satisfies\n\\begin{align*} \n&\\lim_{k\\to\\infty}\\dist_H^\\ell(X_k,X^*)=0,\\\\\n&\\dist_H^\\ell(X_{k+1},X^*)\\le\\ell\\dist_H^\\ell(X_k,X^*)\\quad\\forall\\,k\\in\\mathbbm{N},\\\\\n&\\dist_H^\\ell(X_k,X^*)\\le\\tfrac{\\ell^k}{1-\\ell}\\dist_H^\\ell(X_1,X_0).\n\\end{align*}\n\\item [d)] We have $X^*\\in\\mathcal{K}_c(\\mathbbm{R}^d)$, and if $X_0\\in\\mathcal{K}_c(\\mathbbm{R}^d)$, then\nthe sequence $\\{X_k\\}_{k\\in\\mathbbm{N}}$ in part c) satisfies \n$\\{X_k\\}_{k\\in\\mathbbm{N}}\\subset\\mathcal{K}_c(\\mathbbm{R}^d)$.\n\\item [e)] If $X\\in\\mathcal{K}(\\mathbbm{R}^d)$ satisfies $F(X)\\subset X$, then $X^*\\subset X$.\nConversely, if $X\\subset F(X$), then $X\\subset X^*$.\n\\item [f)] We have the a priori estimate $\\|X^*\\|_\\ell\\le\\frac{1}{1-\\ell}\\|V\\|_\\ell$.\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\na) For all $x,x'\\in\\mathbbm{R}^d$, we compute\n\\[\\dist^\\ell(F(x),F(x'))\n=\\sup_{v\\in V}\\inf_{v'\\in V}\\|Cx+v-Cx'-v'\\|_\\ell\n\\le\\ell\\|x-x'\\|_\\ell.\\]\n\nb) For any $X\\in\\mathcal{K}(\\mathbbm{R}^d)$, we have $\\mathcal{F}(X)=F(X)\\in\\mathcal{K}(\\mathbbm{R}^d)$ \nby Corollary 2.20 and Theorem 2.68 in \\cite{Hu:97}.\nIf $X\\in\\mathcal{K}_c(\\mathbbm{R}^d)$, then $\\mathcal{F}(X)=F(X)=CX+V$ is a Minkowski sum of\ntwo convex sets, and hence $\\mathcal{F}(X)\\in\\mathcal{K}_c(\\mathbbm{R}^d)$,\nsee \\cite[page 47]{Schneider}.  \nFor any $X,X'\\in\\mathcal{K}(\\mathbbm{R}^d)$, we compute\n\\begin{align*}\n&\\dist^\\ell(\\mathcal{F}(X),\\mathcal{F}(X'))\n=\\sup_{x\\in X,\\ v\\in V}\\inf_{x'\\in X',\\ v'\\in V}\\|Cx+v-Cx'-v'\\|_\\ell\\\\\n&\\le\\sup_{x\\in X}\\inf_{x'\\in X'}\\|Cx-Cx'\\|_\\ell\n\\le\\ell\\dist^\\ell(X,X').\n\\end{align*}\n\nc) By Proposition 1.6 in \\cite{Hu:97}, the space $(\\mathcal{K}(\\mathbbm{R}^d),\\dist_H^\\ell)$ \nis complete.\nApplying the contraction mapping principle (Theorem 1.A in \\cite{Zeidler:86})\nto $\\mathcal{F}$ in this situation yields the desired statement.\n\nd) If $X_0\\in\\mathcal{K}_c(\\mathbbm{R}^d)$, then $\\{X_k\\}_{k\\in\\mathbbm{N}}\\subset\\mathcal{K}_c(\\mathbbm{R}^d)$\nfollows by induction from part b).\nThe choice $X_0=\\{0\\}$ induces a sequence $\\{X_k\\}_{k\\in\\mathbbm{N}}\\subset\\mathcal{K}_c(\\mathbbm{R}^d)$, \nwhich, by part c), satisfies $\\lim_{k\\to\\infty}\\dist_H^\\ell(X_k,X^*)=0$.\nAccording to Theorem 1.8.3 in \\cite{Schneider},\nthe space $(\\mathcal{K}_c(\\mathbbm{R}^d),\\dist_H^\\ell)$ is complete, so $X^*\\in\\mathcal{K}_c(\\mathbbm{R}^d)$.\n\ne) If $F(X)\\subset X$, then by induction, we have $F^{k+1}(X)\\subset F^k(X)$ \nfor all $k\\in\\mathbbm{N}$, so the desired statement follows from part c).\nThe proof of the opposite inclusion is analogous.\n\nf) We use part c) to compute\n\\begin{align*}\n&\\dist^\\ell(X^*,0)\n\\le\\dist^\\ell(X^*,F(0))+\\dist^\\ell(F(0),0)\\\\\n&=\\dist^\\ell(F(X^*),F(0))+\\dist^\\ell(V,0)\n\\le\\ell\\dist^\\ell(X^*,0)+\\|V\\|_\\ell,\n\\end{align*}\nsubtract $\\ell\\dist^\\ell(X^*,0)$ and divide by $1-\\ell$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Approximation of $X^*$ via minimization} \\label{AVOP}\n\nFor technical reasons, we will have to consider the situation when\n$F$ is inflated by an $\\varepsilon$-ball.\nThis yields a perturbed mapping\n\\[F_\\varepsilon:\\mathbbm{R}^d\\to\\mathcal{K}_c(\\mathbbm{R}^d),\\quad F_\\varepsilon(x):=Cx+B_\\varepsilon(V).\\]\n\nThe following auxiliary result discusses the interplay between contraction, \nnested dynamics and overapproximation.\n\n\\begin{lemma} \\label{nested:lemma}\nLet $X\\in\\mathcal{K}(\\mathbbm{R}^d)$ with $F(X)\\subset X$, and let $R>0$.\nThen we have $F(B^\\ell_{\\ell^{-1}R}(X))\\subset B^\\ell_R(X)$.\nThe same statement holds for the mapping $F_\\varepsilon$.\n\\end{lemma}\n\n\\begin{proof}\nThis follows from the computation\n\\begin{align*}\n&\\dist^\\ell(F(B^\\ell_{\\ell^{-1}R}(X)),X)\n\\le\\dist^\\ell(F(B^\\ell_{\\ell^{-1}R}(X)),F(X))\\\\\n&\\le\\ell\\dist^\\ell(B^\\ell_{\\ell^{-1}R}(X),X)\n\\le R.\n\\end{align*}\nSince $F_\\varepsilon$ shares all properties of $F$, the same proof applies to $F_\\varepsilon$.\n\\end{proof}\n\n\nWe construct an approximation to $X^*$ by solving an optimization problem, \nwhich uses the property proved in Proposition \\ref{inf:reach} part e) \nas a constraint.\n\n\\begin{proposition}\\label{vacuum:approximation}\nThe optimization problem\n\\begin{equation}\\left.\\begin{aligned} \n&\\min_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\mathcal{B}_{X^*}\\\\\n&\\mathcal{B}_{X^*}:=\\{b\\in\\mathbbm{R}^N:F(Q_{A,b})\\subset Q_{A,b}\\neq\\emptyset\\}\n\\end{aligned}\\right\\}\\label{op1}\\end{equation}\npossesses a unique solution $b^*\\in\\mathbbm{R}^N$. \nThis $b^*$ is given by \n\\[b^*_i=\\inf_{b\\in\\mathcal{B}_{X^*}}b_i\\quad\\text{for}\\quad i\\in\\{1,\\ldots,N\\}\\]\nand satisfies the error bound\n\\[X^*\\subset Q_{A,b^*}\\subset B^\\ell_{\\ell^{-1}R_A^{X^*}}(X^*)\\]\nwith $R_A^{X^*}$ as in Lemma \\ref{not:worse:than:ell}.\n\\end{proposition}\n\n\\begin{remark}\na) Note that Problem \\eqref{op1} selects the smallest polytope\nin the collection $\\{Q_{A,b}:b\\in\\mathcal{B}_{X^*}\\}$ with respect to inclusion.\nThe setup of the problem also guarantees that the vector $b^*$ is a particularly \nnice representation of the polytope $Q_{A,b^*}$, see Section 2.2 \nof \\cite{Rieger:17}.\n\nb) The number $R_A^{X^*}$ is defined in Lemma \\ref{not:worse:than:ell} and\nbounded by the a-priori estimate in Proposition \\ref{inf:reach} part f).\n\nc) It is at this stage not obvious that Problem \\eqref{op1} is a\ndisjunctive program.\nThis will be established in the next section.\n\\end{remark}\n\n\\begin{proof}[Proof of Proposition \\ref{vacuum:approximation}]\nWe clearly have $\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*))\\neq\\emptyset$.\nLemma \\ref{not:worse:than:ell} implies\n\\[\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*))\\subset B^\\ell_{\\ell^{-1}R_A^{X^*}}(X^*),\\]\nand by Proposition \\ref{inf:reach} part c), Lemma \\ref{nested:lemma} \nand Theorem \\ref{projector}, we have\n\\[F(\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*)))\n\\subset F(B^\\ell_{\\ell^{-1}{R_A^{X^*}}}(X^*))\n\\subset B_{R_A^{X^*}}^\\ell(X^*)\n\\subset\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*)).\\]\nIn particular, we find\n\\[\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*))\\in\\{Q_{A,b}:b\\in\\mathcal{B}_{X^*}\\},\\]\nso $\\mathcal{B}_{X^*}\\neq\\emptyset$.\nAccording to Proposition \\ref{inf:reach} part e), we have\n$X^*\\subset(\\cap_{b\\in\\mathcal{B}_{X^*}}Q_{A,b})$,\nso by Lemma \\ref{intersections}, the vector $b^*\\in\\mathbbm{R}^N$ given by \n$b^*_i:=\\inf_{b\\in\\mathcal{B}}b_i$ satisfies\n\\[X^*\\subset(\\cap_{b\\in\\mathcal{B}_{X^*}}Q_{A,b})=Q_{A,b^*}.\\]\nFrom the definition of $\\mathcal{B}_{X^*}$, we obtain\n\\begin{align*}\n&F(Q_{A,b^*})\n=F(\\cap_{b\\in\\mathcal{B}_{X^*}}Q_{A,b})\n=C(\\cap_{b\\in\\mathcal{B}_{X^*}}Q_{A,b})+V\n\\subset(\\cap_{b\\in\\mathcal{B}_{X^*}}(CQ_{A,b})+V)\\\\\n&\\subset(\\cap_{b\\in\\mathcal{B}_{X^*}}(CQ_{A,b}+V))\n=\\cap_{b\\in\\mathcal{B}_{X^*}}F(Q_{A,b})\n\\subset(\\cap_{b\\in\\mathcal{B}_{X^*}} Q_{A,b})\n=Q_{A,b^*},\n\\end{align*}\nso we have $b^*\\in\\mathcal{B}_{X^*}$ as well.\nBy the above and by Lemma \\ref{not:worse:than:ell}, we conclude \n$b^*=\\argmin_{b\\in\\mathcal{B}_{X^*}}\\mathbbm{1}^Tb$ and\n\\[X^*\\subset Q_{A,b^*}\\subset\\pi_{\\mathcal{G}_A}(B_{R_A^{X^*}}^\\ell(X^*))\n\\subset B^\\ell_{\\ell^{-1}R_A^{X^*}}(X^*).\\]\n\\end{proof}\n\nThe unique minimizers of the perturbed problems approximate\nthe unique minimizer of the original problem.\n\n\\begin{proposition}\\label{eps:converge}\nFor any $\\varepsilon>0$, the optimization problem\n\\begin{equation}\\left.\\begin{aligned}\n&\\min_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\mathcal{B}_{X^*_\\varepsilon}\\\\\n&\\mathcal{B}_{X^*_\\varepsilon}:=\\{b\\in\\mathbbm{R}^N:F_\\varepsilon(Q_{A,b})\\subset Q_{A,b}\\neq\\emptyset\\}\n\\end{aligned}\\right\\}\\label{op1eps}\\end{equation}\npossesses a unique solution $b^*_\\varepsilon\\in\\mathbbm{R}^N$.\nThis $b^*_\\varepsilon$ is given by \n\\[b^*_{\\varepsilon,i}=\\inf_{b\\in\\mathcal{B}_{X^*_\\varepsilon}}\nb_i\\quad\\text{for}\\quad i\\in\\{1,\\ldots,N\\},\\]\nand we have\n\\[\\lim_{\\varepsilon\\searrow 0}b_\\varepsilon^*=b^*.\\]\n\\end{proposition}\n\n\\begin{proof}\nSince $F_\\varepsilon$ shares all properties of $F$ for every $\\varepsilon>0$, we can apply \nProposition \\ref{vacuum:approximation} to the mapping $F_\\varepsilon$\nto obtain existence and uniqueness of the solutions $b^*_\\varepsilon$.\nIt remains to show the convergence statement.\n\nThe inclusion\n$F(Q_{A,b_\\varepsilon^*})\\subset F_\\varepsilon(Q_{A,b_\\varepsilon^*})\\subset Q_{A,b_\\varepsilon^*}$\nimplies $b_\\varepsilon^*\\in\\mathcal{B}_{X^*}$, so $b^*\\le b_\\varepsilon^*$ holds by \nProposition \\ref{vacuum:approximation}, and hence \n$Q_{A,b^*}\\subset Q_{A,b_\\varepsilon^*}$.\nLet $R_1=R_1(\\varepsilon):=\\frac{c_{2,\\ell}\\ell\\varepsilon}{1-\\ell}$ and define\n$X(\\varepsilon):=B^\\ell_{\\ell^{-1}R_1}(Q_{A,b^*})$. \nLemma \\ref{nested:lemma} gives\n\\begin{align*}\n&F_\\varepsilon(X(\\varepsilon))=F_\\varepsilon(B^\\ell_{\\ell^{-1}R_1}(Q_{A,b^*}))\n=F(B^\\ell_{\\ell^{-1}R_1}(Q_{A,b^*}))+B_\\varepsilon(0)\\\\\n&\\subset B_{R_1}^\\ell(Q_{A,b^*})+B_{c_{2,\\ell}\\varepsilon}^\\ell(0)\n=B_{R_1+c_{2,\\ell}\\varepsilon}^\\ell(Q_{A,b^*})\n\\subset B^\\ell_{\\ell^{-1}R_1}(Q_{A,b^*})\n=X(\\varepsilon).\n\\end{align*}\nNow let $R_2=R_2(\\varepsilon):=R_A^{X(\\varepsilon)}$ with notation as in \nLemma \\ref{not:worse:than:ell}.\nUsing Lemma \\ref{not:worse:than:ell}, Lemma \\ref{nested:lemma}\nand Theorem \\ref{projector}, we obtain\n\\begin{align*}\nF_\\varepsilon(\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon))))\n\\subset F_\\varepsilon(B_{\\ell^{-1}R_2}^\\ell(X(\\varepsilon)))\n\\subset B_{R_2}^\\ell(X(\\varepsilon))\n\\subset\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon))),\n\\end{align*}\nso $\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon)))\\in\\{Q_{A,b}:b\\in\\mathcal{B}_{X^*_\\varepsilon}\\}$,\nand by minimality of $b_\\varepsilon^*$, we have\n\\[Q_{A,b^*}\\subset Q_{A,b^*_\\varepsilon}\\subset\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon))).\\]\nLet $L_A>0$ be the Lipschitz constant of the mapping $\\pi_{\\mathcal{G}_A}$.\nBy the above, and since $Q_{A,b^*}\\in\\mathcal{G}_A$, we obtain\n\\begin{align*}\n&\\dist^\\ell(Q_{A,b^*_\\varepsilon},Q_{A,b^*})\n\\le\\dist^\\ell(\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon))),Q_{A,b^*})\\\\\n&=\\dist^\\ell(\\pi_{\\mathcal{G}_A}(B_{R_2}^\\ell(X(\\varepsilon))),\\pi_{\\mathcal{G}_A}(Q_{A,b^*}))\n\\le L_A\\dist^\\ell(B_{R_2}^\\ell(X(\\varepsilon)),Q_{A,b^*}),\n\\end{align*}\nwhich implies\n\\begin{align*}\n&\\lim_{\\varepsilon\\searrow 0}\\dist^\\ell(Q_{A,b^*_\\varepsilon},Q_{A,b^*})=0\n\\end{align*}\nand hence the desired convergence statement.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Disjunctive programs} \\label{CC}\n\nWe assume that the values $\\sigma_V(a_i)$ for $i\\in\\{1,\\ldots,N\\}$ \nof the support function of the sets $V$ are available.\nThis is not a strong requirement, because in many applications, \nthe set $V$ has a very simple shape.\nWe use the notation\n\\begin{align*}\n&P_0:=\\{p\\in\\mathbbm{R}^N:A^Tp=0,\\ \\mathbbm{1}^Tp=1,\\ p\\ge 0\\},\\\\\n&P_i:=\\{p\\in\\mathbbm{R}^N,\\ A^Tp=C^Ta_i,\\ p\\ge 0\\},\\quad i\\in\\{1,\\ldots,N\\},\n\\end{align*}\nand we develop representation of the sets $\\mathcal{B}_{X^*_\\varepsilon}$\nwhich is accessible to linear optimization techniques.\nIf $\\varepsilon=0$, then $F_\\varepsilon=F$, $B_\\varepsilon(V)=V$ and $\\mathcal{B}_{X^*_\\varepsilon}=\\mathcal{B}_{X^*}$.\n\n\\begin{proposition} \\label{lpc}\nConsider an arbitrary vector $b\\in\\mathbbm{R}^N$ and $\\varepsilon\\ge 0$.\n\\begin{itemize}\n\\item [a)] For $X\\in\\mathcal{K}(\\mathbbm{R}^d)$, the inclusion \n$X\\subset Q_{A,b}$ holds if and only if we have $\\sigma_{X}(a_i)\\le b_i$ \nfor all $i\\in\\{1,\\ldots,N\\}$.\n\\item [b)] The following statements are equivalent:\n\\begin{itemize}\n\\item [i)] $Q_{A,b}\\neq\\emptyset$;\n\\item [ii)] $p^Tb\\ge 0$ for all $p\\in\\mathbbm{R}^N$ with $A^Tp=0$ and $p\\ge 0$;\n\\item [iii)] $p^Tb\\ge 0$ for all $p\\in P_0$.\n\\item [iv)] $p^Tb\\ge 0$ for all $p\\in\\ext(P_0)$.\n\\end{itemize}\n\\item [c)] If we have $Q_{A,b}\\neq\\emptyset$, then the following statements\nare equivalent:\n\\begin{itemize}\n\\item [i)] $F_\\varepsilon(Q_{A,b})\\subset Q_{A,b}$;\n\\item [ii)] $\\max\\{(C^Ta_i)^Tx:x\\in Q_{A,b}\\}\\le b_i-\\sigma_{B_\\varepsilon(V)}(a_i)$\n$\\forall\\,i\\in\\{1,\\ldots,N\\}$;\n\\item [iii)] $\\min\\{(p-e_i)^Tb:p\\in P_i\\}\\le-\\sigma_{B_\\varepsilon(V)}(a_i)$ \n$\\forall\\,i\\in\\{1,\\ldots,N\\}$;\n\\item [iv)] $\\min\\{(p-e_i)^Tb:p\\in\\ext(P_i)\\}\\le-\\sigma_{B_\\varepsilon(V)}(a_i)$\n$\\forall\\,i\\in\\{1,\\ldots,N\\}$.\n\\end{itemize}\n\\end{itemize}\n\\end{proposition}\n\n\\begin{proof}\nStatement a) is obvious.\n\nb) The equivalence between i) and ii) is the version of the Farkas lemma \ngiven in Proposition 1.7 of \\cite{Ziegler}.\nElementary arguments show that statement ii) is equivalent with statement iii).\nSince the set $P_0$ is a compact polytope, statement iii) is equivalent \nwith statement iv).\n\nc) We have $F_\\varepsilon(Q_{A,b})\\subset Q_{A,b}$ if and only if \n\\[a_i^T(Cx+v)\\le b_i\\quad\\forall\\,x\\in Q_{A,b},\\ \\forall\\,v\\in B_\\varepsilon(V),\\\n\\forall\\,i\\in\\{1,\\ldots,N\\},\\]\nwhich can be rewritten as\n\\[(C^Ta_i)^Tx\\le b_i-a_i^Tv\\quad\\forall\\,x\\in Q_{A,b},\\ \\forall\\,v\\in B_\\varepsilon(V),\\\n\\forall\\,i\\in\\{1,\\ldots,N\\}.\\]\nThis establishes the equivalence of statements i) and ii).\nSince $Q_{A,b}$ is nonempty and bounded, the strong duality theorem \nfor linear programming as presented in Theorem 4.13 of \\cite{Theobald:13}\nguarantees that \n\\begin{align*}\n&\\max\\{(C^Ta_i)^Tx:x\\in Q_{A,b}\\}=\\min\\{p^Tb:p\\in P_i\\}\n=\\min\\{p^Tb:p\\in\\ext(P_i)\\}\n\\end{align*}\nis finite.\nHence statement ii) is equivalent with statements iii) and iv).\n\\end{proof}\n\nWe state Problems \\eqref{op1} and \\eqref{op1eps} as a disjunctive programs \nto highlight their structural properties.\n\n\\begin{theorem}\nProblems \\eqref{op1} and \\eqref{op1eps} are equivalent with the problem\n\\begin{equation}\\label{LP:ext:1}\n\\left.\\begin{aligned}\n&\\min_b\\,\\mathbbm{1}^Tb&&\\\\\n&\\text{subject to}&0&\\le p^Tb\\quad\\forall\\,p\\in P_0\\\\\n&&\\min\\{(p-e_1)^Tb:p\\in\\ext(P_1)\\}&\\le-\\sigma_{B_\\varepsilon(V)}(a_1),\\\\\n&&&\\vdotswithin{\\le}\\\\\n&&\\min\\{(p-e_N)^Tb:p\\in\\ext(P_N)\\}&\\le-\\sigma_{B_\\varepsilon(V)}(a_N)\n\\end{aligned}\\right\\}\n\\end{equation}\nwith $\\varepsilon=0$ in the case of Problem \\eqref{op1}.\n\\end{theorem}\n\nDisjunctive programs are, in general, hard to solve, see \\cite{Balas}.\nTo compute the desired solution $b^*$, we will construct a dual-type\nproblem that is just an ordinary linear program.\n\n\n\n\n\n\n\n\n\\section{Perturbed dual LPs}\n\nIn the following, we will formulate a linear program, which is related\nto the dual of Problem \\eqref{LP:ext:1} with $\\varepsilon>0$ in the sense of \\cite{Balas}.\n\n\\begin{remark}\nIt is, in general, not possible to compute the solution $b^*$ of \nProblem \\ref{LP:ext:1} with $\\varepsilon=0$ by following \\cite{Balas} directly:\n\na) If $\\interior(X^*)\\neq\\emptyset$, then $\\interior(Q_{A,b^*})\\neq\\emptyset$,\nwhich implies $p^Tb^*>0$ for all $p\\in P_0$ by Proposition 36 in \\cite{Rieger:17}.\nThe dual as defined in \\cite{Balas} and similar works involves a constraint\nof type $p^Tb\\le 0$ for some $p\\in P_0$, which means that $b^*$ is dual infeasible \nin this setting.\n\nb) The dual problem from \\cite{Balas} may have more than one maximizer, \nso it is not obvious how to recover $b^*$ from a dual solution.\n\\end{remark}\n\nThese facts motivate us to construct a dual problem following the general idea from\n\\cite{Balas}, but omitting the constraints $p^Tb\\ge 0$ for all $p\\in P_0$, and to\nuse a perturbation argument to show uniqueness of the dual maximizer.\n\n\\begin{proposition}\\label{locmin}\nFor any $\\varepsilon>0$, the global minimizer $b^*_\\varepsilon\\in\\mathbbm{R}^N$ \nof Problem \\eqref{op1eps} satisfies\n\\begin{align}\n&\\max\\{(C^Ta_i)^Tx:x\\in Q_{A,b^*_\\varepsilon}\\}=b_{\\varepsilon,i}^*-\\sigma_{B_\\varepsilon(V)}(a_i)\\quad\n\\forall\\,i\\in\\{1,\\ldots,N\\},\\label{primal:eq}\\\\\n&\\min\\{(p-e_i)^Tb^*_\\varepsilon:p\\in\\ext(P_i)\\}=-\\sigma_{B_\\varepsilon(V)}(a_i)\\quad\\forall\\,i\\in\\{1,\\ldots,N\\},\n\\label{dual:eq}\n\\end{align}\nand the global minimizer $b^*\\in\\mathbbm{R}^N$ of Problem \\eqref{op1} satisfies\n\\[\\min\\{(p-e_i)^Tb^*:p\\in\\ext(P_i)\\}=-\\sigma_{V}(a_i)\\quad\\forall\\,i\\in\\{1,\\ldots,N\\}.\\]\nIn particular, we have $b^*_\\varepsilon\\in\\Omega_\\varepsilon$ and $b^*\\in\\Omega$, where\n\\begin{align*}\n&\\Omega_\\varepsilon:=\\{b\\in\\mathbbm{R}^N:(e_i-p)^Tb\\le\\sigma_{B_\\varepsilon(V)}(a_i)\\ \\forall\\,p\\in\\ext(P_i),\\ i\\in\\{1,\\ldots,N\\}\\},\\\\\n&\\Omega:=\\{b\\in\\mathbbm{R}^N:(e_i-p)^Tb\\le\\sigma_{V}(a_i)\\ \\forall\\,p\\in\\ext(P_i),\\ i\\in\\{1,\\ldots,N\\}\\}.\n\\end{align*}\n\\end{proposition}\n\n\\begin{proof}\nSince $b^*_\\varepsilon\\in\\mathcal{B}_{X^*_\\varepsilon}$, \nwe have $Q_{A,b^*_\\varepsilon}\\neq\\emptyset$, \nand hence $F_\\varepsilon(Q_{A,b^*_\\varepsilon})\\neq\\emptyset$, \nas well as $F_\\varepsilon(Q_{A,b^*_\\varepsilon})\\subset Q_{A,b^*_\\varepsilon}$, which is,\nby Proposition \\ref{lpc} part c), equivalent with\n\\begin{equation}\\label{forall}\n\\max\\{(C^Ta_i)^Tx:x\\in Q_{A,b^*_\\varepsilon}\\}\\le b^*_{\\varepsilon,i}-\\sigma_{B_\\varepsilon(V)}(a_i)\\quad\n\\forall\\,i\\in\\{1,\\ldots,N\\}.\n\\end{equation}\nAssume that there exist $j\\in\\{1,\\ldots,N\\}$ and $\\delta>0$ with\n\\begin{equation}\\label{forj}\n\\max\\{(C^Ta_j)^Tx:x\\in Q_{A,b^*_\\varepsilon}\\}\\le b^*_{\\varepsilon,j}-\\sigma_{B_\\varepsilon(V)}(a_j)-\\delta.\n\\end{equation}\nThen for $b^\\delta:=b^*_\\varepsilon-\\delta e_j$, inequalities \\eqref{forall} \nand \\eqref{forj} yield\n\\begin{align*}\n\\sigma_{F_\\varepsilon(Q_{A,b^*_\\varepsilon})}(a_i)\n&=\\max\\{(C^Ta_i)^Tx:x\\in Q_{A,b^*_\\varepsilon}\\}+\\sigma_{B_\\varepsilon(V)}(a_i)\\\\\n&\\le b^\\delta_i\\quad\\forall\\,i\\in\\{1,\\ldots,N\\},\n\\end{align*}\nso $\\emptyset\\neq F_\\varepsilon(Q_{A,b^*_\\varepsilon})\\subset Q_{A,b^\\delta}$ follows from\nProposition \\ref{lpc} part a).\nBy monotonicity, we have\n\\[\\sigma_{F_\\varepsilon(Q_{A,b^\\delta})}(a_i)\n\\le\\sigma_{F_\\varepsilon(Q_{A,b^*_\\varepsilon})}(a_i)\n\\le b_i^\\delta\\quad\\forall\\,i\\in\\{1,\\ldots,N\\},\\]\nwhich is equivalent with $F_\\varepsilon(Q_{A,b^\\delta})\\subset Q_{A,b^\\delta}$.\nHence $b^\\delta\\in\\mathcal{B}_{X^*_\\varepsilon}$, but we have \n$\\mathbbm{1}^Tb^\\delta<\\mathbbm{1}^Tb^*_\\varepsilon$,\nwhich is a contradiction.\nAll in all, we have proved equation \\eqref{primal:eq},\nand equation \\eqref{dual:eq} follows from the strong duality theorem \nof linear programming.\nEquation \\eqref{dual:eq}, in turn, implies that\n\\[(p-e_i)^Tb^*_\\varepsilon\\ge-\\sigma_{B_\\varepsilon(V)}(a_i)\\quad\\forall\\,p\\in\\ext(P_i),\\ \n\\forall\\,i\\in\\{1,\\ldots,N\\},\\]\nwhich shows that $b^*_\\varepsilon\\in\\Omega_\\varepsilon$.\nThe same arguments work in the case $\\varepsilon=0$.\n\\end{proof}\n\n\nThe following result shows that the set $Q_{A,b^*_\\varepsilon}$ can be computed \nby solving a linear programming problem for $b^*_\\varepsilon$.\n\n\\begin{proposition} \\label{perturbed:unique}\nFor any $\\varepsilon>0$, the unique solution $b^*_\\varepsilon$ of the perturbed disjunctive\nprogram \\eqref{op1eps} is the unique solution of the linear program\n\\[\\max_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\Omega_\\varepsilon.\\]\n\\end{proposition}\n\nThe need to consider an arbitrarily small inflation of the set $V$ \narises from the following proof, in which we need that small perturbations $b$\nof the point $b^*_\\varepsilon$ satisfy $Q_{A,b}\\neq\\emptyset$.\n\n\\begin{proof}\nAssume that there exists $b_*\\in\\Omega_\\varepsilon\\setminus\\{b^*_\\varepsilon\\}$ with \n$\\mathbbm{1}^Tb_*\\ge\\mathbbm{1}^Tb^*_\\varepsilon$. \nSince $b^*_\\varepsilon\\in\\mathcal{B}_{X^*_\\varepsilon}$, we have\n\\[\\emptyset\\neq\\interior B_\\varepsilon(V)\n\\subset\\interior F_\\varepsilon(Q_{A,b^*_\\varepsilon})\\subset\\interior Q_{A,b^*_\\varepsilon},\\]\nand by Proposition 37 in \\cite{Rieger:17}, there exists \n$\\delta>0$ such that $p^Tb^*_\\varepsilon\\ge\\delta$ for all $p\\in\\ext(P_0)$. \nBy H\\\"older inequality, the vector \n$\\bar{b}:=b^*_\\varepsilon+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}(b^*_\\varepsilon-b_*)$\nsatisfies\n\\[p^T\\bar{b}\n=p^Tb^*_\\varepsilon+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}p^T(b^*_\\varepsilon-b_*)\n\\ge\\delta-\\delta\\|p\\|_1\n=0\\quad\\forall p\\in\\ext(P_0),\\]\nso $Q_{A,\\bar{b}}\\neq\\emptyset$ by part b) of Proposition \\ref{lpc}.\nSince $b_*\\in\\Omega_\\varepsilon$ and by Proposition \\ref{locmin}, for every $i\\in\\{1,\\ldots,N\\}$,\nthere exists $p\\in\\ext(P_i)$ such that\n\\[(e_i-p)^Tb_*\\le\\sigma_{B_\\varepsilon(V)}(a_i)\\quad\\text{and}\\quad\n(p-e_i)^Tb^*_\\varepsilon=-\\sigma_{B_\\varepsilon(V)}(a_i).\\]\nFrom this we conclude that\n\\begin{align*}\n(p-e_i)^T\\bar{b}\n&=(p-e_i)^T(b^*_\\varepsilon+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}(b^*_\\varepsilon-b_*))\\\\\n&=(1+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty})(p-e_i)^Tb^*_\\varepsilon-\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}(p-e_i)^Tb_*\\\\\n&\\le-(1+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty})\\sigma_{B_\\varepsilon(V)}(a_i)\n+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}\\sigma_{B_\\varepsilon(V)}(a_i)\n=-\\sigma_{B_\\varepsilon(V)}(a_i),\n\\end{align*}\nand hence that $F_\\varepsilon(Q_{A,\\bar{b}})\\subset Q_{A,\\bar{b}}$ according to\nProposition \\ref{lpc} part c).\nAll in all, we have $\\bar{b}\\in\\mathcal{B}_{X^*_\\varepsilon}$, but \n\\[\\mathbbm{1}^T\\bar{b}\n=\\mathbbm{1}^T(b^*_\\varepsilon+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}(b^*_\\varepsilon-b_*))\n=(1+\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty})\\mathbbm{1}^Tb^*_\\varepsilon\n-\\tfrac{\\delta}{\\|b^*_\\varepsilon-b_*\\|_\\infty}\\mathbbm{1}^Tb_*\n\\le\\mathbbm{1}^Tb^*_\\varepsilon,\\]\nwhich is impossible, because the point $b^*_\\varepsilon$ is the unique global minimum \nof Problem \\eqref{op1}.\n\\end{proof}\n\n\n\n\n\n\n\n\\section{The unperturbed dual LP}\n\nNow we conclude that the approximation $Q_{A,b^*}$ to $X^*$ we wish to compute\nis indeed given by the unique solution of the unperturbed dual linear program.\n\n\\begin{theorem}\nThe unique solution $b^*$ of the disjunctive program \\eqref{op1} \nis the unique solution of the linear program\n\\[\\max_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\Omega.\\]\n\\end{theorem}\n\n\\begin{proof}\nBy Proposition \\ref{perturbed:unique}, for any $\\varepsilon>0$, the unique solution \n$b^*_\\varepsilon$ of the disjunctive program \\eqref{op1eps} is the unique solution \nof the linear program\n\\begin{equation} \\label{loc:1}\n\\max_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\Omega_\\varepsilon.\n\\end{equation}\nBy Proposition \\ref{locmin}, we have $b^*\\in\\Omega$, so the linear program\n\\begin{equation} \\label{loc:2}\n\\max_b\\,\\mathbbm{1}^Tb\\quad\\text{subject to}\\quad b\\in\\Omega\n\\end{equation}\nis feasible. \nSince $\\Omega\\subset\\Omega_\\varepsilon$, the value of Problem \\eqref{loc:2} is bounded \nby the value of Problem \\eqref{loc:1} with $\\varepsilon=1$, so \n$\\argmax_{b\\in\\Omega}\\mathbbm{1}^Tb\\neq\\emptyset$.\nCorollary 3.1 from \\cite{Robinson} yields that if \n$\\tilde{b}\\in\\argmax_{b\\in\\Omega}\\mathbbm{1}^Tb$ and there exists $\\varepsilon_0>0$ with\n$\\argmax_{b\\in\\Omega_\\varepsilon}\\mathbbm{1}^Tb\\neq\\emptyset$ for $\\varepsilon\\in(0,\\varepsilon_0]$, then\nthere exist $\\tilde{b}_\\varepsilon\\in\\argmax_{b\\in\\Omega_\\varepsilon}\\mathbbm{1}^Tb$ \nwith\n\\[\\tilde{b}=\\lim_{\\varepsilon\\searrow 0}\\tilde{b}_\\varepsilon.\\]\nIn the present situation, we have $b^*_\\varepsilon=\\argmax_{b\\in\\Omega_\\varepsilon}\\mathbbm{1}^Tb$,\nso using Proposition \\ref{eps:converge}, we conclude that\n\\[\\argmax_{b\\in\\Omega}\\mathbbm{1}^Tb=\\lim_{\\varepsilon\\searrow 0}b^*_\\varepsilon=b^*.\\]\n\\end{proof}\n\n\n\n\n\n\\subsection*{Acknowledgement}\nWe thank Andrew Eberhard for pointing us towards the term \\emph{disjunctive\nprogramming}, which we would otherwise never have found in the maze of mathematical\nliterature.\n\n\n\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRecently, the sequence-to-sequence (S2S) approach in automatic speech recognition (ASR) has received a considerable amount of interest, due to the ability to jointly train all components towards a common goal which reduces complexity and error propagation compared to traditional hybrid systems. \nTraditional systems divide representation into different levels in the acoustic model, in particular separating global features (such as channel and speaker characteristics) and local features (on the phoneme level). \nThe language model and acoustic model are trained with different loss functions and then combined during decoding.\nIn contrast, neural S2S models perform a direct mapping from audio signals to text sequences based on dynamic interactions between two main model components, an encoder and a decoder, which are jointly trained towards maximizing the likelihood of the generated output sequence. \nThe neural encoder reads the audio features into high-level representations, which are then fed into an auto-regressive decoder which attentively generates the output sequences~\\cite{bahdanau2014neural,bahdanau2016end}.\n\nIn this context, we aim at reconsidering acoustic modeling within end-to-end models. Previous approaches in general had long short-term memory neural networks (LSTM)~\\cite{hochreiter1997long} or time-delay neural networks~\\cite{waibel1989tdnn} operating on top of frame-level features to learn sequence-level representation. These neural networks are able to capture long range and local dependencies between different timesteps. \n\nRecently, self-attention has been shown to efficiently represent different structures including text~\\cite{bahdanau2014neural}, images~\\cite{velivckovic2017graph}, and even acoustic signals~\\cite{sperber2018self} with impressive results. The Transformer model using self-attention achieved the state-of-the-art in mainstream NLP tasks~\\cite{vaswani2017attention}. The attractiveness of self-attention networks originates from the ability to establish a direct connection between any element in the sequence. Self-attention is able to scale with the length of the input sequence without any limiting factor such as, e.g., the kernel size of CNNs, or the vanishing gradient problem of LSTMs. Moreover, the self-attention network is also computationally advantageous compared to recurrent structures because intermediate states are no longer connected recurrently, leading to more efficient batched operations. As a result, self-attention networks can be reasonably trained with many layers leading to state-of-the-art performance in various tasks~\\cite{devlin2018bert}.\nSelf-attention and the Transformer have been exploratorily applied to ASR, but so far with unsatisfactory results. \\cite{sperber2018self} found that self-attention in the encoder (acoustic model) was not effective, but combined with an LSTM brought marginal improvement and greater interpretability, while\n\\cite{dong2018speech} did not find any notable improvement using the Transformer in which the encoder combines self-attention with convolution\/LSTM compared to other model architectures.\\\\\nIn this work, we show that the Transformer requires little modification to adapt on the speech recognition task. Specifically, we exploit the advantages of self-attention networks for ASR such that both our acoustic encoder and character-generating decoder are constructed without any recurrence or convolution. This is the first attempt to propose this system architecture to the best of our knowledge, and we show that a competitive end-to-end ASR model can be achieved solely using standard training techniques from general S2S systems.\n\nOur contributions are as follows. First, we show that depth is an important factor to acquire competitive end-to-end ASR models with the Transformer. Second, in order to facilitate training of very deep configurations, we propose a variation of stochastic depth for the Transformer inspired by the Stochastic Residual Network for image classification~\\cite{huang2016deep}. \n\nWe discovered that its ability to regularize is the key contribution to obtain the state-of-the-art result among end-to-end ASR models for the standard 300h Switchboard (SWB) benchmark. \nThis result is achieved using a total of 48 Transformer layers across the encoder and decoder.~\\footnote{Our source code and final model are available at \\textit{https:\/\/github.com\/quanpn90\/NMTGMinor\/tree\/audio-encoder\/}}\n\n\\begin{figure}[htb]\n\\vspace{-1em}\n\\centering\n\\includegraphics[width = 0.45\\textwidth]{figures\/transformer_speech.PNG}\n\\caption{\\label{fig:model} A diagram of transformation from acoustic features to character-level transcriptions. The red connections represent the residual connections, which are rescaled according to Equation~\\ref{eq:stochastic_res} for stochastic Transformers.}\n\\end{figure}\n\n\\section{Model Description}\n\n\\subsection{Encoder-Decoder with Attention}\nThe main components of the model include an encoder, which consumes the source sequence and then generates a high-level representation, and a decoder generating the target sequence. \nThe decoder models the data as a conditional language model - the probability of the sequence of discrete tokens is decomposed into an ordered product of distributions conditioned on both the previously generated tokens and the encoder representation.\n\nBoth encoder and decoder are neural networks and require neural components that are able to learn the relationship between the time steps in the input and output sequence. The decoder also requires a mechanism to condition on specific components of the encoder representation. \nFor the Transformer, attention or its common variation multi-head attention, is the core of the model in place of recurrence.\n\n\\subsection{Multi-Head Attention}\nFundamentally, attention refers to the method of using a content-based information extractor from a set of queries $Q$, keys $K$ and values $K$. The retrieval function is based on similarities~\\cite{luong2015effective} between the queries and the keys, and in turn returns the the weighted sum of the values as following:\n\\begin{equation}\n    \\text{Attention}(Q, K, V) = \\text{softmax}(QK^T)V\n\\end{equation}\nRecently, \\cite{vaswani2017attention} improves dot-product attention by scaling the queries before hand and introducing sub-space projection for keys, queries and values into $n$ parallel heads, in which $n$ attention operations are performed with corresponding heads. The result is the concatenation of attention outputs from each head.\nNotably, unlike recurrent connections which use single states with gating mechanism to transfer data or convolution connections linearly combining local states limited in a kernel size, self-attention aggregates the information in \\textit{all} time-steps without any intermediate transformation.\n\n\\subsection{Layer Architecture}\n\nThe overall architecture is demonstrated in Figure~\\ref{fig:model}. The encoder and decoder of the Transformers are constructed by layers, each of which contains self-attentional sub-layers coupled with feed-forward neural networks. \n\n\nTo adapt the encoder to long speech utterances, we follow the reshaping practice from~\\cite{sperber2018self} by grouping consecutive frames into one step. \nSubsequently we combine the input features with sinusoidal positional encoding~\\cite{vaswani2017attention}. While directly adding acoustic features to the positional encoding is harmful, potentially leading to divergence during training~\\cite{sperber2018self}, we resolved that problem by simply projecting the concatenated features to a higher dimension before adding ($512$, as other hidden layers in the model). In the case of speech recognition specifically, the positional encoding offers a clear advantage compared to learnable positional embeddings~\\cite{gehring2017convolutional}, because the speech signals can be arbitrarily long with a higher variance compared to text sequences. \n\nThe Transformer encoder passes the input features to a self-attention layer followed by a feed-forward neural network with 1 hidden layer with the ReLU activation function. Before these sub-modules, we follow the original work to include residual connections which establishes short-cuts between the lower-level representation and the higher layers. The presence of the residual layer massively increases the magnitude of the neuron values which is then alleviated by the layer-normalization~\\cite{ba2016layer} layers placed after each residual connection.\n\nThe decoder is the standard Transformer decoder in the recent translation systems~\\cite{vaswani2017attention}. The notable difference between the decoder and the encoder is that to maintain the auto-regressive nature of the model, the self-attention layer of the decoder must be masked so that each state has only access to the past states. Moreover, an additional attention layer using the target hidden layer layers as queries and the encoder outputs as keys and values is placed between the self-attention and the feed-forward layers. Residual and layer-normalization are setup identically to the encoder.\n\nThis particular design of the Transformer has various advantages compared to previously proposed RNNs and CNNs networks. First, computation of each layer and sub-module can be efficiently parallelized over both mini-batch and time dimensions of the input. Second, the combination of residual and layer normalization is the key to enable greater depth  configurations to be trainable, which is the main reason for the performance breakthrough in recent works in both MT and natural language processing~\\cite{devlin2018bert,pham2018wmt}. \n\n\\subsection{Stochastic Layers}\nThe high density of residual connections is the reason why Transformer is favourably trained with many layers. However, deep models in general suffer from either overfitting due to more complex architectures and optimization difficulty~\\cite{bapna2018training}. Studies about residual networks have shown that during training the network consists of multiple sub-networks taking different paths through shortcut connections~\\cite{veit2016residual}, and thus there are redundant layers. Motivated by the previous work of~\\cite{huang2016deep}, we propose to apply stochastic residual layers into our Transformers. The method resembles Dropout~\\cite{srivastava2014dropout}, in which the key idea is the layers are randomly dropped during training.\nThe original residual connection of an input $x$ and its corresponding neural layer $F$ has the following form:\n\\begin{equation}\nR(x) = \\text{LayerNorm}(F(x) + x)\n\\label{eq:res}\n\\end{equation}\nIn equation~\\ref{eq:res}, the inner function $F$ is either self-attention, feed-forward layers or even decoder-encoder attention. The layer normalization as in~\\cite{ba2016layer} keeps the magnitude of the hidden layers from growing large. Stochastic residual connections fundamentally apply a mask $M$ on the function $F$, as follows:\n\\begin{equation}\nR(x) = \\text{LayerNorm}(M * F(x) +  x)\n\\label{eq:res}\n\\end{equation}\nMask $M$ only takes $0$ or $1$ as values, generated from a Bernoulli distribution similar to dropout~\\cite{srivastava2014dropout}. When $M=1$, the inner function $F$ is activated, while it is skipped when $M=0$. These stochastic connections enables more sub-network configurations to be created during training, while during inference the full network is presented, causing the effect of ensembling different sub-networks, as analyzed in~\\cite{veit2016residual}. \nIt is non-trivial regarding how to the parameter $p$ for dropping layers, since the amount of residual connections for the Transformer is considerable. \n\nIn other words, the lower the layer is, the lower the probability $p$ is required to be set. As a result, $p$ values are set with the following policy:\n\\begin{itemize}\n   \n    \\item Sub-layers inside each encoder or decoder layer share the same mask, so each mask decides to drop or to keep the whole layer (including the sub-layers inside). This way we have one hyper-parameter $p$ for each layer.\n    \\item As suggested by~\\cite{huang2016deep}, the lower layers of the networks handle raw-level acoustic features on the encoder side, and the character embeddings on the decoder side. Therefore, lower layers $l$ have lower probability linearly scaled by their depth according to equation~\\ref{eq:p} with $p$ is the global-level parameter and $L$ is the total number of layers.~\\footnote{Our early experiments with a constant $p$ for all connections provide evidence that dropping lower-level representations is less tolerable than dropping higher-level representations.} \n\\end{itemize}\nLastly, since the layers are selected with probability $1-p_l$ during training and are always presented during inference, we scale the layers' output by $\\frac{1}{1-p_l}$ whenever they are not skipped. Therefore, each stochastic residual connection has the following form during \\textit{training} (the scalar is removed during testing):\n\\begin{equation}\np_l = \\frac{l}{L} (1 - p)\n\\label{eq:p}\n\\end{equation}\n\\begin{equation}\nR(x) = \\text{LayerNorm}(M * F(x) * \\frac{1}{1-p_l} +  x)\n\\label{eq:stochastic_res}\n\\end{equation}\n\n\n\\section{Experimental Setup}\n\n\\subsection{Data}\nOur experiments were conducted on the Switchboard-1 Release 2 (LDC97S62) corpus which contains over 300 hours of speech. The Hub5'00 evaluation data (LDC2002S09) was used as our test set. All the models were trained on 40 log mel filter-bank features which are extracted and normalized per conversation. We also adopted a simple down-sampling method in which we stacked 4 consecutive features vectors to reduce the length of input sequences by a factor of 4. Beside the filter-bank features, we did not employ any auxiliary features. We followed the approach from \\cite{ko2015audio} to generate a speech perturbation training set. Extra experiments are also conducted on the TED-LIUM 3~\\cite{hernandez2018ted} dataset which is more challenging due to longer sequences.\n\n\\subsection{Implementation Details}\nOur hyperparameter search revolves around the~\\textit{Base} configuration of the machine translation model in the original Transformer paper~\\cite{vaswani2017attention}. For all of our experiments in this work, the embedding dimension $d$ is set to $512$ and the size of the hidden state in the feed-forward sub-layer is $1024$. The mini-batch size is set so that we can fit our model in the GPU, and we accumulate the gradients and update every $25000$ characters. Adam~\\cite{kingma2014adam} with adaptive learning rate over the training progress:\n\\begin{equation}\n    lr = init\\_lr * d^{-0.5} * min (step^{-0.5}, step * warmup^{-1.5})\n\\end{equation}\nin which the init\\_lr is set to $2$, and we warm up the learning rate for $8000$ steps. Dropout~\\cite{srivastava2014dropout} (applied before residual connection and on the attention weights) is set at $0.2$. We also apply character dropout~\\cite{gal2016theoretically} with $p=0.1$ and label smoothing~\\cite{szegedy2016rethinking} with $\\epsilon=0.1$.\n\\section{Results}\n\\begin{table}[ht]\n\\caption{The performance of deep self-attention networks with and without stochastic layers on Hub5'00 test set with 300h SWB training set.}\n\t\\label{tab:swb}\n\t\\vspace{-0.2cm}\t\n\t\\setlength{\\tabcolsep}{6pt}\n\t\\centering\n\t\\begin{tabular}{lccc}\n\t\t\\toprule\n        \\textbf{Layers} & \\textbf{\\#Param} & \\textbf{SWB} & \\textbf{CH}\\\\\n        \\midrule\n        04Enc-04Dec & 21M & 20.8 & 33.2 \\\\ \n        08Enc-08Dec & 42M & 14.8 & 25.5 \\\\ \n        12Enc-12Dec & 63M & 13.0 & 23.9 \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 13.1 & 23.6 \\\\ \n        24Enc-24Dec & 126M & 12.1 & 23.0 \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 11.7 & 21.5 \\\\ \n        \\;\\; \\textit{+Speed Perturbation} & & 10.6 & 20.4 \\\\\n        48Enc-48Dec & 252M & - & - \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 11.6 & 20.9 \\\\\n                    48Enc-48Dec (half-size) & 63M & - & - \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 12.5 & 22.9\\\\   \n        \\midrule\n        08Enc-08Dec (big) & 168M & 13.8 & 25.1 \\\\\n        \\midrule\n        24Enc-12Dec & 113M & 13.3 & 23.7 \\\\     \n        \\;\\; \\textit{+Stochastic Layers} & & 11.9 & 21.6\\\\\n        36Enc-8Dec & 113M & 12.4  & 22.6 \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 11.5 & 20.6\\\\\n        36Enc-12Dec & 113M & 12.4 & 22.6 \\\\\n        \\;\\; \\textit{+Speed Perturbation} &  & 11.2 & 20.6 \\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 11.3 & 20.7\\\\\n        \\;\\; \\textit{+Both} &  & \\textbf{10.4} & \\textbf{18.6} \\\\\n        40Enc-8Dec & 109M & -- & --\\\\\n        \\;\\; \\textit{+Stochastic Layers} & & 11.9 & 21.4\\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{-0.0cm}\n\\end{table}\n\n\\begin{table}[ht]\n\\caption{Comparing our best model to other hybrid and end-to-end systems reporting on Hub5'00 test set with 300h SWB training set. }\n\\label{tab:swb2}\n\t\\vspace{-0.2cm}\t\n\t\\setlength{\\tabcolsep}{4pt}\n\t\\centering\n\t\\begin{tabular}{lccc}\n\t\t\\toprule\n        \\textbf{Hybrid\/End-to-End Models} & \\textbf{Tgt Unit} & \\textbf{SWB} & \\textbf{CH}\\\\\n        \\midrule\n        TDNN~~~+LFMMI \\cite{povey2016purely}  & Phone & 10.0 & 20.1 \\\\\n        BLSTM +LFMMI \\cite{povey2016purely} & Phone & \\textbf{9.6}  & 19.3 \\\\\n        \\midrule\n        CTC+CharLM \\cite{maas2015lexicon} & Char & 21.4 & 40.2 \\\\\n        LSTM w\/attention~\\cite{bahdanau2014neural} & Char & 15.8 & 36.0 \\\\\n        Iterated-CTC +LSTM-LM \\cite{zweig2017advances} & Char & 14.0 & 25.3 \\\\\n       \n        Seq2Seq ~~~~~~~~+LSTM-LM \\cite{zeyer2018improved} & BPE  & 11.8 & 25.7 \\\\\n        Seq2Seq ~~~~+Speed Perturbation \\cite{weng2018improving} & Char & 12.2 & 23.3 \\\\\n        CTC-A2W +Speed Perturbation \\cite{yu2018multistage} & Word & 11.4 & 20.8 \\\\\n        \\midrule\n       \n       \n        36Enc-12Dec (Ours)  & Char & 10.4 & 18.6 \\\\\n        48Enc-12Dec (Ours)  & Char & 10.7 & 19.4 \\\\\n        60Enc-12Dec (Ours)  & Char & 10.6 & 19.0 \\\\\n        Ensemble  & & 9.9 & \\textbf{17.7} \\\\        \n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{-0.0cm}\n\\end{table}\n\nThe experiment results on SWB testsets are shown in Table~\\ref{tab:swb}. A shallow configuration (i.e $4$ layers) is not sufficient for the task, and the WER reduces from $20.8\\%$ to $12.1\\%$ on the SWB test as we increase the depth from $4$ to $24$. \nThe improvement is less significant between $12$ and $24$ (only $5\\%$ relative WER), which seems to be a symptom of overfitting.\n\n\n\nOur suspicion of overfitting is confirmed by the addition of stochastic networks. At $12$ layers, the stochastic connections only improve the CH performance by a small margin, while the improvement was substantially greater on the $24$ layer setting. Following this trend, the stochastic $48$-layer model keeps improving on the CH test set, showing the model's ability to generalize better. \n\nArguably, the advantage of deeper models is to offer more parameters, as shown in the second column. We performed a contrastive experiment using a shallow model of 8 layers, but doubling the model size so that its parameter count is larger than the deep $24$-layer model. The performance of this model is considerable worse than the $24$ layer model, demonstrating deeper networks with smaller size are more beneficial than a wider yet shallower configuration. Reversely, we found that the 48-layer model with half size is equivalent with the $12$-layer variant, possibly due to over-regularization~\\footnote{We did not change dropout values for this model, so each layer's hidden layers are dropped more severely}.\n\nOur second discovery is that the encoder requires deeper networks than the decoder for the Transformer. This is inline with the previous work from~\\cite{zhang2017very} who increases depth for the CNN encoder.\nAs shown above, the encoder has learn representations starting from audio features, while the decoder handles the generation of character sequences, conditionally based on the encoder representation. \nThe difference in modalities suggest different configurations. \nHolding the total number of layers as $48$, we shift depth to the encoder.\nOur result with a much shallower decoder, only $8$ layers, but with $40$ encoder layers is as good as the $24$-layer configuration. \nMore stunningly, we were able to obtain our best result with the $36-12$ configuration with $20.6\\%$ WER, which is competitive with the best previous end-to-end work using data augmentation.\n\nThird, it was revealed that the combination of our regularization techniques (dropout, label-smoothing and stochastic networks) are additive with data augmentation, which further improved our result to $18.1\\%$ with the $36-12$ setup. This model, as far as we know, establishes the state-of-the-art result for the SWB benchmark among end-to-end ASR models, as shown in table~\\ref{tab:swb2}. Comparing to the best hybrid models with similar data constraints, our models outperformed on the CH test set while remaining competitive on the SWB test set without any additional language model training data. This result suggests the strong generalizability of the Stochastic Transformer. \n\nFinally, the experiments with similar depth suggest that self-attention performs competitively compared to LSTMs~\\cite{hochreiter1997long} or TDNNs~\\cite{waibel1989tdnn}. The former benefits strongly from building deep residual networks, in which our main finding shows that depth is crucial for using self-attention in the regimen of ASR. \n\n\n\n\n\n\n\n\\subsection{On TED-LIUM dataset}\n\n\\begin{table}[ht]\n\\caption{The Transformer results on the TED-LIUM test set using TED-LIUM 3 training set.}\n\\label{tab:ted}\n\t\\vspace{-0.2cm}\t\n\t\\setlength{\\tabcolsep}{4pt}\n\t\\centering\n\t\\begin{tabular}{lccc}\n\t\t\\toprule\n        \\textbf{Models} &  Test WER \\\\\n        CTC~\\cite{hernandez2018ted}   & 17.4       \\\\\n        CTC\/LM + speed perturbation~\\cite{hernandez2018ted}  & 13.7       \\\\\n        \\midrule\n        12Enc-12Dec (Ours)        & 14.2       \\\\\n        Stc. 12Enc-12Dec (Ours)   & 12.4       \\\\\n        Stc. 24Enc-24Dec (Ours)   & 11.3       \\\\\n        Stc. 36Enc-12Dec (Ours)   & \\textbf{10.6}  \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\vspace{-0.0cm}\n\\end{table}\n\nTable~\\ref{tab:ted} shows our result on TED-LIUM (version 3) dataset. \nWith a similar configuration to the SWB models, we outperformed a strong baseline which uses both an external language model trained on larger data than the available transcription and speed perturbation, using our model with 36 encoder layers and 12 decoder layers. This result continues the trend that these models benefit from a deeper encoder, and together with the stochastic residual connections we further improved WER by $21.8\\%$ relatively, from $14.2$ to $11.1\\%$. \nGiven the potential of the models~\\footnote{We did not have enough time for a thorough hyper-parameter search by the time of submission}, it is strongly suggested that better results can be obtained by further hyper-parameter optimization. \n\n\n\n\\section{Related Work}\nThe idea of using self-attention as the main component of ASR models has been investigated in various forms. \\cite{sperber2018self} combines self-attention with LSTMs, while \\cite{salazar2019self} uses self-attention as an alternative in CTC models. A variation of the Transformer has been applied to ASR with additional TDNN layers to downsample the acoustic signal~\\cite{dong2018speech}. Though self-attention has provided various benefits such as training speed or model interpretability, previous works have not been able to point out any enhancement in terms of performance. Our work provides a self-attention-only model and showed that with high capacity and regularization, such a network can exceed previous end-to-end models and approach the performance of hybrid systems. \n\\section{Conclusion}\nDirectly mapping from acoustics to text transcriptions is a challenging task for the S2S model.    \nTheoretically, self-attention can be used alternatively to TDNNs or LSTMs for acoustic modeling, and here we are the first demonstrate that the Transformer can be effective for ASR with the key is to setup very deep stochastic models. State-of-the-art results among end-to-end models on $2$ standard benchmarks are achieved and our networks are among the deepest configurations for ASR.\nFuture works will involve developing the framework under more realistic and challenging conditions such as real-time recognition, in which latency and streaming are crucial. \n\\section{Acknowledgements}\nThe work leading to these results has received funding from the European Union under grant agreement N\\textsuperscript{\\underline{o}}~825460 and the Federal Ministry of Education and Research (Germany) \/ DLR Projekttr\u00e4ger Bereich Gesundheit under grant agreement.  N\\textsuperscript{\\underline{o}}~01EF1803B. We are also grateful to have very useful comments from Elizabeth Salesky. \n\n\\bibliographystyle{IEEEtran}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\n\nOne of the most fundamental attributes of deep networks, and the reason for\ndriving its empirical success, is the ``Depth Efficiency\" result which states\nthat deeper models are exponentially more expressive than shallower models of\nsimilar size. Formal studies of Depth Efficiency include the early work on\nboolean or thresholded circuits~\\citep{Sipser83,Yao89,Hastad91,Hajnal:1993kl},\nand the more recent studies covering the types of networks used in practice\n\\citep{Pascanu:2013ue,Montufar:2014tb,Eldan:2015uc,expressive_power,\ngeneralized_decomp,Telgarsky:2016wk,Safran:2016te,Raghu:2016wn,BenPoole:2016vy}.\nWhat makes the Depth Efficiency attribute so desirable, is that it brings\nexponential increase in expressive power through merely a polynomial change in\nthe model, i.e. the addition of more layers. Nevertheless, depth is merely one\namong many architectural attributes that define modern networks. The deep\nnetworks used in practice consist of architectural features defined by various\nschemes of connectivity, convolution filter defined by size and stride, pooling\ngeometry and activation functions. Whether or not those relate to expressive\nefficiency, as depth has proven to be, remains an open question.\n\nIn order to study the effect of network design on expressive efficiency we\nshould first define \"efficiency\" in broader terms. Given two network\narchitectures~$A$ and~$B$, we say that architecture~$A$ is expressively\nefficient with respect to architecture~$B$, if the following two conditions\nhold: \\emph{(i)}~any function ${\\mathbf h}$ realized by~$B$ of size~$r_B$ can be realized\n(or approximated) by $A$ with size $r_A \\in{\\mathcal O}(r_B)$; \\emph{(ii)}~there exist a\nfunction ${\\mathbf h}$ realized by~$A$ with size $r_A$, that cannot be realized (or\napproximated) by~$B$, unless  $r_B \\in\\Omega(f(r_A))$ for some super-linear\nfunction~$f$. The exact definition of the sizes $r_A$ and $r_B$ depends on the\nmeasurement we care about, e.g. the number of parameters, or the number of\n``neurons''. The nature of the function~$f$ in condition~\\emph{(ii)} determines\nthe type of efficiency taking place~--~if~$f$ is exponential then\narchitecture~$A$ is said to be exponentially efficient with respect to\narchitecture~$B$, and if~$f$ is polynomial so is the expressive efficiency.\nAdditionally, we say $A$ is \\emph{completely efficient} with respect to $B$, if\ncondition (ii) holds not just for some specific functions (realizable by $A$),\nbut for all functions other than a negligible set.\n\nIn this paper we study the efficiency associated with the architectural\nattribute of convolutions, namely the size of convolutional filters (receptive\nfields) and more importantly its proportion to their stride. We say that a\nnetwork architecture is of the \\emph{non-overlapping} type when the size of the\nlocal receptive field in each layer is equal to the stride. In that case, the\nsets of pixels participating in the computation of each two neurons in the same\nlayer are completely separated. When the stride is smaller than the receptive\nfield we say that the network architecture is of the \\emph{overlapping} type. In\nthe latter case, the \\emph{overlapping degree} is determined by the \\emph{total}\nreceptive field and stride projected back to the input layer~--~the implication\nbeing that for the overlapping architecture the total receptive field and stride\ncan grow much faster than with the non-overlapping case.\n\nAs several studies have shown, non-overlapping convolutional networks do have\nsome theoretical merits. Namely, non-overlapping networks are\nuniversal~\\citep{expressive_power,generalized_decomp}, i.e. they can approximate\nany function given sufficient resources, and in terms of optimization, under\nsome conditions they actually possess better convergence guaranties than\noverlapping networks. Despite the above, there are only few instances of\nstrictly non-overlapping networks used in practice (e.g. \\citet{tmm,\nvan2016wavenet}), which raises the question of \\textbf{why are non-overlapping\narchitectures so uncommon?} Additionally, when examining the kinds of\narchitectures typically used in recent years, which employ a mixture of both\noverlapping and non-overlapping layers, there is a trend of using ever smaller\nreceptive fields, as well as non-overlapping layers having an ever increasing\nrole~\\citep{NiN,Springenberg:2014tx,Szegedy:2014tb}. Hence, the most common\nnetworks used practice, though not strictly non-overlapping, are increasingly\napproaching the non-overlapping regime, which raises the question of \\textbf{why\nhaving just slightly overlapping architectures seems sufficient for most tasks?}\n\nIn the following sections, we will shed some light on these questions by \nanalyzing the role of overlaps through a surrogate class of convolutional\nnetworks called Convolutional Arithmetic\nCircuits~(ConvACs)~\\citep{expressive_power}~--~instead of non-linear activations\nand average\/max pooling layers, they employ linear activations and product\npooling. ConvACs, as a theoretical framework to study ConvNets, have been the\nfocused of several works, showing, amongst other things, that many of the\nresults proven on this class are typically transferable to standard ConvNets as\nwell~\\citep{generalized_decomp,inductive_bias}. Though prior works on ConvACs\nhave only considered non-overlapping architectures, we suggest a natural\nextension to the overlapping case that we call Overlapping ConvACs. In our\nanalysis, which builds on the known relation between ConvACs and tensor\ndecompositions, we prove that overlapping architectures are in fact completely\nand exponentially more efficient than non-overlapping ones, and that their\nexpressive capacity is directly related to their \\emph{overlapping degree}.\nMoreover, we prove that having even a limited amount of overlapping is\nsufficient for attaining this exponential separation. To further ground our\ntheoretical results, we demonstrate our findings through experiments with\nstandard ConvNets on the CIFAR10 image classification dataset.\n\n\\section{Overlapping Convolutional Arithmetic Circuits}\n\\label{sec:overlapping_convac}\n\nIn this section, we introduce a class of convolutional networks referred to as\nOverlapping Convolutional Arithmetic Circuits, or Overlapping ConvACs for short.\nThis class shares the same architectural features as standard ConvNets,\nincluding some that have previously been overlooked by similar attempts to model\nConvNets through ConvACs, namely, having any number of layers and unrestricted\nreceptive fields and strides, which are crucial for studying overlapping\narchitectures. For simplicity, we will describe this model only for the case of\ninputs with two spatial dimensions, e.g. color images, and limiting the\nconvolutional filters to the shape of a square.\n\n\\begin{wrapfigure}{r}{0.5\\textwidth} \n\\centering\n\\includegraphics[width=\\linewidth]{figures\/gc_layer_new}\n\\caption{An illustration of a GC Layer.}\n\\label{fig:gc_layer}\n\\end{wrapfigure}\n\nWe begin by presenting a broad definition of a Generalized Convolutional~(GC)\nlayer as a fusion of a $1{\\times}1$ linear operation with a pooling\nfunction~--~this view of convolutional layers is motivated by the\nall-convolutional architecture~\\citep{Springenberg:2014tx}, which replaces all\npooling layers with convolutions with stride greater than 1. The input to a\nGC layer is a 3-order tensor (multi-dimensional array), having width\nand height equal to $H^{(\\text{in})} \\in {\\mathbb N}$ and depth $D^{(\\text{in})} \\in {\\mathbb N}$,\nalso referred to as channels, e.g. the input could be a 2D image with RGB color\nchannels. Similarly, the output of the layer has width and height equal to\n$H^{(\\text{out})} \\in {\\mathbb N}$ and $D^{(\\text{out})} \\in {\\mathbb N}$ channels, where\n$H^{(\\text{out})} = \\frac{H^{(\\text{in})}}{S}$ for $S \\in {\\mathbb N}$ that is referred\nto as the \\emph{stride}, and has the role of a sub-sampling operation. Each\nspatial location $(i,j)$ at the output of the layer corresponds to a 2D window\nslice of the input tensor of size $R \\times R \\times D^{(\\text{in})}$, extended\nthrough all the input channels, whose top-left corner is located exactly at\n$(i\\cdot S, j\\cdot S)$, where $R \\in {\\mathbb N}$ is referred to as its \\emph{local\nreceptive field}, or filter size. For simplicity, the parts of window slices\nextending beyond the boundaries have zero value. Let\n${\\mathbf y} \\in {\\mathbb R}^{D^{(\\text(out)}}$ be a vector representing the channels at some\nlocation of the output, and similarly, let\n${\\mathbf x}^{(1)},\\ldots,{\\mathbf x}^{(R^2)} \\in {\\mathbb R}^{D^{(\\text{in})}}$ be the set of vectors\nrepresenting the slice, where each vector represents the channels at its\nrespective location inside the $R \\times R$ window, then the operation of a\nGC layer is defined as follows:\n\\begin{equation*}\n    {\\mathbf y} = g(W^{(1)}{\\mathbf x}^{(1)}+{\\mathbf b}^{(1)}, \\ldots, W^{(R^2)}{\\mathbf x}^{(R^2)}+{\\mathbf b}^{(R^2)}),\n\\end{equation*}\nwhere $W^{(1)},\\ldots,W^{(R^2)} \\in {\\mathbb R}^{D^{(out)} \\times D^{(in)}}$\nand ${\\mathbf b}^{(1)}, \\ldots, {\\mathbf b}^{(R^2)} \\in {\\mathbb R}^{D^{(out)}}$ are referred to as the\nweights and biases of the layer, respectively, and\n$g:{\\mathbb R}^{D^{(out)}} \\times \\cdots \\times {\\mathbb R}^{D^{(out)}} \\to {\\mathbb R}^{D^{(out)}}$ is\nsome point-wise pooling function. See fig.~\\ref{fig:gc_layer} for an\nillustration of the operation a GC layer performs.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.75\\linewidth]{figures\/gc_network}\n\\caption{An illustration of a Generalized Convolutional Network.}\n\\label{fig:gc_network}\n\\end{figure}\n\nWith the above definitions, a GC network is simply a sequence of $L$ GC layers,\nwhere for $l \\in [L] \\equiv \\{1,\\ldots,L\\}$, the $l$'th layer is specified by a\nlocal receptive field $R^{(l)}$, a stride $S^{(i)}$, $D^{(l)}$ output channels,\nparameters $\\theta^{(l)}$, and a pooling function $g^{(l)}$. For classification\ntasks, the output of the last layer of the network typically has $1{\\times}1$\nspatial dimensions, i.e. a vector, where each output channel\n$y \\in [Y] \\equiv [D^{(L)}]$ represents the score function of the $y$'th class,\ndenoted by ${\\mathbf h}_y$, and inference is perform by $y^* = \\arg\\max_y h_y(X)$.\nOftentimes, it is common to consider the output of the very first layer of a\nnetwork as a low-level feature representation of the input, which is motivated\nby the observation that these learned features are typically shared across\ndifferent tasks and datasets over the same domain (e.g. edge and Gabor filters\nfor natural images). Hence, we treat this layer as a separate fixed\n``zeroth'' convolutional layer referred to as the \\emph{representation} layer,\nwhere the operation of the layer can be depicted as applying a set of fixed\nfunctions $\\{f_d:{\\mathbb R}^s\\to{\\mathbb R}\\}_{d=1}^M$ to the window slices denoted by\n${\\mathbf x}_1,\\ldots,{\\mathbf x}_N \\in {\\mathbb R}^s$, i.e. the entries of the output tensor of this layer\nare given by $\\{f_d({\\mathbf x}_i)\\}_{d \\in [M], i\\in [N]}$. With these notations, the\noutput of a GC network can be viewed as a function ${\\mathbf h}_y({\\mathbf x}_1,\\ldots,{\\mathbf x}_N)$. The\nentire GC network is illustrated in fig.~\\ref{fig:gc_network}.\n\nGiven a non-linear\npoint-wise activation function $\\sigma(\\cdot)$ (e.g. ReLU), then setting all\npooling functions to average pooling followed by the activation, i.e.\n$g({\\mathbf x}^{(1)},\\ldots,{\\mathbf x}^{(R^2)})_c=\\sigma\\left(\\sum_{i=1}^{R^2} x^{(i)}_c\\right)$\nfor $c \\in [D^{(\\text{out})}]$,\ngive rise to the common all-convolutional network with $\\sigma(\\cdot)$\nactivations, which served as the initial motivation for our formulation.\nAlternatively, choosing instead a product pooling function, i.e.\n$g({\\mathbf x}^{(1)},\\ldots,{\\mathbf x}^{(R^2)})_c = \\prod_{i=1}^{R^2} x^{(i)}_c$ for\n$c \\in [D^{(\\text{out})}]$, results in an Arithmetic Circuit, i.e.\na circuit containing just product and sum operations, hence it is referred to\nas a Convolutional Arithmetic Circuit, or ConvAC. It is important to emphasize\nthat ConvACs, as originally introduced by \\citet{expressive_power}, are\ntypically described in a very different manner, through the language of tensor\ndecompositions (see app.~\\ref{app:convac} for background). Since vanilla ConvACs\ncan be seen as an alternating sequence of $1{\\times}1$ convolutions and\nnon-overlapping product pooling layers, then the two formulations coincide when\nall GC layers are non-overlapping, i.e. for all $l \\in [L]$, $R^{(l)}=S^{(l)}$.\nIf, however, some of the layers are overlapping, i.e. there exists $l \\in [L]$\nsuch that $R^{(l)} > S^{(l)}$, then our formulation through GC layers diverges,\nand give rise to what we call \\emph{Overlapping ConvACs}.\n\nGiven that our model is an extension of the ConvACs framework, it inherits many\nof its desirable attributes. First, it shares most of the same traits as modern\nConvNets, i.e. locality, sharing and pooling. Second, it can be shown to form a\nuniversal hypotheses space~\\citep{expressive_power}. Third, its underlying\noperations lend themselves to mathematical analysis based on measure theory and\ntensor analysis~\\citep{expressive_power}. Forth, through the concept of\ngeneralized tensor decompositions~\\citep{generalized_decomp}, many of the\ntheoretical results proven on ConvACs could be transferred to standard ConvNets\nwith ReLU activations. Finally, from an empirical perspective, they tend to work\nwell in many practical settings, e.g. for optimal classification with missing\ndata~\\citep{tmm}, and for compressed networks~\\citep{simnets2}.\n\nWhile we have just established that the non-overlapping GC Network with a\nproduct pooling function is equivalent to vanilla ConvACs, one might wonder if\nusing overlapping layers instead could diminish what these overlapping networks\ncan represent. We show that not only is it not the case, but prove the more\ngeneral claim that a network of a given architecture can realize exactly the\nsame functions as networks using smaller local receptive fields, which includes\nthe non-overlapping case.\n\\begin{proposition}\\label{prop:nothing_to_lose}\n    Let $A$ and $B$ be two GC Networks with a product pooling function. If\n    the architecture of $B$ can be derived from $A$ through the removal of\n    layers with $1{\\times}1$ stride, or by decreasing the local receptive field\n    of some of its layers, then for any choice of parameters for $B$, there\n    exists a matching set of parameters for $A$, such that the function\n    realized by $B$ is exactly equivalent to~$A$. Specifically, $A$ can\n    realize any non-overlapping network with the same order of strides\n    (excluding $1{\\times}1$ strides).\n\\end{proposition}\n\\begin{proof}[Proof sketch]\n    This follows from two simple claims: (i)~a GC layer can produce an output\n    equivalent to that of a GC layer with a smaller local receptive field, by\n    ``zeroing'' its weights beyond the smaller local receptive field; and (ii)\n    GC layers with $1{\\times}1$ receptive fields can be set such that\n    their output is equal to their input, i.e. realize the identity function.\n    With these claims, the local receptive fields of $A$ can be effectively\n    shrank to match the local receptive fields of $B$, and any additional layers\n    of $A$ with stride $1{\\times}1$ could be set such that they are realizing\n    the identity mapping, effectively ``removing'' them from $A$. See\n    app.~\\ref{app:proofs:nothing_to_lose} for a complete proof.\n\\end{proof}\nProposition~\\ref{prop:nothing_to_lose} essentially means that overlapping\narchitectures are just as expressive as non-overlapping ones of similar\nstructure, i.e. same order of non-unit strides. As we recall, this satisfies the\nfirst condition of the efficiency property introduced in sec.~\\ref{sec:intro},\nand does so regardless if we measure the size of a network as the number of\nparameters, or the number of ``neurons''\\footnote{We take here the broader\ndefinition of a ``neuron'', as any one of the scalar values comprising the\noutput array of an arbitrary layer in a network. In the case the output array is\nof width and height equal to $H$ and $C$ channels, then the number of such\n``neurons'' for that layer is $H^2 \\cdot C$.}. In the following section we will\ncover the preliminaries required to show that overlapping networks actually lead\nto an increase in expressive capacity, which under some settings results in an\nexponential gain, proving that the second condition of expressive efficiency\nholds as well.\n\n\\section{Analyzing Expressive Efficiency Through Grid Tensors}\n\\label{sec:efficiency_analysis}\n\nIn this section we describe our methods for analyzing the expressive efficiency\nof overlapping ConvACs that lay the foundation for stating our theorems.\nA minimal background on tensor analysis required to follow our work can be found\nin sec.~\\ref{sec:efficiency_analysis:pre}, followed by presenting our\nmethods in sec.~\\ref{sec:efficiency_analysis:bounds}.\n\n\\subsection{Preliminaries}\n\\label{sec:efficiency_analysis:pre}\n\nIn this sub-section we cover the minimal background on tensors analysis required\nto understand our analysis. A tensor ${\\mathcal A} \\in{\\mathbb R}^{M_1 \\otimes \\cdots \\otimes M_N}$\nof order $N$ and dimension $M_i$ in each mode $i \\in [N] \\equiv \\{1,\\ldots,N\\}$,\nis a multi-dimensional array with entries ${\\mathcal A}_{d_1,\\ldots,d_N} \\in {\\mathbb R}$ for all\n$i \\in [N]$ and $d_i \\in [M_i]$. For simplicity, henceforth we assume that all\ndimensions are equal, i.e. $M \\equiv M_1 = \\ldots = M_N$. One of the central\nconcepts in tensor analysis is that of \\emph{tensor matricization}, i.e.\nrearranging its entries to the shape of a matrix. Let\n$P \\mathbin{\\mathaccent\\cdot\\cup} Q=[N]$ be a disjoint partition of its indices, such that\n$P = \\{p_1,\\ldots,p_{|P|}\\}$ with $p_1< \\ldots< p_{|P|}$,\nand $Q = \\{q_1, \\ldots, q_{|Q|}\\}$ with $q_1 < \\ldots < q_{|Q|}$. The\nmatricization of ${\\mathcal A}$ with respect to the partition\n$P \\mathbin{\\mathaccent\\cdot\\cup} Q$, denoted by $\\mat{{\\mathcal A}}_{P,Q}$, is the $M^{|P|}$-by-$M^{|Q|}$ matrix\nholding the entries of ${\\mathcal A}$, such that for all $i \\in [N]$ and $d_i \\in [M]$ the \nentry $A_{d_1, \\ldots, d_N}$ is placed in row index\n${1 + \\sum_{t=1}^{|P|} (d_{p_t} - 1) M^{|P| - t}}$ and column index\n${1 + \\sum_{t=1}^{|Q|} (d_{q_t} - 1) M^{|Q| - t}}$. Lastly, the tensors we\nstudy in this article originate by examining the values of some given function\nat a set of predefined points and arranging them in a tensor referred to as\nthe \\emph{grid tensor} of the function. Formally, let\n$f:{\\mathbb R}^s \\times \\ldots \\times {\\mathbb R}^s \\to {\\mathbb R}$ be a function, and let\n$\\{{\\mathbf x}^{(1)}, \\ldots, {\\mathbf x}^{(M)} \\in {\\mathbb R}^s\\}$ be a set of vectors called\n\\emph{template vectors}, then the grid tensor of $f$ is denoted by\n${\\mathcal A}(f) \\in {\\mathbb R}^{M \\otimes \\ldots \\otimes M}$ and defined by\n${\\mathcal A}(f)_{d_1,\\ldots,d_N} = f({\\mathbf x}^{(d_1)}, \\ldots, {\\mathbf x}^{(d_N)})$ for all\n$d_1,\\ldots,d_N \\in [M]$.\n\n\\subsection{Bounding the Size of Networks via Grid Tensors}\n\\label{sec:efficiency_analysis:bounds}\n\nWe begin with a discussion on how to have a well-defined measure of efficiency.\nWe wish to compare the efficiency of non-overlapping ConvACs to overlapping\nConvACs, for a fixed set of $M$ representation functions (see\nsec.~\\ref{sec:overlapping_convac} for definitions). While all functions\nrealizable by non-overlapping ConvACs with shared representation functions lay\nin the same function subspace (see \\citet{expressive_power}), this is not the\ncase for overlapping ConvACs, which can realize additional functions outside the\nsub-space induced by non-overlapping ConvACs. We cannot therefore compare both\narchitectures directly, and need to compare them through an auxiliary objective.\nFollowing the work of \\citet{generalized_decomp}, we instead compare\narchitectures through the concept of grid tensors, and specifically, the grid\ntensor defined by the output of a ConvAC, i.e. the tensor ${\\mathcal A}({\\mathbf h})$ for\n${\\mathbf h}({\\mathbf x}_1,\\ldots,{\\mathbf x}_N)$. Unlike with the ill-defined nature of directly comparing\nthe functions of realized by ConvACs, \\citet{generalized_decomp} proved that\nassuming the fixed representation functions are linearly independent, then there\nexists template vectors ${\\mathbf x}^{(1)},\\ldots,{\\mathbf x}^{(M)}$, for which any\nnon-overlapping ConvAC architecture could represent all possible grid tensors\nover these templates, given sufficient number of channels at each layer. More\nspecifically, if $F_{ij} = f_i({\\mathbf x}^{(j)})$, then these template vector are\nchosen such that $F$ is non-singular. Thus, once we fix a set of linearly\nindependent representation functions, we can compare different ConvACs, whether\noverlapping or not, on the minimal size required for them to induce the same\ngrid tensor, while knowing such a finite number always exists.\n\nOne straightforward direction for separating between the expressive efficiency\nof two network architectures A and B is by examining the ranks of their\nrespective matricized grid tensors. Specifically, Let ${\\mathcal A}({\\mathbf h}^{(A)})$ and\n${\\mathcal A}({\\mathbf h}^{(B)})$ denote the grid tensors of A and B, respectively, and let $(P,Q)$\nbe a partition of $[N]$, then we wish to find an upper-bound on the rank of\n$\\mat{{\\mathcal A}({\\mathbf h}^{(A)})}_{P,Q}$ as a function of its size on one hand, while showing\non the other hand that $\\rank{\\mat{{\\mathcal A}({\\mathbf h}^{(B)})}_{P,Q}}$ can be significantly\ngreater. One benefit of studying efficiency through a matrix rank is that not\nonly we attain separation bounds for exact realization, but also immediately\ngain access to approximation bounds by examining the singular values of the\nmatricized grid tensors. This brings us to the following lemma, which connects\nupper-bounds that were previously found for non-overlapping\nConvACs~\\citep{inductive_bias}, with the grid tensors induced by them (see\napp.~\\ref{app:proofs:preliminaries} for proof):\n\\begin{lemma} \\label{lemma:mat_rank_bound}\n    Let $h_y({\\mathbf x}_1,\\ldots,{\\mathbf x}_N)$ be a score function of a non-overlapping\n    ConvAC with a fixed set of $M$ linearly independent and continuous\n    representation functions, and $L$ GC layers. Let $(P,Q)$ be a partition\n    dividing the spatial dimensions of the output of the representation layer\n    into two equal parts, either along the horizontal or vertical axis, referred\n    to as the ``left-right'' and ``top-bottom'' partitions, respectively. Then,\n    for any template vectors such that $F$ is non-singular and for any choice of\n    the parameters of the network, it holds that\n    $\\rank{\\mat{{\\mathcal A}({\\mathbf h}_y)}_{P,Q}} \\leq D^{(L-1)}$.\n\\end{lemma}\n\nLemma~\\ref{lemma:mat_rank_bound} essentially means that it is sufficient to show\nthat overlapping ConvACs can attain ranks super-polynomial in their size to\nprove they are exponentially efficient with respect to non-overlapping ConvACs.\nIn the next section we analyze how the overlapping degree is related to the\nrank, and under what cases it leads to an exponentially large rank.\n\n\\section{The Expressive Efficiency of Overlapping Architectures}\n\\label{sec:overlaps_efficiency}\n\nIn this section we analyze the expressive efficiency of overlapping\narchitectures. We begin by defining our measures\nof the overlapping degree that will used in our claims, followed by presenting\nour main results in sec.~\\ref{sec:main_results}. For the sake of\nbrevity, an additional set of results, in light of the recent work by\n\\citet{inductive_bias} on ``Pooling Geometry'', is deferred to\napp.~\\ref{app:pooling_geometry}.\n\n\\subsection{The Overlapping Degree of a Network}\n\\label{sec:overlapping_degree}\n\n\\begin{wrapfigure}{r}{0.5\\textwidth} \n\\vspace{-3mm}\n\\centering\n\\includegraphics[width=\\linewidth]{figures\/total_properties}\n\\caption{Illustrating the total receptive field and total stride\n         attributes for the $L$'th layer, which could be seen as the projected\n         receptive field and stride with respect to the input layer. Together,\n         they capture the overlapping degree of a network.}\n\\label{fig:total_properties}\n\\vspace{-3mm}\n\\end{wrapfigure}\n\n\nTo analyze the efficiency of overlapping architectures, we will first formulate\nmore rigorously the measurement of the overlapping degree of a given\narchitecture. As mentioned in sec.~\\ref{sec:intro}, we do so by defining the\nconcepts of the \\emph{total receptive field} and \\emph{total stride} of a given\nlayer $l\\in[L]$, denoted by $T_R^{(l)}$ and $T_S^{(l)}$, respectively.\nBoth measurements could simply be thought of as projecting the accumulated local\nreceptive fields (or strides) to the the first layer, as illustrated in\nfig.~\\ref{fig:total_properties}, which represent a type of global statistics\nof the architecture. However, note that proposition~\\ref{prop:nothing_to_lose}\nentails that a given architecture could have a smaller \\emph{effective} total\nreceptive field, for some settings of its parameters. This leads us to define\nthe $\\alpha$-minimal total receptive field, for any $\\alpha\\in{\\mathbb R}_+$, as the\nsmallest effective total receptive field still larger than $\\alpha$, which we\ndenote by $T_R^{(l,\\alpha)}$. The exact definitions of the above concepts\nare formulated as follows:\n\\begin{align}\n    \\label{eq:total_stride}\n    T_S^{(l)} \\equiv T_S^{(l)}(S^{(1)},\\ldots,S^{(l)})\n        &\\equiv \\begin{cases}\n            \\prod_{i=1}^l S^{(i)} & l \\geq 1\\\\\n            1 & l = 0\n        \\end{cases} \\\\\n    \\label{eq:total_receptive}\n    T_R^{(l)} \\equiv T_R^{(l)}(R^{(1)},S^{(1)},\\ldots,R^{(l)},S^{(l)})\n        &\\equiv R^{(l)} \\cdot T_S^{(l-1)} +\n         \\sum\\nolimits_{k=1}^{l-1}\\left(R^{(k)}-S^{(k)}\\right)\\cdot T_S^{(k-1)} \\\\\n    \\label{eq:minimal_receptive}\n    T_R^{(l,\\alpha)}\n        \\equiv T_R^{(l,\\alpha)}(R^{(1)},S^{(1)},\\ldots,R^{(l)},S^{(l)})\n        &\\equiv \\smashoperator[r]{\\argmin_{\\substack{\n            \\forall i\\in[l], S^{(i)} \\leq t_i \\leq R^{(i)} \\\\\n            T_R^{(l)}(t_1,S^{(1)},\\ldots,t_l,S^{(l)}) > \\alpha}}}\n         \\quad\\,\\,T_R^{(l)}(t_1,S^{(1)},\\ldots,t_l,S^{(l)})\n\\end{align}\nwhere we omitted the arguments of $T_S^{(l-1)}$ and $T_S^{(k-1)}$ for\nthe sake of visual compactness.\n\nNotice that for non-overlapping networks the total receptive field always\nequals the total stride, and that only at the end of the network, after the\nspatial dimension collapses to $1{\\times}1$, does the the total receptive field\ngrow to encompass the entire size of the representation layer. For overlapping\nnetworks this is not the case, and the total receptive field could grow much\nfaster. Intuitively, this means that values in regions of the input layer that\nare far apart would be combined by non-overlapping networks only near the last\nlayers of such networks, and thus non-overlapping networks are effectively\nshallow in comparison to overlapping networks. Base on this intuition, in the\nnext section we analyze networks with respect to the point at which their total\nreceptive field is large enough.\n\n\\subsection{Main Results}\\label{sec:main_results}\n\nWith all the preliminaries in place, we are ready to present our main result:\n\\begin{theorem}\\label{thm:main_overlaps}\n    Assume a ConvAC with a fixed representation layer having $M$ output channels\n    and both width and height equal to $H$, followed by $L$ GC layers, where the\n    $l$'th layer has a local receptive field $R^{(l)}$, a stride $S^{(l)}$, and\n    $D^{(l)}$ output channels. Let $K \\in [L]$ be a layer with a total receptive\n    field $T_R^{(K)} \\equiv T_R^{(K)}(R^{(1)},S^{(1)},\n    \\ldots,R^{(K)},S^{(K)})$, such that $T_R^{(K)}>\\frac{H}{2}$.\n    Then, for any choice\n    of parameters, except a null set (with respect to the Lebesgue\n    measure), and for any template vectors such that $F$ is non-singular, the\n    following equality holds:\n    \\begin{align}\\label{eq:lower_bound}\n        \\rank{\\mat{{\\mathcal A}({\\mathbf h}_y)}_{P,Q}} \\geq D^{\\left\\lfloor \\frac{H - T_R^{(K, \\left\\lfloor \\nicefrac{H}{2} \\right\\rfloor)}}{T_S^{(K)}} +1 \\right\\rfloor \\cdot \\left\\lceil \\frac{H}{T_S^{(K)}} \\right\\rceil}\n    \\end{align}\n    where $(P,Q)$ is either the ``left-right'' or the ``top-bottom'' partitions\n    and ${D \\equiv \\min\\{M,D^{(K)},\\frac{1}{2} \\min_{1\\leq l\\leq K} D^{(l)}\\}}$.\n\\end{theorem}\n\\begin{proof}[Proof sketch]\n    Because the entries of the matricized grid tensors are polynomials in the\n    parameters, then according to a lemma by \\citet{tmm}, if there is a single\n    example that attains the above lower-bound on the rank, then it occurs\n    almost everywhere with respect to the Lebesgue measure on the Euclidean\n    space of the parameters.\n\n    Given the last remark, the central part of our proof is simply the\n    construction of such an example. First we find a set of parameters for the\n    simpler case where the first GC layer is greater than a quarter of the\n    input, satisfying the conditions of the theorem. The motivation behind the\n    specific construction is the pairing of indices from each side of\n    the partition, such that they are both in the same local receptive field,\n    and designing the filters such that the output of each local application of\n    them defines a mostly diagonal matrix of rank $D$, with respect to\n    these two indices. The rest of the parameters are chosen such that the\n    output of the entire network results in a product of the entries of these\n    matrices. Under matricization, this results in a matrix who is\n    equivalent\\footnote{Two matrices are equivalent if one could be converted\n    to the other by elementary row or column operations.} to a\n    Kronecker product of mostly diagonal matrices. Thus, the matricization rank\n    is equal to the product of the ranks of these matrices, which results in the\n    exponential form of eq.~\\ref{eq:lower_bound}.\n    Finally, we extend the above example to the general case, by realizing the\n    operation of the first layer of the above example through multiple layers\n    with small local receptive fields. See app.~\\ref{app:proofs:preliminaries}\n    for the definitions and lemmas we rely on, and see\n    app.~\\ref{app:proofs:main_overlaps} for a complete proof.\n\\end{proof}\nCombined with Lemma~\\ref{lemma:mat_rank_bound}, it results in the following\ncorollary:\n\\begin{corollary}\n    Under the same setting as theorem~\\ref{thm:main_overlaps}, and for all\n    choices of parameters of an overlapping ConvAC, except a negligible set,\n    any non-overlapping ConvAC that realizes (or approximates) the same grid\n    tensor must be of size at least:\n    \\begin{equation*}\n        D^{\\left\\lfloor \\frac{H - T_R^{(K, \\left\\lfloor \\nicefrac{H}{2} \\right\\rfloor)}}{T_S^{(K)}} +1 \\right\\rfloor \\cdot \\left\\lceil \\frac{H}{T_S^{(K)}} \\right\\rceil} .\n    \\end{equation*}\n\\end{corollary}\n\nWhile the complexity of the generic lower-bound above might seem\nincomprehensible at first, its generality gives us the tools to analyze\npractically any kind of feed-forward architecture. As an example, we can analyze\nthe lower bound for the well known GoogLeNet\narchitecture~\\citep{Szegedy:2014tb}, for which the lower bound equals $32^{98}$,\nmaking it clear that using a non-overlapping architecture for this case is\ninfeasible. Next, we will focus on specific cases for which we can derive more\nintelligible lower bounds.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.8\\linewidth]{figures\/start_with_big_conv}\n\\caption{A network architectures beginning with large local receptive fields\n    greater than $\\nicefrac{N}{2}$ and at least $M$ output channels. According\n    to theorem~\\ref{thm:main_overlaps}, for almost all choice of parameters\n    we obtain a function that cannot be approximated by a non-overlapping\n    architecture, if the number of channels in its next to last layer is\n    less than $M^{\\frac{H^2}{2}}$.}\n\\label{fig:start_with_big_conv}\n\\end{figure}\n\nAccording to theorem~\\ref{thm:main_overlaps}, the lower bound depends on the\nfirst layer for which its total receptive field is greater than a quarter of the \ninput. As mentioned in the previous section, for non-overlapping networks this\nonly happens after the spatial dimension collapses to $1{\\times}1$, which\nentails that both the total receptive field and total stride would be equal to\nthe width $H$ of the representation layer, and substituting this values in\neq.~\\ref{eq:lower_bound} results simply in $D$~--~trivially meaning that to\nrealize one non-overlapping network by another non-overlapping network, the next\nto last layer must have at least half the channels of the target network.\n\nOn the other extreme, we can examine the case where the first GC layer has a\nlocal receptive field $R$ greater than a quarter of its input, i.e.\n$R > \\nicefrac{H}{2}$. Since the layers following the first GC layer do not\naffect the lower bound in this case, it applies to any arbitrary sequence of\nlayers as illustrated in fig.~\\ref{fig:start_with_big_conv}. For simplicity\nwe will also assume that the stride $S$ is less than $\\nicefrac{H}{2}$, and that\n$\\frac{H}{2}$ is evenly divided by $S$. In this case the $\\frac{H}{2}$-minimal\nreceptive field equals to $\\frac{H}{2} + 1$, and thus the lower bound results in\n$D^{\\frac{H^2}{2S}}$. Consider the case of $D = M$ and $S=1$, then a\nnon-overlapping architecture that satisfies this lower bound is of the order of\nmagnitude at which it could already represent any possible grid tensor. This\ndemonstrate our point from the introduction, that through a a polynomial change\nin the architecture, i.e. increasing the receptive field, we get an exponential\nincrease in expressivity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{figures\/alt_conv_pool_net}\n\\caption{The common network architecture of alternating $B{\\times}B$ ``conv''\n    and $2{\\times}2$ ``pooling'' layers. If $B \\leq \\nicefrac{H}{5}{+}1$ and\n    $D^{(l)} \\geq 2M$ for all $1 \\leq l < L$, then the lower bound of\n    theorem~\\ref{thm:main_overlaps} for this network results in\n    $M^{\\frac{(2B - 1)^2}{4}}$.\n}\n\\label{fig:alt_conv_pool_net}\n\\end{figure}\n\nThough the last example already demonstrates that a polynomially sized\noverlapping architecture could lead to an exponential separation, in practice,\nemploying such large convolutions is very resource intensive. The common best\npractice is to use multiple small local receptive fields of size\n$B \\times B$, where the typical values are $B=3$ or $B=5$, separated by a\n$2 \\times 2$ ``pooling'' layers, i.e. layers with both stride and local\nreceptive field equal to $2 \\times 2$. For simplicity, we assume that $H = 2^L$\nfor some $L \\in {\\mathbb N}$. See fig.~\\ref{fig:alt_conv_pool_net} for an illustration of\nsuch a network. Analyzing the above network with theorem~\\ref{thm:main_overlaps}\nresults in the following proposition:\n\\begin{proposition}\\label{prop:common_case}\n    Consider a network  comprising a sequence of GC blocks, each block begins\n    with a layer whose local receptive field is $B {\\times} B$ and its stride\n    $1{\\times}1$, followed by a layer with local receptive field $2 {\\times} 2$\n    and stride $2 {\\times} 2$, where the output channels of all layers are at\n    least $2M$, and the spatial dimension of the representation layer is\n    $H {\\times} H$ for $H{=}2^L$. Then, the lower bound describe by\n    eq.~\\ref{eq:lower_bound} for the above network is greater than or equal to:\n    \\begin{align*}\n    \\tau(B,H) &\\equiv M^{\\frac{(2B-1)^2}{2} \\cdot \\left(1+\\frac{2B-2}{H}\\right)^{-2}} \n    = M^{\\frac{H^2}{2}\\cdot \\left(1 + \\frac{H-1}{2B-1} \\right)^{-2}},\n    \\end{align*}\n    whose limits are $\\lim_{B\\to\\infty} \\tau(B,H) = M^\\frac{H^2}{2}$ and\n    $\\lim_{H\\to\\infty} \\tau(B,H) = M^\\frac{(2B-1)^2}{2}$. Finally, assuming\n    $B \\leq \\frac{H}{5} + 1$, then $\\tau(B,H) \\geq M^{\\frac{(2B-1)^2}{4}}$.\n\\end{proposition}\n\\begin{proof}[Proof sketch]\n    We first find a closed-form expression for the total receptive field and\n    stride of each of the $B{\\times}B$ layers in the given network. We\n    then show that for layers whose total receptive field is greater than\n    $\\frac{H}{2}$, its $\\alpha$-minimal total receptive field, for\n    $\\alpha{=}\\frac{H}{2}$, is equal to $\\frac{H}{2}{+}1$. We then use the above\n    to find the first layer who satisfies the conditions of\n    theorem~\\ref{thm:main_overlaps}, and then use our closed-forms expressions\n    to simplify the general lower bound for this case. See\n    app.~\\ref{app:proofs:common_case} for a complete proof.\n\\end{proof}\nIn particular, for the typical values of $M=64$, $B=5$, and $H \\geq 20$, the\nlower bound is at least $64^{20}$, which demonstrates that even having a small\namount of overlapping already leads to an exponential separation from the\nnon-overlapping case. When $B$ grows in size, this bound approaches the earlier\nresult we have shown for large local receptive fields encompassing more than a\nquarter of the image. When $H$ grows in size, the lower bound is dominated\nstrictly by the local receptive fields. Also notice that based on\nproposition~\\ref{prop:common_case}, we could also derive a respective lower\nbound for a network following VGG style architecture~\\citep{Simonyan:2014ws},\nwhere instead of a single convolutional layer before every ``pooling'' layer, we\nhave $K$ layers, each with a local receptive field of $C \\times C$. Under this\ncase, it is trivial to show that the bound from\nproposition~\\ref{prop:common_case} holds for $B = K \\cdot (C-1) + 1$, and under\nthe typical values of $C=3$ and $K=2$ it once again results in a lower bound of\nat least $64^{20}$.\n\n\\section{Experiments}\\label{sec:exp}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\linewidth]{figures\/exp_channels}~~~~\\includegraphics[width=0.49\\linewidth]{figures\/exp_params}\n\n\\includegraphics[width=0.49\\linewidth]{figures\/exp_color_channels}~~~~\\includegraphics[width=0.49\\linewidth]{figures\/exp_color_params}\n\n\\caption{Training accuracies of standard ConvNets on CIFAR-10 with data\n    augmentations, where the results of spatial augmentations presented at the\n    top row, and color augmentations at the bottom row. Each network follows the\n    architecture of proposition~\\ref{prop:common_case}, with with receptive\n    field $B$ and using the same number of channels across all layers, as\n    specified by the horizontal axis of left plot. We plot the same results with\n    respect to the total number of parameters in the right plot.}\n\\label{fig:exp}\n\\end{figure}\n\nIn this section we show that the theoretical results of\nsec.~\\ref{sec:main_results} indeed hold in practice. In other words, there\nexists tasks that require the highly expressive power of overlapping\narchitectures, on which non-overlapping architectures would have to grow\nby an exponential factor to achieve the same level of performance. We\ndemonstrate this phenomenon on standard ConvNets with ReLU activations that\nfollow the same architecture that was outlined in\nproposition~\\ref{prop:common_case}, while varying the number of channels and the\nsize of the receptive field of the $B{\\times}B$ ``conv'' layers. The only change\nwe made, was to replace the $2{\\times}2$-``pooling'' layers of the convolutional\ntype, with the standard $2{\\times}2$-max-pooling layers, and using the same\nnumber of channels across all layers. This was done for the purpose of having\nall the learned parameters located only at the (possibly) overlapping layers.\nMore specifically, the network has 5 blocks, each starting with a $B{\\times}B$\nconvolution with $C$ channels, stride $1 {\\times} 1$, and ReLU activation, and\nthen followed by $2 {\\times} 2$ max-pooling layer. After the fifth\n``conv-pool'', there is a final dense layer with 10 outputs and softmax\nactivations.\n\nWe train each of these networks for classification over the\nCIFAR-10 dataset, with two types of data augmentation schemes: (i) spatial\naugmentations, i.e. randomly translating (up to 3 pixels in each direction) and\nhorizontally flipping each image, and (ii) color augmentations following\n\\citet{Dosovitskiy:2014tu}, i.e. randomly adding a constant shift (at most\n$\\pm 0.3$) to the hue, saturation, and luminance, for each attribute separately,\nand in addition randomly sampling a multiplier (in the range $[0.5, 1.5]$) just\nto the saturation and luminance. Though typically data augmentation is only used\nfor the purpose of regularization, we employ it for the sole purpose of raising\nthe hardness of the regular CIFAR-10 dataset, as even small networks can already\noverfit and effectively memorize its small dataset. We separately test both\nthe spatial and color augmentation schemes to emphasize that our empirical\nresults cannot be explained simply by spatial-invariance type arguments.\nFinally, the training itself is carried out for 300 epochs with\nADAM~\\citep{Kingma:2014us} using its standard hyper-parameters, at which point\nthe loss of the considered networks have stopped decreasing. We report the\ntraining accuracy over the augmented dataset in fig.~\\ref{fig:exp}, where for\neach value of the receptive field $B$, we plot its respective training\naccuracies for variable number of channels $C$. The source code for reproducing\nthe above experiments and plots can be found at \\githuburl{OverlapsAndExpressiveness}.\n\nIt is quite apparent that the greater $B$ is chosen, the less channels are\nrequired to achieve the same accuracy. Moreover, for the non-overlapping case of\n$B{=}1$, more than 2048 channels are required to reach the same performance of\nnetworks with $B {>} 2$ and just 64 channels under the spatial\naugmentations~--~which means effectively exponentially more channels were\nrequired. Even more so, under the color augmentations, we were not able to train\nnon-overlapping networks to reach even the smallest overlapping network\n($B=2$ and $C=16$). In terms of total number of parameters, there is a clear\nseparation between the overlapping and the non-overlapping types, and we once\nagain see more than an order of magnitude increase in the number of parameters\nbetween an overlapping and non-overlapping architectures that achieve similar\ntraining accuracy. As a somewhat surprising result, though based only on our\nlimited experiments, it appears that for the same number of parameters, all\noverlapping networks attain about the same training accuracy, suggesting perhaps\nthat having the smallest amount of overlapping already attain all the benefits\noverlapping provides, and that increasing it further does not affect the\nperformance in terms of expressivity.\n\nAs final remark, we also wish to acknowledge the limitations of drawing\nconclusions strictly from empirical experiments, as there could be alternative\nexplanations to these observations, e.g. the effects overlapping has on the\noptimization process. Nevertheless, our theoretical results suggests this is\nless likely the case.\n\n\\section{Discussion} \\label{sec:discussion}\n\nThe common belief amongst deep learning researchers has been that depth is one\nof the key factors in the success of deep networks~--~a belief formalized\nthrough the depth efficiency conjecture. Nevertheless, depth is\none of many attributes specifying the architecture of deep networks, and each\ncould potentially be just as important. In this paper, we studied the effect\noverlapping receptive fields have on the expressivity of the network, and found\nthat having them, and more broadly denser connectivity, results in an\nexponential gain in the expressivity that is orthogonal to the depth.\n\nOur analysis sheds light on many trends and practices in contemporary design\nof neural networks. Previous studies have shown that non-overlapping\narchitectures are already universal~\\citep{expressive_power}, and even have\ncertain advantages in terms of optimization~\\citep{Brutzkus:2017wp}, and yet,\nreal-world usage of non-overlapping networks is scarce. Though there could be\nmultiple factors involved, our results clearly suggest that the main culprit is\nthat non-overlapping networks are significantly handicapped in terms of\nexpressivity compared to overlapping ones, explaining why the former are so\nrarely used. Additionally, when examining the networks that are commonly used in\npractice, where the majority of the layers are of the convolutional type with\nvery small receptive field, and only few if any fully-connected\nlayers~\\citep{Simonyan:2014ws,Springenberg:2014tx,He:2016ib},\nwe find that though they are obviously overlapping, their overlapping degree is\nrather low. We showed that while denser connectivity can increase\nthe expressive capacity, even in the most common types of modern architectures\nalready exhibit exponential increase in expressivity, without relying on\nfully-connected layers. This could partly explain that somewhat surprising\nobservation, as it is probable that such networks are sufficiently expressive\nfor most practical needs simply because they are already in the exponential\nregime of expressivity. Indeed, our experiments seems to suggests the same,\nin which we saw that further increases in the overlapping degree beyond the most\nlimited overlapping case seems to have insignificant effects on performance~--~a \nconjecture not quite proven by our current work, but one we wish to investigate\nin the future.\n\nThere are relatively few other works which have studied the role of receptive\nfields in neural networks. Several empirical works~\\citep{Li:2005kc,\nCoates:2011wo,Krizhevsky:2012wl} have demonstrated similar behavior, showing\nthat the classification accuracy of networks can sharply decline as the degree\nof overlaps is decreased, while also showing that gains from using very large\nlocal receptive fields are insignificant compared to the increase in\ncomputational resources. Other works studying the receptive fields of neural\nnetworks have mainly focused on how to learn them from the\ndata~\\citep{Coates:2011tl,Jia:2012uz}. While our analysis has no direct\nimplications to those specific works, it does lay the ground work for\npotentially guiding architecture design, through quantifying the expressivity of\nany given architecture. Lastly, \\citet{Luo:2016vj} studied the \\emph{effective\ntotal receptive field} of different layers, a property of a similar nature to\nour total receptive field, where they measure the the degree to which each input\npixel is affecting the output of each activation. They show that under common\nrandom initialization of the weights, the effective total receptive field has a\ngaussian shape and is much smaller than the maximal total receptive field. They\nadditionally demonstrate that during training the effective total receptive\nfield grows in size, and suggests that weights should be initialized such that\nthe initial effective receptive field is large. Their results strengthen our\ntheory, by showing that trained networks tend to maximize their effective\nreceptive field, taking full potential of their expressive capacity.\n\nTo conclude, we have shown both theoretically and empirically that overlapping\narchitectures have an expressive advantage compared to non-overlapping ones. Our\ntheoretical analysis is grounded on the framework of ConvACs, which we extend to\noverlapping configurations. Though are proofs are limited to this specific case,\nprevious studies~\\citep{generalized_decomp} have already shown that such results\ncould be transferred to standard ConvNets as well, using most of the same\nmathematical machinery. While adapting our analysis accordingly is left for\nfuture work, our experiments on standard ConvNets~(see sec.~\\ref{sec:exp})\nalready suggest that the core of our results should hold in this case as well.\nFinally, an interesting outcome of moving from non-overlapping architectures to\noverlapping ones is that the depth of a network is no longer capped at\n$\\log_2 \\left(\\textit{input size} \\right)$, as has been the case in the models\ninvestigated by \\citet{expressive_power}~--~a property we will examine in future\nworks\n\n\\newcommand{This work is supported by Intel grant ICRI-CI \\#9-2012-6133, by ISF Center grant 1790\/12 and by the European Research Council (TheoryDL project).}{This work is supported by Intel grant ICRI-CI \\#9-2012-6133, by ISF Center grant 1790\/12 and by the European Research Council (TheoryDL project).}\n\\ifdefined\\COLT\n\t\\acks{This work is supported by Intel grant ICRI-CI \\#9-2012-6133, by ISF Center grant 1790\/12 and by the European Research Council (TheoryDL project).}\n\\else\n\t\\ifdefined\n\t\t\\subsubsection*{Acknowledgments}\n\t\tThis work is supported by Intel grant ICRI-CI \\#9-2012-6133, by ISF Center grant 1790\/12 and by the European Research Council (TheoryDL project).\n\t\\fi\n\\fi\n\n\\small{\n\\ifdefined\n\\bibliographystyle{plainnat}\n\\fi\n\\ifdefined\\NIPS\n\\bibliographystyle{plainnat}\n\\fi\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\nThe launch of the {\\em Submillimeter  Wave Astronomy Satellite} \\citep[SWAS;][]{2000ApJ...539L..87M} made it possible to observe emission from the ground-state transition ($\\mathrm{1_{10}}$$\\rightarrow$$\\mathrm{1_{01}}$) of ortho-$\\mathrm{H_2^{16}O}$ and its isotopomer ortho-$\\mathrm{H_2^{18}O}$, to determine the abundance of water vapour, i.e., column density $\\mathrm{N_{o-H_2O}}$, in various regions in the interstellar medium, e.g., dense and diffuse interstellar gas clouds, circumstellar envelopes, planetary atmospheres, and comets \\citep[e.g.,][]{2000ApJ...539L..87M, 2000ApJ...539L.101S, 2000ApJ...539L..93S, 1995Icar..117..162E, 2000ApJ...539L.147B, 2000ApJ...539L.143G, 2000ApJ...539L..87M}. In the future, the {\\em Heterodyne Instrument for the Far Infrared} (HIFI) on {\\em Herschel} will observe even more transitions of ortho- and para-$\\mathrm{H_2O}$ (o\/p-$\\mathrm{H_2O}$) like $\\mathrm{2_{12}}$$\\to$$\\mathrm{1_{01}}$ (1669.904 GHz), $\\mathrm{2_{21}}$$\\to$$\\mathrm{2_{12}}$ (1661.015 GHz), $\\mathrm{3_{03}}$$\\to$$\\mathrm{2_{12}}$ (1716.774 GHz), $\\mathrm{3_{12}}$$\\to$$\\mathrm{3_{03}}$ (1097.357 GHz), $\\mathrm{3_{21}}$$\\to$$\\mathrm{3_{12}}$ (1162.910 GHz), $\\mathrm{1_{11}}$$\\to$$\\mathrm{0_{00}}$ (1113.342 GHz), $\\mathrm{2_{02}}$$\\to$$\\mathrm{1_{11}}$ (967.924 GHz),  $\\mathrm{2_{11}}$$\\to$$\\mathrm{2_{02}}$ (752.029 GHz) and  $\\mathrm{2_{20}}$$\\to$$\\mathrm{2_{11}}$ (1228.801 GHz); some in absorption while others in emission.\\\\\nOne of the main goals of the SWAS mission is to determine where and whether $\\mathrm{H_2O}$ and $\\mathrm{O_2}$ are the major reservoirs of oxygen through the interstellar medium. SWAS observations have determined the gaseous water abundance in warm dense gas (T $\\gtrsim$ 300 K and n($\\mathrm{H_2}$) $\\gtrsim$ $10^3$ \\mbox{$\\mathrm{cm}^{-3}$}) to be $10^{-5}$ relative to $\\mathrm{H_2}$, in good agreement with chemical models for such conditions.\nHowever, in cold (T $\\lesssim$ 30 K), dense clouds the abundance of gaseous water is $\\sim$100 to 1000 times below the predictions of cold-cloud gas-phase chemical models. It has been suggested that -- toward cold clouds -- gaseous $\\mathrm{H_2O}$ exists only near the cloud surface. Indeed, closer to the surface than an $\\mathrm{A_V}$ of a few mag, $\\mathrm{H_2O}$ is photo-dissociated by the ambient galactic UV field. Deeper into the cloud, i.e., $\\mathrm{A_V}$ of 4 $\\sim$8 mag (depending on density and UV intensity), $\\mathrm{H_2O}$ may rapidly deplete onto dust grains \\citep{2000ApJ...539L.129B, 2001A&A...378.1024C, 2001A&A...370..557V}. Although the derivation of the column density from {\\em absorption} observations is straightforward (column density is simply proportional to the optical depth in the line, see \\cite{2004ApJ...605..247P}) this is not the case for {\\em emission} observations. The analysis to determine the $\\mathrm{H_2O}$ abundance now crucially depends on the physical properties of the gas through the collisional rate coefficients. Therefore, accurate constraints on the gas densities and temperatures are needed.\\\\% This is nearly the case.\\\\\nExtensive SWAS observations of the Orion A molecular cloud show that gaseous $\\mathrm{H_2O}$ correlates with CN, a surface tracer, rather than with $\\mathrm{C^{18}O}$, a volume tracer \\citep{2005AdSpR..36.1027M}. This result has been interpreted as evidence that gaseous water resides only near the surface. However, caution is needed when relying purely on single transition observations to draw such a conclusion in view of the complex rotational level structure of the $\\mathrm{H_2O}$ molecule. In particular, there is a fundamental difference between an optically thin and an effectively optically thin line. The latter case implies a strong coupling between line photons and water molecules. A full radiative transfer calculation is needed to address this problem, since the observed intensities of molecular emission depend on a complex competition between radiative and collisional processes. Moreover, the excitation of $\\mathrm{H_2O}$ also differs from that of other molecules, since both collisions and infrared radiation from warm dust influence the level populations \\citep{1983ApJ...275..145T}.\\\\\nThe intent of this work is to show that it is not straightforward to retrieve accurate information, e.g., column density, from single transition observations of $\\mathrm{H_2O}$ due to the complex level structure of this molecule.\n\n\\section{\\label{sec:}Basic model description}\nThe results presented here were obtained by application of the numerical code of \\cite{2005A&A...440..559P}, described further in \\cite{2006A&A...453..615P}. The interested reader is referred to these papers for a description of the underlying algorithms. The radiative transfer of o\/p-$\\mathrm{H_2O}$ is solved by means of a multi-zone escape probability method in three dimensions. By using a multi-zone formalism, the medium is divided into different zones, i.e., gridcells, each with a value for the abundance of the species (e.g., $\\mathrm{H_2O}$), the density of the medium, and the temperature of gas and dust. Besides this, the cloud is characterized by a fixed total column density. The statistical equilibrium equation for a multilevel sytem can be written as\n\\begin{equation}\n\\sum_{i>j}(n_iA_{ij}~+~nn_i\\gamma_{ij}~+~n_iB_{ij}J)~=~\\sum_{i<j}(nn_i\\gamma_{ij}~+~n_iB_{ij}J),\n\\label{eq:stateq}\n\\end{equation}\nwhere the radiative de-excitation is given by the Einstein A coefficients, $\\mathrm{B_{ij}}$ are the Einstein coefficients for absorption and stimulated emission, $\\gamma_{ij}$ are the collisional rate coefficients, $J$ the mean photon radiation field, $n_i$ the population density of the $i$th level, and $n$ the density of the medium. \nThe level populations depend on the radiation field while the radiation field depends on the level populations everywhere. \nBy introducing the concept of a multi-zone escape probability \\citep{2005A&A...440..559P, 2006A&A...453..615P} the statistical equilibrium equations are decoupled from the radiative transfer equations. Eq. \\ref{eq:stateq} can be rewritten as\n\\begin{equation}\n\\sum_{i>j}(n_iA_{ij}\\beta(\\tau_{ij})~+~nn_i\\gamma_{ij})~=~\\sum_{i<j}(nn_i\\gamma_{ij}),\n\\label{eq:levelpop}\n\\end{equation}\nbecause the net absorptions, corrected for stimulated emission, are equal to those photons that do not escape, i.e.,\n\\begin{equation}\n(n_jB_{ji}~-~n_iB_{ij})_{i>j}J~=~\\sum_{i>j}n_i[1~-~\\beta(\\tau_{ij})]A_{ij},\n\\label{eq:approx}\n\\end{equation}\nwhere $\\beta(\\tau)$ is the probability that a photon formed at optical depth $\\tau$ in a certain direction escapes the cloud along that direction. Therefore, \n\\begin{equation}\nn_{cr}~=~\\frac{\\sum_{i>j}\\beta(\\tau_{ij})A_{ij}}{\\sum_{i>j}\\gamma_{ij}}\n\\label{Eq:ncr}\n\\end{equation}\nNote that because of the large Einstein A coefficients of the water molecule critical densities are of the order of $10^8$-$10^9$ \\mbox{$\\mathrm{cm}^{-3}$} in the optically thin case.\\\\ \nThe background radiation field P($\\nu_{ij}$) in the code consists of two terms: the 2.7K microwave background and the infrared emission of dust at a temperature $T_{\\mathrm{d}}$. This is,\n\\begin{equation}\nP(\\nu_{ij}) = B(\\nu_{ij}, T=2.7K)\\ +\\ (1-e^{-{\\tau_{dust}}})B(\\nu_{ij},T_d)\n\\end{equation}\nThe intensity of transition i$\\rightarrow$j (i$>$j) is then given by\n\\begin{equation}\nI_{ij,\\mathrm{total}} = {1\\over 4\\pi}\\int_0^r\\Lambda_{ij,\\mathrm{local}}dr,\n\\label{eq:int_total}\n\\end{equation}\nwith \n\\begin{equation}\n\\Lambda_{ij,\\mathrm{local}} = n_iA_{ij}h\\nu_{ij}\\beta(\\tau_{ij})\\{[S(\\nu_{ij})-P(\\nu_{ij})]\/S(\\nu_{ij})\\},\n\\end{equation} \nS($\\nu_{ij}$) is the source function at frequency $\\nu_{ij}$. \\\\\nCollisional rate coefficients for inelastic collisions between o\/p-$\\mathrm{H_2O}$ and He \\citep{1993ApJS...85..181G}, and for collisions between o\/p-$\\mathrm{H_2O}$ and both o-$\\mathrm{H_2}$ and p-$\\mathrm{H_2}$ \\citep{1996ApJS..107..467P} are adopted. We adopt the expression for the ortho-to-para ratio (OPR) of $\\mathrm{H_2}$, in thermal equilibrium, defined by \n\\begin{equation}\n\\mathrm{OPR}= {{(2I_\\mathrm{o} + 1)\\sum(2J + 1)\\exp\\left(-{E_\\mathrm{o}(J,K_\\mathrm{a},K_\\mathrm{c})\\over kT}\\right)}\\over{(2I_\\mathrm{p} + 1)\\sum(2J + 1)\\exp\\left(-{E_\\mathrm{p}(J,K_\\mathrm{a},K_\\mathrm{c})\\over kT}\\right)}}\\,, \n\\label{eq:OPR}\n\\end{equation}\nwhere $I_\\mathrm{o}$ and $I_\\mathrm{p}$ are the total nuclear spin, corresponding to wether the hydrogen nuclear spins are parallel ($I_\\mathrm{o}$ = 1, $\\uparrow$$\\uparrow$) or anti-parallel ($I_\\mathrm{p}$ = 0, $\\uparrow$$\\downarrow$). The sum in the numerator (denominator) extends over all ortho (para) levels $({J},K_\\mathrm{a},K_\\mathrm{c})$, \\citet{1987A&A...187..419M}. The code has been tested extensively against (analytical) benchmark problems presented at the radiative transfer workshop held in Leiden (2004), see \\citet{2006A&A...453..615P}. It is found that the level populations are completely consistent with the solutions of other Monte Carlo and ALI codes, as presented in \\citet{2005dmu..conf..431V}\\\\\n\\begin{table\n\\begin{minipage}[b]{\\columnwidth}\n\\renewcommand{\\footnoterule}{}\n\\caption{Model parameters}\n\\label{tab:model}\n\\centering\n\\begin{tabular}{lcccccc}\n\\hline\\hline  \nModel & $\\mathrm{n(H_2)}$ & size & ${\\mathrm{X(H_2O)}^{a}}$ & ${\\mathrm{N({H_2})}^{b}}$ & $\\mathrm{T_{gas}}$ & $\\mathrm{T_{dust}}$\\\\\n &  $[$\\mbox{$\\mathrm{cm}^{-3}$}$]$ & $[$\\mbox{pc}$]$ & &$[$\\mbox{$\\mathrm{cm}^{-2}$}$]$ & $[$K$]$&$[$K$]$\\\\\n\\hline\nI & $10^4$-$10^6$ & 0.002-0.2 & $10^{-10}$-$10^{-6}$ & 6$\\times$$10^{21}$ & 50 & no \\\\\nII & $10^4$-$10^6$ & 0.002-0.2 & $10^{-10}$-$10^{-6}$ & 6$\\times$$10^{21}$ & 50 & 15\\\\\nIII & $10^4$-$10^6$ & 0.002-0.2 & $10^{-10}$-$10^{-6}$ & 6$\\times$$10^{21}$ & 50 & 25\\\\\nIV & $10^4$-$10^6$ & 0.002-0.2 & $10^{-10}$-$10^{-6}$ & 6$\\times$$10^{21}$ & 50 & 50\\\\\nV & $10^4$-$10^6$ & 0.002-0.2 & $10^{-10}$-$10^{-6}$ & 6$\\times$$10^{21}$ & 30 & 15\\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n$^{a}$ Abundance of $\\mathrm{H_2O}$; $^{b}$ Total column density through the centre\n\\end{table}\n\\section{Model results}\n\n\\begin{figure\n\\includegraphics[width=9cm]{N6e21_nodustCMB.ps}\n\\caption{The intensity of the ortho-$\\mathrm{H_2O}$ ground state transition for a homogeneous sphere with $\\mathrm{H_2}$ densities of $10^4$({\\em top}), $10^5$({\\em middle}), $10^6$({\\em bottom}) $\\mathrm{cm^{-3}}$ and temperature of the gas of 50 K. The dust emission as well as the CMB radiation are ignored ({\\em i.e., model I}). In every case, the total column density ($\\mathrm{H_2}$) is kept constant. Lines are plotted for an abundance of $\\mathrm{H_2O}$, relative to $\\mathrm{H_2}$, from $10^{-10}$ ({\\em upper line\/red}) to $10^{-6}$ ({\\em lower line\/light blue}) as function of $\\mathrm{H_2}$ column density along the line of sight, where 2 $\\times$ $10^{19}$ $\\mathrm{cm^{-2}}$ is at the edge, and 6 $\\times$ $10^{21}$ $\\mathrm{cm^{-2}}$ through the center of the cloud. $Y$-axis in units of \\mbox{$\\mathrm{erg\\ s^{-1}\\ sr^{-1}}$}. The position of the $\\tau$=1 surface is displayed with a cross for X($\\mathrm{H_2O}$)=$10^{-9}$-$10^{-6}$.}\n\\label{fig:N6e21_nodustCMB}\n\\end{figure}\n\n\n\\begin{figure\n\\includegraphics[width=9cm]{N6e21_1e4.ps}\n\\caption{The intensity of the ortho-$\\mathrm{H_2O}$ ground state transition for a homogeneous sphere with density ($\\mathrm{H_2}$) of $10^4$ $\\mathrm{cm^{-3}}$ for model II ({\\em top}), III ({\\em middle}) and IV ({\\em bottom}). Lines are plotted for an abundance of $\\mathrm{H_2O}$, relative to $\\mathrm{H_2}$, from $10^{-10}$ ({\\em upper line\/red}) to $10^{-6}$ ({\\em lower line\/light blue}) as function of $\\mathrm{H_2}$ column density along the line of sight, where 2 $\\times$ $10^{19}$ $\\mathrm{cm^{-2}}$ is at the edge, and 6 $\\times$ $10^{21}$ $\\mathrm{cm^{-2}}$ through the center of the cloud. $Y$-axis in units of \\mbox{$\\mathrm{erg\\ s^{-1}\\ sr^{-1}}$}.}\n\\label{fig:N6e21_1e4.ps}\n\\end{figure}\n\n\n\n\n\\begin{figure\n\\includegraphics[width=9cm]{N6e21_Tg30Td15.ps}\n\\caption{The intensity of the ortho-$\\mathrm{H_2O}$ ground state transition in case of a homogeneous sphere with densities ($\\mathrm{H_2}$) of $10^4$({\\em top}), $10^5$({\\em middle}), $10^6$({\\em bottom}) $\\mathrm{cm^{-3}}$ and temperatures for gas and dust of 30 K and 15 K, respectively ({\\em i.e., model V}). In every case, the total column density ($\\mathrm{H_2}$) is kept constant. Lines are plotted for an abundance of $\\mathrm{H_2O}$, relative to $\\mathrm{H_2}$, from $10^{-10}$ ({\\em upper line\/red}) to $10^{-6}$ ({\\em lower line\/light blue}) as function of $\\mathrm{H_2}$ column density along the line of sight, where 2 $\\times$ $10^{19}$ $\\mathrm{cm^{-2}}$ is at the edge, and 6 $\\times$ $10^{21}$ $\\mathrm{cm^{-2}}$ through the center of the cloud. $Y$-axis in units of \\mbox{$\\mathrm{erg\\ s^{-1}\\ sr^{-1}}$}.}\n\\label{fig:N6e21_Tg30Td15}\n\\end{figure}\n\n\n\\begin{figure\n\\includegraphics[width=9cm]{l2.ps}\n\\caption{Shown here is the population of the $\\mathrm{1_{01}}$ level of o-$\\mathrm{H_2O}$ as function of radius. Top, middle, bottom panel are the results in case X($\\mathrm{H_2O}$) = $10^{-10}$, $10^{-9}$, $10^{-8}$, respectively. Dashed, dotted and dashed-dotted lines represent the results of model I, III and IV, respectively.}\n\\label{fig:levelpop}\n\\end{figure}\n\n\\begin{figure\n\\includegraphics[width=9cm]{mean.ps}\n\\caption{Conversion of the results of model I ({\\em top}) and IV ({\\em bottom}) into surface area weighted values. In each panel the dashed-dotted, dotted, and dashed curve represent the results in case the density n($\\mathrm{H_2}$) is $10^6$, $10^5$ and $10^4$ \\mbox{$\\mathrm{cm}^{-3}$}, respectively. $Y$-axis in units of \\mbox{$\\mathrm{erg\\ s^{-1}\\ sr^{-1}}$}.}\n\\label{fig:mean}\n\\end{figure}\nWe calculate, in the case of a homogeneous sphere and as a function of impact parameter, the surface brightness for the $\\mathrm{1_{10}}$$\\rightarrow$$\\mathrm{1_{01}}$ ground-state transition of ortho-$\\mathrm{H_2O}$. The density and abundance of $\\mathrm{H_2O}$ are the main parameters of the model. All the models have a constant total column density N($\\mathrm{H_2}$) of 6 $\\times$ $10^{21}$ \\mbox{$\\mathrm{cm}^{-2}$} through the center of the cloud, corresponding to a total $A_V$ of $\\sim$3 mag. As a result, the physical size of the cloud is inversely proportional to the density of the medium. The density ranges from $10^4$ to $10^6$ \\mbox{$\\mathrm{cm}^{-3}$} covering the relevant range for dense molecular clouds  such as the Orion ridge. \nThe temperatures, ranging from 30 to 50K for the gas and from 15 to 50K for the dust, were chosen to represent the mean observed temperatures towards star forming molecular clouds such as the Orion ridge which have been the focus of the SWAS effort. In model I we ignore the emission from dust and CMB, whereas in model IV a dust temperature of 50K is assumed, in order to assess the influence of the dust and temperature, respectively, on the excitation of the water molecule. The temperatures in model V are a factor of $\\sim$2 lower than in the other models since to maintain a temperature of the gas of 50K throughout the cloud one needs a strong UV radiation field, which is not always the case. Note that the gas and dust temperatures are independent of cloud depth. The intent of the paper is to illustrate the excitation and radiative transfer effects assuming a 'simple' cloud model, not to model a realistic cloud. A Galactic dust-to-gas ratio of $10^{-2}$ by mass is assumed. We adopt the dust opacities of \\cite{1994A&A...291..943O} (Column 5 of their Table 1). In all the models, except for model I, the dust optical depth $\\tau_{\\mathrm{dust}}$ through the center of the cloud at the frequency of the ground state transition of o-$\\mathrm{H_2O}$, i.e., $\\mathrm{\\tau_{\\nu=556.936GHz}}$ or $\\mathrm{\\tau_{\\lambda=538\\mu m}}$, is $10^{-3}$. Within each model the water abundance, X($\\mathrm{H_2O}$), ranges from $10^{-10}$ to $10^{-6}$. The parameters for the different models are shown in Table \\ref{tab:model}. Throughout the models, a velocity dispersion of 1 \\mbox{$\\mathrm{km\\ s^{-1}}$} is adopted, typical for a cloud with moderate turbulence. Note that the results presented in this paper depend on the adopted velocity dispersion. A higher (lower) velocity dispersion will decrease (increase) the optical depth for a given transition, hence having an impact on the excitation of the molecule.\\\\\nFig. \\ref{fig:N6e21_nodustCMB}--\\ref{fig:N6e21_Tg30Td15} present the basic results of this work. We plot, as a function of N($\\mathrm{H_2}$), i.e., different impact parameter, the $\\mathrm{1_{10}}$$\\rightarrow$$\\mathrm{1_{01}}$ line intensity above the continuum per unit column density of o-$\\mathrm{H_2O}$, see Eq. \\ref{eq:int_total}. The results for model I, in which we ignore the contribution of dust and CMB on the excitation of the water molecule are displayed in Fig. \\ref{fig:N6e21_nodustCMB}. The position of the $\\tau$=1 surface is denoted by a cross for X($\\mathrm{H_2O}$)=$10^{-9}$-$10^{-6}$. Models II, III and IV behave in a similar way as in model I in the case n($\\mathrm{H_2}$)$\\ge$$10^5$$\\mathrm{cm^{-3}}$. For this reason, we only plot the outcome in the low density case for models II, III and IV in Fig. \\ref{fig:N6e21_1e4.ps}. The results for model V are plotted in Fig. \\ref{fig:N6e21_Tg30Td15}. \\\\\nThe following trends can be identified, which will be discussed in Sect. \\ref{sec:discussion}. \\\\\nFirst, in the case of n($\\mathrm{H_2}$) $\\ge$ $10^5$ $\\mathrm{cm^{-3}}$ and X($\\mathrm{H_2O}$) $\\lesssim$ $10^{-8}$, i.e., $\\tau$$<$10, a linear relationship holds between the number of photons escaping the cloud and the impact parameter, i.e., I\/$\\mathrm{N_{H_2O}}$ is constant. However, this relationship breaks down at high optical depth, i.e., $\\mathrm{H_2O}$ $>$ $10^{-8}$, for all the models. Second, in all the models, except in model I, absorption occurs in the low density case when abundances are low, i.e., X($\\mathrm{H_2O}$) $<$ $10^{-8}$, which becomes more apparent as the dust temperature increases. However, the amount of absorption is moderate as the self-reversal in the center of the line is small. Third, when the $\\mathrm{H_2O}$ abundance exceeds $10^{-7}$, in all the models, the ratio of the intensity to the column density decreases near the edge of the cloud (N($\\mathrm{H_2}$)$\\sim$5$\\times$$10^{20}$$\\mathrm{cm^{-2}}$). Fourth, for high optical depth, i.e., X($\\mathrm{H_2O}$)$\\gtrsim$$10^{-7}$, and n($\\mathrm{H_2}$)=$10^6$ $\\mathrm{cm^{-3}}$, I\/$\\mathrm{N_{H_2O}}$ decreases with increasing column density. Fifth, lowering the gas and dust temperatures by a factor of $\\sim$2 (model V) does not lead to significant differences in the shapes of the curves. That is, models II-IV and model V experience similar complicating radiative transfer effects.\n\n\\section{\\label{sec:discussion}Discussion}\nThe asymmetry of the water molecule causes the rotational levels to split into a number of different ladders, so called 'K-ladders', characterized by different values of the projection of the angular momentum onto the principal axes of the molecule ($J_{K_-K_+}$). Radiative transitions occur rapidly between levels in each ladder but are much slower between levels in different ladders. This leads to a spectrum more complex than linear or symmetric top molecules, e.g., CO, $\\mathrm{NH_3}$.\nHence, it is not straightforward to disentangle the different processes that contribute to the observed spectrum. We now describe the different effects that play a role in the interpretation of the figures. In this, the 'edge' and 'centre' of the cloud refers to an impact parameter of 1 and 0, respectively. Note that the use of spherical models leads to angular non-trivial re-distribution of line photons. The same holds for continuum photons within the line profile frequency range. \\\\\nFirst, one can see in Figs. \\ref{fig:N6e21_nodustCMB} and \\ref{fig:N6e21_Tg30Td15} that the curves as function of total column density ($\\mathrm{H_2}$) are constant for X($\\mathrm{H_2O}$) $\\lesssim$ $10^{-8}$ and n($\\mathrm{H_2}$)$\\ge$$10^5$$\\mathrm{cm^{-3}}$. In this limit, collisional de-excitation and scattering effects are negligible. Eventually every photon produced in the cloud will escape the cloud with few interactions with the surrounding medium. Note that the number of scatterings $N$ to escape depends on the optical depth. In this regime $\\tau$ $\\ll$ 1, therefore few photons are scattered and $N$ $\\approx$ $\\tau$. With increasing optical depth, i.e., X($\\mathrm{H_2O}$) $\\gtrsim$ $10^{-8}$, more effects have to be taken into account. All models show a drop near the edge of the cloud.\n Because of the increasing optical depth, line-scattering effects become important. Thus, line photons then tend to escape in the direction with the lowest optical depth rather than tangentially to the cloud surface, causing the dip near the edge. However, towards the centre of the cloud, the optical depth increases with orders of magnitude. The photons will undergo numerous scatterings for $\\tau$ $\\gg$ 1, with $N$ $\\approx$ $\\tau^2$, and eventually will escape in the line wings.\\\\\nSecond,\nat densities as low as $10^4$ $\\mathrm{cm^{-3}}$, and abundances not exceeding $10^{-8}$ (modest optical depth), the line is strongly subthermally excited, and radiatively colder than the dust background. Hence, the line appears in absorption. The decrease in intensity\/N($\\mathrm{H_2O}$) shown in Fig. \\ref{fig:N6e21_1e4.ps} (red and green curves) now indicates that lines of sight through the cloud center are no longer contributing evenly to the emissivity across their entire column. The line is not strictly in absorption yet, but it has developed an intensity dip around line center. Thus, the presence of dust causes the trend in the intensity per column in this regime to decrease \\citep{1983ApJ...275..145T}. This behaviour is not seen in case n($\\mathrm{H_2}$) $\\gtrsim$ $10^5$ $\\mathrm{cm^{-3}}$, as in this regime collisions are the dominant process in the excitation of the water molecule, thereby nullifying the effect of dust emission\n The influence of dust on the excitation\/level populations of water is plotted in Fig. \\ref{fig:levelpop} where the relative population of the $\\mathrm{1_{10}}$ level is displayed. One notices that for warmer dust level $\\mathrm{1_{10}}$ is more populated. In essence, dust continuum emission will tend to drive the level populations towards a Boltzmann distribution at the temperature of the dust. For a given density, the effects of radiative excitation by dust continuum emission is more pronounced for higher dust temperatures (e.g., higher continuum intensities).\\\\\nThird, the effect of photon trapping is to lower the density at which LTE is approached, i.e., after each absorption, the gas has a chance to collisionally de-excite the species and return the excitation energy to the thermal bath of the gas, see Eq. \\ref{Eq:ncr}. For optically thin gas, the critical density of the ground state transition at 557 GHz of o-$\\mathrm{H_2O}$ is $\\sim$$10^8$ \\mbox{$\\mathrm{cm}^{-3}$} at 50 K. The optical depth, through the centre of the cloud, varies from 0.1 to $10^3$ when the abundance rises from $10^{-10}$ to $10^{-6}$ in all the models. Hence, for high abundances, i.e., X($\\mathrm{H_2O}$) $\\gtrsim$$10^{-7}$, the effective critical density drops to $10^5$-$10^6$ \\mbox{$\\mathrm{cm}^{-3}$}, since for high optical depth $\\beta(\\tau)$$\\sim$$1\/\\tau$. Collisional de-excitation processes then become important in the regime where n($\\mathrm{H_2}$) $\\gtrsim$ $10^5$ \\mbox{$\\mathrm{cm}^{-3}$}, and X($\\mathrm{H_2O}$) $\\gtrsim$ $10^{-7}$. It is seen in Fig. \\ref{fig:N6e21_nodustCMB} and \\ref{fig:N6e21_Tg30Td15} that for n($\\mathrm{H_2}$)=$10^6$$\\mathrm{cm^{-3}}$ and X($\\mathrm{H_2O}$)=$10^{-6}$ the I\/$\\mathrm{N_{H_2O}}$ drops as function of impact parameter. In this part of parameter space collisional de-excitation is important and the probability that line photons are lost to the thermal bath through collisional de-excitation during one of the many scattering events is high.\\\\% Note that the optical depth for the $\\mathrm{2_{12}}$$\\to$$\\mathrm{1_{01}}$ transition at 1669.904 GHz is comparable with $\\tau_{\\mathrm{1_{10}} \\to \\mathrm{1_{01}}}$ due to the larger Einstein A coefficient. The optical depth of the higher transitions are orders of magnitude lower.\\\\\nFourth, calculations are performed for model V (Fig. \\ref{fig:N6e21_Tg30Td15}) with gas temperatures a factor of 2 lower relative to the temperatures used in model II. We find that the shape of the curves are not affected by such a change in gas temperature. However, it affects the distribution of the level populations and thus the absolute intensity in the lines. Hence, temperature variations cannot dispense of the radiative transfer effects studied in this work.\\\\\nTo summarize, we plot in Fig. \\ref{fig:mean}, as a function of abundance, the average intensity emanating from the cloud for model I and IV, i.e., \n\\begin{equation}\n{{\\int I_{{1_{10}}\\rightarrow{1_{01}}}(b)2\\pi b\\ db}\\over{\\int N_{\\mathrm{H_2O}}(b)2\\pi b\\ db}}, \n\\end{equation}\nwith b the impact parameter. One notices a drop by a factor of $\\sim$2--5 in case n($\\mathrm{H_2}$) is $10^5$--$10^6$ \\mbox{$\\mathrm{cm}^{-3}$}, respectively. Note that with the assumption of an effectively optically thin line one would underestimate the water column by these same factors.\\\\\n\n\\section{Astrophysical implications}\nThe intensity of the ground-state transition of o-$\\mathrm{H_2O}$ is driven by a combination of the ambient gas and dust temperatures on the one side and by the density of the surrounding medium on the other side. It is this interplay, together with the complex structure of the molecule that drives the level populations. To interpret existing SWAS and future HIFI data a clear sense of the information content of the water\n lines is needed. \\\\\nSWAS observations of the lowest rotational transition of o-$\\mathrm{H_2^{16}O}$ of the Orion A molecular cloud show that gaseous water correlates much better with the near surface tracer CN than with the volume tracer $\\mathrm{C^{18}O}$, as presented in \\citet{2005AdSpR..36.1027M}. Through these observations -- in which it is assumed that the ground-state transition of ortho-$\\mathrm{H_2O}$ is effectively optically thin -- one concludes that water is a surface tracer. This is plausible from a chemical point of view in which photo-dissociation destroyes the water molecule near the surface. Further inwards the cloud the water abundance reaches its equilibrium value through photodesorption of $\\mathrm{H_2O}$-ice and photodestruction of $\\mathrm{H_2O}$-gas until it freezes-out onto dust grains deeper into the cloud. However, we have shown, as seen in Fig. \\ref{fig:mean}, that for $\\tau$$>$10 and n$>$$10^5$$\\mathrm{cm^{-3}}$ the effectively optically thin assumption no longer holds. Hence, under these conditions one is limited to observing the '$\\tau$=10' surface and cannot use water to trace the cloud's volume, even when it is present, i.e., not frozen out (Cernicharo, private communication). Therefore, as CN is a surface tracer and the water intensity originates from a layer of gas with an optical thickness of 1 -- depending on local excitation conditions this layer is a surface layer -- the CN intensity correlates much better with the $\\mathrm{H_2O}$ intensity and not with the volume tracer $\\mathrm{C^{18}O}$. Thus, the anti-correlation of $\\mathrm{H_2O}$ with $\\mathrm{C^{18}O}$ is partly due to optical depth effects, and is not neccesarily a result of chemical changes. As a consequence, the presence of water past the $\\tau$=1 surface cannot be ruled out. \\\\\nWe would also like to point out here that the most interesting aspect of the correlation of the water line intensity with the CN line intensity is the fact that both are observed to vary by a factor $\\sim$100. Theoretically, the CN abundance is expected to scale with density squared \\citep{2005ApJ...632..302B}, indicating the importance of density variations over the Orion molecular cloud. Given the results presented in this paper, we surmise that these density variations will hamper the interpretation of the water observations.\\\\ \nIn order to deduce the total water column along the line of sight, additional information is needed from other -- effectively optically thin-- lines, which will be observed with future missions such as Herschel\/HIFI.\n\n\\begin{acknowledgements}\nWe are grateful to Ted Bergin and Gary Melnick for sending an early version of the manuscript. We also thank Floris van der Tak for helpful discussions and suggestions which have improved the paper and the anonymous referee for his\/her constructive comments.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sect:intro}\n\nSpectral measures for the representation graphs for the irreducible representations of the exceptional compact Lie group $G_2$ and its maximal torus $\\mathbb{T}^2$, and for nimrep graphs associated to the $G_2$ modular invariants, were studied in \\cite{evans\/pugh:2012i}. Here we consider the McKay graphs for finite subgroups of $G_2$ and study their spectral measures.\n\nThe spectral measure of a self-adjoint operator $a$ in a unital $C^{\\ast}$-algebra $A$ with state $\\varphi$ is a compactly supported probability measure $\\nu_a$ on the spectrum $\\sigma(a) \\subset \\mathbb{R}$ of $a$, uniquely determined by its moments\n\\begin{equation} \\label{eqn:moments_sa_operator}\n\\varphi(a^m) = \\int_{\\sigma(a)} x^m \\mathrm{d}\\nu_a (x),\n\\end{equation}\nfor all non-negative integers $m$.\n\nThe self-adjoint operators we consider here are the adjacency matrices of the McKay graphs for finite subgroups of $G_2$. The characters of the rank two Lie group $G_2$ are functions on $\\mathbb{T}^2$, and it is convenient to write the spectral measures for these operators as measures $\\varepsilon$ over the torus $\\mathbb{T}^2$. However, $\\mathbb{T}^2$ has dimension one greater than $\\sigma(a) \\subset \\mathbb{R}$, so that there is an infinite family of pullback measures $\\varepsilon$ over $\\mathbb{T}^2$ for any spectral measure $\\nu_a$. The details of the relation between the measures $\\varepsilon$ and $\\nu_a$ are given in Section \\ref{sect:measures-different_domains}.\n\nIn order to remove this ambiguity, we also consider joint spectral measures, that is, measures over the joint spectrum $\\sigma(a,b) \\subset \\sigma(a) \\times \\sigma(b) \\subset \\mathbb{R}^2$ of commuting self-adjoint operators $a$ and $b$. The abelian $C^{\\ast}$-algebra $B$ generated by $a$, $b$ and the identity 1 is isomorphic to $C(X)$, where $X$ is the spectrum of $B$. Then the joint spectrum is defined as $\\sigma(a,b) = \\{ (a(x), b(x)) | \\, x \\in X \\}$. In fact, one can identify the spectrum $X$ with its image $\\sigma(a,b)$ in $\\mathbb{R}^2$, since the map $x \\mapsto (a(x), b(x))$ is continuous and injective, and hence a homomorphism since $X$ is compact \\cite{takesaki:2002}.\nIn the case where the operators $a$, $b$ act on a finite-dimensional Hilbert space, this is the set of all pairs of real numbers $(\\lambda_a,\\lambda_b)$ for which there exists a vector $\\phi$, $||\\phi||=1$, such that $a\\phi = \\lambda_a \\phi$, $b\\phi = \\lambda_b \\phi$.\nThen there exists a compactly supported probability measure $\\widetilde{\\nu}_{a,b}$ on $\\sigma(a,b)$, which is uniquely determined by its cross moments\n\\begin{equation} \\label{eqn:cross_moments_sa_operators}\n\\varphi(a^m b^n) = \\int_{\\sigma(a,b)} x^m y^n \\mathrm{d}\\widetilde{\\nu}_{a,b} (x,y),\n\\end{equation}\nfor all non-negative integers $m$, $n$.\nSuch joint spectral measures specialize to the spectral measures $\\nu_a$ (respectively $\\nu_b$) by integrating over all $y$ for which $(\\lambda_a,y) \\in \\sigma(a,b)$ (respectively all $x$ for which $(x,\\lambda_b) \\in \\sigma(a,b)$).\nAs discussed in Section \\ref{sect:measures-different_domains}, such a measure uniquely defines a measure over $\\mathbb{T}^2$ invariant under an action of the Weyl group of $G_2$. In this paper we determine the joint spectral measure for each finite subgroup $\\Gamma$ of $G_2$ for all non-conjugate embeddings of $\\Gamma$ into the fundamental representations of $G_2$.\n\nThe paper is organised as follows. In Section \\ref{sect:preliminaries} we present some preliminary material, including a discussion on the relation between spectral measures over certain different domains in Section \\ref{sect:measures-different_domains} and a summary of relevant results for $G_2$ and its nimreps from \\cite{evans\/pugh:2012i}.\n\nIn Section \\ref{sect:subgroupsG2} we discuss the finite subgroups of $G_2$, including their embeddings into the fundamental representations of $G_2$. We also give a general discussion of their spectral measures. In Sections \\ref{sect:II1}-\\ref{sect:IP5} we discuss each finite subgroup of $G_2$ individually, including determining all non-conjugate embeddings into the fundamental representations of $G_2$. For each such embedding we construct their McKay graphs, some of which have appeared before in \\cite{he:2003}, and we determine their joint spectral measures.\n\nSpectral measures associated to the compact Lie groups $A_1 = SU(2)$ and $A_2 = SU(3)$ and their maximal tori, nimrep graphs associated to the $SU(2)$ and $SU(3)$ modular invariants, and the McKay graphs for finite subgroups of $SU(2)$ and $SU(3)$ were studied in \\cite{banica\/bisch:2007, evans\/pugh:2009v, evans\/pugh:2010i}.\nSpectral measures associated to the compact Lie group $C_2$ are studied in \\cite{evans\/pugh:2012iii}, whilst spectral measures associated to other compact rank two Lie groups and their maximal tori are studied in \\cite{evans\/pugh:2012iv}.\n\n\n\n\\section{Preliminaries} \\label{sect:preliminaries}\n\\subsection{Spectral measures over different domains} \\label{sect:measures-different_domains}\n\nThe Weyl group of $G_2$ is the dihedral group $D_{12}$ of order 12.\nAs a subgroup of $GL(2,\\mathbb{Z})$, $D_{12}$ is generated by matrices $T_2$, $T_6$, of orders 2, 6 respectively, given by\n\\begin{equation} \\label{T2,T6}\nT_2 = \\left( \\begin{array}{cc} 0 & -1 \\\\ -1 & 0 \\end{array} \\right), \\qquad T_6 = \\left( \\begin{array}{cc} 0 & 1 \\\\ -1 & 1 \\end{array} \\right),\n\\end{equation}\nwhere the action of $D_{12}$ on $\\mathbb{T}^2$ is given by $T(\\omega_1,\\omega_2) = (\\omega_1^{a_{11}}\\omega_2^{a_{12}},\\omega_1^{a_{21}}\\omega_2^{a_{22}})$, for $T = (a_{il}) \\in D_{12}$. This action leaves $\\chi_{\\mu}(\\omega_1,\\omega_2)$ invariant, for any $\\mu \\in P_{++} = \\{ (\\mu_1,\\mu_2) \\in \\mathbb{N}^2 | \\, \\mu_1 \\geq \\mu_2 \\}$, the interior of the Weyl alcove for $G_2$.\nAny $D_{12}$-invariant measure $\\varepsilon_{\\mu}$ on $\\mathbb{T}^2$ yields a pushforward probability measure $\\nu_{\\mu}$ on $I_{\\mu} = \\chi_{\\mu}( \\mathbb{T}^2)\\subset \\mathbb{R}$ by\n\\begin{equation} \\label{eqn:measures-T2-Ij_G2}\n\\int_{I_{\\mu}} \\psi(x) \\mathrm{d}\\nu_{\\mu}(x) = \\int_{\\mathbb{T}^2} \\psi(\\chi_{\\mu}(\\omega_1,\\omega_2)) \\mathrm{d}\\varepsilon_{\\mu}(\\omega_1,\\omega_2),\n\\end{equation}\nfor any continuous function $\\psi:I_{\\mu} \\rightarrow \\mathbb{C}$, where $\\mathrm{d}\\varepsilon_{\\mu}(\\omega_1,\\omega_2) = \\mathrm{d}\\varepsilon_{\\mu}(g(\\omega_1,\\omega_2))$ for all $g \\in D_{12}$.\nThere is a loss of dimension here, in the sense that the integral on the right hand side is over the two-dimensional torus $\\mathbb{T}^2$, whereas on the right hand side it is over the interval $I_{\\mu}$. Thus there is an infinite family of pullback measures $\\varepsilon_{\\mu}$ over $\\mathbb{T}^2$ for any measure $\\nu_{\\mu}$ on $I_{\\mu}$, that is, any $\\varepsilon_{\\mu}$ such that $\\varepsilon_{\\mu}(I_{\\mu}^{-1}[x]) = \\nu_{\\mu}(x)$ for all $x \\in I_{\\mu}$ will yield the probability measure $\\nu_{\\mu}$ on $I_{\\mu}$ as a pushforward measure by (\\ref{eqn:measures-T2-Ij_G2}).\nAs in \\cite{evans\/pugh:2012i}, we instead work with an intermediate probability measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ which lives over the joint spectrum $\\mathfrak{D}_{\\lambda,\\mu} \\subset I_{\\lambda} \\times I_{\\mu} \\subset \\mathbb{R}^2$, for $\\lambda,\\mu \\in P_{+}$, where there is no loss of dimension.\n\nA fundamental domain $C$ of $\\mathbb{T}^2$ under the action of the dihedral group $D_{12}$ is illustrated in Figure \\ref{fig:fund_domain-G2inT2}, where the axes are labelled by the parameters $\\theta_1$, $\\theta_2$ in $(e^{2 \\pi i \\theta_1},e^{2 \\pi i \\theta_2}) \\in \\mathbb{T}^2$, which is a quotient of the fundamental domain of $\\mathbb{T}^2\/S_3$ illustrated in Figure \\ref{fig:fund_domain-A2inT2} (see \\cite{evans\/pugh:2009v}) by the $\\mathbb{Z}_2$-action given by the matrix -1.\nNote that in Figure \\ref{fig:fund_domain-G2inT2}, the lines $\\theta_1=0$ and $\\theta_2=0$ are also boundaries of copies of the fundamental domain $C$ under the action of $D_{12}$, whereas in Figure \\ref{fig:fund_domain-A2inT2} they are not boundaries of copies of the fundamental domain under the action of $S_3$. The torus $\\mathbb{T}^2$ contains 12 copies of $C$, so that\n\\begin{equation} \\label{eqn:measureT2=12C}\n\\int_{\\mathbb{T}^2} \\phi(\\omega_1,\\omega_2) \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2) = 12 \\int_{C} \\phi(\\omega_1,\\omega_2) \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2),\n\\end{equation}\nfor any $D_{12}$-invariant function $\\phi:\\mathbb{T}^2 \\rightarrow \\mathbb{C}$ and $D_{12}$-invariant measure $\\varepsilon$ over $\\mathbb{T}^2$. The only fixed point of $\\mathbb{T}^2$ under the action of $D_{12}$ is the point $(1,1)$.\n\n\\begin{figure}[tb]\n\\begin{minipage}[t]{7.9cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-fund_domain-A2inT2.eps}\\\\\n \\caption{\\small A fundamental domain of $\\mathbb{T}^2\/S_3$.} \\label{fig:fund_domain-A2inT2}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{7.9cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-fund_domain-G2inT2.eps}\\\\\n \\caption{\\small A fundamental domain $C$ of $\\mathbb{T}^2\/D_{12}$.} \\label{fig:fund_domain-G2inT2}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\nLet $x_{\\lambda} = \\chi_{\\lambda}(\\omega_1,\\omega_2)$ and let $\\Psi_{\\lambda,\\mu}$ be the map $(\\omega_1,\\omega_2) \\mapsto (x_{\\lambda},x_{\\mu})$. We denote by $\\mathfrak{D}_{\\lambda,\\mu}$ the image of $\\Psi_{\\lambda,\\mu}(C) (= \\Psi_{\\lambda,\\mu}(\\mathbb{T}^2))$ in $\\mathbb{R}^2$.\nNote that we can identify $\\mathfrak{D}_{\\lambda,\\mu}$ with $\\mathfrak{D}_{\\mu,\\lambda}$ by reflecting about the line $x_{\\lambda} = x_{\\mu}$.\nThen the joint spectral measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ is the measure on $\\mathfrak{D}_{\\lambda,\\mu}$ uniquely determined by its cross-moments as in (\\ref{eqn:cross_moments_sa_operators}).\nThen there is a unique $D_{12}$-invariant pullback measure $\\varepsilon$ on $\\mathbb{T}^2$ such that\n\\begin{equation} \\label{eqn:measures-T2-D_G2}\n\\int_{\\mathfrak{D}_{\\lambda,\\mu}} \\psi(x_{\\lambda},x_{\\mu}) \\mathrm{d}\\widetilde{\\nu}_{\\lambda,\\mu}(x_{\\lambda},x_{\\mu}) = \\int_{\\mathbb{T}^2} \\psi(\\chi_{\\lambda}(\\omega_1,\\omega_2),\\chi_{\\mu}(\\omega_1,\\omega_2)) \\mathrm{d}\\varepsilon_{\\lambda,\\mu}(\\omega_1,\\omega_2),\n\\end{equation}\nfor any continuous function $\\psi:\\mathfrak{D}_{\\lambda,\\mu} \\rightarrow \\mathbb{C}$.\n\nAny probability measure on $\\mathfrak{D}_{\\lambda,\\mu}$ yields a probability measure on the interval $I_{\\lambda}$, given by the pushforward $(p_{\\lambda})_{\\ast}(\\widetilde{\\nu}_{\\lambda,\\mu})$ of the joint spectral measure $\\widetilde{\\nu}_{\\lambda,\\mu}$ under the orthogonal projection $p_{\\lambda}$ onto the spectrum $\\sigma(\\lambda)$ (see \\cite{evans\/pugh:2012i} for more details).\nSince the spectral measure $\\nu_{\\lambda}$ over $I_{\\lambda}$ is also uniquely determined by its (one-dimensional) moments $\\widetilde{\\varsigma}_m = \\int_{I_{\\lambda}} x_{\\lambda}^m \\mathrm{d}\\nu_{\\lambda}(x_{\\lambda})$ for all $m \\in \\mathbb{N}$, one could alternatively consider the moments in (\\ref{eqn:cross_moments_sa_operators}) with $n=0$ to determine the measure $\\nu_{\\lambda}$ over $I_{\\lambda}$.\n\nLet\n\\begin{equation} \\label{def:Dl}\nC_k^W = \\{ (e^{2 \\pi i q_1\/3(k+4)}, e^{2 \\pi i q_2\/3(k+4)}) \\in \\mathbb{T}^2 | \\; q_1,q_2 = 0, 1, \\ldots, 3k+11; \\, q_1 + q_2 \\equiv 0 \\textrm{ mod } 3 \\}\n\\end{equation}\nwhich is the support (over $\\mathbb{T}^2$) of the spectral measure of the nimrep graph $\\mathcal{A}_k(G_2)$ associated to the trivial $G_2$ modular invariant at level $k$.\nThe following $G_2$-invariant measures will be useful later, c.f. \\cite{evans\/pugh:2010i}.\n\\begin{Def} \\label{def:4measures}\nLet $\\omega = e^{2 \\pi i\/3}$, $\\tau = e^{2 \\pi i\/n}$. We define the following measures on $\\mathbb{T}^2$:\n\\begin{itemize}\n\\item[(1)] $\\mathrm{d}_m \\times \\mathrm{d}_n$, where $\\mathrm{d}_k$ is the uniform measure on the $k^{\\mathrm{th}}$ roots of unity, for $k \\in \\mathbb{N}$.\n\\item[(2)] $\\mathrm{d}^{(n)}$, the uniform measure on $C_n^W$ for $n \\in \\mathbb{N}$.\n\\item[(3)] $\\mathrm{d}^{((n))}$, the uniform measure on the $S_3$-orbit of the points $(\\tau, \\tau)$, $(\\overline{\\omega} \\, \\overline{\\tau}, \\omega)$, $(\\omega, \\overline{\\omega} \\, \\overline{\\tau})$, for $n \\in \\mathbb{Q}$, $n \\geq 2$.\n\\item[(4)] $\\mathrm{d}^{(n,k)}$, the uniform measure on the $S_3$-orbit of the points $(\\tau \\, e^{2 \\pi i k}, \\tau)$, $(\\tau, \\tau \\, e^{2 \\pi i k})$, $(\\overline{\\omega} \\, \\overline{\\tau}, \\omega \\, e^{2 \\pi i k})$, $(\\omega \\, e^{2 \\pi i k}, \\overline{\\omega} \\, \\overline{\\tau})$, $(\\overline{\\omega} \\, \\overline{\\tau} \\, e^{-2 \\pi i k}, \\omega \\, e^{-2 \\pi i k})$, $(\\omega \\, e^{-2 \\pi i k}, \\overline{\\omega} \\, \\overline{\\tau} \\, e^{-2 \\pi i k})$, for $n,k \\in \\mathbb{Q}$, $n > 2$, $0 \\leq k \\leq 1\/n$.\n\\end{itemize}\n\\end{Def}\n\nThe sets $\\mathrm{Supp}(\\mathrm{d}^{((n))})$, $\\mathrm{Supp}(\\mathrm{d}^{(n,k)})$ are illustrated in Figures \\ref{fig:poly-15}, \\ref{fig:poly-16} respectively, where $\\mathrm{Supp}(\\mathrm{d}\\mu)$ denotes the set of points $(\\theta_1,\\theta_2) \\in [0,1]^2$ such that $(e^{2 \\pi i \\theta_1}, e^{2 \\pi i \\theta_2})$ is in the support of the measure $\\mathrm{d}\\mu$. The white circles in Figure \\ref{fig:poly-16} denote the points given by the measure $\\mathrm{d}^{((n))}$. The cardinality $|\\mathrm{Supp}(\\mathrm{d}_m \\times \\mathrm{d}_n)|$ of $\\mathrm{Supp}(\\mathrm{d}_m \\times \\mathrm{d}_n)$ is $mn$, whilst $|\\mathrm{Supp}(\\mathrm{d}^{(n)})| = |D_n| = 3n^2$ was shown in \\cite[Section 7.1]{evans\/pugh:2009v}. For $n > 2$ and $0 < k < 1\/n$, $|\\mathrm{Supp}(\\mathrm{d}^{((n))})| = 18$, whilst $|\\mathrm{Supp}(\\mathrm{d}^{(n,k)})| = 36$. The cardinalities of the other sets are $|\\mathrm{Supp}(\\mathrm{d}^{(n,0)})| = |\\mathrm{Supp}(\\mathrm{d}^{(n,1\/n)})| = 18$ for $n > 2$, and $|\\mathrm{Supp}(\\mathrm{d}^{((2))})| = 9$.\nSome relations between these measures are given in \\cite[Section 2]{evans\/pugh:2010i}.\n\n\\begin{figure}[tb]\n\\begin{minipage}[t]{7.5cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{fig-poly-15.eps}\\\\\n \\caption{$\\mathrm{Supp}(\\mathrm{d}^{((n))})$} \\label{fig:poly-15}\n\\end{center}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{7.5cm}\n\\begin{center}\n  \\includegraphics[width=55mm]{fig-poly-16.eps}\\\\\n \\caption{$\\mathrm{Supp}(\\mathrm{d}^{(n,k)})$} \\label{fig:poly-16}\n\\end{center}\n\\end{minipage}\n\\end{figure}\n\n\n\\subsection{Spectral measures for $G_2$} \\label{sect:spec_measure-G2}\n\nHere we review the results, determined in \\cite{evans\/pugh:2012i}, for the spectral measures for $G_2$.\nLet $\\rho_1$, $\\rho_2$ denote the fundamental representations of $G_2$ of dimensions 7, 14 respectively.\nThe restrictions of the characters $\\chi_{\\rho_j}$ of the fundamental representations of $G_2$ to $\\mathbb{T}^2$ yield maps from the torus to the interval $I_j = \\chi_{\\rho_j}(\\mathbb{T}^2) \\subset \\mathbb{R}$:\n\\begin{align*}\n\\chi_{\\rho_1}(\\omega_1,\\omega_2) & = 1 + \\omega_1 + \\omega_1^{-1} + \\omega_2 + \\omega_2^{-1} + \\omega_1\\omega_2^{-1} + \\omega_1^{-1}\\omega_2 \\\\\n& = 1 + 2\\cos(2\\pi\\theta_1) + 2\\cos(2\\pi\\theta_2) + 2\\cos(2\\pi(\\theta_1-\\theta_2)), \\\\\n\\chi_{\\rho_2}(\\omega_1,\\omega_2) & = \\chi_{\\rho_1}(\\omega_1,\\omega_2) + 1 + \\omega_1\\omega_2 + \\omega_1^{-1}\\omega_2^{-1} + \\omega_1^2\\omega_2^{-1} + \\omega_1^{-2}\\omega_2 + \\omega_1\\omega_2^{-2} + \\omega_1^{-1}\\omega_2^2 \\\\\n= \\chi_{\\rho_1}&(\\omega_1,\\omega_2) + 1 + 2\\cos(2\\pi(\\theta_1+\\theta_2)) + 2\\cos(2\\pi(2\\theta_1-\\theta_2)) + 2\\cos(2\\pi(\\theta_1-2\\theta_2)),\n\\end{align*}\nwhere $\\omega_j = e^{2\\pi i \\theta_j} \\in \\mathbb{T}$ for $\\theta_j \\in [0,1]$, $j=1,2$.\n\nLet\n\\begin{equation} \\label{eqn:x,y-G2}\nx := \\chi_{\\rho_1}(\\omega_1,\\omega_2), \\qquad y := \\chi_{\\rho_2}(\\omega_1,\\omega_2),\n\\end{equation}\nand denote by $\\Psi$ the map $\\Psi_{(1,0),(1,1)}: (\\omega_1,\\omega_2) \\mapsto (x,y)$.\nThe image $\\mathfrak{D} = \\Psi(C)$ of the fundamental domain $C$ of $\\mathbb{T}^2\/D_{12}$ (illustrated in Figure \\ref{fig:fund_domain-G2inT2}) under $\\Psi$ is illustrated in Figure \\ref{fig:DomainD-G2}, where the boundaries of $C$ given by $\\theta_1 = 2\\theta_2$, $\\theta_1 = -\\theta_2$, $\\theta_1 = \\theta_2$ yield the curves $c_1$, $c_2$, $c_3$ respectively. These curves are given by \\cite{uhlmann\/meinel\/wipf:2007}\n\\begin{align*}\nc_1: && y & = -5(x+1)+2(x+2)^{3\/2}, \\qquad x \\in [-2,7], \\\\\nc_2: && y & = -5(x+1)-2(x+2)^{3\/2}, \\qquad x \\in [-2,-1], \\\\\nc_3: && 4y & = x^2+2x-7, \\hspace{28mm} x \\in [-1,7].\n\\end{align*}\nThe fixed point $(1,1)$ of $\\mathbb{T}^2$ under the action of $D_{12}$ maps to 7, 14 in the intervals $I_1$, $I_2$ respectively.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=60mm]{Fig-DomainD2-G2.eps}\\\\\n \\caption{The domain $\\mathfrak{D} = \\Psi(C)$.} \\label{fig:DomainD-G2}\n\\end{center}\n\\end{figure}\n\nUnder the change of variables (\\ref{eqn:x,y-G2}) the Jacobian is given by \\cite{evans\/pugh:2012i}\n\\begin{equation} \\label{eqn:J[theta]-D12}\n\\begin{split}\nJ & = 8 \\pi^2 (\\cos(2 \\pi (2\\theta_1 + \\theta_2)) + \\cos(2 \\pi (\\theta_1 - 3\\theta_2)) + \\cos(2 \\pi (3\\theta_1 - 2\\theta_2)) \\\\\n& \\qquad - \\cos(2 \\pi (\\theta_1 + 2\\theta_2)) - \\cos(2 \\pi (3\\theta_1 - \\theta_2)) - \\cos(2 \\pi (2\\theta_1 - 3\\theta_2))).\n\\end{split}\n\\end{equation}\nThe Jacobian is real and vanishes in $\\mathbb{T}^2$ only on the boundaries of the images of the fundamental domain $C$ under $D_{12}$.\nAgain, $J^2$ can be written in terms of the $D_{12}$-invariant elements $x$, $y$ as $J^2 = (4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)$ (see also \\cite{uhlmann\/meinel\/wipf:2007}), which is non-negative since $J$ is real. We write $J$ in terms of $x$ and $y$ as\n\\begin{eqnarray} \\label{eqn:J[x,y]-D12}\n|J| & = & 4 \\pi^2 \\sqrt{(4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)}.\n\\end{eqnarray}\n\n\n\n\n\n\n\n\n\\subsection{Spectral measures for nimrep graphs associated to $G_2$ modular invariants} \\label{sect:spec_measure-nimrepsG2}\n\nSuppose $G$ is the nimrep associated to a $G_2$ braided subfactor at some finite level $k$ with vertex set $G_0$.\nWe define a state $\\varphi$ on $\\ell^2(G_0)$ by $\\varphi( \\, \\cdot \\, ) = \\langle \\,\\cdot \\, e_{\\ast}, e_{\\ast} \\rangle$, where $e_{\\ast}$ is the basis vector in $\\ell^2(G_0)$ corresponding to the distinguished vertex $\\ast$ with lowest Perron-Frobenius weight.\n\nIf we consider the nimrep graphs $G_{\\lambda}$, $G_{\\mu}$, which have joint spectrum $\\mathfrak{D}_{\\lambda,\\mu}$, then the $m,n^{\\mathrm{th}}$ cross moment $\\varsigma_{m,n} = \\varphi(G_{\\lambda}^m G_{\\mu}^n) = \\int_{\\mathfrak{D}_{\\lambda,\\mu}} x^m y^n \\mathrm{d}\\widetilde{\\nu}(x,y)$, where $x=x_{\\lambda}$, $y=x_{\\mu}$, is given by $\\langle G_{\\lambda}^m G_{\\mu}^n e_{\\ast}, e_{\\ast} \\rangle$.\nLet $\\beta_{\\lambda}^{(\\nu)} = \\chi_{\\lambda}(t_{\\nu})$ be the eigenvalues of $G_{\\lambda}$, where $t_{\\nu}=(\\exp(\\xi(\\nu_1+1)),\\exp(-3\\xi (\\nu_2+1)))$ for $\\xi = 6\\pi i\/(k+4)$, with corresponding eigenvectors $\\psi^{(\\nu)}$ (note that the eigenvectors of $G_{\\lambda}$ are the same for all $\\lambda$). Each eigenvalue $\\beta_{\\lambda}^{(\\mu)}$ is also given by a ratio of the $S$-matrix for $G_2$ at level $k$, $\\beta_{\\lambda}^{(\\mu)} = S_{\\lambda\\mu}\/S_{0\\mu}$, where $\\mu \\in \\mathrm{Exp}(G) \\subset P^k_{+} = \\{ (\\lambda_1,\\lambda_2) | \\, \\lambda_1,\\lambda_2 \\geq 0; \\lambda_1 + 2\\lambda_2 \\leq k \\}$ are given by the modular invariant $Z$. Then $G_{\\lambda}^m G_{\\mu}^n = \\mathcal{U} \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast}$, where $\\Lambda_{\\lambda}$ is the diagonal matrix with the eigenvalues $\\beta_{\\lambda}^{(\\nu)}$ on the diagonal, and $\\mathcal{U}$ is the matrix whose columns are given by the eigenvectors $\\psi^{(\\nu)}$, so that\n\\begin{equation}\\label{eqn:moments-nimrep-G2}\n\\varsigma_{m,n} \\;\\; = \\;\\; \\langle \\mathcal{U} \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast} e_{\\ast}, e_{\\ast} \\rangle \\;\\; = \\;\\; \\langle \\Lambda_{\\lambda}^m \\Lambda_{\\mu}^n \\mathcal{U}^{\\ast} e_{\\ast}, \\mathcal{U}^{\\ast} e_{\\ast} \\rangle \\;\\; = \\;\\; \\sum_{\\nu} (\\beta_{\\lambda}^{(\\nu)})^m (\\beta_{\\mu}^{(\\nu)})^n |\\psi^{(\\nu)}_{\\ast}|^2,\n\\end{equation}\nwhere $\\psi^{(\\nu)}_{\\ast} = \\mathcal{U}^{\\ast} e_{\\ast}$ is the entry of the eigenvector $\\psi^{(\\nu)}$ corresponding to the distinguished vertex $\\ast$.\nThen there is a $D_{12}$-invariant measure $\\varepsilon$ over $\\mathbb{T}^2$ such that\n$$\\varsigma_{m,n} = \\int_{\\mathbb{T}^2} \\chi_{\\lambda}(\\omega_1,\\omega_2)^m \\chi_{\\mu}(\\omega_1,\\omega_2)^n \\mathrm{d}\\varepsilon(\\omega_1,\\omega_2),$$\nfor all $\\lambda$, $\\mu$.\n\nNote from (\\ref{eqn:moments-nimrep-G2}) that the measure $\\varepsilon$ is a discrete measure which has weight $|\\psi^{(\\nu)}_{\\ast}|^2$ at the points $g(t_{\\nu}) \\in \\mathbb{T}^2$ for $g \\in D_{12}$, $\\nu \\in \\mathrm{Exp}(G)$, and zero everywhere else. Thus the measure $\\varepsilon$ does not depend on the choice of $\\lambda$, $\\mu$, so that the spectral measure over $\\mathbb{T}^2$ is the same for any pair $(G_{\\lambda},G_{\\mu})$, even though the corresponding measures over $\\mathfrak{D}_{\\lambda,\\mu} \\subset \\mathbb{R}^2$, and indeed the subsets $\\mathfrak{D}_{\\lambda,\\mu}$ themselves, are different for each such pair. The same result holds for the spectral measure over $\\mathbb{T}^2$ of a finite subgroup of $G_2$.\n\n\n\n\n\n\n\\section{The finite subgroups of $G_2$} \\label{sect:subgroupsG2}\n\nThe classification of finite subgroups of $G_2$ is due to \\cite{wales:1970, cohen\/wales:1983} (see also \\cite{greiss:1995, he:2003}).\n\nThe reducible (i.e. block-diagonalizable) finite subgroups of $G_2$ are the finite discrete subgroups of $SU(2) \\times SU(2)$ and $SU(3)$ \\cite{wales:1970}. These subgroups are thus well known, and the corresponding spectral measures can be obtained from \\cite{banica\/bisch:2007, evans\/pugh:2009v, evans\/pugh:2010i}.\n\nThe irreducible finite subgroups of $G_2$, of which there are seven up to conjugacy in $G_2$ (or equivalently, up to conjugacy in $GL(V)$, where $V$ is the natural 7-dimensional module for $O(7,\\mathbb{C})$ \\cite[Corollary 1]{greiss:1995}), can be further classified into two types, primitive and imprimitive, where a linear group $\\Gamma \\subset GL(V)$ is imprimitive if there is a non-trivial decomposition $V=\\bigoplus_i V_i$ such that $\\Gamma$ permutes the $V_i$. There are two imprimitive finite subgroups and five primitive ones.\nThese finite subgroups are listed in Table \\ref{Table:subgroupsG2}, where type denotes whether an irreducible subgroup is primitive (P) or imprimitive (I).\n\n\\renewcommand{\\arraystretch}{1}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c|c|c|} \\hline\nSubgroup $\\Gamma \\subset G_2$ & Type & $|\\Gamma|$ \\\\\n\\hline\\hline finite subgroups of $SU(2) \\times SU(2)$, $SU(3)$ & - & - \\\\\n\\hline $PSL(2;7) \\cong GL(3;2) \\cong \\Sigma (168) \\subset SU(3)$ & I & 168 \\\\\n\\hline $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ & I & 1344 \\\\\n\\hline $PGL(2;7)$ & P & 336 \\\\\n\\hline $PSL(2;8)$ & P & 504 \\\\\n\\hline $PSL(2;13)$ & P & 1092 \\\\\n\\hline $PU(3;3) \\cong G_2(2)'$ & P & 6048 \\\\\n\\hline $G_2(2)$ & P & 12096 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Finite subgroups of $G_2$.} \\label{Table:subgroupsG2}\n\\end{center}\n\\end{table}\n\nThe McKay graph $\\mathcal{G}^{\\rho}_{\\Gamma}$ is the the fusion graph of the irreducible representation $\\rho$ of $\\Gamma$ acting on the irreducible representations of $\\Gamma$.\nThis graph determines the Bratteli diagram for the tower of relative commutants of the subfactor $P^{\\Gamma} \\subset (M_n \\otimes P)^{\\Gamma}$, where $n$ is the dimension the representation $\\rho$ and $P$ is the type $\\mathrm{II}_1$ factor $\\bigotimes_{n=1}^{\\infty} M_n$ \\cite[$\\S$VI]{wassermann:1988}. This graph is not however the principal graph of this subfactor as it is not bipartite. The prinicpal graph is rather an unfolded version of the McKay graph $\\mathcal{G}^{\\rho}_{\\Gamma}$, with adjacency matrix given by $\\left( \\begin{array}{cc} 0 & \\Delta^{\\rho}_{\\Gamma} \\\\ \\Delta^{\\rho}_{\\Gamma} & 0 \\end{array} \\right)$, where $\\Delta^{\\rho}_{\\Gamma}$ is the adjacency matrix of the (folded) graph $\\mathcal{G}^{\\rho}_{\\Gamma}$.\n\nWe will consider the (joint) spectral measure for the McKay graphs $\\mathcal{G}^j_{\\Gamma} := \\mathcal{G}^{\\varrho_j}_{\\Gamma}$ associated to a finite subgroup $\\Gamma \\subset G_2$, where $\\varrho_j$ are the restrictions of the fundamental representations $\\rho_j$ of $G_2$ to $\\Gamma$, $j=1,2$.\nWe will consider all possible embeddings of the subgroup in $G_2$. Any two such embeddings are conjguate in $G_2$ if and only if they afford the same character on the seven-dimensional representation $\\rho_1$ \\cite[Corollary 1]{greiss:1995}. In some cases there is more than one non-conjugate embedding of the subgroup in $G_2$.\nIn these cases the restricted representation $\\varrho_1$ is not necessarily irreducible.\nIn all cases, even for irreducible $\\varrho_1$, the restriction $\\varrho_2$ of the 14-dimensional representation is not necessarily irreducible.\n\nWe use the following methods to determine embeddings $\\varrho_1$ of $\\Gamma$ in $G_2$.\nFirst, take a seven-dimensional (not necessarily irreducible) representation $\\gamma_1$ of $\\Gamma$.\nThe Kronecker square of $\\rho_1$ decomposes into irreducible representations of $G_2$ as $\\rho_1^2 = \\mathrm{id}_{G_2} + \\rho_1 + \\rho_2 + \\lambda_{(2,0)}$, where $\\lambda_{(2,0)}$ has dimension 21. The Kronecker square of $\\varrho_1$ is obtained by restricting this decomposition to $\\Gamma$, and we see that $\\varrho_1$ appears in the decomposition of $\\varrho_1^2$ into irreducible representations of $\\Gamma$. Thus, if $\\gamma_1$ is not contained in the decomposition of $\\gamma_1^2$ into irreducible representations of $\\Gamma$, we can eliminate $\\gamma_1$ as a possible restriction $\\varrho_1$ of $\\rho_1$. The decomposition of $\\gamma_1^2$ into irreducible representations can be obtained using the character table for $\\Gamma$, by decomposing the character $\\chi_{\\gamma_1^2} = \\chi_{\\gamma_1}^2$ of $\\gamma_1^2$ into the characters of the irreducible representations $\\lambda$ of $\\Gamma$: $\\chi_{\\gamma_1}^2 = \\sum_{\\lambda} a_{\\lambda} \\chi_{\\lambda}$, where $a_{\\lambda} = \\langle \\gamma_1^2, \\lambda \\rangle\/|\\Gamma| = \\sum_{g \\in \\Gamma} \\chi_{\\gamma_1^2}(g)\\chi_{\\lambda}(g)\/|\\Gamma|$.\n\nWe next consider the eigenvalues of the representation matrices of $\\Gamma$. If the elements in a conjugacy class $C_n$ of $\\Gamma$ have order $n$, then $(C_n)^n = Z(\\Gamma)$, where $Z(\\Gamma)$ is the center of $\\Gamma$. If the center is trivial, then the eigenvalues $\\xi$ of the matrices representing these elements must satisfy $\\xi^n = 1$ (this is the case for $PSL$, $PGL$ and $PU(3;3)$). Since $\\chi_{\\lambda}(\\Gamma_j)$ is the sum of the eigenvalues $\\xi$, it is usually possible to write down the complete set of eigenvalues from the information provided by the character table of $\\Gamma$ and the fact that the eigenvalues must be powers of $n^{\\mathrm{th}}$ roots of unity.\nWhere there is some ambiguity, we can pin down the correct choice for the set of eigenvalues from the following considerations. Suppose there is ambiguity regarding the eigenvalues of the conjugacy class $C_{mn}$ whose elements have order $mn$, $m,n \\in \\mathbb{N}$. If there is only one conjugacy class $C_n$ whose elements have order $n$, then for $g \\in C_{mn}$, $g^m \\in C_n$, and since $g, g^m$ commute, their corresponding representation matrices can be simultaneously diagonalised, and thus the eigenvalues of $C_n$ must be $m^{\\mathrm{th}}$ powers of those for $C_{mn}$. Suppose now that there is more than one conjugacy class $C_n^{(j)}$ whose elements have order $n$. Since $g^m$ are all conjugate for conjugate $g$, we see that there exists a $j$ such that $g^m \\in C_n^{(j)}$ for all $g \\in C_{mn}$. It turns out in all the cases considered here that there is only one consistent choice of $j$ such that the eigenvalues of $C_n^{(j)}$ are $m^{\\mathrm{th}}$ powers of those for $C_{mn}$ for all (irreducible) representations.\n\nAs was shown in \\cite[$\\S$4]{evans\/pugh:2009v}, the eigenvalues of the representation matrices of $\\Gamma$ can be written in the form $\\chi_{\\varrho_j}(C) = \\mathrm{Tr}(\\varrho_j(g))$, where $g$ is any element of the conjugacy class $C$ of $\\Gamma$.\nEvery element $g \\in \\Gamma$ is conjugate to an element $d$ in the maximal torus of $G_2$, i.e. $\\varrho_j(h^{-1}gh) = \\varrho_j(d) = (\\varrho_j|_{\\mathbb{T}^2})(t_1,t_2)$ for some $(t_1,t_2) \\in \\mathbb{T}^2$, for $j=1,2$, where $\\varrho_j|_{\\mathbb{T}^2}$ is given by \\cite{evans\/pugh:2012i}\n\\begin{align}\n(\\rho_1|_{\\mathbb{T}^2})&(t_1,t_2) = \\textrm{diag}(D(t_1), D(t_2^{-1}), D(t_1^{-1}t_2), 1), \\label{eqn:restrict_rho1G2_to_T2} \\\\\n(\\rho_2|_{\\mathbb{T}^2})&(t_1,t_2) \\nonumber \\\\\n&= \\textrm{diag}(D(t_1), D(t_2^{-1}), D(t_1^{-1}t_2), D(1), D(t_1t_2), D(t_1^2t_2^{-1}), D(t_1^{-1}t_2^2)), \\label{eqn:restrict_rho2G2_to_T2}\n\\end{align}\nwhere $D(t_i) = \\left( \\begin{array}{cc} \\mathrm{Re}(t_i) & -\\mathrm{Im}(t_i) \\\\ \\mathrm{Im}(t_i) & \\mathrm{Re}(t_i) \\end{array} \\right)$ for $t_i \\in \\mathbb{T}$.\nNow $\\mathrm{Tr}(\\varrho_j(g)) = Tr(\\varrho_j(d)) = \\Phi_j(t_1,t_2)$, thus the eigenvalues of $\\varrho_j(g)$ are all of the form $\\Phi_j(\\omega_1,\\omega_2)$ for $\\omega_1,\\omega_2 \\in \\mathbb{T}$, and hence its spectrum is contained in the interval $I_j$.\nAs shown in \\cite[Sections 3,4]{evans\/pugh:2012i} the spectrum of the fundamental representation $\\rho_j$ of $G_2$, and its restriction to $\\mathbb{T}^2$, is the whole of the interval $I_j = \\chi_{\\rho_j}(\\mathbb{T}^2)$, for $j=1,2$.\nThus the support of the spectral measure $\\mu_{\\Delta_j}$ of $\\Delta_j = \\Delta_{\\mathcal{G}^j_{\\Gamma}}$, the adjacency matrix of $\\mathcal{G}^j_{\\Gamma}$, is contained in $I_j$ when $\\Gamma$ is $G_2$ or one of its finite subgroups.\nThen for $\\Gamma \\subset G_2$, the eigenvalues of every group element in $\\varrho_1$ are necessarily of the form $\\mathcal{E}_{t_1,t_2} := \\{ 1,t_1,t_1^{-1},t_2,t_2^{-1},t_1t_2^{-1},t_1^{-1}t_2 \\}$, where $t_i \\in \\mathbb{T}$. Thus, by \\cite[Proposition]{king\/toumazet\/wybourne:1999}, if the eigenvalues of the group elements in the representation $\\gamma_1$ have this form then $\\gamma_1$ is a restriction $\\varrho_1$ of the seven-dimensional fundamental representation $\\rho_1$ of $G_2$ to $\\Gamma$.\n\nWe now turn to consider the possible restrictions $\\varrho_2$ of the fourteen-dimensional fundamental representation $\\rho_2$ of $G_2$ to $\\Gamma$, for fixed $\\varrho_1$.\nBy dimension considerations one can determine the possible candidates for $\\varrho_2$ from the Kronecker square of $\\varrho_1$.\nLet $\\gamma_2$ be such a candidate, i.e. a fourteen-dimensional representation such that $\\varrho_1^2 = \\mathrm{id}_{\\Gamma} + \\varrho_1 + \\gamma_2 + \\lambda$, where $\\lambda$ is (necessarily) some 27-dimensional representation of $\\Gamma$.\nWe make a choice of pair $(t_1^{C},t_2^{C})$ from the set $X_C$ of eigenvalues of group elements (from the conjugacy class $C$) in $\\varrho_1$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$.\nNote that the choice of $(t_1^C,t_2^C)$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ is not unique. However, any other pair $(\\tilde{t}_1^C,\\tilde{t}_2^C)$ such that $\\mathcal{E}_{\\tilde{t}_1^C,\\tilde{t}_2^C} = X_C$ will appear in the orbit of $(t_1^C,t_2^C)$ under the action of the Weyl group $D_{12}$ of $G_2$, where the action of $D_{12}$ on $\\mathbb{T}^2$ is given in Section \\ref{sect:measures-different_domains}. Thus $\\Phi_j(\\tilde{t}_1^C,\\tilde{t}_2^C) = \\Phi_j(t_1^C,t_2^C)$ for $j=1,2$.\nThen one checks that $\\Phi_2(t_1^C,t_2^C) = \\chi_{\\gamma_2}(C)$ for each conjugacy class $C$, in which case $\\gamma_2$ is indeed a restriction $\\varrho_2$ of the fourteen-dimensional fundamental representation $\\rho_2$ of $G_2$ to $\\Gamma$.\n\n\\subsection{Spectral measures for finite subgroups of $G_2$} \\label{sect:spec_measure-subgroupsG2}\n\nThere is an $S$-matrix, with rows, columns labelled by the irreducible characters, conjugacy classes respectively of $\\Gamma$, which simultaneously diagonalizes the representations of $\\Gamma$ \\cite{kawai:1989}. The entries of this matrix for the trivial representation 0 are given by $S_{0,C} = \\sqrt{|C|} \\chi_0(C)\/\\sqrt{|\\Gamma|} = \\sqrt{|C|}\/\\sqrt{|\\Gamma|}$, for conjugacy class $C$.\nThen the $m,n^{\\mathrm{th}}$ moment $\\varsigma_{m,n}$ is given over $\\mathfrak{D}$ by (c.f. \\cite[Section 4]{evans\/pugh:2009v} for the case of finite subgroups of $SU(2)$)\n\\begin{equation} \\label{eqn:moments-subgroupG2}\n\\varsigma_{m,n} \\; = \\; \\int_{\\mathfrak{D}} x^m y^n \\mathrm{d}\\nu(x,y) \\; = \\; \\sum_C \\frac{|C|}{|\\Gamma|} \\chi_{\\varrho_1} (C)^m \\chi_{\\varrho_2} (C)^n.\n\\end{equation}\nThere is an analogous statement to (\\ref{eqn:moments-subgroupG2}) for the joint spectral measure $\\nu_{\\lambda,\\mu}$ over $\\mathfrak{D}_{\\lambda,\\mu}$ for any irreducible representations $\\lambda$, $\\mu$ of $\\Gamma$. The weight on the right hand side will again be $|C|\/|\\Gamma|$, since the same $S$-matrix simultaneously diagonalises all the representations of $\\Gamma$. Since this weight does not depend on the representations $\\lambda$, $\\mu$, we see that the $D_{12}$-invariant pullback measure $\\varepsilon$ over $\\mathbb{T}^2$ will be the same for any joint spectral measure $\\nu_{\\lambda,\\mu}$. This is analogous to the situation for nimrep graphs discussed in Section \\ref{sect:spec_measure-nimrepsG2}.\n\nWe wish to compute `inverse' maps $\\widetilde{\\Psi}: \\mathfrak{D} \\rightarrow \\mathbb{T}^2$ such that $\\Psi \\circ \\widetilde{\\Psi} = \\mathrm{id}$.\nThe following equation can easily be checked by substituting in $x=\\Phi_1(\\omega_1,\\omega_2)$, $y=\\Phi_2(\\omega_1,\\omega_2)$:\n$$(\\omega_j+\\omega_j^{-1})^3 + (1-x)(\\omega_j+\\omega_j^{-1})^2 + (y-2)(\\omega_j+\\omega_j^{-1}) +2y-x^2+2x-1 = 0,$$\nwhere $j=1,2$.\nSolving this cubic in $\\omega_j+\\omega_j^{-1} = 2\\cos(\\vartheta_j)$, we obtain for $l \\in \\{ 0,1,2 \\}$\n$$\\vartheta_j(l) = \\cos^{-1}\\left( \\frac{1}{6} \\left( x-1 + 2^{-1\/3} \\epsilon_l P + 2^{1\/3} \\overline{\\epsilon_l} (x^2-2x+7-3y)P^{-1} \\right) \\right)$$\nwhere $2^{1\/3}$ takes a real value, $\\epsilon_l = e^{2 \\pi i l\/3}$ and $P = (2x^3+21x^2-30x+7-45y-9xy + \\sqrt{(4x^3-x^2-2x-10xy-y^2-10y+7)(x^2+2x-7-4y)} \\, )^{1\/3}$. We note that for the roots of a cubic equation it does not matter whether the square root in $P$ is taken to be positive or negative.\nThen we set $\\Psi_{l,l'}(x,y) = (e^{\\vartheta_1(l)i},e^{\\vartheta_2(l')i})$, and we have that $\\Psi(\\Psi_{l,l'}(x,y)) = (x,y)$ for some $l,l' \\in \\{ 0,1,2 \\}$. The particular choice of pair $l,l'$ such that the equality $\\Psi \\circ \\Psi_{l,l'} = \\mathrm{id}$ is satisfied depends on $x,y$, but it is easy to check (eg. using Mathematica) whether a given choice satisfies this equality for any of the examples we consider.\nWe present in Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)} the values of the eigenvalues $(\\chi_{\\varrho_1}(C), \\chi_{\\varrho_2}(C)) = (x,y) \\in \\mathfrak{D}$ which will appear for the finite subgroups of $G_2$, and (the orbits under $D_{12}$ of) the corresponding points $(\\theta_1,\\theta_2) \\in [0,1]^2$ such that $\\Psi(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2}) = (x,y)$.\n\n\\renewcommand{\\arraystretch}{1.4}\n\n\\begin{table}[tbp]\n\\begin{center}\n\\begin{tabular}{|c|c|} \\hline\n$(x,y) \\in \\mathfrak{D}$ & Orbit of $(\\theta_1,\\theta_2) \\in [0,1]^2$ \\\\\n\\hline $(7,14)$ & $(0,0)$ \\\\\n\\hline $(-2,5)$ & $\\left(\\frac{1}{3},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{3}\\right)$ \\\\\n\\hline $(-1,-2)$ & $\\left(0,\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{2}\\right), \\left(\\frac{1}{2},0\\right)$ \\\\\n\\hline $(1,-1)$ & $\\left(0,\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{1}{3}\\right), \\left(\\frac{1}{3},0\\right), \\left(0,\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{2}{3}\\right), \\left(\\frac{2}{3},0\\right)$ \\\\\n\\hline $(3,2)$ & $\\left(0,\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{1}{4}\\right), \\left(\\frac{1}{4},0\\right), \\left(0,\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{3}{4}\\right), \\left(\\frac{3}{4},0\\right)$ \\\\\n\\hline $(-1,2)$ & $\\left(\\frac{1}{4},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{4}\\right)$ \\\\\n\\hline $(2,1)$ & $\\left(\\frac{1}{6},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{1}{6}\\right), \\left(\\frac{1}{6},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{1}{6}\\right)$ \\\\\n\\hline $(-1,1)$ & $\\left(\\frac{1}{6},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{5}{6}\\right), \\left(\\frac{5}{6},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{6}\\right),$ \\\\\n& $\\left(\\frac{1}{2},\\frac{1}{6}\\right), \\left(\\frac{1}{3},\\frac{1}{2}\\right), \\left(\\frac{5}{6},\\frac{1}{3}\\right), \\left(\\frac{1}{2},\\frac{5}{6}\\right), \\left(\\frac{2}{3},\\frac{1}{2}\\right), \\left(\\frac{1}{6},\\frac{2}{3}\\right)$ \\\\\n\\hline $(0,0)$ & $\\left(\\frac{1}{7},\\frac{3}{7}\\right), \\left(\\frac{3}{7},\\frac{2}{7}\\right), \\left(\\frac{2}{7},\\frac{6}{7}\\right), \\left(\\frac{6}{7},\\frac{4}{7}\\right), \\left(\\frac{4}{7},\\frac{5}{7}\\right), \\left(\\frac{5}{7},\\frac{1}{7}\\right),$ \\\\\n& $\\left(\\frac{3}{7},\\frac{1}{7}\\right), \\left(\\frac{2}{7},\\frac{3}{7}\\right), \\left(\\frac{6}{7},\\frac{2}{7}\\right), \\left(\\frac{4}{7},\\frac{6}{7}\\right), \\left(\\frac{5}{7},\\frac{4}{7}\\right), \\left(\\frac{1}{7},\\frac{5}{7}\\right)$ \\\\\n\\hline $(1,0)$ & $\\left(\\frac{1}{8},\\frac{3}{8}\\right), \\left(\\frac{3}{8},\\frac{1}{4}\\right), \\left(\\frac{1}{4},\\frac{7}{8}\\right), \\left(\\frac{7}{8},\\frac{5}{8}\\right), \\left(\\frac{5}{8},\\frac{3}{4}\\right), \\left(\\frac{3}{4},\\frac{1}{8}\\right),$ \\\\\n& $\\left(\\frac{3}{8},\\frac{1}{8}\\right), \\left(\\frac{1}{4},\\frac{3}{8}\\right), \\left(\\frac{7}{8},\\frac{1}{4}\\right), \\left(\\frac{5}{8},\\frac{7}{8}\\right), \\left(\\frac{3}{4},\\frac{5}{8}\\right), \\left(\\frac{1}{8},\\frac{3}{4}\\right)$ \\\\\n\\hline $(-1,0)$ & $\\left(\\frac{1}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{3}{8}\\right), \\left(\\frac{3}{8},\\frac{7}{8}\\right), \\left(\\frac{7}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{2},\\frac{5}{8}\\right), \\left(\\frac{5}{8},\\frac{1}{8}\\right),$ \\\\\n& $\\left(\\frac{1}{2},\\frac{1}{8}\\right), \\left(\\frac{3}{8},\\frac{1}{2}\\right), \\left(\\frac{7}{8},\\frac{3}{8}\\right), \\left(\\frac{1}{2},\\frac{7}{8}\\right), \\left(\\frac{5}{8},\\frac{1}{2}\\right), \\left(\\frac{1}{8},\\frac{5}{8}\\right)$ \\\\\n\\hline $(-p,p+q+1)$ & $\\left(\\frac{1}{9},\\frac{4}{9}\\right), \\left(\\frac{4}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{8}{9}\\right), \\left(\\frac{8}{9},\\frac{5}{9}\\right), \\left(\\frac{5}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{9}\\right),$ \\\\\n& $\\left(\\frac{4}{9},\\frac{1}{9}\\right), \\left(\\frac{1}{3},\\frac{4}{9}\\right), \\left(\\frac{8}{9},\\frac{1}{3}\\right), \\left(\\frac{5}{9},\\frac{8}{9}\\right), \\left(\\frac{2}{3},\\frac{5}{9}\\right), \\left(\\frac{1}{9},\\frac{2}{3}\\right)$ \\\\\n\\hline $(-q,1-p)$ & $\\left(\\frac{1}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{2}{9}\\right), \\left(\\frac{2}{9},\\frac{8}{9}\\right), \\left(\\frac{8}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{7}{9}\\right), \\left(\\frac{7}{9},\\frac{1}{9}\\right),$ \\\\\n& $\\left(\\frac{1}{3},\\frac{1}{9}\\right), \\left(\\frac{2}{9},\\frac{1}{3}\\right), \\left(\\frac{8}{9},\\frac{2}{9}\\right), \\left(\\frac{2}{3},\\frac{8}{9}\\right), \\left(\\frac{7}{9},\\frac{2}{3}\\right), \\left(\\frac{1}{9},\\frac{7}{9}\\right)$ \\\\\n\\hline $(p+q,1-q)$ & $\\left(\\frac{2}{9},\\frac{5}{9}\\right), \\left(\\frac{5}{9},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{7}{9}\\right), \\left(\\frac{7}{9},\\frac{4}{9}\\right), \\left(\\frac{4}{9},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{2}{9}\\right),$ \\\\\n& $\\left(\\frac{5}{9},\\frac{2}{9}\\right), \\left(\\frac{1}{3},\\frac{5}{9}\\right), \\left(\\frac{7}{9},\\frac{1}{3}\\right), \\left(\\frac{5}{9},\\frac{7}{9}\\right), \\left(\\frac{2}{3},\\frac{4}{9}\\right), \\left(\\frac{2}{9},\\frac{2}{3}\\right)$ \\\\\n\\hline $(0,-1)$ & $\\left(\\frac{1}{12},\\frac{5}{12}\\right), \\left(\\frac{5}{12},\\frac{1}{3}\\right), \\left(\\frac{1}{3},\\frac{11}{12}\\right), \\left(\\frac{11}{12},\\frac{7}{12}\\right), \\left(\\frac{7}{12},\\frac{2}{3}\\right), \\left(\\frac{2}{3},\\frac{1}{12}\\right),$ \\\\\n& $\\left(\\frac{5}{12},\\frac{1}{12}\\right), \\left(\\frac{1}{3},\\frac{5}{12}\\right), \\left(\\frac{11}{12},\\frac{1}{3}\\right), \\left(\\frac{7}{12},\\frac{11}{12}\\right), \\left(\\frac{2}{3},\\frac{7}{12}\\right), \\left(\\frac{1}{12},\\frac{2}{3}\\right)$ \\\\\n\\hline $\\left(\\frac{1+\\sqrt{13}}{2},1\\right)$ & $\\left(\\frac{1}{13},\\frac{4}{13}\\right), \\left(\\frac{4}{13},\\frac{3}{13}\\right), \\left(\\frac{3}{13},\\frac{12}{13}\\right), \\left(\\frac{12}{13},\\frac{9}{13}\\right), \\left(\\frac{9}{13},\\frac{10}{13}\\right), \\left(\\frac{10}{13},\\frac{1}{13}\\right),$ \\\\\n& $\\left(\\frac{4}{13},\\frac{1}{13}\\right), \\left(\\frac{3}{13},\\frac{4}{13}\\right), \\left(\\frac{12}{13},\\frac{3}{13}\\right), \\left(\\frac{9}{13},\\frac{12}{13}\\right), \\left(\\frac{10}{13},\\frac{9}{13}\\right), \\left(\\frac{1}{13},\\frac{10}{13}\\right)$ \\\\\n\\hline $\\left(\\frac{1-\\sqrt{13}}{2},1\\right)$ & $\\left(\\frac{2}{13},\\frac{7}{13}\\right), \\left(\\frac{7}{13},\\frac{5}{13}\\right), \\left(\\frac{5}{13},\\frac{11}{13}\\right), \\left(\\frac{11}{13},\\frac{6}{13}\\right), \\left(\\frac{6}{13},\\frac{8}{13}\\right), \\left(\\frac{8}{13},\\frac{2}{13}\\right),$ \\\\\n& $\\left(\\frac{7}{13},\\frac{2}{13}\\right), \\left(\\frac{5}{13},\\frac{7}{13}\\right), \\left(\\frac{11}{13},\\frac{5}{13}\\right), \\left(\\frac{6}{13},\\frac{11}{13}\\right), \\left(\\frac{8}{13},\\frac{6}{13}\\right), \\left(\\frac{2}{13},\\frac{8}{13}\\right)$ \\\\\n\\hline\n\\end{tabular}\n\\caption{$(x,y) \\in \\mathfrak{D}$ and (the orbits of) the corresponding points $(\\theta_1,\\theta_2) \\in [0,1]^2$. Here $p=2\\cos(4\\pi\/9)$, $q=2\\cos(8\\pi\/9)$} \\label{Table:subgroupsG2-orbits(theta1,theta2)}\n\\end{center}\n\\end{table}\n\n\\renewcommand{\\arraystretch}{1}\n\n\n\\section{Group $PSL(2;7) \\cong GL(3;2) \\cong \\Sigma (168)$} \\label{sect:II1}\n\nThe subgroup $PSL(2;7)$ of $G_2$ is an irreducible imprimitive group of order 168 which is isomorphic to the group $GL(3;2)$, and also to the subgroup $\\Sigma(168)$ of $SU(3)$ which was considered in \\cite{evans\/pugh:2010i}.\n\nThe group $PSL(2;7)$ has irreducible real representations $\\Sigma_d$ of dimensions $d = 1,6,7,8$, and two complex conjugate irreducible representations $\\Sigma_3, \\Sigma_3^{\\ast}$ of dimension 3. Its character table is given in Table \\ref{table:Character_table-II1} \\cite{littlewood:1934}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $(C_7,C_7^2,C_7^4)$ & $(C_7^3,C_7^5,C_7^6)$ & $(C_4,C_4^3)$ & $(C_3,C_3^2)$ \\\\\n\\hline $|C|$ & 1 & 21 & 24 & 24 & 42 & 56 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_3$ & 3 & -1 & $w$ & $\\overline{w}$ & 1 & 0 \\\\\n\\hline $\\Sigma_3^{\\ast}$ & 3 & -1 & $\\overline{w}$ & $w$ & 1 & 0 \\\\\n\\hline $\\Sigma_6$ & 6 & 2 & -1 & -1 & 0 & 0  \\\\\n\\hline $\\Sigma_7$ & 7 & -1 & 0 & 0 & -1 & 1  \\\\\n\\hline $\\Sigma_8$ & 8 & 0 & 1 & 1 & 0 & -1  \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for group $PSL(2;7)$, where $w = \\eta + \\eta^2 + \\eta^4 = (-1+i\\sqrt{7})\/2$, $\\eta = e^{2\\pi i\/7}$.} \\label{table:Character_table-II1}\n\\end{center}\n\\end{table}\n\nThere are two non-conjugate embeddings of $PSL(2;7)$ in $G_2$ \\cite{king\/toumazet\/wybourne:1999}, given by $\\varrho_1^{(1)} = \\Sigma_7$ and $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$.\nThe McKay graph $\\mathcal{G}^{\\varrho_1^{(1)}}_{PSL(2;7)}$ for $\\varrho_1^{(1)}$ is given in \\cite[Figure 1]{he:2003}. We reproduce it in Figure \\ref{Fig-McKay_Graph-II1-rho1} for completeness, along with the McKay graph $\\mathcal{G}^{\\varrho_1^{(2)}}_{PSL(2;7)}$ for $\\varrho_1^{(2)}$. We use the notation $n$, $n^{\\ast}$ to label the vertices corresponding to the irreducible representations $\\Sigma_n$, $\\Sigma_n^{\\ast}$ respectively.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=110mm]{Fig-McKay_Graph-II1-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PSL(2;7)}$, $i=1,2$.} \\label{Fig-McKay_Graph-II1-rho1}\n\\end{center}\n\\end{figure}\n\nThe eigenvalues of the representation matrices are given in \\cite[Tables 4a,b]{king\/toumazet\/wybourne:1999}.\nThe decomposition of the Kronecker square of $\\varrho_1^{(i)}$ into irreducibles is given by\n$$(\\varrho_1^{(1)})^2 = \\mathrm{id} + \\varrho_1^{(1)} + \\Sigma_3 + \\Sigma_3^{\\ast} + 2\\Sigma_6 + \\Sigma_7 + 2 \\Sigma_8, \\qquad\n(\\varrho_1^{(2)})^2 = \\mathrm{id} + \\varrho_1^{(2)} + \\Sigma_1 + 2\\Sigma_3 + 2\\Sigma_3^{\\ast} + 2\\Sigma_6 + 2 \\Sigma_8,$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2^{(i)}$, which are given by $\\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ and $\\Sigma_6 + \\Sigma_8$ for both $i=1,2$.\nHowever, as discussed in Section \\ref{sect:subgroupsG2}, since $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$, where $(t_1^{C},t_2^{C})$ is a pair from the set of eigenvalues of group elements from the conjugacy class $C$, from knowledge of the eigenvalues from \\cite{king\/toumazet\/wybourne:1999} we see that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;7)$ is given by $\\varrho_2 := \\varrho_2^{(i)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ for both $i=1,2$.\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\rho_2}(C) \\in [-2,14]$ for $PSL(2;7)$ are given in Table \\ref{table:(x,y)-II1}, along with the values of $J^2\/64\\pi^4$ for the corresponding pairs $(\\theta_1,\\theta_2) \\in [0,1]^2$ obtained from Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $(C_7,C_7^2,C_7^4)$ & $(C_7^3,C_7^5,C_7^6)$ & $(C_4,C_4^3)$ & $(C_3,C_3^2)$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(C) \\in [-2,7]$ & 7 & -1 & 0 & 0 & -1 & 1  \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(C) \\in [-2,7]$ & 7 & -1 & 0 & 0 & 3 & 1  \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -2 & 0 & 0 & 2 & -1 \\\\\n\\hline $J^2(\\theta_1,\\theta_2)\/64\\pi^4$ & 0 & 0 & 49\/4 & 49\/4 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;7)$, $j=1,2$.} \\label{table:(x,y)-II1}\n\\end{center}\n\\end{table}\n\nLet $\\Omega(\\theta_1,\\theta_2) := \\Phi_1(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2})^m \\Phi_2(e^{2\\pi i \\theta_1},e^{2\\pi i \\theta_2})^n \\in \\mathfrak{D}$ and $\\Omega^W(\\theta_1,\\theta_2)$ its orbit under $W=D_{12}$, $\\Omega^W(\\theta_1,\\theta_2) := \\sum_{g \\in D_{12}} \\Omega(g(\\theta_1,\\theta_2))\/12$.\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-II1}, \\ref{table:(x,y)-II1} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n$$\\varsigma_{m,n} = \\frac{1}{168} \\Omega^W(0,0) + \\frac{21}{168} \\Omega^W(0,1\/2) + \\frac{56}{168} \\Omega^W(0,1\/3) + \\frac{42}{168} \\Omega' + \\frac{24+24}{168} \\Omega^W(1\/7,3\/7),$$\nwhere $\\Omega'$ is $\\Omega^W(1\/4,1\/2)$ for $\\varrho_1^{(1)}$, and $\\Omega^W(0,1\/4)$ for $\\varrho_1^{(2)}$.\nIt is easy to see that $\\Omega^W(0,0) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\mathrm{d}_1 \\times \\mathrm{d}_1$ and $3\\Omega^W(0,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\mathrm{d}_1 \\times \\mathrm{d}_1)$.\nNow $12\\Omega^W(1\/7,3\/7) = 4 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, as illustrated in Figure \\ref{Fig-OmegaWII1}$(a)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/7,3\/7)\/64\\pi^4) = 49\/4$ for all $g \\in D_{12}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWII1}\\\\\n \\caption{The orbits of $(a)$ $(1\/7,3\/7)$, \\mbox{$(b)$ $(0,1\/4) \\, \\bullet$ and $(1\/4,1\/2) \\, \\ast$,} $(c)$ $(0,1\/3)$.} \\label{Fig-OmegaWII1}\n\\end{center}\n\\end{figure}\n\nThe orbits of $(0,1\/4)$, $(1\/4,1\/2)$ are illustrated in Figure \\ref{Fig-OmegaWII1}$(b)$, represented by $\\bullet$, $\\ast$ respectively. Both orbits lie on the boundary of the fundamental domains of $\\mathbb{T}^2\/D_{12}$, however, only the orbit of $(1\/4,1\/2)$ lies on the boundary of the fundamental domains of $\\mathbb{T}^2\/S_3$, illustrated in Figure \\ref{fig:fund_domain-A2inT2}. Then $6K(0,1\/4)\\,\\Omega^W(0,1\/4) = 16 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) K(\\theta_1,\\theta_2) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$ where $K$ is an $S_3$-invariant function on $\\mathbb{T}^2$ which is zero on the boundaries of the fundamental domains of $\\mathbb{T}^2\/S_3$. Such an $S_3$-invariant function is given by the square $\\widetilde{J}^2$ of the Jacobian which appeared for the $A_2$ spectral measures in \\cite{evans\/pugh:2009v, evans\/pugh:2010i}, which is given by $\\widetilde{J}(\\theta_1,\\theta_2) = 4\\pi^2(\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))$. Now $|\\widetilde{J}(0,1\/4)| = 8\\pi^2$, thus we take $K = 16\\widetilde{J}^2\/64\\pi^4 = \\widetilde{J}^2\/4\\pi^4$, and we have $K(0,1\/4) = 16$. We can thus also obtain an expression for $\\Omega^W(1\/4,1\/2)$, where from Figure \\ref{Fig-OmegaWII1}$(b)$ we see that $6\\Omega^W(1\/4,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( (16 - K(\\theta_1,\\theta_2)) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 - 4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\right)$.\n\nFinally, the orbit of $(\\theta_1,\\theta_2)=(0,1\/3)$ is illustrated in Figure \\ref{Fig-OmegaWII1}$(c)$, and thus $6\\Omega^W(0,1\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\, (9 \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - 3 \\, \\mathrm{d}^{(1)})$, and we obtain\n\n\\begin{Thm} \\label{thm:measureII1}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of the projective special linear group $PSL(2;7)$ over the finite field $\\mathbb{F}_7$ into the fundamental representations of $G_2$ is\n\\begin{equation} \\label{eqn:measureII1}\n\\mathrm{d}\\varepsilon = \\frac{1}{672\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{24} K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{2} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{28} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{6} \\, \\mathrm{d}^{(1)},\n\\end{equation}\nwhere $K' = 16-K$ for the embedding of $PSL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, where $K(\\theta_1,\\theta_2) = (\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))^2$, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\\begin{Rem}\nNote that measure in Theorem \\ref{thm:measureII1} for the second embedding $\\varrho_1^{(2)}$ of $PSL(2;7)$ in $G_2$ is precisely that for $\\Sigma (168) \\subset SU(3)$ given in \\cite[Theorem 16]{evans\/pugh:2010i}. However, (\\ref{eqn:measureII1}) has a neater expression than that given in \\cite{evans\/pugh:2010i}, because here we were able to use the Jacobian $J$ for $G_2$ which is also 0 along the diagonal, whereas the Jacobian for $SU(3)$ (essentially $K$ in Theorem \\ref{thm:measureII1}) is non-zero along the diagonal. \\end{Rem}\n\n\n\\section{Group $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$} \\label{sect:II2}\n\nThe subgroup $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ of $G_2$ is an irreducible imprimitive group of order 1344.\nIt has eleven irreducible representations (nine real and two complex conjugate representations) and its character table is given in Table \\ref{table:Character_table-II2} (see \\cite{littlewood:1934, he:2003}),\nwhere elements in $C_4$, $C_4^{\\prime}$, $C_4^{\\prime}$, $C_7$, $C_7^{\\prime}$, $C_6$, $C_3$ are of cycle type $(C_4,C_4^3)$, $(C_4^{\\prime},C_4^{\\prime3})$, $(C_4^{\\prime\\prime},C_4^{\\prime\\prime3})$, $(C_7,C_7^2,C_7^4)$, $(C_7^3,C_7^5,C_7^6)$, $(C_6,C_6^5)$, $(C_3,C_3^2)$ respectively.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_2^{\\prime\\prime}$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_7$ & $C_7^{\\prime}$ & $C_6$ & $C_3$ \\\\\n\\hline $|C|$ & 1 & 7 & 42 & 42 & 84 & 168 & 168 & 192 & 192 & 224 & 224 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_3$ & 3 & 3 & -1 & -1 & -1 & 1 & 1 & $w$ & $\\overline{w}$ & 0 & 0 \\\\\n\\hline $\\Sigma_3^{\\ast}$ & 3 & 3 & -1 & -1 & -1 & 1 & 1 & $\\overline{w}$ & $w$ & 0 & 0 \\\\\n\\hline $\\Sigma_6$ & 6 & 6 & 2 & 2 & 2 & 0 & 0 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)}$ & 7 & -1 & -1 & 3 & -1 & -1 & 1 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{(1)\\prime}$ & 7 & -1 & 3 & -1 & -1 & 1 & -1 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{(2)}$ & 7 & 7 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\Sigma_8$ & 8 & 8 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_{14}$ & 14 & -2 & 2 & 2 & -2 & 0 & 0 & 0 & 0 & 1 & -1 \\\\\n\\hline $\\Sigma_{21}$ & 21 & -3 & -3 & 1 & 1 & 1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}'$ & 21 & -3 & 1 & -3 & 1 & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$, where $w = \\eta + \\eta^2 + \\eta^4 = (-1+i\\sqrt{7})\/2$, $\\eta = e^{2\\pi i\/7}$.} \\label{table:Character_table-II2}\n\\end{center}\n\\end{table}\n\nThere are five non-conjugate seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, $\\gamma_1^{(2)} = \\Sigma_1 + \\Sigma_6$, $\\gamma_1^{(3)} = \\Sigma_7^{(1)}$, $\\gamma_1^{(4)} = \\Sigma_7^{(1)\\prime}$ and $\\gamma_1^{(5)} = \\Sigma_7^{(2)}$.\nThese all satisfy the condition that $\\gamma_1^{(i)}$ appears in the decomposition of $(\\gamma_1^{(i)})^2$.\nWe thus consider the eigenvalues of the representation matrices to determine which of the $\\gamma_1^{(i)}$ are embeddings of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-II2} for representations of dimension less than or equal to 7. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$. The additional information that is needed is to note that the eigenvalues for group elements in $(C_4,C_4^3)$ square to those for elements in $C_2$, those for $(C_4^{\\prime},C_4^{\\prime3})$ square to those for $C_2^{\\prime}$, those for $(C_4^{\\prime\\prime},C_4^{\\prime\\prime3})$ square to those for $C_2^{\\prime\\prime}$, whilst those for $(C_6,C_6^5)$ square to those for $(C_3,C_3^2)$ and also cube to those for $C_2$.\nThese observations follow from \\cite{littlewood:1934} and the fact that, for example, it is impossible to choose eigenvalues for group elements in $(C_6,C_6^5)$ which cube to those for elements in $C_2^{\\prime}$ or $C_2^{\\prime\\prime}$ for all irreducible representations.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_3$ & $\\Sigma_3^{\\ast}$ & $\\Sigma_6$ \\\\\n\\hline \\hline $C_1$ & 1 & $(1,1,1)$ & $(1,1,1)$ & $(1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & 1 & $(1,1,1)$ & $(1,1,1)$ & $(1,1,1,1,1,1)$ \\\\\n\\hline $C_2^{\\prime}$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime\\prime}$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_4$ & 1 & $(1,-1,-1)$ & $(1,-1,-1)$ & $(1,1,1,1,-1,-1)$ \\\\\n\\hline $C_4^{\\prime}$ & 1 & $(1,i,-i)$ & $(1,i,-i)$ & $(1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & 1 & $(1,i,-i)$ & $(1,i,-i)$ & $(1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_7$ & 1 & $(\\eta,\\eta^2,\\eta^4)$ & $(\\eta^3,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_7^{\\prime}$ & 1 & $(\\eta^3,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^4)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & 1 & $(1,\\mu^2,\\mu^4)$ & $(1,\\mu^2,\\mu^4)$ & $(1,1,\\mu^2,\\mu^2,\\mu^4,\\mu^4)$ \\\\\n\\hline $C_3$ & 1 & $(1,\\omega,\\omega^2)$ & $(1,\\omega,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n$\\;$ \\\\\n\\begin{tabular}{|c||c|c|c|} \\hline\n & $\\Sigma_7^{(1)}$ & $\\Sigma_7^{(1)\\prime}$ & $\\Sigma_7^{(2)}$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2^{\\prime}$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime\\prime}$ & $(1,1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_4$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,1,-1,-1,i,-i)$ \\\\\n\\hline $C_4^{\\prime}$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & $(1,1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_7^{\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,1,1,\\mu^2,\\mu^2,\\mu^4,\\mu^4)$ \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ for irreducible representations of dimension $\\leq 7$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$ and $\\eta = e^{2\\pi i\/7}$.} \\label{table:evalues-II2}\n\\end{center}\n\\end{table}\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, we see that there is no choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ for $i=2,3$ when $C = C_2^{\\prime}$ and for $i=4$ when $C = C_2^{\\prime\\prime}$. However, such a choice does exist for $i=1,5$ for all conjugacy classes $C$, thus we set $\\varrho_1^{(1)} = \\gamma_1^{(5)}$, $\\varrho_1^{(2)} = \\gamma_1^{(1)}$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in Table \\ref{table:t1,t2-II2}.\nThe McKay graphs $\\mathcal{G}^{\\varrho_1^{(1)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$, $\\mathcal{G}^{\\varrho_1^{(2)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$ for $\\varrho_1^{(2)}$ for $\\varrho_1^{(1)}$, $\\varrho_1^{(2)}$ are given in Figure \\ref{Fig-McKay_Graph-II2-rho1}. We use the notation $n$, $n^{\\ast}$, $n^{(i)\\prime}$ to label the vertices corresponding to the irreducible representations $\\Sigma_n$, $\\Sigma_n^{\\ast}$, $\\Sigma_n^{(i)\\prime}$ respectively. Since both McKay graphs are not connected, we see that $\\varrho_1^{(i)}$ is not a faithful representation, $i=1,2$.\nNote that the McKay graph for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ given in \\cite[Figure 1]{he:2003} is not the McKay graph for a restriction of the fundamental seven-dimensional representation, as claimed there, but rather for the irreducible representation $\\Sigma_7^{(1)}$ (or equivalently the representation $\\Sigma_7^{(1)\\prime}$).\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=115mm]{Fig-McKay_Graph-II2-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PSL(2;7) \\rtimes \\mathbb{Z}_2^3}$, $i=1,2$.} \\label{Fig-McKay_Graph-II2-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1^{(i)}$ into irreducibles is given by\n$$(\\varrho_1^{(1)})^2 = \\mathrm{id} + \\varrho_1^{(1)} + \\Sigma_3 + \\Sigma_3^{\\ast} + 2\\Sigma_6 + \\Sigma_7^{(2)} + 2 \\Sigma_8, \\qquad\n(\\varrho_1^{(2)})^2 = \\mathrm{id} + \\varrho_1^{(2)} + \\Sigma_1 + 2\\Sigma_3 + 2\\Sigma_3^{\\ast} + 2\\Sigma_6 + 2 \\Sigma_8,$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2^{(i)}$, which are given by $\\gamma_2^{(1)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ and $\\gamma_2^{(2)} = \\Sigma_6 + \\Sigma_8$ for both $i=1,2$.\nHowever, since $\\chi_{\\varrho_2^{(i)}}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:t1,t2-II2} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ is given by $\\varrho_2 = \\gamma_2^{(1)} = \\Sigma_3 + \\Sigma_3^{\\ast} + \\Sigma_8$ for both $i=1,2$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n& \\multicolumn{2}{|c|}{$\\varrho_1^{(1)}$} & \\multicolumn{2}{|c|}{$\\varrho_1^{(2)}$} & & \\\\\n$C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(1)}}(C)$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(1)}}(C)$ & $\\chi_{\\gamma_2^{(2)}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 \\\\\n\\hline $C_2$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 \\\\\n\\hline $C_2^{\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_2^{\\prime\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_4$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & 2 \\\\\n\\hline $C_4^{\\prime}$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 0 \\\\\n\\hline $C_4^{\\prime\\prime}$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 0 \\\\\n\\hline $C_7$ & $(\\eta,\\eta^5)$ & 0 & $(\\eta,\\eta^5)$ & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime}$ & $(\\eta^2,\\eta^3)$ & 0 & $(\\eta^2,\\eta^3)$ & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(1,\\mu^2)$ & -1 & $(1,\\mu^2)$ & -1 & -1 & -1 \\\\\n\\hline $C_3$ & $(1,\\omega)$ & -1 & $(1,\\omega)$ & -1 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1^{(i)}$, $i=1,2$, and corresponding values of $\\chi_{\\varrho_2^{(i)}}(C)$.} \\label{table:t1,t2-II2}\n\\end{center}\n\\end{table}\n\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\rho_2}(C) \\in [-2,14]$ for $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ are given in Table \\ref{table:(x,y)-II1}, along with the values of $J^2\/64\\pi^4$ for the corresponding pairs $(\\theta_1,\\theta_2) \\in [0,1]^2$ obtained from Table \\ref{Table:subgroupsG2-orbits(theta1,theta2)}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_2^{\\prime\\prime}$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_7$ & $C_7^{\\prime}$ & $C_6$ & $C_3$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(\\Gamma_j) \\in [-2,7]$ & 7 & 7 & -1 & -1 & -1 & -1 & -1 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(\\Gamma_j) \\in [-2,7]$ & 7 & 7 & -1 & -1 & -1 & 3 & 3 & 0 & 0 & 1 & 1 \\\\\n\\hline $\\chi_{\\varrho_2}(\\Gamma_j) \\in [-2,14]$ & 14 & 14 & -2 & -2 & -2 & 2 & 2 & 0 & 0 & -1 & -1 \\\\\n\\hline $J^2(\\theta_1,\\theta_2)\/64\\pi^4$ & 0 & 0 & 0 & 0 & 0 & 8 & 8 & 49\/4 & 49\/4 & 9 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$, $j=1,2$.} \\label{table:(x,y)-II2}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-II2}, \\ref{table:(x,y)-II2} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1+7}{1344} \\Omega^W(0,0) + \\frac{42+42+84}{1344} \\Omega^W(0,1\/2) + \\frac{224+224}{1344} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{168+168}{1344} \\Omega' + \\frac{192+192}{1344} \\Omega^W(1\/7,3\/7),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}, and $\\Omega'$ is $\\Omega^W(1\/4,1\/2)$ for $\\varrho_1^{(1)}$, and $\\Omega^W(0,1\/4)$ for $\\varrho_1^{(2)}$. Thus the joint moments are precisely those for the two embeddings of $PSL(2;7)$ in $G_2$, and we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ into the fundamental representations of $G_2$ is given by (\\ref{eqn:measureII1}),\nwhere $K' = 16-K$ for the embedding of $PSL(2;7) \\rtimes \\mathbb{Z}_2^3$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7^{(2)}$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1 + \\Sigma_3 + \\Sigma_3^{\\ast}$, and where $K$ is again given by $-4K(\\omega_1,\\omega_2) = (\\omega_1\\omega_2 - \\omega_1^{-1}\\omega_2^{-1} - \\omega_1^2\\omega_2^{-1} + \\omega_1^{-2}\\omega_2 + \\omega_1\\omega_2^{-2} - \\omega_1^{-1}\\omega_2^2)^2$ for $\\omega_1,\\omega_2\\in\\mathbb{T}$.\n\\end{Thm}\n\n\n\n\n\\section{Group $PGL(2;7)$}\n\nThe subgroup $PGL(2;7)$ of $G_2$ is an irreducible primitive group of order 336.\nIt has nine irreducible representations, all real, and its character table is given in Table \\ref{table:Character_table-IP3} \\cite{collins:1990}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_4$ & $C_7$ & $C_6$ & $C_3$ \\\\\n\\hline $|C|$ & 1 & 21 & 28 & 42 & 42 & 42 & 48 & 56 & 56 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_1^{\\prime}$ & 1 & 1 & -1 & -1 & -1 & 1 & 1 & -1 & 1 \\\\\n\\hline $\\Sigma_6^{(1)}$ & 6 & -2 & 0 & 0 & 0 & 2 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_6^{(2)}$ & 6 & 2 & 0 & $\\sqrt{2}$ & $-\\sqrt{2}$ & 0 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_6^{(2)\\prime}$ & 6 & 2 & 0 & $-\\sqrt{2}$ & $\\sqrt{2}$ & 0 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_7$ & 7 & -1 & 1 & -1 & -1 & -1 & 0 & 1 & 1 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & -1 & -1 & 1 & 1 & -1 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_8$ & 8 & 0 & 2 & 0 & 0 & 0 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_8^{\\prime}$ & 8 & 0 & -2 & 0 & 0 & 0 & 1 & 1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PGL(2;7)$.} \\label{table:Character_table-IP3}\n\\end{center}\n\\end{table}\n\nThere are eight non-conjugate seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_6^{(1)}$, $\\gamma_1^{(2)} = \\Sigma_1 + \\Sigma_6^{(2)}$, $\\gamma_1^{(3)} = \\Sigma_1 + \\Sigma_6^{(2)\\prime}$, $\\gamma_1^{(4)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(1)}$, $\\gamma_1^{(5)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(2)}$, $\\gamma_1^{(6)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(2)\\prime}$, $\\gamma_1^{(7)} = \\Sigma_7$ and $\\gamma_1^{(8)} = \\Sigma_7^{\\prime}$.\nThe decomposition of the Kronecker squares of the $\\gamma_1^{(i)}$ are given by:\n\\begin{align*}\n(\\gamma_1^{(1)})^2 & = \\mathrm{id} + \\gamma_1^{(1)} + \\Sigma_1^{\\prime} + 2\\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(2)})^2 & = \\mathrm{id} + \\gamma_1^{(2)} + 2\\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(4)})^2 & = \\mathrm{id} + \\gamma_1^{(4)} + \\Sigma_1 + 2\\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(5)})^2 & = \\mathrm{id} + \\Sigma_1 + 3\\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(7)})^2 & = \\mathrm{id} + \\gamma_1^{(7)} + \\Sigma_1 + \\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7^{\\prime} + \\Sigma_8 + \\Sigma_8^{\\prime}, \\\\\n(\\gamma_1^{(8)})^2 & = \\mathrm{id} + \\gamma_1^{(8)} + \\Sigma_1 + \\Sigma_6^{(1)} + \\Sigma_6^{(2)} + \\Sigma_6^{(2)\\prime} + \\Sigma_7 + \\Sigma_8 + \\Sigma_8^{\\prime},\n\\end{align*}\nwhere $\\mathrm{id} = \\Sigma_1$.\nNote, we have omitted the decompositions for $\\gamma_1^{(3)}$, $\\gamma_1^{(6)}$, since from the character table we see that these representations are essentially the same as $\\gamma_1^{(2)}$, $\\gamma_1^{(5)}$ respectively.\nThen we see that $\\gamma_1^{(i)}$ does not appear in the decomposition of $(\\gamma_1^{(i)})^2$ for $i=5,6$, therefore they are not embeddings of $PGL(2;7)$ in $G_2$.\nFrom dimension considerations, we see that candidates $\\gamma_2^{(j)}$ for $\\varrho_2$ are $\\gamma_2^{(1)}=\\Sigma_6^{(1)}+\\Sigma_8$, $\\gamma_2^{(2)}=\\Sigma_6^{(1)}+\\Sigma_8^{\\prime}$, $\\gamma_2^{(3)}=\\Sigma_6^{(2)}+\\Sigma_8$, $\\gamma_2^{(4)}=\\Sigma_6^{(2)}+\\Sigma_8^{\\prime}$, $\\gamma_2^{(5)}=\\Sigma_6^{(2)\\prime}+\\Sigma_8$ and $\\gamma_2^{(6)}=\\Sigma_6^{(2)\\prime}+\\Sigma_8^{\\prime}$, where $1 \\leq j \\leq 6$ for $\\gamma_1^{(i)}$ when $i=1,4,7,8$, and $3 \\leq j \\leq 6$ when $i=2$.\nThen with $(x_i^C,y_j^C) = (\\chi_{\\gamma_1^{(i)}}(C),\\chi_{\\gamma_2^{(j)}}(C))$, we see that $(x_i^C,y_j^C) \\not \\in \\mathfrak{D}$ for any candidate $\\gamma_2^{(j)}$ in the cases $i=1,2,3,7$ when $C=C_2^{\\prime}$. Thus $\\gamma_1^{(i)}$ cannot define an embedding of $PGL(2;7)$ in $G_2$ for $i=1,2,3,7$.\nWe also see that $(x_8^C,y_j^C) \\not \\in \\mathfrak{D}$ for $j=3,4,5,6$.\n\nThus we have candidates $(\\gamma_1^{(i)},\\gamma_2^{(j)})$ for $(\\varrho_1,\\varrho_2)$ when $i=4,8$, where $1 \\leq j \\leq 6$ for $i=4$ and $j \\in \\{ 1,2 \\}$ for $i=8$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_1^{\\prime}$ & $\\Sigma_6^{(1)}$ & $\\Sigma_6^{(2)}$ & $\\Sigma_7$ \\\\\n\\hline \\hline $C_1$ & 1 & 1 & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & 1 & 1 & $(1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime}$ & 1 & -1 & $(1,1,1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1)$ \\\\\n\\hline $C_8$ & 1 & -1 & $(1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^7)$ & $(-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_8^{\\prime}$ & 1 & -1 & $(1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu^3,\\nu^5)$ & $(-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_4$ & 1 & 1 & $(1,1,i,i,-i,-i)$ & $(1,1,-1,-1,i,-i)$ & $(1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & 1 & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & 1 & -1 & $(1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ \\\\\n\\hline $C_3$ & 1 & 1 & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\begin{tabular}{|c||c|c|c|} \\hline\n & $\\Sigma_7^{\\prime}$ & $\\Sigma_8$ & $\\Sigma_8^{\\prime}$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1,1)$ \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,-1,-1,-1,-1)$ \\\\\n\\hline $C_2^{\\prime}$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,1,1,1,1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1,-1)$ \\\\\n\\hline $C_8$ & $(1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_8^{\\prime}$ & $(1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ \\\\\n\\hline $C_4$ & $(1,-1,-1,i,i,-i,-i)$ & $(1,1,-1,-1,i,i,-i,-i)$ & $(1,1,-1,-1,i,i,-i,-i)$ \\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu^2,\\mu^2,\\mu^4,\\mu^4,\\mu^5)$ & $(1,-1,\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$ \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ & $(1,1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PGL(2;7)$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$ and $\\eta = e^{2\\pi i\/7}$.} \\label{table:evalues-IP3}\n\\end{center}\n\\end{table}\n\nWe now consider the eigenvalues of the representation matrices to determine which of the remaining $\\gamma_1^{(i)}$ are embeddings of $PSL(2;7)$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-IP3}. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PGL(2;7)$. The additional information that is needed is to note that the eigenvalues for group elements in $C_4$ square to those for elements in $C_2$, those for $C_8$, $C_8^{\\prime}$ square to those for $C_4$, those for $C_6$ square to those for $C_3)$ and also cube to those for $C_2^{\\prime}$.\nThese observations follow from the fact that, for example, it is impossible to choose eigenvalues for group elements in $C_4$ which square to those for elements in $C_2^{\\prime}$ for all irreducible representations.\nNote also that we have omitted the eigenvalues for matrices in the irreducible representation $\\Sigma_6^{(2)\\prime}$. These eigenvalues are identical to those for $\\Sigma_6^{(2)}$, except for elements in the conjugacy classes $C_8$, $C_8^{\\prime}$, where for $\\Sigma_6^{(2)\\prime}$ the eigenvalues for elements in $C_8$, $C_8^{\\prime}$ respectively are given by those for elements in $C_8^{\\prime}$, $C_8$ respectively in the representation $\\Sigma_6^{(2)}$.\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, $i=4,8$, we see that there is a choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$, for all conjugacy classes $C$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in Table \\ref{table:t1,t2-IP3}. Thus we set $\\varrho_1^{(1)} = \\Sigma_7^{\\prime}$, $\\varrho_1^{(2)} = \\Sigma_1^{\\prime} + \\Sigma_6^{(1)}$.\nThe McKay graphs $\\mathcal{G}^{\\varrho_1^{(1)}}_{PGL(2;7)}$, $\\mathcal{G}^{\\varrho_1^{(2)}}_{PGL(2;7)}$ for $\\varrho_1^{(1)}$, $\\varrho_1^{(2)}$ are given in Figure \\ref{Fig-McKay_Graph-IP3-rho1}, where we use the same notation as previously.\nNote that the graph given in \\cite[Figure 2]{he:2003} is not the McKay graph for the restriction $\\varrho_1$ of the fundamental seven-dimensional representation of $G_2$ as claimed in \\cite{he:2003}, but is rather the McKay graph for the seven-dimensional representation $\\Sigma_7$, which as shown above does not define an embedding of $PGL(2;7)$ in $G_2$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=110mm]{Fig-McKay_Graph-IP3-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1^{(i)}}_{PGL(2;7)}$, $i=1,2$.} \\label{Fig-McKay_Graph-IP3-rho1}\n\\end{center}\n\\end{figure}\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|} \\hline\n& \\multicolumn{2}{|c|}{$\\varrho_1^{(1)}$} & \\multicolumn{2}{|c|}{$\\varrho_1^{(2)}$} & & & & \\\\\n$C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(1)}}(C)$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(1)}}(C)$ & $\\chi_{\\gamma_2^{(2)}}(C)$ & $\\chi_{\\gamma_2^{(3)}}(C)$ & $\\chi_{\\gamma_2^{(4)}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1)$ & 14 & $(1,1)$ & 14 & 14 & 14 & 14 & 14 \\\\\n\\hline $C_2$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & -2 & -2 & 2 & 2 \\\\\n\\hline $C_2^{\\prime}$ & $(1,-1)$ & -2 & $(1,-1)$ & -2 & 2 & -2 & 2 & -2 \\\\\n\\hline $C_8$ & $(i,\\nu^7)$ & 0 & $(-1,\\nu)$ & 0 & 0 & 0 & $\\sqrt{2}$ & $\\sqrt{2}$ \\\\\n\\hline $C_8^{\\prime}$ & $(i,\\nu^5)$ & 0 & $(-1,\\nu)$ & 0 & 0 & 0 & $-\\sqrt{2}$ & $-\\sqrt{2}$ \\\\\n\\hline $C_4$ & $(-1,i)$ & 2 & $(1,i)$ & 2 & 2 & 2 & 0 & 0 \\\\\n\\hline $C_7$ & $(\\eta,\\eta^5)$ & 0 & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(-1,\\mu)$ & 1 & $(-1,\\mu)$ & 1 & -1 & 1 & -1 & 1 \\\\\n\\hline $C_3$ & $(1,\\omega)$ & -1 & $(1,\\omega)$ & -1 & -1 & -1 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1^{(i)}$, $i=1,2$, and corresponding values of $\\chi_{\\varrho_2^{(i)}}(C)$.} \\label{table:t1,t2-IP3}\n\\end{center}\n\\end{table}\n\nSince $\\chi_{\\varrho_2^{(i)}}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:t1,t2-II2} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PGL(2;7)$ is given by $\\varrho_2 = \\gamma_2^{(2)} = \\Sigma_6^{(1)} + \\Sigma_8^{\\prime}$ for both $i=1,2$.\nThe values of $x^{(i)} = \\chi_{\\varrho_1^{(i)}}(C) \\in [-2,7]$, $y = \\chi_{\\varrho_2}(C) \\in [-2,14]$ for $PGL(2;7)$ are given in Table \\ref{table:(x,y)-IP3}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_2$ & $C_2^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_4$ & $C_7$ & $C_6$ & $C_3$ \\\\\n\\hline $\\chi_{\\varrho_1^{(1)}}(C) \\in [-2,7]$ & 7 & -1 & -1 & 1 & 1 & -1 & 0 & -1 & 1 \\\\\n\\hline $\\chi_{\\varrho_1^{(2)}}(C) \\in [-2,7]$ & 7 & -1 & -1 & -1 & -1 & 3 & 0 & -1 & 1 \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -2 & -2 & 0 & 0 & 2 & 0 & 1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PGL(2;7)$, $j=1,2$.} \\label{table:(x,y)-IP3}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP3}, \\ref{table:(x,y)-IP3} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{336} \\Omega^W(0,0) + \\frac{21+28}{336} \\Omega^W(0,1\/2) + \\frac{56}{336} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{56}{336} \\Omega^W(1\/6,1\/2) + \\frac{48}{336} \\Omega^W(1\/7,3\/7) + \\frac{42}{336} \\Omega' + \\frac{42+42}{336} \\Omega'',\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1} and $(\\Omega',\\Omega'')$ is $(\\Omega^W(1\/4,1\/2),\\Omega^W(1\/8,3\/8))$ for $\\varrho_1^{(1)}$, and $(\\Omega^W(0,1\/4),\\Omega^W(1\/8,1\/2))$ for $\\varrho_1^{(2)}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWIP3}\\\\\n \\caption{The orbits of $(a)$ $(1\/6,1\/2)$, $(b)$ $(1\/8,3\/8) \\, \\bullet$ and $(1\/8,1\/2) \\, \\ast$.} \\label{Fig-OmegaWIP3}\n\\end{center}\n\\end{figure}\n\nNow $12\\Omega^W(1\/6,1\/2) = 4 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_6 \\times \\mathrm{d}_6$, as illustrated in Figure \\ref{Fig-OmegaWIP3}$(a)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/6,1\/2)\/64\\pi^4) = 9$ for all $g \\in D_{12}$.\n\nThe orbits of $(1\/8,3\/8)$, $(1\/8,1\/2)$ are illustrated in Figure \\ref{Fig-OmegaWIP3}$(b)$, represented by $\\bullet$, $\\ast$ respectively.\nNeither orbit gives a linear combination of the measures in Definition \\ref{def:4measures}, thus we have $12\\Omega^W(1\/8,3\/8) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},e^{3\\pi i\/4})} \\right)$, $12\\Omega^W(1\/8,1\/2) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)} \\right)$.\n\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$ and $\\mathrm{d}_1 \\times \\mathrm{d}_1$ supported by the other points have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for the non-conjugate embeddings of $PGL(2;7)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{1344\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{24} K' \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{4} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 \\\\\n& \\quad + \\frac{1}{9} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{23}{504} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{48} \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},t)},\n\\end{split}\n\\end{equation}\nwhere $K' = 16-K$ for the embedding of $PGL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)} = \\Sigma_7^{\\prime}$ and $K' = K$ for the embedding given by $\\varrho_1^{(2)} = \\Sigma_1^{\\prime} + \\Sigma_6$, with $K$ as in Theorem \\ref{thm:measureII1}, whilst $t = e^{3\\pi i\/4}$ for the embedding of $PGL(2;7)$ in $G_2$ given by $\\varrho_1^{(1)}$ and $t=-1$ for the embedding given by $\\varrho_1^{(2)}$,\nand where $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\delta_x$ is the Dirac measure at the point $x$.\n\\end{Thm}\n\n\n\n\n\\section{Group $PSL(2;8)$}\n\nThe subgroup $PSL(2;8)$ of $G_2$ is an irreducible primitive group of order 504.\nIt has nine irreducible representations, all real, five of which have dimension less than or equal to 7. The character table for $PSL(2;8)$ is given in\nTable \\ref{table:Character_table-IP2} \\cite{james\/liebeck:2001} (the orders of the elements in each conjugacy class can be obtained from \\cite{lopez_pena\/majid\/rietsch:2010}).\nThe spectral measure for the first three seven-dimensional representations are equal, since the conjugacy classes $C_{9}$, $C_{9}^{\\prime}$, $C_{9}^{\\prime\\prime}$ each have the same order.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_9$ & $C_9^{\\prime}$ & $C_9^{\\prime\\prime}$ & $C_2$ & $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ \\\\\n\\hline $|C|$ & 1 & 56 & 56 & 56 & 56 & 63 & 72 & 72 & 72 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_7^{(1)}$ & 7 & 1 & $-p$ & $-q$ & $p+q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)\\prime}$ & 7 & 1 & $p+q$ & $-p$ & $-q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(1)\\prime\\prime}$ & 7 & 1 & $-q$ & $p+q$ & $-p$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_7^{(2)}$ & 7 & -2 & 1 & 1 & 1 & -1 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_8$ & 8 & -1 & -1 & -1 & -1 & 0 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_9$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(2\\pi\/7)$ & $2\\cos(4\\pi\/7)$ & $2\\cos(6\\pi\/7)$ \\\\\n\\hline $\\Sigma_9^{\\prime}$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(4\\pi\/7)$ & $2\\cos(6\\pi\/7)$ & $2\\cos(2\\pi\/7)$ \\\\\n\\hline $\\Sigma_9^{\\prime\\prime}$ & 9 & 0 & 0 & 0 & 0 & 1 & $2\\cos(6\\pi\/7)$ & $2\\cos(2\\pi\/7)$ & $2\\cos(4\\pi\/7)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;8)$, where $p=2\\cos(4\\pi\/9)$, $q=2\\cos(8\\pi\/9)$.} \\label{table:Character_table-IP2}\n\\end{center}\n\\end{table}\n\nWe thus have four candidates for the restriction $\\varrho_1$ of the fundamental representation $\\rho_1$ of $G_2$ to $PSL(2;8)$. The Kronecker squares of $\\Sigma_7^{(1)}$, $\\Sigma_7^{(2)}$ decompose into irreducibles as\n$$(\\Sigma_7^{(1)})^2 = \\mathrm{id} + \\Sigma_7^{(1)} + \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(2)} + \\Sigma_9 + \\Sigma_9^{\\prime} + \\Sigma_9^{\\prime\\prime}, \\qquad\n(\\Sigma_7^{(2)})^2 = \\mathrm{id} + \\Sigma_7^{(1)} + \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(1)\\prime\\prime} + \\Sigma_9 + \\Sigma_9^{\\prime} + \\Sigma_9^{\\prime\\prime},$$\nwhere $\\mathrm{id} = \\Sigma_1$. The irreducible representation $\\Sigma_7^{(2)}$ does not appear in the decomposition of its Kronecker square into irreducibles, and therefore does not give an embedding of $PSL(2;8)$ into $G_2$.\nThe other three seven-dimensional representations give non-conjugate embeddings of $PSL(2;8)$ into $G_2$. We will fix $\\varrho_1 = \\Sigma_7$ for the remainder of this section.\n\nThe McKay graph for $\\varrho_1$ is given in Figure \\ref{Fig-McKay_Graph-IP2-rho1}, where we use the same notation as previously.\nNote that the graph given in \\cite[Figure 2]{he:2003} is not the McKay graph for the restriction $\\varrho_1$ of the fundamental seven-dimensional representation of $G_2$ as claimed in \\cite{he:2003}, but is rather the McKay graph for the seven-dimensional representation $\\Sigma_7^{(2)}$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-McKay_Graph-IP2-rho1}\\\\\n \\caption{The McKay graph $\\mathcal{G}^{\\varrho_1}_{PSL(2;8)}$.} \\label{Fig-McKay_Graph-IP2-rho1}\n\\end{center}\n\\end{figure}\n\nFrom the decomposition of the Kronecker square of $\\varrho_1$ and dimension considerations there is only one possibility for the fourteen-dimensional representation $\\varrho_2$, that is, $\\varrho_2 = \\Sigma_7^{(1)\\prime} + \\Sigma_7^{(2)}$.\nThe values of $x = \\chi_{\\varrho_1}(C) \\in [-2,7]$, $y = \\chi_{\\varrho_2}(C) \\in [-2,14]$ for $PSL(2;8)$ are given in Table \\ref{table:(x,y)-IP2}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_9$ & $C_9^{\\prime}$ & $C_9^{\\prime\\prime}$ & $C_2$ & $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ \\\\\n\\hline $\\chi_{\\varrho_1}(C) \\in [-2,7]$ & 7 & 1 & $-p$ & $-q$ & $p+q$ & -1 & 0 & 0 & 0 \\\\\n\\hline $\\chi_{\\varrho_2}(C) \\in [-2,14]$ & 14 & -1 & $p+q+1$ & $1-p$ & $1-q$ & -2 & 0 & 0 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{$\\chi_{\\varrho_j}(C)$ for group $PSL(2;8)$, $j=1,2$.} \\label{table:(x,y)-IP2}\n\\end{center}\n\\end{table}\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP2} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{504} \\Omega^W(0,0) + \\frac{56}{504} \\Omega^W(0,1\/3) + \\frac{63}{504} \\Omega^W(0,1\/2) + \\frac{72+72+72}{504} \\Omega^W(1\/7,3\/7) \\\\\n& \\quad + \\frac{56}{504} \\Omega^W(1\/9,4\/9) + \\frac{56}{504} \\Omega^W(1\/9,1\/3) + \\frac{56}{504} \\Omega^W(2\/9,5\/9),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\nNow $J^2(g(1\/9,4\/9)\/64\\pi^4) = 9(3+2\\cos(\\pi\/9)+2\\cos(2\\pi\/9))\/4 =: a_1$, $J^2(g(1\/9,1\/3)\/64\\pi^4) = 9(3-2\\cos(2\\pi\/9)+2\\sin(\\pi\/18))\/4 =: a_2$ and $J^2(g(2\/9,5\/9)\/64\\pi^4) = 9(3-2\\cos(\\pi\/9)-2\\sin(\\pi\/18))\/4 =: a_3$, for all $g \\in D_{12}$.\nThus $a_1 \\Omega^W(1\/9,4\/9) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9\/2))}$, $a_2 \\Omega^W(1\/9,1\/3) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9\/4))}$ and $a_3 \\Omega^W(2\/9,5\/9) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{((9))}$, as illustrated in Figure \\ref{Fig-OmegaWIP2} since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain.\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$, $\\mathrm{d}_1 \\times \\mathrm{d}_1$ and $\\mathrm{d}^{(1)}$ supported by the other points above have all appeared in the previous sections, so we obtain:\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-OmegaWIP2}\\\\\n \\caption{The orbits of $(1\/9,4\/9)\\textcolor{red}{\\scriptscriptstyle{\\bullet}}$, $(1\/9,1\/3)\\textcolor{blue}{\\scriptscriptstyle{\\bullet}}$, $(2\/9,5\/9)\\textcolor{green}{\\scriptscriptstyle{\\bullet}}$.} \\label{Fig-OmegaWIP2}\n\\end{center}\n\\end{figure}\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PSL(2;8)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{448\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{6} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 + \\frac{1}{6} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{5}{126} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{18} \\, \\mathrm{d}^{(1)} \\\\\n& \\quad + \\frac{1}{384\\pi^4} a_3^{-1} J^2 \\, \\mathrm{d}^{((9))} + \\frac{1}{384\\pi^4} a_1^{-1} J^2 \\, \\mathrm{d}^{((9\/2))} + \\frac{1}{384\\pi^4} a_2^{-1} J^2 \\, \\mathrm{d}^{((9\/4))},\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}^{((n))}$ is as in Definition \\ref{def:4measures}, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\n\n\\section{Group $PSL(2;13)$}\n\nThe subgroup $PSL(2;13)$ of $G_2$ is an irreducible primitive group of order 1092. It has nine irreducible representations, all real, and its character table is given in Table \\ref{table:Character_table-IP1} \\cite{cohen:1998}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|} \\hline\n$C$ & $C_1$ & $C_{13}$ & $C_{13}^{\\prime}$ & $C_2$ $C_7$ & $C_7^{\\prime}$ & $C_7^{\\prime\\prime}$ & $C_6$ & $C_3$ &  \\\\\n\\hline $|C|$ & 1 & 84 & 84 & 91 & 156 & 156 & 156 & 182 & 182 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_7$ & 7 & $p_-$ & $p_+$ & -1 & 0 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & $p_+$ & $p_-$ & -1 & 0 & 0 & 0 & -1 & 1 \\\\\n\\hline $\\Sigma_{12}$ & 12 & -1 & -1 & 0 & $r_1$ & $r_2$ & $r_3$ & 0 & 0 \\\\\n\\hline $\\Sigma_{12}^{\\prime}$ & 12 & -1 & -1 & 0 & $r_2$ & $r_3$ & $r_1$ & 0 & 0 \\\\\n\\hline $\\Sigma_{12}^{\\prime\\prime}$ & 12 & -1 & -1 & 0 & $r_3$ & $r_1$ & $r_2$ & 0 & 0 \\\\\n\\hline $\\Sigma_{13}$ & 13 & 0 & 0 & 1 & -1 & -1 & -1 & 1 & 1 \\\\\n\\hline $\\Sigma_{14}$ & 14 & 1 & 1 & -2 & 0 & 0 & 0 & 1 & -1 \\\\\n\\hline $\\Sigma_{14}^{\\prime}$ & 14 & 1 & 1 & 2 & 0 & 0 & 0 & -1 & -1 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PSL(2;13)$, where $p_{\\pm} = (1\\pm\\sqrt{13})\/2$, $r_j=-2\\cos(2j\\pi\/7)$.} \\label{table:Character_table-IP1}\n\\end{center}\n\\end{table}\n\nOnly three of the irreducible representation have dimension less than or equal to 7. These are the identity representation, and two seven-dimensional representations $\\Sigma_7$, $\\Sigma_7^{\\prime}$ whose character values only differ on the two conjugacy classes $C_{13}$, $C_{13}^{\\prime}$ whose elements have order 13. Here $\\chi_{\\Sigma_7}(g) = p,q$ for $g \\in C_{13},C_{13}^{\\prime}$ respectively, whilst $\\chi_{\\Sigma_7^{\\prime}}(g) = q,p$ respectively, where $p=(1+\\sqrt{13})\/2=1+\\zeta+\\zeta^3+\\zeta^4+\\zeta^9+\\zeta^{10}+\\zeta^{12}$, $q=(1-\\sqrt{13})\/2=1+\\zeta^2+\\zeta^5+\\zeta^6+\\zeta^7+\\zeta^8+\\zeta^{11}$, for $\\zeta = e^{2\\pi i\/13}$.\nThus the spectral measure for both seven-dimensional representations are equal, since $C_{13}$, $C_{13}^{\\prime}$ both have the same order, and both give embeddings of $PSL(2;13)$ in $G_2$.\nThe McKay graph for the fundamental seven-dimensional representation is given in \\cite[Figure 2]{he:2003}, and is reproduced (twice) in Figure \\ref{Fig-McKay_Graph-IP1-rho1} for completeness, where we use the same notation as previously. The figure on the right hand side illustrates the resemblance of $\\mathcal{G}^{\\varrho_1}_{PSL(2;13)}$ with the McKay graph of $G_2$ itself.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=60mm]{Fig-McKay_Graph-IP1-rho1} \\hspace{15mm} \\includegraphics[width=60mm]{Fig-McKay_Graph-IP1-rho1-2}\\\\\n \\caption{Two presentations of the McKay graph $\\mathcal{G}^{\\varrho_1}_{PSL(2;13)}$.} \\label{Fig-McKay_Graph-IP1-rho1}\n\\end{center}\n\\end{figure}\n\nThe set $X_C$ of eigenvalues of group elements in each conjugacy class for $\\varrho_1 = \\Sigma_7$ are given in Table \\ref{table:evalues-IP1}, along with a choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|c|c|} \\hline\n$C$ & $X_C$ & $(t_1^C,t_2^C)$ & $\\chi_{\\varrho_1}(C)$ & $\\chi_{\\varrho_2}(C)$ & $\\chi_{\\Sigma_{14}}(C)$ & $\\chi_{\\Sigma_{14}^{\\prime}}(C)$ \\\\\n\\hline \\hline $C_1$ & $(1,1,1,1,1,1,1)$ & $(1,1)$ & 7 & 14 & 14 & 14 \\\\\n\\hline $C_{13}$ & $(1,\\zeta,\\zeta^3,\\zeta^4,\\zeta^9,\\zeta^{10},\\zeta^{12})$ & $(\\zeta,\\zeta^4)$ & $(1+\\sqrt{13})\/2$ & 1 & 1 & 1 \\\\\n\\hline $C_{13}^{\\prime}$ & $(1,\\zeta^2,\\zeta^5,\\zeta^6,\\zeta^7,\\zeta^8,\\zeta^{11})$ & $(\\zeta^2,\\zeta^7)$ & $(1-\\sqrt{13})\/2$ & 1 & 1 & 1 \\\\\n\\hline $C_2$ & $(1,1,1,-1,-1,-1,-1)$ & $(1,-1)$ & -1 & -2 & -2 & 2\\\\\n\\hline $C_7$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_7^{\\prime\\prime}$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(\\eta,\\eta^5)$ & 0 & 0 & 0 & 0 \\\\\n\\hline $C_6$ & $(1,-1,-1,\\mu,\\mu^2,\\mu^4,\\mu^5)$ & $(-1,\\mu)$ & -1 & 1 & 1 & -1 \\\\\n\\hline $C_3$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,\\omega)$ & 1 & -1 & -1 & -1\\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Choice of eigenvalues $(t_1^C,t_2^C)$ for $\\varrho_1$ and corresponding values of $\\chi_{\\varrho_2}(C)$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$, $\\eta = e^{2\\pi i\/7}$ and $\\zeta = e^{2\\pi i\/13}$.} \\label{table:evalues-IP1}\n\\end{center}\n\\end{table}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{13} + \\Sigma_{14} + \\Sigma_{14}^{\\prime},$$\nwhere as before the notation $\\Sigma_n$ denotes an irreducible representation of $PSL(2;13)$ of dimension $n$.\nFrom dimension considerations there are thus two candidates for the fourteen-dimensional representation $\\varrho_2$, which are given by the two fourteen-dimensional irreducible representations.\nHowever, since $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$, we see from Table \\ref{table:evalues-IP1} that the decomposition of the fundamental fourteen-dimensional representation into irreducible representations of $PSL(2;13)$ is given by $\\varrho_2 = \\Sigma_{14}$, not $\\Sigma_{14}^{\\prime}$.\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:evalues-IP1} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{1092} \\Omega^W(0,0) + \\frac{91}{1092} \\Omega^W(0,1\/2) + \\frac{182}{1092} \\Omega^W(0,1\/3) + \\frac{182}{1092} \\Omega^W(1\/6,1\/2) \\\\\n& \\quad + \\frac{156+156+156}{1092} \\Omega^W(1\/7,3\/7) + \\frac{84}{1092} \\Omega^W(1\/13,4\/13)  + \\frac{84}{1092} \\Omega^W(2\/13,7\/13),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=55mm]{Fig-OmegaWIP1}\\\\\n \\caption{The orbit of $(1\/13,4\/13)$ and $(2\/13,7\/13)$.} \\label{Fig-OmegaWIP1}\n\\end{center}\n\\end{figure}\n\nThe orbit of the points $(1\/13,4\/13)$, $(2\/13,7\/13)$, illustrated in Figure \\ref{Fig-OmegaWIP1}, do not give a linear combination of the measures in Definition \\ref{def:4measures}, thus we have $12\\Omega^W(1\/13,4\/13) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(\\zeta,\\zeta^4)} \\right)$ and $12\\Omega^W(2\/13,7\/13) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( \\sum_{g \\in D_{12}} \\delta_{g(\\zeta^2,\\zeta^7)} \\right)$.\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$, $\\mathrm{d}_2 \\times \\mathrm{d}_2$, $\\mathrm{d}_1 \\times \\mathrm{d}_1$ and $\\mathrm{d}^{(1)}$ supported by the other points above have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PSL(2;13)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{448\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{4} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 + \\frac{1}{9} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\\\\n& \\quad - \\frac{22}{819} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 - \\frac{1}{12} \\, \\mathrm{d}^{(1)} + \\frac{7}{192} \\sum_{g \\in D_{12}} (\\delta_{g(\\zeta,\\zeta^4)} + \\delta_{g(\\zeta^2,\\zeta^7)})\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity, $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$, $\\delta_x$ is the Dirac measure at the point $x$ and $\\zeta = e^{2 \\pi i\/13}$.\n\\end{Thm}\n\n\n\\section{Group $PU(3;3) \\cong G_2(2)'$}\n\nThe subgroup $PU(3;3)$ of $G_2$ is an irreducible primitive group of order 6048.\nIt has fourteen irreducible representations (seven real repreentations, one quaternionic representation $\\Sigma_6$, and three pairs of complex conjugate representations) and its character table is given in Table \\ref{table:Character_table-IP4} \\cite{conway\/curtis\/norton\/parker\/wilson:1985}.\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|@{\\hspace{1.5mm}}c@{\\hspace{1.5mm}}||@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_2$ & $C_4$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_{12}$ & $C_{12}^{\\prime}$ & $C_6$ & $C_3^{\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_7$ & $C_7^{\\prime}$ \\\\\n\\hline $|C|$ & 1 & 56 & 63 & 63 & 63 & 378 & 504 & 504 & 504 & 672 & 756 & 756 & 864 & 864 \\\\\n\\hline \\hline $\\Sigma_1$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_6$ & 6 & -3 & -2 & -2 & -2 & 2 & 1 & 1 & 1 & 0 & 0 & 0 & -1 & -1 \\\\\n\\hline $\\Sigma_7$ & 7 & -2 & 3 & $-1-2i$ & $-1+2i$ & 1 & $-1-i$ & $-1+i$ & 0 & 1 & $-i$ & $i$ & 0 & 0 \\\\\n\\hline $\\Sigma_7^{\\ast}$ & 7 & -2 & 3 & $-1+2i$ & $-1-2i$ & 1 & $-1+i$ & $-1-i$ & 0 & 1 & $i$ & $-i$ & 0 & 0 \\\\\n\\hline $\\Sigma_7^{\\prime}$ & 7 & -2 & -1 & 3 & 3 & -1 & 0 & 0 & 2 & 1 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{14}$ & 14 & 5 & -2 & 2 & 2 & 2 & -1 & -1 & 1 & -1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}$ & 21 & 3 & 1 & $-3-2i$ & $-3+2i$ & -1 & $-i$ & $i$ & 1 & 0 & $i$ & $-i$ & 0 & 0 \\\\\n\\hline $\\Sigma_{21}^{\\ast}$ & 21 & 3 & 1 & $-3+2i$ & $-3-2i$ & -1 & $i$ & $-i$ & 1 & 0 & $-i$ & $i$ & 0 & 0 \\\\\n\\hline $\\Sigma_{21}^{\\prime}$ & 21 & 3 & 5 & 1 & 1 & 1 & 1 & 1 & -1 & 0 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{27}$ & 27 & 0 & 3 & 3 & 3 & -1 & 0 & 0 & 0 & 0 & 1 & 1 & -1 & -1 \\\\\n\\hline $\\Sigma_{28}$ & 28 & 1 & -4 & $-4i$ & $4i$ & 0 & $i$ & $-i$ & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{28}^{\\ast}$ & 28 & 1 & -4 & $4i$ & $-4i$ & 0 & $-i$ & $i$ & -1 & 1 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{32}$ & 32 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & $\\frac{1+i\\sqrt{7}}{2}$ & $\\frac{1-i\\sqrt{7}}{2}$ \\\\\n\\hline $\\Sigma_{32}^{\\ast}$ & 32 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & $\\frac{1-i\\sqrt{7}}{2}$ & $\\frac{1+i\\sqrt{7}}{2}$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $PU(3;3)$.} \\label{table:Character_table-IP4}\n\\end{center}\n\\end{table}\n\nThere are two non-conjugate real seven-dimensional representations, $\\gamma_1^{(1)} = \\Sigma_1 + \\Sigma_6$ and $\\gamma_1^{(2)} = \\Sigma_7^{\\prime}$.\nThese both satisfy the condition that $\\gamma_1^{(i)}$ appears in the decomposition of $(\\gamma_1^{(i)})^2$.\nWe thus consider the eigenvalues of the representation matrices to determine which of the $\\gamma_1^{(i)}$ are embeddings of $PU(3;3)$ in $G_2$.\nThese eigenvalues are given in Table \\ref{table:evalues-IP4} for the representations $\\Sigma_1$, $\\Sigma_6$ and $\\Sigma_7^{\\prime}$. As described in Section \\ref{sect:subgroupsG2}, these eigenvalues can be determined from the character table of $PU(3;3)$. The additional information that is needed is to note that the eigenvalues for group elements in $C_4$, $C_4^{\\prime}$ and $C_4^{\\prime\\prime}$ all square to those for elements in $C_2$, those for $C_8$, $C_8^{\\prime}$ square to those for $C_4$, $C_4^{\\prime}$ respectively, those for $C_6$ square to those for $C_3$ and also cube to those for $C_2$, those for $C_{12}$ square to those for $C_6$ and cube to those for $C_4$ whilst those for $C_{12}^{\\prime}$ also square to those for $C_6$ but cube to those for $C_4^{\\prime}$ (see \\cite{conway\/curtis\/norton\/parker\/wilson:1985}).\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|c||c|c|c||c|} \\hline\n & $\\Sigma_1$ & $\\Sigma_6$ & $\\Sigma_7^{\\prime}$ & $(t_1^C,t_2^C)$ \\\\\n\\hline \\hline $C_1$ & 1 & $(1,1,1,1,1,1)$ & $(1,1,1,1,1,1,1)$ & $(1,1)$ \\\\\n\\hline $C_3$ & 1 & $(\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$ & $(1,\\omega,\\omega,\\omega,\\omega^2,\\omega^2,\\omega^2)$  & $(\\omega,\\omega^2)$ \\\\\n\\hline $C_2$ & 1 & $(1,1,-1,-1,-1,-1)$ & $(1,1,1,-1,-1,-1,-1)$  & $(1,-1)$ \\\\\n\\hline $C_4$ & 1 & $(-1,-1,i,i,-i,-i)$ & $(1,1,1,i,i,-i,-i)$  & $(1,i)$ \\\\\n\\hline $C_4^{\\prime}$ & 1 & $(-1,-1,i,i,-i,-i)$ & $(1,1,1,i,i,-i,-i)$  & $(1,i)$ \\\\\n\\hline $C_4^{\\prime\\prime}$ & 1 & $(1,1,i,i,-i,-i)$ & $(1,-1,-1,i,i,-i,-i)$  & $(-1,i)$ \\\\\n\\hline $C_{12}$ & 1 & $(\\xi,\\xi^2,\\xi^5,\\xi^7,\\xi^{10},\\xi^{11})$ & $(1,\\xi,\\xi^4,\\xi^5,\\xi^7,\\xi^8,\\xi^{11})$  & $(\\xi,\\xi^5)$ \\\\\n\\hline $C_{12}^{\\prime}$ & 1 & $(\\xi,\\xi^2,\\xi^5,\\xi^7,\\xi^{10},\\xi^{11})$ & $(1,\\xi,\\xi^4,\\xi^5,\\xi^7,\\xi^8,\\xi^{11})$  & $(\\xi,\\xi^5)$ \\\\\n\\hline $C_6$ & 1 & $(\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$ & $(1,\\mu,\\mu,\\mu^2,\\mu^4,\\mu^5,\\mu^5)$  & $(\\mu,\\mu^2)$ \\\\\n\\hline $C_3^{\\prime}$ & 1 & $(1,1,\\omega,\\omega,\\omega^2,\\omega^2)$ & $(1,1,1,\\omega,\\omega,\\omega^2,\\omega^2)$  & $(1,\\omega)$ \\\\\n\\hline $C_8$ & 1 & $(i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$  & $(-1,\\nu)$ \\\\\n\\hline $C_8^{\\prime}$ & 1 & $(i,-i,\\nu,\\nu^3,\\nu^5,\\nu^7)$ & $(1,-1,-1,\\nu,\\nu^3,\\nu^5,\\nu^7)$  & $(-1,\\nu)$ \\\\\n\\hline $C_7$ & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$  & $(\\eta,\\eta^5)$ \\\\\n\\hline $C_7^{\\prime}$ & 1 & $(\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$ & $(1,\\eta,\\eta^2,\\eta^3,\\eta^4,\\eta^5,\\eta^6)$  & $(\\eta,\\eta^5)$ \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Eigenvalues of group elements in each conjugacy class of $PU(3;3)$ for the irreducible representations $\\Sigma_1$, $\\Sigma_6$ and $\\Sigma_7^{\\prime}$, where $\\omega = e^{2\\pi i\/3}$, $\\mu = e^{2\\pi i\/6}$, $\\eta = e^{2\\pi i\/7}$, $\\nu = e^{2\\pi i\/8}$ and $\\xi = e^{2\\pi i\/12}$} \\label{table:evalues-IP4}\n\\end{center}\n\\end{table}\n\nFrom considering the set of eigenvalues $X_C$ for group elements in $C$ in the representation $\\gamma_1^{(i)}$, we see that there is no choice of $(t_1^C,t_2^C) \\in X_C$ such that $\\mathcal{E}_{t_1^C,t_2^C} = X_C$ for $i=1$ when $C=C_{12},C_{12}^{\\prime}$. However, such a choice does exist for all conjugacy classes $C$ for $i=2$, thus we have $\\varrho_1 = \\gamma_1^{(2)}$. We present one such choice of eigenvalues $(t_1^C,t_2^C)$ in the final column of Table \\ref{table:evalues-IP4}.\nThe McKay graph $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$ is given in \\cite[Figure 2]{he:2003}, and we reproduce it here in Figure \\ref{Fig-McKay_Graph-IP4-rho1} for completeness.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=70mm]{Fig-McKay_Graph-IP4-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$.} \\label{Fig-McKay_Graph-IP4-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{14} + \\Sigma_{27},$$\nwhere $\\mathrm{id} = \\Sigma_1$.\nThus the fourteen-dimensional representation $\\varrho_2$ must be $\\Sigma_{14}$, and we note that $\\chi_{\\varrho_2}(C) = \\Phi_2(t_1^C,t_2^C)$ as required.\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP4} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{6048} \\Omega^W(0,0) + \\frac{56}{6048} \\Omega^W(1\/3,2\/3) + \\frac{63}{6048} \\Omega^W(0,1\/2) + \\frac{672}{6048} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{63+63}{6048} \\Omega^W(0,1\/4) + \\frac{378}{6048} \\Omega^W(1\/4,1\/2) + \\frac{504}{6048} \\Omega^W(1\/6,1\/3) \\\\\n& \\quad + \\frac{864+864}{6048} \\Omega^W(1\/7,3\/7) + \\frac{756+756}{6048} \\Omega^W(1\/8,1\/2) + \\frac{504+504}{6048} \\Omega^W(1\/12,5\/12),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=135mm]{Fig-OmegaWIP4}\\\\\n \\caption{The orbits of $(a)$ $(1\/6,1\/3)$, $(b)$ $(1\/12,5\/12)$.} \\label{Fig-OmegaWIP4}\n\\end{center}\n\\end{figure}\n\nNow $2\\Omega^W(1\/3,2\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\, (3 \\mathrm{d}^{(1)} - \\mathrm{d}_1 \\times \\mathrm{d}_1)$.\nThe points $\\circ$ in Figure \\ref{Fig-OmegaWIP4}$(a)$ give the measure $3 \\, \\mathrm{d}^{(1)}$ whilst the points $\\diamond$ in Figure \\ref{Fig-OmegaWIP4}$(a)$ give the measure $4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\mathrm{d}_1 \\times \\mathrm{d}_1$, thus we see that $6\\Omega^W(1\/6,1\/3) = \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) \\left( 12 \\, \\mathrm{d}^{(2)} - 3 \\, \\mathrm{d}^{(1)} - 4 \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 + \\mathrm{d}_1 \\times \\mathrm{d}_1 \\right)$.\n\nFinally, $12\\Omega^W(1\/12,5\/12) = 18 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}^{(4)}$, as illustrated in Figure \\ref{Fig-OmegaWIP4}$(b)$ since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/12,5\/12)\/64\\pi^4) = 12$ for all $g \\in D_{12}$.\n\nThe measures $J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7$, $(24-K) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4$, $\\mathrm{d}_3 \\times \\mathrm{d}_3$ and $\\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)}$ supported by the other points have all appeared in the previous sections, so we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $PU(3;3)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{672\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{144} (24-K) \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 + \\frac{1}{6} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{12} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 \\\\\n& \\quad + \\frac{1}{168} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}^{(4)} + \\frac{1}{6} \\, \\mathrm{d}^{(2)} - \\frac{1}{12} \\, \\mathrm{d}^{(1)} + \\frac{1}{48} \\sum_{g \\in D_{12}} \\delta_{g(e^{\\pi i\/4},-1)},\n\\end{split}\n\\end{equation}\nwhere $K(\\theta_1,\\theta_2) = (\\sin(2\\pi(\\theta_1+\\theta_2))-\\sin(2\\pi(2\\theta_1-\\theta_2))-\\sin(2\\pi(2\\theta_2-\\theta_1)))^2$, $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity, $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$, and $\\delta_x$ is the Dirac measure at the point $x$.\n\\end{Thm}\n\n\n\\section{Group $G_2(2)$} \\label{sect:IP5}\n\nThe subgroup $G_2(2) = G_2(\\mathbb{F}_2)$ of $G_2 = G_2(\\mathbb{C})$ is an irreducible primitive group of order 12096. It is the group $G_2$ defined over the Galois field $\\mathbb{F}_2$.\nIt has sixteen irreducible representations (fourteen real and two complex conjugate representations), and its character table is given in\n\\cite{he:2003} (the orders of the elements in each conjugacy class can be obtained from \\cite{koca\/koc:1994}).\n\n\\begin{table}[tb]\n\\begin{center}\n\\begin{tabular}{|@{\\hspace{1.5mm}}c@{\\hspace{1.5mm}}||@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|@{\\hspace{1mm}}c @{\\hspace{1mm}}|@{\\hspace{1mm}}c@{\\hspace{1mm}}|} \\hline\n$C$ & $C_1$ & $C_3$ & $C_2$ & $C_4$ & $C_2^{\\prime}$ & $C_4^{\\prime}$ & $C_4^{\\prime\\prime}$ & $C_6$ & $C_3^{\\prime}$ & $C_{12}$ & $C_{12}^{\\prime}$ & $C_{12}^{\\prime\\prime}$ & $C_8$ & $C_8^{\\prime}$ & $C_7$ & $C_6^{\\prime}$ \\\\\n\\hline $|C|$                & 1  & 56 & 63 & 126 & 252 & 252 & 378 & 504 & 672 & 1008 & 1008 & 1008 & 1512 & 1512 & 1728 & 2016 \\\\\n\\hline $\\Sigma_1$           & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1  & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\\\\n\\hline $\\Sigma_1'$          & 1  & 1  & 1  & 1  & -1 & -1 & 1  & 1  & 1  & -1 & -1 & 1 & -1 & 1 & 1 & -1 \\\\\n\\hline $\\Sigma_6$           & 6  & -3 & -2 & -2 & 0  & 0  & 2  & 1  & 0  & $i\\sqrt{3}$ & $-i\\sqrt{3}$ & 1 & 0 & 0 & -1 & 0 \\\\\n\\hline $\\Sigma_6^{\\ast}$    & 6  & -3 & -2 & -2 & 0  & 0  & 2  & 1  & 0  & $-i\\sqrt{3}$ & $i\\sqrt{3}$ & 1 & 0 & 0 & -1 & 0 \\\\\n\\hline $\\Sigma_7$           & 1  & -2 & -1 & 3  & -1 & 3  & -1 & 2  & 1  & 0 & 0 & 0 & 1 & -1 & 0 & -1 \\\\\n\\hline $\\Sigma_7'$          & 1  & -2 & -1 & 3  & 1  & -3 & -1 & 2  & 1  & 0 & 0 & 0 & -1 & -1 & 0 & 1 \\\\\n\\hline $\\Sigma_{14}$        & 14 & 5  & -2 & 2  & -2 & 2  & 2  & 1  & -1 & -1 & -1 & -1 & 0 & 0 & 0 & 1 \\\\\n\\hline $\\Sigma_{14}'$       & 14 & -4 & 6  & -2 & 0  & 0  & 2  & 0  & 2  & 0 & 0 & -2 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{14}''$      & 14 & 5  & -2 & 2  & 2  & -2 & 2  & 1  & -1 & 1 & 1 & -1 & 0 & 0 & 0 & -1 \\\\\n\\hline $\\Sigma_{21}$        & 21 & 3  & 5  & 1  & 3  & -1 & 1  & -1 & 0  & -1 & -1 & 1 & 1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{21}'$       & 21 & 3  & 5  & 1  & -3 & 1  & 1  & -1 & 0  & 1 & 1 & 1 & -1 & -1 & 0 & 0 \\\\\n\\hline $\\Sigma_{27}$        & 27 & 0  & 3  & 3  & 3  & 3  & -1 & 0  & 0  & 0 & 0 & 0 & -1 & 1 & -1 & 0 \\\\\n\\hline $\\Sigma_{27}'$       & 27 & 0  & 3  & 3  & -3 & -3 & -1 & 0  & 0  & 0 & 0 & 0 & 1 & 1 & -1 & 0 \\\\\n\\hline $\\Sigma_{42}$        & 42 & 6  & 2  & -6 & 0  & 0  & -2 & 2  & 0  & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{56}$        & 56 & 2  & -8 & 0  & 0  & 0  & 0  & -2 & 2  & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n\\hline $\\Sigma_{64}$        & 64 & -8 & 0  & 0  & 0  & 0  & 0  & 0  & -2 & 0 & 0 & 0 & 0 & 0 & 1 & 0 \\\\\n\\hline\n\\end{tabular} \\\\\n\\caption{Character table for $G_2(2)$.} \\label{table:Character_table-IP5}\n\\end{center}\n\\end{table}\n\nThe group $G_2(2)$ has only two seven-dimensional real representations, both of which are irreducible. Of these two, only $\\Sigma_7$ has character values in $[-2,7]$ for all $g \\in G_2(2)$, thus this is the restriction $\\varrho_1$ of the seven-dimensional fundamental representation $\\rho_1$ of $G_2$ to $G_2(2)$.\nThe McKay graph for $\\varrho_1$ is given in Figure \\ref{Fig-McKay_Graph-IP5-rho1}.\nThis graph is a $\\mathbb{Z}_2$-orbifold of the McKay graph $\\mathcal{G}^{\\varrho_1}_{PU(3;3)}$ for $PU(3;3)$.\nNote that the McKay graph given in \\cite[Figure 2]{he:2003} is not that for $\\varrho_1$, but rather for the other irreducible seven-dimensional representation $\\Sigma_7'$ of $G_2(2)$.\n\n\\begin{figure}[tb]\n\\begin{center}\n  \\includegraphics[width=70mm]{Fig-McKay_Graph-IP5-rho1}\\\\\n \\caption{The McKay graphs $\\mathcal{G}^{\\varrho_1}_{G_2(2)}$.} \\label{Fig-McKay_Graph-IP5-rho1}\n\\end{center}\n\\end{figure}\n\nThe decomposition of the Kronecker square of $\\varrho_1$ into irreducibles is given by\n$$\\varrho_1^2 = \\mathrm{id} + \\varrho_1 + \\Sigma_{14} + \\Sigma_{27},$$\nthus the fourteen-dimensional representation $\\varrho_2$ is given by the irreducible representation $\\Sigma_{14}$.\nWe note that the eigenvalues of the representation matrices for $C_4, C_4^{\\prime}, C_4^{\\prime\\prime}$ all square to those for $C_2$, those for $C_6$ square to those for $C_3$, those for $C_6^{\\prime}$ square to those for $C_3^{\\prime}$, those for $C_8$ square to those for $C_4$, those for $C_8^{\\prime}$ square to those for $C_4^{\\prime\\prime}$, and those for $C_{12}, C_{12}^{\\prime}, C_{12}^{\\prime\\prime}$ all square to those for $C_6$.\n\n\nThen from (\\ref{eqn:moments-subgroupG2}) and Tables \\ref{table:Character_table-IP5} and \\ref{Table:subgroupsG2-orbits(theta1,theta2)}, we see that\n\\begin{align*}\n\\varsigma_{m,n} & = \\frac{1}{12096} \\Omega^W(0,0) + \\frac{56}{12096} \\Omega^W(1\/3,2\/3) + \\frac{63+252}{12096} \\Omega^W(0,1\/2) + \\frac{672}{12096} \\Omega^W(0,1\/3) \\\\\n& \\quad + \\frac{126+252}{12096} \\Omega^W(0,1\/4) + \\frac{378}{12096} \\Omega^W(1\/4,1\/2) + \\frac{504}{12096} \\Omega^W(1\/6,1\/3) \\\\\n& \\quad + \\frac{1728}{12096} \\Omega^W(1\/7,3\/7) + \\frac{1512}{12096} \\Omega^W(1\/8,1\/2) + \\frac{1512}{12096} \\Omega^W(1\/8,3\/8) \\\\\n& \\quad + \\frac{1008+1008+1008}{12096} \\Omega^W(1\/12,5\/12) + \\frac{2016}{12096} \\Omega^W(1\/6,1\/2),\n\\end{align*}\nwhere $\\Omega^W(\\theta_1,\\theta_2)$ is as in Section \\ref{sect:II1}.\nThe measures given by these points have all appeared in the previous sections.\nNote however that $12(\\Omega^W(1\/8,1\/2) + \\Omega^W(1\/8,3\/8)) = 8 \\int_{\\mathbb{T}^2} \\Omega(\\theta_1,\\theta_2) (J^2\/64\\pi^4) \\, \\mathrm{d}_8 \\times \\mathrm{d}_8$, since the Jacobian $J=0$ along the boundaries of the orbit of the fundamental domain, whilst $J^2(g(1\/8,1\/2))\/64\\pi^4 = J^2(g(1\/8,3\/8))\/64\\pi^4 = 8$ for all $g \\in D_{12}$.\nThus we obtain:\n\n\\begin{Thm}\nThe joint spectral measure (over $\\mathbb{T}^2$) for all embeddings of $G_2(2)$ into the fundamental representations of $G_2$ is\n\\begin{equation}\n\\begin{split}\n\\mathrm{d}\\varepsilon & = \\frac{1}{768\\pi^4} J^2 \\, \\mathrm{d}_8 \\times \\mathrm{d}_8 + \\frac{1}{1344\\pi^4} J^2 \\, \\mathrm{d}_7 \\times \\mathrm{d}_7 + \\frac{1}{1152\\pi^4} J^2 \\, \\mathrm{d}_6 \\times \\mathrm{d}_6 + \\frac{1}{12} \\, \\mathrm{d}_4 \\times \\mathrm{d}_4 \\\\\n& \\quad + \\frac{1}{12} \\, \\mathrm{d}_3 \\times \\mathrm{d}_3 - \\frac{1}{72} \\, \\mathrm{d}_2 \\times \\mathrm{d}_2 - \\frac{1}{252} \\, \\mathrm{d}_1 \\times \\mathrm{d}_1 + \\frac{1}{768\\pi^4} J^2 \\, \\mathrm{d}^{(4)} + \\frac{1}{12} \\, \\mathrm{d}^{(2)} - \\frac{1}{24} \\, \\mathrm{d}^{(1)},\n\\end{split}\n\\end{equation}\nwhere $\\mathrm{d}_m$ is the uniform measure over $m^{\\mathrm{th}}$ roots of unity and $\\mathrm{d}^{(k+4)}$ is the uniform measure on the points in $C_k^W$.\n\\end{Thm}\n\n\n\n\n\n\n\n\n\n\\bigskip \\bigskip\n\n\\begin{footnotesize}\n\\noindent{\\it Acknowledgement.}\n\nThe second author was supported by the Coleg Cymraeg Cenedlaethol.\n\\end{footnotesize}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $S$ be a complex manifold and $D$ be a (reduced) hypersurface $D$, referred to as a \\emph{divisor} in the sequel. \nIn the landmark paper \\cite{Sai80}, Kyoji Saito introduced the sheaves of $\\O_S$-modules of logarithmic differential forms and logarithmic vector fields on $S$ along $D$.\nLogarithmic vector fields are tangent to $D$ at any smooth point of $D$; logarithmic differential forms have simple poles and form a complex under the usual differential.\nSaito's clean algebraically flavored definition encodes deep geometric, topological, and representation theoretic information on the singularities that is yet only partly understood.\nThe precise target for his theory was the Gau\\ss-Manin connection on the base $S$ of the semiuniversal deformation of isolated hypersurface singularities, a logarithmic connection along the discriminant $D$.\nSaito developed mainly three aspects of his logarithmic theory in loc.~cit.: Free divisors, logarithmic stratifications, and logarithmic residues.\nMany fascinating developments grew out of Saito's paper, a few of which we highlight in the following brief overview.\n\nA divisor is called free if the sheaf of logarithmic vector fields, or its dual, the sheaf of logarithmic $1$-forms, is a vector bundle; in particular, normal crossing divisors are free.\nNot surprisingly, discriminants of isolated hypersurface singularities are free divisors (see \\cite[(3.19)]{Sai80}). \nSimilar results were shown for isolated complete intersection singularities (see \\cite[\\S6]{Loo84}) and space curve singularities (see \\cite{vST95}).\nBoth the reflection arrangements and discriminants associated with finite unitary reflection groups are free divisors (see \\cite{Ter80b}).\nMore recent examples include discriminants in certain prehomogeneous vector spaces (see \\cite{GMS11}) whose study led to new constructions such as a chain rule for free divisors (see \\cite[\\S4]{BC12}).\nFree divisors can be seen as the extreme case opposite to isolated singularities: Unless smooth, free divisors have Cohen--Macaulay singular loci of codimension $1$.\nThe freeness property is closely related to the complement of the divisor being a $K(\\pi,1)$-space (see \\cite[(1.12)]{Sai80}, \\cite{Del72}), although these two properties are not equivalent (see \\cite{ER95}). \nEven in special cases, such as that of hyperplane arrangements, freeness is not fully understood yet. \nFor instance, Terao's conjecture on the combinatorial nature of freeness for arrangements is one of the central open problems in arrangement theory.\n\nSaito's second topic, the so-called logarithmic stratification of $S$, consists of immersed integral manifolds of logarithmic vector fields along $D$.\nContrary to what the terminology suggests, the resulting decomposition of $S$ is not locally finite, in general.\nSaito attached the term holonomic to this additional feature: a point in $S$ is holonomic if a neighborhood meets only finitely many logarithmic strata.\nAlong any logarithmic stratum, the pair $(D,S)$ is analytically trivial which turns holonomicity into a property of logarithmic strata.\nThe logarithmic vector fields are tangent to the strata of the canonical Whitney stratification; the largest codimension up to which all Whitney strata are (neccessarily holonomic) logarithmic strata is called the holonomic codimension (see \\cite[p.~221]{DM91}).\nHolonomic free divisors were later called Koszul--free divisors.\n\nIn case of a normal crossing divisor, the complex of logarithmic differential forms computes the cohomology of the complement of $D$ in $S$, an ingredient of Deligne's mixed Hodge structure (see \\cite{Del71}).\nThe natural question, for which free divisors the same holds true for the complex of logarithmic differential forms is referred to as the logarithmic comparison theorem, or, for short, by the LCT (see \\cite{Tor07} for a survey).\nFor free divisors, this property turned out to be related to homogeneity properties of the singularities. \nIndeed, an explicit class of hypersurfaces for which the LCT holds true is that of (weakly) locally quasihomogeneous divisors (see \\cite{CNM96,Nar08}).\nMoreover, it is conjectured that the LCT implies strong Euler homogeneity of $D$, which has been proved only for Koszul free divisors, and in dimension $\\dim S\\le3$ (see \\cite{CMNC02,GS06}).\nFor strongly Koszul--free divisors $D$ (see \\cite[Def.~7.1]{GS10}), the logarithmic comparison theorem holds true exactly if $D$ is strongly Euler homogeneous and $-1$ is the minimal integer root of all local $b$-functions (see \\cite[Cor.~4.3]{CN05} and \\cite[Cor.~1.8]{Tor04}).\nFor isolated quasihomogeneous singularities, the LCT is equivalent to the vanishing of certain graded parts of the Milnor algebra (cee \\cite{HM98}); there are related Hodge-theoretic properties in the non-quasihomogeneous case (\\cite{Sch10}).\nThe study of the LCT lead to a variant of $D$-module theory over the ring of logarithmic differential operators along a free divisor (see \\cite{CU02,CN05,Nar08}).\nA key player in this context is the $D$-module $M^{\\log D}$ defined by the ideal of logarithmic vector fields, considered as differential operators of order one (see \\cite{CU02}); Saito-holonomicity of $D$ implies holonomicity of $M^{\\log D}$ in the $D$-module-sense; but the converse is false.\n\nMuch less attention has been devoted to Saito's logarithmic residues, the main topic in this paper.\nIt was Poincar\\'e who first defined a residual $1$-form of a rational differential $2$-form on $\\mathds{C}^2$ (see \\cite{Poi87}).\nLater, the concept was generalized by de Rham and Leray to residues of closed meromorphic $p$-forms with simple poles along a smooth divisor $D$; these residues are holomorphic $(p-1)$-form on $D$ (see \\cite{Ler59}).\nThe construction of Deligne's mixed Hodge structure uses, again holomorphic, residues of logarithmic differential forms along normal crossing divisors (see \\cite{Del71}).\nNotably, in Saito's generalization to arbitrary singular divisors $D$, the residue of a logarithmic $p$-form becomes a \\emph{meromorphic} $(p-1)$-form on $D$, or on its normalization $\\widetilde D$. \nUsing work of Barlet~\\cite{Bar78}, Aleksandrov linked Saito's construction to Grothendieck duality theory: The image of Saito's logarithmic residue map is the module of regular differential forms on $D$ (see \\cite[\\S4, Thm.]{Ale88} and \\cite{Bar78}).\nWith Tsikh, he suggested a generalization for complete intersection singularities based on multilogarithmic differential forms depending on a choice of generators of the defining ideal (see \\cite{AT01}). \nRecently, he approached the natural problem of describing the mixed Hodge structure on the complement of an LCT divisor in terms of logarithmic differential forms (see \\cite{Ale12}).\nIn Dolgachev's work, one finds a different sheaf of logarithmic differential forms which is a vector bundle exactly for normal crossing divisors and whose reflexive hull is Saito's sheaf of logarithmic differential forms (see \\cite{Dol07}).\nAlthough his approach to logarithmic residues using adjoint ideals has a similar flavor to ours, he does not reach the conclusion of our main Theorem~\\ref{10} (see Remark~\\ref{49b}).\n\n\\medskip\n\nWhile most constructions in Saito's logarithmic theory and its generalizations have a dual counterpart, a notion of a dual logarithmic residue associated to a vector field was not known to the authors.\nThe main motivation and fundamental result of this article is the construction of a dual logarithmic residue (see Section~\\ref{13}).\nThis turned out to have surprising applications including a proof of a conjecture of Saito, that was open for more than 30 years.\nSaito's conjecture is concerned with comparing logarithmic residues of $1$-forms, that is, certain meromorphic functions on $\\widetilde D$, with holomorphic functions on $\\widetilde D$.\nThe latter can also be considered as a weakly holomorphic function on $D$, that is, functions on the complement of the singular locus $Z$ of $D$, locally bounded near points of $Z$.\nWhile any such weakly holomorphic function is the residue of some logarithmic $1$-form, the image of the residue map can contain functions which are not weakly holomorphic.\nThe algebraic condition of equality was related by Saito to a geometric and a topological property as follows (see \\cite[(2.13)]{Sai80}).\n\n\\begin{thm}[Saito]\\label{28}\nFor a divisor $D$ in a complex manifold $S$, consider the following conditions:\n\\begin{enumerate}[(A)]\n\\item\\label{28a} The local fundamental groups of the complement $S\\backslash D$ are Abelian.\n\\item\\label{28b} In codimension $1$, that is, outside of an analytic subset of codimension at least $2$ in $D$, $D$ is normal crossing.\n\\item\\label{28c} The residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$.\n\\end{enumerate}\nThen the implications \\eqref{28a} $\\Rightarrow$ \\eqref{28b} $\\Rightarrow$ \\eqref{28c} hold true.\n\\end{thm}\n\nSaito asked whether the the converse implications in Theorem~\\ref{28} hold true.\nThe first one was later established by L\\^e and Saito~\\cite{LS84}; it generalizes the Zariski conjecture for complex plane projective nodal curves proved by Fulton and Deligne (see \\cite{Ful80,Del81}).\n\n\\begin{thm}[L\\^e--Saito]\nThe implication \\eqref{28a} $\\Leftarrow$ \\eqref{28b} in Theorem~\\ref{28} holds true.\n\\end{thm}\n\nOur duality of logarithmic residues turns out to translate condition \\eqref{28c} in Theorem~\\ref{28} into the more familiar equality of the Jacobian ideal and the conductor ideal.\nA result of Ragni Piene~\\cite{Pie79} proves that such an equality forces $D$ to have only smooth components if it has a smooth normalization. \nThis is a technical key point which leads to a proof of the missing implication in Theorem~\\ref{28}.\n\n\\begin{thm}\\label{10}\nThe implication \\eqref{28b} $\\Leftarrow$ \\eqref{28c} in Theorem~\\ref{28} holds true: If the residue of any logarithmic $1$-form along $D$ is a weakly holomorphic function on $D$ then $D$ is normal crossing in codimension $1$.\n\\end{thm}\n\n\\begin{rmk}\\label{49b}\nSaito~\\cite[(2.11)]{Sai80} proved Theorem~\\ref{10} for plane curves.\nIf $D$ has holonomic codimension at least $1$ (as defined above), this yields the general case by analytic triviality along logarithmic strata (see \\cite[\\S3]{Sai80}).\nUnder this latter hypothesis, Theorem~\\ref{10} follows also from a result of Dolgachev (see \\cite[Cor.~2.2]{Dol07}).\nHowever, for example, the equation $xy(x+y)(x+yz)=0$ defines a well-known free divisor with holonomic codimension $0$.\n\\end{rmk}\n\nThe preceding results and underlying techniques serve to address two natural questions:\nThe algebraic characterization of condition~\\eqref{28c} through Theorem~\\ref{10} raises the question about the algebraic characterizations of normal crossing divisors.\nEleonore Faber was working on this question at the same time as the results presented here were developed.\nShe considered freeness as a first approximation for being normal crossing and noticed that normal crossing divisors satisfy an extraordinary condition:\nThe ideal of partial derivatives of a defining equation is radical. \nShe proved the following converse implications (see \\cite{Fab11,Fab12}).\n\n\\begin{thm}[Faber]\nConsider the following condition:\n\\begin{enumerate}[(A)]\\setcounter{enumi}{3}\n\\item\\label{28e} At any point $p\\in D$, there is a local defining equation $h$ for $D$ such that the ideal $\\mathcal{J}_h$ of partial derivatives is radical.\n\\item\\label{28f} $D$ is normal crossing.\n\\end{enumerate}\nThen the following holds:\n\\begin{asparaenum}\n\\item If $D$ is free then condition~\\eqref{28e} decends to all irreducible components of $D$.\n\\item Conditions~\\eqref{28e} and \\eqref{28f} are equivalent if $D$ is locally a plane curve or\na hyperplane arrangement, or \nif its singular locus is Gorenstein.\n\\end{asparaenum}\n\\end{thm}\n\nMotivated by Faber's problem we prove the following \n\n\\begin{thm}\\label{16}\nExtend the list of conditions in Theorem~\\ref{28} as follows:\n\\begin{enumerate}[(A)]\\setcounter{enumi}{5}\n\\item\\label{28d} The Jacobian ideal $\\mathcal{J}_D$ of $D$ is radical.\n\\item\\label{28g} The Jacobian ideal $\\mathcal{J}_D$ of $D$ equals the conductor ideal $\\mathcal{C}_D$ of the normalization $\\tilde D$.\n\\end{enumerate}\nThen condition~\\eqref{28d} implies condition~\\eqref{28b}.\nIf $D$ is a free divisor then conditions~\\eqref{28b}, \\eqref{28d} and \\eqref{28g} are equivalent.\n\\end{thm}\n\n\\begin{rmk}\\label{9}\nNote that $\\mathcal{J}_h$ is an $\\O_S$-ideal sheaf depending on a choice of local defining equation whereas its image $\\mathcal{J}_D$ in $\\O_D$ is intrinsic to $D$.\nIn particular, condition~\\eqref{28e} implies condition~\\eqref{28d}.\n\\end{rmk}\n\nWe obtain the following algebraic characterization of normal crossing divisors.\n\n\\begin{thm}\\label{38}\nFor a free divisor with smooth normalization, any one of the conditions~\\eqref{28a}, \\eqref{28b}, \\eqref{28c}, \\eqref{28d}, or \\eqref{28g} implies condition \\eqref{28f}.\n\\end{thm}\n\n\\begin{rmk}\nThe implication \\eqref{28d} $\\Rightarrow$ \\eqref{28f} in Theorem~\\ref{38} improves Theorem A in \\cite{Fab12} (see Remark~\\ref{9}), which is proved using \\cite{Pie79} like in the proof of our main result.\nProposition~C in \\cite{Fab12} is the implication \\eqref{28c} $\\Rightarrow$ \\eqref{28f} in Theorem~\\ref{38}, for the proof of which Faber uses our arguments.\n\\end{rmk}\n\nAs remarked above, free divisors are characterized by their singular loci being (empty or) maximal Cohen--Macaulay.\nIt is natural to ask when the singular locus of a divisor is Gorenstein.\nThis question is answered by the following\n\n\\begin{thm}\\label{40}\nA divisor $D$ has Gorenstein singular locus $Z$ of codimension $1$ if and only if $D$ is locally the product of a quasihomogeneous plane curve and a smooth space.\nIn particular, $D$ is locally quasihomogeneous and $Z$ is locally a complete intersection.\n\\end{thm}\n\n\\begin{rmk}\nTheorem~\\ref{40} complements a result of Kunz--Waldi \\cite[Satz~2]{KW84} saying that a Gorenstein algebroid curve has Gorenstein singular locus if and only if it is quasihomogeneous.\n\\end{rmk}\n\n\\section{Logarithmic modules and fractional ideals}\\label{30}\n\nIn this section, we review Saito's logarithmic modules, the relation of freeness and Cohen--Macaulayness of the Jacobian ideal, and the duality of maximal Cohen--Macaulay fractional ideals.\nWe switch to a local setup for the remainder of the article.\n \nLet $D$ be a reduced effective divisor defined by $\\mathcal{I}_D=\\O_S\\cdot h$ in the smooth complex analytic space germ $S=(\\mathds{C}^n,0)$.\nDenote by $h\\colon S\\to T=(\\mathds{C},0)$ a function germ generating the ideal $\\mathcal{I}_D=\\O_S\\cdot h$ of $D$.\nWe abbreviate by\n\\[\n\\Theta_S:=\\Der_\\mathds{C}(\\O_S)=\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\n\\]\nthe $\\O_S$-module of vector fields on $S$.\nRecall Saito's definition \\cite[\\S1]{Sai80} of the $\\O_S$-modules of logarithmic differential forms and of logarithmic vector fields.\n\n\\begin{dfn}[Saito]\\label{33}\n\\begin{align*}\n\\Omega^p(\\log D)&:=\\{\\omega\\in\\Omega^p_S(D)\\mid d\\omega\\in\\Omega_S^{p+1}(D)\\}\\\\\n\\Der(-\\log D)&:=\\{\\delta\\in\\Theta_S\\mid dh(\\delta)\\in\\mathcal{I}_D\\}\n\\end{align*}\n\\end{dfn}\n\nThese modules are stalks of analogously defined coherent sheaves of $\\O_S$-modules (see \\cite[(1.3),(1.5)]{Sai80}).\nIt is obvious that each of these sheaves $\\L$ is torsion free and normal, and hence reflexive (see \\cite[Prop.~1.6]{Har80}).\nMore precisely, $\\Omega^1(\\log D)$ and $\\Der(-\\log D)$ are mutually $\\O_S$-dual (see \\cite[(1.6)]{Sai80}).\nNormality of a sheaf $\\L$ means that $\\L=i_*i^*\\L$ where $i\\colon S\\setminus Z\\hookrightarrow S$ denotes the inclusion of the complement of the singular locus of $D$.\nIn case of $\\L=\\Der(-\\log D)$, this means that $\\delta\\in\\Der(-\\log D)$ if and only if $\\delta$ is tangent to $D$ at all smooth points.\nIn addition, $\\Omega^\\bullet(\\log D)$ is an exterior algebra over $\\O_S$ closed under exterior differentiation and $\\Der(-\\log D)$ is closed under the Lie bracket.\n\n\\begin{dfn}\nA divisor $D$ is called free if $\\Der(-\\log D)$ is a free $\\O_S$-module.\n\\end{dfn}\n\nThe definition of $\\Der(-\\log D)$ can be rephrased as a short exact sequence of $\\O_S$-modules\n\\begin{equation}\\label{1}\n\\SelectTips{cm}{}\\xymatrix{\n0&\\mathcal{J}_D\\ar[l]&\\Theta_S\\ar[l]_-{dh}&\\Der(-\\log D)\\ar[l]&0\\ar[l]\n}\n\\end{equation}\nwhere the Jacobian ideal $\\mathcal{J}_D$ of $D$ is defined as the Fitting ideal \n\\[\n\\mathcal{J}_D:=\\mathcal{F}^{n-1}_{\\O_D}(\\Omega_D^1)=\\ideal{\\frac{\\partial h}{\\partial x_1},\\dots,\\frac{\\partial h}{\\partial x_n}}\\subset{\\O_D}.\n\\]\nNote that $\\mathcal{J}_D$ is an ideal in $\\O_D$ and pulls back to $\\ideal{h,\\frac{\\partial h}{\\partial x_1},\\dots,\\frac{\\partial h}{\\partial x_n}}$ in $\\O_S$.\nWe shall consider the singular locus $Z$ of $D$ equipped with the structure defined by $\\mathcal{J}_D$, that is,\n\\begin{equation}\\label{21}\n\\O_Z:=\\O_D\/\\mathcal{J}_D.\n\\end{equation}\nNote that $Z$ might be non-reduced.\nThere is the following intrinsic characterization of free divisors in terms of their singular locus (see \\cite[\\S1 Thm.]{Ale88} or \\cite[Prop.~2.4]{Ter80a}).\n\n\\begin{thm}\\label{19}\nThe following are equivalent:\n\\begin{enumerate}\n\\item\\label{19a} $D$ is a free divisor.\n\\item\\label{19b} $\\mathcal{J}_D$ is a maximal Cohen--Macaulay $\\O_D$-module.\n\\item\\label{19c} $D$ is smooth or $Z$ is Cohen--Macaulay of codimension $1$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nIf $dh(\\Theta_S)$ does not minimally generate $\\mathcal{J}_D$, then $D\\cong D'\\times(\\mathds{C}^k,0)$, $k>0$, by the triviality lemma~\\cite[(3.5)]{Sai80}.\nBy replacing $D$ by $D'$, we may therefore assume that \\eqref{1} is a minimal resolution of $\\mathcal{J}_D$ as $\\O_S$-module.\nThus, the equivalence of \\eqref{19a} and \\eqref{19b} is due to the Auslander--Buchsbaum formula.\nBy Lemma~\\ref{25} below, $\\mathcal{J}_D$ has height at least $1$ and the implication~\\eqref{19b} $\\Leftrightarrow$ \\eqref{19c} is proved in \\cite[Satz~4.13]{HK71}.\n\\end{proof}\n\n\\begin{cor}\\label{27}\nAny $D$ is free in codimension $1$.\n\\end{cor}\n\n\\begin{proof}\nBy Theorem~\\ref{19}, the non-free locus of $D$ is contained in $Z$ and equals\n\\[\n\\{z\\in Z\\mid\\depth\\O_{Z,z}<n-2\\}\\subset D.\n\\]\nBy Scheja~\\cite[Satz~5]{Sch64}, this is an analytic set of codimension at least $2$ in $D$.\n\\end{proof}\n\nWe denote by $Q(-)$ the total quotient ring.\nThen $\\mathcal{M}_D:=Q(\\O_D)$ is the ring of meromorphic functions on $D$.\n\n\\begin{dfn}\nA fractional ideal (on $D$) is a finite $\\O_D$-submodule of $\\mathcal{M}_D$ which contains a non-zero divisor.\n\\end{dfn}\n\n\\begin{lem}\\label{25}\n$\\mathcal{J}_D$ is a fractional ideal.\n\\end{lem}\n\n\\begin{proof}\nBy assumption $D$ is reduced, so $\\O_D$ satisfies Serre's condition $R_0$.\nThis means that $Z\\subset D$ has codimension at least $1$.\nIn other words, $\\mathcal{J}_D\\not\\subseteq\\mathfrak{p}$ for all $\\mathfrak{p}\\in\\Ass(\\O_D)$ where the latter denotes the set of (minimal) associated primes of $\\O_D$.\nBy prime avoidance, $\\mathcal{J}_D\\not\\subseteq\\bigcup_{\\mathfrak{p}\\in\\Ass\\O_D}\\mathfrak{p}$.\nBut the latter is the set of zero divisors of $\\O_D$ and the claim follows.\n\\end{proof}\n\n\\begin{prp}\\label{18}\nThe $\\O_D$-dual of any fractional ideal $\\mathcal{I}$ is again a fractional ideal $\\mathcal{I}^\\vee=\\{f\\in\\mathcal{M}_D\\mid f\\cdot\\mathcal{I}\\subseteq\\O_D\\}$.\nThe duality functor \n\\[\n-^\\vee=\\Hom_{\\O_D}(-,\\O_D)\n\\]\nreverses inclusions. \nIt is an involution on the class of maximal Cohen--Macaulay fractional ideals.\n\\end{prp}\n\n\\begin{proof}\nSee \\cite[Prop.~(1.7)]{JS90a}.\n\\end{proof}\n\nFor lack of reference, we prove the following \n\n\\begin{lem}\\label{39}\nLet $A$ be a Gorenstein ring whose normalization $B$ is a finite $A$-module. \nThen $B$ is a reflexive $A$-module.\n\\end{lem}\n\n\\begin{proof}\nWe apply a standard characterization of reflexive modules (see \\cite[Prop.~1.4.1.(b)]{BH93}).\nTo this end let $\\mathfrak{p}\\in\\Spec A$ and let $\\mathfrak{q}\\in\\Spec B$ with $\\mathfrak{q}\\cap A=\\mathfrak{p}$.\nBy incomparability for integral extensions, $\\mathfrak{q}$ induces a maximal ideal in $B_\\mathfrak{p}$ and a minimal prime ideal in $B_\\mathfrak{p}\/\\mathfrak{p} B_\\mathfrak{p}$.\nIn particular, $B_\\mathfrak{p}$ has finitely many maximal ideals, the localization at one of them being $B_\\mathfrak{q}$.\nBy the Chinese remainder theorem, $B_\\mathfrak{q}\/\\mathfrak{p} B_\\mathfrak{q}$ is a direct factor of $B_\\mathfrak{p}\/\\mathfrak{p} B_\\mathfrak{p}$, and hence $B_\\mathfrak{q}$ is a finite $A_\\mathfrak{p}$-module by Nakayama's lemma.\nBy incomparability and going up for integral extensions, $\\dim B_\\mathfrak{q}=\\dim A_\\mathfrak{p}$.\nBy hypothesis, $B$ satisfies Serre's conditions $R_1$ and $S_2$.\n\nAfter these preparations, two cases need to be considered (see loc.~cit.):\n\\begin{asparaenum}\n\n\\item If $\\depth A_\\mathfrak{p}\\ge2$ then $\\dim B_\\mathfrak{q}=\\dim A_\\mathfrak{p}\\ge 2$.\nHence, $\\depth_{A_\\mathfrak{p}}B_\\mathfrak{q}=\\depth_{B_\\mathfrak{q}} B_\\mathfrak{q}\\ge 2$ by $S_2$ for $B$ and invariance of depth under finite extensions (see \\cite[IV-18, Prop.~12]{Ser65}).\nBut then also $\\depth_{A_\\mathfrak{p}}B_\\mathfrak{p}\\ge2$.\n\n\\item If $\\depth A_\\mathfrak{p}\\le1$ then also $\\dim A_\\mathfrak{p}\\le 1$ by $S_2$ for $A$.\nThus, $\\dim B_\\mathfrak{q}\\le 1$ and $B_\\mathfrak{q}$ is regular by $R_1$ for $B$.\nIn particular, $B_\\mathfrak{q}$, and hence $B_\\mathfrak{p}$, is a maximal Cohen--Macaulay ring.\nBy invariance of Cohen--Macaulayness under finite extensions (see \\cite[IV-18, Prop.~11]{Ser65})\\footnote{The proof in loc.~cit.~applies verbatim to the case where $B$ is semilocal.}, $B_\\mathfrak{p}$ is then also a maximal Cohen--Macaulay $A_\\mathfrak{p}$-module.\nAs $A_\\mathfrak{p}$ is Gorenstein, this implies that $B_\\mathfrak{p}$ is reflexive.\n\n\\end{asparaenum}\n\\end{proof}\n\n\\section{Logarithmic residues and duality}\\label{13}\n\nIn this section, we develop the dual picture of Saito's residue map and apply it to find inclusion relations of certain natural fractional ideals and their duals.\n\nLet $\\pi\\colon\\widetilde D\\to D$ denote the normalization of $D$.\nThen $\\mathcal{M}_D=\\mathcal{M}_{\\widetilde D}:=Q(\\O_{\\widetilde D})$ and $\\O_{\\widetilde D}$ is the ring of weakly holomorphic functions on $D$ (see \\cite[Exc.~4.4.16.(3), Thm.~4.4.15]{JP00}).\nLet\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n\\Omega^p(\\log D)\\ar[r]^-{\\rho^p_D}&\\Omega_D^{p-1}\\otimes_{\\O_D}\\mathcal{M}_D\n}\n\\]\nbe Saito's residue map \\cite[\\S2]{Sai80} which is defined as follows:\nBy \\cite[(1.1)]{Sai80}, any $\\omega\\in\\Omega^p(\\log D)$ can be written as\n\\begin{equation}\\label{4}\n\\omega=\\frac{dh}{h}\\wedge\\frac{\\xi}{g}+\\frac{\\eta}{g},\n\\end{equation}\nfor some $\\xi\\in\\Omega_S^{p-1}$, $\\eta\\in\\Omega^p_S$, and $g\\in\\O_S$, which restricts to a non-zero divisor in $\\O_D$.\nThen \n\\begin{equation}\\label{34}\n\\rho_D^p(\\omega):=\\frac{\\xi}{g}\\vert_D\n\\end{equation}\nis well defined by \\cite[(2.4)]{Sai80}.\nWe shall abbreviate $\\rho_D:=\\rho_D^1$ and denote its image by \n\\[\n\\mathcal{R}_D:=\\rho_D(\\Omega^1(\\log D)).\n\\]\nUsing this notation, condition~\\eqref{28c} in Theorem~\\ref{28}, that the residue of any $\\omega\\in\\Omega^1(\\log D)$ is weakly holomorphic, can be written as $\\O_{\\widetilde D}=\\mathcal{R}_D$.\n\nThe following result of Saito~\\cite[(2.9)]{Sai80} can be considered as a kind of approximation of our main result Theorem~\\ref{10}; in fact, it shall be used in its proof.\nCombined with his freeness criterion~\\cite[(1.8)]{Sai80} it yields Faber's characterization of normal crossing divisors in Proposition~B of \\cite{Fab12}.\n\n\\begin{thm}\\label{60}\nLet $D_i=\\{h_i=0\\}$, $i=1,\\dots,k$, denote irreducible components of $D$.\nThen the following conditions are equivalent:\n\\begin{enumerate}\n\\item\\label{60a} $\\Omega^1(\\log D)=\\sum_{i=1}^k\\O_S\\frac{dh_i}{h_i}+\\Omega^1_S$.\n\\item\\label{60b} $\\Omega^1(\\log D)$ is generated by closed forms.\n\\item\\label{60d} The $D_i$ are normal and intersect transversally in codimension $1$.\n\\item\\label{60c} $\\mathcal{R}_D=\\bigoplus_{i=1}^k\\O_{D_i}$.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{exa}\nThe Whitney umbrella (which is not a free divisor) satisfies the conditions in Theorem~\\ref{28}, but not those in Theorem~\\ref{60}. \n\\end{exa}\n\nThe implication~\\eqref{60d} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60} will be used in the proof of Theorem~\\ref{10} after a reduction to nearby germs with smooth irreducible components. \nIts proof essentially relies on parts \\eqref{48b} and \\eqref{48c} of the following \n\n\\begin{exa}\\label{48}\\\n\\begin{asparaenum}\n\n\\item\\label{48a} Let $D=\\{xy=0\\}$ be a normal crossing of $2$ irreducible components.\nThen $\\frac{dx}{x}\\in\\Omega^1(\\log D)$ and \n\\[\n(x+y)\\frac{dx}{x}=y\\frac{d(xy)}{xy}+dx-dy\n\\]\nshows that\n\\[\n\\rho_D\\left(\\frac{dx}{x}\\right)=\\frac{y}{x+y}\\Big\\vert_D.\n\\]\nOn the components $D_1=\\{x=0\\}$ and $D_2=\\{y=0\\}$ of the normalization $\\widetilde D=D_1\\coprod D_2$, this function equals $1$ and $0$ respectively and is therefore not in $\\O_D$.\nBy symmetry, this yields\n\\[\n\\mathcal{R}_D=\\O_{\\widetilde D}=\\O_{D_1}\\times\\O_{D_2}=\\mathcal{J}_D^\\vee\n\\]\nsince $\\mathcal{J}_D=\\ideal{x,y}_{\\O_D}$ is the maximal ideal in $\\O_D=\\mathds{C}\\{x,y\\}\/\\ideal{xy}$.\nThis observation will be generalized in Proposition~\\ref{22}.\n\n\\item\\label{48b} Conversely, assume that $D_1=\\{h_1=x=0\\}$ and $D_2=\\{h_2=x+y^m=0\\}$ are two smooth irreducible components of $D$.\nConsider the logarithmic $1$-form\n\\[\n\\omega=\\frac{ydx-mxdy}{x(x+y^m)}=y^{1-m}\\left(\\frac{dh_1}{h_1}-\\frac{dh_2}{h_2}\\right)\\in\\Omega^1(\\log(D_1+D_2))\\subset\\Omega^1(\\log D).\n\\]\nIts residue $\\rho_D(\\omega)\\vert_{D_1}=y^{1-m}\\vert_{D_1}$ has a pole along $D_1\\cap D_2$ unless $m=1$.\nThus, if $\\O_{\\widetilde D}=\\mathcal{R}_D$ then $D_1$ and $D_2$ must intersect transversally.\n\n\\item\\label{48c} Assume that $D$ contains $D'=D_1\\cup D_2\\cup D_3$ with $D_1=\\{x=0\\}$, $D_2=\\{y=0\\}$, and $D_3=\\{x-y=0\\}$.\nConsider the logarithmic $1$-form\n\\[\n\\omega=\\frac{1}{x-y}\\left(\\frac{dx}{x}-\\frac{dy}{y}\\right)\\in\\Omega^1(\\log D')\\subset\\Omega^1(\\log D).\n\\]\nIts residue $\\rho_D(\\omega)\\vert_{D_1}=-\\frac{1}{y}\\vert_{D_1}$ has a pole along $D_1\\cap D_2\\cap D_3$ and hence $\\O_{\\widetilde D}\\subsetneq\\mathcal{R}_D$.\n\n\\end{asparaenum}\n\\end{exa}\n\n\\medskip\n\nAfter these preparations, we shall now approach the construction of the dual logarithmic residue.\nBy definition, there is a short exact residue sequence\n\\begin{equation}\\label{2}\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\Omega^1_S\\ar[r]&\\Omega^1(\\log D)\\ar[r]^-{\\rho_D}&\\mathcal{R}_D\\ar[r]&0.\n}\n\\end{equation}\nApplying $\\Hom_{\\O_S}(-,\\O_S)$ to \\eqref{2} gives an exact sequence\n\\begin{equation}\\label{44}\n\\SelectTips{cm}{}\\xymatrix@C-1em{\n0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\Ext^1_{\\O_S}(\\mathcal{R}_D,\\O_S)\\ar[l]&\\Theta_S\\ar[l]&\\Der(-\\log D)\\ar[l]&0\\ar[l].\n}\n\\end{equation}\nThe right end of this sequence extends to the short exact sequence \\eqref{1}.\nFor the hypersurface ring $\\O_D$, the change of rings spectral sequence\n\\begin{equation}\\label{41}\nE^{p,q}_2=\\Ext_{\\O_D}^p(-,\\Ext^q_{\\O_S}(\\O_D,\\O_S))\\underset{p}\\Rightarrow\\Ext_{\\O_S}^{p+q}(-,\\O_S)\n\\end{equation}\ndegenerates because $E^{p,q}_2=0$ if $q\\neq 1$ and hence \n\\begin{equation}\\label{43}\n\\Ext_{\\O_S}^1(-,\\O_S)\\cong E^{0,1}_2\\cong\\Hom_{\\O_D}(-,\\O_D)\\cong-^\\vee.\n\\end{equation}\nTherefore, the second term in the sequence~\\eqref{44} is $\\mathcal{R}_D^\\vee$.\nThis motivates the following key technical result of this paper, describing the dual of Saito's logarithmic residue.\n\n\\begin{prp}\\label{22}\nThere is an exact sequence\n\\begin{equation}\\label{36}\n\\SelectTips{cm}{}\\xymatrix{\n0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\mathcal{R}_D^\\vee\\ar[l]&\\Theta_S\\ar[l]_-{\\sigma_D}&\\Der(-\\log D)\\ar[l]&0\\ar[l]\n}\n\\end{equation}\nsuch that $\\sigma_D(\\delta)(\\rho_D(\\omega))=dh(\\delta)\\cdot\\rho_D(\\omega)$.\nIn particular, $\\sigma_D(\\Theta_S)=\\mathcal{J}_D$ as fractional ideals.\nMoreover, $\\mathcal{J}_D^\\vee=\\mathcal{R}_D$ as fractional ideals.\n\\end{prp}\n\n\\begin{proof}\nThe spectral sequence \\eqref{41} applied to $\\mathcal{R}_D$ is associated with\n\\[\n\\RHom_{\\O_S}(\\Omega_S^1\\hookrightarrow\\Omega^1(\\log D),h\\colon\\O_S\\to\\O_S).\n\\]\nExpanding the double complex $\\Hom_{\\O_S}(\\Omega_S^1\\hookrightarrow\\Omega^1(\\log D),h\\colon\\O_S\\to\\O_S)$, we obtain the following diagram of long exact sequences:\n\\begin{equation}\\label{3}\\small\n\\SelectTips{cm}{}\\xymatrix@C-2em{\n&0\\ar[d]&0\\ar[d]\\\\\n\\Ext_{\\O_S}^1(\\mathcal{R}_D,\\O_S)\\ar[d]^-{0}&\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\\ar[l]\\ar[d]^-{h}&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]\\ar[d]^-{h}&0\\ar[l]\\\\\n\\Ext_{\\O_S}^1(\\mathcal{R}_D,\\O_S)&\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)\\ar[l]\\ar[d]&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]\\ar[d]&0\\ar[l]\\ar[d]\\\\\n&\\Hom_{\\O_S}(\\Omega^1_S,\\O_D)\\ar[d]&\\Hom_{\\O_S}(\\Omega^1(\\log D),\\O_D)\\ar[l]\\ar[d]&\\mathcal{R}_D^\\vee\\ar[l]_-{\\rho_D^\\vee}\\ar[d]^-\\alpha&0\\ar[l]\\\\\n&0&\\Ext^1_{\\O_S}(\\Omega^1(\\log D),\\O_S)\\ar[l]&\\Ext^1_{\\O_S}(\\mathcal{R}_D,\\O_S)\\ar[l]\\ar[d]\\\\\n&&&0\n}\n\\end{equation}\nWe can define a homomorphism $\\sigma_D$ from the upper left $\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)$ to the lower right $\\mathcal{R}_D^\\vee$ by a diagram chasing process and we find that $\\delta\\in\\Theta_S=\\Hom_{\\O_S}(\\Omega^1_S,\\O_S)$ maps to \n\\[\n\\sigma_D(\\delta)=\\ideal{h\\delta,\\rho_D^{-1}(-)}\\vert_D\\in\\mathcal{R}_D^\\vee\n\\]\nand that \\eqref{36} is exact.\n\nBy comparison with the spectral sequence, one can check that $\\alpha$ is the change of rings isomorphism \\eqref{43} applied to $\\mathcal{R}_D$, and that $\\alpha\\circ\\sigma_D$ coincides with the connecting homomorphism of the top row of the diagram, which is the same as the one in \\eqref{44}.\n \nLet $\\rho_D(\\omega)\\in\\mathcal{R}_D$ where $\\omega\\in\\Omega^1(\\log D)$.\nFollowing the definition of $\\rho_D$ in \\eqref{34}, we write $\\omega$ in the form \\eqref{4}.\nThen we compute\n\\begin{align}\n\\nonumber\\sigma_D(\\delta)(\\rho_D(\\omega))&=\\\\\\label{5}\n\\ideal{h\\delta,\\omega}\\vert_D&=\ndh(\\delta)\\cdot\\frac{\\xi}{g}\\vert_D+h\\cdot\\frac{\\ideal{\\delta,\\eta}}{g}\\vert_D=\ndh(\\delta)\\cdot\\rho_D(\\omega)\n\\end{align}\nwhich proves the first two claims.\n\nFor the last claim, we consider the diagram dual to \\eqref{3}:\n\\[\\small\n\\SelectTips{cm}{}\\xymatrix@C-2em{\n&&0\\ar[d]&0\\ar[d]\\\\\n&0\\ar[r]&\\Hom_{\\O_S}(\\Theta_S,\\O_S)\\ar[r]\\ar[d]^-{h}&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)\\ar[r]\\ar[d]^-{h}&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\ar[d]^-{0}\\\\\n&0\\ar[r]&\\Hom_{\\O_S}(\\Theta_S,\\O_S)\\ar[r]\\ar[d]&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)\\ar[r]\\ar[d]&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\\\\n0\\ar[r]&\\mathcal{J}_D^\\vee\\ar[d]^\\beta\\ar[r]^-{dh^\\vee}&\\Hom_{\\O_S}(\\Theta_S,\\O_D)\\ar[r]\\ar[d]&\\Hom_{\\O_S}(\\Der(-\\log D),\\O_D)\\\\\n&\\Ext_{\\O_S}^1(\\mathcal{J}_D,\\O_S)\\ar[r]&0\n}\n\\]\nAs before, we construct a homomorphism $\\rho'_D$ from the upper right $\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)$ to the lower left $\\mathcal{J}_D^\\vee$ such that $\\beta\\circ\\rho'_D$ coincides with the connecting homomorphism of the top row of the diagram, where $\\beta$ is the change of rings isomorphism \\eqref{43} applied to $\\mathcal{J}_D$.\nBy the diagram, $\\omega\\in\\Omega^1(\\log D)=\\Hom_{\\O_S}(\\Der(-\\log D),\\O_S)$ maps to\n\\[\n\\rho'_D(\\omega)=\\ideal{h\\omega,dh^{-1}(-)}\\vert_D\\in \\mathcal{J}_D^\\vee\n\\]\nwhich gives a short exact sequence\n\\begin{equation}\\label{6}\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\Omega^1_S\\ar[r]&\\Omega^1(\\log D)\\ar[r]^-{\\rho'_D}&\\mathcal{J}_D^\\vee\\ar[r]&0\n}\n\\end{equation}\nsimilar to the sequence \\eqref{2}.\nUsing \\eqref{4} and \\eqref{5}, we compute \n\\[\n\\rho'_D(\\omega)(\\delta(h))=\n\\rho'_D(\\omega)(dh(\\delta))=\n\\ideal{h\\omega,\\delta}\\vert_D=\n\\rho_D(\\omega)\\cdot dh(\\delta)=\n\\rho_D(\\omega)\\cdot \\delta(h)\n\\]\nfor any $\\delta(h)\\in \\mathcal{J}_D$ where $\\delta\\in\\Theta_S$.\nHence, $\\rho'_D=\\rho_D$ and the last claim follows using \\eqref{2} and \\eqref{6}.\n\\end{proof}\n\n\\begin{cor}\\label{37}\nThere is a chain of fractional ideals\n\\[\n\\mathcal{J}_D\\subseteq\\mathcal{R}_D^\\vee\\subseteq\\mathcal{C}_D\\subseteq\\O_D\\subseteq\\O_{\\widetilde D}\\subseteq\\mathcal{R}_D\n\\]\nin $\\mathcal{M}_D$ where $\\mathcal{C}_D=\\O_{\\widetilde D}^\\vee$ is the conductor ideal of $\\pi$.\nIn particular, $\\mathcal{J}_D\\subseteq\\mathcal{C}_D$.\n\\end{cor}\n\n\\begin{proof}\nBy Lemma~\\ref{25}, $\\mathcal{J}_D$ is a fractional ideal contained in $\\mathcal{R}_D^\\vee$ by Proposition~\\ref{22}.\nBy \\cite[(2.7),(2.8)]{Sai80}, $\\mathcal{R}_D$ is a finite $\\O_D$-module containing $\\O_{\\widetilde D}$ and hence a fractional ideal.\nThe remaining inclusions and fractional ideals are then obtained using Proposition~\\ref{18}.\n\\end{proof}\n\n\\begin{cor}\\label{24}\nIf $D$ is free then $\\mathcal{J}_D=\\mathcal{R}_D^\\vee$ as fractional ideals.\n\\end{cor}\n\n\\begin{proof}\nIf $D$ is free then the Ext-module in the exact sequence \\eqref{36} disappears and $\\sigma_D$ becomes surjective.\nThen the claim is part of the statement of Proposition~\\ref{22}.\n\\end{proof}\n\nBy Corollary~\\ref{37}, the inclusion $\\O_{\\widetilde D}\\subset R_D$ always holds.\nFor a free divisor $D$, the case of equality is translated into more familiar terms by the following\n\n\\begin{cor}\\label{42}\nIf $D$ is free then $\\mathcal{R}_D=\\O_{\\widetilde D}$ is equivalent to $\\mathcal{J}_D=\\mathcal{C}_D$.\n\\end{cor}\n\n\\begin{proof}\nBy the preceding Corollary~\\ref{24} and the last statement of Proposition~\\ref{22}, the freeness of $D$ implies that $\\mathcal{R}_D$ and $\\mathcal{J}_D$ are mutually $\\O_D$-dual.\nBy Lemma~\\ref{39} and the definition of the conductor, the same holds true for $\\O_{\\widetilde D}$ and $\\mathcal{C}_D$. \nThe claim follows.\n\\end{proof}\n\n\\section{Algebraic normal crossing conditions}\n\nIn this section, we prove our main Theorem~\\ref{10} settling the missing implication in Theorem~\\ref{28}.\nWe begin with some general preparations.\n\n\\begin{lem}\\label{35}\nAny map $\\phi\\colon Y\\to X$ of analytic germs with $\\Omega^1_{Y\/X}=0$ is an immersion.\n\\end{lem}\n\n\\begin{proof}\nThe map $\\phi$ can be embedded in a map $\\Phi$ of smooth analytic germs:\n\\[\n\\SelectTips{cm}{}\\xymatrix{\nY\\ar@{^(->}[r]\\ar[d]^-\\phi&T\\ar[d]^-\\Phi\\\\\nX\\ar@{^(->}[r]&S.\n}\n\\]\nSetting $\\Phi_i=x_i\\circ\\Phi$ and $\\phi_i=\\Phi_i+\\mathcal{I}_Y$ for coordinates $x_1,\\dots,x_n$ on $S$ and $\\mathcal{I}_Y$ the defining ideal of $Y$ in $T$, we can write $\\Phi=(\\Phi_1,\\dots,\\Phi_n)$ and $\\phi=(\\phi_1,\\dots,\\phi_n)$ and hence\n\\begin{equation}\\label{26}\n\\Omega^1_{Y\/X}=\\frac{\\Omega^1_Y}{\\sum_{i=1}^n\\O_Yd\\phi_i}=\\frac{\\Omega^1_T}{\\O_Td\\mathcal{I}_Y+\\sum_{i=1}^n\\O_Td\\Phi_i}.\n\\end{equation}\nWe may choose $T$ of minimal dimension so that $\\mathcal{I}_Y\\subseteq\\mathfrak{m}_T^2$ and hence $d\\mathcal{I}_Y\\subseteq\\mathfrak{m}_T\\Omega^1_T$.\nNow \\eqref{26} and the hypothesis $\\Omega^1_{Y\/X}=0$ show that $\\Omega^1_T=\\sum_{i=1}^n\\O_Td\\Phi_i+\\mathfrak{m}_T\\Omega^1_T$ which implies that $\\Omega^1_T=\\sum_{i=1}^n\\O_Td\\Phi_i$ by Nakayama's Lemma.\nBut then $\\Phi$ and hence $\\phi$ is a closed embedding as claimed.\n\\end{proof}\n\n\\begin{lem}\\label{7}\nIf $\\mathcal{J}_D=\\mathcal{C}_D$ and $\\widetilde D$ is smooth then $D$ has smooth irreducible components.\n\\end{lem}\n\n\\begin{proof}\nBy definition, the ramification ideal of $\\pi$ is the Fitting ideal\n\\[\n\\mathcal{R}_\\pi:=\\mathcal{F}^0_{\\O_{\\widetilde D}}(\\Omega^1_{\\widetilde D\/D}).\n\\]\nAs a special case of a result of Ragnie Piene~\\cite[Cor.~1, Prop.~1]{Pie79} (see also \\cite[Cor.~2.7]{OZ87}), \n\\[\n\\mathcal{C}_D\\mathcal{R}_\\pi=\\mathcal{J}_D\\O_{\\widetilde D}\n\\]\nBy hypothesis, this becomes\n\\[\n\\mathcal{C}_D\\mathcal{R}_\\pi=\\mathcal{C}_D\n\\]\nsince $\\mathcal{C}_D$ is an ideal in both $\\O_D$ and $\\O_{\\widetilde D}$.\nBy Nakayama's lemma, it follows that that $\\mathcal{R}_\\pi=\\O_{\\widetilde D}$ and hence that $\\Omega^1_{\\widetilde D\/D}=0$.\n\nSince $\\widetilde D$ is normal, irreducible and connected components coincide.\nBy localization to a connected component $\\widetilde D_i$ of $\\widetilde D$ and base change to $D_i=\\pi(\\widetilde D_i)$ (see \\cite[Ch.~II, Prop.~8.2A]{Har77}), we obtain $\\Omega^1_{\\widetilde D_i\/D_i}=0$.\nThen the normalization $\\widetilde D_i\\to D_i$ is an immersion by Lemma~\\ref{35} and hence $D_i=\\widetilde D_i$ is smooth.\n\\end{proof}\n\nWe are now ready to prove our main results.\n\n\\begin{proof}[Proof of Theorem~\\ref{10}]\nIn codimension $1$, $D$ is free by Corollary~\\ref{27} and hence $\\mathcal{J}_D=\\mathcal{C}_D$ by Corollary~\\ref{42} and our hypothesis.\nMoreover, $\\widetilde D$ is smooth in codimension $1$ by normality. \nBy our language convention, this means that there is an analytic subset $A\\subset D$ of codimension at least $2$ such that, for $p\\in D\\setminus A$, $\\mathcal{J}_{D,p}=\\mathcal{C}_{D,p}$ and $\\widetilde D$ is smooth above $p$.\nFrom Lemma~\\ref{7} we conclude that the local irreducible components $D_i$ of the germ $(D,p)$ are smooth. \nThe hypothesis $\\mathcal{R}_D=\\O_{\\widetilde D}$ at $p$ then reduces to the equality $\\mathcal{R}_{D,p}=\\bigoplus\\O_{D_i}$.\nThus, the implication \\eqref{60d} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60} yields the claim. \n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{16}]\nIn order to prove that \\eqref{28d} implies \\eqref{28b}, we may assume that $Z$ is smooth and reduce to the case of a plane curve as in the proof of Theorem~\\ref{40}. \nThen the Mather--Yau theorem~\\cite{MY82} applies (see \\cite[Prop.~9]{Fab12} for details).\n\nNow assume that $D$ is free and normal crossing in codimension $1$. \nBy the first assumption and Theorem~\\ref{19}, $Z$ is Cohen--Macaulay of codimension $1$ and, in particular, satisfies Serre's condition $S_1$.\nBy the second assumption, $Z$ also satisfies Serre's condition $R_0$.\nThen $Z$ is reduced, and hence $\\mathcal{J}_D$ is radical, by Serre's reducedness criterion.\nThis proves that \\eqref{28b} implies \\eqref{28d} for free $D$.\n\nThe last equivalence then follows from Theorems~\\ref{28} and \\ref{10} and Corollary~\\ref{42}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem~\\ref{38}]\nBy Theorems~\\ref{28}, \\ref{10}, and \\ref{16}, we may assume that $\\mathcal{J}_D=\\mathcal{C}_D$.\nThen Lemma~\\ref{7} shows that the irreducible components $D_i=\\{h_i=0\\}$, $i=1,\\dots,m$, of $D$ are smooth, and hence normal.\nIt follows that \n\\[\n\\mathcal{R}_D=\\O_{\\widetilde D}=\\bigoplus_{i=1}^m\\O_{\\widetilde D_i}=\\bigoplus_{i=1}^m\\O_{D_i}.\n\\]\nBy the implication \\eqref{60a} $\\Leftarrow$ \\eqref{60c} in Theorem~\\ref{60}, this is equivalent to \n\\begin{equation}\\label{51}\n\\Omega^1(\\log D)=\\sum_{i=1}^m\\O_S\\frac{dh_i}{h_i}+\\Omega^1_S.\n\\end{equation}\nOn the other hand, Saito's criterion \\cite[(1.8) i)]{Sai80} for freeness of $D$ reads\n\\begin{equation}\\label{52}\n\\bigwedge^n\\Omega^1(\\log D)=\\Omega^n_S(D).\n\\end{equation}\nCombining \\eqref{51} and \\eqref{52}, it follows immediately that $D$ is normal crossing (see also \\cite[Prop.~B]{Fab12}):\n\nAs $\\O_S$-module and modulo $\\Omega_S^n$, the left hand side of \\eqref{52} is, due to \\eqref{51}, generated by expressions \n\\begin{gather}\\label{53}\n\\frac{d h_{i_1}\\wedge\\dots\\wedge d h_{i_k}\\wedge dx_{j_1}\\wedge\\dots\\wedge dx_{j_{n-k}}}{h_{i_1}\\cdots h_{i_k}},\\\\ \n\\nonumber1\\le i_1<\\cdots<i_k\\le m, \\quad 1\\le j_1<\\cdots<j_{n-k}\\le n\\ge k\n\\end{gather}\nwhereas the right hand side of \\eqref{52} is generated by\n\\[\n\\frac{d x_1\\wedge\\dots\\wedge d x_n}{h_1\\cdots h_m}.\n\\]\nIn order for an instance of \\eqref{53} to attain the denominator of this latter expression, it must satisfy $k=m$, and, in particular, $m\\le n$.\nFurther, comparing numerators, $d h_1\\wedge\\dots\\wedge d h_m\\wedge dx_{j_1}\\wedge\\dots\\wedge dx_{j_{n-m}}$ must be a unit multiple of $d x_1\\wedge\\dots\\wedge d x_n$.\nIn other words, choosing $i_1,\\dots,i_m$ such that $\\{i_1,\\dots,i_m,j_1,\\dots,j_{n-m}\\}=\\{1,\\dots,n\\}$, \n\\[\n\\frac{\\partial(h_1,\\dots,h_m)}{\\partial(x_{i_1},\\dots,x_{i_m})}\\in\\O_S^*.\n\\]\nBy the implicit function theorem, $h_1,\\dots,h_m,x_{j_1},\\dots,x_{j_{n-m}}$ is then a coordinate system and hence $D$ is a normal crossing divisor as claimed.\n\\end{proof}\n\n\\section{Gorenstein singular locus}\n\nIn this section, we describe the canonical module of the singular locus $Z$ in terms of the module $\\mathcal{R}_D$ of logarithmic residues.\nThen, we prove Theorem~\\ref{40}.\n\nThe complex of logarithmic differential forms along $D$ relative to the map $h\\colon S\\to T:=(\\mathds{C},0)$ defining $D$ is defined as $\\Omega^\\bullet(\\log h):=\\Omega^\\bullet(\\log D)\/\\frac{dh}h\\wedge\\Omega^{\\bullet-1}(\\log D)$ (see \\cite[\\S22]{GMS09} or \\cite[Def.~2.7]{DS12}).\n\n\\begin{prp}\\label{20}\nThe $\\O_Z$-module $\\mathcal{R}_h:=\\mathcal{R}_D\/\\O_D$ fits into an exact square\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n&0\\ar[d]&0\\ar[d]&0\\ar[d]\\\\\n0\\ar[r]&\\O_S\\ar[r]^-h\\ar[d]^-{dh}&\\O_S\\ar[r]\\ar[d]^-{\\frac{dh}{h}}&\\O_D\\ar[r]\\ar[d]&0\\\\\n0\\ar[r]&\\Omega^1_S\\ar[r]\\ar[d]&\\Omega^1(\\log D)\\ar[r]^-{\\rho_D}\\ar[d]&\\mathcal{R}_D\\ar[r]\\ar[d]&0\\\\\n0\\ar[r]&\\Omega^1_{S\/T}\\ar[r]\\ar[d]&\\Omega^1(\\log h)\\ar[r]\\ar[d]&\\mathcal{R}_h\\ar[r]\\ar[d]&0\\\\\n&0&0&0\n}.\n\\]\nIf $D$ is free then $Z$ is Cohen--Macaulay with canonical module $\\omega_Z=\\mathcal{R}_h$; in particular, $Z$ is Gorenstein if and only if $\\mathcal{R}_D$ is generated by $1$ and one additional generator.\n\\end{prp}\n\n\\begin{proof}\nWe set $\\omega_\\emptyset:=0$ in case $D$ is smooth and assume $Z\\ne\\emptyset$ in the following.\nThe exact square arises from the residue sequence \\eqref{2} using the Snake Lemma.\nDualizing the short exact sequence\n\\[\n\\SelectTips{cm}{}\\xymatrix{\n0\\ar[r]&\\mathcal{J}_D\\ar[r]&\\O_D\\ar[r]&\\O_Z\\ar[r]&0,\n}\n\\]\none computes\n\\begin{equation}\\label{29}\n\\Ext_{\\O_D}^1(\\O_Z,\\O_D)=\\mathcal{J}_D^\\vee\/\\O_D=\\mathcal{R}_D\/\\O_D=\\mathcal{R}_h.\n\\end{equation}\nwhich shows that $\\mathcal{R}_h$ is an $\\O_Z$-module.\nIf $D$ is free then, by Theorem~\\ref{19}, $Z$ is Cohen--Macaulay of codimension $1$ and $\\omega_Z=\\mathcal{R}_h$ by \\eqref{29}.\n\\end{proof}\n\nPart~\\eqref{56e} of the following proposition can also be found in Faber's thesis (see \\cite[Prop.~1.29]{Fab11}).\n\n\\begin{prp}\\label{56}\nLet $D$ be a free divisor.  \nThen the following statements hold true:\n\\begin{asparaenum} \n\\item\\label{56e} $\\frac{dh}h$ is part of an $\\O_S$-basis of $\\Omega^1(\\log D)$ if and only if $D$ is Euler homogeneous.\n\\item\\label{56d} $\\frac{dh}h$ is part of an $\\O_S$-basis of $\\Omega^1(\\log D)$ if and only if $1$ is part of a minimal set of $\\O_D$-generators of $\\mathcal{R}_D$.\n\\item\\label{56g} $\\mathcal{R}_D$ is a cyclic $\\O_D$-module if and only if $D$ is smooth.\n\\end{asparaenum}\n\\end{prp}\n\n\\begin{proof}\\\n\\begin{asparaenum} \n\n\\item This is immediate from the existence of a dual basis and the fact that $\\chi\\in\\Der(-\\log D)$ is an Euler vector field exactly if $\\ideal{\\chi,\\frac{dh}h}=\\frac{\\chi(h)}h=1$. \nThen $\\chi$ can be chosen as a member of some basis.\n\n\\item We may assume that $D\\cong D'\\times S'$ with $S'=(\\mathds{C}^r,0)$ implies $r=0$.\nIndeed, a basis of $\\Omega^1(\\log D)$ is the union of bases of $\\Omega^1(\\log D')$ and $\\Omega_{S'}^1$.\nThis assumption is equivalent to $\\Der(-\\log D)\\subseteq\\mathfrak{m}_S\\Theta_S$, where $\\mathfrak{m}_S$ denotes the maximal ideal of $\\O_S$. \nDually, this means that no basis element of $\\Omega^1(\\log D)$ can lie in $\\Omega_S^1$.\n\nConsider a basis $\\omega_1,\\dots,\\omega_n$ of $\\Omega^1(\\log D)$ with $\\omega_1=\\frac{dh}h$ and set $\\rho_i:=\\rho_D(\\omega_i)$ for $i=1,\\dots,n$.\nIf $\\rho_1=1$ is not a member of some minimal set of generators of $\\mathcal{R}_D$ then $\\rho_1=\\sum_{i=2}^na_i\\rho_i$ with $a_i\\in\\mathfrak{m}_S$. \nThus, the form $\\omega_1':=\\omega_1-\\sum _{i=2}^na_i\\omega_i$ can serve as a replacement for $\\omega_1$ in the basis. \nBut, by construction, $\\rho_D(\\omega'_1)=0$ which implies that $\\omega'_1\\in\\Omega_S$ by \\eqref{2}.\nThis contradicts our assumption on $D$.\n \nConversely, suppose that $\\frac{dh}h$ is not a member of any $\\O_S$-basis $\\omega_1,\\dots,\\omega_n$ of $\\Omega^1(\\log D)$.\nThen, by Nakayama's lemma, there are $a_i\\in\\mathfrak{m}_S$ such that $\\frac{dh}h=\\sum_{i=1}^na_i\\omega_i$.\nApplying $\\rho_D$, this gives $1=\\rho_D(\\frac{dh}h)=\\sum_{i=1}^na_i\\rho_i\\in\\mathfrak{m}_D\\mathcal{R}_D$.\nAgain by Nakayama's lemma, this means that $1$ is not a member of any minimal set of $\\O_D$-generators of $\\mathcal{R}_D$.\n\n\\item If $\\mathcal{R}_D\\cong\\O_D$ then also $\\mathcal{J}_D\\cong\\O_D$ by Corollary~\\ref{24} then $D$ must be smooth by Lipman's criterion~\\cite{Lip69}.\nAlternatively, it follows that $\\mathcal{J}_h+\\O_S\\cdot h=\\O_S$ and hence also $\\mathcal{J}_h=\\O_S$ (see Remark~\\ref{9}); so $D$ is smooth by the Jacobian criterion.\n\n\\end{asparaenum} \n\\end{proof}\n\nAs observed by Faber~\\cite[Rmk.~54]{Fab12}, condition~\\eqref{56d} of Proposition~\\ref{56} is satisfied given $\\mathcal{R}_D=\\O_{\\widetilde D}$ and one obtains\n\n\\begin{cor}[Faber]\nAny free divisor $D$ with $\\mathcal{R}_D=\\O_{\\widetilde D}$ is Euler homogeneous.\n\\end{cor}\n\n\\begin{proof}[Proof of Theorem~\\ref{40}]\nAssume that $Z$ is Gorenstein of codimension $1$ in $D$.\nThe preimage $\\mathcal{J}'_D$ of $\\mathcal{J}_D$ in $\\O_S$ is then a Gorenstein ideal of height $2$.\nAs such, it is a complete intersection ideal by a theorem of Serre (see \\cite[Cor.~21.20]{Eis95}), and hence generated by two of the generators $h,\\partial_1(h),\\dots,\\partial_n(h)$.\nThen the triviality lemma~\\cite[(3.5)]{Sai80} shows that $D$ is either smooth or $D\\cong C\\times(\\mathds{C}^{n-2},0)$ and $C\\subset(\\mathds{C}^2,0)$ a plane curve.\nThen also $C$ has Gorenstein singular locus and is hence quasihomogeneous by \\cite{KW84}.\nAlternatively, the last implication follows from Propositions~\\ref{20} and \\ref{56} using that quasihomogeneity of $C$ follows from Euler homogeneity of $C$ by Saito's quasihomogeneity criterion~\\cite{Sai71} for isolated singularities.\nFinally, $D$ is quasihomogeneous and $Z$ is a complete intersection.\nThe converse is trivial.\n\\end{proof}\n\n\\section*{Acknowledgements}\n\nWe are grateful to Eleonore Faber and David Mond for helpful discussions and comments.\nThe foundation for this paper was laid during a ``Research in Pairs'' stay at the ``Mathematisches Forschungsinstitut Oberwolfach'' in summer 2011.\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure}[!t]\n\\centering\n\\begin{center}\n\\begin{tabular}{C{5.0cm}C{5.0cm}}\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_a} &\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_b} \\\\\n    \\centering (a) FCN & (b) DilatedFCN \\\\ \\\\\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_c} &\n    \\includegraphics[height=3.8cm]{figs\/framework_img0_d} \\\\\n    \\centering (c) Encoder-Decoder & (d) EfficientFCN \n\\end{tabular}\n\\end{center}\n\\caption{Different architectures for semantic segmentation. (a) the original FCN with output stride (OS)=32. (b). DilatedFCN based methods sacrifice efficiency and exploit the dilated convolution with stride 2 and 4 in the last two stages to generate high-resolution feature maps. (c)Encoder-Decoder methods employ the U-Net structure to recover the high-resolution feature maps. (d) Our proposed EfficientFCN with codebook generation and codeword assembly for high-resolution feature upsampling in semantic segmentation.}\n\\label{fig:framework_img0}\n\\end{figure}\n\nSemantic segmentation or scene parsing is the task of assigning one of the pre-defined class \nlabels to each pixel of an input image. It is a fundamental yet challenging task in computer vision.\nThe Fully Convolutional Network (FCN) \\cite{long2015fully}, as shown in\nFig.~\\ref{fig:framework_img0}(a), for the first time demonstrates the success of exploiting a fully convolutional network in semantic segmentation, which adopts a DCNN as the feature encoder (\\textit{i}.\\textit{e}., ResNet\\cite{he2016deep}) to extract high-level semantic feature maps and then applies a convolution layer to generate the dense prediction. \nFor the semantic segmentation, high-resolution feature maps are critical for achieving accurate\nsegmentation performance since they contain fine-grained structural information to delineate\ndetailed boundaries of various foreground regions. In addition, due to the lack of large-scale\ntraining data on semantic segmentation, transferring the weights pre-trained on ImageNet can greatly\nimprove the segmentation performance. Therefore, most state-of-the-art semantic segmentation methods\nadopt classification networks as the backbone to take full advantages of ImageNet pre-training. The resolution of feature maps in the original\nclassification model is reduced with consecutive pooling and strided\nconvolution operations to learn high-level\nfeature representations. The output stride of the final feature map is 32\n(OS=32), where the fine-grained structural information is discarded. \nSuch low-resolution feature maps cannot fully meet the requirements of semantic segmentation where detailed spatial information is needed. \nTo tackle this problem, many works exploit dilated convolution (or atrous convolution) to enlarge the receptive field (RF) while maintaining the resolution of high-level feature maps.\nState-of-the-art dilatedFCN based\nmethods\\cite{yu2017dilated,chen2017deeplab,Zhang_2018_CVPR,he2019adaptive,Zhang_2019_CVPR} (shown in\nFig.~\\ref{fig:framework_img0}(b)) have demonstrated that removing the downsampling\noperation and replacing convolution with the dilated convolution\nin the later blocks can achieve superior performance, resulting in \nfinal feature maps of output stride 8 (OS=8). Despite the superior performance and no extra parameters introduced by dilated convolution,  the high-resolution feature representations require high computational complexity and memory consumption. \nFor instance, for an input image with 512$\\times$512 and the ResNet101 as the backbone encoder, the computational complexity of the encoder increases from 44.6 GFlops to 223.6 GFlops when adopting the dilated convolution with the strides 2 and 4 into the last two blocks.\n\n\n\nAlternatively, as shown in Fig.~\\ref{fig:framework_img0}(c), the encoder-decoder based methods (\n\\textit{e}.\\textit{g}.\\ \\cite{ronneberger2015u}) exploit using a decoder to gradually upsample and generate the high-resolution feature maps by aggregating multi-level feature representations from the backbone (or the encoder). These encoder-decoder based methods can obtain high-resolution\nfeature representations efficiently. However, on one hand, the fine-grained structural details are already lost in the topmost high-level feature maps of OS=32. Even with the skip connections, lower-level high-resolution feature maps cannot provide abstractive enough features for achieving high-performance segmentation. \nOn the other hand, existing decoders mainly utilize the bilinear upsampling or deconvolution operations to increase the resolution of the high-level feature maps. These operations are conducted in a\nlocal manner. The feature vector at each location of the upsampled\nfeature maps is recovered from a limited receptive filed. Thus, although the encoder-decoder models are generally faster and more memory friendly than dilatedFCN based methods, their performances generally cannot compete with those of the dilatedFCN models.\n\nTo tackle the challenges in both types of models, we propose the EfficienFCN\n(as shown in Fig.~\\ref{fig:framework_img0}(d)) with the Holistically-guided Decoder (HGD) to bridge the\ngap between the dilatedFCN based methods and the encoder-decoder based methods. Our network can adopt any widely\nused classification model without dilated convolution as the encoder (such as ResNet models) to\ngenerate low-resolution high-level feature maps (OS=8). Such an encoder is both computationally and\nmemory efficient than those in DilatedFCN model.\nGiven the multi-level feature maps from the last three blocks of the encoder, the proposed holistically-guided decoder takes the advantages of both high-level but low-resolution (OS=32) and also mid-level high-resolution feature maps (OS=8, OS=16) for achieving high-level feature upsampling with semantic-rich features. Intuitively, the higher-resolution feature maps contain more fine-grained structural information, which is beneficial for spatially guiding the feature upsampling process; the lower-resolution feature maps contain more high-level semantic information, which are more suitable to encode the global context effectively. Our HGD therefore generates a series of holistic codewords in a codebook to summarize different global and high-level aspects of the input image from the low-resolution feature maps (OS=32). Those codewords can be properly assembled in a high-resolution grid to form the upsampled feature maps with rich semantic information. Following this principle, the HGD generates assembly coefficients from the mid-level high-resolution feature maps (OS=8, OS=16) to guide the linear assembly of the holistic codewords at each high-resolution spatial location to achieve feature upsampling.\nOur proposed EfficientFCN with holistically-guided decoder achieves high  segmentation accuracy on three popular public benchmarks, which demonstrate the efficiency and effectiveness of our proposed decoder.\n\nIn summary, our contributions are as follows.\n\\begin{itemize}\n    \\item We propose a novel holistically-guided decoder, which can efficiently generate the high-resolution feature maps considering holistic contexts of the input image.\n   \n\\item Because of the light weight and high performance of the proposed holistically-guided decoder, our EfficientFCN can adopt the encoder without any dilated convolution but still achieve superior performance.   \n        \n   \\item Our EfficientFCN achieves competitive (or better) results compared\n       with the state-of-the-art dilatedFCN based methods on the PASCAL\n       Context, PASCAL VOC, ADE20K datasets, with 1\/3 fewer FLOPS.\n\\end{itemize}\n\n\n\n\n\\section{Related Work}\nIn this section, we review recent FCN-based methods for semantic segmentation. Since the successful demonstration of FCN \\cite{long2015fully} on semantic segmentation, many methods were proposed to improve the performance the FCN-based methods, which mainly include two categories of methods: the dilatedFCN-based methods and the encoder-decoder architectures.\n\n\\noindent \\textbf{DilatedFCN.} \nThe Deeplab V2 \\cite{chen2017deeplab,chen2017rethinking} proposed to exploit dilated convolution in\nthe backbone to learn a high-resolution feature map, which increases the\noutput stride from 32 to 8. However, the dilated convolution in the last two layers of the backbone adds huge extra computation and leaves large\nmemory footprint. Based on the dilated convolution backbone, many works \\cite{Zhang_2019_CVPR,fu2019dual,Fu_2019_ICCV,he2019dynamic} continued to apply different strategies as the segmentation heads to acquire the context-enhanced feature maps. PSPNet \\cite{zhao2017pyramid} utilized the Spatial Pyramid Pooling (SPP) module to increase the receptive field. EncNet \\cite{Zhang_2018_CVPR} proposed an encoding layer to predict a feature re-weighting vector from the global context and selectively high-lights \nclass-dependent feature maps. \nCFNet \\cite{Zhang_2019_CVPR} exploited an aggregated co-occurrent feature (ACF) module to\naggregate the co-occurrent context by the pair-wise similarities in the feature space.\nGated-SCNN\\cite{takikawa2019gated} proposed to use a new gating mechanism to connect the intermediate layers and a new loss function that exploits the duality\nbetween the tasks of semantic segmentation and semantic\nboundary prediction.\nDANet \\cite{fu2019dual} proposed to use two attention modules with the self-attention mechanism to aggregate features from\nspatial and channel dimensions respectively.\nACNet \\cite{Fu_2019_ICCV} applied a dilated ResNet as the backbone and combined the encoder-decoder strategy for the \nobservation that the global context from high-level features\nhelps the categorization of some large semantic confused\nregions, while the local context from lower-level visual features helps to generate sharp boundaries or clear details.\nDMNet \\cite{he2019dynamic} generated a set of dynamic filters of different\nsizes from the multi-scale neighborhoods for handling the scale variations of objects for semantic segmentation.\nAlthough these works further improve the performances on different benchmarks, these proposed heads \nstill adds extra computational costs to the already burdensome encoder.\n\n\n\\noindent \\textbf{Encoder-Decoder.} \nAnother type of methods focus on efficiently acquire the high-resolution semantic feature maps via the encoder-decoder architectures. Through\nthe upsampling operations and the skip connections, the encoder-decoder\narchitecture \\cite{ronneberger2015u} can gradually recover the high-resolution feature maps for \nsegmentation. DUsampling \\cite{tian2019decoders} designed a data-dependent upsampling module based on fully \nconnected layers for constructing the high-resolution feature maps from the low-resolution \nfeature maps. FastFCN \\cite{wu2019fastfcn} proposed a Joint Pyramid Upsampling (JPU) method via multiple \ndilated convolution to generate the high-resolution feature maps.\nOne common drawback of these methods is that the feature at each location of the upsampled high-resolution feature maps is \nconstructed via only local feature fusion. Such a property limited limits their performance in \nsemantic segmentation, where global context is important for the final performance.\n\n\n\\section{Proposed Method}\nIn this section, We firstly give a thorough analysis of the classical\nencoder-decoder based methods. \nThen, to tackle the challenges in the classical encoder-decoder methods, \nwe propose the EfficientFCN, which is based on the traditional ResNet as the\nencoder backbone \nnetwork for semantic segmentation. In our EfficientFCN, the\nHolistically-guided Decoder (HGD)\nis designed to recover the high-resolution (OS=8) feature maps from three\nfeature maps in the last three blocks from the ResNet encoder backbone.\n\\begin{figure}[tb]\n    \\centering\n    \\subfloat{\\includegraphics[width=11.0cm]{figs\/framework_img1}}\n\\caption{Illustration of the proposed EfficientFCN model. It consists of three main components. Multi-scale feature fusion fuses multi-scale features to obtain OS=8 and OS=32 multi-scale feature maps. Holistic codebook generation results in a series of holistic codewords summarizing different aspects of the global context. High-resolution feature upsampling can be achieved by codeword assembly.}\n\\label{fig:framework_img1}\n\\end{figure}\n\\subsection{Overview}\nIn state-of-the-art DCNN backbones, the high-resolution mid-level feature maps\nat earlier stages can better encode fine-grained structures, while the\nlow-resolution high-level features at the later stages are generally more\ndiscriminative for category prediction. \nThey are essential and complementary information sources for achieving\naccurate semantic segmentation.\nTo combine the strengths of both features, encoder-decoder structures were\ndeveloped to reconstruct high-resolution feature maps that have semantic-rich\ninformation from the multi-scale feature maps. \nTo generate the final feature map $\\tilde{f}_8$ of OS=8 for mask prediction, a\nconventional three-stage encoder-decoder first upsamples the deepest encoder\nfeature map $f_{32}$ of OS=32 to generate OS=16 feature maps $f_{16}$. The\nOS=16 feature maps $e_{16}$ of the same size from the encoder is either\ndirectly concatenated as $[f_{16}; e_{16}]$ (U-Net \\cite{ronneberger2015u}) or summed as\n$f_{16}+e_{16}$ followed by some $1\\times 1$ convolution to\ngenerate the upsampled OS=16 feature maps, $\\tilde{f}_{16}$. The same\nupsampling + skip connection procedure repeats again for $\\tilde{f}_{16}$ to\ngenerate $\\tilde{f}_{8}$. The upsampled features $\\tilde{f}_8$ contain both\nmid-level and high-level information to some extent and can be used to\ngenerate the segmentation masks.\nHowever, since bilinear upsampling and deconvolution layer in the classical decoder are local operations\nwith limited receptive field. We argue that they are incapable of exploring\nimportant global context of the input image, which are crucial for achieving\naccurate segmentation. \nAlthough there were some existing attempts \\cite{Zhang_2018_CVPR,hu2018squeeze} on using\nglobal context to re-weight the contributions of different channels of the\nfeature maps either in the backbone \\cite{Zhang_2018_CVPR} or in the upsampled feature\nmaps \\cite{hu2018squeeze}. This strategy only scales each feature channel but\nmaintains the original spatial size and structures. Therefore, it is incapable\nof generating high-resolution semantic-rich feature maps to improve the\nrecovery of fine-grained structures.\nTo solve this drawback, we propose a novel Holistically-guided Decoder (HGD),\nwhich decomposes the feature upsampling task into the generation of a series\nof holistic codewords from high-level feature maps to capture global contexts,\nand linearly assembling codewords at each spatial location for semantic-rich\nfeature upsampling. Such a decoder can exploit the global contextual information\nto effectively guide the feature upsampling process and is able to recover\nfine-grained details. \nBased on the proposed HGD, we present the EfficientFCN (as shown in\nFig.~\\ref{fig:framework_img0}(d)) with an efficient encoder free of dilated\nconvolution for efficient semantic segmentation. \n\\subsection{Holistically-guided Decoder for Semantic-rich Feature Upsampling}\nTo take advantages of both low-resolution high-level feature maps of size\nOS=32 and the high-resolution mid-level feature maps of sizes OS=8 and OS=16,\nsince the high-level feature maps have already lost most structural details\nbut are semantic-rich to encode categorical information, we argue that\nrecovering detailed structures from them is quite challenging and also\nnon-necessary. Instead, we propose to generate a series of holistic codewords\nwithout any spatial order from the high-level feature maps to capture\ndifferent aspects of the global context. On the other hand, the mid-level\nhigh-resolution feature maps have maintained abundant image structural\ninformation. But they are from relatively shallower layers and cannot encode\naccurate enough categorical features  for final mask prediction. However, they\nwould still be representative enough for guiding the linear assembly of the\nsemantic-rich codewords for high-resolution feature upsampling. Our proposed\nHolistically-guided Decoder therefore contains three main components:\nmulti-scale feature fusion, holistic codebook generation, and codeword\nassembly for high-resolution feature upsampling.\n\n\\noindent \\textbf{Multi-scale features fusion.}\nGiven the multi-scale feature maps from the encoder, although we can directly\nencode the holistic codewords from the high-level OS=32 feature maps and also\ndirectly generate the codeword combination coefficients from the mid-level\nOS=16 and OS=8 feature maps, we observe the fusion of multi-scale feature maps\ngenerally result in better performance. \nFor the OS=8, OS=16, OS=32 feature maps from the encoder, we first adopt\nseparate $1\\times 1$ convolutions to compress each of their channels to 512\nfor reducing the follow-up computational complexity, obtaining $e_{8}, e_{16},\ne_{32}$, respectively. The multi-scale fused OS=32 feature maps $m_{32}$ are\nthen obtained by downsampling $e_{8}, e_{16}$ to the size of OS=32 and\nconcatenating them along the channel dimension with $e_{32}$ as $m_{32} = [\n    e_{8}^\\downarrow;  e_{16}^\\downarrow; e_{32}] \\in \\mathbb{R}^{1536\\times\n(H\/32) \\times (W\/32)}$, where $^\\downarrow$ represents bilinear downsampling,\n$[\\cdot;\\cdot]$ denotes concatenation along the channel dimension, and $H$ and\n$W$ are the input image's height and width, respectively. We can also obtain\nthe multi-scale fused OS=8 feature maps, $m_8 = [ e_{8}; e_{16}^\\uparrow;\ne_{32}^\\uparrow] \\in \\mathbb{R}^{1536\\times (H\/8) \\times (W\/8)}$, in a similar\nmanner.\n\n\\noindent \\textbf{Holistic codebook generation.}\nAlthough the mutli-scale fused feature maps $m_{32}$ are created to integrate\nboth high-level and mid-level features, their small resolutions make them lose\nmany structural details of the scene. On the other hand, because $e_{32}$ is\nencoded from the deepest layer, $m_{32}$ is able to encode rich categorical\nrepresentations of the image. We therefore propose to generate a series of\nunordered holistic codewords from $m_{32}$ to implicitly model different\naspects of the global context.\nTo generate $n$ holistic codewords, a codeword base map $B \\in\n\\mathbb{R}^{1024 \\times (H\/32)\\times}$ $^{(W\/32)}$ and $n$ spatial weighting\nmaps $A \\in \\mathbb{R}^{n \\times (H\/32)\\times (W\/32)}$ are first computed from\nthe fused multi-scale feature maps $m_{32}$ by two separate $1\\times 1$\nconvolutions. \nFor the bases map $B$, we denote $B(x,y) \\in \\mathbb{R}^{1024}$ as the 1024-d\nfeature vector at location $(x,y)$; for the spatial weighting maps $A$, we use\n$A_i \\in \\mathbb{R}^{(H\/32) \\times (W\/32)}$ to denote the $i$th weighting map.\nTo ensure the weighting maps $A$ are properly normalized, the softmax function is adopted to \nnormalize all spatial locations of each channel $i$ (the $i$-th spatial feature map) as\n\\begin{align}\n            \\tilde{A}_i(x,y) = \\frac{\\exp(A_i(x,y))}{\\sum_{p,q}\n            \\exp(A_i(p,q))}.\n\\end{align}\nThe $i$-th codeword $c_i\\in \\mathbb{R}^{1024}$ can be obtained as the weighted\naverage of all codeword bases $B(x,y)$, \\textit{i}.\\textit{e}.,\n\\begin{align}\n            c_i = \\sum_{p,q} \\tilde{A}_i(p,q) B(p,q).\n\\end{align}\nIn other words, each spatial weighting map $\\tilde{A}_i$ learns to linearly\ncombine all codeword bases $B(x,y)$ from all spatial locations to form a\nsingle codeword, which captures certain aspect of the global context. The $n$\nweighting maps eventually result in $n$ holistic codewords $C = [c_1, \\cdots,\nc_n] \\in \\mathbb{R}^{1024 \\times n}$ to encode high-level global features.\n\n\\noindent \\textbf{Codeword assembly for high-resolution feature upsampling.} \nThe holistic codewords can capture various global contexts of the input image.\nThey are perfect ingredients for reconstructing the high-resolution\nsemantic-rich feature maps as they are encoded from the high-level features\n$m_{32}$. However, since their structural information have  been mostly\nremoved during codeword encoding, we turn to use the OS=8 multi-scale fused\nfeatures $m_8$ to predict the linear assembly coefficients of the $n$\ncodewords at each spatial location for creating a high-resolution feature map.\nMore specifically, we first create a raw codeword assembly guidance feature\nmap $G \\in \\mathbb{R}^{1024\\times (H\/8)\\times (W\/8)}$ to predict the assembly\ncoefficients at each spatial location, which are obtained by applying a\n$1\\times 1$ convolution on the multi-scale fused features $m_8$. However, the\nOS=8 fused features $m_8$ have no information on the holistic codewords as they\nare all generated from $m_{32}$. We therefore consider the general codeword\ninformation as the global average vector of the codeword based map $\\bar{B} \\in\n\\mathbb{R}^{1024}$ and location-wisely add it to the raw assembly guidance\nfeature map to obtain the novel guidance feature map $\\bar{G} = G \\oplus \\bar{B}$, where $\\oplus$ represents\nlocation-wise addition. Another $1\\times 1$ convolution applied on the\nguidance feature map $\\bar{G}$ generates the linear assembly weights of the $n$\ncodewords $W \\in \\mathbb{R}^{n \\times (H\/8) \\times (W\/8)}$ for all $(H\/8)\n\\times (W\/8)$ spatial locations. By reshaping the weighting map $W$ as an $n\n\\times (HW\/8^2)$ matrix, the holistically-guided upsampled feature\n$\\tilde{f}_8$ can be easily obtained as\n\\begin{align}\n            \\tilde{f}_8 = W^\\top C.\n\\end{align}\nGiven the holistically-guided upsampled feature map $\\tilde{f}_8$, we reconstruct the final \nupsampled feature map  $\\hat{f}_8$ by concatenating the feature map $\\tilde{f}_8$ with the\nguidance feature map $G$. \nSuch an upsampled feature map $\\hat{f}_8$ takes advantages of both $m_8$\nand $m_{32}$, and contains semantic-rich and also structure-preserved features\nfor achieving accurate segmentation. \n\n\\noindent {\\bf Final segmentation mask.} Given the upsampled feature map\n$\\hat{f}_8$, a $1\\times 1$ convolution can output a segmentation map of\nOS=8, which is further upsampled back to the original resolution $H\n\\times W$ as the final segmentation mask.\n\n\n\n\n\n\\section{Experiments}\nIn this section, we introduce the implementation details, training strategies and evaluation metrics of our experiments.\nTo evaluate our proposed EfficientFCN model, we conduct comprehensive experiments\non three public datasets PASCAL Context \\cite{mottaghi2014role}, PASCAL VOC 2012 \\cite{everingham2010pascal}\nand ADE20K \\cite{zhou2017scene}. To further evaluate the contributions of individual components in our model, we conduct detailed ablation studies on the PASCAL Context dataset. \n\\subsection{Implementation Details}\n\\noindent\n\\textbf{Network Structure.}\\\nDifferent with the dilatedFCN based methods, which remove the stride of the last two blocks of\nthe backbone networks and adopt the dilated convolution with the dilation rates $2$ and $4$,\nwe use the original ResNet \\cite{he2016deep} as our encoder backbone network. \nThus the size of the output feature maps from the last ResBlock is $32\\times$ smaller than that of\nthe input image. After feeding the encoder feature maps into our proposed holistic-guided decoder,\nthe classification is performed on the output upsampled feature map $\\hat{f}_8$. \nThe ImageNet \\cite{russakovsky2015imagenet} pre-trained weights are utilized to initialize the encoder network.\n\n\\noindent\n\\textbf{Training Setting.}\\\nA poly learning rate policy \\cite{chen2017deeplab} is used in our experiments. We set the initial learning rates as $0.001$\nfor PASCAL Context \\cite{mottaghi2014role}, $0.002$ for PASCAL VOC 2012\n\\cite{everingham2010pascal} and \nADE20K \\cite{zhou2017scene}. The power of poly learning rate policy is set as $0.9$. The optimizer\nis stochastic gradient descent (SGD) \\cite{bottou2010large} with momentum $0.9$ and weight\ndecay $0.0001$.\nWe train our EfficientFCN for $120$ epochs on PASCAL Context, $80$ epochs on PASCAL 2012 and $120$ epochs on ADE20K, respectively. We set the crop size to $512\\times 512$ on PASCAL Context and PASCAL 2012. Since the average image size is larger than other two datasets, we use $576 \\times 576$ as the crop size on ADE20K. For data augmentation, we only randomly flip the input image and scale it randomly in the range $[0.5, 2.0]$.\n\n\\noindent\n\\textbf{Evaluation Metrics.}\\\nWe choose the standard evaluation metrics of pixel accuracy (pixAcc) and mean Intersection of Union (mIoU) as the evaluation metrics in our experiments. Following the best practice \\cite{Zhang_2018_CVPR,he2019adaptive,fu2019dual}, \nwe apply the strategy of averaging the network predictions in multiple scales for evaluation. For\neach input image, we first randomly resize the input image with a scaling factor sampled uniformly\nfrom [0.5, 2.0] and also randomly horizontally flip the image. These predictions are then averaged to generate the final prediction.\n\\begin{table}[tb]\n\\centering\n\\centering\n\\caption{Comparisons with classical encoder-decoder methods.}\n\\label{table:ablation_encoder_decoder}\n\\setlength{\\tabcolsep}{0.5mm}{\n\\begin{tabular}{llllll}\n\\hline\n\\textbf{Method} & \\textbf{Backbone} & \\textbf{OS} &\\textbf{mIoU\\%} &\n\\textbf{Parameters (MB)} &\\textbf{GFlops (G)}\\\\\\hline\\hline\nFCN-32s  &  ResNet101 & 32 & 43.3 & 54.0 & 44.6 \\\\\ndilatedFCN-8s  & dilated-ResNet101 & 8 & 47.2 &54.0 & 223.6 \\\\\nUNet-Bilinear  & ResNet101 & 8 & 49.3 & 60.7 & 87.9 \\\\\nUNet-Deconv  & ResNet101 & 8  & 49.1 & 62.8 & 93.2\\\\\n\\hline\nEfficientFCN & ResNet101 & 8  & 55.3 & 55.8 & 69.6\\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\subsection{Results on PASCAL Context}\nThe PASCAL Context dataset consists of 4,998 training images and 5,105 testing images for scene parsing. It is a complex and challenging dataset based on PASCAL VOC 2010 with more annotations and fine-grained scene classes, which includes 59 foreground classes and one background class. \nWe take the same experimental settings and evaluation strategies following previous works \\cite{Zhang_2018_CVPR,Zhang_2019_CVPR,fu2019dual,Fu_2019_ICCV,he2019dynamic}.\nWe first conduct ablation studies on this dataset to demonstrate the\neffectiveness of each individual module design of our proposed EfficientFCN\nand then compare our model with state-of-the-art methods. The ablation studies are conducted with a ResNet101 encoder backbone.\n\\begin{table}[tb]\n\\centering\n\\begin{minipage}[t]{0.44\\textwidth}\n\\caption{Results of using different numbers of scales for multi-scale fused feature $m_{32}$ to generate the holistic codewords.}\n\\label{table:ablation_ms_codebook}\n\\setlength{\\tabcolsep}{1mm}{\n\\begin{tabular}{c|cccc}\n\\hline\n        {} & {$\\{32\\}$} & {$\\{16, 32\\}$} & {$\\{8, 16, 32\\}$} \\\\ \\hline\\hline\n        pixAcc &80.0 & 80.1 & 80.3  \\\\\n        mIoU & 54.8 & 55.1 & 55.3   \\\\\n\\hline\n\\end{tabular}}\n\\end{minipage}\n\\begin{minipage}[t]{0.11\\textwidth}\n\\end{minipage}\n\\begin{minipage}[t]{0.44\\textwidth}\n\\centering\n\\caption{Results of using different numbers of scales for multi-scale fused feature $m_8$ to estimate codeword assembly coefficients.}\n\\label{table:ablation_ms_assembly}\n\\setlength{\\tabcolsep}{1mm}{\n\\begin{tabular}{c|cccc}\n\\hline\n        {} & {$\\{8\\}$} & {$\\{8, 16\\}$} & {$\\{8,16,32\\}$} \\\\ \\hline\\hline\n        pixAcc &78.9 & 80.0 & 80.3  \\\\\n        mIoU & 47.9 & 52.1 & 55.3   \\\\\n\\hline\n\\end{tabular}}\n\\end{minipage}\n\\end{table}\n\n\\noindent \\textbf{Comparison with the classical encoder-decoders.} For the classical encoder-decoder based methods, the feature upsampling is achieved via either bilinear interpolation or\ndeconvolution. We implement two classical encoder-decoder based methods, which include the\nfeature upsampling operation (bilinear upsampling or deconvolution) and the skip-connections. \nTo verify the effectiveness of our proposed HGD,  these two methods are trained and tested \non the PASCAL Context dataset with the same training setting as our model.\nThe results are shown in Table \\ref{table:ablation_encoder_decoder}. Although the classical \nencoder-decoder methods have similar computational complexities, their performances \nare generally far inferior than our EfficientFCN. The key reason is that their upsampled \nfeature maps are recovered in a local manner. The simple bilinear interpolation or \ndeconvolution cannot effectively upsample the OS=32 feature maps even with the skip-connected \nOS=8 and OS=16 feature maps. In contrast, our proposed HGD can effectively upsample the \nhigh-resolution semantic-rich feature maps not only based on the fine-grained structural \ninformation in the OS=8 and OS=16 feature maps but also from the holistic semantic information from the OS=32 feature maps.  \n\n\\noindent \\textbf{Multi-scale features fusion.} We conduct two experiments on multi-scale features fusion to verify their effects on semantic codebook generation and codeword assembly for feature upsampling. In our holistically-guided decoder, the semantic codewords are generated based on the OS=32 multi-scale fused feature maps $m_{32}$ and the codewords assembly coefficients are predicted from the OS=8 multi-scale fused feature maps $m_8$.\nFor the codeword generation, we conduct three experiments to generate the semantic codebook from\nmulti-scale fused features with different numbers of scales. As shown in Table\n\\ref{table:ablation_ms_codebook}, when reducing the number of fusion scales from 3 to 2 and \nfrom 2 to 1, the performances of our EfficientFCN slightly decrease. The phenomenon is reasonable as the deepest feature maps contain more categorical information than the OS=8 and OS=16 feature maps. \nFor the codeword assembly coefficient estimation, the similar experiments are conducted, where results are shown in Table \\ref{table:ablation_ms_assembly}. However, different from the above results, when fewer scales of feature maps are used to form the multi-scale fused OS=8 feature map $m_8$, the performances of our EfficientFCN show significant drops.\nThese results demonstrate that although the OS=8 feature maps contain more fine-grained structural information, the semantic features from higher-level feature maps are essential for guiding the recovery of the semantic-rich high-resolution feature maps.\n\\begin{table}[!t]\n\\centering\n\\caption{Ablation study of the number of the semantic codewords.}\n\\label{table:ablation_n_codewords}\n\\setlength{\\tabcolsep}{2mm}{\n\\begin{tabular}{c|cccccc}\n\\hline\n       \n        {} & {32} & {64} & {128}  & {256} &{512} &{1024}\\\\ \\hline\\hline\n        pixAcc &79.9 & 80.1 & 80.1 & 80.3 & 80.3 &80.1 \\\\\n        mIoU & 54.5 & 54.9 & 55.0 &55.3 & 55.5 & 55.1  \\\\\n        GFLOPS & 67.9 & 68.1 & 68.6 & 69.6 & 72.1 & 78.9 \\\\\n\\hline\n\\end{tabular}}\n\\end{table}\n\\begin{table}[!t]\n\\centering\n\\caption{Segmentation results of state-of-the-art methods on PASCAL Context and ADE20K validation dataset.}\n\\label{table:pascal-context}\n\\centering\n\\begin{tabular}{lllll}\n\\hline\n    \\textbf{Method} & \\textbf{Backbone} &\\begin{tabular}{c} \n        \\textbf{mIoU\\%}\\\\ (PASCAL Context) \\end{tabular} & \\begin{tabular}{c} \n        \\textbf{mIoU\\%} \\\\ (ADE20K) \\end{tabular} & \\textbf{GFlops} \\\\\\hline\\hline\n    DeepLab-v2 \\cite{chen2017deeplab} &  Dilated-ResNet101-COCO & 45.7 & - & $>$223\\\\\n    RefineNet \\cite{RefineNet} &  Dilated-ResNet152 & 47.3 & - & $>$223\\\\\n    MSCI \\cite{MSCI} &  Dilated-ResNet152 & 50.3 & - & $>$223\\\\\n    PSPNet \\cite{zhao2017pyramid} & Dilated-ResNet101 & - & 43.29 & $>$223 \\\\\n    SAC \\cite{zhang2017scale} & Dilated-ResNet101 & - & 44.30 & $>$223  \\\\\n    EncNet \\cite{Zhang_2018_CVPR} &  Dilated-ResNet101 & 51.7 & 44.65 & 234\\\\\n    DANet \\cite{fu2019dual} &  Dilated-ResNet101 & 52.6 & - & $>$223 \\\\ \n    APCNet \\cite{he2019adaptive} &  Dilated-ResNet101 & 54.7 & 45.38 & 245\\\\ \n    CFNet \\cite{Zhang_2019_CVPR} &  Dilated-ResNet101 & 54.0 & 44.89 & $>$223 \\\\ \n    ACNet \\cite{Fu_2019_ICCV} & Dilated-ResNet101 & 54.1 & \\textbf{45.90} & $>$223\\\\ \n    APNB \\cite{zhu2019asymmetric} & Dilated-ResNet101 & 52.8 & 45.24 & $>$223  \\\\ \n    DMNet \\cite{he2019dynamic} & Dilated-ResNet101 & 54.4 & 45.50 & 242\\\\\n\\hline\n    Ours & ResNet101 & \\textbf{55.3} & {45.28} & \\textbf{70} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{comment}\n\\begin{table}[!t]\n\\caption{Segmentation results of state-of-the-art methods on ADE20K validation set.}\n\\label{table:ade20k}\n\\centering\n\\begin{tabular}{lll}\n\\hline\n\\textbf{Method} & \\textbf{Backbone} & \\textbf{mIoU\\%} \\\\\\hline\\hline\nPSPNet \\cite{zhao2017pyramid} & Dilated-ResNet101 & 43.29 \\\\\nEncNet \\cite{Zhang_2018_CVPR} & Dilated-ResNet101 & 44.65 \\\\\nSAC \\cite{zhang2017scale} & Dilated-ResNet101 & 44.30  \\\\\nDSSPN \\cite{DSSPN} & Dilated-ResNet101 & 43.68 \\\\\nAPCNet \\cite{he2019adaptive} & Dilated-ResNet101 & 45.38 \\\\\nCFNet \\cite{Zhang_2019_CVPR} & Dilated-ResNet101 & 44.89 \\\\\nDMNet \\cite{he2019dynamic} & Dilated-ResNet101 & 45.50  \\\\ \nACNet \\cite{Fu_2019_ICCV} & Dilated-ResNet101 & 45.90 \\\\ \nAPCNet \\cite{he2019adaptive} & Dilated-ResNet101 & 45.38 \\\\\nCCNet \\cite{Huang_2019_ICCV} & Dilated-ResNet101 & 45.22 \\\\ \nAPNB \\cite{zhu2019asymmetric} & Dilated-ResNet101 & 45.24 \\\\ \n\\hline\nOurs & ResNet101 & \\textbf{45.28} \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\\end{comment}\n\n\n\n\\begin{figure}[!t]\n\\centering\n\\begin{center}\n\\begin{tabular}{C{2.2cm}C{2.2cm}C{2.2cm}C{2.2cm}C{2.2cm}}\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig0_e} \\\\\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig1_e} \\\\\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_a} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_b} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_c} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_d} &\n    \\includegraphics[width=2.20cm]{figs\/exp1_fig2_e} \\\\\n   \n   \n   \n   \n   \n    \\centering (a) & (b) & (c) & (d) & (e) \\\\\n\\end{tabular}\n\\end{center}\n\\caption{(a) Input images from the PASCAL Context and ADE20K dataset. (b-e) Different weighting maps $\\tilde{A}_i$ for creating the holistic codewords.}\n\\label{fig:weighting_maps}\n\n\\begin{center}\n\\begin{tabular}{C{2.50cm}C{2.50cm}C{2.50cm}C{2.50cm}}\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004877_our} \\\\\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2008_004203_our} \\\\\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_img} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_gt} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_fcn} &\n    \\includegraphics[width=2.50cm]{figs\/figs_pcontext\/2010_004980_our} \\\\\n    \\centering (a) Image & (b) GT & (c) Baseline & (d) EfficientFCN \\\\\n\\end{tabular}\n\\end{center}\n\\caption{Visualization results from the PASCAL Context dataset.}\n\\label{fig:vis}\n\\end{figure}\n\n\\noindent \\textbf{Number of holistic codewords.} We also conduct  experiments to survey the\neffectiveness of the number of codewords in our predicted semantic codebook for feature upsampling.\nAs shown in Table \\ref{table:ablation_n_codewords}, as the number of the semantic codewords\nincreases from 32 to 512, the performance improves 1\\% in terms of mIoU on PASCAL Context.\nHowever, when the number of the semantic codewords further increases from 512 to 1024, the performance\nhas a slight drop, which might be caused by the additional parameters. The larger model capacity\nmight cause model to overfit the training data. In addition, since the assembly coefficients of the\nsemantic codewords are predicted from the OS=8 multi-scale fused feature $m_8$, the increased number\nof the semantic codewords also leads to significantly more extra computational cost. Thus, to balance\nthe performance and also the efficiency, we set the number of the holistic codewords as 256 for the\nPASCAL Context and PASCAL VOC 2012 datasets. Since PASCAL Context only has 60 classes and we observe\nthe number of codewords needed is approximately 4 times than the number of classes. We therefore set\nthe number of codewords as 600 for ADE20K, which has 150 classes.\n\n\\noindent \\textbf{Importance of the codeword information transfer for accurate assembly coefficient estimation.} \nThe key of our proposed HGD is how to linearly assemble holistic codewords at each spatial location\nto form high-resolution upsampled feature maps based on the feature maps $m_8$. In our HGD, although\nthe OS=8 features have well maintained structural image information, we argue that directly using OS=8 features to predict codeword assembly coefficients are less effective since they have no information about the codewords. \nWe propose to transfer the codeword information as the average codeword basis,\nwhich is location-wisely added to the OS=8 feature maps. To verify this\nargument, we design an experiment that removes the additive information\ntransfer, and only utilizes two $1\\times 1$ convolutions with the same output\nchannels on the OS=8 feature maps $m_8$ for directly predicting assembly\ncoefficients. The mIoU of this implementation is 54.2\\%, which has a clear performance drop if there\nis no codeword information transfer from the codeword generation branch to the codeword coefficient prediction branch.\n\n\n\\noindent \\textbf{Visualization of the weighting maps and example results.} \nTo better interpret the obtained holistic codewords, we visualize the weighting maps $\\tilde{A}$ for\ncreating the holistic codewords in Fig.~\\ref{fig:weighting_maps}, where each column shows one\nweighting map $\\tilde{A}_i$ for generating one holistic codeword. Some weighting maps focus on summarizing  foreground objects or regions to create holistic codewords, while some other weighting maps pay attention to summarizing background contextual regions or objects as the holistic codewords. The visualization shows that the learned codewords implicitly capture different global contexts from the scenes.\nIn Fig.~\\ref{fig:vis}, we also visualize some predictions by the baseline\nDilatedFCN-8s and by our EfficientFCN, where our model significantly improves the visualized results with the proposed HGD.\n\n\n\n\n\\begin{table*}[!t]\n\\small\n\\centering\n\\caption{Results of each category on PASCAL VOC 2012 test set. Our\n        EfficientFCN obtains 85.4 \\% without MS COCO dataset pre-training and 87.6\\% with MS COCO dataset pre-training. (For each\n    columns, the best two entries are filled in gray color. )}\n\\label{table:pascal_voc_2012}\n\\resizebox{\\textwidth}{!}{%\n\\begin{tabular}{l|cccccccccccccccccccc|c}\n\\hline\n\\textbf{Method}    & \\textbf{aero} & \\textbf{bike} & \\textbf{bird} & \\textbf{boat} & \\textbf{bottle} & \\textbf{bus}  & \\textbf{car}  & \\textbf{cat}  & \\textbf{chair} & \\textbf{cow}  & \\textbf{table} & \\textbf{dog}  & \\textbf{horse} & \\textbf{mbike} & \\textbf{person} & \\textbf{plant} & \\textbf{sheep} & \\textbf{sofa} & \\textbf{train} & \\textbf{tv}   & \\textbf{mIoU\\%} \\\\ \\hline\\hline\n\\textbf{FCN} \\cite{long2015fully}       & 76.8          & 34.2          & 68.9\n& 49.4          & 60.3            & 75.3          & 74.7          & 77.6\n& 21.4           & 62.5          & 46.8           & 71.8          & 63.9\n& 76.5           & 73.9            & 45.2           & 72.4           & 37.4\n& 70.9           & 55.1          & 62.2   \\\\\n\\textbf{DeepLabv2} \\cite{chen2017deeplab} & 84.4          & 54.5          & 81.5\n& 63.6          & 65.9            & 85.1          & 79.1          & 83.4\n& 30.7           & 74.1          & 59.8           & 79.0            & 76.1           & 83.2           & 80.8            & 59.7           & 82.2           & 50.4          & 73.1           & 63.7          & 71.6  \\\\\n\\textbf{CRF-RNN} \\cite{CRF-RNN}   & 87.5          & 39.0          & 79.7          & 64.2          & 68.3            & 87.6          & 80.8          & 84.4          & 30.4           & 78.2          & 60.4           & 80.5          & 77.8           & 83.1           & 80.6            & 59.5           & 82.8           & 47.8          & 78.3           & 67.1          & 72.0  \\\\\n\\textbf{DeconvNet} \\cite{DeconvNet} & 89.9          & 39.3          & 79.7          & 63.9          & 68.2            & 87.4          & 81.2          & 86.1          & 28.5           & 77.0          & 62.0           & 79.0          & 80.3           & 83.6           & 80.2            & 58.8           & 83.4           & 54.3          & 80.7           & 65.0            & 72.5   \\\\\n\\textbf{DPN} \\cite{DPN}       & 87.7          & 59.4          & 78.4          & 64.9          & 70.3            & 89.3          & 83.5          & 86.1          & 31.7           & 79.9          & 62.6           & 81.9          & 80.0           & 83.5           & 82.3            & 60.5           & 83.2           & 53.4          & 77.9           & 65.0            & 74.1   \\\\\n\\textbf{Piecewise} \\cite{Piecewise} & 90.6          & 37.6          & 80.0\n& 67.8          & 74.4            & 92            & 85.2          & 86.2\n& 39.1           & 81.2          & 58.9           & 83.8          & 83.9\n& 84.3           & 84.8            & 62.1           & 83.2           & 58.2\n& 80.8           & 72.3          & 75.3    \\\\\n\\textbf{ResNet38} \\cite{ResNet38}  & 94.4          & 72.9          & 94.9\n& 68.8          & 78.4            & 90.6          & 90.0          & 92.1\n& 40.1           & 90.4          & 71.7           & 89.9          & 93.7\n& \\bgGray 91.0           & 89.1            & 71.3           & 90.7           & 61.3          & 87.7           & 78.1          & 82.5     \\\\\n\\textbf{PSPNet} \\cite{zhao2017pyramid}    & 91.8          & 71.9          & 94.7          & 71.2          & 75.8            & 95.2          & 89.9          & 95.9          & 39.3           & 90.7          & 71.7           & 90.5          & 94.5           & 88.8           & 89.6            & 72.8           & 89.6           & \\bgGray {64.0}          & 85.1           & 76.3          & 82.6   \\\\\n\\textbf{EncNet} \\cite{Zhang_2018_CVPR}    & 94.1          & 69.2          & \\bgGray\\textbf{96.3} & \\bgGray 76.7          & \\bgGray \\textbf{86.2}   & 96.3          & 90.7          & 94.2          & 38.8           & 90.7          & 73.3           & 90.0          & 92.5           & 88.8           & 87.9            & 68.7           & 92.6           & 59.0          & 86.4           & 73.4          & 82.9            \n            \\\\\n            \\textbf{APCNet} \\cite{he2019adaptive}      & 95.8 &\\bgGray 75.8 & 84.5  & 76.0 & 80.6 &\n            \\bgGray 96.9 & 90.0 & 96.0 & \\bgGray\\textbf{42.0} & \\bgGray 93.7\n            &\\bgGray 75.4 & 91.6 & 95.0 & 90.5 &\n             89.3 &  75.8 & 92.8 & 61.9 &  88.9 & \\bgGray 79.6 & 84.2\n            \\\\\n            \\textbf{CFNet} \\cite{Zhang_2019_CVPR}      & 95.7 & 71.9 &\\bgGray\n            95.0  &\\bgGray 76.3 & \\bgGray 82.8 &\n            94.8 & 90.0 & 95.9 & 37.1 & 92.6 & 73.0 & \\bgGray 93.4 & 94.6 & 89.6 &\n            88.4 & 74.9 & \\bgGray \\textbf{95.2} &  63.2 & \\bgGray \\textbf{89.7} & 78.2 & 84.2\n            \\\\\n            \\textbf{DMNet} \\cite{he2019dynamic}        & \\bgGray 96.1 &\n            \\bgGray\\textbf{77.3} & 94.1 & 72.8 & 78.1 & \n            \\bgGray\\textbf{97.1} & \\bgGray \\textbf{92.7} & \\bgGray 96.4 & 39.8 & 91.4 & \\bgGray 75.5 & 92.7 & \\textbf{95.8} &\n            \\bgGray {91.0} & \\bgGray {90.3} & \\bgGray {76.6} & \\bgGray 94.1 & 62.1 & 85.5 & 77.6 & \\bgGray 84.4 \n            \\\\ \\hline\n            \\textbf{Ours}      & \\bgGray \\textbf{96.4} & {74.1} &\n            92.8 & \\bgGray 75.6 & 81.9 &\\bgGray 96.9 \n            & \\bgGray {92.6}  & \\bgGray \\textbf{97.1} & \\bgGray 41.6 & \\bgGray \\textbf{95.4} \n            & 72.9 & \\bgGray \\textbf{93.9} & \\bgGray \\textbf{95.9} \n            & {90.6} &\\bgGray \\textbf{ 90.6} &\\bgGray \\textbf{77.2}  & 94.0 &\n            67.5 &\\bgGray 89.3 &\n            \\bgGray \\textbf{79.8} & \\bgGray \\textbf{85.4} \\\\\n             \\hline\n            \\multicolumn{22}{c}{\\textbf{With COCO Pre-training}}\\\\\n            \\hline\n            \n            \\textbf{CRF-RNN}~\\cite{CRF-RNN} & 90.4 & 55.3 & 88.7 & 68.4 & 69.8 & 88.3 & 82.4 & 85.1 & 32.6 & 78.5 & 64.4 & 79.6 & 81.9 & 86.4 & 81.8 & 58.6 & 82.4 & 53.5 & 77.4 & 70.1 & 74.7 \\\\\n           \n            \\textbf{Piecewise}~\\cite{Piecewise} & 94.1 & 40.7 & 84.1 & 67.8 & 75.9 & 93.4 & 84.3 & 88.4 & 42.5 & 86.4 & 64.7 & 85.4 & 89.0 & 85.8 & 86.0 & 67.5 & 90.2 & 63.8 & 80.9 & 73.0 & 78.0 \\\\\n           \n           \n            \\textbf{DeepLabv2}~\\cite{chen2017deeplab} & 92.6 & 60.4 & 91.6 & 63.4 & 76.3 & 95.0 & 88.4 & 92.6 & 32.7 & 88.5 & 67.6 & 89.6 & 92.1 & 87.0 & 87.4 & 63.3 & 88.3 & 60.0 & 86.8 & 74.5 & 79.7  \\\\\n            \\textbf{RefineNet}\\cite{RefineNet}  & 95.0 & 73.2 & 93.5 & 78.1 & 84.8 & 95.6 & 89.8 & 94.1 & 43.7 & 92.0 & 77.2 & 90.8 & 93.4 & 88.6 & 88.1 & 70.1 & 92.9 & 64.3 & 87.7 & 78.8 & 84.2 \\\\\n            \\textbf{ResNet38}\\cite{ResNet38}  &  96.2 & 75.2 & \\cellcolor[gray]{0.92} \n            95.4 & 74.4 & 81.7 & 93.7 & 89.9 & 92.5 & \\cellcolor[gray]{0.92}  48.2 & 92.0 & 79.9\n            & 90.1 & 95.5 & 91.8 & 91.2 &  73.0 & 90.5 & 65.4 & 88.7 & 80.6 & 84.9\\\\\n            \\textbf{PSPNet}~\\cite{zhao2017pyramid} & {95.8} & {72.7} & {95.0}\n            & {78.9} & {84.4} & 94.7 & \\cellcolor[gray]{0.92} {92.0} & {95.7} & {43.1} &\n            {91.0} & \\bf \\cellcolor[gray]{0.85}{80.3} & {91.3} & {96.3} & {92.3} & {90.1} & {71.5}\n            & \\cellcolor[gray]{0.92} {94.4} & \\cellcolor[gray]{0.92} {66.9} & {88.8} & \\bf \\cellcolor[gray]{0.85}{82.0} & {85.4} \\\\\n            \\textbf{DeepLabv3}\\cite{chen2017rethinking} & \\cellcolor[gray]{0.92}  96.4 & 76.6\n            & 92.7 & 77.8 & \\cellcolor[gray]{0.92} {87.6} & 96.7 & 90.2 & 95.4 &  47.5 &\n            \\cellcolor[gray]{0.92}  93.4 & 76.3 & 91.4 & \\bf \\cellcolor[gray]{0.85}{97.2} &  91.0 & \\bf \\cellcolor[gray]{0.85}{92.1} &\n            71.3 & 90.9 & \\cellcolor[gray]{0.92} {68.9} & \\cellcolor[gray]{0.92} {90.8} & 79.3 & 85.7 \\\\ \n            \\textbf{EncNet}\\cite{Zhang_2018_CVPR}  & 95.3 & 76.9 & 94.2 &\n            \\cellcolor[gray]{0.92}  80.2 & 85.2 & 96.5 & 90.8 & 96.3 &  47.9\n            &  93.9 & \\cellcolor[gray]{0.92}  80.0 & 92.4 & \\cellcolor[gray]{0.92}  96.6 & 90.5 &  91.5 & 70.8 &  93.6 & 66.5 & 87.7 & 80.8 & 85.9 \n           \n            \\\\ \n            \\textbf{CFNet} \\cite{Zhang_2019_CVPR}      &\\bf \\cellcolor[gray]{0.85} 96.7 &\\cellcolor[gray]{0.92}  79.7 &\n            94.3  & 78.4 & 83.0 & \\bf \\cellcolor[gray]{0.85} 97.7 & 91.6 &\\cellcolor[gray]{0.92}  96.7 &\\bf \\cellcolor[gray]{0.85} 50.1\n            &\\cellcolor[gray]{0.92}  95.3 & 79.6 & \\bgGray 93.6 &\\bf \\cellcolor[gray]{0.85} 97.2 &\\cellcolor[gray]{0.92} \n            94.2 &\\cellcolor[gray]{0.92}  91.7 & \\bgGray {78.4} &\\bf \\cellcolor[gray]{0.85}  95.4 & \\bgGray\n            \\textbf{69.6} & 90.0 & 81.4 & \\cellcolor[gray]{0.92}  87.2\n            \\\\ \\hline\n            \\textbf{Ours} & \\bgGray {96.6} & \\bf \\cellcolor[gray]{0.85} {80.6} &\n            \\bf \\cellcolor[gray]{0.85} 96.1 & \\bf \\cellcolor[gray]{0.85}{82.3} &\\bf \\cellcolor[gray]{0.85} 87.8 &\\bf \\cellcolor[gray]{0.85} 97.7 \n            & \\bf \\cellcolor[gray]{0.85} {94.4}  & \\bf \\cellcolor[gray]{0.85} {97.3} &  47.1 & \\bgGray \\textbf{96.3} \n            & {77.9} & \\bgGray \\textbf{94.8} & \\bgGray\n            \\textbf{97.2} \n            &\\bgGray \\textbf{94.3} & 91.1 & \\bf \\cellcolor[gray]{0.85} 81.0  & 94.3 & 61.5\n            &\\bf \\cellcolor[gray]{0.85} 91.6 &\\bf \\cellcolor[gray]{0.85} {83.5} & \\bgGray \\textbf{87.6}\n            \\\\ \\hline\n\\end{tabular}}\n\\end{table*}\n\n\\noindent\n\\textbf{Comparison with state-of-the-art methods.} \nTo further demonstrate the effectiveness of our proposed EffectiveFCN with the holistically-guided decoder, the comparisons with state-of-the-art methods are shown in Table \\ref{table:pascal-context}. The dilatedFCN based methods dominate semantic segmentation. However, our work is still able to achieve the best results compared to the dilatedFCN based methods on the PASCAL Context validation set without using any dilated convolution and has significantly less computational cost. Because of the efficient design of our HGD, our EfficientFCN only has 1\/3 of the computational cost of state-of-the-arts methods but can still achieve the best performance. \n\n\\subsection{Results on PASCAL VOC}\nThe original PASCAL VOC 2012 dataset consists of 1,464 images for training, 1,449 for validation, and 1,456 for testing, which is a major benchmark dataset for semantic object segmentation. It includes 20 foreground objects classed and one background class. The augmented training set of 10,582 images, namely train-aug, is adopted as the training set following the previous experimental set in \\cite{Zhang_2019_CVPR}.\nTo further demonstrate the effectiveness of our proposed HGD. We adopt all the best strategies of HGD design and compare it with state-of-the-art methods on the test set of PASCAL-VOC 2012, which is evaluated on the official online server. As shown in Table \\ref{table:pascal_voc_2012}, the dilatedFCN based methods dominate the top performances on the PSCAL VOC benchmark. However, our EfficientFCN with a backbone having no dilated convolution can still achieve the best results among all the ResNet101-based methods. \n\\subsection{Results on ADE20K}\nThe ADE20K dataset consists of 20K images for training, 2K images for\nvalidation, and 3K images for testing, which were used for ImageNet Scene\nParsing Challenge 2016. This dataset is more complex and challenging with 150 labeled classes and more diverse scenes. As shown in Table \\ref{table:pascal-context}, our\nEfficientFCN achieves the competitive performance than the dilatedFCN based\nmethods but has only 1\/3 of their computational cost.\n\\section{Conclusions}\nIn this paper, we propose the EfficientFCN model with the holistically-guied decoder for achieving efficient and accurate semantic segmentation. The novel decoder is able to reconstruct the high-resolution semantic-rich feature maps from multi-scale feature maps of the encoder. \nBecause of the superior feature upsampling performance of the HGD, our EfficientFCN, with much fewer parameters and less computational cost, achieves competitive or even better performance compared with state-of-the-art dilatedFCN based methods. \n\n\n\\subsection*{Acknowledgements}\nThis work is supported in part by SenseTime Group Limited, in part by the General Research Fund through the Research Grants Council of Hong Kong under Grants CUHK 14202217 \/ 14203118 \/ 14205615 \/ 14207814 \/ 14213616 \/ 14208417 \/ 14239816, in part by CUHK Direct Grant.\n\n\\clearpage\n\n\n\n{\\small\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\n\\section{Introduction}\n\nThe discovery and characterization of primeval galaxies constitute some of the biggest challenges in current observational and theoretical cosmology\\footnote{In the following we assume cosmological parameters compatible with \\emph{Planck} results, i.e. a $\\Lambda$CDM model with total matter, vacuum and baryonic densities in units of the critical density $\\Omega_{\\Lambda}= 0.692$, $\\Omega_{m}= 0.308$, $\\Omega_{b}= 0.0481$, Hubble constant $\\rm H_0=100\\,{\\rm h}\\,{\\rm km}\\,{\\rm s}^{-1}\\,{\\rm Mpc}^{-1}$ with ${\\rm h}=0.678$, spectral index $n=0.967$, $\\sigma_{8}=0.826$ \\citep[][]{planck:2013_xvi_parameters}.}.\n\nDeep optical\/near infrared (IR) surveys \\citep{Dunlop13,Madau14,Bouwens:2015} have made impressive progresses in identifying galaxies well within the Epoch of Reionization ($z\\simeq6$). Such surveys yield key information about the star formation (SF) of hundreds of galaxies in the early Universe. They also allow to statistically characterize galaxies in terms of their UltraViolet (UV) luminosity up to $z\\sim10$ \\citep{Bouwens:2015}. However -- using these surveys broad band alone -- little can be learned about other properties as their gas and dust content, metallicity, interactions with the surrounding environment \\citep[e.g.][]{Barnes:2014PASP}, feedback \\citep[e.g.][]{Dayal14}, and outflows \\citep{gallerani:2016outflow}. \n\nTo obtain a full picture of these systems, optical\/IR surveys must be complemented with additional probes. Information on the metal content and energetics of the interstellar medium (ISM) can be obtained with observations of Far IR (FIR) fine structure lines, and in particular the \\hbox{[C~$\\scriptstyle\\rm II $]}~{\\small$\\left(^{2}P_{3\/2} \\rightarrow\\,^{2}P_{1\/2}\\right)$} line at 157.74~$\\mu$m. The \\hbox{[C~$\\scriptstyle\\rm II $]}~line is the dominant coolant of the ISM being excited in different ISM phases, as the diffuse cold neutral medium (CNM), warm neutral medium (WNM), high density photodissociation regions (PDRs), and -- to a lower extent -- ionized gas \\citep[][]{Tielens:1985ApJ,Wolfire:1995ApJ,Abel:2006MNRAS,Vallini:2013MNRAS}. As \\hbox{[C~$\\scriptstyle\\rm II $]}~emission can be enhanced by shocks, it has been suggested as a good outflow tracer\\\\ (e.g. \\citealt{maiolino:2012,kreckel:2014apj,cicone:2015aa,janssen:2016arxiv}), and can thus in general be used to study feedback processes in galaxies.\n\nObservationally, the \\hbox{[C~$\\scriptstyle\\rm II $]}~line is a promising probe as it is often the brightest among FIR emission lines, accounting for up to $\\sim1\\%$ of the total IR luminosity of galaxies \\citep[e.g.][]{Crawford:1985ApJ,Madden:1997ApJ}. It has been successfully used to probe the low-$z$ ISM \\citep[e.g.][]{delooze:2014aa}. The unprecedented sensitivity of the Atacama Large Millimeter\/Submillmeter Array (ALMA) makes it possible for the first time to use \\hbox{[C~$\\scriptstyle\\rm II $]}~emission to characterize high-$z$ galaxies. Before the ALMA advent, in fact, detections were limited to a handful of QSO host galaxies, and rare galaxies with extreme SF rates \\citep[$SFR\\simeq10^3{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$, e.g.][]{maiolino:2005AA,debreuck:2011,Carilli:2013ARA&A,gallerani:2012aa,cicone:2015aa}.\n\nHowever, for \\quotes{normal} star forming galaxies ($\\lsim10^{2}{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$) at $z\\sim 6-7$ early ALMA searches for \\hbox{[C~$\\scriptstyle\\rm II $]}~lines have mostly yielded upper limits (e.g. \\citealt{ouchi2013} \\citealt{kanekar2013}; \\citealt{ota:2014apj,schaerer:2015}). The situation has changed recently with a number of robust \\hbox{[C~$\\scriptstyle\\rm II $]}~detections (e.g. \\citealt{maiolino:2015arxiv,capak:2015arxiv}; \\citealt{Willott:2015arXiv15,knudsen:2016arxiv}).\n\nIn many cases the high-$z$ \\hbox{[C~$\\scriptstyle\\rm II $]}~line luminosity is fainter than expected from the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation found in local galaxies \\citep{delooze:2014aa}. To explain such \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~\\emph{deficit}, some efforts have been devoted to model the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission from high-$z$ galaxies \\citep{nagamine:2006ApJ,Vallini:2013MNRAS,munoz:2014MNRAS,vallini:2015,olsen:2015apj}. In brief, these theoretical works show that the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit can be ascribed to different effects:\n\\begin{itemize}\n\\item[(a)] Lower metallicity of high-$z$ galaxies \\citep{Vallini:2013MNRAS,munoz:2014MNRAS,vallini:2015}, in particular supported by observations of lensed galaxies \\citep{knudsen:2016arxiv}.\n\\item[(b)] Suppression of \\hbox{[C~$\\scriptstyle\\rm II $]}~line around star forming regions \\citet{Vallini:2013MNRAS}, typically observed as a displacement of the \\hbox{[C~$\\scriptstyle\\rm II $]}~ with respect to the UV emitting region, as seen e.g. in BDF3299 \\citep{maiolino:2015arxiv} and in some of the \\citet{capak:2015arxiv} galaxies. This would be a signature of stellar feedback heating\/ionizing the putative \\hbox{[C~$\\scriptstyle\\rm II $]}-emitting gas.\n\\item[(c)] Suppression of \\hbox{[C~$\\scriptstyle\\rm II $]}~line by the increased CMB temperature in the WNM\/CNM component \\citep[][]{pallottini:2015_cmb,vallini:2015}, similarly to what observed for dust emission \\citep{dacunha:2013apj}.\n\\end{itemize}\n\nSimulating the ISM of early galaxies at sufficient resolution and including feedback effects might shed light on these questions. Feedback prescriptions are particularly important as such process regulates the amount of (dense) gas likely to radiate most of the power detected with FIR lines. Several studies have explored optimal strategies to include feedback in galaxy simulations.\n\nFor some works, the interest is in the comparison between different kind of stellar feedback prescription, as modelled via thermal and\/or kinetic energy deposition in the gas from supernovae (SN), winds \\citep[][]{agertz:2012arxiv,fire:2014mnras,barai:2015mnras,agertz:2015apj}, and radiation pressure \\citep[][]{wise:2012radpres,ceverino:2014}; other analyses focus on implementing complex chemical networks in simulations \\citep{tomassetti:2015MNRAS,maio:2015,bovino:2015arxiv,richings:2016,grassi_dust:2016}, radiative transfer effect \\citep{petkova:2012mnras,roskar:2014,rosdahl:2015mnras,maio:2016mnras}, or aim at removing tensions between different coding approaches \\citep[][]{agora:2013arxiv}.\n\nThus, we can improve galaxy simulations by providing theoretical expectations for \\hbox{[C~$\\scriptstyle\\rm II $]}~that should be compared with state-of-the-art data. Such a synergy between theory and observations, in turn, can guide the interpretation of upcoming ALMA data and drive future experiments of large\nscale \\hbox{[C~$\\scriptstyle\\rm II $]}~mapping \\citep{Gong:2012ApJ,silva:2015apj,bin:2015mapping, pallottini:2015_cmb}, which would led to a statistical characterization of the high-$z$ galaxy population. In the present work we simulate a $z\\sim6$ galaxy typically detected in \\hbox{[C~$\\scriptstyle\\rm II $]}~with ALMA current observations.\n\nThe paper is structured as follows. In Sec. \\ref{sec_numerical} we detail the numerical model used to set-up the zoom-in simulation, and describe the adopted ${\\rm {H_2}}$~star formation prescription (Sec. \\ref{sec_model_sf}), mass and energy inputs from the stellar populations (Sec. \\ref{sec_stellar_inputs}) and feedback (including SN, winds and radiation pressure Sec. \\ref{sezione_blast} -- see also App. \\ref{app_rad_press} and App. \\ref{app_blastwave}). The results are discussed in Sec. \\ref{sec_result}, where we analyze star formation history and feedback effects in relation to ISM thermodynamics (Sec. \\ref{sec_sfr_result}) and its structural properties. The expected \\hbox{[C~$\\scriptstyle\\rm II $]}~emission and other observational properties of high-$z$ galaxies are discussed in Sec. \\ref{sec_final_results}. Conclusions are given in Sec. \\ref{sec_conclusioni}.\n\n\n\\section{Numerical simulations}\\label{sec_numerical}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{ccccccc}\n\\hline\n~ & $m_{dm}$ & $m_{b}$ & $\\Delta_{x}^{\\rm max}$ & $\\Delta_{x}^{\\rm min}$ & $\\Delta_{x}^{\\rm min}$ at $z=6$\\\\\n~ & \\multicolumn{2}{c}{${\\rm M}_{\\odot}\/{\\rm h}$} & \\multicolumn{2}{c}{${\\rm kpc}\/{\\rm h}$} & pc\\\\\n\\hline\n{\\tt cosmo} & $3.4\\times 10^{7}$ & $-$                & $78.1$ & $78.1$ & $2.5\\times10^3$\\\\\n{\\tt zoom}  & $6.7\\times 10^{4}$ & $1.2\\times 10^{4}$ & $9.7$  & $0.1$  & $32.1$\\\\\n\\end{tabular}\n\\caption{Resolution set-up for the cosmological run ({\\tt cosmo}) and subsequent zoom-in ({\\tt zoom}) simulation. $m_{dm}$ and $m_{b}$ are in units of ${\\rm M}_{\\odot}\/{\\rm h}$ and indicate the dark matter (DM) and baryon mass resolution, respectively; $\\Delta_{x}^{\\rm max}$ and $\\Delta_{x}^{\\rm min}$ indicate the coarse grid and minimum available refinement scale, respectively. Both scales are reported in comoving ${\\rm kpc}\/{\\rm h}$. For $\\Delta_{x}^{\\rm min}$ we also report also the physical pc scale at $z=6$. For the {\\tt cosmo} run, no refinement is used, and for the {\\tt zoom}, we indicate the increased resolution of the zoomed halo due to the multi-mass approach and the AMR.\n\\label{tagella_res}}\n\\end{table}\n\nWe carry out our simulation using a customized version of the adaptive mesh refinement (AMR) code \\textlcsc{ramses} \\citep[][]{Teyssier:2002}. \\textlcsc{ramses} is an octree-based code that uses Particle Mesh N-body solver for the dark matter (DM) and an unsplit 2nd-order MUSCL\\footnote{MUSCL: Monotone Upstream-centred Scheme for Conservation Laws} scheme for the baryons. Gravity is accounted by solving the Poisson equation on the AMR grid via a multi-grid scheme with Dirichlet boundary conditions on arbitrary domains \\citep{guillet:2011Jcoph}. For the present simulation we choose a refinement based on a Lagrangian mass threshold-based criterion.\n\nChemistry and heating\/cooling processes of the baryons are implemented with \\textlcsc{grackle} 2.1\\footnote{See also \\url{https:\/\/grackle.readthedocs.org\/}} \\citep{bryan:2014apjs}, the standard library of thermo-chemical processes of the {\\tt AGORA} project \\citep{agora:2013arxiv}. Via \\textlcsc{grackle}, we follow the \\hbox{H}~and \\hbox{He}~primordial network and tabulated metal cooling and photo-heating rates calculated with \\textlcsc{cloudy} \\citep{cloudy:2013}. Cooling includes also inverse Compton off the cosmic microwave background (CMB), and heating from a redshift-dependent ionizing UV background \\citep[][UVB]{Haardt:2012}. Since ${\\rm {H_2}}$~gas phase formation is not accounted for, we do not include the cooling contribution of such species.\n\nBecause of stellar feedback (Sec \\ref{sec_stellar_inputs} and \\ref{sezione_blast}), the gas can acquire energy both in thermal and kinetic form. The distinction is considered by following the gas evolution of the standard thermal energy and a \\quotes{non-thermal} energy \\citep{agertz:2012arxiv}. Such approach is one of the possible scheme used to solve the over-cooling problem that affect galaxy-scale simulations \\citep[see][ and references therein]{dale:2015new}. The non-thermal energy mimics turbulence, i.e. it is not affected by cooling. The non-thermal energy variation is due to gas advection ($v\\nabla v$), work ($PdV$), and dissipation \\citep{agertz:2015apj}. Following \\citet{maclow1999turb} we assume a dissipation time scale proportional to the size of the cell (injection scale) and inversely proportional to the Mach number\\footnote{While the distinction in thermal and non-thermal is similar to previous works \\citep[e.g.][]{agertz:2015apj}, we note that usually the time scale for dissipation is fixed to $10\\,\\rm Myr$.}. Since the dynamical time is essentially set by the free-fall time, the dissipation time can be written as $t_{\\rm diss} = 9.785 (l_{\\rm cell}\/100\\,{\\rm pc})\/(v_{\\rm turb}\/10\\,{\\rm km}\\,{\\rm s}^{-1}) \\rm Myr$. Then, the non-thermal energy loss due to dissipation can be written as $\\dot{e}_{\\rm nth} = -e_{\\rm nth}\/t_{\\rm diss}$ \\citep[][see eq. 2]{teyssier:2013mnras}. As noted in \\citet{teyssier:2013mnras}, such scheme for non-thermal energy and its dissipation gives results qualitatively similar to a delayed cooling approach \\citep{stinson:2006mnras}.\n\n\\subsection{Initial conditions}\n\nThe initial conditions (IC) used for the suite are generated with \\textlcsc{music} \\citep{hahn:2011mnras}. \\textlcsc{music} produces IC on nested grid using a real-space convolution approach \\citep[cf.][]{bertschinger:1995astro}. The adopted Lagrangian perturbation theory scheme is perfectly suited to produce IC for multi-mass simulations and -- in particular -- zoom simulations. To generate the ICs, the transfer functions are taken from \\citep{eisenstein:1998apj}.\n\nTo set-up the zoom-in simulation, we start by carrying out a cosmological DM-only run. The simulation evolves a volume $V^{\\rm cosmo}=(20\\,{\\rm Mpc}\/{\\rm h})^{3}$ from $z=100$ to $z=6$ with DM mass resolution of $m_{dm}^{\\rm cosmo} = 3.4\\times 10^{7} \/{\\rm h}\\,{\\rm M}_{\\odot}$. The resolution of the coarse grid is $\\Delta x^{\\rm cosmo} = 78.1 \/{\\rm h}\\,{\\rm kpc}$, and we do not include additional levels of refinement. Using \\textlcsc{hop} \\citep{eisenstein_hop_1998apj} we find the DM halo catalogue at $z=6$. The cumulative halo mass function extracted from the catalogue is in agreement with analytical expectations \\citep[e.g.][]{sheth:1999mnras}, within the precision of halo-finder codes \\citep[e.g.][]{knebe:2013arxiv}.\n\nFrom the catalogue we select a halo with DM mass $M_{\\rm h} \\simeq 10^{11}\/{\\rm h}\\,{\\rm M}_{\\odot}$ (resolved by $\\simeq5\\times10^{4}$ DM particles), whose virial radius is $r_{\\rm vir}\\simeq 15\\,{\\rm kpc}$ at $z=6$. Using \\textlcsc{hop} we select the minimum ellipsoid enveloping $10\\,r_{\\rm vir}$, and trace it back to $z=100$. As noted in \\citet[][]{onorbe:2014mnras}, this is usually sufficient to avoid contamination\\footnote{A posteriori, we have checked that the halos in the zoom-in region have a contamination level $\\lsim0.1\\%$.}. At $z=100$ the trace back volume is $V^{\\rm zoom}\\simeq(2.1\\,{\\rm Mpc}\/{\\rm h})^{3}$. Using \\textlcsc{music} we recalculate the ICs, by generating 3 additional level of refinement. For such multi-mass set-up, the finer DM resolution is $m_{dm}^{\\rm zoom} = 6.7\\times 10^{4} \/{\\rm h}\\,{\\rm M}_{\\odot}$, that corresponds to a spatial resolution of $\\Delta x^{\\rm zoom} = 9.7 \/{\\rm h}\\,{\\rm kpc}$. We note that because of the traced back volume, our simulation is expected to probe not only the target halo, but also its satellites and environment, similar to other works (e.g. \\citealt{fiacconi:2015}, where the target halo is chosen at $z\\simeq 3$).\n\nIn the zoom-in simulation $\\Delta x^{\\rm zoom}$ corresponds to our coarse grid resolution, and we allow for 6 additional refinement levels, based on a Lagrangian mass threshold-based criterion. At $z=6$, the baryonic component of the selected halo has a mass resolution of $m_{b} = 1.8\\times 10^{4}{\\rm M}_{\\odot}$ and a physical resolution of $\\Delta x^{\\rm min} = 31.9\\,{\\rm pc}$. For convenience, a summary of the resolution outline can be found in Tab. \\ref{tagella_res}. Note that the refined cell of our simulations have mass and size typical of molecular clouds \\citep[MC, e.g.][]{gorti:2002apj,federrath:2013}.\n\nIn the present paper we refer to metallicity ($Z$) as the sum of all the heavy element species without differentiating among them, and assume solar abundance ratios \\citep{asplund:2009ara&a}. In the IC, the gas is characterized by a mean molecular weight $\\mu = 0.59$, and has metallicity floor $Z=Z_{\\rm floor}>0$. The metallicity floor mimics the pre-enrichment of the halo at high-$z$, when we do not have the resolution to follow precisely star formation and gas enrichment. We set $Z_{\\rm floor}=10^{-3}{\\rm Z}_{\\odot}$, a level that is compatible with the metallicity found at high-$z$ in cosmological simulations for diffuse enriched gas \\citep{dave:2011mnras,pallottini:2014_sim,maio:2015}. Note that such low metallicity only marginally affects the gas cooling time, but is above the critical metallicity for formation of Population III stars. Additionally, a posteriori, we have found that the metallicity floor contribute for only $\\lsim 0.2\\%$ of the total metal mass produced by stars by $z=6$ in the refined region.\n\n\\subsection{Star formation model}\\label{sec_model_sf}\n\nWe model star formation (SF) by assuming a ${\\rm {H_2}}$~dependent Schmidt-Kennicutt relation \\citep{schmidt:1959apj,kennicutt:1998apj}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/kmt09_test.pdf}\n\\caption{\n${\\rm {H_2}}$~fraction ($f_{\\rm H2}$) as a function of gas density ($n$) obtained using the \\citetalias{krumholz:2009apj} model (eqs. \\ref{eq_fh2_full}). Different solid lines correspond to different metallicity ($Z$) of the gas. Horizontal dotted grey lines mark $f_{\\rm H2}$ values of $0.5$ and $1$. Vertical dashed lines indicate the critical density $n_{c}$ where $f_{\\rm H2}=0.5$ for different $Z$; these critical density values are obtained as a fit (eq. \\ref{eq_critical_density}) to the \\citetalias{krumholz:2009apj} model (eq. \\ref{eq_fh2_full}). See the text for details.\n\\label{fig_kmt_test}}\n\\end{figure}\n\\begin{subequations}\\label{eq_sfr_tot}\n\\be\\label{eq_sfr1}\n\\dot{\\rho}_{\\star}=f_{\\rm H2} \\rho\/t_{\\rm sf}\\,\n\\ee\nwhere $\\dot{\\rho}_{\\star}$ is the local SF rate ($SFR$) density, $f_{\\rm H2}$ the molecular hydrogen fraction, $\\rho$ the gas density and $t_{\\rm sf}$ the SF time scale. In eq. \\ref{eq_sfr1} we assume the SF time scale to be proportional to the free-fall time, i.e.\n\n\\be\\label{eq_sfr2}\nt_{\\rm sf} = \\zeta_{\\rm sf}^{-1} \\sqrt{3\\pi\/(32\\,G\\rho)} \\,,\n\\ee\n\\end{subequations}\nwhere $\\zeta_{\\rm sf}$ describes the SF efficiency and it is treated as a parameter in the present work \\citep[cf.][see discussion in Sec. \\ref{sez_sfr_efficiency}]{semenov:2015}. To calculate $f_{\\rm H2}$ we adopt the \\citetalias{krumholz:2009apj} model \\citep{krumholz:2008apj,krumholz:2009apj,mckee:2010apj}. Such model considers ${\\rm {H_2}}$~formation on dust grains by computing radiative transfer on a idealized MC and assumes equilibrium between formation and dissociation rate of ${\\rm {H_2}}$. The solution for $f_{\\rm H2}$ can be approximated as\n\n\\begin{subequations}\\label{eq_fh2_full}\n\\begin{align}\nf_{\\rm H2} &= \\left(1 -0.75\\,s\/(1+0.25\\,s) \\right)\\Theta(2-s)\\\\\ns          &= \\ln\\left(1+0.6\\,\\chi +0.01\\chi^{2}\\right) \/0.6\\,\\tau_{\\rm uv}\\label{eq_dust_optical_depth}\\\\\n\\chi       &= 71\\, \\left(\\sigma_{d,21}\/\\mathcal{R}_{-16.5}\\right)\\,\\left((G\/G_{0})\/(n\/{\\rm cm}^{-3})\\right)\\,,\\label{eq_chi_full}\n\\end{align}\nwhere $\\Theta$ is the Heaviside function, $\\tau_{\\rm uv}$ the dust optical depth of the cloud, $\\sigma_{d}^{-21}=\\sigma_{d}\/10^{-21}{\\rm cm}^{-2}$ is the dust absorption cross section \\citep{li_draine:2001apj}, $\\mathcal{R}\/10^{-16.5}{\\rm cm}^{3}\\,{\\rm s}^{-1} $ is the formation rate coefficient of ${\\rm {H_2}}$~on dust grains \\citep{wolfire:2008apj}, $G$ is the FUV flux in the Habing band ($6-13.6\\,{\\rm eV}$) normalized to the average Milky Way (MW) value $G_{0}$ \\citep{habing:1968,draine:1978apjs}, and $n$ is the hydrogen number density. As in \\citetalias{krumholz:2009apj}, we calculate the dust optical depth by linearly rescaling the MW value, i.e. $\\tau_{\\rm uv} = 10^{-21}{\\rm cm}^{-2} N_{H}\\, Z\/{\\rm Z}_{\\odot} \/\\mu$, where $N_{H}$ is the hydrogen column density and $\\mu$ the mean molecular weight. In the simulation, the column density is calculated as $N_{H}= n\\,l_{\\rm cell}$; because of the mass threshold-based criterion used as a refinement in AMR, we expect $l_{\\rm cell} \\propto n^{-1\/3}$, thus $N_{H} \\propto n^{2\/3}$.\n\nNote that both $\\sigma_{d}$ and $\\mathcal{R}$ are proportional to the dust mass, that we assume to be proportional to the metallicity. Then the ratio between $\\sigma_{d}$ and $\\mathcal{R}$ is independent of $Z$. Additionally, eq. \\ref{eq_fh2_full} can be simplified by assuming pressure equilibrium between the CNM and WNM. In this case, eq. \\ref{eq_chi_full} turns out to be independent on $G\/G_{0}$ and can be written as \\citep{krumholz:2009apj}\n\\be\\label{eq_sfr_last}\n\\chi = 0.75\\,\\left(1+3.1\\,(Z\/{\\rm Z}_{\\odot})^{0.365}\\right)\\,.\n\\ee \n\\end{subequations}\nAs shown in \\citep{krumholz:2011apj}, for $Z\\gsim10^{-2}{\\rm Z}_{\\odot}$ such approximation gives ${\\rm {H_2}}$~fractions compatible with those resulting from a full non-equilibrium radiative transfer calculations.\n\nIn Fig. \\ref{fig_kmt_test} we plot $f_{\\rm H2}$ from the \\citetalias{krumholz:2009apj} model as a function of the gas density. Different solid lines refer to different metallicity. At a fixed metallicity, the molecular fraction as a function of density vanishes for low values of $n$; it steeply rises up to $f_{\\rm H2} \\sim 0.8$ in one density dex and asymptotically reaches $f_{\\rm H2} = 1$. The critical density where the gas can be considered molecular ($f_{\\rm H2}=0.5$) is roughly inversely proportional to the metallicity, i.e. $n_{c} \\sim 25 (Z\/{\\rm Z}_{\\odot})^{-1}{\\rm cm}^{-3}$ \\citep[see also][]{agertz:2012arxiv}. We note that when detailed chemistry calculations are performed, such critical density depends on the chemical network and the assumptions regarding gas shielding from external radiation and clumpiness. As a consequence, the actual critical density can be higher that the one predicted by the \\citetalias{krumholz:2009apj} model \\citep[e.g.][]{bovino:2015arxiv}.\n\nBecause of the particular shape of the $f_{\\rm H2}(n)$ relation, the adopted SF law (eqs. \\ref{eq_sfr_tot}--\\ref{eq_fh2_full}) is roughly equivalent to a prescription based on a density threshold criterion:\n\\begin{subequations}\\label{eqs_sfr_equivalence}\n\\be\n\\dot{\\rho}_{\\star}=\\Theta(n - n_{c}) m_p\\,n \/t_{\\rm sf}\\,,\n\\ee\nwhere $m_p$ is the proton mass and the critical density\n\\be\\label{eq_critical_density}\nn_{c} \\simeq 26.45 \\, (Z\/{\\rm Z}_{\\odot})^{-0.87} {\\rm cm}^{-3}\\,\n\\ee\n\\end{subequations}\nis calculated as a fit to the $f_{\\rm H2}$ \\citetalias{krumholz:2009apj} model. In Fig. \\ref{fig_kmt_test}, we show $n_c$ for various metallicities (dashed vertical lines).\n\nEqs. \\ref{eqs_sfr_equivalence} are not used to calculate the $SFR$ in the simulation. However, being simpler, such formulation can be used to enhance our physical intuition of the adopted SF law\\footnote{As a consequence of the rough equivalence, it is not necessary to manually prevent SF in underdense regions, by imposing that an overdensity $\\Delta>200$ is needed to form stars. At the start of the simulation ($z=100$), the mean density of the gas is $\\sim 0.1\\,m_p\\,{\\rm cm}^{-3}$, while the \\quotes{effective} SF threshold would be $n_{c} \\sim 10^4{\\rm cm}^{-3}$ for gas at $Z=Z_{\\rm floor}$.\\label{footnote_sfr_equivalence}} in analyzing the results. As noted in \\citet[][]{hopkins:2013arxiv}, the morphology of a galaxy is very sensitive to the minimum density of the cells that are able to form star.\n\nDuring the simulation, eqs. \\ref{eq_sfr_tot} are solved stochastically, by drawing the mass of the new star particles from a Poisson distribution \\citep{rasera:2006,dubois:2008,pallottini:2014_sim}. We impose that no more than half of a cell mass can be converted into a star particle in each event. This prescription ensures the numerical stability of the code \\citep{dubois:2008}. This is also consistent with the picture that nearly half of the mass in a MC is Jeans unstable \\citep{federrath:2013}.\n\nWe allow SF only if the mass of a new star particle is at least equal to the baryon mass resolution. This avoids numerical errors for the star particle dynamics and enables us to treat the particle as a stellar population with a well sampled initial mass function (IMF). Additionally, the SF law is driven by ${\\rm {H_2}}$~formation on dust grains, we do not allow gas to form stars if the dust temperature is larger than $\\simeq2\\times 10^{3}$, because of dust sublimation (see Sec. \\ref{sezione_blast} and App. \\ref{app_rad_press} for the details on the dust prescriptions).\n\nFor the present work we assume a SF efficiency $\\zeta_{\\rm sf}=10\\%$, in accordance with the average values inferred from MC observations \\citep[][see also \\citealt{agertz:2012arxiv}]{murray:2011apj}. Note that varying the parameters for the SF law should lead to similar $SFR$ once feedback are properly included, although the galaxy morphology can be different \\citep{hopkins:2013arxiv}.\n\n\\subsection{Mass and energy inputs from stars}\\label{sec_stellar_inputs}\n\nBecause of the finite mass resolution, it is necessary to introduce (according to eqs. \\ref{eq_sfr_tot}--\\ref{eq_sfr_last}) ``star particles'' to represent stellar populations. To this aim, we adopt a \\citet{kroupa:2001} IMF\n\\begin{subequations}\n\\begin{align}\n\\Phi(m)\\propto & \\left[m^{-\\alpha_{1}} \\Theta(m_{1}-m)\\right.\\label{eq_imf}\\\\\n+& \\left. m^{-\\alpha_{2}} \\Theta(m-m_{1}) m_{1}^{\\alpha_{2}-\\alpha_{1}} \\right]\\,,\\nonumber\n\\end{align}\nwhere $\\alpha_{1}= 1.3$, $\\alpha_{2}= 2.3$, $m_{1} = 0.5\\,{\\rm M}_{\\odot}$, and $m$ is in the range $[10^{-1}-10^{2}]{\\rm M}_{\\odot}$. The proportionality constant is chosen such that\n\\be\n\\int_{ 0.1\\,{\\rm M}_{\\odot}}^{100\\,{\\rm M}_{\\odot}} m\\Phi\\,{\\rm d}m=1\\, .\n\\ee\n\\end{subequations}\n\nOnce formed, stars affect the environment with chemical, mechanical and radiative feedback. These stellar inputs are parameterized by the cumulative fraction of the returned gas mass, metals and energy \\citep[e.g.][]{salvadori:2008mnras,debennassuti2014mnras,salvadori:2015}. Mass and energy inputs are conveniently expressed per unit stellar mass formed ($M_{\\star}$).\n\nChemical feedback depends on the return fraction ($R$) and the yield ($Y$):\n\\begin{subequations}\\label{eqs_stellar_inputs}\n\\begin{align}\\label{eqs_def_R_Y}\n  R(t_{\\star})\t=&\\int_{m(t_{\\star}) }^{100\\,{\\rm M}_{\\odot}} (m-w) \\Phi\\,{\\rm d}m\\\\\n  Y(t_{\\star})\t=&\\int_{m(t_{\\star}) }^{100\\,{\\rm M}_{\\odot}} m_{Z} \\Phi\\,{\\rm d}m\\,,\n\\end{align}\nwhere $w(m,Z_{\\star})$ and $m_{Z}(m,Z_{\\star})$ are the stellar remnant and the metal mass produced for a star of mass $m$ and metallicity $Z_{\\star}$ \\citep[e.g.][]{woosley:1995apjs,vandenhoek:1997a&as}, and $m(t_{\\star})$ is the minimum stellar mass with lifetime\\footnote{Stellar lifetimes are roughly independent of metallicity for $Z_{\\star}>10^{-4}{\\rm Z}_{\\odot}$ \\citep[][see eq. 3]{raiteri:1996eq3}.} shorter than $t_{\\star}$, the time elapsed from the creation of the stellar particle (i.e. the \\quotes{burst age}).\n\nThis approach is used both in zoom galaxy simulations \\citep[e.g.][]{agora:2013arxiv} and cosmological simulations \\citep[e.g.][hereafter \\citetalias{pallottini:2014_sim}]{pallottini:2014_sim}. Compared to cosmological simulations, though, zoom simulations have typically a better spatial and -- consequently -- time resolution (e.g. $\\Delta t\\sim 10^{-2}\\,\\rm Myr$ vs $\\Delta t\\sim \\rm Myr$). Thus, here we can follow the gradual release of both gas and metals in the ISM.\n\nThe mechanical energy input includes SN explosions and winds, either by OB or AGB stars in young ($< 40\\,\\rm Myr$) or evolved stellar populations:\n\\begin{align}\\label{eqs_def_mec_energy}\n  \\epsilon_{\\rm sn}(t_{\\star}) =&\\int_{m(t_{\\star})>8\\,{\\rm M}_{\\odot}}^{40\\,{\\rm M}_{\\odot} }\te_{\\rm sn}\\Phi\\,{\\rm d}m,\\\\\n  \\epsilon_{\\rm w}(t_{\\star})  =&\\int_{m(t_{\\star})}^{100\\,{\\rm M}_{\\odot} }\t \te_{\\rm w}\\Phi\\,{\\rm d}m\\,,\n\\end{align}\nwhere $e_{\\rm sn}=e_{\\rm sn}(m,Z)$ and $e_{\\rm w}=e_{\\rm w}(m,Z)$ are the energy released by SN and stellar winds in units of $10^{51}{\\rm erg}\\equiv{\\rm 1 foe}$; we have further assumed that only stars with $8 \\leq m\/{\\rm M}_{\\odot}\\leq40$ can explode as SN.\n\nRadiative energy inputs can be treated within a similar formalism. The cumulative energy $\\epsilon_{12}$ associated to the spectral range $(\\lambda_{1}, \\lambda_{2})$ can be written as\n\\begin{align}\\label{eqs_def_rad_energy}\n  \\epsilon_{\\rm 12}(t_{\\star}) \t=&\\int_0^{t_{\\star}}\\int_{m(t)}^{100\\,{\\rm M}_{\\odot} } L_{12}\\Phi\\,{\\rm d}m\\,{\\rm d}t\\\\\n  L_{\\rm 12}(t) \t\t=& \\int_{\\lambda_{1}}^{\\lambda_{2}}L_{\\lambda}{\\rm d}{\\lambda}\\,,\n\\end{align}\n\\end{subequations}\nwhere $L_{\\lambda}=L_{\\lambda}(m,Z_{\\star})$ is the luminosity per unit wavelength and mass. For convenience, we express the radiation energy in units of ${\\rm foe}$, as for the mechanical energy (eqs. \\ref{eqs_def_mec_energy}). In the following we specify $\\epsilon_{\\rm 12}$ in eq. \\ref{eqs_def_rad_energy}, by separately considering ionizing radiation ($\\lambda_{1}=0$, $\\lambda_{2}=912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$) denoted by $\\epsilon_{\\rm ion}$, and the soft UV band, $\\epsilon_{\\rm uv}$, defined as the range ($\\lambda_{1}=912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$, $\\lambda_{2}=4000\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$).\n\nIn eqs. \\ref{eqs_stellar_inputs}, the quantities $w$, $m_{Z}$, $e_{\\rm sn}$, $e_{\\rm w}$, and $L_{\\lambda}$ can be calculated from stellar evolutionary models. We adopt the {\\tt padova} \\citep{padova:1994} stellar tracks for metallicities $Z_{\\star}\/{\\rm Z}_{\\odot} = 0.02,\\, 0.2,\\, 0.4,{\\rm and}\\, 1$ to compute the chemical\\footnote{Similarly to \\citet{agora:2013arxiv}, when computing the yields in eq. \\ref{eqs_def_R_Y}, we assume that the metal mass is linked to the oxygen and iron masses via $m_{Z}= 2.09\\,m_{\\rm O} + 1.06\\,m_{\\rm Fe}$, as appropriate for \\citet{asplund:2009ara&a} abundances.}, mechanical and radiative inputs using \\textlcsc{starburst99} \\citep{starburst99:1999,starburst99:2010apjs}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/stellar_inputs.pdf}\n\\caption{\nStellar inputs (cumulative fraction) as a function of stellar age ($t_{\\star}$). Shown are the return fraction ($R$), metal yield ($Y$), SN mechanical energy ($\\epsilon_{\\rm sn}$), wind mechanical energy ($\\epsilon_{\\rm w}$), ionizing radiation energy ($\\epsilon_{\\rm ion}$), and UV radiation energy ($\\epsilon_{\\rm uv}$).\nThe fractions are given per unit stellar mass formed; energies are expressed in units of $10^{51}{\\rm erg}\\equiv{\\rm foe}$.\nCumulative fractions are indicated with a different colours, as indicated in the legend: the shaded regions cover the $0.02\\leq Z_{\\star}\/{\\rm Z}_{\\odot}\\leq1$ metallicity range; dark lines denote single metallicity {\\tt padova} stellar tracks \\citep{padova:1994}.\nTo guide the eye, the SN explosion period is bracketed by vertical dashed lines; in the upper axis we report the value of $m(t_{\\star})$, the minimum stellar mass corresponding to the stellar lifetime $t_{\\star}$. For definitions, see eqs. \\ref{eqs_stellar_inputs}.\n\\label{fig_gamete_tables}}\n\\end{figure}\n\nIn Fig. \\ref{fig_gamete_tables} we plot $R$, $Y$, $\\epsilon_{\\rm sn}$, $\\epsilon_{\\rm w}$, $\\epsilon_{\\rm ion}$ and $\\epsilon_{\\rm uv}$ as a function of $t_{\\star}$. For each curve the shaded regions denote the $0.02\\leq Z_{\\star}\/{\\rm Z}_{\\odot}\\leq1$ metallicity range; single $Z_{\\star}$ tracks are indicated with dark lines. The time interval during which massive stars can explode as SN ($0.8 \\lsim \\log t_{\\star}\/\\rm Myr\\lsim 1.6$) is highlighted with vertical dashed lines, and the upper axis is labelled with the corresponding stellar mass.\n\nNote that the OB stars contribution ($\\log t_{\\star}\/\\rm Myr\\lsim 0.8$) to $\\epsilon_{\\rm w}$, $Y$ and $R$ is roughly proportional to $t_{\\star}$ and $Z_{\\star}$ (see also \\citealt{agertz:2012arxiv}, in particular eqs. 4). As in the simulation the metallicity floor is set to $Z_{\\rm floor}=10^{-3}{\\rm Z}_{\\odot}$, we slightly overestimate the wind contribution for low $Z_{\\star}$.\n\nFinally, note that the change of behavior of $\\epsilon_{\\rm uv}$ at $\\log t_{\\star}\/\\rm Myr\\lsim 2$ is due to the ionizing ($\\lambda\\leq\\,912\\,\\textrm{A\\kern -1.3ex\\raisebox{0.6ex}{$^\\circ$}}$) photon production suppression. At late times ($\\log t_{\\star}\/\\rm Myr\\gsim 1.6$), AGB stars give a negligible mechanical energy contribution ($\\epsilon_{\\rm w}\\simeq{\\rm constant}$) but return mass and metals to the gas ($R$, $Y$).\n\n\\subsection{Stellar feedback}\\label{sezione_blast}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/feedback_fractions.pdf}\n\\caption{\nExample of the adopted feedback model. Fractional energy evolution for a single SN explosion ($E_{0}=1\\,{\\rm foe}$) in a gas characterized by $n=1\\,{\\rm cm}^{-3}$ and $Z=10^{-3}{\\rm Z}_{\\odot}$ as a function of the time interval from the explosion $\\Delta t$. We plot the total ($f=f_{\\rm th}+f_{\\rm kn}$), thermal ($f_{\\rm th}$), and kinetic ($f_{\\rm th}$) energy fraction acquired by the gas (see eq. \\ref{sn_energy_gas_equation}) with solid black, dashed red and dotted blue lines, respectively.\nShaded regions indicate different stages of the SN evolution, i.e. the energy conserving Sedov-Taylor (ST) stage, shell formation (SF) stage, and pressure driven snowplow (PDS).\nIn the adopted formalism, the initial energy $E_{0}$ is a function of the stellar input (see Sec. \\ref{sec_stellar_inputs} and eq. \\ref{eqs_def_mec_energy}), e.g. $E_{0}=[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})] M_{\\star}$. The full model is presented in App. \\ref{app_blastwave}.\n\\label{fig_blastwave_sketch}}\n\\end{figure}\n\nEqs. \\ref{eqs_stellar_inputs} provide us with the energy produced by stars in different forms. The next step is to understand what fraction of that energy is eventually deposited in the ISM. Consider a stellar population of initial mass $M_{\\star}$, metallicity $Z_{\\star}$ and age $t_{\\star}$ residing in a gas cell with volume $V_{\\rm cell}$. In our scheme, when the simulation evolves for a time $\\Delta t$, the chemical feedback act as follows:\n\n\\begin{subequations}\n\\begin{align}\n  \\rho \t&= \\rho + \\left[R(t_{\\star}+\\Delta t)-R(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\\\\n  Z    \t&= Z \t+ \\left[Y(t_{\\star}+\\Delta t)-Y(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\,,\n\\end{align}\nwhere $\\rho$ and $Z$ are the the gas density and metallicity and $R$ and $Y$ are taken from eqs. \\ref{eqs_def_R_Y}. Note that chemical enrichment is due both to the SN and AGB winds.\n\n\\subsubsection{Supernova explosions}\n\nFor the mechanical feedback, let us first consider the case of SNe. At each SN event the specific energy of the gas changes as\n\n\\begin{align}\\label{sn_energy_gas_equation}\n e_{\\rm th} &= e_{\\rm th}+ f_{\\rm th} \\left[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\\\\n e_{\\rm nth}&= e_{\\rm nth}+ f_{\\rm kn} \\left[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})\\right] M_{\\star}\/V_{\\rm cell}\\,,\n\\end{align}\n\\end{subequations}\nwhere $e_{\\rm th}$ and $e_{\\rm nth}$ are the thermal and non-thermal energy densities, and $f_{\\rm th}$ and $f_{\\rm kn}$ are the fractions of thermal and kinetic energy deposited in the ISM. Thus, $e_{\\rm nth}$ accounts for the momentum injection by SN and $e_{\\rm th}$ for the thermal pressure part.\n\nIn the present work, we have developed a novel method to compute such quantities. The method derives $f_{\\rm th}$ and $f_{\\rm kn}$ from a detailed modelling of the subgrid blastwave evolution produced by the SN explosion. We calculate $f_{\\rm th}$ and $f_{\\rm kn}$ by evaluating the shock evolution at time $\\Delta t$, the time step of the simulation\\footnote{The underlying assumption is that the shock fronts exit the cell in $\\lsim\\Delta t$. This is quite consistent because the shock is expected to be supersonic, and the sound crossing time is larger or comparable with the simulation time step $\\Delta t$, dictated by the Courant-Friedrichs-Lewy conditions.}.\n\nThe adopted blastwave model is based on \\citet[][hereafter \\citetalias{ostriker:1988rvmp}]{ostriker:1988rvmp}, and it accounts for the evolution of the blast through its different evolutionary stages (energy conserving, momentum conserving, etc.). While each stage is self-similar, the passage from one stage to the next is determined by the cooling time. Thus, $f_{\\rm th}$ and $f_{\\rm kn}$ depends on the blastwave evolutionary stage. The latter, in turn depends on the gas density, cooling time, and the initial energy of the blast ($E_{0}=[\\epsilon_{\\rm sn}(t_{\\star}+\\Delta t)-\\epsilon_{\\rm sn}(t_{\\star})] M_{\\star}$, in eq. \\ref{eqs_def_mec_energy}).\n\nThe model details are presented in App. \\ref{app_blastwave}. As an example, in Fig. \\ref{fig_blastwave_sketch}, we show the energy evolution for a single SN explosion ($E_{0}=1\\,{\\rm foe}$) in a gas characterized by $n=1\\,{\\rm cm}^{-3}$ and $Z=10^{-3}{\\rm Z}_{\\odot}$. The total energy $E(t)$ is constant in the Sedov-Taylor (ST) stage, it decrease down to $0.5\\,E_{0}$ during the shell formation (SF) stage, and it evolves as $\\Delta t^{-2\/7}$ in the pressure driven snowplow (PDS) stage (see eq. \\ref{eq_energy_shock}). In the ST stage most of the energy is thermal, i.e. $f_{\\rm kn}\/f_{\\rm th}\\simeq 0.4$; however, in the SF stage $f_{\\rm kn}$ increases, since part of the thermal energy is radiated away and some is converted into kinetic form\\citep[e.g.][]{cox:1972apj,cioffi:1988apj}. Finally, during the PDS stage the ratio of thermal to kinetic is $f_{\\rm kn}\/f_{\\rm th}\\simeq 2$ (see eqs. 6.14 in \\citetalias{ostriker:1988rvmp}).\n\nIn this particular example -- a $1\\,{\\rm foe}$ SN exploding in a $n=1\\,{\\rm cm}^{-3}$ cell -- by assuming a simulation time step of $\\Delta t\\simeq 10^{-2}\\rm Myr$, we find that the blastwave is in the PDS stage, and the gas receives (via eqs. \\ref{sn_energy_gas_equation}) a fraction of energy $f_{\\rm th} \\simeq 8\\%$ and $f_{\\rm kn} \\simeq 16\\%$ in thermal and kinetic form, respectively. During $\\Delta t$, about $\\simeq 75$\\% of the initial SN energy has been either radiated away or lost to work done by the blastwave to exit the cell. The model is in broad agreement with other more specific numerical studies \\citep[e.g.][]{cioffi:1988apj,walch:2015mnras,martizzi:2015mnras}.\n\n\\subsubsection{Stellar winds}\n\nStellar winds are implemented in a manner paralleling the above scheme for SNe. The energy variation can be calculated via eq. \\ref{eqs_def_mec_energy}, where $\\epsilon_{\\rm sn}$ is substituted with $\\epsilon_{\\rm w}$, given in eqs. \\ref{sn_energy_gas_equation}. Then, $f_{\\rm th}$ and $f_{\\rm kn}$ for winds are calculated via a stage scheme similar to SN. The main difference in the efficiency factors calculation depends on the mode of energy production, i.e. impulsive for SNe, continuous for winds. The complete scheme is detailed in App. \\ref{app_blastwave}.\n\nThe efficiency of SN is greatly increased when the gas is pre-processed by stellar winds \\citep{walch:2015mnras,fierlinger:2016}, since the energy loss process is highly non-linear \\citep[][see Fig. 8]{fierlinger:2016}. For example, when a SN explodes in the lower density bubble produced by the stellar progenitor wind, the adiabatic phase lasts longer and consequently $f_{\\rm kn}$ and $f_{\\rm th}$ increase considerably. \n\n\\subsubsection{Radiation pressure}\\label{sec_rad_press}\n\nFinally, we account for radiation pressure from stars. The coupling of the gas with the radiation can be expressed in terms of $\\dot{p}_{rad}$, the rate of momentum injection \\citep{krumholz:2009radpress,hopkins:2011mnras,krumholz:2012radpress,wise:2012radpres,agertz:2012arxiv}, and accounts for the contribution from ionization, and from dust UV heating and IR-trapping\n\n\\begin{subequations}\n\\begin{align}\\label{eq_rad_moment_injection}\n\\dot{p}_{rad} =&  (L_{\\rm ion}\/c)(1-\\exp(-\\tau_{\\rm ion})) \\\\\n              +&   (L_{\\rm uv}\/c)((1-\\exp(-\\tau_{\\rm uv})) +f_{\\rm ir} )\\,,\\nonumber \n\\end{align}\nwhere $c$ is the speed of light, $\\tau_{\\rm ion}$ the hydrogen optical depth to ionizing radiation, and $f_{\\rm ir}$ is the term accounting for the IR-trapping. $L_{\\rm ion}$ and $L_{\\rm uv}$ are calculated by integration of the stellar tracks (eqs \\ref{eqs_def_rad_energy}). The calculation of $\\tau_{\\rm uv}$ is modelled in Sec. \\ref{sec_model_sf} (eq. \\ref{eq_dust_optical_depth} and related text). We compute $\\tau_{\\rm ion}$ and $f_{\\rm ir}$ according to the physical properties of the gas, as detailed in App. \\ref{app_rad_press}. Note that we do not assume, as sometimes done, $\\tau_{\\rm ion}\\sim \\tau_{\\rm uv}\\gg1$, i.e. we allow for the possibility that some LyC photons can escape.\n\nIn smoothed particle hydrodynamics (SPH) codes, radiation pressure (eq. \\ref{eq_rad_moment_injection}) can be implemented as a \\quotes{kick} \\citep[e.g.][]{hopkins:2011mnras,barai:2015mnras}. Namely, a velocity $\\Delta v = \\dot{p}_{rad}\\Delta t\/m_b$ is directly added to some of the SPH particles of mass $m_b$ near the photon source. The particles that receive kicks are statistically chosen according to a probability $\\mathcal{P}_{\\rm kick}$, and with kick direction $\\hat{v}$ that is sampled from a random distribution. Considering the specific kinetic energy of the SPH particles, we would have\n\\be\ne_k = 0.5\\,\\left\\langle m_b (\\mathbf{v} + \\Delta v \\mathcal{P}_{\\rm kick}\\mathbf{\\hat{v}} )^{2} \\right\\rangle\/V_{cell}\n\\ee\nwhere $\\mathbf{v}$ is the original particle velocity, the $\\langle\\,\\rangle$ operator indicates the particles sum weighted by the SPH kernel, and $V_{cell}$ is the kernel volume. Thus, because of the kick, the increase of energy density would be\\footnote{In eq. \\ref{eq_red_press_energy_increase}, when going from the first to the second line, note that first terms gives a null contribution, as $\\mathbf{v}$ is ordered motion, while the kicks are randomly oriented via $\\mathbf{\\hat{v}}$, and that, by definition, $\\langle \\mathcal{P}_{\\rm kick}\\rangle = 1$.}\n\n\\begin{align}\\label{eq_red_press_energy_increase}\n\\Delta e_k &= \\langle m_b\\, \\mathcal{P}_{\\rm kick} (\\Delta v\\mathbf{v}\\mathbf{\\hat{v}} + 0.5(\\Delta v)^{2} \\rangle\/V_{cell} \\nonumber\\\\\n           &= 0.5\\, m_b \\, (\\Delta v)^2\/V_{cell} \\\\\n           &= 0.5\\,(\\dot{p}_{rad}\\Delta t)^2\/(m_b\\,V_{cell}) \\nonumber\\,,\n\\end{align}\n\\end{subequations}\nwhere $\\dot{p}_{rad}$ can be calculated via eq. \\ref{eq_rad_moment_injection}, and eq. \\ref{eq_red_press_energy_increase} can be directly cast into the AMR formalism. Additionally, because of our approximate treatment of IR-trapping (see App. \\ref{app_rad_press}), we force energy conservation: $V_{cell} \\Delta e_k \\leq \\Delta t\\,(L_{\\rm ion} + L_{\\rm uv})$, i.e. the deposited energy must not exceed the radiative input energy. Finally, we recall here that non-thermal energy is dissipated with a time scale $t_{\\rm diss}$, as described in the beginning of Sec. \\ref{sec_numerical}.\n\n\\section{Results}\\label{sec_result}\n\nAt $z=6$ ($t \\simeq 920\\, \\rm Myr$), the simulated zoom-in region contains a group of 15 DM haloes that host galaxies. We target the most massive halo ($M_{\\rm h} = 1.8\\times 10^{11}{\\rm M}_{\\odot}$) that hosts \\quotes{\\emph{Dahlia}}, which is a galaxy characterized by a stellar mass of $M_{\\star}=1.6\\times 10^{10}{\\rm M}_{\\odot}$, therefore representative of a typical LBG galaxy at that epoch. {Dahlia} has 14 satellites located within $\\simeq 100 \\,{\\rm kpc}$ from its centre. The six largest ones have a DM mass in the range $M_{\\rm h} = 2.5\\times 10^{9}{\\rm M}_{\\odot} - 1.2\\times 10^{10}{\\rm M}_{\\odot}$, and they host stars with total mass $M_{\\star}\\lsim 10^{9}{\\rm M}_{\\odot}$. Additionally, there are eight smaller satellites ($M_{\\rm h} \\simeq 10^{7}{\\rm M}_{\\odot}$), with $M_{\\star}\\simeq 10^{5}{\\rm M}_{\\odot}$.\n\n\\subsection{Overview}\\label{sec_res_barions}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtdensmap_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rttemp_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtpress_27.pdf}\n\\vspace{.3pt}\n\n\\includegraphics[width=0.99\\textwidth,height=0.33\\textwidth]{plots_pdf\/maps\/landscape_completo_type3_zoomed_rtmetal_27.pdf}\n\\caption{\n(Caption next page.) %\n\\label{fig_mappe_hydro}\n}\n\\end{figure*}\n\\addtocounter{figure}{-1}\n\\begin{figure*}\n\\caption{(Previous page.) %\nMaps of the simulated galaxy {Dahlia} $z=6$. From left to right we plot subsequent zooms on the galaxy. From top to bottom we plot the density ($n$), temperature ($T$), pressure ($P$) and metallicity ($Z$). Each map is obtained\\textsuperscript{\\ref{footnote_pymses}} by mass-averaging the physical quantity along the line of sight piercing the field of view and centred on {Dahlia}. In all panels the physical scale is indicated as an inset. Movies of {Dahlia} can be found at \\url{https:\/\/www.researchgate.net\/profile\/Andrea_Pallottini}.\n}\n\\end{figure*}\n\nWe start by looking at the overall properties of {Dahlia} on decreasing scales. In the following we refer to Fig. \\ref{fig_mappe_hydro}, which shows the simulated density ($n$), temperature ($T$), total (thermal+kinetic) pressure ($P$), and metallicity ($Z$) maps\\footnote{Most of the maps of this paper are obtained with a customized version of \\textlcsc{pymses} \\citep{labadens:2012aspc}, a visualization software that implements optimized techniques for the AMR grid of \\textlcsc{ramses}.\\label{footnote_pymses}}\nat $z=6$.\n\n\\subsubsection{Environment (scale $\\simeq 160$~kpc)}\n\n{Dahlia} sits at the centre of a cosmic web knot and accretes mass from the intergalactic medium (IGM) mainly via 3 filaments of length $\\simeq 100\\,{\\rm kpc}$, confirming previous findings \\citep[][]{dekel:2009nat}. These overdense filaments ($n\\simeq 10^{-2}{\\rm cm}^{-3}$) are slightly colder ($T\\simeq10^{3.5}{\\rm K}$) than the IGM ($\\langle T \\rangle\\simeq10^{4.5}{\\rm K}$) as a consequence of their shorter radiative cooling time ($t_{\\rm cool}\\propto n^{-1}$). Along these cold streams, pockets of shock-heated ($T\\gsim10^{4.5}{\\rm K}$) gas produced by both structure formation and feedback (SN and winds) are visible.\n\nThe galaxy locations can be pinpointed from the metallicity map, showing a dozen of metal-polluted regions. The size of the metal bubbles ranges from $\\simeq20$ kpc in the case of {Dahlia} to a few ${\\rm kpc}$ for the satellites. Bubble sizes increase with the total stellar mass (see \\citetalias{pallottini:2014_sim}, in particular Fig. 13), and age of the galaxy stellar population. \n\nOn these scales, the pressure is dominated by the thermal component ($P\\simeq P_{\\rm th}\\sim 10^4{\\rm K}\\,{\\rm cm}^{-3}$); higher values of pressure, associated to non-thermal feedback effects (e.g. gas bulk motion), are confined around star forming regions, again traced by the metallicity distribution.\n\n\\subsubsection{Circumgalactic medium (scale $\\simeq 50$~kpc)}\\label{sec_CGM}\n\nTo investigate the circumgalactic medium (CGM), we zoom in a region within $\\sim 3\\, r_{\\rm vir} = 47.5$ kpc from {Dahlia}'s centre. On these scales, we can appreciate the presence of several {Dahlia}'s satellites, i.e. extended (few ${\\rm kpc}$) structures that are $\\sim 100$ times denser than the filament in which they reside. Two of these density structures are particularly noticeable. These are located at a distance of $\\sim10~{\\rm kpc}$ from the centre in the upper left and lower left part of the map, respectively. By looking at the metallicity distribution, we find that both satellites reside within their own metal bubble, which is separated from Dahlia's one. This clearly indicates an in-situ star formation activity.\n\nAdditionally, the density map shows about $20$ smaller ($\\sim 10-100\\,{\\rm pc}$) overdense clumps ($n\\gsim 10\\,{\\rm cm}^{-3}$). The ones within {Dahlia}'s metal bubble are enriched to $Z\\simeq {\\rm Z}_{\\odot}$. This high $Z$ value is indicative of in-situ self-pollution, which possibly follows an initial pre-enrichment phase from {Dahlia}. Clumps outside {Dahlia} metal bubble have on average an higher density ($n\\sim 10^{2}{\\rm cm}^{-3}$). Since these clumps are unpolluted, they have not yet formed stars, as the effective density threshold for star formation is $\\sim 25\/(Z\/{\\rm Z}_{\\odot}){\\rm cm}^{-3}$ (see eq. \\ref{eq_critical_density} and Sec. \\ref{sec_model_sf}). Such clumps represent molecular cloud complexes caught in the act of condensing as the gas streams through the CGM \\citep{ceverino:2016MNRAS}. Such clumps have gas mass in the range $10^5 - 10^6 {\\rm M}_{\\odot}$, and are not DM-confined, as the DM density field is flat on their location.\n\nStar forming regions are surrounded by an envelope of hot ($T\\simeq 10^{5.5}{\\rm K}$), diffuse ($n\\gsim 10^{-2}\\,{\\rm cm}^{-3}$) and mildly enriched ($Z\\sim 10^{-2}{\\rm Z}_{\\odot}$) gas produced by SN explosions and winds. In the centre of star forming regions, instead, the gas can cool very rapidly due to the high densities\/metallicities. Nevertheless, these regions are highly pressurized due to bulk motions mostly driven by radiation pressure (see Fig. \\ref{fig_feedback_vs_time}).\n\n\\subsubsection{ISM (scale $ \\simeq 10$~kpc)}\\label{sec_small_scale}\n\nThe structure of Dahlia's ISM emerges once we zoom in a region $\\sim 0.5\\, r_{\\rm vir}$ from its centre. In the inner region ($\\simeq 2\\,{\\rm kpc}$), a counterclockwise disk spiral pattern is visible, since the field of view is perpendicular to the rotation plane of the galaxy (see \\citealt{gallerani:2016outflow} for the analysis of the velocity field of {Dahlia}). The presence of disks in these early systems has already been suggested by other studies. For example, \\citet{feng:2015apj} show that already at $z \\sim 8$ nearly $70\\%$ of galaxies with $M_{\\star}\\simeq 10^{10}{\\rm M}_{\\odot}$ have disks (see also Sec. \\ref{sec_final_results}).\n\nThe spiral central region and the spiral arms are dense ($n\\simeq 10^{2}{\\rm cm}^{-3}$) and cold ($T\\simeq 10^3{\\rm K}$), and the active SF produces a large in-situ enrichment ($Z\\simeq {\\rm Z}_{\\odot}$). Winds and shocks from SN have no effect in the inner part of the galaxy, because of the high density and short cooling time of the gas; this implies that metals remain confined within $\\sim 2\\,{\\rm kpc}$.\n Within spiral arms radiation pressure induced bulk motions largely dominate the total pressure, which reaches values as high as $P\\gsim 10^{6.5}{\\rm K}\\,{\\rm cm}^{-3}$. The imprint of SN shocks is evident in the temperature map in regions with $T\\gsim 10^5{\\rm K}$. Shock driven outflows originated in spiral arms travel outward in the CGM, eventually reaching the IGM if outflow velocities exceed the escape velocity ($\\sim 100\\,{\\rm km}\\,{\\rm s}^{-1}$, see Fig. 4 in \\citealt{gallerani:2016outflow}).\n\nOutflows are either preferentially aligned with the galaxy rotation axis, or they start at the edge of the disk. However, when spherically averaged, infall and outflow rates are nearly equal ($\\sim 30\\,{\\rm M}_{\\odot}\/{\\rm yr}$ at $z\\sim6$, \\citealt{gallerani:2016outflow}), and the system seems to self-regulate \\citep[see also][]{dekel:2014}.\n\nOutside the disk, clumps with density $n\\simeq 10^{2}{\\rm cm}^{-3}$ are also present and are actively producing stars. These isolated star forming MCs are located at a distance $\\gsim 2\\,{\\rm kpc}$ from the centre, and show up as spots of high pressure ($P\\gsim 10^7{\\rm K}\\,{\\rm cm}^{-3}$); some of this MCs are completely disrupted by internal feedback and they can be recognized by the low metallicity ($Z\\sim 10^{-3}{\\rm Z}_{\\odot}$): this is consistent with the outcome of numerical simulations of multiple SN explosions in single MC \\citep[e.g.][]{kortgen:2016}.\n\n\\subsubsection{Radial profiles}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/profile_sph_profile_dahlia_27.pdf}\n\\caption{\nDensity ($n$, blue), metallicity ($Z$, red) and molecular hydrogen density ($n_{\\rm H2}$, yellow) radial profile ($r$) with respect to {Dahlia} centre. The profiles are spherically averaged, as indicated by the $\\langle\\,\\rangle_{V}$ operator, and the upper axis shows the radial distance $r$ as a function of the virial radius of {Dahlia} ($r_{\\rm vir}$).\n\\label{fig_sph_profile}\n}\n\\end{figure}\n\nFig. \\ref{fig_sph_profile} shows spherically averaged density, metallicity, and ${\\rm {H_2}}$~density profiles for the gas. The density profile rapidly decreases from $n\\sim 30\\,{\\rm cm}^{-3}$ at $r\\sim 0$ to $n\\sim 0.1\\,{\\rm cm}^{-3}$ at $r\\sim 6\\,{\\rm kpc} (\\sim 0.5\\,r_{\\rm vir})$, and then flattens at larger distances. Such profile is consistent with the average profile of $z=4$ galaxies presented in \\citetalias{pallottini:2014_sim}. There we claimed that the density profile is universal once rescaled to the halo virial radius (see also \\citealt{liang:2016}). Superposed to the mean density profile, local peaks are clearly visible: they result from individual clumps\/satellites, as discussed above. \n\nThe central metallicity is close to the solar value, but by $r\\sim12\\,{\\rm kpc}\\sim r_{\\rm vir}$ it has already dropped to $Z=Z_{floor}$. Within $0\\lsim r\/{\\rm kpc}\\lsim6$, the metallicity gradient closely tracks the density profile, while for $6\\lsim r\/{\\rm kpc}\\lsim15$ the decrease is steeper. \\citet{pallottini:2014cgmh} find that the metallicity profile is not universal, however it usually extend up to few virial radii, as for {Dahlia}; further insights can be obtained by analyzing the $n$-$Z$ relation (Sec. \\ref{sec_eos}).\n\nIn Fig. \\ref{fig_sph_profile} we note that the $Z$ gradient found in {Dahlia} at $z = 6$ is slightly steeper than the one inferred from observations of $z\\sim 3$ galaxies: i.e. we find $\\Delta Z\/r \\sim -0.1\\, {\\rm dex}\/{\\rm kpc}$ while the observed ones are $\\sim 0\\, {\\rm dex}\/{\\rm kpc}$ \\citep{wuyts:2016} and $\\sim +0.1\\, {\\rm dex}\/{\\rm kpc}$ \\citep{troncoso:2013arxiv1311}. This suggests that the metallicity profile evolve with cosmic time and that the flattening is likely caused by stellar feedback, which in our Dahlia may occur in the following Gyr of the evolution. However, to prove such claim we should evolve the simulation to $z\\sim3$.\n\nThe ${\\rm {H_2}}$~profile is spiky, and each peak marks the presence of a distinct SF region\\footnote{We remind that the profiles are volume-weighted, thus the plotted $n_{\\rm H2}$ accounts for the fact that ${\\rm {H_2}}$~is present only in a fraction of the gas at a given radius.}. In {Dahlia} ${\\rm {H_2}}$~is mainly concentrated within $r\\lsim 0.5\\,{\\rm kpc}$ and it is distributed in the disk-like structure seen in Fig. \\ref{fig_mappe_hydro} (see Sec. \\ref{sec_final_results}). The location of the other peaks correspond to the satellites, which are mostly co-located with metallicity peaks. With increasing metallicity, in fact, lower densities are needed to form ${\\rm {H_2}}$~(eq. \\ref{eq_critical_density}).\n\n\\subsection{Star formation and feedback history}\\label{sec_sfr_result}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/sameplot_dahlia_27.pdf}\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/sfr_dahlia_7.pdf}\n\\caption{\n{\\bf Left}: {Dahlia} and satellites cumulative stellar masses ($M_{\\star}$, upper left panel) and star formation rates ($SFR$, lower left panel) as a function of cosmic time ($t$). For each galaxy, individual $M_{\\star}$ and $SFR$ are plotted with a solid line, coloured accordingly to the total dark matter mass ($M_{\\rm h}$) of the host halo at $z=6$. For both $M_{\\star}$ and $SFR$, {Dahlia}'s tracks are plotted with a blue line, and the totals ({Dahlia}+satellites) are in black. {\\bf Right:} $SFR$ as a function of cosmic time, with individual galaxies defined by the merger history up to $z\\simeq8.5$. Note the different $M_{\\rm h}$ colourbar scale with respect to the left panel. \n\\label{fig_sfr_smf_energy}\n}\n\\end{figure*}\n\nWe analyze the SF history of {Dahlia} and its major satellites by plotting in Fig. \\ref{fig_sfr_smf_energy} the cumulative stellar mass ($M_{\\star}$) and star formation rate ($SFR$) vs. time\\footnote{The $SFR$ is averaged in steps of $\\simeq 3\\,\\rm Myr$. We have checked that smaller steps do not alter the following analysis.}.\n\nFor the whole galaxy sample, the time averaged ($\\pm$ r.m.s.) specific star formation is $\\langle{\\rm sSFR}\\rangle= (16.6 \\pm 32.8)\\,{\\rm Gyr}^{-1}$. This mean value is comparable to that obtained by previous simulations of high-$z$ galaxies \\citep{wise:2012radpres} and broadly in agreement with $z\\sim 7$ observations \\citep{Stark:2013ApJ}. At early times the $sSFR$ reaches a maximum of $\\sim 100\\,{\\rm Gyr}^{-1}$, while a minimum of $3.0\\,{\\rm Gyr}^{-1}$ is found during the late time evolution. Both the large ${\\rm sSFR}$ range and maximum at early times are consistent with simulations by \\citet{shen:2014}. At late times, the $sSFR$ is in agreement with analytical calculation \\citep{behroozi:2013apj}, and with $z=7$ observations \\citep{gonzalez:2010}, although we note {Dahlia} has a larger stellar mass with respect to the galaxies in the sample ($M_{\\star}\\simeq 5\\times 10^9{\\rm M}_{\\odot}$).\n\nAt all times, {Dahlia} dominates both the stellar mass and star formation rate, whose mean value is $\\langle SFR\\rangle \\simeq (35.3 \\pm 32.7)\\,{\\rm M}_{\\odot}\/{\\rm yr}$. Its stellar mass grows rapidly, and it reaches $M_{\\star}\\sim 10^{9}{\\rm M}_{\\odot}$ by $t\\simeq 400\\,{\\rm Myr}$ ($z=11$), i.e. after $\\simeq 120\\,{\\rm Myr}$ from the first star formation event. Such rapid mass build-up is due to merger-induced SF, that plays a major role at high-$z$ \\citep{poole:2016MNRAS,behroozi:2013apj,salvadori2010MNRAS}. The $SFR$ is roughly constant from $z\\sim 11$ to $z\\sim 8.5$ and reaches a maximum of $\\simeq 130\\, {\\rm M}_{\\odot}\/{\\rm yr}$ at $z\\sim 6.7$. With respect to observations of $z\\sim6$ LBG galaxies \\citep[e.g.][]{stanway:2003MNRAS,stark:2009apj} the $SFR$ and $M_{\\star}$ of {Dahlia} are above the mean values, but still consistent within one sigma. Additionally, the combination of $SFR$, $M_{\\star}$, and $Z_{\\star}$ for {Dahlia} are compatible with the fundamental mass metallicity relation observed in local galaxies \\citep{mannucci:2010mnras}.\n\nThe total stellar mass in satellites is $M_{\\star}\\sim 10^{9}{\\rm M}_{\\odot}$. Typically, SF starts with a burst, generating $\\sim 10^{7.5} {\\rm M}_{\\odot}$ of stars during the first $\\simeq 20\\,\\rm Myr$. Then the $SFR$ exponentially declines and becomes intermittent with a bursty duty cycle of $\\sim100\\,\\rm Myr$. This process can be explained as follows. As an halo forms, at its centre the density of the gas slowly rises. When the density is higher than the critical density of ${\\rm {H_2}}$~formation (eq. \\ref{eq_critical_density}), the gas in the inner region is converted into stars in few free-fall times. Then feedback, and in particularly coherent SN explosions ($t_{\\star}\\gsim 10\\,\\rm Myr$, see Fig. \\ref{fig_gamete_tables}), quenches $SFR$, and the star formation activity becomes self-regulated. As mergers supply fresh gas, the $SFR$ suddenly goes out of equilibrium and becomes bursty again. Note that self-regulation is possible only for major satellites, since smaller ones ($M_{\\rm h}\\lsim 10^8{\\rm M}_{\\odot}$) cannot retain a large fraction of their gas following feedback events due to their shallow potential wells (see \\citetalias{pallottini:2014_sim}).\n\nNote that the duty cycle and the amplitude of the burst are fairly in agreement with observations of $M_{\\star}\\sim10^8-10^{10}{\\rm M}_{\\odot}$ galaxies at $z\\lsim0.3$ \\citep{kauffmann:2014mnras}. Furthermore, in our satellites we find that the typical behavior of the burst phases -- starburst - quiescent - post-starburst -- is qualitatively similar to what found by \\citet{read:2016mnras}, that simulate the evolution of a $M_{\\star}\\simeq 10^9{\\rm M}_{\\odot}$ galaxy for $\\simeq 1\\, {\\rm Gyr}$ (see also \\citealt[][]{teyssier:2013mnras,read:2016mnras_b} for further specific studies on the bursty nature of this kind of galaxies).\n\nSince individual galaxies are defined as group of star particles in the same DM halo at $z=6$, the SF history accounts for the sum of all the stars that formed in different progenitors of the considered halo. For comparison, in the right panel of Fig. \\ref{fig_sfr_smf_energy} we plot the $SFR$ of individual halos defined by their merger history at $z=8.7$. Galaxies with active SF at $300-550$ Myr merge into {Dahlia} at a later time, thus they do not appear individually in the left panel of Fig. \\ref{fig_sfr_smf_energy}.\n\nSuperimposed to the global trend, the SF history of {Dahlia} and its satellites fluctuates on time scales of $\\sim 10\\,\\rm Myr$, corresponding to the time scale of energy deposition by feedback \\citep[see e.g.][]{torrey:2016arxiv}.\n\n\\subsubsection{Star formation efficiency}\\label{sez_sfr_efficiency}\n\n\\begin{figure}\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/eos\/std_eos_ele_semenov_dahlia_27.pdf}\n\\caption{Effective star formation efficiency ($\\zeta_{\\rm sf}\\,f_{\\rm H2}$) vs density ($n$) at $z=6$. The distribution is ${\\rm {H_2}}$~mass weighted; we consider gas within $3\\, r_{\\rm vir} = 47.5$ kpc from {Dahlia} centre.\n\\label{fig_cfr_semenov}}\n\\end{figure}\n\n$\\zeta_{\\rm sf}\\,f_{\\rm H2}$ represents the quantity of gas converted in stars within a free-fall time (see eq. \\ref{eq_sfr_tot}). In Fig. \\ref{fig_cfr_semenov} we plot the effective star formation efficiency ($\\zeta_{\\rm sf}\\,f_{\\rm H2}$) as a function of gas density, weighted by the ${\\rm {H_2}}$~mass fraction at $z=6$. Most of the ${\\rm {H_2}}$~is contained in the range $n=10-100 {\\rm cm}^{-3}$, and the effective efficiency $\\zeta_{\\rm sf}\\,f_{\\rm H2}$ varies from $10^{-3}$ to $10^{-1}$. Since $\\zeta_{\\rm sf}= \\mathrm{const.} =0.1$, the spread is purely due to the dependence of $f_{\\rm H2}$ on density and metallicity (see Fig. \\ref{fig_kmt_test}). Note that by construction $\\zeta_{\\rm sf}\\,f_{\\rm H2}\\leq0.1$, and the plot does not show values very close to such limit, since gas with higher effective efficiency is converted into stars within a few free-fall times (eq. \\ref{eq_sfr2}).\n\nInterestingly, our ${\\rm {H_2}}$-based star formation criterion is reminiscent of a density threshold one, as below $n \\simeq 3\\, {\\rm cm}^{-3}$ the efficiency drops abruptly (eqs. \\ref{eqs_sfr_equivalence}). However, an important difference remains, i.e. in the present model at any given density the efficiency varies considerably as a result of the metallicity dependence. The relation between efficiency and density is also similar to that found by \\citet{semenov:2015} (\\citetalias{semenov:2015}). This is striking because these authors use a star formation efficiency that depends on the turbulent velocity dispersion of the gas, with no notion of the local metallicity. This comparison is discussed further in Sec. \\ref{sec_conclusioni}.\n\n\\subsubsection{Feedback energy deposition}\\label{sec_feedback_res}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/otf_ist_dahlia.pdf}\n\\caption{Rate of energy deposition in the gas, ${\\rm d}E\/{\\rm d}t$, by feedback processes as a function of cosmic time. Different contributions (SN, wind and radiation) are plotted with a different colour, and we additionally distinguish between the kinetic (thick lines) and thermal (thin lines) energy variation. By definition, radiation pressure has no thermal contribution. Note the jump at $t\\simeq500\\,\\rm Myr$ due to the onset of radiation pressure by AGB stars. The upper axis indicates the corresponding redshift.\n\\label{fig_feedback_vs_time}\n}\n\\end{figure}\n\nAs discussed in Sec. \\ref{sezione_blast}, only a small fraction of the available energy produced by stars can couple to the gas. During the simulation, we find that the time average efficiency of the conversion is $f \\sim 0.1\\%$, regardless of the feedback type. These low efficiencies imply that energy is mostly dissipated within MCs where the stars reside and produce it. For SN and winds, such small efficiency is a consequence of the short cooling times in MCs (see also App. \\ref{app_blastwave}). For radiation pressure the efficiency is limited by the relatively small dust optical depths (see also App. \\ref{app_rad_press}).\n\nNote that, typically in simulations \\citep[e.g.][]{wise:2012radpres,agertz:2012arxiv}, energy from stars is directly deposited in the gas, and then dissipation (mostly by radiative losses) occurs during the hydrodynamical time step. Within our scheme, instead, the deposited energy is already dissipated within high density cells, where cooling is important. Nevertheless, this does not appear to determine major differences in, e.g., $SFR$ history and ISM thermodynamics, as discussed in Sec. \\ref{sec_eos}.\n\nIn Fig. \\ref{fig_feedback_vs_time} we plot the energy deposition rate in the gas by various feedback processes as a function of time. Most evidently, \\emph{radiation dominates the energy budget at all times}: $\\dot E_{rad} \\simeq 10^{2} \\dot E_{SN}\\simeq 10^3 \\dot E_{w}$. The ratios of these energy rates somewhat reflect the stellar inputs shown in Fig. \\ref{eqs_stellar_inputs}, although this is not a trivial finding, given that the interplay among different feedback types is a highly non-linear process.\n\nAs expected, the energy deposition rate behaves as $\\dot E \\propto SRF^q$, with $q \\simgt 1$, apart from fluctuations and jumps as the one at $t\\simeq 500\\,\\rm Myr$. The scaling can be understood by simple dimensional arguments. Assume that most of the energy is deposited by radiation pressure. In the optically thick limit, we can combine eqs. \\ref{eq_red_press_energy_increase} and \\ref{eq_rad_moment_injection} to write $\\dot E_{rad} \\Delta t \\simeq (L_{\\rm uv} \\Delta t)^2 \/ (M_{g}\\,c^2)$, where $M_{g}$ is the gas mass accelerated by radiation, and we neglect ionizing radiation. Then, using \\ref{eqs_def_rad_energy}, we can write $\\dot E_{rad} \\propto SFR\\,(M_{\\star}\/M_{g})$. Initially, $M_{g}\\simeq M_{\\star}$, thus $\\dot E_{rad} \\propto SFR$. Once the gas mass is expelled from the star forming region or converted into stars, $M_{g}\\ll M_{\\star}$. Thus the deposition rate increases faster than the $SFR$ and it is very sensitive to the amount of gas mass around the sources.\n\nThe previous argument holds until the gas remains optically thick. This is warranted by AGB metal\/dust production which becomes important after for stellar ages $t_{\\star}\\sim 100\\,\\rm Myr$ (see Fig. \\ref{fig_gamete_tables}). When combined with the parallel increase of UV photons by the same sources, it is easy to interpret the rapid increase of the radiative feedback efficiency at $t\\simeq500\\,\\rm Myr$, i.e. after $\\simeq 200 \\,\\rm Myr$ from the first star formation events in {Dahlia}. We checked this interpretation by looking at the IR-trapping recorded on the fly during the simulation. We find that on average $f_{\\rm ir}\\simeq 10^{-2}$ for $t\\lsim 500\\, \\rm Myr$, and $f_{\\rm ir}\\simeq 0.1$ at later times, thus confirming our hypothesis.\n\nThe energy deposition rates for different feedback types are highly correlated in time (Pearson coefficients $\\gsim 0.7$). This is partially due to the fact that the same stellar population inputs wind, radiation and supernova energy in the gas. Additionally, as we have just seen for the case of AGB star, different types of feedback are mutually dependent. For example, radiation pressure is more effective when the gas is metal and dust enriched by SN and AGB stars; winds and SN can more efficiently couple with low density gas (longer cooling time).\n\nNote that short and intense peaks in energy deposition rate correspond to the complete disruption of multiple MCs. This occurs following strong SF events in small satellites ($M_{\\rm h} \\sim 10^{7}{\\rm M}_{\\odot}$) that cannot retain the gas and sustain a continuous star formation activity.\n\nFinally, we remind that, when compared with observational\/analytical constraints, the $SFR$ and $M_{\\star}$ of {Dahlia} are higher then the mean, but still consistent within one sigma. We caution that this might imply a somewhat weak feedback prescription.\n\n\\subsubsection{Feedback effects on ISM thermodynamics}\\label{sec_eos}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_dahlia_27.pdf}\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_pressure_dahlia_27.pdf}\n\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/eos_metal_dahlia_27.pdf}\n\\includegraphics[width=0.485\\textwidth]{plots_pdf\/eos\/metal_vs_density_dahlia_27.pdf}\n\\caption{\nEquation of State of the gas within $\\simeq47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia} centre at $z=6$. Each EOS consists in a mass- or metal-weighted probability distribution function (PDF) as specified by the colourbar. We plot the PDF in the $n$-$T$ plane ({\\bf upper left panel}), in the $n$-$P$ plane ({\\bf upper right panel}), the metal-mass weighted PDF in the $n$-$T$ plane ({\\bf lower left panel}), and mass-weighted relation between gas $n$ and $Z$ ({\\bf lower left right}). Mean relations and r.m.s. dispersions are overplotted with solid black and dashed lines, respectively. In the upper horizontal axis of each panel, we indicate the overdensity ($\\Delta$) corresponding to $n$. The density range of rarefied, diffuse and dense phases used in the text are indicated. For the panels on the left, the rarefied gas is additionally divided in \\emph{photo-ionized} ($T<10^{4.5}{\\rm K}$) and \\emph{shock-heated} ($T\\geq10^{4.5}{\\rm K}$). See Tab. \\ref{tagella_eos_riassunto} for a summary of the total values.\n\\label{fig_eos_1}\n}\n\\end{figure*}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lcccc}\n\\hline\\hline\n& mass & rarefied & diffuse & dense \\\\\n\\hline\nGas & $1.3 \\times 10^{10}{\\rm M}_{\\odot}$ & $44\\%$ & $34\\%$ & $22\\%$ \\\\\nMetals & $ 4.2\\times 10^5{\\rm M}_{\\odot}$ & $5\\%$ & $25\\%$ & $70\\%$\\\\\n${\\rm {H_2}}$ & $ 3.6\\times 10^8{\\rm M}_{\\odot}$ & $0\\%$ & $1\\%$ & $99\\%$\\\\\n\\hbox{C~$\\scriptstyle\\rm II $} & $ 2.2\\times 10^5{\\rm M}_{\\odot}$ & $4\\%$ & $22\\%$ & $74\\%$\\\\\n\\end{tabular}\n\\caption{\nSummary of the gas masses for total, metal, \\hbox{C~$\\scriptstyle\\rm II $}, and ${\\rm {H_2}}$~within $\\sim 47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia} center. In the table, we report also the fraction that is contained in different gas phases\\textsuperscript{\\ref{footnote_phases}}: \\emph{rarefied} ($\\log(n\/{\\rm cm}^3)\\leq -1$), \\emph{diffuse} ($-1<\\log(n\/{\\rm cm}^3)\\leq 1$) and \\emph{dense} ($\\log(n\/{\\rm cm}^3)> 1$). Discussion about gas and metal mass is found in Sec. \\ref{sec_eos}; analysis of ${\\rm {H_2}}$~and \\hbox{C~$\\scriptstyle\\rm II $}~is in Sec. \\ref{sec_final_results} (see also App. \\ref{sez_cloudy_model} for \\hbox{C~$\\scriptstyle\\rm II $}~calculation).\n\\label{tagella_eos_riassunto}}\n\\end{table}\n\nFeedback leaves clear imprints in the ISM thermodynamics. For convenience, we classify ISM phases according to their density: we define the gas to be in the \\emph{rarefied}, \\emph{diffuse}, and \\emph{dense} phase if $n \\leq 0.1\\,{\\rm cm}^{-3}$, $0.1 \\leq n\/{\\rm cm}^{-3}\\leq 10$, $n > 10\\,{\\rm cm}^{-3}$, respectively\\footnote{Compared to the definitions used in \\citet{klessen:2014review}, the rarefied corresponds to the warm and hot ionized medium, the diffuse phase to the cold and warm neutral medium and the dense phase to the molecular gas.\\label{footnote_phases}}.\n\nWe focus at $z=6$ and consider the gas in a region within $\\simeq 47.5\\,{\\rm kpc}$ $(3\\, r_{\\rm vir})$ from {Dahlia}'s centre, essentially the scale of the CGM described in Sec. \\ref{sec_CGM}. This region contains a total gas mass of $1.3\\times 10^{10} {\\rm M}_{\\odot}$, and metal mass of $4.2\\times 10^5{\\rm M}_{\\odot}$ (additional data in Tab. \\ref{tagella_eos_riassunto}).\n\nFig. \\ref{fig_eos_1} shows the Equation of State (EOS, or phase diagram) of the gas. The fraction of gas in the rarefied, diffuse and dense phases is $44\\%$, $34\\%$ and $22\\%$; these phases contain $5\\%$, $25\\%$ and $70\\%$ of the metals, respectively. Thus, while the gas mass is preferentially located in the lower density phases, metals are mostly found in dense gas, i.e. star forming regions\/MC. Additionally only $\\sim 30\\%$ of the considered volume shows $Z>10^{-3}{\\rm Z}_{\\odot}=Z_{\\rm floor}$, i.e. it has been polluted by stars in the simulation. We note that the EOS in the $n$-$T$ plane is fairly consistent with the one found in other high-$z$ galaxy simulations \\citep[e.g. see Fig. 5 in][]{wise:2012radpres}. Comparison between the EOS in the $n$-$T$ and $n$-$P$ plane highlights the relative importance of different feedback types.\n\nThe \\emph{rarefied} gas is characterized by long cooling times. Thus, once engulfed by shocks, such phase becomes mildly enriched ($\\langle Z\\rangle \\sim10^{-2}{\\rm Z}_{\\odot}$) and remains hot ($T\\sim10^{6}{\\rm K}$). The enriched rarefied gas preferentially populates the $n\\simeq 10^{-3}{\\rm cm}^{-3}$ and $T\\simeq10^{6.5}{\\rm K}$ region of the phase diagram. However, part of the rarefied gas has $T\\simeq10^{4}{\\rm K}$. This gas component has a temperature set by the equilibrium between adiabatic cooling and the photo-heating by the UV background; it feeds the accretion onto Dahlia, but it is not affected by stellar feedback. As such it is not central in the present analysis.\n\nThe \\emph{dense} gas is mostly unaffected by shocks and it is concentrated in the disk. Typically, such gas has $n\\sim 10^2 {\\rm cm}^{-3}$ and $T\\sim10^{2}{\\rm K}$, thus a thermal pressure $P_{\\rm th}\/k \\sim 10^4 {\\rm cm}^{-3}\\,{\\rm K}$ is expected. However, the total gas pressure is $P\/k \\sim 10^7 \\,{\\rm cm}^{-3}$ K (see the $P$-$n$ EOS). The extra contribution is provided in kinetic form by radiation pressure, thanks to the strong coupling with the gas allowed by the high optical depth of this phase. This leads to the important implication that the central structure of {Dahlia} is radiation-supported (see also Sec. \\ref{sec_final_results}).\n\nThe \\emph{diffuse} gas acts as an interface between the dense disk gas and the rarefied gas envelope. Diffuse gas is found both in hot ($T\\sim10^{5}{\\rm K}$) and cold ($T\\sim10^{3}{\\rm K}$) states. The cold part has a sufficiently high mean metallicity, $Z\\sim 0.1{\\rm Z}_{\\odot}$, to allow an efficient cooling of the gas. This is highlighted by the metal-weighted EOS, where we can see that most of the metals present in the diffuse phase are cold.\n\nNote that the phase diagram also shows evidence for the classical 2-phase medium shape for pressures around $P\/k \\sim 10^3 {\\rm cm}^{-3}\\,{\\rm K}$, while at higher (and lower) pressures only one stable phase is allowed; nevertheless, at any given pressure a range of densities can be supported. Such situation, though, is highly dynamic and does not correspond to a true thermal equilibrium.\n\nA final remark is that by $z=6$ a $n-Z$ correlation is already in place, although considerable scatter is present. The relation gets steeper at large densities, and at the same time the scatter decreases. Such relation arises from the superposition of the analogous relation for metal bubbles of individual galaxies ({Dahlia} and satellites). The scatter instead results from the fact that the slope of the $n-Z$ relation depends on the $SFR$ history (for an in-depth analysis see \\citetalias{pallottini:2014_sim}). The average $n-Z$ relation found is consistent with the results from $z\\simeq3$ galaxies \\citep{shen:2014}.\n\n\\subsection{Additional ISM properties}\\label{sec_final_results}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_surf_dens_kmt_H2_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_surf_dens_CII_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_rt_emission_CII_map_27.pdf}\n\\vspace{.5pt}\n\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_surf_dens_kmt_H2_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_surf_dens_CII_27.pdf}\n\\includegraphics[width=0.32\\textwidth]{plots_pdf\/maps\/landscape_c_edgeon_rt_emission_CII_map_27.pdf}\n\\caption{\nFace-on ({\\bf upper panels}) and edge-on ({\\bf lower panels}) $z=6$ {Dahlia} surface maps for ${\\rm {H_2}}$~density ($\\Sigma_{\\rm H2}\/(\\msun\\,{\\rm kpc}^{-2})$ {\\bf left panels}), \\hbox{C~$\\scriptstyle\\rm II $}~density ($\\Sigma_{\\rm CII}\/(\\msun\\,{\\rm kpc}^{-2})$ {\\bf middle panels}), and \\hbox{[C~$\\scriptstyle\\rm II $]}~brightness ($S_{\\rm [CII]}\/(\\lsun\\,{\\rm kpc}^{-2})$ {\\bf left panels}). The scale is $10$~kpc, as in the right-most panels of Fig. \\ref{fig_mappe_hydro} (Sec. \\ref{sec_small_scale}). Note that lower limits for the maps are drawn for visualization purposes ($\\log(\\Sigma_{\\rm H2}\/(\\msun\\,{\\rm kpc}^{-2}))\\simeq \\log(S_{\\rm [CII]}\/(\\lsun\\,{\\rm kpc}^{-2})) \\simeq 5$, $\\log(\\Sigma_{\\rm CII}\/(\\msun\\,{\\rm kpc}^{-2}))\\simeq 2$). Additionally, an average of the maps is plotted in Fig. \\ref{fig_mappe_results_profili}.\n\\label{fig_mappe_tutte}\n}\n\\end{figure*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.49\\textwidth]{plots_pdf\/dahlia_profiles.pdf}\n\\caption{\nRadially-averaged profiles for face-on (full colour) and edge-on (transparent and hatched) views of Dahlia at $z=6$. {\\bf Upper left:} \n${\\rm {H_2}}$~surface density; {\\bf upper right:} stellar surface density; {\\bf lower left:} \\hbox{C~$\\scriptstyle\\rm II $}~surface density; {\\bf lower right:} \\hbox{[C~$\\scriptstyle\\rm II $]}~surface brightness.\n\\label{fig_mappe_results_profili}\n}\n\\end{figure}\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lccc}\n\\hline\n~ & \\multicolumn{2}{c}{$r_{1\/2}\/{\\rm kpc}$} & approximate\\\\\n~ & face-on & edge-on & value at $r_{1\/2}$\\\\\n\\hline\t\t\t\n${\\rm {H_2}}$\t& 0.59 & 0.36 &\t$10^{8.23}{\\rm M}_{\\odot}$\\\\\n\\hbox{C~$\\scriptstyle\\rm II $}\t& 0.64 & 0.38 &\t$10^{5.14}{\\rm M}_{\\odot}$\\\\\nstars\t& 0.37 & 0.23 &\t$10^{9.89}{\\rm M}_{\\odot}$\\\\\n\\hbox{[C~$\\scriptstyle\\rm II $]}\t& 0.60 & 0.36 &\t$10^{7.25}{\\rm L}_{\\odot}$\\\\\n\\end{tabular}\n\\caption{\nSummary of the effective radius ($r_{1\/2}$) for the ${\\rm {H_2}}$, \\hbox{[C~$\\scriptstyle\\rm II $]}, \\hbox{C~$\\scriptstyle\\rm II $}~and stellar component in {Dahlia} at $z=6$ for the face-on and edge-on case and corresponding values for the mass\/luminosity. For each entry, $r_{1\/2}$ is defined as the radius including half of the mass\/luminosity. The quoted radii have a reference error of $\\pm 0.015\\,{\\rm kpc}$. Note that the approximate values of masses\/luminosity at $r_{1\/2}$ are insensitive to the orientation of the projection (face-on\/edge-on). The full profiles are shown in Fig. \\ref{fig_mappe_results_profili}.\n\\label{tagella_halflightradius}}\n\\end{table}\n\nWe conclude our analysis by inspecting the distribution of two key ISM species, molecular hydrogen and \\hbox{C~$\\scriptstyle\\rm II $}, along with the expected surface brightness of the corresponding $158\\mu$m \\hbox{[C~$\\scriptstyle\\rm II $]}~line. The surface maps of these quantities in Dahlia ($z=6$) are shown in Fig. \\ref{fig_mappe_tutte} for the face-on and edge-on view cases. For reference, in Fig. \\ref{fig_mappe_results_profili} we additionally plot the radially-averaged profiles of the same quantities, and in Tab. \\ref{tagella_halflightradius} we give their typical radial scales.\n\n\\subsubsection{Molecular Hydrogen}\n\n{Dahlia} has a total ${\\rm {H_2}}$~mass $M_{\\rm H2}\\simeq 3.6\\times 10^8{\\rm M}_{\\odot}$, that is mainly concentrated in a disk-like structure of radius $\\simeq 0.6\\,{\\rm kpc}$ and scale height $\\simeq 200\\,{\\rm pc}$, with a sharp cut off beyond these scales\\footnote{Such scales are calculated by using the principal component analysis of the ${\\rm {H_2}}$~distribution around the galaxy.}. The disk has mean surface density $\\langle\\Sigma_{\\rm H2}\\rangle \\simeq 10^{7.5}\\msun\\,{\\rm kpc}^{-2}$, that is approximately constant with radius and presents perturbed spiral arms along which the density is enhanced by a factor $\\simeq 3$. The spiral arms are less pronounced than in a more massive, MW-like galaxy (see \\citetalias{semenov:2015} and \\citealt{ceverino:2015}). This trend with mass has already been pointed out by \\citet{ceverino:2010MNRAS}.\n\nThe disk is composed by dense ($n\\gsim 25\\,{\\rm cm}^{-3}$), enriched ($Z\\simeq 0.5\\,{\\rm Z}_{\\odot}$), radiation-pressure supported gas, as already discussed. It is fed by frequent mergers driving fresh gas to the centre, and supports a star formation rate per unit area of $\\simeq 15\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}\\,{\\rm kpc}^{-2}$, i.e. more than 1000 times the Milky Way value. Fragmentation of the disk is relatively weak \\citep[cfr.][]{mayer:2016}, as indicated by smooth surface density map, and also paralleling the flat metallicity profile in the inner $\\simeq\\,{\\rm kpc}$.\nFor the fragmentation of the ${\\rm {H_2}}$~component, we caution that this result has been obtained assuming a uniform UV interstellar field; stronger fragmentation in the ${\\rm {H_2}}$~distribution may occur when accounting for local radiation sources: Lyman-Werner photons from these sources might in fact locally dissociate the ${\\rm {H_2}}$~by generating pockets of \\HI~in the distribution.\n\nWhile most of the ${\\rm {H_2}}$~gas resides in the disk, we can clearly distinguish 3 clumps of molecular gas both in the face-on and edge-on maps. These clumps are located few kpc away from the centre, and are characterized by sizes of $\\sim 150\\,{\\rm pc}$ and $M_{\\rm H2} \\sim 5\\times10^6 {\\rm M}_{\\odot}$. Such clumps are Jeans-unstable and form stars as they infall and stream through the CGM, as it can be appreciated by comparing ${\\rm {H_2}}$~and stellar mass profiles (Fig. \\ref{fig_mappe_results_profili}). \nThe stellar mass profiles also highlight the presence of 3 stellar clumps at $r\\sim 1\\,{\\rm kpc}$ with no associated ${\\rm {H_2}}$. These \\quotes{older} clumps share the same nature of the previous ones, but the ${\\rm {H_2}}$~has been already consumed and\/or dispersed by the star formation activity that produced the stars present at $z=6$.\n\n\\subsubsection{Singly ionized Carbon}\n\nThe \\hbox{C~$\\scriptstyle\\rm II $}~abundance is calculated by post-processing the simulation outputs with the photoionization code \\textlcsc{cloudy} (\\citealt{cloudy:2013}, and see App. \\ref{sez_cloudy_model}). The result is shown in Fig. \\ref{fig_mappe_tutte}. {Dahlia} contains a \\hbox{C~$\\scriptstyle\\rm II $}~mass of $M_{\\rm CII} = 2.2\\times 10^5 {\\rm M}_{\\odot}$, accounting for $\\sim 50\\%$ of the total metals produced. About 74\\% of the \\hbox{C~$\\scriptstyle\\rm II $}~mass is located in the dense phase, $22\\%$ in the diffuse phase, $4\\%$ in the rarefied phase. Note that the \\hbox{C~$\\scriptstyle\\rm II $}~mass phase distribution differs only for $\\lsim 10\\%$ from the $Z$ distribution (see Tab. \\ref{tagella_eos_riassunto}). The difference arises because shock-heated gas can be collisionally excited to higher ionization states. Thus, to first order, we expect the \\hbox{C~$\\scriptstyle\\rm II $}~spatial distribution to follow the metallicity one.\n\nThe face-on \\hbox{C~$\\scriptstyle\\rm II $}~surface density has a central maximum ($\\Sigma_{\\rm CII} \\sim 10^5\\msun\\,{\\rm kpc}^{-2}$), it gradually decreases to up to $\\simeq 1.2\\,{\\rm kpc}$, and drastically drops to $\\Sigma_{\\rm CII} \\lsim 10^2\\msun\\,{\\rm kpc}^{-2}$ beyond that radius (see also Fig. \\ref{fig_mappe_results_profili}). Thus, most of the \\hbox{C~$\\scriptstyle\\rm II $}~is located into the disk, but a more extended envelope containing a sizable fraction of mass exists. On top of this smooth distribution, there are \\hbox{C~$\\scriptstyle\\rm II $}~enhancements corresponding to the ${\\rm {H_2}}$~clumps described above. \n\nThe \\hbox{C~$\\scriptstyle\\rm II $}~profile is similar for edge-on and face-on case. However, the edge-on has a higher \\hbox{C~$\\scriptstyle\\rm II $}~central density and a steeper slope. While the higher central value is obviously due to the larger column density encountered along the disk, the sharp drop is related to metal transport. As most of the star formation activity is located in the disk, metals above it can be only brought by outflows which become progressively weaker with distance. Metal outflows originating from the centre are preferentially aligned with the rotation axis, and the pollution region starting from the edge is stretched by the disk rotation and by tidal interaction with satellites.\n\n\\subsubsection{Emission from singly ionized carbon}\n\nWe finally compute the expected \\hbox{[C~$\\scriptstyle\\rm II $]}~line emission using the same prescriptions of \\citet{Vallini:2013MNRAS,vallini:2015}, as detailed in App. \\ref{sez_cloudy_model}. Note that for the present work we assume uniform UV interstellar radiation. This approximation is valid in the MW, where variations around the mean field value are limited to a factor of 3. The results are plotted in Fig. \\ref{fig_mappe_tutte}.\n\nWithin $1\\,{\\rm kpc}$ from the centre the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission structure closely follows the \\hbox{C~$\\scriptstyle\\rm II $}~distribution, and we find $S_{\\rm [CII]}\/{\\rm L}_{\\odot} \\simeq 200\\, \\Sigma_{\\rm CII}\/{\\rm M}_{\\odot}$. At larger radii the \\hbox{[C~$\\scriptstyle\\rm II $]}~surface brightness suddenly drops, although the peaks associated with ${\\rm {H_2}}$~clumps are preserved. This result holds both for the face-on and edge-on cases. \n\nSuch behavior can be understood as follows. Take a typical MC with $n = 10^2{\\rm cm}^{-3}$, $Z={\\rm Z}_{\\odot}$, and total mass $M$. Its \\hbox{[C~$\\scriptstyle\\rm II $]}~\nluminosity is $L_{[\\rm CII]}\/{\\rm L}_{\\odot} \\simeq 0.1 (M\/{\\rm M}_{\\odot})$ \\citep[]{vallini:2016a,goicoechea:2015apj}. Also, the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is $\\propto Z\\,n$ for $n\\lsim 10^3$, i.e. the critical density for \\hbox{C~$\\scriptstyle\\rm II $}~collisional excitation by H atoms \\citep{Vallini:2013MNRAS}. Then, \n\\be\\label{eq_stima_luminosita}\nL_{[\\rm CII]} \\simeq 0.1\\, \\left({n\\over 100\\, {\\rm cm}^{-3}}\\right) \\left({Z\\over {\\rm Z}_{\\odot}}\\right) \\left({M\\over {\\rm M}_{\\odot}}\\right)\\,{\\rm L}_{\\odot}\\,.\n\\ee\nIn the central kpc, where $n \\simeq 10^2{\\rm cm}^{-3}$ and $Z\\simeq {\\rm Z}_{\\odot}$, the luminosity depends only on the molecular mass contained in the disk, and the same holds even for ${\\rm {H_2}}$~clumps outside the disk. The envelope is instead more diffuse ($n\\lsim 10{\\rm cm}^{-3}$) and only mildly enriched ($Z\\lsim10^{-1}{\\rm Z}_{\\odot}$). As a result, its \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity per unit mass is lower. \n\nThe emission from this diffuse component is further suppressed by the CMB \\citep[][]{dacunha:2013apj,pallottini:2015_cmb,vallini:2015}. Namely, for gas with $n\\lsim 0.1\\,{\\rm cm}^{-3}$, the upper levels of the \\hbox{[C~$\\scriptstyle\\rm II $]}~transition cannot be efficiently populated through collisions, thus the spin temperature of the transition approaches the CMB one, and to a first order the gas cannot be observed in emission.\n\nIn summary, $\\simeq 95\\%$ of {Dahlia} \\hbox{[C~$\\scriptstyle\\rm II $]}~emission comes from dense gas located in the ${\\rm {H_2}}$~disk. Indeed, the \\hbox{[C~$\\scriptstyle\\rm II $]}~half light radius coincides with the ${\\rm {H_2}}$~half mass radius, i.e. $0.59\\,{\\rm kpc}$ ($0.36\\,{\\rm kpc}$) in the face-on (edge-on) case (see also Tab. \\ref{tagella_halflightradius}). Within such radius, the molecular gas has a mass $M_{\\rm H_{2}}\\simeq 1.69\\times10^8{\\rm M}_{\\odot}$ and the luminosity is $L_{\\rm CII}\\simeq 1.78\\times10^7{\\rm M}_{\\odot}$, i.e. with a \\hbox{[C~$\\scriptstyle\\rm II $]}-${\\rm {H_2}}$~scaling ratio consistent within $15\\%$ from the simple estimate in eq. \\ref{eq_stima_luminosita}.\n\nDahlia has a total \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity $L_{\\rm CII} \\simeq 3.5\\times 10^{7}{\\rm L}_{\\odot}$; this is fainter than expected on the basis of the local \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation \\citep[$L_{\\rm CII}\\sim 10^8 - 10^9 {\\rm L}_{\\odot}$, i.e.][]{delooze:2014aa}. However, at high-$z$, such relation seems to hold only for a small subset of the observed galaxies \\citet[i.e.][]{capak:2015arxiv,Willott:2015arXiv15}. The majority of the observed galaxies show a strong \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit, when considering both detections \\citep[e.g. BDF3299, A383-5.1][]{maiolino:2015arxiv,knudsen:2016arxiv} and upper limits \\citep[e.g. Himiko, IOK1, MS0451-H][]{ouchi2013,ota:2014apj,knudsen:2016arxiv}.\n\nFor {Dahlia}, the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$~deficit depends on multiple factors. The main contribution from \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is in the ${\\rm {H_2}}$~disk, that on average has $\\langle Z\\rangle\\simeq 0.5 {\\rm Z}_{\\odot}$, i.e. slightly lower then solar. Additionally, the gas in the disk is efficiently converted in stars ($SFR\\simeq100{\\rm M}_{\\odot}\/{\\rm yr}$) and has $\\langle n\\rangle\\simeq 25\\,{\\rm cm}^{-3}$, thus the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission is hindered (eq. \\ref{eq_stima_luminosita}). Finally, there is a marginal contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~from the diffuse and rarefied phase: $\\simeq30\\%$ of \\hbox{C~$\\scriptstyle\\rm II $}~is locked in a the low density and metallicity gas that gives a negligible contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~emission, particularly because of CMB suppression.\n\n\\section{Summary and discussion}\\label{sec_conclusioni}\n\nWith the aim of characterizing the internal properties of high-$z$ galaxies, we have performed an AMR zoom-in simulation of \\quotes{Dahlia}, a $z\\simeq6$ galaxy with a stellar mass of $M_{\\star}=1.6\\times10^{10}{\\rm M}_{\\odot}$, therefore representative of LBGs at that epoch. We follow the zoom-in region with a gas mass resolution of $10^{4}{\\rm M}_{\\odot}$ and a spatial resolution of $30\\,{\\rm pc}$.\n\nThe simulation contains a rich set of physical processes. We use a star formation prescription based on a ${\\rm {H_2}}$~dependent Schmidt-Kennicutt relation. The ${\\rm {H_2}}$~abundance is computed from the \\citetalias{krumholz:2009apj} model (Fig. \\ref{fig_kmt_test}). Using stellar evolutionary models \\citep{padova:1994,starburst99:1999}, we include chemical, radiative and mechanical energy inputs, accounting for their time evolution and metallicity dependence on the stellar population properties (Fig. \\ref{fig_gamete_tables}). We include feedback from SN, winds and radiation pressure with a novel, physically motivated coupling scheme between gas and stars. We also compute \\hbox{C~$\\scriptstyle\\rm II $}~abundance and the $158\\mu$m \\hbox{[C~$\\scriptstyle\\rm II $]}~emission, by post-processing the outputs with \\textlcsc{cloudy} \\citep{cloudy:2013}, and a FIR~emission model drawn from radiative transfer numerical simulations \\citep{Vallini:2013MNRAS,vallini:2015}.\n\nThe main results can be summarized as follows:\n\n\\begin{itemize}\n\n\\item[\\bf 1.] {Dahlia} sits at the centre of a cosmic web knot, and accretes mass from the intergalactic medium mainly via 3 filaments of length $\\simeq 100\\,{\\rm kpc}$ (Fig. \\ref{fig_mappe_hydro}). Dahlia has $\\sim 6$ major satellites ($M_{\\star}\\lsim 10^{9}{\\rm M}_{\\odot}$) and is surrounded by $\\sim 10$ minor ones ($M_{\\star}\\sim 10^{5}{\\rm M}_{\\odot}$). The latter represent molecular cloud (MC) complexes caught in the act of condensing as the gas streams through the circumgalactic medium (Fig. \\ref{fig_sph_profile}). {Dahlia} dominates both the stellar mass ($M_{\\star}\\sim 10^{10}{\\rm M}_{\\odot}$) and the SFR of the galaxy ensemble ($SFR\\simeq 100\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}$, Fig. \\ref{fig_sfr_smf_energy}).\n\n\\item[\\bf 2.] Only a small fraction of the available energy produced by stars couples to the gas, as energy is mostly dissipated within MCs where the stars reside. Radiation dominates the feedback energy budget by a factor $> 100$ (Fig. \\ref{fig_feedback_vs_time}). \n\n\\item[\\bf 3.] By $z=6$ {Dahlia} forms a ${\\rm {H_2}}$~disk of mass of $M_{\\rm H2}= 3.6\\times 10^{8}{\\rm M}_{\\odot}$, effective radius $0.6\\,{\\rm kpc}$, and scale height $200\\,{\\rm pc}$ (Fig. \\ref{fig_mappe_tutte}). The disk is dense ($n\\gsim 25\\,{\\rm cm}^{-3}$), enriched ($Z\\simeq 0.5\\,{\\rm Z}_{\\odot}$), and it is fed by frequent mergers driving fresh gas to the centre, and supports a star formation rate per unit area of $\\simeq 15\\,{\\rm M}_{\\odot}\\,{\\rm yr}^{-1}\\,{\\rm kpc}^{-2}$. \n\n\\item[\\bf 4.] The disk is mostly unaffected by SN shocks, and it is pressure-supported by radiation. SN\/winds drive hot metal outflows (Fig. \\ref{fig_eos_1}), that are either preferentially aligned with the galaxy rotation axis, or start at the edge of the disk.\n\n\\item[\\bf 5.] The total \\hbox{[C~$\\scriptstyle\\rm II $]}~luminosity of {Dahlia} is $10^{7.55}{\\rm L}_{\\odot}$, and $\\simeq 95\\%$ of the emission is co-located with the ${\\rm {H_2}}$~disk (Fig. \\ref{fig_mappe_results_profili}). The diffuse, enriched material surrounding {Dahlia} contains $30\\%$ of the \\hbox{C~$\\scriptstyle\\rm II $}~mass, but it negligibly contributes to the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission (Fig. \\ref{fig_mappe_tutte}) due to its low density ($n\\simeq 10\\,{\\rm cm}^{-3}$) and metallicity ($Z\\simeq10^{-1}{\\rm Z}_{\\odot}$). {Dahlia} is under-luminous with respect to the local \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation; however, its luminosity is consistent with upper limits derived for most $z\\sim6$ galaxies. \n\\end{itemize}\n\nWe find clear indications that the SF subgrid prescription might considerably affect the \\hbox{[C~$\\scriptstyle\\rm II $]}-$SFR$ relation and the ISM structure, as noted also by \\citep{hopkins:2013arxiv}. This is because stars form in gas of different densities depending on the chosen prescription. \nIn our simulation gas is converted into stars with an efficiency $\\zeta_{\\rm sf}\\,f_{\\rm H2}$, where the ${\\rm {H_2}}$~fraction is computed from the \\citetalias{krumholz:2009apj} model and we set $\\zeta_{\\rm sf}=0.1$. In \\citetalias{semenov:2015} the SF follows a \\textit{total} (i.e. not molecular) density Schmidt-Kennicutt relation. Further the SF efficiency depends on the free-fall time and the turbulent eddy turnover time. The SF relation is derived from an empirical fit to MC simulations \\citep{padoan:2012}, with no notion of the local metallicity. \n\nInterestingly, although the approaches are considerably different, the resulting efficiencies are compatible: in \\citetalias{semenov:2015} the bulk of the star forming gas has $n\\sim 10^{1.5}{\\rm cm}^{-3}$, as in {Dahlia} (Fig. \\ref{fig_cfr_semenov}). However, with respect to \\citetalias{semenov:2015}, Dahlia misses part of the very dense, star forming gas, and its corresponding contribution to \\hbox{[C~$\\scriptstyle\\rm II $]}~from $Z\\sim{\\rm Z}_{\\odot}$ MCs with $n\\sim10^{3}{\\rm cm}^{-3}$. These MC are expected to have high \\hbox{[C~$\\scriptstyle\\rm II $]}~fluxes (see eq. \\ref{eq_stima_luminosita}), but their abundance might be low \\citep{padoan:2012}.\nFurther investigation is needed before we draw any solid conclusion. To this aim, we plan to upgrade our simulations to a more sophisticated non-equilibrium ${\\rm {H_2}}$~evolution model. This is because the chemical equilibrium assumed in \\citetalias{krumholz:2009apj} does not hold in low-metallicity regimes. \n\nAnother important caveat is that we have assumed a uniform UV background. Instead, discrete sources (stellar clusters) might have a strong impact on star formation. For example, Lyman-Werner photons might locally dissociate the ${\\rm {H_2}}$~by generating pockets of \\HI~in the gas distribution. Thus, unshielded (low dust column density) gas in the disk would contribute only marginally to the SFR.\n\nFurthermore, a uniform UVB assumption likely leads to inaccurate computation of the ISM thermodynamic state. We find that $Z\\simeq 10^{-3}{\\rm Z}_{\\odot}$ gas with $n\\gsim 10^{2}\\,{\\rm cm}^{-3}$ has $T\\simeq 10^{4}$ (Fig. \\ref{fig_eos_1}), with the temperature been set by the UVB heating. However, such gas should be likely able to self-shield from the impinging UVB, whereas internal radiation sources could still play a role \\citep[e.g.][]{gnedin:2010}. \n\nFinally, local FUV flux variations can change the \\hbox{[C~$\\scriptstyle\\rm II $]}~emission from individual regions of the galaxy. Also, very high FUV fluxes can photoevaporate MC on short time scales ($\\lsim t_{\\rm ff}$ for gas with $Z\\sim 10^{-2}{\\rm Z}_{\\odot}$, \\citealt[][]{vallini:2016a}). This effect are particularly important, as it might be responsible for the displacement between the \\hbox{[C~$\\scriptstyle\\rm II $]}~and the UV emitting region observed in BDF3299 \\citep{maiolino:2015arxiv}, and in some of the \\citet{capak:2015arxiv} galaxies. To solve these problems, a multi-frequency radiative transfer computation must be coupled to the present simulations. This work is ongoing and will be presented elsewhere. \n\n\n\n\\section*{Acknowledgments}\nWe are grateful to the participants of \\emph{The Cold Universe} program held in 2016 at the KITP, UCSB, for discussions during the workshop. \nWe acknowledge the {\\tt AGORA} project members and the {\\tt DAVID} group for stimulating discussion. \nWe thank the authors and the community of \\textlcsc{pymses} for their work. \nWe thank B. Smith for support in implementing \\textlcsc{grackle}. \nThis research was supported in part by the National Science Foundation under Grant No. NSF PHY11-25915.\nS.S. was supported by the European Research Council through a Marie-Skodolowska-Curie Fellowship, project PRIMORDIAL-700907.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\n\n\n    Support Vector Machines (SVMs) are one of the most widely used algorithms for classification problems. Originally proposed in the works of Boser et al~\\cite{boser1992training} and Cortes and Vapnik~\\cite{cortes1995support}, they can be defined as learning machines which construct an $n$-dimensional decision boundary surface (also called a hyperplane) that optimally separates data into positive and negative classes by maximizing the margin of separation between them. \n    \n    Contrary to artificial neural networks, which provide a local minimum solution to the optimization problem, SVMs provide a unique globally optimal solution for the margin separation problem, which is addressed through the application of a kernel-based learning method. In this context, a kernel is understood as a similarity function that is applied to each data point to map the original non-linear observations into a higher dimensional space where the observations may become linearly separable. A wide range of different kernels have been proposed in the literature, targeting specific classification problems~\\cite{Haykin08}. The Gaussian kernel (also referred to as the Radial Basis Function kernel (RBF)), is probably the most widely used kernel demonstrating 'state of the art' performance in a variety of  classification problems~\\cite{Bhattacharyya11}.  \\\\\n\\indent More recently, new mappings between non-linear separable observations and higher dimensional feature spaces have been proposed with the purpose of extending the capabilities of SVMs towards theoretically feasible quantum-inspired machine learning algorithms~\\cite{schuld2019quantum,schuld2019machine,havlivcek2019supervised,mehta2019high,killoran2019strawberry,killoran2018continuous,srinivasan2018learning,adhikary2019supervised,bartkiewicz2019experimental}.  For instance, under a quantum theoretical perspective, by mapping data into coherent states which are a superposition of eigen-functions of a quantum harmonic oscillator with minimum Heisenberg  uncertainty, the RBF kernel can be understood as the inner product of two coherent states  \\cite{kubler2019quantum}. The coherent state of this harmonic oscillator comprises the following properties:\n\\begin{enumerate*}[label=({\\it \\roman*})]\n  \\item It is obtained by the displacement operators on the ground state;\n  \\item It is an eigenfunction of the annihilation operator;\n  \\item It satisfies the minimum uncertainty relation, i.e., $\\Delta(\\mathbf{x})=\\Delta(\\mathbf{p})=\\sigma\/\\sqrt{2}$, in which $\\Delta(\\mathbf{x})$ and $\\Delta(\\mathbf{p})$ are respectively the variance of the position and momentum of the harmonic oscillator;\n  \\item It is over-complete.\n\\end{enumerate*}\n\\\\\n\\indent \nThe over-completeness property implies an arbitrary function can be expressible as a linear combination of kernel functions in a \"reproducing Hilbert space\" \\cite{combescure2012coherent}. Any of the first three above-mentioned properties lead to a definition of generalized coherent states, although property ({\\it iv}) is necessary for the definition of coherent states. For example, while a Gazeau-Klauder coherent state is defined by property ({\\it ii}) and fulfils property ({\\it iii}), displacement-type coherent states are obtained by displacement operators on reference states  \\cite{ali2000coherent}.\n\\indent \nRecently, Schuld and Killoran proposed to map data from the original space to a feature space by using squeezed coherent states \\cite{schuld2019quantum}. Squeezed  states are coherent states that saturate the Heisenberg uncertainty principle in such a way that the variance of position and momentum depend on a so-called squeezed parameter. Therefore, the reduced uncertainty is one of its quadrature components, while increased uncertainty is the latter, i.e., $\\Delta(\\mathbf{x})=\\exp{\\zeta}\/\\sqrt{2}$ and $\\Delta(\\mathbf{p})=\\exp{-\\zeta}\/\\sqrt{2}$, where $\\zeta$ is the  squeezing parameter. The squeezing parameter controls uncertainty via a quadrature component, while the third  property of coherent states are preserved.\\\\\n\\indent \nGiven the large number of kernel functions currently being proposed in the literature, the question naturally arises as to which kernel function to apply. In SVM-based classification problems, the appropriate choice of a kernel is fundamental, however, the current 'trial-and-error' nature of selecting the best kernel poses significant challenges, especially when one considers kernels that can support both classical and quantum-inspired machine learning algorithms, which renders the kernel choice problem an open research question \\cite{Ali06}.\\\\\n\\indent To address this problem (and taking as basis the work of Schuld and Killoran on squeezed coherent states~\\cite{schuld2019quantum}), we propose a generalised meta-kernel from which the RBF kernel (and other kernels) can be derived by using a deformed Weyl-Heisenberg (dW-H) algebra, dependent on a parameter $\\alpha \\in \\mathbb{R}$. By applying the associated displacement operator on the reference state, the non-linear coherent state is generated by considering the specific value of the parameter, i.e. $\\alpha=2$, $\\alpha=0$, and $\\alpha=-2$, $SU(2)$, W-H, and $SU(1,1)$-coherent state are respectively recovered. \\cite{dehdashti2013coherent,dehdashti2015realization,castillo2019polynomial}.  \nThe choice of $\\alpha$ allows a specific kernel function to be defined. Therefore, the theory of coherent states can be seen as providing a meta-kernel from which kernel functions can be derived. To the best of our knowledge, no such theory of meta-kernels presently exists.\n \n By means of a feature mapping, data is mapped into the feature space, represented by the deformed coherent space. Schematically, this is illustrated in Figure \\ref{fig1}. Geometrically, the feature space constructed by the dW-H coherent state is a surface of revolution with constant curvature, i.e., the surfaces associated with $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ are respectively a positive compact surface, and negative surface, while $\\alpha=0$ produces a flat surface. Therefore, a kernel function defined in any one of the configurations is an inner product of two elements on the related surface. Through this process, our dW-H algebra acts like a meta-theory from which  a new class of two parameter non-linear kernel functions can be derived.\n\\begin{figure}[t]\n  \\centering\n   \n  \\includegraphics[scale=.55]{fig1.pdf}\n  \\caption{Schematic representation of the SVM method based on the Weyl-Heisenberg algebra, showing the mapping of data into the feature space, represented by the dW-H coherent state.}\n  \\label{fig1}\n\\end{figure}\n\n\nThe paper will proceed as follows. In Section \\ref{coherent_states}, we provide a brief introduction to dW-H coherent states which are then expressed in kernel functions. We also describe the geometric properties of the feature spaces in which these kernel functions are defined. In Section \\ref{test_design}, a test design is formulated for an illustrative evaluation of the introduced kernel functions, from the standpoint of enrichment of Gaussian strategies in SVM classification. \nAccompanying empirical results are presented, along with visualisations that aid descriptions of relevant observations.  Section \\ref{discussion} discusses the benefits of the algebra within SVM classification based on the results of the empirical evaluation.\n\\section{Deriving kernel functions from deformed Coherent States}\n\\label{coherent_states}\nA supervised machine learning (ML) classification problem can be formalised in the following way. Given a set of $N$ training examples $\\{ (x_{1}, y_{1}), (x_{2}, y_{2}), \\cdots, (x_{N}, y_{N})\\}$, where $x_{i}$ corresponds to the $ith$ training example, each training example is represented by a set of input features, and $y_{i}$ corresponds to the 'ground truth' label of the training example $x_{i}$, the objective of ML is to learn a model, $h(x)$, that represents the training set. Ideally, the outcome is to generate the model that is most capable of correctly predicting the class labels of unseen instances. \n\n\nOne way of predicting unseen examples is through the application of a similarity function, a \\textit{kernel}, between the unseen input instance ${x^{\\prime}}$ and each of the training inputs, ${x_i}$, learned during the training phase.\n\nKernel methods, $K(x, x^{\\prime})$, use the inner product between any two inputs $x, x^{\\prime} \\in \\mathcal{X}$, as distance measures in order to construct models that capture the properties of a data distribution. These distance measures can be defined in a feature space $\\mathcal{X}$, depending on whether the data is linear, or non-linearly separable.\n\nThe left side of Figure \\ref{fig1} schematically represents this process. One can define a complex Hilbert space as the feature space, where a feature mapping $\\phi: \\mathcal{X}\\rightarrow \\mathcal{F}$, in which $\\mathcal{F}$ is a complex Hilbert state,   $\\phi: x \\rightarrow |\\phi(x) \\rangle $, implies a kernel function can be defined as $K(x, x^{\\prime} ) =\n \\langle\\phi(x), \\phi(x^{\\prime}) \\rangle$. By operating a dW-H algebra on the reference state, the feature space is constructed with deformed coherent states. These coherent states depend on the attributed sign of the parameter $\\alpha$, meaning `positivity' or `negativity' defines an $\\alpha-SU(2)$ coherent state or $\\alpha-SU(1,1)$ coherent state respectively; while $\\alpha=0$ stands for a harmonic oscillator coherent state ( see Figure \\ref{fig1}). For the sake of simplicity, we define dW-H coherent states for positive and negative values, separately. The first is titled the $\\alpha-SU(2)$ coherent state, defined as follows:\n \\begin{eqnarray}\\label{eq3}\n|x;z,\\alpha\\rangle\n&=&\\left[1+\\tan^{2}\\left(z\\sqrt{\\frac{\\alpha}{2}}\\right)\\right]^{-k}\\sum_{m=0}^{2k} (-1)^{m} e^{-imx}\\nonumber\\\\\n&\\times&\\tan^{m}\\left(z\\sqrt{\\frac{\\alpha}{2}}\\right) \\sqrt{\\frac{(2k)!}{m!(2k-m)!}}|k,m\\rangle,\n\\end{eqnarray} \n in which $\\alpha \\geq 0$, and  $k \\in N$. The second is named the $\\alpha-SU(1,1)$ coherent state:\n \\begin{eqnarray}\\label{eq2}\n|x;z,\\alpha\\rangle&=&\\left[1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)\\right]^{k}\\sum_{m=0}^{\\infty} (-1)^{m} e^{-imx}\\nonumber\\\\\n&\\times&\\tanh^{m}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right) \\sqrt{\\frac{\\Gamma[2k+m]}{m!\\Gamma[2k]}}|k,k+m\\rangle,\n\\end{eqnarray} \nwhere $\\alpha\\leq 0$, and $k\\in N$. \n Note that in the case of $\\alpha = \\pm 2$, coherent states (\\ref{eq3}) and (\\ref{eq2}), respectively reduce $SU(2)$ and $SU(1,1)$ coherent states. It was also shown that if $\\alpha $ approaches zero, both coherent states (\\ref{eq3}) and (\\ref{eq2}) reduce to the harmonic oscillator coherent states  \\cite{dehdashti2013coherent} , i.e.,\n \\begin{eqnarray}\n |z\\rangle=e^{-|z|^{2}}\\sum_{m=0}^{\\infty} \\frac{z^{m} }{\\sqrt{m!}}|m\n \\rangle.\n \\end{eqnarray}\n \\indent Hence,  by considering a multi-dimensional input set in a data set of vectors $\\mathbf{x} = (x_{1},\\cdots,x_{N})^{T} \\in \\mathbb{R}^{N}$, one can define the joint state of $N$ deformed coherent states,\n \\begin{align*}\n & \\phi: (x_{1},\\cdots,x_{N}) \\rightarrow \\\\ & |x_{1},z,\\alpha\\rangle\\otimes |x_{2},z,\\alpha\\rangle \\otimes \\cdots \\otimes |x_{N},z,\\alpha\\rangle. \\numberthis\n \\end{align*}{}\n Therefore, the kernel  is defined as the following:\n \\begin{eqnarray}\n K(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N}\\langle x_{i};z,\\alpha | \n x_{i}^{\\prime};z,\\alpha \\rangle.\n \\end{eqnarray}\nIn the case of $\\alpha-SU(2)$ feature space, the kernel is obtained as follows:\\\\\n\\begin{tcolorbox}[boxsep=2pt,left=2pt,right=2pt,top=4pt,bottom=4pt]\n\\bf{$\\bf{\\alpha-SU(2)}$ Kernel Function}\n\\begin{eqnarray}\nK(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N} \\left[\\frac{1+\\tan^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)e^{i(x_{i}-x^{\\prime}_{i})}}{1+\\tan^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)}\\right]^{2k},\n\\end{eqnarray}\n\\end{tcolorbox}\nMoreover, in the case of $\\alpha-SU(1,1)$ feature space, the kernel is given as follows:\n\\begin{tcolorbox}[boxsep=2pt,left=2pt,right=2pt,top=4pt,bottom=4pt]\n\\bf{$\\bf{\\alpha-SU(1,1)}$ Kernel Function}\n\\begin{eqnarray}\nK(\\mathbf{x},\\mathbf{x^{\\prime}})= \\prod_{i=1}^{N} \\left[\\frac{1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)}{1-\\tanh^{2}\\left(z\\sqrt{\\frac{|\\alpha|}{2}}\\right)e^{i(x_{i}-x^{\\prime}_{i})}}\\right]^{2k}.\n\\end{eqnarray}\n\\end{tcolorbox}\n\\indent For understanding the role of $\\alpha$ and $k$, we study the geometrical properties of the above-mentioned feature spaces.  We can define the line element of the feature space, by using the Fubini\u2013Study metric \\cite{bengtsson2017geometry}, that is,\n\\begin{eqnarray}\nds^{2}=\\| d|x;z,\\alpha\\rangle\\|^{2}-|\\langle x;z,\\alpha|d|x;z,\\alpha\\rangle|^{2},\n\\end{eqnarray}\n By using the above definition, the metric of $\\alpha-SU(2)$ feature space is obtained by\n\\begin{eqnarray}\\label{eq11}\nds^{2}=k\\alpha dz^{2}+\\frac{k}{2} \\sin^{2} \\left(z\\sqrt{2\\alpha}\\right)dx^{2},\n\\end{eqnarray}\nwhich describes a positive constant curvature with the scalar Ricci $R=4\/k$. This is in fact a surface of revolution conforming with a sphere \\cite{carinena2005central}.\nBy using the same method, the metric of the $\\alpha-SU(1,1)$ feature space is given by\n\\begin{eqnarray}\\label{eq10}\nds^{2}=k|\\alpha| dz^{2}+\\frac{k}{2} \\sinh^{2} \\left(z\\sqrt{2|\\alpha|}\\right)dx^{2}.\n\\end{eqnarray}\nwhich describes a negative constant curvature with the scalar Ricci $R=-4\/k$, conformal with pseudo-spheres \\cite{carinena2005central}. Figure \\ref{parametric_feature_spaces} shows topological categories of feature spaces associated with $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ coherent states.\n\\begin{figure}[t]\n  \\centering\n   \n  \\includegraphics[scale=0.6]{fig2.pdf}\n  \\caption{Spatial definition of feature spaces for kernel functions $\\alpha-SU(2)$ \\textbf{(A.)} And $\\alpha-SU(1,1)$ \\textbf{(B.)} Under parametric configuration $\\alpha = 2$ And $k = 1$}\n  \\label{parametric_feature_spaces}\n\\end{figure}\nAs seen above, two differing categories of the deformed coherent states are defined with different topologies: one ($\\alpha-SU(2)$) is constructed on a truncated Hilbert space, forming a compact constant curvature feature space, conforming with a sphere. The other ($\\alpha-SU(1,1)$) is built on an infinite Hilbert space that leads to a feature space conforming with a pseudo-sphere, with negative constant curvature.\n\n\nIn the next section, we present a illustrate the empirical effectiveness of the meta-theoretically derived kernel functions in different experimental settings.\n\n\\section{Empirical evaluation of the kernel functions}\\label{test_design}\n\nIn order to assess the effectiveness of the proposed kernel functions, we conducted a series of experiments using different well known synthetic datasets of the literature, namely 1) Python's \\textit{scikit-learn} library: the circles, moons, and 2) the iris dataset. \nThe circles and the moons dataset involves binary features in a binary classification problem. The iris dataset is a multiclass classification task with three classes and four features.\n\n\nSince the goal of SVMs is to find the maximum-margin hyperplane, a set of parameters are needed to control the error between these margins (Figure~\\ref{fig:svm_hyper} shows this optimization in the high-dimensional feature space). For the RBF kernel, two parameters play a major role in this optimization process:\n\\begin{itemize}\n\\item Hyperparameter $C$: is a regularization parameter that controls the trade-off between the decision boundary and mis-classification term. It basically controls how much mis-classifications are tolerable during the optimization problem.\n\\item $\\gamma$, which controls the non-linearity of the decision boundary. It defines how far influences the calculation of plausible line of separation. A low $\\gamma$ takes into consideration far away points to influence the decision boundary; a high $\\gamma$ considers only points that are close to the decision boundary.\n\\end{itemize}\n\\begin{figure}[t]\n   \n   \n    \\centering\n    \\includegraphics[scale=.6]{fig3.pdf}\n    \\caption{Kernel Trick. A kernel is applied to each data point to map the original non-linear observations into a higher dimensional space where the observations may become linearly separable through a hyperplane.}\n    \\label{fig:svm_hyper}\n\\end{figure}\nWhen considering the proposed meta-kernel, we have a set of new parameters that extend the current RBF kernel with a set of new non-linear functions that are based on the  dW-H algebra. These extra parameters also enable the construction of a feature space over a surface of revolution with constant curvature. Theoretically, this could lead to significant improvements when the dataset is distributed along revolution surfaces. The parameters for both $SU(1,1)$ and $SU(2)$ kernels are the following:\n\\begin{itemize}\n    \\item Parameter $k$ is related to the curvature of the feature surface and  controls the non-linearity of the decision boundary in such way that high values of the parameter consider points that are near to the decision boundary whilst low values cause points further away to influence the decision boundary. Figures \\ref{vis_su11} and \\ref{fig:vis_su2} illustrate the rule of the parameter $k$.   \n    \\item By considering a fixed curvature, i.e., $k=const.$, the product of parameters $z$ and $\\sqrt{\\alpha}$, as an extra parameter   $z\\sqrt{\\alpha}$, controls the decision boundary as well. Figure   \\ref{vis_su11} and \\ref{fig:vis_su2} indicates a schematic behaviour of parameters $\\alpha$ and $k$, for $\\alpha-SU(1,1)$ and $\\alpha-SU(2)$ respectively.  \n\\end{itemize}\n\\begin{figure}[t]\n \n   \n    \\centering\n  \\includegraphics[scale=0.55]{fig4.pdf}\n  \\caption{Shape of the  kernel function obtained by $\\alpha-SU(1,1)$-coherent state  for different strength hyperparameters $\\alpha$ and $k$, while $z=1$. The input $x$ is fixed at $(0, 0)$ and $x^{\\prime}$ is varied.}\n  \\label{vis_su11}\n\\end{figure}\n\\begin{figure}[t]\n  \n  \n\\centering\n  \\includegraphics[scale=.55]{fig5.pdf}\n  \\caption{Shape of the  kernel function obtained by $\\alpha-SU(2)$-coherent state  for different strength hyperparameters $\\alpha$ and $k$, while $z=1$. The input $x$ is fixed at $(0, 0)$ and $x^{\\prime}$ is varied.}\n  \\label{fig:vis_su2}\n\\end{figure}\n\nFigure~\\ref{fig:svc_classification} illustrates different decision boundaries that can be computed using the different kernels. One can see the different non-linearity properties of the $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ kernels compared to the standard RBF kernel. \n\n\n\\begin{figure}[t]\n   \n\\centering\n  \\includegraphics[scale=.7]{fig6.pdf}\n  \\caption{Decision boundaries computed using different kernels for the synthetic datasets \\textit{Moons} and \\textit{Circles}.}\n  \\label{fig:svc_classification}\n\\end{figure}\n\nFor evaluation purposes, the parameters that provided the best results in terms of precision were found using a grid search approach (for more details on the evaluation code and experimental setup. For the synthetic datasets, Moons and Circles, 1000 samples were generated, where 70\\% were used for the training process and 30\\% were used for the evaluation task. To make the classification task more challenging, we applied noise factors of 0.3 and 0.1 to these datasets, respectively. Learning curves were analysed to ensure unbiased results and no overfitting.  Table~\\ref{tab:results} summarises the results.\n\n\\begin{figure}[]\n   \n\\centering\n  \\includegraphics[scale=.7]{table.pdf}\n  \\caption{Results obtained using the proposed kernels $\\alpha$-$SU(1,1)$ and $\\alpha$-$SU(2)$ for different datasets. Note that for the Moons and Circles synthetic dataset, we generated 1000 samples of which $70\\%$ were used for training and the remaining $30\\%$ for test. These datasets were generated using a noise factor of 0.3 and 0.1, respectively.}\n\\label{tab:results}\n\\end{figure}\n\n\n The proposed meta-kernels $\\alpha$-SU(1,1) and $\\alpha$-SU(2) exhibit state of the art performance when compared with the RBF kernel. \n These kernels provide a significant advantage for data points distributed over curved surfaces. Given that it is hard to find benchmark datasets with those characteristics, the results presented in Table~\\ref{tab:results} suggest an cautiously encouraging first step.\n\n\n\n\\section{Discussion} \\label{discussion}\n\n\nIn SVM-based classification problems, the appropriate choice of a kernel is fundamental to achieve high classification performance. \nHowever, the current 'trial-and-error' nature of selecting the best kernel poses significant challenges, especially when one considers kernels that can support both classical and quantum-inspired machine learning algorithms~\\cite{Ali06}. \nIn this section, a visual analysis is provided of how different kernels are derived from the $\\alpha$ and $k$ parameters of both $\\alpha-SU(1,1)$ and $\\alpha-SU(2)$ kernels in order to promote a clear discrimination between kernel functions. \n\nThe $\\alpha$ parameter controls allows a specific kernel function is derived from  deformed Weyl-Heisenberg (dW-H) algebra. \nA value of $\\alpha=0$ is a flat surface. When $\\alpha$ is small, it contributes to an almost flat surface, when maintaining $k$ low. On the other hand, when $\\alpha$ is high, it contributes to squeeze the function towards its center. Figure~\\ref{vis_su11} shows the impact of parameter $\\alpha$ in in the $\\alpha-SU(1,1)$ kernel.\nRegarding the $\\alpha-SU(2)$ kernels, the parameter $\\alpha$ generates a kernel that maps the non-linear observations into a higher dimensional space that `folds' the data in the feature space. A high value in $\\alpha$ and $k$ squeeze  these folds towards the center of the distribution of the geodesic distances between the data points as visualized in Figure~\\ref{fig:vis_su2}.\n\n\n\n\n\n\nIn terms of the empirical evaluation of the kernels, Table~\\ref{tab:results} indicates the the best results were obtained with low values of $\\alpha = 0.1$, $1.0 \\leq k \\leq 2.0$, with the $z$ parameters in the range: $ 0.2 \\leq z \\leq 4.6$. \n\n\n\\section{Conclusion}\nIn this paper, \nby using the theory of non-linear coherent states, we put forward a meta-kernel approach for deriving kernel functions for use in ML. \nMore specifically, data is mapped into a feature space which is defined as a deformed coherent state as defined by a deformed Weyl-Heisenberg algebra.\nThis algebra unifies the well-known $SU(2)$, Weyl-Heisenberg, and $SU(1,1)$ groups, through a common  parameter $\\alpha$.\nIn addition, by studying tgeometrical properties of feature space constructed on the dW-H coherent state, we showed that the meta-kernel function applies associated  surfaces of revolution as feature spaces identified with non-linear coherent states.  \nAn empirical investigation compares the $\\alpha-SU(2)$ and $\\alpha-SU(1,1)$ kernels derived from the meta-kernel which shows performance similar to the Radial Basis kernel.\n\nKernel functions drive developments in the field of machine learning and the meta-kernel function presented in this paper opens new theoretical avenues for the definition and exploration of kernel functions.\\\\\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgement}\nThis research was supported by the Asian Office of Aerospace Research and Development (AOARD) grant: FA2386-17-1-4016.\n\n\\appendices\n\\section{Geometrical Properties of Feature surfaces}\nThe  Christoffel symbols of the second kind according to definition are give by\n\\begin{eqnarray}\n\\Gamma_{ij}^{k}=\\frac{1}{2}g^{kl}\\left[\\partial_{i}g_{jl}+\\partial_{j}g_{il}-\\partial_{l}g_{ij}\\right]\n\\end{eqnarray}\nin which $g_{ij}$ is the $(i,j)$th component of the metric, $g^{ij}=g_{ij}^{-1}$ and $\\partial_{i}$ is an abbreviation of $\\frac{\\partial}{\\partial x_{i}}$. Also, according to standard notation, the Einstein summation convention is applied, i.e.,  summation over a set of indexed terms in a formula, e.g. $g^{ij}g_{jk}=g^{i1}g_{1k}+g^{i2}g_{2k}$. By using the metric (\\ref{eq11}), non-zero components of \nthe  Christoffel symbols of the second kind are respectively given by:\n\\begin{eqnarray}\n\\Gamma_{xz}^{x}&=&\\sqrt{2\\alpha}\\cot (\\sqrt{2\\alpha}\\ z)\\nonumber\\\\\n\\Gamma_{xx}^{z}&=&-\\frac{1}{2\\sqrt{2\\alpha}}\\sin (2\\sqrt{2\\alpha}\\ z)\n\\end{eqnarray}\nAlso, according to definition, the Ricci tensor is given by\n\\begin{eqnarray}\nR_{ij}=\\partial_{k}\\Gamma_{ij}^{k}-\\partial_{i}\\Gamma_{kj}^{k}+\n\\Gamma_{ij}^{k}\\Gamma_{kl}^{l}-\n\\Gamma_{ik}^{l}\\Gamma_{lj}^{k}.\n\\end{eqnarray}{}\nHence,\nthe non-zero Ricci tensors are given by:\n\\begin{eqnarray}\nR_{xx}= \\sin^{2} \\left[\\sqrt{2\\alpha} z\\right], R_{zz}=2\\alpha. \n\\end{eqnarray}\nThe Ricci scalar, which gives the curvature, is obtained by\n\\begin{eqnarray}\nR=\\frac{4}{k}\n\\end{eqnarray}\n\\indent In the case of metric (\\ref{eq10}), non-zero components are given by\n\\begin{eqnarray}\n\\Gamma_{xz}^{x}&=&\\sqrt{2\\alpha}\\coth (\\sqrt{2\\alpha}\\ z)\\nonumber\\\\\n\\Gamma_{xx}^{z}&=&\\frac{-1}{2\\sqrt{2\\alpha}}\\sinh (2\\sqrt{2\\alpha}\\ z)\n\\end{eqnarray}\nand the Ricci tensor are given by\n\\begin{eqnarray}\nR_{xx}=\\sinh^{2}(z\\sqrt{2\\alpha}), R_{zz}=-2\\alpha \n\\end{eqnarray}\nThe Ricci scalar, which gives the curvature, is obtained by\n\\begin{eqnarray}\nR=-\\frac{4}{k}\n\\end{eqnarray}\n\n\n\n\n\n\n\\bibliographystyle{unsrt} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nWaves can exhibit wavefront dislocations or vortices, i.e. phase \nsingularities of the complex wavefuntion, where the modulus of the wave\nvanishes, and around which the phase of the wave changes by a multiple of \n2$\\pi$~\\cite{Berry:royal:1974}. \nIn~\\cite{Irvine:NatPhys:2008}, it has been shown that Maxwell's equations in fact admit solutions where \nthese lines of dislocation or vortex lines are tied into knots embedded in light fields.\nHere, the term ``knot'' refers to an embedding of a circle $S^1$ into a three-sphere $S^3$~\\cite{knot_book}.\nBoth theoretical~\\cite{Shabtay:OC:2003} as well as \nexperimental advances~\\cite{Padgett:NJP:2005,Whyte:NJP:2005,Dennis:Nature:2010,Shanblatt:OE:11} in three-dimensional light shaping allowed to \nthe experimental realization of knotted vortex lines embedded in optical fields. \nApart from optics, the study of knotted topological defect lines and their dynamics has fascinated scientists from diverse settings, including\nclassical fluid dynamics~\\cite{Moffatt:JFM:1969,Moffatt:nature:1990}, excitable media~\\cite{Paul:PRE:2003,Sutcliffe:PRL:2016}, \nchiral nematic colloids~\\cite{Tkalec:science:2011} to semiconductors~\\cite{Babaev:PRL:2002,Babaev:PRB:2009}.\n\nLord Kelvin speculated in 1867~\\cite{Kelvin:1867} that atoms can be described by vortex tubes in the aether.\nIn fact, almost two decades ago it was suggested that (nontrivial) knots might exist as stable solitons or Hopfions\nin three-dimensional field theories~\\cite{Faddeev:Nature:1997}. A Skyrme model served as an example for a field theory \nwhich admits stable knot solitons~\\cite{Paul:PRL:1998}. \n\nIn the context of spinor Bose-Einstein condensates (BECs)~\\cite{TinLun:PRL:98,Machida:JphysJpn:1998}, the\nexistence of knots with nonzero Hopf charge was studied~\\cite{Niemi:PRB:2002,Ueda:PRL:08,Ueda:PTPS:2010} and recently realized\nexperimentally~\\cite{Mottonen:NatPhys:2016}. However, there is work~\\cite{Speight:JGP:2010} that casts doubt on existence of\nstable Hopfions in two-component Ginzburg-Landau type systems. \n\nContrasting these studies, \nnumerical investigations on the robustness and centre of mass motion of {\\em knotted vortex lines} embedded in a {\\em single}\ncomponent BEC have been undertaken~\\cite{Barenghi:PRE:2012,Barenghi:JOP:2014}. \nSuch formations are typically unstable, since there is no topological stabilization mechanism and reconnections of \nvortex lines are allowed due to the occurrence of the quantum stress tensor in the hydrodynamic formulation of the Gross-Pitaevskii Equation. \nNucleation~\\cite{Frisch:PRL:1992} and reconnection of vortex lines~\\cite{Koplik:PRL:1993,Bewley:PNAS:2008,Barenghi:PhysFluid:2012} and vortex line bundles~\\cite{Barenghi:PRL:2008} \nand subsequent emission of sound waves~\\cite{Leadbeater:PRL:2001} have been theoretically studied extensively. \n\nIn this paper, we follow up on the idea formulated in~\\cite{Ruostekoski:PRA:2005}, and \ndiscuss a general experimental scheme to create a two-component BEC which contains a knotted vortex line in one \nof its components. \nWe suggest using a light field containing a knotted vortex line as probe field of a Raman-pulse that drives a \ncoherent two-photon Raman transition of three-level atoms with $\\Lambda$-level configuration [cf.~\\reffig{fig:lambda_scheme}(a)]. \nPreviously, similar methods have been used to create dark solitons and vortices~\\cite{Wright:PRL:1997,Dum:prl:98,Ruostekoski:PRL:2004,Ruostekoski:PRA:2005,Andersen:PRL:2006} \nand have been experimentally realized using microwave~\\cite{Cornell:PRL::1999_2,Cornell:PRL::1999} and more recently optical coupling~\\cite{Schmiegelow:ARXIV:2015}. \nThe pump beam will be the mentioned knotted (stationary) light beam, whereas \nthe control beam is a co-propagating plane wave to remove fast oscillations in the $z$-direction~\\cite{Ruostekoski:PRA:2005}.\nThe large controllability of the pulse parameters (i.e. strength and duration of the Rabi-pulse) \nallows for a large controllability of the excitation. \nThe dynamics of two-component BECs can be monitored in real time via in situ measurements without ballistic expansion~\\cite{Cornell:PRL:2000}. \nUsing numerical methods, we study excitation and subsequent dynamics of specific examples of knotted matter waves for experimentally feasible parameters. \n\nThis paper thus paves the way for experimentally accessing many of the \nphenomena discussed only theoretically in, e.g.~\\cite{Barenghi:PRE:2012,Barenghi:JOP:2014,Irvine:arXiv:2015},\nand if such system were realized experimentally, it would give controlled experimental access to reconnection of vortex lines,  \nsubsequent emission of sound waves and more generally quantum turbulence. \n\n\\section{Model}\\label{sec:artificial_light}\n\\subsection{Equations of motion for light-matter wave coupling}\n\nConsider a three-level atom in $\\Lambda$-configuration with ground states $\\ket{a}$ and $\\ket{b}$, which \nare off-resonantly coupled to an excited state $\\ket{e}$ with detuning $\\Delta$ and spatially dependent Rabi-frequencies\n$\\Omega_a({\\bf r})$ and $\\Omega_b(z)$. Assuming $\\Delta\\gg\\Omega_a,\\Omega_b$, the excited state can be \nadiabatically eliminated, and the dynamics of the condensate wave function confined in a trap $V({\\bf r})$ \ncan be described~\\cite{Dum:prl:98,Ruostekoski:PRA:2005} using~\\refeqs{eq:eqmo1}{eq:eqmo2}\n\\begin{eqnarray}\n\\fl i\\hbar\\partial_t\\psi_a&=\\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V({\\bf r}) + Ng_{aa} |\\psi_a|^2+Ng_{ab}|\\psi_b|^2\\right)\\psi_a+\\frac{\\hbar\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\Delta}\\psi_b\\label{eq:eqmo1}\\\\\n\\fl i\\hbar\\partial_t\\psi_b&=\\left(-\\frac{\\hbar^2}{2m}\\nabla^2 + V({\\bf r}) + Ng_{ba} |\\psi_a|^2+Ng_{bb}|\\psi_b|^2 -\\hbar\\delta \\right)\\psi_b+\\frac{\\hbar\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\Delta}\\psi_a\n \\label{eq:eqmo2}\n\\end{eqnarray}\nwith $g_{kj}=4\\pi\\hbar^2 a_{kj}\/m$, $a_{kj}$ denoting the scattering length between the species and \n\\begin{equation}\nV({\\bf r})= \\frac{1}{2}m\\omega^2 r_\\perp^2+\\frac{1}{2}m\\omega_z^2 z^2.\n\\end{equation}\n\\reffig{fig:lambda_scheme}(a) depicts the level scheme of three-level atoms in $\\Lambda$-type configuration.\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\textwidth]{fig1.eps}\n\\caption{\n(color online) (a) Schematic sketch of a Raman-type transition with $\\Lambda$-configuration. The state $\\ket{a}$ is\noff-resonantly coupled to state $\\ket{b}$ with two-photon detuning $\\delta$.\nThe knotted light field $\\mathcal{E}$ proportional to $\\Omega_a({\\bf r})$ off-resonantly couples state $\\ket{a}$ to $\\ket{e}$ with detuning $\\Delta$. \nThe final state $\\ket{b}$ then reflects the involved structure of the light field $\\mathcal{E}$ associated with $\\Omega_a({\\bf r})$.\n(b) depicts an isosurface of the intensity (purple) of the light-field $\\mathcal{E}$ at a low isointensity value and a slice of its \nphase in the $(x,y)$-plane. Additional to the vortex lines in the center of the beam, the usual diffraction cones become visible.\n(c) Illustration of the situation at $t=0$ for the parameters discussed in~\\ref{sec:trefoil}.\nThe isodensity of the cigar-shaped wave function $\\psi_a$ (yellow) is subject to the knotted light field (purple) from Fig. (b).\n\\label{fig:lambda_scheme}}\n\\end{center}\\end{figure}\nThe Rabi-frequencies $\\Omega_k$, $k=a,b$ relate to their corresponding electric fields via\n\\begin{equation}\n  \\Omega_k=\\frac{dE_k}{\\hbar},\n \\label{eq:Omega_a}\n\\end{equation}\nwhere the transition dipole element $d$ has been introduced. \nAs electric fields, we consider a monochromatic, quasi-linearly polarized light field with \nwavevector $k_0$. \nThen, its slowly varying envelope~$\\mathcal{E}$ is defined by \n\\begin{equation}\n E({\\bf r},t)=\\sqrt{\\frac{\\omega_0\\mu_0}{2k_0}}\\mathcal{E}({\\bf r})e^{i\\left(k_0z-\\omega_0t\\right)}+{\\rm c.c.},\n \\label{eq:sve}\n\\end{equation}\nwhere ${\\rm c.c}$ denotes the complex conjugate. \nWithin the paraxial approximation, the slowly varying envelope $\\mathcal{E}$ of the beam is governed by \n\\begin{equation}\n2ik_0\\partial_z \\mathcal{E}({\\bf r}_{\\perp},z) = -\\nabla^2_{\\perp}\\mathcal{E}({\\bf r}_{\\perp},z),\n\\label{eq:paraxial}\n\\end{equation}\nwhere ${\\bf r}_\\perp=(x,y)$, and $\\nabla^2_\\perp$ denotes the transverse Laplacian.\n\nSuch a system can be realized by considering the two hyperfine ground states $\\ket{a}=\\ket{S_{1\/2},F=2,M_F=-1}$, $\\ket{b}=\\ket{S_{1\/2},F=2,M_F=1}$ \nof ${}^{87}$Rb, that are off-resonantly coupled to an excited state manifold. \nThen, the scattering lengths between the species $a_{kj}$ are given by $a_{ba}=5.5$nm, and the ratio $a_{aa}$:$a_{ba}$:$a_{bb}$ is given by $1.03$:$1$:$0.97$~\\cite{Cornell:PRL:1998}. \nThe motivation for assuming cylindrical symmetry of the trapping potential $V({\\bf r})$ is associated with \nthe specific form of the paraxial wave equation~\\refeq{eq:paraxial}. \nWhereas we use different beams for trapping and Raman transition, the aspect ratio $\\omega_z\/\\omega$ \nfor our otherwise independent cigar shaped trap cannot be chosen arbitrarily.\nInstead, in order to accommodate the optical knot in the BEC (see~\\reffig{fig:lambda_scheme}(c)), \nwe have to make sure that extent of the BEC in the z-direction is large enough. \nThe aspect ratio of the optical beam (and the optical knot) can be expressed using the ratio between the Rayleigh \nlength $z_r=k_0\\sigma^2$ and the width $\\sigma$ of the light field in the \n$(x,y)$-plane:\n\\begin{equation}\n \\frac{z_r}{\\sigma}=\\frac{k_0\\sigma^2}{\\sigma}=k_0\\sigma.\n \\label{eq:aspect_ratio}\n\\end{equation}\nThe basic idea towards imprinting knotted vortex lines into BECs is depicted in~\\reffig{fig:lambda_scheme}(a).\nConsider the case where all atoms are initially in state $\\ket{a}$ [$\\psi_b(t=0)=0$].\nThen, assuming $\\Omega_i\\ll \\Delta$ and $\\delta\\approx 0$ the \nRabi pulse coherently transfers population from $\\ket{a}$ to $\\ket{b}$ as \nillustrated in according to~\\refeqs{eq:eqmo1}{eq:eqmo2}. The light field is depicted in~\\reffig{fig:lambda_scheme}(b) and together with the density of $\\psi_a$ in~\\reffig{fig:lambda_scheme}(c).\nThe laser associated with $\\Omega_b$ is chosen to be a simple plane wave, which coherently co-propagates with $\\mathcal{E}$, \nso that the fast variations $\\sim e^{ik_0z}$ in~\\refeq{eq:sve} are canceled whenever products $\\Omega_a\\Omega_b^*$, as in~\\refeqs{eq:eqmo1}{eq:eqmo2}, occur.\nThe explanation as to why such a setup allows to imprint the phase of the structured light field $\\Omega_a$ onto the condensate is the following.\nFor short time and two-photon detuning $\\delta=0$, the small change $\\delta\\psi_b$ to $\\psi_b$ is \ngiven by \n\\begin{equation}\n \\delta\\psi_b \\approx-\\frac{i}{\\hbar}\\frac{\\hbar\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\Delta}\\psi_a(t=0)\\delta t.\n \\label{eq:simple_argument}\n\\end{equation}\nThus, we may conclude that at least for small times, it should be possible to populate state $\\ket{b}$ \nwith a given phase profile using our light field. \nOnce the phase profile $\\theta$ has been imprinted ($\\mathcal{E}=|\\mathcal{E}|e^{i\\theta}$) onto \nthe condensate, the atoms will display motion according to ${\\bf v}=\\hbar\\nabla\\theta\/m$~\\cite{Ruostekoski:PRA:2005}. \nTo realize such a scenario, the intensity of the involved \nlaser beams must be sufficiently strong, such that the time-scales of the imprinting are small compared to\ntime-scales of the dynamics of the condensate.\nWe will use numerical methods and realistic experimental \nparameters to extend this simple idea beyond the perturbative limit and \nstudy its subsequent dynamics. \n\n\n\\subsection{Knotted Light Field}\\label{sec:knotted_light_field}\n\nLaguerre-Gaussian (LG) functions  $\\mathrm{LG}_{l,p}$ \nform a basis set for solutions for the paraxial wave equation~\\refeq{eq:paraxial}, and are given by the expression\n\\begin{eqnarray}\n  \\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z)=&\\sqrt{\\frac{p!}{\\pi (|l|+p)!}}\\frac{r_\\perp^{|l|}e^{il\\varphi}}{\\sigma^{|l|+1}}\\frac{(1-iz\/z_r)^p}{(1+iz\/z_r)^{p+|l|+1}}\\nonumber\\\\\n   & \\times e^{-r_{\\perp}^{2}\/2\\sigma^2(1+iz\/z_r)} L^{|l|}_p \\left( \\frac{r_{\\perp}^{2}}{\\sigma^2\\left[1+\\left(z\/z_r\\right)^2\\right]} \\right)\n   \\label{eq:LG_modes}\n\\end{eqnarray}\nwith $r_{\\perp}^2=x^2+y^2$ and $L^{|l|}_p$ being the associated Laguerre polynomials. \n\nWe seek a linear superposition of LG-modes, $\\sum_{l,p}a_{lp}\\mathrm{LG}_{l,p}$, \nthat describe light-beams containing knotted vortex lines. \nTo this end, we will review the method proposed in~\\cite{Dennis:Nature:2010}, which uses Milnor polynomials~\\cite{Milnor_book} as an ansatz for complex light fields\nin the shape of torus knots to determine appropriate amplitudes $a_{lp}$, and rescale for application to our setup.\n\nThe basic idea~\\cite{King:thesis,Dennis:Nature:2010} is to parametrize an $N$-strand braid as the roots of the polynomial \n\\begin{equation}\n p_h^{N,n}(u)=\\prod_{j=0}^{N-1} \\left[u-s_j(h)\\right]\n\\end{equation}\nHere, $h$ denotes the height of the periodic braid and $N$ denotes the number of strands or roots of $p_h(u)$ in $u$.\nLet us choose $N=2$ and~\\cite{King:thesis} \n\\begin{equation}\n s_j(h)=\\cos(h_j)+i\\sin(h_j), \\quad h_j=(h-2\\pi j\/n)n\/2.\n\\end{equation}\nThe projection of the braid onto the $(h=0)$-plane leads to a circle. The parameter $n$ represents the number of braid crossings. \nWe will consider the cases $n=2,3$ in the following. \nA small computation allows us to find an explicit expression for the Milnor polynomial $p_h$\nin the variables $u$ and $\\exp(ih)=:v$,\n\\begin{equation}\n p_h^{2,n}=u^2-v^n\n \\label{eq:milnor}\n\\end{equation}\nWe can now imagine a cylinder containing the braid, and ``glue'' top and bottom surfaces together to obtain a knot. In fact, one can show~\\cite{Alexander:PNAS:1923} that\nany knot can be represented as a closure of a braid. \nAn easy way to deform our cylinder into a torus and to make it explicitly dependent on ${\\bf{r}^\\prime}=(x^\\prime,y^\\prime,z^\\prime)$ \nis to write $u$ and $v$ as an inverse \nstereographic projection from three-dimensional space to a \nunit three-sphere, $\\mathbb{R}^3\\rightarrow\\mathbb{S}^3$, i.e.\n\\begin{eqnarray}\nu&=\\frac{r^{\\prime 2}-1+2iz^{\\prime}}{r^{\\prime 2}+1},\\\\\nv&=\\frac{2(x^{\\prime}+iy^{\\prime})}{r^{\\prime 2}+1}\\label{eq:stereo}\n\\end{eqnarray}\nHere, $r^\\prime=\\sqrt{x^{\\prime 2}+y^{\\prime 2}+z^{\\prime 2}}$ and the units $\\bf{r}^\\prime$ are non-dimensional. \nOne can easily show that $|u|^2+|v|^2=\\Re(u)^2+\\Im(u)^2+\\Re(v)^2+\\Im(v)^2=1$, \nand thus~\\refeq{eq:stereo} indeed represents a parametrisation of a three-sphere. \nSince we aim to describe an actual light field, i.e. a solution to the paraxial wave equation, \ninstead of considering the Milnor polynomial~\\refeq{eq:milnor} $p_h$ as it is, it is reasonable to get rid of the denominator in \n$p_h$ and to consider~\\cite{Dennis:Nature:2010,King:thesis} \n\\begin{eqnarray}\n\\xi_a({\\bf r}^\\prime)&=\\left(u^2-v^n\\right)\\left(r^{\\prime 2}+1\\right)^n.\n\\label{eq:ansatz_light}\n\\end{eqnarray}\nFor $n=2,{\\ }3$,~\\refeq{eq:ansatz_light} \ndescribes a Hopf link and a trefoil knot, respectively, which \nrepresent the simplest non-trivial examples of a link and a knot, respectively.\nThe latter two will be used as exemplary fields in the following. \n\nIn order to use~\\refeq{eq:ansatz_light} to find appropriate coefficients $a_{l,p}$, \nlet us rescale the light field~\\refeq{eq:LG_modes} by $R_s$ to nondimensional units $(x^\\prime,y^\\prime,z^\\prime)$, such that \n\\begin{equation}\n {\\bf r}_\\perp={\\bf r}^\\prime_\\perp R_s,\\quad\n \\frac{z}{z_r}=\\frac{z\/R_s}{k_0R_s\\sigma^2\/R_s^2}=\\frac{z^\\prime}{(k_0R_s) (\\sigma^2\/R_s^2)},\n\\end{equation}\nto equate the light field to our knot in the same dimensionless units. \nEffectively, $R_s$ scales the abstract unit sphere to have a transverse extent of approximately $R_s$ with respect to the laser beam.\nHence, there are two length scales of our system, $R_s$ and $\\sigma$, that are associated with the nodal lines of the knot and the transverse extent of the beam. \nAs we will see, the ratio $w=\\sigma\/R_s$ will play a crucial role as an important degree of freedom, that allows us to change the width of the \nbeam relative to the positions of the nodal lines of the knot. \nTo find appropriate superpositions of ${\\mathrm{LG}}$-modes, it is sufficient to restrict considerations to the $(x,y)$-plane only.\nLet us equate~\\refeq{eq:ansatz_light} with a superposition of the above-mentioned rescaled version of~\\refeq{eq:LG_modes}: \n\\begin{eqnarray}\n\\xi_a({\\bf r}_\\perp^\\prime,z^\\prime=0) = \\sum_{l,p} a_{l,p}(w)\\mathrm{LG}_{l,p}^{w,k_0R_sw^2}\\left({\\bf r}_\\perp^\\prime,z^\\prime=0\\right)\\sqrt{\\pi}e^{r_{\\perp}^{\\prime 2}\/2w^2}w.\n\\label{eq:determine_alp}\n \\end{eqnarray}\nComparing different powers in $r_\\perp^\\prime$ allows us to determine the finite number of coefficients $a_{l,p}$ uniquely,\nwhich depend only on the real number $w$. Whereas this ansatz can be used for some knots, there are counterexamples, and \nthere is no rigorous general proof as to why and in which cases this ansatz leads to success. \nA large value of $w$ ensures that the vortex lines \nare actually embedded in the beam, and not chopped off.\nFurthermore, for finite $w$, additional vortex lines in the shape of hairpins appear for larger $z$-values, which leads to the fact that \n$w$ must be chosen large enough. On the other hand, if $w$ is too large, \nthe polynomial increase in the polynomials describing the knots will not be attenuated quickly enough\nby the Gaussian, and thus intensity variations become huge, which is undesirable for our setup. \nIt is possible~\\cite{Dennis:Nature:2010} to further optimize these coefficients to separate vortices with regions of \nlarger intensity. \nOnce appropriate coefficients $a_{l,p}$ have been found, the light field can be \nwritten down as superposition of LG modes by rescaling back\ninto physical units. For the sake of clarity, let us introduce an auxiliary function $f$ defined as \n$f(r_\\perp\/\\sigma,z\/z_r)\/\\sigma:=\\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z)$. Then, we find \n\\begin{eqnarray}\n  \\mathcal{E}_a({\\bf r}_\\perp,z)&=\\frac{A}{R_s}\\sum_{l,p}a_{l,p}(w)\\mathrm{LG}_{l,p}^{w,k_0R_sw^2}({\\bf r}_\\perp^\\prime,z^\\prime),\\\\\n  &=\\frac{A}{R_s}\\sum_{l,p}a_{l,p}(w)\\frac{1}{w}f\\left(\\frac{r_\\perp^\\prime}{w},\\frac{z^\\prime}{k_0R_sw^2}\\right),\\\\\n  &=A\\sum_{l,p}a_{l,p}(w)\\frac{1}{\\sigma}f\\left(\\frac{r_\\perp}{\\sigma},\\frac{z}{z_r}\\right)\\\\\n  &=A\\sum_{l,p}a_{l,p}(w)\\mathrm{LG}_{l,p}^{\\sigma,z_r}({\\bf r}_\\perp,z).\n  \\label{eq:superpos}\n\\end{eqnarray}\nHere, the amplitude $A$ of the light field has been introduced, which gives the intensity of the beam the right value in \nappropriate units. \nNote, that~\\refeq{eq:superpos} is no longer dependent on the choice of $R_s$. \n\n\\subsection{Rescaling}\n\nLet us rescale~\\refeqs{eq:eqmo1}{eq:eqmo2} by \nintroducing spatial $r^\\prime$ and time $t^\\prime$ coordinates rescaled \nby oscillator length $a_0=\\sqrt{\\hbar\/m\\omega}$ and trapping frequency $\\omega$, respectively,\n\\begin{eqnarray}\n t^\\prime&=\\omega t,\\\\\n r^\\prime&=\\frac{r}{a_0},\\\\\n \\psi^\\prime&=a_0^{3\/2}\\psi.\n\\end{eqnarray}\nThen, after multiplying~\\refeqs{eq:eqmo1}{eq:eqmo2} by $1\/m\\omega^2\\sqrt{a_0}$, ~\\refeqs{eq:eqmo1}{eq:eqmo2} become\n\\begin{eqnarray}\n \\fl i\\partial_{t^\\prime}\\psi_a^\\prime&=\\left(-\\frac{1}{2}\\nabla^{\\prime 2} +  \\frac{r_\\perp^{\\prime 2}}{2} + \\gamma_z^2  \\frac{z^{\\prime 2}}{2} + \\kappa_{aa} |\\psi_a^\\prime|^2+\\kappa_{ab}|\\psi_b^\\prime|^2\\right)\\psi_a^\\prime\n +\\frac{\\Omega_a({\\bf r}^\\prime)\\Omega_b^*(z^\\prime)}{8\\omega\\Delta}\\psi_b^\\prime\\label{eq:rescaled_eqmo1}\\\\\n \\fl i\\partial_{t^\\prime}\\psi_b^\\prime&=\\frac{\\Omega_a^*({\\bf r}^\\prime)\\Omega_b(z^\\prime)}{8\\omega\\Delta}\\psi_a^\\prime+ \n \\left(-\\frac{1}{2}\\nabla^{\\prime 2} +  \\frac{r_\\perp^{\\prime 2}}{2} + \\gamma_z^2  \\frac{z^{\\prime 2}}{2} + \\kappa_{ba} |\\psi_a^\\prime|^2+\\kappa_{bb}|\\psi_b^\\prime|^2-\\frac{\\delta}{\\omega}\\right)\\psi_b^\\prime\n \\label{eq:rescaled_eqmo2}\n\\end{eqnarray}\nwhere we left away the prime for our non-dimensional time and space variables and introduced $\\gamma_z=\\omega_z\/\\omega$, and $\\kappa_{kj}=4\\pi a_{kj}N\/a_0$.\nWe can express as $\\kappa_{kj}=a_0^2\/2a_h^2$ using the healing length $a_h=\\sqrt{8\\pi a_{kj} N\/a_0^3}$.  \nSince the aspect ratio $\\gamma_z$ of the trapping frequencies is typically small, it is more convenient for numerical studies to \nrescale the elongated $z$-axis as follows:\n\\begin{eqnarray}\n z^{\\prime\\prime}&=\\gamma_z z^\\prime,\\\\\n \\psi^{\\prime\\prime}_i&=\\frac{1}{\\sqrt{\\gamma_z}}\\psi_i^\\prime.\n\\end{eqnarray}\nThis rescaling finally yields our equations of motion:\n\\begin{eqnarray}\n\\fl   i\\partial_t\\psi_a&=\\left(-\\frac{1}{2}\\left[\\nabla^2_\\perp + \\gamma_z^2\\partial_z^2 \\right] + \\frac{r^2}{2} + \\gamma_z\\kappa_{ja} |\\psi_a|^2+\\gamma_z\\kappa_{jb}|\\psi_b|^2\\right)\\psi_a\n  +\\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\omega\\Delta}\\psi_b\\label{eq:rescaled_eqmo_final1}\\\\\n\\fl  i\\partial_t\\psi_b&=\\left(-\\frac{1}{2}\\left[\\nabla^2_\\perp + \\gamma_z^2\\partial_z^2 \\right] + \\frac{r^2}{2} + \\gamma_z\\kappa_{ja} |\\psi_a|^2+\\gamma_z\\kappa_{jb}|\\psi_b|^2-\\frac{\\delta}{\\omega}\\right)\\psi_b\n  +\\frac{\\Omega_a^*({\\bf r})\\Omega_b(z)}{8\\omega\\Delta}\\psi_a\n  \\label{eq:rescaled_eqmo_final2}\n\\end{eqnarray}\nwhere we have dropped the primes for convenience. \nFurthermore, applying the same rescaling to the paraxial wave equation~[\\refeq{eq:paraxial}], we find the following expression in \nour rescaled units:\n\\begin{equation}\n2ik_0\\partial_z \\mathcal{E}({\\bf r}_{\\perp},z) = -\\nabla^2_{\\perp}\\mathcal{E}({\\bf r}_{\\perp},z).\n\\label{eq:paraxial_rescaled}\n\\end{equation}\nHere, $k_0$ has been redefined and stands for $k_0=2\\pi a_0\\gamma_z\/\\lambda$. \nThen, the functions defined in~\\refeq{eq:LG_modes} remain solutions to~\\refeq{eq:paraxial_rescaled} in the new coordinates, \nand using our modified $k_0$, we can still write the Rayleigh length as $z_r=k_0\\sigma^2$, where \n$z_r$ is measured in units of $a_0\/\\gamma_z$ and $\\sigma$ in units of $a_0$.\n\nWe have $m=1.44\\times 10^{-25}$kg for ${}^{87}$Rb, so that a trapping frequency of $\\omega=2\\pi\\times 10$Hz leads to \nan order of magnitude of $a_0=3.4\\mu$m for the typical transverse extent of the BEC. \nAn order of magnitude estimate for the off-diagonal coupling terms is given by \n\\begin{equation}\n \\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{8\\omega\\Delta}\\approx \\frac{\\Omega_a({\\bf r})\\Omega_b^*(z)}{\\Delta}0.016 {\\mathrm s} \\approx \\Range{2d2}{2d6}\n\\end{equation}\nfor $\\Omega_i\\approx \\Range{1}{10}$MHz and $\\Omega_i\/\\Delta=\\Range{0.01}{0.1}$. \nThe required tight focusing of the beam leads to the fact, that we only need \nmoderate beam powers to achieve adequate Rabi-frequencies. \nIn the following, we set the two-photon detuning $\\delta=0$ to achieve optimal transfer. \n\nFinally, we need to find an estimate for the Rabi pulse duration $t_d$. \nTo this end, consider the simple case, where $\\Omega_a$ and $\\Omega_b$ describe driving by plane waves. \nWe seek to find the optimal time for maximal transfer of atoms from $\\ket{a}$ to $\\ket{b}$. \nGiven that the ``off-diagonal'' terms in the coupled equations~\\refeqs{eq:rescaled_eqmo_final1}{eq:rescaled_eqmo_final2} are much larger than the diagonals, we can approximate \nthe population $N_b$ in state $\\ket{b}$ as \n\\begin{equation}\nN_b\\sim 1-\\cos^2\\left(\\frac{\\Omega_a\\Omega_b^*}{8\\Delta \\omega}t\\right).\n\\end{equation}\nThis expression allows us to find the optimal (non-dimensional) time duration of the Rabi pulse $t_d$ that leads to a complete transfer of population:\n\\begin{equation}\n t_d=\\frac{\\pi\/2}{\\Omega_a\\Omega_b^*\/8\\Delta \\omega}.\n \\label{eq:t_d}\n\\end{equation}\nOn the other hand, this consideration does not apply to our case due to the spatial dependence of the Rabi frequency $\\Omega_a({\\bf r})$. \nDue to spatial dependence of the intensity variations, we do not really have an optimal pulse duration \n$t_d$, and our choice of $t_d$ is a compromise between on one hand \nhaving sufficiently large transfer of atoms to state $\\ket{b}$\nand on the other hand avoiding population being transferred back from\nstate $\\ket{b}$ to state $\\ket{a}$ in positions where the intensity is large and thus the transfer \ndynamics is faster.\nHence, we have to choose shorter pulse durations than what~\\refeq{eq:t_d} suggests. \nFurthermore, due to the nodal lines in the pump beam, it is never possible to use this arrangement for \na total conversion of atomic population from $\\ket{a}$ to $\\ket{b}$. \nWith these aspects kept in mind, we may still use~\\refeq{eq:t_d} as a rough estimate for the order of magnitude for our choice of $t_d$.\n\n\\section{Numerical Investigation}\n\nIn the previous section, we elaborated on how to inscribe complex knotted vortex lines into matter waves. Whereas the \nsimple argument of~\\refeq{eq:simple_argument} allows us to conclude that our setup should work at least in the perturbative regime for \nshort time-scales and small amounts of atoms in state $\\ket{b}$, we cannot infer what happens beyond this perturbative regime. \nIn this section, we aim to numerically study excitation and decay of our highly excited states into more elementary unknots or \nvanishing of the latter by collision and annihilation of vortex lines. \nWe will do so by considering specific examples. However, these examples are by no means an exhaustive treatment of the\nthe huge potential manifold of realizations possible. To thoroughly understand the dynamics remains \nan elusive goal, which goes beyond the scope of this paper. \n\n\\subsection{Hopf-link}\n\nOne of the simplest torus knots is the so-called Hopf-link, which corresponds to setting $n=2$ in~\\refeq{eq:ansatz_light} and is \ndepicted e.g. by the green isodensity surface in~\\reffig{fig:hopf_projection}(b). Using the method outlined in~\\ref{sec:knotted_light_field} and~\\refeq{eq:determine_alp}, we find the following superposition \ncoefficients of Laguerre-Gaussian polynomials for a Hopf-link:\n\\begin{eqnarray}\n  \\frac{\\Omega_a\\Omega_b^*}{8\\omega\\Delta} &= A[(1-2w^2+2w^4)\\mathrm{LG}_{00}+(2w^2-4w^4)\\mathrm{LG}_{01}+2w^4\\mathrm{LG}_{02}\\nonumber\\\\\n  &-4\\sqrt{2}w^2\\mathrm{LG}_{20}]\\Theta(t_{d}-t).\n\\end{eqnarray}\nTo this end, let us consider the dynamics for $\\kappa_{ab}=4\\pi a_{ab} N\/a_0=1887$, $\\gamma_z=0.125$. \nWe set the two-photon detuning to $\\delta=0$.\nThis can be achieved by using typical parameters of ${}^{87}$Rb with $\\omega=2\\pi\\times 10 $Hz, $\\omega_z=2.5\\pi $Hz, $N=93000$, \n$a_{ab}=5.5$nm, $\\lambda=780$nm, which amounts to typical \nunits of length scales of $a_0=3.4\\mu$m in the $(x,y)$-plane and $a_0\/\\gamma_z=27.2\\mu m$ in the $z$-direction.\nThomas-Fermi like dynamics, where the nonlinearity is large compared to the broadening due to the Laplacian, \nis preferable to ensure robustness of the vortex cores and avoid the trivial broadening of the latter. \nFor that reason, we have to choose a sufficiently large value for $\\kappa_{ij}$.\nFor the other $a_{ij}$, we assume the ratio $a_{aa}$:$a_{ab}$:$a_{bb}$ to be given by $1.03$:$1$:$0.97$~\\cite{Cornell:PRL:1998}.\nWith respect to our Rabi pulse, we choose \n$A=450$, $w=1.5$, $\\sigma=0.675$ corresponding to $2.3\\mu$m, \nand a pulse duration of $t_{d}=0.002$ corresponding to $31.8\\mu$s.\n\nWe use the common Fourier split-step method~\\cite{Agrawal_Kivshar:2003} and adaptive Runge-Kutta algorithm to fourth order for the time-step for numerical computation. \nParts of the code were written using~\\cite{XMDS}. To find the appropriate state $\\psi_a(t=0)$, we used imaginary time-evolution (e.g.~\\cite{Tosi:PRE:2000}).\nIt is crucial to use a variable time-step, since the dynamics during the time duration of the Rabi-pulse $t_d$ has to be temporally \nfully resolved, and thus the time-step has to be much smaller than $t_d$ within $t_d$. After $t>t_d$, the time-step can be chosen \nmuch larger, since it only needs to resolve the characteristic time-scale of the BEC dynamics. We used time-steps varying between $\\Delta t=\\Range{d-8}{5d-4}$ and\na spatial resolution of $\\Delta x=0.05-0.06$ with $220^3$ points.\n\n\\reffig{fig:hopf_projection}(a-d) depict an isodensity surface of the condensate wave function $\\psi_b$ with small value,  \nwhich serves as an illustration of the knotted vortex core, and its phase profile in different ways, shortly after the imprinting occurred.\nHere, the initial wave function $\\psi_a(t=0)$ was computed as the ground state of the trap and $\\psi_b(t=0)=0$. \n\nA basic qualitative anticipation of the dynamics can be found by \nprojecting the knot orthogonally to the beam- or $z$-axis and considering the momentum associated with \nthe (unit-)areas, as elaborated in~\\cite{Ricca:Proc:2013}. Consider the projection in the $(x,y)$-plane, as \nshown in~\\reffig{fig:hopf_projection}(c). \nThe index $I_j$ assigned to each of the areas $R_j$ enclosed by the vortex lines $\\gamma$ can be found by evaluating the sum~\\cite{Ricca:Proc:2013}\n\\begin{equation}\n I_j=\\sum_{{\\bf \\rho}\\cap {\\bf \\gamma}} \\mathrm{sign} \\left( {\\bf e_z} {\\bm \\rho} \\times {\\bf t} \\right),\n \\label{eq:I_j}\n\\end{equation}\nwhich runs over all intersections of the chosen vector ${\\bf\\rho}$ with the projected vortex line $\\gamma$ for a given region $R_j$. \nHere, ${\\bm \\rho}$ denotes an arbitrary vector pointing from the inside to the outside of the enclosed area \nand $\\mathrm{sign}$ is the usual sign-function taking the values $\\mathrm{sign}(\\ast)=\\pm 1$. \nFurthermore, the vector $\\bf{t}$ is a tangent vector to the vortex line curve (denoted as $\\gamma$) at the point where ${\\bm \\rho}$ crosses $\\gamma$. \nThe direction of $\\bf t$ is given by the orientation of the arcs, which is determined by the phase~[see~\\reffig{fig:hopf_projection}(c)].\nThe values assigned to the regions in~\\reffig{fig:hopf_projection}(c) correspond to the values computed by~\\refeq{eq:I_j}.\nThe associated momenta of a region $R_j$ can be found by~\\cite{Ricca:Proc:2013}\n\\begin{equation}\n \\left({\\bf p}_z\\right)_j=\\oint_{R_j} {\\bm \\omega} d^2r I_j,\n\\end{equation}\nwhere $\\bm \\omega$ denotes the vorticity. \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig2.eps}\n\\caption{\n(color online) (a) Illustration of the phase of $\\psi_b$ in the $(x,y)$-plane after short time evolution. (b) Additional to the phase in the $(x,y)$-plane from Fig. (a), \nthe green surface represents a low-value isodensity surface of $|\\psi_b|^2$ around the \nvortex line imprinted on the BEC. (c) Projection of Fig (b) into the $(x,y)$-plane and associated indices according to~\\refeq{eq:I_j}. Expected dynamics should thus be, that \nthe central region moves faster in the $z$-direction relative to its neighbouring regions. \n(d) shows again the isosurface from Fig. (b), however, the coloring of the isosurface illustrates the value of the phase at each position of the isosurface. \n\\label{fig:hopf_projection}}\n\\end{center}\\end{figure}\n\nWhat one can deduce from these arguments is a movement in the positive $z$-direction with the central region traveling fastest [\\reffig{fig:hopf_projection}(c)]. \nLet us now look at the numerically computed dynamics. \nSnapshots of the latter are shown in~\\reffig{fig:hopf_to_unknot} (see also movies \\href{hopf_1.mp4}{\\textit{hopf\\_1.mp4}} and \\href{hopf_2.mp4}{\\textit{hopf\\_2.mp4}}).\nClearly, there is an overall center-of-mass mass motion towards the positive $z$-direction, as expected. \nHowever, reconnection of vortex lines as well as finite size effects of the condensate leading to sharp gradients in the density lead to a very involved dynamics, \nwhich we were not able to predict or understand in simple terms.\nSurprisingly, the formation decays into two unknots that propagate into the positive and negative $y$-direction, respectively. \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig3.eps}\n\\caption{\n(color online) Dynamics of the condensate wavefunction $\\psi_b$, whose vortex lines form a Hopf-link. Both upper and lower row show the dynamics in timesteps of \n$\\Delta t=0.1$ (corresponding to $\\Delta t=1.6$ms) starting from $t=0.05$ (corresponding to $t=0.8$ms). \nBoth rows use the same isosurface for $|\\psi_b|^2$, however, in the lower row we cropped the box at a smaller value, so that the ``boundary'' of\nthe condensate does not obfuscate the dynamics of the vortex lines. Additional to that, we used the phase to color the isosurface in the lower row. \nWe can see an overall drift the positive $z$-direction, which can be understood \nfrom~\\reffig{fig:hopf_projection}. \nSee also the movies \\href{hopf_1.mp4}{\\textit{hopf\\_1.mp4}} and \\href{hopf_2.mp4}{\\textit{hopf\\_2.mp4}} for the full propagation dynamics.\n\\label{fig:hopf_to_unknot}}\n\\end{center}\\end{figure}\n\n\\subsection{Trefoil}\\label{sec:trefoil}\n\nIn this section we will study the dynamics of a trefoil imprinted on a BEC. The dynamics is generic, quite different initial conditions lead to similar results,\nthe basics of which can be understood in the simple terms described before. \n\nIn this case, let $w=1.2$. Instead of using the prefactors $a_{l,p}(w)$ from~\\refeq{eq:determine_alp}, \nwe use the optimized prefactors given in~\\cite{King:thesis,Dennis:Nature:2010} of the Laguerre-Gaussian modes for the light field.  \n\\begin{equation}\n \\fl \\frac{\\Omega_a\\Omega_b^*}{8\\omega\\Delta} = A( 1.51\\mathrm{LG}_{00} - 5.06\\mathrm{LG}_{10} + 7.23\\mathrm{LG}_{20}- 2.03\\mathrm{LG}_{30} - 3.97\\mathrm{LG}_{03} )\\Theta(t_{d}-t).\n\\end{equation}\nTo this end, let us use $\\sigma=0.78$ corresponding to $2.65\\mu$m and $A=700$. \nFurthermore, let us choose $\\kappa = 3753$, $\\gamma_z = 0.05$. \nThis can be realized using again $\\omega=2\\pi\\times 10 $Hz, $\\omega_z=1\\pi $Hz,\n$N=1.85\\times10^5$, $a_{ab}=5.5$nm and $\\lambda=780$nm. \nWith respect to our Rabi pulse, we use\na pulse duration of $t_{d}=0.004$ corresponding to $63.7\\mu$s.\n\n\\reffig{fig:trefoil_projection}(a) illustrates the phase of $\\psi_b$ and (b) its vortex core shortly after imprinting. \nProjecting onto the $(x,y)$-plane and assigning values according to~\\refeq{eq:I_j} leads to~\\reffig{fig:trefoil_projection}(c).\nAgain, we can expect that the overall knot will propagate in the positive $z$-direction, and the central part carries the largest momentum.\n\\reffig{fig:trefoil_projection}(d) illustrates the expected reconnection dynamics according to the rules found in e.g.~\\cite{Koplik:PRL:1993}.\nThe arrows indicate the direction of ${\\bf t}$, which is again determined by the phase. The inset (grey dashed box) illustrates how \nthe vortex lines should reconnect. If we apply this simple rule to all three crossings, we can expect that the green vortex line evolves into what is depicted by the solid \nblack line, i.e. a decay into two unkots.\n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\columnwidth]{fig4.eps}\n\\caption{\n(color online) (a) Illustration of the phase of $\\psi_b$ in the $(x,y)$-plane after short time evolution. (b) Additional to the phase in the $(x,y)$-plane from Fig. (a), \nthe green line represents a low-value isodensity surface of $|\\psi_b|^2$ around the \nvortex line imprinted on the BEC. (c) Projection of Fig (b) into the $(x,y)$-plane and associated indices. \nSince the indices reflect the momenta of the regions, we expect that \nthe central region moves faster in the $z$-direction relative to its neighboring regions. \n(d) shows the expected breakup of the green vortex lines \ninto the solid black lines. The black arrows illustrate the orientation, which is determined by the phase. The grey dashed inset illustrates \nthe expected product of collision and reconnection of the vortex lines according to, e.g.~\\cite{Koplik:PRL:1993}.\n\\label{fig:trefoil_projection}}\n\\end{center}\\end{figure}\n\nLet us now consider the dynamics, which is shown in~\\reffig{fig:vortex_expelled} (see also the movie \n\\href{trefoil_1.mp4}{\\textit{trefoil\\_1.mp4}} and \\href{trefoil_2.mp4}{\\textit{trefoil\\_2.mp4}} for full dynamics).\nWe see that the vortex lines of this specific trefoil knot first reconnect [see~\\reffig{fig:vortex_expelled}], and the \ncentral regions travels fastest into the positive $z$-direction according to our expectation~\\reffig{fig:trefoil_projection}(b--c).\nAfter that, the central region expels an unknot or vortex ring which decouples from the rest of the knot, as shown in~\\reffig{fig:vortex_expelled}. \nThe single unknot propagates to the positive $z$-direction faster than the remaining part of the knot. \nInterestingly, our dynamics differs from the usual dynamics \nof a clear breakup into two unknots as expected from [\\reffig{fig:trefoil_projection}(c), solid black line] \nand what has been found in~\\cite{Barenghi:PRE:2012,Irvine:arXiv:2015}. \nDifferences to previous observations are due to finite size effects of the cloud, that in our case \nthe knot has a large aspect ratio, which with our rescaling basically means that the dynamics in the $z$-direction is much slower, smallness of the knot, and finally that we \nactually consider two coupled fields (dynamics of $\\psi_a$ not shown). \n\n\\begin{figure}\\begin{center}\n\\includegraphics[width=\\textwidth]{fig5.eps}\n\\caption{\n(color online) Dynamics of the condensate wavefunction $\\psi_b$, whose vortex lines form a trefoil knot. Both upper and lower row show the same dynamics in timesteps of \n$\\Delta t=0.1$ (corresponding to $\\Delta t=0.8$ms) starting from $t=0.05$ (corresponding to $t=0.8$ms). \nBoth rows use the same isosurface for $|\\psi_b|^2$, however, in the lower row we cropped the box at a smaller value, so that the ``boundary'' of\nthe condensate does not obfuscate the dynamics of the vortex lines. Additional to that, we used the phase to color the isosurface in the lower row. \nUpon evolution, vortex cores reconnect, which leads to a decay into a vortex ring being expelled from \nthe central region, which can be understood from~\\reffig{fig:trefoil_projection}).\nSee also the movie \\href{trefoil_1.mp4}{\\textit{trefoil\\_1.mp4}} and \\href{trefoil_2.mp4}{\\textit{trefoil\\_2.mp4}} for the full propagation dynamics.\n\\label{fig:vortex_expelled}}\n\\end{center}\\end{figure}\n\n\\section{Conclusions}\nRecently there has been a growing interest in the dynamics of knotted vortex lines in BECs. \nHowever, thus far there has been no actual proposal on how to excite such matter waves in a controlled fashion. \nIn this work, we have presented a setup with physically realistic parameters to create such knotted vortex lines in ultracold matter waves. \nWe combined recent theoretical and experimental results from complex light shaping which allowed us to create knotted vortex lines embedded in light fields. We \nuse the latter as a probe field for three-level atoms in a $\\Lambda$-type setup to inscribe the nodal lines to BECs. \nThe setup is quite generic, and allows investigation of a large variety of potential dynamics. \nThe finite time span to populate state $\\psi_b$ reduces production of soundwaves. \nThe finiteness of the condensate, the smallness of the knot as well as the reconnections of the vortex lines give\nrise to a very involved dynamics. \nThe velocity in the $z$-direction of the vortex knot should not be large.  \nThis velocity can be controlled by the ratio between the trapping frequencies $\\gamma_z=\\omega_z\/\\omega$ relative to the condensate.  \nThe probe field which determines $\\Omega_a$ should not be too spatially broad, otherwise the value of $\\gamma_z$ required to fully embed the knotted vortex line \nbecomes impractical. We have assumed that the light field can be treated within the paraxial approximation, although we note that, in principle, \nthis condition can  be relaxed to non-paraxial knotted light fields~\\cite{Dennis:OL:11}. \nDue to the tight focusing of the optical beam, only moderate powers of the light fields that determine the Rabi-frequencies $\\Omega_i$ are required. \nThus, the setup should work generically for a large range of parameters. \nWe have shown two illustrative examples of a trefoil knot and a Hopf-link, and discussed the dynamics using already well-established techniques. \nThe data presented in this paper is available online at~\\cite{Maucher:data}.\n\n\\section{Acknowledgements} \nThis work was funded by the Leverhulme Trust Research Programme Grant RP2013-K-009, SPOCK: Scientific Properties Of Complex Knots. \nWe would like to thank P. M. Sutcliffe, J. L. Helm, T. P. Billam, D. Sugic and M. Dennis for stimulating discussions. \n\n\\section*{References}\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nDetailed understanding of non-contact friction and energy transfer processes in nanostructures is of great importance, both \nfrom the conceptual and practical viewpoints.\nExisting theoretical studies, starting with the seminal paper by Pendry \\cite{Pendry1}, mostly consist of calculations of friction coefficients, i.e. friction force between two parallel dielectric plates (e.g. supported graphenes) in uniform relative motion which is experimentally not easily measured (e.g. current drag in one graphene caused by  current flow in another \none) \\cite{Theory1,Theory2,Theory3,Theory4,Theory5,Theory6,Theory7}.\n\nWhile the experiments with two slabs in parallel relative motion with constant velocity are difficult to perform, we suggest here \nthat for the same systems experiments with slabs in relative oscillatory motion with fixed or variable frequency might be easier to \nperform, and could lead to new and interesting observations. \nRecently a similar approach has been realized experimentally \\cite{QF_exp_osc1,QF_exp_theory_osc,QF_exp_osc2,QF_exp_osc3}.\nIn these experiments the system (usually an AFM tip above the surface) oscillates at some characteristic \nfrequency. These oscillations are then, because of various dissipation mechanisms (which includes quantum friction), damped.       \nOur model is based on a slightly different concept; one of the slabs, e.g. the AFM tip, is driven with variable frequency. This means that the friction can \nbe deduced from the energy dissipated in one oscillating cycle. \nIn this paper we provide a general theoretical description of such processes, expecting that this method might become a useful tool to study dynamical properties of \nlow-dimensional systems \\cite{Munez}.\n\n\nThe main objective of this paper is therefore a theoretical description of these phenomena in systems consisting \nof two non-touching polarizable media, specifically conservative (van der Waals or Casimir) and \ndissipative forces (quantum friction) between two quasi-twodimensional (q2D crystals) in \nrelative parallel and oscillatory motion. While the case of slabs in parallel uniform motion has been extensively studied \\cite{Pendry1,Persson1,Volokitin1,\nVolokitin2,Theory4,Pendry2,Philbin}, here we develop an analogous theory describing interaction of atomically thick slabs (q2D crystals) in oscillatory motion. \n\n\n           \nIn Sec.\\ref{Sec2} the expressions for van der Waals and dissipative energies and forces are derived for such a q2D system in a very general case,\nfor variable slab temperatures and dynamical properties characterized by their surface response functions $D_1$ and \n$D_2$, and for variable oscillating frequencies and amplitudes. \nWe assume 2D translational invariance and neglect retardation for the slab distances in consideration. \nFor the sake of clarity and comparison, in Appendix\\ref{AppA} we derive analogous results for the \ncase of parallel uniform motion, recovering but also generalizing some earlier results \\cite{trenjePRB,PFK}. \n    \n  \n\nIn Sec.\\ref{DerofD} we derive general expressions for surface response functions $D_i$ for multilayer slabs, later to \nbe specified  for monolayers of a substance like graphene or silicene adsorbed on dielectric substrates. Surface response functions \n$D_1$ and $D_2$ will be the key ingredients in the expressions describing dissipative and \nreactive processes in Sec.\\ref{Sec2} and Sec.\\ref{DerofD}.  \nIn Sec.\\ref{DerofD} we also show how to calculate surface response functions $D_i$ for a specific case of q2D crystals on a dielectric substrate\nThe expression for the surface excitation propagator of a system of \ntwo coupled slabs is also derived.\n\n      \nIn Sec.\\ref{nabnak} we present the models used to describe the q2D crystal  and substrate dynamical response. \nWe study the specific case of a graphene monolayer on a dielectric substrate, which is chosen to be ionic  \ncrystal SiO$_2$. \nThe substrate is considered as a homogenous semiinfinite ionic crystal SiO$_2$ with the appropriate dielectric function in \nthe longwavelength limit. Graphene monolayer dynamical  response is determined from \nfirst principles. Also some computational details are specified. \n\n\nIn Sec.\\ref{resuuult}  general expressions of previous sections are applied to the system of \ntwo slabs, where each slab represents a graphene($E_{Fi}$)\/SiO$_2$ system, and where graphene doping is characterized \nby Fermi energy $E_F$ relative to the Dirac point. \n\n   \nIn Sec.\\ref{cmspe} we demonstrate how  the spectra of electronic excitations in one slab \nand in two coupled slabs depend on graphene doping $E_F$.\n \n\nThe form of these coupled ecitations is responsible for the behaviour of the atractive forces and dissipation.\nWe first discuss in Sec.\\ref{weakvdW} the modification of van der Waals force for oscillating in comparison \nwith the static slabs. Van der Waals energies depend on two factors. They increase with the increased \ngraphene doping, but are reduced for  the asymmetric doping when excitations in two slabs are off-resonance. \nDynamical vdW energy shows unusual behavior: it starts as  plateau,  and then decreases.\nThis is, because the fast Dirac plasmon in one slab for low driving frequencies $\\omega_0<\\omega_p$, still  perfectly follows Doppler shifted  charge density fluctuations in another slab. \nFor larger driving frequencies this is not the case and vdW energies decrease.    \nFinally,  for small or zero doping the $\\pi\\rightarrow\\pi^*$  and $\\pi\\rightarrow\\sigma$ excitations \ncause linear weakening of the dynamical vdW energy.   \n       \nIn Sec.\\ref{DISsub} we calculate and discuss how dissipated power depends on various parameters: \ndriving amplitude $\\rho_0$ and frequency  $\\omega_0$, on the separations between slabs $a$ and on the \nsubstrate. We find simple $\\rho^2_0$ dependence, while the $\\omega_0$ dependence is determined \nby the intensity of resonant coupling between hybridized Dirac plasmons and substrate TO phonons.\nWe found that in realistic grahenes (in comparison with Drude model when excitation of undamped Dirac plasmons provides \nunrealistically strong $2\\omega_p$ peak in the dissipated power)  the  dissipation power peak is strongly reduced and  red shifted.\nWe also explain why the substrate substantially reduces dissipated power peak.  \nFor larger separations $a$ additional peaks appear in dissipated power originating from the excitations of hybridized substrate phonons.     \n\nIn Sec.\\ref{DISdop} we explore how the dissipated power depends on graphene dopings.\nWe show that if one graphene is pristine ($E_F=0$) it causes the disappearance of strong $2\\omega_p$ peak \nin the dissipated power. Moreover, for larger separations the doping causes shifts, appearance  and disappearance  of many peaks originating \nfrom resonant coupling between hybridized substrate phonons and Dirac plasmons.\n\n \nIn Sec.\\ref{Sec5} we present the conclusions.\n\n\n\n\n\n\\section{General theory: Oscillating slabs}\n\\label{Sec2}\n\n\n\\subsection{Van der Waals energy and force}\nIn Appendix \\ref{vdWenergyforce} we have derived van der Waals energy and force between two slabs \nin uniform relative motion in some detail because it will help us to treat a \nsimilar problem of two oscillating slabs. \n\nWe shall later assume that the slabs consist of graphene monolayers with variable doping, deposited on dielectric slabs of thickness $\\Delta$ \ndescribed by local  dielectric functions $\\epsilon(\\omega)$, as shown in Fig.\\ref{Fig1}. The left slab mechanically oscillates with frequency $\\omega_0$  and \namplitude ${\\hbox{\\boldmath $\\rho$}}_0$ relative to the right slab. Again we calculate the diagram in Fig.\\ref{FigA1} as in the \\ref{vdWenergyforce}, but now  the slab parallel coordinates change in time \nas \n\\begin{equation}\n{\\hbox{\\boldmath $\\rho$}}-{\\hbox{\\boldmath $\\rho$}}_1\\rightarrow{\\hbox{\\boldmath $\\rho$}}-{\\hbox{\\boldmath $\\rho$}}_1-{\\hbox{\\boldmath $\\rho$}}_0(\\sin\\omega_0t-\\sin\\omega_0t_1)\n\\label{kukuli}\n\\end{equation}\nso that instead of (\\ref{nabij3}) we have\n\\begin{equation}\n\\begin{array}{c}\nE_{c}=\\int^{\\infty}_{-\\infty}dt_1\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\ne^{-i{\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0(\\sin\\omega_0t-\\sin\\omega_0t_1)}\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty} dzdz_1dz_2dz_3 \nS_1({\\bf Q},z,z_1,t-t_1)V({\\bf Q},z,z_3)\n\\nonumber\\\\\n\\nonumber\\\\\nD_2({\\bf Q},z_3,z_2,t-t_1)V({\\bf Q},z_2,z_1).  \n\\end{array}\n\\label{bijem3}\n\\end{equation}\n\n\\begin{figure}[t]\n\\includegraphics[width=0.9\\columnwidth]{Fig1.pdf}\n\\caption{Geometry of the system.}\n\\label{Fig1}\n\\end{figure}   \nIf we use \n\\[\ne^{iz\\sin\\phi}=\\sum^{\\infty}_{m=-\\infty}J_m(z)e^{im\\phi} \n\\]\nwhere $J_m$ are Bessel functions, after Fourier transformation in $\\omega$ space, using expressions (\\ref{Def1}--\\ref{Def3}), (\\ref{impsv}) and \nintegration over $z$ coordinates we obtain  \n\\begin{equation}\n\\begin{array}{c}\nE_{c}=\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}\n\\sum^{\\infty}_{m,m'=-\\infty}J_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)J_{m'}({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\n\\left[2n_1(\\omega)+1\\right]e^{i(m-m')\\omega_0t}\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\nImD_1({\\bf Q},\\omega)ReD_2({\\bf Q},\\omega+m\\omega_0).\\hspace{3cm}\n\\end{array}\n\\end{equation}\nHere we have also used the fact that $ImD_2({\\bf Q},\\omega)$ is an antisymmetric function of \n$\\omega$ and does not contribute to integration. \nWe see that the energy  oscillates in time with frequencies $(m-m')\\omega_0$. \nIf we assume to measure energies on a time scale $\\Delta t>T$, where $T=\\frac{2\\pi}{\\omega_0}$ is the maximal \nduration of one cycle, then we can average over $T$\n\\begin{equation}\n\\frac{1}{T}\\int^{T}_0dte^{i(m-m')\\omega_0t}=\\delta_{mm'},\n\\label{Timeaverage}\n\\end{equation}\nand find the result independent of time: \n\\[\n\\begin{array}{c}\nE_{c}=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}\n\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\ \\left\\{[2n_1(\\omega)+1]\nImD_1({\\bf Q},\\omega)ReD_2({\\bf Q},\\omega+m\\omega_0)+\\right.\n\\nonumber\\\\\n\\nonumber\\\\\n\\left.[2n_2(\\omega)+1]ImD_2({\\bf Q},\\omega)ReD_1({\\bf Q},\\omega+m\\omega_0)\\right\\},\n\\end{array}\n\\]\nwhere the expression in curly brackets is fully analogous to the one in (\\ref{jura66}), but now $\\omega'\\rightarrow\\omega_m=\\omega+m\\omega_0$.\nInclusion of higher order processes follows the same procedure as for the parallel motion in \\ref{vdWenergyforce}. After \nintegration over the coupling constant, we obtain the result analogous to (\\ref{jujujuju7}) \n\\begin{eqnarray}\nE_{c}=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\label{vdwFIN}\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}A({\\bf Q},\\omega,\\omega_m)\\hspace{3cm}\n\\nonumber\n\\end{eqnarray}\nwhere $A$ is given by (\\ref{jalko}) and (\\ref{jujucka}), with $\\omega_m=\\omega+m\\omega_0$.\n\nAgain, the limiting cases can be obtained from Sec.\\ref{vdWenergyforce}.\nFor $\\omega_0=0\\ (\\omega'=\\omega)$ and ${\\hbox{\\boldmath $\\rho$}}_0=0$ we find the well known result for van der \nWaals interaction when the slabs are at rest \\cite{vdW2007,Pedro}:    \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int^{\\infty}_{0}\\frac{d\\omega}{2\\pi}\\ sgn\\omega\\times\\hspace{2cm} \n\\nonumber\\\\\n\\nonumber\\\\\n\\hspace{3cm}Im\\ln\\left[1-e^{-2Qa}D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)\\right]\n\\nonumber\n\\end{eqnarray}\nFor finite frequency $\\omega_0$ and $D_1=D_2=D$ we find: \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\n\\sum^{\\infty}_{m=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}sgn\\omega\\ \nIm \\ln\\left[1-e^{-2Qa}D({\\bf Q},\\omega)D({\\bf Q},\\omega_m)\\right].\n\\nonumber\n\\end{eqnarray}\nWe notice that the frequency integrals are the same as \nin (\\ref{jujujuju7}--\\ref{sinko}). Also, the attractive van der Waals force \nbetween two oscillating slabs is given by \n\\begin{eqnarray}\nF_{\\perp}(a)=-\\frac{dE_{c}(a)}{da}=\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}Qe^{-2Qa}\n\\sum^{\\infty}_{n=0}(2-\\delta_{m0})J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\nonumber\\\\\n\\nonumber\\\\\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}B({\\bf Q},\\omega,\\omega_m) \n\\end{eqnarray}\nwhere the function $B$ is given by (\\ref{prcko}) and (\\ref{prdf99}).   \nThe same holds for the $\\omega_0\\rightarrow 0$ or $D_1=D_2=D$ limits when the \nexpressions for $B$ become (\\ref{jurec}) or (\\ref{kurec}), respectively.\n\\subsection{Dissipated power}\nWe can perform the calculation of the dissipated power for two slabs oscillating parallel to each \nother with amplitude ${\\hbox{\\boldmath $\\rho$}}_0$ and frquency $\\omega_0$ in analogy with the previous treatment of \ntwo slabs in uniform relative motion in Sec.\\ref{jurniga}. Again, we have to transform the parallel \ncoordinates in the left slabs as in (\\ref{kukuli}).\nThen (\\ref{losse5}), after integration over $t_1$ becomes \n\\begin{equation}\n\\begin{array}{c}\nP_{12}(t)=-i\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int\\frac{d\\omega}{2\\pi}\\sum^{\\infty}_{m,m'=-\\infty}\n\\\\\n\\\\\n(-1)^{m+m'}e^{i(m'-m)\\omega_0t}(m'\\omega_0-\\omega)\\ \nJ_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)J_m'({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\\\\n\\\\\nS_1({\\bf Q},|\\omega|,z,z_1)\\otimes V({\\bf Q},z,z_3)\\otimes \n\\\\\n\\\\\nD_2({\\bf Q},m'\\omega_0-\\omega,z_3,z_2)\n\\otimes V({\\bf Q},z_2,z_1) \n\\end{array}\n\\end{equation}\nWe see that the energy transfer rate is time dependent and oscillates \nwith frequency $(m'-m)\\omega_0$. Again, from (\\ref{Timeaverage}) we see \nthat for time intervals large with respect to the oscillation period $T$ the \nterms $m\\ne m'$ do not contribute and the energy transfer rate is\n\\begin{equation}\n\\begin{array}{c}\nP_{12}=\n-i\\hbar\\int\\frac{d{\\bf Q}}{(2\\pi)^2}\\int\\frac{d\\omega}{2\\pi}\\sum^{\\infty}_{m=-\\infty}(m\\omega_0-\\omega)\\ \nJ^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\\\\n\\\\\nS_1({\\bf Q},|\\omega|,z,z_1)\\otimes V({\\bf Q},z,z_3)\\otimes \n\\\\\n\\\\\nD_2({\\bf Q},m\\omega_0-\\omega,z_3,z_2)\n\\otimes V({\\bf Q},z_2,z_1) \n\\end{array}\n\\label{loss11}\n\\end{equation}    \nIf we now use (\\ref{Def1}), the definitions (\\ref{Def2}) and (\\ref{Def3}) \nof the surface correlation function and the surface excitation propagator, respectively, and the connection (\\ref{impsv}) between the surface \ncorrelation function and the imaginary part of \nsurface excitation propagator, equation (\\ref{loss11}) can be written as\n\\begin{eqnarray}\nP_{12}=-\\frac{\\hbar}{\\pi}\\sum^{\\infty}_{m=-\\infty}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{3cm} \n\\nonumber\\\\\n\\label{loss12}\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ \\omega_m\\ [2n_1(\\omega)+1]\\ ImD_1({\\bf Q},\\omega) ImD_2({\\bf Q},\\omega_m).\\hspace{1cm}  \n\\nonumber\n\\end{eqnarray}    \nEvaluating (\\ref{loss12}) we have used the fact that the real part of the function under summation and integration is \nodd and the imaginary part is an even function of $n$ and $\\omega$.\n$P_{12}$ is the energy transferred from the left to the right slab.\nNow we have to repeat the discussion in Sec.\\ref{jurniga} and substract the part of this energy which \nwill be reversibly returned to the left slab. The same arguments, leading to (\\ref{losse17}), will give \nthis energy to be\n\\begin{eqnarray}\nP'_{12}=\\hbar\\sum^{\\infty}_{n=-\\infty}\\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_n({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\hspace{2cm} \n\\nonumber\\\\\n\\label{loss13}\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ \\omega\\ [2n_1(\\omega)+1]\\ Im D_1({\\bf Q},\\omega) Im D_2({\\bf Q},\\omega_n).  \n\\nonumber\n\\end{eqnarray}  \nExpression (\\ref{loss13}) represents the energy transferred from the \nleft to right but which will be reversibly returned, as shown in Fig.\\ref{FigA3}b. Therefore the energy \nwhich is irreversibly transferred from the left to the right, i.e. the dissipated power, is  \n\\begin{equation}\n\\begin{array}{c}\nP_{1}=P_{12}-P'_{12}=\n2\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0) \n\\\\\n\\\\\n\\int\\frac{d\\omega}{2\\pi}\\ [2n_1(\\omega)+1] ImD_1({\\bf Q},\\omega) ImD_2({\\bf Q},\\omega_m).\n\\end{array}\n\\label{diss1}\n\\end{equation}\nAnalogous calculation would give the energy dissipated in the process where the charge \nfluctuation in the right slab induces fluctuations in the left slab. We have to \nexchange $1$ and $2$ in (\\ref{diss1}) and replace $m\\rightarrow-m$. \nRepeating the steps in (\\ref{jutro}) the final result becomes: \n\n\n\\begin{eqnarray}\nP=P_1+P_2=\\hspace{3cm}\n\\nonumber\\\\\n\\nonumber\\\\\n4\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\hspace{2cm}\n\\nonumber\\\\\n\\label{njunja}\\\\\n\\left[n_1(\\omega)-n_2(\\omega_m)\\right]ImD_1({\\bf Q},\\omega)ImD_2({\\bf Q},\\omega_m).\n\\nonumber\n\\end{eqnarray}\nThis expression is analogous to (\\ref{jutro}). For $T=0$  $2n(\\omega)+1\\rightarrow sgn\\omega$ and \n(\\ref{njunja}) can be written as \n\\begin{eqnarray}\nP=\\hspace{5cm}\n\\label{njunja1}\\\\\n\\nonumber\\\\\n4\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\n\\int^{m\\omega_0}_{0}\\frac{d\\omega}{2\\pi}\\hspace{2cm}\n\\nonumber\\\\\n\\nonumber\\\\\nImD_1({\\bf Q},\\omega)ImD_2({\\bf Q},m\\omega_0-\\omega).\n\\nonumber\n\\end{eqnarray}\nAdding higher order terms (\\ref{Eka1},\\ref{Eka2})  we obtain the energy \ndissipated per unit time: \n\\begin{eqnarray}\nP=2\\hbar\\sum^{\\infty}_{m=1}m\\omega_0\\ \\int\\frac{d{\\bf Q}}{(2\\pi)^2}e^{-2Qa}J^2_m({\\bf Q}{\\hbox{\\boldmath $\\rho$}}_0)\\times\n\\nonumber\\\\\n\\label{losshop}\n\\\\\n\\hspace{3cm}\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}C({\\bf Q},\\omega,\\omega_m)\n\\nonumber\n\\end{eqnarray}\nwhere $C$ is given by (\\ref{kaka}). Limiting casses \nare also obtained from (\\ref{losshop}). For $\\omega_0=0$ and\/or for $\\rho_0=0$ \nobviously $P=0$.\n\\section{Derivation of the slab surface excitation propagators $D_{1,2}({\\bf Q},\\omega)$}\n\\label{DerofD}\nThe main quantities which appear in the formula for van der Waals interaction $E_c$ or \ndissipated power $P$ are the surface excitation propagators $D_{1}({\\bf Q},\\omega)$ and $D_{2}({\\bf Q},\\omega)$ of the left (first) and right (second) slab, respectively.\nThe derivation of $D_1$ and $D_2$ is analogous for both slabs, so here we shall derive just one surface \nexcitation propagator $D$. The structure of the monolayer-substrate composite (e.g. graphene on \nSiO$_2$) is shown in Fig.\\ref{Fig2}. The slab consists of the graphene monolayer adsorbed at some small distance $h$ (e.g. $h=0.4$nm) above the substrate of macroscopic thickness $\\Delta$. The \ndielectric, e.g. the SiO$_2$ slab is placed in the region $-\\Delta-h\\le z\\le-h$ and the graphene \nlayer occupies $z=0$ plane. The same model system is used in Refs.\\cite{Ivan1,Ivan2} where \nthe authors explore plasmon-phonon hybridization, stopping power and wake effect produced \nby the proton moving parallel to the composite. The unit cell for such huge nanostructure would \nconsist of hundreds of atoms, so it is impossible to perform full {\\em ab initio} ground state and structure optimization calculation. Moreover,  an {\\em ab initio} calculation of the response function would be even more demanding so we need an approximation for the response function calculation. The easiest (and probably the best) approximation is to treat \nthe SiO$_2$ slab as a homogeneous dielectric described by some local dielectric function $\\epsilon_S(\\omega)$ and to consider graphene as a purely 2D system described \nby the response function $R({\\bf Q},\\omega)$, as sketched in Fig.\\ref{Fig2}.    \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=4.0cm,height=7cm]{Fig2.pdf}\n\\caption{(color online) Simplified model where the SiO$_2$ substrate is shown as a homogenous dielectric slab described by the local dielectric function $\\epsilon_S(\\omega)$ and graphene is described by 2D response function $R({\\bf Q},\\omega)$. $D({\\bf Q},\\omega)$ is the surface excitation propagator of the substrate\/graphene composite.}\n\\label{Fig2}\n\\end{figure}\nIn order to derive the surface excitation \npropagator $D({\\bf Q},\\omega)$ we start from its definition:\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=v_Q\\int^0_{-\\infty}\\ dzdz' e^{-Q(z+z')}R({\\bf Q},\\omega,z,z')= \n\\nonumber\\\\\n\\label{defD}\\\\\n\\frac{1}{v_Q}\\left\\{W({\\bf Q},\\omega,z=0,z'=0)-v_Q\\right\\};\\  i=1,2.\n\\nonumber\n\\end{eqnarray}\nwhich connects the surface excitation propagator with the screened Coulomb interaction $W({\\bf Q},\\omega,z=0,z'=0)$ at $z=z'=0$ surface.\nHere $R({\\bf Q},\\omega,z,z')$ represents the nonlocal dielectric function of graphene\/dielectric composite which we assume occupies the region $z,z'\\le 0$. \n\nIt is well known  \\cite{gr2D1,gr2D2,gr2D3,gr2D4} that physical properties of a graphene \nmonolayer in the low ($Q,\\omega$) region can be described to a very good approximation \nassuming the monolayer to be strictly twodimensional, so that the nonlocal independent \nelectron response function can be written as\n\\begin{equation}\nR^{0}({\\bf Q},\\omega,z,z')=R^{0}({\\bf Q},\\omega)\\delta(z)\\delta(z') \n\\end{equation}\nwhere we assume that the graphene lies in the $z=0$ plane and the response function \n$R^{0}({\\bf Q},\\omega)$ can be derived from first \nprinciples, as decribed in Sec.\\ref{nabnak}. \nDynamically screened response function $R({\\bf Q},\\omega)$ in RPA is given as \na series of terms\n\\begin{equation}\nR({\\bf Q},\\omega)=R^{0}+R^{0}v_QR^{0}+...=\\frac{R^0({\\bf Q},\\omega)}{1-v_QR^0({\\bf Q},\\omega)}.\n\\label{RPARR}\n\\end{equation} \nIf we assume for the moment that there is no dielectric in the system (e.g. $\\epsilon_S(\\omega)=1$) then the screened Coulomb \ninteraction is simply given by \n\\begin{equation}\nW({\\bf Q},\\omega,z=0,z'=0)=v_Q+v_QR({\\bf Q},\\omega)v_Q.\n\\label{scrCint}\n\\end{equation}\nUsing the definition (\\ref{defD}) the surface excitation propagator becomes\n\\begin{equation}\nD({\\bf Q},\\omega)=v_QR({\\bf Q},\\omega).\n\\end{equation}\nWhen the dielectric slab is introduced, the external charges and charge density fluctuations in the graphene layer do not interact via the bare Coulomb \ninteraction $v_Q$ but via the Columob interaction modified by the presence of \nthe dielectric slab \\cite{Laplace}     \n\\begin{equation}\nv_Q\\rightarrow\\tilde{v}_Q(\\omega)=v_Q\\left[1+D_S({\\bf Q},\\omega)\\right],\n\\label{tildeV11}\n\\end{equation}\nwhere the substrate surface excitation propagator is\n\\begin{equation}\nD_S({\\bf Q},\\omega)=D_S(\\omega)\\frac{1-e^{-2Q\\Delta}}{1-D^2_S(\\omega)e^{-2Q\\Delta}}e^{-2Qh}\n\\label{labina}\n\\end{equation}\nand\n\\begin{equation}\nD_S(\\omega)=\\frac{1-\\epsilon_S(\\omega)}{1+\\epsilon_S(\\omega)}\n\\end{equation}   \nrepresents the surface excitation propagator of a semiinfinite \n($\\Delta\\rightarrow\\infty$, $h=0$) dielectric.\nThis causes that the screened Colulomb interaction (\\ref{scrCint}) becomes the function \nof $\\tilde{v}_Q(\\omega)$\n\\begin{equation}\nW\\rightarrow\\tilde{W}=\\tilde{v}_Q(\\omega)+\\tilde{v}_Q(\\omega)\\tilde{R}({\\bf Q},\\omega)\\tilde{v}_Q(\\omega),\n\\label{modifiedW}\n\\end{equation}\nwhere, because charge density fluctuations inside graphene also interact via $\\tilde{v}_Q(\\omega)$, the screened response function is modified as\n\\begin{equation}\n\\tilde{R}({\\bf Q},\\omega)=\\frac{R^0({\\bf Q},\\omega)}{1-\\tilde{v}_Q(\\omega)R^0({\\bf Q},\\omega)}.\n\\label{curoni}\n\\end{equation}\nFinally, after inserting (\\ref{modifiedW}) into (\\ref{defD}) we obtain the surface excitation propagator in the presence of \nthe dielectric\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=\\frac{1}{v_Q}\\left\\{\\tilde{v}_Q(\\omega)\\tilde{R}_i({\\bf Q},\\omega)\\tilde{v}_Q(\\omega)+\\right.\n\\label{newD}\\\\\n\\hspace{3cm}\\left.\\tilde{v}_Q(\\omega)-v_Q\\right\\}.\n\\nonumber\n\\end{eqnarray}\nwhich can be rewritten in a more transparent form as\n\\begin{eqnarray}\nD({\\bf Q},\\omega)=\\hspace{5cm}\n\\nonumber\\\\\n\\nonumber\\\\\n\\frac{D_S({\\bf Q},\\omega)+v_QR({\\bf Q},\\omega)+2v_QR({\\bf Q},\\omega)D_S({\\bf Q},\\omega)}{1-v_QR({\\bf Q},\\omega)D_S({\\bf Q},\\omega)}.\n\\end{eqnarray}\nThe spectrum of coupled excitations in a single slab can be calculated from\n\\begin{equation}\nS({\\bf Q},\\omega)=-\\frac{1}{\\pi}ImD({\\bf Q},\\omega).\n\\end{equation}\nFor the coupled slabs described by their surface excitations propagators \n$D_1$ and $D_2$, separated by the distance $a$, in a similar way we can derive \nthe propagator $\\tilde{D}$ for the coupled system \n\\begin{equation}\n\\tilde{D}({\\bf Q},\\omega)=\n\\frac{D_1({\\bf Q},\\omega)+D_2({\\bf Q},\\omega)+2D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)}{1-e^{-2Qa}D_1({\\bf Q},\\omega)D_2({\\bf Q},\\omega)}\n\\end{equation}\nand the excitation spectrum of this system is\n\\begin{equation}\n\\tilde{S}({\\bf Q},\\omega)=-\\frac{1}{\\pi}Im\\tilde{D}({\\bf Q},\\omega).\n\\end{equation}\n\\section{Description of substrate and graphene dynamical response}\n\\label{nabnak}\nThe results in Sec.\\ref{DerofD} are quite general and can be applied to a monolayer \nof any material on any dielectric substrate.\nNow we shall specify the dielectric substrate to be the homogenous \nlayer of ionic crystal SiO$_2$.\n\nDielectric properties (or dynamical response) of bulk ionic crystals in the long-wavelength limit can \nbe described in terms of their optical phonons at the $\\Gamma$ point. More complex polar crystals such as SiO$_2$ possess a multitude of different optical \nphonons of different symmetries and polarizations. However, here we suppose  that \nSiO$_2$ posses  two well-defined, non-dispersing transverse\noptical (TO) phonon modes at the frequencies $\\omega_{TO1}$\nand $\\omega_{TO2}$ with the corresponding damping rates\n$\\gamma_{TO1}$ and $\\gamma_{TO2}$, giving rise to a \ndielectric function of the form \\cite{Ivan1,Ivan2}\n\\begin{eqnarray}\n\\epsilon_S(\\omega)=\\epsilon_{\\infty}+(\\epsilon_i-\\epsilon_{\\infty})\\frac{\\omega^2_{TO2}}{\\omega^2_{TO2}-\\omega^2-i\\omega\\gamma_{TO2}}+\n\\nonumber\\\\\n(\\epsilon_0-\\epsilon_i)\\frac{\\omega^2_{TO1}}{\\omega^2_{TO1}-\\omega^2-i\\omega\\gamma_{TO1}} ,\n\\label{dielectric}\n\\end{eqnarray}\nwhere $\\epsilon_0$, $\\epsilon_i$, and $\\epsilon_{\\infty}$ represent  \nthe dielectric constant for SiO$_2$ at the zero, intermediate, and very large\nfrequencies. This dielectric function will be inserted in the expression (\\ref{labina}) for the substrate surface excitation propagator $D_S({\\bf Q},\\omega)$. \n\nThe graphene response function $R({\\bf Q},\\omega)$ is given by (\\ref{curoni}) in terms of the noninteracting response function\n\\begin{equation}\nR^{0}({\\bf Q},\\omega)=L\\ R^{0}_{{\\bf G}=0{\\bf G}'=0}({\\bf Q},\\omega)\n\\label{Chi02D}\n\\end{equation}\nwhere the 3D Fourier transform of independent electron response function is given \nby \\cite{PRB13} \n\\begin{eqnarray}\nR^{0}_{{\\bf G}{\\bf G}'}({\\bf Q},\\omega)=\\hspace{5cm}\n\\nonumber\\\\\n\\frac{2}{\\Omega}\\sum_{{\\bf K}\\in S.B.Z.}\\sum_{n,m}\\ \\frac{f_n({\\bf K})-f_m({\\bf K}+{\\bf Q})}\n{\\hbar\\omega+i\\eta+E_n({\\bf K})-E_m({\\bf K}+{\\bf Q})}\\times\n\\label{Resfun0}\\\\\n\\rho_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})\\ \\rho^*_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G'}),\n\\nonumber\n\\end{eqnarray}  \nwhere $f_{n{\\bf K}}=[e^{(E_{n{\\bf K}}-E_F)\/kT}+1]^{-1}$ is the Fermi-Dirac distribution at \ntemperature $T$. The charge vertices in (\\ref{Resfun0}) have the form \n\\begin{equation}\n\\rho_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})=\n\\int_\\Omega\\ d{\\bf r}e^{-i({\\bf Q}+{\\bf G}){\\bf r}}\\ \\phi^*_{n{\\bf K}}({\\bf r})\\phi_{n{\\bf K}+{\\bf Q}}({\\bf r})\n\\label{Matrel}\n\\end{equation}\nwhere ${\\bf Q}$ is the momentum transfer vector parallel to the $x-y$ plane, ${\\bf G}=({\\bf G}_\\parallel,G_z)$ are $3D$ reciprocal lattice vectors and \n${\\bf r}=({\\hbox{\\boldmath $\\rho$}},z)$ is a $3D$ position vector. Integration in (\\ref{Matrel}) is performed over the normalization volume $\\Omega=S\\times L$, where $S$ is the \nnormalization surface and $L$ is the superlattice constant in $z$ direction (separation between graphene layers is superlattice arrangement). \nPlane wave expansion of the wave function has the form \n\\[\n\\phi_{n{\\bf K}}({\\hbox{\\boldmath $\\rho$}},z)=\\frac{1}{\\sqrt{\\Omega}}e^{i{\\bf K}{\\hbox{\\boldmath $\\rho$}}}\\ \\sum_{\\bf G}C_{n{\\bf K}}({\\bf G})e^{i{\\bf G}{\\bf r}},\n\\]\nwhere the coefficients $C_{n{\\bf K}}$ are obtained by solving the Local Density Approximation-Kohn Sham (LDA-KS) equations selfconsistently as will be discussed below.\nHowever, this straightforward calculation of graphene response functions $R({\\bf Q},\\omega)$ is not \nsufficient if we want to investigate the hybridization between the Dirac plasmon and Fuchs-Kliewer (FK) phonons at dielectric surfaces. Namely, due to \nthe very low energy of FK phonons ($\\sim 50$meV) the crossing of their dispersion relations with Dirac plasmon occurs for very small wave vectors ($Q<0.001$a.u.). \nOn the other hand even for very dense $K$-point mesh sampling, as for example $601\\times 601\\times 1$ used in this calculation, \nthe minimum transfer wave vector $Q$ which can be reached (e.g. $Q=0.0026$a.u.$^{-1}$ in this calculation) is still bigger than FK phonon-Dirac \nplasmon crossing wave vector. Therefore we have to find the way how to calculate $R({\\bf Q},\\omega)$ for a denser Q-point mesh \nin the optical $Q\\approx 0$ limit. One possible way is that instead of calculating response function $R^{0}({\\bf Q},\\omega)$ we calculate the optical \n($Q=0$) conductivity $\\sigma(\\omega)$. The optical conductivity in graphene can be written as \\cite{gr2D3}    \n\\begin{equation}\n\\sigma(\\omega)=\\sigma^{\\mathrm{intra}}(\\omega)+\\sigma^{\\mathrm{inter}}(\\omega),       \n\\label{curren1}\n\\end{equation}\nwhere \n\\begin{equation}\n\\sigma^{\\mathrm{intra}}(\\omega)=\\frac{i\\rho_0}{\\omega+i\\eta_{\\mathrm{intra}}}\n\\label{curren2}\n\\end{equation}\nis intraband or Drude conductivity and where \n\\begin{equation}\n\\rho_0=-\\frac{2}{\\Omega}\\sum_{{\\bf K},n}\\frac{\\partial f^i_n({\\bf K})}{\\partial E_n({\\bf K})}\n|j^{x}_{n{\\bf K},n{\\bf K}}({\\bf G}=0)|^2\n\\label{curren3}\n\\end{equation}\nrepresents the effective number of charge carriers. \nThe interband conductivity is  \n\\begin{eqnarray}\n\\sigma^{\\mathrm{inter}}(\\omega)=\n\\frac{-2i}{\\omega\\Omega}\\sum_{{\\bf K},n\\neq m}\\ \\frac{\\hbar\\omega}{E_n({\\bf K})-E_m({\\bf K})}\\times\n\\nonumber\\\\\n\\frac{f^i_n({\\bf K})-f^i_m({\\bf K})}\n{\\hbar\\omega+i\\eta_{\\mathrm{inter}}+E_n({\\bf K})-E_m({\\bf K})}\\times\n\\label{curren4}\\\\\nj^{x}_{n{\\bf K},m{\\bf K}}({\\bf G}=0)\\ [j^{x}_{n{\\bf K},m{\\bf K}}({\\bf G}'=0)]^*\n\\nonumber\n\\end{eqnarray}\nwhere the current vertices are given by\n\\begin{equation}\nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf G})=\n\\int_\\Omega\\ d{\\bf r}e^{-i({\\bf Q}+{\\bf G}){\\bf r}}\\ \nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf r}),\n\\label{curren5}\n\\end{equation}\nand \n\\begin{eqnarray}\nj^{\\mu}_{n{\\bf K},m{\\bf K}+{\\bf Q}}({\\bf r})=\n\\frac{\\hbar e}{2im}\n\\left\\{\\phi_{n{\\bf K}}^*({\\bf r})\\partial_\\mu\\phi_{m{\\bf K}+{\\bf Q}}({\\bf r})\\right.\\hspace{2cm}\n\\label{curren6}\\\\\n\\hspace{2cm}-\\left.[\\partial_\\mu\\phi_{n{\\bf K}}^*({\\bf r})]\\phi_{m{\\bf K}+{\\bf Q}}({\\bf r})\\right\\}.\n\\nonumber\n\\end{eqnarray}\nIn the optical $Q\\approx 0$ limit the  independent electron response function can be written in terms of optical \nconductivities (\\ref{curren1}) as \\cite{Zoran}\n\\begin{equation}\nR^0({\\bf Q}\\approx 0,\\omega)=L\\ \\frac{Q^2}{i\\omega}\\sigma(\\omega).\n\\label{chi-sigma}\n\\end{equation}\nFinally, the RPA or screened response function $R({\\bf Q},\\omega)$ can be obtained from (\\ref{chi-sigma}) \nusing (\\ref{RPARR}). \n\nIn the calculation of Sec.\\ref{resuuult} we shall assume the \ngraphene response to be isotropic in the small $({\\bf Q},\\omega)$ limit. This means \nthat the graphene response functions and the corresponding surface excitation functions are functions of \n$Q$ and not of ${\\bf Q}$.      \n\n\\subsection{Computational details}\n\\label{Comp}\nThe first part of the calculation consists of determining the KS ground state of the single layer graphene and the corresponding wave functions \n$\\phi_{n{\\bf K}}({\\hbox{\\boldmath $\\rho$}},z)$ and energies $E_n({\\bf K})$. For graphene unit cell constant we use the experimental value of $a=4.651\\ \\mathrm{a.u.}$ \\cite{lattice}, and superlattice unit cell constant (separation of graphene layers) is $L=5a$. For calculating KS wave functions and energies we use a plane-wave self-consistent field DFT code (PWSCF) within the QUANTUM ESPRESSO (QE) package \\cite{QE}. The core-electron interaction was approximated by the norm-conserving pseudopotentials \\cite{normcon}, and the exchange correlation (XC) potential by the Perdew-Zunger local density approximation (LDA) \\cite{lda1}. To calculate the ground state electronic density we use $21\\times21\\times1$ Monkhorst-Pack K-point \nmesh \\cite{MPmesh} of the first Brillouin zone (BZ) and for the plane-wave cut-off energy we choose 50 Ry. \nThe second part of calculation consists of determining the independent electron response function (\\ref{Resfun0}) and conductivity \n(\\ref{curren1}--\\ref{curren4}). In order to achieve better resolution in the long wavelength ($Q\\approx 0$) and low energy ($\\omega\\approx 0$) \nlimit the response function (\\ref{Resfun0},\\ref{Matrel}) and conductivity (\\ref{curren1}--\\ref{curren6}) are evaluated from the wave functions $\\phi_{n{\\bf K}}({\\bf r})$ and energies $E_n({\\bf K})$ calculated for the $601\\times601\\times1$ Monkhorst-Pack K-point mesh  which coresponds to 361801 K-points in the first Brillouin zone (1BZ). Band summations ($n,m$) in (\\ref{Resfun0}), (\\ref{curren3}) and (\\ref{curren4}) are performed over 30 bands. In the calculation we use two kinds of damping parameters: $\\eta_{\\mathrm{intra}}=10$meV for transitions within the same bands ($n\\leftrightarrow n$), and $\\eta_{\\mathrm{inter}}=50$meV for transitions between different bands ($n\\leftrightarrow m$). \nFor bulk SiO$_2$ dielectric function given by (\\ref{dielectric}) we use the following \nparameters: $\\epsilon_0=3.9$, $\\epsilon_i=3.05$, $\\epsilon_{\\infty}=2.5$, \n$\\omega_{TO1}=55.6$ meV, $\\omega_{TO2}=138.1$ meV, $\\gamma_{TO1}=5.368$ meV and $\\gamma_{TO2}=8.947$ meV taken from Ref.\\cite{Dielectricpar}. \nFor the gap between graphene and the SiO$_2$ surface, we take $h=4\\AA[7.55$ a.u.$]$ \\cite{height}.\n\n\n\\section{Results for graphene monolayers on SiO$_2$ substrates}\n\\label{resuuult}\nTheoretical expressions derived in Sec.\\ref{Sec2} (and in Appendix \\ref{AppA}) are quite \ngeneral, i.e. are valid for any pair of crystal slabs described by their response \nfunctions, while the corresponding surface excitation functions derived in \nSec.\\ref{DerofD} are valid for any 2D adsorbed monolayer on any dielectric \nsubstrate. In this section we shall apply these results to calculate reactive and \ndissipative response of various combinations of slabs consisting of graphene monolayers with variable doping on SiO$_2$ substrate, using the dynamical surface response functions of \nthese materials given in Sec.\\ref{DerofD}.          \n\nBefore proceeding with detailed calculations a few general comments are \nin order. Though the derived expressions for van der Waals and dissipated \npower (\\ref{vdwFIN}) and (\\ref{losshop}), respectively, include temperature \ndependence, in the systems studied here inclusion of finite temperature leads to practically \nno effects, therefore all results will be reported for $T=0$. \nThe dependence of these two physical properties on the two parameters, the distance \nbetween the slabs $a$ and the oscillation amplitude $\\rho_0$, can be analyzed if \nwe recognize in the expressions  (\\ref{vdwFIN}) and (\\ref{losshop}) the function     \n\\begin{equation}\nf_m(x)=\\int^{2\\pi}_0\\frac{d\\phi}{2\\pi}J^2_m(x\\cos\\phi),\n\\label{fm}\n\\end{equation}\nwhich is possible because of the assumed isotropy of graphene response. The function \n$f_m(x)$ is shown in Fig.\\ref{Fig3} for first four $m$'s, where $x=Q\\rho_0$. \n\\begin{figure}[t]\n\\includegraphics[width=6cm,height=5cm]{Fig3.pdf}\n\\caption{Function $f_m(x)$ for  m=0 (blue solid line), $m=1$ (black solid line), $m=2$ (black dashed line) and $m=3$ (black dashed-dotted line). Vertical dashed line denotes the maximum argument $x_{cut}$ defined by parameters ($a$ and $\\rho_0$) used in the calculation.}\n\\label{Fig3}\n\\end{figure} \nAnother important  factor in (\\ref{vdwFIN}) and (\\ref{losshop}) is $e^{-2Qa}$  which defines the \ncutoff wave vector $Q_c$, depending on the slab separation $a$. \nThe separations we shall consider in this calculation are $a=10-50$nm which defines the cutoff wave \nvector $Q_c\\approx 0.05a.u.$. On the other hand, the ampitudes which will be considered are \n$\\rho_0\\approx 0.1-1$nm. This finally provides the maximum argument $Q$ of the functions (\\ref{fm}) which is $x_{cut}\\approx 1$. From Fig.\\ref{Fig3} is obvious that up to $x_{cut}$ only the $m=0$ \nand $m=1$ terms will contribute. Moreover, for $x<x_{cut}$ the Bessels functions can be approximated \nas $J_0\\approx 1-\\frac{x^2}{4}$ and $J_m(x)\\approx \\frac{x^m}{2^mm!};\\ m>1$ and therefore  \n\\begin{equation}\nf_0\\approx 1-\\frac{x^2}{4};\\ \\ \\ f_1(x)\\approx \\frac{x^2}{8}. \n\\label{approx}\n\\end{equation}\nIn Fig.\\ref{Fig3} we see that approximation (\\ref{approx}) is valid almost up \nto $x_{cut}$.\n\n\\subsection{Spectra of coupled modes}\n\\label{cmspe}\n \nIn this section we shall first discuss the spectra of coupled plasmon\/phonon excitations \nin one and two graphene\/SiO$_2$ slabs separated by distance $a$ in order to understand \nthe dominant dissipation mechanisms.\n\nFig.\\ref{Fig4}(a) shows the spectrum of surface  excitations $S(Q,\\omega)=-Im D(Q,\\omega)$ \nin graphene(200meV)\/SiO$_2$ slab (as shown in Fig.\\ref{Fig2}) and Fig.\\ref{Fig4}(b) in the system \nwhich consists of two graphene\/SiO$_2$ slabs (as shown in Fig.\\ref{Fig1}) separated by distance $a=5$nm. In the lonwavelength limit the SiO$_2$ surface suports two surface polar (FK) TO phonons with \nflat dispersions and the doped graphene contains a Dirac plasmon with square root dispersion. \nCoupling between these modes results in three branches, as shown in Fig.\\ref{Fig4}(a). For larger \n$Q$ the first and second flat branches are phononlike, i.e. their induced \nelectrical fields mostly come from polarization modes on the dielectric\nsurface. On the other hand, the third square root branch is \nplasmon-like, i.e. its induced electrical field mostly comes from charge density \noscillations localised in the graphene layer. However, in the $Q\\rightarrow 0$ limit \nthe strong hybridization (avoided crossings) between these modes occur and they \npossess mixed plasmon-phonon character. When another slab is brought in the vicinity the \nthree modes in each slab interact which results in the mode splitting and formation of \nsix coupled modes as shown in Fig.\\ref{Fig4}(b).    \nFigure \\ref{Fig4}(c) shows the spectrum of surface  excitations in the \ngraphene($0$meV)\/SiO$_2$ slab. Because the pristine graphene does not support Dirac plasmon \nthe spectrum consist just of two weak phonon branches $\\omega_{TO1}$ and $\\omega_{TO2}$ damped \nby $\\pi\\rightarrow\\pi^*$ excitations. The spectrum of surface  excitations in two equal \ngraphene(0meV)\/SiO$_2$ slabs separated by 5nm (not shown here) is very similar to the one \nshown in Fig.\\ref{Fig4}(c) which indicates weak interaction between phonons \nin the two slabs. This could be the consequence of strong screening of FK phonons \nby graphene adlayers which reduces the range of their induced electrical \nfield. Figure \\ref{Fig4}(d) shows the spectrum in the system which consists of two different \nslabs, graphene($0$meV)\/SiO$_2$ and graphene($200$meV)\/SiO$_2$, separated by $5$nm. \nOne can notice interesting hybridization between the Dirac plasmon and two phonons in one slab and two phonons in another slab giving five branches. \n\\begin{figure*}[tt]\n\\includegraphics[width=1.0\\columnwidth]{Fig4a.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4c.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4b.jpg}\n\\includegraphics[width=1.0\\columnwidth]{Fig4d.jpg}\n\\caption{(Color online) The spectra of surface excitations in (a) graphene(200meV)\/SiO$_2$ single \nslab (as shown in Fig.\\ref{Fig2}), (b) in the system consisting of two equal graphene(200meV)\/SiO$_2$ slabs, (as shown in Fig.\\ref{Fig1}) separated by distance 5nm, (c) single \ngraphene(0meV)\/SiO$_2$ slab and (d) in the system consisting of two unequal slabs, \ngraphene(200meV)\/SiO$_2$ and graphene(0meV)\/SiO$_2$, separated by distance 5nm.} \n\\label{Fig4}\n\\end{figure*}\nIn the next section we shall explore how particular plasmon-phonon modes contribute to the \ndissipated power in two oscillating slabs. \n\n\\subsection{Modification of van der Waals force}\n\\label{weakvdW}\nVan der Waals energy and attractive force are usually calculated and measured for static \nobjects. Here we show how their relative oscillating motion can reduce this attraction, \nwhich can be relevant not only from the theoretical standpoint but \nalso in some experimental situations and applications. This phenomenon is present \nalso in the case of parallel motion, as shown in the Appendix \\ref{AppA}, but \nthis situation would be more difficult to realize in practice. \n\nMaking use of the approximation (\\ref{approx}) for the lowest order \nterms of the functions $f_0$ and $f_1$ given by (\\ref{fm}) we can \nrewrite the expression (\\ref{vdwFIN}) for the van der Waals energy as  \n\\begin{eqnarray}\nE_{c}(a)=\\frac{\\hbar}{2}\\int\\frac{QdQ}{2\\pi}\n\\int^{\\infty}_{-\\infty}\\frac{d\\omega}{2\\pi}\\hspace{3cm}\n\\label{java}\\\\\n\\nonumber\\\\\n\\left\\{ \\left[1-\\frac{Q^2\\rho_0^2}{4}\\right]A(Q,\\omega,\\omega)+\\frac{1}{4}Q^2\\rho_0^2\\ A(Q,\\omega,\\omega-\\omega_0)\\right\\}\n\\nonumber\n\\end{eqnarray}\nwhere $A$ is given by (\\ref{jalko}) and (\\ref{jujucka}).  \nIn the $T\\rightarrow 0$ limit and neglecting higher order terms $A$ reduces to\n\\begin{eqnarray}\nA(Q,\\omega,\\omega')=\\hspace{5cm}\n\\nonumber\\\\\ne^{-2Qa}sgn\\omega\\left\\{ImD_1(Q,\\omega)ReD_2(Q,\\omega')+(1\\leftrightarrow 2)\\right\\}\n\\nonumber\n\\end{eqnarray}\nWe see that for $\\rho_0\\rightarrow 0$ the van der Waals energy reduces \nto the standard result for the static case, and for $\\rho_0\\ne 0$ and \n$\\omega_0\\ne 0$ the lowest order corrections scale with $\\rho^2_0$.    \nFrom (\\ref{java}), and also from (\\ref{njunja2}), we see that the slab separation $a$ (because of exponential \nfactor $e^{-2Qa}$) reduces the wave vector range to $Q<1\/2a$.\n\nFig.\\ref{Fig5} shows van der Waals energies $E_c$ of two variously doped, unsupported full conductivity \n(\\ref{curren1}--\\ref{curren4}) graphenes as functions of the driving frequency $\\omega_0$. \nThe driving amplitude is $\\rho_0=20$nm and separation between slabs is $a=10$nm. \nFor the case of two heavily and equally doped graphenes $1-1$eV (thick black solid line)\nthe 'static' ($\\omega_0=0$) van der Waals energy is the largest in comparison with other doping combinations. This is reasonable considering that \nthen except of $\\pi$ and $\\pi+\\sigma$ plasmons (and corresponding electron-hole \nexcitations) the graphenes support strong Dirac plasmons which are all in resonance. \nTherefore, the charge density fluctuation in one slab $ImD_1(\\omega)$ resonantly induces electrical field in another slab \n$ReD_2(\\omega)$ to which it couples, and \nvice versa.  \nAs the driving frequency $\\omega_0$ increases the  fluctuation and the induced field do not match any more, i.e. $ImD_1(\\omega)$ and $ReD_2(\\omega+n\\omega_0)$ become \nDoppler shifted and vdW energy is expected to decrease. \nHowever, the vdW energy first exhibits a wide plateau until $\\omega_0<50$THz. \nWe performed a  separate vdW energy calculation for two unsupported \nDrude (\\ref{curren1},\\ref{curren2}) graphenes (not shown here) and noticed that it shows the same \nfeatures as presented in Fig.\\ref{Fig5}. This suggests that Dirac plasmons are responsible for all characteristic \nfeatures in vdW energy (for larger dopings).\nTherefore, the plateau arises probably because the Dirac plasmon fluctuation in one slab, e.g. at $\\omega_p$, can be efficiently screened by induced \nplasmon field in another slab which is not necessarily at the same frequency $\\omega_p$. \nMoreover, graphene, regardless of doping, exhibits perfect screening $ReD(Q\\approx 0,\\omega\\approx 0)\\approx -1$ \\cite{Duncan2012} causing that \nthe static point charge feels image potential. This causes that $E_c$ shows almost identical plateau \nfor the case of differently doped graphenes $1-0.2$eV (black solid line) and $1-0eV$ \n(thin black solid line). \nAs the doping difference increases plateau energy decreases which is \nreasonable because of plasmon resonance breakdown. \nFor larger $\\omega_0>50$THz  the Dirac plasmon in one slab does not match any more the \nperfect screening regime in another one, resulting in a rapid decrease or weakening of vdW energy.\nIn the case of weakly doped graphenes, such as the combinations $0.2-0.2$eV (red dashed line) and $0.2-0$eV  (thin red dashed \nlines), the 'static' $\\omega_0\\approx 0$  van der Waals energy reduces in comparison with the heavy\ndoping (combinations with $1$eV) cases. This is reasonable considering that Dirac plasmon spectral weight decreases with \ndoping. Additionally, it can be noted that for lower doping the vdW plateau shifts to $\\omega_0<25$THz. \nThis is because the perfect screening frequency region can be roughly estimated as \n$ReD(\\omega<\\omega_p)\\approx-1$, so, as the plasmon energy decreases the frequency interval whithin which fluctuations are perfectly screened becomes narrower.   \nIt is interesting to notice that for some frequencies (e.g. $\\omega_0>100$THz) the resonant but low doping vdW energy (e.g.  $0.2-0.2$eV case) \novercomes the heavily doped but off resonance vdW energy (such as the cases $1-0.2$eV and $1-0$eV). \nThe static $\\omega_0=0$ vdW energy of pristine graphenes $0-0$eV (blue dashed dotted line) is the weakest and shows smooth decreasing, almost \nlinear behaviour. In this case there  are no Dirac plasmons in the graphenes spectra. Therefore,  only resonant coupling between $\\pi\\rightarrow\\pi^*$ electron-hole \nexcitations, $\\pi$ and $\\pi+\\sigma$ plasmons contribute to the vdW energy. As the frequency $\\omega_0$ increases the overlap between these \nelectronic excitations decreases causing smooth and linear vdW energy weakening.   \nThe same linear behaviour (for $\\omega_0>50$THz) can be noticed for doping combinations $0.2-0.2$eV and  $0.2-0$eV  \nwhich proves that for lower dopings the dominant vdW energy weakening mechanism  becomes off-resonant coupling \nbetween $\\pi\\rightarrow\\pi^*$ electron-hole excitations, $\\pi$ and $\\pi+\\sigma$ plasmons. \n\nIt should be noted here that such designed (graphene based) slabs might enable modification of attraction between slabs,  e.g. controlled 'sticking' \nand 'un-sticking' of two slabs. For example, two heavily doped graphenes ($1-1$eV case in Fig.\\ref{Fig5}) are strongly bound, however \nbinding energy between pristine graphenes ($0-0$eV case achieved, e.g. simply by electrostatic gating) is reduced more than \ntwice. Moreover, for larger $\\omega_0$ (and fixed doping) the dynamical binding energy is substantially reduced, leading to 'un-sticking' \nof two slabs, and vice versa, their 're-sticking' by reducing the driving frequency.      \n\n\n\\begin{figure}[t]\n\\includegraphics[width=1.0\\columnwidth]{Fig5.pdf}\n\\caption{(Color online). Van der Waals energies $E_c$ of two variously doped, unsupported full conductivity  \n(\\ref{curren1}--\\ref{curren4}) graphenes as functions of driving frequency $\\omega_0$.\nThe left-right graphene dopings are $1-1$eV (thick black solid),  $1-0.2$eV (black solid), 1-1eV (thin black \nsolid), $0.2-0.2$eV (red dashed), $0.2-0$eV (thin red dashed), $0-0$eV (blue dashed-dotted), as also denoted in the figure.\nSeparation between graphenes is $a=10$nm and oscillating amplitude is $\\rho_0=20$nm. } \n\\label{Fig5}\n\\end{figure}\n\n\n\\subsection{Dissipated power - substrate dependence}\n\\label{DISsub}\nIn this section we shall explore how the dissipation power in two oscillating \nslabs depends on the conductivity model we use to describe graphene and how \nsubstrate influences the dissipation power. \n\n\nIn order to facilitate the analysis of the results we shall again use the \napproximation (\\ref{approx}). The lowest order term \nwhich contributes in (\\ref{losshop}) is $f_1$, and from Fig.\\ref{Fig3} it is \nobvious that, for $x<x_{cut}$, the higher order terms ($m=2,3,...$) do not \ncontribute and $f_1$ can be freely approximated by (\\ref{approx}) (red dotted line).\nFurthermore, because the higher order processes (see Fig.\\ref{FigA2}) included in (\\ref{losshop}) \nweakly influence the power $P$ it can be calculated using equation (\\ref{njunja1}) which includes only the lowest order process. \nTherefore the formula for the dissipated power can be rewritten as\n\\begin{eqnarray}\nP=\\frac{\\hbar\\omega_0\\rho^2_0}{4\\pi}\\ \\int\\ Q^3dQ e^{-2Qa}\\hspace{2cm}\n\\label{njunja2}\\\\\n\\nonumber\\\\\n\\hspace{2cm}\\int^{\\omega_0}_{0}\\frac{d\\omega}{2\\pi}\\ ImD_1(Q,\\omega)ImD_2(Q,\\omega_0-\\omega).\n\\nonumber\n\\end{eqnarray}\nThis suggests that the dissipated power, within the parameter space used in this \ninvestigation (for $x<x_{cut}$), behaves as $P\\sim\\rho^2_0$. Also Eq.\\ref{njunja2} \nsuggests that the resonant condition (maximum in $P$) will occur when the \ndriving frequencies satisfy the condition\n\\begin{equation}\n\\omega_0=n_1\\omega_i+n_2\\omega_j;\\ \\ n_1,n_2=1,2,3,...     \n\\end{equation} \nwhere $\\omega_i=\\omega_p$, $\\omega_{TO1}$ and $\\omega_{TO2}$ are the frequencies of hybridized Dirac plasmons and TO phonons, respectively. \n\n\nFigs \\ref{Fig6} show the dissipated power  $P(\\omega_0)$ for two oscillating graphene \nmonolayers, calculated in several approximations: unsupported graphene (no substrate) \nusing Drude expression (\\ref{curren1}--\\ref{curren2}) for the conductivity (blue thin line), and using full conductivity (\\ref{curren1}--\\ref{curren2},\\ref{curren4}) (red dashed line), as well as for graphenes on semiinfinite ($\\Delta\\rightarrow\\infty$) SiO$_2$ substrates with full expression for \nconductivity (black solid line). Both graphene monolayers are doped so that $E_{F1}=E_{F2}=200$meV. \nIn Fig.\\ref{Fig6}(a) the separation between slabs and oscillation amplitude are \n$a=10$nm and $\\rho_0=0.1$nm, respectively. We see that in the Drude model $P$ shows a strong \npeak which comes from the excitation of undamped Dirac plasmons. \nIn the full conductivity model plasmon peak is strongly suppressed and \ninterband $\\pi\\rightarrow\\pi^*$ excitations become the dominant dissipation mechanism. The \nfingerprint of $\\pi\\rightarrow\\pi^*$ excitations in Fig.\\ref{Fig6}(a) is \nlinear $P(\\omega_0)$ behaviour starting at about $200$THz, where we also added cyan dashed lines to guide the eye. It can also be noted that the plasmon peak is red shifted which is reasonable considering that $\\pi\\rightarrow\\pi^*$ transitions push Dirac plasmon dispersion toward lower energies.       \n\n\nIn the presence of the substrate  dissipation is additionally reduced by almost a \nfactor of three. This is because for smaller separations ($a=10$nm) the modes with higher wave \nvectors (e.g. $Q\\approx 0.01a.u.$), which is in this case only the Dirac \nplasmon, dominantly contribute to $P$. In this wavevector region the Dirac \nplasmon already has high enough frequency ($\\omega\\approx 60$THz) that the dynamical part \nof the substrate screening in not active and the substrate dielectric function can be \napproximated by $\\epsilon_S(\\omega)\\approx\\epsilon_{\\infty}$. This causes the reduction of substrate screened Coulomb interaction $\\tilde{v}_Q(\\omega)=\\frac{2}{1+\\epsilon_S(\\omega)}v_Q$ \n(see Eq.\\ref{tildeV11}) and then (considering Eq.\\ref{newD}) reduction of the plasmon intensity, which finally causes the reduction of $P$. Reduction of the screened Coulomb interaction (\\ref{tildeV11}) also causes the reduction of the plasmon frequency which can also be noted.  \n\n \nFig.\\ref{Fig6}(b) shows the dissipated power $P(\\omega_0)$ for the same set of parameters as \nin Fig.\\ref{Fig6}(a) except that the separation between slabs is increased to $a=50nm$.\nAs expected, from the discussion in Sec.\\ref{weakvdW}, $P$ is reduced by about four \norders of magnitude and plasmon peaks are shifted toward lower frequencies. The latter is also expected considering that for larger separations the modes with smaller $Q$ contribute, and here \nthe Dirac plasmon has lower energy. We can notice qualitative difference between $P$ in Figs.\\ref{Fig6}(a) and (b) for the case when substrate is present (black lines). \nIn Fig.\\ref{Fig6}(b) $P$ possesses additional structures (two additional peaks at \n$\\omega_{TO1}+\\omega_{TO2}$ and $2\\omega_{TO2}$) which are not present in Fig.\\ref{Fig6}(a). This is because for larger $a$ the modes with smaller wave vectors (e.g. $Q\\approx 0.002$a.u.) \nstart contributing to $P$, and this is exactly the region where plasmon\/phonon hybridization \noccurs (as ilustrated in Fig.\\ref{Fig4}(a)), so the additional peaks at $\\omega_{TO1}+\\omega_{TO2}$ \nand $2\\omega_{TO2}$ represent the resonant dissipation to two phonon modes.\n\nFigs.\\ref{Fig6}(c) and (d) show the dissipated power $P$ for the same parameters \nas in Figs.\\ref{Fig6}(a) and (b), respectively, except that the oscillation amplitude is \nincreased to $\\rho_0=1$nm. $P$ in Figs.\\ref{Fig6}(c) \nand (d) are qualitatively the same and exactly hundred times \nlarger than $P$ in Figs.\\ref{Fig6}(a) and (b). This confirms $P\\sim\\rho^2_0$ \nbehaviour of the dissipated power with amplitude as predicted by Eq.\\ref{njunja2}. \n\\begin{figure*}[tt]\n\\includegraphics[width=1.0\\columnwidth]{Fig6a.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig6b.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig6c.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig6d.pdf}\n\\caption{(Color online). Dissipated power in two oscillating unsupported Drude \n(\\ref{curren1},\\ref{curren2}) graphenes (blue thin line), unsupported full conductivity \n(\\ref{curren1},\\ref{curren2},\\ref{curren4}) graphenes (red dashed line) and full conductivity graphenes deposited on semiinfinite ($\\Delta\\rightarrow\\infty$) SiO$_2$ substrates (black solid line). \nThe separations between slabs and oscillation amplitudes are (a) $a=10$nm, $\\rho_0=0.1$nm, (b) $a=50$nm, $\\rho_0=0.1$nm, (c) $a=10$nm, $\\rho_0=1$nm and (d) $a=50$nm, $\\rho_0=1$nm. Both graphenes are doped such that $E_{F1}=E_{F2}=200$meV.} \n\\label{Fig6}\n\\end{figure*}\n\n\\subsection{Dissipated power - graphene doping and distance dependence}\n\\label{DISdop}\n\nIn this section we shall explore the dissipated power for two \noscillationg slabs for different graphene dopings.   \n\nFig.\\ref{Fig7}(a) shows the dissipated power in two oscillating graphenes deposited \non semiinfinite ($\\Delta\\rightarrow\\infty$) SiO$_2$ substrates where the graphene \ndopings $E_{F1}-E_{F2}$ are $0-0$meV (blue thin line) $0-200$meV (red dashed line) and \n$200-200$meV (black solid line). The separations between slabs and oscillation amplitude \nare $a=10$nm and $\\rho_0=0.1$nm, respectively. \n\nIf both graphenes are doped $P$ shows the plasmon peak at \nabout $2\\omega_p=100$THz, and starting at about $200$THz it increases \nlinearly, which is the consequence of interband $\\pi\\rightarrow\\pi^*$ excitations, as \nalready observed in Fig.\\ref{Fig6}. \nHowever, if one doped graphene is replaced by pristine graphene ($E_F=0$), which does \nnot support the Dirac plasmon (as shown in Fig.\\ref{Fig4}(c)), the Dirac plasmon in doped \ngraphene can no longer resonantly transfer energy to the Dirac plasmon in another graphene \nand $P$ loses the plasmon peak at $2\\omega_p$. \nHowever, the visible step remains (at about $75$THz) which is the consequence of energy \ntransfer between Dirac plasmon in the doped graphene and $\\pi\\rightarrow\\pi^*$ excitations \nin the undoped one. In this case (small $a$ and larger $Q$) phonons are still very weak \nand do not represent important dissipation channel. When both graphenes are pristine the \nonly dissipation comes from the resonant energy transfer between $\\pi\\rightarrow\\pi^*$ \nexcitations in different graphenes, resulting in the strictly linear behaviour of \n$P$.  \n\nFig.\\ref{Fig7}(b) shows the dissipated power $P$ for the same parameters as in \nFig.\\ref{Fig7}(a) except that the separation between slabs is increased \nto $a=50nm$. As we have already discussed in Fig.\\ref{Fig6}(a), in this case the modes with \nsmaller wave vectors $Q$ contribute and the dissipated power $P$ gets additional \nstructures coming from resonant phonon excitations. For the case $E_{F1}-E_{F2}=200-200$meV (black solid line) (coupling between modes in Fig.\\ref{Fig4}a) the dissipated power shows three \npeaks at $\\omega_{TO1}+\\omega_{TO2}\\approx 40$THz, $2\\omega_{TO2}\\approx 60$THz and \n$2\\omega_{p}\\approx 75$THz. For the case $E_{F1}-E_{F2}=0-200$meV (red dashed line) there \nis a possibility for resonant coupling between two phonons in the slab with pristine graphene and \nthree hybridized plasmon\/phonon modes in the slab with doped graphene (coupling beteen modes in Fig.\\ref{Fig4}a and modes in Fig.\\ref{Fig4}c). The three peaks correspond to resonant \ncouplings at $\\omega_{TO1}+\\omega_{TO2}$, $2\\omega_{TO2}$ and $\\omega_{TO2}+\\omega_p$, \nas denoted in Fig.\\ref{Fig7}(b). When both graphenes are pristine, i.e. \n$E_{F1}-E_{F2}=0-0$meV (thin solid blue line) the dominant dissipation channels \nbecome the resonant coupling between phonons in both slabs (coupling between \nmodes in Fig.\\ref{Fig4}(c)). The three peaks correspond to resonant couplings at $2\\omega_{TO1}$, $\\omega_{TO1}+\\omega_{TO2}$ and $2\\omega_{TO2}$, as denoted in Fig.\\ref{Fig7}(b). \nOf course, in all three cases $P$ shows linear behaviour for larger \n$\\omega_0$ coming from the resonant $\\pi\\rightarrow\\pi^*$ excitations in both slabs. \nFigs.\\ref{Fig7}(c) and (d) show the same as Figs.\\ref{Fig7} (a) and (b), except \nthat the oscillation amplitude is increased to $\\rho_0=1$nm.\nAs in Figs.\\ref{Fig6}, $P$ is qualitatively similar and exactly \nhundred times larger than $P$ in Figs.\\ref{Fig7}(a) and (b). \nThis again confirms the $P\\sim\\rho^2_0$ behaviour. \n\nThis strong dependence of dissipated power on graphene \ndoping suggests many opportunities for applications. \n\n\n\\begin{figure*}[tt]\n\\includegraphics[width=1.0\\columnwidth]{Fig7a.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig7b.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig7c.pdf}\n\\includegraphics[width=1.0\\columnwidth]{Fig7d.pdf}\n\\caption{(Color online). Dissipated power for two oscillating graphenes deposited on semiinfinite ($\\Delta\\rightarrow\\infty$) SiO$_2$ substrates where the graphene dopings $E_{F1}-E_{F2}$ are $0-0$meV (blue thin line) $0-200$meV (red dashed line) and  $200-200$meV (black solid line). The separations between slabs and oscillation amplitudes are (a) $a=10$nm, $\\rho_0=0.1$nm, (b) $a=50$nm, $\\rho_0=0.1$nm, (c) $a=10$nm, $\\rho_0=1$nm and (d) $a=50$nm, $\\rho_0=1$nm. The graphene response is \ncalculated using full conductivity expression (\\ref{curren1}--\\ref{chi-sigma}).} \n\\label{Fig7}\n\\end{figure*}\n\n\n\n\\section{Conclusions}\n\\label{Sec5}\n\nIn this paper we have provided a complete theoretical description of van der Waals and friction \nforces for two slabs in relative oscillatory motion which \nincludes variable temperatures in two slabs, their dynamical properties, and dependence on characteristic oscillation amplitude \nand frequency. In Appendix we also provide, for comparison, analogous expressions for the slabs in parallel uniform motion. \n\nWe applied this formulation to explore van der Waals and friction forces between two oscillating slabs, each consisting of atomically thick \ncrystal (e.g. graphene) adsorbed on a dielectric substrate (SiO$_2$). \nWe explore dependence of these forces on  osillator characteristics such  as driving amplitude \n$\\rho_0$ and frequeny $\\omega_0$, but also on slab separation $a$, on graphene doping $E_F$ and on substrate properties.   \nWe show how the spectra of coupled electronic\/phononic excitations in slabs determine the energy transfer \nprocesses in this system. \n\nWe show that, in general, as the driving frequency $\\omega_0$ increases the vdW energy first shows an unusual plateau, and then decreases. \nWe propose the idea of controlling  the 'sticking' and 'un-sticking' of slabs by tuning the \ngraphene dopings $E_{Fi}$ and driving frequency $\\omega_0$.\n\n\n\nWe also found a simple $\\rho^2_0$ dependence of both the vdW force  and dissipated power. \nThe dissipated power between Drude model graphenes, as function of $\\omega_0$, shows  unrealistically strong \n$2\\omega_p$ peak. However, in a realistic graphene (whose dielectric properties are calculated \nfrom first principles) this peak is strongly reduced and red shifted. We also explain why the substrate substantially reduces dissipated power peak $2\\omega_p$.  \nFor larger separations $a$ additional peaks appears in dissipation power originating from the excitations of hybridized substrate phonons.     \n\nWe showed that if one graphene is pristine ($E_F=0$) it causes the disappearance of the strong $2\\omega_p$ \npeak in the dissipated power. Moreover, for larger separations $a$ the doping causes shifts, appearance  and disappearance  of many peaks \noriginating from resonant coupling between hybridised electronic\/phononic excitations in graphene\/substrate slabs.\n\n\nObviously, when present, the Dirac plasmons  are the dominant channels through which the energy between slabs can be \ntransferred, so  the studied model system strongly supports the possibility to control the energy or heat transfer between \nthe slabs by tuning the graphene doping, e.g. by  electrostatic gating.\nMore radically, for zero doping $E_F=0$ the energy transfer can be locked, and vice \nversa.    \n\nIn conclusion, it is expected that studies of energy transfer processes in the case of \nosillating slabs, based on our complete theoretical description, will provide supplementary and more \npractical approach as compared to those in parallel uniform motion.   \n\n\n\n\\section*{Acknowledgments}\nTwo of the authors (V. D. and M. \\v S.) are grateful for the hospitality at the Donostia International Physics \nCenter where this work was finalized, and for useful discussions to J. Pendry, A. A. Lucas, S. Silkin  and I. Kup\\v ci\\' c. V. D. acknowledges the support of the University of the Basque Country and the Spanish Ministerio de Ciencia y Tehnologia.\nV. D. also acknowledges the support of QuantiXLie Centre of Excellence, a project\ncofinanced by the Croatian Government and European Union through the\nEuropean Regional Development Fund - the Competitiveness and Cohesion\nOperational Programme (Grant KK.01.1.1.01.0004).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{\\refname}}{}{}{}\n\n\\usepackage{lettrine}\n\\usepackage{paralist}\n\n\\usepackage{titlesec}\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\rhead{\\thepage}\n\\usepackage{graphicx}\n\\restylefloat{figure}\n\\usepackage{url}\n\\usepackage{amssymb}\n\\newcommand{\\jeff}[1]{\\noindent\\underline{\\textbf{#1}}}\n\\newenvironment{Figure}\n  {\\par\\medskip\\noindent\\minipage{\\linewidth}}\n  {\\endminipage\\par\\medskip}\n \\newcommand{\\bigskip \\hrule \\bigskip}{\\bigskip \\hrule \\bigskip}\n\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}{Corollary}[theorem]\n\\newtheorem{lemma}[theorem]{Lemma}\n\\theoremstyle{definition}\n\\newtheorem{definition}{Definition}[section]\n\\theoremstyle{remark}\n\\newtheorem*{remark}{Remark}\n\\newcommand{\\efficientheader}[1]{\\noindent \\underline{\\textbf{#1}}}\n\\newcommand{\\boxedgraphic}[2]{$\\vcenter{\\hbox{\\boxed{\\includegraphics[width=#1]{#2}}}}$}\n\n\\setcounter{tocdepth}{4}\n\\setcounter{secnumdepth}{4}\n\\setlength{\\parskip}{1mm}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\n\\title{Augmenting Supervised Learning by Meta-learning Unsupervised Local Rules}\n\n\n\\author{%\n  Jeffrey Cheng \\thanks{Please find my most recent contact information at \\texttt{http:\/\/jeffreyscheng.com\/}.} \\\\\n   Department of Computer Science\\\\\n  University of Pennsylvania\\\\\n  Philadelphia, PA 19104\\\\\n  \\texttt{jeffch@seas.upenn.edu}\\\\\n  \\And\n  Ari Benjamin\\\\\n  Department of Bioengineering\\\\\n  University of Pennsylvania\\\\\n  Philadelphia, PA 19104\\\\\n  \\texttt{aarrii@seas.upenn.edu}\\\\\n   \\AND\n   Benjamin Lansdell\\\\\n   Department of Bioengineering\\\\\n   University of Pennsylvania\\\\\n   Philadelphia, PA 19104\\\\\n   \\texttt{lansdell@seas.upenn.edu}\n   \\And\n   Konrad Paul Kording\\\\\n   Department of Bioengineering\\\\\n   University of Pennsylvania\\\\\n   Philadelphia, PA 19104\\\\\n   \\texttt{kording@upenn.edu}\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\n    The brain performs unsupervised learning and (perhaps) simultaneous supervised learning. This raises the question as to whether a hybrid of supervised and unsupervised methods will produce better learning. Inspired by the rich space of Hebbian learning rules, we set out to directly learn the unsupervised learning rule on local information that best augments a supervised signal. We present the Hebbian-augmented training algorithm (\\textbf{HAT}) for combining gradient-based learning with an unsupervised rule on pre-synpatic activity, post-synaptic activities, and current weights. We test \\textbf{HAT}'s effect on a simple problem (Fashion-MNIST) and find consistently higher performance than supervised learning alone. This finding provides empirical evidence that unsupervised learning on synaptic activities provides a strong signal that can be used to augment gradient-based methods.\n    \n    We further find that the meta-learned update rule is a time-varying function; thus, it is difficult to pinpoint an interpretable Hebbian update rule that aids in training.  We do find that the meta-learner eventually degenerates into a non-Hebbian rule that preserves important weights so as not to disturb the learner's convergence.\n\\end{abstract}\n\n\\section{Prior Work and the Local Meta-Learning Setting}\n\nBackpropagation achieves great performance in neural net optimization, but might not be biologically plausible because most problems are not explicitly phrased as classification with true labels, because neurons only know local signals (e.g. synaptic density, ACh levels, current), and because backpropagation uses the computational graph, a separate data structure with no known biological basis.  \n\nAlthough some supervised training schemes are more biologically plausible (e.g. contrastive Hebbian learning\\cite{seung} and equilibrium propagation\\cite{scellier}), it's currently unknown whether the behavior of all neurons is accurately encapsulated by these models.  We speculate that some local, unsupervised learning occurs in the brain and demonstrate that the addition of local, unsupervised rules to standard backpropagation actually improves the speed and robustness of learning. \n\n\\subsection{Local Learning Rules}\n\nWe begin by defining a local learning rule.  Consider two adjacent neurons $i, j$ with weight $w_{ij}$: given an impulse traversing $i,j$ with activations $v_i, v_j$, a local learning rule computes updates $\\Delta w_{ij}$ using local data $v_i, w_{ij}, v_j$.  Note that by this definition, a local learning rule is unsupervised at face value.\n\nMany neuroscientists have hypothesized specific functions that describe the brain's true (unsupervised) local learning rule.  Most such rules involve using the correlation of activations as part of the update rule.  Examples include Hebb's Rule ~\\cite{hebb}, Oja's Rule  ~\\cite{oja-1982}, the Generalized Hebbian Algorithm ~\\cite{gha}, and nonlineear Hebbian rules ~\\cite{nonlinear-hebbian}.\n\nIt is not obvious which of these rules (if any) describe the true behavior of neurons.  We employ \\emph{meta-learning} (learning how to learn) as an investigative tool.\n\n\\subsection{The Meta-Learning Framework}\n\nOptimization functions are algorithms too; it stands to reason that we can learn the best optimization function.  In the meta-learning framework, one model $A$ learns a task (e.g. Fashion-MNIST) while another model $B$ learns how to optimize $A$.  \n\nMeta-learning has achieved great results in finding robust optimization schemes.  Andrychowicz et. al. used meta-learning to find the best gradient-based optimization function ($B$ learns to update $A$ using $A$'s gradients)  \\cite{l2lgbg}, and Chen et. al. used meta-learning to find the best gradient-free optimization function ($B$ learns to update $A$ using only the sequence of $A$'s losses).  \\cite{l2lwgbg}  Finally, Metz et al. demonstrated a fully differentiable architecture for learning to learn unsupervised local rules and demonstrate better-than-random performance on a few-shot basis. \\cite{metz}\n\nIf $B$ consistently converges to some stable rule, we take it as strong evidence that this rule may occur in biological brains as well.  We therefore wish to extend Metz's approach to learning semi-supervised local rules not only to improve performance but also to investigate the functional form of the meta-learned update rule.\n\n\\section{The Hebbian-Augmented Training Algorithm}\n\nThe \\textbf{Hebbian-Augmented Training} algorithm (\\textbf{HAT}) is an algorithm that trains the neural net $L$ twice per sample: using local, unsupervised rules on the forward pass and using backpropagation-based gradient descent on the backward pass.  \n\nFormally, we create 2 multilayer perceptrons: a learner $L(\\cdot\\mid\\phi_L)$ with parameters $\\phi_L$ and a meta-learner $M(v_i, v_j, w_{ij}\\mid \\phi_M)$ with parameters $\\phi_M$, which takes inputs $v_i, w_{ij}, v_j$ and returns $\\Delta w_{ij}$.  For a single sample $(\\vec{x}, \\vec{y})$, we train $L$ without supervision using $M$ and $\\vec{x}$; we \\textit{simultaneously} train $L$ and $M$ with supervision using $A$ and $\\vec{y}$.\n\n\\subsection{Phase 1: The Forward Pass}\n\nOn the forward pass, we compute activations for each layer.  For a given layer $\\ell$, we now have the inputs, outputs, and current weights -- all of the inputs of local learning rule.  We can then apply the outputs of meta-learner $M$ to update the weights of layer $\\ell$.  We then \\textit{recompute} the activations of layer $\\ell$ using the new weights.  This process is done efficiently by convolution (for details, see Appendix A).  We compute the activations of the first layer $\\ell_1$, update $\\ell_1$, compute the activations of the second layer $\\ell_2$, update $\\ell_2$, and so on until we compute the predicted Weights $\\hat{\\vec{y}}$ and update $\\ell_{|L|}$.\n\n\\subsection{Phase 2: The Backward Pass}\n\nOn the backward pass, we backpropagate.  Since we \\textit{recomputed} the activations of each layer using weights updated by $M$, the weights of $M$ are upstream of the weights of $L$ in the computational graph; thus, a single iteration of the backpropagation algorithm will compute gradients for both $M$ and $L$.  Given a gradient $\\nabla_p$ for each parameter $p\\in \\phi_L\\cup \\phi_M$, we then perform a supervised update $p\\gets p + A(p, \\nabla_p)$.  The key insight is that the convolution of the meta-learner over the weights of the learner forms a fully differentiable framework $M\\leadsto L \\leadsto \\vec{y}$.\n\n\\begin{algorithm}[H]\n\\caption{Hebbian-Augmented Training Algorithm}\\label{alg:R}\n\\begin{algorithmic}[1]\n\\Procedure{Train-Example}{$L, M, A, \\vec{x}, \\vec{y}$}\n    \\State $\\vec{v}_0\\gets \\vec{x}$\\Comment{Let $\\vec{v}_i$ represent the impulse in layer $i$}\n    \\For{weights $W_\\ell, \\ell\\in [1...|L|]$} \\Comment{Forward pass}\n        \\State $\\hat{v}_{\\ell+1}=\\sigma(W_\\ell\\times \\vec{v}_{\\ell} + b_\\ell)$ \\Comment{$\\hat{v}_{\\ell+1}$ is a placeholder output as input to $M$}\n        \\State $W_\\ell\\gets W_\\ell + M(\\vec{v}_\\ell, W_\\ell, \\vec{v}_{\\ell+1})$\\Comment{Updates weight using local rule $M$}\n        \\State $\\vec{v}_{\\ell+1}\\gets \\sigma(W_\\ell\\times \\vec{v}_{\\ell} + b_\\ell)$\\Comment{Propagate $\\hat{v}_{\\ell+1}$ as actual layer output}\n    \\EndFor\n    \\State Backpropagate loss $H(\\vec{v}_{|L|}, \\vec{y})$.\n    \\For{layer weight $W_\\ell$ in $L$ and $M$}\\Comment{Backward pass}\n        \\State $W_\\ell \\gets A\\left(\\frac{\\partial H}{\\partial W_\\ell}\\right)$\\Comment{Apply gradient update using optimizer $A$}\n    \\EndFor\n    \\State\\Return $L, M$\\Comment{Return updated learner and updated meta-learner}\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{\\textbf{HAT} Improves Performance on Fashion-MNIST}\n\nWe hypothesize that the \\textbf{HAT} algorithm will have three positive effects.\n\\begin{itemize}\n    \\item \\textbf{HAT} will train the learner $L$ faster since there are twice as many updates.  In ordinary backpropagation the metadata generated from the forward pass is computed and wasted; in \\textbf{HAT}, the metadata is computed and used to generate a (potentially) useful update.\n    \\item \\textbf{HAT} will improve the convergence of $L$.  The second update should introduce some stochasticity in the loss landscape since it is not directly tied to gradient descent, which may lead $L$ into better local optima.\n    \\item \\textbf{HAT} will improve the performance of $L$ when some examples are not labeled.  Backpropagation has no ability to learn from just the input $\\vec{x}$, while \\textbf{HAT} is able to perform the unsupervised update.\n\\end{itemize}\n\nWe generate two learning curves to test these hypotheses: one with respect to time and one with respect to the proportion of labeled examples.  The charts below represent the aggregated learning curves of 100 pairs $(L_i, M_i)$.\n\n\\begin{figure}[H]\n    \\centering\n    \\boxedgraphic{0.45\\textwidth}{images\/experiment_1_median.png}\n    \\boxedgraphic{0.45\\textwidth}{images\/experiment_2_median.png}\n    \\caption{The effect of \\textbf{HAT}'s on median accuracy curves.}\n\\end{figure}\n\n\\vspace*{0.2cm}\n\nWe find that the effects of \\textbf{HAT} on training are clearly positive. The median accuracy of the neural nets trained by \\textbf{HAT} is clearly increased along the learning curve, and the \\textbf{HAT}-group neural nets reach a higher asymptotic value than the control group.  We do note that the two learning curves seem to inflect around the same point -- \\textbf{HAT} does not seem to cause a faster convergence, just a better one.  We attribute this to the meta-learner's convergence; it may take the meta-learner up to 0.5 epochs to start to have positive effects.\n\nOne potential concern with adding unsupervised meta-learner updates is that after the convergence of the base learner $L$, the meta-learner's continued output of non-zero updates might ``bounce'' the base learner out of an optimum.  Remarkably, we see in the above plot that the performance of the \\textbf{HAT}-trained neural nets is quite stable for the entire 18 epochs of post-convergence duration.\n\nTo our surprise, we find that \\textbf{HAT} is more effective when there are more labels, even though the self-supervised component of the algorithm is designed to take advantage of scarce labels.  We attribute this to slow convergence of the meta-learner $M$ -- when labels are scarce, the meta-learner may actually converge slower than the learner and thus provide bad update suggestions.\n\n\\section{The Behavior of the Meta-Learned Update Rule}\n\nWe would like insight into why \\textbf{HAT} improves the training of neural nets over vanilla gradient descent.  Thus, we will analyze the functional form of the learned update rule $M$ after it has fully converged.  Recall the setting from experiments 1 and 2: we generate 100 pairs of learners and meta-learners: $(L_i, M_i)$ for $i\\in \\{1,...,100\\}$.  We then investigate the pointwise mean $\\overline{M} $of these meta-learners. \n\nWe first visualize the dependence of the function $\\overline{M}$ on its inputs $(v_i, v_j, w_{ij})$. \n\\begin{figure}[H]\n    \\centering\n    \\boxedgraphic{0.45\\textwidth}{images\/experiment_3_vi_vs_Hebb.png}\n    \\boxedgraphic{0.45\\textwidth}{images\/experiment_3_wij_vs_Hebb.png}\n    \\boxedgraphic{0.45\\textwidth}{images\/experiment_3_vj_vs_Hebb.png}\n    \\caption{These plots show very little dependence of the converged rule on $v_i$ and $w_{ij}$.}\n    \\label{fig:my_label}\n\\end{figure}\n\nWe find that a remarkably linear dependence on $v_j$ explains almost all of the variance in the outputs of the meta-learned update rule.  This indicates that the rule is a ``rich-get-richer'' scheme: neurons that already fired with high magnitude will experience larger incoming weights and thus be encouraged to fire with high activation in the future.\n\nThis linear dependence is surprising since all of the hypothesized rules in neuroscience have a dependence on $v_i\\cdot v_j$.  As a sanity check, we attempted to directly apply this update rule ($\\Delta w_{ij} \\approx 2\\cdot v_j$) without meta-learning to see if we can replicate \\textbf{HAT}'s performance improvement.  However, the results were decisively negative -- \\textbf{HAT} improves performance, but the a priori application of \\textbf{HAT}'s update rule decreases it.  We present three hypotheses:\n\n\\begin{itemize}\n    \\item Perhaps $M$ learns a good update rule while $L$ is training, then learns a degenerate rule once $L$ has converged.  The sole purpose of this degenerate rule would be to not un-learn the important weights that have already converged (thus explaining the rich-gets-richer behavior of the rule $f(\\cdot)=2v_j$).  Thus, analyzing the black-box function at epoch 20 is merely the wrong time -- perhaps observing the meta-learned rule at epoch 1 would be more insightful and useful.\n    \\item Perhaps $M$ learns a good update rule in each run, and these update rules are all complex functions with no good low-order polynomial approximations; however, their pointwise mean (which is itself not a good local update rule) happens to be linear.  Thus, $\\overline{M}$ is the wrong object to analyze and presents behaviors that are not indicative of the results of experiments 1 and 2.\n    \\item Perhaps the learning of $M$ is extremely transient.  For any given point in time, there is a different optimal learning rule, and our exercise in finding a fixed local, unsupervised update rule that is universal across training is futile.\n\\end{itemize}\n\n\\section{Conclusion}\n\nThe \\textbf{HAT} algorithm demonstrates that local, unsupervised signals can provide performance-improving weight updates.  Neural nets under \\textbf{HAT} converge to better asymptotic losses as long as there is sufficient time ($>0.5$ epochs) and a sufficient number of labels ($>20\\%$ of the data is labeled).  The latter finding is surprising since the addition of an unsupervised learning algorithm depends on the presence of labels in order to deliver marginal benefits over gradient descent.\n\nThe underlying form of the learned rule that makes \\textbf{HAT} successful is still a mystery; we find that while the meta-learner may learn a useful update rule during training, the meta-learner does not converge to this useful rule in the long run and instead devolves into a linear function \\textbf{Converged-Rule}.  This converged function preserves fully-converged weights by reinforcing incoming weights for neurons with high activations.\n\n\\subsection{Future Work}\n\nThe discovery that \\textbf{HAT} does not stably converge to a function makes analysis quite difficult.  However, there is potential for future work to do more subtle analyses.\n\nImagine a time $t$ during training in which the meta-learner $M$ has converged to a useful function, but the learner $L$ has not yet finished training.  A follow-up to this thesis might be to discover whether there such a time $t$ exists, what the structure of $M$ at time $t$ is, and how $M$ changes the weights of $L$ at time $t$.  One potential methodology might be to observe the function $f$ not as a 3-dimensional function in $(v_i, w_{ij}, v_j)$ but rather as a 4-dimensional function in $(v_i, w_{ij}, v_j, t)$.  Observing the function along the $t$-axis and checking for phase changes would shed light on whether a single useful update rule is learned during training or whether \\textbf{HAT}'s learning is truly transient and continuous.  If this follow-up were to succeed, then we could have an a priori rule to apply without having to metalearn update rules.\n\nExtracting the local rules from multiple domains could either find that \\textbf{HAT} learns a universal rule or that functional distance between two rules describes the ``difference'' between their originating domains.\n\\vspace*{-1mm}\n\\begin{itemize}\n    \\itemsep-0.4em \n    \\item Suppose we always metalearn the same rule, regardless of problem domain.  \\textbf{Optimal-Hebb} is then a universal learning rule.\n    \\item Suppose \\textbf{Optimal-Hebb} is not universal for all problems.  For local rules $R_A, R_B$ on problems $A,B$, integrating $\\int_{\\mathbb{R}^3} (R_A-R_B)\\cdot dF(v_i, w_{ij}, v_j)$ for input distribution $F$ gives an explicit measure for how similar $A$ and $B$ are.  This provides a systematic way to identify pairs of learning problems that are good candidates for transfer learning.\n\\end{itemize}\n\n\\newpage\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIt is a property of fundamental importance within statistical physics that generic and realistic thermodynamic systems exhibit one particular state -- thermal equilibrium -- which is always approached, irrespective of the initial condition. Yet the important question of which\nmicroscopic conditions are necessary or sufficient for the thermalization of a closed quantum many-body system is still largely unanswered~\\cite{Polkovnikov11}. This is of particular importance, especially because there exists a specific class of isolated quantum systems, termed integrable, for which relaxation to thermal states is prevented due to the presence of an extensive number of (quasi-)local conservation laws~\\cite{Rigol06,Rigol07,Kollar11}.\nSuch particular systems often represent isolated points in the parameter space of physical many-body systems and demand a precise tuning of the microscopic parameters. Nevertheless, these models are very valuable because they often represent fixed points of renormalization group theories and as such contain the low-temperature equilibrium properties of a much wider class of systems. This directly leads to an apparent dilemma in quantum many-body theory which has attracted a lot of interest recently. In particular, beyond equilibrium these integrable models become nongeneric as they fail to thermalize. Instead, they are trapped in extended prethermal states described by nonthermal generalized Gibbs ensembles~\\cite{Polkovnikov11,Rigol06,Rigol07,Rigol08,Kollar11,cazalilla06,cardycalabrese06,Barthel08}. Resolving this dilemma is one of the major challenges for the understanding of the coherent dynamics of quantum many-body systems.\n\nIn this work we address this question for a paradigmatic low-energy model: the Luttinger liquid~\\cite{haldane81,haldane81b,giamarchi04}, representing the fixed point theory of systems of interacting fermionic particles in one dimension at low temperatures. The Luttinger liquid is an integrable theory failing to thermalize but rather exhibiting a description in terms of a generalized Gibbs ensemble~\\cite{cazalilla06,Iucci09,Iucci10}. Here, we will be interested in the nonequilibrium dynamics in the presence of a weak fermionic band curvature, which represents a generic perturbation, irrelevant in the low-energy equilibrium limit, but relevant on intermediate to long time scales in order to drive the crossover towards thermalization. \n\nThe increasing number of cold atom experiments performed under out of equilibrium conditions~\\cite{Greiner2002ux,kinoshita,Hofferberth06,trupke13,schmiedmayer12,nagerl13,meinert14,preiss15,hild14,cheneau12} has driven significant interest in the theoretical understanding of the non-equilibrium dynamics in quantum many-body systems. Importantly, these experiments share a remarkable isolation from the environment, thereby probing the purely coherent unitary time evolution on the experimentally relevant time scales. This has paved the way to experimentally study the constrained relaxational dynamics of quantum systems close to integrability~\\cite{kinoshita,Langen14,agarwal14,schmiedmayernphys12}, showing unconventional properties due to the anticipated (quasi-)local conservation laws.  Although the inherent integrability breaking terms, resulting from, e.g., imperfections in the particle-particle interactions or higher orbital modes, are considered to be weak, they are believed to eventually cause relaxation to thermal states on long-time scales. Yet a full understanding of this process has not been achieved so far.\nWithin the current understanding, however, the thermalization dynamics of quantum many-body systems with weak integrability-breaking perturbations is expected to occur via a two-stage process. Initially, the dynamics of local observables at transient and intermediate time scales are controlled by the corresponding integrable theory\nserving as a metastable attractor for the non-integrable dynamics~\\cite{Moeckel,Kollar11,Stark13}. \nThis trapping in a metastable state has been termed prethermalization~\\cite{berges_pretherm,Moeckel} and is expected to exist for several non-integrable models and models close to integrability~\\cite{Moeckel,Eckstein09,Essler14,Kollar11,Rosch08,Marcuzzi13,Nessi15,Fagotti14,Fagotti15,Babadi2015,Bertini15}. \nIn the quasi-particle picture, prethermalization is associated with the initial formation of well-defined excitations \\cite{Moeckel} which leads to a dephasing of all terms that are not diagonal in quasi-particle modes, i.e. to a projection of the initial density matrix onto the diagonal ensemble in the quasi-particle basis. After this intermediate  quasi-particle formation, the dynamics eventually  crosses over to the thermalization regime, where weak  quasi-particle scattering leads to a slow redistribution of energy and establishes detailed balance between the different modes. This causes asymptotic thermalization on long time scales compatible with the Eigenstate-Thermalization-Hypothesis~\\cite{Srednicki94,Deutsch91,Rigol08,Gibbs,Biroli10}.\n\nIn equilibrium, the fermionic band curvature in the Luttinger liquid, because irrelevant in the renormalization group sense,  does not modify static correlation functions, which are well described by the quadratic Luttinger theory. Importantly, however, the curvature has a strong impact on frequency-resolved fermionic quantities. This has been observed in Coulomb drag experiments \\cite{Debray01,Debray02}, which could not be explained in terms of a quadratic Luttinger theory. In a hydrodynamic representation, the band curvature describes resonant scattering processes between the elementary phononic excitations of the system, such that perturbation theory is plagued by divergences due to the resonant nature of the interactions. Important first approaches to the interacting Luttinger liquid applied a self-consistent Born approximation in order to determine the phonon self-energy on the mass-shell \\cite{andreev80,samokhin98,zwerger06}. However, these works were unable to explain the frequency-dependence of the self-energy, which appeared to be non-negligible for dynamic observables. Using a combination of bosonization and subsequent refermionization a general theory has been developed which has been very successful in determining spectral equilibrium properties such as the dynamic structure factor and the  fermionic spectral function in thermal equilibrium \\cite{pustilnik03,pustilnik07,imambekov09,imambekov09a}. Importantly for the scope of the present work, however, it has not yet been possible to generalize this methodology to systems out of equilibrium. Only recently, these equilibrium results have been recovered by a quantum hydrodynamic approach \\cite{lamacraft15,lamacraft15a}, showing that hydrodynamics is also capable of controlling the resonant phonon interactions.\n\nThe theoretical finding of these works is that the elementary excitations are no longer described in terms of bosonic quasi-particles with exact energy-momentum relation $\\omega=u|q|$ but dissolve into a continuum of excitations. This continuum, however, is energetically confined between two well-defined excitation branches $\\epsilon^-_q<\\omega<\\epsilon^+_q$ (with $\\epsilon^{\\pm}_q\\rightarrow 0$ as $q\\rightarrow0$) at which the spectral weight of the bosonic excitations features algebraic divergences, reflected in corresponding divergences of the dynamical structure factor. This fine structure in the bosonic spectral weight, and equivalently self-energy, makes the development of a general kinetic theory for \\emph{frequency-resolved} observables a very demanding task, which has not yet found a satisfactory solution. However, as will be shown in this work, static properties and their time evolution are nevertheless accessible.\n\nThe goal of this work is to study \\changed{the escape out of the prethermalization regime and the crossover towards thermalization} in Luttinger liquids with \\changed{quadratic} fermionic dispersion on the basis of a hydrodynamic description. Specifically, we aim at formulating a kinetic theory for the momentum distribution of the phononic degrees of freedom \\changed{taking into account the leading nonlinear corrections due to the quadratic dispersion. While in this way we are able to describe the escape out of the prethermalization regime in a controlled way, the final asymptotic thermalization of the system might be modified by the more subleading, off-resonant contributions which we do not consider here.}\nThe kinetic equation describes the time-evolution of the phonon momentum distribution and is\nsuitable in the long-wavelength limit and for weak quenches but still goes beyond the regime of linear response. In turn this kinetic theory gives a valid description  for the fermionic occupation distribution in the vicinity of the Fermi points where the anticipated fine-structure of the bosonic spectral weight only gives subleading contributions.\nThis \"semi-static\" -- and as a consequence tractable -- description, covers the forward time evolution of any static, i.e. frequency independent, observable.\nWe show that the dynamics of precisely these frequency independent observables depend only on the time evolution of the momentum distribution of excitations $n_q$ and can be captured within a kinetic theory. \nThe justification for this approach is the subleading width of the excitation spectrum $|\\epsilon^+_q-\\epsilon^-_q|\\ll u|q|$ compared to the phonon energy for all relevant $q$ (below the Luttinger liquid cutoff), which is equivalent to the statement that even in the presence of the non-linearity the continuum of excitations in the hydrodynamic description is tightly bound to the mass-shell. This condition replaces the common quasi-particle criterion \\cite{kamenevbook} and enables a thorough kinetic description.\n\n\\mh{The applicability of the kinetic equation requires the preformation of well-defined quasi-particles out of the bare particles which occurs during the process of prethermalization before the quasi-particle scattering sets in. We, however, find that close to the integrable point of vanishing fermionic interactions quasi-particle formation becomes very slow shifting the applicability of the theory for weakly interacting fermions to long time scales and far distances. We give quantitative estimates of the corresponding spatio-temporal scales of the breakdown of the kinetic theory. Not too close to the noninteracting point, however, the kinetic equation is well justified and allows us to study the escape out of the prethermalization regime towards thermalization.}\nIn the regime of applicability, the kinetic equation leads in the asymptotic long-time limit to a linearized quantum Boltzmann equation whose attractor is the desired thermal Gibbs state.\nWe find that the thermalization dynamics out of the prethermal state is triggered by short wavelength modes and afterwards progressing algebraically slowly towards longer wavelengths. \nWhether this is a generic feature of weakly-perturbed integrable theories, is an important and interesting question for future work. \n\nThe main result of this work is a spatio-temporal decomposition of correlations in the studied nonlinear Luttinger Liquid, which is illustrated in Fig.~\\ref{fig:QuenchDiag}. \nBy analyzing the equal-time fermionic Green's function $G\\sub{t,x}^{<}$, the Fourier transform of the fermionic occupation distribution, we find three regimes which we term prequench, prethermal, and thermal and which are separated by two crossover scales $x\\sub{th}(t)$ and $x\\sub{pt}(t)$ obeying $x\\sub{th}(t) < x\\sub{pt}(t)$. The crossover scale $x\\sub{pt}(t)=2ut$ sets the light cone~\\cite{cardycalabrese06} with $u$ the sound velocity of the elementary bosonic excitations of the integrable theory. Causality implies that for distances $x \\gg x\\sub{pt}(t)$ the system's properties are not yet influenced by the nonequilibrium protocol, but are rather given by the initial state yielding the notion of the prequench regime. Inside the light cone for distances $x<x\\sub{pt}(t)$ we identify a further crossover scale $x\\sub{th}(t)$ separating the prethermal and thermal spatial regions. For distances $x\\sub{th}(t) \\ll x \\ll x\\sub{pt}(t)$ the system's spatial correlations are controlled by the integrable theory which for long times are determined by the associated generalized Gibbs ensemble. This regime is therefore called prethermal. \nInterestingly, the thermalization dynamics, triggered by the weak fermionic nonlinearity, sets in at even smaller scales $x\\ll x\\sub{th}(t)$. At these distances, the correlations approach their thermal form. However, the associated effective temperature $\\tilde T_t$ is larger than the expected temperature $T$ for the asymptotic fully thermalized state. Instead $\\tilde T_t$ is a dynamical quantity approaching $T$ only algebraically slowly due to macroscopic dynamical slow modes.\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=1\\linewidth]{QuenchDiag}\n  \\caption{Illustration of the spatio-temporal thermalization and prethermalization dynamics in terms of the fermionic Green's function $G^<_{t,x}$. For long distances, $x>2ut, x>x\\sub{th}$, the Green's function is determined by the quasi-particles of the initial state and feature algebraic decay in real-space corresponding to the pre-quench state of the system, modulated by a amplitude decaying as a stretched exponential in time. In the intermediate regime $2ut<x<x\\sub{th}(t)$, the corresponding quasi-particles correspond to the post-quench Hamiltonian but are distributed according to a non-equilibrium distribution function, inducing a prethermal real-space scaling behavior $|G^<_{t,x}|\\sim |x|^{-\\gamma\\sub{Eq}\\gamma\\sub{GGE}}$. On short distances $x<x\\sub{th}(t)$, the Green's function is thermal, $\\sim \\exp{-|x|\/\\xi_T}$, described by an effective temperature $\\tilde{T}_t$ and a corresponding thermal correlation length $\\xi_T$. \nThe scaling $x\\sub{th}(t)\\sim t^{\\alpha_{\\lambda}}$, $\\alpha<1$ implies that there exists a minimal distance $x_m$, for which no clear prethermal regime can be identified since the scattering of quasi-particles is equally fast than the formation of quasi-particles. In this regime, the kinetic theory can not be applied. The short distance regime $x<\\Lambda^{-1}$, for which Luttinger theory is invalid, occupies a negligibly small short time regime.}\n  \\label{fig:QuenchDiag}\n\\end{figure}\n\n\n\n\n\n\n\n\nKinetic equations have been successfully applied to Luttinger liquids with a cosine potential, resulting from particle backscattering in Refs.~\\cite{mitrarosch,Tavora13}. For Luttinger Liquids with cubic interactions a kinetic equation approach has been derived in Ref.~\\cite{buchholdmethod}. The latter makes use of non-perturbative Dyson-Schwinger equations in order to solve the time-evolution of the phonon distribution function in the presence of the RG-irrelevant but resonant interactions. This kinetic equation approach is particularly well suited for Luttinger models close to the ground state, i.e. with a small number of phononic excitations but can also be applied to excited states, as long as the Luttinger criterion is satisfied locally, i.e. as long as the phonon density $n_q<\\Lambda\/|q|$ for all momenta $|q|$. Based on Ref.~\\cite{buchholdmethod}, we can give explicit criteria for the validity of this approach for the fermionic dynamics after we have introduced the quench scenario.\n\nThis paper is organized as follows. We introduce the studied model system, the interacting Luttinger Liquid, in Sec.~\\ref{sec:interactingluttingerliquid}. The main results are summarized in Sec.~\\ref{sec:summaryofresults}. The derivation of the kinetic equations, which is used to solve the complex quantum many-body problem is presented in Sec.~\\ref{sec:kinetic_equation}. It is analyzed and numerically  solved in Sec.~\\ref{sec:thermalization_dynamics}, where we also give the derivation of the main results.\n\n\n\n\n\\section{Interacting Luttinger Liquid}\n\\label{sec:interactingluttingerliquid}\nThe simplest form of an interacting Luttinger Liquid emerges as the effective long-wavelength description of spinless interacting fermions with quadratic (i.e. dispersive) corrections to a perfectly linear dispersion around the Fermi energy \\cite{haldane81,haldane81b,rozhkov05,Proto14a}. Although the fermionic band curvature is irrelevant in the sense of the renormalization group (RG)~\\cite{rozhkov05} and does therefore not modify the static infrared behavior of the fermions, it is visible in dynamic observables, such as the fermionic spectral function or the dynamical structure factor \\cite{imambekov08,zwerger06,affleck06,pustilnik06,Heyl2010,imambekov12}. In this work we will show that in a non-equilibrium situation, the quasi-particle scattering induced by the band curvature leads to a dynamical redistribution of energy and allows the system to relax towards a thermal state. Thus, the system becomes generic. This kind of relaxation is absent for non-dispersive fermions, since the corresponding model, the linear Luttinger model, is integrable. The fermionic band curvature breaks the integrability of the linear model and therefore, even though RG irrelevant, is the leading order term that drives the system away from a prethermal, i.e. GGE-type, dynamical fixed point and towards a thermal one.\n\nThe Luttinger liquid in its fermionic representation is described in terms of left and right moving spinless fermions (labeled with $\\eta=\\pm$), created and annihilated by operators $\\creo{\\psi}{\\eta,x}, \\anno{\\psi}{\\eta,x}$. The Hamiltonian is\n\\begin{equation}H=-\\sum_{\\eta}\\int_x \\creo{\\psi}{\\eta,x}\\left(i\\eta v\\sub{F}\\partial_x+\\frac{1}{2m}\\partial^2_x\\right)\\anno{\\psi}{\\eta,x}+\\frac{1}{2}\\int_{x,x'}g(x-x')\\rho_x\\rho_{x'},\\label{Eq1}\n\\end{equation}\nwith the combined density $\\rho_x=\\rho_{+,x}+\\rho_{-,x}=\\creo{\\psi}{+,x}\\anno{\\psi}{+,x}+\\creo{\\psi}{-,x}\\anno{\\psi}{-,x}$. The interaction, characterized by $g(x-x')$, is supposed to be short ranged in space (decaying faster than algebraic) but has a short distance cutoff of the order of the Luttinger cutoff $\\Lambda^{-1}$. In the long-wavelength limit, particles with a wave-length larger than the effective range of the potential only experience a contact potential, $g(q)=g_0$, where $g_0$ is the interaction strength at zero momentum. In order to regularize the interaction in the ultraviolet (UV) regime, which is required to obtain a non-diverging, quench induced interaction energy, it is cut off at the UV scale $\\Lambda$, i.e. $g(q)=g_0\\theta(\\Lambda-|q|)$. \n\nThe bosonized version of the Hamiltonian in the absence of band curvature describes the well-known Luttinger model\n\\begin{equation}\\label{Eq2}\nH\\sub{LL}=\\int_x uK\\left(\\partial_x\\theta_x\\right)^2+\\frac{u}{K}\\left(\\partial_x\\phi_x\\right)^2\n\\end{equation}\nwith sound velocity $u=\\frac{v\\sub{F}}{K}$ and Luttinger parameter $K=\\left(1+\\frac{g_0}{\\pi v\\sub{F}}\\right)^{-\\frac{1}{2}}$. In addition, due to the fermionic band curvature, a cubic nonlinearity occurs \\cite{affleck06,zwerger06,aristov,imambekov08,lamacraft15}\n\\eq{Eq3}{\nH\\sub{NL}=\\frac{1}{m}\\int_x \\left(\\partial_x\\theta_x\\right)^2\\partial_x\\phi_x,}\nsuch that the complete bosonized Hamiltonian is $H=H\\sub{LL}+H\\sub{NL}$. Due to the linear dispersion of the Luttinger quasi-particles, $H\\sub{NL}$ describes scattering processes on a highly degenerate bosonic manifold, i.e. is governed by a large set of energy conserving scattering processes. This leads to diverging perturbative corrections at any order of perturbation theory. \nThe bosonized fermionic interaction is quadratic in the Luttinger fields, while the band curvature transforms into a cubic nonlinearity $\\propto\\frac{1}{m}$. \n\nIn the following, we will consider a nonequilibrium scenario in terms of an interaction quench. Initially, the system is supposed to be prepared in the ground state of the integrable Luttinger liquid theory at an interaction potential $g^i(x)$.\nDue to the interaction quench, the interaction potential is suddenly switched at time $t=0$ from an initial to a final value\n\\eq{Eq4}{\ng(x)=\\left\\{\\begin{array}{c} g^i(x)\\mbox{ for } t<0\\\\ g^f(x)\\mbox{ for } t>0 \\end{array}\\right.\n}\nand both the quadratic Hamiltonian as well as the non-linearity are modified by this interaction change.\nThe eigenbasis of $H\\sub{LL}$, which is expressed in terms of the physically more transparent phononic creation and annihilation operators $\\creo{a}{q}, \\anno{a}{q}$ according to the canonical Bogoliubov transformation \n\\eq{Bog1}{\n\\theta_x &=& \\theta_0 +\\frac{i}{2}\\int_q\\left(\\frac{2\\pi}{|q|K}\\right)^{1\/2} e^{-iqx-\\frac{|q|}{\\Lambda}}\\left(\\creo{a}{q}-\\anno{a}{-q}\\right),\\\\\n\\phi_x&=&\\phi_0-\\frac{i}{2}\\int_{q}\\left(\\frac{2\\pi K}{|q|}\\right)^{1\/2}\\mbox{sgn}(q)\\ e^{-iqx-\\frac{|q|}{\\Lambda}}\\left(\\creo{a}{q}+\\anno{a}{-q}\\right)\\label{Bog2},} is therefore obviously transformed by the quench. This transformation depends on the interaction via the Luttinger parameter $K$. \n\nThe state of the system before the quench corresponds in general no longer to an equilibrium state after the quench, and the system will consequently undergo a nontrivial time evolution according to the new Hamiltonian. The occupations of bosonic modes after the quench can be computed via the above Bogoliubov transformation. Before the quench, the interacting system is in equilibrium at zero temperature, such that $G^K_{q,t=0}=\\langle \\{\\anno{a}{q},\\creo{a}{q}\\}\\rangle=1$ in the prequench basis. This yields the postquench occupations\n\\begin{eqnarray}\nn_{t=0,q}&=&\\langle \\creo{a}{q}\\anno{a}{q}\\rangle_{t=0}=\\frac{1}{2}\\left[\\frac{\\lambda^2+1}{\\lambda}n_{i,q}+\\frac{\\left(\\lambda-1\\right)^2}{2\\lambda}\\right],\\nonumber\\\\\nm_{t=0,q}&=&\\langle\\creo{a}{q}\\creo{a}{-q}\\rangle_{t=0}=\\frac{1-\\lambda^2}{4\\lambda}\\left(2n_{i,q}+1\\right), \\mbox{ with } \\lambda=\\frac{K\\sub{f}}{K\\sub{i}}.\\ \\ \\ \\ \\ \\ \\ \\ \\ \\label{Occ}\n\\end{eqnarray}\nHere, $n_{i,q}$ is the initial occupation of the bosonic modes and $\\lambda=\\frac{K\\sub{f}}{K\\sub{i}}$ the ratio between the final $K\\sub{f}=\\sqrt{1+\\frac{g\\sub{f}}{\\pi v\\sub{F}}}$ and the initial $K\\sub{i}=\\sqrt{1+\\frac{g\\sub{i}}{\\pi v\\sub{F}}}$ Luttinger parameter. \nIn this work, we focus on a zero temperature initial state, $n_{i,q}=0$ for all $q$. The phonon density after the quench $n_{t,q}> 0$ is always larger than the density before the quench, resulting in a nonzero excitation energy $\\Delta E=\\langle H\\sub{f}\\rangle-\\langle H\\sub{i}\\rangle>0$ generated by the quench. Non-zero off-diagonal occupations $m_{t,q}\\neq0$ indicate that the correlations are not diagonal in the post-quench quasi-particle basis and in order to relax to an equilibrium state, $m_{t,q}$ must decay to zero. In the present setting, we choose $m_{t,q}=e^{-2iu|q|t}\\langle\\creo{a}{q}\\creo{a}{-q}\\rangle_{t}$, such that the off-diagonal occupations remain always real, being either positive or negative, depending on the quench.\n\nIn the phonon basis, \n\\eq{Eq5a}{\nH\\hspace{-1mm}=\\hspace{-1.2mm}\\int_q \\hspace{-1.2mm} u|q|\\creo{a}{q}\\anno{a}{q}\\hspace{-0.5mm}+\\hspace{-1mm}\\int_{q,k}\\hspace{-3mm}\\sqrt{|qk(k+q)|}\\ v(k,q) \\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}+\\mbox{h.c.}\\right),}\nwith the vertex function $v(k,q)=v\\left(\\frac{q}{|q|},\\frac{k}{|k|},\\frac{k+q}{|k+q|}\\right)$, which depends on the signs of the in- and outgoing momenta. In the interaction representation the phonon scattering Hamiltonian is\n\\eq{Eq5b}{\nH\\hspace{-1mm}_I(t)=\\hspace{-1mm}\\int_{q,k}\\hspace{-3mm}\\sqrt{|qk(k+q)|}\\ v(k,q)\\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}e^{iut(|q+k|-|q|-|k|)}+\\mbox{h.c.}\\right).\n}\nInstead of solving the full problem, we aim at extracting the dominant contributions of the nonlinearity which are relevant for intermediate and large times and which drive the crossover towards thermalization.\nIn view of Eq.~\\eqref{Eq5b}, off-resonant processes, for which $|q|+|k| \\not= |k+q|$, will dephase and as a consequence become negligible for the intermediate and long-time evolution of the system~\\cite{zwerger06}. Resonant processes on the other hand, here set by $|q|+|k| = |k+q|$, will at intermediate and long times become relevant in the renormalization group sense, as discussed in Ref.~\\cite{Heyl2015nd}. \n\\changed{The off-resonant processes can be eliminated perturbatively \\cite{Heyl2015nd}, yielding subleading corrections for intermediate and large times, which we will neglect in the following. For the asymptotic thermalization process, these subleading corrections will yield non-universal corrections (i.e. observable in microscopic constants and prefactors). For instance, the presence of off-resonant scattering events will eventually lower the asymptotic temperature compared to a system with purely resonant scattering events.  The influence of off-resonant interactions on the decay rate of the bosonic and fermionic quasi-particles  has been investigated in Ref.~\\cite{Proto14a}. The decay rate extracted from this computation is orders of magnitude lower than the rate due to purely resonant scattering processes.  Furthermore, it has a subleading scaling behavior $\\sim Tq^4$ compared to  $\\sim \\sqrt{q^3T}$ for resonant scattering processes at small momenta $q$ \\cite{zwerger06, andreev80}. Consequently, it is thus no influence on the leading order long time behavior. This allows us for the present purpose to restrict the phonon scattering to the resonant processes alone:}\n\\eq{Eq5}{\nH\\hspace{-1mm}=\\hspace{-1.2mm}\\int_q \\hspace{-1.2mm} u|q|\\creo{a}{q}\\anno{a}{q}\\hspace{-0.5mm}+\\hspace{-1mm}v_0\\int_{q,k}'\\hspace{-3mm}\\sqrt{|qk(k+q)|} \\left(\\creo{a}{q+k}\\anno{a}{q}\\anno{a}{k}+\\mbox{h.c.}\\right),}\nwhere the integral $\\int_{q,k}'$ is performed for momenta $|q+k|=|q|+|k|$ and $v_0=v(1,1)=\\frac{3}{m}\\sqrt{\\frac{\\pi}{K}}$ is the strength of the nonlinearity at resonance \\cite{buchholdmethod,zwerger06}.\n\nAs we are interested in fermionic correlation functions, we switch from an operator based formalism to a field theoretical formulation on the Keldysh contour, which is explained in the appendix \\ref{appendix2}, see also Ref.~\\cite{buchholdmethod}. This allows us to treat both spatial and temporal forward time correlations on an equal footing. We will focus our analysis on the so-called fermionic lesser Green's function \n\\eq{Green}{G^{<}_{t,x}=-i\\langle\\cre{\\psi}{t,x}\\anno{\\psi}{t,0}\\rangle} at equal forward times $t$ from which all fermionic equal time correlations can be deduced. Especially, in terms of a physical interpretation it is the Fourier transform of the fermionic momentum distribution \n\\eq{Mom}{\nn^{\\mbox{\\tiny F}}_{t,q}=i\\int_x e^{iqx}G^<_{t,x}.\n}\nIn the field theory representation, the bosonized fermionic lesser Green's function at equal times is\n\\eq{GF1}{\nG^{<}_{\\eta,t,x}=-i\\langle\\cre{\\psi}{\\eta,-,t,x}\\anno{\\psi}{\\eta,+,t,0}\\rangle=-i\\Lambda\\frac{e^{-i\\eta k\\sub{F}x}}{2\\pi}e^{-\\frac{i}{2}\\mathcal{G}^<_{\\eta,t,x}}.\n}\nHere, $\\cre{\\psi}{\\nu},\\ann{\\psi}{\\nu}$ label Grassmann fields with the index $\\nu=(\\eta,\\gamma,t,x)$ representing right and left movers ($\\eta=\\pm$), the contour variables on the Keldysh plus and minus contour ($\\gamma=\\pm$), the forward time coordinate $t$ and the relative spatial distance $x$. The corresponding lesser  exponent $\\mathcal{G}^<$ is defined as \n\\eq{GF2}{\n\\mathcal{G}^<_{\\eta, t,x}=2i\\log\\left\\langle e^{i\\left(\\eta\\phi_{+,t,0}-\\theta_{+,t,0}-\\eta\\phi_{-,t,x}+\\theta_{-,t,x}\\right)}  \\right\\rangle.\n}\nThe extra index $(\\pm)$ of the Luttinger fields labels position on the plus-minus contour, see appendix. Combining Eq.~\\eqref{GF2} and the Bogoliubov transformation above, one finds that $\\mathcal{G}^{<}_{-\\eta,t,x}=\\mathcal{G}^{<}_{\\eta,t,-x}$. The Green's function of the left movers is the spatially mirrored Green's function of the right movers, and it is sufficient to consider only the Green's function of the right movers \n\\eq{GF3}{\nG^{<}_{t,\\eta x}\\equiv G^{<}_{+,t,\\eta x}=G^{<}_{\\eta,t,x}\n}\nand equivalently for the exponent $\\mathcal{G}^<$. According to the linked cluster theorem, the logarithm in Eq.~\\eqref{GF2} is defined as the sum of all connected diagrams in an expansion of the exponent. As a consequence, it can be expressed to leading order in terms of the full Green's functions, with the next non-vanishing  correction being proportional to the equal-time one-particle irreducible four-point vertex, which is zero in the microscopic theory. Its effective correction remains negligibly small. In particular, the four-point vertex will only contribute to $\\mathcal{O}[(um)^{-4}]$ which is two orders of magnitude smaller than the desired accuracy and its contribution can be safely neglected. The static one-particle irreducible four-point vertex represents a negligible correction for any equilibrium problem since it can only be generated via multiple concatenation of subleading three-point vertices. Especially it is not responsible for the modifications of the dynamic structure factor reported in Refs.~\\cite{pustilnik07,imambekov09,lamacraft15}, since at zero temperature vertex corrections vanish exactly due to causality \\cite{buchholdmethod,forsternelson76}. Consequently, the modifications of the dynamic structure factor happen entirely on the basis of the irreducible two-point vertex, i.e. the phonon self-energy. In the present case, the four-point vertex is exactly zero before the quench since this state corresponds to a zero temperature state as well as immediately after the quench, since a flat quasi-particle distribution in Eq.~\\eqref{Occ} leads to a vanishing vertex correction. \nIn terms of the Luttinger fields and apart from four-point vertex corrections, the exponent for the fermionic Green's function is\n\\eq{GF4}{\n\\mathcal{G}^<_{t,x}=\\sum_{\\alpha,\\beta=\\theta,\\phi}\\left(2\\delta_{\\alpha\\beta}\\hspace{-0.1cm}-\\hspace{-0.1cm}1\\right)\\left[G^K_{\\alpha\\beta,t,0}\\hspace{-0.1cm}\n-\\hspace{-0.1cm}G^K_{\\alpha\\beta,t,x}\\hspace{-0.1cm}+\\hspace{-0.1cm}G^A_{\\alpha\\beta,t,x}\\hspace{-0.1cm}-\\hspace{-0.1cm}\nG^R_{\\alpha\\beta,t,x}\\right]\n,}\nwhere $G^{R\/A}_{\\alpha\\beta}$ is the retarded, advanced Green's function for $\\alpha,\\beta=\\theta,\\phi$ and $G^K_{\\alpha\\beta}$ is the corresponding Keldysh Green's function, i.e. $G^R_{\\alpha\\beta,t,x}=-i\\langle \\alpha_{q,x,t}\\beta_{c,0,t}\\rangle$.\nApplying the Bogoliubov transformation to the phonon basis, the equal time exponent becomes\n\\begin{widetext}\n\\eq{GF5}{\n\\mathcal{G}^<_{t,x}=i\\int_q\\left[\\mbox{$\\frac{\\pi e^{-\\frac{|q|}{\\Lambda}}}{|q|}$}(\\cos(qx)-1)\\left[\\mbox{$\\frac{K^2+1}{K}$}(2n_{t,q}+1)+2\\mbox{$\\frac{K^2-1}{K}$}\\cos(2u|q|t)m_{t,q}\\right]\\right]+2\\arctan(\\Lambda x)+4i\\int_q\\left[\\frac{\\pi e^{-\\frac{|q|}{\\Lambda}}}{|q|}\\sin(|q|x)\\sin(2u|q|t)m_{t,q}\\right].\n}\\end{widetext}\n Here, $n_{t,q}=\\langle \\creo{a}{t,q}\\ann{a}{t,q}\\rangle$ and $m_{t,q}=|\\langle\\ann{a}{t,-q}\\ann{a}{t,q}\\rangle|$ are the equal time normal and anomalous phonon densities, which evolve in time due to phonon scattering. The absence of the quasi-particle self-energy in this expression is caused by the equal time properties of the Green's function and underlines the fact that time-local, i.e. static, observables, even if explicitly forward-time dependent, are not modified by the frequency resolved fine structure of the self-energies once the time dependent distribution $n_{t,q}$ is known. In the remainder of this paper, we will analyze the time evolution of the exponent \\eqref{GF5} after the interaction quench and its implications for the fermionic Green's function \\eqref{GF1}.\n\nConcerning the relevance of the interacting Luttinger model, before closing the section, we would like to mention that only recently pioneering experiments in ultra-cold gases both in and out of equilibrium explored the transient and prethermalization dynamics of systems~\\cite{Hofferberth06,trupke13,schmiedmayer12,Langen14,nagerl13,meinert14,preiss15,hild14,cheneau12,agarwal14,schmiedmayernphys12,schmiedmayernjp13,bloch13,Guan2013} effectively described by a quadratic Luttinger model, the bosonic theory of the Hamiltonian in Eq.~\\eqref{Eq2}. In particular, in Refs.~\\cite{Hofferberth06,trupke13,schmiedmayer12,Langen14} prethermal states in the relative phase of a suddenly split condensate have been identified that have been stable on the experimentally accessible time scales.  For the latter experiments, the cubic nonlinearity studied in the present work constitutes the leading order correction to the quadratic theory in a gradient expansion. Therefore, the framework developed in the subsequent sections to describe the relaxation dynamics in the system, is of direct experimental relevance once the time scales are experimentally accessible to study the escape out of the prethermalization plateau. It is, however, important to note that the concrete experimental setup of the suddenly split condensate requires a further but straightforward extension of the considered model system to include two species of coupled bosonic fields. Moreover, let us emphasize that these experimental systems do not simulate the Luttinger liquid of interacting fermions -- our initial motivation -- but directly the effective bosonic low-energy theory. In this way, it might be possible to obtain experimental access to the dynamics of the bosonic occupation distributions, governed by the kinetic theory formulated below, via time-of-flight imaging.\n\n\\section{Summary of main results}\n\\label{sec:summaryofresults}\n\nBefore formulating and solving the kinetic theory for the interacting Luttinger liquid in detail, we briefly summarize the main results reported in this work. In the subsequent sections, we will then present the detailed calculations. Specifically, the known results on the purely integrable system are reformulated within the present framework in Sec.~\\ref{sec:PT}. The kinetic equation, used to address the presence of the nonlinear phonon scattering, is derived in Sec.~\\ref{sec:kinetic_equation}. This kinetic equation is then solved in Sec.~\\ref{sec:thermalization_dynamics}.\n\nIt is the aim of this work to study the thermalization dynamics of the fermionic equal time Green's function \\eqref{Green}, which is the Fourier transform of the fermionic momentum distribution \\eqref{Mom} and contains the information on quadratic equal time fermion observables.\nWithout loss of generality, we focus on the distribution of the right-movers, i.e., $\\eta=+$.  In the presence of phonon scattering, we determine the time-evolution of $G^<_{t,x}$ via a set of kinetic equations derived later in Sec.~\\ref{sec:kinetic_equation}.\n\nWe find that $G^<_{t,x}$ features two distinct spatio-temporal crossover scales $x\\sub{th}(t)$ and $x\\sub{pt}(t)$, separating three regimes with distinct scaling behavior:\n\\renewcommand{\\arraystretch}{1.5}\n\\begin{center}\n\\begin{tabular}{l l c }\n\t1. & prequench: $\\quad$\t&\t$x\\sub{pt}(t) \\ll |x|$, \\\\\n\t2. & prethermal:\t&\t$x\\sub{th}(t) \\ll |x| \\ll x\\sub{pt}(t)$, \\\\\n\t3. & thermal:\t\t&\t$|x| \\ll x\\sub{th}(t)$. \\\\\n\\end{tabular}\n\\end{center}\n\\renewcommand{\\arraystretch}{1}\nWe find for the associated crossover scales $x_\\mathrm{pt}(t)$ and $x_\\mathrm{th}(t)$:\n\\eq{eq:crossoverScales}{\nx_\\mathrm{pt}(t) = 2ut, \\qquad \\qquad x_\\mathrm{th}(t) =\\frac{x_{\\lambda}}{\\Lambda} \\left( v_0 \\Lambda^2 t \\right)^{\\alpha_\\lambda}.\n}\nThe first crossover at $x\\sub{pt}(t)$ determines the light cone~\\cite{cardycalabrese06} set by the sound velocity $u$ of the phononic elementary excitations and is known from the non-interacting Luttinger model. Two space points a distance $x \\gg x\\sub{pt} (t)$ apart from each other have not been able to exchange information after the quench due to causality. Therefore, the properties at such distances are solely given by the initial condition before the quench such that we term this regime ``prequench''. \\changed{For distances $x<x_{\\text{pt}}$ quasi-particles are starting to form, marking the onset of prethermalization.} \nThe second crossover takes place at $x=x\\sub{th}(t)$ setting the scale for the onset of thermalization due to quasi-particle scattering. The exponent $\\alpha_{\\lambda}$ with $0<\\alpha_{\\lambda}<1$, as well as the dimensionless length $x_{\\lambda}$, depends on the quench parameter $\\lambda$ only and can be determined numerically. The upper bound of $\\alpha_{\\lambda}$ is guaranteed by the subleading nature of the vertex, which forbids ballistic spreading in the thermal region.\n\\changed{The treatment of quasi-particle scattering in terms of a kinetic equation approach is only valid on distances, for which a well-defined prethermal plateau has been established. Given this, we estimate the kinetic equation approach to be valid on distances\n\\eq{dist}{\nx<x_c(t)=x_{\\text{th}}(t)\\exp\\left(-\\tfrac{K^2+1}{|K^2-1|}\\sqrt{\\tfrac{3n_{\\lambda}}{|m_{\\lambda}|}}\\right)\n}\nand in the scattering-less region $x>x_{\\text{th}}$. In the intermediate regime $x_c(t)<x<x_{\\text{th}}$, quasi-particle scattering is as fast as the formation of quasi-particles, such that both effects have no distinguishable time scale. While the results obtained from our approach might not be reliable in this region, $x_{\\text{th}}(t)$ remains the crossover scale below which the non-linearity becomes non-negligible.}\nThe fact that $x\\sub{th}(t)$ has an explicit dependence on the Luttinger cutoff $\\Lambda$ ($\\alpha_{\\lambda}>1\/2$ generally) is not surprising. The non-linearity in the Luttinger model introduces a microscopic energy scale $v_0\\Lambda^2$ which represents the characteristic time scale of the dynamics induced by the non-linearity, i.e. in the present case the thermalization dynamics beyond the quadratic theory. Additionally, the non-linearity breaks the scale invariance of the quadratic model, which is responsible for the fact that all microscopic scales can be eliminated from macroscopic observables in that case. In the absence of scale invariance, however, the microscopic length scale $\\Lambda$ will appear in certain observables, expressing that their explicit value depends on model specific details.\n\nAs we show in our detailed analysis below, we find that this separation into three spatio-temporal regimes -- prequench, prethermal, and thermal -- reflects itself in a remarkable factorization property of the Green's function\n\\eq{eq:factorization2}{\nG^<_{t,x}=G^<_{0,x}Z\\sub{pt}(s\\sub{pt})Z\\sub{th}(s\\sub{th}),}\nwhich holds everywhere except in the vicinity of the crossover scales $x_\\mathrm{th}(t)$ and $x_\\mathrm{pt}(t)$.  Here, we have introduced the following short-hand notations:\n\\begin{equation}\ns\\sub{pt}=\\left\\{\\begin{array}{cl} x &\\mbox{ for } x<x_\\mathrm{pt}(t)\\\\ 2ut& \\mbox{ for } x>x_\\mathrm{pt}(t)\\end{array}\\right. , \\quad s\\sub{th}=\\left\\{\\begin{array}{cc}x&\\mbox{ for } x<x\\sub{th}(t)\\\\\nx\\sub{th}(t)&\\mbox{ for } x>x\\sub{th}(t)\\end{array}\\right. .\n\\end{equation}\nWhile the factorization into $G^<_{0,x}$ and $Z\\sub{pt}$ has been already known for the exact solution of the integrable model~\\cite{cazalilla06}, here, we show that the influence of the nonlinearity can be captured by a further factor in terms of $Z\\sub{th}$. The thermal contribution $Z\\sub{th}(s\\sub{th})$ exhibits interesting spatio-temporal dynamics in particular in the long-time regime $ut \\gg x\\sub{th}(t)$. It is defined as\n\\begin{equation}\n  Z_\\mathrm{th} (s_\\mathrm{th}) = \\exp\\left(-\\frac{K^2+1}{K} \\frac{\\pi \\tilde T_t|s_\\mathrm{th}|}{u}\\right)\n\\end{equation}\nand features two different spatio-temporal regimes.\n\n\n\\emph{(i) thermalized regime:}\nDeep in the thermalized region $|x|\\ll x\\sub{th}(t)$ where $s\\sub{th}=x$, $Z\\sub{th}=\\exp(-|x|\/\\xi_{\\tilde{T}_t})$ exhibits the conventional exponential decay with distance that the system experiences in thermal states with an associated thermal length\n\\begin{equation}\\label{tlength}\n\t\\xi_{\\tilde T_t} = \\frac{K}{1+K^2} \\frac{u}{\\pi \\tilde T_t}.\n\\end{equation}\nThe effective temperature $\\tilde T_t$, however, entering this equation remains a dynamical quantity with\n\\begin{equation}\n  \\tilde T_t=T+u \\Lambda \\Delta_\\lambda(v_0 \\Lambda^2 t)^{-\\mu},\n\\end{equation}\napproaching the temperature $T$ of the final thermal ensemble algebraically slowly. We find that the numerical simulations of the kinetic equation are consistent with an analytical estimate for the exponent $\\mu=2\/3$. Thus, the system in this spatial region appears to be hotter than in the final asymptotic thermal state. The associated excess energy stored at short distances has to be transported to larger distances which, however, is an algebraically slow process since this energy transport in the presence of detailed balance is carried out by dynamical slow modes, emerging as a consequence of exact conservation laws \\cite{Lux13}.\n\n\\emph{(ii) prethermal and prequench regime:} Within the prethermal and prequench region $x\\sub{th}(t) \\ll x$, the amplitude $Z\\sub{th}(s\\sub{th})=Z\\sub{th}(x\\sub{th}(t))$ approaches a space-independent but time-dependent constant quantifying the temporal decay of the prethermal correlations:\n\\begin{equation}\n  Z\\sub{th}(x\\sub{th}(t)) = \\exp[-x\\sub{th}(t)\/\\xi_{\\tilde T_t}].\n\\end{equation}\nBecause $x\\sub{th}(t) \\propto (v_0 \\Lambda^2 t)^{\\alpha_\\lambda}$, we have, remarkably, that this amplitude decays in stretched exponential form. This decay is sub-exponential and thus inherently nonperturbative in nature, highlighting the capabilities of our present approach.\n\n\\section{Dynamics in the absence of phonon scattering}\nIn order to systematically understand the effect of phonon scattering on the relaxation dynamics after the interaction quench, we first determine the dynamics of the exponent $\\mathcal{G}^<_{t,x}$ in the absence of scattering, i.e. for $\\frac{1}{m},v_0\\rightarrow 0$. This quench scenario has been extensively discussed in \\cite{Karrasch12,Kennes13,cazalilla06,Iucci09,Iucci10}, and we will only briefly list the known results in the present formalism in order to make contact to the relaxation dynamics in the presence of phonon scattering, which are discussed subsequently.\n\\subsection{Ground state properties}\nFor a system in the ground state, $n_{t,q}=m_{t,q}=0$ and the exponent evaluates to\n\\eq{GF6}{\n\\mathcal{G}^<_{t,x}=-i\\frac{K^2+1}{2K}\\log(1+\\Lambda^2x^2)+2\\arctan(\\Lambda x),}\nwhich leads to a time-independent fermionic Green's function\n\\eq{GF7}{\nG^<_{t,x}=-\\frac{i\\Lambda}{2\\pi}e^{-ik\\sub{F}x-i\\arctan(\\Lambda x)}\\sqrt{1+\\Lambda^2x^2}^{-\\frac{K^2+1}{2K}}\n,}\nwell known from the literature \\cite{haldane81,giamarchi04}. It features an algebraic decay in space $\\sim x^{-\\frac{K^2+1}{2K}}$ and a power law singularity of the fermionic momentum distribution close to the Fermi momentum $n^{\\mbox{\\tiny F}}_q\\sim|q-k\\sub{F}|^{-\\frac{(K-1)^2}{2K}}$ \\cite{giamarchi04}. \n\\subsection{Quench from the ground state}\\label{sec:PT}\nInitializing the fermions in the ground state and performing an interaction quench leads to constant non-zero phonon densities in the post-quench basis, according to Eq.~\\eqref{Occ}. In the absence of scattering, the phonon densities are constants of motion and remain time independent, $n_{t,q}=n_{0,0}\\equiv n$ and $m_{t,q}=m_{0,0}\\equiv m$. In this situation, only dephasing of the off-diagonal Green's functions induces relaxation and the exponent is\n\\begin{widetext}\\begin{eqnarray}\n\\label{GF8}\n\\mathcal{G}^<_{t,x}&=&2\\arctan(\\Lambda x)-i\\mbox{$\\frac{K^2+1}{2K}$}(2n+1)\\log(1+\\Lambda^2x^2)+im\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+\\Lambda^2(x+2ut)^2}$}\\right)-i\\mbox{$\\frac{K^2-1}{2K}$}m\\left[\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+4u^2t^2\\Lambda^2}$}\\right)+\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x+2ut)^2}{1+4u^2t^2\\Lambda^2}$}\\right)\\right]\\nonumber\\\\\n&=&\\mathcal{G}^<_{0,x}+im\\log\\left(\\mbox{$\\frac{1+\\Lambda^2(x-2ut)^2}{1+\\Lambda^2(x+2ut)^2}$}\\right)-i\\mbox{$\\frac{K^2-1}{2K}$}m\\log\\left[\\mbox{$\\frac{(1+\\Lambda^2(x+2ut)^2)(1+\\Lambda^2(x-2ut)^2)}{(1+4u^2t^2\\Lambda^2)^2(1+x^2\\Lambda^2)^2}$}\\right].\\end{eqnarray}\\end{widetext}\nHere, $\\mathcal{G}^<_{0,x}$ is the exponent corresponding to the prequench state, i.e. the ground state of interacting fermions with the prequench Luttinger parameter $K_i$. Consequently the fermion Green's function \\eqref{GF1} factorizes\n\\eq{GF9}{\nG^<_{t,x}=G^<_{0,x}\\tilde{Z}\\sub{pt}(x,t).\n}\nThe factor $\\tilde{Z}\\sub{pt}$ is defined by Eqs.~\\eqref{GF8} and \\eqref{GF1} and describes the time-dependent modification of the initial zero temperature Green's function due to the quench. In view of the following discussion it is useful to investigate this factor on distances away from the light cone $x=2ut$. For distances $|x|\\ll 2ut$, the temporal factors in Eq.~\\eqref{GF8} cancel each other and $\\tilde{Z}\\sub{pt}(t,x)\\overset{|x|\\ll2ut}{\\rightarrow} Z\\sub{pt}(x)$ looses its time dependence. On the other hand, for distances $|x|\\gg2ut$, the spatial dependence drops out and $\\tilde{Z}\\sub{pt}(t,x)\\overset{|x|\\gg2ut}{\\rightarrow} Z\\sub{pt}(2ut)$. This defines the prethermal amplitude\n\\eq{GF10}{\nZ\\sub{pt}(s)=\\left(\\sqrt{1+\\Lambda^2s^2}\\right)^{\\frac{K^2-1}{2K}\\frac{1-\\lambda^2}{4\\lambda}}.\n}\nThe process associated with the crossover of $Z\\sub{pt}(s)$ from a temporal to a spatial dependence as a function of time is the formation of quasi-particles corresponding to the post-quench Hamiltonian. This is the typical prethermalization scenario in the absence of quasi-particle scattering. For short times, the properties of the system are dominated by the initial state of the system, and the fermion Green's function is only modified by a global amplitude but has the same spatial scaling behavior as for the initial state. The effect of the quadratic Hamiltonian in the time evolution is the dephasing of all terms, which are not diagonal in the basis of the post-quench quasi-particles, leading to a diagonal ensemble in the quasi-particles with a non-equilibrium phonon density. This non-equilibrium distribution of phonons induces a scaling behavior of the fermion Green's function in real space, which is different from the zero and finite temperature cases.\n\nIn the absence of phonon scattering, the diagonal phonon densities $n_{t,q}$ are constants of motion and do not relax, the density matrix $\\rho$ therefore does not approach a Gibbs state but is rather described in the asymptotic limit $t\\rightarrow\\infty$ by a generalized Gibbs ensemble (GGE), which respects the constants of motion and maximizes the entropy. It is given by\n\\eq{GF11}{\n\\rho\\sub{GGE}=Z\\sub{GGE}^{-1}e^{-\\int_q \\nu_q \\hat{n}_q},}\nwhere the Lagrange parameters $\\nu_q=2\\log\\left(\\frac{\\lambda+1}{|\\lambda-1|}\\right)$ depend on the quench parameter and $Z\\sub{GGE}$ is the normalization factor.\n\nThe fermion Green's function for the two different regimes is then\n\\eq{GF12}{\nG^<_{t,x}=G^<_{0,x}\\times\\left\\{\\begin{array}{ll} Z\\sub{pt}(2ut) &\\mbox{ for } |x|\\gg 2ut\\\\\nZ\\sub{pt}(x)& \\mbox{ for } |x|\\ll 2ut\\end{array}\\right. ,\n}\nwith the non-equilibrium scaling behavior \n\\eq{GF13}{G^<_{t,x}\\overset{t\\rightarrow\\infty}{\\sim} |x|^{-\\gamma\\sub{Eq}\\gamma\\sub{GGE}},\n}\nwhere $\\gamma\\sub{Eq}=\\frac{K^2+1}{2K}$ is the equilibrium exponent and $\\gamma\\sub{GGE}=\\frac{\\lambda^2+1}{2\\lambda}=2n+1$ (see, Eq.~\\eqref{Occ}) is the non-equilibrium correction resulting from a non-thermal quasi-particle distribution.\n\\section{Phonon scattering and the kinetic equation}\n\\label{sec:kinetic_equation}\n\nIn the previous sections, we have demonstrated that the forward time evolution of the fermionic equal-time Green's function can be determined solely from the momentum dependent excitation distributions $n_{t,q}, m_{t,q}$. All quadratic, equal-time observables on the other hand can be computed from the fermionic equal-time Green's function via a unitary transformation, such that the knowledge of $n_{t,q}$ and $m_{t,q}$ gives access to the forward time evolution of all the frequency independent quadratic fermion observables. Therefore the time-evolution of this specific set of observables can be captured by the time evolution of the frequency independent and well-defined quantities $n_{t,q}, m_{t,q}$, which does not necessitate the frequency resolved fine structure in the fermionic spectrum. \nIn order to determine the time-evolution of the phonon densities, we derive kinetic equations for the excitation distribution function \\cite{kamenevbook} in the limit of well defined excitations, closely following the steps in Ref.~\\cite{buchholdmethod} and briefly discussing the approximations. \n\nBefore we start with the explicit derivation, we review very briefly the known results for nonlinear Luttinger liquids (c.f. \\cite{imambekov12}) and place the present approach into this context. At zero temperature and without band curvature, long wavelength physics of the interacting fermion model can be exactly mapped to the quadratic Luttinger model and therefore has well-defined, sharp phononic excitations, expressed by a spectral function of the phonons $\\mathcal{A}_{q,\\omega}=i(G^R_{q,\\omega}-G^A_{q,\\omega})=2\\pi \\delta(\\omega-u|q|)$. In the presence of band curvature, however, the phonons themselves interact via a resonant three-point scattering vertex, which leads to a broadening of the spectral function around the mass-shell $\\omega=u|q|$. This broadening can be described in terms of two excitations branches at frequencies $\\omega=\\epsilon^{\\pm}_q$, where $\\epsilon^-_q<u|q|$ labels a solitonic branch and $\\epsilon^+_q>u|q|$ labels a phononic branch (such that $|\\epsilon^{+}_q-\\epsilon^-_q|\/q\\rightarrow 0$ for $q\\rightarrow0$)\\cite{imambekov12,lamacraft15}. The spectral weight of the excitations in the nonlinear Luttinger liquid is distributed continuously between these two branches. Whereas the solitonic branch represents an exact boundary (i.e. no spectral weight is located at frequencies $\\omega<\\epsilon_q^-$), featuring a power law singularity for frequencies above $\\epsilon^-_q$, the phononic branch represents an algebraically sharp boundary (i.e. the spectral weight for frequencies $\\omega>\\epsilon_q^+$ is strongly algebraically suppressed), featuring a power law singularity from both sides \\cite{imambekov12}. While the power-law singularities at the edges of the spectral weight obviously cannot be explained by a frequency independent self-energy, the characteristic width of the spectral weight $\\delta\\omega_q=\\epsilon^+_q-\\epsilon^-_q=\\frac{q^2}{m^*}$ can be captured by an imaginary part of the on-shell value of the self-energy $\\Sigma^R_{q,\\omega=u|q|}$, which determines the renormalized mass $m^*$ \\cite{zwerger06,aristov,lamacraft15,imambekov12}. These results hold for the zero temperature limit of the problem. At finite temperature $T>0$, however, a self-consistent Born-approximation for the on-shell self-energy predicts a scaling of the spectral weight $\\delta\\omega_q\\sim \\sqrt{|q|^3T}$ \\cite{lamacraft13,gangardt13}, which has been also observed in numerical simulations of interacting one-dimensional bosons \\cite{lamacraft13}. For $\\delta\\omega_q\\ll u|q|$, i.e. the width of the spectral weight of the excitations being much smaller than the average excitation energy, the spectral weight is still sharply concentrated at the mass-shell and one can still think (physically) of well defined excitations although the fine structure of the spectral weight is very different from what one is used to for weakly interacting quasi-particles. As a consequence, it is possible to derive a kinetic equation for the excitation densities in this regime, applying the common quasi-particle and local time approximations, and we will implement this approach below. It neglects the specific structure of the spectral weight of nonlinear Luttinger liquids, which is valid for \"static\" variables in the quasi-particle limit $\\delta\\omega_q\\ll u|q|$.\nWe begin by introducing the interaction picture for the Heisenberg operators \n\\eq{Eq21}{\n\\cre{a}{t,q}\\rightarrow \\cre{a}{t,q}e^{-iu|q|t},}\nwhich leaves the Hamiltonian \\eqref{Eq5} unmodified but shifts the spectral weight of diagonal modes to zero frequency and eliminates the phase $\\sim e^{i2u|q|t}$ of off-diagonal correlation functions \\cite{buchholdmethod}. \n The Green's functions in the interaction representation are labeled with a tilde. The Keldysh Green's function in Nambu space is\n\\eq{Eq22}{\ni\\tilde{G}^K_{t,q,\\delta}=\\left(\\hspace{-0.15cm}\\begin{array}{ll}\\langle\\{\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}\\}\\rangle & \\langle\\{\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}\\}\\rangle\\\\\n\\langle\\{\\cre{a}{t+\\frac{\\delta}{2},-q},\\cre{a}{t-\\frac{\\delta}{2},q}\\}\\rangle&\\langle\\{\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}\\}\\rangle\n\\end{array}\\hspace{-0.15cm}\\right),\\ \\ \\ \n}\nwhere  $\\{\\cdot,\\cdot\\}$ is the anti-commutator and we introduced an additional relative time shift $\\delta$ associated with spectral properties of the system. The retarded Green's function is\n\\eq{Eq23}{\ni\\tilde{G}^R_{t,q,\\delta}&=&\\theta(\\delta)\\left(\\hspace{-0.15cm}\\begin{array}{ll}\\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle & \\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}]\\rangle\\\\\n\\langle[\\cre{a}{t+\\frac{\\delta}{2},-q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle&\\langle[\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}]\\rangle\n\\end{array}\\hspace{-0.15cm}\\right)\\nonumber\\\\\n&=&\\theta(\\delta)\\left(\\hspace{-0.15cm}\\begin{array}{cc}\\langle[\\ann{a}{t+\\frac{\\delta}{2},q},\\cre{a}{t-\\frac{\\delta}{2},q}]\\rangle & 0\\\\\n0&\\langle[\\cre{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},q}]\\rangle\n\\end{array}\\hspace{-0.15cm}\\right).}\nThe off-diagonal retarded and advanced Green's functions are exactly zero. This is a consequence of the Hamiltonian, which does not introduce a coupling between the modes $q$ and $-q$, such that the commutator $[\\ann{a}{t+\\frac{\\delta}{2},q},\\ann{a}{t-\\frac{\\delta}{2},-q}]=0$ for all times $t,\\delta$. The anti-hermitian Keldysh Green's function is parametrized according to \\cite{kamenevbook,buchholdmethod}\n\\eq{Eq24}{\n\\tilde{G}^K_{t,q,\\delta}=\\left(\\tilde{G}^R\\circ\\sigma_z\\circ F-F\\circ\\sigma_z\\circ\\tilde{G}^A\\right)_{t,q,\\delta}}\nin terms of the time-dependent, hermitian quasi-particle distribution function $F$ and the Pauli matrix $\\sigma_z$, the latter preserving the symplectic structure of bosonic Nambu space. The $\\circ$ represents matrix multiplication with respect to momentum space and convolution with respect to time. Switching to Wigner coordinates by Fourier transforming the Keldysh Green's function with respect to relative time\n\\eq{Eq25}{\n\\tilde{G}^K_{t,q,\\omega}=\\int_{\\delta}\\tilde{G}^K_{t,q,\\delta}\\ \\ e^{i\\omega\\delta}\n}\nand applying the Wigner approximation, which, due to the RG-irrelevant interactions, is justified in the same regime for which the Luttinger description is applicable \\cite{buchholdmethod,Heating}, we find\n\\eq{Eq26}{\n\\tilde{G}^K_{t,q,\\omega}=\\tilde{G}^R_{t,q,\\omega}\\sigma_zF_{t,q,\\omega}-F_{t,q,\\omega}\\sigma_z\\tilde{G}^A_{t,q,\\omega},} \nwhich is diagonal in momentum and frequency space. Inverting Eq.~\\eqref{Eq26} by multiplying it with $\\left(\\tilde{G}^R\\right)^{-1}$ from the left and $\\left(\\tilde{G}^A\\right)^{-1}$ from the right, yields the kinetic equation for the distribution function\n\\eq{Eq27}{\ni\\partial_tF_{t,q,\\omega}\\hspace{-0.1cm}=\\hspace{-0.1cm}\\sigma_z\\Sigma^R_{t,q,\\omega}F_{t,q,\\omega}\\hspace{-0.1cm}-\\hspace{-0.1cm}F_{t,q,\\omega}\\Sigma^A_{t,q,\\omega}\\sigma_z\\hspace{-0.1cm}-\\hspace{-0.1cm}\\sigma_z\\Sigma^K_{t,q,\\omega}\\sigma_z.\\ \\ \\ \\ \\ \\ \n}\nThe retarded, advanced self-energies $\\Sigma^{R\/A}_{t,q,\\omega}$ are diagonal in Nambu space, while the Keldysh self-energy $\\Sigma^K_{t,q,\\omega}$ consists of non-vanishing diagonal and off-diagonal entries due to the initial off-diagonal occupations $m_{0,q}\\neq0$.\n\nThe kinetic equation for the phonon occupations is obtained by multiplying Eq.~\\eqref{Eq27} on both sides with the spectral function $\\tilde{\\mathcal{A}}_{t,q,\\omega}=i\\left(\\tilde{G}^R_{t,q,\\omega}-\\tilde{G}^A_{t,q,\\omega}\\right)$ and integrating over frequency space. For interacting Luttinger Liquids, the spectral function $\\tilde{\\mathcal{A}}_{t,q,\\omega}$ is very narrowly peaked at the mass shell and the kinetic equation is essentially locked onto $\\omega=0$ in this way (in the interaction picture, the mass shell is at $\\omega=0$). As a consequence, one finds kinetic equations for the diagonal densities\n\\eq{Eq28}{\n\\partial_tn_{t,q}=-\\sigma^R_{t,q}(2n_{t,q}+1)+\\sigma^K_{t,q}\n}\nand the off-diagonal densities\n\\eq{Eq29}{\n\\partial_tm_{t,q}=-2\\sigma^R_{t,q}m_{t,q}-\\Gamma^K_{t,q}.\n}\nThey can be expressed in terms of the imaginary part of the retarded on-shell self-energy \\eq{Nun1}{\\sigma^R_{t,q}=\\frac{1}{2}\\int_{\\omega}\\tilde{\\mathcal{A}}_{t,q,\\omega}\\left(\\Sigma^R_{t,q,\\omega}-\\Sigma^A_{t,q,\\omega}\\right)\\approx\\frac{1}{2}\\left(\\Sigma^R_{t,q,\\omega=0}-\\Sigma^A_{t,q,\\omega=0}\\right)} and the Keldysh on-shell self-energies \\eq{Nun2}{\\sigma^K_{t,q}=\\frac{i}{2}\\int_{\\omega}\\tilde{\\mathcal{A}}_{t,q,\\omega}\\left(\\Sigma^K_{t,q,\\omega}\\right)_{11}\\approx\\frac{i}{2}\\left(\\Sigma^K_{t,q,\\omega=0}\\right)_{11}} and \\eq{Nun3}{\\Gamma^K_{t,q}=\\frac{i}{2}\\int_\\omega\\mathcal{A}_{t,q,\\omega}\\left(\\Sigma^K_{t,q,\\omega}\\right)_{12}\\approx\\frac{i}{2}\\left(\\Sigma^K_{t,q,\\omega=0}\\right)_{12}.} The Keldysh self-energy is always anti-hermitian and therefore purely imaginary in frequency and momentum space, such that Eqs.~\\eqref{Eq28}, \\eqref{Eq29} are real. Since the criterion $|\\epsilon^+_q-\\epsilon^-_q|\\ll u|q|$ is equivalent to $\\sigma^R_{t,q}\\ll u|q|$ at zero and finite temperature equilibrium, we also apply the latter criterion for the present out-of-equilibrium situation in order to estimate the validity of our approach.\n\n\n\\begin{figure}\n\\centering\n  \\includegraphics[width=1\\linewidth]{Diag}\n  \\caption{Diagrammatic illustration of the Dyson-Schwinger equations up to cubic order. Here, $G$ represents the full Green's function, $S^{(3)}$ the bare three-body vertex and $\\Gamma^{(3)}$ the full three-body vertex. For convenience, this displays only the topology of the diagrams, which has not been extended to Keldysh space.}\n  \\label{fig:Diag}\n\\end{figure}\n\nThe phonon scattering terms in Eq.~\\eqref{Eq5} are resonant, i.e. they describe scattering between a continuum of energetically degenerate states, and as a consequence, perturbation theory diverges. In order to determine the self-energies $\\sigma^R_{t,q}, \\sigma^K_{t,q}, \\Gamma^K_{t,q}$, we apply non-perturbative Dyson-Schwinger equations, which are truncated at cubic order. This takes into account renormalization effects of the cubic vertex and yields non-perturbative self-energies. The topology of the corresponding diagrams is shown in Fig.~\\ref{fig:Diag}.\n If we neglect the cubic vertex correction, the Dyson-Schwinger equations reduce to the self-consistent Born approximation \\cite{buchholdmethod}. For an initial state with constant phonon density, as it is the case for the present setup, the vertex correction has been shown to be exactly zero \\cite{buchholdmethod,forsternelson76}, however it obtains a non-zero value in the time-evolution of the system. The kinetic equations \\eqref{Eq28}, \\eqref{Eq29} are solved iteratively, starting at a certain time $t$, the self-energies and vertex correction are computed as functions of the distributions $n_{t,q}, m_{t,q}$. Subsequently $\\partial_tn_{t,q}, \\partial_tm_{t,q}$ are determined,  and used in turn to compute the distributions $n_{t+\\Delta,q}, m_{t+\\Delta,q}$ for an infinitesimally later time. This procedure is repeated in order to determine the time-evolution of the phonon densities and self-energies. A more detailed, technical derivation of the iterative solution for the kinetic equation, self-energies and vertex correction can be found in \\cite{buchholdmethod}.\n\n\\section{Thermalization and Prethermalization Dynamics}\n\\label{sec:thermalization_dynamics}\n\nAs one can see from the kinetic equations in Eq.~\\eqref{Eq28} and Eq.~\\eqref{Eq29}, the diagonal and off-diagonal phonon densities are no longer constants of motion in the presence of phonon scattering and energy is redistributed between the different momentum modes. On a general level, when the system thermalizes, as we will show below, the steady state of the dynamics in the presence of a cubic scattering as in Eq.~\\eqref{Eq5}, is solely determined by the associated temperature $T$ and independent of any further details of the initial nonequilibrium state. Specifically, the diagonal modes acquire a Bose-Einstein distribution $n_{\\infty,q} = n_{t\\rightarrow\\infty,q}=\\left(e^{ u|q|\/T}-1\\right)^{-1}$ whereas the off-diagonal distributions $m_{q}=0$ have to vanish.\n\n\nImportantly, in the resonant approximation, the final temperature $T$ ($k\\sub{B}=1$ in the following) can be computed directly from the initial state as will be shown now. In a closed system, the total energy is conserved. Moreover, the conservation of the kinetic energy is an additional exact feature of the derived kinetic equation. As a consequence, also the interaction energy itself is individually conserved. The latter is not an artifact of the kinetic equation but a feature of the resonant nature of the interactions, which, by definition of resonance, commute with the quadratic part of the Hamiltonian \\eqref{Eq5} already on an operator level. \nThis implies that the relaxation dynamics due to the interactions takes place in closed subsets of degenerate eigenstates of the quadratic Hamiltonian, which would in the absence of phonon scattering only acquire a global phase and were not able to thermalize. \nConsequently, the kinetic energy of the initial ($e_0$) and final state ($e_f$) have to be equal, which yields:\n\\eq{Eq20}{\ne_{0}=un_{\\lambda}\\Lambda^2=\\int_q \\hspace{-0.1cm}u|q| n_{0,q}\\overset{!}{=}\\int_q u |q|n_{\\infty,q}=\\frac{T^2_{\\lambda}\\pi^2}{3u}=e_{f}.\\ \\ \\ \n}\nHere, $n_{0,q}$ is the initial momentum distribution, see Eq.~\\eqref{Occ}, and $n_{\\infty,q} = \\left(e^{\\beta u|q|}-1\\right)^{-1}$ is the final, thermal distribution. This gives:\n\\eq{temperature}{T_{\\lambda}=\\frac{u\\Lambda}{\\pi}\\sqrt{3n_{\\lambda}},} which depends on the details of the quench only through the quench parameter $\\lambda$ such that we denote the temperature via $T\\sub{$\\lambda$}$ in the following. Importantly, this temperature yields a criterion for the applicability of the Luttinger theory for the present quench scenario, since Luttinger theory is only well-defined for temperatures lower than the cutoff $T_{\\lambda}<u\\Lambda$. Evaluating this inequality results in a bound for the quench parameter $\\lambda$, i.e. for $\\frac{1}{15}\\le \\lambda\\le15$, the quench can be described in the framework of Luttinger theory. \n\n\nIn the remainder of this section, we will discuss the time-evolution of the phonon densities according to the kinetic equation and derive the form of the Green's function in Eq.~\\eqref{eq:factorization2}.\n\\subsection{Phonon densities}\nThe time evolution of the phonon densities is determined by the kinetic equations \\eqref{Eq28} and \\eqref{Eq29}. In order to make the time evolution of the phonon densities dimensionless, we rescale the self-energy according to $\\tilde{\\sigma}^{R,K}=\\frac{\\sigma^{R,K}}{v_0\\Lambda^2}$, the momentum $\\tilde{q}=\\frac{q}{\\Lambda}$ and time $\\tau=v_0\\Lambda^2t$. In these units, the time evolved phonon densities depend only on the initial state and are independent of the microscopic details of $v_0$ and $\\Lambda$ \\cite{buchholdmethod}, i.e. in the present setting the time-evolution of the phonon density is completely determined by the quench parameter $\\lambda$, which characterizes the initial state. Additionally, as a consequence of Eq.~\\eqref{Occ}, the dynamics remains invariant under $\\lambda\\rightarrow1\/\\lambda$ and $m_{\\tau,q}\\rightarrow-m_{\\tau,q}$ and we therefore consider only the case $\\lambda>1$.\n\n\\begin{figure*}\n  \\includegraphics[width=1\\linewidth]{OccFull}\n  \\caption{Simulation of the time-evolution of the diagonal phonon density $n_{\\tau,q}$ (left column) and off-diagonal density $m_{\\tau,q}$ (right column) for different quench parameters $\\lambda$. In each row, the individual lines correspond to different times $\\tau=(0,1,2,3,4,5)$. \nLeft column: The total phonon density increases in time (from light to dark green) and the dotted lines represent the corresponding asymptotic density in the limit $\\tau\\rightarrow\\infty$, which is a Bose distribution with the quench dependent temperature $T_{\\lambda}=(0.035, 0.124, 0.24)u\\Lambda$ (from the top to the bottom row). The distribution function is separated into two regimes according to Eq.~\\eqref{Eq37}, with a linear increase in momentum for small momenta and a corresponding thermal distribution for larger momenta. The crossover momentum separating the two regimes is marked with a dot.\nRight column: The off-diagonal phonon density is decreasing in time (from light to dark red), displaying two distinct momentum regimes: For momenta larger than the crossover, $q>q\\sub{th}$, the off-diagonal occupation decreases exponentially in momentum, while it remains close to its initial value $m_{0,q}=m_{\\lambda}$ for momenta smaller than the crossover. While any momentum mode $n_{\\tau,q>0}$ will thermalize at a finite time $\\tau<\\infty$, the zero momentum mode remains pinned to its initial value $n_{\\tau,q=0}=n_{\\tau=0,q=0}$. The latter is not an artifact of the approximation but a consequence of exact fermionic particle number conservation, as outlined in the main text.\n}\n  \\label{fig:OccFull}\n\\end{figure*}\n\nThe time evolution of the phonon densities for three different quench parameters $\\lambda$ is shown in Fig.~\\ref{fig:OccFull}. It features two characteristic regimes, which are separated by a time-dependent crossover momentum $q\\sub{th}(\\tau)$, which turns out to be the inverse thermal length scale $x\\sub{th}(\\tau)=1\/q\\sub{th}(\\tau)$. According to the numerical simulations, $q\\sub{th}(\\tau)$ can be parametrized as $q\\sub{th}(\\tau)=Q_{\\lambda}\\tau^{\\alpha_{\\lambda}}$, where the exponent $\\alpha_{\\lambda}$ and the amplitude $Q_{\\lambda}$ are monotonic functions of the quench parameter (for $\\lambda>1$).  According to Fig.~\\ref{fig:OccFull}, away from the crossover, the phonon distribution can be written as\n\\eq{Eq37}{\nn_{\\tau,q}=\\left\\{\\begin{array}{cl}n_{\\lambda}+c_{\\tau,\\lambda}|q|& \\mbox{ for } |q|<q\\sub{th}(\\tau)\\\\ \\frac{\\tilde{T}_{\\tau,\\lambda}}{u |q|}& \\mbox{ for } |q|>q\\sub{th}(\\tau)\\end{array}\\right. .\n}\nFor small momenta $|q|<q\\sub{th}$, the phonon density increases linearly in momentum, with a time-dependent prefactor $c_{\\tau,\\lambda}$, which has to be computed numerically but is determined solely by the quench parameter. This linear increase is guaranteed by the structure of the cubic vertex, which induces a scaling of the one-loop diagrams $\\sim |q|$ for small momenta $q$. This scaling is imposed by the $U(1)$-symmetry of the action, which forbids a smaller exponent in the scaling of the local vertex as discussed in Ref.~\\cite{buchholdmethod}, where  the same scaling behavior was found although with a different amplitude $c_{\\tau}$ reflecting the driven nature of the system in that case. The very same mechanism guarantees the pinning of the distribution at $q=0$ to its initial value $n_{t,q=0}=n_{t=0,q=0}$, expressed by the constant $n_{\\lambda}$ in Eq.~\\eqref{Eq37}.\\\\\nFor larger momenta $|q|>q\\sub{th}$ fast quasi-particle scattering events have established a local equilibrium and the phonon density is well described by a Bose distribution function $n\\sub{B}(u|q|,\\tilde{T}_{\\tau,\\lambda})=\\left(e^{u|q|\/\\tilde{T}_{\\tau,\\lambda}}-1\\right)^{-1}$, which can be approximated by a classical Rayleigh-Jeans distribution, as in Eq.~\\eqref{Eq37}, for intermediate momenta $q\\sub{th}<q<\\frac{\\tilde{T}_{\\tau,\\lambda}}{u}$. The effective temperature $\\tilde{T}_{\\tau,\\lambda}$ approaches the final temperature $T_{\\lambda}=\\tilde{T}_{\\tau\\rightarrow\\infty,\\lambda}$ asymptotically, following a power law $\\tilde{T}_{\\tau,\\lambda}-T_{\\lambda}\\sim \\tau^{\\mu}$ consistent with $\\mu=2\/3$ for large times, see section Fig.~\\ref{sec:AsymTh}. \nAn important exception is represented by the $q=0$ mode, which does not thermalize. The thermal momentum scale $q\\sub{th}\\sim \\tau^{-\\alpha_{\\lambda}}$ is larger than zero for any finite time $\\tau<\\infty$. As a consequence, for any realistic experiment, there will always exist a small momentum window $q\\in [0,q\\sub{th}(t\\sub{max})]$, which does not thermalize during the runtime of the experiment $t\\sub{max}$. However, even in the limit $\\tau\\rightarrow\\infty$, $n_{\\tau,q=0}$ is pinned to its initial value by the exact $U(1)$-symmetry of the fermionic system. This symmetry corresponds to the exact particle number conservation of the fermionic theory. The occupation $n_{\\tau,q=0}$ of the zero momentum mode is directly related to the variance of the total particle number $n_{\\tau,q=0}\\sim \\langle \\hat{N}^2\\rangle_{\\tau}-\\langle \\hat{N}\\rangle^2_{\\tau}$, where $\\hat{N}$ is the total fermionic number operator \\cite{Heating, mahan}. For particle number conserving dynamics, it is therefore an integral of motion. Consequently, the asymptotic bosonic distribution function in the limit $\\tau\\rightarrow\\infty$ is a perfect Bose-Einstein distribution, with a discontinuity at $q=0$.\n\n\nIn order to express the factor $c_{\\tau,\\lambda}$ in terms of the temperature $\\tilde{T}_{\\tau,\\lambda}$, we equate both forms of the distribution function $n_{\\tau,q}$ in Eq.~\\eqref{Eq37} at the crossover scale $q=q\\sub{th}$. This yields an estimate for the non-equilibrium prefactor\n\\eq{Crosso}{c_{\\tau,\\lambda}=\\frac{\\tilde{T}_{\\tau,\\lambda}}{u q\\sub{th}^2}.}\n\nThe off-diagonal densities $m_{\\tau,q}$ are decaying in the long time limit, with $m_{\\tau,q}\\rightarrow0$ in the limit $\\tau\\rightarrow\\infty$. Their time evolution is shown in Fig.~\\ref{fig:OccFull} for the same quench parameters as used for the diagonal densities. For momenta larger than the crossover scale $q\\sub{th}$, the off-diagonal densities decay exponentially fast in momentum, while they remain close to their initial value $m_{\\lambda}$ for momenta smaller than the crossover. The number of scattering events into the off-diagonal modes is $\\propto\\Gamma^K_{\\tau,q}$, which decreases in time very fast $\\sim m_{\\tau,q}^2$. This stands in contrast to the large number of out-scattering processes, which are given by $\\sigma^R_{\\tau,q}m_{\\tau,q}\\sim n_{\\tau,q}m_{\\tau,q}$ and dominate over the ingoing scattering events for a thermalizing diagonal distribution.\n\n\n\n\n\\subsection{Fermion Green's function}\\label{sec:fac}\n\\begin{figure}\n\\begin{center}\n  \\includegraphics[width=1\\linewidth]{CrossoverMomentum}\n  \\caption{Time evolution of the thermal length scale $T_{\\lambda}x\\sub{th}(\\tau)=\\frac{T_{\\lambda}}{q\\sub{th}(\\tau)}$\nfor three different quench scenarios $\\lambda= (1.1, 1.4, 1.9)$, normalized with the corresponding\nfinal temperature $T_{\\lambda}$. The thermal length scale evolves according to a power law in time $T_{\\lambda}x\\sub{th}(\\tau)=x_{\\lambda}\\tau^{\\alpha_{\\lambda}}$, where both amplitude and exponent depend non-trivially on the quench parameter and the exponent is invariant under a basis transformation from the dimensionless basis to the microscopic basis.\nIn the present example,  $\\alpha_{1.1}= 0.7, \\alpha_{1.4} = 0.85$ and $\\alpha_{1.9} = 0.92$. The exponent is bounded from above $\\alpha_{\\lambda}<1$ due to the subleading nature of the interactions and from below $0<\\alpha_{\\lambda}$ by stability properties. For exponents $\\alpha_{\\lambda}<1$, the thermalization dynamics will always feature a finite spatio-temporal prethermalized region as indicated in Fig.~\\ref{fig:QuenchDiag}.\n }\n  \\label{fig:Cross}\n\\end{center}\n\\end{figure}\nThe fermionic lesser Green's function $G^<_{t,x}$ can be computed using the time-evolved densities according to Eq.~\\eqref{GF5}. The numerically determined fermion Green's function for a quench scenario with $\\lambda=1.6$ are shown in Fig.~\\ref{fig:Cross}, as discussed in the beginning of the section. One can identify three spatio-temporal regimes, with individually different, generic scaling behavior described by Eqs.~\\eqref{eq:factorization2}-\\eqref{tlength}. By exploiting the generic form of the time-evolved phonon densities for interacting Luttinger Liquids, we will in the following derive the form of the fermionic Green's function as given in Eq.~\\eqref{eq:factorization2}.\n\n\nIn order to approximate the contribution from the off-diagonal densities, we exploit the fact that they remain close to their initial value $m_{\\tau,q}\\approx m_{\\lambda}$ for momenta smaller than the crossover $q<q\\sub{th}$ and decay exponentially for larger momenta, yielding a negligible influence on short distances. To account for this behavior, we replace in the corresponding integrals the cutoff $\\Lambda\\rightarrow q\\sub{th}$ by the thermal crossover and approximate $m_{\\tau,q}\\approx m_{\\lambda}$ for small momenta. The result is\n\\begin{eqnarray}\n\\mathcal{G}^<_{t,x}&=&\\mathcal{G}^<_{0,x}+im_{\\lambda}\\log\\left(\\tfrac{1+q\\sub{th}^2(x-2ut)^2}{1+q\\sub{th}^2(x+2ut)^2}\\right)\\label{Eq38a}\\\\\n&&-i\\tfrac{(K^2-1)m_{\\lambda}}{2K}\\log\\left[\\tfrac{(1+q\\sub{th}^2(x+2ut)^2)(1+q\\sub{th}^2(x-2ut)^2)}{(1+4u^2t^2q\\sub{th}^2)^2(1+x^2q\\sub{th}^2)^2}\\right]\\label{Eq38b}\\\\\n&&+i\\tfrac{K^2+1}{K}\\int_q\\frac{2\\pi e^{-\\frac{|q|}{\\Lambda}}}{|q|}(\\cos(qx)-1)(n_{t,q}-n_{\\lambda}).\\label{Eq38c}\n\\end{eqnarray}\nIn this expression, $\\mathcal{G}_{0,x}$ contains again the initial post-quench exponent, the terms proportional to $m_{\\lambda}$ represent the time-dependent contributions stemming from the off-diagonal densities, whereas the first term vanishes for distances away from the prethermal crossover $x\\neq 2ut$ and the second term vanishes for distances $x<1\/q\\sub{th}(\\tau)=x\\sub{th}(\\tau)$ smaller than the thermal length. The latter expresses the fact, that off-diagonal occupations vanish in the asymptotic thermal limit. The term in Eq.~\\eqref{Eq38c} takes into account the deviation of the diagonal phonon occupation from the flat initial distribution. Applying Eq.~\\eqref{Eq37} to \\eqref{Eq38c} with a smooth crossover function $\\sim \\exp(-|q|\/q\\sub{th}), 1-\\exp(-|q|\/q\\sub{th})$, respectively,  amounts to\n\\begin{figure}\n  \\includegraphics[width=1\\linewidth]{FermGreen}\n  \\caption{Numerical results for the absolute value of the fermionic lesser Green's function $G^<_{t,x}$ (green, solid lines) after a quench with $\\lambda=1.6$ for different times $t=\\frac{40}{u\\Lambda}l$, $l=1,...,9$ (magnitude decreasing with $l$). The initial state corresponds to non-interacting fermions ($K=1$) and the interaction was chosen such that $v_0\\Lambda^2=\\frac{u\\Lambda}{4}$. The figure illustrates the two crossovers and the three different spatio-temporal regimes of the Green's function. The red lines are determined according to the factorization \\eqref{eq:factorization2}-\\eqref{tlength} in the different regimes and describe the Green's function very well apart from the crossover regions.\n}\n  \\label{fig:FermGreen}\n\\end{figure}\n\\begin{eqnarray}\n\\mathcal{G}^<_{t,x}&=&\\mathcal{G}^<_{0,x}-i\\tfrac{K^2-1}{2K}m_{\\lambda}\\log\\left[\\tfrac{(1+q\\sub{th}^2(x+2ut)^2)(1+q\\sub{th}^2(x-2ut)^2)}{(1+4u^2t^2q\\sub{th}^2)^2(1+x^2q\\sub{th}^2)^2}\\right]\\label{Eq39a}\\\\\n&&-2\\pi i\\tfrac{K^2+1}{K}\\tfrac{\\tilde{T}_{\\tau,\\lambda}}{u}\\left[\\tfrac{q\\sub{th}^2x^2}{1+q\\sub{th}^2x^2}+|x|\\left(1-\\tfrac{2\\arctan(q\\sub{th}|x|)}{\\pi}\\right)\\right],\\label{Eq39b}\n\\end{eqnarray}\nyielding the form of the fermionic Green's function in Eqs.~\\eqref{eq:factorization2}-\\eqref{tlength}. The prethermal amplitude $Z\\sub{pt}$ is determined by the contribution $\\sim m_{\\lambda}$ in Eq.~\\eqref{Eq39a}, while the thermal amplitude $Z\\sub{th}$ is given by the exponential of the $\\sim \\tilde{T}_{\\tau,\\lambda}$-term in Eq.~\\eqref{Eq39b}.\nThis form of the fermionic Green's function holds away from the crossover lines $|x|=2ut$ and $x=x\\sub{th}(t)$. As shown in Fig.~\\ref{fig:FermGreen}, it is a very good approximation for the fermionic Green's function and illustrates perfectly the different thermalization regimes and their scaling behavior.\n\n\n\\changed{The form of Eqs.~\\eqref{Eq39a}-\\eqref{Eq39b} allow us to estimate the distance $x_c$, below which the kinetic theory is applicable \\mh{and which we have given already in \\eqref{dist}}. One realizes immediately, that the prethermalization described by line \\eqref{Eq39a} is absent for $K=1$, i.e. for a quench to the non-interacting theory. This is due to the absence of a coupling of the single particle sector to the many-body sector of the theory. A clear condition for the applicability of the kinetic theory can be obtained by comparing the time dependent variation of Eqs.~\\eqref{Eq39a},\\eqref{Eq39b} with each other. A well defined prethermal plateau has then been established for $|$\\eqref{Eq39a}$|>||$\\eqref{Eq39b}$|$, which leads to the condition on the distance $x<x_c(t)$, where $x_c(t)$ is given in Eq.~\\eqref{dist}.}\n\nWe want to close the section by a discussion of the way, in which the microscopic scales enter the thermalization dynamics discussed in the present context. For the non-interacting Luttinger Liquid in equilibrium, the microscopic details are completely encoded in the sound velocity $u$ and the Luttinger parameter $K$ as well as the temperature of the system $T\\ge 0$ and the Luttinger cutoff $\\Lambda$. For a non-equilibrium setting in the quadratic Luttinger framework, one has to add the information on the initial state, which in the case of an interaction quench can be summarized in a single quench parameter $\\lambda$. In the presence of interactions, we added the cubic vertex $\\sim v_0$. This lead to the emergence of a new crossover scale $x\\sub{th}(t)$, below which the system is effectively thermal, described by an effective time-dependent temperature $\\tilde{T}_{t,\\lambda}$. In the effective description of a factorizing Green's function, these quantities are sufficient to describe the post-quench dynamics. \n\n\n\n\n\nIn the next section, we will discuss the time dependent temperature and find $\\tilde{T}_{\\lambda,\\tau}=T_{\\lambda}+\\Delta_{\\lambda}\\tau^{-\\mu}$, where $T_{\\lambda}$ is the final temperature of the system, depending on the energy induced by the quench and $\\Delta_{\\lambda}$ the quench-dependent amplitude, while $\\mu$ is a universal exponent. In original units, $\\tilde{T}_{t,\\lambda}=T_{\\lambda}+u\\Lambda\\Delta_{\\lambda}\\left(v_0\\Lambda^2t\\right)^{-\\mu}$. In the simplified picture, these are the only relevant quantities, which show a functional dependence on the nonlinearity $v_0$, naturally containing the limit $v_0\\rightarrow0$, for which the thermal crossover is at zero distance and the temperature is not defined due to the absence of thermalization.\n\n\nThe thermalization dynamics for interacting Luttinger liquids presented so far is not restricted to interaction quenches or global quenches in general, but expected to represent quite generically the relaxation dynamics of Luttinger liquids out of equilibrium. First of all, the dephasing of the off-diagonal modes due to the quadratic Hamiltonian will be present in any setup for which off-diagonal modes have been excited in the initial state and it spreads in space with the light cone $x=2ut$. On the other hand, due to $U(1)$ symmetry and the imposed scaling of the one-loop correction $\\sim q$ for small momenta $q$ \\cite{buchholdmethod}, the change in the diagonal phonon distribution has to scale $\\sim |q|$ as well. The determination of the crossover scale thus proceeds along the same lines as outlined above, and thus separating thermalized short distance modes with occupation $n_{t,q}\\sim 1\/|q|$ from non-thermal long distance modes $n_{t,q}-n_{0,q}\\sim |q|$, leading to a similar three stage process for equilibration as described in the present setup.\n\\subsection{Asymptotic thermalization in the resonant approximation}\\label{sec:AsymTh}\nAfter the quench, momentum modes larger than the temporally decreasing crossover momentum $q\\sub{th}$ establish a local detailed balance between in- and out-scattering processes. This in turn defines the thermalized region in real space, for which, on distances $x<x\\sub{th}(t)=1\/q\\sub{th}(t)$, the fermionic Green's function has the typical thermal form. In this regime, the corresponding momentum modes for $q>q\\sub{th}$ are described by a single, well defined temperature $\\tilde{T}_{t,\\lambda}$ such that $n_{t,q}=n_{\\mbox{\\tiny B}}(u|q|,T)\\approx T\/(u|q|)$. For momenta $q<q\\sub{th}$ the phonon distribution is however smaller than the corresponding thermal distribution $u|q| n_{t,q}<T$ (see Fig.~\\ref{fig:OccFull}) and in order to reach equipartition, energy has to be shifted from the thermal regime to the non-thermalized infrared modes. Consequently, the effective temperature of the large momentum modes is decreasing in time, expressing the energy flow from the large to the low momentum regime, i.e. \n $\\tilde{T}_{t\\rightarrow\\infty,\\lambda}\\rightarrow T_{\\lambda}$ in the limit $x\\sub{th}(t)\\rightarrow\\infty$.\nThe local equipartition of energy in the thermal regime is a consequence of a locally established detailed balance between in- and out-scattering processes. This local detailed balance in combination with exact energy conservation enforces that the energy transport to the long-wavelength modes in the system is performed by a global mechanism, which reveals the presence of dynamical slow modes in the system. They are a consequence of exact conservation laws, i.e. global symmetries, and in the present system emerge as a consequence of exact momentum and energy conservation. These modes are hydrodynamic, gapless modes featuring an algebraic decay of the temperature in time towards its final value.\n\\begin{figure*}\n  \\includegraphics[width=\\linewidth]{MomTherm}\n  \\caption{Momentum dependent temperature $\\tilde{T}_{\\tau,q,\\lambda}$ as defined in Eq.~\\eqref{Eq41} after two distinct quenches. For momenta smaller than the crossover $q\\sub{th}$, the temperature is a momentum-dependent function and can not be seen as a global property. On the other hand, for momenta $q>q\\sub{th}$, the modes are described by the same temperature, indicating the presence of local detailed balance in the momentum regime larger than the crossover. In this regime, the temperature decays algebraically, revealing energy transport from the thermalized to the non-thermalized region, carried by dynamical slow modes. The inset shows the decay of the effective temperature for large times, allowing for numerical estimate $\\mu=2\/3$, which corresponds to the red, dotted line.\n}\n  \\label{fig:MomTherm}\n\\end{figure*}\n\nIn order to determine the asymptotic dynamics in the thermalized regime, we define a momentum and time dependent temperature by inverting the on-shell Bose distribution function \n\\eq{Eq41}{\n\\tilde{T}_{t,\\lambda,q}=\\frac{u|q|}{\\log\\left(\\frac{n_{t,q}+1}{n_{t,q}}\\right).}\n}\nThe time evolution of $\\tilde{T}_{t,\\lambda,q}$ is shown in Fig.~\\ref{fig:MomTherm}. For momenta $q<q\\sub{th}$ it varies as a function of momentum, indicating that the system has not thermalized on this scale and the notion of a temperature is absent. On the other hand, for momenta $q>q\\sub{th}$, $\\tilde{T}_{t,\\lambda,q}$ becomes momentum independent and a global property of the high momentum modes. The decay of $\\tilde{T}_{t,\\lambda}=\\tilde{T}_{t,q>q\\sub{th},\\lambda}$ follows a power law in time, which can be expressed\n\\eq{Eq42}{\n\\tilde{T}_{t,\\lambda}=T_{\\lambda}+u\\Lambda\\Delta_{\\lambda}\\left(v_0\\Lambda^2t\\right)^{-\\mu},\n}\nwhere $\\mu$ is the relaxation exponent associated with the dynamical slow modes. For a one-dimensional system with energy and momentum conserving dynamics $\\mu=2\/3$, since this behavior corresponds to the Kardar-Parisi-Zhang (KPZ) universality class \\cite{KPZ,spohn04,lamacraft13,vanBeijeren,spohn15}. Performing a single parameter fit from the numerical simulations, we find that for large times $\\mu=2\/3$ agrees very well with the numerical data for various different quench scenarios. However, for intermediate times, we find scaling behavior with $\\mu>2\/3$ for some quenches, which might be traced back to the presence of subleading correction terms due to couplings to other diffusive modes \\cite{Lux13,narayan02,Mukerjee}. Numerically a distinction of these possible scaling contributions is only possible for simulation times of multiple decades, such that we cannot exclude a different exponent $\\mu<2\/3$ at the largest times \\cite{Lux13}, which is however not observed in our simulations.\n\n\nWhile the establishment of a local detailed balance, leading to effective thermalization and thermal-like fermionic correlation functions is an effect of local quasi-particle scattering, the asymptotic thermalization dynamics describing energy transport over large distances in momentum space is determined by macroscopic diffusive modes in the system. This is observable by an algebraically decaying temperature towards the final temperature of the system $T_{\\lambda}$. The discussion on the dynamical slow modes remains valid even in the presence of off-resonant scattering processes and therefore the universal properties of the asymptotic thermalization process remain unmodified. However, non-universal properties, such as the final temperature as well as the relaxation rate will be modified by the off-resonant processes. Their precise computation would be a task for numerical simulations.\n\n\\section{Conclusion}\nIn this work, we have analyzed the relaxation dynamics of interacting Luttinger liquids, microscopically represented by one-dimensional interacting fermions with band curvature, after a sudden quench in the fermionic interaction. The theoretical analysis is based on quantum kinetic equations for the phonon distribution function and non-perturbative Dyson-Schwinger equations, which are both well suited to determine the time-evolution of static observables for interacting Luttinger liquids with resonant, cubic interactions, and applicable in a broad parameter regime within the Luttinger framework. The central result is a two-step thermalization procedure including a spatio-temporal prethermalized regime for intermediate distances and times, which leads to fermionic correlation functions described by a generalized Gibbs state on these distances, and corresponds to fast quasi-particle formation after the quench. On smaller distances, a thermalized regime occurs due to the scattering and associated redistribution of energy between the quasi-particle modes. This regime is described by thermal correlation functions with a characteristic thermal correlation length and a thermal quasi-particle distribution with an effective temperature that decays algebraically in time towards its asymptotic value.\n\n This work shows in which way thermalization and prethermalization occur and spread in space for RG-irrelevant, and in this sense weak, integrability breaking interactions. In this setup both thermalization and prethermalization occur locally in space. While the prethermalized region spreads ballistically in space, the thermalized region spreads sub-ballistically due to the subleading, RG-irrelevant nature of the interactions. This allows for a well-defined prethermal regime in time and space, which would not be possible for a constant, momentum independent scattering vertex, for which thermalization would occur immediately on all different length scales.\nThis underpins the statement that typical candidates for clearly observable prethermalized regimes within generic thermalization dynamics are quasi-particle theories with RG irrelevant interactions.\n\n\n\n\\begin{acknowledgments}\nWe acknowledge valuable discussions with Alessio Recati. This research was supported by the\nGerman Research Foundation (DFG) through the Institutional\nStrategy of the University of Cologne within the German\nExcellence Initiative (ZUK 81) and the European Research\nCouncil (ERC) under the European Unions Horizon\n2020 research and innovation programme (grant agreement\nNo 647434)\nas well as the\nDeutsche Akademie der Naturforscher Leopoldina under grant numbers LPDS 2013-07 and LPDR 2015-01.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction\\label{sec:Introduction}}\n\nWe develop an empirical framework to identify and estimate the heterogeneous\neffects of treatments on outcomes of interest, where the treatments\nare the result of agents' interaction (e.g., bargaining,\noligopolistic entry, decisions in the presence of peer effects or\nstrategic effects). Treatments are determined as an equilibrium of\na game and these strategic decisions of players endogenously affect\ncommon or player-specific outcomes. For example, one may be interested\nin the effects of entry of newspapers on local political behavior,\nentry of carbon-emitting companies on local air pollution and health\noutcomes, the presence of potential entrants in nearby markets on\npricing or investment decisions of incumbents, the exit decisions\nof large supermarkets on local health outcomes, or the provision of\nlimited resources when individuals make participation decisions under\npeer effects and their own gains from the treatment.\\footnote{The entry and pollution is our leading example introduced in Section\n\\ref{sec:stylized_ex}; the other examples are discussed in detail\nin Appendix \\ref{sec:Examples}.} As reflected in some of these examples, our framework allows us to\nstudy the \\textit{externalities of strategic decisions}, such as societal\noutcomes resulting from firm behavior. Ignoring strategic interaction\nin the treatment selection process may lead to biased, or at least\nless informative, conclusions about the effects of interest.\n\nWe consider a model in which agents play a discrete game of complete\ninformation, whose equilibrium actions (i.e., a profile of binary\nendogenous treatments) determine a post-game outcome in a nonseparable\nmodel with endogeneity. We are interested in the various treatment\neffects of this model. In recovering these parameters, the setting\nof this study poses several challenges. First, the first-stage game\nposits a structure in which binary dependent variables are simultaneously\ndetermined in threshold crossing models, thereby, making the model,\nas a whole, \\textit{incomplete}. This is related to the problem of\nmultiple equilibria in the game. Second, due to this simultaneity,\nthe selection process for each treatment in the profile does not exhibit\nthe conventional monotonic property \\`a la \\citet{imbens1994identification}.\nFurthermore, we want to remain flexible with other components of the\nmodel. That is, we make no assumptions on the joint distributions\nof the unobservables nor parametric restrictions on the player's payoff\nfunction and how treatments affect the outcome. In addition, we do\nnot impose any arbitrary equilibrium selection mechanism to deal with\nthe multiplicity of equilibria, nor require that players be symmetric.\nIn nonparametric models with multiplicity and\/or endogeneity, identification\nmay be achieved using excluded instruments with large support. Although\nsuch a strong requirement can be met in practice, estimation and inference\ncan still be problematic (\\citet{andrews1998semiparametric}, \\citet{khan2010irregular}).\nThus, we avoid such assumptions for instruments and other exogenous\nvariables.\n\nThe first contribution of this study is to establish that under strategic\nsubstitutability, regions that predict the equilibria of the treatment\nselection process in the first-stage game can present a monotonic\npattern in terms of the number of treatments selected.\\footnote{To estimate payoff parameters, \\citet{berry1992estimation} partly\ncharacterizes equilibrium regions. To calculate the bounds on these\nparameters, \\citet{CT09} simulate their moment inequalities model\nthat are implied by the shape of these regions, especially the regions\nfor multiple equilibria. While their approaches are sufficient for\ntheir analyses, full analytical results are critical for the identification\nanalysis in this current study.} The second contribution of this study is to show, after restoring\nthe \\textit{generalized monotonicity} in the selection process, how\nthe model structure and the data can provide information about treatment\nparameters, such as the average treatment effects (ATEs). We first\nestablish the bounds on the ATE and other related parameters with\npossibly discrete instruments. We also show that tighter bounds on\nthe ATE can be obtained by introducing (possibly discrete) exogenous\nvariables excluded from the first-stage game. This is especially motivated\nwhen the outcome variable is affected by externalities generated by\nthe players. We can derive sharp bounds as long as the outcome variable\nis binary. To deal with the multiple equilibria problem in our analysis,\nwe assume that instruments vary sufficently to offset the effect of\nstrategic substitutability. We provide a simple testable implication\nfor the existence of such instrument variation in the case of mutually\nindependent payoff unobservables. This requirement of variation is\nqualitatively different and substantially weaker than a typical large\nsupport assumption. A marked feature of our analyses is that for the\nsharp bounds on the ATE, player-specific instruments are not necessary.\n\nOur bound analysis\nbuilds on \\citet{VY07} and \\citet{SV11}, which consider point and partial identification in single-agent nonparametric triangular models\nwith binary endogenous variables. Unlike them, however, we allow for multi-agent strategic interaction\nas a key component of the model. Some studies have extended a single-treatment\nmodel to a multiple-treatment setting (e.g., \\citet{heckman2006understanding},\n\\citet{jun2011tighter}), but their models maintain monotonicity in\nthe selection process and none of them allow simultaneity among the\nmultiple treatments resulting from agents' interaction, as we do in\nthis study.\n\nIn interesting recent work, \\citet{heckman2015unordered}, and \\citet{lee2016identifying}\nextend the monotonicity of the selection process in multi-valued treatments\nsettings. \\citet{heckman2015unordered} introduce unordered monotonicity,\nwhich is a different type of treatment selection mechanisms than ours.\n\\citet{lee2016identifying} consider more general non-monotonicity\nand do mention entry games as one example of the treatment selection\nprocesses they allow. However, they assume known payoffs and bypass\nthe multiplicity of equilibria by assuming a threshold-crossing equilibrium\nselection mechanism, both of which we do not assume in this study.\nIn addition, \\citet{lee2016identifying}'s focus is on the identification\nof marginal treatment effects with continuous instruments. In another\nrelated work, \\citet{chesher2014generalized} consider a class\nof generalized instrumental variable models in which our model may\nfall and propose a systematic method of characterizing sharp identified\nsets for admissible structures. This present study's characterization\nof the identified sets is analytical, which helps investigate how the\nidentification is related to exogenous variation in the model and\nto the equilibrium characterization in the treatment selection. Also,\ncalculating the bounds on the treatment parameters using their approach\ninvolves projections of identified sets that may require parametric\nrestrictions. Lastly, \\citet{Han18,Han19c} consider identification\nof dynamic treatment effects and optimal treatment regimes in a nonparametric\ndynamic model, in which the dynamic relationship causes non-monotonicity\nin the determination of each period's outcome and treatment.\n\nWithout triangular structures, \\citet{manski1997monotone}, \\citet{MP00}\nand \\citet{Man13} also propose bounds on the ATE with multiple treatments\nunder various monotonicity assumptions, including an assumption on\nthe sign of the treatment response. We take an alternative approach\nthat is more explicit about treatments interaction while remaining\nagnostic about the direction of the treatment response. Our results\nsuggest that provided there exist exogenous variation excluded from\nthe selection process, the bounds calculated from this approach can\nbe more informative than those from their approach.\n\nIdentification in models for binary games with complete information\nhas been studied in \\citet{Tam03}, \\citet{CT09}, and \\citet{bajari2010identification},\namong others.\\footnote{See also \\citet{galichon2011set} and \\citet{beresteanu2011sharp}\nfor a more general setup that includes complete information games\nas an example.} This present study contributes to this literature by considering post-game outcomes that are often not of players' direct concern.\nAs related work that considers post-game outcomes, \\citet{ciliberto2015market}\nintroduce a model in which firms make simultaneous decisions of entry\nand pricing upon entry. Consequently, their model can be seen as a\nmulti-agent extension of a sample selection model. On the other hand,\nthe model considered in this study is a multi-agent extension of a\nmodel for endogenous treatments. At a more general level, our approach is an attempt to bridge the treatment effect literature and the industrial organization (IO) literature. We are interested in the evaluation of treatments that are the result of agents' strategic interaction, an aspect that is key in the IO literature. To conduct the counterfactual analysis, however, we closely follow the treatment effect literature, instead of the structural approach of the IO literature. For example, \\citet{CT09} and \\citet{ciliberto2015market}\nimpose economic structure and parametric assumptions to recover model primitives for policy analyses. In contrast, our parameters of interest are\ntreatment effects as functionals of the primitives (but excluding\nthe game parameters), and thus, allow our model to remain nonparametric. In addition, as the goal is different, we employ a different approach to partial identification\nunder the multiplicity of equilibria than theirs.\\footnote{Even if we are willing to\nassume a known distribution for the unobserved payoff types, their approach to multiplicity is not applicable\nto the particular setting of this study.}\n\n\n\nTo demonstrate the applicability of our method, we take the bounds\nwe propose to data on airline market structure and air pollution in\ncities in the U.S. Aircrafts and airports land operations are a major\nsource of emissions, and thus, quantifying the causal effect of air\ntransport on pollution is of importance to policy makers.\nWe explicitly allow market structure to be determined endogenously\nas the outcome of an entry game in which airlines behave strategically\nto maximize their profits and where the resulting pollution in this\nmarket is not internalized by the firms. Additionally, we do not impose\nany structure on how airline competition affects pollution and allow\nfor heterogenous effects across firms. In other words, not only do\nwe allow the effect of a different number of firms in the market on\npollution to be nonlinear and not restricted, but also\ndistinguish the identity of the firms. The latter is important if we believe that behavior post-entry differs across\nairlines. For example, different airlines might operate the market with a higher frequency\nor with different types of airplanes, hence affecting pollution in a different\nway.\nTo implement our application, we combine data from two sources. The\nfirst contains airline information from the Department of Transportation,\nwhich we use to construct a dataset of airlines' presence in each\nmarket. We then merge it with air pollution data in each airport from\nair monitoring stations compiled by the Environmental Protection Agency.\nIn our preferred specification, our outcome variable is a binary measure\nof the level of particulate matter in the air.\n\nWe consider three sets of ATE exercises to investigate different aspects\nof the relationship between market structure and pollution in equilibrium.\nThe first simply quantifies the effects of each airline operating\nas a monopolist compared to a situation in which the market is not\nserved by any airline. We find that the effect of each airline on\npollution is positive and statistically significant. We also find\nevidence of heterogeneity in the effects across different airlines.\nThe second set of exercises examines the ATEs of all potential market\nstructures on pollution. We find that the probability of high pollution\nis increasing with the number of airlines in the market, but at a\ndecreasing rate. Finally, the third set of exercises quantifies the\nATE of a single airline under all potential configurations of the\nmarket in terms of its rivals. We observe that in all cases, Delta\nentering a market has a positive effect on pollution and this effect\nis decreasing with the number of rivals. The results from the last\ntwo set of exercises are consistent with the results of a Cournot-competition\noligopolistic model in which incumbents \\emph{accommodate} new entrants\nby reducing the quantity they produce.\n\nThis paper is organized as follows. Section \\ref{sec:stylized_ex}\nsummarizes the analysis of this study using a stylized example. Section\n\\ref{sec:General-Theory} presents a general theory: Section \\ref{subsec:Model}\nintroduces the model and the parameters of interest; Section \\ref{subsec:Geometry}\npresents the generalized monotonicity for equilibrium regions for\nmany players; and Section \\ref{subsec:Partial-Identification} delivers\nthe partial identification results of this study. Section \\ref{sec:Monte-Carlo-Studies}\npresents a numerical illustration and Section \\ref{sec:Empirical-Application}\nthe empirical application on airlines and pollution. In the Appendix,\nSection \\ref{sec:Examples} provides more examples to which our setup\ncan be applied. Section \\ref{sec:Extensions} contains four extensions\nof our main results. Finally, Section \\ref{sec:Proofs} collects the\nproofs of theorems and lemmas.\n\n\\section{A Stylized Example\\label{sec:stylized_ex}}\n\nWe first illustrate the main results of this study with a stylized\nexample. Suppose we are interested in the effects of airline competition\non local air quality (or health). Let $Y_{i}$ denote the binary indicator\nof air pollution in market $i$. For illustration, we assume there\nare two potential airlines. In the next section, we present a general\ntheory with more than two players. Let $D_{1,i}$ and $D_{2,i}$ be\nbinary variables that indicate the decisions to enter market $i$\nby Delta and United, respectively. We allow the decisions $D_{1,i}$\nand $D_{2,i}$ to be correlated with some unobserved characteristics\nof the local market that affect $Y_{i}$. Moreover, since $D_{1,i}$\nand $D_{2,i}$ are equilibrium outcomes of the entry game, we allow\nthem to be outcomes from multiple equilibria. The endogeneity and\nthe presence of multiple equilibria are our key challenges in this\nstudy.\n\nLet $Y_{i}(d_{1},d_{2})$ be the potential air quality had Delta and\nUnited's decisions been $(D_{1},D_{2})=(d_{1},d_{2})$; for example,\n$Y_{i}(1,1)$ is the potential air quality from duopoly, $Y_{i}(1,0)$\nis with Delta being a monopolist, and so on. Let $X_{i}$ be a vector\nof market characteristics that affect $Y_{i}$. Our parameter of interest\nis the ATE, $E[Y_{i}(d_{1},d_{2})-Y_{i}(d_{1}',d_{2}')|X_{i}=x]$,\nwhich captures the effect of market structure on pollution. One interesting\nATE is $E[Y_{i}(1,d_{2})-Y_{i}(0,d_{2})|X_{i}=x]$ for each $d_{2}$,\nwhere we can learn the interaction effects of treatments, e.g., how\nmuch the average effect of Delta's entry is affected by United's entry:\n$E\\left[Y_{i}(1,1)-Y_{i}(0,1)\\right]-E\\left[Y_{i}(1,0)-Y_{i}(0,0)\\right]$\n(suppressing $X_{i}$). In our empirical application (Section \\ref{sec:Empirical-Application}),\nwe consider this and other related parameters in a more realistic\nmodel, where there are more than two airlines.\n\nWe show how we overcome the problems of endogeneity and multiple equilibria\nand how to construct bounds on the ATE using the excluded instruments\nand other exogenous variables. Let $Z_{1,i}$ and $Z_{2,i}$ be cost\nshifters for Delta and United, respectively, which serve as instruments.\nAs a benchmark, we first consider naive bounds analogous to \\citet{manski1990nonparametric}\nusing excluded instruments which satisfy \n\\begin{align}\nY_{i}(d_{1},d_{2}) & \\perp(Z_{1,i},Z_{2,i})|X_{i}\\label{as:manski_IV}\n\\end{align}\nfor all $(d_{1},d_{2})$. To simplify notation, we suppress the index\n$i$ henceforth, let $\\boldsymbol{D}\\equiv(D_{1},D_{2})$ and $\\boldsymbol{Z}\\equiv(Z_{1},Z_{2})$,\nand write $E[\\cdot|w]\\equiv E[\\cdot|W=w]$ for a generic r.v. $W$.\nAs an illustration, we focus on calculating bounds on $E[Y(1,1)|X=x]$.\nNote that \n\\begin{align}\nE[Y(1,1)|x]=E[Y(1,1)|\\boldsymbol{z},x] & =E[Y|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z},x]\\nonumber \\\\\n & +\\sum_{\\boldsymbol{d}^{\\prime}\\neq(1,1)}E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{\\prime}|\\boldsymbol{z},x],\\label{eq:Manski_expand-1}\n\\end{align}\nwhere the first equality is by \\eqref{as:manski_IV}. Manski-type\nbounds can be obtained by observing that the counterfactual term $E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]=\\Pr[Y(1,1)=1|\\boldsymbol{D}=\\boldsymbol{d}^{\\prime},\\boldsymbol{z},x]$\nis bounded above by one and below by zero. By further using the variation\nin $\\boldsymbol{Z}$, which is excluded from $Y(1,1)$, the lower\nand upper bounds on $E[Y(1,1)\\vert x]$ can be written as \n\\begin{align*}\nL_{Manski}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x],\\\\\nU_{Manski}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z}]\\right\\} .\n\\end{align*}\nThe goal of our analysis is to derive tighter bounds than $L_{Manski}(x)$\nand $U_{Manski}(x)$ by introducing further assumptions motivated\nby economic theory.\n\nTo illustrate, we introduce the following semi-triangular model with\nlinear indices. In the next section, we generalize this model by\nintroducing fully nonparametric models that allow continuous $Y$. All the assumptions and results illustrated in the current section are formally stated and proved in the next section. Consider\n\\begin{align}\nY & =1[\\mu_{1}D_{1}+\\mu_{2}D_{2}+\\beta X\\ge\\epsilon],\\label{eq:model_ex1}\\\\\nD_{1} & =1[\\delta_{2}D_{2}+\\gamma_{1}Z_{1}\\ge U_{1}],\\label{eq:model_ex2}\\\\\nD_{2} & =1[\\delta_{1}D_{1}+\\gamma_{2}Z_{2}\\ge U_{2}],\\label{eq:model_ex3}\n\\end{align}\nwhere $(\\epsilon,U_{1},U_{2})$ are continuously distributed unobservables\nthat can be arbitrarily correlated, $(U_{1},U_{2})$ are uniform,\nand assume \n\\begin{align}\n & (\\epsilon,U_{1},U_{2})\\perp(Z_{1},Z_{2})|X,\\label{eq:my_IV}\\\\\n & \\delta_{1}<0\\text{ and }\\delta_{2}<0,\\label{eq:strategic_sub}\\\\\n & sgn(\\mu_{1})=sgn(\\mu_{2}).\\label{eq:mono}\n\\end{align}\nNote that \\eqref{eq:my_IV} replaces \\eqref{as:manski_IV}, \\eqref{eq:strategic_sub}\nassumes strategic substitutability, and \\eqref{eq:mono} is plausible\nin the current example of air quality and entry. Owing to the first\nstage simultaneity,\nthe model \\eqref{eq:model_ex1}--\\eqref{eq:model_ex3} is \\textit{incomplete}, i.e., the model primitives and the\ncovariates do not uniquely predict $(Y,\\boldsymbol{D})$. In this\nmodel, we are \\textit{not} interested in the players' payoff parameters\n$(\\delta_{-s},\\gamma_{s})$ for $s=1,2$, individual parameters $(\\mu_{1},\\mu_{2},\\beta)$\nthat generate the outcome, nor distributional parameters. Instead,\nwe are interested in the ATE as a function of $(\\mu_{1},\\mu_{2},\\beta)$.\nThis is in contrast to \\citet{ciliberto2015market}, where payoff\nand pricing parameters are direct parameters of interest, and thus,\nour identification question and strategy (especially how we deal with\nmultiple equilibria) are different from theirs.\n\nTypically, a standard approach that utilizes instrumental variables\ncompares the reduced-form relationship between the outcome and treatment\nwith the reduced-form relationship between the treatment and instrument.\nWe apply the same idea here by changing the values of $Z_{1}$ and\n$Z_{2}$ and measure the change in $Y$ relative to the change in\n$D_{1}$ and $D_{2}$. To this end, for two realizations $\\boldsymbol{z},\\boldsymbol{z}'$\nof $\\boldsymbol{Z}$, say low and high entry cost for both airlines,\nwe introduce reduced-form objects directly recovered from the data:\n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv\\Pr[Y=1|\\boldsymbol{z},x]-\\Pr[Y=1|\\boldsymbol{z}',x],\\label{eq:h(zzx)-1}\\\\\nh_{\\boldsymbol{d}}(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}',x]\\label{eq:hj-1}\n\\end{align}\nfor $d\\in\\{(0,0),(1,0),(0,1),(1,1)\\}\\equiv\\mathcal{D}$. We show that\n\\eqref{eq:h(zzx)-1}--\\eqref{eq:hj-1} deliver useful information\nabout the outcome index function ($\\mu_{1}D_{1}+\\mu_{2}D_{2}+\\beta X$),\nwhich in turn is helpful in constructing bounds on the ATE. Note that\n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & =h_{11}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{10}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{01}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{00}(\\boldsymbol{z},\\boldsymbol{z}',x)\\nonumber \\\\\n & =\\Pr[Y=1,\\boldsymbol{D}=(1,1)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(1,1)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(1,0)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(1,0)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(0,1)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(0,1)|\\boldsymbol{z}',x]\\nonumber \\\\\n & +\\Pr[Y=1,\\boldsymbol{D}=(0,0)|\\boldsymbol{z},x]-\\Pr[Y=1,\\boldsymbol{D}=(0,0)|\\boldsymbol{z}',x],\\label{eq:h_derive0}\n\\end{align}\nwhere $\\boldsymbol{D}=(1,0)$ and $(0,1)$ are the airlines' decisions\nthat may arise as multiple equilibria. The increase in cost (from\n$\\boldsymbol{z}$ to $\\boldsymbol{z}'$) will make the operation of\nthese airlines less profitable in some markets, depending on the values\nof the unobservables $\\boldsymbol{U}=(U_{1},U_{2})$. This will result\nin a change in the market structure in those markets. Specifically,\nmarkets ``on the margin'' may experience one of the following changes\nin structure as cost increases: (a) from duopoly to Delta-monopoly;\n(b) from duopoly to United-monopoly; (c) from Delta-monopoly to no\nentrant; (d) from United-monopoly to no entrant; and (e) from duopoly\nto no entrant. These changes are depicted in Figure \\ref{fig:As_EQ-1},\nwhere each $R_{d_{1},d_{2}}(\\boldsymbol{z})$ denotes the maximal\nregion that predicts $(d_{1},d_{2})$, given $\\boldsymbol{Z}=\\boldsymbol{z}$.\\footnote{See Section \\ref{subsec:notation} in the Appendix for a formal definition.\nThe figure is drawn in a way that $\\gamma_{1}$ and $\\gamma_{2}$\nare negative.} \n\\begin{figure*}[t]\n\\centering \\begin{subfigure}[t]{0.35\\textwidth} \\centering \\begin{tikzpicture}[scale=0.33]\n\\draw[step=1cm,gray,very thin] (-3,-3) rectangle (5,5); \\draw (-3,-3) node[anchor=north east] {0}; \\draw (5,-3) node[anchor=north] {1}; \\draw (-3,5) node[anchor=east] {1};\n\n\\path [draw=none, fill=gray, opacity=0.3] (3.5,1) rectangle (-3,5); \\path [draw=none, fill=gray, opacity=0.3] (1.7,-3) rectangle (5,3);\n\n\\draw[thick,->] (1.7,3) -- (5,3); \\draw[thick,->] (1.7,3) -- (1.7,-3);  \\draw[thick,->] (3.5,1) -- (-3,1); \\draw[thick,->] (3.5,1) -- (3.5,5);  \n\n\\node [below right, black] at (-3.2,5) {$R_{10}(\\boldsymbol{z})$};\n\\node [below right, black] at (1.4,-0.5) {$R_{01}(\\boldsymbol{z})$};\n\\node [below right, black] at (3.2,5) {$R_{00}(\\boldsymbol{z})$}; \\node [below right, black] at (-3.2,-0.5) {$R_{11}(\\boldsymbol{z})$};\n\n\\draw (-4,6) node[anchor=east] {$U_2$};\n\\draw (6.3,-4) node[anchor=north] {$U_1$};\n\\draw (-3,1) node[anchor=east] {\\small{$\\delta_1+\\gamma_2 z_2$}};\n\\draw (1.7,-3) node[anchor=north] {\\small{$\\delta_2+\\gamma_1 z_1$}};\n\\draw (5,6) node[anchor=east] {\\small{$\\gamma_1 z_1$}};\n\\draw (6.5,3.5) node[anchor=north] {\\small{$\\gamma_2 z_2$}};\n\n\\end{tikzpicture} \\caption{When $\\boldsymbol{Z}=\\boldsymbol{z}$}\n\\end{subfigure\n~ \\begin{subfigure}[t]{0.35\\textwidth} \\centering \\begin{tikzpicture}[scale=0.33]\n\\draw[step=1cm,gray,very thin] (-3,-3) rectangle (5,5); \\draw (-3,-3) node[anchor=north east] {0}; \\draw (5,-3) node[anchor=north] {1}; \\draw (-3,5) node[anchor=east] {1};\n\\path [draw=none, fill=red, opacity=0.3] (1,-1) rectangle (-3,5); \\path [draw=none, fill=red, opacity=0.3] (-1,-3) rectangle (5,0);\n\n\\draw[thick,->] (1.7,3) -- (5,3); \\draw[thick,->] (1.7,3) -- (1.7,-3);  \\draw[thick,->] (3.5,1) -- (-3,1); \\draw[thick,->] (3.5,1) -- (3.5,5); \n\n\\draw[thick,red,dashed,->] (-1,0) -- (5,0); \n\\draw[thick,red,dashed,->] (-1,0) -- (-1,-3);  \n\\draw[thick,red,dashed,->] (1,-1) -- (-3,-1); \n\\draw[thick,red,dashed,->] (1,-1) -- (1,5); \n\n\\node [below right, red] at (-3.2,3) {$R_{10}(\\boldsymbol{z}')$};\n\\node [below right, red] at (0.3,-1) {$R_{01}(\\boldsymbol{z}')$};\n\\node [below right, red] at (1.1,3) {$R_{00}(\\boldsymbol{z}')$}; \\node [below right, red] at (-4,-1) {$R_{11}(\\boldsymbol{z}')$};\n\n\\draw (-4,6) node[anchor=east] {$U_2$};\n\\draw (6.3,-4) node[anchor=north] {$U_1$};\n\\draw[red] (-3,-0.6) node[anchor=east] {\\small{$\\delta_1+\\gamma_2 z'_2$}};\n\\draw[red] (-0.5,-3) node[anchor=north] {\\small{$\\delta_2+\\gamma_1 z'_1$}};\n\\draw[red] (2.3,6) node[anchor=east] {\\small{$\\gamma_1 z'_1$}};\n\\draw[red] (6.5,1) node[anchor=north] {\\small{$\\gamma_2 z'_2$}};\n\n\\end{tikzpicture}\n\n\\caption{When $\\boldsymbol{Z}=\\boldsymbol{z}'$}\n\\end{subfigure}\\caption{Change in Equilibrium Regions with Compensating Strategic Substitutability.\\label{fig:As_EQ-1}}\n\\end{figure*}\n\nThese changes (a)--(e) are a consequence of the monotonic pattern\nof equilibrium regions, which we formally establish in a general setting\nof more than two players in Theorem \\ref{thm:mono_pattern} of Section\n\\ref{subsec:Geometry}.\n\nIn general, besides these five scenarios, there may be markets that\nused to be Delta-monopoly but become United-monopoly and vice versa,\ni.e., markets that exhibit \\textit{non-monotonic} behaviors; see Remark\n\\ref{rem:nonmonotone} below for details. Owing to possible multiple\nequilibria, we are agnostic about these latter types of changes except\nin extreme cases, where one equilibrium is selected with probability\none. We generally do not know the equilibrium selection mechanism\nin play, much less about how such mechanism changes as cost $\\boldsymbol{Z}$\nchanges. The key idea in this study is to overcome the non-monotonicity\nby shifting the cost sufficiently so that there is no market that\nswitches from one monopoly to another. We show that the shift in cost\nthat compensates the strategic substitutability does just that, as\nis depicted in Figure \\ref{fig:As_EQ-1}. In this figure, we assume\n$\\delta_{2}+\\gamma_{1}z_{1}>\\gamma_{1}z_{1}'$ and $\\delta_{1}+\\gamma_{2}z_{2}>\\gamma_{2}z_{2}'$.\nIn other words, we assume \n\\begin{align}\n\\left|\\gamma_{s}(z_{s}'-z_{s})\\right| & \\ge\\left|\\delta_{-s}\\right|\\text{ for }s=1,2.\\label{eq:EQ_S2}\n\\end{align}\nImportantly, we do not require infinite variation in $\\boldsymbol{Z}$.\\footnote{Of course, changing each $Z_{s}$ from $-\\infty$ to $\\infty$ will\ntrivially achieve our requirement of having no market that switches\nfrom one monopoly to another.} In fact, we show that the compensating strategic substitutability\n\\eqref{eq:EQ_S2} is implied by the following condition, which can\nbe tested using the data: there exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$\nsuch that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}'] & >2-\\sqrt 2.\\label{eq:asy2-1}\n\\end{align}\n\nSuppose $\\boldsymbol{z},\\boldsymbol{z}'$ satisfy \\eqref{eq:EQ_S2}.\nThen, by \\eqref{eq:my_IV}, we can derive from \\eqref{eq:h_derive0}\nthat (suppressing $X=x$ for simplicity) \n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}') & =\\Pr[\\epsilon\\leq\\mu_{1}+\\mu_{2},\\boldsymbol{U}\\in\\Delta_{a}\\cup\\Delta_{b}\\cup\\Delta_{e}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq\\mu_{1},\\boldsymbol{U}\\in\\Delta_{a}]+\\Pr[\\epsilon\\leq\\mu_{1},\\boldsymbol{U}\\in\\Delta_{c}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq\\mu_{2},\\boldsymbol{U}\\in\\Delta_{b}]+\\Pr[\\epsilon\\leq\\mu_{2},\\boldsymbol{U}\\in\\Delta_{d}]\\nonumber \\\\\n & -\\Pr[\\epsilon\\leq0,\\boldsymbol{U}\\in\\Delta_{c}\\cup\\Delta_{d}\\cup\\Delta_{e}],\\label{eq:h_derive}\n\\end{align}\nwhere $\\Delta_{i}$ ($i\\in\\{a,...,e\\}$) are disjoint and each $\\Delta_{i}$\ncharacterizes those markets on the margin described above: $\\Delta_{a}$\ncorresponds to the set of $\\boldsymbol{U}$'s that experience (a),\n$\\Delta_{b}$ corresponds to (b), and so on. Once \\eqref{eq:h_derive}\nis derived, it is easy to see that \n\\begin{align}\nsgn\\{h(\\boldsymbol{z},\\boldsymbol{z}',x)\\} & =sgn(\\mu_{1})=sgn(\\mu_{2}),\\label{eq:sign_match}\n\\end{align}\nwhich is formally shown in Lemma \\ref{lem:asy_to_asy_star}(i). See\nSection \\ref{subsec:proof_S=00003D00003D2} in the Appendix for a\nproof in this specific two-player case, which simplifies the argument\nin the general proof. The result \\eqref{eq:sign_match} is helpful\nfor our bound analysis. Again, focus on $E[Y(1,1)|x]$ and suppose\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$. Then, $\\mu_{1}>0$ and $\\mu_{2}>0$,\nand thus, we can derive the lower bound on, e.g., $E[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]$\nin \\eqref{eq:Manski_expand-1} as \n\\begin{align}\nE[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x] & =\\Pr[\\epsilon\\le\\mu_{1}+\\mu_{2}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\nonumber \\\\\n & \\ge\\Pr[\\epsilon\\le\\mu_{1}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\label{eq:ex_tigher_bound-1-1}\\\\\n & =E[Y|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x],\\nonumber \n\\end{align}\nwhich is larger than zero, the previous naive lower bound. Similarly,\nwe can calculate the lower bounds on all $E[Y(1,1)|\\boldsymbol{D}=\\boldsymbol{d},\\boldsymbol{z},x]$\nfor $\\boldsymbol{d}\\neq(1,1)$. Consequently, by \\eqref{eq:Manski_expand-1},\nwe have \n\\begin{align*}\nE[Y(1,1)|x] & \\ge\\Pr[Y=1|\\boldsymbol{z},x],\n\\end{align*}\ni.e., the lower bound on $E[Y(1,1)|x]$ is \n\\begin{align*}\n\\tilde{L}(x) & \\equiv\\sup_{\\boldsymbol{z}}\\Pr[Y=1|\\boldsymbol{z},x].\n\\end{align*}\nNote that $\\tilde{L}(x)\\ge L_{Manski}(x)$. In this case, $\\tilde{U}(x)=U_{Manski}(x)$.\nIn Section \\ref{subsec:Partial-Identification}, we show that $\\tilde{L}(x)$\nand $\\tilde{U}(x)$ are sharp under \\eqref{eq:model_ex1}--\\eqref{eq:mono}.\n\nWe can further tighten the bounds if we have exogenous variables that\nare excluded from the entry decisions, i.e., from the $D_{1}$ and\n$D_{2}$ equations. The existence of such variables is not necessary\nbut helpful in tightening the bounds, and can be motivated by the\nnotion of externalities. That is, there can exist factors that affect\n$Y$ but do not enter the players' first-stage payoff functions. Modify\n\\eqref{eq:my_IV} and assume \n\\begin{align}\n(\\epsilon,U_{1},U_{2}) & \\perp(Z_{1},Z_{2},X),\\label{eq:my_IV2}\n\\end{align}\nwhere conditioning on other (possibly endogenous) covariates is suppressed.\nHere, $X$ can be the characteristics of the local market that\ndirectly affect pollution or health levels, such as weather shocks\nor the share of pollution-related industries in the local economy.\nWe assume that conditional on other covariates, these factors affect\nthe outcome but do not enter the payoff functions, since the airlines\ndo not take them into account in their decisions.\n\nTo exploit the variation in $X$ (in addition to the variation in\n$\\boldsymbol{Z}$), let $(x,\\tilde{x},\\tilde{\\tilde{x}})$ be (possibly\ndifferent) realizations of $X$, and define \n\\begin{equation}\n\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x,\\tilde{x},\\tilde{\\tilde{x}})\\equiv h_{00}(\\boldsymbol{z},\\boldsymbol{z}',x)+h_{10}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{x})+h_{01}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{x})+h_{11}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\tilde{x}}).\\label{eq:h_tilde-1}\n\\end{equation}\nUnder \\eqref{eq:my_IV2} and analogous to \\eqref{eq:sign_match},\nwe can show that if \n\\begin{align}\nsgn\\{\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)\\} & =sgn(-\\mu_{1})=sgn(-\\mu_{2})\\label{eq:sign_match_x}\n\\end{align}\nis positive (negative), then $sgn\\{\\mu_{1}+\\beta(x-x')\\}=sgn\\{\\mu_{2}+\\beta(x-x')\\}$\nis positive (negative). This is formally shown in Lemma \\ref{lem:asy_to_asy_star}(ii).\n\nAs before, suppose $h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$, and thus,\n$\\mu_{1}>0$ and $\\mu_{2}>0$ by \\eqref{eq:sign_match}. Now, if $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$,\nthen $\\mu_{1}+\\beta x<\\beta x'$ and $\\mu_{2}+\\beta x<\\beta x'$.\nTherefore, we can derive \n\\begin{align*}\nE[Y(1,1)|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]= & \\Pr[\\epsilon\\le\\mu_{1}+\\mu_{2}+\\beta x|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x]\\\\\n\\le & \\Pr[\\epsilon\\le\\mu_{1}+\\beta x'|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x']\\\\\n= & \\Pr[Y=1|\\boldsymbol{D}=(1,0),\\boldsymbol{z},x'],\n\\end{align*}\nwhere the second equality also uses \\eqref{eq:my_IV2} and \\eqref{eq:model_ex2}--\\eqref{eq:model_ex3}.\nSimilarly, we have $E[Y(1,1)|\\boldsymbol{D}=(0,1),\\boldsymbol{z},x]\\le\\Pr[Y=1|\\boldsymbol{D}=(0,1),\\boldsymbol{z},x']$,\nand consequently, the upper bound on $E[Y(1,1)\\vert x]$ becomes \n\\[\nU(x)\\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},x']+\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\right\\} \n\\]\nby \\eqref{eq:Manski_expand-1}, and the lower bound is $L(x)=\\tilde{L}(x)$.\nNote that we can further take infimum over $x'$ such that $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$.\n\nTo summarize our illustration, by using two values of $\\boldsymbol{Z}$\nthat satisfy \\eqref{eq:EQ_S2} and $h(\\boldsymbol{z},\\boldsymbol{z}',x)>0$\nand two values of $X$ that satisfy $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x',x',x)<0$,\nour lower and upper bounds, $L(x)$ and $U(x)$, on $E[Y(1,1)\\vert x]$\nachieve \n\\begin{align*}\nL(x) & =\\tilde{L}(x)\\ge L_{Manski}(x),\\\\\nU(x) & \\ge\\tilde{U}(x)=U_{Manski}(x),\n\\end{align*}\nwhere the inequalities are strict if $\\sum_{\\boldsymbol{d}\\neq(1,1)}\\Pr[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x]>0$\nand $\\Pr[Y=0,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},x']>0$.\nWe discuss the sharpness of $L(x)$ and $U(x)$ in the next section.\nSimilarly, we can derive lower and upper bounds on other $E[Y(\\boldsymbol{d})\\vert x]$'s\nfor $\\boldsymbol{d}\\neq(1,1)$, and eventually construct bounds on\nany ATE. The gain from our approach is also exhibited in Figure \\ref{fig:sim1}\nin Section \\ref{sec:Monte-Carlo-Studies}, where we use the same data\ngenerating process as in this section and calculate different bounds\non the ATE, $E[Y(1,1)\\vert x]-E[Y(0,0)\\vert x]$.\n\n\\begin{remark}[Point Identification of the ATE]\\label{rem:Identification-under-Full}\nWhen there exist player-specific excluded instruments with large support,\nwe point identify the ATEs. To invoke an identification-at-infinity\nargument, the following assumptions are instead needed to hold: \n\\begin{align}\n & \\gamma_{1}\\text{ and }\\gamma_{2}\\text{ are nonzero},\\label{eq:infinity1}\\\\\n & Z_{1}|(X,Z_{2})\\text{ and }Z_{2}|(X,Z_{1})\\text{ has an everywhere positive Lebesgue density}.\\label{eq:infinity2}\n\\end{align}\nThese assumptions impose a player-specific exclusion restriction and\nlarge support. Under \\eqref{eq:infinity1}--\\eqref{eq:infinity2},\nwe can easily show that the ATE in \\eqref{eq:ATE} is point identified.\nIn this case, the structure we impose, especially on the outcome function\n(such as the threshold-crossing structure, or more generally Assumption\nM in Section \\ref{subsec:Assumptions} below) is not needed.\n\nThe identification strategy is to exploit the large variation of player\nspecific instruments based on \\eqref{eq:infinity1}--\\eqref{eq:infinity2},\nwhich simultaneously solves the multiple equilibria and the endogeneity\nproblems. For example, to identify $E[Y(1,1)|x]$, consider \n\\begin{align*}\n & E[Y|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]=E[Y(1,1)|\\boldsymbol{D}=(1,1),\\boldsymbol{z},x]\\\\\n & =E[Y(1,1)|\\delta_{2}+\\gamma_{1}z_{1}\\geq U_{1},\\delta_{1}+\\gamma_{2}z_{2}\\geq U_{2},x]\\rightarrow E[Y(1,1)|x],\n\\end{align*}\nwhere the second equation is by \\eqref{eq:my_IV} and $Y(1,1)=\\mu_{1}+\\mu_{2}+\\beta X$,\nand the convergence is by \\eqref{eq:infinity1}--\\eqref{eq:infinity2}\nwith $z_{1}\\rightarrow\\infty$ and $z_{2}\\rightarrow\\infty$. The\nidentification of $E[Y(0,0)|x]$, $E[Y(1,0)|x]$ and $E[Y(0,1)|x]$\ncan be achieved by similar reasoning. Note that $\\boldsymbol{D}=(1,0)$\nor $\\boldsymbol{D}=(0,1)$ can be predicted as an outcome of multiple\nequilibria. However, when either $(z_{1},z_{2})\\rightarrow(\\infty,-\\infty)$\nor $(z_{1},z_{2})\\rightarrow(-\\infty,\\infty)$ occurs, a unique equilibrium\nis guaranteed as a dominant strategy, i.e., $\\boldsymbol{D}=(1,0)$\nor $\\boldsymbol{D}=(0,1)$, respectively.\\end{remark}\n\n\\begin{remark}[Non-Monotonicity of Treatment Selection]\\label{rem:nonmonotone}In\nthe case of a single binary treatment, the standard selection equation\nexhibits monotonicity that facilitates various identification strategies\n(e.g., \\citet{imbens1994identification}, \\citet{heckman2005structural},\n\\citet{VY07} to name a few). Relatedly, \\citet{vytlacil2002independence}\nshows the equivalence between imposing the selection equation with\nthreshold-crossing structure and assuming the local ATE (LATE) monotonicity.\nThis equivalence (and thus, previous identification strategies) is\ninapplicable to our setting due to the simultaneity in the first stage\n\\eqref{eq:main_model2}. To formally state this, let $\\boldsymbol{D}(\\boldsymbol{z})$\nbe a potential treatment vector, had $\\boldsymbol{Z}=\\boldsymbol{z}$\nbeen realized. When cost $\\boldsymbol{Z}=(Z_{1},Z_{2})$ increases\nfrom $\\boldsymbol{z}$ to $\\boldsymbol{z}'$, it may be that some\nmarkets witness Delta entering and United going out of business (i.e.,\n$\\boldsymbol{D}(\\boldsymbol{z})=(0,1)$ and $\\boldsymbol{D}(\\boldsymbol{z}')=(1,0)$),\nwhile other markets witness the opposite (i.e., $\\boldsymbol{D}(\\boldsymbol{z})=(1,0)$\nand $\\boldsymbol{D}(\\boldsymbol{z}')=(0,1)$). The direction of monotonicity\nis reversed in the two groups of markets, and thus, $\\Pr[\\boldsymbol{D}(\\boldsymbol{z})\\ge\\boldsymbol{D}(\\boldsymbol{z}')]\\neq1$\nand $\\Pr[\\boldsymbol{D}(\\boldsymbol{z})\\le\\boldsymbol{D}(\\boldsymbol{z}')]\\neq1$\nwhere the inequality for vectors is pair-wise inequalities, which\nviolates the LATE monotonicity.\\footnote{The same argument applies with a scalar multi-valued treatment $\\tilde{D}\\in\\{1,2,3,4\\}$,\nwhich has a one-to-one map with $\\boldsymbol{D}\\in\\{(0,0),(0,1),(1,0),(1,1)\\}$.\nThen, some markets can experience $\\tilde{D}(\\boldsymbol{z})=2$ and\n$\\tilde{D}(\\boldsymbol{z}')=3$ while others experience $\\tilde{D}(\\boldsymbol{z})=3$\nand $\\tilde{D}(\\boldsymbol{z}')=2$, and thus, it is possible to have\n$\\Pr[\\tilde{D}(\\boldsymbol{z})\\ge\\tilde{D}(\\boldsymbol{z}')]\\neq1$\nand $\\Pr[\\tilde{D}(\\boldsymbol{z})\\le\\tilde{D}(\\boldsymbol{z}')]\\neq1$.} Despite this non-monotonic pattern, Theorem \\ref{thm:mono_pattern}\nbelow restores generalized monotonicity, i.e., monotonicity in terms of\nthe algebra of sets. This generalized monotonicity, combined with\nthe compensating strategic substitutability \\eqref{eq:EQ_S2}, allows\nus to use a strategy analogous to the single-treatment case for our\nbound analysis. This also suggests that we can introduce a generalized\nversion of the LATE parameter in the current framework, although we\ndo not pursue it in this study.\n\nRelated to our study, \\citet{lee2016identifying} introduce a framework\nfor treatment effects with general non-monotonicity of selection,\nand consider the simultaneous treatment selection as one of the examples.\nAlthough they engage in a similar discussion on non-monotonicity,\ntheir approach to gain tractability for identification is different\nfrom ours. When they allow the identity of players being observed\nas in our setting, they show that their treatment measurability condition\n(Assumption 2.1) introduced to restore monotonicity is satisfied,\nprovided they assume a threshold-crossing equilibrium selection mechanism.\nIn contrast, we avoid making assumptions on equilibrium selection,\nbut require compensating variation of instruments. In addition, for\nthis particular example, they assume the first-stage is known (i.e.,\npayoff functions are known), and focus on point identification of\nthe MTE with continuous instruments.\\end{remark}\n\n\\section{General Theory\\label{sec:General-Theory}}\n\n\\subsection{Setup\\label{subsec:Model}}\n\nLet $\\boldsymbol{D}\\equiv(D_{1},...,D_{S})\\in\\mathcal{D}\\subseteq\\{0,1\\}^{S}$\nbe an $S$-vector of observed binary treatments and $\\boldsymbol{d}\\equiv(d_{1},...,d_{S})$\nbe its realization, where $S$ is fixed. We assume that $\\boldsymbol{D}$\nis predicted as a pure strategy Nash equilibrium of a complete information\ngame with $S$ players who make entry decisions or individuals who\nchoose to receive treatments.\\footnote{While this study does not consider mixed strategy equilibria, it may\nbe possible to extend the setup to incorporate mixed strategies, following\nthe argument in \\citet{CT09}.} Let $Y$ be an observed post-game outcome that results from profile\n$\\boldsymbol{D}$ of endogenous treatments. It can be an outcome common\nto all players or an outcome specific to each player. Let $(X,Z_{1},...,Z_{S})$\nbe observed exogenous covariates. We consider a model of a semi-triangular\nsystem: \n\\begin{align}\nY & =\\theta(\\boldsymbol{D},X,\\epsilon_{\\boldsymbol{D}}),\\label{eq:main_model1}\\\\\nD_{s} & =1\\left[\\nu^{s}(\\boldsymbol{D}_{-s},Z_{s})\\geq U_{s}\\right],\\mbox{\\qquad}s\\in\\{1,...,S\\},\\label{eq:main_model2}\n\\end{align}\nwhere $s$ is indices for players or interchangeably for treatments,\nand $\\boldsymbol{D}_{-s}\\equiv(D_{1},...,D_{s-1},D_{s+1},...,D_{S})$.\nWithout loss of generality, we normalize the scalar $U_{s}$ to be\ndistributed as $Unif(0,1)$, and $\\nu^{s}:\\mathbb{R}^{S-1+d_{z_{s}}}\\rightarrow(0,1]$\nand $\\theta:\\mathbb{R}^{S+d_{x}+d_{\\epsilon}}\\rightarrow\\mathbb{R}$\nare unknown functions that are nonseparable in their arguments. We\nallow the unobservables $(\\epsilon_{\\boldsymbol{D}},U_{1},...,U_{S})$\nto be arbitrarily dependent on one another. Although the notation\nsuggests that the instruments $Z_{s}$'s are player\/treatment-specific,\nthey are not necessarily required to be so for the analyses in this\nstudy; see Appendix \\ref{subsec:Common_Z} for a discussion. The exogenous\nvariables $X$ are variables excluded from all the equations for $D_{s}$.\nThe existence of $X$ is not necessary but useful for the bound analysis\nof the ATE. There may be covariates $W$ common to all the equations\nfor $Y$ and $D_{s}$, which is suppressed for succinctness. Implied\nfrom the complete information game, player $s$'s decision $D_{s}$\ndepends on the decisions of all others $\\boldsymbol{D}_{-s}$ in $\\mathcal{D}_{-s}$,\nand thus, $\\boldsymbol{D}$ is determined by a simultaneous system.\nAs before, the model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\nis incomplete because of the simultaneity in the first stage, and\nthe conventional monotonicity in the sense of \\citet{imbens1994identification}\nis not exhibited in the selection process because of simultaneity.\nThe unit of observation, a market or geographical region, is indexed\nby $i$ and is suppressed in all the expressions.\n\nThe potential outcome of receiving treatments $\\boldsymbol{D}=\\boldsymbol{d}$\ncan be written as \n\\begin{align*}\nY(\\boldsymbol{d}) & =\\theta(\\boldsymbol{d},X,\\epsilon_{\\boldsymbol{d}}),\\mbox{\\qquad}\\boldsymbol{d}\\in\\mathcal{D},\n\\end{align*}\nand $\\epsilon_{\\boldsymbol{D}}=\\sum_{\\boldsymbol{d}\\in\\mathcal{D}}1[\\boldsymbol{D}=\\boldsymbol{d}]\\epsilon_{\\boldsymbol{d}}$.\nWe are interested in the ATE and related parameters. Using the average\nstructural function (ASF) $E[Y(\\boldsymbol{d})|x]$, the ATE can be\nwritten as \n\\begin{align}\nE[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}^{\\prime})|x] & =E[\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})-\\theta(\\boldsymbol{d}^{\\prime},x,\\epsilon_{\\boldsymbol{d}^{\\prime}})],\\label{eq:ATE}\n\\end{align}\nfor $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$. Another parameter\nof interest is the average treatment effects on the treated or the untreated:\n$E[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}^{\\prime})|D=\\boldsymbol{d}'',z,x]$\nfor $\\boldsymbol{d}''\\in\\{ \\boldsymbol{d},\\boldsymbol{d}' \\}$.\\footnote{Technically, $\\boldsymbol{d}''$ does not necessarily have to be equal\nto $\\boldsymbol{d}$ or $\\boldsymbol{d}'$, but can take another value.} One might also be interested\nin the sign of the ATE, which in this multi-treatment case is essentially\nestablishing an ordering among the ASF's.\n\nAs an example of the ATE, we may choose $\\boldsymbol{d}=(1,...,1)$\nand $\\boldsymbol{d}'=(0,...,0)$ to measure the cancelling-out effect\nor more general nonlinear effects. Another example would be choosing\n$\\boldsymbol{d}=(1,\\boldsymbol{d}_{-s})$ and $\\boldsymbol{d}'=(0,\\boldsymbol{d}_{-s})$\nfor given $\\boldsymbol{d}_{-s}$, where we use the notation $\\boldsymbol{d}=(d_{s},\\boldsymbol{d}_{-s})$\nby switching the order of the elements for convenience. Sometimes,\nwe instead want to focus on learning about complementarity between\ntwo treatments, while averaging over the remaining $S-2$ treatments.\nThis can be examined in a more general framework of defining the ASF\nand ATE by introducing a partial potential outcome; this is discussed\nin Appendix \\ref{subsec:Partial-ATE}.\n\nIn identifying these treatment parameters, suppose we attempt to recover\nthe effect of a single treatment $D_{s}$ in model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\n\\textit{conditional on} $\\boldsymbol{D}_{-s}=\\boldsymbol{d}_{-s}$,\nand then recover the effects of multiple treatments by transitively\nusing these effects of single treatments. This strategy is not valid\nsince $\\boldsymbol{D}_{-s}$ is a function of $D_{s}$ and also because\nof multiplicity. Therefore, the approaches in the literature with\nsingle-treatment, single-agent triangular models are not directly\napplicable and a new theory is necessary in this more general setting.\n\n\\subsection{Monotonicity in Equilibria\\label{subsec:Geometry}}\n\nAs an important step in the analyses in this study, we establish that\nthe equilibria of the treatment selection process in the first-stage\ngame present a monotonic pattern when the instruments move. Specifically,\nwe consider the regions in the space of the unobservables that predict\nequilibria and establish a monotonic pattern of these regions in terms\nof instruments. The analytical characterization of the equilibrium\nregions when there are more than two players ($S>2$) can generally\nbe complicated (\\citet[p. 1800]{CT09}); however, under a mild uniformity\nassumption (Assumption M1), our result is obtained under strategic\nsubstitutability. Let $\\mathcal{Z}_{s}$ be the support of $Z_{s}$.\nWe make the following assumptions on the first-stage nonparametric\npayoff function for each $s\\in\\{1,...,S\\}$.\n\n\\begin{asSS}For every $z_{s}\\in\\mbox{\\ensuremath{\\mathcal{Z}}}_{s}$,\n$\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$ is strictly decreasing in each\nelement of $\\boldsymbol{d}_{-s}$.\\end{asSS}\n\n\\begin{asM1}For any given $z_{s},z_{s}'\\in\\mathcal{Z}_{s}$, either\n$\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})\\geq\\nu^{s}(\\boldsymbol{d}_{-s},z_{s}')$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$, or $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})\\leq\\nu^{s}(\\boldsymbol{d}_{-s},z_{s}')$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$.\\end{asM1}\n\nAssumption SS asserts that the agents' treatment decisions are produced\nin a game with strategic substitutability. The strictness of the monotonicity\nis not important for our purpose but convenient in making statements\nabout the equilibrium regions. In the language of \\citet{CT09}, we\nallow for heterogeneity in the \\textit{fixed competitive effects}\n(i.e., how each of other entrants affects one's payoff), as well as\nheterogeneity in how each player is affected by other entrants, which\nis ensured by the nonseparability between $\\boldsymbol{d}_{-s}$ and\n$z_{s}$ in $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$; this heterogeneity\nis related to the \\textit{variable competitive effects}. Assumption\nM1 is required in this multi-agent setting, and the uniformity is\nacross $\\boldsymbol{d}_{-s}$. Note that this assumption is weaker\nthan a conventional monotonicity assumption that $\\nu^{s}(\\boldsymbol{d}_{-s},z_{s})$\nis either non-decreasing or non-increasing in $z_{s}$ for all $\\boldsymbol{d}_{-s}$.\nAssumption M1 is justifiable, especially when $z_{s}$ is chosen to\nbe of the same kind for all players. For example, in an entry game,\nif $z_{s}$ is chosen to be each player's cost shifters, the payoffs\nwould decrease in their costs for any given opponents.\n\nAs the first main result of this study, we establish the geometric\nproperty of the equilibrium regions. For $j=0,...,S$, let $\\boldsymbol{R}_{j}(\\boldsymbol{z})\\subset\\mathcal{U}\\equiv(0,1]^{S}$\ndenote the region that predicts all equilibria with $j$ treatments\nselected or $j$ entrants, defined as a subset of the space of the\nentry unobservables $\\boldsymbol{U}\\equiv(U_{1},...,U_{S})$; see\nSection \\ref{subsec:notation} in the Appendix for a formal definition.\nThen, define the region of all equilibria with \\textit{at most} $j$\nentrants as \n\\begin{align*}\n\\boldsymbol{R}^{\\le j}(\\boldsymbol{z}) & \\equiv\\bigcup_{k=0}^{j}\\boldsymbol{R}_{k}(\\boldsymbol{z}).\n\\end{align*}\nAlthough this region is hard to express explicitly in general, it\nhas a simple feature that serves our purpose. For given $j$, choose\n$z_{s},z_{s}'\\in\\mathcal{Z}_{s}$ such that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=(1,...,1)|\\boldsymbol{Z}=(z_{s},\\boldsymbol{z}_{-s})] & >\\Pr[\\boldsymbol{D}=(1,...,1)|\\boldsymbol{Z}=(z_{s}',\\boldsymbol{z}_{-s})]\\label{eq:ps_condi}\n\\end{align}\nfor all $s$. This condition is to merely fix $\\boldsymbol{z},\\boldsymbol{z}'$\nthat change the \\textit{joint propensity score}, and the direction\nof change is without loss of generality. Such $\\boldsymbol{z},\\boldsymbol{z}'$\nexist by the relevance of the instruments, which is assumed below.\nLet $\\mathcal{Z}$ be the support of $\\boldsymbol{Z}\\equiv(Z_{1},...,Z_{S})$.\n\n\\begin{theorem}\\label{thm:mono_pattern}Under Assumptions SS and\nM1 and for $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$ that satisfy\n\\eqref{eq:ps_condi}, we have \n\\begin{equation}\n\\boldsymbol{R}^{\\le j}(\\boldsymbol{z})\\subseteq\\boldsymbol{R}^{\\le j}(\\boldsymbol{z}')\\text{ }\\forall j.\\label{eq:R^j(z)_vs_R^j(z')}\n\\end{equation}\n\n\\end{theorem}\n\nTheorem \\ref{thm:mono_pattern} establishes a generalized version\nof monotonicity in the treatment selection process. This theorem plays\na crucial role in calculating the bounds on the treatment parameters\nand in showing the sharpness of the bounds. Relatedly, \\citet{berry1992estimation}\nderives the probability of the event that the number of entrants is\nless than a certain value, which can be written as $\\Pr[\\boldsymbol{U}\\in\\boldsymbol{R}^{\\le j}(\\boldsymbol{z})]$\nusing our notation. However, his result is not sufficient for our\nstudy and relies on stronger assumptions, such as restricting the\npayoff functions to only depend on the number of opponents.\n\n\\subsection{Main Assumptions\\label{subsec:Assumptions}}\n\nTo characterize the bounds on the treatment parameters, we make the\nfollowing assumptions. Unless otherwise noted, the assumptions hold\nfor each $s\\in\\{1,...,S\\}$.\n\n\\begin{asIN}\\label{as:IN}$(X,\\boldsymbol{Z})\\perp(\\epsilon_{\\boldsymbol{d}},\\boldsymbol{U})$\n$\\forall\\boldsymbol{d}\\in\\mathcal{D}$.\\end{asIN}\n\n\\begin{asE} The distribution of $(\\epsilon_{\\boldsymbol{d}},\\boldsymbol{U})$\nhas strictly positive density with respect to Lebesgue measure on\n$\\mathbb{R}^{S+1}$ $\\forall\\boldsymbol{d}\\in\\mathcal{D}$. \\end{asE}\n\n\\begin{asEX}For each $\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$, $\\nu^{s}(\\boldsymbol{d}_{-s},Z_{s})|X$\nis nondegenerate.\\end{asEX}\n\nAssumptions IN, EX and all the following analyses can be understood\nas conditional on $W$, the common covariates in $X$ and $\\boldsymbol{Z}=(Z_{1},...,Z_{S})$.\nAssumption EX is related to the exclusion restriction and the relevance\ncondition of the instruments $Z_{s}$.\n\nWe now impose a shape restriction on the outcome function $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})$\nvia restrictions on \n\\[\n\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})\\equiv E[\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})|\\boldsymbol{U}=\\boldsymbol{u}]\n\\]\na.e. $\\boldsymbol{u}$. This restriction on the conditional mean is\nweaker than the one directly imposed on $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})$.\nLet $\\mathcal{X}$ be the support of $X$. Recall that we use the\nnotation $\\boldsymbol{d}=(d_{s},\\boldsymbol{d}_{-s})$ by switching\nthe order of the elements for convenience.\n\n\\begin{asM}For every $x\\in\\mathcal{X}$, either $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\geq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$\n$\\forall s$ or $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\leq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$\n$\\forall s$. Also, $Y\\in[\\underline{Y},\\overline{Y}]$.\\end{asM}\n\nAssumption M holds in, but is not restricted to, the leading case\nof binary $Y$ with a threshold crossing model that satisfies uniformity.\n\n\\begin{asM2}(i) $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})=1[\\mu(\\boldsymbol{d},x)\\geq\\epsilon_{\\boldsymbol{d}}]$\nwhere $\\epsilon_{\\boldsymbol{d}}$ is scalar and $F_{\\epsilon_{\\boldsymbol{d}}|\\boldsymbol{U}}=F_{\\epsilon_{\\boldsymbol{d}'}|\\boldsymbol{U}}$\nfor any $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$; (ii) for\nevery $x\\in\\mathcal{X}$, either $\\mu(1,\\boldsymbol{d}_{-s},x)\\geq\\mu(0,\\boldsymbol{d}_{-s},x)$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$ $\\forall s$ or $\\mu(1,\\boldsymbol{d}_{-s},x)\\le\\mu(0,\\boldsymbol{d}_{-s},x)$\n$\\forall\\boldsymbol{d}_{-s}\\in\\mathcal{D}_{-s}$ $\\forall s$.\\end{asM2}\n\nAssumption M$^{*}$ implies Assumption M. The second statement in\nAssumption M is satisfied with binary $Y$.\\footnote{Another example would be when $Y\\in[0,1]$, as in Example \\ref{example3}.}\nThe first statement in Assumption M can be stated in two parts, corresponding\nto (i) and (ii) of Assumption M$^{*}$: (a) for every $x$ and $\\boldsymbol{d}_{-s}$,\neither $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\geq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$, or $\\vartheta(1,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})\\leq\\vartheta(0,\\boldsymbol{d}_{-s},x;\\boldsymbol{u})$\na.e. $\\boldsymbol{u}$; (b) for every $x$, each inequality statement\nin (a) holds for all $\\boldsymbol{d}_{-s}$. For an outcome function\nwith a scalar index, $\\theta(\\boldsymbol{d},x,\\epsilon_{\\boldsymbol{d}})=\\tilde{\\theta}(\\mu(\\boldsymbol{d},x),\\epsilon_{\\boldsymbol{d}})$,\npart (a) is implied by $\\epsilon_{\\boldsymbol{d}}=\\epsilon_{\\boldsymbol{d}'}=\\epsilon$\n(or more generally, $F_{\\epsilon_{\\boldsymbol{d}}|\\boldsymbol{U}}=F_{\\epsilon_{\\boldsymbol{d}'}|\\boldsymbol{U}}$)\nfor any $\\boldsymbol{d},\\boldsymbol{d}'\\in\\mathcal{D}$ and $E[\\tilde{\\theta}(t,\\epsilon_{\\boldsymbol{d}})|\\boldsymbol{U}=\\boldsymbol{u}]$\nbeing strictly increasing (decreasing) in $t$ a.e. $\\boldsymbol{u}$.\\footnote{A single-treatment version of the latter assumption appears in \\citet{VY07}\n(Assumption A-4), which is weaker than assuming $\\tilde{\\theta}(t,\\epsilon)$\nis strictly increasing (decreasing) a.e. $\\epsilon$; see \\citet{VY07}\nfor related discussions.} Functions that satisfy the latter assumption include strictly monotonic\nfunctions, such as transformation models $\\tilde{\\theta}(t,\\epsilon)=r(t+\\epsilon)$\nwith $r(\\cdot)$ being possibly unknown strictly increasing, or their\nspecial case $\\tilde{\\theta}(t,\\epsilon)=t+\\epsilon$, allowing continuous\ndependent variables; and functions that are not strictly monotonic,\nsuch as models for limited dependent variables, $\\tilde{\\theta}(t,\\epsilon)=1[t\\ge\\epsilon]$\nor $\\tilde{\\theta}(t,\\epsilon)=1[t\\ge\\epsilon](t-\\epsilon)$. However,\nthere can be functions that violate the latter assumption but satisfy\npart (a). For example, consider a threshold crossing model with a\nrandom coefficient: $\\theta(\\boldsymbol{d},x,\\epsilon)=1[\\phi(\\epsilon)\\boldsymbol{d}\\beta^{\\top}\\geq x\\gamma^{\\top}]$,\nwhere $\\phi(\\epsilon)$ is nondegenerate. When $\\beta_{s}\\geq0$,\nthen $E[\\theta(1,\\boldsymbol{d}_{-s},x,\\epsilon)-\\theta(0,\\boldsymbol{d}_{-s},x,\\epsilon)|\\boldsymbol{U}=\\boldsymbol{u}]=\\Pr\\left[\\frac{x\\gamma^{\\top}}{\\beta_{s}+\\boldsymbol{d}_{-s}\\beta_{-s}^{\\top}}\\leq\\phi(\\epsilon)\\leq\\frac{x\\gamma^{\\top}}{\\boldsymbol{d}_{-s}\\beta_{-s}^{\\top}}|\\boldsymbol{U}=\\boldsymbol{u}\\right]$,\nand thus, nonnegative a.e. $\\boldsymbol{u}$, and vice versa. Part\n(a) also does not impose any monotonicity of $\\theta$ in $\\epsilon_{\\boldsymbol{d}}$\n(e.g., $\\epsilon_{\\boldsymbol{d}}$ can be a vector).\n\nPart (b) of Assumption M imposes uniformity, as we deal with more\nthan one treatment. Uniformity is required across different values\nof $\\boldsymbol{d}_{-s}$ and $s$. For instance, in the empirical\napplication of this study, this assumption seems reasonable, since\nan airline's entry is likely to increase the expected pollution regardless\nof the identity or the number of existing airlines. On the other hand,\nin Example \\ref{example3} in the Appendix regarding media and political\nbehavior, this assumption may rule out the ``over-exposure'' effect\n(i.e., too much media exposure diminishes the incumbent's chance\nof being re-elected). In any case, knowledge on the direction of the\nmonotonicity is not necessary in this assumption, unlike \\citet{manski1997monotone}\nor \\citet{Man13}, where the semi-monotone treatment response is assumed\nfor possible multiple treatments.\n\nLastly, we require that there exists variation in $\\boldsymbol{Z}$\nthat offsets the effect of strategic substitutability. Similar as\nbefore, using the notation $\\boldsymbol{d}_{-s}=(d_{s'},\\boldsymbol{d}_{-(s,s')})$\nwhere $\\boldsymbol{d}_{-(s,s')}$ is $\\boldsymbol{d}$ without $s$-th\nand $s'$-th elements, note that Assumption SS can be restated as\n$\\nu^{s}(0,\\boldsymbol{d}_{-(s,s')},z_{s})>\\nu^{s}(1,\\boldsymbol{d}_{-(s,s')},z_{s})$\nfor every $z_{s}$. Given this, we assume the following.\n\n\\begin{asEQ}There exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$,\nsuch that $\\nu^{s}(0,\\boldsymbol{d}_{-(s,s')},z_{s}')\\le\\nu^{s}(1,\\boldsymbol{d}_{-(s,s')},z_{s})$\n$\\forall\\boldsymbol{d}_{-(s,s')}$ $\\forall s,s'$.\\end{asEQ}\n\nFor example, in an entry game with $Z_{s}$ being cost shifters, Assumption\nEQ may hold with $z_{s}'>z_{s}$ $\\forall s$. In this example, players\nmay become less profitable with an increase in cost from government\nregulation. In particular, players' decreased profits cannot be overturned\nby the market being less competitive, as one player is absent due\nto unprofitability. Recall that Assumption EQ is illustrated in Figure\n\\ref{fig:As_EQ-1} with $\\nu^{s}(0,z_{s}')=\\gamma_{s}z_{s}'<\\nu^{s}(1,z_{s})=\\delta_{-s}+\\gamma_{s}z_{s}$\nfor $s=1,2$. Assumption EQ is key for our analysis. To see this,\nlet $R_{j}^{M}(\\cdot)$ denote the region that predicts multiple equilibria\nwith $j$ treatments selected or $j$ entrants. In the proof of a\nlemma that follows, we show that Assumption EQ holds if and only if\n$R_{j}^{M}(\\boldsymbol{z})\\cap R_{j}^{M}(\\boldsymbol{z}')=\\emptyset$.\nThat is, we can at least ensure that there is no market where firms'\ndecisions change from one realization of multiple equilibria to another\nrealization of multiple equilibria with the same number of entrants.\nTo the extent of our analysis, this liberates us from concerns about\nthe regions of multiple equilibria and about a possible change in\nequilibrium selection when changing $\\boldsymbol{Z}$.\\footnote{In Section \\ref{subsec:Group}, we discuss an assumption, partial\nconditional symmetry, which can be imposed alternative to Assumption\nEQ.} Assumption EQ has a simple testable sufficient condition, provided\nthat the unobservables in the payoffs are mutually independent.\n\n\\begin{asEQ2}There exist $\\boldsymbol{z},\\boldsymbol{z}'\\in\\mathcal{Z}$,\nsuch that \n\\begin{align}\n\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j-2}|\\boldsymbol{z}'] & >2-\\sqrt{2}.\\label{eq:EQ_suff}\n\\end{align}\nfor all $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}$, $\\boldsymbol{d}^{j-2}\\in\\mathcal{D}^{j-2}$\nand $2\\le j\\le S$.\n\n\\end{asEQ2}\n\nWhen $S=2$, the condition is stated as $\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}']>2-\\sqrt{2}$.\nAs is detailed in the proof, this essentially restricts the sum of\nradii of two circular isoquant curves to be less than the length of\nthe diagonal of $\\mathcal{U}$: $(1-\\Pr[\\boldsymbol{D}=(1,1)|\\boldsymbol{z}])+(1-\\Pr[\\boldsymbol{D}=(0,0)|\\boldsymbol{z}'])<\\sqrt{2}$.\nThis ensures the required variation in Assumption EQ.\n\n\\begin{lemma}\\label{lem:asy_to_asy_star}Under Assumptions SS, M1,\nand $U_{s}\\perp U_{t}$ for all $s\\neq t$, Assumption EQ$^{*}$ implies\nAssumption EQ.\n\n\\end{lemma}\n\nThe mutual independence of $U_{s}$'s (conditional on $W$) is useful\nin inferring the relationship between players' interaction and instruments\nfrom the observed choices of players. The intuition for the sufficiency\nof Assumption EQ$^{*}$ is as follows. As long as there is no dependence\nin unobserved types, \\eqref{eq:EQ_suff} dictates that the variation\nof $\\boldsymbol{Z}$ is large enough to offset strategic substitutability,\nbecause otherwise, the payoffs of players cannot move in the same\ndirection, and thus, will not result in the same decisions. The requirement\nof $\\boldsymbol{Z}$ variation in \\eqref{eq:EQ_suff} is significantly\nweaker than the large support assumption invoked for an identification\nat infinity argument to overcome the problem of multiple equilibria.\n\n\\subsection{Partial Identification of the ATE\\label{subsec:Partial-Identification}}\n\nUnder the above assumptions, we now present a generalized version\nof the sign matching results \\eqref{eq:sign_match} and \\eqref{eq:sign_match_x}\nin Section \\ref{sec:stylized_ex}. We need to introduce additional\nnotation. Let $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}$ denote an equilibrium\nprofile with $j$ treatments selected or $j$ entrants, i.e., a vector\nof $j$ ones and $S-j$ zeros, where $\\mathcal{D}^{j}$ is a set of\nall equilibrium profiles with $j$ treatments selected. For realizations\n$x$ of $X$ and $\\boldsymbol{z},\\boldsymbol{z}'$ of $\\boldsymbol{Z}$,\ndefine \n\\begin{align}\nh(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv E[Y|\\boldsymbol{z},x]-E[Y|\\boldsymbol{z}',x],\\label{eq:h(zzx)}\\\\\nh_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x) & \\equiv E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]\\nonumber \\\\\n & -E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z}',x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}'].\\label{eq:hj}\n\\end{align}\nSince $\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\cdot]=1$,\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)=\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}}h_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x)$.\nLet $\\tilde{\\boldsymbol{x}}=(x_{0},...,x_{S})$ be an $(S+1)$-dimensional\narray of (possibly different) realizations of $X$, i.e., each $x_{j}$\nfor $j=0,...,S$ is a realization of $X$, and define \n\\[\n\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})\\equiv\\sum_{j=0}^{S}\\sum_{\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}}h_{\\boldsymbol{d}^{j}}(\\boldsymbol{z},\\boldsymbol{z}',x_{j}).\n\\]\nFor $1\\le k\\le j$, define a \\textit{reduction} of $\\boldsymbol{d}^{j}=(d_{1}^{j},...,d_{S}^{j})$\nas $\\boldsymbol{d}^{j-k}=(d_{1}^{j-k},...,d_{S}^{j-k})$, such that\n$d_{s}^{j-k}\\le d_{s}^{j}$ $\\forall s$. Symmetrically, for $1\\le k\\le S-j$,\ndefine an \\textit{extension} of $\\boldsymbol{d}^{j}$ as $\\boldsymbol{d}^{j+k}=(d_{1}^{j+k},...,d_{S}^{j+k})$,\nsuch that $d_{s}^{j+k}\\ge d_{s}^{j}$ $\\forall s$. For example, given\n$\\boldsymbol{d}^{2}=(1,1,0)$, a reduction $\\boldsymbol{d}^{1}$ is\neither $(1,0,0)$ or $(0,1,0)$ but not $(0,0,1)$, a reduction $\\boldsymbol{d}^{0}$\nis $(0,0,0)$, and an extension $\\boldsymbol{d}^{3}$ is $(1,1,1)$.\nLet $\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$ and $\\mathcal{D}^{>}(\\boldsymbol{d}^{j})$\nbe the set of all reductions and extensions of $\\boldsymbol{d}^{j}$,\nrespectively, and let $\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})\\equiv\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\{\\boldsymbol{d}^{j}\\}$\nand $\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})\\equiv\\mathcal{D}^{>}(\\boldsymbol{d}^{j})\\cup\\{\\boldsymbol{d}^{j}\\}$.\nRecall $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})\\equiv E[\\theta(\\boldsymbol{d},x,\\epsilon)|\\boldsymbol{U}=\\boldsymbol{u}]$.\nNow, we state the main lemma of this section.\n\n\\begin{lemma}\\label{lem:sign_match_gen}In model \\eqref{eq:main_model1}--\\eqref{eq:main_model2},\nsuppose Assumptions SS, M1, IN, E, EX, and M hold, and $h(\\boldsymbol{z},\\boldsymbol{z}',x)$\nand $h(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$ are\nwell-defined. For $\\boldsymbol{z},\\boldsymbol{z}'$ such that \\eqref{eq:ps_condi}\nand Assumption EQ hold, and for $j=1,...,S$, it satisfies that\\\\\n (i) $sgn\\{h(\\boldsymbol{z},\\boldsymbol{z}',x)\\}=sgn\\left\\{ \\vartheta(\\boldsymbol{d}^{j},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}^{j-1},x;\\boldsymbol{u})\\right\\} $\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$;\\\\\n (ii) for $\\iota\\in\\{-1,0,1\\}$, if $sgn\\{\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})\\}=sgn\\{-\\vartheta(\\boldsymbol{d}^{k},x_{k};\\boldsymbol{u})+\\vartheta(\\boldsymbol{d}^{k-1},x_{k-1};\\boldsymbol{u})\\}=\\iota$\n$\\forall\\boldsymbol{d}^{k-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{k})$\n$\\forall k\\neq j$ ($k\\ge1$), then $sgn\\{\\vartheta(\\boldsymbol{d}^{j},x_{j};\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}^{j-1},x_{j-1};\\boldsymbol{u})\\}=\\iota$\na.e. $\\boldsymbol{u}$ $\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$.\\end{lemma}\n\nParts (i) and (ii) parallel \\eqref{eq:sign_match} and \\eqref{eq:sign_match_x},\nrespectively. Using Lemma \\ref{lem:sign_match_gen}, we can learn\nabout the ATE. First, note that the sign of the ATE is identified\nby Lemma \\ref{lem:sign_match_gen}(i), since $E[Y(\\boldsymbol{d})|x]=E[\\vartheta(\\boldsymbol{d},x;\\boldsymbol{U})]$.\nNext, we establish the bounds on $E[Y(\\boldsymbol{d}^{j})|x]$ for\ngiven $\\boldsymbol{d}^{j}$ for some $j=0,...,S$.\n\nWe first present the bounds using the variation in $\\boldsymbol{Z}$\nonly, i.e., by using Lemma \\ref{lem:sign_match_gen}(i). To this end,\nwe fix $X=x$ and suppress it in all relevant expressions. To gain\nefficiency we define the integrated version of $h$ as \n\\begin{align}\nH(x) & \\equiv E\\left[h(\\boldsymbol{Z},\\boldsymbol{Z}',x)\\left|(\\boldsymbol{Z},\\boldsymbol{Z}')\\in\\mathcal{Z}_{EQ,j}\\text{ }\\forall j=0,...,S-1\\right.\\right],\\label{eq:H}\n\\end{align}\nwhere $\\mathcal{Z}_{EQ,j}$ is the set of $(\\boldsymbol{z},\\boldsymbol{z}')$\nthat satisfy \\eqref{eq:ps_condi} and Assumption EQ given $j$, and\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)=0$ whenever it is not well-defined.\nWe focus on the case $H(x)>0$; $H(x)<0$ is symmetric and $H(x)=0$\nis straightforward. Using Lemma \\ref{lem:sign_match_gen}(i), one\ncan readily show that $L_{\\boldsymbol{d}^{j}}(x)\\le E[Y(\\boldsymbol{d}^{j})|x]\\le U_{\\boldsymbol{d}^{j}}(x)$\nwith \n\\begin{align}\nU_{\\boldsymbol{d}^{j}}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]\\Biggr\\},\\label{eq:U_dj}\\\\\nL_{\\boldsymbol{d}^{j}}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})|\\boldsymbol{z},x]\\Biggr\\}.\\label{eq:L_dj}\n\\end{align}\nWe can simplify these bounds and show that they are sharp under the\nfollowing assumption.\n\n\\begin{asC}(i) $\\mu_{\\boldsymbol{d}}(\\cdot)$ and $\\nu_{\\boldsymbol{d}_{-s}}(\\cdot)$\nare continuous; (ii) $\\mathcal{Z}$ is compact.\\end{asC}\n\nUnder Assumption C, for given $\\boldsymbol{d}^{j}$, there exist vectors\n$\\bar{\\boldsymbol{z}}\\equiv(\\bar{z}_{1},...,\\bar{z}_{S})$ and $\\underline{\\boldsymbol{z}}\\equiv(\\underline{z}_{1},...,\\underline{z}_{S})$\nthat satisfy \n\\begin{equation}\n\\begin{array}{c}\n\\bar{\\boldsymbol{z}}=\\arg\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\max_{\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}],\\\\\n\\underline{\\boldsymbol{z}}=\\arg\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\min_{\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}].\n\\end{array}\\label{eq:max_min_pM}\n\\end{equation}\nThe following is the first main result of this study, which establishes\nthe sharp bounds on $E[Y(\\boldsymbol{d}^{j})|x]$, where $X=x$ is\nfixed in the model.\n\n\\begin{theorem}\\label{thm:sharp}Given model \\eqref{eq:main_model1}--\\eqref{eq:main_model2}\nwith fixed $X=x$, suppose Assumptions SS, M1, IN, E, EX, M$^{*}$,\nEQ and C hold. In addition, suppose $H(x)$ is well-defined and $H(x)\\ge0$.\nThen, the bounds $U_{\\boldsymbol{d}^{j}}$ and $L_{\\boldsymbol{d}^{j}}$\nin \\eqref{eq:U_dj} and \\eqref{eq:L_dj} simplify to \n\\begin{align*}\nU_{\\boldsymbol{d}^{j}}(x) & =\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\bar{\\boldsymbol{z}},x]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})|\\bar{\\boldsymbol{z}},x],\\\\\nL_{\\boldsymbol{d}^{j}}(x) & =\\Pr[Y=1,\\boldsymbol{D}\\in\\mathcal{D}^{\\le}(\\boldsymbol{d}^{j})|\\underline{\\boldsymbol{z}},x],\n\\end{align*}\nand these bounds are sharp.\n\n\\end{theorem}\n\nWith binary $Y$ (Assumption M$^{*}$), sharp bounds on the mean treatment\nparameters can be obtained, which is reminiscent of the findings of\nstudies that consider single-treatment models. However, our analysis\nis substantially different from earlier studies. In a single treatment\nmodel, \\citet{SV11} use the propensity score as a scalar conditioning\nvariable, which summarizes all the exogenous variation in the selection\nprocess and is convenient for simplifying the bounds and proving sharpness.\nHowever, in the context of our paper this approach is invalid, since\n$\\Pr[D_{s}=1|Z_{s}=z_{s},\\boldsymbol{D}_{-s}=\\boldsymbol{d}_{-s}]$\ncannot be written in terms of a propensity score of player $s$ as\n$\\boldsymbol{D}_{-s}$ is endogenous. We instead use the vector $\\boldsymbol{Z}$\nas conditioning variables and establish partial ordering for the relevant\nconditional probabilities (that define the lower and upper bounds)\nwith respect to the joint propensity score $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{Z}=\\boldsymbol{z}]$\n$\\forall\\boldsymbol{d}\\in\\mathcal{D}^{\\ge}(\\boldsymbol{d}^{j})$.\n\nWhen the variation of $X$ is additionally exploited in the model,\nthe bounds will be narrower than the bounds in Theorem \\ref{thm:sharp}.\nWe now proceed with this case, utilizing Lemma \\ref{lem:sign_match_gen}\n(i) and (ii). First, analogous to \\eqref{eq:H}, we define the integrated\nversion of $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$\nas \n\\begin{align*}\n\\tilde{H}(\\tilde{\\boldsymbol{x}}) & \\equiv E\\left[\\tilde{h}(\\boldsymbol{Z},\\boldsymbol{Z}',\\tilde{\\boldsymbol{x}})\\left|(\\boldsymbol{Z},\\boldsymbol{Z}')\\in\\mathcal{Z}_{EQ,j}\\text{ }\\forall j=0,...,S-1\\right.\\right],\n\\end{align*}\nwhere $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})=0$\nwhenever it is not well-defined. Then, we define the following sets\nof two consecutive elements $(x_{j},x_{j-1})$ of $\\boldsymbol{x}$\nthat satisfy the conditions in Lemma \\ref{lem:sign_match_gen}: for\n$j=1,...,S$, define $\\mathcal{X}_{j,j-1}^{0}(\\iota)\\equiv\\{(x_{j},x_{j-1}):sgn\\{\\tilde{H}(\\tilde{\\boldsymbol{x}})\\}=\\iota,x_{0}=\\cdots=x_{S}\\}$\nand for $t\\ge1$, \n\\begin{align*}\n\\mathcal{X}_{j,j-1}^{t}(\\iota) & \\equiv\\{(x_{j},x_{j-1}):sgn\\{\\tilde{H}(\\tilde{\\boldsymbol{x}})\\}=\\iota,(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}^{t-1}(-\\iota)\\mbox{ \\ensuremath{\\forall}}k\\neq j\\}\\cup\\mathcal{X}_{j,j-1}^{t-1}(\\iota),\n\\end{align*}\nwhere the sets are understood to be empty whenever $\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',\\tilde{\\boldsymbol{x}})$\nis not well-defined for any $p_{M^{\\le j}}(\\boldsymbol{z})<p_{M^{\\le j}}(\\boldsymbol{z}')$\n$\\forall j$. Note that $\\mathcal{X}_{j,j-1}^{t}(\\iota)\\subset\\mathcal{X}_{j,j-1}^{t+1}(\\iota)$\nfor any $t$. Define $\\mathcal{X}_{j,j-1}(\\iota)\\equiv\\lim_{t\\rightarrow\\infty}\\mathcal{X}_{j,j-1}^{t}(\\iota)$.\\footnote{In practice, the formula for $\\mathcal{X}_{j,j-1}^{t}$ provides a\nnatural algorithm to construct the set $\\mathcal{X}_{j,j-1}$ for\nthe computation of the bounds. The calculation of each $\\mathcal{X}_{j,j-1}^{t}$\nis straightforward, as it is a search over a two-dimensional space\nfor $(x_{j},x_{j-1})$ once the set $\\mathcal{X}_{j,j-1}^{t-1}$ from\nthe previous step is obtained. Practitioners can employ truncation\n$t\\le T$ for some $T$ and use $\\mathcal{X}_{j,j-1}^{T}$ as an approximation\nfor $\\mathcal{X}_{j,j-1}$.} Then, by Lemma \\ref{lem:sign_match_gen}, if $(x_{j},x_{j-1})\\in\\mathcal{X}_{j,j-1}(\\iota)$,\nthen \n\\begin{align}\n & sgn\\{\\vartheta(\\boldsymbol{d}^{j},x_{j};\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}^{j-1},x_{j-1};\\boldsymbol{u})\\}=\\iota\\text{ a.e. \\ensuremath{\\boldsymbol{u}}}\\text{ }\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j}).\\label{eq:implied_by_lem3.3}\n\\end{align}\nIn conclusion, for bounds on the ATE $E[Y(\\boldsymbol{d}^{j})|x]$,\nwe can introduce the sets $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}')$\nand $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}')$ for\n$\\boldsymbol{d}'\\neq\\boldsymbol{d}^{j}$ as follows: for $\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\mathcal{D}^{>}(\\boldsymbol{d}^{j})$,\n\\begin{align}\n\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(-1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j'+1\\leq k\\leq j,x_{j}=x\\right\\} \\nonumber \\\\\n & \\cup\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j+1\\leq k\\leq j',x_{j}=x\\right\\} ,\\label{eq:script_X_L}\\\\\n\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j'+1\\leq k\\leq j,x_{j}=x\\right\\} \\nonumber \\\\\n & \\cup\\left\\{ x_{j'}:(x_{k},x_{k-1})\\in\\mathcal{X}_{k,k-1}(-1)\\cup\\mathcal{X}_{k,k-1}(0)\\mbox{ for }j+1\\leq k\\leq j',x_{j}=x\\right\\} .\\label{eq:script_X_U}\n\\end{align}\nThe following theorem summarizes our results:\n\n\\begin{theorem}\\label{thm:main} In model \\eqref{eq:main_model1}--\\eqref{eq:main_model2},\nsuppose the assumptions of Lemma \\ref{lem:sign_match_gen} hold. Then\nthe sign of the ATE is identified, and the upper and lower bounds\non the ASF and ATE with $\\boldsymbol{d},\\tilde{\\boldsymbol{d}}\\in\\mathcal{D}$\nare \n\\begin{align*}\nL_{\\boldsymbol{d}}(x) & \\leq E[Y(\\boldsymbol{d})|x]\\leq U_{\\boldsymbol{d}}(x)\n\\end{align*}\nand $L_{\\boldsymbol{d}}(x)-U_{\\tilde{\\boldsymbol{d}}}(x)\\leq E[Y(\\boldsymbol{d})-Y(\\tilde{\\boldsymbol{d}})|x]\\leq U_{\\boldsymbol{d}}(x)-L_{\\tilde{\\boldsymbol{d}}}(x)$,\nwhere for any given $\\boldsymbol{d}^{j}\\in\\mathcal{D}^{j}\\subset\\mathcal{D}$\nfor some $j$, \n\\begin{align*}\nU_{\\boldsymbol{d}^{j}}(x) & \\equiv\\inf_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{ E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}^{j}\\}|\\boldsymbol{z}]\\overline{Y}\\\\\n & +\\sum_{\\boldsymbol{d}^{\\prime}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\mathcal{D}^{>}(\\boldsymbol{d}^{j})}\\inf_{x'\\in\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}')}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x']\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\Biggr\\},\\\\\nL_{\\boldsymbol{d}^{j}}(x) & \\equiv\\sup_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Biggl\\{ E[Y|\\boldsymbol{D}=\\boldsymbol{d}^{j},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{j}|\\boldsymbol{z}]+\\Pr[\\boldsymbol{D}\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}^{j}\\}|\\boldsymbol{z}]\\underline{Y}\\\\\n & +\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})\\cup\\mathcal{D}^{>}(\\boldsymbol{d}^{j})}\\sup_{x'\\in\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}')}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x']\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\Biggr\\}.\n\\end{align*}\n\\end{theorem}\n\nSee Sections \\ref{sec:Monte-Carlo-Studies} and \\ref{sec:Empirical-Application}\nfor concrete examples of the expression of $U_{\\boldsymbol{d}^{j}}(x)$\nand $L_{\\boldsymbol{d}^{j}}(x)$. The terms $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{'}|\\boldsymbol{z}]\\overline{Y}$\nand $\\Pr[\\boldsymbol{D}=\\boldsymbol{d}^{'}|\\boldsymbol{z}]\\underline{Y}$\nappear in the expression of the bounds because Lemma \\ref{lem:sign_match_gen}\ncannot establish an order between $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})$'s\nfor $\\boldsymbol{d}\\in\\mathcal{D}^{j}$, which is related to the complication\ndue to multiple equilibria, which occurs for $\\boldsymbol{d}\\in\\mathcal{D}^{j}$.\nWhen the variation in $\\boldsymbol{Z}$ is only used in deriving the\nbounds, $\\mathcal{X}_{k,k-1}(\\iota)$ should simply reduce to $\\mathcal{X}_{k,k-1}^{0}(\\iota)$\nin the definition of $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{L}(x;\\boldsymbol{d}')$\nand $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{U}(x;\\boldsymbol{d}')$. When\n$Y$ is binary with no $X$, such bounds are equivalent to \\eqref{eq:U_dj}\nand \\eqref{eq:L_dj}. The variation in $X$ given $\\boldsymbol{Z}$\nyields substantially narrower bounds than the sharp bounds established\nin Theorem \\ref{thm:sharp} under Assumption C. However, the resulting\nbounds are not automatically implied to be sharp from Theorem \\ref{thm:sharp},\nsince they are based on a different DGP and the additional exclusion\nrestriction.\n\n\\begin{remark}\\label{rem:mourifie}Maintaining that $Y$ is binary,\nsharp bounds on the ATE with variation in $X$ can be derived assuming\nthat the signs of $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})$\nare identified for $\\boldsymbol{d}\\in\\mathcal{D}$ and $\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})$\nand $x,x'\\in\\mathcal{X}$ via Lemma \\ref{lem:sign_match_gen}. To\nsee this, define \n\\begin{align*}\n\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{U}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x':\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})\\leq0\\text{ a.e. }\\boldsymbol{u}\\right\\} ,\\\\\n\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{L}(x;\\boldsymbol{d}') & \\equiv\\left\\{ x':\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})-\\vartheta(\\boldsymbol{d}',x';\\boldsymbol{u})\\geq0\\text{ a.e. }\\boldsymbol{u}\\right\\} ,\n\\end{align*}\nwhich are identified by assumption. Then, by replacing $\\mathcal{X}_{\\boldsymbol{d}}^{i}(x;\\boldsymbol{d}')$\nwith $\\mathcal{\\tilde{X}}_{\\boldsymbol{d}}^{i}(x;\\boldsymbol{d}')$\n(for $i\\in\\{U,L\\}$) in Theorem \\ref{thm:main}, we may be able to\nshow that the resulting bounds are sharp. Since Lemma \\ref{lem:sign_match_gen}\nimplies that $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')\\subset\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$\nbut not necessarily $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')\\supset\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$,\nthese modified bounds and the original bounds in Theorem \\ref{thm:main}\ndo not coincide. This contrasts with the result of \\citet{SV11} for\na single-treatment model, and the complication lies in the fact that\nwe deal with an incomplete model with a vector treatment. When there\nis no $X$, Lemma \\ref{lem:sign_match_gen}(i) establishes equivalence\nbetween the two signs, and thus, $\\mathcal{X}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')=\\mathcal{\\tilde{X}}_{\\boldsymbol{d}^{j}}^{i}(x;\\boldsymbol{d}')$\nfor $i\\in\\{U,L\\}$, which results in Theorem \\ref{thm:sharp}. Relatedly,\nwe can also exploit variation in $W$, namely, variables that are\ncommon to both $X$ and $\\boldsymbol{Z}$ (with or without exploiting\nexcluded variation of $X$). This is related to the analysis of \\citet{Chi10}\nand \\citet{mourifie2015sharp} in a single-treatment setting. One\ncaveat of this approach is that, similar to these papers, we need\nto additionally assume that $W\\perp(\\epsilon,\\boldsymbol{U})$.\\end{remark}\n\n\\begin{remark}When $X$ does not have enough variation, we can calculate\nthe bounds on the ATE. To see this, suppose we do not use the variation\nin $X$ and suppose $H(x)\\ge0$. Then $\\vartheta(\\boldsymbol{d}^{j},x;\\boldsymbol{u})\\ge\\vartheta(\\boldsymbol{d}^{j-1},x;\\boldsymbol{u})$\n$\\forall\\boldsymbol{d}^{j-1}\\in\\mathcal{D}^{<}(\\boldsymbol{d}^{j})$\n$\\forall j=1,...,S$ by Lemma \\ref{lem:sign_match_gen}(i) and by\ntransitivity, $\\vartheta(\\boldsymbol{d}^{'},x;\\boldsymbol{u})\\ge\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})$\nwith $\\boldsymbol{d}'$ being an extension of $\\boldsymbol{d}$. Therefore,\nwe have \n\\begin{align}\nE[Y(\\boldsymbol{d})|x] & \\le E[Y|\\boldsymbol{D}=\\boldsymbol{d},\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z}]+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{>}(\\boldsymbol{d})}E[Y|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}]\\nonumber \\\\\n & +\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}\\backslash\\mathcal{D}^{\\ge}(\\boldsymbol{d})}E[Y(\\boldsymbol{d}^{j})|\\boldsymbol{D}=\\boldsymbol{d}',\\boldsymbol{z},x]\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{z}].\\label{eq:interim_bd}\n\\end{align}\nWithout using variation in $X$, we can bound the last term in \\eqref{eq:interim_bd}\nby $Y\\in[\\underline{Y},\\overline{Y}]$. This is done above with $\\theta(\\boldsymbol{d},x,\\epsilon)=1[\\mu(\\boldsymbol{d},x)\\geq\\epsilon_{\\boldsymbol{d}}]$\nand $\\vartheta(\\boldsymbol{d},x;\\boldsymbol{u})=F_{\\epsilon|\\boldsymbol{U}}(\\mu(\\boldsymbol{d},x)|\\boldsymbol{u})$.\n\n\\end{remark}\n\n\\section{Numerical Study\\label{sec:Monte-Carlo-Studies}}\n\nTo illustrate the main results of this study in a simulation exercise,\nwe calculate the bounds on the ATE using the following data generating\nprocess: \n\\begin{align*}\nY_{\\boldsymbol{d}} & =1[\\tilde{\\mu}_{\\boldsymbol{d}}+\\beta X\\ge\\epsilon],\\\\\nD_{1} & =1[\\delta_{2}D_{2}+\\gamma_{1}Z_{1}\\ge V_{1}],\\\\\nD_{2} & =1[\\delta_{1}D_{1}+\\gamma_{2}Z_{2}\\ge V_{2}],\n\\end{align*}\nwhere $(\\epsilon,V_{1},V_{2})$ are drawn, independent of $(X,\\boldsymbol{Z})$,\nfrom a joint normal distribution with zero means and each correlation\ncoefficient being $0.5$. We draw $Z_{s}$ ($s=1,2$) and $X$ from\na multinomial distribution, allowing $Z_{s}$ to take two values,\n$\\mathcal{Z}_{s}=\\{-1,1\\}$, and $X$ to take either three values,\n$\\mathcal{X}=\\{-1,0,1\\}$, or fifteen values, $\\mathcal{X}=\\{-1,-\\frac{6}{7},-\\frac{5}{7},...,\\frac{5}{7},\\frac{6}{7},1\\}$.\nBeing consistent with Assumption M, we choose $\\tilde{\\mu}_{11}>\\tilde{\\mu}_{10}$\nand $\\tilde{\\mu}_{01}>\\tilde{\\mu}_{00}$. Let $\\tilde{\\mu}_{10}=\\tilde{\\mu}_{01}$.\nWith Assumption SS, we choose $\\delta_{1}<0$ and $\\delta_{2}<0$.\nWithout loss of generality, we choose positive values for $\\gamma_{1}$,\n$\\gamma_{2}$, and $\\beta$. Specifically, $\\tilde{\\mu}_{11}=0.25$,\n$\\tilde{\\mu}_{10}=\\tilde{\\mu}_{01}=0$ and $\\tilde{\\mu}_{00}=-0.25$.\nFor default values, $\\delta_{1}=\\delta_{2}\\equiv\\delta=-0.1$, $\\gamma_{1}=\\gamma_{2}\\equiv\\gamma=1$\nand $\\beta=0.5$.\n\nIn this exercise, we focus on the ATE $E[Y(1,1)-Y(0,0)\\vert X=0]$,\nwhose true value is $0.2$ given our choice of parameter values. For\n$h(\\boldsymbol{z},\\boldsymbol{z}',x)$, we consider $\\boldsymbol{z}=(1,1)$\nand $\\boldsymbol{z}'=(-1,-1)$. Note that $H(x)=h(\\boldsymbol{z},\\boldsymbol{z}',x)$\nand $\\tilde{H}(x,x',x'')=\\tilde{h}(\\boldsymbol{z},\\boldsymbol{z}',x,x',x'')$,\nsince $Z_{s}$ is binary. Then, we can derive the sets $\\mathcal{X}_{\\boldsymbol{d}}^{U}(0;\\boldsymbol{d}')$\nand $\\mathcal{X}_{\\boldsymbol{d}}^{L}(0;\\boldsymbol{d}')$ for each\n$\\boldsymbol{d}\\in\\{(1,1),(0,0)\\}$ and $\\boldsymbol{d}'\\neq\\boldsymbol{d}$\nin Theorem \\ref{thm:main}.\n\nBased on our design, $H(0)>0$, and thus, the bounds when we use $Z$\nonly are, with $x=0$, \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(0,0)=1\\vert x]\\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1\\vert\\boldsymbol{z},x],\n\\]\nand \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1\\vert\\boldsymbol{z},x]\\le\\Pr[Y(1,1)=1\\vert x]\\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]\\right\\} .\n\\]\nUsing both $\\boldsymbol{Z}$ and $X$, we obtain narrower bounds.\nFor example, when $\\left|\\mathcal{X}\\right|=3$, with $\\tilde{H}(0,-1,-1)<0$,\nthe lower bound on $\\Pr[Y(0,0)=1\\vert X=0]$ becomes \n\\[\n\\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},0]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},-1]\\right\\} .\n\\]\nWith $\\tilde{H}(1,1,0)<0$, the upper bound on $\\Pr[Y(1,1)=1\\vert X=0]$\nbecomes \n\\[\n\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},0]+\\Pr[Y=1,\\boldsymbol{D}\\in\\{(1,0),(0,1)\\}\\vert\\boldsymbol{z},1]+\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},0]\\right\\} .\n\\]\nFor comparison, we calculate the bounds in \\citet{manski1990nonparametric}\nusing $\\boldsymbol{Z}$. These bounds are given by \n\\begin{align*}\n & \\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(0,0)=1\\vert x]\\\\\n & \\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(0,0)\\vert\\boldsymbol{z}]\\right\\} ,\n\\end{align*}\nand \n\\begin{align*}\n & \\max_{\\boldsymbol{z}\\in\\mathcal{Z}}\\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]\\le\\Pr[Y(1,1)=1\\vert x]\\\\\n & \\le\\min_{\\boldsymbol{z}\\in\\mathcal{Z}}\\left\\{ \\Pr[Y=1,\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z},x]+1-\\Pr[\\boldsymbol{D}=(1,1)\\vert\\boldsymbol{z}]\\right\\} .\n\\end{align*}\nWe also compare the estimated ATE using a standard linear IV model\nin which the nonlinearity of the true DGP is ignored:\n\\begin{align*}\nY & =\\pi_{0}+\\pi_{1}D_{1}+\\pi_{2}D_{2}+\\beta X+\\epsilon,\\\\\n\\left(\\begin{array}{c}\nD_{1}\\\\\nD_{2}\n\\end{array}\\right) & =\\left(\\begin{array}{c}\n\\gamma_{10}\\\\\n\\gamma_{20}\n\\end{array}\\right)+\\left(\\begin{array}{cc}\n\\gamma_{11} & \\gamma_{12}\\\\\n\\gamma_{21} & \\gamma_{22}\n\\end{array}\\right)\\left(\\begin{array}{c}\nZ_{1}\\\\\nZ_{2}\n\\end{array}\\right)+\\left(\\begin{array}{c}\nV_{1}\\\\\nV_{2}\n\\end{array}\\right).\n\\end{align*}\nHere, the first stage is the reduced-form representation of the linear\nsimultaneous equations model for strategic interaction. Under this\nspecification, the ATE becomes $E[Y(1,1)-Y(0,0)\\vert X=0]=\\pi_{1}+\\pi_{2}$,\nwhich is estimated via two-stage least squares (TSLS).\n\nThe bounds calculated for the ATE are shown in Figures \\ref{fig:sim1}--\\ref{fig:sim4}.\nFigure \\ref{fig:sim1} shows how the bounds on the ATE change, as\nthe value of $\\gamma$ changes from $0$ to $2.5$. The larger $\\gamma$\nis, the stronger the instrument $\\boldsymbol{Z}$ is. The first conspicuous\nresult is that the TSLS estimate of the ATE is biased because of the\nproblem of misspecification. Next, as expected, Manski's bounds and\nour proposed bounds converge to the true value of the ATE as the instrument\nbecomes stronger. Overall, our bounds, with or without exploiting\nthe variation of $X$, are much narrower than Manski's bounds.\\footnote{Although we do not make a rigorous comparison of the assumptions here,\nnote that the bounds by \\citet{MP00} under the semi-MTR is expected\nto be similar to ours. However, their bounds need to assume the direction\nof the monotonicity.} Notice that the sign of the ATE is identified in the whole range\nof $\\gamma$, as predicted by the first part of Theorem \\ref{thm:main},\nin contrast to Manski's bounds. Using the additional variation in\n$X$ with $\\left|\\mathcal{X}\\right|=3$ decreases the width of the\nbounds, particularly with the smaller upper bounds on the ATE in this\nsimulation design. Figure \\ref{fig:sim2} depicts the bounds using\n$X$ with $\\left|\\mathcal{X}\\right|=15$, which yields narrower bounds\nthan when $\\left|\\mathcal{X}\\right|=3$, and substantially narrower\nthan those only using $\\boldsymbol{Z}$.\n\nFigure \\ref{fig:sim3} shows how the bounds change as the value of\n$\\beta$ changes from $0$ to $1.5$, where a larger $\\beta$ corresponds\nto a stronger exogenous variable $X$. The jumps in the upper bound\nare associated with the sudden changes in the signs of $\\tilde{H}(-1,0,-1)$\nand $\\tilde{H}(0,1,1)$. At least in this simulation design, the strength\nof $X$ is not a crucial factor for obtaining narrower bounds. In\nfact, based on other simulation results (omitted in the paper), we\nconclude that the number of values $X$ can take matters more than\nthe dispersion of $X$ (unless we pursue point identification of the\nATE).\n\nFinally, Figure \\ref{fig:sim4} shows how the width of the bounds\nis related to the extent to which the opponents' actions $D_{-s}$\naffect one's payoff, captured by $\\delta$. We vary the value of $\\delta$\nfrom $-2$ to $0$, and when $\\delta=0$, the players solve a single-agent\noptimization problem. Thus, heuristically, the bound at this point\nwould be similar to the ones that can be obtained when \\citet{SV11}\nis extended to a multiple-treatment setting with no simultaneity.\nIn the figure, as the value of $\\delta$ becomes smaller, the bounds\nget narrower.\n\n\\section{Empirical Application: Airline Markets and Pollution\\label{sec:Empirical-Application}}\n\nAircrafts are a major source of emissions, and thus, quantifying the\ncausal effect of air transport on pollution is of importance to policy\nmakers. Therefore, in this section, we take the bounds proposed in\nSection \\ref{subsec:Partial-Identification} to data on airline market\nstructure and air pollution in cities in the U.S.\n\nIn 2013, aircrafts were responsible for about 3 percent of total U.S.\ncarbon dioxide emissions and nearly 9 percent of carbon dioxide emissions\nfrom the U.S. transportation sector, and it is one of the fastest\ngrowing sources.\\footnote{See \\texttt{https:\/\/www.c2es.org\/content\/reducing-carbon-dioxide-emissions-from-aircraft\/7\/}}\nAirplanes remain the single largest source of carbon dioxide emissions\nwithin the U.S. transportation sector, which is not yet subject to\ngreenhouse gas regulations. In addition to aircrafts, airport land\noperations are also a big source of pollution, making airports one\nof the major sources of air pollution in the U.S. For example, 43\nof the 50 largest airports are in ozone non-attainment areas and 12\nare in particulate matter non-attainment areas.\\footnote{Ozone is not emitted directly but is formed when nitrogen oxides and\nhydrocarbons react in the atmosphere in the presence of sunlight.\nIn United States environmental law, a non-attainment area is an area\nconsidered to have air quality worse than the National Ambient Air\nQuality Standards as defined in the Clean Air Act.}\n\nThere is growing literature showing the effects of air pollution on\nvarious health outcomes (see, \\citet{schlenker2015airports}, \\citet{cg2003},\n\\citet{kms2011}). In particular, \\citet{schlenker2015airports} show\nthat the causal effect of airport pollution on the health of local\nresidents---using a clever instrumental variable approach---is sizable.\nTheir study focuses on the 12 major airports in California and implicitly\nassume that the level of competition (or market structure) is fixed.\nUsing high-frequency data, they exploit weather shocks in the East\ncoast---that might affect airport activity in California through\nnetwork effects---to quantify the effect of airport pollution on\nrespiratory and cardiovascular health complications. In contrast,\nwe take the link between airport pollution and health outcomes as\ngiven and are interested in quantifying the effects of different market\nstructures of the airline industry on air pollution.\\footnote{In this section, we refer to market structure as the particular configuration\nof airlines present in the market. In other words, market structure\nnot only refers to the number of firms competing in a given market\nbut to the actual identities of the firms. Thus, we will regard a\nmarket in which, say, United and American operate as different from\na market in which Southwest and Delta operate, despite both markets\nhaving two carriers.} We explicitly allow market structure to be determined endogenously,\nas the outcome of an entry game in which airlines behave strategically\nto maximize their profits and the resulting pollution in this market\nis not internalized by the firms. Understanding these effects can\nthen help inform the policy discussion on pollution regulation. Given\nthat we treat market structure as endogenous, one cannot simply run\na regression of a measure of pollution on market structure (or the\nnumber of airlines present in a market) to obtain the causal effect,\nif there are unobserved market characteristics that affect both firm\ncompetition and pollution outcomes. For example, if at both ends of\na city-pair there are firms from a high-polluting industry which engage\nin a lot of business travel, it would drive both pollution and the\nentry of airlines in the market. Therefore, we use the method presented\nin Section \\ref{subsec:Partial-Identification}.\n\nIn each market, we assume that a set of airlines chooses to be in\nor out as part of a simultaneous entry game of perfect information,\nas introduced in Section \\ref{subsec:Model}. Therefore, we treat\nmarket structure as our endogenous treatment. We then model air pollution\nas a function of the airline market structure as in equation (\\ref{eq:main_model1}),\nwhere $Y$ is a measure of air pollution at the airport level (including\nboth aircraft and land operation pollution), the vector $\\boldsymbol{D}$\nrepresents the market structure, and $X$ includes market specific\ncovariates that affect pollution directly (i.e., not through airline\nactivity), such as weather shocks or the share of pollution-related\nactivity in the local economy.\\footnote{Note that our definition of market is a city-pair; hence, all of our\nvariables are, in fact, weighted averages over the two cities.} Additionally, we allow for market-level covariates, $\\boldsymbol{W}$,\nwhich affect both the participation decisions and pollution (e.g.,\nthe size of the market as measured by population or the level of economic\nactivity). As instruments, $Z_{s}$, we consider a firm-market proxy\nfor cost introduced in \\citet{CT09}. We discuss the definition and\nconstruction of the variables in detail below.\n\nOur objective is to estimate the effect of a change in market structure\non air pollution, $E[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}')]$. For\nexample, we might be interested in the average effect on pollution\nof moving from two airlines operating in the market to three, or how\nthe pollution level changes on average when Delta is a monopolist\nversus a situation in which Delta faces competition from American.\nFollowing entry, firms compete by choosing their pricing, frequency,\nand which airplanes to operate. Different market structures will have\ndifferent impacts on these variables, which in turn, affect the level\nof pollution. Note that the effects might be asymmetric. That is,\nfor a given number of entrants, their identities are important to\ndetermine the pollution level. For example, when comparing the effect\nof a monopoly on pollution, we find that the airline operating plays\na role.\n\nTo illustrate our estimation procedure, we consider three types of\nATE exercises. The first examines the effects on pollution from a\nmonopolist airline vis-a-vis a market that is not served by any airline.\nThe second set of exercises examine the total effect of the industry\non pollution under all possible market configurations. Finally, the\nthird type of exercises examine how the (marginal) effect of a given\nairline changes when the firm faces different levels of competition.\nNotice that regardless of the exercise we run, we quantify ``reduced-form''\neffects, in that they summarize structural effects resulting from\na given market structure. The idea is that given the market structure,\nprices are determined, and given demand, ultimately the frequency\nof flights in the market is determined, which in fact, causes pollution.\n\nIn the rest of this section, we first describe our data sources, then\nshow results for three different ATE exercises, and conclude with\na brief discussion relating our results to potential policy recommendations.\n\n\\subsection{Data}\n\nFor our analysis, we combine data spanning the period 2000--2015\nfrom two sources: airline data from the U.S. Department of Transportation\nand pollution data from the Environmental Protection Agency (EPA).\n\n\\textbf{Airline Data.} Our first data source contains airline information\nand combines publicly available data from the Department of Transportation's\nOrigin and Destination Survey (DB1B) and Domestic Segment (T-100)\ndatabase. These datasets have been used extensively in the literature\nto analyze the airline industry (see, e.g., \\citet{borenstein89},\n\\citet{berry1992estimation}, \\citet{CT09}, and more recently, \\citet{robertssweeting2013}\nand \\citet{ciliberto2015market}). The DB1B database is a quarterly\nsample of all passenger domestic itineraries. The dataset contains\ncoupon-specific information, including origin and destination airports,\nnumber of coupons, the corresponding operating carriers, number of\npassengers, prorated market fare, market miles flown, and distance.\nThe T-100 dataset is a monthly census of all domestic flights broken\ndown by airline, and origin and destination airports.\n\nOur time-unit of analysis is a quarter and we define a market as the\nmarket for air connection between a pair of airports (regardless of\nintermediate stops) in a given quarter.\\footnote{In cities that operate more than one airport, we assume that flights\nto different airports in the same metropolitan area are in separate\nmarkets.} We restrict the sample to include the top 100 metropolitan statistical\nareas (MSA's), ranked by population at the beginning of our sample\nperiod. We follow \\citet{berry1992estimation} and \\citet{CT09} and\ndefine an airline as actively serving a market in a given quarter,\nif we observe at least 90 passengers in the DB1B survey flying with\nthe airline in the corresponding quarter.\\footnote{This corresponds to approximately the number of passengers that would\nbe carried on a medium-size jet operating once a week.} We exclude from our sample city pairs in which no airline operates\nin the whole sample period. Notice that we do include markets that\nare temporarily not served by any airline. This leaves us with 181,095\nmarket-quarter observations.\n\nIn our analysis, we allow for airlines to have a heterogeneous effect\non pollution, and to simplify computation, in each market we allow\nfor six potential participants: American (AA), Delta (DL), United\n(UA), Southwest (WN), a medium-size airline, and a low-cost carrier.\\footnote{That is, to limit the number of potential market structures, we lump\ntogether all the low cost carriers into one category, and Northwest,\nContinental, America West, and USAir under the medium airline type.} The latter is not a bad approximation to the data in that we rarely\nobserve more than one medium-size or low-cost in a market but it assumes\nthat all low-cost airlines have the same strategic behavior, and so\ndo the medium airlines. Table \\ref{tab:marketstructure} shows the\nnumber of firms in each market broken down by size as measured by\npopulation. As the table shows, market size alone does not explain\nmarket structure, a point first made by \\citet{CT09}.\n\n\\begin{table}[t!]\n\\caption{Distribution of the Number of Carriers by Market Size}\n\\label{tab:marketstructure} \\centering{}%\n\\begin{tabular}{lrrrr}\n\\hline \n & \\multicolumn{3}{c}{\\rule{0ex}{2.5ex}Market size} & \\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}\\# firms  & Large  & Medium  & Small  & Total\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}0  & 7.96  & 8.20  & 8.62  & 8.18\\tabularnewline\n1  & 41.18  & 22.53  & 20.58  & 30.30\\tabularnewline\n2  & 28.14  & 23.41  & 21.25  & 25.04\\tabularnewline\n3  & 12.65  & 20.00  & 16.67  & 16.05\\tabularnewline\n4  & 7.65  & 14.72  & 15.17  & 11.51\\tabularnewline\n5  & 1.98  & 9.90  & 16.48  & 7.80\\tabularnewline\n6+  & 0.52  & 1.23  & 2.21  & 1.12\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}\\# markets  & 79,326  & 64,191  & 37,578  & 181,095\\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\nIn our application, we consider two instruments for the entry decisions.\nThe first is the \\emph{airport presence} of an airline proposed by\n\\citet{berry1992estimation}. For a given airline, this variable is\nconstructed as the number of markets it serves out of an airport as\na fraction of the total number of markets served by all airlines out\nof the airport. A hub-and-spoke network allows firms to exploit demand-side\nand cost-side economies, which should affect the firm's profitability.\nWhile \\citet{berry1992estimation} assumes that an airline's airport\npresence only affects its own profits (and hence, is excluded from\nrivals' profits), \\citet{CT09} argue that this may not be the case\nin practice, since airport presence might be a measure of product\ndifferentiation, rendering it likely to enter the profit function\nof all firms through demand. While an instrument that enters all of\nthe profit functions is fine in our context (see Appendix \\ref{subsec:Common_Z}),\nwe also consider the instrument proposed by \\citet{CT09}, which captures\nshocks to the fixed cost of providing a service in a market. This\nvariable, which they call \\emph{cost}, is constructed as the percentage\nof the nonstop distance that the airline must travel in excess of\nthe nonstop distance, if the airline uses a connecting instead of\na nonstop flight.\\footnote{Mechanically, the variable is constructed as the difference between\nthe sum of the distances of a market's endpoints and the closest hub\nof an airline, and the nonstop distance between the endpoints, divided\nby the nonstop distance.} Arguably, this variable only affects its own profits and is excluded\nfrom rivals' profits.\n\n\\begin{table}[t!]\n\\caption{Airline Summary Statistics}\n\\label{tab:airlinessumstat} \\centering{}%\n\\begin{tabular}{llccccccc}\n\\hline \n\\rule{0ex}{2.5ex}  &  & American  & Delta  & United  & Southwest  & medium  & low-cost  & \\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}Market presence (0\/1)  & mean  & 0.44  & 0.57  & 0.28  & 0.25  & 0.56  & 0.17  & \\tabularnewline\n & sd  & 0.51  & 0.51  & 0.46  & 0.44  & 0.51  & 0.38  & \\tabularnewline\nAirport presence (\\%)  & mean  & 0.43  & 0.56  & 0.27  & 0.25  & 0.39  & 0.10  & \\tabularnewline\n & sd  & 0.17  & 0.18  & 0.16  & 0.18  & 0.14  & 0.08  & \\tabularnewline\nCost (\\%)  & mean  & 0.71  & 0.41  & 0.76  & 0.29  & 0.22  & 0.04  & \\tabularnewline\n & sd  & 1.56  & 1.28  & 1.43  & 0.83  & 0.60  & 0.17  & \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\nTable \\ref{tab:airlinessumstat} presents the summary statistics of\nthe airline related variables. Of the leading airlines, we see that\nAmerican and Delta are present in about half of the markets, while\nUnited and Southwest are only present in about a quarter of the markets.\nAmerican and Delta tend to dominate the airports in which they operate\nmore than United and Southwest. From the cost variable, we see that\nboth American and United tend to operate a hub-and-spoke network,\nwhile Southwest (and to a lesser extent Delta) operates most markets\nnonstop.\n\n\\textbf{Pollution Data.} The second component of our dataset is the\nair pollution data. The EPA compiles a database of outdoor concentrations\nof pollutants measured at more than 4,000 monitoring stations throughout\nthe U.S., owned and operated mainly by state environmental agencies.\nEach monitoring station is geocoded, and hence, we are able to merge\nthese data with the airline dataset by matching all the monitoring\nstations that are located within a 10km radius of each airport in\nour first dataset.\n\nThe principal emissions of aircraft include the greenhouse gases carbon\ndioxide ($\\text{CO}_{2}$) and water vapor ($\\text{H}_{2}\\text{O}$),\nwhich have a direct impact on climate change. Aircraft jet engines\nalso produce nitric oxide ($\\text{NO}$) and nitrogen dioxide ($\\text{NO}_{2}$)\n(which together are termed nitrogen oxides ($\\text{NO}_{\\text{x}}$)),\ncarbon monoxide (CO), oxides of sulphur ($\\text{SO}_{\\text{x}}$),\nunburned or partially combusted hydrocarbons (also known as volatile\norganic compounds or VOC's), particulates, and other trace compounds\n(see, \\citet{FAA2015}). In addition, ozone ($\\text{O}_{3}$) is formed\nby the reaction of VOC's and $\\text{NO}_{\\text{x}}$ in the presence\nof heat and sunlight. The set of pollutants other than $\\text{CO}_{2}$\nare more pernicious in that they can harm human health directly and\ncan result in respiratory, cardiovascular, and neurological conditions.\nResearch to date indicates that fine particulate matter (PM) is responsible\nfor the majority of the health risks from aviation emissions, although\nozone has a substantial health impact too.\\footnote{See \\citet{FAA2015}.}\nTherefore, as our measure of pollution, we will consider both.\n\nOur measure of ozone is a quarterly mean of daily maximum levels in\nparts per million. In terms of PM, as a general rule, the smaller\nthe particle the further it travels in the atmosphere, the longer\nit remains suspended in the atmosphere, and the more risk it poses\nto human health. PM that measure less than 2.5 micrometer can be readily\ninhaled, and thus, potentially pose increased health risks. The variable\nPM2.5 is a quarterly average of daily averages and is measured in\nmicrograms\/cubic meter. For each airport in our sample, we take an\naverage (weighted by distance to the airport) of the data from all\nair monitoring stations within a 10km radius. The top panel of Table\n\\ref{tab:marketsumstats} shows the summary statistics of the pollution\nmeasures.\n\n\\begin{table}[t!]\n\\caption{Market-level Summary Statistics}\n\\label{tab:marketsumstats} \\centering{}%\n\\begin{tabular}{lcc}\n\\hline \n\\rule{0ex}{2.5ex}  & Mean  & Std. Dev.\\tabularnewline\n\\hline \n\\rule{0ex}{2.5ex}Pollution  &  & \\tabularnewline\n\\hspace{2ex}Ozone ($\\text{O}_{3}$)  & .0477  & .0056\\tabularnewline\n\\hspace{2ex}Particulate matter (PM2.5)  & 8.3881  & 2.5287 \\tabularnewline\n\\rule{0ex}{2.5ex}Other controls  &  & \\tabularnewline\n\\hspace{2ex}Market size (pop.)  & 2307187.8  & 1925533.4\\tabularnewline\n\\hspace{2ex}Income (per capita)  & 34281.6  & 4185.5\\tabularnewline\n\\rule{0ex}{2.5ex}\\# of markets  & 181,095  & \\tabularnewline\n\\hline \n\\end{tabular}\n\\end{table}\n\n\\textbf{Other Market-Level Controls.} We also include in our analysis\nmarket-level covariates that may affect both market structure and\npollution levels. In particular, we construct a measure of market\nsize by computing the (geometric) mean of the MSA populations at the\nmarket endpoints and a measure of economic activity by computing the\naverage per capita income at the market endpoints, using data from\nthe Regional Economic Accounts of the Bureau of Economic Analysis.\n\nFinally, as we mentioned in Section \\ref{subsec:Partial-Identification},\nhaving access to data on a variable that affects pollution but is\nexcluded from the airline participation decisions can greatly help\nin calculating the bounds of the ATE. Therefore, we construct a variable\nthat measures the economic activity of pollution related industries\n(manufacturing, construction, and transportation other than air transportation)\nin a given market (MSA) as a fraction of total economic activity in\nthat market, again, using data from the Regional Economic Accounts\nof the Bureau of Economic Analysis.\nOur implicit assumption is as follows. The size of the market, among\nother things, determines whether a firm might enter it, but not the\ntype of economic activity in the cities.\nThe idea is that conditional on the market GDP, a market with a higher\nshare of polluting industries will have a higher level of pollution\nbut this share would not affect the airline market structure.\n\n\\subsection{Estimation and Results}\n\n\nTo simplify the estimation, we discretize all continuous variables\ninto binary variables (taking a value of 0 (1) if the corresponding\ncontinuous variable is below (above) its median). Using the notation\nfrom Section \\ref{subsec:Model}, let the elements of the treatment\nvector $\\boldsymbol{d}=(d_{\\text{DL}},d_{\\text{AA}},d_{\\text{UA}},d_{\\text{WN}},d_{\\text{med}},d_{\\text{low}})$\nbe either 0 or 1, indicating whether each firm is active in the market.\nWe compute the upper and lower bounds on the ATE using the result\nfrom Theorem \\ref{thm:main} and the fact that our $Y$ variable is\nbinary. Specifically, given two treatment vectors $\\boldsymbol{d}$\nand $\\tilde{\\boldsymbol{d}}$ we can bound the ATE \n\\begin{align*}\nL(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\leq E[Y(\\boldsymbol{d})-Y(\\tilde{\\boldsymbol{d}})|x,w]\\leq U(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w)\n\\end{align*}\nwhere the upper bound can be characterized by \n\\begin{align*}\nU(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\equiv\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{z},x,w]+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{j}\\backslash\\{\\boldsymbol{d}\\}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},W=w]\\\\\n & \\quad+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})\\cup\\mathcal{D}^{>}(\\boldsymbol{d})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},X=x'(\\boldsymbol{d}'),W=w]\\\\\n & \\quad-\\text{Pr}[Y=1,\\boldsymbol{D}=\\tilde{\\boldsymbol{d}}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]\\\\\n & \\quad-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{<}(\\tilde{\\boldsymbol{d}})\\cup\\mathcal{D}^{>}(\\tilde{\\boldsymbol{d}})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},X=x''(\\boldsymbol{d}''),W=w]\n\\end{align*}\nfor every $\\boldsymbol{z}$, $x'(\\boldsymbol{d}')\\in\\mathcal{X}_{\\boldsymbol{d}}^{U}(x;\\boldsymbol{d}')$\nfor $\\boldsymbol{d}'\\neq\\boldsymbol{d}$, and $x''(\\boldsymbol{d}'')\\in\\mathcal{X}_{\\tilde{\\boldsymbol{d}}}^{L}(x;\\boldsymbol{d}'')$\nfor $\\boldsymbol{d}''\\neq\\tilde{\\boldsymbol{d}}$. Similarly, the\nlower bound can be characterized by \n\\begin{align*}\nL(\\boldsymbol{d},\\tilde{\\boldsymbol{d}};x,w) & \\equiv\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]\\\\\n & \\quad+\\sum_{\\boldsymbol{d}'\\in\\mathcal{D}^{<}(\\boldsymbol{d})\\cup\\mathcal{D}^{>}(\\boldsymbol{d})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}'|\\boldsymbol{Z}=\\boldsymbol{z},X=x'(\\boldsymbol{d}'),W=w]\\\\\n & \\quad-\\text{Pr}[Y=1,\\boldsymbol{D}=\\tilde{\\boldsymbol{d}}|\\boldsymbol{Z}=\\boldsymbol{z},X=x,W=w]-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{j}\\backslash\\{\\tilde{\\boldsymbol{d}}\\}}\\Pr[\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},W=w]\\\\\n & \\quad-\\sum_{\\boldsymbol{d}''\\in\\mathcal{D}^{<}(\\tilde{\\boldsymbol{d}})\\cup\\mathcal{D}^{>}(\\tilde{\\boldsymbol{d}})}\\text{Pr}[Y=1,\\boldsymbol{D}=\\boldsymbol{d}''|\\boldsymbol{Z}=\\boldsymbol{z},X=x''(\\boldsymbol{d}''),W=w]\n\\end{align*}\nfor every $\\boldsymbol{z}$, $x'(\\boldsymbol{d}')\\in\\mathcal{X}_{\\boldsymbol{d}}^{L}(x;\\boldsymbol{d}')$\nfor $\\boldsymbol{d}'\\neq\\boldsymbol{d}$, and $x''(\\boldsymbol{d}'')\\in\\mathcal{X}_{\\tilde{\\boldsymbol{d}}}^{U}(x;\\boldsymbol{d}'')$\nfor $\\boldsymbol{d}''\\neq\\tilde{\\boldsymbol{d}}$. We estimate the\npopulation objects above using their sample counterparts. We experimented\nwith both measures of pollution discussed earlier and obtain qualitatively\nand quantitatively similar results in all cases, which is not surprising\ngiven that the two pollution measures are highly correlated. In order\nto save space, we only show results using PM2.5 as our outcome variable.\nWe also experimented with several specifications of the covariates,\n$\\boldsymbol{X}$ and $\\boldsymbol{W}$, and instruments, $\\boldsymbol{Z}$.\nIn particular, we tried different discretizations of each variable\n(including allowing for more than two points in their supports and\ndifferent cutoffs). Clearly, there is a limit to how finely we can\ncut the data even with a large sample size such as ours. The coarser\ndiscretization occurs when each covariate (and instrument) is binary\nand it seems to produce reasonable results; hence, we stick with this\ndiscretization in all of our exercises. Again, aiming at the most\nparsimonious model, and after some experimentation, we obtained reasonable\nresults when both $\\boldsymbol{X}$ and $\\boldsymbol{W}$ are scalars\n(share of pollution related industries in the market and total GDP\nin the market, respectively).\n\nWe also compute confidence sets by deriving unconditional moment inequalities\nfrom our conditional moment inequalities and implementing the Generalized\nMoment Selection test proposed by \\citet{as2010}. The confidence\nsets are obtained by inverting the test.\\footnote{For details of this procedure, see \\citet{dm2016}.}\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig2_all_cs_monop_CROP.pdf}\\caption{Effect of a Monopolistic Market Structure}\n\\label{fig:emp_monop} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of a change in market structure\nfrom no airline serving a market to a monopolist serving it. The solid\nblack intervals are our estimates of the identified sets and the thin\nred lines are the 95\\% confidence sets. \\end{small} \n\\end{figure*}\n\n\\textbf{Monopoly Effects.} Here we examine a very simple ATE of a\nchange in market structure from no airline serving a market to a monopolist\nserving it. Intuitively, we want to understand the change in the probability\nof being a high-pollution market when an airline starts operating\non it. Recall that we allow each firm to have different effects on\npollution; hence, we estimate the effects of each one of the six firms\nin our data becoming a monopolist. Thus, we are interested in the\nATEs of the form \n\\[\nE[Y(\\boldsymbol{d}_{\\text{monop}})-Y(\\boldsymbol{d}_{\\text{noserv}})|X,W]\n\\]\nwhere $\\boldsymbol{d}_{\\text{monop}}$ is one of the six vectors in\nwhich only one element is 1 and the rest are 0's, and $\\boldsymbol{d}_{\\text{noserv}}$\nis a vector of all 0's. The results are shown in Figure \\ref{fig:emp_monop},\nwhere the solid black intervals are our estimates of the identified\nsets and the thin red lines are the 95\\% confidence sets. We see that\nall ATEs are positive and statistically significant different from\n0, except for the medium-size carriers. While there no major differences\non the effects of the leader carriers, with the exception of Delta\nwhich seems to induce a higher increase in the probability of high\npollution, the medium and low-cost carriers induce a smaller effect.\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig1_all_cs_num_entrants_CROP.pdf}\\caption{Total Market Structure Effect}\n\\label{fig:emp_numentrants} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of the airline industry under\nall possible market configurations. The solid black intervals are\nour estimates of the identified sets and the thin red lines are the\n95\\% confidence sets. The bars in each cluster correspond to all possible\nmarket configurations, respectively. \\end{small} \n\\end{figure*}\n\n\\textbf{Total Market Structure Effect.} We now turn to our second\nset of exercises. Here, we quantify the effect of the airline industry\non the likelihood of a market having high levels of pollution. To\ndo so, we estimate ATEs of the form \n\\[\nE[Y(\\boldsymbol{d})-Y(\\boldsymbol{d}_{\\text{noserv}})|X,W]\n\\]\nfor all potential market configurations $\\boldsymbol{d}$, and where,\nas before, $\\boldsymbol{d}_{\\text{noserv}}$ is a vector of all 0's.\nFigure \\ref{fig:emp_numentrants} depicts the results. The left-most\nset of intervals corresponds to the 6 different monopolistic market\nstructures, and by construction, coincide with those from Figure \\ref{fig:emp_monop}.\nThe next set corresponds to all possible duopolistic structures, which\nhas 15 possibilities, and so on. Not surprisingly, we observe that\nthe effect on the probability of being a high-pollution market is\nincreasing in the number of firms operating in the market. More interesting\nis the non-linearity of the effect: the effect increases at a decreasing\nrate. This would be consistent with a model in which firms \\emph{accommodate}\nnew entrants by decreasing their frequency, which is analogous to\nthe prediction of a Cournot competition model, as we increase the\nnumber of firms. To further investigate this point, in the next set\nof exercises, we examine the effect of one firm as we change the competition\nit faces.\n\n\\begin{figure*}[!t]\n\\begin{centering}\n\\includegraphics[scale=0.7\n{fig3_all_cs_compet_effect_DL_CROP.pdf}\\caption{Marginal Effect of Delta under Different Market Structures}\n\\label{fig:emp_competDL} \n\\par\\end{centering}\n\\medskip{}\n\n\\begin{small} This plot shows the ATEs of Delta entering the market\ngiven all possible rivals market configurations. The solid black intervals\nare our estimates of the identified sets and the thin red lines are\nthe 95\\% confidence sets. The bars in each cluster correspond to all\npossible market configurations, respectively. \\end{small} \n\\end{figure*}\n\n\\textbf{Marginal Carrier Effect.} In our last set of exercises, we\nare interested in investigating how the marginal effect (i.e., the\neffect of introducing one more firm into the market) changes under\ndifferent configurations of the market structure. Say we are interested\nin the effect of Delta entering the market on pollution, given that\nthe current market structure (excluding Delta) is $\\boldsymbol{d}_{\\text{--DL}}=(d_{\\text{AA}},d_{\\text{UA}},d_{\\text{WN}},d_{\\text{med}},d_{\\text{low}})$.\nThen, we want to estimate \n\\[\nE[Y(1,\\boldsymbol{d}_{\\text{--DL}})-Y(0,\\boldsymbol{d}_{\\text{--DL}})|X,W].\n\\]\n\nFigure \\ref{fig:emp_competDL} shows the identified sets and confidence\nsets of the marginal effect of Delta on the probability of high pollution\nunder all possible market configuration for Delta's rivals. We obtain\nqualitatively similar results when estimating the marginal effects\nof the other five carriers, and hence, we omit the graphs to save\nspace. In the Figure, the left-most exercise is the effect of Delta\nas a monopolist, and coincides, by construction, with the left-most\nexercise in Figure \\ref{fig:emp_monop}. The second exercise (from\nthe left) corresponds to the additional effect of Delta on pollution\nwhen there is already one firm operating in the market, which yields\nfive different possibilities. The next exercise shows the effect of\nDelta when there are two firms already operating in the market yielding\n10 possibilities, and so on. Again, the estimated marginal ATEs in\nall cases are positive and statistically significant. Interestingly,\nalthough we cannot entirely reject the null hypothesis that all the\neffects are the same, it seems that the marginal effect of Delta is\ndecreasing in the number of rivals it faces. Intuitively, this suggests\na situation in which Delta enters the market and operates with a frequency\nthat is decreasing with the number of rivals (again, as we would expect\nin a Cournot competition model) and is consistent with the findings\nin our previous set of exercises.\n\nThe conclusions from the \\emph{total market} and \\emph{marginal} ATEs\nare also interesting from a policy perspective. For example, a merger\nof two airlines in which duplicate routes are eliminated would imply\na decrease in total pollution in the affected markets, but by less\nthan what one would have naively anticipated from removing one airline\nwhile keeping everything else constant.\\footnote{Note that, however, to the extent that a merger alters the way the merged firm behaves post entry, the treatment effects we estimate will not be informative. In other words, our model can only speak to the effects of a merger on pollution that only affects behavior through entry.} In other words, there are\ntwo effects of removing an airline from a market. The first is a direct\naffect: pollution decreases by the amount of pollution by the carrier\nthat is no longer present in the market. However, the remaining firms\nin the market will react strategically to the new market structure.\nIn our exercises, we find that this indirect effect implies an increase\nin pollution. The overall effect is a net decrease in pollution. Moreover,\ngiven the non-linearities of the ATEs we estimate it looks like the\noverall effect, while negative, might be negligible in markets with\nfour or more competitors. While it is unclear that merger analysis, which is typically concerned\nwith price increases post-merge or cost savings of the merging firms,\nshould also consider externalities such as pollution, (social) welfare\nanalysis should. Hence, our findings may serve as guidance to policy\ndiscussion on air traffic regulation.\n\n\\medskip{}\n\n \\bibliographystyle{ecta}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe COVID-19 pandemic has exacerbated the already poor educational experiences of millions of students in Africa who were grappling with educational challenges like poor access to computers, the internet, and teachers. In 2018, the average student-teacher ratio in Sub-Saharan Africa was 35:1 which is higher compared to 14:1  in Europe \\cite{UNESCO2020}. In this context, students struggle to get answers to their questions. Hence, offering quick and accurate answers, outside of the classroom, could improve their overall learning experience. However, it is difficult to scale this support with human teachers.\n\nIn 2020, we developed Kwame \\cite{boateng2021b}, a bilingual AI teaching assistant that provides answers to students' coding questions in English and French for SuaCode, a smartphone-based online coding course \\cite{boateng2019,boateng2021}. Kwame is a deep learning-based question answering system that finds the paragraph most semantically similar to the question via cosine similarity with a Sentence-BERT model. We extended Kwame to work for science education and deployed it as a web app. Specifically, Kwame for Science \\footnote{\\href{http:\/\/kwame.ai\/}{http:\/\/kwame.ai\/}} answers questions of students based on the Integrated Science subject of the West African Senior Secondary Certificate Examination (WASSCE). This is a core subject that covers various aspects of science such as biology, chemistry, physics, earth science, and agricultural science. It is mandatory for senior high school students in the West African Education Council (WAEC) member countries (Ghana, Nigeria, Sierra Leone, Liberia, and The Gambia).\n\nThere are virtual teaching assistants (TA) such as Jill Watson \\cite{goel2016,goel2020}, Rexy \\cite{benedetto2019}, and a physics course TA \\cite{zylich2020} and Curio SmartChat (for K-12 science) \\cite{raamadhurai2019} (see \\cite{boateng2020} for a detailed description of related work). These works are focused on answering logistical questions, except Curio SmartChat. In comparison to Curio SmartChat which is the closest work to ours, our work uses a state-of-the-art language model (Sentence-BERT) relative to theirs (Universal Sentence Encoder). Also, our work is the first to be developed and deployed in the context of high school science education in West Africa.\n\n\\section{Kwame for Science  System Architecture}\nKwame for Science is a Sentence-BERT-based question-answering web app that displays 3 paragraphs as answers along with a confidence score which represents the similarity score in response to science questions (Figure \\ref{fig:kwame4science}). Additionally, it displays the top 5 related past exam questions and their answers in addition to the 3 paragraphs. We used a Sentence-BERT (SBERT) model that was pretrained on a large and diverse set of question-answer pairs. We used the SBERT model as it was with plans for fine-tuning after real-world data collection, especially since exploratory evaluation for our science use case showed it had decent performance.\n\nWhen a user types a question in the web app, our system computes an embedding of the question using the SBERT model. Next, it computes cosine similarity scores with a bank of answers (which are paragraphs from our knowledge source), retrieves, and returns the top 3 answers along with a confidence score and any figures or images referenced in that paragraph to the web app. Additionally, it computes cosine similarity scores with a bank of past exam questions, retrieves, and returns the top 5 related questions and their answers, along with confidence scores. The web app then displays the answers and the related past exam questions that are above a preset similarity score threshold. If no answer is above the threshold, a message is shown saying the question could not be answered using the knowledge source of that subject. We precomputed embeddings for fast real-time retrieval and saved them as indices in ElasticSearch which we hosted on Google Cloud Platform. \n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{kwame4science.png}\n\\caption{Screenshots of Kwame for Science} \n\\label{fig:kwame4science}\n\\end{figure}\n\n\n\\section{Dataset Curation and Preprocessing}\nGiven that our goal was for Kwame to provide answers based on the Integrated Science subject of the WASSCE exam, our training data and knowledge source had to cover the topics in the WASSCE Integrated Science curriculum. We sought to use one of the approved textbooks in Ghana. Unfortunately, their copyrights did not permit such use and the publishers were unwilling to partner with us. Consequently, we searched for free and open-source books and datasets that fulfilled our needs. We came across a middle school science dataset \u2014 Textbook Questions Answering (TQA) \\cite{kembhavi2017} which was curated from the free and open-source textbook, CK-12. Our exploration of the dataset revealed that though it covered several of the WASSCE Integrated Science topics, it lacked others, particularly those related to agricultural science. Consequently, we additionally used a dataset based on Simple Wikipedia to cover those gaps. We used Simple Wikipedia since its explanations were simple and better suited for middle school and high school students compared to regular Wikipedia.\n\nWe parsed the JSON files of the dataset into paragraphs. We also extracted figures that were referenced in the paragraphs so they could be returned to students along with the answers. We then split the paragraphs into groups of 3 sentences, computed embeddings, and indexed them using ElasticSearch to enable fast retrieval and run time. These constituted the answers returned for questions. Furthermore, we augmented our question-answering with curriculum-specific content. In particular, we created question-answer pairs using WASSCE questions that cover exams from 2000 to 2020. The exam has three parts, objectives (multiple-choice), theory, and practicals. Similar to the paragraphs, we computed embeddings of the questions and indexed them using ElasticSearch. These constituted the related past questions (with answers) returned when a question is asked.\n\n\\section{Preliminary Evaluation and Results}\nWe launched the web app in beta on 10th June 2022. Users could provide feedback by upvoting or downvoting answers in response to the question \"Was this helpful?.\" To evaluate Kwame for Science, we used the metrics top 1 and top 3 accuracies. Top 1 accuracy quantifies performance assuming only one answer was returned and voted on. Top 3 accuracy refers to the performance where for each question that received a vote, at least one answer was rated as helpful out of the 3 answers that were returned. The statistics for the deployment between 10th June 2022 and 27th June 2022 (2.5 weeks) are 190 users across 11 countries (6 in Africa), 433 questions with the metrics 71.8\\%  top 1 accuracy (n=117 answers), and 87.5\\%  top 3 accuracy (n=56 questions). The top 3 accuracy result is good, showing that Kwame for Science has a high chance of giving at least one useful answer among the 3. Some challenging cases occurred when there were typos in the spelling of scientific words and the questions were related to topics outside the scope of the knowledge source. Also, some unhelpful answers were cases the returned paragraph was incomplete due to issues with the dataset.\n\n\\section{Conclusion}\nIn this work, we developed and evaluated Kwame for Science which provides instant answers to the Science questions of students across West Africa. Our future work will fine-tune the SBERT model to improve its accuracy. Also, we will make Kwame for Science available in local languages across Africa, and available via offline channels such as SMS, USSD, and toll-free calling. Kwame for Science will enable the delivery of scalable, cost-effective, and quality remote education to millions of people across Africa. \n\\section{Introduction}\nThe COVID-19 pandemic has exacerbated the already poor educational experiences of millions of students in Africa who were grappling with educational challenges like poor access to computers, the internet, and teachers. In 2018, the average student-teacher ratio in Sub-Saharan Africa was 35:1 which is higher compared to 14:1  in Europe \\cite{UNESCO2020}. In this context, students struggle to get answers to their questions. Hence, offering quick and accurate answers, outside of the classroom, could improve their overall learning experience. However, it is difficult to scale this support with human teachers.\n\nIn 2020, we developed Kwame \\cite{boateng2021b}, a bilingual AI teaching assistant that provides answers to students' coding questions in English and French for SuaCode, a smartphone-based online coding course \\cite{boateng2019,boateng2021}. Kwame is a deep learning-based question answering system that finds the paragraph most semantically similar to the question via cosine similarity with a Sentence-BERT model. We extended Kwame to work for science education and deployed it as a web app. Specifically, Kwame for Science \\footnote{\\href{http:\/\/kwame.ai\/}{http:\/\/kwame.ai\/}} answers questions of students based on the Integrated Science subject of the West African Senior Secondary Certificate Examination (WASSCE). This is a core subject that covers various aspects of science such as biology, chemistry, physics, earth science, and agricultural science. It is mandatory for senior high school students in the West African Education Council (WAEC) member countries (Ghana, Nigeria, Sierra Leone, Liberia, and The Gambia).\n\nThere are virtual teaching assistants (TA) such as Jill Watson \\cite{goel2016,goel2020}, Rexy \\cite{benedetto2019}, and a physics course TA \\cite{zylich2020} and Curio SmartChat (for K-12 science) \\cite{raamadhurai2019} (see \\cite{boateng2020} for a detailed description of related work). These works are focused on answering logistical questions, except Curio SmartChat. In comparison to Curio SmartChat which is the closest work to ours, our work uses a state-of-the-art language model (Sentence-BERT) relative to theirs (Universal Sentence Encoder). Also, our work is the first to be developed and deployed in the context of high school science education in West Africa.\n\n\\section{Kwame for Science  System Architecture}\nKwame for Science is a Sentence-BERT-based question-answering web app that displays 3 paragraphs as answers along with a confidence score which represents the similarity score in response to science questions (Figure \\ref{fig:kwame4science}). Additionally, it displays the top 5 related past exam questions and their answers in addition to the 3 paragraphs. We used a Sentence-BERT (SBERT) model that was pretrained on a large and diverse set of question-answer pairs. We used the SBERT model as it was with plans for fine-tuning after real-world data collection, especially since exploratory evaluation for our science use case showed it had decent performance.\n\nWhen a user types a question in the web app, our system computes an embedding of the question using the SBERT model. Next, it computes cosine similarity scores with a bank of answers (which are paragraphs from our knowledge source), retrieves, and returns the top 3 answers along with a confidence score and any figures or images referenced in that paragraph to the web app. Additionally, it computes cosine similarity scores with a bank of past exam questions, retrieves, and returns the top 5 related questions and their answers, along with confidence scores. The web app then displays the answers and the related past exam questions that are above a preset similarity score threshold. If no answer is above the threshold, a message is shown saying the question could not be answered using the knowledge source of that subject. We precomputed embeddings for fast real-time retrieval and saved them as indices in ElasticSearch which we hosted on Google Cloud Platform. \n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{kwame4science.png}\n\\caption{Screenshots of Kwame for Science} \n\\label{fig:kwame4science}\n\\end{figure}\n\n\n\\section{Dataset Curation and Preprocessing}\nGiven that our goal was for Kwame to provide answers based on the Integrated Science subject of the WASSCE exam, our training data and knowledge source had to cover the topics in the WASSCE Integrated Science curriculum. We sought to use one of the approved textbooks in Ghana. Unfortunately, their copyrights did not permit such use and the publishers were unwilling to partner with us. Consequently, we searched for free and open-source books and datasets that fulfilled our needs. We came across a middle school science dataset \u2014 Textbook Questions Answering (TQA) \\cite{kembhavi2017} which was curated from the free and open-source textbook, CK-12. Our exploration of the dataset revealed that though it covered several of the WASSCE Integrated Science topics, it lacked others, particularly those related to agricultural science. Consequently, we additionally used a dataset based on Simple Wikipedia to cover those gaps. We used Simple Wikipedia since its explanations were simple and better suited for middle school and high school students compared to regular Wikipedia.\n\nWe parsed the JSON files of the dataset into paragraphs. We also extracted figures that were referenced in the paragraphs so they could be returned to students along with the answers. We then split the paragraphs into groups of 3 sentences, computed embeddings, and indexed them using ElasticSearch to enable fast retrieval and run time. These constituted the answers returned for questions. Furthermore, we augmented our question-answering with curriculum-specific content. In particular, we created question-answer pairs using WASSCE questions that cover exams from 2000 to 2020. The exam has three parts, objectives (multiple-choice), theory, and practicals. Similar to the paragraphs, we computed embeddings of the questions and indexed them using ElasticSearch. These constituted the related past questions (with answers) returned when a question is asked.\n\n\\section{Preliminary Evaluation and Results}\nWe launched the web app in beta on 10th June 2022. Users could provide feedback by upvoting or downvoting answers in response to the question \"Was this helpful?.\" To evaluate Kwame for Science, we used the metrics top 1 and top 3 accuracies. Top 1 accuracy quantifies performance assuming only one answer was returned and voted on. Top 3 accuracy refers to the performance where for each question that received a vote, at least one answer was rated as helpful out of the 3 answers that were returned. The statistics for the deployment between 10th June 2022 and 27th June 2022 (2.5 weeks) are 190 users across 11 countries (6 in Africa), 433 questions with the metrics 71.8\\%  top 1 accuracy (n=117 answers), and 87.5\\%  top 3 accuracy (n=56 questions). The top 3 accuracy result is good, showing that Kwame for Science has a high chance of giving at least one useful answer among the 3. Some challenging cases occurred when there were typos in the spelling of scientific words and the questions were related to topics outside the scope of the knowledge source. Also, some unhelpful answers were cases the returned paragraph was incomplete due to issues with the dataset.\n\n\\section{Conclusion}\nIn this work, we developed and evaluated Kwame for Science which provides instant answers to the Science questions of students across West Africa. Our future work will fine-tune the SBERT model to improve its accuracy. Also, we will make Kwame for Science available in local languages across Africa, and available via offline channels such as SMS, USSD, and toll-free calling. Kwame for Science will enable the delivery of scalable, cost-effective, and quality remote education to millions of people across Africa. \n\\section{Acknowledgement}\nThis work was supported with grants from ETH for Development (ETH4D) and the MTEC Foundation, both at ETH Zurich.\n\n\\bibliographystyle{splncs04}\n\n\\section{Acknowledgement}\nThis work was supported with grants from ETH for Development (ETH4D) and the MTEC Foundation, both at ETH Zurich.\n\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\n\n\n\n\n\n\n\n\nMachine learning models, especially neural networks, are becoming\nubiquitous in various real-life applications. For example, they are used in medical diagnosis~\\cite{kononenko2001machine}, self-driving cars~\\cite{bojarski2016end} and criminal sentencing~\\cite{compas2016data}. \nMeanwhile, more and more attention has been paid to the fairness issues of these machine learning models~\\cite{verma2018fairness,angell2018themis,ghosh2020justicia,zhang2020white,tse2021adf,ruoss2020learning,salimi2019interventional,buolamwini2018gender,galhotra2017fairness} as discrimination has been discovered in many applications~\\cite{gianfrancesco2018potential,ruoss2020learning,udeshi2018automated,zafar2017fairness}. For instance, machine learning models were used to predict recidivism risk for suspected criminals by computing the likelihood of committing a future crime~\\cite{compas2016data}. Analysis results show that the prediction model was more likely to mislabel black defendants as high recidivism risk and mislabel white defendants as low risk. To minimize such ethical risks, it is crucial to systematically test the fairness of machine learning models, especially neural networks where such issues are typically `hidden' due to the lack of interpretability~\\cite{pei2017deepxplore, tian2018deeptest}.\n\nRecently, multiple efforts have been made in the testing community to first search for (and then guide mitigating) discrimination of machine learning models spanning from traditional ones to neural networks \\cite{zhang2020white,tse2021adf,udeshi2018automated,zafar2017fairness,galhotra2017fairness}. For instance, state-of-the-art fairness testing work utilizes gradient information of the input sample to accelerate search\/generation of discriminative samples~\\cite{zhang2020white,tse2021adf,sg}. Despite being effective, existing research has mostly focused on \\emph{individual discrimination}, i.e., identifying or generating individual discriminatory instances of a machine learning model~\\cite{zhang2020white,tse2021adf,udeshi2018automated,zafar2017fairness,galhotra2017fairness}.\n\\emph{Group discrimination}, which characterizes a model's discrimination against a certain group (whose sensitive features\\footnote{we use ``feature''\/``attribute'' interchangeably} satisfy certain conditions), is another concerning type of discrimination, which has been widely studied~\\cite{galhotra2017fairness, zafar2017fairness, tramer2017fairtest, kleinberg2016inherent}. However, testing against group discrimination has been much less studied so far. Compared to testing of individual discrimination, testing a machine learning model against group discrimination imposes new challenges. First, it is highly non-trivial to effectively enumerate all combinations of sensitive features (especially when the sensitive features have multiple or even continuous values). Second, \\emph{group discrimination can be hidden, i.e., there might be `subtle' group discrimination against\nthose groups whose sensitive features satisfy certain unknown conditions, e.g., male-white of certain age group}. While a prior work~\\cite{kearns2018preventing} similarly addresses discrimination against subgroups defined over conjunctions of protected features in the learning phase, we propose an automatic testing approach to systematically identify such subgroups using interpretable rules and measure such discrimination before model deployment.  \n\n\n\\begin{figure*}[t] \\small\n\\centering \n\\includegraphics[width=0.75\\linewidth]{testsgd_overview.pdf}\n\\caption{An Overview of \\tool{TestSGD}.} \n\\label{fig:tool} \n\\end{figure*}\n\nSpecifically, in this work, we develop an effective method to systematically \\emph{test} a given machine learning model against such hidden \\emph{s}ubtle \\emph{g}roup \\emph{d}iscrimination, namely \\tool{TestSGD}. An overview of \\tool{TestSGD } is shown in Figure~\\ref{fig:tool}, which consists of three main phases: 1) candidate rule set generation, 2) group fairness identification, and 3) discrimination mitigation. In the first phase, \\tool{TestSGD } will automatically generate a candidate set of rules concerning multiple sensitive features. Note that we only consider frequent rule set with sufficient support (which characterize a sufficiently large group). In the second phase, the rule set \\emph{R} effectively partitions the samples into two groups, i.e., $samples_r$ which satisfies the rules and $samples_{\\neg r}$ which does not. The key intuition behind is to develop effective criteria to automatically mine interpretable rules which are practical and relevant in the real-world applications. Then we measure if the model suffers from group discrimination (against the groups partitioned by the rule set) by measuring the group fairness score. \nNote that, solely relying on the training samples might not be enough to accurately measure such a score. We thus propose to apply a standard data augmentation method, i.e., imposing minor perturbation on the available seed samples to generate new samples, and obtain an accurate estimation of the group fairness score (with bounded errors). The testing results of the first two phases are thus the identified subtle group discrimination (characterized by the rules) and their corresponding group fairness score (with bounded errors). For example, we test the model trained on the \\textbf{Crime}~\\cite{crime2009dataset} dataset which predicts whether the violent crimes per population in a specific community is high. The interpretable rule set found by \\tool{TestSGD } shows that it discriminates against communities in which the percentage of Caucasian population is lower than 80\\% and the percentage of females who are divorced is higher than 40\\%, with a 60.7\\% group fairness score, i.e., it is 60.7\\% more likely to predict high crime rate for such a community. In the last phase (optional depending whether the identified discrimination is considered to be harmful), \\tool{TestSGD } leverages the testing results to mitigate the identified subtle group discrimination. That is, to improve group fairness, we generate new samples according to the condition under which discrimination exists and retrain the original model.      \n\n\n\n\n\n \n \\begin{comment}\n Our method works for both models trained on structured data (such as feed-forward neural networks trained on structured dataset) as well as models trained on text data (such as recurrent neural networks for NLP tasks). Furthermore, we apply perturbation-based and replacement-based generation for structured data and textual data generation respectively so that our method can maintain the original data distribution. \n \\end{comment}\n\n\\tool{TestSGD } is implemented as an open-source software~\\cite{website}. We evaluate our \\tool{TestSGD } on 8 models trained on widely adopted datasets including both structured data and text data. The experimental results show \\tool{TestSGD } is effective in identifying and measuring subtle group discrimination. The results also show that \\emph{subtle group discrimination does exist in all of these 8 models and sometimes to a surprising level which has never been revealed before}. For instance, the model trained on the \\textbf{COMPAS}~\\cite{compas2016data} dataset is much less likely to predict Hispanic males older than 40 years old as criminals with high recidivism risk. \nFurthermore, our experiments show that the testing-guided discrimination mitigation is useful.\nThat is, we can mitigate identified subtle group discrimination for all models without sacrificing the accuracy. \n\nIn a nutshell, we summarize our main contributions as follows.\n\\begin{itemize} [leftmargin=*]\n    \\item We propose a method to automatically generate an interpretable rule set to identify \n   \n    subtle group discrimination in neural networks, applicable for both structured and text data;\n    \\item We develop a theoretical bound for accurately sampling and estimating the group fairness score against two groups.  \n   \n    \\item We show that we can generate samples systematically based on the interpretable rule set to mitigate subtle group discrimination.\n\\end{itemize}\n\nThe remainder of this paper is structured as follows. Section~\\ref{sec:background} provides the background on input types and fairness definitions. Section~\\ref{sec:prob_definition} defines our problem.\nWe present the proposed \\tool{TestSGD } framework in Section \\ref{sec:methodology}, which is evaluated in Sections \\ref{sec:evaluation}. Lastly, we review related work in Section~\\ref{sec:related work} and conclude in Section~\\ref{sec:conclusion}.\n\n\\section{Background} \\label{sec:background}\nOur goal is to develop a black-box method to identify subtle group discrimination in a user-provided neural network model. Our method supports neural networks trained on two different kinds of data, i.e., structure data and text data. Our method does not require the inner details of the neural network. That is, the neural network is viewed as a function $M: {R}^{p} \\to {R}^{q}$ which maps an input $x\\in {R}^{p}$  to an output $y \\in {R}^{q}$. Furthermore, we focus on deep feed-forward neural networks and recurrent neural networks. \n\n\\subsection{Input Type}\nFirst of all, we define two different data, i.e., structure data, text data, and their corresponding sensitive features which are used to evaluate the discrimination of the neural networks. \n\nA sample of structured dataset is composed of a set of features, i.e., a feature vector. A feature can be categorical (i.e., with a fixed set of values) or continuous (i.e., with a certain value range). We define the structure data and the corresponding sensitive features as follows. \n\\begin{definition}[Structured Data] \\label{def:structured data}\nA structured data $x$ contains $N$ features $\\{x_1, x_2, \\cdots, x_N\\}$, where $\\forall x_i, x_i \\in L_i$, where $L_i$ is a set of feature values. We write $S=\\{s_1, s_2, \\cdots, s_n\\}$ to denote the set of sensitive features in $x$, where $n<N$. \n\\end{definition}\n\nThe text data is composed of a set of tokens. We define the sensitive feature of text data based on the presence of sensitive terms. Note that there could be different categories of sensitive terms, e.g., terms referring to race, religion, or ethnicity. We define the text data and the corresponding sensitive features as follows. \n\n\\begin{definition}[Text Data] \\label{def:text data}\nA text data $x$ contains a sequence of tokens $\\{x_1, x_2, \\cdots, x_N\\}$. We write $S=\\{s_1, s_2, \\cdots, s_n\\}$ to denote a set of categories of sensitive terms, where $n<N$ and $T$ to denote a set of sensitive terms $\\{t_1, t_2, \\cdots, t_k\\}$, where $t_j \\in s_i$ for some $i$, for all $j \\in [1, k]$ and $t_j \\in x$ .\n\\end{definition}\n\n\n\\subsection{Fairness Definitions} \nTo define our problem, we define fairness and the concept of group fairness score. There are multiple definitions of fairness~\\cite{dwork2012fairness,joseph2016fairness,calders2010three,galhotra2017fairness,kleinberg2016inherent,zafar2017fairness}. Here, we briefly review two well-studied fairness definitions, i.e., individual fairness and group fairness.\n\n\\vspace{1mm}\n\\noindent\\textbf{Individual fairness} focuses on specific pairs of individuals. Intuitively, individual discrimination (unfairness) occurs when two individuals that differ by only certain sensitive feature (such as gender or race) are treated differently, i.e., with a different predicted label. This notion is widely used to search discriminatory instances which differ only in those sensitive characteristics~\\cite{zhang2020white,tse2021adf,udeshi2018automated}. There are also plenty of works on learning models which are more likly to avoid individual discrimination~\\cite{ruoss2020learning}. \n\n\\vspace{1mm}\n\\noindent\\textbf{Group fairness}, also known as statistical fairness, focuses on sensitive groups such as ethnic minority and the parity across different groups based on some statistical measurements~\\cite{galhotra2017fairness, kleinberg2016inherent, biswas2020machine, zafar2017fairness, tramer2017fairtest}. \nA classifier satisfies this metric if the samples in the sensitive group have a positive classification probability or true positive probability that is similar with or equal to that of the insensitive group. \n\nIn this work, we focus on group fairness for its relevance in many neural network applications. In the following, we provide a formal definition of group fairness based on positive classification rate measurement. \n\n\\begin{definition} \\label{def:group fair}\n\\label{def:fairness}\nLet $M$ be a neural network model; $l$ be a (favorable) prediction; and $\\xi$ be a positive constant. Let $G$ be a group identified by certain condition $\\phi$ on sensitive features $S$. $G$ can be defined as a set of samples $\\{x|x \\vDash \\phi\\}$, where $x \\vDash \\phi$ means x satisfies condition $\\phi$. \nWe say $M$ satisfies group fairness, with respect to $\\xi$ and $G$, if and only if \n\\begin{equation}\n\\begin{aligned}\n| \\: P(M(x)=l \\: | \\: x \\in G)-P(M(x)=l \\: | \\: x \\not \\in G) \\: | \\: \\leq \\xi\n\\end{aligned}\n\\end{equation}\n\\end{definition}\n\nNote that, in some cases, the model may be fair overall but unfair under some specific `subtle' conditions. For example, the model is fair considering gender attribute if it approves half of the loans from female or male applicants. However, when we consider both gender and race, the model may show discrimination. For example, it approves loans for far less a percentage of Hispanic female individual, compared to the remaining group. In this setting, we say that the model discriminates against Hispanic females (if we show that the testing results have sufficient statistical confidence). \n\n\n\\section{Problem Definition} \\label{sec:prob_definition}\n\n\nOur problem is to develop a systematic method for identifying subtle group discrimination. That is, given a neural network model $M$ (as well as a constant threshold $\\xi$, we aim to generate a condition $\\phi$ such that $M$ is unfair with respect to the group identified by $\\phi$. The condition $\\phi$ must satisfy the following conditions. (1) It must be constituted by variables representing sensitive features. (2) It must be human-interpretable, so that our analysis result can be presented for human decision making. (3) It must identify a group of non-trivial size. \n\\begin{comment}\n\\begin{itemize}\n    \\item By definition, it must be constituted by variables representing sensitive features. \n    \\item It must be human-interpretable, so that our analysis result can be presented for further human decision making, e.g., whether the discrimination is considered relevant or harmful. \n    \\item Arguably, $\\phi$ must identify a group of non-trivial size, i.e., if $\\phi$ identifies a group of only a few individuals, the discrimination is likely due to lack of sufficient data. \n\\end{itemize}\n\\end{comment}\nIn addition, our method must support both structured data as well as text data. Furthermore, we would like our method to generate results with certain correctness guarantee, e.g., the chance of reporting non-existing discrimination is low. \n\nInspired by rule-based models, which are widely used to learn interpretable models~\\cite{lakkaraju2016interpretable,rivest1987learning}, we generate $\\phi$ in the form of rules (a.k.a.~constraints) which are understandable by human beings and also concrete enough to show model prediction differences between different groups. The rules are constituted by the input features, without relying on any latent variables or representations. We define $\\phi$ to be the conjunction of one or more rules, each of which is constituted by only one sensitive feature. Furthermore, to limit the search space as well as to make sure the generated rules are interpretable, we limit each rule on continuous features to be of the form of a linear inequality, e.g., $age \\geq 30$ is a possible rule but $age~is~multiples~of~7$ is not.\n\nIn order to make sure that the discrimination that we discover is highly likely in the actual system, we propose a sampling based approach to estimate the probability of predicting certain label within a given group. Such a method allows us to generate an estimation with certain level of statistical confidence, i.e., with a bound on the error. Note that it is not straightforward to adopt existing techniques such as hypothesis testing~\\cite{wald1945sequential,wald1948optimum}. This is because the group fairness score is the difference between two estimations (i.e., one for the individual in the group and the other for those not in the group). We solve this problem by establishing a conservative error bound on the difference based on the error bounds for the two estimations.  \n\n\\section{Methodology} \\label{sec:methodology}\nIn this section, we describe the details of our approach. There are two main steps, i.e., learning a rule set and identifying group discrimination based on the learned rule set. The inputs for our method include a machine learning model $M$, its training set $D$, and a set of sensitive features $S$. The output is the subtle group discrimination represented as a rule set characterizing the discriminated group and the corresponding group fairness score. \n\\subsection{Generating Frequent Rule Sets}\\label{sec:rule generation}\n\n\n\nTo identify discrimination against certain group, we first need a way of characterizing a group. In this work, we characterize the groups based on a set of rules, each of which constrains one sensitive feature. In the following, we present how we generate rules for sensitive features of both structured data and text data. \n\nIn terms of categorical features $x_i$ in structured data such as gender or race, in general, a rule can be defined as a subset of the possible feature values $L_i$. For instance, given the sensitive feature of race which has five values, i.e., Caucasian, Black, Hispanic, Asian and other-race, a rule can be any set containing one to four of these five values\nIn total, we have 30 rules. \nFor continuous features $x_i$ in structure data such as age or percentage, there may be too many possible values to enumerate, i.e., the domain of $L_i$ is too large. Thus we apply techniques such as binning to turn continuous features into categorical features. Here, we divide the original value range into $K$ intervals with equal width. Then we consider each interval as a single value and consider a set containing adjacent values as a rule. We set $K$ as 10 in our experiments. For example, we divide age attribute ranging from 0 to 100 into 10 equal intervals.\n\nFor textual dataset, defining rules is not that straightforward. In this work, we define the rules based on the presence of sensitive terms $T$ (refer to Definition~\\ref{def:text data}). For each sensitive category $s$, we define a rule which intuitively means that the text sample contains a term $t$, where $t \\in s$. In this work, we use a set of 48 terms created in~\\cite{dixon2018measuring} as the sensitive terms which can be classified into 4 categories, i.e., gender, race, religion and age. The sensitive terms are shown in Table~\\ref{tab:terms}. For example, when we consider the \\emph{gender} feature for text dataset, there are 14 sensitive terms and thus 14 rules are generated. \n\n\\begin{table}[t] \\small\n\\centering\n\\caption{Identity Sensitive Terms}\n\\label{tab:terms}\n\\begin{tabular}{|c|l|}\n\\hline\n\\textbf{Sensitive Features}      & \\multicolumn{1}{c|}{\\textbf{Identity Terms}}                                                   \\\\ \\hline\n\\multirow{3}{*}{gender}   & lesbian, gay, bisexual, transgender, trans,  \\\\\n                          & queer, lgbt, lgbtq, homosexual, straight, \\\\\n                          & heterosexual, male, female, nonbinary                           \\\\ \\hline\n\\multirow{4}{*}{race}     & african, african american, black, white, \\\\\n                          & european, hispanic, latino, latina, latinx, \\\\\n                          & mexican, canadian, american, asian, indian,   \\\\\n                          & middle eastern, chinese, japanese \\\\ \\hline\n\\multirow{2}{*}{religion} & christian, muslim, jewish, buddhist, catholic, \\\\\n                          & protestant, sikh, taoist, atheist                                                 \\\\ \\hline\n\\multirow{2}{*}{age}      & old, older, young, younger, teenage,    \\\\\n                          & millennial, middle aged, elderly                                                                               \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nOnce a set of rules are identified, we then characterize a group based on a rule set. Each element of a rule set is a rule constraining one sensitive feature. Intuitively, a rule set partitions the input space of $M$ into two disjoint groups, i.e., those who satisfy all the rules in the rule set and the rest. If these two groups have a significant different probability of being predicted favorably by the model $M$, we successfully identify a subtle discrimination.   \n\nNote that a naive approach is to enumerate all possible rules based on one sensitive feature and combine them arbitrarily. Such an approach is both infeasible and undesirable. First, there can be enormous combinations of the rules. Second, not all rule sets are interesting. For instance, a rule set may be $\\{age \\geq 100, gender=Male\\}$. A discrimination found against the group identified by this rule set is likely to be due to the limited data. Furthermore, the discrimination is perhaps not as concerning as discrimination against groups that represent a sizable population.  \n\nWe thus only consider frequent rule sets among all possible combinations of rules. A frequent rule set is a set of rules that are satisfied by a group with a size more than certain threshold. Formally, given a rule set $R$, the \\textit{support} for $R$ is the frequency of the number of samples that satisfy all rules in rule set $R$.\nGiven a \\textit{support threshold} $\\theta$ (i.e., a percentage), we say that $R$ is \\emph{frequent} if its \\textit{support} is no less than $\\theta$. In the following, we present how to identify a set of frequent rule sets for structured and text data.  \n\nFor each sensitive feature $s$, let $R^s$ be the set of rules concerning $s$. A rule set $R$ is composed of rules for each sensitive feature, i.e., $R = \\{r^{s_1}, r^{s_2}, ..., r^{s_n}\\}$ where $r^{s_i} \\in \\{R^{s_i}\\cup\\varnothing\\}$ and $R\\neq\\varnothing $. $R$ is frequent if and only if $support(R) \\geq \\theta$ where $support(R)$ is defined as follows. \n\\begin{equation}\n\\label{equ:support}\n\\mathrm{support}(R)=\\frac{\\#\\{d \\in D | \\forall r \\in R.~d \\vDash r\\}}{\\#D}  \n\\end{equation}\nwhere $\\#X$ of a set $X$ is the number of elements in $X$; and $d \\vDash r$ means that $d$ satisfies $r$. \\\\\n\n\n\\noindent \\emph{Example} Consider the structured dataset \\textbf{Census Income}~\\cite{census1996dataset}. It has three sensitive features, i.e., gender, race, and age. Each feature has a set of values. The following constitutes a rule set\n$$\\{gender = Male, race = White, 40 \\leq  age < 60\\}$$  \\hfill $\\qed$ \\\\\n\\noindent Rule sets for text data are defined differently. Recall that each rule is a proposition on whether the text contains certain sensitive term. Formally, given the set of the categories of sensitive terms $S$, a rule set $R$ is then a set of sensitive terms $\\{r_1, r_2, \\cdots, r_m\\}$, where $r_k \\in s_i$ for some $i$, for all $k \\in [1,m]$ and $m\\leq n$. The support of $R$ is defined as follows.\n\\begin{equation}\n\\label{equ:support}\n\\mathrm{support}(R)=\\frac{\\#\\{d \\in D | \\forall r \\in R.~contains(d,s_r)\\}}{\\#D}  \n\\end{equation}\nwhere $s_r$ is the sensitive category referring to $r$ and $contains(d,s_{r})$ is a proposition which is true if and only if $d$ contains at least one term in the category $s_r$. \\\\\n\n\n\n\\noindent  \\emph{Example} Consider the text dataset \\textbf{Wikipedia Talk Pages}~\\cite{wulczyn2017ex}. We have two categories of sensitive terms, e.g., gender and race. \nFor each category, we have a set of corresponding sensitive terms as shown in Table~\\ref{tab:terms}. \nThe following constitutes a rule set\n$$\\{``bisexual\", ``Caucasian\"\\}$$ \\hfill $\\qed$ \\\\\n\\noindent Algorithm~\\ref{alg:ruleset} shows the exact steps in generating all possible rule sets. At line~1, we first initialize a dictionary $single\\_rules$ and an empty set $rules\\_sets$. During the loop from line 2 to 5, we generate all possible 1-feature rules for each sensitive feature as discussed above. At line 6, we generate a set of all 1-feature rules. Then, we generate all possible rule sets at line 7. Lastly, at line 8, we only keep those rule sets that have a support value no less than $\\theta$. \n\n\\begin{algorithm}[t]\n\\caption{$FrequentRuleSets(D, S, sup\\_thr)$ where $D$ is the training set, $S$ is the sensitive attributes, $\\theta$ is the support threshold}\n\\label{alg:ruleset}\n\\begin{algorithmic}[1]\n\\STATE $single\\_rules \\gets$ $\\{\\}$, $rule\\_sets \\gets \\varnothing$\n\\FOR{each $s$ in $S$}\n\\STATE $rules \\gets \n\\{r_{1}, r_{2}, ...\\}$\n\\STATE $single\\_rule[s]=rules$\n\\ENDFOR\n\\STATE $all\\_single\\_rules \\gets \\{single\\_rule[s] \\cup \\varnothing\\}$ for all $s \\in S$\n\\STATE $rule\\_sets \\gets combinations(all\\_single\\_rules)$\n\\STATE $all\\_rule\\_sets \\gets \\{R\\ for\\ R\\ in\\ rule\\_sets\\ if\\ support(D, R) \\geq \\theta\\}$\n\\RETURN $all\\_rule\\_sets$\n\\end{algorithmic} \n\\end{algorithm}\n \nIn general, given a dataset with $K$ sensitive features, and at most $N$ rules for each sensitive feature, the number of rule sets is $N^{K}$ in the worse case. For example, we have 2 gender-related single rules, 5 race-related 1-feature rules and 10 age-related 1-feature rules, there are 17 rule sets when considering one sensitive attribute, 80 rule sets when considering two sensitive attributes and 100 rule sets when considering all sensitive attributes. So in total, there are 197 possible rule sets. \n\n\\subsection{Identifying Group Fairness}\nFor each group identified by a rule set, we then measure the discrimination against the group. That is, we aim to compute the probability of predicting certain label by $M$ on those samples in the group, and that on those samples not in the group, and measure the difference.\nThe score is the group fairness score, which varies from 0 (i.e., no difference) to 1 (i.e., completely different). Formally, \n\n\\begin{definition}[Group fairness score] \\label{def:score}\nLet $R$ be a rule set. Let $l$ be a (favorable) label. The group fairness score with respect to $R$ and $l$ is $|prob(R, l) - prob(\\neg R, l)|$, where $prob(R,l)$ is the probability of predicting $l$ given samples satisfying $R$, $\\neg R$ identifies samples not satisfying $R$.\n\n\\end{definition}\n\nWe remark that this definition is similar to the CV score~\\cite{calders2010three} and multivariate group discrimination score~\\cite{galhotra2017fairness}. However, the former is limited to binary input types and the latter is limited to categorical input types. In comparison, our fairness score supports both structured data and text data. \\\\ \n\n\\noindent \\emph{Example} Take a model trained on the (structured) \\textbf{Census Income} dataset as an example. The model predicts whether the income of an adult is above \\$50,000 annually, i.e., ``True'' means above the threshold and ``False'' means otherwise. \nAssume the rule set is \n$$\\{gender= Male, race = White, 40 \\leq age < 60\\}$$ \nAssume that the model predicts 28\\% of individuals in this group with ``True'', and 10\\% of the remaining population with ``True''. The model's group fairness score, with respect to the rule set and the prediction, is 18\\%. \\hfill $\\qed$ \\\\\n\n\\noindent \nGiven a rule set, measuring the group fairness score requires us to measure $prob(R,l)$ and $prob(\\neg R,l)$, which is non-trivial since exhaustively enumerating all samples is infeasible due to the enormous input space. On the other hand, measuring it based on a limited number of samples may yield inaccurate results. In the following, we propose an approach to compute group fairness scores with a statistical confidence guarantee. Formally, we would like to measure the group fairness score $f$ within a margin of error $\\epsilon$ under a certain confidence $\\delta$, such that $prob(|f-\\hat{f}| > \\epsilon) < 1-\\delta$, where $\\hat{f}$ is the real group fairness score over all possible samples. \n\nAlgorithm~\\ref{alg:score} shows how we measure the group fairness score. We maintain two sets of samples, i.e., $samples_r$ which contains samples satisfying $R$ and $samples_{\\neg r}$ which contains samples not satisfying $R$. At line 1, we set both $samples_r$ and $samples_{\\neg r}$ to be empty, error margin $\\epsilon$ to be infinity and the number of generated samples as 0. During the loop from line 2 to 16, we keep generating samples and calculating group fairness score until the error margin $\\epsilon$ is no more than the given error threshold $error\\_thr$. From line 3 to line 6, we generate new samples for $samples_r$ and $samples_{\\neg r}$ respectively using a function $Sample$. We remark that the generated samples should follow the original data distribution (i.e., that of the training dataset). We present details on how we sample on structured and text dataset in the next subsection. \n\nAt line 7, we increase $num$ by 1. After generating a sufficient number of samples, we check the error margin $\\epsilon$ from line 9 to 15. We first calculate the probability of predicting $l$ at line 9 and 10 for two sets of samples. Then at line 11, we calculate the error margin $\\epsilon$ on the group fairness score. We explain why it is calculated this way below. If $\\epsilon$ is less than or equal to $error\\_thr$, the stopping criteria is satisfied (as in line 12 and 13). Lastly, at line 17, we return the absolute difference between $\\phi_{r}$ and $\\phi_{\\neg r}$ as the group fairness score. \n\n\\begin{algorithm}[t]\n\\caption{$GroupFairnessScore(D, M, R, sample\\_thr, error\\_thr)$ where $D$ is the training dataset; $M$ is the machine learning model; $R$ is a rule set, $sample\\_thr$ is the number of generated inputs threshold; $error\\_thr$ is error margin threshold}\n\\label{alg:score}\n\\begin{algorithmic}[1]\n\\STATE $samples_r \\gets \\varnothing$, $samples_{\\neg r} \\gets \\varnothing$, $\\epsilon \\gets +\\infty$, $num \\gets 0$\n\\WHILE{$\\epsilon > error\\_thr$}\n\\STATE $x \\gets Sample(D, R)$\n\\STATE $x' \\gets Sample(D, \\neg R)$\n\\STATE $samples_r \\gets samples_r \\cup {x}$\n\\STATE $samples_{\\neg r} \\gets samples_{\\neg r} \\cup {x'}$\n\\STATE $num \\gets num + 1$\n\\IF{$num > sample\\_thr$}\n\\STATE $\\phi_r \\gets \\#\\{i \\in samples_r | M(i)=l\\}\/ num$\n\\STATE $\\phi_{\\neg r} \\gets \\#\\{i \\in samples_{\\neg r} | M(i)=l\\}\/num$\n\\STATE $\\epsilon \\gets z\\times\\sqrt{\\frac{\\phi_r(1-\\phi_r)}{num}} + z\\times\\sqrt{\\frac{\\phi_{\\neg r}(1-\\phi_{\\neg r})}{num}} $\n\\IF{$\\epsilon \\leq error\\_thr$}\n\\STATE break\n\\ENDIF\n\\ENDIF\n\\ENDWHILE\n\\RETURN $f \\gets |\\phi_r-\\phi_{\\neg r}|$\n\\end{algorithmic} \n\\end{algorithm}\n\nIn the above algorithm, we estimate the error margin of the group fairness score based on an estimation of $prob(R,l)$ and $prob(\\neg R,l)$. The complication is that both $prob(R,l)$ and $prob(\\neg R,l)$ carry certain error margin, which may magnify the error margin for the group fairness score. In the following, we prove that line 11 in the above algorithm allows us to conservatively estimate the error margin of the group fairness score.\n\n\\begin{theorem}   \\label{theorem:confidence}\nAssume that $\\phi_r$ satisfies the following \t\n\\begin{equation} \\label{equ:pro}\nprob(|\\phi_r-\\hat{\\phi}_r|>\\epsilon_r) < 1-\\delta_r\n\\end{equation}\nwhere $\\epsilon_r$ and $\\delta_r$ are constants. Similarly, $\\phi_{\\neg r}$ satisfies the following.\n\\begin{equation} \\label{equ:unp}\nprob(|\\phi_{\\neg r}-\\hat{\\phi}_{\\neg r}|>\\epsilon_{\\neg r}) < 1-\\delta_{\\neg r}\n\\end{equation}\nThen the following is satisfied. \n\\begin{equation}   \\label{equ:fair}\nprob(|f-\\hat{f}|>\\epsilon_{r}+\\epsilon_{\\neg r}) < 1-\\delta_{r}\\delta_{\\neg r} \n\\end{equation}\n \\\\\n\\textbf{Proof:} Since $prob(|\\phi_r-\\hat{\\phi_r}|>\\epsilon_r) < 1-\\delta_r$ and $prob(|\\phi_{\\neg r}-\\hat{\\phi_{\\neg r}}|>\\epsilon_{\\neg r}) < 1-\\delta_{\\neg r}$, we have:\n\\begin{equation} \nprob(|\\phi_r-\\hat{\\phi}_r| \\leq \\epsilon_r) \\geq \\delta_r \\nonumber\n\\end{equation}\n\\begin{equation} \nprob(|\\phi_{\\neg r}-\\hat{\\phi}_{\\neg r}| \\leq \\epsilon_{\\neg r}) \\geq \\delta_{\\neg r} \\nonumber\n\\end{equation}\nHence \n\\begin{equation} \n\\begin{aligned}\n&prob(|(\\phi_{r}-\\hat{\\phi_{r}})-(\\phi_{\\neg r}-\\hat{\\phi}_{\\neg r})| \\leq \\epsilon_{r} + \\epsilon_{\\neg r}) \\geq \\\\ \n&prob(|\\phi_r\\!-\\!\\hat{\\phi}_r|\\!\\leq\\!\\epsilon_r) \\cdot prob(|\\phi_{\\neg r}\\!-\\!\\hat{\\phi}_{\\neg r}|\\!\\leq\\!\\epsilon_{\\neg r}) \\geq \\delta_r\\delta_{\\neg r}    \\nonumber\n\\end{aligned}\n\\end{equation} \nand \n\\begin{equation} \nprob(|(\\phi_{r}-\\hat{\\phi_{r}})-(\\phi_{\\neg r}-\\hat{\\phi_{\\neg r}})| > \\epsilon_{r} + \\epsilon_{\\neg r}) < 1- \\delta_r\\delta_{\\neg r}  \\nonumber\n\\end{equation}\n\\begin{equation} \nprob(|(\\phi_{r}-\\phi_{\\neg r})-(\\hat{\\phi}_{r}-\\hat{\\phi}_{\\neg r})| > \\epsilon_{r} + \\epsilon_{\\neg r}) < 1- \\delta_r\\delta_{\\neg r}  \\nonumber\n\\end{equation}\nAccording to Definition~\\ref{def:score}, group fairness score $f=\\phi_{r}-\\phi_{\\neg r}$. Thus \n\\begin{equation} \nprob(|f-\\hat{f}|>\\epsilon_{r}+\\epsilon_{\\neg r}) < 1-\\delta_{r}\\delta_{\\neg r} \\nonumber \n\\end{equation} \\hfill $\\qed$\n\\end{theorem}\nThe above theorem provides a theoretical guarantee on the statistical confidence for the group fairness score estimation. \nThat is, based on the Equation~\\ref{equ:fair}, the fairness level for fairness score $f$ is $\\delta_{r}\\delta_{\\neg r}$ and the margin of error is the sum of two margin of errors as $\\epsilon_{r}+\\epsilon_{\\neg r}$. Each $\\epsilon$ is calculated by:\n\\begin{equation} \\label{equ:error}\n\\epsilon=z\\times\\sqrt{\\frac{\\phi(1-\\phi)}{num}}\n\\end{equation}\nwhere $z$ is the value from the standard normal distribution for a certain confidence level $\\delta$ (e.g., for a 95\\% confidence level, $z=1.96$). So the final margin of error for fairness score $f$ is shown in line 11 of Algorithm~\\ref{alg:score}. \nBased on the result, we derive the stopping criteria, as shown in line 12 and 13 of Algorithm~\\ref{alg:score}.  \n\nThe above shows how we compute the group fairness score for one rule set. Given multiple rule sets, we systematically compute the fairness score for each rule set with Algorithm~\\ref{alg:score}, and then rank the rule sets according to the resultant group fairness score. If the group fairness score of certain rule set is more than a given tolerable threshold, we report that discrimination is identified. \\\\\n\n\\noindent \\emph{Example} Take a model trained on the (structured) \\textbf{Census Income} dataset as an example. We fixed the confidence level to 95\\% and the corresponding $z$-value is 1.96. We set the sampling threshold $sample\\_thr$ as 1000 and the error of margin threshold $error\\_thr$ as 0.05. We are given a rule set \n$$\\{gender = Male, race = White, 40 \\leq  age < 60\\}$$ \nFirst, we sample 1000 inputs as $samples_{r}$ using $Sample$ function that represents white males who are older than 40 but younger than 60. Then we sample another 1000 inputs as $samples_{\\neg r}$ using $Sample$ function that represents the rest individuals. We observe that 283 samples in $samples_{r}$ are labeled as ``True'', while only 91 samples in $samples_{\\neg r}$ are labeled as ``True''. \nSo $\\phi_{r}$ is 28.3\\% and $\\phi_{\\neg r}$ is 9.1\\%. According to Algorithm~\\ref{alg:score}, $\\epsilon_{r}$ is 0.028 and $\\epsilon_{\\neg r}$ is 0.018. So the margin of error $\\epsilon$ for fairness score is 0.046. Since $\\epsilon$ is less than 0.05, we stop sampling. Finally, the group fairness score is computed as 19.2\\% with 90.25\\% confidence. \\hfill $\\qed$\n\n\\subsection{Input Sampling}\\label{sec:generation}\n\n\n\n\nAs discussed above, Algorithm~\\ref{alg:score} requires us to sample inputs with a distribution which is similar to the data distribution of the training dataset. As shown in~\\cite{goodfellow2019research}, modern machine learning models mostly rely on the i.i.d. assumptions. That is, the training and test set are assumed to be generated independently from an identical distribution. It is more likely for machine learning models to predict identically distributed data correctly.\n\nWhile it is impossible to know the actual data distribution, we aim to generate additional samples from a distribution as close as possible to the distribution of the training set. For structured data, instead of generating feature vectors randomly, we generate new samples by adding tiny perturbations on original samples uniformly. The perturbation is added to one randomly selected non-sensitive attribute with randomly selected perturbation direction and the perturbation size is 1 for integer variables or 0.01 for decimal variables. Formally, given the rule set $R$, we first search a seed instance from the dataset $D$ as $seed=\\{x_1, x_2, \\cdots, x_N\\}$, where $\\forall r \\in R.~seed \\vDash r$. Then we randomly select a non-sensitive attribute $x_k$, where $x_k \\notin S$. We perturb $x_k$ as $x_k^{\\prime} = x_k + dir \\cdot s\\_pert$, where $dir \\in [-1, 1]$ and $s\\_pert$ is the perturbation size.\n\nFor text data, we generate new samples by replacing sensitive terms with a different term in the same sensitive term category. For example, when we test the machine learning model trained on the \\textbf{Wikipedia Talk Pages} dataset, given a rule set $\\{``gay\"\\}$, we need to generate additional comments containing the term ``gay''. First, we search all comments containing gender-related sensitive terms such as ``lesbian'' and ``bisexual'', as defined in Table~\\ref{tab:terms}. Then we replace these terms in the original comments with the term ``gay'' to generate new comments. That is, we can generate ``I am a gay'' from an original comment ``I am a lesbian''. The reason why we use text replacement instead of text perturbation, as in the case of structured data, is that perturbing texts with synonyms (as proposed in~\\cite{sato2018interpretable} for adversarial attacks) is ineffective to generate the texts in the desired group. Our text generation method also has the benefit of mitigating the influence of data imbalance which may cause unintended bias~\\cite{dixon2018measuring}. Formally, given the rule set $R=\\{r_1, r_2, \\cdots, r_m\\}$, we first search a seed instance from the dataset $D$ as $seed=\\{x_1, x_2, \\cdots, x_N\\}$, where $\\forall r \\in R.~contains(seed,s_r)$, where $s_r$ is the sensitive category referring to $r$ and $contains(d,s_{r})$ is a proposition which is true if and only if $d$ contains at least one term in the category $s_r$. Then we replace the term $x_i$ to term $r_{j}$, for all $r_{j} \\in R$ and $x_i \\in s_{r_{j}}$.\n\n\n\\section{Implementation and Evaluation} \\label{sec:evaluation}\n\nWe have implemented \\tool{TestSGD } as a self-contained software toolkit based on Tensorflow~\\cite{abadi2016tensorflow} with about 6K lines of Python code. \n\\vspace{1mm}\n\n\n\n\\noindent \\emph{Experiment Subjects} Our experiments are based on 8 models trained with the following benchmark datasets. These datasets have been widely used as evaluation subjects in multiple previous studies on fairness~\\cite{zhang2020white, tse2021adf, galhotra2017fairness, dixon2018measuring, ruoss2020learning, ma2020metamorphic}. \n\n\n\\begin{itemize} [leftmargin=*]\n\\item{\\textbf{Census Income}~\\cite{census1996dataset}: \nThe dataset contains more than 30,000 samples and is used to predict whether the income of an adult is above \\$50,000 annually. The attributes $gender$, $race$ and $age$ are sensitive attributes. }\n\\item{\\textbf{Bank Marketing}~\\cite{moro2014data}: \nThe dataset contains 45,000+ samples and is used to train models for predicting whether the client would subscribe a term deposit. Its sensitive attribute is $age$.}\n\\item{\\textbf{German Credit}~\\cite{credit1994dataset}: \nThis is a small dataset with 600 samples. The task is to assess an individual's credit. The sensitive attributes are $gender$ and $age$.}\n\\item{\\textbf{COMPAS}~\\cite{compas2016data}: \nThis dataset contains 7,000+ samples. The task is to predict whether the recidivism risk score for an individual is high. The sensitive attributes are $gender$, $race$ and $age$.}\n\\item{\\textbf{Crime}~\\cite{crime2009dataset}: \nThis dataset contains almost 2,000 data for communities within the US. The task is to predict whether the violent crimes per population in a specific community is high. Since this dataset records population statistics, their sensitive features are shown in multiple attributes with percentage values. Here, we extract all gender\/race\/age related attributes to learn rule sets.}\n\\item{\\textbf{Law School}~\\cite{anthony2003analysis}: \nThis dataset has more than 20,000 application records and is used to predict whether a student passes the bar exam. The attributes, $race$ and $gender$ are sensitive attributes.}\n\\item{\\textbf{Wiki Talk Pages}~\\cite{wulczyn2017ex}: \nThis is a textual dataset containing more than 100,000 Wikipedia TalkPage comments. The task is to predict whether a given comment is toxic. }\n\\item{\\textbf{IMDB}~\\cite{maas-EtAl:2011:ACL-HLT2011}: \nIMDB dataset contains 50,000 movie reviews. The task is to predict whether a given sentence is a positive review.\n}\n\\end{itemize}\n\n\nFor the first six structured datasets, we train a six-layer feed-forward neural network using the exact same configuration as reported in the previous studies~\\cite{zhang2020white,tse2021adf}.  For the last two textual datasets, we train a convolutional neural network (CNN) combined with long short-term memory (LSTM). The details of trained models are shown in Table~\\ref{tab:models}. The accuracy of the trained models is expectedly similar to what is reported in the previous studies. Table~\\ref{tab:parameters} shows the value of parameters used in our experiment to run \\tool{TestSGD}.\nAll experiments are conducted on a server running Ubuntu 1804 with 1 Intel Core 3.10GHz CPU, 32GB memory and 2 NVIDIA GV102 GPU. To mitigate the effect of randomness, all the results are the average of 3 runs. \n\nWe aim to answer multiple research questions as follows.\n\\vspace{1mm}\n\n\\begin{table}[t] \\small\n\\caption{Parameters of the Experiments}\n\\begin{tabular}{|c|c|c|}\n\\hline\nParameters & Value & Discription \\\\ \\hline\n$\\theta$ & 5\\% & support threshold \\\\ \\hline\nsample\\_thr & 1000 & sampling threshold \\\\ \\hline\n$\\delta$ & 95\\% & confidence level \\\\ \\hline\nerror\\_thr & 0.05 & error margin threshold \\\\ \\hline\nz & 1.96 & z value \\\\ \\hline\ns\\_pert & 1 & perturbation size for integer variables \\\\ \\hline\ns\\_pert & 0.01 & perturbation size for decimal variables \\\\ \\hline\n\\end{tabular}\n\\label{tab:parameters}\n\\end{table}\n\n\\begin{table}[t]\n\\caption{Dataset and Models of Experiments}\n\\label{tab:models}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Dataset}     & \\textbf{Model}                    & \\textbf{Accuracy} \\\\ \\hline\nCensus Income        & Six-layer Fully-connected NN       & 86.13\\%  \\\\ \\hline\nBank Marketing       & Six-layer Fully-connected NN       & 91.62\\%  \\\\ \\hline\nGerman Credit        & Six-layer Fully-connected NN       & 100\\%    \\\\ \\hline\nCOMPAS               & Six-layer Fully-connected NN       & 78.99\\%  \\\\ \\hline\nCrime                & Six-layer Fully-connected NN       & 92.52\\%  \\\\ \\hline\nLaw School           & Six-layer Fully-connected NN       & 95.19\\%  \\\\ \\hline\nWikipedia Talk Pages & CNN Long Short-term memory network & 93.89\\%  \\\\ \\hline\nIMDB                 & CNN Long Short-term memory network & 86.68\\%  \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\noindent \\emph{RQ1: Is our method effective in identifying subtle group discrimination of a given machine learning model?} \n\\begin{table*}[t]\n\\caption{Rule Sets and Fairness Scores for Neural Networks}\n\\label{tab:results}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{|c|cc|cc|cc|}\n\\hline\n\\multirow{3}{*}{\\textbf{Dataset}} & \\multicolumn{2}{c|}{\\textbf{top 1}} & \\multicolumn{2}{c|}{\\textbf{top 2}} & \\multicolumn{2}{c|}{\\textbf{top 3}} \\\\ \\cline{2-7} \n & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Rule Set}}} & \\textbf{Fairness Score} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Rule Set}}} & \\textbf{Fairness Score} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{Rule Set}}} & \\textbf{Fairness Score} \\\\  \n & \\multicolumn{1}{c|}{} & ($\\phi_{r}, \\phi_{\\neg r}$) & \\multicolumn{1}{c|}{} & ($\\phi_{r}, \\phi_{\\neg r}$) & \\multicolumn{1}{c|}{} & ($\\phi_{r}, \\phi_{\\neg r}$) \\\\ \\hline\n\\multirow{2}{*}{Census Income} & \\multicolumn{1}{c|}{gender=male, 40$\\leq$age\\textless{}80,} & 20.2\\% & \\multicolumn{1}{c|}{gender=male, 40$\\leq$age\\textless{}70,} & 19.4\\% & \\multicolumn{1}{c|}{gender=male, 40$\\leq$age\\textless{}80, race=White,} & 18.4\\% \\\\  \n & \\multicolumn{1}{c|}{race=White or Asian-Pac-Islander} & (29.9\\%, 9.7\\%) & \\multicolumn{1}{c|}{race=White or Amer-Indian-Eskimo} & (28.9\\%, 9.5\\%) & \\multicolumn{1}{c|}{Asian-Pac-Islander or Amer-Indian-Eskimo} & (26.9\\%, 8.5\\%) \\\\ \\hline\n\\multirow{2}{*}{Bank Marketing} & \\multicolumn{1}{c|}{\\multirow{2}{*}{10 $\\leq$ age < 90}} & 38.2\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{10 $\\leq$ age < 70}} & 22.8\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{10 $\\leq$ age < 60}} & 20.5\\% \\\\ \n & \\multicolumn{1}{c|}{} & (3.3\\%, 41.5\\%) & \\multicolumn{1}{c|}{} & (26.6\\%, 3.8\\%) & \\multicolumn{1}{c|}{} & (4.7\\%, 25.2\\%) \\\\ \\hline\n\\multirow{2}{*}{German Credit} & \\multicolumn{1}{c|}{\\multirow{2}{*}{gendre = female, 60$\\leq$age\\textless{}70}} & 21.9\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender = female, 60$\\leq$age\\textless{}80}} & 21.8\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender = male, 40$\\leq$age\\textless{}80}} & 15.5\\% \\\\  \n & \\multicolumn{1}{c|}{} & (72.5\\%, 50.6\\%) & \\multicolumn{1}{c|}{} & (70.5\\%, 48.7\\%) & \\multicolumn{1}{c|}{} & (52.6\\%, 47.1\\%) \\\\ \\hline\n\\multirow{2}{*}{COMPAS} & \\multicolumn{1}{c|}{gender = male, age$\\geq$40,} & 62.4\\% & \\multicolumn{1}{c|}{gender = male, 40$\\leq$age\\textless{}60,} & 62.3\\% & \\multicolumn{1}{c|}{gender = male, 50$\\leq$age\\textless{}60,} & 62.3\\% \\\\\n & \\multicolumn{1}{c|}{race = Hispanic or other race} & (20.7\\%, 83.1\\%) & \\multicolumn{1}{c|}{race = Hispanic or other race} & (20.3\\%, 82.6\\%) & \\multicolumn{1}{c|}{race = Hispanic} & (19.5\\%, 81.8\\%) \\\\ \\hline\n\\multirow{2}{*}{Law School} & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender = male, race = Asian or Black}} & 15.0\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender = female, race = Asian or Black}} & 11.1\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender = female, race = Black}} & 10.2\\% \\\\ \n & \\multicolumn{1}{c|}{} & (84.5\\%, 99.5\\%) & \\multicolumn{1}{c|}{} & (88.8\\%, 99.9\\%) & \\multicolumn{1}{c|}{} & (89.7\\%, 99.9\\%) \\\\ \\hline\n\\multirow{2}{*}{Crime} & \\multicolumn{1}{c|}{\\multirow{2}{*}{FemalePctDiv$\\geq$0.4, racePctWhite$\\leq$0.8}} & 60.7\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{ FemalePctDiv$\\geq$0.5, racePctWhite$\\leq$0.8}} & 59.6\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{ FemalePctDiv$\\geq$0.5, racePctWhite$\\leq$0.6}} & 59.5\\% \\\\  \n & \\multicolumn{1}{c|}{} & (83.8\\%, 23.2\\%) & \\multicolumn{1}{c|}{} & (87.0\\%, 27.4\\%) & \\multicolumn{1}{c|}{} & (94.6\\%, 35.1\\%) \\\\ \\hline\n\\multirow{2}{*}{Wiki Talk Pages} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"gay\", \"taoist\"}} & 6.5\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"gay\", \"protestant\"}} & 5.4\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"gay\", \"african american\"}} & 5.1\\% \\\\ \n & \\multicolumn{1}{c|}{} & (13.0\\%, 6.5\\%) & \\multicolumn{1}{c|}{} & (12.9\\%, 7.5\\%) & \\multicolumn{1}{c|}{} & (12.5\\%, 7.4\\%) \\\\ \\hline\n\\multirow{2}{*}{IMDB} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"european\", \"yong\"}} & 6.6\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"white\", \"older\"}} & 6.6\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{\"lgbtq\"}} & 6.5\\% \\\\ \n & \\multicolumn{1}{c|}{} & (56.0\\%, 49.4\\%) & \\multicolumn{1}{c|}{} & (59.1\\%, 52.6\\%) & \\multicolumn{1}{c|}{} & (7.5\\%, 14.0\\%) \\\\ \\hline\n\\end{tabular}}\n\\end{table*}\nTo answer the question, we systematically apply our approach to the above-mentioned models and measure the results. The results are summarized in Table~\\ref{tab:results}. It shows results on the six models trained on structured data and results on the two models trained on text data. These four columns show datasets, rule sets, group fairness scores and model accuracies respectively. The favorable label is ``True'', the meaning of which can be found in the above introduction on the corresponding dataset. Note that for each model, we rank the identified subtle discrimination according to the group fairness score and we report the top 3 worst discrimination only. \n\nWe can observe that subtle discrimination does exist in these models, which were never revealed in the previous studies~\\cite{zhang2020white,tse2021adf,galhotra2017fairness,dixon2018measuring,ma2020metamorphic}. \nFor instance, the model trained on the \\textbf{Bank Marketing} dataset predicts only 3.3\\% of the clients who are older than 10 but younger than 90 would subscribe a term deposit, whilst predicting 41.5\\% of clients older than 90 would subscribe a term deposit. All of the top 3 testing results all show the model discriminates against young clients. \nWe remark that although this is unfair according to the definition, it may have its underlying reasons and it is still up to human experts to decide whether it is actual discrimination. \n\nFor the models trained on the \\textbf{Census Income} dataset, \\textbf{German Credit} dataset and the \\textbf{Law School} dataset, they show relatively mild discrimination. In contrast, the model trained on the \\textbf{COMPAS} dataset shows severe discrimination, with a fairness score of 62.4\\%. That is, for Hispanic or other race male individuals who are older than 40, the model is much less likely to predict the recidivism risk as high. For the remaining individuals, the model predicts 83.1\\% of them have high recidivism risk. Top 2 and top 3 test results also show severe discrimination against older Hispanic or other race male individuals. Similarly, the model trained on the \\textbf{Crime} dataset also shows high discrimination. \nDifferent from the first five structured datasets, samples in this dataset have 10 different sensitive features, each of which is a decimal ranging from 0.0 to 1.0 representing the percentage of certain population. \nAs shown in the top 1 testing result, when the percentage of divorced females is above 40\\% and the percentage of Caucasian is below 80\\%, the model is much more likely to predict that the violent crimes per population in this community is high. All testing results on the model trained on \\textbf{Crime} dataset suggest that the model discriminates against communities with high percentage of divorced females and low percentage of Caucasian. \n\n\\begin{comment}\nOur observation is that there are indeed subtle discrimination in these models, which were never revealed in previous studies~\\cite{zhang2020white,tse2021adf,galhotra2017fairness,dixon2018measuring,ma2020metamorphic}. For instance, the model trained on the \\textbf{Census Income} dataset is more likely to predict the income of an individual above \\$50,000 when this person is a male who is older than 40 years old but less than 60 years old and his race is Caucasian or Asian-Pacific-Islander. The group fairness score 20.2\\%, which means an individual in this group is 20.2\\% more likely to be predicted positively. The model trained on the \\textbf{Bank Marketing} dataset only has one sensitive attribute $age$. This model predicts only 3.3\\% of the clients who are older than 10 but younger than 90 would subscribe a term deposit, whilst predicting 41.5\\% of clients older than 90 would subscribe a term deposit. For the models trained on the \\textbf{German Credit} dataset and the \\textbf{Law School} dataset, they show relatively mild discrimination. In contrast, the model trained on the \\textbf{COMPAS} dataset shows severe discrimination, with a fairness score of 57.7\\%. That is, for male individuals who are older than 40 and are Hispanic or other race, the model is much less likely to predict the recidivism risk as high. For the remaining individuals, the model predicts 83.3\\% of them have high recidivism risk. Similarly, the model trained on the \\textbf{Crime} dataset also shows high discrimination. Different from the first five structured datasets, samples in this dataset have 10 different sensitive features, each of which is a decimal ranging from 0.0 to 1.0. There are two gender-related features: $MalePctDiv$ and $FemalePctDiv$, which means the percentage of males\/females who are divorced; four race-related features: $racePctWhite$, $racePctBlack$, $racePctAsian$ and $racePctHisp$, representing the percentage of population that is Caucasian, African American, Asian heritage or Hispanic heritage respectively; and 4 age-related features: $agePct12t21$, $agePct12t29$, $agePct16t24$ and $agePct65up$, each of which represents the percentage of population in a certain age group. As shown in the last row, when the percentage of divorced females is above 50\\% and the percentage of Caucasian is below 70\\%, the model is much more likely to predict that the violent crimes per population in this community is high.  \n\nWe remark that while there are indeed discrimination according to the group fairness definition, such discrimination may have its underlying reasons and may or may not be problematic. We consider it to be an orthogonal question on whether the model should be improved to reduce such discrimination and how. At the least, our method provides a way so that we are aware the existence of such discrimination. Furthermore, being able to identify the condition under which discrimination exists allows us to judge whether the discrimination is indeed problematic or not, and could potentially reveal problems in the training set. \n\\end{comment}\n\nIn Table~\\ref{tab:results}, the last two rows show the results on models trained on the text data. In general, we observe that the models trained on text dataset show less discrimination. The maximum fairness score for the model trained on the \\textbf{Wikipedia Talk Pages} dataset is 6.5\\%. That is, the model predicts 13.0\\% of comments containing both ``gay'' and ``taoist'' as toxic. For other comments (i.e., those without one of these two terms or both), the model predicts only 6.5\\% of them as toxic. Top 2 and top 3 testing results show that the model discriminates against comments containing both ``gay'' and ``protestant'' and comments containing both ``gay'' and ``african american'' respectively. The model trained on the \\textbf{IMDB} dataset shows a similar level of discrimination. It is more likely to predict reviewers containing ``european'' and ``young'' and reviews containing ``white'' and ``older'' as positive. It also shows discrimination against reviews containing ``lgbtq''.\nOur conjecture on why the level of discrimination is considerably lower on these models is that each sample in these text datasets often has many features and as a result, the influence of each term (including sensitive terms) is distributed. \n\n\\begin{tcolorbox}[fonttitle = \\bfseries]\n  \\textbf{Answer to RQ1:} \\tool{TestSGD } is effective in identifying subtle group discrimination in neural networks.\n\\end{tcolorbox}\n\n\n\n\\begin{comment}\n\\begin{table}[t]\n\\caption{Rule Sets and Fairness Scores for Models Trained on Textual Data}\n\\label{tab:text}\n\\resizebox{.95\\columnwidth}{!}{\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\multirow{2}{*}{\\textbf{Dataset}} & \\multirow{2}{*}{\\textbf{Rule Set}} & \\textbf{Fairness Score} \\\\ \n &  & ($\\phi_{r}, \\phi_{\\neg r}$) \\\\ \\hline\n\\multirow{2}{*}{Wiki Talk Pages} & \\multirow{2}{*}{``gay'', ``taoist''} & 3.7\\% \\\\ \n &  & (12.4\\%, 8.8\\%) \\\\ \\hline\n\\multirow{2}{*}{IMDB} & \\multirow{2}{*}{``European'', ``young''} & 7.2\\% \\\\\n &  & (56.0\\%, 48.8\\%) \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\\end{comment}\n\n\\noindent \\emph{RQ2: Is our method efficient?} To answer this question, we measure the amount of time required to identify the subtle discrimination for each model. The total execution time and the numbers of tested rule sets are shown in Table~\\ref{tab:time}. For all models, the time required to identify the subtle discrimination is less than 20 hours. Furthermore, models trained on structured dataset take considerably less time than those trained on text dataset. That is, models trained on the \\textbf{Census Income}, \\textbf{Bank Marketing}, \\textbf{German Credit}, \\textbf{COMPAS} and \\textbf{Law School} take less than 16 minutes. One exception is the model trained on the \\textbf{Crime} dataset that takes more than 8 hours. The main reason is that it has a large number of rule sets, due to a large number of sensitive features (i.e., 10), all of which are continuous features. In contrast, both models trained on text dataset take more than 9 hours to finish. The main reason is that generating additional samples for such dataset takes much more time in general. We remark that the sampling procedure can be easily parallelized and thus we could significantly reduce the time if it is an issue. \n\n\\begin{table}[t] \\small\n\\centering\n\\caption{Time Taken to Identify the subtle discrimination}\n\\label{tab:time}\n\\begin{tabular}{|c|c|c|}\n\\hline\n\\textbf{Dataset} & \\textbf{Time (seconds)} & \\textbf{\\#rule set} \\\\ \\hline\nCensus Income & 869.35 & 880 \\\\ \\hline\nBank Marketing & 141.52 & 34 \\\\ \\hline\nGerman Credit & 104.85 & 53 \\\\ \\hline\nCOMPAS & 908.5 & 1590 \\\\ \\hline\nLaw School & 18.46 & 17 \\\\ \\hline\nCrime & 29150.01 & 13282 \\\\ \\hline\nWiki Talk Pages & 34982.28 & 732 \\\\ \\hline\nIMDB & 69125.16 & 876 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\nNote that the support threshold $\\theta$ is set to be 5\\% in all the above experiments. Intuitively, it means that each rule must be relevant to 5\\% of the population (although the rule set, which is a conjunction of multiple rules, may impact a smaller population). This hyper-parameter largely determines how many rule sets that we must examine and thus may have an impact on the execution time. We thus conduct additional experiments with different $\\theta$ values, ranging from 1\\% to 50\\%, to evaluate the effect of $\\theta$ on the execution time and the results. The results on two models, i.e., the model on \\textbf{Law School} and the model on \\textbf{COMPAS}, are detailed in Table~\\ref{tab:support thr}.\n\n\\begin{table}[t] \\small\n\\caption{Effect of Different $\\theta$}\n\\label{tab:support thr}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{|c|c|c|c|c|c|}\n\\hline\n\\multirow{2}{*}{\\textbf{Dataset}} & \\multirow{2}{*}{\\textbf{$\\theta$}} & \\textbf{Time} & \\multirow{2}{*}{\\textbf{\\#rule sets}} & \\multirow{2}{*}{\\textbf{Rule Set}} & \\textbf{Fairness} \\\\ \n &  & \\textbf{(seconds)} &  &  & \\textbf{Score} \\\\ \\hline\n\\multirow{10}{*}{Law School} & \\multirow{2}{*}{1\\%} & \\multirow{2}{*}{46.71} & \\multirow{2}{*}{59} & gender=male, & \\multirow{2}{*}{16.3\\%} \\\\ \n &  &  &  & race=Black &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{5\\%} & \\multirow{2}{*}{18.46} & \\multirow{2}{*}{17} & gender=male, & \\multirow{2}{*}{15.0\\%} \\\\ \n &  &  &  & race=Asian or Black &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{10\\%} & \\multirow{2}{*}{17.83} & \\multirow{2}{*}{16} & gender=male, & \\multirow{2}{*}{1.0\\%} \\\\ \n &  &  &  & race=Asian or White &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{20\\%} & \\multirow{2}{*}{17.83} & \\multirow{2}{*}{16} & gender=male, & \\multirow{2}{*}{0.9\\%} \\\\ \n &  &  &  & race=Asian or White &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{50\\%} & \\multirow{2}{*}{6.28} & \\multirow{2}{*}{2} & gender=male, & \\multirow{2}{*}{0.3\\%} \\\\ \n &  &  &  & race=other race &  \\\\ \\hline\n\\multirow{10}{*}{COMPAS} & \\multirow{2}{*}{1\\%} & \\multirow{2}{*}{1175.79} & \\multirow{2}{*}{2063} & gender=male, age$\\geq$40 & \\multirow{2}{*}{62.4\\%} \\\\ \n &  &  &  & race=Hispanic or other race &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{5\\%} & \\multirow{2}{*}{908.50} & \\multirow{2}{*}{1590} & gender=male, age$\\geq$40 & \\multirow{2}{*}{62.4\\%} \\\\ \n &  &  &  & race=Hispanic or other race &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{10\\%} & \\multirow{2}{*}{676.74} & \\multirow{2}{*}{1180} & gender=male, age$\\geq$20 & \\multirow{2}{*}{43.9\\%} \\\\ \n &  &  &  & race=Hispanic or other race &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{20\\%} & \\multirow{2}{*}{0} & \\multirow{2}{*}{0} & \\multirow{2}{*}{NULL} & \\multirow{2}{*}{NULL} \\\\\n &  &  &  &  &  \\\\ \\cline{2-6} \n & \\multirow{2}{*}{50\\%} & \\multirow{2}{*}{0} & \\multirow{2}{*}{0} & \\multirow{2}{*}{NULL} & \\multirow{2}{*}{NULL} \\\\\n &  &  &  &  &  \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\nThe table shows the execution time, the number of rule sets and the worst group fairness score. We can observe that, the larger a $\\theta$ we set, the fewer rule sets, the less execution time and the smaller group fairness score in general. If the threshold $\\theta$ is too low, e.g., 1\\%, we spend a lot of time on testing a huge number of rule sets, which may not be interesting (one such example is $\\{gender=Male, age \\geq 100\\}$). In contrast, if the threshold $\\theta$ is too high, e.g., 20\\% or 50\\%, there may only exists few or even none rule set (as in the case of the model trained on the \\textbf{COMPAS} dataset).  \n\nWe note that different $\\theta$ may result in different discrimination being identified. For the model trained on \\textbf{Law School}, the rule set shows that the model discriminates against black or Asian males the most when $\\theta$ is 5\\%. However, when we set $\\theta$ to be 1\\%, the model is shown to discriminate against black male individuals the most. For the model trained on the \\textbf{COMPAS} dataset, the model discriminates against Hispanic or other race males who is older than 40 years old most when we set $\\theta$ to be 5\\%. However, when we set $\\theta$ higher (i.e., 10\\%), the age range is expanded to be over 20 years in the identified rule set. Such a result is expected as a large $\\theta$ requires us to find discrimination against a large group. What is considered to be a reasonable value for $\\theta$ is a complicated question, which should probably be answered by lawmakers.\n\n\\begin{tcolorbox}[fonttitle = \\bfseries]\n  \\textbf{Answer to RQ2:} \\tool{TestSGD } is reasonably efficient.\n\\end{tcolorbox}\n\n\n\n\n\n\\begin{table*}[t] \\small\n\\centering\n\\caption{Discrimination Mitigation for Neural Networks}\n\\label{tab:mitigation}\n \\resizebox{0.65\\linewidth}{!}{\n\\begin{tabular}{|c|c|cc|cc|}\n\\hline\n\\multirow{3}{*}{\\textbf{Dataset}} & \\multirow{3}{*}{\\textbf{Rule Set}} & \\multicolumn{2}{c|}{\\textbf{Before}} & \\multicolumn{2}{c|}{\\textbf{After}} \\\\ \\cline{3-6} \n &  & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{accuracy}}} & \\textbf{Fairness Score} & \\multicolumn{1}{c|}{\\multirow{2}{*}{\\textbf{accuracy}}} & \\textbf{Fairness Score} \\\\   \n &  & \\multicolumn{1}{c|}{} & ($\\phi_{r}, \\phi_{\\neg r}$) & \\multicolumn{1}{c|}{} & ($\\phi_{r}, \\phi_{\\neg r}$) \\\\ \\hline\n\\multirow{2}{*}{Census Income} & gender=male, 40$\\leq$age<80, & \\multicolumn{1}{c|}{\\multirow{2}{*}{86.1\\%}} & 20.2\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{86.2\\%}} & 10.1\\% \\\\ \n & race=White or Asian-Pac-Islander & \\multicolumn{1}{c|}{} & (29.9\\%, 9.7\\%) & \\multicolumn{1}{c|}{} & (18.9\\%, 8.8\\%) \\\\ \\hline\n\\multirow{2}{*}{Bank Marketing} & \\multirow{2}{*}{10 $\\leq$ age < 90} & \\multicolumn{1}{c|}{\\multirow{2}{*}{91.6\\%}} & 38.2\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{90.6\\%}} & 5.4\\% \\\\ \n &  & \\multicolumn{1}{c|}{} & (3.3\\%, 41.5\\%) & \\multicolumn{1}{c|}{} & (6.9\\%, 12.3\\%) \\\\ \\hline\n\\multirow{2}{*}{German Credit} & \\multirow{2}{*}{gender = female, 60$\\leq$age<70} & \\multicolumn{1}{c|}{\\multirow{2}{*}{100.0\\%}} & 21.9\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{100.0\\%}} & 7.3\\% \\\\ \n &  & \\multicolumn{1}{c|}{} & (72.3\\%, 50.6\\%) & \\multicolumn{1}{c|}{} & (45.9\\%, 53.2\\%) \\\\ \\hline\n\\multirow{2}{*}{COMPAS} & gender = male, age$\\geq$40, & \\multicolumn{1}{c|}{\\multirow{2}{*}{79.0\\%}} & 62.4\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{78.5\\%}} & 4.2\\% \\\\ \n & race = Hispanic or other race & \\multicolumn{1}{c|}{} & (20.7\\%, 83.1\\%) & \\multicolumn{1}{c|}{} & (80.9\\%, 85.1\\%) \\\\ \\hline\n\\multirow{2}{*}{Law School} & \\multirow{2}{*}{gender = male, race = Black} & \\multicolumn{1}{c|}{\\multirow{2}{*}{95.2\\%}} & 15.0\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{95.1\\%}} & 7.5\\% \\\\\n &  & \\multicolumn{1}{c|}{} & (84.5\\%, 99.5\\%) & \\multicolumn{1}{c|}{} & (92.3\\%, 99.8\\%) \\\\ \\hline\n\\multirow{2}{*}{Crime} & \\multirow{2}{*}{FemalePctDiv$\\geq$0.4, racePctWhite$\\leq$0.8} & \\multicolumn{1}{c|}{\\multirow{2}{*}{93.9\\%}} & 60.7\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{98.1\\%}} & 51.4\\% \\\\ \n &  & \\multicolumn{1}{c|}{} & (83.8\\%, 23.2\\%) & \\multicolumn{1}{c|}{} & (90.6\\%, 39.2\\%) \\\\ \\hline\n\\multirow{2}{*}{Wiki Talk Pages} & \\multirow{2}{*}{\"gay\", \"taoist\"} & \\multicolumn{1}{c|}{\\multirow{2}{*}{93.9\\%}} & 6.5\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{95.5\\%}} & 0.4\\% \\\\ \n &  & \\multicolumn{1}{c|}{} & (13.0\\%, 6.5\\%) & \\multicolumn{1}{c|}{} & (8.4\\%, 8.0\\%) \\\\ \\hline\n\\multirow{2}{*}{IMDB} & \\multirow{2}{*}{\"european\", \"yong\"} & \\multicolumn{1}{c|}{\\multirow{2}{*}{86.7\\%}} & 6.6\\% & \\multicolumn{1}{c|}{\\multirow{2}{*}{84. \\%}} & 3.3\\% \\\\ \n &  & \\multicolumn{1}{c|}{} & (56.0\\%, 49.4\\%) & \\multicolumn{1}{c|}{} & (43.7\\%, 40.4\\%) \\\\ \\hline\n\\end{tabular}}\n\\end{table*}\n\n\n\\noindent \\emph{RQ3: Can we mitigate subtle discrimination using our testing results?} To further show the usefulness of our approach, we evaluate whether we can mitigate the identified subtle discrimination using our testing results. The idea is to mitigate the discrimination by retraining. We remark that there are alternative approaches for improving fairness as well~\\cite{kearns2018preventing,salimi2019interventional}. Note that we generate additional instances satisfying the rule set with the sampling approach described in Section~\\ref{sec:generation}. We only select those generated instances with the opposite label. For example, the model trained on \\textbf{COMPAS} is more likely to predict elderly males who are Hispanic or other race with ``False'' label. We can use the $Sample$ function to generate instances satisfying the condition that are labeled as ``True'' according to the original model. Afterward, we retrain the original model with these additional instances and testing the subtle discrimination with respect to the same rule set to see the improvement. Note that we gradually increase the number of additional instances from 50 to 10\\% of original dataset size to achieve the lowest fairness score without decreasing the accuracy of the retrained model. \n\n\nWe only consider the top 1 worst rule sets to mitigate the discrimination. The results are shown in Table~\\ref{tab:mitigation} for six models trained on additional structured data and two models retrained on additional textual data. We can observe that all models show reduced subtle discrimination and almost the same accuracy. The fairness scores for retrained models on \\textbf{Census Income}, \\textbf{German Credit} and \\textbf{Law School} decrease by about half. For the most improvement, the model retrained on the \\textbf{COMPAS} dataset shows much less subtle discrimination as the fairness score decreases by more than 10 times, i.e., from 57.7\\% to 4.2\\%. The fairness score of the model trained on the \\textbf{Crime} dataset decreases from 60.7\\% to 51.4\\%. Relatively, the fairness improvement is not obvious. We believe that it is due to its many continuous sensitive features and the large number of features (i.e., each input contains more than 100 attributes). That is, it would require a lot more additional data to improve fairness. In terms of CNN models, the fairness score decreases from from 6.5\\% to 0.4\\% for the model retrained on \\textbf{Wikipedia Talk Pages} and decreases from 6.6\\% to 3.3\\% for the model retrained on \\textbf{IMDB}. \n\\begin{tcolorbox}[fonttitle = \\bfseries]\n  \\textbf{Answer to RQ3:} \\tool{TestSGD}$ $ is useful in mitigating the identified subtle group discrimination through retraining.\n\\end{tcolorbox}\n\n\n\\noindent \\emph{Comparison with Baselines} We identify the following two baselines from literature which can potentially identify similar group discrimination as our work.\n\\textbf{1) THEMIS~\\cite{galhotra2017fairness}} \ncalculates group discrimination scores over combinations of multiple features (subgroups) by measuring the difference between the maximum and minimum frequencies of two subgroups on randomly generated samples. \nThose subgroups \ncan then be regarded as identified discrimination if the score is higher than a threshold. \\textbf{2) FairFictPlay \\cite{kearns2018preventing}} proposed an in-processing algorithm aiming to improve subgroup fairness. The subgroups are identified with user-provided constraints in the form of conjunctions of Boolean attributes, linear threshold functions, or bounded degree polynomial threshold functions over multiple protected features. \n\n\n\nIn Table~\\ref{tab:baseline}, we show the identified group discrimination with \\tool{TestSGD}, Themis and FairFictPlay respectively, along with the fairness scores,\non the same models trained on structured data (similarly to  \nTable~\\ref{tab:models}). \nWe set the timeout as 24 hours. Note that FairFictPlay uses complex linear functions on all the protected features (which are hard to interpret) to define discriminatory subgroups, thus \nwe do not show the exact concrete linear functions in the table. \nWe have the following observations. 1) Compared to FairFictPlay, \\tool{TestSGD } identifies discrimination with higher scores (more discriminating) while being interpretable. Moreover, \\tool{TestSGD } automatically identifies the discriminated subgroups without any prior knowledge. \n2) Similar to \\tool{TestSGD}, Themis is able to identify discriminated subgroups automatically. However, Themis identifies two subgroups which are maximally different (in terms of being predicted favorably) while \\tool{TestSGD } identifies subgroups which are predicted different from the rest. These two approaches thus produce results that are complementary to each other. Note that Themis does not support text data.   \n\n\n\n\n\\begin{table*}[t]\n\\caption{Comparisons Between \\tool{TestSGD }, Themis and FairFictPlay. `-' means timeout.}\n\\label{tab:baseline}\n\\resizebox{\\linewidth}{!}{\n\\begin{tabular}{|c|cc|cc|cc|}\n\\hline\n\\multirow{2}{*}{\\textbf{Dataset}} & \\multicolumn{2}{c|}{\\textbf{\\tool{TestSGD }}} & \\multicolumn{2}{c|}{\\textbf{Themis}} & \\multicolumn{2}{c|}{\\textbf{FairFictPlay}} \\\\ \\cline{2-7} \n & \\multicolumn{1}{c|}{\\textbf{Rule Set}} & \\textbf{Fairness Score} & \\multicolumn{1}{c|}{\\textbf{Sensitive Attributes' values for Max\/Min Proportion}} & \\textbf{Fairness Score} & \\multicolumn{1}{c|}{\\textbf{Subgroup}} & \\textbf{Fairness Score} \\\\ \\hline\n\\multirow{2}{*}{Census Income} & \\multicolumn{1}{c|}{Gender=male, 40$\\leq$age<80,} & \\multirow{2}{*}{20.20\\%} & \\multicolumn{1}{c|}{{[}gender=Female, 60$\\leq$age\\textless{}70, race=Asian-Pac\\_islander{]} -} & \\multirow{2}{*}{26.6\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{13.9\\%} \\\\\n & \\multicolumn{1}{c|}{race=White or Asian-Pac\\_islander} &  & \\multicolumn{1}{c|}{{[}gender=Male, 10$\\leq$age\\textless{}20, race=White{]}} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\multirow{2}{*}{Bank Marketing} & \\multicolumn{1}{c|}{\\multirow{2}{*}{10$\\leq$age\\textless{}90}} & \\multirow{2}{*}{38.2\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{{[}60$\\leq$age\\textless{}70{]} - {[}10$\\leq$age\\textless{}20{]}}} & \\multirow{2}{*}{8.4\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{7.6\\%} \\\\\n & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\multirow{2}{*}{German Credit} & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender=feamle, 60$\\leq$age\\textless{}70}} & \\multirow{2}{*}{21.9\\%} & \\multicolumn{1}{c|}{{[}gender=Female, 80$\\leq$age\\textless{}90{]} -} & \\multirow{2}{*}{17.1\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{7.0\\%} \\\\\n & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{{[}gender=Male, 10$\\leq$age\\textless{}20{]}} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\multirow{2}{*}{COMPAS} & \\multicolumn{1}{c|}{gender=male, age$\\geq$40,} & \\multirow{2}{*}{62.4\\%} & \\multicolumn{1}{c|}{{[}gender=Female, 10$\\leq$age\\textless{}20, race=Native American{]} -} & \\multirow{2}{*}{67.3\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{22.4\\%} \\\\\n & \\multicolumn{1}{c|}{race=Hispanic or other race} &  & \\multicolumn{1}{c|}{{[}gender=Male, 60$\\leq$age\\textless{}70, race=other race{]}} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\multirow{2}{*}{Law School} & \\multicolumn{1}{c|}{\\multirow{2}{*}{gender=male, race=Asian or Black}} & \\multirow{2}{*}{15.0\\%} & \\multicolumn{1}{c|}{{[}gender=Male, race=White{]} -} & \\multirow{2}{*}{13.5\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{3.7\\%} \\\\\n & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{{[}gender=Female, race=Black{]}} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\multirow{2}{*}{Crime} & \\multicolumn{1}{c|}{\\multirow{2}{*}{FemalePctDiv$\\geq$0.4, racePctWhite$\\leq$0.8}} & \\multirow{2}{*}{60.7\\%} & \\multicolumn{1}{c|}{\\multirow{2}{*}{-}} & \\multirow{2}{*}{-} & \\multicolumn{1}{c|}{\\multirow{2}{*}{Linear Threshold Function}} & \\multirow{2}{*}{38.8\\%} \\\\\n & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{} &  & \\multicolumn{1}{c|}{} &  \\\\ \\hline\n\\end{tabular}}\n\\end{table*}\n\n\\vspace{-1em}\n\\section{Related Work} \\label{sec:related work}\nMany existing works attempted to test discrimination according to different fairness definitions and measurements~\\cite{dwork2012fairness, calders2010three}. In~\\cite{feldman2015certifying}, Feldman \\emph{et al.} provide a fairness definition which is measured according to demographic parity of model predictions. It measures how well the sensitive class can be predicted based on classification accuracy. In~\\cite{hardt2016equality}, Hardt \\emph{et al.} present an alternate definition of fairness based on demographic parity. It requires a decision to be independent of the sensitive attribute. In~\\cite{kusner2017counterfactual}, Kusner \\emph{et al.} define counterfactual discrimination which focuses on single decisions towards an individual. A prediction is counterfactual fair if it is the same in the actual group and a different demographic group. In~\\cite{galhotra2017fairness}, Galhotra \\emph{et al.} propose causal discrimination to measure the fraction of inputs for which model causally discriminates. This definition is similar to counterfactual fairness, but it takes instances of discrimination into account. In~\\cite{kearns2018preventing}, Kearns \\emph{et al.} proposed an in-processing algorithm aiming to improve the fairness of given subgroups, where subgroups are defined as conjunctions of attributes, linear threshold functions, or bounded degree polynomial threshold functions over multiple protected features.\nMost existing works~\\cite{galhotra2017fairness, kleinberg2016inherent, biswas2020machine} use positive classification rate as fairness measurement. \n \nSubsequently, many works focus on individual discrimination to generate individual discriminatory instances~\\cite{zhang2020white,tse2021adf,agarwal2018automated, huchard2018proceedings}. They tried to generated instances which are classified differently after changing sensitive attributes. \nIn~\\cite{agarwal2018automated}, Agarwal \\emph{et al.} present an automated testing approach to generate test inputs to find individual discrimination. In~\\cite{ruoss2020learning}, Ruoss \\emph{et al.} propose a fairness representation framework to generalize individual fairness to multiple notions. It learns a mapping from similar individuals to latent representations.\nHowever, the testing on individual discrimination cannot provide a statistical measurement of fairness.  \n\nSome other existing works attempted to test model discrimination with fairness score measurements. In~\\cite{tramer2017fairtest}, Tramer \\emph{et al.} propose an unwarranted associations framework to detect unfair, discriminatory or offensive user treatment in data-driven applications. It identifies discrimination according to multiple metrics including the CV score, related ratio and associations between outputs and sensitive attributes. In~\\cite{kleinberg2016inherent}, Kleinberg \\emph{et al.} also test multiple discrimination scores and compare different fairness metrics. In~\\cite{galhotra2017fairness}, Galhotra \\emph{et al.} propose a tool called THEMIS to measure software discrimination. It tests discrimination with two fairness definitions, i.e., group discrimination score and causal discrimination score. \nIn~\\cite{adebayo2016iterative}, Adebayo \\emph{et al.} try to determine the relative significance of a model's inputs in determining the outcomes and use it to assess the discriminatory extent of the model. \n\nSome prior work has been done on fairness for text classification tasks as well. In~\\cite{blodgett2017racial}, Blodgett \\emph{et al.} discuss the impact of unfair natural language in NLP and show how statistical discrimination arises in processing applications. \nIn~\\cite{bolukbasi2016man}, Bolukbasi \\emph{et al.} show gender bias in the world embedding and provide a methodology for modifying an embedding to remove gender bias. In~\\cite{dixon2018measuring}, Dixon \\emph{et al.} measure discrimination using a set of common demographic identity terms and propose a method to mitigate the unintended bias by balancing the training data. \n\nCompared with all the above-mentioned existing works, we provide further fairness testing. Instead of measuring the overall discrimination, our approach systematically identifies and measures subtle discrimination. That is, we not only measure statistical discrimination with a confidence guarantee but also offer interpretable rule sets to represent subtle discrimination. \n\nThis work is remotely related to works on applying rule-based models for model explanation. In~\\cite{yang2017scalable}, Yang \\emph{et al.} present an algorithm for building probabilistic rule lists with logical IF-THEN structure.\nIn~\\cite{lakkaraju2016interpretable}, Lakkaraju \\emph{et al.} propose interpretable decision sets to interpret model predictions with high accuracy and high interpretation. Our work leverage such rule-based interpretable structure to present subtle discrimination in models. \n\n\\vspace{-1em}\n\\section{Conclusion} \\label{sec:conclusion}\nIn this work, we focus on testing neural network models against subtle group discrimination and propose a framework to systematically identify interpretable subtle group discrimination based on group fairness measurement with a certain confidence. \nOur extensive evaluation demonstrates that subtle group discrimination in neural networks is common to a surprising level. \nWe also show that it is possible to mitigate such discrimination by utilizing our testing results to generate more data for retraining. \n\n\n\\balance\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFirst-passage percolation (FPP) was introduced by Hammersley and Welsh \\cite{HW} as a model for fluid flow through a porous medium. However, it has since developed into a field of its own, serving for instance as a model for growing interfaces (see \\cite{KS} and connections to other models \\cite{HH}) and competing infections (see \\cite{BS, DH, GM, HP, Hoffman1}). For a survey of recent results, see \\cite{GK}.\n\nWe consider FPP on $(\\ensuremath{\\mathbb{Z}^2},\\ensuremath{\\mathcal{E}^2}),$ the two-dimensional square lattice. $\\mathbb{P}$ will denote a probability measure on the space $\\Omega = \\mathbb{R}^{\\ensuremath{\\mathcal{E}^2}}$ (satisfying some conditions outlined in the next section). An element $\\omega \\in \\Omega$ represents an edge-weight configuration; the passage time across the edge $e$ is denoted $\\omega_e = \\omega(e)$. The passage time between two sites $x,y$ will be called\n\\[\n\\tau(x,y) = \\inf_{\\gamma:x \\to y} \\tau(\\gamma)\\ ,\n\\]\nwhere the infimum is over all (finite) lattice paths from $x$ to $y$ and $\\tau(\\gamma) = \\sum_{e \\in \\gamma} \\omega_e$. \n\n\nIn this paper we study geodesics, (typically self-avoiding) paths in $\\mathbb{Z}^2$ which are everywhere time-minimizing. Precisely, define a finite geodesic from $x$ to $y$ to be a finite lattice path $\\gamma$ from $x$ to $y$ such that $\\tau(\\gamma) = \\tau(x,y)$. Define an infinite geodesic to be an infinite path such that each finite subpath is a finite geodesic. In the mid 90's, Newman \\cite{Newman95} and Licea-Newman \\cite{LN}, along with Wehr \\cite{Wehr} began the rigorous study of infinite geodesics. This was in part motivated by connections between ``bigeodesics'' in FPP and ground states of disordered ferromagnetic spin models \\cite{FLN, newmanbook}. The main questions involve existence of infinite geodesics with asymptotic directions, uniqueness and coalescence of such geodesics, and absence of bigeodesics. After considerable progress on lattice FPP, Howard and Newman gave an essentially complete description for a continuum variant, called Euclidean FPP \\cite{HN}.\n\nThe main theorems proved to date require heavy assumptions on the model, for instance strong moment bounds and so-called curvature inequalities (the establishment of which provides a major open problem in FPP). The main goals of this paper are to prove versions of the current geodesic theorems under minimal assumptions necessary to guarantee their validity. Because the methods of Newman and collaborators involve curvature bounds and concentration inequalities (the latter of which cannot hold under low moment assumptions), we are forced to develop completely new techniques. \n\nOur analysis centers on Busemann functions, which were used and analyzed in papers of Hoffman \\cite{Hoffman1, Hoffman2}. His work was one of the first (along with Garet-Marchand \\cite{GM}) to assert existence of multiple disjoint infinite geodesics under general assumptions, finding at least four almost surely. The methods are notable in their ability to extract any information without knowing the existence of limits for Busemann functions. Indeed, proving the existence of such limits, corresponding to\n\\[\n\\lim_{n \\to \\infty} \\left[ \\tau(x,x_n) - \\tau(y,x_n) \\right]\n\\]\nfor fixed $x,y$ and a deterministic sequence of vertices $(x_n)$ growing to infinity along a ray, provides a major open problem and appears to be an impediment to further analysis of geodesics in the model. Incidentally, in an effort to describe the microstructure of the limiting shape for the model, Newman \\cite{Newman95} was able to show that under strong assumptions, this limit exists in Lebesgue-almost every direction. \n\nOne main aim of the present paper is to develop a framework to overcome the existence of the above limit. We will analyze distributional limits of Busemann functions and relate these back to the first-passage model. The relationship between Busemann functions and geodesics will be preserved in the limit and will provide information about directional geodesics, coalescence, and the structure of geodesic graphs, the latter of which gives nonexistence of certain types of bigeodesics.\n\n\n\n\n\n\\subsection{Main results}\n\nWe will make one of two main assumptions on the passage time distribution. These relate to the degree of independence in the model.  The first deals with i.i.d. passage times:\n\\begin{enumerate}\n\\item[{\\bf A1}] $\\mathbb{P}$ is a product measure whose common distribution satisfies the criterion of Cox and Durrett \\cite{coxdurrett}: if $e_1, \\ldots, e_4$ are the four edges touching the origin,\n\\begin{equation}\\label{eq: jackisherenow}\n\\mathbb{E} \\left[ \\min_{i=1,\\ldots, 4} \\omega_{e_i} \\right]^2 < \\infty\\ .\n\\end{equation}\nFurthermore we assume $\\mathbb{P}(\\omega_e= 0) < p_c=1\/2$, the bond percolation threshold for $\\mathbb{Z}^2$.\n\\end{enumerate}\nCondition \\eqref{eq: jackisherenow} is implied by, for example, the assumption $\\mathbb{E} \\omega_e < \\infty$.\n\nThe other assumption is on distributions that are only translation-invariant. Condition (d) below deals with the limit shape, which is defined in the next paragraph.\n\\begin{enumerate}\n\\item[{\\bf A2}] $\\mathbb{P}$ is a measure satisfying the conditions of Hoffman \\cite{Hoffman2}:\n\\begin{enumerate}\n\\item $\\mathbb{P}$ is ergodic with respect to translations of $\\mathbb{Z}^2$;\n\\item $\\mathbb{P}$ has all the symmetries of $\\mathbb{Z}^2$;\n\\item $\\mathbb{E} \\omega_e^{2+\\varepsilon} < \\infty$ for some $\\varepsilon>0$;\n\\item the limit shape for $\\mathbb{P}$ is bounded.\n\\end{enumerate}\n\\end{enumerate}\nSome of the conditions here can be weakened. For instance, the $2+\\varepsilon$ moment condition can be replaced with a condition of a finite Lorentz-type norm; see \\cite{boivin} for details.\n\nIn each of these settings, a ``shape theorem'' has been proved \\cite{boivin, coxdurrett} for the set of sites accessible from 0 in time $t$.  For $x,y \\in \\mathbb{R}^2$ we set $\\tau(x,y) = \\tau(\\tilde x, \\tilde y)$, where $\\tilde x$ and $\\tilde y$ are the unique points in $\\mathbb{Z}^2$ such that $x \\in \\tilde x + [-1\/2,1\/2)^2$ and $y \\in \\tilde y + [-1\/2,1\/2)^2$. For any $t\\geq 0$ write $B(t)$ for the set of $x$ in $\\mathbb{R}^2$ such that $\\tau(0,x) \\leq t$ and $B(t)\/t = \\{x\/t : x \\in B(t)\\}$.  There exists a deterministic compact convex set $\\mathcal{B}$, symmetric about the axes and with nonempty interior such that for each $\\varepsilon>0$,\n\\[\n\\mathbb{P}\\left( (1-\\varepsilon) \\mathcal{B} \\subseteq B(t)\/t \\subseteq (1+\\varepsilon) \\mathcal{B} \\text{ for all large } t\\right) = 1\\ .\n\\]\nThe statement that $\\mathcal{B}$ has nonempty interior is not explicitly proved in \\cite{boivin} but follows from the maximal lemma stated there.\n\n\\subsubsection{Directional results}\n\nOur first results deal with asymptotic directions for infinite geodesics. Much is known about such questions under various strong assumptions (for instance uniformly positive curvature of $\\mathcal{B}$, exponential moments for $\\mathbb{P}$; see Section~\\ref{sec: global} for a more precise discussion). However, under only {\\bf A1} or {\\bf A2}, very little is known. After initial results by H\\\"aggstr\\\"om-Pemantle \\cite{HP}, Garet-Marchand \\cite{GM} and Hoffman \\cite{Hoffman1}, it was proved by Hoffman \\cite{Hoffman2} that under {\\bf A2}, there exist at least 4 infinite geodesics that are pairwise disjoint almost surely. Nothing is known about the directions of the geodesics; for instance, Hoffman's results do not rule out the case in which the geodesics spiral around the origin.\n\nBelow we will show that under {\\bf A1} or {\\bf A2} there are geodesics that are asymptotically directed in sectors of aperture no bigger than $\\pi\/2$. Under a certain directional condition on the boundary of the limit shape (see Corollary~\\ref{cor: exposed}) we show existence of geodesics with asymptotic direction. To our knowledge, the only work of this type so far \\cite[Theorem~2.1]{Newman95} requires a global curvature assumption to show the existence of geodesics in even one direction.\n\nTo describe the results, we endow $[0,2\\pi)$ with the distance of $S^1$: say that $dist(\\theta_1,\\theta_2) < r$ if there exists an integer $m$ such that $|\\theta_1-\\theta_2 - 2\\pi m| < r$. For $\\Theta \\subseteq [0,2\\pi)$ we say that a path $\\gamma = x_0, x_1, \\ldots$ is {\\it asymptotically directed in} $\\Theta$ if for each $\\varepsilon>0$, $\\arg x_k \\in \\Theta_\\varepsilon \\text{ for all large } k$, where $\\Theta_\\varepsilon = \\{\\theta: dist(\\theta,\\phi) < \\varepsilon \\text{ for some } \\phi \\in \\Theta\\}$. For $\\theta \\in [0,2\\pi)$, write $v_\\theta$ for the unique point of $\\partial \\mathcal{B}$ with argument $\\theta$. Recall that a supporting line $L$ for $\\mathcal{B}$ at $v_\\theta$ is one that touches $\\mathcal{B}$ at $v_\\theta$ such that $\\mathcal{B}$ lies on one side of $L$. If $\\theta$ is an angle such that $\\partial \\mathcal{B}$ is differentiable at $v_\\theta$ (and therefore has a unique supporting line $L_\\theta$ (the tangent line) at this point), we define an interval of angles $I_\\theta$:\n\\begin{equation}\\label{eq: last_night}\nI_\\theta = \\{\\theta' : v_{\\theta'} \\in L_\\theta\\}\\ .\n\\end{equation}\n\n\n\\begin{thm}\\label{thm: sectors}\nAssume either {\\bf A1} or {\\bf A2}. If $\\partial \\mathcal{B}$ is differentiable at $v_\\theta$, then with probability one there is an infinite geodesic containing the origin which is asymptotically directed in $I_\\theta$.\n\\end{thm}\n\nThe meaning of the theorem is that there is a measurable set $\\mathcal{A}$ with $\\mathbb{P}(\\mathcal{A})=1$ such that if $\\omega \\in \\mathcal{A}$, there is an infinite geodesic containing the origin in $\\omega$ which is asymptotically directed in $I_\\theta$. This also applies to any result we state with the phrases ``with probability one there is an infinite geodesic'' or ``with probability one there is a collection of geodesics.''\n\nWe now state two corollaries. A point $x \\in \\partial \\mathcal{B}$ is {\\it exposed} if there is a line that touches $\\mathcal{B}$ only at $x$.\n\n\\begin{cor}\\label{cor: exposed}\nAssume either {\\bf A1} or {\\bf A2}. Suppose that $v_\\theta$ is an exposed point of differentiability of $\\partial \\mathcal{B}$. With probability one there exists an infinite geodesic containing the origin with asymptotic direction $\\theta$.\n\\end{cor}\n\\begin{proof}\nApply Theorem~\\ref{thm: sectors}, noting that $I_\\theta = \\{\\theta\\}$.\n\\end{proof}\n\nIn the next corollary we show that there are infinite geodesics asymptotically directed in certain sectors. Because the limit shape is convex and compact, it has at least 4 extreme points. Angles corresponding to the arcs connecting these points can serve as the sectors.\n\n\\begin{cor}\\label{cor: extreme}\nAssume either {\\bf A1} or {\\bf A2}. Let $\\theta_1\\neq \\theta_2$ be such that $v_{\\theta_1}$ and $v_{\\theta_2}$ are extreme points of $\\mathcal{B}$. If $\\Theta$ is the set of angles corresponding to some arc of $\\partial \\mathcal{B}$ connecting $v_{\\theta_1}$ to $v_{\\theta_2}$, then with probability one there exists an infinite geodesic containing the origin which is asymptotically directed in $\\Theta$.\n\\end{cor}\n\n\\begin{proof}\n\nChoose $\\theta_3 \\in \\Theta$ such that $\\theta_1 \\neq \\theta_3 \\neq \\theta_2$ and $\\mathcal{B}$ has a unique supporting line $L_{\\theta_3}$ at $v_{\\theta_3}$ (this is possible since the boundary is differentiable almost everywhere). \nLet $C$ be the closed arc of $\\partial \\mathcal{B}$ from $v_{\\theta_1}$ to $v_{\\theta_2}$ that contains $v_{\\theta_3}$ and write $D$ for its open complementary arc. We claim $D \\subseteq I_{\\theta_3}^c$. This will prove the corollary after applying Theorem~\\ref{thm: sectors} with $\\theta = \\theta_3$.\n\nFor a contradiction, suppose that $L_{\\theta_3}$ intersects $D$ at some point $v_{\\phi}$ and write $S$ for the segment of $L_{\\theta_3}$ between $v_{\\theta_3}$ and $v_\\phi$. Since $L_{\\theta_3}$ is a supporting line, the set $\\mathcal{B}$ lies entirely on one side of it. On the other hand, since $\\mathcal{B}$ is convex and $v_{\\theta_3}, v_\\phi \\in \\mathcal{B}$, $S \\subseteq \\mathcal{B}$. Therefore $S \\subseteq \\partial \\mathcal{B}$ and must be an arc of the boundary. It follows that one of $v_{\\theta_1}$ or $v_{\\theta_2}$ is in the interior of $S$, contradicting the fact that these are extreme points of $\\mathcal{B}$.\n\\end{proof}\n\n\\begin{rem}\nIf $\\mathbb{P}$ is a product measure with $\\mathbb{P}(\\omega_e=1) = \\vec p_c \\text{ and } \\mathbb{P}(\\omega_e <1) = 0$, where $\\vec p_c$ is the critical value for directed percolation, \\cite[Theorem~1]{AD11} implies that $(1\/2,1\/2)$ is an exposed point of differentiability of $\\mathcal{B}$. Corollary~\\ref{cor: exposed} then gives a geodesic in the direction $\\pi\/4$. Though all points of $\\partial \\mathcal{B}$  (for all measures not in the class of Durett-Liggett \\cite{durrettliggett}) should be exposed points of differentiability, this is the only proven example.\n\\end{rem}\n\n\\begin{rem}\nFrom \\cite[Theorem~1.3]{HM}, for any compact convex set $\\mathcal{C}$ which is symmetric about the axes with nonempty interior, there is a measure $\\mathbb{P}$ satisfying {\\bf A2} (in fact, with bounded passage times) which has $\\mathcal{C}$ as a limit shape. Taking $\\mathcal{C}$ to be a Euclidean disk shows that there exist measures for which the corresponding model obeys the statement of Corollary~\\ref{cor: exposed} in any deterministic direction $\\theta$.\n\\end{rem}\n\n\n\n\\subsubsection{Global results}\\label{sec: global}\n\nIn this section we use the terminology of Newman \\cite{Newman95}. Call $\\theta$ a {\\it direction of curvature} if there is a Euclidean ball $B_\\theta$ with some center and radius such that $\\mathcal{B} \\subseteq B_\\theta$ and $\\partial B_\\theta \\cap \\mathcal{B} = \\{v_\\theta\\}$. We say that $\\mathcal{B}$ has {\\it uniformly positive curvature} if each direction is a direction of curvature and there exists $M< \\infty$ such that the radius of $B_\\theta$ is bounded by $M$ for all $\\theta$.\n\nIn \\cite[Theorem~2.1]{Newman95}, Newman has shown that under the assumptions (a) $\\mathbb{P}$ is a product measure with $\\mathbb{E} e^{\\beta \\omega_e}< \\infty$ for some $\\beta>0$, (b) the limit shape $\\mathcal{B}$ has uniformly positive curvature and (c) $\\omega_e$ is a continuous variable, two things are true with probability one.\n\\begin{enumerate}\n\\item For each $\\theta \\in [0,2\\pi)$, there is an infinite geodesic with asymptotic direction $\\theta$.\n\\item Every infinite geodesic has an asymptotic direction.\n\\end{enumerate}\nAs far as we know, there has been no weakening of these assumptions.\n\nBelow we improve on Newman's theorem. We first reduce the moment assumption on $\\mathbb{P}$ to that of {\\bf A1}. Next we extend the theorem to non-i.i.d. measures. Newman's proof uses concentration inequalities of Kesten \\cite{Kesten} and Alexander \\cite{Alexander}, which require exponential moments on the distribution (and certainly independence). So to weaken the moment assumptions we need to use a completely different method, involving Busemann functions instead.\n\nTo state the theorem, we make slightly stronger hypotheses:\n\\begin{enumerate}\n\\item[{\\bf A1'}] $\\mathbb{P}$ satisfies {\\bf A1} and the common distribution of $\\omega_e$ is continuous.\n\\item[{\\bf A2'}] $\\mathbb{P}$ satisfies {\\bf A2} and $\\mathbb{P}$ has unique passage times.\n\\end{enumerate}\nThe phrase ``unique passage times'' means that for all paths $\\gamma$ and $\\gamma'$ with distinct edge sets, $\\mathbb{P}(\\tau(\\gamma)=\\tau(\\gamma'))=0$.\n\n\\begin{thm}\\label{thm: newman}\nAssume either {\\bf A1'} or {\\bf A2'} and that $\\mathcal{B}$ has uniformly positive curvature.\n\\begin{enumerate}\n\\item With $\\mathbb{P}$-probability one, for each $\\theta$ there is an infinite geodesic with direction $\\theta$.\n\\item With $\\mathbb{P}$-probability one, every infinite geodesic has a direction.\n\\end{enumerate}\n\\end{thm}\n\nThe same method of proof shows the following. \n\n\\begin{cor}\\label{cor: newman2}\nAssume either {\\bf A1'} or {\\bf A2'} and suppose $v_\\theta$ is an exposed point of differentiability of $\\partial \\mathcal{B}$ for all $\\theta$. Then the conclusions of Theorem~\\ref{thm: newman} hold.\n\\end{cor}\n\n\\begin{rem}\nThe proofs of the above two results only require that the set of extreme points of $\\mathcal{B}$ is dense in $\\partial \\mathcal{B}$. In fact, a similar result holds for a sector in which extreme points of $\\mathcal{B}$ are dense in the arc corresponding to this sector.\n\\end{rem}\n\n\n\n\n\n\\subsubsection{Coalescence for geodesics}\\label{subsec: coalesce}\n\nIn this section we describe results for coalescence of infinite geodesics. For this we need some notation. For $S\\subseteq \\mathbb{R}^2$ define the point-to-set passage time \n\\[\n\\tau(x,S) = \\inf_{y \\in S} \\tau(x,y) \\text{ for } x \\in \\mathbb{R}^2\\ .\n\\]\nBy the subadditivity property $\\tau(x,y) \\leq \\tau(x,z) + \\tau(z,y)$ we find\n\\begin{equation}\\label{eq: subadditivity}\n\\tau(x,S) \\leq \\tau(x,y) + \\tau(y,S) \\text{ for } x,y \\in \\mathbb{R}^2\\ .\n\\end{equation}\nA path $\\gamma$ from a point $x \\in \\ensuremath{\\mathbb{Z}^2}$ to a point in \n\\begin{equation}\\label{eq: hatS}\n\\hat S = \\{y \\in \\ensuremath{\\mathbb{Z}^2} : y + [-1\/2,1\/2)^2 \\cap S \\neq \\varnothing\\}\n\\end{equation}\nis called a {\\it geodesic from $x$ to $S$} if $\\tau(\\gamma) = \\tau(x,S)$. Under assumptions {\\bf A1} or {\\bf A2}, one can argue from the shape theorem and boundedness of the limit shape that a geodesic from $x$ to $S$ exists $\\mathbb{P}$-almost surely. However, it need not be unique.  In the case, though, that we assume {\\bf A1'} or {\\bf A2'}, there is almost surely exactly one geodesic from $x$ to $S$. Note that if $\\gamma$ is a geodesic from $x$ to $S$ and $y\\in \\gamma$, then the piece of $\\gamma$ from $x$ to $y$ is a geodesic from $x$ to $y$ and the piece of $\\gamma$ from $y$ to $S$ is a geodesic from $y$ to $S$.\n\nThe set $S$ gives a directed geodesic graph $\\mathbb{G}_S = \\mathbb{G}_S(\\omega)$: $\\langle x,y \\rangle$ is an edge of $\\mathbb{G}_S$ if it is in some geodesic from a point to $S$ and $\\tau(x,S) \\geq \\tau(y,S)$ (we will explain more about this graph in Section~\\ref{subsec: GG}). We say that a sequence of directed graphs $G_n = (\\mathbb{Z}^2,E_n)$ converges to a directed graph $G=(\\mathbb{Z}^2,E)$ if each edge $\\langle x,y \\rangle$ is in only finitely many of the symmetric differences $E_n \\Delta E$. If $x$ and $y$ are vertices of a directed graph $G$, write $x \\to y$ if there is a directed path from $x$ to $y$ in $G$. Last, we say that two infinite directed paths $\\Gamma$ and $\\Gamma'$ {\\it coalesce} if their (edge) symmetric difference is finite.\n\nFor the main theorems on coalescence we need an extra assumption in the case {\\bf A2'}. It allows us to apply ``edge modification'' arguments. Write $\\omega = (\\omega_e,\\check{\\omega})$, where $\\check{\\omega}_f = (\\omega)_{f\\neq e}$. \n\n\\begin{df}\nWe say that $\\ensuremath{\\mathbb{P}}$ has the {\\it upward finite energy property} if for each $\\lambda > 0$ such that $\\mathbb{P}(\\omega_e \\geq \\lambda)>0$,\n\t\t\\begin{equation}\n\t\t\\label{finite_energy_def}\n\t\t\\ensuremath{\\mathbb{P}}\\left(\\omega_e \\geq \\lambda \\, \\big|\\, \\check{\\omega} \\right) >0 \\quad \\text{almost surely}\\ .\n\t\t\\end{equation}\n\\end{df}\nNote that if $\\mathbb{P}$ is a product measure, it has the upward finite energy property.\n\n\\begin{thm}\\label{thm: random_hyperplanes}\nAssume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property. Let $v \\in \\mathbb{R}^2$ be any nonzero vector and for $\\beta \\in \\mathbb{R}$ define\n\\[\nL_\\beta(v) = \\{y \\in \\mathbb{R}^2 : y \\cdot v = \\beta\\}\\ .\n\\]\nThere exists an event $\\mathcal{A}$ with $\\mathbb{P}(\\mathcal{A}) = 1$ such that for each $\\omega \\in \\mathcal{A}$, the following holds. There exists an ($\\omega$-dependent) increasing sequence $(\\alpha_k)$ of real numbers with $\\alpha_k \\to \\infty$ such that $\\mathbb{G}_{L_{\\alpha_k}(v)}(\\omega) \\to G(\\omega)$, a directed graph with the following properties.\n\\begin{enumerate}\n\\item Viewed as an undirected graph, $G$ has no circuits.\n\\item Each $x \\in \\mathbb{Z}^2$ has out-degree 1 in $G$.\n\\item (All geodesics coalesce.) Write $\\Gamma_x$ for the unique infinite path in $G$ from $x$. If $x,y \\in \\mathbb{Z}^2$ then $\\Gamma_x$ and $\\Gamma_y$ coalesce.\n\\item (Backward clusters are finite.) For all $x \\in \\mathbb{Z}^2$, the set $\\{y \\in \\mathbb{Z}^2: y \\to x \\text{ in } G\\}$ is finite.\n\\end{enumerate} \n\\end{thm}\n\nOur last theorem deals with coalescence and asymptotic directions. Before stating it, we discuss some previous results. In 1995, Licea and Newman \\cite{LN} proved that given $\\theta \\in [0,2\\pi)$, all directional geodesics almost surely coalesce except in some deterministic (Lebesgue-null) set $D \\subseteq [0,2\\pi)$. Specifically they showed that under the assumptions (a) $\\mathbb{P}$ is a product measure whose one-dimensional marginals are continuous with finite exponential moments and (b) uniformly positive curvature of $\\mathcal{B}$,\n\\begin{equation}\\label{eq: liceanewman}\n\\text{there exists }D \\subseteq [0,2\\pi) \\text{ with Lebesgue measure zero such that if } \\theta \\in [0,2\\pi) \\setminus D\\ ,\n\\end{equation}\n\\begin{enumerate}\n\\item almost surely, there exists a collection of infinite geodesics $\\{\\gamma_x : x \\in \\mathbb{Z}^2\\}$ such that each $\\gamma_x$ has asymptotic direction $\\theta$ and for all $x,y$, the paths $\\gamma_x$ and $\\gamma_y$ coalesce and \n\\item almost surely, for each $x$, there is a unique infinite geodesic containing $x$ with asymptotic direction $\\theta$.\n\\end{enumerate}\nSince \\cite{LN} it has been an open problem to show that $D$ can be taken to be empty. Zerner \\cite[Theorem~1.5]{newmanbook} proved that $D$ can be taken to be countable. In a related exactly solvable model (directed last-passage percolation, using exponential weights on sites), Coupier has proved \\cite[Theorem~1(3)]{coupier}, building on work of Ferrari-Pimentel \\cite{FP}, that $D$ can be taken to be empty. These results rely on a mapping to the TASEP particle system.\n\nIn part 2 of the next theorem, we improve on \\eqref{eq: liceanewman} in the general case. The result reduces the set $D$ to be empty for existence of coalescing geodesics (item 1 above). It however does not address uniqueness. We reduced the moment condition of \\cite{LN}, extended to non-i.i.d. measures and replaced the global curvature assumption with a directional condition. Without this condition, part 3 gives the existence of coalescing geodesics directed in sectors. For the statement, recall the definition of $I_\\theta$ in \\eqref{eq: last_night}.\n\n\\begin{thm}\\label{thm: exceptional_set}\nAssume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property. Let $\\theta \\in [0,2\\pi)$.\n\\begin{enumerate}\n\\item If $\\partial \\mathcal{B}$ is differentiable at $v_\\theta$ then with probability one there exists a collection $\\{\\gamma_x:x \\in \\mathbb{Z}^2\\}$ of infinite geodesics in $\\omega$ such that \n\\begin{enumerate}\n\\item each $x$ is a vertex of $\\gamma_x$;\n\\item each $\\gamma_x$ is asymptotically directed in $I_\\theta$;\n\\item for all $x,y \\in \\mathbb{Z}^2$, $\\gamma_x$ and $\\gamma_y$ coalesce and\n\\item each $x$ is on $\\gamma_y$ for only finitely many $y$.\n\\end{enumerate}\n\\item If $v_\\theta$ is an exposed point of differentiability of $\\mathcal{B}$ then the above geodesics all have asymptotic direction $\\theta$.\n\\item Suppose $\\theta_1 \\neq \\theta_2$ are such that $v_{\\theta_1}$ and $v_{\\theta_2}$ are extreme points of $\\mathcal{B}$. If $\\Theta$ is the set of angles corresponding to some arc of $\\partial \\mathcal{B}$ connecting $v_{\\theta_1}$ to $v_{\\theta_2}$ then the above geodesics can be taken to be asymptotically directed in $\\Theta$.\n\\end{enumerate}\n\\end{thm}\n\nTheorems~\\ref{thm: random_hyperplanes} and \\ref{thm: exceptional_set} follow from a stronger result. In Sections~\\ref{sec: GG} and \\ref{sec: coalesceG}, we prove that any subsequential limit $\\mu$ defined as in Section~\\ref{sec: mudef} is supported on geodesic graphs with properties 1-4 of Theorem~\\ref{thm: random_hyperplanes}.\n\n\\begin{rem}\nThe finiteness of backward clusters in the graphs produced in the previous two theorems (see item 4 of the first and item 1(d) of the second) is related to nonexistence of bigeodesics. It shows that when constructing infinite geodesics using a certain limiting procedure, it is impossible for doubly infinite paths to arise.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\\subsection{Notation}\n\nWe denote the standard orthonormal basis vectors for \\ensuremath{\\mathbb{R}^2} by ${\\mathbf{e}}_1$ and ${\\mathbf{e}}_2.$  The translation operators $T_{{\\mathbf{e}}_i},~ i = 1,2$\nact on a configuration $\\omega$ as follows: \n$\\left(T_{{\\mathbf{e}}_i} (\\omega) \\right)_{e'} = \\omega_{e'-{\\mathbf{e}}_i}.$ Under any of the assumptions laid out above, the measure $\\mathbb{P}$ is invariant under these translations. Furthermore the passage times have a certain translation-covariance: for $i=1,2$,\n\\begin{equation}\\label{eq: transcov}\n\\tau(x,S)(T_{{\\mathbf{e}}_i}\\omega) = \\tau(x-{\\mathbf{e}}_i,S-{\\mathbf{e}}_i)(\\omega)\\ ,\n\\end{equation}\nwhere $S-{\\mathbf{e}}_i = \\{ x-{\\mathbf{e}}_i : x \\in S\\}$.\n\nWe shall need a function $g:\\mathbb{R}^2 \\to \\mathbb{R}$ which describes the limiting shape $\\mathcal{B}$. It is the norm whose closed unit ball is $\\mathcal{B}$. There are many ways to define it; for instance one can use $g(x) = \\inf\\{\\lambda > 0:  x\/\\lambda \\in \\mathcal{B}\\}$. It follows from the shape theorem that under {\\bf A1} or {\\bf A2},\n\\[\n\\lim_{n \\to \\infty} \\tau(0,nx)\/n = g(x) \\text{ for all } x \\in \\mathbb{R}^2, ~\\mathbb{P}\\text{-almost surely}\\ .\n\\]\nFurthermore, there is convergence in $L^1$:\n\\[\n\\lim_{n \\to \\infty} \\mathbb{E}\\tau(0,nx)\/n = g(x) \\text{ for all } x \\in \\mathbb{R}^2\\ .\n\\]\nIn the case of {\\bf A1} this follows from \\cite[Lemma~3.2]{coxdurrett} and under {\\bf A2} it can be derived from the shape theorem and \\cite[Lemma~2.6]{Hoffman2} (the reader can also see a derivation in the appendix of \\cite{gouere}). We denote the $\\ell^1$ norm on $\\mathbb{R}^2$ by $\\|\\cdot\\|_1$ and the $\\ell^2$ norm by $\\| \\cdot \\|_2 .$ Since the limit shape is bounded and has nonempty interior, there are constants $0 < C_1, C_2 < \\infty$ such that\n\\begin{equation}\\label{eq: normequivalence}\nC_1 \\|x\\|_2 \\leq g(x) \\leq C_2 \\|x\\|_2 \\text{ for all } x \\in \\mathbb{R}^2\\ .\n\\end{equation}\n\nWe recall the fact that under {\\bf A1} or {\\bf A2},\n\\begin{equation}\\label{eq: finitesecondmoment}\n\\mathbb{E} \\tau(x,y)^2 < \\infty \\text{ for all } x,y \\in \\mathbb{R}^2\\ .\n\\end{equation}\nThis was proved in \\cite[Lemma~3.1]{coxdurrett} assuming {\\bf A1} and in the other case it follows directly from the fact that $\\mathbb{E} \\omega_e^{2+\\varepsilon}<\\infty$ for some $\\varepsilon>0$.\n\nWe write $x \\cdot y$ for the standard dot product between $x$ and $y$ in $\\mathbb{R}^2$. \n\n\\begin{center}\n{\\bf For the rest of the paper we assume A1 or A2.}\n\\end{center}\n\n\n\n\\subsection{Structure of the paper}\nIn the next section, we give basic properties of Busemann functions and geodesic graphs. In Section~\\ref{sec: BID} we introduce Busemann increment configurations and construct probability measures on them. Next we reconstruct Busemann functions, and in Section~\\ref{sec: limits} we prove a shape theorem for the reconstruction.  Section~\\ref{sec: GG} begins the study of distributional limits $\\mathbb{G}$ of geodesic graphs, where we show that all paths are asymptotically directed in a sector given by the reconstructed Busemann function. In Section~\\ref{sec: coalesceG} we show coalescence of all paths in $\\mathbb{G}$. We use all of these tools in Section~\\ref{sec: proofs} to prove the main results of the paper.\n\n\n\n\n\n\n\n\n\\section{Busemann functions and geodesic graphs}\n\nIn this section we will give basic properties of Busemann functions and geodesic graphs. These will be carried over through weak limits to a space introduced in the next section. \n\n\n\\subsection{Busemann functions}\n\nFor any $S \\subseteq \\mathbb{R}^2$ and configuration $\\omega$, we define the Busemann function $B_S : \\ensuremath{\\mathbb{Z}^2}\\times \\ensuremath{\\mathbb{Z}^2} \\to \\mathbb{R}$ as\n\\[\nB_S(x,y) = \\tau(x,S) - \\tau(y,S)\\ ,\n\\]\nThis function measures the discrepancy between travel times from $x$ and $y$ to $S$. We list below some basic properties of Busemann functions. One of the most interesting is the additivity property 1. It is the reason that the asymptotic shape for the Busemann function is a half space whereas the asymptotic shape for $\\tau$ is a compact set.\n\n\\begin{prop}\\label{prop: busemannprop1}\nLet $S \\subseteq \\mathbb{R}^2$. The Busemann function $B_S$ satisfies the following properties $\\mathbb{P}$-almost surely for $x,y,z \\in \\ensuremath{\\mathbb{Z}^2}$:\n\\begin{enumerate}\n\\item (Additivity)\n\\begin{equation}\\label{eq: busemannadditivity}\nB_S(x,y) = B_S(x,z) + B_S(z,y)\\ .\n\\end{equation}\n\\item for $i=1,2$,\n\\begin{equation}\\label{eq: busemanntranscov}\nB_S(x,y)(T_{{\\mathbf{e}}_i} \\omega) = B_{S-{\\mathbf{e}}_i}(x-{\\mathbf{e}}_i,y-{\\mathbf{e}}_i)(\\omega)\\ .\n\\end{equation}\nTherefore the finite-dimensional distributions of $B_S$ obey a translation invariance:\n\\[\n\\left( B_S(x,y) \\right) \\underset{d}{=} \\left(B_{S-{\\mathbf{e}}_i}(x-{\\mathbf{e}}_i,y-{\\mathbf{e}}_i) \\right)\\ .\n\\]\n\\item\n\\begin{equation}\\label{eq: bboundtau}\n|B_S(x,y)| \\leq \\tau(x,y)\\ .\n\\end{equation}\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThe first property follows from the definition. The third is a consequence of subadditivity \\eqref{eq: subadditivity} of $\\tau(y,S)$. The second item follows from the statement \\eqref{eq: transcov} for passage times.\n\\end{proof}\n\nThe last property we need regards the relation between geodesics and Busemann functions. Though it is simple, it will prove to be important later.\n\\begin{prop}\\label{prop: busemannpassagetime}\nLet $S \\subseteq \\mathbb{R}^2$ and $x \\in \\ensuremath{\\mathbb{Z}^2}$. If $\\gamma$ is a geodesic from $x$ to $S$ and $y$ is a vertex of $\\gamma$ then $B_S(x,y) = \\tau(x,y)$.\n\\end{prop}\n\n\\begin{proof}\nWrite $\\tau_\\gamma(x,y)$ for the passage time along $\\gamma$ between $x$ and $y$. Since every segment of a geodesic itself a geodesic, $\\tau(x,S)-\\tau(y,S) = \\tau_\\gamma(x,S) - \\tau_\\gamma(y,S) = \\tau_\\gamma(x,y) = \\tau(x,y)$.\n\\end{proof}\n\nUsing this proposition and additivity of the Busemann function we can relate $B_S(x,y)$ to coalescence. If $\\gamma_x$ and $\\gamma_y$ are geodesics from $x$ and $y$ to $S$ (respectively) and they meet at a vertex $z$ then $B_S(x,y) = \\tau(x,z)-\\tau(y,z)$. This is a main reason why Busemann functions are useful for studying coalescence of geodesics.\n\n\n\n\n\n\n\n\n\n\\subsection{Geodesic graphs}\\label{subsec: GG}\n\nFor any $S \\subseteq \\ensuremath{\\mathbb{Z}^2}$ and configuration $\\omega$, we denote the set of edges in all geodesics from a point $v \\in \\ensuremath{\\mathbb{Z}^2}$ to $S$ as $G_S(v)$. We regard each geodesic in $G_S(v)$ as a directed path, giving orientation $\\langle x,y \\rangle$ to an edge if $\\tau(x,S) \\geq \\tau(y,S)$ (the direction in which the edge is crossed), and set $\\vec{G}_S(v)$ to be the union of these directed edges. Let $\\mathbb{G}_S(\\omega)$ be the directed graph induced by the edges in $\\cup_v \\vec{G}_S(v)$. Last, define the configuration $\\eta_S(\\omega)$ of directed edges by\n\\[\n\\eta_S(\\omega)(\\langle x,y \\rangle) = \\begin{cases}\n1 & \\text{if }\\langle x,y \\rangle \\in \\vec{G}_S(v) \\text{ for some } v \\\\\n0 & \\text{otherwise}\n\\end{cases}\\ .\n\\]\nFor $S \\subseteq \\mathbb{R}^2$ we define $\\eta_S(\\omega)$ and $\\mathbb{G}_S(\\omega)$ using $\\hat S$ as in \\eqref{eq: hatS}.\n\n\\begin{prop}\\label{prop: firstGG}\nLet $S \\subseteq \\ensuremath{\\mathbb{R}^2}$. The graph $\\mathbb{G}_S$ and the collection $(\\eta_S)$ satisfy the following properties $\\mathbb{P}$-almost surely.\n\\begin{enumerate}\n\\item Every finite directed path is a geodesic. It is a subpath of a geodesic ending in $S$.\n\\item If there is a directed path from $x$ to $y$ in $\\mathbb{G}_S$ then $B_S(x,y) = \\tau(x,y)$.\n\\item For $i=1,2$,\n\\begin{equation}\\label{eq: GGtranscov}\n\\eta_S(e)(T_{{\\mathbf{e}}_i}\\omega) = \\eta_{S-{\\mathbf{e}}_i}(e-{\\mathbf{e}}_i)(\\omega)\\ .\n\\end{equation}\nTherefore the finite dimensional distributions of $\\eta_S$ obey a translation invariance:\n\\[\n(\\eta_S(e)) \\underset{d}{=} (\\eta_{S-{\\mathbf{e}}_i}(e-{\\mathbf{e}}_i))\\ .\n\\]\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nThe third property follows from translation covariance of passage times \\eqref{eq: transcov}. The second property follows from the first and Proposition~\\ref{prop: busemannpassagetime}.\n\nTo prove the first, let $\\gamma$ be a directed path in $\\mathbb{G}_S$ and write the edges of $\\gamma$ in order as $e_1,\\ldots, e_n$. Write $J \\subseteq \\{1, \\ldots, n\\}$ for the set of $k$ such that the path $\\gamma_k$ induced by $e_1, \\ldots, e_k$ is a subpath of a geodesic from some vertex to $S$. We will show that $n \\in J$. By construction of $\\mathbb{G}_S$, the edge $e_1$ is in a geodesic from some point to $S$, so $1 \\in J$. Now suppose that $k \\in J$ for some $k < n$; we will show that $k+1 \\in J$. Take $\\sigma$ to be a geodesic from a point $z$ to $S$ which contains $\\gamma_k$ as a subpath. Write $\\sigma'$ for the portion of the path from $z$ to the far endpoint $v_k$ of $e_k$ (the vertex to which $e_k$ points). The edge $e_{k+1}$ is also in $\\mathbb{G}_S$ so it is in a geodesic from some point to $S$. If we write $\\hat \\sigma$ for the piece of this geodesic from $v_k$ of $e_k$ to $S$, we claim that the concatenation of $\\sigma'$ with $\\hat \\sigma$ is a geodesic from $z$ to $S$. To see this, write $\\tau_{\\tilde \\gamma}$ for the passage time along a path $\\tilde \\gamma$:\n\\[\n\\tau(z,S) = \\tau_\\sigma(z,v_k) + \\tau_\\sigma(v_k,S) = \\tau_{\\sigma'}(z,v_k) + \\tau_{\\hat \\sigma}(v_k,S)\\ .\n\\]\nThe last equality holds since both the segment of $\\hat \\sigma$ from $v_k$ to $S$ and the segment of $\\sigma$ from $v_k$ to $S$ are geodesics, so they have equal passage time. Hence $k+1 \\in J$ and we are done.\n\\end{proof}\n\nNote that each vertex $x \\notin \\hat S$ has out-degree at least 1 in $\\mathbb{G}_S$. Furthermore it is possible to argue using part 1 of the previous proposition and the shape theorem that there are no infinite directed paths in $\\mathbb{G}_S$. Since we will not use this result later, we omit the proof. Once we take limits of measures on such graphs later, infinite paths will appear.\n\nIf $\\mathbb{P}$ has unique passage times, we can say more about the structure of $\\mathbb{G}_S$. \n\\begin{prop}\\label{prop: secondGG}\nAssume {\\bf A1'} or {\\bf A2'}. The following properties hold $\\mathbb{P}$-almost surely.\n\\begin{enumerate}\n\\item Each vertex $x \\notin \\hat S$ has out-degree 1. Here $\\hat S$ is defined as in \\eqref{eq: hatS}.\n\\item Viewed as an undirected graph, $\\mathbb{G}_S$ has no circuits.\n\\end{enumerate}\n\\end{prop}\n\\begin{proof}\nFor the first property note that every vertex $x\\notin \\hat S$ has out-degree at least 1 because there is a geodesic from the vertex to $S$ and the first edge is directed away from $x$. Assuming $x$ has out-degree at least 2 then we write $e_1$ and $e_2$ for two such directed edges. By the previous proposition, there are two geodesics $\\gamma_1$ and $\\gamma_2$ from $x$ to $S$ such that $e_i \\in \\gamma_i$ for $i=1,2$. If either of these paths returned to $x$ then there would exist a finite path with passage time equal to 0. By the ergodic theorem there would then be infinitely many distinct paths with passage time 0 (with positive probability), contradicting unique passage times. This implies that $\\gamma_1$ and $\\gamma_2$ have distinct edge sets. However, they have the same passage time, again contradicting unique passage times.\n\nFor the second property suppose that there is a circuit in the undirected version of $\\mathbb{G}_S$. Each vertex has out-degree 1, so this is actually a directed circuit and thus a geodesic. But then it has passage time zero, giving a contradiction as above.\n\\end{proof}\n\nProperty 2 implies that $\\mathbb{G}_S$, viewed as an undirected graph, is a forest. It has more than one component if and only if $\\hat S$ has size at least 2. We will see later that after taking limits of measures on these graphs, the number of components will reduce to 1.\n\n\n\n\n\n\n\n\n\n\n\\section{Busemann increment distributions}\\label{sec: BID}\n\nWe are interested in taking limits of measures on Busemann functions and geodesic graphs. \nWe will choose a one-parameter family of lines $L_\\alpha = L + \\alpha \\mathbf{v}$ for $\\mathbf{v}$ a normal vector to $L$ and consider the Busemann functions $B_{L_\\alpha}(x,y)$. The main question is whether or not the limit\n\\begin{equation}\\label{eq: newmanlimit}\n\\lim_{\\alpha \\to \\infty} B_{L_\\alpha}(x,y)\n\\end{equation}\nexists for $x,y \\in \\mathbb{Z}^2$. If one could show this, then one could prove many results about FPP, for instance, that infinite geodesics with an asymptotic direction always exist. Under an assumption of uniformly positive curvature of the limit shape $\\mathcal{B}$ and exponential moments for the common distribution of the $\\omega_e$'s (in the case that $\\mathbb{P}$ is a product measure) Newman \\cite{Newman95} has shown the existence of this limit for Lebesgue-almost every unit vector $\\mathbf{v}$.\n\nWe will try to overcome the difficulty of existence of limits \\eqref{eq: newmanlimit} by enlarging the space to work with subsequential limits in a systematic way. This technique is inspired by work \\cite{AD, ADNS} on ground states of short-range spin glasses.\n\n\n\n\n\n\n\n\n\\subsection{Definition of $\\mu$}\\label{sec: mudef}\n\nWe begin by assigning a space for our passage times. Let $\\Omega_1 = \\mathbb{R}^{\\mathbb{Z}^2}$ be a copy of $\\Omega$. A sample point in $\\Omega_1$ we call $\\omega$ as before. Our goal is to enhance this space to keep track of Busemann functions and geodesic graphs. We will take limits in a fixed direction, so for the remainder of this section, let $\\varpi\\in \\partial \\mathcal{B}$ and let $g_\\varpi$ be any linear functional on $\\mathbb{R}^2$ that takes its maximum on $\\mathcal{B}$ at $\\varpi$ with $g_\\varpi(\\varpi)=1$. The nullspace of $g_\\varpi$ is then a translate of a supporting line for $\\mathcal{B}$ at $\\varpi$.\nFor $\\alpha \\in \\mathbb{R}$, define \n\\[\nL_\\alpha = \\left\\{ x \\in \\mathbb{R}^2: g_\\varpi(x) = \\alpha \\right\\}\\ .\n\\]\nFor future reference, we note the inequality\n\\begin{equation}\\label{eq: gvarpibound}\n\\text{for all } x \\in \\mathbb{R}^2,~ g_\\varpi(x) \\leq g(x)\\ .\n\\end{equation}\nIt clearly holds if $x \\neq 0$. Otherwise since $x\/g(x) \\in \\mathcal{B}$, $1 \\geq g_\\varpi(x\/g(x)) = g_\\varpi(x)\/g(x)$.\n\n\nGiven $\\alpha \\in \\mathbb{R}$ and $\\omega \\in \\Omega_1$, write $B_\\alpha(x,y)(\\omega) = B_{L_\\alpha}(x,y)(\\omega)$. Define the space $\\Omega_2 = (\\mathbb{R}^2)^{\\mathbb{Z}^2}$ with the product topology and Borel sigma-algebra and the {\\it Busemann increment configuration} $B_\\alpha(\\omega) \\in \\Omega_2$ as\n\\begin{align*}\nB_\\alpha(\\omega) = \\big( \\, B_\\alpha(v, v+{\\mathbf{e}}_1),\\, B_\\alpha(v,v+{\\mathbf{e}}_2)\\, \\big)_{v \\in \\ensuremath{\\mathbb{Z}^2}}\\ .\n\\end{align*}\n\nWe also consider directed graphs of geodesics. These are points in a directed graph space $\\Omega_3 = \\{0,1\\}^{\\vec{\\mathcal{E}}^2}$, where $\\vec{\\mathcal{E}}^2$ is the set of oriented edges $\\langle x,y \\rangle$ of $\\mathbb{Z}^2$, and we use the product topology and Borel sigma-algebra. For $\\eta \\in \\Omega_3$, write $\\mathbb{G} = \\mathbb{G}(\\eta)$ for the directed graph induced by the edges $e$ such that $\\eta(e) = 1$. Using the definition from the last section, set\n\\[\n\\eta_\\alpha(\\omega) = \\eta_{L_\\alpha}(\\omega) \\in \\Omega_3 \\text{ and } \\mathbb{G}_\\alpha(\\omega) = \\mathbb{G}(\\eta_\\alpha(\\omega)) \\text{ for } \\alpha \\in \\mathbb{R}\\ .\n\\]\n\n\nSet $\\widetilde \\Omega = \\Omega_1 \\times \\Omega_2 \\times \\Omega_3$, equipped with the product topology and Borel sigma-algebra; \n\\[\n(\\omega, \\Theta,\\eta) = (\\omega(e), \\theta_1(x),\\theta_2(x),\\eta(f) : e \\in \\mathcal{E}^2, x \\in \\ensuremath{\\mathbb{Z}^2},~f \\in \\vec{\\mathcal{E}}^2)\n\\] \ndenotes a generic element of the space $\\widetilde \\Omega.$ Define the map\n\\begin{equation}\\label{eq: phidef}\n\\Phi_\\alpha: \\Omega_1 \\longrightarrow \\widetilde \\Omega  \\text{ by } \\omega \\mapsto (\\omega, B_\\alpha(\\omega),\\eta_\\alpha(\\omega))\\ .\n\\end{equation}\nBecause $\\Phi_\\alpha$ is measurable, we can use it to push forward the distribution $\\ensuremath{\\mathbb{P}}$ to a probability measure $\\mu_\\alpha$ on $\\widetilde \\Omega$. Given the family $(\\mu_\\alpha)$ and $n \\in \\mathbb{N}$, we define the empirical average\n\\begin{equation}\\label{eq: munstar}\n\\mu_n^*\\left( \\cdot \\right) := \\frac{1}{n} \\int_{0}^n \\mu_\\alpha \\left( \\cdot \\right) \\mathrm{d} \\alpha. \n\\end{equation}\nTo prove that this defines a probability measure, one must show that for each measurable $A \\subseteq \\widetilde \\Omega$, the map $\\alpha \\mapsto \\mu_\\alpha(A)$ is Lebesgue-measurable. The proof is deferred to Appendix~\\ref{sec: appendix}.\n\nFrom $B_\\alpha(x,y) \\leq \\tau(x,y)$, the sequence $\\left( \\mu_n^* \\right)_{n=1}^{\\infty}$ is seen to be tight and thus has a subsequential weak limit $\\mu.$ We will call the marginal of $\\mu$ on $\\Omega_2$ a {\\it Busemann increment distribution} and the marginal on $\\Omega_3$ a {\\it geodesic graph distribution}. It will be important to recall the Portmanteau theorem, a basic result about weak convergence. The following are equivalent if $(\\nu_k)$ is a sequence of Borel probability measures on a metric space $X$:\n\\begin{align}\n\\lim_{k \\to \\infty} \\nu_k &\\to \\nu \\text{ weakly } \\nonumber \\\\\n\\limsup_{k \\to \\infty} \\nu_k(A) &\\leq \\nu(A) \\text{ if } A \\text{ is closed} \\label{eq: kallenbergclosed}\\\\\n\\liminf_{k \\to \\infty} \\nu_k(A) &\\geq \\nu(A) \\text{ if } A \\text{ is open} \\label{eq: kallenbergopen}\\ .\n\\end{align}\n(See, for example, \\cite[Theorem~3.25]{Kallenberg}.) Because $\\widetilde \\Omega$ is metrizable, these statements apply.\n\nIn this section and the next, we prove general properties about the measure $\\mu$ and focus on the marginal on $\\Omega_2$. In Sections~\\ref{sec: GG} and \\ref{sec: coalesceG} we study the marginal on $\\Omega_3$ and in Section~\\ref{sec: proofs} relate results back to the original FPP model. It is important to remember that $\\mu$ depends among other things not only on $\\varpi$, but on the choice of the linear functional $g_\\varpi$. We will suppress mention of $\\varpi$ in the notation. Furthermore we will use $\\mu$ to represent the measure and also its marginals. For instance, if we write $\\mu(A)$ for an event $A \\subseteq \\Omega_2$ we mean $\\mu(\\Omega_1 \\times A \\times \\Omega_3)$.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Translation invariance of $\\mu$.}\nWe will show that $\\mu$ inherits translation invariance from \\ensuremath{\\mathbb{P}}. The natural translations $\\tilde{T}_m,~m=1,2$ act on $\\widetilde \\Omega$ as follows:\n\\[\n\\left[ \\tilde{T}_m (\\omega,\\Theta,\\eta) \\right](e,x,f) = \\left( \\omega_{e-{\\mathbf{e}}_m}, \\theta_1 (x - {\\mathbf{e}}_m), \\theta_2 (x - {\\mathbf{e}}_m), \\eta(f-{\\mathbf{e}}_m) \\right)\\ .\n\\]\nHere, for example, we interpret $e-{\\mathbf{e}}_m$ for the edge $e= (y,z)$ as $(y-{\\mathbf{e}}_m,z-{\\mathbf{e}}_m)$.\n\n\n\n\\begin{lem}\n\\label{translate_1}\nFor any $\\alpha \\in \\mathbb{R}$ and $m=1,2$, $\\mu_\\alpha \\circ \\tilde T_m = \\mu_{\\alpha + g_{\\varpi}({\\mathbf{e}}_m)}$.\n\n\\begin{proof}\nLet $A$ be a cylinder event for the space $\\widetilde \\Omega$ of the form\n\\[\nA = \\left\\{ \\omega_{e_i} \\in \\mathbf{B}_i, \\theta_{r_j}(x_j) \\in \\mathbf{C}_j, \\eta(f_k) = a_k : i = 1, \\ldots, l, ~j = 1, \\ldots, m, ~k=1, \\ldots, n \\right\\}\\ ,\n\\]\nwhere each $\\mathbf{B}_i, \\mathbf{C}_j$ is a (real) Borel set with $a_k \\in \\{0,1\\}$, each $r_j \\in \\{1,2\\}$, and each $e_i \\in \\mathcal{E}^2, x_j \\in \\mathbb{Z}^2$ and $f_k \\in \\vec{\\mathcal{E}}^2$.  We will show that for $m=1,2$,\n\\begin{equation}\\label{eq: chazisagooddog}\n\\mu_\\alpha \\left(\\tilde{T}_m^{-1} A\\right) = \\mu_{\\alpha + g_{\\varpi}({\\mathbf{e}}_m)}(A)\\ .\n\\end{equation}\nSuch $A$ generate the sigma-algebra so this will imply the lemma. For $m \\in \\{1,2\\}$,\n\\begin{equation}\n\\label{cylindershift}\n\\tilde{T}_m^{-1}(A) = \\left\\{\\omega_{e_i - {\\mathbf{e}}_m} \\in \\mathbf{B}_i, \\theta_{r_j}(x_j-{\\mathbf{e}}_m) \\in \\mathbf{C}_j, \\eta(f_k-{\\mathbf{e}}_m) = a_k \\right\\}\\ . \\nonumber\n\\end{equation}\nRewriting $\\mu_\\alpha(\\cdot) = \\mathbb{P}(\\Phi_\\alpha^{-1}(\\cdot))$ and using the definition of $\\Phi_\\alpha$ \\eqref{eq: phidef}, \n\\[\n\\mu_\\alpha(\\tilde T_m^{-1}(A)) = \\ensuremath{\\mathbb{P}} \\left( \\omega_{e_i-{\\mathbf{e}}_m} \\in \\mathbf{B}_i, B_\\alpha(x_j - {\\mathbf{e}}_m, x_j - {\\mathbf{e}}_m + {\\mathbf{e}}_{r_j} ) \\in \\mathbf{C}_j, \\eta_\\alpha(f_k-{\\mathbf{e}}_m)(\\omega) = a_k \\right)\\ .\n\\]\nNote that translation invariance of $\\ensuremath{\\mathbb{P}}$ allows to shift the translation by ${\\mathbf{e}}_m$ from the arguments of $\\omega$, $B_\\alpha$ and $\\eta_\\alpha$ to the position of the line $L_\\alpha$.  We have equality in distribution:\n\\[\n\\omega_{e-{\\mathbf{e}}_m} \\underset{d}{=} \\omega_e,~ B_\\alpha(x - {\\mathbf{e}}_m, y - {\\mathbf{e}}_m) \\underset{d}{=} B_\\beta(x,y) \\text{ and } \\eta_\\alpha(e-{\\mathbf{e}}_m) \\underset{d}{=} \\eta_\\beta(e)\\ ,\n\\]\nwhere $\\beta=\\alpha + g_\\varpi ({\\mathbf{e}}_m)$. In fact, using the translation covariance statements \\eqref{eq: transcov}, \\eqref{eq: busemanntranscov} and \\eqref{eq: GGtranscov}, equality of the above sort holds for the joint distribution of the $\\omega$'s, Busemann increments and graph variables appearing in the event $A.$ This proves \\eqref{eq: chazisagooddog}.\n\\end{proof}\n\\end{lem}\n\n\n\n\n\n\n\\begin{prop}\n\t$\\mu$ is invariant under the translations $\\tilde{T}_m$, $m=1,2$.\n\\end{prop}\n\\begin{proof}\n\tLet $f$ be a continuous function (bounded by $D \\geq 0$) on the space $\\widetilde \\Omega,$ and fix $\\epsilon > 0.$  Choose an increasing sequence $(n_k)$ such that $\\mu_{n_k}^* \\to \\mu$ weakly as $k \\to \\infty$. We can then find $k_0$ such that\n$|\\mu(f) - \\mu_{n_k}^*(f)| < \\epsilon\/3$ for $k > k_0.$  \nBy Lemma~\\ref{translate_1}, $\\mu_\\alpha \\circ \\tilde T_m = \\mu_{\\alpha + g_{\\varpi}({\\mathbf{e}}_m)}$ for $m=1,2$. Therefore \n\\begin{align*}\n\t\\left[\\mu_{n_k}^* \\circ \\tilde{T}_m\\right] \\left(f\\right) &= \\frac{1}{n_k}\\int_{g_{\\varpi}({\\mathbf{e}}_m)}^{n_k+g_{\\varpi}({\\mathbf{e}}_m)} \\mu_\\alpha \\left(f \\right)  \\mathrm{d} \\alpha\\\\\n\t\\Rightarrow \\left| \\left[\\mu_{n_k}^* \\circ \\tilde{T}_m\\right] \\left(f\\right) - \\mu_{n_k}^* \\left(f\\right) \\right|  &\\leq\n\t\t\\frac{1}{n_k}\\left| \\int_{0}^{g_{\\varpi}({\\mathbf{e}}_m)} \\mu_\\alpha \\left(f \\right)  \\mathrm{d} \\alpha \\right| +  \\frac{1}{n_k} \\left| \\int_{n_k}^{n_k+g_{\\varpi}({\\mathbf{e}}_m)} \\mu_\\alpha \\left(f \\right)  \\mathrm{d} \\alpha \\right|\\\\\n\t&\\leq \\frac{2 g_{\\varpi}({\\mathbf{e}}_m)D}{n_k} \\rightarrow 0 \\text{ as } k \\to \\infty\\ .\n\\end{align*}\nAs $\\tilde T_m$ is a continuous on $\\widetilde \\Omega$, $(\\mu_{n_k}^* \\circ \\tilde{T}_m) $ converges weakly to $\\mu \\circ \\tilde{T}_m,$ so there exists $k_1>k_0$ such that \n$|\\mu\\circ \\tilde T_m(f) - \\mu_{n_k}^* \\circ \\tilde T_m (f)| < \\epsilon\/3$ for all $k > k_1,$ and $k_2>k_1$ with $2g_{\\varpi}({\\mathbf{e}}_m)D\/n_{k_2} < \\epsilon\/3.$  So $|\\mu(f) - \\mu \\circ \\tilde{T}_m (f)|  < \\epsilon \\text{ for all } \\varepsilon>0$, giving $\\mu = \\mu \\circ \\tilde T_m$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Reconstructed Busemann functions}\nWe wish to reconstruct an ``asymptotic Busemann function\" $f: \\ensuremath{\\mathbb{Z}^2} \\rightarrow \\mathbb{R}$ by summing the Busemann increments of $\\Theta \\in \\Omega_2$. That $\\Theta$ is almost surely curl-free allows the construction to proceed independent of the path we sum over. For this we need some definitions.\n\nGiven $\\Theta \\in \\Omega_2$, $x \\in \\mathbb{Z}^2$ and $z \\in \\mathbb{Z}^2$ with $\\|z\\|_1=1$ we set $\\theta(x,z) = \\theta(x,z)(\\Theta)$ equal to\n\\[\n\\theta(x,z) = \\begin{cases}\n\\theta_1(x) & z = {\\mathbf{e}}_1 \\\\\n\\theta_2(x) & z= {\\mathbf{e}}_2 \\\\\n-\\theta_1(x-{\\mathbf{e}}_1) & z=-{\\mathbf{e}}_1 \\\\\n-\\theta_2(x-{\\mathbf{e}}_2) & z = -{\\mathbf{e}}_2\n\\end{cases}\\ .\n\\]\nFor any finite lattice path $\\gamma$ we write its vertices in order as $x_1, \\ldots, x_n$ and set\n\\[\nf(\\gamma) = f(\\gamma)(\\Theta) = \\sum_{i=1}^{n-1} \\theta(x_i, x_{i+1}-x_i)\\ .\n\\]\n\n\n\\begin{lem}\\label{lem: curlfree}\nWith $\\mu$-probability one, $f$ vanishes on all circuits:\n\\[\n\\mu \\left( f(\\gamma)=0 \\text{ for all circuits } \\gamma \\right) = 1\\ .\n\\]\n\\end{lem}\n\n\\begin{proof}\nPick a circuit $\\gamma$ and let $A\\subseteq \\widetilde \\Omega_2$ denote the event $\\{ \\Theta : f(\\gamma) = 0\\}$. Choose an increasing sequence $(n_k)$ such that $\\mu_{n_k}^* \\to \\mu$ weakly. For fixed $\\gamma$, $f(\\gamma)$ is a continuous function on $\\widetilde \\Omega$, so the event $A$ is closed, giving $\\mu(A) \\geq \\limsup_{k} \\mu_{n_k}^*(A)$ by \\eqref{eq: kallenbergclosed} . However, for each $\\alpha$, by additivity of $B_\\alpha(\\cdot,\\cdot)$ (see \\eqref{eq: busemannadditivity}),\n\\[\n\\mu_\\alpha(A) = \\mathbb{P}\\left(\\sum_{i=1}^n B_\\alpha(x_i,x_{i+1}) = 0\\right) = 1\\ .\n\\]\nThus $\\mu_n^*(A) = 1$ for all $n$ and $\\mu(A)=1$. There are countably many $\\gamma$'s so we are done.\n\\end{proof}\n\nUsing the lemma we may define the reconstructed Busemann function. Fix a deterministic family of finite paths $\\{\\gamma_{x,y}\\}$, one for each pair $(x,y) \\in \\mathbb{Z}^2$ and define \n\\[\nf(x,y) = f(x,y)(\\Theta) := f(\\gamma_{x,y})\\ .\n\\] \nAlthough we use fixed paths $\\gamma_{x,y}$, this is only to ensure that $f$ is a continuous function on $\\widetilde \\Omega$. Actually, for any $\\Theta$ in the $\\mu$-probability one set of Lemma~\\ref{lem: curlfree} and vertices $x,y \\in \\mathbb{Z}^2$ we could equivalently define $f(x,y) = f(\\gamma)$, where $\\gamma$ is any finite lattice path from $x$ to $y$. To see that it would then be well-defined (that is, only a function of $x,y$ and the configuration $\\Theta$) is a standard argument. If we suppose that $\\gamma_1$ and $\\gamma_2$ are finite lattice paths from $x$ to $y$ and $\\Theta$ is given as above, the concatenation of $\\gamma_1$ with $\\gamma_2$ (traversed in the opposite direction) is a circuit and thus has $f$-value zero. However, by definition, this is the difference of $f(\\gamma_1)$ and $f(\\gamma_2)$ and proves the claim.\n\n\nWe now give some properties about asymptotic Busemann functions that come over from the original model. The third says that $f$ retains translation covariance. This will allow us to prove the existence of almost-sure limits using the ergodic theorem in the next section.\n\\begin{prop}\n\\label{bd_f_by_tau_prop}\nThe reconstructed Busemann function satisfies the following properties for $x,y,z \\in \\ensuremath{\\mathbb{Z}^2}$.\n\\begin{enumerate}\n\\item \n\\begin{equation}\\label{eq: additivity}\nf(x,y) + f(y,z) = f(x,z) ~\\mu\\text{-almost surely}\\ . \n\\end{equation}\n\\item For $m=1,2$ \n\\begin{equation}\\label{eq: ftranslation}\nf(x,y)(\\tilde T_m \\Theta) = f(x-{\\mathbf{e}}_m,y-{\\mathbf{e}}_m)(\\Theta) ~\\mu\\text{-almost surely}\\ .\n\\end{equation}\n\\item \\begin{equation}\\label{eq: continuity}\nf(x,y):\\widetilde \\Omega \\to \\mathbb{R} \\text{ is continuous}\\ .\n\\end{equation}\n\\item $f$ is bounded by $\\tau$:\n\\begin{equation}\\label{eq: fboundtau}\n|f(x,y)| \\leq \\tau(x,y)) ~\\mu\\text{-almost surely}\\ .\n\\end{equation}\n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nThe first two properties follow from path-independence of $f$ and the third holds because $f$ is a sum of finitely many Busemann increments, each of which is a continuous function. We show the fourth property. For $x,y \\in \\ensuremath{\\mathbb{Z}^2}$, the event\n\\[\n\\{(\\omega,\\Theta) : |f(x,y)(\\Theta)| - \\tau(x,y)(\\omega) \\leq 0\\}\n\\]\nis closed because $|f(x,y)|-\\tau(x,y)$ is continuous. For every $\\alpha$, \\eqref{eq: bboundtau} gives $|B_\\alpha(x,y)| \\leq \\tau(x,y)$ with $\\mathbb{P}$-probability one, so the above event has $\\mu_\\alpha$-probability one. Taking limits and using \\eqref{eq: kallenbergclosed}, $\\mu(|f(x,y)(\\Theta)| \\leq \\tau(x,y)(\\omega)) = 1$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Expected value of $f$}\nIn this section we compute $\\mathbb{E}_\\mu f(0,x)$ for all $x \\in \\mathbb{Z}^2$. The core of our proof is a argument from Hoffman \\cite{Hoffman2}, which was developed using an averaging argument due to Garet-Marchand \\cite{GM}. The presentation we give below is inspired by that of Gou\\'er\\'e \\cite[Lemma~2.6]{gouere}. In fact, the proof shows a stronger statement. Without need for a subsequence,\n\\[\n\\mathbb{E}_{\\mu_{n}^*}f(0,x) \\to g_\\varpi(x)\\ .\n\\]\n\n\\begin{thm}\\label{thm: expected_value}\nFor each $x \\in \\mathbb{Z}^2$, $\\mathbb{E}_\\mu f(0,x) = g_\\varpi(x)$.\n\\end{thm}\n\n\\begin{proof}\nWe will use an elementary lemma that follows from the shape theorem.\n\\begin{lem}\\label{lem: pointtoplane}\nThe following convergence takes place almost surely and in $L^1(\\mathbb{P})$:\n\\[\n\\frac{\\tau(0,L_\\alpha)}{\\alpha} \\to 1 \\text{ as } \\alpha \\to \\infty\\ .\n\\]\n\\end{lem}\n\n\\begin{proof}\nSince $\\alpha \\varpi \\in L_\\alpha$,\n\\[\n\\limsup_{\\alpha \\to \\infty} \\frac{\\tau(0,L_\\alpha)}{\\alpha} \\leq \\lim_{\\alpha \\to \\infty} \\frac{\\tau(0,\\alpha \\varpi)}{\\alpha} = 1\\ .\n\\]\nOn the other hand, given $\\varepsilon>0$ and any $\\omega$ for which the shape theorem holds, we can find $K$ such that for all $x \\in \\mathbb{R}^2$ with $\\|x\\|_1 \\geq K$, $\\tau(0,x) \\geq g(x)(1-\\varepsilon)$. So if $\\alpha$ is large enough that all $x \\in L_\\alpha$ have $\\|x\\|_1 \\geq K$, then we can use \\eqref{eq: gvarpibound}:\n\\[\n\\tau(0,L_\\alpha) = \\min_{x \\in L_\\alpha} \\tau(0,x) \\geq (1-\\varepsilon) \\min_{x \\in L_\\alpha} g(x) \\geq (1-\\varepsilon)\\alpha\\ .\n\\]\nConsequently, $\\liminf_{\\alpha \\to \\infty} \\tau(0,L_\\alpha)\/\\alpha \\geq 1$, giving almost sure convergence in the lemma.\n\nFor $L^1$ convergence, note $0 \\leq \\tau(0,L_\\alpha)\/\\alpha \\leq \\tau(0,\\alpha \\varpi)\/\\alpha$, so the dominated convergence theorem and $L^1$ convergence of point to point passage times completes the proof.\n\\end{proof}\n\nFor any $x \\in \\mathbb{Z}^2$ and integer $n \\geq 1$, use the definition of $\\mu_n^*$ to write \n\\[\n\\mathbb{E}_{\\mu_n^*}(f(-x,0)) = \\frac{1}{n} \\left[ \\int_0^n \\mathbb{E} \\tau(-x,L_\\alpha)~\\mathrm{d} \\alpha - \\int_0^n \\mathbb{E} \\tau(0, L_\\alpha) ~\\mathrm{d} \\alpha \\right]\\ .\n\\]\nUsing translation covariance of passage times,\n\\[\n\\int_0^n \\mathbb{E} \\tau(-x,L_\\alpha)~\\mathrm{d} \\alpha = \\int_0^n \\mathbb{E} \\tau(0,L_{\\alpha + g_\\varpi(x)})~\\mathrm{d} \\alpha = \\int_{g_\\varpi(x)}^{n+g_\\varpi(x)} \\mathbb{E} \\tau(0,L_\\alpha) ~\\mathrm{d} \\alpha\\ .\n\\]\nTherefore\n\\begin{equation}\\label{eq: almostformula}\n\\mathbb{E}_{\\mu_n^*}(f(-x,0)) = \\frac{1}{n} \\left[ \\int_{n}^{n+g_\\varpi(x)} \\mathbb{E} \\tau(0,L_\\alpha) ~\\mathrm{d} \\alpha - \\int_0^{g_\\varpi(x)} \\mathbb{E} \\tau(0,L_\\alpha) ~\\mathrm{d} \\alpha \\right] \\ .\n\\end{equation}\n\n\n\nChoose $(n_k)$ to be an increasing sequence such that $\\mu_{n_k}^* \\to \\mu$ weakly. We claim that\n\\begin{equation}\\label{eq: momentconvergence}\n\\mathbb{E}_{\\mu_{n_k}^*} f(-x,0) \\to \\mathbb{E}_\\mu f(-x,0)\\ .\n\\end{equation}\nTo prove this, note that for any $R>0$, if we define the truncated variable\n\\[\nf_R(-x,0) = \\text{ sgn} f(-x,0) \\min\\{R, |f(-x,0)|\\}\\ ,\n\\]\nthen continuity of $f$ on $\\widetilde \\Omega$ gives $\\mathbb{E}_{\\mu_{n_k}^*} f_R(-x,0) \\to \\mathbb{E}_\\mu f_R(-x,0)$. To extend this to \\eqref{eq: momentconvergence}, it suffices to prove that for each $\\varepsilon>0$, there exists $R>0$ such that\n\\begin{equation}\\label{eq: limsupcondition}\n\\limsup_{k \\to \\infty} \\mathbb{E}_{\\mu_{n_k}^*} |f(-x,0)| I(|f(-x,0)| \\geq R) < \\varepsilon\\ ,\n\\end{equation}\nwhere $I(A)$ is the indicator of the event $A$. Because $\\mathbb{E}_{\\mu_{n_k}^*} f(-x,0)^2 \\leq \\mathbb{E} \\tau(-x,0)^2 < \\infty$ for all $k$ by \\eqref{eq: finitesecondmoment}, condition \\eqref{eq: limsupcondition} follows from the Cauchy-Schwarz inequality. This proves \\eqref{eq: momentconvergence}. \n\nCombining \\eqref{eq: almostformula} and \\eqref{eq: momentconvergence}, we obtain the formula\n\\begin{equation}\\label{eq: almostdoneagain}\n\\mathbb{E}_\\mu f(-x,0) = \\lim_{k \\to \\infty} \\frac{1}{n_k} \\int_{n_k}^{n_k+g_\\varpi(x)} \\mathbb{E} \\tau(0,L_\\alpha) ~\\mathrm{d} \\alpha = \\lim_{k \\to \\infty} \\int_0^{g_\\varpi(x)} \\frac{\\mathbb{E} \\tau(0,L_{\\alpha + n_k})}{n_k} ~\\mathrm{d} \\alpha\\ .\n\\end{equation}\nBy Lemma~\\ref{lem: pointtoplane}, for each $\\alpha$ between $0$ and $g_\\varpi(x)$,\n\\[\n\\lim_{k \\to \\infty} \\frac{\\mathbb{E} \\tau(0,L_{\\alpha + n_k})}{n_k} = \\lim_{k \\to \\infty} \\frac{\\mathbb{E} \\tau(0,L_{\\alpha + n_k})}{\\alpha + n_k} \\cdot \\frac{\\alpha + n_k}{n_k} = 1\\ .\n\\]\nSo using $\\mathbb{E} \\tau(0,L_{\\alpha + n_k}) \\leq \\mathbb{E} \\tau(0,L_{2n_k})$ for large $k$, we can pass the limit under the integral in \\eqref{eq: almostdoneagain} to get $\\mathbb{E}_\\mu f(0,x) = \\mathbb{E}_\\mu f(-x,0) = g_\\varpi(x)$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Limits for reconstructed Busemann functions}\\label{sec: limits}\n\nIn this section we study the asymptotic behavior of the reconstructed Busemann function $f$. We will see that $f$ is asymptotically a projection onto a line and if the boundary of the limit shape is differentiable at $\\varpi$, we give the explicit form of the hyperplane. Without this assumption we show that the line is a translate of a supporting line for $\\mathcal{B}$ at $\\varpi$.\n\nOne of the advantages of constructing $f$ from our measure $\\mu$ is that we can use the ergodic theorem and translation invariance to show the existence of limits. This gives us almost as much control on the Busemann function as we would have if we could show existence of the limit in \\eqref{eq: newmanlimit}. If we knew this, we would not need differentiability at $\\varpi$ to deduce the form of the random hyperplane for $f$; we could derive it from ergodicity and symmetry. \n\n\n\n\\subsection{Radial limits}\n\nIn this section we will prove the existence of radial limits for $f$. This is the first step to deduce a shape theorem, which we will do in the next section. We extend the definition of $f$ to all of $\\mathbb{R}^2 \\times \\mathbb{R}^2$ in the usual way: $f(x,y)$ is defined as $f(\\tilde x, \\tilde y)$ where $\\tilde x$ and $\\tilde y$ are the unique points in $\\ensuremath{\\mathbb{Z}^2}$ such that $x \\in \\tilde x + [-1\/2,1\/2)^2$ and $y \\in \\tilde y + [-1\/2,1\/2)^2$.\n\n\\begin{prop}\\label{prop: rationallimits}\nLet $q \\in \\mathbb{Q}^2$. Then\n\\[\n\\rho_q := \\lim_{n \\to \\infty} \\frac{1}{n} f(0,nq) \\text{ exists } \\mu\\text{-almost surely}\\ .\n\\]\n\\end{prop}\n\n\\begin{proof}\nChoose $M \\in \\mathbb{N}$ such that $Mq \\in \\mathbb{Z}^2$. We will first show that\n\\begin{equation}\\label{eq: firstrationallimit}\n\\lim_{n \\to \\infty} \\frac{1}{Mn} f(0,nMq) \\text{ exists } \\mu\\text{-almost surely}\\ .\n\\end{equation}\nTo do this, we note that since $\\tau(0,Mq) \\in L^2(\\mu)$ (from \\eqref{eq: finitesecondmoment}), it is also in $L^1$. Using \\eqref{eq: fboundtau}, $f(0,Mq) \\in L^1(\\mu)$ as well. Define the map $\\tilde T_q$ on $\\Omega_2$ as\n\\[\n\\left[ \\tilde T_q \\Theta \\right] (x) = (\\theta_1(x-Mq),\\theta_2(x-Mq))\\ .\n\\]\nThis is a composition of maps $\\tilde T_m$, $m=1,2$, so it is measure-preserving. By \\eqref{eq: additivity} and \\eqref{eq: ftranslation},\n\\[\nf(0,nMq)(\\Theta) = \\sum_{i=1}^n f((i-1)Mq,iMq)(\\Theta) = \\sum_{i=0}^{n-1} f(0,Mq)(\\tilde T_q^{-i} (\\Theta))\\ .\n\\]\nApplying the ergodic theorem finishes the proof of \\eqref{eq: firstrationallimit}.\n\nTo transform \\eqref{eq: firstrationallimit} into the statement of the proposition we need to ``fill in the gaps.'' Choose $M$ as above and for any $n$ pick $a_n \\in \\mathbb{Z}$ such that $a_nM \\leq n < (a_n+1)M$. Then\n\\[\n\\left| \\frac{f(0,nq)}{n} - \\frac{f(0,a_nMq)}{a_nM} \\right| \\leq \\left| \\frac{f(0,a_nMq)}{a_nM} \\right| \\left| 1 - \\frac{a_nM}{n} \\right| + \\frac{1}{n} \\left| f(0,a_nMq) - f(0,nq) \\right|\\ . \n\\]\nThe first term on the right converges to 0. To show the same for the second term we use the fact that $f(x,y) \\in L^1(\\mu,\\Omega_2)$ for all $x,y \\in \\mathbb{R}^2$. Indeed, the difference $f(0,a_nMq)-f(0,nq)$ is equal to $f(nq, a_nMq)$, which has the same distribution as $f(0,(a_nM-n)q)$. For each $\\varepsilon>0$,\n\\[\n\\sum_{n \\geq 1} \\mu(|f(0,(n-a_nM)q)| \\geq \\varepsilon n) \\leq \\frac{1}{\\varepsilon} \\sum_{i=1}^M \\| f(0,-iq)\\|_{L^1(\\mu)} < \\infty\\ .\n\\]\nSo only finitely many of the events $\\{|f(0,a_nMq) - f(0,nq)| \\geq \\varepsilon n\\}$ occur and we are done.\n\\end{proof}\n\n\n\nThe last proposition says that for each $q$ there exists a random variable $\\rho_q = \\rho(q,\\Theta)$ such that $\\mu$-almost surely, the above limit equals $\\rho_q$. Assume now that we fix $\\Theta$ such that this limit exists for all $q \\in \\mathbb{Q}^2$. We will consider $\\rho_q$ as a function of $q$. The next theorem states that $\\rho_q$ represents a random projection onto a vector $\\varrho$.\n\\begin{thm}\\label{thm: pizzapie}\nThere exists a random vector $\\varrho = \\varrho(\\Theta)$ such that\n\\[\n\\mu\\left( \\rho_q = \\varrho \\cdot q \\text{ for all } q \\in \\mathbb{Q}^2 \\right) = 1\\ .\n\\]\nFurthermore $\\varrho$ is translation invariant:\n\\[\n\\varrho(\\tilde T_m \\Theta) = \\varrho(\\Theta) \\text{ for } m=1,2\\ .\n\\]\n\\end{thm}\n\n\\begin{proof}\nWe will show that $q \\mapsto \\rho_q$ is a (random) linear map on $\\mathbb{Q}^2$. Specifically, writing an arbitrary $q \\in \\mathbb{Q}^2$ as $(q_1,q_2)$, we will show that\n\\begin{equation}\\label{eq: blackboard2}\n\\mu\\left(\\rho_q = q_1 \\rho_{{\\mathbf{e}}_1} + q_2 \\rho_{{\\mathbf{e}}_2} \\text{ for all } q \\in \\mathbb{Q}^2\\right) = 1\\ .\n\\end{equation}\nThen, setting $\\varrho = (\\rho_{{\\mathbf{e}}_1},\\rho_{{\\mathbf{e}}_2})$, we will have proved the theorem.\n\nThe first step is to show translation invariance of $\\rho_q$. Given $q \\in \\mathbb{Q}^2$, let $M \\in \\mathbb{N}$ be such that $Mq \\in \\mathbb{Z}^2$. For $m=1,2$, translation covariance implies\n\\begin{eqnarray*}\n|f(0,nMq)(\\tilde T_m \\Theta) - f(0,nMq)(\\Theta)| &=& |f(-{\\mathbf{e}}_m, nMq-{\\mathbf{e}}_m)(\\Theta) - f(0,nMq)(\\Theta)| \\\\\n&\\leq& |f(-{\\mathbf{e}}_m,0)(\\Theta)| + |f(nMq-{\\mathbf{e}}_m,nMq)(\\Theta)|\\ .\n\\end{eqnarray*}\nFurthermore, given $\\delta>0$,\n\\[\n\\sum_n \\mu\\left( |f(nMq-{\\mathbf{e}}_m, nMq)| > \\delta n\\right) \\leq \\sum_n \\mu\\left( |f(0, {\\mathbf{e}}_m)| > \\delta n\\right) \\leq \\frac{1}{\\delta} \\|f(0,{\\mathbf{e}}_m)\\|_{L^1(\\mu)} < \\infty\\ .\n\\]\nTherefore only finitely many of the events $\\{|f(nMq - {\\mathbf{e}}_m,nMq)| > \\delta n\\}$ occur and \n\\[\n\\rho_q(\\tilde T_m\\Theta) = \\lim_{n \\to \\infty} \\frac{f(0,nMq)(\\tilde T_m \\Theta)}{nM} = \\lim_{n \\to \\infty} \\frac{f(0,nMq)(\\Theta)}{nM} = \\rho_q(\\Theta) \\text{ almost surely}\\ .\n\\]\n\nTo complete the proof we show that $q \\mapsto \\rho_q$ is almost surely additive. Over $\\mathbb{Q}$, this suffices to show linearity and thus \\eqref{eq: blackboard2}. Let $q_1,q_2 \\in \\mathbb{Q}^2$ and choose $M \\in \\mathbb{N}$ with $Mq_1,Mq_2 \\in\\mathbb{Z}^2$. By Proposition~\\ref{prop: rationallimits}, for $\\varepsilon>0$, we can pick $N$ such that if $n \\geq N$ then the following hold:\n\\begin{enumerate}\n\\item $\\mu\\left( |(1\/nM)f(0,nMq_1) - \\rho_{q_1} | > \\varepsilon\/2 \\right) < \\varepsilon\/2$ and\n\\item $\\mu\\left( |(1\/nM)f(0,nMq_2) - \\rho_{q_1}| > \\varepsilon\/2 \\right) < \\varepsilon\/2$.\n\\end{enumerate}\nWriting $\\tilde T_{-q}(\\Theta)(x) = \\Theta(x+Mq)$ and using translation invariance of $\\rho_{q_2}$,\n\\begin{eqnarray*}\n&& f(0,nM(q_1+q_2))(\\Theta) - nM\\rho_{q_1}(\\Theta) - nM\\rho_{q_2}(\\Theta) \\\\\n&=& f(0,nMq_1)(\\Theta) - nM\\rho_{q_1}(\\Theta) + f(0,nMq_2)(\\tilde T_{-q_1}^n \\Theta) - nM\\rho_{q_2}(\\tilde T_{-q_1}^n \\Theta)\\ .\n\\end{eqnarray*}\nSo by translation invariance of $\\mu$ and items 1 and 2 above,\n\\begin{eqnarray*}\n&& \\mu(|(1\/nM)f(0,nM(q_1+q_2)) - (\\rho_{q_1}+\\rho_{q_2})| > \\varepsilon) \\\\\n&\\leq& \\mu(|(1\/nM)f(0,nMq_1) - \\rho_{q_1}| > \\varepsilon\/2) + \\mu(|(1\/nM)f(0,nMq_2) - \\rho_{q_2}| > \\varepsilon\/2) < \\varepsilon\\ .\n\\end{eqnarray*}\nThus $(1\/nM)f(0,nM(q_1+q_2))$ converges in probability to $\\rho_{q_1} + \\rho_{q_2}$. By Proposition~\\ref{prop: rationallimits}, this equals $\\rho_{q_1+q_2}$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{A shape theorem} \n\nWe will now upgrade the almost-sure convergence in each rational direction, from Proposition~\\ref{prop: rationallimits}, to a sort of shape theorem for the Busemann function $f$. The major difference is that, unlike in the usual shape theorem of first-passage percolation, the limiting shape of $f$ is allowed to be random.\n\n\\begin{thm}\n\\label{shapetheorem}\nFor each $\\delta>0$,\n\\begin{equation}\n\\label{shape_condition}\n\\mu\\left(|f(0,x) - x \\cdot \\varrho| < \\delta \\|x\\|_1 \\text{ for all } x \\text{ with } \\|x\\|_1 \\geq M \\text{ and all large } M \\right) = 1.\n\\end{equation}\n\\end{thm}\n\nAs in the proofs of the usual shape theorems, we will need a lemma which allows us to compare $f$ in different directions. A result showing that with positive probability, $f(0,x)$ grows at most linearly in $\\|x\\|$ will be sufficient for our purposes. The fourth item of Proposition \\ref{bd_f_by_tau_prop} allows us to derive such a bound by comparison with the usual passage time $\\tau(0,x).$\n\n\\begin{lem}\nThere exist deterministic $K < \\infty$ and $p_g > 0$ depending only on the passage time distribution such that\n\\[\\ensuremath{\\mathbb{P}}\\left( \\sup_{\\substack{x \\in \\ensuremath{\\mathbb{Z}^2}\\\\x \\neq 0}} \\frac{\\tau(0,x)}{\\|x\\|_1} \\leq K\\right) = p_g > 0.  \\]\n\\end{lem}\n\\begin{proof}\nBy the first-passage shape theorem, there exists $\\lambda < \\infty$ and $T, p_g > 0$ such that\n\\[\n\\ensuremath{\\mathbb{P}}\\left(\\forall t \\geq T, \\,B(t)\/t \\supseteq [-\\lambda,\\lambda]^2\\,\\right) = p_g\\ .\n\\]\n(Here we are using \\eqref{eq: normequivalence}.) Choosing $K = T + 2 \/ \\lambda$ completes the proof.\n\n\\end{proof}\n\nThe development of the shape theorem from this point is similar to that of the usual first-passage shape theorem for ergodic passage time distributions.  \n\nWe will say that $z \\in \\mathbb{Z}^2$ is ``good\" for a given outcome if\n\\[ \n\\sup_{\\substack{x \\in \\ensuremath{\\mathbb{Z}^2}\\\\x \\neq z}} \\frac{\\tau(z,x)}{\\|x - z\\|_1} \\leq K\\ . \n\\]\nNote that $\\ensuremath{\\mathbb{P}}(z \\text{ is good}) = p_g>0$ for all $z \\in \\mathbb{Z}^2.$\n\n\\begin{lem}\n\\label{cheesy_lemma}\nLet $\\zeta$ be a nonzero vector with integer coordinates, and let $z_n =n\\zeta.$ Let $(n_k)$ denote the increasing sequence of integers such that $z_{n_k}$ is good. $\\ensuremath{\\mathbb{P}}$-almost surely, $(n_k)$ is infinite and $\\lim_{k \\rightarrow \\infty} (n_{k+1}\/n_k) = 1$.\n\\end{lem}\n\\begin{proof}\nThe ergodic theorem shows that $(n_k)$ is a.s. infinite.  Let $B_i$ denote the event that $z_i$ is good.\nBy another application of the ergodic theorem,\n\\begin{equation} \n\\label{cheesier_still}\n\\frac{k}{n_k} = \\frac{1}{n_k} \\sum_{i=1}^{n_k} \\mathbf{1}_{B_i} \\longrightarrow p_g \\quad \\text{a.s.}\n\\end{equation}\nThus,\n\\[\\frac{n_{k+1}}{n_k} = \\left(\\frac{n_{k+1}}{k+1}\\right) \\left(\\frac{k}{n_k}\\right) \\left(\\frac{k+1}{k}\\right) \\longrightarrow 1 \\quad \\text{a.s.},\\]\nsince the first and second factors converge to $p_g$ and $p_g^{-1}$ by (\\ref{cheesier_still}).\n\n\\end{proof}\n\nIn what follows, we will use the fact that there is a positive density of good sites to show convergence of $f(0,z) \/ \\|z\\|_1$ in all directions. Given the convergence of $f(0,nq) \/ n$ for each rational $q,$ we will find enough good sites along lines close to $nq$ to let us to bound the difference $|f(0,nq) - f(0,z)|.$ To describe this procedure, we need to make several definitions. Call a vector $\\zeta$ satisfying the a.s. event of Lemma \\ref{cheesy_lemma} a good direction. We will extend this definition to $\\zeta \\in \\mathbb{Q}^2$: such a $\\zeta$ will be called a good direction if $m\\zeta$ is, where $m$ is the smallest natural number such that $m\\zeta \\in \\mathbb{Z}^2$. \n\nBy countability, there exists a probability one event $\\Omega''$ on which each $\\zeta \\in \\mathbb{Q}^2$ is a good direction.\nFor each integer $M \\geq 1,$ let $V_M = \\left\\{ x\/M: x \\in \\ensuremath{\\mathbb{Z}^2}\\right\\},$ and let $V = \\cup_{M \\geq 1} V_M.$\nSet $B = \\{z \\in \\mathbb{R}^2: z \\in V, \\, \\|z\\|_1 = 1\\}$ and note that $B$ is dense in the unit sphere of $\\mathbb{R}^2$ (with norm $\\|\\cdot\\|_1$).\nBy Theorem \\ref{thm: pizzapie}, we can find a set $\\hat{\\Omega} \\subseteq \\Omega_2$ with $\\mu (\\hat{\\Omega}) = 1$ such that, for all $\\Theta \\in \\hat{\\Omega},$\n\\begin{equation}\\label{eq: planetolondon}\n\\lim_{n \\rightarrow \\infty} \\frac{1}{n} f(n z_0) (\\Theta) = z_0 \\cdot \\varrho(\\Theta) \\text{ for all } z_0 \\in B\\ .\n\\end{equation}\n\n\\begin{proof}[Proof of Theorem \\ref{shapetheorem}]\nAssume that there exist $\\delta > 0$ and an event $D_\\delta$ with $\\mu(D_\\delta)>0$ such that, for every outcome in $D_\\delta,$ there are infinitely many vertices $x \\in \\ensuremath{\\mathbb{Z}^2}$ with $\\left|f(x) - x \\cdot \\varrho\\right| \\geq \\delta \\|x\\|_1.$ Then $D_\\delta \\cap\\hat{\\Omega} \\cap \\Omega''$ is nonempty and so it contains some outcome $(\\omega,\\Theta,\\eta)$. We will derive a contradiction by showing that $(\\omega,\\Theta,\\eta)$, by way of its membership in these three sets, has contradictory properties.\n\nBy compactness of the $\\ell^1$ unit ball, we can find a sequence $\\{x_n\\}$ in $\\ensuremath{\\mathbb{Z}^2}$ with $\\|x_n\\| \\rightarrow \\infty$ and $y \\in \\mathbb{R}^2$ with $\\|y\\|_1 = 1$ such that $x_n\/\\|x_n\\|_1 \\to y$ and \n\\begin{equation}\n\\label{non_converging_pizza}\n\\left| \\frac{f(x_n)[\\Theta]}{\\|x_n\\|_1} - y \\cdot \\varrho[\\Theta] \\right| > \\frac{\\delta}{2} \\text{ for all } n\\ .\n\\end{equation}\nLet $\\delta' > 0$ be arbitrary (we will ultimately take it to be small). Our first goal is the approximation of $x_n$ by multiples of some element of $B.$\nChoose $z \\in B$ such that $\\|z - y\\|_1 <\\delta'$ and let $\\{n_k\\}$ denote the increasing sequence of integers such that $n_k z$ is good. (Here if $z \\notin \\mathbb{Z}^2$, then $z$ being good means that $Mz$ is good, where $Mz$ was chosen after Lemma~\\ref{cheesy_lemma} to be $\\mathbb{Z}^2$. Therefore $(n_k)$ would then be of the form $(M l_k)$ for some increasing sequence $l_k$.) Note that $n_{k+1}\/n_k \\to 1$ by Lemma \\ref{cheesy_lemma} so we are able to choose a $K > 0$ such that \n\\begin{equation}\\label{eq: planetolondon2}\nn_{k+1} < (1+\\delta') n_k \\text{ and } \\left|\\frac{f(0,n_k z)}{n_k} - \\varrho \\cdot z\\right| \\leq \\delta' \\text{ for all } k > K\\ .\n\\end{equation}\n\nBy the triangle inequality, the left-hand side of (\\ref{non_converging_pizza}) is bounded above by\n\\begin{align}\n\\label{pizza_telescope}\n \\left|\\frac{f(0,x_n)}{\\|x_n\\|_1} - \\frac{f(0,n_k z)}{\\|x_n\\|_1} \\right| +  \\left|\\frac{f(0,n_k z)}{\\|x_n\\|_1} - \\frac{f(0,n_k z)}{n_k}\\right| + \\left| \\frac{f(0,n_k z)}{n_k} - \\varrho \\cdot z  \\right| + \\left| \\varrho \\cdot z - \\varrho \\cdot y\\right|\n\\end{align}\nfor arbitrary $n$ and $n_k.$ Choose some $N_0$ such that\n$\\|x_n - \\|x_n\\|_1\\,y\\|_1 \\leq \\delta' \\|x_n\\|_1$ for all $n > N_0,$ and note that\n\\begin{equation}\\label{ex_to_zee}\n \\left\\|x_n - \\|x_n\\|_1 z\\right\\|_1 \\leq \\left\\|x_n - \\|x_n\\|_1 y \\right\\|_1 + \\|x_n\\|_1\\left\\| y - z \\right\\|_1 \\leq 2 \\|x_n\\|_1 \\delta' \\text{ for } n > N_0\\ .\n\\end{equation}\nFor any $n$, let $k=k(n)$ be the index such that $n_{k+1} \\geq \\|x_n\\|_1 > n_k.$ If $n$ is so large that $k(n) > K$, then $\\|\\, \\|x_n\\|_1 z - n_k z\\|_1 < \\delta' \\|x_n\\|_1.$ Combining this observation with (\\ref{ex_to_zee}) gives\n\\begin{equation}\n\\label{ex_to_zee_2}\n\\|x_n - n_k z \\|_1 \\leq 3 \\delta' \\|x_n\\|_1 \\text{ for } \\|x_n\\|_1 \\in (n_k,n_{k+1}] \\text{ when } k = k(n) > K\\ .\n\\end{equation}\n\n\nFor the remainder of the proof, fix any $n > N_0$ such that $k = k(n) > K$, so that (\\ref{ex_to_zee_2}) holds. We will now control the terms in (\\ref{pizza_telescope}), working our way from right to left. The rightmost term may be bounded by noting\n\\[ \n|  \\varrho \\cdot z - \\varrho \\cdot y| = | \\varrho \\cdot (z - y) | \\leq  \\|z-y\\|_2 \\|\\varrho\\|_2 \\leq \\delta' \\|\\varrho\\|_2\\ .\n\\]\nThe second term from the right is bounded above by $\\delta'$ by \\eqref{eq: planetolondon2}. To bound the third term from the right, note that $n_k < \\|x_n\\|_1 \\leq n_{k+1}$, so by \\eqref{eq: planetolondon2},\n\\begin{align*}\n\\left|\\frac{f(0,n_k z)}{\\|x_n\\|_1} - \\frac{f(0,n_k z)}{n_k}\\right| &= \\left|\\frac{f(0,n_k z)}{n_k}\\right| \\left(1 - \\frac{n_k}{\\|x_n\\|_1} \\right) \\,\\\\\n&\\leq \\left[\\left| \\varrho \\cdot z\\right| +\\delta' \\right] \\left( 1- \\frac{1}{1 + \\delta'}\\right)\\ .\n\\end{align*}\nIt remains to bound the first term of (\\ref{pizza_telescope}). To do this, note that by \\eqref{ex_to_zee_2},\n\\[\n|f(0,x_n) - f(0,n_k z)| = |f(n_k z, x_n)| \\leq \\tau(n_k z, x_n) \\leq K \\|x-n_k z\\|_1 \\leq 3K \\delta' \\|x_n\\|_1\\ .\n\\]\nSo \n\\[ \n\\left|\\frac{f(0,x_n)}{\\|x_n\\|_1} - \\frac{f(0,n_k z)}{\\|x_n\\|_1}\\right| \\leq 3K \\delta'\\ .\n\\]\n\nApplying our estimates for each term in (\\ref{pizza_telescope}) to the left side of (\\ref{non_converging_pizza}) gives\n\\[\n\\frac{\\delta}{2} \\leq 3K \\delta' + (|\\varrho \\cdot z|+\\delta') \\left( 1- \\frac{1}{1 + \\delta'}\\right) + \\delta' + \\delta' \\|\\varrho\\|_2\\ .\n\\]\nBecause this holds for all $\\delta' >0,$ and because the right-hand side goes to zero as $\\delta' \\rightarrow 0,$ we have derived a contradiction and proved the theorem.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{General properties of $\\varrho$}\n\nIn this short section we study the random vector $\\varrho$. In the case that $\\partial \\mathcal{B}$ is differentiable at $\\varpi$, the vector $\\varrho$ is deterministic and we give the explicit form. \n\nThe main theorem of the section is below. It says that the line\n\\[\nL_\\varrho : = \\{x \\in \\mathbb{R}^2 : \\varrho \\cdot x =1\\}\n\\]\nis $\\mu$-almost surely a supporting line for $\\mathcal{B}$ at $\\varpi$. \n\n\\begin{thm}\n\\label{rational_directions}\nWith $\\mu$-probability one, $\\varrho \\cdot \\varpi = 1$ and $\\varrho \\cdot x \\leq 1$ for all $x \\in \\mathcal{B}$. Thus $L_\\varrho$ is a supporting line for $\\mathcal{B}$ at $\\varpi$.\n\\end{thm}\n\nThis theorem has an important corollary. It follows directly from the fact that there is a unique supporting line for $\\mathcal{B}$ at points of differentiability of $\\partial \\mathcal{B}$.\n\\begin{cor}\n\\label{cor: diff}\nIf $\\partial \\mathcal{B}$ is differentiable at $\\varpi$ then\n\\[ \n\\mu\\big( \\varrho = (g_\\varpi({\\mathbf{e}}_1),g_\\varpi({\\mathbf{e}}_2)) \\big) = 1\\ .\n\\]\n\\end{cor}\n\n\n\\begin{proof}[Proof of Theorem~\\ref{rational_directions}]\n\nUsing Theorem~\\ref{thm: expected_value}, we first find the expected value of $\\varrho \\cdot y$ for $y \\in \\mathbb{R}^2$. We simply apply the dominated convergence theorem with the bound $|f(0,my)| \\leq \\tau(0,my)$. Letting $y_m \\in \\mathbb{Z}^2$ be such that $my \\in y_m + [-1\/2,1\/2)^2$,\n\\[\n\\mathbb{E}_\\mu (\\varrho \\cdot y) = \\lim_{m \\to \\infty} \\frac{1}{m}\\mathbb{E}_\\mu f(0,my) = \\lim_{m \\to \\infty} g_\\varpi(y_m\/m) = g_\\varpi(y)\\ .\n\\]\nThe theorem follows from this statement and\n\\begin{equation}\\label{eq: nachosgrande}\n\\mu\\left(x \\cdot \\varrho \\leq g(x) \\text{ for all } x \\in \\mathcal{B}\\right)=1\\ .\n\\end{equation}\nIndeed, assuming this, we have\n\\[\n\\mu(\\varrho\\cdot \\varpi \\leq 1)=1 \\text{ and } \\mathbb{E}_\\mu(\\varrho \\cdot \\varpi) = g_\\varpi(\\varpi) = 1\\ ,\n\\]\ngiving $\\varrho \\cdot \\varpi = 1$ with $\\mu$-probability one. To prove \\eqref{eq: nachosgrande}, first take $x \\in \\mathbb{Q}^2 \\cap \\mathcal{B}$. Then by \\eqref{eq: fboundtau}, for all $n$, $f(nx) \\leq \\tau(nx)$ with $\\mu$-probability one. Dividing by $n$ and taking limits with Proposition~\\ref{prop: rationallimits} and the shape theorem we get $x \\cdot \\varrho \\leq g(x)$. For non-rational $x \\in \\mathcal{B}$ we extend the inequality by almost sure continuity of both sides in $x$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Geodesic graphs}\\label{sec: GG}\n\nIn this section we study the behavior of $\\mu$ on $\\Omega_3$. Given $\\eta \\in \\Omega_3$ recall from Section~\\ref{sec: mudef} the definition of the geodesic graph $\\mathbb{G}$ of $\\eta$ as the directed graph induced by the edges $e$ for which $\\eta(e)=1$. In this section we prove a fundamental property about infinite directed paths in this graph which relates them to the asymptotic Busemann function constructed from $\\Theta$.\n\n\n\\subsection{Basic properties}\n\nWe begin by showing that properties of $\\eta_\\alpha$ from Section~\\ref{subsec: GG} carry over to $\\eta$. We use some new notation. We say that $y\\in \\mathbb{Z}^2$ is connected to $z \\in \\mathbb{Z}^2$ in $\\mathbb{G}$ (written $y \\to z$) if there exists a sequence of vertices $y=y_0, y_1, \\ldots, y_n = z$ such that $\\eta(\\langle y_k,y_{k+1}\\rangle) = 1$ for all $k=0, \\ldots, n-1$. We say that a path in $\\mathbb{G}$ is a geodesic (for the configuration $(\\omega,\\Theta,\\eta)$) if it is a geodesic in $\\omega$.\n\n\\begin{prop}\\label{prop: firstGG2}\nWith $\\mu$-probability one, the following statements hold for $x,y,z \\in \\ensuremath{\\mathbb{Z}^2}$.\n\\begin{enumerate}\n\\item Each directed path in $\\mathbb{G}$ is a geodesic.\n\\item If $x \\to y$ in $\\mathbb{G}$ then $f(x,y) = \\tau(x,y)$.\n\\item If $x \\to z$ and $y \\to z$ in $\\mathbb{G}$ then $f(x,y) = \\tau(x,z) - \\tau(y,z)$.\n\\item There exists an infinite self-avoiding directed path starting at $x$ in $\\mathbb{G}$.\n\\end{enumerate}\n\\end{prop}\n\n\n\\begin{proof}\nThe third item follows directly from the second and additivity of $f$ (from \\eqref{eq: additivity}). For the first item, if $\\gamma$ is a deterministic finite directed path, write $A_\\gamma$ for the event that all edges of $\\gamma$ are edges of $\\mathbb{G}$ and\n\\[\nB_\\gamma = A_\\gamma^c \\cup \\left( A_\\gamma \\cap \\{\\gamma \\text{ is a geodesic}\\}\\right)\\ .\n\\]\nThe event in question equals the intersection over all finite $\\gamma$'s of $B_\\gamma$, so it suffices to show that for each $\\gamma$, $\\mu(B_\\gamma)=1$.\n\nBy part 1 of Proposition~\\ref{prop: firstGG}, for all $\\alpha \\in \\mathbb{R}$ the $\\mathbb{P}$-probability that all directed paths in $\\mathbb{G}_\\alpha(\\omega)$ are geodesics is 1. By pushing forward to $\\widetilde \\Omega$, for each $\\alpha,~ \\mu_\\alpha(B_\\gamma) = 1$ and thus $\\mu_n^*(B_\\gamma) = 1$ for all $n$. Once we show that $B_\\gamma$ is a closed event, we will be done, as we can then apply \\eqref{eq: kallenbergclosed}. To show this we note that the event that a given finite path is a geodesic is a closed event. Indeed, letting $\\gamma_1$ and $\\gamma_2$ be finite paths, the function $\\tau(\\gamma_1) - \\tau(\\gamma_2)$ is continuous on $\\widetilde \\Omega$. Therefore the event $\\{\\omega \\in \\Omega_1 : \\tau(\\gamma_1) \\leq \\tau(\\gamma_2)\\}$ is closed. We then write\n\\[\n\\{\\gamma_1 \\text{ is a geodesic}\\} = \\bigcap_{\\gamma_2} \\{\\tau(\\gamma_1) \\leq \\tau(\\gamma_2)\\}\\ ,\n\\]\nwhere the intersection is over all finite paths $\\gamma_2$ with the same endpoints as those of $\\gamma_1$. Thus $\\{\\gamma_1 \\text{ is a geodesic}\\}$ is closed. Since $A_\\gamma$ depends on finitely many edge variables $\\eta(e)$, it is closed and its complement is closed. Therefore $B_\\gamma$ is closed and we are done.\n\nFor item 2, we write $\\gamma_{xy}$, any path from $x$ to $y$ in $\\mathbb{G}$, in order as $x=x_0, x_1, \\ldots, x_n = y$ and use additivity of $f$:\n\\[\nf(x,y) = \\sum_{i=0}^{n-1} f(x_i,x_{i+1})\\ .\n\\]\nFor each $i$, $x_i \\to x_{i+1}$, and by item 1, $\\gamma_{xy}$ is a geodesic. This means that we only need to show that if $x$ and $y$ are neighbors such that $\\eta(\\langle x,y \\rangle) = 1$ then $f(x,y) = \\omega_{\\langle x,y \\rangle}$, the passage time of the edge between $x$ and $y$. By part 2 of Proposition~\\ref{prop: firstGG}, for each $\\alpha$, with $\\mathbb{P}$-probability one, if $\\eta_\\alpha(\\langle x,y \\rangle) = 1$ then $B_\\alpha(x,y) = \\omega_{\\langle x,y \\rangle}$. By similar reasoning to that in the last item,\n\\[\n\\{\\eta(\\langle x,y \\rangle) = 0\\} \\cup \\left( \\{ \\eta(\\langle x,y \\rangle) = 1\\}  \\cap \\{f(x,y) = \\omega_{\\langle x,y \\rangle} \\} \\right)\n\\]\nis closed and since it has $\\mu_\\alpha$-probability 1 for all $\\alpha$, it also has $\\mu$-probability one.\n\nWe now argue for item 4. By translation-invariance we can just prove it for $x=0$. For $n \\geq 1$ let $A_n \\subseteq \\Omega_3$ be the event that there is a self-avoiding directed path starting at 0 in $\\mathbb{G}$ that leaves $[-n,n]^2$. We claim that $\\mu(A_n) = 1$ for all $n$. Taking $n \\to \\infty$ will prove item 4. \n\nFor each $\\alpha>0$ so large that $[-n,n]^2$ is contained on one side of $L_\\alpha$, let $\\gamma$ be a geodesic from $0$ to $L_\\alpha$. This path is contained in $\\mathbb{G}_\\alpha$. We may remove loops from $\\gamma$ so that it is self-avoiding, and still a geodesic. It will also be directed in the correct way: as we traverse the path from 0, each edge will be directed in the direction we are traveling. So for all large $\\alpha>0$, with $\\mathbb{P}$-probability one, there is a self-avoiding directed path starting at 0 in $\\mathbb{G}_\\alpha$ that leaves $[-n,n]^2$. Thus $\\mu_\\alpha(A_n) = 1$ for all large $\\alpha$ and $\\mu_{n_k}^*(A_n) \\to 1$ as $k \\to \\infty$. The indicator of $A_n$ is continuous on $\\widetilde \\Omega$, as $A_n$ depends on $\\eta(f)$ for finitely many edges $f$, so $\\mu(A_n)=1$.\n\\end{proof}\n\n\n\n\n\\begin{prop}\\label{prop: secondGG2}\nAssume {\\bf A1'} or {\\bf A2'}. With $\\mu$-probability one, the following statements hold.\n\\begin{enumerate}\n\\item Each vertex in $\\ensuremath{\\mathbb{Z}^2}$ has out-degree 1 in $\\mathbb{G}$. Consequently from each vertex $x$ emanates exactly one infinite directed path $\\Gamma_x$.\n\\item Viewed as an undirected graph, $\\mathbb{G}$ has no circuits. \n\\end{enumerate}\n\\end{prop}\n\n\\begin{proof}\nFor $x \\in \\mathbb{Z}^2$, let $A_x\\subseteq \\widetilde \\Omega$ be the event that $\\eta(\\langle x,y\\rangle) = 1$ for only one neighbor $y$ of $x$. Note that the indicator of $A_x$ is a bounded continuous function, so since $\\mu_\\alpha(A_x) = 1$ for all $\\alpha$ such that $x$ is not within Euclidean distance $1$ of $L_\\alpha$ (from part 1 of Proposition~\\ref{prop: secondGG} -- here $\\hat S$ is contained in the set of vertices within distance 1 of $L_\\alpha$) it follows that $\\mu(A_x)=1$. For each $z$ that is not a neighbor of $x$, $\\eta(\\langle x,z \\rangle)=0$ with $\\mu_\\alpha$-probability one for all $\\alpha$. This similarly implies that in $\\mathbb{G}$ with $\\mu$-probability one, there is no edge between $x$ and such a $z$.\n\nTo prove the second statement, fix any circuit $\\mathcal{C}$ in $\\mathbb{Z}^2$ and let $A_\\mathcal{C}$ be the event that each edge of $\\mathcal{C}$ is in $\\mathbb{G}$. Because there are no circuits in $\\mathbb{G}_\\alpha$ with $\\mathbb{P}$-probability one, we have $\\mu_n^*(A_\\mathcal{C})=0$ for all $n$. The indicator of $A_\\mathcal{C}$ is a continuous function on $\\widetilde \\Omega$, so we may take limits and deduce $\\mu(A_\\mathcal{C})=0$. There are a countable number of circuits, so we are done. \n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Asymptotic directions}\nRecall the definition $L_\\varrho = \\{x \\in \\mathbb{R}^2: x \\cdot \\varrho = 1\\}$ for the vector $\\varrho = \\varrho(\\Theta)$ of Theorem~\\ref{thm: pizzapie}. Set \n\\begin{equation}\\label{eq: on_skype}\nJ_\\varrho = \\{ \\theta : L_\\varrho \\text{ touches } \\mathcal{B} \\text{ in direction } \\theta\\}\\ .\n\\end{equation}\nThe main theorem of this subsection is as follows.\n\\begin{thm}\\label{thm: nachostheorem}\nWith $\\mu$-probability one, for all $x \\in \\ensuremath{\\mathbb{Z}^2}$, the following holds. Each directed infinite self-avoiding path in $\\mathbb{G}$ which starts at $x$ is asymptotically directed in $J_{\\varrho}$.\n\\end{thm}\n\n\n\\begin{proof}\nWe will prove the theorem for $x=0$. Assuming we do this, then using translation invariance of $\\mu$ and $\\varrho$ it will follow for all $x$.\n\nLet $\\varepsilon_k = 1\/k$ for $k \\geq 1$ and $\\delta>0$. We will show that if $S_0= \\{x \\in \\mathbb{Z}^2 : 0 \\to x$ in $\\mathbb{G}\\}$ then\n\\begin{equation}\\label{eq: nachos1}\n\\text{ for each } k \\geq 1,~ \\mu(\\arg x \\in (J_\\varrho)_{\\varepsilon_k} \\text{ for all but finitely many }  x \\in S_0) > 1-\\delta\\ .\n\\end{equation}\nHere we write $(J_\\varrho)_{\\varepsilon_k}$ for all angles $\\theta$ with $dist(\\theta,\\theta')<\\varepsilon_k$ for some $\\theta' \\in J_\\varrho$. The line $L_\\varrho$ only touches $\\mathcal{B}$ in directions in $J_\\varrho$ so by convexity, $v_\\theta \\cdot \\varrho < 1$ for all $\\theta \\notin J_\\varrho$. Since the set of angles not in $(J_\\varrho)_{\\varepsilon_k}$ is compact in $[0,2\\pi)$ (using the metric $dist$), we can find a random $a \\in (0,1)$ with $v_\\theta \\cdot \\varrho < 1-a$ for all $\\theta \\notin (J_\\varrho)_{\\varepsilon_k}$. We can then choose $a$ to be deterministic such that\n\\begin{equation}\\label{eq: nachos2}\n\\mu\\left( v_\\theta \\cdot \\varrho < 1-a \\text{ for all } \\theta \\notin (J_\\varrho)_{\\varepsilon_k} \\right) > 1-\\delta\/3\\ .\n\\end{equation}\n\nBy the shape theorem there exists $M_0$ such that $M \\geq M_0$ implies\n\\[\n\\mathbb{P}( \\tau(0,x) \\geq g(x)(1-a\/2)  \\text{ for all } x \\text{ with } \\|x\\|_1 \\geq M) > 1-\\delta\/3\\ .\n\\]\nThe marginal of $\\mu$ on $\\Omega_1$ is $\\mathbb{P}$ so this holds with $\\mu$ in place of $\\mathbb{P}$. By part 2 of Proposition~\\ref{prop: firstGG2},  \n\\begin{equation}\\label{eq: nachos3}\n\\mu( f(x) \\geq g(x)(1-a\/2) \\text{ for all } x \\text{ with } \\|x\\|_1 \\geq M \\text{ and } 0 \\to x) > 1-\\delta\/3\\ .\n\\end{equation}\nChoose $C>0$ such that $\\|x\\|_1 \\leq Cg(x)$ for all $x \\in \\mathbb{R}^2$. This is possible by \\eqref{eq: normequivalence}. By Theorem~\\ref{shapetheorem}, there exists $M_1 \\geq M_0$ such that $M \\geq M_1$ implies\n\\[\n\\mu\\left( |f(x) - x\\cdot\\varrho|< \\frac{a}{2C} \\|x\\|_1 \\text{ for all } x \\text{ with } \\|x\\|_1 \\geq M \\right) > 1-\\delta\/3\\ .\n\\]\nThis implies that for $M \\geq M_1$,\n\\begin{equation}\\label{eq: nachos4}\n\\mu\\left( |f(x) - x\\cdot\\varrho| < \\frac{a}{2} g(x) \\text{ for all } x \\text{ with } \\|x\\|_1 \\geq M \\right) > 1-\\delta\/3\\ .\n\\end{equation}\n\nWe claim that the intersection of the events in \\eqref{eq: nachos2}, \\eqref{eq: nachos3} and \\eqref{eq: nachos4} implies the event in \\eqref{eq: nachos1}. Indeed, take a configuration in the intersection of the three events for some $M \\geq M_1$. For a contradiction, assume there is an $x \\in S_0$ with $\\arg x \\notin (J_\\varrho)_{\\varepsilon_k}$ and $\\|x\\|_1 \\geq M$. Then \n\\[\n(x\/g(x)) \\cdot \\varrho < 1-a \\text{ by \\eqref{eq: nachos2}}\\ .\n\\]\nHowever, since the event in \\eqref{eq: nachos3} occurs and $\\|x\\|_1 \\geq M$,\n\\[\nf(0,x) \\geq g(x)(1-a\/2)\\ .\n\\]\nLast, as the event in \\eqref{eq: nachos4} occurs,\n\\[\nf(0,x) < x \\cdot \\varrho + \\frac{a}{2} g(x)\\ .\n\\]\nCombining these three inequalities,\n\\[\ng(x)(1-a\/2) \\leq x \\cdot \\varrho + (a\/2)g(x) < g(x)(1-a) + (a\/2)g(x)\\  ,\n\\]\nor $g(x)(1-a\/2) < g(x)(1-a\/2)$, a contradiction. This completes the proof.\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Coalescence in $\\mathbb{G}$}\\label{sec: coalesceG}\n\nIn this section we prove that all directed infinite paths coalesce in $\\mathbb{G}$. Recall that under either {\\bf A1'} or {\\bf A2'}, for $x \\in \\mathbb{Z}^2$, $\\Gamma_x$ is the unique infinite directed path in $\\mathbb{G}$ starting at $x$.\n\n\\begin{thm}\\label{thm: Gcoalescethm}\nAssume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property. With $\\mu$-probability one, for each $x,y \\in \\mathbb{Z}^2$, the paths $\\Gamma_x$ and $\\Gamma_y$ coalesce.\n\\end{thm}\n\nThe proof will be long, so we first explain the main ideas. We apply the technique of Licea-Newman \\cite{LN}, whose central tool is a Burton-Keane type argument \\cite{burtonkeane}. We proceed by contradiction, so suppose there are vertices $x, y$ such that $\\Gamma_x$ and $\\Gamma_y$ do not coalesce. By results of the last section, they cannot even intersect. We show in Sections~\\ref{sec: building_blocks} and~\\ref{sec: Bprime} that there are many triples of non-intersecting paths $\\Gamma_{x_1}, \\Gamma_{x_2}$ and $\\Gamma_{x_3}$ such that $\\Gamma_{x_2}$ is ``shielded'' from all other infinite paths in $\\mathbb{G}$. To do this, we must use the information in Theorem~\\ref{thm: nachostheorem} about asymptotic directions. A contradiction comes in Section~\\ref{sec: contradiction} from translation invariance because when $\\Gamma_{x_2}$ is shielded, the component of $x_2$ in $\\mathbb{G}$ has a unique least element in a certain lexicographic-like ordering of $\\mathbb{Z}^2$. This is a different concluding argument than that given in \\cite{LN}, where these shielded paths are used for a Burton-Keane ``lack of space'' proof.\n\n\nWe now give the proof. For the entirety we will assume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property.\n\n\n\n\n\n\n\n\n\n\\subsection{Constructing ``building blocks''}\\label{sec: building_blocks}\nAssume for the sake of contradiction that there are disjoint $\\Gamma_x$'s in $\\mathbb{G}$. Then for some vertex $z_0,$ the event $A_0(z_0)\\subseteq \\widetilde \\Omega$ has positive $\\mu$-probability, where \n\\[\nA_0(z_0) = \\{\\Gamma_{z_0} \\text{ and } \\Gamma_0 \\text{ share no vertices}\\}\\ .\n\\]  \nWe begin with a geometric lemma. It provides a (random) line such that with probability one, any path that is asymptotically directed in $J_\\varrho$ (from \\eqref{eq: on_skype}) intersects this line finitely often. We will need some notation which is used in the rest of the proof.\n\nLet $\\varpi'$ be a vector with \n\\begin{equation}\\label{eq: varpiprimedef}\n\\arg \\varpi' \\in \\{j \\pi \/ 4,\\, j = 0, \\ldots,\\,7\\} \\text{ and } \\|\\varpi'\\|_\\infty = 1\\ ,\n\\end{equation}\nwhere $\\|\\cdot\\|_\\infty$ is the $\\ell^\\infty$ norm. (A precise value of $j$ will be fixed shortly.)\nDefine (for $N \\in \\mathbb{N}$) $L'_N = \\{z \\in \\mathbb{R}^2: \\, \\varpi' \\cdot z = N\\}.$ For such an $N$ and for $x \\in \\ensuremath{\\mathbb{Z}^2},$ write $x \\prec L'_N$ if $\\varpi' \\cdot x < N$ and $x \\succ L'_N$ if $\\varpi' \\cdot x > N.$ The symbols $\\preceq$ and $\\succeq$ are interpreted in the obvious way. We use the terms ``far side of $L'_N$\" and ``near side of $L'_N$\" for the sets of $x \\in \\mathbb{R}^2$ with $x \\succ L'_N$ and $x \\prec L'_N,$ respectively.  Note that any lattice path $\\gamma$ intersecting both sides of $L'_N$ contains a vertex $z \\in L'_N.$\n\n\n\n\\begin{lem}\\label{lem: varpilemma}\nThere is a measurable choice of $\\varpi'$ as in \\eqref{eq: varpiprimedef} such that with $\\mu$-probability one, the following holds. For each vertex $x$ and each integer $N$, \n\\[\n\\Gamma_x \\cap \\{z \\in \\mathbb{Z}^2 : z \\preceq L'_N\\} \\text{ is finite}\\ .\n\\] \nIn other words, $\\Gamma_x$ eventually lies on the far side of $L'_N$ for all $x$ and $N$.\n\\end{lem}\n\n\\begin{proof}\nThe limit shape $\\mathcal{B}$ is convex and compact, so it has an extreme point $p$. Because it is symmetric with respect to the rotation $R$ of $\\mathbb{R}^2$ by angle $\\pi\/2$, the points $p_i = R^i p$, $i=1, \\ldots, 3$ are all extreme points of $\\mathcal{B}$. $J_\\varrho$ is an interval of angles corresponding to points of contact between $\\mathcal{B}$ and one of its supporting lines, so it is connected (in the topology induced by $dist$) and must lie between (inclusively) $\\arg p_i$ and $\\arg p_{i+1}$ for some $i=0, \\ldots, 3$ (here we identify $p_4=p_0$). Therefore $\\mathrm{diam} ~J_\\varrho \\leq \\pi\/2$ almost surely and contains at most three elements of the set $\\{j\\pi\/4 : j = 0, \\ldots, 7\\}$ (and they must be consecutive). Choose five of the remaining elements to be consecutive and label them $j_1\\pi\/4, \\ldots, (j_1+4)\\pi\/4$. The interval $[j_1\\pi\/4,(j_1+4)\\pi\/4]$ defines a half-plane $H$ in $\\mathbb{R}^2$ and since the distance between this interval and $J_\\varrho$ is positive (measured with $dist$), for all sufficiently small $\\varepsilon>0$, the sector\n\\[\n\\{x \\in \\mathbb{R}^2 : x \\neq 0 \\text{ and } dist(\\arg x, \\phi) < \\varepsilon \\text{ for some } \\phi \\in J_\\varrho\\}\n\\]\nis contained in $H^c$. This implies the statement of the lemma for a (random) $\\varpi'$ equal to the normal to $H$. Since $\\varpi'$ can be chosen as a measurable function of $\\varrho$ (which is clearly Borel measurable on $\\widetilde \\Omega$), we are done.\n\\end{proof}\n\n\nFor the rest of the proof, fix a deterministic $\\varpi'$ as in \\eqref{eq: varpiprimedef} that satisfies Lemma~\\ref{lem: varpilemma} with positive probability on the event $A_0(z_0)$. (This is possible because there are only eight choices for $\\varpi'$.) Let $A_0'(0,z_0)$ be the intersection of $A_0(z_0)$ and the event in the lemma. On $A_0'(0,z_0)$, $\\Gamma_0$ and $\\Gamma_{z_0}$ eventually cease to intersect $L'_0.$ In particular, they each have a last intersection with $L'_0.$ Since there are only countably many possible pairs of such last intersections, we see that some pair $(y, y')$ in $L'_0$ occurs with positive probability; that is, $\\mu(A(y,y'))>0$, where $A(y,y')$ is defined by the conditions\n\\begin{enumerate}\n\\item[I.] $\\Gamma_y \\cap \\Gamma_{y'} = \\varnothing;$\n\\item[II.] $\\Gamma_y$ intersects $L'_0$ only at $y$; $\\Gamma_{y'}$ intersects $L'_0$ only at $y'$ and\n\\item[III.] $\\Gamma_u \\cap L_N'$ is nonempty and bounded for $u=y,y'$ and all integers $N \\geq 0$.\n\\end{enumerate}\n(Note that condition III follows directly from the preceding lemma because $\\Gamma_u$ contains infinitely many vertices.) By translation invariance, there exists $z \\in L_0'$ with $\\mu(A(0,z))>0$.\n\nFix \n\\begin{equation}\\label{eq: chicken_alfredo}\n\\varsigma = \\text{ a nonzero vector with the smallest integer coordinates normal to } \\varpi'\n\\end{equation}\n(it will be a rotation of either (0,1) or (1,1) by a multiple of $\\pi\/2$). Defining $\\tilde T_\\varsigma : \\widetilde \\Omega \\to \\widetilde \\Omega$ as the translation by $\\varsigma$ (that is, $\\tilde T_1^{a_1} \\circ \\tilde T_2^{a_2}$, where $\\varsigma = a_1{\\mathbf{e}}_1 + a_2 {\\mathbf{e}}_2$),\n\\[ \n\\mathbf{1}_{A(0,z)}\\left((\\omega,\\Theta,\\eta)\\right) = \\mathbf{1}_{A(\\varsigma,z+\\varsigma)}\\left(\\widetilde{T}_{\\varsigma}(\\omega,\\Theta,\\eta)\\right)\\ .\n\\]\nSince $\\mu$ is invariant under the action of $\\widetilde{T}_{\\varsigma},$  the ergodic theorem implies\n\\begin{equation}\n\\label{poincare}\n\\frac{1}{N}\\sum_{j=0}^{N-1}\\mathbf{1}_{A(j\\varsigma,z+j\\varsigma)}\\left((\\omega,\\Theta,\\eta)\\right) = \\frac{1}{N}\\sum_{j=0}^{N-1}\\mathbf{1}_{A(0,z)}\\left(\\widetilde{T}_{\\varsigma}^j (\\omega,\\Theta,\\eta)\\right) \\rightarrow g(\\omega,\\Theta,\\eta),\n\\end{equation}\nwhere $g$ is a function in $L^1(\\mu);$ the convergence is both $\\mu$-almost sure and in $L^1(\\mu),$ so $\\int g \\, \\mathrm{d} \\mu = \\mu(A(0,z))>0.$ Using this in (\\ref{poincare}) gives infinitely many $j$ with \n\\begin{equation}\n\\label{two_pair}\n\\mu\\left(A(0,z) \\cap A(j\\varsigma, z + j\\varsigma) \\right) > 0.\n\\end{equation}\nWe fix $j > \\|z\\|_1$ to ensure $\\Gamma_{j\\varsigma}$ and $\\Gamma_{z + j\\varsigma}$ are outside the region bounded by $L'_0,$ $\\Gamma_0,$ and $\\Gamma_z.$\n\nWhat is the significance of the event in (\\ref{two_pair})?  When it occurs, we are guaranteed that there is a line $L_0'$ and four directed paths remaining on its far side apart from their initial vertices.  We claim that at least three of them never intersect. Indeed, ordering the paths using the direction of $\\varsigma$, we are guaranteed that the ``first two\" paths do not intersect each other, nor do the ``last two.\"  But if the middle two paths ever intersect, they would merge beyond that point and the three remaining paths could not touch.\n\nFor $x_1,x_2 \\in L_0'$, let $B(0,x_1,x_2)$ be the event that $\\Gamma_0, \\Gamma_{x_1}$ and $\\Gamma_{x_2}$ (a) never intersect, (b) stay on the far side of $L_0'$ except for their initial vertices and (c) intersect $L_N'$ in a bounded set for each $N \\geq 1$. Then the above implies \n\\[\nB(0,z,j\\varsigma) \\cup B(0,z,z+j\\varsigma) \\supseteq A(0,z) \\cap A(j\\varsigma,z + j\\varsigma)\\ .\n\\]\nTherefore we may choose $x_1,x_2 \\in L_0'$ such that the portion of $L_0'$ from 0 to $x_2$ contains $x_1$ and so that $\\mu(B(0,x_1,x_2)) > 0$. The vertices $x_1$ and $x_2$ are fixed for the rest of the proof.\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Constructing $B'$}\\label{sec: Bprime}\n\nOur next step is to refine $B(0,x_1,x_2)$ to a positive probability subevent $B'(x^*;N,R)$ on which no paths $\\Gamma_z$ with $z \\preceq L'_N$ (outside of some large polygon) merge with $\\Gamma_{x_1}.$ We will need to pull events back from $\\widetilde \\Omega$ to $\\Omega_1$ to do an edge modification and this will present a considerable difficulty. Our strategy is reminiscent of that in \\cite{AD}. In the first subsection we give several lemmas that we will need. In the next subsection we will define $B'$ and show it has positive probability.\n\n\n\n\n\n\\subsubsection{Lemmas for $B'$}\n\nWe wish to construct a barrier of high-weight edges on the near side of some $L'_N$. Set\n\\[ \n\\lambda_0^+ = \\sup \\left\\{ \\lambda > 0: \\, \\ensuremath{\\mathbb{P}}\\left( \\omega_e \\in [\\lambda,\\infty)\\right) > 0 \\right\\}\\ .\n\\]  \nBecause we do not wish to assume $\\lambda_0^+ = \\infty,$ our barrier will occupy some wide polygon (in the case that $\\lambda_0^+ = \\infty,$ many of the complications which we address below can be neglected; we direct the interested reader to \\cite{LN}).\nTo control the exit of our directed paths from the polygon, we will need a lemma about weak angular concentration of paths:\n\n\\begin{lem}\n\\label{path_concentration}\nFor $x_1$, $x_2$, and $\\varpi'$ as above, define $B_G(0,x_1,x_2)$ to be the subevent of $B(0,x_1,x_2)$ on which, for all $\\varepsilon > 0,$ there are infinitely\nmany values of $N \\in \\mathbb{N}$ such that the first intersections $\\zeta_N$ and $\\zeta'_N$ of $\\Gamma_0$ and $\\Gamma_{x_2}$ (respectively) with $L'_N$ satisfy $dist(\\arg \\zeta_N, \\arg \\zeta'_N) < \\varepsilon.$\nThen $\\mu\\left(B_G (0,x_1,x_2) \\mid \\, B(0,x_1,x_2)\\right) = 1.$\n\\end{lem}\n\\begin{proof}\nAssume for the sake of contradiction that \n\\begin{equation}\n\\label{paths_spreading}\n\\mu\\left(B_G^c(0,x_1,x_2) \\cap B(0,x_1,x_2)\\right) > 0.\n\\end{equation}\nFor $z \\in \\mathbb{Z}^2$, denote by $\\zeta_N(z)$ the first point of intersection of $\\Gamma_z$ with $L'_N.$  On the event in (\\ref{paths_spreading}), for all but finitely many $N \\in \\mathbb{N}$, we have $dist(\\arg \\zeta_N(0), \\arg \\zeta_N(x_2)) > \\varepsilon \/ 2.$  Taking $\\varsigma$ as before (fixed in \\eqref{eq: chicken_alfredo}) and translating the event in (\\ref{paths_spreading}) by multiples of $\\varsigma$, we see by the ergodic theorem that with positive $\\mu$-probability infinitely many such translates occur.\n\nSo given any finite $b > 0,$ we can find an event of positive $\\mu$-probability on which we have at least $b$ directed paths in $\\mathbb{G}$ which never return to $L'_0$ and such that the first intersections of neighboring paths with lines $L'_N$ stay at least an angle $\\varepsilon$ apart. This is in contradiction with the fact that all directed infinite paths are asymptotically confined to a sector.\n\\end{proof}\n\nThe next lemma is a modification of the usual first-passage shape theorem.\n\\begin{lem}\n\\label{l1_shape}\nThere exists a deterministic $c^+ < \\lambda_0^+$ such that, $\\ensuremath{\\mathbb{P}}$-a.s., \n\\[\n\\lim_{M \\to \\infty} \\sup_{\\|x\\|_1 \\geq M} \\tau(0,x)\/\\|x\\|_1 < c^+\\ .\n\\]\n\\end{lem}\n\n\\begin{proof}\nBecause either {\\bf A1'} or {\\bf A2'} hold, $\\mathbb{E}(\\tau_e) < \\lambda_0^+$. For any $z \\in \\mathbb{Z}^2$, choose a deterministic path $\\gamma_z$ with number of edges equal to $\\|z\\|_1$. For $x \\in \\mathbb{Q}^2$ and $n \\geq 1$ with $nx \\in \\mathbb{Z}^2$,\n\\[\n\\mathbb{E} \\tau(0,nx) \\leq \\mathbb{E} \\tau(\\gamma_{nx}) = n \\|x\\|_1 \\mathbb{E}\\tau_e\\ , \\text{ so } g(x) \\leq \\|x\\|_1 \\mathbb{E} \\tau_e\\ .\n\\] \nThis extends to all $x \\in \\mathbb{R}^2$ by continuity, so the shape theorem gives the result.\n\\end{proof}\n\nWe need a lemma to pull events back from $\\widetilde \\Omega$ to $\\Omega_1$. Fix an increasing sequence $(n_k)$ such that $\\mu^*_{n_k} \\to \\mu$ weakly.\n\\begin{lem}\n\\label{toalphas}\nLet $E \\subseteq \\widetilde \\Omega$ be open with $\\mu(E) > \\beta$. There exists $C_\\beta>0$ and $K_0$ such that for $k\\geq K_0$, the Lebesgue measure of the set $\\{ \\alpha \\in [0,n_k]: \\, \\mu_\\alpha (E)  > \\beta\/2 \\}$ is at least $C_\\beta\\, n_k$.\n\\end{lem}\n\\begin{proof}\nCall the Lebesgue measure of the above set $\\lambda.$ Since $E$ is open, \\eqref{eq: kallenbergopen} allows us to pick $K_0$ such that if $k \\geq K_0$ then $\\mu^*_{n_k}(E) > \\beta$. For such $k$, we can write\n\\[\n\\frac{1}{n_k} \\left( \\lambda + (n_k - \\lambda) \\beta\/2\\right) \\geq \\mu_{n_k}^*(E) > \\beta,~ \\text{giving } \\lambda > \\frac{n_k \\beta}{2 (1 - \\beta\/2)}\\ .\n\\]\nSetting $C_\\beta := \\beta (2 - \\beta)^{-1}$ completes the proof.\n\\end{proof}\n\n\nThe last lemma is based on \\cite[Lemma~3.4]{AD} and will be used in the edge-modification argument. To push the upward finite energy property forward from $\\Omega_1$ to $\\widetilde \\Omega$ we need concrete lower bounds for probabilities of modified events. We write a typical element of $\\Omega_1$ as $\\omega = (\\omega_e, \\check{\\omega}),$ where $\\check{\\omega} = (\\omega_f)_{f \\neq e}.$ We say an event $A \\subseteq \\Omega_1$ is {\\it $e$-increasing} if, for all $(\\omega_e,\\check{\\omega}) = \\omega \\in A$ and $r > 0,$ $(\\omega_e + r, \\check{\\omega}) \\in A.$\n\n\\begin{lem}\n\\label{lem: edge_modification}\nLet $\\lambda > 0$ be such that $\\mathbb{P}\\left(\\omega_e \\geq \\lambda \\right) > 0.$ For each $\\vartheta>0$ there exists $C = C(\\vartheta,\\lambda) > 0$ such that for all edges $e$ and all $e$-increasing events $A$ with $\\mathbb{P}(A) \\geq \\vartheta$,\n\\[\n\\mathbb{P}\\left( A, ~\\omega_e \\geq \\lambda\\right) \\geq C ~\\mathbb{P}\\left(A\\right)\\ .\n\\]\n\\end{lem}\n\n\\begin{proof}\nIf $\\mathbb{P}(A,~ \\omega_e < \\lambda) \\leq (1\/2) \\mathbb{P}(A)$ then \n\\begin{equation}\\label{eq: on_skype_again}\n\\mathbb{P}(A,~\\omega_e \\geq \\lambda) \\geq (1\/2) \\mathbb{P}(A)\\ .\n\\end{equation}\nOtherwise, we assume that\n\\begin{equation}\\label{eq: newassumption}\n\\mathbb{P}(A,~ \\omega_e < \\lambda) \\geq (1\/2) \\mathbb{P}(A)\\ .\n\\end{equation}\n\nWe then need to define an extra random variable. Let $\\omega_e'$ be a variable such that, given $\\check{\\omega}$ from $\\omega \\in \\Omega_1$, it is an independent copy of the variable $\\omega_e$. In other words, letting $\\mathbb{Q}$ be the joint distribution of $(\\omega, \\omega_e')$ on the space $\\Omega_1 \\times \\mathbb{R}$, for $\\mathbb{Q}$-almost every $\\check{\\omega}$,\n\\begin{itemize}\n\\item $\\omega_e'$ and $\\omega_e$ are conditionally independent given $\\check{\\omega}$ and\n\\item the distributions $\\mathbb{Q}(\\omega_e \\in \\cdot \\mid \\check{\\omega})$ and $\\mathbb{Q}(\\omega_e' \\in \\cdot \\mid \\check{\\omega})$ are equal.\n\\end{itemize}\n(This can be defined, for instance, by setting $\\mathbb{Q}(A \\times B) = \\int_A \\mathbb{P}(\\omega_e \\in B \\mid \\check{\\omega}) ~\\mathrm{d} \\mathbb{P}(\\omega)$ for Borel sets $A \\subseteq \\Omega_1$ and $B \\subseteq \\mathbb{R}$.)\n\n\nWe now write $\\mathbb{P}(A, \\omega_e \\geq \\lambda)$ as\n\\begin{align}\n\\mathbb{Q} [ (\\omega_e, \\check{\\omega})\\in A,\\, \\omega_e \\in [\\lambda,\\infty)] & \\geq \\mathbb{Q}\\left[(\\omega_e,\\check{\\omega}) \\in A,\\, \\omega_e \\in [\\lambda,\\infty),\\,\\omega_e' \\in [0,\\lambda)\\right]\\nonumber\\\\\n&=\\mathbb{E}_{\\mathbb{Q}} \\left[ \\mathbf{1}_{(\\omega_e,\\check{\\omega}) \\in A}\\, \\mathbf{1}_{ \\omega_e \\in [\\lambda,\\infty)}\\, \\mathbf{1}_{\\omega_e' \\in [0,\\lambda)}\\right]\\nonumber\\\\\n&\\geq\\mathbb{E}_{\\mathbb{Q}} \\left[ \\mathbf{1}_{(\\omega_e',\\check{\\omega}) \\in A}\\, \\mathbf{1}_{ \\omega_e \\in [\\lambda,\\infty)}\\, \\mathbf{1}_{\\omega_e' \\in [0,\\lambda)}\\right]\\label{switcheroo}\\\\\n&=\\mathbb{E}_{\\mathbb{Q}} \\left[ \\mathbf{1}_{(\\omega_e',\\check{\\omega})\\in A}\\, \\mathbf{1}_{\\omega_e'\\in [0,\\lambda)}\\,  \\mathbb{E}_{\\mathbb{Q}}\\left(\\mathbf{1}_{\\omega_e \\in [\\lambda,\\infty)}\\, \\mid \\check{\\omega},\\omega_e' \\right)\\right]. \\label{condintready}\n\\end{align}\nIn (\\ref{switcheroo}), we have used that $A$ is $e$-increasing. Using conditional independence in (\\ref{condintready}),\n\\begin{equation}\\label{eq: something_something}\n\\ensuremath{\\mathbb{P}}(A, ~\\omega_e \\geq \\lambda) \\geq \\mathbb{E}_{\\mathbb{Q}} \\left[ \\mathbf{1}_{(\\omega_e',\\check{\\omega})\\in A}\\, \\mathbf{1}_{\\omega_e' \\in [0,\\lambda)}\\,  \\mathbb{E}_{\\mathbb{Q}}\\left(\\mathbf{1}_{\\omega_e \\in [\\lambda,\\infty)}\\, \\mid \\check{\\omega}\\right)\\right]\\ .\n\\end{equation}\nBy the upward finite energy property, \n\\[\n\\mathbb{E}_{\\mathbb{Q}}(\\mathbf{1}_{\\omega_e \\in [\\lambda,\\infty)} \\mid \\check{\\omega}) = \\mathbb{E}(1_{\\omega_e \\in [\\lambda,\\infty)}\\mid \\check{\\omega}) > 0 ~~\\mathbb{Q} \\text{-almost surely}\\ ,\n\\]\nso choose $c>0$ such that \n\\[\n\\mathbb{Q}\\left[ \\mathbb{E}_{\\mathbb{Q}}( \\mathbf{1}_{\\omega_e \\in [\\lambda,\\infty)} \\mid \\check{\\omega}) \\geq c \\right] \\geq 1-(\\vartheta\/4)\\ .\n\\]\nNote that this choice of $c$ depends only on $\\lambda$ and $\\vartheta$. By \\eqref{eq: newassumption} and the assumption $\\mathbb{P}(A) \\geq \\vartheta$, the right side is at least $1-(1\/2)\\mathbb{P}(A,~\\omega_e< \\lambda)$, implying\n\\[\n\\mathbb{Q}\\left[ (\\omega_e',\\check{\\omega}) \\in A, ~\\omega_e'\\in [0,\\lambda),~ \\mathbb{E}_{\\mathbb{Q}}(\\mathbf{1}_{\\omega_e \\in [\\lambda,\\infty)} \\mid \\check{\\omega}) \\geq c\\right] \\geq (1\/2) \\mathbb{P}(A,~\\omega_e < \\lambda)\\ .\n\\]\nCombining with \\eqref{eq: something_something}, we find $\\mathbb{P}(A,~\\omega_e\\geq \\lambda) \\geq (c\/2)\\mathbb{P}(A,~\\omega_e < \\lambda)$. We finish the proof by writing\n\\[\n\\ensuremath{\\mathbb{P}}(A) = \\ensuremath{\\mathbb{P}}(A, \\omega_e < \\lambda) + \\ensuremath{\\mathbb{P}}(A, \\omega_e \\geq \\lambda)\\\\\n\\leq \\left[\\frac{2}{c}+1 \\right] \\ensuremath{\\mathbb{P}}(A,\\omega_e \\geq \\lambda)\\ .\n\\]\nObserving this inequality and \\eqref{eq: on_skype_again}, we set $C = \\min\\{1\/2, c\/(2+c))\\}$.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Defining $B'$}\n\nWe begin with the definition of the ``barrier event'' $B'$. For an integer $R > N,$ let \n\\[\nS(R,N) = \\{y \\in \\mathbb{Z}^2 : 0 \\leq y \\cdot \\varpi' \\leq N,~ |y \\cdot \\varsigma| \\leq R\\}\\ .\n\\] \nFor any vertex $x^* \\in S(R,N) \\cap L_N'$, define $B'(x^*;R,N)$ by the condition\n\\begin{equation}\\label{eq: B_prime_def}\n\\text{for all } z \\in \\mathbb{Z}^2 \\setminus S(R,N) \\text{ with } z \\preceq L_N',~ \\Gamma_z \\cap \\Gamma_{x^*} = \\varnothing\\ .\n\\end{equation}\n\n\\begin{prop}\n\\label{B_prime}\nThere exist values of $R,N$ and $x^*$ such that $\\mu(B'(x^*;R,N)) > 0.$\n\\end{prop}\nOur strategy is to pull back cylinder approximations of $B(0,x_1,x_2)$ to $\\Omega_1$ to find events that depend on $\\mathbb{G}$ in the vicinity of $0, x_1$ and $x_2.$ We will find a subevent which is monotone increasing in the weights of edges lying in $S(R,N)$ between the pulled-back versions of $\\Gamma_0$ and $\\Gamma_{x_2}.$ When we look at the subevent on which all of these weights are large (``edge modification\"), the pullback of $\\Gamma_{x_1}$ will be unchanged (past $S(R,N)$), and no pullback of any $\\Gamma_z$ can intersect it if $z \\preceq L'_N$ and $z \\notin S(R,N).$ We will then choose $x^*$ to be a certain point on $\\Gamma_{x_1} \\cap L'_N$. The constants $N$ and $R$ will be chosen to guarantee that the pullback of $\\Gamma_{x_1}$ is so isolated. Pushing forward the subevent to $\\widetilde{\\Omega}$ will complete the proof.\n\n\\begin{proof}\nWe will first fix some parameters to prepare for the main argument. Recall the definition of $c^+$ from Lemma~\\ref{l1_shape} and let\n\\[ \n\\lambda^+ := \\min\\{\\lambda_0^+, \\, 2 c^+\\}\\ ,\n\\]\nand put $\\delta^+ := \\lambda^+ - c^+>0$ (giving $\\lambda^+ = 2c^+$ when $\\lambda_0^+=\\infty$). Choose once and for all some \n\\begin{equation}\\label{eq: epsilondef}\n\\varepsilon < \\frac{\\delta^+}{16\\lambda^+},\n\\end{equation}\nsuch that also\n\\begin{equation}\\label{eq: pizzapie34}\n\\limsup_{\\|x\\|_1 \\rightarrow \\infty}\\,\\, \\sup_{y: \\, \\|y - x\\|_1 \\leq \\varepsilon \\|x\\|_1} \\frac{\\tau(0,y)}{\\|x\\|_1} < \\lambda^+ - \\frac{7 \\delta^+}{8}\\quad \\mu\\text{-a.s.}\n\\end{equation}\nThis follows from Lemma~\\ref{l1_shape} because if $\\|y\\|_1$ is large, $\\|y-x\\|_1 \\leq \\varepsilon \\|x\\|_1$ gives $\\tau(0,y)\/\\|x\\|_1 \\leq (\\tau(0,y)\/\\|y\\|_1)(1+\\varepsilon) < c^+(1+\\varepsilon)$. Fix $\\beta > 0$ with $\\mu(B(0,x_1,x_2)) > \\beta.$  \n\nThe majority of the proof will consist of defining a few events in sequence, the second of which we will pull back to the space $\\Omega_1$ to do the edge modification. We will need to choose further parameters to ensure that each of these events has positive probability. For an arbitrary outcome in $\\widetilde{\\Omega}$ and $N\\geq 0$, denote by $r_0(N)$ and $r_2(N)$ the segments of $\\Gamma_0$ and $\\Gamma_{x_2}$ up to their first intersections with $L'_N$ (if they exist) and let $w_N$ denote the midpoint of the segment of $L'_N$ lying between these first intersections. The first event $B^\\circ(R,N,\\varepsilon)$ is defined by the conditions (for $R,N \\geq 1$)\n\\begin{enumerate}\n\\item $\\Gamma_0, \\Gamma_{x_1}$ and $\\Gamma_{x_2}$ never intersect,\n\\item they stay on the far side of $L'_0$ except for their initial vertices,\n\\item $\\Gamma_0$ and $\\Gamma_{x_2}$ intersect $L'_N$ and their first intersection points are within $\\ell^1$ distance $\\varepsilon N$ of each other,\n\\item for $i=0,2$, $\\tau(r_i(N)) < (\\lambda^+-7\\delta^+\/8)\\|w_N\\|_1$ and\n\\item $\\Gamma_0$ and $\\Gamma_{x_2}$ do not touch any $x \\preceq L'_N$ with $x \\notin S(R,N)$.\n\\end{enumerate}\nSee Figure~\\ref{fig: b_circ} for a depiction of the event $B^\\circ(R,N,\\varepsilon)$.\n\n\\begin{figure}[h]\n\\caption{The event $B^\\circ(R,N,\\varepsilon).$ The solid dots represent the first intersection points of $\\Gamma_0$ and $\\Gamma_{x_2}$ with $L'_N$. They are within $\\ell^1$ distance $\\varepsilon N$ of each other.}\n\\label{fig: b_circ}\n\\centering\n\\includegraphics[scale=0.65]{B_circ_9_11.pdf}\n\\end{figure}\n\nWe claim that there exists $N_0$ and $R_0$ such that \n\\begin{equation}\\label{eq: pizzapie44}\n\\mu(B^\\circ (R_0,N_0,\\varepsilon)) > 0\\ .\n\\end{equation}\nWe also need $N_0$ to satisfy a technical requirement. It will be used at the end of the proof:\n\\begin{equation}\\label{eq: technical_condition}\n\\|x_2\\|_1 \\leq \\varepsilon N_0\\ .\n\\end{equation}\n\nTo pick $N_0$, first choose $N_1>0$ so large that if $N \\geq N_1$ then\n\\begin{equation}\\label{eq: pizzapie45}\n\\ensuremath{\\mathbb{P}}\\left( \\forall z,z' \\text{ with } \\|z\\|_1 \\geq N, \\text{ and } \\frac{\\|z-z'\\|_1}{\\|z\\|_1} \\leq \\varepsilon,~ \\frac{\\tau(0,z')}{\\|z\\|_1} < (\\lambda^+ - \\frac{7 \\delta^+}{8}) \\right) > 1 - \\beta \/ 4\\ ,\n\\end{equation}\nand $\\|x_2\\|_1 \\leq \\varepsilon N$. This is possible by \\eqref{eq: pizzapie34}. Write $E_0(N)$ for the event in \\eqref{eq: pizzapie45} and $E_{x_2}(N)$ for $E_0(N)$ translated so that $0$ is mapped to $x_2$. Then $\\mathbb{P}(B(0,x_1,x_2) \\cap E_0(N) \\cap E_{x_2}(N)) > \\beta\/2$. By Lemma~\\ref{path_concentration}, we can then choose $N_0 \\geq N_1$ such that\n\\begin{equation}\\label{eq: pizzapie46}\n\\mu(B(0,x_1,x_2) \\cap E_0(N_0) \\cap E_{x_2}(N_0) \\cap C(0,x_2;N_0)) > 0\\ ,\n\\end{equation}\nwhere $C(0,x_2;N_0)$ is the event that $\\Gamma_0$ and $\\Gamma_{x_2}$ intersect $L_{N_0}'$ and their first intersection points are within $\\ell^1$ distance $\\varepsilon N_0$ of each other. On the event in \\eqref{eq: pizzapie46}, the endpoints of the $r_i(N_0)$'s are within distance $\\varepsilon N_0$ of $w_{N_0}$ and since they are on $L'_{N_0}$, their $\\ell^1$ distance from $0$ or $x_2$ is at least $N_0$. Therefore $\\tau(r_i(N_0)) < (\\lambda^+ - 7 \\delta^+\/8)\\|w_{N_0} \\|_1$ for $i=0,2$. This shows that the intersection of four of the five events in the definition of $B^\\circ(R,N_0,\\varepsilon)$ occurs with positive probability. For the fifth, recall that on $B(0,x_1,x_2)$, the paths $\\Gamma_0$, $\\Gamma_{x_1}$ and $\\Gamma_{x_2}$ contain only finitely many vertices $z \\preceq L'_{N_0}$. Thus we can choose $R_0$ large enough (depending on $N_0$) to satisfy condition 5 and complete the proof of \\eqref{eq: pizzapie44}.\n\nFix these $R=R_0$ and $N=N_0$ from now on. The next event we define is a cylinder approximation of the first event. It will be needed to pull back to $\\Omega_1$. For $M>0$ and $x \\in \\ensuremath{\\mathbb{Z}^2},$ let $\\Gamma^M_x$ be the finite path formed by starting at $x$ and then passing along out-edges of $\\mathbb{G}$ until we first reach a vertex of $\\mathbb{R}^2 \\setminus (-M,M)^2$. (Note that by this definition, $\\Gamma_x^M = \\{x\\}$ whenever $x \\notin (-M,M)^2$.) We define $B^\\circ_M(R,N,\\varepsilon)$ with the same conditions as $B^\\circ(R,N,\\varepsilon),$\nexcept replacing the paths $\\Gamma_{(\\cdot)}$ by the segments $\\Gamma_{(\\cdot)}^M$. In addition, however, we impose the restriction that, writing\n\\[\n\\partial M = [-M,M]^2 \\setminus (-M,M)^2\\ ,\n\\]\nwe have\n\\begin{equation}\\label{eq: neweq}\n\\Gamma_y^M \\cap \\partial M \\subseteq \\{ z \\in \\mathbb{R}^2 : z \\succ L'_N\\},~ y=0,x_2\\ .\n\\end{equation}\nOf course, if $\\Gamma_0^M$ (etc.) does not intersect $L'_N,$ then $B^\\circ_M$ does not occur.  Then $B^\\circ_M(R,N,\\varepsilon)$ is open for all $M$ and we claim that\n\\begin{equation}\\label{eq: cylinderapprox}\nB^\\circ (R,N,\\varepsilon) = \\cup_{M_0 = 1}^{\\infty} \\cap_{M=M_0}^{\\infty} B^\\circ_M(R,N,\\varepsilon)\\ .\n\\end{equation}\nAssuming we show this, then there exists some $M_0$ such that $\\mu(\\cap_{M=M_0}^{\\infty} B^\\circ_M(R,N,\\varepsilon)) > 0$ and so there is some $\\beta'$ with\n\\begin{equation}\\label{eq: aftercylinder}\n\\mu(B^\\circ_M(R,N,\\varepsilon)) > \\beta' \\text{ for all }M \\geq M_0\\ .\n\\end{equation}\n\nTo prove \\eqref{eq: cylinderapprox}, note that the right side is the event that $B^\\circ_M(R,N,\\varepsilon)$ occurs for all $M$ bigger than some random $M_0$. Suppose that an outcome is in the left side. Then the paths $\\Gamma_0$, $\\Gamma_{x_1}$ and $\\Gamma_{x_2}$ are disjoint and remain on the far side of $L_0'$ (except for their first vertices), so the same is true for each $\\Gamma_{(\\cdot)}^M$ for all $M \\geq 1$. Also $\\Gamma_0^M$ and $\\Gamma_{x_2}^M$ do not touch any $x \\preceq L_N'$ with $x \\notin S(R,N)$ for all $M \\geq 1$. Because $\\Gamma_0$ and $\\Gamma_{x_2}$ intersect $L_N'$, so do $\\Gamma_0^M$ and $\\Gamma_{x_2}^M$ for all $M$ bigger than some random $M_1$. Their first intersection points are the same as those of $\\Gamma_0$ and $\\Gamma_{x_2}$, so for $M \\geq M_1$, their first intersection points with $L'_N$ are within $\\ell^1$ distance $\\varepsilon N$ of each other. Further, the passage times of the segments up to $L'_N$ are strictly bounded above by $(\\lambda^+-7\\delta^+\/8)\\|w_N\\|_1$. Last, because $\\Gamma_0$ and $\\Gamma_{x_2}$ do not touch any $x \\preceq L'_N$ with $x \\notin S(R,N)$, they share only finitely many vertices with $\\{z \\in \\mathbb{Z}^2 : z \\preceq L'_N\\}$ and so must eventually lie on the far side of $L'_N$. This allows us to further increase $M_1$ to an $M_0$ such that if $M \\geq M_0$ then in addition \\eqref{eq: neweq} holds.\n\nSuppose conversely that the right side of \\eqref{eq: cylinderapprox} occurs. Then for all $M$ bigger than some random $M_0$, the six events comprising $B^\\circ_M(R,N,\\varepsilon)$ occur. In particular, the paths $\\Gamma_0$, $\\Gamma_{x_1}$ and $\\Gamma_{x_2}$ are disjoint and stay on the far side of $L_0'$ except for their first vertices (parts 1 and 2 of $B^\\circ(R,N,\\varepsilon)$). Furthermore $\\Gamma_0$ and $\\Gamma_{x_2}$ cannot touch any $x \\preceq L_N'$ with $x \\notin S(R,N)$ (part 5). For $M \\geq M_0$, the paths $\\Gamma_0^M$ and $\\Gamma_{x_2}^M$ intersect $L_N'$, with their first intersection points within distance $\\varepsilon N$ of each other (with passage time strictly bounded above by $(\\lambda^+ - 7\\delta^+\/8)\\|w\\|_1$). These are the same first intersection points as $\\Gamma_0$ and $\\Gamma_{x_2}$, so parts 3 and 4 of $B^\\circ(R,N,\\varepsilon)$ occur.\n\nWe now pull the cylinder approximation $B^\\circ_M(R,N,\\varepsilon)$ back to $\\Omega_1$ using Lemma~\\ref{toalphas}. Because this is an open event and satisfies \\eqref{eq: aftercylinder} for $M\\geq M_0$, we can find an $M$-dependent number $K_0$ such that if $k \\geq K_0$, then there is a set $\\Lambda_{M,k}$ of values of $\\alpha \\in [0,n_k]$ which has Lebesgue measure at least $C_{\\beta'} n_k$, on which $\\mu_\\alpha(B^\\circ_M(R,N,\\varepsilon)) > \\beta'\/2$. Pull back to $\\Omega_1,$ setting $B_M^\\alpha:= \\Phi_\\alpha^{-1}(B^\\circ_M(R,N,\\varepsilon)),$ where $\\Phi_\\alpha$ was defined in \\eqref{eq: phidef}. (Here we have suppressed mention of $R,N,\\varepsilon$ in the notation, as they are fixed for the remainder of the proof.) Then \n\\begin{equation}\\label{eq: pizzapie77}\n\\ensuremath{\\mathbb{P}}(B_M^\\alpha) > \\beta'\/2 \\text{ for all } \\alpha \\in \\Lambda_{M,k} \\text{ if } M \\geq M_0 \\text{ and } k \\geq K_0(M)\\ .\n\\end{equation}\nWe henceforth restrict to values of $M,$ $\\alpha$ and $k$ such that (\\ref{eq: pizzapie77}) holds. In the end of the proof we will take $k \\to \\infty$ and then $M \\to \\infty$. In particular then we will be thinking of \n\\[\n\\alpha \\gg M \\gg N\\ , \n\\]\nthe latter of which is fixed. Some of the remaining definitions will only make sense for such $\\alpha$, $M$ and $N$ but this does not affect the argument.\n\nNext we define the third of our four events, now working on $\\Omega_1$. Let $s^\\alpha_{y}$ be the geodesic from $y \\in \\ensuremath{\\mathbb{Z}^2}$ to $L_\\alpha$ (recall this was defined for $\\varpi$ and not $\\varpi'$), and $s^\\alpha_y(M)$ the path $s^\\alpha_y$ up to its first intersection with $\\mathbb{R}^2 \\setminus (-M,M)^2$. If $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ intersect $L_N'$ then write $r_i^\\alpha(M),$ $i=0,2$ for the portions up to the first intersection point. As before, let $w^\\alpha_N$ be the midpoint of the segment of $L_N'$ between these two intersection points. Let $\\mathcal{R}_1^\\alpha(M)$ be the closed connected subset (in $\\mathbb{R}^2$) of $\\{x \\in \\mathbb{R}^2: x \\succeq L'_0\\}$ with boundary curves $s_0^\\alpha(M)$, $s_{x_2}^\\alpha(M)$, $L_0'$ and $\\partial M$. Similarly let $\\mathcal{R}_2^\\alpha(M)$ be the closed connected subset of $\\mathcal{R}_1^\\alpha(M)$ with the following boundary curves: the portions of $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ after their last intersections with $L_N'$, the segment of $L_N'$ between these intersections and last, $\\partial M$. Note that when \\eqref{eq: neweq} holds, $\\mathcal{R}_2^\\alpha(M)$ is contained in $\\{z \\in \\mathbb{R}^2: z \\succeq L'_N\\}$. See Fig.~\\ref{my_first_table} for an illustration of these definitions.\n\n\\begin{figure}[h]\n\\caption{The regions $\\mathcal{R}_1^\\alpha(M)$ and $\\mathcal{R}_2^\\alpha(M)$. The left figure shows $\\mathcal{R}_1^\\alpha(M)$ in green. It has boundary curves $L'_0$, $\\partial M$, $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$. The right figure shows $\\mathcal{R}_2^\\alpha(M) \\subseteq \\mathcal{R}_1^\\alpha(M)$ in green. It has boundary curves $L'_N$, $\\partial M$, and the pieces of $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ from their last intersections with $L'_N$. Note that $\\mathcal{R}_2^\\alpha(M)$ is contained in the far side of $L'_N$ by \\eqref{eq: neweq}.}\n\\label{my_first_table}\n\\centering\n\\begin{tabular}{cc}\n\\includegraphics[scale=0.40]{R_region_1_9_11.pdf}\n\\includegraphics[scale=0.40]{R_region_2.pdf}\n\\end{tabular}\n\\end{figure}\n\nThe event $\\hat B_M^\\alpha \\subseteq \\Omega_1$ is then defined by the following conditions:\n\\begin{itemize}\n\\item $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ intersect $L'_0$ only once, are disjoint, and do not touch any $y \\preceq L'_N$ with $y \\notin S(R,N)$.\n\\item $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ intersect $L'_N$ and their first intersection points are within $\\ell^1$ distance $\\varepsilon N$ of each other; the paths $r_i^\\alpha(M)$ satisfy $\\tau(r_i^\\alpha(M)) < (\\lambda^+ -7 \\delta^+\/8) \\|w^\\alpha_N\\|_1,$ for $i=0,2$.\n\\item $s_y^\\alpha(M) \\cap \\partial M \\subseteq \\{z \\in \\mathbb{R}^2 : z \\succ L'_N\\}$ for $y=0,x_2$,\n\\item there is a vertex $X^* \\in L'_N \\cap S(R,N)$ such that $s_{X^*}^\\alpha(M)$ is disjoint from $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ but is contained in $\\mathcal{R}_2^\\alpha(M)$, and \n\\item the portions of $s_0^\\alpha,$ $s_{X^*}^\\alpha$ and $s_{x_2}^\\alpha$ beyond $[-M,M]^2$ do not contain a vertex of $S(R,N)$;\n\\end{itemize}\n\nWe claim there is an $M_0' \\geq M_0$ such that\n\\begin{equation}\\label{eq: pizzapie99}\n\\mathbb{P}(\\hat B_M^\\alpha) > \\beta'\/4 \\text{ for all } M \\geq M_0'\\ .\n\\end{equation}\nVerifying this requires us to define an auxiliary event. Let $H_{M} \\subseteq \\Omega_1$ denote the event that no geodesic from any point in $S(R,N)$ returns to $S(R,N)$ after its first intersection with $\\partial M.$  Then $\\ensuremath{\\mathbb{P}}(H_M) \\rightarrow 1$ as $M \\rightarrow \\infty.$  So for any $M$ larger than some $M_0' \\geq M_0$, $\\mathbb{P}(H_M) > 1-\\beta'\/4$, giving\n\\[ \n\\ensuremath{\\mathbb{P}}(B_M^\\alpha \\cap H_M) > \\beta'\/4 \\text{ for all } M \\geq M_0'\\ .\n\\]\nTo finish the proof of \\eqref{eq: pizzapie99} we show that $B_M^\\alpha \\cap H_M \\subseteq \\hat B_M^\\alpha$. Note that the first three conditions of $\\hat B_M^\\alpha$ are immediately implied by $B_M^\\alpha$; they are the analogues on $\\Omega_1$ of the conditions that make up $B_M^\\circ(N,R,\\varepsilon)$ (each $\\Gamma_{(\\cdot)}^M$ is replaced by $s_{(\\cdot)}^\\alpha(M)$). For the fourth condition, note that when $B_M^\\alpha$ occurs, $s_0^\\alpha(M)$, $s_{x_1}^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ stay on the far side of $L_0'$ (aside from their initial vertices) and stop when they touch $\\partial M$. Therefore by planarity, $s_{x_1}^\\alpha(M)$ is contained in $\\mathcal{R}_1^\\alpha(M)$. In particular, if we choose $X^*$ to be the last intersection point of $s_{x_1}^\\alpha(M)$ with $L'_N$, then $s_{X^*}^\\alpha(M)$ is trapped in $\\mathcal{R}_2^\\alpha(M)$. We can see this as follows. The last vertex of $s_{X^*}^\\alpha(M)$ is clearly in this region because it must be in $\\mathcal{R}_1^\\alpha(M) \\cap \\partial M$ and this equals $\\mathcal{R}_2^\\alpha(M) \\cap \\partial M$. Proceeding backward along $s_{X^*}^\\alpha(M)$ from this final vertex, the path can only leave $\\mathcal{R}_2^\\alpha(M)$ if it (a) leaves $[-M,M]^2$ (b) crosses $s_0^\\alpha(M)$ or $s_{x_2}^\\alpha(M)$ or (c) crosses $L'_N$. Because none of these can happen, the fourth condition holds. As for the fifth, it is implied by $H_M$, so we have proved \\eqref{eq: pizzapie99}.\n\nOur fourth and final event will fix some random objects to be deterministic so that we can apply the edge modification lemma. On the event $\\hat B_M^\\alpha$, let $U$ denote the (random) closed connected subset of $[-M,M]^2$ with boundary curves $L'_0$, $L'_N$, $r_0^\\alpha(M)$ and $r_2^\\alpha(M)$. Note that $U \\subseteq S(R,N)$. Furthermore we note that on $\\hat B_M^\\alpha$, $U \\cap \\mathcal{R}_2^\\alpha(M)$ is contained in $L_N'$. This is because $\\mathcal{R}_2^\\alpha(M) \\subseteq \\{z : z \\succeq L'_N\\}$, whereas $U\\subseteq \\{z : z \\preceq L'_N\\}$. Last, define $U_\\mathcal{E}$ to be the random set of edges with both endpoints in $U$ and which are not edges in $s_0^\\alpha(M),s_{x_2}^\\alpha(M), L_0'$ or $L_N'$. See Figure~\\ref{fig: bddcase} for an illustration of these definitions. \n\n\\begin{figure}[h]\n\\caption{Illustration of definitions on $\\hat B_M^\\alpha$. The region $U$ is in blue and is contained in $S(R,N)$ (not pictured). It is bounded by curves $L'_0$, $L'_N$, $r_0^\\alpha(M)$ and $r_2^\\alpha(M)$. The path $s_{x^*}$ begins at the final intersection point of the dotted path with $L'_N$.}\n\\label{fig: bddcase}\n\\centering\n\\includegraphics[scale=0.65]{bddcase2.pdf}\n\\end{figure}\n\nOn $\\hat{B}_M^\\alpha,$ there are at most $2^{64NR}$ possibilities for $U$ and $U_\\mathcal{E}$ and at most $2R$ choices for $X^*.$ So there exist some deterministic $U',$ $U_{\\mathcal{E}}',$ and $x^*$ such that, if we define\n\\[\n\\tilde{B}_M^\\alpha:= \\hat{B}^\\alpha_M \\cap \\{U = U', \\, U_\\mathcal{E} = U'_{\\mathcal{E}}\\}\\cap \\{X^* = x^*\\}\\ , \n\\]\nthen\n\\begin{equation}\\label{eq: nachos_bellgrande}\n\\ensuremath{\\mathbb{P}}(\\tilde{B}_M^\\alpha) > 2^{-2 - 64NR} \\beta' \/ 2R \\text{ for } M \\geq M_0' \\text{ and }\\alpha \\in \\Lambda_{M,k}\\ . \n\\end{equation}\nThe meaning of the event $\\{X^*=x^*\\}$ is that the deterministic point $x^*$ satisfies the conditions in the fourth and fifth items of the description of $\\hat B_M^\\alpha$.\n\nIn the rest of the proof we perform the edge modification and push forward to $\\widetilde \\Omega$. To apply Lemma~\\ref{lem: edge_modification} we need to verify that $\\tilde B_M^\\alpha$ is $e$-increasing for all $e \\in U'_{\\mathcal{E}}$. For this purpose, suppose that $\\omega \\in \\tilde B_M^\\alpha$ and that $\\omega'$ is another configuration such that $\\omega_e' \\geq \\omega_e$ for some fixed $e \\in U'_{\\mathcal{E}}$ but $\\omega_f' = \\omega_f$ for all other $f \\neq e$. By construction, $e$ is not an edge of $s_0^\\alpha(M)$, $s_{x^*}^\\alpha(M)$ or $s_{x_2}^\\alpha(M)$ ($e \\notin s_{x^*}^\\alpha(M)$ since $e$ is contained in $U_\\mathcal{E}$, which does not meet $L_N'$, so is not in $\\mathcal{R}_2^\\alpha(M) \\supseteq s_{x^*}^\\alpha(M)$). Furthermore because $s_0^\\alpha$, $s_{x^*}^\\alpha$ and $s_{x_2}^\\alpha$ do not re-enter $S(R,N)$ after leaving $[-M,M]^2$ and all edges of $U'_{\\mathcal{E}}$ have both endpoints in $S(R,N)$, $e$ cannot be on these paths either. This means that \n\\[\ns_y^\\alpha(\\omega) = s_y^\\alpha(\\omega') \\text{ for } y=0,x^*,x_2 \\text{ and } U(\\omega) = U(\\omega'),~U_{\\mathcal{E}}(\\omega) = U_{\\mathcal{E}}(\\omega')\\ .\n\\]\nSo the fifth condition of $\\hat B_M^\\alpha$ occurs in $\\omega'$. The paths $s_y^\\alpha(M)$ are then equal in $\\omega$ and $\\omega'$, so conditions 1, the first part of 2, and 3 and 4 hold in $\\omega'$. As $e$ is not on any of these paths, their passage times are the same in $\\omega'$. This gives the second part of condition 2 of $\\hat B_M^\\alpha$ and shows that $\\tilde B_M^\\alpha$ is $e$-increasing. \n\n\n\n\nNow we conclude the proof in a slightly different manner depending on whether or not $\\lambda_0^+$ is finite; we focus first on the case that $\\lambda_0^+ < \\infty.$ We will use Lemma~6.6, but several times in sequence, appending events onto $\\hat B_M^\\alpha$. Precisely we note for reference that if $e_1, \\ldots, e_j$ are edges and $a_1, \\ldots, a_j \\in \\mathbb{R}$ then\n\\[\n\\hat B_\\alpha^M \\cap \\left[ \\cap_{i=1}^j \\{ \\omega_{e_i} \\geq a_i \\} \\right] \\text{ is } e\\text{-increasing for } e \\in U'_{\\mathcal{E}}\\ .\n\\]\nUsing Lemma~\\ref{lem: edge_modification} once for each edge $e \\in U'_{\\mathcal{E}}$ and the upper bound $|U'_{\\mathcal{E}}| \\leq 32 N R$, we can find some constant $C_{N,R}$ such that, defining\n\\[ \nB_M'^\\alpha := \\tilde{B}_M^\\alpha \\cap \\left\\{ \\forall e \\in U'_{\\mathcal{E}}, \\, \\omega_e \\geq \\lambda^+ - \\delta^+ \/ 4 \\right\\}\\ ,\n\\]\nwe have\n\\[ \n\\ensuremath{\\mathbb{P}} \\left( B_M'^\\alpha \\right) > C_{N,R} > 0 \\text{ for all }M \\geq M_0' \\text{ and } \\alpha \\in \\Lambda_{M,k} \\text{ when } k \\geq K_0(M)\\ .\n\\]\n(For the first application of the lemma we use $\\vartheta = 2^{-2-64NR}\\beta'\/2R$, for the second, a smaller $\\vartheta$, and so on.)\n\n\nWe claim that on $B_M'^\\alpha,$ no $z \\in \\ensuremath{\\mathbb{Z}^2} \\cap [-M,M]^2$ with $z \\preceq L'_N$ and $z \\notin S(R,N)$ has $s_z^\\alpha(M) \\cap s_{x^*}^\\alpha(M) \\neq \\varnothing$. We argue by first estimating the passage time between vertices from $L_0'$ to $L_N'$ in $U'$. For any outcome in $B_M'^\\alpha,$ given vertices $x \\in U' \\cap L'_0$ and $y \\in U' \\cap L'_N,$ there is a path from $x$ to $y$ formed by moving along $L'_0$ to $0,$ taking $r_0^\\alpha$ to $L'_N,$ and moving similarly along $L'_N$ to $y.$  This gives\n\\begin{equation}\n\\label{fastish_path}\n\\tau(x,y) < (\\lambda^+ - 7\\delta^+\/8)\\|w_N^\\alpha\\|_1+ (N\\varepsilon + \\|x_2\\|_1)\\lambda^+ .\n\\end{equation}\nUsing the choice of $\\varepsilon$ from \\eqref{eq: epsilondef} and condition \\eqref{eq: technical_condition} to bound the right side of (\\ref{fastish_path}),\n\\begin{equation}\n\\label{whoa_so_fast}\n\\tau(x,y) \\leq (\\lambda^+ - 3 \\delta^+ \/ 4) \\|w_N^\\alpha\\|_1.\n\\end{equation}\nSuppose now that a point $z$ exists as in the claim. Since $s_0^\\alpha(M)$ and $s_{x_2}^\\alpha(M)$ do not touch any $y \\notin S(R,N)$ with $y \\preceq L'_N$ (see item 1 in the definition of $\\hat B_M^\\alpha$), \n\\[\n\\mathcal{R}_1^\\alpha(M) \\cap \\{y  : y \\preceq L'_N\\} \\subseteq S(R,N)\\ .\n\\]\nThis implies $z \\notin \\mathcal{R}_1^\\alpha(M)$, whereas $x^* \\in \\mathcal{R}_1^\\alpha(M)$. As $s_z^\\alpha(M)$ cannot touch $s_0^\\alpha(M)$ or $s_{x_2}^\\alpha(M)$ (else it would merge with one of them) it would have to enter $\\mathcal{R}_1^\\alpha(M)$ through $L_0'$ and pass through all of $U'$ from $L'_0$ to $L'_N$, thus taking only edges of $U'_{\\mathcal{E}}.$  The portion $\\gamma'$ of $\\gamma$ from its first intersection with $L_0'$ to its first intersection with $L_N'$ would then satisfy\n\\begin{align*}\n\\tau(\\gamma') &\\geq \\left(\\lambda^+ - \\delta^+ \/ 4 \\right) \\left[ \\|w^\\alpha_N\\|_1 - \\|x_2\\|_1 - N \\varepsilon\\right]\\\\\n\t&\\geq (\\lambda^+ - \\delta^+ \/ 4) \\|w^\\alpha_N\\|_1 - 2 \\|w^\\alpha_N\\|_1 \\varepsilon \\lambda^+\\\\\n\t&\\geq (\\lambda^+  - 3 \\delta^+ \/ 8) \\|w^\\alpha_N\\|_1,\n\\end{align*}\nin contradiction with the estimate of (\\ref{whoa_so_fast}). This establishes the claim.\n\nFor the final step in the case that $\\lambda_0^+<\\infty$, note that by the previous claim, the pushforward, $\\Phi_\\alpha (B_M'^\\alpha)$, is a sub-event of $B_M'= B_M'(x^*;R,N),$ defined exactly as the event $B'=B'(x^*;R,N)$ in \\eqref{eq: B_prime_def} except with $\\Gamma_{x^*}$ and $\\Gamma_z$ replaced by the truncated paths $\\Gamma_{x^*}^M$ and $\\Gamma_z^M$ and considering only $z \\in [-M,M]^2$. Thus \n\\[\n\\mu_\\alpha(B_M') \\geq C_{N,R} \\text{ for all } M \\geq M_0',~ k \\geq K_0(M) \\text{ and } \\alpha \\in \\Lambda_{M,k}\\ ,\n\\]\nwith $\\Lambda_{M,k} \\subseteq [0,n_k]$ of Lebesgue measure at least $C_{\\beta'}n_k$. As the indicator of $B_M'$ is continuous,\n\\[\n\\mu(B_M') = \\lim_{k \\to \\infty} \\mu_{n_k}^*(B_M') \\geq C_{N,R} C_{\\beta'}\\ .\n\\]\nLast, \n\\[\n\\mu(B') = \\mu(B_M' \\text{ for infinitely many }M) \\geq C_{N,R} C_{\\beta'} > 0\\ ,\n\\]\ncompleting the proof in the case $\\lambda_0^+ < \\infty$.\n\nIf $\\lambda_0^+ = \\infty,$ we are no longer guaranteed the estimate (\\ref{whoa_so_fast}), since the passage time of a path taking $N \\varepsilon$ steps along $L'_N$ is not necessarily bounded above by $N \\varepsilon \\lambda^+.$ However, writing $\\tilde E$ for the set of edges with an endpoint within $\\ell^1$ distance 1 of $U'$ but not in $U'_{\\mathcal{E}}$ and noting\n\\[\nA_C := \\{\\text{for all } e \\in \\tilde E,~ \\tau_e \\leq C\\}\n\\]\nsatisfies $\\mathbb{P}(A_C) \\to 1$ as $C \\to \\infty$ independently of $k$ and $M$, we can choose $C_{\\text{big}}$ such that \n\\[\n\\mathbb{P}(\\tilde B_M^\\alpha \\cap A_{C_{\\text{big}}}) > 0\n\\]\nindependently of $k$ and $M$. This event is still monotone increasing in the appropriate edge variables. In particular, we can modify the edges in $U'_{\\mathcal{E}}$ to be each larger than $2C_{\\text{big}} |\\tilde E|$ and the rest of the proof follows as in the case $\\lambda_0^+< \\infty$.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Deriving a contradiction}\\label{sec: contradiction}\n\n\nGiven that the event $B'(x^*;R,N)$ of the preceding section has positive probability, we now derive a contradiction, proving that all paths in $\\mathbb{G}$ must merge. The next lemma is an example of a mass-transport principle. (See \\cite{BLS, Haggstrom1, Haggstrom2} for a more comprehensive treatment.)\n\n\\begin{lem}\n\\label{mass_transport}\nLet $m: \\ensuremath{\\mathbb{Z}^2} \\times \\ensuremath{\\mathbb{Z}^2} \\rightarrow [0,\\infty)$ be such that $m(x,y) = m(x+z,y+z)$ for all $x,y,z \\in \\mathbb{Z}^2.$\nThen\n\\[\n\\forall x \\in \\ensuremath{\\mathbb{Z}^2}, \\quad \\sum_{y \\in \\ensuremath{\\mathbb{Z}^2}} m(x,y) = \\sum_{y \\in \\ensuremath{\\mathbb{Z}^2}} m(y,x)\\ .\n\\]\n\\end{lem}\n\\begin{proof}\nWrite\n\\begin{align*}\n\\sum_{y \\in \\ensuremath{\\mathbb{Z}^2}} m(x,y) = \\sum_{z \\in \\ensuremath{\\mathbb{Z}^2}} m(x,x+z) = \\sum_{z \\in \\ensuremath{\\mathbb{Z}^2}} m(x-z,x) = \\sum_{y \\in \\ensuremath{\\mathbb{Z}^2}} m(y,x)\\ .\n\\end{align*}\n\\end{proof}\nGiven a realization of $\\mathbb{G}$ and $x \\in \\mathbb{Z}^2$, order the set \n\\begin{equation}\\label{eq: backclusterdef}\nC_x = \\{ y \\in \\ensuremath{\\mathbb{Z}^2}: y \\to x \\text{ in } \\mathbb{G}\\}\n\\end{equation}\nusing a dictionary-type ordering where $y$ precedes $y'$ if either $\\varpi' \\cdot y < \\varpi' \\cdot y'$ or if both $\\varpi' \\cdot y = \\varpi' \\cdot y'$ and $y \\cdot \\varsigma < y' \\cdot \\varsigma$ (where $\\varsigma$ was fixed in \\eqref{eq: chicken_alfredo}); clearly this defines a total ordering. If there is a least element $y$ under this ordering, we will call $y$ the progenitor of $x$ (relative to $\\mathbb{G}$).\nWe define the $\\mathbb{G}$-dependent function $m_\\mathbb{G}$ on pairs of vertices $x,y$ by\n\\[ m_{\\mathbb{G}}(x,y) = \\begin{cases}\n\t1 &\\text{if $y$ is the progenitor of $x$}\\\\\n\t0 &\\text{otherwise},\n\t\\end{cases} \\]\nand let $m(x,y):= \\mathbb{E}_\\mu(m_{\\mathbb{G}}(x,y)).$ Note that $m(x,y) = m(x+z,y+z)$ by the fact that $\\mathbb{G}$ has a translation-invariant distribution.\n\nSince each $x$ can have at most one progenitor, \n\\begin{equation}\n\\label{small_mass}\n\\sum_{y\\in \\mathbb{Z}^2} m(x,y) \\leq 1 \\text{ for all } x\\in \\mathbb{Z}^2\\ .\n\\end{equation}\nOn the other hand, if $B'(x^*; R,N)$ occurs, then $\\Gamma_z$ cannot intersect $\\Gamma_{x^*}$ if $z \\preceq L'_N$ and $z \\notin S(R,N).$ Therefore, on this event, there is some vertex $y \\in S(R,N)$ which is the progenitor of infinitely many vertices of $\\Gamma_{x^*}.$ In particular,\n\\begin{equation}\n\\label{big_mass}\n\\sum_{y \\in \\mathbb{Z}^2} m(y,x) = \\infty.\n\\end{equation}\nThe contradiction implied by (\\ref{small_mass}), (\\ref{big_mass}) and Lemma \\ref{mass_transport} gives $\\mu(B'(x^*;R,N))=0$. However this contradicts the previous section and completes the proof of Theorem~\\ref{thm: Gcoalescethm}.\n\n\n\n\n\n\n\n\n\n\n\\subsection{Absence of backward infinite paths}\n\n\nIn this section, we move on from Theorem~\\ref{thm: Gcoalescethm} to show that because all paths in $\\mathbb{G}$ coalesce, all paths in the ``reverse\" direction terminate. That is, recalling the definition of $C_x$ in \\eqref{eq: backclusterdef},\n\n\\begin{thm}\n\\label{no_back_path}\nFor each $x \\in \\ensuremath{\\mathbb{Z}^2},$ $|C_x| < \\infty$ with $\\mu$-probability one.\n\\end{thm}\n\n\\begin{rem}\nThe proof below applies to the following general setting. Suppose $\\nu$ is a translation-invariant probability measure on directed subgraphs of $\\mathbb{Z}^2$ and there is a line $L \\subseteq \\mathbb{R}^2$ such that $\\nu$-almost surely (a) each $x$ has exactly one forward path and it is infinite (b) all forward paths coalesce and (c) each forward infinite path emanating from a vertex on $L$ intersects it finitely often. Then all backward clusters are finite $\\nu$-almost surely.\n\\end{rem}\n\nWe assume that, contrary to the theorem, there exists $x \\in \\ensuremath{\\mathbb{Z}^2}$ with\n$\\mu(|C_x|=\\infty)>0$ for the remainder of this section to derive a contradiction. Using Lemma~\\ref{lem: varpilemma}, choose a deterministic $\\varpi'$ with argument in $\\{j \\pi\/4 : j = 0, \\ldots, 7\\}$ such that with positive $\\mu$-probability on $\\{|C_x| = \\infty\\}$, each $\\Gamma_z$ eventually lies on the far side of each $L'_N$. Note that this event is translation-invariant, so by conditioning on it, we may assume that it occurs with probability 1 (and $\\mu$ is still translation-invariant).\n\n\\begin{clam}\nThere exist vertices $z \\neq z'$ in $L'_0$ such that\n\\begin{equation}\n\\label{triple_pt}\n \\mu\\left(|C_z| = \\infty,\\, |C_{z'}| = \\infty, \\, \\Gamma_z \\cap L'_0 = \\{z\\}, \\, \\Gamma_{z'} \\cap L'_0 = \\{z'\\}\\right) > 0\\ .\n\\end{equation}\n\\end{clam}\n\\begin{proof}\nBy translation-invariance, we may assume that the $x$ with $\\mu(|C_x|=\\infty)>0$ satisfies $x \\prec L'_0.$ $\\mu$-almost surely, $\\Gamma_x$ has a last intersection with $L'_0.$ There are countably many choices for such a last intersection, so there exists a vertex $z \\in L'_0$ such that\n\\[\n\\mu\\left( |C_z| = \\infty, \\, \\Gamma_z \\cap L'_0 = \\{z\\}\\right) > 0\\ .\n\\]\nTranslating by $\\varsigma$ (chosen from \\eqref{eq: chicken_alfredo}), the ergodic theorem gives $z, z'$ satisfying (\\ref{triple_pt}).\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{no_back_path}.]\nGiven an outcome in the event in (\\ref{triple_pt}), $\\Gamma_{z}$ and $\\Gamma_{z'}$ almost surely merge. So there is some random $z_{\\mathbb{G}} \\in \\ensuremath{\\mathbb{Z}^2}$ which is the first intersection point of $\\Gamma_{z}$ and $\\Gamma_{z'}$ (``first\" in the sense of both the ordering in $\\Gamma_z$ and in the ordering of $\\Gamma_{z'}$). Again $z_{\\mathbb{G}}$ can take only countably many values, and so there is a $z_0$ which occurs with positive probability; call the intersection of the event in (\\ref{triple_pt}) with the event $\\{z_{\\mathbb{G}} = z_0\\}$ by the name $B.$\n\nWe now consider the graph $\\mathbb{G}$ as an undirected graph, in which vertices $x$ and $y$ are adjacent if $\\langle x,y \\rangle$ or $\\langle y,x \\rangle$ are in $\\mathbb{G}$ (we abuse notation by using the same symbol for both the directed and undirected versions of $\\mathbb{G}$). We define an encounter point of the undirected $\\mathbb{G}$ to be a vertex whose removal splits $\\mathbb{G}$ into at least three infinite components. Note that $B \\subseteq \\{z_0$ is an encounter point$\\}$; by translation invariance, we see that there is a uniform $c_t > 0$ such that the probability of any fixed vertex to be an encounter point is at least $c_t.$\n\nWe are now in the setting of Burton-Keane \\cite{burtonkeane}. To briefly synopsize, the number of points on the boundary of $[-M,M]^2$ must be at least the number of encounter points within. In particular, the number of encounter points is surely bounded above by $8M$. But since each point within has probability at least $c_t$ to be an encounter point, the expected number of encounter points within $[-M,M]^2$ is at least $c_t M^2.$ This is a contradiction for large $M.$\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proofs of main theorems}\\label{sec: proofs}\n\n\\subsection{Proof of Theorem~\\ref{thm: sectors}}\\label{sec: sectors}\nSuppose that $\\partial \\mathcal{B}$ is differentiable at $v_\\theta = \\varpi$ and construct the measure $\\mu$ as in Section~\\ref{sec: mudef}. Using the notation of Theorem~\\ref{thm: nachostheorem}, we set \n\\[\nL_\\varrho = \\{x \\in \\mathbb{R}^2 : x \\cdot \\varrho = 1\\}\\ .\n\\]\nFrom the theorem, we deduce that with $\\mu$-probability 1, $\\Gamma_0$ is asymptotically directed in $J_\\varrho$. But by the assumption of differentiability, $J_\\varrho = I_\\theta$ with $\\mu$-probability 1 and thus\n\\begin{equation}\\label{eq: ptonboyz1}\n\\mu\\left( \\Gamma_0 \\text{ is asymptotically directed in } I_\\theta \\right) = 1\\ .\n\\end{equation}\nBy Proposition~\\ref{prop: firstGG2}, each finite piece of $\\Gamma_0$ is a geodesic, so $\\Gamma_0$ is an infinite geodesic. Define $\\hat \\Omega \\subseteq \\Omega_1$ as the set\n\\[\n\\hat \\Omega = \\{\\omega \\in \\Omega_1 : \\mu(\\Gamma_0 \\text{ is asymptotically directed in }I_\\theta \\mid \\omega) = 1\\}\\ .\n\\]\nThe inner probability measure is the regular conditional probability measure. The set $\\hat \\Omega$ is measurable and because the marginal of $\\mu$ on $\\Omega_1$ is $\\mathbb{P}$, it satisfies $\\mathbb{P}(\\hat \\Omega)=1$. Further, for each $\\omega \\in \\hat \\Omega$ there is an infinite geodesic from 0 which is asymptotically directed in $I_\\theta$.\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm: newman}}\\label{sec: newman}\n\nIn this section we assume either {\\bf A1'} or {\\bf A2'}. Assume that the limit shape $\\mathcal{B}$ has uniformly positive curvature. Then the boundary $\\partial \\mathcal{B}$ cannot contain any straight line segments. This implies that the extreme points $ext(\\mathcal{B})$ are dense in $\\partial \\mathcal{B}$. Choose some countable set $D \\subseteq ext(\\mathcal{B})$ that is dense in $\\partial \\mathcal{B}$. For any $\\theta_1$ and $\\theta_2$ with $0<dist(\\theta_1,\\theta_2) < \\pi$, let $I(\\theta_1,\\theta_2)$ be the set of angles corresponding to the shorter closed arc of $\\partial \\mathcal{B}$ from $v_{\\theta_1}$ to $v_{\\theta_2}$. By Corollary~\\ref{cor: extreme}, for each $\\theta_1,\\theta_2 \\in D$ with $0<dist(\\theta_1,\\theta_2)<\\pi$, with probability one there is an infinite geodesic from 0 asymptotically directed in $I(\\theta_1,\\theta_2)$. The collection of such sets of angles is countable, so there exists an event $\\Omega'\\subseteq \\Omega_1$ such that $\\mathbb{P}(\\Omega') = 1$ and for each $\\omega \\in \\Omega'$,\n\\begin{enumerate}\n\\item for each $\\theta_1, \\theta_2 \\in D$ such that $dist(\\theta_1,\\theta_2) < \\pi$, there exists an infinite geodesic containing 0 and asymptotically directed in $I(\\theta_1,\\theta_2)$ and\n\\item for each $x,y \\in \\mathbb{Z}^2$ there is exactly one geodesic from $x$ to $y$.\n\\end{enumerate}\nWe claim that for each $\\omega \\in \\Omega'$, both statements of the theorem hold: for each $\\theta$ there is an infinite geodesic with asymptotic direction $\\theta$ and each infinite geodesic has a direction. \n\nTo prove the first statement, let $\\omega \\in \\Omega'$ and $\\theta \\in [0,2\\pi)$. For distinct angles $\\theta_1$ and $\\theta_2$ such that $0 < dist(\\theta_i,\\theta)< \\pi$ we write $\\theta_1 >_\\theta \\theta_2$ if $I(\\theta_1,\\theta)$ contains $\\theta_2$. Because $D$ is dense in $\\partial \\mathcal{B}$, we can find two sequences $(\\theta_n^1)$ and $(\\theta_n^2)$ such that (a) $0 < dist(\\theta_n^i,\\theta)<\\pi$ for all $n$ and $i$, (b) for $i=1,2$, $dist(\\theta_n^i,\\theta) \\to 0$ as $n \\to \\infty$ and (c) for each $i=1,2$ and $n$, $\\theta_n^j >_\\theta \\theta_{n+1}^j$. Let $v_n$ be the point $nv_\\theta$ and let $\\gamma_n$ be the geodesic from $0$ to $v_n$. Define $\\gamma$ as any subsequential limit of $(\\gamma_n)$. By this we mean a path $\\gamma$ such that for each finite subset $E$ of $\\mathbb{R}^2$, the intersection $\\gamma_n \\cap E$ equals $\\gamma \\cap E$ for all large $n$. We claim that $\\gamma$ has asymptotic direction $\\theta$.\n\nLet $\\varepsilon>0$ and choose $N$ such that $dist(\\theta,\\theta_N^j)<\\varepsilon$ for $j=1,2$. Because $\\omega \\in \\Omega'$, for $j=1,2$, we can choose an infinite geodesic $\\gamma_N^j$ containing 0 with asymptotic direction in $I(\\theta_N^j,\\theta_{N+1}^j)$. Write $P$ for the union of $\\gamma_N^1$ and $\\gamma_N^2$. This complement of $P$ in $\\mathbb{R}^2$ consists of two open connected components (as $P$ cannot contain a circuit). Because both paths are directed away from $\\theta$, exactly one of these two components contains all but finitely many of the $nv_\\theta$'s. Let $C_1$ be the union of $P$ with this component and let $C_2$ be the other component.\n\nChoose $N_0$ so that $nv_\\theta \\in C_1$ for all $n \\geq N_0$. We claim now that each finite geodesic $\\gamma_n$ for $n \\geq N_0$ is contained entirely in $C_1$. If this were not true, $\\gamma_n$ would contain a vertex $z$ in $C_2$ and therefore it would cross $P$ to get from $z$ to $v_n$. Then if $w$ is any vertex on $\\gamma_n \\cap P$ visited by $\\gamma_n$ after $z$, then there would be two different geodesics from $0$ to $w$ and this would contradict unique passage times. Therefore, as $\\gamma_n$ is contained in $C_1$ for all large $n$, so must $\\gamma$. This implies that $\\gamma$ is asymptotically directed in the set of angles within distance $\\varepsilon$ of $\\theta$ (for each $\\varepsilon>0$) and therefore has asymptotic direction $\\theta$.\n\nTo prove the second statement choose $\\omega \\in \\Omega'$ and let $\\gamma$ be an infinite geodesic. If $\\gamma$ does not have an asymptotic direction then, writing $x_n$ for the $n$-th vertex of $\\gamma$, we can find an angle $\\phi \\in [0,2\\pi)$ such that $\\phi$ is a limit point of $\\{\\arg x_n : n \\geq 1\\}$ (under the metric $dist$) but $(\\arg x_n)$ does not converge to $\\phi$. So there exists a number $\\varepsilon$ with $0<\\varepsilon<\\pi$ and a subsequence $(x_{n_k})$ of $(x_n)$ such that for each $m$, $dist(\\arg x_{n_{2m}}, \\phi) < \\varepsilon\/2$ but $dist(\\arg x_{n_{2m+1}}, \\phi) > \\varepsilon$. By the first part of the theorem we can find infinite geodesics $\\gamma_1$ and $\\gamma_2$ from $0$ such that $\\gamma_1$ has asymptotic direction $\\phi + 3\\varepsilon\/4$ and $\\gamma_2$ has asymptotic direction $\\phi - 3\\varepsilon\/4$. Now it is clear that if we write $P$ for the union of $\\gamma_1$ and $\\gamma_2$ then $\\gamma$ must both contain infinitely many vertices of $P$ and infinitely many vertices of $P^c$. This again contradicts unique passage times.\n\n\n\n\\begin{proof}[Proof of Corollary~\\ref{cor: newman2}]\nIf $\\theta$ is an exposed point of differentiability then by Corollary~\\ref{cor: exposed}, with probability one there exists an infinite geodesic from 0 in each rational direction. Then the proof above goes through with minor modifications.\n\\end{proof}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm: random_hyperplanes}}\n\nAssume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property. Let $v \\in \\mathbb{R}^2$ be nonzero and $\\varepsilon>0$. We will prove that the statement of the theorem holds with probability at least $1-\\varepsilon$. Choose $\\varpi \\in \\partial \\mathcal{B}$ to be parallel to $v$ and construct a measure $\\mu$ as in Section~\\ref{sec: mudef}. Let $(n_k)$ be an increasing sequence such that $\\mu_{n_k}^* \\to \\mu$ weakly.\n\nWe will define a double sequence of cylinder events that approximate the events in the theorem. For $m \\leq n$, a configuration $\\eta \\in \\Omega_3$ and $x,y \\in [-m,m]^2 \\cap \\mathbb{Z}^2$, we say that $x$ is $n$-connected to $y$ ($x \\to_n y$) if there exists a directed path from $x$ to $y$ whose vertices stay in $[-n,n]^2$. We say that $x$ and $y$ are $n$-connected $(x \\leftrightarrow_n y)$ if there is an undirected path connecting $x$ and $y$ in $[-n,n]^2$. For $m \\leq n$ write $A_{m,n} \\subseteq \\Omega_3$ for the event that\n\\begin{enumerate}\n\\item all vertices $v \\in [-m,m]^2$ have exactly one forward neighbor in $\\mathbb{G} \\cap [-n,n]^2$,\n\\item there is no undirected circuit contained in $[-m,m]^2$,\n\\item for all vertices $v,w \\in [-m,m]^2$, there exists $z \\in [-n,n]^2$ such that $v \\to_n z$ and $w \\to_n z$ and\n\\item for all vertices $v \\in [-m,m]^2$ there is no $z \\in [-n,n]^2 \\setminus (-n,n)^2$ such that $z \\to_n v$.\n\\end{enumerate}\n\nWe claim that for any $m$ there exists $n(m) \\geq m$ such that $\\mu(A_{m,n(m)}) > 1-\\varepsilon\/4^{m+2}$. To prove this, let $\\hat \\Omega \\subseteq \\widetilde \\Omega$ be the event that (a) all vertices have one forward neighbor in $\\mathbb{G}$, (b) $\\mathbb{G}$ has no undirected circuits, (c) for all $x,y \\in \\mathbb{Z}^2$, $\\Gamma_x$ and $\\Gamma_y$ coalesce and (d) $|C_x| < \\infty$ for all $x \\in \\mathbb{Z}^2$. By Proposition~\\ref{prop: secondGG2}, Theorem~\\ref{thm: Gcoalescethm} and Theorem~\\ref{no_back_path}, the $\\mu$-probability of $\\hat \\Omega$ is 1. Therefore conditions 1 and 2 above have probability 1 for all $m$ and $n$. For any configuration in $\\hat \\Omega$ and $m \\geq 1$ we can then choose a random and finite $N(m) \\geq m$ to be minimal so that conditions 3 and 4 hold for all $n \\geq N(m)$. Taking $n(m)$ so large that $\\mu(N(m) \\geq n(m)) \\leq \\varepsilon\/4^{m+1}$ completes the proof of the claim.\n\nWe now pull $A_{m,n(m)}$ back to $\\Omega_1$, using the fact that it is a cylinder event in $\\Omega_3$ and thus its indicator function is continuous. There is an $m$-dependent number $K_0(m)$ such that if $k \\geq K_0(m)$ then $\\mu_{n_k}^*(A_{m,n(m)}) > 1-\\varepsilon\/4^{m+2}$. By definition of $\\mu_{n_k}^*$ in \\eqref{eq: munstar} and $\\Phi_\\alpha$ in \\eqref{eq: phidef}, the set $\\Lambda_{m,k}$ of values of $\\alpha \\in [0,n_k]$ such that $\\mathbb{P}(\\Phi_\\alpha^{-1}(A_{m,n(m)})) > 1-\\varepsilon\/2^{m+2}$ has Lebesgue measure at least $n_k(1-2^{-(m+2)})$. \n\nThe next step is to construct a deterministic sequence $(a_m)_{m \\geq 1}$ of real numbers such that\n\\begin{equation}\\label{eq: clambake}\na_m \\to \\infty \\text{ and } \\mathbb{P}\\left( \\cap_{j=1}^m \\Phi_{a_m}^{-1} (A_{j,n(j)}) \\right) \\geq 1-\\varepsilon\/2 \\text{ for all } m\\ .\n\\end{equation}\nWe do this by induction on $m$. For $m=1$, let $a_1$ be any number in the set $\\Lambda_{1,K_0(1)}$. By definition then $\\mathbb{P}(\\Phi_{a_1}^{-1}(A_{1,n(1)})) \\geq 1-\\varepsilon\/2$. Assuming that we have fixed $a_1, \\ldots, a_m$, we now define $a_{m+1}$. Let $k$ be such that $k \\geq \\max \\{K_0(1), \\ldots, K_0(m+1)\\}$ and $n_k \\geq 3a_m$ and consider $\\Lambda_{1,k}, \\ldots, \\Lambda_{m+1, k}$ as above. The intersection of these sets has Lebesgue measure at least $3n_k\/4$ so choose $a_{m+1}$ as any element of the nonempty set $(3a_m\/2,n_k] \\cap \\left[ \\cap_{i=1}^{m+1} \\Lambda_{i,k}\\right]$. For this choice,\n\\[\n1- \\mathbb{P}\\left(\\cap_{j=1}^{m+1} \\Phi_{a_{m+1}}^{-1}(A_{j,n(j)}) \\right) \\leq \\sum_{j=1}^\\infty \\varepsilon \/2^{j+2} = \\varepsilon\/4\\ .\n\\]\nAs $a_{m+1} \\geq 3a_m\/2$, the condition $a_m \\to \\infty$ holds and we are done proving \\eqref{eq: clambake}.\n\nFrom \\eqref{eq: clambake}, we deduce $\\mathbb{P}(A) \\geq 1-\\varepsilon\/2$, where\n\\[\nA = \\{\\cap_{j=1}^m \\Phi_{a_m}^{-1}(A_{j,n(j)}) \\text{ occurs for infinitely many } m\\}\\ .\n\\]\nWe complete the proof by showing that the statement of the theorem holds for any $\\omega\\in A$. Fix such an $\\omega$ and a random subsequence $(a_{m_k})$ of $(a_m)$ such that $\\omega \\in \\cap_{j=1}^{m_k}\\Phi_{a_{m_k}}^{-1}(A_{j,n(j)})$ for all $k$. By extracting a further subsequence, we may assume that $\\mathbb{G}_{L_{a_{m_k}}(\\varpi)}$ converges to some graph $G$. The event $\\Phi_{\\alpha}^{-1}(A_{j,n(j)})$ is exactly that the graph $\\mathbb{G}_{L_\\alpha(\\varpi)}$ satisfies the conditions of $A_{j,n(j)}$ above, so in particular, it has no undirected circuits in $[-j,j]^2$, all directed paths starting in $[-j,j]^2$ coalesce before leaving $[-n(j),n(j)]^2$, no directed paths connect $[-n(j),n(j)]^2 \\setminus (-n(j),n(j))^2$ to $[-j,j]^2$, and all vertices in $[-j,j]^2$ have one forward neighbor in $[-n(j),n(j)]^2$. On the subsequence $(a_{m_k})$, the events $\\Phi_{a_{m_k}}^{-1}(A_{1,n(1)})$ occur for all $k$, so $G$ must satisfy the conditions of $A_{1,n(1)}$ as well. The same is true for $A_{j,n(j)}$ for all $j$, so $G$ satisfies the conditions of the theorem.\n\n\n\n\n\\subsection{Proof of Theorem~\\ref{thm: exceptional_set}}\n\nThis theorem follows directly from results of the previous sections. Assume either {\\bf A1'} or both {\\bf A2'} and the upward finite energy property. For the first part of the theorem, suppose that $\\partial \\mathcal{B}$ is differentiable at $v_\\theta$. Choose $\\varpi = v_\\theta$ and construct the measure $\\mu$ as in Section~\\ref{sec: mudef}. Given $(\\omega,\\Theta,\\eta) \\in \\widetilde \\Omega$, let $\\mathbb{G}(\\eta)$ be the geodesic graph associated to $\\eta$. By Theorems~\\ref{thm: nachostheorem}, \\ref{thm: Gcoalescethm} and \\ref{no_back_path}, with $\\mu$-probability one, all directed paths in $\\mathbb{G}$ are asymptotically directed in $I_\\theta$, they coalesce, and no vertex $x$ has $|C_x|$ infinite. Call this event $A$ and define\n\\[\n\\hat \\Omega = \\{\\omega \\in \\Omega_1 : \\mu(A \\mid \\omega) = 1\\}\\ .\n\\]\n$\\mu(\\cdot \\mid \\omega)$ is the regular conditional probability measure. $\\hat \\Omega$ is a measurable set and satisfies $\\mathbb{P}(\\hat \\Omega)=1$ since the marginal of $\\mu$ on $\\Omega_1$ is $\\mathbb{P}$. Further, for each $\\omega \\in \\hat \\Omega$, the theorem holds.\n\nFor the other two parts of the theorem we simply argue as in the proof of Corollaries~\\ref{cor: exposed} and \\ref{cor: extreme}. In the former case we just notice that if $v_\\theta$ is also exposed, then $I_\\theta = \\{\\theta\\}$. In the latter case, we find a point $v_\\theta$ on the arc joining $v_{\\theta_1}$ to $v_{\\theta_2}$ at which $\\partial \\mathcal{B}$ is differentiable. The set $I_\\theta$ contains only angles associated to points on the arc and we are done.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\t\n\nHigh-resolution satellite images from most of the Earth's surface have been available for over a decade, and in recent years advanced deep learning methods have been applied to them. Mnih et. al. started this field with a classifier that detects roads \\cite{mnih}. More recently, deep learning models and satellite imagery have been used for more complicated tasks such as crop yield prediction in the U.S. \\cite{cropyield} and poverty prediction in Africa \\cite{jean}. \\par\n\nWe believe there are still many unexplored questions and potential applications for this field of study.\nOne such problem is using satellite images to enhance existing house price prediction models, which historically only used hand-selected features (e.g. number of rooms or floor area). We show that a vision-based model can learn important relevant features from the neighborhood of a house (e.g. distance from green fields or highways) by processing a single satellite image, and then use those features to improve the estimation accuracy.\n\nAlthough high-resolution satellite images contain an abundance of information that might be correlated with house prices, such data are highly unstructured and thus challenging to extract meaningful insights from. Although deep learning models such as convolutional neural networks (CNN) could in principle be trained to directly estimate prices from satellite imagery, the scarcity of training data makes the application of these techniques challenging. Therefore, we use transfer learning techniques to overcome this problem in that we will use knowledge gained while solving one problem and apply it to our different but related problem. We will start with a CNN model that has been trained on ImageNet \\cite{imagenet}, a large image classification data set, to identify low-level image features such as edges and corners that are common to many vision tasks. Next, we will build on the knowledge gained from this image classification task and fine-tune the model on a new task.\n\nWe combine a dataset of house prices (from LA County's property assessment dataset \\footnote{\\href{https:\/\/data.lacounty.gov\/Parcel-\/Assessor-Parcels-Data-2017\/vak5-2hqh}{https:\/\/data.lacounty.gov\/Parcel-\/Assessor-Parcels-Data-2017\/vak5-2hqh}}) and a dataset of satellite images (collected via Google Static Maps API \\footnote{\\href{https:\/\/developers.google.com\/maps\/documentation\/static-maps\/intro}{https:\/\/developers.google.com\/maps\/documentation\/static-maps\/intro}}) to accomplish this objective.\nThe input to our model is a $640\\times640$ image (with the house located at the center) and a feature vector (number of bedrooms, etc.). The output is a single number, the value of the house in dollars. We feed the image to the pretrained CNN whose last two layers have been deleted after pretraining, and get a feature vector for the image. We then combine the two feature vectors using several fully-connected layers and get the output.\n\nFinally, we evaluate our estimation performance using $R^2$ and mean-squared error (MSE) metrics. The results will be compared to two baseline models that only use hand-selected features.\n\nThe inspiration for this project came from Zillow's Home Value Prediction competition \\footnote{\\href{https:\/\/www.kaggle.com\/c\/zillow-prize-1}{https:\/\/www.kaggle.com\/c\/zillow-prize-1}}, which is challenging the data science community to help push the accuracy of the house price estimations further by utilizing any additional source of information \\footnote{We even tried reducing Zillow's model's estimation error (which is what the competition is about) using satellite images, but it proved to be a challenging task. Zillow's model already performs exceptionally well, and improving a blackbox model without having access to its details is difficult. We are not sure if Zillow's model uses any vision-based features or not.}. \n\n\\section{Related work}\nTraditionally, house price prediction models used hand-selected features (e.g. number of rooms or floor area). Feed-forward neural networks and various decision-tree-based models \\cite{dectree} outperformed other approaches. However, as \\cite{Chopra} mentions, the parameters that determine the price of a house are twofold: one that is intrinsic to the house and one that indicates the \"desirability\" of the house. Traditional models can only learn the former, but the latter usually depends on the location of the house \\cite{Kockelman}. In particular, social and economic metrics such as crime rate, pollution levels and distance from important locations (e.g. train station, hospitals) can affect the housing price \\cite{Bency}. Therefore, a model that is able to capture the resulting spatial correlations can potentially outperform other models. Spatial auto-regressive (SAR) models are one approach to capturing the existing spatial correlation in housing prices. These models rely on a spatial contiguity matrix that is usually hand-designed with the help of a domain expert, but as \\cite{Chopra} showed, can also be learned by algorithms. Non of these models use vision-related features. More recently, \\cite{Bency} proposed a model that uses deep neural networks to extract information from satellite images, combined them with an SAR model and other house features. This approach resulted in a $57\\%$ reduction in RMSE compared to a model with SAR but without satellite images.\n\nIt is worth noting that these papers use different datasets which in most cases are gathered by the authors of the paper. This makes it difficult to compare their results in a completely fair manner \\footnote{We also use a dataset that we collected.} However, it can be seen that the focus of the research has been moving towards models that can capture features from outside the building itself, which has generally improved the accuracy of estimators.\n\n\\section{Dataset and Features}\n\nOur dataset consists of two main parts: 1. real estate properties, and 2. high-resolution satellite images. Details of each part are explained in this section.\n\n\\subsection{Real Estate Properties}\nWe collected properties data from LA County's property assessment data which is available online\\footnote{\\href{https:\/\/data.lacounty.gov\/Parcel-\/Assessor-Parcels-Data-2017\/vak5-2hqh}{https:\/\/data.lacounty.gov\/Parcel-\/Assessor-Parcels-Data-2017\/vak5-2hqh}}. This dataset consists of more than 2 million real-estate properties in LA county assessed between 2006 and 2017. Each entry had 50 features, floor area, number of bedrooms, year built, use type, latitude and longitude to name a few. Preprocessing was done on the data, details of which will be discussed later in this paper.\n\n\\subsection{High-Resolution Satellite Images}\nTo better predict the price error, we combined our main dataset with satellite images of houses, collected via Google Static Maps API \\footnote{\\href{https:\/\/developers.google.com\/maps\/documentation\/static-maps\/intro}{https:\/\/developers.google.com\/maps\/documentation\/static-maps\/intro}} using the longitude and latitude of each house. This API gives access to different zoom levels between 1 (the Earth) and 24 (the most detailed). In this study, we set the zoom level to 19, and retrieved images with size 640x640 which is the highest resolution available for the free tier users\\footnote{We planned to try different zoom levels, but the limit of 25,000 images per day for the free tier made image gathering very time-consuming}. Figure \\ref{fig:sample} shows a few examples. We believe this zoom level provides a suitable amount of information about the neighborhood while showing the shape of the building itself.\n\n\\begin{figure}\n\\includegraphics[width=0.5\\textwidth]{sample}\n\\centering\n\\caption{Examples of satellite images. Each image is $640\\times640$ pixels.}\n\\label{fig:sample}\n\\end{figure}\n\n\\subsection{Data Preprocessing}\n\nBefore beginning analysis, the LA County dataset was preprocessed. Eight columns contained unrelated information such as assessor's ID and parcel ID, which were removed. One redundant column was also removed. With each property, came three price values: total land value, personal property value, and total value. In addition to residential properties, the data included rows for non-residential properties as well as empty lots. Since our aim is to study housing price estimations, we focused our analysis on entries for which personal property value was non-zero, and used this column as our labels. Consequently, non-residential rows were removed. Finally, we ended up with 40 features per property, among which 23 were categorical parameters (e.g. use type). We replaced categories with integer values. Each column in the dataset was then normalized to have zero mean and standard deviation of $1$. Table \\ref{table:dataset} shows a summary of the size of our dataset.\n\n\\begin{table}{}\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\nTrain Set Size & Validation Set Size&Test Set Size&Number of Features \\\\\n\\hline\n48,548&2,698&2,698&40\\\\\n\\hline\n\\end{tabular}\n\\vspace{6px}\n\\caption{Train\/validation\/test size of the dataset}\n\\label{table:dataset}\n\\end{table}\n\n\\section{Methods}\n\n\\subsection{Baseline Models}\n\nWe started our analysis by building two baseline models: decision-tree-based models and neural networks. For the former, we analyzed multiple tree-based estimators. Extra tree regressor \\cite{extratrees} demonstrated the best performance and was chosen as our baseline. Moreover, we trained a neural network on our features dataset. Performance of these models are provided in the Experiments and Results section.\n\n\n\\subsection{Inception-v3}\n\nSince we do not have enough data to train vision-based convolutional neural networks, we use pretrained models to encode images. We use Inception-v3 model \\cite{inception}, trained on ImageNet, to construct 2048-dimensional vectors for our images. The process is as follows. First, JPEG satellite images are converted into RGB matrices, resized to $299\\times299\\times3$ and normalized to have values in $[0, 1]$ to match the expected input of Inception-v3. Feeding the data into the model, we then save resulting vectors in a binary file to be used later in our network. This time-consuming process generated around 8 gigabytes of processed data. We do this in order to make the training process faster and be able to iterate more quickly in hyperparameter tuning part of the project.\n\n\\subsection{Cost Function}\nWe use mean-squared error (MSE) as our cost function:\n$\nJ= \\frac{1}{m} \\sum_{i=1}^{m} L(\\hat{y}^{(i)}, y^{(i)})=\\frac{1}{m} \\sum_{i=1}^{m}(\\hat{y}^{(i)}-y^{(i)})^2\n$. \nNote that we report our results using $R^2$ (not MSE), but the two metrics are nicely related in a way that minimizing MSE is equivalent to maximizing $R^2$:\n$\nR^2=1-\\frac{MSE}{S}\n$\nwhere $S$ is the variance of the test set's labels.\n\n\n\\section{Experiments and Results}\n\n\\subsection{Experiments}\nAll numerical results are provided in table \\ref{table:results}.\nWe have studied to see if images alone are able to make estimations about prices. Encodings are fed into fully-connected neural network layers and trained to minimize mean squared error (MSE) loss. \n\nFrom the results we got, it is apparent that by themselves, images are not able to predict prices. This can be attributed to the difficulty of estimations of house size from images. It is not possible for a model to decide on the size of a house by just looking at a satellite picture, as is may include several other buildings too.\n\nNext step was to train models on both features and images to see if combining the two can improve our estimations. We feed our two feature vectors to two separate dense networks and we concatenate the results in the final layer. We will describe the details of this model in the Network Architecture section.\n\n\\begin{table}{}\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\nModel& Train $R^2$ & Dev $R^2$& Test $R^2$& Train MSE& Dev MSE& Test MSE\\\\\n\\hline\nExtra Tree (baseline) &\n0.99\n&0.86&\n0.71&\n0.000&\n0.112&\n0.002\\\\\n\\hline\nNeural Net F (baseline) &\n0.96\n& 0.84\n&0.85\n&  0.037\n&  0.136\n&  0.001\\\\\n\\hline\nNeural Net I &\n$0.000$&-0.0001&-0.038&\n 1.062\n & 0.859\n & 0.010\\\\\n\\hline\nNeural Net F+I &\n  0.98\n&  0.94\n&  0.93\n&  0.011\n&  0.044\n&  0.001\n\\\\\n\\hline\n\\end{tabular}\n \\vspace{0.6em}\n\\caption{Summary of results. MSE (mean-square error) is calculated using normalized labels. F is features, I is satellite images.}\n\\label{table:results}\n\\end{table}\n\\subsection{Interpretation of the Results}\nWe see that the F+I model outperforms all other models by at least $10\\%$ in $R^2$ metric. Intuitively, the cost of a house is approximately floor area $\\times$ price per square foot. While features provide the model with the first factor, images help it improve its estimation for the second. We ranked data points based on how much F+I improves estimations compared to F model. Top 5 positive (F+I corrected underestimation) and top 4 negative improvements (F+I improved overestimation) are shown in figures \\ref{fig:top} and \\ref{fig:bottom} respectively.\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{top}\n\\centering\n\\caption{Baselines underestimated the price of these houses and F+I corrected them.}\n\\label{fig:top}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{bottom}\n\\centering\n\\caption{Baselines overestimated the price of these houses and F+I corrected them.}\n\\label{fig:bottom}\n\\end{figure}\n\nAn interesting observation is that our dataset -despite all the preprocessing that we have done- contained a few empty lands with no buildings\\footnote{Perhaps due to changes that occurred between the time LA County has assessed properties and the time Google has captured their images}. The fact that these data points are present in figures \\ref{fig:top} and \\ref{fig:bottom} shows that our model is more robust to noise in the data.\nAfter evaluating figure \\ref{fig:top}, we think it shows that our model realizes that cars are an indicator of higher prices (since it shows higher accessibility or low distance from city center), so it adds to the estimated price.\n\n\\subsection{Network Architecture}\nWe have tried many different architectures and hyperparameters, and in this section only describe the best results. Also, due to lack of space we only describe our model for F+I model.\nWe use the architecture described in figure \\ref{fig:architecture}. The hyper parameters are mentioned in table \\ref{table:hyperparameters}. Our model, especially on image side is very prone to over-fitting, so we use L2 regularization and dropout to prevent that. The learning rate (lr) is decreased after each batch is processed, using the following formula:\n$lr = \\frac{lr}{1 + \\alpha \\times t}$. We use a relatively big batch size because we were training on a GPU\\footnote{NVIDIA Tesla K80}. Larger batch sizes (but smaller than available memory size) help increase the utilization of GPU and generally speed up the training process.\n\n\\begin{table}{}\n\\centering\n\\begin{tabular}{|c|c|}\n\\hline\nL2 Regularization Parameter & $0.1$\\\\\n\\hline\nFirst Dropout (drop probability)&$0.3$ \\\\\n\\hline\nFirst Dropout (drop probability)&$0.2$ \\\\\n\\hline\nActivation Function  & ReLU\\\\\n\\hline\nBatch Size& $1024$ \\\\\n\\hline\nOptimizer& Adam(learning rate=$0.0005$, $\\beta_1=0.9$, $\\beta_2=0.999$)\\\\\n\\hline\nEpochs& $200$\\\\\n\\hline\nLearning Rate Decay &$\\alpha=0.0001$ \\\\\n\\hline\n\\end{tabular}\n \\vspace{0.6em}\n\\caption{Hyperparameters for F+I model}\n\\label{table:hyperparameters}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[width=0.6\\textwidth]{architecture}\n\\centering\n\\caption{F+I neural network architecture}\n\\label{fig:architecture}\n\\end{figure}\n\n\\section{Conclusion\/Future Work}\nTo sum up, by adding satellite images to our prediction model, we were able to improve $R^2$ by ~10\\% and reduce MSE by ~60\\% compared to the baseline. The results are promising. We believe the following future steps are worth further analysis.\n\n\n Using different image zoom levels or a combination of them: Lower zoom levels will be able to capture more details about the neighborhood as they cover a wider range around the property, while a higher zoom level may estimate based on more details from the house, for example the materials used to build it.\n Fine tuning last layers of  the Inception-v3 network: Inception-v3 encodes images of size 299x299x3 to 2048-dimensional vectors. Thus, valuable information may be omitted by this process. As a future step, we can take outputs of earlier layers of the pretrained model or fine tune parameters of the last few convolutional layers. This will allow us to capture more relevant features from the images.\n Adding data from different locations to the dataset to test how generalizable the model is from one location to another: Since public data of property transactions are widely available today, analysis can continue on larger datasets consisting of more counties. This will add to inputs' variance, and potentially make the model more robust. We can then examine if our approach can be generalized.\n\n\n\\medskip\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Hubbard model is a cornerstone of condensed matter physics.  As a paradigmatic model of strongly-correlated electrons  \\cite{ANDERSON1196}, it is simple to formulate yet rich in behavior. In two dimensions (relevant e.g. to cuprate superconductors) observed behaviors include, but are not limited to, antiferromagnetism, unconventional metallic behavior characterized by a pseudogap and deviations from Fermi liquid theory~\\cite{tremblay2006review,Alloul2014,Gunnarsson2015,Wu2017,Schafer}, as well as stripe orders~\\cite{PhysRevB.93.035126,Zheng1155} closely competing with superconducting states at low temperatures~\\cite{PhysRevX.10.031016}. \nBut despite decades of effort, a comprehensive understanding of the phase diagram of the two-dimensional Hubbard model has not yet been fully reached.\nTherefore any solutions of the Hubbard model, whether obtained analytically or by accurate and controlled numerical techniques, are of great value.\n\nThe most reliable and comprehensive solutions of the Hubbard model obtained so far have been mainly in (quasi-) one dimension \\cite{PhysRevLett.20.1445} and in infinite dimensions (infinite lattice coordination number)~\\cite{PhysRevB.45.6479,PhysRevLett.62.324,RevModPhys.68.13}.\nIn one dimension, solutions can be obtained by integrability~\\cite{1D_Hubbard_book} and bosonisation methods~~\\cite{Giamarchi_book}, \nas well as numerically with matrix product state (MPS) tensor network methods \\cite{PhysRevLett.69.2863,SCHOLLWOCK201196}. \nThe latter can also reliably treat quasi-one-dimensional ladder or cylindrical geometries with a small transverse size. \nIn the limit of infinite dimensions the Hubbard model is again numerically tractable due to the fact that dynamical mean-field theory (DMFT) becomes exact \\cite{RevModPhys.68.13}. Using this technique, the phase diagram of the infinite-dimensional Hubbard model can be\nmapped out and a detailed understanding of the interaction-driven Mott insulator to metal transition \nhas been established (for reviews, see Refs.~\\cite{RevModPhys.68.13,RevModPhys.78.865,\nDMFT@25,georges_lectures,rozenberg_lectures}).\n\nWhile they provide useful insights into the physics of the two-dimensional Hubbard model, these limiting cases also have peculiarities which limit the generality of the \nconclusions that can be drawn from their study. In the limit of infinite dimensions, the metallic state is found to be a Fermi liquid, with interactions affecting one-particle properties in a local \n(momentum-independent) manner only. Hence, in this limit, the feedback of long-wavelength collective modes or even short-range spatial correlations \non quasiparticle properties is entirely absent. \nThese effects are important, in particular close to a critical point. In two dimensions, they are, for example, responsible \nfor the formation of a pseudogap \\cite{Preuss1997,Macridin2006,Kyung2006,Gunnarsson2015,Scheurer2018}.\nIn contrast, in one dimension the low-energy excitations consist {\\it only} of bosonic collective modes, associated with charge \nand spin degrees of freedom.\nThere is no notion of a Fermi liquid and the metallic behavior of the Hubbard model is a Luttinger liquid which lacks coherent quasiparticles \nand displays spin-charge separation~\\cite{Giamarchi_book}.  \n\nIn this work we perform a controlled and accurate numerical study of the ground state of the Hubbard model on the Bethe lattice with a finite coordination number $z$, focusing on the case $z=3$. This is an infinite lattice that has a tree structure, where every site is connected to the same number of other sites ($z$) but there are no loops. We show a portion of this lattice in Fig. \\ref{fig:BL intro}.\n\\begin{figure}\n\t\\includegraphics[scale=0.3]{BL_large_4_gen}\n\t\\caption{A portion of the infinite Bethe lattice with coordination number $z = 3$. The figure depicts four ``generations\" of the tree structure, with a particular site chosen to be the ``center\" site. Note that in the actual Bethe lattice there is no special site, and generations can be counted from any of the sites.}\n\t\\label{fig:BL intro}\n\\end{figure}\nThis lattice provides an intermediate case between one dimension ($z=2$) and infinite dimensions ($z = \\infty$), with the key virtue that it admits controlled solutions via tensor network methods, including away from half filling and in the presence of strong interactions. \n\nExact solutions of models on the Bethe lattice have a long history in statistical mechanics~\\cite{Baxter_book}, \nstarting with the pioneering article of Hans Bethe~\\cite{Bethe_1935}. \nSolutions on the finite coordination number Bethe lattice provide a better approximation to thermodynamic quantities than the mean-field approximation \n(corresponding to the infinite dimensional limit)~\\cite{Baxter_book,Bethe_1935,PhysRevLett.74.809}. \nModels studied on the Bethe lattice include classical and quantum spin models~\\cite{PhysRevB.80.144415,PhysRevB.77.214431,NAGY2012542,PhysRevB.86.195137,PhysRevB.88.035138,PhysRevB.89.054426,LIU20141}, spin glass systems \\cite{Parisi_BL_spin_glass,PhysRevLett.56.1082,doi:10.1002\/pssb.200541282,PhysRevB.78.134424}, the Bose Hubbard model \\cite{PhysRevB.80.014524}, and models of Anderson localization  \\cite{Abou_Chacra_1973,Mirlin_1991,PhysRevLett.78.2803,PhysRevB.100.094201,dupont2020dirty}. The fermionic Hubbard model on the finite version of the $z=3$ Bethe lattice (known as a Cayley tree) has also been studied previously using a variant of the density matrix renormalization group (DMRG) algorithm \\cite{Lepetit2000}, but only the case of half filling was studied (which is a charge insulator) and only local ground state quantities given (energy, staggered magnetization and its fluctuations, and neighboring spin correlations). Since that time, there have been significant advances in DMRG and related algorithms for infinite one-dimensional systems, which we generalize here to the Bethe lattice and use to obtain our results. Notably, there has been no previous study of metallic states on the finite connectivity Bethe lattice, to the best of our knowledge. \n\nWe determine the full phase diagram of the fermionic Hubbard model on the $z=3$ Bethe lattice, allowing for a two-site unit cell, and establish \nthe nature of the doping-driven Mott insulator to metal transition (MIT). We find that this transition is first-order, and that for every value of interaction strength there is a region of forbidden density. Therefore, in the interaction-density plane the model exhibits phase separation at low doping levels.  \nWe find that, for all allowed values of the density, the doped metallic ground-state does not display magnetic long-range order. \n\nImportantly, we also demonstrate that the doped metal hosts coherent quasiparticles at all studied values of the interaction strength, $U$, from weak to strong coupling, and determine the behavior of the quasiparticle weight as a function of $U$. This answers in the affirmative the question of whether Fermi liquid behaviour \napplies as soon as the peculiar kinematic constraints of one dimension are alleviated, and also provides a concrete description of a Fermi liquid ground state with tensor networks, both of which are key motivations for our work. Generally, it is difficult for tensor networks to accurately describe interacting metallic states above one dimension, although much progress has been made in this direction \\cite{PhysRevB.81.165104,PhysRevB.93.045116,PhysRevB.100.195141,mortier2020resolving}. Our work provides an alternate route to this agenda, avoiding the computational challenges of two dimensional tensor networks, while going beyond the restrictions of one dimensional physics. \n\nWe obtain our results by generalizing a recently developed MPS method, variational uniform MPS (VUMPS) \\cite{PhysRevB.97.045145}, to tree tensor network (TTN) states \\cite{PhysRevA.74.022320,PhysRevB.82.205105,doi:10.1063\/1.4798639}, which we dub the variational uniform tree state algorithm (VUTS). We further introduce the fermionic version of the VUTS algorithm, using the swap-gate method of Refs. \\cite{PhysRevB.81.165104,Orus_review}. The VUTS algorithm works directly in the thermodynamic limit, which is important in the study of models on the Bethe lattice. The alternative is to study the finite Cayley tree and perform a finite-size scaling analysis. However, the number of boundary sites on the Cayley tree is always more than half of the total, and therefore finite-size effects are unusually strong and can even lead to conclusions that do not hold on the Bethe lattice \\cite{Baxter_book,PhysRevB.87.085107,OSTILLI20123417}. Working directly in the infinite-size limit is therefore important for models on trees. All previous works studying quantum models on the (infinite) Bethe lattice using tensor networks have used a variant of the infinite time-evolving block decimation (iTEBD) algorithm \\cite{PhysRevLett.98.070201}. In the one-dimensional case, the VUMPS algorithm has been found to be much more efficient than other methods that work in the thermodynamic limit, such as iTEBD or earlier infinite DMRG algorithms \\cite{PhysRevB.97.045145}, and indeed we find its extension in the form of the VUTS algorithm we develop to be very efficient. Our method scales as ${\\mathcal O}(\\chi^{z+1})$ where $\\chi$ is the bond dimension of the tensor network being optimized. For $z=3$ this scaling is significantly better than that of the most modern and accurate projected entangled pair states (PEPS) algorithms that scale as ${\\mathcal O}(\\chi^{10})$ for PEPS bond dimension $\\chi$ (assuming the boundary MPS bond dimension scales as $\\chi^2$) \\cite{PhysRevB.92.035142,PhysRevB.94.035133,PhysRevB.94.155123,PhysRevB.98.235148,PhysRevX.9.031041,PhysRevB.100.195141}, but is more challenging than the ${\\mathcal O}(\\chi^3)$ scaling of DMRG. However, the steeper scaling is mitigated by the fact that the typical bond dimension required to reach an accurate solution generally decreases as one goes to higher dimensions and larger coordination numbers due to the monogamy of entanglement and more mean-field-like properties of the wavefunction.\nThe accuracy of tensor network methods is often measured by the \\emph{truncation error}, which measures the typical loss of fidelity incurred during the truncation step of the optimization algorithm. In this work, for a bond dimension of $\\chi = 100$, we are able to achieve a truncation error of less than $10^{-3}$ in the most computationally challenging part of the phase diagram (most entangled ground state), and a truncation error of less than $10^{-7}$ in the best cases. This level of accuracy allows us to measure long-distance correlation functions well enough to extract information about quasiparticle coherence in the metallic phase, which demonstrates that our method can be used to reliably study critical phases of matter on the Bethe lattice.\n\nThis work suggests promising\nfuture directions for studying the behavior of strongly correlated electrons in a controlled setup. Because Fermi liquid behavior is a rather generic feature of metallic states, the present study allows to establish a controlled platform which can be used to study how Fermi liquid behaviour can be broken by further perturbations to the model considered here, or in other fermionic models. In the concluding section of this article, we discuss possible routes towards achieving this goal. If successful, tensor network solutions of correlated electrons on the $z=3$ Bethe lattice could provide a new platform for studying non-Fermi liquids~\\cite{RevModPhys.73.797,doi:10.1146\/annurev-conmatphys-031016-025531} in a controlled and accurate manner. Other potential applications are the study of the interaction-driven MIT in frustrated fermionic systems, and the study of fermions on closely related tree-like lattices, such as the Husimi cactus on which the Heisenberg model and other spin models have been shown to display spin liquid phases~\\cite{Chandra_1994}. We elaborate on all these directions and others at the end of the paper.\n\nThis paper is organized as follows. In section \\ref{sec:VUTS} we describe the general VUTS method, applicable to generic Hamiltonians. In section \\ref{sec: Hubbard model} we define the Hubbard model on the Bethe lattice and show the phase diagram obtained from the VUTS solution. Section~\\ref{sec:quasiparticles} discusses the calculation of the quasiparticle weight from the occupation function, as well as Luttinger's theorem. Finally, in Sec. \\ref{sec:discussion} we summarize and discuss future directions.  \n\n\n\n\n\n\n\\section{Variational uniform tree state algorithm}\n\\label{sec:VUTS}\n\nIn this section we introduce the variational uniform tree state algorithm (VUTS), a generalization of the variational uniform matrix product state algorithm (VUMPS) \\cite{PhysRevB.97.045145}, for optimizing infinite tree tensor network (TTN) states. We start with a Bethe lattice of quantum degrees of freedom. For simplicity we focus on the algorithm for coordination number $z = 3$, which is the value for the model studied in this paper, but the extension to general $z$ is straightforward. We use an infinite TTN as our ansatz to approximate quantum states on the Bethe lattice. For $z = 3$, the infinite TTN with a 1-site unit cell consists of an order 4 tensor $A  \\in \\mathbb{C}^{\\chi \\times \\chi \\times \\chi \\times d}$, with one physical leg ($s$) which runs over the physical degrees of freedom $1,...,d$, and three virtual legs ($l_0,l_1,l_2)$ that run over virtual degrees of freedom $1,...,\\chi$. The virtual legs of neighboring tensors connect to each other, forming the same geometry as the Bethe lattice, as shown in the tensor network diagram in Fig. \\ref{fig:TTN Bethe lattice unlabeled}.\n\\begin{figure}\n\t\\includegraphics[scale=0.25]{tree_TN_Bethe_lattice_unlabeled}\n\t\\caption{A finite portion of the infinite tree tensor network (TTN) state describing the many-body wave function on the Bethe lattice. The physical legs (green dashed lines) form the nodes of the lattice, while the virtual legs (straight black lines) form the edges. In this case, a single tensor $A$ comprises the state, and the unit cell is just a single site.}\n\t\\label{fig:TTN Bethe lattice unlabeled}\n\\end{figure}\n\\\\\n\\indent \nThe Hamiltonians we will focus on here are isotropic and have an equivalence between all sites (the analog of translational invariance for hypercubic lattices). The ground state will potentially break this isotropy completely, and break the site equivalence down to a non-trivial unit cell. In this paper, we allow for the state to be fully anisotropic between the different directions emanating from a given site, but for simplicity we focus exclusively on the case when the unit cell consists of no more than two sites (generalizing to arbitrary unit cells is straightforward). The infinite TTN state we study therefore has a 2-site unit cell, and is parameterized by a set of $2z$ tensors $A_{i,m}$, where $i = 0,1$ labels the location in the unit cell, and $m = 1, \\dots, z$ labels the direction of the gauge (defined below). Again, each tensor has one physical index $s_i = 1, \\dots, d$ and three virtual ``link\" indices $l_0,l_1,l_2$.\n\\\\\n\\indent\nBecause the TTN has no loops, it is straightforward to work in the \\emph{canonical gauge}, i.e. the gauge where the tensors are constrained to be orthonormal bases when viewed as a matrix from two link indices $l_n,l_k$ and the physical index $s_i$ to the remaining link index $l_m$. This constraint on the tensors is very useful for making the variational optimization faster and more stable, and is standard in a wide variety of tensor network algorithms, particularly in 1D algorithms like VUMPS and DMRG. The constraint on the tensors is written as\n\\begin{equation}\n\\displaystyle\\sum_{s_i,l_n,l_k} \\bar{A}^{s_i,l'_m,l_n,l_k}_{i,m} A^{s_i,l_m,l_n,l_k}_{i,m} = \\mathbb{1}^{l'_m,l_m}_{i,m},\n\\label{eq:canonical gauge}\n\\end{equation}\nwhere we have introduced the notation $\\bar 0 = 1, \\bar 1 = 0$ for the unit cell indices and use $\\bar{A}$ to denote the complex conjugation of $A$. The matrices $\\mathbb{1}_{i,m}$ are identities.  Diagrammatically, Eq. (\\ref{eq:canonical gauge}) is equivalent to Fig.~\\ref{fig:canonical gauge}. The arrows on the links denote the gauge of the $A$ tensors (the outgoing link is the direction of the gauge). Any TTN can be brought into the form where the tensors obey Eq. (\\ref{eq:canonical gauge}) (or equivalently Fig. \\ref{fig:canonical gauge}) by inserting a particular set of ``gauge transformations\"\nonto the link degrees of freedom, i.e. inserting a particular set of resolutions of the identity $X X^{-1}$ (where $X$ is an invertible matrix) onto the links of the TTN. Note that the gauge transformation does not affect the observables of the system, and any TTN can be transformed into the canonical gauge efficiently.\n\\begin{figure}\n    \\includegraphics[scale=0.25]{gauge_condition_1}\n    \\caption{Diagrammatic version of the gauge conditions of Eq. (\\ref{eq:canonical gauge}). The bonds labelled $k,n,m$ have link indices $l_k,l_n,l_m$ respectively (and the uncontracted $m$ bond on the ket has link index $l_m'$). The unlabelled dashed bond is the physical degree of freedom with index $s_i$.}\n\t\\label{fig:canonical gauge}\n\\end{figure}\n\\begin{figure}\n\t\\includegraphics[scale=0.45]{tree_TN_Bethe_lattice_with_centers_2}\n\t\\caption{The same portion of the Bethe lattice as in Fig. \\ref{fig:TTN Bethe lattice unlabeled}, but with the bonds labeled with $m = 0,1,2$, which have link indices $l_0,l_1,l_2$ respectively. Additionally, each tensor is labeled with subscripts $i,m$, where $i = 0,1$ is the unit cell index and $m$ is the direction of the gauge. In the top diagram, the gauge center $C_2$ is shown on a bond labelled by $2$. The next equality shows that the $C_2$ tensor can be absorbed into the $A_{1,2}$ tensor to put the gauge center on the site tensor, creating $A_{1,C}$.}\n\t\\label{fig:TTN Bethe lattice with gauge centers}\n\\end{figure}\n\\\\\n\\indent\nExamples of the TTN state with a 2-site unit cell in the canonical gauge are shown in Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. In the top diagram, the gauge center is $C_2$. Here, $C_2$ represents the projection of the infinite wavefunction of the system onto the finite-sized Hilbert space of that link of the network. The center matrices $C_m$ constitute invertible gauge transformations relating the $A$ tensors to each other via $A_{i,m} C_m = A_{i,n} C_{n}$, where $i = 0,1$ and $n \\neq m$. The center matrices $C_m$ also contain important information like the entanglement spectrum between the two infinite halves of the system split by that link. Additionally, the gauge center can be absorbed onto an $A$ tensor, defining the center site tensors $A_{i,C} = A_{i,m} C_m$ for any $m$. This is shown in the lower diagram of Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. In this case, $A_{i,C}$ would represent the infinite wavefunction of the system projected onto a single site (and again, different spectra of that tensor relate to entanglement spectra of different bipartitions of the lattice).\n\\\\\n\\indent\nWe describe the algorithm for the case of $H = \\sum_{\\<i,j\\>} h_{i,j}$, where $h_{i,j}$ is a two-site operator that acts on nearest neighbors only. The case of longer-range operators is  treatable using techniques like those described in Appendix~C of Ref. \\cite{PhysRevB.97.045145}. The total energy is given by $E = \\sum_{\\<i,j\\>} \\<\\psi|h_{i,j}|\\psi\\>$, and we want to minimize $E$, treating the tensor elements of our TTN as variational parameters. As in VUMPS (and many related tensor network ground state methods), VUTS proceeds in three main steps that are iterated until convergence:\n\\begin{enumerate}\n  \\item Compute the projected Hamiltonians (the Hamiltonian projected into the basis corresponding to the virtual degrees of freedom of the network) to turn the global optimization into a local optimization problem.\n  \\label{alg:projected Hamiltonian}\n  \\item Find the optimized tensors by minimizing the energy of the projected Hamiltonian.\n  \\label{alg:optimize tensors}\n  \\item Update the tensor network with the new optimized tensors.\n  \\label{alg:update network}\n\\end{enumerate}\n\\indent\nTo begin, say we are interested in optimizing a 1-site projected wavefunction $A_{i,C}$, as defined in Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. Step \\ref{alg:projected Hamiltonian}  requires computing infinite sums of local Hamiltonian terms, projected into the basis of our gauged TTN (defined by the tensors $A_{i,m}$), for each of the $z=3$ infinite subtrees connected to $A_{i,C}$. In order to perform the infinite sum, we focus on summing the energy contributions of a single subtree. An example for the series that needs to be summed for the $m=2$ direction in order to optimize the $A_{0,C}$ tensor is shown in Fig. \\ref{fig:H_{1,2} series}. We define the results of these summations as the matrices $H_{i,m}$. The summation can be carried out by making use of the fact that the sum is a geometric series. However, care has to be taken to project out infinite energy contributions to keep the series convergent (i.e. keep the norm of the solution $H_{i,m}$ from diverging). The procedure of performing the summation and projecting out the infinite energy contributions is a generalization of the one in Appendix D of Ref. \\cite{PhysRevB.97.045145}, and we discuss it in more detail in Appendix \\ref{subsec:summing Hamiltonian terms}.\n\\begin{figure}\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n          \\includegraphics[width=1\\linewidth]{H_1_2_series_example}\n          \\caption{}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{0.38\\textwidth}\n          \\includegraphics[width=1\\linewidth]{h_tensor}\n          \\caption{}\n          \\label{fig:h_i_m}\n        \\end{subfigure}\n\t\\caption{(a) The first few terms of the series for $H_{1,2}$, the projected Hamiltonian contribution for one of the three branches of the infinite Bethe lattice connected to the center site tensor $A_{0,C}$ (on the $i=0$ sublattice). (b) Definition of the $h_{i,m}$ tensors used in (a), which are a sum of two local Hamiltonian environment tensors sitting on two branches of the Bethe lattice. Note that the tensor labeled $h$ represents the two-site operator term $h_{i,j}$, which the Hamiltonian is made up of.}\n\t\\label{fig:H_{1,2} series}\n\\end{figure}\n\\\\\n\\indent \nOnce the environment tensors are found, we can proceed to step \\ref{alg:optimize tensors} of the algorithm, which we begin by optimizing $A_{i,C}$. This is done by finding the ground state of the Hamiltonian projected onto the sublattice site $i$, a standard procedure in VUMPS and DMRG. The eigenvalue equation for $A_{i,C}$ is shown diagrammatically in Fig. \\ref{fig:H_Ac}, and is solved iteratively (using a Hermitian eigensolver such as Lanczos). To find the TTN ground state, we obtain the eigenvector with the smallest eigenvalue. As in the VUMPS algorithm, in addition to optimizing $A_{i,C}$, we also optimize $C_m$. $A_{i,C}$ and $C_m$ are then used to solve for $A_{i,m}$, which make up the updated infinite TTN state (see next paragraph). The eigenvalue equation for $C_m$ is shown diagrammatically in Fig. \\ref{fig:H_C}.\n\\begin{figure}\n\t\\includegraphics[scale=0.35]{Ac_eig_eqn.pdf}\n\t\\caption{The eigenvalue equation for $A_{i,C}$. The sum over $m$ is a sum over the contributions from each leg of the $A_{i,C}$ tensor.}\n\t\\label{fig:H_Ac}\n\\end{figure}\n\\begin{figure}\n\t\\includegraphics[scale=0.45]{C_eig_eqn.pdf}\n\t\\caption{The eigenvalue equation for $C_{m}$.}\n\t\\label{fig:H_C}\n\\end{figure}\n\\\\\n\\indent\nFinally, once $A_{i,C}, C_m$ for all $i = 0,1, m = 0,1,2$ are optimized, we can proceed to step \\ref{alg:update network} of the algorithm and solve for our new $A_{i,m}$ tensors by minimizing\n\\begin{equation}\n\\epsilon_{i,m}\n=\n\\min_{A_{i,m}^{\\dagger} A_{i,m} \\, = \\, \\mathbb{1}_{i,m}} || A_{i,C} - A_{i,m} C_m ||. \n\\label{eq:new A tensors}\n\\end{equation}\nThis minimization problem can be solved optimally using techniques described in Eqs. (18)-(22) of Ref. \\cite{PhysRevB.97.045145}. The new $A_{i,m}$ we obtain constitute our updated TTN, and steps \\ref{alg:projected Hamiltonian}-\\ref{alg:update network} are repeated until convergence. Convergence is achieved when the largest error found in Eq. (\\ref{eq:new A tensors}), $\\epsilon_{\\text{prec}} \\equiv \\max\\{\\epsilon_{i,m}\\}$, falls below a chosen threshold (e.g. $\\epsilon_{\\text{prec}} < 10^{-12}$).\n\\\\\n\\indent\nThe VUTS algorithm with a 1-site update, as we describe here, scales as $O(\\chi^{z+1})$  which becomes $O(\\chi^4)$ for the $z=3$ Bethe lattice and $O(\\chi^3)$ for $z=2$, thus reducing to the scaling of VUMPS in the $z=2$ case. Additionally, a 2-site update can be formulated, analogous to the 2-site DMRG algorithm which is commonly used. This requires a slight modification of the algorithm where ground states of 2-site and 1-site projected Hamiltonians are computed (as opposed to 1-site and 0-site projected Hamiltonians in the version of the algorithm described above). This can lead to improved convergence since a larger local Hilbert space is explored, but has a higher computational cost of ${\\mathcal O}(\\chi^5)$ for $z=3$. We use this technique at lower bond dimensions in more challenging parts of the phase diagram (near the phase transition), and switch to the 1-site algorithm later in the calculation to reach higher bond dimensions. To dynamically change the bond dimensions, we use a generalization of the subspace expansion procedure described in Appendix B of Ref. \\cite{PhysRevB.97.045145}.\n\\\\\n\\indent \nFor fermionic models like the Hubbard model, we need to use a fermionic version of VUTS. We use the method outlined in Refs. \\cite{PhysRevB.81.165104,Orus_review}. Every tensor is now endowed with a fermion parity $Z_2$ quantum number and is parity-preserving. When two tensor legs cross on a planar projection of a tensor diagram, a fermionic swap gate is placed at the crossing. In order to employ this method, we need to use a fixed ordering convention for the legs of the tensors $A_{i,m}, A_{i,C}, C_m$, which must be kept consistent in all of the diagrams in the calculation. We address the associated subtleties and details of this approach in Appendix \\ref{subsec:fermions}. Other symmetries beyond the $Z_2$ parity can also be used, such as $U(1)$ particle number conservation (to fix the filling), $U(1)$ spin projection symmetry in the z-direction, and also the spin $SU(2)$ symmetry. The inclusion of these symmetries makes the tensors more sparse and should therefore make the tensor operations more efficient, allowing us to reach larger bond dimensions. In this work, we only employ parity quantum numbers, and leave the use of additional symmetries for future work.\n\n\n\n\n\n\n\n\n\\section{Model and phase diagram} \n\\label{sec: Hubbard model}\n\nThe Hubbard Hamiltonian is given by\n\\begin{align}\nH = - t \\sum_{\\langle i,j \\rangle} \\sum_{\\sigma = \\uparrow, \\downarrow}c_{i, \\sigma}^{\\dagger} c_{j, \\sigma}  + U \\sum_i n_{i, \\uparrow} n_{i, \\downarrow}  - \\mu \\sum_{i, \\sigma} n_{i, \\sigma},\n\\label{eq:interacting Hamiltonian}\n\\end{align}\nwhere the site index $i$ now runs over the Bethe lattice and $n_{i,\\sigma} = c_{i, \\sigma}^{\\dagger} c_{i, \\sigma}$ is the on-site density for an electron of spin $\\sigma$. We set $t = 1$ and vary $U \\geq 0$ and $\\mu$. Since the model is particle-hole symmetric, we only need to consider $\\delta \\mu \\equiv \\mu - \\frac{U}{2} \\geq 0$. To obtain the phase diagram, we compute the ground state of the Hubbard model using the fermionic VUTS algorithm for several values of the bond dimension, $\\chi$. The details of the numerical calculation are given in Appendix \\ref{sec:numerical details}. We perform extrapolations in $\\chi$, the results of which we describe below. The additional plots of the finite $\\chi$ results and details of the extrapolations are in Appendix \\ref{sec:phase diagram more plots}. \n\\\\\n\\indent \nAt half filling, $\\delta \\mu = 0$, the system is a charge insulator with antiferromagnetic order for all $U > 0$, as in one dimension. To illustrate this we compute the staggered magnetization of the bipartite sublattice, $m_s \\equiv \\abs{(\\< \\vec{S}_A \\> - \\< \\vec{S}_B \\>)\/2}$. The extrapolated values $m_s(\\chi\\to\\infty)$ are shown in Fig. \\ref{fig:ms extrapolation}. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{ms_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the staggered magnetization $m_s$ of the insulating state at half filling, plotted as a function of $U$. Note the logarithmic scale used for $m_s$. The error bars are the discrepancy in the extrapolation with and without the last data point (for the points where they are absent they are smaller than the data points).}\n\t\\label{fig:ms extrapolation}\n\\end{figure}\nWe can see that for any $U>0$ the magnetization is non-zero, tending to zero as $U \\rightarrow 0$. From general mean-field theory considerations we expect $m_s \\sim e^{-\\frac{c}{U}}$ for small $U$. \nHowever, we did not attempt to confirm this functional form numerically by systematic calculations at very small values of $U$.\n\\\\\n\\indent \nTo illustrate the charge gap at half filling, we compute the on-site occupation $\\< n_{i} \\> = \\sum_{\\sigma} \\< n_{i,\\sigma} \\>$ as a function of $\\delta \\mu$. We show this in Fig. \\ref{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50} for a single value of $U = 6$ and $\\chi = 50$. All other $U$ and $\\chi$ look qualitatively similar.\n\\begin{figure}\n\t\\begin{subfigure}[c]{0.45\\textwidth}\n          \\includegraphics[width=0.85\\linewidth]{occ_vs_delmu_both_branches_U_6_chi_50}\n          \\caption{$U = 6, \\chi = 50$.}\n        \\end{subfigure}\n\t\\begin{subfigure}[c]{0.45\\textwidth}\n          \\includegraphics[width=0.85\\linewidth]{energy_vs_delmu_both_branches_U_6_chi_50}\n          \\caption{$U = 6, \\chi = 50$.}\n        \\end{subfigure}\n\t\\caption{(a) The occupation versus $\\delta \\mu$ for $U = 6$ and $\\chi = 50$ for the both metallic and insulating branches. All other values of $U,\\chi$ look similar. (b) The energy per site of both branches in (a). \n\tTheir crossing point, $\\delta \\mu_c$ is indicated by a dashed line in both (a) and (b).}\n\t\\label{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50}\n\\end{figure}\nWe see that there are two branches of VUTS solutions: insulating ($\\< n_{i} \\> = 1$) and metallic ($\\< n_{i} \\> > 1$). The insulating branch exists for $\\delta\\mu\\leq\\delta\\mu_1$, and the metallic branch for $\\delta\\mu\\geq\\delta\\mu_2$: these \ntwo values, $\\delta\\mu_{1}$ and $\\delta\\mu_{2}$ (with $\\delta\\mu_{2} < \\delta\\mu_{1}$), are spinodal values limiting the meta-stability of the insulating and metallic solutions, respectively (see Appendix \\ref{sec:phase diagram more plots} for details).\nFor each value of $\\delta \\mu$ the ground state is the branch with the lower energy. The energies of the two branches cross at a particular value, which we define to be $\\delta\\mu_c (\\chi)$. At $\\delta\\mu_c (\\chi)$, the ground state changes from insulating for $\\delta\\mu < \\delta \\mu_c$ to metallic for $\\delta\\mu > \\delta \\mu_c$. The occupation undergoes a finite jump, $\\delta n(\\delta \\mu_c, \\chi) = \\< n_i(\\delta \\mu_c, \\chi) \\> - 1$, indicating that this is a first order metal-insulator transition. We estimate the size of the jump in the real system by the $\\chi \\to \\infty$ extrapolated values, which are shown in Fig. \\ref{fig:jump in occupation extrapolation}. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{jump_in_n_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the jump in density $\\delta n_c$ at the first-order transition occuring at $\\delta \\mu_c$ plotted as a function of $U$. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points.}\n\t\\label{fig:jump in occupation extrapolation}\n\\end{figure}\nWe can see they remain finite for all $U$, meaning that this is a true first-order transition, and not an artifact of finite bond dimension. Using a derivation based on the Maxwell construction (detailed in Appendix~\\ref{sec:phase diagram more plots}), \nit can be shown that the total charge gap\nis given by $\\Delta_c (\\chi) = 2 \\, \\delta\\mu_c (\\chi)$. In order to obtain the charge gap $\\Delta_c$ for the real system, we extrapolate $\\Delta_c (\\chi)$ in $\\chi$. The result is shown in Fig. \\ref{fig:charge gap extrapolation}.\n\\begin{figure}\n\t\\includegraphics[scale=0.8]{charge_gap_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the charge gap $\\Delta_c$ at half filling plotted as a function of $U$. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points.}\n\t\\label{fig:charge gap extrapolation}\n\\end{figure}\nWe can see that the extrapolated gap shows an exponential-like behavior at small $U$ similar to the one-dimensional case, though the precise behavior is difficult to extract numerically. At large $U$ the gap crosses over to a more linear dependence on $U$. \n\\\\\n\\indent \nThe first-order transition we observe implies that for every $U$ there is a range of forbidden density. Hence, in the $(U,n)$ plane the model exhibits phase separation. Our numerical method works in the grand canonical ensemble (we do not fix particle number per unit cell), so we cannot observe this phase separation directly. However, VUTS, similar to other variational tensor network methods like DMRG and VUMPS, can get ``stuck\" in local minima. We use this fact to find both branches of solutions near the transition point, even when they are meta-stable (i.e. not the lowest energy states), by way of hysteresis in the numerical algorithm (see Appendix \\ref{sec:phase diagram more plots} for a detailed explanation). The resulting branches, shown in Fig. \\ref{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50}, can be continued to find the spinodal values of the first-order transition (see Appendix \\ref{sec:phase diagram more plots}).\n\\\\\n\\indent \nThe metallic ground-state has no magnetic order for any value of density: we find that the staggered magnetization vanishes once $\\delta \\mu$ crosses $\\delta \\mu_c$. To illustrate the typical magnetization behavior we observe, in Fig. \\ref{fig:ms of metallic phase} we show the staggered magnetization $m_s$ as a function of $\\delta \\mu$ across $\\delta \\mu_c$ for $U = 6$ at a large but fixed $\\chi = 90$.\n\\begin{figure}\n\t\\includegraphics[scale=0.8]{ms_vs_delmu_U_6}\n\t\\caption{The staggered magnetization $m_s$ versus the chemical potential $\\delta \\mu$ for $U = 6$ and $\\chi = 90$. The critical point $\\delta \\mu_c$ is indicated by a dashed line. We see the drop from $m_s > 0$ to $m_s = 0$, indicating the first-order transition from the antiferromagnetic insulator to the paramagnetic metal.}\n\t\\label{fig:ms of metallic phase}\n\\end{figure}\nOnce $\\delta \\mu$ becomes large enough to drive the system metallic, the staggered magnetization $m_s$ immediately drops to a very small value, which is zero within our error tolerance. All other values of $U$ and $\\chi$ behave similarly.  As $\\chi$ increases, the small value of magnetization in the metal decreases further, although the behavior is not monotonic, as shown in Appendix \\ref{sec:phase diagram more plots}. Also, in Appendix \\ref{sec:numerical details} we describe the strategy we use to make sure we do not bias the magnetization of the metallic solution with our ansatzes.\n\\\\\n\\indent \nIt is interesting to compare our results on the $z=3$ Bethe lattice to those established for the doping-driven MIT \nin the $z=\\infty$ limit where DMFT becomes exact. \nOnly a few studies~\\cite{camjayi_rozenberg_2006,wang_millis_2009,fratino_tremblay_prb_2017} consider this transition while also taking into account phases with magnetic long-range order. As in our results, an antiferromagnetic insulator is found at half-filling for a range of chemical potentials, as well as a non-magnetic metallic solution which can be stabilized for values of the chemical potential above a spinodal value $\\delta \\mu_{2}$. \nFurthermore, a magnetic metallic solution is found to exist in a narrow range of chemical potentials, which connects the magnetic insulator and the non-magnetic metal. \nThis may appear to differ from our findings, but it should be emphasized that all these studies consider only non-zero temperatures. As temperature is lowered, it is reported in Refs~\\cite{camjayi_rozenberg_2006,wang_millis_2009} that the magnetic metallic solution appears to exist only in an increasingly narrow interval of chemical potentials, and Ref.\\cite{camjayi_rozenberg_2006} suggested that at low temperature the MIT is a first-order transition between the magnetic insulator and the non-magnetic metal, with a forbidden range of density corresponding to phase separation. Although, to the best of our knowledge, this has not yet been fully established directly at $T=0$ for $z=\\infty$, this conclusion is consistent with our findings on the finite coordination number lattice. In contrast, on fully frustrated $z=\\infty$ lattices, which do not allow for long-range magnetic order (e.g. on the fully connected lattice with random hopping), it is established that the \ndoping-driven MIT is second order at $T=0$ and becomes first-order only at finite temperature~\\cite{RevModPhys.68.13,moeller_1995,kotliar_2002,werner_2007}. \n\\\\\n\\indent \nTo conclude this section, we mention briefly the numerical accuracy of the data presented here. As noted in the Introduction, the numerical accuracy in tensor networks is generally measured by the truncation error, denoted by $\\epsilon_{\\rho}$. We show here the scaling of $\\epsilon_{\\rho}$ in the metallic state, which is the state with the largest entanglement and therefore the most computationally challenging for tensor networks. \nIn Fig. \\ref{fig:truncation error metal n = 1.2} we plot our estimate of $\\epsilon_\\rho$ at a fixed density of $\\<n_i\\> = 1.2$ as a function of $\\chi$ for various $U$, and also as a function of $U$ at the largest values $\\chi = 90,100$.\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{truncation_error_all_U_vs_chi_metal_n_1_2}\n          \\caption{}\n        \\end{subfigure}\\hspace{0.01\\textwidth}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{truncation_error_at_chi_90_and_100_vs_U_metal_at_n_1_2}\n          \\caption{}\n        \\end{subfigure}\n\t\\caption{Our estimate for the truncation error $\\epsilon_\\rho$ in the metallic phase for $\\<n_i\\> = 1.2$ (a) as a function of $\\chi$ and (b) as a function of $U$ for the two largest $\\chi$. Note that since (a) is a log-log plot the linear form indicates an algebraic relationship.}\n\t\\label{fig:truncation error metal n = 1.2}\n\\end{figure}\nWe can see that the decay with $\\chi$ is algebraic, as expected for a gapless system. As a function of $U$, $\\epsilon_\\rho$ increases initially and then potentially saturates, although the large $U$ behavior is undetermined. Notably, we can see that $\\epsilon_\\rho < 10^{-3}$ for all $U$ and $\\chi = 100$, which is a high level of accuracy. We discuss the behavior of $\\epsilon_\\rho$ at other points in the phase diagram in Appendix \\ref{sec:numerical details}. \n\n\n\n\n\n\n\n\n\n\\section{Quasiparticles}\n\\label{sec:quasiparticles}\n\nIn this section we address the existence of quasiparticles in the system. We do this by computing the quasiparticle weight $Z$ from the ``momentum\" distribution function of the ground state, which in turn is obtained from real-space correlation functions. \nOne peculiarity of the Bethe lattice, is that correlations between any two degrees of freedom sitting on individual nodes of the lattice have a maximal finite correlation length, even at criticality, due to the geometry of the lattice. However, algebraically-decaying correlations reappear after a change of basis to single-particle states that are weighted sums of all nodes of a given generation emanating from a chosen center site. These bases of states unveil the traditional long-range criticality present in gapless states on the Bethe lattice. Below, we introduce a subset of these weighted states called the \\emph{symmetric} states, which we focus on in the rest of this section. From these symmetric states we define a quantum number which plays an analogous role on the Bethe lattice to quasi-momentum on the hypercubic lattice, despite the absence of conventional translation invariance.\n\n\n\\subsection{Single-particle basis of symmetric states}\n\nFor $U = 0$, the free particle Hamiltonian was diagonalized in Ref. \\cite{PhysRevB.63.155110}, using the \\emph{symmetric} set of single-particle states. These are given as follows. Choose any site to be labeled as the origin, with site label $0$. Then consider all permutations of the nodes at each generation $l$ from the center. The symmetric states are those which are invariant under all such permutations. Their creation operators are given by\n\\begin{equation}\n\\tilde{c}^{\\dagger}_{0,\\sigma} \\; \\equiv \\; c^{\\dag}_{0,\\sigma}\n\\label{eq:single particle states l 0}\n\\end{equation}\nand \n\\begin{equation}\n\\tilde{c}^{\\dagger}_{l,\\sigma} \\; \\equiv \\; \\frac{1}{\\sqrt{z(z-1)^{l-1}}} \\displaystyle\\sum_{\\eta_1 = 0}^{z-1} \\; \\displaystyle\\sum_{\\eta_2 \\neq \\eta_1 } \\dots \\displaystyle\\sum_{\\eta_l \\neq \\eta_{l-1}} c^{\\dag}_{\\eta_1 + \\eta_2 + \\dots + \\eta_l,\\sigma}\n\\label{eq:single particle states}\n\\end{equation}\nfor $l > 0$. The collection of $\\eta_i$ denotes a unique path from the origin to the $l$-th generation (this is the usual notation for nodes on the Bethe lattice). The state $\\tilde{c}^{\\dagger}_{l,\\sigma} |\\text{vacuum}\\>$ is the symmetric combination of all the singly-occupied spin-$\\sigma$ states of the $l$-th generation of the tree. These states form an orthonormal subset of all the states on the Bethe lattice, but for $U = 0$ they are the only relevant ones. In the symmetric state basis, the free particle Hamiltonian maps onto fermions hopping on an infinite half-chain, with the first hopping amplitude equal to $\\sqrt{z}$ and all the rest equal to $\\sqrt{z-1}$ (remember $t$ has been set to one). The conjugate variable that replaces momentum is an angle $\\theta \\in \\[0,\\pi\\]$, and the band energy is given by $\\epsilon(\\theta) = 2 \\sqrt{z-1} \\, \\cos\\theta$. Note that the energy of a regular one-dimensional band is obtained by replacing $\\theta$ with a momentum $k$ and $z$ with $2$. The single-particle wavefunctions $\\psi_l(\\theta)$ that diagonalize the Hamiltonian are given by\n\\begin{align}\n\\begin{split}\n\\psi_0(\\theta) &= \\sqrt{\\frac{2}{\\pi}} \\frac{\\sqrt{z(z-1)}\\sin(\\theta)}{\\sqrt{z^2 - 4(z-1) \\cos^2(\\theta)}},\n\\\\ \n\\psi_{l \\neq 0}(\\theta) &= \\sqrt{\\frac{2}{\\pi}} \\sin(l \\cdot \\theta + \\gamma(z,\\theta)),\n\\\\\n\\gamma(z,\\theta) &= \n\\begin{cases}\n\\arcsin\\left(\\frac{z \\sin (\\theta )}{\\sqrt{z^2-4 (z-1) \\cos ^2(\\theta )}}\\right),        & \\hspace{-2mm} \\theta \\in \\[0,\\frac{\\pi}{2}\\) \\\\\n\\pi -\\arcsin\\left(\\frac{z \\sin (\\theta )}{\\sqrt{z^2-4 (z-1) \\cos ^2(\\theta)}}\\right),        & \\hspace{-2mm} \\theta \\in \\[\\frac{\\pi}{2},\\pi\\].\n\\end{cases}\n\\end{split}\n\\label{eq:real space wavefunctions n U=0 exact}\n\\end{align}\n\\\\\n\\indent \nOnce $U \\neq 0$, the symmetric states are no longer enough to describe the system. Indeed, if we simply consider a Mott-like state with one particle sitting on each of the sites at generation $l = 1$ away from the center site, that state cannot be written using only the single-particle symmetric states associated with the same center site. \nIn general, an arbitrary multi-particle Fock state cannot be constructed as a tensor product of the symmetric single-particle states. Therefore, to construct it one must employ states from other symmetry sectors. The interacting ground states we find numerically in this work therefore contain states from various symmetry sectors. However, excitations above the ground state can occur in any of these sectors, and we do not have to consider all of them. In order to tractably answer the question of existence of quasiparticles, we choose to focus on excitations in the symmetric sector. How quasiparticles in different symmetry sectors are related to each other is an interesting question for future work \\cite{Eckstein_thanks}.\n\n\n\\subsection{$\\theta$-distribution function for $U = 0$}\n\nThe $\\theta$-distribution function is calculated from the equal-time correlation functions of symmetric single-particle excitations, $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma} \\>$, where $0$ labels a chosen center site and $l$ labels the generation away from the center site. These can be computed as\n\\begin{equation}\n\\begin{split}\n\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\> \n& = \\int\\displaylimits_{- \\infty}^{0} d{\\omega} ~ \\mathcal{A}_{0l}(\\omega)\n\\\\ & = \\int\\displaylimits_{-2 \\sqrt{z-1}}^{\\mu} d\\varepsilon \\; \\sqrt{\\frac{z}{2 \\pi}} \\frac{\\psi_l\\(\\arccos(-\\frac{\\varepsilon}{2 \\sqrt{z-1}})\\)}{\\sqrt{z^2 - \\varepsilon^2}},\n\\end{split}\n\\label{eq:correlation function U=0 exact}\n\\end{equation}\nwhere $\\mathcal{A}_{0l}(\\omega)$ is the probability of inserting an electron with frequency ${\\omega}$ at the center site and observing it at generation $l$ at the same frequency. The occupation function in $\\theta$-space is defined as\n\\begin{equation}\nn_{\\sigma}(\\theta, \\theta') \\equiv \\< \\tilde{c}_{\\theta,\\sigma}^{\\dag} \\tilde{c}_{\\theta',\\sigma}\\>,\n\\label{eq:occupation function definition}\n\\end{equation}\nwhere \n\\begin{equation}\n\\td c_{\\theta, \\sigma} = \\lim\\limits_{L \\to \\infty}\\sqrt{\\frac{\\pi}{L+1}} \\sum_{d = 0}^{L} \\psi_d(\\theta) \\, \\td c_{d,\\sigma} \n\\label{eq:theta transform c operators}\n\\end{equation}\nare the $\\theta$-transforms of the symmetric state operators $\\td c_{d,\\sigma}$. The key difference with the usual calculation in hypercubic lattices is that $\\< \\td c^{\\dag}_{d,\\sigma} \\td c_{d',\\sigma}\\>$ is not simply a function of $\\abs{d-d'}$, due to the fact that the symmetric states are defined relative to a chosen center site. This is illustrated in Fig. \\ref{fig:BL 4 generations}, where we show that $\\< \\td c^{\\dag}_{2,\\sigma} \\td c_{4,\\sigma}\\>$ contains correlations of length $2,4$ and $6$.\n\\begin{figure}\n\t\\includegraphics[scale=0.4]{BL_large_4_gen_colored}\n\t\\caption{Four generations of the $z = 3$ infinite Bethe lattice emanating from a given center site. The inner shell is at generation $d = 2$, and the outer shell is at generation $d' = 4$. Choosing a given site on the inner shell (colored red), there are three different groups of sites on the outer shell (colored orange, green and purple) that contribute different-length correlations.}\n\t\\label{fig:BL 4 generations}\n\\end{figure}\nTherefore, $n_{\\sigma}(\\theta, \\theta')$ is not diagonal in $\\theta$. One way to understand this is to think of the entire symmetric sector parametrized by $\\theta$ as the $k = 0$ space on the hypercubic lattice, i.e. the one that is fully symmetric under translations. However, within this sector there is no additional symmetry of the Bethe lattice that requires the total $\\theta$ to be conserved in a scattering process, and therefore $n_{\\sigma}(\\theta, \\theta')$ is not diagonal \\cite{Eckstein_thanks}. Carefully inserting Eq. (\\ref{eq:theta transform c operators}) into Eq. (\\ref{eq:occupation function definition}) and rewriting the result in terms of $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ gives\n\\begin{equation}\n\\begin{split}\n& n_{\\sigma}(\\theta, \\theta') = \n\\lim\\limits_{L \\to \\infty}\\frac{\\pi}{L+1} \\sum_{d,d' = 0}^{L} \\psi_d(\\theta) \\psi_{d'}(\\theta') \\, \n\\frac{z-2}{\\sqrt{z (z-1)}}\n\\\\ &\n\\(\\frac{\\sqrt{z}(z-2)}{(z-1)^{3\/2}}\\)^{\\delta_{\\min(d,d'),0}} \n\\sum_{r=0}^{\\min(d,d')} \\(\\frac{z-1}{z-2}\\)^{\\delta_{r,0} + \\delta_{r,\\min(d,d')}} \n\\\\ & \n\\sqrt{\\frac{z}{z-1}}^{\\delta_{d,d'} \\delta_{r,0} - \\delta_{d,0} \\delta_{d',0}} \n\\, \\< \\td c^{\\dag}_{0,\\sigma} \\td c_{\\abs{d-d'}+2r,\\sigma} \\>. \n\\end{split}\n\\label{eq:n_theta final}\n\\end{equation}\nDetails of the derivation of Eq. (\\ref{eq:n_theta final}) are in Appendix \\ref{sec:occupation function}. We plot the exact $n_{\\sigma}(\\theta, \\theta')$ for $U = 0$ at half-filling in Fig. \\ref{fig:n_theta_theta_prime_U_0_delmu_0_exact}.\n\\begin{figure}\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.9]{n_theta_theta_prime_U_0_n_i_1_exact}\n          \\caption{}\n\t  \\label{fig:n_theta_theta_prime_U_0_delmu_0_exact}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_U_0_exact}\n          \\caption{}\n          \\label{fig:n_theta_U_0_exact}\n        \\end{subfigure}\n\t\\caption{(a) The occupation function $n_{\\sigma}(\\theta, \\theta')$ for the half-filled case at $U = 0$. Aside from the expected step-function along the diagonal, there is non-trivial off-diagonal structure. (b) The diagonal component $n_{\\sigma}(\\theta)$ for $U = 0$ and densities $\\<n_i\\> = 1,1.2$, corresponding to values of $\\theta_F = \\pi\/2$ and $\\theta_F \\approx 1.81$, respectively. Calculating the occupation from Eq. (\\ref{eq:n_theta final}) requires a large distance cutoff $L$, which is chosen here to be (a) $L = 100$ and (b) $L = 200$. This introduces an artificial correlation length, causing the step function to be (slightly) smoothed out.}\n\t\\label{fig:n_2D_and_1D_U_0_exact}\n\\end{figure}\n\\\\\n\\indent\nIn this work, we focus exclusively on the diagonal component of the occupation function, $n_{\\sigma}(\\theta) \\equiv n_{\\sigma}(\\theta, \\theta)$. This tells us the occupation of excitations that preserve total $\\theta$ when scattering with each other. We leave the detailed study of the full occupation function $n_{\\sigma}(\\theta, \\theta')$ for future work. We plot the exact $n_{\\sigma}(\\theta)$ in Fig. \\ref{fig:n_theta_U_0_exact} for $U = 0$ and densities $\\<n_i\\> = 1, 1.2$. We can see the expected behavior of $n_{\\sigma} = \\Theta(\\theta_F - \\theta)$, where $\\Theta(x)$ is the Heaviside step function and $\\theta_F$ is the $\\theta$-analog of the ``Fermi momentum\" (the step function in Fig. \\ref{fig:n_theta_U_0_exact} is slightly smoothed out due to a finite $L$ in Eq. (\\ref{eq:n_theta final})).\n\\\\\n\\indent \nIn order to compute the correlation function $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ from the VUTS numerical solution, \nwe use the fact that even in the interacting ground state, $\\< c^{\\dag}_{i,\\sigma} c_{j,\\sigma} \\>$ is still only a function of the distance $\\abs{i-j}$ (and $\\sigma$). We can then compute $\\< c^{\\dag}_{0,\\sigma} c_{l,\\sigma} \\>$ for an arbitrary branch to write\n\\begin{equation}\n\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>\n= \\sqrt{z^{1-\\delta_{l,0}} (z-1)^{l-1 + \\delta_{l,0}}} \\< c^{\\dag}_{0,\\sigma} c_{l,\\sigma}\\>. \n\\label{eq:correlation function symmetric states}\n\\end{equation}\nNote that the difference here from the previous considerations of this paragraph is that one of the reference points has been set to the center site. We plot $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ measured with VUTS in Fig. \\ref{fig:corr function U=0 real space} for $U = 0$ and densities $\\< n_i \\> = 1, 1.2$, along with the exact results. \n\\begin{figure}[!htbp]\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{corr_func_U_0_n_1}\n          \\caption{$U = 0, \\<n_i\\> = 1$.}\n          \\label{fig:corr function U=0 n = 1}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{corr_func_U_0_n_1_2}\n          \\caption{$U = 0, \\<n_i\\> = 1.2$.}\n          \\label{fig:corr function U=0 n = 1.2}\n        \\end{subfigure}\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{xi_vs_chi_U_0}\n          \\caption{}\n          \\label{fig:xi vs chi}\n        \\end{subfigure}\n\t\\caption{The correlation function $\\< \\td c^{\\dag}_{0,\\uparrow} \\td c_{r,\\uparrow}\\>$ ($\\sigma = \\downarrow$ gives the same) for densities (a) $\\<n_i\\> = 1$ and (b) $\\<n_i\\> = 1.2$ for the non-interacting case $U=0$. We show the $\\chi = 50,100$ results along with the exact solution of Eq. (\\ref{eq:correlation function U=0 exact}). Also shown are the functions $\\frac{1}{r} \\, \\psi_r(\\theta_F) \\, \\psi_0(\\theta_F)$ with (a) $\\theta_F = \\pi\/2$ and (b) $\\theta_F \\approx 1.81$, which are excellent fits beyond a short distance scale, showing Friedel oscillations due to the Fermi surface singularity. The insets show that the $1\/r$ decay is fit perfectly over a larger distance by the exact solution, while the finite $\\chi$ solutions display an exponential decay at large distances.\n\t\t(c) The correlation length extracted from the long distance behavior of $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{r,\\sigma}\\>$, plotted versus $1\/\\chi$. The slope for both densities is $\\xi(\\chi) \\sim \\chi^{0.84}$.}\n\t\\label{fig:corr function U=0 real space}\n\\end{figure}\nThe finite $\\chi$ results are close to the exact ones, but, as expected, the correlation function at large enough distances decays exponentially with a finite correlation length $\\xi(\\chi)$. We can measure $\\xi(\\chi)$ by fitting $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ to an exponential at large $l$. The results are shown in Fig. \\ref{fig:xi vs chi}, where the polynomial fit gives $\\xi(\\chi) \\sim \\chi^{0.84}$ for both densities.\n\\\\\n\\indent \nWhen calculating $n_{\\sigma}(\\theta)$ from Eq. (\\ref{eq:n_theta final}), in practice we must choose a finite value of $L$. As long as we take $L$ large enough, the correlation length $\\xi(\\chi)$ will act as the long-distance cutoff and the value of $L$ will not have any effect. Our results are obtained using bond dimensions up to $\\chi = 100$, for which the induced correlation length is $\\xi(\\chi) \\lesssim 40$. We find that a value of $L \\sim 400$ is large enough for all $\\chi$ we study. The finite $\\xi(\\chi)$ smoothes out the step function in $n_\\sigma(\\theta)$, so we estimate $\\theta_F$ from the location of the maximum of $\\abs{n'_{\\sigma}(\\theta)}$. In Fig. \\ref{fig:n_theta near theta_F at U=0 and n=1.2} we show $n_{\\sigma}(\\theta)$ for $\\<n_i\\> = 1.2$ near $\\theta_F$. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{n_theta_near_theta_F_U_0_n_1_2}\n\t\\caption{Plots of $n_{\\sigma}(\\theta)$ near $\\theta_F$ for $U = 0$ at density $\\<n_i\\> = 1.2$ and a range of $\\chi$. The value of $L$ used here from Eq. (\\ref{eq:n_theta final}) is $L = 400$.}\n\t\\label{fig:n_theta near theta_F at U=0 and n=1.2}\n\\end{figure}\nThe finite slope at $\\theta_F$ diverges as a power law in $\\chi$, as we show in Fig. \\ref{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}. \n\\begin{figure}\n\t\\includegraphics[scale=0.9]{nprime_theta_at_theta_F_vs_chi_all_U_n_1_2}\n\t\\caption{The value of $\\abs{n'_{\\sigma}(\\theta_F)}$ vs $1\/\\chi$ for various $U$ at density $\\<n_i\\> = 1.2$. The straight lines on the log-log plot are fit by $\\abs{n'_{\\sigma}(\\theta_F)} \\sim \\chi ^{{\\alpha}}$, with $0.5 < {\\alpha} < 1$.}\n\t\\label{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}\n\\end{figure}\nThis indicates that $\\xi(\\chi)$ is the only low-energy scale in the problem, and the state is truly gapless in the $\\chi \\to \\infty$ limit.\n\\\\\n\\indent\nIn summary, in this section we show how to compute the diagonal part of the occupation function in the non-interacting case, using VUTS and \na careful extrapolation in the bond dimension, and achieve excellent agreement with the analytical result.\n\n\n\n\\subsection{Quasiparticles in the interacting system}\n\nNow we turn on interactions. In Fig. \\ref{fig:n_theta near theta_F all U and n=1.2} we plot $n_{\\sigma}(\\theta)$ near $\\theta_F$ for $\\<n_i\\> = 1.2$ and $U = 5$ at various $\\chi$ as well as for various $U$ at $\\chi = 70$.\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_near_theta_F_U_5_n_1_2}\n          \\caption{$\\<n_i\\> = 1.2, U = 5$.}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_near_theta_F_all_U_n_1_2}\n          \\caption{$\\<n_i\\> = 1.2, \\chi = 70$.}\n        \\end{subfigure}\n\t\\caption{Plots of $n_{\\sigma}(\\theta)$ near $\\theta_F$ at density $\\<n_i\\> = 1.2$, showing the dependence on (a) $\\chi$ for fixed $U = 5$, and on (b) $U$ for fixed $\\chi = 70$. The value of $L$ from Eq. (\\ref{eq:n_theta final}) is $L = 400$.}\n\t\\label{fig:n_theta near theta_F all U and n=1.2}\n\\end{figure}\nThe occupation shows a form similar to that of the free case, albeit with a reduced size of the step at $\\theta_F$. The value of the slope at $\\theta_F$ diverges for all $U$, as we show in Fig. \\ref{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}. The interacting system is therefore also gapless in the $\\chi \\to \\infty$ limit, as expected.\n\\\\\n\\indent \nThe quasiparticle weight, $Z$, of the symmetric state excitations is defined as \n\\begin{equation}\nZ \\equiv \\(\\lim\\limits_{\\theta \\to \\theta_F^-} - \\lim\\limits_{\\theta \\to \\theta_F^+} \\) n_{\\sigma}(\\theta).\n\\label{eq:Z definition}\n\\end{equation}\nFor a Fermi liquid $n_{\\sigma}(\\theta)$ has a step at $\\theta_F$ and $Z > 0$, while for a Luttinger liquid the occupation function has a higher order non-analyticity that scales as $n_{\\sigma}(\\theta - \\theta_F) \\sim \\abs{\\theta - \\theta_F}^{\\gamma} \\sign(\\theta - \\theta_F)$ for some $\\gamma < 1$ and therefore $Z = 0$. Our goal is to find the true thermodynamic value of $Z$ to distinguish between these two scenarios. Of course, for a finite $\\chi$ Eq. (\\ref{eq:Z definition}) will always give zero. However, we can define a quantity $Z(\\chi)$ whose limit will give $Z$ in the $\\chi \\to \\infty$ limit. We define this as\n\\begin{equation}\nZ(\\chi) \\equiv n_{\\sigma}\\(\\theta_F - \\frac{\\pi}{2 \\, \\xi (\\chi)} \\) - n_{\\sigma}\\(\\theta_F + \\frac{\\pi}{2 \\, \\xi (\\chi)} \\),\n\\label{eq:Z finite chi definition}\n\\end{equation}\nwhich satisfies the desired property because $\\xi(\\chi)\\rightarrow \\infty$ with increasing $\\chi$. We choose a spacing of $\\Delta \\theta = \\pi\/\\xi(\\chi)$ around $\\theta_F$ because that is roughly the resolution one expects from a finite correlation length, and therefore the convergence in $1\/\\chi$ should be fastest. We plot $Z(\\chi)$ vs $\\chi$ for $\\<n_i\\> = 1.2$ and a range of $U$ in Fig. \\ref{fig:Z vs chi for all U at n=1.2}. \n\\begin{figure}\n\t\\includegraphics[scale=0.9]{Z_vs_chi_all_U_n_1_2}\n\t\\caption{$Z(\\chi)$ at density $\\<n_i\\> = 1.2$, shown with linear fits.}\n\t\\label{fig:Z vs chi for all U at n=1.2}\n\\end{figure}\nThe results show that the extrapolated $Z$ is (a) very close to the expected value of $Z = 1$ for the free theory and (b) finite for all $U$ we study. We plot $Z$ as a function of $U$ in Fig. \\ref{fig:Z vs U for all n}, where we can see that it decreases as a function of $U$, but seems to saturate to a finite value. The saturation value is an increasing function of doping $\\<n_i\\> - 1$, as \nillustrated by Fig.~\\ref{fig:Z at U =20 vs n-1} where we plot the value of $Z(U=20)$ as a function \nof doping. \n\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n    \t\\includegraphics[scale=0.8]{Z_vs_U_all_n}\n        \\caption{}\n\t    \\label{fig:Z vs U for all n}\n        \\end{subfigure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n    \t\\includegraphics[scale=0.8]{Z_U_20_vs_n_minus_1}\n    \t\\caption{}\n\t    \\label{fig:Z at U =20 vs n-1}\n        \\end{subfigure}\n    \t\\caption{(a) The extrapolated values of $Z$ at densities $\\<n_i\\> = 1.2,1.3,1.4$, plotted as a function of $U$. The first two blue points are covered by the orange ones. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points. We can see that the curves are close to saturation at $U = 20$. (b) The values of $Z$ at $U = 20$, which seem close to the saturated values, as a function of $\\<n_i\\> - 1$.}\n\\end{figure}\nWe also address the question of Luttinger's theorem for $n_{\\sigma}(\\theta)$. We check that $\\theta_F$ is independent of $U$ (the dependence on $\\chi$ is negligible) for various densities  in the range $\\<n_i\\> \\in (1.1,1.4)$. From this we conclude that Luttinger's theorem holds for all values of the density and interaction strength. \n\\\\\n\\indent\nThe decrease of $Z$ with increasing $U$ as well as the saturation at large $U$ to a value which increases with doping are both qualitatively consistent with results established in the $z=\\infty$ (DMFT) limit and with slave boson approaches (for general lattices) ~\\cite{RevModPhys.68.13,4bosons}.\nA distinctive aspect of these theories, however, is that the effective mass of \nquasiparticles is related to $Z$ by $m^*\/m=1\/Z$. On the technical level, this is due to the locality of the self-energy, while physically this reflects the inability of these approaches to capture the feedback of short-range order and collective modes \ninto the physics of quasiparticles. \nIn contrast, in the present case, since the connectivity is kept finite, we would expect this feedback to be present. \nIt is therefore an outstanding question for future work to explore whether the dispersion of quasiparticles is \nrenormalized in a  different manner than $Z$ itself, and in particular whether it is affected by \nshort-range antiferromagnetic correlations at low doping level. This is left for future work since it requires \nan extension of our algorithm to the study of excited states.\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn summary, we have introduced a new numerical algorithm,\n(fermionic) VUTS, to study quantum (fermionic) models on the Bethe lattice. \nWe apply it to the Hubbard model for coordination number $z = 3$, allowing for a two-site unit cell, obtain the $T = 0$ phase diagram and study the doping-induced Mott transition.   \nWe find an antiferromagnetic insulating phase at half filling and a paramagnetic metallic phase for the doped system, which are separated by a first-order insulator to metal phase transition. \nThe model displays phase separation at low doping, with a range of forbidden densities. These conclusions were reached by allowing for a two-site unit cell. We cannot exclude that phases with more complex charge or \nmagnetic ordering exist when allowing for a larger unit cell, which we leave as an open question for future work. By studying the diagonal component of the occupation function for momenta of the symmetric single-particle sector, we find that the quasiparticle weight is non-zero, \nconsistent with the existence of a Fermi liquid ground state for fermions on the Bethe lattice. We find that this Fermi liquid state obeys Luttinger's theorem.\n\nAn interesting direction in which to extend this work would be to further characterize this Fermi liquid state. \nOne would like to know, for example, what happens near $\\theta_F$ to the off-diagonal $n_\\sigma(\\theta,\\theta')$ when interactions are turned on. \nIt is also interesting to look at some of the other symmetry sectors, say the ones that leave each of the $z$ sub-trees connected to the center site invariant, \nand see whether they have quasiparticles and how the quasiparticle weight depends on the sector.\nThese questions are the Bethe lattice version of the important physical question of `momentum dependence' of quasiparticle properties on the Fermi surface of hypercubic lattices.\n\nIn the one-dimensional VUMPS algorithm, it has been shown that low-lying excitations above the ground-state can be accurately computed~\\cite{PhysRevB.97.235155}. \nAn obvious question is how to extend these ideas to VUTS (this is another potential advantage of this method over imaginary time evolution).\nExtension of the algorithm to excited states would allow to characterize the effective mass (dispersion) of the quasiparticles, paving the road for a study of \nhow short-range (e.g. antiferromagnetic) correlations affect quasiparticle properties. This is a very important question for the physics of strongly correlated electron systems. \nA breakdown of the $m^*\/m=1\/Z$ relation would signal that this feedback is indeed present, in contrast to the infinite connectivity limit. \nFinally, studying the energy dependence of the quasiparticle lifetime, as well as the interactions between quasiparticles would be a comprehensive study of Landau \nFermi liquid theory on the Bethe lattice. \n\nIt would also be interesting to look at the entanglement structure of the Fermi liquid state, as compared to a Luttinger liquid. Since the entanglement spectrum is easily obtained on the one-dimensional and Bethe lattices from the singular value decomposition of a single bond tensor, this question could in principle be easily answered. \n\nWe also propose that the finite $z$ Bethe lattice can be used as a computationally tractable platform for the study of how quasiparticles can be destroyed \nand Fermi liquid behaviour breaks down when considering other fermionic hamiltonians on this lattice. \nOne route to explore this, which connects to the feedback of long-wavelength collective modes or short-range spatial correlations on quasiparticle properties, is to study the vicinity of a quantum critical point. \nAnother route is to study microscopic models that are tailor-engineered to have incoherent excitations, such as the one of Ref.~\\cite{PhysRevB.86.045128}, or the multi-channel Kondo lattice.     \n\nStudying the interplay between frustration and strong correlations is another promising direction for future research. \nIntroducing a frustrating next-nearest-neighbor hopping term has been found with DMFT to yield an interaction-driven metal-insulator transition on the infinite $z$ Bethe lattice~\\cite{rozenberg1994,RevModPhys.68.13}. An interesting question is whether this transition can also be found at finite $z$. \nThe VUTS algorithm could also be extended to other lattices with a tree-like structure, such as the Husimi cactus. \nThe study of spin models on such lattices have revealed spin-liquid ground-states\\cite{Chandra_1994,PhysRevB.93.075154} (see also Ref.~\\cite{Udagawa_2019} for considerations on the spin-ice model), \nopening the question of how these models behave upon doping. \n\nFinally, increasing the temperature to a non-zero value is another interesting direction. Intuitively, some finite-temperature properties may be less sensitive to the differences between tree lattices and two- or three-dimensional hypercubic lattices. This could be done using the purification method \\cite{PhysRevLett.93.207205,PhysRevLett.93.207204,PhysRevB.72.220401} which has been formulated on the Bethe lattice in Ref. \\cite{PhysRevB.100.125121}.\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe thank Julien Agier, Martin Claassen, Michel Ferrero, Gabriel Kotliar, Chris Laumann, Sung-Sik Lee, Roderich Moessner, Marcelo Rozenberg, Steve White and in particular Martin Eckstein and Riccardo Rossi for useful discussions. We thank Ruben Verresen for comments on the manuscript. All tensor network calculations and the VUTS code were implemented with the ITensor Library (C++ version 3.1) \\cite{itensor}. The Flatiron Institute is a division of the Simons Foundation.\n\n\n\n\n\\bibliographystyle{apsrev4-1}\n\n\\section{Introduction}\n\nThe Hubbard model is a cornerstone of condensed matter physics.  As a paradigmatic model of strongly-correlated electrons  \\cite{ANDERSON1196}, it is simple to formulate yet rich in behavior. In two dimensions (relevant e.g. to cuprate superconductors) observed behaviors include, but are not limited to, antiferromagnetism, unconventional metallic behavior characterized by a pseudogap and deviations from Fermi liquid theory~\\cite{tremblay2006review,Alloul2014,Gunnarsson2015,Wu2017,Schafer}, as well as stripe orders~\\cite{PhysRevB.93.035126,Zheng1155} closely competing with superconducting states at low temperatures~\\cite{PhysRevX.10.031016}. \nBut despite decades of effort, a comprehensive understanding of the phase diagram of the two-dimensional Hubbard model has not yet been fully reached.\nTherefore any solutions of the Hubbard model, whether obtained analytically or by accurate and controlled numerical techniques, are of great value.\n\nThe most reliable and comprehensive solutions of the Hubbard model obtained so far have been mainly in (quasi-) one dimension \\cite{PhysRevLett.20.1445} and in infinite dimensions (infinite lattice coordination number)~\\cite{PhysRevB.45.6479,PhysRevLett.62.324,RevModPhys.68.13}.\nIn one dimension, solutions can be obtained by integrability~\\cite{1D_Hubbard_book} and bosonisation methods~~\\cite{Giamarchi_book}, \nas well as numerically with matrix product state (MPS) tensor network methods \\cite{PhysRevLett.69.2863,SCHOLLWOCK201196}. \nThe latter can also reliably treat quasi-one-dimensional ladder or cylindrical geometries with a small transverse size. \nIn the limit of infinite dimensions the Hubbard model is again numerically tractable due to the fact that dynamical mean-field theory (DMFT) becomes exact \\cite{RevModPhys.68.13}. Using this technique, the phase diagram of the infinite-dimensional Hubbard model can be\nmapped out and a detailed understanding of the interaction-driven Mott insulator to metal transition \nhas been established (for reviews, see Refs.~\\cite{RevModPhys.68.13,RevModPhys.78.865,\nDMFT@25,georges_lectures,rozenberg_lectures}).\n\nWhile they provide useful insights into the physics of the two-dimensional Hubbard model, these limiting cases also have peculiarities which limit the generality of the \nconclusions that can be drawn from their study. In the limit of infinite dimensions, the metallic state is found to be a Fermi liquid, with interactions affecting one-particle properties in a local \n(momentum-independent) manner only. Hence, in this limit, the feedback of long-wavelength collective modes or even short-range spatial correlations \non quasiparticle properties is entirely absent. \nThese effects are important, in particular close to a critical point. In two dimensions, they are, for example, responsible \nfor the formation of a pseudogap \\cite{Preuss1997,Macridin2006,Kyung2006,Gunnarsson2015,Scheurer2018}.\nIn contrast, in one dimension the low-energy excitations consist {\\it only} of bosonic collective modes, associated with charge \nand spin degrees of freedom.\nThere is no notion of a Fermi liquid and the metallic behavior of the Hubbard model is a Luttinger liquid which lacks coherent quasiparticles \nand displays spin-charge separation~\\cite{Giamarchi_book}.  \n\nIn this work we perform a controlled and accurate numerical study of the ground state of the Hubbard model on the Bethe lattice with a finite coordination number $z$, focusing on the case $z=3$. This is an infinite lattice that has a tree structure, where every site is connected to the same number of other sites ($z$) but there are no loops. We show a portion of this lattice in Fig. \\ref{fig:BL intro}.\n\\begin{figure}\n\t\\includegraphics[scale=0.3]{BL_large_4_gen}\n\t\\caption{A portion of the infinite Bethe lattice with coordination number $z = 3$. The figure depicts four ``generations\" of the tree structure, with a particular site chosen to be the ``center\" site. Note that in the actual Bethe lattice there is no special site, and generations can be counted from any of the sites.}\n\t\\label{fig:BL intro}\n\\end{figure}\nThis lattice provides an intermediate case between one dimension ($z=2$) and infinite dimensions ($z = \\infty$), with the key virtue that it admits controlled solutions via tensor network methods, including away from half filling and in the presence of strong interactions. \n\nExact solutions of models on the Bethe lattice have a long history in statistical mechanics~\\cite{Baxter_book}, \nstarting with the pioneering article of Hans Bethe~\\cite{Bethe_1935}. \nSolutions on the finite coordination number Bethe lattice provide a better approximation to thermodynamic quantities than the mean-field approximation \n(corresponding to the infinite dimensional limit)~\\cite{Baxter_book,Bethe_1935,PhysRevLett.74.809}. \nModels studied on the Bethe lattice include classical and quantum spin models~\\cite{PhysRevB.80.144415,PhysRevB.77.214431,NAGY2012542,PhysRevB.86.195137,PhysRevB.88.035138,PhysRevB.89.054426,LIU20141}, spin glass systems \\cite{Parisi_BL_spin_glass,PhysRevLett.56.1082,doi:10.1002\/pssb.200541282,PhysRevB.78.134424}, the Bose Hubbard model \\cite{PhysRevB.80.014524}, and models of Anderson localization  \\cite{Abou_Chacra_1973,Mirlin_1991,PhysRevLett.78.2803,PhysRevB.100.094201,dupont2020dirty}. The fermionic Hubbard model on the finite version of the $z=3$ Bethe lattice (known as a Cayley tree) has also been studied previously using a variant of the density matrix renormalization group (DMRG) algorithm \\cite{Lepetit2000}, but only the case of half filling was studied (which is a charge insulator) and only local ground state quantities given (energy, staggered magnetization and its fluctuations, and neighboring spin correlations). Since that time, there have been significant advances in DMRG and related algorithms for infinite one-dimensional systems, which we generalize here to the Bethe lattice and use to obtain our results. Notably, there has been no previous study of metallic states on the finite connectivity Bethe lattice, to the best of our knowledge. \n\nWe determine the full phase diagram of the fermionic Hubbard model on the $z=3$ Bethe lattice, allowing for a two-site unit cell, and establish \nthe nature of the doping-driven Mott insulator to metal transition (MIT). We find that this transition is first-order, and that for every value of interaction strength there is a region of forbidden density. Therefore, in the interaction-density plane the model exhibits phase separation at low doping levels.  \nWe find that, for all allowed values of the density, the doped metallic ground-state does not display magnetic long-range order. \n\nImportantly, we also demonstrate that the doped metal hosts coherent quasiparticles at all studied values of the interaction strength, $U$, from weak to strong coupling, and determine the behavior of the quasiparticle weight as a function of $U$. This answers in the affirmative the question of whether Fermi liquid behaviour \napplies as soon as the peculiar kinematic constraints of one dimension are alleviated, and also provides a concrete description of a Fermi liquid ground state with tensor networks, both of which are key motivations for our work. Generally, it is difficult for tensor networks to accurately describe interacting metallic states above one dimension, although much progress has been made in this direction \\cite{PhysRevB.81.165104,PhysRevB.93.045116,PhysRevB.100.195141,mortier2020resolving}. Our work provides an alternate route to this agenda, avoiding the computational challenges of two dimensional tensor networks, while going beyond the restrictions of one dimensional physics. \n\nWe obtain our results by generalizing a recently developed MPS method, variational uniform MPS (VUMPS) \\cite{PhysRevB.97.045145}, to tree tensor network (TTN) states \\cite{PhysRevA.74.022320,PhysRevB.82.205105,doi:10.1063\/1.4798639}, which we dub the variational uniform tree state algorithm (VUTS). We further introduce the fermionic version of the VUTS algorithm, using the swap-gate method of Refs. \\cite{PhysRevB.81.165104,Orus_review}. The VUTS algorithm works directly in the thermodynamic limit, which is important in the study of models on the Bethe lattice. The alternative is to study the finite Cayley tree and perform a finite-size scaling analysis. However, the number of boundary sites on the Cayley tree is always more than half of the total, and therefore finite-size effects are unusually strong and can even lead to conclusions that do not hold on the Bethe lattice \\cite{Baxter_book,PhysRevB.87.085107,OSTILLI20123417}. Working directly in the infinite-size limit is therefore important for models on trees. All previous works studying quantum models on the (infinite) Bethe lattice using tensor networks have used a variant of the infinite time-evolving block decimation (iTEBD) algorithm \\cite{PhysRevLett.98.070201}. In the one-dimensional case, the VUMPS algorithm has been found to be much more efficient than other methods that work in the thermodynamic limit, such as iTEBD or earlier infinite DMRG algorithms \\cite{PhysRevB.97.045145}, and indeed we find its extension in the form of the VUTS algorithm we develop to be very efficient. Our method scales as ${\\mathcal O}(\\chi^{z+1})$ where $\\chi$ is the bond dimension of the tensor network being optimized. For $z=3$ this scaling is significantly better than that of the most modern and accurate projected entangled pair states (PEPS) algorithms that scale as ${\\mathcal O}(\\chi^{10})$ for PEPS bond dimension $\\chi$ (assuming the boundary MPS bond dimension scales as $\\chi^2$) \\cite{PhysRevB.92.035142,PhysRevB.94.035133,PhysRevB.94.155123,PhysRevB.98.235148,PhysRevX.9.031041,PhysRevB.100.195141}, but is more challenging than the ${\\mathcal O}(\\chi^3)$ scaling of DMRG. However, the steeper scaling is mitigated by the fact that the typical bond dimension required to reach an accurate solution generally decreases as one goes to higher dimensions and larger coordination numbers due to the monogamy of entanglement and more mean-field-like properties of the wavefunction.\nThe accuracy of tensor network methods is often measured by the \\emph{truncation error}, which measures the typical loss of fidelity incurred during the truncation step of the optimization algorithm. In this work, for a bond dimension of $\\chi = 100$, we are able to achieve a truncation error of less than $10^{-3}$ in the most computationally challenging part of the phase diagram (most entangled ground state), and a truncation error of less than $10^{-7}$ in the best cases. This level of accuracy allows us to measure long-distance correlation functions well enough to extract information about quasiparticle coherence in the metallic phase, which demonstrates that our method can be used to reliably study critical phases of matter on the Bethe lattice.\n\nThis work suggests promising\nfuture directions for studying the behavior of strongly correlated electrons in a controlled setup. Because Fermi liquid behavior is a rather generic feature of metallic states, the present study allows to establish a controlled platform which can be used to study how Fermi liquid behaviour can be broken by further perturbations to the model considered here, or in other fermionic models. In the concluding section of this article, we discuss possible routes towards achieving this goal. If successful, tensor network solutions of correlated electrons on the $z=3$ Bethe lattice could provide a new platform for studying non-Fermi liquids~\\cite{RevModPhys.73.797,doi:10.1146\/annurev-conmatphys-031016-025531} in a controlled and accurate manner. Other potential applications are the study of the interaction-driven MIT in frustrated fermionic systems, and the study of fermions on closely related tree-like lattices, such as the Husimi cactus on which the Heisenberg model and other spin models have been shown to display spin liquid phases~\\cite{Chandra_1994}. We elaborate on all these directions and others at the end of the paper.\n\nThis paper is organized as follows. In section \\ref{sec:VUTS} we describe the general VUTS method, applicable to generic Hamiltonians. In section \\ref{sec: Hubbard model} we define the Hubbard model on the Bethe lattice and show the phase diagram obtained from the VUTS solution. Section~\\ref{sec:quasiparticles} discusses the calculation of the quasiparticle weight from the occupation function, as well as Luttinger's theorem. Finally, in Sec. \\ref{sec:discussion} we summarize and discuss future directions.  \n\n\n\n\n\n\n\\section{Variational uniform tree state algorithm}\n\\label{sec:VUTS}\n\nIn this section we introduce the variational uniform tree state algorithm (VUTS), a generalization of the variational uniform matrix product state algorithm (VUMPS) \\cite{PhysRevB.97.045145}, for optimizing infinite tree tensor network (TTN) states. We start with a Bethe lattice of quantum degrees of freedom. For simplicity we focus on the algorithm for coordination number $z = 3$, which is the value for the model studied in this paper, but the extension to general $z$ is straightforward. We use an infinite TTN as our ansatz to approximate quantum states on the Bethe lattice. For $z = 3$, the infinite TTN with a 1-site unit cell consists of an order 4 tensor $A  \\in \\mathbb{C}^{\\chi \\times \\chi \\times \\chi \\times d}$, with one physical leg ($s$) which runs over the physical degrees of freedom $1,...,d$, and three virtual legs ($l_0,l_1,l_2)$ that run over virtual degrees of freedom $1,...,\\chi$. The virtual legs of neighboring tensors connect to each other, forming the same geometry as the Bethe lattice, as shown in the tensor network diagram in Fig. \\ref{fig:TTN Bethe lattice unlabeled}.\n\\begin{figure}\n\t\\includegraphics[scale=0.25]{tree_TN_Bethe_lattice_unlabeled}\n\t\\caption{A finite portion of the infinite tree tensor network (TTN) state describing the many-body wave function on the Bethe lattice. The physical legs (green dashed lines) form the nodes of the lattice, while the virtual legs (straight black lines) form the edges. In this case, a single tensor $A$ comprises the state, and the unit cell is just a single site.}\n\t\\label{fig:TTN Bethe lattice unlabeled}\n\\end{figure}\n\\\\\n\\indent \nThe Hamiltonians we will focus on here are isotropic and have an equivalence between all sites (the analog of translational invariance for hypercubic lattices). The ground state will potentially break this isotropy completely, and break the site equivalence down to a non-trivial unit cell. In this paper, we allow for the state to be fully anisotropic between the different directions emanating from a given site, but for simplicity we focus exclusively on the case when the unit cell consists of no more than two sites (generalizing to arbitrary unit cells is straightforward). The infinite TTN state we study therefore has a 2-site unit cell, and is parameterized by a set of $2z$ tensors $A_{i,m}$, where $i = 0,1$ labels the location in the unit cell, and $m = 1, \\dots, z$ labels the direction of the gauge (defined below). Again, each tensor has one physical index $s_i = 1, \\dots, d$ and three virtual ``link\" indices $l_0,l_1,l_2$.\n\\\\\n\\indent\nBecause the TTN has no loops, it is straightforward to work in the \\emph{canonical gauge}, i.e. the gauge where the tensors are constrained to be orthonormal bases when viewed as a matrix from two link indices $l_n,l_k$ and the physical index $s_i$ to the remaining link index $l_m$. This constraint on the tensors is very useful for making the variational optimization faster and more stable, and is standard in a wide variety of tensor network algorithms, particularly in 1D algorithms like VUMPS and DMRG. The constraint on the tensors is written as\n\\begin{equation}\n\\displaystyle\\sum_{s_i,l_n,l_k} \\bar{A}^{s_i,l'_m,l_n,l_k}_{i,m} A^{s_i,l_m,l_n,l_k}_{i,m} = \\mathbb{1}^{l'_m,l_m}_{i,m},\n\\label{eq:canonical gauge}\n\\end{equation}\nwhere we have introduced the notation $\\bar 0 = 1, \\bar 1 = 0$ for the unit cell indices and use $\\bar{A}$ to denote the complex conjugation of $A$. The matrices $\\mathbb{1}_{i,m}$ are identities.  Diagrammatically, Eq. (\\ref{eq:canonical gauge}) is equivalent to Fig.~\\ref{fig:canonical gauge}. The arrows on the links denote the gauge of the $A$ tensors (the outgoing link is the direction of the gauge). Any TTN can be brought into the form where the tensors obey Eq. (\\ref{eq:canonical gauge}) (or equivalently Fig. \\ref{fig:canonical gauge}) by inserting a particular set of ``gauge transformations\"\nonto the link degrees of freedom, i.e. inserting a particular set of resolutions of the identity $X X^{-1}$ (where $X$ is an invertible matrix) onto the links of the TTN. Note that the gauge transformation does not affect the observables of the system, and any TTN can be transformed into the canonical gauge efficiently.\n\\begin{figure}\n    \\includegraphics[scale=0.25]{gauge_condition_1}\n    \\caption{Diagrammatic version of the gauge conditions of Eq. (\\ref{eq:canonical gauge}). The bonds labelled $k,n,m$ have link indices $l_k,l_n,l_m$ respectively (and the uncontracted $m$ bond on the ket has link index $l_m'$). The unlabelled dashed bond is the physical degree of freedom with index $s_i$.}\n\t\\label{fig:canonical gauge}\n\\end{figure}\n\\begin{figure}\n\t\\includegraphics[scale=0.45]{tree_TN_Bethe_lattice_with_centers_2}\n\t\\caption{The same portion of the Bethe lattice as in Fig. \\ref{fig:TTN Bethe lattice unlabeled}, but with the bonds labeled with $m = 0,1,2$, which have link indices $l_0,l_1,l_2$ respectively. Additionally, each tensor is labeled with subscripts $i,m$, where $i = 0,1$ is the unit cell index and $m$ is the direction of the gauge. In the top diagram, the gauge center $C_2$ is shown on a bond labelled by $2$. The next equality shows that the $C_2$ tensor can be absorbed into the $A_{1,2}$ tensor to put the gauge center on the site tensor, creating $A_{1,C}$.}\n\t\\label{fig:TTN Bethe lattice with gauge centers}\n\\end{figure}\n\\\\\n\\indent\nExamples of the TTN state with a 2-site unit cell in the canonical gauge are shown in Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. In the top diagram, the gauge center is $C_2$. Here, $C_2$ represents the projection of the infinite wavefunction of the system onto the finite-sized Hilbert space of that link of the network. The center matrices $C_m$ constitute invertible gauge transformations relating the $A$ tensors to each other via $A_{i,m} C_m = A_{i,n} C_{n}$, where $i = 0,1$ and $n \\neq m$. The center matrices $C_m$ also contain important information like the entanglement spectrum between the two infinite halves of the system split by that link. Additionally, the gauge center can be absorbed onto an $A$ tensor, defining the center site tensors $A_{i,C} = A_{i,m} C_m$ for any $m$. This is shown in the lower diagram of Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. In this case, $A_{i,C}$ would represent the infinite wavefunction of the system projected onto a single site (and again, different spectra of that tensor relate to entanglement spectra of different bipartitions of the lattice).\n\\\\\n\\indent\nWe describe the algorithm for the case of $H = \\sum_{\\<i,j\\>} h_{i,j}$, where $h_{i,j}$ is a two-site operator that acts on nearest neighbors only. The case of longer-range operators is  treatable using techniques like those described in Appendix~C of Ref. \\cite{PhysRevB.97.045145}. The total energy is given by $E = \\sum_{\\<i,j\\>} \\<\\psi|h_{i,j}|\\psi\\>$, and we want to minimize $E$, treating the tensor elements of our TTN as variational parameters. As in VUMPS (and many related tensor network ground state methods), VUTS proceeds in three main steps that are iterated until convergence:\n\\begin{enumerate}\n  \\item Compute the projected Hamiltonians (the Hamiltonian projected into the basis corresponding to the virtual degrees of freedom of the network) to turn the global optimization into a local optimization problem.\n  \\label{alg:projected Hamiltonian}\n  \\item Find the optimized tensors by minimizing the energy of the projected Hamiltonian.\n  \\label{alg:optimize tensors}\n  \\item Update the tensor network with the new optimized tensors.\n  \\label{alg:update network}\n\\end{enumerate}\n\\indent\nTo begin, say we are interested in optimizing a 1-site projected wavefunction $A_{i,C}$, as defined in Fig. \\ref{fig:TTN Bethe lattice with gauge centers}. Step \\ref{alg:projected Hamiltonian}  requires computing infinite sums of local Hamiltonian terms, projected into the basis of our gauged TTN (defined by the tensors $A_{i,m}$), for each of the $z=3$ infinite subtrees connected to $A_{i,C}$. In order to perform the infinite sum, we focus on summing the energy contributions of a single subtree. An example for the series that needs to be summed for the $m=2$ direction in order to optimize the $A_{0,C}$ tensor is shown in Fig. \\ref{fig:H_{1,2} series}. We define the results of these summations as the matrices $H_{i,m}$. The summation can be carried out by making use of the fact that the sum is a geometric series. However, care has to be taken to project out infinite energy contributions to keep the series convergent (i.e. keep the norm of the solution $H_{i,m}$ from diverging). The procedure of performing the summation and projecting out the infinite energy contributions is a generalization of the one in Appendix D of Ref. \\cite{PhysRevB.97.045145}, and we discuss it in more detail in Appendix \\ref{subsec:summing Hamiltonian terms}.\n\\begin{figure}\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n          \\includegraphics[width=1\\linewidth]{H_1_2_series_example}\n          \\caption{}\n        \\end{subfigure}\n        \\begin{subfigure}[b]{0.38\\textwidth}\n          \\includegraphics[width=1\\linewidth]{h_tensor}\n          \\caption{}\n          \\label{fig:h_i_m}\n        \\end{subfigure}\n\t\\caption{(a) The first few terms of the series for $H_{1,2}$, the projected Hamiltonian contribution for one of the three branches of the infinite Bethe lattice connected to the center site tensor $A_{0,C}$ (on the $i=0$ sublattice). (b) Definition of the $h_{i,m}$ tensors used in (a), which are a sum of two local Hamiltonian environment tensors sitting on two branches of the Bethe lattice. Note that the tensor labeled $h$ represents the two-site operator term $h_{i,j}$, which the Hamiltonian is made up of.}\n\t\\label{fig:H_{1,2} series}\n\\end{figure}\n\\\\\n\\indent \nOnce the environment tensors are found, we can proceed to step \\ref{alg:optimize tensors} of the algorithm, which we begin by optimizing $A_{i,C}$. This is done by finding the ground state of the Hamiltonian projected onto the sublattice site $i$, a standard procedure in VUMPS and DMRG. The eigenvalue equation for $A_{i,C}$ is shown diagrammatically in Fig. \\ref{fig:H_Ac}, and is solved iteratively (using a Hermitian eigensolver such as Lanczos). To find the TTN ground state, we obtain the eigenvector with the smallest eigenvalue. As in the VUMPS algorithm, in addition to optimizing $A_{i,C}$, we also optimize $C_m$. $A_{i,C}$ and $C_m$ are then used to solve for $A_{i,m}$, which make up the updated infinite TTN state (see next paragraph). The eigenvalue equation for $C_m$ is shown diagrammatically in Fig. \\ref{fig:H_C}.\n\\begin{figure}\n\t\\includegraphics[scale=0.35]{Ac_eig_eqn.pdf}\n\t\\caption{The eigenvalue equation for $A_{i,C}$. The sum over $m$ is a sum over the contributions from each leg of the $A_{i,C}$ tensor.}\n\t\\label{fig:H_Ac}\n\\end{figure}\n\\begin{figure}\n\t\\includegraphics[scale=0.45]{C_eig_eqn.pdf}\n\t\\caption{The eigenvalue equation for $C_{m}$.}\n\t\\label{fig:H_C}\n\\end{figure}\n\\\\\n\\indent\nFinally, once $A_{i,C}, C_m$ for all $i = 0,1, m = 0,1,2$ are optimized, we can proceed to step \\ref{alg:update network} of the algorithm and solve for our new $A_{i,m}$ tensors by minimizing\n\\begin{equation}\n\\epsilon_{i,m}\n=\n\\min_{A_{i,m}^{\\dagger} A_{i,m} \\, = \\, \\mathbb{1}_{i,m}} || A_{i,C} - A_{i,m} C_m ||. \n\\label{eq:new A tensors}\n\\end{equation}\nThis minimization problem can be solved optimally using techniques described in Eqs. (18)-(22) of Ref. \\cite{PhysRevB.97.045145}. The new $A_{i,m}$ we obtain constitute our updated TTN, and steps \\ref{alg:projected Hamiltonian}-\\ref{alg:update network} are repeated until convergence. Convergence is achieved when the largest error found in Eq. (\\ref{eq:new A tensors}), $\\epsilon_{\\text{prec}} \\equiv \\max\\{\\epsilon_{i,m}\\}$, falls below a chosen threshold (e.g. $\\epsilon_{\\text{prec}} < 10^{-12}$).\n\\\\\n\\indent\nThe VUTS algorithm with a 1-site update, as we describe here, scales as $O(\\chi^{z+1})$  which becomes $O(\\chi^4)$ for the $z=3$ Bethe lattice and $O(\\chi^3)$ for $z=2$, thus reducing to the scaling of VUMPS in the $z=2$ case. Additionally, a 2-site update can be formulated, analogous to the 2-site DMRG algorithm which is commonly used. This requires a slight modification of the algorithm where ground states of 2-site and 1-site projected Hamiltonians are computed (as opposed to 1-site and 0-site projected Hamiltonians in the version of the algorithm described above). This can lead to improved convergence since a larger local Hilbert space is explored, but has a higher computational cost of ${\\mathcal O}(\\chi^5)$ for $z=3$. We use this technique at lower bond dimensions in more challenging parts of the phase diagram (near the phase transition), and switch to the 1-site algorithm later in the calculation to reach higher bond dimensions. To dynamically change the bond dimensions, we use a generalization of the subspace expansion procedure described in Appendix B of Ref. \\cite{PhysRevB.97.045145}.\n\\\\\n\\indent \nFor fermionic models like the Hubbard model, we need to use a fermionic version of VUTS. We use the method outlined in Refs. \\cite{PhysRevB.81.165104,Orus_review}. Every tensor is now endowed with a fermion parity $Z_2$ quantum number and is parity-preserving. When two tensor legs cross on a planar projection of a tensor diagram, a fermionic swap gate is placed at the crossing. In order to employ this method, we need to use a fixed ordering convention for the legs of the tensors $A_{i,m}, A_{i,C}, C_m$, which must be kept consistent in all of the diagrams in the calculation. We address the associated subtleties and details of this approach in Appendix \\ref{subsec:fermions}. Other symmetries beyond the $Z_2$ parity can also be used, such as $U(1)$ particle number conservation (to fix the filling), $U(1)$ spin projection symmetry in the z-direction, and also the spin $SU(2)$ symmetry. The inclusion of these symmetries makes the tensors more sparse and should therefore make the tensor operations more efficient, allowing us to reach larger bond dimensions. In this work, we only employ parity quantum numbers, and leave the use of additional symmetries for future work.\n\n\n\n\n\n\n\n\n\\section{Model and phase diagram} \n\\label{sec: Hubbard model}\n\nThe Hubbard Hamiltonian is given by\n\\begin{align}\nH = - t \\sum_{\\langle i,j \\rangle} \\sum_{\\sigma = \\uparrow, \\downarrow}c_{i, \\sigma}^{\\dagger} c_{j, \\sigma}  + U \\sum_i n_{i, \\uparrow} n_{i, \\downarrow}  - \\mu \\sum_{i, \\sigma} n_{i, \\sigma},\n\\label{eq:interacting Hamiltonian}\n\\end{align}\nwhere the site index $i$ now runs over the Bethe lattice and $n_{i,\\sigma} = c_{i, \\sigma}^{\\dagger} c_{i, \\sigma}$ is the on-site density for an electron of spin $\\sigma$. We set $t = 1$ and vary $U \\geq 0$ and $\\mu$. Since the model is particle-hole symmetric, we only need to consider $\\delta \\mu \\equiv \\mu - \\frac{U}{2} \\geq 0$. To obtain the phase diagram, we compute the ground state of the Hubbard model using the fermionic VUTS algorithm for several values of the bond dimension, $\\chi$. The details of the numerical calculation are given in Appendix \\ref{sec:numerical details}. We perform extrapolations in $\\chi$, the results of which we describe below. The additional plots of the finite $\\chi$ results and details of the extrapolations are in Appendix \\ref{sec:phase diagram more plots}. \n\\\\\n\\indent \nAt half filling, $\\delta \\mu = 0$, the system is a charge insulator with antiferromagnetic order for all $U > 0$, as in one dimension. To illustrate this we compute the staggered magnetization of the bipartite sublattice, $m_s \\equiv \\abs{(\\< \\vec{S}_A \\> - \\< \\vec{S}_B \\>)\/2}$. The extrapolated values $m_s(\\chi\\to\\infty)$ are shown in Fig. \\ref{fig:ms extrapolation}. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{ms_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the staggered magnetization $m_s$ of the insulating state at half filling, plotted as a function of $U$. Note the logarithmic scale used for $m_s$. The error bars are the discrepancy in the extrapolation with and without the last data point (for the points where they are absent they are smaller than the data points).}\n\t\\label{fig:ms extrapolation}\n\\end{figure}\nWe can see that for any $U>0$ the magnetization is non-zero, tending to zero as $U \\rightarrow 0$. From general mean-field theory considerations we expect $m_s \\sim e^{-\\frac{c}{U}}$ for small $U$. \nHowever, we did not attempt to confirm this functional form numerically by systematic calculations at very small values of $U$.\n\\\\\n\\indent \nTo illustrate the charge gap at half filling, we compute the on-site occupation $\\< n_{i} \\> = \\sum_{\\sigma} \\< n_{i,\\sigma} \\>$ as a function of $\\delta \\mu$. We show this in Fig. \\ref{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50} for a single value of $U = 6$ and $\\chi = 50$. All other $U$ and $\\chi$ look qualitatively similar.\n\\begin{figure}\n\t\\begin{subfigure}[c]{0.45\\textwidth}\n          \\includegraphics[width=0.85\\linewidth]{occ_vs_delmu_both_branches_U_6_chi_50}\n          \\caption{$U = 6, \\chi = 50$.}\n        \\end{subfigure}\n\t\\begin{subfigure}[c]{0.45\\textwidth}\n          \\includegraphics[width=0.85\\linewidth]{energy_vs_delmu_both_branches_U_6_chi_50}\n          \\caption{$U = 6, \\chi = 50$.}\n        \\end{subfigure}\n\t\\caption{(a) The occupation versus $\\delta \\mu$ for $U = 6$ and $\\chi = 50$ for the both metallic and insulating branches. All other values of $U,\\chi$ look similar. (b) The energy per site of both branches in (a). \n\tTheir crossing point, $\\delta \\mu_c$ is indicated by a dashed line in both (a) and (b).}\n\t\\label{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50}\n\\end{figure}\nWe see that there are two branches of VUTS solutions: insulating ($\\< n_{i} \\> = 1$) and metallic ($\\< n_{i} \\> > 1$). The insulating branch exists for $\\delta\\mu\\leq\\delta\\mu_1$, and the metallic branch for $\\delta\\mu\\geq\\delta\\mu_2$: these \ntwo values, $\\delta\\mu_{1}$ and $\\delta\\mu_{2}$ (with $\\delta\\mu_{2} < \\delta\\mu_{1}$), are spinodal values limiting the meta-stability of the insulating and metallic solutions, respectively (see Appendix \\ref{sec:phase diagram more plots} for details).\nFor each value of $\\delta \\mu$ the ground state is the branch with the lower energy. The energies of the two branches cross at a particular value, which we define to be $\\delta\\mu_c (\\chi)$. At $\\delta\\mu_c (\\chi)$, the ground state changes from insulating for $\\delta\\mu < \\delta \\mu_c$ to metallic for $\\delta\\mu > \\delta \\mu_c$. The occupation undergoes a finite jump, $\\delta n(\\delta \\mu_c, \\chi) = \\< n_i(\\delta \\mu_c, \\chi) \\> - 1$, indicating that this is a first order metal-insulator transition. We estimate the size of the jump in the real system by the $\\chi \\to \\infty$ extrapolated values, which are shown in Fig. \\ref{fig:jump in occupation extrapolation}. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{jump_in_n_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the jump in density $\\delta n_c$ at the first-order transition occuring at $\\delta \\mu_c$ plotted as a function of $U$. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points.}\n\t\\label{fig:jump in occupation extrapolation}\n\\end{figure}\nWe can see they remain finite for all $U$, meaning that this is a true first-order transition, and not an artifact of finite bond dimension. Using a derivation based on the Maxwell construction (detailed in Appendix~\\ref{sec:phase diagram more plots}), \nit can be shown that the total charge gap\nis given by $\\Delta_c (\\chi) = 2 \\, \\delta\\mu_c (\\chi)$. In order to obtain the charge gap $\\Delta_c$ for the real system, we extrapolate $\\Delta_c (\\chi)$ in $\\chi$. The result is shown in Fig. \\ref{fig:charge gap extrapolation}.\n\\begin{figure}\n\t\\includegraphics[scale=0.8]{charge_gap_extrapolated_vs_U}\n\t\\caption{The extrapolated values of the charge gap $\\Delta_c$ at half filling plotted as a function of $U$. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points.}\n\t\\label{fig:charge gap extrapolation}\n\\end{figure}\nWe can see that the extrapolated gap shows an exponential-like behavior at small $U$ similar to the one-dimensional case, though the precise behavior is difficult to extract numerically. At large $U$ the gap crosses over to a more linear dependence on $U$. \n\\\\\n\\indent \nThe first-order transition we observe implies that for every $U$ there is a range of forbidden density. Hence, in the $(U,n)$ plane the model exhibits phase separation. Our numerical method works in the grand canonical ensemble (we do not fix particle number per unit cell), so we cannot observe this phase separation directly. However, VUTS, similar to other variational tensor network methods like DMRG and VUMPS, can get ``stuck\" in local minima. We use this fact to find both branches of solutions near the transition point, even when they are meta-stable (i.e. not the lowest energy states), by way of hysteresis in the numerical algorithm (see Appendix \\ref{sec:phase diagram more plots} for a detailed explanation). The resulting branches, shown in Fig. \\ref{fig:n and energy vs delmu for the two branches at U = 6 and chi = 50}, can be continued to find the spinodal values of the first-order transition (see Appendix \\ref{sec:phase diagram more plots}).\n\\\\\n\\indent \nThe metallic ground-state has no magnetic order for any value of density: we find that the staggered magnetization vanishes once $\\delta \\mu$ crosses $\\delta \\mu_c$. To illustrate the typical magnetization behavior we observe, in Fig. \\ref{fig:ms of metallic phase} we show the staggered magnetization $m_s$ as a function of $\\delta \\mu$ across $\\delta \\mu_c$ for $U = 6$ at a large but fixed $\\chi = 90$.\n\\begin{figure}\n\t\\includegraphics[scale=0.8]{ms_vs_delmu_U_6}\n\t\\caption{The staggered magnetization $m_s$ versus the chemical potential $\\delta \\mu$ for $U = 6$ and $\\chi = 90$. The critical point $\\delta \\mu_c$ is indicated by a dashed line. We see the drop from $m_s > 0$ to $m_s = 0$, indicating the first-order transition from the antiferromagnetic insulator to the paramagnetic metal.}\n\t\\label{fig:ms of metallic phase}\n\\end{figure}\nOnce $\\delta \\mu$ becomes large enough to drive the system metallic, the staggered magnetization $m_s$ immediately drops to a very small value, which is zero within our error tolerance. All other values of $U$ and $\\chi$ behave similarly.  As $\\chi$ increases, the small value of magnetization in the metal decreases further, although the behavior is not monotonic, as shown in Appendix \\ref{sec:phase diagram more plots}. Also, in Appendix \\ref{sec:numerical details} we describe the strategy we use to make sure we do not bias the magnetization of the metallic solution with our ansatzes.\n\\\\\n\\indent \nIt is interesting to compare our results on the $z=3$ Bethe lattice to those established for the doping-driven MIT \nin the $z=\\infty$ limit where DMFT becomes exact. \nOnly a few studies~\\cite{camjayi_rozenberg_2006,wang_millis_2009,fratino_tremblay_prb_2017} consider this transition while also taking into account phases with magnetic long-range order. As in our results, an antiferromagnetic insulator is found at half-filling for a range of chemical potentials, as well as a non-magnetic metallic solution which can be stabilized for values of the chemical potential above a spinodal value $\\delta \\mu_{2}$. \nFurthermore, a magnetic metallic solution is found to exist in a narrow range of chemical potentials, which connects the magnetic insulator and the non-magnetic metal. \nThis may appear to differ from our findings, but it should be emphasized that all these studies consider only non-zero temperatures. As temperature is lowered, it is reported in Refs~\\cite{camjayi_rozenberg_2006,wang_millis_2009} that the magnetic metallic solution appears to exist only in an increasingly narrow interval of chemical potentials, and Ref.\\cite{camjayi_rozenberg_2006} suggested that at low temperature the MIT is a first-order transition between the magnetic insulator and the non-magnetic metal, with a forbidden range of density corresponding to phase separation. Although, to the best of our knowledge, this has not yet been fully established directly at $T=0$ for $z=\\infty$, this conclusion is consistent with our findings on the finite coordination number lattice. In contrast, on fully frustrated $z=\\infty$ lattices, which do not allow for long-range magnetic order (e.g. on the fully connected lattice with random hopping), it is established that the \ndoping-driven MIT is second order at $T=0$ and becomes first-order only at finite temperature~\\cite{RevModPhys.68.13,moeller_1995,kotliar_2002,werner_2007}. \n\\\\\n\\indent \nTo conclude this section, we mention briefly the numerical accuracy of the data presented here. As noted in the Introduction, the numerical accuracy in tensor networks is generally measured by the truncation error, denoted by $\\epsilon_{\\rho}$. We show here the scaling of $\\epsilon_{\\rho}$ in the metallic state, which is the state with the largest entanglement and therefore the most computationally challenging for tensor networks. \nIn Fig. \\ref{fig:truncation error metal n = 1.2} we plot our estimate of $\\epsilon_\\rho$ at a fixed density of $\\<n_i\\> = 1.2$ as a function of $\\chi$ for various $U$, and also as a function of $U$ at the largest values $\\chi = 90,100$.\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{truncation_error_all_U_vs_chi_metal_n_1_2}\n          \\caption{}\n        \\end{subfigure}\\hspace{0.01\\textwidth}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{truncation_error_at_chi_90_and_100_vs_U_metal_at_n_1_2}\n          \\caption{}\n        \\end{subfigure}\n\t\\caption{Our estimate for the truncation error $\\epsilon_\\rho$ in the metallic phase for $\\<n_i\\> = 1.2$ (a) as a function of $\\chi$ and (b) as a function of $U$ for the two largest $\\chi$. Note that since (a) is a log-log plot the linear form indicates an algebraic relationship.}\n\t\\label{fig:truncation error metal n = 1.2}\n\\end{figure}\nWe can see that the decay with $\\chi$ is algebraic, as expected for a gapless system. As a function of $U$, $\\epsilon_\\rho$ increases initially and then potentially saturates, although the large $U$ behavior is undetermined. Notably, we can see that $\\epsilon_\\rho < 10^{-3}$ for all $U$ and $\\chi = 100$, which is a high level of accuracy. We discuss the behavior of $\\epsilon_\\rho$ at other points in the phase diagram in Appendix \\ref{sec:numerical details}. \n\n\n\n\n\n\n\n\n\n\\section{Quasiparticles}\n\\label{sec:quasiparticles}\n\nIn this section we address the existence of quasiparticles in the system. We do this by computing the quasiparticle weight $Z$ from the ``momentum\" distribution function of the ground state, which in turn is obtained from real-space correlation functions. \nOne peculiarity of the Bethe lattice, is that correlations between any two degrees of freedom sitting on individual nodes of the lattice have a maximal finite correlation length, even at criticality, due to the geometry of the lattice. However, algebraically-decaying correlations reappear after a change of basis to single-particle states that are weighted sums of all nodes of a given generation emanating from a chosen center site. These bases of states unveil the traditional long-range criticality present in gapless states on the Bethe lattice. Below, we introduce a subset of these weighted states called the \\emph{symmetric} states, which we focus on in the rest of this section. From these symmetric states we define a quantum number which plays an analogous role on the Bethe lattice to quasi-momentum on the hypercubic lattice, despite the absence of conventional translation invariance.\n\n\n\\subsection{Single-particle basis of symmetric states}\n\nFor $U = 0$, the free particle Hamiltonian was diagonalized in Ref. \\cite{PhysRevB.63.155110}, using the \\emph{symmetric} set of single-particle states. These are given as follows. Choose any site to be labeled as the origin, with site label $0$. Then consider all permutations of the nodes at each generation $l$ from the center. The symmetric states are those which are invariant under all such permutations. Their creation operators are given by\n\\begin{equation}\n\\tilde{c}^{\\dagger}_{0,\\sigma} \\; \\equiv \\; c^{\\dag}_{0,\\sigma}\n\\label{eq:single particle states l 0}\n\\end{equation}\nand \n\\begin{equation}\n\\tilde{c}^{\\dagger}_{l,\\sigma} \\; \\equiv \\; \\frac{1}{\\sqrt{z(z-1)^{l-1}}} \\displaystyle\\sum_{\\eta_1 = 0}^{z-1} \\; \\displaystyle\\sum_{\\eta_2 \\neq \\eta_1 } \\dots \\displaystyle\\sum_{\\eta_l \\neq \\eta_{l-1}} c^{\\dag}_{\\eta_1 + \\eta_2 + \\dots + \\eta_l,\\sigma}\n\\label{eq:single particle states}\n\\end{equation}\nfor $l > 0$. The collection of $\\eta_i$ denotes a unique path from the origin to the $l$-th generation (this is the usual notation for nodes on the Bethe lattice). The state $\\tilde{c}^{\\dagger}_{l,\\sigma} |\\text{vacuum}\\>$ is the symmetric combination of all the singly-occupied spin-$\\sigma$ states of the $l$-th generation of the tree. These states form an orthonormal subset of all the states on the Bethe lattice, but for $U = 0$ they are the only relevant ones. In the symmetric state basis, the free particle Hamiltonian maps onto fermions hopping on an infinite half-chain, with the first hopping amplitude equal to $\\sqrt{z}$ and all the rest equal to $\\sqrt{z-1}$ (remember $t$ has been set to one). The conjugate variable that replaces momentum is an angle $\\theta \\in \\[0,\\pi\\]$, and the band energy is given by $\\epsilon(\\theta) = 2 \\sqrt{z-1} \\, \\cos\\theta$. Note that the energy of a regular one-dimensional band is obtained by replacing $\\theta$ with a momentum $k$ and $z$ with $2$. The single-particle wavefunctions $\\psi_l(\\theta)$ that diagonalize the Hamiltonian are given by\n\\begin{align}\n\\begin{split}\n\\psi_0(\\theta) &= \\sqrt{\\frac{2}{\\pi}} \\frac{\\sqrt{z(z-1)}\\sin(\\theta)}{\\sqrt{z^2 - 4(z-1) \\cos^2(\\theta)}},\n\\\\ \n\\psi_{l \\neq 0}(\\theta) &= \\sqrt{\\frac{2}{\\pi}} \\sin(l \\cdot \\theta + \\gamma(z,\\theta)),\n\\\\\n\\gamma(z,\\theta) &= \n\\begin{cases}\n\\arcsin\\left(\\frac{z \\sin (\\theta )}{\\sqrt{z^2-4 (z-1) \\cos ^2(\\theta )}}\\right),        & \\hspace{-2mm} \\theta \\in \\[0,\\frac{\\pi}{2}\\) \\\\\n\\pi -\\arcsin\\left(\\frac{z \\sin (\\theta )}{\\sqrt{z^2-4 (z-1) \\cos ^2(\\theta)}}\\right),        & \\hspace{-2mm} \\theta \\in \\[\\frac{\\pi}{2},\\pi\\].\n\\end{cases}\n\\end{split}\n\\label{eq:real space wavefunctions n U=0 exact}\n\\end{align}\n\\\\\n\\indent \nOnce $U \\neq 0$, the symmetric states are no longer enough to describe the system. Indeed, if we simply consider a Mott-like state with one particle sitting on each of the sites at generation $l = 1$ away from the center site, that state cannot be written using only the single-particle symmetric states associated with the same center site. \nIn general, an arbitrary multi-particle Fock state cannot be constructed as a tensor product of the symmetric single-particle states. Therefore, to construct it one must employ states from other symmetry sectors. The interacting ground states we find numerically in this work therefore contain states from various symmetry sectors. However, excitations above the ground state can occur in any of these sectors, and we do not have to consider all of them. In order to tractably answer the question of existence of quasiparticles, we choose to focus on excitations in the symmetric sector. How quasiparticles in different symmetry sectors are related to each other is an interesting question for future work \\cite{Eckstein_thanks}.\n\n\n\\subsection{$\\theta$-distribution function for $U = 0$}\n\nThe $\\theta$-distribution function is calculated from the equal-time correlation functions of symmetric single-particle excitations, $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma} \\>$, where $0$ labels a chosen center site and $l$ labels the generation away from the center site. These can be computed as\n\\begin{equation}\n\\begin{split}\n\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\> \n& = \\int\\displaylimits_{- \\infty}^{0} d{\\omega} ~ \\mathcal{A}_{0l}(\\omega)\n\\\\ & = \\int\\displaylimits_{-2 \\sqrt{z-1}}^{\\mu} d\\varepsilon \\; \\sqrt{\\frac{z}{2 \\pi}} \\frac{\\psi_l\\(\\arccos(-\\frac{\\varepsilon}{2 \\sqrt{z-1}})\\)}{\\sqrt{z^2 - \\varepsilon^2}},\n\\end{split}\n\\label{eq:correlation function U=0 exact}\n\\end{equation}\nwhere $\\mathcal{A}_{0l}(\\omega)$ is the probability of inserting an electron with frequency ${\\omega}$ at the center site and observing it at generation $l$ at the same frequency. The occupation function in $\\theta$-space is defined as\n\\begin{equation}\nn_{\\sigma}(\\theta, \\theta') \\equiv \\< \\tilde{c}_{\\theta,\\sigma}^{\\dag} \\tilde{c}_{\\theta',\\sigma}\\>,\n\\label{eq:occupation function definition}\n\\end{equation}\nwhere \n\\begin{equation}\n\\td c_{\\theta, \\sigma} = \\lim\\limits_{L \\to \\infty}\\sqrt{\\frac{\\pi}{L+1}} \\sum_{d = 0}^{L} \\psi_d(\\theta) \\, \\td c_{d,\\sigma} \n\\label{eq:theta transform c operators}\n\\end{equation}\nare the $\\theta$-transforms of the symmetric state operators $\\td c_{d,\\sigma}$. The key difference with the usual calculation in hypercubic lattices is that $\\< \\td c^{\\dag}_{d,\\sigma} \\td c_{d',\\sigma}\\>$ is not simply a function of $\\abs{d-d'}$, due to the fact that the symmetric states are defined relative to a chosen center site. This is illustrated in Fig. \\ref{fig:BL 4 generations}, where we show that $\\< \\td c^{\\dag}_{2,\\sigma} \\td c_{4,\\sigma}\\>$ contains correlations of length $2,4$ and $6$.\n\\begin{figure}\n\t\\includegraphics[scale=0.4]{BL_large_4_gen_colored}\n\t\\caption{Four generations of the $z = 3$ infinite Bethe lattice emanating from a given center site. The inner shell is at generation $d = 2$, and the outer shell is at generation $d' = 4$. Choosing a given site on the inner shell (colored red), there are three different groups of sites on the outer shell (colored orange, green and purple) that contribute different-length correlations.}\n\t\\label{fig:BL 4 generations}\n\\end{figure}\nTherefore, $n_{\\sigma}(\\theta, \\theta')$ is not diagonal in $\\theta$. One way to understand this is to think of the entire symmetric sector parametrized by $\\theta$ as the $k = 0$ space on the hypercubic lattice, i.e. the one that is fully symmetric under translations. However, within this sector there is no additional symmetry of the Bethe lattice that requires the total $\\theta$ to be conserved in a scattering process, and therefore $n_{\\sigma}(\\theta, \\theta')$ is not diagonal \\cite{Eckstein_thanks}. Carefully inserting Eq. (\\ref{eq:theta transform c operators}) into Eq. (\\ref{eq:occupation function definition}) and rewriting the result in terms of $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ gives\n\\begin{equation}\n\\begin{split}\n& n_{\\sigma}(\\theta, \\theta') = \n\\lim\\limits_{L \\to \\infty}\\frac{\\pi}{L+1} \\sum_{d,d' = 0}^{L} \\psi_d(\\theta) \\psi_{d'}(\\theta') \\, \n\\frac{z-2}{\\sqrt{z (z-1)}}\n\\\\ &\n\\(\\frac{\\sqrt{z}(z-2)}{(z-1)^{3\/2}}\\)^{\\delta_{\\min(d,d'),0}} \n\\sum_{r=0}^{\\min(d,d')} \\(\\frac{z-1}{z-2}\\)^{\\delta_{r,0} + \\delta_{r,\\min(d,d')}} \n\\\\ & \n\\sqrt{\\frac{z}{z-1}}^{\\delta_{d,d'} \\delta_{r,0} - \\delta_{d,0} \\delta_{d',0}} \n\\, \\< \\td c^{\\dag}_{0,\\sigma} \\td c_{\\abs{d-d'}+2r,\\sigma} \\>. \n\\end{split}\n\\label{eq:n_theta final}\n\\end{equation}\nDetails of the derivation of Eq. (\\ref{eq:n_theta final}) are in Appendix \\ref{sec:occupation function}. We plot the exact $n_{\\sigma}(\\theta, \\theta')$ for $U = 0$ at half-filling in Fig. \\ref{fig:n_theta_theta_prime_U_0_delmu_0_exact}.\n\\begin{figure}\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.9]{n_theta_theta_prime_U_0_n_i_1_exact}\n          \\caption{}\n\t  \\label{fig:n_theta_theta_prime_U_0_delmu_0_exact}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_U_0_exact}\n          \\caption{}\n          \\label{fig:n_theta_U_0_exact}\n        \\end{subfigure}\n\t\\caption{(a) The occupation function $n_{\\sigma}(\\theta, \\theta')$ for the half-filled case at $U = 0$. Aside from the expected step-function along the diagonal, there is non-trivial off-diagonal structure. (b) The diagonal component $n_{\\sigma}(\\theta)$ for $U = 0$ and densities $\\<n_i\\> = 1,1.2$, corresponding to values of $\\theta_F = \\pi\/2$ and $\\theta_F \\approx 1.81$, respectively. Calculating the occupation from Eq. (\\ref{eq:n_theta final}) requires a large distance cutoff $L$, which is chosen here to be (a) $L = 100$ and (b) $L = 200$. This introduces an artificial correlation length, causing the step function to be (slightly) smoothed out.}\n\t\\label{fig:n_2D_and_1D_U_0_exact}\n\\end{figure}\n\\\\\n\\indent\nIn this work, we focus exclusively on the diagonal component of the occupation function, $n_{\\sigma}(\\theta) \\equiv n_{\\sigma}(\\theta, \\theta)$. This tells us the occupation of excitations that preserve total $\\theta$ when scattering with each other. We leave the detailed study of the full occupation function $n_{\\sigma}(\\theta, \\theta')$ for future work. We plot the exact $n_{\\sigma}(\\theta)$ in Fig. \\ref{fig:n_theta_U_0_exact} for $U = 0$ and densities $\\<n_i\\> = 1, 1.2$. We can see the expected behavior of $n_{\\sigma} = \\Theta(\\theta_F - \\theta)$, where $\\Theta(x)$ is the Heaviside step function and $\\theta_F$ is the $\\theta$-analog of the ``Fermi momentum\" (the step function in Fig. \\ref{fig:n_theta_U_0_exact} is slightly smoothed out due to a finite $L$ in Eq. (\\ref{eq:n_theta final})).\n\\\\\n\\indent \nIn order to compute the correlation function $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ from the VUTS numerical solution, \nwe use the fact that even in the interacting ground state, $\\< c^{\\dag}_{i,\\sigma} c_{j,\\sigma} \\>$ is still only a function of the distance $\\abs{i-j}$ (and $\\sigma$). We can then compute $\\< c^{\\dag}_{0,\\sigma} c_{l,\\sigma} \\>$ for an arbitrary branch to write\n\\begin{equation}\n\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>\n= \\sqrt{z^{1-\\delta_{l,0}} (z-1)^{l-1 + \\delta_{l,0}}} \\< c^{\\dag}_{0,\\sigma} c_{l,\\sigma}\\>. \n\\label{eq:correlation function symmetric states}\n\\end{equation}\nNote that the difference here from the previous considerations of this paragraph is that one of the reference points has been set to the center site. We plot $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ measured with VUTS in Fig. \\ref{fig:corr function U=0 real space} for $U = 0$ and densities $\\< n_i \\> = 1, 1.2$, along with the exact results. \n\\begin{figure}[!htbp]\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{corr_func_U_0_n_1}\n          \\caption{$U = 0, \\<n_i\\> = 1$.}\n          \\label{fig:corr function U=0 n = 1}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{corr_func_U_0_n_1_2}\n          \\caption{$U = 0, \\<n_i\\> = 1.2$.}\n          \\label{fig:corr function U=0 n = 1.2}\n        \\end{subfigure}\n\t\\begin{subfigure}[c]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{xi_vs_chi_U_0}\n          \\caption{}\n          \\label{fig:xi vs chi}\n        \\end{subfigure}\n\t\\caption{The correlation function $\\< \\td c^{\\dag}_{0,\\uparrow} \\td c_{r,\\uparrow}\\>$ ($\\sigma = \\downarrow$ gives the same) for densities (a) $\\<n_i\\> = 1$ and (b) $\\<n_i\\> = 1.2$ for the non-interacting case $U=0$. We show the $\\chi = 50,100$ results along with the exact solution of Eq. (\\ref{eq:correlation function U=0 exact}). Also shown are the functions $\\frac{1}{r} \\, \\psi_r(\\theta_F) \\, \\psi_0(\\theta_F)$ with (a) $\\theta_F = \\pi\/2$ and (b) $\\theta_F \\approx 1.81$, which are excellent fits beyond a short distance scale, showing Friedel oscillations due to the Fermi surface singularity. The insets show that the $1\/r$ decay is fit perfectly over a larger distance by the exact solution, while the finite $\\chi$ solutions display an exponential decay at large distances.\n\t\t(c) The correlation length extracted from the long distance behavior of $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{r,\\sigma}\\>$, plotted versus $1\/\\chi$. The slope for both densities is $\\xi(\\chi) \\sim \\chi^{0.84}$.}\n\t\\label{fig:corr function U=0 real space}\n\\end{figure}\nThe finite $\\chi$ results are close to the exact ones, but, as expected, the correlation function at large enough distances decays exponentially with a finite correlation length $\\xi(\\chi)$. We can measure $\\xi(\\chi)$ by fitting $\\< \\td c^{\\dag}_{0,\\sigma} \\td c_{l,\\sigma}\\>$ to an exponential at large $l$. The results are shown in Fig. \\ref{fig:xi vs chi}, where the polynomial fit gives $\\xi(\\chi) \\sim \\chi^{0.84}$ for both densities.\n\\\\\n\\indent \nWhen calculating $n_{\\sigma}(\\theta)$ from Eq. (\\ref{eq:n_theta final}), in practice we must choose a finite value of $L$. As long as we take $L$ large enough, the correlation length $\\xi(\\chi)$ will act as the long-distance cutoff and the value of $L$ will not have any effect. Our results are obtained using bond dimensions up to $\\chi = 100$, for which the induced correlation length is $\\xi(\\chi) \\lesssim 40$. We find that a value of $L \\sim 400$ is large enough for all $\\chi$ we study. The finite $\\xi(\\chi)$ smoothes out the step function in $n_\\sigma(\\theta)$, so we estimate $\\theta_F$ from the location of the maximum of $\\abs{n'_{\\sigma}(\\theta)}$. In Fig. \\ref{fig:n_theta near theta_F at U=0 and n=1.2} we show $n_{\\sigma}(\\theta)$ for $\\<n_i\\> = 1.2$ near $\\theta_F$. \n\\begin{figure}\n\t\\includegraphics[scale=0.8]{n_theta_near_theta_F_U_0_n_1_2}\n\t\\caption{Plots of $n_{\\sigma}(\\theta)$ near $\\theta_F$ for $U = 0$ at density $\\<n_i\\> = 1.2$ and a range of $\\chi$. The value of $L$ used here from Eq. (\\ref{eq:n_theta final}) is $L = 400$.}\n\t\\label{fig:n_theta near theta_F at U=0 and n=1.2}\n\\end{figure}\nThe finite slope at $\\theta_F$ diverges as a power law in $\\chi$, as we show in Fig. \\ref{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}. \n\\begin{figure}\n\t\\includegraphics[scale=0.9]{nprime_theta_at_theta_F_vs_chi_all_U_n_1_2}\n\t\\caption{The value of $\\abs{n'_{\\sigma}(\\theta_F)}$ vs $1\/\\chi$ for various $U$ at density $\\<n_i\\> = 1.2$. The straight lines on the log-log plot are fit by $\\abs{n'_{\\sigma}(\\theta_F)} \\sim \\chi ^{{\\alpha}}$, with $0.5 < {\\alpha} < 1$.}\n\t\\label{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}\n\\end{figure}\nThis indicates that $\\xi(\\chi)$ is the only low-energy scale in the problem, and the state is truly gapless in the $\\chi \\to \\infty$ limit.\n\\\\\n\\indent\nIn summary, in this section we show how to compute the diagonal part of the occupation function in the non-interacting case, using VUTS and \na careful extrapolation in the bond dimension, and achieve excellent agreement with the analytical result.\n\n\n\n\\subsection{Quasiparticles in the interacting system}\n\nNow we turn on interactions. In Fig. \\ref{fig:n_theta near theta_F all U and n=1.2} we plot $n_{\\sigma}(\\theta)$ near $\\theta_F$ for $\\<n_i\\> = 1.2$ and $U = 5$ at various $\\chi$ as well as for various $U$ at $\\chi = 70$.\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_near_theta_F_U_5_n_1_2}\n          \\caption{$\\<n_i\\> = 1.2, U = 5$.}\n        \\end{subfigure}\\hspace{0.01\\textwidth}%\n\t\\begin{subfigure}[]{0.49\\textwidth}\n          \\includegraphics[scale=0.8]{n_theta_near_theta_F_all_U_n_1_2}\n          \\caption{$\\<n_i\\> = 1.2, \\chi = 70$.}\n        \\end{subfigure}\n\t\\caption{Plots of $n_{\\sigma}(\\theta)$ near $\\theta_F$ at density $\\<n_i\\> = 1.2$, showing the dependence on (a) $\\chi$ for fixed $U = 5$, and on (b) $U$ for fixed $\\chi = 70$. The value of $L$ from Eq. (\\ref{eq:n_theta final}) is $L = 400$.}\n\t\\label{fig:n_theta near theta_F all U and n=1.2}\n\\end{figure}\nThe occupation shows a form similar to that of the free case, albeit with a reduced size of the step at $\\theta_F$. The value of the slope at $\\theta_F$ diverges for all $U$, as we show in Fig. \\ref{fig:nprime_theta_at_theta_F_vs_chi_all_U_and_n1_2}. The interacting system is therefore also gapless in the $\\chi \\to \\infty$ limit, as expected.\n\\\\\n\\indent \nThe quasiparticle weight, $Z$, of the symmetric state excitations is defined as \n\\begin{equation}\nZ \\equiv \\(\\lim\\limits_{\\theta \\to \\theta_F^-} - \\lim\\limits_{\\theta \\to \\theta_F^+} \\) n_{\\sigma}(\\theta).\n\\label{eq:Z definition}\n\\end{equation}\nFor a Fermi liquid $n_{\\sigma}(\\theta)$ has a step at $\\theta_F$ and $Z > 0$, while for a Luttinger liquid the occupation function has a higher order non-analyticity that scales as $n_{\\sigma}(\\theta - \\theta_F) \\sim \\abs{\\theta - \\theta_F}^{\\gamma} \\sign(\\theta - \\theta_F)$ for some $\\gamma < 1$ and therefore $Z = 0$. Our goal is to find the true thermodynamic value of $Z$ to distinguish between these two scenarios. Of course, for a finite $\\chi$ Eq. (\\ref{eq:Z definition}) will always give zero. However, we can define a quantity $Z(\\chi)$ whose limit will give $Z$ in the $\\chi \\to \\infty$ limit. We define this as\n\\begin{equation}\nZ(\\chi) \\equiv n_{\\sigma}\\(\\theta_F - \\frac{\\pi}{2 \\, \\xi (\\chi)} \\) - n_{\\sigma}\\(\\theta_F + \\frac{\\pi}{2 \\, \\xi (\\chi)} \\),\n\\label{eq:Z finite chi definition}\n\\end{equation}\nwhich satisfies the desired property because $\\xi(\\chi)\\rightarrow \\infty$ with increasing $\\chi$. We choose a spacing of $\\Delta \\theta = \\pi\/\\xi(\\chi)$ around $\\theta_F$ because that is roughly the resolution one expects from a finite correlation length, and therefore the convergence in $1\/\\chi$ should be fastest. We plot $Z(\\chi)$ vs $\\chi$ for $\\<n_i\\> = 1.2$ and a range of $U$ in Fig. \\ref{fig:Z vs chi for all U at n=1.2}. \n\\begin{figure}\n\t\\includegraphics[scale=0.9]{Z_vs_chi_all_U_n_1_2}\n\t\\caption{$Z(\\chi)$ at density $\\<n_i\\> = 1.2$, shown with linear fits.}\n\t\\label{fig:Z vs chi for all U at n=1.2}\n\\end{figure}\nThe results show that the extrapolated $Z$ is (a) very close to the expected value of $Z = 1$ for the free theory and (b) finite for all $U$ we study. We plot $Z$ as a function of $U$ in Fig. \\ref{fig:Z vs U for all n}, where we can see that it decreases as a function of $U$, but seems to saturate to a finite value. The saturation value is an increasing function of doping $\\<n_i\\> - 1$, as \nillustrated by Fig.~\\ref{fig:Z at U =20 vs n-1} where we plot the value of $Z(U=20)$ as a function \nof doping. \n\n\\begin{figure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n    \t\\includegraphics[scale=0.8]{Z_vs_U_all_n}\n        \\caption{}\n\t    \\label{fig:Z vs U for all n}\n        \\end{subfigure}\n\t\\begin{subfigure}[]{0.49\\textwidth}\n    \t\\includegraphics[scale=0.8]{Z_U_20_vs_n_minus_1}\n    \t\\caption{}\n\t    \\label{fig:Z at U =20 vs n-1}\n        \\end{subfigure}\n    \t\\caption{(a) The extrapolated values of $Z$ at densities $\\<n_i\\> = 1.2,1.3,1.4$, plotted as a function of $U$. The first two blue points are covered by the orange ones. The error bars, which are the discrepancy in the extrapolation with and without the last data point, are smaller than the data points. We can see that the curves are close to saturation at $U = 20$. (b) The values of $Z$ at $U = 20$, which seem close to the saturated values, as a function of $\\<n_i\\> - 1$.}\n\\end{figure}\nWe also address the question of Luttinger's theorem for $n_{\\sigma}(\\theta)$. We check that $\\theta_F$ is independent of $U$ (the dependence on $\\chi$ is negligible) for various densities  in the range $\\<n_i\\> \\in (1.1,1.4)$. From this we conclude that Luttinger's theorem holds for all values of the density and interaction strength. \n\\\\\n\\indent\nThe decrease of $Z$ with increasing $U$ as well as the saturation at large $U$ to a value which increases with doping are both qualitatively consistent with results established in the $z=\\infty$ (DMFT) limit and with slave boson approaches (for general lattices) ~\\cite{RevModPhys.68.13,4bosons}.\nA distinctive aspect of these theories, however, is that the effective mass of \nquasiparticles is related to $Z$ by $m^*\/m=1\/Z$. On the technical level, this is due to the locality of the self-energy, while physically this reflects the inability of these approaches to capture the feedback of short-range order and collective modes \ninto the physics of quasiparticles. \nIn contrast, in the present case, since the connectivity is kept finite, we would expect this feedback to be present. \nIt is therefore an outstanding question for future work to explore whether the dispersion of quasiparticles is \nrenormalized in a  different manner than $Z$ itself, and in particular whether it is affected by \nshort-range antiferromagnetic correlations at low doping level. This is left for future work since it requires \nan extension of our algorithm to the study of excited states.\n\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn summary, we have introduced a new numerical algorithm,\n(fermionic) VUTS, to study quantum (fermionic) models on the Bethe lattice. \nWe apply it to the Hubbard model for coordination number $z = 3$, allowing for a two-site unit cell, obtain the $T = 0$ phase diagram and study the doping-induced Mott transition.   \nWe find an antiferromagnetic insulating phase at half filling and a paramagnetic metallic phase for the doped system, which are separated by a first-order insulator to metal phase transition. \nThe model displays phase separation at low doping, with a range of forbidden densities. These conclusions were reached by allowing for a two-site unit cell. We cannot exclude that phases with more complex charge or \nmagnetic ordering exist when allowing for a larger unit cell, which we leave as an open question for future work. By studying the diagonal component of the occupation function for momenta of the symmetric single-particle sector, we find that the quasiparticle weight is non-zero, \nconsistent with the existence of a Fermi liquid ground state for fermions on the Bethe lattice. We find that this Fermi liquid state obeys Luttinger's theorem.\n\nAn interesting direction in which to extend this work would be to further characterize this Fermi liquid state. \nOne would like to know, for example, what happens near $\\theta_F$ to the off-diagonal $n_\\sigma(\\theta,\\theta')$ when interactions are turned on. \nIt is also interesting to look at some of the other symmetry sectors, say the ones that leave each of the $z$ sub-trees connected to the center site invariant, \nand see whether they have quasiparticles and how the quasiparticle weight depends on the sector.\nThese questions are the Bethe lattice version of the important physical question of `momentum dependence' of quasiparticle properties on the Fermi surface of hypercubic lattices.\n\nIn the one-dimensional VUMPS algorithm, it has been shown that low-lying excitations above the ground-state can be accurately computed~\\cite{PhysRevB.97.235155}. \nAn obvious question is how to extend these ideas to VUTS (this is another potential advantage of this method over imaginary time evolution).\nExtension of the algorithm to excited states would allow to characterize the effective mass (dispersion) of the quasiparticles, paving the road for a study of \nhow short-range (e.g. antiferromagnetic) correlations affect quasiparticle properties. This is a very important question for the physics of strongly correlated electron systems. \nA breakdown of the $m^*\/m=1\/Z$ relation would signal that this feedback is indeed present, in contrast to the infinite connectivity limit. \nFinally, studying the energy dependence of the quasiparticle lifetime, as well as the interactions between quasiparticles would be a comprehensive study of Landau \nFermi liquid theory on the Bethe lattice. \n\nIt would also be interesting to look at the entanglement structure of the Fermi liquid state, as compared to a Luttinger liquid. Since the entanglement spectrum is easily obtained on the one-dimensional and Bethe lattices from the singular value decomposition of a single bond tensor, this question could in principle be easily answered. \n\nWe also propose that the finite $z$ Bethe lattice can be used as a computationally tractable platform for the study of how quasiparticles can be destroyed \nand Fermi liquid behaviour breaks down when considering other fermionic hamiltonians on this lattice. \nOne route to explore this, which connects to the feedback of long-wavelength collective modes or short-range spatial correlations on quasiparticle properties, is to study the vicinity of a quantum critical point. \nAnother route is to study microscopic models that are tailor-engineered to have incoherent excitations, such as the one of Ref.~\\cite{PhysRevB.86.045128}, or the multi-channel Kondo lattice.     \n\nStudying the interplay between frustration and strong correlations is another promising direction for future research. \nIntroducing a frustrating next-nearest-neighbor hopping term has been found with DMFT to yield an interaction-driven metal-insulator transition on the infinite $z$ Bethe lattice~\\cite{rozenberg1994,RevModPhys.68.13}. An interesting question is whether this transition can also be found at finite $z$. \nThe VUTS algorithm could also be extended to other lattices with a tree-like structure, such as the Husimi cactus. \nThe study of spin models on such lattices have revealed spin-liquid ground-states\\cite{Chandra_1994,PhysRevB.93.075154} (see also Ref.~\\cite{Udagawa_2019} for considerations on the spin-ice model), \nopening the question of how these models behave upon doping. \n\nFinally, increasing the temperature to a non-zero value is another interesting direction. Intuitively, some finite-temperature properties may be less sensitive to the differences between tree lattices and two- or three-dimensional hypercubic lattices. This could be done using the purification method \\cite{PhysRevLett.93.207205,PhysRevLett.93.207204,PhysRevB.72.220401} which has been formulated on the Bethe lattice in Ref. \\cite{PhysRevB.100.125121}.\n\n\n\n\n\n\\section*{Acknowledgments}\n\nWe thank Julien Agier, Martin Claassen, Michel Ferrero, Gabriel Kotliar, Chris Laumann, Sung-Sik Lee, Roderich Moessner, Marcelo Rozenberg, Steve White and in particular Martin Eckstein and Riccardo Rossi for useful discussions. We thank Ruben Verresen for comments on the manuscript. All tensor network calculations and the VUTS code were implemented with the ITensor Library (C++ version 3.1) \\cite{itensor}. The Flatiron Institute is a division of the Simons Foundation.\n\n\n\n\n\\bibliographystyle{apsrev4-1}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe ATLAS experiment \\cite{ATLAS} at CERN was equipped with the Insertable B-Layer (IBL) \\cite{IBL} between the existing inner pixel layer and a new beam pipe during the long shutdown one to improve the tracking performance. The planar and 3D sensors of the IBL are exposed to a high flux of ionizing radiation because of the close position to the interaction point. The planar sensors are designed to withstand a fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^{2}$. They are produced in n$^+$-in-n sensor technology and have a pixel pitch of $250\\,$\\textmu m $\\times$ $50\\,$\\textmu m \\cite{sensor1, sensor2}. The 80 columns and 336 rows of pixels are read out with the Front-End-I4 readout chip \\cite{FE-I4}.\n\nWhile operating the detector, the radiation damage of the sensors will increase their leakage current and depletion voltage, whereas the charge collection efficiency decreases. In order to continue to obtain a high particle detection efficiency, the bias voltage needs to be increased gradually. Voltages up to $1000\\,$V can be applied to the sensors causing an increase in power consumption. The higher power consumption produces heat which needs to be dissipated by the detectors cooling system to prevent thermal runaway of the sensors' leakage current. Therefore, a higher detection efficiency at lower bias voltage is desirable.\n\nNew pixel implantation shapes were designed in Dortmund to achieve electrical field strength maxima in the pixel and thus increase charge collection and particle detection efficiency at lower voltages after irradiation \\cite{REINER}.\n\nIn the proceeding \\emph{Lab and test beam results of irradiated silicon sensors with modified {ATLAS} pixel implantations} \\cite{meinPaper}, results of these proton and neutron irradiated modules were presented and showed incongruent results. This paper now examines the hypothesis that an annealing process caused the observed differences. \n\n\n\\section{Design of the Pixel Cell}\nThe baseline for the new designs is the IBL pixel design (see figure \\ref{fig:design}, label V0). In the \\mbox{$250\\,$\\textmu m $\\times$ $50$ \\textmu m} pixel cell, the standard n$^+$-implantation is located centrally with rounded corners to create a homogeneous electrical field in the pixel. Moderated p-spray is applied to isolate neighboring pixel cells. In the upper region of the pixel cell the bump bond pad is visible which is the connection to the readout chip. The bias dot with connection to the bias grid is positioned at the other end.\n\nDifferent n$^+$-implantation shapes are realized in the REINER$^2$\\note[2]{\\textbf{RE}designed, \\textbf{IN}novativ, \\textbf{E}xciting and \\textbf{R}ecognizable} pixel designs V1 to V6 (see figure \\ref{fig:design}): For the pixel designs V1 and V4, the n$^+$-implantation is divided in four uniform segments isolated with p-spray. The corners of the n$^+$- implantation of pixel design V1 are rounded, while they are rectangular for pixel design V4. Further division of the n$^+$-implantation are used for pixel designs V2 and V3. No isolation between the segments is used here due to reduced space. A narrowed shape of the metal layer and the n$^+$-implantation are realized in the pixel designs V5 and V6. While in V6 the region for moderated p-spray is the same as for the standard pixel, the section is increased to the inner part of the pixel cell for design V5.\n\nThe connection to the bias dot and bias grid is the same for all pixel designs and causes a loss in efficiency as mentioned in \\cite{effVerlust1, effVerlust2}. This will not be taken into account in this proceeding as it focuses on the influence of the different pixel shapes. \n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.6\\textwidth]{PixelLayoutQuer.png}\n\\caption{\\label{fig:design} Schematic drawing of the pixel structure and the shape of its components. The $\\text{n}^+$-implantation is shown in blue, the metalization in grey. The areas with high p-spray dose correspond to the nitride-openings as indicated in green. The bump bond pad is depicted in orange \\cite{meinPaper}.}\n\\end{figure}\n\n\\section{Sensors and Modules}\nThe REINER pixel sensor permits measurement of all the described pixel types on one sensor. It is produced in n$^+$-in-n sensor technology with a bulk thickness of $200\\,$\\textmu m. The pixel matrix of 80 columns and 336 rows is read out with the FE-I4 readout chip via bump bonds.\n\nThe sensor is divided in eight structures consisting of ten columns and 336 rows of the same pixel design. The outermost structures on the sensor comprise the standard IBL pixel design (labeled as 05 and V0). In between these standard structures, structures with modified pixel designs (V1 to V6) are placed (see figure \\ref{fig:sensor}, left). Each structure has its own p$^+$-implant and two HV pads. All structures are surrounded by thirteen guard rings beneath the second to last pixel column and the last pixel column (see figure \\ref{fig:sensor}, right). In this way, measurements of the structures independent from each other are possible.\n\nTo compare the performance of the different pixel designs, several sensors are bump bonded to FE-I4 readout chips and tested before and after irradiation in irradiation facilities. The results presented in this work are obtained with modules irradiated with neutrons at the TRIGA reactor in Ljubljana \\cite{Ljubljana} and at the Sandia Annular Core Research Reactor$^4$\\note[4]{https:\/\/www.sandia.gov\/research\/facilities\/annular\\_core\\_research\\_reactor.html}. They are irradiated with neutrons to target fluences of $1\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and $5\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=1.0\\textwidth]{P-Seite2.png}\n\\caption{\\label{fig:sensor} Left: P-side of a REINER pixel sensor with six modified structures and two IBL standard structures (05 and V0). Right: Magnification of the area between two structures. The second to last and last column of each structure are beneath guard rings \\cite{meinPaper}.}\n\\end{figure}\n\n\\section{Test Beam Measurements}\nThe sensors are investigated in test beam measurements at the SPS-Beam Line H6 at CERN with a pion beam energy of $120\\,$GeV and at the beam line 22 at DESY \\cite{DESYPaper} with an electron beam energy of $5\\,\\text{GeV}$. With the EUDET-type beam telescopes ACONITE at CERN and DURANTA at DESY the hit detection efficiency can be measured with a spatial resolution of less than $5\\,$\\textmu m (CERN) and $10\\,$\\textmu m (DESY) \\cite{resolution}. A telescope consists of two arms, each equipped with three Mimosa 26 modules \\cite{mimosa26}. In between the two arms, a cooled box holds the Devices Under Test (DUTs). The box is flushed with nitrogen and cooled using a chiller (CERN) or dry ice (DESY). A non-irradiated IBL-type module is used as a timing reference sensor.\n\nWith this setup the REINER modules are investigated at different positions, bias voltages and tunings of the readout chip. For irradiated REINER modules high efficiencies around $97\\,\\%$ can be reached for all pixel designs with a high bias voltage. Therefore, operation with a lower bias voltage where differences in the performance of the pixel designs occur is desirable.\n\nThe traversing particles produce hits in the telescope modules and the DUTs which are reconstructed to tracks using the software EUTelescope$^5$\\note[5]{http:\/\/eutelescope.web.cern.ch\/}. To get unbiased tracks the hit information of the DUTs is not used for track fitting. About $200\\,\\text{k}$ to $500\\,\\text{k}$ events are collected in one \"run\".\n\nThe hit detection efficiency, as one part of the analysis of the performance of the DUTs, is computed with the software tool TBMon2$^6$\\note[6]{https:\/\/gitlab.cern.ch\/tbmon2}.\n\nAll reconstructed particles measured by the reference sensor and not in a masked, edge or noisy region of the investigated sensor are used for efficiency calculation and called \"tracks\". If a \"track\" is also measured by the investigated sensor, it is defined as a \"hit\". The hit detection efficiency of a sensor $\\epsilon$ is calculated by the quotient of \"hits\" and \"tracks\":\n\n\\begin{equation}\n\\epsilon = \\frac{n_\\text{hits}}{n_\\text{tracks}}, \\quad \\sigma_\\epsilon = \\sqrt{\\frac{\\epsilon \\cdot (1-\\epsilon)}{n_\\text{tracks}}}\n\\end{equation}\n\nTo summarize the efficiencies for runs taken under the same conditions the mean efficiency weighted with the number of tracks is calculated. The fluctuation of the efficiency from run to run is determined with the Clopper-Pearson confidence interval \\cite{clopper-pearson} with a confidence level of $\\gamma=95\\,\\%$. For the lower interval limit the $\\frac{1-\\gamma}{2}$-quantile and for the upper limit the $\\frac{1+\\gamma}{2}$-quantile are calculated \\cite{Andreas}.\n\nFor the following section the efficiency is calculated by using only the four innermost pixel columns of every structure to neglect the influence of guard rings.\n\n\\section{Previous Results}\nThe measurements presented in \\cite{meinPaper} showed that for non-irradiated modules the hit detection efficiency for the different pixel designs is consistent. But for modules irradiated with neutrons or protons the hit detection efficiencies are not consistent. Even the two neutron irradiated sensors R1, irradiated at Sandia, and R3, irradiated in Ljubljana, to the same target fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and measured at the same voltage and tuning ($3200\\,$e threshold and a ToT response of $6\\,$ at a reference charge of $20\\,$ke) show dissimilar results (see figure \\ref{fig:bild1}, left). For the module R1, irradiated at Sandia, the pixel designs with narrowed n$^+$-implantation V5 and V6 reach the highest efficiencies at $400\\,$V. No efficiency of the pixel design V0 was measured for module R1 as the beam spot was focused on the left side of the module and did not cover pixel design V0.\n\nFor the Ljubljana irradiated module R3, the pixel design with narrowed n$^+$-implantation V6 has the lowest efficiency while all other pixel designs have similar efficiencies of approx. $50\\,\\%$.\n\nThese diverging results might have been caused by the different neutron energy spectra of the two irradiation facilities while another hypothesis is a different temperature of the modules during irradiation leading to annealing of defects. In Ljubljana the maximum temperature of the modules was $45^\\circ$C while they reached about $100^\\circ$C in the irradiation at Sandia (see figure \\ref{fig:bild1}, right).\n\nTo test the annealing hypothesis, the modules R3 and R9 were irradiated with neutrons in Ljubljana to different target fluences and were now annealed in several steps at $80^\\circ$C.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.48\\textwidth]{R1_R3_comparison_3200_iWORID.png}\n\\includegraphics[width=.48\\textwidth]{TempertureProfileIrradiation_IWORID.png}\n\\caption{\\label{fig:bild1} Left: Efficiencies of the different pixel designs for two sensors irradiated with neutrons to a target fluence of $5\\cdot 10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ at $400\\,$V (Tuning: $3200\\,$e threshold, $6\\,$ToT at $20\\,$ke). R1 is irradiated at Sandia while R3 is irradiated in Ljubljana. Right: Temperatur profile during the neutron irradiation of the module R1 at Sandia.}\n\\end{figure}\n\n\\section{First Annealing Results}\nFor the annealing procedure the climate chamber is preheated to $80^\\circ$C before the module is inserted. The temperature of the module is monitored and it is not biased during the annealing. The cooling down afterwards takes place at room temperature and the module is put back in the freezer for storage. After each annealing step IV scans and test beam measurements are performed at DESY.\n \nTo determine if differences in the efficiency occur with annealing at all, the intention was to anneal a module for a long time. Therefore, the first annealing step of module R3 was three hours at $80^\\circ$C. The following annealing steps lasted for two hours each.\nThe results of the hit detection efficiency for the different pixel designs for module R3 after several annealing steps are presented in figure \\ref{fig:bild2}. For these measurements at $300\\,$V the module was tuned to a threshold of $1600\\,$e and a ToT response of $6\\,$ at a reference charge of $20\\,$ke. The standard pixel designs 05 and V0 reach the highest efficiencies for the non-annealed case while the pixel designs with narrowed n$^+$-implantation V5 and V6 are less efficient compared to all other designs. This behavior appears to be independent on the threshold as figure \\ref{fig:bild1} on the left shows where the sensor was measured at a threshold of $3200\\,$e while in figure \\ref{fig:bild2} the module was tuned to a threshold of $1600\\,$e.\n\nWith the first annealing step of three hours the efficiency drops for all pixel designs except for the designs with a narrowed n$^+$-implantation. Compared to the standard pixel design, the hit detection efficiency drops by $8\\,\\%$ while for pixel design V5 the efficiency increases by more than $12\\,\\%$. After this first annealing step, the efficiency of pixel design V5 is even higher than the non-annealed results of the standard designs. With further annealing steps the efficiency of the standard pixel designs is almost constant, but for pixel designs V5 and V6 where the efficiency increases.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.7\\textwidth]{Efficiency_R3_vergleich_0h_3h_5h_7h_80C_1600e_300V_IWORID.png}\n\\caption{\\label{fig:bild2}\nEfficiencies of the different pixel designs for the sensor R3 after various annealing steps at $300\\,$V (Tuning: $1600\\,$e threshold, $6\\,$ToT at $20\\,$ke).}\n\\end{figure}\n\n\\newpage\n\nThese results confirm the hypothesis that annealing caused the narrowed pixel designs to be more efficient compared to the standard pixel design. To determine if this effect is dependent on the particle fluence and to see when the effect becomes relevant, the module R9 was irradiated with neutrons in Ljubljana to a target fluence of $1\\cdot10^{15}\\,\\text{n}_\\text{eq}\/\\text{cm}^2$ and measured with a shorter annealing step time of five minutes at $80^\\circ$C. The results of the hit detection efficiency for the module R9 at $100\\,$V, tuned to a theshold of $1600\\,$e and a ToT response of $6\\,$ at a reference charge of $20\\,$ke, are presented in \\mbox{figure \\ref{fig:bild3}}.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.7\\textwidth]{Efficiency_R9_vergleich_0min_5min_10min_80C_1600e_100V_IWORID.png}\n\\caption{\\label{fig:bild3} Efficiencies of the different pixel designs for the sensor R9 after various annealing steps at $100\\,$V (Tuning: $1600\\,$e threshold, $6\\,$ToT at $20\\,$ke).}\n\\end{figure}\n\nFor the non-annealed measurement the pixel designs with narrowed n$^+$-implantation have the lowest efficiency. Because the beam spot is focused on the left part of the sensor, there is no data point for the pixel design V0. After the first annealing step an increase in efficiency is visible for the pixel designs with a narrowed n$^+$-implantation and the standard pixel design. However, after the second annealing step the efficiency decreases for all designs. Compared to the non-annealed results, the efficiency after the second annealing step is still higher for the standard pixel design and the designs with narrowed n$^+$-implantation.\n\nIt is known, that annealing performed in a short time scale is able to reduce the effective doping concentration in the sensor material and more signal at the same voltage can be measured. This process is called beneficial annealing. In contrast, reverse annealing degrades sensor properties in the long term. For more information on annealing effects see \\cite{Moll}. Beneficial annealing can explain the higher efficiencies after the first annealing step of the module R9 for the shorter annealing time. But with the second step the reverse annealing becomes dominant and the efficiency drops for all designs. In case of the module R3 only the reverse annealing can be observed because of the long term annealing of three hours at $80^\\circ$C.\n\nThe observed changes of the hit detection efficiency with annealing are caused by at least two different effects. With annealing the depletion voltage and the charge collection efficiency change. Thus, if one of these effects depend on the pixel design, different hit detection efficiencies might be observed. Also charge multiplication could explain the higher efficiencies, which was also observed in \\cite{Milovanoivi,Kramberger1}, where neutron irradiated n-in-p micro-strip sensors showed higher electrical fields and therefore charge multiplication close to the strips.\n\n\\section{Summary and Outlook}\nThe results of the Ljubljana neutron irradiated module R3 indicate that long term annealing caused a higher hit detection efficiency for the pixel designs with narrowed n$^+$-implantation compared to all other designs. As a result of this, the higher detection efficiency observed for pixel designs with narrowed n$^+$-implantation on the neutron irradiated sensor R1 at Sandia (see \\cite{meinPaper}) are explained as being due to the higher temperature during the irradiation.\n\nAs this effect was not visible for the module R9, which was irradiated to a lower fluence and annealed in shorter steps, it is not clear if it depends on the neutron fluence. Therefore, the module R9 needs to be further investigated with additional annealing steps.\n\nFor the module R3 further annealing steps will be performed to find the maximum efficiency of pixel design V5 and further research of the performance after reaching the maximum efficiency.\n\nThe higher hit detection efficiencies might be explained with charge multiplication which is observed in n-in-p micro-strip detectors after neutron irradiation and long term annealing \\cite{Milovanoivi,Kramberger1}. To understand if the higher efficiency is caused by charge amplification, which was the intention of the REINER pixel designs, TCT measurements at different voltages and annealing steps are planned for irradiated modules. With this method the pixel designs can be investigated in the range of \\textmu m to see which pixel parts might cause charge multiplication.\n\n\\acknowledgments\nThe authors would like to thank the team at the Sandia Annular Core Research Reactor, especially M. Hoeferkamp and S. Seidel, and the team at the TRIGA reactor in Ljubljana, especially V. Cindro, for their help with irradiation of the sensors.\n\nMany thanks to all participants of the ATLAS ITk pixel test beam campaigns, especially those who develop and maintain the corresponding hardware and software.\n\nThe presented work is carried out within the framework of Forschungsschwerpunkt FSP 103 and supported by the Bundesministerium f\\\"ur Bildung und Forschung BMBF under grant 05H15PECA9.\n\nThis project has received funding from the European Union's Horizon 2020 Research and Innovation programme under Grant Agreement no. 654168.\n\nThe measurements leading to these results have been performed at the Test Beam Facility at DESY Hamburg (Germany), a member of the Helmholtz Association (HGF).\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\hspace{-0.6cm}.\\hspace{0.3cm}#1}}    %\n\\newcommand{\\zd}{z^{\\delta}}                                         %\n\\newcommand{\\be}{B_{\\Omega}(z;X)}                                    %\n\\newcommand{\\ca}{C_{\\Omega}(z;X)}                                    %\n\\newcommand{\\ko}{K_{\\Omega}(z;X)}                                    %\n\\newcommand{\\tzd}{\\tilde z^{\\delta}}                                 %\n\\newcommand{\\tjd}{\\tilde \\zeta^{\\delta}}                             %\n\\newcommand{\\jd}{\\zeta^{\\delta}}                                     %\n\\newcommand{\\ld}{\\lambda_{\\delta}}                                   %\n\\newcommand{\\tj}{\\tilde \\zeta_1}                                     %\n\\newcommand{\\tz}{\\tilde z_1}                                         %\n\\newcommand{\\Cb}{C_b(z_0,z^{\\delta_0})}                              %\n\\newcommand{\\jdb}{(d\\delta^{\\frac 1\\eta},0,- b\\delta\/2)}             %\n\\newcommand{\\dbar}{\\overline{\\partial}}                              %\n\\newcommand{\\oo}{\\overline \\Omega}                                   %\n\\newcommand{\\ct}{\\Bbb{C}^3}                                          %\n\\newcommand{\\al}{{\\alpha}}                                           %\n\\newcommand{\\ov}{\\overline}                                          %\n\\newcommand{\\dd}{\\partial \\overline{\\partial}}                       %\n\\newcommand{\\de}{\\delta^{\\frac 1\\eta}}                               %\n\\newcommand{\\dee}{\\delta^{-\\frac 2\\eta}}                             %\n\\newcommand{\\bo}{b\\Omega}                                            %\n\\newcommand{\\od}{\\Omega_\\delta}                                      %\n\\newcommand{\\gd}{g_{\\tz,\\delta}}                                     %\n\\newcommand{\\gdd}{g_\\delta}                                          %\n\\newcommand{\\qd}{Q_{\\gamma\\delta}(\\tz)}                              %\n\\newcommand{\\qad}{Q_{a\\delta}(\\tz)}                                  %\n\\newcommand{\\qdd}{Q_{\\gamma\\delta}(\\tzd)}                            %\n\\newcommand{\\rd}{R_{\\gamma\\delta}(\\tz)}                              %\n\\newcommand{\\rbd}{R_{b\\delta}(\\tzd)}                                 %\n\\newcommand{\\qbd}{Q_{b\\delta}(\\tzd)}                                 %\n\\newcommand{\\rdd}{R_{\\gamma\\delta}(\\tjd)}                            %\n\\newcommand{\\rcd}{R_{c\\delta}(\\jd)}                                  %\n\\newcommand{\\ol}{\\overline L}                                        %\n\\newcommand{\\Pd}{\\Phi_{\\tzd}}                                        %\n\\newcommand{\\td}{\\tau(\\tzd,\\delta)}                                  %\n\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lem}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{defn}[thm]{Definition}\n\n\\numberwithin{equation}{section}\n\n\n\\begin{document}\n\n\\title {Necessary conditions for H\\\"older regularity gain of $\\dbar$ equation in  $\\mathbb C^3$}\n\n\\author{Young Hwan You \\thanks{Department of Mathematics, Indiana University East, IN 47374, USA. E-mail:youy@iue.edu\n\\newline 2010 \\textit{Mathematics Subject Classification} Primary 32F45; Secondary 32T25.}}\n\n\\date{}\n\n\\maketitle\n\n\\begin{abstract}\n\\noindent Suppose that a smooth holomorphic curve  $V$ has order of contact $\\eta$ at a point $w_0$ in the boundary of a pseudoconvex domain $\\Omega$ in $\\mathbb{C}^3.$\nWe show that the maximal gain in H\\\"older regularity for solutions of the $\\bar{\\partial}$-equation is at most $\\frac{1}{\\eta}.$ \n\\end{abstract}\n\n\\section{Introduction}\\label{sec1}\n\n\nLet $\\Omega$ be a given domain in $\\mathbb{C}^n$ and $\\alpha$ be a $\\bar{\\partial}$-closed form of type $(0,1)$ in $\\Omega$. The $\\bar{\\partial}$-problem consists of finding a solution $u$ of $\\bar{\\partial} u = \\alpha$ that satisfies certain boundary regularity estimates as measured by  either $L^2$ or $L^p$ norms or in H\\\"older norms.\n\n\nWhen $\\Omega$ is strongly pseudoconvex, in the $L^2$-sense, Kohn \\cite{FK,K1,K2} showed that for any $s\\geq 0$, there is a canonical solution of $\\bar{\\partial}u = \\alpha$ such that \n\\begin{equation}\\label{kohnestimate}\n|||u|||_{s+\\epsilon} \\leq C \\left\\|\\alpha\\right\\|_s \\quad \\mbox{and} \\quad \\ u \\perp A(\\Omega) \\cap L^2(\\Omega),\n\\end{equation}\nwith $\\epsilon = \\frac{1}{2}.$ \n(We say $u$ is the canonical solution if $u \\perp A(\\Omega) \\cap L^2(\\Omega).$)\nHere, $\\left\\|\\cdot\\right\\|^2_s$ is the $L^2$-Sobolev norm of order $s$ and the norm $|||\\cdot |||_{s+\\epsilon}$ measures tangential derivatives near the boundary of order $s+\\epsilon$ in the tangential directions. Kohn showed that if $U$ satisfies\n$\\square U = (\\bar{\\partial}\\bar{\\partial}^* + \\bar{\\partial}^*\\bar{\\partial})U = \\alpha,$ and if $\\bar{\\partial}\\alpha = 0,$ then\n$u = \\bar{\\partial}^*U$ is the canonical solution of $\\bar{\\partial}u = \\alpha.$\nTo prove regularity for this solution, Kohn proved the a priori estimate\n\\begin{equation}\\label{subellipticestimate}\n|||\\phi|||_{\\epsilon}^2 \\leq C(\\left\\|\\bar{\\partial} \\phi \\right\\|^2 + \\left\\|\\bar{\\partial}^* \\phi\\right\\|^2 + \\left\\|\\phi\\right\\|^2)\n\\end{equation}  \nwith $\\epsilon = \\frac{1}{2}.$\nHere, $\\phi \\in C_{(0, 1)}^{\\infty}(W) \\cap \\mbox{Dom}(\\bar{\\partial}) \\cap \\mbox{Dom}(\\bar{\\partial}^*) $ is compactly supported in the neighborhood $W$ of the boundary point $w_0.$ Using this estimate and a bootstrap argument, Kohn proved (\\ref{kohnestimate}). Stein and Greiner \\cite{GS} later extended (\\ref{kohnestimate}) to similar estimates in $L^p$ and H\\\"older spaces. For example,  if $\\left\\|\\cdot\\right\\|_{\\Lambda^s (\\Omega)}$ is the H\\\"older norm of degree $s$, then Stein and Greiner proved that $u$ satisfies \n\\begin{equation} \\label{holder_estimate}\n\\left\\|u\\right\\|_{\\Lambda^{s+\\epsilon}(\\Omega)} \\leq C \\left\\|\\alpha\\right\\|_{\\Lambda^s (\\Omega)},  \n\\end{equation}\nwith $\\epsilon = \\frac{1}{2}.$\n\n Kohn extended his $L^2$ results to when $\\Omega$ is a regular finite 1-type pseudoconvex domain in $\\mathbb{C}^2.$ To define a regular finite 1-type, we measure the order of contact of a given holomorphic curve at $w_0 \\in b\\Omega.$ Let $V$ be a one-dimensional {\\it smooth} variety parametrized by $\\zeta \\rightarrow \\gamma(\\zeta) = (\\gamma_1(\\zeta), \\cdots, \\gamma_n(\\zeta)),$ where $\\gamma(0) = w_0$ and $\\gamma'(0) \\neq 0.$ We define the order of contact of the curve by $\\nu_o(R\\circ\\gamma),$ where $R$ is a defining function of $\\Omega$ and $\\nu_o(g)$ is just the order of vanishing (an integer at least equal to $2$) of $g$ at $0.$ We then define the type, $T_{\\Omega}^{reg}(w_0) = \\sup\\{\\nu_o (R \\circ \\gamma) ;  \\mbox{all} \\ \\gamma \\ \\mbox{with} \\ \\gamma(0) = w_0, \\gamma'(0) \\neq 0\\}.$ Further, we can define the regular type of $\\Omega$ by \n$T^{reg}(\\Omega) = \\sup \\{T_{\\Omega}^{reg}(w_0) ; w_0 \\in b\\Omega\\}.$\n Kohn \\cite{K} proved that if $\\Omega$ is a regular finite 1-type pseudoconvex domain in $\\mathbb{C}^2$, then (\\ref{kohnestimate}) holds for $\\epsilon = \\frac{1}{T^{reg}(\\Omega)}.$ Similarly, Nagel-Rosay-Stein-Wainger \\cite{NRSW} showed that (\\ref{holder_estimate}) also holds for the same $\\epsilon$.\n\n\nIn order to discuss similar estimates in $\\mathbb{C}^n,$ it is important to consider the order of contact of {\\it singular curves}. We define the order of contact of a holomorphic curve parametrized by $\\zeta \\rightarrow \\gamma(\\zeta),$ with $\\gamma(0) = w_0,$ by  $C_{\\Omega}(\\gamma, w_0) = \\frac{\\nu_o (R \\circ \\gamma)}{\\nu_o(\\gamma)},$ where $\\nu_o(\\gamma)=\\min\\{\\nu_o (\\gamma_k);  k =1, \\cdots, n\\}.$ Define the type of point $w_0$ by $T_{\\Omega}(w_0)= \\sup \\{C_{\\Omega}(\\gamma, w_0); \\mbox{all} \\ \\gamma \\ \\mbox{with} \\ \\gamma(0)= w_0 \\}$ and finally, the type of $\\Omega$ is $T_{\\Omega}= \\sup \\{T_{\\Omega}(w_0); w_0 \\in b\\Omega \\}.$ In the case of the $L^2$-norm, Catlin \\cite{C3} showed that if there is a curve $V$ parametrized by $\\gamma$ through $w_0 \\in b\\Omega$, where $\\Omega \\subset \\mathbb{C}^n$ and (\\ref{subellipticestimate}) holds, then $\\epsilon \\leq \\frac{1}{C_{\\Omega}(\\gamma, w_0)}.$ In H\\\"older norms, McNeal \\cite{Mc} proved  that if, with an additional assumption, $\\Omega$ admits a holomorphic support function at $w_0 \\in b\\Omega$ and (\\ref{holder_estimate}) holds, then  $\\epsilon \\leq \\frac{1}{C_{\\Omega}(\\gamma, w_0)}$. \n\n\n\n\nThere is the third notion of type, the ``Bloom-Graham\" type, $T_{BG}(w_0).$ It turns out that $T_{BG}(w_0)$ is the maximal order of contact of smooth $(n-1)$-dimensional complex submanifold. Thus, it follows that for any $w_0 \\in b\\Omega,$  $T_{BG}(w_0) \\leq T_{\\Omega}^{reg}(w_0) \\leq T_{\\Omega}(w_0).$ Krantz \\cite{Kr} showed that if $T_{BG}(w_0) = m,$ then $\\epsilon \\leq \\frac{1}{m}.$  \\\\\n\n\nIn this paper we present geometric conditions that must hold if H\\\"older estimate of order $\\epsilon$ is valid in a neighborhood of $w_0 \\in b\\Omega$ in $\\mathbb{C}^3.$ The main result is the following theorem: \n\n\\begin{thm} \\label{main_theorem}\nLet $\\Omega = \\{R(w) < 0\\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3.$ Suppose that there is a $1$-dimensional smooth  analytic variety $V$ passing through $w_0$ such that for all $w \\in V$, $w$ sufficiently close to $w_0$, $$|R(w)| \\leq C|w-w_0|^\\eta,$$\nwhere $\\eta >0.$\n{\\it If there exists neighborhood $W$ of $w_0$ so that for all $\\alpha \\in L_{\\infty}^{0,1} ({\\Omega})$ with $\\bar{\\partial}\\alpha = 0$, there is a $u \\in \\Lambda^{\\epsilon} (W \\cap \\overline{\\Omega})$ and $C>0$ such that $\\bar{\\partial}u =\\alpha$ and $$ {\\lVert u \\rVert}_{{\\Lambda^\\epsilon}(W \\cap \\overline{\\Omega})} \\leq C{\\lVert \\alpha \\rVert}_{L_{\\infty}(\\Omega)},$$\nthen $\\epsilon \\leq \\frac{1}{\\eta}$}.\n\\end{thm}\n\n\n\\begin{cor} \n$\\epsilon \\leq \\frac{1}{T_{\\Omega}^{reg}(w_0)}.$\n\\end{cor}\n\n\n\\begin{rem} \\\n\\begin{enumerate}[\\normalfont i)]\n   \\item If $T_{BG}(w_0) = +\\infty$, Krantz's result \\cite {Kr} holds for any $m > 0$ and we conclude \\\\ $\\epsilon \\leq \\frac{1}{m} \\leq \\frac{1}{\\eta}$ for large $m$. Thus we can assume $T_{BG}(w_0) = m < \\infty.$ Furthermore, since $\\epsilon \\leq \\frac{1}{m}$, we can assume $m < \\eta$ in the rest of this paper.\n   \\item Theorem \\ref{main_theorem} improves the results by Krantz \\cite{Kr} and  McNeal \\cite{Mc} in the sense that we obtain sharp result since $\\eta > m$ and do not assume the existence of a holomorphic support function. Note that the existence of holomorphic support function is satisfied for restricted domains (see the Kohn-Nirenberg Domain\\cite{KN}).\n \n\\end{enumerate}\n\n\\end{rem}\n\n\nTo prove Theroem \\ref{main_theorem}, the key components are the complete analysis of the local geometry near $w_0 \\in b\\Omega$ (Section \\ref{special coordinates}) and the construction of a bounded holomorphic function with large nontangential derivative near the boundary point (Section \\ref{Sec4}). In Section \\ref{special coordinates}, we construct  special holomorphic coordinates about $w_0$ which are  adapted to both Bloom-Graham type and the order of contact of $V$. Then, we use the truncation technique developed in \\cite{C} to deal with two dimensional slices of the domain. In Section \\ref{Sec4}, by using the holomorphic function constructed by Catlin \\cite{C2} on  two dimensional slice, we construct a bounded holomorphic function $f$ with a large nontangential  derivative defined locally up to the boundary in $\\mathbb{C}^3$.  Finally, in Section \\ref{Sec5}, we prove Theorem   \\ref{main_theorem} by using the constructed holomorphic function.\n\\\\\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Special coordinates}\\label{special coordinates}\n\n\n Let $\\Omega$ be a smoothly bounded pseudoconvex domain in $\\mathbb C^3$ with a smooth defining function $R$ and let $w_0 \\in b\\Omega.$ Since $dR(w_0) \\neq 0$, clearly we can assume that $\\frac{\\partial R}{\\partial w_3}(w) \\neq 0$ for all $w$ in a small neighborhood $W$ about $w_0.$ Furthermore, we may assume that $w_0 = 0.$ In Theorem \\ref{special coordinate}, we construct a special coordinate near $w_0$ which changes the given smooth holomorphic curve into the $z_1$ axis and have a nonzero term along the $z_2$ axis when $z_1 = 0.$\n  \n\n\n\n\n\n\\begin{thm} \\label{special coordinate}\n\nLet $\\Omega = \\{ w ; R(w)< 0  \\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3$ and let $T_{BG}(0)= m$, where $0 \\in b \\Omega  $. Suppose that there is a smooth  $1$-dimensional complex analytic variety  $V$ passing through $0$ such that for all $w \\in V, w $ sufficiently close to $0$,  \n    \\begin{equation}\n    |R(w)| \\leq C|w|^{\\eta},  \\label{condition of order of contact}\n    \\end{equation}\nwhere $\\eta > 0$. Then there is a holomorphic coordinate system $(z_1, z_2, z_3)$ about $0$ with $w = \\Psi(z)$ such that  \n\n\\begin{enumerate}[{\\normalfont (i)}]\n \n  \\item  $r(z)= R \\circ \\Psi(z) = \\mbox{\\normalfont{Re}}{z_3} + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta a_{\\alpha, \\beta}{z'}^{\\alpha} {\\bar z}'^{\\beta}  + \\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}),$ \\label{special coordinate 1}\n\n  \\item   $|r(t,0,0)| \\lesssim |t|^{\\eta}$ \\label{changed order of contact}\n  \n          \n  \\item  $ {a_{0, \\alpha_2, 0,\\beta_2}} \\neq 0$ with $\\alpha_2+\\beta_2 = m$ for some $\\alpha_2 > 0, \\beta_2 > 0,$ \\label{nonzero term}\n\\end{enumerate}\nwhere $z' = (z_1, z_2),$ and $z = (z_1, z_2, z_3).$\n\n\\end{thm} \n\n\n\n\n\n\nNote that $\\eta$ is a positive integer since $V$ is a smooth 1-dimensional complex analytic variety. \nTo construct the special coordinate in Theorem \\ref{special coordinate}, we start  with a similar coordinate about $0$ in $\\mathbb{C}^3$ as in Proposition 1.1 in \\cite{C2}. \n\n\n\n\n\n\n\\begin{prop} \\label{proposition 1}\n\nLet $T_{BG}(0) = m$ and $\\Omega = \\{ w \\in \\mathbb{C}^3 ; R(w)< 0  \\}$.\nThen there is a holomorphic coordinate system  $  u = (u_1, u_2, u_3)$ with $ w = \\widetilde{\\Psi}(u)$ such that the function $\\tilde{R}$, given by $\\widetilde{R}(u) = R\\circ \\widetilde{\\Psi}(u)$, satisfies \n\n \\begin{equation}\\label{coordinate 1} \n   \\widetilde{R}(u) = \\mbox{{\\normalfont Re}}{u_3} + \\sum_ {\\substack{ |\\alpha|+|\\beta| = m \\\\ |  \\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta} {u'}^{\\alpha} {\\bar u}'^{\\beta} + \\mathcal{O}(|u_3||u|+|u'|^{\\eta +1}), \n \\end{equation}\nwhere $u'=(u_1,u_2),$ and where $b_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta$ with $|\\alpha| + |\\beta| = m.$\n\n\\end{prop}\n\n\n\\begin{proof} \nBloom and Graham \\cite{BG} showed that $T_{BG}(w_0) = m$  if and only if there exists coordinate with $w_0$ equal to the origin in $\\mathbb{C}^3$ and $b_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta$ with $|\\alpha| + |\\beta| = m$ such that  \n\n\\begin{equation*}\nR(w) = \\mbox{Re}{w_3} + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}} b_{\\alpha, \\beta} {w'}^{\\alpha} {\\bar w}'^{\\beta} + \\mathcal{O}(|w_3||w|+|w'|^{m+1}),\n\\end{equation*}\nwhere $\\alpha = (\\alpha_1, \\alpha_2), \\beta = (\\beta_1, \\beta_2) \\ \\mbox{and} \\  w' = (w_1, w_2).$ \n\nNow assume that we have defined $\\phi^{l} : \\mathbb{C}^3 \\rightarrow \\mathbb{C}^3 $ so that there exist numbers $b_{\\alpha, \\beta}$ for $|\\alpha|, |\\beta|>0$ \nand $|\\alpha|+|\\beta| < l + 1$ with $l > m$ so that $ R_l = R\\circ \\phi^{l}$ satisfies\n\n\\begin{equation} \\label{coordinate 2}\n  R_l(v) = \\mbox{{\\normalfont {Re}}}{v_3} + \\sum\\limits_{\\substack{|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0, |\\beta| > 0}}^{l} b_{\\alpha, \\beta}  \n           {v'}^{\\alpha}\\bar{v}'^{\\beta} +  \\mathcal{O}(|v_3||v|+|v'|^{l+1}),  \n\\end{equation}\nwhere $v' = (v_1, v_2)$ and $v = (v_1, v_2, v_3).$\n\nIf we define  $$\\phi^{l+1} (u) = \\biggl( u_1, u_2, u_3-\\sum_{|\\alpha|=l+1} {\\frac{2}{\\alpha!}}{\\frac{\\partial^{l+1} R_l}{\\partial {v'}^\\alpha}}(0){u'}^\\alpha  \\biggr),$$ then $R_{l+1} = {R_l}\\circ \\phi^{l+1} = R \\circ \\phi^l \\circ \\phi^{l+1}$ satisfies the similar form of (\\ref{coordinate 2}) with $l$ replaced by $l+1$. Therefore, if we take $\\widetilde{\\Psi} = \\phi^l \\circ \\cdots \\circ \\phi^{\\eta}$, then $\\widetilde{R} = R\\circ \\widetilde{\\Psi}$ satisfies \n$$ \\widetilde{R}(u) = \\mbox{Re}{u_3} + \\sum\\limits_{\\substack{  |\\alpha|+|\\beta| =m \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta} {u'}^{\\alpha} {\\bar u}'^{\\beta} + \\mathcal{O}(|u_3||u|+|u'|^{\\eta +1}).$$ \n\\end{proof}\n\n\n\n\nFrom now on, without loss of generality, we may assume that $\\widetilde{R}$ is $R$ by Proposition \\ref{proposition 1}.\n\n\n\n\n\\begin{lem} \\label{parametrization lemma}\nLet $\\gamma = (\\gamma_1,\\gamma_2,\\gamma_3) :\\mathbb{C} \\to V$ be a local parametrization of a one-dimensional smooth complex analytic variety $V$.\nIf   $|R(w)| \\lesssim |w|^{\\eta}$ for $w \\in V$, then we can assume\n    $\\gamma =(\\gamma_1,\\gamma_2, 0) $  (i.e.,   $\\gamma_3$ vanishes to order at least $\\eta$).\n\\end{lem} \n\n\n\\begin{proof}We show $\\gamma_3 $ vanishes to order at least $\\eta$.\nSince $\\gamma(0)= 0$, we know $\\gamma_3$ vanishes to some order $l$.\nIf we suppose  $l<\\eta$,  then $\\gamma_3(t)= a_l {t}^l+\\mathcal{O}(t^{l+1})$, where $a_l \\neq 0$. Then\n\\begin{align*}\n        R(\\gamma(t))&= \\mbox{\\normalfont Re}{\\gamma_3} + \\sum_ {\\substack{|\\alpha|+|\\beta|=m  \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha,\\beta}\\gamma_1^{\\alpha_1} {{\\bar \\gamma}_1}^{\\beta_1}\\gamma_2^{\\alpha_2} {{\\bar \\gamma}_2}^{\\beta_2}+\\mathcal{O}(|\\gamma_3||\\gamma|+|\\gamma|^{\\eta +1})  \\\\  \n                    &= \\biggl( {\\frac{a_l}{2}}{t^l}+{\\frac{\\bar a_l}{2}}{\\bar{t}^l} \\biggr) +  \\biggl( \\sum_ {\\substack{j + k = m  \\\\ j > 0,                                              k > 0}}^\\eta c_{jk} t^j \\bar{t}^k \\biggr) + \\mathcal{O}(|t|^{l+1}).\n\\end{align*}\n\nNote that the first parenthesis consists of order $l$ pure terms and the summation part consists of the mixed terms. The first one is essentially $|t|^l$ with $l<\\eta$, so if we want to improve on the order of contact, then some terms of the summation part must cancel it. However, it is impossible because the summation part has all mixed terms. This contradicts our assumption $|r \\circ \\gamma(t)| \\lesssim |t|^\\eta$. Therefore, $\\gamma_3$ vanishes to order at least $\\eta.$ \n\n\\end{proof}\n\n\nLet  $ A(u_1, u_2) = \\sum\\limits_{\\substack {|\\alpha|+|\\beta| = m \\\\ |\\alpha| > 0,  |\\beta| > 0}} b_{\\alpha, \\beta} {u}'^{\\alpha} {\\bar u}'^{\\beta}$ be the homogeneous polynomial part of order $m$ in the summation part of (\\ref{coordinate 1}). In the following lemma, we show that there is some nonzero mixed term along some direction in $\\mathbb{C}^2$.\n\n\n\\begin{lem} \\label{z_2 direction} \nConsider $A(hz, z)$ for all $h, z \\in \\mathbb{C}.$ Then there is some $h \\in \\mathbb{C}$ such that\n\\begin{equation*}\n\\frac{\\partial^m A}{{\\partial z^{j}}{\\partial {{\\bar z}^k}}}(0,0) \\neq 0, \\ \\text{ for }\\\nj, k >0.\n\\end{equation*} \n\\end{lem}\n\n\n\n\\begin{proof}\nSuppose that for all $h,$ $A(hz, z) = P(h)z^m + \\overline{P(h)z^m}.$ Since $A(hz, z)$ is a polynomial in $z, {\\bar z}, h$ and ${\\bar h}$ and  $\\frac{\\partial^m A}{\\partial z^m} = m! P(h)$, $P(h)$ is a polynomial.\nLet $P(h) = \\sum a_{j,k} h^j {\\bar h}^k.$ Now, we have $A(hz, z)= \\sum a_{j,k} h^j {\\bar h}^k z^m + \\sum {\\bar a}_{j,k} {\\bar h}^j {h}^k {\\bar z}^m.$ Since $u_1 = hz$ and $u_2 = z,$ we have $h = \\frac{u_1}{u_2}$ and $z = u_2$. Therefore, $A(u_1, u_2) = \\sum a_{j,k} (\\frac{u_1}{u_2})^j ({\\frac{{\\bar{u}_1}}{{\\bar{u}_2}}})^k u_2^m + \\sum \\bar{a}_{j,k} ({\\frac{\\bar{u}_1}{{\\bar u}_2}})^j ({\\frac{u_1}{u_2}})^k {\\bar u_2}^m.$ This forces $j$ and $k$ to be $0$ because $A(u_1, u_2)$ is a polynomial. Therefore, we have $A(hz, z)= a_{0,0}z^m + \\bar{a}_{0, 0} \\bar{z}^m.$ This means $A(u_1, u_2)= a_{0,0}{u_2} ^m + \\bar{a}_{0, 0} {\\bar u_2}^m.$\nHowever, this contradicts $b_{\\alpha, \\beta} \\neq 0 $  for some $\\alpha, \\beta$ with $|\\alpha|, |\\beta| > 0 $ and $|\\alpha|+|\\beta|=m $ in (\\ref{coordinate 1}). \n\\end{proof}\n\n\nNow, we prove Theorem \\ref{special coordinate}\n\n\n\n\\begin{proof}[Proof of Theorem \\ref{special coordinate}]\nWe may assume ${\\gamma_1}' (0) \\neq 0$, and hence, after reparametrization, we can write $\\gamma(t) = (t, \\gamma_2(t), 0)$. Now, define \n\\begin{equation*}\n  u = \\Psi_1 (v) = (v_1, v_2 + \\gamma_2(v_1), v_3).\n\\end{equation*}\nSince $\\gamma_2(t) = \\mathcal{O}(|t|)$ is holomorphic, (\\ref{coordinate 1}) means \n\\begin{align*}\n  r_1 (v)   &= R \\circ \\Psi_1 (v)  \n            = \\mbox{Re}{v_3}+\\sum_ {\\substack{|\\alpha|+|\\beta|=m  \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta b_{\\alpha, \\beta}                                                             v_1^{\\alpha_1}{\\bar v}_1^{\\beta_1}(v_2+\\gamma_2(v_1))^{\\alpha_2} {\\overline {(v_2+\\gamma_2(v_1)) }}^{\\beta_2} \n              + E_1 (v) \\\\\n            &=\\mbox{Re}{v_3}+\\sum_ {\\substack{|\\alpha|+|\\beta|=m  \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta c_{\\alpha, \\beta}                                                              v_1^{\\alpha_1}{\\bar v_1}^{\\beta_1}v_2^{\\alpha_2} {\\bar v_2}^{\\beta_2}+ E_1(v), \\  \\text{ where }\\ E_1(v) = \\mathcal{O}(|v_3||v|+|v'|^{\\eta +1}).\n\\end{align*}\nNote that $T_{BG} = m$ means $c_{\\alpha, \\beta} \\neq 0$ for some $\\alpha, \\beta > 0$ with $|\\alpha|+|\\beta| = m.$ Now, we fix $h$ in lemma \\ref{z_2 direction} and  define \n\n\\begin{equation*}\n       v = \\Psi_2 (z) = (z_1 + h z_2,  z_2, z_3).\n\\end{equation*}\nThen, we have \n \\begin{align}  \n    r(z) &= r_1 \\circ \\Psi_2(z) = R \\circ \\Psi_1 \\circ \\Psi_2 (z) \\nonumber \\\\\n         &= \\mbox{Re}z_3 +\\sum_ {\\substack{|\\alpha|+|\\beta|=m  \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta c_{\\alpha, \\beta}                                                              (z_1+hz_2)^{\\alpha_1}{\\overline {(z_1+hz_2)}}^{\\beta_1}z_2^{\\alpha_2} {\\bar z_2}^{\\beta_2}+ E_1(z), \\label{eqn2}  \\\\\n         &=\\mbox{Re}z_3 + \\sum_ {\\substack{|\\alpha|+|\\beta|=m  \\\\ |\\alpha| > 0, |\\beta| > 0}}^\\eta a_{\\alpha, \\beta}                                                              z_1^{\\alpha_1}{\\bar z_1}^{\\beta_1}z_2^{\\alpha_2} {\\bar z_2}^{\\beta_2}+ E_1(z), \\label{eqn3}\n\\end{align}\nwhere $a_{\\alpha, \\beta}$ is a polynomial of $h$ and ${\\bar h},$ and where $E_1(z) = \\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}).$\nLet $\\Psi = \\Psi_1 \\circ \\Psi_2.$ Then we have $r(z) = R \\circ \\Psi$ and (\\ref{eqn3}) shows (\\ref{special coordinate 1}) of Theorem \\ref{special coordinate}. Furthermore, since $|r(t, 0, 0)| = |R \\circ \\Psi(t, 0, 0)|= |R(\\gamma(t))| \\lesssim |t|^\\eta,$ this proves part (\\ref{changed order of contact}). For (\\ref{nonzero term}), if we consider $r(0, z_2, 0)$ and (\\ref{eqn2}), we have\n\\begin{equation*} \n  r(0, z_2, 0) = A(hz_2, z_2) + \\sum\\limits_ {\\substack{|\\alpha|+|\\beta| = m +1  \\\\  |\\alpha| > 0,   |\\beta| > 0}}^\\eta                                                       c_{\\alpha,\\beta}{(hz_2)}^{\\alpha_1}{\\overline{(hz_2)}}^{\\beta_1}z_2^{\\alpha_2} {\\bar z}_2^{\\beta_2} +\\mathcal{O}(|z_2|^{\\eta +1}).\n\\end{equation*}\nThen Lemma \\ref{z_2 direction} means\n\\begin{equation*}\n \\frac{\\partial^m r }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0) = \\frac{\\partial^m A }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0,0) \\neq 0\n\\end{equation*} \nfor some $\\alpha_2, \\beta_2 > 0$ with $\\alpha_2 + \\beta_2 = m .$\nSince $\\frac{\\partial^m r }{{\\partial {z_2}^{\\alpha_2}}{\\partial {\\bar z}_2}^{\\beta_2}}(0) = \\alpha_2 ! \\beta_2 ! a_{0, \\alpha_2, 0, \\beta_2}$ in (\\ref{eqn3}), this completes the proof. \n\\end{proof}\n\n\n\n\nCatlin \\cite{C2} constructed a bounded holomorphic funtion with a large derivative near a finite type point in the boundary of pseudoconvex domain in $\\mathbb{C}^2$.  To construct a similar function in $\\mathbb{C}^3$, we will use the function constructed by Catlin. In order to achieve this goal, as a first step, we need to consider two dimensional slice with respect to the $z_2$ and $z_3$ variables when $z_1$ is fixed at some point. For this, we consider the representative terms in the summation part of (\\ref{special coordinate 1}) of Theorem \\ref{special coordinate}.\n\nLet \n\\begin{align*}\n \\Gamma &= \\{(\\alpha, \\beta);  a_{\\alpha, \\beta} \\neq 0, m \\leq |\\alpha|+|\\beta| \\leq \\eta \\ \\mbox{and} \\ |\\alpha|, |\\beta| > 0 \\} \\\\\n      S &= \\{(p, q); \\alpha_1 + \\beta_1 = p, \\alpha_2 + \\beta_2 = q \\ \\mbox{for some} \\ (\\alpha, \\beta) \\in \\Gamma  \\} \\cup  \\{(\\eta, 0)\\}.\n\\end{align*}\nThen there is an positive integer $N$ such that $(p_\\nu, q_\\nu) \\in S$ for $\\nu = 0,\\cdots, N $ and $\\eta_\\nu, \\lambda_\\nu > 0$ for $\\nu = 1, \\cdots, N$ satisfying\n \n\\begin{enumerate}[(1)]\n\t\\item $(p_0, q_0)=(\\eta, 0),  (p_{N}, q_{N})= (0, m) , \\lambda_N = m, \\eta_1 = \\eta,$ \\label{condition1}\n\t\\item $p_0 > p_1 > \\cdots > p_{N}$ and $ q_0 < q_1 < \\cdots < q_{N},$ \\label{condition2}\n\t\\item $\\lambda_1 < \\lambda_2 <\\cdots < \\lambda_N$ and  $\\eta_1 > \\eta_2 > \\cdots > \\eta_N,$ \\label{condition3}\n\t\\item $\\frac{p_{\\nu-1}}{\\eta_\\nu} + \\frac{q_{{\\nu-1}}}{\\lambda_\\nu} = 1$ and $\\frac{p_{\\nu}}{\\eta_\\nu} + \\frac{q_{\\nu}}{\\lambda_\\nu}=1$ \\label{condition4} and\n\t\\item $a_{\\alpha, \\beta}=0$ if $\\frac{\\alpha_1 + \\beta_1}{\\eta_\\nu}+ \\frac{\\alpha_2 + \\beta_2}{\\lambda_\\nu} < 1$ for each $\\nu = 1,\\cdots,N.$\\label{condition5}\n\\end{enumerate}\n\nNote that if $1 \\leq l \\leq m,$ then $q_{\\nu - 1} < l \\leq q_\\nu$ for some $\\nu = 1, \\cdots, N.$\nLet $L_\\nu$ be the line segment from $(p_{\\nu - 1}, q_{\\nu - 1})$ to $(p_{\\nu}, q_{\\nu})$ for each $\\nu = 1, \\cdots, N$ and set $L =\\ L_1 \\cup L_2 \\cup \\cdots \\cup L_{N}.$ Define\n\n\\begin{itemize}\n\t\\item $\\Gamma_L = \\{(\\alpha, \\beta) \\in \\Gamma; \\alpha+\\beta \\in L \\}.$\n\t\\item $t_l = \\begin{cases}\n\t             \\eta & \\text{if $l = 0$} \\\\\n\t             \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu} \\biggr) & \\text{if  $q_{\\nu-1} < l \\leq q_\\nu$ for some $\\nu$.}  \n\t            \\end{cases}$\n\\end{itemize}\nNote that $(p_{\\nu-1}, q_{\\nu-1}), (t_l, l)$ and $(p_\\nu, q_\\nu)$ are collinear points in the first quadrant of the plane and $\\eta_\\nu$ and $\\lambda_\\nu$ are  the $x$, $y$-intercepts of the line. \\\\\n\nNow, we want to show that for each element $(p_\\nu, q_\\nu)$ with $\\nu = 1, \\cdots, N$, there is some $(\\alpha, \\beta)$ allowing a mixed term in the $z_2$ variable. To show this, we need to use a variant of the notations and the results from  Lemma 4.1 and Proposition 4.4 in \\cite{C}.\nFor $t$ with $0<t<1$ and each $\\nu = 1, \\cdots, N$, define a family of a truncation map $H_t^\\nu : \\mathbb{C}^3 \\rightarrow \\mathbb{C}^3$ by \n$$H_t^\\nu (z_1,z_2,z_3) = (t^{(1\/{\\eta_\\nu})}z_1 ,t^{(1\/{\\lambda_\\nu})}z_2 ,t z_3 ) .$$\nSet $r_t^\\nu = t^{-1}({H_t^\\nu}^{*} r)$ and ${\\widetilde r}^\\nu = \\lim_{t \\to 0}r_t^\\nu .$ Note that $${\\widetilde r}^\\nu(z)  = \\mbox{Re} {z_3} + \\sum_ {\\substack{\\frac{\\alpha_1+\\beta_1}{\\eta_\\nu} + \\frac{\\alpha_2+\\beta_2}{\\lambda_\\nu}=1 \\\\ (\\alpha, \\beta) \\in \\Gamma_L}} a_{\\alpha_1,\\alpha_2 ,\\beta_1,\\beta_2} z_1^{\\alpha_1} {\\bar {z_1}}^{\\beta_1}z_2^{\\alpha_2} {\\bar {z_2}}^{\\beta_2}.$$\n\nLet $r$ and ${\\widetilde r}^\\nu$ be a defining function of $\\Omega$ and ${\\widetilde \\Omega}_\\nu$ near $0$. Observe that if $\\Omega$ is pseudoconvex, then ${\\widetilde \\Omega}_\\nu$ must also be pseudoconvex, for ${\\widetilde r}^\\nu$ equals the limit in the $C^{\\infty}$-topology of $r_t^\\nu$, which for each $t$ is the defining function of a pseudoconvex domain.\nFor a fixed $(z_1, z_2)$, choose $z_3$ so that ${\\widetilde r}^\\nu(z_1,z_2,z_3)=0$.  Let that point $z$. \nSince the Hessian of ${\\widetilde r}^\\nu$ is nonnegative  in the tangential directions at $z$, it follows that the Hessian of ${\\widetilde r}^\\nu$\nis nonnegative at $z$. This means ${\\widetilde r}^\\nu$ is plurisubharmonic.\n\n\n\\begin{lem}  \n\\label{existence of mixed term in z_2}\nConsider $r$ in (\\ref{special coordinate 1}) of Theorem \\ref{special coordinate}. Then for each $\\nu = 1, \\cdots, N,$ there is $(\\alpha^\\nu, \\beta^\\nu) \\in \\Gamma_L$ with $\\alpha_2^\\nu > 0, \\beta_2^\\nu > 0$\nand  $\\alpha^\\nu + \\beta^\\nu = (p_\\nu, q_\\nu).$\n\n\\end{lem}\n\n\n\\begin{proof}\nConsider ${\\widetilde r}^\\nu$, which is plurisubharmonic. Now, consider $${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1} = \\lim\\limits_{t \\to 0} t^{-1}({H_t^{\\nu+1}}^* {\\widetilde r^\\nu }).$$  This is also  plurisubharmonic.  Since $(p_\\nu, q_\\nu)$ is the unique point with $L_\\nu \\cap L_{\\nu + 1}$ (i.e., $\\frac{p_{\\nu}}{\\eta_\\nu} + \\frac{q_{{\\nu}}}{\\lambda_\\nu} = 1$ and $\\frac{p_{\\nu}}{\\eta_{\\nu+1}} + \\frac{q_{\\nu}}{\\lambda_{\\nu+1}}=1$), we have \n \n\\begin{equation}\\label{formoftrucated}\n {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1} = \\mbox{Re}{z_3} +  \\sum_ {\\substack{\\alpha + \\beta = (p_\\nu, q_\\nu) \\\\ (\\alpha, \\beta) \\in \\Gamma_L}}                                 a_{\\alpha_1,\\alpha_2 ,\\beta_1,\\beta_2} z_1^{\\alpha_1} {\\bar z}_1^{\\beta_1}z_2^{\\alpha_2} {\\bar z}_2^{\\beta_2}. \n                      \n\\end{equation}\nIn particular, $(\\alpha, \\beta) \\in \\Gamma_L$ means $|\\alpha|, |\\beta| > 0.$\nSuppose that ${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}$ has no terms with both $\\alpha_2 > 0$ and $\\beta_2>0$ in (\\ref{formoftrucated}) (i.e., no mixed terms in $z_2$ variable). Thus\n$${\\widetilde{(\\widetilde{r^\\nu})}}^{\\nu+1}  = \\mbox{Re}{z_3} + P_{q_\\nu}(z_1){z_2}^{q_\\nu} +\\overline{P_{q_\\nu}(z_1){z_2}^{q_\\nu}}$$  \nwhere $P_{q_\\nu}(z_1) = \\sum\\limits_{\\alpha_1 +\\beta_1 = p_\\nu} c_{\\alpha_i,\\beta_i} {z_1}^{\\alpha_1}  {\\bar z_1}^{\\beta_1}$ with $\\beta_1 > 0$.  \nBy the plurisubharmonicity of ${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}$, \n            $${\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{11} {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{22}- {\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{12}{\\widetilde {({\\widetilde r}^\\nu)}}^{\\nu+1}_{21} = - \\lvert {q_\\nu} \\frac{\\partial {P_{q_\\nu}}}{\\partial{\\bar z_1}}(z_1) {z_2}^{q_\\nu -1}   \\rvert^2 \\geq 0,$$\nwhere  ${{\\widetilde{(\\widetilde{r^\\nu})}}^{\\nu+1}}_{ij} = \\frac{\\partial^{2}{\\widetilde{({\\widetilde r}^\\nu)}}^{\\nu+1}}{{\\partial z_i}{\\partial \\bar{z_j}}} $ for $i, j = 1, 2.$           \nTherefore, we have $\\frac{\\partial {P_{q_\\nu}}}{\\partial{\\bar{z_1}}}(z_1) = 0.$ This means $P_{q_\\nu} (z_1)$ is holomorphic. \nThis contradicts the fact that $P_{q_\\nu}(z_1) = \\sum\\limits_{\\alpha_1 +\\beta_1 = p_\\nu} c_{\\alpha_i,\\beta_i} {z_1}^{\\alpha_1}  {{\\bar {z_1}}^{\\beta_1}}$ with $\\beta_1 > 0$.  \n\\end{proof} \n\n\n\nNow, we define these special terms with respect to the $z_2$ variable. Let\n\\begin{equation*}\n\\Lambda = \\{(\\alpha, \\beta) \\in \\Gamma_L ; \\alpha+\\beta = (p_\\nu, q_\\nu), \\alpha_2 > 0, \\beta_2 > 0, \\nu = 1, \\cdots, N \\}.\n\\end{equation*}  \nThen we represent the expression of $r$ in terms of these terms. \\\\\n\n\n\n\\begin{prop}\\label{final expression of r}\n The defining function $r$  can be expressed as \n\\begin{equation} \\label{repre of r}\n r(z) = \\mbox{\\normalfont Re}{z_3} + \\sum_ {\\Gamma_L - \\Lambda} a_{\\alpha, \\beta} {z'}^{\\alpha} {\\bar z}'^{\\beta} + {\\sum_{\\nu = 1}^N}\\sum_{\\substack{\\alpha_2 + \\beta_2= q_\\nu  \\\\ \\alpha_2 > 0, \\beta_2 > 0}}                   M_{\\alpha_2, \\beta_2}(z_1) {z_2}^{\\alpha_2} {\\bar z}_2^{\\beta_2}+E_2(z),\n\\end{equation} \nwhere $M_{\\alpha_2, \\beta_2}(z_1) = \\sum\\limits_{\\alpha_1+\\beta_1 = p_\\nu} a_{\\alpha, \\beta} z_1^{\\alpha_1} \\bar{z}_1^{\\beta_1}$  and  $E_2(z) = \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).$\n\n\\end{prop}\n\n\n\n\\begin{proof}\nBy theorem \\ref{special coordinate}, we have \n\\begin{equation} \\label{rform}\n  r(z) = {\\mbox{Re}}{z_3}+ \\sum_{\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta}+ \\sum_{\\Gamma-\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta}+\\mathcal{O}(|z_3||z|+|z'|^{\\eta +1}).   \n\\end{equation}\nSuppose that $(k, l) = (\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta)\\in \\Gamma-\\Gamma_L.$ Then, we consider two cases; $1\\leq l \\leq m$ and $m < l < \\eta.$ If $1\\leq l \\leq m$, there is a unique $\\nu =1, \\cdots, N$ so that $q_{\\nu-1} < l \\leq q_\\nu$ and  $t_l = \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu} \\biggr).$ Since $(k, l) = (\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta) \\in \\Gamma -\\Gamma_L,$  $ \\frac{k}{\\eta_\\nu}+\\frac{l}{\\lambda_\\nu} > 1.$ This gives $t_l =  \\eta_\\nu \\biggl(1 - \\frac{l}{\\lambda_\\nu}\\biggr) < k.$ Since $k$ is an integer, $[t_l]+1 \\leq k.$ Thus, we have $|z_1|^k|z_2|^l \\leq |z_1|^{[t_l]+1}|z_2|^l$ for each $l = 1, \\cdots, m.$   \nOn the other hand, if $(k, l)=(\\alpha_1 + \\beta_1, \\alpha_2 + \\beta_2)$ for some $(\\alpha, \\beta)\\in \\Gamma-\\Gamma_L$ and $m < l < \\eta,$ then $|z_1|^k|z_2|^l \\leq |z_1|^k|z_2|^{m+1} \\leq |z_2|^{m+1}$ for small $z_1$ and $z_2.$\nSince $|z'|^{\\eta +1} \\approx |z_1|^{\\eta +1} + |z_2|^{\\eta +1},$ it follows that\n$\\sum_{\\Gamma -\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar{z'}}^{\\beta}+\\mathcal{O}(|z_3||z|+|z'|^{\\eta +1})= \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).$ Therefore, $r(z)$ in (\\ref{rform}) is represented as \n\\begin{equation} \\label{r form 2}\n{\\mbox{Re}}{z_3}+ \\sum_{\\Gamma_L} a_{\\alpha, \\beta} {z'}^{\\alpha}{\\bar z}'^{\\beta} + \\mathcal{O}(|z_3||z|+\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |z_1|^{[t_l]+1}|z_2|^l+|z_2|^{m+1}).\n\\end{equation}\nNow, apply $\\Gamma_L = (\\Gamma_L - \\Lambda) \\cup \\Lambda$ for the second part of summation in (\\ref{rform}). \n\\end{proof}\n\n\n\\begin{rem}\\label{sizeofM} \\\n\\begin{enumerate}[\\normalfont i)] \n\t\\item $M_{\\alpha_2, \\beta_2}(z_1)$ is not identically zero  for $\\alpha_2 + \\beta_2 = q_\\nu$ and the homogeneous polynomial is of order $p_\\nu$ for each $\\nu           = 1, \\cdots, N-1.$\n\t\\item If  $\\nu = N,$ then $|M_{\\alpha_2, \\beta_2}(z_1)|$ is a nonzero constant for all $\\alpha_2, \\beta_2 > 0$ with $\\alpha_2+\\beta_2 = m = q_N$ since $ p_N = 0.$\n\t\\item Since $M_{\\alpha_2, \\beta_2}(z_1)$ is a homogeneous polynomial of order $p_\\nu, \\nu =1, \\cdots, N,$ in $z_1$-variable, there are $\\theta_0 \\in [0, 2\\pi]$          and a small constant $c > 0$ such that $|M_{\\alpha_2, \\beta_2}(\\tau e^{i\\theta})| \\neq 0$ for all $|\\theta - \\theta_0| < c$ and $0 < \\tau \\leq 1.$   \n\t       In particular, if we take $d = e^{i\\theta_0}$ and $\\tau = \\delta^{\\frac{1}{\\eta}}$ we have $|M_{\\alpha_2, \\beta_2}(d \\delta^{\\frac{1}{\\eta}})| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}$ for all             $\\alpha_2+ \\beta_2 = q_\\nu$ with all              $\\nu = 1, \\cdots, N.$\n\\end{enumerate}\n\\end{rem}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{The construction of bounded holomorphic function with large derivative near the boundary}\\label{Sec4}\n\nLet $z_1 = d\\delta^{\\frac{1}{\\eta}}.$ Then, we get a complex two dimensional slice. After the holomorphic coordinate change as Proposition 1.1 in \\cite{C2}, we can define a bounded holomorphic function with a large nontangential derivative as in \\cite{C2} on the slice. In this section, first, we construct a holomorphic coordinate system in $\\mathbb{C}^3$ to exactly fit the holomorphic coordinate system as in proposition 1.1 of \\cite{C2} when $z_1$ is fixed as $d\\delta^{\\frac{1}{\\eta}}$. Second, we show that the holomorphic function defined on the slice is also well-defined on a family of slices along the small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}.$ To show the well-definedness of the holomorphic function up to boundary in $\\mathbb{C}^3,$ we need the estimates of derivatives.   Let's denote $ U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}} = U \\cap \\{(d\\delta^{\\frac{1}{\\eta}}, z_2, z_3)\\}$ and let $ \\widetilde{e}_\\delta = (d\\delta^{\\frac{1}{\\eta}}, 0, e_\\delta)$ satisfy $r(\\widetilde{e}_\\delta) = 0.$ Since $\\frac{\\partial r}{\\partial z_3}(0) \\neq 0,$ clearly $\\frac{\\partial r}{\\partial z_3}(\\widetilde{e}_\\delta) \\neq 0.$ We start with the similar argument as Proposition 1.1 in \\cite{C2}.\\\\\n\n\n\\begin{prop} \\label{coordinatechangeinc2}\nFor $ \\widetilde{e}_\\delta \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}},$ there exists a holomorphic coordinate system  $ (z_2, z_3)= \\Phi_{\\widetilde{e}_\\delta}(\\zeta '') = (\\zeta_2 , \\Phi_3 (\\zeta'')))$ such that in the new coordinate $\\zeta'' = (\\zeta_2, \\zeta_3)$ defined by\n\\begin{equation}\n\\Phi_{\\widetilde{e}_\\delta}(\\zeta '')= \\biggl(\\zeta_2, \\  e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}(\\widetilde{e_\\delta}) \\biggr)^{-1}                                                             \\biggl(\\frac{\\zeta_3}{2}- \\sum_{l=2}^m c_l({\\widetilde e}_\\delta) \\zeta_2^l                        - \\frac{\\partial r}{\\partial  z_2} ({\\widetilde e}_\\delta){\\zeta_2}  \\biggr)   \\biggr),  \\label{catlin version coordinate chanage}\n\\end{equation}\nthe function $\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}(\\zeta'')$ satisfies \n\\begin{equation} \\label{catlin_defining_expression}\n\\rho(d\\delta^{\\frac{1}{\\eta}},\\zeta'')= \\mbox{\\normalfont{Re}}\\zeta_3 + \\sum\\limits_{\\substack{j+k=2\\\\j, k > 0}}^{m} a_{j,k}(\\widetilde{e_\\delta}) \\zeta_2^j {\\bar \\zeta_2}^k + \\mathcal{O}(|\\zeta_3||\\zeta''|+|\\zeta_2|^{m+1} ),\n\\end{equation}\nwhere $z'' = (z_2, z_3)$.\n\\end{prop} \n\n\n\\begin{proof} For $ {\\widetilde e}_\\delta \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}},$ define\n\\begin{equation}\\label{rho2changeofcoordinate}\n \\Phi_{\\widetilde{e}_\\delta}^1(w'') =\\biggl( w_2, \\ e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}({\\widetilde e}_\\delta) \\biggr)^{-1}\\biggl(\\frac{w_3}{2} - \\frac{\\partial r}{\\partial z_2} ({\\widetilde e}_\\delta)w_2 \\biggr)\\biggr). \n\\end{equation}\nThen we have \n\\begin{equation}\\label{rho2expression}\n\\rho_2(d\\delta^{\\frac{1}{\\eta}}, w'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}^1(w'') = \\mbox{Re} w_3 + \\mathcal{O}(|w''|^2),\n\\end{equation} \nwhere $w'' = (w_2, w_3).$\nNow assume that we have defined $\\Phi_{\\widetilde{e}_\\delta}^{l-1} : \\mathbb{C}^2 \\rightarrow \\mathbb{C}^2 $ so that there exist numbers $a_{j, k}$ for $j, k > 0$ and $j+k < l$ so that $\\rho_l(d\\delta^{\\frac{1}{\\eta}}, w'') = r(d\\delta^{\\frac{1}{\\eta}}, z'')\\circ \\Phi_{\\widetilde{e}_\\delta}^{l-1}(w'')$ satisfies\n$$\\rho_l(d\\delta^{\\frac{1}{\\eta}}, w'') = \\mbox{Re}w_3 + \\sum_{\\substack{j+k=2\\\\j, k > 0}}^{l-1} a_{j,k}({\\widetilde e}_\\delta) w_2^j {\\bar w_2}^k + \\mathcal{O}(|w_3||w''|+|w_2|^l),  $$\nwhere $w'' = (w_2, w_3).$\nIf we define $\\Phi_{\\widetilde{e}_\\delta}^l = \\Phi_{\\widetilde{e}_\\delta}^{l-1}\\circ \\phi^l,$ where \n\\begin{equation}\\label{lthchangeofvariable}\n\\phi^l(\\zeta'') = \\biggl(\\zeta_2, \\  \\zeta_3 - \\frac{2}{l!} \\frac{\\partial^l \\rho_l}{\\partial w_2^l}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\zeta_2^l \\biggr).\n\\end{equation}   \nthen \n\\begin{equation}\\label{rholexpression}\n\\rho_{l+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')= \\rho_l \\circ \\phi^l (\\zeta'')  = r(d\\delta^{\\frac{1}{\\eta}}, z'')\\circ \\Phi_{\\widetilde{e}_\\delta}^l(\\zeta'')\n\\end{equation}  satisfies\n$$\\rho_{l+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\mbox{Re}\\zeta_3 + \\sum\\limits_{\\substack{j+k=2\\\\j, k > 0}}^{l} a_{j,k}({\\widetilde e}_\\delta) \\zeta_2^j {\\bar \\zeta_2}^k + \\mathcal{O}(|\\zeta_3||\\zeta''|+|\\zeta_2|^{l+1}),$$ where $\\zeta'' = (\\zeta_2, \\zeta_3).$\nTherefore, if we choose $\\Phi_{\\widetilde{e}_\\delta} = \\Phi_{\\widetilde{e}_\\delta}^m = \\Phi_{\\widetilde{e}_\\delta}^{m-1} \\circ \\phi^m = \\cdots = \\Phi_{\\widetilde{e}_\\delta}^1 \\circ \\phi^2 \\circ \\cdots \\circ \\phi^m,$ then $\\rho = \\rho_{m+1} = \\rho_m \\circ \\phi^m = r \\circ \\Phi_{\\widetilde{e}_\\delta}.$ \nThis shows (\\ref{catlin version coordinate chanage}) and (\\ref{catlin_defining_expression}), where $ c_l({\\widetilde e}_\\delta)$ is defined by\n\\begin{equation}\\label{clexpression}\n c_l({\\widetilde e}_\\delta) = \\frac{1}{l!}\\frac{\\partial^l \\rho_l}{\\partial w_2^l}(d\\delta^{\\frac{1}{\\eta}}, 0, 0).   \n\\end{equation}\n\\end{proof}\n\nAs in \\cite{C2}, we set\n\\begin{equation}\\label{def of Al}\nA_l ({\\widetilde e}_\\delta) = \\mbox{max} \\{|a_{j,k}({\\widetilde e}_\\delta)|; j+k = l \\}, \\hspace{0.3in} l=2, \\cdots, m \n\\end{equation}\nand \n\\begin{equation}\\label{taudef}\n\\tau({\\widetilde e}_\\delta, \\delta) = \\min \\biggl\\{\\biggl(\\frac{\\delta}{A_l({\\widetilde e}_\\delta)} \\biggr)^{1\/l}; 2 \\leq l \\leq m \\biggr\\}\n\\end{equation}\nAs we will see later (Remark \\ref{Amnonzero}), we have $A_m ({\\widetilde e}_\\delta) \\neq 0$ since $ |A_m ({\\widetilde e}_\\delta)| \\geq c_m > 0,$ where $\\delta > 0$ is sufficiently small. This means $$\\tau({\\widetilde e}_\\delta, \\delta) \\lesssim \\delta^{\\frac{1}{m}}.$$\nDefine\n\\begin{equation}\\label{rbox}\n R_\\delta ({\\widetilde e}_\\delta)= \\{\\zeta'' \\in \\mathbb{C}^2; |\\zeta_2| < \\tau({\\widetilde e}_\\delta, \\delta), |\\zeta_3| < \\delta \\}.\n\\end{equation} \\\\\n\n\nBefore estimating the derivative of $r$, we estimate the size of $e_\\delta$.\nSince $r({\\widetilde e}_\\delta)= 0,$  Taylor's theorem in $z_3$ about $e_\\delta$  gives\n$$r(d\\delta^{\\frac{1}{\\eta}}, 0, z_3)= 2\\mbox{Re}\\biggl( \\frac{\\partial r}{\\partial z_3}(d\\delta^{\\frac{1}{\\eta}}, 0, e_\\delta)(z_3 - e_\\delta) \\biggr)+ \\mathcal{O}(|z_3 - e_\\delta|^2).$$\nIf we take $z_3 = 0,$ then $|r(d\\delta^{\\frac{1}{\\eta}}, 0, 0)|= \\left|2\\mbox{Re}\\biggl( \\frac{\\partial r}{\\partial z_3}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)(- e_\\delta) \\biggr)+ \\mathcal{O}(|e_\\delta|^2)\\right| \\approx |e_\\delta|$ since $|e_\\delta| \\ll 1$ and $|\\frac{\\partial r}{\\partial z_3}| \\approx 1$ near $0.$ Therefore \\ref{changed order of contact}) of Theorem \\ref{special coordinate} means $|e_\\delta| \\lesssim \\delta.$ \\\\\n\n\n\n\n\n\n\n\n\n\n\n\\begin{lem}\\label{rderivative}\nLet $ l = 1, 2, \\cdots, m$ and let $\\alpha_2^\\nu$ and $\\beta_2^\\nu$ be positive numbers as given in Lemma \\ref{existence of mixed term in z_2} for $\\nu = 1, \\cdots, N.$ Then the function $r$ satisfies\n\\begin{enumerate}[\\normalfont (i)]\n\t\\item $\\biggl|\\frac{\\partial^l r}{{\\partial z_2^{\\alpha_2}}{\\partial {\\bar{z}_2}}^{\\beta_2}}({\\widetilde e}_\\delta)\\biggr| \\lesssim \\delta^{\\frac{t_l}{\\eta}},$\n\t       \\quad \\text{where $\\alpha_2, \\beta_2 \\geq 0.$}   \\label{rderivative1}\n\t\\item $\\biggl| \\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }({\\widetilde e}_\\delta)\\biggr| \\approx                       \\delta^{\\frac{p_\\nu}{\\eta}},$ \\quad \\text{where $\\alpha_2^\\nu > 0$ and $\\beta_2^\\nu > 0.$}\\label{rderivative2}\n\\end{enumerate}\n\\end{lem}\n\n\n\\begin{proof}\nBy (\\ref{r form 2}) and $t_l < [t_l]+1$, we have \n\\begin{equation}\\label{rdifferentiation}\n  \\biggl|\\frac{\\partial^l r}{{\\partial z_2^{\\alpha_2}}{\\partial   {\\bar{z}_2}}^{\\beta_2}}({\\widetilde e}_\\delta)\\biggr| \\lesssim \\delta^{\\frac{t_l}{\\eta}} + |e_\\delta| + \\delta^{\\frac{[t_l]+1}{\\eta}} \\lesssim \\delta^{\\frac{t_l}{\\eta}}.\n\\end{equation}  \nFor (\\ref{rderivative2}), note that if $l = q_\\nu,$ then $t_l = p_\\nu$. Therefore,\n(\\ref{repre of r}) gives\n\\begin{equation*}\n |M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| - C_1(|e_\\delta| + \\delta^{\\frac{p_\\nu + 1}{\\eta}}) \\leq \\biggl|\\frac{1}{{\\alpha_2^\\nu}!{\\beta_2^\\nu}!}\\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }(\\widetilde{e_\\delta})\\biggr|  \\leq |M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| + C_1(|e_\\delta| + \\delta^{\\frac{{p_\\nu} + 1}{\\eta}})\n\\end{equation*}\nfor some constant $C_1.$\nSince Remark \\ref{sizeofM} means $|M_{\\alpha_2^\\nu, \\beta_2^\\nu}(d\\delta^{\\frac{1}{\\eta}})| \\approx \\delta^{\\frac{p_\\nu}{\\eta}},$  we have $$\\biggl|\\frac{\\partial^{q_\\nu} r }{{\\partial {z_2}^{\\alpha_2^\\nu}}{\\partial {\\bar z_2}^{\\beta_2^\\nu}} }({\\widetilde e}_\\delta)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$$ \n\\end{proof}\n\n\n\n\n\n\\begin{lem} \\label{rhoderivative}\nLet $\\rho_l, \\phi^l$ and $\\Phi^l$ be given as in (\\ref{rho2changeofcoordinate})-(\\ref{rholexpression}) for  $l=2, \\cdots, m+1$ and $\\alpha_2^\\nu$ and $\\beta_2^\\nu$ be positive numbers as given in Lemma \\ref{existence of mixed term in z_2} for $\\nu = 1, \\cdots, N.$  Then\n\\begin{enumerate}[\\normalfont (i)] \n  \\item $\\biggl|\\frac{\\partial^k \\rho_l}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial {{\\bar \\zeta}_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}} \\quad \\text{for each } \\ k = 1, \\cdots, m.$\n  \\item $\\biggl|\\frac{\\partial^{q_\\nu} \\rho_l }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr|  \\approx \\delta^{\\frac{p_\\nu}{\\eta}} \\quad \\text{for each } \\ \\nu = 1, \\cdots, N.$\n\\end{enumerate}  \n  \nIn particular, $|c_l({\\widetilde e}_\\delta)| \\lesssim \\delta^{\\frac{t_l}{\\eta}},$ where $c_l({\\widetilde e}_\\delta)$ is given in (\\ref{clexpression}).\n\\end{lem} \n\n\n\\begin{proof}\nBy induction, we prove both (i) and (ii). For part (i), let $l = 2.$  Since $\\rho_2(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = r(d\\delta^{\\frac{1}{\\eta}}, z'') \\circ \\Phi_{\\widetilde{e}_\\delta}^1 (\\zeta''),$ by chain rule and Lemma \\ref{rderivative}, we have\n\\begin{equation*}\n\\biggl|\\frac{\\partial^k \\rho_2}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \n          \\lesssim \\biggl|\\frac{\\partial^k r}{{\\partial z_2^{\\alpha_2}}{\\partial {\\bar z_2}^{\\beta_2}}}({\\widetilde e}_\\delta)\\biggr| \n           + \\biggl|\\frac{\\partial r}{\\partial z_2}({\\widetilde e}_\\delta) \\biggr| \n          \\lesssim \\delta^{\\frac{t_k}{\\eta}}+\\delta^{\\frac{t_1}{\\eta}} \\lesssim \\delta^{\\frac{t_k}{\\eta}}.\n\\end{equation*}\nfor all $ k = 1, \\cdots, m.$\nThis proves for the case $l = 2.$ \nNow, by induction, we assume\n$$\\biggl|\\frac{\\partial^k \\rho_l}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}}$$ for all $k = 1, \\cdots, m$ and $l = 2, \\cdots, j.$ \\\nNote that\n\\begin{equation}\\label{rholrealexpression}\n\\rho_{j+1}(d\\delta^{\\frac{1}{\\eta}}, \\zeta_2, \\zeta_3) = \\rho_j(d\\delta^{\\frac{1}{\\eta}},\\ \\zeta_2,\\ \\zeta_3 - 2 c_j({\\widetilde e}_\\delta)\\zeta_2^j).\n\\end{equation}\nIf  $k < j,$  the inductive assumption gives\n$$\\biggl|\\frac{\\partial^k \\rho_{j+1}}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial {\\bar \\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| = \\biggl| \\frac{\\partial^k \\rho_{j}}{{\\partial w_2^{\\alpha_2}}{\\partial {\\bar w_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{t_k}{\\eta}}.$$\nNow, let $k = j.$  If $\\alpha_2 > 0$ and $\\beta_2 > 0,$ we have the same result as the previous one. Otherwise, $\\frac{\\partial^j \\rho_{j+1}}{\\partial \\zeta_2^j}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) = \\frac{\\partial^j \\rho_{j}}{\\partial w_2^{j}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) - 2j!c_j({\\widetilde e}_\\delta)\\frac{\\partial \\rho_j}{\\partial w_3}(d\\delta^{\\frac{1}{\\eta}}, 0, 0) = 0.$  \nIf $k > j,$ the inductive assumption gives\n\\begin{equation*}\n\\biggl|\\frac{\\partial^k \\rho_{j+1}}{{\\partial \\zeta_2^{\\alpha_2}}{\\partial \\bar{\\zeta_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr|  \\lesssim \\biggl| \\frac{\\partial^k \\rho_{j}}{{\\partial w_2^{\\alpha_2}}{\\partial \\bar{w_2}^{\\beta_2}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| + |c_j({\\widetilde e}_\\delta)| \\lesssim \\delta^{\\frac{t_k}{\\eta}}+\\delta^{\\frac{t_j}{\\eta}} \\lesssim \\delta^{\\frac{t_k}{\\eta}}.\n\\end{equation*}  \n\n\nFor part (ii), let $l = 2$ and  apply the chain rule again to $\\rho_2,$  we have \n\\begin{equation*}\n\\biggl|\\frac{\\partial^{q_\\nu} r}{{\\partial z_2^{\\alpha_2^\\nu}}{\\partial \\bar{z_2}^{\\beta_2^\\nu}}}(\\widetilde{e_\\delta})\\biggl| - C\\biggl|\\frac{\\partial r}{\\partial z_2}(\\widetilde{e_\\delta}) \\biggr| \\leq \\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr|  \\leq \\biggl|\\frac{\\partial^{q_\\nu} r}{{\\partial z_2^{\\alpha_2^\\nu}}{\\partial \\bar{z_2}^{\\beta_2^\\nu}}}(\\widetilde{e_\\delta})\\biggl| + C\\biggl|\\frac{\\partial r}{\\partial z_2}(\\widetilde{e_\\delta}) \\biggr| \n\\end{equation*}\nfor some constant $C.$  Then, Lemma \\ref{rderivative} means \n\\begin{equation}\\label{lowerbound2}\n\\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_1}{\\eta}}  \\lesssim  \\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim  \\delta^{\\frac{p_\\nu}{\\eta}} + \\delta^{\\frac{t_1}{\\eta}}.\n\\end{equation}\nSince $1 < q_\\nu$ for each $\\nu = 1, \\cdots, N,$ it gives $p_\\nu = t_{q_\\nu} < t_1.$ Therefore, we have \n\\begin{equation*}\n\\biggl|\\frac{\\partial^{q_\\nu} \\rho_2}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.\n\\end{equation*}\nThis proves the statement for the case  $l = 2.$ By induction,  assume\n$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_l}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$\nFirst, consider the case when $q_\\nu \\leq l.$ Since  $\\alpha_2^\\nu > 0$ and $\\beta_2^\\nu > 0,$  by the similar argument as in the proof of (i) and the by inductive assumption, we have    $$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial  \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| = \\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial {{\\bar w}_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{t_{q_\\nu}}{\\eta}} = \\delta^{\\frac{{p_\\nu}}{\\eta}} .$$   \nNow, consider the case when $q_\\nu > l,$ If we take the derivative of $\\rho_{l+1}$ in (\\ref{rholrealexpression}) about $\\zeta_2,$ the derivative related to the third component involves $c_l(\\widetilde{e_\\delta}).$ Therefore, we have\n\\begin{align*}\n\\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial \\bar{w_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| - C'|c_l(\\widetilde{e_\\delta})|  \\leq \\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| & \\leq \\biggl| \\frac{\\partial^{q_\\nu} \\rho_{l}}{{\\partial w_2^{\\alpha_2^\\nu}}{\\partial \\bar{w_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\\\ \n& + C'|c_l(\\widetilde{e_\\delta})|\n\\end{align*}\nfor some constant $C'$.   \nTherefore, the inductive assumption and part (i) means \n\\begin{equation}\\label{lowerboundl}\n\\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_l}{\\eta}} \\lesssim \\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\lesssim \\delta^{\\frac{p_\\nu}{\\eta}} - \\delta^{\\frac{t_l}{\\eta}}\n\\end{equation}\nSince $q_\\nu > l$, it means $p_\\nu = t_{q_\\nu} < t_l.$ Thus, we have \n$\\biggl|\\frac{\\partial^{q_\\nu} \\rho_{l+1}}{{\\partial \\zeta_2^{\\alpha_2^\\nu}}{\\partial \\bar{\\zeta_2}^{\\beta_2^\\nu}}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)\\biggr| \\approx \\delta^{\\frac{p_\\nu}{\\eta}}.$\n\n\\end{proof}\n\nFinally, we show that the derivatives of $\\rho$ can be bounded from below.\n\n\n\n\\begin{rem}\\label{Amnonzero}\nTake $\\nu = N.$ Since $\\biggl|\\frac{\\partial^{q_\\nu} \\rho }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr| \\approx |A_m ({\\widetilde e}_\\delta)| ,$ Lemma \\ref{rhoderivative} means $|A_m (\\widetilde{e_\\delta})| \\approx 1.$\n\\end{rem} \n\n\nNow, we recall some facts in \\cite{C2} before showing the holomorphic function defined in the complex two dimensional slice(i.e $z_1$ is fixed) is well-defined when we move $z_1$ in a small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}.$\n\n\\begin{thm}[\\bf Catlin]\\label{existenceofholomorphic}\nSuppose the defining function $\\rho$ for a pseudoconvex domain in $b\\Omega \\subset \\mathbb{C}^2$ has the following form:  $$\\rho(\\zeta) = \\mbox{\\normalfont{Re}}\\zeta_2 + \\sum_{\\substack{j+k=2 \\\\ j,k > 0 }}^m a_{j,k}{\\zeta_1}^j{\\bar{\\zeta_1}}^k + \\mathcal{O}(|\\zeta_2||\\zeta|+|\\zeta_1|^{m+1}).$$\nSet $$A_l = \\max \\{|a_{j, k}| ; j+k =l \\},  \\qquad l = 2, \\cdots, m.$$ and   $$J_\\delta(\\zeta) = (\\delta^2 + |\\zeta_2|^2 + \\sum\\limits_{k=2}^m (A_k)^2 |\\zeta_1|^{2k} )^{\\frac{1}{2}}.$$\nDefine\n$$\\Omega_{a, \\delta}^{\\epsilon_0} = \\{\\zeta ; |\\zeta_1| < a , |\\zeta_2|<a, \\rho(\\zeta) < \\epsilon J_\\delta(\\zeta) \\} \\quad \\text{for any small constant $a, \\epsilon_0 > 0.$}$$\nIf we have $|A_m| \\geq c_m > 0$ for some positive constant $c_m,$\nthen there exist small constants $a, \\epsilon_0 > 0$ so that for any sufficiently small $\\delta>0,$ there is a $L^2$ holomorphic function $f \\in A(\\Omega_{a, \\delta}^{\\epsilon_0})$ satisfying $\\biggl|\\frac{\\partial f}{\\partial\\zeta_2}(0, -\\frac{b\\delta}{2}) \\biggr| \\geq \\frac{1}{2\\delta}$ for some small constant $b$. Moreover, the values  $a$ and $\\epsilon_0$  depend only on the constant $c_m$  and  $C_{m+1} = \\left\\| \\rho \\right\\|_{C^{m+1}(U)},$ where $U$ is a small neighborhood of $0.$\n\\end{thm}\n \nThe result stated in \\cite{C2} applies to a more restricted situation, but a careful examination of the proof  actually implies the above result. To apply theorem \\ref{existenceofholomorphic} to the complex two dimensional slice, we consider the pushed out domain about ${\\widetilde e}_\\delta.$ Let $\\Phi_{\\widetilde{e}_\\delta}$ be the map associated with ${\\widetilde e}_\\delta$ as in (\\ref{coordinatechangeinc2}). Set ${U'' \\big|}_{z_1 = d\\delta^{\\frac{1}{\\eta}}} = \\{\\zeta''=(\\zeta_2, \\zeta_3) ; \\Phi_{\\widetilde{e}_\\delta} (\\zeta'') \\in U \\big|_{z_1 = d\\delta^{\\frac{1}{\\eta}}}\\}.$  For all small $\\delta,$  define \n\\begin{equation} \\label{def of Jdelta}\n J_\\delta (\\zeta'') = \\biggl(\\delta^2 + |\\zeta_3|^2 + \\sum_{k = 2}^{m} (A_k ({\\widetilde e}_\\delta))^2 |\\zeta_2|^{2k}  \\biggr)^{\\frac{1}{2}}\n\\end{equation} \nand the pushed-out domain with respect to the slice \n\\begin{equation} \\label{working domain} \n   \\Omega_{a, \\delta}^{\\epsilon_0} = \\{(\\zeta_2, \\zeta_3) ; |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\  \\rho (d\\delta^{\\frac{1}{\\eta}}, \\zeta'') < {\\epsilon_0} J_\\delta(\\zeta'') \\}.  \n\\end{equation} \nBy Theorem \\ref{existenceofholomorphic}, we have a $L^2$ holomorphic function $f$ in $ \\Omega_{a, \\delta}^{\\epsilon_0}$ satisfying\n\\begin{equation}\\label{large derivative}\n \\biggl| \\frac{\\partial f}{\\partial \\zeta_3} ( 0, -\\frac{b\\delta}{2})\\biggr| \\geq \\frac{1}{2\\delta}.\n\\end{equation}\nIn order to show the well-definedness of the holomorphic function $f$ when $z_1$ moves in a small neighborhood of $z_1 = d\\delta^{\\frac{1}{\\eta}}$, we use $\\Phi_{\\widetilde{e}_\\delta} $ given as in (\\ref{coordinatechangeinc2}) and define\n\\begin{equation*}\n\\Phi(\\zeta_1, \\zeta_2, \\zeta_3) = (\\zeta_1, \\zeta_2, \\Phi_3 (\\zeta)), \n\\end{equation*}\nwhere $\\Phi_3 (\\zeta)$ is defined by\n\n\\begin{equation}\\label{phitobeused}\n \\Phi_3 (\\zeta) = e_\\delta + \\biggl(\\frac{\\partial r}{\\partial z_3}({\\widetilde e}_\\delta) \\biggr)^{-1}\\biggl(\\frac{\\zeta_3}{2}- \\sum_{l=2}^m c_l({\\widetilde e}_\\delta)\\zeta_2^l                                 - \\frac{\\partial r}{\\partial  z_2} ({\\widetilde e}_\\delta){\\zeta_2}  \\biggr)   \n\\end{equation}\nand define\n\n\\begin{equation}\\label{rhotobeused} \n  \\rho(\\zeta_1, \\zeta_2, \\zeta_3) = r(z_1, z_2, z_3)\\circ \\Phi(\\zeta_1, \\zeta_2, \\zeta_3).\n\\end{equation}\nIn particular, when we fix $z_1 = d\\delta^{\\frac{1}{\\eta}},$  we have the holomophic function $f$ defined in the slice  $\\Omega_{a, \\delta}^{\\epsilon_0}$ satisfying (\\ref{large derivative}).  Now, we consider the domain given by the family of the pushed out domains of the slice along with $\\zeta_1$ axis and the domain in the new coordinate of $\\Omega$ by $\\Phi$. \nDefine $$\\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0} = \\{\\zeta \\in \\mathbb{C}^3; |\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\  \\rho (d \\delta^{\\frac{1}{\\eta}}, \\zeta'') < {\\epsilon_0} J_\\delta(\\zeta'') \\}$$ and \n$${\\Omega}_{a, \\delta, \\zeta_1} = \\{\\zeta \\in \\mathbb{C}^3; |\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a, |\\zeta_3| < a \\ \\mbox{and} \\  \\rho (\\zeta_1, \\zeta'') < 0 \\} $$ for some small $c > 0$ only depending on $\\epsilon_0.$ \nSince the holomorphic function $f(\\zeta_2, \\zeta_3)$ defined in $\\Omega_{a, \\delta}^{\\epsilon_0}$ is independent of $\\zeta_1,$  $f$ is the well-defined holomophic function in $\\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}.$    \nWe want to show $f$ is well-defined holomorphic function in ${\\Omega}_{a, \\delta, \\zeta_1}.$ Therefore, it is enough to show ${\\Omega}_{a, \\delta, \\zeta_1} \\subset \\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ for the well-definedness of $f$ in ${\\Omega}_{a, \\delta, \\zeta_1}$. More specifically, \n\\begin{align*}\n{\\Omega}_{a, \\delta, \\zeta_1} \\subset \\Omega_{a, \\delta, \\zeta_1}^{\\epsilon_0}\n            \n             &\\  \\Leftrightarrow  \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') -  \\rho (\\zeta_1, \\zeta'') < {\\epsilon_0}J_\\delta(\\zeta''),\n\\end{align*}\nwhere $\\zeta'' = (\\zeta_2, \\zeta_3)$ and $|\\zeta_1-d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a$. \n\n\n\n\\begin{prop} \\label{well-defined property}\nGiven any small $\\epsilon \\leq {\\epsilon_0},$ there is a small $c > 0$ such that if $|\\zeta_1-d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a,$ then\n       $$|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') -  \\rho (\\zeta_1, \\zeta'')| \\lesssim {\\epsilon}J_\\delta (\\zeta'').$$       \n\\end{prop}\n\n\nBefore proving Proposition \\ref{well-defined property}, we note that from the standard interpolation method, we have the following fact: Let $(p_1, q_1), (p, q)$ and $(p_2, q_2)$ be collinear points in the first quadrant of the plane, and $ p_1 \\leq p \\leq p_2, q_2 \\leq q \\leq q_1.$ Then, we have\n$$|\\zeta_1|^{p}|\\zeta_2|^q \\leq |\\zeta_1|^{p_1}|\\zeta_2|^{q_1} + |\\zeta_1|^{p_2}|\\zeta_2|^{q_2}$$ for sufficiently small $\\zeta_1 , \\zeta_2 \\in \\mathbb{C}$. In particular, this means that if $(\\alpha, \\beta) \\in \\Gamma_L,$ then \n\\begin{equation} \\label{interpolation}\n|\\zeta_1|^{\\alpha_1 + \\beta_1} |\\zeta_2|^{\\alpha_2 + \\beta_2} \\lesssim |\\zeta_1|^{p_{\\nu-1}} |\\zeta_2|^{q_{\\nu-1}} + |\\zeta_1|^{p_{\\nu}} |\\zeta_2|^{q_{\\nu}}  \n\\end{equation} for some $\\nu = 1, \\cdots, N.$ \n \n\\begin{proof}[Proof of Proposition \\ref{well-defined property}]\nDefine \n $${J_\\delta}^\\nu(\\zeta'') = \\delta + |\\zeta_3| + \\sum_{\\nu = 1}^N {\\delta^{\\frac{p_\\nu}{\\eta}}}|\\zeta_2|^{q_\\nu}.$$\n\nIn order to show the proposition, it is enough to show ${J_\\delta}^\\nu(\\zeta'') \\lesssim J_\\delta (\\zeta'')$ and $|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta_2, \\zeta_3) -  \\rho (\\zeta_1, \\zeta_2, \\zeta_3)| \\lesssim {\\epsilon}{J_\\delta}^\\nu (\\zeta''),$ where $|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}, |\\zeta_2| < a \\ \\mbox{and} \\ |\\zeta_3| < a.$ \nBy (\\ref{def of Al}) and $a_{j,k}(\\widetilde{e_\\delta}) = j!k! \\frac{\\partial^{j+k} \\rho}{{\\partial {\\zeta_2}^j}{\\partial {\\bar{\\zeta_2}}^k}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0),$ we have $$|\\frac{\\partial^{j+k} \\rho}{{\\partial {\\zeta_2}^j}{\\partial {\\bar{\\zeta_2}}^k}}(d\\delta^{\\frac{1}{\\eta}}, 0, 0)| \\lesssim |A_l(\\widetilde{e_\\delta})| $$ for $ j+k = l$ with $l =2,\\cdots,m.$\nTherefore, Lemma \\ref{rhoderivative} means that \n$$ \\delta^{\\frac{p_\\nu}{\\eta}} \\approx \\biggl|\\frac{\\partial^{q_\\nu} \\rho }{{\\partial {\\zeta_2}^{\\alpha_2^\\nu}}{\\partial {\\bar \\zeta_2}^{\\beta_2^\\nu}} }(d\\delta^{\\frac{1}{\\eta}}, 0, 0) \\biggr| \\lesssim |A_{q_\\nu}(\\widetilde{e_\\delta})|,$$\nwhere $\\alpha_2^\\nu + \\beta_2^\\nu = q_\\nu, \\alpha_2^\\nu \\ \\mbox{and} \\ \\beta_2^\\nu > 0.$ This shows ${J_\\delta}^\\nu(\\zeta'') \\lesssim J_\\delta (\\zeta'').$ \\\\\n\nLet's estimate $|\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') -  \\rho (\\zeta_1, \\zeta'')|$. Let $D_1$ denote the differential operator either $\\frac{\\partial}{\\partial \\zeta_1}$ or $\\frac{\\partial}{\\partial {\\overline \\zeta}_1}.$ Then,\n\\begin{equation} \\label{originalestimate}\n|\\rho (\\zeta_1, \\zeta'')- \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')| \n        \\leq c \\delta^{\\frac{1}{\\eta}} \\max\\limits_{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}} |D_1 \\rho (\\zeta_1, \\zeta'')|.\n\\end{equation}\nLet's estimate $ D_1 \\rho (\\zeta_1, \\zeta'').$ By (\\ref{r form 2}), (\\ref{phitobeused})  and (\\ref{rhotobeused}), we know\n\\begin{align*}\n\\rho(\\zeta_1, \\zeta'') &= \\mbox{Re}(\\Phi_3 (\\zeta) ) + \\sum_{\\Gamma_L} a_{\\alpha,                                                                                                              \\beta}{\\zeta_1}^{\\alpha_1}{\\bar{\\zeta}_1}^{\\beta_1}{\\zeta_2}^{\\alpha_2}{\\bar{\\zeta}_2}^{\\beta_2} +                                                                 \\mathcal{O}(|\\Phi_3(\\zeta)||(\\zeta_1, \\zeta_2, \\Phi_3(\\zeta))|    \\\\\n                                &\\ \\qquad +\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} |\\zeta_1|^{[t_l]+1}|\\zeta_2|^l  + |\\zeta_2|^{m+1}).\n\\end{align*}\nSince $|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}| < c\\delta^{\\frac{1}{\\eta}}$ and $\\Phi_3$ is independent of $\\zeta_1$, we have\n\\begin{equation} \\label{rhozeta1derivative}\n |{D_1 \\rho}(\\zeta_1, \\zeta'')|\n          \\lesssim \\sum_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1 -1}{\\eta}} |\\zeta_2|^{\\alpha_2 + \\beta_2} + |\\Phi_3 (\\zeta)| + \\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]}{\\eta}}|\\zeta_2|^l .  \n\\end{equation}\nCombining (\\ref{originalestimate}) with (\\ref{rhozeta1derivative}), we obtain\n\\begin{equation}\\label{finalestimatetobeused}   \n|\\rho (\\zeta_1, \\zeta'')- \\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'')|  \\lesssim c\\biggl( \\sum_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1}{\\eta}}|\\zeta_2|^{\\alpha_2 + \\beta_2} + |\\Phi_3 (\\zeta)| \\nonumber + \\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l  \\biggr)           \n\\end{equation}\n\n\n\\noindent With $\\zeta_1 = d\\delta^{\\frac{1}{\\eta}}$, (\\ref{interpolation}) means $\\sum\\limits_{\\Gamma_L}{\\delta}^{\\frac{\\alpha_1 + \\beta_1}{\\eta}}|\\zeta_2|^{\\alpha_2 + \\beta_2} \\lesssim {J_\\delta}^\\nu (\\zeta'').$ \nAlso, (\\ref{clexpression}) and Lemma \\ref{rhoderivative} gives $|\\Phi_3(\\zeta)| \\lesssim |e_\\delta| + |\\zeta_3| + \\sum_{l = 1}^m |c_l(\\widetilde{e_\\delta})||\\zeta_2|^l \\lesssim \\delta + |\\zeta_3| + \\sum_{l = 1}^m \\delta^{\\frac{t_l}{\\eta}}|\\zeta_2|^l.$ Since $(t_l, l ) \\in L_\\nu$  for some $\\nu = 1, \\cdots, N,$ again, (\\ref{interpolation}) gives $|\\Phi_3(\\zeta)| \\lesssim {J_\\delta}^\\nu (\\zeta'').$\nFurthermore, since $\\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l  \\lesssim \\delta^{\\frac{t_l}{\\eta}}|\\zeta_2|^l$, the same argument as before gives $\\sum_{\\nu = 1}^N \\sum_{l = q_{\\nu - 1}}^{q_\\nu} \\delta^{\\frac{[t_l]+1}{\\eta}}|\\zeta_2|^l \\lesssim {J_\\delta}^\\nu (\\zeta'').$ \n\\end{proof} \n\n\\vspace{0.5cm}\n\nNow, we know that there is a holomorphic function $f(\\zeta_1, \\zeta_2, \\zeta_3)= f(\\zeta_2, \\zeta_3)$ defined on ${\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ such that\n\\begin{enumerate}[i)]\n\t\\item  ${\\Omega}_{a, \\delta, \\zeta_1} \\subset {\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$\n\n\t\\item  $\\biggl| \\frac{\\partial f}{\\partial \\zeta_3} (0, -\\frac{b\\delta)}{2}\\biggr| \\geq \\frac{1}{2\\delta}$ for a small constant $b > 0.$ \n\\end{enumerate}\n\nWithout loss of generality, we can assume ${\\Omega}_{a, \\delta, \\zeta_1} \\subset {\\Omega}_{a, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}} \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0}.$ For the boundedness of $f$ in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}},$ we follow the same argument as Chapter 7 (p 462) in \\cite{C2}. Before showing the boundedness, we define a polydisc $P_{a_1} ({\\zeta''_0})$ by\n$$P_{a_1} (\\zeta''_0) = \\{\\zeta'' = (\\zeta_2, \\zeta_3); |\\zeta_2 - {\\zeta_2^0}| < \\tau (\\widetilde{e_\\delta}, a_1 J_\\delta (\\zeta''_0)) \\ \\mbox{and} \\\n |\\zeta_3 - {\\zeta_3^0}| <  a_1 J_\\delta (\\zeta''_0)\\}, $$ \nwhere $\\zeta''_0 = (\\zeta_2^0, \\zeta_3^0)$ and $a_1 > 0.$\n\n\n\n\n\n\n\\begin{thm}\\label{boundedholomorphicfunction}\n$f$ is bounded holomorphic function in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$ such that \n\\begin{equation}\\label{largederivative}\n\\biggl| \\frac{\\partial f}{\\partial \\zeta_3} \\biggl(0, -\\frac{b\\delta}{2}\\biggr)\\biggr| \\geq \\frac{1}{2\\delta} \\ \\text{for a small constant $b > 0$}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nSince $f$ is a $L^2$ holomorphic function in ${\\Omega}_{a, \\delta, \\zeta_1}^{\\epsilon_0}$ with (\\ref{largederivative}), it is enough to show $f$ is bounded in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$. Let $(\\zeta_2^0, \\zeta_3^0) \\in \\{\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\frac{\\epsilon_0}{2}J_\\delta(\\zeta''), |\\zeta_2| < \\frac{3a}{4}, |\\zeta_3|<\\frac{3a}{4}\\} \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0}.$ By the similar property as (iii) of Proposition 4.3 in \\cite{C2}, if $\\zeta''_0 = (\\zeta_2^0, \\zeta_3^0) \\in \\{\\rho(d\\delta^{\\frac{1}{\\eta}}, \\zeta'') = \\frac{\\epsilon_0}{2}J_\\delta(\\zeta''), |\\zeta_2| < \\frac{3a}{4}, |\\zeta_3|<\\frac{3a}{4}\\}$, then\n$$ P_{a_1} (\\zeta''_0) \\subset {\\Omega}_{a, \\delta, \\zeta}^{\\epsilon_0},$$\nfor some small constant $a_1 > 0.$ We can apply the same argument as Chapter 7 (p 462) in \\cite{C2} to obtain $|f(\\zeta_2^0, \\zeta_3^0)| \\lesssim 1$. \nFor all others  points on the boundary and interior of  ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon_0}{2}}$, we can choose the polydics with fixed radius which is contained in ${\\Omega}_{{a}, \\delta, \\zeta_1}^{{\\epsilon_0}}$ and apply the same argument as Chapter 7 in \\cite{C2}.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem 1.1} \\label{Sec5}\n\nIn this section, we prove our main theorem. Before proving the Theorem, let's recall the notations for H\\\"older norm and H\\\"older space. For $U \\in \\mathbb{C}^n$, we denote by  ${\\lVert u \\rVert}_{L_{\\infty}(U)}$ the essential supremum of $u \\in L_{\\infty}(U)$ in $U$. For a real $0 < \\epsilon < 1$, set\n \n $${\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)} = {\\lVert u \\rVert}_{L_{\\infty}(U)} + \\mbox{sup}_{z,w \\in U} \\frac{|u(w)-u(z)|}{|w-z|^\\epsilon}, $$ \n $$ \\Lambda^{\\epsilon} (U) = \\{ u : {\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)} < \\infty \\} $$\nIn here, ${\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U)}$ denote the H\\\"older norm of order $\\epsilon$. \n\nBy theorem \\ref{special coordinate}, we can assume $\\Omega = \\{z \\in \\mathbb{C}^3; r(z)< 0\\}$ and restate Theorem \\ref{main_theorem}: \n\n\n\\begin{thm}\nLet $\\Omega = \\{ r(z) < 0\\}$ be a smoothly bounded pseudoconvex domain in $\\mathbb{C}^3,$ where $r$ given by theorem \\ref{special coordinate}.  Furthermore,\nif there exists a neighborhood $U$ of $0$ so that for all $\\alpha \\in L_{\\infty}^{0,1} ({\\Omega})$ with $\\bar{\\partial}\\alpha = 0$, there is a $u \\in \\Lambda_{\\epsilon} (U \\cap \\overline{\\Omega})$ and $C>0$ such that $\\bar{\\partial}u =\\alpha$ and \n\n\\begin{equation} \\label{holder estimate in z}\n{\\lVert u \\rVert}_{\\Lambda^{\\epsilon}(U \\cap \\overline{\\Omega})} \\leq C{\\lVert \\alpha \\rVert}_{L_{\\infty}(\\Omega),}\n\\end{equation}\nthen $\\epsilon \\leq \\frac{1}{\\eta}$.\n\\end{thm}\n\n\n\n\n\n\\begin{proof}Let us consider $U' = \\{(\\zeta_1, \\zeta_2, \\zeta_3) ; \\Phi(\\zeta_1, \\zeta_2, \\zeta_3) \\in U \\}$ and $\\rho = r \\circ \\Phi$ as (\\ref{phitobeused}) and (\\ref{rhotobeused}). Let's choose $\\beta =\\bar{\\partial}(\\phi(\\frac{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}|}{c\\delta^{\\frac{1}{\\eta}}})\\phi(\\frac{|\\zeta_2|}{a\/2})\\phi(\\frac{|\\zeta_3|}{a\/2})f(\\zeta_2, \\zeta_3))$, where \n\n\\begin{displaymath}\n\\phi (t) = \\left \\{\n     \\begin{array}{lr}\n       1 & ,  |t| \\leq \\frac{1}{2}\\\\\n       0 & ,  |t| \\geq \\frac{3}{4}\n     \\end{array}\n   \\right.\n\\end{displaymath}  \nNote that $f$ is the well-defined bounded holomorphic function in ${\\Omega}_{\\frac{a}{2}, \\delta, \\zeta_1}^{\\frac{\\epsilon}{2}}$ by Theorem \\ref{boundedholomorphicfunction}.\nIf we define $\\alpha = (\\Phi^{-1})^* \\beta,$ then $\\bar{\\partial}(\\Phi^* u) = \\Phi^* \\bar{\\partial} u = \\Phi^* \\alpha = \\beta$. Therefore, if we set $U_1 = \\Phi^* u =u\\circ \\Phi$, (\\ref{holder estimate in z}) means \n\\begin{equation} \\label{holder estimate in zeta}\n{\\lVert U_1 \\rVert}_{\\Lambda^{\\epsilon}(U' \\cap \\overline{\\Omega})} \\leq C{\\lVert \\beta \\rVert}_{L_{\\infty}} \n\\end{equation}\nIn here, we note that the definition of $\\beta$ means\n\\begin{equation}\\label{supnorminofbeta}\n{\\lVert \\beta \\rVert}_{L^\\infty} \\lesssim \\delta^{-\\frac{1}{\\eta}}\n\\end{equation}\nNow, let $h(\\zeta_1, \\zeta_2, \\zeta_3) = U_1(\\zeta_1, \\zeta_2, \\zeta_3) -  \\phi(\\frac{|\\zeta_1 - d\\delta^{\\frac{1}{\\eta}}|}{c\\delta^{\\frac{1}{\\eta}}})\\phi(\\frac{|\\zeta_2|}{a\/2})\\phi(\\frac{|\\zeta_3|}{a\/2})f(\\zeta_2, \\zeta_3).$  Then $\\bar{\\partial} U_1 = \\beta$ means $h$ is holomorphic.\nSet  $ q_1^\\delta(\\theta)= (d\\delta^{\\frac{1}{\\eta}}+\\frac{4}{5}c\\delta^{\\frac{1}{\\eta}} e^{i\\theta}, 0, -\\frac{b\\delta}{2}) \\ \\mbox{and} \\ q_2^\\delta(\\theta) = ( d\\delta^{\\frac{1}{\\eta}}+\\frac{4}{5}c\\delta^{\\frac{1}{\\eta}}e^{i\\theta}, 0, -b\\delta)$, where $\\theta \\in \\mathbb{R}$.\nFrom now on, we estimate the lower bound and upper bound of the integral \n\n\\begin{equation*}\n H_{\\delta} = \\biggl| \\frac{1}{2\\pi} \\int_0^{2\\pi} [h(q_1^\\delta(\\theta))-h(q_2^\\delta(\\theta))] d\\theta \\biggr|. \n\\end{equation*}\nFrom the definition of $\\phi,$ (\\ref{holder estimate in zeta}),  and (\\ref{supnorminofbeta}) we have \n\\begin{equation} \\label{upperbound}  \n    H_{\\delta} = \\biggl|\\frac{1}{2\\pi} \\int_0^{2\\pi} [U_1(q_1^\\delta (\\theta))-U_1(q_2^\\delta (\\theta))] d\\theta \\biggr| \\lesssim \\delta^{\\epsilon} {\\lVert \\beta \\rVert}_{L^\\infty}  \\lesssim \\delta^{\\epsilon-\\frac{1}{\\eta}}   \n\\end{equation}     \n\n\n\n\nOn the other hand, for the lower bound estimate, we start with an estimate of the holomorphic function $f$ with a large nontangential derivative we constructed in theorem \\ref{boundedholomorphicfunction}.  The Taylor's theorem of $f$ in $\\zeta_3$ and Cauchy's estimate means  \n\n $$f(0, \\zeta_3) = f(0, -\\frac{b\\delta}{2}) + \\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(\\zeta_3 + \\frac{b\\delta}{2})\n      + \\mathcal{O}(|\\zeta_3 + \\frac{b\\delta}{2}|^2). $$\nNow, if we take $\\zeta_3 = -b\\delta$, we have\n $$ f(0, -b\\delta) - f(0, -\\frac{b\\delta}{2})= \\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(-\\frac{b\\delta}{2})\n      + \\mathcal{O}(\\delta^2).$$ \nSince   $|\\frac{\\partial{f}}{\\partial{z_3}} (0, -\\frac{b\\delta}{2} )| \\geq \\frac{1}{2\\delta},$  we know\n\\begin{equation} \\label{contradictionequation}\n  |f(0, -b\\delta) - f(0, -\\frac{b\\delta}{2})| = \\biggl|\\frac{{\\partial{f}}}{{\\partial{\\zeta_3}}}(0, -\\frac{b\\delta}{2})(-\\frac{b\\delta}{2})\n      + \\mathcal{O}(\\delta^2)\\biggr| \\gtrsim 1 \n\\end{equation} \nfor all sufficiently small $\\delta > 0$.\nReturning to the lower bound estimate of $H_{\\delta},$ the Mean Value Property, (\\ref{holder estimate in zeta}),  (\\ref{supnorminofbeta}), and (\\ref{contradictionequation}) give\n\\begin{align}\n    H_{\\delta} &=  \\biggl| \\frac{1}{2\\pi} \\int_0^{2\\pi}  [h(q_1^\\delta (\\theta))) \n        -h(q_2^\\delta (\\theta)) ]d\\theta \\biggr| = \\left|h(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2}  )- h(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta) )\\right|  \\nonumber \\\\\n     &= \\left|U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2}) - f(0, -\\frac{b\\delta}{2})- U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta) + f(0, -b\\delta)\\right| \\nonumber \\\\\n     &\\geq \\left|f(0, -b\\delta)-f(0, -\\frac{b\\delta}{2})| -|U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -\\frac{b\\delta}{2})-U_1(d\\delta^{\\frac{1}{\\eta}}, 0, -b\\delta)\\right| \\nonumber \\\\\n     &\\gtrsim 1 - \\delta^{\\epsilon-\\frac{1}{\\eta}} \\label{lowerbound} \n \\end{align}\nIf we combine (\\ref{upperbound}) with (\\ref{lowerbound}), we have \n      \n\\begin{equation} \\label{last_estimate}\n           1 \\lesssim \\delta^{\\epsilon-\\frac{1}{\\eta}}. \n\\end{equation}\nIf we assume $\\epsilon > \\frac{1}{\\eta}$ and  $\\delta \\rightarrow 0$, (\\ref{last_estimate}) will be a contradiction. Therefore,  $\\epsilon \\leq \\frac{1}{\\eta}.$ \n\\\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nModern advancement of machine learning is heavily driven by the advancement of computational power and techniques. Nowadays, it is not unusual to train a single model using hundreds of computational devices such as GPUs. As a result, scaling up training algorithms in the distributed setting has attracted intensive interests over the years. One important direction is communication efficient distributed training, which enhances the scalability of the training system by reducing the communication cost. Example techniques include quantization~\\citep{pmlr-v70-zhang17e,Wangni2018-ux}, decentralization~\\citep{Lian2017-ni, Koloskova*2020Decentralized, NIPS2018_8028}, and asynchronous communication~\\citep{DBLP:journals\/corr\/ZhengMWCYML16, NIPS2015_6031}.\n\nOne widely used strategy for alleviating the communication overhead is gradient compression, Before communication, the original gradient $\\bm{g}$ will be compressed into $\\mathcal{C}_{\\omega}[\\bm{g}]$, where $\\mathcal{C}_{\\omega}[\\cdot]$ {\\footnote{$\\mathcal{C}_{\\omega}[\\cdot]$ could also include randomness.}} is the compress operator. As a result the communication volume could be greatly reduced. However, this gradient compression could slow down the convergence speed because important information might get lost during the compression. To recover this information lost, error-compensated compression strategy was proposed: Instead of compressing the gradient at $t$-th iteration directly, we would first add back the compression error from the last step and then do the compression. Recent studies \\citep{martinmemory}\nobserved that by using error-compensated compression, the asymptotic convergence speed remains unchanged for \\textbf{SGD} even using 1-bit compression.\n\n\nOn the other hand, many state-of-the-art models have to be trained using a more complicated variant, \\textbf{Adam} \\citep{adam}. For example, to train models such as BERT, one has to resort to the \\textbf{Adam} optimizer, since training it with vanilla\/momentum \\textbf{SGD} has been shown to be less effective. Unfortunately, we find that error-compensated compression does not work for \\textbf{Adam}, because \\textbf{Adam} is non-linearly dependent on the gradient which affects the error compensation mechanism (see Section~\\ref{sec:moti-convergence} and~\\ref{intuition:why_adam_fails} for more details).\n\nIn this paper, we first analyze the limitation of directly applying existing compression technique to \\textbf{Adam}. One of our key findings is that Adam's variance (the non-linear term) becomes stable at early stage of training (Section~\\ref{sec:moti-variance}). This motivates us to design a new 2-stage algorithm, {{{\\textbf{1-bit Adam}}}}, which uses \\textbf{Adam} (warmup stage) to ``pre-condition'' a communication compressed momentum \\textbf{SGD} algoirthm (compression stage). We provide theoretical analysis on communication compressed momentum \\textbf{SGD}, which is the core component of {{{\\textbf{1-bit Adam}}}}. We design a custom collective primitive using MPI to transfer the $5\\times$ communication volume reduction (achieved by our algorithm) into actual runtime speedup, which is hard to accomplish using existing DL framework libraries. Experiments with BERT-Base, BERT-Large, SQuAD 1.1 and ResNet-18 training tasks on up to 256 GPUs show that {{{{\\textbf{1-bit Adam}}}}} converges as fast as uncompressed \\textbf{Adam}, and runs up to $3.3\\times$ faster than uncompressed algorithms.\n\n{\\bf (Contributions)}\nWe make the following contributions:\n\\vspace{-0.3cm}\n\\begin{itemize}\n\\item We propose a new algorithm, {{{\\textbf{1-bit Adam}}}}, a communication efficient momentum \\textbf{SGD} algorithm pre-conditioned with \\textbf{Adam} optimizer, which to the best of our knowledge is the first work that apply a pre-conditioned strategy for compressed momentum \\textbf{SGD}. We present theoretical analysis on the convergence of {{{\\textbf{1-bit Adam}}}}, and show that it admits the same asymptotic convergence rate as the uncompressed one.\n\n\\item We conduct experiments on large scale ML tasks that are currently challenging for \\textbf{SGD} to train. We show that on both BERT pre-training, SQuAD fine-tuning and ResNet-18, {{{\\textbf{1-bit Adam}}}} is able to achieve the same convergence behaviour and final accuracy as \\textbf{Adam}, together with up to $5\\times$ less communication volume and $3.3\\times$ faster end-to-end throughput (including the full-precision warmup stage). To our best knowledge, this is the first distributed learning algorithm with communication compression that can train a model as demanding as BERT.\n\n\\item We implement a custom collective communication primitive using Message Passing Interface (MPI) to provide a scalable and efficient communication system for 1-bit Adam. The communication as well as the 1-bit Adam optimizer has been open sourced in a deep learning optimization library called DeepSpeed\\footnote{https:\/\/github.com\/microsoft\/DeepSpeed, https:\/\/www.deepspeed.ai\/}.\n\\end{itemize}\n\n\\section{Related Work}\n\\paragraph{Communication-efficient distributed learning:}\nTo further reduce the communication overhead, one promising direction is to compress the variables that are sent between different workers ~\\citep{NIPS2019_8694,NIPS2019_9473}. Previous work has applied a\nrange of techniques such as quantizaiton,\nsparsification, and sketching\n~\\citep{Alistarh2017-yh,Agarwal2018-hg,Spring2019-ep,Ye2018-mf}.\nThe compression is mostly assumed to be unbiased ~\\citep{Wangni2018-ux,pmlr-v80-shen18a,pmlr-v70-zhang17e,NIPS2017_6749,NIPS2018_7519}.\nA general theoretical analysis of centralized compressed parallel \\textbf{SGD} can be found in ~\\citet{Alistarh2017-yh}. Beyond this, some biased compressing methods are also proposed and proven to be quite efficient in reducing the communication cost. One example is the  \\textbf{1-bit SGD} ~\\citep{1-bitexp}, which compresses the entries in gradient vector into $\\pm 1$ depends on its sign. \n\n\\paragraph{Error-compensated compression:}\nThe idea of using error compensation for compression is proposed in ~\\citet{1-bitexp}, where they find that by using error compensation the training could still achieves a very good speed even using $1$-bit compression. Recent study indicates that this strategy admits the same asymptotic convergence rate as the uncompressed one~\\citep{martinmemory}, which means that the influence of compression is trivial. More importantly, by using error compensation, it has been proved that we can use almost any compression methods~\\citep{martinmemory}, whereas naive compression could only converge when the compression is unbiased (the expectation of the compressed tensor is the same as the original).This method can be combined with decentralized training \\citep{ec_decentralize}, local SGD \\citep{ec_local}, accelerated algorithms \\citep{ec_linearly}.  Due to the promising efficiency of this method, error compensation has been applied into many related area ~\\citep{NIPS2019_9321,9051706,NIPS2019_8694,8884924,NIPS2019_9473,NIPS2019_8598,NIPS2019_9610,NIPS2019_9571} in order to reduce the communication cost. \n\n\\paragraph{\\textbf{Adam}:} \\textbf{Adam}~\\citep{Kingma2015AdamAM} has shown\npromising speed for many deep learning tasks, and also admits a very good robustness to the choice of the hyper-parameters, such as learning rate. \nIt can be viewed as an adaptive method that scales the learning rate with the magnitude of the gradients on each coordinate when running \\textbf{SGD}. Beyond \\textbf{Adam},  many other strategies that that shares the same idea of changing learning rate dynamically was studied. For example,  \\citet{JMLR:v12:duchi11a} (\\textbf{Adagrad}) and \\citep{rmsprop} (\\textbf{RESprop}),  use the gradient, instead of momentum, for updating the parameters;  \\textbf{Adadelta}~\\citep{DBLP:journals\/corr\/abs-1212-5701} changes the variance term of \\textbf{Adam} into a non-decreasing updating rule; \\citet{luo2018adaptive} proposed \\textbf{AdaBound} that gives both upper bound and lower bound for the variance term. In \\citet{adam_theoretical,adam_liu2020adam} authors develop a novel analysis for the convergence rate of \\textbf{Adam}. \n\n\\section{Motivation and Insights}\n\\subsection{Communication overhead affects the efficiency of distributed training}\n\\label{sec:moti-profile}\nTo demonstrate the opportunity for communication compression, we conduct performance profiling experiments that measures the impact of communication time with respect to the total training time per step. Here we use BERT-Large pre-training task as an example (sequence length 128, detailed training parameters can be found at Section~\\ref{sec:bert-eval}), since BERT and transformer models in general are the state-of-the-art approaches in natural language processing and many other areas. We evaluate two different kinds of clusters: the first cluster has 4 NVIDIA Tesla V100 GPUs per node, and different nodes are connected by 40 Gigabit Ethernet (effective bandwidth is 4.1 Gbps based on iperf benchmark); the second cluster has 8 V100 GPUs per node, and different nodes are connected by 100 Gigabit InfiniBand EDR (effective bandwidth is close to theoretical peak based on microbenchmark). We perform BERT-Large pre-training using the two clusters with different number of nodes and GPUs, batch sizes, and gradient accumulation steps. And we measure the average latency of forward, backward (allreduce and everything else), and step function calls. Table~\\ref{table_comm_overhead} presents the profiling results.\n\nResults show that allreduce communication contributes to a great portion of the training time per step, up to 94\\% and 75\\% for our experiments on two different kinds of inter-node networks. As expected, communication overhead is proportionally larger when the number of nodes is larger, when the batch size\/gradient accumulation step is smaller, and when the network bandwidth is lower. These are the situations where communication compression could provide the most benefit.\n\n\\begin{table*}\n  \\footnotesize\n  \\caption{BERT-Large pre-training sequence 128 profiling results.}\\label{table_comm_overhead}\n  \\centering\n  \\begin{tabular}{rrrrrrrrrrr}\n  \\hline\n  Cluster& Num.& Num.& Batch& Batch& Grad& Forward& Backward& Backward& Step& allreduce\\% \\\\\n  Network& node& GPU& size per& size& accum.& (ms)& allreduce& everything& (ms)&  \\\\\n  Type& & & GPU& & step& & (ms)& else (ms)& &  \\\\\n  \\hline\n  Ethernet& 16& 64& 1& 64& 1& 36.65& 2205.86& 33.63& 74.96& \\textbf{94\\%} \\\\\n  Ethernet& 16& 64& 16& 1024& 1& 35.71& 2275.43& 60.81& 75.59& 93\\% \\\\\n  Ethernet& 16& 64& 16& 4096& 4& 137.80& 2259.36& 243.72& 74.92& 83\\% \\\\\n  Ethernet& 8& 32& 16& 512& 1& 37.91& 2173.35& 60.71& 75.63& 93\\% \\\\\n  Ethernet& 4& 16& 16& 256& 1& 36.94& 2133.24& 62.82& 76.85& 92\\% \\\\\n  Ethernet& 2& 8& 16& 128& 1& 34.95& 1897.21& 61.23& 75.26& 92\\% \\\\\n  Ethernet& 1& 4& 16& 64& 1& 35.99& 239.76& 59.95& 74.21& 58\\% \\\\\n  \\hline\n  InfiniBand& 8& 64& 1& 64& 1& 25.36& 316.18& 23.25& 58.49& \\textbf{75\\%} \\\\\n  InfiniBand& 8& 64& 16& 1024& 1& 32.81& 336.40& 59.99& 57.79& 69\\% \\\\\n  InfiniBand& 8& 64& 16& 4096& 4& 131.04& 339.52& 237.92& 56.91& 44\\% \\\\\n  InfiniBand& 4& 32& 16& 512& 1& 33.45& 297.28& 56.81& 57.98& 67\\% \\\\\n  InfiniBand& 2& 16& 16& 256& 1& 32.86& 183.74& 56.49& 58.60& 55\\% \\\\\n  InfiniBand& 1& 8& 16& 128& 1& 32.74& 28.18& 59.73& 57.29& 16\\% \\\\\n  \\hline\n  \\end{tabular}\\vspace{-0.1cm}\n\\end{table*}\n\n\\subsection{Basic compression affects \\textbf{Adam}'s convergence}\n\\label{sec:moti-convergence}\nGiven the great opportunity for communication compression, we investigate whether existing Error-Compensated gradient compression strategy can be applied to \\textbf{Adam}, an important optimization algorithm for large model distributed training. We implement a basic compression strategy for \\textbf{Adam} based on the compression-based \\textbf{SGD} approach~\\citep{martinmemory}, where we perform error-compensated 1-bit compression over the gradient, and update both the momentum and variance based on the compressed gradient. We compare the BERT-Large pre-training (sequence 128) training loss when using vanilla \\textbf{Adam} and \\textbf{Adam} with our basic compression strategy in Figure~\\ref{fig:moti_loss}.\n\nResults show that basic compression based on existing work greatly affects the convergence speed for Adam. The main reason is that \\textbf{Adam} is non-linearly dependent to the gradients (see Section~\\ref{intuition:why_adam_fails} for more details).  This motivates us to look for new compression strategy that overcomes the non-linear gradient dependency challenge, and at the same time achieves the same convergence speed as \\textbf{Adam}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{moti_loss.pdf}\n\\caption{Training loss for BERT-Large pre-training using vanilla Adam and Adam with error compensated gradient compression.}\\label{fig:moti_loss}\\vspace{-0.1cm}\n\\end{figure}\n\n\\subsection{\\textbf{Adam}'s variance becomes stable during training}\n\\label{sec:moti-variance}\nUnlike \\textbf{SGD}, which  directly uses the gradient $\\bm{g}$ to update the model $\\bm{x}$, \\textbf{Adam} uses two auxiliary variables $\\bm{m}$ and $\\bm{v}$ for the update. The mathematical updating rule of  original \\textbf{Adam} can be summarized as:\n\\begin{align*}\n\\bm{m}_{t+1} =& \\beta_1\\bm{m}_t + (1-\\beta_1)\\bm{g}_t\\\\\n\\bm{v}_{t+1} =&  \\beta_2\\bm{v}_t + (1-\\beta_2)(\\bm{g}_t)^2,\\numberthis\\label{alg:v}\\\\\n\\bm{\\bm{x}}_{t+1} =&  \\bm{x}_t - \\gamma\\frac{\\bm{m}_{t+1}}{\\sqrt{\\bm{v}_{t+1} + \\eta}}\n\\end{align*}\nHere $\\bm{x}_t$ is the model at $t$-iteration, $\\bm{g}_t = \\nabla F(\\bm{x}_t;\\bm{\\zeta}_t)$ is the stochastic gradient, $\\gamma$ is the learning rate, $\\eta$ usually is a very small constant, $\\beta_1$ and $\\beta_2$ are decaying factor that controls the speed of forgetting history information. Notice here we disable the bias correction term in the original \\textbf{Adam}, which is consistent with exact optimizer for training BERT \\citep{bert}.\n\nHere we refer $\\bm{m}_t$ as the momentum term and $\\bm{v}_t$ as the variance term. Notice that when $\\bm{v}_t$ is changed into a constant $\\bm{v}$, then \\textbf{Adam} becomes equivalent to \\textbf{Momentum SGD} under a coordinate-dependent learning rate $\\frac{\\gamma}{\\sqrt{\\bm{v}} + \\eta}$.\n\n\nTo investigate the non-linear gradient dependency challenge, we analyze \\textbf{Adam}'s variance during BERT-Large pre-training (sequence 128). At each step, we fuse the variance of all parameters, and calculate the norm of the fused variance. Figure~\\ref{fig:moti_var_norm_log} presents this fused variance norm at each step. Results show that the variance norm becomes stable after around $23K$ steps. This motivates our approach {{{\\textbf{1-bit Adam}}}} to ``freeze'' the Adam variance after it becomes stable, and then use it as a precondition during 1-bit compression stage.\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{moti_var_norm_log.pdf}\n\\caption{Norm of fused variance for BERT-Large pre-training using vanilla Adam. The y-axis is in log scale.}\\label{fig:moti_var_norm_log}\\vspace{-0.1cm}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{{{{\\textbf{1-bit Adam}}}} Algorithm}\nIn this section, we start with some background introduction for error compensated compression and why it is incompatible with \\textbf{Adam}. Then we give full description of {{{\\textbf{1-bit Adam}}}}.\n\n\\paragraph*{Problem setting} In this paper, we focus on the following optimization task and rely on the following notions and definitions:\n\\vspace{-0.3cm}\n\\begin{equation}\n\\min_{\\bm{x}\\in\\mathcal{R}^d}\\quad f(\\bm{x}) = \\frac{1}{n} \\sum_{i=1}^n \\underbrace{\\mathbb{E}_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}F(\\bm{x}; \\bm{\\bm{\\zeta}}^{(i)})}_{:=f_i(\\bm{x})},\\label{eq:main}\n\\end{equation}\nwhere $d$ is the dimension of the input model $\\bm{x}$, $\\mathcal{D}_i$ is the data distribution of individual data sample $\\bm{\\zeta}^{(i)}$ on the $i$-th worker, $F(\\bm{x};\\bm{\\zeta})$ is the loss function.\n\n\\paragraph{Notations and definitions}\nThroughout this paper, we use the following notations:\n\\begin{itemize}\n\\item $\\nabla f(\\cdot)$ denotes the gradient of a function $f$.\n\\item $f^{*}$ denotes the optimal value of the minimization problem \\eqref{eq:main}.\n\\item $f_i(\\bm{x}) := \\mathbb{E}_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}F(\\bm{x}; \\bm{\\zeta}^{(i)})$.\n\\item $\\|\\cdot\\|$ denotes the $\\ell_2$ norm for vectors and the spectral norm for matrices.\n\\item $\\|X\\|_A:=\\text{Tr}(X^{\\top}AX)$.\n\\item $\\bm{C}_{\\omega}(\\cdot)$ denotes the randomized compressing operator, where $\\omega$ denotes the random variable. One example is the randomized quantization operator, for example, $\\bm{C}_{\\omega}(0.7) = 1$ with probability $0.7$ and $\\bm{C}_{\\omega}(0.7) = 0$ with probability $0.3$. \n\\item { $\\sqrt{\\cdot}$ denotes the square root of the argument. In this paper if the argument is a vector, then it returns a vector taking the element-wise square root.}\n\\item $(\\bm{x})^2$ denotes the element-wise square operation if $\\bm{x}$ is a vector.\n\\item $\\frac{\\bm{a}}{\\bm{b}}$ or $\\bm{a}\/\\bm{b}$ denotes the element-wise division operation if both $\\bm{a}$ and $\\bm{b}$ are vectors and their dimension matches.\n\\end{itemize}\n\n\n\\subsection{Why error compensation works for \\textbf{SGD}}\nFor  \\textbf{SGD} , since the update is linearly dependent to the gradient, using error compensation could potentially remove the side-effect of the history compression error. The updating rule of \\textbf{vanilla SGD} follows\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma \\bm{g}_t  = \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s.\\numberthis\\label{intuition:sgd_eq1}\n\\end{align*}\nWhen directly compressing the gradient without error compensation, the updating rule becomes\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma C_\\omega[\\bm{g}_t] =  \\bm{x}_t - \\gamma (\\bm{g}_t-\\bm{\\delta}_t)\\\\\n= &\\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\underbrace{\\gamma\\sum_{s=0}^t \\bm{\\delta}_s}_{\\text{history compression error}}.\\numberthis\\label{intuition:sgd_eq2}\n\\end{align*}\nAs we can see in \\eqref{intuition:sgd_eq2}, the history compression error would get accumulated and therefore slow down the convergence rate. Moreover, previous work \\citep{Alistarh2017-yh} indicates that when using biased compression operator, the training convergence cannot be guaranteed. \n\nNow if we apply error compensation at each compression step, the updating rule becomes\n\\begin{align*}\n\\bm{x}_{t+1} =& \\bm{x}_t - \\gamma C_\\omega[\\bm{g}_t + \\bm{\\delta}_{t-1}] =  \\bm{x}_t - \\gamma (\\bm{g}_t-\\underbrace{\\bm{\\delta}_t +\\bm{\\delta}_{t-1}}_{\\text{error cancellation}})\\\\\n=& \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\gamma\\sum_{s=0}^t(\\bm{\\delta}_s - \\bm{\\delta}_{s-1})\\\\\n=& \\bm{x}_0 - \\gamma\\sum_{s=0}^t\\bm{g}_s + \\gamma\\bm{\\delta}_t.\\numberthis\\label{intuition:sgd_eq3}\n\\end{align*}\n\n\nThis demonstrates that by using error compensation, each step's compression error would get cancelled in the next step instead of getting accumulated over steps. To make the error compensation work correctly, it is necessary that we ensure an error cancellation term $\\bm{\\delta}_t +\\bm{\\delta}_{t-1}$ in the updating rule. Below we are going to see that this cannot be achieved for \\textbf{Adam}.\n\n\n\\subsection{Why \\textbf{Adam} cannot be combined with error compensation}\\label{intuition:why_adam_fails}\nAs we can see, \\textbf{Adam} is non-linearly dependent to the gradient, and this non-linearity is widely believed to be essential for the superiority of \\textbf{Adam}. Below we are going to first intuitively explain why error compensation works well for \\textbf{SGD}, and then discuss two major reasons why this non-linearity makes \\textbf{Adam} incompatible with error compensation.\n\n\n\n\n\\paragraph{Difficulty for estimating the variance term $\\bm{v}$.} Notice that for \\textbf{Adam}, it is necessary to communicate the gradient $\\bm{g}_t$ or momentum $\\bm{m}_t$, and the variance term can be updated using $\\bm{g}_t$. However, when using error-compensated gradient to update $\\bm{v}_t$, the updating rule follows:\n\\begin{align*}\n\\bm{v}_{t+1} = &\\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(C_\\omega[\\bm{g}_t + \\bm{\\delta}_{t-1}]  \\right)^2\\\\\n=& \\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(\\bm{g}_t + \\bm{\\delta}_{t-1} - \\bm{\\delta}_t  \\right)^2\\\\\n= & \\beta_2 \\bm{v}_t + (1-\\beta_2)\\left(\\bm{g}_t   \\right)^2  +   \\underbrace{\\left( \\bm{\\delta}_{t-1} - \\bm{\\delta}_t  \\right)^2}_{\\text{non-linear error correction}} + 2 \\langle \\bm{g}_t,\\bm{\\delta}_{t-1} - \\bm{\\delta}_t\\rangle.\n\\end{align*}\nHere the quadratic term $\\left( \\bm{\\delta}_{t-1} - \\bm{\\delta}_t  \\right)^2$ cannot be cancelled by itself, therefore it will be hard to get an accurate estimation of $\\bm{v}_t$ with history error being cancelled.\n\\paragraph{Difficulty for setting the correction factor.} Another problem is that for  \\textbf{SGD} , when applying error compensation under a time varying learning rate $\\gamma_t$, we need to compensate the history error using\n\\begin{align*}\nC\\left[ \\bm{g}_t + \\frac{\\gamma_t}{\\gamma_{t-1}}\\bm{\\delta}_{t-1}\\right],\n\\end{align*} instead of adding back $\\bm{\\delta}_{t-1}$ directly. In this case,\nif we view $\\frac{\\gamma}{\\sqrt{\\bm{v}_t} + \\eta}$ as a coordinate-dependent learning rate, which makes \\textbf{Adam} equivalent to  \\textbf{Momentum SGD} with time-varying learning rate, we need to apply the scale factor according to \n\\begin{align*}\n\\bm{m}_{t+1} = C_\\omega\\left[\\beta_1\\bm{m}_t + (1-\\beta_1)\\bm{g}_t + \\frac{\\sqrt{\\bm{v}_{t-1}} + \\eta}{\\sqrt{\\bm{v}_{t}} + \\eta}\\bm{\\delta}_{t-1}\\right].\n\\end{align*}\nThe problem is that we cannot get the value of $\\bm{v}_{t}$ after the compression, which makes it impossible to set the scale factor for error compensation.\n\n\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[\\scriptsize \\textbf{Gather step}: Each worker sends its $i$-th chunk to worker $i$.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{gather.pdf}\n\\end{minipage}\n}\\quad\n\\subfigure[\\scriptsize \\textbf{Average step}: Each worker averages all chunks it receives.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{average.pdf}\n\\end{minipage}%\n}\\quad\n\\subfigure[\\scriptsize \\textbf{Scatter step}: Each worker receives the $i$-th chunk from worker $i$.]{\n\\begin{minipage}[t]{0.3\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{scatter.pdf}\n\\end{minipage}%\n}%\n\\centering\n\\caption{Efficient system design for communication (compressed\\_allreduce)}\\label{allreduce}\\vspace{-0.1cm}\n\\label{fig:allreduce}\n\\end{figure*}\n\n\\subsection{{{{\\textbf{1-bit Adam}}}}}\\label{alg:description}\nBased on our findings (Section~\\ref{sec:moti-variance}) that \\textbf{Adam}'s variance term becomes stable at an early stage, we propose {{{\\textbf{1-bit Adam}}}} summarized in Algorithm \\ref{alg:de_ec}. First we use vanilla \\textbf{Adam} for a few epochs as a warm-up. After the warm-up stage, the compression stage starts and we stop updating the variance term $\\bm{v}$ and use it as a fixed precondition. At the compression stage, we communicate based on the momentum applied with error-compensated 1-bit compression. The momentums are quantized into 1-bit representation (the sign of each element). Accompanying the vector, a scaling factor is computed as $\\frac{\\text{magnitude of compensated gradient}}{\\text{magnitude of quantized gradient}}.$ This scaling factor ensures that the compressed momentum has the same magnitude as the uncompressed momentum. This 1-bit compression could reduce the $97\\%$ communication cost of the original for float32 type training and $94\\%$ for float16 type training. \n\n\n\n\n\n\n\n\\begin{algorithm}[t!]\\caption{{{{\\textbf{1-bit Adam}}}}}\n\\begin{algorithmic}[1]\n\\footnotesize\n\\STATE {\\bfseries Initialize}: $\\bm{x}_0$, learning rate $\\gamma$,  initial error $\\bm{\\delta} = \\boldsymbol{0}$, $\\bm{m}_0 = \\boldsymbol{0}$, $\\bm{v}_0 = \\boldsymbol{0}$, number of total iterations $T$,  warm-up steps $T_{w}$, two decaying factor $\\beta_1$ and $\\beta_2$ for \\textbf{Adam}.\n\n\\STATE Running the original \\textbf{Adam} for $T_{w}$ steps, then store the variance term (defined as $\\bm{v}_t$ in \\eqref{alg:v}) $\\bm{v}_{_{ T_w}}$.\n\\FOR {$t=T_w,\\ldots,T$}\n\n\\STATE \\textbf{(On $i$-th node)}\n\\STATE  Randomly sample $\\bm{\\xi}_t^{(i)}$  and compute local stochastic gradient $\\bm{g}_t^{(i)} := \\nabla F_i(\\bm{x}_t^{(i)}, \\bm{\\xi}_t^{(i)})$.\n\n\\STATE Update the local momentum variable $\\bm{m}_{t-1}$ according to\n$\n\\bm{m}_t^{(i)} =  \\beta_1\\bm{m}_{t-1} + (1 - \\beta_1)\\bm{g}_t^{(i)}.\n$\n\\STATE  Compress $\\bm{m}_t^{(i)}$ into $\\hat{\\bm{m}}_t^{(i)} = \\bm{C}_\\omega\\left[\\bm{m}_t^{(i)} + \\bm{\\delta}_{t-1}^{(i)}\\right]$, and update the compression error by $\\bm{\\delta}_t^{(i)} = \\bm{m}_t^{(i)} + \\bm{\\delta}_{t-1}^{(i)} - \\hat{\\bm{m}}_t^{(i)}$.\n\\STATE Send the  $\\hat{\\bm{m}}_t^{(i)}$ to the server.\n\\STATE \\textbf{(On server)}\n\\STATE Take the average over all $\\hat{\\bm{m}}_t^{(i)}$ it receives and compress it into\n$\n\\overline{\\bm{m}}_t =\\bm{C}_\\omega\\left[ \\frac{1}{n}\\sum_{i=1}^n\\hat{\\bm{m}}_t^{(i)} + \\overline{\\bm{\\delta}}_{t-1}\\right],\n$\n and update the compression error accordingly by $\\overline{\\bm{\\delta}}_t = \\frac{1}{n}\\sum_{j=1}^n \\bm{C}_\\omega\\left[\\bm{m}_t^{(i)}\\right] + \\overline{\\bm{\\delta}}_{t-1} - \\overline{\\bm{m}}_t$.\n \\STATE Send $\\overline{\\bm{m}}_t$ to all the workers.\n \\STATE \\textbf{(On $i$-th node)}\n \\STATE Set $\\bm{m}_t = \\overline{\\bm{m}}_t$ , and update local model $\\bm{x}_{t+1} = \\bm{x}_t - \\gamma \\bm{m}_t\/\\sqrt{\\bm{v}_{_{\\tiny T_w}}}$.\n\\ENDFOR\n\\STATE {\\bfseries Output}: $\\bm{x}$.\n\\end{algorithmic}\\label{alg:de_ec}\n\\end{algorithm}\n\n\n\\section{Theoretical Analysis}\nNotice that for {{{\\textbf{1-bit Adam}}}}, we only use original \\textbf{Adam} at warm-up, and then we essentially run error-compensated momentum \\textbf{SGD} with coordinate-dependent learning rate $\\frac{\\gamma}{\\sqrt{\\bm{v}_{_{T_w}}}}$. Therefore here we consider the \\textbf{Adam}-based warm-up phase as a way to find a good precondition variance term $\\bm{v}_{_{T_w}}$ to be used in the compression phase. Below we are going to introduce the convergence rate for the compression phase after warm-up. We first introduce some necessary assumptions, then we present the theoretical guarantee of the convergence rate for {{{\\textbf{1-bit Adam}}}}.\n\n\n\n\n\n\n\n\n\n\\begin{assumption}\\label{ass:global}\nWe make the following assumptions:\n\\begin{enumerate}\n\\item \\textbf{Lipschitzian gradient:} $f(\\cdot)$ is assumed to be  with $L$-Lipschitzian gradients, which means\n  \\begin{align*}\n  \\|\\nabla f(\\bm{x}) - \\nabla f(\\bm{y}) \\| \\leq L \\|\\bm{x} - \\bm{y} \\|,\\quad \\forall \\bm{x},\\forall \\bm{y},\n  \\end{align*}\n \\item\\label{ass:var} \\textbf{Bounded variance:}\nThe variance of the stochastic gradient is bounded\n\\begin{align*}\n\\mathbb E_{\\bm{\\zeta}^{(i)}\\sim\\mathcal{D}_i}\\|\\nabla F(\\bm{x};\\bm{\\zeta}^{(i)}) - \\nabla f(\\bm{x})\\|^2 \\leq \\sigma^2,\\quad\\forall \\bm{x},\\forall i.\n\\end{align*}\n\\item \\textbf{Bounded magnitude  of error for $\\mathcal{C}_{\\omega}[\\cdot]$:}\nThe magnitude of worker's local errors $\\bm{\\delta}_t^{(i)}$  and the server's global error $\\overline{\\bm{\\delta}}_t$, are assumed to be bounded by a constant $\\epsilon$\n\\begin{align*}\n\\sum_{k=1}^n\\mathbb E_{\\omega} \\left\\|\\bm{\\delta}_t^{(i)}\\right\\|\\leq \\frac{\\epsilon}{2},\\quad\n\\sum_{i=1}^n\\mathbb E_{\\omega}\\left\\|\\overline{\\bm{\\delta}}_t\\right\\|\\leq  \\frac{\\epsilon}{2},\\quad\\forall t,\\forall i.\n\\end{align*}\n\\end{enumerate}\n\\end{assumption}\n\n\nNext we present the main theorem for {{{\\textbf{1-bit Adam}}}}.\n\\begin{theorem}\\label{theo:global}\n Under Assumption~\\ref{ass:global}, for {{{\\textbf{1-bit Adam}}}}, we have the following convergence rate\n \\begin{align*}\n   &\\left(1-\\frac{\\gamma L}{v_{\\min}} - \\frac{2\\gamma^2 L^2}{(1-\\beta)^2v_{\\min}^2} \\right)\\sum_{t=0}^T \\mathbb E\\|\\nabla f(\\bm{x}_t)\\|^2_{V}\\\\\n    \\leq & \\frac{2\\mathbb E f(\\bm{x}_{0}) - 2\\mathbb Ef(\\bm{x}^*)}{\\gamma}   + \\frac{6\\gamma^2L^2\\epsilon^2 T}{(1-\\beta)^2v_{\\min}^3}  +  \\frac{L\\gamma \\sigma^2T}{nv_{\\min}} + \\frac{2\\gamma^2L^2\\sigma^2  T}{n(1-\\beta)^2v_{\\min}^2},\\numberthis\\label{main:theo:eq}\n\\end{align*}\nwhere $V= \\text{diag}\\left(1\/\\bm{v}_{T_w}^{(1)},1\/\\bm{v}_{T_w}^{(2)},\\cdots,1\/\\bm{v}_{T_w}^{(d)}\\right)$ is a diagonal matrix spanned by $\\bm{v}_{_{T_w}}$ and $v_{\\min} = \\min\\{\\bm{v}_{T_w}^{(1)},\\bm{v}_{T_w}^{(2)},\\cdots,\\bm{v}_{T_w}^{(d)}\\}$ is the mimimum value in $\\bm{v}_{T_w}$\n\\end{theorem}\n\nGiven the generic result in Theorem~\\ref{theo:global}, we obtain the convergence rate for {{{\\textbf{1-bit Adam}}}} with appropriately chosen learning rate $\\gamma$.\n\n\n\\begin{corollary}\\label{coro:global}\nUnder Assumption~\\ref{ass:global}, for {{{\\textbf{1-bit Adam}}}}, choosing\n$\n\\gamma = \\frac{1}{4L(v_{\\min})^{-1} + \\sigma\\sqrt{\\frac{ T}{n}} + \\epsilon^{\\frac{2}{3}} T^{\\frac{1}{3}}(v_{\\min})^{-1} },\n$\nwe have the following convergence rate\n\\begin{align*}\n\\frac{1}{Tv_{\\min}}\\sum_{t=0}^{T-1}\\mathbb{E}\\|\\nabla f(\\bm{x}_t)\\|^2_V \\lesssim \\frac{\\sigma}{\\sqrt{nT}} + \\frac{\\epsilon^{\\frac{2}{3}}}{T^{\\frac{2}{3}}} + \\frac{1}{ T},\n\\end{align*}\nwhere we treat $f(\\bm{x}_1) - f^*$, $\\beta$ and $L$ as constants.\n\\end{corollary}\n\n\nThis result suggests that: {{{\\textbf{1-bit Adam}}}} essentially admits the same convergence rate as distributed \\textbf{SGD} in the sense that both of them admit the asymptotical convergence rate $O(1\/\\sqrt{nT})$, which means we can still achieve linear speedup w.r.t. the number of workers $n$.\n\n\n\\section{Efficient system design for compressed communication}\nNVIDIA NCCL is an efficient and widely used communication library that has been tightly integrated in DL frameworks like PyTorch and TensorFlow. However, NCCL library cannot be used directly for performing communication based on 1-bit compression. This is because the collective communication primitives like Allreduce and Allgather are at a higher level of abstraction and can only perform data movement and\/or simple operations like sum, min, max etc. In addition, NCCL library (before v2.7) did not expose either an Alltoall primitive or any point-to-point (send\/recv) communication primitives that can be used to implement an Alltoall. Thus for {{{\\textbf{1-bit Adam}}}}, we designed a custom collective primitive using Message Passing Interface (MPI). We call it ``compressed allreduce'' and it has three phases as shown in Figure~\\ref{fig:allreduce}: 1) The gather step, which we have implemented using the MPI\\_Alltoall (personalized exchange) primitive, 2) The average step, where {{{\\textbf{1-bit Adam}}}} computes the average of compressed local momentums, and 3) The scatter step, which we implement using MPI\\_Allgather. We develop two versions of compressed allreduce: 1) CUDA-Aware version that exploits GPUDirect features and requires CUDA-Aware libraries like MVAPICH2-GDR and 2) Basic version that can be used with any MPI library but copies data between GPU and CPU buffers. The CUDA-Aware version works only on systems with InfiniBand whereas the basic version can run on any system with Ethernet interconnect. \n\n\\section{Experiments}\n\nWe evaluate {{{{\\textbf{1-bit Adam}}}}} and existing approaches using BERT-Base, BERT-Large, SQuAD 1.1 and ResNet-18 training tasks on up to 256 GPUs. We show that {{{{\\textbf{1-bit Adam}}}}} converges as fast as uncompressed \\textbf{Adam}, and runs up to 3.3 times faster than uncompressed algorithms under limited bandwidth.\n\n  \n  \n\n\\subsection{BERT pre-training and fine-tuning}\n\\label{sec:bert-eval}\n\\paragraph{Dataset and models} We evaluate the convergence and performance of {{{\\textbf{1-bit Adam}}}} and uncompressed \\textbf{Adam} for BERT-Base ($L=12$, $H=768$, $A=12$, $110M$ params) and BERT-Large ($L=24$, $H=1024$, $A=16$, $340M$ params) pre-training tasks. We use the same dataset as \\citet{bert}, which is a concatenation of Wikipedia and BooksCorpus with $2.5B$ and $800M$ words respectively. We use the GLUE fine-tuning benchmark\\citep{glue} to evaluate the convergence of the BERT models trained by \\textbf{Adam} and {{{\\textbf{1-bit Adam}}}}.\n\nIn addition, we also evaluate the convergence and performance of {{{\\textbf{1-bit Adam}}}} for SQuAD 1.1 fine-tuning task\\footnote{https:\/\/rajpurkar.github.io\/SQuAD-explorer\/} using a pre-trained BERT model checkpoint from HuggingFace\\footnote{https:\/\/github.com\/huggingface\/transformers}.\n\n\\paragraph{Hardware} We use the two clusters described in Section~\\ref{sec:moti-profile}. We use up to 256 GPUs for pre-training tasks and up to 32 GPUs for fine-tuning tasks.\n\n\\paragraph{Training parameters} For BERT pre-training, the learning rate linearly increases to $4\\times 10^{-4}$ as a warmup in the first $12.5K$ steps, then decays into $0.99$ of the original after every $520$ steps. We set the two parameters in Algorithm~\\ref{alg:de_ec} as $\\beta_1 = 0.9$ and $\\beta_2 = 0.999$ for {{{\\textbf{1-bit Adam}}}} and \\textbf{Adam}. For convergence test, we set total batch size as $4K$ for BERT-Base and BERT-Large. For performance test, we test different batch sizes. Table~\\ref{table_bert_steps} summarizes the total number of steps for BERT sequence length 128 and 512 phases, together with the number of warmup steps for {{{\\textbf{1-bit Adam}}}}.\n\nFor GLUE benchmarks we use original \\textbf{Adam} optimizer and perform single-task training on the dev set. We search over the hyperparameter space with batch sizes $\\in\\{8,16\\}$ and learning rates $\\in\\{1\\times 10^{-5},3\\times 10^{-5},5\\times 10^{-5},8\\times 10^{-5}\\}$. Other setting are the same as pre-training task.\n\nFor SQuAD fine-tuning we use the same parameters as published by HuggingFace (batch size = $24$, learning rate=$3e-5$, dropout=$0.1$, 2 epochs), except that we increase the batch size to $96$ (using $32$ GPUs). The first $400$ steps out of total $1848$ steps are used as the warmup stage for {{{\\textbf{1-bit Adam}}}}.\n\n\\begin{table}[t]\n  \\footnotesize\n  \\caption{Number of steps for BERT pre-training tasks.}\\label{table_bert_steps}\n  \\centering\n  \\begin{tabular}{lll}\n  \\hline\n  & Seqlen 128& Seqlen 512\\\\\n  & (warmup)& (warmup)\\\\\n  \\hline\n  BERT-Base \\textbf{Adam}& $118K$ (N\/A)& $22K$ (N\/A) \\\\\n  BERT-Base {{{\\textbf{1-bit Adam}}}}& $118K$ ($16K$)& $22K$ ($1.5K$) \\\\\n  BERT-Large \\textbf{Adam}& $152K$ (N\/A)& $10K$ (N\/A) \\\\\n  BERT-Large {{{\\textbf{1-bit Adam}}}}& $152K$ ($23K$)& $10K$ ($1.5K$) \\\\\n  \\hline\n  \\end{tabular}\\vspace{-0.1cm}\n\\end{table}\n\n\\paragraph{Convergence results}\nFigure~\\ref{fig:bert} presents the sample-wise convergence results. We use the BertAdam \\citep{bert} optimizer as the uncompressed baseline. For both BERT-Base and BERT-Large and for both sequence length phases, we find that {{{\\textbf{1-bit Adam}}}} provides the same convergence speed as baseline, while the communication volume is reduced into $6\\%$ of the original during the compression stage.\n\nTable~\\ref{table1} presents the GLUE results using the checkpoints from our pre-training experiments. {{{\\textbf{1-bit Adam}}}} achieves similar accuracy compared to the uncompressed baseline and the numbers reported in previous work.\n\nFor SQuAD 1.1 fine-tuning task using checkpoint from  HuggingFace, {{{\\textbf{1-bit Adam}}}} achieves similar F1 score (93.32) compared to the score reported by HuggingFace (93.33) using same number of samples and trainig parameters. \n  \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{large_loss.pdf}\n\\caption{Epoch-wise convergence speed for BERT-Large pre-training sequence length 128. {{{\\textbf{1-bit Adam}}}} and \\textbf{Adam} also achieve the same convergence speed for BERT-Base pre-training.}\\label{fig:bert}\\vspace{-0.1cm}\n\\end{figure}\n\n\\begin{table*}[t]\n\\footnotesize\n  \\caption{GLUE development set results. BERT-Base\/Large(original) results are from \\citet{bert}. BERT-Base\/Large (uncompressed) results use the full-precision \\textbf{BertAdam} with the same training parameters as the {{{\\textbf{1-bit Adam}}}} case. BERT-Base\/Large (compressed) are the results using {{{\\textbf{1-bit Adam}}}}. The scores are the median scores over 10 runs.}\\label{table1}\n  \\centering\n  \\begin{tabular}{lccccccc}\n  \\hline \n  \\textbf{Model}& RTE& MRPC& CoLA & SST-2& QNLI& QQP& MNLI-(m\/mm) \\\\\n  \\hline \n  BERT-Base (original) & 66.4 & 84.8 & 52.1  & 93.5 & 90.5& 89.2& 84.6\/83.4\\\\\n  BERT-Base (uncompressed) & 68.2 & 84.8 & 56.8  & 91.8 & 90.9& 90.9& 83.6\/83.5\\\\\n  BERT-Base (compressed) & 69.0& 84.8 & 55.6   & 91.6 & 90.8& 90.9& 83.6\/83.9\\\\\n  \\hline\n  BERT-Large (original) & 70.1& 85.4 & 60.5   & 94.9 & 92.7& 89.3& 86.7\/85.9\\\\\n  BERT-Large (uncompressed) & 70.3& 86.0 & 60.3   & 93.1 & 92.2& 91.4& 86.1\/86.2\\\\\n  BERT-Large (compressed) & 70.4& 86.1 & 62.0  & 93.8 & 91.9& 91.5& 85.7\/85.4\\\\\n  \\hline\n  \\end{tabular}\\vspace{-0.1cm}\n\\end{table*}\n\n\\begin{figure*}[t]\n\\centering\n\\subfigure[Bert-Large pre-training, batch size = number of GPUs $\\times$ 16]{\n\\begin{minipage}[t]{0.33\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_bert_accum1.pdf}\\label{fig:e2e-1}\n\\end{minipage}\n}\n\\subfigure[Bert-Large pre-training, batch size = 4K]{\n\\begin{minipage}[t]{0.33\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_bert_bsz4k.pdf}\\label{fig:e2e-2}\n\\end{minipage}\n}\n\\subfigure[SQuAD fine-tuning, batch size = number of GPUs $\\times$ 3]{\n\\begin{minipage}[t]{0.28\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{eval_squad_accum1.pdf}\\label{fig:e2e-3}\n\\end{minipage}%\n}\n\\centering\n\\caption{Scalability of {{{\\textbf{1-bit Adam}}}} for BERT-Large pre-training sequence length 128 and SQuAD 1.1 fine-tuning on V100 GPUs. \\textbf{Adam} lines represent the throughput at {{{\\textbf{1-bit Adam}}}}'s warmup stage (i.e., baseline \\textbf{Adam}'s throughput). {{{\\textbf{1-bit Adam}}}} lines represent the throughput at compression stage. Annotations represent the highest speedup achieved in each figure. Note that this is the speedup between warmup and compression stage. The end-to-end speedup also depends on the percentage of warmup.}\\label{fig:e2e}\\vspace{-0.1cm}\n\\end{figure*}\n\n\n\\paragraph{Performance results}\nComputed as 1\/(warmup ratio + (1 - warmup ratio)\/16) for FP16 training, {{{\\textbf{1-bit Adam}}}} offers up to 5x less end-to-end communication volume for BERT-Base and BERT-Large. This leads to to 3.3x higher throughput for BERT-Large sequence length 128 pre-training and up to 2.9x higher throughput for SQuAD fine-tuning. This end-to-end throughput improvement is enabled by the 5.48x (Figure~\\ref{fig:e2e-1}) and 6.17x (Figure~\\ref{fig:e2e-3}) speedup observed during the compression stage. Figure~\\ref{fig:e2e-2} shows that {{{\\textbf{1-bit Adam}}}} also provides better scalability: \\textbf{Adam}'s throughput reaches peak at 32 GPUs on Ehternet, while {{{\\textbf{1-bit Adam}}}}'s throughput keeps increasing until 128 GPUs. It is also worth mentioning that {{{\\textbf{1-bit Adam}}}} on Ethernet (4.1 Gbps effective bandwidth, 4 GPUs per node) is able to achieve comparable throughput as \\textbf{Adam} on InfiniBand (near 100 Gbps effective bandwidth, 8 GPUs per node), which demonstrates {{{\\textbf{1-bit Adam}}}}'s efficiency considering the hardware differences.\n\n\n\n\\subsection{ResNet on CIFAR10}\\label{resnet}\n\n\n\\begin{figure}[t]\n\\centering\n\\subfigure[Training loss]{\n\\begin{minipage}[t]{0.35\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{resnet_loss.pdf}\n\\end{minipage}\n}\n\\subfigure[Testing accuracy]{\n\\begin{minipage}[t]{0.25\\linewidth}\n\\centering\n\\includegraphics[width=1\\textwidth]{resnet_acc.pdf}\n\\end{minipage}\n}\n\\centering\n\\caption{Epoch-wise convergence speed for ResNet-18.}\\label{fig:resnet}\\vspace{-0.1cm}\n\\end{figure}\n\nTo further evaluate the convergence speed of {{{\\textbf{1-bit Adam}}}} and related works, we train CIFAR10 using ResNet-18\\citep{7780459}. The dataset has a training set of 50000 images and a test set of 10000 images, where each image is given one of the 10 labels. We run the experiments on $8$ 1080Ti GPUs where each GPU is used as one worker. The batch size on each worker is $128$ and the total batch size is $1024$.\n\nWe evaluate five implementations for comparison: 1) Original \\textbf{SGD}. 2) Original \\textbf{Adam} \\citep{adam}. 3) {{{\\textbf{1-bit Adam}}}} where we use $13$ out of $200$ epochs as warmup. 4) {{{\\textbf{1-bit Adam}}}}\\textbf{(32-bits)} where we do not compress the momentum while still freezing the varaince. 5) \\textbf{Adam(1-bit Naive)} where we compress the gradient instead of momentum, and don't freeze the variance. We set the learning rate as $1\\times 10^{-1} $ for \\textbf{SGD} and $1\\times 10^{-4}$ for the other 4 cases. For all five cases, the learning rate is decayed into $10\\%$ of the original after every $100$ epochs.\n\n\n\n\n\n\n\nAs illustrated in Figure~\\ref{fig:resnet}, {{{\\textbf{1-bit Adam}}}} achieves similar convergence speed as \\textbf{Adam} and {{{\\textbf{1-bit Adam}}}}\\textbf{(32-bits)}. \\textbf{SGD} has a slightly slower convergence speed while \\textbf{Adam(1-bit Naive)} is much worse. This and Section~\\ref{sec:moti-convergence} demonstrate that existing compression method doesn't work for \\textbf{Adam}. In the supplementary materials we further compare {{{\\textbf{1-bit Adam}}}} with other related works using ResNet-18.\n\n\\section{Conclusions}\nIn this paper, we propose an error-compensated \\textbf{Adam} preconditioned momentum SGD algorithm, {{{\\textbf{1-bit Adam}}}}, which provides both communication efficiency and \\textbf{Adam}'s convergence speed. Our theoretical analysis demonstrates that ${{\\textbf{1-bit Adam}}}$ admits a linear speed w.r.t the number of workers in the network, and is robust to any compression method. We validate the performance of {{{\\textbf{1-bit Adam}}}} empirically  on BERT, SQuAD and ResNet training tasks on up to 256 GPUs. Results show that {{{\\textbf{1-bit Adam}}}} converges as fast as uncompressed \\textbf{Adam}, reduces communication volume by up to 5x, and runs up to 3.3 times faster than uncompressed algorithms.\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n      An American Mathematical Monthly problem posed within relatively recent memory [{\\bf{1}}] sought the\nevaluation\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi\/2}\\left\\{\\rule{0mm}{4mm}\\log(\\,2\\sin(x)\\,)\\right\\}^{2}dx\\,=\\,\\frac{\\,\\,\\,\\pi^{\\,3}\\,}{\\,24\\,}\\;.\n\\end{equation}\nOne mode of solution depended upon integration of an analytic function around the periphery $\\,\\Omega\\,$ of a semi-infinite\nvertical strip with no singularities enclosed, the quadrature having thus a null outcome\\footnote{Both\ncontour $\\,\\Omega\\,,$ a vertical rectangle of unlimited height, and the notion of integrating an analytic function thereon so\nas to obtain a null result, imitate a similar ploy utilized in [{\\bf{2}}] on behalf of (2) and still further attributed there\nto Ernst Lindel{\\\"{o}}f.} known in advance on the strength of Cauchy's theorem.\nEvaluation (1) emerged automatically by setting to zero the real part of that integral,\\footnote{Two solutions for (1) were submitted by\nthe undersigned, one involving contour integration in the manner suggested, and the other based upon a Fourier series.\nThe Bernoulli recurrence (5) emerged as a spontaneous by-product of an ancillary, null-quadrature calculation upon\nthat same contour $\\,\\Omega\\,,$ initially aimed only at evaluating the log-sine integrals (4).  This note embodies the content\nof that collateral calculation, slightly rephrased so as to highlight the newly recovered Bernoulli number sum identity.}\nwhereas the complementary requirement that the imaginary part likewise vanish brought into play, and successfully so, the known quadratures\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\log(\\,\\sin(x)\\,)\\,dx\\,=\\,-\\,\\pi\\log(2)\n\\end{equation}\nand\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}x\\log(\\,\\sin(x)\\,)\\,dx\\,=\\,-\\frac{\\,\\,\\,\\pi^{\\,2}\\,}{\\,2\\,}\\log(2)\\;.\n\\end{equation}\nWith (2) and (3) in plain view, a temptation arose to provide for them, too, an {\\em{ab initio}} verification, and,\nmore even than that, to evaluate the entire hierarchy of log-sine integrals\\footnote{If that were the only goal then we should\nassuredly stop dead in our tracks, simply because, on the one hand, {\\bf{MATHEMATICA}} provides all such evaluations\non demand, with great aplomb, and this even in its symbolic mode, while, on the other, a relatively painless\nderivation can be based upon a Fourier series, one which emerges in its turn from the power series for $\\,\\log(1-z)\\,$\nwhen argument $\\,z\\,$ is forced to lie upon the unit circle.  This Fourier series underlies in addition an essentially zinger\nverification of (1).  All such manifold benefits of the Fourier option are sketched in an Appendix.  Moreover, it goes without\nsaying that both contour-based (14) and Fourier-based (27) evaluations of (4), even though they may be of secondary interest\nin the present context, do stand in complete agreement.} \n\\begin{equation}\nI_{n}\\,=\\,\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx\n\\end{equation}\nas the power of $\\,x\\,$ roams over all non-negative integers $\\,n\\,=\\,0,\\,1,\\,2,\\,3,\\,\\ldots\\,{\\rm{ad\\;inf}}\\,.$\nNot only was this fresh ambition, digressive and self-indulgent though it may have been, easy to satisfy via quadrature\non the same contour as before, but it also exposed to view once more the fundamental Bernoulli number recurrence\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n-1}  \\left(\\!\\!\\begin{array}{c}\n                                     n \\\\\n                                     k\n                                  \\end{array}\\!\\!\\right)\nB_{k}\\,=\\,0\n\\end{equation}\nwhich is valid for $\\,n\\geq 2\\,$ and, together with the initial condition $\\,B_{0}\\,=\\,1\\,$ and the self-consistent choice\n$\\,B_{1}\\,=\\,-1\/2\\,,$ is adequate to populate the entire Bernoulli ladder, complete with null entries at all odd\nindices beyond $\\,k\\,=\\,1\\,,$ {\\em{viz.,}} $\\,B_{2l+1}\\,=\\,0\\,$ whenever $\\,l\\geq1\\,.$  Source material on the\nBernoulli numbers and the related Bernoulli polynomials is ubiquitous, and can be sampled, for example, in [\\textbf{3-5}].\nReferences [\\textbf{6,7}] provide a valuable overview all at once of their\nmathematical properties and historical genesis in com-\n\\newpage\n\\mbox{   }\n\\newline\n\\newline\n\\newline\nputing sums of finite progressions of successive integers raised to\nfixed positive powers.  Equally valuable is online Reference [{\\bf{8}}], which cites a rich literature and\ncovers besides a vast panorama of diverse mathematical knowledge.\n\n\nBernoulli identity (5), which is the principal object of our present concern, emerges thus by setting to\nzero the imaginary part of the analytic quadrature (6), below, around contour $\\,\\Omega\\,,$\nwith the corresponding null value requirement on its real part providing an evaluation of the general term from sequence (4),\nlisted in (14).  No claim whatsoever is made here as to any ultimate novelty in outcome (14), which is available\nin symbolic form at any desired index $\\,n\\,$ through routine demand from \\textbf{MATHEMATICA}.  Outcome (14),\nexpressed here as a finite sum of Riemann zeta functions at odd integer arguments, continues to attract the attention\nof contemporary research focused upon polylogarithms [\\textbf{9-12}].  But the formulae thus\nmade available are subordinated in [\\textbf{11,12}] and elsewhere to the task of evaluating a variety of dissimilar\nquantities, and appear to be tangled in thickets of notation.  From this\nstandpoint, formula (14) (and its identical twin (27) derived in an even more elementary fashion)\nmay perhaps still provide the modest service of a stand-alone, encapsulated result, easily derived and\neasily surveyed.  In particular, the canonical method of derivation evolved in [\\textbf{9}] and repeatedly\nalluded to in [\\textbf{11,12}] requires rather strenuous differentiations of Gamma function ratios,\nand results finally in a recurrence on the individual $\\,I_{n}\\,$ (or else an equivalent generating function).\nTo be sure, while the work in [\\textbf{9}] is immensely elegant, it is at the same time immensely more intricate\nthan either of our independent derivations culminating in (14) and (27).\n\n\n    On the other hand, it does appear to have escaped previous notice that the Bernoulli recurrence (5), which\nis ancient and foundational in its own right, should likewise re\\\"{e}merge (via (16)) from the same\nquadrature around contour $\\,\\Omega\\,$ when one insists that the\ncorresponding imaginary part also vanish.  And, just as is the case with (14), formula (16), too, emerges\nfrom a contact with Riemann zeta functions, but evaluated this time at even integer arguments, which latter\ncircumstance, by virtue of the celebrated Euler connection, opens the portal to entry by the similarly\nindexed Bernoulli numbers.  It is of course none of our purpose here to compete with, let alone to supplant\nin any way the standard derivations of (5).  Rather, we seek merely to highlight its re\\\"{e}mergence in what surely\nmust be conceded to be an unexpected setting. \n\n   We round out this note with an appendix wherein contour integration cedes place to the more elementary\nsetting of a Fourier series on whose basis (14) is recovered yet again (as (27)) through repeated integration by\nparts.  That same Fourier series provides moreover an exceedingly short and simple confirmation of (1),\ncomplementary to the contour integral method, an option to which allusion has already been made in Footnote 3.\nOf course, at this point, no further light can, nor need be shed upon (5) \\textit{per se}.\n\\vspace{-4mm}\n\n\n    \n\\section{Null Quadratures on Contour $\\,\\Omega$}\n\\vspace{-2mm}\n\n       Guided by the cited example in [{\\bf{2}}], we consider for $\\,n\\,\\geq\\,0\\,$ the sequence of numbers\n\\begin{equation}\nK_{n}\\,=\\,\\int_{\\,\\Omega}z^{n}\\log\\left(1-e^{\\,2\\,i\\,z}\\right)\\,dz\\,=\\,0\\;\\,,\n\\end{equation}\nall of them annulled by virtue of closed contour $\\,\\Omega\\,$ being required to\nlie within a domain of analyticity for $\\,\\log\\left(1-e^{\\,2\\,i\\,z}\\right)\\,$\nin the plane of complex $z=x+iy.$  Save for quarter-circle indentations of vanishing radius\n$\\,\\delta\\,$ around $\\,z\\,=\\,0\\,$ and $\\,z\\,=\\,\\pi\\,,$ contour $\\,\\Omega\\,$\nbounds a semi-infinite vertical strip, with a left leg having $\\,x\\,=\\,0\\,$\n\\newpage\n\\mbox{   }\n\\newline\n\\newline\n\\newline\nfixed and descending from $\\,y\\,=\\,\\infty\\,$ to $\\,y\\,=\\,\\delta\\,$ (quadrature\ncontribution $\\,L_{n}\\,$), and a right leg at a fixed $\\,x\\,=\\,\\pi\\,$ ascending from\n$\\,y\\,=\\,\\delta\\,$ to $\\,y\\,=\\,\\infty\\,$ (quadrature contribution $\\,R_{\\,n}\\,$),\nlinked at their bottom by a horizontal segment with $\\,y\\,=\\,0\\,$ and\n$\\,\\delta\\,\\leq\\,x\\,\\leq\\,\\pi-\\,\\delta\\,$ (quadrature contribution $\\,H_{n}\\,$).\nIn what follows it will be readily apparent that the limit $\\,\\delta\\!\\downarrow\\!0+\\,$\nmay be enforced with full impunity, a gesture whose {\\em{fait accompli}} status will\nbe taken for granted.  Likewise passed over without additional\ncomment will be the fact that no contribution is to be sought from contour completion\nby a retrograde horizontal segment $\\,\\pi\\,\\geq\\,x\\,\\geq\\,0\\,$ at\ninfinite remove, $\\,y\\,\\rightarrow\\,\\infty\\,.$\n\n\n     We now find\n\\begin{equation}\nL_{n}\\,=\\,-\\,i^{\\,n+1}\\int_{\\,0}^{\\,\\infty}y^{\\,n}\\log\\left(1-e^{-2\\,y}\\right)\\,dy\\;\\,,\n\\end{equation}\n\\begin{equation}\nR_{\\,n}\\,=\\,+\\,i\\int_{\\,0}^{\\,\\infty}\\left(\\,\\pi\\,+\\,i\\,y\\,\\right)^{n}\\log\\left(1-e^{-2\\,y}\\right)\\,dy\\;\\,,\n\\end{equation}\nand\n\\begin{eqnarray}\nH_{n}    &   =  &  \\int_{\\,0}^{\\,\\pi}x^{n}\\left[\\rule{0mm}{4mm}\\log(2)\\,-\\,\n                         \\frac{\\,i\\,\\pi\\,}{\\,2\\,}\\,+\\,i\\,x\\,+\\,\\log(\\,\\sin(x)\\,)\\,\\right]\\,dx     \\nonumber  \\\\\n         &   =  &  \\frac{\\,\\pi^{\\,n+1}\\,}{\\,n\\,+\\,1\\,}\\log(2)\\,-\\,i\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,2\\,(\\,n\\,+\\,1\\,)\\,}\\,+\\,\ni\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,\\,n\\,+\\,2\\,}\\,+\\,\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx\\;\\,.\n\\end{eqnarray}\nSeries expansion of the logarithm further gives\n\\begin{equation}\nL_{n}\\, = \\, +\\,i^{\\,n+1}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l\\,}\\int_{\\,0}^{\\,\\infty}y^{\\,n}e^{-2\\,l\\,y}\\,dy \\,=\\,+\\,\ni^{\\,n+1}\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n+1}\\,}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,n+2}\\,} \\,,\n\\end{equation}\nthe interchange in summation and integration being legitimated by Beppo Levi's monotone convergence theorem, and similarly\n\\begin{equation}\nR_{\\,n}\\, = \\, -\\,i\\sum_{k\\,=\\,0}^{n}\\left(\\!\\!\\begin{array}{c}\n                                                 n \\\\\n                                                 k\n                                         \\end{array}\\!\\!\\right)\\pi^{\\,n-k}\\,i^{\\,k}\n\\frac{\\,k\\,!\\,}{\\,2^{\\,k+1}\\,}\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,k+2}\\,}\\;\\,,\n\\end{equation}\nin both of which there insinuates itself the Riemann zeta function \n\\begin{equation}\n\\zeta\\,(s)\\,=\\,\\sum_{l\\,=\\,1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,s}\\,}\n\\end{equation}\nat a variety of its argument values $\\,s.$\\footnote{This canonical\ndefinition implies a guarantee of series convergence, assured by the requirement that $\\,\\Re\\,s\\,>\\,1\\,.$\nA robust arsenal of knowledge exists for continuing $\\,\\zeta(s)\\,$ across the entire plane of\ncomplex variable $\\,s\\,=\\,\\sigma\\,+\\,i\\,t\\,,$ with a simple pole emerging at $\\,s\\,=\\,1\\,.$}\nSo armed, we proceed next to set\n\\begin{equation}\nK_{n}\\,=\\,L_{n}\\,+\\,H_{n}\\,+\\,R_{\\,n}\\,=\\,0\n\\end{equation}\nand remark that, regardless of the parity of index $\\,n\\,,$ $\\,L_{n}\\,$ {\\em{per se}}\nis always absorbed by the contribution from the highest power $\\,y^{\\,n}\\,$ within the\nintegrand for $\\,R_{\\,n}\\,.$  This circumstance accounts for the imminent appearance of\nthe floor function affecting the highest value\nof summation index $\\,k\\,$ in Eqs. (14)-(16) and (19) below.\n\\newpage\n\\mbox{   }\n\\newline\n\n\n      A requirement that the real part of (13) vanish provides now the following string of valuable\nlog-sine quadrature formulae\n\\begin{eqnarray}\n\\int_{\\,0}^{\\,\\pi}x^{n}\\log(\\,\\sin(x)\\,)\\,dx     &    =  &    -\\,\\frac{\\,\\pi^{\\,n+1}\\,}{\\,n\\,+\\,1\\,}\\log(2)\\,+         \\nonumber   \\\\\n        &      &  \\rule{-2.3cm}{0mm}   +\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n+1}}\\!\\sum_{\\,k\\,=\\,1}^{\\lfloor n\/2 \\rfloor}\\,(-1)^{\\,k}\\,  \n                                          \\frac{\\,(2\\pi)^{n\\,-\\,2k\\,+\\,1}\\,} {\\,(n\\,-\\,2k\\,+\\,1\\,)\\,!\\,}\\,\\zeta\\,(\\,2k+1\\,)  \\;\\,,\n\\end{eqnarray}\nof which the first two, at $\\,n\\,=\\,0\\,$ and $\\,n\\,=\\,1\\,,$ with the sum on the\nright missing, validate (2) and (3), and are in any event widely tabulated.  And again, as was\nfirst stated in Footnote 3, Eq. (14) is consistently reaffirmed by {\\bf{MATHEMATICA}},\neven when harnessed in its symbolic mode.  We note in passing the self-evident fact that,\nunlike the corresponding prescriptions found in [\\textbf{9,10}],\nformula (14) is fully explicit, needing to rely neither upon a generating function nor\na recurrence, even though, naturally, such recurrence arrives at a final rendezvous with identically\nthe same result.\n\n\n     A close prelude to identity (5) follows next from the co\\\"{e}xisting  requirement that\nthe imaginary part of (13) vanish.  This requirement takes the initial form\n\\begin{eqnarray}\n-\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,2\\,(\\,n\\,+\\,1\\,)\\,}\\,+\\,\\frac{\\,\\pi^{\\,n+2}\\,}{\\,\\,n\\,+\\,2\\,}\\,-  \\rule{5.2cm}{0mm}  &      &       \\nonumber \\\\\n-\\,\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\left(\\!\\!\\begin{array}{c}\n                                                 n \\\\\n                                                 2k\n                                         \\end{array}\\!\\!\\right)\\pi^{\\,n-2k}(-1)^{\\,k}\n\\frac{\\,(2k)\\,!\\,}{\\,2^{\\,2k\\,+\\,1}\\,}\\,\\zeta\\,(\\,2k+2\\,) \\rule{0.0cm}{0mm}   &   =  & 0  \\rule{8mm}{0mm}      \n\\end{eqnarray}\nand is subsequently moulded into the shape\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor }\\left(\\!\\!\\begin{array}{c}\n                                                 n \\\\\n                                                 2k\n                                         \\end{array}\\!\\!\\right)\n\\frac{\\,B_{2k\\,+\\,2}\\,}{\\,(\\,k\\,+\\,1\\,)(\\,2k\\,+\\,1\\,)\\,}\\,=\\,\\frac{\\,n\\,}{\\,(\\,n\\,+\\,1\\,)(\\,n\\,+\\,2\\,)\\,}    \n\\end{equation}\non taking note of Euler's\ncelebrated connection [{\\bf{3-8}}]\n\\begin{equation}\n\\zeta\\,(2k)\\,=\\,(-1)^{\\,k\\,+\\,1}(2\\,\\pi)^{2k}\\frac{\\,B_{2k}\\,}{\\,2\\,(2k)\\,!\\,} \\;\\;\\;(\\,k\\,=\\,1\\,,\\,2\\,,\\,3\\,,\\,\\ldots\\,)\n\\end{equation}\nallowing us to displace attention from the even-argument values of Riemann's zeta\nto the correspondingly indexed Bernoulli numbers $\\,B_{2k}\\,.$\n\\parindent=0.25in\n\n\n\\section{Recurrence Reduction}\n\n     Recurrence (16) is not quite yet in the desired form (5), but it is easily steered\ntoward this goal.  That process begins by noting that\n\\begin{equation}\n\\left(\\!\\!\\begin{array}{c}\n                    n \\\\\n                   2k\n          \\end{array}\\!\\!\\right)\n\\frac{\\,1\\,}{\\,(\\,k\\,+\\,1\\,)(\\,2k\\,+\\,1\\,)\\,}\\,=\\,\n\\left(\\!\\!\\begin{array}{c}\n                \\,n+2 \\\\\n                 2k+2\n          \\end{array}\\!\\!\\right)\\frac{\\,2\\,}{\\,(\\,n\\,+\\,1\\,)(\\,n\\,+\\,2\\,)\\,}\\;,\n\\end{equation}\n\\newpage\n\\mbox{   }\n\\newline\n\\newline\n\\newline\nwhereupon (16) becomes\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{\\lfloor \\frac{n-1}{2} \\rfloor}\\left(\\!\\!\\begin{array}{c}\n                                                             \\,n+2 \\\\\n                                                              2k+2\n                                                        \\end{array}\\!\\!\\right)B_{2k\\,+\\,2}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\;.\n\\end{equation}\nNow the advance of index $\\,2k\\,$ in steps of two means that it reaches a maximum value $\\,M=n-1\\,$ when $\\,n\\,$ is odd,\nand one offset instead by two below $n,$ $M=n-2,$ when $\\,n\\,$ is even.  At the same time the accepted {\\mbox{null value of odd-index}}\nBernoulli numbers starting with $\\,B_{3}=0\\,$ means that we are free, and self-consistently so, to intercalate all missing indices in steps of one\nand to entertain a common maximum $\\,M=n-1\\,,$ regardless of the parity of $n.$  Altogether then, (19) re{\\\"{e}}merges as\n\\begin{equation}\n\\sum_{k\\,=\\,2}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n                                 \\,n+2 \\\\\n                                    k\n                              \\end{array}\\!\\!\\right)B_{k}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\;,\n\\end{equation}\nor else\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n                                 \\,n+2 \\\\\n                                    k\n                              \\end{array}\\!\\!\\right)B_{k}\\,=\\,\\frac{\\,n\\,}{\\,2\\,}\\,+\\,\\left\\{\\left(\\!\\!\\!\\begin{array}{c}\n                                 \\,n+2 \\\\\n                                    0\n                              \\end{array}\\!\\!\\right)B_{0}\\,+\\,\\left(\\!\\!\\!\\begin{array}{c}\n                                 \\,n+2 \\\\\n                                    1\n                              \\end{array}\\!\\!\\right)B_{1}\\right\\}\\;.\n\\end{equation}\nBut now we find that\n\\begin{equation}\n\\left(\\!\\!\\!\\begin{array}{c}\n               \\,n+2 \\\\\n                  0\n          \\end{array}\\!\\!\\right)B_{0}\\,+\\,\\left(\\!\\!\\!\\begin{array}{c}\n               \\,n+2 \\\\\n                  1\n          \\end{array}\\!\\!\\right)B_{1}\\,=\\,1\\,-\\,\\frac{\\,n+2\\,}{\\,2\\,}\\,=\\,-\\frac{\\,n\\,}{\\,2\\,}\\,,\n\\end{equation}\nwith the effect of reducing (21) to just\n\\begin{equation}\n\\sum_{k\\,=\\,0}^{n+1}\\left(\\!\\!\\!\\begin{array}{c}\n                                 \\,n+2 \\\\\n                                    k\n                              \\end{array}\\!\\!\\right)B_{k}\\,=\\,0\\;,\n\\end{equation}\nwhich is nothing other than (5).\n\n\n\\section{Appendix:  A Fourier Series Grace Note}\n\n\n    A somewhat more pedestrian derivation of (14) rests upon consideration of the power series\n\\begin{equation}\n\\log\\,(\\,1-z\\,)\\,=\\,-\\,\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,z^{\\,l}\\,}{\\,l\\,}\n\\end{equation}\nalong the unit circle $\\,z\\,=\\,e^{\\,i\\,\\vartheta}.$\nSeparation into real and imaginary parts emerges as a pair of Fourier series\n\\begin{equation}\n\\log\\left(\\rule{0mm}{5mm}2\\left|\\,\\sin\\,\\left\\{\\frac{\\,\\vartheta\\,}{\\,2\\,}\\right\\}\\right|\\right)\\,=\\,\n-\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,\\cos\\,(l\\,\\vartheta)\\,}{\\,l\\,}\n\\end{equation}\nand\n\\begin{equation}\n\\left\\{\\rule{0mm}{5mm}\\frac{\\,\\vartheta-\\pi\\,}{\\,2\\,}\\,,\\,{\\rm{mod}}\\,2\\,\\pi\\right\\}\\,\n=\\,-\\sum_{l\\,=\\,1}^{\\infty}\\,\\frac{\\,\\sin\\,(l\\,\\vartheta)\\,}{\\,l\\,}\\;\\;,\n\\end{equation}\nof which the second is of no interest {\\em{vis-\\`{a}-vis}} our immediate objective.  We repress all scruples\nhenceforth as to the divergence of series (25) whenever $\\,\\vartheta\\,=\\,0\\;\\,{\\rm{mod}}\\;2\\,\\pi\\,.$\n\n\n\\newpage\n\\mbox{   }\n\\newline\n\n\n      Repeated integration by parts {\\em{vis-\\`{a}-vis}} the first of these Fourier series,\nwhen multiplied by the argument power $\\,\\vartheta^{\\,n}\\,,$\nadvances by $\\,\\cos\\rightarrow\\sin\\rightarrow\\cos\\,$ couplets, with end-point contributions\narising only on the second beat, and the argument powers falling\nin steps of two.\\footnote{In particular, this quadrature cadence provides a\nmotivation, alternative to that previously given,\nas to why it is that the floor function affects the upper index cutoff\n$\\,\\lfloor n\/2 \\rfloor\\,$ in both (14) and (27), allowing for unit growth in that\ncutoff only when $\\,n\\,$ {\\em{per se}} advances by two.}  One assembles in this manner the general formula\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\,\\vartheta^{\\,n}\\log(\\,\\sin(\\vartheta)\\,)\\,d\\,\\vartheta \\, = \\,                         \n-\\,\\frac{\\,\\pi^{\\,n\\,+\\,1}\\,}{\\,n\\,+\\,1\\,}\\,\\log(2)\\,+\\,\\frac{\\,n\\,!\\,}{\\,2^{\\,n\\,+\\,1}\\,}\n\\,\\sum_{\\,k\\,=\\,1}^{\\lfloor n\/2 \\rfloor}\\,(-\\,1)^{\\,k}\\,  \n\\frac{\\,(\\,2\\,\\pi\\,)^{\\,n\\,-\\,2k\\,+\\,1}\\,}\n{\\,(\\,n\\,-\\,2k\\,+\\,1\\,)\\,!\\,}\\,\\zeta\\,(\\,2k\\,+\\,1\\,\n\\end{equation}\nholding good unrestrictedly for $\\,n\\,$ even or odd, and agreeing in every respect with (14).  The only\nwrinkle to notice, perhaps, is that the sequence of integrations by parts which underlies (27) terminates,\nat each summation index $\\,l\\,$ in (25), with a term proportional to either\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\cos(2l\\vartheta)\\,d\\,\\vartheta\\,=\\,0\n\\end{equation}\nin the event that $\\,n\\,$ is even, or\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\vartheta \\cos(2l\\vartheta)\\,d\\,\\vartheta\\,=\\,0\n\\end{equation} \notherwise.  Equation (28) is of course obvious whereas (29), while equally true and\nwelcome as such, is, at first blush, mildly surprising.  All in all the derivation which underlies (14) is\nfar smoother and less apt to inflict bookkeeping stress, even if it is (27) which seems to rest\non a more elementary underpinning.\n\n    It would be truly disappointing were we not able to utilize (25) so as to give an\nessentially one-line, zinger-style proof of (1).  This anticipation is readily met simply\nby squaring both sides of (25), with summation indices $\\,l\\,$ and $\\,l\\,'\\,$ figuring\nnow on its right, and noting that when, as here, both $\\,l\\,\\geq\\,1\\,$ and\n$\\,l\\,'\\,\\geq\\,1\\,,$\n\\begin{equation}\n\\int_{\\,0}^{\\,\\pi}\\cos(\\,2\\,l\\,\\vartheta\\,)\\cos(\\,2\\,l\\,'\\vartheta\\,)\\,d\\,\\vartheta\\,=\\,\\frac{\\,\\pi\\,}{\\,2\\,}\\,\n\\delta^{\\,l}_{\\,l\\,'}\\;\\,,\n\\end{equation}\nwith $\\,\\delta^{\\,l}_{\\,l\\,'}\\,$ being the Kronecker delta, unity when its indices match, and zero otherwise.\nIt follows immediately that\n\\begin{equation}\nI\\,=\\,\\frac{\\,1\\,}{\\,2\\,}\n\\int_{\\,0}^{\\,\\pi}\\left\\{\\rule{0mm}{4mm}\\,\\log(\\,2\\,\\sin(\\vartheta)\\,)\\,\\right\\}^{\\,2}\\!d\\vartheta\\,=\\,\n\\frac{\\,\\pi\\,}{\\,4\\,}\\sum_{l=1}^{\\infty}\\frac{\\,1\\,}{\\,l^{\\,2}\\,}\\,=\\,\\frac{\\;\\;\\pi^{\\,3}\\,}{\\,24\\,}\\;\\,,\n\\end{equation}\nand we are done.\n\n\n\\parindent=0.0in\n\n\n\\section{References}\n\n1.\tOmran Kouba, Problem No. 11639, {\\bf{The American Mathematical Monthly}}, Vol. 119, No. 4, April 2012, p. 345.\n\n2.\tLars V. Ahlfors, {\\bf{Complex Analysis}}, McGraw-Hill Book Company, Inc., New York, 1953, pp. 130-131.\n\\newpage\n\\mbox{    }\n\\newline\n\\newline\n\\newline\n3.\tTom M. Apostol, {\\bf{Introduction to Analytic Number Theory}}, Springer-Verlag, New York, 1976, p. 266.\n\n4.\tHans Rademacher, {\\bf{Topics in Analytic Number Theory}}, Springer-Verlag, New York, 1973, p. 16.\n\n5.\tHerbert S. Wilf, {\\bf{Mathematics for the Physical Sciences}}, John Wiley \\& Sons, Inc., New York, 1962, pp. 114-116.\n\n6.\tTom M. Apostol, {\\bf{A Primer on Bernoulli Numbers and Polynomials}}, Mathematics Magazine, Vol. 81, No. 3, June 2008, pp. 178-190.\n\n7.\tOmran Kouba, {\\bf{Lecture Notes:  Bernoulli Polynomials and Applications}}, arXiv:1309.7560v1 [math.CA] 29 Sep 2013.\n\n8.\tEric W. Weisstein, {\\bf{Bernoulli Number}}, from {\\em{MathWorld}}--A Wolfram Web Resource available\n@ {\\bf{http:\/\/mathworld.wolfram.com\/BernoulliNumber.html}}\n\n9.\tL. Lewin, {\\bf{On the evaluation of log-sine integrals}}, {\\em{The Mathematical Gazette}}, Vol. 42, 1958, pp. 125-128.\n\n10.\tL. Lewin, {\\bf{Polylogarithms and associated functions}}, North Holland, 1981.\n\n11.\tJonathan M. Borwein and Armin Straub, {\\bf{Special values of generalized log-sine integrals}},\n{\\em{Proceedings of ISSAC 2011 (36th International Symposium on Symbolic and Algebraic Computation)}}, 2011, pp. 43-50.\n\n\n12.\tJonathan M. Borwein and Armin Straub, {\\bf{Mahler measures, short walks and log-sine integrals}}, {\\em{Theoretical Computer Science\n(Special issue on Symbolic and Numeric Computation)}}, Vol. 479, No. 1, 2013, pp. 4-21.\n\n\n\n\n\\end{document}\n \t\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{}\n\n\n\\tableofcontents\n\n\\section{Introduction}\n\nA large class of interesting and useful asymptotically locally anti-de Sitter (AlAdS) spacetimes have been constructed by starting with AdS in Poincar\\'e coordinates, in which the spacetime is foliated by slices on which the metric is conformal to the Minkowski metric $\\eta_{ab}$, and replacing $\\eta_{ab}$ with any Ricci-flat metric $\\gamma_{ab}$.  Thus~$(D-1)$-dimensional vacuum solutions to Einstein's equations straightforwardly give rise to new $D$-dimensional solutions to Einstein's equation with a negative cosmological constant.  For example, taking $\\gamma_{ab}$ to be the Schwarzchild black hole yields a black cigar \\cite{Chamblin:1999by}, and $\\gamma_{ab}$ was taken to be a vacuum $pp$-wave in \\cite{Chamblin:1999cj} to construct a wave  in the far field of an AdS-brane spacetime.  Furthermore, the AlAdS solutions so generated are of particular interest in light of the AdS\/CFT correspondence~\\cite{Maldacena:1997re,Gubser:1998bc,Witten:1998qj}, since they are dual to a large-$N$, strongly coupled conformal field theory (CFT) that lives on the spacetime~$\\gamma_{ab}$ (or a conformally rescaled version thereof).  For instance, \\cite{Engelhardt:2013jda,Engelhardt:2014mea} took~$\\gamma_{ab}$ to be a vacuum Kasner metric in order to study a cosmological singularity by computing the entanglement entropy and Wightman functions of the CFT\\footnote{In fact, in~\\cite{Engelhardt:2014mea} the CFT lived on a singularity-free conformally rescaled version of Kasner.}.\n\nIn this paper we generalize this construction and show how non-vacuum $(D-1)$-dimensional spacetimes can be used to give AlAdS spacetimes with nonzero stress-energy. Specifically, if $\\gamma_{ab}$ is a solution to the Einstein equations in $(D-1)$-dimensions with stress-energy tensor $\\widehat{T}_{ab}$, we show that replacing $\\eta_{ab}$ on the Poincar\\'e slices with $\\gamma_{ab}$ gives a $D$-dimensional AdS solution with stress-energy $T_{ab}$ that satisfies\n\\be\nT_{\\mu\\nu} = \\widehat{T}_{\\mu\\nu},\n\\ee\nwhere~$\\mu,\\nu = 0,\\ldots,D-1$.  By appropriate choice of~$\\widehat{T}_{ab}$, we use this construction to find new AlAdS spacetimes that have a physically sensible stress-energy.  We also show that these spacetimes can be ``solitonized'' \\cite{Haehl:2012tw} by adding a compact dimension that shrinks smoothly to zero in the AdS bulk.  A variation of this non-vacuum construction was performed in \\cite{Cvetic:2000gj,Lu:2000xc,Park:2001jh}, which studied getting  supergravity gauge fields ``on the brane'' by doing a Kaluza-Klein reduction of a supergravity theory in the higher dimensional AdS spacetime. A related\nconstruction starting with supergravity fields in ten dimensions was used to explore properties of time dependent boundaries in references\n \\cite{Das:2006dz,Das:2006pw,Awad:2007fj,Awad:2008jf}.\n The relation between their higher and lower dimensional matter theories differs from what we find here, as will be clarified in Section~\\ref{derive}. \n\nAs a special example, we will take~$\\gamma_{ab}$ to be a Friedman-Robertson-Walker (FRW) cosmology\\footnote{From the CFT side, such solutions can be thought of as a generalization of those in~\\cite{Koyama:2001rf}, which took the boundary metric to be a conformally flat FRW geometry.}.  The cosmological stress-energy is an isotropic perfect fluid with energy density $\\tilde{\\rho} (t)$ and pressure $ \\tilde{p} (t) $, which are related by an equation of state $\\tilde{p} (t) = w \\tilde{\\rho} (t) $. We will show that in the AdS spacetime, the fluid is still isotropic on the cosmological slices with the same equation of state $p= w\\rho $ and that the pressure in the AdS radial direction is given by $p_y= (3w - 1)\\rho \/2 $. The pressures and density decay towards the AdS boundary as well as in time as the universe expands.  A case of special interest is a free, massless scalar field which in a $(D-1)$-dimensional FRW spacetime has the equation of state $\\tilde{p} = \\tilde{\\rho} $, that is, $w=1$. Hence the scalar field generates a $D$-dimensional AdS cosmology which is isotropic in all spatial directions and has corresponding equation of state $p=\\rho$.  According to the AdS\/CFT prescription, such a scalar field in the bulk AdS is dual to to a scalar operator in the CFT with vanishing expectation value but nonzero source.\n    \nWe will focus on FRW metrics with negatively curved spatial slices, in which case~$\\gamma_{ab}$ approaches the future Milne wedge of Minkowski space at late time, so long as $w>-1\/3$.  The resulting AlAdS solution therefore approaches either the Poincar\\'e patch of AdS or the AdS soliton at late times, which we interpret as an approach to equilibrium.  We use these solutions to perturbatively study the approach to equilibrium of the boundary stress tensor and the ADM charges. \nInterestingly, we find that the latter \\textit{decrease} to their equilibrium values at late times, with the time dependent correction proportional to the dimensionless density parameter of the universe $\\Omega$. \nFor example, the mass of the solitonized cosmology decays as\n\\be\n {\\cal M} = \\left( 1-{1\\over 2} \\Omega \\right)  {\\cal M}^{(0)} + \\cdots,\n \\ee\n where ${\\cal M}^{(0)} $ is the mass of the static soliton and~$\\cdots$ stands for subleading terms at late times.\nIn the context of spacetimes approaching the AdS soliton, this result is consistent with the energy conjecture of \\cite{Horowitz:1998ha} that the AdS soliton is the lowest energy spacetime with the prescribed asymptotic structure.\n\nA second application of our cosmological AdS solutions will be to compute the behavior of the entanglement entropy $S$ of a spherical region in the CFT as the spacetime evolves to equilibrium.  We use the covariant  prescription of~\\cite{Hubeny:2007xt}, which is a generalization of the static prescription~\\cite{Ryu:2006ef}.  This states that the entanglement entropy~$S_\\mathcal{R}$ of a region~$\\mathcal{R}$ of a holographic CFT is related to the area of a special bulk surface~$\\Sigma$.  In general,~$S_\\mathcal{R}$ is UV-divergent, but it can be regulated and the behavior of this regulated entropy~$S_\\mathrm{ren}$ is studied.  We find that at late times,~$S_\\mathrm{ren}$ decays as a power law in the proper time of an asymptotically static observer.\n\nOur results add to and complement the substantial body of work in the literature on vacuum AlAdS spacetimes in which the metric on the AdS boundary is time dependent.  For instance,~\\cite{Fischler:2013fba} constructed an elegant solution in which the metric on each Poincar\\'e slice is a de Sitter cosmology.  Several studies in the general category of holographic cosmology apply coordinate transformations to AdS black holes to produce cosmological boundaries \\cite{Lidsey:2009xz,Erdmenger:2012yh,Ghoroku:2012vi,Banerjee:2012dw,\nCompere:2008us,Binetruy:1999hy,Kajantie:2008hh,Apostolopoulos:2008ru}, and resulting metrics have been analyzed as describing an expanding boost-invariant plasma \\cite{Janik:2005zt,Janik:2006ft,Kajantie:2008jz,Culetu:2009xm,Pedraza:2014moa}.\nSignificant analytical work has also been done on out of equilibrium thermal properties of field theories using various AdS black hole spacetimes, including \\cite{Janik:2010we,Lamprou:2011sa,Figueras:2009iu,\nTetradis:2009bk,Heller:2011ju,Fischetti:2012ps,Fischetti:2012vt,Beuf:2009cx}.  Discussion and further references can be found in \\cite{Marolf:2013ioa}.  Our work adds a set of new \nnon-vacuum AlAdS spacetimes which allow a wide range of boundary metrics.\n  \nThis paper is organized as follows.  Section \\ref{derive} contains the derivation of the new AlAdS solutions, as well as an analysis of the scalar field and perfect fluid cases.  In section \\ref{sec:bst}, the leading time dependent corrections to the boundary stress tensor and the ADM charges for a solitonic cosmology in an open universe are found.  In section \\ref{entropy} the perturbation to the entanglement entropy is calculated, and section \\ref{conclusion} contains discussion and concluding remarks.  Unless otherwise specified, we take Newton's constant~$G_N = 1$. \n\n\n\n\\section{AdS and AdS soliton cosmologies}\\label{derive}\n\nWe start by considering AlAdS spacetimes of the general form\n\\be\n\\label{metric}\nds_D^2 = dy^2 + e^{2y\/l}\\gamma_{\\mu\\nu } (x^\\alpha ) dx^\\mu dx^\\nu,\n\\ee\nwhere~$l$ is the AdS length and as before~$\\mu,\\nu = 0,1,\\dots, D-1$.  For $\\gamma_{\\mu\\nu} = \\eta_{\\mu\\nu}$ this is AdS in Poincar\\'e coordinates with cosmological constant given by \n\\be\n\\Lambda = -{(D-1)(D-2)\\over 2l^2},\n\\ee\nso we will refer to hypersurfaces~$y = \\mathrm{const.}$ as ``Poincar\\'e slices''.  As mentioned above, it is well known that the Einstein equations with cosmological constant $\\Lambda$ are still satisfied for any Ricci-flat $\\gamma_{\\mu\\nu}(x^\\rho)$.  In the particular case where~$\\gamma_{\\mu\\nu}$ is a cosmological metric, we refer to \\eqref{metric} as an AdS cosmology.  \n\nIn general, spacetimes of the form~\\eqref{metric} with~$\\gamma_{\\mu\\nu} \\neq \\eta_{\\mu\\nu}$ suffer from a singularity at the Poincar\\'e horizon~$y \\to -\\infty$.  This singularity can be resolved by introducing an additional compact direction~$v$:\n\\be\n\\label{solmetric}\nds_D^2 = {dy^2 \\over F(y) }  + e^{2y\/l}\\left( F(y) dv^2 + \\gamma_{\\mu\\nu } (x^\\rho ) dx^\\mu dx^\\nu  \\right),\n\\ee\nwhere $F(y) = 1- e^{ -(D-1)(y-y_+)\/l }$, and now~$\\mu,\\nu= 0,1,\\dots,D-2$.    The metric~\\eqref{solmetric} is capped off at~$y = y_+$, so that the Poincar\\'e horizon (and its possibly singular behavior) is removed.  Regularity at this cap fixes the period of~$v$ to be\n\\be\\label{period}\nv \\sim v + \\frac{4\\pi l e^{-y_+\/l}}{D-1}.\n\\ee\nNow, if $\\gamma_{\\mu\\nu } = \\eta_{\\mu\\nu}$, then~\\eqref{solmetric} is the usual AdS soliton metric~\\cite{Horowitz:1998ha}\\footnote{Although this is no longer Poincar\\'e AdS, we will continue to call surfaces of~$y = \\mathrm{const.}$,~$v = \\mathrm{const.}$ Poincar\\'e slices.}.  However, it was noted in \\cite{Haehl:2012tw} that the Einstein equations with negative cosmological constant $\\Lambda$ will still be satisfied for any Ricci-flat~$\\gamma_{\\mu\\nu}$.  In analogy with~\\eqref{metric}, if~$\\gamma_{\\mu\\nu}$ is a cosmological metric, we will refer to~\\eqref{solmetric} as an AdS soliton cosmology.\n\nThe solutions~\\eqref{metric} and~\\eqref{solmetric} provide a simple construction of AlAdS spacetimes with any desired Ricci-flat boundary metric~$\\gamma_{\\mu\\nu}$\\footnote{Technically, the boundary of~\\eqref{solmetric} is~$\\gamma_{\\mu\\nu}$ cross the circle direction~$v$.}.  Our goal is to generalize the above results to isotropic FRW cosmological metrics $\\gamma_{\\mu\\nu}$; as such cosmologies are not (in general) Ricci-flat, we will require the introduction of matter fields.\n\n\n\n\\subsection{Massless Scalar Field}\n\nWe obtain a direct analogue of the results for vacuum metrics by considering a free massless scalar field $\\phi$ in the AdS and AdS soliton spacetimes above.  The full $D$-dimensional Einstein-massless scalar equations are\n\\be\n\\label{einstein}\nG_{ab}=-\\Lambda g_{ab}+8\\pi T_{ab},\\qquad \\nabla^2\\phi=0,\n\\ee\nwhere~$G_{ab}$ is the Einstein tensor and \n\\be\nT_{ab}={1\\over 8\\pi}[(\\nabla_a\\phi)\\left(\\nabla_b\\phi\\right)-{1\\over 2}g_{ab}g^{cd}(\\nabla_c\\phi)\\left(\\nabla_d\\phi\\right)]\n\\ee\nis the stress-energy of a free massless scalar field in any dimension.  \nConsider a lower-dimensional metric $\\gamma_{\\mu\\nu}(x^\\rho)$ and scalar field configuration $\\phi(x^\\mu)$ that solve the Einstein-scalar equations\n\\be\n\\label{slice}\n{\\widehat G}_{\\mu\\nu} = 8\\pi{\\widehat T}_{\\mu\\nu}, \\qquad \\widehat\\nabla^2\\phi=0,\n\\ee\nwhere hatted objects are computed with respect to the metric~$\\gamma_{\\mu\\nu}$. Furthermore, let\n\\be\\label{smetricdef}\ns_{\\mu\\nu}= e^{2y\/l} \\gamma_{\\mu\\nu}\n\\ee\nbe the induced metric on a Poincar\\'e slice. Finally, we pause to note that the scalar field stress energy satisfies the important property that from the full~$D$-dimensional point of view, the induced stress tensor on each Poincar\\'e slice is equal to the lower-dimensional stress tensor of the scalar field on~$\\gamma_{\\mu\\nu}$:\n\\be\n\\label{Tcondition}\nT_{\\mu\\nu} = \\widehat{T}_{\\mu\\nu}.\n\\ee\n\nNow, consider first the metric (\\ref{metric}).  The $D$-dimensional Ricci tensor for $g_{ab}$ is related to the Ricci tensor of $s_{\\mu\\nu}$ by\n %\n\\be\n\\label{ricci}\nR_{\\mu\\nu} [g] = R_{\\mu\\nu} [s] - \\frac{D-1}{l^2} \\, s_{\\mu\\nu}, \\quad R_{yy} = \\frac{D-1}{l^2}.\n\\ee\n When these components are assembled into the $D$-dimensional Einstein tensor and substituted into the left hand side of the Einstein field equation \\eqref{einstein}, one sees that the terms which do not involve the curvature of $s_{\\mu\\nu}$ are equal to the cosmological constant term on the right hand side. If $\\gamma_{\\mu\\nu}$ is Ricci flat, then the metric (\\ref{metric}) is a solution with $T_{ab}=0$. If instead $\\gamma_{\\mu\\nu}$ is a solution to (\\ref{slice})\n with nonzero ${\\widehat T}_{\\mu\\nu}$,\n  then it is then straightforward to show that the metric~\\eqref{metric} constructed from~$\\gamma_{\\mu\\nu}$ will satisfy the full equations of motion \\eqref{einstein}, with the full bulk scalar field taken to be~$\\phi(x^\\mu)$ (which, in particular, is independent of~$y$). The additional nonzero component of the\n  stress-energy tensor is $8\\pi T_{yy}=-{1\\over 2} s^{\\mu\\nu} \\nabla_\\mu \\phi \\nabla_\\nu \\phi$. The construction with the solitonized metric~\\eqref{solmetric} proceeds in a similar way, and one finds that $T_{yy}$ is the same and $T_{vv} = g_{vv} T_{yy}$.\n\nOne may naturally ask if such a straightforward foliation can be extended to other types of matter as well.  For instance, one might hope to replace the scalar field with a Maxwell field and obtain multi-black hole solutions analogous to those of~\\cite{Kastor:1992nn}.  This is not the case: a key ingredient in the proof was the property \\eqref{Tcondition} of the scalar field stress-energy. \nThis property holds for the massless scalar field stress tensor but not {\\it e.g.} for Maxwell fields, or even for a scalar field with nonzero potential $V(\\phi)$.\n\nA massless scalar field that depends only on time can serve as the source for an FRW cosmology on the Poincar\\'e slices of either \\eqref{metric} or the soliton metric \\eqref{solmetric}.  For instance, setting $d\\hat s^2=\\gamma_{\\mu\\nu}dx^\\mu dx^\\nu$ and specializing to $4$-dimensional cosmologies with flat spatial sections we have\n\\be\nd\\hat s^2= -dt^2 + \\left({t\\over t_0}\\right)^{2\/3}(dx^2+dy^2+dz^2),\\quad \\phi=-\\sqrt{{2\\over 3}} \\, \\ln\\left({t\\over t_0}\\right),\n\\ee\nwith corresponding stress tensor equal to that of a perfect fluid obeying the stiff matter equation of state $\\tilde{p}=\\tilde{\\rho}$.  From the holographic perspective, the AdS\/CFT dictionary tells us that the bulk scalar field is dual to a scalar operator in the CFT.  To be specific, the near-boundary behavior of a massless scalar field in AdS takes the form\n\\be\n\\phi(y) = \\left(\\phi_0 + \\cdots\\right) + e^{-(D-1)y\/l} \\left(\\phi_{(D-1)} + \\cdots \\right),\n\\ee\nwhere~$\\phi_0$ and~$\\phi_{(D-1)}$ are independent parameters that are fixed by the boundary conditions, and~$\\cdots$ represent subleading terms in~$e^{-y\/l}$.  The coefficient~$\\phi_0$ should be interpreted as the source of a scalar operator~$\\mathcal{O}$ of dimension~$D-1$, whose expectation value is~$\\left\\langle \\mathcal{O} \\right\\rangle = \\phi_{(D-1)}$.  Our solutions correspond to the special case~$\\phi_{(D-1)} = 0$.  Note that this is unconventional: the operator~$\\mathcal{O}$ is being sourced, but nevertheless has a zero expectation value.\n\n\n\\subsection{Perfect Fluid Matter}\n\nNoting that property~\\eqref{Tcondition} of the scalar field stress tensor was the key element in the above construction, we may extend the range of AdS and AdS soliton cosmologies by considering more general types of stress-energy that satisfy this condition.  We will shortly focus on perfect fluids, but we begin by assuming just  that the metric $\\gamma_{\\mu\\nu}$ satisfies Einstein's equation on a Poincar\\'e slice with some stress-energy ${\\widehat T}_{\\mu\\nu}$.   We can then analyze the content of the full $D$ dimensional Einstein equations,\n beginning with the AdS type metrics (\\ref{metric}), in the following way.  \n\nUsing the relations between the components of the Ricci tensor (\\ref{ricci}) as in the previous subsection, we find that\n the AdS-type metric \\eqref{metric} solves the Einstein equation~\\eqref{einstein} with stress-energy given by\n\\be\n\\label{fullads}\nT_{\\mu\\nu} = {\\widehat T}_{\\mu\\nu},\\qquad T_{yy}={1\\over D-3}\\, T ,\n\\ee\nwhere $T=s^{\\mu\\nu}\\,T_{\\mu\\nu}$. \n A similar analysis for the AdS soliton-type metric \\eqref{solmetric} shows that the Einstein equation \\eqref{einstein} is solved with stress-energy given by\n\\be\n\\label{fullsoliton}\nT_{\\mu\\nu} = {\\widehat T}_{\\mu\\nu},\\qquad T_{yy}={1\\over D-4} \\, g_{yy}T, \\qquad T_{vv}={1\\over D-4} \\, g_{vv}T.\n\\ee\nFor example, one could embed a textbook example of a four-dimensional spherical static star into AdS. According to\n(\\ref{fullsoliton}) the pressures in the radial AdS and compact soliton directions of this cigar-star will be equal to each other,\nbut different from the radial pressure in the Poincar\\'e plane.\n\nWe now specialize to the case of AdS and AdS soliton cosmologies, taking\nthe metric $\\gamma_{\\mu\\nu}$ to have the FRW form\n\\be\n\\label{cosmometric}\nd\\hat s^2 =  -dt^2 + a^2 (t)\\,  d\\Sigma _k ^2,\n\\ee\nwhere $d\\Sigma _k ^2 $ is  a metric on a space with constant curvature $k=0,\\pm 1$.  We also restrict our attention to $4$-dimensional Poincar\\'e slices, so that the AdS cosmologies \\eqref{metric} have overall dimension $D=5$ and the AdS soliton cosmologies \\eqref{solmetric} have dimension $D=6$. \n Finally, we assume that the stress-energy ${\\widehat T}_{\\mu\\nu}$ on the slice has the perfect fluid form\n\\be\n{\\widehat T}_{\\mu\\nu} = (\\hat\\rho+\\hat p)\\hat{u}_\\mu \\hat{u}_\\nu +\\hat p \\gamma_{\\mu\\nu},\n\\ee\nwith $\\gamma^{\\mu\\nu}\\hat{u}_\\mu \\hat{u}_\\nu=-1$ and equation of state $\\hat p=w\\hat\\rho$.  Note that the strong energy condition requires~$w \\geq -1\/3$.  Important special cases are $w=0$ for dust, $w=1\/3$ for radiation, and $w=1$ for the massless free scalar field; indeed, note that such a stress tensor obeys the condition~\\eqref{Tcondition}\\footnote{Values of $w$ different from~1 could be obtained from an interacting scalar field, but as such interactions would require the introduction of a scalar potential, they are not compatible with our ansatz.  We will keep $w$ general, with the understanding that this is a bulk, hydrodynamic description.}.\n\nThe cosmological scale factor on the Poincar\\'e slices evolves according to the Friedmann equations\n\\be\n\\label{frweqs}\nd(\\hat \\rho\\,  a^3 ) = - \\hat p\\,  d(a^3 ) , \\qquad\n\\left( {\\dot a\\over a}\\right)^2 ={8\\pi \\hat\\rho\\over 3}-{k\\over a^2}.\n\\ee\nThe full stress energy tensor $T_{ab}$ for AdS cosmologies, given by \\eqref{fullads}, now has the form of an anisotropic fluid, with a distinct equation of state parameter for the pressure in the $y$-direction.  Moreover, the energy density and pressures depend on the radial coordinate~$y$, as well as on time.  One finds that the energy density is given by $\\rho= e^{-2y\/l}\\hat\\rho$, while the pressures  tangent to the Poincar\\'e slices satisfy an equation of state in the $D$-dimensions of the same form as in $(D-1)$, namely $p=w\\rho$. In the $y$-direction one finds that $p_y = w_y\\rho$ with\n\\be\nw_y ={ (3w -1)\\over 2}.\n\\ee\nWith a soliton, equation \\eqref{fullsoliton} implies that the pressure in the compact $v$-direction is equal to $p_y$, so also\n$p_v= w_y\\rho$. To summarize, the stress-energy for the AdS soliton cosmology is\n\\be\n\\rho= e^{-2y\/l}\\hat\\rho (t), \\quad p=w\\rho, \\quad p_y =p_v = { (3w -1)\\over 2} \\,  \\rho.\n\\ee\nSome observations are as follows.  For $w=1$, which corresponds to the massless scalar field discussed above, $w_y=1$ as well so\nthe pressure in the full spacetime is isotropic.  For radiation ($w=1\/3$), the stress-energy on the Poincar\\'e slices is traceless and the pressure orthogonal to the slices vanishes, so the stress tensor remains traceless. For $w<1\/3$, the orthogonal pressure is negative.\n\n\n\\subsection{Open AdS and AdS Soliton Cosmologies}\n\nWe will be particularly interested in AdS and AdS soliton cosmologies with open ($k=-1$) FRW universes on the Poincar\\'e slices.  In this case, provided that the equation of state parameter is in the range $w>-1\/3$ (that is, that the strong energy condition holds), the energy density $\\hat\\rho$ will fall off faster than $1\/a^2$ and at late times the scale factor will grow linearly in time.  At sufficiently late times, the metric $\\gamma_{\\mu\\nu}$ on the Poincare slices then approaches\n\\be\nd\\hat s^2_\\mathrm{late} =  -dt^2 + t^2\\,  d\\Sigma _{-1} ^2,\n\\ee\nwhich is flat spacetime in Milne coordinates.\nThe full AdS and AdS soliton cosmological metrics \\eqref{metric} and \\eqref{solmetric} then respectively approach the AdS or AdS soliton metrics at late times.  The late-time behavior of these cosmologies can therefore be thought of as an approach to equilibrium; in particular, the CFT dual can be thought of as an expanding isotropic plasma equilibrating at late time.  The solutions~\\eqref{metric} and~\\eqref{solmetric} correspond to the plasma being in a deconfined or confined phase, respectively.\n\n\n\n\\section{ADM Mass and Boundary Stress Tensor for AdS Soliton Cosmologies}\n\\label{sec:bst} \n\nFrom the field theoretic side, the late-time behavior of the AdS cosmology with open spatial slices described above is interpreted as a relaxation of the CFT to the vacuum state.  This relaxation can be studied by computing the late-time behavior of CFT observables. \nAs  a first examination of the properties of these AdS cosmologies, we look at how the cosmological expansion impacts the boundary stress tensor and the\nADM mass and tensions of the AdS soliton (which corresponds to the confined phase of the dual field theory; see e.g.~\\cite{Mateos:2007ay}).  In general we lack a definition of the ADM charges that will apply at the boundary $y=\\infty$ with a time dependent boundary metric.  However, as we will see the special case of an open cosmology with matter obeying the strong energy condition~$w>-1\/3$ allows for a perturbative computation of how the ADM charges of the soliton approach their static values at late times.\n\nThe static AdS soliton has negative ADM mass, reflecting the negative Casimir energy of the boundary field theory with a  compact direction, and is conjectured to be the lowest energy solution among spacetimes with these asymptotics \\cite{Horowitz:1998ha}.\nIn addition to its mass, the AdS soliton has nonzero ADM tensions \\cite{El-Menoufi:2013pza,El-Menoufi:2013tca}.  The tension along the compact $v$-direction in (\\ref{solmetric})  is found to be large and positive, while the other three spatial tensions have negative values, such that the trace of the ADM charges (sum of the mass and the tensions) vanishes. One can think of the static soliton solution as an equilibrium configuration. In this section we will compute the approach to equilibrium of the boundary stress tensor, as well as the mass and tensions for an open AdS soliton cosmology. We will see that the mass decreases to the static soliton value, a result that is consistent with the minimum mass conjecture with matter obeying the strong energy condition.\n\nAs noted above, the FRW boundary metric does not have a time-translation symmetry and therefore the ADM mass is not defined in the usual sense.  However, at late times the FRW cosmologies with negatively curved  spatial slices approach Minkowski spacetime.  We can then define a time dependent ADM mass in this late time limit by writing the metric as static AdS plus time dependent perturbations that decay to zero. These perturbations to the metric determine the late time corrections to the asymptotic constant value of the mass of the soliton. \n\n\nConsider an AdS soliton cosmology \\eqref{solmetric} with an open FRW metric \n\\be\n\\label{openfrw}\nd\\hat s^2 = - dt^2 + a^2 (t) \\left( d\\chi ^2 + \\sinh ^2 \\chi \\, d\\Omega_{(2)} ^2 \\right)\n\\ee\non the Poincar\\'e slices.  We assume that $w>-1\/3$, so that in the late time limit $a(t)\\simeq t$.  Define new coordinates on the slices according to\n\\be\n\\label{coordtrans}\nT =a(t ) \\cosh \\chi,  \\quad R= a(t) \\sinh \\chi.\n\\ee\nNote that since $R\/T =\\tanh \\chi$ it follows that $R\/T \\leq 1$ with equality when $\\chi \\rightarrow \\infty$.\nIn terms of these new coordinates the AdS soliton cosmology has the form\n\\be\n\\label{latemetric}\nds^2 = {dy^2 \\over F(y) } + e^{2y\/l} \\left[\\, F(y)\\,  dv^2 -dT^2 (1-\\delta \\tilde{g}_{TT} ) +dR^2 (1+ \\delta \\tilde{g}_{RR} ) + 2 \\, \\delta \\tilde{g}_{TR}\\,  dR\\, dT  +\nR^2 d\\Omega_{(2)}  ^2 \\right]\n\\ee\nwhere \n$F(y)$ is given in (\\ref{solmetric}) and the functions $\\delta \\tilde{g}_{TT}$, $\\delta \\tilde{g}_{RR}$ and $\\delta \\tilde{g}_{TR}$, which give the deviance of the metric on the Poincar\\'e slices from flat, may be written as\n\\be\n\\label{fndef}\n\\delta \\tilde{g}_{TT} = \\Omega \\, \\frac{1}{1-(R\/T)^2}, \\quad\n \\delta \\tilde{g}_{RR} =  \\Omega \\, \\frac{(R\/T)^2}{1-(R\/T)^2}, \\quad \\delta \\tilde{g}_{TR} = \\Omega \\, \\frac{R\/T}{1-(R\/T)^2}.\n\\ee\nHere $\\Omega$ is the dimensionless density parameter of the open FRW metric,\n\\be\\label{omegadef}\n\\Omega = {8\\pi \\hat \\rho \\over 3 H^2} = \\left(1-  {1\\over \\dot{a} ^2 }  \\right),\n\\ee\nand $H=\\dot a\/a$ is the Hubble parameter. For an open universe $\\Omega<1$ and approaches zero in the far future. Hence, the metric \\eqref{latemetric} approaches the AdS soliton at late times.  We emphasize that the expressions in \\eqref{fndef} are exact up to this point.\n\n\nTo proceed further, the density parameter  $\\Omega$  must be expressed in terms of the asymptotically Minkowski coordinates $(T,R)$, which requires the expression for the scale factor $a(t)$ at late times.  To obtain this expression, first we substitute the equation of state~$p=w \\rho$ into the Friedman equations \\eqref{frweqs}, which allows the energy density to be solved for in terms of the scale factor, giving\n\\be\\label{laterho}\n\\hat\\rho (t)=  {3\\bar{\\Omega}_* H_*^2\\over 8\\pi (H_*a(t))^{3(1+w)}} \\  ,  \\mbox{ where }  \\bar{\\Omega}_*  \\equiv {\\Omega_* \\over (1- \\Omega_*  )^{3(w+1)\/2 }}\n\\ee\nand $H_*$ and $\\Omega_*$ are the Hubble and density parameters evaluated at a fiducial time $t=t_*$.\nThe density and scale factor evaluated at $t_*$ are given by  $\\hat{ \\rho}_* = 3\\Omega_* H_*^2\/8\\pi$ and  $a_* = 1\/ ( H_*  \\sqrt{1-\\Omega_* }) $ respectively.   The equation for the scale factor then reduces to\n\\be\n\\dot a^2 = 1 +{ \\bar\\Omega_*H_*^2 a_*^{3(1+w)}\\over a^{1+3w}}.\n\\ee\nFor $w>-1\/3$, this reduces in the limit of large scale factor to $\\dot a^2 \\simeq1$, giving $a(t)\\simeq t$ in the late time limit.  Including a subleading correction of the form~$a(t) \\simeq t + \\alpha t^\\beta$ yields\n\\begin{subequations}\n\\label{latea}\n\\bea\na(t)  &\\simeq t - {\\bar\\Omega_* \\over  6wH_* ( H_* t ) ^{3w } }, \\quad  w\\neq 0, \\\\\na(t) &\\simeq t + {\\bar\\Omega_* \\over 2 H_*  } \\ln \\left( {t \\over H_* } \\right) ,\\quad  w = 0,\n\\eea\n\\end{subequations}\nwhich by~\\eqref{omegadef} yield\n\\be\\label{omegat}\n\\Omega (t)  \\simeq {\\bar\\Omega_*\\over  (H_* t) ^{(1+3w )}}.\n\\ee\n\nThe expressions~\\eqref{latea} can be inverted and combined with the transformation to~$(R,T)$ coordinates to yield the coordinate transformation from~$t$ to~$(R,T)$, valid at late times, including terms up to order $R^2 \/ T^2$,\n\\begin{subequations}\n\\bea\nt&\\simeq T+{\\bar\\Omega_*\\over 6wH_*(H_*T)^{3w}}-{R^2\\over 2T},\\qquad w\\neq 0, \\\\\nt&\\simeq T -{\\bar\\Omega_*\\over 2H_*} \\ln\\left( {T \\over H_* } \\right)   -{R^2\\over 2T},\\qquad w= 0,\n\\eea\n\\end{subequations}\nwhich then give $\\Omega$ as a function of $T$ and $R$:\n\\begin{subequations}\n\\label{lateomega}\n\\bea\n\\Omega &\\simeq  {\\bar\\Omega_*\\over ( H_* T) ^{3w+1 } } \n\\left(1- {(1+3w)\\bar\\Omega_*\\over 6w (H_* T ) ^{3w +1 } }  + {(1+3w ) R^2 \\over 2T^2  } \\right), \\quad  w\\neq 0, \\\\\n\\Omega &\\simeq  {\\bar\\Omega_*\\over H_* T} \\left( 1  +\n  {\\hat\\Omega_* \\over 2 H_* T } \\ln\\left( {T \\over H_* } \\right)    +{ R^2 \\over 2 T^2} \\right), \\quad  w = 0.\n\\eea\n\\end{subequations}\nThis is our desired result.\n\n\nWe are now prepared to compute the leading late time corrections to the boundary stress tensor density, which we will denote by\n $\\tau _{\\mu\\nu}$. Let $K _{\\mu\\nu}$ be the extrinsic curvature of the  AdS boundary.  In the boundary stress tensor formalism a boundary\n action is defined that includes an integral over $K$ plus\n geometrical counterterms that are constructed from the metric on the boundary $s_{\\mu\\nu}$, defined in equation \\eqref{smetricdef}.\n  These terms include a cosmological constant,\n  the scalar curvature of $s_{\\mu\\nu}$, and potentially higher derivative counter terms as needed. \n The stress tensor density  results from \nvarying the boundary action with respect to $s^{\\mu\\nu}$. The coefficients of the counter terms are\n chosen to cancel divergences that occur in $\\tau_{\\mu\\nu}$ and are dimension dependent. One finds the result \\cite{Balasubramanian:1999re,Myers:1999psa,de Haro:2000xn}\n\\be\n\\label{bst}\n8\\pi  \\tau_{\\mu\\nu} = \\sqrt{-s} \\left( K _{\\mu\\nu} - Ks_{\\mu\\nu}  +{ D-2 \\over l} s_{\\mu\\nu}  +{1\\over D-3} G _{\\mu\\nu}  [s] + \\cdots \\right),\n\\ee\nwhere the $\\cdots$ indicate higher derivative terms in the Riemann tensor of $s_{\\mu\\nu}$, which we will show are subdominant at late times.\nWe work with the boundary stress tensor density because the volume element of the late time metric changes at leading order, and \nalso because this is the appropriate quantity to integrate to get the ADM charges.\n\n In the case of the static AdS soliton, the metric on the boundary is flat and the terms in (\\ref{bst})\n  depending on the curvature of $s_{\\mu\\nu}$ all vanish.  This is no longer true for\n the cosmological AdS spacetimes. \n The Einstein tensor  term in (\\ref{bst}) \n contributes a time-dependent piece to $\\tau_{\\mu\\nu}$ which goes to zero at late times like the energy density $\\hat\\rho$ in (\\ref{laterho}).  Additional time dependence\n in $\\tau_{\\mu\\nu}$ comes from the volume element in (\\ref{bst}) which goes like\n %\n \\be\\label{volchange}\n \\sqrt{-s} = \\left(1-{1\\over 2} \\, \\Omega \\right) \\sqrt{-s_{(0)} },\n\\ee\nwhere $\\sqrt{-s_{(0)} }$ denotes the volume element in the static AdS soliton, and the late time behavior of the density parameter\n$\\Omega$ is given in (\\ref{omegat}).\nComparing the decay rates of $\\hat\\rho$ and $\\Omega$, one finds that the \ncontribution of  $G_{\\mu\\nu}$ to the boundary stress tensor density is subdominant at late times compared to that of the volume element.  The contributions of higher derivative terms in (\\ref{bst}) will decay\n even more rapidly. The  leading contributions to the boundary stress tensor density are then readily found by combining\n  the results for the static AdS soliton in \\cite{El-Menoufi:2013pza} with  equation \\eqref{volchange} giving\n \n  %\n  \\be\\label{solitonbst}\n  \\tau_{\\mu\\nu  } \n  = {e^{5y_+\/l} \\over 16\\pi   l} \\left( 1-{1\\over 2} \\, \\Omega \\right) \\mathrm{diag} (-1, 1 ,1, 1, -4),\n  \\ee\n  %\n  where the coordinates are ordered according to $(t, x_1, x_2 , x_3 , v)$. \nHence the decaying time dependent corrections to the static values of $\\tau_{\\mu\\nu  } $ are simply proportional to $\\Omega$, the density parameter of the cosmology.\n \n  \nWe now use the results above to determine the ADM mass and tensions for the AdS soliton cosmologies.  Comparison with \\cite{El-Menoufi:2013pza} shows that the integrands of the ADM charges in AdS coincide with the first three terms in the boundary stress tensor in equation (\\ref{bst}), and the components of  $\\tau_{\\mu\\nu} $ above then just need to be integrated to obtain the ADM charges. In the static coordinates, the density parameter $\\Omega$ depends on $R$ as well as $T$, so the integrand is not a constant. This does not mean that  $R=0$ is a special point, since any location in the homogeneous open cosmology could equally well be chosen as the origin. For the static AdS soliton, the ADM charges are made finite by taking the planar geometry to be periodically identified, with {\\ $-L_j \/2 \\leq x^j \\leq L_j \/2$. For notational brevity let the asymptotic volume be $V= L_1 L_2 L_3 L_v$ where  $L_v$ is the range of compact coordinate $v$ given in (\\ref{period}). In the limit that the plane is infinite, the relevant energy is the mass per unit volume obtained by dividing the total mass by $V$, and similarly for the spatial tensions.\n  \nFinally, it is important to note that the static radial coordinate $R$ has the range $0\\leq R\\leq T$, with the upper limit corresponding to $\\chi \\rightarrow \\infty$ in the coordinate transformation (\\ref{coordtrans}).\nThe integrals for the ADM charges are then over a box of length $L< T$, and at the end we divide out the volume of the box.\nDefine the spatial average of the density parameter $\\Omega$ at time $T$ by\n\\be\n\\label{avomega}\n\\ev{\\Omega} \n= {\\bar\\Omega_*\\over  V  ( H_* T) ^{3w+1 }  } \\int   dx_1 dx_2 dx_3 dv \\, \\Omega (R , T).\n\\ee\nIn the late time limit, we substitute the approximate expression for~$\\Omega$ given in (\\ref{lateomega}).\nFor the general case $w\\neq 0$, this yields\n\\be\\label{intomega}\n\\ev{\\Omega}  = {\\bar\\Omega_*\\over V ( H_* T) ^{3w+1 }  } \\left(1- {c_1\\over T  ^{3w +1 } } + { c_2L^2 \\over T^2 }\\right),\n\\ee \nwhere the coefficients $c_1 , c_2$ can be read off of the expansion of $\\Omega$ in equation (\\ref{lateomega}). One sees that\nthe terms proportional to $c_1$ and $c_2$ make increasingly small contributions and so will be dropped in subsequent formulae.  This also allows us to treat the cases~$w = 0$,~$w \\neq 0$ simultaneously, since the leading-order term in~\\eqref{intomega} is identical to that obtained in the special case\n $w=0$.\n\nFollowing the conventions of past work ({\\it e.g.} \\cite{El-Menoufi:2013pza}), we give the ADM tension rather than a pressure, where tension is simply minus the pressure\\footnote{This convention\nis natural in asymptotically flat static spacetimes where the gravitational tension can be shown to be positive \\cite{Traschen:2003jm}.}.\n Assembling the pieces, at late times the mass and tensions of the soliton in the metric \\eqref{latemetric} are\n\\begin{subequations}\n\\label{admcharges}\n\\bea\n{\\cal M} =  {\\cal T} _j &  = -{ V  \\over 16 \\pi  l } e^{5y_+\/l}\\left( 1 -{1\\over 2}\\ev{\\Omega}  \\right) \\  , \\quad j=1,2,3, \\\\\n{\\cal T}_v &=   {4V  \\over  16 \\pi l }e^{5y_+\/l}  \\left( 1 -{1\\over 2} \\ev{\\Omega}  \\right).\n\\eea\n\\end{subequations}\nThe expressions for the ADM charges have the same structure as the components of the boundary stress tensor, relaxing to the \nequilibrium values like $\\ev{\\Omega} $. \nSince~$\\ev{\\Omega} > 0$, the mass of the AdS soliton cosmology decreases as $ \\ev{\\Omega} $ goes to zero, approaching \nits negative static value at late times, consistent with the energy bound conjectured in \\cite{Horowitz:1998ha}.  The tension ${\\cal T}_v$ around the compact dimension increases to its static positive value, while the trace ${\\cal M} + {\\cal T}_v +\\Sigma _j {\\cal T}_j $ vanishes\nthroughout the relaxation process.\n\n\n\n\\section{Entanglement Entropy}\n\\label{entropy}\n\nThe new AdS cosmological solutions allow us to  compute how the entanglement entropy of a region in the dual CFT  approaches equilibrium.  To perform the computation, we use the holographic prescription \\cite{Ryu:2006ef,Hubeny:2007xt}, which proposes that the entanglement entropy of a region~$\\mathcal{R}$ (called the entangling region) in the boundary CFT is equal to\n\\be\n\\label{sa}\nS_\\mathcal{R} = \\frac{\\mathrm{Area}\\left[\\Sigma\\right]}{4G_N},\n\\ee\nwhere~$\\Sigma$ (referred to as the entangling surface) is the minimal-area extremal surface in the bulk spacetime anchored to~$\\partial\\mathcal{R}$ and homologous to~$\\mathcal{R}$. Note that in this section we have restored Newton's constant $G_N$.  We will also keep~$w$ general, though we emphasize that only the case~$w = 1$ (wherein the bulk matter is a scalar field) has a well-understood CFT dual.\n\nParametrizing~$\\Sigma$ as~$X^a(\\sigma^i)$, with~$\\sigma^i$ coordinates on~$\\Sigma$, $i=1,..., D-2$, the area functional is\n\\be\n\\label{eq:area}\nA = \\int \\sqrt{h} \\, d^{D-2} \\sigma,\n\\ee\nwhere~$h$ is the determinant of the induced metric on the surface\n\\be\n\\label{eq:inducedh}\nh_{ij} = g_{ab} \\partial_i X^a \\partial_j X^b.\n\\ee\n\nIn general, extremizing~\\eqref{eq:area} to obtain the entangling surface is difficult to accomplish analytically, and the AdS soliton cosmologies\nare no exception.  However, we can make progress  by working in the non-solitonized AdS cosmology~\\eqref{metric} and noting that the calculations performed there should approximate those in the AdS soliton cosmology, as long as the relevant surfaces do not extend too deeply into the spacetime.  The boundary metric is then\n\\be\n\\label{eq:bndryflat}\nds^2_\\partial = -dT^2 (1-\\delta \\tilde{g}_{TT}) + dR^2 (1+\\delta \\tilde{g}_{RR}) - 2\\delta \\tilde{g}_{TR} \\, dR \\, dT + R^2 d\\Omega_{(2)}^2,\n\\ee\nand the full metric is given in (\\ref{metric}) with $\\gamma_{\\mu\\nu}$ equal to $ds^2_\\partial$.\nWorking in pure AdS has the significant advantage that the extremal surface is known for a spherical entangling region on the boundary \\cite{Ryu:2006ef}. This allows us to use perturbative techniques to compute the time dependent correction to the area as the metric approaches the static AdS spacetime in the future.\n\nIn order to compute the late-time behavior of the entanglement entropy we work to first order  in powers of $R\/T$ in~$\\delta \\tilde{g}_{TT}$,~$\\delta \\tilde{g}_{RR}$, and~$\\delta \\tilde{g}_{TR}$, given in equations \\eqref{fndef} and \\eqref{lateomega}.  We take the boundary of the entangling region to be a sphere of radius~$R_0$ at some time~$T_0$; the corresponding entangling surface~$\\Sigma$ in pure AdS was found in \\cite{Ryu:2006ef}.  We may then perturb off of this solution to compute the leading correction to the area.  There are two natural options for how this sphere should evolve in time: (i) the sphere can be of fixed proper size in the asymptotically static coordinates so that~$R_0$ is held constant as~$T_0$ advances; or (ii) the sphere can be comoving, so\nthat fluid elements on the boundary of the sphere follow geodesics, and~$R_0$ grows like~$a(t)$.  We will discuss both choices below.\n\n\n\\subsection{Zeroth Order Solutions}\n\nAt zeroth order, the boundary metric~\\eqref{eq:bndryflat} is just Minkowski space.  Parametrizing the surface by~$z \\equiv l e^{-y\/l}$ and the coordinates on the sphere, the area functional~\\eqref{eq:area} is\n\\be\nA  = 4\\pi l^3 \\int _\\epsilon ^1 dx \\, {(1-x^2 )^{1\/2} \\over x^3 },\n\\ee\nwhere $\\epsilon = z_\\mathrm{cut} \/ R_0 $ and $z_\\mathrm{cut}$ is a UV cutoff to regulate the integral.  The corresponding entangling surfaces were calculated in~\\cite{Ryu:2006ef} and are given by \n\\be\n\\label{zerosurf}\n\\Sigma_{0}: \\quad z^2 + R^2 = R_0 ^2  \\  , \\quad T = T_0,\n\\ee\nwith area \n\\be\n\\label{zeroarea}\nA^{(0)} = l^3 \\left[ {A_\\mathrm{static} \\over  2 z_\\mathrm{cut} ^2} - \\pi \\ln\\left( \\frac{A_\\mathrm{static} }{ \\pi z_\\mathrm{cut}^2 }\\right)-\\pi \\right],\n\\ee\nwhere~$A_\\mathrm{static} = 4\\pi R_0 ^2$ is the area of~$\\partial\\Sigma _0 $.  The first term in the above expression denotes the usual area law growth of the entanglement entropy, while the coefficient of the logarithmically divergent term provides a UV-independent measure of the entanglement entropy.\n\n\n\\subsection{First Order Corrections: Approach to Equilibrium}\n\nNow, consider corrections to~\\eqref{zeroarea} which arise both from perturbations to the metric and to the surface~$X^a$. Write each as a zeroth order piece plus a perturbation,\n\\be\n\\label{realdeal}\ng_{ab} = g_{ab}^{(0)} + \\delta g_{ab} \\  , \\quad \\  X^a (\\sigma_i ) = X^a _{(0)} + \\delta X^a .\n\\ee\nTo first order  the volume element on the surface becomes\n\\be\n\\label{deltaa}\nh = h^{(0)} \\left( 1 + \\mathrm{Tr}\\left[ \\delta g_{ab}\\partial_i X_{(0)}^a \\partial_j X_{(0)}^b + 2 g_{ab} ^{(0)} \\partial_i \\delta X^a \\partial_j X_{(0)}^b\\right]\\right).\n\\ee\nHowever, the second term in the trace is a variation of the surface in the background metric, and so this integrates to zero since the background surface is extremal. Thus the first-order change in the area is governed by the perturbation to the metric:\n\\be\n\\label{deltaatwo}\n\\delta A  ={1\\over 2} \\int \\sqrt{h^{(0)} } \\, \\mathrm{Tr}\\left[ \\delta g_{ab}\\partial_i X_{(0)}^a \\partial_j X_{(0)}^b\\right] \\, d^{d-1} \\sigma.\n\\ee\nThe final step is to substitute the expressions for the metric perturbations (\\ref{fndef}) into the metric equations (\\ref{metric}), (\\ref{eq:bndryflat}).\n Using $R^\\prime (z) = -z\/R $ on\nthe zeroth order surface \\eqref{zerosurf}, the induced metric in the perturbed spacetime is given by\n\\be\n\\label{latethree}\n(g_{ab} ^{(0)} +\\delta g_{ab} ) \\partial_i X_{(0)}^a \\partial_j X_{(0)}^b d\\sigma^i d\\sigma^j|_{\\Sigma_0} \n= {l^2   \\over z^2 }  \\left\\{  \\left(  {R_0^2 \\over R^2 } +  {z^2\\Omega \\over  T_0^2}  \\right)  dz^2\n+ R^2 d\\Omega_{(2)} ^2 \\right\\},\n\\ee\nwhere  $R=\\sqrt{ R_0 ^2 -z^2 }$. Using this expression in (\\ref{deltaatwo}) and substituting $\\Omega$ from (\\ref{lateomega}) gives\n the first-order correction to the area of the entangling surface\n\\bea\n\\label{latearea}\n\\delta A &=    { 4\\pi l^3   \\bar{\\Omega}_* \\over  ( H_* T_0 ) ^{3w+1} } \\left( { R_0 ^2 \\over  2T_0^2} \\right) \\int _\\epsilon ^1 dx {(1-x^2 )^{3\/2} \\over x } \\\\\n\t\t &=  { l^3  \\bar{\\Omega}_*  \\over 4 (H_* T_0 ) ^{3w+3}}  H_*^2 A_\\mathrm{static}   \\left( \\ln\\left( {A_\\mathrm{static}\\over \\pi  z_\\mathrm{cut} ^2}\\right) -{4\\over 3} \\right).\n\\eea\nThis result is valid at sufficiently late times such that $H_* T_0 \\gg 1$ and   $T_0 \\gg R_0 $. \n\nThe entanglement entropy, including the leading late time contribution, follows from substituting\n(\\ref{latearea}) and (\\ref{zeroarea}) into the entropy-area relation in equation (\\ref{sa}). \n  The conversion from area to entropy contains the prefactor $l^3\/G^{(5)}_N$, which can be translated into the parameters of the dual CFT.  According to the AdS\/CFT correspondence, the solutions~\\eqref{metric} in~$D = 5$ are dual to an~$\\mathcal{N} = 4$ supersymmetric Yang-Mills theory on the FRW spacetime \\eqref{cosmometric}.   Following  the discussion in \\cite{Ryu:2006ef}, we consider ${\\cal N} =4 \\ SU(N)$ SYM theory on AdS$_5 \\times S^5$, \n  in which case the AdS radius, the ten dimensional Newton's constant, and  the five dimensional Newton's constant are identified with the string coupling, string tension, and $N$ according to $l^4 = 4\\pi g_s  (\\alpha^\\prime)^2 N$, $G^{(10)}_N =  8\\pi^6 g_s^2  (\\alpha^\\prime)^4 $, and  \n$G^{(5)}_N =G_N^{(10)} \/ l^5$.  This gives~$l^3\/G^{(5)}_N = 2N^2\/\\pi $, and the entanglement entropy is then\n\\be\\label{eq:Stot}\nS = \\frac{N^2 }{2\\pi }\\left[\\frac{A_\\mathrm{static}}{2z_\\mathrm{cut}^2} - \n\\pi  \\ln  \\left( {A_\\mathrm{static}\\over \\pi  z_\\mathrm{cut} ^2}\\right) -\\pi\n+  \\frac{   \\bar{\\Omega}_*  H_*^2  A_\\mathrm{static}}{4( H_* T )^{3w+3}   }\\left(   \\ln  \\left( { 2A_\\mathrm{static}\\over \\pi  z_\\mathrm{cut} ^2}\\right) - {4\\over 3}       \\right) + \\cdots\n\\right],\n\\ee\nwhere $\\cdots$ denotes terms that are subleading at late time,  and the subscript on $T$ has been dropped for simplicity. The coefficient of the logarithmic term is invariant under rescalings of the cutoff, so it serves as a regularized measure~$S_\\mathrm{ren}$ of the entanglement entropy.  One finds\n\\be\n\\label{eq:Sren}\n\\delta S_\\mathrm{ren} \\simeq \\frac{N^2 }{8\\pi} \\frac{   \\bar{\\Omega}_*  H_*^2  A_\\mathrm{static}}{( H_* T )^{3w+3}}.\n\\ee\nNote that this is positive, which means that~$S$ \\textit{decreases} to its equilibrium value.  This behavior differs markedly from that of quenches in CFTs~\\cite{Calabrese:2004eu,AbajoArrastia:2010yt,Albash:2010mv,\nBalasubramanian:2010ce,Caceres:2012em,\nAlishahiha:2014cwa,Alishahiha:2014jxa}, wherein the entanglement entropy grows until is saturates.  There is a temptingly simple and compelling physical reason for the decrease of $S$ found in this calculation: in the Cartesian coordinates~$T,R$ there is a nonzero radial flux proportional to $g_{TR}$, so the decrease of the entropy in the ball $R \\leq R_0$ can be interpreted as due to an energy flow out of the ball.  In particular, in the quasi-particle picture of entanglement entropy propagation~\\cite{Calabrese:2004eu}, entanglement is carried by entangled particle pairs; a new flow of such particles out of the entangling ball~$R \\leq R_0$ leading to a decrease in entanglement entropy is consistent with this picture.  Alternatively, note that at late time, our bulk solution approach the Minkowski vacuum, and therefore the CFT evolves from an excited state to the zero-temperature vacuum state.  We would therefore naturally expect probes of correlation (such as entanglement entropy) to decay in the late time limit\\footnote{We thank Juan Pedraza for this observation.}.\n\nThe time dependent contribution decays as a power law, and the time scale for the decay is set by the Hubble parameter $H_*$. The power  depends on the equation of state. For example, for dust the correction goes to zero like $T^{-3} $, and for a free massless scalar field like $T^{-6} $. The time dependence in $\\delta S$ is analogous to the result of \\cite{Engelhardt:2013jda}, in which the entropy of a strip in a vacuum-Kasner AdS spacetime was found to have a power law behavior, in both cases a reflection of the time evolution of the cosmology.\n\nTurning to the amplitude of $\\delta S_\\mathrm{ren}$, we see that this is set by an interesting combination of factors. At sufficiently late times $t_* \\Omega_* \\ll 1$, so that $H_* ^2    \\bar{\\Omega}_*  \\simeq 8 \\pi G_N ^{(5)}\\hat\\rho_* \/ 3 $.\nHence the dimensionless combination $H_* ^2    \\bar{\\Omega}_*  A_\\mathrm{static}$  has the interpretation of the non-vacuum energy, measured in Planck units, that is\ncontained in a shell of width the Planck length that surrounds the sphere. That is, the entangling modes of the perturbation \nact like they are concentrated on the boundary of the sphere. This is a reflection of the fact that the change in the area of the \n extremal surface comes from the metric perturbations near the surface.\n\n \n\\subsection{The Cosmological View}\n\nAn alternative way to interpret the time evolution of the entanglement entropy is to take the boundary sphere to be comoving, so that\npoints on the boundary sphere follow geodesics. \n In the cosmological coordinates \\eqref{openfrw} this means that  the sphere is at a fixed coordinate $\\chi= \\chi _0$.  The extremal surface $\\Sigma_0$ does not lie within a slice of constant cosmological time, but it does intersect the boundary at a constant time, as can be seen by \nevaluating  \\eqref{coordtrans} at $z=0$.  \nTransforming the zeroth order surface (\\ref{zerosurf}) to  the cosmological coordinates gives\n %\n \\be\\label{cosmosurf}\n \\Sigma_0 : \\quad   a(t) \\cosh \\chi = a( t_b )  \\cosh \\chi_0 \\  , \\quad z^2   + \\cosh^2 \\chi_0   \\tanh ^2 \\chi  = a(t_b )^2 \\sinh ^2 \\chi _0.\n \\ee\nLet\n\\be\nA_\\mathrm{geod}(t_b )  = 4\\pi a( t_b )^2 \\sinh ^2 \\chi _0\n\\ee\nbe the proper area of the comoving  sphere on the boundary at $t_b$.\n Then in terms of the cosmological coordinates the zeroth order area (\\ref{zeroarea}) becomes\n\\be\\label{areacosmo}\nA^{(0)} (t_b ) =   l^3 \\left[ { A_\\mathrm{geod}\\over 2  z_\\mathrm{cut} ^2}  - \\pi \\ln \\left({  A_\\mathrm{geod} \\over \\pi  z_\\mathrm{cut}^2 } \\right) -\\pi \n \\right] \n\\ee\nand the time dependent correction (\\ref{latearea}) is\n\\be\\label{dacosmo}\n\\delta A = l^3    \\bar{\\Omega}_*  H_*^2 (4\\pi a_*^2 \\sinh ^2 \\chi _0 ) \\left(  {a_* \\over  a(t_b ) }   \\right)^{3w+1}    \n \\left( \\ln\\left( { A_\\mathrm{geod} \\over 2\\pi z_\\mathrm{cut}^2 }\\right) -{2\\over 3} \\right).\n\\ee\nHence the time dependent piece redshifts to zero as \n $(1+z_\\mathrm{b} )^{-(3w+1)} $ where  $1+z_b = a(t_b ) \/ a_* $ is the cosmological redshift.\nThe power in the decaying term is different than for the static sphere (\\ref{latearea}) because $A_\\mathrm{geod} $ increases\n as $a^2$. \n \n So far the expressions for the area (\\ref{areacosmo}) and (\\ref{dacosmo}) are just translations from\nthe asymptotically static coordinates to the cosmological coordinates. \nThe difference \nfrom the previous section, in which the area of the boundary sphere is held constant,\n comes when one follows the time evolution by considering increasing values of the boundary time $t_b$.\nThe area of the boundary co-moving sphere increases like $a^2 (t_b )$, so  \n although (the UV-independent part of) $\\delta A$ is positive and decreasing to zero, the total entropy increases with $t_b$. This brings up the important issue of \n the range of validity of the expressions in cosmological time. As discussed in\n section \\ref{entropy}, if the results are to be good approximations to the results in a solitonized spacetime, one needs to\n restrict to surfaces that do not penetrate too deeply into the bulk, which precludes taking~$R \\sim t_b \\sinh \\chi _0$ too large. This means\n that the validity of (\\ref{latearea}) is restricted to times that are not so large that the proper radius of the boundary sphere\n approaches the length scale set by the soliton, that is, we need $ t_b \\sinh \\chi _0 \\ll l e^{-y_+ \/l}$. The situation here\n  is similar to that in \\cite{Engelhardt:2014mea}.\n \n \n\n\\section{Discussion}\n\\label{conclusion}\n\nIn this paper, we have shown how to construct AdS cosmologies that satisfy the Einstein equations with a nonzero stress tensor and negative cosmological constant.  Our solutions were built as foliations of lower-dimensional solutions; the induced metric on each of these hypersurfaces itself satisfies Einstein's equations with a nonzero stress tensor.  This construction has the advantage that the boundary metric of the AlAdS solution is just (conformal to) the induced metric on each hypersurface.  Therefore, this construction offers us significant freedom in constructing AlAdS spacetimes with a boundary metric of our choosing.\n\nThe particular AdS cosmologies that we have constructed take the stress tensor to be that of a perfect fluid obeying the strong energy condition; for the equation of state~$w = 1$, the fluid is sourced by a massless noninteracting scalar field.  Moreover, we have focused on the specific case in which the spatial slices of the FRW cosmologies are negatively curved, as such FRW cosmologies then approach the Milne patch of Minkowski space at late times.  The AdS cosmology constructed from these slices therefore approaches the Poincar\\'e patch of AdS at late times, while the AdS soliton cosmology approaches the static AdS soliton.\n\nSuch solutions are especially interesting because they allow us to perturbatively calculate the behavior of physically relevant quantities at late times.  For instance, we have calculated the late-time perturbation to the ADM mass of the AdS soliton cosmology, and have found this perturbation to  be\n\\be\n\\label{deltam}\n\\delta{\\cal M}  =  -{ \\Omega {\\cal M}\\over 2},\n\\ee\nwhere~${\\cal M}$ is the unperturbed mass and~$\\Omega$ is the dimensionless density parameter of the FRW cosmology, which goes to zero at late times in the solutions we are interested in.  Since ${\\cal M}$ is negative for the soliton this implies that the mass \\textit{decreases} to the mass of the static AdS soliton.  Hence this result is consistent with the energy conjecture of \\cite{Horowitz:1998ha} that the AdS soliton is the lowest energy spacetime with the prescribed asymptotic structure.  We found the ADM tensions to be modified in a similar manner to~${\\cal M}$.\n\nMoreover, our solutions also have immediate applicability to large-$N$, strongly coupled CFTs via the AdS\/CFT correspondence.  Indeed, our AdS cosmologies are dual to CFTs living on an FRW cosmology, while the massless, noninteracting scalar field in the bulk is dual to a scalar operator in the CFT with zero expectation value but nonzero source.  This atypical behavior can be tied to the fact that our AdS cosmologies are singular at the Poincar\\'e horizon.  This singularity is removed by ``solitonizing'': introducing a compactified direction in the bulk that caps off the geometry.  This cap amounts to putting the CFT in a confined phase. \n\nAs a probe of the behavior of the CFT on these FRW spacetimes, we study the entanglement entropy~$S$ of a sphere of constant radius.  Using perturbative techniques to find the leading time dependent correction to the entropy, we find that the regulated entanglement entropy $S_\\mathrm{ren}$ decays as a power law to its equilibrium value.  The power depends on the equation of state of the fluid.  Note that this decay to equilibrium is starkly different from the behavior of entanglement entropy after a quench, when the entanglement entropy \\textit{grows} to its equilibrium (thermal) value.  However, note that our solutions at late time approach Poincar\\'e AdS, which has zero temperature; this is drastically different from the end state of a quench, which in the bulk is usually modelled by the injection of energy, forming a black hole of finite temperature.\n\nSeveral issues and questions are raised by these examples.  First, can these solutions be generalized to include planar black holes in the bulk spacetimes?  Such solutions could conceivably be used to model the approach of a CFT on an FRW cosmology to thermal equilibrium with a nonzero temperature~$T$, much as here we have modeled the approach to equilibrium at~$T = 0$.\nSecond, for reasons of tractability our entropy calculations have been perturbative analyses in the AdS cosmology.  It would also be interesting to study the entanglement entropy of the CFT at all times, in both the AdS cosmologies and especially in the AdS soliton cosmologies, to see if there is any behavior that was not captured by our perturbative methods.  Such calculations would most likely be numerical, so we leave them for the future.  \n\n\n\n\\section*{Acknowledgements}\n\nThe authors with to thank Netta Engelhardt and Don Marolf for useful discussions.  We also thank Juan Pedraza for useful comments on an earlier version of this paper.  This project was supported in part by the National Science Foundation under Grant No PHY11-25915, by FQXi grant FRP3-1338, and by funds from the University of California.\n\n\n\n\n\\bibliographystyle{JHEP}\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Omitted Proofs in \\Cref{sec:composition-theorems}}\n\\label{app:CLT}\n\n\n\n\n\nWe first collect the basic properties of $\\otimes$ listed in \\Cref{sub:composition_theorem}.\n\\begin{proposition}\\label{prop:tensor_properties}\n\tThe product $\\otimes$ defined in \\Cref{def:product} has the following properties:\n\t\\begin{enumerate}\n\t\\setlength\\itemsep{0.15em}\n\t\\setcounter{enumi}{-1}\n\t\\item The product $\\otimes$ is well-defined.\n\t\\item The product $\\otimes$ is commutative and associative.\n\t\\item If $g_1\\geqslant g_2$, then $f\\otimes g_1 \\geqslant f\\otimes g_2$.\n\t\\item $f \\otimes \\Id = \\Id \\otimes f = f$.\n\n\t\\item $(f\\otimes g)^{-1} = f^{-1}\\otimes g^{-1}$.\n\n\n\t\\item For GDP, $G_{\\mu_1} \\otimes G_{\\mu_2} \\otimes \\cdots \\otimes G_{\\mu_n} = G_{\\mu}$, where $\\mu = \\sqrt{\\mu_1^2+\\cdots+\\mu_n^2}$.\n\t\\end{enumerate}\n\\end{proposition}\nProperty 0 and 2 are already proved in \\Cref{app:self}. So we only prove 1,3,4,5 here.\n\\begin{proof}[Proof of Properties (1,3,4,5)]\n\tWe will assume $f=T(P,P'), g =T(Q,Q')$ in the entire proof. The upshot is that\n\\[T(P,P')\\otimes T(Q,Q') = T(P\\times Q,P'\\times Q').\\]\n\\begin{enumerate}\n\t\\item Commutativity:\n\t\t$$f\\otimes g = T(P,P')\\otimes T(Q,Q') = T(P\\times Q,P'\\times Q') \\stackrel{(a)}{=} T(Q\\times P,Q'\\times P') =T(Q,Q') \\otimes T(P,P') = g\\otimes f.$$\n\t\tIn step $(a)$, we switch the order of the components of the product, which obviously keeps the trade-off function unchanged.\n\n\t\tAssociativity:\n\t\tLet $h=T(R,R')$.\n\t\t\\begin{align*}\n\t\t\t(f\\otimes g)\\otimes h &= T(P\\times Q,P'\\times Q') \\otimes T(R,R') =  T(P\\times Q\\times R,P'\\times Q'\\times R')\\\\\n\t\t\tf\\otimes (g\\otimes h) &= T(P,P')\\otimes T(Q\\times R,Q'\\times R') =  T(P\\times Q\\times R,P'\\times Q'\\times R')\n\t\t\\end{align*}\n\t\tSo $(f\\otimes g)\\otimes h = f\\otimes (g\\otimes h)$.\n\t\\item Let $R$ be an arbitrary degenerate distribution, i.e. $R$ puts mass 1 on a single point. Then $\\Id = T(R,R)$ and\n\t\t\\[f\\otimes \\Id = T(P\\times R, P'\\times R) = T(P,P') = f.\\]\n\t\\item\n\t\tBy \\Cref{lem:symmetry}, taking the inverse amounts to flipping the arguments of $T(\\cdot,\\cdot)$.\n\t\t\\begin{align*}\n\t\t\t(f\\otimes g)^{-1} = T(P'\\times Q',P\\times Q) = T(P',P)\\otimes T(Q',Q) = f^{-1}\\otimes g^{-1}.\n\t\t\\end{align*}\n\t\\item Let $\\bmu=(\\mu_1,\\mu_2)\\in\\R^2$ and $I_2$ be the $2\\times 2$ identity matrix. Then\n\t\t\\begin{align*}\n\t\t\tG_{\\mu_1}\\otimes G_{\\mu_2} &= T\\big(\\N(0,1),\\N(\\mu_1,1)\\big)\\otimes T\\big(\\N(0,1),\\N(\\mu_1,1)\\big)\\\\\n\t\t\t&= T\\big(\\N(0,1)\\times \\N(0,1),\\N(\\mu_1,1)\\times \\N(\\mu_2,1)\\big)\\\\\n\t\t\t&= T\\big(\\N(0,I_2),\\N(\\bmu,I_2)\\big)\n\t\t\\end{align*}\n\t\tAgain we use the invariance of trade-off functions under invertible transformations. $\\N(0,I_2)$ is rotation invariant, So we can rotate $\\N(\\bmu,I_2)$ so that the mean is $(\\sqrt{\\mu_1^2+\\mu_2^2},0)$. Continuing the calculation\n\t\t\\begin{align*}\n\t\t\tG_{\\mu_1}\\otimes G_{\\mu_2} &=T\\big(\\N(0,I_2),\\N(\\bmu,I_2)\\big)\\\\\n\t\t\t&= T\\big(\\N(0,1)\\times \\N(0,1),\\N(\\sqrt{\\mu_1^2+\\mu_2^2},1)\\times \\N(0,1)\\big)\\\\\n\t\t\t&= T\\big(\\N(0,1),\\N(\\sqrt{\\mu_1^2+\\mu_2^2},1)\\big)\\otimes T\\big(\\N(0,1),\\N(0,1)\\big)\\\\\n\t\t\t&= G_{\\sqrt{\\mu_1^2+\\mu_2^2}}\\otimes \\Id\\\\\n\t\t\t&= G_{\\sqrt{\\mu_1^2+\\mu_2^2}}.\n\t\t\\end{align*}\n\\end{enumerate}\n\\end{proof}\n\n\n\n\t\n\nThe following proposition explains why our central limit theorems need $f_n$ to approach $\\Id$.\n\n\\begin{proposition} \\label{prop:trivial_limit}\n\tFor any trade-off function $f$ that is not $\\Id$,\n\t$$\\lim_{n\\to+\\infty} f^{\\otimes n}(\\alpha)=0, \\quad\\forall\\alpha\\in(0,1].$$\n\tIn fact, the convergence is exponentially fast.\n\\end{proposition}\n\\begin{proof}[Proof of \\Cref{prop:trivial_limit}]\n\tFor any trade-off function $f$, let $P,Q$ be probability measures such that $\\F(P,Q)=f$. The existence is guaranteed by \\Cref{prop:trade-off}. It is well-known that $1-\\mathrm{TV}(P,Q)$ is the minimum sum of type I and type II error, namely,\n\t$$1-\\mathrm{TV}(P,Q) = \\min_{\\alpha\\in[0,1]} \\alpha+f(\\alpha).$$\n\tWe claim that the following limit suffices to prove the theorem:\n\t\\begin{equation}\\label{eq:dtv}\n\t\t\\lim_{n\\to\\infty} \\mathrm{TV}(P^n,Q^{ n})=1.\n\t\\end{equation}\n\tTo see why it suffices, recall that by definition $\\F(P^n,Q^{ n})=f^{\\otimes n}$. Hence\n\t$$1-\\mathrm{TV}(P^n,Q^{ n}) = \\min_{\\alpha\\in[0,1]} \\alpha+f^{\\otimes n}(\\alpha).$$\n\tLet $\\alpha_n$ be the type i error that achieves minimum in the above equation, i.e.\n\t$$\\alpha_n+f^{\\otimes n}(\\alpha_n) = 1-\\mathrm{TV}(P^n,Q^{ n}).$$\n\tThe total variation limit \\eqref{eq:dtv} implies $\\alpha_n\\to0$ and $f^{\\otimes n}(\\alpha_n)\\to0$. For each $n$, consider the piecewise linear function that interpolates $(0,1),(\\alpha_n,f^{\\otimes n}(\\alpha_n))$ and $(1,0)$, which will be denoted by $h_n$. By the convexity of $f^{\\otimes n}$ we know that $f^{\\otimes n}\\leqslant h_n$ in $[0,1]$. It suffices to show that $h_n(\\alpha)\\to 0,\\forall \\alpha\\in(0,1]$. Since $\\alpha_n\\to0$, for large enough\t$n$, $h_n(\\alpha)$ is evaluated on the lower linear segment of $h_n$. So $h_n(\\alpha)\\leqslant h_n(\\alpha_n)\\leqslant f^{\\otimes n}(\\alpha_n) \\to 0$. This yields the desired limit of $f^{\\otimes n}$.\\\\\n\tNow we use Hellinger distance $H^2(P,Q) := \\E_Q\\big[(1-\\sqrt{\\frac{P}{Q}})^2\\big]$ to show the total variation limit \\eqref{eq:dtv}.\\\\\n\tAn elementary inequality relating total variation and Hellinger distance is\n\t$$\\frac{1}{2}H^2(P,Q)\\leqslant \\mathrm{TV}(P,Q)\\leqslant H(P,Q).$$\n\tAnother nice property of Hellinger distance is it tensorizes in the following sense:\n\t$$1-\\frac{H^2(P^n,Q^{ n})}{2} = \\Big(1-\\frac{H^2(P,Q)}{2}\\Big)^n.$$\n\t$f$ is not the diagonal $\\alpha\\mapsto1-\\alpha$, so $P\\neq Q$. Hence $\\mathrm{TV}(P,Q)> 0$. By the second inequality in the sandwich bound, $H^2(P,Q)>0$. By the tensorization property, $H^2(P^n,Q^{ n})\\to 2$. By the first inequality in the sandwich bound and that $\\mathrm{TV}$ is bounded by 1 we have\n\t$$ \\frac{1}{2} H^2(P^n,Q^{ n})\\leqslant \\mathrm{TV}(P^n,Q^{ n})\\leqslant1.$$This shows $\\mathrm{TV}(P^n,Q^{ n})\\to 1$ and completes the proof.\n\\end{proof}\n\n\n\n\nNow we set out the journey to prove the Berry-Esseen style central limit theorem \\ref{thm:Berry}. We first restate the theorem.\n\\berryrep*\nOur approach is to consider the log-likelihood ratio between the distributions of the composition mechanism on neighboring datasets. This log-likelihood ratio can be reduced to the sum of \\textit{independent} components that each correspond to the log-likelihood ratio of a trade-off function in the tensor product. This reduction allows us to carry over the classical Berry--Esseen bound to Theorem~\\ref{thm:Berry}.\n\nAs the very first step, let's better understand the functionals $\\mathrm{kl},\\kappa_2$ and $\\bar{\\kappa}_3$ used in the statement of the theorem. We focus on symmetric $f$ with $f(0)=1$, although some of the following discussion generalizes beyond that subclass. Recall that\n\\begin{align*}\n\t\\mathrm{kl}(f) &= -\\int_0^1\\log |f'(x)|\\diff x\\\\\n\n\t\\kappa_2(f)&=\\int_0^1\\big(\\log |f'(x)|\\big)^2\\diff x\\\\% \\int_0^1\\big(\\log g(x)-\\mu\\big)^2\\diff x\\\\\n\t\\bar{\\kappa}_3(f)&=\\int_0^1\\big|\\log |f'(x)|+\\mathrm{kl}(f)\\big|^3\\diff x\n\n\n\\end{align*}\nFirst we finish the argument mentioned in \\Cref{sub:a_berry_esseen_type_of_clt} that these functionals are well-defined and take values in $[0,+\\infty]$. For $\\kappa_2$ and $\\bar{\\kappa}_3$, as well as the non-central version $\\kappa_3$, the argument is easy because the integrands are non-negative.\n\nFor $\\mathrm{kl}$, the only possible singularities of the integrand is 0 and 1. If 1 is singular then $\\log |f'(x)|\\to -\\infty$ near 1. This is okay because the functionals are allowed to take value $+\\infty$. We need to rule out the case when 0 is a singularity and $\\int^{\\epsilon}_0\\log |f'(x)|\\diff x=+\\infty$. That cannot happen because $\\log |f'(x)|\\leqslant |f'(x)|-1$ and $|f'(x)| = -f'(x)$ is integrable in $[0,1]$ as it is the derivative of $-f$, an absolute continuous function. \nNon-negativity of $\\mathrm{kl}$ follows from Jensen's inequality.\n\nIn the discussion of \\Cref{prop:fdiv}, we showed that $\\mathrm{kl}(T(P,Q)) = D_{\\mathrm{KL}}(P\\|Q)$. This explains the name of this functional. In fact, $\\kappa_2$ also corresponds to a divergence called \\textit{exponential divergence} (\\cite{eguchi1985differential}).\n\nWe introduce a notation that will be useful in the calculation below. For a trade-off function $f$, let $Df$ be a function with the following expression:\n\\[Df(x) = |f'(1-x)| = -f'(1-x).\\]\nIn fact, this is the density introduce in the proof of \\Cref{prop:trade-off}.\n\nBy a simple change of variable, the three functionals can be re-written as\n\\begin{align*}\n\t\\mathrm{kl}(f) &= -\\int_0^1\\log Df(x)\\diff x\\\\\n\n\t\\kappa_2(f)&=\\int_0^1\\big(\\log Df(x)\\big)^2\\diff x\\\\% \\int_0^1\\big(\\log g(x)-\\mu\\big)^2\\diff x\\\\\n\t\\bar{\\kappa}_3(f)&=\\int_0^1\\big|\\log Df(x)+\\mathrm{kl}(f)\\big|^3\\diff x\n\n\n\\end{align*}\n\nThe following ``shadows'' of the above functionals will appear in the proof:\n\\begin{align*}\n\t\\mathrm{lk}(f) &:= \\int_0^1Df(x)\\log Df(x)\\diff x\\\\\n\t\\tilde{\\kappa}_2(f)&:=\\int_0^1Df(x)\\big(\\log Df(x)\\big)^2\\diff x\\\\%\\int_0^1\\big(\\log g(x)-\\tilde{\\mu}\\big)^2g(x)\\diff x\\\\\n\t\\tilde{\\kappa}_3(f)&:=\\int_0^1Df(x)\\big|\\log Df(x)-\\mathrm{lk}(f)\\big|^3\\diff x\n\\end{align*}\nThese functionals are also well-defined on $\\T$ and take values in $[0,+\\infty]$. The argument is similar to that of $\\mathrm{kl}, \\kappa_2$ and $\\bar{\\kappa}_3$.\n\n\nThe following calculations turn out to be useful in the proof.\n\\begin{proposition}\\label{prop:functionals}\n\tSuppose $f\\in\\T^S$ and $f(0)=1$. Then\n\t\\begin{align*}\n\t\t\\mathrm{kl}(f) &= \\mathrm{lk}(f)\\\\\n\t\n\t\t\\kappa_2(f)&= \\tilde{\\kappa}_2(f)\\\\\n\t\t\\bar{\\kappa}_3(f)&=\\tilde{\\kappa}_3(f).\n\t\\end{align*}\n\\end{proposition}\n\\begin{proof}\n\tOur approach, taking $\\kappa_2$ as example, is to show $\\kappa_2(f^{-1}) = \\tilde{\\kappa}_2(f)$. By definition of symmetry, $f^{-1} = f$ and hence the desired result follows. First observe for $f\\in\\T^S$ with $f(0)=1$, $f^{-1}$ agrees with the ordinary function inverse, hence we can apply calculus rule as follows:\n\t\\[Df^{-1}(x) = -\\frac{\\diff f^{-1}}{\\diff x}(1-x) = \\frac{-1}{f'(f^{-1}(1-x))}.\\]\n\tWe only prove $\\kappa_2(f^{-1})= \\tilde{\\kappa}_2(f)$ here and the other two identities can be proved similarly.\n\t\\begin{align*}\n\t\t\\kappa_2(f^{-1}) &= \\int_0^1\\big(\\log Df^{-1}(x)\\big)^2\\diff x \\\\\n\t\t&= \\int_0^1\\big(-\\log\\big[-f'(f^{-1}(1-x))\\big]\\big)^2\\diff x\\\\\n\t\t&= \\int_0^1\\log^2\\big[-f'(f^{-1}(1-x))\\big]\\diff x\n\t\\end{align*}\n\tLet $y=f^{-1}(1-x)$, then $f'(y)\\diff y=-\\diff x$, and $x=0$ corresponds to $y=0$, $x=1$ corresponds to $y=1$.\n\t\\begin{align*}\n\t\t\\kappa_2(f^{-1})\n\t\t&= \\int_0^1\\log^2 [-f'(y)]\\cdot\\big(-f'(y)\\big)\\diff y&& (y=f^{-1}(1-x))\\\\\n\t\t&= \\int_0^1\\log^2 [-f'(1-z)]\\cdot\\big(-f'(1-z)\\big)\\diff z&& (z=1-y)\\\\\n\t\t&= \\int_0^1Df(z)\\big(\\log Df(z)\\big)^2\\diff z\\\\\n\t\t&= \\tilde{\\kappa}_2(f).\n\t\\end{align*}\n\\end{proof}\nWe remark that by properly extending the definition of the shadow functionals, identities like $\\mathrm{kl}(f^{-1}) = \\mathrm{lk}(f)$ holds for general trade-off function $f$.\n\n\n\n\n\nBefore we finally start the proof, let's recall Berry-Esseen theorem for random variables. Suppose we have $n$ independent random variables $X_1,\\ldots, X_n$ with $\\E X_i = \\mu_i, \\Var X_i = \\sigma_i^2, \\E|X_i-\\mu_i|^3 = \\rho_i^3$. Consider the normalized random variable\n$$S_n := \\frac{\\sum_{i=1}^n X_i-\\mu_i}{\\sqrt{\\sum_{i=1}^n \\sigma^2_i}}.$$\nDenote its cdf by $F_n$. Then\n\\begin{theorem}[Berry-Esseen]\\label{thm:BerryRV}\nThere exists a universal constant $C>0$ such that\n\t\\[\\sup_{x\\in\\R}|F_n(x)-\\Phi(x)|\\leqslant C\\cdot \\frac{\\sum_{i=1}^n \\rho_i^3}{\\big(\\sum_{i=1}^n\\sigma_i^2\\big)^{\\frac{3}{2}}}.\\]\n\\end{theorem}\nTo the best of our knowledge, the best $C$ is 0.5600 due to \\cite{shevtsova2010improvement}.\n\n\\bigskip\n\nNow we proceed to the proof of \\Cref{thm:Berry}.\n\\begin{proof}[Proof of \\Cref{thm:Berry}]\nFor simplicity let\n\\[\\boldsymbol{f} := f_{1}\\otimes f_{2} \\otimes \\cdots \\otimes f_{n}.\\]\n\nFirst let's find distributions $P_0$ and $P_1$ such that $\\F(P_0,P_1)=\\boldsymbol{f}$.\n\nFirst, by symmetry, if $f_i(0)<1$, then $f_i'(x)=0$ in some interval $[1-\\ep,1]$ for some $\\ep>0$, which yields $\\mathrm{kl}(f_i)=+\\infty$. So we can assume $f_i(0)$ for all $i$.\n\nRecall that $Df_{i}(x) = -f'_{i}(1-x)$. Let $P$ be the uniform distribution on $[0,1]$ and $Q_{i}$ be the distribution supported on $[0,1]$ with density $Df_{i}$. These are the distributions constructed in the proof of \\Cref{prop:trade-off}. Since $f_i$ are all symmetric and $f_i(0)=1$, the supports of $P$ and all $Q_{i}$ are all exactly $[0,1]$, and we have $\\F(P,Q_{i})=f_{i}$. Hence by definition $\\boldsymbol{f} = \\F(P^{ n},Q_{1}\\times\\cdots\\times Q_{n})$.\n\nNow let's study the hypothesis testing problem $P^{ n}$ vs $Q_{1}\\times\\cdots\\times Q_{n}$. Let\n\\[L_{i}(x) := \\log \\frac{\\diff Q_{i}}{\\diff P}(x) = \\log Df_{i}(x)\\]\nbe the log likelihood ratio. \nSince both hypotheses are product distributions, Neyman-Pearson lemma implies that the optimal rejection rules of this testing problem must be a threshold function of the quantity $\\sum_{i=1}^n L_{i}$.\nWe need to study $\\sum_{i=1}^n L_{i}(x_i)$ under both the null and the alternative hypothesis, i.e. when $(x_1,\\ldots, x_n)$ comes from $P^{ n}$ and $Q_{1}\\times\\cdots\\times Q_{n}$. From here we implement the following plan: first find the quantities that exhibit central limit behavior, then express $\\alpha$ and $\\boldsymbol{f}(\\alpha)$ in terms of these quantities.\n\nFor further simplification, let\n$$T_n := \\sum_{i=1}^n L_{i}.$$\nAs we turn off the $x_i$ notation, we should bear in mind that $T_n$ has different distributions under $P^{ n}$ and $Q_{1}\\times\\cdots\\times Q_{n}$, but it is an independent sum in both cases.\n\nIn order to find quantities with central limit behavior, it suffices to normalize $T_n$ under both distributions. The mysterious functionals we introduced are specifically designed for this purpose.\n\\begin{align*}\n\t\\E_P[L_{i}] &= \\int_0^1 \\log Df_{i}(x_i)\\diff x_i = -\\mathrm{kl}(f_{i}),\\\\\n\t\\E_{Q_{i}}[L_{i}] &= \\int_0^1 Df_{i}(x_i)\\log Df_{i}(x_i)\\diff x_i = \\mathrm{lk}(f_{i}) = \\mathrm{kl}(f_{i}).\n\\end{align*}\nIn the last step we used \\Cref{prop:functionals}. With the bold vector notation,\n\\begin{align*}\n\t\\E_{P^n}[T_n]  &= \\sum_{i=1}^n -\\mathrm{kl}(f_{i}) = -\\|\\boldsymbol{\\mathrm{kl}}\\|_1,\\\\\n\t\\E_{Q_{1}\\times\\cdots\\times Q_{n}}[T_n] &= \\sum_{i=1}^n \\mathrm{kl}(f_{i})= \\|\\boldsymbol{\\mathrm{kl}}\\|_1.\n\\end{align*}\nSimilarly for the variances:\n\\begin{align*}\n\n\t\\Var_{P}[L_{i}] &=\\E_P[L_{i}^2] - \\big(\\E_P[L_{i}]\\big)^2 = \\kappa_2(f_{i}) - \\mathrm{kl}^2(f_{i})\n\t\n\t,\\\\\n\n\t\\Var_{Q_{i}}[L_{i}] &= \\E_{Q_{i}}[L_{i}^2] - \\big(\\E_{Q_{i}}[L_{i}]\\big)^2 = \\tilde{\\kappa}_2(f_{i}) - \\mathrm{lk}^2(f_{i})= \\kappa_2(f_{i}) - \\mathrm{kl}^2(f_{i}).\n\t\n\\end{align*}\n\\[\n\t\\Var_{P^n}[T_n] =\\Var_{Q_{1}\\times\\cdots\\times Q_{n}}[T_n] = \\sum_{i=1}^n \\kappa_2(f_{i}) - \\mathrm{kl}^2(f_{i}) = \\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2.\n\\]\nIn order to apply Berry-Esseen theorem (for random variables) we still need the centralized third moments:\n\\begin{align*}\n\t\\E_P|L_i-\\E_P[L_i]|^3 &=\\int_0^1\\big|\\log Df_i(x)+\\mathrm{kl}(f_i)\\big|^3\\diff x = \\bar{\\kappa}_3(f_i),\\\\\n\t\\E_{Q_i}|L_i-\\E_{Q_i}[L_i]|^3 &=\\int_0^1Df_i(x)\\big|\\log Df_i(x)-\\mathrm{lk}(f_i)\\big|^3\\diff x = \n\t\\tilde{\\kappa}_3(f_i)= \\bar{\\kappa}_3(f_i).\n\\end{align*}\nLet $F_n$ be the cdf of $\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}$ under $P^{ n}$, and $\\tilde{F}^{(n)}$ be the cdf of $\\frac{T_n-\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1-\\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}$ under $Q_{1}\\times\\cdots\\times Q_{n}$. By Berry-Esseen Theorem \\ref{thm:BerryRV},\n\\begin{align}\\label{eq:closetonormal}\n\t\\sup_{x\\in\\R}|F_n(x)-\\Phi(x)|\n\t&\\leqslant C\\cdot\\frac{\\|\\boldsymbol{\\bar{\\kappa}_3}\\|_1}{\\big(\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2\\big)^{\\frac{3}{2}}}=\\gamma\n\n\\end{align}\nand similarly $\\sup_{x\\in\\R}|\\tilde{F}^{(n)}(x)-\\Phi(x)|\\leqslant\\gamma$.\n\n\nSo we find the quantities that exhibit central limit behavior. Now let's relate them with $\\boldsymbol{f}$. Consider the testing problem $(P^{ n},Q_{1}\\times\\cdots\\times Q_{n})$. For a fixed $\\alpha\\in[0,1]$, let the optimal rejection rule (potentially randomized) at level $\\alpha$ be $\\phi$.\nBy Neyman-Pearson lemma, $\\phi$ must be a thresholding on $T_n$. An equivalent form that highlights the central limit behavior is the following:\n$$\\phi=\n\\left\\{\n\\begin{array}{ll}\n1, \t\t& \\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}>t, \\\\\np, & \\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}=t,\\\\\n0, & \\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}<t\n\\end{array}\n\\right.\n$$\nHere $t\\in\\R\\cup\\{\\pm\\infty\\}$ and $p\\in[0,1]$ are parameters uniquely determined by the condition $\\E_{P^n}[\\phi]=\\alpha$. With this form $\\E_{P^n}[\\phi]$ can be easily spelled out in terms of $F_n$:\n\\begin{align*}\n\t\\E_{P^n}[\\phi] &= P^n\\Big[\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}>t\\Big] + p \\cdot P^n\\Big[\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}=t\\Big]\\\\\n\t&= 1-F_n(t) + p \\cdot [F_n(t)-F_n(t^-)].\n\\end{align*}\nHere $F_n(t^-)$ is the left limit of the function $F_n$ at $t$.\nSimple algebra yields\n\\[\n\t1-\\alpha = 1- \\E_{P^n}[\\phi]= (1-p)F_n(t)+pF_n(t^-)\n\\]\nand consequently the inequality\n\\[\n\tF_n(t^-)\\leqslant 1-\\alpha\\leqslant F_n(t).\n\\]\nFor $\\E_{Q_{1}\\times\\cdots\\times Q_{n}}[\\phi]$ it is helpful to introduce another letter $\\tau := t-\\mu$. In the theorem statement $\\mu$ was defined to be $\\frac{2\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}$\nso we have the equivalence\n\\begin{equation}\\label{eq:equiv}\n\t\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}>t\n\t\\Leftrightarrow \\frac{T_n-\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1-\\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}\n\t>\n\t\\tau.\n\\end{equation}\nWith this extra notation we have\n\\begin{align*}\n\t1-\\boldsymbol{f}(\\alpha)\n\t=&\\,\\, \\E_{Q_{1}\\times\\cdots\\times Q_{n}}[\\phi]\\\\\n\t=&\\,\\, Q_{1}\\times\\cdots\\times Q_{n}\\Big[\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}>t\\Big]+\\\\\n\t&\\,\\, p \\cdot Q_{1}\\times\\cdots\\times Q_{n}\\Big[\\frac{T_n + \\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}=t\\Big]\n\t&& \\text{(Def. of $\\phi$)}\\\\\n\t=&\\,\\, Q_{1}\\times\\cdots\\times Q_{n}\\Big[\\frac{T_n-\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1-\\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}>\\tau\\Big]+\\\\\n\t&\\,\\, p \\cdot Q_{1}\\times\\cdots\\times Q_{n}\\Big[\\frac{T_n-\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1-\\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}=\\tau\\Big]\n\t&& \\text{ (By \\eqref{eq:equiv}) }\\\\\n\t=&\\,\\, 1-\\tilde{F}^{(n)}(\\tau) + p \\cdot [\\tilde{F}^{(n)}(\\tau)-\\tilde{F}^{(n)}(\\tau^-)].\n\\end{align*}\nSimilar algebra as before yields\n\\[\\boldsymbol{f}(\\alpha) = (1-p)\\tilde{F}^{(n)}(\\tau)+p\\tilde{F}^{(n)}(\\tau^-)\\]\nand hence\n\\[\n\\tilde{F}^{(n)}(\\tau^-)\\leqslant \\boldsymbol{f}(\\alpha)\\leqslant \\tilde{F}^{(n)}(\\tau).\n\\]\nSo far we have\n\\begin{align}\n\tF_n(t^-)\\leqslant 1-\\alpha\\leqslant F_n(t),\\\\\n\\tilde{F}^{(n)}(\\tau^-)\\leqslant \\boldsymbol{f}(\\alpha)\\leqslant \\tilde{F}^{(n)}(\\tau).\\label{eq:sana}\n\\end{align}\nIn \\eqref{eq:closetonormal} we show $F_n$ and $\\tilde{F}^{(n)}$ are within distance $\\gamma$ to the cdf of standard normal, so\n\\[\n\t\\Phi(t)-\\gamma\\leqslant F_n(t^-)\\leqslant 1-\\alpha\\leqslant F_n(t)\\leqslant \\Phi(t)+\\gamma\n\\]\nand hence\n\\begin{equation}\\label{eq:mina}\n\t\\Phi^{-1}(1-\\alpha-\\gamma)\\leqslant t\\leqslant \\Phi^{-1}(1-\\alpha+\\gamma).\n\\end{equation}\nUsing \\eqref{eq:sana} and \\eqref{eq:mina}, \n\\begin{align*}\n\t\\boldsymbol{f}(\\alpha)&\\leqslant\\tilde{F}^{(n)}(\\tau)\\\\\n\t&\\leqslant \\Phi(\\tau)+\\gamma\\\\\n\t&= \\Phi(t-\\mu)+\\gamma\\\\\n\t&\\leqslant\\Phi(\\Phi^{-1}(1-\\alpha+\\gamma)-\\mu)+\\gamma\\\\\n\t&=G_{\\mu}(\\alpha-\\gamma)+\\gamma\n\\end{align*}\nSimilarly we can show that $\\boldsymbol{f}(\\alpha)\\geqslant G_{\\mu}(\\alpha+\\gamma)-\\gamma$. The proof is now complete.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nNext we prove the asymptotic version. Recall that our goal is\n\\asymprep*\n\\begin{proof}[Proof of \\Cref{thm:CLT}]\n\tWe will first construct pointwise convergence $f_{n1}\\otimes f_{n2} \\otimes \\cdots \\otimes f_{nn}\\to G_{2K\/s}$ and then conclude uniform convergence from a general theorem.\n\n\tApply Berry-Esseen Theorem \\ref{thm:Berry} to the $n$-th row of the triangular array and we have\n\t\\begin{equation*}\\label{eq:sandwich}\n\t\tG_{\\mu_n}(\\alpha+\\gamma_n)-\\gamma_n\\leqslant f_{n1}\\otimes f_{n2} \\otimes \\cdots \\otimes f_{nn}(\\alpha)\\leqslant G_{\\mu_n}(\\alpha-\\gamma_n)+\\gamma_n.\n\t\\end{equation*}\n\tHere $\\mu_n$ and $\\gamma_n$ are the counterparts of $\\mu$ and $\\gamma$ defined in \\Cref{thm:Berry} when applied to $f_{n1},\\ldots,f_{nn}$. Namely,\n\t\\begin{align*}\n\t\t\\mu_n=&\\,\\,\\frac{2\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}^{(n)}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2}},\\\\\n\t\t\\gamma_n=&\\,\\,0.56\\cdot\\frac{\\|\\boldsymbol{\\bar{\\kappa}_3}^{(n)}\\|_1}{\\big(\\|\\boldsymbol{\\kappa_2}^{(n)}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2\\big)^{\\frac{3}{2}}}\n\t\\end{align*}\n\tHere the bold vector notation with a superscript $(n)$ denotes the vector for the $n$-th row. For example, $\\boldsymbol{\\mathrm{kl}}^{(n)} = \\big(\\mathrm{kl}(f_{n1}),\\ldots, \\mathrm{kl}(f_{nn})\\big)$.\n\n\tBy the sandwich inequality, pointwise convergence of $f_{n1}\\otimes f_{n2} \\otimes \\cdots \\otimes f_{nn}$ follows from the two limits\n\t\\begin{equation}\\label{eq:wendy}\n\t\tG_{\\mu_n}(\\alpha+\\gamma_n)-\\gamma_n\\to G_{2K\/s}(\\alpha), \\quad G_{\\mu_n}(\\alpha-\\gamma_n)+\\gamma_n\\to G_{2K\/s}(\\alpha).\n\t\\end{equation}\n\tTo prove these, let's first show $\\gamma_n\\to 0$ and $\\mu_n\\to2K\/s$.\n\n\n\n\tReformulating the assumptions in bold vector notations, we have\n\t\\[||\\boldsymbol{\\mathrm{kl}}^{(n)}||_1\\to K,\\quad ||\\boldsymbol{\\mathrm{kl}}^{(n)}||_\\infty\\to 0,\\quad ||\\boldsymbol{\\kappa_2}^{(n)}||_1\\to s^2,\\quad ||\\boldsymbol{\\kappa_3}^{(n)}||_1\\to 0.\\]\n\tIn addition to these, it suffices to show\n\t\\begin{equation}\\label{eq:ning}\n\t\t\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2\\to0 ~~\\text{ and }~~ \\|\\boldsymbol{\\bar{\\kappa}_3}^{(n)}\\|_1\\to0.\n\t\\end{equation}\n\tFor the first half, notice that $\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2=\\langle \\boldsymbol{\\mathrm{kl}}^{(n)}, \\boldsymbol{\\mathrm{kl}}^{(n)}\\rangle\\leqslant \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty\\cdot \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_1\\to0$. In fact, $\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty\\to0$ is not only sufficient but also necessary, because $\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty\\leqslant\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2$.\n\n\tNext we use the assumptions to show $\\|\\boldsymbol{\\bar{\\kappa}_3}^{(n)}\\|_1\\to0$.\n\tWe need a lemma\n\t\\begin{lemma} \\label{lem:cubicmoments}\n\tFor a trade-off function $f$,\n\t\t$$\\bar{\\kappa}_3(f)\\leqslant \\kappa_3(f) + 3\\mathrm{kl}(f)\\cdot\\kappa_2(f)+3\\mathrm{kl}^2(f)\\cdot \\sqrt{\\kappa_2(f)}+\\mathrm{kl}^3(f).$$\n\t\\end{lemma}\n\t\\begin{proof}[Proof of \\Cref{lem:cubicmoments}]\n\t\t\\begin{align*}\n\t\t\t\\bar{\\kappa}_3(f)=&\\phantom{+}\\int_0^1\\big|\\log Df(x)+\\mathrm{kl}(f)\\big|^3\\diff x\\\\\n\t\t\t\\leqslant&\\phantom{+} \\int_0^1\\big(\\big|\\log Df(x)\\big|+\\big|\\mathrm{kl}(f)\\big|\\big)^3\\diff x\\\\\n\t\t\t\\leqslant&\\phantom{+} \\int_0^1\\big|\\log Df(x)\\big|^3\\diff x+3\\mathrm{kl}(f)\\cdot\\int_0^1\\big|\\log Df(x)\\big|^2\\diff x\\\\\n\t\t\t&+3\\mathrm{kl}^2(f)\\cdot\\int_0^1\\big|\\log Df(x)\\big|\\diff x + \\mathrm{kl}^3(f)\\\\\n\t\t\t\\leqslant&\\phantom{+} \\kappa_3(f) + 3\\mathrm{kl}(f)\\cdot\\kappa_2(f)+3\\mathrm{kl}^2(f)\\cdot \\sqrt{\\kappa_2(f)}+\\mathrm{kl}^3(f).\n\t\t\\end{align*}\n\t\tIn the last step we used Jensen's inequality.\n\t\\end{proof}\n\tApply \\Cref{lem:cubicmoments} to each $f_{ni}$ and sum them up:\n\t\\begin{align*}\n\t\t\\|\\boldsymbol{\\bar{\\kappa}_3}^{(n)}\\|_1 \n\t\t&\\leqslant \\|\\boldsymbol{\\kappa_3}^{(n)}\\|_1 + 3\\textstyle\\sum_i \\mathrm{kl}(f_{ni})\\cdot \\kappa_2(f_{ni})+ 3\\textstyle\\sum_i \\mathrm{kl}(f_{ni})\\cdot \\sqrt{\\kappa_2(f_{ni})} \\cdot \\mathrm{kl}(f_{ni}) + \\textstyle\\sum_i \\mathrm{kl}(f_{ni})\\cdot \\mathrm{kl}^2(f_{ni}).\n\t\\end{align*}\n\tUsing $|\\sum a_i b_i|\\leqslant |\\sum a_i| \\cdot \\max |b_i|$ and Cauchy-Schwarz inequality yields\n\t\\begin{align*}\n\t\t\\|\\boldsymbol{\\bar{\\kappa}_3}^{(n)}\\|_1 \n\t\t&\\leqslant \\|\\boldsymbol{\\kappa_3}^{(n)}\\|_1 + 3 \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty\\cdot \\|\\boldsymbol{\\kappa_2}^{(n)}\\|_1  + 3\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty \\cdot \\Big(\\textstyle\\sum_i \\sqrt{\\kappa_2(f_{ni})} \\cdot \\mathrm{kl}(f_{ni})\\Big) + \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty^2 \\cdot \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_1\\\\\n\t\t&\\leqslant \\|\\boldsymbol{\\kappa_3}^{(n)}\\|_1 + 3 \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty\\cdot \\|\\boldsymbol{\\kappa_2}^{(n)}\\|_1  + 3\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty \\cdot \\sqrt{\\|\\boldsymbol{\\kappa_2}^{(n)}\\|_1 \\cdot \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2} + \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_\\infty^2 \\cdot \\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_1.\n\t\\end{align*}\n\tBy the assumptions\n\t\\[||\\boldsymbol{\\mathrm{kl}}^{(n)}||_1\\to K,\\quad ||\\boldsymbol{\\mathrm{kl}}^{(n)}||_\\infty\\to 0,\\quad ||\\boldsymbol{\\kappa_2}^{(n)}||_1\\to s^2,\\quad ||\\boldsymbol{\\kappa_3}^{(n)}||_1\\to 0\\]\n\tand $\\|\\boldsymbol{\\mathrm{kl}}^{(n)}\\|_2^2\\to0$ which we just proved,\n\tit's easy to see that all four terms goes to 0 as $n$ goes to infinity.\n\n\n\n\n\n\tThe two limits \\eqref{eq:ning} we have just proved imply $\\mu_n\\to2K\/s$ and $\\gamma_n\\to 0$. Given these, convergence \\eqref{eq:wendy} is easy once we notice that $G_{\\mu}(\\alpha)=\\Phi(\\Phi^{-1}(1-\\alpha)-\\mu)$ is continuous in both $\\alpha$ and $\\mu$.\n\n\tIf the readers are concerned with $1-\\alpha-\\gamma_n$ exceeding $[0,1]$, then observe that when $\\alpha\\in(0,1)$, $1-\\alpha-\\gamma_n$ eventually ends up in $(0,1)$ where $\\Phi^{-1}$ is well-defined and continuous. So the only concern is at 0 and 1. If $\\alpha=0$, $\\Phi^{-1}(1-\\alpha-\\gamma_n)\\to+\\infty$ so $G_{\\mu_n}(0+\\gamma_n)-\\gamma_n\\to1 = G_{2K\/s}(0)$. A similar argument works for $\\alpha=1$.\n\n\tAnyway, we have shown pointwise convergence. Uniform convergence is again a direct consequence of \\Cref{lem:uniform}.\n\tThe proof is now complete.\n\\end{proof}\n\n\n\nNext we explain the effect of tensoring $f_{0,\\delta}$.\n\n\t\\begin{equation}\n\tf\\otimes f_{0,\\delta}(\\alpha) =\n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t(1-\\delta)\\cdot f(\\frac{\\alpha}{1-\\delta}), \t\t& 0\\leqslant \\alpha \\leqslant 1-\\delta \\\\\n\t\t0, & 1-\\delta\\leqslant \\alpha\\leqslant 1.\n\t\t\\end{array}\n\t\t\\right.\\tag{\\ref{prop:ruibbit}}\n\t\\end{equation}\n\\begin{proof}[Proof of \\Cref{prop:ruibbit}]\n\tFirst, $f_{0,\\delta}$ is the trade-off function of two uniform distributions $f_{0,\\delta} = T\\big(U[0,1],U[\\delta,1+\\delta]\\big)$.\n\tTo see this, observe that any optimal test $\\phi$ for $U[0,1]$ vs $U[\\delta,1+\\delta]$ \n\tmust have the following form:\n\t$$\\phi(x)=\n\t\\left\\{\n\t\\begin{array}{ll}\n\t1, \t\t&  x\\in(1,1+\\delta]\\\\\n\tp, & x\\in[\\delta,1],\\\\\n\t0, & x\\in[0,\\delta)\n\t\\end{array}\n\t\\right.\n\t$$\n\tThat is, we know it must be from $U[0,1]$ if we see something in $[0,\\delta)$, and must be from $U[\\delta,1+\\delta]$ if we see something in $(1,1+\\delta]$. Otherwise the only thing we can do is random guessing. It's easy to see that the errors of such $\\phi$ linearly interpolates between $(0,1-\\delta)$ and $(1-\\delta,0)$, i.e. type I and type II error add up to $1-\\delta$. On the other hand, by definition, $f_{0,\\delta}(\\alpha) = \\max{\\{1-\\delta-\\alpha,0\\}}$. So they indeed agree with each other.\n\n\tNow suppose $f=T(P,Q)$. By definition of tensor\tproduct, $f\\otimes f_{0,\\delta} = T(P\\times U[0,1], Q\\times U[\\delta,1+\\delta])$. If the optimal test for $P$ vs $Q$ at level $\\alpha$ is $\\phi_\\alpha$, then an optimal test for $P\\times U[0,1]$ vs $Q\\times U[\\delta,1+\\delta]$ must be of the following form:\n\t$$\\tilde{\\phi}_\\alpha(\\omega, x)=\n\t\\left\\{\n\t\\begin{array}{ll}\n\t1, \t\t&  x\\in(1,1+\\delta]\\\\\n\t\\phi_\\alpha(\\omega), & x\\in[\\delta,1],\\\\\n\t0, & x\\in[0,\\delta)\n\t\\end{array}\n\t\\right.\n\t$$\n\tThe errors are\n\t\\begin{align*}\n\t\t\\E_{P\\times U[0,1]}[\\tilde{\\phi}_\\alpha] &= P\\big[x\\in(1,1+\\delta]\\big]+ P\\big[x\\in[\\delta,1]\\big]\\cdot \\E_{P}[\\phi_\\alpha(\\omega)] \\\\\n\t\t&= 0+(1-\\delta)\\alpha=(1-\\delta)\\alpha\\\\\n\t\t1-\\E_{Q\\times U[\\delta,1+\\delta]}[\\tilde{\\phi}_\\alpha] &= 1-P\\big[x\\in(1,1+\\delta]\\big]- P\\big[x\\in[\\delta,1]\\big]\\cdot \\E_{Q}[\\phi_\\alpha(\\omega)] \\\\\n\t\t&=1- \\delta- (1-\\delta)\\big(1-f(\\alpha)\\big) = (1-\\delta)f(\\alpha)\n\t\\end{align*}\n\tThis completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\\DPCLTrep*\n\\begin{proof}[Proof of \\Cref{thm:DPCLT}]\nAs in the main body, we first apply rules $f_{\\ep,\\delta} = f_{\\ep,0}\\otimes f_{0,\\delta}$ and $f_{0,\\delta_1}\\otimes f_{0,\\delta_2} = f_{0,1-(1-\\delta_1)(1-\\delta_2)}$ to get\n\\begin{align*}\n\tf_{\\ep_{n1},\\delta_{n1}}\\otimes \\cdots \\otimes f_{\\ep_{nn},\\delta_{nn}}\n\t&= \\big(f_{\\ep_{n1},0}\\otimes \\cdots \\otimes f_{\\ep_{nn},0}\\big)\\otimes \\big(f_{0,\\delta_{n1}}\\otimes \\cdots \\otimes f_{0,\\delta_{nn}}\\big)\\\\\n\t&= \\big(\\underbrace{f_{\\ep_{n1},0}\\otimes \\cdots \\otimes f_{\\ep_{nn},0}}_{f^{(n)}}\\big)\\otimes f_{0,\\delta^{(n)}}\n\\end{align*}\nwith $\\delta^{(n)} = 1-\\prod_{i=1}^n(1-\\delta_{ni})$.\nFor the second factor, let's first prove the limit $\\delta^{(n)}\\to 1-\\mathrm{e}^{-\\delta}$.\nChanging the product into sum, we have\n\\[\\log (1-\\delta^{(n)}) = \\textstyle\\sum_{i=1}^n\\log(1-\\delta_{ni})\\]\nThe limit almost follows from the Taylor expansion $\\log (1+x) = x+o(x)$, but we need to be a little more careful as the number of summation terms also goes to infinity. \nSince $\\max_{1\\leqslant i\\leqslant n} \\delta_{ni}\\to 0$, we can assume for large $n$, $\\delta_{ni}<r$ for some $r$ such that when $|x|<r$, the following Taylor expansion holds for some constant $C$:\n\\[|\\log(1-x)+x|\\leqslant Cx^2.\\]\nWith this,\n\\begin{align*}\n\t\\big|\\textstyle\\sum_{i=1}^n \\log(1-\\delta_{ni})+\\delta_{ni}\\big|\n\t\\leqslant C\\cdot\\textstyle\\sum_{i=1}^n\\delta_{ni}^2\\leqslant C\\cdot\\max_{i} \\delta_{ni}\\cdot \\textstyle\\sum_i \\delta_{ni} \\to 0.\n\\end{align*}\nTherefore, $\\log (1-\\delta^{(n)}) = \\textstyle\\sum_{i=1}^n\\log(1-\\delta_{ni})$ has the same limit as $\\sum_{i=1}^n\\delta_{ni}$. In other words, $\\log (1-\\delta^{(n)})\\to \\delta$, or equivalently, $\\delta^{(n)}\\to 1-\\mathrm{e}^{-\\delta}$.\n\nFor a fixed $x\\in[0,1]$, $f_{0,\\delta^{(n)}}(x) = \\max\\{0,1-\\delta^{(n)}-x\\}$ is continuous in $\\delta^{(n)}$. Hence we have the pointwise limit $f_{0,\\delta^{(n)}}\\to f_{0,1-\\mathrm{e}^{-\\delta}}$.\n\nFor the first factor $f^{(n)} = f_{\\ep_{n1},0}\\otimes \\cdots \\otimes f_{\\ep_{nn},0}$, we will apply \\Cref{thm:CLT}. Let's check the conditions.\n\nBy the continuity of the function $x\\mapsto x\\tanh\\tfrac{x}{2}$ at 0, the assumption $\\max_{1\\leqslant i\\leqslant n} \\ep_{ni}\\to 0$ implies\n\\[\\max_{1\\leqslant i \\leqslant n}\\mathrm{kl}(f_{ni})=\\max_{1\\leqslant i \\leqslant n} \\ep_{ni}\\tanh\\tfrac{\\ep_{ni}}{2}\\to0.\\]\nNext, we show\n\\[\\sum_{i=1}^n \\mathrm{kl}(f_{ni})=\\sum_{i=1}^n \\ep_{ni}\\tanh\\frac{\\ep_{ni}}{2}\\to K=\\frac{\\mu^2}{2}.\\]\nPreparing for the same Taylor expansion trick, let $n$ be large enough so that Taylor expansion $|\\tanh x -x|\\leqslant Cx^2$ applies to all $\\delta_{ni}$.\n\\begin{align*}\n\t\\bigg|\\sum_{i=1}^n \\ep_{ni}\\tanh\\frac{\\ep_{ni}}{2} -\\sum_{i=1}^n \\frac{\\ep_{ni}^2}{2}\\bigg| &= \\sum_{i=1}^n \\ep_{ni}\\Big|\\tanh\\frac{\\ep_{ni}}{2}-\\frac{\\ep_{ni}}{2}\\Big|\\\\\n\t&\\le C\\cdot \\sum_{i=1}^n \\ep_{ni}\\cdot \\ep_{ni}^2\\\\\n\t&\\le C\\cdot \\max_{1\\leqslant i \\leqslant n} \\ep_{ni}\\cdot \\sum_{i=1}^n\\ep_{ni}^2\\to0.\n\\end{align*}\nSo $\\sum_{i=1}^n \\ep_{ni}\\tanh\\frac{\\ep_{ni}}{2}$ and $\\sum_{i=1}^n \\tfrac{\\ep_{ni}^2}{2}$ has the same limit, which by our assumption is $\\mu^2\/2$.\n\nFor second moment, $\\sum_{i=1}^n \\kappa_2(f_{ni})=\\sum_{i=1}^n \\ep_{ni}^2$ has limit $\\mu^2$. That is, $s$ in \\Cref{thm:CLT} is equal to $\\mu$.\n\nFor third moment,\n\\[\\sum_{i=1}^n \\kappa_3(f_{ni})=\\sum_{i=1}^n \\ep_{ni}^3 \\leqslant \\big(\\max_{1\\le i\\le n} \\ep_{ni} \\big)\\cdot\\sum_{i=1}^n \\ep_{ni}^2\\to 0.\\]\n\nAll four conditions of \\Cref{thm:CLT} check, so we can conclude the limit of $f^{(n)}$ is the GDP trade-off function with parameter $2K\/s = s = \\mu$.\n\nThe last step is to combine the two limits $f^{(n)}\\to G_\\mu$ and $f_{0,\\delta^{(n)}}\\to f_{0,1-\\mathrm{e}^{-\\delta}}$. By \\Cref{prop:ruibbit},\n\n\\begin{equation*}\nf^{(n)}\\otimes f_{0,\\delta^{(n)}}(\\alpha) =\n\t\\left\\{\n\t\\begin{array}{ll}\n\t(1-\\delta^{(n)})\\cdot f^{(n)}(\\frac{\\alpha}{1-\\delta^{(n)}}), \t\t& 0\\leqslant \\alpha \\leqslant 1-\\delta^{(n)}, \\\\\n\t0, & 1-\\delta^{(n)}\\leqslant \\alpha\\leqslant 1.\n\t\\end{array}\n\t\\right.\n\\end{equation*}\n\\Cref{lem:uniform} tells us $f^{(n)} $  uniformly converges to $ G_\\mu$, so we have the limit\n\\[f^{(n)}(\\tfrac{\\alpha}{1-\\delta^{(n)}})\\to G_{\\mu}(\\tfrac{\\alpha}{1-(1-\\mathrm{e}^{-\\delta})})\\]\nThis implies the pointwise limit\n\\[\n\tf_{\\ep_{n1},\\delta_{n1}}\\otimes \\cdots \\otimes f_{\\ep_{nn},\\delta_{nn}} = f^{(n)}\\otimes f_{0,\\delta^{(n)}}\\to G_\\mu\\otimes  f_{0,1-\\mathrm{e}^{-\\delta}}.\n\\]\nAgain, uniform convergence comes for free via \\Cref{lem:uniform}.\n\\end{proof}\n\n\nThe next two corollaries are Berry-Esseen style central limit theorems for the composition of pure $\\ep$-DP. Given the existence of \\Cref{thm:fast}, these results are relatively loose, but might be good enough if we have large $n$. Nonzero $\\delta$ is allowed following a similar argument as in \\Cref{thm:DPCLT}.\n\\begin{corollary}\\label{cor:e0CLT}\n\tSet $t_i = \\tanh \\frac{\\ep_i}{2}$ and\n\t\\begin{align*}\n\t\t\\mu=&\\,\\,\\frac{2\\sum_{i=1}^n \\ep_it_i}{\\big(\\sum_{i=1}^n \\ep_i^2 (1-t_i^2)\\big)^{1\/2}},\\\\\n\t\t\\gamma=&\\,\\,0.56\\cdot \\frac{\\sum_{i=1}^n \\ep_i^3(1-t_i^4)}{\\big(\\sum_{i=1}^n \\ep_i^2 (1-t_i^2)\\big)^{3\/2}}.\n\t\\end{align*}\n\tThen for any $\\alpha\\in[0,1]$,\n\t\\begin{equation*}\n\t\tG_\\mu(\\alpha+\\gamma)-\\gamma\\leqslant f_{\\ep_1,0}\\otimes f_{\\ep_2,0} \\otimes \\cdots \\otimes f_{\\ep_n,0}(\\alpha)\\leqslant G_\\mu(\\alpha-\\gamma)+\\gamma.\n\t\\end{equation*}\n\\end{corollary}\n\nIn order to highlight the $1\/\\sqrt{n}$ convergence rate, we also derive an easy version in the homogeneous case.\n\\begin{corollary}\\label{cor:e0Berry}\n\tLet $\\mu = 2\\sqrt{n}\\sinh\\frac{\\ep}{2},\n\t\t\\gamma = \\frac{0.56}{\\sqrt{n}}\\cdot\\frac{\\cosh \\ep}{\\cosh\\frac{\\ep}{2}}$.\n\tThen\n\t\\[\n\t\tG_\\mu(\\alpha+\\gamma)-\\gamma\\leqslant f_{\\ep,0}^{\\otimes n}(\\alpha)\\leqslant G_\\mu(\\alpha-\\gamma)+\\gamma.\n\t\\]\n\\end{corollary}\nTo see $\\gamma = O(1\/\\sqrt{n})$, note that for the limit to be meaningful, $\\ep$ has to be $o(1)$, which implies $\\frac{\\cosh \\ep}{\\cosh\\frac{\\ep}{2}}\\approx 1$.\n\nBoth of them rely on evaluating the moment functionals on $f_{\\ep,0}$. We summarize the results in the following lemma:\n\\begin{lemma} \\label{lem:epfunctionals}\n\tLet $t = \\tanh \\frac{\\ep}{2}$. Then\n\t\\begin{align*}\n\t\t\\mathrm{kl}(f_{\\ep,0}) = \\ep t, \\quad \\kappa_2(f_{\\ep,0}) = \\ep^2, \\quad \\kappa_3(f_{\\ep,0}) = \\ep^3, \\quad \\bar{\\kappa}_3(f_{\\ep,0}) = \\ep^3(1-t^4).\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of \\Cref{lem:epfunctionals}]\n\tFor convenience, let $p = \\frac{\\mathrm{e}^{\\ep}}{\\mathrm{e}^{\\ep}+1}$ and $q = 1-p = \\frac{1}{\\mathrm{e}^{\\ep}+1}$. We have\n\t\\begin{align*}\n\t\tp &= \\frac{\\mathrm{e}^{\\ep}}{\\mathrm{e}^{\\ep}+1} = \\frac{\\mathrm{e}^\\frac{\\ep}{2}}{\\mathrm{e}^\\frac{\\ep}{2}+\\mathrm{e}^{-\\frac{\\ep}{2}}} = \\frac{\\cosh \\frac{\\ep}{2}+\\sinh \\frac{\\ep}{2}}{2\\cosh \\frac{\\ep}{2}} = \\frac{1}{2}(1+t)\\\\\n\t\tq &= \\frac{1}{\\mathrm{e}^{\\ep}+1} = \\frac{\\mathrm{e}^{-\\frac{\\ep}{2}}}{\\mathrm{e}^\\frac{\\ep}{2}+\\mathrm{e}^{-\\frac{\\ep}{2}}} = \\frac{\\cosh \\frac{\\ep}{2}-\\sinh \\frac{\\ep}{2}}{2\\cosh \\frac{\\ep}{2}} = \\frac{1}{2}(1-t).\n\t\\end{align*}\n\tThe log likelihood ratio is $-\\ep$ with probability $p$ and $\\ep$ with probability $q$, so\n\t\\begin{align*}\n\t\t\\mathrm{kl}(f_{\\ep,0}) &= -[(-\\ep)\\cdot p + \\ep\\cdot q] = \\ep(p-q) = \\ep t,\\\\\n\t\t\\kappa_2(f_{\\ep,0}) &= \\ep^2(p+q) = \\ep^2,\\\\\n\t\t\\kappa_3(f_{\\ep,0}) &= \\ep^3(p+q) = \\ep^3\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\t\t\\bar{\\kappa}_3(f_{\\ep,0}) &= p|-\\ep+\\ep(p-q)|^3+ q|\\ep+\\ep(p-q)|^3 \\\\\n\t\t&= 8\\ep^3\\cdot pq(p^2+q^2)\\\\\n\t\t&=2\\ep^3\\cdot (1-t^2)\\cdot \\frac{1}{2}(1+t^2)\\\\\n\t\t&=\\ep^3(1-t^4).\n\t\\end{align*}\n\\end{proof}\n\n\\begin{proof}[Proof of \\Cref{cor:e0CLT} ]\n\tFollows directly from \\Cref{thm:Berry} and \\Cref{lem:epfunctionals}.\n\\end{proof}\n\n\\begin{proof}[Proof of \\Cref{cor:e0Berry}]\n\tAll we need to do is to simplify the expression of $\\mu$ and $\\gamma$ assuming all $\\ep_i = \\ep$ and hence $t_i = t = \\tanh \\frac{\\ep}{2}$. \n\t\\begin{align*}\n\t\t\\mu&=\\frac{2n \\ep t}{\\sqrt{n\\ep^2 (1-t^2)}}\\\\\n\t\t&=2\\sqrt{n}\\cdot\\frac{t}{1-t^2} = 2\\sqrt{n}\\sinh\\frac{\\ep}{2}\\\\\n\t\t\\gamma&=0.56\\cdot \\frac{n \\ep^3(1-t^4)}{\\big(n \\ep^2 (1-t^2)\\big)^{3\/2}}\\\\\n\t\t&=\\frac{0.56}{\\sqrt{n}}\\cdot\\frac{1+t^2}{\\sqrt{1-t^2}}\\\\\n\t\t&=\\frac{0.56}{\\sqrt{n}}\\cdot\\frac{\\cosh \\ep}{\\cosh\\frac{\\ep}{2}}.\n\t\\end{align*}\n\tThe proof is complete.\n\n\n\n\n\\end{proof}\n\n\n\n\n\\bigskip\n\n\n\n\n\n\\section{Technical Details in \\Cref{sec:fDP}}\n\\label{app:fDP}\nThe Neyman--Pearson lemma is a useful tool in our paper. The following statement is adapted from \\cite{lehmann2006testing}.\n\\begin{theorem} [Neyman-Pearson lemma]\\label{thm:NPlemma}\nLet $P$ and $Q$ be probability distributions on $\\Omega$ with densities $p$ and $q$, respectively. For the hypothesis testing problem $H_0: P$ vs $H_1: Q$, a test $\\phi:\\Omega\\to[0,1]$ is the most powerful test at level $\\alpha$ if and only if there are two constants $h\\in[0,+\\infty]$ and $c\\in[0,1]$ such that $\\phi$ has the form\n\t\\[\n\t\\phi(\\omega)=\n\t\\left\\{\n\t\\begin{array}{ll}\n\t1, \t\t& \\text{if } p(\\omega)>hq(\\omega), \\\\\n\tc, \t\t& \\text{if } p(\\omega)=hq(\\omega), \\\\\n\t0, \t\t& \\text{if } p(\\omega)<hq(\\omega),\n\t\\end{array}\n\t\\right.\n\t\\]\n\tand $\\E_P[\\phi]=\\alpha$.\n\\end{theorem}\n\nNow, we use the Neyman--Pearson lemma to prove the proposition which gives sufficient and necessary conditions for trade-off functions.\n\\tradeoffthm*\nIn the entire appendix, from \\ref{app:fDP} to \\ref{app:SGD}, we will use $\\T$ to denote the class of trade-off functions, and $\\T^S$ the subclass of symmetric trade-off functions.\n\\begin{proof}[Proof of \\Cref{prop:trade-off}]\n\n\t``only if'':\n\tSuppose $f=T(P_0,P_1)$. It is obviously non-increasing. The randomized testing rule that blindly rejects with probability $p$ achieves $(p,1-p)$ errors. It is suboptimal at level $p$, so $f(p)\\leqslant 1-p$.\n\n\tConvexity follows from randomizing over two rejections rules. For given $\\alpha,\\alpha',t$, all in $[0,1]$, let $\\phi$ and $\\phi'$ be the rejection rules achieving errors $(\\alpha,f(\\alpha))$ and $(\\alpha,f(\\alpha))$ respectively. The rejection rule $\\phi_t = t\\phi+(1-t)\\phi'$ achieves errors $t\\alpha+(1-t)\\alpha'$ and $tf(\\alpha)+(1-t)f(\\alpha')$. It is suboptimal at level $t\\alpha+(1-t)\\alpha'$, so we have\n\t\\[f(t\\alpha+(1-t)\\alpha')\\leqslant t\\alpha+(1-t)\\alpha'.\\]\n\n\tAs we remarked in the footnote, continuity in $(0,1]$ follows from properties we have proved.\n\tAt 0 it requires a closer look at Neyman-Pearson lemma. Suppose $\\alpha_n\\to0$. Without loss of generality we can assume $\\alpha_n$ is decreasing. We want to show $f(\\alpha_n)\\to f(0)$. Neyman-Pearson lemma tells us the optimal test $\\phi_n$ at level $\\alpha_n$ must have the form\n\t$$\\phi_n(\\omega)=\n\\left\\{\n\\begin{array}{ll}\n1, \t\t& \\frac{p_1(\\omega)}{p_0(\\omega)}>h_n\\\\\nc_n, & \\frac{p_1(\\omega)}{p_0(\\omega)}=h_n\\\\\n0, & \\frac{p_1(\\omega)}{p_0(\\omega)}<h_n\n\\end{array}\n\\right.\n$$\nfor some $c_n\\in[0,1]$ and $h_n\\in[0,+\\infty]$. The fact that $\\alpha_n$ is decreasing implies that $\\phi_n(\\omega)$ is monotone decreasing in $n$ except on a measure zero set, so it has a pointwise limit $\\phi(\\omega)$. Furthermore, $\\phi(\\omega)$ must be in the same form as $\\phi_n$. Again by Neyman-Pearson lemma, $\\phi$ must be the optimal test at level $\\E_{P_0}[\\phi]$. By dominated convergence theorem, $\\E_{P_i}[\\phi] = \\lim_{n\\to\\infty}\\E_{P_i}[\\phi_n]$ for $i=0,1$. When $i=0$, this translates to $\\E_{P_0}[\\phi] = \\lim_{n\\to\\infty} \\alpha_n = 0$. So $\\phi$ is at level 0. When $i=1$, we have\n\\[\\E_{P_1}[\\phi] = \\lim_{n\\to\\infty}\\E_{P_1}[\\phi_n] = \\lim_{n\\to\\infty} 1-f(\\alpha_n)\\]\nwhere the second equality follows from the optimality of $\\phi_n$. So\n\\[\\lim_{n\\to\\infty} f(\\alpha_n) = 1-\\E_{P_1}[\\phi] = f(0).\\]\nHere the second equality follows from the optimality of $\\phi$. In fact, this argument works not just for 0 but for arbitrary $\\alpha\\in[0,1]$.\n\n\n\n\t``if'':\n\tGiven $f$, we need to find $P,Q$. The common measurable space is the unit interval $[0,1]$. $P$ is the uniform distribution. $Q$ has density $-f'(1-x)$ on $[0,1)$ and an atom at 1 with $Q[\\{1\\}] = 1-f(0)$. In fact, $Q$ is constructed to have cdf $f(1-x)$, with the slight twist that the cdf is reset to be 1 at 1, because cdf has to be right continuous.\n\n\tIt's easy to verify that $Q$ is indeed a probability distribution on $[0,1]$ using the properties of $f$.\n\n\n\n\tThe likelihood ratio is simply $-f'(1-x)$ when $x<1$. At 1, it is 0 if $f(0)=1$ and $+\\infty$ if $f(0)<1$. By convexity of $f$ it is non-decreasing, so the likelihood ratio rejection regions have the form $[h,1]$. Type I error is $P[h,1] = 1-h$. Type II error is $Q[0,h) = f(1-h)$.\n\n\n\n\n\n\n\n\n\\end{proof}\nAn equivalent object to our trade-off function is the ``testing region'' used in \\cite{KOV}. For a trade-off function $f$, we define a special version of epigraph of $f$ as\n\\[\\epi(f):= \\{(\\alpha,\\beta)\\mid \\alpha\\in[0,1], f(\\alpha)\\leqslant \\beta \\leqslant 1-\\alpha\\}.\\]\nLet $P,Q$ be distributions on $\\Omega$. Recall that for a testing rule $\\phi:\\Omega\\to[0,1]$, we denote its type I and type II errors by\n$\\alpha_\\phi = \\E_{P}[\\phi], \\quad \\beta_\\phi = 1 - \\E_{Q}[\\phi]$.\nIt is easy to see that $\\epi(f)$ consists of all achievable type I and type II error pairs that is better than random guessing. Namely\n\t\\[\\epi(f)= \\{(\\alpha_\\phi,\\beta_\\phi)\\mid \\phi:\\Omega\\to[0,1] \\text{ measurable, } \\alpha_\\phi+\\beta_\\phi\\leqslant 1\\}.\\]\nThis means considering $f$ and $\\epi(f)$ are equivalent. For more information on the testing region, see Chapter 12 of \\cite{itlectures}.\n\n\nNext we justify the default assumption of symmetry.\n\\symmrep*\n\\begin{lemma}\\label{lem:symmetry}\n\tIf $f=\\F(P,Q)$, then $f^{-1}=\\F(Q,P)$.\n\\end{lemma}\n\\begin{proof}[Proof of \\Cref{lem:symmetry}]\n\tThe lemma is best illustrated from the epigraph point of view. It is an immediate consequence of the following claim: $(\\alpha,\\beta)\\in \\epi(f)$ if and only if $(\\beta,\\alpha)\\in \\epi(f^{-1})$, which is to say, $f(\\alpha)\\leqslant \\beta \\leqslant 1-\\alpha$ if and only if $f^{-1}(\\beta)\\leqslant \\alpha \\leqslant 1-\\beta$. In order to show this, the definition of $f^{-1}$, together with continuity of $f$, implies that $f(\\alpha)\\leqslant \\beta\\Leftrightarrow f^{-1}(\\beta)\\leqslant \\alpha$. This justfies the claim and hence the lemma.\n\\end{proof}\n\\begin{proof}[Proof of \\Cref{prop:symmetry}]\n\tLet $S,S'$ be neighboring datasets. Since $M$ is $f$-DP, we know that\n\t\\begin{equation}\\label{eq:hubei}\n\t\t\\F\\big(M(S),M({S'})\\big)\\geqslant f,\\quad\\F\\big(M(S'),M({S})\\big)\\geqslant f.\n\t\\end{equation}\n\tIt follows easily from definition that if $f,g\\in\\T$ satisfy $g\\geqslant f$, then $g^{-1}\\geqslant f^{-1}$.\n\tSo by \\Cref{lem:symmetry} and the second inequality in \\eqref{eq:hubei},\n\t$$\\F\\big(M(S),M({S'})\\big) = \\F\\big(M(S'),M({S})\\big)^{-1} \\geqslant f^{-1}.$$\n\tTogether with the first inequality in \\eqref{eq:hubei}, we see for all neighboring datasets we have\n\t$$\\F\\big(M(S),M({S'})\\big)\\geqslant \\max\\{f,f^{-1}\\}.$$\n\n\tIt is straightforward to verify that if $f $ and $ g$ are both convex,\n\tcontinuous, non-increasing and below $\\Id$ then $\\max\\{f,g\\}$ also satisfy these properties. By \\Cref{prop:trade-off}, we have $\\max\\{f,g\\}\\in\\T$. So $f^\\mathrm{S}=\\max\\{f,f^{-1}\\}$ is in $\\T$. The proof is complete.\n\n\\end{proof}\n\nRecall that \\Cref{eq:Gmu} states that\n\\[\n\t\\F\\big(\\N(0,1), \\N(\\mu,1)\\big)(\\alpha) = \\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big).\n\\]\n\\begin{proof}[Proof of \\Cref{eq:Gmu}]\n\tWhen $\\mu\\geqslant 0$, likelihood ratio of $\\N(\\mu,1)$ and $\\N(0,1)$ is $\\frac{\\varphi(x-\\mu)}{\\varphi(x)} = \\mathrm{e}^{\\mu x - \\frac{1}{2}\\mu^2}$, a monotone increasing function in $x$. So the likelihood ratio tests must be thresholds: reject if the sample is greater than some $t$ and accept otherwise. Assuming $X\\sim \\N(0,1)$, the corresponding type I and type II errors are\n\t\\[\\alpha(t) = \\P[X>t] = 1-\\Phi(t),\\quad \\beta(t) = \\P[X+\\mu\\leqslant t] = \\Phi(t-\\mu).\\]\n\tSolving $\\alpha$ from $t$, $t=\\Phi^{-1}(1-\\alpha)$. So\n\t\\[G_\\mu(\\alpha) = \\beta(\\alpha) = \\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big).\\]\n\\end{proof}\n\n\\postrep*\n\\begin{proof}[Proof of Lemma~\\ref{lem:post}]\nThe idea is that whatever can be done with the processed outcome can also be done with the original outcome. Formally, if an optimal test $\\phi:Z\\to[0,1]$ for the problem $\\mathrm{Proc} (P)$ vs $\\mathrm{Proc} (Q)$ at level $\\alpha$ can achieve type II error $\\beta = \\F\\big(\\mathrm{Proc} (P),\\mathrm{Proc} (Q)\\big)(\\alpha)$, then it is easy to verify that $\\phi\\circ\\mathrm{Proc} :Y\\to[0,1]$ has the same errors $\\alpha,\\beta$ for the problem $P$ vs $Q$. The optimal error $T(P,Q)(\\alpha)$ can only be smaller than $\\beta$.\n\\end{proof}\n\n\n\n\n\n\n\nThe next result is a generalization of \\Cref{eq:Gmu}, together with the interesting inverse.\n\nLet $P$ be a probability distribution on $\\R$ with density $p$, cdf $F:\\R\\to[0,1]$, quantile $F^{-1}:[0,1]\\to[-\\infty,+\\infty]$ and $\\xi$ be a random variable from the distribution $P$. Then we have\n\\begin{proposition}\\label{prop:logconcave}\n\t$\\F(\\xi,t+\\xi)(\\alpha) = F(F^{-1}(1-\\alpha)-t)$ holds for every $t>0$ if and only if the density $p$ is log-concave.\n\\end{proposition}\nIn particular, normal density is log-concave, so the expression of $G_\\mu$ is a special case.\n\\begin{proof}[Proof of \\Cref{prop:logconcave}]\n\tFor convenience let\n\t\\begin{equation*}\\label{eq:logconcave}\n\t\tf_t(\\alpha) := F(F^{-1}(1-\\alpha)-t).\n\t\\end{equation*}\n\t``if'': This is the easier direction. Fix $t>0$ and consider the log likelihood ratio of $\\xi$ and $t+\\xi$:\n\t$$\\mathrm{llk} = \\log p(x-t)-\\log p(x).$$\n\tllk is increasing in $x$ because of log-concavity, so according to Neyman-Pearson lemma, the optimal rejection rule must have the form $1_{[h,+\\infty)}$. Hence by a similar calculation as of Gaussian case, the trade-off function indeed has the form $f_t$.\n\n\t``only if'': We are given that $\\F(\\xi,t+\\xi) = f_t$ holds for every $t>0$, and we want to show that $p$ is log-concave. Now that $f_t$ is a trade-off function  for every $t>0$, it must be convex. By chain rule$$f_t'(\\alpha) = (-1)\\cdot\\frac{p(F^{-1}(1-\\alpha)-t)}{p(F^{-1}(1-\\alpha))}.$$\n\tFix any $t>0$, convexity implies $f_t'(\\alpha)$ is increasing in $\\alpha$ for any $\\alpha\\in[0,1]$. Setting $x = F^{-1}(1-\\alpha)$, we know $\\frac{p(x-t)}{p(x)}$ is increasing in $x$ for all $x\\in\\R$, hence also $\\log p(x-t)-\\log p(x)$.\n\n\n\n\n\n\n\n\tFor convenience let $g=\\log p$. We know $g(x-t)-g(x)$ is increasing in $x, \\forall t>0$. Equivalently, $g'(x-t)-g'(x)>0, \\forall x, \\forall t>0$, which means $g'(x)$ is decreasing, i.e. $g=\\log p$ is concave. The proof is complete.\n\\end{proof}\n\n\nNext we prove results presented in \\Cref{sub:a_primal_dual_connection_with_}.\n\\ftoDPrep*\n\\begin{proof}[Proof of \\Cref{prop:ftoDP}]\n    The tangent line of $f$ with slope $k$ has equation $y=kx-f^*(k)$, so when $k=-\\mathrm{e}^\\ep$ the equation is\n    \\[y = -\\mathrm{e}^\\ep x-f^*(-\\mathrm{e}^\\ep).\\]\n    Compare it to $f_{\\ep,\\delta}$, we see $1-\\delta = -f^*(-\\mathrm{e}^\\ep)$. By symmetry, the collection $\\{f_{\\ep,1+f^*(-\\mathrm{e}^{\\ep})}\\}_{\\ep\\geqslant0}$ envelopes the function $f$.\n\\end{proof}\n\\GDPtoDPrep*\n\\begin{proof}[Proof of \\Cref{corr:GDPtoDP}]\n    By \\Cref{prop:ftoDP}, $\\mu$-GDP is equivalent to $(\\ep,1+G_\\mu^*(-\\mathrm{e}^{\\ep}))$-DP, so it suffices to compute the expression of $G_\\mu^*(-\\mathrm{e}^\\ep)$.\n\n    Recall that $G_\\mu(x) = \\Phi\\big(\\Phi^{-1}(1-x)-\\mu\\big)$.\n    By definition,\n    \\[G_\\mu^*(y) = \\sup_{x\\in[0,1]} yx-\\Phi\\big(\\Phi^{-1}(1-x)-\\mu\\big).\\]\n    Let $t = \\Phi^{-1}(1-x)$. Equivalently, $x = 1-\\Phi(t)=\\Phi(-t)$. Do the change of variable and we have\n    \\[G_\\mu^*(y) = \\sup_{t\\in\\R} ~y\\Phi(-t)-\\Phi(t-\\mu).\\]\n    From the shape of $G_\\mu$ we know the supremum must be achieved at the unique critical point. Setting the derivative of the objective function to be zero yields\n    \\begin{align*}\n        \\frac{\\diff}{\\diff t} \\big[y\\Phi(-t)-\\Phi(t-\\mu)\\big] &= 0\\\\\n        -y\\varphi(-t) - \\varphi(t-\\mu) &=0\\\\\n        y\\mathrm{e}^{-\\frac{1}{2}t^2}+\\mathrm{e}^{-\\frac{1}{2}(t-\\mu)^2}&=0\\\\\n        y+\\mathrm{e}^{\\mu t-\\frac{1}{2}\\mu^2}&=0\n    \\end{align*}\n    So $t = \\frac{\\mu}{2}+\\frac{1}{\\mu}\\log(-y)$. Plug this back in the expression of $G_\\mu^*$ and we have\n    \\[G_\\mu^*(y) = y\\Phi\\Big(-\\frac{\\mu}{2}-\\frac{1}{\\mu}\\log(-y)\\Big)-\\Phi\\Big(-\\frac{\\mu}{2}+\\frac{1}{\\mu}\\log(-y)\\Big).\\]\n    When $y=-\\mathrm{e}^\\ep$,\n    \\[G_\\mu^*(-\\mathrm{e}^\\ep) = -\\mathrm{e}^\\ep\\Phi\\Big(-\\frac{\\mu}{2}-\\frac{\\ep}{\\mu}\\Big)-\\Phi\\Big(-\\frac{\\mu}{2}+\\frac{\\ep}{\\mu}\\Big).\\]\n    $1+G_\\mu^*(-\\mathrm{e}^{\\ep})$ agrees with the stated formula in \\Cref{corr:GDPtoDP}. The proof is complete.\n\\end{proof}\n\n\n\n\nThe rest of the section is devoted to group privacy results. The main theorem is\n\n\\groupthm*\n\nFor convenience we define an operation $\\mathbin{\\hat{\\circ}}$, which is function composition with a slight twist. For $f,g\\in\\T$,\n\\[f\\mathbin{\\hat{\\circ}} g (x) := f\\big(1-g(x)\\big).\\]\n$f^{\\mathbin{\\hat{\\circ}} k}$ is defined iteratively:\n\\[f^{\\mathbin{\\hat{\\circ}} k} = \\underbrace{f\\mathbin{\\hat{\\circ}}\\cdots\\mathbin{\\hat{\\circ}} f}_k.\\]\nNotice that $f\\mathbin{\\hat{\\circ}} g = 1-(1-f)\\circ(1-g)$, so $f^{\\mathbin{\\hat{\\circ}} k} = 1-(1-f)^{\\circ k}$.\n\\begin{lemma}\n\tThe operation $\\mathbin{\\hat{\\circ}}$ has the following properties for $f,g\\in\\T$:\n\t\\begin{enumerate}\n\t\t\\item[(a)] $f\\mathbin{\\hat{\\circ}} g \\in\\T$.\n\t\t\\item[(b)] $(f\\mathbin{\\hat{\\circ}} g)^{-1}=(g^{-1})\\mathbin{\\hat{\\circ}} (f^{-1}) $. In particular, if $f\\in\\T^S$, then $f^{\\mathbin{\\hat{\\circ}} k}\\in\\T^S$.\n\t\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n\\begin{enumerate}\n\t\\item[(a)]\n\tBy \\Cref{prop:trade-off}, it suffices to check the four properties for $f\\mathbin{\\hat{\\circ}} g$. Monotonicity and continuity are obvious. Convexity follows by the well-known fact that decreasing convex function composed with a concave function is convex. Finally, because $f(x)\\leqslant 1-x,g(x)\\leqslant 1-x$, we have\n\t\\[f\\mathbin{\\hat{\\circ}} g (x) = f\\big(1-g(x)\\big)\\leqslant 1-\\big(1-g(x)\\big) = g(x)\\leqslant 1-x.\\]\n\t\\item[(b)]\n\tRecall that $f^{-1}(y) = \\inf\\{x\\in[0,1]:f(x)\\leqslant y\\}$. We have\n\t\\begin{align*}\n\t\t\\big[(g^{-1})\\mathbin{\\hat{\\circ}} (f^{-1})\\big](y)\n\t\t=g^{-1}\\big(1-f^{-1}(y)\\big)=\\inf\\{x\\in[0,1]:g(x)\\leqslant 1-f^{-1}(y)\\}.\n\t\\end{align*}\n\tFor any two numbers $x,y\\in[0,1]$, we have the following equivalence chain:\n\t$$g(x)\\leqslant 1-f^{-1}(y) \\Leftrightarrow f^{-1}(y)\\leqslant 1-g(x) \\Leftrightarrow f(1-g(x))\\leqslant y\\Leftrightarrow f\\mathbin{\\hat{\\circ}} g (x)\\leqslant y.$$\n\tSo\n\t\\begin{align*}\n\t\t\\big[(g^{-1})\\mathbin{\\hat{\\circ}} (f^{-1})\\big](y)\n\t\t&=\\inf\\{x\\in[0,1]:f\\mathbin{\\hat{\\circ}} g (x)\\leqslant y\\} = (f\\mathbin{\\hat{\\circ}} g)^{-1}(y).\n\t\\end{align*}\n\tThat is, $(g^{-1})\\mathbin{\\hat{\\circ}} (f^{-1}) = (f\\mathbin{\\hat{\\circ}} g)^{-1}$.\n\tThe proof is complete.\n\\end{enumerate}\n\\end{proof}\n\n\n\n\n\\Cref{thm:group} is an immediate consequence of the following lemma:\n\\begin{lemma} \\label{lem:group}\n\tSuppose $\\F(P,Q)\\geqslant f,\\F(Q,R)\\geqslant g$, then $\\F(P,R)\\geqslant g\\mathbin{\\hat{\\circ}} f$.\n\\end{lemma}\n\\begin{proof}\n\tFix $\\alpha\\in[0,1]$. Suppose $\\phi$ is the optimal testing rules of the problem $P$ vs $R$ at the level of $\\alpha$. Then we know the type I error $\\E_P[\\phi]=\\alpha$ and the type II error achieves the optimal value, i.e.\n\t\\[1-\\E_R[\\phi]=\\F(P,R)(\\alpha).\\]\n\n\t$\\phi$ is suboptimal as a testing rule for the problem $Q$ vs $R$, so the type I and II errors must be above the trade-off function $g$. That is,\n\t\\[1-\\E_R[\\phi]\\geqslant \\F(Q,R)(\\E_Q[\\phi]) \\geqslant g(\\E_Q[\\phi]).\\]\n\tSimilarly, $\\phi$ is also suboptimal for the problem $P$ vs $Q$. So $1-\\E_Q[\\phi]\\geqslant f(\\E_P[\\phi]) = f(\\alpha)$. Equivalently,\n\t\\[\\E_Q[\\phi]\\leqslant 1-f(\\alpha).\\]\n\tPut them together\n\t\\begin{align*}\n\t\\F(P,R)(\\alpha) &= 1-\\E_R[\\phi]\\\\\n\t&\\geqslant g(\\E_Q[\\phi])\\\\\n\t&\\geqslant g\\big(1-f(\\alpha)\\big) \\quad\\quad(g \\text{ is decreasing})\\\\\n\t&=g\\mathbin{\\hat{\\circ}} f(\\alpha).\n\t\\end{align*}\n\tThis completes the proof.\n\\end{proof}\n\n\n\\begin{proof}[Proof of \\Cref{thm:group}]\n\tSuppose $S$ and $S'$ are $k$-neighbors, i.e. there exist datasets $S = S_0, S_1, \\ldots, S_k = S'$ such that $S_i$ and $S_{i+1}$ are neighboring or identical for all $i = 0, \\ldots, k-1$. By privacy of $M$, we know $T\\big(M(S_i),M(S_{i+1})\\big)\\geqslant f$. Iteratively apply \\Cref{lem:group} and we have\n\\[ T\\big(M(S),M(S_2)\\big)\\geqslant f\\mathbin{\\hat{\\circ}} f, \\quad T\\big(M(S),M(S_3)\\big)\\geqslant f^{\\group3} \\quad \\ldots \\quad T\\big(M(S),M(S')\\big)\\geqslant f^{\\mathbin{\\hat{\\circ}} k}.\\]\nWe know that $f^{\\mathbin{\\hat{\\circ}} k} = 1-(1-f)^{\\circ k}$, so the $f$-DP part of the claim is done.\n\nThe GDP part of the claim follows by an easy formula: $G_\\mu\\mathbin{\\hat{\\circ}} G_{\\mu'} = G_{\\mu+\\mu'}$.\nTo see this, recall that $G_\\mu(\\alpha)=\\Phi(\\Phi^{-1}(1-\\alpha)-\\mu)$.\n\\begin{align*}\n\tG_\\mu\\mathbin{\\hat{\\circ}} G_{\\mu'}(\\alpha) = G_\\mu\\big(1-G_{\\mu'}(\\alpha)\\big) = \\Phi\\big(\\Phi^{-1}(G_{\\mu'}(\\alpha))-\\mu\\big) = \\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu-\\mu'\\big) = G_{\\mu+\\mu'}(\\alpha).\n\\end{align*}\n\\end{proof}\nIn fact, similar conclusion holds for any log-concave noise. See \\Cref{prop:logconcave}.\n\n\\grouplimit*\nAs What makes it even more interesting is the convergence occurs with very small $k$. In \\Cref{fig:group} we set $\\ep=0.5$ and $f = 1-(1-f_{\\ep,0})^{\\circ 2}$. So the blue curve in the last panel is $1-(1-f)^{\\circ 2} = 1-(1-f_{\\ep,0})^{\\circ 4}$. Next we set $\\mu=k\\ep = 4\\cdot 0.5 = 2$. It turns out these numbers are good enough for the condition $k\\ep\\to\\mu$, because the predicted limit $\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)$ (orange curve in the last panel) is almost indistinguishable from the blue curve $1-(1-f_{\\ep,0})^{\\circ 4}$.\n\\begin{figure}[!htp]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.8\\linewidth]{.\/figures\/group.pdf}\n\t\\end{center}\n\t\\captionof{figure}{Group privacy corresponds to function composition. Here $f = 1-(1-f_{\\ep,0})^{\\circ 2}$ with $\\ep=0.5$, so the blue curve in the last panel is $1-(1-f)^{\\circ 2} = 1-(1-f_{\\ep,0})^{\\circ 4}$. Orange curve is the predicted limit $T\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(2,1)\\big)$. The distinction is almost invisible even when $k$ is only 4.}\n\t\\label{fig:group}\n\\end{figure}\n\n\n\n\\begin{lemma} \\label{lem:lap}\n\tThe trade-off function between Laplace distributions has expression\n\\begin{align*}\n\t\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)(\\alpha)=\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\t1-\\mathrm{e}^{\\mu}\\alpha, \t\t& \\alpha < \\mathrm{e}^{-\\mu}\/2, \\\\\n\t\t\\mathrm{e}^{-\\mu}\/4\\alpha, & \\mathrm{e}^{-\\mu}\/2\\leqslant\\alpha\\leqslant1\/2,\\\\\n\t\t\\mathrm{e}^{-\\mu}(1-\\alpha), & \\alpha>1\/2.\n\t\\end{array}\n\t\\right.\n\\end{align*}\n\\end{lemma}\n\n\\begin{figure}[!htp]\n\t\\begin{center}\n\t\t\\includegraphics[width=0.7\\linewidth]{.\/figures\/Laplace_app.pdf}\n\t\\end{center}\n\t\\captionof{figure}{\n\tGraph of $\n\t\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)$ with $\\mu=1$. It agrees with the reciprocal function in the middle.\n\t}\n\t\\label{fig:laplace}\n\\end{figure}\nThe graph of this function with $\\mu = 1$ is illustrated in \\Cref{fig:laplace}. In general, it consists of two symmetric line segments: $(0,1)$ connecting $(\\mathrm{e}^{-\\mu}\/2, 1\/2)$ and $(1\/2,\\mathrm{e}^{-\\mu}\/2)$ connecting $(1,0)$. Then $(\\mathrm{e}^{-\\mu}\/2, 1\/2)$ is connected to $(1\/2,\\mathrm{e}^{-\\mu}\/2)$ by the reciprocal function. It is easy to check that this function is $C^1$, i.e. has continuous derivative.\n\\begin{proof}[Proof of \\Cref{lem:lap}]\nLet $F$ be the cdf of $\\mathrm{Lap}(0,1)$. By \\Cref{prop:logconcave},\n$$\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)(\\alpha) = F\\big(F^{-1}(1-\\alpha)-\\mu\\big).$$\nEasy calculation yields\n\\begin{align*}\n\tF(x)=\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\\mathrm{e}^{x}\/2, \t\t& x \\leqslant 0, \\\\\n\t1-\\mathrm{e}^{-x}\/2, & x > 0.\n\t\\end{array}\n\t\\right.\n\\end{align*}\nSo we must expect to divide into several categories. We will refer to the above two expressions as negative and positive regimes.\n\nWhen $\\alpha>1\/2$, we are in negative regime. Solving $\\mathrm{e}^{x}\/2 = 1-\\alpha$ gives us $F^{-1}(1-\\alpha) = \\log 2(1-\\alpha)<0$. An additional $-\\mu$ keeps us in negative regime, so\n\\[F\\big(F^{-1}(1-\\alpha)-\\mu\\big) = \\exp\\big(F^{-1}(1-\\alpha)-\\mu\\big)\/2 = \\mathrm{e}^{\\log 2(1-\\alpha)-\\mu}\/2 = \\mathrm{e}^{-\\mu}(1-\\alpha).\\]\nWhen $\\alpha\\leqslant1\/2$, solving $1-\\mathrm{e}^{-x}\/2 = 1-\\alpha$ gives us $F^{-1}(1-\\alpha) = -\\log 2\\alpha\\geqslant0$. If $-\\log 2\\alpha - \\mu \\leqslant 0$, i.e. $\\mathrm{e}^{-\\mu}\/2\\leqslant \\alpha$, we are in negative regime and\n\\[F\\big(F^{-1}(1-\\alpha)-\\mu\\big) = \\mathrm{e}^{-\\log 2\\alpha-\\mu}\/2 = \\mathrm{e}^{-\\mu}\/4\\alpha.\\]\nIf $-\\log 2\\alpha - \\mu > 0$, i.e. $\\alpha<\\mathrm{e}^{-\\mu}\/2$, we are in positive regime and\n\\[F\\big(F^{-1}(1-\\alpha)-\\mu\\big) = 1-\\mathrm{e}^{\\log 2\\alpha+\\mu}\/2 = 1-\\mathrm{e}^{\\mu}\\alpha.\\]\nThe proof is complete.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of \\Cref{prop:group_limit}]\nFor simplicity assume $\\mu=1$. All arguments carry over for general $\\mu$.\n\nLet $f_n = 1-f_{\\ep,0} = 1-f_{1\/n,0}$. Fix $x_0$ and let $x_{n,k} = f_n^{\\circ k}(x_0) = (1-f_{\\ep,0})^{\\circ k}(x_0)$. We are interested in showing\n\\[\\lim_{n\\to\\infty}1-x_{n,n} = \\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(1,1)\\big)(x_0).\\]\nFirst we make a general observation: the sequence $\\{x_{n,k}\\}$ is increasing in $k$ for any $n$. This is because $f_{\\ep,0}(x) \\geqslant 1-x$ and hence $f_n(x)\\geqslant x$.\n\n\\newcommand{\\frac{1}{1+\\mathrm{e}^{\\frac{1}{n}}}}{\\frac{1}{1+\\mathrm{e}^{\\frac{1}{n}}}}\nLet $\\theta_n = \\frac{1}{1+\\mathrm{e}^{\\frac{1}{n}}}$. By the expression of $f_{\\ep,0}$, we obtain the following two dynamics:\n\\begin{align*}\n\\begin{array}{rcll}\n\tf_n(x) &=& \\mathrm{e}^{\\frac{1}{n}}x, & \\text{ if } x\\leqslant \\theta_n,\\\\\n\t1-f_n(x) &=& \\mathrm{e}^{-\\frac{1}{n}}(1-x),& \\text{ if } x\\geqslant \\theta_n.\n\\end{array}\n\\end{align*}\nThe sequence $\\{x_{n,k}\\}$ evolves according to one of the two formula, potentially different for eack $k$. We will refer to $x\\leqslant \\theta_n$ case as \\textit{linear dynamics} and $x\\geqslant \\theta_n$ case as \\textit{flip linear regime} for evident reason.\nFor any $x_0$ and $n$, since $\\{x_{n,k}\\}$ is increasing in $k$, there exists a moment such that linear dynamics governs before and flip linear dynamics governs after. Extreme cases are one of the dynamics governs from $k=0$ to $n$.\nWe divide the analysis into three cases depending on the initial location $x_0$:\n\\begin{itemize}\n\t\\item[(a)] $x_0< \\frac{1}{2\\mathrm{e}}$. In this case, for large enough $n$, the linear dynamics governs all the time. To see this, notice that $\\theta_n$ increases to $\\frac{1}{2}$ as $n\\to\\infty$. So for large enough $n$, $x_0< \\frac{1}{\\mathrm{e}}\\cdot \\theta_n$. It's easy to see that $x_{n,k}$ never exceeds $\\theta_n$. Hence $x_{n,n} = \\mathrm{e} x_0$.\n\t\\item[(b)] $x_0\\geqslant\\frac{1}{2} = \\sup_n \\theta_n$. $x_{n,k}$ is born above threshold and remains above forever. Flip linear dynamics governs all the $n$ steps, so $1-x_{n,n} =\\mathrm{e}^{-1}(1-x_0)$.\n\t\\item[(c)] $\\frac{1}{2e}\\leqslant x < \\frac{1}{2}$. Let $t$ be the time of dynamics change. More precisely,\n\t\\begin{equation}\\label{eq:group_threshold}\n\t\tt-1 = \\max\\{k:\\mathrm{e}^{\\frac{k}{n}}x\\leqslant \\theta_n.\\}\n\t\\end{equation}\n\tand\n\t\\[x_{n,t} = \\mathrm{e}^{\\frac{1}{n}}x_{n,t-1}=\\mathrm{e}^{\\frac{t}{n}}x_0,\\quad 1-x_{n,n} = \\mathrm{e}^{-\\frac{n-t}{n}}(1-x_{n,t}).\\]\n\tTaking $n\\to\\infty$ in \\eqref{eq:group_threshold} (using $\\liminf$ and $\\limsup$ when necessary), we know $\\mathrm{e}^{\\frac{t}{n}}\\to\\frac{1}{2x_0}$. So\n\t\\[\\lim_{n\\to\\infty}1-x_{n,n} = \\lim_{n\\to\\infty}\\mathrm{e}^{\\frac{t}{n}-1}(1-x_{n,t}) = \\lim_{n\\to\\infty}\\mathrm{e}^{\\frac{t}{n}-1}(1-\\mathrm{e}^{\\frac{t}{n}}x_0) = \\mathrm{e}^{-1}\\cdot\\frac{1}{2x_0}(1-\\frac{1}{2}) = \\mathrm{e}^{-1}\\cdot\\frac{1}{4x_0}.\\]\n\\end{itemize}\nCollecting all three cases, we have\n\\begin{align*}\n\t\\lim_{n\\to\\infty}1-x_{n,n} =\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\t1-\\mathrm{e} x_0, \t\t& x_0< \\frac{1}{2\\mathrm{e}}, \\\\\n\t\t\\mathrm{e}^{-1}\\cdot\\frac{1}{4x_0}, & \\mathrm{e}^{-\\mu}\/2\\leqslant\\alpha\\leqslant1\/2,\\\\\n\t\t\\mathrm{e}^{-1}(1-x_0), & x_0\\geqslant\\frac{1}{2}.\n\t\\end{array}\n\t\\right.\n\\end{align*}\nBy \\Cref{lem:lap}, this agrees with $\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(1,1)\\big)$.\nUniform convergence comes for free for trade-off functions once we have pointwise convergence. This is a direct consequence of \\Cref{lem:uniform} below, which will be used multiple times in this paper.\n\\end{proof}\n\t\\begin{lemma} \\label{lem:uniform}\n\t\tLet $f_n:[a,b]\\to\\R$ be a sequence of non-increasing functions. If $f_n$ has pointwise limit $f:[a,b]\\to\\R$ where $f$ is continuous on $[a,b]$, then the limit is uniform.\n\t\\end{lemma}\n\n\tThis is an easy variant of P\\'{o}lya's theorem (\\cite{polya1920zentralen}. See also Theorem 2.6.1 in \\cite{lehmann2004elements}). For completeness, we provide a proof.\n\t\\begin{proof}[Proof of \\Cref{lem:uniform}]\n\tWe are going to show that for every $\\ep>0$, there exists $N$ such that\n\t$$|f_n(x)-f(x)|< \\ep, \\quad\\forall x\\in[a,b], \\forall n\\geqslant N.$$\n\tSince $f$ is continuous on a closed interval, it is uniformly continuous. So for a fixed $\\ep>0$, we can find $\\delta>0$ such that whenever $x,y\\in[a,b]$ satisfies $|x-y|<\\delta$, we have $|f(x)-f(y)|<\\ep\/2$.\\\\\n\tThen we can divide $[a,b]$ into small intervals $a=x_0<x_1<\\cdots<x_{m-1}<x_m=b$ such that each interval is shorter than $\\delta$. For these $m+1$ points we can find $N$ such that\n\t\\begin{equation}\\label{eq:lsy}\n\t\t|f_n(x_i)-f(x_i)|< \\frac{\\ep}{2}, \\quad\\forall 0\\leqslant i\\leqslant m, \\forall n\\geqslant N\n\t\\end{equation}\n\tWe claim this $N$ works for our purpose. For any $x\\in[a,b]$, there exists a sub-interval that contains it, namely $[x_i,x_{i+1}]$. By monotonicity of $f_n$ we have\n\t\\begin{equation}\\label{eq:liu}\n\t\tf_n(x_{i+1})-f(x)\\leqslant f_n(x)-f(x)\\leqslant f_n(x_i)-f(x).\n\t\\end{equation}\n\tNow for $n\\geqslant N$,\n\t\\begin{align*}\n\t\tf_n(x_i)-f(x) &= [f_n(x_i)-f(x_i)]+[f(x_i)-f(x)]\\\\\n\t\t\t\t\t&< \\frac{\\ep}{2} + [f(x_i)-f(x)] &&\\text{(By \\eqref{eq:lsy})}\\\\\n\t\t\t\t\t&<\\frac{\\ep}{2} +\\frac{\\ep}{2} =\\ep. &&(\\text{uniform continuity of $g$})\n\t\\end{align*}\n\tThis shows the right hand side of \\eqref{eq:liu} is less than $\\ep$. A similar argument for the left hand side yields\n\t$$|f_n(x)-f(x)|<\\ep,$$\n\twhich justifies the choice of $N$.\n\t\\end{proof}\n\n\n\n\\section{Omitted Proofs in \\Cref{sec:application_in_sgd}}\n\\label{app:SGD}\nOur first goal is to prove\n\\mixturerep*\nFirst we point out that the functional $\\chi^2_+$ is computing a variant of $\\chi^2$-divergence. Recall that $\\chi^2$-divergence is an $F$-divergence with $F(t) = (t-1)^2$. We define $\\chi^2_+$-divergence to be the $F$-divergence with $F(t) = (t-1)^2_+ =\n\t\\left\\{\n\t\\begin{array}{ll}\n\t\t0, &t\\leqslant1,\\\\\n\t\t(t-1)^2, & t>1.\n\t\\end{array}\n\t\\right.$\nAs in \\Cref{app:relation}, let $z_f = \\inf\\{x\\in[0,1]:f(x)=0\\}$ be the first zero of $f$. \n\\begin{proposition} \\label{prop:chiplus}\n\tFor a pair of distributions $P$ and $Q$ such that $T(P,Q)=f$ is a symmetric trade-off function with $f(0)=1$,\n\t\\[\\chi^2_+(P\\|Q) = \\chi^2_+(f).\\]\n\\end{proposition}\n\\begin{proof}[Proof of \\Cref{prop:chiplus}]\n\tBy \\Cref{prop:fdiv}, when $f=T(P,Q)$, $\\chi^2_+(P\\|Q)$ can be computed via the following expression:\n$$\\chi^2_+(P\\|Q)=\\int_0^{z_f} \\big({\\big|f'(x)\\big|}^{-1}-1\\big)_+^2\\cdot \\big|f'(x)\\big| \\diff x + F(0)\\cdot(1-f(0))+\\tau_F\\cdot (1-z_f)$$\nwhere $F(0) = \\lim_{p\\to 0^+} {F(t)} = 0, \\tau_F=\\lim_{p\\to+\\infty} \\frac{F(t)}{t} = +\\infty$. Since we assume $f$ is symmetric, $z_f= f(0)=1$. This also implies that $f^{-1}$ is the ordinary function inverse, i.e. $f(f(x))=x$. \nLet $y = f^{-1}(x) = f(x)$. Then $\\diff y = f'(x) \\diff x$. On the other hand, $x=f(y),\\diff x = f'(y)\\diff y$. $x=1$ corresponds to $y=0$ and $x=0$ corresponds to $y=1$, so\n\\begin{align*}\n\t\\chi^2_+(P\\|Q)\n\t&=\\int_0^{1} \\big({\\big|f'(x)\\big|}^{-1}-1\\big)_+^2\\cdot \\big|f'(x)\\big| \\diff x\\\\\n\t&=\\int_0^{1} \\Big({\\Big|\\frac{\\diff y}{\\diff x}\\Big|}^{-1}-1\\Big)_+^2\\cdot \\big|f'(x)\\big| \\diff x\\\\\n\t&=-\\int_1^{0} \\big({\\big|f'(y)\\big|}-1\\big)_+^2\\diff y\\\\\n\t&=\\int_0^{1} \\big({\\big|f'(y)\\big|}-1\\big)_+^2\\diff y\\\\\n\t&=\\chi^2_+(f).\n\\end{align*}\n\\end{proof}\nWe need some more calculation tools to prove \\Cref{thm:mixtureSGD}.\n\\begin{lemma}\\label{lem:functionals2}\n\tLet $f\\in\\T^S$ with $f(0)=1$ and $x^*$ be its unique fixed point. Then\n\t\\begin{align*}\n\t\t\\chi^2_+(f)&= \\int_0^{x^*}(f'(x)+1)^2\\diff x\\\\\n\t\t\\mathrm{kl}(f) &= \\int_0^{x^*}\\big(|f'(x)|-1\\big)\\log |f'(x)|\\diff x \\\\\n\t\n\t\t\\kappa_2(f)&= \\int_0^{x^*}\\big(|f'(x)|+1\\big)\\big(\\log |f'(x)|\\big)^2\\diff x \\\\\n\t\t\\bar{\\kappa}_3(f)&=\\int_0^{x^*}\\big|\\log |f'(x)|+\\mathrm{kl}(f)\\big|^3+|f'(x)|\\cdot\\big|\\log |f'(x)|-\\mathrm{kl}(f)\\big|^3\\diff x\\\\\n\t\t\\kappa_3(f)&=\\int_0^{x^*}\\big(|f'(x)|+1\\big)\\big(\\log |f'(x)|\\big)^3\\diff x.\n\t\\end{align*}\n\\end{lemma}\n\n\\begin{proof}[Proof of \\Cref{lem:functionals2}]\n\tFirst we observe that $f'(x)\\leqslant -1$ for $x\\leqslant x^*$ and $f'(x)\\geqslant -1$ for $x\\geqslant x^*$. This means the integrand involved in $\\chi^2_+$ is 0 in $[x^*,1]$ and hence proves the first identity.\n\n\tThe rest of the proof is entirely based on a trick we used above.\n\tLet $y = f^{-1}(x) = f(x)$. Then $x=f(y),\\diff x = f'(y)\\diff y$. Since $x^*$ is the fixed point of $f$, $x=x^*$ corresponds to $y=x^*$. $x=1$ corresponds to $y=0$ and $x=0$ corresponds to $y=1$.\n\t\\begin{align*}\n\t\t-\\int_{x^*}^1\\log |f'(x)|\\diff x\n\t\t&=\\int_{x^*}^1\\log |f'(x)|^{-1}\\diff x\\\\\n\t\t&=\\int_{x^*}^0\\log \\Big|\\frac{\\diff x}{\\diff y}\\Big| \\cdot f'(y)\\diff y\\\\\n\t\t&=\\int^{x^*}_0\\log |f'(y)| \\cdot |f'(y)|\\diff y.\n\t\\end{align*}\n\tSo\n\t\\begin{align*}\n\t\t\\mathrm{kl}(f) &= -\\int_0^{1}\\log |f'(x)|\\diff x \\\\\n\t\t&=-\\int^{x^*}_0\\log |f'(x)|\\diff x-\\int_{x^*}^1\\log |f'(x)|\\diff x\\\\\n\t\t&=-\\int^{x^*}_0\\log |f'(x)|\\diff x+\\int^{x^*}_0\\log |f'(x)| \\cdot |f'(x)|\\diff x\\\\\n\t\t&= \\int_0^{x^*}\\big(|f'(x)|-1\\big)\\log |f'(x)|\\diff x.\n\t\\end{align*}\n\tThe rest of identities can be proved in exactly the same way.\n\\end{proof}\n\n\n\\begin{lemma} \\label{lem:CpMoments}\n\tSuppose $f\\in\\T^S$ and $f(0)=1$. $x^*$ is its unique fixed point. Let $g(x) = -f'(x)-1 = |f'(x)|-1$. Then\n\t\\begin{align*}\n\t\t\\mathrm{kl}(C_p(f)) &= p\\int_0^{x^*}g(x)\\log \\big(1+pg(x)\\big)\\diff x \\\\\n\t\t\\kappa_2(C_p(f))&= \\int_0^{x^*}\\big(2+pg(x)\\big)\\big[\\log \\big(1+pg(x)\\big)\\big]^2\\diff x \\\\\n\t\t\\kappa_3(C_p(f))&=\\int_0^{x^*}\\big(2+pg(x)\\big)\\big[\\log \\big(1+pg(x)\\big)\\big]^3\\diff x.\n\t\\end{align*}\n\\end{lemma}\n\\begin{proof}[Proof of \\Cref{lem:CpMoments}]\n\tWe prove for kl and the rest are similar. Let $x_p^*$ be the fixed point of $C_p(f)$. By \\Cref{lem:functionals2},\n\t\\[\n\t\t\\mathrm{kl}(C_p(f)) = \\int_0^{x^*_p}\\big(|C_p(f)'(x)|-1\\big)\\log |C_p(f)'(x)|\\diff x.\n\t\\]\n\tFrom the expression of $C_p(f)$ \\eqref{eq:Cp_expression} we know $\\log |C_p(f)'(x)|=0$ in the interval $[x^*,x^*_p]$, and $C_p(f) = f_p = pf+(1-p)\\Id$ in the interval $[0,x^*]$. So\n\t\\begin{align*}\n\t\t\\mathrm{kl}(C_p(f)) \n\t\t&= \\int_0^{x^*}\\big(|f_p'(x)|-1\\big)\\log |f_p'(x)|\\diff x.\n\t\\end{align*}\n\tIn the interval $[0,x^*]$, $g(x)=|f'(x)|-1\\geqslant 0$. $f_p'(x) = pf'(x)+(1-p)(-1) = p(f'(x)+1)-1 = -pg(x)-1$, so $|f_p'(x)| = pg(x)+1$. When plugged in to the expression above, we have\n\t\\[\n\t\\mathrm{kl}(C_p(f)) = p\\int_0^{x^*}g(x)\\log \\big(1+pg(x)\\big)\\diff x.\n\t\\]\n\\end{proof}\n\\begin{proof}[Proof of \\Cref{thm:mixtureSGD}]\n\tIt suffices to compute the limits in \\Cref{thm:CLT}, namely\n\t$$T\\cdot \\mathrm{kl}(C_p(f)), \\,\\,T\\cdot \\kappa_2(C_p(f)) \\text{ and } T\\cdot \\kappa_3(C_p(f)).$$\n\tSince $T\\sim p^{-2}$, we can consider $p^{-2}\\mathrm{kl}(C_p(f))$ and so on.\n\n\tAs in \\Cref{lem:CpMoments}, let $x^*$ be the unique fixed point of $f$ and $g(x) = -f'(x)-1 = |f'(x)|-1$. Note that $g(x)\\geqslant 0$ for $x\\in[0,x^*]$. The assumption expressed in terms of $g$ is simply\n\t$$\\int_0^1 g(x)^4\\diff x<+\\infty.$$\n\tIn particular, it implies $g(x)^k$ are integrable in $[0,x^*]$ for $k=2,3,4$. In addition, $\\chi^2_+(f) = \\int_0^{x^*}g(x)^2\\diff x$ by \\Cref{lem:functionals2}.\n\n\n\n\tFor the functional $\\mathrm{kl}$, by \\Cref{lem:CpMoments},\n\t\\begin{align}\n\t\t\\lim_{p\\to0^+}\\frac{1}{p^2}\\,\\mathrm{kl}(C_p(f))\n\t\t&= \\lim_{p\\to0^+}\\int_0^{x^*}g(x)\\cdot \\frac{1}{p}\\log \\big(1+pg(x)\\big)\\diff x\\tag{$*$}\\label{eq:sgd1}\\\\\n\t\t&= \\int_0^{x^*}g(x)\\cdot \\lim_{p\\to0^+}\\frac{1}{p}\\log \\big(1+pg(x)\\big)\\diff x\\nonumber\\\\\n\t\t&=\\int_0^{x^*}g(x)^2\\diff x=\\chi^2_+(f)\\nonumber\n\t\\end{align}\n\tChanging the order of the limit and the integral in \\eqref{eq:sgd1} is approved by {dominated convergence theorem}. To see this, notice that $\\log(1+x)\\leqslant x$.\n\n\n\n\n\n\n\n\tThe integrand in \\eqref{eq:sgd1} satisfies\n\t\\begin{align*}\n\t\t0\\leqslant g(x)\\cdot \\frac{1}{p}\\log \\big(1+pg(x)\\big)\\leqslant g(x)^2.\n\t\\end{align*}\n\tWe already argued that $g(x)^2$ is integrable, so it works as a dominating function and the limit is justified. When $p\\sqrt{T}\\to p_0$, we have\n\t\\[T\\cdot \\mathrm{kl}(C_p(f))\\to p_0^2\\cdot\\chi^2_+(f).\\]\n\tSo the constant $K$ in \\Cref{thm:CLT} is $p_0^2\\cdot\\chi^2_+(f)$.\n\n\tFor the functional $\\kappa_2$ we have\n\t\\begin{align*}\n\t\t\\frac{1}{p^2}\\kappa_2(C_p(f)) &= \\int_0^{x^*}\\big(2+pg(x)\\big)\\Big[\\frac{1}{p}\\log \\big(1+pg(x)\\big)\\Big]^2\\diff x.\n\t\\end{align*}\n\tBy a similar dominating function argument,\n\t\\begin{align*}\n\t\t\\lim_{p\\to0^+}\\frac{1}{p^2}\\,\\kappa_2(C_p(f)) = 2\\int_0^{x^*}g(x)^2\\diff x=2\\chi^2_+(f).\n\t\\end{align*}\n\tAdding in the limit $p\\sqrt{T}\\to p_0$, we know $s^2$ in \\Cref{thm:CLT} is $2p_0^2\\cdot\\chi^2_+(f)$. Once again, we have $s^2 = 2K$.\n\n\tThe same argument involving $g(x)^4$ applies to the functional $\\kappa_3$ and yields\n\t$$\\lim_{p\\to0^+}\\frac{1}{p^3}\\,\\kappa_3(C_p(f)) = 2\\int_0^{x^*}g(x)^3\\diff x.$$\n\tNote the different power in $p$ in the denominator. It means $\\kappa_3(C_p(f)) = o(p^2)$ and hence $T\\cdot \\kappa_3(C_p(f))\\to 0$ when $p\\sqrt{T}\\to p_0$.\n\n\tHence all the limits in \\Cref{thm:CLT} check and we have a $G_\\mu$ limit where\n\t\\[\\mu = 2K\/s = s = \\sqrt{2p_0^2\\cdot\\chi^2_+(f)} = p_0\\cdot\\sqrt{2\\chi^2_+(f)}.\\]\n\tThis completes the proof.\n\\end{proof}\n\\chirep*\n\\begin{proof}[Proof of \\Cref{lem:chi2GDP}]\n\tWe use \\Cref{prop:chiplus} as the tool. Obviously $P=\\N(\\mu,1)$ and $Q=\\N(0,1)$ satisfy the conditions there. So it suffices to compute $\\chi^2_+(\\N(\\mu,1)\\|\\N(0,1))$. Recall that $\\chi^2_+$ is the $F$-divergence with $F(t) = (t-1)_+^2$, so $\\chi^2_+(P\\|Q) = \\E_Q\\big[( \\frac{P}{Q}-1)_+^2\\big]$. Let $\\varphi$ and $\\Phi$ be the density function and cdf of the standard normal. We have\n\t\\begin{align*}\n\t\t\\chi^2_+(G_\\mu) &= \\chi^2_+(\\N(\\mu,1)\\|\\N(0,1))\\\\\n\t\t&= \\E_{x\\sim \\N(0,1)}\\Big[\\Big(\\frac{\\varphi(x-\\mu)}{\\varphi(x)}-1\\Big)_+^2\\Big]\\\\\n\t\t&= \\int_{\\mu\/2}^{+\\infty}\\Big(\\frac{\\varphi(x-\\mu)}{\\varphi(x)}-1\\Big)^2\\cdot \\varphi(x)\\diff x\\\\\n\t\t&= \\int_{\\mu\/2}^{+\\infty}\\Big(\\frac{\\varphi(x-\\mu)}{\\varphi(x)}\\Big)^2\\cdot \\varphi(x)\\diff x -2 \\int_{\\mu\/2}^{+\\infty}\\varphi(x-\\mu)\\diff x+\\int_{\\mu\/2}^{+\\infty}\\varphi(x)\\diff x\\\\\n\t\t&= \\underbrace{\\int_{\\mu\/2}^{+\\infty}e^{2\\mu x-\\mu^2}\\cdot \\varphi(x)\\diff x}_{I}-2(1-\\Phi(\\mu\/2))+\\Phi(-\\mu\/2)\\\\\n\t\t&=I+3\\Phi(-\\mu\/2)-2.\n\t\\end{align*}\n\tFor the integral $I$,\n\t\\begin{align*}\n\t\tI &= \\int_{\\mu\/2}^{+\\infty}e^{2\\mu x-\\mu^2}\\cdot \\varphi(x)\\diff x\\\\\n\t\t&= \\int_{\\mu\/2}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}}\\cdot e^{2\\mu x-\\mu^2-x^2\/2}\\diff x\\\\\n\t\t&= \\int_{\\mu\/2}^{+\\infty}\\frac{1}{\\sqrt{2\\pi}}\\cdot e^{-(x-2\\mu)^2\/2}\\cdot e^{\\mu^2}\\diff x\\\\\n\t\t&= e^{\\mu^2}\\cdot P[\\N(2\\mu,1)\\geqslant \\mu\/2]\\\\\n\t\t&= e^{\\mu^2}\\cdot\\Phi(3\\mu\/2)\n\t\\end{align*}\n\tThis completes the proof.\n\\end{proof}\n\\SGDlimitrep*\n\\begin{proof}[Proof of \\Cref{thm:SGDlimit}]\n\tCombining \\Cref{thm:sgdcompo,thm:mixtureSGD} and \\Cref{lem:chi2GDP}, it suffices to check $\\int_0^1(f'(x)+1)^4\\diff x<+\\infty$ when $f(x) = G_a(x) = \\Phi(\\Phi^{-1}(1-x)-a)$. Let $y = \\Phi^{-1}(1-x)$. We have $\\varphi(y)\\diff y = -\\diff x$. Hence\n\t\\[G_a'(x) = \\varphi(y-a) \\cdot \\frac{\\diff y}{\\diff x} = -\\frac{\\varphi(y-a)}{\\varphi(y)} = -\\mathrm{e}^{ay-\\frac{a^2}{2}}.\\]\n\tThe integral is\n\t\\begin{align*}\n\t\t\\int_0^1(G_a'(x)+1)^4\\diff x\n\t\t&=\\int_{-\\infty}^{+\\infty}(-\\mathrm{e}^{ay-\\frac{a^2}{2}}+1)^4\\varphi(y)\\diff y,\n\t\\end{align*}\n\twhich is just a linear combination of moment generating functions of the standard normal and hence finite.\n\\end{proof}\n\\functionalGmurep*\n\\begin{proof}[Proof of \\Cref{lem:functionalGmu}]\n\tWe will use \\Cref{lem:CpMoments}. It's easy to show the fixed point of $G_\\mu$ is $x^* = \\Phi(-\\mu\/2)$. So\n\t\\begin{align*}\n\t\t\\mathrm{kl}\\big(C_p(G_\\mu)\\big) &=\n\t\tp\\int_0^{\\Phi(-\\mu\/2)}\\big(-G_\\mu'(x)-1\\big)\\log \\big(1+p(-G_\\mu'(x)-1)\\big)\\diff x\n\t\\end{align*}\n\tUsing the same change of variable $y = \\Phi^{-1}(1-x) = -\\Phi^{-1}(x)$, we have\n\t\\begin{align*}\n\t\t\\mathrm{kl}\\big(C_p(G_\\mu)\\big) &=\n\t\tp\\int^{+\\infty}_{\\mu\/2}\\Big(\\frac{\\varphi(y-\\mu)}{\\varphi(y)}-1\\Big)\\log \\Big(1+p\\Big(\\frac{\\varphi(y-\\mu)}{\\varphi(y)}-1\\Big)\\Big)\\varphi(y)\\diff y\\\\\n\t\t&=p\\int_{\\mu\/2}^{+\\infty} Z(y)\\cdot\\big(\\varphi(y-\\mu)-\\varphi(y)\\big)\\diff y.\n\t\\end{align*}\n\tThe rest can be proved similarly.\n\\end{proof}\n\\SGDBerryrep*\n\\begin{proof}[Proof of \\Cref{thm:SGDBerry}]\n\tFollows from plugging in the expressions above into \\Cref{thm:Berry}.\n\\end{proof}\n\\section{Conversion from $f$-DP to Divergence Based DP}\n\\label{app:relation}\nAs the title suggests, the central question of this section is the conversion from $f$-DP to divergence based DP. It boils down to the conversion from trade-off functions to various divergences. We first introduce the most general tool, and then give explicit formula for a large class of divergences, including {R\\'enyi} divergence. At the end we prove the claim we made in \\Cref{sec:conn-with-blackw} that privacy notion based on R\\'enyi divergence does not induce a strictly larger order than the Blackwell order.\n\n\nSuppose we have a ``divergence'' $D(\\cdot\\|\\cdot)$, which takes in a pair of probability distributions on a common measurable space and outputs a number. We say $D$ satisfies data processing inequality if $D\\big(\\mathrm{Proc}(P)\\|\\mathrm{Proc}(Q)\\big)\\geqslant D(P\\|Q)$ for any post-processing Proc.\n\\begin{proposition} \\label{thm:functional}\n\tIf $D(\\cdot\\|\\cdot)$ satisfies data processing inequality, then there exists a functional $l_D:\\T\\to\\R$ that computes $D$ through the trade-off function:\n\t$$D(P\\|Q) = l_D\\big(\\F(P,Q)\\big).$$\n\\end{proposition}\n\\begin{proof}\n\tIt's almost immediate from the following\n\t\\begin{lemma} \\label{lem:onlyonce}\n\tIf $\\F(P',Q')\\geqslant\\F(P,Q)$, then $D(P'\\|Q') \\leqslant D(P\\|Q)$. In particular, $\\F(P,Q)=\\F(P',Q')$ implies $D(P\\|Q) = D(P'\\|Q')$.\n\t\\end{lemma}\n\tTo see why the lemma holds, notice by Blackwell's theorem, $\\F(P',Q')\\geqslant\\F(P,Q) $ implies that there is a Proc such that $P'=\\mathrm{Proc}(P),Q'=\\mathrm{Proc}(Q)$, and by data processing inequality, $D(P'\\|Q') \\leqslant D(P\\|Q)$.\n\n\tThe lemma implies the existence of $l_D$ because we can define $l_D(f) = D(P\\|Q)$ through any pair $P,Q$ such that $T(P,Q)=f$. This definition is independent of the choice of $P$ and $Q$.\n\\end{proof}\nAn immediate corollary is\n\\begin{corollary}\\label{cor:wellbeing}\n\tIf two trade-off functions $f,g$ satisfy $f\\geqslant g$, then $l_D(f)\\le l_D(g)$.\n\\end{corollary}\n\\paragraph{Example: $F$-divergence}\nLet $P,Q$ be a pair of distributions with density $p$ and $q$ with respect to some common dominating measure. For a convex function $F:(0,+\\infty)\\to\\R$ such that $F(1) = 0$, the $F$-divergence $D_F(P\\|Q)$ is defined as (see \\cite{liese2006divergences})\n\\begin{align*}\n\tD_F(P\\|Q) &= \\int_{\\{pq>0\\}} F\\Big(\\frac{p}{q}\\Big) \\diff Q + F(0)Q[p=0]+\\tau_F P[q=0]\n\\end{align*}\nwhere $F(0) = \\lim_{t\\to 0^+} {F(t)}$ and $\\tau_F:=\\lim_{t\\to+\\infty} \\frac{F(t)}{t}$.\nWe further set the rules $F(0) \\cdot 0 = \\tau_F\\cdot 0=0$ even if $F(0)=+\\infty$ or $\\tau_F=+\\infty$.\n\\begin{proposition} \\label{prop:fdiv}\nLet $z_f = \\inf\\{x\\in[0,1]:f(x)=0\\}$ be the first zero of $f$. The functional $l_F:\\T\\to\\R$ that computes $F$-divergence has expression\n$$l_F(f) = \\int_0^{z_f} F\\big({\\big|f'(x)\\big|}^{-1}\\big)\\cdot \\big|f'(x)\\big| \\diff x + F(0)\\cdot(1-f(0))+\\tau_F\\cdot (1-z_f).$$\nIn particular, when $f\\in\\T^S$ and $f(0)=1$, we have\n\\begin{equation}\\label{eq:fdiv}\n\tl_F(f) = \\int_0^1 F\\big({\\big|f'(x)\\big|}^{-1}\\big)\\cdot \\big|f'(x)\\big| \\diff x.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}[Proof of \\Cref{prop:fdiv}]\n\tFor a given trade-off function $f$, in order to determine $l_F(f)$, it suffices to find $P,Q$ such that $f=T(P,Q)$ and then use the property $l_F(f) = D_F(P\\|Q)$. Such a pair is constructed in the proof of \\Cref{prop:trade-off}: $P$ is the uniform distribution on $[0,1]$ and $Q$ has density $|f'(1-x)|$ on $[0,1)$ and an atom at $1$ with $Q[\\{1\\}] = 1-f(0)$. When we set the dominating measure $\\mu$ to be Lebesgue in $[0,1)$ and have an atom at 1 with measure 1, the densities $p$ and $q$ have expressions\n\t\\[p(x) = \n\t\\left\\{\n\t\\begin{array}{ll}\n\t\t1, \t\t& x\\in[0,1), \\\\\n\t\t0, & x=1.\n\t\\end{array}\n\t\\right.\n\t\\quad\\text{ and }\\quad q(x) = \n\t\\left\\{\n\t\\begin{array}{ll}\n\t\t|f'(1-x)|, \t\t& x\\in[0,1), \\\\\n\t\t1-f(0), & x=1.\n\t\\end{array}\n\t\\right.\\]\n\tReaders should keep in mind that the value at 1 matters because the base measure $\\mu$ has an atom there. For a trade-off function $f$, its derivative $f'(x)$ never vanishes before $f$ hits zero, i.e. $f'(x)>0$ for $x<z_f$ and $f'(x)=0$ for $x\\ge z_f$. Equivalently, $\\{q>0\\} = (1-z_f,1]$ and $\\{q=0\\} = [0,1-z_f]$.\n\tSo\n\t\\begin{align*}\n\t\tD_F(P\\|Q)\n\t\t&= \\int_{\\{pq>0\\}} F\\Big(\\frac{p}{q}\\Big) \\diff Q + F(0)Q[p=0]+\\tau_F P[q=0]\\\\\n\t\t&= \\int_{1-z_f}^1 F(|f'(1-x)|^{-1}) \\cdot |f'(1-x)| \\diff x+ F(0)\\cdot(1-f(0)) + \\tau_F\\cdot (1-z_f)\\\\\n\t\t&=\\int^{z_f}_0 F(|f'(x)|^{-1}) \\cdot |f'(x)| \\diff x+ F(0)\\cdot(1-f(0)) + \\tau_F\\cdot (1-z_f).\n\t\\end{align*}\n\tStarting from the second line, the integral is Lebesgue integral. Now the proof is complete.\n\\end{proof}\nBecause of the generality of $F$-divergence, \\Cref{eq:fdiv} has broad applications. Many important divergences can be computed via a simple formula. Below are some of the examples.\n\\begin{itemize}\n\t\\item \\textbf{Total variation distance} corresponds to $F(t) = \\frac{1}{2}|t-1|$. Easy calculation yields\n\t\\[l_{\\mathrm{TV}}(f) = \\frac{1}{2}\\int_0^1 \\big|1+f'(x)\\big|\\diff x.\\]\n\t\\item \\textbf{KL divergence} corresponds to $F(t) = t\\log t$. We have\n\t\\[l_{\\mathrm{KL}}(f) =-\\int_0^1 \\log\\big|f'(x)\\big|\\diff x.\n\n\n\n\n\n\n\t\\]\n\tThis functional plays an important role in our central limit theorem. We call it $\\mathrm{kl}(f)$ there.\n\n\n\n\n\n\t\\item \\textbf{{Power} divergence} of order $\\alpha$ corresponds to $F_\\alpha(t) =\\frac{t^\\alpha-\\alpha(t-1)-1}{\\alpha(\\alpha-1)}$. The corresponding functional is\n\t\\begin{align*}\n\t\tl_{F_\\alpha}(f) = \n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\frac{1}{\\alpha(\\alpha-1)}\\Big(\\int_0^1 \\big|f'(x)\\big|^{1-\\alpha}\\diff x - 1\\Big), \t\t& z_f = 1, \\\\\n\t\t\t+\\infty, & z_f<1.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{align*}\n\t\\item \\textbf{{R\\'enyi} divergence} of order $\\alpha$ is defined as\n\t\\[D_\\alpha(P\\|Q) = \\tfrac{1}{\\alpha-1}\\log \\big(\\E_P(\\tfrac{p}{q})^{\\alpha-1}\\big)=\\tfrac{1}{\\alpha-1}\\log\\int p^\\alpha q^{1-\\alpha}.\\]\n\tIt is related to power divergence of order $\\alpha$ by\n\t\\begin{equation}\\label{eq:renyipower}\n\t\tD_\\alpha(P\\|Q) = \\frac{1}{\\alpha-1}\\cdot \\log\\big(\\alpha(\\alpha-1)D_{F_\\alpha}(P\\|Q)+1\\big).\n\t\\end{equation}\n\tSo the corresponding functional, which we denote by $l_\\alpha^{\\text{R{\\'e}nyi}}$, has expression\n\t\\begin{equation}\\label{eq:renyi_functional}\n\t\tl_\\alpha^{\\text{R{\\'e}nyi}}(f)=\n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\t\\frac{1}{\\alpha-1}\\log \\int_0^1 \\big|f'(x)\\big|^{1-\\alpha}\\diff x, \t\t& z_f = 1, \\\\\n\t\t\t+\\infty, & z_f<1.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\n\\end{itemize}\n\\begin{proof}[Proof of \\Cref{eq:renyipower}]\n\t\\begin{align*}\n\t\tD_{F_\\alpha}(P\\|Q) &= \\int q\\cdot F_\\alpha\\Big(\\frac{p}{q}\\Big)\\\\\n\t\t&= \\int q\\cdot \\frac{(\\frac{p}{q})^\\alpha - \\alpha(\\frac{p}{q}-1)-1}{\\alpha(\\alpha-1)}\\\\\n\t\t&=\\frac{1}{\\alpha(\\alpha-1)}\\cdot \\int p^\\alpha q^{1-\\alpha} + 0 -\\frac{1}{\\alpha(\\alpha-1)}\\\\\n\t\t&=\\frac{1}{\\alpha(\\alpha-1)}\\Big(e^{(\\alpha-1)D_\\alpha(P\\|Q)}-1\\Big).\n\t\\end{align*}\n\tSolving for $D_\\alpha(P\\|Q)$ yields \\eqref{eq:renyipower}.\n\\end{proof}\nIntroduced in \\cite{renyi}, a mechanism $M$ is said to be $(\\alpha, \\epsilon)$-{R\\'enyi} differentially private (RDP) if\nfor all neighboring pairs $S, S'$ it holds that\n\\begin{equation}\\label{eq:RDP}\nD_\\alpha(M(S) \\| M(S')) \\le \\epsilon,\n\\end{equation}\nA few other DP definitions, including zero concentrated differential privacy (zCDP) \\cite{concentrated2} and truncated concentrated differential privacy (tCDP) \\cite{tcdp}, are defined through imposing bounds in the form of \\eqref{eq:RDP} with certain collections of $\\alpha$. The following proposition provides the general conversion from $f$-DP to RDP via \\eqref{eq:renyi_functional}.\n\\begin{proposition} \\label{lem:}\n\tIf a mechanism is $f$-DP, then it is $\\big(\\alpha,l_\\alpha^{\\text{R{\\'e}nyi}}(f)\\big)$-RDP for any $\\alpha>1$.\n\\end{proposition}\n\nSpecializing to the most important subclass, we have a simple expression.\n\\begin{corollary}\n\tIf a mechanism is $\\mu$-GDP, then it is $(\\alpha,\\frac{1}{2}\\mu^2\\alpha)$-RDP for any $\\alpha>1$.\n\\end{corollary}\n\\begin{proof}\nBy the property of $l_\\alpha^{\\text{R{\\'e}nyi}}$, we know $l_\\alpha^{\\text{R{\\'e}nyi}}(G_\\mu) = D_\\alpha\\big(\\N(0,1)\\|\\N(\\mu,1)\\big)$.\n\tEasy calculation shows $D_\\alpha\\big(\\N(0,1)\\|\\N(\\mu,1)\\big) = \\frac{1}{2}\\mu^2\\alpha$. Readers can refer to Proposition 7 in \\cite{renyi} for a detailed derivation.\n\\end{proof}\n\n\nThe functional $l_D$ allows a consistent, easy conversion from an $f$-DP guarantee to all divergence based DP guarantees. The above conversion to RDP is, among all, the most useful example. On the other hand, conversion from divergence, either to trade-off function or to other divergences, often requires case by case analysis, sometimes significantly non-trivial. What's worse is that it is often hard to tell whether a given conversion between divergences is improvable or already lossless. For conversion between $F$-divergences, a systematic approach called joint range is developed in \\cite{harremoes2011pairs}, but it is still significantly more complicated than \\Cref{eq:fdiv}. On the other hand, \\Cref{thm:functional} means conversion from trade-off to divergence is lossless and unimprovable.\n\nThis fine-grainedness of trade-off function (see also \\Cref{sec:conn-with-blackw}) is somewhat expected: it summarizes the indistinguishability of a pair of distribution by a \\textit{function}, which is an infinite dimensional object. In contrast, divergences usually just summarize by a number, which is obviously less informative by a function.\n\nConnecting back to informativeness, we argue that, even when we consider $\\{D_\\alpha(P\\|Q):\\alpha>1\\}$ as an infinite dimensional object, it still does not induce Blackwell's order. In the language of \\Cref{sec:conn-with-blackw}, $\\mathrm{Ineq}({\\preceq_{\\textup{R\\'enyi}}})\\supsetneq \\mathrm{Ineq}({\\preceq_{\\mathrm{Blackwell}}})$. In other words, there are two pairs of distributions, one easier to distinguish than the other in the R\\'enyi sense, but not in the Blackwell sense.\n\n\nLet $P_\\ep$ and $Q_{\\ep}$ denote Bernoulli distributions with success probabilities $\\frac{\\mathrm{e}^\\ep}{1+\\mathrm{e}^\\ep}$ and $\\frac{1}{1+\\mathrm{e}^\\ep}$, respectively.\n\\begin{proposition}\\label{prop:renyi_fail}\nThere exists $\\ep>0$ such that the following two statements are both true:\t\n\\begin{enumerate}\n\\item[(a)]\nFor all $\\alpha>1$, $D_\\alpha(P_\\ep \\| Q_\\ep) \\leqslant D_\\alpha\\big(\\N(0,1) \\| \\N(\\ep,1)\\big)$;\n\\item[(b)] $\\mathrm{TV}(P_\\ep, Q_\\ep)> \\mathrm{TV}\\big(\\N(0,1), \\N(\\ep,1)\\big)$.\n\\end{enumerate}\n\\end{proposition}\nSurprisingly, although the whole collection of {R\\'enyi} divergences asserts that the pair $\\big(\\N(0,1) ,\\N(\\ep,1)\\big)$ is easier to distinguish than $(P_\\ep , Q_\\ep)$, one can nevertheless achieve smaller summed type I and type II errors when trying to distinguish $P_\\ep, Q_\\ep$.\nFact (a) equivalently says that $\\big(\\N(0,1) ,\\N(\\ep,1)\\big)\\preceq_{\\text{R\\'enyi}} (P_\\ep \\| Q_\\ep)$, while fact (b) excludes the possibility that $\\big(\\N(0,1) ,\\N(\\ep,1)\\big)\\preceq_{\\mathrm{Blackwell}} (P_\\ep , Q_\\ep)$, since otherwise data processing inequality of the total variation distance would imply $\\mathrm{TV}(P_\\ep, Q_\\ep)\\leqslant \\mathrm{TV}\\big(\\N(0,1), \\N(\\ep,1)\\big)$.\n\nWe point out that (a) in \\Cref{{prop:renyi_fail}} in fact holds for all $\\ep\\geqslant0$, which is proved in \\cite{concentrated2}, partially based on numerical evidence. Our proof is entirely analytical.\n\\begin{proof}[Proof of \\Cref{prop:renyi_fail}]\n\t\\begin{align*}\n\t\tD_\\alpha(P_\\ep\\|Q_\\ep)\n\t\t&= \\frac{1}{\\alpha-1}\\log (p^\\alpha q^{1-\\alpha}+q^\\alpha p^{1-\\alpha})\\\\\n\t\t&= \\frac{1}{\\alpha-1}\\log\\, \\frac{\\mathrm{e}^{\\ep\\alpha}+\\mathrm{e}^{\\ep(1-\\alpha)}}{1+\\mathrm{e}^\\ep}.\\\\\n\t\t&= \\frac{1}{\\alpha-1}\\log\\, \\frac{\\mathrm{e}^{\\ep(\\alpha-\\frac{1}{2})}+\\mathrm{e}^{\\ep(\\frac{1}{2}-\\alpha)}}{\\mathrm{e}^{-\\frac{\\ep}{2}}+\\mathrm{e}^{\\frac{\\ep}{2}}}\\\\\n\t\t&= \\frac{1}{\\alpha-1}\\log\\, \\frac{\\cosh \\ep(\\alpha-\\frac{1}{2})}{\\cosh \\frac{\\ep}{2}}\n\t\\end{align*}\n\tNow we claim that $\\cosh x \\cdot \\mathrm{e}^{-\\frac{1}{2}x^2}$ is monotone decreasing for $x\\geqslant0$. To see this, simply take the derivative\n\t$$\\big(\\cosh x \\cdot \\mathrm{e}^{-\\frac{1}{2}x^2}\\big)' = \\sinh x\\cdot \\mathrm{e}^{-\\frac{1}{2}x^2} + \\cosh x \\cdot (-x) \\cdot \\mathrm{e}^{-\\frac{1}{2}x^2} = (\\tanh x - x)\\cdot\\cosh x\\cdot \\mathrm{e}^{-\\frac{1}{2}x^2}.$$\n\tIt is easy to show $\\tanh x\\leqslant x$ for $x\\geqslant0$. Hence the derivative is always non-positive, which justifies the claimed monotonicity.\n\tSince $\\alpha>1,\\ep\\geqslant0$, we have $\\ep(\\alpha-\\frac{1}{2})\\geqslant\\frac{\\ep}{2}\\geqslant 0$. By the monotonicity,\n\t\\begin{align*}\n\t\t\\cosh \\ep(\\alpha-\\frac{1}{2}) \\cdot \\mathrm{e}^{-\\frac{1}{2}\\ep^2(\\alpha-\\frac{1}{2})^2} &\\leqslant \\cosh \\frac{\\ep}{2} \\cdot \\mathrm{e}^{-\\frac{1}{2}\\cdot (\\frac{\\ep}{2})^2}.\n\t\\end{align*}\n\tThat is,\n\t\\[\n\t\t\\frac{\\cosh \\ep(\\alpha-\\frac{1}{2})}{\\cosh \\frac{\\ep}{2}} \\leqslant \\mathrm{e}^{\\frac{1}{2}\\ep^2\\alpha(\\alpha-1)}.\n\t\\]\n\tSo for any $\\ep\\geqslant0$,\n\t\\begin{align*}\n\t\tD_\\alpha(P_\\ep\\|Q_\\ep) = \\frac{1}{\\alpha-1}\\cdot \\log\\, \\frac{\\cosh \\ep(\\alpha-\\frac{1}{2})}{\\cosh \\frac{\\ep}{2}} \\leqslant \\frac{1}{2}\\ep^2\\alpha = D_\\alpha\\big(\\N(0,1)\\|\\N(\\ep,1)\\big).\n\t\\end{align*}\n\tFor the second part, easy calculation and Taylor expansion yields\n\t\\begin{align*}\n\t\t\\mathrm{TV}(P_\\ep,Q_\\ep) &= \\frac{\\mathrm{e}^\\ep-1}{\\mathrm{e}^\\ep+1} = \\tanh \\frac{\\ep}{2} = \\frac{\\ep}{2} +o(\\ep^2),\\\\\n\t\t\\mathrm{TV}\\big(\\N(0,1),\\N(\\ep,1)\\big) &= 1-2\\Phi(-\\frac{\\ep}{2}) = \\int_{-\\frac{\\ep}{2}}^{\\frac{\\ep}{2}}\\varphi(x)\\diff x \\\\\n\t\t&= \\varphi(0)\\cdot\\ep+o(\\ep^2) = \\frac{1}{\\sqrt{2\\pi}}\\cdot\\ep+o(\\ep^2).\n\t\\end{align*}\n\tSince $\\frac{1}{\\sqrt{2\\pi}}<\\frac{1}{2}$, for small enough $\\ep$, $\\mathrm{TV}\\big(\\N(0,1),\\N(\\ep,1)\\big)< \\mathrm{TV}(P_\\ep,Q_\\ep)$.\n\\end{proof}\n\nIn summary, \\Cref{thm:functional} and \\Cref{prop:fdiv} provide general tools to losslessly convert a trade-off function to divergences and hence justifies the fine-grainedness of trade-off functions. This is complementary to the informativeness argument in \\Cref{sec:conn-with-blackw}.\n\n\n\\section{Proof of \\Cref{thm:fast}}\n\\label{app:fast}\nThis section is devoted to the proof of \\Cref{thm:fast}. \nSince we always assume $\\delta=0$, it is dropped from the subscript and we use $f_\\ep$ to denote $f_{\\ep,0}$. As in the proof of \\Cref{thm:Berry}, the first step is to express $f_\\ep^{\\otimes n}$ in the form\n$$1-f_\\ep^{\\otimes n}(\\alpha) = F_n\\big[x_n-F_n^{-1}(1-\\alpha)\\big]$$\nwith $F_n\\to\\Phi$ and $x_n\\to 1$. Then we show both convergences have rate $1\/n$. \n\\begin{proof}[Proof of \\Cref{thm:fast}]\n\tLet's find $F_n$ first. Fix $\\ep$ and let $p = \\tfrac{1}{1+e^\\ep}, q = 1-p = p\\cdot e^\\ep$. Recall that $f_\\ep^{\\otimes n} = T\\big(B(n,p),B(n,q)\\big)$ and we know that it is the linear interpolation of points given by binomial tails.\n\tThe main goal here is to avoid the linear interpolation.\n\t\n\tFor the simple hypothesis testing problem $B(n,p)$ vs $B(n,q)$, we know via Neyman-Pearson that every optimal rejection rule $\\phi$ must have the following form:\n\t$$\n\t\t\\phi(x)=\\left\\{\n\t\t\\begin{array}{ll}\n\t\t1, \t\t& \\text{if } x>k, \\\\\n\t\t0, \t\t& \\text{if } x<k, \\\\\n\t\t1-c, \t\t& \\text{if } x=k.\n\t\t\\end{array}\n\t\t\\right.\n\t$$\n\tIt rejects (i.e. decides that the sample comes from $B(n,q)$) if it sees something greater than $k$, accepts if it sees something smaller than $k$, and reject with probability $1-c$ if it sees $k$. Such tests are parameterized by $(k,c)$ where $k\\in\\{0,1,\\ldots, n\\}$ and $c\\in[0,1)$.\n\n\tThe corresponding type I and type II errors are denoted by $\\alpha_{(k,c)}$ and $\\beta_{(k,c)}$. \n\tLet $X\\sim B(n,p)$ and $Y\\sim U[0,1]$ be independent random variables. We have\n\t\\begin{align*}\n\t\t\\alpha_{(k,c)} &= \\E_{x\\sim B(n,p)}[\\phi(x)] = \\E[\\phi(X)]\\\\\n\t\t&= \\P[X>k] + (1-c)\\P[X=k] \\\\\n\t\t&= \\P[X>k] + \\P[Y>c]\\cdot \\P[X=k] \\\\\n\t\t&= \\P[X+Y>k+c]\\\\\n\t\t\\beta_{(k,c)}&= \\E_{x\\sim B(n,q)}[1-\\phi(x)]= \\E[1-\\phi(n-X)]\\\\\n\t\t&=\\P[n-X<k]+c\\cdot \\P[n-X = k]\\\\\n\t\t&=\\P[X>n-k]+\\P[Y>1-c]\\cdot \\P[X = n- k]\\\\\n\t\t&= \\P[X+Y>n+1-k-c]\n\t\\end{align*}\n\t$X+Y$ supports on $[0,n+1]$ and has a piecewise constant density. As a consequence, the cdf $F_{X+Y}$ is a bijection between $[0,n+1]$ and $[0,1]$. So for a fixed type I error $\\alpha\\in[0,1]$, the optimal testing rule $(k,c)$ is uniquely determined by the formula\n\t$$k+c = F_{X+Y}^{-1} (1-\\alpha).$$\n\tAnd we have for the trade-off function:\n\t$$1-f_\\ep^{\\otimes n}(\\alpha) = F_{X+Y}\\big( n+1 - F_{X+Y}^{-1} (1-\\alpha)\\big).$$\n\tNow we proceed to write $F_{X+Y}$ in a form that reveals its central limit behavior. First notice $\\E[X+Y] = np+\\tfrac{1}{2}, \\Var[X+Y] = \\Var[X] + \\Var[Y] = npq+\\tfrac{1}{12}$. For simplicity denote this variance by $\\sigma^2$. Let $F_n$ be the normalized cdf of $X+Y$, i.e.\n\t$$F_n(x) = P\\Big[\\tfrac{X+Y - \\E[X+Y]}{\\sqrt{\\Var[X+Y]}}\\leqslant x\\Big] = F_{X+Y}\\Big[np+\\tfrac{1}{2} + x\\sigma\\Big].$$\n\tSimple algebra yields\n\t\\begin{equation}\\label{eq:fast1}\n\t1-f_\\ep^{\\otimes n}(\\alpha) = F_n\\Big[\\tfrac{n(q-p)}{\\sigma}-F_n^{-1}(1-\\alpha)\\Big].\n\t\\end{equation}\n\tIt's easy to show that $\\tfrac{n(q-p)}{\\sigma}\\to 1$ and $F_n\\to\\Phi$ pointwise. However, we need to show that the convergence rates are both $1\/n$, which is technically involved, especially for the convergence of $F_n$. In view of this, we pack the conclusions into the following lemmas, and provide the proofs later:\n\t\\begin{lemma} \\label{lem:pqlimit}\n\tWith $\\ep = 1\/\\sqrt{n}$ and $p,q,\\sigma$ defined as above,\n\t\n\t\\[\\frac{n(q-p)}{\\sigma} = 1-\\frac{1}{8n}+o(n^{-1}).\\]\n\t\\end{lemma}\n\tAs a consequence, there exists $C>0$ such that\n\t\\begin{equation}\\label{eq:fast2}\n\t\\big|\\tfrac{n(q-p)}{\\sigma}-1\\big|\\leqslant\\tfrac{C}{n}.\n\t\\end{equation}\n\t\\begin{lemma}\\label{prop:ch.f.}\n\t\tThere is a positive number $C$ such that $|F_n(x)-\\Phi(x)|\\leqslant \\tfrac{C}{n}$ holds for $n\\geqslant 2$.\n\t\\end{lemma}\n\tSince $\\Phi(x)\\geqslant F_n(x)-\\frac{C}{n}$, setting $x = F_n^{-1}(1-\\alpha)$ yields\n\t\\[\\Phi\\big(F_n^{-1}(1-\\alpha)\\big)\\geqslant F_n\\big(F_n^{-1}(1-\\alpha)\\big)-\\tfrac{C}{n} = 1-\\alpha-\\tfrac{C}{n}.\\]\n\tHence\n\t\\begin{equation}\\label{eq:fast3}\n\t\tF_n^{-1}(1-\\alpha)\\geqslant \\Phi^{-1}\\big(1-\\alpha-\\tfrac{C}{n}\\big).\n\t\\end{equation}\n\tWith (\\ref{eq:fast1}--\\ref{eq:fast3}) and \\Cref{prop:ch.f.} we have\n\t\\begin{align*}\n\t\t1-f_\\ep^{\\otimes n}(\\alpha) &= F_n\\Big[\\tfrac{n(q-p)}{\\sigma}-F_n^{-1}(1-\\alpha)\\Big]\\\\\n\t\t&\\leqslant \\Phi\\Big[\\tfrac{n(q-p)}{\\sigma}-F_n^{-1}(1-\\alpha)\\Big]+\\tfrac{C}{n}\\\\\n\t\t&\\leqslant \\Phi\\Big[1+\\tfrac{C}{n}-F_n^{-1}(1-\\alpha)\\Big]+\\tfrac{C}{n}\\\\\n\t\t&\\leqslant \\Phi\\Big[1+\\tfrac{C}{n}-\\Phi^{-1}(1-\\alpha-\\tfrac{C}{n})\\Big]+\\tfrac{C}{n}.\n\t\\end{align*}\n\tThe function $\\Phi$ is $\\frac{1}{\\sqrt{2\\pi}}$-Lipschitz, so\n\t\\[1-f_\\ep^{\\otimes n}(\\alpha)\\leqslant \\Phi\\Big[1-\\Phi^{-1}(1-\\alpha-\\tfrac{C}{n})\\Big]+\\tfrac{1}{\\sqrt{2\\pi}}\\cdot\\tfrac{C}{n}+\\tfrac{C}{n}.\\]\n\tBy blowing up the current $C$ and using the symmetry of standard normal, we have\n\t\\begin{align*}\n\t\tf_\\ep^{\\otimes n}(\\alpha) &\\geqslant 1-\\Phi\\Big[1-\\Phi^{-1}(1-\\alpha-\\tfrac{C}{n})\\Big]-\\tfrac{C}{n}\\\\\n\t\t&= \\Phi\\Big[\\Phi^{-1}(1-\\alpha-\\tfrac{C}{n}) - 1 \\Big]- \\tfrac{C}{n}\\\\\n\t\t& = G_1(\\alpha+\\tfrac{C}{n})-\\tfrac{C}{n}.\n\t\\end{align*}\n\tSimilarly, we can show the upper bound\n\t\\[f_\\ep^{\\otimes n}(\\alpha)\\leqslant G_1(\\alpha-\\tfrac{C}{n})+\\tfrac{C}{n}.\\]\n\tThe proof is now complete.\n\\end{proof}\nNext we show \\Cref{lem:pqlimit} and \\Cref{prop:ch.f.}.\n\\begin{proof}[Proof of \\Cref{lem:pqlimit}]\n\tThe proof is basically careful Taylor expansion. We will frequently use the assumption that $\\ep = 1\/\\sqrt{n}$. First we factor the objective as\n\t\\begin{align*}\n\t\t\\frac{n(q-p)}{\\sigma} &= 2\\sqrt{n}(q-p)\\cdot\\frac{\\sqrt{n}}{2\\sigma} = \\frac{2(q-p)}{\\ep}\\cdot\\frac{\\sqrt{n}}{2\\sigma}\n\t\\end{align*}\n\tand consider Taylor expansions of the two factors separately.\n\n\tFor the first factor, recall that\n\t\\[q-p = \\frac{e^\\ep-1}{e^\\ep+1} = \\frac{e^{\\tfrac{\\ep}{2}}-e^{-\\tfrac{\\ep}{2}}}{e^{\\tfrac{\\ep}{2}}+e^{-\\tfrac{\\ep}{2}}} = \\tanh \\tfrac{\\ep}{2}.\\]\n\tUsing the Taylor expansion $\\tanh x  = x-x^3\/3+o(x^4)$, we have\n\t\\begin{equation}\\label{eq:pq1}\n\t\t\\tfrac{2(q-p)}{\\ep} = \\tanh \\tfrac{\\ep}{2} \\,\\big\/\\, \\tfrac{\\ep}{2} = 1-\\tfrac{1}{3}(\\tfrac{\\ep}{2})^2+o(\\ep^3) = 1-\\tfrac{1}{12n} + o(n^{-3\/2}).\n\t\\end{equation}\n\tFor the second one, since $p+q=1$, we have $4pq=(p+q)^2-(p-q)^2 = 1-(q-p)^2$.\n\tA shorter expansion shows $q-p = \\tanh\\tfrac{\\ep}{2} = \\tfrac{\\ep}{2} +o(\\ep^2)$, and hence\n\t\\begin{align*}\n\t\t4pq= 1-\\big(\\tfrac{\\ep}{2}+o(\\ep^2)\\big)^2 = 1-\\tfrac{\\ep^2}{4}+o(\\ep^3)= 1-\\tfrac{1}{4n}+o(n^{-3\/2}).\n\t\\end{align*}\n\tRecall that $\\sigma$ is defined to be $\\sqrt{npq+\\frac{1}{12}}$. Using the above expansion of $4pq$, we have\n\t\\begin{align*}\n\t\t\\frac{\\sqrt{n}}{2\\sigma} &= \\sqrt{\\frac{n}{4\\sigma^2}} = \\sqrt{\\frac{n}{4npq+\\tfrac{1}{3}}} = \\big(4pq+\\tfrac{1}{3n}\\big)^{-1\/2} = \\big(1+\\tfrac{1}{12n}+o(n^{-3\/2})\\big)^{-1\/2}\n\t\\end{align*}\n\tSince $(1+x)^{-1\/2} = 1-\\tfrac{1}{2}x+o(x)$, we have\n\t\\begin{equation}\\label{eq:pq2}\n\t\t\\frac{\\sqrt{n}}{2\\sigma} = 1-\\tfrac{1}{2}\\big(\\tfrac{1}{12n}+o(n^{-3\/2})\\big)+o(n^{-1}) = 1-\\tfrac{1}{24n}+o(n^{-1}).\n\t\\end{equation}\n\tCombining the expansions \\eqref{eq:pq1} and \\eqref{eq:pq2},\n\t\\begin{align*}\n\t\t\\frac{n(q-p)}{\\sigma} &= \\frac{2(q-p)}{\\ep}\\cdot\\frac{\\sqrt{n}}{2\\sigma}\\\\\n\t\t&=\\Big(1-\\frac{1}{12n}+o(n^{-3\/2})\\Big)\\cdot\\Big(1-\\frac{1}{24n}+o(n^{-1})\\Big)\\\\\n\t\t&=1-\\frac{1}{8n}+o(n^{-1}).\n\t\\end{align*}\n\tThe proof is complete.\n\\end{proof}\nThen we move on to the more challenging \\Cref{prop:ch.f.}.\n\\begin{proof}[Proof of \\Cref{prop:ch.f.}]\n\tThe proof is inspired by Problem 6 on page 305 of \\cite{uspensky1937introduction}. Though involved, the idea is not hard: reduce the bound on cdfs to a bound on characteristic functions (ch.f. for short) by an appropriate Fourier inversion, then control the ch.f. by careful Taylor expansion.\n\n\tRecall that $\\ep,p,q,\\sigma$ depend on $n$ via\n\t\\[\\ep = \\frac{1}{\\sqrt{n}}, \\quad p = \\frac{1}{1+e^\\ep}, \\quad q=\\frac{e^\\ep}{1+e^\\ep},\\quad \\sigma = \\sqrt{npq+\\tfrac{1}{12}}.\\]\n\tRandom variables $X\\sim B(n,p),Y\\sim U[0,1]$. $F_n$ is the normalized cdf of $X+Y$. More precisely, since\n\t$$\\E[X+Y] = np+\\frac{1}{2}, \\quad \\Var[X+Y] = \\Var\\, X + \\Var\\, Y = npq+\\tfrac{1}{12} = \\sigma^2,$$\n\t$F_n$ is the cdf of $\\sigma^{-1}(X+Y-\\frac{1}{2}-np)$. Our goal is to show that $\\sup_{x\\in\\R}|F_n(x)-\\Phi(x)|=O(\\frac{1}{n})$.\n\n\tFirst let's compute the characteristic function (ch.f. for short) $\\varphi_n$ of the distribution $F_n$.\n\t\\begin{align*}\n\t\t\\varphi_n(t)\n\t\t&= \\E[e^{it\\sigma^{-1}(X+Y-\\frac{1}{2}-np)}]\\\\\n\t\t&= e^{-i{np}t\/\\sigma}\\cdot\\E[e^{it\/\\sigma(X+Y-\\frac{1}{2})}]\\\\\n\t\t&= e^{-i{np}t\/\\sigma}\\cdot \\varphi_{X}(t\/\\sigma)\\cdot \\varphi_{Y-\\frac{1}{2}}(t\/\\sigma).\n\t\\end{align*}\n\tEasy calculation shows that the ch.f. of $X$ is $(pe^{it}+q)^n$ and that of $Y-\\tfrac{1}{2}$ is $\\tfrac{\\sin t\/2}{t\/2}$. So\n\t\\begin{align*}\n\t\t\\varphi_n(t)\n\t\t&= e^{-i{np}t\/\\sigma}\\cdot(pe^{it\/\\sigma}+q)^n \\cdot \\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}\\\\\n\t\t&= (pe^{iqt\/\\sigma}+qe^{-ipt\/\\sigma})^n\\cdot \\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}.\n\t\\end{align*}\n\tThe base $pe^{iqt\/\\sigma}+qe^{-ipt\/\\sigma}$ is a convex combination of two complex numbers on the unit circle, so we have $|\\varphi_n(t)|\\leqslant \\tfrac{|\\sin t\/2\\sigma|}{|t\/2\\sigma|} \\leqslant \\min\\{\\tfrac{2\\sigma}{|t|}, 1\\}$.\n\n\tNow let's connect back to cdf. We need some form of Fourier inversion formula. \n\tLet $\\varphi(t) = e^{-t^2\/2}$ be the ch.f. of the standard normal.\n\t\\begin{lemma} \\label{lem:chf}\n\t\tWe have the following inversion formula\n\t\t$$F_n(x) - \\Phi(x) = -\\frac{1}{2\\pi i}\\int_{-\\infty}^{+\\infty}e^{-itx}\\cdot\\frac{\\varphi_n(t)-\\varphi(t)}{t} \\diff t.$$\n\t\\end{lemma}\n\tThe integrand is integrable over $\\R$ because: (1) At infinity $|\\varphi_n(t)|=O(\\frac{1}{t})$, so the integrand has modulus $O(\\frac{1}{t^2})$; (2) When $t\\to0$,\n\t$$\\tfrac{\\varphi_n(t)-\\varphi(t)}{t} = \\tfrac{\\varphi_n(t)-1-\\varphi(t)+1}{t}=\\tfrac{\\varphi_n(t)-\\varphi_n(0)}{t}-\\tfrac{\\varphi(t)-\\varphi(0)}{t}\\to\\varphi_n'(0)-\\varphi'(0) = \\E_{Z\\sim F_n} [Z]$$\n\tis a finite number. So the integrand is continuous at 0.\n\n\t\\Cref{lem:chf} makes it possible to control $F_n(x) - \\Phi(x)$ by controlling $\\varphi_n(t)-\\varphi(t)$.\n\t\\begin{align*}\n\t\t2\\pi|F_n(x) - \\Phi(x)| \\leqslant&\\phantom{+}\\int^{+\\infty}_{-\\infty}\\frac{|\\varphi_n(t)-\\varphi(t)|}{|t|}\\diff t\\\\\n\t\t\\leqslant&\\phantom{+}\n\t\t\\int_{|t|\\leqslant r\\sigma}\\frac{|\\varphi_n(t)-\\varphi(t)|}{|t|} \\diff t &&(I_1)\\\\\n\t\t&+\\int_{|t|>r\\sigma}\\frac{|\\varphi_n(t)|}{|t|} \\diff t &&(I_2)\\\\\n\t\t&+\\int_{|t|>r\\sigma}\\frac{|\\varphi(t)|}{|t|} \\diff t &&(I_3)\n\t\\end{align*}\n\tIt suffices to find some constant $r$ such that all three integrals are $O(\\frac{1}{n})$. This is done via the following three lemmas.\n\t\\begin{lemma} \\label{lem:I1}\n\t\tThere exist universal constants $r>0, C>0$ such that when $|t|\\leqslant r\\sigma$,\n\t\\[|\\varphi_n(t)-\\varphi(t)|\\leqslant Ce^{-\\tfrac{t^2}{8}}\\cdot\\big(\\tfrac{t^2}{n}+\\tfrac{|t|^3}{n}+\\tfrac{t^4}{n}\\big).\\]\n\t\\end{lemma}\n\tConsequently,\n\t\\begin{align*}\n\t\tI_1 &= \\int_{|t|\\leqslant r\\sigma}\\frac{|\\varphi_n(t)-\\varphi(t)|}{|t|} \\diff t\\\\\n\t\t&\\leqslant \\int_\\R Ce^{-\\tfrac{t^2}{8}}\\cdot\\big(\\tfrac{|t|}{n}+\\tfrac{t^2}{n}+\\tfrac{|t|^3}{n}\\big)\\diff t=O(\\tfrac{1}{n}).\n\t\\end{align*}\n\t\\begin{lemma} \\label{lem:I2}\n\t\tFor $r<\\pi$, we have\n\t\t$I_2\\leqslant (2+\\frac{48}{r^2})\\cdot\\frac{1}{n}$.\n\t\\end{lemma}\n\t\\begin{lemma} \\label{lem:I3}\n\t\tFor $n\\geqslant 2$, $I_3\\leqslant\\frac{10}{r^2}\\cdot \\frac{1}{n}\\cdot e^{-0.1r^2n}$ holds for any positive $r$.\n\t\\end{lemma}\n\tSo we can select a small enough $r$ such that all three estimates hold, which implies $I_1=O(\\frac{1}{n}), I_2=O(\\frac{1}{n})$ and $I_3\\ll\\frac{1}{n}$. In summary,\n\t\\[|F_n(x) - \\Phi(x)|\\leqslant \\frac{1}{2\\pi}(I_1+I_2+I_3) = O(\\frac{1}{n}).\\]\n\tAssuming correctness of \\Cref{lem:chf,lem:I1,lem:I2,lem:I3}, the proof of \\Cref{thm:fast} is complete.\n\\end{proof}\nThe rest is to prove \\Cref{lem:chf,lem:I1,lem:I2,lem:I3}. We deal with the three integrals first, and then come back to inversion formula.\n\\begin{proof}[Proof of \\Cref{lem:I1}]\n\tLet $w=pe^{itq\/\\sigma}+qe^{-itp\/\\sigma}$. Then $\\varphi_n(t) = w^n\\cdot \\frac{\\sin t\/2\\sigma}{t\/2\\sigma}$. We have\n\t\\begin{align}\\label{eq:twoterms}\n\t\t|\\varphi_n(t)-\\varphi(t)| &= |w^n\\cdot \\frac{\\sin t\/2\\sigma}{t\/2\\sigma} - e^{-\\frac{1}{2}t^2}|\\notag\\\\\n\t\t&\\leqslant |w^n- e^{-\\frac{1}{2}t^2}| + |w|^n \\cdot \\Big|1-\\frac{\\sin t\/2\\sigma}{t\/2\\sigma}\\Big|\n\t\\end{align}\n\tAll we need is a positive $r$ such that when $|t|\\leqslant r\\sigma$, both of the above terms are small. We are going to shrink $r$ as we need from time to time.\n\n\tFirst, on the disk $|z|\\leqslant r$ we have Taylor expansion \n\t$e^z=1+z+\\tfrac{1}{2}z^2+\\tfrac{1}{6}z^3+O(|z|^4)$.\n\tSo for $|t|\\leqslant r\\sigma$ we have\n\t\\begin{align}\\label{eq:taylor}\n\t\tpe^{itq\/\\sigma} &= p\\big(1+\\tfrac{itq}{\\sigma}+\\tfrac{1}{2}\\big(\\tfrac{itq}{\\sigma}\\big)^2+\\tfrac{1}{6}\\big(\\tfrac{itq}{\\sigma}\\big)^3+O(\\tfrac{t^4}{\\sigma^4})\\big)\\notag\\\\\n\t\tqe^{-itp\/\\sigma} &= q\\big(1-\\tfrac{itp}{\\sigma}+\\tfrac{1}{2}\\big(\\tfrac{itp}{\\sigma}\\big)^2-\\tfrac{1}{6}\\big(\\tfrac{itp}{\\sigma}\\big)^3+O(\\tfrac{t^4}{\\sigma^4})\\big)\\notag\\\\\n\t\tw &= 1-\\tfrac{t^2}{2\\sigma^2}\\cdot (pq^2+qp^2) + \\tfrac{t^3}{6\\sigma^3}\\big(-ipq^3-qp^3(-i)\\big)+O(\\tfrac{t^4}{\\sigma^4})\\notag\\\\\n\t\t&=1-\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} + \\tfrac{t^3}{6\\sigma^3}\\cdot ipq(p-q)+O(\\tfrac{t^4}{\\sigma^4}).\n\t\\end{align}\n\tObviously this implies $w = 1-\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} + o(\\tfrac{t^2}{\\sigma^2})$ (we will return to the more delicate \\eqref{eq:taylor} soon).\n\tSince $npq \\geqslant pq \\geqslant \\frac{1}{4} - \\frac{1}{16n} \\geqslant \\frac{3}{16}$, we have\n\t\\[\\tfrac{pq}{\\sigma^2}  = \\tfrac{pq}{npq+\\frac{1}{12}}=\\tfrac{npq}{npq+\\frac{1}{12}}\\cdot \\tfrac{1}{n}\\geqslant \\tfrac{1}{n} \\cdot \\tfrac{3}{16}\/(\\tfrac{3}{16}+\\tfrac{1}{12}) = \\tfrac{9}{13n}>\\tfrac{2}{3n}.\\]\n\tThat is, the quadratic term is more than $\\tfrac{t^2}{3n}$. We can tune $r$ so that the little $o$ remainder is even smaller, i.e. $|w-1+\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2}|<\\tfrac{t^2}{12n}$. This implies\n\t$$|w|<1-\\tfrac{t^2}{3n}+\\tfrac{t^2}{12n} = 1-\\tfrac{t^2}{4n}\\leqslant e^{-\\tfrac{t^2}{4n}}.$$\n\tOne consequence is we can bound the second term in \\eqref{eq:twoterms}. By Taylor expansion again, $\\frac{\\sin x}{x} = 1+O(x^2)$, so\n\t\\begin{equation}\\label{eq:first}\n\t\t|w|^n \\cdot \\big|1-\\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}\\big| \\leqslant e^{-\\tfrac{t^2}{4}} \\cdot O(\\tfrac{t^2}{\\sigma^2}) = e^{-\\tfrac{t^2}{4}} \\cdot O(\\tfrac{t^2}{n}). \n\t\\end{equation}\n\tThe first term in \\eqref{eq:twoterms} requires a more careful analysis. Let $z = e^{-\\tfrac{t^2}{2n}}$ and $\\gamma = e^{-\\tfrac{t^2}{4n}}$. Our goal is $|w^n-z^n|$. We have proved $|w|<\\gamma$, while $|z|=\\gamma^2<\\gamma$ is obviously true. We have\n\t\\[|w^n-z^n|\\leqslant |w^n-w^{n-1}z|+\\cdots +|wz^{n-1}-z^n|\\leqslant n|w-z|\\cdot \\gamma^{n-1}.\\]\n\tWithout loss of generality assume $n\\geqslant 2$, then $\\gamma^{n-1} = e^{-\\frac{t^2}{4}\\cdot \\frac{n-1}{n}}\\leqslant e^{-\\tfrac{t^2}{8}}$. That is,\n\t\\begin{equation}\\label{eq:a}\n\t\t|w^n- e^{-\\frac{1}{2}t^2}| \\leqslant n|w-e^{-\\frac{t^2}{2n}}|\\cdot e^{-\\frac{1}{8}t^2}.\n\t\\end{equation}\n\n\tFor $n|w-e^{-\\frac{t^2}{2n}}|$ we need \\eqref{eq:taylor} again. First decompose it as\n\t\\begin{align}\\label{eq:b}\n\t\tn|w-e^{-\\frac{t^2}{2n}}| \\leqslant n\\big|w-1+\\tfrac{t^2}{2n}\\big| + n\\big|e^{-\\frac{t^2}{2n}} - 1+\\tfrac{t^2}{2n}\\big|.\n\t\\end{align}\n\tSince $|p-q|=\\tfrac{e^\\ep-1}{e^\\ep+1}\\leqslant e^\\ep-1 = O(\\ep) = O(\\tfrac{1}{\\sqrt{n}})$ and $\\sigma^{-1} = O(\\frac{1}{\\sqrt{n}})$, we have\n\t\\[w = 1-\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} + \\tfrac{t^3}{6\\sigma^3}\\cdot ipq(p-q)+O(\\tfrac{t^4}{\\sigma^4}) = 1-\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} +O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2}).\\]\n\tNote that neither of two ``big $O$'' dominate each other, because $t$ can be as small as 0 and as large as $r\\sigma = O(\\sqrt{n})$.\n\tUsing the more delicate expansion of $w$ and that $\\sigma^2 = npq+\\frac{1}{12}$, we have\n\t\\begin{align}\\label{eq:c}\n\t\tn\\big|w-1+\\tfrac{t^2}{2n}\\big| &= n\\big|\\tfrac{t^2}{2n}-\\tfrac{pq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} +O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2})\\big|\\notag\\\\\n\t\t&= \\big|\\tfrac{t^2}{2}-\\tfrac{npq}{\\sigma^2}\\cdot\\tfrac{t^2}{2} +O(\\tfrac{t^3}{n})+O(\\tfrac{t^4}{n})\\big|\\notag\\\\\n\t\t&= \\big|\\tfrac{t^2}{2}\\cdot \\tfrac{1}{12\\sigma^2} +O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2})\\big|\\notag\\\\\n\t\t&=O(\\tfrac{t^2}{n^2})+O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2}).\n\t\\end{align}\n\tBy Taylor expansion again, we can tune $r$ so that when $|t|\\leqslant r\\sigma$, we have\n\t\\begin{equation}\\label{eq:d}\n\t\tn\\big|e^{-\\frac{t^2}{2n}} - 1+\\tfrac{t^2}{2n}\\big| = n\\cdot O(\\tfrac{t^4}{n^2}) = O(\\tfrac{t^4}{n}).\n\t\\end{equation}\n\tNow plug \\eqref{eq:c} and \\eqref{eq:d} back into \\eqref{eq:b}, and then into \\eqref{eq:a} to get\n\t\\[|w^n- e^{-\\frac{1}{2}t^2}| \\leqslant e^{-\\frac{1}{8}t^2}\\cdot\\big(O(\\tfrac{t^2}{n^2})+O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2})\\big).\\]\n\tThis ends the analysis of the first term of \\eqref{eq:twoterms}. Combining with the estimate of the first term, we have\n\t\\[|\\varphi_n(t)-\\varphi(t)| \\leqslant e^{-\\frac{1}{8}t^2}\\cdot\\big(O(\\tfrac{t^2}{n^2})+O(\\tfrac{t^3}{n^2})+O(\\tfrac{t^4}{n^2})\\big).\\]\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\end{proof}\n\\begin{proof}[Proof of \\Cref{lem:I2}]\n\tFor this integral we only care about the modulus of $|\\varphi_n(t)|$. Let's simplify it first.\n\t\\begin{align*}\n\t\t|\\varphi_n(t)| &= |pe^{iqt\/\\sigma}+qe^{-ipt\/\\sigma}|^n \\cdot \\Big|\\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}\\Big|\n\t\\end{align*}\n\tLet $\\theta = t\/\\sigma$. Using $|z|^2 = z\\bar{z}$, we have\n\t\\begin{align*}\n\t\t|pe^{iq\\theta}+qe^{-ip\\theta}|^2\n\t\t&= \\big(pe^{iq\\theta}+qe^{-ip\\theta}\\big) \\cdot \\big(pe^{-iq\\theta}+qe^{ip\\theta}\\big)\\\\\n\t\t&= p^2+q^2+pq(e^{i\\theta}+e^{-i\\theta})\\\\\n\t\t&= 1-2pq+2pq \\cos \\theta\\\\\n\t\t&= 1-4pq\\sin^2 \\tfrac{\\theta}{2}.\n\t\\end{align*}\n\tSo \n\t\\[|\\varphi_n(t)| = \\big(1-4pq\\sin^2 \\tfrac{t}{2\\sigma}\\big)^{n\/2} \\cdot \\Big|\\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}\\Big|.\\]\n\tWe see from this expression that the integrand of $I_2$ is an even function. Therefore,\n\t\\begin{align*}\n\t\\frac{1}{2}I_2 =& \\int_{r\\sigma}^{+\\infty}\\tfrac{|\\varphi_n(t)|}{t} \\diff t\\\\\n\t\t=& \\int_{r\\sigma}^{+\\infty}\\tfrac{1}{t} \\big(1-4pq\\sin^2 \\tfrac{t}{2\\sigma}\\big)^{n\/2} \\cdot \\Big|\\tfrac{\\sin t\/2\\sigma}{t\/2\\sigma}\\Big|\\diff t\\\\\n\t\t=& \\int_{r\/2}^{+\\infty}\\tfrac{1}{t^2} \\big(1-4pq\\sin^2 t\\big)^{n\/2} \\cdot |\\sin t|\\diff t\n\t\\end{align*}\n\tIn the last step we do a change of variable $s = t\/2\\sigma$ and rename $s$ to $t$. Next, we break down the integral at $k\\pi$, and upper bound the $\\frac{1}{t^2}$ factor by its value at the left end of the interval, so that the rest of the integrand is periodic.\n\t\\begin{align*}\n\t\t\\frac{1}{2}I_2\n\t\t\\leqslant&\\phantom{+}\n\t\t\\int_{r\/2}^{\\pi}\\tfrac{1}{t^2} \\big(1-4pq\\sin^2 t\\big)^{n\/2} \\cdot |\\sin t|\\diff t\\\\\n\t&+\\sum_{k=1}^{+\\infty}\\int_{k\\pi}^{(k+1)\\pi}\\tfrac{1}{t^2} \\big(1-4pq\\sin^2 t\\big)^{n\/2} \\cdot |\\sin t|\\diff t\\\\\n\t\t\\leqslant&\\phantom{+} \\Big(\\tfrac{4}{r^2}+\\sum_{k=1}^{+\\infty}\\tfrac{1}{k^2\\pi^2}\\Big)\\underbrace{\\int_{0}^{\\pi} \\big(1-4pq\\sin^2 t\\big)^{n\/2} \\cdot \\sin t\\diff t}_{J}\n\t\\end{align*}\n\tThe integral $J$ can be estimated as follows:\n\n\n\n\n\n\t\\begin{align*}\n\t\n\t\n\t\tJ\n\t\t&=\\int_{0}^{\\pi} \\big(1-4pq\\sin^2 t\\big)^{n\/2} \\cdot \\sin t\\diff t\\\\\n\t\t&= -\\int_{0}^{\\pi} \\big(1-4pq(1-\\cos^2 t)\\big)^{n\/2} \\diff\\cos t\\\\\n\t\t&= \\int_{-1}^{1} \\big(1-4pq(1-x^2)\\big)^{n\/2} \\diff x\\\\\n\t\n\t\t&= 2\\int_{0}^{1} (1-4pq + 4pqx^2)^{n\/2} \\diff x.\n\t\\end{align*}\n\tWe have seen that $1-4pq = (p-q)^2 = \\tanh^2 \\frac{\\ep}{2}$. It is easy to show that $\\tanh x \\leqslant x$ for $x\\geqslant0$. So\n\t$$\n\t\tpq = \\tfrac{1}{4}(1-\\tanh^2 \\tfrac{\\ep}{2})\\geqslant \\tfrac{1}{4}(1-\\tfrac{\\ep^2}{4}) = \\tfrac{1}{4} - \\tfrac{1}{16n}.\n\t$$\n\tSince $0\\leqslant x \\leqslant 1$, we have \n\t\\[1-4pq + 4pqx^2 \\leqslant \\tfrac{1}{4n}+(1- \\tfrac{1}{4n})x^2.\\]\n\tHence\n\t\\begin{align*}\n\t\tJ \\leqslant 2\\int_{0}^{1} \\Big(\\tfrac{1}{4n}+(1-\\tfrac{1}{4n})x^2\\Big)^{n\/2} \\diff x.\n\t\\end{align*}\n\tIt's easy to check that $\\tfrac{1}{4n-1}$ and 1 are the two roots of the quadratic equation $\\tfrac{1}{4n}+(1-\\tfrac{1}{4n})x^2 = x$. So we have $\\tfrac{1}{4n}+(1-\\tfrac{1}{4n})x^2\\leqslant x$ between the two roots, i.e. for $x\\in[\\tfrac{1}{4n-1},1]$. For the rest of the interval, we upper bound the integrand by 1. That is,\n\t\\begin{align*}\n\t\t\\int_{0}^{1} \\Big(\\tfrac{1}{4n}+(1-\\tfrac{1}{4n})x^2\\Big)^{n\/2} \\diff x\n\t\t\\leqslant\\int_{0}^{\\tfrac{1}{4n-1}} 1\\diff x + \\int_{\\tfrac{1}{4n-1}}^1 x^{n\/2}\\diff x\\leqslant \\tfrac{1}{4n-1} +\\tfrac{1}{n\/2+1} \\leqslant \\tfrac{3}{n}.\n\t\\end{align*}\n\tSo we have $J\\leqslant \\frac{6}{n}$. Returning to $I_2$, with the well-known identity $\\sum_{k=1}^{+\\infty}\\tfrac{1}{k^2} = \\frac{\\pi^2}{6}$, we have\n\t\\begin{align*}\n\t\tI_2&\\leqslant 2\\Big(\\tfrac{4}{r^2}+\\sum_{k=1}^{+\\infty}\\tfrac{1}{k^2\\pi^2}\\Big)\\cdot J\\\\\n\t\t&= \\big(\\tfrac{8}{r^2}+\\tfrac{\\pi^2}{6}\\cdot\\tfrac{2}{\\pi^2}\\big)\\cdot\\tfrac{6}{n}\\\\\n\t\t&= \\big(2+\\tfrac{48}{r^2}\\big)\\cdot\\tfrac{1}{n}\n\t\\end{align*}\n\tThe estimate of $I_2$ is complete.\n\\end{proof}\n\\begin{proof}[Proof of \\Cref{lem:I3}]\n\tFirst notice the following simple facts:\n\t\\begin{enumerate}\n\t \t\\item When $t>r\\sigma$, we have $\\frac{1}{t}\\leqslant t\\cdot \\frac{1}{r^2\\sigma^2}$.\n\t \t\\item $\\sigma^2>0.2n$ for any $n$.\n\t\\end{enumerate}\n\tThe second follows from a bound we derive in the proof of \\Cref{lem:I2}: $pq\\geqslant \\tfrac{1}{4} - \\tfrac{1}{16n}$. In fact,\n\t\\[\\sigma^2 = npq+\\tfrac{1}{12} \\geqslant \\tfrac{n}{4} - \\tfrac{1}{16}+\\tfrac{1}{12} = \\tfrac{n}{4} - \\tfrac{1}{48} > \\tfrac{n}{5}.\\]\n\tUsing these two facts, we can bound $I_3$ as follows:\n\t\\begin{align*}\n\t\tI_3 \n\t\t&= \\int_{|t|>r\\sigma}\\frac{|\\varphi(t)|}{|t|} \\diff t\\\\\n\t\t& = 2\\int_{r\\sigma}^{+\\infty}\\frac{1}{t}e^{-\\frac{t^2}{2}}\\diff t\\\\\n\t\t&\\leqslant \\frac{2}{r^2\\sigma^2}\\cdot\\int_{r\\sigma}^{+\\infty}te^{-\\frac{t^2}{2}}\\diff t\\\\\n\t\t&= \\frac{2}{r^2\\sigma^2}\\cdot e^{-\\frac{t^2}{2}}\\Big|^{r\\sigma}_{+\\infty}\\\\\n\t\t&= \\frac{2}{r^2\\sigma^2}\\cdot e^{-\\frac{r^2\\sigma^2}{2}}\\\\\n\t\t&\\leqslant \\frac{10}{r^2}\\cdot \\frac{1}{n}\\cdot e^{-0.1r^2n}\n\t\\end{align*}\n\tThe estimate of $I_3$ is complete.\n\\end{proof}\n\nWe are done with the three integrals. Before we dive into the proof of the inversion formula \\ref{lem:chf}, we make a few observations.\n\nFirst, one cannot hope to obtain this lemma by showing \n$$F_n(x)=\\tfrac{1}{2\\pi}\\int_{-\\infty}^{+\\infty}-e^{-itx}\\cdot\\tfrac{\\varphi_n(t)}{it} \\diff t$$\nand a similar expression for $\\Phi(x)$ separately because this alternative integrand is not even integrable. To see this, notice $\\varphi_n(0)=1$, so the integrand $\\approx \\frac{1}{t}$ around 0.\n\nInversion formula \\ref{lem:chf} has the same form as Lemma 3.4.19 of \\cite{durrett2019probability}. However, the ch.f.s are assumed to be (absolutely) integrable there, while $\\varphi_n$ is not. To see this, recall that Fourier inversion tells us that if the ch.f. is absolutely integrable, then the probability distribution has continuous density (see e.g. \\cite{durrett2019probability}, Theorem 3.3.14). This is not true for $X+Y$ because its density is piecewise constant. So $\\varphi_n$ cannot be in $L^1(\\R)$. There seems to be no shortcut, so let's work out our own proof.\n\\begin{proof}[Proof of \\Cref{lem:chf}]\n\tApplying the general inversion formula (see e.g. \\cite{durrett2019probability} Theorem 3.3.11) to $F_n$, we have\n\t\\begin{align*}\n\tF_n(x)-F_n(a)&=\\frac{1}{2\\pi}\\lim_{T\\to+\\infty}\\int^T_{-T}\\frac{e^{-ita}-e^{-itx}}{it}\\cdot\\varphi_n(t)\\diff t\n\t\\end{align*}\n\t$\\varphi_n$ is continuous and decays in the rate $\\tfrac{1}{|t|}$, so the integrand is dominated by $O(t^{-2}\\wedge 1)$ and hence the limit on $T$ is equal to the Lebesgue integral. That is,\n\t\\begin{equation}\\label{eq:FnInversion}\n\t\tF_n(x)-F_n(a)=\\frac{1}{2\\pi}\\int^{+\\infty}_{-\\infty}\\frac{e^{-ita}-e^{-itx}}{it}\\cdot\\varphi_n(t)\\diff t.\n\t\\end{equation}\n\tSimilarly,\n\t\\[\n\t\\Phi(x)-\\Phi(a)=\\frac{1}{2\\pi}\\int^{+\\infty}_{-\\infty}\\frac{e^{-ita}-e^{-itx}}{it}\\cdot\\varphi(t)\\diff t.\n\t\\]\n\tNote that in \\eqref{eq:FnInversion}, we cannot let $a\\to-\\infty$ and use Riemann-Lebesgue lemma because $\\frac{\\varphi_n(t)}{t}$ is not integrable, as discussed before the proof. However, subtracting the two formula yields\n\t\\begin{equation}\\label{eq:ligoat}\n\t\t\\big(F_n(x)-\\Phi(x)\\big)-\\big(F_n(a)-\\Phi(a)\\big)=\\frac{1}{2\\pi}\\int^{+\\infty}_{-\\infty}\\frac{e^{-ita}-e^{-itx}}{it}\\cdot\\big(\\varphi_n(t)-\\varphi(t)\\big)\\diff t\n\t\\end{equation}\n\tConsider the part involving $a$\n\t\\[\\int^{+\\infty}_{-\\infty}e^{-ita}\\cdot\\frac{\\varphi_n(t)-\\varphi(t)}{it}\\diff t.\\]\n\tWe argued right after introducing \\Cref{lem:chf} that $\\frac{\\varphi_n(t)-\\varphi(t)}{it} \\in L^1(\\R)$, so by Riemann-Lebesgue lemma we have the limit\n\t\\[\\lim_{a\\to-\\infty}\\int^{+\\infty}_{-\\infty}e^{-ita}\\cdot\\frac{\\varphi_n(t)-\\varphi(t)}{it}\\diff t = 0.\\]\n\tTake the limit $a\\to-\\infty$ on both sides of \\eqref{eq:ligoat} and we have\n\t\\begin{align*}\n\t\tF_n(x)-\\Phi(x)&=\\frac{1}{2\\pi}\\int^{+\\infty}_{-\\infty}\\frac{-e^{-itx}}{it}\\cdot\\big(\\varphi_n(t)-\\varphi(t)\\big)\\diff t\\\\\n\t\t&=-\\frac{1}{2\\pi i}\\int_{-\\infty}^{+\\infty}e^{-itx}\\cdot\\frac{\\varphi_n(t)-\\varphi(t)}{t} \\diff t.\n\t\\end{align*}\n\tThe proof is now complete.\n\\end{proof}\n\\section{Omitted Details in \\Cref{sec:subsampling}}\n\\label{app:property}\nWe begin this appendix with a small example showing our subsampling theorem is generically unimprovable.\n\\paragraph{Tightness}\n\\label{par:example_and_tightness}\n\n\nConsider the mechanism $\\widetilde{M}$ that randomly releases one individual's private information in the dataset. The privacy analysis is easy: without loss of generality we can assume two neighboring datasets differ in the first individual.  Effectively we are trying to distinguish uniform distributions over $\\{1,2,\\ldots, n\\}$ and $\\{1',2,\\ldots, n\\}$. It's not hard to see that the trade-off function of these two uniform distributions is $f_{0,1\/n}$, i.e. $(\\ep,\\delta)$-DP with $\\ep=0, \\delta=1\/n$. This is exact --- the adversary has tests that achieve every point on the curve.\n\n\nOur \\cref{thm:subsample} yields the same result, showing its tightness. To see this, let $M$ be the identity map that takes in one individual and outputs his\/her entire private information. Then $\\widetilde{M} = M\\circ\\mathtt{Sample}_{\\frac{1}{n}}$. Privacy of $M$ is described by $f\\equiv 0$. By \\Cref{thm:subsample}, $\\widetilde{M}$ is $C_{1\/n}(f)$-DP. \\Cref{fig:subsample} shows that $C_{1\/n}(f)=f_{0,1\/n}$.\n\n\n\nNext we show the following two equations:\n\t\\begin{align*}\n\t\t\\ep'&=\\log(1-p + pe^\\ep),\\\\\n\t\t\\delta'&=p\\big(1+f^*(-e^{\\ep})\\big)\n\t\\end{align*}\n\tcan be re-parameterized into \n\t\\begin{equation}\n\t\t\\delta'=1+f_p^*(-e^{\\ep'})\\tag{\\ref{eq:fp}}\n\t\\end{equation}\n\twhere $f_p = pf+(1-p)\\Id$.\n\\begin{proof}[Proof of \\Cref{eq:fp}]\n\t\n\tSince $\\ep\\mapsto\\log(1-p + pe^\\ep)$ maps $[0,+\\infty)$ to $[0,+\\infty)$ monotonically, we can solve $\\ep$ from $\\ep'$ and plug into $\\delta'$. We have\n\t\\begin{align*}\n\t\t\\frac{1}{p}(1-e^{\\ep'}) = 1-e^{\\ep} \\quad\\text{ and }\\quad\n\t\t\\delta'=p\\big(1+f^*(-e^{\\ep})\\big)\n\t\t&=p\\big(1+f^*(\\tfrac{1}{p}(1-e^{\\ep'})-1)\\big).\n\t\\end{align*}\n\tLet $y = -e^{\\ep'}$ and it suffices to show for any $y\\leqslant-1$,\n\t\\begin{equation}\\label{eqn:ning}\n\t\t1+f_p^*(y) = p\\big(1+f^*(\\tfrac{1}{p}(1+y)-1)\\big).\n\t\\end{equation}\n\tTo see this, expand $f_p^*$ as follows\n\t\\begin{align*}\n\t\tf_p^*(y) &= \\sup_{x} yx-f_p(x)\\\\\n\t\t&=\\sup_{x} yx-pf(x)-(1-p)(1-x)\\\\\n\t\t&=p-1  + \\sup_{x} (y+1-p)x -pf(x)\\\\\n\t\t&=p-1  + p\\cdot\\sup_{x} (\\tfrac{1}{p}(1+y)-1)x -f(x)\\\\\n\t\t&=p-1 +pf^*(\\tfrac{1}{p}(1+y)-1)\n\t\\end{align*}\n\t\\eqref{eqn:ning} follows directly.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\nNext we provide the general tool mentioned in \\Cref{sub:proof_of_subsample_theorems} that convert collections of $(\\ep,\\delta)$-DP guarantee in the form of \\eqref{eq:fp} to some $f$-DP.\n\nThe symmetrization operator $\\mathrm{Symm}:\\T\\to\\T^S$ maps a general trade-off function to a symmetric trade-off function.  It's defined as follows:\n\\begin{definition} \\label{def:symm}\n\tFor $f\\in\\T$, let $\\bar{x} = \\inf\\{x\\in[0,1]:-1\\in\\partial f(x)\\}$. The symmetrization operator $\\mathrm{Symm}:\\T\\to\\T^S$ is defined as\n\t\\[\\mathrm{Symm}(f):= \\left\\{\n\t\\begin{array}{ll}\n\t\\min\\{f,f^{-1}\\}^{**}, &\\text{ if }\\,\\, \\bar{x}\\leqslant f(\\bar{x}),\\\\\n\t\\max\\{f,f^{-1}\\}, &\\text{ if }\\,\\, \\bar{x}>f(\\bar{x}).\n\t\\end{array}\n\t\\right.\\]\n\\end{definition}\n\n\\begin{proposition}\\label{prop:asymm_env}\n\tLet $f\\in\\T$, not necessarily symmetric. Suppose a mechanism is $(\\ep,1+f^*(-e^{\\ep}))$-DP for all $\\ep\\geqslant 0$, then it is $\\mathrm{Symm}(f)$-DP.\n\\end{proposition}\n\n\nRecall from basic convex analysis that double convex conjugate $f^{**}$ is the greatest convex lower bound of $f$. If $f$ itself is convex then $f^{**}=f$. For $f$ symmetric , $f=f^{-1}$. By convexity of $f$, we have $\\mathrm{Symm}(f)=f$ in both cases. So \\Cref{prop:ftoDP} is a special case of \\Cref{prop:asymm_env}. The first half of \\Cref{prop:asymm_env} is \\Cref{prop:asymm_for_proof}, the part we used in the proof of our subsampling theorem.\n\n\\begin{figure}[h]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]\n{.\/figures\/Symm.pdf}\n  \\captionof{figure}{Action of Symm. Left panel: $\\bar{x}\\leqslant f(\\bar{x})$. Right panel: $\\bar{x} > f(\\bar{x})$. For both panels the effective parts (red bars on $x$-axes) are $[0,\\bar{x}]$ of $f$ and $[f(\\bar{x}),1]$ of $f^{-1}$. No overlap in the left panel since $\\bar{x} < f(\\bar{x})$, so interpolate with straight line; overlap in the right panel so the max is taken.}\n  \\label{fig:symm}\n\\end{figure}\n\nFrom \\Cref{fig:symm} it's not hard to see that\n\\begin{align*}\n\t\\min\\{f,f^{-1}\\}^{**}(x) =\n\t\\left\\{\n\t\\begin{array}{ll}\n\tf(x),&x\\in[0,\\bar{x}], \\\\\n\t\\bar{x}+f(\\bar{x})-x, & x\\in[\\bar{x},f(\\bar{x})], \\\\\n\tf^{-1}, & x\\in[f(\\bar{x}),1].\n\t\\end{array}\n\t\\right.\n\\end{align*}\n\n\n\n\\begin{proof}[Proof of \\Cref{prop:asymm_env}]\n\t$M$ being $\\big(\\ep,\\delta(\\ep)\\big)$-DP means that for any neighboring datasets $S$ and $S'$,\n\t\\[T\\big(M(S),M(S')\\big)(x)\\geqslant-e^\\ep x+1-\\delta(\\ep).\\]\n\tFix $x\\in[0,1]$. Since the DP condition holds for all $\\ep\\geqslant 0$, the lower bound still holds when we take the supremum over $\\ep\\geqslant 0$. In other words, $M$ is ${f}_\\mathrm{env}$-DP with\n\t$${f}_\\mathrm{env}(x) = \\max\\{0,\\,\\,\\sup_{\\ep\\geqslant0}1-\\delta(\\ep)-e^\\ep x\\}.$$\n\tBy \\Cref{prop:symmetry} $M$ is also $\\max\\{{f}_\\mathrm{env},{f}_\\mathrm{env}^{-1}\\}$-DP. The proof will be complete if we can show $\\max\\{{f}_\\mathrm{env},{f}_\\mathrm{env}^{-1}\\} = \\mathrm{Symm}(f)$.\n\n\t\\begin{figure}[h]\n\t  \\centering\n\t  \\includegraphics[width=0.7\\linewidth]\n\t{.\/figures\/Symm2.pdf}\n\t  \\captionof{figure}{Symm explained. Left panel: $\\bar{x}\\leqslant f(\\bar{x})$. Right panel: $\\bar{x} > f(\\bar{x})$. }\n\t  \\label{fig:symm2}\n\t\\end{figure}\n\n\tWe achieve this by first showing:\n\t\\[{f}_\\mathrm{env}(x) =\n\t\\left\\{\n\t\\begin{array}{ll}\n\tf(x),&x\\in[0,\\bar{x}], \\\\\n\t\\bar{x}+f(\\bar{x})-x, & x\\in[\\bar{x},\\bar{x}+f(\\bar{x})], \\\\\n\t0, & x\\in[\\bar{x}+f(\\bar{x}),1].\n\t\\end{array}\n\t\\right.\\]\n\tFrom \\Cref{fig:symm2} it is almost obvious. We still provide the argument below.\n\n\tPlug in $\\delta(\\ep) = 1+f^*(-e^{\\ep})$ and change the variable $y=-e^\\ep$:\n\t\\begin{align*}\n\t\t\\sup_{\\ep\\geqslant0}[-e^\\ep x+1-\\delta(\\ep)]\n\t\t&=\\sup_{\\ep\\geqslant0}[-e^\\ep x-f^*(-e^{\\ep})]\\\\\n\t\t&=\\sup_{y\\leqslant-1}[y x-f^*(y)]\n\t\\end{align*}\n\tFrom convex analysis we know if $y\\in\\partial f(x)$ then $yx = f(x)+f^*(y)$. By definition of $\\bar{x}$, if $x\\leqslant\\bar{x}$, then at least one subgradient $y\\in\\partial f(x)$ is no greater than $-1$. So this specific $y x-f^*(y) = f(x)$ is involved in the supremum, i.e. $\\sup_{y\\leqslant-1}[y x-f^*(y)] = f(x)$. This justifies the expression for the first segment.\n\n\tWhen $x>\\bar{x}$, the supremum is always attained at $y=-1$. In fact, if we let $l_y(x) = y x-f^*(y)$, then\n\t\\begin{lemma} \\label{lem:ning}\n\t \t$l_y(x)\\leqslant l_{-1}(x)$ when $y\\leqslant -1$ and $x>\\bar{x}$.\n\t\\end{lemma}\n\t\\begin{proof}[Proof of \\Cref{lem:ning}]\n\t$l_y$ is the supporting linear function of $f$ with slope $y$. It suffices to show that $l_y(x)$ is monotone increasing in $y$. To see this, change the variable from the slope $y$ to the supporting location $u$. As $f$ is convex, $y=f'(u)$ is increasing in $u$. In terms of $u$, $l_y(x) = f(u) + f'(u)(x-u)$. Taking derivative with respect to $u$:\n\t\\[\\frac{\\partial}{\\partial u }\\, l_y(x) = f'(u) + f''(u)(x-u) + f'(u)\\cdot(-1) = f''(u)(x-u).\\]\n\t$y\\leqslant -1$ corresponds to location $u\\leqslant \\bar{x}$ and hence $u<x$. So we see that $\\frac{\\partial}{\\partial u }\\, l_y(x)= f''(u)(x-u) \\geqslant 0$. $l_y(x)$ is increasing in $u$, and hence increasing in $y$, completing the proof of the lemma.\n\t\\end{proof}\n\tSo the supremum is attained at $y=-1$. The value is $\\sup_{y\\leqslant -1}\\big[yx-f^*(y)\\big] = l_{-1}(x)$. Support function with slope $-1$ must support $f$ at $\\bar{x}$.\n\tThe location yields expression $l_{-1}(x) = f(\\bar{x}) - (x-\\bar{x})$. This justifies the expression for the second segment. The third one is simply the result of thresholding at 0.\n\n\tIt's straightforward to verify that\n\t\\[\n\t\tf_\\mathrm{env}^{-1}(x) =\n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t\\bar{x}+f(\\bar{x})-x,&x\\in[0,f(\\bar{x})], \\\\\n\t\tf^{-1}(x), & x\\in[f(\\bar{x}),1].\n\t\t\\end{array}\n\t\t\\right.\n\t\\]\n\tObviously $\\max\\{{f}_\\mathrm{env},{f}_\\mathrm{env}^{-1}\\}=\\mathrm{Symm}(f)$. When the intervals $[0,\\bar{x}]$ and $[f(\\bar{x}),1]$ are disjoint, $f$ and $f^{-1}$ are effective separately, and the linear interpolation fills the blank. On the other hand when they intersect, max is taken and the linear function is never effective.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\nWe conclude this appendix with the proof of the classical privacy amplification by subsampling theorem. It primarily follows \\cite{jon}, but is written so that potential generalization and improvement are in reach.\n\\DPsubsamplerep*\n\\begin{proof}[Proof of \\Cref{lem:DPsubsample}]\n\n\tLet $S$ and $S'$ be neighboring datasets, each with $n$ individuals. Without loss of generality, assume $S$ and $S'$ differ in the first individual. We are ultimately interested in $\\widetilde{M}(S)$ and $\\widetilde{M}(S')$. They are generated as follows.\n\n\tLet $I\\subseteq[n]$ be any size $m$ subset of the index set $[n]=\\{1,2,\\ldots, n\\}$. $S_I$ and $S_I'$ denote the $m$ individuals indexed by $I$ in corresponding datasets, both of which can be the input of $M$. When $I$ is uniformly sampled from the $n\\choose m$ subsets of $[n]$ of cardinality $m$, $M(S_I)$ is $\\widetilde{M}(S)$.\n\n\n\n\tLet $\\phi$ be an arbitrary rejection rule. For a fixed $I\\subseteq [n]$, when $\\phi$ is used for the problem $M(S_I)$ vs $M(S'_I)$, the corresponding type I and type II errors are\n\t\\begin{equation}\\label{eq:defS}\n\t\t\\alpha_I:=\\E[\\phi(M(S_I))]\\quad\\text{ and }\\quad\\beta_I:=1-\\E[\\phi(M(S'_I))]\n\t\\end{equation}\n\trespectively. The expectations are over the randomness of $M$.\n\n\tWhen $\\phi$ is used for $\\widetilde{M}(S)$ vs $\\widetilde{M}(S')$, the type I error $\\alpha$ and type II error $\\beta$ satisfy\n\t\\[\n\t\t\\alpha=\\E[\\phi(\\widetilde{M}(S))]= \\E_I\\,\\E[\\phi(M(S_I))] = \\E_I[\\alpha_I]\n\t\\]\n\tand\n\t\\begin{align*}\n\t\t\\beta=1-\\E[\\phi(\\widetilde{M}(S'))]= \\E_I\\big[1-\\E[\\phi(M(S'_I))]\\big] = \\E_I[\\beta_I].\n\t\\end{align*}\n\tUltimately we are going to show that $\\beta$ has a lower bound in terms of $\\alpha$.\n\n\tThis is possible because for each $I$, $\\beta_I$ has a lower bounded in terms of $\\alpha_I$, whose form depends on whether the ``difference'' individual 1 is sampled.\n\n\tWhen $1\\not\\in I$, $S_I = S'_I$. By definition \\eqref{eq:defS}, $\\alpha_I+\\beta_I = 1$.\n\tWhen $1\\in I$, $S_{I}$ and $S'_{I}$ are neighbors in $X^m$, so $(\\ep,\\delta)$ privacy of $M$ becomes effective here. Let $k = -e^\\ep, b = 1-\\delta$. From \\Cref{thm:privacy_testing} we know $\\beta_I\\geqslant k\\alpha_I+b$.\n\n\tSo we should separate these cases $1\\in I$ and $1\\notin I$ and average them respectively.\n\tDefine\n\t\\begin{align*}\n\t\tA&=\\E[\\alpha_I|I\\ni 1],\n\t\t\\quad\n\t\tB=\\E[\\beta_I|I\\ni 1]\n\t\\end{align*}\n\tand\n\t\\begin{align*}\n\t\t\\bar{A}&=\\E[\\alpha_I|I\\not\\ni 1] \n\t\t\\quad\n\t\t\\bar{B}=\\E[\\beta_I|I\\not\\ni 1]\n\t\\end{align*}\n\tLet $p=m\/n$. Then $P[I\\ni 1]=p, P[I\\not\\ni 1]=1-p$. Further averages of these quantities give us $\\alpha$ and $\\beta$:\n\t$$\\alpha = pA+(1-p)\\bar{A}, \\quad \\beta = pB+(1-p)\\bar{B}.$$\n\tLinearity passes through expectations, so we have $\\bar{A}+\\bar{B} = 1$ and\n\t\\[B = \\E[\\beta_I|I\\ni 1] \\geqslant \\E[k\\alpha_I+b|I\\ni 1] = k\\E[\\alpha_I|I\\ni 1]+b = kA+b.\\]\n\t\n\n\n\n\n\n\n\n\n\n\n\n\n\n\tThis inequality $B\\geqslant kA+b$ comes from the difference between $S$ and $S'$. The smart observation is, there are a lot more neighboring datasets we can exploit. For example $S_I$ and $S_{\\tilde{I}}'$ when $I = \\{1,2,\\ldots, m\\}, \\tilde{I} = \\{2,3,\\ldots, m+1\\}$. They share individuals $2,3,\\ldots, m$ and differ in the one left. This yields another inequality $B\\geqslant k\\bar{A}+b$. We fill in the details Jon Ullman omits in his lecture note.\n\t\n\tFor $I\\ni1$, we can replace 1 with any of the $n-m$ indices not in $I$ and obtain $n-m$ different subsets $I^{(1)},\\ldots, I^{(n-m)}$. Note that none of these contains 1. In another word, we have the following correspondence:\n\t\\[\n\t\t\\begin{array}{ccc}\n\t\t\t\\ni 1 &  & \\not\\ni1 \\\\\n\t\t\tI & \\longleftrightarrow & I^{(1)} \\\\\n\t\t\t& \\vdots &\\\\\n\t\t\tI & \\longleftrightarrow & I^{(n-m)}\n\t\t\\end{array}\n\t\\]\n\tEach $S_{I^{(j)}}$ is a neighbor of $S'_I$, so by the privacy of $M$, we have $\\beta_I\\geqslant k\\alpha_{I^{(j)}}+b$. Sum these up for each $I\\ni1$ and the $n-m$ replacements of $I$, we have\n\t\\[(n-m)\\cdot\\sum_{I\\ni 1} \\beta_I\\geqslant m\\cdot\\sum_{I\\not\\ni 1} k\\alpha_I+b.\\]\n\tThe right hand side factor $m$ comes from a different counting: if $I\\not\\ni1$ then each of the $m$ items in $I$ could have been 1 before the replacement. So each $I\\not\\ni1$ appears $m$ times in the summation.\n\n\tMultiply by $\\frac{n}{m(n-m)}\\cdot\\binom{n}{m}^{-1}$,\n\t\\[\\frac{n}{m}\\cdot\\bigg(\\sum_{I\\ni 1} \\beta_I\\bigg)\\cdot \\binom{n}{m}^{-1}\\geqslant b+ k \\cdot \\frac{n}{n-m}\\cdot\\bigg(\\sum_{I\\not\\ni 1} \\alpha_I\\bigg)\\cdot \\binom{n}{m}^{-1}.\\]\n\tBy the simple Bayes' rule, this is $B\\geqslant k\\bar{A}+b$.\n\n\tRemember we want a lower bound of $\\beta$. The best possible lower bound we can ge t via these relations is the minimum of the following linear program:\n\t\\begin{align*}\n\t\t\\min_{A,B,\\bar{A},\\bar{B}} \\quad & \\beta\\\\\n\t\t\\textrm{s.t.}\\quad\n\t\t& pB+(1-p)\\bar{B}=\\beta\\\\\n\t\t& pA+(1-p)\\bar{A}=\\alpha\\\\\n\t\t& \\bar{A}+\\bar{B} = 1 \\\\\n\t\t& B\\geqslant kA+b\\\\\n\t\t& B\\geqslant k\\bar{A}+b \\\\\n\t\t& A,B,\\bar{A},\\bar{B}\\in [0,1]\n\t\\end{align*}\n\t\\begin{lemma} \\label{lem:LP}\n\t\tThe minimum of the above linear program is no less than $p(k\\alpha+b)+(1-p)(1-\\alpha)$.\n\t\\end{lemma}\n\t\\begin{proof}[Proof of \\Cref{lem:LP}]\n\t\tWe are going to remove the $A,B,\\bar{A},\\bar{B}\\in [0,1]$ constraint and find the exact minimum of the relaxation, which is a lower bound of the original minimum.\n\n\t\tWe have $\\beta+\\alpha = p(A+B)+(1-p)(\\bar{A}+\\bar{B})=p(A+B)+(1-p)$. So equivalently we can try to solve\n\t\t\\begin{align*}\n\t\t\t\\min_{A,B,\\bar{A}} \\quad & A+B\\\\\n\t\t\t\\textrm{s.t.}\\quad\n\t\t\t& pA+(1-p)\\bar{A}=\\alpha\\\\\n\t\t\t& B\\geqslant kA+b\\\\\n\t\t\t& B\\geqslant k\\bar{A}+b\n\t\t\\end{align*}\n\t\tWhen $A=\\bar{A}=\\alpha$, the lower bound they impose on $B$ is $k\\alpha+b$. Notice that $k=-\\mathrm{e}^\\ep\\leqslant -1$. The two consequences are:\n\t\t\\begin{enumerate}\n\t\t\t\\setlength\\itemsep{-0.3em}\n\t\t\t\\item $A>\\alpha$ is worse than $A=\\alpha$, because the convex combination equality requires $\\bar{A}<\\alpha$. $B$ has to increase anyway. Both $A$ and $B$ increase, so the objective $A+B$ also increases.\n\t\t\t\\item If $A$ is decreased from $\\alpha$ by some amount, $B$ has to increase by $\\mathrm{e}^\\ep$ times that amount, which is not worth it.\n\t\t\\end{enumerate}\n\t\tSo the minimum of the relaxed linear program is achieved at $A=\\bar{A}=\\alpha, B=k\\alpha+b,\\bar{B} = 1-\\alpha$, thereby inducing the claimed lower bound.\n\t\\end{proof}\n\n\tSo $\\beta\\geqslant p(k\\alpha+b)+(1-p)(1-\\alpha)$. Changing back to $\\ep,\\delta$, we have\n\t\\[\\beta\\geqslant p(-e^\\ep\\alpha+1-\\delta)+(1-p)(1-\\alpha) = -[p\\mathrm{e}^\\ep+1-p]\\alpha+1-p\\delta = -\\mathrm{e}^{\\ep'}\\alpha+1-\\delta'.\\]\n\tBy \\Cref{thm:privacy_testing}, $\\widetilde{M}$ is $(\\ep',\\delta')$-DP.\n\n\n\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{A Self-contained Proof of the Composition Theorem}\n\\label{app:self}\nIn this section we prove the well-definedness of $\\otimes$ and Composition \\Cref{thm:n_steps}.\n\n\nWe begin with the setting of the key lemma, which compares indistinguishability of two pairs of \\textit{randomized algorithms}. Let $K_1, K_1': Y \\to Z_1$ and $K_2, K_2': Y \\to Z_2$ be two pairs of randomized algorithms. Suppose the following is true for these four algorithms: for each fixed input $y\\in Y$, testing problem $K_1(y) $ vs $ K_1'(y)$ is harder than $K_2(y) $ vs $ K_2'(y)$. In mathematical language, let $f_i^y = T\\big(K_i(y), K_i'(y)\\big)$ (See the left panel of \\Cref{fig:comparison}). The above assumption amounts to saying $f_1^y\\geqslant f_2^y$. So far we have fixed the input $y$. In the two pairs of testing problems, if the input of the null comes from $P$ and the input of the alternative comes from $P'$, then intuitively both testing problems become easier than when inputs are fixed, because now the inputs also provide information. Formally, the observation comes from input-output joint distribution $\\big(P, K_i(P)\\big)$ or $\\big(P', K_i'(P')\\big)$ (with a little abuse of notation). Let $f_i = T\\big( (P, K_i(P)), (P', K_i'(P')) \\big), i=1,2$ be the trade-off functions of the joint testing problems (See the right panel of \\Cref{fig:comparison}). As discussed, we expect that $f_1\\leqslant f_1^y, f_2\\leqslant f_2^y$ for all $y$. But what about $f_1$ and $f_2$? Which joint testing problem is harder? The following lemma answers the question.\n\n\\begin{lemma} \\label{lem:comparison}\n\tIf $f_1^y\\geqslant f_2^y$ for all $y\\in Y$, then $f_1\\geqslant f_2$.\n\\end{lemma}\n\n\\input{figures\/comparisonTikZ}\n\nLet's first use the lemma to show the well-definedness of $\\otimes$ and the composition theorem. Its own proof comes afterwards.\nRecall that in \\Cref{def:product}, $f\\otimes g$ is defined as $T(P\\times P',Q\\times Q')$ if $f=T(P,Q), g = T(P',Q')$. To show this definition does not depend on the choice of $P,Q$ and $P',Q'$, it suffices to verify that when $f = T(P,Q) = T(\\tilde{P},\\tilde{Q})$, we have $T(P\\times P', Q \\times Q') = T(\\tilde{P} \\times P', \\tilde{Q} \\times Q')$. The following lemma is slightly stronger than what we need, but will be useful later.\n\\begin{lemma} \\label{lem:welldefined}\nIf $T(P,Q) \\geqslant T(\\tilde{P},\\tilde{Q})$, then\n\\[ T(P\\times P', Q \\times Q') \\geqslant T(\\tilde{P} \\times P', \\tilde{Q} \\times Q').\\]\nAs a consequence, if the assumption holds with an equality, then so does the conclusion.\n\\end{lemma}\n\\input{figures\/ordering}\n\\begin{proof}\n\tIn order to fit it into the setting of \\Cref{lem:comparison}, let the algorithms output a random variable independent of the input $y$. See \\Cref{fig:well-definedness}. The input-output joint distributions are just product distributions, so by the comparison \\cref{lem:comparison},\n\n\\[ T(P\\times P', Q \\times Q') \\geqslant T(\\tilde{P} \\times P', \\tilde{Q} \\times Q').\\]\n\n\tWhen $T(P,Q) = T(\\tilde{P},\\tilde{Q})$, we can apply the lemma in both directions and conclude that \n\\[ T(P\\times P', Q \\times Q') = T(\\tilde{P} \\times P', \\tilde{Q} \\times Q').\\]\n\tThe proof is complete.\n\\end{proof}\n\n\nNow that we have justified the definition of the composition tensor $\\otimes$, \\cref{lem:welldefined} can be written in a concise way:\n\\begin{equation}\\label{eq:ordering}\n\tg_1\\geqslant g_2\\Rightarrow f\\otimes g_1\\geqslant f\\otimes g_2.\n\\end{equation}\nThis is actually the second property we listed after the definition of $\\otimes$.\n\nFor composition theorem, we prove the following two steps version:\n\\begin{lemma} \\label{lem:two_steps}\n\tSuppose in a two-step composition, the two components $M_1:X\\to Y, M_2:X\\times Y \\to Z$ satisfy\n\t\t\\begin{enumerate}\n\t\t\t\\item $M_1$ is $f$-DP;\n\t\t\t\\item $M_2(\\cdot,y):X\\to Z$ is $g$-DP for each fixed $y\\in Y$.\n\t\t\\end{enumerate}\n\t\tThen the composition $M:X\\to Y\\times Z$ is $f\\otimes g$-DP.\n\\end{lemma}\n\\input{figures\/comparison_proof}\n\\begin{proof}[Proof of \\Cref{lem:two_steps}]\n\tLet $Q,Q'$ be distributions such that $g = \\F(Q,Q')$. Fix a pair of neighboring datasets $S$ and $S'$ and set everything as in \\Cref{fig:comparison_proof}. The input $y$ is an element in the output space of $M_1$. Arrows to the left correspond to the mechanism $M_2$, while arrows to the right ignore the input $y$ and output $Q,Q'$ respectively.\n\n\n\tHere $f_1^y$ in \\Cref{lem:comparison} is $T\\big(M_2(S,y),M_2(S',y)\\big)\\geqslant g$, so the condition in \\Cref{lem:comparison} checks. Consequently,\n\\begin{align*}\n\t\\F\\big(M(S),M(S')\\big)&\\geqslant \\F\\big(M_1(S)\\times Q,M_1(S')\\times Q'\\big) &&\\text{(\\Cref{lem:comparison})}\\\\\n\t&= \\F\\big(M_1(S),M_1(S')\\big) \\otimes \\F(Q,Q')&&\\text{(Def. of $\\otimes$)}\\\\\n\t&= \\F\\big(M_1(S),M_1(S')\\big) \\otimes g\\\\\n\t&\\geqslant f\\otimes g&&\\text{(Privacy of $M_1$ and \\eqref{eq:ordering})}\n\\end{align*}\nThe proof is complete.\n\\end{proof}\n\nNow we prove \\Cref{lem:comparison}. The proof is basically careful application of Neyman-Pearson Lemma \\ref{thm:NPlemma}.\n\n\\begin{proof}[Proof of \\Cref{lem:comparison}]\n\tIn order to further simplify the notations, for $i=1,2$, let $\\mu_i$ and $\\mu_i'$ be the joint distributions $\\big(P,K_i(P)\\big)$ and $\\big(P',K_i'(P')\\big)$ respectively. Then $f_1 = \\F(\\mu_1,\\mu_1'), f_2 = \\F(\\mu_2,\\mu_2')$ and we need to show that the testing problem $\\mu_1$ vs $\\mu_1'$ is harder than $\\mu_2$ vs $\\mu_2'$.\n\n\tConsider the testing problem $\\mu_1$ vs $\\mu_1'$. For $\\alpha\\in[0,1]$, let $\\phi_1:Y\\times Z_1\\to[0,1]$ be the optimal rejection rule at level $\\alpha$. By definition of trade-off function, the power of this test is $1-f_1(\\alpha)$.\n\tFormally,\n\t\\[\\E_{\\mu_1}[\\phi_1] = \\alpha, \\quad \\E_{\\mu_1'}[\\phi_1] = 1-f_1(\\alpha).\\]\n\tIt suffices to construct a rejection rule $\\phi_2:Y\\times Z_2\\to[0,1]$ for the problem $\\mu_2$ vs $\\mu_2'$, at the same level $\\alpha$ but with greater power, i.e.\n\t$$\\E_{\\mu_2} [\\phi_2] = \\alpha~~ \\text{ and  }~~ \\E_{\\mu_2'} [\\phi_2]\\geqslant \\E_{\\mu_1'}[\\phi_1] = 1-f_1(\\alpha).$$\n\tIf such $\\phi_2$ exists, then by the sub-optimality of $\\phi_2$ for the problem $\\mu_2$ vs $\\mu_2'$,\n\t\\[1-f_2(\\alpha)\\geqslant \\E_{\\mu_2'} [\\phi_2] \\geqslant 1-f_1(\\alpha),\\]\n\twhich is what we want.\n\n\tFor $y\\in Y$, let $\\phi_1^y:Z_1\\to[0,1]$ be the slice of $\\phi_1$ at $y$, i.e. $\\phi_1^y(z_1) = \\phi_1(y,z_1)$. This is a rejection rule for the problem $K_1(y)$ vs $K_1'(y)$, sub-optimal in general. The type I error is\n\t\\[\\alpha^y := \\E_{z_1\\sim K_1(y)}[\\phi_1^y(z_1)].\\]\n\tThe power is\n\t\\[\\E_{z_1\\sim K_1'(y)}[\\phi_1^y(z_1)]\\leqslant 1-f_1^y(\\alpha^y).\\]\n\tThe last inequality holds because $f_1^y=T\\big(K_1(y),K_1'(y)\\big)$ and that $\\phi_1^y$ is sub-optimal for this problem.\n\tLet $\\phi_2^y:Z_2\\to[0,1]$ be the optimal rejection rule for the testing $K_2(y)$ vs $K_2'(y)$ at level $\\alpha^y$. Construction of $\\phi_2:Y\\times Z_2\\to[0,1]$ is simply putting together these slices $\\phi_2^y$. Formally, $\\phi_2(y,z_2) = \\phi_2^y(z_2)$. Its level is $\\alpha$ because $\\alpha^y$ are averaged in terms of the same distribution $P$. More precisely,\n\t\\begin{align*}\n\t\t\\E_{\\mu_2}[\\phi_2] &= \\E_{y\\sim P}\\big[\\E_{z_2\\sim K_2(y)}[\\phi_2^y(z_2)]\\big] &&\\text{(Construction of $\\phi_2$)}\\\\\n\t\t&= \\E_{y\\sim P}[\\alpha^y]&&\\text{($\\phi_2^y$ has level $\\alpha^y$)}\\\\\n\t\t&= \\E_{y\\sim P}\\big[\\E_{z_1\\sim K_1(y)}[\\phi_1^y(z_1)]\\big]&&\\text{(Def. of $\\alpha^y$)}\\\\\n\t\t&= \\E_{\\mu_1}[\\phi_1] = \\alpha.\n\t\\end{align*}\n\tLet's compute its power:\n\t\\begin{align*}\n\t\t\\E_{\\mu_2'}[\\phi_2] &= \\E_{y\\sim P'}\\big[\\E_{z_2\\sim K_2'(y)}[\\phi_2^y(z_2)]\\big]\\\\\n\t\t&= \\E_{y\\sim P'}\\big[1-f^y_2(\\alpha^y)\\big] &&\\text{($\\phi_2^y$ is optimal)}\\\\\n\t\t&\\geqslant \\E_{y\\sim P'}\\big[1-f^y_1(\\alpha^y)\\big]&&\\text{($f^y_1\\geqslant f^y_2$)}\\\\\n\t\t&\\geqslant\\E_{y\\sim P'}\\big[\\E_{z_1\\sim K_1'(y)}[\\phi_1^y(z_1)]\\big] &&\\text{($\\phi_1^y$ is sub-optimal)}\\\\\n\t\t&=\\E_{\\mu_1'}[\\phi_1] = 1-f_1(\\alpha).&&\\text{(Optimality of $\\phi_1$ for $\\mu_1$ vs $\\mu_1'$)}\n\t\\end{align*}\n\tSo $\\phi_2$ constructed this way does have the desired level and power. The proof is complete.\n\\end{proof} \n\\subsection{Central Limit Theorems for Composition}\n\\label{sub:a_berry_esseen_type_of_clt}\n\n\n\n\\newcommand{\\boldsymbol{\\mathrm{kl}}}{\\boldsymbol{\\mathrm{kl}}}\n\\newcommand{\\boldsymbol{\\kappa_2}}{\\boldsymbol{\\kappa_2}}\n\\newcommand{\\boldsymbol{\\kappa_3}}{\\boldsymbol{\\kappa_3}}\n\\newcommand{\\boldsymbol{\\bar{\\kappa}_3}}{\\boldsymbol{\\bar{\\kappa}_3}}\n\n\nIn this subsection, we identify a central limit theorem type phenomenon of composition in the $f$-DP framework. Our main results (\\Cref{thm:Berry} and \\Cref{thm:CLT}), roughly speaking, show that trade-off functions corresponding to small privacy leakage accumulate to $G_\\mu$ for some $\\mu$ under composition. Equivalently, the privacy of the composition of many ``very private'' mechanisms is best measured by GDP in the limit. This identifies GDP as the focal privacy definition among the family of $f$-DP privacy guarantees, including $(\\epsilon,\\delta)$-DP. More precisely, \\emph{all} privacy definitions that are based on a hypothesis testing formulation of ``indistinguishability'' converge to the guarantees of GDP in the limit of composition. We remark that \\cite{sommer2018privacy} proved a conceptually related central limit theorem for random variables corresponding to the privacy loss. This theorem is used to reason about the non-adaptive composition for $(\\epsilon,\\delta)$-DP. In\ncontrast, our central limit theorem is concerned with the optimal hypothesis testing trade-off functions for the composition theorem. Moreover, our theorem is applicable in the setting of composition, where each mechanism is informed by prior interactions with the same database.\n\n\nFrom a computational viewpoint, these limit theorems yield an efficient method of approximating the composition of general $f$-DP mechanisms. This is very appealing for analyzing the privacy properties of algorithms that are comprised of many building blocks in a sequence. For comparison, the exact computation of privacy guarantees under composition can be computationally hard \\cite{complexity} and, thus, tractable approximations are important. Using our central limit theorems, the computation of the exact overall privacy guarantee $f_1\\otimes\\cdots\\otimes f_n$ in \\Cref{thm:n_steps} can be reduced to the evaluation of a single mean parameter $\\mu$ in a GDP guarantee. We give an exemplary application of this powerful technique in \\Cref{sec:application_in_sgd}.\n\n\n\n\n\nExplicitly, the mean parameter $\\mu$ in the approximation depends on certain functionals of the trade-off functions\\footnote{Although the trade-off function satisfies $f'(x) \\le 0$ almost everywhere on $[0, 1]$, we prefer to use $|f'(x)|$ instead of $-f'(x)$ for aesthetic reasons.}:\n\\begin{align*}\n\t\\mathrm{kl}(f) &:= -\\int_0^1\\log |f'(x)|\\diff x\\\\\n\n\t\\kappa_2(f)&:=\\int_0^1\\log^2 |f'(x)| \\diff x\\\\% \\int_0^1\\big(\\log g(x)-\\mu\\big)^2\\diff x\\\\\n\t\\kappa_3(f)&:=\\int_0^1\\big|\\log |f'(x)|\\big|^3\\diff x\\\\% \\int_0^1\\big(\\log g(x)-\\mu\\big)^2\\diff x\\\\\n\t\\bar{\\kappa}_3(f)&:=\\int_0^1\\big|\\log |f'(x)|+\\mathrm{kl}(f)\\big|^3\\diff x.\n\n\n\\end{align*}\nAll of these functionals take values in $[0,+\\infty]$, and the last is defined for $f$ such that $\\mathrm{kl}(f) < \\infty$. In essence, these functionals are calculating moments of the log-likelihood ratio of $P$ and $Q$ such that $f=T(P,Q)$. In particular, all of these functionals are 0 if $f(x) = \\Id(x) = 1 - x$, which corresponds to zero privacy leakage. As its name suggests, $\\mathrm{kl}(f)$ is the Kullback--Leibler (KL) divergence of $P$ and $Q$ and, therefore, $\\mathrm{kl}(f) \\ge 0$. Detailed elaboration on these functionals is deferred to \\Cref{app:CLT}.\n\n\n\nIn the following theorem, $\\boldsymbol{\\mathrm{kl}}$ denotes the vector $\\big(\\mathrm{kl}(f_{1}),\\ldots, \\mathrm{kl}(f_{n})\\big)$ and $\\boldsymbol{\\kappa_2}, \\boldsymbol{\\kappa_3},\\boldsymbol{\\bar{\\kappa}_3}$ are defined similarly; in addition, $\\|\\cdot\\|_1$ and $\\|\\cdot\\|_2$ are the $\\ell_1$ and $\\ell_2$ norms, respectively.\n\\begin{restatable}{theorem}{berryrep} \\label{thm:Berry}\nLet $f_1,\\ldots, f_n$ be symmetric trade-off functions such that $\\kappa_3(f_i) < \\infty$ for all $1 \\le i \\le n$. Denote\n\\[\n\\mu:= \\frac{2\\|\\boldsymbol{\\mathrm{kl}}\\|_1}{\\sqrt{\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2}}\\,\\, ~ \\text{and} \\,\\, ~ \\gamma:=\\frac{0.56\\|\\boldsymbol{\\bar{\\kappa}_3}\\|_1}{\\big(\\|\\boldsymbol{\\kappa_2}\\|_1 - \\|\\boldsymbol{\\mathrm{kl}}\\|_2^2\\big)^{3\/2}}\n\\]\nand assume $\\gamma < \\frac12$. Then, for all $\\alpha\\in[\\gamma,1-\\gamma]$, we have\\footnote{We can extend $G_{{\\mu}}$ to be 1 in $(-\\infty,0)$ and 0 in $(1,+\\infty)$ so that the assumption that $\\alpha\\in[\\gamma,1-\\gamma]$ can be removed.}\n\\begin{equation}\\label{eq:lower_upper}\nG_\\mu(\\alpha+\\gamma)-\\gamma\\leqslant f_{1}\\otimes f_{2} \\otimes \\cdots \\otimes f_{n}(\\alpha)\\leqslant G_\\mu(\\alpha-\\gamma)+\\gamma.\n\\end{equation}\n\n\\end{restatable}\n\n\nLoosely speaking, the lower bound in \\eqref{eq:lower_upper} shows that the composition of $f_i$-DP mechanisms for $i = 1, \\ldots, n$ is approximately $\\mu$-GDP and, in addition, the upper bound demonstrates that the tightness of this approximation is specified by $\\gamma$. In the case where all $f_i$ are equal to some $f\\neq \\Id$, the theorem reveals that the composition becomes blatantly non-private as $n \\to \\infty$ because $\\mu \\asymp \\sqrt{n} \\to \\infty$. More interesting applications of the theorem, however, are cases where each $f_i$ is close to the ``perfect privacy'' trade-off function $\\Id$ such that collectively $\\mu$ is convergent and $\\gamma$ vanishes as $n \\to \\infty$ (see the example in \\Cref{sec:application_in_sgd}). For completeness, the condition $\\kappa_3(f_i) < \\infty$ (which implies that the other three functionals are also finite) for the use of this theorem excludes the case where $f_i(0) < 1$, in particular, $f_{\\epsilon,\\delta}$ in $(\\epsilon, \\delta)$-DP with\n$\\delta > 0$. We introduce an easy and general technique in \\Cref{sub:_ep_0_dp_} to deal with this issue.\n\n\n\n\nFrom a technical viewpoint, Theorem~\\ref{thm:Berry} can be thought of as a Berry--Esseen type central limit theorem.\nThe detailed proof, as well as that of \\Cref{thm:CLT}, is provided in \\Cref{app:CLT}.\n\n\n\n\nNext, we present an asymptotic version of \\Cref{thm:Berry} for composition of $f$-DP mechanisms. In analogue to classical central limit theorems, below we consider a triangular array of mechanisms  $\\{M_{n1},\\ldots, M_{nn}\\}_{n=1}^{\\infty}$, where $M_{ni}$ is $f_{ni}$-DP for $1 \\le i \\le n$.\n\n\n\n\\begin{restatable}{theorem}{asymprep} \\label{thm:CLT}\nLet $\\{f_{ni}: 1\\leqslant i \\leqslant n\\}_{n=1}^{\\infty}$ be a triangular array of symmetric trade-off functions and assume the following limits for some constants $K \\ge 0$ and $s > 0$ as $n \\to \\infty$:\n\n\t\\begin{enumerate}\n\t\t\\item[\\textup{1.}] $\\sum_{i=1}^n \\mathrm{kl}(f_{ni})\\to K;\n\t\t\\item[\\textup{2.}] $\\max_{1\\leqslant i\\leqslant n} \\mathrm{kl}(f_{ni}) \\to 0;$\n\t\t\\item[\\textup{3.}] $\\sum_{i=1}^n \\kappa_2(f_{ni})\\to s^2;\n\t\t\\item[\\textup{4.}] $\\sum_{i=1}^n \\kappa_3(f_{ni})\\to 0$.\n\t\\end{enumerate}\nThen, we have\n\t$$\\lim_{n\\to \\infty} f_{n1}\\otimes f_{n2} \\otimes \\cdots \\otimes f_{nn} (\\alpha) = G_{2K\/s}(\\alpha)$$\nuniformly for all $\\alpha \\in [0,1]$.\n\\end{restatable}\n\nTaken together, this theorem and \\Cref{thm:n_steps} amount to saying that the composition $M_{n1} \\otimes \\ldots \\otimes M_{nn}$ is asymptotically ${2K\/s}$-GDP. In fact, this asymptotic version is a consequence of Theorem~\\ref{thm:Berry} as one can show $\\mu \\to 2K\/s$ and $\\gamma \\to 0$ for the triangular array of symmetric trade-off functions. This central limit theorem implies that GDP is the \\emph{only} parameterized family of trade-off functions that can faithfully represent the effects of composition. In contrast, neither $\\epsilon$- nor $(\\epsilon,\\delta)$-DP can losslessly be tracked under composition---the parameterized family of functions $f_{\\epsilon,\\delta}$ cannot represent the trade-off function that results from the limit under composition.\n\n\n\n\n\n\nThe conditions for use of this theorem are reminiscent of Lindeberg's condition in the central limit theorem for independent random variables. The proper scaling of the trade-off functions is that both $\\mathrm{kl}(f_{ni})$ and $\\kappa_2(f_{ni})$ are of order $O(1\/n)$ for most $1 \\le i \\le n$. As a consequence, the cumulative effects of the moment functionals are bounded. Furthermore, as with Lindeberg's condition, the second condition in Theorem~\\ref{thm:CLT} requires that no single mechanism has a significant contribution to the composition in the limit.\n\n\n\n\nIn passing, we remark that $K$ and $s$ satisfy the relationship $s = \\sqrt{2K}$ in all examples of the application of \\Cref{thm:CLT} in this paper, including \\Cref{thm:DPCLT} and \\Cref{thm:mixtureSGD} as well as their corollaries. As such, the composition is asymptotically $s$-GDP. A proof of this interesting observation or the construction of a counterexample is left for future work.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Composition and Limit Theorems}\n\\label{sec:composition-theorems}\n\n\n\\newcommand{\\mathrm{kl}}{\\mathrm{kl}}\n\\newcommand{\\mathrm{lk}}{\\mathrm{lk}}\n\\renewcommand{\\mathrm{Proc}}{\\mathrm{Proc}}\n\nImagine that an analyst performs a sequence of analyses on a private dataset, in which each analysis is informed by prior analyses on the same dataset. Provided that every analysis alone is private, the question is whether all analyses collectively are private, and if so, how the privacy degrades as the number of analyses increases, namely under composition. It is essential for a notion of privacy to gracefully handle composition, without which the privacy analysis of complex algorithms would be almost impossible.\n\n\nNow, we describe the composition of two mechanisms. For simplicity, this section writes $X$ for the space of datasets and abuse notation by using $n$ to refer to the number of mechanisms in composition\\footnote{As will be clear later, the use of $n$ is consistent with the literature on central limit theorems.}. Let $M_1: X \\to Y_1$ be the first mechanism and $M_2: X \\times Y_1 \\to Y_2$ be the second mechanism. In brief, $M_2$ takes as input the output of the first mechanism $M_1$ in addition to the dataset. With the two mechanisms in place, the joint mechanism $M: X \\to Y_1 \\times Y_2$ is defined as\n\\begin{equation}\\label{eq:M_2_comp}\nM(S) = (y_1, M_2(S, y_1)),\n\\end{equation}\nwhere $y_1 = M_1(S)$.\\footnote{Alternatively, we can write $M(S) = (M_1(S), M_2(S, M_1(S)))$, in which case it is necessary to specify that $M_1$ should be run only once in this expression.} Roughly speaking, the distribution of $M(S)$ is constructed from the marginal distribution of $M_1(S)$ on $Y_1$ and the conditional distribution of $M_2(S, y_1)$ on $Y_2$ given $M_1(S) = y_1$. The composition of more than two mechanisms follows recursively. In general, given a sequence of mechanisms $M_i: X \\times Y_1 \\times \\cdots \\times Y_{i-1} \\to Y_i$ for $i=1,2,\\ldots, n$, we can recursively define the joint mechanism as their composition:\n\\[\nM: X \\to Y_1 \\times \\cdots \\times Y_n.\n\\]\nPut differently, $M(S)$ can be interpreted as the trajectory of a Markov chain whose initial distribution is given by $M_1(S)$ and the transition kernel $M_i(S, \\cdots)$ at each step.\n\n\nUsing the language above, the goal of this section is to relate the privacy loss of $M$ to that of the $n$ mechanisms $M_1, \\ldots, M_n$ in the $f$-DP framework. In short, Section~\\ref{sub:composition_theorem} develops a general composition theorem for $f$-DP. In Sections~\\ref{sub:a_berry_esseen_type_of_clt}, we identify a central limit theorem phenomenon of composition in the $f$-DP framework, which can be used as an approximation tool, just like we use the central limit theorem for random variables. This approximation is extended to and improved for $(\\epsilon, \\delta)$-DP in Section~\\ref{sub:_ep_0_dp_}.\n\n\n\n\n\n\n\n\n\n\\subsection{A General Composition Theorem}\n\\label{sub:composition_theorem}\n\n\nThe main thrust of this subsection is to demonstrate that the composition of private mechanisms is closed and tight\\footnote{\\Cref{sub:group_privacy} shows that $f$-DP is ``closed and tight'' in a similar sense, in terms of the guarantees of group privacy.} in the $f$-DP framework. This result is formally stated in Theorem~\\ref{thm:n_steps}, which shows that the composed mechanism remains $f$-DP with the trade-off function taking the form of a certain product.  To define the product, consider two trade-off functions $f$ and $g$ that are given as $f = \\F(P, Q)$ and $g = \\F(P', Q')$ for some probability distributions $P, P', Q, Q'$.\n\n\n\n\n\\begin{definition} \\label{def:product}\nThe tensor product of two trade-off functions $f = \\F(P, Q)$ and $g = \\F(P', Q')$ is defined as\n$$f\\otimes g := \\F(P\\times P', Q\\times Q').$$\n\\end{definition}\nThroughout the paper, write $f \\otimes g (\\alpha)$ for $(f \\otimes g) (\\alpha)$, and denote by $f^{\\otimes n}$ the $n$-fold tensor product of $f$. The well-definedness of $f^{\\otimes n}$ rests on the associativity of the tensor product, which we will soon illustrate.\n\nBy definition, $f\\otimes g$ is also a trade-off function. Nevertheless, it remains to be shown that the tensor product is well-defined: that is, the definition is independent of the choice of distributions used to represent a trade-off function. More precisely, assuming $f = \\F(P, Q) = \\F(\\tilde P, \\tilde Q)$ for some distributions $\\tilde P, \\tilde Q$, we need to ensure that\n\\begin{equation*}\\label{eq:tensor_welldef}\nT(P\\times P', Q \\times Q') = T(\\tilde{P} \\times P', \\tilde{Q} \\times Q').\n\\end{equation*}\nWe defer the proof of this intuitive fact to \\Cref{app:self}.  Below we list some other useful properties\\footnote{These properties make the class of trade-off functions a \\textit{commutative monoid}. Informally, a monoid is a group without the inverse operator.} of the tensor product of trade-off functions, whose proofs are placed in \\Cref{app:CLT}.\n\\begin{enumerate}\n\\setlength\\itemsep{0.15em}\n\\item The product $\\otimes$ is commutative and associative.\n\\item If $g_1\\geqslant g_2$, then $f\\otimes g_1 \\geqslant f\\otimes g_2$.\n\\item $f \\otimes \\Id = \\Id \\otimes f = f$, where the identity trade-off function $\\Id(x)=1-x$ for $0 \\le x \\le 1$.\n\\item $(f\\otimes g)^{-1} = f^{-1}\\otimes g^{-1}$. See the definition of inverse in \\eqref{eq:inver_f}.\n\\end{enumerate}\nNote that $\\Id$ is the trade-off function of two identical distributions. Property 4 implies that when $f,g$ are symmetric trade-off functions, their tensor product $f\\otimes g$ is also symmetric.\n\n\n\n\n\n\nNow we state the main theorem of this subsection. Its proof is given in \\Cref{app:self}.\n\\begin{theorem} \\label{thm:n_steps}\nLet $M_i(\\cdot,y_1,\\cdots,y_{i-1})$ be $f_i$-DP for all $y_1 \\in Y_1, \\ldots, y_{i-1}\\in Y_{i-1}$. Then the $n$-fold composed mechanism $M:X\\to Y_1\\times\\cdots\\times Y_n$ is $f_1\\otimes\\cdots\\otimes f_n$-DP.\n\\end{theorem}\n\n\nThis theorem shows that the composition of mechanisms remains $f$-DP or, put differently, composition is closed in the $f$-DP framework. Moreover, the privacy bound $f_1\\otimes\\cdots\\otimes f_n$ in Theorem~\\ref{thm:n_steps} is \\textit{tight} in the sense that it cannot be improved in general. To see this point, consider the case where the second mechanism completely ignores the output of the first mechanism. In that case, the composition obeys\n$$\n\\begin{aligned}\nT\\big(M(S),M(S')\\big) &= T\\big(M_1(S)\\times M_2(S),M_1(S')\\times M_2(S')\\big) \\\\\n&= T\\big(M_1(S),M_1(S')\\big)\\otimes T\\big(M_2(S),M_2(S')\\big).\n\\end{aligned}\n$$\nNext, taking neighboring datasets such that $T\\big(M_1(S),M_1(S')\\big) = f_1$ and $T\\big(M_2(S),M_2(S')\\big) = f_2$, one concludes that $f_1 \\otimes f_2$ is the tightest possible bound on the two-fold composition. For comparison, the advanced composition theorem for $(\\ep, \\delta)$-DP does not admit a single pair of optimal parameters $\\epsilon, \\delta$ \\cite{boosting}. In particular, no pair of $\\ep,\\delta$ can exactly capture the privacy of the composition of $(\\ep,\\delta)$-DP mechanisms. See \\Cref{sub:_ep_0_dp_} and \\Cref{fig:comp} for more elaboration.\n\n\n\n\n\n\n\nIn the case of GDP,  composition enjoys a simple and convenient formulation due to the identity\n\\[\nG_{\\mu_1} \\otimes G_{\\mu_2} \\otimes \\cdots \\otimes G_{\\mu_n} = G_{\\mu},\n\\]\nwhere $\\mu = \\sqrt{\\mu_1^2+\\cdots+\\mu_n^2}$. This formula is due to the rotational invariance of Gaussian distributions with identity covariance. We provide the proof in \\Cref{app:CLT}. The following corollary formally summarizes this finding.\n\n\n\n\\begin{corollary}\\label{cor:gdp_comp}\nThe $n$-fold composition of $\\mu_i$-GDP mechanisms is $\\sqrt{\\mu_1^2+\\cdots+\\mu_n^2}$-GDP.\n\\end{corollary}\n\nOn a related note, the pioneering work \\cite{KOV} is the first to take the hypothesis testing viewpoint in the study of privacy composition and to use Blackwell's theorem as an analytic tool therein. In particular, the authors offered a composition theorem for $(\\ep,\\delta)$-DP that improves on the advanced composition theorem \\cite{boosting}. Following this work, \\cite{complexity} provided a self-contained proof by essentially proving the ``$(\\ep,\\delta)$ special case'' of Blackwell's theorem. In contrast, our novel proof of \\Cref{thm:n_steps} only makes use of the Neyman--Pearson lemma, thereby circumventing the heavy machinery of Blackwell's theorem. This simple proof better illuminates the essence of the composition theorem.\n\n\n\n\n\n\\input{CLT\/Berry}\n\\input{CLT\/e0CLT}\n\n\n\n\n\n\n\n\\subsection{Composition of $(\\ep,\\delta)$-DP: Beating Berry--Esseen}\n\\label{sub:_ep_0_dp_}\n\n\\newcommand{\\mathrm{Bern}}{\\mathrm{Bern}}\n\n\n\nNow, we extend central limit theorems to $(\\ep,\\delta)$-DP. As shown by \\Cref{thm:privacy_testing}, $(\\ep, \\delta)$-DP is equivalent to $f_{\\epsilon, \\delta}$-DP and, therefore, it suffices to approximate the trade-off function $f_{\\ep_{1},\\delta_{1}}\\otimes \\cdots \\otimes f_{\\ep_{n},\\delta_{n}}$ by making use of the composition theorem for $f$-DP mechanisms. As pointed out in Section~\\ref{sub:a_berry_esseen_type_of_clt}, however, the moment conditions required in the two central limit theorems (Theorems~\\ref{thm:Berry} and \\ref{thm:CLT}) exclude the case where $\\delta_i > 0$.\n\nTo overcome the difficulty caused by a nonzero $\\delta$, we start by observing the useful fact that\n\\begin{equation}\\label{eq:decomposition}\nf_{\\ep,\\delta} = f_{\\ep,0}\\otimes f_{0,\\delta}.\n\\end{equation}\nThis decomposition, along with the commutative and associative properties of the tensor product, shows\n$$f_{\\ep_{1},\\delta_{1}}\\otimes \\cdots \\otimes f_{\\ep_{n},\\delta_{n}} = \\big(f_{\\ep_{1},0}\\otimes \\cdots \\otimes f_{\\ep_{n},0}\\big)\\otimes \\big(f_{0,\\delta_{1}}\\otimes \\cdots \\otimes f_{0,\\delta_{n}}\\big).$$\nThis identity allows us to work on the $\\epsilon$ part and $\\delta$ part separately. In short, the $\\epsilon$ part $f_{\\ep_{1},0}\\otimes \\cdots \\otimes f_{\\ep_{n},0}$ now can be approximated by $G_{\\sqrt{\\ep_1^2+\\cdots+\\ep_n^2}}$ by invoking Theorem~\\ref{thm:CLT}. For the $\\delta$ part, we can iteratively apply the rule\n\\begin{equation}\\label{eq:delta}\nf_{0,\\delta_1}\\otimes f_{0,\\delta_2} = f_{0,1-(1-\\delta_1)(1-\\delta_2)}\n\\end{equation}\nto obtain $f_{0,\\delta_{1}}\\otimes \\cdots \\otimes f_{0,\\delta_{n}} = f_{0, 1 - (1 - \\delta_1)(1 - \\delta_2) \\cdots (1 - \\delta_n)}$. This rule is best seen via the interesting fact that $f_{0,\\delta}$ is the trade-off function of shifted uniform distributions $T\\big(U[0,1],U[\\delta,1+\\delta]\\big)$.\n\n\nNow, a central limit theorem for $(\\epsilon, \\delta)$-DP is just a stone's throw away. In what follows, the privacy parameters $\\ep$ and $\\delta$ are arranged in a triangular array $\\{(\\ep_{ni},\\delta_{ni}):1\\leqslant i\\leqslant n\\}_{n=1}^{\\infty}$.\n\\begin{restatable}{theorem}{DPCLTrep}\\label{thm:DPCLT}\nAssume\n$$\\sum_{i=1}^n \\ep_{ni}^2 \\to \\mu^2, \\quad \\max_{1\\leqslant i\\leqslant n} \\ep_{ni}\\to 0, \\quad \\sum_{i=1}^n \\delta_{ni} \\to \\delta, \\quad \\max_{1\\leqslant i\\leqslant n} \\delta_{ni}\\to 0$$\nfor some nonnegative constants $\\mu,\\delta$ as $n \\to \\infty$. Then, we have\n$$f_{\\ep_{n1},\\delta_{n1}}\\otimes \\cdots \\otimes f_{\\ep_{nn},\\delta_{nn}} \\to G_{\\mu}\\otimes f_{0, 1 - \\mathrm{e}^{-\\delta}}$$\nuniformly over $[0, 1]$ as $n \\to \\infty$.\n\\end{restatable}\n\\begin{remark}\nA formal proof is provided in \\Cref{app:CLT}. The assumptions concerning $\\{\\delta_{ni}\\}$ give rise to $1 - (1 - \\delta_{n1})(1 - \\delta_{n2}) \\cdots (1 - \\delta_{nn}) \\to 1 - \\mathrm{e}^{-\\delta}$. In general, tensoring with $f_{0,\\delta}$ is equivalent to scaling the graph of the trade-off function $f$ toward the origin by a factor of $1-\\delta$. This property is specified by the following formula, and we leave its proof to \\Cref{app:CLT}:\n\t\\begin{equation}\\label{prop:ruibbit}\n\tf\\otimes f_{0,\\delta}(\\alpha) =\n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\t(1-\\delta)\\cdot f(\\frac{\\alpha}{1-\\delta}), \t\t& 0\\leqslant \\alpha \\leqslant 1-\\delta \\\\\n\t\t0, & 1-\\delta\\leqslant \\alpha\\leqslant 1.\n\t\t\\end{array}\n\t\t\\right.\n\t\\end{equation}\nIn particular, $f\\otimes f_{0,\\delta}$ is symmetric if $f$ is symmetric. Note that \\eqref{eq:decomposition} and \\eqref{eq:delta} can be deduced by the formula above.\n\\end{remark}\n\nThis theorem interprets the privacy level of the composition using Gaussian and uniform distributions. Explicitly, the theorem demonstrates that, based on the released information of the composed mechanism, distinguishing between any neighboring datasets is at least as hard as distinguishing between the following two bivariate distributions:\n\\[\n\\N(0, 1)\\times U[0, 1]  \\text{ versus }  \\N(\\mu, 1)\\times U[1 - \\mathrm{e}^{-\\delta}, 2 - \\mathrm{e}^{-\\delta}].\n\\]\nWe note that for small $\\delta$, $\\mathrm{e}^{-\\delta}\\approx 1-\\delta$. So $U[1 - \\mathrm{e}^{-\\delta}, 2 - \\mathrm{e}^{-\\delta}]\\approx U[\\delta,1+\\delta]$.\n\nThis approximation of the tensor product $f_{\\ep_{n1},\\delta_{n1}}\\otimes \\cdots \\otimes f_{\\ep_{nn},\\delta_{nn}}$ using simple distributions is important from the viewpoint of computational complexity. Murtagh and Vadhan \\cite{complexity} showed that, given a collection of $\\{(\\ep_i,\\delta_i)\\}_{i=1}^n$, finding the smallest $\\ep$ such that $f_{\\ep,\\delta}\\leqslant f_{\\ep_{1},\\delta_{1}}\\otimes \\cdots \\otimes f_{\\ep_{n},\\delta_{n}}$ is \\#P-hard\\footnote{\\#P is a complexity class that is ``even harder than'' NP (i.e.~a polynomial time algorithm for any \\#P-hard problem would imply P=NP).  See, e.g.,  Ch. 9.~of \\cite{arora2009computational}.} for any $\\delta$. From the dual perspective (see \\Cref{sub:a_primal_dual_connection_with_}), this negative result is equivalent to the\n\\#P-hardness of evaluating the convex conjugate $\\big(f_{\\ep_{1},\\delta_{1}}\\otimes \\cdots \\otimes f_{\\ep_{n},\\delta_{n}}\\big)^*$ at any point. For completeness, we remark that \\cite{complexity} provided an FPTAS\\footnote{An approximation algorithm is called a fully polynomial-time approximation scheme (FPTAS) if its running time is polynomial in both the input size and the inverse of the relative approximation error. See, e.g., Ch.~8.~of \\cite{vazirani2013approximation}.} to approximately find the smallest $\\epsilon$ in $O(n^3)$ time for a \\textit{single} $\\delta$. In comparison, Theorem~\\ref{thm:DPCLT} offers a \\textit{global} approximation of the tensor product in $O(n)$ time using a closed-form expression, subsequently enabling an analytical approximation of the smallest\n$\\epsilon$ for each $\\delta$.\n\n\n\n\n\n\\begin{figure}[!htp]\n\\centering\n  \\includegraphics[width=0.75\\linewidth]{.\/figures\/edCLT.pdf}\n  \\captionof{figure}{Left: Tensoring with $f_{0,\\delta}$ scales the graph towards the origin by a factor of $1-\\delta$. Right: 10-fold composition of $(1\/\\sqrt{10},0)$-DP mechanisms, that is, $f_{\\ep,0}^{\\otimes n}$ with $n=10, \\ep=1\/\\sqrt{n}.$ The dashed curve corresponds to $\\ep=2.89,\\delta = 0.001$. These values are obtained by first setting $\\delta = 0.001$ and finding the smallest $\\ep$ such that the composition is $(\\ep,\\delta)$-DP. Note that the central limit theorem approximation to the true trade-off curve is almost perfect, whereas the tightest possible approximation via $(\\ep,\\delta)$-DP is substantially looser.}\n  \\label{fig:comp}\n\\end{figure}\n\n\n\n\nThat being said, \\Cref{thm:DPCLT} remains silent on the approximation error in applications with a moderately large number of $(\\epsilon, \\delta)$-DP mechanisms. Alternatively, we can apply \\Cref{thm:Berry} to obtain a non-asymptotic normal approximation to $f_{\\ep_{1},0}\\otimes \\cdots \\otimes f_{\\ep_{n},0}$ and use $\\gamma$ to specify the approximation error. It can be shown that $\\gamma = O(1\/\\sqrt{n})$ under mild conditions (\\Cref{cor:e0Berry}). This bound, however, is not sharp enough for tight privacy guarantees if $n$ is not too large (note that $1\/\\sqrt{n} \\approx 0.14$ if $n = 50$, for which exact computation is already challenging, if possible at all). Surprisingly, the following theorem establishes a $O(1\/n)$ bound, thereby ``beating'' the classical Berry--Esseen bound.\n\\begin{theorem} \\label{thm:fast}\nFix $\\mu > 0$ and let $\\ep = \\mu\/\\sqrt{n}$. There is a constant $c > 0$ that only depends on $\\mu$ satisfying\n\\[\nG_{\\mu}\\left( \\alpha+\\tfrac{c}{n} \\right) - \\tfrac{c}{n} \\leqslant f_{\\ep,0}^{\\otimes n}(\\alpha)\\leqslant G_{\\mu}\\left( \\alpha-\\tfrac{c}{n} \\right) + \\tfrac{c}{n}\n\\]\nfor all $n\\geqslant 1$ and $c\/n \\le \\alpha \\le 1 - c\/n$.\n\\end{theorem}\n\n\n\nAs with \\Cref{thm:DPCLT}, this theorem can be extended to approximate DP ($\\delta \\ne 0$) by making use of the decomposition \\eqref{eq:decomposition}. Our simulation studies suggest that $c \\approx 0.1$ for $\\mu = 1$, which is best illustrated in the right panel of \\Cref{fig:comp}. Despite a fairly small $n = 10$, the difference between $G_1$ and its target $f_{\\ep,0}^{\\otimes n}$ is less than 0.013 in the pointwise sense. Interestingly, numerical evidence suggests the same $O(1\/n)$ rate in the inhomogeneous composition provided that $\\ep_1, \\ldots, \\ep_n$ are roughly the same size. A formal proof, or even a quantitative statement of this observation, constitutes an interesting problem for future investigation.\n\n\nIn closing this section, we highlight some novelties in the proof of \\Cref{thm:fast}. Denoting $p_{\\epsilon} = \\frac{1}{1+\\mathrm{e}^\\ep}$ and $q_{\\epsilon} = \\frac{\\mathrm{e}^\\ep}{1+\\mathrm{e}^\\ep}$, \\cite{KOV} presented a very useful expression (rephrased in our framework):\n\\[\nf_{\\ep,0}^{\\otimes n} = \\F\\big(B(n,p_{\\ep}),B(n, q_{\\ep})\\big),\n\\]\nwhere $B(n, p)$ denotes the binomial distribution with $n$ trials and success probability $p$. However, directly approximating $f_{\\ep,0}^{\\otimes n}$ through these two binomial distributions is unlikely to yield \nan $O(1\/n)$ bound because the Berry--Esseen bound is rate-optimal for binomial distributions. Our analysis, instead, rests crucially on a certain smoothing effect that comes for free in testing between the two distributions. It is analogous to the continuity correction for normal approximations to binomial probabilities. See the technical details in \\Cref{app:CLT}.\n\n\n\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\n\nIn this paper, we have introduced a new framework for private data analysis that we refer to as $f$-differential privacy, which generalizes $(\\ep, \\delta)$-DP and has a number of attractive properties that escape the difficulties of prior work. This new privacy definition uses trade-off functions of hypothesis testing as a measure of indistinguishability of two neighboring datasets rather than a few parameters as in prior differential privacy relaxations. Our $f$-DP retains an interpretable hypothesis testing semantics and is expressive enough to losslessly reason about composition, post-processing, and group privacy by virtue of the informativeness of trade-off functions. Moreover, $f$-DP admits a central limit theorem that identifies a simple and single-parameter family of privacy\ndefinitions as focal: Gaussian differential privacy. Precisely, all hypothesis testing based definitions of privacy converge to Gaussian differential privacy in the limit under composition, which implies that Gaussian differential privacy is the unique such definition that can tightly handle composition. The central limit theorem and its Berry--Esseen variant give a tractable analytical approach to tightly analyzing the privacy cost of iterative methods such as SGD. Notably, $f$-DP is \\emph{dual} to $(\\ep, \\delta)$-DP in a constructive sense, which gives the ability to import results proven for $(\\epsilon,\\delta)$-DP. This powerful perspective allows us to obtain an easy-to-use privacy amplification by subsampling theorem for $f$-DP, which in particular significantly improves on the\nstate-of-the-art counterpart in the $(\\ep, \\delta)$-DP setting. \n\n\n\n\nWe see several promising directions for future work using and extending the $f$-DP framework. First, \\Cref{thm:fast} can possibly be extended to the inhomogeneous case where trade-off functions are different from each other in the composition. Such an extension would allow us to apply the central limit theorem for privacy approximation with strong finite-sample guarantees to a broader range of problems. Second, it would be of interest to investigate whether the privacy guarantee of the subsampled mechanism in \\Cref{thm:subsample} can be improved for some trade-off functions. Notably, we have shown in \\Cref{app:property} that this bound is tight if the trade-off function $f = 0$, that is, the original mechanism is blatantly non-private. Third, the notion of $f$-DP naturally has a \\textit{local} realization where the obfuscation of the sensitive information is applied at the individual record level. In this setting, what are the fundamental limits of estimation with local $f$-DP\nguarantees \\cite{duchi2018minimax}? In light of \\cite{duchi2018right}, what is the correct complexity measure in local $f$-DP estimation? If it is not the Fisher information, can we identify an alternative to the Fisher information for some class of trade-off functions? Moreover, we recognize that an adversary in differentially private learning may set different pairs of target type I and type II errors. For example, an adversary that attempts to control type I and II errors at 10\\% and 10\\%, respectively, can behave very differently from one who aims to control the two errors at 0.1\\% and 99\\%, respectively. An important question is to address the trade-offs between resources such as privacy and statistical efficiency and target type I and type II errors in the framework of $f$-DP. \n\nFinally, we wish to remark that $f$-DP can possibly offer a mathematically tractable and flexible framework for minimax estimation under privacy constraints\n(see, for example, \\cite{cai2019cost,bun2018fingerprinting,dwork2015robust}). Concretely, given a candidate estimator satisfying $(\\epsilon, \\delta)$-DP appearing in the upper bound and a possibly loose lower bound under the $(\\ep, \\delta)$-DP constraint, we can replace the $(\\epsilon, \\delta)$-DP constraint by the $f$-DP constraint where $f$ is the tightest trade-off function characterizing the estimation procedure. As is clear, the $f$-DP constraint is more stringent than the $(\\ep, \\delta)$-DP constraint by recognizing the primal-dual conversion (see \\Cref{prop:ftoDP}). While the upper bound remains the same as the estimator continues to satisfy the new privacy constraint, the lower bound can be possibly improved due to a more stringent constraint. It would be of great interest to investigate to what extent this $f$-DP based approach can reduce the gap between upper and lower bounds minimax estimation under privacy constraints.\n\n\nUltimately, the test of a privacy definition lies not just in its power and semantics, but also in its ability to usefully analyze diverse algorithms. In this paper, we have given convincing evidence that $f$-DP is up to the task. We leave the practical evaluation of this new privacy definition to future work.\n\n\n\n\n\n\n\\subsection{Gaussian Differential Privacy}\n\\label{sub:gaussian_differential_privacy}\n\n\n\nThis subsection introduces a parametric family of $f$-DP guarantees, where $f$ is the trade-off function of two normal distributions. We refer to this specialization as Gaussian differential privacy (GDP). GDP enjoys many desirable properties that lead to its central role in this paper. Among others, we can now precisely define the trade-off function with a single parameter. To define this notion, let\n\\[\nG_\\mu:=\\F\\big(\\N(0,1), \\N(\\mu,1)\\big)\n\\]\nfor $\\mu \\ge 0$. An explicit expression for the trade-off function $G_\\mu$ reads\n\\begin{equation}\\label{eq:Gmu}\n\tG_\\mu(\\alpha) = \\Phi\\big(\\Phi^{-1}(1-\\alpha)-\\mu\\big),\n\\end{equation}\nwhere $\\Phi$ denotes the standard normal CDF. For completeness, we provide a proof of \\eqref{eq:Gmu} in \\Cref{app:fDP}. This trade-off function is decreasing in $\\mu$ in the sense that $G_\\mu \\leqslant G_{\\mu'}$ if $\\mu \\ge \\mu'$. We now define GDP:\n\\begin{definition} \\label{def:GDP}\nA mechanism $M$ is said to satisfy $\\mu$-Gaussian Differential Privacy ($\\mu$-GDP) if it is $G_\\mu$-DP. That is,\n\\[\nT\\big(M(S),M(S')\\big) \\ge G_\\mu\n\\]\nfor all neighboring datasets $S$ and $S'$.\n\\end{definition}\nGDP has several attractive properties. First, this privacy definition is fully described by the single mean parameter of a unit-variance Gaussian distribution, which makes it easy to describe and interpret the privacy guarantees. For instance, one can see from the right panel of \\Cref{fig:DPvsGDP} that $\\mu \\le 0.5$ guarantees a reasonable amount of privacy, whereas if $\\mu \\geqslant 6$, almost nothing is being promised. Second, loosely speaking, GDP occupies a role among all hypothesis testing based notions of privacy that is similar to the role that the Gaussian distribution has among general probability distributions. We formalize this important point by proving central limit theorems for $f$-DP in \\Cref{sec:composition-theorems}, which, roughly speaking, says that $f$-DP converges to GDP under composition in the limit. Lastly, as shown in the remainder of this subsection, GDP \\emph{precisely} characterizes the Gaussian mechanism, one of the most fundamental building blocks of differential privacy.\n\n\n\n\n\n\n\n\n\nConsider the problem of privately releasing a univariate statistic $\\theta(S)$ of the dataset $S$. Define the sensitivity of $\\theta$ as\n\\[\n\\mathrm{sens}(\\theta) = \\sup_{S, S'} | \\theta(S) - \\theta(S') |,\n\\]\nwhere the supremum is over all neighboring datasets. The Gaussian mechanism adds Gaussian noise to the statistic $\\theta$ in order to obscure whether $\\theta$ is computed on $S$ or $S'$. The following result shows that the Gaussian mechanism with noise properly scaled to the sensitivity of the statistic satisfies GDP.\n\\begin{theorem}\\label{thm:g_mech}\nDefine the Gaussian mechanism that operates on a statistic $\\theta$ as $M(S) = \\theta(S) + \\xi$, where $\\xi \\sim \\N(0, \\mathrm{sens}(\\theta)^2\/\\mu^2)$. Then, $M$ is $\\mu$-GDP.\n\\end{theorem}\n\\begin{proof}[Proof of Theorem~\\ref{thm:g_mech}]\nRecognizing that $M(S), M(S')$ are normally distributed with means $\\theta(S), \\theta(S')$, respectively, and common variance $\\sigma^2 = \\mathrm{sens}(\\theta)^2\/\\mu^2$, we get\n\\begin{align*}\n\\F\\big(M(S),M(S')\\big) = \\F\\big( \\N(\\theta(S), \\sigma^2), \\N(\\theta(S'), \\sigma^2)\\big) = G_{|\\theta(S)-\\theta(S')|\/\\sigma}.\n\\end{align*}\nBy the definition of sensitivity, $|\\theta(S)-\\theta(S')|\/\\sigma \\le \\mathrm{sens}(\\theta)\/\\sigma = \\mu$. Therefore, we get\n\\[\n\\F\\big(M(S),M(S')\\big) = G_{|\\theta(S)-\\theta(S')|\/\\sigma} \\geqslant G_\\mu.\n\\]\nThis completes the proof.\n\n\n\n\n\n\\end{proof}\n\n\n\nAs implied by the proof above, GDP offers the tightest possible privacy bound of the Gaussian mechanism. More precisely, the Gaussian mechanism in \\Cref{thm:g_mech} satisfies\n\\begin{equation}\\label{eq:f_inf}\nG_{\\mu}(\\alpha) = \\inf_{\\text{neighboring }S, S'} \\, \\F \\big(M(S),M(S') \\big)(\\alpha),\n\\end{equation}\nwhere the infimum is (asymptotically) achieved at the two neighboring datasets such that $|\\theta(S) - \\theta(S')| = \\mathrm{sens}(\\theta)$ \\textit{irrespective} of the type I error $\\alpha$. As such, the characterization by GDP is precise in the pointwise sense. In contrast, the right-hand side of \\eqref{eq:f_inf} in general is not necessarily a convex function of $\\alpha$ and, in such case, is not a trade-off function according to \\Cref{prop:trade-off}. This nice property of Gaussian mechanism is related to the log-concavity of Gaussian distributions. See \\Cref{prop:logconcave} for a detailed treatment of log-concave distributions.\n\n\n\n\n\n\n\n\\subsection{A Primal-Dual Perspective}\n\\label{sub:a_primal_dual_connection_with_}\n\n\nIn this subsection, we show that $f$-DP is equivalent to an infinite \\textit{collection} of $(\\ep,\\delta)$-DP guarantees via the convex conjugate of the trade-off function. As a consequence of this, we can view $f$-DP as the \\textit{primal} privacy representation and, accordingly, its \\textit{dual} representation is the collection of $(\\ep,\\delta)$-DP guarantees. Taking this powerful viewpoint, many results from the large body of $(\\epsilon, \\delta)$-DP work can be carried over to $f$-DP in a seamless fashion. In particular, this primal-dual perspective is crucial to our analysis of ``privacy amplification by subsampling'' in \\Cref{sec:subsampling}. All proofs are deferred to \\Cref{app:fDP}.\n\n\n\nFirst, we present the result that converts a collection of $(\\epsilon, \\delta)$-DP guarantees into an $f$-DP guarantee.\n\\begin{proposition}[Dual to Primal] \\label{prop:DPtof}\nLet $I$ be an arbitrary index set such that each $i\\in I$ is associated with $\\ep_i\\in[0, \\infty)$ and $\\delta_i\\in[0,1]$. A mechanism is $(\\ep_i,\\delta_i)$-DP for all $i\\in I$ if and only if it is $f$-DP with\n$$f = \\sup_{i\\in I} f_{\\ep_i,\\delta_i}.$$\n\\end{proposition}\nThis proposition follows easily from the equivalence of $(\\ep,\\delta)$-DP and $f_{\\ep,\\delta}$-DP. We remark that the function $f$ constructed above remains a symmetric trade-off function.\n\nThe more interesting direction is to convert $f$-DP into a collection of $(\\ep,\\delta)$-DP guarantees. Recall that the convex conjugate of a function $g$ defined on $(-\\infty, \\infty)$ is defined as\n\\begin{equation}\\label{eq:conjugate}\ng^*(y) = \\sup_{-\\infty < x < \\infty} y x - g(x).\n\\end{equation}\nTo define the conjugate of a trade-off function $f$, we extend its domain by setting $f(x) = \\infty$ for $x < 0$ and $x > 1$. With this adjustment, the supremum is effectively taken over $0 \\le x \\le 1$.\n\n\n\\begin{restatable}[Primal to Dual]{proposition}{ftoDPrep} \\label{prop:ftoDP}\n\tFor a symmetric trade-off function $f$, a mechanism is $f$-DP if and only if it is $\\big(\\ep,\\delta(\\ep)\\big)$-DP for all $\\ep\\geqslant 0$ with $\\delta(\\ep)=1+f^*(-\\mathrm{e}^{\\ep})$.\n\\end{restatable}\n\n\\begin{figure}[!ht]\n\\centering\n  \\includegraphics[width=0.6\\linewidth]\n{.\/figures\/envelope.pdf}\n  \\captionof{figure}{Each $(\\ep,\\delta(\\ep))$-DP guarantee corresponds to two supporting linear functions (symmetric to each other) to the trade-off function describing the complete $f$-DP guarantee. In general, characterizing a privacy guarantee using only a subset of $(\\ep,\\delta)$-DP guarantees (for example, only those with small $\\delta$) would result in information loss.}\n  \\label{fig:envelope}\n\\end{figure}\n\n\nFor example, taking $f=G_\\mu$, the following corollary provides a lossless conversion from GDP to a collection of $(\\ep,\\delta)$-DP guarantees. This conversion is exact and, therefore, any other $(\\ep,\\delta)$-DP guarantee derived for the Gaussian mechanism is implied by this corollary. See \\Cref{fig:envelope} for an illustration of this result.\n\\begin{restatable}{corollary}{GDPtoDPrep}\\label{corr:GDPtoDP}\n    A mechanism is $\\mu$-GDP if and only if it is $\\big(\\ep,\\delta(\\ep)\\big)$-DP for all $\\ep\\geqslant0$, where\n    \\[\n    \\delta(\\ep)= \\Phi\\Big( -\\frac{\\varepsilon}{\\mu} +\\frac{\\mu}{2} \\Big)-\n    \\mathrm{e}^{\\varepsilon}\\Phi\\Big(- \\frac{\\varepsilon}{\\mu} - \\frac{\\mu}{2} \\Big).\n    \\]\n\\end{restatable}\nThis corollary has appeared earlier in \\cite{balle2018improving}. Along this direction, \\cite{balle2018privacy} further proposed ``privacy profile'', which in essence corresponds to an infinite collection of $(\\ep,\\delta)$. The notion of privacy profile mainly serves as an analytical tool in \\cite{balle2018privacy}.\n\nThe primal-dual perspective provides a useful tool through which we can bridge the two privacy definitions. In some cases, it is easier to work with $f$-DP by leveraging the interpretation and informativeness of trade-off functions, as seen from the development of composition theorems for $f$-DP in \\Cref{sec:composition-theorems}. Meanwhile, $(\\ep,\\delta)$-DP is more convenient to work with in the cases where the lower complexity of two parameters $\\epsilon,\\delta$ is helpful, for example, in the proof of the privacy amplification by subsampling theorem for $f$-DP. In short, our approach in \\Cref{sec:subsampling} is to first work in the dual world and use existing subsampling theorems for $(\\ep,\\delta)$-DP, and then convert the results back to $f$-DP using a slightly more advanced version of \\Cref{prop:ftoDP}.\n\n\n\n\n\n\n\\section{$f$-Differential Privacy and Its Basic Properties}\n\\label{sec:fDP}\n\n\n\n\n\nIn Section~\\ref{sub:trade_off_function}, we give a formal definition of $f$-DP. Section~\\ref{sub:gaussian_differential_privacy} introduces Gaussian differential privacy, a special case of $f$-DP. In Section~\\ref{sec:conn-with-blackw}, we highlight some appealing properties of this new privacy notation from an information-theoretic perspective. Next, Section~\\ref{sub:a_primal_dual_connection_with_} offers a profound connection between $f$-DP and $(\\epsilon, \\delta)$-DP. Finally, we discuss the group privacy properties of $f$-DP.\n\n\n\n\n\n\nBefore moving on, we first establish several key pieces of notation from the differential privacy literature.\n\\begin{itemize}\n\\item \n{\\bf Dataset.} A dataset $S$ is a collection of $n$ records, each corresponding to an individual. Formally, we write the dataset as $S = (x_1,\\ldots, x_n)$, and an individual $x_i \\in X$ for some abstract space $X$. Two datasets $S' = (x'_1,\\ldots, x'_n)$ and $S$ are said to be \\textit{neighbors} if they differ in exactly one record, that is, there exists an index $j$ such that $x_i = x_i'$ for all $i \\ne j$ and $x_j \\ne x_j'$.\n\n\n\n\\item {\\bf Mechanism.}  A mechanism $M$ refers to a randomized algorithm that takes as input a dataset $S$ and releases some (randomized) statistics $M(S)$ of the dataset in some abstract space $Y$. For example, a mechanism can release the average salary of individuals in the dataset plus some random noise.\n\n\n\\end{itemize}\n\n\n\\input{fDP\/tradeoff}\n\\input{fDP\/GDP}\n\\input{fDP\/post}\n\\input{fDP\/dual}\n\\input{fDP\/group}\n\n\n\n\n\\subsection{Group Privacy}\\label{sub:group_privacy}\n\\newcommand{\\mathbin{\\hat{\\circ}}}{\\mathbin{\\hat{\\circ}}}\n\n\nThe notion of $f$-DP can be extended to address privacy of a \\textit{group} of individuals, and a question of interest is to quantify how privacy degrades as the group size grows. To set up the notation, we say that two datasets $S, S'$ are $k$-neighbors (where $k \\ge 2$ is an integer) if there exist datasets $S = S_0, S_1, \\ldots, S_k = S'$ such that $S_i$ and $S_{i+1}$ are neighboring or identical for all $i = 0, \\ldots, k-1$. Equivalently, $S,S'$ are $k$-neighbors if they differ by at most $k$ individuals. Accordingly, a mechanism $M$ is said to be $f$-DP for \\textit{groups of size $k$} if\n\\[ T\\big(M(S),M(S')\\big)\\geqslant f\\]\nfor all $k$-neighbors $S$ and $S'$.\n\nIn the following theorem, we use $h^{\\circ k}$ to denote the $k$-fold iterative composition of a function $h$. For example, $h^{\\circ 1} = h$ and $h^{\\circ 2}(x) = h(h(x))$.\n\\begin{restatable}{theorem}{groupthm} \\label{thm:group}\nIf a mechanism is $f$-DP, then it is $\\left[1-(1-f)^{\\circ k}\\right]$-DP for groups of size $k$. In particular, if a mechanism is $\\mu$-GDP, then it is $k\\mu$-GDP for groups of size $k$.\n\\end{restatable}\nFor completeness, $1-(1-f)^{\\circ k}$ is a trade-off function and, moreover, remains symmetric if $f$ is symmetric. These two facts and \\Cref{thm:group} are proved in \\Cref{app:fDP}. As revealed in the proof, the privacy bound $1-(1-f)^{\\circ k}$ in general cannot be improved, thereby showing that the group operation in the $f$-DP framework is \\textit{closed} and \\textit{tight}. In addition, it is easy to see that $1 - (1-f)^{\\circ k} \\leqslant 1 - (1-f)^{\\circ (k-1)}$ by recognizing that the trade-off function $f$ satisfies $1-f(x)\\geqslant x$. This is consistent with the intuition that detecting changes in groups of $k$ individuals becomes easier as the group size increases.\n\nAs an interesting consequence of \\Cref{thm:group}, the group privacy of $\\epsilon$-DP in the limit corresponds to the trade-off function of two Laplace distributions. Recall that the density of $\\mathrm{Lap}(\\mu,b)$ is $\\frac1{2b}\\mathrm{e}^{-|x-\\mu|\/b}$.\n\\begin{restatable}{proposition}{grouplimit}\n\\label{prop:group_limit}\nFix $\\mu \\ge 0$ and set $\\epsilon = \\mu\/k$. As $k \\to \\infty$, we have\n\\[\n1-(1-f_{\\ep,0})^{\\circ k} \\to \\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big).\n\\]\nThe convergence is uniform over $[0,1]$.\n\\end{restatable}\n\nTwo remarks are in order. First, $\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)$ is not equal to $f_{\\ep,\\delta}$ for any $\\ep,\\delta$ and, therefore, $(\\ep,\\delta)$-DP is not expressive enough to measure privacy under the group operation. Second, the approximation in this theorem is very accurate even for small $k$. For example, for $\\mu=1,k=4$, the function $1-(1-f_{\\ep,0})^{\\circ k}$ is within $0.005$ of $\\F\\big(\\mathrm{Lap}(0,1),\\mathrm{Lap}(\\mu,1)\\big)$ uniformly over $[0, 1]$. The proof of \\Cref{prop:group_limit} is deferred to \\Cref{app:fDP}.\n\n\n\n\n\\subsection{Post-processing, Blackwell's Information Ordering, and Connection with Divergence-Based Definitions}\n\n\\subsection{Post-Processing and the Informativeness of $f$-DP}\n\\label{sec:conn-with-blackw}\n\nIntuitively, a data analyst cannot make a statistical analysis more disclosive only by processing the output of the mechanism $M$. This is called the post-processing property, a natural requirement that any notion of privacy, including our definition of $f$-DP, should satisfy.\n\n\n\n\nTo formalize this point for $f$-DP, denote by $\\mathrm{Proc}: Y \\to Z$ a (randomized) algorithm that maps the input $M(S) \\in Y$ to some space $Z$, yielding a new mechanism that we denote by $\\mathrm{Proc} \\circ M$. The following result confirms the post-processing property of $f$-DP.\n\\begin{proposition}\\label{prop:post_process}\nIf a mechanism $M$ is $f$-DP, then its post-processing $\\mathrm{Proc}\\circ M$ is also $f$-DP.\n\\end{proposition}\nProposition~\\ref{prop:post_process} is a consequence of the following lemma. Let $\\mathrm{Proc}(P)$ be the probability distribution of $\\mathrm{Proc}(\\zeta)$ with $\\zeta$ drawn from $P$. Define $\\mathrm{Proc}(Q)$ likewise.\n\\begin{restatable}{lemma}{postrep} \\label{lem:post}\nFor any two distributions $P$ and $Q$, we have\n$$\\F\\big(\\mathrm{Proc}(P),\\mathrm{Proc}(Q)\\big)\\geqslant \\F(P,Q).$$\n\\end{restatable}\n\nThis lemma means that post-processed distributions can only become more difficult to tell apart than the original distributions from the perspective of trade-off functions. While the same property holds for many divergence based measures of indistinguishability such as the {R\\'enyi} divergences\\footnote{See \\Cref{app:relation} for its definition and relation with trade-off functions.} used by the concentrated differential privacy family of definitions \\cite{concentrated,concentrated2,renyi,tcdp}, a consequence of the following theorem is that trade-off functions offer the most informative measure among all. This remarkable inverse of \\Cref{lem:post} is due to Blackwell (see also Theorem 2.5 in \\cite{KOV}).\n\\begin{theorem}[\\cite{blackwell1950comparison}, Theorem 10]\\label{thm:blackwell}\nLet $P,Q$ be probability distributions on $Y$ and $P',Q'$ be probability distributions on $Z$. The following two statements are equivalent:\n\\begin{enumerate}\n\\item[(a)] $\\F(P,Q)\\leqslant \\F(P',Q')$.\n\\item[(b)] There exists a randomized algorithm $\\mathrm{Proc}: Y \\to Z$ such that $\\mathrm{Proc}(P)=P',\\mathrm{Proc}(Q)=Q'$.\n\\end{enumerate}\n\\end{theorem}\n\\newcommand{\\mathrm{Ineq}}{\\mathrm{Ineq}}\nTo appreciate the implication of this theorem, we begin by observing that post-processing induces an order\\footnote{This is in general not a partial order.} on pairs of distributions, which is called the Blackwell order (see, e.g., \\cite{raginsky2011shannon}). Specifically, if the above condition (b) holds, then we write $(P,Q)\\preceq_{\\mathrm{Blackwell}}(P',Q')$ and interpret this as ``$(P,Q)$ is easier to distinguish than $(P',Q')$ in the Blackwell sense''. Similarly, when $\\F(P,Q)\\leqslant \\F(P',Q')$, we write $(P,Q)\\preceq_{\\mathrm{tradeoff}}(P',Q')$ and interpret this as ``$(P,Q)$ is easier to distinguish than $(P',Q')$ in the testing sense''. In general, any privacy measure used in defining a privacy notion induces an order $\\preceq$ on pairs of distributions. Assuming the post-processing property for the privacy notion, the induced order $\\preceq$ must be consistent with $\\preceq_{\\mathrm{Blackwell}}$. Concretely, we\ndenote by $\\mathrm{Ineq}(\\preceq) = \\{(P,Q; P', Q'): (P, Q) \\preceq (P', Q')\\}$ the set of all comparable pairs of the order $\\preceq$. As is clear, a privacy notion satisfies the post-processing property if and only if the induced order $\\preceq$ satisfies $\\mathrm{Ineq}({\\preceq})\\supseteq \\mathrm{Ineq}({\\preceq_{\\mathrm{Blackwell}}})$. \n\n\nTherefore, for any reasonable privacy notion, the set $\\mathrm{Ineq}({\\preceq})$ must be large enough to contain $\\mathrm{Ineq}({\\preceq_{\\mathrm{Blackwell}}})$. However, it is also desirable to have a not too large $\\mathrm{Ineq}({\\preceq})$. For example, consider the privacy notion based on a trivial divergence $D_0$ with $D_0(P\\|Q) \\equiv 0$ for any $P,Q$. Note that $\\mathrm{Ineq}({\\preceq_{D_0}})$ is the largest possible and, meanwhile, it is not informative at all in terms of measuring the indistinguishability of two distributions.\n\n\nThe argument above suggests that going from the ``minimal'' order $\\mathrm{Ineq}({\\preceq_{\\mathrm{Blackwell}}})$ to the ``maximal'' order $\\mathrm{Ineq}({\\preceq_{D_0}})$ would lead to information loss. Remarkably, $f$-DP is the most informative differential privacy notion from this perspective because its induced order $\\preceq_{\\mathrm{tradeoff}}$ satisfies $\\mathrm{Ineq}({\\preceq_{\\mathrm{tradeoff}}}) = \\mathrm{Ineq}({\\preceq_{\\mathrm{Blackwell}}})$. In stark contrast, this is not true for the order induced by other popular privacy notions such as R\\'enyi differential privacy and $(\\epsilon,\\delta)$-DP. We prove this claim in \\Cref{app:relation} and further justify the informativeness of $f$-DP by providing general tools that can losslessly convert $f$-DP guarantees into divergence based privacy guarantees.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Trade-off Functions and $f$-DP}\n\\label{sub:trade_off_function}\n\n\nAll variants of differential privacy informally require that it be hard to \\textit{distinguish} any pairs of neighboring datasets based on the information released by a private a mechanism $M$. From an attacker's perspective, it is natural to formalize this notion of  ``indistinguishability'' as a hypothesis testing problem for two neighboring datasets $S$ and $S'$:\n\\[\nH_0: \\text{the underlying dataset is } S \\quad\\text{ versus }\\quad H_1: \\text{the underlying dataset is } S'.\n\\]\nThe output of the mechanism $M$ serves as the basis for performing the hypothesis testing problem. Denote by $P$ and $Q$ the probability distributions of the mechanism applied to the two datasets, namely $M(S)$ and $M(S')$, respectively. The fundamental difficulty in distinguishing the two hypotheses is best delineated by the \\textit{optimal} trade-off between the achievable type I and type II errors. More precisely, consider a rejection rule $0 \\le \\phi \\le 1$, with type I and type II error rates defined as\\footnote{A rejection rule takes as input the released results of the mechanism. We flip a coin and reject the null hypothesis with probability $\\phi$.}\n\\[\n\\alpha_\\phi = \\E_{P}[\\phi], \\quad \\beta_\\phi = 1 - \\E_{Q}[\\phi],\n\\]\nrespectively. The two errors satisfy, for example, the constraint is well known to satisfy\n\\begin{equation}\\label{eq:tv_norm}\n\\alpha_{\\phi} + \\beta_{\\phi} \\ge 1 - \\mathrm{TV}(P, Q),\n\\end{equation}\nwhere the total variation distance $\\mathrm{TV}(P, Q)$ is the supremum of $|P(A) - Q(A)|$ over all measurable sets $A$. Instead of this rough constraint, we seek to characterize the fine-grained trade-off between the two errors. Explicitly, fixing the type I error at \\textit{any} level, we consider the minimal achievable type II error. This motivates the following definition.\n\\begin{definition}[trade-off function] \\label{def:trade-off}\nFor any two probability distributions $P$ and $Q$ on the same space, define the trade-off function $\\F(P, Q): [0, 1] \\rightarrow [0, 1]$ as\n\\[\n\\F(P, Q)(\\alpha) = \\inf \\left\\{ \\beta_\\phi: \\alpha_\\phi \\leqslant \\alpha \\right\\},\n\\]\nwhere the infimum is taken over all (measurable) rejection rules.\n\\end{definition}\nThe trade-off function serves as a clear-cut boundary of the achievable and unachievable regions of type I and type II errors, rendering itself the \\textit{complete} characterization of the fundamental difficulty in testing between the two hypotheses. In particular, the greater this function is, the harder it is to distinguish the two distributions. For completeness, we remark that the minimal $\\beta_{\\phi}$ can be achieved by the likelihood ratio test---a fundamental result known as the Neyman--Pearson lemma, which we state in the appendix as \\Cref{thm:NPlemma}.\n\n\n\n\nA function is called a trade-off function if it is equal to $T(P, Q)$ for some distributions $P$ and $Q$. Below we give a necessary and sufficient condition for $f$ to be a trade-off function. This characterization reveals, for example, that $\\max\\{f, g\\}$ is a trade-off function if both $f$ and $g$ are trade-off functions.\n\\begin{restatable}{proposition}{tradeoffthm}\\label{prop:trade-off}\nA function $f: [0, 1] \\rightarrow [0, 1]$ is a trade-off function if and only if $f$ is convex, continuous\\footnote{Convexity itself implies continuity in $(0,1)$ for $f$. In addition, $f(\\alpha) \\ge 0$ and $f(\\alpha) \\le 1-\\alpha$ implies continuity at 1. Hence, the continuity condition only matters at $x = 0$.}, non-increasing, and $f(x) \\leqslant 1-x$ for $x \\in [0,1]$.\n\\end{restatable}\n\n\n\n\n\n\nNow, we propose a new generalization of differential privacy built on top of trade-off functions. Below, we write $g \\ge f$ for two functions defined on $[0, 1]$ if $g(x) \\ge f(x)$ for all $0 \\le x \\le 1$, and we abuse notation by identifying $M(S)$ and $M(S')$ with their corresponding probability distributions. Note that if $T(P,Q) \\ge T(\\widetilde P, \\widetilde Q)$, then in a very strong sense, $P$ and $Q$ are harder to distinguish than $\\widetilde P$ and $\\widetilde Q$ at \\textit{any} level of type I error.\n\n\\begin{definition}[$f$-differential privacy] \\label{def:kstable}\nLet $f$ be a trade-off function. A mechanism $M$ is said to be $f$-differentially private if\n\\[\nT\\big(M(S), M(S')\\big) \\ge f\n\\]\nfor all neighboring datasets $S$ and $S'$.\n\\end{definition}\n\n\n\nA graphical illustration of this definition is shown in Figure~\\ref{fig:intro_def}. Letting $P$ and $Q$ be the distributions such that $f = \\F(P, Q)$, this privacy definition amounts to saying that a mechanism is $f$-DP if distinguishing any two neighboring datasets based on the released information is at least as difficult as distinguishing $P$ and $Q$ based on a single draw. In contrast to existing definitions of differential privacy, our new definition is parameterized by a function, as opposed to several real valued parameters (e.g.~$\\epsilon$ and $\\delta$). This functional perspective offers a complete characterization of ``privacy'', thereby avoiding the pitfall of summarizing statistical information too early. This fact is crucial to the development of a composition theorem for $f$-DP in \\Cref{sec:composition-theorems}. Although this completeness comes at the cost of increased complexity, as we will see in Section~\\ref{sub:gaussian_differential_privacy}, a simple family of trade-off functions can often closely capture privacy loss in many scenarios.\n\n\\begin{figure}[!htp]\n\\centering\n\\includegraphics[width=.6\\linewidth]{.\/figures\/nonfDP.pdf}\n\\caption{Three different examples of $T\\big(M(S),M(S')\\big)$. Only the dashed line corresponds to a trade-off function satisfying $f$-DP.}\n\\label{fig:intro_def}\n\\end{figure}\n\n\n\n\n\n\n\nNaturally, the definition of $f$-DP is symmetric in the same sense as the neighboring relationship, which by definition is symmetric. Observe that this privacy notion also requires\n\\[\nT\\big(M(S'), M(S)\\big) \\ge f\n\\]\nfor any neighboring pair $S, S'$. Therefore, it is desirable to restrict our attention to ``symmetric'' trade-off functions. Proposition~\\ref{prop:symmetry} shows that this restriction does not lead to any loss of generality.\n\\begin{restatable}{proposition}{symmrep}\\label{prop:symmetry}\nLet a mechanism $M$ be $f$-DP. Then, $M$ is $f^{\\mathrm{S}}$-DP with $f^{\\mathrm{S}} = \\max\\{f, f^{-1}\\}$, where the inverse function is defined as\\footnote{\n\\Cref{eq:inver_f} is the standard definition of the left-continuous inverse of a decreasing function. When $f$ is strictly decreasing and $f(0)=1$ and hence bijective as a mapping, \\eqref{eq:inver_f} corresponds to the inverse function in the ordinary sense, i.e. $f(f^{-1}(x)) = f^{-1}(f(x)) = x$. However, this is not true in general.}\n\\begin{equation}\\label{eq:inver_f}\nf^{-1}(\\alpha):=\\inf\\{t \\in [0,1]:f(t) \\leqslant \\alpha\\}\n\\end{equation}\nfor $\\alpha \\in[0,1]$.\n\\end{restatable}\n\nWriting $f = \\F(P,Q)$, we can express the inverse as $f^{-1} = \\F(Q, P)$, which therefore is also a trade-off function. As a consequence of this, $f^{\\mathrm{S}}$ continues to be a trade-off function by making use of \\Cref{prop:trade-off} and, moreover, is \\textit{symmetric} in the sense that \n\\[\nf^{\\mathrm{S}} = (f^{\\mathrm{S}})^{-1}.\n\\] \nImportantly, this symmetrization gives a tighter bound in the privacy definition since $f^{\\mathrm{S}} \\geqslant f$. In the remainder of the paper, therefore, trade-off functions will always be assumed to be symmetric unless otherwise specified. We prove \\Cref{{prop:symmetry}} in \\Cref{app:fDP}.\n\n\n\nWe conclude this subsection by showing that $f$-DP is a generalization of $(\\ep,\\delta)$-DP. This foreshadows a deeper connection between $f$-DP and $(\\epsilon, \\delta)$-DP that will be discussed in Section~\\ref{sub:a_primal_dual_connection_with_}. Denote\n\\begin{equation}\\label{eq:fed}\nf_{\\ep, \\delta}(\\alpha) = \\max\\left\\{ 0,1 - \\delta - \\mathrm{e}^\\ep \\alpha, \\mathrm{e}^{-\\ep}(1-\\delta-\\alpha) \\right\\}\n\\end{equation}\nfor $0 \\le \\alpha \\le 1$, which is a trade-off function. \\Cref{fig:DPvsGDP} shows the graph of this function and its evident symmetry. The following result is adapted from \\cite{wasserman_zhou}.\n\\begin{proposition}[\\cite{wasserman_zhou}] \\label{thm:privacy_testing}\nA mechanism $M$ is $(\\ep, \\delta)$-DP if and only if $M$ is $f_{\\ep, \\delta}$-DP.\n\\end{proposition}\n\n\n\n\\begin{figure}[!htp]\n\\centering\n  \\includegraphics[width=.75\\linewidth]\n{.\/figures\/DPvsGDP.pdf}\n  \\captionof{figure}{Left: $f_{\\ep,\\delta}$ is a piecewise linear function and is symmetric with respect to the line $y = x$. It has (nontrivial) slopes $-\\mathrm{e}^{\\pm\\ep}$ and intercepts $1-\\delta$. Right: Trade-off functions of  unit-variance Gaussian distributions with different means. The case of $\\mu=0.5$ is reasonably private, $\\mu=1$ is borderline private, and $\\mu=3$ is basically non-private: an adversary can control type I and type II errors simultaneously at only 0.07. In the case of $\\mu=6$ (almost coincides with the axes), the two errors both can be as small as 0.001.}\n  \\label{fig:DPvsGDP}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\n\n\nModern statistical analysis and machine learning are overwhelmingly applied to data concerning \\emph{people}. Valuable datasets generated from personal devices and online behavior of billions of individuals contain data on location, web search histories, media consumption, physical activity, social networks, and more. This is on top of continuing large-scale analysis of traditionally sensitive data records, including those collected by hospitals, schools, and the Census. This reality requires the development of tools to perform large-scale data analysis in a way that still protects the \\emph{privacy} of individuals represented in the data.\n\nUnfortunately, the history of data privacy for many years consisted of ad-hoc attempts at ``anonymizing'' personal information, followed by high profile de-anonymizations. This includes the release of AOL search logs, de-anonymized by the \\textit{New York Times} \\cite{aol}, the Netflix Challenge dataset, de-anonymized by Narayanan and Shmatikov \\cite{netflix}, the realization that participants in genome-wide association studies could be identified from aggregate statistics such as minor allele frequencies that were publicly released \\cite{gwas}, and the reconstruction of individual-level census records from aggregate statistical releases \\cite{census}.\n\nThus, we urgently needed a rigorous and principled privacy-preserving framework to prevent breaches of personal information in data analysis. In this context, \\textit{differential privacy} has put private data analysis on firm theoretical foundations \\cite{DMNS06,approxdp}. This definition has become tremendously successful: in addition to an enormous and growing academic literature, it has been adopted as a key privacy technology by Google \\cite{rappor}, Apple \\cite{apple}, Microsoft \\cite{microsoft}, and the US Census Bureau \\cite{census}. The definition of this new concept involves privacy parameters $\\epsilon \\ge 0$ and $0 \\le \\delta \\le 1$.\n\\begin{definition}[\\cite{DMNS06,approxdp}]\\label{def:dpintro}\nA randomized algorithm $M$ that takes as input a dataset consisting of individuals is $(\\ep, \\delta)$-differentially private (DP) if for any pair of datasets $S, S'$ that differ in the record of a single individual, and any event $E$,\n\\begin{equation}\\label{eq:dp_ine}\n\\P\\left[ M(S)\\in E \\right] \\leqslant \\mathrm{e}^\\ep \\P \\left[ M(S')\\in E \\right] + \\delta.\n\\end{equation}\nWhen $\\delta = 0$, the guarantee is simply called $\\epsilon$-DP.\n\\end{definition}\n\nIn this definition, datasets are \\textit{fixed} and the probabilities are taken \\textit{only} over the randomness of the mechanism\\footnote{A randomized algorithm $M$ is often referred to as a mechanism in the differential privacy literature.}. In particular, the event $E$ can take any measurable set in the range of $M$. To achieve differential privacy, a mechanism is necessarily randomized. Take as an example the problem of privately releasing the average cholesterol level of individuals in the dataset $S = (x_1, \\ldots, x_n)$, each $x_i$ corresponding to an individual. A privacy-preserving mechanism may take the form\\footnote{Here we identify the individual $x_i$ with his\/her cholesterol level.}\n\\[\nM(S) = \\frac1n (x_1 + \\cdots + x_n) + \\text{noise}.\n\\]\nThe level of the noise term has to be sufficiently large to mask the \\textit{characteristics} of any individual's cholesterol level,\nwhile not being too large to distort the population average for accuracy purposes. Consequently, the probability distributions of $M(S)$ and $M(S')$ are close to each other for any datasets $S, S'$ that differ in only one individual record.\n\n\n\n\nDifferential privacy is most naturally defined through a hypothesis testing problem from the perspective of an attacker who aims to distinguish $S$ from $S'$ based on the output of the mechanism. This statistical viewpoint was first observed by \\cite{wasserman_zhou} and then further developed by \\cite{KOV}, which is a direct inspiration for our work. In short, consider the hypothesis testing problem\n\\begin{equation}\\label{eq:hyp_into}\nH_0: \\text{the underlying dataset is } S \\quad\\text{ versus }\\quad H_1: \\text{the underlying dataset is } S'\n\\end{equation}\nand call Alice the only individual that is in $S$ but not $S'$. As such, rejecting the null hypothesis corresponds to the detection of absence of Alice, whereas accepting the null hypothesis means to detect the presence of Alice in the dataset. Using the output of an $(\\ep, \\delta)$-DP mechanism, the power\\footnote{The power is equal to 1 minus the type II error.} of any test at significance level $0 < \\alpha < 1$ has an upper bound\\footnote{A more precise bound is given in \\Cref{thm:privacy_testing}.} of $\\mathrm{e}^\\ep\\alpha+\\delta$. This bound is only slightly larger than $\\alpha$ provided that $\\epsilon,\\delta$ are small and, therefore, \\textit{any} test is essentially powerless. Put differently, differential privacy with small privacy parameters protects against any inferences of the presence of Alice, or any other individual, in the dataset.\n\n\nDespite its apparent success, there are good reasons to want to relax the original definition of differential privacy, which has led to a long line of proposals for such relaxations. The most important shortcoming is that $(\\ep,\\delta)$-DP does not tightly handle composition. Composition concerns how privacy guarantees degrade under repetition of mechanisms applied to the same dataset, rendering the design of differentially private algorithms \\textit{modular}. Without compositional properties, it would be near impossible to develop complex differentially private data analysis methods. Although it has been known since the original papers defining differential privacy \\cite{DMNS06,approxdp} that the composition of an $(\\epsilon_1,\\delta_1)$-DP mechanism and an $(\\epsilon_2,\\delta_2)$-DP mechanism yields an $(\\epsilon_1+\\epsilon_2,\\delta_1+\\delta_2)$-DP mechanism, the corresponding upper bound $\\mathrm{e}^{\\epsilon_1 + \\epsilon_2}\\alpha +  \\delta_1 + \\delta_2$ on the power of any test at\nsignificance level $\\alpha$ no longer tightly characterizes the trade-off between significance level and power for the testing between $S$ and $S'$. In \\cite{boosting}, Dwork, Rothblum, and Vadhan gave an improved composition theorem, but it fails to capture the correct hypothesis testing trade-off. This is for a fundamental reason: $(\\epsilon,\\delta)$-DP is mis-parameterized in the sense that the guarantees of the composition of $(\\epsilon_i,\\delta_i)$-DP mechanisms cannot be characterized by any pair of parameters $(\\epsilon,\\delta)$. Worse, given any $\\delta$, finding the parameter $\\ep$ that most tightly approximates the correct trade-off between significance level and type II error for a composition of a\nsequence of differentially private algorithms is  computationally hard \\cite{complexity}, and so in practice, one must resort to approximations. Given that composition and modularity are first-order desiderata for a useful privacy definition, these are substantial drawbacks and often continue to push practical algorithms with meaningful privacy guarantees out of reach.\n\n\nIn light of this, substantial recent effort has been devoted to developing relaxations of differential privacy for which composition can be handled exactly. This line of work includes several variants of ``concentrated differential privacy'' \\cite{concentrated,concentrated2}, ``R\\'enyi differential privacy'' \\cite{renyi}, and ``truncated concentrated differential privacy'' \\cite{tcdp}. These definitions are tailored to be able to exactly and easily track the ``privacy cost'' of compositions of the most basic primitive in differential privacy, which is the perturbation of a real valued statistic with Gaussian noise. \n\nWhile this direction of privacy relaxation has been quite fruitful, there are still several places one might wish for improvement. First, these notions of differential privacy no longer have hypothesis testing interpretations, but are rather based on studying divergences that satisfy a certain information processing inequality. There are good reasons to prefer definitions based on hypothesis testing. Most immediately, hypothesis testing based definitions provide an easy way to interpret the guarantees of a privacy definition. More fundamentally, a theorem due to Blackwell (see \\Cref{thm:blackwell}) provides a formal sense in which a tight understanding of the trade-off between type I and type II errors for the hypothesis testing problem of distinguishing between $M(S)$ and $M(S')$ contains only more information than any divergence between the distributions $M(S)$ and $M(S')$ (so long as the divergence satisfies the information processing inequality).\n\nSecond, certain simple and fundamental primitives associated with differential privacy---most notably, \\emph{privacy amplification by subsampling} \\cite{KLNRS}---either fail to apply to the existing relaxations of differential privacy, or require a substantially complex analysis \\cite{wang2018subsampled}. This is especially problematic when analyzing privacy guarantees of stochastic gradient descent---arguably the most popular present-day optimization algorithm---as subsampling is inherent to this algorithm. At best, this difficulty arising from using these relaxations could be overcome by using complex technical machinery. For example, it necessitated Abadi et al.~\\cite{deep} to develop the numerical \\emph{moments accountant} method to sidestep the issue.\n\n\\subsection{Our Contributions}\n\\label{sec:our-contributions-1}\nIn this work, we introduce a new relaxation of differential privacy that avoids these issues and has other attractive properties. Rather than giving a ``divergence'' based relaxation of differential privacy, we start fresh from the hypothesis testing interpretation of differential privacy, and obtain a new privacy definition by allowing the \\textit{full} trade-off between type I and type II errors in the simple hypothesis testing problem \\eqref{eq:hyp_into} to be governed by some function $f$. The functional privacy parameter $f$ is to this new definition as $(\\ep, \\delta)$ is to the original definition of differential privacy. Notably, this definition that we term $f$-differential privacy ($f$-DP)---which captures $(\\ep, \\delta)$-DP as a special case---is accompanied by a powerful and elegant toolkit\nfor reasoning about composition. Here, we highlight some of our contributions:\n\n\n\\smallskip\n\\noindent {\\bf An Algebra for Composition.} We show that our privacy definition is \\emph{closed} and \\textit{tight} under composition, which means that the trade-off between type I and type II errors that results from the composition of an $f_1$-DP mechanism with an $f_2$-DP mechanism can always be \\emph{exactly} described by a certain function $f$. This function can be expressed via $f_1$ and $f_2$ in an algebraic fashion, thereby allowing for losslessly reasoning about composition. In contrast, $(\\epsilon,\\delta)$-DP or any other privacy definition artificially restricts itself to a small number of parameters. By allowing for a \\emph{function} to keep track of the privacy guarantee of the mechanism, our new privacy definition avoids the pitfall of premature summarization\\footnote{To quote Susan Holmes \\cite{evil}, ``premature\n  summarization is the root of all evil in statistics.''} in intermediate steps and, consequently, yields a comprehensive delineation of the overall privacy guarantee. See more details in \\Cref{sec:composition-theorems}.\n\n\\smallskip\n\n\\noindent{\\bf A Central Limit Phenomenon.} We define a single-parameter family of $f$-DP that uses the type I and type II error trade-off in distinguishing the standard normal distribution $\\N(0,1)$ from $\\N(\\mu,1)$ for $\\mu \\ge 0$. This is referred to as Gaussian differential privacy (GDP). By relating to the hypothesis testing interpretation of differential privacy \\eqref{eq:hyp_into}, the GDP guarantee can be interpreted as saying that determining whether or not Alice is in the dataset is at least as difficult as telling apart $\\N(0,1)$ and $\\N(\\mu,1)$ based on one draw. Moreover, we show that GDP is a ``canonical'' privacy guarantee in a fundamental sense: for any privacy definition that retains a hypothesis testing interpretation, we prove that the privacy guarantee of\ncomposition with an appropriate scaling converges to GDP in the limit. This central limit theorem type of result is remarkable not only because of its profound theoretical implication, but also for providing a computationally tractable tool for analytically approximating the privacy loss under composition. Figure~\\ref{fig:contribution} demonstrates that this tool yields surprisingly accurate approximations to the exact trade-off in testing the hypotheses \\eqref{eq:hyp_into} or substantially improves on the existing privacy guarantee in terms of type I and type II errors. See \\Cref{sub:gaussian_differential_privacy} and \\Cref{sec:composition-theorems} for a thorough discussion.\n\n\n\n\n\\begin{figure}[!htp]\n\\centering\n  \\includegraphics[width=0.85\\linewidth]{.\/figures\/contribution.pdf}\n  \\captionof{figure}{Left: Our central limit theorem based approximation (in blue) is very close to the composition of just $10$ mechanisms (in red). The tightest possible approximation via an $(\\ep,\\delta)$-DP guarantee (in back) is substantially looser. See \\Cref{fig:comp} for parameter setup. Right: Privacy analysis of stochastic gradient descent used to train a convolutional neural network on MNIST \\cite{lecun-mnisthandwrittendigit-2010}. The $f$-DP framework yields a  privacy guarantee (in red) for this problem that is significantly better than the optimal $(\\epsilon,\\delta)$-DP guarantee (in black) that is derived from the moments accountant (MA) method \\cite{deep}. Put simply, our analysis shows that stochastic gradient descent releases less sensitive information than expected in the literature. See \\Cref{sec:application_in_sgd} for more plots and details.}\n  \\label{fig:contribution}\n\\end{figure}\n\n\n\n\\smallskip\n\\noindent {\\bf A Primal-Dual Perspective.} We show a general duality between $f$-DP and infinite collections of $(\\epsilon,\\delta)$-DP guarantees. This duality is useful in two ways. First, it allows one to analyze an algorithm in the framework of $f$-DP, and then convert back to an $(\\epsilon,\\delta)$-DP guarantee at the end, if desired. More fundamentally, this duality provides an approach to import techniques developed for $(\\epsilon,\\delta)$-DP to the framework of $f$-DP. As an important application, we use this duality to show how to reason simply about privacy amplification by subsampling for $f$-DP, by leveraging existing results for $(\\epsilon,\\delta)$-DP. This is in contrast to divergence based notions of privacy, in which reasoning about amplification by subsampling is difficult.\n\n\n\n\n\n\n\\smallskip\n\nTaken together, this collection of attractive properties render $f$-DP a mathematically coherent, computationally efficient, and versatile framework for privacy-preserving data analysis. To demonstrate the practical use of this hypothesis testing based framework, we give a substantially sharper analysis of the privacy guarantees of noisy stochastic gradient descent, improving on previous special-purpose analyses that reasoned about divergences rather than directly about hypothesis testing \\cite{deep}. This application is presented in \\Cref{sec:application_in_sgd}.\n\n\n\n\n\n\n\n\n\\iffalse\n\n\n\n\n\n\\ar{In the middle of a first pass on an intro along the lines that Weijie and I discussed earlier this week. In progress}\n\n\n\n\n\\input{phil}\n\nOver the last decade, differential privacy \\cite{DMNS06} has become established as the gold standard for privacy in statistical analyses, seeing wide deployment both in industry \\cite{google, apple, microsoft} and in government, as part of the statistical release procedure for the US 2020 census \\cite{census}. Informally, differential privacy is a \\emph{stability} property on a randomized algorithm, which requires that on input datasets that differ in just a single individual's data record, the algorithm must induce ``similar'' distributions on outputs. Technically, it requires a probability 1 bound on the log-odds ratio when comparing the distribution induced when any given individual's data is included in the dataset, and when their data is not included.\n\n\\ar{Need to change definition to notation from the paper}\n\\begin{definition}[Privacy Loss Definition of Differential Privacy \\cite{DMNS06}]\n\\label{def:privloss}\nFixing a randomized algorithm $M:X\\rightarrow Y$. Define the \\emph{privacy loss random variable}  (log odds ratio) for $M$ and a pair of inputs $x, x' \\in X$ $Z$ as follows: Let $\\ell:Y\\rightarrow \\mathbb{R}$ be $f(y) = \\log \\frac{\\Pr[M(x) = y]}{\\Pr[M(x') = y]}$\\footnote{Here $\\Pr[M(x) = y]$ refers to either the probability density function or probability mass function of $M(x)$ depending on whether $M$ induces a discrete or continuous distribution over $Y$.}. Then the privacy loss random variable $Z(M, x, x')$ is distributed as $f(y)$ when $y \\sim M(x)$.\n\n$M$ is $\\epsilon$-differentially private if for all pairs of \\emph{neighboring} datasets $x, x' \\in X$:\n$$\\Pr[Z(M, x, x') \\leq \\epsilon] = 1$$\n\\end{definition}\nWhat exactly constitutes a pair of ``neighboring'' datasets is domain specific, but it usually means two datasets that differ only in the addition or removal of the data corresponding to a single individual.\n\nAs first formalized by Wasserman and Zhou \\cite{WZ10}, differential privacy can also be viewed as controlling the trade-off between the rate of false positives (type I errors) and false negatives (type II errors), for any hypothesis test whose goal is to distinguish whether an individual's data record was an input to a differentially private computation or not. This leads to an alternative (although equivalent) definition of differential privacy, in terms of testing regions (formally defined in Section \\ref{?}):\n\n\\begin{definition}[Testing Region Definition of Differential Privacy \\cite{WZ10}]\n\\label{def:hyptest}\n\\ar{This definition seems wrong -- fix}\nGiven a pair of distributions $P, Q$, their \\emph{testing region} $K_{P,Q} \\subseteq [0,1]^2$ is the set of achievable pairs of type-1 and type-2 error rates, achieved by any hypothesis test designed to distinguish between $P$ and $Q$ from a single sample. Define $K_\\epsilon$ to be the following testing region:\n$$K_\\epsilon = \\{(\\alpha,\\beta) \\in [0,1]^2 : \\alpha \\geq 1 - e^\\epsilon \\beta\\ \\ \\mathrm{and} \\ \\ \\beta \\geq 1 - e^\\epsilon \\alpha \\}$$\n\\ar{Seems like it should be instead:\n$$\nK_\\epsilon = \\{(\\alpha,\\beta) \\in [0,1]^2 : \\alpha \\leq e^\\epsilon(1- \\beta)\\ \\ \\mathrm{and} \\ \\ \\alpha \\geq e^{-\\epsilon}(1-\\beta) \\}$$\n}\n\nAn algorithm $M$ is $\\epsilon$-differentially private if and only if, for any pair of neighboring datasets $x, x'$, the corresponding testing region $K_{M(x),M(x')} \\subseteq K_\\epsilon$.\n\\end{definition}\n\nThe success of differential privacy can be attributed to the fact that it provides strong, formal, interpretable semantic guarantees for individuals, while being flexible enough to accommodate a wide range of statistical analysis tasks, including \\ar{examples...}. What makes differential privacy amenable to numerous algorithmic and statistical tasks is the fact that it satisfies closer under post-processing, and the fact that its guarantees gracefully degrade under \\emph{composition}. Post-processing gives an information-processing like guarantee:\n\n\\begin{proposition}[Postprocessing \\cite{DMNS06}]\n\\label{prop:post}\nIf $M:X\\rightarrow Y$ is $\\epsilon$ differentially private, and $f:Y\\rightarrow Y'$ is an arbitrary randomized map, then $f \\circ M : X \\rightarrow Y'$ is also $\\epsilon$ differentially private.\n\\end{proposition}\n\nComposition rules guarantee that if multiple differentially private algorithms are run in an adaptive sequence on the same dataset, their privacy guarantees degrade gracefully rather than failing catastophically:\n\n\\begin{proposition}[Simple Composition \\cite{DMNS06}]\n\\label{prop:comp}\nLet $M_1 : X \\rightarrow Y_1$ and let $M_2 : Y_1 \\times X \\rightarrow Y_2$ be such that $M_1$ is $\\epsilon_1$ differentially private, and $M_2(y_1,\\cdot)$ is $\\epsilon_2$ differentially private for all $y_1 \\in Y_1$. Define $M:X \\rightarrow Y_1 \\times Y_2$ as the algorithm that first samples $y_1 \\sim M_1(x)$ and then samples $y_2 = M_2(y_1,x)$, and outputs $(y_1,y_2)$. Then $M$ is $(\\epsilon_1 + \\epsilon_2)$-differentially private.\n\nIn particular, the adaptive composition of $k$ $\\epsilon$-differentially private algorithms is $k\\epsilon$-differentially private.\n\\end{proposition}\n\nThese two properties allow the design of differentially private algorithms to be \\emph{modular}, facilitating the development of complex algorithmic techniques. The \\emph{linear} degradation in the privacy parameter given in Proposition \\ref{prop:comp} is often too severe to provide meaningful privacy guarantees in practice, however. A substantial focus in the differential privacy literature has therefore been to understand exactly how privacy loss degrades under composition, and how to relax the definition of differential privacy to better quantify this degradation. To foreshadow our contribution, we remark that thus far, the literature has primarily focused on relaxations Definition \\ref{def:privloss}, the privacy loss formulation of differential privacy. We instead propose a natural relaxation of Definition \\ref{def:hyptest}, the testing region definition of differential privacy, and show that this relaxation has a number of appealing properties.\n\n\\subsection{Relaxations of Differential Privacy}\nBecause of the requirement in Definition \\ref{def:privloss} that the privacy loss random variable be bounded with probability 1, the simple linear manner in which the privacy loss accumulates under composition in Proposition \\ref{prop:comp} cannot be improved upon. It also precludes certain simple mechanisms, such as the perturbation of statistics with Gaussian noise (instead, the canonical perturbation mechanism for achieving $\\epsilon$-differential privacy is to add noise sampled from a Laplace distribution). For these reasons, one of the first relaxations of differential privacy proposed was ``approximate differential privacy'', which is parameterized by two parameters: $\\epsilon$ and $\\delta$ \\cite{}. Up to constants, approximate differential privacy is equivalent to requiring that the privacy loss random variable be bounded with high probability: i.e. that for all neighboring datasets $x, x' \\in X$, $\\Pr[Z(M, x, x') \\leq \\epsilon] \\geq 1-\\delta$ \\cite{KS10}. This is still the most popular relaxation, and yields a number of advantages above and beyond the exact definition. It also retains most of the nice properties of the exact definition. Principle among these are:\n\\begin{enumerate}\n\\item The privacy loss parameter $\\epsilon$ can now be shown to accumulate sub-linearly with the number of composed algorithms. In particular, for sufficiently small $\\epsilon$ and for a fixed $\\delta$, the adaptive composition of $k$ $\\epsilon$-differentially private algorithms can be shown to be approximately differentially private with $\\epsilon' = O(\\sqrt{k\\log 1\/\\delta}\\epsilon)$ \\cite{DRV10}.\n\\item Perturbation with Gaussian noise provides approximate differential privacy \\cite{}.\n\\item Approximate differential privacy continues to be closed under post-processing, and various powerful algorithmic techniques like ``privacy amplification by subsampling'' \\cite{KLNRS08} continue to apply.\n\\item Finally, approximate differential privacy continues to have a canonical ``extremal'' mechanism \\cite{KOV15,MV16} --- i.e. for every pair $(\\epsilon,\\delta)$, there is a ``least private'' mechanism, from which all other $(\\epsilon,\\delta)$-approximately differentially private mechanisms can be derived via appropriate post-processing. This is useful when analyzing attacks, amongst other things.\n\\end{enumerate}\n\nHowever, approximate differential privacy has two serious disadvantages: First: although it is possible to give simple \\emph{asymptotically} tight composition theorems that show that the privacy loss degrades sublinearly under composition \\cite{DRV10,KOV15}, these composition theorems are not exact --- and for practical applications, tightly understanding the constants involved in composition is crucial to providing meaningful privacy guarantees. Moreover, computing the \\emph{exact} composition parameters for approximately differentially private algorithms is computationally hard \\cite{MV17}. Second, the parametrization of approximate differential privacy is cumbersome: although in principle, it allows mechanisms which produce arbitrarily disclosive outputs with probability $\\leq \\delta$, in practice, most natural mechanisms like perturbation with Gaussian noise do not have this property. Instead, they are approximately differentially private for an entire curve of parameters: for every $\\delta \\geq 0$, the Gaussian perturbation mechanism is $(f(\\epsilon),\\delta)$-approximately differentially private for some function $f(\\epsilon)$. Reporting a single pair of parameters $(\\epsilon,\\delta)$ along this curve loses information, and this lossy parameterization tends to result in needless dependencies on $\\delta$ parameters under composition.\n\n\\subsubsection{The ``CDP'' Family of Relaxations}\nDwork and Rothblum \\cite{DR16} introduced a relaxation of differential privacy called ``Concentrated Differential Privacy'' (CDP) that has inspired a flurry of related definitions \\cite{BS16,Mir18,tzdp} that aim to address some of the shortcomings of approximate differential privacy. Informally, concentrated differential privacy requires that the privacy loss random variable $Z(M,x,x')$ be sub-gaussian. The original definition of \\cite{DR16} had the disadvantage that it did not satisfy a post-processing inequality like Proposition \\ref{prop:comp}, but Bun and Steinke introduced a closely related definition, termed ``zCDP'' formulated via \\emph{Renyi Divergences} that did \\cite{BS16}. The main advantage of CDP and zCDP is that composition can be analyzed exactly (without any loss in constants) by simple additive rules. Moreover, the Gaussian mechanism satisfies CDP\/zCDP with easily computed parameters (by design), and so the definition is particularly well suited to algorithms that are designed as simply sequential composition and post-processing of Gaussian perturbations of data. Unfortunately, these definitions are not compatible with other standard techniques that are part of the algorithmic tool-kit of differential privacy: most notably, ``amplification by sub-sampling''. Roughly speaking, if a differentially private (or approximately differentially private) algorithm is run on a random subsample of a $p$ fraction of the dataset, then its privacy parameter is improved by a factor of $p$. This is particularly useful in the analysis of algorithms like stochastic gradient descent, which already operate by randomly sub-sampling data points. Because zCDP does not support amplification by subsampling, the tight analysis of algorithms like stochastic gradient descent requires complex and computationally intensive workarounds, like the ``moments accountant method'' of \\cite{accountant}. In part to correct this issue, Bun et al. \\cite{BDRS18} introduced ``truncated concentrated differential privacy'' (tCDP), which informally requires that the privacy random variable be sub-gaussian up to a set boundary specified by a certain number of standard deviations, but can be less concentrated than a Gaussian beyond this boundary (The formal definition is given in terms of Renyi divergences, as with zCDP \\cite{BS16}). Mironov gave a related definition he termed Renyi Differential Privacy \\cite{RDP}. Truncated concentrated differential privacy and Renyi differential privacy can both be used to analyze privacy amplification under subsampling \\cite{RDP-subsample,BRDS18}, but the analysis is complex. This series of definitional modifications has also led to a definition with reasonable properties, but one that is complicated to understand: for example, the canonical perturbation distribution for tCDP is no longer Guassian as it is for CDP --- it is ``sinh-Normal''.\n\n\n\n\\subsection{Our Contributions}\n\\label{sec:our-contributions}\n\nWe give a simple relaxation of differential privacy that begins from Definition \\ref{def:hyptest} --- the hypothesis testing definition of differential privacy, rather than the privacy loss definition. Informally, our definition, ``Gaussian Differential Privacy'' requires that the testing region $K_{M(x),M(x')}$ that results from running a mechanism $M$ on two neighboring datasets lie within the testing region corresponding to the problem of distinguishing two unit variance Gaussians with shifted means (We give all formal definitions in Section~\\ref{sec:main-results}.). We show that this simple relaxation has many desirable properties:\n\\begin{enumerate}\n\\item Like CDP\/zCDP, it has a simple linear adaptive composition rule, and can be used to describe the privacy of the Gaussian mechanism without nay loss of parameters It also offers a ``group privacy'' guarantee. Like zCDP it is closed under post-processing.\n\\item Like tCDP and Renyi differential privacy (but unlike CDP\/zCDP), it is compatible with privacy amplification by subsampling.\n\\item Like approximate differential privacy  (but unlike the CDP family of definitions, as far as we are aware), there is a simple, direct \\emph{transfer theorem} \\cite{DFHPRR15a,DFHPRR15b,BNSSSU16} which states that algorithms which privately and accurately estimate expectations \\emph{in sample} also provide accurate estimates \\emph{out of sample} (i.e. on the distribution from which the data were drawn). Together with the adaptive composition theorem, this provides a general, robust method of performing certain kinds of post-selection inference.\n\\item \\ar{More?}\n\\end{enumerate}\n\n\n\\wjs{define GDP in an informal way as a preview for the reader.}\n\\djs{We should define GDP and $f$-DP informally in the intro. An attempt is}\n\n... the idea of ``differential'' privacy lies in preventing the adversary's ability to do the following:\n\\begin{quote}\n\tGiven the random output of $M$, tell whether the input is $x$ or $x'$\n\\end{quote}\n\\begin{definition}[Informal] \\label{def:GDP_informal}\n\tA mechanism\/randomized algorithm $M$ is $\\mu$-GDP if the above question is harder than telling apart $N(0,1)$ and $N(\\mu,1)$ from a single sample.\n\\end{definition}\nIn principle, the pair of distributions $N(0,1)$ and $N(\\mu,1)$ could be replaced by any pair of distributions. This leads to the general definition of $f$-DP ...\n\n\n\n\nWe derive most of our results in a more general framework, for a family of privacy definitions that can be parameterized by an \\emph{arbitrary} hypothesis testing region --- and Gaussian differential privacy falls out as a special case. Moreover, we show that in a strong sense, Gaussian differential privacy is the inevitable definition that we must end up with if we start with a hypothesis-testing-region definition of privacy (like differential privacy), and consider its behavior under composition. We prove a ``central limit theorem'' that shows that under general conditions, hypothesis testing definitions of privacy converge in the limit under sequential composition to gaussian differential privacy.\n\\fi\n\n\n\\subsection*{Acknowledgements}\n\\label{sec:acknowledgements}\n\nThis work was supported in part by NSF grants CCF-1763314 and CNS-1253345, and by the Sloan Foundation, and by a sub-contract for the DARPA Brandeis program.\n\n\n{\\small\n\\bibliographystyle{alpha}\n\n\\section{Application: Privacy Analysis of Stochastic Gradient Descent}\n\\label{sec:application_in_sgd}\nOne of the most important algorithms in machine learning and optimization is stochastic gradient descent (SGD). This is an iterative optimization method used to train a wide variety of models, for example, deep neural networks. SGD has also served as an important benchmark in the development of private optimization: as an iterative algorithm, the tightness of its privacy analysis crucially depends on the tightness with which composition can be accounted for. The analysis also crucially requires a privacy amplification by subsampling argument. \n\nThe first asymptotically optimal analysis of differentially private SGD was given by \\cite{bassily2014private}. Because of the inherent limits of $(\\epsilon,\\delta)$-DP, however, this original analysis did not give meaningful privacy bounds for realistically sized datasets. This is in part what motivated the development of divergence based relaxations of differential privacy. Unfortunately, these relaxations cannot be directly applied to the analysis of SGD due to the lack of a privacy amplification by subsampling theorem. In response, Abadi et al.~\\cite{deep} circumvented this challenge by developing the moments accountant---a numeric technique tailored specifically to repeated application of subsampling, followed by a Gaussian mechanism---to give privacy bounds for SGD that are strong enough to give non-trivial guarantees when training deep neural networks on real datasets. But this analysis is ad-hoc in the sense that it uses a tool designed specifically for the analysis of SGD. \n\nIn this section, we use the general tools we have developed so far to give a simple and improved analysis of the privacy of SGD. In particular, the analysis rests crucially on the compositional and subsampling properties of $f$-DP.\n\n\n\\subsection{Stochastic Gradient Descent and Its Privacy Analysis}\n\\label{sub:noisy_sgd}\nThe private variant of the SGD algorithm is described in \\Cref{alg:dpsgd}. As we will see, from the perspective of its privacy analysis, it can simply be viewed as a repeated composition of Gaussian mechanisms operating on subsampled datasets.\n\\begin{algorithm}[htb]\n\t\\caption{\\texttt{NoisySGD}}\\label{alg:dpsgd}\n\t\\begin{algorithmic}[1]\n\t\\State{\\bf Input:} Dataset $S = (x_1,\\ldots,x_n)$, loss function $L(\\theta, x)$.\n\t\\Statex \\hspace{1.35cm}Parameters: initial state $\\theta_0$, learning rate $\\eta_t$, batch size $m$, time horizon $T$,\n\t\\Statex \\hspace{3.55cm}noise scale $\\sigma$, gradient norm bound $C$.\n\t\n\t\t\\For{$t = 1, \\ldots, T$}\n\t\t\\State {\\bf Subsampling:}\n\t\t\\Statex {\\hspace{0.55cm}Take a uniformly random subsample $I_t\\subseteq \\{1, \\ldots, n\\}$ with batch size $m$}\n\t\t\\Comment{$\\mathtt{Sample}_m$ in \\Cref{sec:subsampling}}\n\t\t\\For{$i\\in I_t$}\n\t\t\t\\State {\\bf Compute gradient:}\n\t\t\t\\Statex \\hspace{1.1cm} {$v_t^{(i)} \\gets \\nabla_{\\theta} L(\\theta_t, x_i)$}\t\t\n\t\t\t\\State {\\bf Clip gradient:}\n\t\t\t\\Statex \\hspace{1.1cm} {$\\bar{v}_t^{(i)} \\gets v_t^{(i)} \/ \\max\\big\\{1, \\|v_t^{(i)}\\|_2\/C\\big\\}$}\n\t\t\n\t\t\n\t\t\\EndFor\n\t\t\\State {\\bf Average, perturb, and descend:}\n\t\t\\Statex \\hspace{0.55cm} {$\\theta_{t+1} \\gets \\theta_{t} - \\eta_t \\Big(\\frac{1}{m} \\sum_i\\bar{v}_t^{(i)}+\\N(0, \\frac{4\\sigma^2 C^2}{m^2} I)\\Big)$}\n                 \\Comment{$I$ is an identity matrix}\n\t\t\\EndFor\n\t\t\\State {\\bf Output} $\\theta_T$\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\nTo analyze the privacy of \\texttt{NoisySGD}, we start by building up the privacy properties from the inner loop. Let $V$ be the vector space where parameter $\\theta$ lives in and $M:X^m\\times V\\to V$ be the mechanism that executes lines 4-7 in \\Cref{alg:dpsgd}. Here $m$ denotes the batch size. In effect, what $M$ does in iteration $t$ can be expressed as\n$$M(S_{I_t},\\theta_t) = \\theta_{t+1},$$\nwhere $S_{I_t}$ is the subset of the dataset $S$ indexed by $I_t$. Next, we turn to the analysis of the subsampling step (line 3) and use $\\widetilde{M}$ to denote its composition with $M$, that is, $\\widetilde{M} = M\\circ \\mathtt{Sample}_m $. Taken together, $\\widetilde{M}$ executes lines 3-7 and maps from $X^n\\times V$ to $V$.\n\nThe mechanism we are ultimately interested in\n\\begin{align*}\n\t\\mathrm{\\texttt{NoisySGD}}:X^n&\\to V\\times V\\times\\cdots\\times V\\\\\n\tS&\\mapsto(\\theta_1,\\theta_2,\\ldots, \\theta_T)\n\\end{align*}\nis simply the composition of $T$ copies of $\\widetilde{M}$. To see this fact, note that the trajectory $(\\theta_1,\\theta_2,\\ldots, \\theta_T)$ is obtained by iteratively running\n\\[\n\\theta_{j+1} = \\widetilde{M}(S,\\theta_j)\n\\]\nfor $j = 0, \\ldots, T-1$. Let $M$ be $f$-DP. Straightforwardly, $\\widetilde{M}$ is $C_{m\/n}(f)$-DP by \\Cref{thm:subsample}. Then, from the composition theorem (\\Cref{thm:n_steps}), we can immediately prove that \\texttt{NoisySGD} is $C_{m\/n}(f)^{\\otimes T}$-DP.\n\nHence, it suffices to give a bound on the privacy of $M$. For simplicity, we now focus on a single step and drop the subscript $t$. Recognizing that changing one of the $m$ data points only affects one $v^{(i)}$, the sensitivity of $\\frac{1}{m} \\sum_i\\bar{v}_t^{(i)}$ is at most $\\frac{2C}{m}$ due to the clipping operation. Making use of \\Cref{thm:g_mech}, adding Gaussian noise $N(0, \\sigma^2 \\cdot\\frac{4C^2}{m^2} I)$ to the average gradient renders this step $\\frac1{\\sigma}$-GDP. Since that the gradient update following the gradient averaging step is deterministic, we conclude that $M$ satisfies $\\frac1{\\sigma}$-GDP.\n\n\nIn summary, the discussion above has proved the following theorem:\n\\begin{theorem} \\label{thm:sgdcompo}\n\\Cref{alg:dpsgd} is $C_{m\/n}(G_{\\sigma^{-1}})^{\\otimes T}$-DP.\n\\end{theorem}\nTo clear up any confusion, we remark that this $C_{m\/n}(G_{\\sigma^{-1}})^{\\otimes T}$-DP mechanism does not release the subsampled indices.\n\nThe use of \\Cref{thm:sgdcompo} requires the evaluation of $C_{m\/n}(G_{\\sigma^{-1}})^{\\otimes T}$. However, numerical computation of this tensor product is computationally cumbersome. As a matter of fact, the moment's accountant technique applied to the present problem is basically equivalent to direct computing $C_{m\/n}(G_{\\sigma^{-1}})^{\\otimes T}$. In contrast, our central limit theorems provide an entirely different tool by analytically approximating $C_{m\/n}(G_{\\sigma^{-1}})^{\\otimes T}$ in a way that becomes nearly exact as $T$ grows. The next two subsections presents two such results, corresponding to our two central limit theorems (\\Cref{thm:Berry} and \\Cref{thm:CLT}), respectively. An asymptotic privacy analysis of \\texttt{NoisySGD} is given in \\Cref{sub:asymptotic_analysis_of_noisy_sgd} by developing a general limit theorem for composition of subsampled mechanisms. A Berry--Esseen type analysis is shown in \\Cref{sub:berry_esseen_for_privacy_of_sgd}.\n\n\n\\begin{figure}[!tp]\n\\centering\n  \\includegraphics[width=\\linewidth]\n{.\/figures\/compareGoogle.pdf}\n  \\captionof{figure}{Comparison of the GDP bounds derived from our method, and the $(\\ep,\\delta)$-DP bounds derived using the moments accountant \\cite{deep}. All three experiments run \\Cref{alg:dpsgd} on the entire MNIST dataset with $n=60,000$ data points, batch size $m=256$, learning rates $\\eta_t$ set to 0.25, 0.15, and 0.25, respectively, and clipping thresholds $C$ set to 1.5, 1.0, 1.5, respectively. The red lines are obtained via \\Cref{thm:SGDlimit}, while the blue dashed lines are produced by the tensorflow\/privacy library. See \\url{https:\/\/github.com\/tensorflow\/privacy} for the detail of the experiments.}\n  \\label{fig:compareSGD}\n\\end{figure}\n\n\n\n\\subsection{Asymptotic Privacy Analysis}\n\\label{sub:asymptotic_analysis_of_noisy_sgd}\nIn this subsection, we first consider the limit of $C_p(f)^{\\otimes T}$ for a general trade-off function $f$, then plug in $f = G_{\\sigma^{-1}}$ for the analysis of \\texttt{NoisySGD}. The more general approach is useful for analyzing other iterative algorithms. \n\nRecall from \\Cref{sec:subsampling} that a $p$-subsampled $f$-DP mechanism is $C_p(f)$-DP, where $C_p(f)$ is defined as\n\\[\n\tC_p(f)(x)=\n\t\t\\left\\{\n\t\t\\begin{array}{ll}\n\t\tf_p(x),&x\\in[0,x^*] \\\\\n\t\tx^*+f_p(x^*)-x, & x\\in[x^*,f_p(x^*)]\\\\\n\t\tf_p^{-1}(x), &x\\in[f_p(x^*),1],\n\t\t\\end{array}\n\t\t\\right.\n\\]\nwhere $x^*$ is the unique fixed point of $f$. We will let the sampling fraction $p$ tend to 0 as $T$ approaches infinity.\nIn the following, $a_+^2$ is a short-hand for $(\\max\\{a,0\\})^2$\n\\begin{restatable}{theorem}{mixturerep} \\label{thm:mixtureSGD}\n\tSuppose $f$ is a symmetric trade-off function such that $f(0)=1$ and $\\int_0^1(f'(x)+1)^4\\diff x<+\\infty$. Furthermore, assume $p\\sqrt{T} \\to p_0$ as $T\\to\\infty$ for some constant $p_0 > 0$. Then we have the uniform convergence\n\t$$C_p(f)^{\\otimes T}\\to G_{p_0\\sqrt{2\\chi^2_+(f)}}$$\nas $T \\to \\infty$, where\n\t\\[\\chi^2_+(f)=\\int_0^1\\big(|f'(x)|-1\\big)_+^2\\diff x.\\]\n\\end{restatable}\nThis theorem has implications for the design of iterative private mechanisms involving subsampling as a subroutine. One way to bound the privacy of such a mechanism is to let the sampling ratio $p$ go to zero as the total number of iterations $T$ goes to infinity. The theorem says that the correct scaling between the two values is $p\\sim 1\/\\sqrt{T}$ and, furthermore, gives an explicit form of the limit.\n\n\n\n\n\nIn order to analyze \\texttt{NoisySGD}, we need to compute the quantity $\\chi^2_+(G_\\mu)$. This can be done by directly working with its definition. In \\Cref{app:SGD}, we provide a different approach by relating $\\chi^2_+(f)$ to $\\chi^2$-divergence.\n\\begin{restatable}{lemma}{chirep} \\label{lem:chi2GDP}\nWe have\t\\[\\chi^2_+(G_\\mu) = \\mathrm{e}^{\\mu^2}\\cdot\\Phi(3\\mu\/2)+3\\Phi(-\\mu\/2)-2.\\]\n\\end{restatable}\nWhen using SGD to train large models, we typically perform a very large number of iterations, so it is reasonable to consider the parameter regime in which $n\\to\\infty,T\\to\\infty$. The batch size can also vary with these quantities. The following theorem is a direct consequence of \\Cref{thm:sgdcompo,thm:mixtureSGD} and \\Cref{lem:chi2GDP}.\n\\begin{restatable}{corollary}{SGDlimitrep} \\label{thm:SGDlimit}\n\tIf $m\\sqrt{T}\/n \\to c$, then \\textup{\\texttt{NoisySGD}} is asymptotically $\\mu$-GDP with\n\t\\[\\mu = \\sqrt{2}c\\cdot\\sqrt{\\mathrm{e}^{\\sigma^{-2}}\\cdot\\Phi(1.5\\sigma^{-1})+3\\Phi(-0.5\\sigma^{-1})-2}.\\]\n\\end{restatable}\nFirst, we remark that the condition in the theorem is consistent with the analysis of private SGD in \\cite{bassily2014private}, which considers $m = 1$ and $T = O(n^2)$. We also note that in the deep learning literature, the quantity $\\frac{m}{n}\\cdot\\sqrt{T}$ is generally quite small. The convention in this literature is to reparameterize the number of gradient steps $T$ by the number of ``epochs'' $E$, which is the number of sweeps of the entire dataset. The relationship between these parameters is that  $E = Tm\/n$. In this reparameterization, our assumption is that $Em\/n\\to c^2$. Concretely, the AlexNet \\cite{krizhevsky2012imagenet} sets the parameters as $m=128, E\\approx90$ on the ILSVRC-2010 dataset with $n\\approx 1.2\\times 10^6$, leading to $Em\/n < 0.01$. Many other prominent implementations\\footnote{See the webpage of Gluon CV Toolkit \\cite{he2018bag,zhang2019bag} for a collection of such hyperparameters on computer vision\n  tasks.} also lead to a small value of $Em\/n$. \n\n\n\n\n\n\n\n\n\n\\subsection{A Berry--Esseen Privacy Bound}\n\\label{sub:berry_esseen_for_privacy_of_sgd}\nNow, we apply the Berry--Esseen style central limit theorem (\\Cref{thm:Berry}) for the privacy analysis of \\texttt{NoisySGD}, with the advantage of giving sharp privacy guarantees. However, the disadvantage is that the expressions it yields are more unwieldy: they are computer evaluable, so usable in implementations, but do not admit simple closed forms.\n\nThe individual components in \\Cref{thm:Berry} have the form $C_p(G_\\mu)$ with $p = m\/n, \\mu = \\sigma^{-1}$. It suffices to evaluate the moment functionals on $C_p(G_\\mu)$. This is done in the following lemma.\n\\begin{restatable}{lemma}{functionalGmurep}\\label{lem:functionalGmu}\n\tLet $Z(x) = \\log(p\\cdot \\mathrm{e}^{\\mu x-\\mu^2\/2}+1-p)$ and $\\varphi(x) = \\frac{1}{\\sqrt{2\\pi}}\\mathrm{e}^{-x^2\/2}$ be the density of the standard normal distribution. Then\n\t\\begin{align*}\n\t\t\\mathrm{kl}\\big(C_p(G_\\mu)\\big) &= p\\int_{\\mu\/2}^{+\\infty} Z(x)\\cdot\\big(\\varphi(x-\\mu)-\\varphi(x)\\big)\\diff x\\\\\n\t\n\t\t\\kappa_2\\big(C_p(G_\\mu)\\big)&= \\int_{\\mu\/2}^{+\\infty} Z^2(x)\\cdot\\big(p\\varphi(x-\\mu)+(2-p)\\varphi(x)\\big)\\diff x\\\\\n\t\t\\bar{\\kappa}_3\\big(C_p(G_\\mu)\\big)&=\\int_{\\mu\/2}^{+\\infty} \\big|Z(x)-\\mathrm{kl}\\big(C_p(G_\\mu)\\big)\\big|^3\\cdot(p\\varphi(x-\\mu)+(1-p)\\varphi(x))\\diff x\\\\\n\t\t&+\\int_{\\mu\/2}^{+\\infty} \\big|Z(x)+\\mathrm{kl}\\big(C_p(G_\\mu)\\big)\\big|^3\\cdot\\varphi(x)\\diff x.\n\t\\end{align*}\n\\end{restatable}\nWe can plug these expressions into \\Cref{thm:Berry} and get\n\\begin{restatable}{corollary}{SGDBerryrep} \\label{thm:SGDBerry}\nLet $p = m\/n, \\mu = \\sigma^{-1}$ and\n\t\\begin{align*}\n\t\t\\tilde{\\mu}=\\frac{2\\sqrt{T}\\cdot\\mathrm{kl}\\big(C_p(G_\\mu)\\big)}{\\sqrt{\\kappa_2\\big(C_p(G_\\mu)\\big) - \\mathrm{kl}^2\\big(C_p(G_\\mu)\\big)}}, \\quad\t\t\\gamma=\\frac{0.56}{\\sqrt{T}}\\cdot \\frac{\\bar{\\kappa}_3\\big(C_p(G_\\mu)\\big)}{\\big(\\kappa_2\\big(C_p(G_\\mu)\\big) - \\mathrm{kl}^2\\big(C_p(G_\\mu)\\big)\\big)^{\\frac{3}{2}}}.\n\t\\end{align*}\n\\textup{\\texttt{NoisySGD}} is $f$-DP with\n\t\\begin{equation*}\n\t\tf(\\alpha) = \\max\\{G_{\\tilde{\\mu}}(\\alpha+\\gamma)-\\gamma,0\\}.\n\t\\end{equation*}\n\\end{restatable}\nWe remark that $G_{\\tilde{\\mu}}$ can be set to 0 in $(1,+\\infty)$ so that $f$ is well-defined when $\\alpha>1-\\gamma$.\n\n\n\n\\section{Amplifying Privacy by Subsampling}\n\\label{sec:subsampling}\n\n\\newcommand{\\mathtt{Sample}}{\\mathtt{Sample}}\n\\newcommand{\\widetilde{M}}{\\widetilde{M}}\n\\newcommand{S\\cup\\{n\\}}{S\\cup\\{n\\}}\n\\newcommand{S}{S}\n\nSubsampling is often used prior to a private mechanism $M$ as a way to \\textit{amplify} privacy guarantees. Specifically, we can construct a smaller dataset $\\tilde S$ by flipping a fair coin for each individual in the original dataset $S$ to decide whether the individual is included in $\\tilde{S}$. This subsampling scheme roughly shrinks the dataset by half and, therefore, we would expect that the induced mechanism applied to $\\tilde S$ is about twice as private as the original mechanism $M$. Intuitively speaking, this privacy amplification is due to the fact that every individual enjoys perfect privacy if the individual is not included in the resulting dataset $\\tilde S$, which happens with probability 50\\%.\n\nThe claim above was first formalized in \\cite{KLNRS} for $(\\ep,\\delta)$-DP. Such a privacy amplification property is, unfortunately, no longer true for the most natural previous relaxations of differential privacy aimed at recovering precise compositions (like concentrated differential privacy (CDP) \\cite{concentrated,concentrated2}). Further modifications such as truncated CDP \\cite{tcdp} have been introduced primarily to remedy this deficiency of CDP---but at the cost of extra complexity in the definition. Other relaxations like {R\\'enyi} differential privacy \\cite{renyi} can be shown to satisfy a form of privacy amplification by subsampling, but both the analysis and the statement are complex \\cite{wang2018subsampled}.\n\n\nIn this section, we show that these obstacles can be overcome by our hypothesis testing based relaxation of differential privacy. Explicitly, our main result is a simple, general, and easy-to-interpret subsampling theorem for $f$-DP. Somewhat surprisingly, our theorem significantly improves on the classical subsampling theorem for privacy amplification in the $(\\ep,\\delta)$-DP framework \\cite{jon}. Note that this classical theorem continues to use $(\\epsilon,\\delta)$-DP to characterize the subsampled mechanism. However, $(\\epsilon,\\delta)$-DP is simply not expressive enough to capture the amplification of privacy.\n\n\n\n\n\\subsection{A Subsampling Theorem}\n\\label{sec:subsample_theorems}\n\n\n\n\nGiven an integer $1 \\le m \\le n$ and a dataset $S$ of $n$ individuals, let $\\mathtt{Sample}_m(S)$ be a subset of $S$ that is chosen uniformly at random among all the $m$-sized subsets of $S$. For a mechanism $M$ defined on $X^m$, we call $M\\big(\\mathtt{Sample}_m(S)\\big)$ the subsampled mechanism, which takes as input an $n$-sized dataset. Formally, we use $M \\circ \\mathtt{Sample}_m$ to denote this subsampled mechanism. To clear up any confusion, note that intermediate result $\\mathtt{Sample}_m(S)$ is not released and, in particular, this is different from the composition in \\Cref{sec:composition-theorems}.\n\n\n\nIn brief, our main theorem shows that the privacy bound of the subsampled mechanism in the $f$-DP framework is given by an operator acting on trade-off functions. To introduce this operator, write the convex combination $f_p:= pf + (1-p)\\Id$ for $0 \\le p \\le 1$, where $\\Id(x) = 1-x$. Note that the trade-off function $f_p$ is asymmetric in general.\n\\begin{definition} \\label{def:Cp}\nFor any $0 \\le p \\le 1$, define the operator $C_p$ acting on trade-off functions as\n\\[C_p(f) := \\min\\{f_p, f_p^{-1}\\}^{**}.\\]\nWe call $C_p$ the $p$-sampling operator.\n\\end{definition}\nAbove, the inverse $f_p^{-1}$ is defined in \\eqref{eq:inver_f}. The biconjugate $\\min\\{f_p, f_p^{-1}\\}^{**}$ is derived by applying the conjugate as defined in \\eqref{eq:conjugate} twice to $\\min\\{f_p, f_p^{-1}\\}$. For the moment, take for granted the fact that $C_p(f)$ is a symmetric trade-off function.\n\n\nNow, we present the main theorem of this section.\n\n\\begin{theorem}\\label{thm:subsample}\nIf $M$ is $f$-DP on $X^m$, then the subsampled mechanism $M\\circ\\mathtt{Sample}_m$ is $C_p(f)$-DP on $X^n$, where the sampling ratio $p=\\frac{m}{n}$.\n\\end{theorem}\n\n\nAppreciating this theorem calls for a better understanding of the operator $C_p$. In effect, $C_p$ performs a two-step transformation: symmetrization (taking the minimum of $f_p$ and its inverse $f_p^{-1}$) and convexification (taking the largest convex lower envelope of $\\min\\{f_p, f_p^{-1}\\}$). The convexification step is seen from convex analysis that the biconjugate $h^{**}$ of any function $h$ is the greatest convex lower bound of $h$. As such, $C_p(f)$ is convex and, with a bit more analysis, \\Cref{prop:trade-off} ensures that $C_p(f)$ is indeed a trade-off function. As an aside, $C_p(f) \\le \\min\\{f_p, f_p^{-1}\\} \\le f_p$. See \\Cref{fig:subsample} for a graphical illustration.\n\n\n\n\nNext, the following facts concerning the $p$-sampling operator qualitatively illustrate this privacy amplification phenomenon.\n\\begin{enumerate}\n\\item If $0\\leqslant p\\leqslant q\\leqslant 1$ and $f$ is symmetric, we have $f=C_1(f)\\leqslant C_q(f)\\leqslant C_p(f)\\leqslant C_0(f)= \\Id$. That is, as the sampling ratio declines from 1 to 0, the privacy guarantee interpolates monotonically between the original $f$ and the perfect privacy guarantee $\\Id$. This monotonicity follows from the fact that $g \\ge h$ is equivalent to $g^{-1} \\ge h^{-1}$ for any trade-off functions $g$ and $h$.\n\n\\item\nIf two trade-off functions $f$ and $g$ satisfy $f \\ge g$, then $C_p(f) \\ge C_p(g)$. This means that if a mechanism is more private than the other, using the same sampling ratio, the subsampled mechanism of the former remains more private than that of the latter, at least in terms of lower bounds.\n\n\\item For any $0 \\le p \\le 1$, $C_p(\\Id) = \\Id$. That is, perfect privacy remains perfect privacy with subsampling.\n\\end{enumerate}\n\nExplicitly, we provide a formula to calculate $C_p(f)$ for a symmetric trade-off function $f$. Letting $x^*$ be the unique fixed point of $f$, that is $f(x^*)= x^*$, we have\n\\begin{equation}\\label{eq:Cp_expression}\n\t\tC_p(f)(x)=\n\t\t\t\\left\\{\n\t\t\t\\begin{array}{ll}\n\t\t\tf_p(x),&x\\in[0,x^*] \\\\\n\t\t\tx^*+f_p(x^*)-x, & x\\in[x^*,f_p(x^*)]\\\\\n\t\t\tf_p^{-1}(x), &x\\in[f_p(x^*),1].\n\t\t\t\\end{array}\n\t\t\t\\right.\n\t\\end{equation}\nThis expression is almost self-evident from the left panel of \\Cref{fig:subsample}. Nevertheless, a proof of this formula is given in \\Cref{app:property}. This formula, together with \\Cref{thm:subsample}, allows us to get a closed-form characterization of the privacy amplification for $(\\epsilon, \\delta)$-DP.\n\\begin{corollary}\\label{cor:ep_d}\nIf $M$ is $(\\epsilon, \\delta)$-DP on $X^m$, then the subsampled mechanism $M\\circ\\mathtt{Sample}_m$ is $C_p(f_{\\epsilon, \\delta})$-DP on $X^n$, where\n\\begin{equation}\\label{eq:ep_d_sump_for}\nC_p(f_{\\epsilon, \\delta})(\\alpha) = \\max\\left\\{f_{\\epsilon', \\delta'}(\\alpha), 1 - p\\delta - p \\, \\frac{\\mathrm{e}^{\\ep} - 1}{\\mathrm{e}^{\\ep} + 1} - \\alpha \\right\\}.\n\\end{equation}\nAbove, $\\epsilon'= \\log(1-p + p \\mathrm{e}^\\ep), \\delta' = p\\delta$, and $p = \\frac{m}{n}$.\n\\end{corollary}\n\n\n\n\n\n\n\\begin{figure}[!htp]\n\\centering\n  \\includegraphics[width=.7\\linewidth]\n{.\/figures\/subsample_main.pdf}\n  \\captionof{figure}{The action of $C_p$. Left panel: $f=G_{1.8}, p=0.35$. Right panel: $\\ep = 3,\\delta=0.1,p=0.2$. The subsampling theorem \\ref{thm:subsample} results in a significantly tighter trade-off function compared to the classical theorem for $(\\ep,\\delta)$-DP.}\n  \\label{fig:subsample}\n\\end{figure}\n\n\nFor comparison, we now present the existing bound on the privacy amplification by subsampling for $(\\ep,\\delta)$-DP. To be self-contained, \\Cref{app:property} gives a proof of this result, which primarily follows \\cite{jon} .\n\n\\begin{restatable}[\\cite{jon}]{lemma}{DPsubsamplerep}\\label{lem:DPsubsample}\nIf $M$ is $(\\ep,\\delta)$-DP, then $M\\circ\\mathtt{Sample}_m$ is $(\\ep',\\delta')$-DP with $\\ep'$ and $\\delta'$ defined in \\Cref{cor:ep_d}.\n\\end{restatable}\n\nUsing the language of the $f$-DP framework, \\Cref{lem:DPsubsample} states that $M\\circ\\mathtt{Sample}_m$ is $f_{\\epsilon',\\delta'}$-DP. \\Cref{cor:ep_d} improves on \\Cref{lem:DPsubsample} because, as is clear from \\eqref{eq:ep_d_sump_for},\n\\[\nC_p(f_{\\epsilon, \\delta}) \\ge f_{\\epsilon',\\delta'}.\n\\]\nThe right panel of \\Cref{fig:subsample} illustrates \\Cref{lem:DPsubsample} and our \\Cref{cor:ep_d} for $\\epsilon = 3, \\delta = 0.1$, and $p = 0.2$. In effect, the improvement is captured by the shaded triangle enclosed by $C_p(f_{\\epsilon, \\delta})$ and $f_{\\epsilon',\\delta'}$, revealing that the minimal sum of type I and type II errors in distinguishing two neighboring datasets with subsampling can be significantly lower than the prediction of \\Cref{lem:DPsubsample}. This gain is only made possible by the flexibility of trade-off functions in the sense that $C_p(f_{\\epsilon, \\delta})$ \\textit{cannot} be expressed within the $(\\epsilon, \\delta)$-DP framework. The unavoidable loss in the $(\\epsilon,\\delta)$-DP representation of the subsampled mechanism is compounded when analyzing the composition of many private mechanisms. \n\n\nIn the next subsection, we prove \\Cref{thm:subsample} by making use of \\Cref{lem:DPsubsample}. Its proof implies that \\Cref{thm:subsample} holds for any subsampling scheme for which \\Cref{lem:DPsubsample} is true. In particular, it holds for the subsampling scheme described at the beginning of this section, that is, independent coin flips for every data item. \n\n\n\n\n\n\\subsection{Proof of the Subsampling Theorem}\n\\label{sub:proof_of_subsample_theorems}\nThe proof strategy is as follows. First, we convert the $f$-DP guarantee into an infinite collection of $(\\ep,\\delta)$-DP guarantees by taking a dual perspective that is enabled by \\Cref{prop:ftoDP}. Next, by applying the classical subsampling theorem (that is, \\Cref{lem:DPsubsample}) to these $(\\ep,\\delta)$-DP guarantees, we conclude that the subsampled mechanism satisfies a new infinite collection of $(\\ep,\\delta)$-DP guarantees. Finally, \\Cref{prop:DPtof} allows us to convert these new privacy guarantees back into an $\\tilde{f}$-DP guarantee, where $\\tilde f$ can be shown to coincide with $C_p(f)$.\n\n\n\\begin{proof}[Proof of \\Cref{thm:subsample}]\nProvided that $M$ is $f$-DP, from \\Cref{prop:ftoDP} it follows that $M$ is $\\big(\\ep,\\delta(\\ep)\\big)$-DP with $\\delta(\\ep) = 1+f^*(- \\mathrm{e}^{\\ep})$ for all $\\ep\\geqslant 0$. Making use of \\Cref{lem:DPsubsample}, the subsampled mechanism $M \\circ \\mathtt{Sample}_m$ satisfies the following collection of $(\\ep',\\delta')$-DP guarantees for all $\\ep\\geqslant 0$:\n\\[\n\\ep' =\\log(1 - p + p\\mathrm{e}^\\ep),\\quad \\delta' =p\\big(1+f^*(-\\mathrm{e}^{\\ep})\\big).\n\\]\nEliminating the variable $\\epsilon$ from the two parametric equations above, we can relate $\\ep'$ to $\\delta'$ using\n\\begin{equation}\\label{eq:fp}\n\\delta' = 1+f_p^*(-\\mathrm{e}^{\\ep'}),\n\\end{equation}\nwhich is proved in \\Cref{app:property}. The remainder of the proof is devoted to showing that $(\\epsilon', \\delta')$-DP guarantees for all $\\ep' \\ge 0$ is equivalent to the $C_p(f)$-DP guarantee.\n\n\nAt first glance, \\eqref{eq:fp} seems to enable the use of \\Cref{prop:ftoDP}. Unfortunately, that would be invalid because $f_p$ is asymmetric. To this end, we need to extend \\Cref{prop:ftoDP} to general trade-off functions. To avoid conflicting notation, let $g$ be a generic trade-off function, not necessarily symmetric. Denote by $\\bar{x}$ be the smallest point such that $g'(x)=-1$, that is, $\\bar{x} = \\inf\\{x\\in[0,1]:g'(x)=-1\\}$.\\footnote{For simplicity, the proof assumes differentiable trade-off functions. If $g$ is not differentiable, use the definition $\\bar{x} = \\inf\\{x\\in[0,1]: -1 \\in \\partial g(x) \\}$ instead. This adjustment applies to other parts of the proof.} As a special instance of \\Cref{prop:asymm_env} in the appendix, the following result serves our purpose.\n\\begin{proposition}\\label{prop:asymm_for_proof}\nIf $g(\\bar{x})\\geqslant\\bar{x}$ and a mechanism $M$ is $(\\ep,1+g^*(-\\mathrm{e}^{\\ep}))$-DP for all $\\ep\\geqslant 0$, then $M$ is $\\min\\{g,g^{-1}\\}^{**}$-DP.\n\\end{proposition}\n\n\nThe proof of the present theorem would be complete if \\Cref{prop:asymm_for_proof} can be applied to the collection of privacy guarantees in \\eqref{eq:fp}for $f_p$. To use \\Cref{prop:asymm_for_proof}, it suffices to verify the condition $f_p(\\bar{x})\\geqslant \\bar{x}$ where $\\bar{x}$ is the smallest point such that $f_p'(x)=-1$. Let $x^*$ be the (unique) fixed point of $f$. To this end, we collect a few simple facts:\n\t\\begin{itemize}\n\t\n\t\t\\item First, $f'(x^*) = -1$. This is because the graph of $f$ is symmetric with respect to the $45^\\circ$ line passing through the origin.\n\t\t\\item Second, $\\bar{x}\\leqslant x^*$. This is because $f_p'(x^*)=pf'(x^*)+(1-p)\\Id'(x^*)=-1$ and, by definition, $\\bar{x}$ can only be smaller.\n\t\\end{itemize}\nWith these facts in place, we get\n\\[f_p(\\bar{x})\\geqslant f_p(x^*) \\geqslant f(x^*) = x^* \\geqslant \\bar{x}\\]\nby recognizing that $f_p$ is decreasing and $f_p \\ge f$. Hence, the proof is complete.\n\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}                                        \\label{sec:Introduction}\n    Since the discovery in 1995 of a planet orbiting 51 Peg \\citep{1995Natur.378..355M}, over 1800 planets in around 1100 planetary systems have been found \\citep[][]{2011A&A...532A..79S}\\footnote{{http:\/\/exoplanet.eu\/}}: this number increases steadily.  Furthermore, close to 470 multiple planetary systems have been detected, some of which are highly complex \\citep[e.g.,][]{2013Sci...340..587B}.\n    \n    The search for exoplanets has been following two different, but complementary, paths: the detection of exoplanets with increasingly\nlower masses, and the characterization of these exoplanets and their atmospheres. On the detection side, the radial velocity {\\citep[e.g.,][]{2011exop.book...27L,2013A&A...549A.109B}} and transit methods {\\citep[e.g., ][]{2010Sci...327..977B,2011exop.book...55W}} have been the most prolific. One of the most important resul?ts of planet detection surveys is the ubiquity of planets around solar-type stars {\\citep[e.g.,][]{2010Sci...330..653H}}.\n    \n    On the characterization side, the current frontier of exoplanet characterization has been pushed toward the study of exoplanet atmospheres, both from a composition and from a dynamics point of view. To overcome this difficult challenge, several indirect techniques have been developed. Transmission spectroscopy relies on observing the host star spectrum as it is filtered by a planet atmosphere during a transit \\citep[e.g.,][]{2002ApJ...568..377C,2014arXiv1403.4602K}. Occultation photometry and spectroscopy measure the occultation (secondary eclipse) depth of the star+planet light curve at different wavelengths to derive the planet's thermal \\citep[e.g.,][]{2005Natur.434..740D,2010A&A...513A..76S,2014arXiv1410.2241S} and reflected \\citep{2009A&A...501L..23A-eprintv1,2013AN....334..188R} signatures. The detection of the exoplanet emission spectra was also possible through high-resolution spectroscopy \\citep{2012Natur.486..502B, 2012ApJ...753L..25R, 2014A&A...561A.150D}. The measurement of phase variations relies on the detection of the flux variation along the planet's orbit as it alternately presents its day and\nnight hemisphere to us  \\citep[e.g.,][]{2009ApJ...703..769K-eprintv1,2011ApJ...740...61K}. These techniques represent the current front line of exoplanet characterization and are limited only by flux measurement precision\nthat they impose as a result of the low planet-star flux ratio (e.g., in the most favorable cases, for a Jupiter sized planet with a 3 day period orbit $F_{\\rm Planet}\/F_{Star}\\approx10^{-4}$ in the visible and $F_{\\rm Planet}\/F_{Star}\\approx10^{-3}$ in the IR). \n \n    {\\cite{1999ApJ...522L.145C} attempted to recover the reflected spectrum of the giant planet orbiting $\\tau$ Boo, which paved the way for detecting reflected light in the\noptical. To do so, they performed a $\\chi^2$ evaluation of simulated star+planet spectra against high-resolution observations obtained\nwith the HIRES spectrometer at  the Keck observatory. Although unable to detect the reflected signal from planet $\\tau$ Boo b, they were able to set an upper limit on the planet's maximum planet-to-star flux ratio of about $5\\times10^{-5}$. In the same year, \\cite{1999Natur.402..751C} made an attempt at detecting the reflected signal of the same planet using a least-squares deconvolution technique, also without conclusive results. More recently, other attempts with alternative methods have been made \\citep[e.g.,][]{2003MNRAS.346L..16L, 2010A&A...514A..23R}, albeit all results have been inconclusive about a definitive detection of a reflected exoplanet signal. Nonetheless, all these attempts are of great scientific values because they allow establishing upper limits on the planet-to-star flux ratio.} \n    \n    More recently, researchers were able to collect measurements of the albedo of several exoplanets in an effort to constrain current atmosphere models \\citep[e.g.,][]{2011ApJ...729...54C,2014arXiv1405.3798D} and infer their composition \\citep[e.g., the presence of clouds in the atmosphere of HD 189733 b as done by ][]{2014arXiv1403.6664B}. An interesting result is the observation of \\object{HD 189733 b,} where researchers were able to infer the planet's color from albedo measurements \\citep{2013ApJ...772L..16E}. Nonetheless, {several of these} results are still subject to some discussion because it has been shown that the blue excess in the planet's transmission spectrum might also be the result of stellar activity \\citep[][]{2014A&A...568A..99O}. \n    \n    {The detection of sodium absorption in the transmission spectrum of \\object{HD209458} b \\citep{2002ApJ...568..377C} paved the way for the detection of spectral features in exoplanet atmospheres. As new facilities were developed and analysis techniques perfected with time, other elements or even molecules were detected  \\cite[e.g.,][]{2008ApJ...673L..87R, 2010Natur.465.1049S, 2013MNRAS.436L..35B, 2014ApJ...791L...9M, 2014arXiv1404.3769B, 2014Natur.509...63S}.}\n    \n    In this paper we apply the technique described in \\cite{2013MNRAS.436.1215M} to HARPS observations of 51 Peg {to try to retrieve} the reflected spectrum of its planetary companion. \\object{51 Peg} (\\object{HD217014}) is a solar-type star, slightly more massive than our Sun \\cite[${M_{51\\;Peg}}\/{M_{\\odot}}\\approx 1.04$,][]{2013A&A...556A.150S}, at a distance of approximately 15.6 parsec from us \\citep{2007A&A...474..653V}. {With a minimum mass slightly under half the mass of Jupiter} and an orbital period slightly longer than days, \\object{51 Peg b} is the prototype of a hot Jupiter, giant gas planets in close orbits \\citep{1995Natur.378..355M}. The combination of the host brightness \\citep[$V_{Mag} = 5.46$,][]{2009ApJ...694.1085V}, the giant planet's large dimensions, and the short-period planetary orbit yield a relatively high planet-star flux ratio and make this planetary system an excellent candidate for testing the method presented in \\cite{2013MNRAS.436.1215M}. \n    \n    In Sect. \\ref{sec:Principle} we describe the principle behind our method. In Sect. \\ref{sec:Method} we describe the method and its application to our data. The results are presented in Sect. \\ref{sec:Results} and are discussed in Sect. \\ref{sec:Discussion}. We conclude in Sect. \\ref{sec:Conclusions}.\n    \n        \n  \\section{Principle behind the method}                               \\label{sec:Principle}\n    \\cite{2013MNRAS.436.1215M} showed that the cross-correlation function (hereafter CCF) can be used to mathematically enhance the S\/N of our observations to a level where the extremely low S\/N planetary signal can be recovered.  The CCF of a spectrum with a binary mask \\citep{1996A&AS..119..373B} has been extensively tested in detecting exoplanets with the radial velocities method. Briefly, this technique corresponds to mapping the degree of similarity between the stellar spectrum and a binary mask (representing the stellar type), which increases the S\/N of the data by a factor proportional to the square root of the number of spectral lines identified in the mask.\n    \n    As discussed in  \\cite{2013MNRAS.436.1215M}, we expect the planet's signature to replicate the stellar signal, scaled down as a result of geometric (e.g., planet size) and atmospheric (e.g., albedo) factors. {The planetary albedo measures the fraction of incident light that is reflected by the planet atmosphere. Several albedo definitions exist \\cite[e.g.,][]{1999ApJ...513..879M,2002MNRAS.330..187C,2010eapp.book.....S}, but in our study we only considered the geometric albedo $A_{\\rm g}$, which is defined as the ratio of a planet's flux measured at opposition ($\\alpha = 0$) by the flux of a Lambertian disk at the same distance and with the same radius as the planet. This allows us to define the planet\/star flux ratio as\n    \\begin{equation}\n      \\frac{F_{planet}(\\alpha)}{F_{Star}} = A_{\\rm g} \\, g(\\alpha) \\left[\\frac {R_{planet}}{a} \\right ]^{2}\n      \\label{eq:FluxesRatio}\n    ,\\end{equation}\n    where $A_{\\rm g}$ is the planet's geometric albedo, $\\alpha$ the orbital phase, $g(\\alpha)$ the phase function, and $R_{planet}$ and $a$ the planetary radius and orbital semi major axis.}.\n    \n    \n    \n    For \\object{51 Peg b}, assuming an albedo of 0.3 and a planetary radius of $1.2\\;\\rm R_{Jup}$, Eq. \\ref{eq:FluxesRatio} will yield a maximum planet-star flux ratio of $\\approx\\,2\\times10^{-5}$.\nTo detect the planet's reflected signal under these conditions, we consequently need data with a combined S\/N of at least $165\\,000$ for a 3 $\\rm \\sigma_{noise}$ detection. A typical G2 stellar CCF-mask used in HARPS has over 4000 spectral lines. This means that in principle a spectrum with a S\/N of about 2600 will contain enough information to allow us to build a CCF with the necessary S\/N to detect the light spectrum of 51 Peg reflected on its planet (at 3-$\\sigma$ level). A lower S\/N will suffice if the albedo or the planetary radius are higher (according to Eq. \\ref{eq:FluxesRatio}).\n    \n\n  \\section{Method}\\label{sec:Method}\n \n    \\subsection{Data}                                       \\label{sec:Data}\n      Our data were collected with the HARPS spectrograph at ESO's 3.6 m telescope at La Silla-Paranal Observatory as part of the ESO program {091.C-0271}. It consists of {90} spectra observed in seven different nights (\\textit{2013-06-08},  \\textit{2013-06-25}, \\textit{2013-08-02}, \\textit{2013-08-04}, \\textit{2013-09-05}, \\textit{2013-09-09,} and \\textit{2013-09-30}), which amounts to about 12.5 h of observing time. These observations were split into several carefully selected time windows in which the planet could be observed close to superior conjunction (i.e., when the day side of the planet faces us) to maximize the planet's flux (maximum phase). These time windows were computed from the ephemeris provided by \\cite{2006ApJ...646..505B}.\n      \n      The obtained  spectra have a S\/N on the $\\rm 50^{th}$ order ($\\sim$5560\\AA) that varies between 122 and 388. The spectra cover the wavelength range from about 3781\\AA \\;to 6910\\AA. More detailed information can be found in Table \\ref{tab:SNRanges}.\n      \n      \\begin{table}\n        \\caption{Available data for the individual nights.}\n        \\centering\\begin{tabular}{c c c c}\n          \\hline\\\\[-.5em]\n          Night &       Number of               &       Total exposure        &       S\/N range\\\\\n                &       Spectra &       [seconds]       &       \\\\\n          \\hline\\\\[-.5em]\n          2013-06-08    &       3       &       1360    &       243 - 296   \\\\\n          2013-06-25    &       10      &       5260    &       273 - 351   \\\\\n          2013-08-02    &       2       &       1092    &       145 - 151   \\\\\n          2013-08-04    &       20      &       10000   &       215 - 311   \\\\\n          2013-09-05    &       4       &       2400    &       191 - 248   \\\\\n          2013-09-09    &       13      &       7810    &       122 - 265   \\\\\n          2013-09-30    &       39      &       17100   &       179 - 388   \\\\\n          \\hline\\\\[-.5em]\n        \\end{tabular}\n        \\label{tab:SNRanges}\n      \\end{table}\n\n      Despite the brightness of the target, some cloudy nights decreased the expected S\/N (e.g., S\/N$\\sim$150 after an exposure of 600s on 2013-08-02, while on 2013-09-30 we managed to obtain $\\sim$390 after 450s).\n      \n    \\subsection{Data reduction}\n      To reduce the data and create the CCFs for each spectrum, the \\textit{HARPS DRS} \\citep{2003Msngr.114...20M} was used. The data were reduced using the default settings and were then fed to the CCF calculation recipe, used with a weighted $G2$ mask \\citep{2002A&A...388..632P}. \n      \n      Selecting an optimized CCF computation step was of critical importance. During the detection process, the CCFs need to be shifted in radial velocity (see below), which implies an interpolation between consecutive pixels. The errors in this interpolation can be minimized and its precision increased by selecting the smallest possible step. On the other hand, the computing time increases as the step size decreases. Therefore we settled for a {$\\rm 50\\; m\\; s^{-1}$} step, a good compromise between computing time and high precision. \n      \n      The CCF width also requires particular attention because we require a window wide enough to cover the planet's orbital path. Since the expected planet semi-amplitude is about 130 $\\rm km\\; s^{-1}$ \\citep[e.g.,][]{2013ApJ...767...27B}, we selected a window of 175 $\\rm km\\; s^{-1}$ on each side of the stellar radial velocity (centred on the stellar CCF). This allowed covering the planet's full orbit while leaving on each side of the corresponding RV a continuum section large enough to estimate the noise level.\n\n    \\subsection{Calculating the best orbital solution}  \\label{sec:Orbital}\n\n      Our initial ephemerides for the orbit of \\object{51 Peg b} were taken from \\cite{2006ApJ...646..505B}. However, the obtained HARPS data allow us to derive precise radial velocities\\footnote{The radial velocities derived with the HARPS DRS pipeline can be found in the online version.} that can be combined with other measurements from the literature, so that we cover a baseline of RV measurements spanning almost 20 years with {different facilities}{(see Table \\ref{tab:Measurements})}.\n\n    \\begin{table}\n          \\centering\\caption{\\object{} Radial velocity data for 51 Peg used to derive the orbital parameters. }\n          \\centering\\resizebox{\\columnwidth}{!}{%\n            \\begin{tabular}{l c c c}\n              \\hline\\\\[-.5em]\n              Instrument                & Number        & $RV_{\\rm System}$ & Reference\\\\\n                                        & of points     & {[$\\rm km\\; s^{-1}$]}          & \\\\\n              \\hline\\\\\n              ELODIE@OHP        &       153             & -33.2516732         & (1)\\\\\n              KECK, AAT, Lick   &       256             & -0.0020280    &(2)\\\\\n              \\hline\n            \\end{tabular\n          }\n          {\\tablebib{(1) \\citet{2004A&A...414..351N}; (2) \\citet{2006ApJ...646..505B}.}\\tablefoot{$RV_{\\rm system}$ corresponds to the radial velocity of the system as measured by the corresponding instrument.}}\n          \\label{tab:Measurements}\n        \\end{table}\n\n      We have thus re-derived the orbital parameters of 51\\,Peg\\,b. These were computed using the code $YORBIT$ \\citep{2011A&A...535A..54S}. This combined dataset allowed us to derive a precise set of orbital parameters for the star and its planetary companion. The derived orbital parameters can be found in Table \\ref{tab:51PegYORBIT}, where the value of $RV_{\\rm system}$ was set to the HARPS value\\footnote{In the fit a different zero point was fitted to each instrument's radial velocities.}. During the fitting process, the eccentricity was fixed to zero because the obtained value was not statistically significant. \n      \n      \n      \n        \\begin{table}\n          \\centering\\caption{Basic orbital parameters for \\object{51 Peg b} as fitted by $YORBIT$}\n          \\centering\\begin{tabular}{l l}\n            \\hline\\\\[-.5em]\n            Parameters  & Value \\\\\n            \\hline\\\\\n            $RV_{\\rm system}$ {[$\\rm km\\; s^{-1}$]}     &       $-33.152$       \\\\\n            Period {[days]}                                             &       $4.231$ \\\\\n            e                                                                   &       $0.0 (fixed)$                \\\\\n            a { [AU]}                                                           &       $0.052$ \\\\\n            $k_{Star}$ {[$\\rm m\\; s{-1}$]}                                                      &       $55.2$  \\\\\n            $m_2\\;sin(i)$ $\\rm [M_{\\rm Jup}]$                           &       $0.450$ \\\\\n            $\\omega$ {[degrees]}                                                &       $0.0 (fixed)$                \\\\\n            $t_{0}$ {[BJD-2400000]}             \n&       $56021.256$     \\\\[.5em]        \n            \\hline\\\\[-.5em]\n          \\end{tabular}\n          \\label{tab:51PegYORBIT}\n        \\end{table}\n\n                \n      \\subsection{Recovery of the planet signal: methodology}         \\label{sec:PlanetRecovery}\n        We extracted the planet signal from the stellar noise with the technique described in \\cite{2013MNRAS.436.1215M}. In brief, after the CCFs of each observation has been computed, the signal can be recovered in two steps: \n        \\begin{itemize}\n          \\item[step 1:]\\label{itm:template} the CCFs are normalized with a stellar template and\n          \\item[step 2:]\\label{itm:stacking} the individual CCFs resulting from step 1 are stacked after correcting for planetary\nvelocity.\\\\\n        \\end{itemize}\n        {Especial care in this process was taken for step 1, which consisted in dividing (normalizing) each of the star+planet CCFs by a carefully built stellar CCF template that represents the stellar signal as accurately as\npossible. This permitted us to remove the stellar signal so that only the planetary signal and noise were left.} For this template we also needed to ensure the highest possbile S\/N so that the division by this template would not introduce significant additional noise. To this end, we constructed two different templates from our observations:\n        \\begin{description}\n          \\item[\\textit{Template \\#1}] This template was constructed by stacking all the CCFs in our sample, after correcting each for the stellar radial velocity induced by the planet. This option implies that the template spectrum also includes a contribution {of the planetary reflected light spectrum}. Despite this fact, since the planet RV varies very rapidly, the planet's minute signal is diluted amidst the noise and can in principle be neglected.\\\\\n\n          \\item[\\textit{Template \\#2}] This template was constructed by stacking the CCFs of the data collected close to the expected inferior conjunction ephemerids (i.e., the position of the planet in its orbit when its night side faces towards Earth, $\\rm 0.9 < \\phi < 0.1${, where $\\phi$ is the orbital phase}). With this template, the contamination of possible reflected light incoming from the planet is minimized. The downside of this selection is that the number of available spectra for constructing the template is more limited than with template \\#1 and therefore might introduce non-negligible noise in the data. For comparison, since only 20 CCFs in our sample are available for template \\#2, its S\/N is $\\sim$40\\% of the S\/N of template \\#1. \n        \\end{description}\n\n        To stack the CCFs after normalization by the stellar template (step 2), we discarded observations in which the planetary and stellar signals were spectroscopically blended or close in velocity. To avoid this, we only considered observations (after correcting for the planet's RV computed assuming an elliptical orbit with the orbital parameters in Table \\ref{tab:51PegYORBIT}) in which the radial velocity difference between the planet and the star exceeds $\\rm 8 \\times FWHM_{Star}$ (eight times the FWHM of the stellar CCF). For a more detailed explanation see \\citet{2013MNRAS.436.1215M}. Figure \\ref{fig:Phases} shows the orbital phases of our data (black circles) and of the observations used to recover the planetary signal (red stars) {at maximum detection significance.}\n        \n        \\begin{figure}\n          \\includegraphics[width = \\columnwidth]{.\/TemplatePoints_All.pdf}\n          \\caption{Orbital phases of our data (black circles) and of the observations used to recover the planetary signal (red stars) {at maximum detection significance}. Phase zero corresponds to the inferior conjunction phase, i.e., the position on the orbit where the planetary night side faces Earth. The green dotted line corresponds to the orbital fit from the parameters in Table \\ref{tab:51PegYORBIT}.}\n          \\label{fig:Phases}\n        \\end{figure}\n        \n        For each individual CCF, the expected radial velocity of the planet was computed from the Keplerian fit presented in Table \\ref{tab:51PegYORBIT}. The different processed CCFs were then re-centered by subtracting the planetary radial velocity from all of them (i.e., centering them around the velocity of the planet at any given moment) and co-adding them to increase their S\/N. \n        \n        After the individual CCFs were stacked, a Gaussian curve was fitted and its amplitude was used to compute the detection significance. To discard fits with no physical meaning, {two} constrains were set in place:\n        \\begin{itemize}\n          \\item $\\rm FWHM_{Planet} > 0.9\\; FWHM_{Star}$ - {This lower limit takes into account noise that might decrease the width of the planet's CCF. For instance, if the star's convective envelope were to be tidally locked to its planetary companion, the planet would \"see\" its host star unaffected by the star's rotation and therefore the planetary CCF might be an unbroadened version of the star's \\citep[for further details we refer to][]{1999ApJ...522L.145C}. This is clearly not the case for the 51 Peg system, as the stellar rotation period \\citep[$\\sim$ 21 days, see][]{2010MNRAS.408.1666S} is much longer than the planetary orbital period ($\\sim$ 4 days), and therefore this effect should be negligible.}\n          \\item $\\rm FWHM_{Planet} < 4\\; FWHM_{Star} $ - Planetary rotation might cause the planetary CCF to be broadened relatively to the CCF of the host star while allowing for enough of a continuum for noise estimation. This limit was computed from the extreme case of a planet with twice the size of Jupiter but the same rotation period and observed edge-on. Nonetheless, we expect the rotation period of a close-in giant planet to be much lower due to tidal interaction\nwith its star. \n        \\end{itemize}\n        \n        The significance of this detection $D$ is then defined by \n        \\begin{equation}\n          D = \\left|\\frac{A}{\\rm \\sigma_{noise}}\\right|\n          \\label{eq:DetectSig}\n        ,\\end{equation}\n        where $A$ is the amplitude of a Gaussian fit to the planet's signal and $\\rm \\sigma_{noise}$ is the continuum noise on both sides of the signal. {We define the continuum noise as the standard deviation of the pixel intensity of the stacked CCFs at a separation of more than $\\rm 2\\times FWHM_{Planet}$ from the center of the detected signal.} \n        \n        As mentioned above, the planet's real mass, and therefore its real orbital velocity semi-amplitude $k_{\\rm Planet}$, is not known. To pinpoint this real semi-amplitude, we computed the detection significance over a range of planetary orbital semi-amplitudes ({75-275 $\\rm km\\; s^{-1}$}) centered on the semi-amplitude ($k_{\\rm Planet}\\sim$ 133 $\\rm km\\; s^{-1}$) calculated from the minimum mass recovered from the radial velocity fit presented in Table \\ref{tab:51PegYORBIT}.\n        \n    \\subsection{Characterizing the planet}\n      When stacking the CCFs, we can expect that the maximum detection significance occurs when the correct velocity shift is used in each individual CCF that we combine. In particular, this is expected to occur if we are able to input the correct semi-amplitude radial velocity signal of the planet. A significant detection of the planetary signal (and its radial velocity semi-amplitude) will then allow inferring the planet-star mass ratio $q$ from\n      \\begin{equation}\n         q \\equiv \\frac{M_{\\rm Planet}}{M_{\\rm Star}} =  \\frac{k_{\\rm Star}}{k_{\\rm Planet}}\n        \\label{eq:MassRatio}\n      ,\\end{equation}\n      where $k_{\\rm Planet}$ is the most significant amplitude as delivered from the analysis of the previous section, and $k_{\\rm Star}$ as derived in Sect. \\ref{sec:Orbital}. With $q$, and given the derived value for the stellar mass ($1.04\\,M_\\odot$), we can compute the real mass for the planet. Together with the minimum mass, this value allows deriving the planetary orbital inclination.\n    \n     \n    \\begin{figure*}\n        \\hfill\\includegraphics[width = \\columnwidth, page =1 ]{.\/51Peg_All_Final.pdf}\\hfill\n        \\includegraphics[width = \\columnwidth, page =1 ]{.\/51Peg_Transit_Final.pdf}\\hfill\n      \\caption{Detection significance as a function of $k_{\\rm Planet}$. The red line corresponds to the $k_{\\rm Planet}$ value for maximum detection. The maximum detection occurs for similar $k_{\\rm Planet}$ values for both templates. The amplitude values set to zero corresponds to $k_{\\rm Planet}$ values for which no Gaussian fit with our restrictions could be achieved. \\textit{Left panel:} using template \\#1; \\textit{right panel:} using template \\#2.}\n      \\label{fig:AmpK2}\n    \\end{figure*} \n    \n    \n    \\section{Results}                                           \\label{sec:Results}\n    \n    \n     \\begin{figure} \n        \\includegraphics[width = \\columnwidth, page =2 ]{.\/51Peg_All_Final.pdf}\n      \\caption{Detected signals as a function of $k_{\\rm Planet}$ over the selected velocity range. With decreasing distance to the maximum detection, the signal becomes better defined and the continuum noise less dispersed.}\n      \\label{fig:DetectionGRid}\n    \\end{figure}\n    \n\n    Following the method described in Sect. \\ref{sec:Method}, we calculated the detection significance {for evenly distributed radial velocity semi-amplitudes of the planet {75 $\\rm km\\; s^{-1}$ < $k_{\\rm Planet}$ < 275 $\\rm km\\; s^{-1}$},, with a step of 0.05 $\\rm km\\; s^{-1}$}. For each $k_{\\rm Planet}$ we stacked the individual CCFs after correcting for the corresponding radial velocity\\footnote{{The constraints mentioned in Sect. \\ref{sec:PlanetRecovery}\ncause the number of spectra used for constructing each combined CCF to vary with the assumed value of $k_{\\rm Planet}$ (it increases with $k_{\\rm Planet}$). However, the total S\/N of the resulting CCF is always above the S\/N threshold for detection of the observed signal (with an amplitude of $\\rm 6\\pm0.4 \\times 10^{-5}$).}}. {We then performed a simple Gaussian fit to the resulting stacked CCF, using the restrictions discussed in Sect. \\ref{sec:Method}. }The significance of the detection (D) was then derived using Eq. \\ref{eq:DetectSig}. This process was repeated for the two different template options presented in Sect. \\ref{sec:Method}.   \n     \n    {Template \\#1 yields a maximum detection significance of 3.7-$\\rm \\sigma_{noise}$ for {$k_{\\rm Planet}$ = 132 $\\rm km\\; s^{-1}$}, while with Template \\#2 a maximum significance of 5.6-$\\rm \\sigma_{noise}$ is computed for {$k_{\\rm Planet}$ = 133 $\\rm km\\; s^{-1}$}. To be conservative, the error bars on the value derived using template \\#2 were computed from the 2-$\\sigma$ uncertainty in the amplitude of the Gaussian fitted to the CCF of the planet at maximum detection significance, yielding $k_{\\rm Planet} = 133^{+19}_{-20}$ {$\\rm km\\; s^{-1}$}. Using the same procedure to compute the error bars for the results using template \\#1 would lead to $k_{\\rm Planet} = 132^{+2}_{-11}$ {$\\rm km\\; s^{-1}$}. However, for this latter case, a simple visual inspection of Fig. 2 shows that the significance curve is relatively flat for values between about 120 and 150 $\\rm km\\; s^{-1}$ (but always above 3-$\\rm \\sigma_{noise}$ in this range). We thus decided to adopt all the values of $k_{\\rm Planet}$ with a significance higher than 3.0-$\\rm \\sigma_{noise}$, that is, $k_{\\rm Planet} = 132^{+19}_{-15}$ {$\\rm km\\; s^{-1}$}. This of course lowers the significance of our detection to 3.0-$\\rm \\sigma_{noise}$ (and not 3.7-$\\rm \\sigma_{noise}$ as derived with the maximum value).}\n    \n    Following the constraints presented in Sect. \\ref{sec:PlanetRecovery}, the planetary CCF for maximum significance was constructed from stacking 25 observations of the available {90} ($\\sim$27\\%). Although template \\#2 yields a higher detection significance, we decided to use template \\#1 for the remainder of the paper since it has a much higher S\/N because many more CCFs were used to construct it. Therefore it will represent the stellar signal more reliably and is less prone to introduce additional noise into the CCF (including correlated noise that is difficult to quantify).\n    \n{Table \\ref{tab:GaussParams} shows the parameters for the recovered CCF derived from the Gaussian fit using template \\#1. In this table, the amplitudes of the star (0.48) and planet ($\\rm 6\\pm0.4 \\times 10^{-5}$) CCFs correspond to the depth of the CCF when the the continuum has been normalized to 1. The planet-to-star flux ratio will then be given by the ratio of these two quantities. The significance of the detection corresponds to the maximum significance for 75 $\\rm km\\; s^{-1}$ < $k_{\\rm Planet}$ < 275 $\\rm km\\; s^{-1}$ and was computed by dividing the amplitude of the planetary CCF by the noise on the wings of the CCF. The FWHM is the full width at half maximum of the fitted Gaussian, 7.43 $\\rm km\\; s^{-1}$ for the star, $\\rm 22.6 \\pm 3.6$ $\\rm km\\; s^{-1}$ for the planet. }\n\n    Figure \\ref{fig:DetectionGRid} shows for several assumed values of $k_{\\rm Planet}$ over the adopted range a section of the stacked CCFs centered on the radial velocity  where the planetary signal would be expected from a Keplerian fit to the observations using the parameters in Table \\ref{tab:51PegYORBIT} and the selected $k_{\\rm Planet}$. {The plot illustrates how the recovered CCF changes when we co-add spectra for different assumed semi-amplitude values. When a value for the semi-amplitude close to 132 $\\rm km\\; s^{-1}$ is assumed, the recovered signal is well defined and its wings are less noisy; this is closer to the distribution of the expected Gaussian shape of a CCF. However, as the assumed value of the semi-amplitude departs from 132 $\\rm km\\; s^{-1}$, the CCFs of the {individual} observations will be co-added to each other, albeit imperfectly. Therefore, in these cases a signal is also expected to be detected, but spread out across the {continuum} and of lower significance because the wings additionally contribute to noise. When close enough to the best-fit value of the semi-amplitude, a signal above the noise level can still be seen. This signal might seem to be of similar amplitude as the one detected for the correct value of orbital semi-amplitude, but it will have a lower significance due to the increased noise in the wings. This can be seen in the panels for $k_{\\rm Planet}$ = 115 and 120 $\\rm km\\; s^{-1}$, which show signals that might seem to\nbe of similar amplitude to the one in the central panel, but because of the increased noise in the wings, these signals are of lower significance according to Eq. \\ref{eq:DetectSig} (see Fig. \\ref{fig:AmpK2}).} \n     \n     \\begin{figure} \n        \\includegraphics[width = \\columnwidth, page =3 ]{.\/51Peg_All_Final.pdf}\n      \\caption{Detected signals as a function of $k_{\\rm Planet}$ , but for random sections of the normalized CCF where the planetary signal is not expected to be found. }\n      \\label{fig:DetectionGRidNull}\n    \\end{figure}\n    \n    To test whether the observed signal might be a spurious combination of random noise, we repeated the process, but instead of using a Keplerian function to compute the expected radial velocity signal of the planet, we attributed a random radial velocity within the range [-50, 80] $\\rm km\\; s^{-1}$ to each\nCCF. The result of that analysis can be seen in Fig. \\ref{fig:DetectionGRidNull}. They show that no significant signal is detected, as expected. This test gives us confidence that the detected CCF of the planet is not a mere artifact. {The dips that appear in the panels of Fig. \\ref{fig:DetectionGRidNull} appear at random positions, and their FWHM is always lower than the FWHM of the star. For instance, the FWHM of the center dip in the 115 $\\rm km\\; s^{-1}$ panel of Fig. \\ref{fig:DetectionGRidNull} has a FWHM of only 3.8 $\\rm km\\; s^{-1}$. This strongly suggests that these peaks are purely caused by noise, especially as they appear randomly positioned. The constraints that we have specified in Sect. \\ref{sec:Method} ensure these dips as discarded as nonphysical and are not confused with the planetary CCF.}\n    \n    {To verify whether our analysis data-processing was sound, we simulated 100 sets of noiseless CCF functions simulating idealized observations of the  star + planet signal for random values of the planetary semi-amplitude $k_{\\rm Planet}$ in the range [100, 180] $\\rm km\\; s^{-1}$. These sets of simulated observations consisted of {90} star+planet CCFs (the number of observations we have), where the planet radial velocity is computed from one of the randomly selected $k_{\\rm Planet}$ through Eq.\\,\\ref{eq:MassRatio}. Each simulated CCF was built by positioning the stellar Gaussian at the observed velocity of the star at each given moment (obtained from the real CCFs) and co-adding the planet Gaussian with an {amplitude} of $5\\times 10^{-5}$ (a value similar to the expected for our test subject) . The goal was to verify whether the injected signal was always fully recovered by the reduction process. The results show that the injected signal was successfully recovered in all simulations. The obtained values of $k_{\\rm Planet}$ were always the values that were injected into the expected figure, with a standard deviation of only 0.11\\%. Similar results were obtained for the {amplitude}, which was always recovered with a standard deviation lower than 0.01\\%. This test shows that that the data-analysis pipeline works correctly and can be used safely to retrieve the planetary signal.}\n        \n    \\begin{figure} \n        \\includegraphics[width = \\columnwidth, keepaspectratio =true, page = 1]{.\/51Peg_Final_CCF.png}\\hfill\n      \\caption{Planet signal for maximum detection and fitted Gaussian curve with the Levenberg-Marquardt method.}\n      \\label{fig:PlanetCCF}\n    \\end{figure}\n\n    The planetary CCF with the highest recovered significance is plotted in Fig. \\ref{fig:PlanetCCF} together with a Gaussian fit using a Levenberg-Marquardt algorithm. After removing the planetary CCF and fitting it with a Gaussian, we used the residuals of the fit to define the error bars. To do so, we injected into the residuals a Gaussian curve with the same parameters as the detected planetary CCF after\nsubtracting the detected Gaussian, but at different radial velocities. This procedure was repeated 10\\,000 times, and in each we recovered the signal injected by fitting a Gaussian profile. The standard deviation of the recovered FWHM and {amplitude} values were then considered to be the 1-$\\sigma$ errors as listed in Table\\,\\ref{tab:GaussParams}. \n\n        \n    \\begin{table}\n      \\caption{{Comparison of the stellar and planetary CCF parameters. }}\n      \\centering\\begin{tabular}{l c c l}\n        \\hline\\\\[-.5em]  \n        Parameter       &       Star    &       {Planet}        &\\\\\n                                &               &                       &\\\\\n        \\hline\\\\\n        {Amplitude}     &       0.48    &       6.0$\\pm$0.4 &\\hspace{-1.em}$\\rm \\times$ $10^{-5}$\\\\\n        Significance $[\\rm \\sigma_{noise}]$ &--&        3.7$\\pm$0.2&\\\\\n        FWHM {[$\\rm km\\; s^{-1}$]}      &       7.43    &       \\hspace{-.5em}22.6$\\pm$3.6&\\\\\n        \\hline\\\\\n      \\end{tabular}\n      {\\tablefoot{Comparison of the stellar and planetary CCF parameters. For the star signal, we present the median value of the amplitude and FWHM of its CCF for all observations. For the planetary CCF, we present the values of the amplitude and FWHM of its Gaussian fit and its detection significance. In both cases, the level of the CCF continuum flux has been set to one.}}\n      \\label{tab:GaussParams} \n    \\end{table}\n\n\n\n  \\section{Discussion}                                          \\label{sec:Discussion}\n    {Our results suggest that we were able to successfully detect the light spectrum of 51\\,Peg reflected on its hot Jupiter companion. The results also indicate that the best-fit semi-amplitude of the orbital motion of the planet is $k_{\\rm Planet} = 132^{+19}_{-15}$ $\\rm km\\; s^{-1}$.}.\n        \n    {The planetary mass cannot be lower than its real mass (corresponding to an orbital inclination of 90 $^{\\circ}$), which places a higher limit on the planetary orbital semi-amplitude of $k_{\\rm Planet} \\sim 133$ $\\rm km\\; s^{-1}$. Combining the detected value of the semi-amplitude with this assumption, Eq. \\ref{eq:MassRatio} yields a real mass for \\object{51 Peg b} of $0.46^{+0.06}_{-0.01}\\;\\rm M_{Jup}$ for a stellar mass of $\\rm 1.04\\; M_{\\odot}$ \\citep{2013A&A...556A.150S}. By comparing this with the derived minimum mass of $m_2\\;sin\\;i = 0.45 \\;\\rm M_{Jup}$ (Table \\ref{tab:51PegYORBIT}), we can infer an orbital inclination of $80^{+10}_{-19}$ degrees. This result is compatible with the results obtained independently\nby \\citet{2013ApJ...767...27B} - $79.6^{\\circ}<I<82.2^{\\circ}$  for a $1\\;\\sigma$ confidence level. }\n    \n    {These results suggest that we were able to successfully recover the reflected planetary signature, but the noise present in the data hinders constraining parameters like the planet's geometric albedo. It is interesting, however, to verify which possible values of the albedo we would recover if we assumed plausible  values for the radii of the planet according to Eq. \\ref{eq:FluxesRatio} and assumed that the planet-star flux ratio is given by the ratio of the amplitudes of the the detected planet and stellar CCFs. Note that the scatter of radii observed for hot-Jupiter planets of a given mass does not allow us to simply use the derived planetary mass to estimate a reliable value for the planetary radius \\cite[e.g.,][]{2010RPPh...73a6901B, 2010SSRv..152..423F}.}\n    \n    {The results show that the obtained $A_{\\rm g}$ is above unity for a radius of 1.2 $\\rm R_{Jup}$ . Furthermore, if the planet's reflectivity is assumed to be Lambertian (i.e., the planetary surface reflects isotropically in all directions), it has been shown that the planetary albedo will have at most\na value of $\\text{}\\text{two thirds}$ \\citep{1975lpsa.book.....S}. Higher values imply that the planetary surface or atmosphere would be strongly backscattering. It has been shown that cloudy planets in the solar system tend to be close to Lambertian \\citep[e.g., Venus - see ][]{2011exop.book..111T}, while rocky planets tend to be more uniformly bright than a Lambert's sphere \\citep[e.g., Earth and Moon - see][]{2011exop.book..111T}. Moreover, literature values suggest that hot Jupiters usually have albedos lower than $\\sim$0.3  \\citep[][]{2011ApJ...729...54C}. However, several works \\citep[e.g., the case of HD 189733b,][]{2013ApJ...772L..16E,2013MNRAS.432.2917P} reported albedo values as high as 0.4-0.5. Such high values may be explained by the scattering of condensates in the high atmosphere \\citep{2003ApJ...588.1121S}. }\n\n{For our measurements, the observations used to recover the planetary signal were obtained at an average orbital phase $g(\\alpha)=0.87$. Thus, for a perfectly reflecting Lambert sphere, that is, $A_{\\rm g} = 2\/3$, Eq. \\ref{eq:FluxesRatio} yields a radius for the planet of 1.6$\\pm$0.2 $\\rm R_{Jup}$ (the error bar denotes the error found in the detected CCF amplitude). For an albedo of 0.5, the planetary radius will increase to 1.9$\\pm$0.3 $\\rm R_{Jup}$. Such high radii have been observed in other hot Jupiters of similar mass and orbital period as 51 Peg\\,b \\citep[e.g., Kepler-12\\,b,][]{2011ApJS..197....9F}, making our assumptions plausible.}\n\n    The estimates discussed here are based on a {3-sigma detection, however}. As we can see in Figure\\,\\ref{fig:PlanetCCF}, the detection we obtained shows a strongly correlated noise signature, which is difficult to take into account when fitting the observed signal with a Gaussian function. We can thus not fully trust the derived parameters for the detected planetary CCF. This is crucial for the FWHM of the recovered CCF, which might be used to estimate the planetary rotation velocity \\citep[e.g.,][]{2002A&A...392..215S}. Such a high CCF width, when compared with the stellar CCF, would imply an extremely rapid rotation for the planet ($\\sim$18 $\\rm km\\; s^{-1}$), much higher than expected for a planet assumed to be tidally locked to its host star ($\\sim$ 2 $\\rm km\\; s^{-1}$). Nonetheless, because of our low confidence in the recovered CCF parameters, any conclusion regarding the planetary rotation velocity would be too speculative. \n    \n    An alternative explanation for the broadening might be an imperfect orbital solution. But our target has been followed for many years, and the star's orbital parameters are very well constrained. Furthermore, the errors in the stellar RV are on the order of the $\\rm m\\; s^{-1}$, which transposed into the planet RV domain would yield at most errors in RV of {2-3} $\\rm km\\; s^{-1}$. This is insufficient to explain such broadening.\n    \n    {Even {though there is no stellar (visual) companion to 51 Peg reported in the literature }\\citep[e.g., see ][]{2002ApJ...566.1132L}, we explored the possibility that the observed signal might be caused by the spectrum of a contaminating background star. We used the Besan\\c{c}on model \\citep{2003A&A...409..523R} to determine the probability of a stellar contaminant close to 51 Peg, whose spectrum would cause the magnitude of the observed signal. Given the area of the CCF for the (here suggested) planetary signal ($10^{-4}$) and the magnitude of 51 Peg, the contaminant should be of magnitude 16 or brighter (brighter if the contaminant does not fall entirely within the HARPS fibre, which has a diameter of 1 arcsec). According to the Besan\\c{c}on model, there are only 760 FGK and M stars per square degree at the position of \\object{51 Peg b} that possess a magnitude brighter than this value\\footnote{We only chose FGK and M stars since other spectral types would not produce a CCF detectable by the HARPS reduction, which used a G2 template for the cross-correlation.}  . If we conservatively assume that any contaminant at a distance smaller than 2.6 arcsec were able to produce the observed signal \\citep[for details see][]{2013A&A...550A..75C}, we can expect a probability of a stellar contaminant of at most only 0.1\\%. We can thus reasonably confidently discard the possibility that the signal is produced by a background stellar contaminant. }\n    \n    {Note that the sky brightness has a magnitude per arcsec$^2$, in V band, of 21.8 on a new moon night, and of 20.0 on a full moon night \\citep[e.g., see ][]{2013A&A...550A..75C}, which excludes this as a possible source of contamination.}\n    \n    {A small offset can also be seen on the center of the CCF\nrecovered with the highest detection significance, which can be expected to be centered around zero for the real $k_{\\rm Planet}$. Several explanations can be found for this offset: \n      \\begin{inparaenum}[i)]\n        \\item it might be physically induced, for instance by high-speed winds from the day to the night side in the planetary atmosphere \\citep[ see ][]{2012ApJ...751..117M};\n        \\item it might be an observational bias that results from stacking observations of the planet all on the same side of the superior conjunction while not at full phase (see Fig. \\ref{fig:Phases});\n        \\item and it might be a higher contribution of noise in one of the sides of the CCF.\n      \\end{inparaenum}\n    However, we consider that this offset is not significant because of the high noise.}\n    \n    To test how much the observed signal can be affected by this noise, we performed the following test. For each observed CCF we injected an artificial signal (a pure Gaussian) with an amplitude of $5\\times10^{-5}$ (similar to the retrieved planetary signal), but at a different position in velocity (at +60\\,km\\,s$^{-1}$ when compared to the planet). We then ran our data-reduction process and tried to retrieve this signal. As can be seen in Fig.\\,\\ref{fig:SimulatedSignal}, we recovered the injected CCF with a similar amplitude of $4.6\\times10^{-5}$. However, the FWHM of the fitted Gaussian is $\\sim\\;$27.6\\,km\\,s$^{-1}$, significantly larger than the injected FWHM value (7.43\\,km\\,s$^{-1}$ as observed in the stellar CCF). This test was repeated for different signal intensities, and we found that when the signal strength is much higher than the noise, it is recovered with parameters close to the injected values. Nonetheless, when the signal strength is close to the noise level, the recovered signal appears broadened. This test shows that the parameters of the recovered CCF are strongly affected by the noise.\n    \n    \\begin{figure}\n     \\includegraphics[width = \\columnwidth]{.\/SimulatedSignal60km.pdf} \n     \\caption{Simulated signal injected at +60\\,km\\,s$^{-1}$ in relation to the position of the center of the planetary CCF.}\n     \\label{fig:SimulatedSignal}\n    \\end{figure}\n\nIn this particular simulation we show that the obtained FWHM is much broader than the injected one. Interestingly, the value for the FWHM we recovered in our data for the planetary signal (Table\\,\\ref{tab:GaussParams}) is also much higher than the value observed for the stellar CCF. This shows that the FWHM we obtain cannot be considered reliable to derive the rotational velocity of the planet or the velocity of its atmospheric winds, for instance \\citep[e.g.,][]{2010Natur.465.1049S}.\n \n\n    \n  \\section{Conclusions}                                         \\label{sec:Conclusions} \n  \n  We presented the first application of the technique described by \\cite{2013MNRAS.436.1215M}. We were able to {find evidence for the reflected light from} the hot-Jupiter 51 Peg\\,b with\nthis technique. {This result is encouraging} and constitutes a very valuable proof of concept. Our method can be used to recover an exoplanet's spectroscopic reflected signature from among the stellar noise, despite the extremely low planet-to-star flux ratio.\n  \n  {Although only a 3 $\\rm \\sigma_{noise}$ detection was possible, the obtained data allow us to place constraints on the planetary mass ($\\rm 0.46^{+0.06}_{-0.01}\\;M_{Jup}$) and orbital inclination ($\\rm 80^{+10}_{-19}$ degrees), assuming\nthe detection is real. As presented above, reasonable values for the albedo (0.5) are found if we assume that \\object{51 Peg b} is an inflated hot Jupiter with a radius of about $1.9\\;\\rm R_{Jup}$. Strongly inflated planets with similar mass and orbital period have been reported in the literature (with radii measured with transits).}\n    \n  Of particular interest is the fact that this detection was possible with data collected with a currently existing observing facility on a 4-meter-class telescope (the HARPS spectrometer at the 3.6 m ESO telescope). Additional data will also allow additionally constraining of some of the inferred parameters by increasing the detection significance and allowing for a better fit to the detected planetary CCF. \n  \n  These encouraging results clearly show a bright future for this type of studies when next-generation instruments (e.g., ESPRESSO at the VLT) and telescopes (e.g., ESO's E-ELT) become available to the community. The sheer increase in precision and collecting power will allow for the detection of reflected light from smaller planets, planets on orbits with longer periods, or an increase in detail for larger planets like \\object{51 Peg b}.\n \n \n\\begin{acknowledgements}\n  This work was supported by the European Research Council\/European Community under the FP7 through Starting Grant agreement number 239953. P.F. and N.C.S. acknowledge support by  Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (FCT) through Investigador FCT contracts of reference IF\/01037\/2013 and IF\/00169\/2012, respectively, and POPH\/FSE (EC) by FEDER funding through the program ``Programa Operacional de Factores de Competitividade - COMPETE''. S.G.S, acknowledges the support from FCT in the form of the fellowships SFRH\/BPD\/47611\/2008. {We further acknowledge the support from Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (Portugal) through FEDER funds in program COMPETE, as well as through national funds, in the form of the grants with references RECI\/FIS-AST\/0176\/2012 (FCOMP-01-0124-FEDER-027493), RECI\/FIS-AST\/0163\/2012 (FCOMP-01-0124-FEDER-027492), and UID\/FIS\/04434\/2013}. {We acknowledge the referee for the several and interesting points raised and for drawing our attention to important shortcomings of our analysis.}\n\\end{acknowledgements}\n\n\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Baselines and Feature Analysis}\n\\label{sec:baselines}\nIn this section, we extract several features following previous work~\\cite{DBLP:conf\/webdb\/BhatiaM12, DBLP:conf\/acl\/DingCLZ08} and adopt different ML methods to build baseline models. In addition to reporting baseline performance, we also perform feature importance analysis to identify key factors in user intent prediction.\n\n\\subsection{Features}\n\\label{subsec:features}\n\n\\begin{table*}[ht]\n\t\\centering\n\t\\footnotesize\n\t\t\\caption{Features extracted for user intent prediction in information-seeking conversations.}\n\t\t\\label{tab:features}\n\t\t\\vspace{-0.3cm}\n\t\t\\begin{tabular}{ l l l l }\n\t\t\t\\toprule\n\t\t\tFeature Name\t & Group & Description & Type \\\\\n\t\t\t\\midrule\n\t\t\tInitial Utterance Similarity     & Content     & Cosine similarity between the utterance and the first utterance of the dialog  & Real   \\\\\n\t\t\tDialog Similarity          & Content     & Cosine similarity between the utterance and the entire dialog                  & Real   \\\\ \n\t\t\tQuestion Mark              & Content     & Does the utterance contain a \\textit{question mark}                                      & Binary \\\\\n\t\t\tDuplicate                  & Content     & Does the utterance contain the keywords \\textit{same}, \\textit{similar}                           & Binary \\\\\n\t\t\t5W1H                        & Content     & Does the utterance contain the keywords \\textit{what}, \\textit{where}, \\textit{when}, \\textit{why}, \\textit{who}, \\textit{how}        & One-hot vector  \\\\ \\hline\n\t\t\tAbsolute Position          & Structural  & Absolute position of an utterance in the dialog                            & Numerical \\\\\n\t\t\tNormalized Position       & Structural  & Normalized position of an utterance in the dialog (AbsPos divided by \\# utterances)  & Real\\\\\n\t\t\tUtterance Length              & Structural  & Total number of words in an utterance after stop words removal             & Numerical \\\\\n\t\t\tUtterance Length Unique        & Structural  & Unique number of words in an utterance after stop words removal            & Numerical \\\\\n\t\t\tUtterance Length Stemmed Unique & Structural  & Unique number of words in an utterance after stop words removal and stemming & Numerical \\\\ \n\t\t\tIs Starter                & Structural  & Is the utterance made by the dialog starter                                & Binary\\\\ \\hline\n\t\t\tThank                       & Sentiment   & Does the utterance contain the keyword \\textit{thank}                               & Binary \\\\ \n\t\t\tExclamation Mark           & Sentiment   & Does the utterance contain an \\textit{exclamation mark}                             & Binary \\\\\n\t\t\tFeedback             & Sentiment   & Does the utterance contain the keyword \\textit{did not}, \\textit{does not}                  & Binary \\\\\n\t\t\tSentiment Scores           & Sentiment   & Sentiment scores of the utterance computed by VADER~\\cite{DBLP:conf\/icwsm\/HuttoG14} (positive, neutral, and negative)  & Real \\\\\n\t\t\tOpinion Lexicon            & Sentiment   & Number of positive and negative words from an opinion lexicon              & Numerical \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\n\\end{table*}\n\n\nWe extract three groups of features to detect user intent in information-seeking conversations, including content, structural, and sentiment features. An overview of the features is provided in Table~\\ref{tab:features}. Although MSDialog is derived from forum data, it is considered as a dialog dataset~\\cite{Qu2018UserIntent}. Thus, we refrain from developing features that can only be extracted from the metadata of the forum, such as user authority level or answer votes, so that our method can be applicable to dialog systems. The features are designed to capture the content and sentiment characteristics of the utterances as well as the structural information of the dialogs. \n\n\\textbf{Content features}. We build a TF-IDF representation of utterances and compute the cosine similarity of the given utterance with the dialog initial utterance (which typically is the question that initiates the QA dialog), and the entire dialog. These features are meant to capture the relevance level of the given utterance to the dialog in a general way. In addition, the presence of question marks is a strong indicator that the current utterance contains a question. Moreover, we assume that 5W1H keywords (what, where, when, why, who, and how) can suggest the type of the question.\n\n\\textbf{Structural features}. The position of an utterance in a dialog can reveal crucial information about user intent. Intuitively, answers tend to be at even number positions in a dialog, while user feedback and follow up questions tend to be at odd number positions. In addition, we include the utterance length with and without duplication removal and stemming. We analyzed the data and found that utterances containing positive feedback are relatively short, while utterances containing questions or answers tend to be long as they typically contain details. Finally, if the given utterance is generated by the information seeker (dialog starter), it is more likely to contain user related questions or user feedback. The structural features not only provide individual characteristics for utterances, but also evaluate the utterances on a dialog level.\n\n\\textbf{Sentiment features}. We expect sentiment features to be useful in identifying user feedback and gratitude expressions. In information-seeking conversations, positive and negative sentiments do not necessarily determine the feedback type. However, we expect them to be correlated to some extent. We also include classic indicators of sentiment, such as the presence of ``thank'', ``does not\/did not'' and exclamation marks. In addition, we use VADER~\\cite{DBLP:conf\/icwsm\/HuttoG14} to compute the positive\/negative\/neutral sentiment scores. We also count the number of positive and negative words using an opinion lexicon~\\cite{Liu:2005:OOA:1060745.1060797}.\n\n\n\n\\subsection{Methods and Evaluation Metrics}\n\\label{subsec:baseline-methods}\n\n\\subsubsection{Methods}\nFor each utterance, we extract a set of features as described in Section \\ref{subsec:features}. To apply traditional ML methods with features to this task, we need to transform this multi-label classification problem to multi-class classification. Three transformation strategies are typically used: binary relevance, classifier chains, and label powerset. Binary relevance does not consider the label correlations and label powerset generates new labels for every label combination. So we choose classifier chains as the transformation strategy for traditional ML methods. This strategy performs binary classifications for each label and take predictions for previous labels as extra features. This transformation strategy is the best fit for our task as it considers the label dependency without explicitly generating new labels for every label combination. We adopt classic ML methods, including Naive Bayes classifier, SVM, random forest, and AdaBoost as baseline classifiers. In addition, we use ML-kNN, which supports multi-label classification by nature.\n\n\n\n\n\n\\subsubsection{Metrics}\nDue to the nature of the intent prediction task, we adopt metrics suitable for multi-label classification problems.\n\n\\textbf{Accuracy} (Acc). It is known as the intersection over the union (IoU) in the multi-label classification settings. Accuracy is defined as the number of correctly predicted labels divided by the union of predicted and true labels for every utterance. For example, if the model predicts ``\\textit{PA+CQ}'' for the given utterance while the ground truth is ``\\textit{PA+IR}'', then the accuracy is $\\frac{1}{3}$. The reported performance is the average metric over all utterances.\n\n\\textbf{Precision, recall and F1 score}. Precision is defined as the number of correctly predicted labels divided by predicted labels. Recall is defined as the number of correctly predicted labels divided by true labels. F1 is their harmonic mean. These metrics provide an overall performance evaluation for all utterances.\n\n\\subsection{Main Experiments and  Results}\n\\label{subsec:baseline-exp}\n\\subsubsection{Experimental Setup}\n\\label{subsubsec:baseline-exp-setup}\nWe split the labeled subset of MSDialog into training, validation, and test sets. Table~\\ref{tab:set-stats} gives the statistics of the three sets. We have to point out that although this dataset is not that large, the annotation cost for such fine-grained user intent in conversations is quite high (estimated around \\$1,700 on MTurk according to~\\cite{Qu2018UserIntent}). This dataset size is also larger than the data used in related work~\\cite{DBLP:conf\/webdb\/BhatiaM12, Surendran2006Dialog}. The models are trained with scikit-multilearn\\footnote{\\url{http:\/\/scikit.ml\/}} and scikit-learn\\footnote{\\url{http:\/\/scikit-learn.org\/stable\/}} on the training set. We tune the hyper-parameters on the validation set based on accuracy and report the performance on the test set.\n\n\\begin{table}[htbp]\n\\footnotesize\n\\centering\n\\caption{Statistics of training, validation, and testing sets}\n\\label{tab:set-stats}\n\\vspace{-0.3cm}\n\\begin{tabular}{@{}llll@{}}\n\\toprule\nItem                        & Train & Val & Test  \\\\ \\midrule\n\\# Utterances               & 8,064 & 986        & 970   \\\\\nMin. \\# Turns Per Dialog    & 3     & 3          & 3     \\\\\nMax. \\# Turns Per Dialog    & 10    & 10         & 10    \\\\\nAvg. \\# Turns Per Dialog    & 4.58  & 4.48       & 4.43  \\\\\nAvg. \\# Words Per Utterance & 70.42 & 67.53      & 68.64 \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\\subsubsection{Baseline Results}\n\\label{subsubsec:baselien-results}\n\nThe baseline results are presented in Table~\\ref{tab:baseline-results}. Two ensemble methods, random forest and AdaBoost achieve the best overall performance of all baseline classifiers. AdaBoost achieves the best accuracy while random forest achieves the best F1\\ignore{\\jtcomment{check consistency of spelling ''F1'' or ''F-1''}} score. Surprisingly, ML-kNN performs relatively poorly despite its nature of an adapted algorithm for multi-label classification.\n\n\\begin{table}[htbp]\n\\footnotesize\n\\centering\n\\caption{Experiment results for baseline classifiers}\n\\label{tab:baseline-results}\n\\vspace{-0.3cm}\n\\begin{tabular}{@{}lllll@{}}\n\\toprule\nMethods    & Acc & Precision       & Recall          & F1              \\\\ \\midrule\nML-kNN     & 0.4715                & 0.6322          & 0.4471          & 0.5238          \\\\ \nNaiveBayes & 0.4870                & 0.5563          & 0.4988          & 0.5260          \\\\\nSVM        & 0.6342                & 0.7270          & 0.5847          & 0.6481          \\\\\nRandForest & 0.6268                & \\textbf{0.7657} & 0.5903          & \\textbf{0.6667} \\\\\nAdaBoost   & \\textbf{0.6399}       & 0.7247          & \\textbf{0.6030} & 0.6583          \\\\\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Additional Feature Importance Analysis}\n\\label{subsec:feature-importance}\n\n\\subsubsection{Feature Group Analysis}\n\\label{subsubsec:feature-group}\nWe use one of the best baseline classifiers, random forest, and different combinations of feature groups to analyze the feature importance on a group level. The hyper-parameters are set to the best ones tuned on all features .\n\n\nFor using a single feature group, structural features is the most important feature group as presented in Table~\\ref{tab:feature-group}. Structural features and content features are significantly more important than sentiment features. We expect the sentiment features to capture the sentiment in user feedback but they might not be able to effectively discriminate other user intent. Structural features provide better performance than content features.  We believe that this can be explained by the fact that hand-crafted content features cannot capture the complex user intent dynamics in human information-seeking conversations. \n\n\\begin{table}[htbp]\n\\footnotesize\n\\centering\n\\caption{Experiment results for different feature groups}\n\\label{tab:feature-group}\n\\vspace{-0.3cm}\n\\begin{tabular}{lllll}\n\\toprule\nGroup(s)       & Acc & Precision       & Recall          & F1             \\\\ \\midrule\nContent                & 0.5272                                                          & 0.6097          & 0.4821          & 0.5384          \\\\\nStructural             & \\textbf{0.5809}                                                 & \\textbf{0.6871} & \\textbf{0.5434} & \\textbf{0.6068} \\\\\nSentiment              & 0.3306                                                          & 0.4087          & 0.3222          & 0.3603          \\\\ \\hline\nCon+Str   & 0.6081                                                          & 0.7393          & 0.5640          & 0.6399          \\\\\nCon+Sen    & 0.5577                                                          & 0.6523          & 0.5179          & 0.5774          \\\\\nStr+Sent & \\textbf{0.6110}                                                 & \\textbf{0.7569} & \\textbf{0.5672} & \\textbf{0.6485} \\\\ \\hline\nAll                    & \\textbf{0.6268}                                                 & \\textbf{0.7657} & \\textbf{0.5903} & \\textbf{0.6667} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nFor combinations of two feature groups, content+structural features and structural+sentiment features achieve comparable results. However, structural+sentiment features achieve slightly higher results on all metrics. The performance of using two groups of features is higher than using one of these two feature groups individually. Thus, combining structural features with another feature group boosts the performance of using structural features alone. Interestingly, content+sentiment features is unable to outperform the structural features alone. The results of using all features is the highest among all settings, confirming that all feature groups contribute to the performance of user intent prediction. \n\n\n\\subsubsection{Feature Importance Scores}\n\\label{subsubsec:feature-scores}\n\n\n\nIn the previous section, we evaluated the feature importance on a group level. In this section we focus on individual features to provide a more fine-grained analysis. We use random forest to output individual feature importance scores.\\footnote{\\url{https:\/\/scikit-learn.org\/stable\/auto_examples\/ensemble\/plot_forest_importances.html}} As described in Section~\\ref{subsec:baseline-methods}, we used classifier chains to transform this multi-label classification problem. This method expands the feature space by including previous label predictions as new features for the current label prediction. This makes it not appropriate to evaluate original features. Thus, we use the Label Powerset method as the data transformation strategy for this section. The relative feature importance scores are presented in Table~\\ref{tab:feature-importance}. This analysis can identify key factors in user intent prediction.\n\n\\begin{table}[htbp]\n\\footnotesize\n\\centering\n\\caption{Individual feature importance from a random forest classifier with relative importance scores. ``Str'', ``Con'', ``Sen'' refer to ``Structural'', ``Content'', ``Sentiment'' respectively.} \n\\vspace{-0.3cm}\n\\label{tab:feature-importance}\n\\begin{tabular}{llll|llll}\n\\toprule\nRank & Feature                    & Group      & Impt & Rank & Feature               & Group     & Impt \\\\ \\midrule\n1    & AbsPos              & Str & 1.0        & 13   & Lex(Pos) & Sen & 0.2814     \\\\\n2    & InitSim    & Con    & 0.9745     & 14   & Lex(Neg) & Sen & 0.2337     \\\\\n3    & NormPos             & Str & 0.8684     & 15   & Thank                      & Sen & 0.1607     \\\\\n4    & Starter                      & Str & 0.8677     & 16   & How                        & Con   & 0.08074    \\\\\n5    & DlgSim               & Con    & 0.6778     & 17   & Dup                  & Con   & 0.06908    \\\\\n6    & SenScr(Neu)      & Sen  & 0.6465     & 18   & What                       & Con   & 0.06576    \\\\\n7    & SenScr(Pos)     & Sen  & 0.5601     & 19   & ExMark           & Sen & 0.06424    \\\\\n8    & Len                & Str & 0.5335     & 20   & When                       & Con   & 0.05989    \\\\\n9    & LenUni        & Str & 0.4381     & 21   & Feedback              & Sen & 0.02859    \\\\\n10   & LenStem & Str & 0.4354     & 22   & Where                      & Con   & 0.02356    \\\\\n11   & SenScr(Neg)     & Sen  & 0.3495     & 23   & Why                        & Con   & 0.0232     \\\\\n12   & QuestMark                    & Con    & 0.3003     & 24   & Who                        & Con   & 0.01423    \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nWe summarize our observations as follows:\n(1) Structural features including ``Absolute Position'', ``Normalized\nPosition'', ``Is Starter'' are ranked in the top-5 in terms of feature importance. Moreover, other structural features, such as various forms of utterance length are observed to be relatively informative in general. This confirms the results in Section~\\ref{subsubsec:feature-group} that the structural feature group is the most important one.\n(2) ``Initial Utterance Similarity'' and ``Dialog Similarity'' are content features that can be highly informative for identifying user intent. Both features are indicators of how closely the utterance connects with the information-seeking process. Other content features, such as ``5W1H'', however, contribute little to predicting user intent.\n(3) Some sentiment features are relatively important in identifying user intent, such as positive and neutral sentiment scores. However, some other sentiment features contribute little to the task, such as the existence of exclamation marks and ``thank''.\n(4) We observe that features ranked from the 15\\textsuperscript{th} to the last one in Table~\\ref{tab:feature-importance} are all ``keyword features''. These features are based on a simple rule that whether the given utterance contains pre-defined keywords. For example, the ``5W1H'' feature looks for ``what\/where\/when\/why\/who\/how'' in the given utterance and the ``Feedback'' feature looks for ``did not\/does not''. The major drawback of manual feature engineering is amplified in this task due to the complexity and diversity of human information-seeking conversations.\n\n\n\n\n\n\\section{Conclusions} \n\\label{sec:conclusion}\n\n\n\nIn this paper, we studied two approaches to predict user intent in information-seeking conversations. First we use different ML methods with a rich feature set, including the content, structural, and sentiment features. We perform thorough feature importance analysis on both group level and individual level, which shows that structural features contribute most in this prediction task. Given findings from feature analysis, we construct enhanced neural classifiers to incorporate context information for user intent prediction. The enhanced neural model without feature engineering outperforms the baseline models by a large margin. Our findings can provide insights in the important factors of user intent prediction in information-seeking conversations. Future work will consider other methods for user intent prediction. Utilizing the predicted user intent to rank or generate conversation responses in an information-seeking setting is also interesting to explore.\n\n\n\n\\section{Task Definition and Dataset}\n\\label{sec:task-and-data}\n\\subsection{Task Definition}\n\\label{subsec:task}\nThe research problem of user intent prediction in information-seeking conversations is defined as follows. The input of the system is an information-seeking dialog dataset $\\mathcal{D} = \\{(\\mathcal{U}_i, \\mathcal{Y}_i)\\}_{i=1}^N$ and a set of user intent labels $ \\mathcal{L} = \\{l_1, l_2, \\dots, l_c\\} $. A dialog $ \\mathcal{U}_i = \\{u_i^1, u_i^2, \\dots, u_i^{k}\\} $ contains multiple turns of utterances. $u_i^{k}$ is the utterance at the $k$-th turn of the  $i$-th dialog. $\\mathcal{Y}_i$ consists of annotated user intent labels $ \\{\\mathbf{y}_i^1, \\mathbf{y}_i^2, \\dots, \\mathbf{y}_{i}^k\\} $, where $\\mathbf{y}_i^k = \\{y_i^{k(1)}, y_i^{k(2)}, \\dots, y_i^{k(c)}\\}$. Here $y_i^{k(m)}, \\dots, y_i^{k(n)} = 1$ denotes that the utterance $u_i^{k}$ in dialog $\\mathcal{U}_i$ is labeled with user intent \\{$l_m, \\dots, l_n $\\}. Given an utterance $u_i^{k}$ and other utterances in dialog $\\mathcal{U}_i$, the goal is to predict the user intent $\\mathcal{Y}_i$ of this utterance. The challenge of this task lies in the complexity and diversity of human information-seeking conversations, where one utterance often expresses multiple user intent~\\cite{Trippas:2018:IDS:3176349.3176387}.\n\\subsection{Dataset}\n\\label{subsec:data}\n\nWe use the MSDialog dataset that consists of technical support dialogs for Microsoft products. \nThe data was crawled from the well-moderated Microsoft Community forum\\footnote{\\url{https:\/\/answers.microsoft.com}} which contains high-quality technical support dialogs between users and agents.\nThe agents include Microsoft staff, community moderators, MVPs, or other experienced product users. Very rich structured data were collected with the dialogs, including the question popularity, answer vote, affiliation of information providers, etc. We choose this dataset because of its information-seeking nature and the well annotated user intent. \nAlthough MSDialog has limitations such as a narrow subject domain and forum-style language, no other openly available dialog datasets with the same detailed annotation exist. We believe this should be a first step to predict user intent in an information-seeking setting.\n\nThe dataset contains two sets, a complete set that consists of all the crawled dialogs and a labeled subset that contains only dialogs with user intent annotation. A taxonomy of 12 labels presented in Table~\\ref{tab:taxonomy} were developed in~\\citet{Qu2018UserIntent} based on work by ~\\citet{DBLP:conf\/webdb\/BhatiaM12} to characterize the user intent in information-seeking conversations. We also present the user intent distribution in Table~\\ref{tab:taxonomy}. The complete set consists of 35,000 multi-turn QA dialogs in the technical support domain. Over 2,000 dialogs were selected for user intent annotation on MTurk with the following criteria: (1) With 3 to 10 turns. (2) With 2 to 4 participants. (3) With at least one correct answer selected by the community. (4) Falls into one of the categories of following major Microsoft products: Windows, Office, Bing, and Skype. The inter-rater agreement score was used to ensure the annotation quality. One utterance can be labeled with more than one user intent. A comparison of statistics between the complete set and the labeled subset is presented in Table~\\ref{tab:compare-stats}. \n\n\n\n\n\n\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\caption{User intent taxonomy and distribution in MSDialog}\n\\label{tab:taxonomy}\n\\vspace{-0.3cm}\n\\begin{tabular}{llll}\n\\toprule\n\\hspace{-0.08in} Code & \\hspace{-0.1in} Label \\hspace{-0.1in} & Description \\hspace{-0.1in}                              \\hspace{-0.1in} & \\%              \\\\ \\hline\n\\hspace{-0.08in} OQ   & \\hspace{-0.18in} Original Question     \\hspace{-0.15in} & The first question from the user to initiate the dialog. \\hspace{-0.15in} & 13              \\\\\n\\hspace{-0.08in} RQ   & \\hspace{-0.18in} Repeat Question       \\hspace{-0.15in} & Other users repeat a previous question.                  \\hspace{-0.15in} & 3                 \\\\\n\\hspace{-0.08in} CQ   & \\hspace{-0.18in} Clarifying Question   \\hspace{-0.15in} & User or agent asks for clarifications.                   \\hspace{-0.15in} & 4                \\\\\n\\hspace{-0.08in} FD   & \\hspace{-0.18in} Further Details       \\hspace{-0.15in} & User or agent provides more details.                     \\hspace{-0.15in} & 14                  \\\\\n\\hspace{-0.08in} FQ   & \\hspace{-0.18in} Follow Up Question    \\hspace{-0.15in} & User asks for follow up questions about relevant issues. \\hspace{-0.15in} & 5                \\\\\n\\hspace{-0.08in} IR   & \\hspace{-0.18in} Information Request   \\hspace{-0.15in} & Agent asks for information from users.                   \\hspace{-0.15in} & 6                 \\\\\n\\hspace{-0.08in} PA   & \\hspace{-0.18in} Potential Answer      \\hspace{-0.15in} & A potential answer or solution provided by agents.       \\hspace{-0.15in} & 22              \\\\\n\\hspace{-0.08in} PF   & \\hspace{-0.18in} Positive Feedback     \\hspace{-0.15in} & User provides positive feedback for working solutions.   \\hspace{-0.15in} & 6             \\\\\n\\hspace{-0.08in} NF   & \\hspace{-0.18in} Negative Feedback     \\hspace{-0.15in} & User provides negative feedback for useless solutions.   \\hspace{-0.15in} & 4     \\\\\n\\hspace{-0.08in} GG   & \\hspace{-0.18in} Greetings\/Gratitude   \\hspace{-0.15in} & Greetings or expressing gratitude.                       \\hspace{-0.15in} & 22               \\\\\n\\hspace{-0.08in} JK   & \\hspace{-0.18in} Junk                  \\hspace{-0.15in} & No useful information in the utterance.                 \\hspace{-0.15in}  & 1           \\\\\n\\hspace{-0.08in} O    & \\hspace{-0.18in} Others                \\hspace{-0.15in} & Utterances cannot be categorized using other classes.    \\hspace{-0.15in} & 1            \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\n\n\\begin{table}[ht]\n\\footnotesize\n\\centering\n\\caption{Statistics of MSDialog (complete \\& labeled subset)}\n\\label{tab:compare-stats}\n\\vspace{-0.3cm}\n\\begin{tabular}{@{}lll@{}}\n\\toprule\nItems                       & Complete set        & Labeled subset \\\\ \\midrule\n\\# Dialogs                  & 35,000              & 2,199    \\\\\n\\# Utterances               & 300,000             & 10,020   \\\\\n\\# Words (in total)         & 24,000,000          & 653,000  \\\\\nAvg. \\# Participants        & 3.18                & 2.79     \\\\\nAvg. \\# Turns Per Dialog    & 8.94                & 4.56     \\\\\nAvg. \\# Words Per Utterance & 75.91               & 65.16    \\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\nIn order to test the generalization performance of our findings, we use a small portion of UDC that is annotated with the same user intent types. This dataset also consists of multi-turn information-seeking conversations in a technical support domain between an information seeker and provider. However, UDC is generated from internet relay chat (IRC) and contains a significant amount of typos, Internet language, and abbreviations. In addition, UDC contains shorter utterances and more turns per dialog. This part of experiment is presented in Section~\\ref{subsec:udc}.\n\n\n\n\\subsection{Data Preprocessing}\n\\label{subsec:data-transformation}\n\n\n\nThe purpose of this classification task is to identify and predict user intent so that CAs can process the information accordingly to satisfy the users' information needs. However, utterances which were labeled \\textit{Greetings\/Gratitude}, \\textit{Junk}, and \\textit{Others} do not contribute to the purpose of providing information about QA related user intent. Therefore, we remove occurrences of these labels. Note that we only remove these labels if there are more than one label of the given utterance. For example, if the annotation for the given utterance is \\textit{GG+OQ}, we transform the annotation into \\textit{OQ}. If the annotation is just \\textit{GG}, no transformation is needed. This reduces the number of unique label combinations from 316 to 152. \n\nIn addition, some label combinations of user intent labels are quite rare in the data. As indicated in Figure~\\ref{fig:label-dist}, the top frequent label combinations have hundreds of occurrences in the data (e.g. \\textit{PA}, \\textit{OQ}, \\textit{PF}, \\textit{FD+PA}, \\textit{FD}), while the least frequent labels only have exactly one occurrence (e.g. \\textit{CQ+FD+IR+RQ}, \\textit{CQ+FD+FQ+PF}). These rare label combinations are very likely due to minor annotation errors or noise with MTurk. Annotation quality assurance was performed based on the dialog-level inter-rater agreement~\\cite{Qu2018UserIntent} to keep the complete dialog intact and thus may result in minor noise on an utterance level. We also plot the cumulative distribution of label combinations for better illustration in Figure~\\ref{fig:label-dist-incre}. The most frequent 32 label combinations constitute 90\\% of total label combination occurrences as marked in the figure. All 12 user intent labels are individually present in these 32 most frequent combinations except for \\textit{Others}. For the rest of the label combinations, we randomly sample one of the labels from each combination as the user intent label for the given utterance. For example, if the annotation for the given utterance is \\textit{CQ+FD+IR+RQ}, we transform it into a single label by randomly sampling one of the four labels, such as \\textit{CQ}. Therefore, the total number of label combinations in the data was reduced to 33 (including \\textit{Others}). \nWe adopted this setting since these rare label combinations are very likely due to minor annotation errors. \nIn addition, it would be very difficult to learn a prediction model for these label combinations with few instances.\n\n\n\n\n\\begin{figure}[h!]\n  \\begin{subfigure}[t]{.23\\textwidth}\n  \\centering\n    \\includegraphics[width=.9\\textwidth]{img\/label_dist.pdf}\n    \\caption{Label combination occurrence from the most frequent to the least frequent.}\n    \\label{fig:label-dist}\n  \\end{subfigure}\\hfill\n  \\begin{subfigure}[t]{.23\\textwidth}\n  \\centering\n    \\includegraphics[width=.9\\textwidth]{img\/label_dist_incre.pdf}\n    \\caption{Cumulative occurrence probability added from the most frequent to the least frequent. The marked point is $(32, 0.9)$.}\n    \\label{fig:label-dist-incre}\n  \\end{subfigure}\n  \\vspace{-0.3cm}\n  \\caption{Label combination distribution}\n\\end{figure}\n\n\nFor UDC, we observe a similar label combination distribution. So we preprocess the data in the same way. We have 34 label combinations for the UDC with 27 of them overlapping with MSDialog.\n\\section{Introduction}\n\\label{sec:intro}\n\n\nMany companies have launched conversational assistants (CA) such as Amazon Echo, Google Home, Microsoft Cortana, etc. These devices allow users to issue voice commands to the CA for goal oriented tasks or to conduct simple question answering (QA) tasks, such as adding calendar events or asking for news.\nThis trend has led many researchers in the information retrieval (IR) and natural language processing (NLP) community to pay more attention to conversational search. For examples, SIGIR'18, ICTIR'17, and EMNLP'18 all have workshops\\footnote{\\url{https:\/\/sites.google.com\/view\/cair-ws\/cair-2018}}\\textsuperscript{,}\\footnote{\\url{http:\/\/sigir.org\/ictir2017\/sessions\/search-oriented-conversational-ai-scai\/}}\\textsuperscript{,}\\footnote{\\url{https:\/\/scai.info\/}} on conversational search.\n\n\n\\if0\nHowever, most CAs are not yet capable of handling multi-turn information-seeking conversations, where users have multiple rounds of information exchange with CAs to retrieve answers. This kind of conversations can be best described as a joint effort between human users and CAs in an exploratory manner. Different from single-turn QA, conversational QA requires CAs to proactively ask or confirm before presenting an answer~\\cite{DBLP:journals\/corr\/LiMCRW16a} and expect user feedback after presenting an answer. Despite the effort of all the major internet companies, iterative answer finding with CAs has not yet been achieved. A main reason is the failure to model the conversation bout the information need both before and after an answer has been given. \nThe key to the solution is to accurately detect and predict user intent in this interactive information-seeking process. Concretely, CAs would be capable to improve previous answers if it can correctly process critical information provided by the users, such as relevance judgment (feedback) and clarifications of the information need. Moreover, we expect CAs to request more information proactively when they are not confident to provide an answer. \n\\fi\n\nHowever, most CAs are not yet capable of handling multi-turn information-seeking conversations, where users have multiple rounds of information exchange with CAs to retrieve or specify answers. \nOne reason is the difficulty to model the conversation about the information need both before and after an answer has been given. Thus an important step in modeling conversational interactions is to accurately detect and predict user intent in this interactive information-seeking process. More specifically, CAs should be capable of improving previous answers if they can correctly process critical information provided by the users, such as relevance judgments (feedback) and clarifications of the information need. Thus, CAs need to elicit more information proactively when they are not confident, before providing an answer~\\cite{Trippas:2018:IDS:3176349.3176387}.\n\nThe Learn-IR workshop\\footnote{\\url{https:\/\/task-ir.github.io\/wsdm2018-learnIR-workshop\/}} at WSDM'18 highlighted the significant research need for user intent analysis and prediction in an interactive information-seeking process. To address this research demand, we conducted experiments on user intent prediction using the MSDialog~\\cite{Qu2018UserIntent} data. This data collection consists of multi-turn information-seeking dialogs in the technical support domain. The dialogs are typically initiated by an information seeker who asks technical issues about Microsoft products, such as ``How do I downgrade from Windows 10 to Windows 7?''. This kind of question is non-factoid and often requires multiple rounds of conversational interactions. The answers are provided by Microsoft staff and other experienced product users such as ``Microsoft Most Valuable Professionals'' (MVPs).\nThese human agents (information providers) also explicitly ask for feedback on their provided answers and thus keep the users engaged. The MSDialog data is annotated with a set of 12 \\textit{user intent types}~\\cite{DBLP:conf\/webdb\/BhatiaM12, Qu2018UserIntent}. There are different definitions of ``user intent'' in our field. In this paper, user intent refers to a taxonomy of utterances in information-seeking conversations (Table~\\ref{tab:taxonomy}). Each utterance of the MSDialog was annotated with the user intent types using Amazon Mechanical Turk (MTurk)\\footnote{\\url{https:\/\/www.mturk.com\/}}. MSDialog provides a high-quality resource to show how information-seeking conversations are structured between humans. In addition to MSDialog, we also used a portion of Ubuntu Dialog Corpus (UDC)~\\cite{Lowe2015The} which was annotated with the same user intent types in~\\cite{Qu2018UserIntent} to further validate our findings.\n\nThe purposes of user intent prediction are threefold. First, it is necessary for CAs to accurately identify user intent in information-seeking conversations. Only in this way can CAs process the information accordingly and use it to provide answers and adjust previous answers. Similar to customer service over phones, routing user questions to different sub-modules in a conversational retrieval system is only possible if the user intent is correctly identified. Second, the CAs need to learn and imitate the behavior of human agents. \nBy identifying user intent in information-seeking conversations, we expect the CA to learn the use of different intent and when to issue requests for more information or details spontaneously. Finally, user intent prediction models can be used to automatically annotate more dialog utterances for data analysis and other tasks such as conversational answer finding.\n\n\nPrevious work typically focused on dialog act classification for open-domain conversations~\\cite{Stolcke2000Dialogue, LiuEMNLP17, Khanpour2016DialogueAC}. In human-computer chitchat, the goal of the CA is to generate responses that are as realistic as possible with the primary purpose of entertaining. In contrast to chatting, users initiate information-seeking conversations for specific information needs. Human behaviors in chatting and information-seeking conversations can be very different due to the fundamentally distinct purposes. In addition, the Dialog State Tracking Challenges (DSTC)\\footnote{\\url{https:\/\/www.microsoft.com\/en-us\/research\/event\/dialog-state-tracking-challenge\/}} focus on goal oriented conversations. These tasks are typically tackled with slot filling~\\cite{Zhang:2016:JMI:3060832.3061040, DBLP:conf\/aaai\/YanDCZZL17}. In information-seeking conversations, slot filling is not suitable because of the diversity of information needs. User intent analysis and prediction are needed for an information-seeking setting.\n\nWe conduct experiments using two different approaches to predict user intent in information-seeking conversations. Firstly, we extract rich features to capture the content, structural, and sentiment \t\ncharacteristics of utterances and learn models with traditional machine learning (ML) methods. Secondly, we use the implicit representation learning in neural architectures to predict user intent without feature engineering. We then incorporate context information into neural models for enhanced performance. \n\n\\if0\nOur contributions can be summarized as follows:\n\n\\textbf{(1) We created a conversation dataset called \\textit{MSDialog}.} It consists of multi-turn information-seeking dialogs in the technical support domain. We also annotated utterance level user intent in conversations with MTurk. We will release this data with annotated user intent labels to the research community. \n\n\\textbf{(2) We identified the key factors in user intent prediction.} We designed and extracted rich features to predict user intent in information-seeking conversations. These features can be classified into three groups: content, structural, and sentiment. We performed in-depth feature importance analysis on both group level and individual level. \n\n\\textbf{(3) We designed enhanced neural classifiers to predict user intent without explicit feature engineering.} We conducted experiments with several variations of neural models with different incorporated information. We proved that neural models can achieve comparable performances without feature engineering. Moreover, neural models achieved statistically significant improvements over traditional methods after incorporating context information.\n\\fi\n\nOur contributions can be summarized as follows. (1) We extract rich features including feature groups related to content, structures, and sentiment to predict user intent in information-seeking conversations. We perform an in-depth feature importance analysis on both group and individual level to identify the key factors in this task. (2) We design several variations of neural classifiers to predict user intent without explicit feature engineering. We show that neural models can achieve comparable performance compared to feature engineering based methods. Moreover, neural models achieve statistically significant improvements over traditional methods after incorporating context information. (3) Our experiments show that the trained model achieves good generalization performance on another open benchmark information-seeking conversation dataset (UDC). The code of the implemented user intent prediction models will be released to the research community.\\footnote{\\url{https:\/\/github.com\/prdwb\/UserIntentPrediction}}\n\nThe rest of our paper is organized as follows. In Section~\\ref{sec:relatedwork}, we present related work regarding utterance type classification, conversational search, and multi-turn question answering. In Section~\\ref{sec:task-and-data}, we formulate the research question of user intent prediction in information-seeking conversations. We also describe the data creation and annotating process. In Section~\\ref{sec:baselines}, we extract rich features and learn traditional ML models for user intent prediction. We also perform feature importance analysis in this section. In Section~\\ref{sec:neural-models}, we introduce various enhanced neural classifiers for user intent prediction. Section~\\ref{sec:conclusion} presents the conclusion and future work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Enhanced Neural Classifiers}\n\\label{sec:neural-models}\nWe expected the content of an utterance to be a good indicator of user intent types compared to other features. \\ignore{\\lycomment{This is an overclaim. How about structure features like the ``Absolute Position'' ? You can say ``the content and sequential information of utterances is important for user intent type prediction''. or ''to the best of our knowledge...''} }However, as shown in Section~\\ref{subsec:feature-importance}, the hand-crafted content features are unable to capture the complex characteristics of human information-seeking conversations. Thus, in this section we adopt neural architectures to automatically learn representations of utterances without feature engineering.\n\n\\subsection{Our Approach}\n\\label{subsec:our-approach}\n\\subsubsection{Base Models}\n\\label{subsubsec:cnn}\nGiven the previous success in modeling text sequences using CNN and bidirectional LSTM (BiLSTM)~\\cite{GRAVES2005602}, we choose these two architectures as our base models. Although utterances are grouped as dialogs, the base models consider utterances independently.\n\nGiven an utterance $u_i^{k} = \\{w_1, w_2, \\dots, w_m\\}$ (the $k$-th utterance in the $i$-th dialog) that contains $m$ tokens, we first transform the sequence of tokens into a sequence of token indices $\\bm{S} = \\{s_1, s_2, \\dots, s_m\\}$. Then we pad the sequence $\\bm{S}$ to a fixed length $n$ (the max sequence length). \nBoth CNN and BiLSTM start with an embedding layer initiated with pre-trained word embeddings. Preliminary experiments indicated that using MSDialog (complete set) to train word embeddings is more effective than using GloVe~\\cite{pennington2014glove} in terms of final model performance. The embedding layer maps each token in the utterances to a word embedding vector with a dimension of $d$.\n\n\nWe focus on the CNN model following previous work~\\cite{DBLP:conf\/emnlp\/Kim14} here, because it achieves better performance in our experiments. After the embedding layer, filters with the shape $(f, d)$ are applied to a window of $f$ words. $f$ is also referred to as the filter size. Concretely, a convolution operation is denoted as\n\\begin{footnotesize}\n\\begin{equation}\\label{eqn:conv}\n\\begin{aligned}\nc_i = \\sigma(\\mathbf{w} \\cdot \\mathbf{e}_{i:i+f-1} + b)\n\\end{aligned}\n\\end{equation}\n\\end{footnotesize}\nWhere $c_i$ is the feature generated by the $i$-th filter with weights $\\mathbf{w}$ and bias $b$. This filter is applied to an embedding matrix, which is the concatenation from the $i$-th to the $(i+f-1)$-th embedding vectors. An non-linearity function (ReLU) is also applied. This operation is applied to every possible window of words and generates a feature map $\\mathbf{c} = \\{c_1, c_2, \\dots, c_{n-f+1}\\}$. More filters are applied to extract features of the utterance content. Max pooling are applied to select the most salient feature of a window of $f'$ features by taking the maximum value $\\hat{c_i} = max\\{\\mathbf{c}_{i:i+f'-1}\\}$. $f'$ denotes the max pooling kernel size. A dropout layer is applied after the pooling layer for regularization.\n\n\nAfter the last convolutional layer, we perform global max pooling by taking the maximum value $\\hat{c} = max\\{\\mathbf{c}\\}$ for the feature map $\\mathbf{c}$ at this step. This operation reduces the dimension of the tensor to one. This tensor is further transformed to an output tensor of shape $(1, l)$, where $l$ is the number of user intent labels (12 for our task). Sigmoid activation is applied to each value of the output tensor to squash the value to a confidence level between 0 and 1. A threshold is chosen to determine whether the given label present in the final prediction. If the model is not confident of predicting any label, the label of the highest confidence level is the prediction. We tuned the threshold with the validation data.\n\n\\subsubsection{Incorporate Context Information}\n\\label{subsubsec:cnn-context}\n\nAs shown in the previous work~\\cite{Qu2018UserIntent}, user intent follows clear flow patterns in information-seeking conversations. The user intent of a given utterance is closely related to the utterances around it, which compose the \\textit{context} for the given utterance. Incorporating context information into neural models is easier compared to that for traditional ML methods shown in Section~\\ref{sec:baselines}. We consider two ways as follows.\n\n\n\\textbf{Direct Expansion}. The most straightforward way to incorporate context information is to expand the given utterance with its context. Concretely, the expanded utterance for $u_i^k$ is $\\hat{u}_i^k = u_i^{k-1} \\oplus u_i^{k} \\oplus u_i^{k+1} $, where $\\oplus$ is the concatenate operator. $\\hat{u}_i^k$ is considered as the given utterance in base models.\n\n\\textbf{Context Representation}. Given an expanded utterance $\\hat{u}_i^k$ as input, the neural architecture first segments it into three original utterances of $u_i^{k - 1}$, $u_i^{k}$, and $u_i^{k + 1}$. We apply convolution operations and max pooling to the utterances separately as shown in Figure~\\ref{fig:cnn-context-rep}. After global pooling following the last convolutional layer, the three one-dimensional tensors are concatenated for final predictions. This approach extracts features from the given utterance and its context separately. Thus, we are able to learn the importance of the given utterance and its context by tuning context-specific hyper-parameters, such as the number of filters for context utterances. \n\n\n\\begin{figure}[ht]\n  \\begin{subfigure}[t]{.281\\textwidth}\n  \\centering\n    \\includegraphics[width=.99\\textwidth]{img\/CNN-Context-Rep_crop.pdf}\n    \\vspace{-0.14in}\n    \\caption{CNN-Context-Rep}\n    \\label{fig:cnn-context-rep}\n  \\end{subfigure}\\hfill\n  \\begin{subfigure}[t]{.189\\textwidth}\n  \\centering\n    \\includegraphics[width=.99\\textwidth]{img\/CNN-Feature_crop.pdf}\n    \\vspace{-0.14in}\n    \\caption{CNN-Feature}\n    \\label{fig:cnn-feature}\n  \\end{subfigure}\n    \\vspace{-0.3cm}\n  \\caption{Architectures for enhanced neural classifiers. Components marked orange are extra information incorporated into the base CNN model (black). The utterance in bold is the current utterance. Predicted labels are marked blue.}\n\\end{figure}\n\n\n\\subsubsection{Incorporate Extra Features}\n\\label{subsubsec:cnn-feature}\nWe found many useful features for user intent prediction such as structural features in the feature importance analysis in Section~\\ref{subsec:feature-importance}.\nHowever, nearly half of the features can not be exploited by only looking at a single utterance. For example, normalized\/absolute utterance position and utterance similarity with the dialog\/initial utterance cannot be captured without a holistic view over the entire dialog. Some of the uncaptured features are highly informative. This motivates us to incorporate hand-crafted features into the neural architectures. As shown in Figure~\\ref{fig:cnn-feature}, all hand-crafted features described in Section~\\ref{subsec:features} are incorporated into neural architectures at the last dense layer. The feature vector is concatenated with the neural representation of the utterance before making final predictions.\n\nFinally, we combine two base models with various extra components to produce several systems for comparison as follows:\n\n\\begin{itemize}[leftmargin=*]\n    \\item \\textbf{CNN}. The base CNN model that consists of three convolutional layers with the same filter size.\n   \n    \\item \\textbf{CNN-Feature}. The CNN model that incorporates extra hand-crafted features at the last layer.\n    \\item \\textbf{CNN-Context}. The CNN model that incorporates context information with direct expansion.\n    \\item \\textbf{CNN-Context-Rep}. The CNN model that incorporates context information with context representation.\n    \\item \\textbf{BiLSTM}. The BiLSTM model that represents the given utterance both in the ordinary order and the reverse order. \n    \\item \\textbf{BiLSTM-Context}. The BiLSTM model that incorporates context information with direct expansion.\n\\end{itemize}\n\n\n\n\\subsection{Experiments and Evaluation}\n\\label{subsec:exp}\n\\subsubsection{Neural Baselines}\n\\label{subsubsec:neural-baselines}\nIn addition to the base model of \\textbf{BiLSTM} and \\textbf{CNN}, we further introduce two commonly used neural models for text classification as baselines. For both new baselines, we modify the models to generate multi-label predictions.\n\n\\textbf{CNN-MFS}. The CNN model with multiple filter sizes as described by~\\citet{DBLP:conf\/emnlp\/Kim14} is a pioneer model to apply neural networks to text classification. This model uses different filter sizes of 3, 4, and 5 to generate feature maps of different window sizes. \n\n\\textbf{Char-CNN}. \\citet{DBLP:conf\/nips\/ZhangZL15} introduced a character-level CNN for text classification. There are two variants of the model, a large one and a small one, depending on the numbers of convolutional filters and dense layer units. We report the performance on the small model as it achieves better results in our task.\n\\subsubsection{Experimental Setup}\n\\label{subsubsec:exp-setup}\nWe use the same data and metrics as in baseline experiments in Section~\\ref{sec:baselines}. All models are implemented with TensorFlow\\footnote{\\url{https:\/\/www.tensorflow.org\/}} and Keras\\footnote{\\url{https:\/\/keras.io\/}}. Hyper-parameters are tuned with the validation data. We found that setting (convolutional filters, dropout rate, dense layer units, max sequence length, convolutional filters for context, and dense layer units for context) to (1024, 0.6, 256, 800, 128, 128) respectively turned out to be the best setting for our best performing model CNN-Conext-Rep. The convolutional filter size and pooling size are set to (3, 3). All models are trained with a NVIDIA Titan X GPU using Adam~\\cite{DBLP:journals\/corr\/KingmaB14}. The initial learning rate is 0.001. The parameters of Adam, $\\beta_1$ and $\\beta_2$ are 0.9 and 0.999 respectively. The batch size is 128. For the word embedding layer, we trained word embeddings with Gensim\\footnote{\\url{https:\/\/radimrehurek.com\/gensim\/}} with CBOW model using MSDialog (complete set). The dimension of word embedding is 100. Word vectors are set to trainable.\n\\subsubsection{Evaluation Results}\n\\label{subsubsec:evaluation-results}\nWe select the two strongest feature based classifiers from Section~\\ref{sec:baselines} as feature based baselines in addition to neural baselines. They are random forest with the best F1 score and AdaBoost with the best accuracy. The performance comparison of models is presented in Table~\\ref{tab:neural-results}.\n\n\\begin{table}[ht]\n\\centering\n\\footnotesize\n\\caption{Results comparison. The significance test can only be performed on accuracy. In a multi-label classification setting, accuracy gives a score for each individual sample, while other metrics evaluate the performance over all samples. $\\ddagger$ means statistically significant difference over the best baseline with $p < 10^{-4}$ measured by the Student's paired t-test.}\n\\label{tab:neural-results}\n\\vspace{-0.3cm}\n\\begin{tabular}{@{}llllll@{}}\n\\toprule\nMethod Types                                                                        & Methods         & Accuracy & Precision       & Recall          & F1             \\\\ \\midrule\n\\multirow{2}{*}{\\begin{tabular}[c]{@{}l@{}}Feature based \\\\ Baselines\\end{tabular}} & Random Forest   & 0.6268                                                          & \\textbf{0.7657} & 0.5903          & \\textbf{0.6667} \\\\\n                                                                                    & AdaBoost        & \\textbf{0.6399}                                                 & 0.7247          & \\textbf{0.6030} & 0.6583          \\\\ \\midrule\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}l@{}}Neural \\\\ Baselines\\end{tabular}}     & BiLSTM           & 0.5515                                                          & 0.6284          & 0.5274          & 0.5735          \\\\\n                                                                                    & CNN             & \\textbf{0.6364}                                                 & 0.7152          & \\textbf{0.6054} & \\textbf{0.6558} \\\\\n                                                                                    & CNN-MFS  & 0.6342                                                          & \\textbf{0.7308} & 0.5919          & 0.6541          \\\\\n                                                                                    & Char-CNN        & 0.5419                                                          & 0.6350          & 0.4940          & 0.5557          \\\\ \\midrule\n\\multirow{4}{*}{\\begin{tabular}[c]{@{}l@{}}Neural\\\\ Classifiers\\end{tabular}}       & BiLSTM-Context   & 0.6006                                                          & 0.6951          & 0.5640          & 0.6227          \\\\\n                                                                                    & CNN-Feature     & 0.6509                                                          & 0.7619          & 0.6110          & 0.6781          \\\\\n                                                                                    & CNN-Context     & 0.6555                                                          & 0.7577          & 0.6070          & 0.6740          \\\\\n                                                                                    & CNN-Context-Rep & \\textbf{0.6885}$^\\ddagger$                                                 & \\textbf{0.7883} & \\textbf{0.6516} & \\textbf{0.7134} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nThe base CNN model without feature engineering achieves similar results with the strongest feature based baselines. The performance of the base CNN model is better than the base BiLSTM model. CNN-MFS takes advantage of different filter sizes and also achieves comparable results. Char-CNN, however, performs poorly in this task. Char-CNN does not use pre-trained word embeddings because it learns features from a character-level which would require much more training data.\n\nBiLSTM performs poorly in this task. Even though BiLSTM-Context has a major improvement over BiLSTM, it has inferior results compared to the base CNN model. Compared to non-factoid question answering or chatting, utterances in information-seeking conversations tend to be longer. BiLSTM(-Context) tries to model a holistic sequence dependency and thus performs poorly on handling these long utterances.\n\nThe best result of the feature based baselines is slightly higher than neural baselines. This can be accounted for by the lack of information in neural models. Even though we assume that most of the content and sentiment features can be learned by neural models, the neural models have no access to most of the structural features. Thus, we incorporate all the features to neural models to produce CNN-Feature model. This model outperforms all baseline classifiers. This confirms that incorporating dialog-level information can be beneficial to predicting user intent. \n\nBoth CNN-Context and CNN-Context-Rep outperform baseline models and CNN-Feature without explicit feature engineering. These results demonstrate the effectiveness of the implicit feature learning of neural architectures. CNN-Context-Rep performs better than CNN-Context. This indicates that incorporating high-level features of context information learned by neural architectures is better than directly capturing the raw context information. In a multi-label classification setting, accuracy produces a score for each individual sample, while precision\/recall\/F1 evaluate the performance over all samples. Thus, accuracy is the only metric that is suitable for significance tests. Our best model, CNN-Context-Rep achieves statistically significant improvement over the best baseline with $p < 10^{-4}$ measured by the Student's paired t-test.\n\n\\subsection{Generalization on Ubuntu Dialogs}\n\\label{subsec:udc}\n\n\n\\if0\nWe also annotated 200 dialogs from Ubuntu Dialog Corpus (UDC)~\\cite{Lowe2015The}. MSDialog and UDC share a lot of similarities. Both datasets are derived from technical support domain. They both consist of multi-turn information-seeking conversations. They are also different in many ways. UDC is generated from internet relay chat (IRC) and thus contains a significant amount of typos, internet language, and abbreviations. In addition, Ubuntu dialogs have shorter utterances and more turns. We merged consecutive utterances from the same user as a single utterance to try to match the utterance length in MSDialog. Given their similarities and differences, we would like to know whether the model trained on MSDialog data can generalize well on Ubuntu dialogs.\n\nThe sampled UDC data contains 159 dialogs with 3,185 utterances. They were annotated under the exact same settings of MSDialog. We also perform the same data preprocessing steps as mentioned in Section~\\ref{subsec:data-transformation}. Concretely, after removing \\textit{GG}, \\textit{JK}, and \\textit{O}, we select the most frequent 32 label combinations from the remaining 95 label combinations. They cover 94\\% of total label combination occurrences. For the rest of the label combination, we randomly sample one of the labels from the combination as the user intent label for the given utterance. After preprocessing, we have 34 label combinations for Ubuntu dialogs. 27 of them overlap with MSDialog. \n\\fi\n\n\n\n\n\nIn this section, we would like to evaluate the generalization performances of different methods on other data in addition to MSDialog. We train different models on MSDialog and test them on the Ubuntu Dialog Corpus (UDC). We select the two best performing feature based classifiers (random forest and AdaBoost) and the best neural model (CNN-Context-Rep) to test the generalization performance. Although the number of annotated Ubuntu dialogs is limited, it is sufficient to demonstrate the predicting performance. We split the annotated UDC data into validation and test sets with an equal size. We train the model on MSDialog data only and tune the hyper-parameters on the UDC validation set. The performance on the UDC test set is presented in Table~\\ref{tab:udc-results}. \n\n\\begin{table}[ht]\n\\footnotesize\n\\centering\n\\caption{Testing performance on UDC of different models trained with MSDialog. The significance test can only be performed on accuracy. $\\ddagger$ means statistically significant difference over both strongest feature based baselines with $p < 0.01$ measured by the Student's paired t-test.}\n\\label{tab:udc-results}\n\\vspace{-0.3cm}\n\\begin{tabular}{lllll}\n\\toprule\nMethods         & Accuracy & Precision       & Recall          & F1             \\\\ \\midrule\nRandom Forest   & 0.4405                                                         & \\textbf{0.6781} & 0.4077          & 0.5092          \\\\\nAdaBoost        & 0.4430                                                         & 0.5913          & 0.4187          & 0.4902          \\\\\nCNN-Context-Rep & \\textbf{0.4708}$^\\ddagger$                                                & 0.5647          & \\textbf{0.5129} & \\textbf{0.5375} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\nThe generalization results on UDC are not as good as MSDialog. Although MSDialog and UDC both consist of multi-turn information-seeking dialogs from the technical support domain, the drastically different language style adds difficulty for model generalization and transferring. In this challenging setting, CNN-Context-Rep still achieves statistically significant improvement over both baselines in terms of accuracy with $p < 0.01$ measured by the Student's paired t-test.\n\n\n\\subsection{Hyper-parameter Sensitivity Analysis}\n\nWe further analyze the impact of two hyper-parameters on CNN-Context-Rep: the number of convolutional filters for the given utterance and the max sequence length. The choices for number of convolutional filters are (64, 128, 256, 512, 1024). We tune the max sequence length in (50, 100, 200, \\dots, 1000). As presented in Figure~\\ref{fig:tune}, the performance gradually increases as the number of filters increases. The best performance is at 1,024 filters. This confirms our expectation that more convolutional filters can extract richer features and thus produce better results. In addition, the performance fluctuates as the max sequence length increases. Performance with larger ($\\geqslant800$) max sequence length are better in general.\n\n\n\n\\subsection{Case Study}\n\nTable 10 gives examples of the predictions that different systems fail to make. In the first utterance, the agent asks for the user's iOS version before providing a potential answer, which is a very common pattern of agents' responses. Our CNN-Context-Rep is able to identify the \\textit{Information Request} in the utterance while AdaBoost cannot. In the second utterance, both models fail to predict the \\textit{Negative Feedback}. This might be due to the fact that the feedback is not explicitly expressed. In addition, it could be relevant that the number of feedback utterances in the training data is relatively limited compared to questions and answers, which makes it more difficult to predict positive\/negative feedback.\n\n\\begin{table}[ht]\n\\footnotesize\n\\centering\n\\caption{Two utterances with their ground-truth and predicted user intent labels. Bold font indicates mispredicted content or labels. ``Ours'' refers to CNN-Context-Rep.}\n\\label{tab:case-study}\n\\vspace{-0.3cm}\n\\begin{tabular}{llll}\n\\toprule\n\\multicolumn{4}{l}{\\begin{tabular}[c]{@{}l@{}}Hello. Welcome to Skype Community! \\textbf{Please provide us the iOS version of} \\\\ \\textbf{your iPad}. The required iOS version for iPad is iOS 8 or higher and for the new \\\\Skype on iOS requires iOS 9 or higher. For more information, click here. Hope \\\\ this helps. Let me know if you need further assistance. Thank you!\\end{tabular}} \\\\ \\hline\n\\multicolumn{1}{l|}{Ground truth: \\textbf{IR}, PA}                                                            & \\multicolumn{1}{l|}{Ours: \\textbf{IR}, PA}                                                            & \\multicolumn{1}{l|}{AdaBoost: PA}                                                            & Actor: agent                                                           \\\\ \\hline\\hline\n\\multicolumn{4}{l}{\\begin{tabular}[c]{@{}l@{}}After modified the Windows entry, value of regedit, \\textbf{the error also happened}. \\\\ When I use C++ for creating another new Microsoft::Office::Interop::PowerPoint::\\\\Application  instance, the COMException is throwed.\\end{tabular}}                                           \\\\ \\hline\n\\multicolumn{1}{l|}{Ground truth: FD, \\textbf{NF}}                                                            & \\multicolumn{1}{l|}{Ours: FD}                                                                & \\multicolumn{1}{l|}{AdaBoost: FD}                                                                & Actor: user                                                            \\\\ \\hline\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\section{Related Work}\n\\label{sec:relatedwork}\n\n\nOur work is closely related to utterance classification, conversational search, and multi-turn question answering.\n\n\n\\if0\n\\textbf{Utterance type classification}. Utterance type classification as well as text classification has been widely studied in both natural language processing (NLP) and IR communities. \\citet{Stolcke2000Dialogue} performed dialog acts classification with a statistical approach on SwitchBoard corpus~\\cite{Godfrey1997Switchboard}, which consists of human-human chit chats conversations. While \\citet{Olney:2003:UCA:1118894.1118895} classified students' utterances in an intelligent tutoring system (AutoTutor~\\cite{Graesser2001AutoTutor}) with cascaded finite\nstate transducers. \n\\citet{Surendran2006Dialog} conducted dialog acts tagging on the HCRC MapTask corpus~\\cite{Thompson1993The} with a combined method with SVM and Hidden Markov Model. \nRecently, \\citet{DBLP:conf\/webdb\/BhatiaM12, bhatia2014summarizing} focused on forum post classification for applications in information extraction and summarizing. Recent advancement in deep learning has made it possible to use neural architectures for text classification. Related research is conducted on both word level~\\cite{DBLP:journals\/corr\/KingmaB14, DBLP:conf\/aaai\/LaiXLZ15} and character-level~\\cite{DBLP:conf\/nips\/ZhangZL15, DBLP:conf\/eacl\/SchwenkBCL17}. Specifically, such methods are applied to intent determination in medical dialog systems~\\cite{DBLP:conf\/iva\/DattaBOWB16}. In addition to utterance classification based solely on text, extensive research has been done with special interests in speech transcription. These work studied the impact of speech recognition quality~\\cite{Homma2016} and how to perform classification without speech transcription~\\cite{speech-utterance-classification, Wang2006}. \n\\fi\n\n\n\n\n\n\\textbf{Utterance Classification}. Utterance classification is well studied in the NLP and IR domain. Many different classification techniques such as\nstatistical approaches~\\cite{Stolcke2000Dialogue}, SVM, or Hidden Markov Models~\\cite{Surendran2006Dialog}\nhave been used for different applications including\nhuman-human chatting~\\cite{Stolcke2000Dialogue},\nstudent's utterance~\\cite{Olney:2003:UCA:1118894.1118895}, or forum post classifications~\\cite{DBLP:conf\/webdb\/BhatiaM12}.\nHowever, recent advances in deep learning allow us to use neural architectures for utterance classification both on the word~\\cite{DBLP:conf\/emnlp\/Kim14} and character level~\\cite{DBLP:conf\/nips\/ZhangZL15}. \nThese new deep learning techniques have been applied in medical dialog systems~\\cite{DBLP:conf\/iva\/DattaBOWB16}. In this paper, we focus on user intent prediction in information-seeking conversations. This specific utterance classification task presents unique challenges because of the complexity and diversity of human information-seeking conversations.\n\n\n\n\n\n\n\n\n\n\n\n\n\\ignore{\\jtcomment{This section may need to be revised and differently structured. I'll try to come back to this later to help rewrite it. How are DA related to user intent?}}\n\n\n\\textbf{Conversational Search}. Searching via conversational interactions with IR systems is an increasingly popular research topic in both industry and academia~\\cite{Zhang2018Rec, Shiga2017modelling, Croft:1987:IRN:35053.35054}. \n\\citet{Oddy1977Information} introduced man-machine IR through dialogs without explicitly formulating queries. \\citet{belkin1995merit} modeled the human-computer interaction in information-seeking as dialogs. Moreover, information-seeking via conversations is especially important in exploratory search~\\cite{Marchionini:2006:ESF:1121949.1121979, DBLP:series\/synthesis\/2009White}, where users are unfamiliar with the domain they are searching and would rely on effective interactions with retrieval systems. More recently, \\citet{Radlinski:2017:TFC:3020165.3020183} identified key properties in conversational IR systems. \\citet{Trippas:2018:IDS:3176349.3176387} conducted lab-based observational studies on conversational search and identified that it is more interactive than traditional search and new information-seeking models are needed. In our work, we focus on an essential study in conversational search, which is to predict user intent in this information-seeking setting. Our findings can help build conversational search systems that can provide enhanced searching experience using predicted user intent. \n\n\n\\textbf{Multi-turn Question Answering}. Early research on multi-turn question answering dates back to AutoTutor~\\cite{Graesser2001AutoTutor}, which can simulate human tutors to assist college students to learn computer science. Recent work has focused on single turn QA on factoid questions~\\cite{Yin2015Neural} and other open-domain questions (e.g. WikiQA~\\cite{Yang2015WikiQA}). The Ubuntu Dialog Corpus~\\cite{Lowe2015The} and MSDialog~\\cite{Qu2018UserIntent} provide large scale multi-turn QA dialogs in the technical support domain. \\citet{DBLP:conf\/acl\/WuWXZL17} and \\citet{Yang2018RespRank} used these datasets to perform conversation response ranking for non-factoid questions. Our work focuses on a specific research need for multi-turn QA in the information-seeking setting. The performance of multi-turn QA could be improved if the user intent is correctly identified. \n\n\\section{Additional Figure}\nWe include an additional figure to illustrate our findings from above. Figure~\\ref{fig:tune} shows the results of the hyper-parameter sensitivity analysis.\n\\begin{figure}[ht]\n\t\\centering\n\t\\begin{subfigure}[b]{0.24\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{img\/filter_new.pdf}\n\t\t\\label{fig:nb_filter}\n\t\\end{subfigure}\n\t\\hspace{-0.1in}\n\t\\begin{subfigure}[b]{0.24\\textwidth}\n        \\includegraphics[width=\\textwidth]{img\/max_len_new.pdf}\n\t\t\\label{fig:max_len}\n\t\\end{subfigure}\n\t\\vspace{-1cm}\n\t\\caption{Performance of CNN-Context-Rep with different number of convolutional filters and max sequence length.}\n\t\\label{fig:tune}\n\\end{figure}\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe idea that future asset performance is a continuation of past performance is the cornerstone of momentum trading strategies. In the past 20 years, momentum has become widely accepted by both academics and practitioners as one of the strongest and most persistent factors that explain assets returns \\cite{Harvey}. Even though momentum has been know since the 1930's \\cite{Antonacci}, the first rigorous analysis is due to Jegadeesh and Titman \\cite{JT1993}. \n\nJegadeesh and Titman \\cite{JT1993} construct a market neutral (long\/short) portfolios of stocks based on past returns and holds the portfolio for different horizons. For instance, one takes the long position for the top decile of the best performing stocks and the short position for the bottom decile of the worst performing stocks, rebalancing the portfolio every month. One then checks if such a long\/short portfolio results in significant positive returns. \n\nSince the work of  Jegadeesh and Titman \\cite{JT1993}, momentum has been extended to different asset classes, portfolios, and other world markets. Momentum has been reported in international equity markets by \\cite{Doukas,Forner,Nijman,Rouwenhorst}; in industries by \\cite{Lewellen,Moskowitz}; in indexes by \\cite{Bhojraj,Asness1997};  and in commodities by \\cite{Erb,Moskowitz2011}.  Single risky asset momentum is analyzed in \\cite{Daniel,Barberis,Hong,Berk,Johnson,Ahn,Liu,Sagi} to cite a few. Recently, momentum has also been studied linking its performance to business cycles and regimes by \\cite{Chordia,Kim2013,Griffin,Guidolin,Barroso}.\n\nOn the theoretical aspect, behavioral finance is generally evoked to explain momentum \\cite{Daniel,Barberis,Hong}. In general, theoretical studies suggest that if stock prices overreact or under-react to information, then the trading decision based on the past performances will occur. Daniel, Hirshleifer, and Subrahmanyam \\cite{Daniel}  propose that under-reaction and overreaction are consequences of investor's overconfidence on the inside information and self-attribution bias. Barberis, Shleifer and Vishny \\cite{Barberis} connect overreaction of stock prices to investor's attitude towards a series good or bad news, and under-reaction of stock prices  to investor's attitude towards information such as earning announcement. In Hong and Stein's \\cite{Hong} model,  the investors are categorized into two groups, namely, the news watchers and the momentum traders, which lead to under-reaction at short horizons and overreaction at long horizons. Generally speaking, this direction of studies describe investors as Bayesian optimizers: the investor observes or receives  information at each investment time period, and updates his\/her investment decision according to his\/her belief. \nThese behavioral models predict that under-reaction implies positive short-lag autocorrelation, and that overreaction implies negative long-lag autocorrelation. \n\nAlternatively to under and overreaction \\cite{Daniel,Barberis,Hong} there are other causes that have been cited as possible explanation to the abnormal momentum return. Lewellen suggests that the lead-lag cross-serial correlation should explain cross-sectional momentum \\cite{Lewellen}. Conrad and Kaul point to the cross-sectional variation of asset returns \\cite{Conrad}. Chordia and Shivakumar and others \\cite{Chordia,Kim2013,Griffin,Guidolin,Barroso} study business cycles and suggest that time-varying expected returns can explain momentum.\n\nThe traditional long\/short momentum  strategy mixes portfolio construction with the technical rule (past performance, as in technical analysis \\cite{Brock,Griffioen}) used to select the assets to be included in the portfolio. The portfolio construction introduces a relative performance effect since the investor needs to build a portfolio by selecting the assets based on the technical rule and assign a proportion of the total wealth to each of the asset selected. To highlight this portfolio aspect, momentum is sometimes called ``cross-sectional momentum\" to contrast with \"time-series momentum\" introduced by Moskowitz, Ooi and Pedersen \\cite{Moskowitz2011}. Time series momentum is the study of the technical rule outside a portfolio (apply momentum on individual assets): basically one of the simplest trend following strategies, and therefore a building block for more complex strategies. Moskowitz, Ooi and Pedersen perform an extensive study using over 25 years of index, futures and forward contracts' data and show that every asset they have analyzed (58 in total) present strong time series momentum.  \n\nSimilar to Moskowitz, Ooi and Pedersen, we focus on the momentum of individual assets. We study the technical rule (moving average of past returns) for one asset, therefore avoiding the portfolio effect that is important for cross-section momentum. This work adds to the paper of \\cite{Moskowitz2011} by looking at the information ratio of the time series momentum strategy. Our work also contributes to the literature of linking momentum to cycles\/regimes \\cite{Chordia,Kim2013,Griffin,Guidolin,Barroso}. However, contrary to the previous studies, we do not associate economical episodes to the regimes. Our approach is to divide and transform the data in a way such that the final asset returns are as close as possible to stationary. We believe that our work is new in this respect.\n\nWe study momentum by looking at the risk adjusted performance measured by the information ratio (IR) as a function of the look-back lag used to construct the portfolio. Our main new contribution from a mathematical point of view, is to present in close form the risk associated with the momentum strategy. Previous works \\cite{Lo1990,Lewellen,Moskowitz2011} calculate the same expression for the average return as given here, however they do not calculate the standard deviation of the strategy. Furthermore, we analyze the stability of the results across time as non-stationary effects become important in explaining the results. We find that both autocorrelation and mean drift of the random process are important in the final performance of the strategy. In particular, for look-back periods up to 4 months, the most important effect is the autocorrelation; and for look-back periods larger than 4 months to 1 year, the drift. However, in contrast with previous studies, we find that the mean drift is the most important factor after 1975.\n\nIn section \\ref{sec:stationaryProcess}, we briefly introduce the notion of stationary process that is relevant for this article and describe how we find stationary regimes. In section 3, we present our momentum strategy and find the theoretical average performance, theoretical standard deviation and information ratio. Section 4 and 5 presents empirical results. We first look at stationary data comparing it to our theoretical formulas and then at non-stationary data. We confirm the general findings that momentum works for look-back (portfolio formation period) periods of up to 1 year after which the performance degrades finding a minimum for look-back periods around 2 years. Furthermore, we show that the performance improves again for look-back periods that are longer than 2 years, indicating an oscillatory (wave like) market behavior. Some formulae derivations are gathered in Appendix.\n\n\\section{Stationary Process} \n\\label{sec:stationaryProcess}\n\nThe strict definition of a stationary process is that the joint probability distribution of all random variables is invariant under time shifts or translations. Equivalently, the probability density depends only on the time difference since the time origin is not relevant \\cite{Feller,Gardiner}. However, it is most common to use the weak-sense stationary definition. That is, one requires that the data first moment and covariance do not change in time. In particular, the variance exists and the covariance depends only on the time difference.\n\nWe take the view that financial time series are not weak-sense stationary in general. This includes many of the common transformation of the price time series, such as log-returns. Most of the literature does not discuss the effect of non-stationary data; however, some studies show measurable and important effects \\cite{Seemann,McCauley}.\n\nWe follow \\cite{Seemann} and assume that the data has patches or regimes of stationary periods. Therefore, our time series can be described by Figure \\ref{fig:cartoon}. The rectangles and the circles in Figure \\ref{fig:cartoon} represent intervals of the time series that are stationary. \n\nOne example of such assumption  is the intraday FX data studied in \\cite{Seemann}, another example is  the distribution of trading volumes (or number of trades) for stocks discussed in \\cite{Silva}. It has been shown by \\cite{Seemann,Silva} that every day the patterns repeat. As such, though the process is not stationary inside of the day (clearly since the volatility changes from parts of the day to other parts of the day), we can have a stationary process of all the first $5$ minutes (for instance) across different days.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=\\textwidth]{StationaryCartoon.jpg}\n\\caption{Cartoon representation of stationary patches in a give time\nseries \\label{fig:cartoon}}\n\\end{figure}\n\nAnother possibility is that every day of the week represents a different stochastic process. That is, all Mondays are represented by a probability distribution, all Tuesdays by another, and so forth \\cite{Tsiakas}. In fact, the actual probability distribution might even have a different functional form and the relation between all the days can be arbitrary. In other words, all Mondays could be correlated with each other and those could be correlated to Tuesdays and so forth.\n\nIn general, we do not postulate a periodic structure, nevertheless, we expect the presence of some repetition (Fig. \\ref{fig:cartoon}) in order to have enough data to perform ensemble averages. Furthermore, in order to practically detect such periods, we shall simplify the problem by detecting periods with constant mean and constant variance. We assume that the autocorrelation is well-behaved enough to preserve the time independent property.\n\nTo detect periods with constant mean (drift), we use the ``Breaks for Additive Season and Trend\" (BFAST) algorithm \\cite{Verbesselt}. BFAST first starts by using the Loess regression to decompose the time series into seasonal trends and irregular components. Thereafter, the algorithm performs a loop where it uses the methodology developed by  \\cite{Bai} until the number and the position of the breakpoints are unchanged. Intuitively, the algorithm  finds the trend by fitting a piecewise linearity where the breakpoints (changes from one linearity to the next) are found at the same time to the linear fit. From an implementation standpoint, we use the R package ``bfast\" in our empirical studies. \n\nBesides BFAST, there are several other algorithms and we cite a few but we postpone the comparison for a future study. Particularly interesting is the recent work of \\cite{Matteson} which is able to detect a change in the probability distribution non parametrically. There is also \\cite{Fukuda} which presents a heuristic algorithm that introduces a $t$-test type of statistic. For other alternatives, see also \\cite{McCauley,Seemann} and \\cite{Guidolin} for models that include regime switching explicitly.\n\n\\section{Model} \n\nThe momentum strategy intends to extract predictability of future price returns from past price returns. Here we define price return by the following expression: \n\\begin{equation} r_{i} = \\ln (S_{i}\/S_{i-1}), \n\\end{equation} \n\n\\noindent where $S_{i}$ is the price in period $i$.\n\nIn order to understand momentum strategies, we use a proxy algorithm that should represent the general characteristics of any given momentum strategy that can be implemented by buying past winners and\/or taking short positions in past losers \\cite{JT1993}. The momentum strategy we select is:\n\n\n\\begin{equation} \n\\left\\{ \\begin{array}{lll} m_{n}(N) =\n\\displaystyle\\sum_{i=n-N+1}^{n} r_i\/N, & ~ & ~\\\\ \\\\ \\mbox{if} ~ ~ m_{n}(N) > 0 &\n\\mbox{ Buy } & m_{n}(N) ~~\\mbox{Shares} \\\\ \\\\ \\mbox{if}~~ m_{n}(N) < 0 &\\mbox{\nShort } & m_{n}(N) ~~\\mbox{Shares}\n\\end{array}\\right. \n\\label{eq:strategy} \n\\end{equation} \n\n\\noindent where $m$ is a simple moving average, and $N$ is the look-back period used to calculate the moving average. This same set of rules was used in \\cite{Asness2013,Kim,Moskowitz2011,Lewellen,Lo1990} and other prior studies.\\footnote{There is no guarantee that such an algorithm represents well momentum strategies, however we defer the question of how to represent a class of strategies for future study. For now, we consider that our algorithm is able to capture the main mathematical features present in a general momentum strategy.} \n\nIn order to implement the algorithm (Eq. (\\ref{eq:strategy})) described above, we first reduce the effect of volatility clustering. We do so by dividing the returns by the average absolute returns of the past $p$ periods (Eq. (\\ref{eq:normalizeData})). This transformation avoids look-ahead bias and creates a strategy that can be implemented realistically.\n\n\\begin{equation}\\label{eq:normalizeData} \nX_t = \\frac{r_t}{\\sum_{i=1}^{i=p} \\mid r_{t-i}\/p \\mid} \n\\end{equation}\n\nThe average return for our momentum strategy (Eq. (\\ref{eq:strategy})) is given by\n\n\\begin{equation}\\label{eq:R} \n\\left \\langle R \\right \\rangle = \\frac{1}{T-N+1}\n\\sum_{t=N}^{T} m_{t-1}(N) X_{t} \n\\end{equation}\n\n\\noindent where $N$ is the look-back period used to calculate $m_n(N)$, and $T$ is the total length of our data series (for instance the total number of weeks). By using Eq. (\\ref{eq:strategy}), one can rewrite  Eq. (\\ref{eq:R}) as\n\n\\begin{eqnarray} \n\\left \\langle R \\right \\rangle &=& \\frac{1}{T-N+1}\n\\frac{1}{N}\\sum_{t=N}^{T}\\sum_{i=t-N}^{t-1}X_i X_t \\\\ \\nonumber &=&\n\\frac{1}{T-N+1} \\frac{1}{N}\\left[ \\sum_{t=N}^{T} X_t X_{t-N} + \\sum_{t=N}^{T} X_t\nX_{t-N+1} + \\cdots +\\sum_{t=N}^{T} X_t X_{t-1} \\right]\\\\ \\nonumber &=&\\frac{1}{N}\n\\sum_{i=1}^{N} \\left \\langle X_{t} X_{t-i} \\right \\rangle \n\\end{eqnarray}\n\n\\noindent where $<\\ >$ stands for average (sometimes represented by $E[\\  \\ ]$). The last equality is only true if the process for $X$ is such that the product $X_T X_{T-\\tau}$ is equal in probability to $X_{T-1}X_{T-1-\\tau}$, and hence it depends only on $\\tau$.\n\nIt was pointed in section \\ref{sec:stationaryProcess} that in order to exactly model non-stationary data, we need to know how the different stationary patches relate to each other because the moving average $m(N)$ crosses different patches (Fig.\\ref{fig:cartoon}). Due to this complexity, we restrict our theoretical analysis to stationary patches before working with the non-stationary data.\n\n\\subsection{Risk and Return for stationary random variables}\n\nFor stationary random variable $X_t$, the expected return (Eq. (\\ref{eq:R})) can be expressed as an average of auto-covariance as follows:\n\n\\begin{equation}\\label{eq:theAve} \n\\left \\langle R \\right \\rangle = \\frac{1}{N}\n\\sum_{i=1}^{N} \\left \\langle X_{t} X_{t-i} \\right \\rangle = \\mu^{2} + \\frac{V}{N}\n\\sum_{i=1}^{N} \\rho(t,t-i) \n\\end{equation}\n\n\\noindent where $\\rho$ is the autocorrelation function, $V$ the variance and $\\mu$ the mean of the stationary stochastic process $X$. Notice that the result (Eq. (\\ref{eq:theAve})) is independent of functional form of the distribution of $X$.\n\nThe variance $Var(R)$ of the strategy in Eq. (\\ref{eq:strategy}) is given by:\n\n\\begin{equation} \\label{eq:risk1} \nVar(R) = \\left \\langle \\left(\\frac{X_t}{N}\n\\sum_{i=1}^{i=N} X_{t-i}\\right)^{2} \\right \\rangle- \\left \\langle\n\\frac{X_t}{N}\\sum_{i=1}^{i=N} X_{t-i} \\right \\rangle^{2}. \n\\end{equation}\n\nThe first term in Eq. (\\ref{eq:risk1}) relates to the autocorrelation of the squared return and the cross-correlation with the squared return (similar to the leverage effect). The first term can be re-written as:\n\n\\begin{equation} \\label{eq:risk2} \n\\sum_{i=1}^{N} \\left \\langle X_t^{2}\nX_{t-i}^{2}\\right \\rangle + \\sum_{i,j=1,i \\neq j}^{N} \\left \\langle X_t^{2}\nX_{t-i}X_{t-j}\\right \\rangle. \n\\end{equation}\n\nWe further simplify Eq. (\\ref{eq:risk1}) by assuming a multivariate Gaussian distribution. Thus, the correlations are linear correlations and that the marginal distributions are Gaussian. Although empirical financial data are not described by a Gaussian distribution, the weekly normalized returns obtained according to Eq. (\\ref{eq:normalizeData}) are well approximated by a Gaussian distribution.\n\nWe can therefore calculate the variance  of our momentum strategy by using the characteristic function of the multivariate Gaussian distribution. Performing the right order of differentiation and enforcing that the variance $V$ and the drift $\\mu$ of the returns are constants and that the autocorrelation depends only on time lags, we have the variance of our strategy given by:\n\n\\begin{eqnarray}\\label{eq:theVar} \nVar(R) &=& \\frac{1}{N^{2}}\\bigg[ N V^{2} + N \\mu^{2} V\n+ N^{2} V \\mu^{2} \\\\ \\nonumber &+&V^{2} \\big(\\sum_{i=1}^{N} \\rho(t,t-i)\\big)^{2} +V^{2}\\sum_{i,j=1,i\\neq\nj}^{N} \\rho(t-i,t-j)\\\\\n\\nonumber &+&\\mu^{2} V \\Big(2\\sum_{i=1}^{N} \\rho(t,t-i) +\\sum_{i,j=1,i\\neq\nj}^{N}\\big(\\rho(t,t-j)+\\rho(t-i,t-j)+\\rho(t,t-i)\\big)\\Big) \\bigg]  \n\\end{eqnarray}\n\n\\noindent where $\\rho(t,t-i)$ is the correlation coefficient of the returns from time $t$ and $t-i$. Details of the calculation can be found in the Appendix. In the next section we will derive useful asymptotic limits of Eq. (\\ref{eq:theVar}).\n\n\\subsection{Limits and interpretations}\n\nIn contrast to prior studies \\cite{Kim,Lewellen,Lo1990,Moskowitz2011} we do not look only at the return of the strategy . We use the variance (Eq. (\\ref{eq:theVar})) to calculate the Sharpe or Information Ratio (IR) defined here by the ratio between the average return and the standard deviation. The general expression is fairly complex but two limiting cases are enough to help us understand how the IR depends on the parameters.\n\nIn case I, all the autocorrelations are zero: $\\rho(t,t-i)=0$. This is equivalent to say that the log-returns are independent and identically distributed (iid) Gaussian random variables. The Information Ratio (IR) is given by:\n\n\n\\begin{equation}\\label{eq:case1} \n\\text{IR} =\n\\frac{\\mu^{2}}{\\sqrt{V\\mu^{2}+\\frac{V^{2}}{N}+\\frac{\\mu^{2}V}{N}}} \n\\ \\xrightarrow[N \\to \\infty]{} \\\n\\frac{\\left|\\mu\\right|}{\\sqrt{V}}, \n\\end{equation} \n\n\\noindent where we also show the behavior when $N \\rightarrow \\infty$. $N \\rightarrow \\infty$ is the limit of long (or short) and hold since $m(N)$ converges to $\\mu$. It is interesting that the optimal point is when $N \\rightarrow \\infty$; in this case, the best information ratio is in fact what one would expect for a given process: mean over standard deviation. Any other $N$ gives worst results. Hence, as expected, if  the given process is iid, the moving average $m(N)$ provides one way to estimate $\\mu$. A cartoon representation of the IR as a function of $N$ for case I is given in Fig. \\ref{fig2}.\n\nIn case II, we assume that $\\mu=0$. Thus, all performance comes from autocorrelation. The IR is given by\n\n\\begin{equation}\\label{eq:case2} \\text{IR} =\n\\frac{\\sum_{i=1}^{N}\\rho(t,t-i)}{\\sqrt{N+(\\sum_{i=1}^{N}\\rho(t,t-i))^{2}+(\\sum_{i,j=1,i\\neq\nj}^{N}\\rho(t-j,t-i))}} \\end{equation}\n\n\\noindent where the exact shape of IR as a function of $N$ depends on the way $\\rho$ is a function of $N$. Practically it is very unlikely that $\\sum_{i=1}^{N} \\rho(t,t-i)$ grows fast enough to dominate the $\\sqrt{N}$ term in the denominator. More surprising is that Eq. (\\ref{eq:case2}) does not depend on the variance $V$, in other words, the IR of the strategy is the same for very large or small $V$. The expression is also useful practically since one can calculate IR for a given correlation value. For instance, for $N=2$, $\\rho(t,t-1) = 0.05$ and $\\rho(t,t-2) = 0.02$, IR $ \\approx 0.0422$ per period, that means that if we are dealing with weekly data, for instance, IR $=\\sqrt{52}*0.0422= 0.3$ per year.\n\n\\begin{figure}[htbp] \\centering\n\\includegraphics[width=\\textwidth]{case1} \\caption{Increasing red line: IR as function of lookback lag $N$ for case I ($\\rho=0$, Eq. (\\ref{eq:case1})) normalized to have maximum of $1$. Decreasing blue line: Normalized IR for case II\n($\\mu=0$, Eq. (\\ref{eq:case2})) generated using simulated data from an hypotetical process with $\\rho \\neq 0$ for lag 1 to lag 5. \\label{fig2}} \n\\end{figure}\n\nIn order to illustrate case II, we look at a moving average process ($MA$ i.e. of the kind $x(t) = a_{1} \\epsilon_{1}(t)+a_{2} \\epsilon_{2}(t-1)+..., \\epsilon_{i}=N(0,\\sigma)$) with autocorrelation $\\rho \\neq 0$ from lag 1 to lag 5. The general shape of case II is presented in Figure \\ref{fig2} (case II, blue line). We have chosen to show an arbitrary $MA$ process because it is the only process from the $ARMA$ family where the $\\rho$ term in the numerator can grow fast enough before $1\/\\sqrt{N}$ in the denominator takes over. This creates the ``hump'' in the graph. The hump is located where the $MA$ process has autocorrelation different from zero and the size of the hump depends on the autocorrelation intensity.\n\nIn summary, case I (red line) is increasing with the value of $N$, while  case II (blue line)  increases initially and then decreases slowly (Figure \\ref{fig2}).\n\nIt is clear that if we have a pure case II, there could be an optimal $N$ above which we get worst risk adjusted performance. Therefore, it is generally not advantageous to use a momentum type strategy unless the best $N$ is selected. \n\nThe dependence of the IR on $N$ is different for case I and case II: the first case grows and the second decreases in the limit of large $N$.\nNormally, empirical data has IR as a hybrid of case I and case II, that indicates that a pure momentum strategy will transit from a case II dominated performance to a case I dominated performance with the increase of $N$. Large $N$ shows that case I is targeted: momentum estimates the drift. Keep in mind  that if the sum of the autocorrelations happens to be positive (negative), case I is shifted up (down) due to the case II contribution.\n\nFinally, we remind the readers that our discussion here assumes a stationary process. That is clearly not the case in practice. We assume that there are patches or periods where the data is approximately stationary. Next, we will look at data by approximately finding these stationary patches. We will conclude by looking at the full non-stationary data set and present a simple model that describes momentum for non-stationary data.\n\n\n\\section{Empirical stationary analysis}\n\\label{sec:E_stationary} \n\nWe use the Dow-Jones Industrial Average (DJIA) Index from 05\/1896 to 02\/2013 to perform our empirical studies.\\footnote{We present the study with the DJIA index but we also looked at the S\\&P 500 index from 1950 to the present with virtually the same results, therefore we will not show them here.} The daily DJIA index values are downloaded from Federal Reserve Economic Data (FRED) webpage.\\footnote{Data download uses \"quantmod\" library in R \\cite{R}} We convert the daily index values to weekly index values and calculate weekly log-returns. We choose to work with weekly data because we get at least 4 times more data than the traditional monthly values and we also reduce considerably any market micro-structure issue from daily data.\n\nMost prior studies have a minimum holding period of one month even when using weekly data \\cite{Kim}. In contrast with most prior studies,  we  re-balance our position weekly (holding period of one week), which maximizes data usage. All our analyses are done using the  software {\\bf R} \\cite{R}.  A simple sample code pertinent to this article can be found in \\cite{Rpubs}.\n\nWe take the view that financial log-returns are generally not stationary \\cite{Guidolin,McCauley,Seemann}. Notice that this assumption is different from several prior studies, take for instance \\cite{Lehmann,Lo1988,Lo1990} where the results are obtained with 25 years of data (1962 to 1986). This study is more in line with studies that advocate business cycles dependence such as \\cite{Chordia}.\n\n\n\\begin{figure}[htbp] \\centering\n\\includegraphics[width=\\textwidth]{transAlgosMean} \\caption{Log cumulative\nreturn of weekly DJIA from 1995 to 2012 (points). The intervals between green and red vertical dash lines were obtained by\nusing the BFAST algorithm which finds these intervals by fiting piecewise linear functions to the data. The linear fits are represented by the solid red lines. \\label{fig4}} \\end{figure}\n\n\nIn order to find stationary periods, we first transform the data using Eq.\n(\\ref{eq:normalizeData}) to obtain approximately Gaussian data with constant volatility, and then we search for periods of constant drift. We use the ``Breaks for Additive Season and Trend''\" (BFAST) algorithm to find the constant drift periods. Here we are interested in the trend component which is a piecewise linear function where the breakpoints (change from a linear function to the next) are determined by BFAST as well. \n\nWe apply BFAST to the log DJIA index sampled monthly. We choose a monthly index because running the algorithm is faster and because we want periods that are few years long in order to have enough data points (weeks) within each regime. Our average regime length is of 2.2 years with a maximum of approximately 5 years, and a minimum of nearly 1 year.\n\nAfter applying BFAST to find the stationary patches, we ignore any regime with less than 1.3 years (70 weeks). We are left with 47 regimes (patches).  We plot the log cumulative return for DJIA from 1995 to 2013 together with the fitted trends in Figure \\ref{fig4}. Each regime is defined between dashed green line and the consecutive dashed blue line. The inset shows the data from 1900 to 2013.\n\nIn general, linear regression in Figure \\ref{fig4} shows a very good fit. Some patches present a good adherence with the liner fit while others present a larger fluctuation around the linear fit. The goodness of the fit validates the method used in this work. Patches with larger fluctuations correspond to troubled economical moments as in 1997, 2002 and 2008.\n\nWe calculate the empirical performance of the strategy (Eq. (\\ref{eq:strategy})) for look-back periods ranging from $N=1$ to $N=43$ weeks within each regime. Figure \\ref{fig5} shows the rescaled log-DJIA plotted together with the maximum annualized (weekly values multiplied by $\\sqrt{52}$)  information ratio. The color coded bar graph refers to case I and case II in Figure \\ref{fig2}. If the maximum IR is for look-back periods $N>20$ weeks we classify it to be a case I (red) otherwise a case II (blue) period. That is, the red bars imply a graph IR versus lag $N$ that is visually closer to case I than case II (Figure \\ref{fig2}). In addition to Figure \\ref{fig5},  Table \\ref{table1} includes the standard error for the IR within each regime.  \n\nFrom Figure \\ref{fig5}, we notice that  case I is more prominent for post 1975 data. This is due to two reasons. First,  the DJIA increases quite steadily from 1980 to 2000 which implies a large drift effect. Second, after 1975, the autocorrelation effect in Eq. (\\ref{eq:theAve}) is mostly negative ($\\sum_{i} \\rho(t,t-i) < 0$) or at best insignificant. Consequently, there is a higher IR with  large $N$ (better to buy and hold - case I) compared to  small $N$ (case II - cumulative autocorrelation). \n\nFigure \\ref{fig6} and Table \\ref{table1} present the first lag autocorrelation within each regime. The bars are again color coded to match Figures \\ref{fig2} and \\ref{fig5}. Notice that from 1900 to 1975, most bars are positive, and after 1975 the bars are mostly negative. The sign of the first lag autocorrelation survives even if we calculate the autocorrelation over all 4080 weeks up to 1975 and all 1988 weeks after 1975. The autocorrelation for all weekly returns before 1975 is 0.17 and after 1975 it becomes $-0.04$. Therefore, even though most of the autocorrelation values per regime are marginally significant (Table \\ref{table1}) there is a likely change on the sign of the autocorrelation of weekly returns in 1975.\n\nPriviously, work by Lo and MacKinelay  \\cite{Lo1988,Lo1990} hint to the autocorreltion regime change we reported here (Figure \\ref{fig6}). They show that for an index of large cap stocks, the autocorrelation for the sub-period (1975$-$1985) is not significantly positive, where as, it is significantly positive from 1962 to 1988. See also the work of Froot and Perold \\cite{Froot}, which study the first lag autocorrelation for returns from 15 minutes to 1 week for stock indices with data up to 1990. In agreement with our results, they show that the large positive autocorrelation decreases substantially after 1970 going negative (see Figure 5 in \\cite{Froot}).\n\nTo further compare our results to Lo and MacKinelay \\cite{Lo1988,Lo1990}, we calculate the first four lag autocorrelations of the weekly log-returns from July 6, 1962 to December 31, 1987 for the DJIA index. Our values have the same sign as reported by  Lo and MacKinelay \\cite{Lo1990} but different magnitudes. We get 1.1\\%, 1.7\\%, 5.1\\%, $-$2.8\\% from lags 1 to 4 whereas they have 7.4\\%, 0.7\\%, 2.1\\% and $-$0.5\\% (for the value-weighted CRSP index, see Table 1 of \\cite{Lo1990}). We attribute these different values to the fact that, we use the DJIA and they look at the value-weighted Center for research in security prices (CRSP) index.\n\nFurthermore, the autocorrelation values shown in Figure \\ref{fig6} and Table \\ref{table1} are calculated with the DJIA rescaled log-returns (Eq. (\\ref{eq:normalizeData})). Therefore we expect an additional difference especially on the magnitude of the autocorrelation. We find that the first four autocorrelation of our rescaled DJIA index are  5.7\\%,1.6\\%,0.7\\% and -3.2\\%  which agree in sign with both Lo and MacKinelay (7.4\\%, 0.7\\%, 2.1\\% and $-$0.5\\% ) and with the un-scaled DJIA log-returns (1.1\\%, 1.7\\%, 5.1\\%, $-$2.8\\%) .  The positive sign of the first autocorrelation only shows that by using data from (1962$-$1988) we are missing the regime change around 1975. In other words, by taking the data from 1962 to 1988, we are mixing our 11 regimes in such a way that the first lag autocorrelation becomes positive. That is, the pre-1975 data ``dominates\" the post-1975 data when the autocorrelation of the entire period (1962 to 1988) is calculated.\n\nFor completeness, it is worth mentioning that \\cite{Bondt1989,Lehmann} report that individual weekly stock returns are negatively autocorrelated using data from virtually the same period (1962 to 1986) as in \\cite{Lo1990}. This apparent dilemma (since indices are made of individual stocks) was addressed by \\cite{Lo1990}. They show that it is possible for the autocorrelation of the index to be positive even though the autocorrelation of the component stocks is negative if the stocks cross-autocorrelation (lead-lag relation among stocks) is large and positive. \n\nOur autocorrelation results in Figure \\ref{fig6} indicate a fundamental change for at least large cap stocks that start in 1975. Not only is the first lag autocorrelation of DJIA weekly returns negative but  assuming that first lag autocorrelation for individual stocks is still negative (see \\cite{Gutierrez} with data from 1983 to 2003), cross-autocorrelation between stocks should be negative or insignificant after 1975.\n\n\n\\begin{figure}[htbp] \\centering\n\\includegraphics[width=\\textwidth]{case3} \\caption{Time series representation of stationary periods. Bar graph shows the annualized IR values for each patch\/regime.\nThe red bars refer to case I and blue bars refer case II. We also show the log of the rescalled DJIA index (black line) in arbitrary units. \\label{fig5}}\n\\end{figure}\n\n\\begin{figure}[ht] \n\\centering \\includegraphics[height=8cm, width=\\textwidth]{autoCorrPlot}\n\\caption{First lag autocorrelation for rescaled DJIA log-returns within each regime. Red bars correspond to case I and blue to case II using the same convention that was used in Figure \\ref{fig5} \\label{fig6}} \n\\end{figure}\n\n\n\n\\begin{figure}[htbp] \\centering\n\\includegraphics[width=\\textwidth]{case4} \\caption{Average IR vs. look-back lag\nover the 47 patches determined by BFAST algorithm. Circles represent the data and\nthe red line is the best fit theoretical IR given by Eqs. (\\ref{eq:theAve}) and\n(\\ref{eq:theVar}).\n\\label{fig:aveSpec}} \n\\end{figure}\n\n\\clearpage\n\\begin{table}\n\\centering\n\\small\n\\begin{tabular}{rrrrrr}\n  \\hline\n Regimes& \\# Weeks & Acf(1) & Acf SE(95\\%)  & Max IR (Year) & IR SE(95\\%)  \\\\ \n  \\hline\n1896-11-06 to 1899-01-27 & 117.00 & 0.15 & 0.18 & 1.72 & 1.33 \\\\ \n  1899-03-30 to 1900-09-28 & 79.00 & 0.12 & 0.23 & 1.61 & 1.62 \\\\ \n  1901-01-04 to 1903-07-31 & 135.00 & -0.08 & 0.17 & 0.74 & 1.24 \\\\ \n  1903-10-02 to 1905-11-24 & 113.00 & -0.01 & 0.19 & 2.72 & 1.36 \\\\ \n  1906-02-02 to 1907-07-26 & 78.00 & 0.01 & 0.23 & 0.91 & 1.63 \\\\ \n  1907-10-04 to 1909-11-26 & 113.00 & 0.07 & 0.19 & 1.33 & 1.36 \\\\ \n  1910-02-04 to 1911-09-29 & 87.00 & 0.17 & 0.21 & 1.22 & 1.55 \\\\ \n  1912-03-29 to 1915-06-25 & 151.00 & 0.31 & 0.16 & 0.74 & 1.17 \\\\ \n  1915-10-01 to 1917-06-29 & 92.00 & 0.02 & 0.21 & 0.16 & 1.50 \\\\ \n  1917-08-31 to 1919-03-28 & 83.00 & 0.08 & 0.22 & 0.67 & 1.58 \\\\ \n  1919-05-29 to 1921-08-26 & 118.00 & 0.08 & 0.18 & 1.37 & 1.33 \\\\ \n  1921-11-04 to 1923-04-27 & 78.00 & 0.19 & 0.23 & 1.39 & 1.63 \\\\ \n  1923-06-29 to 1926-01-29 & 136.00 & 0.02 & 0.17 & 2.10 & 1.24 \\\\ \n  1926-07-30 to 1928-08-31 & 110.00 & 0.10 & 0.19 & 1.26 & 1.38 \\\\ \n  1928-11-30 to 1930-05-29 & 79.00 & 0.16 & 0.23 & 1.14 & 1.62 \\\\ \n  1930-08-01 to 1932-02-26 & 83.00 & 0.03 & 0.22 & 1.96 & 1.58 \\\\ \n  1932-04-29 to 1934-03-29 & 100.00 & 0.00 & 0.20 & 0.76 & 1.44 \\\\ \n  1934-06-01 to 1936-08-28 & 118.00 & -0.16 & 0.18 & 1.46 & 1.33 \\\\ \n  1936-10-30 to 1938-04-29 & 79.00 & 0.03 & 0.23 & 1.51 & 1.62 \\\\ \n  1938-07-01 to 1940-03-29 & 92.00 & -0.09 & 0.21 & 0.08 & 1.50 \\\\ \n  1942-04-02 to 1943-09-24 & 78.00 & 0.09 & 0.23 & 2.52 & 1.63 \\\\ \n  1944-02-04 to 1946-06-28 & 126.00 & -0.02 & 0.18 & 1.38 & 1.28 \\\\ \n  1946-10-04 to 1949-08-26 & 152.00 & 0.06 & 0.16 & 0.65 & 1.17 \\\\ \n  1949-12-30 to 1953-01-30 & 162.00 & -0.01 & 0.16 & 0.53 & 1.13 \\\\ \n  1953-05-29 to 1955-10-28 & 127.00 & 0.08 & 0.18 & 2.33 & 1.28 \\\\ \n  1955-12-30 to 1957-08-30 & 88.00 & 0.07 & 0.21 & 0.84 & 1.54 \\\\ \n  1957-11-01 to 1959-11-27 & 109.00 & 0.15 & 0.19 & 2.60 & 1.38 \\\\ \n  1960-04-29 to 1961-12-29 & 88.00 & -0.04 & 0.21 & 1.02 & 1.54 \\\\ \n  1962-05-04 to 1966-03-25 & 204.00 & 0.10 & 0.14 & 0.96 & 1.01 \\\\ \n  1966-07-29 to 1968-07-26 & 105.00 & 0.20 & 0.20 & 1.69 & 1.41 \\\\ \n  1968-11-01 to 1970-06-26 & 86.00 & 0.09 & 0.22 & 1.26 & 1.56 \\\\ \n  1970-09-04 to 1972-09-29 & 109.00 & 0.18 & 0.19 & 1.46 & 1.38 \\\\ \n  1972-12-01 to 1974-06-28 & 83.00 & -0.15 & 0.22 & 0.19 & 1.58 \\\\ \n  1974-08-30 to 1976-04-30 & 88.00 & -0.24 & 0.21 & 1.01 & 1.54 \\\\ \n  1976-07-02 to 1978-02-24 & 87.00 & -0.18 & 0.21 & 1.48 & 1.55 \\\\ \n  1978-06-30 to 1980-01-25 & 83.00 & -0.06 & 0.22 & 0.68 & 1.58 \\\\ \n  1980-08-01 to 1982-08-27 & 109.00 & 0.05 & 0.19 & 0.40 & 1.38 \\\\ \n  1982-12-03 to 1985-09-26 & 148.00 & -0.07 & 0.16 & -0.05 & 1.19 \\\\ \n  1985-11-29 to 1987-08-28 & 92.00 & 0.05 & 0.21 & 2.90 & 1.50 \\\\ \n  1987-10-30 to 1990-06-29 & 140.00 & -0.24 & 0.17 & 0.44 & 1.22 \\\\ \n  1991-08-30 to 1994-03-31 & 136.00 & 0.06 & 0.17 & 0.53 & 1.24 \\\\ \n  1994-07-01 to 1996-12-27 & 131.00 & -0.12 & 0.17 & 2.57 & 1.26 \\\\ \n  1997-07-03 to 1999-01-29 & 83.00 & -0.15 & 0.22 & 0.33 & 1.58 \\\\ \n  1999-04-30 to 2002-05-31 & 162.00 & -0.12 & 0.16 & 0.07 & 1.13 \\\\ \n  2002-08-02 to 2007-07-27 & 261.00 & -0.07 & 0.12 & 0.08 & 0.89 \\\\ \n  2007-09-28 to 2009-03-27 & 79.00 & 0.06 & 0.23 & 1.52 & 1.62 \\\\ \n  2009-05-29 to 2013-02-01 & 193.00 & -0.03 & 0.14 & 0.04 & 1.04 \\\\ \n   \\hline\n\\end{tabular}\n\\caption{First lag autocorrelation and IR per regime.} \n\\label{table1}\n\\end{table}\n\\clearpage\n\nResults from the autocorrelation (Figure \\ref{fig6}) are further sustained by Figure \\ref{fig:aveSpec}. Figure \\ref{fig:aveSpec} shows the average of IR vs $N$ (look-back) over all 47 regimes. In addition to the ensemble average, we show the theoretical IR that is constructed by dividing the theoretical average performance  (Eq. (\\ref{eq:theAve})) by the theoretical standard deviation of the performance (Eq. (\\ref{eq:theVar})). \n\nThe parameters for theoretical model are found by least square fitting the empirical data points. The model uses the empirical standard deviation ($1.5$)  of our renormalized data and 11 fitting parameters: the drift $\\mu$ and the first 10 autocorrelations. Clearly, we are over-fitting since the number of parameters is large giving that we have only 43 data points; however, our goal here is not to find a robust model but to qualitatively show the potential of the theoretical model. \n\nThe average IR shows that for very short look-back, 1 or 2 weeks, we have a large positive autocorrelation effect (case II). The decrease from the peak in two weeks is much faster then the natural case II would predict (Figure \\ref{fig2}). Based on our fit, what explains the fast decrease from week 3 to 7 is a progression of negative autocorrelation which pull the average IR down from its maximum in two weeks. The influence of autocorrelation can be detected all the way to week 12 with a quick alternation of what looks like a ``peak\" of positive autocorrelation at approximate 10 weeks ($\\approx$2.5 months) in the mist of negative autocorrelation. For look-back periods of more than 16 weeks ($\\approx$4 months) the IR curves clearly resembles case I, where if we add more look-back weeks we do better.\n\nIt is very unlikely that the high autocorrelation effect for the first 12 weeks is significant in the recent history, considering that most first lag autocorrelations are negative (at best neutral) after 1975 (Figure \\ref{fig6} and Table \\ref{table1}). However, it is true that on average both case I and case II (Figure \\ref{fig2}) can be present and equally important in an average stationary regime. What is interesting is that even though both appear to be equally significant, there is a clear transition from one (case II for small $N$) to the other (case I for large $N$) as we change $N$. Due to such transition, one can classify a momentum strategy based on the portfolio formation period (look-back $N$). The 3 to 4 months look-back leads to momentum of the kind associated to under-reaction since there seams to be significantly positive autocorrelation \\cite{JT2001} before 4 months. On the other side, momentum for look-back periods larger than 4 months is mostly due to the natural drift present in the asset. In the literature this drift is sometimes associated with macroeconomic variables and business cycles  \\cite{Chordia,Kim2013,Griffin,Guidolin,Barroso,Conrad} and often perceived as orthogonal to behavioral models \\cite{Chordia,Conrad}.\n\nFinally, we note that for large lag $N$, the average IR in Figure \\ref{fig:aveSpec} disagrees with the theoretical fit. It is difficult to draw conclusions since we stop at 10 months and the number of data points become small. In order to capture large $N$ effects (which show overreaction \\cite{Bondt1989}) and also in order to be closer to the way a momentum strategy is typically tested in applications, we will construct IR versus lag $N$ for all 6068 weeks of data in the next section.\n\n\\section{Empirical non-stationary analysis}\n\\label{sec:NE_stationary}\nIn this section we apply the momentum strategy of Eq. (\\ref{eq:strategy}) to all DJIA data. We do not try to break the data into stationary periods, but we still normalize the weekly log-returns using Eq. (\\ref{eq:normalizeData}). The goal here is to present the effect of non-stationary data to the performance of momentum strategies. Since we normalize the log-returns using Eq. (\\ref{eq:normalizeData}), we still expect the data to have constant variance. Nevertheless, the data will not have constant drift nor constant autocorrelation. Therefore, we do not expect this section to conform with case I or case II in Figure \\ref{fig2}.\n\nFigure \\ref{fig:notStat} shows the information ratio (IR) versus $N$ number of weeks used to construct the moving average (Eq. (\\ref{eq:strategy})). Since we have 6068 weeks (data points) we present the IR for our momentum strategy that looks back up to 400 weeks (almost 8 years). In agreement with our expectations, the curve of the IR dependence on the look-back period is not even qualitatively similar to case I or case II. However, we can still use results of the previous section as guide to develop a model here.\n\nWe learned that both autocorrelation and drift are important to explain the IR of a momentum strategy (Figure \\ref{fig:aveSpec}). We know that case I is more important than case II for large time lags $N$ (portfolio formation look-back lags) since the IR in case I increases with lag and in case II it decreases (Figure \\ref{fig2}). Considering this, we postulate that the oscillations are mostly due to changes in the drift of case I. That leads us to assume that the theoretical model for the normalized log-returns is given by:\n\n\\begin{equation}\n\\label{eq:nonSmodel}\n r_m(t) = \\mu + A\\:{\\rm sgn}( \\sin(2 \\pi t\/T))+\\epsilon,\n\\end{equation}\n\n\\noindent where $\\mu$ stands for the average growth rate, $A$ for the amplitude of our square wave, $T$ for the period and $\\epsilon$ for the noise. Figure \\ref{fig:notStat2} shows the cumulative return of our model (with and without noise added) together with the DJIA data.\n\nWe take the parameter $\\mu$ to be equal to the empirical growth rate of the weekly re-normalized DJIA ($\\mu=0.075$ per week). The parameter $A$ and $T$ are selected to fit the empirical IR oscillation in Figure \\ref{fig:notStat}. We find that $T$ is approximately equal to 3.5 years (180 weeks) and that $A\\approx 2\\times\\mu$. The root mean square $\\sigma$ of the noise $\\epsilon$ is equal to the empirical noise level of the re-normalized DJIA weekly returns over the entire period ($\\sigma = 1.5\/week$). \n\nThe model in Eq. (\\ref{eq:nonSmodel}) shows the long term reversals reported by \\cite{Bondt1985,Bondt1987,Bondt1989,Fama} for stocks and indices. The period of 3.5 years implies a mean reversion of about 1.75 years which is within the prior reported value of 1.5 to 5 years \\cite{Bondt1989}. In terms of correlations, our model has a small positive autocorrelation that progressively goes negative and back up again: oscillating with the period $T$ even when $\\epsilon$ is independent and identically distributed noise . However since the empirical noise level is between 7 to 20 times larger than $\\mu \\pm A$ such correlations are not easy to detect by calculating the autocorrelation function. Consider Figure \\ref{fig:notStat2} Monte Carlo generated sample curve for our model (blue line): notice that it is impossible to recognize the periodic oscillations. \n\nIn order to compute the IR versus the lag length $N$ for our theoretical model (Eq. (\\ref{eq:nonSmodel})), we perform Monte Carlo (MC) simulations (500 realizations). We consider two types of noise $\\epsilon$. The first one is an iid Gaussian random variable with zero drift and variance equal to the empirical variance of the re-normalized data. This is the simplest case and assumes that all the performance for a momentum strategy is only due to the drift ($\\mu$). The second one is a MA process with zero mean and variance as the Gaussian iid random variable, but with best fit positive autocorrelation for lag 1 (0.05) and lag 20 (0.08). The second model takes in consideration that short lags might be strongly influenced by autocorrelation (case II situation) and therefore both case I and case II co-exist as in Figure \\ref{fig:aveSpec}. \n\nThe MA process is selected by first choosing the non-zero autocorrelation lags and then by fitting the model to the empirical data (IR vs N curve) to find the numerical coefficients, which imply a correlation of 0.05 and 0.08 for lag 1 and lag 20.  We select the non-zero lags by using the intuition developed for the stationary model and by keeping the number of non-zero lags minimal. From Eq. (\\ref{eq:case2}) and Figure \\ref{fig2}, we know that a MA model generates a localized hump in the graph IR versus lag $N$ in the case of stationary data. The location of the hump is centered at the location of the autocorrelation. Assuming that this approximately valid here, the IR vs N  for uncorrelated $\\epsilon$ can be shifted up in the most relevant places to get a better fit. Therefore, by looking at the empirical IR vs N of Figure \\ref{fig:notStat}, we postulate that the most relevant lags should be around lag 1 and lag 20, since we need to shift the theoretical IR vs N generated with $\\epsilon$ iid Gaussian (dashed red) at least at lag 1 and lag 20, to produce the IR vs N from the MA generated $\\epsilon$ (solid blue).\n\nFigure \\ref{fig:notStat} shows the empirical IR versus lag $N$ (circles) and 2 theoretical lines (dashed red and solid blue). The best fit is achieved with $\\epsilon$ drawn form a $MA$ process (solid blue line). The worst fit is with $\\epsilon$ drawn from an iid Gaussian noise (red dashed line). The oscillations are well captured  by the uncorrelated $\\epsilon$, especially for very large lags $N$, however it underestimates the IR ratio for small lags. We have shown (Fig. \\ref{fig6})  that before 1975 the autocorrelation is significantly positive, and since we are in effect averaging the data by treating the 100 years of the DJIA as stationary, we need to include large positive short range autocorrelation to account for data before 1975. The result is the much better fit of the solid blue line in Figure \\ref{fig:notStat}.\n\nBased on Figure \\ref{fig:aveSpec} one could expect to find a small autocorrelation effect on the IR for lag $N \\approx 20$, which is not what we find by fitting the IR here (Figure \\ref{fig:notStat}). However, this is misleading, since we would be comparing approximately stationary data to non-stationary data. Furthermore, even if some regimes have \"large\" lag 20 autocorrelation, this autocorrelation would have been averaged out. That is, the difference between Figure \\ref{fig:aveSpec} and Figure \\ref{fig:notStat}, is that Figure \\ref{fig:aveSpec} is constructed by taking an average over all the IR versus lag $N$ curves created within our stationary patches. In other words, all IR for $N  \\approx 20$ that have a hump are averaged with all IR which do not show a hump. Since at $N \\approx 20$ the contribution to IR of the autocorrelation hump is of order $1\/\\sqrt{20}$ and the contribution of drift is of order $1$ (Eq. (\\ref{eq:theVar})), the average is dominated by the drift. The result is that, the effect of the autocorrelation for lags larger than 15 in Fig. \\ref{fig:aveSpec} should not be evident. \n\nThe model of Eq. (\\ref{eq:nonSmodel}) with $\\epsilon$ drawn from a MA process not only fits the empirical IR curve well (Figures \\ref{fig:notStat} and \\ref{fig:notStat2}), but it also agrees well with prior studies in the literature. It shows a large positive first lag autocorrelation of the normalized weekly returns which agree with \\cite{Lo1988,Lo1990}. It also shows a large autocorrelation around 1\/2 year, which suggests that autocorrelation is the main driver of traditional momentum returns for look-back of 3 to 12 months and that the drift plays as a much less important role \\cite{JT2001,JT2002}. Our finding here however suggests, that the results in the literature are mostly due to data prior to 1975.\n\n\\begin{figure}[ht] \n\\begin{minipage}[b]{0.45\\linewidth}\n \\centering \\includegraphics[height=10cm, width=\\textwidth]{case5}\n  \\caption{Rescaled DJIA index, model and Monte Carlo simulations. Solid red line shows the rescaled log-DJIA data. The dashed green line shows the model of Eq. (\\ref{eq:nonSmodel}) without noise (integrated square wave). The blue line shows one Monte Carlo realization of Eq. (\\ref{eq:nonSmodel}).} \\label{fig:notStat2} \\end{minipage} \\hspace{0.1cm} \\begin{minipage}[b]{0.45\\linewidth} \\centering\n\\includegraphics[height=11cm, width=\\textwidth]{NonStationarySpecRevisited} \\caption{IR versus lag for 100 years of DJIA index. Dashed red line shows the IR calculated via Monte Carlo method for model in Eq. (\\ref{eq:nonSmodel}) with $\\epsilon$ drawn from IID gaussian distribution. Solid blue line corresponds to theoretical line from Eq. (\\ref{eq:nonSmodel}) with $\\epsilon$ from a MA process with autocorrelation at lag 1 and lag 20. Black circles corresponds to the DJIA rescaled data.} \\label{fig:notStat} \n\\end{minipage} \n\\end{figure}\n\n\\section{Conclusions}\n\nIn the present work we perform statistical analysis on a momentum strategy and find a closed form formula for the expected value and the variance. We are therefore able to present an analytic expression for the Information Ratio. The innovation here allows one to compute and discuss the risk adjusted performance of a momentum strategy.\n\nThe theory developed here assumes that the time series are patches stationary. One key point is to find the proper stationary patch. Once this condition is satisfied, we get good agreement between theory and data as discussed in section \\ref{sec:E_stationary}. Notwithstanding, when a longer time series is considered, the stationary assumption may no longer be valid. In this scenario, we proposed a stochastic model in section \\ref{sec:NE_stationary}. This model agrees well with the data by introducing a periodic change in the average growth rate of the DJIA. \n\nSummarizing our empirical results: in section \\ref{sec:E_stationary}, we find that both autocorrelation and drift can be important in describing the risk adjusted performance of a time series momentum strategy. More originally, we show that the information ratio appears to present 2 phases. The first phase is for short look-back periods and the second for long look-back periods. The first phase is mainly driven by autocorrelation where as the second phase by average return (drift). Furthermore, there is a behavioral change at 1975, were the first lag autocorrelation of weekly returns go from mainly positive before 1975 to mainly negative after 1975.\n\nIn section \\ref{sec:NE_stationary}, we find an oscillatory IR for long portfolio formation periods which we associate with periodic cycles in the market. However, we emphasize that the results in section \\ref{sec:NE_stationary} are inherently unstable since we use 100 years of DJIA data. That means that if we repeat the same study for different sub-samples, we will find different quantitative and sometimes qualitative results. It is true that for 100 years ($\\approx 6000$ weeks) of data our results are statistically significant with little doubt. In fact the exact extreme case is the study done in section \\ref{sec:E_stationary}, where we have $\\approx 100$ weeks per patch and therefore the error bars are  $\\approx 10$ times larger. This is one of the challenges of empirical finance: to keep significance we may end up mixing stationary patches that creates an average behavior which may hide the true mechanism. On the other hand if we look at less data we may have to compromise significance which makes us more reluctant to believe in the observed effects. Balancing these two is a great challenge when dealing with non-stationary data.\n\n\\newpage\n\n\\section{Appendix}\n\nThe characteristic function for the multivariate Gaussian random variable is\n\n\\begin{equation}\n\\phi(s) =  \\exp\\Big( i\\sum_{i} s_{i} \\mu_{i} -\\frac{1}{2} \\sum_{i} V_{ii} s_{i}^{2} + \\frac{1}{2} \\sum_{i \\neq j} s_{i} s_{j} V_{ij}\\Big )\n\\end{equation}\nWe will calculate each term of Eq. (\\ref{eq:risk2}) separately. The summand of the first term is\n\\begin{equation}\n\\label{eq:riskA1}\n\\left \\langle X_t^{2} X_{t-i}^{2}\\right \\rangle =\\left \\langle X_{k}^{2} X_{i}^{2}\\right \\rangle = \\frac{\\partial^{2}}{\\partial s_{k}^{2}} \\frac{\\partial^{2}}{\\partial s_{i}^{2}} \\phi(s) |_{s=0}\n\\end{equation}\nwhere we have simplified our notations by taking current time $t$ as $k$; and\npast times being labeled as $i$, $j$ or $q$. The double partial derivatives in (\\ref{eq:riskA1}) result into:\n\n\\begin{eqnarray}\n\\label{eq:cal1}\nV_{ii} V_{kk} \\phi(s) \\\\ \\nonumber\n -V_{ii} [i \\mu_{k}-V_{kk} s_{k}-\\sum_{j} s_{j}V_{ij}]^{2} \\phi(s) \\\\ \\nonumber\n+V_{ik}^{2} \\phi(s) \\\\ \\nonumber\n-V_{ik}\n [i\\mu_{k}-V_{kk}s_{k}-\\sum_{j}S_{j}V_{kj}][i\\mu_{i}-V_{ii}s_{i}-\\sum_{j} s_{j}V_{ij}]\\phi(s) \\\\ \\nonumber\n+V_{ik}^{2} \\phi(s) \\\\ \\nonumber\n-V_{ik}\n [i\\mu_{k}-V_{kk}s_{k}-\\sum_{j} s_{j} V_{kj}][i\\mu_{i}-V_{ii}s_{i}-\\sum_{j} s_{j}V_{ij}]\\phi(s) \\\\ \\nonumber\n-V_{ik}\n [i\\mu_{k}-V_{kk}s_{k}-\\sum_{j} s_{j} V_{kj}][i\\mu_{i}-V_{ii}s_{i}-\\sum_{j} s_{j}V_{ij}]\\phi(s) \\\\ \\nonumber\n-V_{ik}\n [i\\mu_{k}-V_{kk}s_{k}-\\sum_{j} s_{j} V_{kj}][i\\mu_{i}-V_{ii}s_{i}-\\sum_{j} s_{j}V_{ij}]\\phi(s) \\\\ \\nonumber\n-V_{kk}\n[i\\mu_{i}-V_{ii} s_{i}-\\sum_{j} s_{j} V_{ij}]^{2} \\phi(s) \\\\ \\nonumber\n+[i \\mu_{k} - V_{kk} s_{k} - \\sum_{j} s_{j} V_{kj}]^{2} [i \\mu_{i} - V_{ii} s_{i} - \\sum_{j} s_{j} V_{ij}]^{2} \\phi(s) \\nonumber\n\\end{eqnarray}\n\n\\noindent where $j$ is the index of  the summation. Let $s=0$, thus\nEq. (\\ref{eq:cal1}) is reduced to:\n\n\\begin{equation}\n\\left \\langle X_{k}^{2} X_{i}^{2}\\right \\rangle=V_{ii}V_{kk}+2V_{ik}^{2}+V_{ii}\\mu_{k}^{2}+4V_{ik}\\mu_{i}\\mu_{k}\n+\\mu_{i}^{2}V_{kk}+\\mu_{i}^{2}\\mu_k^{2}\n\\end{equation}\n\n\\noindent Now, the summand of the second term of Eq. (\\ref{eq:risk2}) is given by\n\n\\begin{equation}\n\\label{eq:riskA2}\n\\left \\langle X_t^2X_{t-i}X_{t-j}\\right \\rangle = \\left \\langle X_{k}^{2} X_{i}X_{j}\\right \\rangle = \\\\\n\\frac{\\partial^{2}}{\\partial s_{k}^{2}} \\frac{\\partial}{\\partial s_{i}} \\frac{\\partial}{\\partial s_{j}} \\phi(s) |_{s=0}\n\\end{equation}\nNotice that we use $j$ to indicate a different time lag in Eq. (\\ref{eq:riskA2}). The partial derivatives in Eq. (\\ref{eq:riskA2}) result into:\n\n\\begin{eqnarray}\n\\label{eq:cal2}\n+V_{ij}V_{kk}\\phi(s)\\\\ \\nonumber\n-V_{ij} [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}]^{2} \\phi(s) \\\\ \\nonumber\n+ V_{ik}V_{kj} \\phi(s) \\\\ \\nonumber\n-V_{ik} [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}] [i \\mu_{j} - V_{jj} s_{j} - \\sum_{q} s_{q} V_{jq}] \\phi(s) \\\\ \\nonumber\n+V_{ik}V_{kj} \\phi(s)\\\\ \\nonumber\n-V_{jk} [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}] [i \\mu_{i} - V_{ii} s_{i} - \\sum_{q} s_{q}V_{iq}] \\phi(s) \\\\ \\nonumber\n-V_{jk} [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}] [i \\mu_{i} - V_{ii} s_{i} - \\sum_{q} s_{q}V_{iq}] \\phi(s) \\\\ \\nonumber\n-V_{ik} [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}] [i \\mu_{j} - V_{jj} s_{j} - \\sum_{q} s_{q} V_{jq}] \\phi(s) \\\\ \\nonumber\n-V_{kk} [i \\mu_{i} - V_{ii} s_{i} - \\sum_{q} s_{q} V_{iq}] [i \\mu_{j} - V_{jj} s_{j} - \\sum_{q} s_{q} V_{jq}] \\phi(s) \\\\ \\nonumber\n+ [i \\mu_{k} - V_{kk} s_{k} - \\sum_{q} s_{q} V_{kq}]^{2} [i \\mu_{j} - V_{jj} s_{j} - \\sum_{q} s_{q} V_{jq}] \\\\ \\nonumber\n\\times [i \\mu_{i} - V_{ii} s_{i} - \\sum_{q} s_{q} V_{iq}]  \\phi(s) \\\\ \\nonumber\n\\end{eqnarray}\n\n\\noindent Let $s=0$, then (\\ref{eq:cal2}) becomes\n\n\\begin{equation}\nV_{ij}V_{kk}+2V_{ik}V_{jk}+2V_{kj}\\mu_{i}\\mu_{k}\n+V_{ij}\\mu_{k}^{2}+2V_{ik}\\mu_{j}\\mu_{k}\n+V_{kk}\\mu_{j}\\mu_{i}+\\mu_{k}^{2} \\mu_{i} \\mu_{j}\n\\end{equation}\n\n\\noindent Hence, the first term of (\\ref{eq:risk1}), i.e. the square of strategy return (R), is given by\n\n\\begin{eqnarray}\n\\label{eq:cal3}\n\\left \\langle R^{2} \\right \\rangle &=& \\frac{1}{N^2}\\Big(\n\\sum_{i} \\big(V_{ii}V_{kk}+2V_{ik}^{2}+V_{ii}\\mu_{k}^{2}+4V_{ik}\\mu_{i}\\mu_{k}\n+\\mu_{i}^{2}V_{kk}+\\mu_{i}^{2}\\mu_k^{2}\\big)\\nonumber \\\\ \n&+&\\sum_{i,j, i \\neq j} \\big(V_{ij}V_{kk}+2V_{ik}V_{jk}+2V_{kj}\\mu_{i}\\mu_{k}\n+V_{ij}\\mu_{k}^{2}+2V_{ik}\\mu_{j}\\mu_{k}+ \nV_{kk}\\mu_{j}\\mu_{i}+\\mu_{k}^{2} \\mu_{i} \\mu_{j}\\big)\\Big)\n\\end{eqnarray}\n\n\\noindent Now,  we compute the second term of Eq. (\\ref{eq:risk1}), i.e. the square of the average, and we get\n\n\n\\begin{eqnarray}\n\\label{eq:cal4}\n\\left \\langle R \\right \\rangle^{2} &=& \\frac{1}{N^2}\\Big(\\sum_{i} \\big(V_{ij} +\\mu_{i}\\mu_{j}\\big)\\Big)^{2} \\nonumber \\\\&=&\n\\frac{1}{N^2}\\Big(\\sum_{i} \\big(V_{ik}^{2} + 2V_{ki}\\mu_{k}\\mu_{i}+\\mu_{k}^{2}\\mu_{i}^{2}\\big)  \\nonumber \\\\\n&+&\\sum_{i,q,i\\neq q} \\big(V_{ki}V_{kq}+V_{ki}\\mu_{k}\\mu_{q}+V_{kq}\\mu_{k}\\mu_{q}\n+\\mu_{k}^{2}\\mu_{i}\\mu_{q}\\big)\\Big)\n\\end{eqnarray}\n\\noindent where we use the same simplifying notation ($i$,$j$,$k$) to\nrepresent the time lags.\n\nFinally, from  Equations (\\ref{eq:cal3}) and (\\ref{eq:cal4}), we obtain\nEq. (\\ref{eq:risk1}) as the following:\n\n\\begin{eqnarray}\n\\label{eq:finalGeneralVar}\nVar(R) &=& \\frac{1}{N^2}\\Big(\\sum_{i} \\big(V_{ii}V_{kk}+V_{ik}^{2}+V_{ii}\\mu_{k}^{2}+2V_{ki}\\mu_{i}\\mu_{k}+V_{kk}\\mu_{i}^{2}\\big) \\\\ \\nonumber\n&+&\\sum_{i,j,i\\neq j} \\big(V_{ij}V_{kk}+V_{ki}V_{kj}+V_{kj}\\mu_{i}\\mu_{k}+V_{ij}\\mu_{k}^{2}+V_{kk}\\mu_{i}\\mu_{j} +V_{ik}\\mu_j\\mu_k\\big)\\Big) \\nonumber\n\\end{eqnarray}\nNow, assuming  that the variance $V$ and the drift $\\mu$ are constants and that the autocorrelation depends only on time lag, (\\ref{eq:finalGeneralVar}) can be converted to Eq. \\ref{eq:theVar} .\n\n\\begin{eqnarray}\n\\label{eq:theVarApp} Var(R) &=& \\frac{1}{N^{2}}\\bigg[ N V^{2} + N \\mu^{2} V\n+ N^{2} V \\mu^{2} \\\\ \\nonumber &+&V^{2} \\big(\\sum_{i=1}^{N} \\rho(t,t-i)\\big)^{2} +V^{2}\\sum_{i,j=1,i\\neq\nj}^{N} \\rho(t-i,t-j)\\\\\n\\nonumber &+&\\mu^{2} V \\Big(2\\sum_{i=1}^{N} \\rho(t,t-i) +\\sum_{i,j=1,i\\neq\nj}^{N}\\big(\\rho(t,t-j)+\\rho(t-i,t-j)+\\rho(t,t-i)\\big)\\Big) \\bigg]  \n\\end{eqnarray}\n\n\\noindent where we convert back to the time lag notation used in the main text by reverting the $i,j,k$ notation. For example, we take $V_{ki} = V \\rho(t,t-i)$,$V_{ij} = V \\rho(t-i,t-j)$ and $V_{kk}=V$.\n\n\\section*{Acknowledgments}\n\nACS would like to thank the participants of R\/Finance 2013 for comments and the organizers for financial support to present a preliminary version of this paper.\nFFF acknowledges financial support from Funda\\c{c}\\~ao Amparo a Pesquisa do Estado de S\\~ao Paulo (FAPESP) and Funda\\c{c}\\~ao Instituto de F\\'isica Te\\'orica (FIFT) for hospitality. We thank Constantin Unanian for detailed comments.\n\n\n\\section{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe recent impressive advances of machine learning and in particular deep learning methods in solving several challenging tasks, came along with a vulnerability that questions the security of several applications. In fact, as shown in ~\\citet{Szegedy2013}, small and visually imperceptible perturbations of images, called \\emph{adversarial examples}, can lead to highly confident misclassification. Moreover, these \nadversarial examples can generalize across different networks ~\\citep{papernot2016practical, Universal_adv}, which gives the \\emph{adversary} the ability to fool artificial systems without any prior knowledge on their architecture and makes models deployed for real-world applications an easy target of adversarial attacks.\n\n\nA growing body of work is now devoted to designing adversarial attacks, quantifying the robustness of classifiers  and building effective defense mechanisms ~\\citep{Fawzi2015, Kurakin2016, papernot2016distillation, DeepFool, Ensemble_adv}. A standard approach known as \\emph{adversarial training}, suggested in \\citet{Goodfellow2015_adv}, consists in augmenting the training set with its adversarial version. More recent methods \\citep{mixup} try to encourage linear behavior between training examples, thus building some robustness to adversarial examples.\n\nIn this paper, we suggest a new approach to building some resistance to adversarial attacks by training a discriminator to distinguish latent representations of real samples from the ones of adversarial examples. The discriminator gets intermediate features of a classifier as input. At the same time, the classifier is trained not only to correctly classify the training data but also to fool the discriminator when fed an adversarial example. This enforces invariance of the hidden representation with respect to the nature of the input i.e whether it is a real sample or an adversarial examples. The classifier ends up filtering out the adversarial \\emph{noise} at the level of the hidden representation, leading the adversarial example to be classified as its corresponding real sample.    \n\nWe perform experiments on the MNIST datasets to validate the proof of concept. Our results suggest that our procedure compares well to standard adversarial training, and that combining the two methods may further improve robustness.  \n\n\n\\section{Related work}\n\n \nThe idea of augmenting a network with a discriminator to enforce hidden representation invariance is at the heart of the work \\citet{Ganin2016domainadaptation} on domain adaptation, where the network learns features that adapt to different domains for the same task.\nThis idea has been exploited in several recent works.\n\\citet{metzen2017detecting} uses this kind of architecture in the context of {\\it detection}: the discriminator, trained separately from the classifier, is merely  used as detector of adversarial attacks. \nBy contrast,  the role of the discriminator in our approach is to teach the classifier to build robust features, in a procedure similar in spirit to the GAN framework ~\\citep{Goodfellow2014GAN}.\n\n\nFader nets \\citep{lample2017fader} use an adversarial loss for an encoder to extract the invariant patterns that are orthogonal to some image attributes. This helped disentangling these attributes factors  and allowed controlling them in a generative model setting.\n\n\n\\citet{nuisances} takes the view from information theory  to a regularizing term added to the classifier loss, which filters out the information of a given class of  `nuisances', i.e variations of the inputs that are  irrelevant to the task. This regularizer takes the form of mutual information, which can be estimated by means of a discriminator. Our work can be viewed as a generalization of this proposal to the case of adversarial perturbations.\n\n\n\\section{Approach}\n\n\\subsection{Adversarial perturbations} \n\nConsider a classifier $C_\\theta \\colon \\mathcal{X} \\rightarrow \\mathcal{Y}$ \ntrained by minimizing  the average of a given loss function $\\mathcal{L}_\\theta(x, y)$ over labeled examples $(x, y)$ in a training dataset. \nAn \\emph{adversary} willing to fool the classifier may design input perturbations $\\bar{x} = x+\\delta(x)$ so as to maximize the loss: \n\\begin{equation} \\label{adv} \n\\delta(x) = \\argmax_{\\delta \\in \\mathcal{C}} \\mathcal{L}_\\theta(x+\\delta, y) \n\\end{equation}\nwhere $\\mathcal{C}$ encodes a set of constraints controlling the amplitude of the perturbations -- e.g small enough to remain quasi-imperceptible to a human observer. As shown in ~\\citet{Szegedy2013}, deep neural networks are highly unstable under such adversarial perturbations. \n\n \n\nThe simplest methods to generating perturbations defined in (\\ref{adv}) are first order approaches,  which linearly expand the loss \n$\\mathcal{L}_\\theta(x+\\delta, y)  \\simeq \\mathcal{L}_\\theta(x, y) + \\langle \\delta, \\nabla_x \\mathcal{L}_\\theta(x, y)\\rangle$. If the constraints $\\mathcal{C}$ take the form $\\|\\delta\\|_{\\infty} \\leq \\epsilon$ for some $\\epsilon >0$, \nEqu. (\\ref{adv}) yields the so-called  fast gradient method for generating adversarial examples ~\\citep{Goodfellow2015_adv}: \n\\begin{equation} \\label{sign_grad}\nx^{adv}  = x + \\epsilon \\mbox{sign} \\left(\\nabla_x \\mathcal{L}_ \\theta(x, y)\\right)\n\\end{equation}\nOther generation methods have been proposed, which include ~\\citet{Kurakin2016, DeepFool}. In our preliminary experiments, we use the procedure (\\ref{sign_grad}) to generate adversarial examples {\\it on-the-fly} during training. \n\n\n\\subsection{Augmented network}\nAssuming the classifier has $L$ layers, we choose a layer $\\ell$, which  effectively splits the network into an encoder part $E$ (layers $1, \\cdots \\ell$) and a residual classification part $R$ (layers $\\ell, \\ell+1, \\cdots L$). We supplement the network with a discriminator $D$ branched off from $\\ell$.  If  $\\ell$ has $k$ units, $D \\colon \\mathbb{R}^k \\rightarrow [0,1]$  takes as input the feature at $\\ell$ and produces an output interpreted as the probability of the feature coming from a real (as opposed to adversarial) input. \nGiven an input image $x \\in \\mathbb{R}^d$ we thus have the following notation: \n\\begin{itemize} \n\\item $z = E(x)$ is the feature vector at layer $\\ell$\n\\item $R(z) = R(E(x)) = C_\\theta(x)$ is the output of the original classifier  \n\\item $D(z)$ is the output of the discriminator, and the probability that $z$ is the representation of a sample of the real data. \n\\end{itemize} \nThe architecture is shown in Fig \\ref{fig:model_}. \n\n\\begin{figure}[h]\n\\centering \n\\includegraphics[width=9cm]{model_.pdf}\n\\caption{Model architecture}\n\\label{fig:model_}\n\\end{figure}\n\n\n\n\\subsection{Adversarially Augmented Training} \n\nAt each iteration, we forward-pass a mini-batch of real data $(x, y)$, then generate and forward pass the corresponding mini-batch of adversarial examples $(x^{adv}, y)$.\nClassifier and discriminator are trained simultaneously. \nThe discriminator is trained as a binary classifier where $z$ has tag $t=1$ \nwhen  $z = E(x)$ and $t=0$ when $z=E(x^{adv})$. The classifier is trained to classify each real input correctly (which involves both encoder and residual) and to fool the discriminator (which involves only the encoder). Training the classifier on the discriminator's response should enforce an invariance across real samples and their adversarial versions at the level of the latent representation $z$.\nThis can be combined with standard adversarial training for the correct classification of each adversarial input. \n\n\nLet $\\theta_{enc}, \\theta_{res}$ and $\\theta_{disc}$ the parameters of the encoder, the residual classifier and the discriminator respectively, and $\\theta=(\\theta_{enc}, \\theta_{res})$.  We use the following losses: \n\n\n\n{\\bf Classification Loss}. For each labeled sample $(x, y)$, the classifier outputs class probabilities $P_\\theta(\\cdot |x)$.  We use the cross-entropy loss for the classification task.  \nThis loss can be extended to include classical adversarial training \\citep{Goodfellow2015_adv}:  \n$$ \\mathcal{L}_{\\theta}(x,x^{adv}, y) = - \\alpha \\log P_\\theta(y|x) - (1-\\alpha)\\log P_\\theta(y|x^{adv}),$$\nwhere $\\alpha \\in [0, 1]$ is a convex mixing parameter (we used $\\alpha = 0.5$)\n\n\n{\\bf Discriminator Loss}. For each feature $z$ with $t$, the discriminator outputs the probability $D(z)=P_{\\theta_{disc}}(t=1|z)$.  We use binary cross-entropy to train the discriminator to distinguish the real features from the adversarial ones:\n$$\n\\mathcal{L}_{\\theta_{disc}}(z, t) = -  \\log P_{\\theta_{disc}}(t|z)  \n$$\n\n{\\bf Encoder Adversarial Loss}. We use the cross-entropy loss to train (the encoder part of) the classifier on the discriminator response, so as to make the adversarial representation be classified as real by the discriminator: \n$$ \n\\mathcal{L}^{adv}_{\\theta_{enc}}(x^{adv}) = -\\beta \\log D(E(x^{adv})),\n$$\nwhere $\\beta > 0$ is a factor controlling the importance of the adversarial loss i.e how much we enforce invariance (we used $\\beta = 1$)\n\n\n\n\n\n\\section{Experiments -- Proof of concept}\n{\\bf Setup.}\nThe reported experiments are based on a 3-layer feed-forward network using leaky ReLU as activations. The number of hidden units of its hidden layers are respectively 512, 256, 128. The discriminator is fed the output of the second hidden layer. The adversarial examples were generated using $\\epsilon = 0.25$.\n\nThe discriminator is a simple shallow network with 128 hidden units, a ReLU activation and a 0.5-dropout.\n\n{\\bf Training.}\nAt each iteration, we update the discriminator, and the classifier with adversarially augmented loss. One can try multiple updates of the discriminator for one update of the classifier as suggested in \\citet{Goodfellow2014GAN}, but we noticed that a single update gave already the expected training behavior.\n\nFigure \\ref{fig:model_curves} gives the classification accuracies reached by the classifier and the discriminator for real and adversarial examples in the training and validation dataset, as a function of the training epochs.  \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=13cm]{final_wkp_curves.pdf}\n\\caption{Mnist Adversarially Augmented Training}\n\\label{fig:model_curves}\n\\end{figure}\n\n\nTable \\ref{tab:mnist_test} gives the test performance of the same model trained with 4 different approaches: simple training of the classifier,  standard adversarial training (SAT), adversarially augmented training (A2T) and finally the adversarially augmented adversarial training  (A3T). \n\\begin{table}[h]\n\\centering\n\\begin{tabular}{ r || c | c }\n   \t\t\t\t\t& Accuracy on real & Accuracy on adversarial \\\\ \\hline\n  \t\t\t\t\tSimple training & 98.18\\% & 25.37\\% \\\\\n  \t\t\t\t\tAT ($\\alpha=\\frac12, \\beta=0$)  & 98.89\\% & 94.45\\% \\\\\n\n       \t\t\t\tA2T ($\\alpha=\\beta=1$) \\hspace{3mm} & 97.77\\% & 89.74\\%\\\\\n  \t   A3T ($\\alpha=\\frac12, \\beta=1$) & 98.72\\% &  {\\bf 96.10\\%} \\\\\n  \t\t\t\t\t\\hline  \n\t\t\t\t\t\\end{tabular}\n                    \\vspace{5mm}\n\t\t\t\t\t\\caption{Test accuracies for MNIST on real and adversarial samples ($\\epsilon=0.1$, 10 runs)}\n\\label{tab:mnist_test}\n\\end{table}\n\n\n\\vspace*{-0.7cm}\n\\section{Discussion}\n\n\n\nWe proposed a new  approach to enforce invariance of a classifier's features with respect to adversarial perturbations. We proved the efficacy of the method with some proof of concept experiments.\nMore work needs to be done to test the sensitivity of the model to the hyperparameters and the adversarial generation method.  A crucial point will also be  to test its scalability to larger datasets and architectures.  \n\nThe existence of adversarial examples  suggests that neural networks, despite their impressive generalization abilities in perceptual tasks, do not learn the true underlying concepts defining the correct labels. One could hope that a procedure seeking feature invariance would push the network to not only ignore adversarial perturbations but also to focus more on relevant patterns in real data. We conjecture that our approach is one step in that direction. \n\n\n\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{\\label{Intro} Introduction}\nGinzburg-Landau (GL) theories with several complex scalar matter fields minimally \ncoupled to one gauge field are of interest in a wide variety of condensed matter \nsystems and beyond. This includes such apparently disparate systems as the two-Higgs \ndoublet model \\cite{lee1973}, superconducting low temperature phases of light atoms \nsuch as hydrogen \\cite{Ashcroft1999,Neilnew} under extreme enough pressures to \nproduce liquid metallic states, and effective theories for easy-plane quantum \nantiferromagnets  \\cite{senthil2003,motrunich2004,sachdev2004}. Well known cases \nof  multicomponent systems are represented by  multiband  superconductors \\cite{suhl} \nlike $MgB_2$ where there are two order parameters corresponding to Cooper pairs\nmade up of electrons living on different sheets of Fermi surface. In that case however \ncondensates are not independently conserved and the $U(1)\\times U(1)$ symmetry is broken \nto $U(1)$, so the main results of this paper do not apply to multiband superconductors.\nIn contrast, in the projected liquid metallic state of  hydrogen \\cite{Ashcroft1999,Neilnew}, \nwhich appears being close to a realization in high pressure experiments \\cite{Datchi2000,Bonev},\nthe scalar fields represent Cooper pairs of electrons and protons. This excludes, on symmetry \ngrounds, the possibility of inter-flavor pair tunneling, i.e. there is no intrinsic  Josephson \ncoupling between  different species of the condensate. This sets it apart from systems with \nmulti-flavor {\\it electronic} condensates arising out of superconducting order parameters \noriginating on multiple-sheet Fermi surfaces, such as is the case in $MgB_2$. For the latter \nsystem, Josephson coupling in internal order parameter space {\\it cannot} be ruled out on \nsymmetry grounds, and must therefore be included in the description. This is so because the \nJosephson coupling represents a singular perturbation and can never be ignored on sufficiently \nlong length scales. This is otherwise well known from studies of extremely layered \nsuperconductors \\cite{clem}, where the critical sector is that of the 2D$xy$\\  model in the \nabsence of Josephson coupling, while any amount of interlayer phase-coupling (in an extended \nsystem) produces a critical sector belonging to the 3D$xy$\\  universality class.  {\\it It is \nprecisely the lack of Josephson coupling in certain, but by no means\nall systems with multiple \nflavor order parameters, that opens up the possibility of novel and interesting critical \nphenomena.} However, even in inter-band Josephson-coupled condensates, interesting physics \narises at finite length scales \\cite{egor2002,frac}.\n\nA two-component action with no Josephson coupling in $(2+1)$ dimensions, with matter fields \noriginating in a bosonic representation of spin operators, is also claimed to be the critical \nsector of a field theory separating a N{\\'e}el state and a paramagnetic (valence bond ordered) \nstate of a two dimensional quantum antiferromagnet at zero temperature with easy-plane \nanisotropy\\cite{senthil2003,sachdev2004}. This happens because, although the \neffective description of the antiferromagnet involves an  {\\it a priori} compact gauge field, \nit must be supplemented by Berry-phase terms in order to properly describe $S=1\/2$ spin \nsystems \\cite{haldane1988,read_sachdev}. Berry-phase terms in turn cancel the effects of \nmonopoles at the critical point \\cite{senthil2003,sachdev2004}. Hence, an effective \ndescription in terms of two complex scalar matter fields coupled to one {\\it non-compact} gauge \nfield suffices to describe the non-trivial quantum critical point separating a state with \nbroken {\\it internal} $SU(2)$ {\\it symmetry} and a paramagnetic $SU(2)$-symmetric state with \nbroken {\\it external symmetry} (lattice translational invariance). The latter state is the \nvalence-bond ordered state. Critical behavior separating states differing in this manner is \nnot captured by the Landau-Wilson-Ginzburg paradigm \\cite{senthil2003,senthil_science2004}, \nand requires a description of a phase transition without a local order parameter. An example \nof such a description is the well known Kosterlitz-Thouless phase transition taking place in \nthe 2D$xy$\\  model \\cite{KT1974}. The difference from the Kosterlitz-Thouless case and the quantum \ncritical behavior described above is that while the low temperature phase of the 2D$xy$\\  model is \na Gaussian fixed line, this is not so for either side of the quantum critical point of the \neasy-plane quantum antiferromagnet \\cite{senthil2003,motrunich2004,senthil_science2004,sachdev2004}. \nWe also mention that another example of a multicomponent system with no inter-component Josephson \neffect are spin-triplet superconductors which are well known to allow a variety of topological \ndefects and phase transitions \\cite{volovik}. Some of the topics we discuss below are \nrelated to the models of spin-triplet paired electrons \\cite{ssf}.\n\nSince the condensates described above in the context of light atoms and easy-plane \nquantum antiferromagnets are gauge-charged condensates, the order parameter flavors \nare all coupled to each other via a non-compact gauge field. This \ncoupling is vastly different from the Josephson coupling in the sense that \nwhile an $N$-flavor order parameter condensate with no coupling \nbetween different species in general will have $N$ phase transitions, a \nJosephson coupling between a pair of order parameter species will \ncollapse the two independent phase transitions they undergo with no coupling, \ndown to one. Josephson coupling between all pairs of order parameter species \nwill collapse all $N$ phase transitions down to a single one, namely an \ninverted 3D$xy$\\  transition. On the other hand, $N$ order parameter species\ncoupled to one and the same gauge field will still undergo in general \n$N$ phase transitions, namely one {\\it inverted} 3D$xy$\\  transition where \na Higgs phenomenon takes place, followed by $N-1$ 3D$xy$\\  transitions as the coupling \nconstants are increased beyond the Higgs\/3D$xy$\\  critical point \\cite{sachdev2004,smiseth2004}. \n\n\nA special feature is presented by the important case $N=2$. Here, it turns out \nthat the {\\it dual} description of the theory is isomorphic to  the starting \npoint \\cite{motrunich2004,sachdev2004,smiseth2004}. Normally, in $d=2+1$, a gauge \ntheory dualizes into a global theory and vice versa. In contrast a $U(1)\\times U(1)$ gauge theory \ndualizes into another $U(1)\\times U(1)$ gauge theory, i.e. the theories are {\\it self-dual}. \nIn general the theory has two separate critical points, one inverted 3D$xy$\\  and one 3D$xy$\\  \ncritical point \\cite{smiseth2004}. For the special case where the bare phase stiffnesses\nof the two matter fields are equal, as they naturally are in the case of easy-plane \nquantum antiferromagnets in the absence of an external magnetic field \n\\cite{motrunich2004,sachdev2004}, another interesting feature appears. \nIn this case, there is only one critical point separating two phases described by \n{\\it self-dual field theories}. This cannot be either an inverted 3D$xy$\\  or a 3D$xy$\\  \nfixed point. Self-duality also precludes the possibility of a $Z(2)$ universality \nclass although the exponent $\\nu$ that we find for this case appears to be close to \nthe Ising value (while $\\alpha$ is not). This phase transition therefore defines a \nnew universality class, namely that of the $d=2+1$ $U(1)\\times U(1)$ self-dual gauge theory. \n\nWhat happens to such multi-component charged condensates in three dimensions in the \nabsence of Josephson coupling between the order parameter components, but in the \npresence of an external magnetic field, has been recently studied in\nRefs. \\onlinecite{BSA},\\onlinecite{SSBS}  for the case $N=2$, with particular emphasis \non applications to liquid metallic hydrogen. In this paper, we extend on this and consider \nin detail the effects of tuning the external magnetic field and temperature when also \n$N \\geq 3$. New features appear compared to the $N=2$ case, because composite vortices \nconsisting of non-trivial windings in all order parameter components can now undergo partial\ndecompositions by tearing vortices  of individual order parameter components off the\ncomposite vortices, one after the other. We provide a dual picture of these processes: \n{\\it i)} as a vortex loop proliferation in the background of a composite vortex lattice, \nand {\\it ii)} as a  metal-insulator transition  in a system \nconsisting of several ``colors of electric charges'' in a multi color dielectric \nbackground. The new concept of ``color charge''  will be introduced and \nexplained in  detail in this paper. {\\it It allows us \nto determine the universality class, and the partially broken\nsymmetries of the partial decomposition\ntransitions taking place in multi flavor superconductors in an external magnetic field. }\n We also show that the  number of colors $N_{\\rm color}$ of dual charges exceeds\nthe number of field components (flavors) for $N>3$.\n\nThe outline of the paper is as follows. The first six sections of the paper deal with results\nin zero external magnetic field. In Sections VII and VIII  we present results \nin finite magnetic field. Readers who wish to consult results on finite magnetic\nfield may proceed directly to Section VII. \n\nIn Section \\ref{model}, we introduce the model and the main approximation we will use to \nstudy the model, as well as the duality transform that will be used extensively, along \nwith the explicit vortex representation of the model. In Section \\ref{Neutral_modes}, we \nexplicitly transform the action for the $N=2$ case into an action consisting of two parts: \n{\\textit{i})} one charged vortex mode with vortex interactions mediated by a massive vector \nfield, and {\\textit{ii})} one neutral vortex mode with vortex interactions mediated by a \ngauge field. In Section \\ref{gauge_corr}, we compute gauge field correlators and dual gauge \nfield correlators in terms of vortex correlators. {\\it This explicitly identifies the mechanism \nby which a thermally driven vortex loop proliferation destroys the Higgs phase (Meissner effect) \nand dual Higgs phase} \\cite{hove2000,smiseth2004}. Gauge field correlators are useful in \ncharacterizing the charged fixed point of the $N$-flavor London model \\cite{hove2000,smiseth2004}, \nwhile dual gauge field correlators are also useful in characterizing the $N-1$ neutral fixed points \n\\cite{smiseth2004}. In Section \\ref{mc_simsN=2}, we present large-scale Monte Carlo (MC) simulations \nfor the case $N=2$, computing critical exponents at the neutral and charged fixed points, as well \nas the mass of the gauge field as\na function of temperature. The neutral fixed point is found to be in the 3D$xy$\\  universality class, \nwhile the charged fixed point is shown to be in the inverted 3D$xy$\\  universality class. We also \nconsider in detail the case when the two bare phase stiffnesses of the model are identical, \nshowing that the resulting one fixed point is in a new universality class distinct from\nthe 3D$xy$\\  and inverted 3D$xy$\\  universality classes. In Section \\ref{mc_simsN=3}, we  present \ncorresponding results for the case $N=3$. In Section \\ref{N=2_extfield}, we outline the phases \nto expect for the case $N=2$ when an external magnetic field is applied. We also present\nresults from large-scale Monte-Carlo simulations revealing a novel phase transition\nin the 3D$xy$\\  universality class inside the Abrikosov vortex lattice phase at low magnetic\nfields when temperature is increased. In Section \\ref{N>2_extfield}, \nwe do the same when $N > 2$, emphasizing the qualitatively new features compared to the case \n$N=2$. We also introduce  a useful ``color charge\" picture of the various partial\ndecomposition transitions of the composite vortex lattice that we encounter\nfor the case when $N \\geq 3$. In Section \\ref{Concl}, we summarize our results. In \nAppendix \\ref{App:charged_neutral}, we identify charged an neutral vortex modes for \ngeneral $N$. In Appendix \\ref{App:vortex}, we derive the vortex representation\nfor the general-$N$ case. In Appendices \\ref{App:gauge} and \\ref{App:dualgauge}, \nwe derive expressions for gauge field correlators and dual gauge field correlators,\nrespectively. In Appendix \\ref{App:josephson} we generalize our dual representation\nfor arbitrary $N$ to also include inter-flavor Josephson coupling.  In Appendix \n\\ref{App:bkt}, we consider Kosterlitz-Thouless transitions for the \ngeneral-$N$ case in two spatial dimensions at finite temperature. \n\n\n\n\\section{\\label{model} Model and dual action}\nFor an  analysis of the possible phase transitions in a  GL model of $N$ individually \nconserved bosonic  matter fields, each coupled to one and the same  $U(1)$ non-compact \ngauge field, we study a version of the  $N$-flavor GL theory in $(2+1)$ dimensions \n{\\it with no Josephson coupling terms} between order parameter components. Moreover,\nwe ignore mixed gradient terms, such that there is  no Andreev-Bashkin effect \\cite{AB}.\nThe model  is defined by $N$ complex scalar fields \n$\\{\\Psi_0^{(\\alpha)}(\\ve r) \\ | \\ \\alpha=1,\\dots,N\\}$ coupled \nthrough the charge $e$ to a fluctuating gauge field $\\ve{A}(\\ve r)$, with the action\n\\begin{equation}\n\\begin{split}\n\\label{gl_action}\nS =& \\int d^3\\ve r \\left[\\sum_{\\alpha=1}^N \\frac{|(\\nabla -{\\rm i} e\\ve{A}(\\ve r))\\Psi_0^{(\\alpha)}(\\ve r)|^2}{2 M^{(\\alpha)}}\\right. \\\\\n&+ \\left.V(\\{\\Psi_0^{(\\alpha)}(\\ve r)\\}) + \\frac{1}{2}(\\nabla\\times\\ve{A}(\\ve r))^2\\right],\n\\end{split}\n\\end{equation}\nwhere $M^{(\\alpha)}$ is the mass of the condensate species\n$\\alpha$. \nAssuming that the individual condensates are conserved, the potential \n$V(\\{\\Psi_0^{(\\alpha)}(\\ve r)\\})$ must be function of \n$|\\Psi_0^{(\\alpha)}(\\ve r)|^2$ only. \nIn this paper, we focus on  the critical phenomena and phase diagram of Eq. (\\ref{gl_action})\nin zero as well as finite external magnetic field, and for these purposes the model \nin Eq. (\\ref{gl_action}) will be studied in the phase-only approximation \n$\\Psi_0^{(\\alpha)}(\\ve r) = |\\Psi^{(\\alpha)}_0|\\exp [ {\\rm i}\\theta^{(\\alpha)}(\\ve r) ]$ \nwhere $|\\Psi^{(\\alpha)}_0|$ is a constant, i.e. we freeze out amplitude fluctuations \nof {\\it each individual matter  field}. The model we study is therefore the generalization \nto arbitrary $N$ of the frozen-amplitude one gap lattice superconductor model also known \nas the London superconductor model \\cite{dasgupta1981}. \n\nOne may well ask what confidence one should put in the phase only approximation\nfor all fields when the bare phase stiffness of each individual condensate is \nvery different, such as is the case in LMH. The answer is that one can be quite \nconfident that this is a useful and reasonable approximation. Consider first\nthe case $N=2$. We use the phase only approximation with confidence for considering \nthe criticality here. It certainly works at the lowest critical temperature.  After \nthat point, we are left with a one-component superconductor. What the field with the \nlowest phase stiffness does above the lowest critical temperature is not of interest, \nit is only the remaining field with criticality at higher temperature that matters. \nHence, significantly above the lowest critical temperature, we may still apply the \nphase only approximation for the remaining one-component case. For this field, we may \nuse the phase only approximation up to the highest critical temperature with the same \nconfidence as we can use the phase only approximation for the field with the lowest \nphase stiffness up to and slightly above the lowest critical temperature. The same \nargument can  be repeated for arbitrary $N$:  We can use the phase only approximation for \nthe fields up to and slightly above their respective critical temperatures. After \nthat it is immaterial what they do, it is only the remaining components that matter. \n\n\\subsection{Basic properties of the model}\nVarying \\eq{gl_action} with respect to ${\\bf A}$, we obtain the equation for the \nsupercurrent\n\\begin{eqnarray} \n{\\bf J} \n&=& \n \\sum_{\\alpha=1}^N\n\\frac{{\\rm i} e  }{2 M^{(\\alpha)}}\n\\left\\{{\\Psi_0^{(\\alpha)}}^* \\nabla \\Psi_0^{(\\alpha)}-\n\\Psi_0^{(\\alpha)} \\nabla {\\Psi_0^{(\\alpha)}}^* \\right\\}\n\\nonumber \\\\&&\n-2 e^2 \\left( \\frac{|\\Psi_0^{(\\alpha)}|^2}{M^{(\\alpha)}} \n\\right) {\\bf A}. \\\n\\label{AA}\n\\end{eqnarray}\n Vortex excitations in such an $N$-flavor GL model carry fractional flux.\n Consider a vortex\n where the phase  $\\theta^{(\\eta)}(\\ve r)$  has a $2\\pi$\n winding around a vortex core, while other phases do\n not have nontrivial windings. Expressing ${\\bf A}$\nfrom  \\eq{AA}, and integrating \nalong a path around the vortex core at a distance larger than the\n magnetic penetration length, \n we obtain an expression for the magnetic \n flux encompassed by the path given by \n \\begin{eqnarray}\n \\Phi^{(\\eta)}=\\oint {\\bf A} d {\\bf l} =\n\\Phi_0 \n\\frac{|\\Psi_0^{(\\eta)}|^2}{M^{(\\eta)}}\n\\left[ \\sum_{\\alpha=1}^N\\frac{\n|\\Psi_0^{(\\alpha)}|^2}{M^{(\\alpha)}}\n\\right]^{-1},\n\\label{Phi}\n \\end{eqnarray}\n where $\\Phi_0 = 2.07 \\cdot 10^{-15} {\\rm Tm}^2$ is the flux quantum. As it will be clear\n from a discussion following \\eq{potential_0} (see \\eq{potential}), such a vortex has \n a logarithmically divergent energy\\cite{frac,smiseth2004}. Only a {\\it composite}\n vortex where all phases $\\theta^{(\\alpha)}$ have $2\\pi n$ winding around the core \n carries integer flux and has finite energy. As detailed below,  the composite\n vortices are responsible for the magnetic properties of the\n system at low temperatures while thermal excitations\nin the form of loops of  individual fractional-flux vortices are\nresponsible for the critical properties of the system in the\nabsence of an external field.\n\n{\\it Note that since each individual amplitude is frozen, this model will \nbe different from the case where only the sums of the squares of the \namplitudes are frozen} \\cite{Nogueira1}. The latter is usually referred to \nas the $N$-component scalar QED $(NSQED)$ \\cite{Coleman_Weinberg,HLM}, or \nthe CP${}^{N-1}$  model \\cite{Hikami}. (As far as critical properties are \nconcerned, the NSQED model and the CP${}^{N-1}$ model\nhave been shown to belong to the same universality class \\cite{Hikami}). \nWe strongly emphasize that we must distinguish our model from NSQED \nand CP${}^{(N-1)}$, and will consequently be referring to it\nas the $N$-flavor London superconductor (NLS) model.   \nThe NLS is in fact the natural model to consider for the physical\nsystems mentioned in the introduction, in particular\npertaining to the superconducting mixtures of metallic\nphases of light atoms. As we shall see, the NLS model\nhas physics which sets it distinctly apart from the\nNSQED and the CP${}^{N-1}$ \nmodels, and it does not have critical properties in the same\nuniversality class as they do. This becomes particularly\napparent in the large-$N$ limit, as we shall see in section \\ref{vort_rep}. \n\n\\subsection{Separation of variables}\nBefore we proceed further, it is useful to give another form of the action. \nFor brevity we introduce the bare phase stiffness of the matter field with\nflavor index $\\alpha$ defined as $|\\psi^{(\\alpha)}|^2 =\n|\\Psi^{(\\alpha)}_0|^2\/M^{(\\alpha)}$. \nThen  \\eq{gl_action} may be rewritten\nin terms of {\\it one} charged and $N-1$ neutral modes as follows (details of this \nare found in Appendix \\ref{App:charged_neutral}). We have \n$S = \\int d^3 \\ve r {\\cal{L}}$, with \n\\begin{eqnarray}\n{\\cal L} & = & \\frac{1}{2 \\Psi^2}  \\left( \n\\sum_{\\alpha = 1}^N |\\psi^{(\\alpha)}|^2 \\nabla \\theta^{(\\alpha)}  - e \\Psi^2 \\ve A \n\\right)^2  + \\frac{1}{2}(\\nabla\\times\\ve{A})^2 \\nonumber \\\\\n& &+  \\frac{1}{4 \\Psi^2}  \\sum_{\\alpha,\\beta=1}^N |\\psi^{(\\alpha)}|^2  |\\psi^{(\\beta)}|^2 \n \\left( \\nabla (\\theta^{(\\alpha)}- \\theta^{(\\beta)}) \\right)^2,\n \\label{charge_neutral}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{defpsi2}\n \\Psi^2 \\equiv \\sum_{\\alpha=1}^N |\\psi^{(\\alpha)}|^2. \n \\end{eqnarray}\nThe first term in \\eq{charge_neutral} represents the charged mode\n coupling to the gauge \nfield $\\ve A$, and the remaining terms are the $N-1$  neutral modes which do not couple \nto $\\ve A$. This means that they have gauge charge equal to zero. We will come back to \nthis in Section \\ref{Neutral_modes}. This form \\eq{charge_neutral} will be useful later \nwhen we discuss finite field effects in section \\ref{N>2_extfield}. We \nalso stress that $\\Psi$ in the above expression should not be confused with \n$\\Psi_0^{(\\alpha)}$ defined in \\eq{gl_action}.\n\nCounting degrees of freedom  in \\eq{charge_neutral} requires\ncare.  The case $N=1$  yields the well known answer that \na phase variable (which is not a gauge invariant quantity) is higgsed into \na massive vector field by coupling to the vector potential. In the case $N=2$,  the \nsituation is different in the  sense that one can form a gauge invariant \nquantity by subtracting phase gradients. Thus the $U(1)\\times U(1)$ system may \nbe viewed as possessing {\\it i}) a local $U(1)$ gauge symmetry associated with the phase \nsum which is  coupled to the vector potential and thus yields  a massive vector field, \nand {\\it ii}) a global  $U(1)$ symmetry which is associated with a phase difference where \nthere is no coupling to the vector potential. These charged and neutral modes \nare naturally described by the first and third terms in \\eq{charge_neutral}, respectively. \nFor  $N=3$, the situation is  principally different from both the $N=1$ and $N=2$ cases. That \nis, in \\eq{charge_neutral} for $N=3$, we find one term describing the charged mode  (the \nfirst term) and {\\it three} terms describing gauge-invariant neutral phase combinations.\n\nThe two neutral modes in Eq. (\\ref{charge_neutral}), in the $N=3$ case,\ncannot be properly described by only two terms, for\ntopological reasons. A vortex excitation produces a zero in the \norder parameter space, thus making the superconductor  multiply connected.\nA vortex with a non-trivial phase winding in any of the three components\nwould result in non-trivial contributions to \ntwo of three phase-difference terms in Eq. (\\ref{charge_neutral}).\n{\\it Hence, for $N=3$ an elementary vortex {\\it i.e.} with nontrivial\nwinding only in one of the phases excites two neutral modes}.\nIn general, when all $|\\psi^{(\\alpha)}|$ differ, the bare phase stiffnesses of two neutral modes\nexcited by each of the three possible elementary vortices, \nare different. Thus, the neutral modes in the system are described by  \nthree  phase-difference terms in Eq. (\\ref{charge_neutral}).\n{\\it These three \nterms are not independent when the condition of single-valuedness of each of\nthe $N$ order parameter components is enforced}, namely that individual phases\nmay change only by integer multiples of $2\\pi$ around zeroes of the order parameters.\n\nUsing  Eq. (\\ref{charge_neutral}) as opposed to Eq.  (\\ref{gl_action}), has \nadvantages, because the neutral and charged modes are explicitly identified.\nThis facilitates a discussion of the critical properties of the $N$-flavor system.\nMoreover, Eq. (\\ref{charge_neutral}) will allow us to identify various \nstates of {\\it partially} broken symmetry which emerge if an $N$-flavor system is \nsubjected to external magnetic field \\cite{BSA}. We will come back to these \npoints in detail in Sections \\ref{N=2_extfield} and \\ref{N>2_extfield}.\n\n \n\n\\subsection{The Villain approximation}\nThe theory Eq. (\\ref{gl_action}) is discretized on a $d=3$ dimensional\ncubic lattice with spacing\n$a=1$ and size $L^3$, and in the phase only approximation the action reads\n\\begin{eqnarray}\nS &=& \\sum_{\\ve r}\\left[-\n  \\beta\\sum_{\\alpha=1}^N|\\psi^{(\\alpha)}|^2\\sum_{\\mu=1}^3\\cos(\\Delta^\\mu\\theta^{(\\alpha)}(\\ve\n  r) - eA^\\mu(\\ve r)) \\right. \\nonumber\\\\\n& &+ \\left. \\frac{\\beta}{2}(\\nabla\\times\\ve{A}(\\ve r))^2\\right]. \n\\end{eqnarray}\nHere, we have included the inverse temperature coupling $\\beta = 1\/T$. The symbol $\\Delta^\\mu$ denotes the lattice difference\noperator in direction $\\mu$ in Euclidean space and the position vector $\\ve r$ runs over\nall points on the lattice. The partition function in the Villain approximation is \n\\begin{equation}\n\\begin{split}\n  \\label{villain}\nZ =& \\int_{-\\infty}^{\\infty}\\mathcal{D}\\ve A\\prod_{\\gamma=1}^N\\int_{-\\pi}^{\\pi}\\mathcal{D}\\theta^{(\\gamma)}\\prod_{\\eta=1}^N\\sum_{\\ \\ve n^{(\\eta)}}\\exp(-S) \\\\\nS =& \\sum_{\\ve r}\\left[\\sum_{\\alpha=1}^N\\frac{\\beta|\\psi^{(\\alpha)}|^2}{2}(\\Delta\\theta^{(\\alpha)} - e\\ve{A} +2\\pi \\ve n^{(\\alpha)})^2\\right. \\\\\n&\\left.+\\frac{\\beta}{2}(\\Delta\\times\\ve{A})^2\\right],\n\\end{split}\n\\end{equation}\nwhere  $\\ve n^{(\\alpha)}(\\ve r)$ are integer vector fields\nensuring $2\\pi$ periodicity, and the lattice position index vector\n$\\ve r$ is suppressed. {\\it Here, we stress the importance of keeping \ntrack of the $2\\pi$ periodicity of the individual phases}. For $N=1$ it has been\nshown that thermal fluctuations in this model excite topological\ndefects in form of closed vortex loops. At the critical temperature\nthe system undergoes a vortex loop\nproliferation phase transition\\cite{kleinert_book,tesanovic1999,nguyen1999}. \n\n\\subsection{\\label{vort_rep}Vortex representation}\n\nIn the following, we transform the model Eq. (\\ref{villain}) into a theory of interacting vortex \nloops of different flavors. The procedure is described in detail in Appendix \\ref{App:vortex}. \nThe kinetic energy terms are linearized by introducing $N$ auxiliary\nfields $\\ve v^{(\\alpha)}(\\ve r)$. \nApplying the Poisson summation formula and integrating over $\\ve n^{(\\alpha)}(\\ve r)$ constrains the \nfields $\\ve v^{(\\alpha)}(\\ve r)$ to take only integer values $\\ve{\\hat v}^{(\\alpha)}(\\ve r)$. Integration over \nall $\\theta^{(\\alpha)}(\\ve r)$ produces the local constraints $\\Delta\\cdot\\ve{\\hat v}^{(\\alpha)}(\\ve r)=0$, \nwhich are fulfilled by replacing \n$\\ve{\\hat v}^{(\\alpha)}(\\ve r)$ with $\\Delta\\times\\ve{\\hat h}^{(\\alpha)}(\\ve r)$ \nwhere $\\ve{\\hat h}^{(\\alpha)}(\\ve r)$ are integer-valued fields. \nBy applying the Poisson summation once more and summing over \nall $\\ve{\\hat h}^{(\\alpha)}(\\ve r)$, the fields $\\ve{\\hat  h}^{(\\alpha)}(\\ve r)$ \ntake continuous values $\\ve h^{(\\alpha)}(\\ve r)$ and the integer-valued vortex \nfields $\\ve m^{(\\alpha)}(\\ve r)$ are introduced. We recognize $\\ve h^{(\\alpha)}(\\ve r)$ as \nthe dual gauge fields of the theory. To preserve the gauge symmetry of\n$\\ve h^{(\\alpha)}(\\ve r)$ each vortex field of flavor index $\\alpha$\nis constrained by the condition \n\\begin{equation}\n\\label{closedloops}\n\\Delta\\cdot\\ve m^{(\\alpha)}(\\ve r)=0. \n\\end{equation}\nHence, the vortex fields form closed loops. At this stage, the action reads\n\\begin{equation}\n\\begin{split}\n  \\label{dual1}\nS =&\\sum_{\\ve r}\\left[\\sum_{\\alpha=1}^N\\frac{(\\Delta\\times\\ve\n    h^{(\\alpha)})^2}{2\\beta|\\psi^{(\\alpha)}|^2} - {\\rm i} e\\ve\n  A\\cdot\\left(\\sum_{\\alpha=1}^N\\Delta\\times\\ve h^{(\\alpha)}\\right) \\right.\\\\\n&+ \\left.2\\pi{\\rm i}\\sum_{\\alpha=1}^N \\ve m^{(\\alpha)}\\cdot\\ve h^{(\\alpha)} + \\frac{\\beta}{2}(\\Delta\\times\\ve A)^2\\right],\n\\end{split}\n\\end{equation}\nwhere the vortex fields $\\ve m^{(\\alpha)}(\\ve r)$ are constrained by Eq. (\\ref{closedloops}). We integrate \nout the gauge field $\\ve A(\\ve r)$ and get a theory in the dual gauge fields  $\\ve h^{(\\alpha)}(\\ve r)$ \nand the vortex fields $\\ve m^{(\\alpha)}(\\ve r)$ \\cite{smiseth2004}\n\\begin{equation}\n\\begin{split}  \n\\label{dual2}\nS = \\sum_{\\ve r}&\\left[2\\pi{\\rm i}\\sum_{\\alpha=1}^N \\ve m^{(\\alpha)}\\cdot\\ve h^{(\\alpha)}+ \n\\sum_{\\alpha=1}^N\\frac{(\\Delta\\times\\ve h^{(\\alpha)})^2}{2\\beta|\\psi^{(\\alpha)}|^2}\\right.\\\\\n+&\\left. \\frac{e^2}{2\\beta}\\left(\\sum_{\\alpha=1}^N\\ve h^{(\\alpha)}\\right)^2\\right].\n\\end{split}\n\\end{equation}\nThis  generalizes to arbitrary $N$ the results of Peskin \\cite{Peskin}, and Thomas and \nStone \\cite{Thomas_Stone}. In Appendix \\ref{App:josephson} we generalize this result even \nfurther by including inter-flavor Josephson coupling.\n\nWhen $N \\geq 2$ there is an important difference from the $N=1$ case, which gives \nrise to entirely novel physics. Note how it is the {\\it algebraic sum} of the dual photon fields \nin \\eq{dual2} that is massive. This differs from the case $N=1$, where $e$ produces one massive dual \nphoton with bare mass $e^2\/2$, and the model describes a vortex field $\\ve m(\\ve r)$ interacting through a \n\\textit{massive} dual vector field $\\ve h(\\ve r)$. However, when $N\\ge 2$,  since $\\Delta\\cdot\\ve m^{(\\alpha)}(\\ve r)=0$, \na gauge transformation $\\ve h^{(\\alpha)}(\\ve r)\\to \\tilde{\\ve h}^{(\\alpha)}(\\ve r) = \\ve h^{(\\alpha)}(\\ve r) + \\Delta\ng^{(\\alpha)}(\\ve r)$ for $\\alpha\\in [1,\\dots ,N]$ \nleaves the action in \\eq {dual2} invariant if one of the gauge fields, say $\\tilde{\\ve h}^{(\\eta)}(\\ve r) $ \ncompensates the sum in the last term in the action with $\\Delta g^{(\\eta)}(\\ve r) =\n-\\sum_{\\gamma\\neq\\eta}\\Delta  g^{(\\gamma)}(\\ve r)$. \nThus, even in the presence of a gauge charge $e$, such that the direct model is a gauge \ntheory, the dual description is such that the individual dual photon fields are also \ngauge fields. \n\n\nIntegrating out the dual gauge fields we get a generalized theory\nof vortex fields of $N$ flavors interacting through the potential $D^{(\\alpha,\\eta)}(\\ve r)$\n\\begin{equation}\n  \\label{vortex_action}\n\\begin{split}\nZ &= \\prod_{\\alpha=1}^N ~ \\sum_{\\ve m^{(\\gamma)}} ~\\delta_{\\Delta\\cdot\\ve m^{(\\gamma)},0} ~ {\\rm e}^{-S_{\\ve V}} \\\\\nS_{\\rm V} &= \\pi^2 \\sum_{\\ve r,\\ve r^\\prime}\\sum_{\\alpha,\\eta}\\ve m^{(\\alpha)}(\\ve r)D^{(\\alpha,\\eta)}(\\ve r - \\ve r^\\prime)\\ve m^{(\\eta)}(\\ve r^\\prime),\n\\end{split}\n\\end{equation}\nwhere $\\delta_{x,y}$ is the Kronecker-delta, and the discrete Fourier\ntransform of the vortex interaction potential is\n$\\widetilde{D}^{(\\alpha,\\eta)}(\\ve q)$, given by \\cite{smiseth2004}\n\\begin{equation}\n\\label{potential}\n\\frac{\\widetilde{D}^{(\\alpha,\\eta)}(\\ve q)}{2 \\beta|\\psi^{(\\alpha)}|^2} = \n\\frac{\\lambda^{(\\eta)}}{|\\ve Q_{\\ve q}|^2 + m_0^2} + \\frac{\\delta_{\\alpha,\\eta} \n- \\lambda^{(\\eta)}}{|\\ve Q_{\\ve q}|^2},\n\\end{equation}\nwhere $\\lambda^{(\\alpha)} \\equiv |\\psi^{(\\alpha)}|^2\/\\Psi^2$, and\n$\\Psi^2$ is given by Eq. (\\ref{defpsi2}). Here, the bare mass $m_0$ is\nthe inverse bare screening length given by $m^2_0 = e^2 \\Psi^2$, and $|\\ve Q_{\\ve q}|^2 =\n\\sum_{\\mu=1}^3 (2\\sin(q^\\mu\/2))^2$ is the \nFourier representation of the lattice Laplace operator, where\n$q^\\mu=2\\pi n^\\mu\/L$ with $n^\\mu\\in [1,\\dots,L]$. Note that\n$\\sum_{\\alpha} \\lambda^{(\\alpha)} =1$. Note also that when $e^2=0$,\nthe interaction matrix reduces to \n\\begin{equation}\n\\label{potential_0}\n\\widetilde{D}^{(\\alpha,\\eta)}(\\ve q) = 2 \\beta|\\psi^{(\\alpha)}|^2\n \\frac{\\delta_{\\alpha,\\eta}}{|\\ve Q_{\\ve q}|^2}.\n\\end{equation}\nThis means that when there is no charge coupling the matter fields to \na {\\it fluctuating} gauge field, there is no interaction between\nvortices of different flavors. \nThis simple case corresponds to Eq. (\\ref{gl_action}) \nrepresenting a system of $N$ decoupled 3D$xy$\\  models. Also note that for vortices\nof different flavors, $\\eta \\neq \\alpha$, when $e \\neq 0$,  the interaction matrix\ntends to vanish when the inter-vortex distance is much smaller\nthen the effective penetration length $\\lambda = 1\/m_0$. It follows from\nthe fact that when the inter-vortex distance is much smaller than\n$\\lambda$, the vortices interact as if $\\ve A$ does not screen, i.e.\nas if $\\ve A$  does not fluctuate. In this case, it is clear that the \naction we describe is simply that of $N$ decoupled 3D$xy$\\  models, i.e. \ninter-flavor interactions vanish, cf. Eq. (\\ref{potential_0}).\nFor instance, for the case $N=2$,\nthere will be no interactions between vortices of condensate\n$\\Psi_0^{(1)}$ and \nvortices of the condensate $\\Psi_0^{(2)}$ unless we allow the gauge field to \nfluctuate. In the extreme type-II limit where $\\lambda \\to \\infty$ only intra-flavor interactions between \nvortices will exist (see also Ref. \\onlinecite{npb}). \n\n\n\nThe first term of the vortex interaction potential Eq. (\\ref{potential}) is a Yukawa \nscreened potential, {\\it while the second term mediates long range Coulomb \ninteractions between vortex fields}. If $N=1$ \nthe latter cancels out exactly and we are left with the well studied vortex theory of the GL model \nwhich has a charged fixed point for $e \\neq 0$ \\cite{Herbut_Tesanovic1996,hove2000}. For $N\\ge2$ \nwe find a theory of vortex loops of $N$ flavors interacting through long range Coulomb with an \nadditive screened part. If the number of species $N$ grows to infinity and $\\Psi^2 \\to \\infty$, \nthe vortex interaction receives the \ndominant contribution from a diagonal unscreened $N \\times N$ Coulomb matrix. \nBut there are physical situations where off-diagonal interactions play an\nimportant role even in the large-$N$ limit (to be discussed below).\nOne can also observe from Eq. (\\ref{Phi}) that in the $N \\to \\infty$ limit when all components \nhave similar stiffness the magnetic flux enclosed by elementary vortices also tends to zero. \nThus, for $N \\to \\infty$ the physics of the model is governed by\nneutral modes only.\n\nThe energy density of one straight vortex line of flavor $\\alpha$ in a distance $\\ve r$ \nlarger than the effective penetration depth $\\lambda$ is found by integrating along the \nline using the last term in the potential \\eq{potential} only\\cite{Fossheim_Sudbo_book}. \nThis produces an energy term  of the form $D({\\ve{r}}) \\sim \\ln( |\\ve r|)$, and shows that \nsuch a vortex has logarithmically divergent energy. \n\nThe large-$N$ limit of the NLS serves to illustrate how different the physics is from the \nlarge-$N$ limit of the NSQED model and the CP${}^{N-1}$ model \\cite{HLM,Hikami}. In the \nlarge-$N$ expansion of the NSQED model, only one charged fixed point is found (which is \ninfrared stable provided $2N > 365$), with critical exponent $1\/\\nu = 1+48\/N+\\dots$ in $D=3$ \n\\cite{HLM}. This is consistent with the results found in the large-$N$ limit of the\nCP${}^{(N-1)}$ model \\cite{Hikami}. The origin of the difference between these results and \nthe results we find for the NLS model is easily traced  to the following fact. The treatment \nof the NSQED model in Ref. \\onlinecite{HLM} is strictly speaking correct only in the case \nof type-I superconductivity, since they find that for physical values of $N$, only a first \norder phase transition from a superconductor to a normal metal takes place (no infrared stable \n{\\it fixed point} is found for physical values of $N$). This is correct only for values of the \nGinzburg-Landau parameter $\\kappa < 0.8\/\\sqrt{2}$, as has been shown\nin recent large-scale MC simulations  \\cite{Mo2002} and in earlier\nanalytical treatments \\cite{kleinert_book}.   \nThe transitions discussed below where neutral modes appear do not \nsignificantly  depend on whether the system is type-I or type-II. Our results are therefore best \nthought of as generalizations to arbitrary $N$ of the problem studied many years ago by Dasgupta \nand Halperin on the frozen-amplitude $N=1$ lattice superconductor model \\cite{dasgupta1981}. It \nis this fact that in the present model the modulus of each component is fixed, along with the \n{ precise absence of internal Josephson coupling} between matter field species, that brings out \nthe novel physics we shall describe, namely the {\\it charge-neutral superfluid modes arising out \nof $N$ charged condensate fields}. \n\n\\subsection{Dual field theory}\nStarting from \\eq{dual2} the above vortex system may be formulated as a field theory, introducing $N$ complex \nmatter fields $\\phi^{(\\alpha)}$ for each vortex species, minimally coupled to the dual \ngauge fields $\\bf{h}^{(\\alpha)}$. This generalizes the dual theory for $N=1$ \nin Refs. \\onlinecite{Thomas_Stone,kleinert_book}. The theory reads \\cite{smiseth2004} \n(for a comment on the case of general $N$, see also bottom of page 42, \nRef. \\onlinecite{sachdev2004})\n\\begin{equation}\n\\begin{split}\nS_{\\rm{dual}}  =& \\sum_{\\ve r} \\left[ \\sum_{\\alpha=1}^N \\left( m_\\alpha^2 |\\phi^{(\\alpha)}|^2 + \n|(\\Delta - {\\rm i} \\ve h^{(\\alpha)})\\phi^{(\\alpha)}|^2 \\right.\\right.\\\\\n&+ \\left.\\frac{(\\Delta\\times\\ve h^{(\\alpha)})^2}{2\\beta|\\psi^{(\\alpha)}|^2} \\right) \n +  \\frac{e^2}{2\\beta} \\left( \\sum_{\\alpha=1}^N\\ve h^{(\\alpha)}\n \\right)^2 \\\\\n&+ \\left.\\sum_{\\alpha,\\eta} g^{(\\alpha, \\eta)} |\\phi^{(\\alpha)}|^2 |\\phi^{(\\eta)}|^2 \\right].\n\\end{split}\n\\label{S_dual}\n\\end{equation}\nHere, we have added chemical potential (core-energy) terms for the vortices, as well \nas steric  short-range repulsion interactions between vortex elements. In the $N=1$ \ncase, a RG treatment of the term $\\frac{e^2}{2\\beta}\\ve h^2$ yields \n\\begin{equation}\n\\label{rg_flow}\n\\frac{\\partial e^2}{\\partial \\ln l} = e^2,\n\\end{equation}\nand hence this term\nscales up, suppressing the dual vector field $\\ve h$. The charged \ntheory in $d=2+1$ therefore dualizes into a $|\\phi|^4$ theory and vice versa \\cite{hove2000}.\nCorrespondingly, for $N \\geq 2$, \\eq{rg_flow} suppresses $\\sum_{\\alpha=1}^N\\ve h^{(\\alpha)}$, but\nnot each individual dual gauge field. For the particular case $N=2$, assuming the same \nto hold, we end up with a gauge theory of two complex matter fields coupled minimally to \none gauge field, which was also precisely the starting point. Thus the theory is \nself-dual for $N=2$ \\cite{motrunich2004,sachdev2004}. \n\n\\section{\\label{Neutral_modes} Charged and neutral vortex modes}\nIn this section, we present a straightforward method of \nidentifying charged and neutral vortex modes for the\nmodel Eq. (\\ref{gl_action}). Consider first the case $N=2$,\nwhen the action Eq. (\\ref{dual2}) reads\n\\begin{widetext}\n\\begin{eqnarray}\nS = \\sum_{\\ve r} \\left \\{ 2 \\pi {\\rm i} \\left[ \\ve m^{(1)}\\cdot \\ve h^{(1)} +  \\ve m^{(2)}\\cdot \\ve h^{(2)} \\right]\n+\\frac{1}{2 \\beta} \n\\left[ \\frac{(\\ve \\nabla \\times \\ve h^{(1)})^2}{|\\psi^{(1)}|^2} \n     + \\frac{(\\ve \\nabla \\times \\ve h^{(2)})^2}{|\\psi^{(2)}|^2} \\right]\n + \\frac{e^2}{2 \\beta} \\left( \\ve h^{(1)} + \\ve h^{(2)} \\right)^2 \\right \\}.\n\\label{dual2_N=2} \n\\end{eqnarray}\n\\end{widetext}\nFrom this we identify the massive linear combination of the dual gauge\nfields $\\ve h^{(\\alpha)}$, \nnamely ${\\cal H} = \\ve h^{(1)} + \\ve h^{(2)}$. If a neutral vortex mode exists\nin the system, this implies the existence also of a gauge field in the \nproblem, which we will denote by $\\ve{\\cal A}$. We therefore write $\\ve h^{(\\alpha)}$ \nas linear combinations of ${\\cal H}$ and ${\\cal A}$ as follows\n\\begin{eqnarray}\n\\ve h^{(\\alpha)} & = & \\Gamma^{(\\alpha)} {\\mathcal H} + \\Lambda^{(\\alpha)} {\\mathcal A}.\n\\end{eqnarray} \nWe insert this into Eq. (\\ref{dual2_N=2}) and demand that cross-terms between\n${\\cal H}$ and ${\\cal A}$ vanish, thus obtaining the following set of equations\ndetermining the coefficients $ (\\Gamma^{(\\alpha)}, \\Lambda^{(\\alpha)} )$\n\\begin{eqnarray}\n\\Gamma^{(1)} + \\Gamma^{(2)} &=&  1, \\nonumber \\\\\n\\Lambda^{(1)} + \\Lambda^{(2)} & = & 0, \\nonumber \\\\\n\\Gamma^{(1)} \\Lambda^{(1)}\/|\\psi^{(1)}|^2 + \n\\Gamma^{(2)} \\Lambda^{(2)}\/|\\psi^{(2)}|^2 & = & 0.\n\\end{eqnarray}\nThus, we have $\\Gamma^{(\\alpha)} = |\\psi^{(\\alpha)}|^2\/\\Psi^2$,\nwhere $\\Psi^2 =  |\\psi^{(1)}|^2 + |\\psi^{(2)}|^2$, which yields the following expression for the gauge field ${\\cal A}$\n\\begin{eqnarray}\n{\\cal A} = \\frac{1}{\\Lambda^{(1)}} \n\\frac{ |\\psi^{(2)}|^2  \\ve h^{(1)} - |\\psi^{(1)}|^2 \\ve h^{(2)}}{\\Psi^2}.\n\\end{eqnarray}\nSince we have three equations and four unknowns, we may choose $\\Lambda^{(1)}$ freely, \nand determine it by simplifying the prefactor in ${\\cal A}$ to get\n$\\Lambda^{(1)} = 1\/\\Psi^2 = - \\Lambda^{(2)}$,\nwhence we have\n\\begin{eqnarray}\n\\label{neutralgaugefield}\n{\\cal A} =  |\\psi^{(2)}|^2  \\ve h^{(1)} - |\\psi^{(1)}|^2  \\ve h^{(2)}.\n\\end{eqnarray}\nInverting the relations for ${\\cal H}$ and ${\\cal A}$, we have\n\\begin{eqnarray}\n\\ve h^{(1)} & =  & \n(|\\psi^{(1)}|^2 {\\cal H} + {\\cal A})\/\\Psi^2, \\nonumber \\\\\n\\ve h^{(2)} & =  & \n(|\\psi^{(2)}|^2 {\\cal H} - {\\cal A})\/\\Psi^2.\n\\end{eqnarray}\nInserting this back into Eq. (\\ref{dual2_N=2}), collecting terms, and redefining\nthe fields ${\\cal H}\/\\Psi^2 \\to {\\cal H}$ and ${\\cal A}\/\\Psi^2 \\to  {\\cal A}$,\nwe have the action $S =  S_{{\\cal H}} + S_{{\\cal A}}$ where\n\\begin{eqnarray}\nS_{{\\cal H}}  & = & \\sum_{\\ve r} \\left \\{ 2 \\pi {\\rm i}  {\\cal H} \\cdot \\ve m^{(+)}  \n +  \\frac{1}{2 \\beta_{{\\cal H}}}  \\left[(\\nabla \\times {\\cal H})^2 + m_0^2  {\\cal H}^2 \\right] \\right \\}, \\nonumber \\\\\nS_{{\\cal A}} & = & \\sum_{\\ve r} \\left \\{ 2 \\pi {\\rm i}  {\\cal A} \\cdot \\ve m^{(-)} \n                   + \\frac{1}{2 \\beta_{{\\cal A}}}  (\\nabla \\times {\\cal A})^2 \\right \\}, \n\\label{Sdual_AH}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n\\ve m^{(+)}  &=&    |\\psi^{(1)}|^2  \\ve m^{(1)} + |\\psi^{(2)}|^2  \\ve m^{(2)}, \\nonumber \\\\\n\\ve m^{(-)}  &=&  \\ve m^{(1)} - \\ve m^{(2)}, \\nonumber \\\\\n\\frac{1}{2 \\beta_{{\\cal H}}}  &=&    \\frac{ (|\\psi^{(1)}|^2 + |\\psi^{(2)}|^2)}{2 \\beta}, \\nonumber \\\\\n\\frac{1}{2 \\beta_{{\\cal A}}}  &=&  \\frac{(1\/|\\psi^{(1)}|^2 + 1\/|\\psi^{(2)}|^2 )}{2\\beta}, \n\\label{defs_neutral}\n\\end{eqnarray}\nand $ m_0^2 = e^2\\Psi^2$. The action in Eq. (\\ref{Sdual_AH}), which is equivalent to Eq. (\\ref{dual2_N=2}), therefore describes a\n vortex mode $\\ve m^{(+)}$ interacting with itself via a screened anti Biot-Savart interaction mediated by the \n massive vector field ${\\cal H}$, and the\nvortex mode $\\ve m^{(-)}$ interacting with itself via an \nunscreened anti Biot-Savart interaction mediated by  the gauge field\n${\\cal A}$. Hence, the former vortex mode is charged, the latter is\nneutral. In Appendix  \n\\ref{App:charged_neutral}, we present an alternative method of identifying\ncharged and neutral modes for general $N$.\n\n\n\n\n\\section{\\label{gauge_corr} Gauge field correlators}\nGauge field correlation functions are useful objects to study when considering \nthe critical properties of gauge theories. The main reason is that they \nprovide non-local gauge invariant order parameters for the theories,\nwhich in turn enable reliable determination of critical exponents, including anomalous \nscaling dimensions. {\\it Moreover, these correlators explicitly identify the \nmechanism by which the Meissner effect is destroyed in type-II superconductors:\nThe  mass of the gauge field $\\ve A$, and hence the Higgs phase (equivalently \nthe Meissner phase) is destroyed by a thermally driven vortex loop proliferation \nof the charged vortex mode} \\cite{tesanovic1999,nguyen1999,hove2000,smiseth2004}.\n\n\nIn this section, we study in detail the direct gauge field \ncorrelation function, as well as various combinations of dual gauge field correlation \nfunctions, in order to gain insights into the nature of the critical points \nEq. (\\ref{gl_action}) can exhibit.\n\n\\subsection{$\\ve A$-field correlator and Higgs mass}\nWe first consider the propagator for the gauge field $\\ve A$, which provides information \nabout at which of the critical points the Higgs phenomenon takes place, and where the \nremaining (neutral) fixed points appear. We present  compact expressions for \nthe general-$N$ case, in later sections we present explicit numerical results for \nthe cases  $N=2$ and $N=3$. \n\nWe compute the correlation function $\\langle \\ve A(\\ve r)\\cdot\\ve\nA(0)\\rangle$ in terms of vortex correlators in the standard way by starting from the action \nEq. (\\ref{dual1}), {\\it prior to integrating out the gauge field $\\ve A$},\nadding source terms containing currents $\\ve J$ minimally coupled to $\\ve A$, \nand performing functional derivations with respect to the currents that are subject \nto the constraint $\\ve \\nabla \\cdot \\ve J = 0$, after which the currents are \nset to zero.  The details of the computations required to compute the \n$\\ve A$-field correlator are given in Appendix \\ref{App:gauge}. The\ndiscrete Fourier transform of the gauge field propagator is\n$\\mathcal{G}_{\\ve A}(\\ve q) = \\langle\\ve A_{\\ve q}\\cdot\\ve A_{-\\ve q}\\rangle$. We find \n\\begin{equation}\n  \\mathcal{G}_{\\ve A}(\\ve q)  = \\frac{2\/\\beta}{|\\ve Q_{\\ve q}|^2 + m_0^2}\\left(1 +\\frac{2\\pi^2\\beta\n  e^2}{|\\ve Q_{\\ve q}|^2 }\\frac{G^{(+)}(\\ve q)}{|\\ve Q_{\\ve  q}|^2 + m_0^2}\\right),\n \\label{A_gaugeprop}\n\\end{equation}\nwhere we have defined the correlation function of the charged vortex\nmode as\n\\begin{eqnarray}\nG^{(+)}(\\ve q) & = & \n\\langle (\\sum_{\\alpha=1}^N|\\psi^{(\\alpha)}|^2\\ve m^{(\\alpha)}_{\\ve q})\n\\cdot(\\sum_{\\eta=1}^N|\\psi^{(\\eta)}|^2\\ve m^{(\\eta)}_{-\\ve q})\\rangle.\n\\label{Gpluss_corr}\n\\end{eqnarray}\nNotice in Eq. (\\ref{A_gaugeprop}), that the $\\ve A$-field  correlator is only affected \nby the gauge-charged vortex mode $\\sum_{\\alpha=1}^N\n|\\psi^{(\\alpha)}|^2\\ve m^{(\\alpha)}_{\\ve q}$ via the coupling constant $m_0^2 \\propto e^2$. \n\n\n\nEq. (\\ref{A_gaugeprop}) is useful in MC simulations, \nin conjunction with scaling forms to be presented below, for extracting \nthe gauge field mass and the anomalous scaling dimension of the gauge\nfield. The correlation length  \n$\\xi_{\\ve A}$ that appears in a scaling Ansatz for the $\\ve A$-field correlator\n\\begin{eqnarray}\n{\\cal{G}}_{\\ve A}(\\ve x) = \\frac{1}{|\\ve x|^{D-2+\\eta_{\\ve A}}} \n{\\cal{G}}_{\\pm} \\left( \\frac{|\\ve x|}{\\xi_{\\ve A}} \\right),\n\\end{eqnarray} \nis related to the mass of the gauge field via \n$m_{\\ve A} = \\xi_{\\ve A}^{-1}$.\nHere, $\\eta_{\\ve A}$ is the anomalous scaling dimension of \nthe gauge field $\\ve A$. \nConsequently, the gauge field\npropagator \\eq{A_gaugeprop} has the general structure\\cite{kajantie2004}\n\\begin{equation}\n  \\label{def_gaugemass}\n  \\mathcal{G}_{\\ve A}(\\ve q) \\sim \\frac{1}{|\\ve Q_{\\ve q}|^2 + \\Sigma_{\\ve A}(\\ve q)},\n\\end{equation}\nwhere, close to the critical point \n\\begin{equation}\n  \\label{ansatz_gaugemass}\n  \\Sigma_{\\ve A}(\\ve q) = m_{\\ve A}^2 + C|\\ve q|^{2-\\eta_{\\ve A}} + \\mathcal{O}(|\\ve q|^\\delta),\n\\end{equation}\n$C$ is a constant and $\\delta > 2-\\eta_{\\ve A}$. By taking the $\\ve q\\to 0$ limit of\nthe Eqs. (\\ref{def_gaugemass}) and (\\ref{ansatz_gaugemass}) we may\nextract the gauge mass from MC simulations. From the relation $\\ve\nB = \\Delta\\times\\ve A$ the gauge mass is identified as the inverse magnetic\npenetration depth $\\lambda$. The masses of dual gauge fields are\ndefined in a similar fashion.\n\nLet us make a remark concerning how  a charged fixed\npoint ($\\eta_{\\ve A} = 1$) could be distinguished from a neutral fixed\npoint ($\\eta_{\\ve A} = 0$) by gauge mass measurements. The magnetic\npenetration length  \nis related to the {\\it superconducting} coherence length $\\xi$\nvia \\cite{Herbut_Tesanovic1996,hove2000}\n\\begin{eqnarray}\n\\lambda^{-1} \\sim \\xi^{\\frac{2-d}{2-\\eta_{\\ve A}}} \\sim |T-T_c|^{\\frac{\\nu(d-2)}{2-\\eta_{\\ve A}}},\n\\end{eqnarray}\nwhere  $\\nu$ is the critical exponent of the coherence length in the \nsuperconductor, i.e. $\\nu = 0.67155(3)$\\cite{Hasenbusch}, and $d$ is dimensionality. Therefore, \nwe see that when $\\eta_{\\ve A}=0$, we have \\cite{Herbut_Tesanovic1996,hove2000}\n\\begin{eqnarray}\n\\lambda \\sim \\sqrt{\\xi}  \\sim |T-T_c|^{-\\frac{\\nu}{2}},\n\\end{eqnarray}\nwhile when $\\eta_{\\ve A}=1$, we have\n\\begin{eqnarray}\n\\lambda \\sim \\xi \\sim |T-T_c|^{-\\nu}.\n\\end{eqnarray}\nHence, the gauge mass $m_{\\ve A}=\\lambda^{-1}$ plotted as a function\nof temperature in the critical regime should for $\\eta_{\\ve A}=1$ give a curve with {\\it\n  positive curvature},\nwhile for $\\eta_{\\ve A}=0$ it should give a curve with {\\it negative curvature}.\n\nThe compact expression Eq. (\\ref{A_gaugeprop}) is valid for arbitrary\nnumber of matter field \nflavors $N$, and generalizes the expression obtained in\n\\cite{hove2000}. \nNote that if $e^2=0$, we have trivially that \\eq{A_gaugeprop} reduces to\n\\begin{equation}\n  \\mathcal{G}_{\\ve A}(\\ve q)  = \\frac{2\/\\beta}{|\\ve Q_{\\ve q}|^2},\n\\end{equation}\nwhich is always massless.\nIn Sections \\ref{mc_simsN=3} \nand \\ref{mc_simsN=3} we will use large-scale MC simulations to study in detail the case \n$N=2$ and $N=3$, respectively. The main feature of Eq. (\\ref{A_gaugeprop}) is that at low temperatures, \nwe may in the very simplest approximation entirely ignore the vortex\ncorrelation function $ G^{(+)}(\\ve q)$\nsuch that $\\mathcal{G}_{\\ve A}(\\ve q)$ is obviously massive with photon mass given by the bare \nmass $m_0$ of the problem. Actually, in the low-temperature regime, we have  \n$G^{(+)}(\\ve q) \\sim \\ve q^2$ which in the long-wavelength limit exactly cancels the factor \n$1\/|\\ve Q_{\\ve q}|^2$, rendering the propagator massive. \n\nHowever, at the superconducting critical temperature, vortex loops proliferate \\cite{tesanovic1999,nguyen1999,hove2000,loops_transition,qed3} \nresulting in vortex condensation and hence $\\lim_{\\ve q \\to 0}\nG^{(+)}(\\ve q) \\sim \\rm{const}$. \nNow, the term inside the  brackets in Eq. (\\ref{A_gaugeprop}) will diverge, dominating \nthe behavior of the $\\ve A$-field correlator, such that\n$\\mathcal{G}_{\\ve A}(\\ve q) \\sim 1\/\\ve q^2$.\nThus, the Higgs mass is destroyed. {\\it Note that the amplitudes\nof the matter fields play no role in this, since they are entirely\nfrozen in the present London approximation. It is the condensation of \ntopological defects of the matter fields, i.e. vortex loops, that are responsible \nfor bringing the Higgs mass to zero, not the vanishing of the amplitudes} \\cite{loops_transition}.\nTherefore, we may view the divergence of the penetration length (the correlation length in the \n$\\ve A$-field propagator), as a manifestation of the vortex loop blowout in the \nsystem. {\\it Vortex loops} have dual counterparts in the current loops of the \nmatter fields $\\Psi_0^{(\\alpha)}(\\ve r)$ in Eq. (\\ref{gl_action}). Conversely therefore, \nwe  may also  view the Higgs-mass, i.e. the Meissner effect in the superconductor, \nas a manifestation of blowout of super-current loops upon entering the low-temperature \nphase. Again, the amplitudes of the matter fields\n$\\Psi_0^{(\\alpha)}(\\ve r)$ play no special role\nhere, other than that they have to be non-zero across the Higgs transition \n\\cite{tesanovic1999,nguyen1999,hove2000,loops_transition,qed3}.\n\n\\begin{widetext}\n\\subsection{Dual gauge field correlators}\nThe details of the computations required for finding the dual gauge field\ncorrelation functions in terms of vortex fields are found in Appendix \\ref{App:dualgauge}. We find\nthe following \"Dysons's equation\" for the gauge field correlator\n\\begin{eqnarray}\n\\langle  \\ve h^{(\\alpha)}_{\\ve q}  \\cdot \\ve h^{(\\beta)}_{-\\ve q} \\rangle\n = \\widetilde{D}^{(\\alpha,\\beta)} (\\ve q) \n- \\pi^2 \\widetilde{D}^{(\\alpha,\\eta)} (\\ve q) \\widetilde{D}^{(\\beta,\\kappa)} (\\ve q) \n\\langle \\ve m^{(\\eta)}_{\\ve q} \\cdot  \\ve m^{(\\kappa)}_{-\\ve q} \\rangle, \n\\label{dual_h_individual}\n\\end{eqnarray}\n\\end{widetext}\nwhere we have used the fact that the trace of the transverse projection operator is \ngiven by  $\\rm{Tr} \\left[ P_{T}^{\\mu\\nu} \\right] = 2$,  the matrix elements \n$\\widetilde{D}^{(\\alpha,\\eta)} (\\ve q) $ are defined\nin Eq. (\\ref{potential}), and a summation over the indices $(\\eta,\\kappa) \\in [1,\\dots,N]$\nis understood. These results are valid for all $N$. \n\nTo obtain more explicit expressions, we will work out in detail what we obtain for $N=2$. As we have \nseen above, in this case it is natural to use  Eq. (\\ref{dual_h_individual}) to \nform correlation functions of the combination $\\ve h^{(1)}+ \\ve h^{(2)}$. We will, for completeness\nalso consider the combination and \n$\\ve h^{(1)}- \\ve h^{(2)}$ and $|\\psi^{(2)}|^2 \\ve h^{(1)} - |\\psi^{(1)}|^2 \\ve h^{(2)}$. We also \nuse the fact that the interaction matrix $\\widetilde{D}^{(\\alpha,\\beta)}(\\ve q)$ is symmetric, and \nintroduce the definitions\n\\begin{eqnarray}\n\\ve h^{(\\pm)}_{\\ve q} & \\equiv & \\ve h^{(1)}_{\\ve q} \\pm  \\ve h^{(2)}_{\\ve q}                  \\nonumber \\\\\na^{(\\pm)}         & \\equiv & \\widetilde{D}^{(1,1)}(\\ve q) \\pm \\widetilde{D}^{(1,2)}(\\ve q) \\nonumber \\\\\nb^{(\\pm)}         & \\equiv & \\widetilde{D}^{(2,2)}(\\ve q) \\pm \\widetilde{D}^{(1,2)}(\\ve q). \n\\label{ab1_eqs}\n\\end{eqnarray}\nIt is enlightening at this stage to introduce the expressions for $\\widetilde{D}^{(\\alpha,\\beta)}(\\ve q)$, \nas follows\n\\begin{eqnarray}\n\\frac{\\widetilde{D}^{(1,1)}(\\ve q)\\Psi^2}{2 \\beta |\\psi^{(1)}|^2  |\\psi^{(2)}|^2 } & = & \n\\frac{1}{|\\ve Q_{\\ve q}|^2} + \\frac{|\\psi^{(1)}|^2 }{|\\psi^{(2)}|^2}\n\\frac{1}{|\\ve Q_{\\ve q}|^2 + m_0^2}, \\nonumber \\\\\n\\frac{\\widetilde{D}^{(2,2)}(\\ve q)\\Psi^2}{2 \\beta |\\psi^{(1)}|^2  |\\psi^{(2)}|^2 } & = & \n \\frac{1}{|\\ve Q_{\\ve q}|^2} + \\frac{|\\psi^{(2)}|^2 }{|\\psi^{(1)}|^2}\n\\frac{1}{|\\ve Q_{\\ve q}|^2 + m_0^2}, \\nonumber \\\\\n\\frac{\\widetilde{D}^{(1,2)}(\\ve q)\\Psi^2}{2 \\beta |\\psi^{(1)}|^2  |\\psi^{(2)}|^2 } & = & \n-\\frac{1}{|\\ve Q_{\\ve q}|^2} + \\frac{1}{|\\ve Q_{\\ve q}|^2 + m_0^2},\n\\label{D_eqs}\n\\end{eqnarray}\nwhere $\\Psi^2 = |\\psi^{(1)}|^2 + |\\psi^{(2)}|^2$. Using Eqs. (\\ref{D_eqs}) in Eqs. (\\ref{ab1_eqs}), we find\n\\begin{eqnarray}\na^{(+)} & \\equiv & \\frac{2 \\beta |\\psi^{(1)}|^2}{|\\ve Q_{\\ve q}|^2 + m_0^2}, \\nonumber \\\\ \nb^{(+)} & \\equiv & \\frac{2 \\beta |\\psi^{(2)}|^2}{|\\ve Q_{\\ve q}|^2 + m_0^2},\n\\label{ab_pluss}\n\\end{eqnarray}\nand $a^{(-)}$ and $b^{(-)}$ given by\n\\begin{eqnarray}\n\\frac{a^{(-)}\\Psi^2}{2 \\beta |\\psi^{(1)}|^2 |\\psi^{(2)}|^2} & \\equiv & \\frac{2}{|\\ve Q_{\\ve q}|^2} +  \\frac{|\\psi^{(1)}|^2\/|\\psi^{(2)}|^2-1}{|\\ve Q_{\\ve q}|^2 + m_0^2}, \\nonumber \\\\ \n\\frac{b^{(-)}\\Psi^2}{2 \\beta |\\psi^{(1)}|^2 |\\psi^{(2)}|^2} & \\equiv & \\frac{2}{|\\ve Q_{\\ve q}|^2} +  \\frac{|\\psi^{(2)}|^2\/|\\psi^{(1)}|^2-1}{|\\ve Q_{\\ve q}|^2 + m_0^2}, \n\\label{ab_minus}\n\\end{eqnarray}\nwhere $m_0^2 = e^2\\Psi^2$. \n{\\it Notice how the  unscreened part of the interactions cancel out in $(a^{(+)}, b^{(+)})$\nbut not in $(a^{(-)}, b^{(-)})$}. This is the origin of the qualitatively different behavior\nwe will find for the $\\ve h^{(+)}_{\\ve q}$ and   $\\ve h^{(-)}_{\\ve q}$ correlators.\nNotice also how the expressions \nsimplify when $|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2  $, when the screened part of the interactions \nappearing in $a^{(-)}, b^{(-)}$ vanish,  such  that $a^{(-)}= b^{(-)}$.\n\nWe may now write the correlation functions of the two relevant linear combinations\nof dual gauge fields as follows\n\\begin{widetext}\n\\begin{eqnarray}\n\\mathcal{G}^{(\\pm)}_{\\ve h}(\\ve q) & \\equiv & \\langle  \\ve h^{(\\pm)}_{\\ve q} \\cdot \\ve h^{(\\pm)}_{-\\ve q}  \\rangle\n= a^{(\\pm)} + b^{(\\pm)} \n -  \\pi^2 \\langle (a^{(\\pm)} \\ve m^{(1)}_{\\ve q} \\pm  b^{(\\pm)} \\ve m^{(2)}_{\\ve q}) \\cdot\n(a^{(\\pm)}\\ve m^{(1)}_{-\\ve q} \\pm  b^{(\\pm)} \\ve m^{(2)}_{-\\ve q}) \\rangle.\n\\label{hpm_corr_eq}\n\\end{eqnarray}\n\\end{widetext}\nUsing Eqs. (\\ref{ab_pluss}), and (\\ref{hpm_corr_eq}),\nwe find the surprisingly compact expression, valid for all $N$\n\\begin{eqnarray}\n\\mathcal{G}^{(+)}_{\\ve h}(\\ve q) = \\frac{2\\beta\\psi^2}{|\\ve Q_{\\ve q}|^2 + m_0^2}\n\\left(1  -  \\frac{2\\pi^2\\beta}{\\psi^2}\\frac{G^{(+)}(\\ve q)}{|\\ve Q_{\\ve q}|^2 + m_0^2}\n  \\right),\n\\label{hp_corr}\n\\end{eqnarray}\nwhere we have again introduced $G^{(+)}(\\ve q)$ appearing in Eq. (\\ref{Gpluss_corr}).\nIn fact, this result could have been written down using the known result for\nthe charged case for $N=1$ \\cite{hove2000}, in combination with Eq. (\\ref{Sdual_AH}),\nconsidering the part of Eq. (\\ref{Sdual_AH}) only pertaining to the massive vector\nfield ${\\cal H}$. This provides a nice consistency check on the general expression\nfor the dual gauge field correlators, as well as on the interaction\nmatrix $\\widetilde D^{(\\alpha,\\eta)}(\\ve q)$. In the low- and\nhigh-temperature phase, the vortex correlator $G^{(+)}(\\ve q)$\nbehaves as $\\sim \\ve q^2$ and $\\sim c(T)$, respectively. In either case, the \ndual gauge field correlator $\\mathcal{G}^{(+)}_{\\ve h}(\\ve q)$ is always massive.  \n\n\n\n\nConsider the correlation function of the combination of dual gauge fields\n$\\mathcal{A} = |\\psi^{(2)}|^2 \\ve h^{(1)} - |\\psi^{(1)}|^2 \\ve\nh^{(2)}$ which couples to the gauge-neutral vortex mode in \\eq{Sdual_AH}. In principle \nwe may follow the routes used in the above\ncalculations, but by now we realize that a quick way of obtaining the\nresults is to use \\eq{Sdual_AH} in combination with the\nknown results for the case $N=1$ in the {\\it neutral} case \\cite{hove2000}.\nWe define\n\\begin{eqnarray}\n{\\mathcal G}_{{\\mathcal A}}(\\ve q) \\equiv  \\langle {\\mathcal A}_{\\ve q}  {\\mathcal A}_{-\\ve q}  \\rangle,\n\\end{eqnarray}\nand find immediately, using the results of Ref. \\onlinecite{hove2000} along \nwith the definitions in \\eq{defs_neutral}\n\\begin{eqnarray}\n{\\mathcal G}_{\\mathcal A}(\\ve q) = \\frac{2 \\beta_{{\\mathcal A}}}{|\\ve Q_{\\ve q}|^2} \n\\left(1 - \\frac{2 \\pi^2 \\beta_{{\\mathcal A}} G^{(-)}(\\ve q)}{|\\ve Q_{\\ve q}|^2} \\right),\n\\label{dual_corr3}\n\\end{eqnarray}\nwhere\n\\begin{equation}\nG^{(-)}(\\ve q) =  \n\\langle (\\ve m^{(1)}_{\\ve q}  - \\ve m^{(2)}_{\\ve q}) \\cdot (\\ve\nm^{(1)}_{-\\ve q}  -\n\\ve m^{(2)}_{-\\ve q}) \\rangle\n\\end{equation}\nis the correlation function of the gauge-neutral vortex mode.\n\nIn the long wave length limit the behavior of $G^{(-)}(\\ve q)$ gives rise\nto a dual Higgs mechanism. This comes about because  \nthe $G^{(-)}(\\ve q)$ correlation function is always $\\sim \\ve q^2$ at long \nwavelengths, but has a non-analytic coefficient in front of the $\\ve q^2$ \nterm given by the helicity modulus of the gauge-neutral mode \n$\\ve m^{(1)} - \\ve m^{(2)}$. This serves to cancel the $1\/\\ve q^2$ term \nin the $ \\mathcal{G}_{\\mathcal A}(\\ve q)$ correlation function exactly.\nThis cancellation, originating in the vanishing of the helicity\nmodulus of the gauge-neutral mode, is responsible for producing a dual \nHiggs mass $m_\\mathcal{A}$ in $\\mathcal{G}_{\\mathcal A}(\\ve q) $. Higher order terms\ndetermine the actual value of the dual Higgs mass.  \nThus, we see that while $\\ve h^{(1)} + \\ve h^{(2)}$ is always massive,\n$|\\psi^{(2)}|^2\\ve h^{(1)} - |\\psi^{(1)}|^2\\ve h^{(2)}$ plays the role of a gauge degree of freedom\nwhich provides a dual counterpart to $\\ve A$ in \\eq{gl_action}.\nThis is a manifestation of the self-duality of the theory which we have \nalluded to above \\cite{motrunich2004,sachdev2004,smiseth2004}.\n\n\nNotice that the existence of a dual Meissner effect arising out of\n\\eq{dual_corr3} is a substantially more subtle effect than \nthe direct Meissner effect coming out of \\eq{A_gaugeprop}.  \nThe correlator of the gauge-neutral mode\nhas the property  \n\\begin{equation}\n \\label{g_min}\n G^{(-)}(\\ve q) =  C_2 \\ve q^2 + C_4 \\ve q^4 + \\mathcal{O}(\\ve q^6),\n\\end{equation}\n for all \ntemperatures, in analogy with the vortex correlator of the 3D$xy$\\  model\nfor the case $N=1$. It is the non-analytic behavior of the\ncoefficient $C_2$, involving the {\\it helicity modulus} of the\ngauge-neutral mode, which is responsible for {\\it producing} \na dual Higgs mass as the gauge-neutral mode proliferates. To obtain a \ndual Meissner effect, a subtle cancellation is required, namely \nthat at some critical temperature $T_{\\rm c1}$, we must have \n\\begin{eqnarray}\n\\label{criterion2}\n1-\\frac{2\\pi^2 \\beta |\\psi^{(1)}|^2|\\psi^{(2)}|^2C_2(T_{c1})}{|\\psi^{(1)}|^2\n  + |\\psi^{(2)}|^2} = 0,\n\\end{eqnarray}\nwhere we have used the expression for $\\beta_{{\\mathcal A}}$ from \\eq{defs_neutral}.\nIt is important to note that while the actual value of the dual Higgs mass \nis influenced by the higher order terms in \\eq{g_min}, the {\\it criterion} for obtaining a dual Higgs \nphenomenon is only determined by the cancellation among the terms of\norder $1\/\\ve q^2$ terms in \\eq{dual_corr3}. \nThis differs from the mechanism that destroys the Higgs mass in the $\\ve A$ \ncorrelator, since there no such subtle cancellations are required, it suffices \nthat the correlator $G^{(+)}(\\ve q) $ changes behavior from a constant to \n$\\sim \\ve q^2$ in the long-wavelength limit. \n\n\nWe finally consider the correlation function of $\\ve h^{(-)}$. Applying the results \nfrom \\eq{hpm_corr_eq}, we find \\cite{smiseth2004} \n\\begin{widetext}\n\\begin{equation}\n\\mathcal{G}_{\\ve h}^{(-)}(\\ve q) =\n\\frac{8 \\beta \\lambda^{(1)}\\lambda^{(2)}\\Psi^2}{|\\ve Q_{\\ve q}|^2 }\n \\left\\{ 1 - \\frac{2\\pi^2\\beta\\lambda^{(1)}\\lambda^{(2)}\\Psi^2\n   G^{(-)}(\\ve q)}{|\\ve Q_{\\ve q}|^2}\n- \\frac{2\\pi^2\\beta(\\lambda^{(1)} - \\lambda^{(2)}) G^{(m)}(\\ve q)}{|\\ve Q_{\\ve q}|^2 + m_0^2} \n\\right\\}  + (\\lambda^{(1)} - \\lambda^{(2)})^2\\mathcal{G}_{\\ve h}^{(+)}(\\ve q),\n\\label{hm_dual}\n\\end{equation}\n\\end{widetext}\nwhere $\\lambda^{(\\alpha)} = |\\psi^{(\\alpha)}|^2\/\\Psi^2$, $\\Psi^2 =\n  |\\psi^{(1)}|^2 + |\\psi^{(2)}|^2$, and the mixed gauge-neutral and gauge-charged vortex field\ncorrelator is given by\n\\begin{equation}\nG^{(\\rm m)}(\\ve q) = \n\\langle (\\ve m^{(1)}_{\\ve q}  - \\ve m^{(2)}_{\\ve q}) \\cdot\n(\\sum_{\\alpha=1}^2|\\psi^{(\\alpha)}|^2\\ve m^{(\\alpha)}_{-\\ve q}) \\rangle.\n\\end{equation}\nNote that for the case $N=1$, such that either $\\lambda^{(1)}$ or $\\lambda^{(2)}$\nvanishes, then the remaining $\\lambda^{(\\eta)}=1$, only the last term in \\eq{hm_dual} \nsurvives, and $\\mathcal{G}^{(-)}_{\\ve h}(\\ve q)$ correctly reduces to\n$\\mathcal{G}^{(+)}_{\\ve h}(\\ve q)$\nin \\eq{hp_corr}.\nIn the long wave length limit, it is the second term in the curly\nbrackets in \\eq{hm_dual} that \ndominates, giving rise to a dual Higgs mechanism. Notice again how it\nis the vortex correlator $G^{(-)}(\\ve q)$ which determines the fate of the\nmassless dual gauge field $\\ve h^{(1)} - \\ve h^{(2)}$, just like in\n\\eq{dual_corr3}. This is particularly evident for the case $|\\psi^{(1)}|=|\\psi^{(2)}|$,\nwhen \\eq{hm_dual} reduces to\n\\begin{eqnarray}\n \\mathcal{G}^{(-)}_{\\ve h}(\\ve q) \n= \\frac{4 \\beta |\\psi^{(1)}|^2 }{|\\ve Q_{\\ve q}|^2}\\left(1  \n- \\pi^2\\beta |\\psi^{(1)}|^2  \\frac{G^{(-)}(\\ve q)}{|\\ve Q_{\\ve q}|^2}\n  \\right).\n\\label{H_corr_iso}\n\\end{eqnarray}\nThis correlator for $N=2$, $e \\neq 0$ has precisely the same form\nas the dual gauge field correlator for the case $N=1, e=0$, which\nexhibits a dual Higgs phenomenon \\cite{hove2000}. \n\n\nSubstituting $\\lambda^{(\\alpha)} =\n|\\psi^{(\\alpha)}|^2\/\\Psi^2$ in \\eq{hm_dual}, we see that the criterion\nfor destroying the dual Higgs mass is precisely the same as the\ncriterion we arrived at in \\eq{criterion2}. \nThus, whether we compute the correlator \nin \\eq{hm_dual} or that in \\eq{dual_corr3} to establish the \nexistence of a dual Higgs phase does not matter. \nFurthermore, for $N=2, e \\neq 0$, \n$\\ve m^{(1)} - \\ve m^{(2)}$ behaves  as vortices for $N=1, e=0$, {\\it i.e. it is \na superfluid mode arising out of superconducting condensates.} A nonzero  $m_{\\cal A}$ \nfor the dual gauge field ${\\cal A}$ is {\\it produced} by disordering $\\theta^{(1)}$ \nat a critical temperature $T_{\\rm c1}$ while a nonzero  $m_{\\ve A}$\nfor the gauge \nfield $\\ve A$ is {\\it destroyed} by disordering $\\theta^{(2)}$ at a critical temperature \n$T_{\\rm c2}$. \n\n\n\\section{\\label{mc_simsN=2}Monte Carlo simulations, $N=2$}\nSince the bare interaction between vortices is dominated at\nlong distances by an unscreened part, it is of interest to study the \ncharacter of the phase transition associated with the generation of a Higgs \nmass for the gauge field ${\\bf{A}}$. For the $N=1$ case, it is \nknown that the vortex tangle of the 3D$xy$\\  model is incompressible and the dual \ntheory is a gauge theory such that $\\langle \\phi \\rangle \\neq 0$ is \nprohibited. For the charged case, the vortex tangle is compressible, the dual \ntheory only has global symmetry,  and hence vortex condensation and \n$\\langle \\phi \\rangle \\neq 0$ is possible. The introduction of charge \ndestabilizes the 3D$xy$\\  fixed point. \n\nTo investigate what happens for the case $N=2$, MC simulations have \nbeen carried out for the action Eq. (\\ref{vortex_action}) on a three\ndimensional lattice of size $L\\times L\\times L$ for two different cases. \nIn the first case we simulate with unequal bare stiffnesses \n$|\\psi^{(1)}|^2 = 1\/2$ and $|\\psi^{(2)}|^2 = 1$, $e^2 = 1\/4$ and\n$m_0^2 = 3\/8$. The bare stiffnesses have been chosen to have well-separated bare energy scales\nassociated with the twist of the two types of phases. In the second\ncase we use equal phase stiffnesses\n$|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 = 1$, $e^2 = 1\/4$ and $m_0^2 = 1\/2$.\nThe values for $m_0$\nhave been chosen such that they are of order the lattice spacing in the problem to\navoid difficult finite-size effects. One MC update consists of inserting a unitary \nvortex loop of random direction and species according to the Metropolis algorithm.\n\nTo calculate the critical exponents $\\alpha$ and $\\nu$ we performed\nfinite size scaling (FSS) analysis with bootstrap error estimates of the \nthird moment of the action\\cite{m3} $M_3=\\langle(S_{\\rm V}-\\langle S_{\\rm\n  V}\\rangle)^3\\rangle\/L^3$ where $S_{\\rm V}$ is given in \\eq{vortex_action}. The peak \nto peak value of this quantity scales with system size $L$ as $L^{(1+\\alpha)\/\\nu}$, whereas \nthe width between the peaks scales as $L^{-1\/\\nu}$. The advantage of this is that \nasymptotically correct behavior is reached for practical system sizes. \n\nTo characterize the phase transitions further, we consider the correlation functions given in \nEqs. (\\ref{A_gaugeprop}), (\\ref{Gpluss_corr}), and (\\ref{hp_corr}). In the Higgs phase the gauge \nfield mass $m_{\\ve A}$ scales according to the Ansatz\\cite{kajantie2004} given by \nEqs. (\\ref{def_gaugemass}) and (\\ref{ansatz_gaugemass})\n\\begin{eqnarray}\n\\label{prop_ansatz}\n\\mathcal{G}_{\\ve A}(\\ve q)^{-1}\\frac{2}{\\beta} = m_{\\ve A}^2 + C |\\ve\nq|^{2-\\eta_{\\ve A}} + \\mathcal{O}(|\\ve q|^\\delta),\n\\end{eqnarray}\nwith a corresponding Ansatz for $\\mathcal{G}^{(+)}_{\\ve h}(\\ve q)$. The masses of  \n$\\ve A$ and $\\sum_{\\alpha=1}^{N} \\ve h^{(\\alpha)}$ are therefore\ndefined through the $\\ve q\\to 0$ limit of the respective Ans\\\"{a}tze\n\\begin{eqnarray}\nm_{\\ve A}^2 & \\equiv & \\lim_{\\ve q\\to0} \\frac{2}{\\beta\\mathcal{G}_{\\ve A}(\\ve q)} \\nonumber \\\\\nm_{\\Sigma\\ve h}^2 & \\equiv & \\lim_{\\ve q\\to0}\\frac{2\\beta\\psi^2}{\\mathcal{G}^{(+)}_{\\ve h}(\\ve q)}.\n\\label{gauge_field_masses}\n\\end{eqnarray}\nThe gauge field masses are found by measuring vortex correlators\nfollowed by a fit for small $\\ve q$ to their respective Ans\\\"{a}tze.\n\nWe briefly review the $N=1$ GL-model. The dual field theory of the neutral \nfixed point is a charged theory describing an incompressible vortex tangle \n\\cite{hove2000}. The leading behavior of the vortex correlator\n$G^{(+)}(\\ve q)\\sim\\langle \\ve m_{\\ve q}\\ve \\cdot \\ve m_{-\\ve q}\\rangle$ is\\cite{hove2000}\n\\begin{equation}\n\\lim_{\\ve q\\to 0}G^{(+)}(\\ve q) \\sim \\begin{cases}\n  [1-C_2(T)]\\ve q^2 & ; T<T_{c} \\\\\n  \\ve q^2-C_3(T)|\\ve q|^{2+\\eta_{\\ve h}} & ; T=T_{\\rm c} \\\\\n  \\ve q^2+C_4(T)\\ve q^4 & ;T>T_{\\rm c},\n\\end{cases}\n\\end{equation}\nwhere $\\eta_{\\ve h}$ is the anomalous scaling dimension of the dual\ngauge field $\\ve h$. For $T < T_{\\rm c}$ the mass of the dual gauge field given by\nEqs. (\\ref{hp_corr}) and (\\ref{gauge_field_masses}) (with $N=1$ and $e=0$)\nis zero, however for $T>T_{\\rm c}$ the $\\ve q^{-2}$ terms in\n\\eq{hp_corr} \ncancel out exactly and the mass $m_{\\ve h}$ attains an expectation value. \nAt the charged fixed point of the GL model, the effective field theory\nof the vortices is a neutral theory. The vortex tangle is compressible\nwith a scaling Ansatz for the vortex correlator \n\\begin{equation}\n\\label{n1_corr}\n\\lim_{\\ve q\\to 0}G^{(+)}(\\ve q)\\sim \\begin{cases} \\ve q^2 & ; T<T_{\\rm c} \\\\\n|\\ve q|^{2-\\eta_{\\ve A}} & ; T=T_{\\rm  c} \\\\\nc(T) & ; T> T_{\\rm c},\n\\end{cases}\n\\end{equation}\nwhere $c(T)$ is a nonzero constant. Consequently, from Eqs. (\\ref{A_gaugeprop}), (\\ref{hp_corr}), and\n(\\ref{gauge_field_masses}) (with $N=1$ and $e\\neq 0$), the mass $m_{\\ve A}$ drops to zero at $T_{\\rm c}$, \nand the mass of the dual vector field $m_{\\ve h}$ is finite for all temperatures and has \na kink at $T_{\\rm c}$ \\cite{hove2000}. Renormalization group arguments yield \n$\\eta_{\\ve  A}=4-d$ where $d$ is the dimensionality \\cite{HLM,Bergerhoff,Herbut_Tesanovic1996}, \nwhich has recently been verified numerically \\cite{hove2000,kajantie2004}.   \n\n\n\\subsection{Critical exponents $\\alpha$ and $\\nu$, $|\\psi^{(1)}| < |\\psi^{(2)}|$}\nWe observe two anomalies in the specific heat at $T_{\\rm c1}$ and $T_{\\rm c2}$ where \n$T_{\\rm c1}< T_{\\rm c2}$. We find $T_{\\rm c1}$ and $T_{\\rm c2}$ from scaling of the second moment of \nthe action $\\langle (S_{\\rm V} - \\langle S_{\\rm V} \\rangle )^2\\rangle\/L^3$\nto be $T_{\\rm c1} = 1.4(6)$ and $T_{\\rm c2} = 2.7(8)$. The $M_3$ FSS\nplots for system sizes $L=4, 6, 8, 10, 12, 14, 16, 20, 24$ are shown\nin \\fig{m3}. \n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.34}{\\rotatebox{-90.0}{\\includegraphics{m3scaling.ps}}}} \n\\caption{\\label{m3} The FSS of the peak to peak value of\n  the third moment $\\Delta M_3$ labeled ($\\Box$) and (+) for\n  $T_{\\rm c1}$ and $T_{\\rm c2}$, respectively. The scaling of\n  the width between the peaks $\\Delta\\beta$ is labeled ($\\blacktriangle$) \n  and ($\\times$) for $T_{\\rm c1}$ and $T_{\\rm c2}$ respectively. The \n  lines are power law fits to the data for $L>6$ used to extract \n  $\\alpha$ and $\\nu$.}\n\\end{figure}\nFrom the scaling we conclude that both anomalies are in fact critical points, and \nwe obtain $\\alpha = -0.02 \\pm 0.02$ and $\\nu=0.67\\pm\n0.01$ for $T_{\\rm c1}$ and $\\alpha = -0.03 \\pm 0.02$ and $\\nu = 0.67\\pm 0.01$ for\n\t $T_{\\rm c2}$. These values are consistent with those of the\n3D$xy$\\  and the {\\it inverted} 3D$xy$\\  universality classes found with high\nprecision in Refs. \\onlinecite{Kleinert1999,Hasenbusch,Lipa}.\n \n\\subsection{Vortex correlator, Higgs mass, and anomalous scaling dimension, $|\\psi^{(1)}| < |\\psi^{(2)}|$}\n\nThe vortex correlators for the $N=2$ case are\nsampled in real space and $G^{(+)}(\\ve q)$ given in\n\\eq{Gpluss_corr} is found by a discrete Fourier\ntransformation. At the lower transition $T_{\\rm c1}$ the leading\nbehavior is $G^{(+)}(\\ve q)\\sim \\ve q^2$ on both sides of the\ntransition. Consequently, due to Eqs. (\\ref{A_gaugeprop}),\n(\\ref{hp_corr}), and (\\ref{gauge_field_masses}), $m_{\\ve A}$ and\n$m_{\\Sigma \\ve h}$ are finite in this regime. This shows that the \nvortex tangle is incompressible and that the anomalous scaling\ndimension $\\eta_{\\ve A}=0$, which corresponds to a neutral fixed\npoint. Fig. \\ref{G_q} shows the correlator $G^{(+)}(\\ve q)$ around $T_{\\rm\n  c2}$.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.34}{\\rotatebox{-90.0}{\\includegraphics{G_q.ps}}}} \n\\caption{\\label{G_q} $G^{(+)}(\\ve q)$ for $N=2$ and $L=32$, plotted\n  for temperatures \n$T=2.86 > T_{\\rm{c2}}$, $T=2.76\\simeq T_{\\rm c2}$, and $T=2.63 < T_{\\rm{c2}}$,  \n$\\lim_{\\ve q \\to 0} G^{(+)}(\\ve q) \\sim c(T)$,  $\\sim |\\ve q|$, and\n  $\\sim \\ve q^2$, \nrespectively. The $q\\to 0$ behavior of the correlator matches\n  precisely the signature of a changed fixed point given in \\eq{n1_corr}.}\n\\end{figure}\n Below $T_{\\rm c2}$ the dominant behavior is $G^{(+)}(\\ve q)\\sim\n\\ve q^2$ whereas $G^{(+)}(\\ve q)\\sim c(T)$\nabove the transition. At the critical point $G^{(+)}(\\ve q)\\sim |\\ve q|$, indicating\n$\\eta_{\\ve A}=1$. Accordingly $m_{\\ve A}$ is finite below the\ntransition and zero for $T \\ge T_{\\rm c2}$.\n\nFor each coupling we fit $\\mathcal{G}_{\\ve A}(\\ve q)^{-1}$ for $|\\ve\nQ_{\\ve q}| < 0.9$ using system sizes\n$L=8,12,20,32$ to Eq. (\\ref{prop_ansatz}). The results for $m_{\\ve A}$,\nand $m_{\\Sigma \\ve h}$ which is found in a similar fashion, are given in\n\\fig{gauge_mass}. The system exhibits Higgs mechanism\nwhen $m_{\\ve A}$ drops to zero at $T_{\\rm c2}$ with an anomaly in\n$m_{\\Sigma\\ve h}$ due to vortex condensation. Furthermore $m_{\\ve A}$\nhas a kink at $T_{\\rm c1}$ due to ordering of the phase difference\n$\\theta^{(1)}-\\theta^{(2)}$ with the phase stiffness\n$|\\psi^{(1)}|^2|\\psi^{(2)}|^2\/(2|\\psi^{(1)}|^2 + 2|\\psi^{(2)}|^2)$,\nconfirm \\eq{charge_neutral}\\cite{frac}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.34}{\\rotatebox{-90.0}{\\includegraphics{gauge_mass.ps}}}} \n\\caption{\\label{gauge_mass}\n The mass $m_{\\ve A}$ ($\\blacklozenge$) and\n  $1-m_0\/m_{\\Sigma\\ve h}$ (+) found from Eqs. (\\ref{A_gaugeprop}) and (\\ref{hp_corr}). \nTwo  non-analyticities can be seen in $m_{\\ve A}$ at $T_{\\rm c1}$ and\n  $T_{\\rm c2}$, corresponding a neutral fixed point and a charged\n  Higgs fixed point, respectively. An abrupt increase in $m_{\\Sigma\\ve h}$ due \n  to vortex\n  condensation is located at $T_{\\rm c2}$. } \n\\end{figure}\nThe anomalies in $m_{\\ve A}$ and \n$m_{\\Sigma \\ve h}$ coincide precisely with $T_{\\rm c2}$ and $T_{\\rm c1}$. \nNote also how $m_{\\Sigma\\ve h}$ changes abruptly at $T_{\\rm c2}$.\nThis is due to a sudden change in screening by \n$\\sum_{\\alpha=1}^N\\ve h^{(\\alpha)}$, giving an abrupt increase in $m_{\\Sigma\\ve h}$. This is \nconsistent with the flow equation Eq. (\\ref{rg_flow}). Note that the mass of the\nalgebraic sum of the dual fields appears in Eq. (\\ref{dual2}) after \nintegrating out the gauge field $\\ve A$.  \n\n\nWe may understand the  transitions  as follows. Above  $T_{\\rm{c2}}$,  $\\bf{A}$ is \nmassless, giving a compressible vortex tangle which accesses configurational entropy \nbetter than an incompressible one. Below $T_{\\rm{c2}}$, ${\\bf{A}}$ is massive and \nmerely renormalizes $|\\Psi|^4$ terms in Eq. (\\ref{gl_action}). The theory is \neffectively a $|\\Psi |^4$ theory in this regime. Thus, the remaining proliferated \nvortex species originating in the matter fields with lower bare stiffnesses form  vortex \ntangles  as if they originated in a neutral superfluid. For the general $N$ case, a Higgs \nmass is generated at the highest critical temperature, after which ${\\bf{A}}$\nrenormalizes the $|\\Psi|^4$ term, such that the Higgs  fixed point is followed by $N-1$ \nneutral fixed points as the temperature is lowered.\n\nThe picture that emerges from the above discussion of the gauge field and the dual gauge field \ncorrelators is the following. Below $T_{\\rm c1}$ there is one massless \"photon\", namely \n$|\\psi^{(2)}|^2\\ve h^{(1)} - |\\psi^{(1)}|^2\\ve h^{(2)}$, while $\\ve A$ is massive.  Above $T_{\\rm\n  c1}$ and  below $T_{\\rm c2}$, \nboth $|\\psi^{(2)}|^2\\ve h^{(1)} - |\\psi^{(1)}|^2\\ve h^{(2)}$ and $\\ve A$ are massive, while above\n$T_{\\rm c2}$,  \n$|\\psi^{(2)}|^2\\ve h^{(1)} - |\\psi^{(1)}|^2\\ve h^{(2)}$ is massive and $\\ve A$ is massless. \n\n\\subsection{Critical exponents $\\alpha$ and $\\nu$, $|\\psi^{(1)}| = |\\psi^{(2)}|$}\n\nA special case is obviously presented by the case $|\\psi^{(1)}| = |\\psi^{(2)}|$ since then\n$T_{\\rm c1} = T_{\\rm c2} \\equiv T_{\\rm c}$, and we have a transition directly from a low-temperature phase with \none massless dual gauge  field $|\\psi^{(2)}|^2\\ve h^{(1)}- |\\psi^{(1)}|^2\\ve h^{(2)} = \n|\\psi^{(1)}|^2(\\ve h^{(1)}- \\ve h^{(2)})$ to a high-temperature phase with one massless direct gauge \nfield $\\ve A$. This is the remarkable self-duality observed in Refs. \\onlinecite{motrunich2004,sachdev2004,smiseth2004}.\n\nThe second moment of the action with $|\\psi^{(1)}|^2 =\n|\\psi^{(2)}|^2 = 1$, $e^2=1\/4$ and $m_0^2 = 1\/2$ exhibits one anomaly at\n$T_{\\rm c}=2.7(8)$. Scaling plots of the third moment of the action \nare shown in Fig. \\ref{scaling_eqpsi}. FSS yields $\\alpha=0.03 \\pm 0.04$ and\n$\\nu=0.60\\pm0.02$.  The numerical value for $\\nu$ is in agreement with the value found in\nRef. \\onlinecite{motrunich2004}, $\\nu = 0.60\\pm0.05$. Note that our\nresult for $\\alpha$ and $\\nu$ is not in agreement with hyper scaling.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.34}{\\rotatebox{-90.0}{\\includegraphics{m3scaling_eqpsi.ps}}}} \n\\caption{\\label{scaling_eqpsi} The FSS of the peak to peak value of\n  the third moment $\\Delta M_3$ labeled (+) for\n  $T_{\\rm c}$ and for $|\\psi^{(1)}| = |\\psi^{(2)}|$.  The scaling\n  of the width between the peaks $\\Delta\\beta$ is labeled ($\\circ$). The lines\nare power law fits to the data for $L>8$ used to extract $\\alpha$ and $\\nu$.}\n\\end{figure}\n\n\n\\subsection{Vortex correlator and Higgs mass, $|\\psi^{(1)}| = |\\psi^{(2)}|$}\n\nThe mass of the gauge field $m_{\\ve A}$ was found by fitting $\\mathcal{G}_{\\ve A}(\\ve q)^{-1}$ data from system sizes\n$L=8,12,20,32$ to Eq. (\\ref{prop_ansatz}). The mass $m_{\\Sigma \\ve h}$\nwas found similarly. The results are presented in \\fig{mass_A_eqpsi}.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.34}{\\rotatebox{-90.0}{\\includegraphics{gauge_mass_eqpsi.ps}}}} \n\\caption{\\label{mass_A_eqpsi}\n The mass $m_{\\ve A}$ ($\\blacklozenge$) and\n  $1-m_0\/m_{\\Sigma\\ve h}$ (+) found from Eqs. (\\ref{A_gaugeprop}) and (\\ref{hp_corr}),\nfor $|\\psi^{(1)}| = |\\psi^{(2)}|$. One  non-analyticity can be seen in $m_{\\ve A}$ \nat $T_{\\rm c}$, corresponding to a fixed point which is\nnot in the 3D$xy$\\  or inverted 3D$xy$\\  universality class. \nAn abrupt increase in $m_{\\Sigma\\ve h}$ due to vortex\n  condensation is located at $T_{\\rm c}=2.7(8)$. } \n\\end{figure}\n\n\\subsection{Discussion}\n\nThe result for the exponents $\\alpha$ and $\\nu$ at $T_{\\rm c}$ for  $|\\psi^{(1)}| = |\\psi^{(2)}|$ \n shows that when the 3D$xy$\\  and inverted \n3D$xy$\\  critical points collapse onto each other, then instead of  a simple superposition,\none gets a new fixed point which is in a different universality class. This result is \nfar from obvious. Naively one would perhaps have guessed from Eq. (\\ref{Sdual_AH}) that \nfor $N=2$ one has two \ndecoupled vortex modes, one neutral mode exhibiting a phase transition in the 3D$xy$\\  universality \nclass and one charged mode exhibiting a phase transition in the inverted 3D$xy$\\  universality class. \nAt $|\\psi^{(1)}| = |\\psi^{(2)}|$ a naive guess would be that one would have two such phase \ntransitions superimposed on each other, giving $\\alpha$ and $\\nu$ values in the 3D$xy$\\  universality \nclass. However, there is a principal distinction from the case when $|\\psi^{(1)}| \\ne |\\psi^{(2)}|$. \nIn the latter case the upper phase transition is always a charged critical point\nbecause the neutral mode is not developed. Thus at the upper\ntransition the interaction of vortices is of short range, while\nat the lower transition there is a proliferation of vortices\nwith long range interaction. However, in the case $|\\psi^{(1)}| = |\\psi^{(2)}|$, \nthen  below the single phase transition {\\it both} types of vortices have neutral vorticity \nalong with charged vorticity and  thus this phase transition\ncan not be mapped onto a superposition of a neutral and a charged fixed\npoints. \n\nAlso, it is the fact that the system is self-dual at this point that invalidates\nthe naive superposition conjecture, since the 3D$xy$\\  and inverted 3D$xy$\\  \nphase transitions do not describe  phase transitions of a self-dual system. \nEven though the value of $\\nu$ appears to be in good agreement with\nthe 3D Ising value, we observe that the 3D Ising model is not self-dual either, and \nthe new type of critical point for $|\\psi^{(1)}| = |\\psi^{(2)}|$ can \ntherefore not be in the 3D Ising universality class.\nThe origin of the novel exponents is therefore essentially topological,\nshowing that when the vortex loop blowouts of the neutral and \ncharged modes are not well separated, they interact in a \nnon-trivial fashion. There will therefore exist a crossover regime\nparametrized by the field $|\\psi^{(1)}|^2 - |\\psi^{(2)}|^2$\nwhere the exponents $\\alpha$ and $\\nu$ change from 3D$xy$\\ \nvalues to the new values we find here (see Fig. 7 of Ref. \\onlinecite{motrunich2004}).  In principle, \nit is possible to compute the relevant crossover \nexponents in order to shed further light on this new self-dual \nuniversality class.\n\n\n\\section{\\label{mc_simsN=3}Monte Carlo simulations, $N=3$}\n\nIn the model Eq. (\\ref{vortex_action}) with $N=3$ vortex flavors we expect\nin general one charged critical point associated with the condensation of the\ncharged vortex mode and two neutral critical points where neutral\nvortex modes proliferate. To study the phases of this model we have \nperformed MC simulations with the action given in Eq. (\\ref{vortex_action}) \nwith bare phase stiffnesses given in \\tab{tab:n3_stiffnesses}. We have \napplied the same methods for calculating the critical exponents $\\alpha$ \nand $\\nu$ as well as gauge masses as we did for the $N=2$ case.\n\n\nIt is useful to  give the superfluid modes specifically  for the $N=3$ case (see \nAppendix \\ref{App:charged_neutral} for details of the derivation for the general-$N$ \ncase). Using Eq. (\\ref{charge_neutral}), we have for this case\n \\begin{widetext}\n \\begin{eqnarray}\n\\label{gl4}\nS & = &\n\\int d^3\\ve r \\left[\\frac{1}{\\Psi^2} \\Biggl( \\frac{|\\psi^{(1)}|^2}{2} \\nabla\\theta^{(1)} +\n\\frac{|\\psi^{(2)}|^2}{2} \\nabla \\theta^{(2)} + \\frac{|\\psi^{(3)}|^2}{2} \\nabla \\theta^{(3)}- \ne \\Psi^2 {\\bf A}\\Biggr)^2 + V(\\{\\psi^{(\\alpha)}\\}) + \\frac{1}{2}(\\nabla\\times\\ve{A})^2 \\right.\\nonumber \\\\\n&+&\\left.\\frac{|\\psi^{(1)}|^2 |\\psi^{(2)}|^2}{2\\Psi^2}(\\nabla(\\theta^{(1)}-\\theta^{(2)}))^2 \n + \\frac{|\\psi^{(1)}|^2 |\\psi^{(3)}|^2}{2\\Psi^2}(\\nabla(\\theta^{(1)}-\\theta^{(3)}))^2 \n + \\frac{|\\psi^{(2)}|^2 |\\psi^{(3)}|^2}{2\\Psi^2}(\\nabla(\\theta^{(2)}-\\theta^{(3)}))^2\\right]. \n\\label{varsep}\n\\end{eqnarray}\nHere, we have defined $\\Psi^2 = |\\psi^{(1)}|^2 + |\\psi^{(2)}|^2+|\\psi^{(3)}|^2 $.\nIn the regime of short penetration length, the combination of phase gradients which\nis  coupled to the gauge field $\\ve A$ can be gauged away at length scales of the\norder of the penetration length $\\lambda = 1\/e \\Psi$. \nThe remaining gradient terms for the neutral  modes are given by\n\\begin{eqnarray}\nS_{\\rm n}&=&  \n\\int d^3\\ve r\\left[\\frac{|\\psi^{(1)}|^2  |\\psi^{(2)}|^2}\n{2 \\Psi^2}(\\nabla (\\theta^{(1)}-\\theta^{(2)}))^2\\right.  \n+\\frac{|\\psi^{(1)}|^2 |\\psi^{(3)}|^2}\n{2 \\Psi^2}(\\nabla (\\theta^{(1)}-\\theta^{(3)}))^2 \n+\\left.\\frac{|\\psi^{(2)}|^2 |\\psi^{(3)}|^2 }\n{2 \\Psi^2}(\\nabla (\\theta^{(2)}-\\theta^{(3)}))^2\\right].\n\\label{neutral}\n\\end{eqnarray}\nThis action could be inferred also directly from Eq. (\\ref{vortex_action}).\nFor the case $N=3$, we write the action in the vortex representation as \n\n\\begin{eqnarray}\nS_V & = & \\sum_{\\ve q} \\sum_{\\eta=1}^3 \\sum_{\\alpha=1}^3  2 \\pi^2 \\beta |\\psi^{(\\alpha)}|^2  \\ve m^{(\\alpha)}_{\\ve q}\n \\left( \\frac{\\lambda^{(\\eta)} }{|\\ve Q_{\\ve q}|^2 + m_0^2}\n+ \\frac{\\delta_{\\alpha,\\eta}-\\lambda^{(\\eta)}}{|\\ve Q_{\\ve q}|^2}\\right)  \\ve m^{(\\eta)}_{\\ve q} \n\\end{eqnarray}\nwhich when written out takes the form\n\\begin{equation}\n\\begin{split}\n\\frac{S_V}{2 \\pi^2 \\beta\/\\Psi^2} = &  \\sum_{\\ve q} \\left\\{ \n\\frac{\n(\\sum_{\\alpha} |\\psi^{(\\alpha)}|^2  \\ve m^{(\\alpha)}_{\\ve q} ) \\cdot\n(\\sum_{\\eta}  |\\psi^{(\\eta)} |^2  \\ve m^{(\\eta)}_{-\\ve q} )\n}\n{|\\ve Q_{\\ve q}|^2 + m_0^2} \n + \n\\frac{\n|\\psi^{(1)}|^2  |\\psi^{(2)} |^2 (\\ve m^{(1)}_{\\ve q} - \\ve m^{(2)}_{\\ve q})\n\\cdot (\\ve m^{(1)}_{-\\ve q} - \\ve m^{(2)}_{-\\ve q})\n}\n{|\\ve Q_{\\ve q}|^2}  \\right. \\\\\n + &    \\left.\n\\frac{\n |\\psi^{(1)}|^2  |\\psi^{(3)} |^2 (\\ve m^{(1)}_{\\ve q} - \\ve m^{(3)}_{\\ve q})\n\\cdot (\\ve m^{(1)}_{-\\ve q} - \\ve m^{(3)}_{-\\ve q})\n}\n{|\\ve Q_{\\ve q}|^2} \n + \n\\frac{\n|\\psi^{(2)}|^2  |\\psi^{(3)} |^2 (\\ve m^{(2)}_{\\ve q} - \\ve m^{(3)}_{\\ve q})\n\\cdot (\\ve m^{(2)}_{-\\ve q} - \\ve m^{(3)}_{-\\ve q})\n}\n{|\\ve Q_{\\ve q}|^2} \\right\\} .\n\\end{split}\n\\label{neutral_vortex}\n\\end{equation}  \n\\end{widetext}\nThe three last terms in Eq. (\\ref{neutral_vortex}) are nothing but the vortex representation \nof Eq. (\\ref{neutral}). Notice also how all cross-terms between different vortex species\ncancel out for arbitrary bare phase stiffnesses when $m_0^2 =0$. \n\nThus, for  the case $N=3$, we have three phase variables yielding three neutral gauge \ninvariant combinations  of phase differences. This amounts to two true neutral modes, \nthe remaining degree of freedom is associated with the composite charged mode, which \nabsorbs $\\ve A$ and yields a massive vector field  via the Higgs mechanism. If all three \nbare phase stiffnesses $|\\psi^{(1)}|$, $|\\psi^{(2)}|$, and $|\\psi^{(3)}|$ are different, \nthis yields one charged inverse 3D$xy$\\  critical point where the Meissner effect sets in, and \ntwo neutral 3D$xy$\\  critical points at lower temperatures,  all separate. Consider now \n$|\\psi^{(1)}| = |\\psi^{(2)}| < |\\psi^{(3)}|$.  \nThen the charged mode proliferates at the highest critical temperature where the Meissner-effect \nsets in, and the two neutral modes proliferate simultaneously at a lower temperature.\nThe highest transition is still an inverted 3D$xy$\\  transition, the lower one is a neutral \n3D$xy$\\  critical point. {\\it Note how this is dramatically different from the case $N=2$, when \nthe original neutral 3D$xy$\\  critical point was collapsed on top of the inverted 3D$xy$\\  critical \npoint, resulting in a new universality class of the phase transition, essentially due \nto the self-duality of the $N=2$ system. It is also evident that collapsing a neutral \nand a charged fixed point is quite different from collapsing two neutral fixed points.} \n\nFor the case $|\\psi^{(1)}| = |\\psi^{(2)}| < |\\psi^{(3)}|$, in terms of the masses \nof $\\ve A$ and the two  dual gauge fields associated with the neutral modes, $m_{\\ve A}$ \nis non-zero below the {\\it upper} critical temperature, while the two dual gauge fields become \nmassive above the {\\it lower} critical temperature.\nIn this case, the degenerate lower critical point is therefore\na 3D$xy$\\  critical point, while the upper critical point is an inverted 3D$xy$\\  critical point. \n\nA further interesting possibility is to set $|\\psi^{(1)}| <\n|\\psi^{(2)}| = |\\psi^{(3)}|$.\nConsider the masses of $\\ve A$ and the two dual gauge fields associated with the neutral mode in this \ncase. At the lower critical temperature, one neutral vortex mode proliferates\nin a 3D$xy$\\  transition, generating a mass to the dual gauge field (thus breaking\none dual gauge symmetry). This mode is therefore dual-higgsed out of the problem at\nhigher temperatures. The gauge field $\\ve A$ becomes massive {\\it below} the upper critical \ntemperature, while the dual gauge field associated with the remaining neutral mode\nbecomes massive {\\it above} the same upper critical temperature. Hence, the\nsituation at the upper critical point corresponds precisely to the case \n$N=2$, $|\\psi^{(1)}| = |\\psi^{(2)}|$, for which we have already seen that a \nnon-3D$xy$\\  critical point emerges.  When all bare stiffnesses are equal, \n$|\\psi^{(1)}| = |\\psi^{(2)}| = |\\psi^{(3)}|$, all three fixed point collapse. \nWe present MC simulations for the three cases given in\n\\tab{tab:n3_stiffnesses}, of which the case  $|\\psi^{(1)}| <\n|\\psi^{(2)}| < |\\psi^{(3)}|$ is the most pertinent \nto mixtures of superconducting condensates of for instance hydrogen and deuterium, \nor hydrogen and tritium. \n\n\n\n \n\\begin{table}[h]\n\\label{Sweeps}\n\\caption{\\label{tab:n3_stiffnesses}Phase stiffnesses\n  $|\\psi^{(\\alpha)}|$ and bare masses $m_0^2$ for the $N=3$ MC\n  simulations. In all cases the charge $e=1\/2$.}\n\\begin{tabular}{ccccc}\n\\hline\nCase & $|\\psi^{(1)}|^2$ & $|\\psi^{(2)}|^2$ & $|\\psi^{(3)}|^2$ & $m_0^2$  \\\\\n\\hline\n1    &   1\/3              &       2\/3        &     4\/3          &  7\/12  \\\\ \n2    &   1\/2              &       1\/2        &     4\/3          &  7\/12  \\\\ \n3    &   7\/9              &       7\/9        &     7\/9          &  7\/12  \\\\ \n\\hline\n\\end{tabular} \n\\end{table}\n\n\n\n\\subsection{Critical exponents $\\alpha$ and $\\nu$, $|\\psi^{(1)}| <\n  |\\psi^{(2)}| < |\\psi^{(3)}|$}\n\nMC simulations are performed for a $N=3$ system with bare phase\nstiffnesses $|\\psi^{(1)}|^2 =1\/3$, $|\\psi^{(2)}|^2=2\/3$,\n$|\\psi^{(3)}|^2=4\/3$ and system sizes $L=4,6,8,10,12,14,16$. \nWe sample the second moment of the action Eq. (\\ref{vortex_action}) and\nfind three anomalies for temperatures $T_{\\rm c1}$, $T_{\\rm c2}$, and\n$T_{\\rm c3}$, which from FSS are found to be  $T_{\\rm c1}=0.98$,\n$T_{\\rm c2}=1.92$, and $T_{\\rm c3}=3.63$.  \n\nFrom a FSS analysis of the third moment of the action, \nwe have measured the critical exponents $\\alpha$ \nand $\\nu$. The FSS plots are given in \\fig{m3_fss_n3_1}. We find $\\alpha =\n-0.03 \\pm 0.02$ and $\\nu = 0.65 \\pm 0.02$ for $T_{\\rm c1}$, $\\alpha =\n-0.02 \\pm 0.02$ and $\\nu = 0.66 \\pm 0.01$ for $T_{\\rm c2}$, and $\\alpha =\n-0.01 \\pm 0.03$ and $\\nu = 0.69 \\pm 0.02$ for $T_{\\rm c3}$. These values\nare consistent with the values for the 3D$xy$\\  and the inverted 3D$xy$\\  universality classes.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.6}{\\rotatebox{-90.0}{\\includegraphics{m3scaling_n3_1.ps}}}} \n\\caption{\\label{m3_fss_n3_1} FSS of the peak to peak value of the third moment of action \n $\\Delta M_3$ for $N=3$ with $|\\psi^{(1)}|^2 =1\/3$, $|\\psi^{(2)}|^2=2\/3$, \n$|\\psi^{(3)}|^2=4\/3$ labeled ($\\blacktriangle$), ($\\vartriangle$), and ($\\bullet$), \n for $T_{\\rm c1}$, $T_{\\rm c2}$ and $T_{\\rm c3}$, respectively. The scaling of the \n width between the peaks $\\Delta\\beta$ ($\\circ$), ($\\blacksquare$), and labeled \n ($\\Box$), for $T_{\\rm c1}$, $T_{\\rm c2}$, and $T_{\\rm c3}$ respectively. The lines \n are power law fits to the data for $L>6$ used to extract $\\alpha$ and $\\nu$.\n}\n\\end{figure}\n\n\n\n\\subsection{Vortex correlator, Higgs mass, and anomalous scaling\n dimension, $|\\psi^{(1)}| < |\\psi^{(2)}| < |\\psi^{(3)}|$}\nIn the Higgs phase, we expect the gauge field correlator\n$\\mathcal{G}_{\\ve A}(\\ve q)$ in Eq. (\\ref{A_gaugeprop}) \nto scale according to the Ansatz Eq. (\\ref{prop_ansatz}). For each coupling \nwe fit $\\mathcal{G}_{\\ve  A}(\\ve q)^{-1}$ from the MC simulations \nfor system sizes $L=8,12,20$ and estimate the gauge field mass $m_{\\ve A}$. \n\nThe results for the vortex correlator $G^{(+)}({\\ve q})$ in Eq. (\\ref{Gpluss_corr}) and \nthe Higgs mass Eq. (\\ref{gauge_field_masses}) are given in \\fig{Gq_m3_1}. Note how the \n$\\ve q$-dependence of $G^{(+)}({\\ve q})$ changes when the temperature is \nvaried from above to below $T_{\\rm{c3}}$ from $G^{(+)}({\\ve q}) \\sim \\rm{const}$ \nto $G^{(+)}({\\ve q}) \\sim q^2$, respectively. Note also how the $\\ve q$-behavior\nof the vortex correlator remains unchanged when the temperature is varied through \n$T_{\\rm{c2}}$ and $T_{\\rm{c1}}$, i.e. it remains $G^{(+)}({\\ve q}) \\sim q^2$.\nThis reflects the fact that the field $\\ve A$ has been higgsed out of the problem\nat $T_{\\rm{c3}}$ such that the vortex tangle is incompressible below this \ntemperature. From Eq. (\\ref{gauge_field_masses}) it is therefore clear that a Higgs \nmass is generated at $T_{\\rm{c3}}$ by the establishing of a charged superconducting\nmode. Moreover, when the two additional neutral superfluid modes are established \nat $T_{\\rm{c2}}$ and $T_{\\rm{c1}}$, this adds to the total superfluid density and \nhence leads to kinks in the London penetration length and thereby $m_{\\ve A}$.\n\n\nPrecisely at $T_{\\rm{c3}}$, $m_{\\ve A}$ vanishes, and the scaling\nAnsatz given by \\eq{n1_corr}\nmay be used to extract $\\eta_{\\ve A}$. From \\fig{Gq_m3_1} and\n$G^{(+)}(\\ve q)$ at $T_{\\rm{c3}}$, we extract $\\eta_{\\ve A}=1$, from\nwhich we  \nconclude that the critical point at $T_{\\rm{c3}}$ is an inverted 3D$xy$\\  critical point. \nLikewise, from the $G^{(+)}(\\ve q) \\sim q^2$ behavior at $T_{\\rm{c1}}$ and $T_{\\rm{c2}}$ \nwe conclude that these two critical points feature $\\eta_{\\ve A}=0$ and hence represent \n3D$xy$\\  critical points.\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.60}{\\rotatebox{-90.0}{\\includegraphics{mGq_n3_1.ps}}}} \n\\caption{\\label{Gq_m3_1} Results for the vortex correlator Eq. (\\ref{Gpluss_corr}), \nand the Higgs mass Eq. (\\ref{gauge_field_masses}) for the case $|\\psi^{(1)}|^2 =1\/3 ,|\\psi^{(2)}|^2=2\/3, \n|\\psi^{(3)}|^2=4\/3$. The upper panel shows \n$G^{(+)}({\\ve q})$ as a function of $|\\ve Q_{\\ve q}|$ for seven temperatures starting from \nabove: Above and close to $T_{\\rm {c3}}$,  above and close to $T_{\\rm {c2}}$, \nabove and close to $T_{\\rm {c1}}$, and below $T_{\\rm c1}$. Above $T_{\\rm{c3}}$, the vortices are seen to \nhave condensed, $G^{(+)}({\\ve q}) \\sim \\rm{const}$ while for all temperatures \nbelow $T_{\\rm{c3}}$, including above and below $T_{\\rm {c1}}$ and \n$T_{\\rm {c2}}$, $G^{(+)}({\\ve q}) \\sim q^2$ for small $\\ve q$. The lower panel \nshows the Higgs mass as a function of temperature, showing the onset of Meissner \neffect at $T_{\\rm{c3}}$, and the additional anomalies at $T_{\\rm {c1}}$ and \n$T_{\\rm {c2}}$ due to the appearance of additional neutral\nmodes at these temperatures.\n}\n\\end{figure}\n\n\n\n\\subsection{Critical exponents $\\alpha$ and $\\nu$, $|\\psi^{(1)}| =\n  |\\psi^{(2)}| < |\\psi^{(3)}|$}\n\nMC simulations have been performed for a $N=3$ system with bare phase\nstiffnesses $|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 = 1\/2$ and\n$|\\psi^{(3)}|^2=4\/3$ and system sizes $L=4,6,8,10,12,14,16$.\nBy measuring the second moment of the action Eq. (\\ref{vortex_action})\nwe find two anomalies for the temperatures $T_{\\rm c1}$ and $T_{\\rm\n  c2}$, which from FSS are found to be  $T_{\\rm c1}=1.46$ and $T_{\\rm c2}=3.63$. \nFrom a FSS analysis of the third moment of the action \nwe have measured the critical exponents $\\alpha$ \nand $\\nu$. The FSS plots are given in \\fig{m3_fss_n3_2}. We find $\\alpha =\n-0.03 \\pm 0.02$ and $\\nu = 0.65 \\pm 0.02$ for $T_{\\rm c1}$, and $\\alpha =\n-0.03 \\pm 0.03$ and $\\nu = 0.68 \\pm 0.02$ for $T_{\\rm c2}$. These values\nare consistent with the values for the 3D$xy$\\  and the inverted 3D$xy$\\  universality classes.\n\n\n\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.6}{\\rotatebox{-90.0}{\\includegraphics{m3scaling_n3_2.ps}}}} \n\\caption{\\label{m3_fss_n3_2} FSS of the peak to peak value of the third moment of action \n $\\Delta M_3$ for $N=3$ for $|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 =\n 1\/2$ and $|\\psi^{(3)}|^2 = 4\/3$, labeled ($\\blacktriangle$), and ($\\bullet$), \n for $T_{\\rm c1}$ and $T_{\\rm c2}$, respectively. The scaling of the \n width between the peaks $\\Delta\\beta$ labeled ($\\circ$),\n ($\\blacksquare$), for $T_{\\rm c1}$ and $T_{\\rm c2}$ respectively. The lines \n are power law fits to the data for $L>6$ used to extract $\\alpha$ and $\\nu$.}\n\\end{figure}\n\n\n\\subsection{Vortex correlator, Higgs mass, and anomalous scaling\n dimension, $|\\psi^{(1)}| = |\\psi^{(2)}| < |\\psi^{(3)}|$}\n\nLike the previous case, we extract the gauge field mass by fitting the\ngauge field correlators for small $\\ve q$ to the Ansatz Eq. (\\ref{prop_ansatz}) for\nsystem sizes $L=8,12,20$. \n\nThe results for the vortex correlator $G^{(+)}({\\ve q})$ in Eq. (\\ref{Gpluss_corr}) and the Higgs mass \ndefined in \\eq{gauge_field_masses}  are\ngiven in \\fig{Gq_m3_2}. Note how the $\\ve q$-dependence of\n$G^{(+)}({\\ve q})$ changes when the temperature is \nvaried from above to below $T_{\\rm{c2}}=3.63$ from $G^{(+)}({\\ve q}) \\sim \\rm{const}$ \nto $G^{(+)}({\\ve q}) \\sim q^2$, respectively. Note also how the $\\ve q$-behavior\nof the vortex correlator remains unchanged when the temperature is varied through \n$T_{\\rm{c1}}=1.46$, i.e. it remains $G^{(+)}({\\ve q}) \\sim q^2$.\nThis reflects the fact that the field $\\ve A$ has been higgsed out of the problem\nat $T_{\\rm{c2}}=3.63$ such that the vortex tangle is incompressible below this \ntemperature. From Eq. (\\ref{gauge_field_masses}) it is therefore clear that a Higgs \nmass is generated at $T_{\\rm{c2}}$ by the establishing of a charged superconducting\nmode. Moreover, when the two additional neutral superfluid modes are established \nat $T_{\\rm{c2}}$ this adds to the total superfluid density and \nhence leads to a kink in the London penetration length and thereby $m_{\\ve A}$.\n\n\nPrecisely at the charged transition $T_{\\rm{c2}}$, $m_{\\ve A}$ vanishes\nand we find the gauge field correlator has the form $G^{(+)}(\\ve q)\n\\sim |\\ve q|^{2-\\eta_{\\ve A}}$. From the $G^{(+)}(\\ve\nq)$ data in \\fig{Gq_m3_2} we extract $\\eta_{\\ve A}=1$, from which we \nconclude that the critical point at $T_{\\rm{c2}}$ is an inverted 3D$xy$\\  critical point. \nLikewise, from the behavior of $G^{(+)}(\\ve q) \\sim q^2$ at $T_{\\rm{c1}}$\nconclude that this critical point features $\\eta_{\\ve A}=0$ and hence represents \na 3D$xy$\\  critical point.\n\n\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.60}{\\rotatebox{-90.0}{\\includegraphics{mGq_n3_2.ps}}}} \n\\caption{\\label{Gq_m3_2} Results for the vortex correlator Eq. (\\ref{Gpluss_corr}), \nand the Higgs mass Eq. (\\ref{gauge_field_masses}) for the case\n$|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 = 1\/2$ and $|\\psi^{(3)}|^2 = 4\/3$. The upper panel shows \n$G^{(+)}({\\ve q})$ as a function of $|\\ve Q_{\\ve q}|$ for five temperatures starting from \nabove: Above and close to $T_{\\rm {c2}}$, \nabove and close to $T_{\\rm {c1}}$, and below $T_{\\rm c1}$. Above $T_{\\rm{c2}}$, the vortices are seen to \nhave condensed, $G^{(+)}({\\ve q}) \\sim \\rm{const}$ while close to\n$T_{\\rm{c2}}$, $G^{(+)}({\\ve q}) \\sim |\\ve q|$. For all temperatures \nbelow $T_{\\rm{c2}}$, including above and below $T_{\\rm {c1}}$, \n$G^{(+)}({\\ve q}) \\sim q^2$ for small $\\ve q$. The lower panel \nshows the Higgs mass as a function of temperature, showing the onset of Meissner \neffect at $T_{\\rm{c2}}$, and an additional anomaly at $T_{\\rm {c1}}$ due to the appearance of additional neutral\nmodes at this temperature. }\n\\end{figure}\n\n\n\n\\subsection{Critical exponents $\\alpha$ and $\\nu$, $|\\psi^{(1)}| =\n  |\\psi^{(2)}| = |\\psi^{(3)}|$}\n\nMC simulations are performed for a $N=3$ system with equal bare phase\nstiffnesses $|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 = |\\psi^{(3)}|^2 = 7\/9$\nand system sizes $L=4,6,8,10,12,14,16$.\nFrom measurements of the second moment of the action Eq. (\\ref{vortex_action})\nwe find one anomaly for temperature the $T_{\\rm c}$, which from FSS is  \nfound to be  $T_{\\rm c}=2.19$. \nFrom a FSS analysis of the third moment of the action we have measured\nthe critical exponents $\\alpha$  \nand $\\nu$. The FSS plots are given in \\fig{m3_fss_n3_3}. We find $\\alpha =\n0.02 \\pm 0.03$ and $\\nu = 0.59 \\pm 0.02$. The values appear not to\nagree with hyper scaling. They are \\textit{not} consistent with the \n3D$xy$\\  universality class. \n\nThe above values for $\\alpha$ and $\\nu$  are however in agreement with \nthose found for the case $N=2$, $|\\psi^{(1)}| = |\\psi^{(2)}|$. We \nobserve, based on the numerical results for the two cases\n$N=2$, $|\\psi^{(1)}| = |\\psi^{(2)}|$ and\n$N=3$, $|\\psi^{(1)}| = |\\psi^{(2)}|=|\\psi^{(3)}|$ compared to\nthe other cases that we have considered, that collapsing two neutral\ncritical points in the 3D$xy$\\  universality class leads to a single\ncritical point also in the 3D$xy$\\  universality class. On the other\nhand, it appears that collapsing\n$N-1$ neutral critical points in the 3D$xy$\\  universality class\n{\\it and one charged fixed point in the inverted} 3D$xy$\\  universality class\nleads to an $N$-fold degenerate single critical point in a universality class \n(which in principle depends on $N$) which is not that of the 3D$xy$\\  or inverted \n3D$xy$\\  type. For $N=2$, we may define the universality class as that of a 3D \nself-dual $U(1)\\times U(1)$ gauge theory, while it is less clear what it is for other \n$N \\geq 3$. \n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.6}{\\rotatebox{-90.0}{\\includegraphics{m3scaling_n3_3.ps}}}} \n\\caption{\\label{m3_fss_n3_3} FSS of the peak to peak value of the third moment of action \n $\\Delta M_3$ for $N=3$ for $|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 =\n |\\psi^{(3)}|^2 = 7\/9$ labeled ($\\blacktriangle$). The scaling of the  \n width between the peaks $\\Delta\\beta$ labeled ($\\circ$). The lines \n are power law fits to the data for $L>6$ used to extract $\\alpha$ and $\\nu$.}\n\\end{figure}\n\n\n\n\\subsection{Vortex correlator, Higgs mass, and anomalous scaling\n dimension, $|\\psi^{(1)}| = |\\psi^{(2)}| = |\\psi^{(3)}|$}\n\n\nWe extract the gauge field mass $m_{\\ve A}$ by fitting the \ngauge field correlators for small $\\ve q$ to the Ansatz Eq. (\\ref{prop_ansatz}) for\nsystem sizes $L=8,12,20,32$. \n\nThe results for the vortex correlator $G^{(+)}({\\ve q})$ in Eq. (\\ref{Gpluss_corr}) and the Higgs mass \ndefined in Eq. (\\ref{gauge_field_masses}) are given in \\fig{Gq_m3_3}. \nNote how the $\\ve q$-dependence of $G^{(+)}({\\ve q})$ changes when the temperature is \nvaried from above to below $T_{\\rm{c}}=2.20$ from $G^{(+)}({\\ve q}) \\sim \\rm{const}$ \nto $G^{(+)}({\\ve q}) \\sim q^2$, respectively.  From\nEq. (\\ref{gauge_field_masses}) it is therefore clear that a Higgs  \nmass is generated at $T_{\\rm{c}}=2.19$ by the establishing of a charged superconducting\nmode. From $G^{(+)}({\\ve q})$ measurements at $T_c$ we find the\nanomalous scaling dimension to be $\\eta_{\\ve A}=1$.\n\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.60}{\\rotatebox{-90.0}{\\includegraphics{mGq_n3_3.ps}}}} \n\\caption{\\label{Gq_m3_3} Results for the vortex correlator Eq. (\\ref{Gpluss_corr}), \nand the Higgs mass Eq. (\\ref{gauge_field_masses}) for the case\n$|\\psi^{(1)}|^2 = |\\psi^{(2)}|^2 = |\\psi^{(3)}|^2 = 7\/9$. The upper panel shows \n$G^{(+)}({\\ve q})$ as a function of $|\\ve Q_{\\ve q}|$ for temperatures\nabove and close to $T_{\\rm {c}}$, and below $T_{\\rm c}$. Above\n$T_{\\rm{c}}$, the vortices have condensed, $G^{(+)}({\\ve q}) \\sim\n\\rm{const}$. \nBelow $T_{\\rm{c}}$, $G^{(+)}({\\ve q}) \\sim q^2$ for small $\\ve q$. The lower panel \nshows the Higgs mass as a function of temperature, showing the onset of Meissner \neffect at $T_{\\rm{c}}$.}\n\\end{figure}\n\n\n\n\\subsection{General $N$}\n\nThe critical properties of the $N$-component system are governed solely by excitations of \nvortex loops with fractional flux. That is, in the $N=2$ case, $T_{\\rm c1}$ is governed by \nproliferation of the vortex loops with phase windings ($\\Delta \\theta^{(1)}=2\\pi, \\Delta\n\\theta^{(2)}=0$), while  $T_{\\rm c2}$ marks the onset of proliferation of the loops of  \nvortices with windings ($\\Delta \\theta^{(1)}=0, \\Delta \\theta^{(2)}=2\\pi$). Remarkably, \nfor general $N$, below the temperature $T_{{\\rm c}N-1}$, where $T_{\\rm\n  c1} < \\cdots < T_{{\\rm c}N-1}< T_{{\\rm c}N}$, topological excitations \nwith nontrivial windings only in one phase has a logarithmically divergent energy\n\\cite{frac,smiseth2004}. Moreover,  the composite vortex loops \n($\\Delta \\theta^{(1)}=2\\pi, \\Delta \\theta^{(2)}=2 \\pi$) which in contrast have finite \nenergy per unit length, do not play a role as far as  critical properties  are concerned. \n\nFor the\ncase $N=2$, the \ncritical point at $T_{\\rm c2}>T_{\\rm c1}$ is a charged fixed point. Proliferation of the vortex loops \n($\\Delta \\theta^{(1)}=2\\pi, \\Delta \\theta^{(2)}=0$) at $T_{\\rm c1}$  eliminates the neutral mode. \nOn the other hand, the composite vortices ($\\Delta \\theta^{(1)}=2\\pi, \\Delta \\theta^{(2)}=2\\pi$) \ndo not feature neutral vorticity at any temperature and thus can be mapped onto vortices in a \n$N=1$ superconductor with bare phase stiffness $|\\psi^{(1)}|^2+|\\psi^{(2)}|^2$. \nA characteristic temperature of proliferation of such vortex loops is higher than $T_{\\rm c2}$, \nwhich excludes the composite vortices from  the sector of  critical fluctuations in the system.\nThe same argument applies to the $N>2$ case.\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.8}{\\rotatebox{0.0}{\\includegraphics{conclusions_.eps}}}} \n\\caption{  \\label{transitions}  (Color online). Phase transitions in the $N$-flavor London superconductor \nwith different bare stiffnesses of the $N$ order parameter components. The green line is \nthe gauge field mass $m_{\\ve A}$. At the highest temperature the system becomes superconducting \nvia a phase transition in the inverted 3D$xy$\\  universality class.  At the lower transitions \nthe system develops composite neutral superfluid modes in the superconducting state via a \nseries of $N-1$ phase transitions, all in the 3D$xy$\\  universality class.} \n\\end{figure}\n\n\nSummarizing the previous two sections, the resulting schematic phase diagram of the \n$N$-flavor London superconductor in the absence of external field is presented in \n\\fig{transitions}. Assuming the bare stiffnesses have been chosen to have \nwell separated bare energy scales associated with the twist of phases of every flavor, \nwe find $N$ distinct critical points. At the highest critical temperature, the charged \nvortex mode condenses and the gauge field acquires a mass, driving the system into a \nsuperconducting phase. For lower critical temperatures, neutral vortex loops condense \nand the system develops superfluid modes. Hence, in zero magnetic field there are \n$N-1$ superfluid modes arising in a superconducting state. \n\n\n\n\n\n\n\n\n\n\\section{\\label{N=2_extfield} $N=2$ system in an external magnetic field, lattice and \nsub-lattice melting, and metallic superfluidity}\n\nWe next discuss the situation  when the system is subjected to an external magnetic field.\n{ Two important aspects  of the physics to be described below, are\n{\\it{ i)}} three dimensionality and {\\it{ii)}}  a significant difference in the bare \nstiffnesses of the condensates.}\nAs discussed recently  \\cite{BSA,SSBS}, when an external magnetic field is applied to a \nthree dimensional type-II \n$N$-component superconductor,  it  changes its properties much more dramatically than in the\nordinary  $N=1$ case.\nThe composite charged vortices have finite energy per unit length and\ncouple to the magnetic field, \nand hence are relevant for magnetic properties.\nIf the bare stiffnesses of the fields are different,\n the existence of composite purely charged vortices\nresults in a particularly rich phase diagram \nwith several novel phases and phase transitions. \nNote that in the following two chapters we denote a constituent vortex originating in a \n$2\\pi$ phase-winding in $ \\theta^{(\\alpha)}$ a \\textit{type-$\\alpha$\n  vortex}, where  $\\alpha\\in [1,\\dots,N]$. \n\n\n\\subsection{$N=2$ system in external field at $T=0$}\nIn the presence of an external magnetic field, but in the absence of thermal \nfluctuations, the formation of an Abrikosov lattice of non-composite\nvortices is forbidden because these defects have a logarithmically\ndivergent energy\\cite{frac,smiseth2004}, cf. discussion following Eq. (\\ref{potential_0}). \nIn a type-II $N$-component system, the system forms a\nlattice of  \n{\\it composite vortices} for which $\\Delta \\theta^{(\\alpha)}=2\\pi$ for every \n$\\alpha\\in [1,\\dots,N]$. A schematic picture of the resulting \nlattice of composite vortices in an $N=2$ superconductor is  shown in Fig. \\ref{lattice}. \nIn the discussion below we consider the type-II limit, but not extreme type-II since \nthe interaction between vortices of different species is depleted at the length scales \nsmaller than the penetration length, cf. Eq. (\\ref{potential}) and the discussion following\nEq. (\\ref{potential_0}). We do not discuss effects of this depletion assuming \na moderately short penetration length scale.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.09}{\\rotatebox{0.0}{\\includegraphics{tzero.eps}}}} \n\\caption{ \\label{lattice} (Color online). A type-II, $N=2$ system at zero temperature in external magnetic \nfield forms a lattice of composite Abrikosov vortices. A composite\nvortex may be viewed as co-centered type-1 (red) and type-2 (blue) vortices $(\\Delta \\theta^{(1)}=2\\pi, \\Delta \\theta^{(2)}=2\\pi)$.}\n\\end{figure}\n\n\n\\subsection{Effects of low-temperature fluctuations on field-induced composite vortices}\nIn this subsection, we will consider the effects of thermal\nfluctuations, and how it affects the Abrikosov vortex  \nlattice of composite vortices defined above.\n\n\n\\subsubsection{Thermal generation of loop-like splitting of line vortices}\nAt finite temperature, the $N$-component system subjected to a magnetic field $\\ve B$ will \nexhibit thermal excitations in the form of vortex loops with fractional flux similar to the \n${\\ve B}=0$ discussion in the first part of this paper. \nWe observe that since the field-induced composite vortices are logarithmically bound\n\\cite{frac,smiseth2004}, thermal fluctuations will induce a {\\it local splitting} of composite \nvortices {\\it in a configuration of two half-loops connected to  a straight line} \n\\cite{BSA,SSBS} as shown in Fig. \\ref{fluct1}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.09}{\\rotatebox{0.0}{\\includegraphics{thermalfl.eps}}}} \n\\caption{\\label{fluct1}  (Color online). Low-temperature fluctuations\nin the $N=2$ system subjected to a magnetic field. Thermal fluctuations generate closed \nloops of composite fractional flux vortices and  {\\it local} splitting of field-induced \ncomposite vortex lines. The type-1 vortices (red) are the vortices of the component with the \nlowest  bare phase stiffness.  When these vortices are viewed as world-lines of bosons, \nthey constitute the ``lighter\" of the vortex species. These ``light\" vortices fluctuate \nmore strongly than the ``heavier\"  type-2 vortices (blue).} \n\\end{figure}\nWe observe that every branch of  a ``split loop\" formed on a field-induced  vortex \nline features neutral as well as charged vorticity. The interaction between these two \nbranches is mediated by a neutral vortex mode exclusively associated\nwith the phase \ndifference $\\gamma=\\theta^{(1)}-\\theta^{(2)}$. The \\textit{screened charged mode} does not contribute  \nto the interaction between the two branches.\nThis is implicit in\nEq. (\\ref{potential}) as follows. The\nvortex segments of different flavors do not interact at short distances\nmuch smaller than $\\lambda = m_0^{-1}$, where the charge, or $m_0^2$ appearing \nin the interaction matrix \\eq{potential}, can be ignored. On such length scales, the \nscreened part of the interaction matrix is essentially unscreened,\nand is canceled by  the inter-flavor interaction, which is \nunscreened on all length scales. Hence, as also discussed in Section \\ref{model}, \nthe intra-vortex interaction is strongly reduced at length scales smaller than \n$\\lambda$.\n\nMoreover, in terms of the field $\\gamma=\\theta^{(1)}-\\theta^{(2)}$, two split branches of a composite \nfield-induced  vortex have opposite vorticities ($\\Delta\\gamma=2\\pi$ on one branch and \n$\\Delta\\gamma=-2\\pi$ on another branch). On the other hand, such a loop emits two integer flux \nvortices at its top and bottom, which do {\\it not} feature neutral vorticity. So the process of \nsuch a thermal local splitting of a field-induced line may be mapped onto a thermally generated \nproliferation of  {\\it closed} vortex loops in the artificial phase field $\\gamma$ as those in \nthe neutral $N=1$ model in absence of magnetic field \\cite{tesanovic1999,nguyen1999}. Hence, somewhat \ncounterintuitively such a splitting transition\nshould be in the  3D$xy$\\  universality class \\cite{BSA,SSBS} .  This transition, being topological in its origin, should not be \nconfused with the topological Kosterlitz-Thouless transition known to occur in planar systems.\n\n\n\\subsubsection{Melting}\nApart from  the splitting of composite vortices and generation of closed vortex loops, the thermal \nfluctuations will produce one more competing process. That is, the lattice of composite vortices\ncan be mapped onto an ordinary vortex lattice in a one-component superconductor. \nSufficiently strong thermal fluctuations drive {\\it a first-order melting transition} of the \nfield-induced Abrikosov lattice \\cite{Hetzel1992,Fossheim_Sudbo_book,nguyen1999}. A counterpart to \nthis effect for the case $N=2$ when $|\\psi^{(1)}| \\ne |\\psi^{(2)}|$ is much more complicated. \nWe next consider this process in the regimes of low and high magnetic fields, separately.\n\n\n\n\n\\subsection{Sublattice melting in low magnetic fields}\nConsider the case of weak magnetic field (much smaller than the upper critical magnetic \nfield for which superconductivity is essentially destroyed) for the situation where \n$|\\psi^{(1)}| \\ll |\\psi^{(2)}|$. Introducing a characteristic temperature \nassociated with a melting of the type-2 vortex\nlattice in the absence of the condensate \n$\\Psi_0^{(1)}$, then at sufficiently low magnetic field this melting temperature will be much \nhigher than the characteristic temperature of thermal decomposition of a composite vortex line \ninto two individual vortex lines. Thus, the first transition that would be encountered upon heating \nthe system, is the thermal splitting of field-induced composite\nvortices into separate type-1 and type-2 vortices.\nThis would be accompanied by a  proliferation of closed loops of type-1 vortices, {\\it while the \nvortices of type-2 will remain arranged in a lattice}. We will denote this phase transition as \n{\\it sublattice melting} \\cite{BSA,SSBS}. The critical temperature of\nthis phase transition is denoted $T_{\\rm SLM}$ (see \\fig{fd}). A schematic picture of the sublattice vortex liquid \nis given in \\fig{sl}. As discussed above, upon thermal decomposition of the composite  vortices, \nthe emerging individual vortices can be mapped onto positively and negatively electrically \ncharged strings which logarithmically interact with each other. \n\n{\\it Quite remarkably, the Abrikosov lattice order for \nthe component with the highest phase stiffness survives the decomposition \ntransition}, for the following reason.  The dominant interaction between individual vortices \nis the long-ranged interaction mediated by neutral vorticity, cf. Eq. (\\ref{potential}).\nThis permits a  mapping of such vortices onto positively and negatively charged strings. Upon thermal \ndecomposition, the effective long-range Coulomb interaction\nmediated by the neutral mode  is screened without affecting \nthe charged modes.\nConsider the case when $|\\psi^{(1)}|\\ll|\\psi^{(2)}|$.\nThen the stiffness $|\\psi^{(2)}|$ is large enough to keep\nthe type-2 vortices arranged in a lattice while the stiffness $|\\psi^{(1)}|$  \nis too weak to constrain type-1 vortices to the lattice. Thus, the\n``light\" type-1 vortex lines are in their molten phase. This is the physical origin of the sublattice melting process. \nThe situation is illustrated in \\fig{sl}. We emphasize that the existence of the regime \nof sublattice melting follows from the fact that the stiffness of the\nneutral mode, which keeps composite \nvortices bound at low temperatures, is always smaller than the smallest stiffness of the individual \ncondensates, namely\n\\begin{eqnarray}\nJ_{\\rm neutral}=\\frac{|\\psi^{(1)}|^2|\\psi^{(2)}|^2}{|\\psi^{(1)}|^2+|\\psi^{(2)}|^2} < |\\psi^{(1)}|^2.\n\\end{eqnarray}\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.09}{\\rotatebox{0.0}{\\includegraphics{sublatticel.eps}}}} \n\\caption{ \\label{sl} (Color online). A vortex liquid of type-1 vortices (red) immersed\n  in a background of a type-2 vortex lattice (blue) in the $N=2$ system in the regime \n$|\\psi^{(1)}| \\ll |\\psi^{(2)}|$. This is the type-1 vortex sublattice melting.\nThere is a temperature region in low magnetic \nfield when ``light\" vortices are decoupled and form a liquid. ``Light\" vortex loops are \nproliferated, while ``heavy\" vortices form a lattice immersed a liquid of ``heavy\" vortex \nloops. Both heavy and light vortices carry only a fraction of magnetic flux quantum in \nthis state.} \n\\end{figure}\n\n\n\n\n\\subsection{Composite vortex lattice melting in strong magnetic fields}\n\nIt is known  from the $N=1$ system that an increase in magnetic field suppresses the melting\ntemperature of the vortex lattice\\cite{Fossheim_Sudbo_book}. Thus, an important \nand characteristic  feature of the phase diagram of the $N=2$ system\nis that the composite vortex lattice melting curve should at some point cross the \ndecomposition curve. Thus, the phase diagram should feature a composite vortex liquid \nphase in the low-temperature, high-magnetic field corner. A schematic picture of this \nphase is given in Fig. \\ref{liquid}\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.09}{\\rotatebox{0.0}{\\includegraphics{liquid.eps}}}} \n\\caption{\\label{liquid} (Color online). Liquid of composite vortices in the $N=2$ model\nimmersed in a liquid of non-proliferated vortex loops.\nIt is realized for $|\\psi^{(1)}| \\ll |\\psi^{(2)}|$ in strong magnetic fields.} \n\\end{figure}\n\n\n\n\n\\subsection{Vortex line plasma in the $N=2$ model}\n\nIf the temperature is raised either at strong or weak magnetic fields,\na  situation arises \nwhere all field-induced composite vortices are decomposed and disordered. In addition, closed loops \nhave proliferated\\cite{tesanovic1999,nguyen1999,Fossheim_Sudbo_book}.\nA schematic picture of this state is shown in Fig. \\ref{normal}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.09}{\\rotatebox{0.0}{\\includegraphics{normal.eps}}}} \n\\caption{\\label{normal} (Color online). Plasma  of fractional vortices in the $N=2$ model in the regime \n$|\\psi^{(1)}| \\ll |\\psi^{(2)}|$ at high temperatures.} \n\\end{figure}\nThe resulting phase diagram of the $N=2$ GL model featuring the various transitions described \nabove, is shown in Fig. \\ref{fd}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.049}{\\rotatebox{0.0}{\\includegraphics{fd.eps}}}} \n\\caption{\\label{fd} (Color online). A schematic phase diagram of different phases of vortex matter \nand phase transition lines in the $N=2$ model in the regime \n$|\\psi^{(1)}| \\ne |\\psi^{(2)}|$. At temperatures $T_{\\rm\n  M}^{\\rm c}$, $T_{\\rm M}^{\\rm 2}$, and $T_{\\rm SLM}$ the melting of\nthe composite vortex lattice, the sublattice of heavy vortices and the\nsublattice of the light vortices occurs, respectively. At $T_{\\rm LP}$\nthe composite vortices decompose. The temperatures $T_{\\rm L}^{\\rm 1}$ \nand $T_{\\rm L}^{\\rm 2}$ denote temperatures where a phase transition via a\nproliferation of vortex loops would take place in the absence of a magnetic \nfield in models with bare phase stiffnesses $|\\tilde{\\psi}^{(\\alpha)}|(T)$ \nequal to $|\\psi^{(\\alpha)}|(B,T)$ (where $\\alpha=1,2$), if the effect of a magnetic were to be \ntaken into account only via the depletion of the modulus of the order \nparameter.} \n\\end{figure}\n\n\n\n\n\\subsection{Physical interpretation of the external field-induced\n  phases of the $N=2$ model}\n\nWe next discuss the  physical interpretation of the various phases that appear as a result \nof the  above described vortex matter transitions. The resulting phases, which exhibit\nsome quite unusual properties, come about as a result of the interplay between the topology \nof the system and thermal fluctuations. This is rather remarkable, given the \nthree-dimensionality of the systems we consider.\n\n\n\\subsubsection{Vortex lattice melting and the disappearance of superconductivity.}\nConsider first the melting transition of an interacting ensemble of composite Abrikosov \nvortices. This phase transition, which is of first order \\cite{Hetzel1992}, corresponds\nto the lines $T_{\\rm M}^{\\rm c}(B)$ and $T_{\\rm M}^2(B)$\nshown in Fig. \\ref{fd}. It is only the  gauge-charged mode that couples to the external field, \nwhile the neutral mode does not. The charged mode at low temperature forms an Abrikosov vortex \nlattice with a melting temperature that is suppressed with increasing magnetic field\n\\cite{Nelson1988,Houghton1989,Blatter1994,Fossheim_Sudbo_book}. The melting temperature of the \nAbrikosov vortex lattice can be suppressed below the temperature where the neutral mode proliferates \nand where the composite vortex lines decompose. For $N=1$, it is \nknown that when the Abrikosov  lattice melts, superconductivity is lost also along the \ndirection of the magnetic field \\cite{Nonomura1998,nguyen1999}. The situation in the \n$N=2$ model is much more complex, since then there still exists a superfluid mode (the gauge neutral \nmode) which is decoupled from external magnetic field. {\\it Thus, upon  melting of the Abrikosov \nlattice we arrive at emergent effective neutral superfluidity  existing in a system of charged \nparticles  \\cite{BSA}}. This  is a genuinely novel state of condensed matter, and moreover one \nwhich should be realizable in liquid metallic  states of light atoms at in principle experimentally \naccessible pressures in the range of $400 {\\rm GPa}$ \\cite{BSA}.\n\nSo in the absence of an external magnetic \nfield, the system thermally excites only fractional flux vortices in the forms of loops, with phase \nwindings only in individual condensates, and these fluctuations are responsible for critical \nproperties. In contrast, the purely charged vortices (i.e. the composite one-flux-quantum vortices \nwith no neutral super-flow) are not relevant in the absence of  external field and  the system is \neither a superconductor (below $T_{\\rm c2}$) or a superconductor with neutral mode (below $T_{\\rm c1}$). \n{ \\it Thus, the effect of a sufficiently strong magnetic field essentially inverts the temperatures \nof the transitions by  melting the lattice of charged modes at  $T_{\\rm M}^{\\rm c}(B)$ while \nleaving neutral modes intact.} \n\nThe phase transition from a {\\it superconducting superfluid} phase where the neutral mode is superfluid \n{\\it and} the Abrikosov vortex lattice is intact such that longitudinal superconductivity (parallel to \nthe magnetic field) exists\\cite{nguyen1999,Nonomura1998}, to a {\\it metallic superfluid} phase where \nthe system is superfluid, but  longitudinal superconductivity is lost due to the melting of the vortex \nlattice, can be mapped onto a lattice melting transition in the $N=1$ model\n, because it is governed only by composite vortices and neutral modes are not involved. Thus it is a \nfirst order phase transition \\cite{Hetzel1992,Fossheim_Sudbo_book}. \n\n\\subsubsection{Decomposition. The disappearance of superfluidity.}\nAnalogously, the physical meaning of the  sublattice \nmelting transition $T_{\\rm SLM}$  (see \\fig{fd}) is a transition from a {\\it superconducting superfluid} \nto an {\\it ordinary \none-gap superconductor}, because a disordering of the phase $\\theta^{(1)}$\ndestroys the massless neutral boson associated with the gauge invariant phase\ndifference $\\theta^{(1)}-\\theta^{(2)}$.\n\nIf we heat the system further, the ordinary superconductivity will disappear via disordering \nof the phase $\\theta^{(2)}$ when we reach the melting transition of the remaining sublattice \nof ``heavy\" vortices at $T_{\\rm M}^2$.\n\nThe system features one more phase transition. That is a transition from the metallic superfluid \nto a normal fluid, which has a purely topological origin. That is, from the vortex matter point of \nview, this manifests itself as a decomposition of a liquid of composite vortices to a ``plasma\" of \nindividual vortices at the characteristic temperature $T_{\\rm LP}$, and such a transition has no \ncounterpart in an $N=1$ superconductor. A schematic diagram of the resulting physical phases is \nshown in Fig. \\ref{fd2}.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.04}{\\rotatebox{0.0}{\\includegraphics{fd2.eps}}}} \n\\caption{\\label{fd2} (Color online). A schematic phase diagram of physical states appearing in the \n$N=2$ model in the regime $|\\psi^{(1)}| \\ne |\\psi^{(2)}|$ as a consequence \nof vortex matter phase transitions. Increasing the magnetic field suppresses the \nmelting transition of the composite vortex lattice formed by the charged mode\nbelow the proliferation line for the neutral mode, which does not couple to magnetic \nfield. In the absence of disorder (pinning of vortices), superconductivity only remains \n{\\it along} the direction of the magnetic field, provided that the vortex system remains \nin a lattice phase. When the composite vortex lattice melts, the system looses ability \nto carry dissipation-less charge currents, but at large enough magnetic field, the neutral \nmode should still be superfluid above the melting temperature \\cite{BSA}. Thus, we have a first \norder phase transition from a {\\it superconducting superfluid to a  metallic superfluid}. \nThe neutral mode proliferates through a second order phase transition in the 3D$xy$\\  \nuniversality class. Therefore, at large enough magnetic fields, a 3D$xy$\\  anomaly \nin the specific heat should appear inside the vortex liquid phase. The separation\nbetween the first order specific heat anomaly due to vortex lattice melting and the \n3D$xy$\\  anomaly due to loop-proliferation should increase with increasing magnetic field. \nAt low magnetic fields one has another   phase transition inside the Abrikosov \nvortex lattice phase, from a {\\it superconducting superfluid to a one-gap (``ordinary\")\nsuperconducting state}. \n} \n\\end{figure}\n\n\n\\subsubsection{A direct SSF $\\to$ NSF transition}\n\nWe note also the possibility of an existence of a phase transition directly from a \nsuperconducting superfluid (SSF) to a metallic normal fluid (NF),  shown in \nFig. \\ref{direct_transition}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.05}{\\rotatebox{0.0}{\\includegraphics{othertr.eps}}}} \n\\caption{\\label{direct_transition} (Color online). A direct phase transition from SSF to NF phase \n(shown with red arrow).} \n\\end{figure}\nWhat is remarkable is that this  resembles (while indeed being a different type of \ntransition) the type of direct phase transition from a low temperature phase with \nHiggs mass and superfluid density of the neutral mode, to a phase with zero Higgs \nmass and zero helicity modulus of the neutral mode that we would find in zero magnetic\nfield when  the bare phase stiffnesses of Eq. (\\ref{gl_action}) are equal, i.e.\n$|\\psi^{(1)}| = |\\psi^{(2)}|$ \\ \\ \\cite{sachdev2004,motrunich2004,smiseth2004}.\nThe SSF phase features one massless dual Higgs photon, and one massive Higgs photon \nwhile the NF phase features one massless photon and one massive dual Higgs photon. \nThese phases are therefore {\\it self-dual}, in analogy to the situation encountered \nwhen Eq. (\\ref{gl_action}) is viewed as a quantum anti-ferromagnet with easy-plane \nanisotropy \\cite{sachdev2004,motrunich2004}. However, there is a significant difference. \nThe novel critical point encountered in the case of the quantum antiferromagnet was a \nresult of a superposition of a 3D$xy$\\  and an inverted 3D$xy$\\  critical point. In the $2$-flavor \nLondon model in finite magnetic field, the crossing point is a superposition between a 3D$xy$\\  \ncritical line and first order phase transition line, so it  should have a different character \nthan the self-dual critical point discussed in Ref. \\onlinecite{motrunich2004}.\nIt can, however, not be a 3D$xy$\\  or inverted 3D$xy$\\  critical point, since neither of \nthese are critical points between self-dual phases. \n\nThus, both for high and low magnetic fields we have genuinely novel physics in that for a three \ndimensional system, {\\it i)} a critical phenomenon takes place inside either the vortex liquid \nphase (high magnetic fields) or the vortex lattice phase (low magnetic fields), and {\\it ii)} we \nhave three new equilibrium states which have no counterpart in the $N=1$ case. Moreover,\nby the discussion given above, it is clear that  the crossing point between the vortex\nlattice melting line and the neutral mode proliferation line warrants further study. \nThis is best left for a separate computational analysis\n\n\n\\subsection{Monte-Carlo results, finite magnetic field, $N=2$}\n\nWe now present large-scale Monte Carlo results for the case $N=2$ in finite magnetic field\nat low magnetic fields when the temperature is varied \\cite{SSBS}. We consider the model \nbased on \\eq{gl_action} for $N=2$ on an $L^3$ lattice (with $L$ up $96$) with periodic \nboundary conditions for coupling constants $|\\psi^{(1)}|^2=0.2$,\n$|\\psi^{(2)}|^2=2$, and $e^2=1\/10$. The ratio  $|\\psi^{(2)}|^2\/|\\psi^{(1)}|^2 = 10$ brings out \none second order phase transition at $T_{\\rm SLM}(B)$ in the 3D$xy$\\  universality class well below the \nmelting temperature $T_{\\rm M}^2$ of the vortex lattice. In LMH $|\\psi^{(2)}|^2\/|\\psi^{(1)}|^2 \\approx 2000$, \nbut the physical picture remains. For real estimates of $T_{\\rm{SLM}}$ and $T_{\\rm M}^2$ in LMH, \nsee Ref. \\onlinecite{Ashcroft1999}. The Metropolis algorithm with local updating is used in \ncombination with Ferrenberg-Swendsen reweighting. The external magnetic field \n$\\ve B$ studied is $B^x=B^y=0$, $B^z = 2\\pi\/32$, thus there are $32$ plaquettes in the  \n$(x,y)$-plane per flux-quantum. This is imposed by splitting the gauge field into a static \npart $\\ve A_0$ and a fluctuating part $\\ve A_{\\rm fluct}$. The former is kept fixed to \n$(A_0^x,A_0^y(\\ve r),A_0^z) = (0, 2\\pi xf,0)$ where $f=1\/32$ is the magnetic filling fraction, \non top of which the latter field is free to fluctuate. Together with periodic boundary conditions \non $\\ve A_{\\rm fluct}$,  the constraint $\\oint_C(\\ve A_{0}+\\ve A_{\\rm fluct}) d\\ve l=2\\pi fL^2$, \nwhere $C$ is a contour enclosing the system in the (x,y)-plane, is ensured. It is imperative to \nfluctuate $\\ve A$, otherwise type-$1$ and type-$2$ vortices do not interact \\cite{frac,smiseth2004}. \nTo investigate the transition at $T_{\\rm{SLM}}$ we have performed finite size scaling (FSS) of the third \nmoment of the action. The simulations are done by using vortices directly \\cite{smiseth2004}, but \nwith a finite magnetic induction $B^z=2\\pi\/32$. \n\nWe compute the specific heat $C_V$ and the third moment of the action. To probe the structural order \nof the vortex system we compute the planar structure function $S^{(\\alpha)}(\\ve k_\\perp)$ of the \n{\\it local vorticity} $\\ve n^{(\\alpha)}(\\ve r) \n= \\left( \\ve \\nabla \\times  \\left[ \\ve \\nabla \\theta^{(\\alpha)} -e ~  \\ve  A  \\right] \\right)\/2 \\pi$, \ngiven by\n\\begin{equation}\nS^{(\\alpha)}(\\ve k_\\perp) = \n\\frac{1}{(f L^3)^2}~ \\langle | \\sum_{\\ve r} ~ n_z \\f{\\alpha}(\\ve r) ~ e^{i \\ve k_\\perp \\cdot \\ve r_\\perp} |^2 \\rangle,\n\\end{equation}\nwhere $\\ve r$ runs over dual lattice sites and $\\ve k_\\perp$ is perpendicular to $\\ve B$. This function \nwill exhibit sharp peaks for the characteristic Bragg vectors $\\ve K$ of the type-$\\alpha$ vortex lattice and will \nfeature a ring-structure in its corresponding liquid of type-$\\alpha$ vortices. The signature of vortex \nsub-lattice melting will be a transition from a six-fold symmetric Bragg-peak structure to a ring \nstructure in $S^{(1)}(\\ve K)$ while the peak structure remains intact in $S^{(2)}(\\ve K)$. Furthermore, \nwe compute the {\\it vortex co-centricity} $N_{\\rm{co}}$ of type-$1$ and type-$2$  vortices, defined as \n$N_{\\rm co} \\equiv  N^{+}_{\\rm{co}} - N^{-}_{\\rm{co}}$, where\n\\begin{equation}\nN^{\\pm}_{\\rm{co}}\n \\equiv \\frac{\\sum_{\\ve r}|n_z\\f 2(\\ve r)|\\delta_{n_z\\f 1(\\ve r),\\pm\n n_z\\f 2(\\ve r)}}{\\sum_{\\ve r}|n_z\\f 2(\\ve r)|},\n\\end{equation}\nwhere $\\delta_{i,j}$ is the Kronecker-delta. The reason for considering $N_{\\rm co}$ \nis that we then eliminate the effect of random overlap of vortices in the high-temperature \nphase $T > T_{\\rm SLM}$ due to vortex-loop proliferation, and focus on the {\\it compositeness} \nof field-induced vortices. \n\nThe quantity $N_{\\rm{co}}$ is the fraction of type-$2$ vortex segments that are co-centered \nwith type-$1$ vortices, providing a measure of the extent to which vortices of type-$1$ and \ntype-$2$ form a {\\it composite} vortex system. Hence, it probes the splitting processes \nvisualized  in Fig. \\ref{fluct1}. The results are shown in Fig. \\ref{MC_results}. \n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.55}{\\rotatebox{0.0}{\\includegraphics{ExtField_2.eps}}}} \n\\caption{ \\label{MC_results} (Color online). \n  MC results for $N=2$ $|\\psi^{(1)}|^2=0.2$, $|\\psi^{(2)}|^2=2$, and $e=1\/\\sqrt{10}$. \n  Panel a: $C_V$ (black) and  $N_{\\rm co}$  (green). The $C_V$-anomaly at $T_{\\rm{SLM}}=0.37$, \n  where type-$1$ vortices proliferate, matches the point at which $N_{\\rm co}$ drops to\n  zero. Thus, type-$1$ vortices are torn off type-$2$\n  vortices. The remnant of the zero-field anomaly in $C_V$ is \n  seen as a hump at $T \\sim 3.6$. Panel b:   \n  $S^{(1)}(\\ve K)$ (red) and $S^{(2)}(\\ve K)$ (blue) for\n  the particular Bragg vector $\\ve K =(\\pi\/4,-\\pi\/4)$. \n  $S^{(1)}(\\ve K)$ vanishes continuously at $T_{\\rm{SLM}}$,\n  while $S^{(2)}(\\ve K)$ vanishes discontinuously at $T_{\\rm M}^2=2.34$. \n  Panel c: FSS plots of the $M_3$ from which the exponents \n  $\\alpha=-0.02 \\pm 0.05$ and $\\nu=0.67 \\pm 0.03$\n  is extracted, showing that the sub-lattice melting is a  \n  3D$xy$\\  phase transition. Panels d, e, f, and g: plots of \n  $S^{(2)}(\\ve k_\\perp)$ for the temperatures $T_{\\rm d}=0.35$,\n  $T_{\\rm e}=0.4$, $T_{\\rm f}=1.66$, and $T_{\\rm g}=2.85$,\n  respectively. At $T_{\\rm d}$, $T_{\\rm e}$, and $T_{\\rm f}$, \n  the vortex lattice remains intact. The vortex lattice melts at \n  $T_{\\rm M}^2$ to give a vortex liquid ring pattern at $T_{\\rm g}$.}  \n\\end{figure}\n\nAt $T_{\\rm{SLM}}$, $C_V$ has a pronounced peak associated with the 3D$xy$\\  transition, \nand a broader less pronounced peak which is the finite field remnant of the zero-field \ninverted 3D$xy$\\  transition \\cite{nguyen1999}. Scaling of $M_3$ at $T_{\\rm{SLM}}$ shown in the inset \nc in Fig. \\ref{MC_results} yields the critical exponents $\\alpha =-0.02\\pm0.05$ and \n$\\nu=0.67\\pm0.03$ in agreement with the 3D$xy$\\  universality class. A novel result is that \n$S^{(1)}(\\ve K)$ vanishes {\\it continuously} as the temperature approaches $T_{\\rm{SLM}}$ \nfrom below, precisely the hallmark of the decomposition transition that separates \nthe two types of vortex states depicted in Figs. \\ref{fluct1} and \\ref{sl}.\nA related feature is the {\\it vanishing} of $N_{\\rm co}$ at \n$T_{\\rm{SLM}}$ as a function of temperature, discussed in detail below. The \nfirst-order melting transition takes place at  $T_{\\rm M}^2$, where $S^{(2)}(\\ve K)$ \nvanishes discontinuously. This is the temperature at which the translational invariance \nis restored through melting of the type-$2$ vortex lattice. In the temperature interval \n$T < T_{\\rm{SLM}}$ the system {features superconductivity and superfluidity simultaneously} \n\\cite{BSA}, since there is long-range order both in the charged and the neutral vortex modes. \nIn the temperature interval $T_{\\rm{SLM}} < T < T_{\\rm M}^2$ long-range order in the neutral mode \nis destroyed by loop-proliferation of type-$1$ vortices, hence superfluidity is lost \n\\cite{BSA}. However, {longitudinal one-component superconductivity} is retained along the \ndirection of the external magnetic field. For $T > T_{\\rm M}^2$ superconductivity is also lost, \nhence this is the normal metallic state, which is a two-component vortex liquid. \n\nThe most unusual and surprising feature is the continuous variation of $S^{(1)}(\\ve K)$ with \ntemperature, even at $T_{\\rm{SLM}}$ where it vanishes.\nThe explanation for this is the \nproliferation of type-$1$ vortices (which destroys the neutral superfluid mode) in the \nbackground of a composite vortex lattice, which the type-$1$ vortices essentially do not see, cf. Fig. \n\\ref{splitting}. As far as the composite neutral Bose field $\\theta^{(1)}-\\theta^{(2)}$ \nis concerned, {\\it it is precisely as if the composite vortex lattice were not present at all}. \nHence, $S^{(1)}(\\ve K)$ vanishes for a completely different reason than  $S^{(2)}(\\ve K)$, \nnamely due to {\\it critical fluctuations, i.e. vortex-loop proliferation} in the condensate \ncomponent with lowest bare stiffness. Such a phase transition does not completely restore \nbroken translational invariance associated with a vortex lattice, since for the type-$2$ vortices \n{\\it quite remarkably, the vortex lattice order survives the decomposition transition}, due to \ninteraction between heavy vortices mediated by charged modes. The vanishing of $N_{\\rm{co}}$ \nis particularly \ninteresting, and finds a natural explanation within the framework of the above discussion. \nThat is, for $T \\ll T_{\\rm{SLM}}$, we have $N_{\\rm{co}} \\approx 1$, so the vortex system \nconsists practically exclusively of composite vortices. As the temperature increases, thermal \nfluctuations induce excursions such as those illustrated in  \nin Fig. \\ref{splitting}, which reduces $N_{\\rm co}^{\\rm{+}}$ from its low-temperature value, \nreaching a {\\it minimum} at $T_{\\rm{SLM}}$ and then {\\it increase} for $T > T_{\\rm{SLM}}$. \n\nWe may view the splitting process as a type-$1$ closed vortex loop superposed on a vortex lattice of \n(slightly) fluctuating composite vortices. An important point to notice is that a \ntype-$\\alpha$ vortex does not interact with a composite vortex by means of a neutral mode. \nThis follows from a topological argument that two split branches will feature \nnontrivial winding in the composite neutral field $\\theta^{(1)}-\\theta^{(2)}$, \nwhile a composite vortex line does not. Hence, the splitting transition may be \nviewed as {\\it a type-$1$ vortex loop-proliferation in a neutral superfluid}. \nThis is illustrated in Fig. \\ref{splitting}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.13}{\\rotatebox{0.0}{\\includegraphics{superim5.eps}}}} \n\\caption{ \\label{splitting} (Color online). \nDetailed illustration of the low-temperature thermal fluctuations in a vortex lattice of composite \nvortices. A local excursion of  a type-$1$ vortex  away from the composite vortex lattice may be \nviewed as a type-$1$ bound vortex loop superposed on the composite vortex lattice. \nThe composite vortex line does not interact with a vortex with nontrivial winding in\n$\\Delta \\gamma = \\Delta(\\theta^{(1)}-\\theta^{(2)})$. A splitting \nof the composite vortex lattice \nmay be thus viewed as a  {\\it zero-field} vortex-loop proliferation of type-$1$ vortices; \na 3D$xy$\\  phase transition universality \\cite{tesanovic1999,nguyen1999,Fossheim_Sudbo_book}.  }\n\\end{figure}\nThus, we may utilize the well-known results for the critical properties of the 3D$xy$\\  model \nfor neutral superfluids described as a vortex-loop proliferation \n\\cite{tesanovic1999,nguyen1999,Fossheim_Sudbo_book}. \nThis ``vortex sublattice melting\" phase transition is therefore in the  3D$xy$\\  universality \nclass \\cite{tesanovic1999,nguyen1999,Fossheim_Sudbo_book}, not a first order melting \ntransition. The resulting phase is one where superfluidity is lost and longitudinal \nsuperconductivity retained in the component $\\Psi_0^{(2)}$.\n\nConversely, $N_{\\rm co}^{\\rm{-}}$ remains essentially zero until  $T_{\\rm{SLM}}$, thereafter \nincreasing monotonically. For temperatures above, but close to $T_{\\rm{SLM}}$, fluctuations \nin vortices originating in $\\Delta \\theta^{(2)}$ are still small, so the variations in \n$N_{\\rm{co}}=N_{\\rm co}^{\\rm{+}}-N_{\\rm co}^{\\rm{-}}$ reflect thermal fluctuations \nin vortices originating in $\\Delta \\theta^{(1)}$. The increase of $N_{\\rm co}^{\\rm{\\pm}}$ \nmeans that type-$1$ vortex loops  are thermally generated, and thus tend to {\\it randomly} \noverlap more with the moderately fluctuating type-$2$ vortices. At their first order melting \ntransition, type-$2$ vortices fluctuate only slightly. {\\it Thus, the vanishing of $N_{\\rm{co}}$ \nabove $T_{\\rm{SLM}}$ reflects the increase in the density of thermally generated type-$1$ vortex \nloops in the background of a slightly fluctuating type-$2$ vortex lattice}.\n\n\n\\subsection{Graphical representation of phase disordering transitions\n  in the $N=2$ model.}\nIn Fig.  \\ref{2ph1} we present a schematic picture \nof configuration of the order parameters phases $\\theta^{(1)}$ and $\\theta^{(2)}$ in various points in \nphysical space, when vortex matter drives the system into one of the above discussed \nsuperconducting and superfluid states. \n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.25}{\\rotatebox{0.0}{\\includegraphics{n-2_new3.eps}}}} \n\\caption{ \\label{2ph1} (Color online). Phases of the order parameters in the  various \nstates for $N=2$. In the upper left  panel, both $\\theta^{(1)}$ and $\\theta^{(2)}$ \nare ordered, this is the superconducting superfluid state. In the upper right panel,\nneither of the phases $\\theta^{(1)}$ and $\\theta^{(2)}$ are ordered, however the \ncombination $\\theta^{(1)} - \\theta^{(2)}$ exhibits long-range order, this is the \nmetallic superfluid state. In the lower left panel, $\\theta^{(1)}$ is disordered\nand $\\theta^{(2)}$ is ordered. In this case, the neutral superfluid mode is \ndestroyed and we are left with one charged superconducting mode, this is the \nanalog of the one-gap superconducting state. In the lower right panel,\nneither of the phases $\\theta^{(1)}$ and $\\theta^{(2)}$ are ordered and\nthe combination $\\theta^{(1)} - \\theta^{(2)}$ does not exhibit long-range order, this is the \nmetallic normal fluid state. The states illustrated in the upper left and lower right and left\npanels exist at zero as well as finite magnetic fields. The state illustrated in the upper right\npanel only exists at finite magnetic fields.} \n\\end{figure}\n\n\n\\section{\\label{N>2_extfield} The $N >2$ model in external magnetic field}\nWe next consider the novel features that are encountered, compared to the $N=1$-\nand $N=2$ cases, when an $N>2$ system is subjected to an external magnetic field. \nThe new features are  due to the fact that we have more than one neutral vortex \nmode, and that a vortex with phase winding in any single phase field $\\theta^{(\\alpha)}$ \nwill excite {\\it $N-1$ neutral modes} (see Appendix \\ref{App:charged_neutral}).\nWe consider first the case $N=3$, followed by  the case $N=4$. En route we \nintroduce the useful concept of ``color charge'' which  facilitates a discussion\nof the universality class of the phase transitions that occur in multi-component\nsuperconductors in external magnetic field when composite vortices decompose\ndue to thermal fluctuations.  \n \n\n\\subsection{Decomposition transitions, $N=3$} \n \nWe stress that the system is not \nmapped onto a $U(1)\\times U(1)$ neutral system because the neutral modes remain topologically \ncoupled, as a consequence of multiple connectedness of space introduced by the vortex core. \nIt  manifests itself  in the fact that any  {\\it single} phase variable \n$\\theta^{(\\alpha)}; \\alpha \\in [1,2,3]$ excites {\\it two} neutral  modes, as illustrated in \ndetail below.\n\nWe introduce bare phase stiffnesses for the neutral modes\nin \\eq{neutral} as follows\n\\begin{eqnarray}\nJ^{12}&=&\\frac{|\\psi^{(1)}|^2 |\\psi^{(2)}|^2}{\\Psi^2}, \\nonumber \\\\\nJ^{23}&=&\\frac{|\\psi^{(1)}|^2 |\\psi^{(3)}|^2}{\\Psi^2}, \\nonumber \\\\\nJ^{13}&=&\\frac{|\\psi^{(2)}|^2 |\\psi^{(3)}|^2}{\\Psi^2}.\n\\label{stiffnesses}\n\\end{eqnarray}\nHence, a vortex with phase windings $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$,\ncan be mapped onto two co-centered vortices in a two-component neutral superfluid with bare stiffnesses \n$J^{12}$ and $J^{13}$. Thus, at a distance larger than the penetration length, such a vortex interacts with \na vortex  $(\\Delta\\theta^{(1)}=-2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$ like two vortices in \na neutral superfluid with bare phase stiffness $\\tilde{J}=J^{12}+J^{13}$.\n\nIntra-vortex interaction, e.g. of the vortex\n$(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$\nwith a vortex $(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=0)$\nor with a vortex  $(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=2\\pi)$\nis more complicated.\nIt can most conveniently be described by introduction of  the ``color charge\" concept,\nwhich we explain in the next subsection, \\ref{subsection_color}. \n\nFirst, however, we observe that only a composite vortex  $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$\nhas finite energy.  The key feature of a system with \n$|\\psi^{(1)}|\\neq|\\psi^{(2)}|\\neq|\\psi^{(3)}|$, is\nthat  the three elementary constituent vortices are bound with\ndifferent strength to such \na composite vortex. For example, when\n$|\\psi^{(1)}|\\ll|\\psi^{(2)}|\\ll|\\psi^{(3)}|$, the  neutral modes excited by a vortex $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$\nhave bare phase stiffnesses $(J^{12},J^{13}) \\ll J^{23}$.  This in turn implies that in the composite \nvortex $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$, the \nconstituent elementary vortex $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$ is \nmost loosely bound. Thus, in contrast to the $N=2$ case, the effect of\nthermal fluctuations for $N=3$ is a two step transition. In\nthe general $N$ case the process of stripping a composite vortex of \nits $N$ constituent vortices is an $N-1$-step process occurring successively, starting at $T_{\\rm c1}$ \nand progressing up through $T_{\\rm c2}$ up to $T_{{\\rm c}N-1}$, at which point the vortex system is \nfully decomposed. \n\n\nFor $N=3$, at a low temperature determined by the smallest bare phase stiffness $|\\psi^{(1)}|$\nand by $J^{12}$ and $J^{13}$, \nthere should therefore take place a partial decomposition of the vortex\n$(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$\ninto two vortices $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0) +\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$,\nillustrated in \\fig{3spl_2}. Then, upon increasing \nthe temperature there should take place a phase transition, also illustrated in \\fig{3spl_2},\ninto a fully decomposed state defined by the phase windings\n$(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0) +\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)\n\\to(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0) +\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=0)+\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=2\\pi)$.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.13}{\\rotatebox{0.0}{\\includegraphics{3spl2.eps}}}} \n\\caption{\\label{3spl_2} (Color online). A two-step decomposition transition in the\n$N=3$ model in external magnetic field. \nThe ``lightest\" vortex component, originating \nin the order parameter component with the lowest bare phase stiffness, tears itself \nloose from the composite Abrikosov vortex  of the two stiffer order parameter \ncomponents at $T_{\\rm Spl}^{1}$. At a higher temperature $T_{\\rm Spl}^{2}$, \nthe ``next-to-lightest\" vortex component \n(green vortex), originating in the order parameter component with the next-to-lowest bare \nphase stiffness, tears itself loose from the vortex  of the \nstiffest order parameter component in the background of proliferated vortices\noriginating in non-trivial phase-windings of the phase with lowest phase stiffness\n(red vortex).} \n\\end{figure}\n\nWe also stress that apart from the neutral mode\n $(\\theta^{(2)}-\\theta^{(3)})$, which provides an attractive\n interaction for the vortex pair\n$(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=0)+\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=2\\pi)$,\nthe first of these vortices excites the neutral mode  $(\\theta^{(1)}-\\theta^{(2)})$\nwhich consists of oppositely directed currents in the condensates\n$\\Psi_0^{(1)}$ and $\\Psi_0^{(2)}$. The second vortex\nexcites the mode $(\\theta^{(1)}-\\theta^{(3)})$\nwhich is associated with oppositely directed currents in condensates\n$\\Psi_0^{(1)}$ and $\\Psi_0^{(3)}$. These modes, apart from giving the pair \n$(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=0)+\n(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=2\\pi)$\nlogarithmically divergent energy per unit length\\cite{frac,smiseth2004}, also\nyield some repulsive interaction in this pair, because both the modes\n$(\\theta^{(1)}-\\theta^{(2)})$ and $(\\theta^{(1)}-\\theta^{(3)})$\nfeature unscreened currents of condensate $\\Psi_0^{(1)}$. However,\nsuch a repulsive interaction in this pair is negligibly small in the \nconsidered regime $|\\psi^{(1)}|\\ll|\\psi^{(2)}|\\ll|\\psi^{(3)}|$, \ncompared to the interactions mediated by the mode $(\\theta^{(2)}-\\theta^{(3)})$.\n\n\n\n\\subsection{\\label{subsection_color}Color electric charge}\nFormally, the partial decomposition process can be described by introducing \nthe concept of ``color electric charges''. That is, we may introduce e.g. ``+red\", \n``+green\" and ``+blue\"  charges associated with $2\\pi$ windings in $(\\theta^{(1)}-\\theta^{(2)})$, \n$(\\theta^{(1)}-\\theta^{(3)})$ and $(\\theta^{(2)}-\\theta^{(3)})$,\nrespectively (see Tab. \\ref{tab:colors_n3}). If we\nhave a $-2\\pi$ winding in $(\\theta^{(1)}-\\theta^{(2)})$, $(\\theta^{(1)}-\\theta^{(3)})$\nor $(\\theta^{(2)}-\\theta^{(3)})$,  that would correspond to `` $-$red\", ``$-$green\" and ``$-$blue\"  \ncolor electric charges, respectively. We stress once more that in order to preserve \nsingle-valuedness of the order parameters, the $\\pm 2\\pi$ gains in phase differences may only \ncome as $\\pm 2\\pi$ gains in individual phases. For example if we would have \n$(\\Delta\\theta^{(1)}=3\\pi\/4 ,\\Delta\\theta^{(2)}=-5\\pi\/4 )$ then $(\\theta^{(1)}-\\theta^{(2)})$  \nwould change by $2\\pi$. However, such a configuration would be unphysical because individual \norder parameters $\\Psi_0^{(1)}$ and $\\Psi_0^{(2)}$ would loose their single-valuedness. Then, \na vortex $(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$ which excites  \ntwo neutral modes associated with  $(\\theta^{(1)}-\\theta^{(2)})$ and $(\\theta^{(1)}-\\theta^{(3)})$ \nwith stiffnesses  $J^{12}$ and $J^{13}$, respectively, may be viewed as a color charged string \nwith color charge ``$+$red\" and ``$+$green\". \n\nThe regime $|\\psi^{(1)}|\\ll|\\psi^{(2)}|\\ll|\\psi^{(3)}|$, i.e. when\n$J^{12}\\ll J^{13}\\ll J^{23}$,  corresponds to the situation where red\nelectric charges are much weaker than green charges, which in turn are much\nweaker than the blue charges. The blue charge then dominates the  \nbinding of the vortices $(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=0)$\nand $(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=2\\pi)$. The tightly bound \ncomposite vortex $(\\Delta\\theta^{(1)}=0,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$ then \nhas electric charge $(-\\rm{red},-\\rm{green})$ which loosely binds it with \n$(+\\rm{red},+\\rm{green})$ color charged vortex \n$(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=0,\\Delta\\theta^{(3)}=0)$\ninto a color charge neutral \nfinite energy  one-flux-quantum vortex \n$(\\Delta\\theta^{(1)}=2\\pi,\\Delta\\theta^{(2)}=2\\pi,\\Delta\\theta^{(3)}=2\\pi)$.\n\n\nIn Fig. \\ref{bridge_vortex_charge}, we illustrate how to connect the\nvortex picture to \nthe picture of color charges, for the case $N=3$.  For $N=3$, \neach type of vortex is a bound state of two color charges.\n\\begin{table}\n\\caption{\\label{tab:colors_n3}A color charge is defined as a $ \\pm 2\\pi$\n  winding in the mode $(\\theta^{(\\alpha)}-\\theta^{(\\eta)})$ with the\n  following mapping for $N=3$. }\n\\begin{ruledtabular}\n\\begin{tabular}{rccc}\n  Mode: & $(\\theta^{(1)}-\\theta^{(2)})$ &\n  $(\\theta^{(1)}-\\theta^{(3)})$ & $(\\theta^{(2)}-\\theta^{(3)})$\\\\\n  Color charge:        & red  & green  & blue\\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.14}{\\rotatebox{0.0}{\\includegraphics{illustr.eps}}}} \n\\caption{ \\label{bridge_vortex_charge}  (Color online). \nThe connection between vortex illustrations and bound states of color charges, for \nthe case $N=3$. Each type of vortex is a bound state of 2 color\ncharges. The radius of each black circle is for graphical\nconvenience is used to differentiate between ``heavy\" and\n``light\" vortices. The order parameter component  with the lowest bare\nphase stiffness is taken to be the vortex with the ``smallest \ndiameter\". A vortex originating in a non-trivial phase-winding in $\\theta^{(\\alpha)}$\nis denoted a type-$\\alpha$ vortex. {\\it The color of the vortex on the top\nof the Figure should not be confused with the color of the electric charges\nin the ``dual\" charge representation}.} \n\\end{figure}\n\n\nA schematic picture of the low-temperature composite vortex lattice phase, the partial \ndecomposition transition in the color electric charge representation, and the complete \ndecomposition transition,  are given in Figs.  \\ref{colorcharges_31}, \\ref{colorcharges_32}, \nand \\ref{colorcharges_33}.\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.27}{\\rotatebox{0.0}{\\includegraphics{CC1.eps}}}} \n\\caption{ \\label{colorcharges_31}  (Color online). A color charge representation for $N=3$ of the \ncomposite vortex lattice phase illustrated also in the left part of Fig. \\ref{3spl_2}.\nThis low-temperature phase, where the fluctuations of the composite vortex lines only \ninvolve small excursions of each of the constituent vortex lines away from the\nmain composite object, may be viewed as a 3-color dielectric phase. \nThe main composite vortex line is ``anchored\" on the thickest vortex. We assume we \nare at temperatures and fields such that the thickest vortex line does not undergo a \nmelting transition. Each constituent vortex may be viewed as a bound state of certain \ncombinations of color charges. The composite vortex line may, on the other hand, be \nviewed as bound states of $\\pm {\\rm red}$, $\\pm {\\rm green}$, and $\\pm {\\rm blue}$ \ncharges, as indicated by the dotted ellipses. This is a three-color dielectric \n``insulating\" phase. We strongly emphasize that the above illustration is meant \nto illustrate what the situation is in a typical cross section  along the lines, \n{\\it which are not rigid straight vortex lines}.} \n\\end{figure}\nFor the purposes of determining the universality class of this partial decomposition, \n{\\it which is a genuine phase transition}, it may also be viewed as follows. The \ncompletely intact composite vortex is color charge neutral in the sense that it contains \na positive and negative electric charge of each color. The fluctuation snapshot  in the \nleft part of Fig. \\ref{3spl_2} can be viewed as a completely intact composite vortex line \nwith a small type-1 vortex loop superimposed on it as shown on Fig. \\ref{superim}.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.15}{\\rotatebox{0.0}{\\includegraphics{superim.eps}}}} \n\\caption{ \\label{superim} (Color online). A local thermally driven detachment of the\n  type-$1$ vortex line (red color) \nfrom the composite vortex line can be viewed as a superposition of thermally created \n{\\it closed type-$1$ vortex loop} on a completely composite vortex line. Both processes are \ntopologically equivalent because  the vorticity of the type-$1$ constituent vortex in the \ncomposite vortex line is exactly canceled by a superimposed type-$1$ vortex ring\nwith the opposite vorticity. We stress that if left segment of a vortex loop has a \ncounterclockwise vorticity then the opposite right segment has a clockwise vorticity.\n} \n\\end{figure}\n The type-$1$ vortex \nloop which carries a red and green electric \ncharges does not interact with the  composite vortex\nwhich is color charge neutral. This can also\nbe seen by examining the vortex interaction matrix given in \\eq{potential}. \nThat is, when we add up the\nneutral-mode-mediated interactions between the type-$1$ vortex\nwith all three  type-$1,2,3$ vortices in the composite vortex \n$(\\Delta \\theta^{(1)}=2\\pi,\\Delta \\theta^{(2)}=2\\pi, \\Delta \\theta^{(3)}=2\\pi)$\n{\\it they add  to zero}. The situation in the right \npart of Fig. \\ref{3spl_2} is topologically equivalent\nto a completely intact composite vortex \nline superimposed with one segment of an {\\it unbound} vortex loop.\n{\\it Therefore the transition may be viewed as an Onsager vortex loop proliferation \ntransition \\cite{Onsager1949,tesanovic1999,nguyen1999}\n of type-$1$ vortex loops in the background of a \ncolor charge neutral composite vortex lattice \\cite{BSA,SSBS}.} \nThat is because type-$1$ vortex loops, from the point of view of superfluid modes, \ncannot ``see'' color charge neutral vortex lines, so this is equivalent to a type-$1$ \nvortex-loop proliferation transition in  the complete absence of a composite\ncolor charge neutral vortex lattice. \nSince these vortices excite neutral modes, this transition, which is the first stage in \ndecomposing a color charge neutral composite  vortex line, is a vortex\nloop proliferation \ntransition in the 3D$xy$\\  universality class \\cite{BSA,SSBS}. \nIn the color charge representation given in Fig. \\ref{colorcharges_32},\nthis also means that the first-stage partial decomposition transition of three \ncharged fluctuating {\\it line objects} with different charges (but such that the \nalgebraic sum of their charges add up to zero) is also a $2$-color metal-insulator\nphase transition in the 3D$xy$\\  universality class, {\\it involving flexible color\nline charges}\n\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{colorcharges.eps}}}} \n\\caption{ \\label{colorcharges_32}  (Color online). A color charge representation for $N=3$ of the \npartial decomposition transition \nIn the color charge representation, this \nmay be viewed as a $\\rm{red}$-$\\rm{green}$ dielectric-metal transition, \nwhile the $\\rm{blue}$ dielectric phase remains intact. As explained in the text, this \npartial 2-color metal-insulator transition involving fluctuating line charges, is in \nthe 3D$xy$\\  universality class. \nThe dotted ellipse indicates which color is involved in forming the remaining \ndielectric phase.} \n\\end{figure}\n\nThe usefulness of the color charge representation becomes particularly clear when we \ngo on to describe the second stage of the decomposition transition, illustrated in \nFig. \\ref{3spl_2}. This  transition may be viewed as a proliferation of type-$2$ vortex \nloops in the background of liberated type-$1$ vortex loops, all superimposed on a background\ncomposite vortex lattice. The situation therefore is more complicated than in the first \nstage illustrated in Fig. \\ref{3spl_2}, since that was a proliferation of type-$1$ loops \nin  vacuum.\n\nHowever, let us view this transition in the color charge picture, illustrated in going from \nFig. \\ref{colorcharges_32} to Fig. \\ref{colorcharges_33}. This is a metal-insulator transition \nfor the $\\rm{blue}$-charge sector in the background of coexistent $\\rm{red}$ and $\\rm{green}$ \nmetallic phases. \nHowever, $\\rm{red}$ and $\\rm{green}$ charges cannot screen $\\rm{blue}$ \ncharges, while these charges eliminate the neutral modes associated with them \n(that is the ones with bare stiffnesses $J^{12}$ and $J^{13}$). Therefore, this is \na metal-insulator transition for the sector of the $\\rm{blue}$ charges, while $\\rm{red}$ \nand $\\rm{green}$ charges screen themselves and do not affect this transition.\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{CC3.eps}}}} \n\\caption{ \\label{colorcharges_33} (Color online). A color charge representation of the complete \ndecomposition transition also illustrated in Fig. \\ref{3spl_2}.  \nIn the color charge representation, this may be viewed as a $\\rm{blue}$  \ndielectric-metal transition, in the background of a $\\rm{red}$-$\\rm{green}$ metallic \nphase. As explained in the text, this complete ``1-color\" metal-insulator transition \ninvolving fluctuating line charges is in the same universality class as the\npartial decomposition transition, namely the 3D$xy$\\  universality class. \n} \n\\end{figure}\n\nIn the color charge representation given in Fig. \\ref{colorcharges_33}, this also means that \nthe second-stage decomposition transition depicted in Fig. \\ref{3spl_2} is a $1$-color \nmetal-insulator transition in the 3D$xy$\\  universality class, involving flexible \ncolor charged strings.\n\n\n\n\\subsection{Decomposition transitions, $N=4$} \n\nIn Figs. \\ref{colorcharges_41}, \\ref{colorcharges_42}, \\ref{colorcharges_43}, and \n\\ref{colorcharges_44} we illustrate the partial decomposition\ntransitions for the case $N=4$ in the color charge picture. That is, we  introduce \nas in the case $N=3$ case  ``$\\pm$ red\", ``$\\pm$ green\" and ``$\\pm$ blue\"  charges \nassociated with $\\pm 2\\pi$ windings in $(\\theta^{(1)}-\\theta^{(2)})$, \n$(\\theta^{(1)}-\\theta^{(3)})$ and $(\\theta^{(2)}-\\theta^{(3)})$, respectively. \nIn addition, we introduce ``$\\pm$ yellow\", ``$\\pm$ violet\", and ``$\\pm$ orange\" \ncharges associated with $\\pm 2\\pi$ windings in $(\\theta^{(1)}-\\theta^{(4)})$, \n$(\\theta^{(2)}-\\theta^{(4)})$ and $(\\theta^{(3)}-\\theta^{(4)})$,\nrespectively (see Tab. \\ref{tab:colors_n4}).\nTherefore, the case $N=4$ features one new aspect which was absent in the case $N=3$, \nnamely that for $N=4$ we need more color charges than number of order parameter\ncomponents in order to completely cover all the possible ways that a \nneutral mode $(\\theta^{(\\alpha)}-\\theta^{(\\beta)})$ can be excited.   \nEach type-$\\alpha$ vortex is a bound state of three color charges,\nas indicated in Fig. \\ref{colorcharges_41}.\n\nThe low-temperature phase is a $4$-composite color charge neutral vortex system, which\nmay alternatively be viewed as a $6$-color dielectric phase, Fig. \\ref{colorcharges_41}. \nThe first-stage partial decomposition involves a type-$1$ vortex tearing itself off the\n$2$-composite vortex, i.e. a  vortex loop proliferation of type-$1$ loops carrying \n$+\\rm{red}, +\\rm{green},+\\rm{yellow}$ color charges in the background\nof color charge neutral \nobjects. So this is a phase transition in the 3D$xy$\\  universality class. It may \nalternatively be viewed as a $3$-color metal insulator transition in the\n$\\rm{red},\\rm{green},\\rm{yellow}$ line-charge sectors, Fig. \\ref{colorcharges_42},\nleaving a $3$-color ($\\rm{violet,blue,orange}$) dielectric phase. \nThe second-stage decomposition process involves a type-$2$ vortex tearing itself off \na $3$-composite vortex in the background of a system of proliferated  type-$1$ loops. \nDue to screening of $\\rm{red},\\rm{green}$ and $\\rm{yellow}$ charges, this transition \nmay be viewed in a simplified manner. It may be considered as the first-stage \ndecomposition in a $N=3$ system consisting of type-$2$, type-$3$, and type-$4$ \nvortices, involving $\\rm{violet}, \\rm{blue}$, and $\\rm{orange}$ charges. This \nwe have already argued is a 3D$xy$\\  (type-$2$) vortex loop proliferation transition, \nwhen we considered the $N=3$ case. Alternatively, it may be viewed as a $2$-color \nmetal-insulator transition in the $\\rm{violet}$ and $\\rm{blue}$ line-charge sectors, \nFig. \\ref{colorcharges_43}, leaving a $1$-color ($\\rm{orange}$) dielectric phase. \nThe third-stage decomposition may be viewed as a type-$3$ vortex loop proliferation \nin the background of type-$1$ and type-$2$ \nproliferated vortex lines. Due to screening of $\\rm{violet}$ and $\\rm{blue}$ charges \nthis may be viewed in a simplified manner. It may be considered as a vortex loop \nproliferation of loops carrying $\\rm{orange}$ charges in the background  of an \n$\\rm{orange}$-neutral vortex lattice, or vacuum. This is a vortex loop \nproliferation in the 3D$xy$\\  universality class \\cite{BSA,SSBS}. Alternatively, it may be viewed  \nas a $1$-color ($\\rm{orange}$) metal-insulator transition, (see Fig. \\ref{colorcharges_44}),\nleaving the $6$-color dielectric phase completely destroyed. \n\n\\begin{table}\n\\caption{\\label{tab:colors_n4} A color charge is defined as a $ \\pm 2\\pi$\n  winding in the mode $(\\theta^{(\\alpha)}-\\theta^{(\\eta)})$ with the\n  following mapping for $N=4$. }\n\\begin{ruledtabular}\n\\begin{tabular}{rccc}\n  Mode: & $(\\theta^{(1)}-\\theta^{(2)})$ &\n  $(\\theta^{(1)}-\\theta^{(3)})$ & $(\\theta^{(2)}-\\theta^{(3)})$ \\\\\n  Color charge:        & red  & green  &  blue \\\\\n\\hline\n  Mode: & $(\\theta^{(1)}-\\theta^{(4)})$ & $(\\theta^{(2)}-\\theta^{(4)})$ & $(\\theta^{(3)}-\\theta^{(4)})$\\\\\n  Color charge: & yellow & violet & orange \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{C4.eps}}}} \n\\caption{ \\label{colorcharges_41}  (Color online). \nA color charge representation for $N=4$ of the composite vortex lattice phase. \nThis low-temperature phase, where the fluctuations of the composite vortex lines only \ninvolve small excursions of each of the constituent vortex lines away from the\nmain composite object, may be viewed as a ``6-color dielectric\" phase. Moreover,\nfor $N=4$, each type-$\\alpha$ vortex in the $N=4$ case is a bound state of $3$ color charges.\nThe dotted ellipses indicate which colors are involved in forming the dielectric phase.}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{C4-2.eps}}}} \n\\caption{ \\label{colorcharges_42}  (Color online). A color charge representation for $N=4$ of the \nfirst partial decomposition transition. In the color charge representation, \nthis may be viewed as a $\\rm{red}$-$\\rm{green}$-$\\rm{yellow}$  dielectric-metal transition, \nwhile the $\\rm{violet}$-$\\rm{blue}$-$\\rm{orange}$ dielectric phase remains intact. As explained \nin the text, this partial 3-color metal-insulator transition involving fluctuating line charges, \nis in the 3D$xy$\\  universality class. The arrows indicate which three colors are involved in the\nmetal-insulator transition. The dotted ellipses indicate which colors are involved in forming \nthe remaining dielectric phase.} \n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{C4-3.eps}}}} \n\\caption{ \\label{colorcharges_43}  (Color online). A color charge representation for $N=4$ of the \nsecond partial decomposition transition. In the color charge representation, \nthis may be viewed as a $\\rm{violet}$-$\\rm{blue}$  dielectric-metal transition, \nwhile the $\\rm{orange}$ dielectric phase remains intact. As explained in the text, this \npartial 2-color metal-insulator transition involving fluctuating line charges, is in \nthe 3D$xy$\\  universality class. The arrows indicate which two colors are involved in the\nmetal-insulator transition. The dotted ellipse indicates which color is involved in \nforming the remaining dielectric phase.} \n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.28}{\\rotatebox{0.0}{\\includegraphics{C4-4.eps}}}} \n\\caption{ \\label{colorcharges_44}  (Color online). A color charge representation for $N=4$ of the \nthird and complete  decomposition transition. In the color charge \nrepresentation, this may be viewed as an $\\rm{orange}$  dielectric-metal transition, \nwhile the color-dielectric phase is completely destroyed. As explained in the text, this \npartial 1-color metal-insulator transition involving fluctuating line charges, is in \nthe 3D$xy$\\  universality class. The arrow indicates which  color is involved in the\nmetal-insulator transition.} \n\\end{figure}\n\n\\subsection{General $N$ }\nIn the general $N$-component case, the number of color charges that needs to be introduced\nto give an equivalent description as the above, may be counted as follows, starting from the \nthird term in Eq. (\\ref{charge_neutral}). Each combination $\\theta^{(\\alpha)}-\\theta^{(\\beta)}$ \nis given a color. We start with one phase, the one with the lowest bare stiffness say $\\theta^{(1)}$, \nand \nintroduce a non-trivial phase winding $\\pm 2 \\pi$ in this phase. This excites $N-1$ neutral \nmodes since $N-1$ gauge-invariant phase\ndifferences which involve $\\theta^{(1)}$ can be formed. Introducing non-trivial \nphase-windings in the next phase $\\theta^{(2)}$, the one with the next-to-lowest \nbare stiffness say,  will also excite $N-1$ neutral modes, but only $N-2$ {\\it new} neutral \nmodes. Non-trivial phase-windings in the third phase \n$\\theta^{(3)}$ will excite $N-3$ new neutral modes, and \nso on.  The number of different colors $N_{\\rm color}$ we will have to introduce \nfor a theory with $N$ flavors of scalar fields is therefore given by \n$N_{\\rm color}(N) = (N-1) +(N-2)+(N-3) + \\dots + 2 + 1 = N(N-1)\/2$,\ni.e.$N_{\\rm color}(2) = 1$, $N_{\\rm color}(3) = 3$, \n$N_{\\rm color}(4) = 6\n$, and $N_{\\rm color}(5)=10$. A completely composite vortex, which we denote as an $N$-composite \nvortex, consists of $N$ constituent vortices originating in nontrivial phase windings in each of the \nindividual phases $\\theta^{(\\alpha)}, \\alpha \\in [1,\\dots,N]$. Since a non-trivial phase winding in  any phase \n$\\theta^{(\\eta)}$, $\\eta \\in [1,\\dots,N]$, excites $N-1$ neutral modes $\\theta^{(\\eta)} - \\theta^{(\\alpha)}$, \nit is clear that a  type-$\\eta$ vortex may be viewed as a bound state of $N-1$ \ncolor charges. The particular combination of $N-1$ color charges, out of the \ntotal collection of $N(N-1)\/2$ color charges, that will enter the $N-1$-body bound state\nin each vortex, will depend on $\\eta$. {\\it The  $N$-composite vortex is a \ncolor charge neutral object.} \n\nSmall fluctuations in the $N$-composite vortex may therefore be viewed\n as a dielectric insulating phase of an $N(N-1)\/2$-component\ndielectric. The first stage in the $N-1$ stage decomposition process of the $N$-composite \nvortex line, where a type-$1$ vortex tears itself off the $N$-composite vortex, \nis therefore a metal-insulator transition where $N-1$ color charges of the $N(N-1)\/2$-colors \ndielectric system {\\it simultaneously} undergo a metal insulator transition, in the \n3D$xy$\\  universality class. The next stage, where a type-$2$  vortex tears itself off the \nremaining $N-1$-composite vortex in the background of a system of proliferated type-$1$ \nloops, is phase where $N-2$ color charges {\\it simultaneously} undergo a metal insulator \ntransition in the 3D$xy$\\  universality class by the same argument as used for the $N=3$ case, \nand so on. The complete decomposition of the $N$-composite vortex proceeds in $N-1$ steps of \nmetal-insulator transitions for color charges, where step number  $1 \\leq {\\cal N} \\leq N-1$ \nmay be viewed as either a type-${\\cal N}$ vortex tearing itself off an $N-{\\cal N}-1$-composite \nvortex line, or equivalently a {\\it simultaneous} metal-insulator transition  for $N-{\\cal N}$ \nnew color charges that have not been involved in the previous ${\\cal N}-1$ metal-insulator \ntransitions. All the $N-1$ partial decomposition transitions, or metal-insulator transitions \nfor color charges, are in the 3D$xy$\\  universality class. \n\nWe should emphasize that, as follows from  Eq. (\\ref{charge_neutral}) in the limit  \n$N\\to\\infty$, the strength of each of the electric charges goes to zero. At the same time \nthe number of colors of electric charges $N_c$ tends to infinity.\nFrom Eq. (\\ref{charge_neutral}), it follows that even in the limit $N\\to\\infty$,\nthe energy binding of a type-$\\alpha$ vortex to a color charge neutral composite vortex \nis finite, even though the strength of each individual color charge tends to zero.\n \n\n\\subsection{Graphical representation of phase disordering transitions\n  in the $N=3$ model.} \nIn Fig.  \\ref{ph1}, we illustrate graphically the various  phase disordering transitions and \npartial symmetry restorations discussed in the previous section.\n\\begin{figure}[htb]\n\\centerline{\\scalebox{0.20}{\\rotatebox{0.0}{\\includegraphics{n_3v4.eps}}}} \n\\caption{ \\label{ph1} (Color online). A schematic plot of states in the $N=3$ system. The upper left panel \nshows the phases \nof the condensate order parameters that are involved. The upper right panel shows the state where \nall three phases $\\theta^{(1)}$,  $\\theta^{(2)}$, and $\\theta^{(3)}$ are ordered individually. This \nstate is the low-temperature (ground) state and features one superconducting charged mode and two \nsuperfluid neutral modes. The middle left panel is a phase where $\\theta^{(1)}$ is disordered, while \n$\\theta^{(2)}$ and $\\theta^{(3)}$ are ordered. Thus, this is a state which features one charged \nsuperconducting mode and one neutral superfluid mode. The middle right panel illustrates a state where \nall of the phases $\\theta^{(1)}$,  $\\theta^{(2)}$, and $\\theta^{(3)}$ are individually disordered. However,\nthe differences $\\theta^{(1)} - \\theta^{(2)}$ and $\\theta^{(2)} -\\theta^{(3)}$ (and therefore also \n$\\theta^{(1)} - \\theta^{(3)}$) feature long-range order. This is therefore a state which is normal \nmetallic, but nevertheless features two neutral superfluid modes. The bottom left panel illustrates \na state where all of the phases $\\theta^{(1)}$,  $\\theta^{(2)}$, and $\\theta^{(3)}$ are individually \ndisordered. Only the phase difference  $\\theta^{(2)} - \\theta^{(3)}$ exhibits long-range order. This\nis a normal metallic state featuring one neutral superfluid mode. The bottom right panel illustrates\na state where all of the phases $\\theta^{(1)}$,  $\\theta^{(2)}$, and $\\theta^{(3)}$ are individually \ndisordered and where none of the phase differences $\\theta^{(1)} - \\theta^{(2)}$ and $\\theta^{(2)}-\\theta^{(3)}$ \nand  $\\theta^{(1)} -\\theta^{(2)}$ feature long-range order. This is therefore a state which is normal \nmetallic and normal fluid (no neutral superfluid modes). The states illustrated in the upper left, \nmiddle left, and lower right panel exist at zero as well as finite magnetic fields. The states \nillustrated in the middle right and lower left panels only exist at finite magnetic fields. } \n\\end{figure}\n\n\n\n\n\\section{Applications}\n\nIn this Section, we will briefly mention some possible applications of the results obtained\nin this paper. Emphasis will be on the case $N=2$, but some results for the cases\n$N=3$ and even $N=4$ may find applications in  mixtures of\nsuperconducting condensates in the not too distant future.  \n\n\\subsection{Applications of results for $N=2$}\n\nThe  results  for $N=2$ are expected to apply to two-component superconductivity which could be \nachieved in  metallic states of light atoms\\cite{Ashcroft1999,Neilnew}, such as electronic \nand protonic condensates in liquid metallic hydrogen (LMH) under extreme pressure. Estimates \nexist for $T_{\\rm c2}$ for such systems, $T_{\\rm c2} \\approx 160 {\\rm K}$ \n\\cite{Ashcroft1999}. A rough estimate for $T_{\\rm c1}$ follows from the mass ratio of the electronic \nand protonic condensate, $T_{\\rm c1} \\approx 0.1 {\\rm K}$. Hence,  at $T_{\\rm c1}$ one should \nobserve an extra low-temperature 3D$xy$\\  specific heat anomaly, as well as an anomaly in the \nLondon penetration depth. An even more promising candidate is the the system $CH_x$  \n\\cite{Neilnew} where there are predictions of liquid metallic states at considerably lower \npressures than those required to achieve LMH. In such multi-flavor \nsuperconductors it is the case $|\\psi^{(1)}| \\neq |\\psi^{(2)}|$ which appears most \nnaturally \\cite{smiseth2004}. \n\nHere, it is appropriate to remark briefly on the microscopic origins of\nsuperconductivity\nin the projected liquid metallic phase of hydrogen\n \\cite{Ashcroft1999,Neilnew}.\nThe proton is four times lighter than a ${}^4He$ atom.\nIt is well known that ${}^4He$\nat normal conditions\nis a classic {\\it permanent liquid}, because of high zero-point energy\nand  weak ordering energies.\nIndeed zero-point energies of\nprotons in a dense environment\nare also high, and at increasing compression\nthere is a shift of electron density from intra-molecular\nregions to inter-molecular,\nand with it a progressive decline in the effective inter-proton\nattractions.\nBecause of this there is also a decline of ordering energies from\ninteractions relative to protonic zero-point energies.\nThe existence of {\\it a melting point maximum}\nas a function of pressure\nin hydrogen, as well as\n a range of densities where hydrogen may take up \\textit{a fluid\n   phase} in its ground state  was\nsuggested in Ref. \\onlinecite{Ashcroft1999}.\nAnother important circumstance\nis that en route hydrogen should undergo  an insulator-metal\ntransition and therefore the resulting phase\nshould be the liquid metallic hydrogen, a\ntranslationally invariant two-component fermionic liquid. There is\npreliminary experimental evidence\nthat a melting point maximum may indeed exist \\cite{Datchi2000} and it\nhas received recent powerful\nbacking in {\\it ab initio}\ncalculations \\cite{Bonev}. Experimentally a 12.4 fold compression of\nhydrogen has already been achieved at\naround 320 ${\\rm GPa}$ \\cite{Datchi2000,Bonev}. Estimates suggest that\nLMH should appear at 13.6 fold\ncompression at pressure in the\nvicinity of 400 ${\\rm GPa}$  \\cite{Bonev}, whereas hydrogen alloys may\nexhibit metallic behavior at significantly\nlower pressures \\cite{Neilnew}. A predicted key feature of LMH at low\ntemperature is the {coexistence\nof superconductivity of proton-proton and electron-electron Cooper\npairs} \\cite{Ashcroft1999}.\n\n\nThe special point $|\\psi^{(1)}| = |\\psi^{(2)}|$  has a physical realization when Eq. (\\ref{gl_action}) \nis viewed as an effective  field theory of a quantum antiferromagnet with easy plane anisotropy, which \nfacilitates a suppression of topological defects in the form of \"hedgehog\"-configurations which appear \nin $O(3)$-symmetric models \\cite{motrunich2004,sachdev2004}. (More generally, it may be viewed as a \nfield theory of an $O(3)$ model where \"hedgehogs\" are suppressed by some unspecified mechanism, not \nnecessarily limited to easy-plane anisotropy). \n\nWe conclude this subsection on the $N=2$ case with a remark on how these results relate to \nmulti-flavor {\\it electronic} condensates. To describe this case, we need to include a \nJosephson coupling between the matter field species. The details of how to give a vortex \nrepresentation of the $N$-flavor London model in the presence of inter-flavor Josephson \ncoupling, is given in Appendix \\ref{App:josephson}. Had a Josephson coupling term \nbetween condensate species been introduced in the theory for $N \\geq 2$, this would \nhave have altered the dual theory in a completely non-perturbative way, and would tend \nto lock the phases of individual condensate fields to each other. (For a dual representation \nof this case, where the non-perturbative character of the Josephson coupling is brought \nout in a particularly clear way, see Appendix \\ref{App:josephson}). As a result, the transitions \nwe describe here would collapse to one, namely the charged Higgs fixed point, which is\nin the inverted 3D$xy$\\  universality class.\n\n\\subsection{Applications of results for $N=3,4$}\n\nMixtures of superconducting condensates in LMH can be extended to include  also the \nhydrogen-isotopes deuterium and tritium \\cite{thanksneil}. Tritium is a $S=1\/2$ fermion, so this \nmay give \nrise to a superconducting condensate via forming spin-singlet Cooper pairs, just as \nin the protonic case. Hence, our results for $N=3$ could be applicable to to the mixtures of \nliquid metallic hydrogen-tritium at extremely high pressures. Another possibility is to \ninclude  deuterium as a new component. Including  deuterium as a new component in \naddition to hydrogen means that we have an $N=3$-component mixture of superconducting \ncondensates consisting of electrons, protons, \nand deuterons (deuterium nuclei), all of which in principle can undergo a metal-superconductor \ntransition. Compared to the situation for $N=2$, the situation is complicated by at least \ntwo circumstances. Firstly, deuterons are bosons, and secondly they have spin $S=1$. This \nmeans that the electrons and protons become superconducting via forming Cooper pairs, while \nthe deuterons undergo a metal-superconducting transition via Bose-Einstein Condensation.\nExtending this to the case of having {both} tritium and deuterium in addition to\nhydrogen, might provide a realization of the case $N=4$.\n\n\n\n\n\\section{\\label{Concl} Summary }\nWe have  analyzed  the $N$-flavor London superconductor model\ncoupled to one gauge field with no Josephson coupling between the\nmatter field components. The dual theory is an $N$-flavor GL theory coupled to $N$\ndual gauge fields, where the sum of all dual gauge fields is massive at all\ncouplings. There are $N-1$\ncharge-neutral superfluid modes and one charged superconducting mode\nin this model. We have given a prescription for how to identify the\n$N-1$ neutral modes for arbitrary  $N$.\n\n\nFor $N=2$, a case which should apply to a superconducting state of\nliquid metallic hydrogen, as well as for the case $N=3$, we have\nperformed large scale MC simulations computing \\textit{i)} critical\nexponents $\\alpha$ and $\\nu$, \\textit{ii)} gauge field and dual gauge\nfield correlators, \\textit{iii)} the corresponding masses, and\n\\textit{iv)} critical couplings using FSS. For $N=2$ and $|\\psi^{(1)}|\\neq |\\psi^{(2)}|$, \nwe  find one low-temperature critical point in the 3D$xy$\\  universality class at $T_{\\rm c 1}$, \nand one critical point in the inverted 3D$xy$\\  universality class at $T_{\\rm c2} > T_{\\rm c1}$.\nFor $N=2$ with $|\\psi^{(1)}| = |\\psi^{(2)}|$ we find one critical point with non-3D$xy$\\  \nvalues of $\\alpha$ and $\\nu$ \\cite{motrunich2004}. We propose these to be critical exponents in the universality\nclass of a self-dual gauge theory. For $N=3$ and all $|\\psi^{(\\alpha)}|$ unequal, we find two  fixed points\nin the 3D$xy$\\  universality class at $T_{\\rm c1}$ and $T_{\\rm c2}$, and one\nfixed point in the inverted 3D$xy$\\  universality class at $T_{\\rm c3} (> T_{\\rm c2} >\nT_{\\rm c1})$. All critical points therefore exhibit 3D$xy$\\  values\nof $\\alpha$ and $\\nu$ when all bare phase stiffnesses\n$|\\psi^{(\\alpha)}|$ are different. In the case $N=3$ with\n$|\\psi^{(1)}| = |\\psi^{(2)}| < |\\psi^{(3)}|$ we find two critical\npoints. The critical point at $T_{\\rm c1} < T_{\\rm c2}$ is found to be\nin the 3D$xy$\\  universality class, and the critical point at $T_{\\rm c2}$\nis in the inverted 3D$xy$\\  universality class. For the case $N=3$ with\nequal phase stiffnesses we find one critical point which is non-3D$xy$\\ .\n\n\nIn this context, we have also noted that collapsing two neutral critical points \nin the 3D$xy$\\  universality class leads to a single critical point also in the \n3D$xy$\\  universality class. This follows from an argument implying that collapsing\nany number of neutral critical points in the 3D$xy$\\  universality class leads to a \nsingle critical point also in the 3D$xy$\\  universality class. On the other hand, it \nappears that collapsing $N-1$ neutral critical points in the 3D$xy$\\  universality \nclass {\\it and one charged fixed point in the inverted} 3D$xy$\\  universality class \nleads to a single critical point in a universality class (which in principle \ncould depend on $N$) which is not that of the 3D$xy$\\  or inverted 3D$xy$\\  type. For \n$N=2$, we may define the universality class as that of a 3D self-dual \n$U(1) \\times U(1)$ \ngauge theory, while it is less clear what it is for other $N \\geq 3$. The \nnumerical values we have obtained for the critical exponents $\\alpha$ and \n$\\nu$ the two cases {\\it{i)}} $N=2$, $|\\psi^{(1)}| = |\\psi^{(2)}|$ and \n{\\it{ii)}} $N=3$, $|\\psi^{(1)}| = |\\psi^{(2)}| = |\\psi^{(3)}|$ are \nremarkably similar, indicating that the values of the critical exponents \nare at most weakly dependent on $N$. \n\n\nIn an external magnetic field at low temperature, the ground state of\nthe system is an Abrikosov vortex lattice of composite vortices. \nHowever the effect of thermal fluctuations alters the physics significantly.\nWe discuss in detail that \nin the low-field regime and when the  bare stiffnesses of the condensates all\ndiffer, we find that a 3D$xy$\\  vortex sub-lattice melting transition takes place. Upon thermal\ndecomposition of field induced composite vortex lines, the\nconstituent vortices originating in the condensates with lowest bare stiffness disorder,\nwhile the ones originating in the stiffer condensates remain arranged in an Abrikosov\nvortex lattice. When  such a transition occurs, an $N=2$ system looses superfluid\nproperties, but remains superconducting. \nIn contrast at high magnetic fields, the neutral mode \ndisappears via the melting transition\nof the lattice of composite vortices. \nThis is  a transition from the superconducting state to a non-superconducting metallic superfluid state \\cite{BSA}.\nInside this novel metallic superfluid phase, at a temperature much lower\nthan the zero-field metal-superconductor transition, we find  a\nsuperfluid-normal fluid 3D$xy$\\ \nphase transition associated with a neutral mode-driven proliferation of\nvortex loops nucleating\non field-induced vortex lines.\nThese various transitions  should in principle be detectable in flux-noise\nexperiments.\nWe extend the discussion for $N>2$, where we find phase transitions\nassociated with partial decomposition of composite vortices, yielding\nseveral unusual states\nof partial symmetry breakdown.\nThe universality class and partial symmetry breakdowns\nare identified by mapping the system \nto an ensemble of electrically charged strings \nwhere for the $N$-component system there are $N(N-1)\/2$ \nreplicas of electric charges of different color.\n\n\nThe sublattice melting and partial decomposition transitions are \nof purely topological origin.\nIt involves what can be viewed as vortices and anti-vortices (positively \nand negatively charged objects). The existence of positively and negatively \ncharged objects implies that the physics conceptually in some sense is \nsimilar to what occurs in a Kosterlitz-Thouless (KT) transition in two \ndimensions. In spite of being a decomposition of positively and negatively \ncharged composite objects, {\\it such a transition can  be mapped onto a \nproliferation of vortex loops in a vacuum}, {\\it i.e} a phase transition \nin the 3D$xy$\\  universality class. Another principal difference between this \ntype of phase transition compared to the KT transition, is that in the 2D \nKT transition, vortices and anti-vortices (positively and negatively charged \nobjects) are thermally generated. This can not happen in three dimensions\nsince any  vortex line  in 3D has an infinite energy in an infinite sample.\nIn the 3D transition which we consider, the neutral bound states of\nthe charged objects are introduced by an external magnetic field.\nThus we deal with a novel type of transition which in a very unusual \nform involves concepts both from two- and three- dimensional physics.                                                                                                                                                                     \n\n\n\\begin{acknowledgments}\nThis work was funded by the Norwegian University of Science and Technology through the \nNorwegian High Performance Computing Program, by the Research Council of Norway, Grant\nNos. 157798\/432, 158518\/431, and 158547\/431 (NANOMAT), by STINT and the Swedish Research \nCouncil, and by the US National Science Foundation DMR-0302347. We acknowledge helpful \ndiscussions and communications with  K. B\\o rkje, H. Kleinert, O. Motrunich, F. S. Nogueira, \nS. Sachdev, Z. Tesanovic, and A. Vishwanath. E. B. thanks L. D. Faddeev and A. J. Niemi\nfor many discussions of multigap models. A. S. thanks H. Kleinert and the Institut f{\\\"u}r \nTheoretische Physik der Freie Universit{\\\"a}t Berlin, for hospitality while part of this \nwork was being completed. In  particular, we wish to thank N. W. Ashcroft for many \nenlightening discussions and for collaboration on Ref. \\onlinecite{BSA}. \n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\n\nAmorphous solids, including but not limited to, glasses, granular matter, colloidal suspensions, foams and polymers,  yield to external shear~\\cite{bonn2017yield, nicolas2018deformation}. The yielding point, typically characterized by a maximum on the stress-strain curve, is essentially the end-point of the solid regime, as the material transitions to  plastic flow thereafter.\nUnderstanding the nature of yielding in amorphous solids, in the context of statistical mechanics, has been becoming an active research area. Recent theories attempt to explain yielding as a depinning transition~\\cite{nicolas2018deformation}, a first-order non-equilibrium phase transition~\\cite{jaiswal2016mechanical, kawasaki2016macroscopic}, a spinodal point of the glass state of equation~\\cite{rainone2015following, parisi2017shear}, or phase transitions in the universality class of a random field Ising model~\\cite{ozawa2018random}. However, treating non-linear effects is intellectually challenging~\\cite{hentschel2011athermal, biroli2016breakdown}, and consequently their impact on yielding remains unclear to a large extent.\n\nNon-linear elastic responses, in particular {\\it shear hardening}, as evidenced by a rapid increase of the shear modulus before yielding, are not uncommon in amorphous solids. Shear hardening has been widely observed in polymers~\\cite{treloar1975physics}, and more recently  in dense hard sphere (HS) colloidal glasses close to jamming~\\cite{jin2018stability, urbani2017shear}. In both cases, hardening is accompanied by entropy vanishing caused by structural constraints:\nthe number of allowed configurations tends to zero approaching the maximum stretch limit in polymer chains, and the jamming limit in HSs. In this study, we discover a new type of shear hardening in amorphous solids, where the entropy increases with strain, caused by shear-induced rejuvenation. This non-linearity also leads to  a negative correlation between the yielding strain and the degree of annealing, in sharp contradiction to previous results~\\cite{rainone2015following, jin2017exploring, ozawa2018random}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\linewidth]{cartoon_v2.pdf}\n   \\caption{\\textbf{A schematic representation of the holographic duality.} The gravitational models live in $(3+1)$ dimensions while effective field theories\/amorphous solid simulations are in $(2+1)$ dimensions.\n    \\label{cart}\n\\end{figure}\n\nLet us sketch out our strategy. (i)\nFirst, a zero-temperature effective field theory (EFT) for non-linear elasticity is constructed, based on shift-symmetric Goldstone fields, the phonons, as fundamental building blocks~\\cite{PhysRevD.102.069901}. The theory also takes advantage of the fact that amorphous solids are typically rotationally invariant (isotropic), and assumes that the solid is homogeneous at low energy, which is true for systems at large spatial scales and\/or slow dynamics, such as granular matter under quasi-static shear. The non-linear elasticity is implemented in the form of an effective potential (cf. strain energy function), from which the stress-strain curve, as well as the onset of instability (yielding), can be calculated.\n\n(ii) To  overcome the difficulty in treating dissipative (finite temperature) effects in the EFT framework, we map the EFT, defined in a $(2+1)-$dimensional flat spacetime with coordinates $(x, y, t)$, onto a gravitational dual model~\\cite{Baggioli:2020qdg} in asymptotically anti-de Sitter spacetime $(x, y, t, u)$, according to the  holographic duality (or gauge-gravity duality) ~\\cite{Ammon:2015wua, Baggioli:2019rrs, Roy:2020vku,PhysRevLett.120.171602, PhysRevLett.120.195301}, where the extra dimension $u$ represents the energy scale (see Fig.~\\ref{cart}). The computation of the entropy becomes particularly simple now, since it boils down to the estimation of the black hole entropy given by the famous Bekenstein - Hawking Area law~\\cite{hawking1971gravitational}. Such a holographic approach has been already successfully applied to study the dynamics of strongly coupled fluids (e.g., Quark-Gluon plasma) \\cite{CasalderreySolana:2011us} and novel strongly correlated materials (e.g., Cuprates) \\cite{Hartnoll:2016apf}.\n\n(iii) To test the predictions from the EFT and the gravitational theory, we perform numerical simulations on soft sphere (SS) models of granular matter. Consistent with the theoretical setup, shear is applied quasi-statically, thermal motions are neglected (for macroscopic grains, the thermal energy is much smaller than  the inter-particle contact energy), and plastic effects are irrelevant. We consider a common type of shear, planar shear (such as simple or pure shear), where the material is unaffected along the dimensions perpendicular to the plane and the effective rheology is two dimensional (see Supplementary Materials (SM) Sec.~S6). The spatial dimensionality is thus reduced to $d=2$ in the theories. The simulations are performed in both 3D (main text) and 2D (SM { Secs.~S7 and S8}). Below, we discuss our results in detail.\n\n\n\n\n\\begin{figure*}[ht!]\n    \\centering\n      \\includegraphics[width=\\linewidth]{mainFig2_Theory.pdf}\n    \\caption{\\textbf{Predictions from the EFT and the gravity theory}.\n    \\textbf{(A)} Stress-strain curves extracted from the EFT. The Poisson's ratio is fixed to $\\mathfrak{r} = 97\\%$, nearly the limit of incompressibility. The non-linear shear exponent $\\nu$ is varied from $1.3$ to $3.1$ (from \\textbf{purple} to \\textbf{red}) in steps of $0.2$. The empty circles locate the breaking points. \\textbf{(B)} The relation between the non-linear exponent $\\nu$ and the  { breaking point} strain $\\epsilon_{\\rm Y}$ (\\textbf{red}) or the {breaking point} stress $\\sigma_{\\rm Y}$ (\\textbf{blue}) for $\\mathfrak{r}=97\\%$. \\textbf{(C)} The entropy-stress curves for $\\nu = 1.00, 1.50, 3.00, 3.75$ (from \\textbf{purple} to \\textbf{red}),\n    from the gravity theory at $T=0.1$, with the large stress scaling Eq.~(\\ref{eq:s0}) indicated (dashed lines). \\textbf{(Inset)} The theoretical relation $\\xi = 2\/(3+\\frac{1}{3}\\nu^2)$.}\n    \\label{fig:max}\n\\end{figure*}\n\n\\section*{Results}\n\n\\subsection*{Correlation between yielding and non-linear elasticity predicted by an EFT}\n\nThe EFT builds on two scalar fields $   \\phi^I(t,\\bf{x})$, whose fluctuations $\\delta \\phi^I(t,\\bf{x})$ represent the displacement fields in the solid ($I=1,2$). The stress-strain curves are obtained from a power-law scalar potential $\\mathcal{V}$, which plays the same role of the non-linear strain-energy function in standard elasticity theories (see Materials and Methods and SM Secs.~S1 and S2 for details). The assumptions used in the theory include: (i) rotational invariance (amorphicity), (ii) homogeneity at large scales (larger than the granularity scale of the solid), and (iii) absence of plastic effects. The validity of these assumptions is tested by a direct comparison between theoretical and simulation results.\n\n\nThe only input to the theory is a power-law form of the stress-strain relation (see Fig.~\\ref{fig:max}(A)),\n\\begin{eqnarray}\n\\sigma(\\epsilon)\\,\\sim \\,\\epsilon^{\\nu},\n\\label{eq:nonlinear_scaling1}\n\\end{eqnarray}\nwhich corresponds to a concrete choice of the potential $\\mathcal{V}$. Here, $\\nu = 2A$ is an exponent characterizing the non-linear elasticity, which cannot be determined directly from the effective theory, but rather should be considered  as a phenomenological parameter. A complete microscopic model, able for example to describe the dependence on  initial conditions, would be needed in order to determine directly such an exponent from theory. To continue, an important quantity defined in the theory is the Poisson's ratio (the negative ratio of transverse to axial strains),\n\\begin{eqnarray}\n\\mathfrak{r}=\\frac{B(B-1)-A}{B(B-1)+A},\n\\label{poisso1}\n\\end{eqnarray}\nwhich is related to the unstrained bulk modulus $K_0$ and the shear modulus $G_0$ via $\\mathfrak{r} = (K_0-G_0)\/(K_0+G_0)$ in two dimensions (notations with subscript $_0$ are defined at zero strain). Because there are only two independent parameters $A$ and $B$ (see Materials and Methods), the model is completely fixed by a given combination of $\\nu$ and $\\mathfrak{r}$.\n\n\nAn interesting prediction is the dependence of {\\it breaking point} $\\{ \\epsilon_{\\rm Y}, \\sigma_{\\rm Y}\\}$, defined by a global instability where the speed of sound vanishes, on the non-linear elasticity. Since the theory is constructed in the hydrodynamic (long-wavelength) limit, this instability is naturally associated with the yielding of the whole system, instead of local plastic rearrangements. Analytic formulas for $ \\epsilon_{\\rm Y}$ and $\\sigma_{\\rm Y}$ are derived and  presented in SM Sec.~S2. For a fixed, large $\\mathfrak{r}$ (i.e., $K_0 \\gg G_0$), corresponding to a nearly incompressible material, $\\nu$  is negatively correlated  with $\\epsilon_{\\rm Y}$, and\npositively correlated with $\\sigma_{\\rm Y}$ (see Fig.~\\ref{fig:max}(B) and SM Sec.S2 for more details). Moreover, in the case of a power-law form of the stress-strain curve as in Eq.~\\eqref{eq:nonlinear_scaling1}, the EFT predicts a power-law correlation between the\n{breaking point} strain $\\epsilon_{\\rm Y}$ and the non-linear exponent $\\nu$ of the type $\\epsilon_{\\rm Y}\\sim \\nu^{-\\kappa}$, where $\\kappa>0$ depends on the specific details of the EFT potential (see SM Sec. S2). Thus, non-linear elasticity increases the strength (maximum stress) of the material, while making it more brittle and having an earlier breaking point.\n\n\n\\subsection*{Scaling of entropy predicted by a gravitational theory}\n\nIn the holographic description~\\cite{Baggioli:2020qdg}, gravity is coupled to a scalar potential $W$, which depends on two bulk fields $\\varphi^I$ (see Materials and Methods and SM Sec.~S3 for details).\nFor a given shear strain $\\epsilon$, the stress $\\sigma$ is read off from the black hole geometry using the holographic dictionary, and a power-law scaling as in Eq.~\\eqref{eq:nonlinear_scaling1} is recovered (see SM Sec.~S4).\nThe breaking point can also be obtained by looking at the gradient instabilities of the gravitational modes and, at least in the decoupling limit, its behaviour as a function of the non-linear exponent $\\nu$ is identical to the EFT results reported above~\\cite{Baggioli:2020qdg}. Let us emphasize that the power-law exponent $\\kappa$ and even the power-law functional form are not expected to be universal but rather dependent  on the details of the concrete model. Nevertheless, in all cases investigated $\\varepsilon_{\\rm Y}$ (or $\\sigma_{\\rm Y}$) is a monotonically decreasing (or increasing) function of the non-linear exponent $\\nu$.\nTherefore, we conclude that the correlations between the breaking point and the non-linear exponent are robust at the qualitative level.\n\nThe zero-temperature entropy $s_{\\text{ath}} = s(T \\to 0)$, computed from the theory, remains non-zero\\cite{PhysRevX.5.041025}, which surprisingly resembles a key feature of amorphous solids. It is well known that the total entropy of an amorphous solid can be decomposed into configurational and vibrational parts, $s = s_{\\rm conf} + s_{\\rm vib}$.\nWhen $T \\to 0$,\nthe vibrational entropy $s_{\\rm vib} \\to 0$, while the configurational entropy $s_{\\rm conf}$ and the total entropy $s$ remain finite, in contrast to crystals where $s \\to 0$.  The finite $s_{\\text{ath}}$ indicates the possibility of  multiple meta-stable states, revealing the glassy nature of black hole systems \\cite{DeGiuli:2020zew,PhysRevB.100.205108, PhysRevX.5.041025}. As an essential result from our theory, the zero-temperature entropy, $s_{\\text{ath}} \\sim s_{\\rm conf}$, scales with the shear stress $\\sigma$ as (for large $\\sigma$),\n\\begin{eqnarray}\ns_\\text{ath} \\sim \\sigma^{\\xi}\\,,\n\\label{eq:s0}\n\\end{eqnarray}\nwhere the exponent $\\xi = 2\/(3+\\nu^2\/3)$. The prefactor vanishes at $T=0$, but the scaling is robust at sufficiently low temperatures (see SM Sec.~S5). Since $\\xi$ is positive, the entropy always increases under shear, implying a rejuvenation effect -- the system becomes  more entropic and less stable under shear. This effect is overlooked by state-following (SF) calculations in mean-field glass theories~\\cite{rainone2015following}, where, by construction, the entropy is kept constant during shear. Importantly, the increase of entropy under shear deformation is in stark contrast with the behavior observed in ordered crystals: e.g. high-density face-centered cubic HS crystals jam, and therefore their (vibrational) entropy vanishes (the configurational entropy $s_{\\rm conf} = 0$ in crystals), under shear~\\cite{jin2021jamming}. It suggests that our  holographic models might share more commonalities with amorphous  rather than crystalline systems, in agreement with recent related considerations \\cite{PhysRevB.100.205108}.\n\n\\begin{figure*}[ht!]\n    \\centerline{    \\includegraphics[width=\\linewidth]{mainFig3_SimHardening.pdf}}\n    \\caption{\\textbf{Correlation between non-linear elasticity and yielding obtained from granular simulations.}\n    \\textbf{(A)} Stress-strain curves for $P_{0}=10^{-2}$ and a few different $\\varphi_{\\rm g}$. The yielding point $\\{ \\epsilon_{\\rm Y}, \\sigma_{\\rm Y}\\}$ is estimated  at $\\sigma_{\\rm Y} = c \\sigma_{\\rm max}$, where $\\sigma_{\\rm max}$ is the maximum stress and $c = 0.98$ (see SM Sec.~S6 for other choices of $c$ and the discussion therein). Fitting the data (solid lines) according to Eq.~(\\ref{eq:nonlinear_scaling1}) gives the exponent $\\nu$ (see SM Sec.~S6 for a discussion on the fitting).\n    \\textbf{(B)} $\\epsilon_{\\rm Y}$ and $\\sigma_{\\rm Y}$ as functions of $\\nu$.\n    \\textbf{(C)} Dependence of $\\Delta \\epsilon_{\\rm Y} = \\epsilon_{\\rm Y}(\\Delta \\hat{\\varphi}_{\\rm g}) - \\epsilon_{\\rm Y}(0) $ on re-scaled degree of annealing $\\Delta \\hat{\\varphi}_{\\rm g} = (\\varphi_{\\rm g} - \\varphi_{\\rm MCT})\/ \\varphi_{\\rm MCT}$, where  $\\varphi_{\\rm MCT}$ is the  mode-coupling theory (MCT) transition density.\n    Besides EFT and simulation results  of athermal SSs ($P_{0}=10$ and $10^{-2}$) obtained in this study, we present as well simulation data of 3D thermal HSs~\\cite{jin2017exploring} and  the SF theoretical result (divided by 5)~\\cite{rainone2015following}. In addition, we plot the theoretical result from the elasto-plastic model (EPM), where $x$-axis represents $(A_{\\rm c} - A)\/A$ with $A$ being the degree of annealing and $A_{\\rm c}$ a critical point~\\cite{ozawa2018random}. Theories are indicated by lines and simulations by line-points. It is clear that  $\\epsilon_{\\rm Y}$ decreases with the degree of annealing due to shear-hardening non-linearity (\\textbf{red}); in all other cases shear-hardening is absent and $\\epsilon_{\\rm Y}$ increases or remains nearly constant (\\textbf{blue}).\n    }\n    \\label{fig:simulation_yielding}\n\\end{figure*}\n\n\n\\begin{figure*}[ht]\n    \\centering\n    \\includegraphics[width=\\linewidth]{mainFig4_SimEntropy.pdf}\n    \\caption{{\\bf\n   \n   \n    {Entropy evolution} of simulated granular matter under shear.} \\textbf{(A)} Jamming density $\\varphi_{\\rm j}$ versus strain $\\epsilon$, for a few different $\\varphi_{\\rm g}$.\n    { \\textbf{(B)} Configurational entropy $s_{\\rm conf}$ from ~Ref.\\cite{berthier2017configurational} and \\textbf{(C)} Edwards entropy $\\Delta s_{\\rm Ed}$  as a  function of stress $\\sigma$, fitted to Eq.~(\\ref{eq:s0}) (solid lines, see SM Sec.~S6 for a discussion on the fitting). The fitting parameters ${\\xi}$ and $\\xi_{\\rm Ed}$ as a function of the non-linear exponent $\\nu$ are plotted in the inset of \\textbf{(C)}.} The error bars represent the standard error of the fitting exponents.\n    \\label{fig:simulation_entropy}}\n\\end{figure*}\n\n\\subsection*{Simulations of a granular matter model}\n\nWe perform computer simulations of a 3D granular matter model, which consists of poly-disperse spheres interacting via frictionless short-range repulsive forces (see Materials and Methods).   The system is compression quenched from initial configurations that are equilibrated at $\\varphi_{\\rm g}$ using an efficient swap algorithm (see Materials and Methods),  to zero temperature where it jams randomly at $\\varphi_{\\rm j}$. Thus $\\varphi_{\\rm g}$  can be understood as a glass transition density, quantifying the degree of annealing (the larger $\\varphi_{\\rm g}$, the deeper annealing). Thanks to the swap algorithm, we are able to prepare ultra-stable states corresponding to deep annealing. This is the key reason to observe significantly stronger non-linear elasticity, compared to previous numerical studies~\\cite{kawasaki2020shear}.\nAdditional data are provided in SM for 2D models {without (Sec.~S7) and with (Sec.~S8) friction}, and systems mechanically annealed by cyclic shear (Sec.~S6) , confirming that the reported behavior is qualitatively insensitive to friction, dimensionality and preparation protocols.\n\nThe above compression quenching procedure generates isotropic configurations at zero temperature and a finite pressure $P_{0}>0$, where $P_{0} = P(\\epsilon =0)$ characterizes the distance to isotropic jamming, $P_{0} \\sim \\varphi_0 - \\varphi_{\\rm j}$. These isotropic configurations serve as unstrained ($\\epsilon = 0$) reference states to athermal quasi-static shear (AQS) with simple strain deformations under constant volume conditions (see Materials and Methods). For a small $P_{0} = 10^{-2}$, the stress-strain curves in Fig.~\\ref{fig:simulation_yielding}(A) display clear non-linear elasticity, following the scaling law Eq.~(\\ref{eq:nonlinear_scaling1}). The exponent  $\\nu > 1$ reveals shear-hardening behavior ($G$ increases with $\\epsilon$), which is equivalent to a dilatancy effect under a constant pressure condition. The stress-strain curve is reversible before yielding in cyclic shear ({Fig.~S13 in SM}), confirming that the observed non-linearity has a dominating elastic origin.\n\nThe yield stress $\\sigma_{\\rm  Y}$ increases, and the yield strain $\\epsilon_{\\rm Y}$ decreases, with $\\nu$ (see Fig.~\\ref{fig:simulation_yielding}(B)), consistent qualitatively with our theoretical predictions in Fig.~\\ref{fig:max}(B). Note that near jamming, the unstrained shear modulus $G_0 \\approx 0$ and the unstrained bulk modulus $K_0$ is finite~\\cite{o2003jamming}, thus we have set the Poisson's ratio $\\mathfrak{r}$ close to one in the theory (see Fig.~\\ref{fig:max}(B)). The negative correlation between the yielding strain $\\epsilon_{\\rm Y}$ and the degree of annealing $\\varphi_{\\rm g}$ is in contradiction with previous simulation~\\cite{jin2017exploring, ozawa2018random}\nand theoretical results~\\cite{ozawa2018random, rainone2015following}, where $\\epsilon_{\\rm Y}$ increases or nearly unchanged with the degree of annealing (see Fig.~\\ref{fig:simulation_yielding}(C)). Note that in all those previous cases, the non-linear shear hardening effect is absent. To confirm this point, additional simulations are performed at larger $P_{0}$, where shear hardening is compensated by strong plasticity, and $\\epsilon_{\\rm Y}$ correspondingly becomes either independent of $\\varphi_{\\rm g}$\n{ (Fig.~S14)}\nor slightly increasing with $\\varphi_{\\rm g}$\n{ (Fig.~S16(B))}.\nWe thus conclude that the left-shifted yielding peak is caused by non-linear corrections to the elasticity, and therefore can not be captured by linear elasticity  theories~\\cite{ozawa2018random}. Shear hardening disappears in  poorly annealed  (small $\\varphi_{\\rm g}$, see Fig.~\\ref{fig:simulation_yielding}(A)) or over-compressed (large $P_{0}$, see~Fig.~S14 { and~S16(B)}) systems, which explains   why it was not observed in many previous simulations~\\cite{o2003jamming, ozawa2018random}.\nDeep annealing and isostaticity (the coordination number $Z=2d$) could be  two key ingredients to  this effect.\n\n\n\n\nNext, we investigate the change of entropy during shear in simulations. Fig.~\\ref{fig:simulation_entropy}(A) shows that the jamming density $\\varphi_{\\rm j}(\\epsilon)$, at which $P$ vanishes upon decompression for the given $\\epsilon$, decreases monotonically with $\\epsilon$. According to the mean-field glass theory, $\\varphi_{\\rm j}$ is positively correlated with the configurational entropy $s_{\\rm conf}$ of glass states~\\cite{parisi2020theory}. Thus, Fig.~\\ref{fig:simulation_entropy}(A) already suggests an increase of entropy induced by shear, consistent qualitatively with our theoretical prediction, Eq.~(\\ref{eq:s0}).\nBecause the direct computation of the zero temperature entropy is\nimpractical for systems of thousands particles~\\cite{martiniani2017numerical},\nwe estimate it indirectly by the following two independent ways. (i) In the first approach, we assume that the zero-temperature configurational entropy at $\\varphi_{\\rm j}$ is proportional to the finite-temperature configurational entropy $s_{\\rm conf}(\\varphi_{\\rm g})$ of the corresponding parent liquid state at $\\varphi_{\\rm g}$.\nUnder this assumption, we\ncollect the data of $s_{\\rm conf}(\\varphi_{\\rm g})$ for the same model from Ref.~\\cite{berthier2017configurational} (estimated by taking the difference between the total entropy $s$ and the vibrational entropy $s_{\\rm vib}$), and $\\varphi_{\\rm j}(\\varphi_{\\rm g})$ from Ref.~\\cite{jin2021jamming}.\nCombining  them with $\\varphi_{\\rm j}(\\epsilon)$ in Fig.~\\ref{fig:simulation_entropy}(A) and $\\sigma(\\epsilon)$ in Fig.~\\ref{fig:simulation_yielding}(A) gives $s_{\\rm conf}(\\sigma)$ in Fig.~\\ref{fig:simulation_entropy}(B).\n{ The numerical value of $\\xi$ is estimated  by fitting the data to Eq.~(\\ref{eq:s0}).\n (ii) In the second approach, we consider the {\\it Edwards entropy}, constructed using the framework of Edwards statistical mechanics of granular matter, which is a generalization of Boltzmann statistical mechanics to non-equilibrium, athermal systems~\\cite{edwards1989theory,RevModPhys.90.015006}. The Edwards entropy is computed based on the fluctuations of the local Voronoi volumes~\\cite{PhysRevLett.101.188001} (see SM Sec.~S6 for details), and has been applied to unstrained, isotropic granular systems, in both simulations~\\cite{PhysRevLett.101.188001,PhysRevE.80.031301, jin2010first} and experiments~\\cite{PhysRevLett.127.018002}.\nHere, we apply the method to anisotropic systems\n under simple shear, using the Lees-Edwards boundary conditions~\\cite{lees1972computer}.\nThe change of Edwards entropy $\\Delta s_{\\rm Ed}$ per particle under shear (where we have taken the jammed states at the lowest jamming density, or the J-point density~\\cite{o2003jamming}, $\\varphi_{\\rm J} \\approx 0.655$~\\cite{jin2021jamming}, as the reference), is plotted in Fig.~\\ref{fig:simulation_entropy}(C) and fitted to Eq.~(\\ref{eq:s0}) to obtain $\\xi_{\\rm Ed}$.\nThe two approaches give close results on the exponents (see Fig.~\\ref{fig:simulation_entropy}(C)-inset). The numerical values of $\\xi$ and $\\xi_{\\rm Ed}$ are of the same order of the theoretical prediction  (Fig.~\\ref{fig:max}(C)-inset), and decay  similarly with the non-linear exponent $\\nu$.} We point out that the agreement remains to be mainly qualitative due to the phenomenological nature of our theories.\n\n\n\\section*{Discussion}\n\nOur results shed lights on the correct and concrete physical interpretation of the theoretical models at hand~\\cite{PhysRevD.102.069901, Baggioli:2020qdg}. In particular, the behavior of the entropy under shear suggests that the holographic models considered \\cite{Baggioli:2020qdg} are phenomenologically closer to amorphous solids rather than crystalline systems.\nOn the other hand, the current\nversions of theories do not incorporate marginal stability and isostaticity, and therefore cannot properly describe\nthe jamming transition. The theories assume that the system is always stable, instead of marginally stable, before the breaking point, and thus are applicable only to the regime away from jamming where the comparison to simulations is made. It would be extremely interesting to incorporate the necessary microscopic information, and extend our theoretical formulation towards the jamming transition to explore its critical properties.\n\nOur simulations show that shear hardening universally exists in 2D\/3D, frictionless\/frinctional, and thermally\/mechanically annealed granular models,  which suggests that the phenomenon could be directly relevant to a number of experimental systems (see Sec.~S9)~\\cite{xing2021x, zhao2022ultrastable, wang2022experimental}.  In particular, ultra-stable shear jammed granular materials were realized in a recent experiment~\\cite{zhao2022ultrastable}, making a direct test of the discussed correlations  possible in the  laboratory.\n\nThe approach presented here can be generalized to study more complex non-linear behaviors.\nFor example, very close to jamming, the stress-strain curve of granular matter typically displays  three consecutive elastic regimes with  shear~\\cite{kawasaki2020shear}: linear ($\\nu=1$), shear softening ($\\nu<1$) and shear hardening ($\\nu>1$). In this study, we mainly focus on the third regime (shear-hardening) at large strains.\nWe expect the entire stress-strain curve to be  captured by a generalized two-potential gravity theory. Further extensions can be made by taking into account the effects of finite temperature, finite shear rates and visco-elasticity, as well as non-equilibrium relaxational dynamics.\n\n\n\n\n\n\n\\section*{Materials and Methods}\n\n\n\n\\subsection*{Effective field theory}\n\nFollowing Ref.~\\cite{Nicolis:2015sra},\nthe EFT description is implemented in terms of  $d=2$ scalar fields ($I=1 \\ldots d$), $   \\phi^I(t,\\textbf{x})\\,=\\,\\langle \\phi^I \\rangle+\\delta \\phi^I(t,\\textbf{x})$, which play the role of co-moving coordinates. The variations from their equilibrium positions $\\delta \\phi^I(t,\\bf{x})$ coincide with the displacement fields used in standard elasticity theory. The EFT action is built only in terms of the derivatives of the fields, reflecting the invariance under the shift symmetry $\\phi^I \\rightarrow \\phi^I+a^I$ with $a^I$ constants. This symmetry follows from identifying the effective fields with phonons, which can be understood as Goldstone bosons for the spontaneously broken translations~\\cite{Leutwyler:1996er} -- fluctuations around the non-trivial vacuum expectation values $\\langle \\phi^I \\rangle = x^I$, with zero energy.\n\nThe only independent scalar objects that can be constructed out of the derivatives of the displacement fields (in $d=2$) are,\n$X\\,\\equiv\\,\\mathrm{Tr}\\,\\left[\\partial_\\mu \\phi^I \\partial^\\mu \\phi^J\\right]$ and\n$Z\\,\\equiv\\,\\mathrm{Det}\\,\\left[\\partial_\\mu \\phi^I \\partial^\\mu \\phi^J\\right]$,\nwhere the index $\\mu=(t,\\bf{x})$ collectively describes the set of spacetime coordinates. The scalar potential then becomes a generic function $\\mathcal{V}\\left(X,Z\\right)$\nand it is the only ingredient that must be provided in the theory. In most of our discussion, we will consider a power-law potential, which is reminiscent of the so-called hyper-elastic models, and takes the form,\n$ \\mathcal{V}\\left(X,Z\\right)\\,=\\,X^A\\,Z^{(B-A)\/2}$.\nHere $A$ and $B$ are two phenomenological model parameters that cannot be fixed without first-principle calculations.\nThe stress tensor of the theory can be derived using the standard quantum field theory variational prescription and the dispersion relation of the low-energy excitations by computing the action for the fluctuations around equilibrium at second order. More technical details are presented in SM Secs.~S1 and S2.\n\n\n\n\n\\subsection*{Gravity theory}\n\nThe gravitational description takes advantage of the so-called holographic duality in which a gravity system in a ($d+2$)-dimensional spacetime is mapped to a many-body system ``defined\" on its ($d+1$)-dimensional boundary. The boundary system is referred to as a ``hologram\" of the bulk. Thanks to the duality, introducing dissipative mechanisms and the effects of temperature becomes easy. More precisely, we use the bottom-up holographic duality in the large $\\mathcal{N}$ limit. Here, the parameter $\\mathcal{N}$ is interpreted as the number of effective degrees of freedom in the dual field theory \\cite{Ammon:2015wua}. Large $\\mathcal{N}$ in the gravity description corresponds to $L \\gg l_p$ with $L$ the AdS length-scale and $l_p$ the Planck scale.\nThis framework is more general than the original, string theory inspired, anti-de Sitter\/conformal field theory (AdS\/CFT) correspondence~\\cite{Maldacena:1997re} and it applies also to systems which are not critical (i.e., without conformal invariance). Concretely, the infrared (IR) physics of our gravitational system is governed by a non-relativistic geometry with an AdS$_2$ fixed point which does not enjoy the conformal group. Interestingly, this type of geometry shares some features with amorphous systems and spin glasses, and in particular, has a finite entropy at zero temperature~\\cite{PhysRevB.100.205108}. Finally, in the large $\\mathcal{N}$ limit (or equivalently $L \\gg l_p$), all quantum loops in the boundary field theory are suppressed by factors of $1\/\\mathcal{N}$ and the corresponding physics is effectively classical. All these assumptions justify the validity of our framework to describe the nonlinear rheology of classical particles in amorphous systems. The framework thus allows us to perform simple and robust computations of\n{ important physical}\nobservables such as the entropy. Our computations are based on the non-linear generalization~\\cite{PhysRevLett.114.251602,Alberte:2015isw} of the known holographic axion model~\\cite{Baggioli:2021xuv}.\n\nThe benchmark model uses a bulk potential form, $W(\\mathcal{X},\\mathcal{Z})\\,=\\,\\mathcal{X}^\\mathfrak{a}\\,\\mathcal{Z}^{(\\mathfrak{b}-\\mathfrak{a})\/2}$, where $\\mathcal{X}=\\frac{1}{2}\\mathrm{Tr}\\left[\\mathcal{I}^{IJ}\\right]$ and $\\mathcal{Z}=\\mathrm{Det}\\left[\\mathcal{I}^{IJ}\\right]$ with $\\mathcal{I}^{IJ}\\,\\equiv\\,\\partial_\\mu \\varphi^I \\partial^\\mu \\varphi^J$. Importantly, the index $\\mu$ here spans a 4-dimensional spacetime $(t,\\textbf{x},u)$ with $\\textbf{x}\\equiv (x,y)$.\nWhile the bulk potential $W(\\mathcal{X},\\mathcal{Z})$ has similar form as the one in EFT, the connection between the bulk potential and the dual EFT potential is very non-local and subtle. To avoid any clutter we will always use different symbols for bulk quantities and EFT ones. The stress tensor of the dual field theory can be extracted from the gravitational action by using the standard holographic dictionary while the entropy by utilizing the famous Bekenstein-Hawking Area law. See SM Secs.~S3-5 for more details.\n\n\n\n\\subsection*{Granular model of 3D frictionless soft spheres}\n\nThe model~{ \\cite{BCNO2016PRL, berthier2016growing, jin2021jamming}} is composed of $N = 8000$ SSs, with a diameter distribution  $P(D) \\sim D^{-3}$, where $D_{\\rm min} \\leq D \\leq D_{\\rm  min}\/0.45$.\nTwo spheres interact via a potential $V(r_{kl}) = \\frac{k_v}{2}(1 - \\frac{r_{kl}}{D_{kl}})^2$, only if their separation $r_{kl}$ is less than their mean diameter $D_{kl} = (D_k + D_l)\/2$; otherwise, $V=0$. The unit of length is the average diameter of all particles, the unit of energy is $10^3 \\times k_v$, and all particles have the same unit mass. Simulation data  are averaged over 96 independent samples.\n\n\\subsection*{Swap algorithm}\n\nThe SS granular configurations are quenched from equilibrium states at $\\varphi_{\\rm g}$~\\cite{berthier2016growing,jin2021jamming}. To prepare these equilibrium configurations, we use HS potential and a very efficient swap Monte Carlo (MC) algorithm~\\cite{BCNO2016PRL}. The HS configurations are generated by the hybrid of two different kinds of moves, the standard moves and the swap MC moves of exchanging the diameters of two randomly picked spheres. The swap moves are accepted only if the resulting configuration does not violate the HS constraint. Such moves help particles breaking out of cages formed by their neighbors and diffusing freely, and hence, facilitate the equilibration procedure. With the aid of this algorithm, we obtain equilibrium HS configurations over a wide range of $\\varphi_{\\rm g}$.\n\n\\subsection*{Simulation protocol of compression quench}\n\nOnce an equilibrium HS configuration is obtained by the swap algorithm, we switch off the temperature and switch to the SS potential. The system is then quenched to jamming density by a series of athermal quasi-static compression and decompression~\\cite{o2003jamming}. If the system is jammed, i.e., the energy per particle is larger than $10^{-13}$, the system is decompressed; otherwise compressed. During each  compression  (decompression) step, we instantaneously inflate (deflate) the spheres to increase (decrease) the packing density by $\\delta \\varphi$, and then minimize the total potential energy using the FIRE algorithm~\\cite{bitzek2006structural}. The energy minimization stops when the averaged residual force per particle is less than $10^{-11}$, which means that the configuration has reached a mechanically stable state.\nThe initial $\\delta \\varphi = 10^{-4}$;\nit is then reduced by a factor of two, whenever the state alters from jammed to unjammed (in the meanwhile, we switch from decompression to compression), or vice versa. This procedure stops when $\\delta \\varphi< 10^{-6}$ and the system is jammed. The jammed configurations are quasi-statically compressed by $\\delta \\varphi = 10^{-5}$ to the target pressure $P_{0}$ to obtain unstrained configurations at $\\epsilon = 0$.\n\n\\subsection*{Simulation protocol of athermal quasi-static shear}\n\nShear is performed under the athermal quasi-static and Lees-Edwards boundary~\\cite{lees1972computer} conditions. Each shear step ($\\delta \\epsilon = 10^{-4}$) involves an affine transformation of coordinates, and then an energy minimization using the FIRE algorithm~\\cite{bitzek2006structural}. The energy minimization stops when the average residual force per particle is less than $10^{-11}$.\nThe pressure $P$ and stress $\\sigma$ (shear is applied in the $xy$ plane) is calculated from the Virial formula,\n\\begin{eqnarray}\nP =  \\frac{1}{3V} \\sum_{\\langle kl \\rangle } \\mathbf{r}_{kl} \\cdot \\mathbf{f}_{kl}\\,,\\quad \\sigma=-\\frac{1}{V}\\sum_{\\langle kl \\rangle } r_{kl,x} f_{kl,y}\\,,\n\\end{eqnarray}\nwhere $\\mathbf{r}_{kl}$ and $ \\mathbf{f}_{kl}$ are the center-to-center vector and force between particles $k$ and $l$ ($r_{kl,x}$ and $ f_{kl,y}$ are the  $x$ and $y$ components), $V$ is the volume of simulation box, and $\\langle kl \\rangle$ stands for all contacting pairs.\n\n\n\n\n\n\n\n\\section*{Aknowledgments}\nWe thank Oriol Pujolas and Alessio Zaccone for providing useful comments and discussions.\n\n\\noindent{\\bf Funding:} M.B. acknowledges the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01). L.L. acknowledges the supports in part by NSFC No.12122513, No.12075298 and No.11991052, and by the CAS Project for Young Scientists in Basic Research YSBR-006. Y.J. acknowledges funding from {Project 12161141007,} Project 11974361 and  Project 11935002  supported by NSFC, and the Key Research Program of Frontier  Sciences, Chinese Academy of Sciences, Grant NO. ZDBS-LY-7017.  The authors from ITP acknowledge funding from Project 12047503 supported by NSFC and the Key Research Program of the Chinese Academy of Sciences, Grant NO. XDPB15.\nThe simulations were performed using the HPC Cluster at ITP-CAS.\n\n\\noindent{\\bf Author contributions:} T.J., L.L. and M.B. performed the theoretical computations.\nD.P. and Y.J. performed the simulations. All the authors contributed to the writing of the manuscript and the discussion of the ideas behind it.\n\n\\noindent{\\bf Competing interests:} Authors declare that they have no competing interests.\n\n\\noindent{\\bf Data and materials availability:} All data are available in the main text or the supplementary materials.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nSymplectic dynamics has been an intense research area for the last fifty or sixty years.\nIt all started with Poincar\\'{e}--Birkhoff Theorem in the first decades of last century.\nArnold conjecture and related questions were pivotal to the revolution in this field, though.\nThe existence of extra fixed points for Hamiltonian symplectomorphisms, statement of Arnold conjecture, has\nbeen proved in most of the instances.\nThe Hamiltonian flows also tend to have special dynamical properties. For instance, the existence of periodic orbits\non a dense set of energy levels (see Hofer and Zehnder \\cite{hoferzehnder}).\nThere is another instance of Hamiltonian rigidity, that is the Conley conjecture (proven in several cases,\nsee \\cite{ginzburggurel}). The conjecture states that a Hamiltonian symplectomorphism has an infinite number of\ngeometrically different periodic orbits. The mantra behind this note is to point out that the removal of the\nHamiltonian condition destroys most of the rigidity in dynamics. The prototypical example is $\\mathds{T}^2$ with the symplectic\n(non Hamiltonian) irrational translation: it is minimal and, therefore, it does not have a single periodic orbit. We will generalize this example.\n\nRigidity in contact dynamics is more subtle.\nHamiltonian contact flows do correspond to a special kind of Hamiltonian symplectic flows.\nTherefore, the existence result of periodic orbits in a generic energy level was conjecturally strengthen to\nall the levels in the convex case. The generalization of this fact is the content of the famous Weinstein conjecture. It has been a powerful engine developing\ncontact topology in the last 30 years. It has been proven in several cases.\nHowever, the behaviour of a discrete contact dynamical system remained vague. Only, recently, an analogue of the Arnold conjecture\nwas stated: the translated points conjecture (see \\cite{sandon}).\nIn this case, our aim is to show that the interplay between the Reeb flow and the Hamiltonian contactomorphisms,\nwhich is the content of the conjecture, is probably the only natural rigid phenomenum in discrete Hamiltonian contact dynamics.\nIn particular, we will show that (positive) Hamiltonian contactomorphisms can be pretty wild from a dynamical point of view.\n\nThe approximation by conjugation method, introduced by Anosov and Katok, is a remarkable technique to\nproduce diffeomorphisms with prescribed properties on the asymptotical distribution of their orbits.\nIn their seminal paper \\cite{anosovkatok}, Anosov and Katok proved the existence of ergodic transitive diffeomorphisms\non any closed manifold. The name of the method comes from the fact that the maps are constructed as limits\nof conjugates $h R_{\\alpha} h^{-1}$ of a non--trivial $\\mathbb{S}^1$--action $\\{R_{\\alpha}: \\alpha \\in \\mathbb{S}^1\\}$ on the manifold.\nThis direct approach was replaced by an abstract one by Fathi and Herman in \\cite{fathiherman}. They applied\nthe Baire category theorem to the closure of the previous set of conjugates to prove the existence of minimal and uniquely ergodic\ndiffeomorphisms in closed manifolds that admit a locally free $\\mathbb{S}^1$--action. A short overview of the conjugation method\ntogether with further applications is found in \\cite{fayadkatok}.\n\nIn this article, the arguments in \\cite{fathiherman} are adapted to the symplectic and contact setting.\nDenote by $\\mathrm{Symp}(M, \\omega)$ the group of symplectomorphisms of a symplectic manifold $(M, \\omega)$\nand $\\mathrm{Symp}_0(M, \\omega)$ the connected component of the identity map. Similarly, denote by $\\mathrm{Cont}(M, \\xi)$\nthe group of contactomorphisms of a contact manifold $(M, \\xi)$ and $\\mathrm{Cont}_0(M, \\xi)$ the connected component\nof the identity. All these groups are known to be $C^1$--closed, hence $C^{\\infty}$--closed, in $\\mathrm{Diff}(M)$.\n\nAs it will be clear from the contents of the article a more ambitious goal would be to adapt the statements\nto the Hamiltonian symplectomorphism group\\footnote{As is well--known, the Hamiltonian contactomorphism group\ncorresponds to the identity component of the contactomorphisms group, i.e. any contact vector field is Hamiltonian.}. This encounters two obstacles:\n\\begin{itemize}\n\\item $\\mathrm{Ham}(M, \\omega)$ is not known to be $C^{\\infty}$--closed in $\\mathrm{Diff}(M)$ as this is the content of the Flux\nconjecture. However, this can be dealt with in particular cases (for instance, assuming that $M$ is simply connected).\n\\item A Hamiltonian $\\mathbb{S}^1$--action is never locally free. Therefore, the conjugation method, particularly simple in \\cite{fathiherman}, needs to be sophisticated \\cite{fayadkatok}.\nThis requires more subtle symplectic topology results to be developed in a forthcoming work.\n\\end{itemize}\n\nA map is called minimal if every orbit is dense and is called uniquely ergodic provided it has\njust one invariant probability measure.\nThe main theorems of this article are:\n\n\\begin{theorem}\\label{thm:symplectic}\nIf a symplectic manifold $(M, \\omega)$ admits a locally free $\\mathbb{S}^1$--action by symplectomorphisms, then\nthere exists $\\psi \\in \\mathrm{Symp}_0(M)$ such that $\\psi$ is minimal.\n\\end{theorem}\n\n\\begin{theorem}\\label{thm:contact}\nIf a contact manifold $(M, \\xi)$ admits a locally free $\\mathbb{S}^1$--action by contactomorphisms, then\nthere exists $\\psi \\in \\mathrm{Cont}_0(M, \\xi)$ such that $\\psi$ is strictly ergodic.\n\\end{theorem}\n\n\nGiven a contact manifold $(M, \\xi)$, $\\xi = \\ker(\\alpha)$ cooriented, a path\/loop of contactomorphisms\n$\\{\\psi_t\\}$ is \\emph{positive} if $\\alpha (\\partial \\psi\/\\partial t) > 0$ everywhere.\nBy definition, this is equivalent to the Hamiltonian $H$ which generates this path\/loop being everywhere positive.\nThe $\\mathbb{S}^1$--action on $M$ is said positive if the loop of contactomorphisms it defines is positive.\n\n\\begin{corollary}\nIf the $\\mathbb{S}^1$--action is positive then $\\psi$ from Theorem \\ref{thm:contact} is generated by a positive path.\n\\end{corollary}\n\\begin{proof}\nThis follows from a general fact for contact manifolds that admit a positive loop.\nThe group of contactomorphisms is locally path connected (see \\cite[Chapter 6]{banyaga}) hence $\\mathrm{Cont}_0(M, \\xi)$ is path connected.\nLet $\\{\\varphi_t\\}_{t = 0}^1$ be a path of contactomorphisms from $\\varphi_0 = \\mathrm{id}$ to $\\varphi_1 = \\psi$.\nDenote by $\\{R_{t}\\}_{t=0}^1$ the loop of contactomorphisms given by the action.\nA computation (see \\cite{casalspresas})\n shows that, if $H, G: M \\times [0,1] \\to \\mathds{R}$ are the Hamiltonians which generate $\\{\\varphi_t\\}$ and $\\{R_t\\}$,\nthe composition path $\\{\\varphi_t \\circ R_{mt}\\}_{t = 0}^1$ is generated by the Hamiltonian\n$$F(p, t) = H(p, t) + m e^{-f_t} G(\\varphi_t^{-1}(p), t),$$\nwhere $\\varphi_t^* \\alpha = e^{f_t} \\alpha$ and $m \\ge 1$ is arbitrary. Incidentally, notice that $G$ is independent of $t$.\nThen, since $G$ is strictly positive, if $m \\ge 1$ is sufficiently large, $F > 0$.\n\\end{proof}\n\nIt will be obvious from the version of the conjugation method used in this note that:\n\\begin{corollary}\\label{cor:id}\nThe diffeomorphisms constructed in Theorems \\ref{thm:symplectic} and \\ref{thm:contact} can be chosen $C^{\\infty}$--close\nto the identity.\n\\end{corollary}\n\n\\textbf{Examples}\n\n{\\bf 1. Prequantum contact manifolds.}\n\nBy a result of Thomas \\cite{thomas}, a contact manifold admits a locally free $\\mathbb{S}^1$--action if and only if\nit is the prequantization $\\mathbb{S}^1$--bundle over a symplectic orbifold (also called circle orbibundle).\n\n{\\bf 2. Manifolds not admitting a positive $\\mathbb{S}^1$--action.}\n\nThe spherical cotangent bundle, $\\mathbb{S} T^* M$, of a manifold $M$ (also referred to as oriented projectivization of $T^* M$)\ncarries a canonical cooriented contact structure. There do not exist positive paths of Legendrian isotopies connecting\ndifferent fibers of $\\mathbb{S} T^* M$ provided the universal cover of $M$ is $\\mathds{R}^n$ (see \\cite{colin}) or, more generally,\nan open manifold (see \\cite{chernov}).\nAs a consequence, $\\mathbb{S} T^* M$ does not admit positive loops.\nHowever, it may admit $\\mathbb{S}^1$--actions: any $\\mathbb{S}^1$--action on $M$ by diffeomorphisms lifts to a $\\mathbb{S}^1$--action on $\\mathbb{S} T^* M$ by contactomorphisms so Theorem \\ref{thm:contact} can be applied.\nThis is the case, for instance, of standard $\\mathds{T}^3$ as it is the spherical cotangent bundle over $\\mathds{T}^2$.\n\n\n{\\bf 3. Symplectic examples.}\n\nIf $M$ admits a symplectic free $\\mathbb{S}^1$--action,\nthen it admits a fibration $H\\colon M \\to \\mathbb{S}^1$ such that the action preserves its fibers. Moreover,\nthere is a $C^{\\infty}$-small perturbation $\\widetilde{\\omega}$ of the original symplectic form for which the action is still symplectic\nand such that $\\ker (\\iota_{X} \\widetilde{\\omega}) = \\ker (dH)$,\nwhere $X$ denotes the infinitesimal generator of the action.\nConsequently, there exists a symplectic fibration $\\widehat{H}\\colon M\/\\mathbb{S}^1 \\to \\mathbb{S}^1$.\nPartial converse results are available \\cite{marisafdez}.\n\n\n\n\n\\section*{Acknowledgements}\nThe authors express their gratitude to Viktor Ginzburg and Daniel Peralta for their useful comments during\nthe preparation of the manuscript.\nThe authors have been supported by the Spanish Research Projects SEV-2015-0554 and MTM2013-42135 and by the\nERC Starting Grant 335079 from Daniel Peralta.\n\n\n\\section{Conjugation method}\\label{sec:conjugationmethod}\n\nThe conjugation method was introduced by Anosov and Katok \\cite{anosovkatok} in 1970.\nThey constructed ergodic diffeomorphisms on every closed manifold as limits of volume preserving periodic transformations.\nLater, Fathi and Herman \\cite{fathiherman} developed the method to prove the existence of\nminimal and uniquely ergodic volume preserving diffeomorphisms in manifolds which admit a locally free $\\mathbb{S}^1$--action.\nKatok had previously announced \\cite{katokrussian} the theorem provided the action is free.\nThis section, following the approach of \\cite{fathiherman}, presents how the conjugation method is used\nto find minimal and uniquely ergodic diffeomorphisms.\n\nGiven a closed manifold $N$, $\\mathrm{Diff}(N)$ denotes the group of diffeomorphisms of $N$ equipped with the $C^{\\infty}$--topology.\nAs our aim is to adapt the method to work for subgroups of $\\mathrm{Diff}(N)$, an abstract subgroup $\\mathcal{G}(N) < \\mathrm{Diff}(N)$ is considered.\nThese subgroups must be defined, at least, for the manifold in consideration, later denoted $M$, and some quotients\nof $M$ over the free action of a finite group.\nIn \\cite{fathiherman}, $\\mathcal{G}(N)$ was set to be equal to $\\mathrm{Diff}(N)$ or $\\mathrm{Diff}_{\\mu}(N)$\n(diffeomorphisms preserving a prescribed measure $\\mu$ of positive $C^{\\infty}$ density).\nIn this work, $\\mathcal{G}(N)$ will later be set to be $\\mathrm{Symp}_0(M, \\omega)$ or $\\mathrm{Cont}_0(M, \\xi)$ and the underlying geometric structure\nwill be preserved by the finite quotients under consideration.\nAlthough the previously cited subgroups are closed, this discussion does not assume $\\mathcal{G}(N)$ is necessarily closed,\n$\\overline{\\mathcal{G}(N)}$ denotes the closure of $\\mathcal{G}(N)$.\nThe topology considered in the group of diffeomorphisms and their subsets will always be the $C^{\\infty}$--topology.\n\nLet $M$ be a closed manifold and, for simplicity, put $\\mathcal{G} = \\mathcal{G}(M)$.\nAssume $M$ admits a locally free $\\mathbb{S}^1$--action on $M$ by diffeomorphisms in $\\mathcal{G}$, i.e.\na group homomorphism $\\mathbb{S}^1 \\rightarrow \\mathcal{G}: \\alpha \\mapsto R_{\\alpha}$ such that for every $x \\in M$ the stabilizer subgroup\n$S_x = \\{\\alpha: R_{\\alpha}(x) = x\\}$ is a discrete subgroup of $\\mathbb{S}^1$ (in particular, it is finite).\nThroughout this article, $\\mathbb{S}^1$ is identified to $\\mathds{R}\/\\mathds{Z}$.\n\nThe union of all stabilizers $S = \\cup_{x \\in M} S_x$ is still a finite subgroup of $\\mathbb{S}^1$.\nIndeed, for any $x \\in M$ consider a small neighborhood $V_x$ of $S_x$ in $\\mathbb{S}^1$ such that any subgroup of $\\mathbb{S}^1$ contained\nin $V_x$ is a subgroup of $S_x$.\nBy continuity one gets that $S_y \\subset V_x$ in some neighborhood of $x$, so $S_y < S_x$.\nThe conclusion follows from the compactness of $M$.\n\nThe goal is to prove the existence of elements of $\\overline{\\mathcal{G}}$ satisfying certain dynamical properties. Let $\\mathcal{A}$ be the subset of\n$\\overline{\\mathcal{G}}$ composed of such elements. The conjugation method aims to prove that $\\mathcal{A}$ is not empty by showing that\nthe elements of $\\mathcal{A}$ are generic in the subgroup\n\\begin{equation}\\label{eq:Cdefinition}\n\\mathcal{C} = \\overline{\\{g R_{\\alpha} g^{-1}: \\alpha \\in \\mathbb{S}^1, g \\in \\mathcal{G}\\}}\n\\end{equation}\nof $\\overline{\\mathcal{G}}$, that is, $\\mathcal{A} \\cap \\mathcal{C}$ is a dense $G_{\\delta}$ subset of $\\mathcal{C}$\n(intersection of countably many open and dense subsets).\nCorollary \\ref{cor:id} is then a trivial consequence of this approach.\nRecall that since $\\mathrm{Diff}(M)$ is a Baire space and $\\mathcal{C}$ is closed in $\\overline{\\mathcal{G}}$ and so in $\\mathrm{Diff}(M)$, $\\mathcal{C}$ is also a Baire space.\nThus, it is enough to find a countable family $\\{A_j\\}_{j \\in J}$ of open and dense subsets of $\\mathcal{C}$\nsuch that\n$$\\bigcap_{j \\in J} A_j = \\mathcal{A} \\cap \\mathcal{C}.$$\n\nIn the proofs below, a family of open and dense subsets $\\{A_i\\}_{i \\in I}$ of $\\mathcal{C}$ is defined.\nIt has a countable coinitial subfamily $\\{A_j\\}_{j \\in J}$, that is, for every $i \\in I$ there is $j \\in J$\nsuch that $A_j \\subset A_i$.\nConsequently, $\\cap_{i \\in I} A_i = \\cap_{j \\in J} A_j$.\nAdditionally, for any $g \\in \\mathcal{G}$ and $i \\in I$, $g A_i g^{-1} \\subset A_{k}$ for some $k \\in I$.\nIn order to prove the density of $A_i$ in $\\mathcal{C}$, one needs to show that every element $g R_{\\alpha} g^{-1}$\nbelongs to the closure of $A_i$. In view of the previous properties on the family $\\{A_i\\}_{i \\in I}$,\nit is enough to show that $R_{\\alpha} \\in \\overline{A_i}$ for any $\\alpha \\in \\mathbb{S}^1$ and $i \\in I$.\n\nA number $\\alpha \\in \\mathds{R}\/\\mathds{Z}$ is said to be \\emph{coprime} with $S$ if $S \\cap \\langle \\alpha \\rangle = \\{0\\}$,\nwhere $\\langle \\alpha \\rangle$ denotes the subgroup of $\\mathbb{S}^1 \\cong \\mathds{R}\/\\mathds{Z}$ generated by $\\alpha$.\nSince the set $S$ of stabilizers is finite and $\\mathds{Q}\/\\mathds{Z}$ is dense in $\\mathbb{S}^1$, it suffices to check that\n$R_{\\alpha} \\in \\overline{A_i}$ for $\\alpha \\in \\mathds{Q}\/\\mathds{Z}$ coprime with $S$ in order to prove it for all $\\alpha \\in \\mathds{R}\/\\mathds{Z}$.\n\n\\subsection{Minimal diffeomorphisms}\n\nThis subsection shows how the conjugation method was applied in \\cite{fathiherman} to obtain diffeomorphisms\nwith all their orbits dense.\n\n\\begin{definition}\nA homeomorphism $f \\colon X \\to X$ is said to be minimal if every orbit is dense, i.e.\n$M = \\overline{\\{f^n(x): n \\in \\mathds{Z}\\}}$ for every $x \\in X$.\n\\end{definition}\nMinimality is a topological property which is easily seen to be equivalent to the non-existence of proper\nclosed subsets invariant under the dynamics.\n\\begin{lemma}\\label{lem:minimalequivalence}\nLet $X$ be a compact topological space.\nA homeomorphism $f\\colon X \\to X$ is minimal if and only if for every open set $U \\subset X$ there exists $n \\in \\mathds{N}$\nsuch that\n$$U \\cup f(U) \\cup \\ldots \\cup f^n(U) = X.$$\n\\end{lemma}\n\\begin{proof}[Sketch of the proof.]\nNotice that the complement of $\\cup_{m \\in \\mathds{Z}} f^m(U)$ in $X$ is closed and $f$--invariant, it must be empty if $f$ is minimal.\nThe open cover $\\{f^m(U)\\}_{m \\in \\mathds{Z}}$ of $X$ has, by compactness, a finite subcover whose elements, after applying $f$ conveniently,\nare forward iterates of $U$.\n\\end{proof}\n\nLet us apply the conjugation method to obtain minimal diffeomorphisms in $\\overline{\\mathcal{G}}$.\nDenote by $\\mathcal{M}$ the set of minimal diffeomorphisms. Consider the sets\n$$W_{U, n} = \\{g \\in \\mathcal{G}: U \\cup g(U) \\cup \\ldots \\cup g^n(U) = M\\},$$\nwhere $U$ ranges over the open subsets of $M$ and $n$ over the positive integers.\nBy definition, $W_{U,n}$ is an open subset of $\\mathrm{Diff}(M)$.\nConsequently, $\\mathcal{M}_U = \\overline{\\mathcal{G}} \\, \\cap \\, \\left(\\cup_{n \\ge 1} W_{U, n}\\right)$ is an open subset of $\\overline{\\mathcal{G}}$.\nBy Lemma \\ref{lem:minimalequivalence}, $\\mathcal{M}$ is the intersection of all open sets $\\mathcal{M}_U$.\nSince $M$ has a countable basis of open sets $\\{U_i\\}_{i\\in \\mathds{N}}$, the subfamily $\\{\\mathcal{M}_{U_i}\\}_{i \\in \\mathds{N}}$ is coinitial\nand $\\mathcal{M} = \\cap_i \\mathcal{M}_{U_i}$. Moreover, $g \\mathcal{M}_U g^{-1} = \\mathcal{M}_{g^{-1}(U)}$.\nThus, in order to check that $\\mathcal{M}_U$ is dense in $\\mathcal{C}$ it is enough to prove that $R_{\\alpha}$ is accumulated by elements\nof $\\mathcal{M}_U$ for every $\\alpha \\in (\\mathds{Q}\/\\mathds{Z}) \\setminus S$.\n\nGiven any $\\alpha = p\/q \\notin S$, $\\mathrm{gcd}(p,q) = 1$, $F_q$ denotes the subgroup of $\\mathbb{S}^1$ generated by $p\/q$.\nThe quotient of $M$ under the action of $F_q$ is a manifold $\\widehat{M} = M\/F_q$. The $\\mathbb{S}^1$--action on $M$ induces,\nby the identification $\\mathbb{S}^1\/F_q \\cong \\mathbb{S}^1$, another locally free $\\mathbb{S}^1$--action on $\\widehat{M}$.\nThis action is given by $\\{\\widehat{R}_{\\beta}: \\beta \\in \\mathbb{S}^1\\}$, where $\\widehat{R}_{q\\alpha}$\nis the projection of $R_{\\alpha}$ onto $\\widehat{M}$.\nThe subgroup $\\widehat{\\mathcal{G}} := \\mathcal{G}(\\widehat{M}) < \\mathrm{Diff}(\\widehat{M})$ must satisfy the following hypothesis:\n\n\\medskip\n\\textbf{(H1)}\nAny element of $\\widehat{\\mathcal{G}}$ has a lift in $\\mathcal{G} = \\mathcal{G}(M)$.\n\\medskip\n\nNext condition must be also fulfilled. These hypothesis are discussed in Section \\ref{sec:proofs}.\nNote that it is an exercise to check them in the case $\\mathcal{G}(N) = \\mathrm{Diff}(N)$ or $\\mathcal{G}(N) = \\mathrm{Diff}_{\\mu}(N)$,\nwhere $\\mu$ is a probability measure on $N$ of positive $C^{\\infty}$ density.\n\n\\medskip\n\\textbf{(H2)} For any locally free $\\mathbb{S}^1$--action on a manifold $N$, and any open $V \\subset N$, there exists an\nelement $f \\in \\mathcal{G}(N)$ such that $f^{-1}(V)$ meets all the orbits of the action.\n\\medskip\n\nFix an open set $U \\subset M$. For the previous $\\alpha = p\/q$, denote by $\\widehat{U}$ the projection of $U$ onto $\\widehat{M}$\nand let $\\widehat{h}$ be the map supplied by (H2) for $N = \\widehat{M}$,\n$V = \\widehat{U}$ and the induced $\\mathbb{S}^1$--action on $\\widehat{M}$.\nThere is a lift $h \\in \\mathcal{G}$ of $\\widehat{h}$ whose existence is guaranteed by (H1). It satisfies\n$h R_{\\alpha} h^{-1} = R_{\\alpha}$ and $h^{-1}(U)$ meets all the orbits of the $\\mathbb{S}^1$--action on $M$ because $\\alpha$\nis coprime with $S$.\n\nThe following lemma generalizes to $\\mathbb{S}^1$--actions a simple fact:\nthe iterates of an open interval under an irrational rotation eventually cover $\\mathbb{S}^1$.\n\n\\begin{lemma}\\label{lem:irrationalminimal}\nFor any $\\beta \\notin \\mathds{Q}\/\\mathds{Z}$, $h R_{\\beta} h^{-1} \\in \\mathcal{M}_U$.\n\\end{lemma}\n\\begin{proof}\nLet $\\gamma$ denote a orbit of the action and $W$ an open subset of $M$, $\\gamma \\cap W \\neq \\emptyset$.\nThere exists $n \\ge 1$ such that $W \\cup R_{\\beta}(W) \\cup \\ldots \\cup R_{n\\beta}(W)$ cover $\\gamma$.\nThe same is true for orbits close to $\\gamma$.\nTake $W = h^{-1}(U)$, then $W$ meets every orbit.\nSince the orbit space of the action is compact, there exists $m \\ge 1$ such that\n$W \\cup R_{\\beta}(W) \\cup \\ldots \\cup R_{\\beta}^m(W)$ contains every orbit $\\gamma$.\nThus $h R_{\\beta} h^{-1} \\in W_{U, m} \\cap \\mathcal{C} \\subset \\mathcal{M}_U$.\n\\end{proof}\n\nTake a sequence of irrational numbers $\\beta_n \\to \\alpha$.\nClearly, $h R_{\\beta_n} h^{-1} \\xrightarrow{\\mathcal{C}^{\\infty}} h R_{\\alpha} h^{-1} = R_{\\alpha}$.\nSince from Lemma \\ref{lem:irrationalminimal}, $h R_{\\beta_n} h^{-1}$ belongs to $\\mathcal{M}_U$ and also, by definition, to $\\mathcal{C}$,\nthe map $R_{\\alpha} \\in \\overline{\\mathcal{M}_U}$ and the conclusion follows.\n\nIn conclusion, the existence of a minimal diffeomorphism in $\\overline{\\mathcal{G}}$ is guaranteed provided (H1) and (H2) are satisfied.\n\n\\subsection{Strictly ergodic diffeomorphisms}\n\nIn this subsection, the conjugation method is applied to obtain a diffeomorphism\nwhose orbits are uniformly distributed along $M$ in a measure theory sense.\n\n\\begin{definition}\nLet $X$ be a compact metric space. A homeomorphism $f\\colon X \\to X$ is called uniquely ergodic if it has only one\ninvariant probability measure. If the invariant measure has full support $f$ is called strictly ergodic.\n\\end{definition}\n\nNext lemma follows from two dynamical facts: the support of an $f$--invariant measure is itself invariant under $f$\nand any compact invariant subset of $X$ admits an invariant measure supported on it.\n\n\\begin{lemma}\nA map is strictly ergodic if and only if it is uniquely ergodic and minimal.\n\\end{lemma}\n\nOnce strict ergodicity has been split into two properties, let us take care of unique ergodicity.\n\n\\begin{proposition}\\label{prop:uecharacterization}\nLet $X$ be a compact metric space and $f\\colon X \\to X$ be a homeomorphism. The following statements are equivalent.\n\\begin{itemize}\n\\item $f$ is uniquely ergodic.\n\\item For every map $u \\in C^0(X, \\mathds{R})$,\n$\\frac{1}{n} \\left(\\sum_{k = 0}^{n-1} u \\circ f^k\\right)$ converges uniformly\nas $n \\to \\infty$ to a constant (equal to $\\int_X u \\, d\\mu$,\nwhere $\\mu$ is the only invariant probability measure).\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nSee Walters \\cite[Theorem 6.19]{walters}\n\\end{proof}\n\nSince the space of real--valued continuous functions over $M$ is separable, Proposition \\ref{prop:uecharacterization}\ncan be used to show that the set of uniquely ergodic diffeomorphisms $\\mathcal{E}$ is a $G_{\\delta}$ subset of $\\mathrm{Diff}(M)$.\nIndeed, let $u \\in C^0(X, \\mathds{R})$, $\\varepsilon > 0$. Define\n$$\\mathcal{E}(u, \\varepsilon) = \\biggl\\{f \\in \\mathcal{C}: \\exists \\, n \\ge 1, c \\in \\mathds{R} \\text{ such that }\n\\biggl|\\biggl| \\frac{1}{n} \\sum_{k = 0}^{n-1} u \\circ f^k - c\\biggr|\\biggr| < \\varepsilon \\biggr\\},$$\nwhere $\\mathcal{C}$, the closure of the set of conjugates of the action, was defined in (\\ref{eq:Cdefinition}).\nTrivially, the sets $\\mathcal{E}(u, \\varepsilon)$ are open.\nFix a dense sequence $\\{u_i\\}$ in $C^0(X, \\mathds{R})$.\n\\begin{lemma}\n$$\\mathcal{E} \\cap \\mathcal{C} = \\bigcap_i \\bigcap_{k \\ge 1} \\mathcal{E}(u_i, 1\/k).$$\n\\end{lemma}\n\\begin{proof}\nSimply notice that if $\\varepsilon_1 > \\varepsilon_0 > 0$ and $\\Vert u_1 - u_0 \\Vert < \\varepsilon_1 - \\varepsilon_0$\nthen $\\mathcal{E}(u_0, \\varepsilon_0) \\subset \\mathcal{E}(u_1, \\varepsilon_1)$.\n\\end{proof}\nFurthermore, note that $g \\, \\mathcal{E}(u, \\varepsilon) \\, g^{-1} = \\mathcal{E}(u \\circ g, \\varepsilon)$.\nThus, in order to show that $\\mathcal{E}$ is dense in $\\mathcal{C}$ it suffices to check that $R_{\\alpha}$ belongs to\n$\\overline{\\mathcal{E}(u, \\varepsilon)}$ for every rational $\\alpha$ not contained in $S$.\n\nHenceforth, assume $u \\in C^0(M, \\mathds{R})$ and $\\varepsilon > 0$ are fixed and the following hypothesis is satisfied.\n\n\\medskip\n\\textbf{(H3)}\nGiven a locally free $\\mathbb{S}^1$--action $\\{R_{\\alpha}\\}_{\\alpha \\in \\mathbb{S}^1}$ on a manifold $N$, an open set $V \\subset N$\nand $\\varepsilon > 0$, there exists an element $f \\in \\mathcal{G}(N)$ such that\n$$\\lambda(\\{\\alpha \\in \\mathbb{S}^1: R_{\\alpha}(y) \\notin f^{-1}(V)\\}) < \\varepsilon$$\nfor every $y \\in N$ ($\\lambda$ denotes the Lebesgue measure in $\\mathbb{S}^1$ of total mass equal to 1).\n\\medskip\n\nThere are some unavoidable bothering technical issues to check this condition\n(see \\cite[Section 6]{fathiherman}).\n\nAs in the minimal case, fix $\\alpha = p\/q \\notin S$, $\\gcd(p, q) = 1$, and write $F_q = \\langle \\alpha \\rangle$. Define\n$$\\widehat{u} = \\frac{1}{q} \\sum_{k = 0}^{q-1} u \\circ R_{k\/q}.$$\nClearly, $\\widehat{u}$ can be seen as a function on the quotient $\\widehat{M} = M\/F_q$.\nConsider an arbitrary $\\eta > 0$ and fix $\\delta > 0$ such that $\\delta (1 + ||u||) < \\eta$.\nTake $y_0 \\in \\widehat{M}$ and note that\n$\\widehat{U} = \\{y \\in \\widehat{M}: |\\widehat{u}(y) - \\widehat{u}(y_0)| < \\delta\\}$\nis a non--empty open subset of $\\widehat{M}$.\nApply (H3) to find $\\widehat{h} \\in \\widehat{\\mathcal{G}}$ such that for every $y \\in \\widehat{M}$,\n$\\lambda(\\{\\alpha \\in \\mathbb{S}^1: \\widehat{R}_{\\alpha}(y) \\notin \\widehat{h}^{-1}(U)\\}) < \\delta$,\nwhere $\\widehat{R}$ denotes the $\\mathbb{S}^1$--action on $\\widehat{M}$ induced by $R$. Then,\n\\[\n\\left|\\int_{\\mathbb{S}^1} \\widehat{u} \\circ \\widehat{h} \\circ \\widehat{R}_{\\beta}(y) d\\beta - \\widehat{u}(y_0)\\right| \\le\n\\delta + \\delta ||\\widehat{u}|| \\le \\delta (1 + ||u||) < \\eta.\n\\]\nfor every $y \\in \\widehat{M}$.\nChoose a lift $h \\in \\mathcal{G} = \\mathcal{G}(M)$ of $\\widehat{h}$, whose existence is guaranteed by (H1). Then,\n$h \\circ R_{\\alpha} = R_{\\alpha} \\circ h$, and, for every $x \\in M$,\n$$\\int_{\\mathbb{S}^1} u \\circ h \\circ R_{\\theta}(x) d\\theta =\n\\int_{\\mathbb{S}^1} \\widehat{u} \\circ \\widehat{h} \\circ \\widehat{R}_{\\beta}(\\widehat{x}) d\\beta,$$\nwhere $\\widehat{x}$ denotes the projection of $x$ onto $\\widehat{M}$.\nIn sum, we have proved the following lemma.\n\n\\begin{lemma}\\label{lem:averagingmap}\nFor any $\\eta > 0$, there exists $h \\in \\mathcal{G}$ and a constant $c \\in \\mathds{R}$ such that\n$$\\left| \\int_{\\mathbb{S}^1} u \\circ h \\circ R_{\\theta}(x) d\\theta - c\\right| < \\eta$$\nholds for every $x \\in M$.\n\\end{lemma}\n\n\\begin{proposition}\n$R_{\\alpha} \\in \\overline{\\mathcal{E}(u, \\varepsilon)}$.\n\\end{proposition}\n\\begin{proof}\nSince an irrational rotation is uniquely ergodic, for any $\\beta \\notin \\mathds{Q}\/\\mathds{Z}$ and $x \\in M$\n$$\\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k = 0}^{n-1} v \\circ R_{\\beta}^k(x) = \\int_{\\mathbb{S}^1} v \\circ R_{\\theta} d\\theta$$\nholds for every $v \\in C^0(M, \\mathds{R})$.\nFurthermore, the convergence is uniform in $x \\in M$.\nIn particular, if $h$ comes from Lemma \\ref{lem:averagingmap}\n\\[\n\\lim_{n \\to \\infty} \\frac{1}{n} \\sum_{k = 0}^{n-1} u \\circ h \\circ R_{\\beta}^k \\left( h^{-1} (x) \\right) =\n\\int_{\\mathbb{S}^1} (u \\circ h) \\circ R_{\\theta}(h^{-1}(x)) d\\theta.\n\\]\nand the convergence is uniform.\nIf $\\eta$ from Lemma \\ref{lem:averagingmap} is chosen sufficiently small and $n$ is large\n\\begin{align*}\n\\left\\Vert \\frac{1}{n} \\sum_{k = 0}^{n-1} u \\circ \\left( h R_{\\beta}^k h^{-1} \\right) - c \\right\\Vert &\n\\le \\left\\Vert \\frac{1}{n} \\sum_{k = 0}^{n-1} u \\circ h \\circ R_{\\beta}^k h^{-1} -\n\\int_{\\mathbb{S}^1} u \\circ h \\circ R_{\\theta} d\\theta \\right\\Vert + \\\\\n & + \\left\\Vert \\int_{\\mathbb{S}^1} u \\circ h \\circ  R_{\\theta} d\\theta - c \\right\\Vert < \\varepsilon,\n\\end{align*}\nso $h R_{\\beta} h^{-1} \\in \\mathcal{E}(u, \\varepsilon)$.\nThe conclusion is obtained taking a sequence of irrational $\\beta_n \\to \\alpha$ because\n$h R_{\\beta_n} h^{-1} \\xrightarrow{\\mathcal{C}^{\\infty}} h R_{\\alpha} h^{-1} = R_{\\alpha}$.\n\\end{proof}\n\nIn conclusion, $\\overline{\\mathcal{G}}$ contains strictly ergodic diffeomorphisms as long as it satisfies (H1) and (H3).\n\n\\section{Proofs of main theorems}\\label{sec:proofs}\n\nThe strategy of the proofs of Theorems \\ref{thm:symplectic} and \\ref{thm:contact} is the\nconjugation method which was explained in Section \\ref{sec:conjugationmethod}.\nThere have been some marked points in the arguments, where hypothesis on the diffeomorphisms subgroups $\\mathcal{G}$\nwere assumed and must be checked separately.\n\n\nTheorem \\ref{thm:symplectic} follows once it is shown that (H1) and (H2) are valid for the connected component of the\ngroup of symplectic diffeomorphisms which contains the identity.\nAnalogously, to prove Theorem \\ref{thm:contact} (H1) and (H3) must be shown to hold true for\nthe connected component of the identity in the group of contactomorphisms.\nNotice, incidentally, that (H3) implies (H2).\n\n\\begin{definition}\nThe action of a group $G < \\mathrm{Diff}(M)$ is said to be $n$--transitive if for every $x_1, \\ldots, x_n$ and\n$y_1, \\ldots, y_n$ there is $g \\in G$ such that $g(x_i) = y_i$, $i = 1\\ldots n$.\n\\end{definition}\nThe following result is well-known, see Boothby \\cite{boothby} for a detailed proof.\n\\begin{theorem}\\label{thm:transitivity}\nThe group of symplectic\/contact diffeomorphisms acts $n$--transitively for any $n \\ge 1$.\nFurthermore, the same is true for the group of Hamiltonian symplectomorphisms and Hamiltonian contactomorphisms.\n\\end{theorem}\n\n\\medskip\n\\textbf{Hypothesis (H1):}\n\\emph{Any element of $\\widehat{\\mathcal{G}} = \\mathcal{G}(\\widehat{M})$ has a lift in $\\mathcal{G} = \\mathcal{G}(M)$.}\n\\medskip\n\nThis is obviously true for contactomorphisms and symplectomorphisms.\nRecall that $\\widehat{M}$ was defined as the quotient of $M$ under the free action of the group generated\nby $R_{p\/q}$.\n\n\n\n\\bigskip\n\\textbf{Hypothesis (H2):}\n\\emph{Given any locally free $\\mathbb{S}^1$--action on a manifold $N$, and any open $V \\subset N$, there exists an\nelement $f \\in \\mathcal{G}(N)$ such that $f^{-1}(V)$ meets all the orbits of the action.}\n\\medskip\n\n\nIt will be now checked that this hypothesis is verified in the case\n$(N^{2n}, \\omega)$ is a symplectic manifold, the action is symplectic and $\\mathcal{G}(N) = \\mathrm{Symp}_0(N, \\omega)$.\n\n\nA Darboux flow box for a symplectic vector field $X$ is a Darboux chart $(U_i, \\theta_i)$,\n$$\\theta_i \\colon U_i \\to Q(\\delta, \\rho) := [-\\delta, \\delta] \\times [-\\rho, \\rho] \\times B^2(\\rho) \\times \\ldots B^2(\\rho) \\subset \\mathds{R}^{2n}$$ such that $\\theta^*_i \\bigl(\\frac{\\partial}{\\partial x_1} \\bigr) = X$.\nHenceforth, $X$ will be the infinitesimal generator of the $\\mathbb{S}^1$--action.\nDarboux flow boxes do exist at any point of $M$.\nThe next statement follows from compactness of $M$.\n\n\\begin{lemma}\\label{lem:flowboxes}\nThere exist $\\varepsilon, r > 0$ and $\\{(U_i, \\vartheta_i)\\}_{i = 1}^m$\na finite set of pairwise disjoint Darboux flow boxes such that\n$\\vartheta_i: U_i \\to Q(\\varepsilon, r + \\varepsilon)$ and the union of the codimension--1 disks\n$$\\bigsqcup_{i=1}^m \\vartheta_i^{-1}(\\{0\\}\\times [-r,r] \\times B^2(r) \\times \\ldots \\times B^2(r))$$\ntouches all the orbits of $X$.\n\\end{lemma}\n\nThis lemma allows to work in a local fashion in $(\\mathds{R}^{2n}, \\omega_0)$ so as to apply the following\nsqueezing result, which will be proved in Section \\ref{sec:packing}.\n\n\\begin{proposition}\\label{prop:folding}\nLet $r > 0$, $D = \\{0\\} \\times [-r, r] \\times B^2(r) \\times \\ldots \\times B^2(r) \\subset \\mathds{R}^{2n}$\nand $\\varepsilon > 0$. For any $\\delta > 0$, there exists a Hamiltonian symplectomorphism $\\psi$\nwith support in\n$Q(\\varepsilon, r) = [-\\varepsilon, \\varepsilon] \\times [-r-\\varepsilon, r + \\varepsilon] \\times B^2(r + \\varepsilon) \\times\n\\ldots B^2(r + \\varepsilon)$ such that $\\psi(D) \\subset P^{2n}(\\delta, \\ldots, \\delta)$.\n\\end{proposition}\n\nApply Lemma \\ref{lem:flowboxes} to obtain $r, \\varepsilon > 0$ and a family of Darboux flow boxes $\\{(U_i, \\vartheta_i)\\}_{i =1}^m$.\nBy Theorem \\ref{thm:transitivity}, there is a Hamiltonian symplectomorphism $\\phi$ such that $\\phi^{-1}(V)$ contains the centers\n$\\vartheta_i^{-1}(\\textbf{0})$ of the flow boxes.\nLet $\\psi$ be the squeezing map given by Proposition \\ref{prop:folding}. Define a Hamiltonian\nsymplectomorphism $\\varphi$ in $N$ which is equal to $\\vartheta_i \\circ \\psi \\circ \\vartheta_i^{-1}$ in\n$U_i$, for any $1 \\le i \\le m$, and to the identity elsewhere.\nThen $(\\phi \\circ \\varphi)^{-1}(V)$ meets all the orbits of the $\\mathbb{S}^1$--action.\n\n\n\\begin{remark}\nNote that the map $\\phi \\circ \\varphi$ in $\\mathrm{Symp}_0(N)$ realizing (H2) is actually a Hamiltonian symplectomorphism.\nUltimately, this boils down to the fact that both $n$--transitivity and Proposition \\ref{prop:folding} are realized\nby Hamiltonian symplectomorphisms.\n\\end{remark}\n\n\\bigskip\n\\textbf{Hypothesis (H3):}\n\\emph{Given a locally free $\\mathbb{S}^1$--action $\\{R_{\\alpha}\\}_{\\alpha \\in \\mathbb{S}^1}$ on a manifold $N$, an open set $V \\subset N$\nand $\\varepsilon > 0$, there exists an element $f \\in \\mathcal{G}(N)$ such that\n$$\\lambda(\\{\\alpha \\in \\mathbb{S}^1: R_{\\alpha}(y) \\notin f^{-1}(V)\\}) < \\varepsilon$$\nfor every $y \\in N$ ($\\lambda$ denotes the Lebesgue measure in $\\mathbb{S}^1$ of total mass 1).}\n\\medskip\n\nAs was said before, there are some technicalities difficulties to overcome in the proof even in the case\n$\\mathcal{G}(N) = \\mathrm{Diff}(N)$ or $\\mathrm{Diff}_{\\mu}(N)$ (see \\cite{fathiherman}). Notice also that (H3) is out of reach\nin the symplectic case. Indeed, the volume is always preserved under symplectic transformations so\nit is not possible to modify $V$ to swallow ``most'' of all the orbits.\n\nHenceforth, assume that $(N^{2n+1}, \\xi)$ is a contact manifold, $R_{\\alpha}$ is a contactomorphism\n for every $\\alpha \\in \\mathbb{S}^1$ and $\\mathcal{G}(N) = \\mathrm{Cont}_0(N, \\xi)$.\nLet $\\{(U_j, \\theta_j\\colon U_j \\to \\theta_j(U_j) \\subset \\mathds{R}^{2n+1})\\}$ be a finite atlas of $N$ by Darboux charts,\nthat is, $\\theta_j\\colon (U_j, \\xi_{|U_j}) \\to (\\mathds{R}^{2n+1}, \\xi_{0})$ is a contactomorphism.\nThe standard contact distribution in $\\mathds{R}^{2n+1}$ is denoted $\\xi_0 = \\ker(dz - \\sum_{k=1}^n y_k dx_k)$.\nThe following packing result is well--known in the field of contact geometry,\na proof accessible to non--experts is presented in Section \\ref{sec:packing}.\nThe term \\emph{cuboid} is here used to name the closure of a domain in $\\mathds{R}^{2n+1}$ enclosed by\n$2n+1$ couples of parallel hyperplanes in general position.\n\n\\begin{lemma}\\label{lem:contactinflating}\nLet $p \\in \\mathds{R}^{2n+1}$, $V_p$ a neighborhood of $p$ and $Y$ be a vector field in $V_p$ such that $Y(p) \\neq 0$.\nThere exists a cuboid $C$ centered at $p$ such that\n\\begin{itemize}\n\\item $C$ is contained in $V_p$,\n\\item $Y$ is transverse to the faces of $C$ and\n\\item for any neighborhood $W$ of $\\partial C$ and any ball $B \\subset C$ centered at $p$\nit is possible to find a Hamiltonian contactomorphism $\\varphi \\colon (\\mathds{R}^{2n+1}, \\xi_0) \\to (\\mathds{R}^{2n+1}, \\xi_0)$ such that:\n\\begin{enumerate}\n\\item $\\mathrm{supp}(\\varphi) \\subset C$.\n\\item $\\varphi(B)$ contains $C \\setminus W$.\n\\end{enumerate}\n\\end{itemize}\n\\end{lemma}\n\nFor every $j$ and every $x \\in U_j$\napply Lemma \\ref{lem:contactinflating} to $\\theta_j(x)$ and the vector field $(\\theta_j)_*(X)$\nto obtain a cuboid $Q^j_x$, centered at $x$, which is further assumed to lie within $\\theta_j(U_j)$.\nDefine $C^j_x = \\theta_j^{-1}(Q^j_x)$ and note that $X$ is transversal to $\\partial C^j_x$.\nSince $N$ is compact and $\\{\\mathrm{int}(C^j_x): x \\in U_j\\}$ is an open cover of $N$,\nthere is a finite set of different points $\\{p_i\\}_{i = 1}^m$ whose associated $C_{p_i}$ cover $N$.\nBy transversality, any orbit of the action meets $\\Delta = \\bigcup_{i = 1}^m \\partial C_{p_i}$ at a finite number of points.\n\n\\begin{lemma}\\label{lem:skeleton}\nIf $W$ is a sufficiently small neighborhood of $\\Delta$ then\n$$\\lambda(\\{\\alpha \\in \\mathbb{S}^1: R_{\\alpha}(y) \\in W\\}) < \\varepsilon$$\nfor every $y \\in N$.\n\\end{lemma}\n\\begin{proof}\nFor any $y_0 \\in N$, the orbit $\\mathcal{O}(y_0) = \\{R_{\\alpha}(y_0): \\alpha \\in \\mathbb{S}^1\\}$ meets $\\Delta$ in finitely many points,\n$R_{\\alpha_1}(y_0), \\ldots, R_{\\alpha_k}(y_0)$. Take a neighborhood $U$ of the union of these points\nsmall enough so that $\\lambda(\\{\\alpha \\in \\mathbb{S}^1: R_{\\alpha}(y) \\in U\\}) < \\varepsilon$ for every $y \\in N$.\nClearly, if $W^{y_0}$ is a sufficiently small neighborhood of $\\Delta$, $W^{y_0} \\cap \\mathcal{O}(y_0) \\subset U$\nand the same is true for orbits of points in a neighborhood of $y_0$. A compactness argument yields the result.\n\\end{proof}\n\nFix $W$ from the previous lemma and note that to conclude (H3) it is enough to inflate $V$ to cover $N \\setminus W$.\nThis strategy splits into two steps. Firstly, choose $s$ different points $\\{q_i\\}$ in $V$. By Theorem \\ref{thm:transitivity}\nthere is a Hamiltonian contactomorphism $\\phi\\colon (N, \\xi) \\to (N, \\xi)$ such that $\\phi(p_i) = q_i$.\nThen, $\\{p_i\\} \\subset \\phi^{-1}(V)$.\nLet $B_i$ be a small ball centered at $p_i$ and contained in $\\phi^{-1}(V)$.\n\nRecall from Lemma \\ref{lem:contactinflating} the properties of $Q^i_x$, which are inherited by\n$C_{p_i}$. In particular, there are Hamiltonian contactomorphisms $\\varphi_i\\colon (N, \\xi) \\to (N, \\xi)$\nsupported in $C_{p_i}$ such that $\\varphi_i(B_i) \\supset C_{p_i} \\setminus W$.\nConsider\n$\\varphi = \\varphi_1 \\circ \\ldots \\circ \\varphi_m.$\n\n\\begin{lemma}\n$$\\varphi \\left( \\cup_{i = 1}^s B_{i} \\right) \\cup W = N.$$\n\\end{lemma}\n\\begin{proof}\nFor a point $p$ in the support of $\\varphi_i$ there are two non--exclusive possibilities:\n$\\varphi_i(p) \\in W$ or $p \\in B_{i}$. The conclusion follows from the fact that any point of $N$\nbelongs to some $C_{p_i}$ hence to the support of one or more $\\varphi_i$.\n\\end{proof}\n\nAs a consequence, $\\varphi \\circ \\phi^{-1} (V)$ contains $N \\setminus W$. Thus, Lemma \\ref{lem:skeleton} concludes (H3).\n\n\\begin{remark}\nNotice that the map $\\varphi \\circ \\psi$ is the composition of two Hamiltonian contactomorphisms.\n\\end{remark}\n\n\\section{Packing lemmas}\\label{sec:packing}\n\nThis section discusses the two packing results (Proposition \\ref{prop:folding} for (H2) in the symplectic case and\nLemma \\ref{lem:contactinflating} for (H3) in the contact case) which eventually led to the proofs of the main theorems.\n\n\\subsection{Contact}\n\nIn the contact case, the goal is to construct a ``box'' within which any small ball may be inflated\n(by a contact transformation) to take up all the space inside but a small margin. This is an easy consequence\nof the basic fact that special dilations preserve the standard contact structure.\n\n\\begin{lemmacontactinflating}\nLet $p \\in \\mathds{R}^{2n+1}$, $V_p$ a neighborhood of $p$ and $Y$ be a vector field in $V_p$ such that $Y(p) \\neq 0$.\nThere exists a cuboid $C$ centered at $p$ such that\n\\begin{itemize}\n\\item $C$ is contained in $V_p$,\n\\item $Y$ is transverse to the faces of $C$ and\n\\item for any neighborhood $W$ of $\\partial C$ and any ball $B \\subset C$ centered at $p$\nit is possible to find a Hamiltonian contactomorphism $\\varphi \\colon (\\mathds{R}^{2n+1}, \\xi_0) \\to (\\mathds{R}^{2n+1}, \\xi_0)$ such that:\n\\begin{enumerate}\n\\item $\\mathrm{supp}(\\varphi) \\subset C$.\n\\item $\\varphi(B)$ contains $C \\setminus W$.\n\\end{enumerate}\n\\end{itemize}\n\\end{lemmacontactinflating}\n\\begin{proof}\nIf $p = (\\textbf{x}_0, \\textbf{y}_0, z_0)$, the affine map\n\\[\n\\tau(\\textbf{x}, \\textbf{y}, z) = (\\textbf{x} - \\textbf{x}_0, \\textbf{y} - \\textbf{y}_0, z - z_0 + \\textbf{x}_0 \\cdot (\\textbf{y} - \\textbf{y}_0))\n\\]\nis a contactomorphism in $(\\mathds{R}^{2n+1}, \\xi_{0})$ which maps $p$ to the origin.\nThus, assume without lose of generality $p = \\textbf{0}$.\n\nThe vector field $V(\\textbf{x}, \\textbf{y}, z) = (\\textbf{x}, \\textbf{y}, 2z)$ is a contact vector field because its flow\n$\\psi_t(\\textbf{x}, \\textbf{y}, z) = (e^t\\textbf{x}, e^t\\textbf{y}, e^{2t}z)$ preserves the standard contact structure $\\xi_{0}$.\nDenote $H_V$ the contact Hamiltonian associated to $V$.\nLet $C_r$ be the cuboid of size $r > 0$ centered at $\\textbf{0}$ and\ngenerated by the set of linear 1--forms $\\{\\lambda_1 = dx_1, \\ldots, \\lambda_{2n+1} = dz\\}$, that is,\n\\[\nC_r = \\{v \\in \\mathds{R}^{2n+1} : |\\lambda_i(v)| < r \\enskip \\forall i\\}.\n\\]\nA computation shows that $V$ points outwards $C_{r}$.\nAs a consequence, the image under the flow $\\psi_t$ of any neighborhood $B$ of the origin eventually covers $C_{r}$.\nNote that the statement remains valid as long as the vector field $V$ points outwards every such cuboid.\nIn case $\\lambda_i(Y(\\textbf{0})) = 0$, replace $\\lambda_i$ by a sufficiently close linear 1--form $\\tilde{\\lambda}_i$.\nFor $r_0 > 0$ sufficiently small, $V$ still point outwards the modified cuboids\n$\\widetilde{C}_r = \\{v \\in \\mathds{R}^{2n+1} : |\\widetilde{\\lambda}_i(v)| < r \\enskip \\forall i\\}$\nand the faces of $\\widetilde{C}_r$,\n$\\partial \\widetilde{C}_r$, are transversal to $Y$ if $r \\le r_0$. Take $C = \\widetilde{C}_{r_0}$.\n\nFix now a ball $B$ centered at \\textbf{0} and a neighborhood $W$ of $\\partial C$. As was noticed before,\n$\\psi_t(B) \\subset C \\setminus W$ for large $t$.\nConsider now a smooth function $H\\colon \\mathds{R}^{2n+1} \\to \\mathds{R}$ equal to $H_V$ inside $C \\setminus W$ which vanishes outside $C$.\nThe contact vector field $V'$ associated to $H$ is then equal to $V$ inside $C \\setminus W$ and vanishes outside $C$.\nConsequently, the flow $\\varphi_t$ generated by $V'$ satisfies the properties in the statement.\n\\end{proof}\n\n\n\\subsection{Symplectic}\n\nThis subsection contains the proof of Proposition \\ref{prop:folding}, that is,\nit is devoted to show how to squeeze a large codimension--1 disk into a small ball in a symplectic fashion.\nDenote by $B^2(r)$ the closed 2--ball of radius $r$ and $P^{2n}(r_1, \\ldots, r_n) = B^2(r_1) \\times \\ldots \\times B^2(r_n)$.\nThe proof presented here adapts the following non--trivial result to answer the question.\n\n\\begin{lemma}\\label{lem:folding}\nGiven $s, \\rho > 0$ there exists $\\eta = \\eta(s, \\rho) > 0$ and a Hamiltonian symplectomorphism $\\phi$\nsuch that $\\phi$ embeds $B^2(\\eta) \\times B^2(s)$ into $B^2(\\rho) \\times B^2(1)$.\nFurthermore, $\\phi$ can be assumed to be supported in $B^2(c \\rho) \\times B^2(s + c)$ for a constant $c > 1$\nindependent of $s, \\rho$.\n\\end{lemma}\n\nThere exist several approaches to this result in the literature. One could use the $h$--principle for isosymplectic embeddings to\nobtain an embedding of the disk $B^2(s)$ and then extend it to a neighborhood thanks to the Symplectic Neighborhood Theorem.\nThe parametric version of this symplectic embedding theorem (see \\cite[Section 12.1]{eliashberg}) would provide the result.\nHowever, we prefer a more hands--on approach using symplectic folding.\nFollowing Schlenk (\\cite[Remark 3.3.1]{schlenk}), the nature of this deformation is local\nand each step of the construction is induced by a Hamiltonian flow.\nIndeed, a careful look through the folding shows that all deformations take place inside an arbitrary neighborhood\nof the figures apart from the stretching in the base, where an extra space proportional to the size of the fibers is required.\nThe constant $c$ in the lemma is introduced to make up for it.\n\nFor our purpose, the extra space around the codimension--1 disk in which the transformation is supported must be\narbitrarily small. The following technical lemma shows how to\nscale properly Lemma \\ref{lem:folding} and to fold the disk in $n-1$ directions.\n\n\\begin{lemma}\\label{lem:sabana}\nGiven $r, \\varepsilon > 0$, for every $0 < \\delta < \\varepsilon$, there exists $\\sigma > 0$ and a Hamiltonian\nsymplectomorphism $\\varphi$ such that $\\varphi$ embeds $P^{2n}(\\sigma, r, \\ldots, r)$\ninto $P^{2n}(\\delta, \\ldots, \\delta)$ and the support of $\\varphi$ is contained in\n$P^{2n}(\\varepsilon, r+\\varepsilon, \\ldots r+\\varepsilon)$.\n\\end{lemma}\n\\begin{proof}\nThe goal is to find $\\sigma$ so that the result of folding in each of the $n-1$ ``thick'' directions the thin polydisk\nis contained into the target cube. The squeezing map of Lemma \\ref{lem:folding} has to be scaled properly.\n\nSet $\\rho_1 = 1$ and consider $\\lambda_1 > 0$ small enough so that the following inequalities are satisfied for\n$\\rho = \\rho_1$ and $\\lambda = \\lambda_1$:\n\\begin{equation}\\label{eq:1}\n\\lambda, \\lambda \\rho < \\delta, \\enskip \\enskip \\enskip \\lambda c, \\lambda c \\rho < \\varepsilon,\n\\end{equation}\nwhere $c$ comes from Lemma \\ref{lem:folding}.\nDefine $s_1 = r \/ \\lambda_1$. Lemma \\ref{lem:folding} yields $\\rho_2 := \\eta(s_1,\\rho_1)$ and $\\phi_{1}$.\nThe map $\\widehat{\\phi}_{1}(x) = \\lambda_1 \\phi_{1}(x\/\\lambda_1)$ defines a Hamiltonian symplectomorphism\nsuch that:\n\\begin{itemize}\n\\item $\\widehat{\\phi}_{1}$ embeds $B^2(\\lambda_1 \\rho_2) \\times B^2(\\lambda_1 s_1)$ into\n$B^2(\\lambda_1 \\rho_1) \\times B^2(\\lambda_1)$ so,\nthe set of inequalities (\\ref{eq:1}) implies $B^2(\\lambda_1 \\rho_2) \\times B^2(r)$ is sent inside $P^4(\\delta, \\delta)$.\n\\item The support of $\\widehat{\\phi}_{1}$ is contained in $B^2(\\lambda_1 c \\rho_1) \\times B^2(\\lambda_1(s_1 + c))$\nwhich, by (\\ref{eq:1}), is in turn contained in $B^2(\\varepsilon) \\times B^2(r + \\varepsilon)$.\n\\end{itemize}\nSetting $\\sigma = \\lambda_1 \\rho_2$ would already prove the lemma for $n = 2$. Let us continue\nthe argument pursuing the general case.\n\nFor $k \\ge 2$, define inductively $\\rho_k = \\eta(s_{k-1}, \\rho_{k-1})$,\nand $\\lambda_k > 0$ smaller than $\\lambda_{k-1}$ and such that inequalities (\\ref{eq:1})\nare satisfied for $\\rho = \\rho_k$ and $\\lambda = \\lambda_k$.\nDefine $\\widehat{\\phi}_{k} = \\lambda_k \\phi_{k}(x\/\\lambda_k)$ and $s_k = r\/\\lambda_k$.\nFor every $k \\ge 1$, the following properties are satisfied:\n\\begin{itemize}\n\\item $\\widehat{\\phi}_{k}$ embeds $B^2(\\lambda_k \\rho_{k+1}) \\times B^2(r) =\nB^2(\\lambda_k \\eta(s_k, \\rho_k)) \\times B^2(\\lambda_k s_k)$ inside $B^2(\\lambda_k \\rho_k) \\times B^2(\\lambda_k)$.\n\\item The support of $\\widehat{\\phi}_{k}$ is contained in $B^2(\\lambda_k c) \\times B^2(\\lambda_k(s_k + c))$\nso, by (\\ref{eq:1})\n\\begin{equation}\\label{eq:2}\n\\mathrm{supp}(\\widehat{\\phi}_{k}) \\subset B^2(\\varepsilon) \\times B^2(r + \\varepsilon).\n\\end{equation}\n\\end{itemize}\nFor any $1 \\le k \\le n-1$, $E_k$ denotes the linear subspace of $\\mathds{R}^{2n} = \\mathds{R}^2 \\times \\ldots \\times \\mathds{R}^2$\nspanned by the $1^{st}$ and $(n-k+1)^{th}$ factors.\nDefine $\\widehat{\\varphi}_{k}$ as the map which acts as $\\widehat{\\phi}_{k}$ in $E_k$ and as the identity in the other directions.\nEvidently, $\\widehat{\\varphi}_{k}$ is again a Hamiltonian symplectomorphism.\nIn view of (\\ref{eq:2}),\nafter a suitable cut-off we can obtain another Hamiltonian symplectomorphism $\\varphi_k$ which coincides with $\\widehat{\\varphi}_k$\nin $P^{2n}(\\delta, r, \\ldots, r)$ and is supported on $P^{2n}(\\varepsilon, r + \\varepsilon, \\ldots, r + \\varepsilon)$.\nDefine $\\sigma = \\lambda_{n-1} \\rho_{n}$. Then,\n\\begin{equation*}\n\\left.\\begin{aligned}\n& P^{2n}(\\sigma, r, r, \\ldots, r) = P^{2n}(\\lambda_{n-1}\\rho_{n}, \\lambda_{n-1}s_{n-1}, r, \\ldots, r)\n\\xhookrightarrow{\\varphi_{n-1}}\\\\\n& \\xhookrightarrow{\\varphi_{n-1}} P^{2n}(\\lambda_{n-1} \\rho_{n-1}, \\lambda_{n-1}, r, \\ldots, r)\\\\\n& P^{2n}(\\lambda_{n-1} \\rho_{n-1}, \\lambda_{n-1}, r, \\ldots, r)\n\\subset P^{2n}(\\lambda_{n-2} \\rho_{n-1}, \\delta, r, \\ldots, r) \\xhookrightarrow{\\varphi_{n-2}} \\ldots \\\\\n& \\qquad \\qquad \\qquad \\qquad \\qquad \\ldots \\xhookrightarrow{\\varphi_{2}} P^{2n}(\\lambda_2 \\rho_2, \\delta, \\ldots, \\delta, r) \\subset\nP^{2n}(\\lambda_1 \\rho_2, \\delta, \\ldots, \\delta, r) \\\\\n& P^{2n}(\\lambda_1 \\rho_2, \\delta, \\ldots, r) \\xhookrightarrow{\\varphi_{1}} P^{2n}(\\lambda_1 \\rho_1, \\delta, \\ldots, \\delta)\n\\subset P^{2n}(\\delta, \\ldots, \\delta).\n\\end{aligned}\\right.\n\\end{equation*}\nThus, in order to conclude the lemma it suffices to define\n$$\\varphi = \\varphi_1 \\circ \\ldots \\circ \\varphi_{n-1}.$$\n\\end{proof}\n\n\\begin{propositionfolding}\nLet $r > 0$, $D = \\{0\\} \\times [-r, r] \\times B^2(r) \\times \\ldots \\times B^2(r) \\subset \\mathds{R}^{2n}$\nand $\\varepsilon > 0$. For any $\\delta > 0$, there exists a Hamiltonian symplectomorphism $\\psi$\nwith support in\n$[-\\varepsilon, \\varepsilon] \\times [-r-\\varepsilon, r + \\varepsilon] \\times B^2(r + \\varepsilon) \\times\n\\ldots B^2(r + \\varepsilon)$ such that $\\psi(D) \\subset P^{2n}(\\delta, \\ldots, \\delta)$.\n\\end{propositionfolding}\n\\begin{proof}\nFix $\\sigma$ from the previous lemma.\nThe Hamiltonian $H(\\textbf{x}, \\textbf{y}) = -x_1 y_1$ induces a flow in $\\mathds{R}^{2n}$ which carries $\\{0\\} \\times [-r, r] \\times A^{2n-2}$ onto\n$\\{0\\} \\times [-\\sigma, \\sigma] \\times A^{2n-2}$, for arbitrary $A^{2n-2}$.\nApplying an appropriate cut-off to $H$ we can assume the flow is supported in\n$[-\\varepsilon, \\varepsilon] \\times [-r-\\varepsilon, r + \\varepsilon]  \\times B^2(r+ \\varepsilon) \\times \\ldots \\times B^2(r+ \\varepsilon)$.\nIt is enough to compose the time--$t$ map of the flow, for large $t > 0$,\nwith $\\varphi$ from Lemma \\ref{lem:sabana} to obtain the desired map.\n\\end{proof}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\partial{\\partial}\n\\def\\rangle{\\rangle}\n\\def\\subset{\\subset}\n\\def\\simeq{\\simeq}\n\\def\\times{\\times}\n\\def\\wedge{\\wedge}\n\\def\\rm cf{\\rm cf}\n\\def{^-\\hspace{-7pt}\\d}{{^-\\hspace{-7pt}\\delta}}\n\\def{-\\hspace{-9pt}\\g}{{-\\hspace{-9pt}\\gamma}}\n\\def{-\\hspace{-9pt}\\b}{{-\\hspace{-9pt}\\beta}}\n\\def\\rightarrow{\\rightarrow}\n\\def\\longrightarrow{\\longrightarrow}\n\\def\\Rightarrow{\\Rightarrow}\n\\def\\leftrightarrow{\\leftrightarrow}\n\\def\\hbox{\\hbox}\n\n\\def\\backslash{\\backslash}\n\\def\\bullet{\\bullet}\n\\def\\forall{\\forall}\n\\def\\infty{\\infty}\n\\def\\mapsto{\\mapsto}\n\\def\\nabla{\\nabla}\n\\def\\otimes{\\otimes}\n\\def\\overline{\\overline}\n\\def\\rightarrow{\\rightarrow}\n\\def\\Rightarrow{\\Rightarrow}\n\\def\\subset{\\subset}\n\\def\\subseteq{\\subseteq}\n\\def\\supset{\\supset}\n\n\n\\def\\times{\\times}\n\\def\\wedge{\\wedge}\n\\def\\widetilde{\\widetilde}\n\n\n\n\\def\\mathop{\\rm Ad}\\nolimits{\\mathop{\\rm Ad}\\nolimits}\n\\def\\mathop{\\rm Aut}\\nolimits{\\mathop{\\rm Aut}\\nolimits}\n\\def\\mathop{\\rm Diff}\\nolimits{\\mathop{\\rm Diff}\\nolimits}\n\\def\\mathop{\\rm Dom}\\nolimits{\\mathop{\\rm Dom}\\nolimits}\n\\def\\mathop{\\rm End}\\nolimits{\\mathop{\\rm End}\\nolimits}\n\\def\\mathop{\\rm Ext}\\nolimits{\\mathop{\\rm Ext}\\nolimits}\n\\def\\mathop{\\rm Hom}\\nolimits{\\mathop{\\rm Hom}\\nolimits}\n\\def\\mathop{\\rm GV}\\nolimits{\\mathop{\\rm GV}\\nolimits}\n\\def\\mathop{\\rm GL}\\nolimits{\\mathop{\\rm GL}\\nolimits}\n\\def\\mathop{\\rm SO}\\nolimits{\\mathop{\\rm SO}\\nolimits}\n\\def\\mathop{\\rm Id}\\nolimits{\\mathop{\\rm Id}\\nolimits}\n\\def\\mathop{\\rm exp}\\nolimits{\\mathop{\\rm exp}\\nolimits}\n\\def\\mathop{\\rm mod}\\nolimits{\\mathop{\\rm mod}\\nolimits}\n\\def\\mathop{\\rm pt}\\nolimits{\\mathop{\\rm pt}\\nolimits}\n\\def\\mathop{\\rm Index}\\nolimits{\\mathop{\\rm Index}\\nolimits}\n\\def\\mathop{\\rm Inf}\\nolimits{\\mathop{\\rm Inf}\\nolimits}\n\\def\\mathop{\\rm Int}\\nolimits{\\mathop{\\rm Int}\\nolimits}\n\\def\\mathop{\\rm Ker}\\nolimits{\\mathop{\\rm Ker}\\nolimits}\n\\def\\mathop{\\rm Rang}\\nolimits{\\mathop{\\rm Rang}\\nolimits}\n\\def\\mathop{\\rm Re}\\nolimits{\\mathop{\\rm Re}\\nolimits}\n\\def\\mathop{\\rm Res}\\nolimits{\\mathop{\\rm Res}\\nolimits}\n\\def\\mathop{\\rm Sign}\\nolimits{\\mathop{\\rm Sign}\\nolimits}\n\\def\\mathop{\\rm sign}\\nolimits{\\mathop{\\rm sign}\\nolimits}\n\\def\\mathop{\\rm Spectrum}\\nolimits{\\mathop{\\rm Spectrum}\\nolimits}\n\\def\\mathop{\\rm spin}\\nolimits{\\mathop{\\rm spin}\\nolimits}\n\\def\\mathop{\\rm Spin}\\nolimits{\\mathop{\\rm Spin}\\nolimits}\n\\def\\mathop{\\rm Sup}\\nolimits{\\mathop{\\rm Sup}\\nolimits}\n\\def\\mathop{\\rm Tor}\\nolimits{\\mathop{\\rm Tor}\\nolimits}\n\\def\\mathop{\\rm Trace}\\nolimits{\\mathop{\\rm Trace}\\nolimits}\n\\def\\Kc_\\Lc{{\\cal K}_{\\cal L}}\n\n\n\n\n\n\\def\\build#1_#2^#3{\\mathrel{\n\\mathop{\\kern 0pt#1}\\limits_{#2}^{#3}}}\n\n\n\\def\\limproj{\\mathop{\\oalign{lim\\cr\n\\hidewidth$\\longleftarrow$\\hidewidth\\cr}}}\n\\def\\limind{\\mathop{\\oalign{lim\\cr\n\\hidewidth$\\longrightarrow$\\hidewidth\\cr}}}\n\n\n\\newcommand{\\FBOX}[2]\n{\\begin{center} \\fcolorbox{black}{white}{\\parbox{#1 cm}{ #2}}\n\\end{center}}\n\n\n\n\n\\numberwithin{equation}{section}\n\\parindent0in\n\n\n\\def{\\bf t}{{\\bf t}}\n\n\\begin{document}\n\n\\title{\\bf  Characteristic classes of foliations via SAYD-twisted cocycles}\n\\author{\n\\begin{tabular}{cc}\nBahram Rangipour \\thanks{Department of Mathematics  and   Statistics,\n     University of New Brunswick, Fredericton, NB, Canada      Email: bahram@unb.ca, \\quad and \\quad ssutlu@ unb.ca }\n     \\quad and  \\quad  Serkan S\\\"utl\\\"u $~^\\ast$\n      \\end{tabular}}\n\n\n\\maketitle\n\n\\abstract{\\noindent   We have previously shown  that  the truncated\nWeil algebra of any Lie algebra is a Hopf-cyclic type complex with\nnontrivial coefficients. In this paper we apply this  result  to\ntransfer   the characteristic classes of  transversely orientable\nfoliations into the cyclic  cohomology of the groupoid action\nalgebra. Our result in codimension 1 matches with  the only existing\nexplicit  computation  done by    Connes-Moscovici. In codimension 2 case,  we carry out a constructive and  explicit computation, by which  we present  the transverse  fundamental class, the Godbillon-Vey class,  and the other four residual  classes as cyclic cocycles on the groupoid action algebra.     The main object in charge  in this new characteristic map\nis a SAYD-twisted cyclic cocycle of the same degree as the\ncodimension. We construct such a  cocycle  by introducing an equivariant Hopf-cyclic cohomology and an equivariant cup product.}\n\n\\tableofcontents\n\n\n\n\n\\section{Introduction}\\label{section-intro}\n\n\n Following Connes-Moscovici \\cite{ConnMosc98},  let\n${\\cal A}_\\Gamma:=C^{\\infty}_c(F^+)\\rtimes \\Gamma$. Here $F^+$ is the oriented\nframe bundle over ${\\mathbb R}^n$, and $\\Gamma$ is a  discrete subgroup of\n$\\mathop{\\rm Diff}\\nolimits^+({\\mathbb R}^n)$,  the group of orientation preserving\n diffeomorphisms of ${\\mathbb R}^n$.\n\n\\medskip\n\n\\noindent For an arbitrary  $\\Gamma$, the cyclic cohomology of ${\\cal A}_\\Gamma$ is not\nknown  \\cite[Sect. III.2]{Conn-book}. However, the Gelfand-Fuks\ncohomology of $\\mathfrak{a}_n$, the Lie algebra of formal vector fields on\n${\\mathbb R}^n$, is finite dimensional and is embedded  in this cohomology\nas a direct summand. In other words, there is a  map, even in  the\nlevel of complexes, which is  a composite of two complicated maps:\n\\begin{equation}\n\\xymatrix{ H_{\\rm GF}(\\mathfrak{a}_n,{\\mathbb C})\\ar[rr]^{\\Phi\\circ{\\cal V}}\\ar[rd]^{ {\\cal V}_{\\rm van Est}}& & HP({\\cal A}_\\Gamma)\\\\\n& H_\\tau(F^+, {\\mathbb C})\\ar[ru]^{\\Phi_{\\rm Connes}}.&\n }\n\\end{equation}\nThe first map  is  a van Est type  map \\cite{ConnMosc98}, which\nlands in the twisted cohomology  computed  by the Bott bicomplex\n\\cite[Prop. III.2.11]{Conn-book}, while the second map is due to\nConnes \\cite[Thm. III.2.14]{Conn-book}. The representatives of the\nGelfand-Fuks cohomology\n classes in $H(\\mathfrak{a}_n,{\\mathbb C})$ are known thanks to the  Vey basis of the\n cohomology of  the truncated Weil algebra \\cite{Godb72}. However it\n is difficult to transfer them to the cyclic cohomology of ${\\cal A}_\\Gamma$.\n  The reader is referred to \\cite{ConnMosc} for a complete account of the computation in codimension 1.\n\n\\medskip\n\n\\noindent The Hopf-cyclic cohomology, invented  by Connes-Moscovici\n for computing a local index formula \\cite{ConnMosc98}, made it\npossible to have another characteristic map\n\\begin{equation}\\label{CM-char}\n\\chi_\\tau: HP({\\cal H}_n, {\\mathbb C}_\\delta)\\rightarrow HP({\\cal A}_\\Gamma).\n\\end{equation}\nHere ${\\cal H}_n$ is the Connes-Moscovici Hopf algebra of codimension $n$  and ${\\mathbb C}_\\delta$ is\n the canonical one dimensional  SAYD module over ${\\cal H}_n$.  One of the  interesting features  of this\n characteristic map  is its   simple  presentation on the level of\n complexes,\n\\begin{equation}\n\\chi_\\tau(h_1\\ot\\cdots\\ot h_n)(a^0, \\ldots, a^n)=\\tau\\big(a^0\nh_1(a^1)\\cdots h_n(a^n)\\big).\n\\end{equation}\nHere $\\tau$ is the canonical trace on ${\\cal A}_\\Gamma$ defined by\n\\begin{equation}\\label{aux-canonical-trace}\n \\tau(fU^\\ast_\\psi) = \\left\\{\\begin{array}{cc}\n  \\displaystyle \\int_{F^+}f\\varpi,  & \\hbox{if}\\quad \\psi = \\mathop{\\rm Id}\\nolimits, \\\\\n  &\\\\\n  0, & \\hbox{otherwise.}\n  \\end{array}\\right.\n\\end{equation}\nIt is also proved  that $HP({\\cal H}_n, {\\mathbb C}_\\delta)$ and $H_{\\rm GF}(\\mathfrak{a}_n, {\\mathbb C})$ are\n canonically isomorphic, although once again this isomorphism is not\n easy to present \\cite{ConnMosc98,MoscRang11}. In view of\n \\eqref{CM-char}, the only obstacle to transfer the\n  characteristic classes of transversely orientable foliations to the cyclic\n cohomology of ${\\cal A}_\\Gamma$ is a basis of the representatives\n of the Hopf-cyclic cohomology classes of  ${\\cal H}_n$.\n  There is an intensive ongoing  study\n\\cite{MoscRang07,MoscRang09,MoscRang11}  to investigate the Hopf-cyclic cohomology of the geometric bicrossed product Hopf algebras\nsuch as ${\\cal H}_n$.   The main idea is\n  to use the bicrossed product construction  to find the smallest\ncomplex by which one can compute the cohomology classes. This\ncomplex is found in \\cite{MoscRang11} and\n is shown to be the codomain  of  the van Est isomorphism \\cite{MoscRang11}. However,\n  the return map from that complex to the Hopf-cyclic complex of the\n Hopf algebra  is still missing.\n\n\\medskip\n\n\\noindent In this paper we develop a new characteristic map, whose source\nis the Hopf-cyclic cohomology of ${\\cal K}:=U(g\\ell_n)$, the enveloping\nalgebra of the general linear Lie algebra $g\\ell_n$.\n The Hopf algebra   ${\\cal K}$ obviously  is not as sophisticated as\n${\\cal H}_n$. Therefore to obtain and transfer the  same classes, by considering the  conservation of work, we would expect\n a characteristic map and a SAYD module more sophisticated\nthan $\\chi_\\tau$ and ${\\mathbb C}_\\delta$\nrespectively.\n\n\\medskip\n\n   \\noindent  In fact,  the first step of our mission  was taken in\n\\cite{RangSutl-II}, where the authors showed that the\n      truncated Weil algebra is  a  Hopf-cyclic complex. As a\n      result, the characteristic classes of transversely orientable\n      foliations can be calculated from $HC({\\cal K}, V)$. Here\n      $V:=S(g\\ell_n^\\ast)\\nsb{2n}$, the algebra of $n$-truncated\n      polynomials on $g\\ell_n$,\n       is a canonical and nontrivial SAYD module over ${\\cal K}$.\n\n\\medskip\n\n\\noindent The other important  piece of this new characteristic map is a SAYD twisted   cyclic $n$-cocycle\n $\\varphi\\in C^n_{\\cal K}({\\cal A}_\\Gamma, V)$.\n Next, we apply  the cup product in Hopf cyclic cohomology introduced in \\cite{KhalRang05-II} by  Khalkhali and the  first named author.  In fact, since  $\\varphi$ is a cyclic cocycle, we use  the explicit\n   formula derived in \\cite{Rang08}\n  to compute    the characteristic classes of foliations as\n   cyclic cocycles in $HC({\\cal A}_\\Gamma)$ via\n\\begin{equation}\\label{cup}\n\\chi_\\varphi: HC^{\\bullet}({\\cal K}, V)\\rightarrow HC^{\\bullet+n}({\\cal A}_\\Gamma),\\quad\n\\chi_\\varphi(x)=x\\cup \\varphi.\n\\end{equation}\n\n\\noindent In order to test  our method  we first carry out the computation\nfor  codimension 1 and observe  that our result matches with the classes\nobtained by  Connes-Moscovici in \\cite{ConnMosc}.  The result of\n\\cite{MoscRang11} shows that  the amount of work in codimension 2 is\nnot comparable with that of  codimension 1.  However, we completely\ndetermine the representatives of all classes in $HC({\\cal K},V)$ in\naddition to an explicit formula for $\\varphi\\in HC^2_{\\cal K}({\\cal A}_\\Gamma, V)$.\nThen \\eqref{cup} yields our desired cyclic cocycles in\n $HC({\\cal A}_\\Gamma)$.\n\n\\medskip\n\n\\noindent Throughout the paper, all vector spaces and their tensor\n products are over ${\\mathbb C}$ unless otherwise is specified. We  use the Sweedler's notation for comultiplication and coaction.\n   We denote the comultiplication of a coalgebra $C$ by $\\Delta: C\\rightarrow C\\otimes C$  and its action on $c\\in C$ by $\\Delta(c) = c\\ps{1} \\otimes c\\ps{2}$. The image of $v\\in V$ under  a left coaction $\\nabla: V\\rightarrow C\\otimes V$   is denoted by $\\nabla(v) = v\\ns{-1} \\otimes v\\ns{0}$, summation   suppressed. By the coassociativity, we simply write\n$\\Delta(c\\ps{1}) \\otimes c\\ps{2} = c\\ps{1} \\otimes \\Delta(c\\ps{2}) = c\\ps{1} \\otimes c\\ps{2} \\otimes c\\ps{3}.$\nUnless stated otherwise, a Lie algebra $\\mathfrak{g}$ is finite dimensional with a basis $\\{X_i\\,|\\,1 \\leq i \\leq n\\}$ and a dual basis $\\{\\theta^i\\,|\\,1 \\leq i \\leq n\\}$. In particular, for $\\mathfrak{g} = g\\ell_n$ we use $\\{Y_i^j\\,|\\,1 \\leq i,j \\leq n\\}$ for a basis and $\\{\\theta^i_j\\,|\\,1 \\leq i,j \\leq n\\}$ for a dual basis. We denote the Weil algebra of $\\mathfrak{g}$ by $W(\\mathfrak{g})$, and $W(\\mathfrak{g})\\nsb{2n}$ stands for the $n$-truncated Weil algebra of $\\mathfrak{g}$. We denote the Kronecker symbol by $\\delta^i_j$.  We also adopt the Einstein summation convention on the repeating indices unless otherwise is stated. Finally, for the sake of  simplicity  we  use\n\\begin{equation*}\nB_{\\sigma(1)}\\ot\\cdots\\ot B_{\\sigma(q)} := \\sum_{\\sigma \\in S_q} {\\rm sgn}(\\sigma) B_{\\sigma(1)}\\ot\\cdots\\ot B_{\\sigma(q)}\n\\end{equation*}\nfor any set of objects $\\{B_1,\\ldots, B_q\\}$. Here $S_q$ is the group of all permutations on $q$ objects and ${\\rm sgn}(\\sigma)$ stands for the signature of $\\sigma$.\n\n\n\n\\section{Preliminaries}\\label{section-prelim}\nIn this section we bring all  material needed  for the sequel\nsections.  The definition of  Hopf-cyclic cohomology and a brief\naccount of the Connes-Moscovici characteristic map is provided in\nthe first  subsection. The basics of the  cyclic cohomology of Lie\nalgebras are recalled in the other subsection.\n\n\n\\subsection{Hopf-cyclic cohomology with coefficients}\n\n\\noindent Let $H$ be a Hopf algebra equipped with a character $\\delta: H\\rightarrow\n{\\mathbb C}$, {\\it i.e.\\\/}\\  an algebra map,\n and a group-like element $\\sigma\\in H$, {\\it i.e.\\\/}\\  $\\Delta(\\sigma)=\\sigma\\otimes \\sigma$ and $\\varepsilon(\\sigma)=1$.\n The pair $(\\delta,\\sigma)$ is called a modular pair in involution (MPI for short) if\n\\begin{equation}\n\\delta(\\sigma)=1, \\quad \\hbox{and}\\quad  S_\\delta^2=\\mathop{\\rm Ad}\\nolimits_\\sigma,\n\\end{equation}\nwhere $\\mathop{\\rm Ad}\\nolimits_\\sigma(h)= \\sigma h\\sigma^{-1}$, for any $h \\in H$, and  $S_\\delta$ is\ndefined by\n\\begin{equation}\\label{aux-twisted-antipode}\nS_\\delta(h)=\\delta(h\\ps{1})S(h\\ps{2}), \\quad h \\in H.\n\\end{equation}\n\n\\medskip\n\\noindent A vector space\n $M$ is called a right-left\nstable-anti-Yetter-Drinfeld module (SAYD for short) over $H$ if it\nis a right\n $H$-module, a left $H$-comodule, and\n\\begin{equation}\\label{aux-SAYD-condition}\n\\nabla(m\\cdot h)= S(h\\ps{3})m\\ns{-1}h\\ps{1}\\otimes m\\ns{0}\\cdot h\\ps{2},\\qquad  m\\ns{0}\\cdot m\\ns{-1}=m,\n\\end{equation}\nfor any $v\\in V$ and $h\\in H$. \\noindent Using $\\delta$ and $\\sigma$ one endows\n$^\\sigma{\\mathbb C}_\\delta:={\\mathbb C}$, the field of complex numbers, with a  right module\nand left comodule structures  over $H$. This way $^\\sigma{\\mathbb C}_\\delta$ is a\nSAYD module over the Hopf algebra $H$ if and only if $(\\delta,\\sigma)$ is an\nMPI.\n\n\\medskip\n\n\\noindent Now let $M$  be  a right-left SAYD module over  $H$ and $C$ an $H$-module coalgebra, that is, $\\Delta(h(c))=h\\ps{1}(c\\ps{1}) \\otimes h\\ps{2}(c\\ps{2})$ for any $h \\in H$ and $c \\in C$. Then we\nhave the graded space\n\\begin{equation}\\label{aux-stand-Hopf-cyclic}\nC_H(C,M) := \\bigoplus_{q\\geq 0} C^q_H(C,M), \\quad C^q_H(C,M):= M\\otimes_H C^{\\otimes q+1}\n\\end{equation}\nwith the coface operators\n\\begin{align}\\label{aux-coface-opt}\n&\\mathfrak{d}_i: C^q_H(C,M)\\rightarrow C^{q+1}_H(C,M), \\quad 0\\le i\\le q+1\\\\\\notag\n&\\mathfrak{d}_i(m\\otimes_H c^0\\ot\\cdots\\ot c^q)= m\\otimes c^0\\ot\\cdots\\ot \\Delta(c^i )\\ot\\cdots\\ot c^q, \\\\\\notag\n&\\mathfrak{d}_{q+1}(m\\otimes_H c^0\\ot\\cdots\\ot c^q)=\\\\\\notag\n& m\\ns{0}\\otimes_H c^0\\ps{2} \\otimes c^1\\ot\\cdots\\ot c^q\\otimes m\\ns{-1}(c^0\\ps{1}),\n\\end{align}\nthe codegeneracy operators\n\\begin{align}\\label{aux-codegeneracy-opt}\n&\\mathfrak{s}_j: C^q_H(C,M)\\rightarrow C^{q-1}_H(C,M), \\quad 0\\le j\\le q-1 \\\\\\notag\n&\\mathfrak{s}_j (m\\otimes_H c^0\\ot\\cdots\\ot h^q)= m\\otimes_H c^0\\ot\\cdots\\ot \\varepsilon(c^{j+1})\\ot\\cdots\\ot\nc^q,\n\\end{align}\nand the cyclic operator\n\\begin{align}\\label{aux-cyclic-opt}\n&\\mathfrak{t}_q: C^q_H(C,M)\\rightarrow C^q_H(C,M),\\\\\\notag\n&\\mathfrak{t}_q(m\\otimes_H c^0\\ot\\cdots\\ot c^q)=m\\ns{0}\\otimes_H c^1\\ot\\cdots\\ot c^q\\otimes m\\ns{-1}(c^0).\n\\end{align}\nThe graded space $C_H(C,M)$ endowed with the above operators forms\na cocyclic module.\n\\noindent Using the above operators one defines the Hochschild coboundary\n$b$ and  the Connes boundary operator $B$,\n\\begin{align}\n&b: C^{q}_H(C,M)\\rightarrow C^{q+1}_H(C,M), \\qquad\nb:=\\sum_{i=0}^{q+1}(-1)^i\\mathfrak{d}_i, \\\\\n&B: C^{q}_H(C,M)\\rightarrow C^{q-1}_H(C,M), \\quad\nB:=\\left(\\sum_{i=0}^{q}(-1)^{qi}\\mathfrak{t}^{i}\\right) \\mathfrak{s}_{q-1}\\mathfrak{t}.\n\\end{align}\nThe cyclic cohomology of $C$ under the symmetry of $H$\nwith coefficients in the SAYD module $M$, which is  denoted by\n$HC(C,M)$, is defined to be the cyclic cohomology of the complex $C_H(C,M)$.\n\n\\medskip\n\n\\noindent For $C = H$, the map\n\\begin{align}\n\\begin{split}\n&{\\cal J}: M\\otimes_H H^{\\otimes (n+1)}\\rightarrow M\\otimes H^{\\otimes n},\\\\\n&{\\cal J}(m\\otimes_H h^0\\ot\\cdots\\ot h^n)=mh^0\\ps{1}\\otimes S(h\\ps{2})\\cdot(h^1\\ot\\cdots\\ot\nh^n).\n\\end{split}\n\\end{align}\nidentifies the standard Hopf-cyclic complex \\eqref{aux-stand-Hopf-cyclic} with\n\\begin{equation}\nC(H,M) := \\bigoplus_{q\\geq 0} C^q(H,M), \\quad C^q(H,M):= M\\otimes H^{\\otimes q}.\n\\end{equation}\nThen the coface operators become\n\\begin{align*}\n&\\mathfrak{d}_i: C^q(H,M)\\rightarrow C^{q+1}(H,M), \\quad 0\\le i\\le q+1\\\\\n&\\mathfrak{d}_0(m\\otimes h^1\\ot\\cdots\\ot h^q)=m\\otimes 1\\otimes h^1\\ot\\cdots\\ot h^q,\\\\\\notag\n&\\mathfrak{d}_i(m\\otimes h^1\\ot\\cdots\\ot h^q)=\\\\\n& m\\otimes h^1\\ot\\cdots\\ot h^i\\ps{1}\\otimes h^i\\ps{2}\\ot\\cdots\\ot h^q, \\quad 1\\le i\\le q \\\\\n&\\mathfrak{d}_{q+1}(m\\otimes h^1\\ot\\cdots\\ot h^q)=m\\ns{0}\\otimes h^1\\ot\\cdots\\ot h^q\\otimes m\\ns{-1},\n\\end{align*}\nthe codegeneracy operators\n\\begin{align*}\n&\\mathfrak{s}_j: C^q(H,M)\\rightarrow C^{q-1}(H,M), \\quad 0\\le j\\le q-1 \\\\\n&\\mathfrak{s}_j (m\\otimes h^1\\ot\\cdots\\ot h^q)= m\\otimes h^1\\ot\\cdots\\ot \\varepsilon(h^{j+1})\\ot\\cdots\\ot\nh^q,\n\\end{align*}\nand the cyclic operator becomes\n\\begin{align*}\n&\\mathfrak{t}: C^q(H,M)\\rightarrow C^q(H,M),\\\\\n&\\mathfrak{t}(m\\otimes h^1\\ot\\cdots\\ot h^q)=m\\ns{0}h^1\\ps{1}\\otimes\nS(h^1\\ps{2})\\cdot(h^2\\ot\\cdots\\ot h^q\\otimes m\\ns{-1}).\n\\end{align*}\n\n\\medskip\n\n\\noindent Let $A$ be a  $H$-module algebra, that is,    a (left) $H$-module\nand\n\\[h(ab)=h\\ps{1} (a) h\\ps{2}(b),\\quad h(1_A)=\\varepsilon(h)1_A, \\quad \\forall h\\in H,  a\\in A.\\]\n   One endows $V\\otimes A^{\\otimes n+1}$ with the action of $H$\n\\begin{equation}\n(v\\otimes a^0\\ot\\cdots\\ot q^q)\\cdot h= mh\\ps{1}\\otimes S(h\\ps{q+2})a^0\\ot\\cdots\\ot S(h\\ps{2})a^q,\n\\end{equation}\nand  forms\n\\begin{equation}\\label{aux-HC(A,V)}\nC^n_H(A,V)=\\mathop{\\rm Hom}\\nolimits_H(V\\otimes A^{\\otimes n+1}, {\\mathbb C})\n\\end{equation}\nas the space of  $H$-linear maps.  It is checked in  \\cite{HajaKhalRangSomm04-II}\n that for any $v \\otimes \\tilde{a}:=v \\otimes a^0 \\ot\\cdots\\ot a^{n+2} \\in V \\otimes A^{\\otimes\\,n+2}$ the  morphisms\n\\begin{align*}\n&(\\partial_i \\varphi)(v\\otimes \\tilde{a}) = \\varphi(v\\otimes a^0\\otimes \\dots\n\\otimes a^i a^{i+1}\\otimes\\dots \\otimes a^{n+1}), \\quad 0 \\leq i <n,\\\\\n&  (\\partial_{n+1}\\varphi)(v\\otimes \\tilde{a}) =\\varphi(v\\ns{0}\\otimes\n(S^{-1}(v\\ns{-1})a^{n+1})a^0\\otimes a^1\\otimes\\dots \\otimes a^{n}),\\\\\n&(\\sigma_i\\varphi)(v\\otimes \\tilde a) = \\varphi(v\\otimes a^0 \\otimes \\dots\n\\otimes a^i\\otimes 1 \\otimes \\dots \\otimes a^{n-1}), \\quad 0\\le i\\le n-1,\\\\\n&(\\tau\\varphi)(v\\otimes \\tilde a) = \\varphi(v\\ns{0}\\otimes (S^{-1}(v\\ns{-1})a^n)\\otimes\na^0\\otimes \\dots \\otimes a^{n-1})\n\\end{align*}\ndefine a cocyclic module structure on $C^n_H(A,V)$, whose cyclic\ncohomology  is denoted by $HC_H(A,V)$.\n\n\\medskip\n\n\\noindent One uses $HC(H,V)$ and $HC_H(A,V)$ to define a cup product\n\\begin{align*}\n&HC^p_H(A,V)\\otimes HC^q_H(H,V)\\rightarrow HC^{p+q}(A),\n\\end{align*}\nwhose definition can be found in \\cite{Rang08,KhalRang05-II}.\n\n\\medskip\n\\noindent As the simplest example,   one notes that the cup product with the 0-cocycle $\\tau \\in C^0_H(A,\\;\n^\\sigma{\\mathbb C}_\\delta)$ defines the Connes-Moscovici characteristic map\n\\cite{ConnMosc98,ConnMosc},\n\\begin{align}\\label{aux-Connes-Moscovici-charac-map}\n\\begin{split}\n& \\chi_\\tau:HC^\\bullet(H, ^\\sigma{\\mathbb C}_\\delta) \\to HC^\\bullet(A)\\\\\n& \\chi_\\tau(h^1 \\ot\\cdots\\ot h^n)(a^0 \\ot\\cdots\\ot a^n)=\\tau(a^0h^1(a^1)\\ldots\nh^n(a^n)).\n\\end{split}\n\\end{align}\n\n\n\n\\subsection{ Lie algebra (co)homology}\nIn this subsection, after recalling the  Lie algebra (co)homology,\nwe summarize  our work in \\cite{RangSutl-II} on the cyclic\ncohomology of Lie algebras with coefficients in SAYD modules.\n\n\\medskip\n\n\\noindent Let  $\\mathfrak{g}$ be a Lie algebra and $V$ be a right  $\\mathfrak{g}$-module.\nThe Lie algebra homology complex is defined to be\n\\begin{equation}\nC(\\mathfrak{g}, V) = \\bigoplus_{q\\geq0}C_q(\\mathfrak{g}, V), \\quad C_q(\\mathfrak{g}, V) := \\wedge^q \\mathfrak{g} \\otimes V\n\\end{equation}\nwith the Chevalley-Eilenberg boundary map\n\\begin{align}\n&\\xymatrix{\\cdots \\ar[r]^{\\partial_{\\rm CE}\\;\\;\\;\\;\\;\\;\\;\\;} & C_2(\\mathfrak{g},V)\n\\ar[r]^{\\partial_{\\rm CE}\\;\\;\\;} & C_1(\\mathfrak{g},V)\n\\ar[r]^{\\;\\;\\;\\;\\;\\;\\;\\partial_{\\rm CE}} & V},\\\\\\notag\n & \\partial_{\\rm CE}(X_0 \\wedge\\dots\\wedge X_{q-1} \\otimes v ) = \\sum_{i= 0}^{q-1} (-1)^i\nX_0 \\wedge\\dots\\wedge \\widehat{X}_i \\wedge\\dots\\wedge X_{q-1} \\otimes v \\cdot X_i + \\\\\\notag\n & \\sum_{0 \\leq i<j \\leq q-1} (-1)^{i+j} [X_i, X_j] \\wedge X_0 \\wedge\\dots\\wedge\n\\widehat{X}_i \\wedge\\dots\\wedge \\widehat{X}_j \\wedge\\dots\\wedge X_{q-1} \\otimes v.\n\\end{align}\nThe homology of the complex $(C(\\mathfrak{g}, V),\\partial_{\\rm CE})$ is called the\nLie algebra homology of $\\mathfrak{g}$ with coefficients in $V$ and it is\ndenoted by $H_{\\bullet}(\\mathfrak{g},V)$.  In a dual fashion one defines  the\nLie algebra cohomology complex\n\\begin{equation}\nW(\\mathfrak{g}, V) = \\bigoplus_{q\\geq0}W^q(\\mathfrak{g}, V),\n \\quad W^q(\\mathfrak{g},V)=\\mathop{\\rm Hom}\\nolimits(\\wedge^q \\mathfrak{g} ,V),\n\\end{equation}\nwhere $\\mathop{\\rm Hom}\\nolimits(\\wedge^q \\mathfrak{g} ,V)$ is the vector space\n of all alternating linear maps on $\\mathfrak{g}^{\\otimes q}$\n with values in $V$. The Chevalley-Eilenberg coboundary\n\\begin{equation}\n\\xymatrix{V\\ar[r]^{d_{\\rm CE}\\;\\;\\;\\;\\;\\;\\;\\;}\n&W^1(\\mathfrak{g},V)\\ar[r]^{d_{\\rm CE}}& W^2(\\mathfrak{g},V)\n\\ar[r]^{\\;\\;\\;\\;\\;\\;d_{\\rm CE}}&\\cdots\\;,}\n\\end{equation}\nis defined by\n \\begin{align}\\label{aux-Chevalley-Eilenberg-coboundary}\n \\begin{split}\n& d_{\\rm CE}(\\alpha)(X_0, \\ldots,X_q)=\\sum_{0 \\leq i<j \\leq q}\n(-1)^{i+j}\\alpha([X_i,X_j], X_0\\ldots \\widehat{X}_i, \\ldots,\n \\widehat{X}_j, \\ldots, X_q)+\\\\\n& ~~~~~~~~~~~~~~~~~~~~~~~\\sum_{i=0}^q (-1)^{i+1}\\alpha(X_0,\\ldots,\n\\widehat{X}_i,\\ldots X_q)\\cdot X_i.\n\\end{split}\n \\end{align}\nAlternatively, we may identify  $W^q(\\mathfrak{g},V)$ with $\\wedge^q\\mathfrak{g}^\\ast \\otimes V$\nand the coboundary $d_{\\rm CE}$ with\n\\begin{align}\\notag\n& d_{\\rm CE}(v)= - \\theta^i \\otimes v \\cdot X_i,\\quad  d_{\\rm CE}(\\beta \\otimes v)=\nd_{\\rm dR}(\\beta)\\otimes v  - \\theta^i \\wedge \\beta \\otimes v \\cdot X_i, \\\\\n& d_{\\rm dR}:\\wedge^p\\mathfrak{g}^\\ast\\rightarrow \\wedge^{p+1}\\mathfrak{g}^\\ast, \\quad d_{\\rm\ndR}(\\theta^i)=-\\frac{1}{2}C^i_{jk}\\theta^j\\wedge\\theta^k\n\\end{align}\nThe cohomology of the complex $(W(\\mathfrak{g},V),d_{\\rm CE})$, the Lie\nalgebra cohomology of $\\mathfrak{g}$ with coefficients in $V,$  is denoted by\n$H^\\bullet(\\mathfrak{g},V).$\n\n\\medskip\n\\noindent In this paper  we are particularly interested in the SAYD\nmodules over the universal enveloping algebra $U(\\mathfrak{g})$ of a Lie\nalgebra $\\mathfrak{g}$. By our results in \\cite{RangSutl-II},  such SAYD\nmodules are in fact obtained from the SAYD modules over the Lie\nalgebra $\\mathfrak{g}.$\n\\begin{definition}[\\cite{RangSutl-II}]\nA vector space $V$ is a left comodule over the Lie algebra $\\mathfrak{g}$\n if there is a linear map\n\\begin{equation}\n\\nabla_{\\mathfrak{g}}: V \\rightarrow \\mathfrak{g} \\otimes V,\\qquad \\nabla_{\\mathfrak{g}}(v)=v\\nsb{-1}\\otimes v\\nsb{0}\n\\end{equation}\nsuch that\n\\begin{equation*}\nv\\nsb{-2}\\wedge v\\nsb{-1}\\otimes v\\nsb{0}=0,\n\\end{equation*}\nwhere\n\\begin{equation*}\nv\\nsb{-2}\\otimes v\\nsb{-1}\\otimes v\\nsb{0}= v\\nsb{-1}\\otimes (v\\nsb{0})\\nsb{-1}\\otimes\n (v\\nsb{0})\\nsb{0}.\n\\end{equation*}\n\\end{definition}\n\\noindent It is clear that  left $\\mathfrak{g}$-comodules and  right\n$S(\\mathfrak{g}^*)$-modules are identical.\n\\begin{definition}[\\cite{RangSutl-II}]\nLet $V$ be a right module and left comodule over a Lie algebra\n$\\mathfrak{g}$. We call $V$ a right-left anti-Yetter-Drinfeld module (AYD\nmodule) over $\\mathfrak{g}$ if\n\\begin{equation}\n\\nabla_{\\mathfrak{g}}(v \\cdot X) = v\\nsb{-1} \\otimes v\\nsb{0} \\cdot X +\n[v\\nsb{-1}, X] \\otimes v\\nsb{0}.\n\\end{equation}\nMoreover, $V$ is called stable if\n\\begin{equation}\nv\\nsb{0} \\cdot v\\nsb{-1} = 0.\n\\end{equation}\nFinally, $V$ is said to be unimodular stable if $V_{-\\delta}$ is stable.\nHere the character $\\delta$ is defined by  $\\delta = \\mathop{\\rm Tr}\\nolimits \\circ \\mathop{\\rm ad}\\nolimits:\\mathfrak{g} \\to\n{\\mathbb C}$ and  $V_{-\\delta}$ is the  deformation of $V$ via\n\\[ v\\triangleleft X:= v\\cdot X-\\delta(X)v.\\]\n\n\n\\end{definition}\n\n\\begin{example}\\label{example-symm-alg-SAYD}{\\rm\n The truncated polynomial algebra $V = S(\\mathfrak{g}^\\ast)\\nsb{2n}$, of a Lie\nalgebra $\\mathfrak{g}$, is a unimodular SAYD module over $\\mathfrak{g}$ with the\ncoadjoint action and the Koszul coaction defined by\n\\begin{equation}\\label{aux-Koszul-ex}\n\\nabla_{K}: V \\to \\mathfrak{g} \\otimes V, \\quad \\nabla_{K}( v)= \\sum_{i=1}^nX_i \\otimes\nv\\theta^i.\n\\end{equation}\n}\\end{example}\n\n\\noindent Via the help of (unimodular) SAYD modules  we generalize  Lie\nalgebra (co)homology complexes. Let us start  with the  Lie algebra\nhomology  by  introducing  the  complex\n\\begin{equation}\\label{aux-complex-C(g,V)}\nC(\\mathfrak{g},V) = \\bigoplus_{i \\geq 0}\\wedge^i\\mathfrak{g}\\otimes V, \\qquad \\partial = \\partial_{\\rm\nCE} + \\partial_{\\rm K}\n\\end{equation}\nwith the Chevalley-Eilenberg boundary and the Koszul coboundary\n\\begin{equation}\n\\partial_{\\rm K}:C_n(\\mathfrak{g},V) \\to C_{n+1}(\\mathfrak{g},V), \\qquad \\partial_{\\rm K}(e \\otimes\nv)= v\\nsb{-1} \\wedge e \\otimes v\\nsb{0}.\n\\end{equation}\n\n\\noindent  Applying the Poincar\\'e duality  in the Lie algebra\ncohomology  to the complex \\eqref{aux-complex-C(g,V)}, see\n\\cite[Prop. 4.4]{RangSutl-II}, we obtain\n\\begin{equation}\nW(\\mathfrak{g},V) = \\bigoplus_{i \\geq 0}\\wedge^i\\mathfrak{g}^\\ast\\otimes V\n\\end{equation}\nwith $d = d_{\\rm CE} + d_{\\rm K}$, where $d_{\\rm CE}:W^n(\\mathfrak{g},V) \\to W^{n+1}(\\mathfrak{g},V)$ is the Chevalley-Eilenberg coboundary and\n\\begin{equation*}\nd_{\\rm K}:W^n(\\mathfrak{g},V) \\to W^{n-1}(\\mathfrak{g},V), \\qquad d_{\\rm K}(\\alpha \\otimes\nv)= \\iota(v\\nsb{-1})(\\alpha) \\otimes v\\nsb{0}.\n\\end{equation*}\nHere $\\iota(X)$ is the contraction with respect to $X$.\n\n\\medskip\n\n \\noindent In particular, we recover  the (truncated) Weil algebra\n\\cite{RangSutl-II}:\n\\begin{equation}\nW(\\mathfrak{g},S(\\mathfrak{g}^\\ast)) = W(\\mathfrak{g}), \\qquad W(\\mathfrak{g},S(\\mathfrak{g}^\\ast)\\nsb{2n}) = W(\\mathfrak{g})\\nsb{2n}.\n\\end{equation}\n\n\n\n\n\n\n\\section{SAYD-twisted cyclic cocycles}\nIn this section we fix $K$ to be a cocommutative Hopf subalgebra of\na Hopf algebra $H$. Let $A$ be an $H$-module algebra, and  $V$ be a\nSAYD module over $K$. We aim to develop a\nmachinery to produce SAYD-twisted cyclic cocycles in $HC_K(A,V)$. In\nthe first subsection we introduce equivariant Hopf-cyclic cohomology\n$HC_K(H,V,N)$, where $N$ is a SAYD module over $H$. In the second\nsubsection we construct a cup product $HC^p_K( H, V,N)\\otimes HC^q_\nH(A,N)\\rightarrow HC^{p+q}_K(A,V)$. In the third and fourth subsection we\napply the results of the first two subsections. This way we produce\n a nontrivial SAYD-twisted cyclic cocycle  over the groupoid action\nalgebra under the symmetry of the general linear Lie algebra with\ncoefficients in the truncated polynomials on this Lie algebra.\n\n\n\n\\subsection{Equivariant Hopf-cyclic cohomology}\n\\label{section-equiv-cohom}   For a SAYD module $N$ over $H$ and a\nright module-left comodule  $V$ over $K$ we define  the graded space\n\\begin{align}\\label{aux-equivariant-complex}\n\\begin{split}\n& C_K(H, V,N)=\\bigoplus_{q \\geq 0}C^q_K(H, V,N), \\\\\n& {\\cal C}^q:=C^q_K(H, V,N):=\\mathop{\\rm Hom}\\nolimits_K(V, N\\otimes_H  H^{\\otimes q+1}).\n\\end{split}\n\\end{align}\nMore precisely,  $\\phi\\in {\\cal C}^q$ if and only  if  for any $u\\in K$\nand  any $v\\in V$\n\\begin{equation}\\label{aux-equivariancy}\n\\phi(v\\cdot u)= \\phi(v)\\cdot u,\n\\end{equation}\nwhere the right  action of $K$ on $N\\otimes_H H^{\\otimes q+1}$ is the usual\ndiagonal action,  {\\it i.e.\\\/}\\ \n\\begin{equation}\n(n\\otimes_H h^0\\ot\\cdots\\ot h^{q})\\cdot u= n\\otimes_H h^0 u\\ps{1}\\ot\\cdots\\ot\nh^{q}u\\ps{q+1}.\n\\end{equation}\nFor $\\phi\\in C^q_K(H, V,N)$ and $v\\in V$,  we use the notation\n\\begin{equation}\\label{aux-notation-phi(m)}\n\\phi(v)=\\phi(v)\\snb{-1}\\otimes_H \\phi(v)\\snb{0} \\ot\\cdots\\ot \\phi(v)\\snb{q}.\n\\end{equation}\nLet us define the morphisms  $\\mathfrak{d}_i:{\\cal C}^q\\rightarrow {\\cal C}^{q +1}$,\n$\\mathfrak{s}_j:{\\cal C}^{q}\\rightarrow{\\cal C}^{q-1} $, and $\\mathfrak{t}_q:{\\cal C}^q\\rightarrow {\\cal C}^q$ as\n\\begin{align}\n\\begin{split}\n&d_i(\\phi)(v)= \\mathfrak{d}_i(\\phi(v)), \\quad 0\\le i\\le q\\\\[.5cm]\n&d_{q+1}(\\phi)(v)= \\mathfrak{d}_{q+1}(\\phi(v\\ns{0}))\\triangleleft S(v\\ns{-1}),\\\\[.5cm]\n&s_j(\\phi)(v)=\\mathfrak{s}_j(\\phi(v)), \\quad 0\\le j\\le q-1,\\\\[.5cm]\n&t_q(\\phi)(v)=\\mathfrak{t}_q(\\phi(v\\ns{0}))\\triangleleft S(v\\ns{-1}),\n\\end{split}\n\\end{align}\nwhere the right action $\\triangleleft$ of $K$ on $N\\otimes_H H^{\\otimes q+1}$ is defined\nby\n\\begin{equation}\n(n\\otimes_H h^0\\ot\\cdots\\ot h^q) \\triangleleft u= n\\otimes_H h^0\\ot\\cdots\\ot h^{q-1}\\otimes h^q u.\n\\end{equation}\nHere  the morphisms  $(\\mathfrak{d}_i,\\mathfrak{s}_j, \\mathfrak{t})$ are the usual morphisms of\nthe cocyclic module  $C_H(H, N)$ defined in\n \\eqref{aux-coface-opt},\\eqref{aux-codegeneracy-opt} and \\eqref{aux-cyclic-opt}.\n\n\\begin{theorem}\nIf $V$  and $N$ are SAYD modules over $K$ and $H$ respectively, then\n the morphisms $d_i,s_j$ and $t$ define  a cocyclic module structure\non $C_K(H, V,N)$.\n\\end{theorem}\n\n\\begin{proof}\nLet us prove that the morphisms $d_i, s_j$, and  $t$  are\nwell-defined. Indeed, it suffices to check\n  that $t$, $d_0$, and $s_n$ are well-defined as the other morphisms are made\n   of these three.  For $d_0 $ and $s_n$ the task is obvious as\n$\\Delta:H\\rightarrow H\\otimes H$ and $\\varepsilon:H\\rightarrow {\\mathbb C}$ are multiplicative respectively.\nLet us check that $t$ is well-defined. We have\n \\begin{align}\n \\begin{split}\n &t(\\phi)(v\\cdot y)=\\tau(\\phi((v\\cdot y) \\ns{0}))\\triangleleft S((v\\cdot y)\\ns{-1})\\\\\n &= \\tau(\\phi(v\\ns{0}\\cdot y\\ps{2}) )\\triangleleft S(( S(y\\ps{1})v\\ns{-1}y\\ps{3})\\\\\n &= \\tau(\\phi(v\\ns{0})\\cdot y\\ps{2} )\\triangleleft S(y\\ps{3}) S(v\\ns{-1})y\\ps{1}\\\\\n  &=\\tau(\\phi(v)\\snb{-1}\\otimes_H \\phi(v)\\snb{0} y\\ps{2} \\ot\\cdots\\ot \\phi(v)\\snb{q} y\\ps{q+2} )\\triangleleft S(y\\ps{q+3}) S(v\\ns{-1})y\\ps{1}\\\\\n  &= \\phi(v)\\snb{-1}\\ns{0}\\otimes_H \\phi(v)\\snb{1} y\\ps{3} \\otimes\\dots\\\\\n  &\\dots \\otimes \\phi(v)\\snb{q} y\\ps{q+2}\\otimes \\phi(v)\\snb{-1}\\ns{-1} \\phi(v)\\snb{0} y\\ps{2} S(y\\ps{q+3}) S(v\\ns{-1})y\\ps{1}\\\\\n  &= \\phi(v)\\snb{-1}\\ns{0}\\otimes_H \\phi(v)\\snb{1} y\\ps{1} \\otimes\\dots\\\\\n  &\\dots\\otimes \\phi(v)\\snb{q} y\\ps{q}\\otimes \\phi(v)\\snb{-1}\\ns{-1} \\phi(v)\\snb{0}  S(v\\ns{-1})y\\ps{q+1}\\\\\n  &=t(\\phi(v))\\cdot y.\n\\end{split}\n \\end{align}\n\\noindent In the second and the sixth equalities we use the fact that  $K$\nis cocommutative.\n\n\\medskip\n\n\\noindent Let us next show that $C_K(H,V,N)$ is a cocyclic module which means that $d_i,$ $s_j$ and $t$ satisfy\n\\begin{equation}\\label{aux-disj-order}\nd_j  d_i = d_i  d_{j-1}, \\, \\, i < j  , \\qquad s_j s_i = s_i\ns_{j+1},  \\, \\,  i \\leq j\n\\end{equation}\n\\begin{equation}\\label{aux-disj-order-II}\ns_j  d_i = \\left\\{ \\begin{matrix} d_i  s_{j-1} \\hfill &i < j\n\\hfill \\cr \\mathop{\\rm Id}\\nolimits_q \\hfill &\\hbox{if} \\ i=j \\ \\hbox{or} \\ i = j+1 \\cr\nd_{i-1}  s_j \\hfill &i > j+1  ;  \\hfill \\cr\n\\end{matrix} \\right.\n\\end{equation}\n\n\\begin{eqnarray}\\label{aux-t-d-order}\nt_{q+1}  d_i  = d_{i-1}  t_q ,\n \\quad && 1 \\leq i \\leq q+1 ,  \\quad t_{q+1}  d_0 =\nd_{q+1} \\\\ \\label{cj} t_{q-1}  s_i = s_{i-1} t_q , \\quad &&\n1 \\leq i \\leq q-1 , \\quad t_q  s_0 = s_{q-1}  t_q^2 \\\\\n\\label{ce} t_q^{q+1} &=& \\mathop{\\rm Id}\\nolimits_q  \\, .\n\\end{eqnarray}\nThe equalities \\eqref{aux-disj-order}, \\eqref{aux-disj-order-II} and \\eqref{cj} follow directly from their counterparts for the operators $\\mathfrak{d}_i, \\mathfrak{s}_j$ and $\\mathfrak{t}$.\n\n\\medskip\n\n\\noindent As for \\eqref{aux-t-d-order}, we check the case $i = q+1$.\nIndeed\n\\begin{align}\n\\begin{split}\n& t_{q+1} (d_{q+1}(\\phi))(v) = \\mathfrak{t}_{q+1}(d_{q+1}(\\phi)(v\\ns{0}))\\triangleleft S((v\\ns{-1}))\\\\\n&= \\mathfrak{t}_{q+1}(\\mathfrak{d}_{q+1}(\\phi(v\\ns{0}\\ns{0}))\\triangleleft S(v\\ns{0}\\ns{-1}))\\triangleleft S(v\\ns{-1})\\\\\n&= \\mathfrak{t}_{q+1}(\\mathfrak{d}_{q+1}(\\phi(v\\ns{0}))\\triangleleft S(v\\ns{-1}))\\triangleleft S(v\\ns{-2})\\\\\n& = \\mathfrak{t}_{q+1}\\left(\\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes_H \\phi(v\\ns{0})\\snb{0}\\ps{2} \\otimes \\phi(v\\ns{0})\\snb{1} \\ot\\cdots\\ot\\right. \\\\\n& \\left. \\phi(v\\ns{0})\\snb{q} \\otimes  \\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{0}\\ps{1}S(v\\ns{-1})\\right)\\triangleleft S(v\\ns{-2})\\\\\n&= \\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes_H \\phi(v\\ns{0})\\snb{1} \\ot\\cdots\\ot \\phi(v\\ns{0})\\snb{q} \\otimes \\\\\n& \\phi(v\\ns{0})\\snb{-1}\\ns{-2}\\phi(v\\ns{0})\\snb{0}\\ps{1}S(v\\ns{-1}) \\otimes \\\\\n& \\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{0}\\ps{2}S(v\\ns{-2}) \\\\\n&= \\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes_H \\phi(v\\ns{0})\\snb{1} \\ot\\cdots\\ot \\phi(v\\ns{0})\\snb{q} \\otimes \\\\\n& \\Delta\\left(\\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{0}S(v\\ns{-1})\\right) \\\\\n& = \\mathfrak{d}_q\\left(\\left[\\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes_H \\phi(v\\ns{0})\\snb{1} \\ot\\cdots\\ot \\phi(v\\ns{0})\\snb{q} \\otimes \\right.\\right.\\\\\n& \\left.\\left.\\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{0}\\right]\\triangleleft S(v\\ns{-1})\\right)\\\\\n& = \\mathfrak{d}_q(\\mathfrak{t}_q(\\phi(v\\ns{0}))\\triangleleft S(v\\ns{-1})) = d_q(t_q(\\phi))(v).\n\\end{split}\n\\end{align}\nA simple calculation yields one to\n\\begin{align*}\n& t_q^{q+1}(\\phi)(v)\\\\\n&= \\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes_H \\phi(v\\ns{0})\\snb{-1}\\ns{-q-1}\n\\phi(v\\ns{0})\\snb{0}S(v\\ns{-1})\\otimes\\dots\\\\\n& \\dots\\otimes\n\\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{q}S(v\\ns{-q-1})\\otimes\\dots\\\\\n&=\\phi(v\\ns{0})\\snb{-1}\\ns{0} \\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\otimes_H \\phi(v\\ns{0})\\snb{0}S(v\\ns{-1})\\\\\n& \\dots\\otimes \\phi(v\\ns{0})\\snb{q}S(v\\ns{-q-1})\\\\\n&=\\phi(v\\ns{0})\\snb{-1}\\otimes_H \\phi(v\\ns{0})\\snb{0}S(v\\ns{-1})\n \\ot\\cdots\\ot \\phi(v\\ns{0})\\snb{q}S(v\\ns{-q-1})\\\\\n&=\\phi(v\\ns{0})\\cdot S(v\\ns{-1})= \\phi(v\\ns{0}\\cdot S(v\\ns{-1}))=\\phi(v)\\\\\n\\end{align*}\nThe last equality is held because of the fact that for any SAYD\nmodule $V$ and any $v \\in V$, $v\\ns{0}\\cdot S^{-1}(v\\ns{-1})=v$, see \\cite[Lemma\n4.9]{KhalRang03}. Here Since $K$ is cocommutative $S=S^{-1}$.\n \\end{proof}\n\n\\noindent The cyclic cohomology of $C_K(H,V,N)$ is denoted by\n$HC_K(H,V,N)$.\n\n\\medskip\n\n\\noindent One notes that by taking $K={\\mathbb C}$ and $V={\\mathbb C}$ one recovers the\nusual Hopf-cyclic cohomology $HC(H,N).$\n\n\n\n\n\n\\subsection{Equivariant characteristic map}\n\\label{subsection-eq-char} Let $V$ and $N$ be  SAYD modules over $K$\nand $ H$ respectively. We define the map\n  \\begin{align}\n &\\Psi: C^q_K( H,V,N)\\otimes C^q_ H(A,N)\\longrightarrow  C^q_K(A,\nV),\\\\\\notag\n &\\Psi(\\phi\\otimes\\psi)(v\\otimes a_0\\ot\\cdots\\ot a_q)\\\\\n &= \\psi\\big( \\phi(v)\\snb{-1}\\otimes  \\phi(v)\\snb{0}(a_0)\\otimes \\phi(v)\\snb{1}( a_1) \\ot\\cdots\\ot \\phi(v)\\snb{q}( a_q)\\big).\n  \\end{align}\nOne may check that $\\Psi$ is a map of cocyclic modules,  where  on the left hand side we consider the product of two cocyclic modules. This is enough  to produce  a generalization of the  cup product in Hopf-cyclic cohomology \\cite{KhalRang05-II,Rang08}.\n\n\\medskip\n\n\\noindent We define a new bicocyclic module by\ntensoring the  cocyclic modules \\eqref{aux-cyclic-opt}, and \\eqref{aux-HC(A,V)}. The new bigraded  module\n in the bidegree  $(p,q)$ is defined by\n\\begin{equation}\\label{c**}{\\cal C}^{p,q}:=\\mathop{\\rm Hom}\\nolimits_K(V, N\\otimes_H H^{\\otimes p+1})\\otimes \\mathop{\\rm Hom}\\nolimits_H(A^{\\otimes q+1},N)\\end{equation} with horizontal structure\n$\\overset{\\ra}{\\partial}_i=\\mathfrak{d}_i\\otimes \\mathop{\\rm Id}\\nolimits$, $\\overset{\\ra}{\\sigma}_j=\\mathfrak{s}\\otimes \\mathop{\\rm Id}\\nolimits$, and $\\overset{\\ra}{\\tau}=\\mathfrak{t}\\otimes \\mathop{\\rm Id}\\nolimits$ and\nvertical structure $\\uparrow\\hspace{-4pt}\\partial_i= \\mathop{\\rm Id}\\nolimits\\otimes\\partial_i$, $\\uparrow\\hspace{-4pt}\\sigma_j=\\mathop{\\rm Id}\\nolimits\\otimes \\sigma_j$, and\n$\\uparrow\\hspace{-4pt}\\tau=\\mathop{\\rm Id}\\nolimits\\otimes \\tau$. Obviously $({\\cal C}^{\\bullet,\\bullet},\\overset{\\ra}{\\partial},\\overset{\\ra}{\\sigma},\\overset{\\ra}{\\tau},\\uparrow\\hspace{-4pt}\\partial,\\uparrow\\hspace{-4pt}\\sigma,\\uparrow\\hspace{-4pt}\\tau)$ defines a bicocyclic module.\n\n\\medskip\n\n\\noindent Now let us define  the  map\n\\begin{align}\\label{acpsi}\n&\\Psi:{\\cal D}^q\\rightarrow \\mathop{\\rm Hom}\\nolimits_K(V\\otimes A^{\\otimes q+1},{\\mathbb C}),\\\\\\notag\n&\\Psi(\\phi\\otimes\\psi)(v\\otimes a_0\\ot\\cdots\\ot a_q)\\\\\\notag\n &= \\psi\\big( \\phi(v)\\snb{-1}\\otimes  \\phi(v)\\snb{0}(a_0)\\otimes \\phi(v)\\snb{1}( a_1) \\ot\\cdots\\ot \\phi(v)\\snb{q}( a_q)\\big).\n\\end{align}\nHere ${\\cal D}^{\\bullet}$ denotes the diagonal complex of the\nbicocyclic module  ${\\cal C}^{\\bullet,\\bullet}$. It is a cocyclic\nmodule whose $q$th component is ${\\cal C}^{q,q}$ and its cocyclic structure\nmorphisms are  $\\partial_i:=\\overset{\\ra}{\\partial}_i\\circ\\uparrow\\hspace{-4pt}\\partial_i$, $\\sigma_j:=\\overset{\\ra}{\\sigma}_j\\circ\\uparrow\\hspace{-4pt}\\sigma_j$, and\n$\\tau:=\\overset{\\ra}{\\tau}\\circ\\uparrow\\hspace{-4pt}\\tau$.\n\n\\begin{proposition} \\label{cyclic-map}\nThe map $\\Psi$ is  a well-defined map of cocyclic modules.\n\\end{proposition}\n\n\\begin{proof}\nLet us  first show that $\\Psi$ is well-defined. Indeed, by using the\nfact that  $\\phi$ is  $K$-linear,  we see\n\\begin{align*}\n&\\Psi(\\phi\\otimes \\psi)\\big(vk\\ps{1} \\otimes  S(k\\ps{q+2})(a^0)\\ot\\cdots\\ot S(k\\ps{2})(a^{q})\\big)\\\\\n&=\\psi( \\phi(vk\\ps{1})\\snb{-1}\\otimes  \\phi(vk\\ps{1})\\snb{0}S(k\\ps{q+2})(a^0)\\ot\\cdots\\ot      \\phi(vk\\ps{1})\\snb{q} S(k\\ps{2})(a^{q}))\\\\\n&= \\psi( \\phi(v)\\snb{-1}\\otimes  \\phi(v)\\snb{0}k\\ps{1}S(k\\ps{2q+2})(a^0)\\ot\\cdots\\ot      \\phi(v)\\snb{q}k\\ps{q+1} S(k\\ps{q+2})(a^{q}))\\\\\n&= \\varepsilon(k)\\psi( \\phi(v)\\snb{-1}\\otimes  \\phi(v)\\snb{0}(a^0)\\ot\\cdots\\ot      \\phi(v)\\snb{q}(a^{q}))\\\\\n&=\\varepsilon(k)\\Psi(\\phi\\otimes \\psi)\\big(v \\otimes  a^0\\ot\\cdots\\ot a^{q}\\big).\n\\end{align*}\n Next, we show that $\\Psi$  commutes with the  cocyclic structure\n morphisms. To this end,  we need only to show the commutativity of\n $\\Psi$ with zeroth coface, the last codegeneracy and the cyclic\n operator, because these  operators generate all cocyclic structure\n morphisms. We check it only  for the cyclic operators and leave the\n rest to the reader.\n\\begin{align*}\n&\\tau \\Big(\\Psi(\\phi\\otimes \\psi)\\Big)\\big(v \\otimes a^0\\ot\\cdots\\ot a^{q}\\big)\\\\\n&= \\Psi(\\phi\\otimes \\psi)\\big(v\\ns{0} \\otimes  S^{-1}(v\\ns{-1})(a^q)\\otimes a^0\\ot\\cdots\\ot a^{q-1}\\big)\\\\\n&=\\psi(\\phi(v\\ns{0})\\snb{-1}\\otimes \\phi(v\\ns{0})\\snb{0}S^{-1}(v\\ns{-1})(a^q)\\otimes \\phi(v\\ns{0})\\snb{1}(a^1)\\otimes\\dots\\\\\n&\\dots\\otimes \\phi(v\\ns{0})\\snb{q}(a^q).\n\\end{align*}\nOn the other hand we have\n\\begin{align*}\n&\\Psi(\\mathfrak{t}\\phi\\otimes \\tau\\psi)\\big(v \\otimes a^0\\ot\\cdots\\ot a^{q}\\big)\\\\\n&= \\tau\\psi( \\mathfrak{t}\\phi(v)\\snb{-1}\\otimes  \\mathfrak{t}\\phi(v)\\snb{0}(a^0)\\ot\\cdots\\ot \\mathfrak{t}\\phi(v)\\snb{q}(a^q))\\\\\n&=\\tau\\psi( \\mathfrak{t}\\phi(v)\\snb{-1}\\otimes  \\mathfrak{t}\\phi(v)\\snb{0}(a^0)\\ot\\cdots\\ot \\mathfrak{t}\\phi(v)\\snb{q}(a^q))\\\\\n&=\\tau\\psi( \\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes  \\phi(v)\\snb{1}(a^0)\\otimes\\dots\\\\\n &~~~~~~~\\dots\\otimes \\phi(v)\\snb{q}(a^{q-1})\\otimes \\phi(v\\ns{0})\\snb{-1}\\ns{-1}\\phi(v\\ns{0})\\snb{0} S(v\\ns{-1})(a^q))\\\\\n & =\\psi\\big( \\phi(v\\ns{0})\\snb{-1}\\ns{0}\\otimes  S^{-1}(\\phi(v\\ns{0})\\snb{-1}\\ns{-1})\\phi(v\\ns{0})\\snb{-1}\\ns{-2}\\phi(v\\ns{0})\\snb{0} S(v\\ns{-1})(a^q)\\otimes \\\\\n &  \\phi(v)\\snb{1}(a^0)\\ot\\cdots\\ot \\phi(v)\\snb{q}(a^{q-1})\\big)\\\\\n &=\\psi(\\phi(v\\ns{0})\\snb{-1}\\otimes \\phi(v\\ns{0})\\snb{0}\n S(v\\ns{-1})(a^q)\\otimes \\phi(v\\ns{0})\\snb{1}(a^1)\\ot\\cdots\\ot \\phi(v\\ns{0})\\snb{q}(a^q).\n\\end{align*}\nSince $K$ is cocommutative $S^2=\\mathop{\\rm Id}\\nolimits$, and hence $$\\tau\\big(\\Psi(\\phi\\otimes \\psi)\\big)=\\Psi(\\mathfrak{t}\\phi\\otimes \\tau\\psi). $$\n\\end{proof}\n\n\\begin{theorem}\nAssume that  $K$ is a cocommutative Hopf subalgebra of  a Hopf\nalgebra $ H$,   ${\\cal A}$  is a $ H$-module algebra,  and  $V$ and $N$\nare SAYD modules over $K$ and $ H$ respectively.  Then  the map\n$\\Psi$ defines a cup product\n\\begin{equation}\\label{aux-equ-cup}\nHC^p_K( H, V,N)\\otimes HC^q_ H(A,N)\\rightarrow HC^{p+q}_K(A,V).\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nLet $[\\phi]\\in HC^p_K(H,V,N)$ and $[\\psi]\\in HC^q_H(A,N)$. Without\nloss of generality we assume that  $\\phi$ and $\\psi$  are\nrespectively cyclic cocycles horizontally and  vertically. This\nimplies\n that $\\phi\\otimes \\psi$ is a $(b,B)$ cocycle of degree $p+q$ in total\n  compex of  ${\\cal C}^{\\bullet,\\bullet}$. On the other hand by the cyclic\n  Eilenberg-Zilber theorem \\cite{KhalRang04}, the total\n  complex of ${\\cal C}^{\\bullet,\\bullet}$ is quasi-isomorphic\n   with ${\\cal D}^{\\bullet}$  via the Alaxander-Witney map $AW$.\n     So, $AW(\\phi\\otimes \\psi)$ is a $(b, B)$ cocycle\n     in ${\\cal D}^{\\bullet}$. Since $\\Psi$ is cyclic, we conclude that\n     $\\Psi(AW(\\phi\\otimes \\psi))$ defines a\n class in $HC^{p+q}_K(A, V)$.\n\\end{proof}\n\\noindent One notes that by setting $K:={\\mathbb C}$ as the trivial  Hopf subalgebra\nand $M={\\mathbb C}$ as the trivial SAYD module over $K$  the above cup\nproduct becomes  the cup product defined in\n\\cite{KhalRang05-II,Rang08}.\n\n\n\n\n\\subsection{Equivariant charactrestic map for\n${\\cal H}_n$}\\label{subsection-H-n} In this subsection we apply our\nequivariant characteristic map we built  in  Subsection\n\\ref{subsection-eq-char} to produce the desired  cyclic cocycle on\nthe groupoid action algebra.\n\n\\medskip\n\n \\noindent Let us first  recall the Connes-Moscovici Hopf algebra\n${\\cal H}_n$ from \\cite{ConnMosc98,ConnMosc}. To this end let $\\mathfrak{h}_n$ be\nthe Lie algebra generated by\n\\begin{equation} \\label{gens}\n\\{X_k , \\, Y_i^j , \\, \\delta_{jk \\,  \\ell_1  \\ldots \\ell_r}^i \\, ;  \\,\ni, j, k, \\ell_1  \\ldots \\ell_r  =1, \\ldots , n , \\, r \\in {\\mathbb N}\\}\n\\end{equation}\nwith relations\n\\begin{align}\\label{aux-relations-CM}\n\\begin{split}\n& [Y_i^j , Y_k^{\\ell}] = \\delta_k^j Y_i^{\\ell} - \\delta_i^{\\ell} Y_k^j, \\quad  [Y_i^j , X_k]=\n\\delta_k^j X_i, \\quad [X_k ,\nX_{\\ell}] = 0,\\\\\n& \\delta_{jk   \\ell_1 \\ldots \\ell_r}^i = [X_{\\ell_r} , \\ldots\n[X_{\\ell_1} , \\delta_{jk}^i] \\ldots ],\\quad [\\delta_{jk  \\ell_1 \\ldots \\ell_r}^i , \\, \\delta_{j'k'   {\\ell}'_1\n\\ldots  {\\ell}'\n_r}^{i'}]= 0,\\\\\n&  [Y_p^q , \\delta_{j_1 j_2 \\,  j_3 \\ldots j_r}^i ]  = \\sum_{s=1}^r \\, \\delta^q_{j_s} \\,\n \\delta^i_{j_1  j_2 \\,  j_3  \\ldots j_{s-1} p j_{s+1} \\ldots j_r}\n  - \\delta_p^i \\, \\delta_{j_1 j_2 \\,  j_3  \\ldots j_r}^q, \\\\\n  & \\delta_{ jk\\ell_1 \\,   \\ldots \\ell_r}^i=  \\delta_{jk\\ell_{\\pi(1)} \\ldots\n  \\ell_{\\pi(r)}}^i, \\quad \\forall \\pi\\in S_r.\n\\end{split}\n\\end{align}\nAs an algebra, ${\\cal H}_n$ is $U(\\mathfrak{h}_n)$ modulo the (Bianchi-type) identities\n\\begin{equation}\n \\delta_{j \\ell   k}^i -  \\delta_{j k   \\ell}^i =  \\delta_{j k}^s \\, \\delta_{s \\ell}^i  -  \\delta_{j \\ell}^s \\, \\delta_{s k}^i .\n\\end{equation}\nThe coalgebra structure of ${\\cal H}_n$ is defined by a Leibniz  rule\nthat makes the  groupoid action algebra ${\\cal A}=\nC^{\\infty}_c(F^+)\\rtimes \\Gamma$ an ${\\cal H}_n$-module algebra.\n\n\\medskip\n\n\\noindent In order to describe the action of ${\\cal H}_n$ explicitly, let us\nfirst identify $F^+$ with ${\\mathbb R}^n\\rtimes {\\rm GL}^+(n,{\\mathbb R})$ and use\nthe local coordinates $(x,y) \\in F^+$. A typical element of the\ngroupoid action algebra is of the form $fU^\\ast_\\phi$, where\n$U^\\ast_\\phi$ stands for $\\phi^{-1} \\in \\Gamma$ and $f\\in\nC^\\infty_c(F^+)$. The elements of ${\\cal H}_n$ act as the following\noperators.\n\\begin{align}\\label{aux-action-Hn-on-A}\n\\begin{split}\n& X_k = y_k^\\mu\\frac{\\partial}{\\partial x^\\mu}, \\qquad X_k(fU^\\ast_\\phi) := X_k(f)U^\\ast_\\phi,\\\\\n& Y_i^j = y_i^\\mu\\frac{\\partial}{\\partial y_\\mu^j}, \\qquad Y_i^j(fU^\\ast_\\phi) := Y_i^j(f)U^\\ast_\\phi,\\\\\n& \\delta_{jk \\,  \\ell_1 \\ldots \\ell_r}^i \\, ( f \\, U_{\\phi}^\\ast) = \\gamma_{jk \\,  \\ell_1 \\ldots \\ell_r}^i (\\phi)\\, f \\, U_{\\phi}^\\ast,\n\\end{split}\n\\end{align}\nwhere\n\\begin{align}\n\\begin{split}\n& \\gamma_{jk \\,  \\ell_1 \\ldots \\ell_r}^i (\\phi) = X_{\\ell_r} \\cdots X_{\\ell_1} \\big(\\gamma_{jk}^i (\\phi)\\big),\\\\\n& \\gamma_{jk}^i (\\phi) (x, y)  = \\left( y^{-1} \\cdot\n{\\phi}^{\\prime} (x)^{-1} \\cdot \\partial_{\\mu} {\\phi}^{\\prime} (x) \\cdot\ny\\right)^i_j \\, y^{\\mu}_k.\n\\end{split}\n\\end{align}\nTherefore, for any $a,b\\in {\\cal A}$ we have the Leibniz rule\n\\begin{align}\n\\begin{split}\n& Y_i^j (a b)  = Y_i^j (a) \\, b \\, + \\, a \\,Y_i^j (b), \\\\\n& X_k(a  b)  =  X_k (a) \\, b \\, + \\, a \\,X_k (b)  + \\delta_{jk}^i (a) \\, Y_{i}^{j} (b),\\\\\n& \\delta_{jk}^i (a  b) =  \\delta_{jk}^i  (a) \\, b  +  a \\, \\delta_{jk}^i  (b).\n\\end{split}\n\\end{align}\nAccordingly, we have\n\\begin{align}\\label{aux-D-Y}\n& \\Delta(Y_i^j) = Y_i^j \\otimes 1+1 \\otimes Y_i^j,\\\\\\label{aux-D-d}\n& \\Delta(\\delta_{jk }^i) = \\delta_{jk}^i \\otimes 1 + 1 \\otimes \\delta_{jk}^i,\\\\\\label{aux-D-X}\n& \\Delta(X_k) = X_k \\otimes 1 + 1 \\otimes X_k + \\delta^i_{jk} \\otimes Y_i^j.\n\\end{align}\nFor simplicity, we will also employ the notation\n\\begin{equation}\n{\\delta^a}_{k  \\ell_1 \\ldots \\ell_r} :=\\delta_{jk  \\ell_1 \\ldots \\ell_r}^i, \\quad Y_a :=Y_i^j\n\\qquad a=\\left(\\begin{matrix}i\\\\j\\end{matrix}\\right).\n\\end{equation}\n\n\\medskip\n\n\\noindent Throughout this subsection we  set  $ {\\cal H}:= {\\cal H}_n$,  the\nConnes-Moscovici Hopf algebra, and ${\\cal K}:=U(\\mathfrak{g}_0)$, where\n$\\mathfrak{g}_0:=g\\ell_n$.  We also let $N={\\mathbb C}_\\delta$ the  SAYD\nmodule over $ {\\cal H}$ where $\\delta: {\\cal H}\\rightarrow {\\mathbb C}$ is the character defined on\nthe generators by\n\\begin{equation}\n\\delta(Y_i^j)=\\delta_i^j, \\qquad \\delta(X_k)=\\delta(\\delta_{jk   \\ell_1 \\ldots \\ell_r}^i)=0, \\qquad 1\\le i,j,k, l_t\\le n\n\\end{equation}\nand is extended  on $ {\\cal H}$ multiplicatively. Finally we set $V=\nS(\\mathfrak{g}_0^\\ast)_{[2n]}$ with the canonical ${\\cal K}$-SAYD module structure\nas recalled in \\eqref{example-symm-alg-SAYD}.\n\n\\medskip\n\n\\noindent Let ${\\cal A}:={\\cal A}_\\Gamma= C^{\\infty}_c(F^+)\\rtimes \\Gamma$ and $\\tau$ be the\ncanonical trace  on ${\\cal A}$ defined in \\eqref{aux-canonical-trace}.\nSince $\\tau\\in C^0_{\\cal H}({\\cal A}, {\\mathbb C}_\\delta)$ and $\\tau$ is a cyclic cocycle\n\\cite{ConnMosc98}, applying the cup product \\eqref{aux-equ-cup} we\nget the  map of cocyclic modules\n\\begin{align}\\label{aux-equivariant-map}\n\\begin{split}\n&\\chi_\\tau^{\\rm eq}:HC^q_{\\cal K}( {\\cal H},V,{\\mathbb C}_\\delta)\\longrightarrow HC^q_{\\cal K}({\\cal A}, V)\\\\\n&\\chi^{\\rm eq}_\\tau(\\phi)(v\\otimes a_0\\ot\\cdots\\ot a_q)= \\tau\\left(\\phi(v)\n\\snb{0}(a_0)\\cdots \\phi(v)\\snb{q}(a_q)\\right).\n\\end{split}\n\\end{align}\nWe conclude this section by the identification of\n$C^q_{\\cal K}({\\cal K},V,{\\mathbb C}_\\delta)$ with $(V^\\ast\\otimes  {\\mathbb C}_\\delta\\otimes {\\cal H}^{\\otimes\\, q})^{\\mathfrak{g}_0}$,\nwhere $V^\\ast=\\mathop{\\rm Hom}\\nolimits_{\\mathbb C}(V, {\\mathbb C})$ and  $\\mathfrak{g}_0$ acts on $V^\\ast\\otimes\n{\\cal H}^{\\otimes\\, q}$ via\n\\begin{align}\n\\begin{split}\n&(\\phi\\otimes {\\bf 1}\\otimes  h^1\\ot\\cdots\\ot h^q)Z=- \\sum^q_{i=1}\n \\phi\\otimes {\\bf 1}\\otimes h^1\\ot\\cdots\\ot \\mathop{\\rm Ad}\\nolimits_Z(h^i)\\ot\\cdots\\ot h^q\\\\\n&+\\phi\\otimes \\delta(Z)\\otimes h^1\\ot\\cdots\\ot h^q + \\phi\\cdot Z\\otimes\n {\\bf 1}\\otimes  h^1\\ot\\cdots\\ot h^q.\n\\end{split}\n\\end{align}\nHere, the action of $\\mathfrak{g}_0$ on $V^\\ast$ is defined by $(\\phi\\cdot\nZ)(v)=-\\phi(v\\cdot Z)$.\n\n\\medskip\n\n\\noindent The aforementioned identification is defined  by the map\n\\begin{align}\\label{iso}\n\\begin{split}\n&{\\cal I}: (V^\\ast\\otimes {\\mathbb C}_\\delta\\otimes  {\\cal H}^{\\otimes\\, q})^{\\mathfrak{g}_0}\\rightarrow C_{\\cal K}^q( {\\cal H}, V,{\\mathbb C}_\\delta)\\\\\n&{\\cal I}(\\phi\\otimes {\\bf 1}\\otimes h^1\\ot\\cdots\\ot h^q)(v)= \\phi(v)\\otimes_{\\cal H} 1_ {\\cal H}\\otimes\nh^1\\ot\\cdots\\ot h^q.\n\\end{split}\n\\end{align}\n\n\\begin{proposition}\nThe map ${\\cal I}$ defined in \\eqref{iso} is an isomorphism of vector spaces.\n\\end{proposition}\n\n\\begin{proof}\nLet us first check that ${\\cal I}$ is well-defined. Indeed,\n\\begin{align*}\n&{\\cal I}(\\phi\\otimes {\\bf 1} \\otimes h^1\\ot\\cdots\\ot h^q)(v\\cdot Z)=\\phi(v\\cdot Z)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot h^q \\\\\n&= -(\\phi\\cdot Z)(v)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot h^q\\\\\n&=  -\\sum^q_{i=1} \\phi(v)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot \\mathop{\\rm Ad}\\nolimits_Z(h^i)\\ot\\cdots\\ot h^q\\\\\n&+\\delta(Z)\\phi(v)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot h^q\\\\\n&=  -\\sum^q_{i=1} \\phi(v)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot \\mathop{\\rm Ad}\\nolimits_Z(h^i)\\ot\\cdots\\ot h^q\\\\\n&+   \\phi(v)\\otimes_ {\\cal H} Z\\otimes h^1\\ot\\cdots\\ot h^q  +\\sum_{i=1}^q \\phi(v)\\otimes_ {\\cal H} 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot Zh^i\\ot\\cdots\\ot  h^q\\\\\n&= (\\phi(v)\\otimes 1_ {\\cal H}\\otimes h^1\\ot\\cdots\\ot h^q)\\cdot Z=  ({\\cal I}(\\phi\\otimes {\\bf 1} \\otimes\nh^1\\ot\\cdots\\ot h^q)(v))\\cdot Z.\n\\end{align*}\nNext,  we introduce an inverse map  for ${\\cal I}$. To this end we fix a\nbasis for $V$, say $\\{v_1, \\ldots, v_m\\}$, with a dual basis\n$\\{\\nu^1, \\ldots, \\nu^m\\}$ for $V^\\ast$, and we define\n\\begin{align}\n&{\\cal I}^{-1}: C_{\\cal K}^q( {\\cal H}, V,{\\mathbb C}_\\delta)\\rightarrow  (V^\\ast\\otimes{\\mathbb C}_\\delta\\otimes   {\\cal H}^{\\otimes\nq})^{\\mathfrak{g}_0}\\\\\\notag\n &{\\cal I}^{-1}(\\phi)=\\\\\\notag\n &\\sum_{i=1}^m \\nu^i \\otimes\n\\phi(v_i)\\snb{-1}\\delta(\\phi(v_i)\\snb{0}\\ps{1})\\otimes\nS(\\phi(v_i)\\snb{0}\\ps{2})\\cdot \\Big(\\phi(v_i)\\snb{1})\\ot\\cdots\\ot\n\\phi(v_i)\\snb{q}\\Big).\n\\end{align}\nIt is straightforward to check that ${\\cal I}^{-1}$ is independent of the\nchoice of bases and is inverse to ${\\cal I}$.\n\\end{proof}\n\n\n\n\\subsection{A SAYD-twisted cyclic  cocycle in codimension\n1}\\label{section-codim-1} In this subsection we  keep the setting of\nSubsection \\ref{subsection-H-n} for $n=1$.  Our aim is to introduce\nan equivariant cyclic 1-cocycle $\\varphi\\in C^1_{\\cal K}({\\cal A},V)$. In order to\napply \\eqref{aux-equivariant-map},   we shall consider the complex\n$C^1_{\\cal K}({\\cal H},V,{\\mathbb C}_\\delta)$, which in turn is identified with the\n$\\mathfrak{g}_0$-invariant subspace $(V^\\ast \\otimes {\\mathbb C}_\\delta \\otimes {\\cal H})^{\\mathfrak{g}_0}$ via\n\\eqref{iso}.\n\n\\medskip\n\n\\noindent Let $\\{R\\}$ be the  basis for $\\mathfrak{g}_0^\\ast$ as the dual basis of\n$\\{Y:=Y^1_1\\}$ for $\\mathfrak{g}_0$. Let also $\\{1, R\\}$ be the basis of\n$V$ and  $\\{1^\\ast,S\\}$ as  the  dual basis for $V^\\ast$.\n\n\\medskip\n\nWe define $ \\varphi_0,\\varphi_1:V \\otimes {\\cal A}^{\\otimes\\;2} \\to {\\mathbb C},$ by\n\\begin{equation}\n \\varphi_0 = \\chi_\\tau^{\\rm eq}(1^\\ast \\otimes {\\bf 1} \\otimes X), \\quad \\varphi_1 =\n\\chi_\\tau^{\\rm eq}(S \\otimes {\\bf 1} \\otimes \\delta_1),\n\\end{equation}\n more precisely\n\\begin{align}\\label{aux-pieces-of-cocycle}\n\\begin{split}\n&\\varphi_0((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1) = \\alpha\\tau(a_0X(a_1)), \\\\\n& \\varphi_1((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1) = \\beta\\tau(a_0\\delta_1(a_1)).\n\\end{split}\n\\end{align}\n\\begin{lemma}\nThe linear maps  $\\varphi_0,\\varphi_1:V \\otimes {\\cal A}^{\\otimes\\;2} \\to {\\mathbb C}$ defined in\n\\eqref{aux-pieces-of-cocycle} are ${\\cal K}$-equivariant.\n\\end{lemma}\n\n\\begin{proof}\nWe have\n\\begin{align}\n\\begin{split}\n& \\varphi_0((\\a1 + \\beta\\theta) \\otimes Y(a_0) \\otimes a_1) + \\varphi_0((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes Y(a_1)) = \\\\\n& \\alpha\\tau(Y(a_0)X(a_1)) + \\alpha\\tau(a_0X(Y(a_1))) = \\\\\n& \\alpha\\tau(Y(a_0)X(a_1)) + \\alpha\\tau(a_0Y(X(a_1))) - \\alpha\\tau(a_0X(a_1)) = \\\\\n& \\alpha\\tau(Y(a_0)X(a_1)) - \\alpha\\tau(Y(a_0)X(a_1)) = 0,\n\\end{split}\n\\end{align}\n the third equality follows from the integration by parts property \\cite{ConnMosc98}\n \\begin{equation}\\label{aux-property-S-tilde}\n\\tau(h(a)b) = \\tau(aS_\\delta(h)(b)), \\quad \\forall h\\in {\\cal H},\n\\forall a,b\\in {\\cal A}.\n\\end{equation}\nSimilarly we have\n\\begin{align}\n\\begin{split}\n& \\varphi_1((\\a1 + \\beta\\theta) \\otimes Y(a_0) \\otimes a_1) + \\varphi_1((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes Y(a_1)) \\\\\n& =\\alpha\\tau(Y(a_0)\\delta_1(a_1)) + \\alpha\\tau(a_0\\delta_1(Y(a_1)))  \\\\\n& =\\alpha\\tau(Y(a_0)\\delta_1(a_1)) + \\alpha\\tau(a_0Y(\\delta_1(a_1))) - \\alpha\\tau(a_0\\delta_1(a_1))  \\\\\n& =\\alpha\\tau(Y(a_0)\\delta_1(a_1)) - \\alpha\\tau(Y(a_0)\\delta_1(a_1)) = 0.\n\\end{split}\n\\end{align}\n\n\\end{proof}\n\n\\begin{lemma}\nThe linear map  $\\varphi_0 - \\varphi_1:V \\otimes {\\cal A}^{\\otimes\\;2} \\to {\\mathbb C}$ is\na Hochschild 1-cocycle.\n\\end{lemma}\n\\begin{proof}\nWe have\n\\begin{align}\n\\begin{split}\n& b(\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1 \\otimes a_2) = (\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_0a_1 \\otimes a_2) \\\\\n& - (\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1a_2) + (\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_2a_0 \\otimes a_1) \\\\\n& - (\\varphi_0 - \\varphi_1)(\\alpha\\theta \\otimes Y(a_2)a_0 \\otimes a_1) \\\\\n& = \\alpha\\tau(a_0a_1X(a_2)) - \\alpha\\tau(a_0X(a_1a_2)) + \\alpha\\tau(a_2a_0X(a_1)) + \\alpha\\tau(Y(a_2)a_0\\delta_1(a_1)) \\\\\n& - \\beta\\tau(a_0a_1\\delta_1(a_2)) + \\beta\\tau(a_0\\delta_1(a_1a_2)) -\n\\beta\\tau(a_2a_0\\delta_1(a_1))=0.\n\\end{split}\n\\end{align}\nHere we have used \\eqref{aux-D-d} and \\eqref{aux-D-X}.\n\\end{proof}\n\n\\begin{proposition}\nThe Hochschild cocycle  $\\varphi_0 - \\varphi_1:V \\otimes {\\cal A}^{\\otimes\\;2} \\to {\\mathbb C}$,\ndefined in \\eqref{aux-pieces-of-cocycle}, is cyclic.\n\\end{proposition}\n\\begin{proof}\nBy using the $\\delta$ invariancy of $\\tau$, \\eqref{aux-D-d} and \\eqref{aux-D-X}\nwe have\n\\begin{align*}\n& t(\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1) = (\\varphi_0 - \\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_1 \\otimes a_0) \\\\\n& - (\\varphi_0 - \\varphi_1)(\\alpha\\theta \\otimes Y(a_1) \\otimes a_0) \\\\\n& = \\alpha\\tau(a_1X(a_0)) + \\alpha\\tau(Y(a_1)\\delta_1(a_0)) - \\beta\\tau(a_1\\delta_1(a_0)) \\\\\n& = - \\alpha\\tau(a_0X(a_1)) + \\beta\\tau(a_0\\delta_1(a_1))=-(\\varphi_0 -\n\\varphi_1)((\\a1 + \\beta\\theta) \\otimes a_0 \\otimes a_1).\n\\end{align*}\n\\end{proof}\n\n\n\\subsection{A SAYD-twisted cyclic  cocycle in codimension\n2}\\label{section-codim-2} Similar to the previous subsection, we\nkeep the setting of Subsection \\ref{subsection-H-n} for $n=2$. Our\ngoal is to find a nontrivial  cyclic 2-cocycle  $\\phi\\in\nC^2_{\\cal K}({\\cal A}, V)$.\n\n\\medskip\n\n\\noindent Similar to the case $n=1$,  we apply\n\\eqref{aux-equivariant-map}\n by considering $C^2_{\\cal K}({\\cal H},V,{\\mathbb C}_\\delta)\n\\cong (V^\\ast \\otimes {\\mathbb C}_\\delta \\otimes {\\cal H}^{\\otimes\\,2})^{\\mathfrak{g}_0}$.\n\n\\medskip\n\n\\noindent Let $\\{R^i_j\\;\\mid 1\\le i,j\\le 2\\}$ be the dual basis of\n$g\\ell_2$ with the pairing $\\langle\nY^j_i,R^k_l\\rangle= \\delta^j_k\\delta^i_l$. We take\n\\begin{equation}\n\\left\\{1, R^i_j, R^k_lR^p_q\\,\\left|\\;\\;\\left(\\begin{array}{c}\n                                      k \\\\\n                                      l\n                                    \\end{array}\n\\right.\\right) \\leq \\left(\\begin{array}{c}\n                                      p \\\\\n                                      q\n                                    \\end{array}\n\\right)\\right\\},\n\\end{equation}\nas  a basis for  $V$ which is simplified by $ \\left\\{1, R^a,\nR^{ab}\\,|\\,a\\leq b\\right\\}$. The  dual basis for $V^\\ast$ is\nexpressed by $ \\left\\{1^\\ast, S_a, S_{ab}\\,|\\,a\\leq b\\right\\}.$\n\n\\medskip\n\n\\noindent We recall from \\cite{RangSutl-II} that the Koszul coaction \\eqref{aux-Koszul-ex} gives rise to a ${\\cal K}$-coaction by the formula\n\\begin{align}\\label{aux-Koszul-K-coaction}\n\\begin{split}\n& \\nabla_{K}: V \\to {\\cal K} \\otimes V,\\\\\n& \\nabla_{K}(1)=1 \\otimes 1 + Y_a \\otimes R^a + \\frac{1}{2!}Y_aY_b \\otimes R^{ab},\\\\\n& \\nabla_{K}(R^a)=1 \\otimes R^a + Y_b \\otimes R^{ab},\\\\\n& \\nabla_{K}(R^{ab})=1 \\otimes R^{ab}.\n\\end{split}\n\\end{align}\n\n\n\\medskip\n\n\\noindent We decompose $V=V_0\\oplus V_1\\oplus V_2 $, where $V_0={\\mathbb C}\\langle\n1\\rangle$, $V_1={\\mathbb C}\\langle R^a \\rangle$, and $V_2={\\mathbb C}\\langle R^{ab}\n\\rangle$. Using this decomposition, any $\\psi\\in \\mathop{\\rm Hom}\\nolimits(V\\otimes {\\cal A}^{q+1},\n{\\mathbb C})$ is decomposed uniquely  as $\\psi=\\psi_0+\\psi_1+\\psi_2$ by\n$\\psi_i= \\psi|_{V_i\\otimes {\\cal A}^{\\otimes q+1}}$.\n\n\\medskip\n\n \\noindent We now consider the linear map  $\\psi: V\\otimes {\\cal A}^{\\otimes 3}\\rightarrow {\\mathbb C}$  with\ncomponents\n\\begin{align}   \\label{aux-vp-0}\n& \\psi_0 :=\n\\chi_\\tau^{\\rm eq}\\left(\\gamma_1 1^\\ast \\otimes {\\bf 1} \\otimes X_{\\sigma(1)} \\otimes X_{\\sigma(2)} +\n\\gamma_2 1^\\ast \\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)}Y_a\\right. \\\\\\notag &\n+\\left.\\gamma_3 1^\\ast \\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b \\otimes Y_a +\\gamma_4 1^\\ast\n\\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes Y_a\\right).\\\\ \\label{aux-vp-2}\n& \\psi_2 := \\chi_\\tau^{\\rm eq}\\left(\\alpha_1S_{ab} \\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}\n\\otimes {\\delta^b}_{\\sigma(2)}\\right. + \\\\\\notag & \\hspace{4cm} \\left.\\alpha_2S_{ab}\n\\otimes {\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}\\right),\n\\end{align}\n\\begin{align}\\notag\n& \\psi_1:=\n\\chi_\\tau^{\\rm eq}\\left(\\beta_1S_a \\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)}\n+ \\beta_2S_a\\otimes {\\bf 1}\\otimes X_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}\\right. \\\\\\notag & +\n\\beta_3S_a \\otimes  {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_b + \\beta_4S_a \\otimes\n{\\bf 1}\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}\\\\\\label{aux-vp-1} & + \\beta_5 S_a\n\\otimes  {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes {\\delta^b}_{\\sigma(2)} + \\beta_6S_a \\otimes {\\bf 1} \\otimes\n{\\delta^b}_{\\sigma(1)}Y_b \\otimes {\\delta^a}_{\\sigma(2)} \\\\\\notag &  + \\beta_7 S_a\\otimes {\\bf 1} \\otimes\n{\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_b + \\beta_8S_a \\otimes {\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)}\n\\otimes {\\delta^a}_{\\sigma(2)}Y_b \\\\\\notag & + \\left.\\beta_9S_a \\otimes {\\bf 1}\\otimes\n\\Delta({\\delta^a}_{\\sigma(1)\\sigma(2)})\\right).\n\\end{align}\nOur aim is to  determine the coefficients $\\alpha_i,\\beta_j,\\gamma_k$, such\nthat  $\\psi$ is a cyclic 2-cocycle. To do so we prove a series of\ntechnical lemmas.\n\n\\begin{lemma}\\label{lemma-psi-eq}\nThe cochain $\\psi$ is ${\\cal K}$-equivariant.\n\n\\end{lemma}\n\n\\begin{proof}\nWe first simply observe that\n\\begin{align}\\label{aux-gl_2-invariance-1}\n &\\delta^j_{\\sigma(1)}B_i \\otimes B_{\\sigma(2)}\n  + \\delta^j_{\\sigma(2)}B_{\\sigma(1)} \\otimes B_i =\n   \\delta_i^j \\left(B_{\\sigma(1)} \\otimes\nB_{\\sigma(2)}\\right).\n\\end{align}\n\n\\noindent Using  \\eqref{aux-relations-CM}, in view of the action of\n$g\\ell_2$ on the Hopf algebra ${\\cal H}$,\n  the equivariancy condition follows from\n\\begin{align*}\n & \\mathop{\\rm ad}\\nolimits(Y_i^j)\\left(\\delta^p_{qr} \\otimes\nY^q_p\\right) = \\delta^j_r\\left(\\delta^p_{qi} \\otimes Y^q_p\\right), \\\\\\notag &\n\\mathop{\\rm ad}\\nolimits(Y_i^j)\\left(\\delta^k_{tr} \\otimes \\delta^t_{ln}\\right) =\n\\\\ & \\delta^j_l\\left(\\delta^k_{tr} \\otimes\n\\delta^t_{in}\\right) - \\delta^k_i\\left(\\delta^j_{tr} \\otimes \\delta^t_{ln}\\right) +\n\\delta^j_r\\left(\\delta^k_{ti} \\otimes \\delta^t_{ln}\\right) + \\delta^j_n\\left(\\delta^k_{tr}\n\\otimes \\delta^t_{li}\\right).\n\\end{align*}\n\n\\end{proof}\n\n\\noindent We first observe that $\\psi \\in C^2_{\\cal K}({\\cal A},V)$ is a\n Hochschild cocycle on $V_2 \\otimes {\\cal A}^{\\otimes\\,n}$.\n\n\\begin{lemma}\\label{lemma-bvp2=0}\nFor any $\\alpha_i,\\beta_j,\\gamma_k$, $(b\\psi)_2=0$.\n\\end{lemma}\n\n\\begin{proof}\nThe result follows directly from the application of the Hochschild\ncoboundary map and the fact that ${\\delta^a}_{k}$ are derivations of\n${\\cal A}$.\n\\end{proof}\n\\noindent On the next move, we determine $\\alpha_i$, $1 \\leq i \\leq 2$, in such a way that $\\psi \\in C^2_{\\cal K}({\\cal A},V)$ is a cyclic cocycle on $V_2 \\otimes {\\cal A}^{\\otimes\\,n}$.\n\n\\begin{lemma}\\label{lemma-tauvp2=vp2}\nFor $\\alpha_1 = \\alpha_2$, we have $(\\tau\\psi)_2 = \\psi_2$.\n\\end{lemma}\n\n\\begin{proof}\nBy definition of the cyclic operator, we have\n\\begin{equation}\\label{aux-vp2-cyclic}\n\\tau\\psi(R^{ab} \\otimes a_0\\otimes a_1 \\otimes a_2) = \\psi_2(R^{ab} \\otimes a_2 \\otimes a_0\\otimes a_1).\n\\end{equation}\nHence, by the integration by parts property \\eqref{aux-property-S-tilde},\n\\begin{align}\n\\begin{split}\n& t\\psi(R^{ab} \\otimes a_0 \\otimes a_1\\otimes a_2) =\\\\\n& \\alpha_1 \\tau\\left(a_0(-{\\delta^a}_{\\sigma(1)}){\\delta^b}_{\\sigma(2)}(a_1)a_2\\right) + \\alpha_1 \\tau\\left( a_0{\\delta^b}_{\\sigma(2)}(a_1)(-{\\delta^a}_{\\sigma(1)})(a_2)\\right) \\\\\n& +  \\alpha_2\\tau\\left( a_0 (-{\\delta^b}_{\\sigma(1)}){\\delta^a}_{\\sigma(2)}(a_1) a_2\\right) + \\alpha_2 \\tau\\left(a_0 {\\delta^a}_{\\sigma(2)}(a_1) (-{\\delta^b}_{\\sigma(1)})(a_2)\\right) \\\\\n& = \\alpha_1 \\tau\\left(a_0{\\delta^b}_{\\sigma(1)}(a_1)  {\\delta^a}_{\\sigma(2)}(a_2)\\right) + \\alpha_2\\tau\\left(a_0  {\\delta^a}_{\\sigma(1)}(a_1) {\\delta^b}_{\\sigma(2)}(a_2)\\right).\n\\end{split}\n\\end{align}\nTherefore, $(\\tau\\psi)_2 = \\psi_2$ if and only if $\\alpha_1 = \\alpha_2$.\n\\end{proof}\n\\noindent  As a result  we set\n \\begin{equation}\\label{aux-alpha}\n \\alpha_1 = \\alpha_2 = r.\n\\end{equation}\n\\medskip\n\n\\noindent On the next step, we find a constraint on $\\beta_j$'s\n such that $\\psi$ is a Hochschild cocycle over $V_1 \\otimes {\\cal A}^{\\otimes\\,q+1}$.\n\n\\begin{lemma}\\label{lemma-bvp1=0}\nWe have $(b\\psi)_1=0$ if and only if $\\{\\beta_j\\,|\\,1 \\leq j \\leq 9\\}$ satisfy the system\n\\begin{align}\\label{system-vp1-Hochschild}\n\\begin{split}\n \\beta_1-\\beta_3+\\beta_7+r &= 0 \\\\\n \\beta_3+\\beta_8+r &= 0 \\\\\n -\\beta_2-\\beta_6+\\beta_8 &= 0 \\\\\n \\beta_4-\\beta_5 &= 0 \\\\\n -\\beta_4-\\beta_6& = 0 \\\\\n -\\beta_5+\\beta_7 &= 0.\n\\end{split}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nLet us remind the reader that we have to use the Koszul coaction  \\eqref{aux-Koszul-K-coaction} in the last coface\noperator.\n\\begin{align}\\label{aux-vp1-Hochschild}\n\\begin{split}\n& b\\psi(R^a \\otimes a_0\\otimes a_1 \\otimes a_2 \\otimes a_3) = \\\\\n& \\psi_1(R^a \\otimes a_0a_1 \\otimes a_2 \\otimes a_3) - \\psi_1(R^a \\otimes a_0\\otimes a_1a_2 \\otimes a_3) \\\\\n& + \\psi_1(R^a \\otimes a_0\\otimes a_1 \\otimes a_2a_3) - \\psi_1(R^a \\otimes a_3a_0\\otimes a_1 \\otimes a_2) \\\\\n& + \\psi_2(R^{ab} \\otimes Y_b(a_3)a_0\\otimes a_1 \\otimes a_2).\n\\end{split}\n\\end{align}\nTherefore, as a result of the tracial property \\cite[Thm. 6]{ConnMosc}\nand the faithfullness \\cite[(3.12)]{ConnMosc} of the trace, we have $(b\\psi)_1 = 0$ if and only if\n\\begin{align}\n\\begin{split}\n& \\beta_1 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_b - \\beta_2 1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^a}_{\\sigma(2)} \\\\\n& -\\beta_3 \\left(1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_b + 1\\otimes {\\delta^b}_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_b\\right) \\\\\n& +\\beta_4 \\left(1\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} + 1\\otimes Y_b \\otimes {\\delta^b}_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)}\\right) \\\\\n& -\\beta_5 \\left(1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^b}_{\\sigma(2)} + 1\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}\\right) \\\\\n& -\\beta_6\\left(1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^a}_{\\sigma(2)} + 1\\otimes Y_b \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}\\right) \\\\\n& +\\beta_7\\left(1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_b + 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^b}_{\\sigma(2)}\\right) \\\\\n& +\\beta_8\\left(1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_b + 1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^a}_{\\sigma(2)}\\right) \\\\\n& + r 1 \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_b + r 1 \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_b = 0.\n\\end{split}\n\\end{align}\nIn other words,\n\\begin{align*}\n& \\left(\\beta_1-\\beta_3+\\beta_7+k\\right) 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_b + (\\beta_3+\\beta_8+k) 1 \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_b \\\\\n& + (-\\beta_2-\\beta_6+\\beta_8) 1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^a}_{\\sigma(2)} + (\\beta_4-\\beta_5) 1\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}\\\\\n& + (-\\beta_4-\\beta_6) 1\\otimes Y_b \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} + (-\\beta_5+\\beta_7) 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_b \\otimes {\\delta^b}_{\\sigma(2)} = 0.\n\\end{align*}\nAccordingly we get the system \\eqref{system-vp1-Hochschild}.\n\\end{proof}\n \\noindent As a result we set $\\beta_j$, $1\\le j\\le 9$  to satisfy  \\eqref{system-vp1-Hochschild}.\n\n \\medskip\n\n\\noindent On the next step we determine $\\beta_j$'s in such a way that $\\psi\n\\in C^2_{\\cal K}({\\cal A},V)$ is a cyclic  cocycle over $(V_1\\oplus V_2) \\otimes\n{\\cal A}^{\\otimes\\,n}$.\n\n\\begin{lemma}\\label{lemma-tauvp1=vp1}\nWe have $(\\tau\\psi)_1=\\psi_1$ if and only if $\\{\\beta_j\\,|\\,1 \\leq j \\leq 9\\}$, satisfy\n\\begin{align}\\label{aux-beta}\n\\begin{split}\n& \\beta_1=\\beta_2=-r, \\quad \\beta_3=\\beta_4=\\beta_5=-\\beta_6=\\beta_7=s, \\\\\n& \\beta_8=-r-s, \\quad \\beta_9 = \\frac{1}{2}r+s.\n\\end{split}\n\\end{align}\n\\end{lemma}\n\n\\begin{proof}\nBy the Koszul coaction, we have\n\\begin{equation*}\nt\\psi(R^a \\otimes a_0\\otimes a_1 \\otimes a_2) = \\psi_1(R^a \\otimes a_2 \\otimes a_0\\otimes a_1) - \\psi_2(R^{ab} \\otimes Y_b(a_2) \\otimes a_0\\otimes a_1).\n\\end{equation*}\nAccordingly,\n\\begin{align}\n\\begin{split}\n& t\\psi(R^a \\otimes a_0 \\otimes a_1 \\otimes a_2) = \\\\\n& \\beta_1\\tau\\left( {\\delta^a}_{\\sigma(1)}(a_0) X_{\\sigma(2)}(a_1)a_2\\right) + \\beta_2\\tau\\left( X_{\\sigma(1)}(a_0) {\\delta^a}_{\\sigma(2)}(a_1)a_2\\right) \\\\\n& + \\beta_3\\tau\\left( {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}(a_0) Y_b(a_1)a_2\\right) + \\beta_4\\tau\\left( Y_b(a_0) {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}(a_1) a_2\\right) \\\\\n&  + \\beta_5\\tau\\left( {\\delta^a}_{\\sigma(1)}Y_b(a_0) {\\delta^b}_{\\sigma(2)}(a_1)a_2\\right) + \\beta_6\\tau\\left( {\\delta^b}_{\\sigma(1)}Y_b(a_0) {\\delta^a}_{\\sigma(2)}(a_1)a_2\\right) \\\\\n& + \\beta_7\\tau\\left( {\\delta^a}_{\\sigma(1)}(a_0) {\\delta^b}_{\\sigma(2)}Y_b(a_1)a_2\\right) + \\beta_8 \\tau\\left(  {\\delta^b}_{\\sigma(1)}(a_0) {\\delta^a}_{\\sigma(2)}Y_b(a_1)a_2\\right) \\\\\n&+ \\beta_9\\tau\\left( \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right)(a_0 \\otimes a_1)a_2\\right) - r\\tau\\left( {\\delta^a}_{\\sigma(1)}(a_0) {\\delta^b}_{\\sigma(2)}(a_1) Y_b(a_2)\\right) \\\\\n&   - r\\tau\\left( {\\delta^b}_{\\sigma(1)}(a_0) {\\delta^a}_{\\sigma(2)}(a_1) Y_b(a_2)\\right).\n\\end{split}\n\\end{align}\nHence $(t\\psi)_1 = \\psi_1$ if and only if\n\\begin{align*}\n& \\beta_1\\left(- 1 \\otimes {\\delta^a}_{\\sigma(1)}X_{\\sigma(2)} \\otimes 1 - 1 \\otimes X_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)}\\right) + \\\\\n& \\beta_2\\big(- 1 \\otimes X_{\\sigma(1)} {\\delta^a}_{\\sigma(2)} \\otimes 1 - 1 \\otimes {\\delta^a}_{\\sigma(2)} \\otimes X_{\\sigma(1)} + 1 \\otimes {\\delta^b}_{\\sigma(1)} {\\delta^a}_{\\sigma(2)} Y_b \\otimes 1 \\\\\n& - 2 \\cdot 1 \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes 1 + 1 \\otimes  {\\delta^a}_{\\sigma(2)} \\otimes {\\delta^b}_{\\sigma(1)}Y_b + 1 \\otimes {\\delta^a}_{\\sigma(2)}Y_b \\otimes {\\delta^b}_{\\sigma(1)} \\\\\n& - 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(1)}\\right) \\big) + \\\\\n& \\beta_3\\big(1 \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} Y_b \\otimes 1 + 1 \\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\\\\n& + 1 \\otimes {\\delta^a}_{\\sigma(1)} Y_b \\otimes {\\delta^b}_{\\sigma(2)} + 1 \\otimes {\\delta^b}_{\\sigma(2)} Y_b \\otimes {\\delta^a}_{\\sigma(1)} \\big) + \\\\\n& \\beta_4\\big(-1 \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b \\otimes 1 + 2 \\cdot 1 \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes 1 -1 \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}\\otimes Y_b\\big) + \\\\\n& \\beta_5\\big(1 \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b \\otimes 1 - 2 \\cdot 1 \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes 1 + 1 \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}\\otimes Y_b+ \\\\\n& + 1 \\otimes {\\delta^b}_{\\sigma(2)}Y_b \\otimes {\\delta^a}_{\\sigma(1)} + 1 \\otimes {\\delta^b}_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)}Y_b + 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\big) + \\\\\n& \\beta_6\\big(1 \\otimes {\\delta^b}_{\\sigma(1)}{\\delta^a}_{\\sigma(2)}Y_b \\otimes 1 - 2 \\cdot 1 \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes 1+ 1 \\otimes {\\delta^b}_{\\sigma(1)}{\\delta^a}_{\\sigma(2)}\\otimes Y_b\\\\\n& + 1 \\otimes {\\delta^a}_{\\sigma(2)}Y_b \\otimes {\\delta^b}_{\\sigma(1)} - 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) + 1 \\otimes {\\delta^a}_{\\sigma(2)} \\otimes {\\delta^b}_{\\sigma(1)}Y_b\\big) +\\\\\n& \\beta_7 \\left(- 1 \\otimes {\\delta^a}_{\\sigma(1)} {\\delta^b}_{\\sigma(2)}Y_b \\otimes 1 - 1 \\otimes  {\\delta^b}_{\\sigma(2)}Y_b \\otimes {\\delta^a}_{\\sigma(1)}\\right) +\\\\\n& \\beta_8\\left(- 1 \\otimes  {\\delta^b}_{\\sigma(1)} {\\delta^a}_{\\sigma(2)}Y_b \\otimes 1 -  1 \\otimes {\\delta^a}_{\\sigma(2)}Y_b \\otimes {\\delta^b}_{\\sigma(1)}\\right) + \\\\\n& \\beta_9\\left(-2 \\cdot 1 \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes 1 - 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right)\\right) \\\\\n&  + r1\\otimes  {\\delta^b}_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)} Y_b + r 1 \\otimes {\\delta^a}_{\\sigma(2)} \\otimes {\\delta^b}_{\\sigma(1)} Y_b \\\\\n& = \\beta_11\\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)} +\\beta_2 1\\otimes X_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} + \\beta_31\\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_b \\\\\n& \\beta_41\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}  + \\beta_51\\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes {\\delta^b}_{\\sigma(2)}+\\beta_6 1\\otimes {\\delta^b}_{\\sigma(1)}Y_b \\otimes {\\delta^a}_{\\sigma(2)} \\\\\n&  + \\beta_71\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_b + \\beta_8 1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}Y_b + \\beta_9 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right).\n\\end{align*}\nCollecting the terms, we obtain the equations\n\\begin{align}\\label{system-vp1-cyclic}\n\\begin{split}\n \\beta_1-\\beta_2 & = 0 \\\\\n \\beta_2-\\beta_3+\\beta_4-\\beta_5+\\beta_6+\\beta_7-\\beta_8 &= 0 \\\\\n \\beta_1-2\\beta_2+2\\beta_4-2\\beta_5-2\\beta_6-2\\beta_9 &= 0 \\\\\n \\beta_2+\\beta_6+\\beta_7+k &=0 \\\\\n \\beta_2-\\beta_3+\\beta_5+\\beta_6-\\beta_8&=0 \\\\\n \\beta_5+\\beta_8+k &=0 \\\\\n \\beta_3+\\beta_5+\\beta_6-\\beta_7&=0\\\\\n \\beta_3-\\beta_4&=0\\\\\n -\\beta_2+\\beta_5-\\beta_6-2\\beta_9&=0.\n\\end{split}\n\\end{align}\nSolving the systems \\eqref{system-vp1-Hochschild} and \\eqref{system-vp1-cyclic}\nwe obtain \\eqref{aux-beta}.\n\\end{proof}\n\\noindent As a result of Lemma \\ref{system-vp1-Hochschild} we set $\\beta_j$,\n$1\\le j\\le 9$ to satisfy \\eqref{aux-beta}.\n\n\\medskip\n\n\\noindent Finally we determine $\\gamma_k$, $1\\leq k \\leq 4$ such that $\\psi \\in\nC^2_{\\cal K}({\\cal A},V)$ is a Hochschild cocycle.\n\n\\begin{lemma}\\label{lemma-bvp0=0}\nWe have $(b\\psi)_0=0$ if and only if $\\{\\gamma_k\\,|\\,1 \\leq k \\leq 4\\}$ satisfy\n\\begin{equation}\\label{aux-gamma}\n\\gamma_1 = \\gamma_2 = r, \\quad \\gamma_3 = \\gamma_4 = s.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nUsing the  Koszul coaction \\eqref{aux-Koszul-K-coaction}, we have\n\\begin{align*}\\label{aux-vp0-Hochschild}\n& b\\psi(1 \\otimes a_0\\otimes a_1 \\otimes a_2 \\otimes a_3) = \\\\\n& \\psi_0(1 \\otimes a_0a_1 \\otimes a_2 \\otimes a_3) - \\psi_0(1 \\otimes a_0\\otimes a_1a_2 \\otimes a_3) \\\\\n& + \\psi_0(1 \\otimes a_0\\otimes a_1 \\otimes a_2a_3) - \\psi_0(1 \\otimes a_3a_0\\otimes a_1 \\otimes a_2) \\\\\n& + \\psi_1(R^a \\otimes Y_a(a_3)a_0\\otimes a_1 \\otimes a_2) - \\frac{1}{2!}\\psi_2(R^{ab} \\otimes Y_bY_a(a_3)a_0\\otimes a_1 \\otimes a_2).\n\\end{align*}\nAs a result,  $(b\\psi)_0 = 0$ if and only if\n\\begin{align*}\n& \\gamma_1 \\Big(- 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_a \\otimes X_{\\sigma(2)} + 1\\otimes X_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_a\\Big) + \\\\\n& \\gamma_2 \\Big(1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)} \\otimes Y_a + 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_a \\otimes  X_{\\sigma(2)}  \\\\\n& + 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_a \\otimes Y_b + 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_bY_a\\Big) + \\\\\n& \\gamma_3 \\Big(- 1\\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_b \\otimes Y_a  - 1\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_a\n\\end{align*}\n\\begin{align}\n\\begin{split}\n& - 1\\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_a - 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_b \\otimes Y_a \\\\\n& - 1\\otimes {\\delta^b}_{\\sigma(2)}Y_b \\otimes {\\delta^a}_{\\sigma(1)} \\otimes Y_a - 1\\otimes {\\delta^b}_{\\sigma(2)} \\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes Y_a \\Big) +\\\\\n& -\\gamma_4 1 \\otimes \\Delta({\\delta^a}_{\\sigma(1)\\sigma(2)}) \\otimes Y_a -r 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)} \\otimes Y_a\\\\\n&  + -r 1\\otimes X_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_a +s 1\\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_b \\otimes Y_a \\\\\n& + s 1\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_a + s 1\\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_a\\\\\n& -s 1\\otimes {\\delta^b}_{\\sigma(1)}Y_b \\otimes {\\delta^a}_{\\sigma(2)} \\otimes Y_a + s 1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_b \\otimes Y_a \\\\\n& + (-r-s) 1\\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}Y_b \\otimes Y_a + (\\frac{r}{2}+s) 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\otimes Y_a \\\\\n& -r  1\\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} \\otimes Y_bY_a - \\frac{r}{2} 1 \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\otimes Y_a = 0.\n\\end{split}\n\\end{align}\nHence we obtain \\eqref{aux-gamma}.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-psi}\nThe cochain  $\\psi: V \\otimes {\\cal A}^{\\otimes\\,3} \\to {\\mathbb C}$ is a\n cyclic 2-cocycle if and only if $\\{\\alpha_i\\,|\\,1 \\leq i \\leq 2\\}$, satisfy \\eqref{aux-alpha}, $\\{\\beta_j\\,|\\,\n1 \\leq j \\leq 9\\}$, fulfill  \\eqref{aux-beta},\n   and $\\{\\gamma_k\\,|\\,1 \\leq k \\leq 4\\}$ satisfy \\eqref{aux-gamma}. The\n   resulting  cocycle is then a SAYD-twisted cyclic cocycle.\n\\end{proposition}\n\n\\begin{proof}\nWe note that  $\\psi$ is a Hochschild cocycle, {\\it i.e.\\\/}\\   $b\\psi =\n(b\\psi)_0+(b\\psi)_1+(b\\psi)_2 =0$,  if and only if\n  $(b\\psi)_t =0$, $t=0,1,2$. We see that $(b\\psi)_2=0$ via  Lemma\n  \\ref{lemma-bvp2=0},  $(b\\psi)_1=0$ via  Lemma\n  \\ref{lemma-bvp1=0}, $(b\\psi)_0=0$ via  Lemma\n  \\ref{lemma-bvp0=0}.\n\n  \\medskip\n\n  \\noindent On the other hand $\\psi$ is cyclic, {\\it i.e.\\\/}\\   $\\tau\\psi =\\psi$,  if and only if\n   $(\\tau\\psi)_t=\\psi_t$, $t = 0,1,2$. Indeed, for $t=1$ Lemma \\ref{lemma-tauvp1=vp1},\nfor $t=2$ Lemma \\ref{lemma-tauvp2=vp2} yields the claims. As for $t=0$ we have\n\\begin{align*}\n& t\\psi(1 \\otimes a_0\\otimes a_1 \\otimes a_2) = \\psi_0(1 \\otimes a_2 \\otimes a_0\\otimes a_1) - \\psi_1(R^a \\otimes Y_a(a_2) \\otimes a_0\\otimes a_1) \\\\\\notag\n& +\\frac{1}{2!}\\psi_2(R^{ab} \\otimes Y_bY_a(a_2) \\otimes a_0\\otimes a_1).\n\\end{align*}\nAccordingly,\n\\begin{align*}\n& t\\psi(1 \\otimes a_0\\otimes a_1 \\otimes a_2) =\n\\eqref{aux-box-1}+\\eqref{aux-box-2}+\n\\eqref{aux-box-3}+\\eqref{aux-box-4}.\n\\end{align*}\n\\FBOX{12}{\n\\begin{align}\\label{aux-box-1}\n\\begin{split}\n &r\\tau\\Big(X_{\\sigma(1)}(a_0)\nX_{\\sigma(2)}(a_1)a_2\\Big) +\n r\\tau\\Big({\\delta^a}_{\\sigma(1)} (a_0) X_{\\sigma(2)}Y_a (a_1)a_2 \\Big) \\\\\n& + s\\tau\\Big({\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_b(a_0) Y_a(a_1)a_2\\Big) +\n s\\tau\\Big({\\delta^a}_{\\sigma(1)\\sigma(2)} (a_0)Y_a (a_1)a_2\\Big)\n\\end{split}\n \\end{align}\n }\n\n\\FBOX{14}{\n \\begin{align}\\label{aux-box-2}\n \\begin{split}\n&  r\\tau\\Big({\\delta^a}_{\\sigma(1)}(a_0) X_{\\sigma(2)}(a_1) Y_a(a_2)\\Big) + r\\tau\\Big( X_{\\sigma(1)} (a_0) {\\delta^a}_{\\sigma(2)}(a_1) Y_a(a_2)\\Big)\\\\\n& - s\\tau\\Big({\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} (a_0) Y_b (a_1) Y_a(a_2)\\Big) -s\\tau\\Big( Y_b (a_0) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} (a_1) Y_a(a_2)\\Big) \\\\\n&  +(r+s) \\tau\\Big(  \\delta^b_{\\sigma(1)}(a_0) {\\delta^a}_{\\sigma(2)}Y_b(a_1) Y_a(a_2) \\Big)-s\\tau\\Big( {\\delta^a}_{\\sigma(1)}Y_b(a_0) \\delta^b_{\\sigma(2)} (a_1) Y_a (a_2)\\Big)\\\\\n& +s\\tau\\Big( \\delta^b_{\\sigma(1)}Y_b (a_0) {\\delta^a}_{\\sigma(2)} (a_1) Y_a (a_2))-s\\tau\\Big( {\\delta^a}_{\\sigma(1)} (a_0) \\delta^b_{\\sigma(2)}Y_b(a_1) Y_a(a_2)\\Big) \\\\\n\\end{split}\n\\end{align}}\n\n\\FBOX{10}{\n \\begin{equation}\\label{aux-box-3}\n(- \\frac{r}{2}-s)\\tau\\Big(  \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right)(a_0\\otimes a_1) Y_a(a_2)\\Big) \\\\\n\\end{equation}}\n\n\\FBOX{14}{\n \\begin{equation}\\label{aux-box-4}\n \\frac{r}{2!} \\tau\\Big( {\\delta^a}_{\\sigma(1)}(a_0) \\delta^b_{\\sigma(2)}(a_1)\nY_bY_a(a_2) \\Big)+ \\frac{r}{2!} \\tau\\Big( \\delta^b_{\\sigma(1)} (a_0)\n{\\delta^a}_{\\sigma(2)}(a_1) Y_bY_a(a_2)\\Big)\n\\end{equation}}\nWe shall put the above expressions into the\n standard form $\\tau(a_0h^1(a_1)h^2(a_2))$.  To this end we use\n the integration by parts property \\eqref{aux-property-S-tilde}. On the computation below, we will employ the  actions\n\\begin{equation}\n(f \\otimes g) \\triangleleft Y := f \\otimes g \\cdot Y,\\qquad (f \\otimes g) \\blacktriangleleft Y := f  \\cdot Y \\otimes g.\n\\end{equation}\nof $\\mathfrak{g}_0$ on ${\\cal H}_n^{\\otimes\\,2}$. We first deal with \\eqref{aux-box-3} by\n\\begin{align}\\label{aux-sum-1}\n\\begin{split}\n& \\tau\\Big( \\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right)(a_0 \\otimes a_1) Y_a(a_2)\\Big) =\\\\\n& -2\\tau\\Big( a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)}(a_1) Y_a(a_2)\\Big) -\\tau\\Big( a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big).\n\\end{split}\n\\end{align}\nAs for \\eqref{aux-box-4} we compute\n\\begin{align}\\label{aux-sum-2}\n\\begin{split}\n& \\frac{1}{2!} \\tau\\Big( {\\delta^a}_{\\sigma(1)} (a_0) \\delta^b_{\\sigma(2)} (a_1) Y_bY_a(a_2)\\Big) + \\frac{1}{2!} \\tau\\Big( \\delta^b_{\\sigma(1)} (a_0) {\\delta^a}_{\\sigma(2)} (a_1) Y_bY_a(a_2)\\Big) = \\\\\n& - \\frac{1}{2} \\tau\\Big( a_0  \\delta^b_{\\sigma(2)}(a_1) {\\delta^a}_{\\sigma(1)}Y_bY_a(a_2) \\Big)- \\frac{1}{2} \\tau\\Big( a_0  {\\delta^a}_{\\sigma(2)}(a_1) \\delta^b_{\\sigma(1)}Y_bY_a(a_2) \\Big)= \\\\\n& \\frac{1}{2} \\tau\\Big( a_0  {\\delta^a}_{\\sigma(1)}(a_1) \\delta^b_{\\sigma(2)}Y_bY_a (a_2)\\Big)+ \\frac{1}{2} \\tau\\Big( a_0  {\\delta^a}_{\\sigma(1)} (a_1) \\delta^b_{\\sigma(2)}Y_aY_b(a_2) \\Big)= \\\\\n& \\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)} (a_1) \\delta^b_{\\sigma(2)}Y_bY_a(a_2)\\Big) + \\frac{1}{2} \\tau\\Big( a_0 {\\delta^a}_{\\sigma(1)} (a_1) \\delta^b_{\\sigma(2)}[Y_a,Y_b](a_2) \\Big)= \\\\\n& \\tau\\Big(a_0  {\\delta^a}_{\\sigma(1)}(a_1) \\delta^b_{\\sigma(2)}Y_bY_a (a_2)\\Big)+\n \\frac{1}{2} \\tau\\Big( a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big).\n\\end{split}\n\\end{align}\nOn the third step we compute \\eqref{aux-box-1} term by term.\n\\begin{align*}\n& \\tau\\Big(X_{\\sigma(1)} (a_0) X_{\\sigma(2)}(a_1) a_2\\Big) =\\\\\n& -\\tau\\Big(a_0 X_{\\sigma(2)} (a_1) X_{\\sigma(1)}(a_2)\\Big) +\n\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}Y_aX_{\\sigma(2)}(a_1) a_2 \\Big) \\\\\n&  + \\tau\\Big(a_0 X_{\\sigma(2)}(a_1) {\\delta^a}_{\\sigma(1)}Y_a(a_2)\\Big)\n + \\tau\\Big(a_0 Y_aX_{\\sigma(2)}(a_1) {\\delta^a}_{\\sigma(1)}(a_2)\\Big).\n\\end{align*}\n\\begin{align*}\n&\\tau\\Big( {\\delta^a}_{\\sigma(1)} (a_0) X_{\\sigma(2)}Y_a (a_1) a_2\\Big) =\\\\\n& -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)} X_{\\sigma(2)}Y_a (a_1)a_2 \\Big)\n -\\tau\\Big(a_0  X_{\\sigma(2)}Y_a (a_1) {\\delta^a}_{\\sigma(1)}(a_2) \\Big).\n\\end{align*}\n\\begin{align*}\n& \\tau\\Big({\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_b (a_0) Y_a (a_1) a_2\\Big) =\\\\\n& \\tau\\Big(Y_b (a_0) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_a (a_1)a_2\\Big) +\\tau\\Big( Y_b (a_0) Y_a (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}(a_2) \\Big) \\\\\n&  +\\tau\\Big( Y_b (a_0) {\\delta^a}_{\\sigma(1)}Y_a (a_1) \\delta^b_{\\sigma(2)} (a_2)\\Big)+ \\tau\\Big(Y_b (a_0) \\delta^b_{\\sigma(2)}Y_a (a_1) {\\delta^a}_{\\sigma(1)}(a_2) \\Big) =\\\\\n&  -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_bY_a (a_1) a_2\\Big) - 2\\tau\\Big( a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)} Y_a (a_1) a_2 \\Big)\\\\\n&  -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_a (a_1) Y_b(a_2)\\Big) -\\tau\\Big(a_0 Y_bY_a (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}(a_2) \\Big)\\\\\n&  -\\tau\\Big(a_0 Y_a (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_b(a_2)\\Big) -2 \\tau\\Big( a_0 Y_a (a_1) {\\delta^a}_{\\sigma(1)\\sigma(2)}(a_2) \\Big)\\\\\n&  -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}Y_bY_a (a_1) \\delta^b_{\\sigma(2)}(a_2)\\Big) -\\tau\\Big( a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\blacktriangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big)\\\\\n&  -\\tau\\Big(a_0{\\delta^a}_{\\sigma(1)}Y_a(a_1) \\delta^b_{\\sigma(2)}Y_b(a_2) \\Big)- \\tau\\Big(a_0 \\delta^b_{\\sigma(2)}Y_bY_a(a_1) {\\delta^a}_{\\sigma(1)}(a_2) \\Big)\\\\\n&  -\\tau\\Big(a_0 \\delta^b_{\\sigma(2)}Y_a (a_1) {\\delta^a}_{\\sigma(1)}Y_b(a_2)\\Big)\n -\\tau\\Big(a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\blacktriangleleft Y_a\\right)(a_1 \\otimes a_2) \\Big).\n\\end{align*}\n\\begin{align*}\n& \\tau\\Big({\\delta^a}_{\\sigma(1)\\sigma(2)} (a_0) Y_a(a_1) a_2\\Big) =\\\\\n&\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)} Y_a (a_1)a_2\\Big)  + \\tau\\Big(a_0 Y_a (a_1)  {\\delta^a}_{\\sigma(1)\\sigma(2)}(a_2) \\Big)\\\\\n&  +\\tau\\Big( a_0\\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\blacktriangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big).\n\\end{align*}\nSumming up, we obtain\n\\begin{align}\\label{aux-sum-3}\n\\begin{split}\n& r\\tau\\Big(X_{\\sigma(1)} (a_0) X_{\\sigma(2)}(a_1) a_2\\Big) +  r\\tau\\Big({\\delta^a}_{\\sigma(1)} (a_0) X_{\\sigma(2)}Y_a (a_1) a_2\\Big)\\\\\n& + s\\tau\\Big({\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_b (a_0) Y_a (a_1)a_2\\Big) + s\\tau\\Big({\\delta^a}_{\\sigma(1)\\sigma(2)}(a_0) Y_a (a_1)a_2\\Big) = \\\\\n& -r\\tau\\Big(a_0 X_{\\sigma(2)} (a_1) X_{\\sigma(1)}(a_2)\\Big) -s\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_a (a_1) Y_b(a_2)\\Big) \\\\\n&+r \\tau\\Big(a_0 X_{\\sigma(2)}(a_1) {\\delta^a}_{\\sigma(1)}Y_a(a_2) \\Big) -s\\tau\\Big(a_0 Y_a (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_b(a_2)\\Big) \\\\\n& -s\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}Y_a (a_1) \\delta^b_{\\sigma(2)}Y_b(a_2)\\Big)\n -s\\tau\\big(a_0 \\delta^b_{\\sigma(2)}Y_a (a_1) {\\delta^a}_{\\sigma(1)}Y_b(a_2)\\Big).\n\\end{split}\n\\end{align}\nFinally, for \\eqref{aux-box-2} we have\n\\begin{align*}\n& \\tau\\Big({\\delta^a}_{\\sigma(1)}(a_0) X_{\\sigma(2)}(a_1) Y_a(a_2)\\Big) =\\\\\n& -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)} X_{\\sigma(2)}(a_1) Y_a(a_2)\\Big)\n -\\tau\\big(a_0 X_{\\sigma(2)}(a_1) {\\delta^a}_{\\sigma(1)} Y_a(a_2) \\Big),\n\\end{align*}\n\\begin{align*}\n&\\tau\\Big( X_{\\sigma(1)} (a_0) {\\delta^a}_{\\sigma(2)} (a_1) Y_a(a_2)\\Big) =\\\\\n& -\\tau\\Big(a_0 X_{\\sigma(1)} {\\delta^a}_{\\sigma(2)}(a_1) Y_a(a_2)\\Big) -\\tau\\Big(a_0  {\\delta^a}_{\\sigma(2)} (a_1) X_{\\sigma(1)}Y_a(a_2)\\Big) \\\\\n&  + \\tau\\Big(a_0 \\delta^b_{\\sigma(1)} {\\delta^a}_{\\sigma(2)}Y_b (a_1) Y_a \\Big)- 2 \\tau\\Big( a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)}(a_1) Y_a(a_2) \\Big) \\\\\n& + \\tau\\Big(a_0 {\\delta^a}_{\\sigma(2)} (a_1) \\delta^b_{\\sigma(1)}Y_bY_a(a_2) \\Big) +\\tau\\Big(a_0 {\\delta^a}_{\\sigma(2)}Y_b(a_1) \\delta^b_{\\sigma(1)}Y_a (a_2)\\Big) \\\\\n&  -\\tau\\Big( a_0\\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right) (a_1\\otimes\na_2)\\Big).\n\\end{align*}\n\\begin{align*}\n&  \\tau\\Big({\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} (a_0) Y_b(a_1) Y_a(a_2)\\Big) =\\\\\n&  \\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} Y_b (a_1) Y_a(a_2)\\Big) + \\tau\\Big(a_0 Y_b (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_a (a_2)\\Big)\\\\\n&  \\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}Y_b (a_1) \\delta^b_{\\sigma(2)}Y_a(a_2) \\Big)+\n \\tau\\Big(a_0 \\delta^b_{\\sigma(2)}Y_b (a_1) {\\delta^a}_{\\sigma(1)}Y_a (a_2)\\Big).\n\\end{align*}\n\\begin{align*}\n& \\tau\\Big( \\delta^b_{\\sigma(1)}(a_0) {\\delta^a}_{\\sigma(2)}Y_b (a_1) Y_a(a_2)\\Big) =\\\\\n&  \\tau\\Big( a_0 \\delta^b_{\\sigma(1)} {\\delta^a}_{\\sigma(2)}Y_b (a_1) Y_a (a_2)\\Big)+\\tau\\Big(a_0 {\\delta^a}_{\\sigma(2)}Y_b(a_1) \\delta^b_{\\sigma(1)}Y_a(a_2)\\Big).\n\\end{align*}\n\\begin{align*}\n&  \\tau\\Big({\\delta^a}_{\\sigma(1)}Y_b (a_0) \\delta^b_{\\sigma(2)}(a_1) Y_a(a_2)\\Big) =\\\\\n& -\\tau\\Big( Y_b (a_0) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} (a_1) Y_a(a_2) \\Big)+ \\tau\\Big(a_0 \\delta^b_{\\sigma(2)}Y_b (a_1) {\\delta^a}_{\\sigma(1)}Y_a(a_2) \\Big)\\\\\n&  + \\tau\\Big(a_0\\delta^b_{\\sigma(2)}(a_1)\n{\\delta^a}_{\\sigma(1)}Y_bY_a (a_2)\\Big)+\n \\tau\\Big(a_0\\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right) (a_1\\otimes a_2)\\Big).\n\\end{align*}\n\\begin{align*}\n& \\tau\\Big(\\delta^b_{\\sigma(1)}Y_b (a_0) {\\delta^a}_{\\sigma(2)} (a_1) Y_a (a_2)\\Big)=\\\\\n&  \\tau\\Big( a_0 \\delta^b_{\\sigma(1)}{\\delta^a}_{\\sigma(2)}Y_b (a_1) Y_a(a_2)\\Big) -2 \\tau\\Big( a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)} (a_1) Y_a (a_2)\\Big)\\\\\n& +\\tau\\Big( a_0 \\delta^b_{\\sigma(1)}{\\delta^a}_{\\sigma(2)}(a_1) Y_bY_a(a_2)\\Big) +\\tau\\Big( a_0{\\delta^a}_{\\sigma(2)}Y_b(a_1) \\delta^b_{\\sigma(1)}Y_a(a_2) \\Big)\\\\\n&  -\\tau\\Big( a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big) + \\tau\\Big(a_0 {\\delta^a}_{\\sigma(2)} (a_1) \\delta^b_{\\sigma(1)} Y_bY_a (a_2)\\Big).\n\\end{align*}\n\\begin{align*}\n&  \\tau\\Big({\\delta^a}_{\\sigma(1)}(a_0) \\delta^b_{\\sigma(2)}Y_b (a_1) Y_a(a_2)\\Big) =\\\\\n& -\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)} \\delta^b_{\\sigma(2)}Y_b(a_1) Y_a(a_2)\\big) - \\tau\\Big(a_0 \\delta^b_{\\sigma(2)}Y_b(a_1) {\\delta^a}_{\\sigma(1)}Y_a(a_2)\\Big).\n\\end{align*}\nTherefore,\n\\begin{align}\\label{aux-sum-4}\n\\begin{split}\n& r\\tau\\Big({\\delta^a}_{\\sigma(1)} (a_0) X_{\\sigma(2)} (a_1) Y_a(a_2) \\Big)+ r\\tau\\Big( X_{\\sigma(1)} (a_0) {\\delta^a}_{\\sigma(2)} (a_1) Y_a (a_2)\\Big) \\\\\n& -s\\tau\\Big( {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}(a_0) Y_b (a_1) Y_a(a_2) \\Big) -s\\tau\\Big( Y_b (a_0) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)} (a_1) Y_a(a_2)\\Big) \\\\\n& +(r+s) \\tau\\Big(  \\delta^b_{\\sigma(1)} (a_0) {\\delta^a}_{\\sigma(2)}Y_b (a_1) Y_a(a_2) \\Big)-s\\tau\\Big( {\\delta^a}_{\\sigma(1)}Y_b (a_0) \\delta^b_{\\sigma(2)} (a_1) Y_a (a_2)\\Big) \\\\\n& +s\\tau\\Big( \\delta^b_{\\sigma(1)}Y_b (a_0) {\\delta^a}_{\\sigma(2)}(a_1) Y_a(a_2) \\Big)-s\\tau\\Big( {\\delta^a}_{\\sigma(1)}(a_0) \\delta^b_{\\sigma(2)}Y_b (a_1) Y_a(a_2)\\Big) \\\\\n&= -r\\tau\\Big(a_0  {\\delta^a}_{\\sigma(2)} (a_1) X_{\\sigma(1)}Y_a(a_2)\\Big) -r\\tau\\Big(a_0 X_{\\sigma(2)} (a_1) {\\delta^a}_{\\sigma(1)} Y_a(a_2)\\Big)\\\\\n& - s\\tau\\Big(a_0  Y_b (a_1) {\\delta^a}_{\\sigma(1)}\\delta^b_{\\sigma(2)}Y_a(a_2)\\Big) - s\\tau\\Big(a_0  \\delta^b_{\\sigma(2)}Y_b(a_1) {\\delta^a}_{\\sigma(1)}Y_a(a_2)\\Big)\\\\\n& +s\\tau\\Big(a_0 {\\delta^a}_{\\sigma(2)}Y_b (a_1) \\delta^b_{\\sigma(1)}Y_a (a_2)\\Big)- r\\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)}(a_1) \\delta^b_{\\sigma(2)}Y_bY_a(a_2)\\Big) \\\\\n& (-r-s) \\tau\\Big( a_0 \\left(\\Delta\\left({\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\triangleleft Y_a\\right)(a_1 \\otimes a_2)\\Big) +(-r-s) \\tau\\Big(a_0 {\\delta^a}_{\\sigma(1)\\sigma(2)}(a_1 ) Y_a(a_2)\\Big).\n\\end{split}\n\\end{align}\nAs a result of the computations \\eqref{aux-sum-1}, \\eqref{aux-sum-2}, \\eqref{aux-sum-3}, and \\eqref{aux-sum-4} we obtain\n\\begin{equation*}\nt\\psi(1 \\otimes a_0\\otimes a_1 \\otimes a_2) = \\psi_0(1 \\otimes a_0 \\otimes a_1\\otimes a_2).\n\\end{equation*}\n\\end{proof}\n\n\n\n\n\\begin{theorem}\\label{theorem-vp}\nThe following cochain  $\\varphi=\\varphi_0+\\varphi_1+\\varphi_2\\in C^2_{\\cal K}({\\cal A}, V)$ is\na SAYD-twisted cyclic cocycle and  cohomologous to $\\psi$ which is\ndefined in Proposition \\ref{prop-psi}.\n\\begin{align}\\label{aux-vp'-2-quick}\n& \\varphi_2 = \\chi_\\tau^{\\rm eq}\\left(S_{ab}\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)} + S_{ab}\\otimes{\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)} \\right) \\\\\\notag\n& \\varphi_1 = \\chi_\\tau^{\\rm eq}\\Big(- S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)} - S_a\\otimes{\\bf 1}\\otimes X_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}\\\\\\label{aux-vp'-1-quick}\n & - S_a\\otimes{\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}Y_b +\\frac{1}{2}S_a\\otimes {\\bf 1}\\otimes \\Delta({\\delta^a}_{\\sigma(1)\\sigma(2)})\\Big) \\\\\\label{aux-vp'-0-quick}\n& \\varphi_0 = \\chi_\\tau^{\\rm eq}\\left(1^\\ast\\otimes{\\bf 1} \\otimes X_{\\sigma(1)} \\otimes X_{\\sigma(2)} + 1^\\ast\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes X_{\\sigma(2)}Y_a\\right),\n\\end{align}\n\\end{theorem}\n\n\\begin{proof}\nAs a result of Proposition \\ref{prop-psi}  we can write $\\psi = r\\varphi+s\\phi$ for a 2-cochain $\\phi = \\phi_0+\\phi_1+\\phi_2$ given by\n\\begin{align}\n& \\phi_2 = 0 \\\\\\notag\n& \\phi_1 = \\chi_\\tau^{\\rm eq}\\left(S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)} \\otimes Y_b + S_a\\otimes{\\bf 1}\\otimes Y_b \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}\\right. \\\\\\notag\n& + S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}Y_b \\otimes {\\delta^b}_{\\sigma(2)} - S_a\\otimes{\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)}Y_b \\otimes {\\delta^a}_{\\sigma(2)}  \\\\\\notag\n& + S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)} \\otimes {\\delta^b}_{\\sigma(2)}Y_b - S_a\\otimes{\\bf 1} \\otimes {\\delta^b}_{\\sigma(1)} \\otimes {\\delta^a}_{\\sigma(2)}Y_b \\\\\n& +S_a\\otimes{\\bf 1}\\otimes \\Delta({\\delta^a}_{\\sigma(1)\\sigma(2)})\\Big) \\\\\n& \\phi_0 = \\chi_\\tau^{\\rm eq}\\left(1^\\ast\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b \\otimes Y_a +1^\\ast\\otimes {\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)} \\otimes Y_a\\right).\n\\end{align}\nWe note that\n\\begin{equation}\n\\phi_1 = \\chi_\\tau^{\\rm eq}\\left(S_a\\otimes{\\bf 1} \\otimes \\Delta\\left({\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b\\right) + S_a\\otimes{\\bf 1}\\otimes \\Delta({\\delta^a}_{\\sigma(1)\\sigma(2)})\\right).\n\\end{equation}\nIt is then straightforward to check that the 1-cochain $\\phi'=\\phi'_0+\\phi'_1+\\phi'_2$ given by\n\\begin{align}\n& \\phi'_2 = 0 \\\\\n& \\phi'_1 = \\chi_\\tau^{\\rm eq}\\left(S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)}{\\delta^b}_{\\sigma(2)}Y_b + S_a\\otimes{\\bf 1} \\otimes {\\delta^a}_{\\sigma(1)\\sigma(2)}\\right) \\\\\n& \\phi'_0 = 0\n\\end{align}\nis an equivariant cyclic 1-cocycle, and that\n\\begin{equation}\nb\\phi' = \\phi.\n\\end{equation}\n\\end{proof}\n\n\n\n\n\\section{The characteristic map with coefficients}\n\nIn this section we construct a new characteristic map  from the\ntruncated Weil complex of the Lie algebra $\\mathfrak{g}_0:=g\\ell_n$ to\n the cyclic complex of the crossed product algebra\n ${\\cal A} = C_c^\\infty(F^+)\\rtimes \\Gamma$, and we\n illustrate  it completely  in codimensions $n=1$ and $n=2$. We\n observe that the resulting cocycles in codimension  1  match  with\n those   in \\cite{ConnMosc98,ConnMosc} by Connes-Moscovici .\n\n\\medskip\n\n\\noindent Such a characteristic map is obtained by composing a series of\nmaps\n\\begin{equation}\n\\xymatrix{ H(W(\\mathfrak{g}_0,V)) \\ar[r]^{\\mathfrak{D}_P} & H(C(\\mathfrak{g}_0,V))\n\\ar[r]^\\cong& HC(U(\\mathfrak{g}_0),V) \\ar[r]^{~~~\\chi_\\varphi} & HC({\\cal A}). }\n\\end{equation}\nAs it is shown in \\cite{RangSutl-II} the truncated Weil algebra is\nidentical with $W(\\mathfrak{g}_0,V)$. The Poincar\\'e isomorphism\n$\\mathfrak{D}_{\\rm P}$  is defined in \\cite[Prop. 4.4]{RangSutl-II}. The middle\nquasi-isomorphism is defined  in \\cite[Thm. 6.2]{RangSutl-II}.\nFinally the map $\\chi_\\varphi$ is given by the cup product, in the sense of\n\\cite{KhalRang05-II,Rang08}, with the SAYD-twisted  cyclic cocycle\n$\\varphi$ defined in Theorem \\ref{theorem-vp}.\n\n\\medskip\n\n\\noindent Let us recall the mentioned cup product from \\cite{Rang08}. Let $C$\nbe a  $H$-module coalgebra and $A$ be an $H$-module algebra that\nare equipped with a mapping\n\\begin{equation}\nC \\otimes A \\to A, \\quad c \\otimes a \\mapsto c(a)\n\\end{equation}\nsatisfying the conditions\n\\begin{equation}\\label{aux-condition-C-A}\n (h \\cdot c)(a) = h \\cdot (c(a)),\\quad  c(ab) =\nc\\ps{1}(a)c\\ps{2}(b), \\quad  c(1) = \\varepsilon(c)1.\n\\end{equation}\nLet also $V$ be a  SAYD module over a Hopf algebra $H$.  One defines\n\\begin{equation}\n\\cup: C_H^p(C,V) \\otimes C_H^q(A,V) \\to C^{p+q}(A)\n\\end{equation}\n for any $\\varphi \\in C_H^q(A,V)$ and any $x = v \\otimes_H c^0 \\ot\\cdots\\ot\nc^p \\in C_H^p(C,V)$,\n\\begin{align}\\label{aux-cup-product}\n& (x \\cup \\varphi)(a_0 \\ot\\cdots\\ot a_{p+q}) := \\\\\\notag\n & \\sum_{\\sigma \\in\n{\\rm Sh}(p,q)} (-1)^\\sigma \\partial_{\\wbar{\\sigma}(p)} \\ldots\n\\partial_{\\wbar{\\sigma}(1)}\\varphi(\\langle \\partial_{\\wbar{\\sigma}(p+q)} \\ldots\n\\partial_{\\wbar{\\sigma}(p+1)} x, a_0 \\ot\\cdots\\ot a_{p+q} \\rangle),\n\\end{align}\nwhere\n\\begin{equation}\n\\langle x, a_0 \\ot\\cdots\\ot a_n \\rangle := v \\otimes_H c^0(a^0) \\ot\\cdots\\ot\nc^n(a^n).\n\\end{equation}\nHere ${\\rm Sh}(p,q)$ is the set of all  $(p,q)$-shuffle\npermutations, and\n\\begin{equation}\n\\wbar{\\sigma}(n) = \\sigma(n)-1.\n\\end{equation}\nWe set\n$$C:= H := {\\cal K}, \\quad V= S(\\mathfrak{g}_0^\\ast)\\nsb{2n}, \\quad A = {\\cal A}.$$\nHere $H$ acts on $C$ via multiplication, on $A$ as  \\eqref{aux-action-Hn-on-A} , and\non $V$ via the coadjoint action.\n This construction yields for any  $\\varphi\\in C^n_{\\cal K}({\\cal A},V)$ a characteristic map\n\\begin{equation}\n\\chi_\\varphi:C^\\ell_{\\cal K}({\\cal K}, V)\\rightarrow C^{\\ell+n}({\\cal A}).\n\\end{equation}\n\n\n\n\\subsection{The characteristic map in codimension 1}\nIn this subsection we apply  the SAYD-twisted cyclic cocycle\n\\eqref{aux-pieces-of-cocycle} to illustrate the new characteristic\nmap\n\\begin{equation}\n\\chi_\\varphi:H(W(g\\ell_1)\\nsb{2})\\longrightarrow\nHC(C^\\infty_c(F^+{\\mathbb R})\\rtimes \\Gamma).\n\\end{equation}\n\n\\noindent In order to verify that the new characteristic map is\ngeometrically meaningful,   we compare it with the Connes-Moscovici\ncomputations for the classes of ${\\cal H}_1$ in \\cite{ConnMosc}. For the\nconvenience of the reader we recall the Hopf-cyclic classes in\n$C({\\cal H}_1,{\\mathbb C}_\\delta)$, namely the the transverse fundamental class\n\\begin{equation}\n{\\rm TF} = X \\otimes Y - Y \\otimes X - \\delta_1Y \\otimes Y \\in C^2({\\cal H}_1, {\\mathbb C}_\\delta)\n\\end{equation}\nand the Godbillon-Vey class\n\\begin{equation}\n{\\rm GV} = \\delta_1 \\in C^1({\\cal H}_1,{\\mathbb C}_\\delta).\n\\end{equation}\n\n\\noindent In view of the characteristic map\n\\eqref{aux-Connes-Moscovici-charac-map}, one  expresses  the\ncharacteristic classes $TF \\in C^2({\\cal A}_\\Gamma)$ and $GV \\in\nC^1({\\cal A}_\\Gamma)$ as\n\\begin{align}\n& TF(a_0 \\otimes a_1 \\otimes a_2)  \\\\\\notag\n & =\\tau(a_0X(a_1)Y(a_2)) -\n\\tau(a_0Y(a_1)X(a_2)) - \\tau(a_0\\delta_1Y(a_1)Y(a_2))\n\\end{align}\nand\n\\begin{equation}\nGV(a_0 \\otimes a_1)  = \\tau(a_0\\delta_1(a_1)).\n\\end{equation}\n\n\\medskip\n\n\\noindent In this subsection we set ${\\cal K} = U(g\\ell_1)$, $V = S(g\\ell_1^\\ast)\\nsb{2}$, and ${\\cal A} = C^\\infty_c(F^+{\\mathbb R})\\rtimes\\Gamma$.\n\n\\medskip\n\n\\noindent The next step is to find the representative cocycles of\n$H(g\\ell_1,V)$. Let $\\{Y\\}$ and $\\{\\theta\\}$ be\n a dual pair of bases for $g\\ell_1$ and $g\\ell_1^\\ast$.\n\n\\medskip\n\n\\noindent By the Vey basis \\cite{Godb72}, the cohomology of\n$W(g\\ell_1)\\nsb{2}$ is spanned by\n\\begin{equation}\n{\\rm TF}:= 1\\in S(g\\ell_1^\\ast)\\nsb{2},\\quad {\\rm GV}:=\\theta\\otimes R\\in\ng\\ell_1^\\ast\\otimes S(g\\ell_1^\\ast)\\nsb{2}\n\\end{equation}\nApplying the Poincar\\'e duality  \\cite[Prop. 4.4]{RangSutl-II},  we\nobtain the classes in $HC(g\\ell_1, V)$\n\\begin{equation}\n\\mathfrak{D}_{\\rm P}(1)= Y\\otimes 1\\in g\\ell_1\\otimes V, \\quad \\mathfrak{D}_{\\rm P}( \\theta\\otimes R)=\nR\\in V.\n\\end{equation}\n\n\\begin{proposition}\nThe Hopf-cyclic cohomology $HC({\\cal K},V)$ is generated by the classes\n\\begin{align}\\label{aux-0-cocycle-codim-1}\n& [R] \\in HC^0({\\cal K},V),\\\\\\label{aux-1-cocycle-codim-1}\n& [1 \\otimes Y + \\frac{1}{2}R \\otimes Y^2] \\in HC^1({\\cal K},V).\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nWe first check that $R \\in C^0({\\cal K},V)$ is a cyclic 0-cocycle. Indeed,\n\\begin{equation}\nb(R) = R \\otimes 1 - R \\otimes 1 = 0, \\qquad \\mathfrak{t}(R) = R.\n\\end{equation}\nIn a similar fashion,\n\\begin{equation*}\nb(1 \\otimes Y) = 1 \\otimes 1 \\otimes Y - 1 \\otimes \\Delta(Y) + 1 \\otimes Y \\otimes 1 + R \\otimes Y \\otimes Y\n\\end{equation*}\nyields\n\\begin{equation}\nb(1 \\otimes Y + \\frac{1}{2}R \\otimes Y^2) = 0.\n\\end{equation}\nMoreover, applying the cyclic map we observe\n\\begin{equation}\n\\mathfrak{t}(1 \\otimes Y + \\frac{1}{2}R \\otimes Y^2) = -1 \\otimes Y - R \\otimes Y^2 + \\frac{1}{2}R \\otimes Y^2 = - (1 \\otimes Y + \\frac{1}{2}R \\otimes Y^2).\n\\end{equation}\nFinally, we apply the quasi-isomorphism\n\\begin{equation}\\label{aux-quasi-inverse-of-anti-symm}\n\\mu:C^\\bullet(U(\\mathfrak{g}),V) \\longrightarrow C_\\bullet(\\mathfrak{g},V),\n\\end{equation}\nwhich, for any Lie algebra $\\mathfrak{g}$, is the left inverse of the anti-symmetrization map, see \\cite{ConnMosc,RangSutl-II}. Then we have\n\\begin{equation}\n\\mu(R) = R, \\qquad \\mu(1 \\otimes Y + \\frac{1}{2}R \\otimes Y^2) = 1 \\otimes Y,\n\\end{equation}\nthe generators of the cohomology $HC(g\\ell_1,V)$. This observation finishes the proof.\n\\end{proof}\n\n\\noindent Let us find the images of the cocycles \\eqref{aux-0-cocycle-codim-1} and \\eqref{aux-1-cocycle-codim-1} under the map\n\\begin{equation}\n\\chi_\\varphi:C_{\\cal K}^\\bullet({\\cal K},V) \\to C^{\\bullet+1}({\\cal A}).\n\\end{equation}\nWe compute\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(\\theta)(a_0 \\otimes a_1) = (\\varphi \\cup (\\theta \\otimes 1))(a_0 \\otimes a_1) \\\\\n& = \\varphi(\\langle\\partial_0(\\theta \\otimes 1),a_0 \\otimes a_1\\rangle) = \\varphi(\\theta \\otimes a_0 \\otimes a_1) = -\\tau(a_0\\delta_1(a_1)),\n\\end{split}\n\\end{align}\nand in the same way,\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2)(a_0 \\otimes a_1\\otimes a_2) = \\\\\n& \\sum_{\\sigma \\in Sh(1,1)} (-1)^\\sigma \\partial_{\\wbar{\\sigma}(1)}\\varphi(\\langle \\partial_{\\wbar{\\sigma}(2)}(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2),a_0 \\otimes a_1\\otimes a_2 \\rangle) = \\\\\n& -\\partial_0\\varphi(\\langle \\partial_1(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2),a_0 \\otimes a_1\\otimes a_2 \\rangle) + \\\\\n& \\partial_1\\varphi(\\langle \\partial_0(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2),a_0 \\otimes a_1\\otimes a_2 \\rangle).\n\\end{split}\n\\end{align}\nThen since\n\\begin{align}\n\\begin{split}\n& \\partial_1(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2) = 1 \\otimes 1 \\otimes 1 \\otimes Y + 1 \\otimes 1 \\otimes Y \\otimes 1 +  \\\\\n& \\frac{1}{2} \\theta \\otimes 1 \\otimes Y^2 \\otimes 1 + \\frac{1}{2}\\theta \\otimes 1 \\otimes 1 \\otimes Y^2 + \\theta \\otimes 1 \\otimes Y \\otimes Y,\n\\end{split}\n\\end{align}\nwe obtain\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2)(a_0 \\otimes a_1\\otimes a_2) = \\\\\n& \\varphi(-1 \\otimes a_0a_1 \\otimes Y(a_2) - 1 \\otimes a_0Y(a_1) \\otimes a_2 - \\frac{1}{2}\\theta \\otimes a_0Y^2(a_1) \\otimes a_2 \\\\\n& - \\frac{1}{2}\\theta \\otimes a_0a_1 \\otimes Y^2(a_2) - \\theta \\otimes a_0Y(a_1) \\otimes Y(a_2)) + \\varphi(1 \\otimes a_0 \\otimes a_1Y(a_2) \\\\\n& + \\frac{1}{2}\\theta \\otimes a_0 \\otimes a_1Y^2(a_2)) \\\\\n& = -\\tau(a_0Y(a_1)X(a_2)) + \\tau(a_0X(a_1)Y(a_2)) + \\frac{1}{2}\\tau(a_0Y^2(a_1)\\delta_1(a_2)) \\\\\n& + \\frac{1}{2}\\tau(a_0\\delta_1(a_1)Y^2(a_2)) + \\tau(a_0Y(a_1)\\delta_1Y(a_2)).\n\\end{split}\n\\end{align}\n\n\\begin{remark}\\rm{\nWe note that\n\\begin{equation}\nb(\\delta_1Y^2) \\in C^2_{{\\cal H}_1}({\\cal H}_1,{\\mathbb C}_\\delta)\n\\end{equation}\nis a cyclic cocycle and\n\\begin{equation}\n\\chi_\\varphi(1 \\otimes Y + \\frac{1}{2}\\theta \\otimes Y^2) = \\chi_\\tau({\\rm TF} + \\frac{1}{2}b(\\delta_1Y^2)).\n\\end{equation}\nHence, we obtain the transverse fundamental class up to a coboundary. Similarly we have\n\\begin{equation}\n\\chi_\\varphi(\\theta) = - \\chi_\\tau({\\rm GV}),\n\\end{equation}\n{\\it i.e.\\\/}\\  we also obtain the Godbillon-Vey class.\n}\\end{remark}\n\n\n\n\\subsection{The characteristic map in codimension 2}\nIn this subsection we exercise the machinery we developed in\nSubsection \\ref{subsection-H-n} for $n=2$. There is no such\ncomputations in the literature that we know of.\n\n\\medskip\n\n\\noindent We keep our conventions as before, {\\it i.e.\\\/}\\   ${\\cal K}:= U(g\\ell_2)$, $V\n:= S(g\\ell_2^\\ast)\\nsb{4}$, ${\\cal H}:={\\cal K}_2$,  and ${\\cal A}:=\nC^\\infty_c(F^+{\\mathbb R}^2)\\rtimes \\Gamma$.\n\n\\medskip\n\n\\noindent Let us recall the Vey basis of the cohomology of the truncated\nWeil algebra $W(g\\ell_2)\\nsb{4}$. To this end, we fix the following notation\n \\begin{align*}\n& c_1 = {\\rm Tr} = R^1_1+R^2_2 \\in S(g\\ell_2^\\ast), \\qquad c_2 = R^1_2R^2_1 \\in S(g\\ell_2^\\ast), \\\\\n& u_1 = \\theta^1_1 + \\theta^2_2, \\qquad u_2 = \\theta^1_1\\wedge\\theta^1_2\\wedge\\theta^2_1, \\qquad \\omega = \\theta^1_1\\wedge\\theta^1_2\\wedge\\theta^2_1\\wedge\\theta^2_2.\n\\end{align*}\nThe  Vey basis \\cite{Godb72}, is then introduced by\n\\begin{equation}\n\\big\\{1,\\, c_1^2\\otimes u_1,\\,c_2\\otimes u_1,\\,c_2\\otimes u_2,\\, c_1^2 \\otimes\n\\omega,\\, c_2 \\otimes \\omega\\big\\}.\n\\end{equation}\nNext, the Poincar\\'e duality yields the 6 cocycles in the complex\n$C(g\\ell_2,V)$:\n\\begin{align*}\n& \\mathfrak{D}_{\\rm P}(1) = 1 \\otimes Y_1^1\\wedge Y_1^2\\wedge Y_2^1\\wedge Y_2^2,\\\\\n& \\mathfrak{D}_{\\rm P}(c_2 \\otimes u_1) =c_2 \\otimes \\left(Y_1^2\\wedge Y_2^1\\wedge Y_2^2 - Y_1^1\\wedge Y_1^2\\wedge Y_2^1\\right),\\, \\\\\n& \\mathfrak{D}_{\\rm P}(c_1^2 \\otimes u_1) = c_1^2 \\otimes \\left(Y_1^2\\wedge Y_2^1\\wedge Y_2^2 - Y_1^1\\wedge Y_1^2\\wedge\nY_2^1\\right) ,\\\\\n& \\mathfrak{D}_{\\rm P}(c_2 \\otimes u_2)=c_2 \\otimes Y_2^2,\\quad \\mathfrak{D}_{\\rm P}(c_1^2 \\otimes \\omega)=c_1^2,\\\\\n&\\mathfrak{D}_{\\rm P}(c_2 \\otimes \\omega) = c_2 .\n\\end{align*}\nWe label $Y_i^j$ as\n\\begin{equation}\nY_1:=Y_1^1, \\;\\; Y_2:= Y_1^2, \\;\\;Y_3:=Y_2^1,\\;\\; Y_4:=Y_2^2.\n\\end{equation}\n\n\\begin{proposition}\nThe Hopf-cyclic cohomology $HC({\\cal K},V)$ is generated by the classes\n\\begin{align}\n\\begin{split}\n& [\\mathscr{TF}] \\in HC^4({\\cal K},V) \\\\\n& [\\mathscr{GV}] :=\\Big[\\sum_{\\sigma \\in S_3}(-1)^\\sigma c_1^2 \\otimes \\Big(Y_{\\sigma(2)}\\otimes Y_{\\sigma(3)}\\otimes Y_{\\sigma(4)}\\\\\n&\\hspace{4cm} - Y_{\\sigma(1)}\\otimes Y_{\\sigma(2)}\\otimes Y_{\\sigma(3)}\\Big)\\Big]\\in HC^3({\\cal K},V), \\\\\n& [\\mathscr{R}_1] := \\Big[\\sum_{\\sigma \\in S_3}(-1)^\\sigma c_2 \\otimes \\Big(Y_{\\sigma(2)}\\otimes Y_{\\sigma(3)}\\otimes Y_{\\sigma(4)}\\\\\n &\\hspace{4cm} - Y_{\\sigma(1)}\\otimes Y_{\\sigma(2)}\\otimes Y_{\\sigma(3)}\\Big)\\Big]\\in HC^3({\\cal K},V), \\\\\n& [\\mathscr{R}_2] := [c_2 \\otimes Y_4] \\in HC^1({\\cal K},V),\\\\\n& [\\mathscr{R}_3] := [c_1^2] \\in HC^0({\\cal K},V),\\\\\n& [\\mathscr{R}_4] := [c_2] \\in HC^0({\\cal K},V) .\n\\end{split}\n\\end{align}\n\\end{proposition}\n\n\\begin{proof}\nIt is straightforward to check that $\\mathscr{R}_1,\\ldots,\\mathscr{R}_4$ and $\\mathscr{GV}$ are Hopf-cyclic cocycles and that\n\\begin{align*}\n& \\mu(\\mathscr{R}_1) = \\mathfrak{D}_{\\rm P}(c_2\\otimes u_1),\\quad \\mu(\\mathscr{R}_2) = \\mathfrak{D}_{\\rm P}(c_2\\otimes u_2),\\\\\n& \\mu(\\mathscr{R}_3) = \\mathfrak{D}_{\\rm P}(c_1^2\\otimes \\omega), \\quad  \\mu(\\mathscr{R}_4) = \\mathfrak{D}_{\\rm P}(c_2\\otimes \\omega),\\\\\n& \\mu(\\mathscr{GV}) = \\mathfrak{D}_{\\rm P}(c_1^2\\otimes u_1).\n\\end{align*}\nOn the other hand,\n\\begin{equation}\\label{aux-TF-in-K}\n[\\mathscr{TF}] = \\Big[\\sum_{\\sigma \\in S_4}(-1)^\\s1 \\otimes Y_{\\sigma(1)}\\otimes Y_{\\sigma(2)}\\otimes Y_{\\sigma(3)}\\otimes Y_{\\sigma(4)}\\Big] \\in E_1^{2,2}(U(g\\ell_2),V)\n\\end{equation}\nis a cyclic cocycle  in the $E_1$ level of the spectral sequence\nthat corresponds to the natural filtration of $V$, \\cite[Thm.\n6.2]{RangSutl-II}.  Hence\n\\begin{equation*}\n\\mu(\\mathscr{TF}) = \\mathfrak{D}_{\\rm P}(1).\n\\end{equation*}\nTherefore, the claim follows from \\cite[Thm. 6.2]{RangSutl-II}.\n\\end{proof}\n\n\\noindent In this paper we do not complete the fundamental  cocycle as we\nknow its counterpart as a cyclic cocycle over ${\\cal A}$ by the following\nargument. Let us recall the characteristic  map\n\\begin{equation}\n\\chi_\\varphi:C^\\bullet({\\cal K},V) \\longrightarrow C^{\\bullet +2}({\\cal A})\n\\end{equation}\nfor the SAYD-twisted cyclic 2-cocycle defined by the Theorem\n\\ref{theorem-vp}. To this end, we first prove a generalization of\n\\cite[Prop. 18]{ConnMosc}. In view of   in \\cite{MoscRang09},\n${\\cal H}_n$ is realized as a bicrossed product Hopf algebra\n ${\\cal U} {\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}{\\cal F}^{\\rm cop}$. Here  ${\\cal F}$\n  is the commutative algebra of regular functions on the group of diffeomorphisms which\n  preserve the  origin and with identity Jacobian at the origin,\n  and ${\\cal U}=U(g\\ell_2^{\\rm affine})$. The coaction involed in this\n  bicrossed product realization  is recalled below\n\\begin{align}\\label{aux-the-left-F^cop-coaction-on-U}\n\\begin{split}\n& \\nabla:{\\cal U} \\to {\\cal F}{^{\\rm cop}} \\otimes {\\cal U}, \\\\\n& X_k \\mapsto 1 \\otimes X_k + \\delta^i_{jk} \\otimes Y_i^j,\\qquad  Y_i^j \\mapsto\n1 \\otimes Y_i^j.\n\\end{split}\n\\end{align}\nIn the following proposition, for any $1 \\leq j \\leq m := n^2+n$,\n\\begin{equation}\n\\nabla^j(Z) = Z\\ns{-j} \\ot\\cdots\\ot Z\\ns{-1} \\otimes Z\\ns{0} \\otimes 1 \\ot\\cdots\\ot 1 \\in {\\cal H}_n^{\\otimes\\,m+1}.\n\\end{equation}\n\n\\begin{proposition}\nThe $m:=n^2 + n$-cochain\n\\begin{equation}\\label{aux-transverse-fundamental-class-in-Hn}\n{\\rm TF} := (-1)^{(m-1)!}\\, \\sum_{\\sigma \\in S_m} (-1)^\\sigma \\nabla^m(Z^{\\sigma(1)}) \\cdots \\nabla(Z^{\\sigma(m)}) \\in {\\cal H}_n^{\\otimes\\,m+1}\n\\end{equation}\nis a cyclic $m$-cocycle whose class $[{\\rm TF}] \\in HC^m({\\cal H}_n)$ corresponds, by the\nConnes-Moscovici characteristic map, to the transverse fundamental class\n$[TF] \\in HC^m({\\cal A})$.\n\\end{proposition}\n\n\\begin{proof}\nLet  $a^i := f^iU^\\ast_{\\psi_i} \\in {\\cal A}$, where  $0 \\leq i \\leq m$,\n$f^i\\in C^\\infty_c(F^+{\\mathbb R}^2)$ and $\\psi\\in \\Gamma$. Without loss of\ngenerality we assume that  $\\psi_m\\ldots\\psi_0 = \\mathop{\\rm Id}\\nolimits$. The cyclic\ncocycle $TF \\in HC^m({\\cal A})$ is given by the $m$-cocycle\n\\begin{align*}\n& \\displaystyle TF(a^0\\ot\\cdots\\ot a^m) = \\int_{F^+{\\mathbb R}^n} a^0 da^1\\cdots\nda^m, \\qquad \\qquad da^i=df^iU^\\ast_{\\psi_i}.\n\\end{align*}\nWe note that in order to prove the claim, we need to find suitable\n$h^0, \\ldots, h^m \\in {\\cal H}_n$ such that\n\\begin{equation}\nTF(a^0\\ot\\cdots\\ot a^m)=\\tau(h^0(a^0)\\cdots h^m(a^m)).\n\\end{equation}\nIndeed,\n\\begin{align}\\label{aux-h^0-to-h^m}\n\\begin{split}\n&\\int_{F^+{\\mathbb R}^n} a^0 da^1\\cdots da^m=\n\\int_{F^+{\\mathbb R}^n}f^0{\\psi_0}^\\ast(df^1)\n\\cdots({\\psi_0}^\\ast\\ldots{\\psi_{m-1}}^\\ast)(df^m) = \\\\\n& \\int_{F^+{\\mathbb R}^n}h^0(f^0){\\psi_0}^\\ast(h^1(f^1))\n\\cdots({\\psi_0}^\\ast\\ldots\n{\\psi_{m-1}}^\\ast)(h^m(f^m)) \\varpi = \\\\\n& \\int_{F^+{\\mathbb R}^n}(\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast \\ot\\cdots\\ot\n {\\psi_0}^\\ast\\ldots{\\psi_{m-1}}^\\ast)\n (h^0 \\ot\\cdots\\ot h^m)(f^0 \\ot\\cdots\\ot f^m)\\varpi,\n\\end{split}\n\\end{align}\nwhere the volume form on the frame bundle is\n\\begin{equation}\\label{aux-volume-form}\n\\varpi = \\bigwedge_{i = 1}^n\\theta^i \\wedge \\bigwedge_{1 \\leq i,j \\leq\nn}\\omega^i_j\n \\quad {\\rm (ordered \\,\\, lexicographically)}\n\\end{equation}\nIn the above computation  we use the following notations.\n\\begin{equation*}\n(h^0 \\ot\\cdots\\ot h^m)(f^0 \\ot\\cdots\\ot f^m) = h^0(f^0)\\ldots h^m(f^m),\n\\end{equation*}\nand similarly for any $g^0,\\ldots,g^m \\in C^\\infty_c(F^+{\\mathbb R}^n)$,\n\\begin{align*}\n(\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast \\ot\\cdots\\ot\n{\\psi_0}^\\ast\\ldots{\\psi_{m-1}}^\\ast)&(g^0 \\ot\\cdots\\ot g^m)\\\\\n&=g^0{\\psi_0}^\\ast(g^1)\\ldots\n{\\psi_0}^\\ast\\ldots{\\psi_{m-1}}^\\ast(g^m).\n\\end{align*}\nHere  $\\psi^\\ast(g)(x,y)= g(\\psi(x), \\psi'(x)\\cdot y)$.\n\\medskip\n\n\n\\noindent For  any $f \\in C^\\infty_c(F^+{\\mathbb R}^n)$ we have\n\\begin{equation}\ndf = \\frac{\\partial f}{\\partial x^i}dx^i + \\frac{\\partial f}{\\partial y_i^j}dy_i^j =  X_i(f)\\theta^i + Y_i^j(f)\\omega^i_j.\n\\end{equation}\nTherefore, for $\\psi_0 \\in \\mathop{\\rm Diff}\\nolimits({\\mathbb R}^n)$, and $f^0, f^1 \\in C^\\infty_c(F^+{\\mathbb R}^n)$ we have\n\\begin{align}\\label{aux-f^0-df^1-formula}\n\\begin{split}\n& f^0{\\psi_0}^\\ast (df^1) =f^0{\\psi_0}^\\ast(X_i(f^1)){\\psi_0}^\\ast(\\theta^i) + f^0{\\psi_0}^\\ast(Y_i^j(f^1)){\\psi_0}^\\ast(\\omega^i_j) \\\\\n& = f^0{\\psi_0}^\\ast(X_i(f^1))\\theta^i + f^0{\\psi_0}^\\ast(Y_i^j(f^1))(\\gamma^i_{jk}(\\psi_0)\\theta^k + \\omega^i_j) \\\\\n& = (\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast) \\big[(1 \\otimes X_k + \\delta^i_{jk} \\otimes Y_i^j)(f^0 \\otimes f^1)\\theta^k + (1 \\otimes Y_i^j)(f^0 \\otimes f^1)\\omega^i_j\\big] \\\\\n& = (\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast) ((X_k)\\ns{-1} \\otimes (X_k)\\ns{0})(f^0 \\otimes f^1)\\theta^k + \\\\\n& (\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast)((Y_i^j)\\ns{-1} \\otimes (Y_i^j)\\ns{0})(f^0 \\otimes\nf^1)\\omega^i_j.\n\\end{split}\n\\end{align}\n\\noindent On the second equality we have used \\cite[(2.16)]{ConnMosc}, and on the third equality we used \\eqref{aux-action-Hn-on-A}. On the forth equality, the left coaction is \\eqref{aux-the-left-F^cop-coaction-on-U}.\n\n\\medskip\n\n\\noindent On the other hand we have\n\\begin{align}\\label{aux-multiplication-f^0-df^1-to-df^m}\n\\begin{split}\n& (-1)^{(m-1)!} \\,\\, f^0{\\psi_0}^\\ast (df^1) {\\psi_0}^\\ast{\\psi_1}^\\ast (df^2) \\ldots {\\psi_0}^\\ast \\cdots {\\psi_{m-1}}^\\ast (df^m)  \\\\\n&= {\\psi_0}^\\ast \\cdots {\\psi_{m-1}}^\\ast (df^m) \\ldots {\\psi_0}^\\ast{\\psi_1}^\\ast (df^2) {\\psi_0}^\\ast (df^1) f^0 \\\\\n& =\\left({\\psi_{m-1} \\cdots \\psi_0}^\\ast(X_i(f^m))\\theta^i \\right. +\\\\\n& \\left. {\\psi_{m-1} \\cdots \\psi_0}^\\ast(Y_i^j(f^m))(\\gamma^i_{jk}(\\psi_{m-1} \\cdots \\psi_1)\\theta^k + \\omega^i_j)\\right) \\\\\n& \\cdots \\\\\n& \\left({\\psi_1\\psi_0}^\\ast(X_i(f^2))\\theta^i + {\\psi_1\\psi_0}^\\ast(Y_i^j(f^2))(\\gamma^i_{jk}(\\psi_2\\psi_1)\\theta^k + \\omega^i_j)\\right) \\cdot \\\\\n& \\left( {\\psi_0}^\\ast(X_i(f^1))\\theta^i + {\\psi_0}^\\ast(Y_i^j(f^1))(\\gamma^i_{jk}(\\psi_1)\\theta^k + \\omega^i_j)\\right) \\cdot f^0  \\\\\n& =\\left(\\mathop{\\rm Id}\\nolimits \\otimes {\\psi_0}^\\ast \\ot\\cdots\\ot {\\psi_0}^\\ast\\ldots{\\psi_{m-1}}^\\ast\\right) \\Big\\{ \\\\\n& \\left[((X_i)\\ns{-m} \\ot\\cdots\\ot (X_i)\\ns{0})\\theta^i + ((Y_i^j)\\ns{-m} \\ot\\cdots\\ot (Y_i^j)\\ns{0})\\omega^i_j\\right] \\cdot \\\\\n& \\cdots\\\\\n& \\big[((X_i)\\ns{-2} \\otimes (X_i)\\ns{-1} \\otimes (X_i)\\ns{0} \\otimes 1 \\ot\\cdots\\ot 1)\\theta^i + \\\\\n & \\hspace{3cm}((Y_i^j)\\ns{-2} \\otimes (Y_i^j)\\ns{-1} \\otimes (Y_i^j)\\ns{0} \\otimes 1 \\ot\\cdots\\ot 1)\\omega^i_j\\big] \\cdot \\\\\n & \\left[((X_i)\\ns{-1} \\otimes (X_i)\\ns{0} \\otimes 1 \\ot\\cdots\\ot 1)\\theta^i\\right. + \\\\\n& \\left. ((Y_i^j)\\ns{-1} \\otimes (Y_i^j)\\ns{0} \\otimes 1 \\ot\\cdots\\ot 1)\\omega^i_j\\right] \\Big\\} \\left(f^0 \\ot\\cdots\\ot f^m\\right).\n\\end{split}\n\\end{align}\n\\noindent On the third equality, we have used the cocycle identity \\cite[(1.16)]{MoscRang09} in order to obtain the expressions in ${\\cal H}_n^{\\otimes\\,m+1}$ in the range of the coaction \\eqref{aux-the-left-F^cop-coaction-on-U}. We reversed the order of the multiplication in \\eqref{aux-multiplication-f^0-df^1-to-df^m} in order to avoid obtaining elements in ${\\cal H}_n^{\\otimes\\,m+1}$ involving $Y_i^j\\delta^p_{qr} \\in {\\cal H}_n$ which do not belong the PBW basis of ${\\cal H}_n$, \\cite[Prop. 3]{ConnMosc}.\n\n\\medskip\n\n\\noindent The coefficient of the volume form \\eqref{aux-volume-form}, which is an element $H \\in {\\cal H}_n^{\\otimes\\,m+1}$, can now be expressed by carrying out the multiplication in \\eqref{aux-multiplication-f^0-df^1-to-df^m}. Let $(Z^1, \\ldots, Z^m) = (X_1,\\ldots, X_n, Y_1^1, \\ldots, Y_n^n)$, where the right hand side is ordered lexicographically. Then\n\\begin{align}\\label{aux-the-element-H}\n\\begin{split}\n& H = \\sum_{\\sigma \\in S_m} (-1)^\\sigma \\nabla^m(Z^{\\sigma(1)}) \\cdots \\nabla(Z^{\\sigma(m)}).\n\\end{split}\n\\end{align}\n\n\\medskip\n\n\\noindent Finally, as a result of \\eqref{aux-h^0-to-h^m},\n\\eqref{aux-multiplication-f^0-df^1-to-df^m} and \\eqref{aux-the-element-H} we have the element\n\\begin{equation}\n{\\rm TF} := (-1)^{(m-1)!}\\, \\sum_{\\sigma \\in S_m} (-1)^\\sigma \\nabla^m(Z^{\\sigma(1)}) \\cdots \\nabla(Z^{\\sigma(m)}) \\in {\\cal H}_n^{\\otimes\\,m+1}\n\\end{equation}\nsuch that for the Connes-Moscovici characteristic map \\eqref{aux-Connes-Moscovici-charac-map} we have\n\\begin{equation}\n\\chi_\\tau({\\rm TF}) = TF \\in C^m({\\cal A}).\n\\end{equation}\n\\end{proof}\n\\noindent Let us illustrate the proposition for $n=1$. We have $(Z^1,Z^2) = (X,Y)$, $m = 1^2+1=2$ and\n\\begin{align}\n\\begin{split}\n& {\\rm TF} = (-1)^{1!}\\sum_{\\sigma \\in S_2}(-1)^\\sigma \\nabla^2(Z^{\\sigma(1)})\\nabla (Z^{\\sigma(2)}) \\\\\n& = -\\Big((X\\ns{-2} \\otimes X\\ns{-1} \\otimes X\\ns{0})(Y\\ns{-1} \\otimes Y\\ns{0} \\otimes 1) - \\\\\n& (Y\\ns{-2} \\otimes Y\\ns{-1} \\otimes Y\\ns{0})(X\\ns{-1} \\otimes X\\ns{0} \\otimes 1)\\Big) \\\\\n& =  - 1 \\otimes Y \\otimes X - 1\\otimes \\delta_1Y \\otimes Y - \\delta_1 \\otimes Y \\otimes Y + 1 \\otimes X \\otimes Y + \\delta_1 \\otimes Y \\otimes Y \\\\\n& = 1 \\otimes X \\otimes Y- 1 \\otimes Y \\otimes X - 1\\otimes \\delta_1Y \\otimes Y.\n\\end{split}\n\\end{align}\nNext we recall the isomorphism\n\\begin{align}\\label{aux-diagonal-isomorphism}\n\\begin{split}\n& \\Psi_{\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}:C^\\bullet({\\cal H}, {\\mathbb C}_\\delta) \\to D^\\bullet({\\cal U},{\\cal F},{\\mathbb C}_\\delta) \\\\\n&\\Psi_{\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}( u^1\\blacktriangleright\\hspace{-4pt}\\vartriangleleft  f^1\\otimes \\ldots\\otimes u^n\\blacktriangleright\\hspace{-4pt}\\vartriangleleft  f^n) = \\\\\n& u^1\\ns{-n}f^1 \\ot\\cdots\\ot u^1\\ns{-1}\\ldots u^n\\ns{-1}f^n \\otimes u^1\\ns{0} \\ot\\cdots\\ot u^n\\ns{0}\n\\end{split}\n\\end{align}\ndefined in \\cite{MoscRang09} that identifies the Hopf-cyclic complex $C^\\bullet({\\cal H},{\\mathbb C}_\\delta)$\nof a Hopf algebra ${\\cal H} = {\\cal U} \\vartriangleright\\hspace{-4pt} \\blacktriangleleft {\\cal F}$ with the diagonal subcomplex\n\\begin{equation}\nD^\\bullet({\\cal U},{\\cal F},{\\mathbb C}_\\delta) := {\\mathbb C}_\\delta \\otimes {\\cal F}^{\\otimes \\, \\bullet} \\otimes {\\cal U}^{\\otimes\\,\\bullet}.\n\\end{equation}\n\\noindent On the other hand, for ${\\cal U} = U(\\mathfrak{g})$ we have the quasi-isomorphism\n\\begin{equation}\\label{aux-mu-map}\n\\mu:{\\mathbb C}_\\delta \\otimes {\\cal F}^{\\otimes \\, \\bullet} \\otimes {\\cal U}^{\\otimes\\,\\bullet} \\to {\\mathbb C}_\\delta \\otimes {\\cal F}^{\\otimes \\, \\bullet} \\otimes \\bigwedge^{\\bullet}\\mathfrak{g},\n\\end{equation}\nwhich is the inverse of the antisymmetrization map on the level of cohomologies.\n\n\\begin{remark}\\rm{\nThe transverse fundamental class $[{\\rm TF}] \\in HC^{n^2+n}({\\cal H}_n,{\\mathbb C}_\\delta)$ defined in \\eqref{aux-transverse-fundamental-class-in-Hn} corresponds to the class\n\\begin{equation}\n[1 \\otimes X_1 \\wedge\\dots\\wedge X_n \\wedge Y_1^1 \\wedge\\dots\\wedge Y_n^n],\n\\end{equation}\nin the total complex  $C^{\\bullet, \\bullet}(\\mathfrak{g}, {\\cal F}, {\\mathbb C}_\\delta)$ \\cite[(3.37)]{MoscRang09}, by the composition of \\eqref{aux-diagonal-isomorphism} and \\eqref{aux-mu-map}.\n}\\end{remark}\n\\noindent On the next move, we introduce the commutative diagram\n\\begin{equation}\\label{aux-diagram-of-characteristic-maps}\n\\xymatrix {\n\\ar[dr]_{\\chi_\\varphi} C_{\\cal K}^j({\\cal K},V) \\ar[r]^{\\wbar{\\chi}_\\varphi} & C^{j+k}({\\cal H}_n,{\\mathbb C}_\\delta) \\ar[d]^{T^\\natural} \\\\\n& C^{j+k}({\\cal A})\n}\n\\end{equation}\ninduced by (a decomposition of) the cup product \\eqref{aux-cup-product} via a cyclic cocycle  $\\varphi \\in C^k_{{\\cal K}}({\\cal A},V)$ in the image of \\eqref{aux-equivariant-map}. Here $T^\\natural:{\\cal H}_n^j \\to C^j({\\cal A})$ is the isomorphism\n\\begin{equation}\nT^\\natural(h^1 \\ot\\cdots\\ot h^j)(a^0 \\ot\\cdots\\ot a^j) = \\tau(a^0h^1(a^1)\\ldots h^j(a^j)),\n\\end{equation}\ndefined in \\cite[(3.12)]{ConnMosc}, onto the space of elementary characteristic $j$-cochains, \\cite[Section 3]{ConnMosc}.\n\n\\medskip\n\n\\noindent We are now ready to prove our claim. On the following proposition, $\\varphi \\in C^2_{{\\cal K}}({\\cal A},V)$ is the cyclic cocycle defined in \\eqref{aux-vp'-2-quick},\\eqref{aux-vp'-1-quick}, \\eqref{aux-vp'-0-quick}.\n\n\\begin{proposition}\nThe cyclic cohomology class $[\\mathscr{TF}] \\in HC^{4}({\\cal K},V)$ is mapped by $\\chi_\\varphi:HC^{4}({\\cal K},V) \\to HC^6({\\cal A}_\\Gamma)$ to the transverse characteristic class $[TF] \\in HC^6({\\cal A}_\\Gamma)$.\n\\end{proposition}\n\n\\begin{proof}\nBy the diagram \\eqref{aux-diagram-of-characteristic-maps} we understand that it is enough to observe $[\\wbar{\\chi}_\\varphi(\\mathscr{TF})] = [{\\rm TF}] \\in HC^6({\\cal H}_n)$. This, in turn, follows from the observation\n\\begin{equation}\n\\mu\\circ \\psi_{\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}([\\wbar{\\chi}_\\varphi(\\mathscr{TF})]) = \\mu\\circ\\psi_{\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}([{\\rm TF}]) = [1 \\otimes X_1 \\wedge X_2 \\wedge Y_1^1 \\wedge\\dots\\wedge Y_2^2],\n\\end{equation}\n thanks to the large kernel of \\eqref{aux-mu-map}. Hence the result follows since $\\mu\\circ \\psi_{\\vartriangleright\\hspace{-4pt} \\blacktriangleleft}$ is an isomorphism on the level of cohomologies.\n\\end{proof}\n\n\\noindent In the following we  present the image of the cyclic cocycles  $\\mathscr{GV}$, $\\mathscr{R}_1$, $\\mathscr{R}_2$, $\\mathscr{R}_3$,\n$\\mathscr{R}_4$ under the characteristic map  $\\chi_\\varphi: C^\\bullet({\\cal K}, V)\\rightarrow C^{\\bullet+2}({\\cal A})$. These are cyclic  cocycles in $C^\\bullet({\\cal A})$.  We do not display the detailed account of the  computation as it is  lengthy and straightforward.\n\n\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(\\mathscr{GV})(a_0 \\ot\\cdots\\ot a_5) = \\sum_{k=1}^3\\sum_{1 \\leq i,j \\leq 2}\\sum_{\\sigma,\\gamma,\\eta \\in S_2}2 \\cdot (-1)^\\sigma(-1)^\\gamma(-1)^{k-1} \\\\\n&\\Big\\{-\\tau\\left(a_0\\delta^i_{i\\eta(1)}(a_1)\\delta^j_{j\\eta(2)}(a_2) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n& + \\tau\\left(a_0\\delta^i_{i\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) \\delta^j_{j\\eta(2)}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0\\delta^i_{i\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^j_{j\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0\\delta^i_{i\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^j_{j\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^i_{i\\eta(1)}(a_2)\\delta^j_{j\\eta(2)}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^i_{i\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^j_{j\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^i_{i\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^j_{j\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^i_{i\\eta(1)}(a_3)  \\delta^j_{j\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^i_{i\\eta(1)}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^j_{j\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_3)\\delta^i_{i\\eta(1)}(a_4)\\delta^j_{j\\eta(2)}(a_5)\\right) \\Big\\}.\n\\end{split}\n\\end{align}\n\n\n\n\\begin{align}\\notag\n& \\chi_\\varphi(\\mathscr{R}_2)(a_0 \\ot\\cdots\\ot a_3)\\\\\\notag\n& = \\sum_{\\sigma \\in S_2}(-1)^\\sigma \\Big\\{-\\tau(a_0\\delta^1_{2\\sigma(1)}(a_1)\\delta^2_{1\\sigma(2)}(a_2)Y_2^2(a_3)) \\\\\\notag\n&- \\tau(a_0\\delta^2_{1\\sigma(1)}(a_1)\\delta^1_{2\\sigma(2)}(a_2)Y_2^2(a_3)) +\\tau(a_0\\delta^1_{2\\sigma(1)}(a_1)Y_2^2(a_2)\\delta^2_{1\\sigma(2)}(a_3))\\\\\\notag\n& + \\tau(a_0\\delta^2_{1\\sigma(1)}(a_1)Y_2^2(a_2)\\delta^1_{2\\sigma(2)}(a_3))-\\tau(a_0Y_2^2(a_1)\\delta^1_{2\\sigma(1)}(a_2)\\delta^2_{1\\sigma(2)}(a_3)) \\\\\n& -\\tau(a_0Y_2^2(a_1)\\delta^2_{1\\sigma(1)}(a_2)\\delta^1_{2\\sigma(2)}(a_3))\\Big\\}.\n\\end{align}\n\\begin{align}\n \\chi_\\varphi(\\mathscr{R}_3)(a_0 \\otimes a_1 \\otimes a_2) = \\sum_{1 \\leq i,j \\leq 2}\\sum_{\\sigma \\in S_2}2 \\cdot (-1)^\\sigma \\tau(a_0\\delta^i_{i\\sigma(1)}(a_1)\\delta^j_{j\\sigma(2)}(a_2)).\n\\end{align}\n\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(\\mathscr{R}_1)(a_0 \\ot\\cdots\\ot a_5) = \\sum_{k=1}^3\\sum_{\\sigma,\\gamma,\\eta \\in S_2}(-1)^\\sigma(-1)^\\gamma(-1)^{k-1} \\Big\\{\\\\\n&-\\tau\\left(a_0\\delta^1_{2\\eta(1)}(a_1)\\delta^2_{1\\eta(2)}(a_2) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0\\delta^2_{1\\eta(1)}(a_1)\\delta^1_{2\\eta(2)}(a_2) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n& + \\tau\\left(a_0\\delta^1_{2\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) \\delta^2_{1\\eta(2)}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n& + \\tau\\left(a_0\\delta^2_{1\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) \\delta^1_{2\\eta(2)}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0\\delta^1_{2\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^2_{1\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0\\delta^2_{1\\eta(1)}(a_1)Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^1_{2\\eta(2)}(a_4) Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0\\delta^1_{2\\eta(1)}(a_1) Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^2_{1\\eta(2)}(a_5)\\right) \\\\\n&+\\tau\\left(a_0\\delta^2_{1\\eta(1)}(a_1)Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^1_{2\\eta(2)}(a_5)\\right)\\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^1_{2\\eta(1)}(a_2)\\delta^2_{1\\eta(2)}(a_3) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^2_{1\\eta(1)}(a_2) \\delta^1_{2\\eta(2)}(a_3)Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_4)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^1_{2\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^2_{1\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^2_{1\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  \\delta^1_{2\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^1_{2\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^2_{1\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1)\\delta^2_{1\\eta(1)}(a_2) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^1_{2\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^1_{2\\eta(1)}(a_3)  \\delta^2_{1\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&-\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^2_{1\\eta(1)}(a_3)  \\delta^1_{2\\eta(2)}(a_4)Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_5)\\right) \\\\\n&+\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^1_{2\\eta(1)}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^2_{1\\eta(2)}(a_5)\\right) \\\\\n&+\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)\\delta^2_{1\\eta(1)}(a_3)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_4)\\delta^1_{2\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0 Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_3)\\delta^1_{2\\eta(1)}(a_4)\\delta^2_{1\\eta(2)}(a_5)\\right) \\\\\n&-\\tau\\left(a_0Y_{\\mu^k(\\sigma(1))}^{\\mu^k(\\sigma(2))}(a_1) Y_{\\mu^k(\\gamma(1))}^{\\mu^k(\\sigma(1))}(a_2)  Y_{\\mu^k(\\gamma(2))}^{\\mu^k(\\sigma(1))}(a_3)\\delta^2_{1\\eta(1)}(a_4)\\delta^1_{2\\eta(2)}(a_5)\\right)\\Big\\}.\n\\end{split}\n\\end{align}\nFinally,\n\\begin{align}\n\\begin{split}\n& \\chi_\\varphi(\\mathscr{R}_4)(a_0 \\otimes a_1 \\otimes a_2) =\\varphi(c_2 \\otimes a_0 \\otimes a_1 \\otimes a_2)\\\\\n&=\\sum_{\\sigma \\in S_2}(-1)^\\sigma  \\Big\\{\\tau(a_0\\delta^1_{2\\sigma(1)}(a_1)\\delta^2_{1\\sigma(2)}(a_2)) + \\tau(a_0\\delta^2_{1\\sigma(1)}(a_1)\\delta^1_{2\\sigma(2)}(a_2))\\Big\\}.\n\\end{split}\n\\end{align}\n\n\\begin{remark}{\\rm\nOne knows that the characteristic map  $\\chi_\\tau:C^\\bullet({\\cal H}, {\\mathbb C}_\\delta)\\rightarrow C^\\bullet({\\cal A})$ is injective \\cite{ConnMosc98}.  Since $\\chi_\\varphi(\\mathscr{TF})$, $\\chi_\\varphi(\\mathscr{GV}),$  $\\chi_\\varphi(\\mathscr{R}_1),$  $\\chi_\\varphi(\\mathscr{R}_2),$  $\\chi_\\varphi(\\mathscr{R}_3),$  and $\\chi_\\varphi(\\mathscr{R}_4)$,  are all in the range of $\\chi_\\tau$, as a byproduct  of our study in this paper,  one  calculates cyclic cocycles representing a basis for    $HP^\\bullet({\\cal H}_2, {\\mathbb C}_\\delta)$.  }\n\\end{remark}\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRed supergiants (RSGs) are the evolved descendants of 8-30$M_\\odot$ stars.   Once their main-sequence\nprogenitor OB stars exhaust the hydrogen in their cores, hydrogen fusion in a shell around the inert\nhelium core results in rapid expansion of their outer layers.  The stars zip across the H-R diagram, cooling and expanding tremendously.  By the time the stars have reached equilibrium at the Hayashi\nlimit, helium has ignited in the cores.  At this point the star has expanded to radii of 500-1800$R_\\odot$, with\neffective temperatures of 3500-4500~K. (For a recent exposition, the reader is referred to \\citealt{LevesqueRSGs}; see also \n\\citealt{Dorda2016} and \\citealt{Sylvia}.)\nUnevolved single stars of higher masses ($>\\sim 30M_\\odot$) do not evolve to the RSG phase, but instead\nspend their He-burning years as Wolf-Rayet (WR) stars after experiencing extensive mass loss, possibly aided by first undergoing a luminous blue variable (LBV) phase.   Stars of lower mass ($<8M_\\odot$) evolve instead to  red {\\it giants}, (not  {\\it supergiants}), and are not considered massive stars, as they are not expected to end as a core-collapse supernova.  Binary interactions may modify this picture, most notably providing a channel to the WR stage by stars that would otherwise become RSGs (see, e.g., \\citealt{1996LIACo..33...55D,sana12,2017IAUS..329....3M}).\n\n\n\n\nAn accurate knowledge of the RSG content of nearby galaxies is useful for testing models of massive star evolution. As \\citet{Kipp} put it, these evolved stages act a ``sort of magnifying glass\" on stellar models, ``revealing relentlessly the faults of calculations of earlier phases.\"  \nFor instance, \\citet{Maeder80} first pointed out that the relative number of RSGs and WRs should be a very sensitive function of metallicity, and indeed \\citet{MasseyARAA} found such an effect observationally, but showed there was remarkably poor quantitative agreement with theoretical predictions.  However, in retrospect, neither the RSG content nor WR numbers were known accurately at that time.  Further complications in interpreting these numbers have been discussed by \\cite{2018ApJ...867..125D} and \\cite{2020MNRAS.497.2201S}. \n\n Another example of how evolved massive stars can help us is in improving our understanding of binary influence on massive star evolution, and its metallicity dependence.  The RSG binary frequency can be compared to that of unevolved massive stars, providing a stringent test on the interaction rates.  About ten RSG binaries are known in the Milky Way \\citep{NeugentRSGBinII}. A spectroscopic survey of cool supergiants in the Small and Large Magellanic Clouds (SMC, LMC)  by \\citet{2015A&A...578A...3G} identified several likely physical binaries.  Directed searches for RSG binaries by \\citet{NeugentRSGBinII}  led to the discoveries of  of 87 in LMC, M31, and M33.  Follow-up work in the LMC identified \\citet{LMCBins} 37 more, and allowed a determination of the RSG binary frequency in the LMC.  In a paper being submitted contemporaneously with this one, \\citet{M31M33RSGBins} has used the sample of RSGs identified here, along with new spectroscopy and {\\it HST} UV data, to establish the RSG binary frequency in M31 and M33, demonstrating a significant metallicity dependence.\n  \nTo support this work, we must first identify relatively complete populations of RSGs in galaxies of various metallicities. Work by \\citet{Yang2019} has recently identified the RSG content in the Small Magellanic Cloud (SMC).  \\citet{LMCBins} subsequently used a similar technique to obtain a census of the RSG population of the LMC. In this paper, we determine a luminosity-limited, spatially complete sample of RSGs for the nearby spiral galaxies M31 and M33.  Both galaxies have long been known to be rich in massive stars; the results of recent spectroscopic survey can be found in \\citet{BigTable}.  \n\nThese galaxies provide a high-metallicity complement to studies in the SMC and LMC, where the young-population metallicities are 25\\% and 50\\% of the solar value \\citep{Russell1990}.  For M31 the metallicity (at 10~kpc) is 1.5-2$\\times$ solar with only a modest galactocentric gradient \\citep{Zaritsky1994,Venn2000,Sanders}, while the metallicity of M33 varies from super-solar in the central regions to SMC-like in the outer regions \\citep{Zaritsky1994,Urbaneja2005}; see further discussion in \\citealt{NeugentM33} and \\citealt{M31M33RSGBins}.\n \nStudies of the RSG content of nearby galaxies have traditionally been hampered by two problems.  The first of these is foreground contamination.  Although such contamination is relatively low ($<$10\\%) for RSGs in the Magellanic Clouds \\citep{Massey2002,NeugentLMC}, it is 50\\% or more for the more distant members of the Local Group, such as M31 and M33 \\citep{MasseyRSGs}. Optical two-color diagrams can help distinguish between foreground dwarfs and extragalactic supergiants \\citep{MasseyRSGs}, although radial velocities obtained at the Ca\\,{\\sc ii} triplet remained the ``gold standard\" (e.g., \\citealt{DroutM33}).  However, with the {\\it Gaia} Data Release 2 (DR2, \\citealt{DR2}), parallaxes and proper motions for many of the stars in the relevant magnitude range allow us to readily reject most foreground interlopers. \n\nThe second problem has been contamination by these galaxies' own asymptotic giant branch (AGB) stars.  These stars are a late evolutionary stage in low- and intermediate-mass stars, consisting of an inert core, and powered by a helium or hydrogen shell, surrounded by an extended envelope.   \\citet{Brunish86} warned that AGB contamination in RSG samples were a potential issue since the higher luminosity AGBs would overlap with lower luminosity RSGs.  In their study of Magellanic Cloud RSGs, \\citet{MasseyOlsen} used a luminosity cutoff $\\log L\/L_\\odot$ of 4.9 dex consistent with \\citet{Brunish86}'s models, but the exact upper luminosity limit of AGBs are poorly determined.  Some ``super-AGB\" stars (the highest mass AGBs) are thought to have luminosities as high as $\\log L\/L_\\odot$ of 5.1 (see, e.g.,   \\citealt{2018A&A...609A.114G}.  However, restricting the sample in this way is unappealing, as RSGs of this luminosity are\ntypically 15$M_\\odot$ objects, while the RSG population extends down luminosities of $\\log L\/L_\\odot$ of 4.0 (i.e., $\\sim 8M_\\odot$).  Fortunately, AGBs and RSGs can be distinguished on the basis of their near-IR {\\it J-K} colors, as AGBs are significantly cooler than RSGs of the same K-band brightness, with corresponding redder colors.  This was nicely demonstrated for a sample of red stars in the SMC by\n\\citet{Yang2019}, following the work of \\citet{2006AA...448...77C,2006A&A...452..195C} and \\citet{2011AJ....142..103B}, and was used successfully by \\citet{LMCBins,UKIRT} in identifying the RSGs of the LMC and two fields of M31.\n\n\n\n\\section{Observations and Photometry}\n\nOur NIR imaging of M31 and M33 are archival in the sense that in both cases the data were taken for other purposes. We describe the characteristics of each data set below, and summarize some of the basic properties of the surveys in Table~1.\n\n\\subsection{M31}\nThe {\\it J} and {\\it K$_s$} images of M31 were taken as part of the Andromeda Optical and Infrared Disk Survey ({\\sc androids}) aimed primarily at obtaining a uniform surface brightness map across 5 deg$^2$ region of disk and bulge.   The project, calibration, and reduction procedure are described by \\citet{2014AJ....147..109S}.\n\nThe images were taken with the wide-field NIR camera WIRCam \\citep{2004SPIE.5492..978P} on the 3.6-m Canada France Hawaii Telescope located atop Maunakea.  The dataset consists of 70 usable {\\it J}- and {\\it K$_s$}-band image pairs.  The data were taken in queue mode during four blocks of time:  nine nights during (UT) 2007 Aug 1-Sep 23 (27 fields); 12 nights during 2009 Aug 6-Oct 28 (12 fields); four nights during 2011 Sep 18-Oct 9 (4 fields); and six nights during 2012 Oct 6-Oct 29 (27 fields).  (In addition, there were images obtained\nfor 3 fields obtained on 2009 Oct 29 and 1 field on 2012-Oct 10 that we judged unusable due to reduction issues.)    Each field was observed in 16 $\\times$ 47~s exposures (i.e., 752 s total) in {\\it J}  and 26 $\\times$ 25~s exposures (i.e., 650 s total) in {\\it K$_s$}.  An equivalent amount of time was spent observing sky: because of crowding and the large angular extent of M31, the sky-subtraction was achieved by nodding the telescope away from the M31 field by one or more degrees.  The nodding strategy is discussed\nextensively in \\citet{2014AJ....147..109S} based upon the first two observing seasons.  A customized \npipeline, described by \\citet{2014AJ....147..109S}, produced the final, stacked images that we used in\nour analysis.  They were made available for our project through co-author S. C. \n\nThe image scale of each image is 0\\farcs3 pixel$^{-1}$.  Although nominally each field covered 21\\farcm5 $\\times$ 21\\farcm5, we found that there were significant issues along the edges, and trimmed each image to the central 4001$\\times$4001 pixels, i.e., 20\\farcm0$\\times$20\\farcm0.  This had no effect on the actual areal coverage, given the large overlap between adjacent frames.  Figure~\\ref{fig:m31cfht} shows the area surveyed, which includes roughly 5 deg$^2$.  The pipeline had provided an accurate world-coordinate-system, vastly simplifying our task of combining photometry from so many overlapping frames.\n\nPhotometry was complicated by the fact that the seeing was so good ($\\sim$0.5-0.6\\arcsec) compared to the image scale (0\\farcs3 pixel$^{-1}$) that the point-spread-functions (PSFs) were generally under-sampled. That, combined with the large quantity of fields, made PSF-fitting an unattractive option.  After numerous experiments, we concluded that we could do little better than standard aperture photometry.  We used {\\it daofind} within the  {\\sc iraf} environment to identify (mostly) stellar objects 3$\\sigma$ above the expected noise based upon the equivalent read-noise and gain.  We then performed aperture photometry with {\\it phot} using a 3-pixel (0\\farcs9) radius aperture; sky was determined locally using a modal definition in an annulus extending from 3\\farcs0 to 6\\farcs0. Attempts to centroid before performing the photometry resulted in many problems, primarily moving fainter detections towards brighter neighboring objects, or moving a detection from a stellar source to a nearby diffraction spike from a very bright star.  Therefore we performed the photometry centered on the {\\it daofind} positions.\n\nThe photometry for each frame and filter were then matched with sources from the 2MASS catalog \\citep{2MASS} in order to determine photometric zero-points.   Although the most heavily saturated stars had been masked out in our images by the pipeline reduction, we found that stars that matched with 2MASS sources brighter than $J=14.3$ or $K_s=13.3$ showed symptoms of non-linearity issues. Restricting the sample\nto 2MASS stars fainter than these limits, and with photometric ratings of ``A\" in the 2MASS catalog, we were then able to calibrate each frame photometrically.  Typically 150-200 stars were used per frame, with zero-point standard deviations of 0.01-0.02~mag, and standard deviations of the mean below 0.005~mag.\n\nThe calibrated {\\it J} photometry from all 70 fields were then combined and averaged, and the same process repeated for the {\\it K$_s$}-band data.   Finally the {\\it J-} and {\\it K$_s$}-band photometry were matched.  Only stars detected in both filters were retained.  We removed anything fainter than $J=17.5$ or $K_s=19.5$ or\nbrighter than $J=14.3$ or $K_s=13.3$.  \n\nThese bright limits are very similar to what we expect for the brightest RSGs in M31; see, for example, Figure 8 in \\citet{UKIRT}.  Nevertheless, we decided to add to this sample any 2MASS stars that were not already in our object list.  In order to allow for the irregular field boundaries, we simply required that a 2MASS star be within 1\\arcmin\\ of a star with existing photometry.  This added about 6,000 stars, bringing the total to 712,716 stellar sources with photometry.  Of course, we expect only a small fraction (a few percent) to be in the color-magnitude range of RSGs after we remove foreground objects. \n\nWe show the photometric errors as a function of magnitude in Figure~\\ref{fig:errors}.   The vast majority of stars have a photometric precision better than 0.02~mags to our faint limit.  Due to the large number of stars (over 700,000) we have plotted only every tenth point for clarity. \n\n\\clearpage\n\\subsection{M33}\n\nThe $J$ and $K_s$ data for M33 were obtained as part of a study of AGB stars in M33 and are described in detail by \\citet{2008A&A...487..131C}.  The data were subsequently used by \\citet{2015MNRAS.447.3973J} as part of their study of the variability of red giant stars in M33.  \n\nThe data were taken under program U\/05B\/7 with the Wide Field Camera (WFCam) on the 3.8-m United Kingdom Infrared Telescope (UKIRT) located on Maunakea\\footnote{When the data reported here were acquired, UKIRT was operated by the Joint Astronomy Centre on behalf of the Science and Technology Facilities Council of the U.K.}.\nThe camera consists of four arrays such that with four pointings the gaps between the arrays can be filled in to prove spatial coverage of  52\\farcm0 $\\times$52\\farcm0 with a native image scale of 0\\farcs4 pixel$^{-1}$.  (The ensemble of four such pointings is referred to as a ``tile.\")  Five such tiles centered on M33 were observed, with some overlap, resulting in coverage of 3 deg$^2$, as shown in Figure~\\ref{fig:m33ukirt}.  The equivalent exposure times were 150~s in $J$ and 270~s in $K_s$. (Details of the dithering scheme are given in \\citealt{2008A&A...487..131C}.)   The data were taken in queue mode on (UT) 2005 Sep 29-30, Oct 24, Nov 5, and Dec 16.\nThe data were reduced through the standard Cambridge Astronomy Survey Unit (CASU) pipeline\\footnote{http:\/\/casu.ast.cam.ac.uk\/}, which produces a chip-by-chip source list with $J$ and $K_s$ values \\citep{2004SPIE.5493..411I} calibrated from 2MASS along with source classification (i.e., stellar vs.\\ non-stellar), and kindly provided to us by the program's P.I., Michael Irwin.  For our analysis we followed the recommendations of the CASU documentation and employed their photometry using a 3\\farcs0 radius aperture; these fluxes had been corrected for sky using a locally interpolated value from their\ntwo-dimensional background-following algorithm.\n\nIn order to\nmake sense out of the 160 source lists (4 arrays $\\times$ 4 pointing per tile $\\times$ 5 tiles in each of the two bandpasses), we first combined all of the measurements on a filter-by-filter basis.  We then averaged the photometry for multiple measurements of the same star.   The averaged $J$ and $K_s$ photometry was\nmatched with each other, requiring agreement in the coordinates to 0\\farcs8 (2 pixels).  In doing so, we paid careful attention to the classification flags.  These are produced by the\nCASU pipeline, with values denoting ``stellar,\" ``non-stellar,\" ``noise,\" ``marginal stellar,\" and ``saturated\"\\footnote{See https:\/\/research.ast.cam.ac.uk\/vdfs\/docs\/catalogues.pdf}.\nWe found that for objects with multiple detections there was not good consistency between the values\nassigned to the same object.  After much experimentation, we found that we obtained the most\nsatisfactory source list if we required only that an object be classified as ``stellar\" or ``marginally stellar\"  in\none of the two filters. \n\nAlthough the original source lists were nominally calibrated with respect to 2MASS photometry, a comparison of\nour final source list with that of 2MASS showed that small corrections were needed to bring them into good\naccord; we adjusted the CASU $J$ magnitudes by +0.023~mag and $K_s$ by -0.019~mag.  These values are\ncomparable to what were found by in analysis of our own UKIRT WFCam data on M31 \\citep{UKIRT}.  Similar to that study, stars brighter than $J=12$ or $K_s=12$ began to deviate systematically from the 2MASS values, although they were not flagged as saturated.  Thus, we removed these stars, and added in any 2MASS stars in the same region that were not included in the edited UKIRT photometry catalog.  The final\nphotometry list included 121,328 stellar objects.  As with M31, we expect only a small fraction to be in the color-magnitude range of RSGs and pass our membership criteria.\n\nWe show the photometric errors as a function of magnitude in Figure~\\ref{fig:errors}.   M33 is a ``cleaner\" field than that of M31, without a large bulge, and with considerably less crowding; hence the photometric errors are smaller, in general.\nAgain we see that the majority of our photometric errors are $<$0.02~mag to our faint limit.\n\n\\section{The Identification of RSGs}\n\nIn this section, we proceed to identify RSGs following the same technique as used by \\citet{LMCBins} for the LMC and for a previous study by a smaller region of M31 by \\citet{UKIRT}.  We begin by using {\\it Gaia} to remove many of the foreground stars, and then define a region in the ($J-K_s$, $K_s$) color-magnitude diagram that contains the RSGs but not the AGBs.  Finally, we sanitize the sample by eliminating visually obvious non-RSGs.\n\n\\subsection{Identifying Foreground Stars with {\\it Gaia}}\n\nPrior to the availabilty of the high-quality proper motions and parallaxes that {\\it Gaia} provides, the only way to definitively differentiate between foreground Galactic red stars and M31\/M33 RSGs was by the use of radial velocities (see, e.g., \\citealt{MasseyRSGs,MasseySilva,DroutM31,Massey2016}). Even so, this method was not effective for stars in the north-east quadrant of M31 as there was too much overlap there between the expected radial velocities of the two groups  due to M31's rotational velocity.  In addition, stars in the Galactic halo will have radial velocities that are similar to the\nsystemic velocities of these galaxies, as much of that motion is actually the reflex motion of the sun (see, e.g., Figure 5 in \\citealt{Stephane}), making it hard to weed out the occasional Galactic halo red giant star.  Broad-band optical photometry also provides a clue, as there is a relatively clean separation between dwarfs and supergiants in $B-V$ at a given $V-R$ \\citep{MasseyRSGs} due to the effects of surface-gravity dependent line-blanketing in the blue, although it is clear from radial velocity studies that the method is not completely reliable (see Figure 3 in \\citealt{Massey2016}). Fortunately, Data Release 2 (DR2) of {\\it Gaia} \\citep{DR2} has revolutionized this process. \n\nNevertheless, the use of DR2 is not completely straight-forward; to do this properly requires taking spatially-dependent zero-point issues into account as well as considering the intrinsic dispersion in parallax and proper motions within a sample.   As with previous applications, we have followed a process similar to that of \\citet{2018A&A...616A..12G}, using a co-variance analysis to separate members from non-members \\citep{ErinLBV,UKIRT}, having first established the distributions of parallaxes and proper motions using stars whose {\\it Gaia} magnitudes and colors makes it highly likely that they are members.  After cleaning the sample, we find average values for the parallax of -0.033~mas,  and 0.028 mas yr$^{-1}$ and -0.014 mas yr$^{-1}$ for the proper motion in right ascension and declination, respectively, for M31.  For M33 the values are -0.081~mas for the parallax, and 0.103 mas yr$^{-1}$ and 0.034 mas yr$^{-1}$ for the proper motions. With our expected values of proper motions and parallax defined from the median of this sample, we then compare the {\\it Gaia} measurements for all the stars in our sample against these values.  A star is considered to be foreground if its proper motion and parallax fall beyond the region that contains 99.5\\% of the comparison sample, while a star is considered to be a definite member if it falls within 75\\% of the comparison sample.  The membership is considered to be uncertain if the star's value falls between this range. \n\nHowever, the \\citet{2018A&A...616A..12G} methodology makes sense really only if the membership scatter is dominated by intrinsic dispersion and not the uncertainties in the individual values.   For our M31\/M33 sources, that is not the case.  Consider simply the proper motions.  Massive stars in M31 have radial velocities that extend from nearly $-600$ km s$^{-1}$ to $-50$ km s$^{-1}$ due to the galaxy's rotational velocity (see, e.g., Figure 2 in \\citealt{Massey2016}). Of course, the space motions perpendicular to the radial velocity will be much smaller than this since M31 is seen mostly edge on (inclination 77$^\\circ$ according to \\citealt{vandenbergh2000}).  Nevertheless, even if the intrinsic dispersion in the plane of the sky were 500 km s$^{-1}$, at the distance of M31 (760~kpc, \\citealt{vandenbergh2000}), this translates to an angular proper motion dispersion of  0.14 mas yr$^{-1}$.  By comparison,\ntypical uncertainties in the proper motions in our sample are 1-2 mas yr$^{-1}$, an order of magnitude larger.\nIn other words,  the intrinsic scatter is negligible compared to the individual measurement uncertainties.  This is even more true for the parallaxes, where a 20~kpc long galaxy arranged along the line of sight at the same distance of M31 would have an intrinsic dispersion of 3.5$\\times10^{-5}$ mas, compared to a typical uncertainty in the parallax measurements of our stars of 1~mas.\n\nThat being the case, we applied an additional test for stars which the covariance procedure flagged as non-members.  If both the parallax and proper motions were within 2.5$\\sigma$ of the expected values (where $\\sigma$ is measurement uncertainty), then our classification was changed from ``non-member\" to ``uncertain.\"  We have color-coded the resulting membership determinations in Figures~\\ref{fig:M31CMD} and \\ref{fig:M33CMD}.   We have also used the\nBesan\\c{c}on Galactic model \\citep{Besancon} to simulate the expected distribution of foreground at the corresponding Galactic longitude and latitudes and covering an area of the same size as our surveys; we see that there is excellent agreement in where in the CMD we find foreground contamination.\n\nThe distribution of membership in the CMDs is very similar for the two galaxies.  There is a virtual ``wall\" of foreground stars (shown in red) at $J-K_s\\sim 0.8-0.9$.  These colors correspond to late K and early M dwarfs  (Table 7.6 of \\citealt{Cox}).  At magnitudes fainter than $K_s\\sim16-16.5$,  the members (shown in black) and foreground stars begin to blend together, and there are increasing number of stars with ambiguous results (green).  Fortunately this is corresponds to $\\log L\/L_\\odot$ of 4, below which we lose interest.  The triangle of fainter ($K_s<15.5$), redder stars (centered at $J-K_s\\sim 1.5$) mostly lack {\\it Gaia} data (as indicated by blue), but these are the stars that we expect to be AGBs (see, e.g., \\citealt{UKIRT}.)   The comparison with the Besan\\c{c}on models confirms that although most of these fainter stars lack {\\it Gaia} data, we are justified in our interpretation that these are indeed members.\n\n\n\n\\subsection{Selecting RSGs from the Color-Magnitude Diagram}\n\nFollowing our work in the LMC \\citep{LMCBins}, and a small portion of M31 \\citep{UKIRT}, we now identify the region in the CMD where RSGs are found, as shown in Figures~\\ref{fig:M31RSGs} and \\ref{fig:M33RSGs}.\nOur ability to separate the RSGs from AGBs stems from the fact that the Hayashi limit shifts to cooler temperatures at lower masses \\citep{HayashiHoshi}; thus the AGBs, which are lower in mass than the RSGs, are found at redder colors.  At the same time, the Hayashi limit shifts to cooler temperatures at higher metallicities.  Thus there is more overlap between the RSGs and AGBs in M31 at faint magnitudes than in M33.\nWe give the adopted boundaries in Table~\\ref{tab:FunFacts}, where we have retained the $K_0$ and $K_1$ notation of \\citet{2006AA...448...77C}.  \n\nThe $K_1$ line denotes the equation separating RSGs from AGBs.  However, for brighter stars, we have expanded the\nallowed region to redder colors as there are clearly some stars in this region, and yet these stars are much brighter than\nexpected for even the brightest AGB stars. (See discussion in \\citealt{LMCBins}.)  As described below, we assume these\nstars are bright RSGs with extra (presumed) circumstellar reddening.   \n\nIn the \\citet{Yang2019} study which pioneered this technique to identify the SMC's RSG population, they defined the low $J-K_s$ cutoff line  ($K_0$) as simply parallel the $K_1$ line.  We followed a similar procedure in \\citet{UKIRT} in our study of two M31 fields.  However, we have since come to realize that this has no physical basis, and runs the risk of excluding K-type RSGs at brighter magnitudes.  Furthermore, it tends to remove any RSG binaries from the sample, as these will have their colors affected by their bluer companions.\nThus in our analysis of the LMC RSG population \\citep{LMCBins}, we used a constant $J-K_s$ cutoff except for the faintest stars, which are still likely contaminated by foreground stars.   We have followed a similar procedure here. \n\n\nHaving make our selection of RSGs based on this CMD, we then performed some manual editing.  Visual inspection of the Local Group Galaxies Survey (LGGS) R-band images \\citep{LGGSI} was used to eliminate obvious galaxies, a few of which were misidentified as potential RSGs.  The CFHT data also included M32 and NGC~205, dwarf elliptical companions to M31, visible in Figure~\\ref{fig:m31cfht} to the south and north-west, respectively.  We eliminated any spurious detections in the nearby area, i.e, in the rectangular areas\n$\\alpha$=00:42:28.6 to 00:42:55.0, $\\delta$=40:49:25 to 40:54:25 (M32), and  $\\alpha$=00:39:55.0 to 00:40:40.0, $\\delta$=41:35:10 to 41:48:15 (NGC~205).  There were also numerous detections near the bulge of M31 which were spurious due to the extra noise introduced by the background, and these were also eliminated from our list; we excluded the rectangular region $\\alpha$=00:42:30.0 to 00:43:00.5 and $\\delta$=41:13:30 to 41:19:00.  M33 had far fewer problems.\n\n\nThe list of RSGs is given in Tables~\\ref{tab:M31RSGs} and \\ref{tab:M33RSGs} for M31 and M33, respectively. \nWe include the IDs, V magnitudes, and any spectral information from the LGGS, using the most recent version \\citep{Massey2016}.  There are 38,817 photometrically identified RSGs in the M31 list, and 7,088 in the M33 list. Of these, 31,444 and 4,819 are within their galaxy's Holmberg radius.  As we emphasize below, one must take care to use proper luminosity cuts in discussing the numbers of RSGs.    Only 238 and 197 had been as RSGs in M31 and M33 respectively in the LGGS.   Additional spectroscopy of both galaxies will be discussed in the companion binary frequency paper \\citep{M31M33RSGBins}.   Prior to finalizing the RSG lists, we used the matches with the LGGS to remove several QSOs that had been identified in \\cite{M31M33QSOs}. In addition, both lists originally contained several WN-type Wolf-Rayet stars; presumably their strong emission lines affected\ntheir $J-K_s$ colors, causing them to appear red.  We removed these, with the exception of J004453.06+412601.7, which is described in the LGGS as ``WN3+M:\".  Whether or not this star is a line-of-sight coincidence between a WR star and an M dwarf, or an actual WR+RSG binary, is discussed in the binary paper \\citep{M31M33RSGBins}. We also removed two emission-lined luminous blue variables from the lists for each galaxies.  There are a handful of remaining stars whose classifications do not agree with their being RSGs: 32 of the M31 photometrically selected RSGs have previously been been classified as OB stars; there are eight such stars in M33.  In some cases, these are listed as having M-type companions; it is possible that the others are binaries as well.   These need to be re-checked spectroscopically.\n\n\n \\section{Transformation to Physical Properties and Comparison with Evolutionary Tracks}\n\\label{Sec-extinct}\n\nHaving identified the RSG populations of M31 and M33, we will next use our photometry to determine the\nphysical properties of effective temperatures and luminosities.  This will allow us to compare the distribution\nof stars in the H-R diagram to that predicted by the evolutionary tracks, and at the same time, it will facilitate the future work\nwe describe in the introduction.  We emphasize that it makes little sense to talk about ``the total number of RSGs\" without specifying some luminosity limit associated with that number, and to determine that we must do these transformations.\n\n\\clearpage\n\\subsection{Reddening Corrections}\nBy far the most uncertain part of this process is what to assume for the extinction towards these stars.  In general, studies of OB stars in both M31 and M33 have revealed only modest reddening.   In many ways, this is most surprising for M31, given that the galaxy is inclined 77$^\\circ$ \\citep{vandenbergh2000} to our line of sight.  Spectroscopy and photometry of stars in four OB associations of M31 by \\citet{MAC86} showed color excesses of $(B-V)$ ranging from 0.08 to 0.24, i.e., $A_V$=0.25-0.75.  Using the LGGS photometry, \\citet{LGGSII} used the ``blue plume\" of OB stars to estimate the average reddening of OB stars M31, finding a very similar value, $E(B-V)=0.13$ ($A_V=0.4$).  The same method yielded\nessentially identical results for M33.   By contrast, an analysis of the Panchromatic Hubble Andromeda Treasury (PHAT) photometry by \\citet{PHATDust} suggest an average extinction of $A_V\\sim$1~mag for stars in the disk of M31.  We discuss this issue further below, in Section~\\ref{Sec-discussion}. \n\nThe reddening situation is further complicated by the fact that RSGs are known to make their {\\it own} dust, and in general show circumstellar extinction in excess of that of OB stars in the same region \\citep{Smoke}.  Model fitting of a sample of M31's RSGs by \\citet{MasseySilva} yielded direct measurements for $A_V$ for a sample of RSGs in M31.  \\citet{UKIRT} noted that these results showed a luminosity dependent to the amount of extinction, and adopted $A_V=0.75$ for stars fainted than $K=14.5$, but a linear increase in $A_V$ for stars brighter than that\\footnote{Note that although we are conducting our study in the NIR, we discuss the extinction in terms of $A_V$ simply for comparison with other studies; the conversion is given in Table~\\ref{tab:FunFacts}.}.   They found that such a procedure was not only consistent with the handful of RSGs with direct $A_V$ measurements, but also resulted in much better agreement with the evolutionary tracks than would have occurred by adopting a constant $A_V$ (see their Figure 10).  Physically this also makes sense: RSGs above some mass limit ($\\sim 20M_\\odot$) flirt with the Eddington limit during their evolution \\citep{Sylvia}, resulting in large episodic mass loss.  \n\nWe have used\nthe same procedure here; the relevant relations are given in Table~\\ref{tab:FunFacts}.    In addition, allowing for very\nred stars ($J-K_s \\sim 1.5$) to be included in the RSG list as long as they were brighter than that expected for AGBs\nmeant that \\citet{LMCBins} had to apply an additional reddening correction for those, with the assumption that their\nintrinsic colors were similar to that of other RSGs.  Thus their location the color excess in $J-K_s$ was computed by\nmoving along a reddening line to center of the RSG distribution.  This resulted in a small number of bright stars having a larger correction for extinction.   We adopt similar relations for M33. The relevant equations are given in Table~\\ref{tab:FunFacts}.  We include the adopted $A_V$ values along with our photometry in Tables~\\ref{tab:M31RSGs} and \\ref{tab:M33RSGs}.\n\n\nOne of the great advantages in conducting our study using NIR photometry is that we are less sensitive to reddening than would otherwise be the case.  The extinction at $K$ is only 12\\% that of $V$.  Thus, even a 1~mag uncertainty in $A_V$ translates to an uncertainty of 0.12~mag in $A_K$.  The impact this has on the luminosity determination must take into account the uncertainty in $J-K_s$ and hence in the effective temperature as well.  Using the relations in Table~\\ref{tab:FunFacts} we find that $\\Delta E(J-K) = 1.46 \\Delta A_K$.  For an uncertainty of 0.12~mag in $A_K$ this translates to an uncertainty 0.18 mag in $E(J-K)$.  This would change the\ntemperature by about 300~K, and the bolometric correction by 0.23~mag.  Thus the overall effect on the luminosity would be 0.15~dex. \n\n\n\n\\subsection{Transformations to Physical Properties}\n\nThere are several steps needed to transform the ($J-K_s$, $K_s$) photometry into effective temperatures and bolometric luminosities.  First, the photometry must be transformed from the 2MASS system to the more\nstandard \\citet{BessellBrett} system, as these will provide the connection with effective temperatures via stellar atmopshere models.  The transformation equations from 2MASS to the standard system are given by \\citet{Carpenter}, and repeated in Table~\\ref{tab:FunFacts}.  Next, the photometry is corrected for reddening and extinction based upon the adopted $A_V$ values derived as explained above.  The relationships between $A_V$, $A_K$, and $E(J-K)$ come from \\citet{schlegel}, and are also summarized in Table~\\ref{tab:FunFacts}.  \n\nThe basic transformation equations between these intrinsic colors and the desired values for the effective temperatures and bolometric corrections are based upon the MARCS stellar atmospheres \\citep{Marcs75,Marcs92,marcs} computed\nat various metallicities specifically for the low surface gravities of RSGs by B.\\ Plez in our collaborative studies of RSGs in the Milky Way \\citep{Levesque2005}, Magellanic Clouds \\citep{EmilyMC}, and M31 \\citep{MasseySilva}.   A linear equation between intrinsic $(J-K)_0$ colors and effective temperature works very well for RSGs.  There are small differences associated with the metallicity of the models.  Following \\citet{UKIRT}, for M31 we adopted a compromise between the solar-metallicity and 2$\\times$ solar metallicity models, as a value of 1.5$\\times$ solar is more in keeping with recent determinations of the metallicity of young stellar population \\citep{Sanders}, with evidence of little or no gradient (see also \\citealt{Zaritsky1994}).  M33 has a stronger metallicity gradient, but the absolute values are very poorly established.  Thus we found there was little benefit in providing a correction based on galactrocentric distance. Instead, we adopted the same transformation equation as used in our recent LMC \nstudy \\citep{LMCBins},  corresponding to 0.5$\\times$ solar, as that is fairly representative of studies of OB stars \\citep{Urbaneja2005}.  (For a few of the differing views of M33's abundances, see \\citealt{Kwitter, Zaritsky1994, 2007A&A...470..865M} and \\citealt{2011ApJ...730..129B}.) \n\nBy contrast, we find that there is negligible dependence on metallicity in the equation for the bolometric correction  (BC$_K$). Again, the equation is given in Table~\\ref{tab:FunFacts}.  Unlike the V-band BCs that many astronomers are familiar with, the K-band BCs have positive values.  Since they are closer to the peak flux of RSGs, K-band corrections also change more slowly than V-band corrections with respect to effective temperature, again minimizing the errors we make by doing our analysis in the NIR. \n\nFinally we combined the de-reddened K values with the BC$_K$ and the true distance modulus (assumed to be 24.40 for M31 and 24.60 for M33, following \\citealt{vandenbergh2000}), yielding the bolometric magnitude.\nWe then converted these to bolometric luminosities relative to the sun, i.e., $\\log L\/L_\\odot$.\n\nThe resulting effective temperatures and luminosities have uncertainties of about 150~K and 0.05~dex, respectively, where we have done error propagation based not only on the photometric errors, but also an uncertainty of 0.5~mag in the adopted values for $A_V$.\n\nThe use of a linear relationship with $K_s$ for determining $A_V$ results in a good match with the\nevolutionary tracks as we see in the next section.  However, it also has a drawback:  a dozen M31 stars have unlikely large extinction values ($A_V=5-10$) and unrealistically high luminosities.  Several of these stars have ambiguous {\\it Gaia} results, but most are considered probable members despite their very bright $K_s$ 2MASS values ($K_s=6-9$).  We have retained these stars in the list. Although they are unlikely to be bona-fide RSGs, they are probably quite interesting IR sources.  None of these overly bright stars are within the field of the LGGS, suggesting they lie well away from regions of star formation.  It is possible that in some cases the membership assessment is wrong.  There is no similar problem for M33.\n\nWe note that of the 38,817 photometrically identified RSGs in M31 list and 7,088 in the M33 list,  only 7585 and 3911 (respectively) have luminosities $\\log L\/L_\\odot\\geq 4.0$.  Of these, 6412 and 2858 are within their galaxy's Holmberg radius.\n\n\\subsection{Comparisons with Evolutionary Tracks}\n\nIn Figure~\\ref{fig:HRDs} we show the location of the RSGs in the H-R diagram along with the evolutionary\ntracks from the Geneva group.  For M31, we have used the tracks of \\citet{Sylvia} computed for solar metallicity ($z=0.014$).\nAlthough a higher metallicity would be more appropriate for M31, such tracks are not yet available.  For M33 we have used the unpublished $z=0.006$ models, kindly made available to us by Cyril Georgy, Sylvia Ekstr\\\"{o}m, and Georges Meynet. This metallicity corresponds roughly to that of the LMC, and is as appropriate as anything as discussed above.  Although the Geneva models are computed only for single-star evolution, we are using them as the standard for our comparisons as they include the effects of rotation, which is important particularly at sub-solar metallicity, and as their new prescription for the mass-loss rates during the RSG phase have been vetted against observations \\citep{UKIRT}.  Binary evolution has little effect on the location of RSGs in the HRD; this is nicely illustrated in the comparison between the single-star Geneva and BPASS binary evolutionary tracks \\citep{2016MNRAS.462.3302E,BPASS2} in Figure 8.3 of \\citet{LevesqueRSGs}. (See also Figure 7 of  \\citealt{2018ApJ...867..155L}.)\n\n\nThe most striking thing about Figure~\\ref{fig:HRDs} is how superb the agreement is between the tracks calculated by stellar evolution theory, and the effective temperatures and luminosities derived from our photometry and the MARCS stellar atmospheres.  The upturn and vertical evolution shown by the tracks is due to the increase in the mean particle mass $\\mu$ in the stellar cores as He is converted into C and O; the luminosity is a steep function of $\\mu$ (see discussion in, e.g., \\citealt{LevesqueRSGs}).  Where this happens in terms of effective temperature is dependent upon many things, but particularly the treatment of convection (see, e.g., Figure 9 in \\citealt{1987A&A...182..243M})\\footnote{We note that all of these complications, and much more, must be dealt with when including binary interactions in evolutionary calculations as well.}.  At the same time, the calculation of the spectral energy distribution in the stellar atmosphere calculations for these bloated stars is fraught with approximations as well.  Thus this agreement between these two complex models seems absolutely remarkable.\n\nThe redwards extent of the tracks shift to higher effective temperatures at larger luminosities: at $\\log L\/L_\\odot$=5.5 ($\\sim 30M_\\odot$) these stars are barely cool enough to be called ``red\" supergiants.   This shift is is mostly due to the effects of mass loss during these late stages.  That is why for the solar-metallicity tracks in M31 the 32$M_\\odot$ stop at $\\log T_{\\rm eff}\\sim3.77$ (roughly the temperature of the sun!)--mass loss causes the evolution to proceed back to higher\ntemperatures.  (See discussion in \\citealt{Sylvia}.)  At the lower metallicities even\nthe 40$M_\\odot$ models will produce RSGs, as shown by the tracks for M33.   Our data are consistent with these results, although the high-extinction stars mentioned above makes this a little less clean for M31 as we would care.    Additional spectroscopy will resolve this.  We note that\nwithout our luminosity-dependent extinction correction, such agreement would be poorer, as shown in Figure 10 of \\citet{UKIRT}.\n\n \n\\section{Resolving the Extinction Puzzle}\n\\label{Sec-discussion}\n\nWe noted back in Section~\\ref{Sec-extinct} that both the PHAT survey and our expectations from the M31's inclination would suggest that the average extinction should be higher in M31 than in M33.  However, photometry of OB stars (both in individual associations by \\citealt{MAC86} and by use of the ``blue plume\" by \\citealt{LGGSII}) suggest essentially identical low extinctions.  \n\nAs an unexpected bonus, we believe we have solved this puzzle.  In composing Tables~\\ref{tab:M31RSGs} and \\ref{tab:M33RSGs} we discovered a large discrepancy in the percentages of matches between our NIR-selected RSGs and the LGGs optical catalog between the two galaxies.\nFor M31 the percentage of RSGs with $\\log L\/L_\\odot>4.0$ that had matches in the LGGS was only 52\\%.\nBy contrast, for M33 the percentage is 87\\%. (We restricted the sample the areas which overlap with the LGGS.)  \n\nWe carefully examined this discrepancy and found that the percentage of matches in M31 were similar to that of M33 in the two extreme outlying LGGS fields (the north-eastern and south-western most one) which are sparse,\nbut after that were mostly poor in all of the other fields.   In Figure~\\ref{fig:matches} we show a grey-scale map of the matches.  The field here roughly corresponds to that shown by the green squares in Figure~\\ref{fig:m31cfht}.   In constructing this map we smoothed over 5\\arcmin $\\times$ 5\\arcmin\\ squares,\nand insisted that there be at least 50 stars in each square; if not, the region is black.  The percentage of\nmatches varied then from 12\\% (dark) to 90\\% (bright).  We find there is an excellent match with the\ndust maps of \\citet{2014ApJ...780..172D}; see in particular their Figure 2.  Thus, we  believe this is telling us is that the prominent OB associations and the blue plume stars are the ones generally on the ``near side\" of M31's disk, and thus least affected by dust.  This is of course a classic example of Malmquist bias, and explains the apparent discrepancy.\n\n\\section{The Spatial Distributions of RSGs}\n\nIn an influential study, \\citet{HumphreysSandage80} discuss the stellar content of M33, identifying blue and red stars across the face of the galaxy, and defining its OB associations\\footnote{Indeed, this paper was instrumental in shaping the course of research for the next 40+ years for P. M., who was an impressionable graduate student at the time. Note that the on-line version does not do the photographic finding charts justice.}.  The blue stars and their associated OB associations clearly followed the spiral arms.  However, the red stars clearly did not.  (Compare their Figure 21 with their Figure 22.)  Although we expect there to be some age difference between the OB stars and RSGs (1-5~Myr vs 10-15~Myr, say), this is not sufficient by far to explain the effect. \\citet{HumphreysSandage80} correctly surmised that many of the red stars were foreground dwarfs in our own Galaxy.  (Their photographic study did not go sufficiently deep to detect AGBs.)  This work was followed by large-area photographic surveys of  M33 by \\citet{1993ApJS...89...85I}, which suffered from a similar problem.  \\citet{MasseyRSGs} devised a method to separate extragalactic RSGs from foreground dwarfs using two-color plots as mentioned above; however his CCD study covered only several small regions of M31 and M33. \\citet{2005AJ....129..729R} performed a wide-field CCD study of M33 aimed primarily at  identifying carbon-rich AGBs. No correction for foreground contamination was made.  They identified a region of their CMD that likely contained RSGs, and in their density plot (their Figure 8) indeed a hint of the spiral pattern can be discerned. \n\nWe have the advantage, of course, of being able to use {\\it Gaia} to remove foreground stars, although again it should be noted that the majority of the stars in the AGB region of the CMD do not have {\\it Gaia} measurements.  However, at these extreme red and faint colors, little foreground contamination is expected, as shown previously in Figures 4 and 5.   We were therefore interested to see what the spatial distribution of RSGs and AGBs looked like in our cleaner samples.\n\nThe results are shown in Figures~\\ref{fig:M31RSGAGB} and \\ref{fig:M33RSGAGB}.  In making these figures, we have restricted the sample to stars within the Holmberg radius ($\\rho<1.0$).  For the RSGs we have imposed $\\log L\/L_\\odot>4.3$ (roughly corresponding to $K_s<15.5$; see Figures~\\ref{fig:M31RSGs} and \\ref{fig:M33RSGs}), and for the AGB stars $K_s<17.0$.  (Even so, only a small subset of the AGB stars are plotted for clarity; 2\\% for M31 and 20\\% for M33.)  As expected, the RSGs very much follow the spiral arms.  In M31 (Figure~\\ref{fig:M31RSGAGB}) the vast majority of RSGs are found in the well-known star-formation ring where most of the HI,  OB associations, and H$\\alpha$ emission are located (see, e.g.,  \\citealt{1966ApJ...144..639R,vdbergh,1994AJ....108.1667D}, respectively).   This suggests that despite our finding in the previous section,\nhuge sections of M31's massive star population do not remain hidden. In M33 (Figure~\\ref{fig:M33RSGAGB}) the RSGs very nicely follow the spiral arm pattern.  By contrast, the AGBs are more uniformly distributed across the disk.  This is as expected given the extreme difference in ages between these two populations, with the RSGs representing 10-20~Myr old, and the AGBs (100~Myr-1~Gyr for the dominate population; see \\citealt{2014A&A...565A...9S}).\n\n\n\\section{Summary and Conclusions}\n\nOur study has identified likely RSGs throughout M31 and M33, using archival NIR data that extend well beyond each galaxy's Holmberg radius.   Using a cut of $\\log L\/L_\\odot$ of 4.0 and restricting the area to within the Holmberg radius of the disk ($\\rho\\leq1$ in Tables~\\ref{tab:M31RSGs} and \\ref{tab:M33RSGs}), we find  \n6412 RSGs in M31 and 2858 RSGs in M33.   By contrast, only a few hundred RSGs are known within our own Galaxy \\citep{LevesqueRSGs}.  Many of these were found through the the HD catalog plus other objective prism studies (e.g., \\citealt{1954ApJ...120..478N,1957ApJ...125..408B,1992AJ....104..821M}; see also \\citealt{1978ApJS...38..309H}). More recently,\nclusters rich in RSGs have been found near the Galactic center; i.e, \\citet{2006ApJ...643.1166F,2007ApJ...671..781D,2010A&A...513A..74N,2011A&A...528A..59N,2016csss.confE.120M}.   Further studies are being carried out in other directions (e.g., \\citealt{2018MNRAS.475.2003D}), but like all Galactic studies, extinction prevents a true Galaxy-wide census, and uncertainties about distances (needed to distinguish RSGs from giants) persist even in the {\\it Gaia} era.   Various authors have suggested that the\nMilky Way's RSG population should number in the thousands (e.g., \\citealt{1987PASP...99..453G,1989IAUS..135..445G,2012A&A...537A..10M}).\nThe number of RSGs we've discovered in M31 and M33 gives credence to these predictions.\n\nUsing the MARCs stellar atmosphere models, we have computed effective temperatures and bolometric luminosities for all of the stars in our sample, and compared those to the location of the Geneva evolutionary tracks.  The agreement is spectacular, and gives us high confidence in the tremendous progress that has been made in both fields.\n\nA comparison of the M31 RSG sample with the LGGS resolves the discrepancy between the average extinction in M31's disk\nestimated from thermal dust emission with the comparatively low reddening found in several OB associations and from the blue plume.  The later samples are likely due to stars on the near side of M31's disk, and suffer lower extinction on average.  Nevertheless, RSGs generally are primarily found in the region where the OB associations and other star-formation indicators are prominent in M31, suggesting that this is not too serious a problem in terms of identifying most of M31's massive star population.  In M33 the RSGs also delineate the spiral arms.  As expected, the fainter, redder AGB stars are much more evenly distributed across the face of these galaxies.\n\nThe identification of these RSG candidates will hopefully provide a valuable resource for spectroscopic studies for years to come.   Empirical confirmation of our assumed extinction as a function of luminosity will require well-fluxed spectrophotometry similar to what we have employed in the past.   Of more immediate use, our catalog of RSGs will be the basis for the contemporaneous study of the RSG binary frequency \\citep{M31M33RSGBins}, and the a detailed comparison of the relative number of WRs and RSGs with stellar evolutionary models in an upcoming paper.\n\n\\acknowledgments\nLowell Observatory sits at the base of mountains sacred to tribes throughout the region. We honor their past, present, and future generations, who have lived here for millennia and will forever call this place home.\n Similarly, K. F. N. and E. M. L. would like to acknowledge that they work  on the traditional land of the first people of Seattle, the Duwamish People past and present, and honor with gratitude the land itself and the Duwamish Tribe. In addition, M. D.  acknowledges the land on which the University of Toronto operates: For thousands of years it has been the traditional land of the Huron-Wendat, the Seneca, and most recently, the Mississaugas of the Credit River. Finally, S. C.  wishes to acknowledge that Queen's University is situated on traditional Anishinaabe and Haudenosaunee territory. We are grateful to be able to live, learn and play on these lands. We thank Ming Yang for correspondence and thoughts on the separation of RSGs from AGBs,\nand Michael Irwin for providing us with the very nice UKIRT M33 data.  We are also grateful to our colleagues Georges Meynet, Sylvia Ekstr\\\"{o}m, and Cyril Georgy for allowing us to use their $z=0.006$ models in advance of publication.   We also appreciate correspondence with Bruce Draine concerning extinction maps of M31.  An anonymous referee made useful suggestions which improved this paper. P. M.'s work was partially supported by the National Science Foundation through AST-1612874. \nK. F. N.'s work was supported by a Cottrell Scholar Award from the Research Corporation for Scientific Advancement granted to E. M. L.  Both\nM. R. D. and S. C.  acknowledge generous funding from the Natural Science and Engineering Research Council of Canada via the Discovery Grant program.\nM. R. D. also acknowledges support from the Canada Research Chairs Program, the Canadian Institute for Advanced Research, and the Dunlap Institute at the University of Toronto.\nThis publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center\/California Institute of Technology, funded by the National Aeronautics and Space Administration and the NSF.  \n\n\n\\clearpage\n\n\\begin{figure}\n\\epsscale{1.3}\n\\plotone{m31cfht.eps}\n\\caption{\\label{fig:m31cfht} M31 survey area.  The outlines of the 70 CFHT fields are shown marked in yellow.  The outlines of the ten fields of the optical Local Group Galaxy Survey are shown for comparison in green.}\n\\end{figure}\n\n\\begin{figure}\n\\epsscale{1.3}\n\\plotone{m33ukirt.eps}\n\\caption{\\label{fig:m33ukirt} M33 survey area.  The outlines of the 5 UKIRT fields are shown marked in yellow.  (The fifth field is nearly coincident with the other one at lower right.) The outlines of the three fields of the optical Local Group Galaxy Survey are shown for comparison in green.}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\epsscale{0.5}\n\\plotone{M31Jerr.eps}\n\\plotone{M31Kerr.eps}\n\\plotone{M33Jerr.eps}\n\\plotone{M33Kerr.eps}\n\\caption{\\label{fig:errors} Photometric errors.  The photometric errors are shown as a function of magnitude.  The red points indicate the stars that were added from 2MASS.  Note that for clarity we have included only 10\\% of the data for the two M31 plots.}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\epsscale{0.7}\n\\plotone{M31CMD.eps}\n\\plotone{m31beas.eps}\n\\caption{\\label{fig:M31CMD}  M31 color-magnitude diagram.  {\\it Upper:} The color-magnitude diagram is shown for M31, with the\npoints color-coded depending upon {\\it Gaia} determined membership.  The black points are the probable members, while the red points are the foreground stars.  The green points represent the stars with ambiguous membership, while the blue points represent the stars with no {\\it Gaia} data. {\\it Lower:} The color-magnitude diagram computed using the Besan\\c{c}on model of the Milky Way \\citep{Besancon} to estimate the foreground contamination.  The simulation covered the same area as our M31 survey, and was centered on M31's location in Galactic longitude and latitude.}\n\\end{figure}\n\n\\clearpage\n\n\\begin{figure}\n\\epsscale{0.75}\n\\plotone{M33CMD.eps}\n\\plotone{m33beas.eps}\n\\caption{\\label{fig:M33CMD}  M33 color-magnitude diagram. {\\it Upper:} The color-magnitude diagram is shown for M33, with the\npoints color-coded depending upon {\\it Gaia} determined membership.  The black points are the probable members, while the red points are the foreground stars.  The green points represent the stars with ambiguous membership, while the blue points represent the stars with no {\\it Gaia} data.  {\\it Lower:} The color-magnitude diagram computed using the Besan\\c{c}on model of the Milky Way \\citep{Besancon} to estimate the foreground contamination.  The simulation covered the same area as our M33 survey, and was centered on M33's location in Galactic longitude and latitude.}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\plotone{M31RSG.eps}\n\\caption{\\label{fig:M31RSGs} M31 color-magnitude diagram with RSGs identified.  The color-magnitude diagram is shown for M31, with the foreground stars removed and the RSGs region indicated by the red border. The RSG sample of stars is shown by the red points.  The two green diagonal lines show where $\\log L\/L_\\odot=4.0$ and 4.3.  The few points within the RSG boundaries not red are background galaxies or in the regions eliminated as described in the text. The short line at upper left shows the size of a reddening vector corresponding to $A_V=1.0$ mag.}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\plotone{M33RSG.eps}\n\\caption{\\label{fig:M33RSGs} M33 color-magnitude diagram with RSGs identified.  The color-magnitude diagram is shown for M31, with the foreground stars removed and the RSGs region indicated by the red border. The RSG sample of stars is shown by the red points.  The two green diagonal lines show where $\\log L\/L_\\odot=4.0$ and 4.3.  The few points within the RSG boundaries not red are background galaxies.  The short line at upper left shows the size of a reddening vector corresponding to $A_V=1.0$ mag.}\n\\end{figure}\n\n\n\n\n\\clearpage\n\\begin{figure}\n\\epsscale{0.7}\n\\plotone{m31hrd.eps}\n\\plotone{m33hrd.eps}\n\\caption{\\label{fig:HRDs} RSGs in the H-R diagram.  The location of the RSGs relative to those of the Geneva evolutionary tracks are shown.  The M31 plot uses the $z=0.014$ metallicity tracks of \\citet{Sylvia}.  The M33 tracks were computed in the same manner by the Geneva group, but correspond to $z=0.006$. There are a few stars with spuriously high luminosity in the M31 sample, as discussed in the text, but generally the distribution of stars matches that expected by evolutionary theory extremely well. }\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\epsscale{1.0}\n\\plotone{ds9.eps}\n\\caption{\\label{fig:matches} Connection between RSGs and LGGS.  This intensity in this image is related to the fraction of matches between our NIR-selected RSG sample the optical counterparts in the LGGS, with brighter regions showing a higher fraction of matches.  The structure is a good match to extinction maps of M31, notably \\citet{2014ApJ...780..172D}.  The area covered corresponds to that of the green boxes in Figure~\\ref{fig:m31cfht}.}\n\\end{figure}\n\n\\clearpage\n\\begin{figure}\n\\epsscale{0.8}\n\\plotone{ds9m31rsgsN.eps}\n\\plotone{ds9m31agb50N.eps}\n\\caption{\\label{fig:M31RSGAGB} Distributions of RSGs and AGBs in M31.  We compare the spatial distributions of stars identified as RSGs and AGBs within the Holmberg radius of M31.  {\\it Upper:} We have restricted the sample to those RSGs with $\\log L\/L_\\odot>4.3$ (roughly corresponding to $K_s<15.5$; see Figure~\\ref{fig:M31RSGs}).  {\\it Lower:} The AGBs are the stars redder than the RSGs sample in Figure~\\ref{fig:M31RSGs} and with $K_s<17.0$.  For clarity only 2\\% of the stars in our photometric sample are plotted in the lower diagram.  Note that we have excluded the region containing M32 and the nuclear region of M31 as described in the text.} \n\\end{figure}\n\n\n\\clearpage\n\\begin{figure}\n\\epsscale{0.8}\n\\plotone{ds9m33rsgN.eps}\n\\plotone{ds9m33agb5N.eps}\n\\caption{\\label{fig:M33RSGAGB} Distributions of RSGs and AGBs in M33.  We compare the spatial distributions of stars identified as RSGs and AGBs within the Holmberg radius of M33.   {\\it Upper:} We have restricted the sample to those RSGs with $\\log L\/L_\\odot>4.3$ (roughly corresponding to $K_s<15.5$; see Figure~\\ref{fig:M33RSGs}).  {\\it Lower:} The AGBs are the stars redder than the RSGs sample in Figure~\\ref{fig:M33RSGs} and with $K_s<17.0$.  For clarity only 20\\% of the stars in our photometric sample are plotted in the lower diagram.}\n\\end{figure}\n\n\n\\clearpage\n\\begin{deluxetable}{l  c c}\n\\tablecaption{\\label{tab:Data} Characteristics of Imaging Data}\n\\tablewidth{0pt}\n\\tablehead{\n&\\colhead{M31}\n& \\colhead{M33} \n}\n\\startdata\nTelescope & 3.6-m CFHT & 3.8-m UKIRT \\\\\nExp. time (s) \\\\\n\\multicolumn{1}{c}{$J$} & 752 & 150 \\\\\n\\multicolumn{1}{c}{$K_s$} & 650 & 270 \\\\\nScale (\\arcsec\/pixel) & 0.3 & 0.4 \\\\\nCoverage (deg$^2$) & 5.0 & 3.0 \\\\\nSeeing (\\arcsec) & 0.5-0.6 & 0.7-1.1 \\\\\nBright limit\\tablenotemark{a} \\\\\n\\multicolumn{1}{c}{$J$} & 14.3 & 12.0 \\\\\n\\multicolumn{1}{c}{$K_s$} & 13.3 & 12.0 \\\\\nNumber of stars &  712,716 &    121,328 \\\\\n\\enddata\n\\tablenotetext{a}{2MASS was used to supplement the list for brighter stars.}\n\\end{deluxetable}\n\n\n \\clearpage\n\\begin{deluxetable}{l l r}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\\label{tab:FunFacts} Adopted and Derived Relations}\n\\tablewidth{0pt}\n\\tablehead{\n&\\colhead{Relation}\n&\\colhead{Source}\n}\n\\startdata\n\\sidehead{Adopted Distance:}\n\\phantom{MakeSomeSpace}& M31: 760~kpc (DM=24.40~mag)& 1 \\\\\n & M33: 830~kpc  (DM=24.60~mag)& 1 \\\\\n\\sidehead{Reddening Relations:}\n&$A_K =0.12 A_V = 0.686 E(J-K)$ & 2 \\\\\n&$E(J-K) = A_V\/5.79 $ & 2 \\\\\n\\sidehead{RSG Photometric Criteria:}\n\\multicolumn{1}{r}{M31:}  \n        &$15.5<K_s \\leq 17.0$: $K_s \\geq K_{s0}$ and $K_s \\leq K_{s1}$ & 3 \\\\\n        &$K_s\\leq15.5$: $J-K_s\\geq 0.984$ and $K_s \\leq K_{s1}$ & 3 \\\\\n                &$K_s\\leq 14.4$ and $(J-K_s)\\leq 1.9$: $J-K_s\\geq 0.984$ & 3 \\\\\n        &$K_{s0}=28.62-13.33(J-K_s)$  & 3,4\\\\\n        &$K_{s1}=31.95-13.33(J-K_s)$  & 3,4\\\\\n\\multicolumn{1}{r}{M33:}\n        &$15.0<K_s \\leq 17.0$: $K_s \\geq K_{s0}$ and $K_s \\leq K_{s1}$ & 3 \\\\\n        &$K_s\\leq15.0$: $J-K_s\\geq 0.963$ and $K_s \\leq K_{s1}$ & 3 \\\\\n                &$K_s\\leq 14.6$ and $(J-K_s)\\leq 1.9$: $J-K_s\\geq 0.917$ & 3 \\\\\n        &$K_{s0}=28.04-13.542(J-K_s)$  & 3,4\\\\\n        &$K_{s1}=31.56-13.542(J-K_s)$  & 3,4\\\\\n\\sidehead{Adopted Extinction:}\n        & $K_s>14.5$: $A_V=0.75$ & 3 \\\\\n        & $K_s\\leq 14.5$ and $K_s\\leq K_{s1}$: $A_V=0.75-1.26\\times(K_s-14.5)$ & 3 \\\\\n        & $K_s\\leq 14.5$ and $K_s\\geq K_{s1}$: $A_V=0.75+5.79\\times\\Delta(J-K_s)$ & 3 \\\\\n        & M31: $\\Delta(J-K_s)=(J-K_s) - (30.29-K_s+0.686 (J-K_s))\/14.02$ & 3 \\\\\n        & M33: $\\Delta(J-K_s)=(J-K_s) - (30.14-K_s+0.686 (J-K_s))\/14.23$ & 3 \\\\\n\\sidehead{Conversion of 2MASS ($J, K_s$) to Standard System ($J, K$):}\n&$K=K_s + 0.044$ & 5\\\\\n&$J-K = (J-K_s+0.011)\/0.972$ & 5\\\\\n\\sidehead{Conversion to Physical Properties (Valid for 3500-4500 K):}\n&M31: $T_{\\rm eff} = 5606.6 - 1713.3 (J-K)_0$ & 3\\\\\n&M33: $T_{\\rm eff} = 5643.5 - 1807.1 (J-K)_0$ & 3\\\\\n&${\\rm BC}_K = 5.495 - 0.73697 \\times T_{\\rm eff}\/1000$ & 3\\\\\n&$K_* = K - A_K$ & \\nodata \\\\\n&$M_{\\rm bol} = K_* + {\\rm BC}_K - DM$ & 1 \\\\\n&$\\log L\/L_\\odot =(M_{\\rm bol}-4.75)\/-2.5$ & \\nodata \\\\\n\\enddata\n\n\\tablerefs{1--\\citealt{vandenbergh2000}; 2--\\citealt{Schlegel1998};\n3--This paper; 4--\\citealt{2006AA...448...77C}; 5--\\citealt{Carpenter}\n}\n\\end{deluxetable}\n\n \\clearpage\n \n\\rotate\n\\begin{deluxetable}{l l l l l l l l l l l l l l l l }\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\\label{tab:M31RSGs} Photometrically Selected RSGs in M31}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{M31RSG\\#}\n& \\colhead{$\\alpha_{2000}$}\n& \\colhead{$\\delta_{2000}$}\n& \\colhead{$\\rho$\\tablenotemark{a}}\n& \\colhead{$K_s$}\n& \\colhead{$\\sigma_{K_s}$}\n& \\colhead{$J-K_s$}\n& \\colhead{$\\sigma_{J-K_s}$}\n& \\colhead{\\#obs}\n& \\colhead{Gaia\\tablenotemark{b}}\n& \\colhead{$A_V$}\n& \\colhead{Teff\\tablenotemark{c}}\n& \\colhead{$\\log L\/L_\\odot$\\tablenotemark{d}}\n& \\multicolumn{3}{c}{LGGS} \\\\ \\cline{13-15}\n&&&&&&&&&&&&\\colhead{ID} & \\colhead{$V$} & \\colhead{Type}\n}\n\\startdata\n6832 & 00 40 50.843 &+41 06 48.30 &0.61& 17.368 & 0.038 & 1.080 & 0.052  &1 &2 & 0.75 & 3850 &3.67 &\\nodata &\\nodata &\\nodata\\\\\n6833 & 00 40 50.845 &+41 45 49.99 &1.61& 15.420 & 0.012 & 1.197 & 0.030 & 2 &2 & 0.75 & 3650 &4.38 &\\nodata &\\nodata &\\nodata\\\\\n6834 & 00 40 50.846 &+41 05 40.98 &0.58 &14.793 & 0.004 & 1.031&  0.006  &1& 0  &0.75  &3950 &4.72 &J004050.87+410541.1  &19.72 &RSG\\\\\n6835 & 00 40 50.865 &+40 39 46.27 &0.47& 17.260 & 0.034&  1.093 & 0.048  &1 &2  &0.75  &3850 &3.70 &J004050.84+403946.3 & 22.07&\\nodata\\\\\n6836 & 00 40 50.874 &+40 46 21.81& 0.39 &17.260 & 0.037 & 1.077 & 0.148  &1 &2  &0.75  &3850 &3.71 &J004050.87+404621.6  &22.59&\\nodata\\\\\n\\enddata\n\\tablecomments{Table 3 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.}\n\\tablenotetext{a}{Galactocentric distance.  Assumes Holmberg radius of 95.3 arcmin, inclination 77.0 deg, and a position angle of the major axis of 35.0 deg.  At a distance of 760 kpc, a $\\rho$ of 1.00 corresponds to 21.07 kpc.}\n\\tablenotetext{b}{Gaia flag: 0=probable member, 1=uncertain membership; 2=no Gaia data.}\n\\tablenotetext{c}{Typical uncertainty 150~K.}\n\\tablenotetext{d}{Typical uncertainity 0.05~dex.}\n\\end{deluxetable}\n\n\\clearpage\n\\rotate\n\\begin{deluxetable}{l l l l l l l l l l l l l l l l }\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\\label{tab:M33RSGs} Photometrically Selected RSGs in M33}\n\\tablewidth{0pt}\n\\tablehead{\n\\colhead{M33RSG\\#}\n&\\colhead{$\\alpha_{2000}$}\n& \\colhead{$\\delta_{2000}$}\n& \\colhead{$\\rho$\\tablenotemark{a}}\n& \\colhead{$K_s$}\n& \\colhead{$\\sigma_{K_s}$}\n& \\colhead{$J-K_s$}\n& \\colhead{$\\sigma_{J-K_s}$}\n& \\colhead{\\#obs}\n& \\colhead{Gaia\\tablenotemark{b}}\n& \\colhead{$A_V$}\n& \\colhead{Teff\\tablenotemark{c}}\n& \\colhead{$\\log L\/L_\\odot$\\tablenotemark{d}}\n& \\multicolumn{3}{c}{LGGS} \\\\ \\cline{13-15}\n&&&&&&&&&&&&\\colhead{ID} & \\colhead{$V$} & \\colhead{Type}\n}\n\\startdata\n854&01 32 19.96 & +30 37 18.6 & 1.04 &17.114 & 0.024&  0.981  &0.038  &4& 2 & 0.75 & 4100 &3.92 &\\nodata &\\nodata &\\nodata \\\\\n855&01 32 20.03 & +30 34 18.0 & 1.01 &16.874 & 0.019&  0.922 & 0.030  &4 &0 & 0.75  &4200 &4.04 &J013220.05+303418.1 & 20.76&\\nodata \\\\\n856&01 32 20.32  &+30 29 37.9 & 0.98 &16.229 & 0.011&  0.924 & 0.018  &4 &0 & 0.75 & 4200 &4.30 &J013220.36+302938.0  &19.69 &RSG\\\\\n857&01 32 20.69 & +31 21 52.2 & 2.23 &16.124&  0.013&  0.884 & 0.019  &2 &0 & 0.75  &4250 &4.36 &\\nodata &\\nodata &\\nodata \\\\\n858&01 32 20.95 & +30 24 16.2&  0.98 &17.397&  0.031&  0.856 & 0.051  &4& 2 & 0.75 & 4300 &3.87&\\nodata &\\nodata &\\nodata \\\\\n\\enddata\n\\tablecomments{Table 4 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.}\n\\tablenotetext{a}{Galactocentric distance.  Assumes Holmberg radius of 30.8 arcmin, inclination 56.0 deg, and a position angle of the major axis of 23.0 deg.  At a distance of 830 kpc, a $\\rho$ of 1.00 corresponds to 7.44 kpc.}\n\\tablenotetext{b}{Gaia flag: 0=probable member, 1=uncertain membership; 2=no Gaia data.}\n\\tablenotetext{c}{Typical uncertainty 150~K.}\n\\tablenotetext{d}{Typical uncertainty 0.05~dex.}\n\\end{deluxetable}\n\n\\clearpage\n\n\t\n\\bibliographystyle{aasjournal}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and statement of the results}\n\n\\noindent\nWe start by recalling the super-symmetric model consisting of coupling gravity with fermionic interaction. We fix a three dimensional closed (compact, without boundary) manifold $M$, then we define the energy functional $\\mathcal{E}$ of this model by\n$$\\mathcal{E}(g,\\psi)=\\int_{M}R_{g}dv_{g}+\\int_{M}\\langle D_{g}\\psi,\\psi \\rangle-\\langle \\psi,\\psi\\rangle  dv_{g},$$\nwhere $g$ is a Riemannian metric on $M$, $\\psi$ is a spinor in the spin bundle $\\Sigma M$ on $M$, $R_g$ is the scalar curvature, $D_{g}$ is the Dirac operator and $\\langle \\cdot, \\cdot \\rangle$ is the compatible Hermitian metric on $\\Sigma M$; we will give the precise definitions in the next section. The functional $\\mathcal{E}$ generalizes the classical Hilbert-Einstein functional and it is invariant under the group of diffeomorphisms of $M$ as well; we address the reader to \\cite{Belg, Fin, Kim}, where it was introduced and studied.\\\\\nAs in the classical case of the Hilbert-Einstein functional, since the group of diffeomorphisms is usually big, in first instance we restrict the functional to a fixed conformal class, namely given a Riemannian metric $g$, we set\n$$[g]=\\left\\{u^{4}g;\\; u\\in C^{2,\\alpha}(M)\\right\\} .$$\nIn this way, the energy functional reads as\n\\begin{equation}\nE(u,\\psi)=\\frac{1}{2}\\left(\\int_{M}u L_{g}u+\\langle D_{g} \\psi,\\psi \\rangle -|u|^{2}|\\psi|^{2} dv_{g}\\right),\n\\end{equation}\nwhere $L_{g}$ is the conformal Laplacian of the metric $g$.\\\\\nThis energy functional can also be seen as the three dimensional version of the super-Liouville equation investigated in \\cite{J1,J2}, which is fundamental in the study of string fermions, see \\cite{Fuk}.\\\\\nBy the first variation of the functional $E$, we see that its critical points satisfy the coupled system\n\\begin{equation}\\label{el}\n\\left\\{\\begin{array}{ll}\nL_{g} u=|\\psi|^{2}u\\\\\n& \\text{on } M .\\\\\nD_{g}\\psi=|u|^{2}\\psi\n\\end{array}\n\\right.\n\\end{equation}\nSince the functional is conformally invariant, one expects compactness to be violated for this problem; moreover, due to the presence of the Dirac operator, it is strongly indefinite. For the later part, the authors studied an effective method, based on a homological approach \\cite{M,MV,MV1}, for general functionals with this feature of being strongly indefinite; here we will focus on the first issue, that is the lack of compactness.\\\\\n\n\\noindent\nWe recall that a $C^{1}$ function $F$ satisfies the Palais-Smale condition (PS) if: for any sequence $x_{k}$ such that $F(x_{k})\\to c$ and $\\nabla F(x_{k})\\to 0$ (such a sequence is then called a (PS) sequence), there exists a converging subsequence.\\\\\nThe (PS) condition is fundamental in the study of problems with variational structure as min-max theorems or Morse type methods, which rely heavily on this condition since it guaranties the convergence of the deformation flow. In several geometric problems though, such condition is violated, mainly because of the conformal invariance. We recall the widely investigated cases of prescribing curvatures as the Yamabe problem or the $Q$-curvature problem (see for instance \\cite{L,B,Br,Ben,Ben2} and the references therein). In the previously stated problems, the lack of compactness is well understood and there is a specific characterization for the (PS) sequences. The first result in this paper concerns the study and the characterization of the (PS) sequences of the functional $E$, in particular we will show the following:\n\\begin{theorem}\\label{first}\nLet us assume that $M$ has a positive Yamabe constant $Y_g(M)$ and let $(u_{n},\\psi_{n})$ be a Palais-Smale sequence for $E$ at level $c$. Then there exist $u_{\\infty}\\in C^{2,\\alpha}(M)$, $\\psi_{\\infty}\\in C^{1,\\beta}(\\Sigma M)$ such that $(u_{\\infty},\\psi_{\\infty})$ is a solution of $(\\ref{el})$, $m$ sequences of points $x_{n}^{1},\\cdots, x_{n}^{m} \\in M$ such that $\\lim_{n\\to \\infty}x_{n}^{k}= x^{k}\\in M$, for $k=1,\\dots,m$ and $m$ sequences of real numbers $R_{n}^{1},\\cdots, R_{n}^{m}$ converging to zero, such that:\n\\begin{itemize}\n\\item[i)]   $\\displaystyle u_{n}=u_{\\infty}+\\sum_{k=1}^{m} v_{n}^{k}+o(1)$  in  $H^{1}(M)$,\n\\item[ii)]  $\\displaystyle \\psi_{n}=\\psi_{\\infty}+\\sum_{k=1}^{m}\\phi_{n}^{k}+o(1)$ in $H^{\\frac{1}{2}}(\\Sigma M)$,\n\\item[iii)] $\\displaystyle E(u_{n},\\psi_{n})=E(u_{\\infty},\\psi_{\\infty})+\\sum_{k=1}^{m}E_{\\mathbb{R}^{3}}(U_\\infty^{k},\\Psi_\\infty^{k})+o(1)$,\n\\end{itemize}\nwhere\n$$v_{n}^{k}=(R_{n}^{k})^{-\\frac{1}{2}}\\beta_{k}\\sigma_{n,k}^{*}(U_\\infty^{k}) ,$$\n$$\\phi_{n}^{k}=(R_{n}^{k})^{-1}\\beta_{k}\\sigma_{n,k}^{*}(\\Psi_\\infty^{k}) ,$$\nwith $\\sigma_{n,k}=(\\rho_{n,k})^{-1}$ and $\\rho_{n,k}(\\cdot)=exp_{x_{n}^k}(R_{n}^k \\cdot)$ is the exponential map defined in a suitable neighborhood of $\\R^{3}$.\nAlso, here $\\beta_{k}$ is a smooth compactly supported function, such that $\\beta_{k}=1$ on $B_{1}(x^{k})$ and $supp(\\beta_{k})\\subset B_{2}(x^{k})$ and $(U_\\infty^{k},\\Psi_\\infty^{k})$ are solutions to our equations (\\ref{el}) on $\\mathbb{R}^3$ with its Euclidian metric $g_{\\R^3}$.\n\\end{theorem}\n\n\\begin{remark}\nThe assumption on $M$ of having a positive Yamabe constant implies in particular that there are no harmonic spinors, namely the Dirac operator $D_g$ has no kernel: this will be used in the proof. In fact, by conformal invariance of the Dirac operator, the vanishing of the kernel is preserved by conformal change. So if the Yamabe constant is positive, then the conformal class of the metric contains a metric with positive scalar curvature, hence using the Schr\\\"{o}dinger-Lichnerowicz formula for this last metric and denoting by $\\Delta_{\\Sigma}$ the connection Laplacian, we have that\n$$D_{g}^{2}=-\\Delta_{\\Sigma}+\\frac{R_g}{4},$$\nwhich implies the vanishing of the kernel of $D_g$.\n\\end{remark}\n\n\\noindent\nHere $H^{1}(M)$ and $H^{\\frac{1}{2}}(\\Sigma M)$ are suitable Sobolev spaces on which the functional $E$ is well defined (see next section).\\\\\n\n\\noindent\nNow, for non-trivial  $(u,\\psi) \\in H^1(M) \\times H^{\\frac{1}{2}}(\\Sigma M)$, we define the functionals\n$$\\tilde{E}(u,\\psi)=\\frac{\\left(\\displaystyle\\int_{M}uL_{g}udv_{g}\\right)\\left(\\displaystyle\\int_{M}\\langle D_g \\psi,\\psi\\rangle dv_{g}\\right)}{\\displaystyle\\int_{M}|u|^{2}|\\psi|^{2}dv_{g}} ,\\quad I(\\psi)=\\frac{\\displaystyle\\int_{M}\\langle D_g \\psi,\\psi\\rangle dv_{g}}{\\displaystyle\\int_{M}|u|^{2}|\\psi|^{2}dv_{g}}.$$\nAlso, we let $P^{-}$ be the projector on $H^{\\frac{1}{2},-}$ (the negative space of $H^{\\frac{1}{2}}(\\Sigma M)$ according to the splitting given by the eigenspinors of the Dirac operator), so that for a given $\\psi \\in H^{\\frac{1}{2}}(\\Sigma M)$\n$$P^{-}(\\psi)=0 \\Longleftrightarrow \\int_{M}\\langle \\psi,\\varphi \\rangle dv_{g}, \\quad \\forall \\varphi\\in H^{\\frac{1}{2},-} .$$\nAs for the Yamabe problem, we define a conformal constant\n$$\\tilde{Y}_g (M)=\\inf\\left\\{\\begin{array}{ll}\n\\tilde{E}(u,\\psi); \\text{ for  $(u,\\psi)\\in H^{1}(M)\\setminus\\{0\\}\\times H^{\\frac{1}{2}}(\\Sigma M)\\setminus \\{0\\}$ s.t. }  I(\\psi)> 0, \\\\\n P^{-}\\left(D_g \\psi-I(\\psi)u^{2}\\psi\\right) =0\\end{array}  \\right\\} .$$\nIndeed, we first recognize that the constant $\\tilde{Y}_g (M)$ only depends on the conformal class of the metric $g$, then we have an Aubin type result by comparing $\\tilde{Y}_g (M)$ with the invariants on the sphere $S^3$ with its standard metric $g_0$. In particular we will show the following\n\\begin{theorem}\\label{second}\nIt holds:\n$$Y_g(M)\\lambda^{+}_g(M)\\leq \\tilde{Y}_g(M) \\leq Y_{g_0}(S^{3})\\lambda^{+}_{g_0}(S^{3})=\\tilde{Y}_{g_0}(S^{3}) .$$\nMoreover, if\n$$\\tilde{Y}_g(M) < \\tilde{Y}_{g_0}(S^{3}) ,$$\nthen problem $(\\ref{el})$ has a non-trivial ground state solution.\n\\end{theorem}\n\\noindent\nHere we have denoted by\n$$\\lambda^{+}_g(M):=\\inf_{\\tilde g \\in [g]}\\lambda_1(\\tilde g)Vol_{\\tilde g}(M)^{\\frac{1}{3}}$$\nthe invariant as defined in \\cite{Ginoux} and $\\lambda_1(g)$ being the smallest positive eigenvalue of $D_g$ on $M$.\\\\\nWe recall that $\\lambda^{+}_{g}(M)$ can be characterized as follows (see \\cite{Am1,Am2}):\n\\begin{align}\n\\lambda^{+}_{g}(M)&=\\inf_{\\psi \\in im_{C^{\\infty}}D_{g}\\setminus\\{0\\}}\\frac{\\displaystyle\\left(\\int_{M}|\\psi|^{\\frac{3}{2}}dv_{g}\\right)^{\\frac{4}{3}}}{\\displaystyle\\left|\\int_{M}\\langle \\psi,D^{-1}_{g}\\psi \\rangle dv_{g}\\right|}\\notag\\\\\n&=\\inf_{\\varphi \\in W^{1,\\frac{3}{2}}(\\Sigma M)\\setminus \\{0\\}} \\frac{\\displaystyle\\left(\\int_{M}|D_{g}\\varphi|^{\\frac{3}{2}}dv_{g}\\right)^{\\frac{4}{3}}}{\\displaystyle\\left|\\int_{M} \\langle \\varphi, D_{g}\\varphi \\rangle dv_{g}\\right|}, \\notag\n\\end{align}\nwhere $im_{C^{\\infty}}D_{g}$ is the image of the operator $D_{g}:C^{\\infty}(\\Sigma M)\\to C^{\\infty}(\\Sigma M)$.\\\\\n\n\\noindent\nFinally, in the last section we will consider a three-dimensional closed manifold $M$ with an isometric group action $G$ acting on $M$, such that the orbits of $G$ have infinite cardinality and we will show that equations $(\\ref{el})$ admit two infinite families of solutions on such a manifold.\n\n\n\n\\section{Conformally invariant operators, spaces of variations and splitting}\n\n\\noindent\nIn this section we will briefly recall some notations and properties of conformally invariant operators involved and we will give the definition of the Sobolev spaces that we are going to use.\\\\\nLet $(M,g)$ be a closed (compact, without boundary) three dimensional Riemannian manifold, we define the conformal Laplacian acting on functions by\n$$L_g u:=-\\Delta_g u+\\frac{1}{8}R_g u ,$$\nwhere $\\Delta_g$ is the standard Laplace-Beltrami operator and $R_g$ is the scalar curvature. The conformal invariance of $L_g$ reads as follows: if $\\tilde g=g_u= u^4g$ is a metric in the conformal class of $g$, then we have\n$$L_{\\tilde g}f=u^{-5}L_g(uf) .$$\nWe will denote by $H^1(M)$ the usual Sobolev space on $M$, and we recall that by the Sobolev embedding theorems there is a continuous embedding $$H^1(M)\\hookrightarrow L^p(M), \\quad 1\\leq p \\leq 6 ,$$\nwhich is compact if $1\\leq p <6$.\\\\\nNow let $\\Sigma M$ the canonical spinor bundle associated to $M$ see \\cite{F}, whose sections are simply called spinors on $M$.\nThis bundle is endowed with a natural Clifford multiplication $\\text{Cliff}$, a hermitian metric and\na natural metric connection $\\nabla^\\Sigma$. The Dirac operator $D_g$ acts on spinors\n$$D_g:C^\\infty(\\Sigma M)\\longrightarrow C^\\infty(\\Sigma M)$$\ndefined as the composition $\\text{Cliff} \\circ \\nabla^\\Sigma$ in the following way\n$$\\nabla^\\Sigma:C^\\infty(\\Sigma M)\\longrightarrow C^\\infty(T^*M\\otimes\\Sigma M) ,$$\n$$\\text{Cliff}:C^\\infty(TM\\otimes\\Sigma M)\\longrightarrow C^\\infty(\\Sigma M), $$\nwhere $T^*M \\simeq TM$ have been identified by means of the metric. We also have a conformal invariance that in our situation, $\\tilde g= u^4g$, reads as follows\n$$D_{\\tilde g}\\psi=u^{-4}D_g(u^2\\psi) .$$\nThe functional space that we are going to define is the Sobolev space $H^{\\frac{1}{2}}(\\Sigma M)$. First we recall that the Dirac operator $D_g$ on a compact manifold is essentially self-adjoint in $L^2(\\Sigma M)$, has compact resolvent and there exists a complete $L^2$-orthonormal basis of eigenspinors $\\{\\psi_i\\}_{i\\in\\mathbb{Z}}$ of the operator\n$$D_g\\psi_i=\\lambda_i \\psi_i ,$$\nand the eigenvalues $\\{\\lambda_i\\}_{i\\in\\mathbb{Z}}$ are unbounded, that is $|\\lambda_i|\\rightarrow\\infty$, as $|i|\\rightarrow\\infty$.\nNow if $\\psi\\in L^{2}(\\Sigma M)$, it has a representation in this basis, namely:\n$$\\psi=\\sum_{i\\in \\mathbb{Z}}a_{i}\\psi_{i}.$$\nLet us define the unbounded operator $|D_g|^{s}: L^{2}(\\Sigma M)\\rightarrow L^{2}(\\Sigma M)$ by\n$$|D_g|^{s}(\\psi)=\\sum_{i\\in \\mathbb{Z}} a_{i}|\\lambda_{i}|^{s}\\psi_{i}.$$\nWe denote by $H^s(\\Sigma M)$ the domain of $|D_g|^{s}$, namely $\\psi\\in H^s(\\Sigma M)$ if and only if\n$$\\sum_{i\\in \\mathbb{Z}} a_{i}^2|\\lambda_{i}|^{2s}<+\\infty .$$\n$H^s(\\Sigma M)$ coincides with the usual Sobolev space $W^{s,2}(\\Sigma M)$ and for $s <0$, $H^s(\\Sigma M)$ is defined as the dual of $H^{-s}(\\Sigma M)$. For $s >0$, we can define the inner product\n$$\\langle u,v\\rangle_{s}=\\langle|D_g|^{s}u,|D_g|^{s}v\\rangle_{L^{2}},$$\nwhich induces an equivalent norm in $H^{s}(\\Sigma M)$; we will take\n$$\\langle u,u\\rangle:=\\langle u,u\\rangle_{\\frac{1}{2}}=\\|u\\|^{2}$$\nas our standard norm for the space $H^{\\frac{1}{2}}(\\Sigma M)$. Even in this case, the Sobolev embedding theorems say that there is a continuous embedding\n$$H^{s}(\\Sigma M) \\hookrightarrow L^p(\\Sigma M), \\quad 1\\leq p \\leq 3 ,$$\nwhich is compact if $1\\leq p <3$.\nFinally, we will decompose $H^{\\frac{1}{2}}(\\Sigma M)$ in a natural way. Let us consider the $L^2$-orthonormal basis of eigenspinors $\\{\\psi_i\\}_{i\\in\\mathbb{Z}}$: we denote by $\\psi_i^-$ the eigenspinors with negative eigenvalue, $\\psi_i^+$ the eigenspinors with positive eigenvalue and $\\psi_i^0$ the eigenspinors with zero eigenvalue; we also recall that the kernel of $D_g$ is finite dimensional. Now we set:\n$$H^{\\frac{1}{2},-}:=\\overline{\\text{span}\\{\\psi_i^-\\}_{i\\in\\mathbb{Z}}},\\quad\nH^{\\frac{1}{2},0}:=\\text{span}\\{\\psi_i^0\\}_{i\\in\\mathbb{Z}}, \\quad\nH^{\\frac{1}{2},+}:=\\overline{\\text{span}\\{\\psi_i^+\\}_{i\\in\\mathbb{Z}}},$$\nwhere the closure is taken with respect to the $H^{\\frac{1}{2}}$-topology. Therefore we have the orthogonal decomposition  $H^{\\frac{1}{2}}(\\Sigma M)$ as:\n$$H^{\\frac{1}{2}}(\\Sigma M)=H^{\\frac{1}{2},-}\\oplus H^{\\frac{1}{2},0}\\oplus H^{\\frac{1}{2},+}.$$\nAlso, we let $P^{+}$ and $P^{-}$ be the projectors on $H^{\\frac{1}{2},+}$ and $H^{\\frac{1}{2},-}$ respectively.\\\\\nFinally, sometimes we will denote by $\\mathcal{H}$ the product space $\\mathcal{H}=H^1(M) \\times H^{\\frac{1}{2}}(\\Sigma M)$.\n\n\n\n\n\\section{Regularity}\n\n\\noindent\nHere we will prove the regularity of weak solutions of the system of equations (\\ref{el}). Due to the critical nonlinearity, the bootstrap argument does not work, and we explicitly note the two equations in (\\ref{el}) are strongly coupled, therefore we cannot apply the existing results for the conformal Yamabe equation and the conformal Dirac one separately; anyway we will proceed as in \\cite{I} and we will be able to prove the following\n\\begin{theorem}\nLet $(u,\\psi)\\in H^1(M) \\times H^{\\frac{1}{2}}(\\Sigma M)$ be a weak solution of the system of equations (\\ref{el}), then $(u,\\psi)\\in C^{2,\\alpha}(M) \\times C^{1,\\beta}(\\Sigma M)$, for some $0<\\alpha,\\beta<1$.\n\\end{theorem}\n\\begin{proof}\nFirst of all, by the Sobolev embedding there is a continuous injection\n$$H^1(M)\\hookrightarrow L^p(M), \\quad 1\\leq p \\leq 6 ,$$\n$$H^{\\frac{1}{2}}(\\Sigma M) \\hookrightarrow L^p(\\Sigma M), \\quad 1\\leq p \\leq 3.$$\nNow let $\\rho,\\eta \\in C^{\\infty}(M)$, with $\\eta=1$ on $supp(\\rho)$ and let us denote $B=supp(\\eta)$. We compute\n\\begin{align}\nL_g(\\rho u)&=-\\Delta_g(\\rho u)+\\frac{1}{8}R_g \\rho u  \\notag\\\\\n&=-\\rho\\Delta_g u-u\\Delta_g \\rho-2g(\\nabla\\rho,\\nabla u)+\\frac{1}{8}R_g \\rho u \\notag\\\\\n&=\\rho L_g( u)-u\\Delta_g \\rho-2g(\\nabla\\rho,\\nabla u) \\notag\\\\\n&=\\eta|\\psi|^2 \\rho u -u\\Delta_g \\rho-2g(\\nabla\\rho,\\nabla u) \\notag ,\n\\end{align}\nand\n$$D_g(\\rho \\psi)=\\rho D_g \\psi+\\nabla\\rho\\cdot\\psi=\\eta u^2 \\rho\\psi+\\nabla\\rho\\cdot\\psi,$$\nwhere we have denoted by $\\cdot$ the Clifford multiplication for brevity. Now, since\n$$u\\in H^1(M),\\qquad\\psi \\in L^3(\\Sigma M),$$\nwe have that also\n$$u\\Delta_g \\rho+2g(\\nabla\\rho,\\nabla u)\\in L^2(M),\\qquad \\nabla\\rho\\cdot\\psi \\in L^3(\\Sigma M).$$\nWe define the two maps:\n$$P_1:W^{2,q}(M)\\longrightarrow L^{q}(M), \\qquad P_1(v)=\\eta|\\psi|^2v,$$\n$$P_2:W^{1,p}(\\Sigma M)\\longrightarrow L^{p}(\\Sigma M), \\qquad P_2(\\phi)=\\eta u^2 \\phi.$$\nBy H\\\"{o}lder's inequality, the previous maps are well defined if $1<q<\\frac{3}{2}$ and $1<p<3$, moreover there are constants depending on $q$ and $p$, such that the operator norms are bounded as follows:\n$$\\|P_1\\|_{op}\\leq C_q \\|\\psi\\|^2_{L^3(\\Sigma B)}, \\qquad \\|P_2\\|_{op}\\leq C_p \\|u\\|^2_{L^6(B)} .$$\nIn this way the operators\n$$L_g-\\eta|\\psi|^2:W^{2,q}(M)\\longrightarrow L^{q}(M), \\qquad 1<q<\\frac{3}{2} ,$$\n$$D_g-\\eta u^2:W^{1,p}(\\Sigma M)\\longrightarrow L^{p}(\\Sigma M), \\qquad 1<p<3 ,$$\nare invertible if $\\|\\psi\\|_{L^3(\\Sigma B)}$ and $\\|u\\|_{L^6(B)}$ are small, which is possible by taking $B$ even smaller. Therefore there are unique solutions $v\\in W^{2,q}(M)$ and $\\phi\\in W^{1,p}(\\Sigma M)$ to the equations\n\\begin{align}\nL_g v-\\eta|\\psi|^2v & = -u\\Delta_g \\rho-2g(\\nabla\\rho,\\nabla u) \\notag ,\\\\\nD_g \\phi-\\eta u^2 \\phi & =\\nabla\\rho\\cdot\\psi \\notag,\n\\end{align}\nif $1<q<\\frac{3}{2}$ and $1<p<3$.\\\\\nNow we will consider the two dual maps, defined as follows:\n$$\\tilde P_1:L^{6}(M)\\longrightarrow W^{-1,6}(M), \\qquad \\tilde P_1(\\tilde v)=\\eta|\\psi|^2 \\tilde v,$$\n$$\\tilde P_2:L^{3}(\\Sigma M)\\longrightarrow W^{-1,3}(\\Sigma M), \\qquad \\tilde P_2(\\tilde \\phi)=\\eta u^2 \\tilde \\phi.$$\nAgain, by H\\\"{o}lder's inequality and Sobolev embedding, the previous maps are well defined and there exist constants, such that the operator norms are bounded as follows:\n$$\\|\\tilde P_1\\|_{op}\\leq C_q \\|\\psi\\|^2_{L^3(\\Sigma B)}, \\qquad \\|P_2\\|_{op}\\leq C_p \\|u\\|^2_{L^6(B)} .$$\nEven in this case, the operators\n$$L_g-\\eta|\\psi|^2:L^{6}(M)\\longrightarrow W^{-1,6}(M) ,$$\n$$D_g-\\eta u^2:L^{3}(\\Sigma M)\\longrightarrow W^{-1,3}(\\Sigma M) ,$$\nare invertible if $\\|\\psi\\|_{L^3(\\Sigma B)}$ and $\\|u\\|_{L^6(B)}$ are small; therefore there are unique solutions $\\tilde v\\in L^6(M)$ and $\\tilde \\phi\\in L^3(\\Sigma M)$ to the equations\n\\begin{align}\nL_g \\tilde v-\\eta|\\psi|^2\\tilde v & =-u\\Delta_g \\rho-2g(\\nabla\\rho,\\nabla u) \\notag ,\\\\\nD_g \\tilde \\phi-\\eta u^2 \\tilde \\phi & =\\nabla\\rho\\cdot\\psi  \\notag .\n\\end{align}\nMoreover, since\n$$ W^{2,q}(M) \\hookrightarrow L^6(M), \\quad \\frac{6}{5}\\leq q <\\frac{3}{2} ,$$\n$$W^{1,p}(\\Sigma M) \\hookrightarrow L^p(\\Sigma M), \\quad \\frac{3}{2}\\leq p < 3 ,$$\nthen by the uniqueness $\\tilde v=v=\\rho u$ and $\\tilde \\phi=\\phi=\\rho \\psi$, under the above conditions on $q$ and $p$. Now, since $\\rho$ and $\\eta$ are smooth functions with arbitrary small supports, we have that $u\\in  W^{2,q}(M)$ and $\\psi \\in W^{1,p}(\\Sigma M)$, provided $\\frac{6}{5}\\leq q <\\frac{3}{2}$ and $\\frac{3}{2}\\leq p < 3$. Therefore, by the Sobolev embedding, we get that $u\\in L^q(M)$ and $\\psi \\in L^p(\\Sigma M)$, for any $1< q,p <\\infty$; and then by plugging them in the initial equations, we have that $u\\in  W^{2,q}(M)$ and $\\psi \\in W^{1,p}(\\Sigma M)$, for any $1< q,p <\\infty$, by the elliptic regularity estimates. Once more, by the Sobolev embedding for the H\\\"{o}lder spaces, we have that there exist $0<\\alpha,\\beta<1$ such that $u\\in C^{0,\\alpha}(M)$ and $\\psi \\in C^{0,\\beta}(\\Sigma M)$; finally by the elliptic regularity again, we get $u\\in C^{2,\\alpha}(M)$ and $\\psi \\in C^{1,\\beta}(\\Sigma M)$.\n\\end{proof}\n\n\n\n\n\n\\section{Classification of the (PS) sequences}\n\n\\noindent\nHere we will prove Theorem (\\ref{first}). We will need many preliminary propositions and lemmata.\n\\begin{proposition}\nIf $\\ker D_{g}=\\{0\\}$, then every (PS) sequence for $E$ is bounded.\n\\end{proposition}\n\\begin{proof}\nLet $(u_{n},\\psi_{n})_{n\\in \\N} \\in H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)$ be a (PS) sequence for $E$, that is\n$$E(u_{n},\\psi_{n})\\to c, \\qquad dE(u_{n},\\psi_{n})\\to 0,  \\text{ in }  H^{-1}(M)\\times H^{-\\frac{1}{2}}(\\Sigma M) .$$\nTherefore, there exists a sequence $(\\varepsilon_n, \\delta_n) \\in  H^{-1}(M)\\times H^{-\\frac{1}{2}}(\\Sigma M)$ such that\n\\begin{equation}\\label{equ}\nL_{g} u_{n}=|\\psi_{n}|^{2}u_{n}+\\varepsilon_{n} ,\n\\end{equation}\n\\begin{equation}\\label{eqp}\nD_g\\psi_{n}=|u_{n}|^{2}\\psi_{n}+\\delta_{n} ,\n\\end{equation}\nwith\n$$\\varepsilon_{n}\\to 0,  \\text{ in } H^{-1}(M) \\quad \\text{ and } \\quad  \\delta_{n}\\to 0,  \\text{ in } H^{-\\frac{1}{2}}(M) .$$\nWe let $z_{n}=(u_{n},\\psi_{n}) \\in \\mathcal{H}$, then\n$$2E(z_{n})-\\langle dE(z_{n}),z_{n} \\rangle = \\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g} .$$\nHence\n\\begin{equation}\\label{lev}\n\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}=2c+o(\\|z_{n}\\|).\n\\end{equation}\nMultiplying (\\ref{equ}) by $u_{n}$ and integrating we have\n$$\\|u_{n}\\|^{2}=\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}+o(\\|u_{n}\\|),$$\nhence\n\\begin{equation}\n\\|u_{n}\\|^{2}=2c+o(\\|z_{n}\\|) .\n\\end{equation}\nNow multiplying (\\ref{eqp}) by $\\psi_{n}^{+}=P^{+}(\\psi_n)$, we find\n\\begin{align}\n\\|\\psi_{n}^{+}\\|^{2}&\\leq C\\int_{M}|u_{n}|^{2}|\\psi_{n}||\\psi_{n}^{+}|dv_{g} +o(\\|\\psi_{n}^{+}\\|)\\notag \\\\\n&\\leq C\\left(\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\right)^{\\frac{1}{2}} \\left(\\int_{M}|u_{n}|^{2}|\\psi^{+}_{n}|^{2}dv_{g}\\right)^{\\frac{1}{2}}+o(\\|\\psi^{+}_{n}\\|)\\notag \\\\\n&\\leq C\\big(2c+o(\\|z_{n}\\|)\\big)^{\\frac{1}{2}}\\|u_{n}\\|_{L^{6}}\\|\\psi_{n}^{+}\\|_{L^{3}}+o(\\|\\psi_{n}\\|)\\notag \\\\\n&\\leq C\\big(2c+o(\\|z_{n}\\|)\\big)\\|\\psi_{n}\\| +o(\\|\\psi_{n}\\|)\\notag.\n\\end{align}\nSimilarly, we have for $\\psi_{n}^{-}=P^{-}(\\psi_n)$ that\n$$\\|\\psi_{n}^{-}\\|^{2}\\leq C\\big(2c+o(\\|z_{n}\\|)\\big)\\|\\psi_{n}\\| +o(\\|\\psi_{n}\\|).$$\nHence,\n$$\\|\\psi_{n}\\|\\leq C\\big(2c+o(\\|z_{n}\\|)\\big)+o(1),$$\nso that\n$$\\|z_{n}\\|\\leq C+o(\\|z_{n}\\|)$$\nand $\\|z_{n}\\|$ is bounded.\n\\end{proof}\n\n\n\\noindent\nFrom the previous proposition, we have that up to a subsequence, $z_{n}\\rightharpoonup z_{\\infty}=(u_{\\infty},\\psi_{\\infty})$ in $\\mathcal{H}$, also $u_{n}\\to u_{\\infty}$ in $L^{p}$ for $p<6$ and $\\psi_{n}\\to\\psi$ in $L^{q}$ for $q<3$.\nWe claim that $z_{\\infty}$ is a week solution to (\\ref{el}). Indeed, let $z_0=(u_{0},\\psi_{0})\\in \\mathcal{H}$, since $(z_{n})$ is a (PS) sequence for $E$, we have\n$$\\int_{M}u_{0} L_{g}u_{n}  dv_{g}=\\int_{M}|\\psi_{n}|^{2}u_{n}u_{0}dv_{g} +o(1),$$\nbut $|\\psi_{n}|^{2}\\in L^{\\frac{3}{2}}$ and $u_{n}\\in L^{6}$, thus $u_{n}|\\psi_{n}|^{2}$ converges weakly to $u_{\\infty}|\\psi_{\\infty}|^{2}$ in $L^{\\frac{6}{5}}$, hence\n$$\\int_{M}|\\psi_{n}|^{2}u_{n}u_{0}dv_{g}\\to \\int_{M}|\\psi_{\\infty}|^{2}u_{\\infty}u_{0}dv_{g} .$$\nAlso, by the weak convergence we have that\n$$\\int_{M}  u_{0}L_{g}u_{n}  dv_{g}\\to \\int_{M} u_{0}L_{g}u_{\\infty}  dv_{g} .$$\nTherefore,\n$$L_g u_{\\infty}=|\\psi_{\\infty}|^{2}u_{\\infty},$$\nand similarly it holds\n$$D_{g}\\psi_{\\infty}=|u_{\\infty}|^{2}\\psi_{\\infty}.$$\nWe let now $v_{n}=u_{n}-u_{\\infty}$ and $\\phi_{n}=\\psi_{n}-\\psi_{\\infty}$, then we have the following\n\\begin{lemma}\nLet $h_{n}=(v_{n},\\phi_{n})$, then\n$$E(h_{n})=E(z_{n})-E(z_{\\infty})+o(1)$$\nand\n$$dE(h_{n})\\to 0, \\text{ in } H^{-1}(M)\\times H^{-\\frac{1}{2}}(\\Sigma M) .$$\n\\end{lemma}\n\\begin{proof}\n\\begin{align}\n2E(z_{n})&=\\int_{M}(v_{n}+u_{\\infty})L_{g}( v_{n}+u_{\\infty})dv_{g} +\\int_{M}\\langle D_g(\\phi_{n}+\\psi_{\\infty}),\\phi_{n}+\\psi_{\\infty}\\rangle dv_{g}\\notag\\\\\n&\\quad -\\int_{M}| v_{n}+u_{\\infty}|^{2}|\\phi_{n}+\\psi_{\\infty}|^{2}dv_{g}\\notag\\\\\n&=2E(h_{n})+2E(z_{\\infty})+\\langle dE(z_{\\infty}),h_{n}\\rangle -\\int_{M}|v_{n}|^{2}|\\psi_{\\infty}|^{2}+2|v_{n}|^{2}\\langle \\phi_{n},\\psi_{\\infty}\\rangle\\notag\\\\\n&\\quad +|v_{\\infty}|^{2}|\\phi_{n}|^{2}+2|\\phi_{n}|^{2}v_{n}u_{\\infty} +2|\\psi_{\\infty}|^{2}u_{\\infty}v_{n}-4v_{n}u_{\\infty}\\langle \\psi_{\\infty}, \\phi_{n}\\rangle dv_{g} \\notag .\n\\end{align}\nNow, we first notice that since $dE(z_{\\infty})=0$, we focus on the remaining terms. By the regularity result in Theorem (3.1), we have that $u_{\\infty}\\in C^{2,\\alpha}(M)$ and $\\psi_{\\infty}\\in C^{1,\\beta}(\\Sigma M)$. Since $v_{n}\\to 0$ strongly in $L^{2}(M)$ and $\\phi_{n}\\to 0$ in $L^{2}(\\Sigma M)$ we have that\n$$ -\\int_{M}|v_{n}|^{2}|\\psi_{\\infty}|^{2}+2|v_{n}|^{2}\\langle \\phi_{n},\\psi_{\\infty}\\rangle+|v_{\\infty}|^{2}|\\phi_{n}|^{2}+2|\\psi_{\\infty}|^{2}u_{\\infty}v_{n}-4v_{n}u_{\\infty}\\langle \\psi_{\\infty}, \\phi_{n}\\rangle dv_{g} \\to 0 .$$\nThe last term is $\\int_{M}2|\\phi_{n}|^{2}v_{n}u_{\\infty}dv_{g}$, but we have that $\\phi_{n}\\to 0$ in $L^{\\frac{5}{2}}(\\Sigma M)$ and $v_{n}\\to 0$ in $L^{5}(M)$ therefore we conclude that\n$$E(z_{n})=E(h_{n})+E(z_{\\infty})+o(1),$$\nand this finishes the energy estimate. Now for the gradient part $dE$, we denote by $d_{u}E$ and $d_{\\psi}E$ the scalar and the spinorial components respectively. We have:\n$$d_{u}E(h_{n})=d_{u}E(u_{n},\\psi_{n})+d_{u}E(u_{\\infty},\\psi_{\\infty})+|\\psi_{n}|^{2}u_{\\infty}-|\\psi_{\\infty}|^{2}u_{n}+2\\langle \\psi_{n},\\psi_{\\infty}\\rangle v_{n} .$$\nBut again, $d_{u}E(u_{\\infty},\\psi_{\\infty})=0$ and since $\\psi_{n}\\to \\psi_{\\infty}$ in $L^{\\frac{12}{5}}(\\Sigma M)$ and $u_{n}\\to u_{\\infty}$ in $ L^{\\frac{6}{5}}(M)$ we have that\n$$|\\psi_{n}|^{2}u_{\\infty}-|\\psi_{\\infty}|^{2}u_{n}\\to 0$$\nin $L^{\\frac{6}{5}}(M)$ hence in $H^{-1}(M)$. We also have that $v_{n}\\to 0$ in $L^{\\frac{12}{5}}(M)$ and $\\psi_{n}\\to \\psi_{\\infty}$ in $L^{\\frac{12}{5}}(\\Sigma M)$, thus\n$\\langle \\psi_{n},\\psi_{\\infty}\\rangle v_{n}\\to 0$ in $L^{\\frac{6}{5}}(M)$, thus in $H^{-1}(M)$. Therefore,\n$$d_{u}E(h_{n})=o(1), \\text{ in } H^{-1}(M).$$\nWe move now to the spinorial part, that is\n$$d_{\\psi}E(h_{n})=d_{\\psi}E(u_{n},\\psi_{n})-d_{\\psi}E(u_{\\infty},\\psi_{\\infty})+ |u_{n}|^{2}\\psi_{\\infty}-|u_{\\infty}|^{2}\\psi_{n}+2u_{n}u_{\\infty}\\phi_{n} .$$\nAgain $d_{\\psi}E(u_{\\infty},\\psi_{\\infty})=0$ and $u_{n}\\to u_{\\infty}$ in $L^{\\frac{6}{2}}(M)$ and $\\psi_{n}\\to \\psi_{\\infty}$ in $L^{\\frac{3}{2}}(\\Sigma M)$. Moreover we have that $\\phi_{n}\\to 0$ in $L^{\\frac{5}{2}}(\\Sigma M)$ and $u_{n}\\to u_{\\infty}$ in $L^{\\frac{15}{4}}(M)$. It follows that\n$$d_{\\psi}E(h_{n})=o(1), \\text{ in } H^{-\\frac{1}{2}}(\\Sigma M) .$$\n\\end{proof}\n\n\n\\noindent\nSo from now on, we will assume that our (PS) sequence $z_{n}=(u_{n},\\psi_{n})$, converges weakly to zero in $H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)$ and strongly in $L^{p}(M)\\times L^{q}(\\Sigma M)$, for $p<6$ and $q<3$.\\\\\nWe assume that $z_{n}$ does not converge to zero in $H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)$ since otherwise the (PS) condition would be satisfied. Now, let us denote by $B_{r}(x)$ the geodesic ball with center in $x\\in M$ and radius $r$, we define the following sets, for a given $\\epsilon_0>0$:\n$$\\Sigma_{1}=\\left\\{x\\in M; \\liminf_{r\\to 0} \\liminf_{n\\to\\infty}\\int_{B_{r}(x)}|u_{n}|^{6}dv_{g}\\geq\\epsilon_{0}\\right\\} ,$$\n$$\\Sigma_{2}=\\left\\{x\\in M; \\liminf_{r\\to 0} \\liminf_{n\\to\\infty}\\int_{B_{r}(x)}|\\psi_{n}|^{3}dv_{g}\\geq\\epsilon_{0}\\right\\} ,$$\n$$\\Sigma_{3}=\\left\\{x\\in M; \\liminf_{r\\to 0} \\liminf_{n\\to\\infty}\\int_{B_{r}(x)}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\geq \\epsilon_{0}\\right\\} .$$\nWe have:\n\\begin{lemma}\nThere exists $\\epsilon_{0}>0$ depending on $M$, such that if $x_{0}\\not \\in \\Sigma_{1}\\cap \\Sigma_{2} \\cap \\Sigma_{3}$, then there exists $r>0$ such that $z_{n}\\to 0$ in $H^{1}(B_{r}(x_{0}))\\times H^{\\frac{1}{2}}(\\Sigma B_{r}(x_{0}))$.\n\\end{lemma}\n\\begin{proof}\nWe will prove this result by contradiction, by assuming that for every $\\epsilon>0$, there exists $x_{0}\\not \\in \\Sigma_{1}\\cap \\Sigma_{2} \\cap \\Sigma_{3}$, such that for every $r>0$, $z_{n}\\not \\to 0$ in $H^{1}(B_{r}(x_{0}))\\times H^{\\frac{1}{2}}(\\Sigma B_{r}(x_{0}))$.\\\\\n\n\\noindent\n\\emph{Case I}: $x_{0}\\not \\in \\Sigma_{1}$.\\\\\nGiven $\\epsilon>0$, there exists $r>0$ such that $\\int_{B_{4r}(x_{0})}|u_{n}|^{6}dv_{g}<\\epsilon$. We first estimate the $\\psi$ component. That is, we consider a smooth cut off function $\\eta$ supported on $B_{4r}(x_{0})$ and equals to $1$ on $B_{2r}(x_{0})$, then by (\\ref{eqp}) we have:\n\\begin{align}\nD_g(\\eta\\psi_{n})&=\\eta D_g\\psi_{n}+\\nabla \\eta \\cdot \\psi_{n}\\notag \\\\\n&=\\eta |u_{n}|^{2}\\psi_{n}+\\nabla \\eta \\cdot \\psi_{n}+\\eta\\delta_{n}, \\notag\n\\end{align}\nwhere $\\|\\delta_{n}\\|_{H^{-\\frac{1}{2}}}\\to 0$. Hence\n\\begin{align}\n\\|\\eta\\psi_{n}\\|_{H^{\\frac{1}{2}}}&\\leq C_{1}\\|\\eta |u_{n}|^{2}\\psi_{n}+\\nabla \\eta \\cdot \\psi_{n}+\\eta\\delta_{n}\\|_{H^{-\\frac{1}{2}}}\\notag\\\\\n&\\leq C_{2}\\left(\\|\\eta|u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}+ \\|\\psi_{n}\\|_{L^{\\frac{3}{2}}}+\\|\\delta_{n}\\|_{H^{-\\frac{1}{2}}}\\right) .\\notag\n\\end{align}\nSince $\\|\\psi_{n}\\|_{L^{\\frac{3}{2}}}\\to 0$, it remains to estimate\n\\begin{align}\n\\|\\eta |u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}&\\leq \\|u_{n}\\|_{L^{6}(B_{4r}(x_{0}))}^{2}\\|\\eta \\psi_{n}\\|_{L^{3}} \\leq C_{3}\\epsilon^{\\frac{1}{3}} \\|\\eta \\psi_{n}\\|_{H^{\\frac{1}{2}}} .\\notag\n\\end{align}\nHence, taking $C_{3}\\epsilon<\\frac{1}{2}$, we deduce that $\\eta \\psi_{n}\\to 0$ in $H^{\\frac{1}{2}}$, yielding\n$$\\|\\eta \\psi_{n}\\|_{L^{3}}\\to 0.$$\nNow we estimate the $u$ component. We consider a smooth cut off function $\\rho$ supported on $B_{2r}(x_{0})$ and equals to $1$ on $B_{r}(x_{0})$, then by (\\ref{equ}) we have\n\\begin{align}\nL_{g} (\\rho u_{n})&=\\rho L_{g} u_{n}- u_{n}\\Delta_{g} \\rho-2g(\\nabla \\rho, \\nabla u_{n})\\notag \\\\\n&=\\rho|\\psi_{n}|^{2}u_{n}- u_{n}\\Delta_{g} \\rho-2\\nabla \\rho \\cdot \\nabla u_{n}+\\rho \\varepsilon_{n}\\notag\n\\end{align}\nwhere $\\varepsilon_{n}\\to 0$ in $H^{-1}(M)$. From elliptic estimates now we have that\n\\begin{align}\n\\|\\rho u_{n}\\|_{H^{1}}&\\leq C\\| \\rho|\\psi_{n}|^{2 }u_{n}- u_{n}\\Delta \\rho+2g(\\nabla \\rho , \\nabla u_{n})+\\rho \\varepsilon_{n}\\|_{H^{-1}}\\notag\\\\\n&\\leq C_{1}\\left(\\|\\rho|\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}+\\|u_{n}\\Delta \\rho\\|_{L^{\\frac{6}{5}}}+2\\|g(\\nabla \\rho , \\nabla u_{n})\\|_{H^{-1}}\\right).\\notag\n\\end{align}\nFirst we estimate $\\|\\rho|\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}$:\n\\begin{align}\n\\|\\rho|\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}&\\leq \\| \\eta \\psi_{n}\\|^{2}_{L^{3}}\\|\\rho u_{n}\\|_{L^{6}} \\leq C_{1}\\| \\eta \\psi_{n}\\|^{2}_{H^{\\frac{1}{2}}}\\|\\rho u_{n}\\|_{H^{1}} . \\notag\n\\end{align}\nFrom the previous estimates, for $n$ big enough we have that $CC_{1}\\| \\eta \\psi_{n}\\|^{2}_{H^{\\frac{1}{2}}}<\\frac{1}{2}$. Thus we have that\n$$\\|\\rho u_{n}\\|_{H^{1}}\\leq \\|u_{n}\\Delta \\rho\\|_{L^{\\frac{6}{5}}}+2\\|g(\\nabla \\rho , \\nabla u_{n})\\|_{H^{-1}} .$$\nNow clearly\n$$\\|u_{n}\\Delta \\rho\\|_{L^{\\frac{6}{5}}}\\leq C\\|u_{n}\\|_{L^{2}},$$\nand the term\n$$\\|g(\\nabla \\rho , \\nabla u_{n})\\|_{H^{-1}}\\leq \\sup_{k\\in H^{1};\\|k\\|_{H^{1}}\\leq 1}\\left|\\int_{M}g(\\nabla\\rho,\\nabla u_{n})kdv_{g}\\right| .$$\nBut\n\\begin{align}\n\\left|\\int_{M} g(\\nabla \\rho , \\nabla u_{n})kdv_{g}\\right|&\\leq \\left|\\int_{M}u_{n}\\left(k\\Delta \\rho+g(\\nabla \\rho, \\nabla k ) \\right)dv_{g}\\right|\\leq C\\|k\\|_{H^{1}}\\|u_{n}\\|_{L^{2}} \\to 0 .\\notag\n\\end{align}\nHence $\\rho u_{n}$ converges to zero in $H^{1}(B_{r}(x_{0}))$ and this leads to a contradiction.\\\\\n\n\n\\noindent\n\\emph{Case II}: $x_{0}\\not \\in \\Sigma_{2}$.\\\\\nGiven $\\epsilon>0$, there exists $r>0$ such that $\\int_{B_{4r}(x_{0})}|\\psi_{n}|^{3}dv_{g}<\\epsilon$. Then again we compute\n\\begin{align}\nL_{g} (\\rho u_{n})&=\\rho (L_{g} u_{n})- u_{n}\\Delta_{g} \\rho -2g(\\nabla \\rho , \\nabla u_{n})\\notag \\\\\n&=\\rho|\\psi_{n}|^{2}u_{n}- u_{n}\\Delta_{g} \\rho-2g(\\nabla \\rho , \\nabla u_{n})+\\rho \\varepsilon_{n} ,\\notag\n\\end{align}\nwhere $\\varepsilon_{n}\\to 0$ in $H^{-1}$. From elliptic estimates now we have that\n\\begin{align}\n\\|\\rho u_{n}\\|_{H^{1}}&\\leq C\\|\\rho|\\psi_{n}|^{2}u_{n}- u_{n}\\Delta_g \\rho+2g(\\nabla \\rho_{1} , \\nabla u_{n})+\\rho \\varepsilon_{n}\\|_{H^{-1}}\\notag\\\\\n&\\leq C_{1}\\left(\\|\\rho|\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}+\\|u_{n}\\Delta_{g} \\rho\\|_{L^{\\frac{6}{5}}}+2\\|g(\\nabla \\rho, \\nabla u_{n})\\|_{H^{-1}}\\right) .\\notag\n\\end{align}\nAgain, we estimate\n\\begin{align}\n\\|\\rho|\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}&\\leq \\| \\psi_{n}\\|^{2}_{L^{3}}\\|\\rho u_{n}\\|_{L^{6}}\\leq C_{1}\\epsilon^{\\frac{2}{3}}\\|\\rho u_{n}\\|_{H^{1}} .\\notag\n\\end{align}\nTaking $C_{1}C\\epsilon<\\frac{1}{2}$, we have that\n$$\\|\\rho u_{n}\\|_{H^{1}}\\leq C_{1}\\left(\\|u_{n}\\Delta_g \\rho\\|_{L^{\\frac{6}{5}}}+2\\|g(\\nabla \\rho , \\nabla u_{n})\\|_{H^{-1}} +\\|\\rho\\varepsilon_{n}\\|_{H^{-1}}\\right),$$\nand as in the previous case we have that\n$$\\|u_{n}\\Delta_g \\rho\\|_{L^{\\frac{6}{5}}}+2\\|g(\\nabla \\rho , \\nabla u_{n})\\|_{H^{-1}} +\\|\\rho\\varepsilon_{n}\\|_{H^{-1}}\\to 0.$$\nHence $\\|\\rho u_{n}\\|_{H^{1}}\\to 0$. Next, we estimate the spinorial component:\n\\begin{align}\n\\|\\eta\\psi_{n}\\|_{H^{\\frac{1}{2}}}&\\leq C_{1}\\|\\eta |u_{n}|^{2}\\psi_{n}+\\nabla \\eta \\cdot \\psi_{n}+\\eta \\delta_{n}\\|_{H^{-\\frac{1}{2}}}\\notag\\\\\n&\\leq C_{2}\\left(\\|\\eta |u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}+ \\|\\psi_{n}\\|_{L^{\\frac{3}{2}}}+\\|\\delta_{n}\\|_{H^{-\\frac{1}{2}}}\\right).\\notag\n\\end{align}\nBut\n$$\\|\\eta |u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}\\leq \\|\\rho u_{n}\\|_{L^{6}}^{2}\\|\\eta \\psi_{n}\\|_{H^{\\frac{1}{2}}}.$$\nUsing the fact that $\\|\\rho u_{n}\\|_{H^{1}}\\to 0$, we have that $z_{n}\\to 0$ in $H^{1}(B_{r}(x_{0}))\\times H^{\\frac{1}{2}}(\\Sigma B_{r}(x_{0}))$, yielding a contradiction.\\\\\n\n\n\\noindent\n\\emph{Case III}: $x_{0}\\not\\in \\Sigma_{3}$.\\\\\nAgain, given $\\epsilon>0$, let $r>0$ so that $\\int_{B_{2r}(x_{0})}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}<\\epsilon$. We have that\n$$\\|\\rho \\psi_{n}\\|_{H^{\\frac{1}{2}}} \\leq C_{2}\\left(\\|\\rho |u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}+o(1)\\right).$$\nBut\n$$\\|\\rho |u_{n}|^{2}\\psi_{n}\\|_{L^{\\frac{3}{2}}}\\leq \\left(\\int_{B_{2r}(x_{0})}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\right)^{\\frac{1}{2}}\\|\\rho u_{n}\\|_{L^{6}} ,$$\nthus\n$$\\|\\rho\\psi_{n}\\|_{H^{\\frac{1}{2}}} \\leq C\\epsilon^{\\frac{1}{2}}\\|\\rho u_{n}\\|_{H^{1}}+o(1).$$\nSimilarly, we have for the  $u$ component,\n$$\\|\\rho u_{n}\\|_{H^{1}}\\leq C \\|\\rho |\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}+o(1),$$\nand\n$$\\|\\rho |\\psi_{n}|^{2}u_{n}\\|_{L^{\\frac{6}{5}}}\\leq \\left(\\int_{B_{2r}(x_{0})}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\right)^{\\frac{1}{2}}\\|\\rho \\psi_{n}\\|_{L^{3}}. $$\nHence\n$$\\|\\rho u_{n}\\|_{H^{1}}\\leq C\\epsilon^{\\frac{1}{2}}\\|\\rho \\psi_{n}\\|_{H^{\\frac{1}{2}}}+o(1).$$\nCombining both the previous inequalities we have $\\rho z_{n}\\to 0$ in $H^1B_{r}(x_{0})\\times H^{\\frac{1}{2}}\\Sigma B_{r}(x_{0})$, leading to a contradiction.\n\\end{proof}\n\n\n\\noindent\nFrom the previous lemma we deduce the following properties.\n\\begin{corollary}\nIf $(z_{n})$ does not satisfy the (PS) condition, then\n$$\\Sigma_{1}= \\Sigma_{2} =\\Sigma_{3}\\not=\\emptyset.$$\n\\end{corollary}\n\n\n\\begin{corollary}\nLet $(z_{n})$ be a (PS) sequence at the level $c$. If $c<\\frac{\\epsilon_{0}}{2}$ then $z_{n}$ converges strongly to zero.\n\\end{corollary}\n\\begin{proof}\nThe proof follows from the boundedness of the (PS) sequences. Indeed, from $(\\ref{lev})$ we have\n$$\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}=2c+o(1).$$\nHence if $2c<\\epsilon_0$, we have that for $n$ big enough,\n$$\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}<\\epsilon_0,$$\nthus $z_{n}\\to 0$.\n\\end{proof}\n\n\n\n\\noindent\nNow, for a given (PS) sequence $(z_{n})$, we define the concentration function $Q_{n}$ for $r> 0$ by\n$$Q_{n}(r)=\\sup_{x\\in M} \\int_{B_{r}(x)}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g} .$$\nWe explicitly notice that one can define equivalently the $\\sup$ on the integrals relative to $\\Sigma_{1}$ and $\\Sigma_{2}$. We choose $\\epsilon>0$ so that $3\\epsilon<\\epsilon_{0}$, then if $\\Sigma_3\\not=0$, we have the existence of $x_{n}\\in M$ and $R_{n}\\to 0$ such that\n$$Q_{n}(R_{n})=\\int_{B_{x_{n}}(R_{n})}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}=\\epsilon .$$\nWithout loss of generality, we can always assume that $x_{n}\\to x_{0}$ and $i(M)\\geq 3$, where $i(M)$ is the injectivity radius of $M$. Also, we define the map $\\rho_{n}(x)=exp_{x_{n}}(R_{n}x)$ for $x\\in \\R^{3}$ such that $R_{n}|x|<3$; we denote also $\\sigma_{n}=\\rho_{n}^{-1}$. We let $B_{R}^{0}$ denote the Euclidian ball centered at zero and with radius $R$. That is,\n$$B_{R}^{0}=\\{x\\in \\R^{3}; |x|< R\\}.$$\nWe can then consider the metric $g_{n}$ on $B_{R}^{0}$ defined by a suitable rescaled of the pull-back of $g$:\n$$g_{n}=R_{n}^{-2}\\rho_{n}^{*}g .$$\nClearly, the two Riemannian patches $(B_{R}^{0},g_{n})$ and $(B_{RR_{n}}(x_{n}),g)$ are conformally equivalent for $n$ large enough and $g_{n}\\to g_{\\R^{3}}$ in $C^{\\infty}(B_{R}^{0})$. We consider now the identification map  (see \\cite{Bour})\n$$(\\rho_{n})_{*}:\\Sigma_{p}(B_{R}^{0},g_{n})\\to \\Sigma_{\\rho_{n}(p)}(B_{RR_{n}}(x_{n}),g),$$\nand we set\n$$\\rho_{n}^{*}(\\varphi)=(\\rho_{n})_{*}^{-1}\\circ \\varphi \\circ \\rho_{n}.$$\nUsing these maps, we can define the spinors $\\Psi_{n}$ on $\\Sigma B_{R}^{0}$ by\n$$\\Psi_{n}=R_{n}\\rho_{n}^{*}\\psi_{n} ,$$\nand from the conformal change of the Dirac operator, we have that\n$$D_{g_{n}}\\Psi_{n}=R_{n}^{2}\\rho_{n}^{*}D_{g}\\psi_{n} .$$\nSo we get:\n$$\\int_{B_{R}^{0}}\\langle D_{g_{n}}\\Psi_{n},\\Psi_{n}\\rangle dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}\\langle D_{g}\\psi_{n},\\psi_{n}\\rangle dv_{g},$$\n$$\\int_{B_{R}^{0}}|\\Psi_{n}|^{3}dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}|\\psi_{n}|^{3}dv_{g}.$$\nNow we consider the $u$ component, that is we define\n$$U_{n}=R_{n}^{\\frac{1}{2}}\\rho_{n}^{*}u_{n} ,$$\nso that by conformal change of the conformal Laplacian, we have:\n$$L_{g_{n}}U_{n}=R_{n}^{\\frac{5}{2}}\\rho^{*}_{n}L_{g}u_{n} .$$\nHence\n$$\\int_{B_{R}^{0}}U_{n}L_{g_{n}}U_{n}dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}u_{n}L_{g}u_{n}dv_{g},$$\n\n\\begin{equation}\\label{e1}\n\\int_{B_{R}^{0}}|U_{n}|^{6}dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}|u_{n}|^{6}dv_{g},\n\\end{equation}\nand\n$$\\int_{B_{R}^{0}}|U_{n}|^{2}|\\Psi_{n}|^{2}dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}.$$\nWe have the following:\n\\begin{lemma}\nLet us set\n$$F_{n}=L_{g_{n}}U_{n}-|\\Psi_{n}|^{2}U_{n}, \\qquad H_{n}=D_{g_{n}}\\Psi_{n}-|U_{n}|^{2}\\Psi_{n} .$$\nThen\n$$F_{n}\\to 0  \\text{ in } H^{-1}_{loc}(\\R^{3}), \\qquad  H_{n}\\to 0   \\text{ in } H^{-\\frac{1}{2}}_{loc}(\\Sigma \\R^{3}) .$$\n\\end{lemma}\n\n\\noindent\nHere the convergence in $H^{-1}_{loc}$ is understood in the sense that for all $R>0$,\n$$\\sup\\left\\{\\langle F_{n}, F\\rangle_{H^{-1},H^{1}}; F\\in H^{1}(\\R^{3}), \\; \\text{supp}(F)\\subset B_{R}^{0},\\; \\|F\\|_{H^{1}}\\leq 1\\right\\}\\to 0,$$\nand similarly for $H_{n}$.\n\\begin{proof}\nWe first notice that by construction, we have that\n$$L_{g_{n}}U_{n}-|\\Psi_{n}|^{2}U_{n}=R_{n}^{\\frac{5}{2}}\\rho_{n}^{*}(L_{g}u_{n}-|\\psi_{n}|^{2}u_{n}) .$$\nHence we get\n$$F_{n}=R_{n}^{\\frac{5}{2}}\\rho_{n}^{*}(\\varepsilon_{n}) ,$$\nand similarly\n$$H_{n}=R_{n}^{2}\\rho_{n}^{*}(\\delta_{n}) .$$\nNow we consider $F\\in H^{1}(\\R^{3})$ such that $supp(F)\\subset B_{R}^{0}$ and $\\|F\\|_{H^{1}}\\leq 1$. Since $R_{n}\\to 0$, then for $n$ big enough we have that:\n\\begin{align}\n\\langle F_{n},F\\rangle_{H^{-1},H^{1}} &=\\int_{B^{0}_{R_{n}^{-1}}}F_{n}F dv_{g_n}\\notag \\\\\n&=\\int_{B^{0}_{R_{n}^{-1}}}\\rho^{*}_{n}(\\varepsilon_{n})R_{n}^{\\frac{5}{2}} F dv_{g_{n}}\\notag\\\\\n&=\\int_{B^{0}_{R_{n}^{-1}}}\\rho^{*}_{n}(\\varepsilon_{n})R_{n}^{-\\frac{1}{2}} F dv_{\\rho^{*}_{n}g}\\notag\\\\\n&=\\int_{B_{1}(x_{n})}\\varepsilon_{n}R_{n}^{-\\frac{1}{2}}\\sigma_{n}^{*}(F) dv_{g} .\\notag\n\\end{align}\nBut we have that $\\|R_{n}^{-\\frac{1}{2}}\\sigma_{n}^{*}(F)\\|_{H^{1}}\\leq C$, hence\n$$\\langle F_{n},F\\rangle_{H^{-1},H^{1}}\\to 0.$$\nA similar estimate holds for $H_{n}$.\n\\end{proof}\n\n\n\\noindent\nNow, let us re recall the spaces\n$$D^{1}(\\R^{3})=\\left\\{u\\in L^{6}(\\R^{3}); |\\nabla u| \\in L^{2}(\\R^{3})\\right\\} $$\nand\n$$D^{\\frac{1}{2}}(\\Sigma\\R^{3})=\\left\\{\\psi \\in L^{3}(\\Sigma\\R^{3});|\\xi|^{\\frac{1}{2}}|\\widehat{\\psi}|\\in L^{2}(\\R^{3})\\right\\},$$\nwhere here $\\widehat{\\psi}$ is the Fourier transform of $\\psi$. We have then the following:\n\n\\begin{lemma}\nFor $\\epsilon>0$ small enough, there exist $U_{\\infty}\\in D^{1}(\\R^{3})$ and $\\Psi_{\\infty}\\in D^{\\frac{1}{2}}(\\Sigma \\R^{3})$ such that $U_{n}\\to U_{\\infty}$ in $H^{1}_{loc}(\\R^{3})$ and $\\Psi_{n}\\to \\Psi_{\\infty}$ in $H^{\\frac{1}{2}}_{loc}(\\Sigma\\R^{3})$. Moreover they satisfy\n\\begin{equation}\\label{eqR3}\n\\left\\{\\begin{array}{ll}\n-\\Delta_{g_{\\R^3}} U_{\\infty}=|\\Psi_{\\infty}|^{2}U_{\\infty} \\\\\n& \\text{ on } \\R^{3} .\\\\\nD_{g_{\\R^3}}\\Psi_{\\infty}=|U_{\\infty}|^{2}\\Psi_{\\infty}\n\\end{array}\n\\right.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nSince the sequence $Z_{n}=(U_n,\\Psi_n)$ is bounded in $H^{1}_{loc}\\times H^{\\frac{1}{2}}_{loc}$, for every $\\beta \\in C^{\\infty}_{0}(\\R^{3})$, we have that $\\beta Z_{n}$ is bounded in $H^{1}\\times H^{\\frac{1}{2}}$, hence there exist $U_{\\infty}$ and $\\Psi_{\\infty}$ such that $U_{n}\\rightharpoonup U_{\\infty}$ in $H^{1}_{loc}$ and $U_{n}\\to U_{\\infty}$ strongly in $L_{loc}^{p}$ for $p<6$. Similarly $\\Psi_{n}\\rightharpoonup \\Psi_{\\infty}$ in $H_{loc}^{\\frac{1}{2}}$ and strongly in $L_{loc}^{p}$ for $p<3$. Now we notice that from (\\ref{e1}), we have that\n$$\\int_{B_{R}^{0}}|U_{n}|^{6}dv_{g_{n}}=\\int_{B_{RR_{n}}(x_{n})}|u_{n}|^{6}dv_{g}.$$\nHence\n\\begin{equation}\\label{e3}\n\\limsup_{n\\to \\infty}\\int_{B_{R}^{0}}|U_{n}|^{6}dv_{g_{n}}\\leq \\sup_{n\\geq 1}\\int_{M}|u_{n}|^{6}dv_{g}<+\\infty ,\n\\end{equation}\nhence $U_{\\infty}\\in L^{6}(\\R^{3})$ and similarly $\\Psi_{\\infty}\\in L^{3}(\\Sigma\\R^{3})$. Also as in the proof of Proposition (4.1), we see that $(U_{\\infty},\\Psi_{\\infty})$ satisfies equation (\\ref{eqR3}); hence\n$$\\int_{\\R^{3}}|\\nabla U_{\\infty}|^{2}dv_{g} <\\infty$$\nand $\\nabla \\Psi_{\\infty}\\in L^{\\frac{3}{2}}(\\Sigma\\R^3)\\subset H^{-\\frac{1}{2}}(\\Sigma\\R^{3})$, which leads to the fact that $U_{\\infty}\\in D^{1}(\\R^{3})$ and $\\Psi_{\\infty}\\in D^{\\frac{1}{2}}(\\Sigma\\R^{3})$. Now, using again Lemma (4.2), we can assume at this stage that $\\Psi_{\\infty}=0$ and $U_{\\infty}=0$ by replacing $\\Psi_{n}$ by $\\Psi_{n}-\\Psi_{\\infty}$ and $U_{n}$ by $U_{n}-U_{\\infty}$. Now let $x\\in \\R^{3}$, then by assumption we have that for $n$ big enough,\n$$\\int_{B_{1}^{0}}|U_{n}|^{2}|\\Psi_{n}|^{2}dv_{g_n}\\leq \\epsilon.$$\nLet $\\beta \\in C^{\\infty}_{0}(\\R^{3})$, then by elliptic regularity, we have that\n\\begin{align}\n\\|\\beta^{2} U_{n}\\|_{H^{1}}&\\leq C\\left(\\|L_{g_{\\R^{3}}}(\\beta^{2}U_{n})\\|_{H^{-1}}+\\|\\beta^{2}U_{n}\\|_{L^{2}}\\right)\\\\\n&\\leq C\\left(\\|L_{g_{n}}(\\beta^{2}U_{n})\\|_{H^{-1}}+\\|(L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta^{2}U_{n})\\|_{H^{-1}}+\\|\\beta^{2}U_{n}\\|_{L^{2}}\\right).\\notag\n\\end{align}\nNow, we have that $\\|\\beta^{2}U_{n}\\|_{L^{2}}\\to 0$, and we want to estimate the term\n$$\\|(L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta^{2}U_{n})\\|_{H^{-1}}.$$\nFirst, we have that for every $F\\in H^{1}$:\n$$\\langle  (L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta^{2}U_{n}),F \\rangle_{H^{-1},H^{1}}=\\langle \\beta U_{n},\\beta (L_{g_{\\R^{3}}}-L_{g_{n}})^{*}F\\rangle,$$\nwhere $ (L_{g_{\\R^{3}}}-L_{g_{n}})^{*}$ is the adjoint of $L_{g_{\\R^{3}}}-L_{g_{n}}$ with respect to the metric $g_{\\R^{3}}$. Now since $g_{n}\\to g_{\\R^{3}}$ in $C^{\\infty}$, we have that\n$$\\|(L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta \\cdot )\\|_{H^{2},L^{2}}\\to 0,$$\nand by duality\n$$\\|\\beta(L_{g_{\\R^{3}}}-L_{g_{n}})^{*}\\|_{L^{2},H^{-2}}\\to 0.$$\nSimilarly, we have also that\n$$\\|\\beta(L_{g_{\\R^{3}}}-L_{g_{n}})^{*}\\|_{H^{2},L^{2}}\\to 0,$$\ntherefore, by interpolation, we have that\n\\begin{equation}\\label{e4}\n\\|\\beta(L_{g_{\\R^{3}}}-L_{g_{n}})^{*}\\|_{H^{1},H^{-1}}\\to 0.\n\\end{equation}\nSo we have that:\n\\begin{align}\n|\\langle  (L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta^{2}U_{n}),F\\rangle_{H^{-1},H^{1}}|&\\leq C\\|\\beta U_{n}\\|_{H^{1}}\\|\\beta (L_{g_{\\R^{3}}}-L_{g_{n}})^{*}F\\|_{H^{-1}}\\notag \\\\\n&\\leq C\\|\\beta(L_{g_{\\R^{3}}}-L_{g_{n}})^{*}\\|_{H^{1},H^{-1}} \\|F\\|_{H^{1}},\\notag\n\\end{align}\nhence\n$$\\|(L_{g_{\\R^{3}}}-L_{g_{n}})(\\beta^{2}U_{n})\\|_{H^{-1}}\\to 0.$$\nIt remains to estimate the term $\\|L_{g_{n}}(\\beta^{2}U_{n})\\|_{H^{-1}}$, but we have that\n$$\\|L_{g_{n}}(\\beta^{2}U_{n})\\|_{H^{-1}}\\leq \\|\\beta^{2}(|\\Psi_{n}|^{2}U_{n}+F_{n})\\|_{H^{-1}}+o(1),$$\nand from Lemma 4.6, we have that $\\beta^{2}F_{n}\\to 0$ in $H^{-1}$, therefore, we have that\n$$\\|\\beta^{2} U_{n}\\|_{H^{1}}\\leq C\\| \\beta^{2}|\\Psi_{n}|^{2}U_{n}\\|_{H^{-1}}+o(1).$$\nNow, if we take $supp(\\beta)\\in B_{1}^{0}$, we have that\n\\begin{align}\n\\|\\beta^{2} U_{n}\\|_{H^{1}}&\\leq C\\| \\beta^{2}|\\Psi_{n}|^{2}U_{n}\\|_{L^{\\frac{6}{5}}(B_{1}^{0})}+o(1)\\notag \\\\\n&\\leq C\\left( \\int_{B_{1}^{0}}|U_{n}|^{2}|\\Psi_{n}|^{2}dv_{g_{n}}\\right)^{\\frac{1}{2}}\\|\\beta^{2}\\Psi_{n}\\|_{L^{3}}+o(1)\\notag\\\\\n&\\leq C\\epsilon^{\\frac{1}{2}}\\|\\beta^{2}\\Psi_{n}\\|_{L^{3}}+o(1) .\\notag\n\\end{align}\nA similar computation can be done to show that\n$$\\|\\beta^{2} \\Psi_{n}\\|_{H^{\\frac{1}{2}}}\\leq  C\\epsilon^{\\frac{1}{2}}\\|\\beta^{2}U_{n}\\|_{L^{6}}+o(1),$$\nand combining these last two estimates we have that\n$$\\|\\beta^{2}U_{n}\\|_{H^{1}}+\\|\\beta^{2}\\Psi_{n}\\|_{H^{\\frac{1}{2}}}\\to 0.$$\n\\end{proof}\n\n\n\\noindent\nIt follows from this lemma in particular, since\n$$\\int_{B_{1}^{0}}|U_{n}|^{2}|\\Psi_{n}|^{2}dv_{g_n}=Q(R_{n})=\\epsilon,$$\nthat also\n$$\\int_{B_{1}^{0}}|U_{\\infty}|^{2}|\\Psi_{\\infty}|^{2}dv_{g_{\\R^{3}}}=\\epsilon,$$\nand hence $U_{\\infty}\\not=0$ and $\\Psi_{\\infty}\\not=0$ and they satisfy equation (\\ref{eqR3}); by the regularity results proved in the previous section, we have that $U_{\\infty}\\in C^{2,\\alpha}(\\R^{3})$ and $\\Psi_{\\infty}\\in C^{1,\\beta}(\\Sigma\\R^{3})$.\\\\\nNow, we assume that $x_{n}\\to x$ and  we consider a cut-off function $\\beta=1$ on $B_{1}(x)$ and $supp(\\beta)\\subset B_{2}(x)$, we define then $v_{n}\\in C^{2,\\alpha}(M)$ and $\\phi_{n}\\in C^{1,\\beta}(\\Sigma M)$ by\n\\begin{equation}\\label{e5}\nv_{n}=R_{n}^{-\\frac{1}{2}}\\beta\\sigma_{n}^{*}(U_\\infty)\n\\end{equation}\nand\n\\begin{equation}\\label{e6}\n\\phi_{n}=R_{n}^{-1}\\beta\\sigma_{n}^{*}(\\Psi_\\infty) .\n\\end{equation}\nWe are going to prove the following\n\\begin{lemma}\nLet  $\\overline{u}_{n}=u_{n}-v_{n}$ and $\\overline{\\psi}_{n}=\\psi_{n}-\\phi_{n}$. Then, up to a subsequence, $\\overline{u}_{n}\\rightharpoonup 0$ in $H^{1}(M)$ and $\\overline{\\psi}_{n}\\rightharpoonup 0$ in $H^{\\frac{1}{2}}(\\Sigma M)$.\n\\end{lemma}\n\\begin{proof}\nWe already have that $u_{n}\\rightharpoonup 0$ and $\\psi_{n}\\rightharpoonup 0$, thus to prove the lemma we only need to show the weak convergence for $v_{n}$ and $\\phi_{n}$: these sequences are bounded in $H^{1}(M)$ and $H^{1\/2}(\\Sigma M)$ respectively, then up to subsequences, they converge to some limit. So if we show that the distributional limit is zero, then the limit in the desired space is also zero. So let $f\\in C^{\\infty}(M)$ and $h\\in C^{\\infty}(\\Sigma M)$. We want to show that\n$$\\int_{M}v_{n}fdv_{g} \\to 0$$\nand\n$$\\int_{M}\\langle \\phi_{n},h\\rangle dv_{g}\\to 0.$$\nWe fix $R>0$, then we have\n\\begin{align}\n\\int_{B_{R_{n}R}(x_{n})}v_{n}fdv_{g}&=R_{n}^{-\\frac{1}{2}}\\int_{B_{R_{n}R}(x_{n})}\\beta \\sigma_{n}^{*}(U_\\infty) fdv_{g}\\notag\\\\\n&=R_{n}^{\\frac{5}{2}}\\int_{B_{R}^{0}}\\rho_{n}^{*}(\\beta) \\rho_{n}^{*}(f)U_\\infty dv_{g_{n}} .\\notag\n\\end{align}\nHence\n$$\\left|\\int_{B_{R_{n}R}(x_{n})}v_{n}f dv_{g}\\right|\\leq CR_{n}^{\\frac{5}{2}} \\|f\\|_{\\infty}\\int_{B_{R}^{0}}|U_\\infty|dv_{g_{\\R^{3}}}.$$\nAlso, for $n$ big enough we have that\n\\begin{align}\n\\int_{M\\setminus B_{R_{n}R}(x_{n})}v_{n}fdv_{g}&=\\int_{B_{3}(x_{n})\\setminus B_{R_{n}R}(x_{n})}v_{n}fdv_{g}\\notag\\\\\n&=R_{n}^{\\frac{5}{2}}\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}\\rho_{n}^{*}(\\beta) \\rho_{n}^{*}(f)U_\\infty dv_{g_{n}} .\\notag\n\\end{align}\nHence, we have\n\\begin{align}\n\\left|\\int_{M\\setminus B_{R_{n}R}(x_{n})}v_{n}fdv_{g}\\right|&\\leq CR_{n}^{\\frac{5}{2}}\\|f\\|_{\\infty}\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|U_\\infty|dv_{g_{\\R^{3}}} \\notag \\\\\n&\\leq C\\|f\\|_{\\infty}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|U_\\infty|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{6}}.\\notag\n\\end{align}\nBased on these last two inequalities we have that\n$$\\left|\\int_{M}v_{n}fdv_{g}\\right|\\leq C\\|f\\|_{\\infty}\\left(R_{n}^{\\frac{5}{2}}\\int_{B_{R}^{0}}|U_\\infty|dv_{g_{\\R^{3}}}+\\left(\\int_{\\R^{3}\\setminus B_{R}^{0}}|U_\\infty|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{6}}\\right).$$\nLetting $n\\to \\infty$ and then $R\\to \\infty$ we get the desired result. A similar inequality holds for $\\phi_{n}$ and $h$.\n\\end{proof}\n\n\n\\noindent\nNow we estimate the differential, that is\n\\begin{lemma}\nWe have\n$$dE(v_{n},\\phi_{n})\\to 0 \\qquad \\text{ and }\\qquad dE(\\overline{u}_{n},\\overline{\\psi}_{n})\\to 0 ,$$\nin $H^{-1}(M)\\times H^{-\\frac{1}{2}}(\\Sigma M)$.\n\\end{lemma}\n\\begin{proof}\nWe set\n$$f_{n}=L_{g}v_{n}-|\\phi_{n}|^{2}v_{n}$$\nand\n$$h_{n}=D_{g}\\phi_{n}-|v_{n}|^{2}\\phi_{n} .$$\nLet $f\\in H^{1}(M)$ and $h\\in H^{\\frac{1}{2}}(\\Sigma M)$, we compute\n\\begin{align}\n\\int_{M}f_{n}f dv_{g}&=R_{n}^{-\\frac{1}{2}}\\left(\\int_{M}(-\\Delta_{g}\\beta)\\sigma_{n}^{*}(\\U) fdv_{g}+2\\int_{M}g(-\\nabla \\beta,\\nabla \\sigma_{n}^{*}(\\U)) fdv_{g} \\right)\\notag\\\\\n&\\quad +R_{n}^{-\\frac{1}{2}}\\int_{M}\\beta L_{g}(\\sigma_{n}^{*}(U_\\infty))fdv_{g}-R_{n}^{-\\frac{5}{2}}\\int_{M}\\beta^{3} |\\sigma_{n}^{*}(\\Psi_\\infty)|^{2}\\sigma_{n}^{*}(U_\\infty) fdv_{g}\\notag\\\\\n&=R_{n}^{-\\frac{1}{2}}\\int_{M}(\\Delta_{g}\\beta) \\sigma_{n}^{*}(\\U) fdv_{g}+2R_{n}^{-\\frac{1}{2}}\\int_{M}g(\\nabla \\beta,\\nabla f) \\sigma_{n}^{*}(\\U)dv_{g} \\notag\\\\\n&\\quad + R_{n}^{-\\frac{5}{2}}\\int_{M}\\beta \\sigma_{n}^{*}(L_{g_{n}}\\U)fdv_{g}-R_{n}^{-\\frac{5}{2}}\\int_{M}\\beta^{3} \\sigma_{n}^{*}(|\\Ps|^{2}\\U)fdv_{g}\\notag\\\\\n&=R_{n}^{-\\frac{1}{2}}\\int_{M}(\\Delta_{g}\\beta) \\sigma_{n}^{*}(\\U) fdv_{g}+2R_{n}^{-\\frac{1}{2}}\\int_{M}g(\\nabla \\beta,\\nabla f) \\sigma_{n}^{*}(\\U)dv_{g} \\notag\\\\\n&\\quad + R_{n}^{-\\frac{5}{2}}\\int_{M}\\beta \\sigma_{n}^{*}\\left((L_{g_{n}}-L_{g_{\\R^{3}}})V\\right)fdv_{g}+R_{n}^{-\\frac{5}{2}}\\int_{M}(\\beta-\\beta^{3}) \\sigma_{n}^{*}(|\\Ps|^{2}\\U) f dv_{g}\\notag\\\\\n&=I_{1}+I_{2}+I_{3}+I_{4} .\\notag\n\\end{align}\nWe estimate first $I_{1}$, so we fix $\\gamma \\in C^{\\infty}(M)$ such that $\\gamma=1$ on $supp(\\beta)$. Then we have\n\\begin{align}\nI_{1}&=R_{n}^{-\\frac{1}{2}}\\int_{M}(\\Delta_{g}\\beta)\\sigma_{n}^{*}(\\U) \\gamma fdv_{g}\\notag \\\\\n&=R_{n}^{-\\frac{1}{2}}\\int_{\\R^{3}}\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U \\rho^{*}_{n}(\\gamma f)dv_{\\rho^{*}_{n}g}\\notag\\\\\n&=R_{n}^{\\frac{5}{2}}\\int_{\\R^{3}}\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U \\rho^{*}_{n}(\\gamma f)dv_{g_{n}}.\\notag\n\\end{align}\nThus\n\\begin{align}\n|I_{1}|&\\leq R_{n}^{2}\\|\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U\\|_{L^{\\frac{6}{5}}(\\R^{3},g_{n})}\\|R_{n}^{\\frac{1}{2}}\\rho^{*}_{n}(\\gamma f)\\|_{L^{6}(\\R^{3},g_{n})}\\notag \\\\\n&\\leq R_{n}^{2}\\|\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U\\|_{L^{\\frac{6}{5}}(\\R^{3},g_{n})}\\|\\gamma f\\|_{L^{6}(M)}\\notag\\\\\n&\\leq R_{n}^{2}\\|\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U\\|_{L^{\\frac{6}{5}}(\\R^{3},g_{n})}\\|f\\|_{H^{1}(M)}. \\notag\n\\end{align}\nBut,\n\\begin{align}\n\\int_{\\R^{3}}|\\rho^{*}_{n}(\\Delta_{g}\\beta)\\U|^{\\frac{6}{5}}dv_{g_{n}}&\\leq C\\int_{R_{n}^{-1}exp^{-1}_{x_{n}}(B_{2}(x))\\setminus R_{n}^{-1}exp^{-1}_{x_{n}}(B_{1}(x))}|\\U|^{\\frac{6}{5}}dv_{g_{\\R^{3}}}\\notag \\\\\n&\\leq C R_{n}^{-\\frac{12}{5}} \\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0}}|\\U|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{5}}.\n\\end{align}\nTherefore,\n$$|I_{1}|\\leq C\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0}}|\\U|^{6}dv_{\\R^{3}}\\right)^{\\frac{1}{6}}\\|f\\|_{H^{1}(M)},$$\nand since $\\U\\in D^{1}(\\R^{3})$, we have that\n$$|I_{1}|\\leq o(1)\\|f\\|_{H^{1}}.$$\nThe estimate for $I_{2}$ is very similar to the one of $I_{1}$, indeed we have\n\\begin{align}\n|I_{2}|&\\leq R_{n}^{\\frac{5}{2}}\\int_{\\R^{3}}|\\rho_{n}^{*}(\\nabla \\beta)||\\U| |\\rho_{n}^{*}(\\nabla f)|dv_{g_{n}}\\notag\\\\\n&\\leq R_{n}^{2}\\|\\rho_{n}^{*}(\\nabla \\beta)|\\U|\\|_{L^{2}(\\R^{3},g_{n})} \\|R_{n}^{\\frac{1}{2}}\\rho_{n}^{*}(\\nabla f)\\|_{L^{2}(\\R^{3},g_{n})}\\notag\\\\\n&\\leq R_{n}^{2}\\|\\rho_{n}^{*}(\\nabla \\beta)|\\U|\\|_{L^{2}(\\R^{3},g_{n})}\\|f\\|_{H^{1}(M)} . \\notag\n\\end{align}\nBut\n\\begin{align}\n\\int_{\\R^{3}}|\\rho^{*}_{n}(\\nabla\\beta)|^{2}|\\U|^{2}dv_{g_{n}}&\\leq C\\int_{R_{n}^{-1}exp^{-1}_{x_{n}}(B_{2}(x))\\setminus R_{n}^{-1}exp^{-1}_{x_{n}}(B_{1}(x))}|\\U|^{2}dv_{\\R^{3}}\\notag \\\\\n&\\leq C R_{n}^{-2} \\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0}}|\\U|^{6}dv_{\\R^{3}}\\right)^{\\frac{1}{3}}, \\label{e2}\n\\end{align}\nwhich allows us to conclude that\n$$|I_{2}|\\leq o(1)\\|f\\|_{H^{1}}.$$\nWe move now to $I_{3}$. We first notice that\n$$I_{3}=\\int_{M} \\left( (L_{g_{n}}-L_{\\R^{3}})\\U\\right)R_{n}^{-\\frac{1}{2}}\\rho_{n}^{*}(\\beta\\gamma f)dv_{g_{n}}.$$\nTherefore,\n$$|I_{3}|\\leq C\\|\\beta\\rho_{n}^{*}((L_{g_{n}}-L_{\\R^{3}})\\U)\\|_{L^{\\frac{6}{5}}(\\R^{3},g_{n})}\\|f\\|_{H^{1}(M)},$$\nbut\n$$\\|\\beta\\rho_{n}^{*}((L_{g_{n}}-L_{\\R^{3}})\\U)\\|_{L^{\\frac{6}{5}}}\\leq $$\n$$\\leq\\|\\beta\\rho_{n}^{*}((L_{g_{n}}-L_{\\R^{3}})\\U)\\|_{L^{\\frac{6}{5}}(B_{R}^{0})}+ \\|\\beta\\rho_{n}^{*}((L_{g_{n}}-L_{\\R^{3}})\\U)\\|_{L^{\\frac{6}{5}}(\\R^{3}\\setminus B_{R}^{0})}.$$\nNow, since $g_{n}\\to g_{\\R^{3}}$ in $C^{\\infty}(B_{R}^{0})$, and since $-\\Delta_{g_{\\R^3}} \\U=|\\Ps|^{2}\\U\\in L^{\\frac{6}{5}}(\\R^{3})$, we have as in (\\ref{e2}), by letting $n \\to \\infty$ and then $R \\to \\infty$ that\n$$|I_{3}|\\leq o(1)\\|f\\|_{H^{1}}.$$\nIt remains now to consider the term $I_{4}$:\n$$I_{4}=\\int_{\\R^{3}}\\rho_{n}^{*}(\\beta-\\beta^{3}) |\\Ps|^{2}\\U R_{n}^{\\frac{1}{2}}\\rho_{n}^{*}(\\gamma f)dv_{g_{n}} .$$\nWe have\n$$|I_{4}|\\leq C\\|\\rho_{n}^{*}(\\beta-\\beta^{3}) |\\Ps|^{2}\\U\\|_{L^{\\frac{6}{5}}(\\R^{3},g_{n})}\\|f\\|_{H^{1}} ,$$\nand\n$$\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta-\\beta^{3})|^{\\frac{6}{5}} |\\Ps|^{\\frac{12}{5}}|\\U|^{\\frac{6}{5}}dv_{g_{n}}\\leq C\\|\\Ps\\|_{L^{3}}^{\\frac{6}{5}}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0}}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R^{3}}}\\right)^{\\frac{3}{5}}.$$\nThen using the fact that $U_{\\infty}\\in D^{1}(\\R^{3})$ and $\\Psi_{\\infty}\\in D^{\\frac{1}{2}}(\\Sigma\\R^{3})$, we have that\n$$|I_{4}|\\leq o(1)\\|f_{1}\\|.$$\nTherefore, we have that $f_{n}\\to 0$ in $H^{-1}(M)$ and a similar convergence holds for $h_{n}\\to 0$ in $H^{-\\frac{1}{2}}(\\Sigma M)$.\\\\\nNext we move to $(\\overline{u}_{n},\\overline{\\psi}_{n})$ and again we fix $f\\in H^{1}(M)$. First we notice that\n\\begin{align}\nd_{u}E(\\overline{u}_{n},\\overline{\\psi}_{n})f&=\\int_{M}fL_{g}\\overline{u}_{n}dv_{g}-\\int_{M}|\\overline{\\psi}_{n}|^{2}\\overline{u}_{n} fdv_{g}\\notag\\\\\n&=d_{u}E(u_{n},\\psi_{n})f-d_{u}E(v_{n},\\phi_{n})f+\\int_{M}A_{n}fdv_{g} ,\\notag\n\\end{align}\nwhere\n$$A_{n}=|\\psi_{n}|^{2}v_{n}-|\\phi_{n}|^{2}u_{n}+2\\langle \\psi_{n},\\phi_{n}\\rangle (u_{n}-v_{n}).$$\nNow, since we already proved that $d_{u}E(u_{n},\\psi_{n})\\to 0$ and $d_{u}E(v_{n},\\phi_{n})\\to 0$, it is enough to show that $A_{n}\\to 0$ in $H^{-1}$. First we have that\n\\begin{align}\n\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\psi_{n}|^{\\frac{12}{5}}|v_{n}|^{\\frac{6}{5}}dv_{g}&\\leq \\|\\psi_{n}\\|_{L^{3}}^{\\frac{12}{5}}\\left(\\int_{M\\setminus B_{RR_{n}}(x_{n})}|v_{n}|^{6}dv_{g}\\right)^{\\frac{1}{5}}\\notag \\\\\n&\\leq C \\|\\psi_{n}\\|_{H^{\\frac{1}{2}}}^{\\frac{12}{5}}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\U|^{6}dv_{g_{n}}\\right)^{\\frac{1}{5}}.\\notag\n\\end{align}\nBut since $\\|\\psi_{n}\\|_{H^{\\frac{1}{2}}}$ is bounded and $dv_{g_{n}}\\leq Cdv_{g_{\\R^{3}}}$, we have that\n$$\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\U|^{6}dv_{g_{n}}\\to 0 ,$$\nas $R\\to \\infty$ uniformly on $n$; hence\n$$\\||\\psi_{n}|^{2}v_{n}\\|_{L^{\\frac{6}{5}}(M\\setminus B_{RR_{n}}(x_{n}))}\\to 0.$$\nSimilarly, we have that\n\\begin{align}\n\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{\\frac{12}{5}}|u_{n}|^{\\frac{6}{5}}dv_{g}&\\leq \\|u_{n}\\|_{L^{6}}^{\\frac{6}{5}}\\left(\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{3}dv_{g}\\right)^{\\frac{4}{5}}\\notag \\\\\n&\\leq C \\|u_{n}\\|_{H^{1}}^{\\frac{6}{5}}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\Ps|^{3}dv_{g_{n}}\\right)^{\\frac{4}{5}}.\\notag\n\\end{align}\nHence the same conclusion holds when $R\\to \\infty$. To finish this estimate on the exterior domain, we consider the mixed terms:\n\\begin{align}\n\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{\\frac{6}{5}}|\\psi_{n}|^{\\frac{6}{5}}|u_{n}|^{\\frac{6}{5}}dv_{g}&\\leq \\|u_{n}\\|_{L^{6}}^{\\frac{6}{5}}\\|\\psi_{n}\\|_{L^{3}}^{\\frac{6}{5}}\\left(\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{3}dv_{g}\\right)^{\\frac{2}{5}}\\notag \\\\\n&\\leq C \\|\\psi_{n}\\|_{H^{\\frac{1}{2}}}^{\\frac{6}{5}} \\|u_{n}\\|_{H^{1}}^{\\frac{6}{5}}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\Ps|^{3}dv_{g_{n}}\\right)^{\\frac{2}{5}} .\\notag\n\\end{align}\nNow\n$$\n\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{\\frac{6}{5}}|\\psi_{n}|^{\\frac{6}{5}}|v_{n}|^{\\frac{6}{5}}dv_{g}\\leq$$ $$\\leq\\|\\psi_{n}\\|_{L^{3}}^{\\frac{6}{5}}\\left(\\int_{M\\setminus B_{RR_{n}}(x_{n})}|\\phi_{n}|^{3}dv_{g}\\right)^{\\frac{2}{5}} \\left(\\int_{M\\setminus B_{RR_{n}}(x_{n})}|v_{n}|^{6}dv_{g}\\right)^{\\frac{1}{5}} $$\n$$\\leq  C \\|\\psi_{n}\\|_{H^{\\frac{1}{2}}}^{\\frac{6}{5}} \\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\Ps|^{3}dv_{g_{n}}\\right)^{\\frac{2}{5}}\\left(\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\U|^{6}dv_{g_{n}}\\right)^{\\frac{1}{5}} .$$\nWe need now an estimate inside the ball $B_{RR_{n}}(x_{n})$, that is\n$$\\int_{B_{RR_{n}}(x_{n})}|A_{n}|^{\\frac{6}{5}}dv_{g}=$$\n$$\n=\\int_{B_{RR_{n}}(x_{n})}||\\psi_{n}|^{2}(v_{n}-u_{n})+ (|\\psi_{n}|^{2}-|\\phi_{n}|^{2})u_{n}+2\\langle \\psi_{n},\\phi_{n}\\rangle (u_{n}-v_{n})|^{\\frac{6}{5}}dv_{g} \\leq $$\n$$\\int_{B_{R}^{0}}\\left(|\\Psi_{n}|^{2}|\\rho_{n}^{*}(\\beta) \\U -U_{n}|+(|\\Psi_{n}|^{2}-|\\rho_{n}^{*}(\\beta) \\Ps|^{2})|U_{n}|  +2|\\Psi_{n}||\\rho^{*}_{n}(\\beta)\\Ps| |U_{n}-\\rho^{*}_{n}(\\beta) \\U|\\right)^{\\frac{6}{5}}dv_{g_{n}}.$$\nHence for $n$ large, we have that\n$$\\int_{B_{RR_{n}}(x_{n})}|A_{n}|^{\\frac{6}{5}}dv_{g}$$\n$$\\leq \\int_{B_{R}^{0}}\\left(|\\Psi_{n}|^{2}| \\U -U_{n}|+ (|\\Psi_{n}|^{2}-|\\Ps|^{2})|U_{n}|+2|\\Psi_{n}||\\Ps||U_{n}- \\U|\\right)^{\\frac{6}{5}}dv_{g_{n}},$$\nand since $\\Psi_{n}\\to \\Ps$ in $H^{\\frac{1}{2}}_{loc}(\\Sigma\\R^{3})$ and $U_{n}\\to \\U$ in $H^{1}_{loc}(\\R^{3})$, we have that\n$$\\int_{B_{RR_{n}}(x_{n})}|A_{n}|^{\\frac{6}{5}}dv_{g}\\to 0,$$\nwhich finishes the proof for $d_{u}E$. The same computations also hold for $d_{\\psi}E$.\n\\end{proof}\n\n\n\n\n\\noindent\nNow we estimate the energy, that is:\n\\begin{lemma}\nWe have\n$$E(\\overline{u}_{n},\\overline{\\psi}_{n})=E(u_{n},\\psi_{n})-E_{\\R^{3}}(\\U,\\Ps)+o(1).$$\n\\end{lemma}\n\\begin{proof}\nWe have\n$$E(\\overline{u}_{n},\\overline{\\psi}_{n})=$$\n$$=\\frac{1}{2}\\left(\\int_{M}(u_{n}-v_{n})L_{g}(u_{n}-v_{n})+\\langle D_{g}(\\psi_{n}-\\phi_{n}),\\psi_{n}-\\phi_{n}\\rangle -|\\psi_{n}-\\phi_{n}|^{2}|u_{n}-v_{n}|^{2}dv_{g}\\right)=$$\n$$=E(u_{n},\\psi_{n})+E(v_{n},\\phi_{n})-dE(v_{n},\\phi_{n})(u_{n},\\psi_{n})+B_{n},$$\nwhere\n$$B_{n}=\\frac{1}{2}\\left(\\int_{M}|u_{n}|^{2}|\\phi_{n}|^{2}+|v_{n}|^{2}|\\psi_{n}|^{2}-2|u_{n}|^{2}\\langle \\psi_{n},\\phi_{n}\\rangle -2|\\psi_{n}|^{2}u_{n}v_{n}+4u_{n}v_{n}\\langle \\psi_{n},\\phi_{n}\\rangle dv_{g}\\right).$$\nWe first notice that since $(u_{n},\\psi_{n})$ is bounded in $H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)$, from Lemma 4.2, we have that $$dE(v_{n},\\phi_{n})(u_{n},\\psi_{n})\\to 0 .$$\nWe are going to estimate the five terms of $B_{n}$, one by one.\n\\begin{align}\n\\int_{M}|u_{n}|^{2}|\\phi_{n}|^{2}&=\\int_{M}|u_{n}|^{2}R_{n}^{-2}|\\beta|^{2}|\\sigma_{n}^{*}\\Ps|^{2}dv_{g}\\notag\\\\\n&=\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2}|U_{n}|^{2}|\\Ps|^{2}dv_{g_{n}}\\notag\\\\\n&=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}-\\int_{\\R^{3}}(|\\U|^{2}-|\\rho_{n}^{*}(\\beta) U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}}.\\notag\n\\end{align}\nFor $n$ large, we have\n$$\\int_{\\R^{3}}(|\\U|^{2}-|\\rho_{n}^{*}(\\beta)U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}}=$$\n$$= \\int_{B_{R}^{0}}(|\\U|^{2}-|U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}}+\\int_{\\R^{3}\\setminus B_{R}^{0}} (|\\U|^{2}-|\\rho_{n}^{*}(\\beta) U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}}.$$\nNow since $U_{n}\\to \\U$ in $H^{1}_{loc}(\\R^3)$, we have that\n$$\\int_{B_{R}^{0}}(|\\U|^{2}-|U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}} \\to 0,$$\nas $n\\to \\infty$. Now\n$$\\left|\\int_{\\R^{3}\\setminus B_{R}^{0}} (|\\U|^{2}-|\\rho_{n}^{*}(\\beta)U_{n}|^{2})|\\Ps|^{2}dv_{g_{n}}\\right|\\leq$$\n$$\\leq C \\|\\Ps\\|_{L^{3}(\\R^{3}\\setminus B_{R}^{0})}^{2}\\left(\\|\\U\\|_{L^{6}(\\R^{3})}^{2}+\\|U_{n}\\|_{L^{6}(B_{3R^{-1}_{n}}^{0}\\setminus B_{R}^{0})}\\right).$$\nFrom (\\ref{e3}),  $\\|U_{n}\\|_{L^{6}(B_{3R^{-1}_{n}}^{0})}$ is bounded, hence we can take $R\\to \\infty$ uniformly in $n$ to get that\n$$\\int_{M}|u_{n}|^{2}|\\phi_{n}|^{2}dv_{g}=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R^{3}}}+o(1).$$\nThe next term that we want to estimate is\n\\begin{align}\n\\int_{M}|v_{n}|^{2}|\\psi_{n}|^{2}dv_{g}&=\\int_{\\R^{3}}|\\U|^{2}|\\rho_{n}^{*}(\\beta)|^{2}|\\Psi_{n}|^{2}dv_{g_{n}}\\notag\\\\\n&=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}+\\int_{\\R^{3}}|\\U|^{2}|\\left(\\rho_{n}^{*}(\\beta)|^{2}|\\Psi_{n}|^{2}-|\\Ps|^{2}\\right)dv_{g_{n}} .\\notag\n\\end{align}\nAgain by splitting the integral in $B_{R}^{0}$ and $\\R^{3}\\setminus B_{R}^{0}$, we get that\n$$\\int_{M}|v_{n}|^{2}|\\psi_{n}|^{2}dv_{g}=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R^{3}}}+o(1).$$\nNext,\n\\begin{align}\n\\int_{M}|u_{n}|^{2}\\langle \\psi_{n},\\phi_{n}\\rangle dv_{g}&=\\int_{\\R^{3}}\\rho_{n}^{*}(\\beta)|U_{n}|^{2}\\langle \\Psi_{n},\\Ps\\rangle dv_{g_{n}}\\notag\\\\\n&=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}+\\int_{\\R^{3}}\\langle \\rho_{n}^{*}(\\beta)|U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}} .\\notag\n\\end{align}\nSimilarly, we have\n$$\\int_{\\R^{3}}\\langle \\rho_{n}^{*}(\\beta)|U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}}=$$\n$$=\\int_{B_{R}^{0}}\\langle |U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}}+\\int_{\\R^{3}\\setminus{B_{R}^{0}}}\\langle \\rho_{n}^{*}(\\beta)|U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}}.$$\nUsing the convergence of $U_{n}$ and $\\Psi_{n}$ in $H^{1}_{loc}(\\R^3)$ and $H^{\\frac{1}{2}}_{loc}(\\Sigma \\R^3)$ respectively, we have that\n$$\\int_{B_{R}^{0}}\\langle |U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}}\\to 0 ,$$\nas $n\\to \\infty$. For the  term in $\\R^{3}\\setminus{B_{R}^{0}}$, we have\n$$\\left|\\int_{\\R^{3}\\setminus{B_{R}^{0}}}\\langle \\rho_{n}^{*}(\\beta)|U_{n}|^{2}\\Psi_{n}-|\\U|^{2}\\Ps,\\Ps\\rangle dv_{g_{n}}\\right|\\leq $$\n$$\\leq C\\|\\Ps\\|_{L^{3}(\\R^{3}\\setminus B_{R}^{0})}\\left(\\|\\Ps\\|_{L^{3}(\\R^{3}\\setminus B_{R}^{0})}\\|\\U\\|_{L^{6}(\\R^{3}\\setminus B_{R}^{0})}^{2}+\\|\\Psi_{n}\\|_{L^{3}(B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0})}\\|U_{n}\\|_{L^{6}(B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0})}^{2}\\right).$$\nThus\n$$\\int_{M}|u_{n}|^{2}\\langle \\psi_{n},\\phi_{n}\\rangle dv_{g}=\\int_{\\R^{3}}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R_{3}}}+o(1) .$$\nThe remaining two terms can be handled in the same way. Therefore, so far we have\n$$E(\\overline{u}_{n},\\overline{\\psi}_{n})=E(u_{n},\\psi_{n})+E(v_{n},\\phi_{n})-2E_{\\R^{3}}(\\U,\\Ps)+o(1) .$$\nNow we estimate $E(v_{n},\\phi_{n})$. First, we have\n\\begin{align}\n\\int_{M}|v_{n}|^{2}\\phi_{n}|^{2}dv_{g}&=\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{4}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}\\notag\\\\\n&=\\int_{B_{R}^{0}}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}+\\int_{\\R^{3}\\setminus B_{R}^{0}}|\\rho_{n}^{*}(\\beta)|^{4}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}, \\notag\n\\end{align}\nbut\n$$\\int_{\\R^{3}\\setminus B_{R}^{0}}|\\rho_{n}^{*}(\\beta)|^{4}|\\U|^{2}|\\Ps|^{2}dv_{g_{n}}\\leq C\\int_{\\R^{3}\\setminus B_{R}^{0}}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R^{3}}}.$$\nHence, if we let $R\\to \\infty$, we have\n$$\\int_{M}|v_{n}|^{2}|\\phi_{n}|^{2}dv_{g}=\\int_{\\R^3}|\\U|^{2}|\\Ps|^{2}dv_{g_{\\R^{3}}}+o(1).$$\nNow,\n\\begin{align}\n\\int_{M}v_{n}L_{g}v_{n}dv_{g}&=\\int_{\\R^{3}}\\rho_{n}^{*}(\\beta) \\U L_{g_{n}}(\\rho_{n}^{*}(\\beta)\\U)dv_{g_{n}}\\notag \\\\\n&=\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2} \\U L_{g_{n}}\\U dv_{g_{n}}+\\int_{\\R^{3}}\\rho_{n}^{*}(\\beta)(-\\Delta_{g_{n}} \\rho_{n}^{*}(\\beta))|\\U|^{2}dv_{g_{n}}\\notag \\\\\n& \\quad -2\\int_{\\R^{3}}\\U \\rho_{n}^{*}(\\beta)g(\\nabla \\rho_{n}^{*}(\\beta),\\nabla \\U) dv_{g_{n}}. \\notag\n\\end{align}\nNow\n$$\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2} \\U L_{g_{n}}\\U dv_{g_{n}}=$$\n$$=\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2}\\U L_{g_{\\R^{3}}}\\U dv_{g_{n}}+\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2} \\U (L_{g_{n}}-L_{g_{\\R^{3}}})\\U dv_{g_{n}}.$$\nClearly,\n$$\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2}\\U L_{g_{\\R^{3}}}\\U dv_{g_{n}}=\\int_{\\R^{3}}|\\nabla \\U|^{2}dv_{g_{\\R^{3}}}+o(1),$$\nand\n$$\\left|\\int_{\\R^{3}}|\\rho_{n}^{*}(\\beta)|^{2} \\U (L_{g_{n}}-L_{g_{\\R^{3}}})\\U dv_{g_{n}}\\right|\\leq $$\n$$ \\leq \\int_{B_{R}^{0}} |\\U| |(L_{g_{n}}-L_{g_{\\R^{3}}})\\U | dv_{g_{n}}+C\\int_{B_{3R_{n}^{-1}}^{0}\\setminus B_{R}^{0}}|\\U| |(L_{g_{n}}-L_{g_{\\R^{3}}})\\U| dv_{g_{n}}.$$\nThe first term converges to zero as $n\\to \\infty$ from (\\ref{e4}). For the second term, we use the fact that $\\nabla \\U\\in L^{2}(\\R^{3})$ hence it converges to zero if we let $R\\to \\infty$ uniformly on $n$. Also\n$$\\left|\\int_{\\R^{3}}\\rho_{n}^{*}(\\beta)(-\\Delta_{g_{n}} \\rho_{n}^{*}(\\beta))|\\U|^{2}dv_{g_{n}}\\right|\\leq C\\|\\U\\|^{2}_{L^{6}(B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0})}=o(1),$$\nand\n$$\\left|\\int_{\\R^{3}}\\U \\rho_{n}^{*}(\\beta)g(\\nabla \\rho_{n}^{*}(\\beta),\\nabla \\U) dv_{g_{n}}\\right|\\leq $$\n$$\\leq C\\|\\nabla \\U\\|_{L^{2}(B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0})}\\|\\U\\|_{L^{6}(B_{3R_{n}^{-1}}^{0}\\setminus B_{\\frac{1}{2}R_{n}^{-1}}^{0})}=o(1).$$\nHence\n$$\\int_{M}v_{n}L_{g}v_{n}dv_{g}=\\int_{\\R^{3}}|\\nabla \\U|^{2}dv_{g_{\\R^{3}}}+o(1),$$\nand similarly\n$$\\int_{M}\\langle D_{g} \\phi_{n},\\phi_{n}\\rangle dv_{g}=\\int_{\\R^{3}}\\langle D_{g_{\\R^{3}}}\\Ps,\\Ps \\rangle dv_{g_{\\R^{3}}}+o(1).$$\nCombining al these estimates, we have that\n$$E(\\overline{u}_{n},\\overline{\\psi}_{n})=E(u_{n},\\psi_{n})-E_{\\R^{3}}(\\U,\\Ps)+o(1).$$\n\\end{proof}\n\n\n\\noindent\nNow we will prove the following energy lower bound for solutions in $\\R^3$.\n\\begin{proposition}\nLet $(U,\\Psi)\\in D^{1}(\\R^{3})\\times D^{\\frac{1}{2}}(\\Sigma \\R^{3})$ be a non trivial solution of\n$$\\left\\{\\begin{array}{ll}\n-\\Delta_{g_{\\R^3}} U=|\\Psi|^{2}U \\\\\n& \\text{ on } \\R^{3} .\\\\\nD_{g_{\\R^3}}\\Psi=|U|^{2}\\Psi\n\\end{array}\n\\right.\n$$\nThen\n$$E_{\\R^{3}}(U,\\Psi)\\geq \\tilde{Y}_{g_0}(S^{3})=: c_{3}.$$\n\\end{proposition}\n\\begin{proof}\nFirst we recall that\n$$Y_{g_{0}}(S^{3})\\leq\\frac{\\displaystyle\\int_{\\R^{3}}|\\nabla U|^{2}dv_{g_{\\R^{3}}}} {\\left(\\displaystyle\\int_{\\R^{3}}|U|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{3}}}.$$\nBut\n\\begin{align}\n\\int_{\\R^{3}}|\\nabla U|^{2}dv_{g_{\\R^{3}}}&=\\int_{\\R^{3}}|U|^{2}|\\Psi|^{2}dv_{g_{\\R^{3}}}\\notag\\\\\n&\\leq \\left(\\int_{\\R^{3}}|U|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{3}} \\left(\\int_{\\R^{3}}|\\Psi|^{3}dv_{g_{\\R^{3}}}\\right)^{\\frac{2}{3}} .\\notag\n\\end{align}\nThus\n$$Y_{g_{0}}(S^{3})\\leq \\left(\\int_{\\R^{3}}|\\Psi|^{3}dv_{g_{\\R^{3}}}\\right)^{\\frac{2}{3}}.$$\nOn the other hand, if we denote by $(u,\\psi)$ the pull-back of $(U,\\Psi)$ by the standard stereographic projection, we have\n$$\\lambda_{g_{0}}^{+}(S^{3})\\leq \\frac{\\displaystyle\\left(\\int_{S^{3}}|D_{g_0}\\psi|^{\\frac{3}{2}}dv_{g_0}\\right)^{\\frac{4}{3}}}{\\displaystyle\\int_{S^{3}}\\langle D_{g_0}\\psi,\\psi\\rangle dv_{g_0}}.$$\nAgain, we have\n$${\\int_{S^{3}}\\langle D_{g_0}\\psi,\\psi\\rangle dv_{g_0}}=\\int_{S^{3}}|u|^{2}|\\psi|^{2}dv_{g_0},$$\nand\n\\begin{align}\n\\int_{S^{3}}|D_{g_0}\\psi|^{\\frac{3}{2}}dv_{g_0}&=\\int_{S^{3}}|u|^{3}|\\psi|^{\\frac{3}{2}}dv_{g_0}\\notag\\\\\n&\\leq \\left(\\int_{S^{3}}|u|^{6}dv_{g_0}\\right)^{\\frac{1}{3}} \\left(\\int_{S^{3}}|u|^{2}|\\psi|^{2}dv_{g_0}\\right)^{\\frac{3}{4}} .\\notag\n\\end{align}\nHence,\n$$\\lambda_{g_{0}}^{+}(S^{3})\\leq \\left(\\int_{S^{3}}|u|^{6}dv_{g_0}\\right)^{\\frac{1}{3}}.$$\nNow, using the Sobolev inequality\n$$Y_{g_{0}}(S^{3})\\left(\\int_{\\R^{3}}|U|^{6}dv_{g_{\\R^{3}}}\\right)^{\\frac{1}{3}}\\leq \\int_{\\R^{3}}|\\nabla U|^{2}dv_{g_{\\R^{3}}},$$\nhence\n$$E_{\\R^3}(U,\\Psi)=\\frac{1}{2}\\int_{\\R^{3}}|U|^{2}|\\Psi|^{2}dv_{g_{\\R^{3}}}\\geq \\frac{1}{2}Y_{g_{0}}(S^{3})\\lambda_{g_{0}}^{+}(S^{3})=c_3.$$\n\\end{proof}\n\n\n\\noindent\n\\begin{proof}  \\emph{of Theorem (\\ref{first}})\\\\\nFrom the previous results, we can re-iterate the process $m$ times, for $(\\overline{u}_{n},\\overline{\\psi}_{n})$ since they satisfy the same assumptions as $(u_{n},\\psi_{n})$, and we will have\n$$E(u_{n},\\psi_{n})=E(u_{\\infty},\\psi_{\\infty})+\\sum_{k=1}^{m}E_{\\R^{3}}(U_\\infty^{k},\\Ps^{k})+o(1),$$\nwhere $(U_\\infty^{k},\\Psi_\\infty^{k})$ are solutions to equations (\\ref{el}) on $\\mathbb{R}^3$. Now using the fact that\n$$\\sum_{k=1}^{m}E_{\\R^{3}}(\\U^{k},\\Ps^{k})\\geq mc_{3} ,$$\nwe stop the process when $c-mc_{3}<\\frac{\\epsilon_{0}}{2}$. Then from Corollary 4.5, we have the existence of $m$ sequences $x_{n}^{1},\\cdots x_{n}^{m}$ such that $x_{n}^{k}\\to x^{k}\\in M$, and $m$ sequences of real numbers $R_{n}^{1},\\cdots, R_{n}^{m}$ converging to zero, such that\n$$u_{n}=u_{\\infty}+v_{n}^{1}+\\cdots +v_{n}^{m}+o(1) \\text{ in } H^{1}(M),$$\n$$\\psi_{n}=\\psi_{\\infty}+\\phi_{n}^{1}+\\cdots +\\phi_{n}^{m}+o(1) \\text{ in } H^{\\frac{1}{2}}(\\Sigma M),$$\nwhere\n$$v_{n}^{k}=(R_{n}^{k})^{-\\frac{1}{2}}\\beta_{k}\\sigma_{n,k}^{*}(U_\\infty^{k}) ,$$\n$$\\phi_{n}^{k}=(R_{n}^{k})^{-1}\\beta_{k}\\sigma_{n,k}^{*}(\\Psi_\\infty^{k}) ,$$\nwith $\\sigma_{n,k}=(\\rho_{n,k})^{-1}$ and $\\rho_{n,k}(\\cdot)=exp_{x_{n}^k}(R_{n}^k \\cdot)$.\nAlso $\\beta_{k}$ are smooth compactly supported functions, such that $\\beta_{k}=1$ on $B_{1}(x^{k})$ and $supp(\\beta_{k})\\subset B_{2}(x^{k})$. Moreover, we have\n$$E(u_{n},\\psi_{n})=E(u_{\\infty},\\psi_{\\infty})+\\sum_{k=1}^{p}E_{\\R^{3}}(\\U^{k},\\Ps^{k})+o(1).$$\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\\section{Existence of a positive solution}\n\n\n\\noindent\nIn this section we will prove Theorem (\\ref{second}): for our convenience we will split the two statements, and we will prove them separately. First we need the following characterization of the first eigenvalue of the Dirac operator: let us fix $u>0$ and let us consider the minimization problem\n$$\\tilde{\\lambda}_u=\\inf\\left\\{I(\\psi); \\text{ for  $\\psi\\in H^{\\frac{1}{2}}(\\Sigma M)$ s.t. }  I(\\psi)> 0, \\; P^{-}\\left(D_g \\psi-I(\\psi)u^{2}\\psi\\right) =0 \\right\\} ,$$\nthen we have\n\n\n\\begin{proposition}\nFor a given $u>0$ and smooth, we have that $\\tilde{\\lambda}_u>0$. Moreover, the minimization problem is achieved and $\\tilde{\\lambda}_u$ is the first eigenvalue for the Dirac operator $D_{g_{u}}$, where $g_u=u^4g$.\n\\end{proposition}\n\\begin{proof}\nLet $\\psi_{n}$ be a minimizing sequence, that is $I(\\psi_{n})\\to \\tilde{\\lambda}_u$.  Without loss of generality, we can assume that $\\int_{M}u^{2}|\\psi_n|^{2}dv_{g}=1$. Then we have\n$$I(\\psi_{n})=\\|\\psi_{n}^{+}\\|^{2}-\\|\\psi_{n}^{-}\\|^{2},$$\nbut\n$$-\\|\\psi_{n}^{-}\\|^{2}=I(\\psi_{n})\\int_{M}u^{2}\\langle \\psi_n,\\psi_n^{-}\\rangle dv_{g},$$\nhence, using Holder's inequality, we have\n$$\\|\\psi_{n}^{-}\\|^{2}\\leq I(\\psi_{n})\\left(\\int_{M}u^{2}|\\psi_n^{-}|^{2}dv_{g}\\right)^{\\frac{1}{2}}.$$\nSince, the projector $P^{-}:H^{\\frac{1}{2}}(\\Sigma M)\\to H^{\\frac{1}{2},-}$ is a pseudo-differential operator of order zero, we have that\n$$\\|\\psi_{n}^{-}\\|^{2}\\leq C I(\\psi_{n}),$$\nthus, we have\n$$\\|\\psi_{n}^{+}\\|^{2}\\leq C I(\\psi_{n}).$$\nTherefore, if $I(\\psi_{n})\\to 0$, we would have that $\\psi_{n}\\to 0$ in $H^{\\frac{1}{2}}(\\Sigma M)$, contradicting the fact that $\\int_{M}u^{2}|\\psi_n|^{2}dv_{g}=1$. Therefore $\\tilde{\\lambda}_u>0$, and any minimizing sequence has $\\|\\psi_{n}\\|\\geq \\delta>0$.\\\\\n\\noindent\nNow we will prove the existence of a minimizer.  We notice that without the condition\n$$\\int_{M}\\langle D_g\\psi-I(\\psi)u^{2}\\psi,\\varphi\\rangle dv_{g}=0, \\quad \\forall \\varphi\\in H^{\\frac{1}{2},-},$$\nwe would be in a classical variational setting allowing us to find a minimizer: unfortunately this is not the case, so we use here the idea in \\cite{Pan}, later on inproved in \\cite{Sul}. First, we claim that\n$$S=\\{\\psi\\in H^{\\frac{1}{2}}(\\Sigma M);P^{-}(D\\psi-I(\\psi)u^{2}\\psi)=0\\} , $$\nis a manifold. Indeed, we consider the operator\n$$G:H^{\\frac{1}{2}}(\\Sigma M)\\to H^{\\frac{1}{2},-}, \\qquad G(\\psi)=P^{-}(D_g\\psi-I(\\psi)u^{2}\\psi),$$\nso that $S=G^{-1}(0)$; therefore, if $DG$ is onto, then $S$ will be indeed a manifold. We compute then $DG$:\n$$DG(\\psi)h=P^{-1}(D_gh-I(\\psi)u^{2}h).$$\nIf we restrict this operator to $H^{\\frac{1}{2},-}$, we have that\n$$\\langle DG(\\psi)h,h\\rangle=-\\|h\\|^{2}-I(\\psi)\\int_{M}u^{2}|h|^{2}dv_{g} .$$\nThus, $DG(\\psi)$ is definite negative and hence invertible thus onto and $S$ is a manifold. Now, using Ekeland's variational principle, see \\cite{Ek}, we can find a minimizing sequence for $\\tilde{\\lambda}_u$ that is a (PS) sequence. We want to show that it is still a (PS) sequence in $H^{\\frac{1}{2}}(\\Sigma M)$. So let $\\psi_{n}$ be a (PS) sequence for $I$ in $S$. We write $DI(\\psi_{n})=\\varepsilon_{n}$. We have that $\\varepsilon_{n}^{T}$ (the tangential component of $\\varepsilon_{n}$ on the tangent space of the manifold $S$) already converges to zero since $\\psi_{n}$ is a (PS) sequence for $I$ on $S$. We want to show that also the normal component converges to zero.\\\\\nAs we did previously, the operator $DG(\\psi_{n}):H^{\\frac{1}{2}}(\\Sigma M)\\to H^{\\frac{1}{2},-}$ is onto, hence it has a left inverse $K_{n}:H^{\\frac{1}{2},-} \\to H^{\\frac{1}{2},-}$. Moreover, since we can always assume that $\\int_{M}u^{2}|\\psi_{n}|^{2}dv_{g}=1$, we have that $\\|K_{n}\\|_{op}\\leq C$. The operator $P_{n}=A_{n}\\circ DG(\\psi_{n})$ is now a projector on $H^{\\frac{1}{2},-}$ parallel to $T_{\\psi_{n}}S$. Indeed, we have that if $h\\in H^{\\frac{1}{2},-}$ then $P_{n}h=h$ and $T_{\\psi_{n}}S=\\ker DG(\\psi_{n})$. We consider then the adjoint of $P_{n}$, denoted by $P_{n}^{*}$: it is a projector of $N_{\\psi}S$  (the orthogonal space to the tangent space) parallel to $H^{\\frac{1}{2},+}$. Now we notice that $\\langle \\varepsilon_{n}, \\varphi \\rangle =0$ for all $\\varphi\\in H^{\\frac{1}{2},-}$, hence $\\varepsilon_{n}\\in H^{\\frac{1}{2},+}$ so we have\n$$\\varepsilon_{n}=P_{n}^{*}\\varepsilon_{n}^{T}.$$\nThus $\\varepsilon_{n}\\to 0$ and $\\psi_{n}$ is a (PS) sequence for $I$ in $H^{\\frac{1}{2}}(\\Sigma M)$.\nThis (PS) sequence then satisfies\n\\begin{align}\n\\|\\psi_{n}^{+}\\|^{2}&\\leq \\int_{M}u^{2}|\\psi^{+}_{n}||\\psi_{n}|dv_{g}+o(\\|\\psi_{n}^{+}\\|)\\leq C +o(\\|\\psi_{n}^{+}\\|),\\notag\n\\end{align}\nand a similar inequality holds for $\\psi_{n}^{-}$ which gives us the boundedness in $H^{\\frac{1}{2}}(\\Sigma M)$. The rest of the proof is classical in order to show that $\\psi_{n}\\to \\psi$ in $H^{\\frac{1}{2}}(\\Sigma M)$ and $\\psi$ satisfies\n$$D_g\\psi=\\tilde{\\lambda}_uu^{2}\\psi.$$\nFinally, since $\\tilde{\\lambda}_u$ is a minimizer then by a conformal change of the metric $g_u= u^{4}g$ we have that $$\\tilde{\\lambda}_u=\\lambda_{1}(D_{g_u}).$$\n\\end{proof}\n\n\\noindent\nNow we prove the following:\n\\begin{proposition}\nIt holds:\n$$Y_g(M)\\lambda^{+}_g(M)\\leq \\tilde{Y}_g(M) \\leq Y_{g_0}(S^{3})\\lambda^{+}_{g_0}(S^{3})=\\tilde{Y}_{g_0}(S^{3}) .$$\n\\end{proposition}\n\\begin{proof}\nWe first notice that using the Sobolev inequality and Proposition (5.1), we have for any non-trivial $(u,\\psi)\\in H^{1}(M) \\times H^{\\frac{1}{2}}(\\Sigma M)$:\n\\begin{align}\n\\tilde{E}(u,\\psi)&\\geq Y_g(M)\\left(\\int_{M}u^{6}dv_{g}\\right)^{\\frac{1}{3}}\\frac{\\displaystyle\\int_{M}\\langle D_g \\psi,\\psi\\rangle dv_{g}}{\\displaystyle\\int_{M}|u|^{2}|\\psi|^{2}dv_{g}}\\notag\\\\\n&\\geq Y_g(M)\\lambda_{1}(g_{u})Vol(g_{u})^{\\frac{1}{3}}\\notag \\\\\n&\\geq Y_g(M)\\lambda_g^{+}(M),\n\\end{align}\nwhich proves the first inequality of the claim. We also recall that we set $g_u=u^4g$. Now for the second inequality, we consider a spinor $\\overline{\\Psi}_{0}\\in (\\Sigma\\R^{3})$ such that $|\\overline{\\Psi}_{0}|=\\frac{1}{\\sqrt{2}}$ and we define the standard spinor\n$$\\Psi_{0}(x)=\\left(\\frac{2}{1+|x|^{2}}\\right)^{\\frac{3}{2}}(1-x)\\cdot \\overline{\\Psi}_{0},$$\nand bubble\n$$U_{0}=\\left(\\frac{2}{1+|x|^{2}}\\right)^{\\frac{1}{2}}.$$\nThen an easy computation shows that $(U_{0},\\Psi_{0})$ is a critical point for $E_{\\R^{3}}$ and\n$$E(U_{0},\\Psi_{0})=\\frac{1}{2}Y_{g_{0}}(S^{3})\\lambda_{g_{0}}^{+}(S^{3}).$$\nNow, we fix $x_{0}\\in M$ and let $\\lambda>0$ be a parameter that we will be tending to zero. We define as in (\\ref{e5})) and (\\ref{e6}), $\\rho_{\\lambda}(x)=exp_{x_{0}}(\\lambda x)$,  and we consider $\\beta$ a cut-off function supported in $B_{2}(x_{0})$ and equals $1$ on $B_{1}(x_{0})$. We can therefore define the functions\n$$\\left\\{\\begin{array}{ll}\nu_{\\lambda}=\\lambda^{-\\frac{1}{2}}\\beta\\sigma_{\\lambda}^{*}(U_{0}),\\\\\n\\\\\n\\psi_{\\lambda}=\\lambda^{-1}\\beta\\sigma_{\\lambda}^{*}(\\Psi_{0}),\n\\end{array}\n\\right.\n$$\nwhere $\\sigma_{\\lambda}=\\rho_{\\lambda}^{-1}$. Similar computations as in Lemma (4.5) show that\n$$dE(u_{\\lambda},\\psi_{\\lambda})\\to 0, \\qquad \\lambda \\to 0 .$$\nAlso we have\n$$\\int_{M}u_{\\lambda}L_{g}u_{\\lambda}dv_{g}=\\int_{\\R^{3}}|\\nabla U_{0}|^{2}dv_{g_{\\R^{3}}}+o(1),$$\n$$\\int_{M}\\langle D_g\\psi_{\\lambda},\\psi_{\\lambda}\\rangle dv_{g}=\\int_{\\R^{3}}\\langle D_{g_{\\R^3}}\\Psi_{0},\\Psi_{0}\\rangle dv_{g_{\\R^{3}}}+o(1),$$\n$$\\int_{M}|u_{\\lambda}|^{2}|\\psi_{\\lambda}|^{2}dv_{g}=\\int_{\\R^{3}}|U_{0}|^{2}|\\Psi_{0}|^{2}dv_{g_{\\R^{3}}}+o(1).$$\nNow, for these test functions we might have the possibility that\n$$P^{-}(D_g\\psi_{\\lambda}-I(\\psi_{\\lambda})|u_{\\lambda}|^{2}\\psi_{\\lambda})\\not =0,$$\nso we want to perturb $\\psi_{\\lambda}$ so that the previous inequality is satisfied. Therefore we have to show that there exists $h\\in H^{\\frac{1}{2},-}$ so that\n$$P^{-}(D_g(\\psi_{\\lambda}+h)-I(\\psi_{\\lambda}+h)|u_{\\lambda}|^{2}(\\psi_{\\lambda}+h))=0.$$\nThis is equivalent to solving\n$$D_g h-P^{-}[I(\\psi_{\\lambda})|u_{\\lambda}|^{2}h]+B(h)=A_{\\lambda},$$\nwhere\n$$A_{\\lambda}=P^{-}(D_g\\psi_{\\lambda}-I(\\psi_{\\lambda})|u_{\\lambda}|^{2}\\psi_{\\lambda}),$$\nand\n$$B(h)=P^{-}\\left([I(\\psi_{\\lambda})-I(\\psi_{\\lambda}+h)]|u_{\\lambda}|^{2}(\\psi_{\\lambda}+h)\\right).$$\nSo we consider the operator $T:H^{\\frac{1}{2},-}\\to H^{\\frac{1}{2},-}$ defined by\n$$T(h)=D_gh-I(\\psi_{\\lambda})P^{-}(|u_{\\lambda}|^{2}h)+B(h).$$\nThen\n$$dT(0)\\varphi=D_g\\varphi-I(\\psi_{\\lambda})P^{-}(|u_{\\lambda}|^{2}\\varphi)-\\langle dI(\\psi_{\\lambda}),\\varphi\\rangle P^{-}(|u_{\\lambda}|^{2}\\psi_{\\lambda}).$$\nThe operator\n$$\\varphi\\mapsto D\\varphi-I(\\psi_{\\lambda})P^{-}(|u_{\\lambda}|^{2}\\varphi)$$\nis negative definite on $H^{\\frac{1}{2},-}$, hence it is invertible for all $\\lambda>0$, and for $\\lambda$ small enough we have that\n$$\\langle dI(\\psi_{\\lambda}),\\varphi\\rangle|u_{\\lambda}|^{2}\\psi_{\\lambda}\\to 0,$$\nhence $dT(0)$ is invertible for $\\lambda$ small enough. Since $A_{\\lambda}\\to 0$ as $\\lambda \\to 0$, we have by the implicit function theorem, the existence of $h_{\\lambda}\\in H^{\\frac{1}{2},-}$ such that $T(h_{\\lambda})=A_{\\lambda}$, moreover $h_{\\lambda}\\to 0$ as $\\lambda\\to 0$. Now we check that\n$$\\int_{M}\\langle D_g(\\psi_{\\lambda}+h_{\\lambda}),(\\psi_{\\lambda}+h_{\\lambda}\\rangle dv_{g}=\\int_{\\R^3}\\langle D_{g_{\\R^3}}\\Psi_{0},\\Psi_{0}\\rangle dv_{g_{\\R^{3}}}+o(1),$$\nand\n$$\\int_{M}|u_{\\lambda}|^{2}|\\psi_{\\lambda}+h_{\\lambda}|^{2}dv_{g}=\\int_{\\R^3}|U_{0}|^{2}|\\Psi_{0}|^{2}dv_{g_{\\R^{3}}}+o(1).$$\nHence, we have\n$$\\tilde{E}(u_{\\lambda},\\psi_{\\lambda}+h_{\\lambda})=Y_{g_{0}}(S^{3})\\lambda_{g_{0}}^{+}(S^{3})+o(1),$$\ntherefore we can conclude that\n$$\\tilde{Y}_g(M) \\leq \\tilde{Y}_{g_0}(S^{3}) .$$\n\\end{proof}\n\n\n\n\\noindent\nNow we consider the original functional $\\tilde{E}(u,\\psi)$ end the minimization problem giving $\\tilde{Y}_g(M)$.\n\\noindent\nWe complete the proof of Theorem (\\ref{second}) by proving the following\n\\begin{proposition}\nIf\n$$\\tilde{Y}_g(M)<\\tilde{Y}_{g_{0}}(S^{3}),$$\nthen the problem (\\ref{el}) has a non-trivial solution with $u>0$.\n\\end{proposition}\n\n\\begin{proof}\nWe introduce here a generalized Nehari manifold. We consider the functional $E$ and we notice first that there exist $t>0$ and $s>0$ small such that if $\\|u\\|=t$ and $\\|\\psi\\|=s$ then $E(u,\\psi)\\geq c>0$. Indeed, we have that\n$$2E(u,\\psi)=\\|u\\|^{2}+\\int_{M}\\langle D_g\\psi,\\psi\\rangle dv_{g}-\\int_{M}|u|^{2}|\\psi|^{2} dv_{g} .$$\nNow using the fact that $\\int_{M}\\langle D\\psi-I(\\psi)u^{2}\\psi,\\varphi\\rangle dv_{g}=0$, by taking $\\varphi=\\psi^{-}$, we have\n\\begin{align}\n\\|\\psi^{-}\\|^{2}&\\leq I(\\psi)\\int_{M}|u|^{2}|\\psi||\\psi^{-}|dv_{g}\\leq CI(\\psi)\\|u\\|_{L^6}^{2}\\|\\psi\\|_{L^3}\\|\\psi^{-}\\|_{L^3},\n\\end{align}\nhence\n$$\\|\\psi^{-}\\|\\leq CI(\\psi)\\|u\\|_{L^6}^{2}\\|\\psi\\|_{L^3} .$$\nTherefore,\n$$\\|\\psi\\|^{2}=\\|\\psi^{+}\\|^{2}+\\|\\psi^{-}\\|^{2}\\leq \\|\\psi^{+}\\|^{2}+CI(\\psi)\\|u\\|_{L^6}^{4}\\|\\psi\\|_{L^3}^{2}.$$\nNow\n\\begin{align}\n2E(u,\\psi)&\\geq t^{2}+s^{2}-C_{1}t^{4}s^{4}-C_{2}t^{2}s^{2}\\notag\\\\\n&\\geq t^{2}+s^{2}-C(t^{8}+s^{8}+t^{4}+s^{4})\\notag\\\\\n&\\geq t^{2}-C(t^{8}+t^{4})+s^{2}-C(s^{8}+s^{4}) .\\notag\n\\end{align}\nHence for $t$ and $s$ small enough, we have that $E(u,\\psi)\\geq c> 0$.\\\\\nMoreover, if we fix $u$ and $\\psi$ such that $\\int_{M}|u|^{2}|\\psi|^{2}dv_{g}=1$, we have that\n$$E(ru,r\\psi)\\leq r^{2}\\|u\\|^{2}-r^{2}\\|\\psi^{-}\\|^{2}-r^{4}\\to -\\infty, \\text{ when } r\\to \\infty.$$\nThis tells us that $E$ has the geometry of mountain pass, so we consider the following, min-max problem\n$$m=\\inf\\left\\{\\max_{t\\geq 0, s\\geq 0}E(tu,s\\psi);I(\\psi)>0;  P^{-}\\left(D_g \\psi-I(\\psi)u^{2}\\psi\\right)=0 \\right\\}.$$\nSo, if $E$ satisfies (PS) at the level $m$ and we disregard the orthogonality condition, then $m$ is a critical value. An easy computation shows that,\n$$\\max_{t\\geq 0, s\\geq 0}E(tu,s\\psi)=\\frac{1}{2}\\tilde{E}(u,\\psi)$$\nTherefore $2m=\\tilde{Y}_g(M)$. Now we consider the Nehari manifold\n$$N=\\left\\{\\begin{array}{ll}(u,\\psi)\\in H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M);\\int_{M}uL_{g}u dv_{g}=\\int_{M}\\langle D_g\\psi,\\psi\\rangle dv_{g}=\\int_{M}|u|^{2}|\\psi|^{2}dv_{g} \\not=0;\\\\\nP^{-}\\left(D_g \\psi-I(\\psi)u^{2}\\psi\\right)=0\n\\end{array}\\right\\} $$\nWe first claim that $N$ is indeed a manifold. We consider the operator\n$$G:H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)\\to \\R\\times \\R \\times H^{\\frac{1}{2},-},$$\ndefined by\n$$G(u,\\psi)=\\left[\\int_{M}uL_{g}u-|u|^{2}|\\psi|^{2}dv_{g},\\int_{M}\\langle D_g\\psi-|u|^{2}\\psi,\\psi\\rangle dv_{g},P^{-}(D_g\\psi-I(\\psi)|u|^{2}\\psi)\\right].$$\nClearly, we have that $N=G^{-1}(0)$ thus, if $DG(u,\\psi)$ is onto for all $(u,\\psi) \\in N$ then $N$ is a manifold. So we fix $(u_{0},\\psi_{0})\\in N$ and we will show invertibility of $DG(u_{0},\\psi_{0})$ restricted to some special subspace. Indeed, we will use the following parametrization $$(tu_{0},s\\psi_{0}+h)=[t,s,h], \\qquad h\\in H^{\\frac{1}{2},-},$$\nand we will express $DG(u_{0},\\psi_{0})$ in the basis of $\\R\\times \\R\\times H^{\\frac{1}{2},-}$. Since $\\int_{M}|u_{0}|^{2}|\\psi_{0}|^{2}dv_{g}\\not=0$, we will assume for the sake of simplicity that $\\int_{M}|u_{0}|^{2}|\\psi_{0}|^{2}dv_{g}=1$. We have then\n$$DG(u_{0},\\psi_{0})[1,0,0]=[0,-2,2P^{-}(D_g\\psi_{0})],$$\n$$DG(u_{0},\\psi_{0})[0,1,0]=[-2,0,0],$$\nand\n$$DG(u_{0},\\psi_{0})[0,0,h]=[2\\langle D_g\\psi_{0}, h \\rangle,0,D_gh-|u_{0}|^{2}h].$$\nNow we notice that the operator $K:h\\to D_gh-|u_{0}|^{2}h$ is negative definite on $H^{\\frac{1}{2},-}$. Indeed,\n$$\\langle Kh,h\\rangle=-\\|h\\|^{2}-\\int_{M}|u_{0}|^{2}h^{2}dv_{g}.$$\nTherefore, $K$ is invertible. Now, given $[a,b,c]\\in \\R\\times \\R \\times H^{\\frac{1}{2},-}$, we want to find $[x_{1},x_{2},w]$ so that\n$DG(u_{0},\\psi_{0},w)=[a,b,c]$. First, we have\n$$x_{1}=-\\frac{b}{2},$$\nand\n$$2x_{1}P^{-}(D_g\\psi_{0})+Kw=c.$$\nHence,\n$$w=K^{-1}(c+bP^{-}(D_g\\psi_{0})).$$\nFinally, we have that\n$$-2x_{2}+2\\langle D_g\\psi_{0},w\\rangle =a,$$\nthus\n$$x_{2}=-\\frac{a}{2}+\\langle D_g\\psi_{0},K^{-1}(c+bP^{-}(D_g\\psi_{0}))\\rangle.$$\nThis proves that $DG(u_{0},\\psi_{0})$ is onto and hence $N$ is a manifold. Moreover, $DG(u_{0},\\psi_{0})$ has a left inverse $A(u_{0},\\psi_{0}):\\R\\times \\R \\times H^{\\frac{1}{2},-}\\to \\R u_{0} \\oplus \\R \\psi_{0}\\oplus H^{\\frac{1}{2},-}$ and $$\\|A(u_{0},\\psi_{0})\\|_{op}\\leq C(\\|u_{0}\\|,\\|\\psi_{0}\\|).$$\nUsing Ekeland's principle now, we have the existence of a minimizing (PS) sequence for $E$ restricted to $N$: we want to show that this is indeed a (PS) sequence for $E$ also in $H^{1}(M)\\times H^{\\frac{1}{2}}(\\Sigma M)$. Let us call such a sequence $(u_{n},\\psi_{n})\\in N$. Then similarly as in the proof of Proposition (5.1), we set $DE(u_{n},\\psi_{n})=\\varepsilon_{n}$. We clearly have that $\\varepsilon_{n}^{T}\\to 0$, since it is the tangential part of the (PS) sequence and it is a (PS) sequence in $N$. Notice now that\n$$P_{n}=A(u_{n},\\psi_{n})\\circ DG(u_{n},\\psi_{n})$$\nis a projector on $\\R u_{0} \\oplus \\R \\psi_{0}\\oplus H^{\\frac{1}{2},-}$ parallel to $T_{(u_{0},\\psi_{0})}N$. Now since\n$$E(u_{n}\\psi_{n})=\\frac{1}{2}\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\to m,$$\nwe have that $\\|u_{n}\\|^{2}\\leq C$. Also, we have\n$$-\\|\\psi^{-}_{n}\\|^{2}=\\int_{M}|u_{n}|^{2}\\langle \\psi_{n},\\psi_{n}^{-}\\rangle dv_{g}.$$\nThus\n\\begin{align}\n\\|\\psi^{-}_{n}\\|^{2}&\\leq \\int_{M}|u_{n}|^{2}|\\psi_{n}||\\psi_{n}^{-}| dv_{g}\\notag\\\\\n&\\leq \\left(\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g}\\right)^{\\frac{1}{2}}\\left(\\int_{M}|u_{n}|^{2}|\\psi_{n}^{-}|^{2}dv_{g}\\right)^{\\frac{1}{2}}\\notag\\\\\n&\\leq C_{1}\\|u_{n}\\|_{L^6}\\|\\psi_{n}^{-}\\|_{L^3}.\\notag\n\\end{align}\nTherefore\n$$\\|\\psi^{-}_{n}\\|\\leq C,$$\nbut we have that\n$$\\|\\psi_{n}^{+}\\|^{2}-\\|\\psi_{n}^{-}\\|^{2}=\\int_{M}|u_{n}|^{2}|\\psi_{n}|^{2}dv_{g},$$\nhence\n$$\\|\\psi_{n}^{+}\\|^{2}\\leq C.$$\nTherefore, we have that $A(u_{n},\\psi_{n})$ is uniformly bounded and so is $P_{n}$. We consider now the operator $P_{n}^{*}$, the adjoint of $P_{n}$. Then $P_{n}^{*}$ is also a projector on $(\\R u_{0} \\oplus \\R \\psi_{0}\\oplus H^{\\frac{1}{2},-})^{\\perp}$ parallel to $\\mathcal{N}_{(u_{n},\\psi_{n})}N$, the normal space of $N$ at the point $(u_{n},\\psi_{n})$. We also notice that\n$$\\varepsilon_{n}\\in (\\R u_{0} \\oplus \\R \\psi_{0}\\oplus H^{\\frac{1}{2},-})^{\\perp},$$\nhence, $\\varepsilon_{n}=P_{n}^{*}\\varepsilon_{n}^{T}$ and so $(u_{n},\\psi_{n})$ is indeed a (PS) sequence for $E$. Therefore, this (PS) sequence is at the energy level $\\frac{1}{2}\\tilde{Y}_g(M)$: from Theorem (\\ref{first}), if $\\tilde{Y}_g(M)<\\tilde{Y}_{g_{0}}(S^{3})$, then this (PS) sequence converges and thus we have a solution to our problem.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\\section{Existence of infinitely many solutions in symmetric manifolds}\n\n\n\\noindent\nHere we will consider a three-dimensional closed manifold $(M,g)$ with an isometric group action $G$ acting on $M$, such that the orbits of $G$ have infinite cardinality. As an example, we can consider the standard sphere $S^{3}\\subseteq\\R^4$ with the action introduced by Ding \\cite{Ding}, that is $G=O(2)\\times O(2)$. Such symmetries where exploited an improved in other settings such as in \\cite{M2,MV2,MVG}.  We will show the following\n\\begin{theorem}\\label{third}\nGiven a manifold $M$ as described above, then $(\\ref{el})$ has two infinite families of solutions.\n\\end{theorem}\n\\begin{proof}\nFirst of all we notice that the functional $E$ satisfies the (PS) condition on the space $H_G:=H_{G}^{1}(M)\\times H_{G}^{\\frac{1}{2}}(\\Sigma M)$,\nwhere $H_{G}^{1}(M)$ and $H_{G}^{\\frac{1}{2}}(\\Sigma M)$ are respectively the subspaces of  $H^{1}(M)$ and $H^{\\frac{1}{2}}(\\Sigma M)$ which are invariant under the action of $G$. In order to prove this claim, let us consider $z_{n}\\in H_{G}$ a (PS) sequence for $E$, then according to the characterization in Theorem (\\ref{first}) above we have that\n$$E(z_{n})=E(z_{\\infty})+\\sum_{k=1}^{m}c_{k}+o(1) ,$$\nwhere $c_{k}=E_{\\R^3}(Z_\\infty^{k})\\geq \\tilde{Y}(S^{3})$, and $Z_\\infty^{k}$ are solutions of equation (\\ref{eqR3}) in $\\R^3$. The main point is that the number of these solutions is finite and that the energy is finite. In particular if $\\{z_{n}\\}$ is a (PS) sequence that concentrates on $x^{1},\\cdots, x^{m}$ then $z_{n}(h\\cdot)$ concentrates at $h\\cdot x^{1}, \\cdots, h\\cdot x^{m}$ for every $h\\in G$. Now, since $z_{n}\\in H_{G}$, then $z_{n}(h\\cdot)=z_{n}$ hence $z_{n}$ concentrates at all the orbits of $x^{1},\\cdots, x^{m}$ under the action of $G$; but the orbits are infinite: therefore the set of concentration needs to be empty and hence the (PS) condition holds.\\\\\nNow we consider the functional $E:H_{G}\\to \\mathbb{R}$ defined by\n$$E(u,\\psi)=\\frac{1}{2}\\left(\\int_{M}uL_{g}udv_g+\\int_{M}\\langle D_g\\psi,\\psi\\rangle dv_g -\\int_{M}|u|^{2}|\\psi|^{2}dv_g\\right).$$\nWe will study the restriction of this functional to the Nehari manifold $N_G$ defined by\n$$N_G=\\left\\{\\begin{array}{ll}(u,\\psi)\\in H_{G};\\int_{M}uL_{g}udv=\\int_{M}|u|^{2}|\\psi|^{2}dv_g=\\int_{M}\\langle D_g\\psi,\\psi\\rangle dv_g\\neq0;\\\\\nP^{-}(D_g\\psi-I(\\psi)|u|^{2}\\psi)=0\\end{array}\\right\\}.$$\nAs in the previous section, $N_G$ is a manifold, moreover critical points of $E_{|N_G}$ are critical points of $E$, as we saw above, and moreover any (PS) sequence of $E_{|N_G}$ is a (PS) sequence of $E$. Therefore, $E_{|N_G}$ satisfies the (PS) condition. So now we want to use the classical min-max theorem on the manifold $N_g$, so we define a collection $\\mathcal{A}$ of sets $A\\subset N_G$ such that $-A=A$ and\n$$c_{k}=\\inf_{A\\in \\mathcal{A};\\gamma(A)\\geq k}\\max_{(u,\\psi)\\in A}E(u,\\psi),$$\nwhere $\\gamma(A)$ denotes the genus of $A$. Now, if we can show that $N_G$ contains sets of arbitrarily high genus, we can show that we have infinitely many solutions. To this aim, we will prove that there exists a continuous $\\mathbb{Z}_{2}$-equivariant map\n$$\\mathcal{T}:\\{-1,1\\}\\times S^{+} \\longrightarrow N_G,$$\nwhere $S^{+}$ is the unit sphere of $H^{\\frac{1}{2},+}$. First, we recall that the generalized Nehari manifold originates from considering the functional $R:\\mathbb{R}^{+}\\times \\mathbb{R}^{+}\\times H^{\\frac{1}{2},-}\\to \\mathbb{R}$, defined by\n$$R(t,s,\\varphi)=E(tu,s(\\psi+\\varphi)).$$\nTherefore, the nonzero critical points of $R$ are in $N_G$. Indeed, if such a critical point exists, then it satisfies\n$$\\left\\{\\begin{array}{lll}\ns^{2}=\\frac{\\displaystyle\\int_{M}uL_{g}udv_g}{\\displaystyle\\int_{M}|u|^{2}|\\psi+\\varphi|^{2}dv_g},\\\\\n\\\\\nt^{2}=\\frac{\\displaystyle\\int_{M}\\langle D_g(\\psi+\\varphi),\\psi+\\varphi)dv_g}{\\displaystyle\\int_{M}|u|^{2}|\\psi+\\varphi|^{2}dv_g},\\\\\n\\\\\nP^{-}(D_g(\\psi+\\varphi)-I(\\psi+\\varphi)|u|^{2}(\\psi+\\varphi))=0.\\\\\n\\end{array}\n\\right.\n$$\nNow, the main issue in solving this system resides in the last equation, which is equivalent to solving\n$$T(\\varphi)+B(\\varphi)=A(\\varphi),$$\nwhere $$T(\\varphi)=P^{-}(D_g\\varphi-I(\\psi)|u|^{2}\\varphi),$$\n$$B(\\varphi)=P^{-}([I(\\psi)-I(\\psi+\\varphi)]|u|^{2}(\\psi+\\varphi)),$$\nand\n$$A(\\varphi)=P^{-}(D_g\\psi-I(\\psi)|u|^{2}\\varphi).$$\nAgain, as in the previous section, the operator $T$ is invertible, so the term that we need to consider here is $B(\\varphi)$. Now, we notice that for some particular choice of $u$ and $\\psi$ we can always find a solution to this system. Indeed, if $u$ is constant and $\\psi\\in H^{\\frac{1}{2},+}$, then we can take $\\varphi=0$, so that we have a unique critical point of $R$ denoted by $(t_{(u,\\psi)},s_{(u,\\psi)},0)$ such that $(t_{(u,\\psi)}u,s_{(u,\\psi)}(\\psi))\\in N_G$. Therefore, we will consider the map $\\mathcal{T}$ defined by $$\\mathcal{T}(1,\\psi)=(t_{(1,\\psi)},s_{(1,\\psi)}\\psi),$$\nwhere\n$$t_{(1,\\psi)}=\\frac{1}{\\|\\psi\\|_{L^{2}}},$$\nand\n$$s_{(1,\\psi)}=\\frac{\\displaystyle\\frac{1}{8}\\int_{M}R_g dv_g}{\\|\\psi\\|_{L^{2}}}.$$\nClearly $\\mathcal{T}(-(\\delta,\\psi))=-\\mathcal{T}(\\delta,\\psi)$ where $\\delta=\\pm 1$ and the map $\\mathcal{T}$ is continuous. Now, since $\\mathcal{T}$ is an equivariant map, and since $S^{+}$ has infinite genus, that is $\\gamma(S^{+})=+\\infty$, we have also that $N_G$ has infinite genus; moreover if $A\\subset S^{+}$ is symmetric and such that $\\gamma(A)=k$, then $\\mathcal{T}(A)\\subset N_G$ satisfies $\\gamma(\\mathcal{T}(A))\\geq k$. Also since $E$ is bounded from below on $N_G$, we have by classical min-max argument, see \\cite{Rab}, that $E_{|N_G}$ has infinitely many critical points, hence $E$ has infinitely many critical points.\\\\\nFinally, in order to find another infinite family of solutions, we argue in a similar way, by noticing that the set $N_{G}$ is invariant under the action of $S^{1}$ on the spinorial part, defined by\n$$\\theta \\cdot (u,\\psi)=(u,e^{i2\\pi \\theta} \\psi).$$\nClearly, $E_{|N_G}$ is also invariant under the this action of $S^{1}$. Therefore we can define the family of sets $\\mathcal{K}$ by saying that a set $A$ belongs to $\\mathcal{K}$ if and only if $ e^{i2\\pi \\theta}A=A$. We define also the min-max levels\n$$\\tilde{c}_{k}=\\inf_{A\\in \\mathcal{K};i_{S^{1}}(A)\\geq k}\\max_{(u,\\psi)\\in A}E(u,\\psi),$$\nwhere $i_{S^{1}}$ is the Faddell-Rabinowitz cohomological index \\cite{Fa}. Then, we use a restriction of the previous map $\\mathcal{T}$, that we denote here by $\\mathcal{G}:S^{+}\\to N_G$, defined by\n$$\\mathcal{G}(\\psi)=\\mathcal{T}(1,\\psi).$$\nWe see that $\\mathcal{G}$ is $S^{1}$-equivariant, hence $i_{S^{1}}(N_G)=+\\infty$ and hence, $E_{|N_G}$ has infinitely many critical points.\n\\end{proof}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe American Cancer Society's estimate for the number of new colon cancer cases in 2020 alone is 147,950, \\cite{siegel2020colorectal}. Among those, African Americans have the highest colorectal cancer incidence and mortality rates of all racial groups in the US, \\cite{wolf2018colorectal}. While the reason for this are not fully understood, research on graph neural networks for colon cancer prediction based on histopathology images has the potential to facilitate better diagnosis , possibly decreased costs for a check up and better prognosis for the patient due to earlier detection. \\\\\nThe histopathology images used to identify cancer grades are microscope images of extracted tissue from a colonscopy. As the colon wall where the tissue is extracted from contains several layers, each sample might be from different layers showing different structures, possibly also different layers in one sample: The inner most layer, the so called Epitheleum, contains glands that lead to a structure that might appear similar to bubbles under the microscope, the lower layers are constituted of muscle and connection layers of different strength.  \\\\\nWhile convolutional neural networks are strong in segmentation capabilities of histopathology images, this work proposes to use a cell graph to represent the image and analyse the cancer level through a graph neural network, simply because the structure of the graph and the evaluation through the network can generalize inspection of colon tissue through all of the above mentioned layers in a natural way. For general CNNs, there is typically a strong reliance on correct image size, image alignment and image scaling. Graph representations and networks have the potential to operate without pixel coordinate and those implicated dependencies.\n\\section{Related Work}\nRelated to this work in terms of \\textbf{medical application} and general colon cancer prediction, a variety of convolutional neural network approaches have been proposed. Recent state of the art for colon cancer grading with convolutional neural networks has been published by \\cite{shaban2020context}. The neural network described consists of a feature module with a CNN as its core that extracts features from the image into a feature cube, which is then further processed by an attention and context block to evaluate the overall grade. \\\\\nClose to this work, \\cite{zhou2019cgc} proposed a graph neural network that has a three layered approach of their adaptive GraphSage model, which contains a LSTM structure for information aggregation over three different graph convolutional layers.\\\\\nGraph neural networks have also been used for other medical applications, publications from \\cite{yu2019st} and \\cite{juarez2019joint} propose to use graph neural network for 2D and or 3D image analysis of medical data, mostly scans.\\\\\nRelated to this work in terms of \\textbf{graph neural network theory and architectures}, various authors have provided detailed benchmarks on aggregation methods and graph neural network architectures tested over common graph data sets such as biomedical graphs or social graphs, see \\cite{dwivedi2020benchmarking}, \\cite{morris2019weisfeiler} and \\cite{xu2018powerful}. \nFrom the overall availability of different methods and architectures with performance slightly different from other graph neural networks depending on the data sets, it becomes clear that there is no one graph neural network that fits all applications. From the different aggregation methods tested on graph neural networks, this work adapts the GraphSage \\cite{hamilton2017inductive} methodology for several reasons: The design to work on specifically large graphs by approximating aggregations, the capability to work in a semi supervised setting, and the large scale adaption and testing of GraphSage among other authors.\\\\\nPrior to this, the author has been working on \\cite{baylearn}, which experiments on edge downsampling methods on the MNIST Superpixel data set using sparse graphs with the intention of applying it to histopathological data. While the down sampling is also of importance to this work, it is used on nodes rather than edges, for dense graph representations rather than sparse, and hence requires a different approach to a solution.\\\\\nCompared to the CGC Net by \\cite{zhou2019cgc}, there are three main differences in this work's approach:  this paper provides neural network structures that are more simple and light weight, hence more easily reproducible in research: the feature segmentation has been trained on a u-net which does not require any backbone and can be trained in a short amount of tine. The graph neural network structure does not contain the adaptive LSTM structure but is rather a simple combination of three convolutional blocks with stabilization and pooling layers. The graph augmentation from cell features is independent of spatial coordinates, locally dense and effective in down sampling to not more than 5000 nodes per image.\n\\section{Cancer grading  prediction}\n\\begin{figure*}\n\\includegraphics[scale=.6]{pipeline4}\n\\caption{Overall carcinoma level prediction pipeline divided into three steps}\n\\label{fig:pipeline}\n\\end{figure*}\n\\subsection{From medical analysis to technical application}\nCancer grading from a medical perspective requires thorough analysis of the cells within a whole tissue, and the challenge lies in interpreting a combination of the cell tissue type, tissue cut direction, cell overall distribution and abnormal structures, in addition complicated by non uniformity in the histopathology process. \\\\\nSome of the observatory skills required to understand the complete image as one context and to differentiate between different grades are acquired by assistants through medical training, yet still hard for a technical application to understand. \\\\\nThose challenges are amongst others:  The understanding of \"empty spaces\", the implicit layer type determination and the overall grading logic for an image that contains partially different cancer level.\n\\subsubsection{Interpretation of space and pattern} \"Blank space\" - in the image visible as \"white\", can either stand for missing tissue due to the extraction method or be a marker for a necrosis. While with the human eye there is a difference between a clean cut and a tissue decaying at the border of the cut, this feature is hard to understand from a machine learning point of view. \\\\\nFurther more glandular structures in the images pose a challenge to traditional visual analysis - \\cite{awan2017glandular} have proposed to separately recognize glandular structures for cancer grading. This work assumes the opposite, namely that the graph representation in combination with the graph neural network can treat different structures for one cancer grade equally. \nThis is enforced by design, as the graph representing the cell structures is locally dense, i.e. each cell is connected to each other cell, and as edge weights are applied in a locally constant manner, this means that the representation of cells around glandular structures can be topologically equivalent to a graph representing a muscular structure, if the number of cells per area is the same. For technical details see section \\ref{sec:featselect} and \\ref{sec:graphaug}.\n\\subsubsection{Comparison to a ground truth pattern}\nAnalysing the tissue layer type and the tissue cut direction, is crucial to differentiate between a normal and abnormal tissue. Cancer grading is not straight forward linearly related to the number of cells per image. Traditional convolutional neural networks work on a pixel wise evaluation of an image. The pipeline steps two and three however only use relational position, not absolute. Whether or not cell A is 5 more pixel to the left or the right should in no manner influence the network. This is only possible by building a graph structure based upon nodes with visual features only and passing the graph through a graph neural network.\n\\subsubsection{Local vs Global evaluation}\nAs a result of the different tissue types and layers per sample, the local cancer grading results for different regions are not in general identical, i.e. one sample might contain different grades at different areas. Which is why the graph neural network uses patching to evaluate the result of each patch locally and then reason on the overall grading system with a linear layer block, see section \\ref{sec:patching}.\n\\subsection{Pipeline for prediction}\nThe complete pipeline from histopathology image to prediction consists of three steps, it starts with cell segmentation, which is then followed by cell graph augmentation, and concluded with level prediction through a graph neural network, see figure \\ref{fig:pipeline} for reference.\\\\\n\\textbf{Step 1: Feature calculation}\nThis work uses a 5-layer U-net trained on cell segmentation images as originally proposed by \\cite{ronneberger2015u}.\nNodes for the graph correspond to features of the cell segmentation provided by the Unet. Similar to \\cite{zhou2019cgc}, 16 different properties per segmented cell are used for each node.\\\\\n\\textbf{Step 2A: Feature Selection}\nDue to the high amount of cells per image, the selected number of cells are downsampled using distribution aware methods. For a fixed number of features, the selection uses random choices per area.\\\\\n\\textbf{Step 2B: Graph Augmentation}\nThose randomly downsampled features are then augmented into a graph. The graph's node have 16 features each and the edges are determined using distribution based edge values. The edge value function is proportional to both the local distribution and the variance in distribution. \\\\\n\\textbf{Step 3: Graph Neural Network}\nThe graph is analysed through a network consisting of three convolutional blocks and a prediction layer, where as each block consists of one GraphSAGE convolution, one pooling and one normalization layer, as shown in detail in figure \\ref{fig:arch}.\n\n\\section{Methodology}\nThe pipeline is designed to be easily reproducible and lightweight during training: the cell segmentation is achieved with a conventional U-net \\cite{ronneberger2015u}, the feature selection and graph augmentation methods have linear complexity with respect to the number of nodes (i.e. cells), the graph neural network acts on vertices and edges in 3 convolutional blocks. In the following, the different technical details of those methods will be described:\n\\subsection{Cell feature calculation}\n\\label{sec:cellfeat}\nA U-net with 5 convolutional layers is used as described by the authors in \\cite{ronneberger2015u}. The resulting mask is processed to extract 16 features per cell instance, as proposed in \\cite{zhou2019cgc}. The first half of those are bounding box color features, the second half are segmentation mask based image features. In the experiments a study on the feature dimension has been executed.\n\\subsection{Spatially aware feature selection}\n\\label{sec:featselect}\nFor a given segmentation map $s(i,j)$ of an image $I$ of dimension $w \\times h $ at pixel position $(i,j)$ , let us define the distribution map as $D(i,j)$, which uses a regular 2D grid of dimension $dxd$ and counts the number of points per box.\nThen for any set of feature points $ F = \\{ f_0,\\dots,f_N \\}$ the corresponding distribution map for M feature points is \n\\begin{equation*}\nD_M(i,j) = \\frac{M}{\\| D \\|} D(i,j)\n\\end{equation*}\nwhere $ \\| \\cdot \\|$ stands for the $L_1$ matrix norm.\nGiven the number of points per area through the distribution map $D_M(i,j)$, the features are augmented into a graph through random selection of nodes per area.\n\n\\begin{figure*}[t]\n\\includegraphics[scale=.6]{graphnet_arch8}\n\\caption{Architecture of graph network with dimensions of vertices where as p stands for the number of patches, f for the feature dimension, e for the embedded feature dimension and n for the number of nodes.}\n\\label{fig:arch}\n\\end{figure*}\n\n\\subsection{Spatially aware graph augmentation}\n\\label{sec:graphaug}\nFor a given set of feature points  $ F = \\{ f_0,\\dots,f_N \\}$ , the edge value between two feature points $f_k,f_m$ and their corresponding coordinates $c_k,c_m$ is defined as:\n\\begin{equation*}\n\\begin{split}\nA_{k,m} &= \\alpha (D_M(c_k)+D_M(c_m)) \\\\\n        &+  \\beta  | D_M(c_k) - D_M(c_m) |\n\\end{split}\n\\end{equation*}\nwhere in this case  $|\\cdot |$ stands for the absolute value and $\\alpha,\\beta \\in [0,1]$ are hyper parameters. \nThen the Matrix $A$ corresponds to the adjacency matrix of the augmented graph.\n\n\\subsection{Cell Graph Neural Network architecture}\nThis part of the pipeline leverages a graph neural network architecture that acts both on nodes and edges, applies pooling and normalization operations and classifies the graph into a category. \\\\\nIt consists of two parts, the graph modification and the graph classification. The graph modification network consists of graph convolution blocks that modify both edge and vertices stepwise. Each convolutional block consists of three graph convolutions that are concatenated, one pooling and one normalizing layer.\\\\\nAs a graph convolution propagation rule, this work uses GraphSAGE \\cite{hamilton2017inductive}: \n\\begin{equation*}\nh_v^k \\leftarrow \\sigma ( \\textbf{W}  \\cdot \\textit{mean} ( \\{ h_v^{k-1} \\} \\cup \\{ h_u^{k-1} , \\forall \\textit{u} \\in \\textsl{N} \\}  ) )  \n\\end{equation*}\nwhere W is the edge weight matrix, N stands for the set of points lying in the direct neighborhood of node v, and h denotes the node's feature with respect to node v and number of neighborhood jumps k. In this work only one hop convolutions will be applied.\n\\subsubsection{Feature embedding}\nMany graph neural network architectures do not work with the original nodes i.e. features as input but upscale the dimensions, such as shown in work by \\cite{errica2019fair}.\nUpscaling of the feature dimensions is done through increasing the output dimensions of the graph convolution respectively. Down scaling of features is executed by applying linear layers with the output dimension as the desired scaled size. \\\\\nWhile each convolutional block contains three convolutions that are being concatenated, this would increase the feature dimensions by a factor equal to the number of concatenations. The feature dimension of each convolutional block is hence down scaled back to the original embedding dimension with a linear layer at each step.\nThe convolutional block structure is simplified compared to the work proposed in \\cite{zhou2019cgc}, as the bi-directional LSTM is not used this way.\n\\subsubsection{GNN normalization and stabilization}\nGraph neural networks suffer from the over smoothening of adjacency values. To fix this, the values of the adjacency matrix $A$ are reassigned after every block in the following manner as originally proposed by \\cite{chen2019multi}:\\\\\n\\begin{equation}\nA'_{i,j} = \n\\begin{cases} \n1-p \\textbf{ if } i = j\\\\ \n\\frac{p}{\\sum_{i, i\\neq j }A_{i,j}} \\textbf{ if } i \\neq  j\\\\ \n\\end{cases}\n\\end{equation}\n\\subsubsection{GNN pooling operations}\nThe graph neural network leverages differential graph pooling Diffpool as originally described in \\cite{ying2018hierarchical}, which computes an assignment matrix seperating nodes into different clusters.\n\\subsubsection{Graph patching}\n\\label{sec:patching}\nGiven the original Graph G corresponding to the image I, the nodes and respective edges are split into patches according to the pixel position along each axis into two sections resulting in four patches. The patches are each treated as individual graphs during the convolutional blocks and are merged into one result at the end of the pipeline with a block of linear layers.\\\\\nFor sake of simplicity in our architecture sketch, it is assumed that the patches have same size or are padded with zeros if not. In practice different number of nodes per patch have been used, as the splitting method does not take number of nodes into account.\\\\\nInstead of typically used batch norm, the graph convolutional layers' outputs are stabilized with \"patch norm\".\n\n\\section{Experiments}\n\\subsection{Preliminaries}\nAll of the experiments are conducted on a single GPU per task. Many of the experiments take less than 20 GB of GPU Memory, some experiments on the single graph with high number of nodes take up to 46 GB. All experiments on work from this paper use less than  10 GB of CPU memory. Graph augmentation was implemented to work on multiple cores.\\\\\nThe neural network has been coded in PyTorch \\cite{paszke2019pytorch} and specifically PyTorch Geometric \\cite{fey2019fast}. For the cell segmentation the code available at \\cite{unetgh} has been used. \n\n\\subsection{Data}\nThe Unet for semantic segmentation of cells is trained on the training set of the \\textbf{colorectal cancer cell data set} CoNSeP \\cite{graham2019hover}, which contains 41 images with segmentation labels of resolution 1000x1000 pixel.\\\\\n The graph neural network is trained on the \\textbf{Extended Colorectal Adenocarcinoma Dataset} (abbreviated in this work with ECA) for Grading of Histology Images \\cite{shaban2020context}, which classifies 300 images from 178 different patients of resolutions around 5000x7000 pixel into three categories: no cancer , low level and high level.\n In order to enable comparisons to scores in previous works, the \\textbf{Colorectal Cancer Grading Dataset} (CRC) ( \\cite{awan2017glandular}) is used, which contains a subset of the Extended Colorectal Adenocarcinoma Dataset, in total 137 images from 38 different patients.\n\n\\subsection{Feature calculation} \nFor the semantic segmentation, the u-net is trained on 100 epochs on the CoNSeP training data set. The training and test split is provided with the data. The binary cross entropy with logits is used for training, together with a learning rate plateau scheduler and Root Mean Square Propagation. After those epochs, it overall achieves a dice test score of 75.60 out of 100. As the dice score heavily punishes segmentation areas larger than the labeled segmentation, this means that the segmented cells potentially are slightly larger than the ground truth. Cells in the training data and test data have a bounding of around 10x10 pixel to 20x20 pixel, with main variations in different images and small size variation only per one image. For a 10x10 sized bounding box, with a mask area smaller than 100 pixel, a dice score of around 75 corresponds to a cell of around 125 pixel area. \n\\begin{table}\n  \\caption{Overview over the ECA (\\cite{shaban2020context}) and CRC datasets (\\cite{awan2017glandular}) used for cancer level prediction; the label split shows the distribution of samples among labels no grade, low grade, high grade and the fold split shows the number of samples per fold 1,2 and 3}\n  \\label{table:data}\n  \\centering\n  \\begin{tabular}{l|l|l}\n    \\toprule\n   Dataset & CRC & ECA  \\\\\n    \n    \\midrule\n   Samples & 137 & 300  \\\\\n   Label Split & 71-31-35 & 120-120-60 \\\\\n   Fold Split & 45-46-46 & 100-100-100  \\\\\n   Patients & 38 & 178  \\\\\n\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\\subsection{Cancer Grading and Graph Network}\n\n\\begin{table*}[ht]\n  \\caption{Overall best accuracy scores for different parameters, Accuracy and standard deviation values are provided in percentage}\n  \\label{table:acctotal}\n  \\centering\n  \\begin{tabular}{lllll}\n    \\toprule\n   Method &    CRC & ECA  \\\\\n    \\midrule\n   \n   \n    CA-CNN \\cite{shaban2020context} & $95.70 \\pm 3.04$ & - \\\\\n    CGC \\cite{zhou2019cgc} & $ 97.00 \\pm 1.10$ & -  \\\\\n    \\midrule \n    this work (CRC) & $99.26 \\pm 1.05$ & - \\\\\n    this work (ECA)& $99.26 \\pm 0.66$ & $99.33 \\pm 0.94$ \\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table*}\n\nIn Table \\ref{table:acctotal}, it is shown that this work improves the accuracy of other works on this data set by around two percent on the CRC data set and the ECA data set.\n\\textbf{Parameter}\nThe normalization layer is being used with p = 0.4. The embedding dimension has been chosen to be 100, in some experiments 120. From the classification head, the first linear layer block contains three linear layers with dimensions 50,25,3 respectively. The patch classification layer contains two linear layers with dimensions 3 and 1. The learning rate has to be adjusted for each specific experiment but on average lies between 0.00005 and 0.00001 at initialization. The learning rate is scheduled with a plateau scheduler with respect to the accuracy\\\\\n\\textbf{Evaluation criteria}\nAs training is executed using 3-fold cross validation, the evaluation for the complete pipeline is given in terms of the average and standard deviation of the last three accuracy values. The loss function used during training of the graph neural network is a smooth L1 loss.\\\\\n\\textbf{Time estimations}\nTraining on the graph neural network including all necessary steps requires around 6-14h per model and configuration, with variations of several hours for  usage of normalization layers and large graph sizes.\n\\subsection{Ablation Studies}\n\\subsubsection{Patching versus one graph}\n\n\\begin{figure}\n\\includegraphics[scale=.5]{patchwholeplot_mini.png}\n\\caption{Comparison of accuracy scores with and without patching for different number of nodes on the CRC dataset}\n\\label{fig:patchwhole}\n\\end{figure}\nIn figure \\ref{fig:patchwhole}, accuracy values for one graph per image vs. 4 graphs per image have been compared.\\\\\nWhile one graph per image yields a straight forward classification score, separating the graph into patches and evaluating on those patches requires a classification head that decides on the overall label based on each patch. Evaluation results for a single graph are unstable over the selection of number of nodes for the augmentation and achieve in general lower accuracy than with four graphs per image.\n\\subsubsection{Different cell segmentation features as input}\n\\begin{figure}\n\\includegraphics[scale=.5]{featdimplot.png}\n\\caption{Change of accuracy for different feature dimensions 8,12,16 of nodes and different graph sizes for training on the ECA dataset}\n\\label{fig:featdim}\n\\end{figure}\nWhile this paper adapted the 16 cell segmentation features as in \\cite{zhou2019cgc}, a set of experiments have been conducted on using smaller amount of cell features from the segmentation. Of those features, the first half of them characterize the bounding box color and black and white values, while the second half of features characterizes the properties of the masked cell. Testing has been done on 8-dim feat input (only looking at bounding box based features) and 12 dim input with restricted segmentation features. While the stability inspected over smaller graphs decreases for lower feature dimensions, for a graph size of 16000 the accuracy values are observed to peak for all feature dimensions and differ by around 2 percent only. \n\n\\section{Conclusion}\nIn this work a stable and efficient graph neural network pipeline has been proposed to evaluate cancer grades. Experiments on parameters have shown that a stable architecture design and efficient pipeline can improve evaluation results.\\\\\nGraph augmentation, just like image augmentation and text augmentation, has significant impact on the quality of the results. Compared to the state of the art, the pipeline accuracy results are improved by around two percent and the overall number of false predictions is decreased. \\\\\nFurther, experiments have been executed with respect to graph size and number of nodes, which shows that downsampling of graphs must not lead to a loss in accuracy. This would also possibly enable testing on devices and hardware with less available memory, as graph neural network training can be much more expensive than training of traditional neural networks. \\\\\nFurther, the author suggest that this pipeline has the potential to be used on other diseases related to histopathological image diagnosis.\n\\section{Acknowledgement}\nThe author would like to thank Dr. Anish Mohammed for this medical advice. Special thanks to Gregory Senay for his support during the writing process.\n \n \n   \n  \n   \n   \n   \n   \n   \n \n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nEquivariant diagrams of spaces and their homotopy colimits have broad applications throughout topology. They are used in \\cite{Jackowsky} for decomposing classifying spaces of finite groups, to study posets of $p$-groups in \\cite{TW}, for splitting Thom spectra in \\cite{Takayasu}, and even in the definition of the cyclic structure on $\\THH$ of \\cite{BHM}. \nIn previous joint work with K. Moi \\cite{Gdiags} the authors develop an extensive theory of equivariant diagrams in a general model category, and they study the fundamental properties of their homotopy limits and colimits.\nIn the present paper we restrict our attention to $G$-diagrams in the category of spaces. The special feature of $G$-diagrams of spaces is the existence of generalized fixed point functors that preserve and reflect equivalences. We study these functors and we use them to generalize the Blakers-Massey Theorem and Quillen's Theorem $B$ to equivariant cubical diagrams. \n\nThroughout the paper $G$ is going to be a discrete group. Let $I$ be a small category with a $G$-action. A $G$-structure on a diagram $X\\colon I\\to \\Top$ is a sort of generalized $G$-action on $X$, which depends on the way $G$ acts on $I$ (see \\ref{defGdiag}). The key feature of a $G$-structure is that it induces a $G$-action on the homotopy limit and on the homotopy colimit of $X$ (\\S\\ref{secGdiags}). These equivariant constructions can be described as derived functors in a suitable model categorical context (\\cite{Gdiags}). In the present paper, we will focus mostly on $G$-diagrams of cubical shape.\nIf $J$ is a finite $G$-set, the poset category of subsets $I=\\mathcal{P}(J)$ ordered by inclusion inherits a $G$-action. A $J$-cube is a diagram $X\\colon \\mathcal{P}(J)\\to C$ equipped with a $G$-structure. There canonical maps\n\\[\\phi\\colon X_{\\emptyset}\\longrightarrow \\holim_{\\mathcal{P}_0(J)}X\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\psi\\colon \\hocolim_{\\mathcal{P}_1(J)}X\\longrightarrow X_{J}\\]\nare $G$-equivariant, where $\\mathcal{P}_0(J)$ and $\\mathcal{P}_1(J)$ are the category $\\mathcal{P}(J)$ respectively with the initial and the final object removed. Given a function $\\nu\\colon\\{H\\leq G\\}\\to\\mathbb{Z}$ which is invariant on conjugacy classes, we say that $X$ is $\\nu$-cartesian if the restriction of $\\phi$ on $H$-fixed points is $\\nu(H)$-connected. Dually, $X$ is $\\nu$-cocartesian if $\\psi$ is $\\nu(H)$-connected on $H$-fixed points. The following generalizes the Blakers-Massey Theorem of \\cite[2.5]{calcII}. We prove it in \\S\\ref{secBM}.\n\n\\begin{theorem*}[Equivariant Blakers-Massey]\nLet $X\\colon\\mathcal{P}(J)\\to\\Top$ be a $J$-cube of spaces, and suppose that for every subgroup $H$ of $G$ and every non-empty $H$-invariant subset $U$ of $J$ the restriction $X|_{\\mathcal{P}(U)}$ is $\\nu^U$-cocartesian. Suppose moreover that these functions satisfy $\\nu^U\\leq \\nu^V$ whenever $U\\subset V$ are non-empty $H$-subsets of $J$. Then $X$ is $\\nu$-cartesian, where $\\nu$ is the function\n\\[\\nu(H)=\\!\\min\\left\\{\\!\\min_{\\{T_\\alpha\\}\\in\\Pa_H(J)}\\{\n\\sum_{\\alpha}\\nu^{T_\\alpha}(H)\\}-|J\/H|+1\\ ,\\ \\min_{\\emptyset\\neq U\\subset J}\\min_{L\\in \\Eff_H(U)}\\big\\{\\conn X^{L}_U-|U\/L|+1\\big\\}\n\\right\\}\\]\n\\end{theorem*}\nHere $\\Pa_H(J)$ is the set of partitions of $J$ by $H$-invariant subsets, and $\\Eff_H(U)$ is roughly the set of proper subgroups $L$ of $H$ for which $U\/L\\neq U\/H$. The first term of the minimum corresponds to the standard range of the Blakers-Massey Theorem \\cite[2.5]{calcII}. The second term is purely equivariant, and it is infinite if $J$ has trivial $G$-action. It comes form the equivariant connectivity of a certain space of natural transformations of diagrams, which is calculated in \\ref{connholim} using a diagramatic obstruction theory argument.\nWe also prove a dual form of this Blakers-Massey Theorem in \\ref{dualBM}.\nIn the same way that the Freudenthal suspension Theorem is an immediate consequence of the Blakers-Massey Theorem for the square\n\\[\\xymatrix@=13pt{X\\ar[r]\\ar[d]&CX\\ar[d]\\\\\nCX\\ar[r]&\\Sigma X\n}\\]\nthe equivariant Freudenthal suspension Theorem follows from the equivariant Blakers-Massey Theorem applied to a certain equivariant cube, whose all but the initial and the final vertex are contractible. Given a finite $G$-set $J$ and a pointed $G$-space $X$, let $\\Sigma^{J}X$ and $\\Omega^JX$ be respectively the suspension and the loop space of $X$ by the permutation representation of $J$. We prove the following Corollary in \\S\\ref{secsusp}.\n\n\n\\begin{cor*}[Equivariant Suspension Theorem, \\cite{Nam}] Let $X$ be a pointed $G$-space. The unit of the $(\\Sigma^{J},\\Omega^{J})$-adjunction restricted on $G$-fixed points $\\eta\\colon X^G\\to (\\Omega^J\\Sigma^{J}X)^G$ is\n\\[\\min\\Big\\{2\\conn X^G+1,\\min_{\\substack{H\\leq G\\\\ J\/H\\neq J\/G}}\\conn X^H\\Big\\}\\]\nconnected.\n\\end{cor*}\n\nAs a second application of the equivariant Blakers-Massey Theorem, we prove an equivariant version of the relative disjunction Theorem of \\cite{GK} for equivariant configuration spaces. Let $M$ be a manifold with a proper $G$-action. The space of configurations of $J$-points in $M$ is the space $\\conf(J,M)$ of injective maps $J\\rightarrowtail M$. This space inherits a $G$-action by conjugation, and it is functorial in the $J$-variable with respect to injective $G$-maps. The following is proved in \\S\\ref{secconf}.\n\n\\begin{cor*}\nLet $J$ be a finite $G$-set and let $J_+$ be the $G$-set $J$ with an added fixed basepoint. The diagram $\\conf(J_+\\backslash (-),M)\\colon \\mathcal{P}(J_+)\\to Top$ has a canonical $G$-structure, and it is $\\nu$-cartesian for the function\n\\[\\nu(H)=\\min\\Big\\{|J|m_H-2|J\/H|+1\\ ,\n\\min_{\\substack{\nL\\leq H\\\\\nJ\/L\\neq J\/H\n}}\\big\\{\\min\\{\\conn M^L+1,m_L\\}-|J\/L|\\big\\}\n \\Big\\}\\]\nwhere $m_H$ is the dimension of the fixed points manifold $M^H$.\n\\end{cor*}\nWe prove in fact a stronger statement involving suitably transverse families of submanifolds of $M$ (Theorem \\ref{intsubman}), and we deduce this Corollary from the case where all of these submanifolds are points.\nIn \\ref{exsharp} we find an example where this range is sharp and it is determined by the second term of the minimum.\n\nWe turn to another classical result in homotopy theory, the celebrated Theorem $B$ of Quillen, from \\cite{Quillen}. This theorem shows that under certain conditions the homotopy fiber of the geometric realization of a functor is itself the geometric realization of a category. It is not immediately clear how to generalize this result equivariantly. The analogous statement for an equivariant functor between categories with $G$-actions can easily be reduced to Theorem $B$ by taking fixed points. In order to achieve an interesting equivariant Theorem $B$, we need to extend it first to higher dimensional cubes. Let $J$ be a finite $G$-set and let $X\\colon \\mathcal{P}(J)\\to Cat$ be a cube of categories. The natural transformations of diagrams of categories from $\\mathcal{P}_0(-)\\colon\\mathcal{P}_0(J)\\to Cat$ to $X$ form a category $\\Hom\\big(\\mathcal{P}_0(-),X\\big)$. A $G$-structure on $X$ induces a $G$-action on the category $\\Hom\\big(\\mathcal{P}_0(-),X\\big)$ by conjugation. The inclusion of objects as natural transformations of constant functors defines a functor $m_\\emptyset\\colon X_\\emptyset\\to \\Hom\\big(\\mathcal{P}_0(-),X\\big)$. Given a natural transformation $\\Phi\\colon \\mathcal{P}_0(-)\\to X$, the over category $m_\\emptyset\/_{\\Phi}$ has an action of the stabilizer group $G_{\\Phi}$. We proved the following result in \\ref{GQuillen}.\n\n\n\\begin{theorem*}\nLet $X\\colon \\mathcal{P}(J)\\to Cat$ be a $J$-cube of categories, which satisfies a certain ``weak Reedy fibrancy condition'' (Definition \\ref{defGreedy}). For every natural transformation $\\Phi\\colon \\mathcal{P}_0(-)\\to X$ the classifying space of the category $m_{\\emptyset}\/_{\\Phi}$ is $G_{\\Phi}$-equivalent to the total homotopy fiber of the cube of spaces $BX$ over $B\\Phi$. In particular if all the categories $m_{\\emptyset}\/_{\\Phi}$ are $G_{\\Phi}$-contractible, $BX\\colon \\mathcal{P}(J)\\to \\Top$ is a homotopy cartesian $J$-cube of spaces.\n\\end{theorem*}\n\nWhen $G$ is the trivial group and $J=1$ is the set with one element this is precisely Quillen's Theorem $B$. If $J=2$ is the set with two elements this is essentially Barwick and Kan's Quillen Theorem $B_2$ for homotopy pullbacks from \\cite{ClarkKan}. To the best of the authors knowledge this is a new result for larger $J$, even for the trivial group. The key for proving this theorem is to define a good model for the homotopy limit of a diagram of categories. Let $I$ be a small category with finite dimensional nerve. Taking the over categories of $I$ defines a diagram of categories $I\/_{(-)}\\colon I\\to Cat$. Given a functor $X\\colon I\\to Cat$, the natural transformations $\\Hom\\big(I\/_{(-)},X\\big)$ form a category, whose nerve is the Bousfield-Kan formula for the homotopy limit of the nerve of $X$. The following is proved in \\S\\ref{secBI}, and its equivariant version in \\S\\ref{secBGI}.\n\n\n\\begin{theoremBI}\nLet $I$ be a category with finite dimensional nerve, and let $X\\colon I\\to Cat$ be a ``Reedy quasi-fibrant diagram'' (Definition \\ref{quasiReedy}). The classifying space $B\\Hom\\big(I\/_{(-)},X\\big)$ is the homotopy limit of the diagram of spaces $BX\\colon I\\to\\Top$.\n\\end{theoremBI}\n\n\n\\subsection*{Acknowledgments}\nThis project is a continuation of the work of \\cite{Gdiags}, and part of a mission aimed at understanding a ``genuine'' context for equivariant homotopy theory. I wish to thank Kristian Moi for the many inspiring conversations we had over the last couple of years.\nI also want to thank Brian Munson and Ismar Volic for useful correspondences.\n\n\n\\tableofcontents\n\n\n\n\\section{Preliminaries on equivariant diagrams}\n\n\n\n\n\\subsection{Equivariant diagrams of spaces and their fixed points}\\label{secGdiags}\n\n\nLet $G$ be a discrete group. A category with $G$-action is a functor $a\\colon G\\to Cat$, where the group $G$ is seen as a category with a unique object $\\ast$. By abuse of notation we will refer to the underlying category $I=a(\\ast)$ as a category with $G$-action.\n\n\n\\begin{defn}[\\cite{vilf}]\\label{defGdiag} Let $I$ be a small category with $G$-action, and let $C$ be a possibly large category. A $G$-structure on a diagram $X\\colon I\\to C$ is a collection of natural transformations $\\phi_g\\colon X\\to X\\circ g$, for every $g$ in $G$, subject to the following axioms.\n\\begin{enumerate}[label=\\roman*)]\n\\item Let $1$ be the unit of $G$. Then $\\phi_1$ is the identity natural transformation on $X$,\n\\item For every $h$ and $g$ in $G$ the diagram \\ \\ \\  $\\vcenter{\\hbox{\\xymatrix@=20pt{X\\ar[dr]_-{\\phi_{hg}}\\ar[r]^-{\\phi_g}&X\\circ g\\ar[d]^-{\\phi_h|_g}\\\\\n&X\\circ h\\circ g}}}$\n\\ \\ \\ commutes. \n\\end{enumerate}\nA diagram $X\\colon I\\to C$ equipped with a $G$-structure is called a $G$-diagram.\nA morphism of $G$-diagrams is a natural transformation of underlying diagrams $f\\colon X\\to Y$ such that the square\n\\[\\xymatrix@R=15pt{\nX\\ar[r]^-f\\ar[d]_{\\phi_g}&Y\\ar[d]^-{\\phi_g}\\\\\nX\\circ g \\ar[r]_-{f|_g}& Y\\circ g\n}\\]\ncommutes for every $g$ in $G$. Here $f|_g$ is the restriction of $f$ along the functor $g\\colon I\\to I$. The resulting category of $G$-diagrams is denoted $C_{a}^I$. We will often abuse the notation and write $g$ for the natural transformation $\\phi_g$.\n\\end{defn}\n\n\n\n\\begin{ex}\\label{overcatGdiag} Let $I$ and $J$ be two categories with $G$-action, and let $F\\colon I\\to J$ be an equivariant functor. The functor $F\/(-)\\colon J\\to Cat$ which sends an object $j$ to the over category $F\/j$ has a natural $G$-structure. The natural transformation $\\phi_g$ is defined by the functors $F\/j\\to F\/gj$ which send an object $(i\\in I, \\alpha\\colon F(i)\\to j)$ to $(gi,g\\alpha\\colon F(gi)=gF(i)\\to gj)$.\nBy applying the classifying space functor we obtain a $G$-diagram  $B(F\/-)\\colon J\\to \\Top$ in the category of compactly generated Hausdorff spaces $\\Top$.\n\\end{ex}\n\nThe category of $G$-diagrams of spaces $\\Top_{a}^I$ is enriched in the category of $G$-spaces $\\Top^G$. Given two $G$-diagrams $X$ and $Y$, the space of all natural transformations of underlying diagrams $\\Hom_I(X,Y)$ inherits a $G$-action by conjugation:\n\\[g\\cdot f=\\big(X\\stackrel{\\phi_{g^{-1}}}{\\xrightarrow{\\hspace*{1cm}}}X\\circ g^{-1}\\stackrel{f|_{g^{-1}}}{\\xrightarrow{\\hspace*{1cm}}}Y\\circ g^{-1}\\stackrel{\\phi_{g}}{\\xrightarrow{\\hspace*{1cm}}}Y\\big)\\]\nThe fixed points space $\\Hom_I(X,Y)^G$ has the set of morphisms of $G$-diagrams $\\Top_{a}^I(X,Y)$ as underlying set. \n\nThe $G$-space $\\Hom_I(X,Y)$ can be defined as a certain equalizer of $G$-spaces, and its construction can be dualized as follows. The $G$-action on $I$ induces a $G$-action on $I^{op}$. Given a $G$-diagram $X\\colon I\\to\\Top$ and a $G$-diagram $Y\\colon I^{op}\\to\\Top$, we define the coequalizer of $G$-spaces\n\\[X\\otimes_I Y=\\colim\\left(\\coprod_{\\substack{\\alpha\\colon i\\rightarrow j\\\\\n\\in \\hom I}}Y_i\\times X_j\\rightrightarrows \\coprod_{i\\in \\Ob I}Y_i\\times X_i\\right)\\]\nThe maps are the standard maps of the Bousfield-Kan formula, see e.g. \\cite[18.3.2]{hirsch}. \nThe $G$-action on the target of the maps sends $(y,x)$ in $Y_i\\times X_i$ to $(\\phi_g(y),\\phi_g(x))$ in $Y_{gi}\\times X_{gi}$. The action on the source space is defined by a similar indexed coproduct.\n\n\\begin{defn}\nLet $I$ be a category with $G$-action and $X\\in \\Top_{a}^I$ a $G$-diagram of spaces. The $G$-homotopy limit and the $G$-homotopy colimit of $X$ are the $G$-spaces defined respectively by\n\\[\\holim_I X=\\Hom_I\\big(B(I\/-),X\\big)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\hocolim_I X=X\\otimes_I\\big( B(-\/I)^{op}\\big) \\]\n\\end{defn}\n\nThe underlying space of the $G$-homotopy limit is the homotopy limit of the underlying diagram, defined via the Bousfield-Kan formula, and dually for the $G$-homotopy colimit.\n\nWe recall that an equivariant map of $G$-spaces $f\\colon X\\to Y$ is an equivalence (in the fixed points model structure) if its restriction to the $H$-fixed points $f\\colon X^H\\to Y^H$ is a weak homotopy equivalence of spaces for every subgroup $H$ of $G$.\nThe vertex $X_i$ of a $G$-diagram $X$ inherits an action of the stabilizer group $G_i$ of the object $i\\in I$, defined by the maps $\\phi_g\\colon X_i\\to X_{gi}=X_i$.\n\n\n\\begin{defn}[\\cite{vilf}]\\label{defeqGdiag}\nA morphism of $G$-diagrams $f\\colon X\\to Y$ is a weak equivalence if for every object $i$ of $I$ the map $f_i\\colon X_i\\to Y_i$ is an equivalence of $G_i$-spaces.\n\\end{defn}\n\nThe homotopical properties of the $G$-homotopy limit and of the $G$-homotopy colimit were studied extensively in \\cite{Gdiags}. In particular both constructions send equivalences of $G$-diagrams to equivalences of $G$-spaces.\n\n\\begin{rem}\\label{holimGrot} Let $G\\ltimes_a I$ be the Grothendieck construction of the functor $a\\colon G\\to Cat$ which defines the $G$-action on $I$.\nA $G$-structure on a diagram $X\\colon I\\to \\Top$ is equivalent to an extension of $X$ to $G\\ltimes_a I$ along the projection map $G\\ltimes_a I\\to I$. This results into an isomorphism of categories $\\Top_{a}^I\\cong \\Top^{G\\ltimes_a I}$ (see \\cite[1.9]{Gdiags}). There is a relationship between the $G$-homotopy limit of a $G$-diagram $X\\in \\Top_{a}^I$, and the homotopy limit of the corresponding diagram $\\overline{X}\\colon G\\ltimes_a I\\to Top$. The latter computes the homotopy fixed points of the former:\n\\[\\holim_{G\\ltimes_a I}\\overline{X}\\simeq\\big(\\holim_I X\\big)^{hG}\\]\nDually, the homotopy colimit of $\\overline{X}$ is described by the homotopy orbits\n\\[\\hocolim_{G\\ltimes_a I}\\overline{X}\\simeq\\big(\\hocolim_I X\\big)_{hG}\\]\nThese equivalences are an immediate consequence of the Fubini Theorems \\cite[26.5]{CS}.\nHomotopy orbits and homotopy fixed points are homotopy invariant with respect to na\\\"{i}ve equivalences of $G$-spaces ($G$-maps whose underlying map is an equivalence of spaces). We see from the formula above that $\\holim_{G\\ltimes_a I}\\overline{X}$ and $\\hocolim_{G\\ltimes_a I}\\overline{X}$ are invariant for the pointwise na\\\"{i}ve equivalences of $G$-diagrams.\nThus the $G$-homotopy limit and the $G$-homotopy colimit retain more equivariant information than the homotopy limit and homotopy colimit of $\\overline{X}$. The categorical (in opposition to homotopy) fixed points of the $G$-homotopy limits and colimits are the focus of the next two sections.\n\\end{rem}\n\nThere is a notion of fixed points of a $G$-diagram. Let $H$ be a subgroup of $G$, and let $I^H$ be the subcategory of $I$ of objects and morphisms that are fixed (strictly) by the $H$-action. Equivalently, $I^H$ is the limit of the functor $H\\to G\\stackrel{a}{\\to}Cat$. Since the $i$-vertex of a $G$-diagram $X\\in \\Top^{I}_a$ has an action of $G_i$, if $i$ belongs to $I^H$ the space $X_i$ has an $H$-action.\n\n\\begin{defn}\\label{deffixeddiag}\nLet $H$ be a subgroup of $G$.\nThe $H$-fixed points diagram of a $G$-diagram $X\\in \\Top^{I}_a$ is the diagram of spaces $X^H\\colon I^H\\to \\Top$ of pointwise fixed points $(X^H)_i=(X_i)^H$. Given a map $\\alpha\\colon i\\to j$ in $I^H$, the corresponding map $X_{i}^H\\to X_{j}^H$ is the restriction of $\\alpha_\\ast\\colon X_i\\to X_j$ on $H$-fixed points.\n\\end{defn}\n\nThe diagram $X^H\\colon I^H\\to \\Top$ is well defined on morphism because for every $h$ in $H$ the diagram\n\\[\\xymatrix@R=15pt{X_{i}\\ar[r]^-{\\alpha_\\ast}\\ar[d]_-{\\phi_h}&X_j\\ar[d]^-{\\phi_h}\\\\\nX_{hi}\\ar[r]_-{(h\\alpha)_\\ast}&X_{hj}\n}\\]\ncommutes by naturality of $\\phi_h$. Hence if $\\alpha$ is a morphism of $I^H$, the map $\\alpha_\\ast\\colon X_i\\to X_j$ is $H$-equivariant and it can be restricted on $H$-fixed points.\n\n\n\\begin{rem}\nIf $f\\colon X\\to Y$ is a morphism of $G$-diagrams, the map $f_i\\colon X_i\\to Y_i$ is $G_i$-equivariant for every object $i$ of $I$. Therefore $f$ restricts to a natural transformation $f^H\\colon X^{H}\\to Y^H$ for every subgroup $H$ of $G$. It is immediate from Definition \\ref{defeqGdiag} that $f$ is an equivalence of $G$-diagrams if and only if $f^H\\colon X^{H}\\to Y^H$ is an equivalence of $I^H$-shaped diagrams of spaces, for every subgroup $H$ of $G$. In this sense, the fixed points diagrams of a $G$-diagram $X$ retain all the homotopical information of $X$.\n\\end{rem}\n\n\\subsection{Equivariant homotopy colimits and fixed points}\n\nLet $G$ be a discrete group, and let $a\\colon G\\rightarrow Cat$ be a $G$-action on a category $I=a(\\ast)$. We study the interaction between the $G$-homotopy colimit of a $G$-diagram and its fixed points diagrams, as defined in \\ref{deffixeddiag}.\n\n\\begin{prop}\\label{resmapF}\nLet $X\\in \\Top^{I}_a$ and $K\\in \\Top^{I^{op}}_{a^{op}}$ be two $G$-diagrams of spaces. There is a natural homeomorphism\n\\[(X\\otimes_{I} K)^G\\cong X^G\\otimes_{I^G} K^G\\]\nIn particular for $K=B(-\/I)^{op}$ this gives a natural homeomorphism\n\\[(\\hocolim_IX)^G\\cong \\hocolim_{I^G}X^G\\]\n\\end{prop}\n\n\\begin{proof}\nSince fixed points commute with colimits, there is a canonical homeomorphism\n\\[(X\\otimes_I K)^G\\cong\\colim\\left((\\coprod_{\\substack{\\alpha\\colon i\\rightarrow j\\\\\n\\in \\hom I}}K_i\\times X_j)^G\\rightrightarrows (\\coprod_{i\\in \\Ob I}K_i\\times X_i)^G\\right)\\]\nFixed points commute with $G$-coproducts, in the sense that the above is homeomorphic to\n\\[(X\\otimes_I K)^G\\cong\\colim\\left(\\coprod_{\\substack{\\alpha\\colon i\\rightarrow j\\\\\n\\in \\hom I^G}}(K_i\\times X_j)^G\\rightrightarrows \\coprod_{i\\in Ob I^G}(K_i\\times X_i)^G\\right)\\]\nFinally, commuting the fixed points and the products we obtain an isomorphism between the equalizer above and $X^G\\otimes_{I^G} K^G$.\n\nWhen $K=B(-\/I^{op})$, the formula above gives a homeomorphism\n\\[(\\hocolim_IX)^G\\cong X^G\\otimes_{I^G} \\big(B(-\/I^{op})\\big)^G\\]\nFor every object $i$ of $I^G$ there are natural homeomorphisms\n\\[\\big(B(i\/I^{op})\\big)^G\\cong B\\big((i\/I^{op})^G\\big)\\cong B\\big(i\/(I^{G})^{op}\\big)\\]\nthat identify $X^G\\otimes_{I^G} \\big(B(-\/I^{op})\\big)^G$ with $\\hocolim_{I^G}X^G$.\n\\end{proof}\n\n\\begin{rem}\nThe description of the fixed points of the $G$-homotopy colimit in terms of fixed points diagrams given in \\ref{resmapF} makes it possible to deduce virtually all the equivariant homotopical properties of the $G$-homotopy colimit functor from the classical homotopical properties of the homotopy colimit. For example, it follows immediately from \\ref{resmapF} that the $G$-homotopy colimits of equivalent $G$-diagrams are equivalent as $G$-spaces, recovering a result of \\cite[6.1]{vilf}. Another example is the equivariant Thomason's Theorem \\ref{GThom} below.\nWe will see in \\S\\ref{fixholim} that the relationship between $G$-homotopy limits and fixed points is more involved. \n\\end{rem}\n\nAn immediate application of Proposition \\ref{resmapF} is the equivariant version of Thomason's Theorem. Let $X\\colon I\\to Cat$ be a $G$-diagram of categories. The Grothendieck construction $I\\wr X$ has an induced $G$-action, defined on objects by\n\\[g\\cdot \\big(i\\in I\\ ,\\ x\\in\\Ob X_i\\big)=\\big(gi\\ ,\\ gx\\in X-{gi}\\big)\\]\nand on morphisms by\n\\[g\\cdot \\big( i\\stackrel{\\alpha}{\\to} j\\ ,\\  \\alpha_{\\ast}x\\stackrel{\\gamma}{\\to} y\\big)=\\big( gi\\stackrel{g\\alpha}{\\longrightarrow} gj\\ ,\\  (g\\alpha)_{\\ast}(gx)=g(\\alpha_\\ast x)\\stackrel{g\\gamma}{\\longrightarrow} gy\\big)\\]\nThe equality $(g\\alpha)_{\\ast}(gx)=g(\\alpha_\\ast x)$ expresses the naturality of $g\\colon X\\to X\\circ g$. Thus the classifying space $B(I\\wr X)$ inherits a $G$-action.\nThe following result was proved in \\cite[2.27]{Gdiags} as a special case of a general equivariant Fubini Theorem. The proof presented here is more direct, reducing it to Thomason's Theorem \\cite{Thomason} by a fixed points argument.\n\n\\begin{cor}[\\cite{Gdiags}]\\label{GThom}\nLet $X\\colon I\\to Cat$ be a $G$-diagram of small categories, and let $BX$ be the $G$-diagram of spaces obtained by composing with the geometric realization. There is a natural equivalence of $G$-spaces\n\\[B(I\\wr X)\\stackrel{\\simeq}{\\longrightarrow}\\hocolim_IBX\\]\n\\end{cor}\n\n\\begin{proof}\nThe map $\\eta_X\\colon B(I\\wr X)\\to\\hocolim_IBX$ defined in \\cite{Thomason} is equivariant. For a subgroup $H$ of $G$, there is a natural isomorphism of categories $(I\\wr X)^H\\cong I^H\\wr X^H$. Under this isomorphism and the homeomorphism of \\ref{resmapF}, the map $\\eta_{X}^H$ corresponds to the map\n\\[\\eta_{X^H}\\colon I^H\\wr X^H\\longrightarrow \\hocolim_{I^H}B(X^H)\\]\nwhich is an equivalence by Thomason's Theorem \\cite{Thomason}.  \n\\end{proof}\n\n\n\\subsection{Equivariant homotopy limits and fixed points}\\label{fixholim}\n\nThe relationship between the fixed points of the $G$-homotopy limit and the homotopy limit of the fixed points diagrams is more involved than it is for $G$-homotopy colimits. Given a pair of $G$-diagrams of spaces  $K,X\\in \\Top^{I}_a$ there is a restriction map\n\\[\\Hom_I(K,X)^G\\longrightarrow \\Hom_{I^G}(K^G,X^G)\\]\nwhich sends a morphism of $G$-diagrams to its restriction on the fixed points diagrams. This is generally far from being an equivalence.\nWhen $K=B(I\/_{-})$ we describe the homotopy fibers of the restriction map $(\\holim_{I}X)^G\\rightarrow \\holim_{I^G}X^G$ in Proposition \\ref{resfixespts} below. It turns out that in order to describe the whole fixed points space $\\Hom_I(K,X)^G$ one needs to consider the natural transformations between the $H$-fixed points diagrams of $K$ and $X$, for every subgroup $H$ of $G$. This mapping spaces for the various subgroups of $G$ are related by means of the twisted arrow category.\n\nLet us recall from \\cite{DK} that the twisted arrow category $Tw(C)$ of a category $C$ has objects the morphisms $f\\colon c\\to d$ in $C$. A morphisms $f\\to f'$ in $Tw(C)$ is a commutative diagram\n\\[\\xymatrix@=13pt{c\\ar[r]^-f\\ar[d]& d\\\\\nc'\\ar[r]_-{f'}&d'\\ar[u]}\\]\nLet  $\\mathcal{O}_G$ be the orbit category of $G$. We use the twisted arrow category $Tw(\\mathcal{O}^{op}_G)$ to glue together the mapping spaces between the fixed point diagrams of $K$ and $X$. An object of $Tw(\\mathcal{O}^{op}_G)$ is an equivariant map $G\/H\\stackrel{f}{\\leftarrow} G\/L$. This induces a functor $f^{\\ast}\\colon I^H\\to I^L$, defined on objects by\n\\[f^{\\ast}i=f(L)\\cdot i\\]\nA similar formula defines $f^\\ast$ on morphisms. Precomposing a diagram with $f^{\\ast}$ leads to a functor $f_!\\colon \\Top^{I^L}\\to \\Top^{I^H}$. Given two $G$-diagrams of spaces $K,X\\in Top^{I}_{a}$ define a functor\n\\[Tw(\\mathcal{O}^{op}_G)^{op}\\longrightarrow \\Top\\]\nby sending an object $G\/L\\stackrel{f}{\\leftarrow} G\/H$ of $Tw(\\mathcal{O}^{op}_G)$ to the space of natural transformations of $I^H$-diagrams $\\Hom_{I^H}(K^H,f_! X^L)$. On morphisms this functor is defined as\n\\[\\vcenter{\\hbox{\\xymatrix{G\/H&\\ar[l]_-f G\/L\\ar[d]_{b}\\\\\nG\/H'\\ar[u]^a&\\ar[l]^-{f'}G\/L'}}} \\longmapsto\n\\vcenter{\\hbox{\\xymatrix@C=20pt{\\Hom_{I^H}(K^H,f_! X^L)&&\\Hom_{I^H}(a_! K^{H'},a_! f'_! b_{\\ast} X^{L'})\\ar[ll]_-{(-)\\circ a(H')}\\\\\n\\Hom_{I^{H'}}(K^{H'},f'_! X^{L'})\\ar@{-->}[u]\\ar[rr]_-{\\res_{a^{\\ast}}}\n&&\\Hom_{I^H}(a_! K^{H'},a_! f'_! X^{L'})\\ar[u]_{b(L)\\circ (-)}\n}}}\\]\nIn the right-hand square, the lower horizontal map restricts a natural transformation along the functor $a^{\\ast}\\colon I^H\\to I^{H'}$. The top horizontal map is precomposition with pointwise multiplication by $a(H')$ in the $G$-structure on $K$, in symbols $a(H')\\colon K^{H}_i\\to K^{H}_{a(H)\\cdot i}= (a_\\ast K^H)_i$. Similarly, the right vertical map composes a natural transformation with the action of a representative of $b(L)$ for the $G$-structure of $X$. Explicitly, a natural transformation $\\Phi\\colon K^{H'}\\to f'_\\ast X^{L'}$ is sent to\n\\[\\xymatrix@C=40pt{K^{H}_i\\ar[r]^-{a(H')}& K^{H'}_{a(H')\\cdot i}\\ar[r]^-{\\Phi_{a(H')i}}&X^{L'}_{f'(L')a(H')i}\\ar[r]^-{b(L)}&X^{L}_{b(L)f'(L')a(H')i}=X^{L}_{f(L)i}\n}\\]\n\n\\begin{prop}\\label{twisted}\nFor every pair of $G$-diagrams $K,X$ in $\\Top^{I}_{a}$, there is a natural homeomorphism\n\\[\\Hom_I(K,X)^G\\cong \\lim_{\\substack{f\\colon G\/L\\to G\/H \\\\\n\\in Tw(\\mathcal{O}^{op}_G)^{op}}}\\Hom_{I^H}(K^H,f_{!}X^L)\\]\nIn particular when $K=B(I\/-)$ this is a homeomorphism\n\\[(\\holim_I X)^G\\cong \\lim_{\\substack{f\\colon G\/L\\to G\/H \\\\\n\\in Tw(\\mathcal{O}^{op}_G)^{op}}}\n\\holim_{I^H}f_{!}X^L\\]\n\\end{prop}\n\n\\begin{proof}\nIn trying to dualize the argument of the proof of \\ref{resmapF}, one encounters the problem that $G$-indexed products do not commute with fixed points. Instead, we express the fixed points of the mapping space as a mapping space on a more complicated category. The functor $a\\colon G\\to Cat$ which defines the $G$-action on $I$ induces a functor $\\overline{a}\\colon \\mathcal{O}^{op}_G\\to Cat$ that sends $G\/H$ to the fixed points category $I^H$, and a $G$-map $G\/H\\stackrel{f}{\\leftarrow} G\/L$ to the functor $f^\\ast\\colon I^H\\to I^L$ described above. Recall the isomorphism of categories $\\Top^{I}_{a}\\cong \\Top^{G\\ltimes_a I}$ of \\ref{holimGrot}, where $G\\ltimes_a I$ is the Grothendieck construction $G\\wr a$. The canonical inclusion $G\\to \\mathcal{O}^{op}_G$ induces a functor $\\Top^{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\to \\Top^{G\\wr a}\\cong\\Top^{I}_{a}$. Theorem \\cite[2.28]{Gdiags} shows that this functor is the left adjoint of a Quillen equivalence\n\\[L\\colon \\Top^{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\rightleftarrows \\Top^{G\\wr a}\\cong\\Top^{I}_{a} \\colon R\\]\nwhere the left-hand category has the projective model structure, and the right-hand category has a model structure where the equivalences are the equivalences of $G$-diagrams of \\ref{defeqGdiag}. This is a sort of Elmendorf Theorem for the category of $G$-diagrams. The functor $R$ sends a $G$-diagram $X$ to the diagram with vertices\n\\[R(X)_{(G\/H,i\\in I^H)}=X^{H}_i\\]\nThe counit of this adjunction is an isomorphism, giving a homeomorphism\n\\[\\Hom(K,X)^G\\cong \\Hom_{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\big(R(K),R(X)\\big)\\]\n(see \\cite[2.28]{Gdiags}). We finish the proof by defining a homeomorphism between the right-hand side and the limit of \\ref{twisted}. An element of  $\\Hom_{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\big(R(K),R(X)\\big)$ is the data of a map $\\Phi_{(H,i)}\\colon K^{H}_i\\to X^{H}_i$ for every subgroup $H$ of $G$ and fixed object $i\\in I^H$, subject to compatibility conditions corresponding to the morphisms of $\\mathcal{O}^{op}_G\\wr \\overline{a}$. For every equivariant map $G\/H\\stackrel{f}{\\leftarrow} G\/L$ define a map\n\\[\\Hom_{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\big(R(K),R(X)\\big)\\longrightarrow \\Hom_{I^H}(K^H,f_{!}X^L)\\]\nby sending a collection of maps $\\Phi$ as above to\n\\[K^{H}_i\\stackrel{\\Phi_{(H,i)}}{\\xrightarrow{\\hspace{1cm}}} X^{H}_i\\stackrel{f(L)}{\\xrightarrow{\\hspace{1cm}}} X^{L}_{f(L)\\cdot i}=(f_! X^L)_i\\] \nNaturality of $\\Phi$ insures that these maps are compatible with the morphisms in $Tw(\\mathcal{O}^{op}_G)^{op}$, and this defines a map\n\\[\\Hom_{\\mathcal{O}^{op}_G\\wr \\overline{a}}\\big(R(K),R(X)\\big)\\longrightarrow \\lim_{f\\in Tw(\\mathcal{O}^{op}_G)^{op}}\\Hom_{I^H}(K^H,f_{!}X^L)\\]\nThe inverse of this map sends a collection of natural transformations $\\Psi_f\\colon K^H\\to f_{\\ast}X^L$ in the limit to\n\\[\\Psi_{\\id_{G\/H}}\\colon K^{H}_i{\\longrightarrow} X^{H}_i\\]\n\\end{proof}\n\n\\begin{rem}\nThe limit over the Twisted arrow category of \\ref{twisted} is in general not a homotopy limit, and hence it is not homotopy invariant. This prevents us from easily deduce homotopical properties of the equivariant mapping space $\\Hom_I(K,X)$ from the homotopical properties of the fixed points diagrams. For example, proving that the $G$-homotopy limit functor preserves equivalences requires a considerable amount of work. The equivariant mapping space $\\Hom_I(K,X)$ does however enjoy many equivariant homotopical properties, which are studied extensively in \\cite{Gdiags} using a model-categorical approach.\n\\end{rem}\n\nWe turn our attention to the connectivity of the restriction map  $(\\holim_{I}X)^G\\rightarrow \\holim_{I^G}X^G$ which will play a key role in the proof of the equivariant Blakers-Massey Theorem \\ref{BM}.\nLet $\\iota\\colon I^G\\rightarrow I$ be the inclusion of the fixed points category. This is a $G$-equivariant functor for the trivial $G$-action on $I^G$. Therefore  the functor $B(\\iota\/-)\\colon I\\to\\Top$ has a natural $G$-structure (see Example \\ref{overcatGdiag}). There is a pointwise injective morphism $B(\\iota\/-)\\rightarrow B(I\/-)$ in $\\Top_{a}^I$, and we denote its pointwise quotient $\\bigslant{B(I\/-)}{B(\\iota\/-)}$. This is indeed a $G$-diagram since $\\Top_{a}^I$ has all colimits. Moreover it is canonically a diagram of based spaces, meaning that there is a canonical natural transformation $\\ast\\rightarrow \\bigslant{B(I\/-)}{B(\\iota\/-)}$.\n\n\\begin{prop}\\label{resfixespts} Let $X\\colon I\\to \\Top$ be a $G$-diagram of spaces and suppose that the simplicial set $NI\/_i$ is finite dimensional for every object $i$ of $I$. Then the restriction map $(\\holim_{I}X)^G\\rightarrow \\holim_{I^G}X^G$ is a fibration. Given a natural transformation $\\ast\\rightarrow X$ in $\\Top_{a}^I$, the fiber of the restriction map over the associated constant natural transformation $B(I^G\/-)\\rightarrow \\ast\\stackrel{}{\\rightarrow} X^G$ is the fixed points space\n\\[\\Hom_\\ast\\Big(\\bigslant{B(I\/-)}{B(\\iota\/-)},X\\Big)^G\\]\nof the space of natural transformations of pointed maps.\n\\end{prop}\n\n\\begin{proof} Since the $G$-action on $I^G$ is trivial, there is a natural homeomorphism\n$\\Hom(B(I^G\/-),X^G)\\cong \\Hom(B(I^G\/-),X|_{I^G})^G$ and the restriction map fits into a commutative diagram\n\\[\\xymatrix@C=50pt{\\Hom_I\\big(B(I\/-),X\\big)^G\\ar[r]^-{res}\\ar[dr]&\n\\Hom_{I^G}\\big(B(I^G\/-),X|_{I^G}\\big)^G\\\\\n&\\Hom_I\\big(B(\\iota\/-),X\\big)^G\\ar[u]_{\\cong}\n}\\]\nThe vertical map is the restriction on $G$-fixed points of an adjunction isomorphism of the kind $\\Hom_D\\big(B(F\/-),Z\\big)\\cong \\Hom_C\\big(B(C\/-),F^{\\ast}Z\\big)$ for functors $C\\stackrel{F}{\\rightarrow }D\\stackrel{Z}{\\rightarrow } \\Top$ (see e.g. \\cite[19.6.6]{hirsch}). We show that the diagonal map $\\Hom_I\\big(B(I\/-),X\\big)\\rightarrow\n\\Hom_I\\big(B(\\iota\/-),X\\big)$ induced by the inclusion $B(\\iota\/-)\\rightarrow B(I\/-)$ is a fibration in $\\Top^G$, and that the point-fiber is the space of natural transformations of \\ref{resfixespts}.\n\nThe diagonal map is a fibration provided $B(\\iota\/-)\\rightarrow B(I\/-)$ is a cofibration in the $G$-projective model structure of $\\Top_{a}^I$, as defined in \\cite[2.6]{Gdiags}. The following argument is analogous to \\cite[18.4.1]{hirsch}.\nThe map $B(\\iota\/-)\\rightarrow B(I\/-)$ is a cofibration if we can solve the lifting problem\n\\[\\xymatrix@=20pt{B(\\iota\/-)\\ar[r]\\ar[d]&E\\ar@{->>}[d]^{\\simeq}\\\\\nB(I\/-)\\ar[r]\\ar@{-->}[ur]_-l&B\n}\\]\nfor every acyclic fibration $E\\rightarrow B$ in $\\Top_{a}^I$. This is a morphism of $G$-diagrams with the property that $E_i\\rightarrow B_i$ is an acyclic fibration of $G_i$-spaces for every object $i$ of $I$. For every $G$-diagram $X$ in $\\Top_{a}^I$ and every object $i$ of $I$ let $L_iX$ be the $G_i$-space\n\\[L_iX=\\colim\\limits_{(j\\rightarrow i)\\neq \\id_i}X_j\\]\nIf we can prove that the relative latching maps \n\\[B(\\iota\/i)\\coprod\\limits_{L_iB(\\iota\/-)}\nL_iB(I\/-)\\longrightarrow B(I\/i)\\]\nare cofibrations of $G_i$-spaces for every $i$, one can build a lift $l$ inductively on the filtration of $I$ defined by the degree function $\\deg=\\dim N(I\/-)\\colon Ob I\\rightarrow\\mathbb{N}$. The construction of $l$  is completely analogous to \\cite[A.6]{Gdiags}. Since the geometric realization functor preserves cofibrations and it commutes with colimits, it is enough to prove that\n\\[N(\\iota\/i)\\coprod\\limits_{L_iN(\\iota\/-)}\nL_iN(I\/-)\\longrightarrow N(I\/i)\\]\nis a cofibration of simplicial $G_i$-sets. A cofibration of simplicial $G_i$-sets is a $G_i$-equivariant map that is levelwise injective,\nthat is a $G_i$-equivariant maps which is a cofibrations of underlying simplicial sets.\nThe maps $L_iN(I\/-)\\rightarrow N(I\/i)$ and $L_iN(\\iota\/-)\\rightarrow N(\\iota\/i)$ are cofibrations of simplicial sets as both $N(I\/-)$ and $N(\\iota\/-)$ are (Reedy) cofibrant diagrams of simplicial sets (see e.g. \\cite[14.8.5]{hirsch}).\nIt follows that the $G_i$-equivariant map from the pushout\n\\[\\xymatrix@R=10pt@C=50pt{L_iN(\\iota\/-)\\ \\ar[d]\\ar@{>->}[r]&N(\\iota\/i)\\ar[d]\\ar@\/^1pc\/[ddr]\\\\\nL_iN(I\/-)\\ \\ar@{>->}[r]\\ar@{>->}@\/_1pc\/[drr]&\nN(\\iota\/i)\\coprod\\limits_{L_iN(\\iota\/-)}\\ar@{-->}[dr]\nL_iN(I\/-)\\\\\n&& N(I\/i)\n}\\]\nis also a cofibration of simplicial sets.\nThis finishes the proof that the restriction map is a fibration.\n\nWe still need to describe the fiber of the restriction map. Since fibers commute with fixed points, equalizers and products, the fiber of the restriction map $\\Hom_I(B(I\/-),X)^G\\rightarrow\\Hom_I(B(\\iota\/-),X)^G$ over $ B(\\iota\/-)\\rightarrow\\ast\\stackrel{x}{\\rightarrow} X$ is the equalizer\n\\[\\lim\\bigg(\\prod_{i\\in \\Ob I}\\fib\\left(\n\\vcenter{\\hbox{\\xymatrix{\\map(B(I\/i), X_i)\\ar[d]\\\\ \\map(B(\\iota\/i), X_i)}}}\n\\right)\\rightrightarrows \\prod_{\\substack{\\alpha\\colon i\\rightarrow j\\\\\n\\in \\hom I}}\\fib\\left(\n\\vcenter{\\hbox{\\xymatrix{\\map(B(I\/i), X_j)\\ar[d]\\\\ \\map(B(\\iota\/i), X_j)}}}\n\\right)\\bigg)^G\\]\nwhere the fibers are taken over the constant maps \n$B(\\iota\/i)\\rightarrow\\ast\\stackrel{x_i}{\\rightarrow} X_i$ and $B(\\iota\/i)\\rightarrow\\ast\\stackrel{x_j}{\\rightarrow} X_j$ respectively. These fibers are naturally isomorphic to the spaces of pointed maps from the quotients, giving a homeomorphism between the equalizer above and\n\\[\\lim\\bigg(\\prod_{i\\in \\Ob I}\\map_\\ast\\left(\\bigslant{B(I\/i)}{B(\\iota\/i)},X_i\\right)\\rightrightarrows \\prod_{\\substack{\\alpha\\colon i\\rightarrow j\\\\\n\\in \\hom I}}\\map_\\ast\\left(\\bigslant{B(I\/i)}{B(\\iota\/i)},X_j\\right)\\bigg)^G\\]\nThis is precisely the mapping space of the statement.\n\\end{proof}\n\n\n\\begin{cor}\\label{connresmap}\nLet $X\\colon I\\to \\Top$ be a $G$-diagram of spaces. Suppose that the simplicial sets $NI\/_i$ are finite dimensional, and that the composition of the canonical maps \\[(\\lim_{I}X)^G\\to(\\holim_{I}X)^G\\to\n\\holim_{I^G}X^G\\]\nis surjective in $\\pi_0$. Then the restriction map $(\\holim_{I}X)^G\\rightarrow \\holim_{I^G}X^G$ is at least\n\\[\\min_{i\\in Ob I}\\min_{\\substack{H\\leq G_i\\\\ \\hom (\\iota^H\/{i})\\subsetneqq \\hom (I^H\/{i})}\n}\\left(\\conn X_{i}^H-\\dim N(I^H\/i)\\right)+1\\]\nconnected, where $\\iota^H\\colon I^G\\rightarrow I^H$ is the inclusion.\n\\end{cor}\n\\begin{rem}\nThe $\\pi_0$ hypothesis of Corollary \\ref{connresmap} is satisfied if for every fixed object $i$ of $I^G$ the connectivity of $X^G_{i}$ is greater or equal to the dimension of $NI\/_{i}^G$, since in this case $\\holim_{I^G}X^G$ is connected (see \\ref{connfixedptsholim}).\n\\end{rem}\n\n\\begin{proof}[Proof of\\ref{connresmap}]\nLet us write $K$ for the quotient of $B(I\/-)$ by the subdiagram $B(\\iota\/-)$. It is cofibrant as the inclusion is a cofibration (cf. \\ref{resfixespts}). The $\\pi_0$ assumption insures that all the homotopy fibers are a space of natural transformation from $K$ to $X$ for some basepoint $\\ast\\rightarrow X$, as in \\ref{resfixespts}. We show in \\ref{connfixedptsholim} that this space of natural transformations is at least\n\\[\\min_{i\\in Ob I}\\min_{\\substack{H\\leq G_i\\\\ K^{H}_i\\neq\\ast}}\\left(\\conn X_{i}^H-\\dim K_{i}^H\\right)\\]\nconnected. The space $K^{H}_i$ is a point precisely when $B(I^H\/{i})=B(\\iota^H\/{i})$. This is the case if the equality is verified in nerve level one, that is when $\\hom (I^H\/{i})=\\hom(\\iota^H\/i)$.\n\\end{proof}\n\n\n\n\n\n\\section{The equivariant Blakers-Massey Theorem and applications}\n\n\\subsection{The equivariant Blakers-Massey Theorem}\\label{secBM}\n\nLet $G$ be a discrete group and let $J$ be a finite $G$-set. We write $\\mathcal{P}(J)$ for the poset category of subsets of $J$ ordered by inclusion. The $G$-action on $J$ induces a $G$-action on the category $\\mathcal{P}(J)$, by sending a subset $U$ of $J$ to its image $g(U)$ under the map $g\\colon J\\to J$. This action restricts to the subposets \n\\[\\mathcal{P}_0(J)=\\mathcal{P}(J)\\backslash\\emptyset\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\mathcal{P}_1(J)=\\mathcal{P}(J)\\backslash J\\]\nA $J$-cube of spaces is a $G$-diagram $X\\colon \\mathcal{P}(J)\\to\\Top$. The initial and final vertices $X_\\emptyset$ and $X_J$ have a $G$-action induced by the $G$-structure of $X$, since $\\emptyset$ and $J$ are fixed objects of $\\mathcal{P}(J)$. Moreover the canonical maps\n\\[X_{\\emptyset}\\rightarrow \\holim_{\\mathcal{P}_0(J)}X\n\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\\n\\hocolim_{\\mathcal{P}_1(J)}X\\rightarrow X_{J}\\]\nare $G$-equivariant with respect on the $G$-actions on the homotopy limit and on the homotopy colimit of \\S\\ref{secGdiags}.\n\n\\begin{defn}\nLet $J$ be a finite $G$-set and let $X\\in\\Top^{\\mathcal{P}(J)}_a$ be a $J$-cube. Given a function $\\nu\\colon \\{H\\leq G\\}\\rightarrow \\mathbb{N}$ which is invariant on conjugacy classes, we say that $X$ is $\\nu$-cartesian if for every subgroup $H$ of $G$ the map of spaces\n\\[X_{\\emptyset}^H\\longrightarrow (\\holim_{\\mathcal{P}_0(J)}X)^H\\]\nis $\\nu(H)$-connected. We say that $X$ is homotopy cartesian if it is $\\nu$-cartesian for every function $\\nu$. Dually, we say that $X$ is $\\nu$-cocartesian if\n\\[(\\hocolim_{\\mathcal{P}_1(J)}X)^H\\longrightarrow X_{J}^H\\]\nis $\\nu(H)$-connected for every subgroup $H$ of $G$, and that $X$ is homotopy cocartesian if it is $\\nu$-cocartesian for every $\\nu$.\n\\end{defn}\n\n\\begin{rem}\nThere is a natural isomorphism of categories $\\mathcal{P}(J)^H\\cong \\mathcal{P}(J\/H)$. Under this isomorphism the fixed points diagram $X^H\\colon \\mathcal{P}(J)^H\\to \\Top$ of a $J$-cube $X$ is a $J\/H$-cube. By the fixed points description of the $G$-homotopy colimit of Proposition \\ref{resmapF}, a $J$-cube $X$ is $\\nu$-cocartesian precisely when the $J\/H$-cube $X^H$ is $\\nu(H)$-cocartesian for every subgroup $H$ of $G$. Because of the failure of $G$-homotopy limit to commute with fixed points, the analogous statement does not hold for cartesian $J$-cubes (cf. \\ref{resfixespts}).\n\\end{rem}\n\n\nGiven a finite $G$-set $J$ and a subgroup $H$ of $G$ we denote by $\\Pa_H(J)$ the set of partitions of $J$ by $H$-subsets. This is the set of coverings $\\{\\emptyset\\neq T_\\alpha\\subset J\\}_{\\alpha}$ of $J$ by non-empty subsets to which the $H$-action on $J$ restricts, and such that $T_\\alpha$ and $T_{\\alpha'}$ are disjoint for $\\alpha\\neq \\alpha'$. For a subset $U\\subset J$ let $\\Eff_H(U)$ be the set of subgroups $L\\leq H$ ``that act effectively on $U$'', defined by\n\\[\\Eff_H(U)=\\left\\{\\begin{array}{ll}\n\\{L\\leq H_U\\}&\\mbox{if } H_U\\neq H\\\\\n\\\\\n\\{L\\leq H \\ |\\ U\/L \\neq U\/H\\}&\\mbox{if } H_U= H\n\\end{array}\\right.\\]\nHere $H_U$ denotes the stabilizer group in $H$ of the object $U$ of $\\mathcal{P}(J)$. This is the largest subgroup of $H$ whose action on $J$ restricts to $U$.\nThe important point of this definition is that the elements of $\\Eff_H(U)$ are all proper subgroups of $H$.\n\n\n\\begin{theorem}\\label{BM}\nLet $X\\in Top^{\\mathcal{P}(J)}_a$ be a $J$-cube, and suppose that for every subgroup $H$ of $G$ and every non-empty $H$-subset $U\\subset J$ the restriction $X|_{\\mathcal{P}(U)}$ is $\\nu^U$-cocartesian for some function $\\nu^U\\colon \\{K\\leq H\\}\\rightarrow \\mathbb{Z}$. Suppose moreover that these functions satisfy $\\nu^U\\leq \\nu^V$ whenever $U\\subset V$ are non-empty $H$-subsets of $J$. Then $X$ is $\\nu$-cartesian, where $\\nu$ is the function\n\\[\\nu(H)=\\!\\min\\left\\{\\!\\min_{\\{T_\\alpha\\}\\in\\Pa_H(J)}\\{\n\\sum_{\\alpha}\\nu^{T_\\alpha}(H)\\}-|J\/H|+1\\ ,\\ \\min_{\\emptyset\\neq U\\subset J}\\min_{L\\in \\Eff_H(U)}\\big\\{\\conn X^{L}_U-|U\/L|+1\\big\\}\n\\right\\}\\]\n\\end{theorem}\n\n\\begin{rem}\nThe first term of the minimum is analogous to the formula of Goodwillie's Blakers-Massey Theorem \\cite[2.5]{calcII} for $n$-cubes. The second term is purely equivariant: if the $G$-action on $J$ is trivial the set of subgroups $\\Eff_H(U)$ is empty. If $J$ is the set with $n$ elements and trivial $G$-action, this is the standard Blakers-Massey Theorem for $n$-cubes of $G$-spaces.  \n\\end{rem}\n\nThe Blakers-Massey Theorem \\ref{BM} has the following dual statement.\n\n\n\\begin{theorem}\\label{dualBM}\nLet $X\\in Top^{\\mathcal{P}(J)}_a$ be a $J$-cube, and suppose that for every subgroup $H$ of $G$ and for every non-empty $H$-subset $U\\subset J$ the cube $X|_{\\mathcal{P}(U)}$ is $\\nu^U$-cartesian. Suppose moreover that the functions $\\nu^U\\colon \\{K\\leq H\\}\\rightarrow \\mathbb{N}$ satisfy $\\nu^U\\leq \\nu^V$ whenever $U\\subset V$ are non-empty $H$-subsets of $J$. Then $X$ is $\\nu$-cocartesian, for\n\\[\\nu(H)\\!=\\!\\!\\!\\min_{\\{T_\\alpha\\}\\in\\Pa_H(J)}\\left\\{\n|J\/H|-1+\\sum_{\\alpha}\\min\\Big\\{\\nu^{T_\\alpha}(H)\\ , \\min_{\\emptyset\\neq U\\subset T_\\alpha}\\min_{L\\in \\Eff_H(U)}\\{\\conn X^{L}_U-|U\/L|+1\\}\\Big\\}\n\\!\\right\\}\\]\n\\end{theorem}\n\n\n\\begin{proof}[Proof of \\ref{BM}]\nFor each subgroup $H$ of $G$, we need to calculate the connectivity of the canonical map $\\phi\\colon X^{H}_\\emptyset\\rightarrow (\\holim_{\\mathcal{P}_0(J)} X)^H$ at any basepoint in $X^{H}_\\emptyset$. Since the empty set is initial in $\\mathcal{P}(J)^H$, a basepoint of $X^{H}_\\emptyset$ defines a basepoint $\\ast\\rightarrow X^H$ for the whole fixed points diagram. There is a commutative diagram\n\\[\\xymatrix@R=15pt@C=40pt{X^{H}_\\emptyset\\ar[r]^-\\phi\\ar[dr]_-\\psi&(\n\\displaystyle\\holim_{\\mathcal{P}_0(J)} X)^H\\ar[d]^-r\\\\\n&\\displaystyle\\holim_{\\mathcal{P}_0(J)^H} X^H\n}\n\\]\nwhere $r$ is the restriction map of Proposition \\ref{resfixespts}, and $\\psi$ is the canonical map for the fixed points diagram $X^H$.\nOur map is as connected as\n\\[\\conn\\phi=\\min\\{\\conn \\psi,\\conn r-1\\}\\]\nThe connectivity of $\\psi$ expresses how cartesian the $J\/H$-cube $X^H$ is, and it is determined by the standard Blakers-Massey Theorem of \\cite[2.5]{calcII}. In order to apply this theorem we need to compare the cocartesianity of the subcubes of $X^H$. Under the canonical isomorphism $\\mathcal{P}(J\/H)=\\mathcal{P}(J)^H$ a non-empty subset $U\\subset J\/H$ corresponds to a non-empty $H$-subsets of $J$. By assumption the cube $X^H$ is $\\nu^U(H)$-cocartesian, and for  subsets $U\\subset V \\subset J\/H$ the inequality $\\nu^U(H)\\leq \\nu^V(H)$ holds. By \\cite[2.5]{calcII} the cube $X^H$ is\n\\[\\min_{\\{T_\\alpha\\}}\\{\n\\sum_{\\alpha}\\nu^{T_\\alpha}(H)\\}-|J\/H|+1\\]\ncartesian,\nwhere $\\{T_\\alpha\\}$ runs over the partitions of $J\/H$, which correspond to the partitions of $J$ by $H$-subsets. This shows that $\\psi$ is as connected as the first term of the minimum of the statement.\n\nNow we calculate the connectivity of the restriction map $r$.\nWe can assume that $\\psi$ is at least $0$-connected, otherwise the connectivity range of \\ref{BM} gives us no information about the cartesianity of $X$. The $\\pi_0$-hypothesis of \\ref{connresmap} is satisfied, since there is a commutative diagram\n\\[\\xymatrix@R=15pt{\\pi_0X^{H}_\\emptyset\\ar[d]\n\\ar[drr]^-{\\pi_0\\psi}\n\\\\\n\\displaystyle\\pi_0(\\lim_{\\mathcal{P}_0(J)} X)^H\\ar[r]&\\displaystyle\\pi_0(\\holim_{\\mathcal{P}_0(J)} X)^H\\ar[r]_-r\n&\\displaystyle\\pi_0\\holim_{\\mathcal{P}_0(J)^H} X^H\n}\n\\]\nand the surjectivity of $\\pi_0\\psi$ implies the surjectivity of the lower composite. By \\ref{connresmap} the connectivity of $r$ is\n\\[\\min_{\\emptyset\\neq U\\subset J}\\min_{\\substack{L\\leq H_U\\\\ \\hom (\\iota^L\/U)\\subsetneqq \\hom \\mathcal{P}(U)^L}\n}\\left(\\conn X_{U}^L-|U\/L|+1\\right)+1\\]\nwhere $\\iota^L\\colon \\mathcal{P}(J)^H\\rightarrow  \\mathcal{P}(J)^L$ is the inclusion. It remains to show that if a subgroup $L$ of $H_U$ satisfies the condition $\\hom (\\iota^L\/U)\\subsetneqq \\hom \\mathcal{P}(U)^L$ , then $L$ belongs to $\\Eff_H(U)$ (in fact, the two conditions are equivalent). If $H_U$ is different from $H$ clearly $L$ belongs to $\\Eff_H(U)$ since it is a proper subgroup of $H$.\nIf $H_U=H$, the category $\\iota^L\/U$ is the full subcategory of $\\mathcal{P}(J)^H$ of objects that are subsets of $U$. This is the same as the full subcategory $\\mathcal{P}(U)^H$ of $\\mathcal{P}(U)^L$. These are different precisely when $\\mathcal{P}(U\/H)\\neq \\mathcal{P}(U\/L)$, that is when $U\/H\\neq U\/L$.\n\\end{proof}\n\n\\begin{proof}[Proof of \\ref{dualBM}]\nFor every subgroup $H$ of $G$ there is a commutative diagram \n\\[\\xymatrix@R=18pt@C=40pt{\\displaystyle(\\hocolim_{\\mathcal{P}_1(J)}X)^H\\ar[dr]\\ar[r]^-{\\cong}& \\displaystyle\\hocolim_{\\mathcal{P}_1(J)^H}X^H\\ar[d]\\\\\n&X^{H}_J}\\]\nwhere the homeomorphism is from Proposition \\ref{resmapF}. Since $\\mathcal{P}_1(J)^H$ is isomorphic to $\\mathcal{P}_1(J\/H)$, we need to determine how cocartesian the $J\/H$-cube $X^H$ is. By the dual Blakers-Massey Theorem \\cite[2.6]{calcII}, it is\n\\[\\min_{\\{T_\\alpha\\}\\in\\Pa_H(J)}\\left\\{\n|J\/H|-1+\\sum_{\\alpha}\\omega^{T_\\alpha}(H)\\right\\}\\]\ncocartesian, if each restriction $X^H|_{\\mathcal{P}(T_\\alpha\/H)}$\nis $\\omega^{T_\\alpha}(H)$-cartesian. To determine $\\omega^{T_\\alpha}(H)$, consider the commutative diagram\n\\[\\xymatrix@R=18pt@C=40pt{\nX_{\\emptyset}^H\\ar[dr]\\ar[r]&\\displaystyle(\\holim_{\\mathcal{P}_0(T_\\alpha)}X)^H\\ar[d]^-{r}\\\\\n&\\displaystyle\\holim_{\\mathcal{P}(T_\\alpha\/H)}X^H\n}\\]\nwhere the vertical map is the restriction. By hypothesis the horizontal map is $\\nu^{T_\\alpha}(H)$-connected. The connectivity of $r$ is calculated using \\ref{connresmap} just as we did in the proof of \\ref{BM}, showing that\n\\[\\omega^{T_\\alpha}(H)=\\min\\Big\\{\\nu^{T_\\alpha}(H)\\ , \\ \\min_{\\emptyset\\neq U\\subset T_\\alpha}\\min_{L\\in \\Eff_H(U)}\\{\\conn X^{L}_U-|U\/L|+1\\}\\Big\\}\\]\n\\end{proof}\n\n\n\\subsection{The equivariant Freudenthal suspension Theorem}\\label{secsusp}\n\nLet $G$ be a discrete group and let $J$ be a finite $G$-set. We let $S^J$ be the permutation representation sphere of $J$, defined as the one-point compactification $\\mathbb{R}[J]^+$. Given a pointed $G$-space $X$, we define its $J$-loop and $J$-suspension respectively as the space of pointed maps and the smash product\n\\[\\Omega^J X=\\map_\\ast(S^J,X)\\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\Sigma^JX=X\\wedge S^J\\]\nwith $G$-action by conjugation on $\\Omega^ JX$ and diagonal on $\\Sigma^ JX$. The pair of functors $\\Sigma^J\\colon \\Top^{G}_\\ast\\leftrightarrows\\Top^{G}_\\ast\\colon \\Omega^J$ forms an adjunction. The equivariant Freudenthal suspension Theorem (\\cite{Nam}, see also \\cite{Lewis},\\cite{Adams}) calculates the connectivity of the unit of this adjunction\n\\[X^G\\longrightarrow (\\Omega^J\\Sigma^ JX)^G\\]\nrestricted to the $G$-fixed points spaces. We show that his theorem is a direct consequence of the equivariant Blakers-Massey Theorem \\ref{BM}. The relationship between the Blakers-Massey theorem and the Freudenthal suspension Theorem is a well-established fact when $J$ has the trivial $G$-action.\n\n\n\n\n\n\n\n\\begin{cor}[\\cite{Nam}]\\label{susp} Let $X$ be a pointed $G$-space. The unit of the $(\\Sigma^{J},\\Omega^{J})$-adjunction $\\eta\\colon X^G\\to (\\Omega^J\\Sigma^{J}X)^G$ is\n\\[\\min\\left\\{2\\conn X^G+1,\\min_{\\substack{H\\leq G\\\\ J\/H\\neq J\/G}}\\conn X^H\\right\\}\\]\nconnected.\n\\end{cor}\n\n\\begin{proof}\nThe idea of the proof is to construct the loop space $\\Omega^J\\Sigma^JX$ as the $G$-homotopy limit of an equivariant cube. Let $J_+$ be the $G$-set $J$ with an added fixed basepoint.\nDefine a $J_+$-cube $\\sigma^JX\\colon \\mathcal{P}(J_+)\\to \\Top_\\ast$ with vertices \\[(\\sigma^JX)_U=\\left\\{\\begin{array}{lll} X &, U=\\emptyset\\\\\nC^UX &, \\emptyset\\neq U\\subsetneqq J_+\\\\\n\\Sigma^JX&, U=J_+\n\\end{array}\\right.\\]\nwhere $C^UX$ is the $U$-fold reduced cone of $X$, defined as\n\\[C^UX=\\hocolim_{\\mathcal{P}(U)}\\big(V\\longmapsto\\left\\{\\begin{array}{cl}X&, V=\\emptyset\\\\\n\\ast&,\\mbox{otherwise}\n\\end{array}\\right.\\big)\\]\nThis homotopy colimit is performed in the category of pointed spaces. The functor $\\sigma^JX$ has a natural $G$-structure. It is defined at the initial and final vertices by the $G$-actions on $X$ and $\\Sigma^JX$ respectively, and at the other vertices by the canonical isomorphism\n\\[C^UX\\longrightarrow C^{gU}X\\]\ninduced by the functor $g\\colon \\mathcal{P}(U)\\to\\mathcal{P}(gU)$. For the group $G=\\mathbb{Z}\/2$ and the $G$-set $J=\\mathbb{Z}\/2$ this is the cube\n\\[\\xymatrix@=12pt{X\\ar[rr]\\ar[dd]\\ar[dr]\n&&CX\\ar@{<-->}[dl]\\ar[dr]\\ar@{|}[d]\\\\\n&CX\\ar[dd]\\ar[rr]&\\ar[d]&C^2X\\ar[dd]\\\\\nCX\\ar@{-}[r]\\ar[dr]&\\ar[r]&C^2X\\ar@{<-->}[dl]\\ar[dr]\\\\\n& C^2X\\ar[rr]&&\\Sigma^{2,1}X\n}\\]\nThe dashed maps denote the $G$-structure between the non-fixed vertices, and $\\Sigma^{2,1}$ is the suspension by the regular representation of $\\mathbb{Z}\/2$.\nThe space $C^UX$ is $G_U$-contractible, since for every subgroup $H$ of $G_U$ Proposition \\ref{resmapF} provides a homeomorphism\n\\[(C^UX)^H\\cong C^{U\/H}(X^H)\\simeq\\ast\\]\nThis shows that the restriction of $\\sigma^JX$ to $\\mathcal{P}_0(J_+)$ is equivalent to the $G$-diagram $\\omega^JX\\colon\\mathcal{P}_0(J_+)\\to\\Top$ defined by\n\\[(\\omega^JX)_U=\\left\\{\\begin{array}{ll} \\ast&, U\\neq J_+\\\\\n\\Sigma^JX &, U=J_+\n\\end{array}\\right.\\]\nvia the obvious pointed map $\\omega^JX\\to\\sigma^JX|_{\\mathcal{P}_0(J_+)}$.\nThe homotopy limit of $\\omega^JX$ is $G$-homeomorphic to $\\Omega^J\\Sigma^JX$, by inspection on the definition of the Bousfield-Kan formula. This gives a commutative diagram of pointed $G$-spaces\n\\[\\xymatrix@C=8pt{X=(\\sigma^JX)_\\emptyset\\ar[rrrrr]\\ar[drrrr]_\\eta\n&&&&&\n\\displaystyle\\holim_{\\mathcal{P}_0(J_+)}\\sigma^JX\\\\\n&&&&\\Omega^J(\\Sigma^JX)\\ar@<-1ex>@{}[r]^-\\cong& \\displaystyle\\holim_{\\mathcal{P}_0(J_+)}\\omega^JX\\ar[u]_{\\simeq}}\\]\nwhere the vertical map is a $G$-equivalence by homotopy invariance of $G$-homotopy limits. The unit map $\\eta$ is then as connected as $\\sigma^JX$ is cartesian.\n\n\nWe show that $\\sigma^JX$ satisfies the condition of the Blakers-Massey Theorem \\ref{BM}, and we show that its cartesianity estimate is the same as the range claimed in \\ref{susp}.\nFor each $G$-subset $U$ of $J_+$, we need to find an estimate $\\nu^U$ for the cocartesianity of the restriction $\\sigma^JX|_{\\mathcal{P}(U)}$. For every subgroup $H$ of $G_U$, there are isomorphisms\n\\[(\\hocolim_{\\mathcal{P}_1(U)}\\sigma^JX)^H\\cong \\hocolim_{\\mathcal{P}_1(U\/H)}(\\sigma^JX)^H\\cong \\hocolim_{\\mathcal{P}_1(U\/H)}\\sigma^{J\/H}X^H\\]\nby the fixed points description of \\ref{resmapF}.\nSince all the vertices of $\\sigma^{J\/H}X^H|_{\\mathcal{P}_1(U\/H)}$ are contractible except for the initial one, there is a natural equivalence \n\\[(\\hocolim_{\\mathcal{P}_1(U)}\\sigma^{J}X)^H\\cong\n\\hocolim_{\\mathcal{P}_1(U\/H)}\\sigma^{J\/H}X^H\\stackrel{\\simeq}{\\longrightarrow} \\Sigma^{|U\/H|-1}X^H\\]\nHence the canonical map $(\\hocolim_{{\\mathcal{P}_1(U)}} \\sigma^JX)^H\\to (\\sigma^JX)^{H}_U$ factors through the equivalence\n\\[(\\hocolim_{\\mathcal{P}_1(U\/H)}\\sigma^{J}X)^H\\stackrel{\\simeq}{\\longrightarrow}\\Sigma^{|U\/H|-1}X^H\\longrightarrow (\\sigma^{J}X)_{U}^H=\\left\\{\\begin{array}{ll}\nC^{U\/H}X^H&, U\\neq J_+\\\\\n\\Sigma^{|J\/H|}X^H &,U= J_+\n\\end{array}\\right.\\]   \nThis map is $(\\conn X^H+|U\/H|)$-connected for $U\\neq J_+$ and it is an equivalence for $U=J_+$. It follows that $\\sigma^JX|_{\\mathcal{P}(U)}$ is $\\nu^U$-cocartesian for the function\n\\[\\nu^U(H)=\\left\\{\\begin{array}{ll}\n\\conn X^H+|U\/H|&, U\\neq J_+\\\\\n\\infty &,U= J_+\n\\end{array}\\right.\\]\nThese functions satisfy $\\nu^U\\leq \\nu^V$ for $H$-subsets $U\\subset V$ of $J_+$, and the equivariant Blakers-Massey Theorem \\ref{BM} applies. The first term of the minimum in the range of \\ref{BM} for the group $H=G$ is\n\\[\\min_{\\{T_\\alpha\\}\\in\\Pa_G(J_+)}\\{\n\\sum_{\\alpha}\\nu^{T_\\alpha}(G)\\}-|J_+\/G|+1=\\min_{\\{J_+\\}\\neq\\{T_\\alpha\\}_{\\alpha\\in A}}\\{\n\\sum_{\\alpha}(|T_\\alpha\/G|)+|A|\\conn X^G\\}-|J_+\/G|+1\\]\nThe trivial partition $\\{J_+\\}$ is removed from the minimum because $\\nu^{J_+}=\\infty$. For any partition $\\{T_\\alpha\\}$ of $J_+$ by $G$-sets, the quotient $J_+\/G$ decomposes as the disjoint union of the quotients $T_\\alpha\/G$. Therefore the sum in the formula above is equal to $|J_+\/G|$. The minimum is thus realized when the size of the partition $|A|$ is minimal. This is the partition with two elements $\\{J,+\\}$, and the quantity above is\n\\[\\min_{\\{J_+\\}\\neq\\{T_\\alpha\\}_{\\alpha\\in A}}\\{|J_+\/G|\n+|A|\\conn X^G\\}-|J_+\/G|+1=2\\conn X^G+1\\]\nThe second term of the minimum in the formula of Theorem \\ref{BM} is \n\\[\\min_{\\emptyset\\neq U\\subset J_+}\\min_{H\\in \\Eff_G(U)}\\conn (\\sigma^JX)^{H}_U-|J_+\/H|+1\\]\nAll the vertices of $\\sigma^JX$ have contractible fixed points except for the initial and the final one. The outer minimum is then realized when $U=J_+$, with value\n\\[\\min_{H\\in \\Eff_G(J_+)}\\conn (S^J\\wedge X)^{H}-|J_+\/H|+1=\\min_{H\\in \\Eff_G(J_+)}\\conn X^H\\]\nThe set $\\Eff_G(J_+)$ is by definition the set of subgroups $H$ of $G$ for which $J_+\/H\\neq J_+\/G$, which is the same condition as $J\/H\\neq J\/G$.\n\\end{proof}\n\n\\subsection{Equivariant intersections of submanifolds and configuration spaces}\\label{secconf}\n\n\nLet $M$ be a manifold and let $P_1,\\dots,P_n$ be a collection of submanifold of $M$. For every integer $2\\leq k\\leq n$, we define inductively what it means for the collection $P_1,\\dots,P_n$ to be $k$-transverse in $M$. Let us denote $\\underline{n}=\\{1,\\dots n\\}$.\n\\begin{enumerate}[label=\\roman*)]\n\\item $P_1,\\dots,P_n$ is $2$-transverse if $P_i$ intersects $P_j$ transversally for every $i\\neq j$.\n\\item $P_1,\\dots,P_n$ is $k$-transverse if it is $(k-1)$-transverse and if for every set $S\\subset \\underline{n}$ with $k$ elements and every $s\\in S$ the submanifold $\\bigcap_{t\\in S\\backslash s}P_t$ intersects $P_s$ transversally.\n\\end{enumerate}\nLet $m$ be the dimension of $M$, and let $d_i$ be the dimension of $P_i$.\nIf the collection of submanifolds $P_1,\\dots,P_n$ is $n$-transverse, the intersection $\\bigcap_{s\\in S}P_s$ is either a submanifold of $M$ of codimension $\\sum_{s\\in S}(m-d_s)$, or it is the empty submanifold.\n\nNow suppose that $G$ is a discrete group, let $J$ be a finite $G$-set, and suppose that $G$ acts properly on a manifold $M$. Let $\\{P_j\\}_{j\\in J}$ be a set of submanifolds indexed on $J$, and suppose that for every group element $g$ the corresponding automorphism of $M$ restricts to a map $g\\colon P_j\\to P_{gj}$. The $G$-action on $M$ defines a $G$-structure on the cube $M\\backslash P_{\\bullet}\\colon\\mathcal{P}(J)\\to \\Top$ which sends a subset $U\\subset J$ to the complement $M\\backslash\\bigcup_{j\\notin U}P_j$. The $G$-structure is defined by the maps\n\\[g\\colon M\\backslash\\bigcup_{j\\notin U}P_j\\longrightarrow M\\backslash \\bigcup_{j\\notin U}g(P_j)=M\\backslash\\bigcup_{j\\notin U}P_{gj}=M\\backslash\\bigcup_{j\\notin gU}P_{j}\\]\n\n\n\\begin{theorem}\\label{intsubman}\nLet $M$ be a manifold with proper $G$-action, and let $\\{P_j\\}_{j\\in J}$ be a set of closed submanifolds as above. Suppose moreover that for every $j\\in J$ and every subgroup $H$ of $G$ the intersection $P_j\\cap M^H$ is a submanifold of $M$, and that the collection of submanifolds $\\{P_j\\cap M^H\\}_{j\\in J}$ is $|J|$-transverse in $M^H$. Then\nthe $J$-cube $M\\backslash P_{\\bullet}\\colon\\mathcal{P}(J)\\to \\Top$ is $\\nu$-cartesian, where $\\nu$ is the function\n\\[\\min\\Big\\{\\sum_{j\\in J}\\big(m_H-d_j(H)\\big)-2|J\/H|+1\\ ,\n\\min_{\\substack{\nL\\leq H\\\\\nJ\/L\\neq J\/H\n}}\\big\\{\\min\\{\\conn M^L+1,m_L-\\max_{j\\in J}d_j(L)\\}-|J\/L|\\big\\}\n \\Big\\}\\]\nHere $d_j(H)$ is the dimension of $P_j\\cap M^H$, and $m_H$ is the dimension of $M^H$.\n\\end{theorem}\n\n\\begin{proof}\nIn order to understand how cartesian $M\\backslash P_{\\bullet}$ is it is sufficient, by the Blakers-Massey theorem, to understand how cocartesian the subcubes of $M\\backslash P_{\\bullet}$ are. Let $H$ be a subgroup of $G$ and let $U$ be an $H$-invariant subset of $J$. The homotopy colimit of the restriction of $M\\backslash P_{\\bullet}$ to $\\mathcal{P}_1(U)$ is $H$-equivalent to\n\\[\\hocolim_{\\mathcal{P}_1(U)}M\\backslash P_{\\bullet}\\stackrel{\\simeq}{\\longrightarrow}\\colim_{\\mathcal{P}_1(U)}M\\backslash P_{\\bullet}\n=\\big(M\\backslash\\bigcup_{j\\notin U}P_j\\big)\\backslash\\bigcap_{u\\in U}P_u\\]\nHence the cube $(M\\backslash P_{\\bullet})|_{\\mathcal{P}(U)}^{H}$ is $\\nu^U(H)$-cocartesian, where $\\nu^U(H)$ is the connectivity of the inclusion \n\\[\\nu^U(H)=\\conn\\Big(\n\\big(M\\backslash\\bigcup_{j\\notin U}P_j\\big)\\backslash\\bigcap_{u\\in U}P_u\\longrightarrow M\\backslash\\bigcup_{j\\notin U}P_j\n\\Big)^H\\]\nof $H$-fixed points. This map is the inclusion of submanifolds\n\\[\\big(M^H\\backslash(\\bigcup_{j\\notin U}P_j)^H\\big)\\backslash(\\bigcap_{u\\in U}P_u)^H\\longrightarrow M^H\\backslash(\\bigcup_{j\\notin U}P_j)^H\\]\nwhich is as connected as the codimension of $(\\bigcap_{u\\in U}P_u)^H$ in $M^H$ minus one. By our transversality assumption $(\\bigcap_{u\\in U}P_u)^H=\\bigcap_{u\\in U}P_u\\cap M^H$ is a submanifold of $M^H$ of codimension $\\sum_{u\\in U}\\big(m_H-d_u(H)\\big)$. The functions\n\\[\\nu^U(H)=\\sum_{u\\in U}\\big(m_H-d_u(H)\\big)-1\\]\nsatisfy $\\nu^U\\leq \\nu^V$ when $U\\subset V$ are both $H$-invariant, and Theorem \\ref{BM} applies. The first term of the minimum of \\ref{BM} is\n\\[\\min_{\\{T_\\alpha\\}_{\\alpha\\in A}\\in\\Pa_H(J)}\\Big\\{\n\\sum_{\\alpha\\in A}\\sum_{u\\in T_\\alpha}\\big(m_H-d_u(H)\\big)-|A|\\Big\\}-|J\/H|+1\\]\nThe double sum is independent of the partition, and this minimum is realized for the finest partition of $J$ by $H$-invariant sets. This is the partition of $J$ in $H$-orbits, and the term above is\n\\[\\sum_{j\\in J}\\big(m_H-d_j(H)\\big)-2|J\/H|+1\\]\nThe second term of the minimum of the Blakers-Massey formula is\n\\[\\min_{\\emptyset\\neq U\\subset J}\\min_{L\\in \\Eff_H(U)}\\big\\{\\conn M^L\\backslash (\\bigcup_{u\\in U}P_u)^L-|U\/L|+1\\big\\}\n\\]\nThe connectivity of the complement of $(\\bigcup_{u\\in U}P_u)^L=\\bigcup_{u\\in U}(P_u\\cap M^L)$ in $M^L$ is \n\\[\\conn M^L\\backslash (\\bigcup_{u\\in U}P_u)^L=\\min\\Big\\{\\conn M^L,\\min_{u\\in U}\\big(m_L-d_u(L)\\big)-1\\Big\\}\\]\nThe double minimum above is  thus realized when $U=J$, with value\n\\[\\min_{L\\in \\Eff_H(J)}\\big\\{\\min\\Big\\{\\conn M^L,m_L-1-\\max_{j\\in J}d_j(L)\\Big\\}-|J\/L|+1\\big\\}\n\\]\nMoreover $\\Eff_H(J)$ is the collection of proper subgroups $L$ of $H$ with $J\/L\\neq J\/H$.\n\\end{proof}\n\nThis Theorem has interesting consequences for equivariant cubes of configuration spaces. What follows is an equivariant version of the multiple disjunction Theorem \\cite{GK} in the easy situation where the submanifolds $P_j$ are points. The techniques of \\cite{GK} can supposedly be extended to equivariant collections of higher dimensional submanifolds.\nGiven a finite $G$-set $J$ and a proper $G$-manifolds $M$, we let $\\conf(J,M)$ be the space of ordered configurations of $|J|$-points in $M$, with $G$ acting by conjugation. Explicitly, a configuration is an injective map $\\underline{x}\\colon J\\to M$, which is sent by the homeomorphism associated to $g\\in G$ to the composite $g\\underline{x}g^{-1}$.\nThe cube $\\conf\\big(J\\backslash(-),M\\big)\\colon \\mathcal{P}(J)\\to Top$ that sends a subset $U$ of $J$ to the configurations of $J\\backslash U$-points has a $G$-structure defined by the maps $g\\colon \\conf\\big(J\\backslash U,M\\big)\\to \\conf\\big(J\\backslash gU,M\\big)$ which send $\\underline{x}\\colon U\\to M$ to\n\\[g\\underline{x}\\colon J\\backslash g(U)\\stackrel{g^{-1}}{\\longrightarrow}J\\backslash U\\stackrel{\\underline{x}}{\\longrightarrow}M\\stackrel{g}{\\longrightarrow}M\\]\n\\begin{cor}\\label{conf}\nLet $J$ be a finite $G$-set and let $J_+$ be the $G$-set $J$ with an added fixed basepoint. The $J_+$-cube $\\conf(J_+\\backslash (-),M)\\colon \\mathcal{P}(J_+)\\to Top$ is $\\nu$-cartesian, where $\\nu$ is the function\n\\[\\nu(H)=\\min\\Big\\{|J|m_H-2|J\/H|+1\\ ,\n\\min_{\\substack{\nL\\leq H\\\\\nJ\/L\\neq J\/H\n}}\\big\\{\\min\\{\\conn M^L+1,m_L\\}-|J\/L|\\big\\}\n \\Big\\}\\]\nand $m_H$ is the dimension of the fixed points manifold $M^H$.\n\\end{cor}\n\n\nBecause of the second term of the minimum the formula is meaningful only when the connectivity of $M$ is greater than the number of points in the configuration. Therefore one cannot expect to make $\\nu$ diverge by increasing the number of orbits of $J$. If the action on $J$ is trivial, the second term is infinite and the range of \\ref{conf} is the classical cartesianity range from embedding calculus (see \\cite{GK}).\nBefore proving Corollary \\ref{conf} we see an example where the range is sharp and it is determined by the second term of the minimum. \n\n\\begin{ex}\\label{exsharp}\nLet us consider the group $G=\\mathbb{Z}\/2$ and the finite $G$-set $J=\\mathbb{Z}\/2$ with action by left multiplication. The space $\\conf(\\mathbb{Z}\/2_+,M)$ of configurations in a manifold with involution $M$ is the space of triples of pairwise distinct elements $(x_+,x_0,x_1)$ in $M$. Such a triple is sent by the non-trivial element of $\\mathbb{Z}\/2$ to $(\\tau x_{+},\\tau x_{1},\\tau x_{0})$, where $\\tau$ denotes the involution of $M$. The $\\mathbb{Z}\/2$-fixed points of this space is described by a natural homeomorphism\n\\[\\conf(\\mathbb{Z}\/2_+,M)^{\\mathbb{Z}\/2}\\cong M^{\\mathbb{Z}\/2}\\times (M\\backslash M^{\\mathbb{Z}\/2})\\]\nThe equivariant cube $\\conf(\\mathbb{Z}\/2_+\\backslash(-),M)$ is the $\\mathbb{Z}\/2_+$-cube\n\\[\\xymatrix@=8pt{\\conf(\\mathbb{Z}\/2_+,M)\\ar[rr]\\ar[dd]\\ar[dr]\n&&\\conf(0_+,M)\\ar@{<-->}[dl]_-{\\tau}\\ar[dr]\\ar@{|}[d]\\\\\n&\\conf(1_+,M)\\ar[dd]\\ar[rr]&\\ar[d]&M\\ar[dd]\\\\\n\\conf(\\mathbb{Z}\/2,M)\\ar@{-}[r]\\ar[dr]&\\ar[r]&M\\ar@{<-->}[dl]_-{\\tau}\\ar[dr]\\\\\n& M\\ar[rr]&&\\ast\n}\\]\nwhere $0$ and $1$ are the elements of $J=\\mathbb{Z}\/2$. The dashed maps denote the $G$-structure on the non-fixed objects of $\\mathcal{P}(\\mathbb{Z}\/2_+)$. By inspection, a point in the homotopy limit of this cube with the initial vertex removed is the data of three paths $\\gamma_+,\\gamma_0,\\gamma_1\\colon I\\to M$, subject to the conditions $\\gamma_+(0)\\neq \\gamma_0(0)$, $\\gamma_0(1)\\neq \\gamma_1(1)$ and $\\gamma_1(0)\\neq \\gamma_+(1)$. The non-trivial element of $\\mathbb{Z}\/2$ sends $(\\gamma_+,\\gamma_0,\\gamma_1)$ to $(\\tau\\overline{\\gamma}_+,\\tau\\gamma_1,\\tau\\gamma_0)$, where the bar denotes the backwards path.\nSuch a triple of paths is fixed by the action precisely when $\\gamma_+$ is a loop in the fixed points manifold $M^{\\mathbb{Z}\/2}$, and $\\gamma_1=\\overline{\\gamma}_0$.\nThe fixed points of this homotopy limit is therefore homeomorphic to the space\n\\[\\big(\\holim_{\\mathcal{P}_0(\\mathbb{Z}\/2_+)}\\conf(\\mathbb{Z}\/2_+\\backslash(-),M)\n\\big)^{\\mathbb{Z}\/2}\\cong \\Big\\{\\big(\\sigma\\in\\Lambda M^{\\mathbb{Z}\/2},\\gamma\\colon I\\to M\\big) \\ |\\ \\gamma(0)\\notin M^{\\mathbb{Z}\/2}, \\gamma(1)\\neq\\sigma(\\ast)\\Big\\}\\]\nHere $\\Lambda M$ is the space of free loops in $M$.\nThe canonical map from $\\conf(\\mathbb{Z}\/2_+,M)^{\\mathbb{Z}\/2}$ to the fixed points of the homotopy limit sends a pair $(x,y)\\in M^{\\mathbb{Z}\/2}\\times (M\\backslash M^{\\mathbb{Z}\/2})$ to the corresponding constant paths. By Corollary \\ref{conf} it is at least $\\nu(\\mathbb{Z}\/2)$-connected.\n\nLet us choose $M$ to be the torus $S^1\\times S^1$ with the involution that swaps the two product factors. The fixed points of the action is a copy of $S^1$ embedded diagonally, and the range of Corollary \\ref{conf} is\n\\[\\nu(\\mathbb{Z}\/2)=\\min\\left\\{1\\cdot 2-2\\cdot 1+1\n\\ , \\ \\min\\{1,2\\}-2\\right\\}=\\min\\big\\{\n1\\ , -1\\big\\}=-1\\]\nLet us show that the cube is not $0$-cartesian. The fixed points of the configuration space is\n\\[\\conf(\\mathbb{Z}\/2_+,S^1\\times S^1)^{\\mathbb{Z}\/2}\\cong S^1\\times \\big((S^1\\times S^1)\\backslash \\Delta\\big)\\simeq S^1\\times (S^1\\vee S^1)\\]\nwhich is $0$-connected. We need to show that the fixed points space of the homotopy limit has non-trivial $\\pi_0$. There is projection map \\[\\big(\\holim_{\\mathcal{P}_0(\\mathbb{Z}\/2_+)}\\conf(\\mathbb{Z}\/2_+\\backslash(-),M)\n\\big)^{\\mathbb{Z}\/2}\\longrightarrow \\Lambda M\\]\nwhich is split by any choice of path in $M\\backslash M^{\\mathbb{Z}\/2}$. Hence $\\pi_0$ of the fixed points of the homotopy limit contains a copy of $\\pi_0\\Lambda M$, which is non-trivial as $M=S^1\\times S^1$ has non-trivial $\\pi_1$. \n\\end{ex}\n\n\n\\begin{proof}[Proof of \\ref{conf}]\nLet $\\underline{x}\\colon J_+\\to M$ be a configuration. Its restrictions $\\underline{x}_{J_+\\backslash U}\\in \\conf(J_+\\backslash U,M)$ for the subsets $U$ of $J_+$ define a basepoint of the diagram $\\conf(J_+\\backslash (-),M)$. Let $\\mathcal{F}_{\\underline{x}}(M)\\colon \\mathcal{P}(J)\\to \\Top$ be the $J$-cube defined by the homotopy fibers of the maps that forget the basepoint\n\\[\\mathcal{F}_{\\underline{x}}(M)_U=\\hof\\big(\\conf(J_+\\backslash U,M)\\longrightarrow \\conf(J\\backslash U,M)\\big)\\]\nover the points $\\underline{x}_{J\\backslash U}$, with the induced canonical maps. The cube $\\mathcal{F}_{\\underline{x}}(M)$ has a $G_{\\underline{x}}$-structure, where $G_{\\underline{x}}$ is the stabilizer group of the configuration $\\underline{x}$, defined by the canonical maps\n\\[\\xymatrix{\n\\mathcal{F}_{\\underline{x}}(M)_U\\ar@{-->}[d]\\ar[r]&\\conf(J_+\\backslash U,M)\n\\ar[r]\\ar[d]^{g}&\n\\conf(J\\backslash U,M)\\ni\\underline{x}_{J\\backslash U} \\ar[d]^{g}\\ar@{|->}@<8ex>[d]\\\\\n\\mathcal{F}_{\\underline{x}}(M)_{gU}\\ar[r]&\\conf(J_+\\backslash gU,M)\n\\ar[r]&\n\\conf(J\\backslash gU,M)\\ni\\underline{x}_{J\\backslash gU}\n}\\]\nThe homotopy fiber of the map $\\conf(J_+,M)\\to \\holim_{\\mathcal{P}_0(J_+)}\\conf(J_+\\backslash (-),M)$ over a configuration $\\underline{x}$ is $G_{\\underline{x}}$-equivalent to the homotopy fiber of the map $\\mathcal{F}_{\\underline{x}}(M)_\\emptyset\\to \\holim_{\\mathcal{P}_0(J)}\\mathcal{F}_{\\underline{x}}(M)$ over the canonical basepoint defined by $\\underline{x}$. Hence $\\nu(H)$ is the minimum of the cartesianities of $\\mathcal{F}_{\\underline{x}}(M)$, for $\\underline{x}$ running through the configurations in $\\conf(J_+\\backslash U,M)$ which are fixed by $H$.\nThe restriction map $\\conf(J_+\\backslash U,M)\\to \\conf(J\\backslash U,M)$ is a fibration of $G_U$-spaces, whose fiber over $\\underline{x}_{J\\backslash U}\\colon J\\backslash U\\to M$ is the manifold $M$ with the image of $\\underline{x}_{J\\backslash U}$ removed. Hence the cube $\\mathcal{F}_{\\underline{x}}(M)$ is equivalent as a $G_{\\underline{x}}$-diagram to the $J$-cube\n\\[F_{\\underline{x}}(M)_U=M\\backslash \\bigcup_{j\\in J\\backslash U}x_j\\]\nThe collection of $0$-dimensional submanifolds $\\{x_j\\}_{j\\in J}$ satisfies the transversality conditions of Theorem \\ref{intsubman}, and the cube $F_{\\underline{x}}(M)=M\\backslash \\underline{x}_{\\bullet}$ is then\n$\\nu(H)$-cartesian.\n\n\\end{proof}\n\n\n\n\n\n\\section{The equivariant Quillen Theorem $B$}\n\nQuillen's Theorem $B$ of \\cite{Quillen} shows that under certain conditions the homotopy fiber of the geometric realization of a functor $F\\colon C\\to D$ over an object $d$ in $D$ is weakly equivalent to the geometric realization of the over category $F\/d$. It is not immediately clear how this result can be generalized to an equivariant context. The analogous statement for an equivariant functor between categories with $G$-actions can easily be reduced to Theorem $B$ by taking fixed points.\nThe generalization presented in this paper gives a categorical model for the total homotopy fiber of the geometric realization of a $J$-cube of categories. When $J$ is the set with one element, one recovers Quillen's Theorem $B$. \n\nThe key to our equivariant Theorem $B$ is a categorical model for the $G$-homotopy limit of the nerve of a suitable $G$-diagram of categories. This result is not limited to diagrams of cubical shape, but it applies more generally to categories which satisfy a suitable finiteness condition (see \\S\\ref{secBI}). Even when $G$ is the trivial group our categorical model for the homotopy limit generalizes Theorem $B$ considerably, from homotopy fibers to all finite homotopy limits. Since this result might be of interest to non-equivariant homotopy theorists we prove it as a separate statement.\n\nThe section is organized as follows. In \\S\\ref{secmodel} we define a categorical model for the homotopy limit of the nerve of a diagram $X\\colon I\\to Cat$, as well as a quasi-fibrancy condition analogous to the hypothesis of Quillen's Theorem $B$. In \\S\\ref{secBI} we prove that the classifying space of the category constructed in \\S\\ref{secmodel} models indeed the homotopy limit of $BX$, if $X$ is quasi-fibrant. We call this result ``Theorem $B^I$''. In Corollary \\ref{higherQ} we use this result to give a categorical model for the total homotopy fiber of a cube of categories. We think of this Corollary as a Theorem $B$ for cubes. Section \\ref{secBGI} contains the generalization of Theorem $B^I$ to $G$-diagrams, and the equivariant Quillen Theorem $B$.\n\n\n\\subsection{Reedy quasi-fibrant diagrams of categories}\\label{secmodel}\n\nLet $I$ be a small category and let  $K,X\\colon I\\to Cat$ be diagrams of small categories. The natural transformations from $K$ to $X$ form a category $\\Hom(K,X)$. An object of this category is a natural transformation $\\Phi\\colon K\\to X$, and a morphism $\\Lambda\\colon \\Phi\\to \\Phi'$ is a pair of natural transformations $\\Lambda_1\\colon K\\to K$ and $\\Lambda_2\\colon X\\to X$ such that the square \n\\[\\xymatrix@=15pt{K\\ar[r]^-{\\Lambda_1}\\ar[d]_-{\\Phi}&K\\ar[d]^-{\\Phi'}\n\\\\ X\\ar[r]_-{\\Lambda_2}&X\n}\\]\ncommutes.\n\n\n\\begin{rem} The category $\\Hom(K,X)$ was introduced in \\cite{Lydakis} where the author shows, among other homotopical properties of this construction, that its nerve is isomorphic to the simplicial mapping space of natural transformations $\\Hom(NK,NX)$. In particular when $K$ is the functor $K=I\/_{(-)}\\colon I\\to Cat$ the nerve of $\\Hom\\big(I\/_{(-)},X\\big)$ is isomorphic to the Bousfield-Kan formula (\\cite{BK}) for the homotopy limit of $NX$. This construction computes the homotopy limit of $NX$ in the standard model structure of simplicial sets only when $NX$ is pointwise fibrant, that is only in the rare situation when the vertices $X_i$ are groupoids. The goal of Theorem $B^I$ is to find a condition on $X\\colon I\\to Cat$, weaker than assuming that $X$ is valued in groupoids, for which the nerve of $\\Hom\\big(I\/_{(-)},X\\big)$ is equivalent to the homotopy limit of $NX$.\n\\end{rem}\n\n\\begin{ex}\\label{modelhpb}\nLet $I$ be the poset $\\bullet\\to\\bullet\\leftarrow\\bullet$. A diagram indexed over this poset is a pullback diagram of categories $C\\stackrel{f}{\\to} D\\stackrel{g}{\\leftarrow} E$. There is an isomorphism of categories  \n\\[\\Hom\\Big((\\bullet\\to\\bullet\\leftarrow\\bullet)\/_{(-)}\\ ,\\ C\\stackrel{f}{\\to} D\\stackrel{g}{\\leftarrow} E\\Big)\\cong f\\!\\downarrow\\! g\\]\nwhere $f\\!\\!\\downarrow\\!\\!g$ is the model for the homotopy pullback of Barwick and Kan \\cite{ClarkKan}. The objects of $f\\!\\!\\downarrow\\!\\!g$ are triples $(c,d,\\gamma)$ consisting of objects $c\\in C$ and $e\\in E$, and a zig-zag of morphisms $\\gamma=(f(c)\\to d \\leftarrow g(e))$ in the category $D$. This can also be described as the Grothendieck construction of the functor $f\/_{(-)}\\times g\/_{(-)}\\colon D\\longrightarrow Cat$.\n\\end{ex}\n\n\n\n\nLet $i\\!<\\!I$ be the full subcategory of the under category $i\/I$ of non-identity maps with source $i$. Given a diagram $X\\colon I\\to Cat$ we define $X_{i<}: (i\\!<\\! I)\\to Cat$ to be the restriction of $X$ along the projection functor $i\\!<\\! I\\to I$ that sends $i\\to j$ to $j$. For every object $i$ of $I$, there is a functor \n\\[m_i\\colon X_i\\longrightarrow \\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)\\]\nthat sends an object $x$ of $X_i$ to the natural transformation $m_i(x)\\colon (i\\!<\\! I)\/_{(-)}\\to X_{i<}$ consisting of the constant functors $m_i(x)_\\alpha\\colon (i\\!<\\! I)\/_{\\alpha}\\to X_{j}$ that send every object to $\\alpha_\\ast x$. For the purpose of this paper, we say that a functor of small categories is a weak equivalence if its nerve is a weak equivalence of simplicial sets.\n\\begin{defn}\\label{quasiReedy}\nA diagram $X\\colon I\\to Cat$ is Reedy quasi-fibrant if for every object $i$ of $I$ the functor\n\\[m_i\/_{(-)}\\colon \\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)\\longrightarrow Cat\\]\nsends every morphism in the category of natural transformations to a weak equivalence of categories.\n\\end{defn}\n\n\nThis condition is reminiscent of the Reedy fibrancy conditions of diagrams indexed over a Reedy category. We explain the relationship between the two conditions more closely. The functor $m_i$ factors through the categorical limit\n\\[m_i\\colon X_i\\longrightarrow\\lim\\limits_{i\\stackrel{\\neq\\id}{\\to}j}X_j=\\Hom(\\ast,X_{i<})\\longrightarrow \\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)\\]\nwhere the second map is induced by the projection $(i\\!<\\! I)\/_{(-)}\\to\\ast$. The first functor is the $i$-matching functor of $X$. The nerves of the $m_i$'s are thickenings of the matching maps of $NX$. The diagram $NX$ would be Reedy fibrant if the matching maps were Kan fibrations. If $X$ is Reedy quasi-fibrant then, by Quillen's Lemma \\cite[p.98]{Quillen} and Thomason's Theorem \\cite{Thomason}, the replacement of $m_i$ by the Grothendieck construction\n\\[NX_i\\simeq N\\big(\\Hom\\mathlarger{\\mathlarger{\\wr}} m_i\/_{(-)}\\big)\\longrightarrow N\\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)\\]\nis a quasi-fibration. In this sense the condition of Theorem $B^I$ is a Reedy quasi-fibrancy condition, hence the terminology.\n\n\\begin{ex}\\label{Reedyhpb}\nIf we return to the diagram of categories $C\\stackrel{f}{\\to} D\\stackrel{g}{\\leftarrow} E$ of Example \\ref{modelhpb} we see that this is Reedy quasi-fibrant precisely when the functors $f\/_{(-)},g\/_{(-)}\\colon D\\to Cat$ send every morphism to a weak equivalence.\n\\end{ex}\n\n\n\\subsection{Finite homotopy limits of categories and higher Quillen's Theorem $B$}\\label{secBI}\n\n\n\nLet $I$ be a small category, and suppose that the nerves of the under categories $N(i\/I)$ are finite dimensional simplicial sets, for every object $i$ of $I$. We call a category $I$ with this property left-finite. Such a category has a canonical degree function $Ob I\\to\\mathbb{N}$ that sends $i$ to the dimension of $N(i\/I)$. This is the length of the longest sequence of non-identity morphisms starting at $i$. \nThe degree function induces a filtration \n\\[I_{\\leq 0}\\subset I_{\\leq 1}\\subset \\dots \\subset I\\]\nwhere $I_{\\leq n}$ is the full subcategory of $I$ of objects of degree less than or equal to $n$. This filtration is finite precisely when $NI$ is itself finite dimensional.\n\nLet  $X\\colon I\\to Cat$ be a diagram of categories. We choose the space\n\\[\\holim_INX:=\\Hom\\big(N(I\/_{(-)}),FNX\\big)\\]\nas a model for the homotopy limit of $NX$, where $NX\\stackrel{\\simeq}{\\to} FNX$ is a pointwise fibrant replacement of the diagram $NX$. The fibrant replacement induces a comparison map $N\\Hom\\big(I\/_{(-)},X\\big)\\to \\holim_INX$.\n\n\\begin{theoremBI}\nLet $I$ be a left-finite category, and let $X\\colon I\\to Cat$ be a Reedy quasi-fibrant diagram of categories (see Definition \\ref{quasiReedy}). There is a weak equivalence\n\\[\\holim_{n\\in \\mathbb{N}^{op}}N\\Hom\\big((I_{\\leq n})\/_{(-)},X_{\\leq n}\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_I NX\\]\nwhere $X_{\\leq n}$ is the restriction of $X$ to $I_{\\leq n}$.\nIn particular if the nerve of $I$ is finite dimensional the map\n\\[N\\Hom\\big(I\/_{(-)},X\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_I NX\\]\nis a weak equivalence of simplicial sets.\n\\end{theoremBI}\n\n\n\\begin{rem}\\label{reduceB}\nLet us see how Theorem $B^I$ relates to Quillen's Theorem $B$. Let us consider a diagram of categories $C\\stackrel{f}{\\to} D\\stackrel{g}{\\leftarrow} E$ on the poset $I=(\\bullet\\to\\bullet\\leftarrow\\bullet)$. Theorem $B^I$ tells us that the nerve of the category\n\\[\\Hom\\Big((\\bullet\\to\\bullet\\leftarrow\\bullet)\/_{(-)}\\ ,\\ C\\stackrel{f}{\\to} D\\stackrel{g}{\\leftarrow} E\\Big)\\cong f\\!\\downarrow\\! g\\]\nis equivalent to the homotopy pullback of $Nf$ and $Ng$, if both functors $f\/_{(-)},g\/_{(-)}\\colon D\\to Cat$ send morphisms to equivalences (see \\ref{modelhpb} for the definition of $f\\!\\downarrow\\! g$). Barwick and Kan arrive to the same conclusion by assuming that just one among the functors $f\/_{(-)}$ and $g\/_{(-)}$ satisfies this property (see \\cite{ClarkKan}). In particular when $E=\\ast$ is the trivial category, their result is equivalent to the original formulation of Quillen's Theorem $B$, from \\cite{Quillen}.\nThe fact that one is able to weaken the quasi-fibrancy condition in the pullback case to only one of the functors $f\/_{(-)}$ and $g\/_{(-)}$ is a special feature of squares. It is completely analogous to the fact that the pullback along a fibration is homotopy invariant, even though the pullback diagram itself is injectively fibrant only when both maps are fibrations (see e.g. \\cite[13.3.9]{hirsch}).\n\\end{rem}\n\n\n\nThe proof of Theorem $B^I$ is by induction on the filtration $I_{\\leq 0}\\subset I_{\\leq 1}\\subset \\dots \\subset I$, by exploiting the fact that the complements $I_n=I_{\\leq n}\\backslash I_{\\leq n-1}$ are discrete categories. The inductive step is based on a Lemma that describes the interaction between natural transformations and Grothendieck constructions, which requires us to set up some notation.\nFor any set of degree $n$ objects $U\\subset I_n$, let $U\\!\\leq\\! I$ be the union of the under categories $u\/I$ for $u\\in U$. Explicitly, its set of objects is\n\\[Ob\\ (U\\!\\leq\\! I)=\\{(u\\in U,\\alpha \\colon u\\to i)\\}\\]\nThe set of morphisms $(u,\\alpha)\\to (v,\\beta)$ is empty if $u$ and $v$ are different, and it is the set of morphisms $(u,\\alpha)\\to (u,\\beta)$ in $u\/I$ otherwise.\nDefine $U\\!<\\! I$ to be the full subcategory of $U\\!\\leq\\! I$ whose objects are non-identity maps.\nGiven a diagram of categories $X\\colon I\\to Cat$, we denote the corresponding restrictions by\n\\[X_{U\\leq}\\colon U\\!\\leq\\! I\\to I\\stackrel{X}{\\to} Cat \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ X_{U<}\\colon U\\!<\\! I\\to I\\stackrel{X}{\\to} Cat\\] where $U\\!\\leq\\! I\\to I$ and  $U\\!<\\! I\\to I$ project onto the target. We recall from \\cite{Thomason} that the Grothendieck construction of a functor $F\\colon C\\to Cat$ is the category $C\\wr F$ with objects pairs $(c\\in C,x\\in \\Ob F(c))$, and where a morphism $(c,x)\\to (d,y)$ is a morphism $\\alpha\\colon c\\to d$ in $C$ together with a morphism $\\delta\\colon \\alpha_\\ast x\\to y$ in the category $F(d)$. \n\n\n\\begin{lemma}\\label{indgrot}\nLet $X\\colon I\\to Cat$ be a diagram of categories, and suppose that $I$ is left-finite. For every subset $U\\subset I_n$, there is a natural isomorphism of categories\n\\[\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\cong \\Big(\\Hom\\big((U\\!<\\! I)\/_{(-)},X_{U<}\\big)\\mathlarger{\\mathlarger{\\wr}}F_U\\Big)\\]\nwhere $F_U\\colon \\Hom\\big((U\\!<\\! I)\/_{(-)},X_{U<}\\big)\\to Cat$ is the functor that sends a natural transformation $\\Phi$ to the category\n\\[F_U(\\Phi)=\\prod_{u\\in U}(m_u)\/_{(\\Phi|_{u<I})}\\]\n\\end{lemma}\n\n\\begin{proof}\nAn object in the Grothendieck construction is a collection of functors $\\{\\Phi_{\\alpha}\\colon (U\\!<\\! I)\/_{\\alpha}\\to X_{i}\\}_{\\alpha}$ natural in the maps $\\alpha\\colon u\\to i$ ranging over the objects of $U\\!<\\! i$, together with  objects $x_u\\in X_u$ for every $u\\in U$, and compatible natural transformations for every $\\alpha\\colon u\\to i$\n\\[\\gamma_{\\alpha}\\colon \\alpha_\\ast x_u\\longrightarrow \\Phi_\\alpha\\]\n Here $\\alpha_{\\ast}x_u\\colon (U\\!<\\! I)\/_{\\alpha}\\to X_{i}$ is the constant functor with value $\\alpha_{\\ast}x_u$. Given such an object $(\\Phi,\\underline{x},\\underline{\\gamma})$, define a natural transformation $\\Psi\\colon (U\\!\\leq\\! I)\/_{(-)}\\to X_{U\\leq}$ as follows.\n\nAn object of $(U\\!\\leq\\! I)\/_{\\alpha}$ is a factorization $\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}}$, and an object of $(U\\!<\\! I)\/_{\\alpha}$ is a similar factorization where the map $u\\to k$ is not an identity. The functor $\\Psi_\\alpha\\colon (U\\!\\leq\\! I)\/_{\\alpha}\\to X_{i}$ is defined on objects by\n\\[\\Psi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}})=\\left\\{\\begin{array}{ll}\\alpha_{\\ast}x_u&\\ , \\mbox{if } (u\\to k)=\\id_u\\\\\n\\Phi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}})&\\ , \\mbox{if } (u\\to k)\\neq \\id_u\n\\end{array}\\right.\\]\nThe point here is that $\\Phi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}})$ is defined precisely when $u\\to k$ is not an identity.\n\\vspace{-.4cm}\nA morphism $\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}}\\to \\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&l\\ar[ur]}}}$ in $(U\\!\\leq\\! I)\/_\\alpha$ is a map $k\\to l$ such that the two relevant triangles commute. Such a morphism is sent to\n\\[\\Psi_\\alpha\\Big(\\!\\!\\vcenter{\\hbox{\n\\xymatrix@C=5pt@R=8pt{u\\ar[ddr]\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[d]\\ar[ur]\\\\\n&l\\ar[uur]}}}\\Big)=\\left\\{\\begin{array}{ll}\n\\id_{\\alpha_{\\ast}x_u}&, \\mbox{if } (u\\to l)=\\id_u\n\\\\\n\\gamma_{\\alpha}\\colon \\alpha_\\ast x_u\\rightarrow \\Phi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&l\\ar[ur]}}})&, \\mbox{if } (u\\to l)\\neq \\id_u  ,\\ (u\\to k)= \\id_u\n\\\\\n\\Phi_\\alpha\\big(\\vcenter{\\hbox{\\xymatrix@=8pt{k\\ar[d]\\\\ l}}}\\big)\\colon \\Phi_\\alpha\\big(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}}\\big)\\to\\Phi_\\alpha\\big(\\vcenter{\\hbox{\\xymatrix@=6pt{u\\ar[dr]\n\\ar[rr]^{\\alpha}&&i\\\\\n&l\\ar[ur]}}}\\big)\n&, \\mbox{if } (u\\to l)\\neq \\id_u ,\\ (u\\to k)\\neq \\id_u\n\\end{array}\\right.\\]\nNotice that if $u\\to l$ is the identity map on $u$, by degree reasons both $u\\to k$ and $k\\to l$ must be identities.\nThis procedure defines a functor \n\\[\\Big(\\Hom\\big((U\\!<\\! I)\/_{(-)},X_{U<}\\big)\\mathlarger{\\mathlarger{\\wr}}F_U\\Big)\\longrightarrow \\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\]\non objects. Extend this on morphisms as follows. Unraveling the definitions of the Grothendieck construction and of the natural transformations category, we see that a morphism $(\\Phi,\\underline{x},\\underline{\\gamma})\\to (\\Phi',\\underline{x}',\\underline{\\gamma}')$ in the left-hand category is a collection of compatible natural transformations $\\lambda_{\\alpha}\\colon \\Phi_\\alpha\\to \\Phi_{\\alpha}'$, for every non-identity map $\\alpha\\colon u\\to i$ with $u\\in U$, together with morphisms $f_u\\colon x_u\\to x_u'$ in $X_u$ for every $u\\in U$, which make the squares\n\\[\\xymatrix{\\alpha_\\ast x_u\\ar[r]^{\\alpha_\\ast f_u}\\ar[d]_{\\gamma_{\\alpha}}&\\alpha_\\ast x'_u\\ar[d]^{\\gamma'_{\\alpha}}\\\\\n\\Phi_\\alpha\\ar[r]_{\\lambda_{\\alpha}}&\\Phi'_{\\alpha}\n}\\]\ncommutative. Such a pair $(\\lambda,\\underline{f})$ induces a morphism $\\Psi\\to \\Psi'$ between the associated natural transformations in $\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)$, defined at a non-identity morphism $\\alpha\\colon u\\to i$ by\n\\[\\Psi_\\alpha=\\Phi_\\alpha\\stackrel{\\lambda_\\alpha}{\\longrightarrow}\\Phi'_\\alpha=\\Psi'_\\alpha\\]\nand at an identity map $\\id_u$ by $f_u\\colon \\Psi_{\\id_u}=\\alpha_\\ast x_u\\to\\alpha_\\ast x'_u=\\Psi'_{\\id_u}$.\nThe resulting functor is an isomorphism of categories. Its inverse sends a natural transformation $\\{\\Psi_\\alpha\\colon (U\\!\\leq\\! I)\/_{\\alpha}\\to X_i\\}_{\\alpha\\colon u\\to i}$ to the triple $(\\Phi,\\underline{x},\\underline{\\gamma})$ consisting of the restrictions $\\Phi_\\alpha\\colon (U\\!<\\! I)\/_{\\alpha}\\to (U\\!\\leq\\! I)\/_\\alpha\\stackrel{\\Psi_\\alpha}{\\to}X_i$ for each $(\\alpha\\colon u\\to i)\\in U\\!<\\! I$, the objects $x_u=(\\Psi_u\\colon\\ast=(U\\!\\leq\\! I)\/_{\\id_u}\\to X_u)$, and the natural transformations $\\gamma_{\\alpha}$ defined at an object $\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}}$ of $(U\\!\\leq\\! I)\/_\\alpha$ by the morphism in $X_i$\n \\[\\alpha_\\ast x_u=\\alpha_\\ast\\Psi_{\\id_u}(\\vcenter{\\hbox{\\xymatrix@=4pt{u\\ar@{=}[dr]\\ar@{=}[rr]&&u\\\\\n&u\\ar@{=}[ur]}}})\n=\\Psi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar@{=}[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&u\\ar[ur]_{\\alpha}}}})\n\\stackrel{}{\\longrightarrow}\n\\Psi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}})=\\Phi_\\alpha(\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}})\\]\nHere the second equality holds by naturality of $\\Psi$, and the arrow is $\\Psi_{\\alpha}$ applied to the morphism $\\vcenter{\\hbox{\\xymatrix@C=9pt@R=7pt{u\\ar[ddr]\\ar@{=}[dr]\n\\ar[rr]_{\\alpha}&&i\\\\\n&u\\ar[d]\\ar[ur]\\\\\n&k\\ar[uur]}}}$\nof $(U\\!\\leq\\! I)\/_{\\alpha}$ induced by the factorization $\\vcenter{\\hbox{\\xymatrix@=5pt{u\\ar[dr]\\ar[rr]^{\\alpha}&&i\\\\\n&k\\ar[ur]}}}$. The inverse can be extended similarly to morphisms.\n\\end{proof}\n\n\n\n\\begin{proof}[Proof of Theorem $B^I$]\nLet $NX\\to FNX$ be a pointwise fibrant replacement of the diagram $NX$.\nWe prove just below, by induction on $n$, that for every subset $U\\subset I_n$ the map\n\\begin{equation}\\label{induction} N\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\cong \\Hom\\big(N(U\\!\\leq\\! I)\/_{(-)},NX_{U\\leq}\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_{U\\leq I}NX_{U\\leq}\n\\end{equation}\nis a weak equivalence. In particular by choosing $U=I_n$ the category $I_n\/I$ is $I_{\\leq n}$, and this is an equivalence \n\\[N\\Hom\\big((I_{\\leq n})\/_{(-)},X_{\\leq n}\\big)\\stackrel{\\simeq}{\\longrightarrow} \\displaystyle\\holim_{I_{\\leq n}} NX_{\\leq n}\\]\nIf $NI$ is finite dimensional, then $I=I_{\\leq d}$ for some integer $d$, and the map $N\\Hom(I\/_{(-)},X)\\to \\displaystyle\\holim_{I} NX$ is an equivalence. When $I$ is infinite, taking the homotopy limit over the maps induced by the filtration gives an equivalence\n\\[\\holim_{n\\in \\mathbb{N}^{op}}N\\Hom\\big((I_{\\leq n})\/_{(-)},X_{\\leq n}\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_{n\\in \\mathbb{N}^{op}}\\holim_{I_{\\leq n}} NX_{\\leq n}\\]\nThe structure maps $\\holim_{I_{\\leq n}} NX_{\\leq n}\\to \\holim_{I_{\\leq n-1}} NX_{\\leq n-1}$ in the right-hand tower are Kan fibrations. Indeed, they are induced by mapping the cofibrations of diagrams of simplicial sets $\\iota_{n}\/_{(-)}\\to I_{\\leq n}$, where $\\iota_n\\colon I_{\\leq n-1}\\to I_{\\leq n}$ is the inclusion, into the fibrant diagram $FNX_{\\leq n}$. Hence the right-hand homotopy limit is equivalent to the categorical limit. Now each $\\Hom\\big(N(I_{\\leq n})\/_{(-)},FNX_{\\leq n}\\big)$ is isomorphic to $\\Hom\\big(Nj_n\/_{(-)},FNX\\big)$, where $j_n\\colon I_{\\leq n}\\to I$ is the inclusion. The right-hand limit is then\n\\[\\lim_{n\\in \\mathbb{N}^{op}}\\Hom\\big(N(I_{\\leq n})\/_{(-)},FNX_{\\leq n}\\big)\\cong \\Hom\\big(\\colim_{n}Nj_n\/_{(-)},FNX\\big)\\cong\\holim_INX\\]\nThe last isomorphism holds as the category $j_{n}\/_{i}$ includes in $j_{n+1}\/_{i}$ for every object $i$ of $I$, with union $\\bigcup_{n\\in\\mathbb{N}}j_{n}\/_{i}=I\/_i$. This will finish the proof of Theorem $B^I$.\n\nWe are left with proving the inductive statement (\\ref{induction}) above.\nThe base induction step $n=0$, relies on the fact that for a subset $U\\subset I_0$, the category $(U\\!\\leq\\! I)$ is discrete, with objects the identity maps $\\id_u$, for $u\\in U$. Therefore the category\n $(U\\!\\leq\\! I)\/_{\\id_u}=\\{\\id_u\\}$ is the one point category. It follows that $\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)$ is the product category\n\\[\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)=\\prod_{u\\in U}X_u\\]\nand the homotopy limit of $NX_{U\\leq}$ is the product\n\\[\\holim NX_{U\\leq}=\\prod_{u\\in U}FNX_u\\]\nSince the product of simplicial sets preserve all equivalences (not only between fibrant objects), the map $N\\prod_{u\\in U}X_u\\to \\prod_{u\\in U}FNX_u$ is an equivalence.\n\nNow suppose that $N\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\to\\holim NX_{U\\leq}$ is an equivalence for every subset $U\\subset I_n$, and let $V$ be a subset of $I_{n+1}$. Let $\\Phi\\colon (V\\!<\\! I)\/_{(-)}\\to X_{V<}$ be a natural transformation, and consider the commutative diagram\n\\[\\xymatrix{\nNF_{V}(\\Phi)\\ar[rr]\\ar[d]&&\\prod\\limits_{v\\in V}\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\ar[d]\\\\\nN \\big(\\Hom\\mbox{\\scalebox{1.3}{$\\wr$}} F_V\\big)\\ar[r]^-{\\cong}\\ar[dr]&N\\Hom\\big((V\\!\\leq\\! I)\/_{(-)},X_{V\\leq}\\big)\\ar[d]\\ar[r]&\\holim NX_{V\\leq}\\ar[d]\\\\\n&N\\Hom\\big((V\\!<\\! I)\/_{(-)},X_{V<}\\big)\\ar[r]^-\\simeq&\\holim NX_{V<}\n}\\]\nThe bottom map is an equivalence by the inductive hypothesis, since $V\\!<\\! I=U\\!\\leq\\! I$ for the subset of objects $U:=(V\\!<\\! I)\\cap I_{n}$ of $I_n$. The isomorphism is from Lemma \\ref{indgrot}. By our assumption on the diagram $X$, the functor $F_V$ sends every morphism to a weak equivalence. Thus by Quillen's Lemma \\cite[p.98]{Quillen} and Thomason's Theorem \\cite{Thomason}, the left-hand vertical sequence is a fiber sequence. Let us turn to the right-hand vertical sequence. Let $\\iota \\colon V\\!<\\! I\\to V\\!\\leq\\! I$ be the inclusion. The map induced by $\\iota$ on homotopy limits is the restriction\n\\[\\holim_{V\\!\\leq\\! I}NX_{V\\leq}\\longrightarrow \\Hom\\big(N\\iota\/_{(-)},FNX_{V\\leq}\\big)\\cong \\holim_{V\\!<\\! I} NX_{V<}\\]\nalong the inclusion $\\iota\/_{(-)}\\to (V\\!\\leq\\! I)\/_{(-)}$. Since this is a cofibration of diagrams of simplicial sets and $FNX_{V\\leq}$ is fibrant, the restriction map is a Kan fibration.\nIts point fiber over $\\Phi$ is the product of total homotopy fibers\n\\[\\prod_{v\\in V}\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\]\nand therefore the right-hand vertical sequence in the diagram above is also a fiber sequence.\nFor finishing our inductive proof, is then enough to show that the map on homotopy fibers\n \\[NF_V(\\Phi)=\\prod_{v\\in V}N\\big( X_v\\stackrel{m_v}{\\to} \\Hom\\big((v\\!<\\! I)\/_{(-)},X_{v<}\\big)\\big)\/_{\\Phi|_{v\\!<\\! I}}\\longrightarrow \\prod_{v\\in V}\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\]\nis an equivalence. The product components of this map factor as\n\\[\\xymatrix{N\\big(X_v\\stackrel{m_v}{\\to}\\Hom\\big((v\\!<\\! I)\/_{(-)},X_{v<}\\big)\\big)\/_{\\Phi|_{v<I}}\\ar[r]\\ar[dr]&\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to N\\Hom\\big((v\\!<\\! I)\/_{(-)},X_{v<}\\big)\\big)\\ar[d]\\\\\n&\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)}\\]\nOur assumption on $X$ says that the functor $m_v\/_{(-)}$ sends every map to a weak equivalence. Hence by Quillen's Theorem $B$ the horizontal map at the top of the triangle is an equivalence. The vertical map is also an equivalence, as by hypothesis of induction for the set $U:=(v\\!<\\! I)\\cap I_{n}$ the map $N\\Hom\\big((v\\!<\\! I)\/_{(-)},X_{v<}\\big)\\big)\\to \\holim NX_{v<}$ is an equivalence.\n\\end{proof}\n\n\nThe following result can be thought of as a generalization of Quillen's Theorem $B$ to higher dimensional cubes, with a higher Quillen Theorem $A$ as a consequence.\n\n\n\\begin{cor}\\label{higherQ}\nLet $n\\geq 1$ be an integer, let $X\\colon\\mathcal{P}(n)\\to Cat$ be a Reedy quasi-fibrant cube of categories, and let $\\Phi\\colon \\mathcal{P}_0(-)\\to X_{\\emptyset <}$ be a natural transformation. The total homotopy fiber of $NX$ over $N\\Phi$ is equivalent to the nerve of the over category $m_{\\emptyset}\/_{\\Phi}$. In particular if the categories $m_{\\emptyset}\/_{\\Phi}$ are contractible $NX$ is homotopy cartesian.\n\\end{cor}\n\\begin{proof}\nLet us recall that the total homotopy fiber of $NX$ over $N\\Phi$ is the homotopy fiber\n\\[\\hofib_\\Phi\\big(NX_{\\emptyset}\\longrightarrow \\holim_{\\mathcal{P}_0(n+1)}NX_{\\emptyset <}\\big)\\]\nover the element in the homotopy limit defined by $N\\Phi$.\nClearly the restriction $X_{\\emptyset <}$ of $X$ to $\\mathcal{P}_0(n+1)$ is also Reedy quasi-fibrant, and by Theorem $B^I$ the total homotopy fiber is equivalent the homotopy fiber of\n\\[\\hofib_\\Phi\\big(NX_{\\emptyset}\\stackrel{Nm_{\\emptyset}}{\\longrightarrow} N\\Hom\\big(\\mathcal{P}_0(-),X_{\\emptyset <}\\big)\\big)\\]\nSince $m_{\\emptyset}\/_{(-)}$ also sends all maps to equivalences, this is equivalent to $Nm_{\\emptyset}\/_{\\Phi}$ by Quillen's Theorem $B$.\n\\end{proof}\n\n\\subsection{The equivariant Quillen Theorem $B$}\\label{secBGI}\n\n\nLet $G$ be a discrete group acting on a category $I$. We generalize the results of the previous section to $G$-homotopy limits of $G$-diagrams of categories $X\\colon I\\to Cat$. As in \\S\\ref{secBI} it is going to be convenient to work with diagrams of simplicial sets instead of diagrams of topological spaces. The only difference is that one needs to perform the suitable fibrant replacements. We say that a morphism of $G$-diagrams of simplicial sets $f\\colon Z\\to Y$ is an equivalence if is an equivalence of $G$-diagrams of topological spaces (as defined in \\ref{defeqGdiag}) after taking geometric realizations.\n\n\\begin{defn}\\label{defHholimsset}\nThe $G$-homotopy limit of a $G$-diagram of simplicial sets $Y\\colon I\\to sSet$ is the $G$-simplicial set of natural transformations\n\\[\\holim_IY=\\Hom\\big(N(I\/_{(-)}),FY\\big)\\]\nwhere $FY$ is a $G$-diagram of simplicial sets with an equivalence $Y\\stackrel{\\simeq}{\\to}FY$, with the property that $(FY)_i$ is a fibrant $G_i$-simplicial set for every object $i$ in $I$.\n\\end{defn}\n\nIt is proved in \\cite[2.6]{Gdiags} that such a replacement $Y\\stackrel{\\simeq}{\\to}FY$ always exists, and that the $G$-homotopy limit functor preserves equivalences of $G$-diagrams of simplicial sets. As simplicial mapping spaces with fibrant target commute with geometric realizations, there is a $G$-equivalence $|\\holim_IY|\\simeq\\holim_I|Y|$ relating this construction with the $G$-homotopy limit of \\S\\ref{secGdiags}.\n\nIf $K,X\\colon I\\to Cat$ are two $G$-diagrams of categories, the category of natural transformations $\\Hom(K,X)$ has an induced $G$-action by conjugation, and the isomorphism of simplicial sets $N\\Hom(K,X)\\cong \\Hom(NK,NX)$ is $G$-equivariant. Composing this isomorphism with a fibrant replacement of $X$ leads to a $G$-equivariant map\n\\[N\\Hom\\big(I\/_{(-)},X\\big)\\longrightarrow \\holim_I NX\\]\nWe extend the quasi-fibrancy condition \\ref{quasiReedy} to the equivariant setting, and we show that this map is an equivalence.\nSuppose that $I$ is left-finite. The fixed point categories $I^H$ are automatically left-finite for every subgroup $H$ of $G$. For every object $i$ of $I^H$, the under category $i\/I$ has an action of $H$, that restricts to the subcategory $i\\!<\\! I$.\nThe restriction of a $G$-diagram $X\\colon I\\to Cat$ to $i\\!<\\! I$ has a canonical structure of $H$-diagram, and the functor $m_i\\colon X_i\\rightarrow \\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)$ of Definition \\ref{quasiReedy} is $H$-equivariant. Let\n\\[m^{H}_i\\colon X^{H}_i\\longrightarrow \\Hom\\big((i\\!<\\! I)\/_{(-)},X_{i<}\\big)^{H}\\] be its restriction to the categories of fixed points.\n\n\\begin{defn}\\label{defGreedy}\nA $G$-diagram of categories $X\\colon I\\to Cat$ is $G$-Reedy quasi-fibrant if for every subgroup $H$ of $G$ and every object $i$ of $I^H$ the functor $m^{H}_i\/_{(-)}$\nsends every morphism to a weak equivalence of categories.\n\\end{defn}\n\n\n\\begin{theoremBIG}\nLet $I$ be a left finite category with $G$-action, and let $X\\colon I\\to Cat$ be a $G$-diagram of categories. If $X$ is $G$-Reedy quasi-fibrant, there is a weak $G$-equivalence\n\\[\\holim_{n\\in \\mathbb{N}^{op}}N\\Hom\\big((I_{\\leq n})\/_{(-)},X_{\\leq n}\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_I NX\\]\nIn particular if the nerve of $I$ is finite dimensional the map $N\\Hom\\big(I\/_{(-)},X\\big)\\stackrel{\\simeq}{\\longrightarrow}\\holim_I NX$\nis a weak $G$-equivalence of simplicial $G$-sets.\n\\end{theoremBIG}\n\nLet $J$ be a finite $G$-set. The following result is our equivariant generalization of Quillen Theorems' $A$ and $B$.\n\n\\begin{cor}\\label{GQuillen}\nLet $X\\colon \\mathcal{P}(J)\\to Cat$ be a $G$-Reedy quasi-fibrant $J$-cube of categories. For every natural transformation $\\Phi\\colon \\mathcal{P}_0(-)\\to X_{\\emptyset<}$ the nerve of the category $m_{\\emptyset}\/_{\\Phi}$ is $G_{\\Phi}$-equivalent to the total homotopy fiber of the cube $NX$ over $N\\Phi$. In particular if all the categories $m_{\\emptyset}\/_{\\Phi}$ are $G_{\\Phi}$-contractible, $BX$ is a homotopy cartesian $J$-cube of spaces.\n\\end{cor}\n\n\\begin{proof}\nIt is immediate from Theorem $B^{I}_G$, using the argument of \\ref{higherQ}.\n\\end{proof}\n\nThe proof of Theorem $B_{G}^I$ is based on the same inductive argument in the proof of Theorem $B^I$. The key ingredient for the induction step is an equivariant analogue of Lemma \\ref{indgrot}. If $Y\\colon I\\to Cat$ is a $G$-diagram of categories, recall from \\ref{GThom} that its Grothendieck construction $I\\wr Y$ has an induced $G$-action.\nGiven a subset $U\\subset I_n$, the $G$-action on $I$ induces a $G_U$-action on the categories $U\\leq I$ and $U<I$, where $G_U$ is the subgroup of $g$ of elements that send $U$ to itself. The functor $F_U\\colon \\Hom\\big((U\\!<\\! I)\/_{(-)},X_{U<}\\big)\\to Cat$ from Lemma \\ref{indgrot} that sends $\\Phi\\colon (U\\!<\\! I)\/_{(-)}\\to X_{U<}$ to \n\\[F_U(\\Phi)=\\prod_{u\\in U}(m_u)\/_{(\\Phi|_{u<I})}\\]\nhas a canonical $G_U$-structure. It is defined by conjugating the $G_U$-action on $U$ indexing the product with the functors\n\\[\\xymatrix{(m_u)\/_{(\\Phi|_{u<I})}\\ar[r]\\ar@{-->}[d]&X_u\\ar[r]\\ar[d]^{g}&\\Hom\\big((u\\!<\\! I)\/_{(-)},X_{u<}\\big)\\ar[d]^g\\\\\n(m_u)\/_{(g\\Phi|_{u<I})}\\ar[r]&X_u\\ar[r]&\\Hom\\big((u\\!<\\! I)\/_{(-)},X_{u<}\\big)\n}\\]\nHence the Grothendieck construction of $F_U$ inherits a $G_U$-action. The following is immediate.\n\\begin{lemma}\nFor every subset $U\\subset I_n$, the isomorphism of categories\n\\[\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\cong \\Big(\\Hom\\big((U\\!<\\! I)\/_{(-)},X_{U<}\\big)\\mathlarger{\\mathlarger{\\wr}} F_U\\Big)\\]\nof Lemma \\ref{indgrot} is $G_U$-equivariant.\n\\end{lemma}\n\n\\begin{proof}[Proof of Theorem $B_{G}^I$]\nFor every group element $g$ of $G$, the automorphism $g$ of $I$ induces an isomorphism of categories $g\\colon i\/I\\to gi\/I$. It follows that the nerves $N(i\/I)$ and $N(gi\/I)$ have the same dimension, and that the degree function $\\deg\\colon Ob I\\to\\mathbb{N}$ is $G$-invariant. Hence the $G$-action restricts to the filtration\n\\[I_{\\leq 0}\\subset I_{\\leq 1}\\subset\\dots\\subset I_{\\leq n}\\subset\\dots\\subset I\\]\nand the $G$-structure on $X\\colon I\\to Cat$ restricts to a $G$-structure on $X_{\\leq n}\\colon I_{\\leq n}\\to Cat$.\nLet $NX\\stackrel{\\simeq}{\\to} FNX$ be a pointwise fibrant replacement of $NX$, like in Definition \\ref{defHholimsset}. We prove by induction on $n$ that for every subset $U\\subset I_n$ the map\n\\[N\\Hom\\big((U\\!\\leq\\! I)\/_{(-)},X_{U\\leq}\\big)\\cong \\Hom\\big(N(U\\!\\leq\\! I)\/_{(-)},NX_{U\\leq}\\big)\\longrightarrow\\holim_{U\\leq I}(FNX)_{U\\leq}\\]\nis a weak $G_U$-equivalence. Once this is established, the same argument in the proof of Theorem $B^I$ finishes the proof of $B_{G}^I$.\n\nFor $n=0$, the category $U\\leq I$ is discrete and the map above is the map of indexed products\n\\[\\prod\\limits_{u\\in U}NX_u\\longrightarrow \\prod\\limits_{u\\in U}FNX_u\\]\nThe fixed points of this map by a subgroup $H\\leq G_U$ is isomorphic to the map\n\\[\\prod\\limits_{[u]\\in U\/H}NX_{u}^{H_{u}}\\longrightarrow \\prod\\limits_{[u]\\in U\/H}FNX_{u}^{H_{u}}\\]\nfor a choice of representatives in each $H$-orbit of $U$, where $H_u$ is the stabilizer group of $u$ in $H$. Each map $NX_{u}^{H_{u}}\\to FNX_{u}^{H_{u}}$ is an equivalence of simplicial sets by assumption, and the map above is an equivalence.\n\nNow suppose that the claim is true for $n$, and let $V$ be a subset of $I_{n+1}$. The sequence\n\\[NF_{V}(\\Phi)\\longrightarrow N\\big(\\Hom\\mathlarger{\\mathlarger{\\wr}} F_V\\big)\\cong N\\Hom\\big((V\\!\\leq\\! I)\/_{(-)},X_{V\\leq}\\big)\\longrightarrow\nN\\Hom\\big((V\\!<\\! I)\/_{(-)},X_{V<}\\big)\\]\ninduced by the restriction map is a fiber sequence of simplicial $G_V$-sets. This is because its restriction on fixed points of a subgroup $H\\leq G_V$ is the sequence\n\\[NF_{V}(\\Phi)^H\\longrightarrow N\\big(\\Hom\\mathlarger{\\mathlarger{\\wr}} F_V\\big)^ H\\cong N \\big(\\Hom^H\\mathlarger{\\mathlarger{\\wr}} F^{H}_V\\big)\\longrightarrow\nN\\Hom\\big((V\\!<\\! I)\/_{(-)},X_{V<}\\big)^H\\]\nwhere the functor $F^{H}_V\\colon N\\Hom\\big((V\\!<\\! I)\/_{(-)},X_{V<}\\big)^H\\to Cat$ sends an $H$-equivariant natural transformation $\\Phi$ to\n\\[F^{H}_V(\\Phi)=\\Big(\\prod_{v\\in V}(m_v)\/_{(\\Phi|_{v<I})}\\Big)^H\\cong \\prod\\limits_{[v]\\in V\/H}m^{H_v}_v\/_{(\\Phi|_{v<I})}\\]\nBy assumption  $m^{H_v}_v\/_{(-)}$ sends every morphism to a weak equivalence, and thus so does $F^{H}_V$. It follows by Lemma \\cite[p.98]{Quillen} and \\cite{Thomason} that $NF^{H}_V$ is indeed the homotopy fiber of the restriction map. The restriction map\n\\[\\holim NX_{V\\leq}\\longrightarrow \\holim NX_{V<}\n\\]\nis a fibration of simplicial $G$-sets by an argument analogous to the one in the proof of Theorem $B^I$. Its fiber is the product of homotopy fibers $\\prod\\limits_{v\\in V}\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)$. Therefore it is sufficient to show that the map on homotopy fibers\n\\[NF_V(\\Phi)\\longrightarrow \\prod\\limits_{v\\in V}\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\]\nis a $G_V$-equivalence. By taking fixed points, this is the case if for every $v\\in V$ the map\n\\[Nm_v\/_{(\\Phi|_{v<I})}\\longrightarrow\\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\]\nis a $G_v$-equivalence. This map factors as\n\\[Nm_v\/_{(\\Phi|_{v<I})}\\to\\hof_{\\Phi|_{v<I}}\\Big( NX_v\\to N\\Hom\\big((v\\!<\\!I)\/_{(-)},X_{v<}\\big)\\Big)\\to \\hof_{\\Phi|_{v<I}}\\big( NX_v\\to \\holim NX_{v<}\\big)\\]\nThe first map is a $G_v$-equivalence, since $m^{H}_v\/_{(-)}$ sends every morphism to a weak equivalence of categories for every subgroup $H$ of $G_v$. The second map is also a $G_v$ equivalence, as the map $N\\Hom\\big((v<I)\/_{(-)},X_{v<}\\big)\\to\\holim NX_{v<}$ is a $G_v$-equivalence by the inductive hypothesis.\n\\end{proof} \n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the 1990s, the nonnormal nature of shear flows and its\nconsequences became well understood and extensively studied by the\nhydrodynamic community (see e.g., Refs.~\\cite{Farrell88,\nButler_Farrell92,Reddy_etal93,Trefethen_etal93, Reddy_Henningson93,\nSchmid_Henningson01,Criminale_etal03, Schmid07, Zhuravlev14}). As a\nresult, a new, so-called bypass concept was developed for explaining\nthe onset and self-sustenance of turbulence in spectrally stable\nshear flows\n\\cite{Farrell_Ioannou94,Gebhardt_Grossmann94,Henningson_Reddy94,Baggett_etal95,Grossmann00,\nChapman02,Eckhardt_etal07,Farrell_Ioannou12,Brandt14}. It is based\non the linear nonnormality-induced, or nonmodal growth of vortical\nperturbations due to the nonnormality (shear), which is the only\nsource of energy for turbulence in such flows\n\\cite{Henningson_Reddy94}. In this case, the role of nonlinear\nprocesses becomes crucial -- they lie at the heart of\nself-sustenance of the turbulence. In the linear regime, without\nnonlinear feedback, the nonmodal growth has a transient character\nand thus itself alone is incapable of self-sustenance.\n\nStudies of turbulence in constant shear flows, or homogeneous shear\nturbulence usually concentrate on dynamics and statistics in\nphysical space. Coherent vortical structures responsible for the\nself-sustenance of the turbulence were identified and described in\n\\cite{Tavoularis_Corrsin81,Rogers_Moin87,Lee_etal90,\nKida_Tanaka94,Pumir_Shraiman95,Waleffe95,Waleffe97}. The dependence\nof the dynamics of these structures, and hence the characteristics\nof turbulence, on the aspect ratio of flow domain was extensively\nanalyzed in \\cite{Pumir96,Sekimoto_etal15}. Compared with these\nstudies, which mostly perform analysis in physical space, the main\nidea of our investigation is the analysis of\nturbulence dynamics in three-dimensional (3D) spectral, or Fourier (${\\bf k}$-) space that allows us:\\\\\n-- to identify harmonics\/modes that play a key role in the\nself-sustaining process of turbulence;\\\\\n-- to define a wavenumber area in Fourier space that is vital in the\nself-sustenance of turbulence;\\\\\n-- to define the range of aspect ratios for which the dynamically\nimportant modes are fully taken into account;\\\\\n-- to reveal a new type of nonlinear cascade process that\nprincipally differs from the canonical ones, i.e., from direct and\ninverse cascades;\\\\\n-- to show that the turbulence is sustained by a subtle interplay\nbetween the linear nonmodal growth and the new type of the nonlinear\ncascade.\n\n\nGenerally, in spectrally stable shear flows, the classical nonlinear\ndirect and inverse cascades alone are in fact not able to ensure the\nself-sustenance of transiently growing perturbations. In the case of\na specific spectrally stable shear flow, it was demonstrated that\nthe turbulence can be self-organized and self-sustained by a subtle\ninterplay of the linear nonmodal and nonlinear processes\n\\cite{Gebhardt_Grossmann94,Baggett_etal95,Waleffe95,\nWaleffe97,Grossmann00,Chapman02, Eckhardt_etal07,\nFarrell_Ioannou12}. In this situation, a continuous energy supply\nfrom shear flow into the turbulence is provided by the linear\nnonmodal growth mechanism owing to an essential positive feedback\nprovided by the nonlinear processes. One can say that subcritical\nturbulence in shear flows falls outside the validity of the\nclassical Kolmogorov phenomenology (i.e., direct and inverse\ncascades). The deviation from the latter occurs especially in the\nregion of small and intermediate wavenumbers of perturbations, where\nthe linear nonnormality-induced energy-exchange and (nonclassical)\nnonlinear processes work in cooperation to ensure the sustenance of\nshear turbulence.\n\nActually, the shear-induced linear nonmodal growth of a perturbation\nharmonic mainly depends on its wavevector orientation and, to a\nlesser degree, on the value (e.g., see Refs.\n\\cite{Farrell87,Farrell88,Butler_Farrell92,Farrell_Ioannou93,\nSchmid_Henningson01,Jimenez13}): the spatial Fourier harmonics that\nhave a certain orientation of the wavevector with respect to shear\nflow draw energy from it and grow, whereas harmonics with other\norientation of the wavevector give energy back to the flow and\ndecay. This anisotropy of the linear energy-exchange processes with\nrespect to wavevector orientation (angle), in turn, leads to\nanisotropy of nonlinear processes in {\\bf k}-space. Specifically, as\nrevealed in \\cite{Chagelishvili_etal02,\nHorton_etal10,Mamatsashvili_etal14,Salhi_etal14}, in HD\/MHD smooth\nshear flows, the main nonlinear process is not a direct or inverse,\nbut rather a nonlinear transverse cascade, that is\nangular\/transverse redistribution of perturbation harmonics in {\\bf\nk}-space. The nonlinear transverse cascade, or transverse cascade\nfor short, represents an alternative to the classical direct and\ninverse cascades in the presence of flow shear.\n\nIn the paper by Horton {\\it et al.} \\cite{Horton_etal10}, the\nnonlinear dynamics of coherent and stochastic vortical perturbations\nin two-dimensional (2D) plane constant shear flow was studied\nnumerically and the existence and importance of the transverse\ncascade for the self-sustaining dynamics of perturbations was\ndemonstrated. Specifically, it was shown that its interplay with\nlinear nonnormality-induced phenomena becomes intricate: it can\nrealize either positive or negative feedback. In the case of the\nformer, the transverse cascade repopulates those quadrants in {\\bf\nk}-space where the shear flow causes linear nonmodal growth. In the\npaper by Mamatsashvili {\\it et al.} \\cite{Mamatsashvili_etal14}, the\ndynamics of 2D perturbations in incompressible MHD flows with\nconstant shear of velocity and uniform magnetic field parallel to\nthe flow was studied. Investigating subcritical turbulence, the\naction of the transverse cascade -- a keystone of the turbulence\nself-sustaining (non-decaying) dynamics in this simple open MHD flow\nsystem -- was described in detail. In this spirit, Salhi {\\it et\nal.} \\cite{Salhi_etal14} studied 3D homogeneous turbulence in\nrotating shear flows, focusing on the anisotropy of the dynamics in\nspectral space, especially, on the nonlinear transverse\nredistribution. A refined analysis was done, employing the\npolarization and helicity scalars. However, due to the rather short\nfinal time, the DNS had no possibility to characterize a saturated\nstate of the turbulence and hence had no access to the complete set\ncharacteristic quantities. Nevertheless, putting the line of\nresearch of paper \\cite{Salhi_etal14} into perspective, it could\nyield important results on the spectral anisotropy of shear\nturbulence in the long-time DNS for which the saturated state is\nalready reached.\n\nIn the present paper, we consider the dynamics of homogeneous shear\nturbulence in spectral space, extending the knowledge and results\ngained from the investigation of a 2D constant shear flow to a more\nrealistic 3D one. An advantage of the homogeneous shear turbulence\nis that it includes the basic effects of shear, while avoiding\ncomplications arising in more realistic cases (e.g., effect of\nboundaries, etc.). The proposed investigation of this simple shear\nflow system allows us to gain insight into the scheme of realization\nof the bypass concept in the case of homogeneous shear turbulence.\n\nThe paper is organized as follows. The physical model and derivation\nof dynamical equations in spectral space is in Sec. II. The direct\nnumerical simulations (DNS) of the turbulence dynamics at different\naspect ratios of the flow domain as well as the analysis of the\nself-sustaining mechanism in Fourier space are presented in Sec.\nIII. A summary and discussion are given in Sec. IV.\n\n\\section{Physical model and equations}\n\nThe motion of an incompressible fluid with constant kinematic\nviscosity, $\\nu$, is governed by the Navier-Stokes equations\n\\begin{equation}\n\\frac{\\partial {\\bf U}}{\\partial t}+\\left({\\bf U}\\cdot \\nabla\n\\right){\\bf U}=-\\frac{\\nabla P}{\\rho}+\\nu\\nabla^2 {\\bf U},\n\\end{equation}\n\\begin{equation}\n\\nabla\\cdot {\\bf U}=0,\n\\end{equation}\nwhere $\\rho,~{\\bf U},~P$ are the density, velocity and pressure,\nrespectively.\n\nEquations (1)-(2) have a stationary equilibrium solution with\nvelocity profile ${\\bf U}_0=(Sy,0,0)$ and spatially constant density\nand pressure ($\\rho_0, P_0=const.$), i.e., a flow in the streamwise\n$x$-direction with a linear shear of velocity in the\nvertical\/shearwise $y$-direction, while the $z$-axis points in the\nspanwise direction. Without loss of generality, the constant shear\nrate $S$ is chosen to be positive. Such an idealized configuration\nof a flow with a linear shear, despite its simplicity, allows us to\ngrasp key effects of the shear on the perturbation dynamics and\nultimately on a resulting turbulent state.\n\nConsider 3D perturbations of the velocity and pressure, ${\\bf u}$\nand $p$, about the background flow: ${\\bf u}={\\bf U}-{\\bf U}_0,\np=P-P_0$. Eqs. (1)-(2) then give the following system of nonlinear\nequations for the perturbations\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t} + Sy \\frac{\\partial}{\\partial x}\n\\right)u_x = - Su_y - \\frac{1}{\\rho_0}\\frac{\\partial p}{\\partial\nx}-\\\\\\frac{\\partial}{\\partial x}u_x^2-\\frac{\\partial}{\\partial\ny}(u_xu_y)-\\frac{\\partial}{\\partial z}(u_xu_z)+\\nu\\nabla^2u_x,\n\\end{multline}\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}  + Sy \\frac{\\partial}{\\partial x}\n\\right)u_y =-\\frac{1}{\\rho_0}\\frac{\\partial p}{\\partial\ny}-\\\\\\frac{\\partial}{\\partial x}(u_xu_y)-\\frac{\\partial}{\\partial\ny}u_y^2-\\frac{\\partial}{\\partial z}(u_yu_z)+\\nu\\nabla^2u_y,\n\\end{multline}\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}  + Sy \\frac{\\partial}{\\partial x}\n\\right)u_z =-\\frac{1}{\\rho_0}\\frac{\\partial p}{\\partial\nz}-\\\\\\frac{\\partial}{\\partial x}(u_xu_z)-\\frac{\\partial}{\\partial\ny}(u_yu_z)-\\frac{\\partial}{\\partial z}u_z^2+\\nu\\nabla^2u_z,\n\\end{multline}\n\\begin{equation}\\label{4}\n\\frac{\\partial u_x}{\\partial x}+\\frac{\\partial u_y}{\\partial\ny}+\\frac{\\partial u_z}{\\partial z}=0.\n\\end{equation}\n\nTo investigate the energy balance in the self-sustained turbulent\nstate, from Eqs. (3)-(6) we derive a dynamical equation for the\nkinetic energy density of perturbations, $E= {\\rho_0{\\bf u}^2}\/2$:\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t} + Sy \\frac{\\partial}{\\partial x}\n\\right)E =-S\\rho_0u_xu_y+\\nabla\\cdot[{\\bf u}\n(p+E)]+\\\\\\nu\\nabla\\cdot(\\rho_0{\\bf u}\\otimes\\nabla{\\bf\nu})-\\rho_0\\nu[\\left(\\nabla u_x\\right)^2+\\left(\\nabla u_y\\right)^2\n+\\left(\\nabla u_z\\right)^2],\n\\end{multline}\nwhich after averaging over an entire flow domain gives\n\\begin{equation}\n\\frac{d}{dt}\\langle E \\rangle=-S\\left\\langle\n\\rho_0u_xu_y\\right\\rangle -\\rho_0\\nu\\langle \\left(\\nabla\nu_x\\right)^2+\\left(\\nabla u_y\\right)^2 +\\left(\\nabla u_z\\right)^2\n\\rangle,\n\\end{equation}\nwhere the angle brackets denote a volume average. The first term on\nthe right hand side of Eq. (8) is the flow shear rate, $S$,\nmultiplied by the volume-averaged Reynolds stress, $\\langle\n\\rho_0u_xu_y \\rangle$, and describes exchange of energy between\nperturbations and the background flow. Note that this term\noriginates from the linear term proportional to shear ($-Su_y$) on\nthe right hand side of Eq. (3). The second -- viscous dissipation --\nterm is negative by definition. The nonlinear terms, represented by\ndivergence in Eq. (7), cancel out in the total energy evolution Eq.\n(8) after volume averaging, because there is no net flux of energy\ninto the flow through the boundaries. Thus, only the Reynolds stress\ncan supply perturbations with energy, extracting it from the\nbackground flow due to shear. In the case of turbulence studied\nbelow, the Reynolds stress ensures energy injection into turbulent\nfluctuations, balancing viscous dissipation. The nonlinear processes\ndo not directly change the total perturbation energy (extracted from\nthe background flow by the stress), but redistribute it among\nFourier harmonics of perturbations with different wavenumbers. In\nthis way, nonlinear processes indirectly contribute to the\nperturbation energy balance. Below, we focus on the nonlinear\nredistribution process in spectral space to grasp the\nself-sustenance scheme of the turbulence.\n\n\\subsection{Spectral representation of the dynamical equations}\n\nWe derive dynamical equations for the quadratic forms of velocity\nperturbations, $(u_x^2$, $u_y^2$, $u_z^2)$, and kinetic energy\ndensity, $E$, in Fourier space by decomposing the perturbations into\nspatial Fourier harmonics\n\\[\nf({\\bf r},t)=\\int \\bar{f}({\\bf k},t)e^{{\\rm i}{\\bf k}\\cdot{\\bf\nr}}dk_xdk_ydk_z\n\\]\nwhere $f\\equiv ({\\bf u},p)$ denotes the perturbations and\n$\\bar{f}\\equiv (\\bar{\\bf u}, \\bar{p})$ -- their corresponding\nFourier transforms. Then, from Eqs. (3)-(6), we get the following\nequations governing the dynamics of perturbation harmonics in\nspectral space (the constant density is set to unity, $\\rho_0=1$)\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_x=-S\\bar{u}_y-{\\rm i}k_x\\bar{p}+Q_x-\\nu\nk^2\\bar{u}_x,\n\\end{equation}\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_y=-{\\rm i}k_y\\bar{p}+Q_y-\\nu k^2\\bar{u}_y,\n\\end{equation}\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_z=-{\\rm i}k_z\\bar{p}+Q_z-\\nu k^2\\bar{u}_z,\n\\end{equation}\n\\begin{equation}\nk_x\\bar{u}_x+k_y\\bar{u}_y+k_z\\bar{u}_z=0,\n\\end{equation}\nwhere $k^2=k_x^2+k_y^2+k_z^2$. These spectral equations contain the\nlinear as well as the nonlinear, ${\\bf Q}({\\bf k},t)=[Q_x({\\bf\nk},t), Q_y({\\bf k},t), Q_z({\\bf k},t)]$, terms that are the Fourier\ntransforms of corresponding ones in the original Eqs. (3)-(6). The\nlatter are given by\n\\[\nQ_i({\\bf k},t)=\\sum_{j}{k_jq_{ij}({\\bf k},t)},~~~~i,j=x,y,z\n\\]\nwhere\n\\[\nq_{ij}({\\bf k},t)=-{\\rm i}\\int\\bar{u}_i({\\bf\nk^{\\prime}},t)\\bar{u}_j({\\bf k}-{\\bf k^{\\prime}},t)d^3{\\bf\nk^{\\prime}}\n\\]\nand describe nonlinear triad interactions among harmonics. From Eqs.\n(9)-(12) one can eliminate the pressure,\n\\[\n\\bar{p}=2{\\rm i}S\\frac{k_x}{k^2}\\bar{u}_y -{\\rm i} \\frac{{\\bf\nk}\\cdot{\\bf Q}}{k^2},\n\\]\nand rewrite them in the following form:\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_x=S\\left(2\\frac{k_x^2}{k^2}-1\n\\right)\\bar{u}_y+\\\\Q_x-\\frac{k_x}{k^2}({\\bf k}\\cdot{\\bf Q})-\\nu\nk^2\\bar{u}_x,\n\\end{multline}\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_y=2S\\frac{k_xk_y}{k^2}\\bar{u}_y+\\\\Q_y-\\frac{k_y}{k^2}({\\bf\nk}\\cdot{\\bf Q})-\\nu k^2\\bar{u}_y,\n\\end{multline}\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\bar{u}_z=2S\\frac{k_xk_z}{k^2}\\bar{u}_y+\\\\Q_z-\\frac{k_z}{k^2}({\\bf\nk}\\cdot{\\bf Q})-\\nu k^2\\bar{u}_z,\n\\end{multline}\nor, for the quadratic forms of the Fourier transforms of the\nvelocity\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\frac{|\\bar{u}_x|^2}{2}=\\\\S\\left(1-2\\frac{k_x^2}{k^2}\\right){\\cal\nH}_k+{\\cal N}_x-\\nu k^2|\\bar{u}_x|^2,\n\\end{multline}\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\frac{|\\bar{u}_y|^2}{2}=2S\\frac{k_xk_y}{k^2}|\\bar{u}_y|^2+{\\cal\nN}_y-\\nu k^2|\\bar{u}_y|^2,\n\\end{equation}\n\\begin{multline}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right)\\frac{|\\bar{u}_z|^2}{2}=\\\\S\\frac{k_xk_z}{k^2}(\\bar{u}_z^{\\ast}\\bar{u}_y+\\bar{u}_z\\bar{u}_y^{\\ast})+{\\cal\nN}_z-\\nu k^2|\\bar{u}_z|^2,\n\\end{multline}\nwhere ${\\cal\nH}_k=-(\\bar{u}_x\\bar{u}_y^{\\ast}+\\bar{u}_x^{\\ast}\\bar{u}_y)\/2$ is\nthe spectral density of the Reynolds stress (with minus sign) and\n${\\cal N}_x, {\\cal N}_y, {\\cal N}_z$ are the modified nonlinear\ntransfer functions,\n\\begin{multline*}\n{\\cal\nN}_i=\\frac{1}{2}(\\bar{u}_i^{\\ast}Q_i+\\bar{u}_iQ_i^{\\ast})-\\\\\\frac{k_i}{2k^2}[\\bar{u}_i^{\\ast}({\\bf\nk}\\cdot{\\bf Q})+\\bar{u}_i({\\bf k}\\cdot{\\bf Q}^{\\ast})],~~~i=x,y,z.\n\\end{multline*}\nUsing Eqs. (16)-(18) together with the incompressibility condition\n(12), we obtain the evolution equation for the spectral energy\ndensity ${\\cal E}_k=(|\\bar{u}_x|^2+|\\bar{u}_y|^2+|\\bar{u}_z|^2)\/2$,\n\\begin{equation}\n\\left(\\frac{\\partial}{\\partial t}-Sk_x\\frac{\\partial}{\\partial\nk_y}\\right){\\cal E}_k=S{\\cal H}_k+{\\cal N}_k-{\\cal D}_k,\n\\end{equation}\nwhich can be viewed as a counterpart of energy density Eq. (7) in\nspectral space. Here ${\\cal D}_k=2\\nu k^2{\\cal E}_k$ describes\nenergy dissipation and the nonlinear transfer term ${\\cal N}_k$ for\nthe spectral energy is the sum of the modified transfer functions\n\\[\n{\\cal N}_k = {\\cal N}_x+{\\cal N}_y+{\\cal N}_z = \\frac{1}{2}[{\\bf\n\\bar{u}^{\\ast}}\\cdot{\\bf Q}+{\\bf \\bar{u}}\\cdot{\\bf Q^{\\ast}}].\n\\]\nEqs. (16)-(19) fully determine the nonlinear dynamics of the\nconsidered system in Fourier space and are the basis for subsequent\nanalysis. According to them, four basic processes underly the\nperturbation dynamics:\n\\begin{enumerate}\n\\item\nThe second terms with $Sk_x\\partial\/\\partial k_y$ in the brackets on\nthe lhs are the fluxes of the corresponding quantities parallel to\nthe $k_y$-axis. These terms are of linear origin, coming from the\nconvective derivative on the left hand sides of the main Eqs.\n(3)-(5) and correspond to the advection by the background flow. In\nother words, the background shear flow makes the spectral quantities\n(Fourier transforms) ``drift'' in {\\bf k}-space: harmonics with\n$k_x>0$ and $k_x<0$ drift in opposite directions along the\n$k_y$-axis at a speed $S|k_x|$, $k_y(t)=k_y(0)-Sk_xt$, whereas the\nones with $k_x=0$ are not advected by the flow. Since $\\int d^3{\\bf\nk}\\partial (k_x{\\cal E}_k)\/\\partial k_y = 0$, this drift only\ntransports harmonics parallel to the $k_y$-axis without changing the\ntotal kinetic energy.\n\\item\nThe first terms on the rhs are associated with flow shear, i.e.,\noriginate from the linear term proportional to the shear rate on the\nrhs of Eq. (3). They describe the direct influence of the background\nflow on the evolution of perturbation harmonics, while the first\nterm on the rhs of Eq. (19) describes the exchange of energy between\nthe background flow and individual harmonics. This term in the\nspectral energy equation is related to the volume-averaged Reynolds\nstress entering Eq. (8) through\n\\[\n-\\langle u_xu_y\\rangle=\\int {\\cal H}_k({\\bf k},t)d^3{\\bf k},\n\\]\nand hence serves as the only source of energy for harmonics. This\nshear-induced nonmodal growth process is linear by nature and has a\ntransient character when nonlinearity is not included\n\\cite{Chagelishvili_etal03,Tevzadze_etal03,Umurhan_Regev04,Jimenez13,Zhuravlev14}.\nThe Reynolds stress spectral density, ${\\cal H}_k({\\bf k},t)$,\ndescribes injection of kinetic energy into turbulent eddies as a\nfunction of wavenumbers. Areas of positive and large values of\n${\\cal H}_k({\\bf k},t)$ correspond to intensive pumping of the\nkinetic energy into turbulent eddies and therefore are dynamically\nimportant. Due to the ``drift'' along the $k_y$-axis, the harmonics\nenter and leave the energy injection areas in spectral space. The\nfirst terms on the rhs of Eqs. (16)-(18) describe the same process\nfor the quadratic forms of perturbation velocity components.\nObviously, nonlinear transfer of kinetic energy to the dynamically\nimportant areas from other areas of {\\bf k}-space -- i.e.,\nrepopulation of the dynamically important areas -- is vital for the\nself-sustenance of turbulence.\n\\item\nThe second terms on the rhs of these equations, ${\\cal N}_x$, ${\\cal\nN}_y$ and ${\\cal N}_z$, describe, nonlinear transfers\n(redistributions), respectively, of the streamwise,\n$|\\bar{u}_x|^2\/2$, shearwise, $|\\bar{u}_y|^2\/2$, and spanwise\n$|\\bar{u}_z|^2\/2$ spectral energies in ${\\bf k}$-space. Their net\neffect in the spectral energy budget over all wavenumbers is zero in\ntime:\n\\begin{equation}\n\\int [{\\cal N}_x({\\bf k},t)+{\\cal N}_y({\\bf k},t)+{\\cal N}_z({\\bf\nk},t)]d^3{\\bf k}=0,\n\\end{equation}\nwhich is a consequence of vanishing of the nonlinear terms in the\ntotal energy evolution equation in physical space (Eq. 8). Although\nnonlinearity does not directly change the total (i.e., summed over\nall wavenumbers) spectral energy, it plays a central role in shear\nflow turbulence. ${\\cal N}_k$ determines transfer, or cascade of the\nspectral energy in {\\bf k}-space. These transfer functions are one\nof the main focus of the present study. We explore them by adopting\nthe approach developed in\n\\cite{Chagelishvili_etal02,Horton_etal10,Mamatsashvili_etal14},\nwhose main idea is performing a full 3D Fourier analysis of\nindividual terms in the dynamical Eqs. (16)-(18), thus allowing for\nthe flow shear-induced anisotropy of spectra and cascades.\n\\item\nThe third terms on the rhs describe the viscous dissipation that\nbecomes important at large wavenumbers $k \\gtrsim\n\\sqrt{S\/\\nu}=k_{\\cal D}$.\n\\end{enumerate}\n\nConcluding this section, the only source for the total perturbation\nenergy is the integral of the stress over an entire spectral space\n$\\int {\\cal H}_kd^3{\\bf k}$ -- the flow energy extraction and\nperturbation growth mechanisms are essentially linear by nature. The\nrole of nonlinearity is to continually repopulate those harmonics in\n{\\bf k}-space that are able to undergo nonmodal growth and in this\nway continually feed the nonlinear state over long times. This\nscenario of a self-sustained state, based on a subtle interplay\nbetween linear and nonlinear processes, is a keystone of the bypass\nconcept of subcritical turbulence in spectrally stable shear flows\n\\cite{Gebhardt_Grossmann94,Baggett_etal95,Grossmann00,Chapman02,Rempfer03,Eckhardt_etal07}.\n\n\\begin{table*}[t]\n\\caption{Properties of the numerical simulations: box aspect ratio,\nnumber of grid points, ratio of the grid cell sizes to the temporal\nmean of the dissipation scale, volume- and time-averaged values of\nthe energy, $\\tilde{E}$, and Reynolds stress, $-\\widetilde{u_xu_y}$,\nin the fully developed turbulence. The Reynolds number, $Re_z=4930$,\nis the same in all the runs.}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n$(A_{xz},A_{yz})$ & $N_x\\times N_y\\times N_z$ & $(\\Delta\nx\\times \\Delta y\\times \\Delta z)\/\\eta$ & $\\tilde{E}$ & -$ \\widetilde{u_xu_y}$ \\\\\n\\hline $(1,1)$ & $128\\times 128\\times 128$ & $1.66\\times 1.66\\times\n1.66$ & 2.37 & 0.71 \\\\ $(3,2)$ &\n$256\\times 128\\times 64$ & $2.15\\times 2.86\\times 2.86$ & 1.36 & 0.38 \\\\\n$(1,2)$ & $128\\times 256\\times 128$ &\n$1.49\\times 1.49\\times 1.49$ & 1.51 & 0.43 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table*}\n\n\n\\section{Results}\n\nIn this section, we present the results of numerical analysis of the\nnonlinear evolution of perturbations. Main emphasis will be on the\nspectral aspect of the dynamics based on the mathematical formalism\noutlined in the previous section. The numerical simulations are\nperformed for the flow in a rectangular domain with size $0 \\leq x\n\\leq L_x, -L_y\/2\\leq y \\leq L_y\/2, 0\\leq z \\leq L_z$ and $N_x\\times\nN_y\\times N_z$ number of grid points. The corresponding resolutions\nin these directions are $\\Delta x = L_x\/N_x, \\Delta y = L_y\/N_y,\n\\Delta z=L_z\/N_z$. In this domain, we solve the main Eqs. (3)-(6)\nusing the code developed by Sekimoto et al. \\cite{Sekimoto_etal15}\nat the School of Aeronautics, Universidad Polit\\'ecnica de Madrid,\nwhich solves the Navier-Stokes equations in the velocity-vorticity\nrepresentation as in \\cite{Kim_etal87}. The streamwise and spanwise\ndirections are periodic, so that the spectral method is applied in\nthese directions with the ``3\/2'' dealiasing rule. The vertical\ndirection is shear-periodic and is numerically realized by adding\n``shift'' factors in the compact-finite-difference matrices which\nguarantee spectral like resolution \\cite{Lele92}. The method does\nnot require the remeshing procedure and therefore does not lead to\nthe loss of enstrophy \\cite{Rogallo81}. To assess how well our\nsimulations are resolved, we compare the dissipation length scale\n$\\eta=(\\nu^3\/\\langle \\varepsilon \\rangle)^{1\/4}$, where\n$\\langle\\varepsilon \\rangle=\\nu \\langle(\\nabla \\times {\\bf\nu})^2\\rangle$ is the volume-averaged dissipation rate of the\nperturbation energy, to the grid cell sizes $(\\Delta x, \\Delta y,\n\\Delta z)$ in the computational box (see Table I). Since in the\n$y$-direction the boundary conditions are shear-periodic\ncorresponding to time-dependent, or drifting $k_y$ of each harmonic,\na standard Fast Fourier Transform (FFT) technique generally cannot\nbe applied for calculating Fourier transforms along this direction\nduring post-processing. We circumvent this by using the method of\nRefs. \\cite{Mamatsashvili_Rice09,Heinemann_Papaloizou09} that allows\nfor the time-variation of the shearwise wavenumber when calculating\nFourier transforms (see Appendix).\n\nAll physical variables are normalized by the dynamical, or shear\ntime $S^{-1}$ and the spanwise size of the box, $L_z$. The latter is\nthe relevant lengthscale in the homogeneous shear turbulence\ndynamics, because it sets the typical length and velocity scales of\nturbulent structures \\cite{Sekimoto_etal15}. The problem has two\ndimensionless free parameters -- aspect ratios of the domain,\n$A_{xz}=L_x\/L_z$, $A_{yz}=L_y\/L_z$. Table I lists the\ncharacteristics of the three simulations with different box aspect\nratios that we performed. The Reynolds number is defined in terms of\nthe box spanwise size, $Re_z=SL_z^2\/\\nu$, and is the same in all the\nruns, $Re_z=4930$. The ratio of the grid cell sizes to the\ntime-averaged dissipation scale varies from 1.49 for the box\n$(A_{xz},A_{yz})=(1,2)$ up to 2.86 for the box\n$(A_{xz},A_{yz})=(3,2)$. This implies that large and intermediate\nlengthscales, where, as we will show, the self-sustaining dynamics\nis concentrated, are well resolved in our simulations. The\nwavenumbers $k_x,k_y,k_z$ are normalized, respectively, by $\\Delta\nk_x=2\\pi\/L_x, \\Delta k_y=2\\pi\/L_y$ and $\\Delta k_z=2\\pi\/L_z$, that\nis, $(k_x\/\\Delta k_x, k_y\/\\Delta k_y, k_z\/\\Delta k_z)\\rightarrow\n(k_x,k_y,k_z)$. Since $\\Delta k_x, \\Delta k_y$ and $\\Delta k_z$ are\nthe grid cell sizes in Fourier space, the normalized streamwise and\nspanwise wavenumbers become integers $k_x,k_z=0,\\pm 1, \\pm 2, ...$,\nwhile $k_y$, although changes with time due to drift, is integer at\ndiscrete moments $t_m=mA_{xz}\/(|k_x|A_{yz})$, where $m$ is a\npositive integer.\n\nA self-sustained turbulent state is achieved in each simulation,\nstarting from initial random perturbations. Our main goal is to\ninvestigate underlying self-sustaining process of the turbulence and\nhow its dynamics depends on the box aspect ratio. This allows us to\nreveal general patterns of the self-organization as well as it\nspecificities for each model parameters. Finally, it should be\nstressed that our study is based on the DNS of turbulence and in\nthis respect is general, not relying on simplifying assumptions used\nin low-order models of self-sustaining processes (e.g.,\n\\cite{Waleffe95,Waleffe97}).\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_1.eps}\n\\caption{(Color online) Transient growth factor of optimal\nharmonics' energy, ${\\cal E}_k(t_d)\/{\\cal E}_k(0)$, as a function of\nwavenumber ratios. It is calculated during the dynamical time\n$t_d=S^{-1}$ with the linearized equations of motion.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_2.eps}\n\\caption{Evolution of (a) the volume-averaged total kinetic energy,\n$\\langle E \\rangle$, and the energy of the basic mode with ${\\bf\nk}_b=(0,0,\\pm 1)$, $2{\\cal E}_k({\\bf k}_b)$, as well as (b) the\nvolume-averaged Reynolds stress, $-\\langle u_xu_y\\rangle$ and the\nstress corresponding to the basic mode, $2{\\cal H}_k({\\bf k}_b)$, in\nthe cubic box.}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=5.6cm]{fig_3a.eps}\n\\includegraphics[width=5.6cm]{fig_3b.eps}\n\\includegraphics[width=5.6cm]{fig_3c.eps}\n\\includegraphics[width=5.6cm]{fig_3d.eps}\n\\includegraphics[width=5.6cm]{fig_3e.eps}\n\\includegraphics[width=5.6cm]{fig_3f.eps}\n\\caption{(Color online) Distribution of the velocity components in\nthe fully developed turbulence (at $t-t_s=33$) in the cubic box.\nShown are $(x,y)$- and $(x,z)$-slices at $z=\\pi$ and $y=0$,\nrespectively. The streamwise component $u_x$ is larger than $u_y$\nand $u_z$. In the $(x,z)$ slices of $u_y$ and especially of $u_x$, a\nsignature of the basic harmonic with large spanwise scale can be\ndiscerned.}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_4.eps}\n\\caption{Evolution of (a) the volume-averaged perturbation kinetic\nenergy (solid line) and the quadratic forms of streamwise, $\\langle\nu_x^2\\rangle\/2$ (dot-dashed line), shearwise\/vertical, $\\langle\nu_y^2\\rangle\/2$ (dashed line) and spanwise, $\\langle u_z^2\\rangle\/2$\n(doted line) velocities as well as (b) the Reynolds stress in the\ncubic box.}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_5a.eps}\n\\includegraphics[width=5.9cm]{fig_5b.eps}\n\\includegraphics[width=5.9cm]{fig_5c.eps}\n\\caption{Distribution of the modes carrying dominant energy in {\\bf\nk}-space at (a) $k_z=0$ (b) $k_z=1$ and (c) $k_z=2$ in the cubic\nbox. Black dots represent the harmonics whose energies grow more\nthan $10 \\%$ of the maximum spectral energy at least once during\nevolution. The biggest black dot in the plot (b) corresponds to the\nbasic mode with the maximum spectral energy at all times. Circles\nrepresent the harmonics whose energies grow from 5\\% to 10\\% of the\nmaximum spectral energy.}\n\\end{figure*}\n\nIt is well-known from the linear nonmodal theory of shear flows that\noptimal perturbations characterized by a maximum nonmodal growth\nduring the dynamical or eddy turnover time, which can be assumed of\nthe order of shear time, $t_d \\simeq S^{-1}$ (or, $t_d \\simeq 1$ in\nnormalized units) are responsible for most of the energy extraction\nfrom the background flow\n\\cite{Schmid_Henningson01,Farrell_Ioannou96}. Figure 1 shows the\ngrowth factor of energy for optimal harmonics calculated from the\nlinearized inviscid equations of motion, i.e., using only the first\nlinear terms on the rhs of Eqs. (13)-(15). These terms, and hence\nthe linear dynamics of harmonics in the inviscid case, depends only\non the ratios of wavenumbers, $k_y\/k_x$ and $k_z\/k_x$. It is seen\nfrom this figure that the optimal growth factor is largest at $1\n\\lesssim k_z\/k_x < 4$ and $0<k_y\/k_x \\lesssim 1$. Consequently, one\ncan encompass as many of these optimal perturbations as possible and\nthus better account for their major role in the energy exchange\nprocesses between turbulence and background flow if the\ncomputational box aspect ratios satisfy the conditions: $A_{xz}\\leq\nA_{yz}$ and $A_{xz}\\geq 1$.\n\nIn agreement with the previous study of homogeneous shear turbulence\nby Pumir \\cite{Pumir96}, in our simulations we found that in\naddition to the non-symmetric ($k_x\\neq 0$) optimal harmonics, a\nharmonic with a large scale spanwise variation, which is uniform in\nthe streamwise and shearwise directions, $k_x=k_y=0, k_z=\\pm 1$,\nplays a central role in the turbulence dynamics. Due to its\nimportance, we call this harmonic the basic mode. Other harmonics,\nwhose energy grows more than 10\\% of the maximum value of the\nspectral energy at least once during the evolution, are considered\nto be actively participating in and forming the self-sustaining\ndynamics of turbulence. In particular, we show that one of the\ncharacteristic features of the homogeneous shear turbulence --\nquasi-periodic bursts of energy and Reynolds stress \\cite{Pumir96}\n-- are determined by a competition between the basic mode and other\ndynamically ``active'' harmonics, whose number generally depends on\nthe box aspect ratio and is usually large.\n\nBelow we focus on three different boxes with the following aspect\nratios: $(A_{xz}, A_{yz})=(1,1)$ (cubic box),  $(3,2)$, $(1,2)$ and\nanalyze in detail the specific spectra and self-sustaining dynamics\nof turbulence in wavenumber\/spectral space by calculating the\nFourier transforms of individual linear and nonlinear terms in\ngoverning equations using the DNS data. In all simulations, the\ntime-averages of the spectral quantities are calculated over an\nentire time of the saturated turbulent state, discarding an initial\ntransient phase, which typically lasts for about $30S^{-1}$.\n\n\\subsection{Aspect ratio $(A_{xz}, A_{yz})=(1,1)$}\n\nAt the early stage of evolution, each harmonic contained in the\ninitial perturbations grows due to the flow nonnormality, leading to\nthe growth of the total energy and Reynolds stress. Then, after\nabout $t_s=30S^{-1}$, the amplitudes of the perturbations become\nsufficiently large in the nonlinear regime and eventually the flow\nsettles into a self-sustained turbulence. In this cubic box, the\nbasic mode\/harmonic with wavenumber ${\\bf k}_b=(0,0,\\pm 1)$,\ndominates over other ones, i.e., its spectral energy, ${\\cal\nE}_k({\\bf k}_b)$, and stress, ${\\cal H}_k({\\bf k}_b)$, are,\nrespectively, maximum of ${\\cal E}_k$ and ${\\cal H}_k$ during an\nentire course of evolution (see also Ref. \\cite{Pumir96}). Figure 2\nshows the evolution of the volume-averaged total energy and stress\nas well as the energy and stress of the basic mode in the saturated\nstate, which display similar patterns of quasi-periodic bursts in\ntime. The volume-averaged Reynolds stress (with negative sign),\n$-\\langle u_xu_y\\rangle$, and the stress of the basic mode, ${\\cal\nH}_k({\\bf k}_b)$ are positive at all times and non-decaying as\nrequired by the self-sustaining process according to Eq. (8). The\nbursts in the total energy and stress closely follow amplifications\n(peaks) of ${\\cal E}_k({\\bf k}_b)$ and stress ${\\cal H}_k({\\bf\nk}_b)$ in the basic mode. This indicates that these bursts are\nrelated to the dynamics of the basic mode, which thus appears to be\nmainly responsible for the energy pumping from the background flow\ninto turbulence. During the bursts, the contribution of ${\\cal\nH}_k({\\bf k}_b)$ to the total Reynolds stress is as much as 60\\% -\n70\\%, while other modes become significant, although still remain\nsmaller than the basic mode, in quiescent intervals between the\nbursts, when the total energy and stresses as well as the energy and\nstress of the basic mode are low. So, for the aspect ratio $(1,1)$,\nmost of the turbulent energy is produced through the interaction of\nthe basic mode with the background shear flow.\n\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_6a.eps}\n\\includegraphics[width=5.9cm]{fig_6b.eps}\n\\includegraphics[width=5.9cm]{fig_6c.eps}\n\\caption{(Color online) Logarithm of the time-averaged spectrum of\nthe kinetic energy, $log_{10}({\\cal E}_k)$, in the cubic box. The\n$(k_x,k_y)$-slices are at (a) $k_z=0$, (b) $k_z=1$ and (c) $k_z=2$.\nThe spectrum has an anisotropic character, having larger power in\nthe $k_y\/k_x<0$ part of Fourier space at a given $k_x$. The spectral\nenergy has maximum at $k_z=1$ and decreases with increasing $k_z$.}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_7.eps}\n\\caption{Integrated in $(k_x,k_y)$-plane time-averaged (a) kinetic\nenergy spectrum, $\\hat{\\cal E}$, and (b) Reynolds stress, $\\hat{\\cal\nH}$ vs. $k_z$.}\n\\end{figure}\n\nThe structure of the velocity components in the fully developed\nturbulence in physical space is presented in Fig. 3. The evolution\nof the volume-averaged perturbation kinetic energy and quadratic\nforms of streamwise, $\\langle u_x^2\\rangle\/2$, shearwise\/vertical,\n$\\langle u_y^2\\rangle\/2$, and spanwise, $\\langle u_z^2\\rangle\/2$,\nvelocities as well as the Reynolds stress are shown in Fig. 4. From\nthese plots it is clear that the streamwise velocity prevails over\nthe shearwise and spanwise ones. In the $(x,z)$ slices of $u_y$ and,\nespecially, of $u_x$, we clearly see the signatures of the dominant\nbasic harmonic with small scale fluctuations due to other harmonics\nsuperimposed on it.\n\nTo understand the role of other modes in the turbulence dynamics, it\nis important to define the main, energy-containing area in {\\bf\nk}-space, which we also refer to as \\emph{the vital area} of\nturbulence. For this purpose we define two sets of harmonics whose\nenergy grows from 5\\% to 10\\% and those larger than 10\\% of the\nmaximum spectral energy at least once during the evolution; the\nsecond set is referred to as ``active modes''. These\nenergy-carrying, or dynamically important harmonics are displayed in\nFig. 5. They have small wavenumbers $|k_x|\\leq 2, |k_y|\\leq 3,\n|k_z|\\leq 3$, which basically form the vital area of turbulence (at\n$k_z=3$, not shown in this figure, mode energies always remain less\nthan 10\\% of the maximum energy). Harmonics with larger wavenumbers\n$|k_x|>2, |k_y|>3, |k_z|>3$ lie outside the vital area and always\nhave energy and stress less than 5\\% of the maximum value,\ntherefore, not playing a major role in the energy exchange process\nbetween the background flow and turbulence. Essentially, only these\nlarge scale modes in the vital area take part in the self-sustaining\ndynamics of turbulence, but the basic mode\/harmonic still remains\ndominant -- it is the most active among the active harmonics. We\nemphasize that the total number of the large scale active modes\n(black dots in Fig. 5) is equal to 36, implying that the dynamics of\nhomogeneous shear turbulence cannot be reduced to low-order models\nof the self-sustaining processes.\n\nFigure 6 shows the time-averaged spectrum of the kinetic energy in\n$(k_x,k_y)$-plane at different $k_z$. The spectrum is anisotropic\ndue to shear -- the isolines have elliptical shape inclined opposite\nto the $k_y$-axis. This fact indicates that on average modes with\n$k_y\/k_x<0$ have more energy than those with $k_y\/k_x>0$ at fixed\n$k_x$. The plots show that the time-averaged spectral kinetic energy\nat fixed $k_z$ is larger for small $k_x$ and $k_y$ and decays at\nleast by two order of magnitude at $\\sqrt {k_x^2+k_y^2} \\approx 2$.\nSo, the active modes are located just within this range of small\nwavenumbers, as also seen in Fig. 5. As mentioned above, the maximum\nof the spectral energy is reached at $k_x=k_y=0, k_z=\\pm 1$\ncorresponding to the basic mode. However, it decays significantly\nalready for the next spanwise modes with $k_z=2$ and further\ndecreases rapidly with increasing $k_z$. This behavior is also seen\nin Fig. 7, which shows the integrated in $(k_x,k_y)$-plane the\ntime-averaged spectral energy, $\\hat{\\cal E}(k_z)=\\int {\\cal E}_k\ndk_xdk_y$, and stress, $\\hat{\\cal H}(k_z)=\\int {\\cal H}_k dk_xdk_y$,\nvs. $k_z$. Both reach a maximum at $k_z=\\pm 1$ and rapidly decrease\nwith $|k_z|$.\n\nThe anisotropic nature of the kinetic energy spectrum, which clearly\ndistinguishes it from the isotropic spectrum in the classical\n(without shear) Kolmogorov phenomenology, arises as a result of the\nspecific action of linear and nonlinear processes in {\\bf k}-space.\nAs we show below, these processes are anisotropic over wavenumbers\ndue to shear, resulting in a new phenomenon -- the transverse\nredistribution of power in spectral space.\n\n\\begin{figure*}\n\\includegraphics[width=0.32\\textwidth]{fig_8a.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8b.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8c.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8d.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8e.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8f.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8g.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8h.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8i.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8j.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8k.eps}\n\\includegraphics[width=0.32\\textwidth]{fig_8l.eps}\n\\caption{(Color online) Maps of the time-averaged energy injecting\nstress [(a), (b), (c)], ${\\cal H}_k$, and nonlinear transfer, [(d),\n(e), (f)] ${\\cal N}_x$, [(g), (h), (i)] ${\\cal N}_y$, [(j), (k),\n(l)] ${\\cal N}_z$ functions in {\\bf k}-space for the cubic box.\nShown are $(k_x,k_y)$-slices of these quantities at $k_z=0$ (left\ncolumn), $k_z=1$ (middle column) and $k_z=2$ (right column). ${\\cal\nH}_k$ is significant in the vital area $|k_x|\\leq 2, |k_y|\\leq 3,\n|k_z|\\leq 2$, on the $k_y\/k_x>0$ side. Hence, the energy injection\ninto turbulence mainly occurs in this region. At $k_z=1$, ${\\cal\nH}_k$ reaches largest values and consequently energy injection is\nmost intensive at this wavenumber. ${\\cal N}_x$, ${\\cal N}_y$ and\n${\\cal N}_z$ transfer, respectively, the streamwise, shearwise and\nspanwise components of the spectral kinetic energy anisotropically,\nor transversely over wavevector angles in Fourier space, away from\nthe regions where they are negative ${\\cal N}_x<0,~{\\cal\nN}_y<0,~{\\cal N}_z<0$ (blue) towards the regions where they are\npositive ${\\cal N}_x>0,~{\\cal N}_y>0, ~{\\cal N}_z>0$ (yellow and\nred).}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_9.eps}\n\\caption{The time-averaged nonlinear transfer functions integrated\nin $(k_x,k_y)-$plane and represented as a function of $k_z$, (a)\n$\\hat{\\cal N}_x$, (b) $\\hat{\\cal N}_y$, (c) $\\hat{\\cal N}_z$ and\ntheir sum (d) $\\hat{\\cal N}_k$ describing nonlinear transfer of\nenergy. These functions transfer corresponding quantities from $k_z$\nat which they are negative to those $k_z$ at which they are\npositive.}\n\\end{figure}\n\n\n\\subsubsection*{Energy injection, nonlinear transfers and their\ninterplay in ${\\bf k}$-space}\n\nTo understand the origin of the anisotropic character of the energy\nspectrum, in Fig. 8 we present the distribution of the time-averaged\nenergy-injecting Reynolds stress spectrum, ${\\cal H}_k$, and the\nnonlinear transfer functions, ${\\cal N}_x, {\\cal N}_y$ and ${\\cal\nN}_z$, in {\\bf k}-space. Since these quantities are symmetric with\nrespect to a change ${\\bf k}\\rightarrow -{\\bf k}$, without loss of\ngenerality, hereafter we concentrate on the upper part ($k_z \\geq\n0$) of {\\bf k}-space. First of all, these plots clearly demonstrate\nthe anisotropic nature of the linear and nonlinear processes in {\\bf\nk}-space -- all these quantities exhibit anisotropy over\nwavenumbers, that is, depend on wavevector angle, or orientation.\n${\\cal H}_k$ is significant primarily in the vital area of spectral\nspace, $|k_x|\\leq 2, |k_y|\\leq 3, |k_z|\\leq 2$, at $k_y\/k_x>0$,\nwhere it supplies the turbulence with energy via nonmodal growth of\nthe dynamically active modes (see below). At $k_z=0$, ${\\cal H}_k$\nis mainly concentrated around $|k_y|,|k_x| \\sim 1$ [Fig. 8(a)], it\nis positive at $k_x\/k_y>0$, supplying harmonics with energy and\nnegative at $k_x\/k_y<0$, returning energy to the flow. On the other\nhand, at $k_z\\geq 1$, the distribution of ${\\cal H}_k$ is changed --\nit is positive and achieves local maximum around $k_x=k_y=0$ at each\n$k_z$ [Figs. 8(b) and 8(c)]. The Reynolds stress spectrum is highest\nand therefore the energy extraction from the background flow into\nturbulence more effectively occurs for modes with $k_{z}=\\pm 1$,\nreaching the maximum at ${\\bf k}_b$ corresponding to the basic mode\n[Figs. 8(b), see also Fig. 7(b)]. Outside the vital area, at $|k_x|>\n2, |k_y|>3, |k_z|>2$, the stress and hence the energy input into\nturbulence rapidly decrease and vanish at large wavenumbers. The\ntotal energy input at $k_z=0$ is also negligible, $\\hat{\\cal\nH}\\approx 0$, as seen from Fig. 7(b), since ${\\cal H}_k$ changes\nsign in $(k_x,k_y)$-plane at this $k_z$ [Fig. 8(a)]. \\emph{\\it Thus,\nthe main energy supply for turbulence is due to large-scale modes\nwith the wavelengths comparable to the box size.} Note that the\nenergy injection into turbulence mediated by the stress occurs over\na range wavenumbers and in this respect differs from the classical\ncases of forced turbulence in flows without shear, where forcing is\napplied at a specific wavenumber or a narrow range of wavenumbers\n(see e.g., \\cite{Biskamp03}).\n\nThe nonlinear transfer functions ${\\cal N}_x, {\\cal N}_y, {\\cal\nN}_z$ [Figs. 8(d)-8(l)] depend on wavevector angle and,\nconsequently, redistribute the streamwise, shearwise and spanwise\nvelocity energies generally not only along wavevector\n(direct\/inverse cascades), but also transverse to wavevector. The\nlatter occurs away from the regions in {\\bf k}-space where the\nrespective transfer functions are negative ${\\cal N}_x,{\\cal\nN}_y,{\\cal N}_z<0$ [blue areas in Figs. 8(d)-8(l)] to regions where\nthey are positive ${\\cal N}_x,{\\cal N}_y,{\\cal N}_z>0$ (yellow and\nred areas); at all other wavenumbers (light green areas) these\nfunctions are small. It is seen from Figs. 8(d)-8(l) that this\ntransverse cascade primarily operates within the vital area of the\nwavenumber space, $|k_x|\\leq 2, |k_y|\\leq 3, |k_z|\\leq 2$, where the\nnonnormality-induced energy exchange between the background flow and\nharmonics is dominant. Outside the vital area at large wavenumbers,\nthe transfer functions become nearly independent of the wavevector\nangle and depend mostly on its magnitude, i.e., the cascade is not\ntransverse anymore but direct. So, outside the vital area the\nsituation is similar to the classical homogeneous isotropic\nturbulence described by Kolmogorov phenomenology. To characterize\ntransfers along $k_z$, we integrate ${\\cal N}_i$ over\n$(k_x,k_y)$-plane, $\\hat{\\cal N}_i(k_z)=\\int {\\cal N}_i dk_xdk_y,\n(i=x,y,z,k)$, and represent them as functions of $k_z$ in Fig. 9.\n\nThe interplay between the energy-injecting linear process and the\ntransverse cascade determines the self-sustaining dynamics of\nturbulence and characteristics of the energy spectrum in the\npresence of shear. Figures 8(d)-8(f) and 9(a) show that ${\\cal N}_x$\nis negative and large by absolute value in the vital area, where the\nstress, ${\\cal H}_k$, is positive and appreciable. Outside the vital\narea, i.e., at larger wavenumbers ($|k_x|>2, |k_y|>3, |k_z| \\geq\n3$), ${\\cal N}_x$ is positive but small and the stress is also\nnegligibly small. In other words, ${\\cal N}_x$ provides a sink, or\nnegative feedback for ${\\cal H}_k$ by taking out energy in the\nstreamwise velocity gained from the background flow and transferring\nit to large wavenumbers and part of it to $k_z=0$, as evident from\nthe dependence of ${\\cal {\\hat N}}_x$ on $k_z$. By contrast, ${\\cal\nN}_y$ is positive in those regions of the vital area where ${\\cal\nH}_k$ is also positive [Figs. 8(g)-8(i) and 9(b)]. As a result, it\nprovides positive feedback at these wavenumbers by generating the\nshearwise energy, $|\\bar{u}_y|^2\/2$, which, in turn, leads to the\ngrowth of the streamwise energy, $|\\bar{u}_x|^2\/2$, due to the\nlinear nonmodal growth mechanism (the first terms of linear origin\non the rhs of Eqs. 13 and 16, while ${\\cal N}_x<0$ at these\nwavenumbers and cannot contribute to the growth). The streamwise\nenergy, which in fact appears to be largest among the other two\ncomponents, is then transferred to larger wavenumbers by ${\\cal\nN}_x$. Note that this positive feedback due to ${\\cal N}_y$ is\ncrucial to the self-sustenance of turbulence and is realized by the\ntransverse cascade, which ensures supply of the shearwise energy for\nharmonics with such wavevector orientations ($k_y\/k_x>0$) that\ncorrespond to positive stress ${\\cal H}_k>0$ and hence with the\ncapability of nonmodal growth. Figures 8(j)-8(l) and 9(c) show\n${\\cal N}_z$, which is responsible for the transfer of energy in the\nspanwise velocity. However, since the Reynolds stress providing\nenergy supply for the turbulence is produced by the product\n(correlation) of $u_x$ and $u_y$, below we concentrate only on\n${\\cal N}_x$ and ${\\cal N}_y$ that govern nonlinear redistribution\nof these velocity components. Figure 9(d) shows the nonlinear\ntransfer function for the energy, $\\hat{\\cal N}_k=\\hat{\\cal\nN}_x+\\hat{\\cal N}_y+\\hat{\\cal N}_z$. It is dominated by $\\hat{\\cal\nN}_x$ and hence the nonlinear transfer of the total energy over\nwavenumbers is similar to that of the streamwise energy.\n\n\\emph{\\it Thus, the transverse cascade of energy appears to be a\ngeneric feature of nonlinear dynamics of perturbations in spectrally\nstable shear flows.} The conventional description of shear flow\nturbulence solely in terms of direct and inverse cascades, which\nleaves such nonlinear transverse cascade out of consideration, might\nbe incomplete and misleading. It should be emphasized that revealing\nthe complete picture of these nonlinear cascade processes has become\nlargely possible due to carrying out the analysis in Fourier space.\nBecause of the shear-induced anisotropy of cascade directions, often\nused energy spectra and transfer functions integrated, or averaged\nover wavevector angles, clearly, are not fully representative of the\nactual, more general (transverse) redistribution of the spectral\nenergies due to nonlinearity in {\\bf k}-space.\n\nWe have described above the linear and nonlinear process in a\ntime-average sense in the fully developed turbulence. To gain a\nbetter insight in its self-sustenance and appearance of the bursts\nin the total energy evolution (Fig. 2), below we analyze the\ntemporal evolution of individual harmonics in the vital area for\nfixed $k_x, k_y, k_z$\\footnote{Although $k_y$ of every harmonic with\n$k_x\\neq 0$ changes with time as a result of drift due to shear\nflow, we can still take Eulerian approach and focus on a fixed $k_y$\nin spectral space.}.\n\nFirst we consider the dynamics of the dominant basic mode with ${\\bf\nk}_b=(0, 0, \\pm 1)$, which corresponds to the maximum energy and\nstress in spectral space during the whole evolution in the case of\nthe cubic box. For this mode, the dynamical Eqs. (16)-(18) are\nreduced to the following system ($S=1$ everywhere below):\n\\begin{equation}\n\\frac{\\partial}{\\partial t} \\frac{|\\bar{u}_x({\\bf\nk}_b,t)|^2}{2}={\\cal H}_k({\\bf k}_b,t)+{\\cal N}_x({\\bf\nk}_b,t)-\\nu|\\bar{u}_x({\\bf k}_b,t)|^2,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial}{\\partial t}\\frac{|\\bar{u}_y({\\bf\nk}_b,t)|^2}{2}={\\cal N}_y({\\bf k}_b,t)-\\nu|\\bar{u}_y({\\bf\nk}_b,t)|^2,\n\\end{equation}\n\\begin{equation}\n\\frac{\\partial}{\\partial t}\\frac{|\\bar{u}_z({\\bf\nk}_b,t)|^2}{2}=-\\nu|\\bar{u}_z({\\bf k}_b,t)|^2.\n\\end{equation}\nNote that ${\\cal N}_z({\\bf k}_b,t)=0$ and Eq. (23) can be solved\nanalytically, $\\bar{u}_z({\\bf k}_b,t)=\\bar{u}_z({\\bf k}_b,0) \\exp\n(-2\\nu t)$. Thus, the spanwise velocity decays exponentially in time\nand the velocity field of the basic harmonic becomes 2D. Figure 10\nshows the evolution of the spectral energies of the streamwise and\nshearwise velocities, $|\\bar{u}_x({\\bf k}_b)|^2\/2$, $|\\bar{u}_y({\\bf\nk}_b)|^2\/2$, and the nonlinear transfer functions ${\\cal N}_x({\\bf\nk}_b)$, ${\\cal N}_y({\\bf k}_b)$ for the basic mode. Because the\nbasic mode is of large scale and the Reynolds number is high, the\ndissipation terms in these equations are negligible. The velocities\nexhibit a sequence of amplifications (bursts) and decays, with the\nstreamwise velocity being always much larger than the shearwise one.\n${\\cal N}_x({\\bf k}_b)$ is always negative and therefore acts as a\nsink in the streamwise velocity evolution Eq. (21), while ${\\cal\nN}_y({\\bf k}_b)$ oscillates irregularly with alternating sign. When\n${\\cal N}_y({\\bf k}_b)$ is positive, it amplifies the shearwise\nvelocity, while when negative causes its decrease, but the\ntime-average of ${\\cal N}_y({\\bf k}_b)$ is positive, though small\n[see also Fig. 8(h)], indicating the growth trend of the shearwise\nvelocity on average. The shearwise velocity generated by ${\\cal\nN}_y({\\bf k}_b)$, in turn, leads to the amplification of the\nstreamwise velocity due to the shear-related first linear term on\nthe rhs of dynamical Eq. (13), or equivalently, to the amplification\nof the streamwise energy due to the stress term ${\\cal H}_k({\\bf\nk}_b)$ in Eq. (21). Note that this is the only cause of\namplification for $|\\bar{u}_x({\\bf k}_b)|$, since the corresponding\nnonlinear term, ${\\cal N}_x({\\bf k}_b)$, is always negative and\ncannot amplify it. Then, $\\bar{u}_y({\\bf k}_b)$ and the amplified\n$\\bar{u}_x({\\bf k}_b)$ jointly give rise to positive stress, ${\\cal\nH}_k({\\bf k}_b)>0$ [Figs. 2(b) and 11(b)]. As a result, the stress\nproduction rate, $d{\\cal H}_k({\\bf k}_b)\/dt$ (calculated by\ncombining Eqs. 13 and 14), closely follows, or correlates with\n${\\cal N}_y({\\bf k}_b)$ [Fig. 11(a), the correlation coefficient is\n$R(d{\\cal H}_k\/dt,{\\cal N}_y)=0.93$,\\footnote{The correlation\ncoefficient $R(a,b)$ between two time-dependent functions $a$ and\n$b$ is defined in a usual way as Pearson's product-moment\ncorrelation coefficient. The time average is done over an entire\nduration of the saturated turbulent state in the simulations.}] and,\nconsequently, the stress itself follows $|\\bar{u}_y({\\bf k}_b)|$\n[Fig. 11(b), $R({\\cal H}_k,|\\bar{u}_y|)=0.97$], i.e., the peaks, or\nbursts and dips in the temporal evolution of these quantities\ncoincide. So, the transfer function ${\\cal N}_y({\\bf k}_b)$ plays a\ncentral role in the dynamics of the basic mode by providing a\npositive feedback to the linear nonmodal growth: when ${\\cal\nN}_y({\\bf k}_b)>0$, it causes the amplification of $\\bar{u}_y({\\bf\nk}_b)$, which via nonmodal growth mechanism leads in turn to the\namplification of $\\bar{u}_x({\\bf k}_b)$ and hence of the\ncorresponding stress. These growth intervals in the evolution of\nthese quantities alternate with the decaying intervals where ${\\cal\nN}_y({\\bf k}_b)<0$. Since for the aspect ratio $(1,1)$, the basic\nmode is the dominant one among all the active modes, these bursts\nalso manifest themselves in the total stress evolution, as is seen\nin Fig. 2(b).\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_10.eps}\n\\caption{Evolution of (a) the streamwise, $|\\bar{u}_x({\\bf\nk}_b)|^2\/2$, and (b) shearwise, $|\\bar{u}_y({\\bf k}_b)|^2\/2$,\nspectral energies as well as the nonlinear transfer functions (c)\n${\\cal N}_x({\\bf k}_b)$ and (d) ${\\cal N}_y({\\bf k}_b)$ for the\nbasic mode with ${\\bf k}_b=(0,0,\\pm 1)$ in the cubic box. The\nstreamwise energy is much larger than the shearwise one due to the\nlinear nonmodal growth process. ${\\cal N}_x({\\bf k}_b)$ is negative\nduring the whole evolution, acting as a sink for the streamwise\nenergy, while ${\\cal N}_y({\\bf k}_b)$ oscillates and, when positive,\ncauses increase in $|\\bar{u}_y|$; its time-averaged value over the\nevolution is positive $4.78\\cdot 10^{-4}$.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_11.eps}\n\\caption{Evolution of (a) the stress production rate, $d{\\cal\nH}_k({\\bf k}_b)\/dt$, and nonlinear transfer term ${\\cal N}_y({\\bf\nk}_b)$ as well as (b) the spectral Reynolds stress, ${\\cal H}_k({\\bf\nk}_b)$, and shearwise velocity, $|\\bar{u}_y({\\bf k}_b)|$ of the\nbasic mode in the cubic box. The stress production rate closely\nfollows ${\\cal N}_y({\\bf k}_b)$ (which is scaled by a factor of 2 to\nmake comparison easier) and, consequently, the amplifications\n(bursts) of both shearwise velocity and stress occur when ${\\cal\nN}_y({\\bf k}_b)$ is positive and decrease when this term is\nnegative.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, height=8cm]{fig_12.eps}\n\\caption{Evolution of (a) the stress production rate and nonlinear\ntransfer term ${\\cal N}_y$, (b) the stress ${\\cal H}_k$ and\nshearwise velocity $|\\bar{u}_y|$ and (c) the stress and ${\\cal N}_x$\nfor the mode ${\\bf k}_1=(0, 1, 1)$. The nonlinear function in (a)\nand the stress in (b) are scaled by 2 and 3, respectively, for the\nsake of comparison.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, height=8cm]{fig_13.eps}\n\\caption{Same as in Fig. 12, but for the non-symmetric mode ${\\bf\nk}_2=(1,1,1)$. The stress in (b) is scaled by a factor of 3. A high\ndegree of correlation exists between the stress production rate and\nthe nonlinear term ${\\cal N}_y$ (a) as well as the stress and\nshearwise velocity (b), implying that the stress amplification is\ndue to the nonmodal growth of $\\bar{u}_x$ from $\\bar{u}_y$.}\n\\end{figure}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth, height=8cm]{fig_14.eps}\n\\caption{Same as in Fig. 13, but for the mode ${\\bf k}_3=(1, -1,\n1)$. The stress in (b) is scaled by a factor of 3. Unlike the modes\nin Figs. 12 and 13, ${\\cal N}_x$ dominates the\nnonnormality\/shear-induced linear term and consequently the stress\nand shearwise velocity do not correlate for this harmonic. The\ncorresponding stress is smaller than those of the other two modes.}\n\\end{figure}\n\nThe same nonmodal linear growth and nonlinear processes underlie the\ndynamics of other symmetric ($k_x=0$) and non-symmetric ($k_x\\neq\n0$) active modes in the vicinity of ${\\bf k}_b$ within the vital\narea (Fig. 5), however, their quantitative character depends on mode\nwavenumbers. To demonstrate this, in Figs. 12-14 we show the\ndynamics for such nearby symmetric mode with ${\\bf k}_1=(0,1,1)$ and\nnon-symmetric modes with ${\\bf k}_2=(1,1,1)$ and ${\\bf\nk}_3=(1,-1,1)$. These figures display the time-history of (a) the\nstress production rate $d{\\cal H}_k\/dt$ and the nonlinear transfer\nterm ${\\cal N}_y$, (b) stress ${\\cal H}_k$ and the absolute value of\nthe shearwise velocity, $|\\bar{u}_y|$, (c) the first shear-induced\nterm of linear origin, $(1-2k_x^2\/k^2){\\cal H}_k$, on the rhs of Eq.\n(16) and the nonlinear transfer term ${\\cal N}_x$. We focus on these\nharmonics with $k_z=1$, because, as noted above, they correspond to\nhigher levels of stress and energy than those at other values of\n$k_z$ (Fig. 7), and hence contribute most to the turbulence\ndynamics. The time-development of the symmetric harmonic with ${\\bf\nk}_1$ (Fig. 12) is similar to that of the basic mode except that it\nhas much less energy and stress. It is again governed by Eqs. (21)\nand (22) (with ${\\bf k}_b$ replaced by ${\\bf k}_1$), but since\n${\\cal N}_z({\\bf k}_1)$ is no longer zero, it generates the spanwise\nvelocity, which, although does not contribute to the stress, takes\nup part of the mode's energy. The stress production rate and ${\\cal\nN}_y({\\bf k}_1)$ evolve nearly similarly with time [Fig. 12(a)],\nthough with a bit lesser correlation $R(d{\\cal H}_k\/dt,{\\cal\nN}_y)=0.78$ than that in the case of the basic mode. Therefore, the\nstress ${\\cal H}_k({\\bf k}_1)$ and the shearwise velocity,\n$|\\bar{u}_y({\\bf k}_1)|$, also display similar behavior in time with\ncoinciding amplification and decay intervals [Fig. 12(b), $R({\\cal\nH}_k,|\\bar{u}_y|)=0.86$]. Figure 12(c) shows that the stress remains\npositive at all times and supplies the mode with energy, while\n${\\cal N}_x({\\bf k}_1)$ is negative and acts as a sink for the\nenergy of the streamwise velocity. This indicates again that ${\\cal\nN}_y$ represents the only source maintaining the shearwise velocity\nand hence the stress.\n\nThe linear nonmodal dynamics for the non-symmetric harmonics is\ndescribed by the terms of linear origin related to shear in the\nabove dynamical equations in Fourier space. First consider the\nnon-symmetric mode with ${\\bf k}_2$ (Fig. 13). For this mode, ${\\cal\nN}_y({\\bf k}_2)$ acts as a source and generates the shearwise\nvelocity, $|\\bar{u}_y({\\bf k}_2)|$. The latter is then nonmodally\namplified due to the first linear shear-induced term on the right\nhand side of Eq. (17), $k_{x2}k_{y2}|\\bar{u}_y({\\bf\nk}_2)|^2\/k_{2}^2$, which is always positive for this mode since\n$k_{x2}k_{y2}>0$. Because ${\\cal N}_y({\\bf k}_2)$ is also positive\non average in time [see Fig. 8(h)], the linear and nonlinear terms\nin this equation cooperate with each other in the growth of\n$|\\bar{u}_y({\\bf k}_2)|$. During its amplification, the shearwise\nvelocity generates and nonmodally amplifies the streamwise velocity,\n$\\bar{u}_x({\\bf k}_2)$, due to the first shear-related linear term\non the rhs of Eq. (13), or equivalently, by the\n$(1-2k_{x2}^2\/k_2^2){\\cal H}_k({\\bf k}_2)$ term on the rhs of Eq.\n(16). As seen from Fig. 13(c), ${\\cal N}_x({\\bf k}_2)$ is negative\nmost of the time (and hence on average) for this harmonic and\ntherefore acts as a sink for $|\\bar{u}_x({\\bf k}_2)|$, opposing its\ngrowth. $\\bar{u}_x$ and $\\bar{u}_y$ velocity components give rise to\nthe stress ${\\cal H}_k({\\bf k}_2)$, whose production rate, as a\nresult, correlates with ${\\cal N}_y({\\bf k}_2)$ in time [Fig. 13(a),\n$R(d{\\cal H}_k\/dt,{\\cal N}_y)=0.84$]. Due to this, the temporal\nevolution of ${\\cal H}_k({\\bf k_2})$ itself closely correlates with\n$|\\bar{u}_y({\\bf k}_2)|$ [Fig. 13(b), $R({\\cal\nH}_k,|\\bar{u}_y|)=0.9$]. So, increase (decrease) of ${\\cal N}_y({\\bf\nk}_2)$ results in the increase (decrease) of the stress production\nrate. Consequently, the increase (decrease) in the shearwise\nvelocity leads to the increase (decrease) in the stress and hence in\nthe energy extraction from the flow. It is clear from Figs. 13(a)\nand 13(b) that the peaks, respectively, in the evolution of ${\\cal\nN}_y({\\bf k}_2)$ and $d{\\cal H}_k({\\bf k}_2)\/dt$ and in the\nevolution of $|\\bar{u}_y({\\bf k}_2)|$ and ${\\cal H}_k({\\bf k}_2)$\ncoincide. We note that these high correlations, leading to the\ngrowth of the Reynolds stress, are related to the fact that\n$\\bar{u}_x({\\bf k}_2)$ grows from $\\bar{u}_y({\\bf k}_2)$ due to\nnonnormality\/shear (first term on the rhs of Eq. 13), whereas\n$\\bar{u}_y({\\bf k}_2)$ is produced by the nonlinear transfer term\n${\\cal N}_y({\\bf k}_2)$. The nonmodal growth mechanism, as for the\nbasic mode, also for this non-symmetric mode ensures that streamwise\nand shearwise velocities correlate such that the resulting Reynolds\nstress remains positive, ${\\cal H}_k({\\bf k}_2)>0$, at all times\n[Fig. 13(b)], which is necessary for the continual extraction of\nenergy from the background flow into the harmonic. This is the\nessence of the interplay between nonlinearity, which generates seed\nshearwise velocity, and the nonmodal growth process that generates\nand amplifies streamwise velocity from the shearwise one and\nultimately the positive stress.\n\nThe situation is different for the mode with ${\\bf k}_3=(1,-1,1)$\n(Fig. 14). The stress and shearwise velocity are not related -- the\ncorresponding correlation coefficients are much smaller than those\nin the previous cases, $R(d{\\cal H}_k\/dt, {\\cal N}_y)=0.17$,\n$R({\\cal H}_k,|\\bar{u}_y|)=0.24$. The first shear-induced linear\nterm on the rhs of Eq. (16), responsible for the nonmodal growth of\nthe streamwise velocity, is dominated by ${\\cal N}_x({\\bf k}_3)$\n[Fig. 14(c)]. As a result, the evolution of $|\\bar{u}_x({\\bf k}_3)|$\nis governed mainly by ${\\cal N}_x({\\bf k}_3)$, and the effect of the\nnonnormality is small. Moreover, in contrast to the harmonic with\n${\\bf k}_2$, the term responsible for the nonmodal growth of the\nshearwise velocity is always negative, $k_{x3}k_{y3}|\\bar{u}_y({\\bf\nk}_3)|^2\/k_{3}^2<0$, and opposes ${\\cal N}_y({\\bf k}_3)$ in the\namplification of $|\\bar{u}_y({\\bf k}_3)|$. In other words, for the\nshearwise velocity, there is no constructive interplay between\nnonnormality-induced linear dynamics and the nonlinear term. So, the\nproduction rate of the streamwise velocity at ${\\bf k}_3$ is much\nreduced compared to that for ${\\bf k}_2$. This leads to the absence\nof correlation between the temporal evolution of the stress\nproduction rate and ${\\cal N}_y({\\bf k}_3)$ [Fig. 14(a)] and hence\nbetween the stress ${\\cal H}_k({\\bf k}_3)$ and shearwise velocity\n[Fig. 14(b)]. Since $|\\bar{u}_x({\\bf k}_3)|$ is determined mainly by\nthe nonlinear transfer term ${\\cal N}_x({\\bf k}_3)$ rather than by\nthe nonmodal dynamics, it is not correlated with $\\bar{u}_y({\\bf\nk}_3)$. As a result, the Reynolds stress produced by these two\nvelocity components is irregularly oscillating between positive and\nnegative values [Fig. 14(b)] in contrast to the basic mode and the\n${\\bf k}_1$ and ${\\bf k}_2$ modes, whose corresponding stresses are\nalways positive owing to the linear nonmodal growth mechanism [see\nfor comparison Figs. 12(b) and 13(b)]. Because of this, the\ntime-averaged value of the stress corresponding to this mode, being\nequal to $0.0011$, is smaller than that for the ${\\bf k}_2$ mode,\nwhich is 0.006. So, due to the constructive interplay between the\nnonlinear and linear terms in the case of the basic mode and ${\\bf\nk}_1$ and ${\\bf k}_2$ modes, their effectiveness in the energy\nextraction from the background flow is higher than that of the ${\\bf\nk}_3$ mode, for which such an interplay is practically absent.\n\nAlthough the dynamics of only three modes, one symmetric and other\ntwo non-symmetric ones with $k_y\/k_x>0$ and $k_y\/k_x<0$ at $k_z=1$,\nwere described above, the other neighboring modes in these regions\nof {\\bf k}-space exhibit a similar dynamics too. The time-averaged\nplots of Figs. 8(a)-8(c) clearly show that in the vital area, the\nReynolds stress of active modes with $k_y\/k_x<0$ on average is\nsmaller than that of ones with $k_y\/k_x>0$, which thus contribute\nmost to the positive stress. They play a key role in supplying\nturbulence with energy, together with the basic and nearby symmetric\nmodes. These modes share a similar dynamics with the ${\\bf k}_2$\nmode and therefore the higher values of the associated stress are\ndue to a constructive interplay between nonmodal growth and\nnonlinear feedback discussed above. From these plots we see that on\naverage ${\\cal N}_y\n> 0$ at $k_y\/k_x\n> 0$ and $k_x=0$, indicating that the transverse cascade continually regenerates\njust these modes, thereby ensuring the self-sustenance of the\nturbulence.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_15.eps}\n\\caption{Evolution of the volume-averaged (a) perturbed total\nkinetic energy, $\\langle E\\rangle$, and (b) the energy ${\\cal\nE}_k({\\bf k}_b)$ of the basic mode with ${\\bf k}_b=(0,0,\\pm 1)$ as\nwell as the maximum energy in {\\bf k}-space, ${\\cal E}_{k,max}$\n(dashed line) for the aspect ratio $(3,2)$. The basic mode\ncorresponds to the maximum energy only during large bursts of the\nspectral energy, while at other times this maximum is achieved by\nother neighboring active modes [bigger black dots in Figs. 16(a) and\n16(b)]. $\\langle E \\rangle$ approximately follows the maximum\nspectral energy and therefore the highest peaks are associated with\nthe dominance of the basic mode at these times.}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_16a.eps}\n\\includegraphics[width=5.9cm]{fig_16b.eps}\n\\includegraphics[width=5.9cm]{fig_16c.eps}\n\\caption{Distribution of the dominant energy-carrying modes in {\\bf\nk}-space at (a) $k_z=0$, (b) $k_z=1$ and (c) $k_z=2$ during the\nwhole run, for the aspect ratio $(3,2)$. The symbols have the same\nmeaning as in Fig. 5. The bigger black dots represent modes that\ncorrespond to the maximum of ${\\cal E}_k$ at least once during the\nevolution, including the basic mode at $k_x=k_y=0, k_z=1$.}\n\\end{figure*}\n\n\\subsection{Aspect ratio $(A_{xz}, A_{yz})=(3,2)$}\n\nThe evolution of the volume-averaged energy in the saturated\nturbulence for the aspect ratio $(3,2)$ is shown in Fig. 15(a). It\nis different from that in the previous case, having weaker and less\npronounced bursts. This difference is due to the specific changes of\nthe dynamics in spectral space as the aspect ratio varies. To make a\ndetailed comparison with the previous case, we performed a similar\nanalysis of energy spectra, linear energy-injecting stress and\nnonlinear transfer terms.\n\nFigure 16 shows the dynamically active, energy-carrying modes during\nthe evolution, as defined above. The number of these modes (209\nblack dots in Fig. 16) is much larger compared to that in the cubic\nbox. The vital area of turbulence containing these harmonics has\nbecome wider in {\\bf k}-space and spans the range $|k_x|\\leq 8,\n|k_y| \\leq 7, |k_z|\\leq 4$. The basic mode ${\\bf k}_b=(0,0,\\pm 1)$\nno longer dominates over the other active modes during the whole\nevolution, but corresponds to the maximum energy in spectral space\nonly during the highest bursts [Fig. 15(b)]. The lower peaks in the\nmaximum spectral energy curve in Fig. 15(b) are due to symmetric and\nnon-symmetric modes that lie in the vicinity of the basic one in the\nvital area; these modes are marked by bigger black dots in Figs.\n16(a) and 16(b) and have $k_z=0,1$. The modes with larger $k_z\\geq\n2$ in the vital area, although being active, never yield maximum of\nthe spectral energy. One can say that the dominance of the basic\nmode is still retained during the strongest bursts, but because\nother active modes also reach comparable maximum energies at\ndifferent times, the bursts in the volume-averaged energy do not\nappear to be as pronounced as in the cubic box, where only the basic\nmode prevails over the other active modes at all times.\n\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_17a.eps}\n\\includegraphics[width=5.9cm]{fig_17b.eps}\n\\includegraphics[width=5.9cm]{fig_17c.eps}\n\\caption{(Color online) Logarithm of the time-averaged kinetic\nenergy spectrum, $log_{10}({\\cal E}_k)$, for the aspect ratio\n$(3,2)$. The slices in $(k_x,k_y)$-plane are at (a) $k_z=0$, (b)\n$k_z=1$ and (c) $k_z=2$. The spectrum is anisotropic, with larger\npower at the $k_y\/k_x<0$ side.}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_18.eps}\n\\caption{Integrated in $(k_x, k_y)$-plane time-averaged (a) kinetic\nenergy spectrum, $\\hat{\\cal E}$, and (b) spectral Reynolds stress,\n$\\hat{\\cal H}$, vs. $k_z$ for the aspect ratio $(3,2)$. Both the\nenergy and stress achieve their maximum at $k_z=\\pm 1$, although the\nenergy at $k_z=0$ is close to that at $k_z=\\pm 1$.}\n\\end{figure}\n\nFigure 17 shows the time-averaged energy spectrum in\n$(k_x,k_y)$-plane at $k_z=0, 1, 2$. This spectrum is more extended\nover the wavenumbers compared to that in the cubic box, but has a\nsimilar shape and type of anisotropy. Although the maximum of energy\nis again at $k_z=1$, modes with $k_z=0$ also have comparable\nenergies, unlike that in the cubic box. As seen from Fig. 16, modes\nwith these two spanwise wavenumbers correspond to the maximum of\nspectral energy at certain times during evolution and hence play a\nmain role in the turbulence dynamics. The spectral energy then\nrapidly decreases with increasing $k_z$. This behavior is also seen\nin Fig. 18 showing the time-averaged spectra of the energy and\nstress integrated in $(k_x,k_y)$-plane, $\\hat{\\cal E}$ and\n$\\hat{\\cal H}$. Both these quantities reach their maximum at\n$k_z=\\pm 1$. $\\hat{\\cal E}$ at $k_z=0$ is close to its maximum\nvalue, while $\\hat{\\cal H}$ at $k_z=0$ is small, because ${\\cal\nH}_k(k_x,k_y,0)$ changes sign in $(k_x,k_y)$-plane, as in the case\nof the cubic box [see Fig. 8(a)]. So, although harmonics with\n$k_z=0$ can reach energies comparable to those with $k_z=1$, in\nfact, they are not as effective in the net energy extraction from\nthe flow as the latter ones, for which ${\\cal H}_k$ is positive over\nthe whole $(k_x,k_y)$-plane at $k_z=1$ [Fig. 19(a)].\n\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_19a.eps}\n\\includegraphics[width=5.9cm]{fig_19b.eps}\n\\includegraphics[width=5.9cm]{fig_19c.eps}\n\\caption{(Color online) Maps of the time-averaged (a)\nenergy-injecting stress, ${\\cal H}_k$, and nonlinear transfer terms,\n(b) ${\\cal N}_x$ and (c) ${\\cal N}_y$ in {\\bf k}-space for the\naspect ratio $(3,2)$. Shown are slices in $(k_x,k_y)$-plane at\n$k_z=1$.}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_20.eps}\n\\caption{Same as in Fig. 15, but for the aspect ratio $(1,2)$. The\nbasic mode corresponds to the maximum energy only during largest\nbursts, while at other times maximum energy is achieved by other\nnearby symmetric active modes [bigger black dots in Figs. 21(a) and\n21(b)]. The volume-averaged energy follows the maximum spectral\nenergy and therefore the highest peaks are manifestation of the\nbasic mode dynamics (growth) at these times.}\n\\end{figure}\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_21a.eps}\n\\includegraphics[width=5.9cm]{fig_21b.eps}\n\\includegraphics[width=5.9cm]{fig_21c.eps}\n\\caption{Distribution of the dominant energy-carrying modes in {\\bf\nk}-space at (a) $k_z=0$, (b) $k_z=1$ and (c) $k_z=2$ for the aspect\nratio $(1,2)$. The symbols have the same meaning as in Fig. 16. The\nmost active modes that contribute to the maximum of the spectral\nenergy at least once during the evolution are only symmetric ones,\nincluding the basic mode at $k_x=k_y=0, k_z=1$.}\n\\end{figure*}\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_22a.eps}\n\\includegraphics[width=5.9cm]{fig_22b.eps}\n\\includegraphics[width=5.9cm]{fig_22c.eps}\n\\caption{(Color online) Logarithm of the time-averaged kinetic\nenergy spectrum, $log_{10}({\\cal E}_k)$ for the aspect ratio\n$(1,2)$. The slices in $(k_x,k_y)$-plane at (a) $k_z=0$, (b) $k_z=1$\nand (c) $k_z=2$ are presented.}\n\\end{figure*}\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig_23.eps}\n\\caption{Same as in Fig. 18, but for the aspect ratio $(1,2)$. Both\nthe energy and stress achieve their maximum again at $k_z=\\pm 1$.}\n\\end{figure}\n\\begin{figure*}\n\\includegraphics[width=5.9cm]{fig_24a.eps}\n\\includegraphics[width=5.9cm]{fig_24b.eps}\n\\includegraphics[width=5.9cm]{fig_24c.eps}\n\\caption{(Color online) Maps of the time-averaged (a)\nenergy-injecting stress ${\\cal H}_k$, and nonlinear transfer terms,\n(b) ${\\cal N}_x$ and (c) ${\\cal N}_y$ in {\\bf k}-space for the\naspect ratio $(1,2)$. Shown are sections of these quantities in\n$(k_x,k_y)$-plane at $k_z=1$.}\n\\end{figure*}\n\nFigure 19 shows the time-averaged stress and the nonlinear transfer\nfunctions in spectral space at slices $k_z=1$: (a) ${\\cal H}_k$, (b)\n${\\cal N}_x$ and (c) ${\\cal N}_y$. Note that all these spectra\nexhibit qualitatively a similar shape and anisotropy as are in the\ncubic box. The maximum of the time-averaged stress comes again at\nthe basic mode with ${\\bf k}_b=(0,0, \\pm 1)$, indicating that in\ntime-average sense it is dominant, however, as mentioned above, the\nbasic mode in fact prevails over other modes only during the high\nburst events [Fig. 15(b)]. The nonlinear transfer term ${\\cal N}_x$,\nis negative in the vital area and therefore acts as a sink for\nstreamwise energy there, but transfers it to intermediate and larger\nwavenumbers where this term is positive. For these modes lying\noutside the vital area, the stress is negligible and therefore they\nget energy mainly due to the nonlinear transfer by ${\\cal N}_x$\nrather than from the flow. On the other hand, ${\\cal N}_y$ exhibits\na noticeable dependence on the wavevector angle (i.e., the\ntransverse cascade): ${\\cal N}_y$ is positive in the first and third\nquadrants [$k_y\/k_x>0$, yellow and red areas in Fig. 19(c)], where\nit creates the shearwise velocity, like that in the the cubic box\n[see Fig. 8(h)]. It is then further amplified jointly by ${\\cal\nN}_y$ and the linear shear-induced term on the rhs of Eq. (17),\nwhich is also positive in this region of spectral space. The\nshearwise velocity, undergoing amplification, induces growth of\nstreamwise velocity due to the nonnormality and ultimately stress.\nAs a result, as seen from Fig. 19(a), majority of the stress occurs\njust in this quadrants of spectral space (at a given $k_z$). So, a\nbasic self-sustaining scheme discussed above in the case of the\ncubic box, extends naturally to this aspect ratio.\n\nFinally, we note that the main distinction between the spectral\ndynamics of turbulence in the cubic and $(A_{xz},A_{yz})=(3,2)$\nboxes is the difference between the number of active,\nenergy-carrying modes and, consequently, the size of the vital area\nin spectral space. Increasing $A_{xz}$ with respect to $A_{yz}$\nintroduces more active symmetric and non-symmetric modes into the\nturbulence dynamics, widening the vital area. Together with the\nbasic mode, more active harmonics with $|k_z|=1$ and $k_z=0$ reach\nthe maximum spectral energy in succession in short time intervals.\nDue to this the bursts in the volume-averaged energy evolution are\nnot as pronounced, or strong as for the cubic box, however, largest\npeaks can still be attributed to the dynamics of the basic mode.\n\n\\subsection{Aspect ratio $(A_{xz},A_{yz})=(1,2)$}\n\nEvolution of the volume-averaged energy in the case of the aspect\nratio $(1,2)$ is shown in Fig. 20. In this figure, we also show the\ntime-evolution of the maximum value of the spectral energy and the\nenergy of the basic mode. Note that the volume-averaged energy well\nfollows the maximum spectral energy: the bursts (peaks) in the\nmaximum spectral energy are related to the basic mode dynamics\n(amplification). Figure 21 displays the active modes (black dots)\nduring the evolution that form the self-sustaining dynamics of\nturbulence for this aspect ratio. Their total number is 86, more\nthan that in the cubic box but less than that for the aspect ratio\n$(3,2)$. The vital area, where these modes lie, spans the range\n$|k_x|\\leq 2, |k_y|\\leq 7, |k_z|\\leq 2$. Note that compared to the\ncubic box (Fig. 5), increasing $A_{yz}$ with respect to $A_{xz}$\nintroduces more active modes with larger $k_y$, but with the same\n$k_x$ and $k_z$, resulting in the extension of the vital area along\nthe $k_y$-axis. Together with the basic mode with ${\\bf\nk}_b=(0,0,\\pm 1)$, the five symmetric modes -- two with $k_z=0$ and\nthree with $k_z=1$ [shown with bigger black dots in Figs. 21(a) and\n21(b)], contribute to the maximum spectral energy, ${\\cal\nE}_{k,max}$. As a result, the peaks in the volume-averaged energy\nappear more pronounced than that in the case $(3,2)$, where larger\nnumber of modes reach comparable energy maxima. However, as evident\nfrom Fig. 20(b), during the largest bursts, the maximum of the\nspectral energy is still due to the basic mode, as in the case of\nthe above two aspect ratios. So, the conclusion drawn above that the\nstronger bursts in the total energy are associated with the basic\nmode dynamics holds also for the aspect ratio $(1,2)$.\n\nThe time-averaged energy spectrum at three different $k_z=0, 1, 2$\nis shown in Fig. 22. It exhibits the same type of anisotropy due to\nshear as those in the above two cases. The time-averaged energy and\nstress spectra integrated in $(k_x,k_y)$-plane is represented as a\nfunction of $k_z$ in Fig. 23. Both spectra reach maximum again at\n$k_z=\\pm 1$ and decrease with $|k_z|$. Compared to the aspect ratio\n$(3,2)$, at $k_z=0$, the integrated spectral energy is smaller than\nthat at $k_z=\\pm 1$ and the integrated stress is again small,\nimplying that a net energy extraction by 2D modes is much less\neffective compared to that by 3D modes. Figure 24 shows the\ntime-averaged spectra of the stress and nonlinear transfer functions\nat $k_z=1$, which are qualitatively similar to those for the other\ntwo aspect ratios considered above and hence the self-sustaining\nprocess works in the same way.\n\n\\section{summary and discussion}\n\nWe considered a spectrally stable flow with constant shear of\nvelocity, where the only energy supply of (subcritical) turbulence\nis due to the flow nonnormality-induced linear growth of\nperturbations. To understand the underlying self-sustenance\nmechanism of homogeneous turbulence in this flow, we performed DNS\nand analyzed the dynamical processes in Fourier space. From the\nNavier-Stokes equations we derived the evolution equations for the\nperturbed velocity components in wavenumber space using Fourier\ntransformation. We explicitly calculated and visualized individual\nlinear and nonlinear terms in these spectral equations and analyzed\nthem in detail qualitatively as well as quantitatively based on the\nDNS data.\n\nThe main results corresponding to the five objectives outlined in\nIntroduction are summarized below:\n\n\n-- \\emph{\\it We first calculated the dynamically important optimal\nharmonics (Fig. 1) using the linear nonmodal analysis and identified\nthe range of aspect ratios, $A_{xz}\\leq A_{yz}$ and $A_{xz}\\geq 1$,\nfor which as many of them as possible are included in the simulation\ndomain.} Afterwards, based on this, we considered three aspect\nratios of the flow domain: cubic $(A_{xz},A_{yz})=(1,1)$, streamwise\nelongated $(A_{xz},A_{yz})=(3,2)$ and shearwise elongated\n$(A_{xz},A_{yz})=(1,2)$ boxes to understand the dependence of the\nturbulence dynamics on the latter.\n\n\n-- The Reynolds stress, which is of linear origin, is responsible\nfor the energy extraction from the flow into turbulence via the\nnonmodal growth of Fourier harmonics of perturbations. In the\nnonlinear regime, we identified a range of wavenumbers in which this\ngrowth is significant and hence the dynamically active modes that\ndetermine the self-sustenance process of the turbulence, carrying\nhigh values of energy and Reynolds stress (Figs. 5, 16, 21). They\nhave length scales comparable to the box size and hence are mainly\nconcentrated at small wavenumbers in Fourier space, which is\nsuitably labeled as \\emph{the vital area of the turbulence}.\n\n\n-- From these active harmonics, those contributing to the maximum of\nthe spectral energy and stress play a key role in the\nself-sustenance process (bigger black dots in Figs. 5, 16, 21). They\nhave smallest spanwise wavenumbers $k_z=0,2\\pi\/L_z$ and their number\ngenerally depends on the box aspect ratio. It was shown that for all\nthe three aspect ratios considered, \\emph{\\it the basic large scale\nmode with the wavenumber ${\\bf k}_b = (0, 0, \\pm 2\\pi\/L_z)$ plays a\nspecial role in the homogeneous shear turbulence dynamics},\nconsistent with previous study by Pumir \\cite{Pumir96}. For the\ncubic box, it is the only mode that yields the maximum spectral\nenergy and Reynolds stress in Fourier space during the entire\nevolution, much larger than that of the nearby active modes, and\nundergoes the largest nonmodal amplification. This amplification\noccurs in bursts, alternating with decaying intervals (Fig. 2).\nDuring these bursts, it contributes up to $70\\%$ of the total\nReynolds stress. \\emph{\\it As a result, the basic mode is\nresponsible for the production of a major fraction of the\nturbulence's total energy and largely determines its behavior -- the\ntotal energy also exhibits bursts closely following those of the\nbasic mode.} Such bursting events in the energy are also observed in\nturbulent boundary layers \\cite{Robinson91,Jimenez_etal05} and the\nhomogeneous shear turbulence model is often invoked to understand\ntheir nature \\cite{Pumir96,Sekimoto_etal15}. The prevalence of the\nbasic mode is somewhat reduced for other two aspect ratios. We found\nthat increasing streamwise or\/and shearwise dimensions of the box\nintroduces more modes, respectively, along these directions in the\nvital area, which grow comparably with the basic mode, also\ncorresponding to the maximum of the spectral energy at certain times\nduring evolution. As a result, their contribution to the total\nenergy increases. The evolution of the latter is now determined by\nthese active modes, rather than by the basic mode only. Due to this,\nthe total energy is characterized by weaker bursts, especially for\nthe aspect ratio $(A_{xz},A_{yz})=(3,2)$, with the largest number of\nthe active modes in the vital area. Nevertheless, the appearance of\nhighest bursts in the total energy and stress appears to be related\nto the basic mode dynamics for these aspect ratios too (Figs. 15,\n20).\n\n\n -- The nonmodal growth process and, therefore the resulting energy\nand Reynolds stress spectra, are anisotropic in Fourier ($\\bf k$-)\nspace, i.e., mainly depend on the orientation of wavevector (Figs.\n8, 19, 24). This anisotropy of the linear processes in shear flows,\nin turn, leads to anisotropy of nonlinear processes in Fourier\nspace: the nonlinear terms, which do not directly draw the\nbackground flow energy, transversely redistribute this energy over\nwavevector angles. \\emph{It was found that this new -- transversal\n-- type of nonlinear redistribution of harmonics, referred to as the\ntransverse cascade, is a generic feature of nonlinear dynamics of\nperturbations in spectrally stable shear flows. It differs from and\nrepresents an alternative to the classical energy cascade processes\n(direct\/inverse) in shear flow turbulence.} Previously, the\ntransverse cascade was also demonstrated to take place and play an\nimportant role in 2D HD and MHD shear flows\n\\cite{Horton_etal10,Mamatsashvili_etal14}. This cooperation of\nanisotropic linear and nonlinear processes, in turn, gives rise to\nan anisotropic energy spectrum in shear flows, which, in general,\ndiffers from the classical Kolmogorov spectrum in homogeneous\nturbulence without background shear, especially at small and\nintermediate wavenumbers. As a result, the transverse cascade may\nnaturally appear to be a keystone of the \\emph{bypass} concept of\nsubcritical turbulence in spectrally stable HD shear flows, which is\nbeing actively discussed among the hydrodynamical community.\nTherefore, the conventional description of shear flow turbulence\nsolely in terms of direct and inverse cascades, which leaves out the\ntransverse cascade, might be incomplete and misleading.\n\n\n-- The vital area in wavenumber space is the primary site of\nactivity of the energy-supplying linear nonmodal growth and the\nnonlinear transverse cascade that lie at the heart of the\nself-sustaining dynamics of turbulence. The course of events\nensuring the self-sustenance of the turbulence is as follows. The\ntransverse cascade continually transfers harmonics to those\nquadrants of the vital area where the shear flow causes linear\nnonmodal growth, i.e., regenerates harmonics that can undergo\namplification due to the flow nonnormality, and thereby closes\nfeedback loop ensuring sustenance of the turbulence. This general\npicture comprises refined, somewhat differing interplay of linear\nand nonlinear processes for the basic mode and active symmetric and\nnon-symmetric modes. These, qualitatively quite simple but\nquantitatively a bit involved schemes are described in detail for\ndifferent aspect ratios in Section III (see also Figs. 10-14).\n\\emph{\\it Thus, we have demonstrated that in the considered constant\nshear flow, the turbulence is maintained by a subtle interplay\nbetween linear and nonlinear processes: the first supplies energy\nfor turbulence via shear-induced nonmodal growth process\n(characterized by the Reynolds stresses) and the second plays an\nimportant role of providing a positive feedback that makes this\ngrowth process long-lived against viscous dissipation, which would\notherwise be transient in the absence of the latter.}\n\n\nThe number of the active harmonics (defined as the harmonics whose\nenergies grow more than $10\\%$ of the maximum spectral energy at\nleast once during the evolution) in the vital area, which are the\nmain participants in the turbulence dynamics, generally depends on\nthe box aspect ratio and is quite large. In the considered here\naspect ratios, it exceeds 35 for the cubic box and can be more than\n200 for longer boxes in the streamwise or shearwise directions (see\nFigs. 5, 16 and 21), indicating fairly complex nature of\nself-sustaining dynamics of turbulence, which apparently cannot be\ndescribed by low-order models. In this paper, performing a full DNS,\nthe realization of the nonlinear transverse cascade and its role in\nthe self-sustenance of homogeneous shear turbulence are demonstrated\nin the most general manner, without truncating the number of modes\nand using other simplifying assumptions usually made in low-order\nmodels of self-sustaining processes (e.g., Refs.\n\\cite{Waleffe95,Waleffe97,Mohelis_etal04}). Nevertheless, one can\nrelate the described here self-sustaining scheme of homogeneous\nshear turbulence in the constant shear flow without (physical)\nboundaries to the self-sustenance scenario in low-order models\nwhich, however, commonly include walls affecting the dynamics (see\ne.g., Refs.~\\cite{Farrell_Ioannou12,Thomas_etal14,Thomas_etal15}).\nNamely, the nonlinear regeneration of streamwise rolls is analogous\nto the production of shearwise velocity due to the nonlinear\ntransfer terms and the generation and nonmodal amplification of\nstreamwise streaks from the former due to shear (nonnormality) is\nsimilar to the amplification of streamwise velocity. One of the main\nresults of the present study is that the self-sustaining process of\nturbulence in spectrally stable shear flows in fact does not require\nthe presence of walls and is intrinsic to the flow system itself,\nbeing governed by the interplay between linear nonnormality-induced\nand the nonlinear transverse cascade processes.\n\nFinally, we would like to emphasize that revealing the new aspects\nof the homogeneous shear turbulence dynamics -- anisotropic energy\nspectra, the nonlinear transverse cascade and the notion of the\nvital area, where the self-sustaining process is concentrated, have\nbecome largely possible owing to analysis in Fourier space. This\nallows us to gain a deeper insight into turbulence dynamics and its\nunderlying sustenance mechanism. At the same time, the performed\nsimulations show that the basic mode, which is central in the energy\nexchange between the shear flow and turbulence, has a large scale\nspanwise variation and is uniform in the streamwise and shearwise\ndirections, $k_x= k_y = 0, k_z = \\pm 2\\pi\/L_z$; it has streamwise\nand shearwise velocities, but zero spanwise velocity. This simple\nconfiguration (geometry) of the basic mode makes it easily\ndescribable in physical space too -- its amplification due to the\nnonnormality can be readily understood from Eqs. (3) and (4) in\nphysical space. The streamwise velocity $u_x$ is generated from the\nshearwise velocity due to shear (linear term -$Su_y$ in Eq. 3). As a\nresult, this two velocities correlate and produce positive stress,\nextracting flow energy. The role of nonlinearity in Eq. (4) is then\nto regenerate the shearwise velocity $u_y$.\n\n\n\\begin{acknowledgments}\nGM, GK and GC are thankful for hospitality at the School of\nAeronautics, Universidad Polit{\\'e}cnica de Madrid, and for\nfinancial support within the Second Multiflow Summer Workshop, 25\nMay-26 June, 2015, funded in part by the Multiflow Program of the\nEuropean Research Council. GM is supported by the Georg Forster\nPostdoctoral Research Fellowship from the Alexander von Humboldt\nFoundation. SD is supported by the China Scholarship Council. We\nthank the Referees for useful comments that improved the\npresentation of our results.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe notion of complexity is widely used in Mathematics and Computer Science in the context of several various abstract objects. The computational complexity of algorithms, the algebraic complexity of polynomials, the Rademacher complexity in the computational learning theory or the social complexity in the social systems are the concepts of great importance in the corresponding fields of science. The present work is devoted to the particular type of complexity -- the analytic complexity of bivariate holomorphic functions. \n\nThe notion of analytic complexity is closely related to Hilbert's 13th problem, which was solved by A.N.~Kolmogorov and V.I.~Arnold in 1957~\\cite{Arnold}. The initial formulation of Hilbert's 13th problem asks whether any continuous function of several variables can be represented as a finite superposition of bivariate functions~\\cite{Vitushkin}. The main purpose of the theory of the analytic complexity is finding similar representations for analytic functions. The objects under consideration in this theory are the {\\it analytic complexity classes.}\n\n\\begin{definition} {\\rm (See~\\cite{BeloshapkaRJMP2007}).\nLet $\\mathcal{O}(U(x_0,y_0))$ denote the set of holomorphic functions in an open neighborhood~$U(x_0,y_0)$ of a point~$(x_0,y_0)\\in \\mathbb{C}^2.$ The class~$Cl_0$ of analytic functions of analytic complexity zero is defined to comprise the functions that depend on at most one of the variables. A function $f(x,y)$ is said to belong to {\\it the class $Cl_n$ of functions with analytic complexity $n>0$} if there exists a point $(x_0,y_0)\\in \\mathbb{C}^2$ and a germ~$\\mathfrak{f}(x,y)\\in\\mathcal{O}(U(x_0,y_0))$ of this function holomorphic at $(x_0, y_0)$ such that $\\mathfrak{f}(x,y)=c(a(x,y)+b(x,y))$ for some germs of holomorphic functions $a,b\\in Cl_{n-1}$ and $c\\in Cl_0$. If~there is no such representation for any finite~$n,$ then the function~$f$ is said to be {\\it of infinite analytic complexity}}.\n\\end{definition}\n\n\\begin{example}\nA generic element of the first complexity class~$Cl_1$ is a function of the form $f_3(f_1(x)+f_2(y)).$ A function in~$Cl_2$ can be represented in the form $f_7\\left(f_5(f_1(x)+f_2(y))+f_6(f_3(x)+f_4(y))\\right),$ \n\\noindent where $f_i(\\cdot)$ are univariate holomorphic functions, $i=1,\\ldots,7$.\n\\end{example}\n\nFor any class of analytic complexity~$Cl_n, n\\in \\mathbb{N}$ there exists a system of differential polynomials with constant coefficients~$\\Delta_n$ which annihilates a~function if and only if it belongs to~$Cl_n.$ \n\\begin{example} (See~\\cite{BeloshapkaRJMP2007}). For a bivariate function $f(x,y)$ consider the differential polynomial\n$$\\Delta_1 (f) = f'_{x}(f'_{y})^2f'''_{xxy}-(f'_{x})^2f'_{y}f'''_{xyy}+f''_{xy}(f'_{x})^2f''_{yy}-f''_{xy}(f'_{y})^2f''_{xx}.$$ \nThis differential polynomial vanishes if and only if its argument $f\\in Cl_1.$\n\\end{example}\n\nThe problem of defining whether a function belongs to an analytic complexity class is equivalent to computing the corresponding differential polynomial. Note that this is the problem of great computational complexity~\\cite{BeloshapkaZametki2019,SadykovCASC2018}, thus a direct approach to its solution appears to be inappropriate.\n\nAn important question is a possible connection between the classes of finite analytic complexity and hypergeometric functions. In this paper we consider hypergeometric functions as solutions of hypergeometric systems in the sense of Horn~\\cite{Horn1889,Sadykov-Tanabe}. We choose a matrix $A \\in \\mathbb{Z}^{m\\times n} = \\left(A_{ij},\\; i=1,\\ldots,m, j = 1,\\ldots,n\\right) $ and a vector of parameters $c = (c_1,\\ldots,c_m) \\in \\mathbb{C}^m$. We denote the rows of this matrix by~{\\bf A$_i,\\, i=1,\\ldots,m.$}\n\n\\begin{definition}{\\rm\nThe {\\it hypergeometric system} (or {\\it Horn system}) Horn$(A,c)$ is the following system of partial differential equations:\n\n\\begin{equation}\nx_j P_j(\\theta) f(x) = Q_j(\\theta) f(x), \\, j = 1,\\ldots, n,\n\\label{eq:HornSystem}\n\\end{equation}\n\n\\noindent where \n\n$$P_j(s)=\\prod\\limits_{i:A_{ij}>0}\\prod\\limits_{l_j^{(i)}=0}^{A_{ij}-1}\\left(\\left<\\textbf{A}_i,s\\right>+c_i+l_j^{(i)}\\right), $$ \n$$Q_j(s)=\\prod\\limits_{i:A_{ij}<0}\\prod\\limits_{l_j^{(i)}=0}^{|A_{ij}|-1}\\left(\\left<\\textbf{A}_i,s\\right>+ c_i+l_j^{(i)}\\right),$$ and $\\theta = (\\theta_1,\\ldots,\\theta_n),\\,\\theta_j=x_j\\frac{\\partial}{\\partial x_j}.$\n}\n\\end{definition}\n\n\\begin{definition} {\\rm\nThe system of equations Horn$(A,c)$ is called {\\it nonconfluent} if $\\sum\\limits_{i=1}^{m}\\textbf{A}_i=0.$\n}\n\\end{definition}\n\n It has been conjectured in~\\cite{SadykovProceedings} that any hypergeometric function has finite analytic complexity.  Hypergeometric systems of equations differ greatly from the differential criteria for the analytic complexity classes, but numerous computer experiments suggest the hypothesis is true in a lot of particular cases~\\cite{DickensteinSadykovDoklady,DickensteinSadykovMatSbornik}. The case of hypergeometric systems with low holonomic rank has been considered~in~\\cite{KrasikovCASC2019}.\n\nThe set of functions of infinite analytic complexity is also a matter of interest. Until recently, all known examples of such functions were the differentially transcendental functions, that is, the functions that are not solutions to any nonzero differential polynomial with constant coefficients. Important examples of  differentially algebraic functions of infinite analytic complexity have been presented in \\cite{StepanovaSbMath,StepanovaJSFU}.   \n\n\\begin{definition}{\\rm\nLet $l_i$ denote the generator of the sublattice $\\{s\\in\\mathbb{Z}^n:\\left<\\textbf{A}_i,s\\right>=0\\}$ and let $k_i$ be the number of elements in the set $\\{\\textbf{A}_1,\\ldots,\\textbf{A}_m\\},$ which coincide with $\\textbf{A}_i.$ Let us define a polygon~$P(A)$ (see~\\cite{SadykovBullSciMath}) as the integer convex polygon whose sides are translations of the vectors $k_i l_i,$ the vectors $\\textbf{A}_1,\\ldots,\\textbf{A}_m$ being the outer normals to its sides. We will say that hypergeometric system Horn$(A,c)$ is {\\it defined by the polygon}~$P(A).$\n}\n\\label{def:OreSatoPolygon}\n\\end{definition}\n\n\\begin{definition} \\rm\nA polygon is called {\\it a zonotope} if it can be represented as the Minkowski sum of segments.\n\\end{definition}\n\nIn this article we investigate the analytic complexity of solutions to hypergeometric systems of equations~(\\ref{eq:HornSystem}) defined by zonotopes.\n\nThe present paper is organised as follows. In Section~\\ref{sec:SystemsDefinedByZonotopes} we investigate particular cases of hypergeometric systems defined by zonotopes and analyze the analytic complexity of their solutions. We formulate and prove an estimate of the analytic complexity for polynomial solutions to such systems in terms of the defining matrices and parameter vectors. In Section~\\ref{sec:Algorithms} we present algorithms for finding the supports of polynomial solutions to hypergeometric systems and estimating the analytic complexity of polynomials. In Section~\\ref{sec:Examples} we consider examples of hypergeometric systems and estimate the analytic complexity of their solutions.\n\nWe use the Wolfram Mathematica package HyperGeometry for solving hypergeometric systems we investigate in this article. The package is available for free public use at {\\small https:\/\/www.researchgate.net\/publication\/318986894\\_HyperGeometry}, the description of available functions is given in~\\cite{SadykovProgramming}. \n\n\n\n\n\\section{Hypergeometric Systems Defined by Zonotopes}\n\\label{sec:SystemsDefinedByZonotopes}\n\nLet us consider the special case of hypergeometric systems defined by zonotopes. Numerous experiments suggest that the analytic complexity of polynomial solutions to such systems can be much lower than its estimate based on the number of their monomials.\n\nThe set of hypergeometric systems defined by zonotopes enjoys the following properties:\n\n\\noindent a) these systems are holonomic for the generic parameter value;\n\n\\noindent b) the holonomic rank of hypergeometric systems (see Theorem~2.5 in~\\cite{DickensteinSadykovAdvances}) is given by \n\n$$\\textrm{rank}(\\textrm{Horn}(A,c))=d_1 d_2-\\sum\\limits_{\\textbf{A}_i, \\textbf{A}_j \\textrm{ lin. dependent}}\\nu_{ij},$$ where $d_j=\\sum\\limits_{\\scriptsize\\begin{array}{c} i=1\\\\A_{ij}>0\\end{array}}^m A_{ij}, j=1,2$ and\n\n$$\\nu_{ij}=\\left\\{\\begin{array}{l}\\textrm{min}(|A_{i1}A_{j2}|,|A_{j1}A_{i2}|), \\textrm{ if } \\textbf{A}_i, \\textbf{A}_j \\textrm{ are in opposite open quadrants of }\\mathbb{Z}^2,\\\\0, \\textrm{ otherwise.}\\end{array}\\right.$$\n\nFor the hypergeometric systems defined by zonotopes there is another formula for computing their holonomic rank (see Proposition~1 in~\\cite{KrasikovCASC2019}), which in some cases may be more suitable;\n\n\\noindent c) the rows of the matrix defining such a system can be united into two matrices $\\hat{A},-\\hat{A};$ \n\n\n\\noindent d) for a hypergeometric system defined by a~zonotope one can always choose parameter values such that any solution to the resulting system is a polynomial (see~\\cite{Sadykov-Tanabe}). Namely, for such a hypergeometric system~$\\text{Horn}(A,c),$ where the matrix $A$ contains~$2k$ rows, let $\\alpha = (\\alpha_1,\\ldots,\\alpha_k)$ be a part of the parameter vector~$c,$ corresponding to the matrix $\\hat{A}$ (see the property (c) above)$, \\beta = (\\beta_1,\\ldots,\\beta_k)$~be a part of this vector, corresponding to~$-\\hat{A}.$ Then the general solution to $\\text{Horn}(A,c)$ is a polynomial if $-\\alpha_i-\\beta_i\\in \\mathbb{N}\\backslash\\{0\\}$ for~$i=1,\\ldots,k.$ \n\nThe simplest case of a zonotope is a parallelogram. The analytic complexity estimate of the solutions to the systems defined by parallelograms is the basis for more complex cases.  \n\n\\begin{proposition} \\rm\n\nThe analytic complexity of a hypergeometric systems defined by a parallelogram cannot exceed~$2.$\n\n\\noindent{\\it Proof.}\nThe solution to the hypergeometric system defined by a parallelogram, has been described in Proposition~4.7 in~\\cite{Sadykov-Tanabe}. For the bivariate system ($n=2$) this formula leads to $$(x_1^{-a_{11}}x_2^{-a_{21}})^{\\alpha_1} \\left(1+x_1^{-a_{11}}x_2^{-a_{21}}\\right)^{-\\alpha_1-\\beta_1}\\cdot (x_1^{-a_{12}}x_2^{-a_{22}})^{\\alpha_2} \\left(1+x_1^{-a_{12}}x_2^{-a_{22}}\\right)^{-\\alpha_2-\\beta_2},$$where $A^{-1}=\\left(\\begin{array}{rr}a_{11} & a_{12} \\\\ a_{21} & a_{22}\\end{array}\\right), c=(\\alpha_1,\\alpha_2,\\beta_1,\\beta_2).$ The monomials $x_1^{-a_{11}}x_2^{-a_{21}}$ and $x_1^{-a_{12}}x_2^{-a_{22}}$ both belong to~$Cl_1,$ thus for any univariate analytic functions $\\phi(\\cdot),\\psi(\\cdot)$ the product $\\phi(x_1^{-a_{11}}x_2^{-a_{21}})\\cdot\\psi(x_1^{-a_{12}}x_2^{-a_{22}})$ belongs to~$Cl_2.$~\\hfill $\\square$\n\\label{prop:parallelogram}\n\\end{proposition}\n\nThe following example shows that the solutions to hypergeometric systems defined by more complex polygons can be of a low analytic complexity.\n\n\\begin{example} {\\it Simple zonotope.}\nLet us consider the hypergeometric system $\\text{Horn}(A_0,$ $c_0)$ defined by the matrix $A_0 = \\left(\\begin{array}{rrrrrr} 1 & -1 & 1 & -1 & 0 & 0 \\\\ 1 & -1 & 0 & 0 & 1 & -1 \\end{array}\\right)^{\\rm T}$ and the parameter vector $c_0=(-23,22,-10,0,-9,0).$ The holonomic rank of this system is equal to~$3.$ The hypergeometric system $\\text{Horn}(A_0,c_0)$ is defined by a zonotope, since rows of $A_0$ correspond to normal vectors to sides of the polygon. Representation of this zonotope in the form of the Minkowski sum of segments is shown in Figure~\\ref{fig:polyOreSatoCoef}\n\n\\begin{figure}[htbp]\n\\begin{minipage}{4cm}\n\\begin{picture}(100,100)\n  \\put(20,0){\\vector(0,1){100}}\n  \\put(0,20){\\vector(1,0){100}}\n \n\n\n \n  \\put(20,45){\\line(1,-1){25}}\n  \\put(45,70){\\line(1,-1){25}}\n  \\put(20,70){\\line(1,0){25}}\n  \\put(70,45){\\line(0,-1){25}}\n  \\put(20,70){\\circle*{3}}\n  \\put(45,20){\\circle*{3}}  \n  \\put(70,45){\\circle*{3}}\n  \\put(45,70){\\circle*{3}}\n  \\put(20,45){\\circle*{3}}\n  \\put(70,20){\\circle*{3}}\n\n\\end{picture}\n\\end{minipage}\n=\n\\begin{minipage}{2cm}\n\\begin{picture}(70,70)\n \n\n\n  \\put(10,35){\\line(1,-1){25}}\n  \\put(35,10){\\circle*{3}}  \n  \\put(10,35){\\circle*{3}}\n\n\\end{picture}\n\\end{minipage}\n+\n\\begin{minipage}{2cm}\n\\begin{picture}(70,70)\n \n\n\n  \\put(10,35){\\line(1,0){25}}\n  \\put(35,35){\\circle*{3}}  \n  \\put(10,35){\\circle*{3}}\n\n\\end{picture}\n\\end{minipage}\n+\n\\begin{minipage}{2cm}\n\\begin{picture}(70,70)\n \n\n\n  \\put(35,10){\\line(0,1){25}}\n  \\put(35,10){\\circle*{3}}  \n  \\put(35,35){\\circle*{3}}\n\n\\end{picture}\n\\end{minipage}\n\\caption{Polygon, defining the system $\\text{Horn}(A_0,c_0),$ and its representation as the Minkowski sum of segments}\n\\label{fig:polyOreSatoCoef}\n\\end{figure}\n\n The support for the system $\\text{Horn}(A_0,c_0)$ polynomial solution is shown in Figure~\\ref{fig:generalZonotopeExample}. \n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{8cm}\n\\begin{picture}(250,250)\n  \\put(20,0){\\vector(0,1){260}}\n  \\put(0,20){\\vector(1,0){260}}\n\n\n  \\put(8,8){\\small $0$}\n  \\put(30,20){\\line(0,1){3}}\n \n  \\put(40,20){\\line(0,1){3}}\n \n  \\put(50,20){\\line(0,1){3}}\n \n  \\put(60,20){\\line(0,1){3}}\n \n  \\put(70,20){\\line(0,1){5}}\n  \\put(68,8){\\small $5$}\n  \\put(80,20){\\line(0,1){3}}\n \n  \\put(90,20){\\line(0,1){3}}\n \n  \\put(100,20){\\line(0,1){3}}\n \n  \\put(110,20){\\line(0,1){3}}\n \n  \\put(120,20){\\line(0,1){5}}\n  \\put(120,8){\\small $10$}\n  \\put(130,20){\\line(0,1){3}}\n \n  \\put(140,20){\\line(0,1){3}}\n  \\put(150,20){\\line(0,1){3}}\n  \\put(160,20){\\line(0,1){3}}\n  \\put(170,20){\\line(0,1){5}}\n  \\put(166,8){\\small $15$}\n  \\put(180,20){\\line(0,1){3}}\n  \\put(190,20){\\line(0,1){3}}\n  \\put(200,20){\\line(0,1){3}}\n  \\put(210,20){\\line(0,1){3}}\n  \\put(220,20){\\line(0,1){5}}\n  \\put(216,8){\\small $20$}\n  \\put(230,20){\\line(0,1){3}}\n  \\put(240,20){\\line(0,1){3}}\n  \\put(250,20){\\line(0,1){3}}\n \n  \\put(20,30){\\line(1,0){3}}\n \n  \\put(20,40){\\line(1,0){3}}  \n \n  \\put(20,50){\\line(1,0){3}}\n \n  \\put(20,60){\\line(1,0){3}}  \n \n  \\put(20,70){\\line(1,0){5}}\n  \\put(8,67){\\small $5$}\n  \\put(20,80){\\line(1,0){3}}  \n \n  \\put(20,90){\\line(1,0){3}}  \n \n  \\put(20,100){\\line(1,0){3}}  \n \n  \\put(20,110){\\line(1,0){3}}  \n \n  \\put(20,120){\\line(1,0){5}}\n  \\put(8,117){\\small $10$}\n  \\put(20,130){\\line(1,0){3}}   \n  \\put(20,140){\\line(1,0){3}} \n  \\put(20,150){\\line(1,0){3}} \n  \\put(20,160){\\line(1,0){3}} \n  \\put(20,170){\\line(1,0){5}} \n  \\put(8,167){\\small $15$}\n  \\put(20,180){\\line(1,0){3}}   \n  \\put(20,190){\\line(1,0){3}} \n  \\put(20,200){\\line(1,0){3}} \n  \\put(20,210){\\line(1,0){3}} \n  \\put(20,220){\\line(1,0){5}} \n  \\put(8,217){\\small $20$}\n  \\put(20,230){\\line(1,0){3}} \n  \\put(20,240){\\line(1,0){3}} \n  \\put(20,250){\\line(1,0){3}}\n  \\put(0,110){\\line(1,0){260}}\n  \\put(120,0){\\line(0,1){260}}\n  \\put(10,250){\\line(1,-1){240}}\n  \\put(10,260){\\line(1,-1){250}}\n  \\put(20,240){\\circle*{4}}\n  \\put(20,250){\\circle*{4}}\n  \\put(30,230){\\circle*{4}}\n  \\put(30,240){\\circle*{4}}\n  \\put(40,220){\\circle*{4}}\n  \\put(40,230){\\circle*{4}}\n  \\put(50,210){\\circle*{4}}\n  \\put(50,220){\\circle*{4}}\n  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\\put(240,30){\\circle*{4}}\n  \\put(240,20){\\circle*{4}}\n  \\put(250,20){\\circle*{4}}\n\n  \\put(20,20){\\circle*{4}}\n  \\put(30,20){\\circle*{4}}\n  \\put(40,20){\\circle*{4}}\n  \\put(50,20){\\circle*{4}}  \n  \\put(60,20){\\circle*{4}}\n  \\put(70,20){\\circle*{4}}\n  \\put(80,20){\\circle*{4}}\n  \\put(90,20){\\circle*{4}}\n  \\put(100,20){\\circle*{4}}\n  \\put(110,20){\\circle*{4}}\n  \\put(120,20){\\circle*{4}}\n  \\put(20,30){\\circle*{4}}\n  \\put(30,30){\\circle*{4}}\n  \\put(40,30){\\circle*{4}}\n  \\put(50,30){\\circle*{4}}  \n  \\put(60,30){\\circle*{4}}\n  \\put(70,30){\\circle*{4}}\n  \\put(80,30){\\circle*{4}}\n  \\put(90,30){\\circle*{4}}\n  \\put(100,30){\\circle*{4}}\n  \\put(110,30){\\circle*{4}}\n  \\put(120,30){\\circle*{4}}\n  \\put(20,40){\\circle*{4}}\n  \\put(30,40){\\circle*{4}}\n  \\put(40,40){\\circle*{4}}\n  \\put(50,40){\\circle*{4}}  \n  \\put(60,40){\\circle*{4}}\n  \\put(70,40){\\circle*{4}}\n  \\put(80,40){\\circle*{4}}\n  \\put(90,40){\\circle*{4}}\n  \\put(100,40){\\circle*{4}}\n  \\put(110,40){\\circle*{4}}\n  \\put(120,40){\\circle*{4}}\n  \\put(20,50){\\circle*{4}}\n  \\put(30,50){\\circle*{4}}\n  \\put(40,50){\\circle*{4}}\n  \\put(50,50){\\circle*{4}}  \n  \\put(60,50){\\circle*{4}}\n  \\put(70,50){\\circle*{4}}\n  \\put(80,50){\\circle*{4}}\n  \\put(90,50){\\circle*{4}}\n  \\put(100,50){\\circle*{4}}\n  \\put(110,50){\\circle*{4}}\n  \\put(120,50){\\circle*{4}}\n  \\put(20,60){\\circle*{4}}\n  \\put(30,60){\\circle*{4}}\n  \\put(40,60){\\circle*{4}}\n  \\put(50,60){\\circle*{4}}  \n  \\put(60,60){\\circle*{4}}\n  \\put(70,60){\\circle*{4}}\n  \\put(80,60){\\circle*{4}}\n  \\put(90,60){\\circle*{4}}\n  \\put(100,60){\\circle*{4}}\n  \\put(110,60){\\circle*{4}}\n  \\put(120,60){\\circle*{4}}\n  \\put(20,70){\\circle*{4}}\n  \\put(30,70){\\circle*{4}}\n  \\put(40,70){\\circle*{4}}\n  \\put(50,70){\\circle*{4}}  \n  \\put(60,70){\\circle*{4}}\n  \\put(70,70){\\circle*{4}}\n  \\put(80,70){\\circle*{4}}\n  \\put(90,70){\\circle*{4}}\n  \\put(100,70){\\circle*{4}}\n  \\put(110,70){\\circle*{4}}\n  \\put(120,70){\\circle*{4}}\n  \\put(20,80){\\circle*{4}}\n  \\put(30,80){\\circle*{4}}\n  \\put(40,80){\\circle*{4}}\n  \\put(50,80){\\circle*{4}}  \n  \\put(60,80){\\circle*{4}}\n  \\put(70,80){\\circle*{4}}\n  \\put(80,80){\\circle*{4}}\n  \\put(90,80){\\circle*{4}}\n  \\put(100,80){\\circle*{4}}\n  \\put(110,80){\\circle*{4}}\n  \\put(120,80){\\circle*{4}}\n  \\put(20,90){\\circle*{4}}\n  \\put(30,90){\\circle*{4}}\n  \\put(40,90){\\circle*{4}}\n  \\put(50,90){\\circle*{4}}  \n  \\put(60,90){\\circle*{4}}\n  \\put(70,90){\\circle*{4}}\n  \\put(80,90){\\circle*{4}}\n  \\put(90,90){\\circle*{4}}\n  \\put(100,90){\\circle*{4}}\n  \\put(110,90){\\circle*{4}}\n  \\put(120,90){\\circle*{4}}\n  \\put(20,100){\\circle*{4}}\n  \\put(30,100){\\circle*{4}}\n  \\put(40,100){\\circle*{4}}\n  \\put(50,100){\\circle*{4}}  \n  \\put(60,100){\\circle*{4}}\n  \\put(70,100){\\circle*{4}}\n  \\put(80,100){\\circle*{4}}\n  \\put(90,100){\\circle*{4}}\n  \\put(100,100){\\circle*{4}}\n  \\put(110,100){\\circle*{4}}\n  \\put(120,100){\\circle*{4}}\n  \\put(20,110){\\circle*{4}}\n  \\put(30,110){\\circle*{4}}\n  \\put(40,110){\\circle*{4}}\n  \\put(50,110){\\circle*{4}}  \n  \\put(60,110){\\circle*{4}}\n  \\put(70,110){\\circle*{4}}\n  \\put(80,110){\\circle*{4}}\n  \\put(90,110){\\circle*{4}}\n  \\put(100,110){\\circle*{4}}\n  \\put(110,110){\\circle*{4}}\n  \\put(120,110){\\circle*{4}}\n\\end{picture}\n\\end{minipage}\n\\caption{The support for the solution of the system $\\text{Horn}(A_0,c_0)$}\n\\label{fig:generalZonotopeExample}\n\\end{figure}\nLet us consider the part of the solution whose support is bounded by the divisors parallel to coordinate axis. This polynomial~$p_0(x,y)$ belongs to the basis of the linear space of solutions to~$\\text{Horn}(A_0,c_0)$. Note that $p_0(x,y)$ contains $90$ monomials (we do not put here the whole expression due to its large size) and the known estimates for polynomials imply that the analytic complexity of~$p_0(x,y)$ does not exceed~$5.$ Indeed, the support of~$p_0(x,y)$ lies in the union of~$10$ lines parallel to $x$ axis. The analytic complexity of polynomial whose support lies on a straight line parallel to axis cannot exceed~$1.$ Then the analytic complexity of the sum of $k$ such polynomials cannot exceed~$1+\\lceil \\log_2 k\\rceil.$  Later we prove that the analytic complexity of~$p_0(x,y)$ is actually equal to~$3.$ \n \\label{ex:SimpleZonotopeSystem}\n\\end{example}\n\nIn general, appending a pair of rows~$(a_i, b_i),(-a_i,-b_i)$ to the matrix defining a hypergeometric system is equivalent to adding a pair of parallel divisors in the exponent space. Let the hypergeometric system $\\text{Horn}(A_0,c_0)$ be defined by a parallelogram, and $p_0(x,y) = \\sum\\limits_{(s,t)\\in S}c_{s,t}\\cdot x^sy^t$ be a polynomial solution of this system, $S$ be its support. Adding a pair of divisors in the exponent space leads to the system with the solution given by\n\n$$p_1(x,y)=\\sum\\limits_{(s,t)\\in S}\\frac{{\\rm\\Gamma}\\left(\\alpha_1 s+\\beta_1 t  + \\gamma_1 + 1\\right)}{{\\rm\\Gamma}\\left(\\alpha_1 s+\\beta_1 t  + \\gamma_1\\right)}\\cdot c_{s,t}\\cdot x^s y^t = \\sum\\limits_{(s,t)\\in S} \\left(\\alpha_1 s + \\beta_1 t + \\gamma_1\\right) x^s y^t$$\n\n$$=\\left(\\alpha_1 \\theta_x + \\beta_1 \\theta_y + \\gamma_1\\right)\\sum\\limits_{(s,t)\\in S}  c_{s,t} x^s y^t = \\left(\\alpha_1 \\theta_x + \\beta_1 \\theta_y + \\gamma_1\\right) p_0(x,y).$$\n\nUsing this formula repetitively we obtain the solution for $k$~pairs of additional divisors:\n\n$$p_k(x,y) = \\left(\\prod\\limits_{j=1}^{k}\\left(\\alpha_j \\theta_x+\\beta_j\\theta_y+\\gamma_j\\right)\\right) p_0(x,y).$$\n\nThus the estimate for the analytic complexity of $p_k(x,y)$ depends on the analytic complexity of $p_0(x,y).$ This dependence is described in detail in the following Proposition and its corollaries. There $\\theta_x=x\\frac{\\partial}{\\partial x}, \\theta_y=y\\frac{\\partial}{\\partial y}$ and $\\alpha_i,\\beta_i,\\gamma_i\\in\\mathbb{C}, i=0,1,\\ldots$\n\n\\begin{proposition} \\rm\nLet $f(x,y)$ be a $Cl_n$ function. Then $$(\\alpha_0 \\theta_x + \\beta_0 \\theta_y + \\gamma_0) f(x,y) \\in Cl_{2n+1}.$$\n{\\it Proof.} The proof of the statement is based on the proof of Proposition 8 in~\\cite{BeloshapkaRJMP2012}. Consider the result of the differential operator $\\alpha_0\\theta_x+\\beta_0\\theta_y$ action on the function $f(x,y).$ Using the induction by $n$ we can prove that this function belongs to~$Cl_{2n}.$\n\n For $n=1$ we can represent $f(x,y)$ in the form $f(x,y)=c(a(x)+b(y)).$ $$(\\alpha_0\\theta_x+\\beta_0\\theta_y) c(a(x)+b(y)) = c'(a(x)+b(y))\\cdot\\left(\\alpha_0 xa'(x)+\\beta_0 yb'(y)\\right),$$ and this function belongs to $Cl_2$ as a product of $Cl_1$ functions. If the statement holds for all $n<N,$ and $f(x,y)$ belongs to $Cl_N,$ which means it can be represented as $f(x,y)=h(f_1(x,y)+f_2(x,y)),$ where $f_1(x,y),f_2(x,y)\\in Cl_{N-1},$ then $$(\\alpha_0\\theta_x+\\beta_0\\theta_y) h(f_1(x,y)+f_2(x,y))=h'(f_1(x,y)+f_2(x,y))\\cdot $$ $$\\cdot\\left((\\alpha_0 \\theta_x + \\beta_0 \\theta_y) f_1(x,y) + (\\alpha_0 \\theta_x + \\beta_0 \\theta_y) f_2(x,y) \\right).$$ Both of the functions $f_1(x,y)$ and $f_2(x,y)$ belong to $Cl_{N-1},$ so the estimate of the analytic complexity for  $(\\alpha_0 \\theta_x + \\beta_0 \\theta_y) f_i(x,y),i=1,2$ is $Cl_{2N-2}.$ Then their sum belongs to $Cl_{2N-1}$ and, after the multiplication of the result by $h'(f_1(x,y)+f_2(x,y))\\in Cl_N,$ the product belongs to $Cl_{2N}.$ Thus we conclude that for any $n,$ if $f(x,y)\\in Cl_n$ then $(\\alpha_0 \\theta_x + \\beta_0 \\theta_y) f(x,y)\\in Cl_{2n}.$  Adding $\\gamma_0 f(x,y) \\in Cl_n$ to this expression we obtain a~function in~$Cl_{2n+1}.$~\\hfill $\\square$\n\\end{proposition}\n\n\\begin{corollary} \\rm\nFor any $f(x,y)\\in Cl_n$ the analytic complexity of $$\\left(\\prod\\limits_{j=1}^{k}\\left(\\alpha_j \\theta_x+\\beta_j\\theta_y+\\gamma_j\\right)\\right)f(x,y)$$ cannot exceed $2^k(n+1)-1.$\n\\end{corollary}\n\n\\begin{corollary} \\rm\nAssume that the analytic complexity of a polynomial solution $p_0(x,y)$ to the hypergeometric system~$\\text{Horn}(A_0,c_0)$ does not exceed $n,\\; S$ is a support of~$p_0(x,y).$ Let the matrix~$A$ be obtained from~$A_0$ by appending~$k$ pairs of vectors $(a_i,b_i),(-a_i,-b_i),$ vector~$c$ be obtained from~$c_0$ by appending~$2k$ elements. Then the analytic complexity of a polynomial solution with the support~$S$ to the hypergeometric system~$\\text{Horn}(A,c)$ does not exceed~$2^k(n+1)-1.$\n\\label{cor:AnalyticComplexityEstimate}\n\\end{corollary}\n\nWhile this estimate is rough when we use it for several additional pairs of divisors ($k > 1$), for $k=1$ it can be quite accurate.\n\n\\vskip.3cm\n{\\it\\noindent Example~\\ref{ex:SimpleZonotopeSystem}. (Continued).}\n Let us use Corollary~\\ref{cor:AnalyticComplexityEstimate} to estimate the analytic complexity of a solution to the system $\\text{Horn}(A_0,c_0).$ To do this, consider the system $\\text{Horn}(\\tilde{A}_0,\\tilde{c}_0),$ defined by the matrix $\\tilde{A}_0= \\left(\\begin{array}{rrrr} 1 & 0 & 0 & -1 \\\\ 0 & 1 & -1 & 0 \\end{array}\\right)^{\\rm T}$ and the vector of parameters $\\tilde{c}_0=(-10,-9, 1, 1).$ This system differs from the original one only by an absence of the pair of divisors with the normal vectors $(1,1)$ and $(-1,-1).$ Thus the support of the solution to the system $\\text{Horn}(\\tilde{A}_0,\\tilde{c}_0)$ coincides with the support of~$p_0(x,y).$  Note that this system is defined by a parallelogram and hence by Proposition~\\ref{prop:parallelogram} the analytic complexity of its solutions cannot exceed~$2.$ Computations show that the basis in the space of solutions to the system~$\\text{Horn}(\\tilde{A}_0,\\tilde{c}_0)$ consists only of one function: $(x-1)^{10}(y-1)^9 \\in Cl_1,$ then $p_0(x,y)\\in Cl_3$ by Corollary~\\ref{cor:AnalyticComplexityEstimate}. Supports of two other solutions to~$\\text{Horn}(A_0,c_0)$ lie on two parallel divisors, so a linear combination of these solutions belongs to~$Cl_3,$ and the general solution to~$\\text{Horn}(A_0,c_0)$ is a function in~$Cl_4.$\n\nThe resulting analytic complexity estimate of solutions to hypergeometric systems defined by zonotopes is formulated in the following theorem.\n\n\\begin{theorem} \\rm\nLet $\\text{Horn}(A,c)$ be a hypergeometric system and~$\\text{Horn}(A,c)$ is defined by a zonotope. Assuming the matrix~$A$ contains~$2k$ rows, consider matrices~$\\hat{A}$ and $-\\hat{A}$ such that the union of their rows coincides with the set of rows of~$A.$ Let $\\alpha$ be a part of the parameter vector~$c,$ corresponding to the matrix $\\hat{A},\\; \\beta$ be a part of this vector, corresponding to~$-\\hat{A},$ and define the vector $\\hat{c}$ with elements $\\hat{c}_i=-\\alpha_i-\\beta_i.$ \n\nIf $\\hat{c}_i \\in \\mathbb{N}\\backslash\\{0\\},i=1,\\ldots,k,$ then the analytic complexity of the general solution to~$\\text{Horn}(A,c)$ does not exceed $$\\min\\left(3\\cdot 2^{k-2}-1+\\lceil \\log_2\\frac{k(k-1)}{2}\\rceil,2+\\lceil\\log_2(\\max\\limits_i\\hat{c}_i+1)\\rceil+\\lceil\\log_2(k-1)\\rceil\\right).$$\n\n\\noindent{\\it Proof.} The condition $\\hat{c}_i \\in \\mathbb{N}\\backslash\\{0\\}$ provides the existence of a polynomial basis in the space of solutions to~$\\text{Horn}(A,c).$ The matrix~$A$ contains $2k$ rows, so supports of the solutions are bounded by $k$~pairs of divisors. The union of these supports is a subset of~$\\frac{k(k-1)}{2}$ parallelogram intersections (it is a sum of an arithmetic progression) and in every intersection the solution belongs to~$Cl_{3\\cdot 2^{k-2}-1}$ (by Corollary~\\ref{cor:AnalyticComplexityEstimate}). \n\nOn the other hand, there is the estimate based on the number of parallel lines connecting the points of the support (see Proposition~4 in~\\cite{BeloshapkaRJMP2012}). While the analytic complexity of any polynomial with the support belonging to a straight line does not exceed~$2,$ the number of these lines for every pair of divisors equals~$\\hat{c}_i+1.$ Thus for any pair of divisors, the part of the solution, belonging to intersections of this pair and any other pairs cannot exceed~$2+\\lceil\\log_2(\\max\\limits_i\\hat{c}_i+1)\\rceil.$ Note that there is no need to use all of~$k$ pairs of divisors to estimate the analytic complexity of the general solution this way, since $k-1$ pairs already bound the whole support of the solution.\n\nTo obtain the estimate from the statement of Theorem, we have to find the estimate based on pairing of $\\frac{k(k-1)}{2}$ parallelogram intersections, then to find the estimate, based on pairing of $k-1$ pairs of the divisors. Minimal of those numbers is the sought estimate.~\\hfill $\\square$\n\\label{thm:ZonotopeAnalyticComplexity}\n\\end{theorem}\n\nLet us order $\\hat{c}_i$ by the ascension and then $v$ be a vector with the elements $v_i=\\min\\left(2+\\lceil\\log_2(\\hat{c}_i+1)\\rceil, 3\\cdot 2^{k-2}-1+\\right.$ $\\left.\\lceil \\log_2(k-i)\\rceil \\right),i=1,\\ldots,k-1.$ To find more accurate value for the analytic complexity estimate from Theorem~\\ref{thm:ZonotopeAnalyticComplexity}, one could use Algorithm~\\ref{alg:SumAnalyticComplexity} from Section~\\ref{sec:Algorithms} using $v$ as an input vector. The accuracy is obtained due to the fact that the vector $v$ provides the decision of better estimate for every pair of divisors, since $\\hat{c}_i$ may have high values not for all of them. \n\n\n\\section{Algorithms of Analytic Complexity Estimation}\n\\label{sec:Algorithms}\n\nTo estimate the analytic complexity of the general solution to the hypergeometric system from Theorem~\\ref{thm:ZonotopeAnalyticComplexity} one can use the following algorithm. \n\n\\begin{algorithm}[H]\n\\DontPrintSemicolon\n  \n  \\KwInput{$c=\\{c_1,c_2,\\ldots,c_n\\}$ - a set of known estimates of the analytic complexity values for bivariate functions~$f_1(x,y),f_2(x,y),\\ldots,f_n(x,y),$ where $(x, y)\\in\\mathbb{C}^2.$}\n  \\KwOutput{$N$ - an estimate for the analytic complexity of the function $\\sum\\limits_{i=1}^n f_i(x,y).$}\n\n  \\While{$c$ {\\rm contains more than}~$1$ {\\rm element}}\n  {\n  \n  find $2$ minimal elements of $c,$ namely, $c_i$ and $c_j.$\n  \n  $c = (c \\cup \\{\\max(c_i,c_j)+1\\}) \\backslash  \\{c_i, c_j\\}.$ \n  \n  }\n  \n  $N \\leftarrow$ only element of~$c.$\n    \n\\caption{Finding the analytic complexity estimate for the sum of functions}\n\\label{alg:SumAnalyticComplexity}\n\\end{algorithm}\nAlgorithm~\\ref{alg:SumAnalyticComplexity} is finite, since at each step the number of elements in~$c$ decreases~by~$1.$\n\nThe following algorithm allows one to find the support of a polynomial solution to a given hypergeometric system defined by a zonotope, provided that such a solution exists. The algorithm is based on Proposition~4.7 in~\\cite{Sadykov-Tanabe}. \n\n\\begin{algorithm}[H]\n\\DontPrintSemicolon\n  \n  \\KwInput{ the matrix $A,$ the parameter vector~$c$ for the hypergeometric system $\\text{Horn}(A,c)$ defined by a zonotope}\n  \\KwOutput{$supp$ - the support for the polynomial solution to~$\\text{Horn}(A,c).$}\n  \n  $supp \\leftarrow \\{\\}$\n  \n  find $\\hat{A}:$ rows$(\\hat{A})\\cup$ rows$(-\\hat{A})$ = rows$(A)$\n  \n  \\For {$(r_i, r_j) \\subset\\text{\\rm rows} (\\hat{A}), i < j$}\n  {\n  \n  $A_{i,j} \\leftarrow (r_i,r_j)^T$\n\n  $\\alpha \\leftarrow$ elements of $c$ corresponding to~$(r_i, r_j)$\n  \n  $\\beta \\leftarrow$ elements of $c$ corresponding to~$(-r_i,-r_j)$\n \n  \\If{$-\\alpha_j-\\beta_j > 0 \\text{ for } j=1,2$}{ \n  $supp = supp \\cup \\text{Supp} \\left( x^{-A_{i,j}^{-1}\\alpha}\\left(1+x^{-A_{i,j}^{-1}e_1}\\right)^{-\\alpha_1-\\beta_1}\\left(1+x^{-A_{i,j}^{-1}e_2}\\right)^{-\\alpha_2-\\beta_2}\\right)$}\n  \\Else{ the general solution to $\\text{Horn}(A,c)$ is not a polynomial}\t\n\n  }\n\\caption{Constructing the support for the polynomial solution to the hypergeometric system}\n\\label{alg:SupportForPolynomialSolution}\n\\end{algorithm}\n\nFor some pairs of rows $r_i, r_j$ solution to the corresponding system defined by a parallelogram is not a polynomial. In this case, part of the basis in the solution space can still consists of polynomials, and their supports can be found by the means of Algorithm~\\ref{alg:SupportForPolynomialSolution}.\n\nThe following algorithm allows one to compute the analytic complexity of any given bivariate polynomial. \n\n\\begin{algorithm}[H]\n\\DontPrintSemicolon\n  \n  \\KwInput{$p(x,y)$ - a polynomial, $x, y\\in\\mathbb{C}.$}\n  \\KwOutput{$N$ - an estimate for the analytic complexity of $p(x,y).$}\n  $result \\leftarrow 0$\n  \n  $short \\leftarrow \\{\\}$\n  \n  $polys \\leftarrow  \\{p_i(x,y)| p(x,y)=\\sum\\limits_i p_i(x,y), \\text{Supp }p_i(x,y)||\\text{Supp }p_j(x,y) \\forall i,j \\} $\n\n   \\For{$p \\in polys$}    \n   { \n      \t$curr = getShort(p)$\n      \t\n      \t\\If{$curr \\not\\subset short$}\n      \t{\n\t\tresult += 1\t      \t\t\n      \t}\n      \t\n      \t$short = short \\cup curr$\n      \t\n   }\n   \n   $N \\leftarrow 2 + \\lceil Log_2(result)\\rceil$\n\n\\caption{Finding the analytic complexity estimate for the polynomial}\n\\end{algorithm}\n\nThe main improvement of this algorithm compared to the existing ones is its ability to distinct the powers of lower degree polynomials included in the original polynomial as summands. Without this feature, even the analytic complexity of the function like~$p(a(x)+b(y))\\in Cl_1,$ where $p(t),a(x),b(y)$ are univariate polynomials, is estimated based on its support, which becomes very complex with the growth of degree of~$p(t).$    \n\nThe input of the function $getShort()$ is a homogeneous polynomial and the output contains elements of its decomposition into the sum of powers. Note that the definition of $polys$ assumes the ambiguity of the representation of the polynomial as the sum of finitely many polynomials with their supports lying in parallel straight lines. Any of such representations give an estimate, but some of them may be better than other ones.\n\n\n\n\n\\section{Examples}\n\\label{sec:Examples}\n\n{\\it\\noindent Example~\\ref{ex:SimpleZonotopeSystem}. (Continuation).} Let us replace the parameter vector~$c_0$ in the system~$\\text{Horn}(A_0,c_0)$ by the vector~$(k,0,0,0,0,0).$ The corresponding system is given~by\n$$\n\\begin{array}{l}\nx \\theta_{x} (\\theta_x + \\theta_y + k) - \\theta_{x} (\\theta_x + \\theta_y), \\\\\ny \\theta_{y} (\\theta_x + \\theta_y + k) - \\theta_{y} (\\theta_x + \\theta_y).\n\\end{array}\n$$\nA basis in its solution space is given by $1,$ $ \\log\\frac{x}{x-1} + \\sum_{j=1}^{k-1} \\frac{(-1)^j}{j(x-1)^j},$\n$ \\log\\frac{y}{y-1} + \\sum_{j=1}^{k-1} \\frac{(-1)^j}{j(y-1)^j},$ so there is no polynomial basis for these parameter values. Nevertheless, the analytic complexity of the general solution is equal to~$1.$ \n\nThe present example shows that the analytic complexity of solutions to hypergeometric systems can be heavily dependent on parameter vectors defining these systems. A resonant choice of their parameters can drastically reduce the analytic complexity of general solutions to such systems.\n\n\\begin{example} {\\it An octagon zonotope.}\nConsider Example 6.8 in~\\cite{Sadykov-Tanabe}. In order to find the analytic complexity of a polynomial solution to the hypergeometric system defined by the matrix \n$$A=\\left(\\begin{array}{rrrrrrrr}1 & -1 & -1 & 1 & -3 & 3 & 2 & -2 \\\\ 2 & -2 & 1 & -1 & -2 & 2 & -1 & 1 \\end{array}\\right)^{\\rm T}$$ and the vector of parameters $c=(3,-5,-2,1,-2,-1,-1,-1)$ we can use the basis of the solutions to this system, computed in the book. There are $3$~solutions whose analytic complexity equal to~$2$, and $28$~solutions in~$Cl_1,$ two of them also belonging to $Cl_0.$ Therefore the analytic complexity of the general solution to this system cannot exceed~$7.$ Note that this estimate is based on a~trivial pairing of the basis functions, but very specific structure of the solution support makes it possible to estimate the analytic complexity not to exceed~$6.$\n\n\nLet us estimate the analytic complexity of the general solution to this system, using Theorem~\\ref{thm:ZonotopeAnalyticComplexity}. The vector $\\hat{c},$ ordered by the ascension, is $(1,2,2,3).$ Then the vector~$v=(3,4,4)$ (it includes only support-based estimates, because of low values of the elements of~$\\hat{c}),$ and, by using Algorithm~\\ref{alg:SumAnalyticComplexity}, we conclude that the general solution belongs to~$Cl_6.$ Note that this estimate coincides with the one we have obtained by hand.\n\nFuthermore, we can estimate the analytic complexity of a solution to any hypergeometric system we obtain by appending a pair of rows to~$A$ (the only condition is that these rows are not collinear to the rows of~$A$). Note that this estimate does not depend much on the difference between new parameters. If this difference is great, it becomes the last element of the ordered vector $\\hat{c},$ and does not affect the new vector~$v,$ the new element of the vector~$v$ is equal to~$2+\\lceil\\log_2(3+1)\\rceil=4,$ and the resulting analytic complexity is $6.$ On the contrary, if this difference is low, for example, if it is equal to~$1,$ the new vector~$\\hat{c}=(1,1,2,2,3),$ the new vector~$v=(3,3,4,4),$ and the analytic complexity is also equal to~$6.$ Thus we conclude that the addition of $2$ rows to the matrix~$A$ does not affect the analytic complexity of the solution to the system.\n\\end{example}\n\n\\begin{example}  {\\it A decagon zonotope.}\nConsider the hypergeometric system~$\\text{Horn}(A_1,c_1),$ defined by the matrix \\begin{equation} A_1=\\left(\\begin{array}{rrrrrrrrrr} -1 & 1 & 0 & 0 & -2 & 2 & 3 & -3 & 3& -3\\\\ 0 & 0 & -1 & 1 & 1 & -1 & 1 & -1 & 2 & -2 \\end{array}\\right)^{\\rm T} \\label{eq:A1Matrix} \\end{equation} and the parameter vector~$c_1 = (-1,0,4,-5,1,-4,-9,6,-4,0).$ The zonotope defining the matrix~$A_1$ is shown in Figure~\\ref{fig:zonotopeComplexExample}. \n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{5cm}\n\\begin{picture}(120,120)\n  \\put(20,0){\\vector(0,1){120}}\n  \\put(0,20){\\vector(1,0){120}}\n \n\n\n  \\put(60,20){\\line(1,2){10}}\n  \\put(70,40){\\line(0,1){10}}\n  \\put(70,50){\\line(-1,3){10}}\n  \\put(60,80){\\line(-2,3){20}}\n  \\put(40,110){\\line(-1,0){10}}\n  \\put(30,110){\\line(-1,-2){10}}\n  \\put(20,80){\\line(1,-3){10}}\n  \\put(30,50){\\line(2,-3){20}}\n  \\put(50,20){\\circle*{3}}\n  \\put(60,20){\\circle*{3}}\n  \\put(70,40){\\circle*{3}}\n  \\put(70,50){\\circle*{3}}\n  \\put(60,80){\\circle*{3}} \n  \\put(40,110){\\circle*{3}} \n  \\put(30,110){\\circle*{3}}\n  \\put(20,90){\\circle*{3}} \n  \\put(20,80){\\circle*{3}}  \n  \\put(30,50){\\circle*{3}}\n\\end{picture}\n\\end{minipage}\n\\caption{The zonotope which defines the matrix~(\\ref{eq:A1Matrix})}\n\\label{fig:zonotopeComplexExample}\n\\end{figure}\n\nThe holonomic rank of the system~$\\text{Horn}(A_1,c_1)$ equals~$34.$ The support to the solution of this system computed by the means of Algorithm~\\ref{alg:SupportForPolynomialSolution} is shown in Figure~\\ref{fig:supportComplexExample}.\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{8cm}\n\\begin{picture}(250,250)\n  \\put(125,0){\\vector(0,1){260}}\n  \\put(0,125){\\vector(1,0){260}}\n\n\n  \\put(119,117){\\small $0$}\n  \\put(5,125){\\line(0,1){3}}\n  \\put(25,125){\\line(0,1){5}}\n  \\put(20,116){\\small $-5$}\n  \\put(45,125){\\line(0,1){3}}\n  \\put(65,125){\\line(0,1){3}}\n  \\put(85,125){\\line(0,1){3}}\n  \\put(105,125){\\line(0,1){3}}\n  \\put(145,125){\\line(0,1){3}}\n  \\put(165,125){\\line(0,1){3}}\n  \\put(185,125){\\line(0,1){3}}\n  \\put(205,125){\\line(0,1){3}}\n  \\put(225,125){\\line(0,1){5}}\n  \\put(222,116){\\small $5$}\n  \\put(245,125){\\line(0,1){3}}\n  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\\put(10,0){\\line(1,1){250}}\n  \\put(20,0){\\line(1,1){240}}\n  \\put(30,0){\\line(1,1){230}}\n  \\put(40,0){\\line(1,1){220}}\n  \\put(0,239){\\line(4,-3){260}}\n  \\put(0,234){\\line(4,-3){260}}\n  \\put(0,229){\\line(4,-3){260}}  \n  \\put(0,224){\\line(4,-3){260}}\n  \\put(0,219){\\line(4,-3){260}}  \n  \\put(95,260){\\line(2,-3){165}}\n  \\put(88,260){\\line(2,-3){174}}\n  \\put(81,260){\\line(2,-3){174}}\n  \\put(74,260){\\line(2,-3){174}}\n  \\put(86,175){\\circle*{3}}\n  \\put(79,175){\\circle*{3}}\n  \\put(72,175){\\circle*{3}}\n  \\put(65,175){\\circle*{3}}\n  \\put(58,175){\\circle*{3}}\n  \\put(100,165){\\circle*{3}}\n  \\put(93,165){\\circle*{3}}\n  \\put(86,165){\\circle*{3}}\n  \\put(79,165){\\circle*{3}}\n  \\put(72,165){\\circle*{3}}  \n  \\put(131,175){\\circle*{3}}\n  \\put(138,175){\\circle*{3}}\n  \\put(145,175){\\circle*{3}}\n  \\put(152,175){\\circle*{3}}\n  \\put(138,165){\\circle*{3}}  \n  \\put(145,165){\\circle*{3}}\n  \\put(152,165){\\circle*{3}}\n  \\put(159,165){\\circle*{3}}  \n  \\put(205,165){\\circle*{3}}\n  \\put(195,165){\\circle*{3}}\n  \\put(185,165){\\circle*{3}}\n  \\put(175,165){\\circle*{3}}\n  \\put(185,175){\\circle*{3}}\n  \\put(195,175){\\circle*{3}}\n  \\put(205,175){\\circle*{3}}\n  \\put(215,175){\\circle*{3}}\n  \\put(131,121){\\circle*{3}}\n  \\put(134,124){\\circle*{3}}\n  \\put(137,127){\\circle*{3}}\n  \\put(137,117){\\circle*{3}}\n  \\put(140,130){\\circle*{3}}\n  \\put(140,120){\\circle*{3}}\n  \\put(143,133){\\circle*{3}}\n  \\put(143,123){\\circle*{3}}\n  \\put(142,112){\\circle*{3}}\n  \\put(146,126){\\circle*{3}}\n  \\put(145,115){\\circle*{3}}     \n  \\put(149,129){\\circle*{3}}\n  \\put(148,118){\\circle*{3}}\n  \\put(151,121){\\circle*{3}}\n  \\put(154,124){\\circle*{3}}\n  \\put(148,108){\\circle*{3}}\n  \\put(151,111){\\circle*{3}}\n  \\put(154,114){\\circle*{3}}\n  \\put(157,117){\\circle*{3}}\n  \\put(160,120){\\circle*{3}}\n\n  \\put(165,125){\\circle*{3}}\n  \\put(153,142){\\circle*{3}}\n  \\put(157,146){\\circle*{3}}\n  \\put(157,136){\\circle*{3}}\n  \\put(161,150){\\circle*{3}}\n  \\put(161,140){\\circle*{3}}\n  \\put(161,130){\\circle*{3}}\n  \\put(165,155){\\circle*{3}}\n  \\put(165,145){\\circle*{3}}\n  \\put(165,135){\\circle*{3}}\n  \\put(169,149){\\circle*{3}}\n  \\put(169,139){\\circle*{3}}\n  \\put(169,129){\\circle*{3}}\n  \\put(173,143){\\circle*{3}}\n  \\put(173,133){\\circle*{3}}\n  \\put(177,137){\\circle*{3}}\n  \\put(125,215){\\circle*{3}}\n  \\put(125,205){\\circle*{3}}\n  \\put(125,195){\\circle*{3}}\n  \\put(125,185){\\circle*{3}}\n  \\put(105,245){\\circle*{3}}\n  \\put(105,235){\\circle*{3}}\n  \\put(105,225){\\circle*{3}}\n  \\put(105,215){\\circle*{3}}\n  \\put(125,165){\\circle*{3}}\n  \\put(125,175){\\circle*{3}}\n  \\put(105,165){\\circle*{3}}\n  \\put(105,175){\\circle*{3}}\n  \\put(105,160){\\circle*{3}}\n  \\put(105,155){\\circle*{3}}\n  \\put(105,150){\\circle*{3}}\n  \\put(105,145){\\circle*{3}}\n  \\put(105,140){\\circle*{3}}\n  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\\put(238,45){\\circle*{3}}\n  \\put(245,35){\\circle*{3}}\n\\end{picture}\n\\end{minipage}\n\\caption{The support for the solution of the system $\\text{Horn}(A_1,c_1)$}\n\\label{fig:supportComplexExample}\n\\end{figure}\n\nPolynomial basis in the solution space to~$\\text{Horn}(A_1,c_1)$ consists of the $4$~monomials $\\frac{x^6}{y^9},\\frac{x^{17\/3}}{y^8},\\frac{x^3}{y^3},\\frac{x^{8\/3}}{y^2}$ and $30$~polynomials $$\\frac{1}{x y^6}+\\frac{5643}{637 x y^5}+\\frac{247095}{8281 x y^4}+\\frac{329460}{8281 x y^3}+\\frac{27455}{286 y^4}+\\frac{82365}{49 y^3}+\\frac{741285}{49 y^2}+\\frac{724812}{7 y},$$ $$\\frac{20 y^3}{63 x}-\\frac{4 y^2}{35 x}+\\frac{4 y^2}{5}-\\frac{18 y}{5}+1,\\frac{3 y^{3\/2}}{380 x}-\\frac{3969 y^{5\/2}}{41990 x}+\\frac{1323 y^{7\/2}}{16796 x}-\\frac{51}{55} y^{3\/2}+ \\sqrt{y},$$ $$-\\frac{11 y^{12}}{115311 x}-\\frac{33 y^{11}}{38437 x}-\\frac{297 y^{10}}{100555 x}-\\frac{24 y^9}{5915 x}+\\frac{3 y^9}{1105}+\\frac{y^8}{26}+\\frac{36 y^7}{143}+y^6, x y^4-\\frac{2}{13} x y^5,$$ $$\\frac{8 y^5}{99 x}+\\frac{4 y^4}{3 x}+\\frac{50y^5}{81}+ y^4, \\frac{1550775 x^{7\/2} y^5}{82808479}-\\frac{31465  x^{9\/2} y^5}{61400001}+ x^{5\/2} y^4-\\frac{5175 x^{7\/2} y^4}{89947},$$ $$\\frac{1547 x^4 y^5}{103455}-\\frac{91 x^4 y^4}{6840}-\\frac{91 x^3 y^5}{1026}+ x^3 y^4, \\frac{806 y^5}{129 x^{8\/3}}+\\frac{84656 y^4}{735 x^{5\/3}}+\\frac{y^4}{x^{8\/3}},\\frac{x^{13\/3}}{y^6}+\\frac{451 x^{13\/3}}{261 y^5},$$ $$\\frac{87 y^5}{82 x^{7\/3}}+\\frac{5220 y^5}{275561 x^{10\/3}}+\\frac{36575 y^4}{2392 x^{4\/3}}+\\frac{y^4}{x^{7\/3}},\\frac{44 y^5}{1183 x^3}+\\frac{33 y^5}{182 x^2}+\\frac{y^4}{x^2},\\frac{x^{16\/3}}{y^8}+\\frac{1378 x^{16\/3}}{451 y^7},$$ $$-\\frac{21}{46} x^{2\/3} y^5+x^{2\/3} y^4+\\frac{119}{286} x^{5\/3} y^4,-\\frac{12}{247} x^{4\/3} y^5+x^{4\/3} y^4-\\frac{364 \\sqrt[3]{x} y^5}{1045},\\frac{2 x}{7 y}+ x,$$ $$\\frac{11985}{299} x^{8\/7} y^{2\/7}+\\frac{14382 x^{8\/7}}{253 y^{5\/7}}+\\frac{x^{8\/7}}{y^{12\/7}},\\frac{1200 x^{2\/7}}{1643 y^{3\/7}}+\\frac{345 x^{9\/7}}{31 y^{3\/7}}+\\frac{x^{9\/7}}{y^{10\/7}},\\frac{x^{10\/3}}{y^4}+\\frac{261 x^{10\/3}}{238 y^3},$$ $$\\frac{114774 x^{6\/7} y^{5\/7}}{28405}+\\frac{1188 x^{6\/7}}{65 y^{2\/7}}+\\frac{x^{6\/7}}{y^{9\/7}},\\frac{x^{10\/7}}{y^{8\/7}}+\\frac{731 x^{3\/7}}{638 \\sqrt[7]{y}}+\\frac{1763 x^{10\/7}}{754 \\sqrt[7]{y}},$$ $$\\frac{x^{4\/7}}{y^{6\/7}}+\\frac{32680 x^{11\/7}}{8613 y^{6\/7}}+\\frac{1558}{261} x^{4\/7} \\sqrt[7]{y},\\frac{169}{150} x^{5\/7} y^{3\/7}+\\frac{x^{5\/7}}{y^{4\/7}}+\\frac{65 x^{12\/7}}{136 y^{4\/7}},$$ $$-\\frac{1}{66} 5 x^2 y^3+\\frac{5}{7} x^2 y^2-\\frac{45}{28} x^2 y+ x^2,x^{11\/5} y^{2\/5}-\\frac{4301 x^{11\/5} y^{7\/5}}{4277}+\\frac{232254 x^{11\/5} y^{12\/5}}{1056419},$$ $$x^{9\/5} y^{3\/5}-\\frac{1287 x^{9\/5} y^{8\/5}}{1634}+\\frac{55913 x^{9\/5} y^{13\/5}}{346408},x^{12\/5} y^{4\/5}-\\frac{68}{19} x^{7\/5} y^{9\/5}-\\frac{116}{231} x^{12\/5} y^{9\/5},$$ $$x^{8\/5} y^{6\/5}+\\frac{5824 x^{13\/5} y^{6\/5}}{432837}-\\frac{1064 x^{8\/5} y^{11\/5}}{2829},\\frac{x^5}{y^7}+\\frac{8 x^5}{15 y^6}-\\frac{21 x^4}{55 y^6}-\\frac{182 x^4}{15 y^5}-\\frac{91 x^4}{24 y^4},$$ $$\\frac{x^{14\/3}}{y^7}+\\frac{828 x^{14\/3}}{85 y^6}-\\frac{585488 x^{11\/3}}{48825 y^5}+\\frac{21758 x^{14\/3}}{23715 y^5}-\\frac{2488324 x^{11\/3}}{35805 y^4}.$$ There are $14$ functions in~$Cl_1$ and $20$~functions in~$Cl_2\\backslash Cl_1$ among these polynomials. \n\nThe analytic complexity estimate of the general solution obtained by the pairing of these functions is~$Cl_7.$ Theorem~\\ref{thm:ZonotopeAnalyticComplexity} gives the following estimate: $\\hat{c} = (2,2,3,3,4), v=(4,4,4,4),$ then the general solution belongs to~$Cl_6.$\n\n\\end{example}\n\nThe following examples present hypergeometric systems defined by non-zonotope polygons, with solutions having low analytic complexity.\n\n\\begin{example} {\\it A pentagon.}\n\\rm The matrix $\\left(\\begin{array}{rrrrrrrrrr} 1 & -1 & 0 & 1 & -1 & 0 & 0\\\\ 1 & 0 & -1 & 0 & 0 & -1 & 1 \\end{array}\\right)^{\\rm T}$ and the vector of parameters $(-4,0,0,-1,-2,-1,-2)$ define the hypergeometric system\n\\begin{equation}\n\\begin{array}{l}\nx (\\theta_x + \\theta_y -4) (\\theta_x - 1) - \\theta_x (\\theta_x - 2), \\\\\ny (\\theta_x + \\theta_y -4) (\\theta_y - 1) - \\theta_y (\\theta_y - 2).\n\\end{array}\n\\label{eq:NonZonotopeSystem}\n\\end{equation}\nThis system is holonomic and its holonomic rank equals~4. The pure basis (see~\\cite{Sadykov-Tanabe}) in its\nsolution space is given by the Taylor polynomials\n$$\nx^2 y^2, \\; 1 - 4 x - 4 y + 12 x y, \\; 6x^2 - 4x^3 + x^4 - 12x^2 y + 4 x^3 y, \\;\n6y^2 - 12x y^2 - 4y^3 + 4x y^3 + y^4.\n$$ The first and the second of these polynomials belong to~$Cl_1,$ the third and the fourth belong to~$Cl_2.$ Thus the general solution is a function in~$Cl_4.$ \n\\begin{figure}\n\\centering\n\\begin{minipage}{4cm}\n\\begin{picture}(60,60)\n  \\put(10,0){\\vector(0,1){60}}\n  \\put(0,10){\\vector(1,0){60}}\n  \\put(54,12){\\small $s$}\n  \\put(12,54){\\small $t$}\n  \\put(0,60){\\line(1,-1){60}}\n  \\put(20,0){\\line(0,1){50}}\n  \\put(30,00){\\line(0,1){40}}\n  \\put(0,20){\\line(1,0){50}}\n  \\put(0,30){\\line(1,0){40}}\n  \\put(10,10){\\circle*{3}}\n  \\put(10,20){\\circle*{3}}\n  \\put(20,10){\\circle*{3}}\n  \\put(20,20){\\circle*{3}}\n  \\put(30,10){\\circle{5}}\n  \\put(30,20){\\circle{5}}\n  \\put(40,10){\\circle{5}}\n  \\put(40,20){\\circle{5}}\n  \\put(50,10){\\circle{5}}\n  \\put(10,30){\\circle*{5}}\n  \\put(20,30){\\circle*{5}}\n  \\put(10,40){\\circle*{5}}\n  \\put(20,40){\\circle*{5}}\n  \\put(10,50){\\circle*{5}}\n  \\put(30,30){\\circle*{7}}\n\\end{picture}\n\\end{minipage}\n\\hskip3cm\n\\begin{minipage}{3cm}\n\\begin{picture}(40,40)\n  \\put(10,0){\\vector(0,1){40}}\n  \\put(0,10){\\vector(1,0){40}}\n  \\put(30,10){\\line(0,1){10}}\n  \\put(30,20){\\line(-1,1){10}}\n  \\put(20,30){\\line(-1,0){10}}\n  \\put(10,10){\\circle*{3}}\n  \\put(10,20){\\circle*{3}}\n  \\put(10,30){\\circle*{3}}\n  \\put(20,10){\\circle*{3}}\n  \\put(20,20){\\circle*{3}}\n  \\put(20,30){\\circle*{3}}\n  \\put(30,10){\\circle*{3}}\n  \\put(30,20){\\circle*{3}}\n\\end{picture}\n\\end{minipage}\n\\caption{a): the supports of solutions to the system~(\\ref{eq:NonZonotopeSystem}); b) polygon defining the system~(\\ref{eq:NonZonotopeSystem})}\n\\end{figure}\n\\label{ex(1,1)(1,0),(0,1),(-1,0)^2,(0,1)^2}\n\\end{example}\n\n\\begin{example} {\\it A trapezoid, high holonomic rank.}\n\\rm\nThe Ore-Sato coefficient $\\varphi(s,t)={\\rm \\Gamma}(s+t){\\rm \\Gamma}(s)^{k-1}{\\rm \\Gamma}(-s)^{k}{\\rm \\Gamma}(-t)$\ndefines the following hypergeometric system with holonomic rank $k:$\n$$\n\\begin{array}{l}\nx \\theta_{x}^{k-1} (\\theta_x + \\theta_y) - (-1)^k \\theta_{x}^k, \\\\\ny (\\theta_x + \\theta_y) + \\theta_{y}.\n\\end{array}\n$$\nA basis in its solution space is given by $\\{ \\log^{j}((y+1)\/x), \\, j=0,\\ldots, k-1 \\}.$ The generating solution equals $\\log^{k-1}((y+1)\/x).$ Thus the general solution to this system belongs to~$Cl_1$ by the conservation principle. This example shows that the analytic complexity of solutions to hypergeometric systems with high holonomic rank can still be low.\n\\label{ex(1,1)(1,0)^(k-1)(-1,0)^k(0,-1)}\n\\end{example}\n\n\\begin{example} {\\it A triangle with no symmetries.}\n\\rm The hypergeometric system\n\\begin{equation}\n\\begin{array}{l}\nx (\\theta_x + \\theta_y -4) (\\theta_x + 2 \\theta_y - 4) -\n(2\\theta_x + 3 \\theta_y - 4) (2\\theta_x + 3 \\theta_y -\n5), \\\\\ny (\\theta_x + \\theta_y -4) (\\theta_x + 2 \\theta_y - 4) (\\theta_x +\n2 \\theta_y - 3) \\\\\n\\hskip2.3cm - (2\\theta_x + 3 \\theta_y - 4) (2\\theta_x + 3\n\\theta_y - 5) (2\\theta_x + 3 \\theta_y - 6)\n\\end{array}\n\\label{eq:TriangleWithNoSymmetries}\n\\end{equation}\nis holonomic and its holonomic rank equals~6. The pure basis in its\nsolution space is given by the Laurent polynomials\n$$\nx^{-4}y^4,\\quad x^{-2}y^3, \\quad x^7 y^{-3}, \\quad  x^8 y^{-4},\n\\quad 3 y^2 + 2 x^{-1} y^2,\n$$\n$$\n6 x^2 + 12 x^3 + x^4 + 4 x^5 y^{-2} + 6 x^6 y^{-2} - 12 x^4 y^{-1}\n- 4 x^5 y^{-1} - 12 x y - 4 x^2 y.\n$$\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}{7cm}\n\\begin{picture}(250,180)\n  \\put(80,0){\\vector(0,1){180}}\n  \\put(0,80){\\vector(1,0){250}}\n  \\put(240,68){\\small $s$}\n  \\put(68,170){\\small $t$}\n  \\put(240,0){\\line(-3,2){240}}\n  \\put(240,7){\\line(-3,2){240}}\n  \\put(240,13){\\line(-3,2){240}}\n  \\put(240,0){\\line(-1,1){120}}\n  \\put(200,60){\\line(-2,1){200}}\n  \\put(200,50){\\line(-2,1){200}}\n  \\put(0,160){\\circle*{3}}\n  \\put(40,140){\\circle*{3}}\n  \\put(220,20){\\circle*{3}}\n  \\put(240,0){\\circle*{3}}\n  \\put(60,120){\\circle{3}}\n  \\put(80,120){\\circle{3}}\n  \\put(100,100){\\circle*{5}}\n  \\put(120,100){\\circle*{5}}\n  \\put(120,80){\\circle*{5}}\n  \\put(140,80){\\circle*{5}}\n  \\put(160,80){\\circle*{5}}\n  \\put(160,60){\\circle*{5}}\n  \\put(180,60){\\circle*{5}}\n  \\put(180,40){\\circle*{5}}\n  \\put(200,40){\\circle*{5}}\n\\end{picture}\n\\end{minipage}\n\n\\caption{The supports of\nsolutions to the system~(\\ref{eq:TriangleWithNoSymmetries})}\n\\end{figure}\n\n\\noindent Here the small filled circles correspond to\nmonomial solutions, the two empty circles indicate the binomial\nsolution and the big filled circles correspond to the remaining\npolynomial solution. The analytic complexity of the general solution to the system~(\\ref{eq:TriangleWithNoSymmetries}) does not exceed~$5.$\n\\label{ex(1,1)(1,2),(-2,-3)}\n\\end{example}\n\n\\paragraph{Acknowledgements.}\n\nThis research was performed in the framework of the state task in the field of scientific activity of the Ministry of Science and Higher Education of the Russian Federation, \ngrant no. FSSW-2020-0008.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction and Background}\n\\label{sec1}\n\\IEEEPARstart {A}{n} index coding problem, consists of a transmitter and a set of $m$ receivers, $R=\\{R_0,R_1,\\ldots,R_{m-1}\\}$. The transmitter has a set of $n$ independent messages, $X=\\{x_0,x_1,\\ldots,x_{n-1}\\}$. Each receiver, $R_k=(\\mathcal{K}_k,\\mathcal{W}_k)$, knows a subset of messages, $\\mathcal{K}_k \\subset X$, called its \\textit{side-information}, and demands to know another subset of messages, $\\mathcal{W}_k \\subseteq \\mathcal{K}_k^\\mathsf{c}$, called its \\textit{Want-set} or \\textit{Demand-set}. The transmitter can take cognizance of the side-information of the receivers and broadcast coded messages, called the index code, over a noiseless channel. The problem of index coding with side-information was introduced by Birk and Kol \\cite{BiK}. An index coding problem is called single unicast \\cite{OnH} if the demand sets of the receivers are disjoint and the size of each demand set is one.\n\n\nIndex coding with side-information is motivated by wireless broadcasting applications. In applications like video-on-demand, during the initial transmission by the transmitter, each receiver may miss a part of data. A na\\\"{i}ve technique is to rebroadcast the entire data again to the receivers. This is an inefficient approach. The index coding problem, therefore aims at reducing the number of transmissions by the transmitter by intelligently using the data already available at the receivers.  \n\n\n\nIn a vector linear index code (VLIC) $x_k \\in \\mathbb{F}_q^{p_k},~ x_k=(x_{k,1},x_{k,2},\\ldots,x_{k, p_k }),~x_{k,j} \\in \\mathbb{F}_q$ for \n$k \\in [0:n-1]$ and $j \\in [1:p_k]$ where $\\mathbb{F}_q$ is a finite field with $q$ elements. In vector linear index coding setting, we refer $x_k \\in \\mathbb{F}_q^{p_k}$ as a message vector or a message and $x_{k,1},x_{k,2},\\ldots,x_{k,p_k} \\in \\mathbb{F}_q$ as the message symbols. An index coding is a mapping defined as\n\\begin{align*}\n\\mathfrak{E}: \\mathbb{F}^{p_0+p_1+\\ldots+p_{n-1}}_q \\rightarrow \\mathbb{F}^N_q,\n\\end{align*}\nwhere $N$ is the length of index code. The index code $\\mathfrak{C}=\\{(c_0,c_1,\\ldots,c_{N-1}) \\}$ is the collection of all images of the mapping $\\mathfrak{E}$. We call the symbols $c_0$,$c_1,\\ldots$,$c_{N-1}$ the code (broadcast) symbols, which are the symbols broadcasted by the transmitter. If $p_0=p_1=\\cdots=p_{n-1}$, then the index code is called symmetric rate vector index code. If $p_0=p_1=\\cdots=p_{n-1}=1$, then the index code is called scalar index code. The index coding problem is to design an index code such that the number of transmissions $N$ broadcasted by the transmitter is minimized and all the receivers get their wanted messages by using the index code symbols (broadcast symbols) broadcasted and their known side-information. \n\nMaleki, Cadambe and Jafar \\cite{MCJ} found the capacity of SUICP(SNCS) with $K$ messages and $K$ receivers, each receiver has a total of $U+D$ side-information, corresponding to the $U$ messages before and $D$ messages after its desired message. In this setting, the $k$th receiver $R_k$ demands the message $x_k$ having the side-information\n\\begin{equation}\n\\label{antidote}\n{\\cal K}_k= \\{x_{k-U},\\dots,x_{k-2},x_{k-1}\\}\\cup\\{x_{k+1}, x_{k+2},\\dots,x_{k+D}\\}.\n\\end{equation}\n\nThe symmetric capacity of this index coding problem setting is:\n\\begin{equation}\n\\label{capacity}\nC=\\left\\{\n                \\begin{array}{ll}\n                  {1 ~~~~~~~~~~~~ \\mbox{if} ~~ U+D=K-1}\\\\\n                  {\\frac{U+1}{K-D+U}} ~~~ \\mbox{if} ~~U+D\\leq K-2. \n                  \\end{array}\n              \\right.\n\\end{equation}\nwhere $U,D \\in$ $\\mathbb{Z},$ $0 \\leq U \\leq D$.\n\nIn the one-sided side-information case, i.e., the cases where $U$ is zero, the $k$th receiver $R_k$ demands the message $x_k$ having the side-information,\n\\begin{equation}\n\\label{antidote1}\n{\\cal K}_k =\\{x_{k+1}, x_{k+2},\\dots,x_{k+D}\\}, \n\\end{equation}\n\\noindent\nfor which \\eqref{capacity} reduces to\n\\begin{equation}\n\\label{capacity1}\nC=\\left\\{\n                \\begin{array}{ll}\n                  {1 ~~~~~~~~~~~~ \\mbox{if} ~~ D=K-1}\\\\\n                  {\\frac{1}{K-D}} ~~~~~~~ \\mbox{if} ~~D\\leq K-2 \n                  \\end{array}\n              \\right.\n\\end{equation}\nsymbols per message.\n \nJafar \\cite{TIM} established the relation between index coding problem and topological interference management problem. The capacity and optimal coding results in index coding can be used in corresponding topological interference management problems. The symmetric and neighboring side-information problems are motivated by topological interference management problems.\n\n\\subsection{Review of Adjacent Independent Row (AIR) matrices}\nIn \\cite{VaR2}, we constructed binary matrices of size $m \\times n (m\\geq n)$ such that any $n$ adjacent rows of the matrix are linearly independent over every field. We refer these matrices as Adjacent Independent Row (AIR) matrices.\n\nThe matrix obtained by Algorithm \\ref{algo2} is called the $(m,n)$ AIR matrix and it is denoted by $\\mathbf{L}_{m\\times n}.$ The general form of the $(m,n)$ AIR matrix is shown in Fig. \\ref{fig1}. It consists of several submatrices (rectangular boxes) of different sizes as shown in Fig.\\ref{fig1}.\n\\begin{figure*}\n\\centering\n\\includegraphics[scale=0.38]{vaddi3.eps}\\\\\n~ $\\mathbf{S}=\\mathbf{I}_{\\lambda_l \\times \\beta_l \\lambda_l}$ if $l$ is even and ~$\\mathbf{S}=\\mathbf{I}_{\\beta_l\\lambda_l \\times \\lambda_l}$ otherwise.\n\\caption{AIR matrix of size $m \\times n$.}\n\\label{fig1}\n~ \\\\\n\\hrule\n\\end{figure*}\nThe description of the submatrices are as follows: Let $c$ and $d$ be two positive integers and $d$ divides $c$. The following matrix denoted by $\\mathbf{I}_{c \\times d}$ is a rectangular matrix.\n\\begin{align}\n\\label{rcmatrix}\n\\mathbf{I}_{c \\times d}=\\left.\\left[\\begin{array}{*{20}c}\n   \\mathbf{I}_d  \\\\\n   \\mathbf{I}_d  \\\\\n   \\vdots  \\\\\n  \n   \\mathbf{I}_d \n   \\end{array}\\right]\\right\\rbrace \\frac{c}{d}~\\text{number~of}~ \\mathbf{I}_d~\\text{matrices}\n\\end{align}\nand $\\mathbf{I}_{d \\times c}$ is the transpose of $\\mathbf{I}_{c \\times d}.$\n\nTowards explaining the other quantities shown in the AIR matrix shown in Fig. \\ref{fig1}, for a given $m$ and $n,$ let $\\lambda_{-1}=n,\\lambda_0=m-n$ and\\begin{align}\n\\nonumber\nn&=\\beta_0 \\lambda_0+\\lambda_1, \\nonumber \\\\\n\\lambda_0&=\\beta_1\\lambda_1+\\lambda_2, \\nonumber \\\\\n\\lambda_1&=\\beta_2\\lambda_2+\\lambda_3, \\nonumber \\\\\n\\lambda_2&=\\beta_3\\lambda_3+\\lambda_4, \\nonumber \\\\\n&~~~~~~\\vdots \\nonumber \\\\\n\\lambda_i&=\\beta_{i+1}\\lambda_{i+1}+\\lambda_{i+2}, \\nonumber \\\\ \n&~~~~~~\\vdots \\nonumber \\\\ \n\\lambda_{l-1}&=\\beta_l\\lambda_l.\n\\label{chain}\n\\end{align}\nwhere $\\lambda_{l+1}=0$ for some integer $l,$ $\\lambda_i,\\beta_i$ are positive integers and $\\lambda_i < \\lambda_{i-1}$ for $i=1,2,\\ldots,l$. The number of submatrices in the AIR matrix is $l+2$ and the size of each submatrix is shown using $\\lambda_i,\\beta_i,$  $i \\in [0:l].$\n\n\t\t\\begin{algorithm}\n\t\t\\caption{Algorithm to construct the AIR matrix $\\mathbf{L}$ of size $m \\times n$}\n\t\t\t\\begin{algorithmic}[2]\n\t\t\t\n\t\t\t\t \\item Let $\\mathbf{L}=m \\times n$ blank unfilled matrix.\n\t\t\t\t\\item [Step 1]~~~\n\t\t\t\t\\begin{itemize}\n\t\t\t\t\\item[\\footnotesize{1.1:}] Let $m=qn+r$ for $r < n$.\n\t\t\t\t\\item[\\footnotesize{1.2:}] Use $\\mathbf{I}_{qn \\times n}$ to fill the first $qn$ rows of the unfilled part of $\\mathbf{L}$.\n\t\t\t\t\\item[\\footnotesize{1.3:}] If $r=0$, Go to Step 3.\n\t\t\t\t\\end{itemize}\n\n\t\t\t\t\\item [Step 2]~~~\n\t\t\t\t\\begin{itemize}\n\t\t\t\t\\item[\\footnotesize{2.1:}] Let $n=q^{\\prime}r+r^{\\prime}$ for $r^{\\prime} < r$.\n\t\t\t\t\\item[\\footnotesize{2.2:}] Use $\\mathbf{I}_{q^{\\prime}r \\times r}^{\\mathsf{T}}$ to fill the first $q^{\\prime}r$ columns of the unfilled part of $\\mathbf{L}$.\n\t\t\t    \\item[\\footnotesize{2.3:}] If $r^{\\prime}=0$, go to Step 3.\t\n\t\t\t\t\\item[\\footnotesize{2.4:}] $m\\leftarrow r$ and $n\\leftarrow r^{\\prime}$.\n\t\t\t\t\\item[\\footnotesize{2.5:}] Go to Step 1.\n\t\t\t\t\\end{itemize}\n\t\t\t\t\\item [Step 3] Exit.\n\t\t\n\t\t\t\\end{algorithmic}\n\t\t\t\\label{algo2}\n\t\t\\end{algorithm}\n\n\nIn \\cite{VaR2}, we gave an optimal length scalar linear index code for one-sided SUICP(SNCS) using AIR encoding matrices. In \\cite{VaR1}, we constructed optimal length $(U+1)$ dimensional VLICs for two-sided SUICP(SNCS) satisfying some conditions on $K,D$ and $U$. The VLIC construction in \\cite{VaR1} does not use AIR matrices. In \\cite{VaR3}, we gave a low-complexity decoding for one-sided SUICP(SNCS) with AIR matrix as encoding matrix. The low complexity decoding method helps to identify a reduced set of side-information for each user with which the decoding can be carried out. By this method every receiver is able to decode its wanted message symbol by simply adding some broadcast symbols.\t\n\n\\subsection{Contributions}\n\nIn a $b$-dimension vector index coding, the transmitter has to wait for $b$-realizations of a given message symbol to perform the index coding. In scalar index codes, the transmitter encodes each realization of the message symbols separately. In a delay critical environment like real time video streaming, it may not be desirable to wait for $b$-realizations of a given message symbol. Hence, in this paper, we focus on reducing the dimension of VLICs without compromising on optimal length of the code.\n\n\\begin{itemize}\n\\item For the two-sided SUICP(SNCS) with arbitrary $K,D$ and $U$, we construct optimal length $\\frac{U+1}{\\text{gcd}(K,D-U,U+1)}$ dimensional VLICs. The proposed construction is independent of field size and works over every field. \n\\item We prove that the constructed VLICs are of minimal dimension if $\\text{gcd}(K-D+U,U+1)$ is equal to $\\text{gcd}(K,D-U,U+1)$. In other words, optimal VLICs of lesser dimension do not exist.\n\\item We give a low-complexity decoding for the two-sided SUICP(SNCS). By this method every receiver is able to decode its wanted message symbol by simply adding some broadcast symbols.\t\n\\item We give generator matrices for the proposed VLICs by using AIR matrices.\n\\end{itemize}\n\nThe SUICP(SNCS) considered in this paper was referred as symmetric neighboring antidote multiple unicast index coding problem by Maleki, Cadambe and Jafar in \\cite{MCJ}.\n\nAll the subscripts in this paper are to be considered $~\\text{\\textit{modulo}}~ K$. \n\nThe paper is organized as follows. In Section \\ref{sec2}, we give a construction of reduced dimension optimal VLICs for two-sided SUICP(SNCS). In Section \\ref{sec3}, we give some special cases of the proposed construction. \n\n\n\\section{Construction of Reduced Dimension Vector Linear Index Codes}\n\\label{sec2}\nFor the $b$ dimensional vector linear index coding, we refer $x_k \\in \\mathbb{F}_q^b$ as a message vector and $x_{k,1},x_{k,2},\\ldots,x_{k,b} \\in \\mathbb{F}_q$ as message symbols.\n\n\\begin{theorem}\n\\label{vcthm1}\nGiven arbitrary positive integers $K,D$ and $U$, consider the two-sided SUICP(SNCS) with $K$ messages $\\{x_0,x_1,\\cdots,x_{K-1}\\}$ and the receivers being $K,$ and the receiver $R_k$ for $k \\in [0:K-1]$ wanting the message $x_k$ and having the side-information given by \n\\begin{align}\n\\label{thm1eq}\n{\\cal K}_k= \\{x_{k-U},\\dots,x_{k-2},x_{k-1}\\}~\\cup \\{x_{k+1},x_{k+2},\\dots,x_{k+D}\\}.\n\\end{align}\n\n\nLet \n\\begin{align}\n\\label{thm1eq2}\nu_a=\\frac{U+1}{a},~~~~&\\Delta_a=\\frac{D-U}{a},~~~~ K_a=\\frac{K}{a},\n\\end{align}\nwhere\n\\begin{align}\n\\label{thm1eq1}\na=\\text{gcd}(K,D-U,U+1).\n\\end{align}\n\nLet $x_k=(x_{k,1},x_{k,2},\\cdots,x_{k,u_a})$ be the message vector wanted by the $k$th receiver, where $x_k \\in \\mathbb{F}_q^{u_a}, x_{k,i} \\in \\mathbb{F}_q$ for every $k \\in [0:K-1]$ and $i \\in [1:u_a]$. \n\nLet\n\\begin{align}\n\\label{thm1eq6}\ny_s=\\sum_{i=1}^{u_a} \\sum_{j=0}^{a-1}& x_{a(s+1-i)+j,i} ~~~~~~~~~s \\in [0:K_a-1].\n\\end{align}\n\nLet $\\mathbf{L}$ be the AIR matrix of size $K_a \\times (K_a-\\Delta_a)$ and $L_s$ be the $s$th row of $\\mathbf{L}$ for every $s \\in [0:K_a-1]$. The optimal length $u_a$ dimensional VLIC for the two-sided SUICP(SNCS) with $K$ messages and $D$ side-information after and $U$ side-information before is given by \n\\begin{align}\n\\label{thm1eq5}\n[c_0~c_1~\\ldots~c_{K_a-\\Delta_a-1}]=\\sum_{s=0}^{K_a-1}y_sL_s.\n\\end{align} \n\\end{theorem}\n\n\\begin{proof}\nWe prove that every receiver $R_k$ for $k \\in [0:K-1]$ decodes its wanted message vector $(x_{k,1},x_{k,2},\\ldots,x_{k,u_a})$. \n\nLet $s=\\left \\lfloor \\frac{k}{a} \\right  \\rfloor$. We have \n\\begin{align}\n\\label{thm1rem}\nk=as+ r~~\\text{for~some}~ r \\in [0:a-1]. \n\\end{align}\n\nLet\n\\begin{align}\n\\label{thm1eq3}\n&z_{s,i}=\\sum_{j=0}^{a-1} x_{as+j,i}, ~~~~s \\in [0:K_a-1]~\\text{and}~i \\in [1:u_a].\n\\end{align}\n\nWe refer $z_{s,i}$ as extended message symbol and $z_s=(z_{s,1}~z_{s,2}~\\ldots,z_{s,u_a})$ as extended message vector. The extended message symbol $z_{s,i}$ comprises of $'a'$ message symbols $x_{as,i},x_{as+1,i},\\ldots,x_{as+a-1,i}$. Each of the $Ku_a$ message symbols $x_{k,i}$ for $k \\in [0:K-1]$ and $i \\in [1:u_a]$ appear exactly once in $K_au_a$ extended message symbols $z_{s,i}$ for $s \\in [0:K_a-1]$ and $i \\in [1:u_a]$.\n\nFrom \\eqref{thm1eq6} and \\eqref{thm1eq3}, we have\n\\begin{align}\n\\label{thm1eq4}\ny_s=\\sum_{i=1}^{u_a} z_{s+1-i,i}~~\\text{for}~~s \\in [0:K_a-1].\n\\end{align}\n\nThe symbol $y_s$ comprises of $u_a$ extended message symbols $z_{s,1},z_{s-1,2},$ $\\ldots,z_{s-u_a-1,u_a}$. The $K_au_a$ symbols $z_{s,i}$ for $s \\in [0:K_a-1]$ and $i \\in [1:u_a]$ appear exactly once in $K_a$ symbols $y_s$ for $s \\in [0:K_a-1]$. By combining \\eqref{thm1eq3} and \\eqref{thm1eq4}, the symbol $y_s$ comprises of $au_a=U+1$ message symbols corresponding to $U+1$ neighboring receivers. \n\\begin{itemize}\n\\item The symbol $y_s$ comprises of $(U+1)$ message symbols one from each of the message vectors $x_t$ for every $t \\in [as-(U+1)+a:as+a-1]$.\n\\item The symbol $y_{s+1}$ comprises of $U+1$ message symbols one from each of the message vectors $x_t$ for every $t \\in [as-(U+1)+2a:as+2a-1]$.\n\\item The symbol $y_{s+\\Delta_a}$ comprises of $U+1$ message symbols one from each of the message vectors $x_t$ for every $t \\in [as-(U+1)+a(\\Delta_a+1):as+a(\\Delta_a+1)-1]$.\n\\item The symbol vector $[y_s~y_{s+1}~\\ldots~y_{s+\\Delta_a}]$ comprises of $(U+1)\\Delta_a$ message symbols from the message vectors $x_t$ for every $t \\in [as-(U+1)+a:as+a(\\Delta_a+1)-1]=[\\underbrace{as-(U+1)+a}_{=g}:\\underbrace{as+D-U+a-1}_{=h}]$. \\\\Note that $h-g=D$.\n\\end{itemize}\n\nFrom the alignment of the symbols $y_s,y_{s+1},\\ldots,y_{s+\\Delta_a}$ and the side-information given in \\eqref{thm1eq}, we can conclude the following:\n\\begin{observation}\n\\label{obs1}\nFor every receiver $R_k$ whose wanted message symbol is present in $y_s$, every other message symbol in $y_s$ is in side-information of $R_k$.\n\\end{observation}\n\\begin{observation}\n\\label{obs2}\nFor every receiver $R_k$ whose wanted message is present in $y_s$, every message symbol in $y_{s+1},y_{s+2},\\ldots,y_{s+\\Delta_a}$ which does not belong to the message vector $x_k$ is in side-information of $R_k$. That is, for every receiver $R_k$ whose wanted message is present in $y_s$, every extended message symbol in $y_{s+1},y_{s+2},\\ldots,y_{s+\\Delta_a}$ which does not belong to the message vector $z_s$ is in side-information of $R_k$ (we refer the extended message symbol $z_{t,i}=\\sum_{j=0}^{a-1} x_{at+j,i}$ is in the side-information of $R_k$ if every message symbol $x_{at+j,i}$ for $j \\in [0:a-1]$ is in the side-information of $R_k$). \n\\end{observation}\n\n\n\nConsider the one-sided SUICP(SNCS) with $K_a$ messages $\\{y_0,y_1,\\cdots,y_{K_a-1}\\}$, $K_a$ receivers and the receiver wanting the message $y_s$ and having the side-information given by ${\\cal K}_s= \\{y_{s+1},y_{s+2},\\dots,y_{s+\\Delta_a}\\},$ for every $s \\in [0:K_a-1]$. Let $\\mathfrak{C}$ be the optimal length index code given by \\eqref{thm1eq5}. In \\cite{VaR2}, we proved that the code $\\mathfrak{C}$ enables the decoding of $K_a$ message $y_0,y_1,\\cdots,y_{K_a-1}$. That is, for every $s \\in [0:K_a-1]$, with linear decoding there exists a linear combination of broadcast symbols $c_0,c_1, \\cdots, c_{K_a-\\Delta_a-1}$ to get the sum of the form  \n\\begin{align}\n\\label{vcsum}\nS_s=y_s+b_s^{(1)}y_{s+a_s^{(1)}}+b_s^{(2)}y_{s+a_s^{(2)}}+\\cdots+\nb_s^{(d)}y_{s+a_s^{(d)}} \n\\end{align} \nwhere $b_s^{(1)},b_s^{(2)},\\ldots,b_s^{(d)} \\in \\mathbb{F}_q,d \\leq \\Delta_a$ and $1 \\leq a_s^{(1)} <a_s^{(2)}< \\ldots <a_s^{(d)} \\leq \\Delta_a$.\n\nIn the given two-sided SUICP(SNCS), we have $k \\in [0:K-1]$, $s=\\left \\lfloor \\frac{k}{a} \\right  \\rfloor \\in [0:K_a-1].$ From \\eqref{thm1eq3}, the message symbol $x_{k,i}$ for $i \\in [1:u_a]$ is present in the extended message symbol $z_{s,i}$. For every message symbol $x_{k,i}$ for $i \\in [1:u_a]$, the decoding is performed as given below.\n\\begin{itemize}\n\\item $R_k$ first decodes the extended message symbol $z_{s,i}$ for $i \\in [1:u_a]$, where the message symbol $x_{k,i}$ is present.\n\\item In $z_{s,i}=\\sum_{j=0}^{a-1} x_{as+j,i}$, every message symbol present is in the side-information of $R_k$. Hence, $R_k$ decodes its wanted message symbol $x_{k,i}$ from $z_{s,i}$.\n\\end{itemize}\n\\begin{step}\n\\label{step1}\nDecoding of $x_{k,u_a}$\n\\end{step}\n\nThe message symbol $x_{k,u_a}$ is present in $z_{s,u_a}$. Receiver $R_k$ first decodes the extended message symbol $z_{s,u_a}$ as follows. From \\eqref{thm1eq4}, we have \n\\begin{align*}\ny_{s+u_a-1}&=\\sum_{i=1}^{u_a} z_{s+u_a-i,i}=z_{s,u_a}+\\sum_{i=1}^{u_a-1} z_{s+u_a-i,i}.\n\\end{align*}\nHence, the extended message symbol $z_{s,u_a}$ is present in $y_{s+u_a-1}$. The symbol $y_{s+u_a-1}$ can be decoded from $S_{s+u_a-1}$. The sum $S_{s+u_a-1}$ obtained from \\eqref{vcsum} is given in \\eqref{vcstep10} and \\eqref{vcstep1}.\n\\begin{figure*}[ht]\n\\begin{align}\n\\label{vcstep10}\nS_{s+u_a-1}=y_{s+u_a-1}+\\underbrace{\\sum_{t=1}^d b_{s+u_a-1}^{(t)}y_{s+u_a-1+a_{s+u_a-1}^{(t)}}}_{\\text{side~information~to}~R_k}.\n\\end{align}\n\\begin{align}\n\\label{vcstep1}\nS_{s+u_a-1}=\\underbrace{\\underbrace{z_{s,u_a}}_{\\text{wanted extended message symbol to} R_k} +\\underbrace{\\sum_{i=1}^{u_a-1} z_{s+u_a-i,i}}_{\\text{side~information~to~receiver~}R_k}}_{y_{s+u_a-1}}+\\underbrace{\\sum_{t=1}^d b_{s+u_a-1}^{(t)}y_{s+u_a-1+a_{s+u_a-1}^{(t)}}}_{\\text{side~information~to}~R_k}.\n\\end{align}\n\\end{figure*}\nIn $S_{s+u_a-1}$, we have \n\\begin{align}\n\\label{vcstep1eq1}\n1 \\leq a_{s+u_a-1}^{(1)}<a_{s+u_a-1}^{(2)}< \\cdots <a_{s+u_a-1}^{(d)}\\leq \\Delta_a. \n\\end{align}\nFrom \\eqref{thm1eq4} and \\eqref{vcstep1eq1}, we have \n\\begin{align*}\ny_{s+u_a-1+t}=\\sum_{i=1}^{u_a}& z_{s+u_a+t-i,i}~\\text{for}~s \\in [0:K_a-1], \\\\&t \\in \\{a_{s+u_a-1}^{(1)},a_{s+u_a-1}^{(2)},\\ldots,a_{s+u_a-1}^{(d)}\\},\n\\end{align*}\nand in $y_{s+u_a-1+t}$, the extended message symbols belonging to $z_s$ are not present. Hence, in $S_{s+u_a-1}$, only one extended message symbol is present which belongs to $z_s.$ From Observation \\ref{obs2}, the symbols $y_{s+u_a-1+t}$ for every $t \\in \\{a_{s+u_a-1}^{(1)},a_{s+u_a-1}^{(2)},\\ldots,a_{s+u_a-1}^{(d)}\\}$ are in the side-information of $R_k$. From Observation \\ref{obs1}, every message symbol present in $y_{s+u_a-1}$ is in side-information of $R_k$ except $x_{k,u_a}$. Hence, by using $S_{s+u_a-1}$, receiver $R_k$ decodes its wanted extended message symbol $z_{s,u_a}$.\nFrom \\eqref{thm1rem} and \\eqref{thm1eq3}, we have \n\\begin{align*}\nz_{s,u_a}&=\\sum_{j=0}^{a-1} x_{as+j,u_a} \\\\&=\\sum_{j=0}^{a-1} x_{k-r+j,u_a}~\\text{where}~r \\in [0:a-1]\\\\&=x_{k,u_a}+\\underbrace{\\sum_{j=0,j \\neq r}^{a-1} x_{k-r+j,u_a}}_{\\text{side~information~to}~R_k}.\n\\end{align*}\n\nHence, $R_k$ decodes its wanted message $x_{k,u_a}$ by using $z_{s,u_a}$ (Note that if $S_{s+u_a-1}$ comprises $h \\geq 2$ extended message symbols for some integer $h$ which belong to $z_s$ and if $R_k$ yet to decode $c$ of them for some integer $c$ such that $2 \\leq c \\leq h$, then the $c$ extended message symbols interfere and $R_k$ can not decode any of the $c$ message symbols).\n \n\n\n\\begin{step}\n\\label{step2}\nDecoding of $x_{k,u_a-1}$\n\\end{step}\n\nThe message symbol $x_{k,u_a-1}$ is present in the extended message symbol $z_{s,u_a-1}$. Receiver $R_k$ first decodes the extended message symbol $z_{s,u_a-1}$ as follows. From \\eqref{thm1eq4}, we have \n\\begin{align*}\ny_{s+u_a-2}&=\\sum_{i=1}^{u_a} z_{s+u_a-1-i,i}\\\\&=z_{s,u_a-1}+\\sum_{i=1,i \\neq u_a-1}^{u_a} z_{s+u_a-1-i,i}.\n\\end{align*}\nHence, the extended message symbol $z_{s,u_a-1}$ is present in $y_{s+u_a-2}$. Receiver $R_k$ uses the sum $S_{s+u_a-2}$ to decode $z_{s,u_a-1}$. The sum $S_{s+u_a-1}$ obtained from \\eqref{vcsum} is given in \\eqref{vcstep20} and \\eqref{vcstep2}.\n\\begin{figure*}\n\\begin{align}\n\\label{vcstep20}\nS_{s+u_a-2}=y_{s+u_a-2}+b_{s+u_a-2}^{(1)}y_{s+u_a-2+a_{s+u_a-2}^{(1)}}+\\underbrace{\\sum_{t=2}^d b_{s+u_a-2}^{(t)}y_{s+u_a-2+a_{s+u_a-2}^{(t)}}}_{\\text{side~information~to}~R_k}.\n\\end{align}\n\\begin{align}\n\\label{vcstep2}\n\\nonumber\nS_{s+u_a-2}=&\\underbrace{\\underbrace{z_{s,u_a-1}}_{\\text{wanted~extended~message~symbol~to}~R_k}+\\underbrace{\\sum_{i=1,i \\neq u_a-1}^{u_a} z_{s+u_a-1-i,i}\n}_{\\text{side~information~to~receiver~}R_k}}_{y_{s+u_a-2}}\\\\& +\n\\underbrace{\\underbrace{b_{s+u_a-2}^{(1)}z_{s-1+a_{s+u_a-2}^{(1)},u_a}}_{\\text{interference~from}~z_s~\\text{if}~a_{s+u_a-2}^{(1)}=1}+\\underbrace{b_{s+u_a-2}^{(1)}\\sum_{i=1}^{u_a-1} z_{s+u_a-1+a_{s+u_a-2}^{(1)}-i,i}}_{\\text{side~information~to}~R_k}}_{y_{s+u_a-2+a_{s+u_a-2}^{(1)}}}+\\underbrace{\\sum_{t=2}^d b_{s+u_a-2}^{(t)}y_{s+u_a-2+a_{s+u_a-2}^{(t)}}}_{\\text{side~information~to}~R_k}.\n\\end{align}\n\\end{figure*}\n\nIn \\eqref{vcstep20}, we have \n\\begin{align}\n\\label{vcstep2eq1}\n1 \\leq a_{s+u_a-2}^{(1)}<a_{s+u_a-2}^{(2)}< \\cdots <a_{s+u_a-2}^{(d)}\\leq \\Delta_a.\n\\end{align}\n\nDepending on the value of $a_{s+u_a-2}^{(1)}$, sum $S_{s+u_a-2}$ comprises of either one extended message symbol or two extended message symbols belonging to $z_s~(z_{s,1},z_{s,2},\\ldots,z_{s,u_a-1},z_{s,u_a})$. If $a_{s+u_a-2}^{(1)}>1$, then $S_{s+u_a-2}$ comprises only one extended message symbol ($z_{s,u_a-1}$) belonging to $z_s$. If $a_{s+u_a-2}^{(1)}=1$, $S_{s+u_a-2}$ comprises two extended message symbols ($z_{s,u_a},z_{s,u_a-1}$) belonging to $z_s.$ Receiver $R_k$ has already decoded the extended message symbol $z_{s,u_a}$ in Step \\ref{step1}. From Observation \\ref{obs2}, the symbols $y_{s+u_a-2+t}$ for every $t \\in \\{a_{s+u_a-2}^{(1)},a_{s+u_a-2}^{(2)},\\ldots,a_{s+u_a-2}^{(d)}\\}$ are in the side-information of $R_k$. From Observation \\ref{obs1}, every message symbol present in $y_{s+u_a-2}$ is in side-information of $R_k$ except $x_{k,u_a-1}$. Hence, by using $S_{s+u_a-2}$, receiver $R_k$ decodes its wanted extended message symbol $z_{s,u_a-1}$. From \\eqref{thm1rem} and \\eqref{thm1eq3}, we have \n\\begin{align*}\nz_{s,u_a-1}&=\\sum_{j=0}^{a-1} x_{as+j,u_a-1} \\\\&=\\sum_{j=0}^{a-1} x_{k-r+j,u_a-1}~\\text{where}~r \\in [0:a-1]\\\\&=x_{k,u_a-1}+\\underbrace{\\sum_{j=0,j \\neq r}^{a-1} x_{k-r+j,u_a-1}}_{\\text{side~information~to}~R_k}.\n\\end{align*}\n\nHence, $R_k$ decodes its wanted message $x_{k,u_a-1}$ by using $z_{s,u_a-1}$.\n\n\n\\textit{Step l.}  Decoding of $x_{k,u_a+1-l}$ for $l \\in [3:u_a]$\n\n\nThe message symbol $x_{k,u_a+1-l}$ is present in the extended message symbol $z_{s,u_a+1-l}$. Receiver $R_k$ first decodes the extended message symbol $z_{s,u_a+1-l}$ as follows. From \\eqref{thm1eq4}, the extended message symbol $z_{s,u_a+1-l}$ is present in $y_{s+u_a-l}$. The symbol $y_{s+u_a-l}$ can be decoded from $S_{s+u_a-l}$. Define $A_j=\\{a_{s+u_a-l}^{(j)}+1,a_{s+u_a-l}^{(j)}+2,\\cdots,u_a\\}$ if $u_a > a_{s+u_a-l}^{(j)}$, else $A_j=\\phi$ for every $j \\in [1:l-1]$. Let $1_A$ be the indicator function such that $1_A(x)=1$ if $x \\in A$, else it is zero. The sum $S_{s+u_a-l}$ obtained from \\eqref{vcsum} is given in \\eqref{vcstepl0} and \\eqref{vcstepl}.\n\\begin{figure*}[ht]\n\\begin{align}\n\\label{vcstepl0}\nS_{s+u_a-l}&=y_{s+u_a-l}+\\sum_{t=1}^{l-1}b_{s+u_a-l}^{(t)}y_{s+u_a-l+a_{s+u_a-l}^{(t)}}++\\underbrace{\\sum_{t=l}^d b_{s+u_a-l}^{(t)}y_{s+u_a-l+a_{s+u_a-l}^{(t)}}}_{\\text{side~information~to}~R_k}.\n\\end{align}\n\\begin{align}\n\\nonumber\nS_{s+u_a-l}=&\\underbrace{\\underbrace{z_{s,u_a+1-l}}_{\\text{wanted~extended~message~symbol~to}~R_k}+ \\underbrace{\\sum\\limits_{i=1, i\\neq u_a+1-l}^{u_a} z_{s+u_a+1-l-i,i}\n}_{\\text{side~information~to}~R_k}}_{y_{s+u_a-l}}+\\underbrace{\\sum_{t=l}^{d}b_{s+u_a-l}^{(t)}y_{s+u_a-l+a_{s+u_a-l}^{(t)}}}_{\\text{side~information~to}~R_k}\\\\&\n+ \\sum_{t=1}^{l-1}\\underbrace{\\underbrace{b_{s+u_a-l}^{(t)}1_{A_t}(l)z_{s,u_a+1-l+a_{s+u_a-l}^{(t)}}}_{\\text{Interference~from}~z_s}+\\underbrace{\\sum_{t=1}^{l-1}b_{s+u_a-l}^{(t)}\\left(y_{s+u_a-l+a_{s+u_a-l}^{(t)}}-1_{A_{t}}(l)z_{s,u_a+1-l+a_{s+u_a-l}^{(t)}}\\right)}_{\\text{side~information~to}~R_k}}_{y_{s+u_a-l+a_{s+u_a-l}^{(t)}}}.\n\\label{vcstepl}\n\\end{align}\n\\end{figure*}\nIn this sum $S_{s+u_a-l}$, $z_{s,u_a+1-l}$ is the required extended message symbol and $1_{A_j}(l)b_{s+u_a-l}^{(j)}z_{s,u_a+1-l+a_{s+u_a-l}^{(j)}}$ for $j \\in [1:l-1]$ is the interference to the receiver $R_k$ from its wanted extended message vector $z_s$. Receiver $R_k$ already knows the $l-1$ extended message symbols $\\{z_{s,u_a}, z_{s,u_a-1},\\cdots,z_{s,u_a+2-l}\\}$ and thus the interference from the extended message vector $z_s$ in the sum $S_{s+u_a-l}$ can be cancelled. From Observation \\ref{obs2}, the symbols $y_{s+u_a-l+t}$ for every $t \\in \\{a_{s+u_a-l}^{(1)},a_{s+u_a-l}^{(2)},\\ldots,a_{s+u_a-l}^{(d)}\\}$ are in the side-information of $R_k$. From Observation \\ref{obs1}, every message symbol present in $y_{s+u_a-l}$ is in side-information of $R_k$ except $x_{k,u_a+1-l}$. Hence, by using $S_{s+u_a-l}$, receiver $R_k$ decodes its wanted extended message symbol $z_{s,u_a+1-l}$. From \\eqref{thm1rem} and \\eqref{thm1eq3}, we have \n\\begin{align*}\nz_{s,u_a-l}&=\\sum_{j=0}^{a-1} x_{as+j,u_a-l}=\\sum_{j=0}^{a-1} x_{k-r+j,u_a-l}\\\\&=x_{k,u_a-l}+\\underbrace{\\sum_{j=0,j \\neq r}^{a-1} x_{k-r+j,u_a-l}}_{\\text{side~information~to}~R_k}.\n\\end{align*}\n\nHence, $R_k$ decodes its wanted message $x_{k,u_a-l}$ by using $z_{s,u_a-l}.$\n\n\nThe decoding continues until the receiver $R_k$ has decoded its $u_a$ wanted extended message symbols $z_{s,1},z_{s,2},\\cdots,z_{s,u_a}$. The receiver $R_k$ decodes the extended message vector $z_s$ successively in the following order:\n\\begin{align*}\nz_{s,u_a}\\rightarrow z_{s,u_a-1}\\rightarrow z_{s,u_a-2}\\rightarrow \\ldots \\rightarrow z_{s,2}\\rightarrow z_{s,1}.\n\\end{align*}  \n\nHence, The receiver $R_k$ decodes the wanted message $x_k$ successively in the following order:\n\\begin{align*}\nx_{k,u_a}\\rightarrow x_{k,u_a-1}\\rightarrow x_{k,u_a-2}\\rightarrow \\ldots \\rightarrow x_{k,2}\\rightarrow x_{k,1}.\n\\end{align*}  \n\nThe extended message symbols wanted by receiver $R_k$ are decoded from $S_i$ as given in Table \\ref{vctable1}. \n\\begin{table*}[ht]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|c|}\n\\hline\n\\textbf{$x_{k,u_a}$} & $x_{k,u_a-1}$ & $\\ldots$ & $x_{k,u_a+1-l}$ & $\\ldots$& $x_{k,2}$ & $x_{k,1}$\\\\\n\\hline\n\\textbf{$z_{s,u_a}$} & $z_{s,u_a-1}$ & $\\ldots$ & $z_{s,u_a+1-l}$ & $\\ldots$& $z_{s,2}$ & $z_{s,1}$\\\\\n\\hline\n\\textbf{$S_{s+u_a-1}$} & $S_{s+u_a-2}$ & $\\ldots$& $S_{s+u_a-l}$ & $\\ldots$ & $S_{s+1}$& $S_s$\\\\\n\\hline \n\\end{tabular}\n\\caption{Sequential decoding of message vector from VLIC}\n\\label{vctable1}\n\\end{table*}\n\nThe number of broadcast symbols in the $u_a$ dimensional VLIC $\\mathfrak{C}$ is $K_a-\\Delta_a$. The rate achieved by $\\mathfrak{C}$ is \n\\begin{align*}\n\\frac{u_a}{K_a-\\Delta_a}=\\frac{\\frac{U+1}{a}}{\\frac{K}{a}-\\frac{D-U}{a}}=\\frac{U+1}{K-D+U}.\n\\end{align*}\nThis is equal to the capacity mentioned in \\eqref{capacity}. Hence, the constructed VLICs are capacity achieving. \n\\end{proof}\n\n\n\\begin{remark}\nThe decoding procedure given in Theorem \\ref{vcthm1} uses successive interference cancellation. That is, in the decoding, the receiver $R_k$ decode its wanted message symbols $(x_{k,1},x_{k,2},\\ldots,x_{k,u_a})$ one at a time starting from $x_{k,u_a}$ and proceeds to decode $$x_{k,u_a-1}\\rightarrow x_{k,u_a-2} \\rightarrow \\ldots \\rightarrow x_{k,1}$$ after cancelling the interference from already decoded message symbols.\n\\end{remark}\n\nIn the Lemma \\ref{lemmaminimal} given below, we prove that the constructed optimal VLICs in Theorem \\ref{vcthm1} are minimal dimensional \nif $\\text{gcd}(K,D-U,U+1)$ is equal to $\\text{gcd}(K-D+U,U+1)$. \n\\begin{lemma}\n\\label{lemmaminimal}\nIf $\\text{gcd}(K,D-U,U+1)=\\text{gcd}(K-D+U,U+1)$, then the constructed VLICs in Theorem \\ref{vcthm1} are minimal dimensional.\n\\end{lemma}\n\\begin{proof}\nLet $a=\\frac{U+1}{\\text{gcd}(K-D+U,U+1)}, b=\\frac{K-D+U}{\\text{gcd}(K-D+U,U+1)}$. We have $\\text{gcd}(a,b)=1$. For SUICP(SNCS), we have\n\\begin{align*}\nC=\\frac{U+1}{K-D+U}=\\frac{\\frac{U+1}{\\text{gcd}(K-D+U,U+1)}}{\\frac{K-D+U}{\\text{gcd}(K-D+U,U+1)}}= \\frac{a}{b}.\n\\end{align*}\nIf $a=1$, then the index code is a scalar linear index code and it is the minimal dimensional index code. Hence, with out loss of generality, we assume $a>1$. Consider there exist an $a-t$ dimensional capacity achieving VLIC for some $t \\in [1:a-1]$. This $a-t$ dimensional VLIC is obtained by mapping $(a-t)K$ message symbols into $b-s$ broadcast symbols for some $s \\in [1:b-1]$. This code is capacity achieving and hence we have,\n\\begin{align}\n\\label{lneweq1}\n\\frac{a-t}{b-s}=\\frac{a}{b}\n\\end{align}\n which gives $as-bt=0.$ From the extended Euclid algorithm and Bezout coefficients, the equation of the form $as-bt=0$ has solutions of the form $s=k\\frac{b}{\\text{gcd}(a,b)}$ and $t=k\\frac{a}{\\text{gcd}(a,b)}$ for $k \\in Z_{>0}$. We have $\\text{gcd}(a,b)=1$ and hence $s=kb \\geq b$ and $t=ka \\geq a$. This is a contradiction. Hence, the dimension used in Theorem \\ref{vcthm1} $$a=\\frac{U+1}{\\text{gcd}(K-D+U,U+1)}=\\frac{U+1}{\\text{gcd}(K,D-U,U+1)}$$ is minimal.\n\\end{proof}\nFor all SUICP(SCNC) not satisfying the condition $\\text{gcd}(K,D-U,U+1)=\\text{gcd}(K-D+U,U+1)$, we conjecture that the constructed optimal VLICs in Theorem \\ref{vcthm1} are minimal dimensional.\n\\subsection{Low Complexity Decoding}\nIn \\cite{VaR3}, we gave a low-complexity decoding for one-sided SUICP(SNCS) with AIR matrix as encoding matrix. The low complexity decoding method helps to identify a reduced set of side-information for each user with which the decoding can be carried out. By this method every receiver is able to decode its wanted message symbol by simply adding some broadcast symbols. The low complexity decoding of one-sided SUICP(SNCS) with $K_a$ messages and $\\Delta_a$ side-information explicitly gives the values of $a_t^{(d)}$ and $b_t^{(d)}$ in \\eqref{vcsum} for $t \\in [0:K_a-1]$ and $d \\in [1:\\Delta_a]$. Hence, from the proof of Theorem \\ref{vcthm1} we get a  low complexity decoding for two-sided SUICP(SNCS) with arbitrary $K,D$ and $U$.\t\n\nThe following examples illustrate the construction of optimal VLICs and low complexity decoding for two-sided  SUICP(SNCS).\n\n\n\\begin{example}\n\\label{vcex1}\nConsider the two-sided SUICP(SNCS) with $K=8, D=2, U=1$. For this SUICP(SNCS), $\\text{gcd}(K,D-U,U+1)=a=1$. The optimal length index code is a two dimensional index code. For this two-sided SUICP(SNCS), we have $K_a=8$ and $\\Delta_a=1$. The AIR matrix of size $\\mathbf{L}_{8 \\times 7}$ is given below.\n\\begin{figure}[ht]\n\\arraycolsep=1pt\n{\n$$\\mathbf{L}_{8 \\times 7}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n 1 & 1 & 1 & 1 &1 & 1 & 1\\\\\n   \\end{array}\\right]$$\n}\n\n\\end{figure}\n\nThe scalar linear index code for the one-sided SUICP(SNCS) with $K_a=8$  and $\\Delta_a=1$ is given by \n\\begin{align*}\n\\mathfrak{C}=\\{&y_0+y_7, ~~~y_1+y_7,~~~y_2+y_7, ~~~y_3+y_7, \\\\& y_4+y_7, ~~~y_5+y_7, ~~~y_6+y_7\\}. \n\\end{align*}\n\nThe VLIC for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{s,1}+x_{s-1,2}$ for $s \\in [0:7]$. The VLIC is given by\n\\begin{align*}\n\\mathfrak{C}^{(2s)}=\\{&x_{0,1}+x_{7,2}+x_{7,1}+x_{6,2}, ~x_{1,1}+x_{0,2}+x_{7,1}+x_{6,2}, \\\\&x_{2,1}+x_{1,2}+x_{7,1}+x_{6,2},~x_{3,1}+x_{2,2}+x_{7,1}+x_{6,2}, \\\\&x_{4,1}+x_{3,2}+x_{7,1}+x_{6,2}, ~x_{5,1}+x_{4,2}+x_{7,1}+x_{6,2}, \\\\&x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}\\}. \n\\end{align*}\n\nLet $\\tau_s$ be the set of broadcast symbols required to decode $y_s$ from $\\mathfrak{C}$. The values of $\\tau_s$ and $S_s$ are given in Table \\ref{vctableex122}.\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{$\\mathcal{W}_s$} & $\\tau_s$ & $S_s$ \\\\\n\\hline\n$y_0$ &$y_0+y_7,y_1+y_7$& $S_0=y_0+y_1$ \\\\\n\\hline\n$y_1$&$y_1+y_7,y_2+y_7$&  $S_1=y_1+y_2$ \\\\\n\\hline \n$y_2$ &$y_2+y_7,y_3+y_7$&$S_2=y_2+y_3$  \\\\\n\\hline\n$y_3$ &$y_3+y_7,y_4+y_7$&$S_3=y_3+y_4$  \\\\\n\\hline\n$y_4$ &$y_4+y_7,y_5+y_7$& $S_4=y_4+y_5$  \\\\\n\\hline\n$y_5$ &$y_5+y_7,y_6+y_7$&$S_5=y_5+y_6$ \\\\\n\\hline\n$y_6$ &$y_6+y_7$&$S_6=y_6+y_7$ \\\\\n\\hline\n$y_7$ &$y_0+y_7$&$S_7=y_7+y_0 $\\\\\n\\hline \n\\end{tabular}\n\\caption{Decoding at each receiver for one-sided SUICP(SNCS) with $K=8,\\Delta_a=1$}\n\\label{vctableex122}\n\\end{table}\n\nReceiver $R_k$ wants to decode $x_{k,1}$ and $x_{k,2}$ for $k \\in [0:7]$. $R_k$ first decode $x_{k,2}$ from $S_{s+1}$, then decodes $x_{k,1}$ from $S_s$ for every $k \\in [0:7]$. The decoding procedure at the receivers $R_0$, $R_1\\ldots R_7$ is shown in Table \\ref{vctableex121}.\n\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$R_k$ & {$\\mathcal{W}_k$} & Sum from which $x_{k,i}$ is decoded \\\\\n\\hline\n$R_0$ & $x_{0,2}$&  $S_1=x_{1,1}+x_{0,2}+x_{2,1}+x_{1,2}$ \\\\\n\\hline \n$R_0$ & $x_{0,1}$ &$S_0=x_{0,1}+x_{7,2}+x_{1,1}+x_{0,2}$  \\\\\n\\hline\n$R_1$ & $x_{1,2}$&  $S_2=x_{2,1}+x_{1,2}+x_{3,1}+x_{2,2}$ \\\\\n\\hline \n$R_1$ & $x_{1,1}$ &$S_1=x_{1,1}+x_{0,2}+x_{2,1}+x_{1,2}$  \\\\\n\\hline\n$R_2$ & $x_{2,2}$&  $S_3=x_{3,1}+x_{2,2}+x_{4,1}+x_{3,2}$ \\\\\n\\hline \n$R_2$ & $x_{2,1}$ &$S_2=x_{2,1}+x_{1,2}+x_{3,1}+x_{2,2}$  \\\\\n\\hline\n$R_3$ & $x_{3,2}$&  $S_4=x_{4,1}+x_{3,2}+x_{5,1}+x_{4,2}$ \\\\\n\\hline \n$R_3$ & $x_{3,1}$ &$S_3=x_{3,1}+x_{2,2}+x_{4,1}+x_{3,2}$  \\\\\n\\hline\n$R_4$ & $x_{4,2}$&  $S_5=x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2}$ \\\\\n\\hline \n$R_4$ & $x_{4,1}$ &$S_4=x_{4,1}+x_{3,2}+x_{5,1}+x_{4,2}$  \\\\\n\\hline\n$R_5$ & $x_{5,2}$&  $S_6=x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}$ \\\\\n\\hline \n$R_5$ & $x_{5,1}$ &$S_5=x_{5,1}+x_{4,2}+x_{6,1}+x_{5,2}$  \\\\\n\\hline\n$R_6$ & $x_{6,2}$&  $S_7=x_{7,1}+x_{6,2}+x_{0,1}+x_{7,2}$ \\\\\n\\hline \n$R_6$ & $x_{6,1}$ &$S_6=x_{6,1}+x_{5,2}+x_{7,1}+x_{6,2}$  \\\\\n\\hline\n$R_7$ & $x_{7,2}$&  $S_0=x_{0,1}+x_{7,2}+x_{1,1}+x_{0,2}$ \\\\\n\\hline \n$R_7$ & $x_{7,1}$ &$S_{7}=x_{7,1}+x_{6,2}+x_{0,1}+x_{7,2}$  \\\\\n\\hline\n\\end{tabular}\n\\caption{Decoding at each receiver for the two-sided SUICP(SNCS) with $K=8,D=2$ and $U=1$}\n\\label{vctableex121}\n\\end{table}\n\nThe capacity of the given two-sided SUICP(SNCS) is $\\frac{2}{7}$. The VLIC $\\mathfrak{C}$ uses seven broadcast symbols to transmit two message symbols per receiver. The rate achieved by $\\mathfrak{C}$ is $\\frac{2}{7}$. Hence, $\\mathfrak{C}^{(2)}$ is an optimal length index code. \n\\end{example}\n\n\\begin{example}\n\\label{vcex2}\nConsider the two-sided SUICP(SNCS) with $K=22, D=7, U=3$. For this SUICP(SNCS), $\\text{gcd}(K,D-U,U+1)=a=2$. The optimal length index code is a two dimensional index code. For this two-sided SUICP(SNCS), we have $K_a=11$  and $\\Delta_a=2$. The AIR matrix of size $\\mathbf{L}_{11 \\times 9}$ is given below.\n\\begin{figure}[ht]\n\\arraycolsep=1pt\n{\n$$\\mathbf{L}_{11 \\times 9}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n   0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n   \\end{array}\\right]$$\n}\n\n\\end{figure}\n\nThe scalar linear index code for the one-sided SUICP(SNCS) with $K_a=11,\\Delta_a=2$ is given by \n\\begin{align*}\n\\mathfrak{C}=\\{&y_0+y_9, ~~~y_1+y_{10},~~~y_2+y_9,~~~y_3+y_{10}, \\\\&y_4+y_9, ~~~y_5+y_{10}, ~~~y_6+y_9,~~~y_7+y_{10}, \\\\&y_8+y_9+y_{10}\\}. \n\\end{align*}\n\nThe VLIC for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{2s,1}+x_{2s+1,1}+x_{2(s-1),2}+x_{2(s-1)+1,2}$ for $s \\in [0:10]$. The two dimensional VLIC for the is given SUICP(SNCS) is given in Table \\ref{vcex2table1}.\n\\begin{table*}[ht]\n\\centering\n\\setlength\\extrarowheight{0.5pt}\n\\begin{tabular}{|c|c|}\n\\hline\n\\textbf{$c_0$} & $\\underbrace{x_{0,1}+x_{1,1}+x_{20,2}+x_{21,2}}_{y_0}+\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}$\\\\\n\\hline\n\\textbf{$c_1$} & $\\underbrace{x_{2,1}+x_{3,1}+x_{0,2}+x_{1,2}}_{y_1}+\\underbrace{x_{20,1}+x_{21,1}+x_{0,2}+x_{1,2}}_{y_{10}}$ \\\\\n\\hline \n\\textbf{$c_2$} & $\\underbrace{x_{4,1}+x_{5,1}+x_{2,2}+x_{3,2}}_{y_2}+\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}$  \\\\\n\\hline\n\\textbf{$c_3$} & $\\underbrace{x_{6,1}+x_{7,1}+x_{4,2}+x_{5,2}}_{y_3}+\\underbrace{x_{20,1}+x_{21,1}+x_{0,2}+x_{1,2}}_{y_{10}}$ \\\\\n\\hline\n\\textbf{$c_4$} & $\\underbrace{x_{8,1}+x_{9,1}+x_{6,2}+x_{7,2}}_{y_4}+\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_{9}}$ \\\\\n\\hline\n\\textbf{$c_5$} & $\\underbrace{x_{10,1}+x_{11,1}+x_{8,2}+x_{9,2}}_{y_5}+\\underbrace{x_{20,1}+x_{21,1}+x_{0,2}+x_{1,2}}_{y_{10}}$ \\\\\n\\hline\n\\textbf{$c_6$} & $\\underbrace{x_{12,1}+x_{13,1}+x_{10,2}+x_{11,2}}_{y_6}+\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}$ \\\\\n\\hline\n\\textbf{$c_7$} & $\\underbrace{x_{14,1}+x_{15,1}+x_{12,2}+x_{13,2}}_{y_7}+\\underbrace{x_{20,1}+x_{21,1}+x_{0,2}+x_{1,2}}_{y_{10}}$ \\\\\n\\hline \n\\textbf{$c_8$} & $\\underbrace{x_{16,1}+x_{17,1}+x_{14,2}+x_{15,2}}_{y_8}+\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}+\n\\underbrace{x_{20,1}+x_{21,1}+x_{0,2}+x_{1,2}}_{y_{10}}$ \\\\\n\\hline \n\\end{tabular}\n\\caption{Vector linear index code for the two-sided SUICP(SNCS) given in Example \\ref{vcex2}}\n\\label{vcex2table1}\n\\end{table*}\n\nLet $\\tau_s$ be the set of broadcast symbols required to decode $y_s$ from $\\mathfrak{C}$. The values of $\\tau_s$ and $S_s$ for $s \\in [0:10]$ are given in Table \\ref{vctableex14}.\n\\begin{table}[ht]\n\\centering\n\\begin{tabular}{|c|c|c|}\n\\hline\n{$\\mathcal{W}_s$} & $\\tau_s$ & $S_s$ \\\\\n\\hline\n$y_0$ &$y_0+y_9,y_2+y_9$& $S_0=y_0+y_3$ \\\\\n\\hline\n$y_1$&$y_1+y_{10},y_3+y_{10}$&  $S_1=y_1+y_3$ \\\\\n\\hline \n$y_2$ &$y_2+y_9,y_4+y_9$&$S_2=y_2+y_4$  \\\\\n\\hline\n$y_3$ &$y_3+y_{10},y_5+y_{10}$&$S_3=y_3+y_5$  \\\\\n\\hline\n$y_4$ &$y_4+y_9,y_6+y_9$& $S_4=y_4+y_6$  \\\\\n\\hline\n$y_5$ &$y_5+y_{10},y_7+y_{10}$&$S_5=y_5+y_7$ \\\\\n\\hline\n$y_6$ &$y_6+y_9,y_7+y_{10},$&$S_6=y_6+y_7+y_8$ \\\\\n&$y_8+y_9+y_{10}$&\\\\\n\\hline\n$y_7$ &$y_7+y_{10},y_8+y_9+y_{10}$&$S_7=y_7+y_8+y_9 $\\\\\n\\hline \n$y_8$ &$y_8+y_9+y_{10}$&$S_8=y_8+y_9+y_{10}$ \\\\\n\\hline\n$y_9$ &$y_0+y_9$&$S_9=y_0+y_9$ \\\\\n\\hline\n$y_{10}$ &$y_1+y_{10}$&$S_{10}=y_1+y_{10} $\\\\\n\\hline \n\\end{tabular}\n\\caption{Decoding at each receiver for one-sided SUICP(SNCS) with $K_a=11,\\Delta_a=2$}\n\\label{vctableex14}\n\\end{table}\n\nReceiver $R_k$ wants to decode $x_{k,1}$ and $x_{k,2}$. $R_k$ first decode $x_{k,2}$ from $S_{s+1}$, then decodes $x_{k,1}$ from $S_s$ for every $k \\in [0:21]$. The receiver $R_{14}$ decodes $x_{14,2}$ from $S_8$ and decodes $x_{14,1}$ from $S_7.$ From Table \\ref{vctableex14}, we have \n\\begin{align*}\n&S_8=y_8+y_9+y_{10}=\\underbrace{x_{16,1}+x_{17,1}+x_{14,2}+x_{15,2}}_{y_8}+\\\\&\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}+\n\\underbrace{x_{20,1}+x_{21,1}+x_{18,2}+x_{19,2}}_{y_{10}}.\n\\end{align*} \n\nIn $S_8$, all other messages except $x_{14,2}$ are in the side information of $R_{14}$. Hence $R_{14}$ decodes $x_{14,2}$ from $S_8$. From Table \\ref{vctableex14}, we have \n\\begin{align*}\n&S_7=y_7+y_8+y_9=\\underbrace{x_{14,1}+x_{15,1}+x_{12,2}+x_{13,2}}_{y_7}+\\\\&\\underbrace{x_{16,1}+x_{17,1}+x_{14,2}+x_{15,2}}_{y_8}+\n\\underbrace{x_{18,1}+x_{19,1}+x_{16,2}+x_{17,2}}_{y_9}.\n\\end{align*} \n\nIn $S_7$, all other messages except $x_{14,1}$ and $x_{14,2}$ are in the side information of $R_{14}$. $R_{14}$ already decoded $x_{14,2}$ and hence it cancels the interference in $S_7$ due to $x_{14,2}$ and then decodes $x_{14,1}$. \n\nThe capacity of the given two-sided SUICP(SNCS) is $\\frac{4}{18}=\\frac{2}{9}$. The VLIC $\\mathfrak{C}$ uses nine broadcast symbols to transmit two message symbols per receiver. The rate achieved by $\\mathfrak{C}$ is $\\frac{2}{9}$.\n\\end{example}\n\\begin{note}\nFor the two-sided SUICP(SNCS) given in Example \\ref{vcex2}, the construction given in \\cite{VaR1} gives four dimensional optimal length VLIC, whereas the construction given in this paper gives two dimensional optimal length VLIC.\n\\end{note}\n\\begin{example}\n\\label{vcex3}\nConsider the two-sided SUICP(SNCS) with $K=24,D=11$ and $U=2$. For this SUICP(SNCS), we have $\\text{gcd}(K,D-U,U+1)=3$. The optimal length index code for this SUICP(SNCS) is scalar linear index codes. For this two-sided SUICP(SNCS), we have $K_a=8,\\Delta_a=3$. AIR matrix of size $8 \\times 5$ is given below.\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{8 \\times 5}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 \\\\\n   0 & 1 & 0 & 0 & 0 \\\\\n   0 & 0 & 1 & 0 & 0 \\\\\n   0 & 0 & 0 & 1 & 0 \\\\\n   0 & 0 & 0 & 0 & 1 \\\\\n   1 & 0 & 0 & 1 & 0 \\\\\n   0 & 1 & 0 & 0 & 1 \\\\\n   0 & 0 & 1 & 1 & 1 \\\\\n  \\end{array}\\right]$$\n}\n\nThe scalar linear code $\\mathfrak{C}$ for the one-sided SUICP(SNCS) with $K_a=8$ and $\\Delta_a=3$ is \n\\begin{align*}\n\\mathfrak{C}&=\\{c_0,c_1.c_2,c_3,c_4\\}\\\\&=\\{y_0+y_5, ~y_1+y_6, ~y_2+y_7, ~y_3+y_5+y_7, ~y_4+y_6+y_7\\}.\n\\end{align*}\n \nThe scalar linear index code $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is given by\n\\begin{align*}\n&\\mathfrak{C}^{(2s)}=\\{c_0,c_1,c_2,c_3,c_4\\}=\\\\\\{&\\underbrace{x_0+x_1+x_2}_{y_0}+\n\\underbrace{x_{15}+x_{16}+x_{17}}_{y_5},\\\\&\\underbrace{x_3+x_4+x_5}_{y_1}+\\underbrace{x_{18}+x_{19}+x_{20}}_{y_6},\\\\&\\underbrace{x_6+x_7+x_8}_{y_2}+\\underbrace{x_{21}+x_{22}+x_{23}}_{y_7},\\\\& \\underbrace{x_9+x_{10}+x_{11}}_{y_3}+\\underbrace{x_{15}+x_{16}+x_{17}}_{y_5}+\\underbrace{x_{21}+x_{22}+x_{23}}_{y_7},\\\\& \\underbrace{x_{12}+x_{13}+x_{14}}_{y_4}+\\underbrace{x_{18}+x_{19}+x_{20}}_{y_6}+\\underbrace{x_{21}+x_{22}+x_{23}}_{y_7}\\}.\n\\end{align*}\n\\end{example}\n\\begin{note}\nFor the two-sided SUICP(SNCS) given in Example \\ref{vcex3}, the construction given in  \\cite{VaR1} gives three dimensional optimal length VLIC, whereas the construction given in this paper gives  optimal length scalar linear index code.\n\\end{note}\n\\subsection{Encoding Matrices for Optimal Vector Linear Codes}\nIn this subsection, we give encoding matrices for optimal VLICs of two-sided SUICP(SNCS).  with $K$ messages, $D$ side-information after and $U$ side-information before the desired message. \n\n\\begin{definition}\n\\label{vcdef5}\nLet $\\mathbf{L}$ be the AIR matrix of size $K_a \\times (K_a-\\Delta_a)$. Let $L_s$ be the $s$th row of $\\mathbf{L}$ for $s \\in [0:K_a-1]$. Then, the $s$th precoding matrix $\\mathbf{V}_s$ is defined as \n$$\\mathbf{V}_{s}=\\left[\\begin{array}{*{20}c}\n   L_s  \\\\\n   L_{s+1} \\\\\n   L_{s+2} \\\\\n    \\vdots \\\\\n   L_{s+u_a-1} \\\\\n   \\end{array}\\right].$$\nThe matrix $\\mathbf{\\tilde{V}}_s$ is defined as \n\\begin{align}\n\\label{precoding}\n\\mathbf{\\tilde{V}}_{s}=\\left.\\left[\\begin{array}{*{20}c}\n   \\mathbf{V}_s  \\\\\n   \\mathbf{V}_s  \\\\\n   \\vdots  \\\\\n   \\mathbf{V}_s \n   \\end{array}\\right]\\right\\rbrace a~\\text{number~of}~ \\mathbf{V}_s~\\text{matrices}\n\\end{align}\n\\end{definition}\n\\begin{lemma}\n\\label{lemma1}\nConsider the two-sided SUICP(SNCS) with $K$ messages, $U$ side-information after and $D$ side-information before the desired message. Let $x_k=(x_{k,1},x_{k,2},\\cdots,x_{k,u_a})$ be the message vector wanted by the $k$th receiver, where $x_k \\in \\mathbb{F}_q^{u_a}, x_{k,i} \\in \\mathbb{F}_q$ for $k \\in [0:K-1]$ and $i \\in [1:u_a]$. For this two-sided SUICP(SNCS), an encoding matrix for optimal length VLIC $\\mathbf{L}^{(2s)}$ of size $Ku_a \\times (K_a-\\Delta_a)$ is given by \n$$\\mathbf{L}^{(2s)}=\\left.\\left[\\begin{array}{*{20}c}\n   \\mathbf{\\tilde{V}}_0  \\\\\n   \\mathbf{\\tilde{V}}_1  \\\\\n   \\mathbf{\\tilde{V}}_2  \\\\\n   \\vdots  \\\\\n   \\mathbf{\\tilde{V}}_{K_a-1}   \\\\\n   \\end{array}\\right]\\right. $$\nwhere $\\mathbf{\\tilde{V}}_k$ is defined in Definition \\ref{vcdef5}.\n\\end{lemma}\n\\begin{proof}\nLet $\\mathbf{x}_k=[x_{k,1}~x_{k,2}~\\ldots~x_{k,u_a}]$. Let $\\mathbf{x}_{[a:b]}=[\\mathbf{x}_a~\\mathbf{x}_{a+1}~\\ldots~\\mathbf{x}_b]$. From Theorem \\ref{vcthm1}, the $u_a$ dimensional VLIC for the given two-sided SUICP(SNCS) is given by \n\\begin{align*}\n[c_0~c_1\\ldots &c_{K_a-\\Delta_a-1}]=\\sum_{s=0}^{K_a-1}y_sL_s\\\\&=\\sum_{s=0}^{K_a-1} \\left(\\sum_{i=1}^{u_a} \\sum_{j=0}^{a-1}x_{a(s+1-i)+j,i}\\right)L_s \\\\&=\\sum_{s=0}^{K_a-1} (\\sum_{j=0}^{a-1} \\left(\\sum_{i=1}^{u_a}x_{a(s+1-i)+j,i} L_s\\right) \\\\&=\\sum_{s=0}^{K_a-1} \\left(\\sum_{j=0}^{a-1} \\mathbf{x}_{as+j}\\mathbf{V}_s \\right) \\\\&= \\sum_{s=0}^{K_a-1} \\mathbf{x}_{[as:as+a-1]} \\mathbf{\\tilde{V}_s}= \\mathbf{x}_{[0:K-1]}\\mathbf{L}^{(2s)}.\n\\end{align*}\n\nThis completes the proof.\n\\end{proof}\n\\begin{example}\n\\label{vcex4}\nConsider the two-sided SUICP(SNCS) with $K=8,D=2,U=1$. The optimal length VLIC for this SUICP is given in Example \\ref{vcex1}. The encoding matrix $\\mathbf{L}_{16 \\times 7}^{(2s)}$ for this optimal length VLIC by using Lemma \\ref{lemma1} is given in Fig \\ref{encodingmatrix2}.\n\\end{example}\n\\begin{figure}\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{16 \\times 7}^{(2s)}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n   1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n   1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   \\end{array}\\right]$$\n}\n\\caption{Encoding matrices for two-sided SUICP(SNCS) given in Example \\ref{vcex4}}\n\\label{encodingmatrix2}\n\\end{figure}\n\\begin{example}\n\\label{vcex5}\nConsider the two-sided SUICP(SNCS) with $K=22,D=7,U=3$. The optimal length VLIC for this SUICP is given in Example \\ref{vcex2}. The encoding matrix $\\mathbf{L}_{44 \\times 9}^{(2s)}$ for this optimal length VLIC by using Lemma \\ref{lemma1} is given in Fig \\ref{encodingmatrix}.\n\\begin{figure}\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{44 \\times 9}^{(2s)}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   \n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   \n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   \n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   \n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   \n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   \n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n   \n   1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n   0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n   1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1\\\\\n   0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n   \n   0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\\\\\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   \\end{array}\\right]~\\mathbf{L}_{24 \\times 5}^{(2s)}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0  \\\\\n   1 & 0 & 0 & 0 & 0  \\\\\n   1 & 0 & 0 & 0 & 0  \\\\\n   0 & 1 & 0 & 0 & 0  \\\\\n   0 & 1 & 0 & 0 & 0  \\\\\n   0 & 1 & 0 & 0 & 0  \\\\\n   0 & 0 & 1 & 0 & 0  \\\\\n   0 & 0 & 1 & 0 & 0  \\\\\n   0 & 0 & 1 & 0 & 0  \\\\\n   0 & 0 & 0 & 1 & 0  \\\\\n   0 & 0 & 0 & 1 & 0  \\\\\n   0 & 0 & 0 & 1 & 0  \\\\\n   0 & 0 & 0 & 0 & 1  \\\\\n   0 & 0 & 0 & 0 & 1  \\\\\n   0 & 0 & 0 & 0 & 1  \\\\\n   1 & 0 & 0 & 1 & 0  \\\\\n   1 & 0 & 0 & 1 & 0  \\\\\n   1 & 0 & 0 & 1 & 0  \\\\\n   0 & 1 & 0 & 0 & 1  \\\\\n   0 & 1 & 0 & 0 & 1  \\\\\n   0 & 1 & 0 & 0 & 1  \\\\\n   0 & 0 & 1 & 1 & 1  \\\\\n   0 & 0 & 1 & 1 & 1  \\\\\n   0 & 0 & 1 & 1 & 1  \\\\\n   \\end{array}\\right]$$\n}\n\\caption{Encoding matrices for two-sided SUICP(SNCS) given in Example \\ref{vcex5} and \\ref{vcex6}}\n\\label{encodingmatrix}\n\\end{figure}\n\\end{example}\n\\begin{example}\n\\label{vcex6}\nConsider the two-sided SUICP(SNCS) with $K=24,D=11,U=2$. The optimal length VLIC for this SUICP is given in Example \\ref{vcex3}. The encoding matrix $\\mathbf{L}_{24 \\times 5}^{(2s)}$ for this optimal length VLIC by using Lemma \\ref{lemma1} is given in Fig \\ref{encodingmatrix}.\n\\end{example}\n\\section{Some special cases of proposed construction}\n\\label{sec3}\nIn this section, we give four special cases of the proposed construction. Consider the two-sided SUICP(SNCS) with $K$ messages, $D$ side-information after and $U$ side-information before the desired message as given in \\eqref{antidote}. We have $a=\\text{gcd}(K,D-U,U+1)$. In Theorem \\ref{vcthm1}, we combine $U+1$ message symbols, one from each of the message vectors $x_t$ for every $t \\in [as-(U+1)+a:as+a-1]$ as given below.\n\\begin{align}\n\\label{thm2eq6}\n\\nonumber\ny_s=\\sum_{i=1}^{u_a} \\sum_{j=0}^{a-1}& x_{a(s+1-i)+j,i} \\\\& \\text{for}~s \\in [0:K_a-1].\n\\end{align}\n\\subsection{Optimal scalar linear index codes for two-sided SUICP(SNCS)}\nIn the construction given in Theorem \\ref{vcthm1}, if $U+1$ divides both $K$ and $D-U$, then we have $a=U+1$ and $u_a=1$. In this case, \\eqref{thm2eq6} reduces to \n\\begin{align}\n\\label{thm2eq7}\n\\nonumber\ny_s&=\\sum_{i=1}^1 \\sum_{j=0}^U x_{(U+1)(s+1-i)+j,i}\\\\&=\\sum_{j=0}^U x_{(U+1)s+j} ~~~ \\text{for}~s \\in [0:K_a].\n\\end{align}\n\nHence, in this case, the index code given by \\eqref{thm1eq5} is an optimal length scalar linear index code.\n\\begin{example}\n\\label{vcex7}\nConsider the two-sided SUICP(SNCS) with $K=14,D=3$ and $U=1$. For this SUICP(SNCS), we have $\\text{gcd}(K,D-U,U+1)=U+1=2$. The optimal length index code for this two-sided SUICP(SNCS) is a scalar linear index code. For this two-sided SUICP(SNCS), we have $K_a=7,\\Delta_a=1$. An optimal length encoding matrix for this one-sided SUICP(SNCS) is a AIR matrix of size $7 \\times 6$. AIR matrix of size $7 \\times 6$ is given below.\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{7 \\times 6}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 1 & 1 & 1 & 1 & 1\\\\\n  \\end{array}\\right]$$\n}\n\nThe scalar linear index code $\\mathfrak{C}$ for this one-sided SUICP(SNCS) is given below.\n\\begin{align*}\n\\mathfrak{C}=\\{c_0,c_1,c_2,c_3,c_4,c_5\\}=\\{&y_0+y_6,~~~y_1+y_6,\\\\&y_2+y_6,~~~y_3+y_6,\\\\&y_4+y_6,~~~y_5+y_6\\}.\n\\end{align*}\n \nThe scalar linear index code $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{2s}+x_{2s+1}$ for $s \\in [0:6]$. The scalar linear index code $\\mathfrak{C}^{(2s)}$ is given below.\n\\begin{align*}\n\\mathfrak{C}^{(2s)}=\\{&x_0+x_1+x_{12}+x_{13},~~~x_2+x_3+x_{12}+x_{13},\\\\&x_4+x_5+x_{12}+x_{13},~~~~x_6+x_7+x_{12}+x_{13},\\\\&x_8+x_9+x_{12}+x_{13},~~~x_{10}+x_{11}+x_{12}+x_{13}\\}.\n\\end{align*}\n\\end{example}\n\\subsection{Optimal $(U+1)$ dimensional VLICs for two-sided SUICP(SNCS)}\nIn the construction given in Theorem \\ref{vcthm1}, if $\\text{gcd}(K,D-U,U+1)=1$, then we have $a=1$ and $u_a=U+1$. In this case, \\eqref{thm2eq6} reduces to \n\\begin{align}\n\\label{thm2eq8}\n\\nonumber\ny_s&=\\sum_{i=1}^{U+1} \\sum_{j=0}^0 x_{(s+1-i)+j,i}\\\\&=\\sum_{i=1}^{U+1} x_{s+1-i,i} ~~\\text{for}~s \\in [0:K-1].\n\\end{align}\n\nHence, in this case, the index code given by \\eqref{thm1eq5} is an optimal length $(U+1)$ dimensional VLIC.\n\n\\begin{example}\n\\label{vcex8}\nConsider the two-sided SUICP(SNCS) with $K=11, D=5, U=2$. For this SUICP(SNCS), $\\text{gcd}(K,D-U,U+1)=a=1$. The optimal length index code is a three dimensional VLIC. For this SUICP(SNCS), we have $K_a=K=11$  and $\\Delta_a=3$. The AIR matrix of size $\\mathbf{L}_{11 \\times 8}$ is given below.\n\n\\begin{small}\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{11 \\times 8}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 0 & 0 & 1 & 0 & 0 & 1 & 0\\\\\n   0 & 1 & 0 & 0 & 1 & 0 & 0 & 1\\\\\n   0 & 0 & 1 & 0 & 0 & 1 & 1 & 1\\\\\n   \\end{array}\\right]$$\n}\n\\end{small}\n\nThe scalar linear index code $\\mathfrak{C}$ for the one-sided SUICP(SNCS) with $K_a=11$ and $\\Delta_a=3$ is given below. \n\\begin{align*}\n\\mathfrak{C}=\\{&y_0+y_8, ~~y_1+y_9,~~y_2+y_{10},\\\\&y_3+y_8, ~~y_4+y_9, ~~y_5+y_{10},\\\\& y_6+y_8+y_{10}, ~y_7+y_9+y_{10}\\}. \n\\end{align*}\n\nThe VLIC $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{s,1}+x_{s-1,2}+x_{s-2,3}$ for $s \\in [0:10]$. The VLIC $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is given below.\n\\begin{align*}\n\\mathfrak{C}^{(2s)}=\\{&\\underbrace{x_{0,1}+x_{10,2}+x_{9,3}}_{y_0}+\\underbrace{x_{8,1}+x_{7,2}+x_{6,3}}_{y_8},\\\\& \\underbrace{x_{1,1}+x_{0,2}+x_{10,3}}_{y_1}+\\underbrace{x_{9,1}+x_{8,2}+x_{7,3}}_{y_9}, \\\\& \\underbrace{x_{2,1}+x_{1,2}+x_{0,3}}_{y_2}+\\underbrace{x_{10,1}+x_{9,2}+x_{8,3}}_{y_{10}},\\\\& \\underbrace{x_{3,1}+x_{2,2}+x_{1,3}}_{y_3}+\\underbrace{x_{8,1}+x_{7,2}+x_{6,3}}_{y_8}, \\\\& \\underbrace{x_{4,1}+x_{3,2}+x_{2,3}}_{y_4}+\\underbrace{x_{9,1}+x_{8,2}+x_{7,3}}_{y_9},\\\\& \\underbrace{x_{5,1}+x_{4,2}+x_{3,3}}_{y_5}+\\underbrace{x_{10,1}+x_{9,2}+x_{8,3}}_{y_{10}}, \\\\& \\underbrace{x_{6,1}+x_{5,2}+x_{4,3}}_{y_6}+\\underbrace{x_{8,1}+x_{7,2}+x_{6,3}}_{y_8}\\\\&+\\underbrace{x_{10,1}+x_{9,2}+\nx_{8,3}}_{y_{10}},\\\\& \\underbrace{x_{7,1}+x_{6,2}+x_{5,3}}_{y_7}+\\underbrace{x_{9,1}+x_{8,2}+x_{7,3}}_{y_9}\\\\& +\\underbrace{x_{10,1}+x_{9,2}+\nx_{9,3}}_{y_{10}}\\}.\n\\end{align*}\n\n\\end{example}\n\\subsection{Optimal scalar linear index codes for one-sided SUICP(SNCS)}\nIn the construction given in Theorem \\ref{vcthm1}, if $U=0$, then we have, $a=\\text{gcd}(K,D-U,U+1)=1$ and $u_a=U+1=1$. In this case, \\eqref{thm2eq6} reduces to \n$y_s=x_s ~~\\text{for}~s \\in [0:K-1].$\n\nHence, in this case, the index code given by \\eqref{thm1eq5} is an optimal length scalar linear index code.\n\\begin{example}\n\\label{vcex9}\nConsider the two-sided SUICP(SNCS) with $K=7,D=2$ and $U=0$. For this SUICP(SNCS), we have $\\text{gcd}(K,D-U,U+1)=1$. The optimal length index code for this SUICP(SNCS) is scalar linear index code. For this SUICP(SNCS), we have $K=K_a=7,D=\\Delta_a=2$. An optimal length encoding matrix for this SUICP(SNCS) is a AIR matrix of size $7 \\times 5$. AIR matrix of size $7 \\times 5$ is given below.\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{7 \\times 5}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 \\\\\n   0 & 1 & 0 & 0 & 0 \\\\\n   0 & 0 & 1 & 0 & 0 \\\\\n   0 & 0 & 0 & 1 & 0 \\\\\n   0 & 0 & 0 & 0 & 1 \\\\\n   1 & 0 & 1 & 0 & 1 \\\\\n   0 & 1 & 0 & 1 & 1 \\\\\n  \\end{array}\\right]$$\n}\n\nThe scalar linear index code $\\mathfrak{C}^{(2s)}$ for this SUICP(SNCS) is given below.\n\\begin{align*}\n\\mathfrak{C}^{(2s)}=\\mathfrak{C}=\\{&x_0+x_5,~~~x_1+x_6,~~~x_2+x_5,\\\\&x_3+x_6,~~~x_4+x_5+x_6\\}.\n\\end{align*}\n\\end{example}\n\\subsection{Optimal $\\frac{U+1}{D-U}$ dimensional VLICs for two-sided SUICP(SNCS)}\nIn the construction given in Theorem \\ref{vcthm1}, if $\\text{gcd}(K,D-U,U+1)=D-U,$ then we have $u_a=\\frac{U+1}{D-U}$, $K_a=\\frac{K}{D-U}$ and $\\Delta_a=1$. Hence, in this case, the AIR matrix used is always of dimension $K_a \\times (K_a-1)$ and the index code given by \\eqref{thm1eq5} is an optimal length $\\frac{U+1}{D-U}$ dimensional VLIC. In \\cite{TRCR}, it is shown that the message probability of error in decoding a message at a particular receiver decreases with a decrease in the number of transmissions used to decode the message among the total of broadcast transmissions. In this case, as the AIR matrix size is always of the form $K_a \\times (K_a-1)$, the low complexity decoding given in \\cite{VaR3} enables each receiver to decode its wanted message symbol by using atmost two broadcast symbols. \n\n\n\\begin{example}\n\\label{vcex10}\nConsider the two-sided SUICP(SNCS) with $K=12,D=5$ and $U=3$. For this SUICP(SNCS), we have $\\text{gcd}(K,D-U,U+1)=D-U=2$. The optimal length index code for this two-sided SUICP(SNCS) is a two dimensional VLIC. For this two-sided  SUICP(SNCS), we have $K_a=6,\\Delta_a=1$. An optimal length encoding matrix for this one-sided SUICP(SNCS) is a AIR matrix of size $6 \\times 5$. AIR matrix of size $6 \\times 5$ is given below.\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{6 \\times 5}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 \\\\\n   0 & 1 & 0 & 0 & 0 \\\\\n   0 & 0 & 1 & 0 & 0 \\\\\n   0 & 0 & 0 & 1 & 0 \\\\\n   0 & 0 & 0 & 0 & 1 \\\\\n   0 & 0 & 0 & 0 & 0 \\\\\n   1 & 1 & 1 & 1 & 1 \\\\\n  \\end{array}\\right]$$\n}\n\nThe scalar linear index code $\\mathfrak{C}$ for this one-sided SUICP(SNCS) is given below.\n\\begin{align*}\n\\mathfrak{C}=\\{c_0,c_1.c_2,c_3,c_4\\}=\\{&y_0+y_5,~~~y_1+y_5,~~~y_2+y_5,\\\\&y_3+y_5,~~~y_4+y_5\\}.\n\\end{align*}\n \nThe scalar linear index code $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{2s,1}+x_{2s-2,2}+x_{2s+1,1}+x_{2s-1,2}$ for $s \\in [0:5]$. \n\n\\end{example}\n\n\\begin{example}\n\\label{vcex11}\nConsider the two-sided SUICP(SNCS) with $K=27,D=8$ and $U=5$. For this SUICP(SNCS), we have $\\text{gcd}(K,D-U,U+1)=D-U=3$. The optimal length index code for this two-sided SUICP(SNCS) is a two dimensional VLIC. For this two-sided SUICP(SNCS), we have $K_a=9,\\Delta_a=1$. An optimal length encoding matrix for this one-sided SUICP(SNCS) is a AIR matrix of size $9 \\times 8$. AIR matrix of size $9 \\times 8$ is given below.\n\\arraycolsep=1pt\n\\setlength\\extrarowheight{-2.0pt}\n{\n$$\\mathbf{L}_{9 \\times 8}=\\left[\\begin{array}{*{20}c}\n   1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n   0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n   1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\\\\n  \\end{array}\\right]$$\n}\n\nThe scalar linear index code $\\mathfrak{C}$ for this one-sided SUICP(SNCS) is given below.\n\\begin{align*}\n\\mathfrak{C}=\\{&y_0+y_8,~~~y_1+y_8,~~~y_2+y_8,~~~y_3+y_8,\\\\&y_4+y_8,~~~y_5+y_8,~~~y_6+y_8,~~~y_7+y_8\\}.\n\\end{align*}\n \nThe two dimensional VLIC $\\mathfrak{C}^{(2s)}$ for the given two-sided SUICP(SNCS) is obtained by replacing $y_s$ in $\\mathfrak{C}$ with $x_{3s,1}+x_{3s-3,2}+x_{3s+1,1}+x_{3s-2,2}+x_{3s+2,1}+x_{3s-1,2}$ for $s \\in [0:8]$. \n\n\\end{example}\n\\section*{Acknowledgment}\nThis work was supported partly by the Science and Engineering Research Board (SERB) of Department of Science and Technology (DST), Government of India, through J.C. Bose National Fellowship to B. Sundar Rajan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nCore collapse supernovae (SN) are powerful astrophysical events that involve the interplay of various physical fields. Next to light and gravitational waves signals, a galactic SN would also produce a prominent neutrino signal leading to the possibility of studying the influence of fundamental physics on astrophysical-scale events.\nNeutrinos are mainly produced by charged current $\\beta$-processes and pair production during the first ten seconds after the onset of the collapse and have energies of order tens of MeV. They carry away $99\\%$  of the gravitational binding energy released in a SN and are expected to play a major role in the dynamics of the SN explosion by revitalising the stalled shock wave and driving nucleosynthesis of heavy elements.\n\nBoth reviving the shock wave and the nucleosynthesis depend on the flavor composition of the neutrinos which is affected by vacuum oscillation and the Mikheyev-Smirnov-Wolfenstein (MSW) matter effect~\\cite{Wolfenstein:1977ue,Mikheev:1986gs,Mikheev:1986wj}. Due to the high neutrino densities of $10^{30-35}$ per $\\textrm{cm}^3$ close to the neutrino sphere (radius $\\sim 50\\,\\textrm{km}$) during the accretion phase, neutrino-neutrino interactions must also be taken into account. However, this makes the neutrino flavour evolution non-linear and results in what is called collective oscillations, where neutrinos of different energies all convert from one flavour to another at the same time. Since collective oscillations significantly affect the flavor composition, they can also have an important impact on the shock wave revival as well as on nucleosynthesis.\n\nThe physics of dense neutrino media are also relevant in the environment of neutron star mergers and in early Universe cosmology. The importance of neutrino-neutrino interactions was first realised by Pantaleone \\cite{Pantaleone:1992eq}. The corresponding quantum kinetic equations were derived by Sigl and Raffelt \\cite{Sigl:1992fn}, and these non-linear equations have since been intensively studied and shown to give rise to collective oscillations such as synchronisation \\cite{Pastor:2001iu} and various forms of bipolar oscillations \\cite{Duan:2005cp,Duan:2006an,Hannestad:2006nj}. Also fast flavor conversions have been found for which the typical conversion scale is proportional to the neutrino density and is not directly related to the scale of the neutrino mass difference \\cite{Sawyer:2005jk,Sawyer:2008zs,Sawyer:2015dsa,Chakraborty:2016lct,Dasgupta:2016dbv,Capozzi:2017gqd,Izaguirre:2016gsx,Abbar:2017pkh}. For recent reviews of collective oscillations and supernova neutrino physics, see e.g.~\\cite{Mirizzi:2015eza,Chakraborty:2016yeg}.\n\nA powerful method to identify models where collective oscillations are present is to use a linearised stability analysis~\\cite{Banerjee:2011fj}. If bipolar oscillations or fast flavour conversions are present, they are seen as unstable modes that grow exponentially. With this method, several different types of collective instabilities have been identified in two-neutrino models. For a homogeneous and isotropic neutrino gas, the simple bipolar oscillations are found for inverted mass ordering (IO), while no instability is found for normal mass ordering (NO)~\\cite{Duan:2005cp,Duan:2006an,Hannestad:2006nj}. When inhomogeneities and anisotropy are allowed, other instabilities are identified such as the multi-zenith angle (MZA) instability, the multi-azimuthal angle (MAA) instability~\\cite{Raffelt:2013rqa,Raffelt:2013isa} and also the fast flavour conversion~\\cite{Sawyer:2005jk,Sawyer:2008zs,Sawyer:2015dsa,Chakraborty:2016lct,Dasgupta:2016dbv,Capozzi:2017gqd,Izaguirre:2016gsx,Abbar:2017pkh}. The MZA and MAA are only found for NO, while fast flavour conversions are largely independent on mixing parameters and are present for both mass orderings.\nThree-neutrino effects have been included in a number of numerical studies~\\cite{Duan:2007sh,Balantekin:2007es,EstebanPretel:2007yq,Gava:2008rp, Fogli:2008fj, Friedland:2010sc,Cherry:2010yc,Mirizzi:2010uz}. Although most analytical work has been done in the limit of two flavors in which muon and tau neutrinos are combined to $\\nu_x$, three flavor effects have also been considered in several papers~\\cite{Duan:2008za,Dasgupta:2007ws, Kneller:2009vd, Dasgupta:2010ae, Airen:2018nvp}. \nMost recently, Airen et al.~\\cite{Airen:2018nvp} did an extensive normal-mode analysis including a discussion of three-neutrino effects. Noting that the neutrino is almost in a flavour eigenstate when the electron background density is large, they neglected the off-diagonal part of the vacuum-contribution to the Hamiltonian describing the flavour evolution as it is usually done (see e.g.~\\cite{Duan:2006an,Duan:2005cp,Hannestad:2006nj, Banerjee:2011fj}). This allowed them to simplify the problem to two-neutrino oscillations and collect the solutions previously described in the literature in one unified framework. In the description of three-neutrino effects, they notice terms coupling the different off-diagonal parts of the density matrix through the vacuum term, but do not pursue them further.\n\nIn our study, we include the full expression of the vacuum term in a two- and four-beam model, and numerically identify the flavour instabilities including one that has evaded attention until now. \nIn \\sref{sec:eom} we introduce the formalism that is used to describe flavor conversion in dense neutrino environments. Then, in \\sref{sec:asol}, we construct an approximated solution to the problem. We apply this solution in \\sref{sec:models} to a system of a neutrino beam colliding with an antineutrino beam and extend this model to a system of four neutrino and antineutrino beams with an nonvanishing intersection angle. The numerical results are interpreted in terms of two-neutrino instabilities in \\sref{sec:PropStates} before we conclude in \\sref{sec:conclusion}.\n\n\n\\section{Equation of Motion}\n\\label{sec:eom}\n\\subsection{Basic description}\nOur goal is to describe the evolution of the flavor composition of a neutrino ensemble in a supernova. For a homogeneous ensemble of collisonless neutrinos with momentum $\\textbf{p}$ and flavor $\\alpha\\in\\{e,\\mu,\\tau\\}$, the flavor composition is characterized by the expectation value of bilinear combinations of neutrino and antineutrino annihilation $a_{\\alpha}^{\\dagger}(\\textbf{p}),b_{\\alpha}^{\\dagger}(\\textbf{p})$ and creation operators $a_{\\alpha}(\\textbf{p}),b_{\\alpha}(\\textbf{p})$ \\cite{Notzold:1987ik,Sigl:1992fn}. This in turn translates into a mean field density matrix formulation of the system due to the relations\n\\begin{gather}\n \\langle a_{\\beta}^{\\dagger}(t,\\textbf{p})a_{\\alpha}(t,\\textbf{p}^{\\prime})\\rangle=(2\\pi)^3\\,\\delta^{(3)}(\\textbf{p}-\\textbf{p}^{\\prime})(\\rho(t,\\textbf{p}))_{\\alpha\\beta},\\\\[0.3cm]\n  \\langle b_{\\alpha}^{\\dagger}(t,\\textbf{p})b_{\\beta}(t,\\textbf{p}^{\\prime})\\rangle=(2\\pi)^3\\,\\delta^{(3)}(\\textbf{p}-\\textbf{p}^{\\prime})(\\overline{\\rho}(t,\\textbf{p}))_{\\alpha\\beta}.\n\\end{gather}\nThe time evolution of the flavor composition of a neutrino ensemble can thus be described by the evolution of the mean field density matrix in three dimensional flavor space\n\\begin{align}\n \\rho(t,\\textbf{p}):=\\begin{pmatrix}\n                                   \\rho_{ee}(t,\\textbf{p}) & \\rho_{e\\mu}(t,\\textbf{p}) & \\rho_{e\\tau}(t,\\textbf{p})\\\\\n                                   \\rho_{\\mu e}(t,\\textbf{p}) & \\rho_{\\mu\\mu}(t,\\textbf{p}) & \\rho_{\\mu\\tau}(t,\\textbf{p})\\\\\n                                   \\rho_{\\tau e}(t,\\textbf{p}) &\\rho_{\\tau\\mu}(t,\\textbf{p})&\\rho_{\\tau\\tau}(t,\\textbf{p})\n                                  \\end{pmatrix},\n\\end{align}\nwhere the diagonal entries encode the flavor occupation numbers and the off-diagonal elements represent flavor correlations. Furthermore we treat neutrinos as ultra-relativistic which implies that their four velocity is $v^{\\mu}=(1,\\textbf{v})$ and hence  $|\\textbf{v}|=1$. For the sake of a more compact notation, we denote antineutrino states by negative phase-space densities  $-\\overline{\\rho}(E)=\\rho(-E)$ with $E=|\\textbf{p}|$.  The direction of motion, the energy, and the $\\nu\/\\overline{\\nu}$-nature are summarized in a mode index $i$. In a physical situation many different modes may exist and it is the interplay among those modes that is the root of collective oscillation effects.\nOur interest is to investigate the time-dependent behavior of the flavor composition of the neutrino ensemble and hence of the density matrix. The density matrix evolves in flavor space for each mode according to Von Neumann's equation\n\\begin{gather}\n i\\,\\partial_t\\rho_i(t,\\textbf{v})=[H_i(t,\\textbf{v}),\\rho_i(t,\\textbf{v})],\n \\label{eq:VonNeumann}\n\\end{gather}\nwhen collisions are neglected and applies to pure as well as to mixed states \\cite{Stirner:2018ojk}. \nThe Hamiltonian $H_i(t,\\textbf{v})$ on the right hand side decomposes into three components that trigger neutrino oscillation\n\\begin{gather}\n H_i(t,\\textbf{v})=H_{i}^{\\text{vac}}+H^{\\lambda}+H_i^{\\nu\\nu}(\\textbf{v}).\n \\label{eq:Hamiltonian}\n\\end{gather}\nApart from terms proportional to the unit matrix, the vacuum term for a neutrino mode of energy $E_i$ has in flavor basis the form\n\\begin{gather}\n H_{i}^{\\text{vac}}:=\\frac{U M^2U^{\\dagger}}{2E_i},\n\\end{gather}\nwhere $M^2:=\\text{diag}(0,\\Delta m_{21}^2,\\Delta m_{31}^2)$ is the matrix of neutrino square mass differences and $U=R_{23}R_{13}R_{12}$ is, up to a CP-violating phase, the neutrino mixing matrix with the rotation matrices $R_{kl}$ between the $k$th and $l$th mass eigenstate $\\Delta m_{kl}:=m_k^2-m_l^2,\\,\\, k,l\\in\\{1,2,3\\}$ with $m_k$ being the mass of neutrino mass eigenstate $\\nu_k$. The rotation matrices are\n\\begin{align}\nR_{12}:=\\begin{pmatrix}\n            c_{12} &s_{12} &0\\\\\n            -s_{12} &c_{12}&0\\\\\n            0& 0& 1\n          \\end{pmatrix},\\,\nR_{13}:=\\begin{pmatrix}\n           c_{13} & 0 & s_{13}\\\\\n           0 & 1 & 0\\\\\n           -s_{13}&0&c_{13}\n          \\end{pmatrix},\\,\nR_{23}:=\\begin{pmatrix}\n          1 & 0 & 0\\\\\n          0 &c_{23} &s_{23}\\\\\n          0 & -s_{23} &c_{23}\n          \\end{pmatrix}.\n\\end{align}\nWe use here and in the following the shorthand $c_{kl}:=\\cos(\\theta_{kl})$ with the mixing angles $\\theta_{12}, \\theta_{23}$ and $\\theta_{13}$. \nNote, that a positive sign of $\\Delta m_{31}^2$  refers to NO while a negative sign refers to IO.\n\n\nThe second term in \\eref{eq:Hamiltonian} describes matter effects which includes the MSW-effect and is proportional to the Fermi coupling constant $G_F$. In the weak eigenstate basis, the interaction of neutrinos with charged leptons of net density $N_{\\alpha}$, $\\alpha\\in\\{e,\\mu,\\tau\\}$ reads\n\\begin{gather}\n H^{\\lambda}:=\\sqrt{2}G_F\\,\\text{diag}(N_e,N_{\\mu},N_{\\tau}).\n\\end{gather}\n\nThe interaction among neutrino modes requires high neutrino densities and also affects the flavor evolution of the density matrix. The description of  interacting neutrino modes is encoded in the last term of \\eref{eq:Hamiltonian} which for a discrete set of $N$ modes with velocities $\\textbf{v}_j$, $j\\in\\{1,...,N\\}$, is given as\n\\begin{gather}\n H_i^{\\nu\\nu} := \\sqrt{2}G_F n_{\\nu}\\sum_{j=1}^N(1-\\textbf{v}_i\\textbf{v}_j)\\,\\rho_j.\n\\end{gather}\nNote that the Hamiltonian depends on the relative angle of the propagation directions of the modes and is proportional to the effective neutrino density $n_{\\nu}:=\\frac{1}{2}(n_{\\nu_e}-n_{\\bar{\\nu}_e}+n_{\\nu_x}-n_{\\bar{\\nu}_x}+n_{\\nu_y}-n_{\\bar{\\nu}_y})$. \n\n\\subsection{Two-Neutrino Limit}\nIn the context of supernovae, it is expected that $\\nu_{\\mu}$ and $\\nu_{\\tau}$ have almost identical number density spectra which allows for a basis transformation in three dimensional flavor space spanned by $\\{|\\nu_e\\rangle,|\\nu_{\\mu}\\rangle,|\\nu_{\\tau}\\rangle\\}$ such that $\\nu_{\\mu}$ and $\\nu_{\\tau}$ unmix \\cite{Dasgupta:2007ws,Dasgupta:2010ae} \n\\begin{gather}\n \\begin{pmatrix}\n  \\nu_e\\\\\n  \\nu_x\\\\\n  \\nu_y\n \\end{pmatrix}:=\n R_{23}^{\\dagger}\n \\begin{pmatrix}\n  \\nu_e\\\\\n  \\nu_{\\mu}\\\\\n  \\nu_{\\tau}\n \\end{pmatrix}.\n\\end{gather}\nIn most works, collective neutrino oscillations are studied in the two-neutrino subspace spanned by $\\{|\\nu_e\\rangle,|\\nu_x\\rangle\\}$. This requires that $|\\nu_y\\rangle$ is proportional to the third mass eigenstate $|m_3\\rangle$ only which can be achieved exactly for $\\theta_{13}=0$ \\cite{Balantekin:1999dx}. Since $\\theta_{13}\\approx 8^{\\circ}$, the two-neutrino limit is only approximatively achievable.\n\nIn this work, we will assume that $\\nu_{\\mu}$ and $\\nu_{\\tau}$  have identical distributions and hence work in the new basis spanned by $|\\nu_e\\rangle, |\\nu_x\\rangle$ and $|\\nu_y\\rangle$ in order to simplify the equations. For \\eref{eq:VonNeumann} this statement is equivalent to set $\\theta_{23}=0$.\n\n\n\\section{Linearized equations}\n\\label{sec:asol}\n\n\\subsection{Three neutrino formalism}\nSince the Von Neumann equation conserves the trace of the density matrix, we subtract the trace and split the entries into the neutrino density of the $i$th mode $g_i$ and parametrize the remaining fractions  as\n\\begin{align}\n \\rho_i(t,\\textbf{v})-\\frac{1}{3}\\text{Tr}\\rho_i\\mathbbm{1}=:\\frac{g_i}{3}\\begin{pmatrix} \\mathfrak{s}_{1}^i(t,\\textbf{v}) & \\mathcal{S}_{ex}^i(t,\\textbf{v}) & \\mathcal{S}_{ey}^i(t,\\textbf{v})\\\\ \\mathcal{S}_{ex}^{\\ast i}(t,\\textbf{v}) &  \\mathfrak{s}_{2}^i(t,\\textbf{v}) & \\mathcal{S}_{xy}^i(t,\\textbf{v}) \\\\ \\mathcal{S}_{ey}^{\\ast i}(t,\\textbf{v}) & \\mathcal{S}_{xy}^{\\ast i}(t,\\textbf{v}) & - \\mathfrak{s}_{1}^i(t,\\textbf{v})- \\mathfrak{s}_{2}^i(t,\\textbf{v}) \\end{pmatrix}.                                                                                       \n\\end{align}\nThe neutrino density $g_i$ is assumed to be independent of time and space and has to fulfill the normalisation condition $\\sum_{i=1}^N|g_i|=2$.  Note that $\\mathfrak{s}_1^i$ and $\\mathfrak{s}_2^j$ have to be real for all modes and that in general $\\mathcal{S}_{\\alpha\\beta}^i\\in\\mathbb{C}$.  \nIn this paper, our focus is on flavor conversion, and hence we are interested in the off-diagonal entries of the density matrix. The differential equation we aim to solve is thus\n\\begin{align}\n\\renewcommand*{\\arraystretch}{1.5}\ni\\partial_t\\: \\frac{g_i}{3}\n \\begin{pmatrix}\n  \\mathcal{S}_{ex}^i(t,\\textbf{v})\\\\\n  \\mathcal{S}_{ey}^i(t,\\textbf{v})\\\\\n  \\mathcal{S}_{xy}^i(t,\\textbf{v})\n \\end{pmatrix}\n=\n\\underbrace{\n\\begin{pmatrix}\n \\left[H_{E_i}^{\\text{vac}},\\rho_i\\right]_{ex}\\\\\n  \\left[H_{E_i}^{\\text{vac}},\\rho_i\\right]_{ey}\\\\\n \\left[H_{E_i}^{\\text{vac}},\\rho_i\\right]_{xy}\n\\end{pmatrix}}_{\\textrm{(I)}}\n+\n\\underbrace{\n\\begin{pmatrix}\n \\left[H^{\\lambda},\\rho_i\\right]_{ex}\\\\\n  \\left[H^{\\lambda},\\rho_i\\right]_{ey}\\\\\n  \\left[H^{\\lambda},\\rho_i\\right]_{xy}\n\\end{pmatrix}}_{\\textrm{(II)}}\n+\n\\underbrace{\n\\begin{pmatrix}\n \\left[H^{\\nu\\nu}(\\textbf{v}),\\rho_i\\right]_{ex}\\\\\n  \\left[H^{\\nu\\nu}(\\textbf{v}),\\rho_i\\right]_{ey}\\\\\n  \\left[H^{\\nu\\nu}(\\textbf{v}),\\rho_i\\right]_{xy}\n\\end{pmatrix}}_{\\textrm{(III)}},\n\\label{eq:Goal}\n\\end{align}\nwith the initial conditions $\\mathfrak{s}^i_1(0,\\textbf{v})=2$, $\\mathfrak{s}^i_2(0,\\textbf{v})=-1$ and $\\mathcal{S}^i_{\\alpha\\beta}(0,\\textbf{v})=0$, assuming that all neutrinos are produced as electron neutrinos and antineutrinos.\n\\paragraph{Vacuum-Term (I):}\nTo obtain the first term of \\eref{eq:Goal}, we calculate the transformed mass matrix and the commutator with the density matrix of mode $i$. It is useful to introduce the oscillation frequency $\\omega_i:=\\Delta m_{21}^2\/(2E_i)$ and the mass-squared difference ratio $\\eta:=\\Delta m_{31}^2\/\\Delta m_{21}^2$ and order the terms with respect to the latter one. Solving the commutator yields\n\\begin{align}\n\\renewcommand*{\\arraystretch}{1.5}\n\\frac{3}{g_i\\omega_i}\n\\textbf{\\textrm{(I)}}\n=\\left[\n \\underline{\\underline{A_0}}\n+\n \\underline{\\underline{B_0}}\\eta\\right]\n  \\begin{pmatrix}\n  \\mathcal{S}_{ex}^i\\\\\n  \\mathcal{S}_{ey}^i\\\\\n  \\mathcal{S}_{xy}^i\n \\end{pmatrix}\n+\\left[\\underline{\\underline{A_1}}+\\underline{\\underline{B_1}}\\eta\\right]\n\\begin{pmatrix}\n \\mathfrak{s}_{1}\\\\\n \\mathfrak{s}_{2}\\\\\n -(\\mathfrak{s}_1+\\mathfrak{s}_2)\n\\end{pmatrix}+\\left[\\underline{\\underline{A_2}}+\\underline{\\underline{B_2}}\\eta\\right]\n\\begin{pmatrix}\n   \\mathcal{S}_{ex}^{\\ast i}\\\\\n  \\mathcal{S}_{ey}^{\\ast i}\\\\\n  \\mathcal{S}_{xy}^{\\ast i}.\n\\end{pmatrix}.\n\\label{eq:VacTermDec}\n\\end{align}\nThe matrices $\\underline{\\underline{A_n}}$ and $\\underline{\\underline{B_n}}$ contain combinations of the mixing angles and read\n\\begin{gather}\n \\underline{\\underline{A_0}}:=\n \\begin{pmatrix}\n  -c_{12}^2+c_{13}^2s_{12}^2 & \\frac{1}{2}S_{12}s_{13} &0\\\\\n  \\frac{1}{2}S_{12}s_{13} & C_{13} s_{12}^2 & \\frac{1}{2}S_{12} c_{13} \\\\\n0 & \\frac{1}{2}S_{12} c_{13} & c_{12}^2-s_{12}^2s_{13}^2\n \\end{pmatrix},\n\\quad\n \\underline{\\underline{B_0}}:=\n \\begin{pmatrix}\n  s_{13}^2 & 0 & 0\\\\\n  0 & -C_{13} & 0\\\\\n 0 & 0 & -c_{13}^2\n \\end{pmatrix},\\nonumber\\\\[0.5cm]\n \\underline{\\underline{A_1}}:=\\begin{pmatrix}\n  -\\frac{1}{2}S_{12}c_{13} & \\frac{1}{2}S_{12}c_{13} & 0\\\\\n  S_{13}s_{12}^2 & \\frac{1}{2}S_{13}s_{12}^2 & 0\\\\\n  \\frac{1}{2}S_{12}s_{13} & S_{12}s_{13}& 0\n \\end{pmatrix}\n,\\quad\n \\underline{\\underline{B_1}}:=\\begin{pmatrix}\n       0 & 0 &0 \\\\\n       -S_{13} & -\\frac{1}{2}S_{13} & 0 \\\\\n       0 & 0 & 0\n      \\end{pmatrix},\\\\[0.5cm]\n \\underline{\\underline{A_2}}:=\\begin{pmatrix}\n       0 & 0 & -\\frac{1}{2}S_{13}s_{12}^2\\\\\n       0 & 0 & 0 \\\\\n       \\frac{1}{2}S_{13}s_{12}^2 & 0 & 0\n      \\end{pmatrix}\n,\\quad\n \\underline{\\underline{B_2}}:=\\begin{pmatrix}\n       0 & 0 & \\frac{1}{2}S_{13}\\\\\n       0 & 0& 0\\\\\n       -\\frac{1}{2}S_{13} & 0 & 0\n      \\end{pmatrix},\\nonumber\n\\end{gather}\nwhere we abbreviate $c_{kl}:=\\cos(\\theta_{kl})$, $s_{kl}:=\\sin(\\theta_{kl})$ for mixing angle $\\theta_{kl}$ and refer by capital $S_{kl}$ to $\\sin(2\\theta_{kl})$. Here, the introduced double underline notation refers to matrices that act on vectors spanned by the off-diagonal density matrix elements. We will use this notation throughout the paper from now on.\n\n\\paragraph{Matter-Term (II):}\nThe matter-term describes the interaction between the neutrino ensemble and the surrounding matter. These interactions are dominated by charged current processes of electrons and electron-neutrinos while the the densities of muons and tau leptons are very small. By defining the effective MSW-potential strength  $\\lambda:=\\sqrt{2}G_F N_e$, the Hamiltonian for the matter effect becomes\n\\begin{align}\nH^{\\lambda}=\n \\begin{pmatrix}\n  \\lambda & 0 & 0\\\\\n  0 &0 & 0\\\\\n  0 & 0 &0\n \\end{pmatrix}.\n\\end{align}\nCalculating the commutator with the density matrix and reading off the upper off-diagonal components we find \n\\begin{align}\n\\label{eq:MatterTerm}\n\\frac{3}{g_i}\\textbf{\\textrm{(II)}}=\n \\begin{pmatrix}\n  \\lambda & 0 & 0\\\\\n  0 & \\lambda & 0 \\\\\n  0 & 0 & 0\n \\end{pmatrix}\n  \\begin{pmatrix}\n  \\mathcal{S}_{ex}^i(t,\\textbf{v})\\\\\n  \\mathcal{S}_{ey}^i(t,\\textbf{v})\\\\\n  \\mathcal{S}_{xy}^i(t,\\textbf{v})\n \\end{pmatrix}.\n\\end{align}\n\n\\paragraph{Neutral-Current-Term (III):} \nThe neutrino-neutrino interaction term includes the commutator of two density matrices of different modes which leads to higher order terms in the flavor cross correlation terms. We omit terms of higher than linear order in the flavor cross correlations $|\\mathcal{S}_i|^2\\ll 1$ which  implies that the flavor densities will not depart a lot from the initial value $\\mathfrak{s}(t,\\textbf{v})\\approx \\mathfrak{s}(0,\\textbf{v}) $ $\\forall t$ . We find  \n\\begin{align}\n \\left[\\rho_j,\\rho_i\\right]\\approx\\frac{g_ig_j}{9}\n \\begin{pmatrix}\n  0 & \\mathfrak{s}_{1}^{[j,}\\mathcal{S}_{ex}^{i]}+\\mathcal{S}_{ex}^{[j,}\\mathfrak{s}_{2}^{i]} & \\mathfrak{s}_{1}^{[j,}\\mathcal{S}_{ey}^{i]}+\\mathcal{S}_{ey}^{[i,}(\\mathfrak{s}_{1}+\\mathfrak{s}_{2})^{j]}  \\\\\n  \\mathcal{S}_{ex}^{\\ast[j,}\\mathfrak{s}_{1}^{i]}+\\mathfrak{s}_{2}^{[j,}(\\mathcal{S}_{ex}^{\\ast})^{i]} & 0 & \\mathfrak{s}_{2}^{[j,}\\mathcal{S}_{xy}^{i]}+\\mathcal{S}_{xy}^{[i,}(\\mathfrak{s}_1+\\mathfrak{s}_2)^{j]}\\\\\n  \\mathfrak{s}_1^{[j,}(\\mathcal{S}_{ey}^{\\ast})^{i]}+\\mathcal{S}_{ey}^{\\ast[j,}(\\mathfrak{s}_1+\\mathfrak{s}_2)^{i]} & \\mathcal{S}_{xy}^{\\ast[j,}\\mathfrak{s}_2^{i]}+\\mathcal{S}_{xy}^{\\ast[j,}(\\mathfrak{s}_1+\\mathfrak{s}_2)^{i]} & 0\n \\end{pmatrix},\n\\end{align}\nwhere we have introduced the following notation\n\\begin{align}\n \\mathcal{S}_a^{[i,}\\mathfrak{s}_b^{j]}:=\\mathcal{S}_a^i\\mathfrak{s}_b^j-\\mathcal{S}_b^j\\mathfrak{s}_a^i.\n\\end{align}\n\nWe conclude that the contributions to the equations for the off-diagonal matrix elements of the density matrix are\n\\begin{align}\n\\frac{3}{g_i}\\textbf{\\textrm{(III)}}=\n \\mu\\sum_{j=1}^N\\frac{g_j}{3}(1-\\textbf{v}^i\\textbf{v}^j)\n            \\begin{pmatrix}\n            \\mathfrak{s}_{1}^{[j,}\\mathcal{S}_{ex}^{i]}+\\mathcal{S}_{ex}^{[j,}\\mathfrak{s}_{2}^{i]}\\\\\n            \\mathfrak{s}_{1}^{[j,}\\mathcal{S}_{ey}^{i]}+\\mathcal{S}_{ey}^{[i,}(\\mathfrak{s}_{1}+\\mathfrak{s}_{2})^{j]} \\\\\n              \\mathfrak{s}_{2}^{[j,}\\mathcal{S}_{xy}^{i]}+\\mathcal{S}_{xy}^{[i,}(\\mathfrak{s}_1+\\mathfrak{s}_2)^{j]}\n            \\end{pmatrix}.\\centering\n\\label{eq:long}\n\\end{align}\n\n\\subsection{Eigenvalue equation}\nHaving introduced the theoretical description of flavor conversion in a dense environment and derived a matrix equation for the flavor correlations, we present an approximated solution for this equation in this section. \n\nThe flavor correlations are often treated as plane waves $\\mathcal{S}_i(t,\\textbf{v})=Q_i(\\textbf{v})e^{-i\\Omega t}$. Flavor conversion occurs if the frequency $\\Omega$ has an imaginary part that leads to an exponentially growing flavor correlation function. Our main focus shall therefore be to find those instabilities. A consequence of this behavior is that we can neglect the constant term of \\eref{eq:VacTermDec}.\n\nDue to the environmental conditions in a SN, we assume from now on a \\textit{large matter potential}. This leads to a suppression of off-diagonal terms in \\eref{eq:VacTermDec}, and we hence neglect the third term of the equation.\\footnote{We have also included the third term of \\eref{eq:VacTermDec} under the assumption that $S$ is purely imaginary or real without changing the results.} We will justify this more carefully in \\sref{sec:PropStates}.\n\n\nWith these simplifications and the plane wave ansatz, \\eref{eq:Goal} reduces to an eigenvalue equation for the frequency $\\Omega$\n\\begin{equation}\n\\begin{aligned}\n \\Omega \n \\begin{pmatrix}\n   Q_{ex}^i\\\\\n   Q_{ey}^i\\\\\n   Q_{xy}^i\n \\end{pmatrix}\n =&\\left[\n \\underline{\\underline{A_0}}\\omega_i+\\underline{\\underline{B_0}}\\eta\\omega_i+\n   \\lambda\\begin{pmatrix}\n  1 & 0 & 0\\\\\n  0 & 1 & 0 \\\\\n  0 & 0 & 0\n \\end{pmatrix}\\right]\n  \\begin{pmatrix}\n   Q_{ex}^i\\\\\n   Q_{ey}^i\\\\\n   Q_{xy}^i\n \\end{pmatrix}\\\\\n &+  \\mu\\sum_{j=1}^Ng_j(1-\\textbf{v}^i\\textbf{v}^j)\n            \\begin{pmatrix}\n             (Q_{ex}^i-Q_{ex}^j)\\\\\n             (Q_{ey}^i-Q_{ey}^j)\\\\\n             0\n            \\end{pmatrix}.\\centering\n\\end{aligned}\n\\end{equation}\nThe density of the $i$th mode $g_i\/3$ appears in every term and hence cancels out. The normalization factor of $1\/3$ of $g_j$ cancels a factor of $3$ arising from the initial conditions of $\\mathfrak{s}_{1,2}^{i,j}$ which we assumed to approximately constant in time. By solving this equation, it can be determined if $\\Omega$ has an imaginary part and linearly unstable solution thus exist.\n\n\\section{Toy systems}\n\\label{sec:models}\nIn order to investigate and explore the developed model, we consider the case of two colliding and four intersecting neutrino beams. These pictures were already used in the literature \\cite{Chakraborty:2016lct,Raffelt:2013isa,Mangano:2014zda,Hansen:2014paa,Duan:2014gfa,Mirizzi:2015fva,Chakraborty:2015tfa,Abbar:2015mca} and allow us to compare our results with the two-neutrino limit. For the mixing angles we take the following values  \\cite{Esteban:2016qun}\n\\begin{align}\n \\theta_{12}= 33.62^{\\circ},\\qquad\n \\theta_{23}= 47.2^{\\circ},\\qquad\n \\theta_{13}= 8.54^{\\circ}.\n\\end{align}\nWe checked that our results are insensitive to changes within the uncertainties.\n\n\\subsection{Two-Beam-System}\nThe simplest model to study is a system of two different neutrino modes that propagate in one dimension and can be interpreted as two colliding beams. Let us assume a left moving antineutrino beam and a right moving neutrino beam denoted by\n\\begin{align}\\underline{Q}^{\\bar{l}}=\n \\begin{pmatrix}\n    \\bar{L}_{ex}\\\\\n  \\bar{L}_{ey}\\\\\n  \\bar{L}_{xy}\n \\end{pmatrix}\n\\quad\\textrm{and}\\quad\n\\underline{Q}^{r}=\n\\begin{pmatrix}\n  R_{ex}\\\\\n R_{ey}\\\\\n R_{xy}\n \\end{pmatrix}.\n\\end{align}\nWe collect them in a six dimensional vector $\\underline{Q}:=(\\underline{Q}^{r},\\underline{Q}^{\\bar{l}})^T$.  \nParameters referring to the left moving antineutrino beam pick up an index $\\bar{l}$ while parameters belonging to the right moving beam are labeled by an index $r$.\nFor the antineutrino beam, we have $\\omega\\to-|\\omega|$ due to our convention, and the neutral-current term $(1-\\textbf{v}_i\\textbf{v}_j)$ becomes $0$ for $i=j$ (parallel moving modes) and $2$ for $i\\neq j$ (oppositely moving modes). This leads to a $6\\times6$ dimensional matrix which we write as a matrix of $3\\times3$ block matrices.  For equal numbers of neutrinos and antineutrinos $-g_{\\bar{l}}=g_r=1$, the equation reduces to\n\\begin{align}\n \\Omega\\, \\underline{Q}=\\left[\n \\begin{pmatrix}\n |\\omega|\\underline{\\underline{A_0}}+|\\omega|\\eta\\underline{\\underline{B_0}}+(\\lambda-2\\mu)\\underline{\\underline{\\Lambda}} & +2\\mu\\underline{\\underline{\\Lambda}}\\\\\n -2\\mu\\underline{\\underline{\\Lambda}} & -(|\\omega|\\underline{\\underline{A_0}}+|\\omega|\\eta\\underline{\\underline{B_0}}-(\\lambda+2\\mu)\\underline{\\underline{\\Lambda}})\n\\end{pmatrix}\\right]\n\\underline{Q},\n\\label{eq:2BE}\n\\end{align}\nwhere $\\underline{\\underline{\\Lambda}}$ is defined as\n\\begin{align}\n\\underline{\\underline{\\Lambda}}:=\n\\begin{pmatrix}\n 1 & 0 & 0\\\\\n 0 & 1 &0 \\\\\n 0& 0 & 0\n\\end{pmatrix}.\n\\end{align}\nSince we have a $6\\times 6$ matrix, we cannot solve the equation analytically, but have to resolve to a numerical solution.\n\n\\subsubsection{Results}\n\n\\begin{figure}[tbp]\n \\centering\n  \\includegraphics[width=1\\textwidth]{2Beam_ex_IO_L100.pdf}\n  \\caption{Imaginary parts of all eigenvalues for the two beam model as a function of the neutrino potential in the two-neutrino limit with IO (green and orange lines) compared with \\eref{eq:2BE} (blue dashed line). }\n  \\label{fig:2NLimitIH}\n\\end{figure}\nBefore exploring new features of the matrix in \\eref{eq:2BE}, we convince ourselves that in the two-neutrino limit ($\\theta_{13}\\approx 0^{\\circ}$), we are able to reproduce the result found in \\cite{Chakraborty:2016lct} \n\\begin{align}\n \\Omega_{2\\nu}=\\sqrt{\\frac{\\Delta m_{31}^2}{2E_i}\\cdot\\left(\\frac{\\Delta m_{31}^2}{2E_i}+4\\mu\\right)}.\n \\label{eq:2NLR}\n\\end{align}\nIn \\fref{fig:2NLimitIH} we show for IO the imaginary parts of the eigenvalues of \\eref{eq:2BE} in the two-neutrino limit ($\\theta_{13}=0$) as orange lines as a function of $\\mu$.  The  two-neutrino result in \\eref{eq:2NLR} is shown by the blue dashed curve which agrees very well with our result. For NO all eigenvalues $\\Omega$ are real and thus no unstable modes exist.\n\nWe now return to the case of three neutrino flavors and calculate numerically the eigenvalues of the matrix \\eref{eq:2BE}. In the presence of a large matter potential of $\\lambda=100\\,\\textrm{km}^{-1}$, a mass square ratio $|\\eta|=33$, and an oscillation frequency $\\omega=0.015\\,\\textrm{km}^{-1}$, we find results identical to the two-neutrino limit for IO. For NO, however, unstable modes are present in the three neutrino case with $\\theta_{13}\\neq 0$. Comparing \\fref{fig:3NNO} and \\fref{fig:2NLimitIH} also shows, that the growth rate for the new instability in NO is smaller than for the instability in the case of IO. The instability for NO depends on the mixing angles $\\theta_{13}$, $\\theta_{23}$ as well as on the mass square ratio $\\eta$. Numerically, we also confirm that the rotation $R_{23}$ does not change the results as expected.\n\n\n\\begin{figure}[tbp]\n \\centering\n  \\includegraphics[width=1\\textwidth]{2Beam_exy_NO_L100.pdf}\n  \\caption{Imaginary parts of all eigenvalues for the two beam model as a function of the neutrino potential with the full three flavor equation for NO.}\n  \\label{fig:3NNO}\n\\end{figure}\n\n\\subsection{Four-Beam-System}\nOur next step is to generalize the two beam picture to a four beam configuration in which an additional free parameter, the intersection angle $\\alpha$, is introduced. We consider a configuration in which a right moving neutrino beam and a left moving antineutrino beam separated by the relative angle interact with a left moving neutrino beam and a left moving antineutrino beam as it is shown in figure \\fref{fig:4BSketch}. Our goal is to investigate whether the instability we have found in the two beam configuration for NO is also present in this model. \nIn the limit of zero relative angle and zero left moving antineutrino and right moving neutrino density, the four beam system coincides with the two beam system discussed before.\n\n\\begin{figure}[tbp]\n \\centering\n  \\includegraphics[width=0.4\\textwidth]{FourBeamscetch.pdf}\n  \\caption{Geometry of the four-beam model.}\n  \\label{fig:4BSketch}\n\\end{figure}\n\nThe generalization of \\eref{eq:2BE} to the case of four intersecting beams with relative angle $\\alpha$ is \n\n\\begin{align}\n \\Omega\\,\\underline{Q}=&\\left\\{|\\omega|\n \\begin{pmatrix}\n  \\underline{ \\underline{A}}+\\eta\\underline{\\underline{B}} & 0 & 0 & 0 \\\\\n  0 & -\\underline{\\underline{A}}-\\eta\\underline{\\underline{B}} & 0 & 0\\\\\n    0&0&\\underline{ \\underline{A}}+\\eta\\underline{\\underline{B}} & 0 \\\\\n  0&0&0 & -\\underline{\\underline{A}}-\\eta\\underline{\\underline{B}}\n \\end{pmatrix}\\right.\n +\\lambda\n \\begin{pmatrix}\n  \\underline{\\underline{\\Lambda}} & 0 & 0 & 0\\\\\n  0 & \\underline{\\underline{\\Lambda}}& 0 & 0\\\\\n  0 & 0&\\underline{\\underline{\\Lambda}} & 0\\\\\n  0 & 0&0 & \\underline{\\underline{\\Lambda}}\n \\end{pmatrix}\\nonumber\\\\\n&\\quad+\\mu\n \\left[2\\begin{pmatrix}\n  g_{\\bar{l}}\\underline{\\underline{\\Lambda}} & 0 & 0 & -g_{\\bar{l}}\\underline{\\underline{\\Lambda}} \\\\\n    0 &   g_{l} \\underline{\\underline{\\Lambda}} & -g_l\\underline{\\underline{\\Lambda}} & 0\\\\\n     0 & -g_{\\bar{r}}\\underline{\\underline{\\Lambda}} & g_{\\bar{r}}\\underline{\\underline{\\Lambda}} &  0 \\\\\n    -g_r\\underline{\\underline{\\Lambda}} &0 & 0&   g_{r} \\underline{\\underline{\\Lambda}} \n \\end{pmatrix}\\right.\\\\\n&\\quad+(1-\\cos\\alpha)\n\\begin{pmatrix}\n  g_{\\bar{r}} \\underline{\\underline{\\Lambda}} & -g_{\\bar{r}}  \\underline{\\underline{\\Lambda}}& 0 & 0 \\\\\n   -g_{r}  \\underline{\\underline{\\Lambda}} &   g_{r}\\underline{\\underline{\\Lambda}}& 0 & 0\\\\\n   0 & 0 &g_{\\bar{l}} \\underline{\\underline{\\Lambda}} & -g_{\\bar{l}}  \\underline{\\underline{\\Lambda}} \\\\\n   0 & 0 &-g_{l}  \\underline{\\underline{\\Lambda}} &   g_{l}\\underline{\\underline{\\Lambda}}\n\\end{pmatrix}\\nonumber\\\\\n &\\quad+(1+\\cos\\alpha)\\left.\\left.\n \\begin{pmatrix}\n  g_l   \\underline{\\underline{\\Lambda}}& 0 &-g_l  \\underline{\\underline{\\Lambda}} & 0\\\\\n 0 &g_{\\bar{l}}  \\underline{\\underline{\\Lambda}} & 0 & -g_{\\bar{l}} \\underline{\\underline{\\Lambda}}\\\\\n  -g_r  \\underline{\\underline{\\Lambda}} & 0& g_r   \\underline{\\underline{\\Lambda}}& 0\\\\\n  0& -g_{\\bar{r}} \\underline{\\underline{\\Lambda}} & 0 &g_{\\bar{r}}  \\underline{\\underline{\\Lambda}}\n \\end{pmatrix}\\right]\\right\\}\\underline{Q},\\nonumber\n\\end{align}\nwhere $\\underline{Q}:=(\\underline{R},\\underline{\\bar{R}},\\underline{L},\\underline{\\bar{L}})^T$ is a twelve dimensional vector with $\\underline{R}:=(R_{ex},R_{ey},R_{xy})^T$ and similarly for the other amplitudes. The neutrino-neutrino term can be decomposed in a symmetric and antisymmetric term with respect to the propagation direction of the mode.\nWe perform a basis transformation $A_{\\pm}:=\\frac{1}{2}(L\\pm R)$ and $\\bar{A}_{\\pm}:=\\frac{1}{2}(\\bar{L}\\pm\\bar{R})$ which can be written in matrix form\n\\begin{align}\n \\begin{pmatrix}\n  \\underline{A}_+\\\\\n  \\underline{\\bar{A}}_+\\\\\n  \\underline{A}_-\\\\\n  \\underline{\\bar{A}}_-\n \\end{pmatrix}:=\n  \\frac{1}{2}\n \\begin{pmatrix}\n  \\mathbbm{1}_3 & 0 & \\mathbbm{1}_3 & 0\\\\\n  0 & \\mathbbm{1}_3 & 0 & \\mathbbm{1}_3\\\\\n  -\\mathbbm{1}_3 & 0 & \\mathbbm{1}_3 & 0\\\\\n  0 & -\\mathbbm{1}_3 & 0 & \\mathbbm{1}_3\n \\end{pmatrix}\n  \\begin{pmatrix}\n  \\underline{R}\\\\\n  \\underline{\\bar{R}}\\\\\n  \\underline{L}\\\\\n  \\underline{\\bar{L}}\n \\end{pmatrix}.\n\\end{align}\nWhile this transformation leaves the vacuum- and matter terms invariant, it changes the three contributions to the neutrino-neutrino term. Setting the neutrino and antineutrino densities to $g_{r}=g_{l}=\\frac{1}{2}$ and $g_{\\overline{r}}=g_{\\overline{l}}=-\\frac{1}{2}$, the neutral current term becomes\n\\begin{align}\\mu\\left[\n \\begin{pmatrix}\n    -\\underline{\\underline{\\Lambda}}&\\underline{\\underline{\\Lambda}}&0 & 0\\\\\n  -\\underline{\\underline{\\Lambda}} & \\underline{\\underline{\\Lambda}} & 0& 0 \\\\\n    0&0&-\\underline{\\underline{\\Lambda}} & -\\underline{\\underline{\\Lambda}}\\\\\n  0 & 0 & \\underline{\\underline{\\Lambda}}& \\underline{\\underline{\\Lambda}} \n \\end{pmatrix}+\n \\frac{(1-\\cos\\alpha)}{2}\n \\begin{pmatrix}\n -\\underline{\\underline{\\Lambda}} &   \\underline{\\underline{\\Lambda}} &   0& 0\\\\\n    -\\underline{\\underline{\\Lambda}} &  \\underline{\\underline{\\Lambda}}&  0 & 0\\\\\n    0 &  0 & -\\underline{\\underline{\\Lambda}} &  \\underline{\\underline{\\Lambda}}\\\\\n   0 &   0 &   -\\underline{\\underline{\\Lambda}}& \\underline{\\underline{\\Lambda}}\\\\\n \\end{pmatrix}+\n (1+\\cos\\alpha)\n \\begin{pmatrix}\n 0&0 & 0 & 0\\\\\n  0&0&   0 &  0\\\\\n   0 & 0 & \\underline{\\underline{\\Lambda}} &  0\\\\\n    0 &  0 &0 &-\\underline{\\underline{\\Lambda}}\\\\\n \\end{pmatrix}\\right],\n\\end{align}\nThe transformation separates the equations into two decoupled blocks corresponding to the symmetric and the antisymmetric amplitudes.\nIn the cases of $\\cos\\alpha=1$ and $\\cos\\alpha=-1$, we recover with the upper block of equations the two beam model.\n\n\\subsubsection{Results}\n\n\\begin{figure}[tbp]\n \\centering\n  \\includegraphics[width=0.8\\textwidth]{4Beam_exy_mu_NO_L100.pdf}\n  \\caption{Instabilities in the four beam configuration with relative angle $\\alpha=0^{\\circ}$ for NO. There exist growing instabilities in both, asymmetric and symmetric modes, but the latter ones grow siginificantly slower than the former ones.}\n  \\label{fig:3NNOTest}\n\\end{figure}\n\nWe solve the eigenvalue equation again numerically with the same parameters as in the previous section. Next to fast growing instabilities for the asymmetric modes, we also find instabilities for the symmetric modes with significantly smaller growth rates, see \\fref{fig:3NNOTest}. As for the two beam model, the instabilities for the symmetric modes depend on both mixing angles and the square mass ratio $\\eta$. In \\fref{fig:2NLimitIHg} we show the behavior of the eigenvalues as a function of the intersection angle $\\alpha$. The fast growing antisymmetric instability vanishes for an intersection angle of $\\alpha\\ge\\pi\/2$ while the imaginary part of an eigenvalue belonging to a symmetric mode still grows. \n\n\n\\begin{figure}[tbp]\n   \\centering\n   \\includegraphics[width=0.47\\textwidth]{4Beam_exy_alpha_NO_L100.pdf}\n   \\includegraphics[width=0.47\\textwidth]{4Beam_exy_alpha_NO_L100_EV11.pdf}\n   \\caption{Angle dependencies of modes in the four beam model for NO and for $\\mu=10\\,\\frac{1}{\\textrm{km}}$. \\emph{Left:} Angle dependence of the imaginary part of the antisymmetric eigenvalues in the four beam model for NO. \\emph{Right:} Angle dependence of the imaginary part of the symmetric eigenvalues in the four beam model for NO.}\n   \\label{fig:2NLimitIHg}\n\\end{figure}\n\n\\section{Analysis in the basis of propagating states}\n\\label{sec:PropStates}\nUntil now, we have performed the calculations in the $(\\nu_e, \\nu_x, \\nu_y)$-basis. However, as it is noted in \\cite{Hansen:2019}, it is beneficial to perform the stability analysis in the basis of propagating states since a diagonal density matrix here correspond to a 'fixed point'.\nUnder the assumption of adiabatic evolution, the basis of propagating states is coinciding with the basis of instantaneous eigenstates in which the Hamiltonian is diagonal.\n\nFor understanding our results, it turns out that we can ignore the $H^{\\nu\\nu}$ term when determining the basis of propagating states. Then the Hamiltonian can be written as\n\\begin{equation}\n  \\label{eq:Hflavour}\n      H = \\omega_i \n  \\begin{pmatrix}\n    s_{12}^2c_{13}^2 & s_{12} c_{13} c_{12} & -s_{12}^2 c_{13} s_{13}\\\\\n    s_{12}c_{12}c_{13} & c_{12}^2 & s_{13} s_{12} c_{12} \\\\\n    -s_{12}^2 c_{13} s_{13} & s_{13} c_{12} s_{12} & s_{12}^2 s_{13}^2 \n  \\end{pmatrix}\n  + \\omega_i \\eta\n  \\begin{pmatrix}\n    s_{13}^2 & 0 & s_{13} c_{13} \\\\\n    0 & 0 & 0\\\\\n    s_{13} c_{13} & 0 & c_{13}^2\n  \\end{pmatrix}  +\n  \\begin{pmatrix}\n    \\lambda & 0 & 0\\\\ 0 & 0 & 0\\\\ 0 & 0 & 0\n  \\end{pmatrix} .\n\\end{equation}\nin the ($\\nu_e,\\nu_x,\\nu_y$)-basis.\n\nIn the limit of $\\lambda \\gg \\omega \\eta$, the diagonalised Hamiltonian is \n\\begin{equation}\n  \\label{eq:Hdiag}\n  H'_{\\rm diag} \\approx\n  \\begin{pmatrix}\n    \\lambda + \\omega_i (s_{12}^2c_{13}^2 + \\eta s_{13}^2) & 0 & 0 \\\\\n    0  & \\omega_i c_{12}^2 & 0 \\\\\n    0 & 0 & \\omega_i ( s_{12}^2 s_{13}^2 + \\eta c_{13}^2)\n  \\end{pmatrix}\n\\end{equation}\nup to terms of order $(s_{13}c_{13}\\omega \\eta)^2\/\\lambda$. This approximation basically corresponds to neglecting the off-diagonal terms of $H^{\\rm vac}_{i}$, and for two-neutrino oscillations it corresponds to the statement that $\\omega_{\\rm eff} \\approx \\lambda - c_{2\\theta} \\omega$ \\cite{Hansen:2019}. The result in \\eref{eq:Hdiag} agree with the diagonal matrix derived in~\\cite{Duan:2008za} in the limit $\\theta_{13}=0$, but we are more interested in non-zero $\\theta_{13}$. When going to the basis of propagating states, we also transform $\\rho_i$ to $\\rho'_i$:\n\\begin{equation}\n  \\label{eq:rhop}\n   \\rho'_i(t,\\textbf{v})-\\frac{1}{3}\\text{Tr}\\rho'_i\\mathbbm{1}=:\\frac{g_i}{2}\n\\begin{pmatrix} \n\\mathfrak{s}_{1}^i(t,\\textbf{v}) & \n\\mathcal{S}_{12}^i(t,\\textbf{v}) & \n\\mathcal{S}_{13}^i(t,\\textbf{v})\\\\\n \\mathcal{S}_{12}^{\\ast i}(t,\\textbf{v}) &  \n\\mathfrak{s}_{2}^i(t,\\textbf{v}) & \n\\mathcal{S}_{23}^i(t,\\textbf{v}) \\\\\n \\mathcal{S}_{13}^{\\ast i}(t,\\textbf{v}) & \n\\mathcal{S}_{23}^{\\ast i}(t,\\textbf{v}) & \n- \\mathfrak{s}_{1}^i(t,\\textbf{v})- \\mathfrak{s}_{2}^i(t,\\textbf{v})\n\\end{pmatrix}\n\\end{equation}\nHere, we already used that the diagonal is unchanged by the transformation to first order in $\\omega\/\\lambda$. The off-diagonal elements transforms to first order as\n\\begin{equation}\n  \\begin{aligned}\n    \\mathcal{S}_{12}^i(t,\\textbf{v}) &\\approx \\mathcal{S}_{ex}^i(t,\\textbf{v}) - \\frac{\\omega}{\\lambda} s_{12} c_{12} c_{13} (\\mathfrak{s}_1^i(t,\\textbf{v}) - \\mathfrak{s}_2^i(t,\\textbf{v}) ), \\\\\n    \\mathcal{S}_{13}^i(t,\\textbf{v}) &\\approx \\mathcal{S}_{ey}^i(t,\\textbf{v}) + \\frac{\\omega}{\\lambda} ( \\eta s_{13} c_{13} + s_{12}^2 s_{13} c_{13} ) (2 \\mathfrak{s}_1^i(t,\\textbf{v}) + \\mathfrak{s}_2^i(t,\\textbf{v}) ), \\\\\n    \\mathcal{S}_{23}^i(t,\\textbf{v}) &\\approx \\mathcal{S}_{xy}^i(t,\\textbf{v}) .\n  \\end{aligned}\n\\end{equation}\nFor the antineutrinos, the transformation is similar, but with opposite signs for the terms proportional to $\\omega\/\\lambda$. \n\nUsing \\eref{eq:Hdiag} to derive the Vacuum-Term (I) and the Matter-Term (II) from \\eref{eq:Goal}, we recover the Matter-Term in \\eref{eq:MatterTerm}. For the Vacuum-Term in \\eref{eq:VacTermDec}, $\\underline{\\underline{A_0}}$ and $\\underline{\\underline{B_0}}$ become diagonal, while the remaining four matrices $\\underline{\\underline{A_1}}$, $\\underline{\\underline{B_1}}$, $\\underline{\\underline{A_2}}$ and $\\underline{\\underline{B_2}}$ vanish.\nThe result of this simplification is that $\\mathcal{S}_{12}^i$, $\\mathcal{S}_{13}^i$ and $\\mathcal{S}_{23}^i$ decouple, and we can consider them one by one -- effectively in two-neutrino setups. \n\nThe vacuum term now gives\n\\begin{equation}\n  \\label{eq:A0B0}\n  \\omega_i (\\underline{\\underline{A_0}} + \\eta \\underline{\\underline{B_0}}) = \\omega_i \\; \\textrm{diag}\\left( - c_{12}^2 + c_{13}^2s_{12}^2 + \\eta s_{13}^2, C_{13} (s_{12}^2 - \\eta ), c_{12}^2 - s_{12}^2s_{13}^2 - \\eta c_{13}^2 \\right) .\n\\end{equation}\nIf the two-neutrino limit is realised by putting $\\theta_{12} = \\theta_{13} = 0$, the diagonal matrix has the entries $-\\omega_i$, $- \\omega_i \\eta$ and $-\\omega_i (\\eta-1)$. This corresponds to three pairs of two-neutrino oscillations with normal ordering if $\\eta > 0$ and inverted ordering for the last two if $\\eta < 0$. However, the measured values of $\\theta_{12}$ and $\\theta_{13}$ are non-zero, and plugging in the values reveals that $- c_{12}^2 + c_{13}^2s_{12}^2 + \\eta s_{13}^2 > 0$ for $\\eta > 0$. Hence \\emph{normal mass ordering gives rise to an effective two-neutrino system with inverted mass ordering}.\nThis is exactly what we have seen in \\fref{fig:3NNO} and \\fref{fig:3NNOTest}: Even in the case of normal mass ordering, an instability is present that belongs to the symmetric modes which is typical for a two-neutrino system with inverted mass ordering.\n\nWith the understanding that the three-neutrino instability we have identified can be interpreted in the two-neutrino approximation by using \n\\begin{equation}\n\\label{eq:omegaeff}\n\\omega_{\\textrm{eff}} =  \\omega( - c_{12}^2 + c_{13}^2s_{12}^2 + \\eta s_{13}^2)  ,\n\\end{equation}\nwe can expand our results by considering previous results on linear stability in the literature. Specifically, in the bulb model, the results from \\cite{Chakraborty:2015tfa} on multi-angle effects and small scale inhomogeneities can be directly applied to the three-neutrino instability by using $\\omega_{\\textrm{eff}}$.\nThe absolute value of $\\omega_{\\textrm{eff}}$ is significantly smaller than $\\omega\\eta$, and the growth rate is thereby also reduced. Although this will delay the flavour conversion, it does not affect the stability as such. When comparing the results for NO and IO in \\cite{Chakraborty:2015tfa}, there are regions of the $(\\mu,\\lambda)$ plane where IO is unstable and NO is stable. Consequently, for a supernova density profile transversing these regions, we expect to see collective oscillations that would only be predicted by a simple two-neutrino analysis if $\\omega_{\\textrm{eff}}$ from \\eref{eq:omegaeff} was properly used.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\nIn this work, we studied self-induced flavor conversion in SNe taking into account all three neutrino flavors. We constructed an approximate solution to the non-linear Von Neumann equation which describes the time dependent propagation of flavor density matrix and includes mixing among different neutrino modes. The ensemble of neutrinos were assumed to be homogeneous and collisonless, and we neglected the CP-violating phase in the mixing matrix. In the vacuum term of the the system Hamiltonian, we dropped terms that remain constant to linear order in $\\mathcal{S}_{\\alpha\\beta}$ and that are suppressed in the environment of a large matter potential. The commutator of different neutrino modes was calculated to first order in $|\\mathcal{S}_{\\alpha\\beta}|$, and only a discrete number of different modes was assumed. Finally, the differential equation was transformed into an eigenvalue problem by making a plane wave ansatz for $\\mathcal{S}_{\\alpha\\beta}$.\n\nWe applied our approximate description to a system of two colliding beams of neutrinos and antineutrinos and found flavor instabilities not only for IO but also for NO. In order to get a deeper understanding, we turned to a model of four colliding neutrino\/antineutrino beams that intersect with an angle $\\alpha$. In this system, we studied symmetric and antisymmetric modes and found that symmetric modes can develop flavor instabilities for both mass orderings. The origin of these instabilities in NO is the value of the mass square ratio $\\eta$ combined with those of the mixing angles $\\theta_{12}$ and $\\theta_{13}$.\n\nFinally, we showed in the framework of propagating states that the new instability for NO can be interpreted as an effective two-neutrino system with $\\omega_{\\textrm{eff}}<0$ corresponding to an effective IO. With this understanding, our results are expected to be valid beyond our simple toy systems as one simply replaces $\\omega$ in any two-neutrino analysis by $\\omega_{\\textrm{eff}}$.\n\nWith a good understanding of all the instabilities that we identify, we can also confirm that the off-diagonal terms of the density matrix do not lead to any new instabilities at least when the CP-violating phase is neglected and equal fluxes of muon and tau neutrinos are assumed.\n\nOur findings show that instabilities that were only expected to exist for IO are also present in NO, albeit with a smaller growth rate. This once again demonstrates how great care must be taken when dealing with collective neutrino oscillations in supernova in order to capture all the relevant physics.\n\n\\section{Acknowledgements}\nThe authors thank Alexei Yu Smirnov for fruitful discussion on the topic of self-induced neutrino conversion. R.S.L.H. was partly funded by the Alexander von Humboldt Foundation.\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn Riemannian geometry, a hypersurface is called isoparametric if itself and all\nits sufficiently close parallel hypersurfaces have constant mean\ncurvature. Since 1938, E. Cartan began to study the isoparametric hypersurfaces of real space forms systematically and\nproved that a hypersurface has constant principal curvatures if and only if it is isoparametric \\cite{C38}.\n\nIn Finsler geometry, the conception of isoparametric hypersurfaces has been firstly introduced in \\cite{HYS16}. Moreover, the analytic definitions of\nlocally isoparametric hypersurfaces\nand $d\\mu$-isoparametric hypersurfaces are given in~\\cite{HDY, DH}.  Let~$(N,F,d\\mu)$ be an\n$n$-dimensional Finsler manifold with volume form~$d\\mu=\\sigma(x)dx$ and $f$ be a non-constant $C^1$ function and smooth on\n$N_f=\\{x\\in N~|~df(x)\\neq 0\\}$.\nSet $J=f(N_f)$. Then $f$ is\nsaid to be \\emph{isoparametric} (resp. \\emph{$d\\mu$-isoparametric}) on $(N,F,d\\mu)$,\nif there exist a smooth function ${a} (t)$ and a continuous function $b (t)$ on $J$ such that\n\\begin{equation}\\label{1.3} \\left\\{\\begin{aligned}\n&F(\\nabla f)=a(f),\\\\\n&\\Delta f=b(f),\n\\end{aligned}\\right.\n\\end{equation}\nwhere $\\nabla f$ denotes the gradient of~$f$ and $\\Delta {f}=\\Delta^{\\nabla f} f$ (resp. $\\Delta {f}=\\Delta_{\\sigma} f$) is\ndefined by (\\ref{l1})(resp. (\\ref{l2})).\nAll the regular level surfaces ${M}_t = {f}^{-1}(t)$ are named an \\emph{($d\\mu$-)isoparametric family}, each of which is called an \\emph{($d\\mu$-)isoparametric hypersurface} on $(N,F,d\\mu)$. A function $f$ only satisfying the first equation\nof (1.1) is said to be \\emph{transnormal}.\n\nSimilar to Riemannian geometry, \\cite{HYS16} proved that\na transnormal function $f$ is isoparametric if and only if each regular level hypersurface of $f$ has constant ($d\\mu$)-mean curvatures. Particularly, if $N$ has constant flag curvature (and constant $\\mathbf{S}$-curvature), then a transnormal function $f$ is ($d\\mu$)-isoparametric if and only if all the principal curvatures of $M_{t}$ are constant. So\nthe relationship between isoparametric hypersurfaces and principal curvatures is worth studying.\n\nThe previous work mainly focused on isoparametric hypersurfaces with constant principal curvatures in Finsler space forms (with constant flag curvature). For some very special Finsler space forms, such as Minkowski spaces (with zero flag curvature) and Funk spaces (with negative constant flag curvature), the isoparametric hypersurfaces have been completely classified in \\cite{HYS16, GM, HYS17}. In~\\cite{HDRY}, the authors studied isoparametric hypersurfaces of Finsler space forms, obtained the Cartan-type formula and some classifications on the number of distinct principal curvatures or their multiplicities.\n\nIn Riemannian geometry, the equivalence between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures is no longer true for more general ambient spaces with nonconstant curvatures. There exist many examples of isoparametric hypersurfaces with (non)constant principal\ncurvatures with nonconstant curvatures \\cite{RC13, GTY}. In \\cite{R19}, a conformally flat metric in $\\mathbb{R}^{n}$ is constructed, which admits a family of nonisoparametric hyperplanes. This shows that there is no necessary relation between the isoparametric hypersurface and the hypersurface with constant principal curvature.\n\nA natural question is whether there exist similar results in Finsler spaces. In this paper, we give some examples of isoparametric hypersurfaces with (non)constant principal curvatures in Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Finsler manifold which admits a family of hypersurface with constant principal curvatures, but each of them is an nonisoparametric hypersurface, that is, its\nsufficiently close parallel hypersurfaces don't have constant mean curvatures or $d\\mu_{BH}$-mean curvatures. So we have the following theorem.\n\\begin{theo}\nFor Finsler spaces with nonconstant flag curvatures, the relation between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures is no longer equivalent. In fact, there exist isoparametric hypersurfaces without constant principal curvatures and nonisoparametric hypersurfaces with constant principal curvatures.\n\\end{theo}\n\\section{Preliminaries}\n\n\\subsection{Finsler manifolds}\nLet~$(N,F)$ be an $n$-dimensional smooth connected Finsler manifold and~$TN$ be the tangent bundle over~$N$ with local coordinates\n$(x,y)$, where~$x=(x^i)$ and~$y=y^i\\frac{\\partial}{\\partial x^{i}}$.\nHere and from now on, we will use the following convention of index ranges unless otherwise stated:\n$$1\\leq i, j, \\cdots \\leq n ;~~~~~~~1\\leq a, b, \\cdots \\leq n-1.$$\nThe fundamental form~$g$ of~$(N,F)$ is\n\\begin{equation*}\ng=g_{ij}(x,y)dx^{i} \\otimes dx^{j}, ~~~~~~~g_{ij}(x,y)=\\frac{1}{2}[F^{2}] _{y^{i}y^{j}}.\n\\end{equation*}\nThe projection~$\\pi : TN\\rightarrow N$ gives rise to the pull-back bundle~$\\pi^{\\ast}TN$. On it there exists a unique\n\\emph{Chern connection}~$\\nabla$ with~$\\nabla\n\\frac{\\partial}{\\partial x^i}=\\omega_{i}^{j}\\frac{\\partial}{\\partial\nx^{j}}=\\Gamma^{i}_{jk}dx^k\\otimes\\frac{\\partial}{\\partial x^{j}}$ satisfying (\\cite {BCS})\n$$dg_{ij}-g_{ik}\\omega^{k}_{j}-g_{kj}\\omega^{k}_{i}=2C_{ijk}(dy^{k}+N^{k}_{l}dx^{l}),$$\n$$~~~~N_j^i:=\\frac{\\partial G^i}{\\partial y^j}=\\Gamma^{i}_{jk}y^k,$$\nwhere $C_{ijk}=\\frac{1}{2}\\frac{\\partial g_{ij}}{\\partial y^k}$ is\ncalled the\\emph{ Cartan tensor} and $G^{i}=\\frac{1}{4}g^{il}\\left\\{[F^{2}]_{x^{k}y^{l}}y^{k}-[F^{2}]_{x^{l}}\\right\\}$ are the \\emph{geodesic coefficients} of $(N,F)$. For $X=X^{i}\\frac{\\partial}{\\partial x^{i}}\\in\\Gamma(TN)$, the covariant derivative of $X$ along $v=v^{i}\\frac{\\partial}{\\partial x^{i}}\\in T_{x}N$ with respect to a reference vector $w\\in T_{x}N\\setminus\\{0\\}$ is defined by\n$$D^{w}_{v}X(x)=\\{v^{j}\\frac{\\partial X^{i}}{\\partial x^{j}}(x)+\\Gamma^{i}_{jk}(w)v^{j}X^{k}(x)\\}\\frac{\\partial}{\\partial x^{i}}.$$\nAnother torsion-free Berwald connection $^{b}\\nabla$ is defined by\n$$^{b}\\omega^{i}_{j}=^{b}\\Gamma^{i}_{jk}dx^{k}=(\\Gamma^{i}_{jk}+L^{i}_{jk})dx^{k}$$\nwhere $L^{i}_{jk}$ is the Landsberg curvature of $(N, F)$. For~$X=X^{i}\\frac{\\partial}{\\partial x^{i}}\\in \\pi^{\\ast}TN$, the \\emph{horizontal covariant derivative} of~$X$ with respect to $^{b}\\nabla$ is defined by\n\\begin{align}\nX^{i}_{~|j}=\\frac{\\delta X^{i}}{\\delta x^{j}}+^b\\Gamma^{i}_{js}X^s,\\label{Z10}\n\\end{align}\nwhere $\\frac{\\delta}{\\delta x^{j}}=\\frac{\\partial}{\\partial x^{j}}-N_j^i\\frac{\\partial}{\\partial y^{i}}$.\n\nLet~${\\mathcal L}:TN\\rightarrow T^{\\ast}N$ denotes the \\emph{Legendre transformation}, satisfying~${\\mathcal L}(\\lambda\ny)=\\lambda {\\mathcal L}(y)$ for all~$\\lambda>0,~y\\in TN$.\nFor a smooth function~$f: N\\rightarrow \\mathbb{R}$, the \\emph{gradient vector} of~$f$ at~$x$ is defined as~$\\nabla f(x)={\\mathcal\nL}^{-1}(df(x))\\in T_{x}N$.\nSet~$N_{f}=\\{x\\in N|df(x)\\neq 0\\}$ and~$\\nabla^{2}f(x)=\\nabla^{\\nabla f}(\\nabla f)(x)$ for~$x\\in N_{f}$. We define the \\emph{Laplacian} of~$f$\nby\n\\begin{equation}\\label{l1}\n\\Delta^{\\nabla f} f=\\textmd{tr}_{g_{_{\\nabla f}}}(\\nabla^{2}f).\n\\end{equation}\nAnd the \\emph{Laplacian} of~$f$ with respect to the volume form~$d\\mu=\\sigma(x)dx=\\sigma(x)dx^{1}\\wedge dx^{2}\\wedge\\cdots\\wedge dx^{n}$ can be represented as\n\\begin{align}\\label{l2}\n\\Delta_{\\sigma} f=\\textmd{div}_{\\sigma}(\\nabla f)=\\frac{1}{\\sigma}\\frac{\\partial}{\\partial x^{i}}(\\sigma g^{ij}(\\nabla f)f_{j})=\\Delta^{\\nabla f} f-S(\\nabla f),\n\\end{align}\nwhere\n\\begin{align}\\label{1.4.19}\nS(x,y)=\\frac{\\partial G^{i}}{\\partial y^{i}}-y^{i}\\frac{\\partial}{\\partial x^{i}}(\\ln \\sigma(x))\n\\end{align}\nis the \\emph{$\\mathbf{S}$-curvature}.\n\nLet $\\phi:M\\to N$ be an embedded hypersurface of~$(N,F)$. For any~$x\\in M$,\nthere exist exactly two unit\\emph{ normal vectors}~$\\mathbf{n}_{\\pm}$. Let~$\\mathbf{n}$ be a given normal vector of $N$.\nSet $\\hat g=\\phi^*g_{\\mathbf{n}}$.\nFrom \\cite{HYS16}, we have the following \\emph{Gauss-Weingarten formulas}\n\\begin{align}\\label{1.1}\n{ D}^{\\textbf{n}}_{X}Y&={ {\\hat{\\nabla}}}_{X}Y+\\hat{h}(X,Y)\\textbf{n},\\\\\nD^{\\textbf{n}}_{X}\\textbf{n}&=-{A}_{\\textbf{n}}X,~~~~~\\quad \\forall X,~Y\\in \\Gamma(TM),\\label{1.2}\n\\end{align}\nwhere $\\hat{\\nabla}$ is the induced connection on $(M,\\hat{g})$ and $\\hat{h}(X,Y):=g_{\\textbf{n}}(\\textbf{n},D^{\\textbf{n}}_{X}Y)$. The eigenvalues of the shape operator ${A}_{\\mathbf{n}}$,\n$\\mu_1,\\mu_2,\\cdots,\\mu_{n-1}$, and~$\\hat{H}_{\\mathbf{n}}=\\sum\\limits_{a=1}^{n-1}\\mu_{a}$ are called the \\emph{ principal\ncurvatures} and the \\emph{anisotropic mean curvature} with respect to~$\\mathbf{n}$, respectively.\n\n\\begin{lemm}\\cite{HYS16} Let $M$ be an embedded hypersurface of $(N, F, d\\mu)$, then\n\\begin{equation}\\label{hn}\nH_{\\textbf{$\\mathbf{n}$}}=\\hat{H}_{\\textbf{$\\mathbf{n}$}}+S(\\textmd{\\textbf{$\\mathbf{n}$}}),\n\\end{equation}\nwhere $H_{\\textbf{$\\mathbf{n}$}}$ is the $d\\mu$-mean curvature induced by variation of the volume in $(N, F, d\\mu)$ (see (3.3) in \\cite{HYS16} for detail).\n\\end{lemm}\nSimilar to Riemannian geometry, we can also give the geometric definitions of isoparametric hypersurface on Finsler manifolds as following.\n\\begin{defi}\\label{d1}\nA hypersurface $M$ on Finsler manifold $(N, F)$ is called isoparametric if it and its sufficiently close parallel hypersurfaces have constant anisotropic mean curvatures.\nMoreover,\na hypersurface $M$ in Finsler manifold $(N, F, d\\mu)$ is named $d\\mu$-isoparametric if it and all its sufficiently close parallel hypersurfaces have constant $d\\mu$-mean curvatures.\n\\end{defi}\n\\begin{remark}\nObviously, when $S(\\textmd{\\textbf{$\\mathbf{n}$}})=\\textmd{constant}$, a hypersurface $M$ is $d\\mu$-isoparametric if and only if it is isoparametric.\n\\end{remark}\n\n\\section{The examples of isoparametric hypersurfaces on Randers manifolds with nonconstant flag curvatures}\n\nIn Riemannian geometry, there are many examples of isoparametric hypersurfaces with (non)constant principal\ncurvatures on some Riemannian manifolds with nonconstant curvature \\cite{RC13, GTY, W82, DRDV12}.\nThe first examples were found by Wang \\cite{W82}, who constructed some inhomogeneous isoparametric\nhypersurfaces with nonconstant principal curvatures in the complex projective space, by projecting some of the\ninhomogeneous hypersurfaces in spheres via the Hopf map. In \\cite{DRDV12}, the examples of isoparametric real hypersurfaces with nonconstant principal curvatures in complex hyperbolic spaces $\\mathbb{C}H^{n}$ are constructed by Lie groups and Lie algebras.\n\nIn this section, we plan to give some examples of isoparametric hypersurfaces with (non)constant principal curvatures in Randers manifolds with nonconstant flag curvatures.\n\nIn section 2, we study the principal curvatures of anisotropic submanifolds and hypersurfaces in a Randers\nspace with the navigation datum. Here we will use this relationship to construct examples.\nLet~$(N, h)$ be an~$n$-dimensional Riemannian manifold and $v$ be a vector\nfield. By navigation problem, $(h, v)$ can define a Randers metric\n\\begin{equation}\\label{c3}\nF=\\frac{\\sqrt{\\lambda h^2+v_{0}^2}-v_{0}}{\\lambda}=\\frac{\\sqrt{\\lambda h_{ij}y^iy^j+(v_{i}y^i)^2}-v_{i}y^i}{\\lambda},\n\\end{equation}\nwhere~$\\lambda=1-b^2, ~b=\\|v\\|_h, ~ v=v^i\\frac{\\partial}{\\partial x^{i}}, ~v_i=h_{ij}v^j$.\nLet~$\\phi: M\\to(N, F)$ be an embedded hypersurface and the unit normal vector field of~$M$ with respect to~$F$ and $h$ be $\\mathbf{n}$ and $\\bar{\\mathbf{n}}$, respectively. From \\cite{HDY}, we know\n\\begin{equation}\n\\mathbf{n}=\\bar{\\mathbf{n}}+v\\label{a3.21}\n\\end{equation}\nand the relationship between the principal curvatures of~$M$ in~$(N, F)$ and $(N, h)$ is as follows.\n\\begin{lemm} \\label{thm0}\\cite{HDY}\nLet~$M$ be a hypersurface in a Randers space~$(N,F,d\\mu_{BH})$ with the navigation datum~$(h, v)$. If~$F$ has isotropic~$\\mathbf{S}$-curvature~$S=(n+1)k(x)F$, then for any unit normal vector field $\\mathbf{n}$, the shape operators of $M$ in Randers space~$(N, F)$ and Riemannian space~$(N, h)$, ${A}_{\\mathbf{n}}$ and $\\bar{A}_{\\bar{\\mathbf{n}}}$,  have the same principal vectors and their principal curvatures satisfy\n\\begin{align}\\label{3.24}\\mu=\\bar{\\mu}+k(x),\\end{align}\nwhere~$\\mu$ and~$\\bar{\\mu}$ are the principal curvatures of~$M$ in Randers space~$(N, F)$ and Riemannian space~$(N, h)$, respectively.\n\\end{lemm}\nLater, Xu and his coworkers got the local correspondence between isoparametric functions\nor isoparametric hypersurfaces for homothetic navigation in a Finsler manifold.\n\\begin{lemm}\\cite{XMYZ}\\label{x1}\nLet $v$ be a homothetic vector field on Finsler manifold $(N, \\bar{F})$, and $F$ the metric\ndefined by navigation from the datum $(\\bar{F}, v )$. Then\nlocally around any point $x_{0}$ with $\\bar{F}(x_{0}, -v(x_{0}))< 1$, a hypersurface is isoparametric\nfor $(\\bar{F}, d\\mu^{\\bar{F}}_{BH})$ if and only if it is isoparametric for $(F, d\\mu^{F}_{BH})$.\n\\end{lemm}\n\n\nAccording to \\cite{H05}, Randers metric $F$  with respect to the Busemann-Hausdorff (BH) measure has constant $\\mathbf{S}$-curvature if and only if $v$ is a homothetic vector field.\nFrom Lemma \\ref{thm0} and Lemma \\ref{x1}, we know that if $v$ is a homothetic vector field, the ($d\\mu^{F}_{BH}$-)isoparametric hypersurface of $(N, h)$ is also ($d\\mu^{F}_{BH}$-)isoparametric\nfor the Randers manifold $(N, F)$, and the principal curvatures of $M$ in $(N, F)$ are all (non)constant\nif and only if its principal curvatures in $(N, h)$ are all (non)constant.\n\n\\begin{remark}\\label{k1}\nBy using homothetic navigation, we can construct ($d\\mu^{F}_{BH}$-)isoparametric hypersurfaces with (non)constant principal\ncurvatures on some Randers manifolds with nonconstant flag curvatures.\n\\end{remark}\nFrom Lemma \\ref{thm0}, Lemma \\ref{x1} and \\cite{DRDV12}, we have the following example.\n\\begin{exam}\nLet $\\mathbb{C}H^{n}$ be a complex hyperbolic space, $AN$ be the connected and simply connected\nsubgroups of $G=SU(1, n)$, $S_{\\varpi}$ be a\nconnected subgroup of $AN$, $M^{r}$ be\nthe tube of radius $r$ around the submanifold $W_{\\varpi}$, where $W_{\\varpi}$ is the orbit $S_{\\varpi}\\cdot o$ of the\ngroup $S_{\\varpi}$ through the point $o\\in\\mathbb{C}H^{n}$. Then for every $r > 0$, $M^{r}$\nis an isoparametric real hypersurface which has, in general, nonconstant principal curvatures. If\ngiven a (locally) homothetic vector field $v$ in $\\mathbb{C}H^{n}$, then by (\\ref{c3}), we can get a Randers metric $F$ with nonconstant flag curvatures.\nSo $M^{r}$ is also\nthe isoparametric hypersurface of $(\\mathbb{C}H^{n}, F)$, and the principal curvatures of $M^{r}$ in $(\\mathbb{C}H^{n}, F)$ are all nonconstant.\n\\end{exam}\n\nIn Riemannian spaces of constant curvatures, a hypersurface is isoparametric if and only if it has constant principal curvatures.\nBut in Finsler manifolds with constant flag curvatures,\nfor any given volume form $d\\mu$ on $(N, F)$, if $F$ doesn't have constant $\\mathbf{S}$-curvature, then\nthe existence of isoparametric hypersurfaces with nonconstant principal curvatures is still an open problem.\n\\section{The example of nonisoparametric hypersurfaces with constant principal curvatures}\nLet~$N=\\mathbb{R}^{n}$, $h=\\sqrt{\\delta_{ij}y^{i}y^{j}}$ and $v=b\\frac{\\partial}{\\partial x^{n}}$\nbe a vector field, where $b$ is a constant and satisfies $b<1$.\nConsider a Randers-Minkowski metric\n\\begin{equation}\nF(y)=\\alpha+\\beta=\\frac{\\sqrt{\\lambda h^2+v^2_{0}}}{\\lambda}-\\frac{v_{0}}{\\lambda},\n\\end{equation}\nwhere\n$ v_{0}=by^n, \\lambda=1-\\delta_{ij}v^{i}v^{j}=1-b^2$.\nSet $\\alpha=\\sqrt{\\alpha_{ij}y^{i}y^{j}}, ~\\beta=b_{i}y^{i}$, then\n\\begin{equation*}\n\\alpha_{ab}=\\frac{\\delta_{ab}}{\\lambda}, ~\\alpha_{an}=0, ~\\alpha_{nn}=\\frac{\\lambda+b^{2}}{\\lambda^2}=\\frac{1}{\\lambda^2},\n\\end{equation*}\n\\begin{equation*}\n\\alpha^{ab}=\\lambda\\delta_{ab}, ~\\alpha^{an}=0, ~\\alpha^{nn}=\\lambda^2,\n\\end{equation*}\n\\begin{equation*}\nb^{a}=\\alpha^{aj}b_{j}=0, ~b^{n}=-\\lambda b, ~\\parallel\\beta\\parallel^{2}_\\alpha=b^{2}.\n\\end{equation*}\nHence\n$$\\alpha_{y^{a}}=\\frac{y^{a}}{\\lambda\\alpha},~\\alpha_{y^{n}}=\\frac{y^{n}}{\\lambda^{2}\\alpha},$$\n\\begin{equation}\\label{gab}\ng_{ab}=\\frac{F}{\\lambda\\alpha}\\delta_{ab}-\\frac{\\beta y^{a}y^{b}}{\\alpha^{3}\\lambda^{2}}, ~g_{an}=-\\frac{by^{a}}{\\lambda^{2}\\alpha}-\\frac{\\beta y^{a}y^{n}}{\\lambda^{3}\\alpha^{3}},\n\\end{equation}\n\\begin{equation*}\ng_{nn}=\\frac{F}{\\lambda^{2}\\alpha}\\left(1-\\frac{(y^{n})^{2}}{\\lambda^{2}\\alpha^{2}}\\right)+(\\frac{y^{n}}{\\lambda^{2}\\alpha}-\\frac{b}{\\lambda})^{2},\n\\end{equation*}\nand\n\\begin{equation*}\ng^{ab}=\\frac{\\alpha}{F}\\lambda\\delta^{ab}+\\frac{b^{2}\\alpha+\\beta}{F^{3}}y^{a}y^{b},\n~g^{an}=\\frac{b\\lambda\\alpha}{F^{2}}y^{a}+\\frac{b^{2}\\alpha+\\beta}{F^{3}}y^{a}y^{n},\n\\end{equation*}\n\\begin{equation*}\ng^{nn}=\\frac{\\lambda^{2}\\alpha}{F}+2\\frac{b\\lambda\\alpha}{F^{2}}y^{n}+\\frac{b^{2}\\alpha+\\beta}{F^{3}}(y^{n})^{2}.\n\\end{equation*}\n\\subsection{A conformally flat Randers manifold}\nDefine a new metric\n\\begin{equation*}\\label{dl1}\n\\widetilde{F}(x, y)=e^{\\rho(x)}F(y)\n\\end{equation*}\nand the corresponding fundamental tensor is\n\\begin{equation}\n\\tilde{g}_{ij}(x, y)=e^{2\\rho(x)}g_{ij}(y),\n\\end{equation}\nwhere $\\rho(x)=\\sum\\limits_{a=1}^{n-1} \\ln(2+cos(\\pi x^{a}))$, $x=(x^{i})\\in \\mathbb{R}^{n}$.\nIt is easy to know that $\\tilde{g}$ is (locally) conformally flat.\nFrom \\cite{SS16}, we get the relation of spray coefficient $\\tilde{G}^{i}$ and $G^{i}$ with respect to $\\widetilde{F}$ and $F$, respectively,\n\\begin{equation}\\label{gg}\n\\tilde{G}^{i}=G^{i}+\\rho_{k}y^{k}y^{i}-\\frac{1}{2}F^{2}\\rho^{i},\n\\end{equation}\nwhere $\\rho_{k}=\\frac{\\partial \\rho}{\\partial x^{k}}, \\rho^{i}=g^{ij}\\rho_{j}$.\nBy $\\rho_{n}=0$, $G^{i}=0$,\n\\begin{equation*}\n\\rho^{a}=g^{ai}\\rho_{i}=g^{ab}\\rho_{b}=\\frac{\\alpha}{F}\\lambda\\rho_{a}+\\frac{ b^{2}\\alpha+\\beta}{F^{3}}y^{a}y^{b}\\rho_{b}\n\\end{equation*}\nand\n\\begin{equation*}\n\\rho^{n}=g^{nb}\\rho_{b}=(\\frac{b\\lambda\\alpha}{F^{2}}y^{b}+\\frac{b^{2}\\alpha+\\beta}{F^{3}}y^{b}y^{n})\\rho_{b}.\n\\end{equation*}\nWe have\n\\begin{equation}\\label{g1}\n\\tilde{G}^{a}=\\rho_{b}y^{b}y^{a}-\\frac{1}{2}F^{2}\\rho^{a}\n=(1-\\frac{b^{2}\\alpha+\\beta}{2F})\\rho_{b}y^{b}y^{a}-\\frac{1}{2}\\lambda\\alpha F \\rho_{a},\n\\end{equation}\n\\begin{equation}\\label{g2}\n\\tilde{G}^{n}=\\rho_{b}y^{b}y^{n}-\\frac{1}{2}F^{2}\\rho^{n}=(1-\\frac{b^{2}\\alpha+\\beta}{2F})\\rho_{b}y^{b}y^{n}\n-\\frac{b\\lambda\\alpha}{2}\\rho_{b}y^{b}.\n\\end{equation}\nThen\nby (\\ref{gg}), it followed that\n\\begin{equation}\\label{n1}\n\\tilde{N}^{i}_{j}=\\frac{\\partial \\tilde{G}^i}{\\partial y^j}\n=\\rho_{k}y^{k}\\delta^i_j+\\rho_{j}y^i\n-\\frac{1}{2}(F^{2})_{y^{j}}\\rho^i+F^2 C^{ik}_{j}\\rho_{k},\n\\end{equation}\nwhere\n\\begin{align}\\label{n2}\n&-2C^{ik}_{j}\\nonumber=\\frac{\\partial g^{ik}}{\\partial y^{j}}\n=(\\frac{\\alpha}{F})_{y^{j}}\\alpha^{ik}-(\\frac{\\alpha}{F^2})_{y^{j}}(b^{i}y^{k}+b^{k}y^{i}),\\\\\n&-\\frac{\\alpha}{F^2}(b^{i}\\delta^{k}_{j}+b^{k}\\delta^{i}_{j})\n+(\\frac{b^{2}\\alpha+\\beta}{F^3})_{y^{j}}y^{i}y^{k}+\\frac{b^{2}\\alpha+\\beta}{F^3}(y^{i}\\delta^{k}_{j}+y^{k}\\delta^{i}_{j}).\n\\end{align}\nFrom \\cite{SS16}, we have\n\\begin{equation*}\n\\tilde{S}(y)=S(y)+F^{2}\\rho^{k}I_{k},\n\\end{equation*}\nwhere $I_{k}=g^{ij}C_{ijk}$ is the Cartan form, $\\tilde{S}$ and $S$ respectively are the $\\mathbf{S}$-curvature of $\\tilde{F}$ and $F$.\nIt is well known that $S=0$ with respect to the Busemann-Hausdorff volume form.\nThen by (\\ref{n2}), $\\rho_{n}=0$ and $\\rho_{k}b^{k}=0$, we have\n\\begin{align}\\label{s1}\n\\tilde{S}\\nonumber=&F^{2}\\rho_{k}C^{ik}_{i}\\\\ \\nonumber\n=&-\\frac{1}{2}F^{2}\\rho_{k}\\left[(\\frac{\\alpha}{F})_{y^{i}}\\alpha^{ik}\n-(\\frac{\\alpha}{F^2})_{y^{i}}b^{i}y^{k}\n+(\\frac{b^{2}\\alpha+\\beta}{F^3})_{y^{i}}y^{i}y^{k}+(n+1)y^{k}\\frac{b^{2}\\alpha+\\beta}{F^3}\\right]\\\\\n=&-\\frac{1}{2}\\rho_{k}\\left[(\\alpha_{y^{i}}\\beta-\\alpha\\beta_{y^{i}})\\alpha^{ik}-b^{i}y^{k}\\frac{\\alpha_{y^{i}}(\\beta-\\alpha)-2\\alpha\\beta_{y^{i}}}{F}\n+(n-1)y^{k}\\frac{b^{2}\\alpha+\\beta}{F}\\right].\n\\end{align}\n\n\\subsection{The geodesics with respect to $\\widetilde{F}$}\nBy (\\ref{dl1}), $\\tilde{F}$ is produced by navigation datum $(\\tilde{h}, \\tilde{v})$,\nwhere $\\tilde{h}=e^{\\rho}h, \\tilde{v}=e^{\\rho}v$. Then\n$\\tilde{v}_{a}=0, \\tilde{v}_{n}=b e^{\\rho}$, $\\tilde{v}^{a}=0, \\tilde{v}^{n}=b e^{-\\rho}$.\nLet $M=\\{x\\in\\mathbb{R}^{n}|x^{n}=x_{0}^{n}\\in\\mathbb{R}\\}$, $\\bar{\\mathbf{n}}$ and $\\tilde{\\mathbf{n}}$ be the unit normal vector field of~$M$ with respect to~$\\tilde{h}$ and $\\tilde{F}$, respectively.\nAccording to \\cite{HDY}, we have\n\\begin{equation*}\\label{3.21}\n\\tilde{\\mathbf{n}}=\\bar{\\mathbf{n}}+\\tilde{v}.\n\\end{equation*}\nFrom \\cite{R19}, we know that $\\bar{\\mathbf{n}}=(0, 0, \\cdots, e^{-\\rho})$, then $\\tilde{\\mathbf{n}}=(1+b)\\bar{\\mathbf{n}}$.\n\nDefine\n\\begin{equation*}\n\\Omega:=\\{(m_{1}, m_{2}, \\cdots, m_{n-1}, x^{n})\\in N|~m_{a}\\in\\mathbb{Z}\\}.\n\\end{equation*}\nLet $p=(m_{1}, m_{2}, \\cdots, m_{n-1}, x_{0}^{n})\\in\\Omega$, $\\{e_{1}, e_{2}, \\cdots, e_{n-1}, \\tilde{\\textbf{n}}\\}$ be a local orthogonal frame in the neighbourhood of $p$ with respect to $\\tilde{F}$. Let $\\gamma$ be an unit-speed geodesic with respect to $\\tilde{F}$ and start at $p$ with initial direction $\\tilde{\\textbf{n}}$. By (\\ref{g1}) and (\\ref{g2}), we know that\n$\\tilde{F}$ is not projectively flat. So the geodesics with respect to $\\tilde{F}$ are not necessarily straight lines.\nSet\n\n$(1)~\\Lambda_{a}: x=(x^{1}, \\cdots, x^{a}, \\cdots, x^{n})\\rightarrow \\hat{x}=(x^{1}, \\cdots, -x^{a}, \\cdots, x^{n})$,\n\n$(2)~\\Psi_{m_{a}}: x=(x^{1}, \\cdots, x^{a}, \\cdots, x^{n})\\rightarrow \\check{x}=(x^{1}, \\cdots, x^{a}+2m_{a}, \\cdots, x^{n})$.\n\nNext we will prove $\\hat{\\gamma}(t)=\\Lambda_{a}(\\gamma(t))$\nand $\\check{\\gamma}(t)=\\Psi_{m_{a}}(\\gamma(t))$ are all geodesics.\nSo $\\hat{\\gamma}(t)$ and $\\check{\\gamma}(t)$ should satisfy\n\\begin{equation}\\label{c1}\n\\ddot{\\gamma}^{i}+\\widetilde{\\Gamma}^{i}_{jk}\\dot{\\gamma}^{k}\\dot{\\gamma}^{j}=0.\n\\end{equation}\nFor $\\Lambda_{a}$, we get\n$$\\ddot{\\hat{\\gamma}}^{a}=-\\ddot{\\gamma}^{a}, ~\\alpha(\\dot{\\hat{\\gamma}})=\\alpha(\\dot{\\gamma}),~F(\\dot{\\hat{\\gamma}})=F(\\dot{\\gamma}),$$\n$$\\rho_{a}(\\hat{\\gamma})=-\\rho_{a}(\\gamma),~\\rho_{b}(\\hat{\\gamma})\\dot{\\hat{\\gamma}}^{b}=\\rho_{b}(\\gamma)\\dot{\\gamma}^{b}.$$\nSo by (\\ref{g1}), we have\n\\begin{align*}\n&\\ddot{\\hat{\\gamma}}^{b}+2\\widetilde{G}^{b}(\\hat{\\gamma}, \\dot{\\hat{\\gamma}})\\\\ \\nonumber\n&=\\ddot{\\hat{\\gamma}}^{b}+2\\left[\\left(1-\\frac{b^{2}\\alpha(\\dot{\\hat{\\gamma}})+\\beta(\\dot{\\hat{\\gamma}})}{2F( \\dot{\\hat{\\gamma}})}\\right)\\rho_{c}(\\hat{\\gamma})\\dot{\\hat{\\gamma}}^{c}\\dot{\\hat{\\gamma}}^{b}-\\frac{1}{2}\\lambda F(\\dot{\\hat{\\gamma}})\\alpha(\\dot{\\hat{\\gamma}})\\rho_{b}(\\hat{\\gamma})\\right]\\\\ \\nonumber\n&=\\left\\{\\begin{aligned}\n      &-(\\ddot{\\gamma}^{b}+2\\widetilde{G}^{b}(\\gamma, \\dot{\\gamma})), ~a=b,\\\\\n      &\\ddot{\\gamma}^{b}+2\\widetilde{G}^{b}(\\gamma, \\dot{\\gamma}), ~~~~~~~a\\neq b.\n\\end{aligned}\\right.\n\\end{align*}\nThen $\\hat{\\gamma}(t)=\\Lambda_{a}(\\gamma(t))$ is a geodesic.\n\nFor $\\Psi_{m_{a}}$, we also get\n$$\\ddot{\\check{\\gamma}}^{a}=\\ddot{\\gamma}^{a},\n~F(\\dot{\\check{\\gamma}})=F(\\dot{\\gamma}), ~\\rho_{a}(\\check{\\gamma})=\\rho_{a}(\\gamma).$$\nThen\n\\begin{align*}\n&\\ddot{\\check{\\gamma}}^{a}+2\\widetilde{G}^{a}(\\check{\\gamma}, \\dot{\\check{\\gamma}})\\\\ \\nonumber\n&=\\ddot{\\check{\\gamma}}^{a}+2\\left[\\left(1-\\frac{b^{2}\\alpha(\\dot{\\check{\\gamma}})+\\beta(\\dot{\\check{\\gamma}})}{2F( \\dot{\\check{\\gamma}})}\\right)\\rho_{c}(\\check{\\gamma})\\dot{\\check{\\gamma}}^{c}\\dot{\\check{\\gamma}}^{a}-\\frac{1}{2}\\lambda\\alpha(\\dot{\\check{\\gamma}})F( \\dot{\\check{\\gamma}})\\rho_{a}(\\check{\\gamma})\\right]\\\\ \\nonumber\n&=\\ddot{\\gamma}^{a}+2\\widetilde{G}^{a}(\\gamma, \\dot{\\gamma}).\n\\end{align*}\nSo $\\hat{\\gamma}(t)=\\Psi_{a}(\\gamma(t))$ is also a geodesic.\n\nFrom the above, $\\tilde{\\gamma}(t)=\\Psi_{m_{a}} \\Lambda_{a}(\\gamma(t))\n=(\\gamma^{1}(t), \\cdots, -\\gamma^{a}(t)+2m_{a}, \\cdots, \\gamma^{n}(t))$\nis also a geodesic with respect to $\\tilde{F}$. But $\\tilde{\\gamma}(t)$ and $\\gamma(t)$ have the same initial condition.\nHence, by the uniqueness of geodesic, we get $\\gamma^{a}(t)=m_{a}\\in\\mathbb{Z}$.\nNote that $\\rho(\\gamma(t))=c\\ln 3$,\nwhere $c$ is the number of even entries of $(m_{1}, m_{2}, \\cdots, m_{n-1})$.\nSince\n\\begin{equation}\\label{rou1}\n\\rho_{a}(\\gamma)=\\frac{-\\pi sin(\\pi x^{a})}{2+cos(\\pi x^{a})}(\\gamma)=0.\n\\end{equation}\nThen by (\\ref{g1}) and (\\ref{g2}), we get\n$$\\widetilde{G}^{a}({\\gamma}, \\dot{\\gamma})=\\widetilde{G}^{n}({\\gamma}, \\dot{\\gamma})=0. $$\nFrom (\\ref{c1}), we have\n\\begin{equation}\n\\ddot{\\gamma}^{n}=0.\n\\end{equation}\nAccording to initial condition, then\n\\begin{equation}\\label{j1}\n\\gamma(t)=(m_{1}, m_{2}, \\cdots, m_{n-1}, x_{0}^{n}+(1+b)e^{-\\rho}t),\n\\end{equation}\nwhere $t$ is arc length.\n\n\\subsection{Parallel hypersurfaces}\n\nFor $x_{0}\\in M$, there exist a neighbourhood $U(x_{0})\\subset M$ and a distance function $r(x)$ to $M$, such that $r(x)=t$.\nThen $\\tilde{\\mathbf{n}}=\\nabla r$ and all the integral curves of $\\nabla r$ are all normal geodesics (see \\cite{HYS16} for detail).\nSo we can define the parallel hypersurfaces of $M$ by\n$$M_{t}=\\{x\\in M~|~r(x)=t, t>0\\}.$$\n\nFor $X=X^{i}\\frac{\\partial}{\\partial x^{i}}\\in TM$, note that\n$X^{n}=0$ and $\\tilde{F}_{y^{j}}X^{j}=F_{y^{j}}X^{j}=\\alpha_{y^{j}}X^{j}=0$.\nThen by $\\alpha(\\tilde{\\mathbf{n}})=\\frac{1}{1-b}e^{-\\rho}, \\beta(\\tilde{\\mathbf{n}})=\\frac{-b}{1-b}e^{-\\rho}$, (\\ref{n1}) and (\\ref{n2}), we get\n\\begin{align*}\nD^{\\tilde{\\mathbf{n}}}_{X}\\tilde{\\mathbf{n}}\n=&(\\tilde{\\mathbf{n}}^{i}_{x^{j}}+\\tilde{N}^{i}_{j}(\\tilde{\\mathbf{n}}))X^{j}\\frac{\\partial}{\\partial x^{i}}\\nonumber\\\\\n=&\\left[\\tilde{\\mathbf{n}}^{i}_{x^{a}}+\\rho_{a}\\tilde{\\mathbf{n}}^i\n+\\frac{1}{2}F^{2}(\\tilde{\\mathbf{n}})\\rho_{a}\\left(\\frac{\\alpha(\\tilde{\\mathbf{n}})}{F(\\tilde{\\mathbf{n}})^2}b^{i}\n-\\frac{b^{2}\\alpha+\\beta}{F(\\tilde{\\mathbf{n}})^3}\\tilde{\\mathbf{n}}^{i}\\right)\\right]X^{a}\\frac{\\partial}{\\partial x^{i}} \\nonumber\\\\\n=&(\\tilde{\\mathbf{n}}^{i}_{x^{a}}+\\rho_{a}\\tilde{\\mathbf{n}}^i)X^{a}\\frac{\\partial}{\\partial x^{i}} \\nonumber\\\\\n=&0.\n\\end{align*}\nBy Weingarten formula (\\ref{1.2}), $M$ has constant principal curvatures $\\tilde{\\theta}=0$ with respect to $\\tilde{F}$.\nFrom \\cite{HYS16} (Lemma 4.4), we have\n\\begin{equation*}\n\\frac{d k_{a}(t)}{dt}=\\tilde{K}(\\tilde{\\mathbf{n}}, e_{a})+k^{2}_{a}(t),\n\\end{equation*}\nwhere $k_{a}(t)$ are the principal curvature of $M_{t}$, $k_{a}(0)=\\mu_{a}$ and $\\tilde{K}$ is the flag curvature of $(N, \\tilde{F})$. Take the trace\n\\begin{equation}\\label{h1}\n\\frac{d}{dt}\\sum\\limits_{a=1}^{n-1}k_{a}(t)=\\widetilde{Ric}(\\tilde{\\mathbf{n}})+\\sum\\limits_{a=1}^{n-1}k_{a}^{2}(t),\n\\end{equation}\nwhere $\\widetilde{Ric}$ is the Ricci curvature of $(N, \\tilde{F})$.\n\nNow we prove that $M_{t}$ is neither isoparametric nor $d\\mu$-isoparametric.\nBy \\cite{SC07}, we have\n\\begin{align}\n\\widetilde{Ric}(y)\n=&Ric(y)-F^{2}\\{2\\rho_{k}J^{k}+g^{ij}\\rho_{i|j}+[\\rho^{j}I_{j}]_{|k}y^k+2\\theta\\rho^{j}I_{j}\\}\\\\ \\nonumber\n&+(n-2)(\\Xi-F^{2}\\parallel d\\rho\\parallel^{2}_{F})+F^{4}\\rho^{j}\\rho^{k}\\{2I_{s}C^{s}_{ij}-[I_{j}]_{y^{k}}-C^{s}_{ij}C^{i}_{ks}\\}, \\nonumber\n\\end{align}\nwhere $Ric(y)$ is the Ricci curvature of $(N, F)$, $J^{k}=g^{ij}L^{k}_{ij}, ~\\theta(y)=\\rho_{k}y^{k},$\n$\\Xi=\\theta^{2}-\\theta_{x^k}y^{k}$, $\\parallel d\\rho\\parallel^{2}_{F}=g^{ij}\\rho_{i}\\rho_{j}.$\nNote that $G^{i}=0$,\n$L_{ijk}=0$, $Ric(y)=0$, $C_{ijn}(\\tilde{\\mathbf{n}})=0$. By (\\ref{gab}), we get\n\\begin{equation*}\n2C_{abc}=-\\frac{(\\delta_{ab}y^{c}+\\delta_{ac}y^{b}+\\delta_{bc}y^{a})\\beta}{\\lambda^{2}\\alpha^3}\n+3\\frac{\\beta y^{a}y^{b}y^c}{\\lambda^3 \\alpha^5},\n\\end{equation*}\n\\begin{equation*}\n2C_{abn}=\\frac{\\delta_{ab}(\\alpha\\beta_{y^{n}}-\\beta\\alpha_{y^{n}})}{\\lambda\\alpha^2}\n-\\frac{y^{a}y^{b}(\\alpha\\beta_{y^{n}}-3\\beta\\alpha_{y^{n}})}{\\lambda^2 \\alpha^4},\n\\end{equation*}\nthen $C_{abc}(\\tilde{\\mathbf{n}})=0$. In conclusion, $C_{ijk}(\\tilde{\\mathbf{n}})=0$.\n\\begin{align*}\n2\\frac{\\partial C_{abc}}{\\partial y^{d}}\n&=-\\frac{(\\delta_{ab}\\delta_{cd}+\\delta_{ac}\\delta_{bd}+\\delta_{bc}\\delta_{ad})\\beta}{\\lambda^{2}\\alpha^3}\\\\\n&-3\\frac{(\\delta_{ab}y^{c}+\\delta_{ac}y^{b}+\\delta_{bc}y^{a})\\beta\\alpha_{y^{d}}}{\\lambda^{2}\\alpha^4}\n+3(\\frac{\\beta y^{a}y^{b}y^c}{\\lambda^3 \\alpha^5})_{y^{d}},\n\\end{align*}\n\\begin{align*}\n2\\frac{\\partial C_{abn}}{\\partial y^{n}}\n&=\\frac{\\delta_{ab}[-\\beta\\alpha\\alpha_{y^{n} y^{n}}-2(\\alpha\\beta_{y^{n}}-\\beta\\alpha_{y^{n}})\\alpha_{y^{n}}]}{\\lambda\\alpha^3}\n-y^{a}y^{b}[\\frac{(\\alpha\\beta_y^{n}-3\\beta\\alpha_y^{n})}{\\lambda^2 \\alpha^4}]_{y^{n}}.\n\\end{align*}\nThen $C_{abc}(\\tilde{\\mathbf{n}})=0, ~2\\frac{\\partial C_{abc}}{\\partial y^{d}}(\\tilde{\\mathbf{n}})=\\frac{(\\delta_{ab}\\delta_{cd}+\\delta_{ac}\\delta_{bd}+\\delta_{bc}\\delta_{ad})be^{2\\rho}}{(1+b)^{2}}$,\n$\\frac{\\partial C_{abn}}{\\partial y^{n}}(\\tilde{\\mathbf{n}})=0$.\n$$\\theta(\\tilde{\\mathbf{n}})=0,\n~F(\\tilde{\\mathbf{n}})=e^{-\\rho}, ~\\Xi(\\tilde{\\mathbf{n}})=0,\n~\\frac{\\alpha(\\tilde{\\mathbf{n}})}{F(\\tilde{\\mathbf{n}})}=\\frac{1}{1-b},~I_{i}(\\tilde{\\mathbf{n}})=0,$$\n$$g^{ab}(\\tilde{\\mathbf{n}})=(1+b)\\delta^{ab},~g^{an}(\\tilde{\\mathbf{n}})=0,~g^{nn}(\\tilde{\\mathbf{n}})=(1+b)^{2},$$\n$$\\parallel d\\rho\\parallel^{2}_{F}(\\tilde{\\mathbf{n}})=g^{ij}(\\tilde{\\mathbf{n}})\\rho_{i}\\rho_{j}\n=(1+b)\\delta^{ab}\\rho_{a}\\rho_{b}=(1+b)\\sum\\limits_{a=1}^{n-1}(\\rho_{a})^2,$$\n$$g^{ij}(\\tilde{\\mathbf{n}})\\rho_{i|j}\n=-(1+b)\\delta^{ab}\\frac{\\pi^{2}+2\\pi^{2}cos(\\pi x^{b})}{[2+cos(\\pi x^{a})]^{2}}\n=-(1+b)\\sum\\limits_{a=1}^{n-1}\\frac{\\pi^{2}+2\\pi^{2}cos(\\pi x^{a})}{[2+cos(\\pi x^{a})]^{2}},$$\n$$[\\rho^{j}I_{j}]_{|k}y^k(\\tilde{\\mathbf{n}})=(\\rho^{j}_{|k}I_{j}(\\tilde{\\mathbf{n}})+\\rho^{j}I_{j|k}(\\tilde{\\mathbf{n}}))\\tilde{\\mathbf{n}}^k\n=\\rho^{j}g^{il}\\frac{\\partial C_{ilj}}{\\partial x^k }\\tilde{\\mathbf{n}}^k=0,$$\n\\begin{align*}\n&\\rho^{j}\\rho^{k}[I_{j}]_{y^{k}}(\\tilde{\\mathbf{n}})\n=\\rho^{j}\\rho^{k}[-2C^{il}_{k}C_{ilj}+g^{il}\\frac{\\partial C_{ijl}}{\\partial y^{k}})](\\tilde{\\mathbf{n}})\\\\\n&=\\rho^{a}\\rho^{b}[g^{cd}\\frac{\\partial C_{bcd}}{\\partial y^{a}}+g^{cn}\\frac{\\partial C_{abc}}{\\partial y^{n}}+g^{nn}\\frac{\\partial C_{abn}}{\\partial y^{n}}](\\tilde{\\mathbf{n}})\\\\ \\nonumber\n&=\\rho^{a}\\rho^{b}\\frac{(\\delta_{ab}\\delta_{cd}+\\delta_{ac}\\delta_{bd}+\\delta_{bc}\\delta_{ad})be^{2\\rho}\\delta^{cd}}{(1+b)}\n=\\sum\\limits_{a=1}^{n-1}(\\rho^{a})^2\\frac{(n+2)be^{2\\rho}}{(1+b)}.\n\\end{align*}\nSo\n\\begin{align}\n&\\widetilde{Ric}(\\tilde{\\mathbf{n}})=e^{-2\\rho}(1+b)\\sum\\limits_{a=1}^{n-1}\\left[\\frac{\\pi^{2}+2\\pi^{2}cos(\\pi x^{a})}{[2+cos(\\pi x^{a})]^{2}}\n-(n-2)(\\rho_{a})^2-(\\rho^{a})^2\\frac{(n+2)b}{(1+b)^2}\\right].\n\\end{align}\nIf $u_{1}=(0, 0, \\cdots, x_{0}^{n})\\in M\\cap\\Omega$ and $u_{2}=(1, 1, \\cdots, x_{0}^{n})\\in M\\cap\\Omega$, we know\n$$\\widetilde{Ric}(\\tilde{\\mathbf{n}})|_{\\gamma_{u_{1}}}=3^{(-2n+1)}(1+b)\\pi^{2}(n-1)>0,$$\nand\n$$\\widetilde{Ric}(\\tilde{\\mathbf{n}})|_{\\gamma_{u_{2}}}=-(1+b)\\pi^{2}(n-1)<0.$$\nMeanwhile, when $t=0$, $k_{a}^{2}|_{\\gamma_{u_{1}}(0)}=k_{a}^{2}|_{\\gamma_{u_{2}}(0)}=0$. Then by (\\ref{h1}),\n$\\frac{d}{dt}\\sum\\limits_{i=1}^{n}k_{a}|_{\\gamma_{u_{1}}(0)}>0$ and $\\frac{d}{dt}\\sum\\limits_{i=1}^{n}k_{a}|_{\\gamma_{u_{2}}(0)}<0$.\nSo the mean curvatures of the parallel hypersurfaces $M_{t}$ is not constant.\nNote that $\\rho_{k}\\tilde{\\mathbf{n}}^{k}=0$ and by (\\ref{s1}), we have\n$$\\tilde{S}(\\tilde{\\mathbf{n}})=0.$$\n\\begin{remark}\nAccording to Definition \\ref{d1}, $M$ with constant principal curvatures is neither isoparametric hypersurface nor $d\\mu$-isoparametric hypersurface.\n\\end{remark}\n\n\\small ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\chapter{Introduction}\n\nRising demands for data and computationally-intensive applications such as deep learning\nand big-data driven applications entail the request for higher performance processors, e.g., multi-core CPUs, Graphics Processor Units (GPUs), and Tensor Processing Units (TPUs) as well as more aggressive computing technologies such as 3D chips. With this rapidly increasing power trend, the problem of thermal management in systems is becoming acute. My research focuses on enabling thermal-aware real-time computing in next-generation cyber-physical systems (CPS), such as self-driving cars driving across the country, drones flying over volcanic areas, and medical devices for humans, where the systems have to timely accomplish their missions even under harsh thermal conditions. This is an important problem because the two requirements, real-time and thermal, are fundamental to the reliable operation of such systems, but hard to be holistically analyzed and optimized by the current state of the art. Solving this problem will bring significant benefits like improving system safety and reliability, adding more intelligent features, and reducing costs for high-volume products. This dissertation presents novel analytical methodologies and system software techniques to analyze and safely guarantee both requirements. \n\nOne of major factors that plays a role in temperature increase of modern systems-on-chips (SoCs) is heat generation due to complex computations. There exist two reasons that directly or indirectly result in temperature increase: 1) the increased number of accelerators and CPU cores, and 2) the high power consumption of each subsystems. Nowadays, different types of accelerators, such as TPUs, FPGAs, DSPs, GPUs, video processing units, and also multiple heterogeneous CPU cores are integrated into SoCs to satisfy various needs of complex applications. The power consumption of active units to perform computation generates heat. There exists heat conduction between these units, which causes temperature increase on other units even when they are idle. Aside from it, the need for high computation which can inevitably necessitate a high clock frequency and a large number of transistors further raises the power consumption of each computing unit.  The power consumption is converted into heat, resulting in a significant increase in temperature due to performing computation on active computing units. Moreover, an increase in temperature has a considerable impact on the growth of leakage current which leads to a rise in static power consumption. An increase in power consumption levels up the device temperature, thereby resulting  in again an increase in power consumption. This detrimental loop causes``thermal runaway''~\\cite{ahmed2016necessary}.\n\nAmbient temperature is one of the key factors that affect the cooling process of mixed-criticality CPS.\nDesigning of cooling packages such as heat sinks and cooling fans can be costly~\\cite{tiwari1998reducing}. There is also an extra challenge in cooling solutions through packaging only to enable thermal safety under various thermal conditions. Active\/passive cooling packages attached to SoCs dissipate the heat generated due to computation. These solutions try to transfer heat from SoCs to the environment. Fans and heat sinks are an example of active and passive cooling packages respectively that blow air to chip  or maximize the hot surface area in contact with the cooling medium surrounding it, e.g., transferring heat to ambient air by heat convection. These mechanisms are only effective when SoCs are exposed to low-temperature ambient air. In harsh ambient temperature, heat flux is reduced substantially, meaning heat convection to the air becomes little,  because the temperature difference between the operating chip and the air is negligible. Therefore, the harsh ambient temperature reduces the the effectiveness of the cooling process while SoCs still generate heat to perform computation. This condition can occur frequently in practice which may lead to burnout of SoC. In addition, for many high-performance embedded systems like medical implants or smartphones, that require thermal safety, packaging solutions are expensive, bulky and inapplicable.\n\nTo protect the processor chip from thermal damage, a set of policies are defined in the thermal governor of the OS for different scenarios. When the chip temperature crosses a trip point, a predefined cooling scenario is performed such as frequency throttling or shutting down of CPU cores~\\cite{choi2004fine, herbert2007analysis, wang2009temperature}. Such thermal countermeasures, however, lead to timing unpredictability in real-time mixed-criticality systems (MCS) since the deadlines of tasks could be unexpectedly violated by reduced processing speed or temporarily unavailable CPU cores. A single delayed response in the pedestrian detection in a self-\ndriving car can lead to irreparable harm. A delay in the control system of the Patriot\nmissiles, used to protect Saudi Arabia during the Gulf War led to injuries and enormous\neconomic damage\\footnote{L'Espresso, Vol. XXXVIII, No. 14, 5 April 1992, p. 167.}.\n\nThis unwanted performance degradation may also affect the timing predictability of integrated accelerators as well. In this dissertation, we focus on integrated GPUs because the  unique  characteristics  of  GPU operations, e.g., kernel execution on the GPU and interactions between  CPU  and  GPU  cores  for  data  transfer, introduce new  challenges  to  thermal-safety  design  and  timing assurance. For instance, miscellaneous operations for GPU segments require CPU intervention. The intermittent performance degradation of CPU cores adds up additional delay for completion of real-time tasks because GPUs are required to spend extra time for miscellaneous operations while other tasks are waiting for access to shared GPUs. This delay due to CPU slowdown leads to performance unpredictability of real-time GPU-using tasks on multi-core SoC platforms. \n\nDespite its importance, the thermal parameter estimation of commercial off-the-shelf (COTS) processors still remains a challenging problem. The practical use of thermally-safe mechanisms remains largely limited due to the fact that it is extremely difficult to obtain a precise thermal model of commercial-processors without using special measurement instruments or access to proprietary information, such as the power traces of micro-architectural units and detailed floorplan maps. \n\nIn this dissertation, we focus on challenges arising from the thermal interference on heterogeneous multi-core SoC platforms. We develop novel frameworks supported by analytical time and thermal models to bound the maximum operating temperature of CPU cores with the assurance of performance predictability in mixed-criticality domains. Additionally, we propose a data-driven thermal parameter estimation scheme that is directly applicable to MCSs built with commercial-off-the-shelf multi-core processors to obtain a precise thermal model without using special measurement instruments or access to proprietary information. The main thesis supported by this dissertation is as follows:\\\\\n\n\\begin{minipage}{5.4in}\n\\textbf{Thesis Statement:} The timing predictability of real-time tasks in dynamic thermal conditions is achievable on multi-core GPU-integrated SoC platforms by designing a novel thermal-aware system framework with the support of analytical timing and thermal models.\n\\end{minipage}\n\\\\\n\n\\section{Thesis Outline and Contributions}\nThe aim of my PhD work is to test the hypothesis that thermal-aware real-time computing\non heterogeneous multi-core CPU and GPU systems can be achieved by novel analytical\nand software support. To do so, my work develops an ensemble framework for real-time\ntasks to support various policies under different circumstances while providing analytical\nfoundations to check both thermal and temporal safety. This dissertation includes the following\nseveral components:\n\\subsection{Chapter 2: Thermal-Aware Servers for Real-Time Tasks on Multi-Core GPU-Integrated Embedded Systems}\nIn Chapter 2, we propose a thermal-aware CPU-GPU framework to handle both real-time CPU-only and GPU-using tasks. Our framework satisfies thermal-safety under the given thermal constraint on CPU and GPU cores and offers bounded response time to real-time tasks. The framework also introduces two novel mechanisms for GPU requests to reduce task response time.  \n\n\\smallskip\\noindent\\textbf{Contributions.} The contributions of this work are as follows:\n\\begin{itemize}\n\\item We propose a thermal-aware CPU-GPU server framework for multi-core GPU-integrated real-time systems. We characterize different timing penalties and present a protocol for CPU and GPU thermal servers.\n\\item For real-time predictability on a GPU, we propose a GPU server design with a variant of the sporadic server policy, where the GPU segments of tasks execute with no thermal violation.\n\\item We propose an enhancement to the waiting queue of the GPU server to mitigate the pessimism of a priority-based queue.\n\\item We introduce a \\textit{miscellaneous operation time reservation} mechanism for deferrable and sporadic CPU servers to reduce CPU-GPU handover delay and remote blocking time.\n\\item We extensively analyze the thermal safety and task schedulability of CPU and GPU servers with various budget replenishment policies.\n\\end{itemize}\n\n\\subsection{Chapter 3: On Dynamic Thermal Conditions in Mixed-Criticality Systems}\n The key contribution of our work in Chapter 3 is showing that the problem of thermal-aware real-time scheduling can be decomposed into thermal schedulability (how much CPU budget is usable under thermal constraints) and timing schedulability (if tasks are schedulable using given budget).\n \n Under harsh ambient temperature, a system may not be able to utilize 100\\% of CPU time even if the CPU runs at the minimum possible frequency with active\/passive cooling packages.  In such a condition, the only option left to ensure timing and thermal guarantees of critical tasks is to secure cooling time by suspending less critical tasks. Our work addresses this problem. \n \n \\smallskip\\noindent\\textbf{Contributions.} The contributions of this work are as follows:\n\\begin{itemize}\n\\item We show that the problem of thermal-aware real-time scheduling can be decomposed into thermal schedulability (how much CPU budget is usable under thermal constraints) and timing schedulability (if tasks are schedulable using given budget). Our thermal schedulability achieves the simplicity in timing analysis by ensuring that the budget is guaranteed to be made available for any execution patterns without violating thermal constraints.\n\n\\item We extensively analyze the thermal safety of a multi-core system and bound the maximum operating temperature that the system can reach. At a specific ambient temperature level, we characterize the worst-case thermal behavior of a system \\cmnt{by  using the notion of idle servers} and also determine the minimum time for the system to transition from one criticality level to a lower level.\n\\item \\new{We introduce the notion of \\textit{idle thermal servers} that allow bounding the maximum operating temperature caused by multiple preemptive active servers scheduled dynamically on a multi-core processor for a given  mixed-criticality taskset.} \n\n\\end{itemize}\n\n\\subsection{Chapter 4: Data-driven Thermal Parameter Estimation for COTS-based Mixed-criticality Systems}\nIn Chapter 4, we propose a fast and accurate scheme to estimate the thermal parameters of COTS multi-core processors for real-time MCS. Our scheme requires only a small number of temperature traces from on-chip thermal sensors which are widely available in today's processors. Our scheme also improves the accuracy of thermal parameters through the ensemble of measurements from different frequency levels and execution patterns. \n\n\\smallskip\\noindent\\textbf{Contributions.} The contributions of this work are as follows:\n\\begin{itemize}\n    \\item We present a thermal estimation scheme that has low computational cost by design. Given that steady-state profiles are much compact than transient-state profiles, our scheme first estimates the thermal parameters of a given system using only steady-state profiles, and then uses transient-state data for calibration purpose. \n    \n    \\item We characterize various sources of errors in thermal parameter estimation, and reduce their negative effects through the multiple refinement stages of our scheme. Our scheme also enables locating errors in the temperature profiles.\n    \\item Our scheme can identify the relative distance between CPU cores and produce an estimated chip floorplan from temperature profiles. It can also estimate the relative power consumption for a given  workload on each CPU core.\n    \\item We present techniques to further improve the accuracy of thermal parameters by exploiting the ensemble of measurement data obtained at various frequency and workload settings.\n\\end{itemize}\n\n\\chapter{Thermal-Aware Servers for Real-Time Tasks on Multi-Core GPU-Integrated Embedded Systems}\n\\input{Chapter2\/_0abstract}\n\\input{Chapter2\/_1intro}\n\\input{Chapter2\/_2related}\n\\input{Chapter2\/_3prelim.tex}\n\\input{Chapter2\/_4model}\n\\input{Chapter2\/_5frame}\n\\input{Chapter2\/_60analysis}\n\\input{Chapter2\/_61schd}\n\\input{Chapter2\/_7eval}\n\\input{Chapter2\/_8CaseStudy}\n\\input{Chapter2\/_9conclud}\n\\section{Introduction}\n\n\n\nHigh temperature in embedded systems with modern systems-on-chips (SoCs) causes several major issues. \nAn increase in temperature has a considerable impact in the growth of leakage current which leads to rise in static power consumption. An increase in power consumption levels up the device temperature, thereby resulting  in again an increase in power consumption. \nThis detrimental loop causes not only rapid battery drain but also  ``thermal runaway''~\\cite{ahmed2016necessary}.  Furthermore, studies show that operating in high temperature reduces the system reliability substantially~\\cite{srinivasan2004impact}.  For instance,  10\\textdegree C to 15\\textdegree C increase in temperature doubles the probability of failure of the underlying electronic devices~\\cite{viswanath2000thermal}. Thermal violation not only reduces the reliability in real-time implantable medical devices, but also causes physical harm~\\cite{chandarli2015response}. Therefore, bounding the maximum temperature is an important issue, especially when real-time requirements have to be satisfied. \n\nThermal management on today's multi-core CPU-GPU integrated platforms with real-time requirements is a challenging problem. Dynamic Thermal Management (DTM) is triggered when thermal violation occurs so that it forces frequency throttling or shutdown of the SoC for cooling purpose. This unwanted performance degradation leads to timing unpredictability in task execution, and real-time tasks may miss their deadlines. Thermal violation avoidance in uni-processor systems has been studied extensively in the literature of real-time systems~\\cite{ chandarli2015response, 4032360}  but it cannot be directly used for multi-core GPU-integrated devices due to heat conduction between processor units. Dynamic Voltage Frequency Scaling (DVFS) techniques to mitigate the heat and power dissipation of processors also has been widely studied in the literature~\\cite{ 4032360, chen2009proactive}. However, aside from a considerable reduction in system reliability over time due to continuous frequency changes~\\cite{iranfar2017thespot, lasance2003thermally,  xiang2010system}, not all embedded devices support DVFS, especially for integrated GPUs.  \n\nDespite the popularity of integrated GPUs in modern multi-core SoCs, state-of-the-art approaches are incapable of simultaneously addressing thermal management and real-time schedulability issues. On the one hand, the GPU access segment of a real-time task has been modeled as a critical section to ensure schedulability~\\cite{GPUSync, elliott2012globally, patel2017}. These approaches, however, are oblivious of thermal constraints so that the system may suffer from heat dissipation and intermittent performance drops by DTM. On the other hand, there are previous studies~\\cite{Ahmed2017, ahmed2011minimizing}  introducing the concept of thermal servers for real-time uni-processor and multi-core platforms. However, their schemes are not ready to use for a multi-core SoC with an integrated GPU. The unique characteristics of GPU operations, e.g., kernel execution on the GPU and interactions between CPU and GPU cores for data transfer, introduce new challenges to thermal server design and schedulability analysis. To the best of our knowledge, there is no prior work that offers both thermal safety and real-time schedulability in  multi-core GPU-integrated embedded systems.\n\nIn this chapter, we propose a thermal-aware CPU-GPU framework to handle both real-time CPU-only and GPU-using tasks. Our framework enhances the notion of thermal servers to satisfy the given thermal constraint on CPU and GPU cores and to offer bounded response time to real-time tasks.\nThe framework also introduces two mechanisms, miscellaneous-operation-time reservation and pre-ordered waiting queue for GPU requests, to reduce task response time. We will show with experimental results that our framework is effective in satisfying both thermal and temporal requirements. \n\n\\smallskip\\noindent\\textbf{Contributions.} The contributions of this chapter are as follows:\n\\begin{itemize}\n\\item We propose a thermal-aware CPU-GPU server framework for multi-core GPU-integrated real-time systems.\n\n We characterize different timing penalties and present a protocol for CPU and GPU thermal servers.\n\\item For real-time predictability on a GPU, we propose a GPU server design with a variant of the sporadic server policy, where the GPU segments of tasks execute with no thermal violation.\n\\item We propose an enhancement to the waiting queue of the GPU server to mitigate the pessimism of a priority-based queue.\n\\item We introduce a \\textit{miscellaneous operation time reservation} mechanism for deferrable and sporadic CPU servers to reduce CPU-GPU handover delay and remote blocking time.\n\\item We extensively analyze the thermal safety and task schedulability of CPU and GPU servers with various budget replenishment policies.\n\\end{itemize}\n\n\n\n\\section{Background on Thermal Behavior}\n\\label{sec:chap2_background}\n\nIn this section, we briefly introduce the thermal model used in this chapter. It depends on power consumption, heat dissipation, and the conductive heat transfer between adjacent power-consuming resource components, which includes CPU cores, CPU peripherals, GPU, caches, and other IPs.\n\n\n\n\\smallskip\\noindent\\textbf{Uni-processor thermal model.} With respect to the power function~\\cite{4212027, 4484694}, the thermal model of a uni-processor is modeled as an RC circuit in the literature~\\cite{4484694, schor2012worst, Skadron}. According to the Fourier's Law~\\cite{wang2010schedulability} and considering $t_0 = 0$, the temperature of a core $\\theta(t)$ after $t$ time units operating at a fixed clock frequency is given by:\\footnote{Details are available in~\\cite{DSouza2017ThermalIO} and~\\cite{Ahmed2017}.}\n\\begin{equation}\n     \\theta(t) = \\alpha + (\\theta(t_0)-\\alpha) e^{\\beta t}\n     \\label{eq:chap2_tempActive}\n\\end{equation}\nwhere $\\alpha > 0$ and $\\beta < 0 $ are constants. \n\nMost operating systems in embedded platforms transition the processor to the sleep state when there is no task execution. Therefore, the power consumption can be assumed to be negligible in such a case. The thermal function in the sleep state (\\textit{aka} cooling phase) where the frequency is switched off can be modeled as:\n\\begin{equation}\n    \\theta(t) = \\theta(t_0) e^{\\beta t}.\n    \\label{eq:chap2_tempCool}\n\\end{equation}\n because in the cooling phase, $\\alpha = 0$.\n \n\\smallskip\\noindent\\textbf{Heterogeneous multi-core thermal model.} With the presence of multiple cores and other power-consuming resources, there is heat dissipation not only to ambient but also between nodes. In this chapter, we only consider CPU and GPU cores as power-consuming nodes because other IPs consume much less power than them, thereby causing negligible thermal effects. \n\nIn such a CPU-GPU integrated system with the lateral thermal conductivity between processing cores, the temperature of a core depends not only on its current temperature and power consumption, but also on those of adjacent cores.  Prior work~\\cite{6629322} showed that the temperature of each core at time $t+\\Delta$ can be modeled with an acceptable accuracy as follows:\n\\[ \\Theta(t+\\Delta) = A \\times P(t+\\Delta) + \\Gamma \\times \\Theta(t)  \\]\nwhere $A$ is  $m \\times m$ matrix, and $\\Gamma$, $P$ and $\\Theta$ are $m \\times 1$ matrices, respectively. One can interpret the thermal model as a function of the current temperature and heat produced by execution and the thermal conductivity between cores. \nHence, according to the thermal composability characteristics of heat transfer~\\cite{Ahmed2017} with the initial temperature of $\\theta(t_0)$, the temperature after $t$ time units is given by:\n\\begin{equation}\n\\theta_i(t_0+t) = \\alpha + (\\theta(t_0)-\\alpha) e^{\\beta t} +  \\sum_{j=1}^{m} \\gamma_{ij} \\theta_j(t_0+t).\n    \\label{eq:chap2_tempCompo}\n\\end{equation} \n\\section{Related Work}\n\\label{sec:chap2_rel}\n\nReal-time GPU management has been studied to handle GPU requests with the goal of improving timing predictability~\\cite{kato2011timegraph, kato2011rgem, kato2012gdev, zhou2015gpes}. To guarantee the schedulability of GPU-using real-time tasks, synchronization-based approaches~\\cite{GPUSync,elliott2012globally,patel2017} that model GPU access segments as critical sections have been developed. The work in \\cite{8046309,kim2018server} introduces a dedicated GPU-serving task as an alternative to the synchronization-based approaches. While these prior studies have established many fundamental aspects on real-time task schedulability with GPUs, thermal violation issues have not been considered. \n\nThere exist extensive studies on bounding the maximum temperature in real-time uni-processor systems~\\cite{chandarli2015response, Youngmoon2018} and non-real-time heterogeneous multi-core systems~\\cite{7092527, Prakash, 7746768, 7372657, 7904613}. In \\cite{chandarli2015response}, the authors proposed a novel scheme that bounds the maximum temperature by analyzing task execution and cooling phases in uni-processor real-time systems. Real-time thermal-aware resource management for uni-processor systems has been developed with the consideration of varying ambient temperature and diverse task-level power dissipation~\\cite{Youngmoon2018}. For non-real-time heterogeneous systems, Singla et. al~\\cite{7092527} proposed a dynamic thermal and power management algorithm to adjust the frequency of GPU and CPU as well as number of active cores by computing the power budget. Prakash et al.~\\cite{Prakash} proposed a control-theory based dynamic thermal management technique for mobile games.\nGong et al.~\\cite{7904613} presented a thermal model on a real-life heterogeneous mobile platform. In~\\cite{7372657} and~\\cite{7746768}, the authors proposed a proactive frequency scheduling for heterogeneous mobile devices to maintain their performance. However, these studies cannot be directly applied to real-time multi-core GPU-integrated systems where the GPU is shared among tasks.\n\n  \nThe notion of periodic thermal-aware servers was proposed in \\cite{ahmed2011minimizing} for uni-processors. In this work, the optimal server utilization has been proved and the budget replenishment period is determined by a heuristic algorithm. Similar to the thermal server, the notion of cool shapers was proposed in~\\cite{kumar2011cool} to satisfy the maximum temperature constraint by throttling task execution. The notion of hot tasks was introduced in~\\cite{Huang2014} to partition lengthy tasks into several chunks to avoid continuous task execution and thermal violation while maximizing throughput. Although all of these aforementioned studies have brought valuable contributions, they have been proposed for uni-processor platforms.  In contrast, our framework addresses the temperature bounding problem for real-time tasks with GPU segments running on modern CPU-GPU integrated SoCs. \n\nRecently, the authors of~\\cite{DSouza2017ThermalIO} introduced  a novel technique for periodic tasks executing on multi-core platform. This technique introduces an Energy Saving (ES) task that runs with the highest priority and captures the sleeping time of CPU cores. The technique can be seen as an alternative to a thermal server because the ES task effectively models the budget-depleted duration of a thermal server. The authors of~\\cite{Ahmed2017} proposed thermal-isolation servers that avoid the thermal interference among tasks in temporal and spatial domains with  thermal composability. These techniques, however, cannot address the challenges of scheduling GPU-using real-time tasks.\n\n \n\n\n\n\n\\section{Framework Design}\n\\label{sec:chap2_frame}\nIn this section, we present our proposed framework. We first give a protocol for CPU and GPU thermal servers, and then explain thermal server design to address the challenges discussed in the previous section. We lastly describe a \\textit{miscellaneous operation time reservation} mechanism to trade-off between server budget and GPU waiting time. \n\n\n\\subsection{Thermal-Aware CPU-GPU Server Protocol}\nOur framework simultaneously bounds the worst-case blocking time for GPU access and the maximum temperature of CPU and GPU cores. The thermal server designed with our framework isolates the thermal conductive effects of each compute node to other nodes. To achieve these properties, we establish the following rules in our framework.\n\n\n\n\\smallskip\\noindent\\textbf{Shared GPU Server}\n\\begin{enumerate}\n    \\item Pending GPU requests are inserted to a priority queue that orders the requests based on the priorities of the corresponding tasks. This rule assures that the GPU request of a high-priority task is blocked by at most one lower-priority request in the queue.\n    \\item To handle the GPU request of a task $\\tau_i$, there must exist at least $E_i$ budget available on the GPU server. This rule is due to the preemptive and non-self-suspending characteristics of the GPU. \n    \\item If there is an insufficient amount of budget to launch a GPU segment, the GPU is locked until having enough budget for that GPU segment. This rule assures that starvation does not happen and the critical section of a higher-priority task waits for just a single critical section of a lower-priority task. With these rules, remote blocking time can be bounded. Later, we will propose an enhancement to these rules.\n\n\\end{enumerate}\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.55\\textwidth}\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{Chapter2\/figs\/frmwrk\/nobusy_wait.pdf}\n  \\caption{}\n  \\label{fig:chap2_busy_waitA}\n\\end{subfigure}\n\\begin{subfigure}{.43\\textwidth}\n  \\centering\n \n  \\includegraphics[width=\\linewidth]{Chapter2\/figs\/frmwrk\/busy_wait}\n   \\caption{}\n  \\label{fig:chap2_busy_waitB}\n\\end{subfigure}\n\\caption{Example task scheduling with an unlimited GPU server budget and a CPU server following the polling server policy a) No busy waiting: $\\tau_1$ finishes at 22. b) Busy waiting: $\\tau_1$ finishes at 14.}\n\\label{fig:chap2_busy_wait}\n\\end{figure}\n\n\\smallskip\\noindent\\textbf{CPU Core Server}\n\\begin{enumerate}\n   \n    \\item Servicing a GPU request boosts the priority of the corresponding task to the highest-priority level. As a result, the normal segments of higher-priority tasks are blocked until the GPU segment completes. If there is not enough budget on the corresponding CPU server (e.g., $M_{i,1}$) or on the GPU server (e.g., $E_i$), no other task can execute on the CPU or the GPU.\n    \\item During data transmission to\/from the GPU, the CPU server ``busy-waits\". The reasoning behind this rule is to reduce the total response time of a task. Fig. \\ref{fig:chap2_busy_wait} illustrates task scheduling with and with or without this rule on a CPU server using the polling server policy. As one can see in this example, without this rule, it takes one additional replenishment period for transferring data to\/from the CPU, because during processing the transferring request on the GPU, the CPU server has no other workload to execute; hence it deactivates until the beginning of the next replenishment period.\n   \n\\end{enumerate}\n\\subsection{GPU Server Design}\nWe now discuss budget replenishment policies for the GPU server and their implications. \nOne may consider both the CPU and GPU servers following the polling server policy. \nIn this case, the CPU-GPU handover delay can be at least three complete replenishment periods of a CPU server.\nFor instance, consider a task holding a GPU lock and its kernel being executed on the GPU. Once the kernel execution completes, the GPU has to wait for the task on the CPU to transfer the results and release the lock. It is possible that at this time, the CPU server budget has been already depleted. Thus, the GPU has to wait for the next period of the CPU server, and this kind of extra delay happens for each sub-segment of the CPU segment, i.e., $M_{i,1}$, $K_i$ and $M_{i,2}$. \nMoreover, since operations on the GPU are non-preemptive and non-suspendable, the GPU server budget has to be large enough that at least one entire GPU sub-segment of $M_{i,1}$, $K_i$ or $M_{i,2}$ can complete its execution within the same period. However, having a large replenishment period would exacerbate the response time of GPU segments especially for small kernels.\n\n\nOne may consider busy waiting between GPU sub-segments so as to fill their execution gaps because such gaps may deactivate the GPU's polling server. This approach, however, not only requires over-provisioning of the GPU budget, but also produces a considerable amount of heat. In the worst case, the extra heat generation due to the busy-waiting on the GPU server may continue over two periods of the CPU server because the CPU server with the polling server policy can be deactivated between GPU sub-segments.\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}{.85\\textwidth}\n  \\centering\n \n \\includegraphics[width=\\textwidth]{Chapter2\/figs\/frmwrk\/def.pdf}\n  \\caption{}\n  \\label{fig:chap2_thermal_bck2bckA}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.85\\textwidth}\n  \\centering\n \n  \\includegraphics[width=\\textwidth]{Chapter2\/figs\/frmwrk\/sprdc.pdf}\n  \\caption{}\n  \\label{fig:chap2_thermal_bck2bckB}\n\\end{subfigure}\n\\caption{Example task scheduling with the GPU server under a) the deferrable server policy and b) the sporadic server policy. For both tasks $\\tau_1$ and $\\tau_2$, $E_i=8$ and $M_{i,1}=M_{i,2}=2$.}\n\\label{fig:chap2_thermal_bck2bck}\n\\end{figure}\n\nOne may suggest the deferrable server policy for the GPU to mitigate the heat generation issue and to minimize the response time of a GPU segment. This approach, however leads to the thermal back-to-back execution phenomenon. Fig.~\\ref{fig:chap2_thermal_bck2bck}a illustrates an example of possible drawbacks of the deferrable server chosen for the GPU. The first jobs of $\\tau_1$ and $\\tau_2$ arrive at the latest moments in the first GPU server period, and the second job of $\\tau_1$ arrives at the beginning of the second GPU server period. Although the tasks are schedulable by the given server budgets, it causes burst heat generation by back-to-back execution, which can potentially lead to thermal violation. In order to avoid this, the budget of the deferrable server for the GPU has to be halved to avoid thermal violation but  it can drastically lower task schedulability.\n\nIn contrast, the sporadic server policy can take the merits of both the polling server and deferrable server policies. If the sporadic server policy is used for the GPU, a GPU segment can execute at any time as long as there is enough budget, and back-to-back heat generation does not occur. Fig.~\\ref{fig:chap2_thermal_bck2bck}b illustrates the previous example with the sporadic server on the GPU. As can be seen, unlike the deferrable server case, the budget replenishment of the GPU server is one period apart from its consumption time, thereby preventing potential thermal violation. \nThe sporadic server on the GPU is also practically effective because the GPU server needs to have a relatively large budget with a long replenishment period due to its non-preemptive nature whereas the CPU server typically has a short replenishment period to reduce task response time. \n\nIn summary, due to the aforementioned reasons, our framework specifically uses the sporadic server policy for the GPU, while all the three policies (polling, deferrable, and sporadic) are allowed for CPU cores. We also set the budget of the GPU sporadic server to be at least as large as one complete GPU segment of any task, i.e., $\\max(E_i)$, because this can reduce the response time of a GPU segment and remote blocking time. The detailed analysis on these delays will be presented in Section~\\ref{sec:chap2_analysis}.\n\nThe thermal server for the GPU is a resource abstraction managed on the CPU side, similar to other real-time GPU management schemes~\\cite{patel2017,8046309,kim2018server,GPUSync,elliott2012globally}. One can implement the GPU thermal server as part of GPU drivers or application-level APIs.\n\n\\subsection{Miscellaneous Operation Time (MOT) Reservation}\n\\label{sec:chap2_Overrun}\nIn order to reduce the CPU-GPU handover delay, we propose an MOT reservation mechanism. \nWith this mechanism, a small portion of the CPU {server budget is} reserved only for miscellaneous operations in a GPU segment, e.g., transferring data to\/from the GPU. The MOT reservation is feasible with the deferrable and sporadic server policies but not with the polling server policy because the polling server is unable to keep unused budget by design. Although this MOT reservation reduces the amount of CPU budget for regular task execution, it guarantees that the GPU does not need to wait for the budget replenishment of the CPU server during the data transmission phase of a GPU segment.\nIt can also reduce the remote blocking time of other tasks. \nThis reserved budget for MOT has to be  the largest amount of the CPU intervention time in all GPU requests. The trade off between the MOT reservation and the reduced budget for regular task execution will be extensively investigated in the evaluation. \n\nFig.~\\ref{fig:chap2_overrun} illustrates an example to highlight the benefit of the MOT reservation mechanism. As one can see in Fig.~\\ref{fig:chap2_overrun}a, $\\tau_3$ has to wait until 10 to transfer its data to the GPU because $\\tau_1$ already has consumed all budget of $v_2$. Similarly, although the result of the GPU kernel of $\\tau_3$ is ready at 27, its result begins to be transferred at 30. These delays cause the remote blocking of $\\tau_2$ lasts for 22 time units although there exists an available amount of the server budget on $v_1$. Designating 2 time units as the MOT budget (see Fig.~\\ref{fig:chap2_overrun}b) leads $\\tau_3$ to transfer its data to the GPU at 7 and finishes its kernel launch at 27; hence $\\tau_2$ initiates its kernel launch accordingly. Consequently, designating some amount of the server budget as MOT reservation leads to reduction in the remote blocking of a GPU-using task from other CPU core.\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}{.85\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/frmwrk\/non-over.pdf}\n  \\caption{}\n  \\label{fig:chap2_overrunA}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.85\\textwidth}\n  \\centering\n \n \\includegraphics[width=\\textwidth]{Chapter2\/figs\/frmwrk\/ovr.pdf}\n  \\caption{}\n  \\label{fig:chap2_overrunB}\n\\end{subfigure}\n\\caption{Data transfer in a GPU request a) without and b) with the MOT reservation mechanism.}\n\\label{fig:chap2_overrun}\n\\end{figure}\n\\section{System Model}\n\\label{sec:chap2_model}\nIn this section, we describe the thermal-aware server as well as the task models used in this chapter and explain the procedure of a kernel launch on a GPU. Then, we characterize the scheduling penalties that arise from the use of a GPU with thermal-aware servers.\n\n\\subsection{Computing Platform}\nWe consider a temperature-constrained embedded platform equipped with an integrated CPU-GPU SoC. The SoC has multiple CPU cores and one GPU core, each running at a fixed clock frequency. Note that such SoC design with a single GPU is popular in today's embedded processors, such as Samsung Exynos 5422 and NVIDIA TX2. The assumption on the fixed operating frequency is particularly suitable for the GPU as DVFS capabilities are not widely supported on embedded on-chip GPUs. The thermal behavior of CPU and GPU cores follows the model described in Section III. For simplicity, we assume that the amount of temperature generated by other SoC components, such as peripherals and caches, is either negligible or acceptably small.\n\n\\subsection{Thermal-Aware Servers}\n\nWe consider one thermal-aware server for each CPU and GPU core.\\footnote{Running multiple thermal servers on each CPU\/GPU core is left for future work.}\nEach server is statically associated with one core and does not migrate to another core at runtime. However, unlike prior work \\cite{Ahmed2017,ahmed2011minimizing}, we do not limit the server to follow only the polling server policy. We will show in the later section that this flexibility brings significant benefit in the schedulability of tasks accessing the GPU. \n\nTo bound the temperature of each core, its corresponding server $v_i$ is modeled as $v_i = (C_i^v,T_i^v) $\nwhere $C_i^v$ is the maximum execution budget and $T_i^v$ is the budget replenishment period of $v_i$. For brevity, we will use ${v_g = (C^g, T^g)}$ to denote the GPU server and ${v_c = (C^c,T^c)}$ for the CPU server. For budget replenishment policies, we consider \\textit{polling}~\\cite{sha1986solutions}, \\textit{deferrable}~\\cite{strosnider1995deferrable}, and \\textit{sporadic servers}~\\cite{sprunt1989aperiodic}. Under the polling server policy, the corresponding server activates periodically and executes ready tasks until its budget is depleted. The budget is fully replenished at the start of the next period. If there is no task ready, the remaining budget is immediately depleted. In contrast, under the deferrable server, any unused budget is preserved until the end of the period. Hence, a task can execute at any time during the server period while the budget is available. The sporadic server also preserves remaining budget, but replenishes the budget sporadically; only the amount of budget consumed is replenished after $T^v$ time units from the time when that budget is used. Let $J^v$ denote the task release jitter relative to the server release. The value of  $J^v$ is $T^v$ under the polling server policy and $T^v - C^v$ under the deferrable and sporadic server policies~\\cite{bernat1999new}.\n\n\\subsection{Task Model}\nThis work considers sporadic tasks with implicit deadlines under \\textit{partitioned fixed-priority preemptive scheduling}, which is widely used in many real-time systems. Each task $\\tau_i$ has been statically allocated to one CPU core (thus to the server of that core) with a unique priority. Tasks are labeled in an increasing order of priority, i.e., $i<j$ implies $\\tau_i$ has lower priority than $\\tau_j$. Without loss of generality, each task can contain at most one GPU segment, but it can be easily extended to multiple GPU segments. A task $\\tau_i$ is modeled as ${\\tau_i = ((C_{i,1},E_{i},C_{i,2}), T_i,s_i)}$, where $C_{i,1}$ and $C_{i,2}$ are the worst-case execution time (WCET) of the normal execution segments of task $\\tau_i$, and $E_i$ is the worst-case time of the GPU access segment. The normal execution segments run entirely on the CPU core and the GPU segment involves GPU operations. Let $T_i$ denote the the minimum inter-arrival time of $\\tau_i$ and $s_i$ indicate whether $\\tau_i$ has a GPU segment, i.e., $s_i=1$ means a GPU-using task. In case $\\tau_i$ executes only on the CPU, $s_i$, $E_i$ and $C_{i,2}$ are all zero. Thus, the accumulated sum of the WCETs of $\\tau_i$ is denoted as\n\\[  C_i = s_i \\times (C_{i,1}+E_i+C_{i,2}) + (1-s_i)\\times C_{i,1}. \\]\nFurthermore, $V(\\tau_i)$ represents the CPU server where $\\tau_i$ is assigned. Tasks are considered as fully compute-intensive and independent from each other during normal segment execution. \nThe only  resource shared among tasks is the GPU and it is modeled as a critical section protected by a suspension-based mutually-exclusive lock (mutex). Note that this approach follows the well-established locking-based real-time GPU access schemes~\\cite{patel2017,GPUSync,elliott2012globally}. We will later present how pending GPU requests are queued in our proposed framework. \n\n\\subsection{GPU Execution Model}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.40]{Chapter2\/figs\/mdl\/gpu_mdl}\n\\caption{Task execution with a GPU segment.}\n\\label{fig:chap2_gpu_model}\n\\end{figure}\nThe GPU has its own memory region, which is assumed to be sufficient enough for the tasks under consideration. \nWe do not consider the concurrent execution of GPU requests from different tasks because of the resulting unpredictability in kernel execution time~\\cite{patel2017,otterness2017evaluation}.\nOnce a task acquires the GPU lock, its GPU segment is handled through the following steps (see Fig.~\\ref{fig:chap2_gpu_model}): \n\\begin{enumerate}\n    \\item \\textit{Data Transfer to the GPU:} The task first copies data needed for the GPU computation, from CPU memory to GPU memory. This can be done by Direct Memory Access (DMA), which requires minimal CPU intervention. If the GPU uses a unified memory model, this step can be omitted.\n    \\item   \\textit{Kernel Launch: } Kernel launches on the GPU. Meanwhile, the task on the CPU side self-suspends and waits for the GPU computation to complete. \n    \\item \\textit {Kernel Notification Signal:} The GPU signals the CPU to notify the completion of kernel execution.\n    \\item \\textit{Data Transfer to the CPU:} The task wakes up and transfers the results from  GPU memory to  CPU memory.\n\\end{enumerate}\n\\cmnt{Finally, the task continues its normal execution segment. }\nIt is worth noting that a GPU kernel {\\em cannot self-suspend} on the GPU in the middle of execution.\\footnote{Although GPU kernel preemption is available on some recent GPU architectures, e.g., Nvidia Pascal~\\cite{pascal}, to the best of our knowledge, the self-suspension of a kernel is not supported in any of today's GPU architectures.}\nOn the other hand, since there is no CPU intervention during kernel execution, the task on the CPU side {\\em self-suspends} to save CPU cycles. As a result, other tasks have a chance to execute or the CPU core sleeps during the kernel execution. \nFor a task $\\tau_i$, the total time for the above four steps consists of two major parts:\n\\begin{itemize}\n    \\item   Miscellaneous operations that require CPU intervention. Let $M_{i,1}$ denote the time for data transfer before the kernel execution and $M_{i.2}$ denote that after the kernel execution.\n    \\item  Pure GPU kernel operations that do not require any CPU intervention denoted as $K_i$.\n\\end{itemize}\nIn such aspect, the GPU segment time of a task $\\tau_i$ is modeled as $E_i= M_{i,1} + K_i + M_{i,2}$. As the GPU is non-suspendable during kernel execution, the GPU server should have enough budget larger than or equal to $E_i$. The CPU server only needs to have budget larger than $M_{i,1}$ or $M_{i,2}$.\n\n\\subsection{Challenges of Thermal-Aware Servers with an Integrated GPU}\n\\label{sec:chap2_challenges}\nAside from the blocking delays coming from the locking-based GPU access approach, e.g., local and remote blocking to acquire the GPU lock~\\cite{patel2017,GPUSync,elliott2012globally}, there are other new challenges faced by thermal-aware servers in a CPU-GPU integrated system. \n\n\n\n\\begin{itemize}\n    \\item \\textit{Server budget depletion}: Task execution in a server is scheduled with respect to the  available budget. When the server budget is depleted, a task has to wait until the budget is replenished.\n    \\item \\textit{Mutual budget availability}: \n   \n    If a task $\\tau_i$ issues a GPU request, both CPU and GPU servers must have enough budget to handle this request. It is worth noting that server budget needs for the CPU and GPU servers are different ($M_{i,1}$ or $M_{i,2}$ vs. $E_i$).\n    \\item \\textit{CPU-GPU handover delay}: Even if both servers are designed to have enough budget, each server may have to wait for the other's budget to be replenished when their interactions are needed, e.g., data transfer between the CPU and the GPU.\n   \n    \\item \\textit{Back-to-back heat generation}: In case of the deferrable server, some tasks can use up all the budget at the end of the budget replenishment period, and at the very beginning of the next period, some other tasks can start to consume the replenished budget. This causes the server to run longer than its budget and generate heat in a back-to-back manner. \n   \n\\end{itemize}\n\n   \n   \n   \n\n\n\n\n\n\n\n\n\\subsection{ Task Schedulability Analysis}\nThe thermal analysis in the previous subsection gives the maximum budget of each CPU\/GPU server that satisfies the thermal constraint of the system. Hereby, we present the schedulability analysis of a task $\\tau_i$ in our framework.\n\nBefore introducing our analysis, we review the existing response time test for independent tasks with no thermal constraints and no shared GPU under hierarchical scheduling~\\cite{saewong2002analysis}, which is \n\\begin{equation}\n\\label{eq:chap2_hierchical}\n W_i^{n+1}   =  C_i + \\sum_{\\begin{subarray}{h} \\tau_h \\in V(\\tau_i) \\\\\n       h > i \\end{subarray} }\\ceil*{\\frac{W_i^n  +J^c}{T_h}}C_h + \\ceil*{\\frac{W^{n}+C^c}{T^c}} (T^c - C^c)\n\\end{equation}\nwhere $J^c$ is the jitter of a task running in a server (see Section~\\ref{sec:chap2_model}) and $W^0 = C_i$. The recursion terminates successfully when $W^{n+1}= W^n$ and  fails when $W^{n+1}> T_i$. This equation considers the budget depletion of a CPU server. The first term is the amount of CPU time used by the task $\\tau_i$, the second term captures the back-to-back execution of each higher-priority task $\\tau_h$,  and the third term captures the amount of interference that the server can generate due to the periodic budget replenishment. However, this equation cannot be used directly for the thermal constraint problem with a  shared GPU.\n \nOur analysis extends the existing response time test by considering the factors discussed in Section V:  (i) local blocking time, (ii) remote blocking time, (iii) back-to-back execution due to remote blocking, (iv) mutual budget availability, (v) CPU-GPU handover delay, (vi) multiple priority inversions, and (vii) CPU and GPU server budget co-depletion.  We take into account the factors (iv) and (v) together in the analysis of the handover delay, the factor (vi) as part of the local blocking time analysis, and the factor (vii) as part of the remote blocking time analysis. By considering all these factors, the following recurrence equation bounds the worst-case response time of a task $\\tau_i$: \n   \n  \\begin{align}\n& W_i^{n+1}   =  C_i + B_i^l + B_i^r +  H_i^{gc}  + \\ceil*{\\frac{W_i^{n}+C^c- s_i (H_i^{gc} +  K_i)}{T^c}} (T^c - C^c)  + \\nonumber \\\\\n &\\sum_{\\begin{subarray}{l} \\tau_h \\in V(\\tau_i) \\\\\n       h > i \\end{subarray}}\\ceil*{\\frac{W_i^n +J^c +(W_h - C_h) - s_i  (H_i^{gc} +  E_i)}{T_h}}C_h  \n  \\end{align}\nwhere $C_i$ is the worst-case execution time of $\\tau_i$, $B_i^l$ is the local blocking time, $B_i^r$ is the remote blocking time, $H^{gc}_i$ is the CPU-GPU handover delay. We will later discuss each of them in details. The recurrence equation terminates when $W^{n+1}_i = W^n_i$, and the task $\\tau_i$ is schedulable if its response time does not exceed its implicit deadline (i.e., $W^n_i <= T_i$).\n\n\nThe second line of the equation captures the delay due to the server budget depletion on the CPU side (an extension of the last term of Eq.~\\ref{eq:chap2_hierchical}). It is worth noting that during the pure GPU kernel execution of $\\tau_i$ ($K_i$), no CPU budget is consumed by $\\tau_i$. Any other tasks can execute on the CPU core if there is a remaining budget. The task $\\tau_i$ only consumes the CPU server budget when it executes  normal segments or the miscellaneous operations (e.g. data copy) of GPU segments. Therefore, $H^{gc}_i $ and $K_i$, which already exist in $W_i^n$ for a GPU-using task $\\tau_i$, are excluded such that only the CPU-consuming parts of this task is affected by the CPU server budget depletion. Any additional delay due to CPU budget replenishments during GPU segment execution will be captured by $H^{gc}_i$.\n\nThe last line captures the preemption time by the normal execution segments of higher-priority tasks (an extension of the second term of Eq.~\\ref{eq:chap2_hierchical}). The fact that a higher-priority task $\\tau_h$ can only preempt the normal execution segments of $\\tau_i$ leads to the deduction of $H^{gc}_i$ and $E_i$ from $W_i^n$. There can be at most one additional job of $\\tau_h$ that has arrived during $\\tau_i$'s GPU segment execution and interfere $\\tau_i$'s normal segments. This holds true if $\\tau_h$ is schedulable, i.e., $W_h\\le T_h$, because other arrivals of $\\tau_h$ during $\\tau_i$'s GPU segment will finish their executions while $\\tau_i$ is self-suspended for its kernel execution on the GPU. The interference from this additional carry-in job of $\\tau_h$ is taken into account by modeling it as a dynamic self-suspending model and adding $W_h-C_h$ to $W_i^n$~\\cite{bletsas2018errata}. \n\nNow, we present the detailed analysis of the delay factors used in our response time test.\n\\subsubsection{\\textbf{CPU-GPU Handover Delay}}\n It captures an extra delay a task  can experience after acquiring the GPU lock because of the factors (iv) and (v).  Fig. \\ref{fig:chap2_handover} illustrates this type of delay decomposed into three parts when the polling server policy is used for a CPU server. \\circled{1} When there is not enough budget available in the GPU server for the execution of a GPU segment, the task has to wait at most the jitter of the GPU server ($T^g-C^g$). \\circled{2} After the GPU budget is replenished, if the CPU server is inactive, the task has to wait for additional $T^c$ time units for the next period of the CPU server. \\circled{3} After the completion of kernel execution on the GPU, at most $T^c$ time units are needed for the CPU server to be activated in order to transfer the results back from the GPU to the CPU. Hence, \n\\begin{equation}\n\\label{eq:chap2_handover}\n    H_i^{gc} = s_i  (T^g - C^g + 2  T^c).\n\\end{equation}\nFor a CPU server with the deferrable and sporadic server policies, \\circled{2} and \\circled{3} change to the jitter of the CPU server (i.e., $T^c - C^c$) because a GPU segment needs to wait for the replenishment of its corresponding CPU server's budget. The handover delay is then given by\n\\begin{equation}\n\\label{eq:chap2_handover_deferrable}\n{H_i^{gc} = s_i  [T^g - C^g + 2  (T^c-C^c)]}.\n\\end{equation}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Chapter2\/figs\/anl\/handover.pdf}\n\\caption{The worst-case scenario for CPU-GPU handover delay with a CPU polling server. When a GPU request with $E_i$ is chosen from the GPU waiting queue for execution, it experiences the delay highlighted in blue boxes.}\n\\label{fig:chap2_handover}\n\\end{figure}\n\n\\subsubsection{\\textbf{Local Blocking}}\n  It occurs when a task $\\tau_i$ is blocked by the lower-priority task on the same core. As in the case of MPCP~\\cite{rajkumar1990real, rajkumar1988real}, a task can be blocked by each of its lower-priority task $\\tau_l$'s GPU segment at most once due to the priority boosting of our framework. To obtain a tight bound, we analyze local blocking time from two different perspectives.\n  \n  On the one hand, the worst-case local blocking time of a task $\\tau_i$ happens when each normal execution segment of $\\tau_i$ is blocked by the GPU segment of each lower-priority task $\\tau_l$ with the amount of ${E_l- K_l = M_{i,1}+M_{i,2}}$, which is the maximum CPU time used by the GPU segment of $\\tau_l$. Hence, the total local blocking time of a task $\\tau_i$ is\n  \\begin{equation}\n  {(1+s_i) \\sum_{ l<i \\And \\tau_l \\in V(\\tau_i) \\And s_i>0} E_l-K_l}\n  \\end{equation}\n  where $(1+s_i)$ indicates the number of normal execution segments of $\\tau_i$. It is worth noting that under the RM policy, the total blocking time can be bounded by just\n  \\begin{equation}\n  {\\sum_{ l<i \\And \\tau_l \\in V(\\tau_i) \\And s_i>0} E_l-K_l}\n  \\end{equation}\n   because each task has only one GPU segment and the period of any lower-priority task $\\tau_l$ is larger than that of $\\tau_i$, which leads to only one blocking time from each $\\tau_l$ during the job execution of $\\tau_i$.  \n \n\n\nOn the other hand, the worst-case local blocking time of $\\tau_i$ can also be bounded by the amount of GPU budget available during one period of $\\tau_i$. The reasoning behind this approach is that lower-priority tasks cannot execute GPU segments more than the available budget on the GPU. Thus, the maximum total blocking time of $\\tau_i$ is bounded by ${(1+s_i)(\\ceil{\\frac{T_i}{T^g}}+1) C^g}$ where ``+1\" is due to the carry-in effect. \n\nUsing these two approaches, the total local blocking time of $\\tau_i$ is bounded by \n\\begin{equation}\n \\begin{split}\nB_i^l  = \n(s_i+1) \\cdot \\min \\left(\\left(\\ceil*{\\frac{T_i}{T^g}}+1\\right) C^g , \\sum_{\\begin{subarray}{l} l<i \\\\ \\tau_l \\in V(\\tau_i) \\\\ s_i>0\\end{subarray}} E_l-K_l\\right).\n \\end{split}\n\\end{equation}\n\n\\subsubsection{\\textbf{Remote Blocking}}\n It occurs when the GPU segment of a task is blocked in the GPU waiting queue due to other GPU requests. \nRecall that the GPU segments of tasks are ordered in a priority queue according to their tasks' original priorities. The response time of a GPU segment of $\\tau_i$ is given by ${W'_i = H_i^{gc} + E_i }$. The reasoning is that after a task acquires the GPU lock, it has to wait $H^{gc}_i$  for the handover delay of data transferring and mutual server synchronization (Eq.~\\ref{eq:chap2_handover}). There is no other delay than $H_i^{gc}$ added to the GPU segment length $E_i$ because our framework sets the GPU budget to be large enough to perform any GPU segment in one GPU period and boosts the priority of the task executing a GPU segment.  Hence, the remote blocking time of $\\tau_i$ is bounded by the following recurrence equation:\n\\begin{equation}\nB_{i}^{r,n+1} =  \\max_{\\begin{subarray}{l} l<i  \\\\ s_i>0\\end{subarray}}{W'_l} + \\sum_{\\begin{subarray}{l} h>i \\\\ s_i>0\\end{subarray}}\\bigg(\\ceil*{\\frac{B_{i}^{r,n}}{T_h}}+1\\bigg). W'_{h}.\n\\end{equation}\nwhere the base is ${B_{i}^{r,0} =  \\max_{\\begin{subarray}{l} l<i  \\\\ s_i>0\\end{subarray}}{W'_l}}$. The first term of the equation captures the waiting time for acquiring the GPU lock due to the currently-running of one GPU segment of a lower- priority task and the second term represents the waiting time for the GPU segments of higher-priority tasks. This analysis is pessimistic because it assumes that the GPU budget is exhausted after the completion of each GPU segment and the GPU segment of the next task has to wait for the amount of $H^{gc}$. It is noteworthy that because of the  non-preemptive GPU resource, the GPU server budget has to be large enough that the GPU segment of any task is able to execute without interruption which leads to enormous remote blocking time due to a considerable data handover  delay. Let $\\Gamma^g$ denote the set of GPU-using tasks in a taskset. For the lowest-priority GPU-using task, ${|\\Gamma^g-1| \\times (T^g-C^g)}$ is the amount of delay only from the GPU server budget replenishment of higher-priority GPU-using tasks. To mitigate this issue, we will present another design of the waiting queue for GPU requests in the later part of this section.\n\n\\subsection{Improvement}\n\\subsubsection{ \\textbf{Miscellaneous Operation Time Reservation Policy}}\nRecall our MOT reservation mechanism presented in Section~\\ref{sec:chap2_Overrun}. When enabled, it ensures that there is always an enough amount of budget for miscellaneous operations (e.g. data copy from\/to GPU); thus, the GPU does not have to wait until the start of the next CPU budget replenishment. \nHence, the CPU-GPU handover delay with the MOT mechanism is\n\\begin{equation}\n    H_i^{gc} = s_i  (T^g - C^g).\n\\end{equation}\n\nThe improvement in the handover delay has also a profound impact on the remote blocking delay by reducing the worst-case response time of a GPU segment. However, it is worth noting that for deferrable CPU servers, their budget needs to be halved as discussed earlier in Section~\\ref{sec:chap2_server} to avoid the thermal back-to-back phenomenon. \\begin{comment}In the next section, we will evaluate the influence of this option from different aspects on schedulability percentage. \\end{comment}\n\n\\subsubsection{\\textbf{Remote Blocking Enhancement}}\nTo address the problem of enormous remote blocking time due to CPU-GPU handover delay, we propose an alternative approach to the GPU waiting queue. This approach implements the queue based on a variant of the first-come first-served (FCFS) policy with a pre-defined bin-packing order. To be more precise, a bin-packing heuristic is employed to determine the number of bins, where the size of each bin is the GPU budget and the length of a GPU  segment is the size of an item to be packed. \nThe total number of items in the bins is $|\\Gamma^g|$ and the number of bins is related to the waiting time for a GPU segment. \nThe reason for employing the FCFS policy is to avoid starvation of jobs with large period because under this policy, jobs with shorter period get serviced only once at any time.  \nIf a small-period job arrives meanwhile, it waits until the rest of waiting jobs get serviced and after finishing all other jobs, it gets serviced according to its position in the bins. Since in this approach all tasks have the same amount of waiting time, it leads to a significant reduction in the waiting time of low-priority GPU-using tasks but a moderate increase in that of high-priority ones. It is worth reminding that the replenishment period of the GPU server is typically much larger than that of the CPU server. The remote blocking time for a GPU-using task $\\tau_i$ under the polling server policy for CPU cores is\n\\begin{equation}\nB_{i}^{r} = s_i \\left[ (|bins|+1) T^g + 2  (|\\Gamma^g|-1)  T^c \\right].\n\\end{equation}\n  In this approach, missing activation points of the CPU polling server can still happen in the transferring time of data from\/to the CPU server and due to the FCFS characteristics of the queue, the total amount is ${2  (|\\Gamma^g|-1)  T^c}$. The remote blocking time of $\\tau_i$ under the deferrable and sporadic CPU server policies without the MOT mechanism is\n\\begin{equation}  \n  {B^r_{i} = s_i \\left(|bins|+1\\right) T^g + 2  (|\\Gamma^g|-1) (T^c -C^c) }.\n\\end{equation}\n  This is because in the worst case, the GPU server waits for the amount of CPU server jitter. The remote blocking time with the MOT mechanism is \n\\begin{equation}  \n  {B^r_{i} = s_i \\left(|bins|+1\\right) T^g}.\n\\end{equation}\n\n To this end, our framework takes a \\textit{hybrid} scheduling scheme that chooses one of the proposed queue implementations which successfully passes the schedulability analysis. \n\\section{Thermal and Schedulability Analysis}\n\\label{sec:chap2_analysis}\nIn this section, we present the schedulability analysis of our framework. We first design our thermal-aware servers for multi-core GPU-integrated  platforms to avoid thermal violation. Then, we analyze  task schedulability with and without the MOT reservation mechanism. \n\\begin{comment}\n\\subsection{Worst thermal execution }\nIn this section, we calculate the steady state condition for server executing sporadic tasks. First, we prove that the worst case execution time happens where all jobs execute consecutively. \n\\begin{proof}\nFor any arbitrary execution of jobs in a period of  $T$, we have \n\n\n\\[ \\theta_{t_0}   = \\theta_s \\]\n\\[ \\theta_{t_1}   =\\alpha + (\\theta_{t_0} - \\alpha) * e^{\\beta t_1}\\]\n\\[ \\theta_{t_2}   = \\theta_{t_1}  * e^{\\beta t_2}\\]\n\\[ \\theta_{t_3}   =\\alpha + (\\theta_{t_2} - \\alpha) * e^{\\beta t_3} \\]\n\\[ \\theta_{t_4}   = \\theta_{t_3}  * e^{\\beta t_4}\\]\n\\[ \\theta_{t_5}   =\\alpha + (\\theta_{t_4} - \\alpha) * e^{\\beta t_5} \\]\n\\[ \\vdots\\]\n\\[ \\theta_{t_n}   = \\theta_{t_{n-1}}  * e^{\\beta t_n} = \\theta_{t_0}\\]\n\nwhere $\\theta_s$ is the steady temperature, $\\sum t_i = T $ for all $n>1$ and $t_i >0$.\n\nSolving the recursion we have,\n\\[  \\theta_{t_n} = e^{\\beta t_{n-1}} (\\alpha-e^{\\beta t_{n-2}}\\cdots  (\\alpha-e^{\\beta t_{2}} (\\alpha-e^{\\beta t_{1}} (\\alpha-\\theta_s ) ) ) ) ) )\\]\nsolving $\\theta_{t_n} = t_{slp}$, we have\n\n\\[\n    \\theta_{s} =\\alpha \\frac{e^{\\beta(t_{1}+t_{2}+\\cdots+t_{n-1}+t_{n})}-e^{\\beta(t_{2}+\\cdots+t_{n-1}+t_{n})}+\\cdots+e^{\\beta(t_{n-1}+t_{n})}-e^{\\beta t_{n}}}{ e^{\\beta t_{1}}e^{\\beta t_{2}}\\cdots e^{\\beta t_{n-1}}e^{\\beta t_{n}}-1}\n    \\]\nso,\n    \\[     \\theta_{s} = \\frac{e^{\\beta t_{n}}(\\alpha-e^{\\beta t_{n-1}}\\cdots (\\alpha-e^{\\beta t_{2}}(\\alpha-\\alpha e^{\\beta t_{1}})))}{e^{\\beta T}-1}\n\\]\nWhere $ t_{wk} = \\sum_{i =1 ,\\, i= i+2 }t_i$ is executing time and  $ t_{slp} = \\sum_{i =2 ,\\, i= i+2 }t_i $ is idle time, respectively. The above formulae shows that the least  amount of total execution time occurs when we have only one execution time cycle. Therefore, the worst case scenario happens when all jobs execute consecutively.\n\\end{proof}\n\n Next, we find the budget of a server for each power\n\\end{comment}\n\n\\subsection{Design of Server Budget}\n\\label{sec:chap2_server}\nIn our framework, task execution is performed within thermal-aware servers. As discussed, the notion of servers is to isolate each compute node from others in terms of the thermal aspect. Accordingly, under any circumstance of task execution, the thermal violation avoidance has to be guaranteed in the server design. The specifications of servers are independent of their running tasks (except for the design of the MOT reservation),  whereas task schedulability does depend on them. In our proposed framework, introducing the thermal-aware servers for both the CPU and the GPU makes the physical characteristics of underlying platforms be transparent of the task schedulability test.\n\n\\subsubsection{Single-Core Platforms}\nFirst, we calculate the ``maximum\" budget that a server can have while limiting the temperature not  to exceed the given thermal constraint in a single-core platform under a given replenishment period. In the worst case of a polling server, the server exhausts all of its budget at the beginning of its period and then sleeps until the beginning of the next replenishment period. Let $t_{wk}$ and $t_{slp}$ denote the active time (i.e., the budget-consuming phase) and the sleeping time (i.e., the cooling phase) of a CPU core server, respectively. Hence, the server period $T$ is \\begin{equation} \n\\label{eq:chap2_1}\nT = t_{wk} + t_{slp}.\n\\end{equation}\nIn the steady state of the system, we are interested in bounding the server's maximum temperature. According to  Eq.~\\ref{eq:chap2_tempActive}, \\[ \\alpha + (\\theta_s - \\alpha ) e^{\\beta t_{wk}} \\leq \\theta_M\\]\nwhere $ \\theta_s$ and $\\theta_M$ are the steady state temperature and the thermal constraint, respectively. Therefore,\n\\begin{equation}  \n\\label{eq:chap2_2}\ne^{\\beta t_{wk}} \\geq  \\frac{ \\theta_M -\\alpha}{\\theta_s - \\alpha } \\Longrightarrow t_{wk} \\leq \\frac{1}{\\beta} \\ln{\\frac{ \\theta_M -\\alpha}{\\theta_s - \\alpha }}.\n \\end{equation}\nOn the other hand, in the cooling phase, according to Eq.~\\ref{eq:chap2_tempCool} to respect the steady state, $ \\theta_M e^{\\beta t_{slp}} = \\theta_s$.\nHence, \n\\begin{equation} \\label{eq:chap2_3}\nt_{slp} = \\frac{1}{\\beta} \\ln{\\frac{\\theta_s}{\\theta_M}}.\n \\end{equation}\n By substituting Eqs.~\\ref{eq:chap2_2} and \\ref{eq:chap2_3} by Eq.~\\ref{eq:chap2_1}, we have\n\\[ \\frac{1}{\\beta}\\ln{ \\frac{ \\theta_M -\\alpha}{\\theta_s - \\alpha }} + \\frac{1}{\\beta} \\ln{\\frac{\\theta_s}{\\theta_M}} \\leq T\n\\]\n\\[  \\frac{ \\theta_M -\\alpha}{\\theta_s - \\alpha } \\times \\frac{\\theta_s}{\\theta_M} \\leq e^{\\beta T }.\n\\]\nTherefore, the worst-case steady state temperature at the beginning of each period is\n\\begin{equation} \\label{eq:chap2_4}\n\\theta_s = \\frac{\\alpha \\theta_{M} e^{\\beta T}}{\\theta_{M} (e^{\\beta T}-1)+\\alpha}.\n\\end{equation}\nAccordingly, the maximum budget for the period $T$ is\n\\begin{equation} \\label{eq:chap2_5}\nt_{wk} = T - \\frac{1}{\\beta} \\ln{\\frac{\\theta_s}{\\theta_M}}.\n\\end{equation}\nConsequently, for a given replenishment period, a server ${v_i = (T - \\frac{1}{\\beta} \\ln{\\frac{\\theta_s}{\\theta_M}}, T)}$ can bound the maximum temperature to $\\theta_M$.\n\nAs one can figure out from the analysis, the maximum budget converges because of $\\alpha$. This means that after some point, an increase in the replenishment period has no effect on the maximum feasible budget. As discussed earlier through our framework design, the budget has to be considered as $\\frac{t_{wk}}{2}$ for a deferrable server due to the \\textit{thermal back-to-back} phenomenon. This phenomenon does not occur in a sporadic server, and it can use the computed budget as is.\n\n\\subsubsection{Homogeneous Multi-core Platforms}\nThe worst case for the budget of polling servers on a multi-core CPU happens when all of them exhaust their budget completely. Therefore, according to Eq. \\ref{eq:chap2_tempCompo} for the composability characteristics of the heat transfer, we have\n \\[    \\alpha + (\\theta_s - \\alpha) e^{\\beta t_{wk}} +  \\sum_{\\begin{subarray}{l}j=1\\\\ j \\neq i \\end{subarray}}^{m} \\gamma_{i,j} \\theta^j_M \\leq \\theta^i_M\\]\nwhere $\\theta^i_M$ is the maximum temperature for the $i$th node. In the worst case, every core may reach its maximum temperature at the same time. Hence,\n\\[  \\underbrace{(1 +  \\sum_{\\begin{subarray}{l}j=1\\\\ j \\neq i \\end{subarray}}^{m} \\gamma_{i,j})}_{\\lambda_i} [ \\alpha + (\\theta_s - \\alpha) e^{\\beta t_{wk}}] \\leq \\theta^{i}_{M}. \\]\nHowever, the geographic location of cores on the chip results in different values of the conduction coefficients although it remains symmetric (i.e., $\\gamma_{i,j} = \\gamma_{j,i}$). Denoting ${\\lambda  = \\max_{1 \\leq i \\leq m}{\\lambda_i}}$, $\\theta_M$ is given by $\\theta_M = \\lambda [ \\alpha + (\\theta_s - \\alpha) e^{\\beta t_{wk}}] $. Similar to the single-core analysis given in the previous subsection, the steady state temperature is \n\\begin{equation} \\label{eq:chap2_10}\n\\theta_s = \\frac{\\alpha \\frac{\\theta_{M}}{ \\lambda} e^{\\beta T}}{\\frac{\\theta_{M}}{\\lambda} (e^{\\beta T}-1)+\\alpha}\n\\end{equation}\n Therefore, the server budget for each compute node $i$ is\n \\begin{equation} \\label{eq:chap2_11}\nC^c = t_{wk} = T - \\frac{1}{\\beta} \\ln{\\frac{\\theta_s \\times \\lambda}{\\theta_M}} .\n\\end{equation}\n\n\\subsubsection{Heterogeneous Multi-Core  GPU-Integrated Platforms}\nHereby, we will determine the budget of servers in the presence of an integrated GPU. Similar to the homogeneous multi-core platform, the worst case happens when all servers exhaust their budgets completely. There is also a GPU segment execution on the GPU which causes extra heat dissipation. Since the GPU runs at a different frequency and its architecture is different from the CPU cores, its heat generation parameters differ from those of the CPU cores. Hence, \n\n\n\\[\\left\\{\n                \\begin{array}{ll}\n                  \\theta^i_M = \\lambda_i [ \\alpha + (\\theta_s - \\alpha) e^{\\beta t_{wk}}]  +  \\gamma_{i,g}[\\alpha^g + (\\theta^g_s - \\alpha^g) e^{\\beta^g t^g_{wk}}] \\\\\n                 \n                \\theta^g_M = \\alpha^g + (\\theta^g_s - \\alpha^g) e^{\\beta^g t^g_{wk}} +  \\underbrace{\\sum_{j=1}^{m} \\gamma_{g,j}}_{\\gamma^g} [ \\alpha + (\\theta_s - \\alpha) e^{\\beta t_{wk}}]  \n                \\end{array}\n              \\right. \n\\]\n\\begin{equation}\n\\label{eq:chap2_double}\n\\Longrightarrow\\left\\{\n                \\begin{array}{ll}\n                  \\lambda \\theta^i_M e^{\\beta t_{slp}} + \\gamma_{i,g} \\theta^g_M e^{\\beta^g t^g_{slp} }= \\theta_s \\\\\n                \\theta^g_M e^{\\beta^g t^g_{slp} }+ \\gamma^g \\theta^i_M e^{\\beta t_{slp}} = \\theta^g_s\n                \\end{array}\n              \\right.             \n\\end{equation}where the symbols with a superscript $g$ represent the corresponding parameters of the GPU. \n\\subsubsection{Miscellaneous Operation Time Reservation}\nTo reduce the remote locking time, some portion of the CPU server budget can be reserved for data transferring from\/to the GPU that needs CPU intervention. The MOT budget has to be large enough to handle the longest data transferring time, therefore \n\\begin{equation} \\label{eq:chap2_overrun}\nC^c  =  t_{wk}  - \\max_{\\forall \\tau_i}{(M_{i,1}, M_{i,2})}\n\\end{equation}\n\nIt is noteworthy that acquiring a lock on the GPU happens when there is enough budget on the GPU server to execute the whole GPU request; hence, no budget reservation is needed on the GPU side.\n\n\\section{Case Study}\nWe have implemented a prototype of our framework and conducted a case study on ODroid-XU4. We show that without our framework, a real-time task experiences unexpectedly large delay due to the thermal violation, but with our framework, the operating temperature is safely bounded within a desired range.\n\nIn the case study, we used three types of applications: a real-time GPU-using task, and non-real-time CPU-only and GPU-using tasks. The non-real-time CPU-only application is run on each of the four big CPU cores with the lowest priority. The non-real-time GPU-using matrix multiplication task is run on one big CPU core with the medium priority. This task randomly generates two matrices and performs the matrix multiplication repetitively using the GPU-accelerated Mali OpenCL library. The size of matrix is set to $512\\times512$ in order not to cause unnecessarily long waiting time to the real-time task. On the other big CPU core, the highway workzone recognition application for autonomous driving~\\cite{lee2013kernel} is run as the real-time GPU-using task with the highest priority. This task is configured to process 8 frames per second and has one GPU segment based on OpenCL. A video consisting of around 800 frames is given as input to this task. To avoid unexpected delay in data fetching, video frames are preloaded into memory as a vector during the initialization of the task and the loaded frames are repeatedly processed at runtime. After the initialization, all tasks are signaled to start their execution together and the CPU fan is turned off during the experiment. Other tasks in the system, including system maintenance and monitoring processes, are assigned to the little cluster cores. The thermal-aware servers are implemented based on the polling server policy with the budget replenishment period of 10 milliseconds. \n\nTwo scenarios are used to show the effectiveness of our proposed framework. The first one is the baseline system with no thermal-aware server. Fig.~\\ref{fig:chap2_casestudy}a shows the response time of each frame (job) of the real-time workzone recognition task, and Fig.~\\ref{fig:chap2_casestudy}b depicts the temperature measurements during the experiment. As one can see, after processing around 1040 frames, the DTM was triggered and it started throttling the CPU frequency from 2.0 GHz to 900 MHz since the CPU temperature had reached the threshold of 95\\textdegree~C. However, the frequency throttling did prevent the operating temperature from rising on both the CPU and the GPU, which caused the OS to power off the big CPU cluster temporarily. This experiment took less than three minutes. In the second scenario, all tasks execute within the thermal-aware servers. As illustrated in Fig.~\\ref{fig:chap2_casestudy}c, the observed response time of each frame was larger compared to the first scenario due to the budget of servers, but all frames were processed before the deadline. Fig.~\\ref{fig:chap2_casestudy}d depicts that it took around 500 seconds to reach the steady state temperature. Moreover, the operating temperature was tightly bounded by the thermal threshold in any circumstance. This result shows the thermal safety and accuracy of our framework in practical settings.\n\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.48\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/casestudy\/cs_time100.pdf}\n  \\caption{}\n  \\label{fig:chap2_casestudyA}\n\\end{subfigure}\n\\begin{subfigure}{.48\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/casestudy\/cs_temp100.pdf}\n  \\caption{}\n  \\label{fig:chap2_casestudyB}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.48\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/casestudy\/cs_time033.pdf}\n  \\caption{}\n  \\label{fig:chap2_casestudyC}\n\\end{subfigure}\n\\begin{subfigure}{.48\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/casestudy\/cs_temp033.pdf}\n  \\caption{}\n  \\label{fig:chap2_casestudyD}\n\\end{subfigure}\n\\caption{a) Response time of the real-time workzone recognition task without our framework. b) Temperature of CPU and GPU without our framework. c) Response time of the real-time task with our framework d) Temperature of CPU and GPU with our framework.}\n\\label{fig:chap2_casestudy}\n\\end{figure}\n\\section{Evaluation}\n\\label{sec:chap2_eval}\n This section gives the experimental evaluation of our framework. First, we explain our implementation on a real platform. Then, we explore the impact of proposed approaches on task schedulability with randomly-generated tasksets based on  practical parameters.\n\\subsection{Implementation}\nWe did our experiments on an ODroid-XU4 development board \\cite{ODROIDXU4} equipped with a Samsung Exynos5422 SoC. There exist two different CPU clusters of little Cortex-A7 and  big Cortex-A15 cores, where each cluster consists of four homogeneous cores. There exists an integrated Mali-T628 GPU on the chip which supports OpenCL 1.1. Built-in sensors with sampling rate of 10 Hz with the precision of 1\\textdegree~C are on each big CPU core and also the GPU to measure the temperature\\footnote{There are no temperature sensors on little cores since the power consumption and heat generation of the little cluster is considerably low.}. The DTM throttles the frequency of the big CPU cluster to 900 MHz when one of its cores reaches the pre-defined maximum temperature. During experiments, the CPU fan is always either turned off or on at its maximum speed and the CPU is set to run at its maximum frequency.\n\nWe stressed the CPU cores of the big cluster and the GPU with different settings by executing \\texttt{sgemm} program of the Mali SDK benchmark \\cite{Mali} to measure the system parameters used in Section~\\ref{sec:chap2_analysis}. We observed that without launching any kernel on the GPU, because of the heat conduction, the temperature on the GPU rises from 40\\textdegree~C (the ambient temperature) to 70\\textdegree~C. However, the GPU has less thermal effect on CPU cores due to the low heat dissipation. The kernel execution on the GPU in the presence of CPU workloads raises the CPU temperature by 5-10\\textdegree~C. \n\nFig.~\\ref{fig:chap2_board}a illustrates the result of the implementation of the CPU polling server with the replenishment period of 1 second\\footnote{We conducted the experiment with large values of replenishment period because of the coarse granularity of the sampling rate of the on-board temperature sensors.} and the maximum temperature bound of 95\\textdegree~C when the CPU fan is off. As one can see, the CPU temperature oscillates between 78\\textdegree~C to 95\\textdegree~C after a long time of the system operating in the steady state.\n\nFigures~\\ref{fig:chap2_board}b depicts the server utilization with respect to the maximum temperature bound. The maximum achievable utilization of the CPU server is only 43\\% when the fan is off and almost 95\\% when the fan is on. The gap in server utilization between these two cases (fan on\/off) decreases as the value of the maximum temperature bound reduces. The steady state temperatures under different maximum temperature bounds remain almost the same regardless of whether the fan is on\/off. \n\nWith our proposed thermal-aware server design, it is possible to bound the operating temperature to any thermal constraint. It is worth noting that the same server utilization value can give different temperature bounds depending on the value of replenishment period used. Fig.~\\ref{fig:chap2_board}d shows the temperature bounds at the invariant server utilization of 30\\% and the replenishment period in the range of $[600, 1600]$ milliseconds. The length of 1600 milliseconds is the maximum feasible server replenishment period at the server utilization of 30\\% that the board can reach when the fan is off. This is because with a larger period value, the CPU exceeds its maximum temperature bound during the active phase.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/board_serverutil.pdf}\n  \\caption{}\n  \\label{fig:chap2_boardA}\n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/board_serverutilization.pdf}\n  \\caption{}\n  \\label{fig:chap2_boardB}\n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/board_30period.pdf}\n  \\caption{}\n  \\label{fig:chap2_boardC}\n\\end{subfigure}\n\\caption{a) Server design with the given maximum temperature of 95\\textdegree C when the CPU fan is off.  b) Server utilization w.r.t the given maximum temperature. c) The maximum observed temperature w.r.t the server replenishment period when the CPU fan is off and CPU utilization is 30\\%.}\n\\label{fig:chap2_board}\n\\end{figure}\n With these results, we have shown that it is possible to satisfy the maximum temperature constraint based on our proposed server design. Next, we will use the measured parameters in our analysis and discuss its effect in taskset schedulability.\n\n\\subsection{Schedulability Experiments}\n\\textbf{Task Generation.} We randomly generate 10,000 tasksets for each experimental setting. The base parameters given in Table \\ref{tab:chap2_spec} and the measured parameters from the board are used for the taskset generation and the server design, respectively. It is worth noting that the GPU parameters are in compliance with the case study of prior work \\cite{kato2011timegraph, kato2012gdev, 8046309, 7010477}. Server budgets are determined according to the maximum temperature need by applying the equations of Section~\\ref{sec:chap2_server} and the measured system parameters. The number of tasks in each taskset is determined based of the uniform distribution in the range of [8, 20]. Then, the utilization of the taskset is partitioned randomly for these tasks in a way that no task has the utilization more than the CPU server utilization. The total WCET of each task (i.e., $C_i$) is calculated based on the task's utilization and its randomly-chosen period. If the task $\\tau_i$ is a CPU-only task, the whole $C_i$ is assigned to $C_{i,1}$ otherwise $C_i$ is divided into $E_i$, $C_{i,1}$ and $C_{i,2}$, according to the random ratio of the GPU segment length to the normal WCET. In this phase, if $E_i$ is more than the GPU server budget, another random ratio is generated. Then, $E_i$ is partitioned randomly into the miscellaneous time ($M_{i,1}+M_{i,2}$) and the pure kernel execution time ($k_i$) according to the ratio of miscellaneous operations given in Table~\\ref{tab:chap2_spec}. The accumulated miscellaneous-operation time is randomly divided into $M_{i,1}$ and $M_{i,2}$. Finally, tasks are assigned to CPU cores by using the worst-fit decreasing (WFD) heuristic for load balancing across cores. When the MOT reservation is used, the MOT budget is determined by the  maximum of $M_{i,1}$ and $M_{i,2}$ for all tasks. \n\\begin{table}[h!]\n\\centering\n \\caption{ Base parameters for taskset generation}\n  \\label{tab:chap2_spec}\n \\begin{tabular}{c | c} \n \\hline\n Parameters & Values \\\\ [0.5ex] \n \\hline\\hline\nNumber of CPU cores & 4 \\\\ \nNumber of tasks & [8, 20]\\\\\nTaskset utilization & [0.4, 1.6] \\\\\nTask period and deadline & [30, 500] ms \\\\\nPercentage of GPU-using tasks & [10, 30] \\% \\\\\nRatio of GPU segment len. to normal WCET & [2, 3]:1  \\\\\nRatio of misc. operations in GPU segment $\\frac{M_{i,1} + M_{i,2}}{E_i}$ & [10, 20]\\% \\\\\nServer Period & 10 ms \\\\\nGPU Period &  20 ms \\\\\n     [1ex] \n \\hline\n \\end{tabular}\n\\end{table}\n\n\\textbf{Results.}\nFigures~\\ref{fig:chap2_analtaskset}a-b depict the percentage of the schedulable tasksets when the CPU fan is on or off with different taskset utilization. The CPU sporadic servers outperform the other CPU server policies as expected because their replenishment budget are as large as the polling server and the remote blocking and CPU-GPU handover delays are as low as those in the deferrable server. Compared to the polling server, the deferrable server with MOT yields a higher percentage of schedulable tasksets especially when taskset utilization is low. Designating some portion of CPU server budget under the deferrable replenishment policy results in improvement in taskset schedulability. However, the percentage of schedulable taskset under the deferrable server drops sharply as taskset utilization increases because of the insufficient amount of server budget (which is just the half of the polling\/sporadic server budget). It is worth noting that when the CPU fan is off, there is a large gap between the two sporadic servers with and without MOT at low taskset utilization. This is due to the advantage of the MOT mechanism that also reduces enhanced remote blocking delay.\n\n\\cmnt{the rate of schedulable tasksets under deferrable policy with\/without overrun option is very low because the budget for polling server is drastically low; hence the budget for deferrable server which is half of its corresponding polling server is negligible. When temperature demands rises, the deferrable has more chance to schedule some tasksets but even in this condition their budgets are still low. }\n\n\n\n\\begin{figure}[ht]\n\\centering\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/ANAL_off.pdf}\n  \\caption{}\n  \\label{fig:chap2_analtasksetA}\n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/ANAL_on.pdf}\n  \\caption{}\n  \\label{fig:chap2_analtasksetB}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/ANAL_enhance_polling.pdf}\n  \\caption{}\n  \\label{fig:chap2_analtasksetC}\n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/ANAL_Gtasks_sprodic.pdf}\n  \\caption{}\n  \\label{fig:chap2_analtasksetD}\n\\end{subfigure}\n\\newline\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n \n\\includegraphics[width=\\textwidth]{Chapter2\/figs\/results\/ANAL_period_poll.pdf}\n  \\caption{}\n  \\label{fig:chap2_analtasksetE}\n\\end{subfigure}\n\\caption{a) and b) The percentage of schedulable tasksets under the given maximum temperature constraint of 95\\textdegree~C when the CPU fan is off and on, respectively. c) Taskset schedulability results with and without the remote blocking enhancement under the polling server policy while the CPU fan is off. d) The percentage of schedulable tasksets w.r.t the ratio of GPU-using tasks.}\n\\label{fig:chap2_analtaskset}\n\\end{figure}\n\nFig.~\\ref{fig:chap2_analtaskset}c shows the effect of the proposed remote blocking enhancement under the polling policy. As one can see, the remote blocking reduces substantially due to our proposed remote blocking enhancement by up to 20\\% especially when there exists less workload in the system.  Fig.~\\ref{fig:chap2_analtaskset}d depicts the impact of the rate of GPU-using tasks on schedulable taskset rate under the CPU sporadic server policy with and without MOT. As the number of GPU-using tasks increases, taskset schedulability goes decreases as more tasks contend for the shared GPU.\n\n\nNext, we investigate the effect of server periods on the schedulability rate. In this experiment, the temperature constraint is fixed to the maximum level and the server period is varied from 5 to 30 milliseconds (see Fig.~\\ref{fig:chap2_analtaskset}e) under the polling server policy. As expected, because of the large CPU-GPU handover delay, the percentage of schedulable tasksets drops significantly as the server replenishment period increases. \n\\section{Summary}\n\nIn this chapter, we proposed a novel thermal-aware framework to bound the maximum temperature of CPU cores as well as an integrated GPU while guaranteeing real-time schedulability. Our framework supports various server policies and provides analytical foundations to check both thermal and temporal safety. Experimental results show that each CPU server policy provided by our framework is effective in bounding the maximum temperature. We proposed the miscellaneous operation time reservation mechanism for the CPU servers in order to improve task schedulability by reducing the CPU-GPU handover delay. We also introduced a remote blocking enhancement technique that employs the bin-packing strategy to reduce the remote blocking caused by other tasks.\n\n\n\\subsection{Thermal Back-to-Back Execution}\nSuppose that the CPU workload runs at the end of a period and the workload of the next period execute at the beginning of the period. It causes burst heat generation by back-to-back execution, which can potentially lead to thermal violation. It is noteworthy that this case has been shown in Corollary~\\ref{col:1}. The worst-case occurrence is when this phenomenon happens repetitively in the steady state. \n\n\n\\smallskip\\noindent\\textbf{Server budget calculation under deferrable server budget replenishment policy.} \nHereby, the maximum replenishment budget for the deferrable server is determined considering this phenomenon. One can model it as a periodic workload with doubled waking (${2 t_{wk}}$) and sleeping (${2 t_{slp}}$) times, and determine the maximum waking time. \n\n\n\\begin{thm}\\label{thm:util}\nThe maximum waking time in thermal back-to-back execution is\n\n$t_{wk} =  \\frac{1}{2\\beta} \\ln{\\frac{\\theta_M(e^{2 \\beta T}-1)+\\alpha}{\\alpha}} $.\n\\end{thm}\n\\begin{proof} \n\\new{\n Considering the period of workload as twice of the previous example, we have $2 t_{wk} + 2 t_{slp}  = 2 T.$\nThe maximum temperature is reached after $2 t_{wk}$ time units. So,  \n${\\theta_M =  \\alpha + (\\theta_L-\\alpha) \\mathrm{e}^{2 \\beta\\, t_{wk}}}$.\n  Similarly to Eq.~21, we have $t_{wk} = \\frac{1}{2\\beta} \\ln{\\frac{\\theta_M-\\alpha}{\\theta_L-\\alpha}}$. Since the minimum  temperature in the steady state is reached after $2 t_{slp}$, $\\theta_M \\mathrm{e}^{2 \\beta\\, t_{slp}} = \\theta_L$ which leads to $t_{slp}  =  \\frac{1}{2\\beta} \\ln{\\frac{\\theta_L}{\\theta_M}}$. Hence,\n \\[\n     t_{slp}  +t_{wk} =T \\Longrightarrow \\frac{1}{2\\beta} \\ln{\\frac{\\theta_M-\\alpha}{\\theta_L-\\alpha}}  + \\frac{1}{2\\beta} \\ln{\\frac{\\theta_L}{\\theta_M}} = T.\n \\]\n Therefore, the minimum temperature in the steady state is ${\\theta_L = \\frac{\\alpha \\theta_M \\mathrm{e}^{2 \\beta\\, T}}{\\theta_M(\\mathrm{e}^{2 \\beta\\, T}-1)+\\alpha}}$. By substituting the value of $\\theta_L$ in $t_{wk}$, the maximum waking time for the period $T$ is obtained. \n}\n \n\\end{proof}\n\\begin{corollar}\nThe maximum achievable utilization of a server under the maximum temperature constraint $\\theta_M$ is $\\frac{\\theta_M}{\\alpha}$. \n\\end{corollar}\n\\begin{proof} \nSince the maximum utilization is obtained when the period converges to 0, the maximum utilization is \n\\[ \\lim_{T \\to 0} u = \\lim_{T \\to 0} \\frac{\\frac{1}{2\\beta} \\theta_M 2\\beta e^{2\\beta T} }{\\theta_M (e^{2\\beta T} -1 ) + \\alpha} = \\frac{\\theta_M}{\\alpha}.  \\]\nThe same approach applies to the polling servers.\n\\end{proof}\n\n\n\n \n\n\n\nUnlike the claim in~\\cite{rai2011worst,yang2013real} which states that the worst-case peak temperature occurs when the system warms up by applying a periodic pattern to be in steady state following by a burst workload, we will show that by employing thermal-aware servers for mixed-criticality tasks, this will never happen. \n\n\\begin{thm}\nThe maximum temperature of a CPU for any workload execution pattern in the ``steady-state\" condition is less than the periodic back-to-back execution pattern.\n\\end{thm}\n\n\\begin{proof} \n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.50\\textwidth]{Chapter3\/figs\/thermal\/theorem3.pdf}\n\\caption{Workload execution pattern in the steady state.}\n\\label{fig:chap3_b2bSteady}\n\\end{figure}\nTo prove that, we follow up on the two scenarios illustrated in Fig.~\\ref{fig:chap3_b2bSteady}. Without loss of generality, it is assumed that the system has been already warmed up. The green line in Fig.~\\ref{fig:chap3_b2bSteady} shows that back-to-back execution pattern continues in the steady-state phase. The orange line depicts the worst-case burst workload that can be executed (according to the Theorem~\\ref{thm:min_waking_time} where all of the replenished budget is exhausted at the beginning of the period). Let $\\theta_1$ and $\\theta_2$ represent the maximum temperature of burst execution and normal back-to-back execution, respectively. Hence, ${\\theta_1 = \\alpha +\\theta _{M}\\,{\\mathrm{e}}^{T\\,\\mathrm{\\beta}}-\\alpha \\,{\\mathrm{e}}^{\\mathrm{\\beta}\\,t_{w}}}$ and ${\\theta_2 = \\alpha +\\theta _{M}\\,{\\mathrm{e}}^{2\\,T\\,\\mathrm{\\beta}}-\\alpha \\,{\\mathrm{e}}^{2\\,\\mathrm{\\beta}\\,t_{w}}}$. Let $\\Delta_{\\theta}$ denote the temperature difference of these scenarios. Therefore,     \n     \\[\n    \\Delta_{\\theta} =  \\theta_2 - \\theta_1= \\theta _{M}\\,{\\mathrm{e}}^{2\\,T\\,\\mathrm{\\beta}}-\\theta _{M}\\,{\\mathrm{e}}^{T\\,\\mathrm{\\beta}}+\\alpha \\,{\\mathrm{e}}^{\\mathrm{\\beta}\\,t_{w}}-\\alpha \\,{\\mathrm{e}}^{2\\,\\mathrm{\\beta}\\,t_{w}}.\n     \\]\n By considering $u = \\frac{\\theta_M}{\\alpha}$ and substituting the maximum waking time determined from Theorem~\\ref{thm:util}, we show that $\\Delta_{\\theta}>0$ which mean\n \\[\n u{\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}} - {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,{\\frac{1}{2\\beta} \\ln{\\frac{\\theta_M(e^{2 \\beta T}-1)+\\alpha}{\\alpha}}}}} + {\\mathrm{e}^{\\,\\mathrm{\\beta}{\\frac{1}{2\\beta} \\ln{\\frac{\\theta_M(e^{2 \\beta T}-1)+\\alpha}{\\alpha}}}}} - u {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} > 0.\n \\]\n\nFor any value of $u\\in[0,1]$, since $(u-1) ( 1+ {\\mathrm{e}^{\\mathrm{\\beta}\\,T}}) ^2  < 0 $, then we have \n\\[u + u {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}} + 2 {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} < 1 + 2 u {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} + {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}}.\\]\n\nBy multiplying $u$ in the inequality, \\new{we have} \n    \\[ \n         u ^2  + 1 - 2 u + u^2 {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}}- 2 (u-1) u {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} < u {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}}  - u  + 1\n        \\]  \nwhich leads to $\n        u - 1 -  u {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} >  - {(\\frac{\\theta_M \\mathrm{e}^{2\\,\\mathrm{\\beta}\\, T} - \\theta_M + \\alpha}{\\alpha})}^{\\frac{1}{2}}$. Therefore,\n      \n\\[\n       u {\\mathrm{e}^{2\\,\\mathrm{\\beta}\\,T}} - \\frac{\\theta_M \\mathrm{e}^{2\\,\\mathrm{\\beta}\\, T} - \\theta_M + \\alpha}{\\alpha} +   {(\\frac{\\theta_M \\mathrm{e}^{2\\,\\mathrm{\\beta}\\, T} - \\theta_M + \\alpha}{\\alpha})}^{\\frac{1}{2}} - u {\\mathrm{e}^{\\mathrm{\\beta}\\,T}} > 0.\n\\]\n\\end{proof}\n\n\n\n\n       \n\n\\subsection{Peak Temperature Analysis on Multi-Core Platforms}\nNow, we extend our analysis to multi-core platforms where each CPU core consists only of one thermal-aware server. We will show that the minimum replenishment budget of polling servers on multi-core CPU happens when all of them exhaust their budget completely at the same time. Afterwards, the maximum available server budget will be determined. \n\n\n\n\n\n\n\n\\begin{thm}\\label{thm:min_waking_time_multi}\nThe minimum value of waking time for a maximum temperature constraint is when the server on each core exhausts all its budget at once, simultaneously. \n\\end{thm}\n\\begin{proof}\nAssume we have a dual-core CPU with the same physical characteristics and surrounding conditions. Both cores have the same periodic step power signal but with a phase difference of $\\phi$. For simplicity we assume $P_S$ is zero and $P=P_D$. The input power of the first core $P_1(t)$ starts at $t=t_0$ while the input power of the second core $P_2(t)$ starts with a phase change at $t=t_0+\\phi$. We consider two cases: for case I, the phase change is zero ($\\phi=0$), and for case II it is positive ($\\phi>0$). The schematic of power signal for the two cases is depicted in Fig.\\ref{fig:chap3_thrm4_power}. The temperature profiles of in-phase and out-of-phase cases are compared in Fig.\\ref{fig:chap3_thrm4_temperature}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.78\\textwidth]{Chapter3\/figs\/thermal\/Theorem4_2.pdf}\n\\caption{Power signal of the cores for (a) case I: in-phase, and (b) case II: out-of-phase.}\n\\label{fig:chap3_thrm4_power}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Chapter3\/figs\/thermal\/Theorem4_3.pdf}\n\\caption{Temperature profiles of in-phase and out-of-phase cases for a (a) two-core and (b) three-core system. (Blue lines represent the temperature of the cores in in-phase and other colors represent temperature of the cores in out-of-phase states.) }\n\\label{fig:chap3_thrm4_temperature}\n\\end{figure}\n\n\\textit{Lemma}. In a multi-core system, if the power signal of each core varies with time in the way described above, the maximum change rate magnitude of temperature is when the powers are in-phase ($\\phi=0$). \n\n\\textit{For a \\new{dual-core CPU}}. According to the developed model, temperatures of the two cores between $t_0$ and $t$ when the power $P_1$ and $P_2$ are constant are as follows: \n\\[\n \\scriptstyle\n\\begin{aligned}\n\\theta  = \\frac{1}{2}{e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}}\\left[ {\\begin{array}{*{20}{c}}\n{{\\theta _0}_1 + {\\theta _0}_2}\\\\\n{{\\theta _0}_1 + {\\theta _0}_2}\n\\end{array}} \\right] + \\frac{1}{2}\\;{e^{{\\ali{\\beta _2}}\\left( {t - {t_0}} \\right)}}\\left[ {\\begin{array}{*{20}{c}}\n{{\\theta _0}_1 - {\\theta _0}_2}\\\\\n{{\\theta _0}_2 - {\\theta _0}_1}\n\\end{array}} \\right] + \\\\\n\\frac{{{B_1}}}{{2{\\ali{\\beta_1}}}}\\left( {{e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}} - 1} \\right)\\left[ {\\begin{array}{*{20}{c}}\n{{P_1} + {P_2}}\\\\\n{{P_1} + {P_2}}\n\\end{array}} \\right] + \n\\frac{{{B_1}}}{{2{\\ali{\\beta_2}}}}\\left( {{e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}} - 1} \\right)\\left[ {\\begin{array}{*{20}{c}}\n{{P_1} - {P_2}}\\\\\n{{P_2} - {P_1}}\n\\end{array}} \\right]\n\\end{aligned}\n\\]\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.95\\textwidth]{Chapter3\/figs\/thermal\/Theorem4_4.pdf}\n\\caption{Comparison of temperature (a) increase and (b) decrease between in-phase and out-of-phase cases.}\n\\label{fig:chap3_thrm4_tempderivative}\n\\end{figure}\n\nAssume that core 1 and 2 start from initial temperatures of $\\theta_{01}$ and $\\theta_{02}$ at time $t_0$ and get to the final temperatures of $\\theta_1$ and $\\theta_2$ at time $t$ as shown in Fig.\\ref{fig:chap3_thrm4_tempderivative} . We want to show that $\\left| {{\\rm{\\Delta }}{\\theta _{in - phase}}} \\right| > \\left| {{\\rm{\\Delta }}{\\theta _{out - of - phase}}} \\right|$  for any time slot between $t_0$ and $t$ where the power remains constant. We calculate the derivative of temperature for the two cases. When there is a power for the in-phase case ($P\\neq0$), the temperate change rate is positive. For in-phase case $P_1+P_2=2P$ and $P_1-P_2=0$. For the out-of-phase case, ${P_1+P_2=P}$ and ${P_1-P_2=-P}$ because $P_1=0$ and $P_2=P$ or vice versa. We have: (I represents in-phase and II represents out-of-phase)\n\n\\[\n\\begin{aligned}\n{\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{I}} - {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}} =\n\\frac{{{B_1}P}}{2}\\left[ {\\begin{array}{*{20}{c}}\n{{e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}} + {e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}}}\\\\\n{{e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}} - {e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}}}\n\\end{array}} \\right]\n\\end{aligned}\n\\]\n\n\n\nSince \\ali{$\\beta_1 > \\beta_2$}  for both cores and $B_1$ and $P$ are positive, then ${\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{I}} \\ge {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}}$. Same conclusion can be drawn if $P_1=P$ and $P_2=0$. We can conclude that ${\\rm{\\Delta }}{\\theta _{in-phase}} \\ge {\\rm{\\Delta }}{\\theta _{out-of-phase}}$ based on the fact that if $f\\ge g$ in $\\left[{a,b}\\right]$, and $f$ and $g$ are integrable in $\\left[{a,b}\\right]$, then $\\mathop \\smallint \\limits_a^b fdx\\ge\\mathop\\smallint\\limits_a^b gdx$. If the power in the in-phase case is zero ($P_1+P_2=0$ and $P_1-P_2=0$), the temperature change rate is negative, and for $P_1=0$ and $P_2=P$ we have: \n\\[\\begin{aligned}\n{\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_I} - {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}} = \\frac{{{B_1}P}}{2}\\left[ {\\begin{array}{*{20}{c}}\n{ - {e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}} + {e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}}}\\\\\n{ - {e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}} - {e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}}}\n\\end{array}} \\right]\n\\end{aligned}\\]\n\nSince \\ali{$\\beta_1 > \\beta_2$}  for both cores ${\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{I}} \\le {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}}$. Same conclusion can be drawn if $P_1=P$ and $P_2=0$. Therefore, in case of an increase in temperature, the in-phase state has the highest temperature change rate, and in case of a decrease in temperature, the in-phase state has the lowest temperature change rate. In conclusion, $\\left| {{\\rm{\\Delta }}{\\theta _{in - phase}}} \\right| \\ge \\left| {{\\rm{\\Delta }}{\\theta _{out - of - phase}}} \\right|$.\n\n\\textit{For \\new{a multi-core CPU}}. For a multi-core system when temperature is increasing, the difference between the temperature derivatives of the in-phase (I) and out-of-phase (II) cases between $t_0$ and $t$ when the power signal does not change is as follows:\n\n\\\n\\begin{aligned}\n{\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_I} - {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}} = \\frac{{{B_1}}}{n}{e^{{\\ali{\\beta_1}}\\left( {t - {t_0}} \\right)}}\\left( {n - {k_1}} \\right)P{\\left[ {\\begin{array}{*{20}{c}}\n1\\\\\n1\\\\\n \\vdots \\\\\n1\n\\end{array}} \\right]_{n \\times 1}} \\\\\n- \\frac{{{B_1}}}{{{k_2}}}{e^{{\\ali{\\beta_2}}\\left( {t - {t_0}} \\right)}}{\\left[ {\\begin{array}{*{20}{c}}\n{{m_{21}}P}\\\\\n{{m_{22}}P}\\\\\n \\vdots \\\\\n{{m_{2n}}P}\n\\end{array}} \\right]_{n \\times 1}} -  \\ldots  - \\frac{{{B_1}}}{{{k_n}}}{e^{{\\ali{\\beta_n}}\\left( {t - {t_0}} \\right)}}{\\left[ {\\begin{array}{*{20}{c}}\n{{m_{n1}}P}\\\\\n{{m_{n2}}P}\\\\\n \\vdots \\\\\n{{m_{nn}}P}\n\\end{array}} \\right]_{n \\times 1}}\n\\end{aligned}\n\\]\n\n${k_i}$, $i = 2,..,n$ are integers which satisfy $1\\le{k_i}\\le n,\\,\\,i = 2,..,n$. For ${k_1}$ we have $1\\le {k_1} < n$ since there is at least one core which is idle when powers are in out-of-phase. $m_{ij}$ can change based on how many cores are active and we have $m_{ij} < k_i$. Since $n > {k_i}$, the first term is positive, and since \\ali{${\\beta _1} \\gg {\\beta _2},...,{\\beta _n}$} and all \\ali{$\\beta$s} are negative, the other terms decrease exponentially to zero and can be neglected compared to the first term. Therefore, ${\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_I} \\ge {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}}$. It can be shown in the same way that when there is no power in the in-phase case, the temperature is decreasing and ${\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_I} \\le {\\left( {\\frac{{d\\theta }}{{dt}}} \\right)_{II}}$. \n\n\\textit{Lemma}. For the described power signals, the maximum temperature for in-phase case is larger than that of the either core for the out-of-phase case (${\\theta _M} \\ge \\theta _M^{'}$). \\\\\nFirst, let's point out that ${\\theta _{ave}} =  - {{\\bf{A}}^{ - 1}}{\\bf{B}}{{\\bf{P}}^\\infty }$ is the same for both cases in the steady state. Therefore, ${\\theta _M} + {\\theta _L} = \\theta _M^{'} + \\theta _L^{'}$. Assume that ${\\theta _M} < \\theta _M^{'}$, then ${\\theta _L} > \\theta _L^{'}$. Assume that the time it takes for the in-phase case temperature to get from ${\\theta _L}$ to ${\\theta _M}$ is $t_w$ and the time it takes to get from ${\\theta _M}$ to ${\\theta _L}$ is $t_s$. Assume these times for the out-of-phase case is $t{'_w}$ to get from $\\theta {'_L}$ to $\\theta {'_M}$ and $t{'_s}$ to get from $\\theta {'_M}$ to $\\theta {'_L}$. If ${\\theta _M} < \\theta _M^{'}$, and ${\\theta _L} > \\theta _L^{'}$, then $t_w$ would be smaller than $t{'_w}$ because the temperature increase rate is largest for the in-phase case. At the same time, ${t_s} < t_s^{'}$ since temperature decrease rate is largest for the in-phase case. Therefore, ${t_w} + {t_s} < t{'_w} + t{'_s}$. But the periods for the two cases are the same ${t_w} + {t_s} = t{'_w} + t{'_s}$.\n\\end{proof}\n\n\n\n\n\n\\noindent\\textbf{Experimental Example }\nTo support our claim, we  measure the temperature of big cores on the Exynos 5422 SoC~\\cite{exynos} using IR camera FLIR A325sc~\\cite{flir} with the sampling rate of 60 frames per second. Four periodic computationally-intensive workloads are ran on four big cores of the board. In all cases, the CPU frequency is set to 1.4 GHz, and each workload executes every 10 seconds with the utilization of $40\\%$. Let $\\phi(i)$ denote the delay of starting time of a workload execution on core $i$. Table~\\ref{tab:chap3_cases} shows the configurations of each case.\n\n\\begin{table}[h!]\n\\centering\n \\caption{ Descriptions of workload executions on big cores.}\n  \\label{tab:chap3_spec}\n    \\begin{tabular}{ |p{1.1cm}|p{8cm}|p{2.3cm}|}\n    \\hline\n     Name & description & workloads delay settings (seconds)\\\\\n    \\hline\n   \\begin{footnotesize} Case 1 \\end{footnotesize}& \\begin{footnotesize}all workloads execute at the same time \\end{footnotesize}&\\begin{tiny}  ${\\phi(1)=\\phi(2)=0}$ ${\\phi(3) = \\phi(4) = 0}$ \\end{tiny}\\\\ \\hline\n   \\begin{footnotesize} Case 2 \\end{footnotesize}& \\begin{footnotesize}workloads on two cores begin after those of other cores \\end{footnotesize}&\\begin{tiny} ${\\phi(1) = \\phi(2) = 0 }$ ${\\phi(3) = \\phi(4)=4}$ \\end{tiny}\\\\\n    \\hline\n    \\begin{footnotesize} Case 3 \\end{footnotesize}& \\begin{footnotesize} workloads on two cores begin 1 second before finishing those of on other cores \\end{footnotesize}&\\begin{tiny}  ${\\phi(1) = \\phi(2) = 0}$ ${\\phi(3) = \\phi(4) = 3} $ \\end{tiny}\\\\ \\hline\n   \\begin{footnotesize} Case 4 \\end{footnotesize} & \\begin{footnotesize}workloads on each core execute with a 2-second overlap \\end{footnotesize}&\\begin{tiny}  ${\\phi(1) = 0 \\ \\phi(2) =2}$   ${\\phi(3) = 4\\  \\phi(4) = 6} $ \\end{tiny}\\\\\n    \\hline\n    \\end{tabular}\n    \\label{tab:chap3_cases}\n\\end{table}\n\nFig.~\\ref{fig:chap3_thrm4_observ} shows our observations of the maximum temperature of the SoC for the described cases in the steady state. As one can see, in Case 1 where all workloads execute synchronously, the maximum temperature of the chip reaches its highest value and its minimum temperature is the lowest among all cases. On contrary, in Case 4 where workloads of different cores have the least overlap, the maximum temperature of the chip fluctuates the least of other cases and it is close to the average temperature in the steady state. In Case 2 where there is no overlap between executions of two pairs of workloads, the rate of temperature rise is lower than the Case 3 where there is 1 second overlap within all workload executions.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.62\\textwidth]{Chapter3\/figs\/thermal\/observe_1400mhz.pdf}\n\\caption{The maximum temperature of big cores in the Exynos 5422 captured by FLIR A325sc IR camera in the operating frequency of 1.4 GHz without heat sink.}\n\\label{fig:chap3_thrm4_observ}\n\\end{figure}\n\n\n\n\n\n\\section{Multiple Server Analysis}\n\nIn this section, we extend our analysis for multiple servers running on the multi-core CPU at each criticality level. We are interested to see if there \\new{are} enough sleeping slacks between active times of servers to cool down the CPU. The cooling time must be large enough such that the CPU does not exceed the maximum temperature constraint under any circumstance. \nHence, we should check if the amount of \\new{the} sleep\\new{ing} time required for cooling is guaranteed for a given period of time. \n\n\nWe propose an {\\it idle thermal server} technique in this regard.\nUnlike regular servers, the idle server does not execute; instead, its budget represents the amount of time that the CPU core needs to be idle in the cooling phase. \\new{\\chk{The reasoning behind this technique is to simplify the modeling of the resulting idle time from the execution of multiple regular servers as a single budget parameter. \\cmnt{Hence, the idle server budget can be determined such that heat dissipation during the idle server's inactive time reaches the same or higher maximum operating temperature than that of running multiple servers under any task execution pattern.}} Hence, the idle server's budget can be determined such that the maximum CPU operating temperature caused by heat dissipation during the idle server's inactive time is the same as or higher than that of running regular servers under any task execution pattern.} If such an idle server is schedulable, one can conclude that the given taskset is thermally schedulable. \\new{The thermal effect of multiple servers is analyzed by the complement signal of periodic idle-server execution, the worst-case behavior of which has been proved by Theorems~\\ref{thm:superposition}-\\ref{thm:min_waking_time_multi}.}\n\n\\cmnt{If this idle server is schedulable, the CPU core will never exceed the maximum temperature constraint.} After analytically finding the relation between the budget and period of the idle server, we check its schedulability. \n\nSince the idle server does not actually exist on the CPU, it is considered as the lowest-priority server in the schedulability test. In our proposed framework, the CPU is not forced to sleep so that the proposed idle server has no effect on the timing schedulability of regular running servers.\n\nThe idle server utilization corresponds to the time during which all regular servers are deactivated. Based on this, we investigate the feasibility of the idle server with its minimum possible period within a valid range. In this work, we focus on designing homogeneous idle servers, i.e., all idle servers on different CPU cores share the same parameters at each criticality level. It is worth mentioning that this does not mean that sleeping time in different cores must happen at the same time. Finding the different idle server settings for each CPU core is beyond the scope of this chapter. \\cmnt{To this end, the thermal model of the multi-core has to change and the formulae will change according to eigen matrix of the CPU cores and their operating power signal functions.} \n\n\\smallskip\\noindent\\textbf{Idle server design}\nFirst, we compute the total utilization of the CPU core $c$ at the critical\\new{ity} level $l$ by\n${u^{l}_c = \\sum_{\\forall i\\ \\mathbb{P}(v_i) = c \\land v_i \\in V^l }\\frac{C_i}{T_i}}$, where $\\mathbb{P}(v_i)$ is the CPU core assigned to $v_i$.  \nThe maximum per-core CPU utilization is then $u^{l}_{max} = \\max_{\\forall c}{u^{l}_c}$. Due to the homogeneity of idle servers on all cores, $u^{l}_{max}$ is considered as the utilization of one core so that each core is supposed to be idle at least $1-u^{l}_{max}$. As a result, we are looking for a server that can be schedulable with the amount of utilization of at least $1-u^{l}_{max}$. Let $u^{l}_{idle}$ denote  this value. According to Theorem~\\ref{thm:util}, the period of the idle server $T^{l}_{idle}$ can be modeled as:\n\\begin{equation}\\label{eq:chap3_tuidle}\nT^{l}_{idle} \\leq \\frac{\\ln({1+\\alpha \\frac{ e^{1-u^{l}_{idle}+\\frac{1}{2\\beta}}-1}{\\theta_M}})}{2\\beta}.\n\\end{equation}\n It is worth noting that the period is determined based on the back-to-back execution under in presence of multiple servers for both polling and deferrable budget replenishment policies since servers can preempt each other and sleeping time does not happen in a contiguous manner. \n $T^{l}_{idle}$ is an increasing function in terms of utilization. As one can figure out from Eq.~\\ref{eq:chap3_tuidle}, an increase in the workload on the CPU core (hence resulting in a decrease in $u^{l}_{idle}$) leads to a decrease in $T^{l}_{idle}$. In other words, under heavier workload, the CPU has to sleep more frequently but in a shorter duration, to satisfy the maximum temperature constraint.\n\n\\smallskip\\noindent\\textbf{Thermal schedulability}\nHereby, we present our proposed thermal schedulability test for a specific utilization of idle servers. The  worst-case response time of each idle server of each core at each criticality level can be obtained as follows\n\\begin{equation}\\label{eq:chap3_resp}\nR^{n+1,l}_{idle,c} = u^{l}_{idle}\\times T^l_{idle} + \\sum_{\\forall i\\ \\mathbb{P}(v_i) = c \\land v_i \\in V^l}\\ceil*{\\frac{R^{n,l}_{idle,c}}{T_i}}C_i\n\\end{equation}\nwhere $R^{n+1,l}_{idle,c}$ denotes the worst-case response time of the idle server on the core $c$ at the critical\\new{ity} level $l$. As shown in Eq.~\\ref{eq:chap3_resp}, the idle server of each core can be preempted by all active servers at $l$. The only unknown parameter in above test is the idle server settings. Next, we discuss the optimal valid range of idle server's setting range to reduce the number of tests for each criticality level.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.62\\textwidth]{Chapter3\/figs\/thermal\/range.pdf}\n\\caption{ Search space for the idle server settings.}\n\\label{fig:chap3_multi-phase}\n\\end{figure}\n\n\\smallskip\\noindent\\textbf{Optimal server setting range}\nAs discussed in Eq.~\\ref{eq:chap3_tuidle}, the period of the idle server is an increasing function in terms of its utilization. Fig.~\\ref{fig:chap3_multi-phase} plots the period with respect to utilization of the idle server. The area of the plot is divided to four regions. We discuss that only the highlighted area needs to be searched for finding the valid settings of idle servers.\n\n\n\\begin{itemize}\n    \\item \\textbf{region \\circled{1}:} In this region, the CPU violates the maximum temperature constraint according to Eq.~\\ref{eq:chap3_tuidle}\n    \\item \\textbf{region \\circled{2}:} The utilization of idle server has to be less than $1-u^l_{max}$. Choosing a setting in this region causes the system to be unschedulabe at this criticality level.\n    \\item \\textbf{region \\circled{3}:} Since idle servers have the lowest priority, they cannot preempt other servers. According Eq.~\\ref{eq:chap3_resp}, in the worst case all servers preempt the idle servers.\n    \\item \\textbf{region \\circled{4}:} Valid settings can be found in this region, but it is unnecessary to search the entire of this region because there exists a valid setting with period of $T'$, a solution is valid on the highlighted range with the same utilization value.\n\\end{itemize}\n\nTherefore, the only period of ${T_{idle} = \\frac{\\ln({1+\\alpha \\frac{ e^{1-u_{idle}+\\frac{1}{2\\beta}}-1}{\\theta_M}})}{2\\beta}}$ within the optimal range of utilization, $u_{idle}$, needs to be checked. \nOne may try to insert $T_{idle}$ into Eq.~\\ref{eq:chap3_resp} and assign $T_{idle}$ to ${R^{n+1,l}_{idle,c}}$ to find a single optimum point. However, because of the ceiling operation in Eq.~\\ref{eq:chap3_resp}, finding the optimum $u_{idle}$ would need an exhaustive search.\n\n\\new{After finding the idle server settings, the critical ambient temperature for each criticality level is computed by using  Eq.~\\ref{eq:chap3_1009} with  ${u = 1-u^l_{idle}}$. The shifting time from criticality level $l+1$ to $l$ can be determined by applying Eq.~\\ref{eq:chap3_1012}.}\n\n\n\n\\subsection{Timing Schedulability Analysis}\n\n\n\\noindent\\textbf{Timing schedulability}\nDue to our separate thermal schedulability analysis,\nwe are able to use the existing response time test developed for independent tasks with no thermal constraints under hierarchical scheduling~\\cite{saewong2002analysis}:\n\\chk{\n\\begin{equation}\n\\label{eq:chap3_hierchical} \\begin{split}\n W_i^{n+1,l}   =  E_i + \\sum_{\\begin{subarray}{h} \\tau_h \\in \\mathbb{V}^l(\\tau_i) \\\\\n       h > i \\end{subarray} }\\ceil*{\\frac{W_i^{n,l}  +J_i + (W_{h}^{l} - E_h)}{D_h}}E_h +\n       \\ceil*{\\frac{W^{n,l}+C_j}{T_j}} (T_j - C_j)\n\\end{split}\n\\end{equation}\n}\nwhere $W_i^{n,l}$ is the worst-case response time of $\\tau_i$ at criticality level $l$, $\\mathbb{V}^l(\\tau_i)$ is the server of $\\tau_i$, $J_j$ is the jitter of a task running in a server $v_j$ (see Sec~\\ref{sec:chap3_model}), and $W^{0,l}_i = E_i$. \n\n\n\n\n\\section{Evaluation}\n\\label{sec:chap3_eval}\nThis section gives the experimental evaluation of our framework. First, we show the model validation by measuring the physical system parameters on a real platform. After the analysis of server characteristics in different ambient temperatures, we present our discussion with a case study.\n\n\\subsection{Experimental Platform}\nThe experimental platform is an ODroid-XU4 development board~\\cite{ODROIDXU4} equipped with a Samsung Exynos5422 SoC. There exist two different CPU clusters of little Cortex-A7 and  big Cortex-A15 cores, where each cluster consists of four homogeneous cores. Built-in sensors with the sampling rate of 10 Hz with the precision of $1\\textdegree$C are on each big CPU core to measure the chip temperature. Note that there are no temperature sensors on little cores since the power consumption and heat generation of the little cluster is considerably low. The DTM throttles the frequency of the big CPU cluster to 900 MHz when one of its cores reaches the pre-defined maximum temperature constraint of $95\\textdegree$C. During experiments, the CPU fan is turned off and the CPU is set to run at 1400 MHz.\n\n\n\n\n\\subsection{Model Validation}\nAccording to Eq.~\\ref{eq:chap3_1003} and~\\ref{eq:chap3_1019}, the characteristic matrices {\\bf{A}} and {\\bf{B}} can be determined by a utilization test at different CPU frequencies. A zero utilization (idle) at different ambient temperatures can reveal the range of the static and consequently the dynamic dissipation. \nAfter finding the thermal parameters of the system and model calibration, the analytical model is validated with the experimental results. For this purpose the CPU temperature is recorded at 90\\% utilization with a period of 1 s in $\\Theta_{amb}$ of $23\\textdegree$C. Then utilization is decreased to 30\\% and the CPU is cooled down until reaching a steady state. Afterwards, the CPU working at 30\\% is placed in the furnace with $\\Theta_{amb}$ of $42\\textdegree$C. The CPU temperature is recorded until the steady state. The same conditions are simulated using the developed model. Fig.~\\ref{fig:chap3_Model_validation} compares the CPU temperature recorded in the experiment at different stages of workloads and ambient temperature with the predicted values by the model. It can be seen that there is a good agreement between the model and the experimental results. \n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.63\\textwidth]{Chapter3\/figs\/results\/Model_Validation.png}\n\\caption{Comparison of the experimental results and the model prediction for CPU temperature at different workloads and ambient temperatures.}\n\\label{fig:chap3_Model_validation}\n\\end{figure}\n\n\n\\subsection{Workload, Period, and Ambient Temperature Relations}\nWe investigate the effect of  ambient temperature, server period, and utilization using Eq.~\\ref{eq:chap3_1009} and Eq.~\\ref{eq:chap3_1010}. Assuming the temperature threshold of $\\Theta_m=95\\textdegree$C, the maximum allowable utilization is plotted in Fig.~\\ref{fig:chap3_u_vs_T_Tamb}a against period and ambient temperatures. It can be seen that for all considered periods, when the ambient temperature is increased from $23\\textdegree$C, the maximum workload decreases almost linearly with the ambient temperature. At higher ambient temperatures lower workloads can be used until the ambient temperature of $69\\textdegree$C where even an idle CPU usage will result in a working temperature equal to the threshold temperature. To better see the effect of period, the maximum workload is plotted against period at different ambient temperatures in Fig.~\\ref{fig:chap3_u_vs_T_Tamb}b. The period has been changed from 10 ms to 30 s. It can be seen that the maximum workload decreases by increasing the value of the period in an almost linear manner. This has been discussed and confirmed in Theorem~\\ref{thm:min_waking_time}. Moreover, it can be concluded that the effect of ambient temperature on the maximum allowable workload is more prominent than the effect of period. \n\\begin{figure}[ht]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.45\\textwidth]{Chapter3\/figs\/results\/u_vs_T_Tamb.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.44\\textwidth]{Chapter3\/figs\/results\/u_vs_T.pdf}}\n\\caption{a) Utilization versus period and ambient temperature. b) Utilization versus period at different ambient temperatures.}\n\\label{fig:chap3_u_vs_T_Tamb}\n\\end{figure}\n\n\\subsection{Shifting Time Analysis}\nAs mentioned before, any change in the working parameters of the system may cause a transient thermal behavior and change the steady state conditions. Here, we discuss the effects of changing workload and also ambient temperature on the thermal response of the CPU cores. In Fig.~\\ref{fig:chap3_Shifting_Tamb}a. shifting times are plotted against the final ambient temperature at different workloads assuming the initial ambient temperature is $23\\textdegree$C or $50\\textdegree$C. It can be seen that at higher workloads it takes less time for the CPU to reach the steady state when ambient temperature is changed from $\\Theta_{amb_i}$ to $\\Theta_{amb_f}$. Also, at all workloads, it takes more time for the CPU to reach the final ambient temperature $\\Theta_{amb_f}$ if it starts from a lower initial ambient temperature $\\Theta_{amb_i}$. Furthermore, for all utilizations, the time it takes for the CPU to reach the steady state when the ambient temperature goes up by $\\Delta \\Theta$ is almost equal to the time it takes when it goes down by the same amount. \n\nIn Fig.~\\ref{fig:chap3_Shifting_Tamb}b. shifting times are plotted against the final workload $u_f$ for different initial workloads $u_i$. It can be seen that it takes more time to reach the steady state of final $u_f$ when starting from a lower workload $u_i$. Also, opposite to the case of ambient temperature, if ${u_i} < {u_f}$, it takes less time to reach the steady state when shifting from $u_i$ to $u_f$ (heating), compared to when shifting from $u_f$ to $u_i$ (cooling).  \n\n\\begin{figure}[t]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.43\\textwidth]{Chapter3\/figs\/results\/Shifting_Time_Tamb.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.43\\textwidth]{Chapter3\/figs\/results\/Shifting_Time_u.pdf}}\n\\caption{Shifting time a) from initial ambient temperature to different final ambient temperatures at various workloads, b) from different initial workloads to different final workloads at an ambient temperature of $23\\textdegree$C.}\n\\label{fig:chap3_Shifting_Tamb}\n\\end{figure}\n\n\n\\subsection{Case Study}\nWe emulate the mixed-critical\\new{ity} Flight Management System (FMS) application~\\cite{Ahmed2017, giannopoulou2013scheduling} which comprises  two criticality levels of \\textit{H} and \\textit{L}. The parameters of the executing real-time tasks are given in ~\\cite{Ahmed2017}\\cmnt{Table~\\ref{tab:spec}}. In our experiment, there exist one high\\new{-}criticality and one \\new{low-criticality} server on each CPU \\new{core}. The budget replenishment period of each thermal-aware server is considered 50 ms under the deferrable budget replenishment policy. The budget for high-criticality and low-criticality servers are 15 ms and 27 ms, respectively. Tasks are assigned to CPU cores by using the worst-fit decreasing (WFD) heuristic for load balancing across cores and are scheduled by the Rate Monotonic (RM) policy.\nSince the amount of the workload in low-criticality level is insignificant to reach the maximum temperature, non-real-time tasks are also assigned to low-criticality servers with the lowest priority level. \n\n\nCritical ambient temperatures have been determined by Eq.~\\ref{eq:chap3_1009}: $24\\textdegree$C for the low-criticality and $40\\textdegree$C for the high-criticality level.\nAs shown in Fig.~\\ref{fig:chap3_furnace}, the experiment has been performed in the furnace. Nordic Semiconductor Thingy:52\u2122 IoT sensor development kit~\\cite{Thingy52} is used to capture the ambient temperature with the sampling rate of 10~Hz. \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{Chapter3\/figs\/results\/furnace2.jpg}\n\\caption{Experimental environment using furnace.}\n\\label{fig:chap3_furnace}\n\\end{figure}\n\nFig.~\\ref{fig:chap3_case_study} shows the experimental and model results for a case study with period of 50 ms. In step I, the CPU is idling for 1800 s, and then in step II, workload with $u=95\\%$ is applied at $\\Theta_{amb}=24\\textdegree$C. The system is left to work with $u=95\\%$ until it reaches steady state conditions and keeps working for about 10000 s. The CPU temperature reaches to a value around 89$\\textdegree$C. Afterwards, in step III, the CPU is placed in the furnace with $\\Theta_{amb}=40\\textdegree$C and the workload is changed to 30\\% at the same time. The temperature increases to 92$\\textdegree$C and remains steady for about 4000 s. The CPU is then taken out of the furnace and left to work with $u = 30\\%$ for a fast cooling. Finally, in step IV, the workload is set to 95\\% at ambient temperature $\\Theta_{amb}=24\\textdegree$C. It can be seen that the developed model matches the experimental results with a good accuracy. The time step in the model is more accurate than the actual temperature sensor and temperature variations for each period can be captured by the developed model. Temperature curve from 6000 s to 6002 s is zoomed for a better comparison of the variations.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{Chapter3\/figs\/results\/Case_Study_Whole_2.pdf}\n\\caption{Experimental and model results for a case study with a period of 50 ms at two ambient temperatures and workloads.}\n\\label{fig:chap3_case_study}\n\\end{figure}\n\\section{Summary}\n\nIn this chapter, we proposed a novel mixed-criticality thermal-aware server framework to bound the maximum temperature of CPU cores in the presence of dynamic ambient temperature. In this framework, the server schedule is flexible -- fully preemptive and priority-based. We investigated the thermal feasibility by analyzing the amount of slack between execution of preemptive thermal servers with the notion of idle servers. We presented a mechanism to optimally search the maximum ambient temperature for every critical\\new{ity} level.  We provided analytical foundations to check thermal safety while temporal safety is guaranteed. Experimental results show that our proposed framework is effective in bounding the maximum temperature at every critical\\new{ity} level.\n\n\n\n\\chapter{On Dynamic Thermal Conditions in Mixed-Criticality Systems}\n\\input{Chapter3\/0_abstract}\n\\input{Chapter3\/1_intro}\n\\input{Chapter3\/2_related}\n\\input{Chapter3\/3_model}\n\\input{Chapter3\/4_frame}\n\n\\input{Chapter3\/5_0_ThermAnal}\n\\input{Chapter3\/5_1_thermCrit}\n\\input{Chapter3\/5_2_1_steady_single.tex}\n\\input{Chapter3\/5_2_2_back2back.tex}\n\\input{Chapter3\/5_2_3_multicore.tex}\n\n\\input{Chapter3\/6_multiserver.tex}\n\\input{Chapter3\/7_timingschd.tex}\n\\input{Chapter3\/8_eval}\n\\input{Chapter3\/9_conclud}\n\n\\section{Introduction}\n\nEnsuring continuous operation with high assurance in the physical environment \nremains a significant challenge to cyber-physical systems (CPS). This is particularly important for safety-critical applications with real-time mixed-criticality components, e.g., automotive, aerospace, manufacturing, and defense systems, where even occasional timing failures of high-critical\\new{ity} components can lead to catastrophic consequences. Various types of unexpected changes in the physical environment may affect the system behavior and contribute to the difficulty of this problem. \n\nAmbient temperature is one of the key factors that affect many mixed-criticality CPS applications. For instance, in automobiles, a report from the National Renewable Energy Laboratory of the US Department of Energy~\\cite{farrington2000impact} indicates that cabin air temperature can reach up to and 82$^{\\circ}$C in Phoenix, Arizona. \\new{The heat generated by the engine worsens the ambient temperature level of nearby electronic control units~\\cite{park2010dynamic}.}\n\\cmnt{The higher ambient temperature due to the affinity to running car engines exacerbates the operating temperature of embedded electronic control units (ECUs)~\\cite{park2010dynamic}.}\n\\cmnt{For the safe operation of automotive computing systems, air conditioning (AC) is required but it cannot always guarantee constant ambient temperature in diverse driving scenarios. Excessive use of AC can also increase carbon monoxide (CO) and nitrogen oxide (NOx) emissions by 71\\% and 81\\%, respectively, and reduce fuel economy by 22\\%~\\cite{bevilacqua1999effect}.}\nAnother example is a fire-containment drone~\\cite{toribio2016fire}. Even with a heat protection shield, the drone's computing system starts warning when the ambient temperature reaches 35$^{\\circ}$C and becomes nonoperational at 40$^{\\circ}$C. This also limits the minimum distance from the drone to a fire hazard.\n\nWhile dynamic ambient temperature is an important problem, most thermal management schemes for real-time embedded systems~\\cite{kumar2011cool,kumar2011system, ahmed2011minimizing,Huang2014,DSouza2017ThermalIO,Ahmed2017} assume fixed, room-level ambient temperature and focus only on the temperature increase caused by the computing system itself. CPS are expected to run in various physical environments; hence, the assumption of room temperature made by prior work limits their practical applicability. There is recent work~\\cite{Youngmoon2018} considering dynamic ambient temperature but it assumes a uni-processor single-criticality system.\n\n\nUnder harsh ambient temperature, a system may not be able to utilize 100\\% of CPU time even if the CPU runs at the minimum possible frequency with active\/passive cooling packages.\nThe operating temperature of the CPU may still reach the maximum thermal constraint, resulting in temporary system shutdown or permanent hardware damage.\nIn such a condition, the only option left to ensure timing and thermal guarantees of more critical tasks is to secure cooling time by suspending less critical tasks.\nIn other words, although DVFS~\\cite{fu2010feedback,ma2015improving, 7372657, 7746768, zhang2010thermal} and cooling packages~\\cite{chantem2010temperature, elghool2017review, ghahremannezhad2018thermal, ghahremannezhad2019effect} can help tolerate high ambient temperature, it is inevitable to consider only partial operations of the system.\n\nIn this work, we aim to design a system that offers different levels of assurance against ambient temperature changes. This is different from the well-known Vestal model~\\cite{vestal2007preemptive}, which focuses on varying assurance of execution time, but it shares the spirit of addressing uncertainties in real-time system design. To avoid confusion, we clarify our mixed-criticality model as follows:\n\n\n\\begin{defn}\n{\\bf Thermally mixed-criticality systems} are the systems that assure ambient temperature changes and heat dissipation from lower-criticality task execution do not adversely affect the real-time schedulability of higher-criticality tasks.\n\\end{defn}\n\nIn the thermally mixed-criticality model, ambient temperature plays a key role in determining the maximum amount of workload that can be executed on the CPU. The following questions still remain unanswered by the  state-of-the-art:\n\\begin{itemize}\n\t\\item  Up to what ambient temperature is the system fully or partially operational? Specifically, for a given criticality level of a mixed-criticality system, can we find the corresponding {\\it critical ambient temperature}, under which real-time tasks with that or higher criticality are guaranteed to meet their deadline at the expense of lower-criticality tasks?\n\t\\item  Can we take into account the effect of dynamic ambient temperature along with heat conduction on a modern multi-core processor? \n\t\\item  If the system moves from a hot to a cold region, how long will it take for the system to cool down and safely resume the operation of low-critical\\new{ity} tasks, without violating the processor thermal constraint?\n\\end{itemize}\n\n\nThis chapter presents a multi-core mixed-criticality scheduling framework with ambient temperature awareness. In our proposed framework, thermal-aware servers are used to bound heat generation at each criticality level and the criticality mode change is triggered by ambient temperature changes.\nThis is the first work to address the aforementioned limitations and provides analytical guarantees on the timely execution of critical components in dynamic thermal conditions. \n\n\\smallskip\\noindent\\textbf{Contributions.} The contributions of this chapter are as follows:\n\\begin{itemize}\n\\item \\new{We show that the problem of thermal-aware real-time scheduling can be decomposed into thermal schedulability (how much CPU budget is usable under thermal constraints) and timing schedulability (if tasks are schedulable using given budget). Our thermal schedulability achieves the simplicity in timing analysis by ensuring that the budget is guaranteed to be made available for any execution patterns without violating thermal constraints.}\n\n\\item We extensively analyze the thermal safety of a multi-core system and bound the maximum operating temperature that the system can reach. At a specific ambient temperature level, we characterize the worst-case thermal behavior of a system \\cmnt{by  using the notion of idle servers} and also determine the minimum time for the system to transition from one criticality level to a lower level.\n\\item \\new{We introduce the notion of \\textit{idle thermal servers} that allow bounding the maximum operating temperature caused by multiple preemptive active servers scheduled dynamically on a multi-core processor for a given  mixed-criticality taskset.} \n\\item We perform a case study on mixed-criticality applications running on an ODROID-XU4 embedded platform, and evaluate our framework and analysis in different ambient temperature levels.\n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:chap3_rel}\n\nThere exist extensive studies on bounding the maximum operating temperature in non-real-time multi-core systems \\cite{fu2012feedback,fu2010feedback,ma2015improving, 7372657, 7746768, zhang2010thermal}, most of which propose adjusting CPU clock speed. The authors of~\\cite{fu2012feedback,fu2010feedback} proposed a feedback loop that dynamically controls processor temperature by either adapting CPU utilization or scaling frequency to satisfy thermal constraints in varying ambient temperature environment. In~\\cite{7372657} and~\\cite{7746768}, the authors proposed proactive frequency scheduling to improve overall system performance under different ambient temperatures. Although average-case performance degradation has been discussed in these papers, they cannot be directly applied to our problem. \n\nThe notion of \\textit{hot} and \\textit{cold} tasks has been introduced  ~\\cite{chantem2010temperature, Huang2014, Youngmoon2018, kumar2011system, jayaseelan2008temperature,   zhang2010thermal} in both real-time and non-real-time systems. They propose scheduling algorithms to interleave hot and cold tasks, adjust the CPU frequency, and force idling time for the CPU to cool down after running hot tasks. In most of them, the scheduling problem is either to find task execution order or to reduce the size of lengthy hot tasks~\\cite{Huang2014} in a fixed schedule. However, DVFS may cause a considerable reduction in system reliability over time~\\cite{iranfar2017thespot, lasance2003thermally,  xiang2010system} and may be unsupported in some embedded devices.\n\nThere exists extensive research on real-time uni-processor systems with stringent thermal constraints~\\cite{chen2009proactive, wang2006delay, chen2007minimization, kumar2011cool,ahmed2014temperature, ahmed2013thermal, ahmed2011minimizing, Huang2014, Youngmoon2018, jayaseelan2008temperature}. In uni-processor systems, thermal dependency between workloads is only  \\textit{temporal}. However, in multi-core systems, because of heat dissipation between cores, there also exists  \\textit{spatial} thermal dependency between the execution of workload on one core and those on other cores. Due to this reason, the work on uni-processor systems cannot be used safely in multi-core real-time systems.\n\n\nThe notion of thermal servers (either by injecting of idle tasks or using thermal-aware servers explicitly) have been proposed in the literature of real-time systems for  uni-processors~\\cite{kumar2011cool,kumar2011system, ahmed2011minimizing,Huang2014} and multi-core platforms~\\cite{DSouza2017ThermalIO,Ahmed2017}. In particular, the authors of~\\cite{DSouza2017ThermalIO} introduced  a novel technique for periodic tasks executing on multi-core platforms. This technique introduces an Energy Saving (ES) task that runs with the highest priority and captures the sleeping time of CPU cores. The technique can be seen as an alternative to a thermal-aware server because the ES task effectively models the budget-depleted duration of a thermal server. The authors of~\\cite{Ahmed2017} proposed thermal-isolation servers that avoid the thermal interference among tasks in temporal and spatial domains with thermal composability. Despite their achievements in isolating the thermal-aware design from real-time schedulability analysis, these techniques assume a fixed schedule for idle task or periodic servers, and are inapplicable to dynamically-scheduled (e.g., priority-based) servers in multi-core platforms. Since priority-based preemptive servers are widely used in many real-time systems, such as real-time virtualization~\\cite{kim2017predictable,Xi_EMSOFT11}, the restriction imposed by these prior thermal servers is a significant limiting factor.\n\n\nMatrix exponential is a well-known approach to solve the first-order linear system of differential equation. In~\\cite{pagani2015matex}, the authors proposed a technique based on the numerical  Newton\u2013Raphson method to solve the thermal equations in the steady state for each power change. \nBased on this, the work in~\\cite{rai2011worst, yang2013real, kumar2011thermally, chantem2009online} finds the worst-case peak temperature by exhaustively searching all possible patterns.\nUnlike prior work, \nthe unique contribution of our work is that \nwe solve the temperature equation for oscillating power signals analytically, by representing them as continuous functions, and analyze the worst-case peak temperature directly for multi-core platforms.\nIt is worth noting that our work not only proves the worst case for peak temperature but also reduces  computational complexity considerably.\n\n\n\n\n\nThere exists some research focusing on varying ambient temperature in real-time systems~\\cite{Youngmoon2018, Huang2014, wang2006delay}. However, there is no discussion in these studies about mixed-criticality systems and the effect of harsh ambient temperature change on the schedulability of critical tasks. \n\nTo the best of our knowledge, there is no prior work on multi-core mixed-criticality systems with the consideration of the effect of ambient temperature change.\n\n\\section{System Model}\n\\label{sec:chap3_model}\n\\subsection{Power Model}\\label{sec:chap3_power_model} The total power consumption of CMOS circuits is modeled as the summation of dynamic and static powers~\\cite{4484694}, i.e., ${P(t) = P_S(t) + P_D(t)}$. Static power $P_S$ depends on the semiconductor technology and the operating temperature caused by current leakage. \nHence, it can be modeled as: $P_S(t) = k_1 \\theta(t) + k_2 $, where $k_1$ and $k_2$ are technology-dependent system constants, and $\\theta(t)$ is the operating temperature \\cite{4212027}. Dynamic power $P_D(t)$ is the amount of power consumption due to the processor operating frequency $f$, modeled as ${P_D = k_0 f^s}$, where $s$ and $k_0$ are system constants that depend on the semiconductor technology.\n\n\\subsection{Thermal-Aware Mixed-Criticality Servers}\nWe consider multiple preemptive thermal-aware servers for each CPU core. Each server is statically associated with one core and does not migrate to another core at run-time. A server $v_i$ is modeled as $v_i = (C_i,T_i,L_i)$, where $C_i$ is the maximum execution budget of the server $v_i$, $T_i$ is its budget replenishment period, and $L_i$ is its criticality level. Servers are ordered in an increasing order of priorities, i.e. $i<j$ implies that a server $v_i$ has lower priority than a $v_j$. At the criticality level of $l$, only the servers with criticality level higher than or equal to $l$ (i.e., $L_i \\geq l$) are eligible to execute.\n\nFor budget replenishment policies, we consider \\textit{polling}~\\cite{sha1986solutions}, \\textit{deferrable}~\\cite{strosnider1995deferrable}, and \\textit{sporadic servers}~\\cite{sprunt1989aperiodic}. \nLet $J_i$ denote the task release jitter relative to the server release. The value of  $J_i$ is $T_i$ under the polling server policy and $T_i - C_i$ under the deferrable and sporadic server policies~\\cite{bernat1999new}.\n\n\\subsection{Task Model}\nThis work considers sporadic tasks with implicit deadlines under \\textit{partitioned fixed-priority preemptive scheduling}, which is widely used in many practical systems. Each task $\\tau_i$ is statically allocated to one thermal server (thus to the corresponding CPU core of that server) with a unique priority. In each server, tasks are labeled in an increasing order of priority, i.e., $i<j$ implies $\\tau_i$ has lower priority than $\\tau_j$. There also exist non-real-time tasks running with the lowest priority level in the server and they execute only if there is no real-time task ready to execute and the server has remaining budget.\nA task $\\tau_i$ is modeled as $\\tau_i=(E_i, D_i)$ where $E_{i}$ is the worst-case execution time (WCET) of task $\\tau_i$, $D_i$ denotes the minimum inter-arrival time of $\\tau_i$ which is implicit deadline of $\\tau_i$.\n\\cmnt{It is noteworthy that the criticality level of each real-time task $\\tau_i$ is at least lower than that of its allocated server $v_j$ (i.e., $CR_i \\leq L_j$)\\cmnt{ whereas non-real-time tasks can always execute under any criticality level if $v_j$ has enough budget and no real-time task runs on the server.} Therefore, at the criticality level of $l$, in a server $v_j$ whose criticality level is at least $l$, real-time tasks whose criticality level is at least $l$ execute.} \n\\new{It is worth mentioning that the simplicity in timing schedulability achieved by our work enables easy adoption of more complex task models. For instance, the analysis for tasks with critical sections under hierarchical scheduling~\\cite{7010477,hosseinimotlagh2019thermal} is directly applicable to our work since we ensure periodic resource budget.}\n\n\\subsection {Criticality Model}\nIn this work, there exists a set of $m$ criticality levels ${L=\\{l_1,l_2,\\dots, l_m\\}}$. At criticality mode $l$, only the servers (and tasks within these servers) with criticality level higher than or equal to $l$ are eligible to execute. \nThus, for each criticality level $l$, there exists a subset of servers $V^l \\subseteq V$ and a subset of taskset $\\Gamma^l \\subseteq \\Gamma$ that execute.\n\n\\begin{defn}\n{\\bf Critical ambient temperature} of a  criticality level $l$ is the maximum ambient temperature that the  system  can  execute  eligible  servers $v \\in V^l$ without  violating  the system's  maximum  temperature  constraint.\n\\end{defn}\n\\noindent Details on how criticality mode changes is given in Sec.~\\ref{sec:chap3_frame}.\n\n\n\n\n\\section{Framework Design}\n\\label{sec:chap3_frame}\nThis section presents the overall design-time and run-time aspects of our framework and how the criticality mode changes. \nWith our framework, all tasks in a system run within thermal-aware servers. A criticality mode change is triggered by ambient temperature, which can be obtained from an off-chip temperature sensor.\nIf the criticality mode changes from a lower criticality level to a higher one, i.e., the critical ambient temperature has been reached, the framework terminates the lower-criticality servers and their tasks immediately. \nThis design guarantees that the lower-criticality tasks have no effect on the  \\textit{thermal} and \\textit{timing} schedulability of tasks running in servers with higher criticality levels. The timing schedulability of tasks refers to the ability to complete their execution by the deadline. Thermal schedulability is defined as follows. \n\\begin{defn}\n{\\bf Thermal schedulability} is to guarantee that under any task execution patterns, the CPU does not exceed the maximum temperature constraint.\n\\end{defn}\n\nWhen the ambient temperature changes from a higher criticality level to a lower one, lower-criticality servers and their tasks do not resume immediately. The reasoning is that it takes time for the CPU to cool down to reach the safe temperature level that the increase in workload (due to resuming the lower-criticality tasks) will not lead to a temperature violation. Therefore, in the transition to a lower criticality level, only higher-criticality servers (and also their tasks) still run, and after reaching the safe temperature, then lower-criticality servers (and their tasks) resume. Let \\textit{shifting time} denote this delay. We will later determine this delay as a function of physical characteristics and system settings.\n\n\\cmnt{Since actual power dissipation is affected by operating temperature as discussed in Sec.~\\ref{sec:chap3_power_model}, we consider the peak power dissipation at each criticality level.\\footnote{Note that each criticality level has a different critical ambient temperature.}}\n\n\n\n\n\\begin{figure}[th]\n\\centering\n\\includegraphics[width=0.72\\textwidth]{Chapter3\/figs\/frmwrk\/trans.pdf}\n\\caption{Criticality mode change diagram}\n\\label{fig:chap3_crit_mode}\n\\end{figure}\n\nThe state diagram of criticality mode changes is illustrated in Fig.~\\ref{fig:chap3_crit_mode}. The criticality mode of the system is determined by the associated servers of the current state (e.g., $V^l$ for level $l$), and the change of the state is triggered by the monitored ambient temperature. If the ambient temperature goes higher than the critical ambient temperature of the highest criticality mode $l_m$, the system will shutdown. An increase in the ambient temperature leads the system state to transition to a higher criticality mode immediately. However, a transition to a lower criticality mode involves a shifting state. Let $S_i$ denote the shifting state from one criticality mode $l_{i+1}$ to its immediate lower mode $l_i$, and $M_i$ represents the state of the criticality mode $l_i$. After staying in $S_i$ for $tm_{Si}$ time units and the ambient temperature is still under the safe level, the system state will change to $M_i$. \n\n\nIn summary, the design-time analysis and the run-time support of our framework are as follows:\n\\begin{enumerate}\n    \\item[\\textbf{Design-time:}]\n    \\item Check the timing schedulability of tasks for each criticality.\n    \\item Find the parameters of thermal-aware servers that ensure thermal schedulability for each criticality.\n    \\item Compute the corresponding critical ambient temperature for each criticality.\n    \\item Compute the shifting time from each criticality level to its immediate lower one.\n\\end{enumerate}\n\\begin{enumerate}\n    \\item[\\textbf{Run-time:}]\n    \\item Monitor ambient temperature.\n    \\item If ambient temperature exceeds the critical ambient temperature of the current criticality level $l_i$, a) switch to a higher criticality $l_{i+1}$, and b) terminate servers with criticality less than $l_{i+1}$ immediately.\n    \\item If ambient temperature decreases below the critical ambient temperature of one lower criticality level $l_{i-1}$, a) switch to the lower criticality $l_{i-1}$, b) wait for \\textit{shifting time}, and c) resume servers with criticality $l_{i-1}$.\n\\end{enumerate}\n\n\n\n\n\n\n\\section{Thermal Analysis}\n\\label{sec:chap3_thermal_analysis}\n\nIn this section, we first develop a generalized thermal model to represent the CPU temperature as a function of a generic input power signal. We show the relation of workload and ambient temperature level under the maximum thermal constraint in a steady state. The shifting time to a lower-criticality level will be discussed. Finally, we prove the worst-case scenario of task arrivals in invariant ambient temperature at a specific criticality level in multi-core platforms. \n\n\n\\subsection{General Thermal Model for Periodic Power Signal}\nThe thermal behavior of the CPU is modeled when a generic periodic power signal generates heat dissipation, which results in temperature oscillations. The temperature function of the CPU is analytically extracted based on the ambient temperature, workload, physical and geometrical properties, and the thermal resistances between the CPU and surroundings.\n\nA generic power signal is a periodic step function:\n\\[P(t) = \\left\\{ {\\begin{array}{*{20}{c}}\n{{P_S}}&{Sleeping\\,time}\\\\\n{{P_S} + {P_D}}&{O.W.}\n\\end{array}} \\right.\\]\n\nFig.~\\ref{fig:chap3_power} illustrates this power signal function where $t_{wk}$ and $t_{slp}$ denote the execution time and \\new{the} sleeping time of \\new{a} periodic workload, $u$ is the CPU utilization (i.e., ${u = \\frac{t_{wk}}{T}}$), and $\\phi$ is the release offset.\n\\begin{figure}[h]\n\\centering\n\\vspace{-5pt}\n\\includegraphics[width=0.80\\textwidth]{Chapter3\/figs\/thermal\/Periodic_Power_Signal.pdf}\n\\caption{A generic periodic power signal.}\n\\vspace{-5pt}\n\\label{fig:chap3_power}\n\\end{figure}\n\nIn order to derive a continuous temperature function, here we represent the periodic step function of the power signal as the following Fourier series: \n\\begin{equation}\\label{eq:chap3_1001}\n    P(t) = {P_S} + {P_D}u + {P_O}\n\\end{equation}\nwhere\n\\[{P_O} = \\sum\\limits_{k = 1}^\\infty  {\\frac{{2{P_D}\\sin \\left( {uk\\pi } \\right)}}{{k\\pi }}\\cos \\left( {\\frac{{2k\\pi }}{T}\\left( {t - \\frac{{uT}}{2} - \\phi } \\right)} \\right)} \\]\n\n\nNote that using this continuous representation of the power signal is advantageous in deriving\na straightforward formula for a temperature equation, compared to iteratively applying the recursion relation with new initial conditions at every period. First, we define ${\\theta(t)  = \\Theta_{CPU}(t) - {\\Theta_{amb}(t)}}$ as the temperature difference of the CPU core and the ambient. Then, the rate of temperature change can be captured by the following differential equation~\\cite{Ahmed2017,DSouza2017ThermalIO}:\n\\begin{equation}\\label{eq:chap3_1002}\n    \\frac{{d\\theta(t) }}{{dt}} = A\\theta(t)  + BP(t)\n\\end{equation}\nwhere $A$ and $B$ are scalar values determined based on the system inner thermal resistance and capacitance. Substituting $P(t)$ from Eq.~\\ref{eq:chap3_1001}, we solve Eq.~\\ref{eq:chap3_1002} for the CPU temperature. Assuming the temperature difference at the initial time ${t_0}$ is ${\\theta _0}$, the temperature of the CPU can be written as:\n\\begin{equation}\\label{eq:chap3_1003}\n\\begin{split}\n    \\theta (t) = {\\theta _0}{e^{A(t - {t_0})}} - \\frac{B}{A}\\left( {{P_S} + {P_D}u} \\right)\\left( {1 - {e^{A(t - {t_0})}}} \\right) + \\\\\n    B\\left( {S(A,{P_D},u,T,\\phi ,t) - S(A,{P_D},u,T,\\phi ,{t_0}){e^{A(t - {t_0})}}} \\right)\n\\end{split}\n\\end{equation}\nwhere $S$ is a function defined as:\n\\begin{multline*}\n    S(\\beta ,P,u,T,\\phi ,t) =  - \\sum\\limits_{k = 1}^\\infty  \\frac{{2PT\\sin \\left( {uk\\pi } \\right)}}{{k\\pi \\left( {{T^2}{\\beta ^2} + 4{k^2}{\\pi ^2}} \\right)}} \\times \\\\\n    \\left( { - T\\beta \\cos \\left( {\\frac{{2k\\pi }}{T}\\left( {t - \\psi } \\right)} \\right) + 2n\\pi \\sin \\left( {\\frac{{2k\\pi }}{T}\\left( {t - \\psi } \\right)} \\right)} \\right)\n\\end{multline*}\nwith $\\psi  = \\frac{{uT}}{2} + \\phi$.\n\nThe parameters $A$ and $B$ in Eq.~\\ref{eq:chap3_1003} are the system characteristics which depend on the thermal resistances, heat capacity, CPU mass, and thermal convection of the ambient. It is worth mentioning that the thermal response of a system with constant power dissipation can be derived as a special case of Eq.~\\ref{eq:chap3_1003} by considering ${u = 1}$:\n\\begin{equation}\\label{eq:chap3_1004}\n\\begin{split}\n    \\theta (t) = {\\theta _0}{e^{A(t - {t_0})}} - \\frac{B}{A}\\left( {{P_S} + {P_D}} \\right)\\left( {1 - {e^{A(t - {t_0})}}} \\right)\n\\end{split}\n\\end{equation}\n\nIf ${t_0 = 0}$, the well-known expression for the constant power dissipation case can be obtained from Eq.~\\ref{eq:chap3_1004}:\n\\begin{equation}\\label{eq:chap3_1005}\n    \\theta (t) = \\alpha  + \\left( {{\\theta _0} - \\alpha } \\right){e^{\\beta t}}\n\\end{equation}\nwhere $\\beta  = A$, and $\\alpha  =  - \\frac{B}{A}\\left( {{P_S} + {P_D}} \\right)$.\n\nFor any ${u}$, the thermal response of the system with a constant power signal reaches the steady state. \nFrom Eq.~\\ref{eq:chap3_1004}:\n\\begin{equation}\\label{eq:chap3_1006}\n    {\\theta _s}(t) = \\alpha  =  - \\frac{B}{A}\\left( {{P_S} + {P_D}} \\right).\n\\end{equation}\n\nFor the case where the CPU utilization is not $100\\%$, the temperature still oscillates in the steady state, but the oscillation stays within a certain range given by the minimum and the maximum steady-state temperatures (${\\theta _L}$ and ${\\theta _M}$). We can derive the steady state thermal response from Eq.~\\ref{eq:chap3_1003}:\n\\begin{equation}\\label{eq:chap3_1007}\n    {\\theta _s}(t) =  - \\frac{B}{A}\\left( {{P_S} + {P_D}u} \\right) + BS(A,{P_D},u,T,\\phi ,t)\n\\end{equation}\nwhere $S(A,{P_D},u,T,\\phi,t)$ represents the oscillation. Based on this general expressions, we will give details on the thermal behavior of multiple CPU cores under transient and steady conditions in Sec.~\\ref{sec:chap3_thermal_model_multicore}. \n\n\n\\subsubsection{Workload, ambient, and maximum temperature relations}\n\n\nIn the steady state condition, the relation between workload, ambient temperature ${\\Theta_{amb}}$, and the maximum operating temperature ${\\Theta_{M}}$ can be determined based on the model developed above. The temperature difference  ${\\theta _M}$ can be written as:\n\\begin{equation}\\label{eq:chap3_1008}\n    {\\theta _M} = {\\Theta _M} - {\\Theta _{amb}} =  - \\frac{B}{A}\\left( {{P_S} + {P_D}\\frac{{1 - {e^{AuT}}}}{{1 - {e^{AT}}}}} \\right)\n\\end{equation}\n\nTherefore, if ${\\Theta_{M}}$ is used as a thermal constraint, the maximum ambient temperature for a given utilization $u$ is expressed as follows:\n\\begin{equation}\\label{eq:chap3_1009}\n    {\\Theta _{amb}} = {\\Theta _M} + \\frac{B}{A}\\left( {{P_S} + {P_D}\\frac{{1 - {e^{AuT}}}}{{1 - {e^{AT}}}}} \\right)\n\\end{equation}\n\nAlso, another useful expression can be derived for the workload based on the constraint maximum temperature, ambient temperature, and the power signal:\n\\begin{equation}\\label{eq:chap3_1010}\nu = \\frac{1}{{AT}}\\ln \\left( {1 + \\frac{A}{B}\\frac{{{\\Theta _M} - {\\Theta _{amb}} + \\left( {{B \\mathord{\\left\/{\\vphantom {B A}} \\right. \\kern-\\nulldelimiterspace} A}} \\right){P_S}}}{{{P_D}}}\\left( {1 - {e^{AT}}} \\right)} \\right)\n\\end{equation}\n\n\\subsubsection{Time shifting and transient analysis}\n\nWe derive average steady-state temperature by removing the oscillations from the above equations.  \nThis is especially useful for transient phase analysis. From Eq.~\\ref{eq:chap3_1003}, the following can be derived:\n\\begin{equation}\\label{eq:chap3_1011}\n\\begin{split}\n    {\\theta _{ave}}(t) = {\\theta _0}{e^{A(t - {t_0})}} - \\frac{B}{A}\\left( {{P_S} + {P_D}u} \\right)\\left( {1 - {e^{A(t - {t_0})}}} \\right)\n    - BS(A,{P_D},u,T,\\phi ,{t_0}){e^{A(t - {t_0})}}\n\\end{split}\n\\end{equation}\n${\\theta _{ave}}(t)$ can be approximated by taking the first few terms of the series for $S(A,{P_D},u,T,\\phi ,{t_0})$. It can be seen from Eq.~\\ref{eq:chap3_1011} that the plateau of ${\\theta _{ave}}$ in the steady state is ${ - \\frac{B}{A}({P_S} + {P_D}u)}$.\n\n\n\nBased on the expression of the  ${\\theta _{ave}}(t)$, shifting time can be calculated which is defined as the time it takes for the system to reach a new steady\\new{-}state condition when system parameters are changed. We calculate the shifting time that the CPUs take to reach to  99\\% of the new steady\\new{-}state condition from an initial temperature difference of ${\\theta _0}$. Assuming that $S_k$ is the value of $S(t_0)$ taking the first $k$ terms, we have:\n\\begin{equation}\\label{eq:chap3_1012}\n    {t_{shift}} = \\frac{1}{A}\\ln \\left( {\\left| {\\frac{{0.01\\frac{B}{A}({P_S} + {P_D}u)}}{{{\\theta _0} + \\frac{B}{A}({P_S} + {P_D}u) + B{S_k}}}} \\right|} \\right)\n\\end{equation}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.64\\textwidth]{Chapter3\/figs\/thermal\/Theta.pdf}\n\\caption{Current, steady state, and average temperature profiles.}\n\\label{fig:chap3_theta}\n\\end{figure}\nFig.~\\ref{fig:chap3_theta} illustrates temperature profile ${\\theta}(t)$, oscillating steady state temperature ${\\theta}_{s}(t)$, and average temperature ${\\theta}_{ave}(t)$.\n\\subsection{General Thermal Model for Multi-Core Platforms}\\label{sec:chap3_thermal_model_multicore}\nSimilar to the single-core case, a general thermal model can be developed for a multi-core platform with a periodic input power signal. In this subsection, we represent the power signal as a continuous function and show its benefit in simplifying the final solution. For a multi-core platform with $n$ cores, we can have $n$ eigenvalues that define the thermal response of the system. Therefore, we use matrix notations to solve the differential equations. After performing a thermal analysis, the rate of CPUs' temperature change can be denoted as:\n\\begin{equation}\\label{eq:chap3_1013}\n    \\frac{{d\\boldsymbol{\\theta}}(t)}{{dt}} = {\\bf{A}}\\boldsymbol{\\theta}(t)  + {\\bf{BP}}(t)\n\\end{equation} \nwhere {\\bf{A}}, an $n \\times n$ matrix, and {\\bf{B}}, a diagonal $n \\times n$ matrix, are determined by the inner thermal resistance and capacitance of the system. ${\\boldsymbol{\\theta}}$ is an $n \\times 1$ matrix of the CPU cores' temperature difference, and {\\bf{P}}$(t)$ is the $n \\times 1$ matrix of CPU cores' power signal functions. If ${\\boldsymbol{\\theta_{0}}}$ is the initial temperature difference matrix at $t_{0}$, the solution of Eq. \\ref{eq:chap3_1013} can be written as:\n\\begin{equation}\\label{eq:chap3_1014}\n    \\boldsymbol{\\theta} (t) = {e^{(t - {t_0}){\\bf{A}}}}{\\boldsymbol{\\theta} _0} + \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{BP}}(s)ds}\n\\end{equation}\nThe first part is the homogeneous solution which is the thermal response due to the initial temperature difference from the ambient. The second part is the non-homogeneous solution caused by the input power signal. The temperature rise due to a power signal is independent of the initial condition. ${e^{(t - {t_0}){\\bf{A}}}}$ is the matrix exponential of ${\\bf{A}}$. For a platform with $n$ CPU cores, we denote the eigenvalues as ${\\beta_{1}}$, ${\\beta_{2}}$, ..., and ${\\beta_{n}}$ \\ali{with ${\\beta_1} \\ge {\\beta_2} \\ge ... \\ge {\\beta_n}$}. To solve Eq.~\\ref{eq:chap3_1014}, we first show that the solution of power signals can be obtained as the sum of the solutions of each signal.\n\n\\begin{thm}\\label{thm:superposition}\n(Superposition) Thermal response due to any combinations of power signals is equivalent to the sum of thermal responses caused by each of those power signals. \n\\end{thm}\n\\begin{proof}\n\nAssume that ${\\boldsymbol{\\theta}}_1$ and ${\\boldsymbol{\\theta}_2}$ are the thermal responses caused by executions ${\\bf{P}}_1(t)$ and ${\\bf{P}}_2(t)$. Also, ${\\boldsymbol{\\theta}_3}$ is the thermal response caused by  ${\\bf{P}}_3(t) = {{\\bf{P}}_1}(t) + {{\\bf{P}}_2}(t)$. From Eq.~13:\n\\[\\frac{{d{\\boldsymbol{\\theta} _1}}(t)}{{dt}} = {\\bf{A}}{\\boldsymbol{\\theta}_1(t)} + {\\bf{B}}{{\\bf{P}}_1}(t)\\text{, and  }\\frac{{d{\\boldsymbol{\\theta}_2(t)}}}{{dt}} = {\\bf{A}}{\\boldsymbol{\\theta}_2(t)} + {\\bf{B}}{{\\bf{P}}_2}(t).\\]\n\nBy adding these two differential equations, we have:\n\\[\\frac{d}{{dt}}\\left( {{\\boldsymbol{\\theta}_1(t)} + {\\boldsymbol{\\theta}_2(t)}} \\right) = {\\bf{A}}\\left( {{\\boldsymbol{\\theta}_1(t)} + {\\boldsymbol{\\theta}_2(t)}} \\right) + {\\bf{B}}\\left( {{{\\bf{P}}_1}(t) + {{\\bf{P}}_2}(t)} \\right)\\]\n\nThis is equivalent to the differential equation of ${\\boldsymbol{\\theta}_3}$:\n\\[\\frac{{d{\\boldsymbol{\\theta}_3(t)}}}{{dt}} = {\\bf{A}}{\\boldsymbol{\\theta}_3(t)} + {\\bf{B}}{{\\bf{P}}_3}(t) = {\\bf{A}}{\\boldsymbol{\\theta}_3(t)} + {\\bf{B}}\\left( {{{\\bf{P}}_1}(t) + {{\\bf{P}}_2}(t)} \\right)\\]\n\nTherefore, $\\boldsymbol{\\theta}_3(t) = \\boldsymbol{\\theta}_1 (t) + \\boldsymbol{\\theta}_2 (t)$.\n\\end{proof}\nLet {\\bf{V}} denote an $n \\times n$ matrix of eigenvectors of {\\bf{A}} where column $i$ is the $i^{th}$ eigenvector. Also, {\\bf{D}} is a diagonal $n \\times n$ matrix in which the diagonal entries are the eigenvalues of {\\bf{A}}. The solutions of the non-homogeneous part in Eq.~\\ref{eq:chap3_1014} can be written as:\n\\begin{equation}\\label{eq:chap3_1015}\n\\begin{split}\n    \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{BP}}(s)ds}  = \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{B}}\\left( {{{\\bf{P}}^\\infty } + {{\\bf{P}}_o}(s)} \\right)ds} \\\\\n    = \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{B}}{{\\bf{P}}^\\infty }ds}  + \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{B}}{{\\bf{P}}_o}(s)ds}\n\\end{split}\n\\end{equation}\nwhere ${{\\bf{P}}^\\infty}$ and ${{{\\bf{P}}_o}(s)}$ are the constant and oscillating part of the power signal matrix, respectively. \n\\[{{\\bf{P}}^\\infty } = [{P_j}^\\infty]_{n\\times 1}  = [{P_S}_{_j} + {P_D}_{_j}{u_j}]_{n\\times 1}\\]\n\\[{{\\bf{P}}_o} = [{P_o}_{_j}] = \\left[\\sum\\limits_{k = 1}^\\infty  {\\frac{{2{P_D}_{_j}}}{{k\\pi }}\\sin \\left( {{u_j}k\\pi } \\right)\\cos \\left( {\\frac{{2k\\pi }}{{{T_j}}}\\left( {s - {\\psi _j}} \\right)} \\right)}\\right]_{n \\times 1} \\]\nwith ${\\psi _j} = \\frac{{{u_j}{T_j}}}{2} + {\\phi _j}$. Since ${{\\bf{P}}^\\infty}$ is constant,\n\\begin{equation}\\label{eq:chap3_1016}\n    \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{B}}{{\\bf{P}}^\\infty }ds}  =  - {{\\bf{A}}^{ - 1}}\\left( {{\\bf{I}} - {e^{(t - {t_0}){\\bf{A}}}}} \\right){\\bf{B}}{{\\bf{P}}^\\infty }\n\\end{equation}\nFor the second integral we have:\n\\begin{equation}\\label{eq:chap3_1017}\n\\begin{aligned}\n   &\\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{B}}{{\\bf{P}}_o}(s)ds}  = {\\bf{V}}{e^{t{\\bf{D}}}}\\int_{{t_0}}^t {{e^{ - s{\\bf{D}}}}{{\\bf{V}}^{ - 1}}{\\bf{B}}{{\\bf{P}}_o}(s)ds} \\\\ &\n   = {\\bf{V}}{e^{t{\\bf{D}}}}.{\\left[ {\\sum\\limits_{i = 1}^n {\\int_{{t_0}}^t {{e^{ - {\\beta _j}s}}V_{ji}^{ - 1}{B_{ii}}{P_{{o_i}}}(s)} } } \\right]_{n \\times 1}}\n\\end{aligned}\n\\end{equation}\n\n\nSimilar to the single core case, we can use $S$ in Eq.~\\ref{eq:chap3_1003} but in a matrix form. We define the matrix ${\\bf{S}}$ as:\n\\[{\\bf{S}}(t) = {\\left[ {{S_{ij}}(t)} \\right]_{n \\times n}} = {\\left[ {S({\\beta _i},{P_{{D_j}}},{u_j},{T_j},{\\phi _j},t)} \\right]_{n \\times n}}\\]\n\nWe have $\\int {{e^{ - {\\beta _j}s}}{P_{{o_i}}}(s)ds}  = {e^{ - {\\beta _j}s}}{S_{ji}}(s)$. Therefore:\n\\begin{multline*}\n\\begin{aligned}\n   &{\\left[ {\\sum\\limits_{i = 1}^n {\\int_{{t_0}}^t {{e^{ - {\\beta _j}s}}V_{ji}^{ - 1}{B_{ii}}{P_{{o_i}}}(s)ds} } } \\right]_{n \\times 1}} \n   &= {\\left[ {\\sum\\limits_{i = 1}^n {{e^{ - {\\beta _j}t}}V_{ji}^{ - 1}{B_{ii}}{S_{ji}}(t) - {e^{ - {\\beta _j}{t_0}}}V_{ji}^{ - 1}{B_{ii}}{S_{ji}}({t_0})} } \\right]_{n \\times 1}}\n\\end{aligned}\n\\end{multline*}\n\nTo write this in a matrix notation, we define ${\\bf{B}}' = diag({\\bf{B}})$ an $n \\times 1$ matrix containing the diagonal entries of ${\\bf{B}}$ such that $B{'_j} = {B_{jj}}$. Then,\n\\begin{multline*}\n\\begin{aligned}\n   &{\\left[ {\\sum\\limits_{i = 1}^n {{e^{ - {\\beta _j}t}}V_{ji}^{ - 1}{B_{ii}}{S_{ji}}(t) - {e^{ - {\\beta _j}{t_0}}}V_{ji}^{ - 1}{B_{ii}}{S_{ji}}({t_0})} } \\right]_{n \\times 1}} \\\\\n   &= {e^{ - t{\\bf{D}}}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}(t)} \\right){\\bf{B}}' - {e^{ - {t_0}{\\bf{D}}}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}({t_0})} \\right){\\bf{B}}'\n\\end{aligned}\n\\end{multline*}\nwhere ${{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}(t)}$ is the Hadamard product of matrices ${{{\\bf{V}}^{ - 1}}}$ and ${{\\bf{S}}(t)}$, which is a matrix of the same dimension of $n \\times n$ where each element $ij$ is the product of counterpart elements $ij$ of  ${{{\\bf{V}}^{ - 1}}}$ and ${{\\bf{S}}(t)}$: ${\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}(t)} \\right)_{ij}} = V_{ij}^{ - 1}{S_{ij}}(t)$. Substituting this in Eq.~\\ref{eq:chap3_1017} gives:\n\\begin{equation}\\label{eq:chap3_1018}\n\\begin{aligned}\n   &{\\bf{V}}{e^{tD}}.{\\left[ {\\sum\\limits_{i = 1}^n {\\int_{{t_0}}^t {{e^{ - {\\beta _j}s}}V_{ji}^{ - 1}{B_{ii}}{P_{{o_i}}}(s)} } } \\right]_{n \\times 1}}\\\\\n   &= {\\bf{V}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}(t) - {e^{(t - {t_0}){\\bf{D}}}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}({t_0})} \\right)} \\right){\\bf{B}}'\n\\end{aligned}\n\\end{equation}\nSubstituting Eq.~\\ref{eq:chap3_1016} and Eq.~\\ref{eq:chap3_1018} in Eq.~\\ref{eq:chap3_1015} results in a general solution for the thermal response of a $n$-core CPU platform with distinct input power signals to each core:\n\\begin{equation}\\label{eq:chap3_1019}\n\\begin{aligned}\n   \\boldsymbol{\\theta} (t) = &{e^{(t - {t_0}){\\bf{A}}}}{\\boldsymbol{\\theta} _0} - {{\\bf{A}}^{ - 1}}\\left( {{\\bf{I}} - {e^{(t - {t_0}){\\bf{A}}}}} \\right){\\bf{B}}{{\\bf{P}}^\\infty } + \\\\\n   &{\\bf{V}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}(t) - {e^{(t - {t_0}){\\bf{D}}}}\\left( {{{\\bf{V}}^{ - 1}} \\circ {\\bf{S}}({t_0})} \\right)} \\right){\\bf{B}}'\n\\end{aligned}\n\\end{equation}\n\nThis expression of ${\\boldsymbol{\\theta}(t)}$, which contains simple matrix operations, can be easily used to compute the general solution for the  transient temperature profile of multi-core CPUs. \n\n\n\n\\subsection{Worst-Case Execution Scenarios}\n\nIt is important to know the workload execution patterns which cause the peak heat dissipation. In this section, based on the model developed in the previous sections, we explore and discuss the worst-case scenarios for workload execution.\n\n\\subsubsection{Consecutive workload execution}\n\nFirst, we prove that the time to reach the maximum temperature is minimized if all workloads execute consecutively. Suppose that for any arbitrary execution of workloads in a period of  $T$, the CPU executes some workload for $t_1$, sleeps for $t_2$, and then wakes up and executes for $t_3$ time units, and these working-sleeping switches occur $n$ times. For the CPU idling time, Eq.~\\ref{eq:chap3_1005} changes to ${\\theta (t) = {\\theta _0}\\ {e^{\\beta t}}}$ if the static power  is ignored (i.e., $\\alpha = 0$). Hence, the temperature change during a period is:\n\\[ \\theta_{t_0}   = \\theta_L \\]\n\\[ \\theta_{t_1}   =\\alpha + (\\theta_{t_0} - \\alpha) * e^{\\beta t_1}\\]\n\\[ \\theta_{t_2}   = \\theta_{t_1}  * e^{\\beta t_2}\\]\n\\[\\vdots\\]\n\\[ \\theta_{t_n}   = \\theta_{t_{n-1}}  * e^{\\beta t_n}\\]\nwhere $\\sum t_i = T $ for all $n > 1$ and $t_i > 0$.\n\nIn the steady state, the temperature of the beginning and the end of each period is equal.  Therefore, considering $\\theta_{t_n} = \\theta_{L}$,  \n\\[\n    \\theta_{L} =\\alpha \\frac{e^{\\beta(t_{1}+t_{2}+\\cdots+t_{n-1}+t_{n})}-e^{\\beta(t_{2}+\\cdots+t_{n-1}+t_{n})}+\\cdots\\cmnt{+e^{\\beta(t_{n-1}+t_{n})}}-e^{\\beta t_{n}}}{e^{\\beta T}-1}\n    \\]\nwhere $ t_{wk} = \\sum_{i =1 ,\\, i= i+2 }t_i$ is the execution time and  ${t_{slp} = \\sum_{i =2 ,\\, i= i+2 }t_i }$ is the idle time, respectively. We prove that the amount of total execution time is minimized when there is only one execution time chunk. In other words, the worst-case thermal scenario happens when all workloads execute consecutively.\n\nFor clarification, suppose the following two scenarios for a given period under the maximum temperature constraint:\n\\begin{itemize}\n    \\item All workloads execute consecutively to reach the maximum temperature for $t_{wk}$ time units; then, the CPU sleeps until the beginning of the next period for $t_{slp}$ time units (Fig.~\\ref{fig:chap3_budget}a).\n    \\item A portion of workloads executes for $t_1$ time units, then the CPU sleeps for $t_2$ \\new{time} units, and the rest of the workloads run until the CPU reaches the maximum temperature at \\new{$t_1+t_2+t_3$}; afterwards the CPU sleeps for $t_4$ \\new{time} units (Fig.~\\ref{fig:chap3_budget}b).\n\\end{itemize}\n\n\\begin{figure}[h]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.40\\textwidth]{Chapter3\/figs\/thermal\/oneBudget.pdf}}\n \\hspace{10pt}\n\\subfloat [][]{\\includegraphics[width=0.40\\textwidth]{Chapter3\/figs\/thermal\/twoBudgets.pdf}}\n\\caption{Temperature change in one period when the CPU operates a)  for $t_{wk}$ time units b)  for $t_1$ and \\new{$t_3$} time units.}\n\\label{fig:chap3_budget}\n\\end{figure}\n\nThe following shows that the budget of a thermal server should be bounded based on the first scenario.\n\\begin{thm}\\label{thm:min_waking_time}\nThe amount of waking time to reach the maximum temperature constraint is minimized when all workloads execute consecutively. \n\\end{thm}\n\\begin{proof}\nWe are interested to prove that ${t_w \\leq t_1+ t_3}$. To prove the statement, we find the relation of the minimum steady state temperature in each scenario, individually. \n To calculate the budget for the first scenario, we first calculate the minimum steady\\new{-}state temperature which is\n \\[\n \\theta_L = \\frac{{\\mathrm{e}}^{\\beta\\,t_{slp}}\\,\\left(\\alpha-\\alpha\\,{\\mathrm{e}}^{\\beta\\,t_{wk}}\\right)}{1-{\\mathrm{e}}^{\\beta\\,t_{wk}}\\,{\\mathrm{e}}^{\\beta\\,t_{slp}}}.    \n \\]\nAccordingly, the maximum temperature in the steady state is \n\n \\[ \n \\theta_{M} = \\alpha-{\\mathrm{e}}^{\\beta\\,t_{wk}}\\,\\left(\\alpha+\\frac{{\\mathrm{e}}^{\\beta\\,t_{slp}}\\,\\left(\\alpha-\\alpha\\,{\\mathrm{e}}^{\\beta\\,t_{wk}}\\right)}{{\\mathrm{e}}^{\\beta\\,t_{wk}}\\,{\\mathrm{e}}^{\\beta\\,t_{slp}}-1}\\right) = \\alpha \\frac{{\\mathrm{e}}^{\\beta\\,t_{wk}}-1}{{\\mathrm{e}}^{\\beta\\,T}-1}.\n \\]\n\nFor the second scenario,\n the maximum temperature is reached after ${t_1+t_2+t_3}$ time units. Therefore, \n\n\\[\n\\theta_{M} = \\alpha \\frac{{\\mathrm{e}}^{\\beta(t_1+t_2+t_3)}-{\\mathrm{e}}^{\\beta(t_2+t_3)}+{\\mathrm{e}}^{\\beta(t_3)}-1}{{\\mathrm{e}}^{\\beta T}-1}\n\\]\n\nIt is worth noting that although the minimum steady-state temperature can be different ($\\theta_L$ vs. $\\theta_{L'}$), in both scenarios the given maximum temperature is the same. Hence,\n\\[\n\\theta_{M} =\\alpha \\frac{{\\mathrm{e}}^{\\beta\\,t_{wk}}-1}{{\\mathrm{e}}^{\\beta\\,T}-1}= \\alpha \\frac{{\\mathrm{e}}^{\\beta(t_1+t_2+t_3)}-{\\mathrm{e}}^{\\beta(t_2+t_3)}+{\\mathrm{e}}^{\\beta(t_3)}-1}{{\\mathrm{e}}^{\\beta T}-1}.\n\\]\nTherefore, we have ${\\mathrm{e}}^{\\beta\\,t_{wk}} = {\\mathrm{e}}^{\\beta(t_1+t_2+t_3)}-{\\mathrm{e}}^{\\beta(t_2+t_3)}+{\\mathrm{e}}^{\\beta(t_3)}$. Now we want to prove that for any value of $t_1$, $t_2$ and $t_3$, $t_1 + t_3 \\geq t_w$.\\\\\n\\textit{Contradiction:} Suppose the hypothesis is not correct. Hence,\n\\[\n  t_1 + t_3 < t_w \\longrightarrow \\beta\\,(t_{1}+ t_{3}) > \\beta\\, t_w \\longrightarrow {\\mathrm{e}}^{\\beta\\,(t_{1}+ t_{3})} \\geq {\\mathrm{e}}^{\\beta\\,t_{wk}}\n\\]\n\\[\n{\\mathrm{e}}^{\\beta\\,(t_{1}+ t_{3})} \\geq {\\mathrm{e}}^{\\beta\\,t_{wk}}= {\\mathrm{e}}^{\\beta(t_1+t_2+t_3)}-{\\mathrm{e}}^{\\beta(t_2+t_3)}+{\\mathrm{e}}^{\\beta(t_3)} \\Longleftrightarrow\n\\]\n\\[\n{\\mathrm{e}}^{\\beta\\,t_{1}} \\geq {\\mathrm{e}}^{\\beta(t_1+t_2)}-{\\mathrm{e}}^{\\beta(t_2)}+1 \\Longleftrightarrow {\\mathrm{e}}^{\\beta\\,t_{1}} - 1 \\geq {\\mathrm{e}}^{\\beta(t_1+t_2)}-{\\mathrm{e}}^{\\beta(t_2)}\n\\]\n\\[\n \\Longleftrightarrow{\\mathrm{e}}^{\\beta\\,t_{1}} - 1 \\geq {\\mathrm{e}}^{\\beta(t_2)}({\\mathrm{e}}^{\\beta(t_1)}-1) \\Longleftrightarrow {\\mathrm{e}}^{\\beta(t_2)} \\geq 1 \\tikz[baseline, x=0.22em, y=0.22em, line width=0.032em]\\draw (0,2.83)--(2.83,0) (0.71,3.54)--(3.54,0.71) (0,0.71)--(2.83,3.54) (0.71,0)--(3.54,2.83);\n\\]\nsince $\\beta < 0 $ and $t_2>0$.\n\\end{proof}\n\n\n\n\n\n\\begin{corollar}\\label{col:1}\nThe maximum temperature reduces when the period and waking time are halved, since it is the special condition where $t_1 = t_3$ and $t_2=t_4$.\n\\end{corollar}\n\\smallskip\\noindent\\textbf{Server budget calculation under polling server budget replenishment policy.} \nNow we calculate the ``maximum\" budget that a server can have while limiting the operating temperature not  to exceed the given thermal constraint in a single-core platform. The worst case for the  polling server happens when it exhausts all of its replenishment budget at the beginning of its period and then it sleeps until the beginning of the next replenishment period. For a server period $T$,\n\\begin{equation} \n\\label{eq:chap3_1}\nT = t_{wk} + t_{slp}.\n\\end{equation}\n\n\n\nIn the steady state of the system, we are interested in bounding the server's maximum temperature. According to Eq.~\\ref{eq:chap3_1005}, $ \\alpha + (\\theta_L - \\alpha ) e^{\\beta t_{wk}} \\leq \\theta_M$. By calculating the minimum temperature at the end of the period and substituting it in this formula, the maximum budget for a period $T$ is given by:\n\n\\begin{equation} \\label{eq:chap3_2}\nt_{wk} <=  \\frac{1}{\\beta} \\ln{\\frac{\\theta_M(e^{\\beta T}-1)+\\alpha}{\\alpha}}.\n\\end{equation}\n\n\\chapter{Data-driven Thermal Parameter Estimation for COTS-based Mixed-criticality Systems}\n\n\\input{Chapter4\/0_abstract\/0_abstract}\n\\input{Chapter4\/1-introduction\/intro}\n\\input{Chapter4\/2-related\/related}\n\\input{Chapter4\/3-Model\/0_model}\n\\input{Chapter4\/3-Model\/1_power}\n\\input{Chapter4\/3-Model\/2_temperature}\n\\input{Chapter4\/3-Model\/3_problem}\n\\input{Chapter4\/4-Errors\/0_Error_correction}\n\\input{Chapter4\/5-Analysis\/0_analysis}\n\\input{Chapter4\/5-Analysis\/1_steady_state}\n\\input{Chapter4\/5-Analysis\/3_tune_steady}\n\\input{Chapter4\/6-floorplan\/1_floorplan_estimate}\n\\input{Chapter4\/6-floorplan\/2_platform_agnostic}\n\\input{Chapter4\/5-Analysis\/2_transient}\n\n\\input{Chapter4\/7-Improvement\/0_improvement}\n\\input{Chapter4\/7-Improvement\/1_multi_experiments}\n\\input{Chapter4\/7-Improvement\/2_multi_freqs}\n\\input{Chapter4\/8-Evaluation\/0_methodology}\n\\input{Chapter4\/8-Evaluation\/1_hardware}\n\\input{Chapter4\/8-Evaluation\/2_steady_state}\n\\input{Chapter4\/8-Evaluation\/3-static_postprocessing}\n\\input{Chapter4\/8-Evaluation\/4_floorplan}\n\\input{Chapter4\/8-Evaluation\/5_power_est}\n\\input{Chapter4\/8-Evaluation\/6_trans_paramter_est}\n\\input{Chapter4\/9-Conclusion\/9_conclusion}\n\n\n\n\\section{\\new{Background on Thermal Behavior}}\n\\label{sec:background}\n\\new{ALI::: change  here ALSO}\nIn this section, we briefly introduce the thermal model used in this paper. It depends on power consumption, heat dissipation, and convective heat transfer between adjacent power-consuming resource components, which includes CPU cores, CPU peripherals, GPU, caches, and other IPs.\n\n\\smallskip\\noindent\\textbf{Power model.} The total power consumption of CMOS circuits is modeled as the summation of dynamic and static powers ($P(t) = P_D(t) + P_S(t)$)\\cite{4484694}. Static power $P_S$ depends on the semiconductor technology and the operating temperature caused by current leakage. Therefore, static power presents when a power resource is switched on. Static power can be modeled as a linear function of temperature. Hence, static power is $P_S(t) = k_1 \\theta(t) + k_2 $, where $k_1$ and $k_2$ are technology-dependent system constants, and $\\theta$ is the operating temperature \\cite{4212027}. Dynamic power $P_D(t)$ is the amount of power consumption due to the processor operating frequency $f$, modeled as $P_D = k_0 f^s$, where $f$ and $k_0$ are system constants that depend on the semiconductor technology. It is worth noting that in this paper, our framework runs processor cores at fixed operating frequency to obtain predictable worst-case task execution time. Consequently, the total power is the function of $\\theta(t)$ because the other factors such as frequency remain invariant during execution. \n\n\n\\smallskip\\noindent\\textbf{Uni-processor thermal model.} With respect to the power function\\cite{4212027, 4484694}, the thermal model of a uni-processor is modeled as an RC circuit in the literature \\cite{4484694, 6200041, Skadron}. According to the Fourier's Law \\cite{wang2010schedulability}, the temperature of a core $\\theta(t)$ at time $t$ can be derived by solving the following differential equation\\footnote{Details are available in~\\cite{DSouza2017ThermalIO} and \\cite{Ahmed2017}.}:\n\\begin{equation}\n     \\theta(t) = \\alpha + (\\theta(t_0)-\\alpha) e^{\\beta t}\n     \\label{eq:tempActive}\n\\end{equation}\nwhere $\\alpha > 0$ and $\\beta < 0 $ are constants. \n\nMost operating systems in embedding platforms transition the processor to the sleep state when there is no task execution. Therefore, the power consumption can be assumed to be negligible in such a case. The thermal function in the sleep state (\\textit{aka} cooling phase) where the frequency is switched off can be modeled as:\n\\begin{equation}\n    \\theta(t) = \\theta(t_0) e^{\\beta t}.\n    \\label{eq:tempCool}\n\\end{equation}\n because in the cooling phase, $a = 0$.\n \n\\smallskip\\noindent\\textbf{Homogeneous multi-core thermal model.} With the presence of multiple cores and other power-consuming resources, there is heat dissipation not only to ambient but also between nodes. In this paper, we only consider CPU cores as power-consuming nodes because other IPs consume much less power than them, thereby causing negligible thermal effects. \n\nIn such a multi-core system with thermal transition between processing cores, the temperature of a core depends not only on its current temperature and power consumption, but also on those of adjacent cores.  Prior work~\\cite{6629322} showed that the temperature of each core at time $t+\\Delta$ can be modeled with acceptable accuracy as follows:\n\\[ \\Theta(t+\\Delta) = A \\times P(t+\\Delta) + \\Gamma \\times \\Theta(t)  \\]\nwhere $A$ is  $m \\times m$ matrix, and $\\Gamma$, $P$ and $\\Theta$ are $m \\times 1$ matrices, respectively. One can interpret the thermal model as a function of the current temperature and heat produced by execution and convection between cores. \nHence, according to the thermal composability characteristics of heat transfer~\\cite{Ahmed2017} with the initial temperature of $\\theta(t_0)$, the temperature after $t$ time units is given by:\n\\begin{equation}\n\\theta_i(t_0+t) = \\alpha + (\\theta(t_0)-\\alpha) e^{\\beta t} +  \\sum_{j=1}^{m} \\gamma_{ij} \\theta_j(t_0+t).\n    \\label{eq:tempCompo}\n\\end{equation} \n\n\\section{Introduction}\nOne of the major concerns in recent embedded systems is the high heat dissipation caused by complex applications running on high-performance multi-core systems-on-chips (SoCs). Ambient temperature in the physical environment is another key factor that increases chip operating temperature. The high temperature also increases power consumption~\\cite{ahmed2016necessary}, reduces the chip reliability~\\cite{srinivasan2004impact, viswanath2000thermal}, and leads to chip burnout. Due to these reasons, many of today's operating systems (OSs) monitor the chip operating temperature using on-chip temperature sensors to check if it is within the safe thermal constraint. \n\nTo protect the processor chip from thermal damage, a set of policies are defined in the thermal governor of the OS for different scenarios.\nWhen the chip temperature crosses a trip point, a predefined cooling scenario is performed such as frequency throttling or shutting down of CPU cores~\\cite{choi2004fine, herbert2007analysis, wang2009temperature}. Such thermal countermeasures, however, lead to timing unpredictability in real-time mixed-criticality systems (MCS) since the deadlines of tasks could be unexpectedly violated by reduced processing speed or temporarily unavailable CPU cores.\nExtensive studies have been have been conducted to prevent the negative impact of such performance disruption, including the techniques based on Dynamic Voltage Frequency Scaling (DVFS)~\\cite{fu2010feedback,ma2015improving, 7372657, 7746768, zhang2010thermal} and forced sleeping during the execution of \\textit{hot} applications~\\cite{chantem2010temperature, Huang2014, Youngmoon2018, kumar2011system, jayaseelan2008temperature, zhang2010thermal}. Moreover, offline analysis has been proposed to guarantee the thermal safety of real-time tasks by bounding the maximum operating temperature in  the steady~\\cite{ mehdi2020dynamic, hosseinimotlagh2019thermal} and transient states~\\cite{pagani2015matex} of a given system. They all assume to have a priori knowledge of precise thermal models for temperature prediction, resource management, and task scheduling, and need accurate simulation tools or extra equipment for validation. Therefore, obtaining thermal parameters of a given system is the fundamental requirement to substantiate these techniques in practice.\n\nDespite its importance, the thermal parameter estimation of commercial off-the-shelf (COTS) processors still remains a challenging problem. \nExisting numerical simulation tools like HotSpot~\\cite{huang2006hotspot} can construct a compact thermal model for modern VLSI devices by using the resistance-capacitance (RC) thermal network to capture the transient temperature and generate the heatmap at each time instant. However, they are not directly applicable to modern COTS multi-core processors because the information required by these tools, such as power traces of micro-architectural units, detailed floorplan maps, and cooling package, is proprietary and not publicly available. An exhaustive search approach for approximating this information through reverse engineering is time-consuming and prone to unacceptably high inaccuracies, especially for transient temperature estimation. \n\nThe latest work~\\cite{rai2015calibration, rai2012power} partially addresses these issues by calibrating thermal parameters without power traces and detailed floorplans. However, it is not applicable to MCS where execution patterns are dynamic due to preemptive, priority-driven, work-conserving task schedulers~\\cite{mehdi2020dynamic, hosseinimotlagh2019thermal, pagani2015matex, Ahmed2017, DSouza2017ThermalIO}. Similar to HotSpot, it uses a time-driven prediction model where the temperature is calculated for each time instant under static workloads. This makes the model not only very slow but also inflexible to capture the correct operating temperature when multiple real-time tasks with different periodicity are concurrently scheduled.\n\nIn this chapter, we propose a fast and accurate scheme to estimate the thermal parameters of COTS multi-core processors for real-time MCS. Our scheme requires only a small number of temperature traces from on-chip thermal sensors which are widely available in today's processors. Our scheme also improves the accuracy of thermal parameters through the ensemble of measurements from different frequency levels and execution patterns. \n\n\\smallskip\\noindent\\textbf{Contributions.} The contributions of this chapter are as follows:\n\\begin{itemize}\n    \\item \n   \n    We present a thermal estimation scheme that has low computational cost by design. Given that steady-state profiles are much compact than transient-state profiles, our scheme first estimates the thermal parameters of a given system using only steady-state profiles, and then uses transient-state data for calibration purpose. \n    \n    \\item We characterize various sources of errors in thermal parameter estimation, and reduce their negative effects through the multiple refinement stages of our scheme. Our scheme also enables locating errors in the temperature profiles.\n    \\item Our scheme can identify the relative distance between CPU cores and produce an estimated chip floorplan from temperature profiles. It can also estimate the relative power consumption for a given  workload on each CPU core.\n    \\item We present techniques to further improve the accuracy of thermal parameters by exploiting the ensemble of measurement data obtained at various frequency and workload settings.\n   \n   \n    \\item The effectiveness and accuracy of our proposed scheme is demonstrated with extensive experimental results on a real ARM embedded platform.\n   \n\\end{itemize}\n\n\\section{Limitations and Errors}\n\\label{sec:chap4_limitations_and_errors}\nBefore continuing our discussion, we must address some limitations to thermal parameter estimation on real-life platforms. Identifying these potential sources of error is critical to our data analysis and comprehension. It allows us to address the noise and limitations of our profile data set by preprocessing raw data and improve the accuracy of thermal parameters through the ensemble of measurements from different frequency levels and workload settings.\n\n\\subsection{Built-in Sensors}\nThe largest sources of error in the raw data are the built-in temperature sensors.\n\n\\subsubsection{Sensor Locations} The data sampled from the CPU's built-in temperature sensors is sensitive to the physical location of the sensors within CPU cores. For example, the thermal sensor for one CPU core may be located near its primary source of heat dissipation (i.e., hotspot) while the thermal sensor for another CPU core may be located further away from the CPU's hotspot. In addition to the proximity of the thermal sensor to the CPU's hotspot, the physical location of the hotspot may vary depending on the type of workload. Thus, distinct thermal footprints of various applications may not be precisely captured by a single built-in sensor. \n\n\\subsubsection{Sensor Sampling Frequency} In CPUs for embedded systems, the sampling rate of temperature sensors is, generally, very low.  This limitation causes inaccuracies in raw data. For instance, the ODroid XU4 board maintains a 10Hz sampling rate for the thermal sensors. If the utilization pattern of a prospective application changes in less than 100~ms, e.g., real-time control tasks activated every 30~ms, the thermal footprint cannot be captured with the built-in sensors. \n\n\\subsubsection{Sampling Precision} The data from on-chip CMOS temperature sensors is subject to quantization. This generates another source of error because the accuracy of temperature measurements is reduced to the granularity of quantization. Hence, the data needs to be processed before and after the estimating procedure to ensure the correctness of our results. For instance, if the sampling precision of all CPU core temperature sensors is 1\\textdegree~C, then according to the superposition law in thermal modeling~\\cite{mehdi2020dynamic}, the magnitude of total temperature error for a quad-core processor can reach up to 4\\textdegree~C.  \n\n\\subsubsection{Sensor Response Function} Due to differences in the construction and architecture of on-chip thermal sensors, their response time may vary. Hence, the sensor data may not represent the actual temperature if the CPU utilization changes in a relatively short time interval. Over time, if there is no fluctuation in CPU utilization, the sensor data will converge to the actual CPU core temperature. Therefore, we can assume that this type of error only affects the transient-state data while the steady-state data is unaffected. \n\n\\subsection{Ambient Temperature} Temperature is a relative value to the ambient temperature in our thermal model. Although it is assumed that the ambient temperature would remain invariant during profiling, in reality, it may change even in a room or a thermal furnace.\n\n\\subsubsection{Varying Ambient Temperature} The ambient temperature in room can change slightly, or even several degrees given circumstances that are difficult to regulate such as poorly insulated walls, drastic changes in the outside temperature, and even heat dissipated from idle electrical outlets. This change can affect the operating temperature of the CPU. Therefore, the fluctuation of the ambient temperature while profiling can introduce noise into the raw data. This type of noise impacts the relative operating temperature. \n\n\\subsubsection{Varying Air Convection} Heat convection may also introduce noise into our data sets (thermal profiles). Forced air convection caused by air conditioning, active cooling package of the CPU or even people just moving around the board can all contribute to fluctuations in the CPU core's heat dissipation. This type of noise will directly affect the heat transfer from the CPU cores to the ambient air, hence the values of diagonal elements of the matrix $\\myb[A]$ change.\n\n\\subsection{Thermal Interference} \nHeat dissipation due to miscellaneous tasks can impact the accuracy of the data set. Even though one expects the operating temperature of CPU cores to approach the ambient temperature when the CPU cores are disabled (e.g., by using the CPU hotplug mechanism in Linux), the operating temperature is much higher than this level.\n\n\\subsubsection{OS Tasks} There are some essential OS service tasks that cannot be terminated while profiling thermal data. Some of them even have predefined CPU affinity, thereby affecting the observed thermal behavior of target CPU cores.\n\n\\subsubsection{Cache Coherent Interconnect (CCI)} Under cache consistency policies, CCI which is near to the CPU cores, generates some heat during task execution which causes an increase in the operating temperature of the CPU cores.\n\n\\subsubsection{Workload on Intellectual Property (IP) Blocks} Running tasks on IP blocks such as integrated GPUs, video encoders\/decoders, and digital signal processors (DSP), can significantly affect the overall heat dissipation. Additionally the static power consumed while IPs are idle can also cause non-trivial heat dissipation. If there is no change in the utilization status of IPs, we can consider the amount of heat dissipation from those IPs as a constant quantity in thermal modeling. Nevertheless, this type of the noise can continuously or temporarily affect the profiling of thermal data. \n\n\n\\section{Related Work}\n\nThere exist well-known numerical tools to estimate the chip operating temperature. For instance, HotSpot solves the system of differential equations using the fourth-order Runge-Kutta numerical method through very fine-granularity iterations. The authors of ~\\cite{7904613, 8442110} constructed the thermal model with the measured power and temperature trace of each subsystem on a real mobile platform.  Power Blurring~\\cite{ziabari2014power} calculates temperature distributions using a matrix convolution technique, in contrast to the finite-element analysis (FEA) used in other simulation tools like HotSpot. To use these tools in practice, one needs to obtain the detailed information of the target device including power traces and floorplans or to arrange special measurement equipment such as a high-precision IR camera. \n\n\n\nThe authors of \\cite{rai2015calibration, rai2012power} proposed a calibration-based method to predict thermal behavior by using an impulse response model. They assume that the power consumption of each application task remains unchanged during execution. Similar to~\\cite{5456979, Ahmed2017}, they employed the Generalized-Pencil-Of-Function (GPOF)~\\cite{hua1989generalized} to calculate the impulse response of each application from utilization and temperature traces. In their work, the thermal effects due to conduction between CPU cores are represented as impulse responses.\nHowever, if an application task is preempted by another task or getting suspended, the impulse function has to switch. This spatio-temporal thermal dependency cannot be captured in their model for preemptively-scheduled tasks on the same CPU core or concurrently-executing tasks on other CPU cores. Due to this reason, despite all the other benefits provided, the applicability of their approach to real-time mixed-criticality systems is limited. In this chapter, we present a thermal model based on matrix exponential, which overcomes the aforementioned limitations of prior work and allows estimating the key thermal parameters that represent the characteristics of the semiconductor technology independent of the specificity of applications. \n\\section{System Model}\nWe consider a homogeneous multi-core processor where each CPU core uses the same microarchitecture. Each core is assumed to have a dedicated temperature sensor that is accessible by the OS at runtime. This assumption can be easily met in many commercial processors such as Intel Core i7 and Samsung Exynos products. Reflecting reality, it is assumed that the following information is not available to use: the chip floorplan, the exact locations of on-chip temperature sensors, and the power traces of the processor.\n\nIn the rest of this section, we introduce the power and temperature model used in this chapter.\n\\subsection{Power Model}\n\n The total power consumption of CMOS circuits at time $t$ is modeled as the summation of dynamic and static powers~\\cite{4484694}, i.e., ${P(t) = P_S(t) + P_D(t)}$. Static power $P_S$ depends on the semiconductor technology and the operating temperature caused by current leakage. \nHence, it can be modeled as: ${P_S(t) = k_1 \\theta(t) + k_2}$, where $k_1$ and $k_2$ are technology-dependent system constants, and $\\theta(t)$ is the operating temperature \\cite{4212027}. Dynamic power $P_D(t)$ is the amount of power consumption due to the processor operating frequency $f$ at time $t$, modeled as ${P_D(t) = k_0 f(t)^s}$, where $s$ and $k_0$ are the system constants that depend on the semiconductor technology.\n\n\n\n\\subsection{Temperature Model}\n\\label{sec:chap4_tempMdl}\nWe consider the temperature model widely used in real-time mixed-criticality systems~\\cite{Ahmed2017,DSouza2017ThermalIO}, which follows the well-known linear time-invariant (LTI) model. Hence, the temperature model for a multi-core CPU with $n$ cores is given by the following equation:\n\\begin{equation}\\label{eq:chap4_tempMdl}\n    [ \\boldsymbol{\\bf{\\theta'}}(t)]_{n \\times 1} = \\boldsymbol{\\bf{A}}_{n \\times n}\\ [\\boldsymbol{\\bf{\\theta}}(t)]_{n \\times 1} + \\boldsymbol{\\bf{B}}_{n \\times n}\\ [\\boldsymbol{\\bf{P}}(t)]_{n \\times 1}\n\\end{equation}\n where $\\myb[\\theta](t)$ is the $n \\times 1 $ matrix of the CPU core operating temperatures relative to the ambient temperature, and $\\myb[P](t)$ is the power consumption of all cores at time $t$. $\\myb[A]$ is an invariant $n \\times n$ matrix and it is based on the characteristics of the semiconductor technology. It quantifies the effect of the conduction between adjacent cores, the convection among all cores, and the difference between ambient and operating temperature. \n\n$\\myb[B]$ is the diagonal $n \\times n$  matrix and it captures the effect of power consumption on the temperature of each core. For homogeneous multi-core CPUs, since the total power of CPU cores are the same (either $P_S$ or $P_S + P_D$), the matrix $\\myb[B]$ can be represented as $b \\times \\myb[I]$, where $\\myb[I]$ is the $n \\times n$ identity matrix. Similar to $\\myb[A]$, the values of $b$ are invariant to the changes in static or dynamic power consumption\n\nHence, the problem will be estimating the values of the matrices $\\myb[A]$ and $\\myb[B]$ ($=b \\times \\myb[I]$) without any prior knowledge or direct measurement of CPU power consumption. We will discuss that it is impossible to estimate the value of $b$ without having any knowledge of CPU power consumption; instead, we estimate $\\myb[B]\\times \\myb[P]$ which matters in calculating the temperature in the LTI model.\n\\subsection{Problem Description}\nGiven a multi-core CPU equipped with on-chip temperature sensors, construct an accurate and fast thermal RC model by the estimation of $\\myb[A]$ and $\\myb[B]\\times \\myb[P]$ of the CPU, exclusively from a limited number of temperature profiles without requiring a priori knowledge of the floorplan, cooling package, and power traces.\n\n\\subsection{Floorplan Estimation}\nHereby, we introduce a breadth-first-search (BFS) algorithm to estimate the topography of the CPU cores in the chip. Later, we use this information to calibrate the thermal term $b\\ P_D$ in the temperature model by using the transient-state data. One of the steps to refine the thermal parameters of a chip is to estimate the parameters complying with the floorplan. We present a mechanism to estimate the relative location of the CPU cores on a given chip. \n\nThe intuition behind our proposed algorithm is that the amount of heat dissipation from a heat source to a closer object is more than that to a farther object. Therefore, we expect that the temperature increase due to heat conduction of adjacent CPU cores is more than that of non-adjacent CPU cores. We introduce a fully-connected weighted graph where the edge weight represents the temperature increase of a CPU core when its neighboring cores are fully utilized. For instance, suppose there is a quad-core CPU where each core is labeled as $C_i$ with $i \\in [1,4]$. The fully-connected graph of this CPU is illustrated in Fig.~\\ref{fig:chap4_001}. The weight of each edge, $w_{i,j}$, represents the temperature increase of a core $C_j$ when another core $C_i$ is fully utilized (i.e., $y_{i,j}-y^0_i$). \nIt is worth noting that $w_{i,j} \\neq w_{j,i}$ not only because of noise but also due to the location of the built-in temperature sensor relative to the hotspot of each CPU core, although the amount of heat dissipation from each homogeneous core is ideally the same.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.22\\textwidth]{Chapter4\/6-floorplan\/figs\/graph1.pdf}\n\\caption{Example of adjacency graph of a quad-core CPU.}\n\\label{fig:chap4_001}\n\\end{figure}\n\nWe are interested in the relative temperature increase of core $C_i$ when it is fully utilized compared to when another core $C_j$ is fully-utilized. Hence, we have ${w'_{i,j} = y_{i,i} - y_{i,j}}$. We also remove self-loops from the graph for simplicity. After this, the graph of Fig.~\\ref{fig:chap4_001} is reduced to a graph where all weights are positive. Next, we convert the bi-directed graph to a uni-directed graph by taking maximum of the edges in the different direction (i,e., ${\\min\\{w'_{i,j},w'_{j,i}\\}}$) as shown in Fig.~\\ref{fig:chap4_002}. It is noteworthy that one can use average, minimum or any reduction operation for this stage. As shown in Fig.~\\ref{fig:chap4_002}, the temperature increase of CPU core $C_4$ when the CPU core $C_1$ is fully utilized is because of heat conduction of CPU cores $C_2$ and $C_3$. In other words, the CPU core $C_4$ is heated up because of transitive heat transfer from $C_1$ to both $C_2$ and $C_3$ and then heat transfer occurs from these CPU cores to the CPU core $C_4$. Therefore, there is only transitive heat conductivity between the CPU core $C_1$ and the core $C_4$. \n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.22\\textwidth]{Chapter4\/6-floorplan\/figs\/graph3.pdf}\n\\caption{Example of adjacency graph of a quad-core CPU.}\n\\label{fig:chap4_002}\n\\end{figure}\n\n\nBy using this graph and also the aforementioned intuition, we estimate the location of the CPU cores on the floorplan. To do this end, we begin from one arbitrary core (let's say $C_1$) and find the minimum value of all weights connected to its corresponding node in the graph. In this example, the maximum weight is 2 for both CPU cores $C_2$ and $C_3$. Next, we find the maximum weight of CPU cores $C_2$ and $C_3$ from the unvisited node list, one at a time. We continue this step until all nodes are visited. If some of the visited CPU cores have the same value (within some error margin), they are connected. In this example, the error margin is 1 so the cores are connected as illustrated in Fig.~\\ref{fig:chap4_004}. It is shown that the core $C_4$ (or at least its built-in temperature sensor) is closer to the core $C_3$ than $C_2$.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.22\\textwidth]{Chapter4\/6-floorplan\/figs\/graph4.pdf}\n\\caption{Final adjacency graph of a quad-core CPU.}\n\\label{fig:chap4_004}\n\\end{figure}\n\nFurthermore, if the temperature profiles of on-chip IPs such as an integrated GPU are available, the same approach can be applied to  locate the corresponding IP \\new{by using the temperature increase when each CPU core is fully-utilized.} In this way, there is only one edge from each CPU core $C_i$ to the IP.\n\n\n\\subsection{Thermal Parameter Estimation with Steady-State Data}\n\\label{sec:chap4_estimate}\n\nWe propose a method to calibrate the thermal parameters based on the floorplan estimation discussed in the previous section. The thermal parameter estimation without considering the floorplan may not be applicable in practice.\\footnote{This is because $\\myb[Y]$ can be inaccurate and the properties of $\\myb[A]$ may be violated if the floorplan is not considered.} Some important properties of $\\myb[A]$ according to heat transfer are as follows:\n\\begin{itemize}\n    \\item $\\myb[A]$ is a symmetric matrix. \n    \\item The elements on diagonal of $\\myb[A]$ are positive.\n    \\item The zero values of non-diagonal elements represent a pair of corresponding CPU cores are not adjacent.\n    \\item All non-zero values of non-diagonal elements are negative.\n\\end{itemize}\n These properties of $\\myb[A]$ guarantees that there exist $n$ real eigenvalues and a set of $n$ eigenvectors, one for each eigenvalue, that are mutually orthogonal. This stage estimates the values of thermal parameters in way that it not only complies with the floorplan but also reduces the effect of noisy data observed data in $\\myb[A]$. \n\nThe matrix $\\Tilde{\\myb[A]}$ is constructed according to the estimated floorplan. If there is no thermal interference between two cores $C_i$ and $C_j$ in the estimated floorplan, the corresponding value in the matrix, i.e., $\\Tilde{a}_{i,j}$, is zero. A single value can be used for all adjacent CPU cores in the matrix $\\Tilde{\\myb[A]}$ as they share the same heat dissipation increase value. \n The gradient descent algorithm will then estimate the non-zero values. For the quad-core CPU exemplified in the previous section, there exist 4 zeros in the matrix $\\Tilde{\\myb[A]}$ because each CPU core has only 2 adjacent CPU cores. The number of distinct non-zero values in $\\Tilde{[A]}$ are six in total because four different values are on the diagonal of $\\Tilde{\\myb[A]}$ and two distinct values are used for the other non-zero elements. It is worth noting that,  depending on the user's accuracy demand, more distinct non-zero values may be used. For the explained example, the matrix $\\Tilde{\\myb[A]}$ will be \n\\[\n\\Tilde{A} = \\begin{bmatrix}\na_1 & a_5 & a_5 & 0\\\\ \na_5 & a_2 & 0 & a_6\\\\ \na_5 & 0 & a_3 & a_5\\\\ \n0 & a_6 & a_5 & a_4\\\\ \n\\end{bmatrix}.\n\\]\nWe propose a gradient descent algorithm to \\new{find the parameters according the estimated floorplan}. For the calibration of the parameters, $\\Tilde{\\myb[A]}$ is compared with the inverse of $\\myb[Y]$, which means that $\\Tilde{\\myb[A]}$ follows the estimated floorplan template and its inverse must be close to the temperature profiles. We propose the cost function as follows\n\\begin{equation}\n\\operatorname*{argmin}_{\\substack{\\text{$a_{i}$} \\\\ \\text{$i \\in [1,r]$}}}  ||\\Tilde{\\myb[A]}^{-1} - Y ||^2_F\n\\label{eq:chap4_cost}\n\\end{equation}\nwhere $r$ is the number of unknown parameters. The sign of each parameter has to be carefully considered in the implementation, taking into account the properties of $\\myb[A]$ discussed before (e.g., diagonal elements should be positive and the rest be non-positive).\n\n\\section{Proposed Scheme}\nIn this section, we introduce our scheme for estimating the thermal parameters of a given multi-core processor. \nThe entire workflow of the proposed scheme is illustrated in Fig.~\\ref{fig:chap4_scheme}. The very first step is profiling steady-state temperature data for a set of designed workloads. Then, the scheme removes noise from the raw data set (\\circled{2}), and performs the floorplan estimation (\\circled{3}). By using the estimated floorplan template and the collected steady-state data, the value of the matrix $\\myb[A]$ is estimated in terms of the power parameters (\\circled{4}). The parameters $\\myb[B] \\times \\myb[P]$ are then estimated by analyzing a subset of the transient-state data at the final stage (\\circled{5}). One of the reasons that we propose dividing analysis into the steady-state and the transient-state stages (\\circled{4} and \\circled{5}) is to cope with the various types of errors that may be introduced during temperature profiling. If one tries to tackle this problem by using both data at the same time, errors in the transient-state data can adversely affect the characterization of the system in steady state because the transient-state data has a much larger number of data points than the steady-state data. Moreover, our proposed scheme has a low computational cost as it requires processing only a few data points in the steady-state stage to obtain the thermal characteristics of the system.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Chapter4\/5-Analysis\/figs\/scheme_new.pdf}\n\\caption{Parameter Estimation Scheme}\n\\label{fig:chap4_scheme}\n\\end{figure}\n\n\n\\subsection{Thermal Analysis and Steady-State Profiling}\n\\label{sec:chap4_SSI}\nThe proposed scheme primarily uses the state-state data for thermal parameter estimation since it is more robust to measurement errors than the transient-state data but still contains the required information about semiconductor technology and power consumption. Hence, before introducing the detailed stages of our scheme, we analyze the thermal model and provide the reasoning behind the steady-state profiling. \n\nOur goal is to estimate the thermal parameters related to the steady-state data. After solving the first order equation of Eq.~\\ref{eq:chap4_tempMdl}, we have\n\n\\begin{equation}\\label{eq:chap4_1}\n    \\boldsymbol{\\bf{\\theta}} (t) =  \\boldsymbol{\\bf{\\theta}}_{chip} + {e^{(t - {t_0}){\\bf{A}}}}{\\boldsymbol{\\theta} _0} + \\int_{{t_0}}^t {{e^{(t - s){\\bf{A}}}}{\\bf{BP}}(s)ds}\n\\end{equation}\nwhere $[\\boldsymbol{\\bf{\\theta}}_{chip}]_{n \\times 1}$ is the total heat dissipation caused by IP blocks on the chip and idle power of the CPU cores. We assume that all IPs generate a constant amount of heat. This term captures the heat conduction between all parts of the chip and the CPU. When the CPU cores are \\new{idle} for a very long time, $\\boldsymbol{\\bf{\\theta}}_{chip}$ can be captured. It is worth noting \\new{that  due to the location CPU cores and their sensors on the floorplan, the steady-state temperature of idle CPU cores are different}. \\cmnt{value is different for each CPU core because of the the location of CPU cores on the floorplan and also the location of each thermal sensor.} \\new{The second term is the homogeneous solution which is the thermal response due to the initial temperature difference from the ambient. The third term is the non-homogeneous solution caused by the input power signal.}\n\n\\new{The total power is the function of temperature at any time instant $t$ because the other factors such as frequency remain invariant during task execution. For homogeneous multi-core CPUs, the power for each core is $P_S(t)$ when CPU is idle, and $P_S(t) + P_D(t)$ when the CPU executes some workload in which all cores are fully utilized. Hence,} \\cmnt{If the power is considered to be fixed,} Eq.~\\ref{eq:chap4_1} can be written as: \n\\begin{equation}\\label{eq:chap4_2}\n\\boldsymbol{\\bf{\\theta}} (t) = \\boldsymbol{\\bf{\\theta}}_{chip} + {e^{(t - {t_0}){\\bf{A}}}} \\boldsymbol{\\theta}_0 -\\bf{A}^{-1}\\ (I-e^{\\bf{A}t}\\ )\\bf{BP}.\n\\end{equation}\n\nIn the steady state, the second term disappears because the steady-state operating temperature depends only on the power consumption (third term) and it is unaffected by initial temperature $\\myb[\\theta]_0$. Suppose $\\boldsymbol{\\bf{\\theta}}^\\infty$ represents the operating temperature in the steady state, then:\n\\begin{equation}\\label{eq:chap4_3}\n\\boldsymbol{\\bf{\\theta}}^\\infty  = \\lim_{t \\rightarrow \\infty}   \\boldsymbol{\\bf{\\theta}} (t) =  \\boldsymbol{\\bf{\\theta}}_{chip}  -\\bf{A}^{-1}\\bf{BP}.\n\\end{equation}\n\nWe are interested in finding the value of matrices $\\myb[A]$ and $\\myb[BP]$. It is worth noting that the only control parameter is the power signal. It means that for each profiling procedure, it is possible to execute workload on any subset of CPU cores or hot-unplug them but the actual value of the power remains unknown. Let $[\\myb[Y_i]]_{n \\times 1}$ denote the operating temperature of the CPU cores when the $i$-th core is fully-utilized and $\\myb[Y]_0$ represent the temperature of the CPU when all CPU cores are idle. Furthermore, let $[\\myb[Y]]_{n \\times n} = [\\myb[Y_1] \\myb[Y_2] \\dots \\myb[Y_n]]^T$ be the matrix of temperature profiles of the CPU in the steady state. For instance, $y_{i,j}$ in the row $i$ and column $j$ of $\\myb[Y]$ is the temperature of the $j$-th CPU core when tasks are executing only on the $i$-th CPU core. Hence, according to Eq.~\\ref{eq:chap4_3}, \n\\begin{equation}\\label{eq:chap4_4}\n\\myb[Y] - [\\myb[Y_0] \\myb[Y_0] ... \\myb[Y_0]]^T_{n \\times n} =  P_D\\ (-\\bf{A}^{-1}\\bf{B}).\n\\end{equation}\n\nWe solve the equation to find the value of $\\myb[A]$. Therefore, \n\\begin{equation}\\label{eq:chap4_5}\n  \\myb[A]=- P_D\\ b\\ (\\myb[Y] - [\\myb[Y_0] \\myb[Y_0] ... \\myb[Y_0]]^T)^{-1}.\n\\end{equation}\n\nBy denoting ${\\Tilde{\\myb[A]} = (\\myb[Y] - [\\myb[Y_0] \\myb[Y_0] ... \\myb[Y_0]]^T)^{-1}}$ and ${\\gamma = b\\ P_D}$, we have\n\\begin{equation}\\label{eq:chap4_6}\n  \\myb[A]=- \\gamma \\Tilde{\\myb[A]}.\n\\end{equation}\n Therefore, by estimating $\\Tilde{\\myb[A]}$ and $\\gamma$, we can determine the value of $\\myb[A]$. $\\Tilde{\\myb[A]}$ can be determined by  profile without any knowledge of power consumption. The value of $\\gamma$ cannot be determined by the information from the steady-state profiles even with temperature profiles encompassing various CPU frequencies. \n \n As shown in Eq.~\\ref{eq:chap4_6}, we only need $n+1$ profiles to estimate the value of $\\Tilde{\\myb[A]}$, i.e., one profile measured when all cores are off, and $n$ profiles when each core is fully utilized one at a time. However, because of errors and limitations discussed in Section~\\ref{sec:chap4_limitations_and_errors}, there may be a considerable \\new{amount of noises} in estimating the value of $\\Tilde{\\myb[A]}$. If the profiles are noiseless, $\\Tilde{\\myb[A]}$ will be a symmetric matrix with positive values on diagonal and non-positive values on non-diagonal elements.  \\new{Zeros at $\\Tilde{a}_{i,j}$ represent} that there is no heat conduction between core $i$ and $j$. It is noteworthy that a non-symmetric $\\Tilde{\\myb[A]}$ caused by noisy data sets can lead to imaginary eigenvectors which is impossible \\new{in practice}. We will later propose several methods to address different types of noises. Moreover, we will extend our analysis to reach a more precise $\\Tilde{\\myb[A]}$ by infusing more data profiles. By using those techniques, it is not necessary to have the exact CPU core combinations of $\\myb[Y]_i$ for profiling purposes, but still possible to benefit from multiple auxiliary data obtained from the same core configuration. \n \n One interesting property of $\\myb[A]$ and $\\Tilde{\\myb[A]}$ is that both have the same eigenvectors. Additionally the eigenvalues of $\\myb[A]$ are $-\\gamma$ times the eigenvalues of $\\Tilde{\\myb[A]}$. We will use these properties to find the value of $\\gamma$ later and justify why it is infeasible to estimate the value of power consumption from thermal profiling.\n \n\n\\subsection{floorplan-agnostic parameter estimation by transient information}\n\\subsection{Thermal Parameter Estimation with Transient-State Data}\n\\label{sec:chap4_trans}\n\nIn this section, we discuss how to estimate the matrix $\\myb[B]$ for homogeneous multi-core platforms. As we mentioned in Sec. \\ref{sec:chap4_tempMdl}, it is impossible to determine $\\gamma$ from the steady-state observations, hence, we must process the transient-state data. \n\nAs discussed in the steady-state section, if the power remains the same, the temperature equation will become Eq.~\\ref{eq:chap4_2}.  To estimate the value of $\\myb[A]$, we only need $\\gamma$  which is commensurate to the power consumption. Substituting Eq.\\ref{eq:chap4_6} in Eq.~\\ref{eq:chap4_2}, we have:\n\n\\begin{equation}\\label{eq:chap4_7}\n\\boldsymbol{\\bf{\\theta}} (t) = \\boldsymbol{\\bf{\\theta}}_{chip} + {e^{(t - {t_0}){\\bf{- \\gamma \\Tilde{A}}}}} \\boldsymbol{\\theta}_0 +\\bf{(\\gamma \\Tilde{A})}^{-1}\\ (I-e^{\\bf{A}t}\\ )\\bf{b I P}.\n\\end{equation}\n\nSuppose that $\\myb[V]$ and $\\myb[\\Lambda]$ are the eigenvectors and eigenvalues of $\\Tilde{\\myb[A]}$, respectively. Therefore, ${e^{(t - {t_0}){\\bf{- \\gamma \\Tilde{\\myb[A]}}}}}$ can be represented as ${\\myb[V]\\ e^{(t - {t_0}){\\bf{- \\gamma \\Lambda}}}\\ \\myb[V]^{-1}}$. From our proposed steady-state scheme, $\\myb[V]$ and $\\lambda$ are determined. We can estimate the value of $\\gamma$ from the transient-state data of a singular observation.  To better calibrate  $\\gamma$, multiple profiles may also be considered. Hence,\n\\begin{equation}\\label{eq:chap4_8}\n\\boldsymbol{\\bf{\\theta}} (t) = \\boldsymbol{\\bf{\\theta}}_{chip} + {e^{-(t - {t_0}){\\bf{\\gamma \\Tilde{A}}}}} \\boldsymbol{\\theta}_0 +\\bf{\\Tilde{A}}^{-1}\\ (I-e^{\\bf{A}(t-t_0)}\\ )\\bf{H}\n\\end{equation}\nwhere $\\myb[H]$ is an $n\\times 1$ input signal whose $i$-th element is $h_i \\in \\{0,1\\}$. It is noteworthy that $h_i = 1$ when $i$-th core is fully utilized. By substituting the eigenvalues and eigenvectors, the equation can be represented as \n\n\\begin{equation}\\label{eq:chap4_9}\n\\begin{aligned}\n\\boldsymbol{\\bf{\\theta}} (t) = \\boldsymbol{\\bf{\\theta}}_{chip} + V\\ e^{-(t - {t_0}){\\bf{ \\gamma \\Lambda}}}\\ V^{-1} \\boldsymbol{\\theta}_0 + \\\\\n\\bf{\\Tilde{A}}^{-1}\\ (I-V\\ e^{-(t - {t_0}){\\bf{\\gamma \\Lambda}}}\\ V^{-1}\\ )\\bf{H}.\n\\end{aligned}\n\\end{equation}\n\nThe only missing part in the temperature model is $\\gamma$ which can be estimated by curve fitting on transient-state temperature. The values of $P_d(t)$, $\\boldsymbol{\\bf{\\theta}}_{chip}$ and $\\bf{\\Tilde{A}}$ change at different frequency levels but the values of $\\myb[\\Lambda]$, $\\myb[V]$ and $b$ remain invariant against frequency change. Although the value of $b\\gamma$ can be estimated and is embedded in the temperature data, it is not possible to estimate the value of power even with temperature information at different frequency levels.\n\nSince $\\gamma$ is a scalar value, we observed that curve fitting on only a few cores can provide good results. The simplest test for estimating the value of $\\gamma$ is when all cores are cooling down. In this case the third term is eliminated and \\new{the temperature equation} for the $i$th core can be reduced to \n\n\\begin{equation}\\label{eq:chap4_10}\n\\boldsymbol{\\bf{\\theta}}^i (t) = \\boldsymbol{\\bf{\\theta}}_{chip}^i + \\sum_{ \\lambda_j \\in \\Lambda} v_{i,j} y_{i,j} e^{-(t - {t_0}){\\bf{ \\gamma \\lambda_j}}}  \\boldsymbol{\\theta}_0^i \n\\end{equation}\nwhere $v_{i,j}$ and $y_{i,j}$ are the elements of the $i$th row and the $j$th column in the matrices $V$ and $V^{-1}$, respectively. \n\\subsection{\\new{Noise Reduction and Anomaly Detection}}\nCompensating for various sources of errors in the steady-state data is critical to accurately estimate the system thermal parameters. In this section, we will discuss two low-cost noise reduction and detection procedures for the steady-state data.\n\n\\subsubsection{Pre-processing for Noise Reduction}\nThe most challenging sources of errors in the steady-state data are the built-in CPU core temperature sensors and the uncontrollable fluctuations in the ambient temperature. Heat dissipation due to all types of \\new{task interference} can be captured in $\\myb[\\theta]_{chip}$. We will show in the evaluation section that this amount is almost invariant for any given operating frequency. Therefore, we focus our discussion on other types of errors. \\new{The accuracy of  the steady-state data} is crucial to calibrating thermal parameters with transient-state data  in the later stage of our scheme. It is also important for estimating \\new{the relative locations of CPU cores on} the floorplan since they depend on the accuracy of this stage.\n\nThe errors that could be found in transient-state data, such as due to the sampling frequency and response time of built-in sensors, do not occur in steady-state data because the steady-state temperature converges to a certain level and remains invariant. To overcome the limitation of sampling precision, we pre-process the steady-state temperature data by taking a moving average. Although the temperature should remain invariant throughout the steady state, the limitation introduced by sensor data quantization may cause fluctuations. Suppose that the precision of built-in temperature sensor is 1\\textdegree~C and the actual steady-state core temperature is 55.1\\textdegree~C. The data will usually reflect a sensor reading of 55\\textdegree~C. However, on less frequent occasions, the data may reflect a sensor reading of 56\\textdegree~C. Aside from the precision concern, some transient noises such as varying air convection, temporary OS tasks, or variant ambient temperature can also be added up in the steady state. To alleviate the effects of the limitation of sensed data precision and also transient noises, one can apply a low-pass filter such as the moving average in an arbitrary time window on the steady-state data before constructing the observation matrix $\\myb[Y]$.\n\n\\subsubsection{\\new{Post-processing for Anomaly Detection}}\n\\label{sec:chap4_anomaly}\nAs we discussed in Sec.~\\ref{sec:chap4_SSI}, our scheme needs steady-state temperature profiles which are obtained when only one of the CPU cores is fully utilized at a time. If there are more profiled data, it is possible to detect if there is any persistent error in the temperature profiles. For instance, if the ambient temperature in one profile is different from that in other profiles, it can be detected and rectified. Our post-processing error detection tests if the observed data $\\myb[Y]_i$ are consistent with other auxiliary profiles that are obtained when more than one CPU core is fully utilized.  If any error is found from the observed data $\\myb[Y]_i$, the corresponding column of $\\myb[Y]$ will be rectified with the correct value.  \n\nWe now present the details. Let $[\\myb[X]_Z]$ denote the steady-state temperature of CPU cores and $Z=\\overline{z_1 z_2 \\dots z_n}$ is the predicate that shows the status of CPU cores in the profile test where $z_i \\in \\{0,1\\}$ represents whether the $i$-th CPU core is busy ($z_i = 1$) or not ($z_i = 0$). We are interested to test primary observed data $Y_i$ by using auxiliary data $\\myb[X]$s to detect the prospective error in $Y$. According to the thermal superposition theorem~\\cite{mehdi2020dynamic}, \n\\begin{equation}\\label{eq:chap4_11}\n\\myb[X]_{Z} = \\sum_{i=1}^n z_i \\times (\\myb[Y]_i -\\myb[Y]_0) + \\myb[Y]_0\n\\end{equation}\n\nSuppose that the available tests for a hypothetical 3-core CPU are $\\myb[Y]_i$ for $i \\in [1,3]$ and the auxiliary test $\\myb[X]_{\\overline{101}}$. Hence, $\\myb[X]_{\\overline{101}} = \\myb[Y]_1 +  \\myb[Y]_3 - \\myb[Y]_0$. In the idle condition where there is no errors in the data, the equation much hold. Let's suppose there is an error data in one of them. We are interested to detect or correct it. By adding an error term $\\epsilon$ to the equation, we have $\\myb[X]_{\\overline{101}} = \\myb[Y]_1 + \\myb[Y]_3 - \\myb[Y]_0 + \\epsilon_{1,3}$. If there is no error in data, one can assume that ${\\epsilon_{1,3} = [0, 0, 0]^T}$. If there is an error in one of the tests, it can be detectable but not correctable. Now suppose that there is another test $\\myb[X]_{\\overline{011}}$ available, hence $\\myb[X]_{\\overline{011}} = \\myb[Y]_2 + \\myb[Y]_3 - \\myb[Y]_0 + \\epsilon_{2,3}$.\n\nNow in this case, it is possible to detect not only the presence of error but also the location of error in the profiles. In our example, \n\n\\textbf{I)} if $\\epsilon_{1,3} = \\epsilon_{2,3} = 0 $, there is no single error in the steady state values of any of $\\myb[Y_1]$, $\\myb[Y]_2$ or $\\myb[Y]_3$.\n\n\\textbf{II)} if $\\epsilon_{1,3} = 0 $ but $\\epsilon_{2,3} \\neq 0$, there is error in either $\\myb[Y]_2$ or $\\myb[X]_{\\overline{011}}$. \n\n\\textbf{III)} if $\\epsilon_{1,3} \\neq 0  $ but $\\epsilon_{2,3} = 0$, there is error in either $\\myb[X]_{\\overline{101}}$ or $\\myb[Y]_1$.\n\n\\textbf{IV)} if $\\epsilon_{1,3}\\neq 0$ and also $\\epsilon_{2,3} \\neq 0$, there is error in the steady state values of $\\myb[Y]_3$.\n\n\n\nIn order to detect and correct the error, we design a test in an n-hypercube format. Each corner of the hypercube represents one measurement setting $Z$, and there is only a one-bit difference in the $Z$ values of two neighboring corners. In this way, it is possible to examine the correctness of each test with its neighbors. It is also possible to spot the error with a sufficient number of tests. \n\n We explain the procedure by making an example. Fig.~\\ref{fig:chap4_err}a illustrates the auxiliary test $\\myb[X]_{\\overline{011}}$. The blue side shows the verification of the $\\myb[Y]_1$ and $\\myb[Y]_2$. If all the data are consistent, one can conclude that there is no single error in $\\myb[Y]_1$ and $\\myb[Y]_2$. If there is an error in one data an additional test is needed that we will explain later.\n\n\\begin{figure}[ht]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.32\\textwidth]{Chapter4\/5-Analysis\/figs\/error1.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.32\\textwidth]{Chapter4\/5-Analysis\/figs\/error2.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.32\\textwidth]{Chapter4\/5-Analysis\/figs\/error3.pdf}}\n\\caption{a) Auxiliary test for detect one error in either $\\myb[Y]_1$ or $\\myb[Y]_2$. b) Auxiliary test for detect one error in either $\\myb[Y]_1$ or $\\myb[Y]_3$.\nc) Second-tier testing of auxiliary tests.}\n\\label{fig:chap4_err}\n\\end{figure}\n\nThe green side in Fig.~\\ref{fig:chap4_err}b shows the verification for $Y_1$ and $Y_3$. If there is no single error, the data on this side is also consistent. \n\n\nNow suppose that both sides are inconsistent, then it is easy to say that  the data of $\\myb[Y]_1$ is faulty because we assume that there is only one error in observation profiles and the edge of $(\\myb[Y]_0,\\myb[Y]_1)$ is the intersection of both sides. Since $\\myb[Y]_0$ is zero base and not faulty, $\\myb[Y]_1$ is noisy. \n\nWe now discuss the case III where  $\\epsilon_{1,3} \\neq 0  $ and $\\epsilon_{2,3} = 0$. To locate the error in the data, we use the auxiliary test that make a side with two of suspicious tests (the yellow side in Fig.~\\ref{fig:chap4_err}c). In this case, we consider $\\myb[X]_{\\overline{111}}$ that can make a side with $\\myb[X]_{\\overline{001}}$, $\\myb[X]_{\\overline{011}}$ and $\\myb[X]_{\\overline{101}}$. If the data of this side is consistent, the error will belong to $\\myb[Y]_1$, otherwise to the test $\\myb[X]_{\\overline{101}}$. \n\n\n\nBy employing the same technique, it is easily possible to extend the proposed method for two errors in the tests by adding another tier to test the auxiliary tests with each other. Additionally, it is possible to conclude that there exists a single error or two errors in the data. It is important to mention that in this chapter, it is assumed that errors in the two tests cannot conceal each other. \n\n\n\n\n\n\n\n\n\\section{Accuracy Enhancement}\nIn this section, we extend our proposed scheme to improve accuracy using additional steady-state data from different core settings and extra data observed at different frequencies.\n\\subsection{Use of Steady-State Data Ensembles}\nWe discuss how to use the ensemble of extra observations at one frequency level to obtain a more accurate thermal observation profile. As discussed in the previous section, our method requires $n+1$ observed steady-state temperature profiles for a $n$-core system to construct the matrix $\\myb[Y]$: one profile when all cores are idle and $n$ profiles when each of CPU core is only fully-utilized. \nNow, we extend our analysis to answer the following questions:\n\\begin{itemize}\n    \\item Is it possible to collect different CPU core usage settings (other than the aforementioned $n+1$ observed data) to construct $\\myb[Y]$?\n    \\item Is it possible to have a more precise $\\myb[Y]$ if there exist multiple instantiations from an identical setting and use all of them?\n    \\item Is it possible to construct $\\myb[Y]$ with more than $n+1$ steady-state data?\n\\end{itemize}\n\nWe are interested in generalizing the construction of $\\myb[Y]$ to include any possible thermal traces of fully-utilized CPU core combinations and the ensemble of more thermal traces.\nIt is noteworthy that if there is no noise in the observed data and also data is not quantized, no extra data or multiple instantiations are needed due to superposition law in Eq.~\\ref{eq:chap4_11}.\n\nNow, suppose we have $m$ profiles such that $n \\leq m \\leq 2^n-1$ and each profiling was done under a different CPU core setting $Z_i$. Based on that, we can construct the predicate matrix $[\\myb[D]]_{m \\times n} = [\\myb[Z]_1 \\myb[Z]_2 \\dots \\myb[Z]_m]^T$ from the auxiliary profiles in $\\myb[U] = [\\myb[X]_{z_1} X_{z_2} \\dots X_{z_m}]^T$. The matrix $\\myb[Y]$ is then generated as follows\n\n\\begin{equation}\n    \\myb[Y] = ((\\myb[D] \\times \\myb[D]^{T})^{-1} \\times \\myb[D] \\times \\myb[U]^T)^{-1}.\n\\label{eq:chap4_ensemble}\n\\end{equation}\n\n\nBecause of the inversion in the equation, the equation works when the number of the profiles is more than the number of CPU cores. It also works when there exist enough orthogonal profiles, meaning that there is at least one profile for each CPU core where the core is idle. The matrix $\\Tilde{\\myb[A]}$ is then calculated as explained in Sec.~\\ref{sec:chap4_estimate}.\n\\subsection{Use of Multi-Frequency Data Ensembles}\n\\label{sec:chap4_multi_frequency_ensembles}\nWe now discuss how to obtain a more accurate $\\myb[A]$ by using additional data collected at multiple frequency levels. As explained in Sec.~\\ref{sec:chap4_tempMdl}, the thermal parameter $\\myb[A]$ is dependent on the semiconductor technology and remains invariant against frequency changes. Therefore, it would be logical to assume that having observed temperature profiles from different frequency levels would lead to an identical $\\myb[A]$. However, due to the term $\\gamma$ in Eq.~\\ref{eq:chap4_6}, $\\Tilde{\\myb[A]}$ is proportional to the power consumption and the clock frequency of CPU cores. Based on this, we extend our scheme to answer the following questions:\n\n\\begin{itemize}\n    \\item Is it possible to have an identical $\\myb[A]$ from different $Y$s which are collected at different frequency levels?\n    \\item Is it possible to estimate the power consumption with respect to different frequency levels?\n\\end{itemize}\n\nTo answer these questions, we propose a thermal parameter estimation approach based on multi-frequency data ensembles. \nLet $\\myb[Y]^i$ denote the temperature profile matrix constructed from the data when the CPU frequency is $f_i$, and $\\gamma_i$ denote its power effect on temperature at the frequency $f_i$. Unlike the method presented in Sec.~\\ref{sec:chap4_estimate} which estimates the parameters in $\\Tilde{\\myb[A]}$, we will calculate a $\\Tilde{\\myb[A]_1}$, which is the base at $f_1$. Additionally, we consider $\\gamma_1 = 1$  and estimate $\\frac{\\gamma_i}{\\gamma_1}, \\forall i >1$. In such case, tracking the values of $\\gamma$ over different frequency levels gives the power consumption of CPU cores proportional to the $\\gamma_1$ of the base frequency level $f_1$, and all $\\myb[Y]$s share the same thermal parameter $\\myb[A]$. Using the same procedure as in Sec.~\\ref{sec:chap4_trans} allows estimating the value of $\\gamma_1$;  hence, all $\\gamma$s can be estimated. It is noteworthy that, even in this case, the actual value of the power consumption cannot be computed, but the relative power consumption at different frequency levels can be obtained. Therefore, one can see the effect of power consumption embedded in the temperature profiles.\n\nWe change the cost model of Eq.~\\ref{eq:chap4_cost} to \n\n\n\\begin{equation}\n\\operatorname*{argmin}_{\\substack{\\text{$a_{j}$} \\\\ \\text{$ j \\in [1,r]$} \\\\ \\text{$\\frac{\\gamma_i}{\\gamma_1}$} \\\\  \\text{$i \\in [1,|f|]$}}}  \\sum_{1}^{|f|} ||\\Tilde{\\myb[A]_1}^{-1} - \\frac{\\gamma_i}{\\gamma_1}\\myb[Y]^i ||^2_F\n\\label{eq:chap4_mfreqs}\n\\end{equation}\n\nThe problem is estimating the values of $\\Tilde{\\myb[A]}_1$ and also $\\frac{\\gamma_i}{\\gamma_1}$. One can expect that $f_i > f_j$ leads to $\\gamma_i < \\gamma_j$ due to dynamic power increase and this can be considered as a constraint in the implementation. \n\\section{Evaluation}\n This section describes the experimental evaluation of our scheme. We explain our implementation on a real embedded platform. We show the results of our proposed method to find the thermal parameters from steady-state temperature data. We also explore the effect of our approaches in the error correction of estimated thermal parameters. Furthermore, we validate our method of finding the location of CPU core on the floorplan. We also show the results of our parameter-based power estimation model and compare it with data collected from on-chip power sensors. \n\n\n\\subsection{Platform} \nWe performed our experiments on an ODroid-XU4 development board \\cite{ODROIDXU4} equipped with a Samsung Exynos5422 SoC based on the ARM big.LITTLE heterogeneous computing architecture. The Exynos CPU package contains two different quad-core CPU clusters of little Cortex-A7 and big Cortex-A15 cores. Built-in temperature sensors with a sampling rate of 10 Hz and precision of 1\\textdegree~C are available for each big CPU core as well as the GPU to measure the operating temperature\\footnote{There is no temperature sensor for little cores since the power consumption and heat dissipation of the little cluster is substantially lower than that of the big cluster.}. The DTM throttles the frequency of the entire big CPU cluster to 900 MHz when one of its cores exceeds the hardware-defined maximum temperature threshold of 95\\textdegree~C. There is no active or passive cooling mechanism enabled on the CPU. The big CPU cluster frequency can be dynamically adjusted within the range of $[0.2, 2.0]$ GHz. However, for each experiment, it was pinned at a fixed frequency in the range of $[0.7,1.4]$ GHz to avoid thermal violations that occur when all CPU cores run fully-utilized beyond 1.4 GHz. Moreover, as the frequency increases, the data incurs a higher magnitude noise compared to the noise observed when operating at lower frequencies. It is worth noting that frequency changes between tests affect all homogeneous cores on the CPU, ensuring that each core in the cluster maintains the same frequency. Also, during each experiment, ambient temperature is regulated at 21\\textdegree~C.\n\\subsection{Thermal Parameter Estimation with Steady-State Data}\n\n\\subsubsection{Idle Steady-State Temperature}\nFirst, we measure the temperature of CPU cores at different frequencies and determine the steady state temperature when all big CPU cores are idle.\nAs we show in Fig.~\\ref{fig:chap4_off}, the temperatures of the big CPU cores during the idle state changes with CPU frequency for two possible reasons: \ni) some parts of big CPU package such as CCI, big CPU controlling unit, cache memory and its peripherals still operate at their designated frequencies and cause heat dissipation, and ii) other IPs on the SoC may operate with big CPU frequency. This temperature increase can also be caused by chip leakage power-induced heat dissipation.\n\n\\subsubsection{Dynamic Temperature} Fig.~\\ref{fig:chap4_temp_increase} depicts the relative temperature increase of each CPU core at different frequency levels when that core is fully-utilized. There is a slight difference in the temperature increase between the CPU cores despite identical workloads and micro-architectures. Although there is noise related to the precision of on-chip thermal sensors, this difference is mainly due to a variation in the location of those sensors on the CPU cores.\n\n\n\\begin{figure}[t]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/static.pdf}\\label{fig:chap4_off}}\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/dynamic.pdf}\\label{fig:chap4_temp_increase}}\n\\caption{(a) Idle steady-state temperature of each core at different frequency levels. (b) Temperature increase of CPU cores due to computation.}\n\\label{fig:chap4_obsrv_temp}\n\\end{figure}\n\n\n\n\n\n\n\\subsection{Noise Detection and Steady-State Data Ensembles}\n\n\n\n\\begin{table}[t]\n\\centering\n \\caption{Descriptions of steady-state traces on big cores.}\n  \\label{tab:chap4_spec}\n    \\begin{tabular}{ |p{2.1cm}|p{4.2cm}|p{4.5cm}|p{2.0cm}|}\n    \\hline\n     Case name & Descriptions & Traces (decimal) & \\# of traces\\\\\n    \\hline\n   \\begin{footnotesize} $CA1$ \\end{footnotesize}& \\begin{footnotesize}one-core experiments \\end{footnotesize}&\\begin{tiny}  ${Z_1, Z_2, Z_4, Z_8}$ \\end{tiny} & 4\\\\ \\hline\n\n   \\begin{footnotesize} $CA3$ \\end{footnotesize}& \\begin{footnotesize}three-core experiments \\end{footnotesize}&\\begin{tiny}  ${Z_7, Z_{11}, Z_{13}, Z_{14}}$ \\end{tiny} & 4\\\\ \\hline\n   \n   \\begin{footnotesize} $CA2$ \\end{footnotesize}& \\begin{footnotesize}two-core experiments \\end{footnotesize}&\\begin{tiny}  ${Z_3, Z_5, Z_6, Z_9, Z_{10}, Z_{12}}$ \\end{tiny} & 6\\\\ \\hline\n\n   \\begin{footnotesize} $CA1,3$ \\end{footnotesize}& \\begin{footnotesize}one-core \\& three-core experiments \\end{footnotesize}&\\begin{tiny}  ${Z_1, Z_2, Z_4, Z_7, Z_{8}, Z_{11}, Z_{13}, Z_{14}}$ \\end{tiny} & 8\\\\ \\hline\n   \n   \\begin{footnotesize} $CA1,2$ \\end{footnotesize}& \\begin{footnotesize}one-core \\& two-core experiments \\end{footnotesize}&\\begin{tiny}  $Z_1, Z_2, Z_3, Z_4, Z_5, Z_6, Z_8, Z_9, Z_{10}, Z_{12}$ \\end{tiny} & 10\\\\ \\hline\n   \n   \\begin{footnotesize} $CA1,2,3,4$ \\end{footnotesize}& \\begin{footnotesize}all experiments \\end{footnotesize}&\\begin{tiny}  ${Z_1, Z_2, \\dots, Z_{15}}$ \\end{tiny} & 15\\\\ \\hline\n  \n    \\hline\n    \\end{tabular}\n    \\label{tab:chap4_cases}\n\\end{table}\n\n\n\\begin{figure}[t]\n\\centering\n\\subfloat [][Core 1]{\\includegraphics[width=0.45\\textwidth]{Chapter4\/8-Evaluation\/figs\/error_1.4_c1.pdf}}\n\\subfloat [][Core 2]{\\includegraphics[width=0.45\\textwidth]{Chapter4\/8-Evaluation\/figs\/error_1.4_c2.pdf}}\\\\\n\\subfloat [][Core 3]{\\includegraphics[width=0.45\\textwidth]{Chapter4\/8-Evaluation\/figs\/error_1.4_c3.pdf}}\n\\subfloat [][Core 4]{\\includegraphics[width=0.45\\textwidth]{Chapter4\/8-Evaluation\/figs\/error_1.4_c4.pdf}}\n\\caption{The error of steady-state temperature of CPU cores by using different cases in construction of $Y$.}\n\\label{fig:chap4_error}\n\\end{figure}\nWe evaluate our scheme in improving the accuracy of thermal parameter estimation by using the ensemble of steady-state profiles from multiple frequency levels. We apply the superposition theorem as in Eq.~\\ref{eq:chap4_11} to estimate the steady-state temperature of CPU cores when different subsets of them are fully-utilized.\nWe use the data collected from a subset of  all possible combinations of CPU cores at the fixed clock frequency of 1.4~GHz.\\footnote{Note that the total number of possible core combinations is $2^4$ in a quad-core system and the total number of selecting a subset of the combinations is $2^{2^4}$.} \nThe details on the subsets used to construct the matrix $\\myb[Y]$ are given in Table~\\ref{tab:chap4_cases}. Some cases contain the least number of orthogonal profiles (i.e., 4 profiles) while others contain more profiles. The case $CA1,2,3,4$ includes all profiles to construct the matrix $\\myb[Y]$. The name of each case indicates the number of fully-utilized CPU cores. For instance, $CA1,2$ includes four profiles of one CPU core and six profiles of two cores. \n\nFig.~\\ref{fig:chap4_error} depicts the error in steady-state temperature of each CPU core under different utilization scenarios ($\\myb[Y]$ from Eq.~\\ref{eq:chap4_ensemble}).\nAs shown in the figure, using more profiles helps reduce the effect of noise in  constructing $\\myb[Y]$. The proposed anomaly detection mechanism presented in Sec.~\\ref{sec:chap4_anomaly} was applied to the profiles and excluded the outliers from being considered for $\\myb[Y]$. For instance, applying this mechanism detected an anomaly in the profile $Z_6$ by comparing it with the other profiles. Hence, while constructing $\\myb[Y]$, we were able to find out that the anomaly was not because of an error in the other profiles but rather due to the error in $Z_6$. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.64\\textwidth]{Chapter4\/8-Evaluation\/figs\/mse_allcases.pdf}\n\\caption{MSE of CPU cores from all setting in different cases.}\n\\label{fig:chap4_mse}\n\\end{figure}\n\nThe mean square errors (MSE) of the temperature model for all CPU cores under all different settings are shown in Fig.~\\ref{fig:chap4_mse}. The $x$ axis is sorted in ascending order in terms of the number of profiles used during the construction of $\\myb[Y]$. Some spikes in the results, e.g., the yellow line at $CA1,2$, are due to that  additional traces contained noisy data. However, the error generally decreases as more profiles are used for the construction of $\\myb[Y]$. This trend indicates that our proposed approach to considering data ensembles can reduce the negative impact of noisy data and improve the accuracy of thermal parameter estimation. \n\n\n\\subsection{Floorplan Estimation}\nWe validate our proposed floorplan estimation method on the Exynos5422 SoC. The temperature increase of each CPU core when only one CPU core is fully-utilized at a time is depicted in Fig.~\\ref{fig:chap4_full_oneByone}.\n\\begin{figure}[t]\n\\centering\n\\subfloat [][core 1]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/increase_core1.pdf}}\n\\subfloat [][core 2]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/increase_core2.pdf}}\\\\\n\\subfloat [][core 3]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/increase_core3.pdf}}\n\\subfloat [][core 4]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/increase_core4.pdf}}\n\\caption{Temperature increase when a)core 1 2)core 2 3)core 3 4)core 4 operates fully-utilized }\n\\label{fig:chap4_full_oneByone}\n\\end{figure}\n\nWe draw the proposed adjacency graph according to data of $1.4$ GHz. Figures~\\ref{fig:chap4_exp_graphs}a-c show the steps to construct the final adjacency graph for the Exynos5422. As we observe in Fig.~\\ref{fig:chap4_exp_graphs}d, the cores $C_3$ and $C_4$ have greater spatial proximity than cores $C_3$ and $C_2$. Although the physical layout of the cores on the CPU package is bi-symmetrical, the primary reason for the asymmetrical layout shown in Fig.~\\ref{fig:chap4_exp_graphs}d is that the location of the built-in temperature sensors on each processor may vary between cores. Additionally the L2 cache memory and peripheral controllers may have an effect on the modeled spatial proximity between the core pairs of ${C_1, C_2}$ and ${C_3,C_4}$ .\n\nUsing the same approach and GPU temperature data, we are also able to locate the embedded GPU by profiling the heat conduction between each CPU cores and the GPU. Fig.~\\ref{fig:chap4_exp_graphs}d depicts the estimated location of the embedded GPU in the Exynos5422. The floorplan estimation is validated with the data reported in~\\cite{7904613}\\footnote{The CPU core labeling in ~\\cite{7904613} is different from the labels in the driver and it is verified with infra-red imaging  captured  by an FLIR A325sc  IR  camera~\\cite{flir}.}.\n\\begin{figure}[t]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.18\\textwidth]{Chapter4\/8-Evaluation\/figs\/exp_graph1.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.2\\textwidth]{Chapter4\/8-Evaluation\/figs\/exp_graph2.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.2\\textwidth]{Chapter4\/8-Evaluation\/figs\/exp_graph3.pdf}}\n\\subfloat [][core 1]{\\includegraphics[width=0.2\\textwidth]{Chapter4\/8-Evaluation\/figs\/exp_gpu.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.22\\textwidth]{Chapter4\/8-Evaluation\/figs\/floorplan.pdf}}\n\\caption{Floorplan estimation of Exynos5422 based on data of 1.4 GHz. (a) The fully-connected graph from the temperature increase data, (b) Graph reduction stage (c) The CPU affinity graph, (d) Estimation of GPU location relative to CPU location, and (e) The actual Exynos5422 floorplan~\\cite{7904613}. }\n\\label{fig:chap4_exp_graphs}\n\\end{figure}\n\n\nWe construct the template of the matrix $\\myb[A]$ to be compatible with the estimated floorplan. The matrix $\\Tilde{\\myb[A]}$ at 1.4 GHz is then computed by using the steady-state data of $CA1,2,3,4$ at different frequency levels:\n\n\\[\n\\Tilde{\\myb[A]} = \\begin{bmatrix}\n0.2961 & -0.1324 & 0 & -0.1194\\\\ \n-0.1324 &  0.3017 &-0.1579 & 0\\\\ \n0 & -0.1579 & 0.3088 & -0.1269\\\\ \n-0.1194 & 0 & -0.1269 & 0.2798\\\\ \n\\end{bmatrix}.\n\\]\n\n\n\\subsection{Relative Power Estimation}\n\\begin{figure}[t]\n\\centering\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/actual_power_freq.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/static_power_temperature.pdf}}\\\\\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/power_c1.pdf}}\n\\subfloat [][]{\\includegraphics[width=0.49\\textwidth]{Chapter4\/8-Evaluation\/figs\/power_c1234.pdf}}\n\\caption{Power data of CPU cores in Exynos5422. (a) Total power of the big cluster with built-in sensors, (b) The leakage power data; the relative power estimates with different frequency ranges used, (c) Comparison between the estimated relative power consumption and the normalized actual power data from built-in power sensors for $CA1$, and (d) Comparison for $CA1,2,3,4$.}\n\\label{fig:chap4_power_est}\n\\end{figure}\n\n\nThe relative power consumption of each CPU core can be estimated while estimating $\\Tilde{\\myb[A]}$, as explained in Sec.~\\ref{sec:chap4_multi_frequency_ensembles}. \nFig.~\\ref{fig:chap4_power_est} illustrates the estimated power consumption. The results are obtained using profiles from different frequency ranges. As depicted in the figure, the estimated relative power closely follows the actual data collected from the built-in power sensors of the XU3 board that is equipped with the same Exynos5422 SoC.\n\n\\subsection{Temperature Prediction}\nWe estimate the parameter $\\gamma_1$ as the base parameter for the clock frequency 1.4 GHz in our experiments. Based on this, the absolute value of other $\\gamma$s can be determined. As discussed in Sec.~\\ref{sec:chap4_trans}, we use the transient-state trace when all CPU cores are cooling down. \n\nAfter estimating the values of $\\gamma$s, we apply our model to predict the operating temperature of CPU cores in two different scenarios. Figures~\\ref{fig:chap4_transData}a-d  show the CPU temperatures when only two CPU cores are fully utilized and Figures~\\ref{fig:chap4_transData}e-h show that when all CPU cores are fully utilized. In each sub-figure, ``data'' means the CPU operating temperature measured from a real platform and ``mdl'' means the temperature predicted by our model. \nAs shown, there are some differences in transient-state temperature values, especially at the earlier part of the experiments. The difference is relatively small when the frequency is low. \nOn the other hand, in steady state, the temperature predicted by the model is very close to the real one and the difference is less than 1.25\\textdegree~C in all cases. Since the steady-state temperature is the metric to test the thermal safety of the system, we expect that our proposed scheme can be effectively used in the thermal-aware design of COTS-based mixed-criticality systems.\n\n\\begin{figure}[t]\n\\centering\n\\subfloat [][core 1]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core1_1010.pdf}}\n\\subfloat [][core 2]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core2_1010.pdf}}\\\\\n\\subfloat [][core 3]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core3_1010.pdf}}\n\\subfloat [][core 4]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core4_1010.pdf}}\\\\\n\\subfloat [][core 1]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core1_1111.pdf}}\n\\subfloat [][core 2]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core2_1111.pdf}}\\\\\n\\subfloat [][core 3]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core3_1111.pdf}}\n\\subfloat [][core 4]{\\includegraphics[width=0.4\\textwidth]{Chapter4\/8-Evaluation\/figs\/trans_core4_1111.pdf}}\n\\caption{The CPU temperature from sensor data and the model when a-d) two cores are fully-utilized $Z_{10}$ e-h) all cores are fully-utilized $Z_{15}$. }\n\\label{fig:chap4_transData}\n\\end{figure}\n\\subsection{Case study on intel processor}\n\\section{Summary} \n\nIn this chapter, we proposed a novel, rapid, and accurate scheme to estimate the thermal parameters of COTS multi-core processors for real-time mixed-criticality systems. By decomposing our estimation scheme into steady-state and transient-state stages, we substantially reduce the number of transient-state profiles needed to estimate the system's thermal parameters. We presented methods to improve the accuracy of our scheme by utilizing additional temperature profiles in the parameter estimation. Our proposed scheme is fast to converge and has a low computational cost for the prediction of chip operating temperature. Hence, it can be even used in an event-driven manner, e.g., at the arrival and the departure of each job of periodic real-time tasks, with negligible memory and computational overhead. We derived the thermal characteristics of a multi-core processor which remain unchanged across different frequency levels. We also showed the effectiveness of our scheme in extracting the relative power consumption information from  temperature profiles. \n\n\n\n\n\n\n\n\\chapter{Conclusions}\nThis dissertation shows that the timing predictability of real-time tasks in dynamic thermal conditions is achievable on multi-core GPU-integrated SoC platforms by designing a novel thermal-aware system framework with the support of analytical timing and thermal models.\n\nWe discussed challenges arising from the thermal interference on heterogeneous multi-core SoC platforms. In Chapter 2, we focused on bounding the temperature increase due to heat conduction among CPU cores as well as integrated computational accelerators like GPUs. We proposed a thermal-aware CPU-GPU framework to handle both real-time CPU-only and GPU-using tasks by designing thermal-aware servers that bound the maximum operating temperature of SoCs with the assurance of performance predictability. In Chapter 3, we introduced a novel multi-core mixed-criticality scheduling framework with ambient temperature awareness to bound heat generation at each criticality level and to trigger the criticality mode change by ambient temperature changes. We tackled the thermal-aware real-time scheduling challenges in dynamic environment conditions by decomposing the analysis into thermal schedulability and timing schedulability. We introduced the notion of \\textit{idle thermal servers} that allow bounding the maximum operating temperature in MCS in way that the temperature increase due to heat dissipation during an inactive time of idle servers is not less than that of running preemptive servers under any task execution pattern. In Chapter 4, we presented our work on the estimation of processor thermal parameters to obtain a precise thermal model without using special measurement instruments or access to proprietary information. We showed that even a small number of temperature traces from on-chip thermal sensors is sufficient to achieve an accurate thermal parameter estimation. Not only can our framework detect anomalies in thermal profiles, but increases accuracy using  multi-frequency data ensembles and takes into account extra observations made at one or a subset of frequency levels. \n\n\\section{Future Work}\nThere are possible extensions and improvements to the work presented in this dissertation. The following suggests several future research directions that can be built upon our work.\n\\subsection{Timing Assurance and Thermal Safety on SoC Platforms}\nBy considering task allocation to CPU and GPU cores, the schedulability of real-time tasks with a given temperature constraint can be further increased.\nWe believe that it is possible to develop an offline mapping algorithm to increase the schedulability of a taskset by determining whether a task executes on the CPU or the GPU according to the system parameters and its execution timing model. \n\nIn our task model, tasks may or may not include concurrent segment which executes on GPU. When there is a GPU segment request, the corresponding task issues a GPU access request. However, it is possible to design an online algorithm to determine if concurrent task segment executes on GPU or CPU. In this way, some jobs of a task can execute only on CPU while the rest executing on the integrated GPU. We believe this technique can reduce the maximum peak temperature of embedded systems and also increase the timing schedulability of tasksets. \n\nThe miscellaneous operations contains a single data transfer to\/from GPUs. However, in practice, for improving performance or overcoming the resource limit of GPUs, data can be divided into several chunks and transmitted to GPUs so that data copy and kernel execution can be overlapped. We believe that updating the GPU execution model and revisiting some of our proposed protocols will enable this feature and the practical applicability of our proposed framework will also increase.\n\nThe thermal behavior of GPU-using tasks may depend significantly on the type of resources used by their kernels. For instance, a GPU kernel frequently accessing local memory may generate much less heat than those using global memory or being computationally-intensive. A potential direction to this issue would be developing an analytical model based on the thermal footprint of real-time applications to bound the maximum peak temperature with acceptable margins. \n\n\\subsection{Dynamic Ambient Temperature-Aware Framework}\nIn our proposed thermal-schedulability analysis for multiple real-time tasks on multi-core SoCs, identical idle servers are assumed for all CPU cores at each criticality level. This assumption can affect the thermal schedulability of a real-time taskset. We believe that extending our analysis to allow different idle server settings per CPU core can improve the range of ambient temperature supported by mixed-criticality levels. Besides, developing our analysis to capture the effect of cooling packages and forced heat convection can be a challenging issue.\n\nIn our proposed framework, a  criticality  mode  change  is  triggered  by  ambient  temperature which is different from the well-known Vestal model~\\cite{vestal2007preemptive}, which focuses on varying assurance of execution timing. It is intriguing to extend our framework such that changes in both ambient temperature and execution time of tasks trigger the criticality mode of MCS.\n\n\\subsection{Thermal Parameter Estimation Scheme}\nWe believe that our proposed scheme can be extended to capture the thermal footprint of different tasks on heterogeneous SoCs by using a minimum number of profiles. We also believe that the idea of  further development of the analysis to capture the effect of cooling packages and forced heat convection lead to the  estimation of operating temperature of SoCs in various CPU cooling conditions at runtime. Additionally, statistical approaches to construct the adjacency matrix of the location of CPU cores and IPs can be a potential intriguing extension.    \n\nThe robustness of our proposed estimation scheme against noise in the thermal profiles has not been discussed through mathematical analysis. To be more precise, although our proposed method estimates the thermal parameters of SoCs with acceptable margins, we did not quantify the degree of noise or errors in the estimation with respect to the amount of data used. Given that it is essential to assess the trustworthy of each thermal profile, we consider this as interesting future work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn this paper, we consider the following nonlinear Schr\\\"odinger equation (NLSE)\n\\begin{equation}\\label{NLSE}\n\t\\left\\{\n\t\\begin{aligned}\n\t\t&i \\partial_t \\psi(\\mathbf{x}, t) = -\\Delta \\psi(\\mathbf{x}, t) + V(\\mathbf{x}) \\psi(\\mathbf{x}, t) + f(|\\psi(\\mathbf{x}, t)|^2) \\psi(\\mathbf{x}, t), \\quad \\mathbf{x} \\in \\Omega, \\  t>0, \\\\\n\t\t&\\psi(\\mathbf{x}, 0) = \\psi_0(\\mathbf{x}), \\quad \\mathbf{x} \\in \\overline{\\Omega},\n\t\\end{aligned}\n\t\\right.\n\\end{equation}\nwhere $ t $ is time, $\\mathbf{x} = (x_1, \\cdots, x_d)^T \\in \\mathbb{R}^d \\  (d=1, 2, 3) $ is the spatial coordinate, $ \\psi:=\\psi(\\mathbf{x}, t) $ is a complex-valued wave function, and $ \\Omega = \\Pi_{i=1}^d (a_i, b_i) \\subset \\mathbb{R}^d $ is a bounded domain equipped with periodic boundary condition. Here, $ V = V(\\mathbf{x}):\\Omega \\rightarrow \\mathbb{R} $ is a \nreal-valued potential and $ f=f(\\rho):[0, \\infty) \\rightarrow \\mathbb{R} $ with $ \\rho=|\\psi|^2 $ being the density describes the nonlinear interaction. We assume that $ V \\in L^\\infty(\\Omega) $ and $ f(|z|^2)z:\\mathbb{C} \\rightarrow \\mathbb{C} $ is locally Lipschitz continuous, and thus both $ V $ and $ f $ may be of low regularity. The NLSE \\cref{NLSE} conserves the mass\n\\begin{equation}\n\tM(\\psi(\\cdot, t)) = \\int_{\\Omega} |\\psi(\\mathbf{x}, t)|^2 \\mathrm{d} \\mathbf{x} \\equiv M(\\psi_0), \\quad t \\geq 0,\n\\end{equation}\nand the energy\n\\begin{eqnarray}\n\tE(\\psi(\\cdot, t))\n\t&&= \\int_{\\Omega} \\left[ |\\nabla \\psi(\\mathbf{x}, t)|^2 + V(\\mathbf{x})|\\psi(\\mathbf{x}, t)|^2 + F(|\\psi(\\mathbf{x}, t)|^2) \\right] \\mathrm{d} \\mathbf{x} \\nonumber \\\\\n\t&&\\equiv E(\\psi_0), \\quad t \\geq 0,\n\\end{eqnarray}\nwhere the interaction energy density $F(\\rho)$ is given as\n\\begin{equation}\\label{eq:F}\n\tF(\\rho) = \\int_0^\\rho f(s) \\mathrm{d} s, \\quad \\rho \\geq 0.\n\\end{equation}\n\nWhen $ V(\\mathbf{x}) = |\\mathbf{x}|^2\/2 $ and $ f(\\rho) = \\rho $, the NLSE \\cref{NLSE} collapses to the nonlinear Schr\\\"odinger equation with harmonic potential and cubic nonlinearity (or smooth potential and nonlinearity) or the Gross-Pitaevskii equation (GPE), which has been widely adopted for modeling and simulation in quantum mechanics, nonlinear optics and Bose-Einstein condensation (BEC) \\cite{review_2013,ESY,NLS}. For the smooth NLSE with sufficiently smooth initial data $ \\psi_0 $, many accurate and efficient numerical methods have been proposed and analyzed in last two decades, including the finite difference method \\cite{FD,bao2013,review_2013,Ant}, the exponential wave integrator \\cite{bao2014,ExpInt,SymEWI}, the time-splitting method \\cite{bao2003JCP,BBD,lubich2008,schratz2016,review_2013,Ant,BCF}, the finite element method \\cite{FEM1,FEM2,FEM3,FEM4,henning2017}, etc. Recently, many works are done to analyze and design numerical methods for the cubic NLSE with low regularity initial data $ \\psi_0 $ and without potential (see \\cite{schratz2016,splitting_low_reg,LRI,LRI_sinum,LRI_error,LRI_general,LRI_fulldisc,LRI_sec} and references therein for other dispersive partial differential equations).\n\nArising from different physics applications, both $ V $ and $ f $ in \\cref{NLSE} may be of low regularity. Typical examples of the low regularity $ L^\\infty $-potential include, in many physical contexts, the square-well potential or step potential, which are discontinuous; in the study of BEC in different trapping shape, the power law potential $ V(\\mathbf{x}) = |\\mathbf{x}|^\\gamma \\ (\\gamma > 0) $ \\cite{poten_power_law,poten_power_law_2}, and in the analysis of Josephson effect and Anderson localization, some disorder potential \\cite{poten_Josephson,poten_anderson}. Low regularity nonlinearity such as $ f(\\rho) = \\rho^\\sigma \\ (\\sigma>0) $ or $ f(\\rho) = \\rho \\ln \\rho $ are considered in, e.g., the Schr\\\"{o}dinger-Poisson-X$ \\alpha $ model \\cite{bao2003,SPXalpha}, the Lee-Huang-Yang correction \\cite{LHY} which is adopted to model and simulate quantum droplets \\cite{QD1,QD2,QD3,QD4}, and the mean-field model for Bose-Fermi mixture \\cite{Had,Cai}.\n\n\nMost numerical methods for the cubic NLSE with smooth potential and nonlinearity can be extended straightforwardly to solve the NLSE \\cref{NLSE} with $ L^\\infty $-potential and\/or locally Lipschitz nonlinearity (different from the singular nonlinearity in \\cite{sinum2019,bao2019,bao2022,bao2022singular})). However, the performance of these methods are quite different from the smooth case and the error analysis of them is a very subtle and challenging question. For \\cref{NLSE} with power-type nonlinearity $ f(\\rho) = \\rho^\\sigma $ and sufficiently smooth potential, the Lie-Trotter time-splitting method is analyzed in \\cite{bao2023,choi2021,ignat2011} with reduced convergence order in $ L^2 $-norm when $ \\sigma<1\/2 $ and in $ H^1 $-norm when $ \\sigma<1 $. The analysis of \\cref{NLSE} with smooth nonlinearity and $ L^\\infty $-potential seems more challenging and the only known convergence result is the one obtained in \\cite{henning2017} for the Crank-Nicolson Galerkin scheme, where first order convergence in time and less than second order convergence in space in $ L^2 $-norm are shown under strong assumptions on the exact solution (among others $ \\partial_t \\psi \\in H^2 $), and a coupling condition between the time step size $ \\tau $ and the mesh size $ h $. Some low regularity integrators or resonance-based Fourier integrators are also proposed to reduce the regularity requirements on both $ V $ and $ \\psi $, while the regularity assumption on $ V $ is still stronger than $ H^1 $ \\cite{zhao2021,alama2022}, which still excludes the popular well potential and step potential widely adopted in physics literatures. The main difficulty comes from the low regularity of solution of the NLSE with $ L^\\infty $-potential and locally Lipschitz nonlinearity, where only $ H^2 $ well-posedness is guaranteed \\cite{kato1987,cazenave2003}, and the low regularity of the potential and the nonlinearity which causes order reduction in local truncation errors and prevent us from obtaining stability estimates in high order Sobolev spaces $ H^\\alpha\\ (\\alpha>d\/2) $ (see \\cite{bao2023,zhao2021,henning2017} for more detailed discussion). Besides, for the NLSE \\cref{NLSE} with purely $ L^\\infty $-potential, how to estimate the spatial discretization is also a challenging problem and it turns out that it is very subtle and challenging to estimate the classical methods including the finite difference method, the pseudospectral method and the finite element method \\cite{henning2017}.\n\nThe main aim of this paper is to establish optimal error bounds for the exponential wave integrator (EWI) applied to the NLSE with $ L^\\infty $-potential and\/or locally Lipschitz nonlinearity. For the semi-discretization in time by the first-order Gautschi-type EWI, we prove an optimal $ L^2 $-error bound at $ O(\\tau) $ with $\\tau>0$ being the time step size, and a uniform $ H^2 $-bound of the numerical solution under the assumption of $ H^2 $-solution of the NLSE. Then we further discretize the semi-discretization in space by using the Fourier spectral method (EWI-FS), and prove an optimal $ L^2 $-error bound at $ O(\\tau + h^2) $ without any coupling condition between $ \\tau $ and the mesh size $ h $. In addition, for $ V \\in W^{1, 4} $ and a little stronger regularity of the nonlinearity, optimal $ H^1 $-error bounds are obtained for the EWI and EWI-FS schemes. Furthermore, when $V$ is of low regularity but the nonlinearity is sufficiently smooth, we propose an extended Fourier pseudospectral method for spatial discretization, and we prove that it has the same error bound as the Fourier spectral method while its computational cost is similar to the standard Fourier pseudospectral method.\n\nOur error bounds greatly improve the previous results for the NLSE with low regularity potential and\/or nonlinearity. Compared with the result in \\cite{bao2023}, to obtain optimal error bounds, we reduce the differentiability requirement on the potential by two orders and on the nonlinearity by one order. Moreover, when $ V \\in L^\\infty $ and $ f $ is smooth as considered in \\cite{henning2017}, we improve the convergence order in $ L^2 $-norm to the optimal first order in time and the optimal second order in space, remove the coupling condition requirement between $ \\tau $ and $ h $ in \\cite{henning2017}, relax the regularity assumption on the exact solution such that it is theoretically guaranteed, and reduce the computational cost in practical implementation.\n\nHere we briefly explain why we can obtain the improved error bounds. In general, compared with the finite difference methods, time-splitting methods and EWI usually need lower regularity requirements on the exact solution to obtain the same order of convergence. Usually, time-splitting methods perform better than EWI in practical computation when the solution\nis smooth, which usually requires that the potential and nonlinearity as well as the initial data are all smooth. The main reason is due to that time-splitting methods are usually structure-preserving scheme, i.e., they preserve mass conservation, time-transverse invariant and dispersion relation in the discretized level \n\\cite{Ant,review_2013}. On the contrary, when the NLSE \\cref{NLSE} is with low regularity potential and\/or nonlinearity, and thus the solution is of low regularity, the first-order Gautschi-type EWI has two major advantages in obtaining optimal error bounds: (i) in obtaining local truncation errors, time-splitting methods need to apply the Laplacian $ \\Delta $ to the equation while the EWI only needs to apply $ \\partial_t $ to the equation, and thus\nthe EWI needs weaker regularity requirement on both potential and nonlinearity; and (ii) a smoothing operator is adopted in the EWI scheme to control the dispersion of  high frequencies and thus it helps to keep the numerical solution in $ H^2 $ at each time step, which makes it possible to obtain the stability estimates in high order Sobolev spaces, while it is a challenging and subtle task to establish $H^2$-bounds of the numerical solution by using the time-splitting methods.\n\nThe rest of the paper is organized as follows. In Section 2, we present a semi-discretization in time by the first-order Gautschi-type EWI and then a full discretization in space by the Fourier spectral\/pseudospectral methods.\nSections 3 and 4 are devoted to the error estimates of the semi-discretization scheme and the full-discretization scheme, respectively. Numerical results are reported in Section 5 to confirm the error estimates. Finally, some conclusions are drawn in Section 6. Throughout the paper, we adopt the standard Sobolev spaces as well as their corresponding norms, and denote by $ C $ a generic positive constant independent of the mesh size $ h $ and time step size $ \\tau $, and by $ C(\\alpha) $ a generic positive constant depending only on the parameter $ \\alpha $. The notation $ A \\lesssim B $ is used to represent that there exists a generic constant $ C>0 $, such that $ |A| \\leq CB $.\n\n\\section{The exponential wave integrator Fourier spectral method}\nIn this secton, we introduce an exponential wave integrator and its spatial discretization to solve the NLSE with low regularity potential and nonlinearity.\nFor simplicity of the presentation, we only carry out the analysis in 1D and take $ \\Omega = (a, b) $. Generalizations to 2D and 3D are straightforward. For $ m \\geq 1 $, we define the periodic Sobolev spaces as (see, e.g., \\cite{feng2022} for the equivalent definition)\n\\begin{equation*}\n\tH_\\text{per}^m(\\Omega) := \\{\\phi \\in H^m(\\Omega) : \\phi^{(k)}(a) = \\phi^{(k)}(b), \\ k=0, \\cdots, m-1\\}.\n\\end{equation*}\n\n\\subsection{Semi-discretization in time by an exponential wave integrator}\nChoose a time step size $ \\tau > 0 $ and denote time steps as $ t_n = n\\tau $ for $ n = 0, 1, \\cdots $. By Duhamel's formula, the exact solution of the NLSE \\cref{NLSE} is given as\n\\begin{eqnarray}\n\\psi(t_{n+1})=&&\\psi(t_{n}+\\tau)= e^{i \\tau \\Delta} \\psi(t_n)\\nonumber\\\\\n\t\t &&- i \\int_0^\\tau e^{i(\\tau - s)\\Delta} \\left [ V \\psi(t_n + s) + f(|\\psi(t_n + s)|^2)\\psi(t_n + s) \\right ] \\mathrm{d} s,\\quad n\\ge0, \\label{eq:Duhamel}\n\\end{eqnarray}\nwhere we abbreviate $\\psi(x,t)$ by $\\psi(t)$ for simplify of notations when there is\nno confusion.\nLet $ \\psin{n}:=\\psin{n}(x) $ be the approximation of $ \\psi(x, t_n) $ for $ n \\geq 0 $. Applying the approximation $ \\psi(t_n + s) \\approx \\psi(t_n) $ for the integrand in \\eqref{eq:Duhamel} and integrating out $ e^{i(\\tau - s)\\Delta} $ exactly, we get a semi-discretization in time by the first-order Gautschi-type EWI as\n\\begin{equation}\\label{EWI}\n\t\\begin{aligned}\n\t\t\\psin{n+1} &= \\Phi^\\tau(\\psin{n}) := e^{i \\tau \\Delta} \\psin{n} - i \\tau \\varphi_1(i \\tau \\Delta) \\left( V \\psin{n} + f(|\\psin{n}|^2)\\psin{n} \\right), \\quad n \\geq 0, \\\\\n\t\t\\psin{0} &= \\psi_0,\n\t\\end{aligned}\n\\end{equation}\nwhere $ \\varphi_1 $ is an entire function defined as\n\\begin{equation*}\n\t\\varphi_1(z) = \\frac{e^z - 1}{z}, \\qquad z \\in \\mathbb{C}.\n\\end{equation*}\nThe operator $ \\varphi_1(i \\tau \\Delta) $ is defined through its action in the Fourier space as\n\\begin{eqnarray}\n\t\\left (\\varphi_1(i \\tau \\Delta) v \\right )(x)\n\t&&= \\sum_{l \\in \\mathbb{Z}} \\varphi_1(- i \\tau \\mu_l^2 ) \\widehat{v}_l e^{i \\mu_l (x-a)}  \\nonumber\\\\\n\t&&= \\widehat{v}_0 + \\sum_{l \\in \\mathbb{Z} \\setminus \\{0\\}} \\frac{1-e^{-i\\tau\\mu_l^2}}{i \\tau \\mu_l^2} \\widehat{v}_l e^{i \\mu_l (x-a)},\n\t\\quad x \\in \\Omega, \\label{eq:phi1_def}\n\\end{eqnarray}\nwhere $\\mu_l = \\frac{2 \\pi l}{b-a}$ for $l\\in{\\mathbb Z}$, and $ \\widehat{v}_l \\ (l \\in \\mathbb{Z}) $ are the Fourier coefficients of the function $ v \\in L^2(\\Omega) $ defined as\n\\begin{equation}\\label{eq:FT}\n\t\\widehat{v}_l = \\frac{1}{b-a} \\int_a^b v(x) e^{-i \\mu_l(x-a)} \\mathrm{d} x, \\quad l \\in \\mathbb{Z}.\n\\end{equation}\nFrom \\eqref{eq:phi1_def}, noting that $ | 1-e^{-i\\theta} | \\leq 2 $ for $ \\theta \\in \\mathbb{R} $, we see that\n\\begin{equation}\n\t\\left | \\widehat{(\\varphi_1(i \\tau \\Delta) v)}_l \\right | \\leq\n\t\\left\\{\n\t\\begin{aligned}\n\t\t&\\frac{2}{\\tau} \\frac{|\\widehat{v}_l|}{\\mu_l^2}, && l \\in \\mathbb{Z} \\setminus \\{0\\}, \\\\\n\t\t&|\\widehat{v}_0|, && l = 0,\n\t\\end{aligned}\n\t\\right.\n\\end{equation}\nwhich implies $ \\varphi_1(i \\tau \\Delta) v \\in H^2_\\text{per}(\\Omega) $ for all $ v \\in L^2(\\Omega) $. Hence, $ \\Phi^\\tau $ is indeed a flow in $ H^2_\\text{per}(\\Omega) $ for any $ V \\in L^\\infty(\\Omega) $.\n\nIn fact, the introduction of the smoothing function $ \\varphi_1(i \\tau \\Delta) $ in\n\\eqref{EWI} is one of the major advantages of the EWI \\cref{EWI} over the time-splitting methods in terms of controlling the dispersion of high frequencies or resonance. With this smoothing function, one can show that the numerical solution\nis in $H^2$ at every time step. For comparison, based on the results in \\cite{bao2023}\nfor time-splitting methods applied to the NLSE with semi-sooth nonlinearity, the numerical solution of the semi-discretization is not in $ H^2 $ in general! The situation is even worse if there is purely $ L^\\infty $-potential.\n\n\n\\subsection{Full discretization by the Fourier spectral method}\nThen we further discretize the semi-discretization \\cref{EWI} in space by the Fourier spectral method to obtain a full-discretization scheme.\nChoose a mesh size $ h=(b-a)\/N $ with $ N $ being a positive integer and denote grid points as\n\\begin{equation*}\n\tx_j = a + jh, \\quad j = 0, 1, \\cdots, N.\n\\end{equation*}\nDefine the index sets\n\\begin{equation*}\n\t\\mathcal{T}_N = \\left\\{-\\frac{N}{2}, \\cdots, \\frac{N}{2}-1 \\right\\}, \\quad \\mathcal{T}_N^0 = \\{0, 1, \\cdots, N\\},\n\\end{equation*}\nand denote\n\\begin{align}\n\t&X_N = \\text{span}\\left\\{e^{i \\mu_l(x - a)}: l \\in \\mathcal{T}_N\\right\\},  \\\\\n\t&Y_N = \\left\\{ v=(v_0, v_1, \\cdots, v_N)^T \\in \\mathbb{C}^{N+1}: v_0 = v_N \\right\\}.\n\\end{align}\nLet $ P_N:L^2(\\Omega) \\rightarrow X_N $ be the standard $ L^2 $-projection onto $ X_N $ and $ I_N: Y_N  \\rightarrow X_N $ be the standard Fourier interpolation operator as\n\\begin{align}\n\t&(P_N u)(x) = \\sum_{l \\in \\mathcal{T}_N} \\widehat u_l e^{i \\mu_l(x - a)}, \\\\\n\t&(I_N v)(x) = \\sum_{l \\in \\mathcal{T}_N} \\widetilde v_l e^{i \\mu_l(x - a)}, \\quad x \\in \\overline{\\Omega} = [a, b],\n\\end{align}\nwhere $ u \\in L^2(\\Omega) $, $ v \\in Y_N $, $ \\widehat{u}_l \\ (l \\in \\mathbb{Z}) $ are the Fourier coefficients of $ u $ defined in \\cref{eq:FT} and $ \\widetilde v_l \\ (l \\in \\mathcal{T}_N) $ are the discrete Fourier transform coefficients defined as\n\\begin{equation}\n\t\\widetilde{v}_l = \\frac{1}{N} \\sum_{j=0}^{N-1} v_j e^{-i \\mu_l (x_j - a)}, \\quad l \\in \\mathcal{T}_N.\n\\end{equation}\nLet $ \\psihn{n}: = \\psihn{n}(x) $ be the approximation of $ \\psi(x, t_n) $ for $ n \\geq 0 $. Then an exponential wave integrator-Fourier spectral method (EWI-FS) for the NLSE  \\eqref{NLSE} is given as\n\\begin{equation}\\label{EWI-FS}\n\t\\begin{aligned}\n\t\t\\psihn{n+1} &= \\Phi_h^\\tau(\\psihn{n}) := e^{i \\tau \\Delta} \\psihn{n} - i \\tau \\varphi_1(i \\tau \\Delta) P_N \\left( V \\psihn{n} + f(|\\psihn{n}|^2)\\psihn{n} \\right), \\quad n \\geq 0, \\\\\n\t\t\\psihn{0} &= P_N \\psi_0.\n\t\\end{aligned}\n\\end{equation}\nNote that $ \\psihn{n} \\in X_N $ for $ n \\geq 0 $ and we have\n\\begin{equation}\\label{psihn_coef}\n\t\\begin{aligned}\n\t\t\\widehat{(\\psihn{n+1})}_l &= e^{-i \\tau \\mu_l^2} \\widehat{(\\psihn{n})}_l - i \\tau \\varphi_1(-i \\tau \\mu_l^2) \\left( \\widehat{(V \\psihn{n})}_l + \\widehat{G(\\psihn{n})}_l \\right), \\quad n \\geq 0, \\\\\n\t\t\\widehat{(\\psihn{0})}_l &= \\widehat{(\\psi_0)}_l, \\quad l \\in \\mathcal{T}_N,\n\t\\end{aligned}\n\\end{equation}\nwhere $ G(\\psihn{n})(x) = G(\\psihn{n}(x)) := f(|\\psihn{n}(x)|^2)\\psihn{n}(x) $ for $ x \\in \\Omega $. We remark here that the EWI-FS is usually implemented by the Fourier pseudospectral method (see, e.g., \\cite{bao2014,feng2022}) in practical computations. Of course, due to the low regularity of the potential and\/or nonlinearity, it is very hard to establish error bounds for the full-discretization by the Fourier pseudospectral method.\n\n\\subsection{Full discretization by an extended Fourier pseudospectral method}\nIn practice, the Fourier spectral method cannot be efficiently implemented. Here, we propose an extended Fourier pseudospectral method when the potential is of low regularity but the nonlinearity is sufficiently smooth, i.e., we adopt the Fourier spectral method to discretize the linear potential and use the Fourier pseudospectral method to discretize the nonlinearity. This full discretization has two advantages:\n(i) we can establish its optimal error bounds, and (ii) the computational cost\nof this discretization is similar to the standard Fourier pseudospectral method.\n\nLet $ \\psihhn{n}_j $ be the numerical approximation of $ \\psi(x_j, t_n) $ for $ j \\in \\mathcal{T}_N^0 $ and $ n \\geq 0 $, and denote $ \\psihhn{n}:=(\\psihhn{n}_0, \\psihhn{n}_1, \\cdots, \\psihhn{n}_N)^T \\in Y_N $. Then an exponential wave integrator-extended Fourier pseudospectral (EWI-EFP) method for the NLSE \\eqref{NLSE} reads\n\\begin{equation}\\label{EWI-EFP}\n\t\\begin{aligned}\n\t\t\\psihhn{n+1}_j\n\t\t&= \\sum_{l \\in \\mathcal{T}_N} e^{-i \\tau \\mu_l^2} \\widetilde{(\\psihhn{n})_l} e^{i \\mu_l(x_j-a)} \\\\\n\t\t&\\quad - i \\tau \\sum_{l \\in \\mathcal{T}_N} \\varphi_1(-i \\tau \\mu_l^2) \\left( \\widehat{\\left (V I_N \\psihhn{n} \\right )_l} + \\widetilde{G(\\psihhn{n})_l} \\right) e^{i \\mu_l(x_j-a)}, \\quad n \\geq 0, \\\\\n\t\t\\psihhn{0}_j &= \\psi_0(x_j), \\quad j \\in \\mathcal{T}_N^0,\n\t\\end{aligned}\n\\end{equation}\nwhere $ G(\\psihhn{n})_j = f(|\\psihhn{n}_j|^2)\\psihhn{n}_j $ for $ j \\in \\mathcal{T}_N^0 $. To compute the Fourier projection coefficients $ \\widehat{\\left (V I_N \\psihhn{n} \\right )_l} $, we use an extended FFT as shown below. Note that $ I_N \\psihhn{n} \\in X_N $ for all $ n \\geq 0 $, and thus we have\n\\begin{equation}\\label{poten_trunc}\n\tP_N(V I_N \\psihhn{n}) = P_N\\left (P_{2N}(V) I_N \\psihhn{n} \\right ), \\quad n \\geq 0.\n\\end{equation}\nMoreover, since $ P_{2N}(V) I_N \\psihhn{n} \\in X_{4N} $ and $ I_{4N} $ is an identity on $ X_{4N} $, we have\n\\begin{equation*}\n\t P_{2N}(V) I_N \\psihhn{n} = I_{4N}\\left (P_{2N}(V) I_N \\psihhn{n} \\right ), \\quad n \\geq 0,\n\\end{equation*}\nwhich plugged into \\cref{poten_trunc} yields\n\\begin{equation}\\label{eq:spectral_discretization_of_potential}\n\tP_N\\left (V I_N \\psihhn{n}\\right ) = P_N I_{4N}\\left (P_{2N}(V) I_N \\psihhn{n} \\right ), \\quad n \\geq 0,\n\\end{equation}\nwhere $ P_{2N}(V) $ can be precomputed numerically or analytically, and thus the right hand side of \\cref{eq:spectral_discretization_of_potential} can be computed exactly and efficiently using the extended FFT: using FFT for $ P_{2N}(V) I_N \\psihhn{n} $ with length $ 4N $ instead of $ N $. As a result, the memory cost is $ O(4N) $ and the computational cost per time step is $ O(4N\\log(4N)) $.\nNote that $ \\psihhn{n} \\ (n \\geq 0) $ obtained by \\cref{EWI-EFP} satisfies\n\\begin{equation}\\label{EWI-FSP_equation}\n\t\\begin{aligned}\n\t\tI_N \\psihhn{n+1} &= e^{i \\tau \\Delta} I_N \\psihhn{n} - i \\tau \\varphi_1(i \\tau \\Delta) \\left( P_N \\left( V I_N \\psihhn{n} \\right) + I_N G (\\psihhn{n}) \\right), \\\\\n\t\tI_N \\psihhn{0} &= I_N \\psi_0, \\qquad n \\geq 0.\n\t\\end{aligned}\n\\end{equation}\n\n\\section{Optimal error bounds for the semi-discretization \\eqref{EWI}}\nIn this section, we establish optimal error bounds in $L^2$-norm and $H^1$-norm\nfor the semi-discretization \\eqref{EWI} of the NLSE \\eqref{NLSE}.\n\n\\subsection{Main results}\nFor the optimal $ L^2 $-norm error bound, we assume that the nonlinearity is locally Lipschitz continuous, i.e., there exists a fixed function $ {C_\\text{Lip}} (\\cdot): {\\mathbb R}^+\\to {\\mathbb R}^+$ such that\n\\begin{equation}\\label{A}\n\t| f(|z_1|^2)z_1 - f(|z_2|^2)z_2 | \\leq {C_\\text{Lip}}(M_0) |z_1 - z_2|, \\quad z_j \\in \\mathbb{C}, \\  |z_j| \\leq M_0, \\quad j=1, 2. \\tag{A}\n\\end{equation}\nAssumption \\cref{A} is satisfied by $ f \\in C^1((0, \\infty)) $ satisfying\n\t\\begin{equation*}\\label{A2}\n\t\t|f(\\rho)| + |\\rho f^\\prime(\\rho)| \\leq L(M_0), \\quad 0 < \\rho \\leq M_0\n\t\\end{equation*}\n\twith $ {C_\\text{Lip}}(M_0) \\sim L(M_0) $ for $ M_0 > 0 $. In particular, \\cref{A} allows\n\t\\begin{enumerate}\n\t\t\\item[(i)] $ f(\\rho) = \\lambda_1 \\rho^{\\sigma_1} +\\lambda_2 \\rho^{\\sigma_2} $ for any $ 0<\\sigma_1<\\sigma_2 $ and $\\lambda_1,\\lambda_2\\in{\\mathbb R}$ with $ {C_\\text{Lip}}(M_0) \\sim |\\lambda_1| M_0^{\\sigma_1} +|\\lambda_2| M_0^{\\sigma_2} $;\n\t\t\\item[(ii)] $ f(\\rho) = \\lambda \\rho^\\sigma \\ln \\rho $ for any $ \\sigma > 0 $\nand $\\lambda\\in {\\mathbb R}$ with $ {C_\\text{Lip}}(M_0) \\sim 1 + M_0^\\sigma + M_0^\\sigma |\\ln M_0| $.\n\t\\end{enumerate}\n\n\\smallskip\nFor the optimal $ H^1 $-norm error bound, we assume\n\\begin{equation}\\label{B}\n\t\\|  f(|v|^2)v - f(|w|^2)w  \\|_{H^1} \\leq C(\\| v \\|_{H^3}, \\| w \\|_{H^2}) \\| v-w \\|_{H^1}, \\  v \\in H^3(\\Omega), w \\in H^2(\\Omega). \\tag{B}\n\\end{equation}\nAssumption \\cref{B} is satisfied by\n\t\\begin{enumerate}\n\t\t\\item[(i)] $ f(\\rho) = \\lambda_1\\rho^{\\sigma_1} +\\lambda_2 \\rho^{\\sigma_2} $ for $ \\sigma_2>\\sigma_1 \\geq 1\/2 $ and $\\lambda_1,\\lambda_2\\in{\\mathbb R}$\nwith $ C(\\cdot,\\cdot) $ depending on $ \\| v \\|_{H^3} $ and $ \\| w \\|_{H^2} $;\n\t\t\\item[(ii)] $ f(\\rho) = \\lambda\\rho^\\sigma \\ln \\rho $ for any $ \\sigma > 1\/2 $\nand $\\lambda\\in{\\mathbb R}$ with $ C(\\cdot,\\cdot) $ depending on $ \\| v \\|_{H^3} $ and $ \\| w \\|_{H^2} $.\n\t\\end{enumerate}\nWe remark here that \\cref{B} implies \\cref{A} by taking $ v(x) \\equiv z_1 $ and $ w(x) \\equiv z_2 $ in \\cref{B}.\n\n\nLet $ T_\\text{max} $ be the maximal existing time of the solution of the NLSE \\cref{NLSE} and take $ 0 < T < T_\\text{max} $ be a fixed time. Define\n\\begin{equation}\\label{definM}\n\tM := \\max\\left\\{\\| \\psi \\|_{L^\\infty([0, T]; H^2)}, \\| \\psi \\|_{L^\\infty([0, T]; L^\\infty)}, \\| \\partial_t \\psi \\|_{L^\\infty([0, T]; L^2)}, \\| V \\|_{L^\\infty} \\right\\}.\n\\end{equation}\n\nLet $ \\psin{n} $ be the numerical approximation obtained by the EWI \\cref{EWI}, then we have\n\\begin{theorem}\\label{thm:error_estimates}\n\tAssume that $ V \\in L^\\infty(\\Omega) $, $ f $ satisfies Assumption \\cref{A} and the exact solution $ \\psi \\in C([0, T]; H_{\\text{\\rm per}}^2(\\Omega)) \\cap C^1([0, T]; L^2(\\Omega)) $, there exists $ \\tau_0>0 $ depending on $ M $ and $ T $ and sufficiently small such that for any $ 0 < \\tau < \\tau_0 $, we have $ \\psin{n} \\in H^2_\\text{\\rm per}(\\Omega) $ for $ 0 \\leq n \\leq T\/\\tau $ and\n\t\\begin{equation}\\label{eq:semi_error_L2}\n\t\t\\begin{aligned}\n\t\t\t&\\| \\psi(\\cdot, t_n) - \\psin{n}  \\|_{L^2} \\lesssim \\tau, \\quad \\| \\psin{n} \\|_{H^2} \\leq C(M), \\\\\n\t\t\t&\\| \\psi(\\cdot, t_n) - \\psin{n}  \\|_{H^1} \\lesssim \\sqrt{\\tau}, \\qquad 0 \\leq n \\leq T\/\\tau.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tMoreover, if $ V \\in W^{1, 4}(\\Omega) \\cap H^1_\\text{\\rm per}(\\Omega) $, $ f $ satisfies \\cref{B} and $ \\psi \\in C([0, T]; H_\\text{\\rm per}^3(\\Omega)) \\cap C^1([0, T]; H^1(\\Omega)) $, we have, for $ 0 < \\tau < \\tau_0 $,\n\t\\begin{equation}\\label{eq:semi_error_H1}\n\t\t\\| \\psi(\\cdot, t_n) - \\psin{n}  \\|_{H^1} \\lesssim \\tau, \\qquad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation}\n\\end{theorem}\n\n\\begin{remark}\n\tRecall that, for the time-splitting methods analyzed in \\cite{bao2023} with $ f(\\rho) = \\rho^\\sigma $, the optimal $ L^2 $-norm error bound in time is obtained for $ V \\in H^2(\\Omega)$ and $\\sigma \\geq 1\/2 $, and the optimal $ H^1 $-norm error bound in time is obtained for $ V \\in H^3(\\Omega)$ and $\\sigma \\geq 1 $. Hence, our results greatly relax the regularity requirements on both the potential and nonlinearity.\n\\end{remark}\n\nIn the following, we shall prove \\cref{thm:error_estimates}. We start with the proof of \\cref{eq:semi_error_L2}, and the proof of \\cref{eq:semi_error_H1} can be obtained by the standard Lady Windermere's fan argument with the established uniform $ H^2 $-bound of the semi-discretization solution in \\cref{eq:semi_error_L2}.\n\nIn the rest of this section, we assume that $ V \\in L^\\infty(\\Omega) $, $ f $ satisfies Assumption \\cref{A} and $ \\psi \\in C([0, T]; H_{\\text{\\rm per}}^2(\\Omega)) \\cap C^1([0, T]; L^2(\\Omega)) $.\n\n\n\\subsection{Local truncation error}\nWe define an operator $ B:L^\\infty(\\Omega) \\rightarrow L^\\infty(\\Omega) $ as\n\\begin{equation}\\label{eq:B_def}\n\tB(v) = V v + f(|v|^2)v, \\quad v \\in L^\\infty(\\Omega).\n\\end{equation}\nand define a constant $ {C_L}(\\cdot):= \\| V \\|_{L^\\infty} + {C_\\text{Lip}}(\\cdot) $ with $ {C_\\text{Lip}}(\\cdot) $ given by Assumption \\cref{A}. For the operator $ B $, we have\n\\begin{lemma}\\label{lem:diff_B}\n\tLet $ v, w \\in L^\\infty(\\Omega) $ satisfying $ \\| v \\|_{L^\\infty} \\leq M_0 $ and $ \\| w \\|_{L^\\infty} \\leq M_0 $, then\n\t\\begin{equation}\n\t\t\\| B(v) - B(w) \\|_{L^2} \\leq {C_L}(M_0) \\| v-w \\|_{L^2}.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tRecalling \\cref{eq:B_def,A}, we have\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\| B(v) - B(w) \\|_{L^2}\n\t\t\t&= \\| V(v-w) + f(|v|^2)v - f(|w|^2)w \\|_{L^2} \\\\\n\t\t\t&\\leq \\| V \\|_{L^\\infty} \\| v-w \\|_{L^2} + {C_\\text{Lip}}(M_0) \\| v - w \\|_{L^2} \\\\\n\t\t\t&= {C_L}(M_0) \\| v-w \\|_{L^2},\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhich completes the proof.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:est_gn}\n\tFor $ 0 \\leq n \\leq T\/\\tau-1 $, define\n\t\\begin{equation}\\label{eq:gn_def}\n\t\t\\begin{aligned}\n\t\t\tg_n(t)\n\t\t\t&:= B(\\psi(t_n+t)) - B(\\psi(t_n)), \\qquad 0 \\leq t \\leq \\tau.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tThen $ g_n \\in C([0, \\tau]; L^2(\\Omega)) \\cap W^{1, \\infty}([0, \\tau]; L^2(\\Omega)) $ satisfies\n\t\\begin{align}\n\t\t&\\| g_n \\|_{L^\\infty([0, \\tau];L^2)} \\leq {C_L}(M) M \\tau, \\label{eq:est_g_n} \\\\\n\t\t&\\| \\partial_t g_n \\|_{L^\\infty([0, \\tau];L^2)} \\leq {C_L}(M)M. \\label{eq:est_partial_t_g_n}\n\t\\end{align}\n\\end{lemma}\n\n\\begin{proof}\n\tUsing \\cref{lem:diff_B}, we have, for $ 0 \\leq s < t \\leq \\tau $,\n\t\\begin{eqnarray}\\label{eq:diff_gn}\n\t\t\\| g_n(t) - g_n(s) \\|_{L^2}\n\t\t&&= \\|  B(\\psi(t_n+t)) - B(\\psi(t_n+s)) \\|_{L^2} \\nonumber\\\\\n\t\t&&\\leq {C_L}(M) \\| \\psi(t_n+t) - \\psi(t_n+s) \\|_{L^2} \\nonumber\\\\\n\t\t&&\\leq {C_L}(M) \\int_s^t \\| \\partial_t \\psi(t_n+\\sigma) \\|_{L^2} \\mathrm{d} \\sigma.\n\t\\end{eqnarray}\n\tFrom \\eqref{eq:diff_gn}, recalling \\cref{A}, one has $ g_n \\in C([0, \\tau]; L^2(\\Omega)) $, and, by using \\cref{lem:diff_B} again, one has\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\| g_n(t) \\|_{L^2}\n\t\t\t&=\\| B(\\psi(t_n+t)) - B(\\psi(t_n)) \\|_{L^2} \\leq {C_L}(M) \\| \\psi(t_n+t) - \\psi(t_n) \\|_{L^2} \\\\\n\t\t\t&\\leq \\tau {C_L}(M) \\| \\partial_t \\psi \\|_{L^\\infty([t_n, t_n+\\tau]; L^2)} \\leq {C_L}(M) M \\tau,\n\t\t\\end{aligned}\n\t\\end{equation*}\t\n\twhich proves \\cref{eq:est_g_n}. Noting \\eqref{eq:diff_gn}, from the standard theory of Sobolev spaces (see, e.g., Proposition 1.3.12 in \\cite{cazenave2003}), we have $ g_n \\in W^{1, \\infty}([0, T]; L^2(\\Omega)) $ and, by letting $ \\varphi(\\sigma) = {C_L}(M) \\| \\partial_t \\psi(t_n+\\sigma) \\|_{L^2} $ for $ 0 \\leq \\sigma \\leq \\tau $,\n\t\\begin{equation*}\n\t\t\\| \\partial_t g_n \\|_{L^\\infty([0, \\tau]; L^2)} \\leq \\| \\varphi \\|_{L^\\infty([0, \\tau])} \\leq {C_L}(M) M,\n\t\\end{equation*}\n\twhich concludes the proof.\n\\end{proof}\n\nSimilar to Lemma 4.8.5 in \\cite{cazenave2003}, we have\n\\begin{lemma}\\label{lem:H2_estimate}\n\tAssume $ \\tau > 0 $ and $ g \\in C([0, \\tau]; L^2(\\Omega)) \\cap W^{1, 1}([0, \\tau]; L^2(\\Omega)) $. If\n\t\\begin{equation}\\label{eq:w_def}\n\t\tw(t) = -i \\int_0^t e^{i(t-s)\\Delta} g(s) \\mathrm{d} s, \\quad t \\in [0, \\tau],\n\t\\end{equation}\n\tthen we have\n\t\\begin{equation}\\label{eq:est_w}\n\t\t\\| \\Delta w \\|_{L^\\infty([0, \\tau]; L^2)} \\leq \\| g \\|_{L^\\infty([0, \\tau]; L^2)} + \\| g(0) \\|_{L^2} + \\| \\partial_t g \\|_{L^1([0, \\tau]; L^2)}.\n\t\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\n\tTaking the time derivative on both sides of \\cref{eq:w_def} and noting that $ g \\in W^{1, 1}([0, \\tau]; L^2(\\Omega)) $, we have for $ 0 \\leq t \\leq \\tau $\n\t\\begin{eqnarray}\\label{eq:partialw}\n\t\t\\partial_t w(t) = - i \\frac{d}{dt} \\int_0^t e^{is\\Delta} g(t-s) \\mathrm{d} s\n\t\t&&= - i  e^{it\\Delta} g(0) - i \\int_0^t e^{is\\Delta} \\partial_t g(t-s) \\mathrm{d} s \\nonumber\\\\\n\t\t&&= - i  e^{it\\Delta} g(0) - i \\int_0^t e^{i(t-s)\\Delta} \\partial_t g(s) \\mathrm{d} s.\n\t\\end{eqnarray}\n\tFrom \\eqref{eq:partialw}, using the isometry property of $ e^{i t \\Delta} $, we have\n\t\\begin{equation}\\label{eq:partialw_est}\n\t\t\\| \\partial_t w(t) \\|_{L^2} \\leq \\| g(0) \\|_{L^2} + \\| \\partial_t g \\|_{L^1([0, \\tau]; L^2)}, \\quad 0 \\leq t \\leq \\tau.\n\t\\end{equation}\n\tNote that $ w $ defined in \\cref{eq:w_def} satisfies the equation\n\t\\begin{equation*}\n\t\ti \\partial_t w = - \\Delta w + g, \\quad 0 \\leq t \\leq \\tau,\n\t\\end{equation*}\n\twhich implies, by using \\cref{eq:partialw_est},\n\t\\begin{equation}\n\t\t\\| \\Delta w(t) \\|_{L^2} \\leq \\| \\partial_t w(t) \\|_{L^2} + \\| g(t) \\|_{L^2} \\leq \\| g(0) \\|_{L^2} + \\| \\partial_t g \\|_{L^1([0, \\tau]; L^2)} + \\| g(t) \\|_{L^2},\n\t\\end{equation}\n\tand the conclusion follows from taking supreme of $ t $ on both sides.\n\\end{proof}\n\nNow we can obtain the following local truncation error estimates.\n\\begin{proposition}[local truncation error]\\label{prop:local_eror}\n\tFor $ 0 \\leq n \\leq T\/\\tau -1 $, we have\n\t\\begin{equation}\\label{eq:local_error}\n\t\t\\| \\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n)) \\|_{H^{\\alpha}} \\leq C(M) \\tau^{2-\\alpha\/2}, \\quad 0 \\leq \\alpha \\leq 2,\n\t\\end{equation}\n\twhere $ C(M) \\sim {C_L} (M)M $.\n\\end{proposition}\n\n\\begin{proof}\n\tRecalling \\eqref{eq:Duhamel} and \\eqref{eq:B_def}, we have\n\t\\begin{equation}\\label{eq:duhamel}\n\t\t\\psi(t_{n+1}) = e^{i \\tau \\Delta} \\psi(t_n) - i \\int_0^\\tau e^{i(\\tau - s) \\Delta}B(\\psi(t_n+s))\\mathrm{d} s, \\quad 0 \\leq n \\leq T\/\\tau -1.\n\t\\end{equation}\n\tBy the construction of the EWI \\cref{EWI} and \\cref{eq:B_def}, we have\n\t\\begin{equation}\\label{eq:EWI}\n\t\t\\Phi^\\tau(\\psi(t_n)) = e^{i \\tau \\Delta} \\psi(t_n) - i \\int_0^\\tau e^{i(\\tau - s)\\Delta} B(\\psi(t_n)) \\mathrm{d} s, \\quad 0 \\leq n \\leq T\/\\tau -1.\n\t\\end{equation}\n\tSubtracting \\cref{eq:EWI} from \\cref{eq:duhamel} and recalling \\cref{eq:gn_def}, we have\n\t\\begin{eqnarray}\\label{eq:local_error_integral}\n\t\t\\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n))\n\t\t&&= - i \\int_0^\\tau e^{i(\\tau - s) \\Delta} (B(\\psi(t_n+s)) - B(\\psi(t_n)))\\mathrm{d} s \\nonumber\\\\\n\t\t&&= - i \\int_0^\\tau e^{i(\\tau - s) \\Delta} g_n(s) \\mathrm{d} s, \\quad 0 \\leq n \\leq T\/\\tau -1.\n\t\\end{eqnarray}\n\tFrom \\eqref{eq:local_error_integral}, using \\cref{eq:est_g_n}, one gets\n\t\\begin{equation}\\label{eq:local_error_L2}\n\t\t\\| \\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n)) \\|_{L^2} \\leq \\int_0^\\tau \\| g_n(s) \\|_{L^2} \\mathrm{d} s \\leq {C_L}(M) M \\tau^2,\n\t\\end{equation}\n\twhich proves \\cref{eq:local_error} for $ \\alpha=0 $. Then we shall establish \\cref{eq:local_error} with $ \\alpha=2 $, and \\cref{eq:local_error} with $ 0 < \\alpha< 2 $ will follow from the Gagliardo-Nirenberg interpolation inequalities. Applying \\cref{lem:H2_estimate} to \\eqref{eq:local_error_integral}, using \\cref{eq:est_partial_t_g_n} and noting $ g_n(0) = 0 $, we have\n\t\\begin{eqnarray}\n\t\t\t&&\\| \\Delta(\\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n))) \\|_{L^2} \\nonumber\\\\\n\t\t\t&&\\leq \\| g_n \\|_{L^\\infty([0, \\tau]; L^2)} + \\| g_n(0) \\|_{L^2} + \\| \\partial_t g_n \\|_{L^1([0, \\tau]; L^2)} \\nonumber\\\\\n\t\t\t&&\\leq {C_L}(M)M \\tau + \\tau \\| \\partial_t g_n \\|_{L^\\infty([0, \\tau]; L^2)} \\leq 2{C_L}(M)M \\tau,\n\t\\end{eqnarray}\n\twhich combined with \\cref{eq:local_error_L2} implies\n\t\\begin{equation}\\label{eq:local_error_H2}\n\t\t\\| \\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n)) \\|_{H^2} \\leq C(M) \\tau, \\quad 0 \\leq n \\leq T\/\\tau -1,\n\t\\end{equation}\n\twhere $ C(M) \\sim {C_L} (M)M $. The conclusion follows from \\cref{eq:local_error_L2,eq:local_error_H2} and the Gagliardo-Nirenberg interpolation inequalities.\n\\end{proof}\n\n\\begin{remark}\n\tIn \\cref{eq:local_error_L2}, the optimal local truncation error in $ L^2 $-norm is obtained with the boundedness of $ \\| \\partial_t B(\\psi(t)) \\|_{L^2} $ (recalling \\cref{lem:est_gn}) instead of $ \\| \\Delta B(\\psi(t)) \\|_{L^2} $ in the time-splitting methods \\cite{bao2023}.\n\\end{remark}\n\n\\subsection{$ L^\\infty $-conditional stability estimate of \\eqref{EWI}}\nThen we shall establish the $ L^\\infty $-conditional stability estimate of the numerical flow \\eqref{EWI}. The key is the following lemma, which can be understood as the smoothing effect of the operator $ \\varphi_1(i \\tau \\Delta) $, which is another major advantage of the EWI \\cref{EWI}.\n\\begin{lemma}\\label{lem:smoothingofphi1}\n\tLet $ v, w \\in L^2(\\Omega) $ and $ 0 < \\tau < 1 $. Then we have\n\t\\begin{equation*}\n\t\t\\| \\varphi_1(i\\tau\\Delta) v - \\varphi_1(i\\tau\\Delta) w \\|_{H^{\\alpha}} \\leq C(\\alpha)\\tau^{-\\alpha\/2} \\| v - w \\|_{L^2}, \\quad 0 \\leq \\alpha \\leq 2,\n\t\\end{equation*}\n\twhere $ C(\\alpha) = 2^\\frac{\\alpha}{2}(1+\\mu_1^{-2})^\\frac{\\alpha}{2} $.\n\\end{lemma}\n\n\\begin{proof}\n\tIt suffices to show that for any $ v \\in L^2(\\Omega) $,\n\t\\begin{equation}\\label{eq:smoothingofphi1}\n\t\t\\| \\varphi_1(i\\tau\\Delta) v \\|_{H^{\\alpha}} \\leq C(\\alpha) \\tau^{-\\alpha\/2} \\| v \\|_{L^2}, \\quad 0 \\leq \\alpha \\leq 2.\n\t\\end{equation}\n\tNote that\n\t\\begin{equation}\\label{eq:expm1}\n\t\t| e^{i \\theta} - 1 | \\leq 2^{\\gamma} \\theta^{1-\\gamma}, \\quad \\theta \\in \\mathbb{R}, \\quad 0 \\leq \\gamma \\leq 1.\n\t\\end{equation}\n\tBy Parseval's identity, using \\cref{eq:expm1} with $ \\gamma = \\alpha\/2 $ and recalling \\eqref{eq:phi1_def}, we have\n\t\\begin{equation*}\n\t\t\\begin{aligned}\n\t\t\t\\| \\varphi_1(i\\tau\\Delta) v \\|_{H^{\\alpha}}^2\n\t\t\t&= \\sum_{l\\in\\mathbb{Z}} (1+\\mu_l^2)^{\\alpha} |\\varphi_1(-i\\tau \\mu_l^2)|^2 | \\hat v_l |^2 \\\\\n\t\t\t&= | \\widehat v_0 |^2 + \\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}} (1+\\mu_l^2)^{\\alpha} \\left| \\frac{e^{i\\tau\\mu_l^2} - 1}{\\tau\\mu_l^2} \\right|^2 |\\widehat v_l|^2 \\\\\n\t\t\t&\\leq | \\widehat v_0 |^2 + 2^{\\alpha} \\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}} (1+\\mu_l^2)^{\\alpha} \\left( \\tau\\mu_l^2 \\right)^{-\\alpha} |\\widehat v_l|^2 \\\\\n\t\t\t&=  | \\widehat v_0 |^2 + 2^{\\alpha} \\tau^{-\\alpha} \\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}} \\left(\\frac{1+\\mu_l^2}{\\mu_l^2}\\right)^{\\alpha}  |\\widehat v_l|^2 \\\\\n\t\t\t&\\leq | \\widehat v_0 |^2 + C(\\alpha)^2 \\tau^{-\\alpha} \\sum_{l\\in\\mathbb{Z}\\setminus\\{0\\}} |\\widehat v_l|^2 \\\\\n\t\t\t&\\leq C(\\alpha)^2\\tau^{-\\alpha} \\sum_{l\\in\\mathbb{Z}} |\\widehat v_l|^2 = C(\\alpha)^2 \\tau^{-\\alpha} \\| v \\|_{L^2}^2,\n\t\t\\end{aligned}\n\t\\end{equation*}\n\twhich proves \\cref{eq:smoothingofphi1} and concludes the proof.\n\\end{proof}\n\nWith \\cref{lem:smoothingofphi1}, we are able to obtain the stability estimate of the numerical flow \\eqref{EWI} up to $ H^2 $ without additional regularity on the potential and nonlinearity.\n\\begin{proposition}[stability estimate]\\label{prop:stability}\n\tLet $ v, w \\in H_\\text{\\rm per}^2(\\Omega) $ such that $ \\| v \\|_{L^\\infty} \\leq M_0 $ and $ \\| w \\|_{L^\\infty} \\leq M_0 $ and let $ 0 < \\tau < 1 $. Then we have, for $ 0 \\leq \\alpha \\leq 2 $,\n\t\\begin{equation*}\n\t\t\\| \\Phi^\\tau(v) - \\Phi^\\tau(w) \\|_{H^\\alpha} \\leq \\| v - w \\|_{H^\\alpha} + C(M_0) \\tau^{1-\\alpha\/2} \\| v - w \\|_{L^2}.\n\t\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n\tRecalling \\cref{EWI} and \\cref{eq:B_def}, we have\n\t\\begin{equation}\\label{eq:Phi}\n\t\t\\Phi^\\tau(u) = e^{i \\tau \\Delta} u - i \\tau \\varphi_1(i \\tau \\Delta) B(u), \\qquad u \\in H^2_\\text{per}(\\Omega).\n\t\\end{equation}\n\tTaking $ u = v $ and $ u = w $ in \\cref{eq:Phi}, subtracting one from the other and using the isometry property of $ e^{it\\Delta} $, \\cref{lem:smoothingofphi1} and \\cref{lem:diff_B}, we have\n\t\\begin{align*}\n\t\t\\| \\Phi^\\tau(v) - \\Phi^\\tau(w) \\|_{H^\\alpha}\n\t\t&\\leq \\| e^{i \\tau \\Delta} v - e^{i \\tau \\Delta} w \\|_{H^\\alpha} + \\tau \\| \\varphi_1(i \\tau \\Delta) (B(v) - B(w)) \\|_{H^\\alpha} \\\\\n\t\t&\\leq \\| v - w \\|_{H^\\alpha} + C(\\alpha) \\tau^{1-\\alpha\/2} \\| B(v) - B(w) \\|_{L^2} \\\\\n\t\t&\\leq \\| v - w \\|_{H^\\alpha} + C(\\alpha) \\tau^{1-\\alpha\/2} {C_L}(M_0) \\| v - w \\|_{L^2}.\n\t\\end{align*}\n\tThe conclusion follows from letting $ C(M_0) = C(\\alpha) {C_L}(M_0) $ with $ \\alpha=2 $.\n\\end{proof}\n\n\\subsection{Proof of the optimal $ L^2 $-error bound \\cref{eq:semi_error_L2}}\nWith the local truncation error estimate in Proposition \\ref{prop:local_eror} and the $ L^\\infty $-conditional stability estimate in Proposition \\ref{prop:stability},\nwe can prove \\cref{eq:semi_error_L2} by mathematical induction.\n\n\\begin{proof}[Proof of \\cref{eq:semi_error_L2} in \\cref{thm:error_estimates}]\n\tDefine the error function $ \\en{n} := \\psi(t_n) - \\psin{n} $ for $ 0 \\leq n \\leq T\/\\tau $. For $ 0 \\leq n \\leq T\/\\tau-1 $ and $ 0 \\leq \\alpha \\leq 2 $, we have\n\t\\begin{eqnarray}\\label{eq:error_propogation}\n\t\t\\| \\en{n+1} \\|_{H^\\alpha}\n\t\t&&= \\| \\psi(t_{n+1}) - \\psin{n+1}  \\|_{H^\\alpha} = \\| \\psi(t_{n+1}) - \\Phi^\\tau (\\psin{n}) \\|_{H^\\alpha} \\nonumber \\\\\n\t\t&&\\leq \\| \\psi(t_{n+1}) - \\Phi^\\tau (\\psi(t_n)) \\|_{H^\\alpha} + \\| \\Phi^\\tau (\\psi(t_n)) - \\Phi^\\tau (\\psin{n}) \\|_{H^\\alpha}.\n\t\\end{eqnarray}\n\tIn the following, we first establish the error bounds in $ L^2 $-norm and $ H^\\frac{7}{4} $-norm together by the mathematical induction, which, in particular, yield the uniform $ L^\\infty $-bound of $ \\psin{n} $. With the uniform $ L^\\infty $-bound, we can obtain the uniform $ H^2 $-bound of $ \\psin{n} $. The error bound in $ H^1 $-norm will follow from the error bound in $ L^2 $-norm and the uniform $ H^2 $-bound by using the standard interpolation inequalities.\n\t\n\tLet $ C_0 := \\max\\{C(1+M), M, 1\\} \\geq 1 $ with $M$ given in \\eqref{definM} and $ C(\\cdot) $ being defined in \\cref{prop:stability} and $C_1:=C(M)$ with $C(\\cdot)$ being defined in \\cref{prop:local_eror}. Let $ 0<\\tau_0<1 $ be chosen such that\n\t\\begin{equation}\\label{eq:tau_0}\n\t\t2 C_0Te^{C_0T}C_1\\tau_0^\\frac{1}{8} \\leq 1\/c,\n\t\\end{equation}\n\twhere $ c $ is the constant given by the Sobolev embedding $ H^\\frac{7}{4} \\hookrightarrow L^\\infty $. We are going to prove that when $ 0 < \\tau < \\tau_0 $, we have, for $ 0 \\leq n \\leq T\/\\tau $,\n\t\\begin{equation}\\label{eq:claim}\n\t\t\\| \\en{n} \\|_{L^2} \\leq e^{C_0T}C_1 \\tau, \\quad \\| \\en{n} \\|_{H^\\frac{7}{4}} \\leq 2 C_0Te^{C_0T}C_1\\tau^\\frac{1}{8}.\n\t\\end{equation}\n\tWe shall prove \\cref{eq:claim} by mathematical induction. When $ n = 0 $, $ \\en{n} = \\psin{0} - \\psi_0 = 0 $, and $ \\cref{eq:claim} $ holds trivially. We assume that \\cref{eq:claim} holds for $ 0 \\leq n \\leq m \\leq T\/\\tau -1 $. Under this assumption, we have, by Sobolev embedding, $ \\tau < \\tau_0 $ and \\cref{eq:tau_0},\n\t\\begin{equation}\\label{eq:Linfty}\n\t\t\\| \\psin{n} \\|_{L^\\infty} \\leq \\| \\psi(t_n) \\|_{L^\\infty} + \\| \\en{n} \\|_{L^\\infty} \\leq M + c \\| \\en{n} \\|_{H^\\frac{7}{4}} \\leq M + 1, \\quad 0 \\leq n \\leq m.\n\t\\end{equation}\n\tTaking $ \\alpha = 0 $ and $ \\alpha=7\/4 $ in \\eqref{eq:error_propogation}, we have for $ 0 \\leq n \\leq T\/\\tau-1 $,\n\t\\begin{align}\n\t\t&\\| \\en{n+1} \\|_{L^2} \\leq \\| \\psi(t_{n+1}) - \\Phi^\\tau (\\psi(t_n)) \\|_{L^2} + \\| \\Phi^\\tau (\\psi(t_n)) - \\Phi^\\tau (\\psin{n}) \\|_{L^2}, \\label{error_eq_1}\\\\\n\t\t&\\| \\en{n+1} \\|_{H^\\frac{7}{4}} \\leq \\| \\psi(t_{n+1}) - \\Phi^\\tau (\\psi(t_n)) \\|_{H^\\frac{7}{4}} + \\| \\Phi^\\tau (\\psi(t_n)) - \\Phi^\\tau (\\psin{n}) \\|_{H^\\frac{7}{4}}. \\label{error_eq_2}\n\t\\end{align}\n\tUsing \\cref{prop:local_eror,prop:stability} with $ \\alpha=0 $ and $ \\alpha=7\/4 $ for \\cref{error_eq_1} and \\cref{error_eq_2}, respectively, and noting \\cref{eq:Linfty}, we have for $ 0 \\leq n \\leq m $,\n\t\\begin{align}\n\t\t&\\| \\en{n+1} \\|_{L^2} \\leq (1+C_0 \\tau) \\| \\en{n} \\|_{L^2} + C_1 \\tau^2, \\label{eq:induction_1}\\\\\n\t\t&\\| \\en{n+1} \\|_{H^\\frac{7}{4}} \\leq \\| \\en{n} \\|_{H^\\frac{7}{4}} + C_0 \\tau^\\frac{1}{8} \\| \\en{n} \\|_{L^2} + C_1 \\tau^{1+\\frac{1}{8}}.  \\label{eq:induction_2}\n\t\\end{align}\n\tFrom \\cref{eq:induction_1}, using the standard discrete Gronwall's inequality, we have\n\t\\begin{equation}\\label{eq:conclusion1}\n\t\t\\| \\en{m+1} \\|_{L^2} \\leq e^{C_0 T} C_1 \\tau.\n\t\\end{equation}\n\tFrom \\cref{eq:induction_2}, using the assumption that \\cref{eq:claim} holds for $ 0 \\leq n \\leq m $, we have\n\t\\begin{equation}\\label{eq:e_H7\/4}\n\t\t\\| \\en{n+1} \\|_{H^\\frac{7}{4}} \\leq \\| \\en{n} \\|_{H^\\frac{7}{4}} + C_0 \\tau^\\frac{1}{8}  e^{C_0 T}C_1\\tau  + C_1 \\tau^{1+\\frac{1}{8}}, \\quad 0 \\leq n \\leq m.\n\t\\end{equation}\n\tSumming over $ n $ from $ 0 $ to $ m $ in \\cref{eq:e_H7\/4}, noting that $ \\en{0}=0 $ and $ C_0 \\geq 1 $, we obtain\n\t\\begin{eqnarray}\\label{eq:conclusion2}\n\t\t\t\\| \\en{m+1} \\|_{H^\\frac{7}{4}}\n\t\t\t&&\\leq C_0 \\tau^\\frac{1}{8}  e^{C_0T}C_1 m\\tau + C_1 m \\tau^{1+\\frac{1}{8}} \\nonumber\\\\\n\t\t\t&&\\leq C_0 T e^{C_0T} C_1 \\tau^{\\frac{1}{8}}+ C_1 T \\tau^{\\frac{1}{8}} \\nonumber\\\\\n\t\t\t&&\\leq 2 C_0 Te^{C_0T} C_1 \\tau^\\frac{1}{8}.\n\t\\end{eqnarray}\n\tCombing \\eqref{eq:conclusion1} and \\eqref{eq:conclusion2}, we prove \\cref{eq:claim} for $ n=m+1 $, and thus for all $ 0 \\leq n \\leq T\/\\tau $ by mathematical induction.\n\t\n\tThen we prove the uniform $ H^2 $ bound of $ \\psin{n} $. We first note that \\cref{eq:Linfty} now holds for any $ 0 \\leq n \\leq T\/\\tau $. Taking $ \\alpha=2 $ in \\eqref{eq:error_propogation}, using \\cref{prop:stability,prop:local_eror} with $ \\alpha=2 $ and \\cref{eq:claim}, we have for $ 0 \\leq n \\leq T\/\\tau - 1 $,\n\t\\begin{eqnarray}\\label{eq:e_H2}\n\t\t\\| \\en{n+1} \\|_{H^2}\n\t\t&&\\leq  \\| \\Psi^\\tau (\\psi(t_n)) - \\Phi^\\tau (\\psi(t_n))  \\|_{H^2} + \\| \\Phi^\\tau (\\psi(t_n)) - \\Phi^\\tau(\\psin{n}) \\|_{H^2} \\nonumber\\\\\n\t\t&&\\leq \\| \\en{n} \\|_{H^2} + C_0 \\| \\en{n} \\|_{L^2} + C_1 \\tau \\nonumber\\\\\n\t\t&&\\leq \\| \\en{n} \\|_{H^2} + C_0 e^{C_0T}C_1 \\tau + C_1 \\tau.\n\t\\end{eqnarray}\n\tSumming \\eqref{eq:e_H2} from $ 0 $ to $ n-1 $, we obtain\n\t\\begin{equation}\\label{uniformH2bound}\n\t\t\\| \\en{n} \\|_{H^2} \\leq C_0 e^{C_0T}C_1 n \\tau + C_1 n \\tau \\leq 2C_0e^{C_0T} C_1 T, \\quad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation}\n\t\n\tFinally, combining \\cref{eq:claim,uniformH2bound}, and using the interpolation inequality for the $ H^1 $-error bound, we prove \\cref{eq:semi_error_L2}.\n\\end{proof}\n\n\\subsection{Proof of the optimal $ H^1 $-error bound \\cref{eq:semi_error_H1}}\nTo prove \\cref{eq:semi_error_H1}, we assume that $ V \\in W^{1, 4}(\\Omega) \\cap H^1_\\text{per}(\\Omega) $, $ f $ satisfies Assumption \\cref{B}, $ \\psi \\in C([0, T]; H_\\text{\\rm per}^3(\\Omega)) \\cap C^1([0, T]; H^1(\\Omega)) $ and $ 0<\\tau <\\tau_0 $ with $ \\tau_0 $ given in \\cref{eq:tau_0}. Under the assumptions above, $ B:H^1_\\text{per}(\\Omega) \\rightarrow H^1_\\text{per}(\\Omega) $ satisfies\n\\begin{equation}\\label{eq:diff_B_H1}\n\t\\| B(v) - B(w) \\|_{H^1} \\leq C(\\| v \\|_{H^3}, \\| w \\|_{H^2}, \\| V \\|_{W^{1, 4}(\\Omega)}) \\| v-w \\|_{H^1}, \\quad v, w \\in H^2(\\Omega).\n\\end{equation}\n\\begin{proof}[Proof of \\cref{eq:semi_error_H1} in \\cref{thm:error_estimates}]\n\tFrom \\eqref{eq:local_error_integral}, using \\cref{eq:diff_B_H1} and the isometry property of $ e^{i t \\Delta} $, and noting that $ \\psi \\in C^1([0, T]; H^1(\\Omega)) $, we have, for $ 0 \\leq n \\leq T\/\\tau-1 $,\n\t\\begin{equation}\\label{eq:semi_local_error_H1}\n\t\t\\| \\psi(t_{n+1}) - \\Phi^\\tau(\\psi(t_n)) \\|_{H^1} \\leq \\int_0^\\tau \\| B(\\psi(t_n+s)) - B(\\psi(t_n)) \\|_{H^1} \\mathrm{d} s \\lesssim \\tau^2.\n\t\\end{equation}\n\tNoting that $ |\\varphi_1(i \\theta)| \\leq 1 $ for $ \\theta \\in \\mathbb{R} $, we have\n\t\\begin{equation}\\label{eq:phi_1_H1}\n\t\t\\| \\varphi_1(i \\tau \\Delta) v\\|_{H^1} \\leq \\| v \\|_{H^1}, \\quad v \\in H^1_\\text{per}(\\Omega),\n\t\\end{equation}\n\twhich implies, by recalling \\cref{eq:Phi} and using \\cref{eq:diff_B_H1} again,\n\t\\begin{eqnarray}\\label{eq:semi_stability_H1}\n\t\t\\| \\Phi^\\tau(\\psi(t_n)) - \\Phi^\\tau(\\psin{n}) \\|_{H^1}\n\t\t&&\\leq \\| \\psi(t_n) - \\psin{n} \\|_{H^1} + \\tau \\| B(\\psi(t_n)) - B(\\psin{n}) \\|_{H^1} \\nonumber\\\\\n\t\t&&\\leq (1 + C \\tau) \\| \\psi(t_n) - \\psin{n} \\|_{H^1}, \\quad  0 \\leq n \\leq T\/\\tau-1,\n\t\\end{eqnarray}\n\twhere $ C $ depends on $ \\| V \\|_{W^{1, 4}} $, $ \\| \\psi(t_n) \\|_{H^3} $ and $ \\| \\psin{n} \\|_{H^2} $, which are uniformly bounded. Then \\cref{eq:semi_error_H1} follows from \\eqref{eq:semi_local_error_H1} and \\eqref{eq:semi_stability_H1} by the standard Lady Windermere's fan argument.\n\\end{proof}\n\n\\section{Optimal error bounds for the full discretization \\eqref{EWI-FS}}\nIn this section, we establish optimal error bounds in $L^2$- and $H^1$-norm for the full-discretization scheme EWI-FS \\eqref{EWI-FS}, and generalize them to the EWI-EFP scheme \\cref{EWI-EFP}.\n\n\\subsection{Main results}\nFor $ \\psihn{n}\\ (0 \\leq n \\leq T\/\\tau) $ obtained by the EWI-FS scheme \\cref{EWI-FS}, we have\n\\begin{theorem}\\label{thm:error_estimates_FS}\n\tAssume that $ V \\in L^\\infty(\\Omega) $, $ f $ satisfies Assumption \\cref{A} and the exact solution $ \\psi \\in C([0, T]; H_{\\text{\\rm per}}^2(\\Omega)) \\cap C^1([0, T]; L^2(\\Omega)) $, there exists $ \\tau_0>0 $ and $ h_0>0 $ depending on $ M $ and $ T $ and sufficiently small such that for any $ 0 < \\tau < \\tau_0 $ and $ 0 < h < h_0 $, we have\n\t\\begin{equation}\\label{eq:full_error_L2}\n\t\t\\begin{aligned}\n\t\t\t&\\| \\psi(\\cdot, t_n) - \\psihn{n} \\|_{L^2} \\lesssim \\tau + h^2, \\quad \\| \\psihn{n} \\|_{H^2} \\leq C(M), \\\\\n\t\t\t&\\| \\psi(\\cdot, t_n) - \\psihn{n} \\|_{H^1} \\lesssim \\sqrt{\\tau} + h,  \\qquad 0 \\leq n \\leq T\/\\tau.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tMoreover, if $ V \\in W^{1, 4}(\\Omega) \\cap H_\\text{\\rm per}^1(\\Omega) $, $ f $ satisfies \\cref{B} and $ \\psi \\in C([0, T]; H_\\text{\\rm per}^3(\\Omega)) \\cap C^1([0, T]; H^1(\\Omega)) $, we have, for $ 0 < \\tau < \\tau_0 $ and $ 0 < h < h_0 $,\n\t\\begin{equation}\\label{eq:full_error_H1}\n\t\t\\| \\psi(\\cdot, t_n) - \\psihn{n} \\|_{L^2} \\lesssim \\tau + h^3, \\quad \\| \\psi(\\cdot, t_n) - \\psihn{n} \\|_{H^1} \\lesssim \\tau + h^2, \\qquad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation}\n\\end{theorem}\n\n\n\\begin{remark}\\label{rem:coupling}\n\tThanks to the strong $ H^2 $-control of the semi-discretization solution in \\cref{eq:semi_error_L2}, there is no coupling condition between $ \\tau $ and $ h $ for all $ 1 \\leq d \\leq 3 $ in \\cref{thm:error_estimates_FS}.\n\\end{remark}\n\n\nIn the following, we shall prove \\cref{thm:error_estimates_FS}. We use different methods to prove \\cref{eq:full_error_L2} and \\cref{eq:full_error_H1}. For the $ L^2 $-norm error bound \\cref{eq:full_error_L2}, we compare the full-discretization solution $ \\psihn{n} $ with the semi-discretization solution $ \\psin{n} $ to avoid the coupling condition between $ \\tau $ and $ h $ when using the inverse inequalities. Then, for the $ H^1 $-norm error bound \\cref{eq:full_error_H1}, we can directly compare the full-discretization solution with the exact solution since we already have control of the full-discretization solution in $ H^2 $-norm.\n\nIn the rest of this section, we assume that $ V \\in L^\\infty(\\Omega) $, $ f $ satisfies Assumption \\cref{A} and $ \\psi \\in C([0, T]; H_{\\text{\\rm per}}^2(\\Omega)) \\cap C^1([0, T]; L^2(\\Omega)) $.\n\n\\subsection{Proof of the optimal $ L^2 $-error bound \\cref{eq:full_error_L2}}\nWe start with the error estimates between the semi-discretization solution $ \\psin{n} $ and the full-discretization solution $ \\psihn{n} $.\n\n\\begin{proposition}\\label{thm:truncation_error}\n\tLet $ 0<\\tau<\\tau_0 $ with $ \\tau_0 $ given in \\cref{thm:error_estimates}. Then there exists $ h_0 $ depending on $ M $ and $ T $ and small enough such that for $ 0<h<h_0 $, we have\n\t\\begin{equation*}\n\t\t\\| P_N \\psin{n} - \\psihn{n} \\|_{L^2} \\leq C(M, T)h^2, \\quad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n\tDefine the error function $ \\ehn{n} := P_N \\psin{n} - \\psihn{n} $ for $ 0 \\leq n \\leq T\/\\tau $. Then $ \\ehn{0} = P_N \\psin{0} - \\psihn{0} = 0 $. Applying $ P_N $ on both sides of \\cref{EWI}, noting that $ P_N $ commutes with $ e^{i \\tau \\Delta} $ and $ \\varphi_1(i \\tau \\Delta) $ and recalling \\cref{eq:B_def}, we have\n\t\\begin{equation}\\label{eq:P_Npsi}\n\t\tP_N \\psin{n+1} = e^{i \\tau \\Delta} P_N \\psin{n} - i \\tau \\varphi_1(i \\tau \\Delta) P_N B(\\psin{n}), \\quad 0 \\leq n \\leq T\/\\tau-1.\n\t\\end{equation}\n\tRecalling \\cref{EWI-FS,eq:B_def}, we have\n\t\\begin{equation}\\label{eq:psi_h}\n\t\t\\psihn{n+1} = e^{i \\tau \\Delta} \\psihn{n} - i \\tau \\varphi_1(i \\tau \\Delta) P_N B(\\psihn{n}), \\quad 0 \\leq n \\leq T\/\\tau-1.\n\t\\end{equation}\n\tSubtracting \\cref{eq:psi_h} from \\cref{eq:P_Npsi}, we have\n\t\\begin{equation}\\label{eq:eh_equation}\n\t\t\\ehn{n+1} = e^{i \\tau \\Delta} \\ehn{n} - i \\tau \\varphi_1(i \\tau \\Delta) P_N (B(\\psin{n}) - B(\\psihn{n})).\n\t\\end{equation}\n\tFrom \\cref{eq:eh_equation}, using the isometry property of $ e^{i t \\Delta} $, the $ L^2 $-projection property of $ P_N $ and \\cref{lem:smoothingofphi1} with $ s=0 $, we have, for $ 0 \\leq n \\leq T\/\\tau - 1 $,\n\t\\begin{eqnarray}\\label{eq:errorh}\n\t\t\t\\| \\ehn{n+1} \\|_{L^2}\n\t\t\t&\\leq&\\| \\ehn{n} \\|_{L^2} + \\tau \\| \\varphi_1(i \\tau \\Delta) P_N ((B(\\psin{n}) - B(\\psihn{n})) \\|_{L^2} \\nonumber\\\\\n\t\t\t&\\leq&\\| \\ehn{n} \\|_{L^2} + \\tau \\| (B(\\psin{n}) - B(\\psihn{n}) \\|_{L^2}\\nonumber \\\\\n\t\t\t&\\leq&\\| \\ehn{n} \\|_{L^2} + \\tau \\| (B(\\psin{n}) - B(P_N \\psin{n}) \\|_{L^2} + \\tau \\| (B(P_N \\psin{n}) - B(\\psihn{n}) \\|_{L^2}.\n\t\\end{eqnarray}\n\tBy \\cref{eq:semi_error_L2}, using Sobolev embedding and the property of $ P_N $, we have\n\t\\begin{equation}\\label{eq:linfty_psin}\n\t\t\\| P_N \\psin{n} \\|_{L^\\infty} \\leq \\tilde c \\| P_N \\psin{n} \\|_{H^2} \\leq \\tilde c \\| \\psin{n} \\|_{H^2} \\leq \\tilde c C(M) =: M_0, \\quad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation}\n\tSimilarly, $ \\| \\psin{n} \\|_{L^\\infty} \\leq M_0 $. From \\eqref{eq:errorh}, noting \\cref{eq:linfty_psin}, using \\cref{lem:diff_B,eq:semi_error_L2} and the standard estimate of the projection error $ \\| \\phi - P_N \\phi \\|_{L^2} \\lesssim h^2 | \\phi |_{H^2} \\  \\forall \\phi \\in H^2_\\text{per}(\\Omega) $, we have, for $ 0 \\leq n \\leq T\/\\tau - 1 $,\n\t\\begin{equation}\\label{eq:error_propagation_full}\n\t\t\\| \\ehn{n+1} \\|_{L^2} \\leq \\| \\ehn{n} \\|_{L^2} + {C_L}(\\max\\{M_0, \\| \\psihn{n} \\|_{L^\\infty}\\}) \\tau \\| \\ehn{n} \\|_{L^2} + C(M) \\tau h^2.\n\t\\end{equation}\n\tThe conclusion then follows from the discrete Gronwall's inequality and the standard induction argument by using the inverse estimates $ \\| \\phi \\|_{L^\\infty} \\lesssim h^{-\\frac{d}{2}}\\| \\phi \\|_{L^2} \\  \\forall \\phi \\in X_N $, where $ d $ is the dimension of the space, i.e., $ d=1 $ in the current case. We omit the details here for brevity.\n\\end{proof}\n\n\\begin{proof}[Proof of \\cref{eq:full_error_L2} in \\cref{thm:error_estimates_FS}]\n\tBy the triangle inequality, for $ 0 \\leq \\alpha \\leq 2 $,\n\t\\begin{equation}\\label{eq:decomp_full}\n\t\t\\|  \\psi(t_n) - \\psihn{n} \\|_{H^\\alpha} \\leq \\| \\psi(t_n) - \\psin{n}  \\|_{H^\\alpha} + \\| \\psin{n} - P_N \\psin{n} \\|_{H^\\alpha} + \\| P_N \\psin{n} - \\psihn{n} \\|_{H^\\alpha}.\n\t\\end{equation}\n\tFrom \\cref{eq:semi_error_L2}, using the interpolation inequalities, we have\n\t\\begin{equation}\\label{eq:semi_error_Hs}\n\t\t \\| \\psi(t_n) - \\psin{n}  \\|_{H^\\alpha} \\lesssim \\tau^{1-\\alpha\/2}, \\quad 0 \\leq \\alpha \\leq 2.\n\t\\end{equation}\n\tFrom \\cref{eq:decomp_full}, using \\cref{eq:semi_error_Hs} and the standard Fourier projection error estimates\n\t\\begin{equation}\n\t\t\\| \\phi - P_N \\phi \\|_{H^\\alpha} \\lesssim h^{2-\\alpha} | \\phi |_{H^2}, \\quad 0 \\leq \\alpha \\leq 2, \\quad \\phi \\in H^2_{\\text{per}}(\\Omega),\n\t\\end{equation}\n\tand noting \\cref{eq:semi_error_L2}, we have\n\t\\begin{equation}\\label{eq:error_decomp_full}\n\t\t\\| \\psi(t_n) - \\psihn{n}  \\|_{H^\\alpha} \\lesssim \\tau^{1-\\alpha\/2} + h^{2-\\alpha} + \\|  P_N \\psin{n} - \\psihn{n} \\|_{H^\\alpha}.\n\t\\end{equation}\n\tFrom \\cref{eq:error_decomp_full}, using the inverse estimate $ \\| \\phi \\|_{H^\\alpha} \\lesssim h^{-\\alpha} \\| \\phi \\|_{L^2} \\  \\forall \\phi \\in X_N $ \\cite{book_spectral_2,book_spectral} and \\cref{thm:truncation_error}, we have\n\t\\begin{equation}\\label{eq:full_error_proof}\n\t\t\\| \\psihn{n} - \\psi(t_n) \\|_{H^\\alpha} \\lesssim \\tau^{1-\\alpha\/2} + h^{2-\\alpha} + h^{-\\alpha}h^2 \\lesssim \\tau^{1-\\alpha\/2} + h^{2-\\alpha}, \\quad 0 \\leq \\alpha \\leq 2,\n\t\\end{equation}\n\twhich proves \\cref{eq:full_error_L2} by taking $ \\alpha=0, 1, 2 $.\n\\end{proof}\n\n\\subsection{Proof of the  optimal $ H^1 $-error bound \\eqref{eq:full_error_H1}}\nTo prove \\cref{eq:full_error_H1}, we assume that $ V \\in W^{1, 4}(\\Omega) \\cap H_\\text{\\rm per}^1(\\Omega) $, $ f $ satisfies Assumption \\cref{B}, $ \\psi \\in C([0, T]; H_\\text{\\rm per}^3(\\Omega)) \\cap C^1([0, T]; H^1(\\Omega)) $ and $ 0<\\tau<\\tau_0 $, $ 0<h<h_0 $. Since we already have the uniform control of $ \\psihn{n} $ in $ H^2 $-norm, \\cref{eq:full_error_H1} can be proved similarly to \\cref{eq:semi_error_H1}, and we just sketch the outline here.\n\n\\begin{proof}[Proof of \\cref{eq:full_error_H1} in \\cref{thm:error_estimates_FS}]\n\tRecalling \\cref{eq:duhamel,EWI-FS}, we have\n\t\\begin{eqnarray}\n\t\t&&P_N \\psi(t_{n+1}) - \\Phi_h^\\tau(P_N \\psi(t_n)) \\nonumber \\\\\n\t\t&&=-i \\int_0^\\tau e^{i (\\tau - s)\\Delta} P_N \\left ( B(\\psi(t_n+s)) - B(P_N \\psi(t_n)) \\right ) \\mathrm{d} s, \\quad 0 \\leq n \\leq \\frac{T}{\\tau}-1,\n\t\\end{eqnarray}\n\twhich implies, by the property of $ e^{it\\Delta} $ and $ P_N $, for $ X=L^2 $ or $ H^1 $, that\n\t\\begin{eqnarray}\n\t\t&&\\| P_N \\psi(t_{n+1}) - \\Phi_h^\\tau(P_N \\psi(t_n)) \\|_{X} \\nonumber \\\\\n\t\t&&\\leq \\int_0^\\tau \\| B(\\psi(t_n+s)) - B(\\psi(t_n)) \\|_{X} + \\| B(\\psi(t_n))- B(P_N \\psi(t_n)) \\|_{X} \\mathrm{d} s. \\label{local_error_decomp_full}\n\t\\end{eqnarray}\n\tFrom \\eqref{local_error_decomp_full}, using \\cref{lem:diff_B} and \\cref{eq:diff_B_H1}, we have, for $ 0 \\leq n \\leq T\/\\tau-1$,\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t&\\| P_N \\psi(t_{n+1}) - \\Phi_h^\\tau(P_N \\psi(t_n)) \\|_{L^2} \\lesssim \\tau^2 + \\tau h^3, \\\\\n\t\t\t&\\| P_N \\psi(t_{n+1}) - \\Phi_h^\\tau(P_N \\psi(t_n)) \\|_{H^1} \\lesssim \\tau^2 + \\tau h^2.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tBesides, recalling \\cref{EWI-FS} and using \\cref{lem:smoothingofphi1,eq:smoothingofphi1,lem:diff_B,eq:diff_B_H1}, we have\n\t\\begin{equation}\n\t\t\\begin{aligned}\n\t\t\t&\\| \\Phi_h^\\tau(P_N \\psi(t_n)) - \\Phi_h^\\tau(\\psihn{n}) \\|_{L^2} \\leq (1+C_1\\tau) \\| P_N \\psi(t_n) - \\psihn{n} \\|_{L^2}, \\\\\n\t\t\t&\\| \\Phi_h^\\tau(P_N \\psi(t_n)) - \\Phi_h^\\tau(\\psihn{n}) \\|_{H^1} \\leq (1+C_2\\tau) \\| P_N \\psi(t_n) - \\psihn{n} \\|_{H^1},\n\t\t\\end{aligned}\n\t\\end{equation}\n\twhere $ C_1 $ depends on $ \\| P_N \\psi(t_n) \\|_{L^\\infty} $ and $ \\| \\psihn{n} \\|_{L^\\infty} $, and $ C_2 $ depends on $ \\| P_N \\psi(t_n) \\|_{H^3} $ and $ \\| \\psihn{n} \\|_{H^2} $ for $ 0 \\leq n \\leq T\/\\tau -1 $. Thus, both $ C_1 $ and $ C_2 $ are under control. Then the proof can be completed by the Lady Windermere's fan argument and the standard projection error estimates of $ P_N $.\n\\end{proof}\n\n\\subsection{Extension to the EWI-EFP \\cref{EWI-EFP}}\n\\Cref{thm:error_estimates_FS} also holds for $ I_N \\psihhn{n}(0 \\leq n \\leq T\/\\tau) $ obtained by the EWI-EFP scheme \\cref{EWI-EFP} when $ f $ is sufficiently smooth. To be precise, we introduce another assumption on the nonlinearity as\n\\begin{equation}\\label{C}\n\tf(|v|^2)v \\in H^\\alpha_\\text{per}(\\Omega), \\quad \\forall v \\in H^\\alpha_\\text{per}(\\Omega). \\tag{C}\n\\end{equation}\n\nFor the optimal $L^2$-norm error bound, we assume that $ f $ satisfies Assumptions \\cref{A} and \\cref{C} with $\\alpha=2$. Two typical examples of $f$ include (i) $ f(\\rho) = \\lambda_1\\rho^{\\sigma_1} + \\lambda_2\\rho^{\\sigma_2} $ with $ \\sigma_2>\\sigma_1 \\geq 1\/2 $ and $\\lambda_1,\\lambda_2\\in{\\mathbb R}$, and (ii) $ f(\\rho) =\\lambda \\rho^\\sigma \\ln \\rho $ with $ \\sigma > 1\/2 $ and $\\lambda\\in{\\mathbb R}$.\n\nFor the optimal $H^1$-norm error bound, we assume, in addition to Assumption \\cref{B}, $ f $ satisfies \\cref{C} with $ \\alpha=3 $ and the discrete counterpart of Assumption \\cref{B}\n\\begin{equation}\\label{B'}\n\t\\| I_N (f(|v|^2)v - f(|w|^2)w)  \\|_{H^1} \\leq C(\\| v \\|_{H^3}, \\| w \\|_{H^2}) \\| v-w \\|_{H^1}, \\quad v, w \\in X_N, \\tag{B'}\n\\end{equation}\nwith two typical examples of $f$: (i) $ f(\\rho) =\\lambda_1 \\rho^{\\sigma_1} +\\lambda_2 \\rho^{\\sigma_2} $ with $ \\sigma_2>\\sigma_1 \\geq 1 $ and $\\lambda_1,\\lambda_2\\in{\\mathbb R}$, and (ii) $ f(\\rho) = \\lambda\\rho^\\sigma \\ln \\rho $ with $ \\sigma > 1 $ and $\\lambda\\in{\\mathbb R}$ (see, e.g., \\cite{bao2023} for the proof).\n\nThen we have the following error bounds for the EWI-EFP scheme \\cref{EWI-EFP}.\n\\begin{corollary}\\label{coro:error_estimates_EFP}\n\tAssume that $ V \\in L^\\infty(\\Omega) $, $ f $ satisfies Assumptions \\cref{A} and \\cref{C} with $\\alpha=2$ and the exact solution $ \\psi \\in C([0, T]; H_{\\text{\\rm per}}^2(\\Omega)) \\cap C^1([0, T]; L^2(\\Omega)) $, there exists $ \\tau_0>0 $ and $ h_0>0 $ depending on $ M $ and $ T $ and sufficiently small such that for any $ 0 < \\tau < \\tau_0 $ and $ 0 < h < h_0 $, we have\n\t\\begin{equation}\\label{eq:full_error_L2_EFP}\n\t\t\\begin{aligned}\n\t\t\t&\\| \\psi(\\cdot, t_n) - I_N \\psihhn{n} \\|_{L^2} \\lesssim \\tau + h^2, \\quad \\| I_N \\psihhn{n} \\|_{H^2} \\leq C(M), \\\\\n\t\t\t&\\| \\psi(\\cdot, t_n) - I_N \\psihhn{n} \\|_{H^1} \\lesssim \\sqrt{\\tau} + h,  \\qquad 0 \\leq n \\leq T\/\\tau.\n\t\t\\end{aligned}\n\t\\end{equation}\n\tMoreover, if $ V \\in W^{1, 4}(\\Omega) \\cap H_\\text{\\rm per}^1(\\Omega) $, $ f $ satisfies Assumptions \\cref{B}, \\cref{B'} and \\cref{C} with $\\alpha=3$ and $ \\psi \\in C([0, T]; H_\\text{\\rm per}^3(\\Omega)) \\cap C^1([0, T]; H^1(\\Omega)) $, we have, for $ 0 < \\tau < \\tau_0 $ and $ 0 < h < h_0 $,\n\t\\begin{equation}\\label{eq:full_error_H1_EFP}\n\t\t\\| \\psi(\\cdot, t_n) - I_N \\psihhn{n} \\|_{L^2} \\lesssim \\tau + h^3, \\  \\| \\psi(\\cdot, t_n) - I_N \\psihhn{n} \\|_{H^1} \\lesssim \\tau + h^2, \\quad 0 \\leq n \\leq T\/\\tau.\n\t\\end{equation}\n\\end{corollary}\nThe proof of \\cref{coro:error_estimates_EFP} follows the proof of \\cref{thm:error_estimates_FS} and we omit it for brevity.\n\n\\section{Numerical results}\nIn this section, we present some numerical examples for the NLSE with either low regularity potential or nonlinearity. In the following, we fix $ \\Omega = (-16, 16) $, $ T=1 $, $ d=1 $ and consider the power-type nonlinearity $ f(\\rho) = -\\rho^\\sigma\\ (\\sigma > 0) $.\n\nLet $ \\psihn{n} (0 \\leq n \\leq T\/\\tau) $ be the numerical solution obtained by the EWI-FS method \\cref{EWI-FS} or the EWI-EFP method \\cref{EWI-EFP}, which will be made clear in each case. Define the error functions\n\\begin{equation*}\n\te^k_{L^2} = \\| \\psi(t_k) - I_N \\psihn{k}  \\|_{L^2}, \\quad e^k_{H^1} = \\| \\psi(t_k) - I_N \\psihn{k} \\|_{H^1}, \\quad 0 \\leq k \\leq n:=T\/\\tau.\n\\end{equation*}\n\n\\subsection{For the NLSE with locally Lipschitz nonlinearity}\nIn this subsection, we only consider the NLSE with the power-type nonlinearity and without potential:\n\\begin{equation}\\label{eq:NLSE_semi-smooth}\n\ti \\partial_t \\psi(x, t) = -\\Delta \\psi(x, t) - |\\psi(x, t)|^{2\\sigma} \\psi(x, t), \\quad x \\in \\Omega, \\quad t>0,\n\\end{equation}\nwhere $ \\sigma > 0 $. Recall that Assumption \\cref{A} is satisfied for any $ \\sigma > 0 $ and Assumption \\cref{B} is satisfied for any $ \\sigma \\geq 1\/2 $. Note that when there is no potential, the extended Fourier pseudospectral method collapses to the standard Fourier pseudospectral method.\n\nTwo types of initial data are considered:\n\n(i) Type I. The $ H^2 $ initial datum\n\t\\begin{equation}\\label{typeI_ini}\n\t\t\\psi_0(x) = x|x|^{0.51} e^{-\\frac{x^2}{2}}, \\quad x \\in \\Omega.\n\t\\end{equation}\n\t\n(ii) Type II. The smooth initial datum\n\t\\begin{equation}\\label{typeII_ini}\n\t\t\\psi_0(x) = x e^{-\\frac{x^2}{2}}, \\quad x \\in \\Omega.\n\t\\end{equation}\n\n\nThe two initial data are specially chosen to demonstrate the influence of the low regularity of $ f $ around the origin. Since both Type I and II initial data are odd functions, the corresponding solutions of the NLSE will satisfy $ \\psi(0, t) \\equiv 0 $ for all $ t \\geq 0 $. The difference of these two initial data lies in the regularity.\n\nWe shall test the convergence order in both time and space for Type I and II initial data. For each initial datum, we choose $ \\sigma = 0.1, 0.2, 0.3, 0.4 $. The 'exact' solutions are computed by the Strang splitting Fourier pseudospectral method with $ \\tau = \\tau_\\text{e} := 10^{-6} $ and $ h = h_\\text{e} := 2^{-9} $.\n\nWe start with the Type I $ H^2 $ initial datum \\cref{typeI_ini}. \\cref{fig:comp_FS_FP_semi_smooth_H2_ini} exhibits the spatial error in $ L^2 $- and $ H^1 $-norm of the EWI-FS (solid lines) and the EWI-EFP (dotted lines) method for $ \\sigma = 0.1 $ with the Type I initial datum. We can observe that the EWI-FS method is second order convergent in $ L^2 $-norm and first order convergent in $ H^1 $-norm. Moreover, we see that there is almost no difference between the spatial error of the EWI-FS method and the EWI-EFP method, which suggests that the Fourier pseudospectral method seems also suitable to discretize the low regularity nonlinearity.\n\n\\cref{fig:conv_dt_semi_smooth_H2_ini} plots the temporal error in $ L^2 $- and $ H^1 $-norm of the EWI for different $ 0<\\sigma<1\/2 $ with Type I initial datum. \\cref{fig:conv_dt_semi_smooth_H2_ini} (a) shows that the temporal convergence is first order in $ L^2 $-norm for all the four $ \\sigma $ and \\cref{fig:conv_dt_semi_smooth_H2_ini} (b) shows the temporal convergence is half order in $ H^1 $-norm for all the four $ \\sigma $.\n\nThe results in \\cref{fig:comp_FS_FP_semi_smooth_H2_ini,fig:conv_dt_semi_smooth_H2_ini} confirm our optimal $ L^2 $-norm error bound for the NLSE with locally Lipschitz nonlinearity, and demonstrate that it is sharp.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.475\\textwidth]{figures\/conv_h_sigma_01_comp_FS_FP_xexp_H2_ini}\n\t\\caption{Comparison of the Fourier spectral and pseudospectral discretization of the nonlinear term in \\cref{eq:NLSE_semi-smooth} with $ \\sigma = 0.1 $ and Type I initial datum \\cref{typeI_ini}.}\n\t\\label{fig:comp_FS_FP_semi_smooth_H2_ini}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_xexp_H2_ini_L2_error}\n\t\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_xexp_H2_ini_H1_error}\n\t\\caption{Temporal errors of the EWI for the NLSE \\cref{eq:NLSE_semi-smooth} with Type I initial datum \\cref{typeI_ini}: (a) $L^2$-norm errors, and (b) $H^1$-norm errors. }\n\t\\label{fig:conv_dt_semi_smooth_H2_ini}\n\\end{figure}\n\nThen we consider the Type II smooth initial datum \\cref{typeII_ini}. \\cref{fig:comp_FS_FP_semi_smooth_smooth_ini} shows the spatial error in $ L^2 $- and $ H^1 $-norm of the EWI-FS (solid lines) and the EWI-EFP (dotted lines) method for $ \\sigma = 0.1 $ with the Type II initial datum. We can observe that the convergence orders in $ H^1 $-norm of the EWI-FS (solid lines) and the EWI-EFP (dashed lines) are almost the same (roughly 2.5), though the value of the error of the EWI-FS is smaller than the EWI-EFP. While the convergence order in $ L^2 $-norm of the EWI-FS method is roughly 3.5, which is almost one order higher than that of the EWI-EFP method. This observation suggests that when the solution has better regularity, the Fourier spectral method outperforms the Fourier pseudospectral method for discretizing the low regularity nonlinearity.\n\n\\cref{fig:conv_dt_semi_smooth_smooth_ini} displays the temporal error in $ L^2 $- and $ H^1 $-norm of the EWI for different $ 0<\\sigma<1 $ with the Type II initial datum. \\cref{fig:conv_dt_semi_smooth_smooth_ini} (a) and (b) show that the temporal convergence is first order in both $ L^2 $- and $ H^1 $-norm for all the four $ \\sigma $. However, currently, we can only prove the first order $ H^1 $-convergence in time under Assumption \\cref{B} which holds only when $ \\sigma \\geq 1\/2 $. Besides, as shown in Figure 5.3 in \\cite{bao2023}, for the time-splitting methods, we can observe first order convergence in $ H^1 $-norm only when $ \\sigma \\geq 1\/2 $, which suggests that the EWI may be better than the time-splitting methods when the nonlinearity if of low regularity.\n\nThe results in \\cref{fig:comp_FS_FP_semi_smooth_smooth_ini,fig:conv_dt_semi_smooth_smooth_ini} confirm our optimal $ H^1 $-norm error bound for the NLSE with low regularity nonlinearity, but also indicates that Assumption \\cref{B} may be relaxed.\n\n\\begin{figure}[htbp]\n\t\\centering {\\includegraphics[width=0.475\\textwidth]{figures\/conv_h_sigma_01_comp_FS_FP_xexp_ini}}\n\t\\caption{Comparison of the Fourier spectral and pseudospectral discretizations of the nonlinear term in \\cref{eq:NLSE_semi-smooth} with $ \\sigma = 0.1 $ and Type II initial datum \\cref{typeII_ini}.}\n\t\\label{fig:comp_FS_FP_semi_smooth_smooth_ini}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_xexp_ini_L2_error}}\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_xexp_ini_H1_error}}\n\t\\caption{Temporal errors of the EWI for the NLSE \\cref{eq:NLSE_semi-smooth} with Type II initial datum \\cref{typeII_ini}: (a) $L^2$-norm errors, and (b) $H^1$-norm errors.}\n\t\\label{fig:conv_dt_semi_smooth_smooth_ini}\n\\end{figure}\n\n\\subsection{For the NLSE with low regularity potential}\nIn this subsection, we only consider the cubic NLSE with low regularity potential as\n\\begin{equation}\\label{eq:NLSE_Linfty_poten}\n\ti \\partial_t \\psi(x, t) = -\\Delta \\psi(x, t) + V(x)\\psi(x, t) - |\\psi(x, t)|^2 \\psi(x, t), \\quad \\mathbf{x} \\in \\Omega, \\quad t>0,\n\\end{equation}\nwhere $ V $ is chosen as either $ V_1 \\in L^\\infty(\\Omega) $ or $ V_2 \\in W^{1, 4}(\\Omega) $ defined as\n\\begin{equation}\\label{eq:potential}\n\tV_1(x) = \\left\\{\n\t\\begin{aligned}\n\t\t&-4, &x \\in (-2, 2) \\\\\n\t\t&0, &\\text{otherwise}\n\t\\end{aligned}\n\t\\right., \\qquad V_2(x) = |x|^{0.76}, \\qquad x \\in \\Omega.\n\\end{equation}\n\nWe shall test the convergence orders for the NLSE \\cref{eq:NLSE_Linfty_poten} with $ V=V_1 $ and $ \\psi_0 \\in H^2(\\Omega) $, and $ V = V_2 $ and $ \\psi_0 \\in H^3(\\Omega) $, respectively. The 'exact' solutions are computed by the EWI-EFP method with $ \\tau = \\tau_\\text{e} := 10^{-6} $ and $ h = h_\\text{e} := 2^{-9} $.\n\nWe start with the spatial error and compare the performance of the extended Fourier pseudospectral method and the standard Fourier pseudospectral (FP) method which can be obtained by replacing $ \\widehat{(V I_N \\psihhn{n})_l} $ with $ \\widetilde{(V \\psihhn{n})_l} $ in \\cref{EWI-EFP}. We remark here that, since the nonlinearity is smooth in \\cref{eq:NLSE_Linfty_poten}, the results of the EWI-FS method are almost the same as those of the EWI-EFP method.\n\n\\cref{fig:Linfty_poten_H2_ini} (a) shows the spatial error in $ L^2 $- and $ H^1 $-norm of the EWI-EFP method (solid lines) and the EWI-FP method (dotted lines) with $ V = V_1 \\in L^\\infty(\\Omega)$ given in \\cref{eq:potential} and $ \\psi_0 \\in H^2(\\Omega) $ given in \\cref{typeI_ini}. We can observe that the EWI-EFP is second order convergent in $ L^2 $-norm and first order convergent in $ H^1 $-norm in space. However, the spatial convergence order of the EWI-FP method is only first order in both $ L^2 $- and $ H^1 $-norm, and the value of the error is much larger. This implies that when discretizing purely $ L^\\infty $-potential, the extended Fourier pseudospectral method is much better than the standard Fourier pseudospectral method. \\cref{fig:Linfty_poten_H2_ini} (b) plots the temporal convergence of the EWI in $ L^2 $- and $ H^1 $-norm with the Type I initial datum. We can observe that the EWI is first order convergent in $ L^2 $-norm and half order convergent in $ H^1 $-norm in time.\n\nThe results in \\cref{fig:Linfty_poten_H2_ini} validate our optimal $ L^2 $-norm error bound for the NLSE with $L^\\infty$-potential and demonstrate that it is sharp.\n\n\\begin{figure}[htbp]\n\t\\centering\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_h_sigma_1_comp_FS_FP_xexp_H2_ini}}\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_sigma_1_xexp_H2_ini}}\n\t\\caption{Convergence tests of the EWI for \\cref{eq:NLSE_Linfty_poten} with $ V=V_1 \\in L^\\infty(\\Omega) $ and $ \\psi_0 \\in H^2(\\Omega) $: (a) spatial errors\nof the Fourier spectral and pseudospectral discretizations for the linear potential,\nand  (b) temporal errors in $L^2$-norm and $H^1$-norm. }\n\t\\label{fig:Linfty_poten_H2_ini}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\t\\centering\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_h_sigma_1_comp_FS_FP_H3_ini}}\n\t{\\includegraphics[width=0.475\\textwidth]{figures\/conv_dt_sigma_1_H3_ini}}\n\t\\caption{Convergence tests of the EWI for \\cref{eq:NLSE_Linfty_poten} with\n$ V=V_2 \\in W^{1, 4}(\\Omega) $ and $ \\psi_0 \\in H^3(\\Omega) $: (a) spatial errors\nof the Fourier spectral and pseudospectral discretizations for the linear potential,\nand  (b) temporal errors in $L^2$-norm and $H^1$-norm. }\n\t\\label{fig:W14_poten_H3_ini}\n\\end{figure}\n\n\\cref{fig:W14_poten_H3_ini} (a) shows the spatial error in $ L^2 $- and $ H^1 $-norm of the EWI-EFP method (solid lines) and the EWI-FP method (dotted lines) with $ V = V_2 \\in W^{1, 4}(\\Omega)$ given in \\cref{eq:potential} and $ \\psi_0 \\in H^3(\\Omega) $ given by $ \\psi_0(x) = (1+|x|^{2.51})e^{-x^2\/2} $. We can observe that the EWI-EFP is third order convergent in $ L^2 $-norm and second order convergent in $ H^1 $-norm in space. However, the spatial convergence order of the EWI-FP method is only $ 1.7 $ order in both $ L^2 $- and $ H^1 $-norm, and the value of the error is much larger. This implies again that the extended Fourier pseudospectral method is much better than the standard Fourier pseudospectral method when the potential is of low regularity. \\cref{fig:W14_poten_H3_ini} (b) plots the temporal convergence of the EWI in $ L^2 $- and $ H^1 $-norm with the $ H^3 $ initial datum. We can observe that the EWI is first order convergent in $ H^1 $-norm in time for $ V \\in W^{1, 4}(\\Omega) $.\n\nThe results in \\cref{fig:W14_poten_H3_ini} validate our optimal $ H^1 $-norm error bound for the NLSE with $ W^{1, 4} $-potential and demonstrate that it is sharp.\n\n\n\n\\section{Conclusions}\nWe established optimal error bounds for the first-order Gautschi-type exponential wave integrator (EWI) applied to the nonlinear Schr\\\"odinger equation (NLSE) with $ L^\\infty $-potential and\/or locally Lipschitz nonlinearity under the assumption of $ H^2 $-solution. For the semi-discretization in time by the first-order Gautschi-type EWI, we proved an optimal $ L^2 $-norm error bound at $ O(\\tau) $ and a uniform $ H^2 $-bound of the numerical solution. For the full discretization obtained from the semi-discretization by using the Fourier spectral method in space,  we proved an optimal $ L^2 $-norm error bound at $ O(\\tau + h^2) $ without any coupling condition between $ \\tau $ and $ h $. For $ W^{1, 4} $-potential and a little stronger regularity assumption for the nonlinearity, under the assumption of $ H^3 $-solution of the NLSE, we proved optimal $ H^1 $-norm error bounds for both the semi-discretization and the full discretization. As a by-product, we proposed an extended Fourier pseudospectral method to implement the full discretization when the potential is of low regularity and the nonlinearity is smooth,\nin which the potential and nonlinearity were discretized by the Fourier\nspectral method and the Fourier pseudospectral method, respectively.\nThe proposed numerical implementation has similar computational cost as the standard\nFourier pseudospectral method, but we can establish rigorous error bounds for this method. On the contrary, one cannot establish optimal error bounds for the\nstandard Fourier pseudospectral method for the NLSE when the potential is of low regularity, e.g. $V\\in L^\\infty$. In the future, we will consider even weaker potential, e.g. $V\\in L^1$, including Coulomb potential and\/or spatial\/temporal\nDirac delta potential.\n\n\n\\bibliographystyle{siamplain}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nIn the Standard Model (SM),\n  the electroweak symmetry breaking is realized by the negative mass term in the Higgs potential,\n  which seems to be artificial because there is nothing to stabilize the electroweak scale.\nIf new physics takes place at a very high energy, e.g. the Planck scale,\n  the mass term receives large corrections which are quadratically sensitive to the new physics scale, \n  so that the electroweak scale is not stable against the corrections.  \nThis is the so-called gauge hierarchy problem.\nIt is well known that supersymmetry (SUSY) can solve this problem. \nSince the mass corrections are completely canceled by the SUSY partners,\n  no fine-tuning is necessary to reproduce the electroweak scale correctly, \n  unless the SUSY breaking scale is much higher than the electroweak scale.\nOn the other hand, since no indication of SUSY particles has been obtained in the large hadron collider (LHC) experiments, \n  one may consider other solutions to the gauge hierarchy problem without SUSY. \n\n\nIn this direction, recently a lot of works have been done in models based on a classically conformal symmetry,\n  where an additional $U(1)$ gauge symmetry, e.g., $U(1)_{B-L}$, is added\n  \\cite{Hempfling:1996ht}-\\cite{Plascencia:2015xwa}. \nThis direction is based on the argument by Bardeen~\\cite{Bardeen:1995kv} that \n  the quadratic divergence in the Higgs mass corrections can be subtracted \n  by a boundary condition of some ultraviolet complete theory, \n  which is classically conformal, and only logarithmic divergences should be considered\n  (see Ref.~\\cite{Iso:2012jn} for more detailed discussions). \nIf this is the case, imposing the classically conformal symmetry to the theory \n  is another way to solve the gauge hierarchy problem. \nSince there is no dimensionful parameter in this class of models,\n   the gauge symmetry must be broken by quantum corrections. \nThis structure fits the model first proposed by Coleman and Weinberg~\\cite{Coleman:1973jx}, \n   where a model is defined as a massless theory and the gauge symmetry is radiatively broken \n   by the Coleman-Weinberg (CW) mechanism, generating  a mass scale through the dimensional transmutation. \n\n\nIn this paper we propose a classically conformal $U(1)_{B-L}$ extended SM with two Higgs doublets.  \nAn SM singlet, $B-L$ Higgs field develops its vacuum expectation value (VEV) by the CW mechanism, \n  and the $U(1)_{B-L}$ symmetry is radiatively broken. \nThis gauge symmetry breaking also generates the mass terms for the two Higgs doublets \n  through quartic couplings between the two Higgs doublets and the $B-L$ Higgs field. \nWe assume the quartic couplings to be all positive but, nevertheless, the electroweak symmetry breaking \n  is triggered through the so-called bosonic seesaw mechanism~\\cite{Calmet:2002rf,Kim:2005qb,Haba:2005jq}, \n  which is analogous to the seesaw mechanism for the neutrino mass generation \n  and leads to a negative mass squared for the SM-like Higgs doublet.  \nA large hierarchy among the quartic Higgs couplings is crucial for the bosonic seesaw mechanism \n  to work at the $U(1)_{B-L}$ symmetry breaking scale. \nAlthough it seems unnatural to introduce the large hierarchy by hand,    \n  we find that  the renormalization group evolutions of the quartic Higgs couplings \n  dramatically reduce the large hierarchy toward high energies. \nTherefore,  once our model is defined at some high energy, say, the Planck scale, \n   the large hierarchy at the $U(1)_{B-L}$ symmetry breaking scale \n   is naturally realized by a mild hierarchy.\nWe also show that the perturabativity of model couplings and the electroweak \n   vacuum stability are maintained up to the Planck scale with a suitable choice \n   of the input parameters. \n From the naturalness of the electroweak scale, we find the $U(1)_{B-L}$ gauge symmetry breaking \n   scale to be $\\lesssim 100$ TeV, which predicts extra heavy Higgs boson masses to be $\\lesssim 2$ TeV. \nSuch heavy Higgs boson can be tested at the LHC in the near future. \n\n\nIn the next section, we will define our model, and discuss the $U(1)_{B-L}$ symmetry breaking \n  as well as the electroweak symmetry breaking by the bosonic seesaw mechanism. \nWe also present the mass spectrum of the model. \nIn Sec.~\\ref{sec:result}, we will analyze the renormalization group evolutions for all couplings of the model  \n  and present our numerical results. \nWe will see that the hierarchy among the quartic Higgs couplings is dramatically reduced \n  toward high energies. \nSec.~\\ref{sec:conclusion} is devoted to conclusion.  \n\n\n\n\\section{Model}\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\\hline\n & $SU(3)_c \\otimes SU(2)_L \\otimes U(1)_Y$ & $U(1)_{B-L}$ \\\\\n\\hline \\hline\n$Q^i$ & (3, 2, 1\/6) & $1\/3$\\\\\n$U^i$ & (3, 1, $2\/3$) & $1\/3$\\\\\n$D^i$ & (3, 1, $-1\/3$) & $1\/3$\\\\\n$L^i$ & (1, 2, $-1\/2$) & $-1$\\\\\n$E^i$ & (1, 1, $-1$) & $-1$\\\\\n$N^i$ & (1, 1, 0) & $-1$\\\\\n$H_1$ & (1, 2, 1\/2) & $0$\\\\\n$H_2$ & (1, 2, 1\/2) & $4$\\\\\n$\\Phi$ & (1, 1, 0) & $2$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Particle contents in our model. \n$i=1,2,3$ is the generation index.}\n\\label{table:1}\n\\end{table}\n\n\nWe consider an extension of the SM with an additional $U(1)_{B-L}$ gauge symmetry. \nThe particle contents of our model are listed in Table \\ref{table:1},\n  where two Higgs doublets ($H_1$ and  $H_2$) and one SM singlet, $B-L$ Higgs field ($\\Phi$) are introduced. \nAs is well known, the introduction of the three right-handed neutrinos ($N^i$, $i=1$, 2, 3)\n is crucial to make the model free from all the gauge and gravitational anomalies. \nIn addition, we impose a classically conformal symmetry to the model, \n  under which the scalar potential is given by\n\\begin{eqnarray}\n\tV &=& \\lambda_1 |H_1|^4 + \\lambda_2 |H_2|^4 + \\lambda_3 |H_1|^2 |H_2|^2 + \\lambda_4 (H_2^\\dagger H_1) (H_1^\\dagger H_2) + \\lambda_\\Phi |\\Phi|^4 \\nonumber \\\\\n\t&& + \\lambda_{H1\\Phi} |H_1|^2 |\\Phi|^2 + \\lambda_{H2\\Phi} |H_2|^2 |\\Phi|^2 + \\left( \\lambda_{\\rm mix} (H_2^\\dagger H_1) \\Phi^2 + h.c. \\right).\n\\end{eqnarray}\nHere, all of the dimensionful parameters are prohibited by the classically conformal symmetry.\nIn this system, the $U(1)_{B-L}$ symmetry must be radiatively broken by quantum effects, i.e., the CW mechanism.\nThe CW potential for $\\Phi$ is described as\n\\begin{eqnarray}\n\tV_\\Phi(\\phi) = \\frac{1}{4} \\lambda_\\Phi(v_\\Phi)\\, \\phi^4\n\t\t\t\t\t\t+ \\frac{1}{8} \\beta_{\\lambda_{\\Phi}}(v_\\Phi)\\, \\phi^4\n\t\t\t\t\t\t\t\\left( \\ln \\frac{\\phi^2}{v_\\Phi^2} -\\frac{25}{6} \\right),\n\\label{CWpotential}\n\\end{eqnarray}\n where $\\Re[\\Phi] = \\phi\/\\sqrt{2}$, and $v_\\Phi=\\langle \\phi \\rangle$ is the VEV of $\\Phi$.\nWhen the beta function $\\beta_{\\lambda_\\Phi}$ is dominated by\n  the $U(1)_{B-L}$ gauge coupling ($g_{B-L}$) and the Majorana Yukawa couplings of right-handed neutrinos ($Y_M$) \n  as shown in Appendix, the minimization condition of $V_\\Phi$ approximately leads to\n\\begin{eqnarray}\n\t\\lambda_\\Phi \\simeq \\frac{11}{6 \\pi^2}\n\t\t\\left( 6 g_{B-L}^4 - {\\rm tr} Y_M^4 \\right),\n\\label{CW_relation}\n\\end{eqnarray}\n  where all parameters are evaluated at $v_\\Phi$.  \nThrough the $U(1)_{B-L}$ symmetry breaking, the mass terms of the two Higgs doublets arise \n  from the mixing terms between $H_{1,2}$ and $\\Phi$, and the scalar mass squared matrix is read as \n\\begin{eqnarray}\n\t-\\mathcal{L}&=& \\frac{1}{2}\n\t\t\t\t\t(H_1, H_2) \\left( \\begin{array}{cc}\n\t\t\t\t\t\\lambda_{H1\\Phi} v_\\Phi^2 & \\lambda_{\\rm mix} v_\\Phi^2 \\\\\n\t\t\t\t\t\\lambda_{\\rm mix} v_\\Phi^2 & \\lambda_{H2\\Phi} v_\\Phi^2\n\t\t\t\t\\end{array} \\right)\n\t\\left( \\begin{array}{c} H_1 \\\\ H_2 \\end{array} \\right) \\nonumber\\\\\n\t&\\approx& \\frac{1}{2}\n\t\t\t\t\t(H'_1, H'_2) \\left( \\begin{array}{cc}\n\t\t\t\t\t\\lambda_{H1\\Phi} v_\\Phi^2 - \\frac{\\lambda_{\\rm mix}^2}{\\lambda_{H2\\Phi}} v_\\Phi^2 & 0 \\\\\n\t\t\t\t\t0 & \\lambda_{H2\\Phi} v_\\Phi^2\n\t\t\t\t\\end{array} \\right)\n\t\\left( \\begin{array}{c} H'_1 \\\\ H'_2 \\end{array} \\right),\n\\label{matrix}\n\\end{eqnarray}\n where we have assumed a hierarchy among the quartic couplings as\n $0 \\leq  \\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$ at the scale $\\mu = v_\\Phi$.\\footnote{\n  In our analyses, we will take boundary conditions as $\\lambda_1(v_\\Phi)=\\lambda_2(v_\\Phi)=\\lambda_H(v_\\Phi)$,\n  for simplicity, where $\\lambda_H$ is a Higgs quartic coupling in the SM.\n  }\nIn the next section, we will show that this hierarchy is dramatically reduced toward high energies \n  in their renormalization group evolutions. \nBecause of this hierarchy,\n  mass eigenstates $H'_1$ and $H'_2$ are almost composed of $H_1$ and $H_2$, respectively.\nHence, we approximately identify $H'_1$ with the SM-like Higgs doublet. \nNote that even though all quartic couplings are positive, \n  the SM-like Higgs doublet obtains a negative mass squared for $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix}^2\/\\lambda_{H2\\Phi}$, \n  and hence the electroweak symmetry is broken. \nThis is the so-called bosonic seesaw mechanism \\cite{Calmet:2002rf,Kim:2005qb, Haba:2005jq}.\n\n\nIn more precise analysis for the electroweak symmetry breaking, we take into account \n   a scalar one-loop diagram through the quartic couplings, $\\lambda_3$ and $\\lambda_4$, \n   shown in Fig.~\\ref{one-loop}, and \n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[scale=0.7,clip]{higgs_corr.eps}\n\\end{center}\n\\caption{Scalar one-loop diagram which contributes to the SM-like Higgs doublet mass.}\n\\label{one-loop}\n\\end{figure}\n the SM-like Higgs doublet mass is given by\n\\begin{eqnarray}\n\tm_h^2 &\\simeq& -\\frac{\\lambda_{H1\\Phi}}{2} v_\\Phi^2 + \\frac{\\lambda_{\\rm mix}^2}{2\\lambda_{H2\\Phi}} v_\\Phi^2\n\t\t\t\t\t+ \\frac{\\lambda_{H2\\Phi}}{16\\pi^2} (2\\lambda_3 + \\lambda_4) v_\\Phi^2 \\nonumber\\\\\n\t\t\t&\\simeq& \\lambda_{H2\\Phi} v_\\Phi^2 \\left[ \n\t\t\t\t\\frac{1}{2} \\left(\\frac{\\lambda_{\\rm mix}}{\\lambda_{H2\\Phi}}\\right)^2\n\t\t\t\t+ \\frac{2\\lambda_3 + \\lambda_4}{16\\pi^2}  \\right],\n\\label{Higgs}\n\\end{eqnarray}\n  where we have omitted the $\\lambda_{H1\\Phi}$ term in the second line, and \n  the observed Higgs boson mass $M_h=125$ GeV is given by $M_h=m_h\/\\sqrt{2}$. \n\n\nIn addition to the scalar one-loop diagram, \n  one may consider other Higgs mass corrections coming from a neutrino one-loop diagram and \n  two-loop diagrams involving the $U(1)_{B-L}$ gauge boson ($Z'$) and the top Yukawa coupling, \n  which are, respectively, found to be~\\cite{Iso:2009ss}\n\\begin{eqnarray}\n\t\t\\delta m_h^2 \\sim \\frac{Y_\\nu^2 Y_M^2 v_\\Phi^2}{16 \\pi^2},\\qquad\n\t\t\\delta m_h^2 \\sim \\frac{y_t^2 g_{B-L}^4 v_\\Phi^2}{(16 \\pi^2)^2},\n\\label{deltam}\n\\end{eqnarray}\n  where $Y_\\nu$ and $y_t$ are Dirac Yukawa couplings of neutrino and top quark, respectively.  \nIt turns out that these contributions are negligibly small\n compared to the scalar one-loop correction in Eq.~(\\ref{Higgs}). \nAs we will discuss in the next section, the quartic couplings $\\lambda_3$ and $\\lambda_4$ \n  should be sizable $\\lambda_{3,4} \\gtrsim 0.15$ in order to stabilize the electroweak vacuum. \nThe neutrino one-loop correction is roughly proportional to the active neutrino mass \n  by using the seesaw relation,  and it is highly suppressed by the lightness of the neutrino mass.\nThe two-loop corrections with the $Z'$ boson is suppressed by a two-loop factor $1\/(16\\pi^2)^2$. \nUnless $g_{B-L}$ is large, the two-loop corrections are smaller than the scalar one-loop correction. \nIn Table~\\ref{table:QC}, we summarize typical orders of magnitude for the three corrections for $v_\\Phi=10$ and $100$ TeV. \nFor the light neutrino mass, we have adopted the seesaw relation, $m_\\nu \\sim (Y_\\nu v_H)^2\/(Y_M v_\\Phi)\\sim 0.1$ eV, \n  with the SM-like Higgs field VEV, $v_H=246$ GeV. \nFor both $v_\\Phi=10$ and $100$ TeV, we have fixed \n   $\\lambda_3=\\lambda_4=0.15$, $g_{B-L}=0.15$ and  $Y_\\nu=2.0 \\times 10^{-6}$, \n   while we have used  $\\lambda_{H 2 \\Phi}=0.01$ ($10^{-4}$) and $Y_M=0.23$ ($0.023$) \n   for $v_\\Phi=10$ ($100$) TeV. \n\n\nThe other scalar masses are approximately given by\n\\begin{eqnarray}\n\tM_\\phi^2 &=& \\frac{6}{11} \\lambda_\\Phi v_\\Phi^2, \\\\\n\tM_H^2 &=& M_A^2 = \\lambda_{H2\\Phi} v_\\Phi^2 + (\\lambda_3 + \\lambda_4) v_H^2, \\\\\n\tM_{H^\\pm}^2 &=& \\lambda_{H2\\Phi} v_\\Phi^2 + \\lambda_3 v_H^2, \n\\end{eqnarray}\n where  $M_\\phi$ is the mass of the SM singlet scalar,\n $M_H$ ($M_A$) is the mass of CP-even (CP-odd) neutral Higgs boson,\n and $M_{H\\pm}$ is the mass of charged Higgs boson. \nThe extra heavy Higgs bosons are almost degenerate in mass.   \nThe masses of the $Z'$ boson and the right-handed neutrinos are given by\n\\begin{eqnarray}\n\tM_{Z'} &=& 2 g_{B-L} v_\\Phi,\n\\label{MZ} \\\\\n\tM_N &=& \\sqrt{2} y_M v_\\Phi\n\t\t\\simeq \\left[ \\frac{3}{2N_\\nu} \\left( 1 - \\frac{\\pi^2 \\lambda_\\Phi}{11 g_{B-L}^4} \\right) \\right]^{1\/4} M_{Z'},\n\\label{MN}\n\\end{eqnarray}\n where we have used ${\\rm tr}Y_M=N_\\nu y_M$, for simplicity,\n and $N_\\nu$ stands for the number of relevant Majorana couplings.\nIn the following analysis, we will take $N_\\nu = 1$ for simplicity,\n  because our final results are almost insensitive to $N_\\nu$. \nIn the last equality in Eq.~(\\ref{MN}), we have used Eq.~(\\ref{CW_relation}).\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\\hline\n$v_\\Phi$ & $10\\,{\\rm TeV}$ & $100\\,{\\rm TeV}$ \\\\ \\hline \\hline\n1-loop with scalar & $\\sim (50\\,{\\rm GeV})^2$ &  $\\sim (50\\,{\\rm GeV})^2$\\\\ \\hline\n2-loop with $Z'$ &  $(\\mathcal{O}(1)\\,{\\rm GeV})^2$ & $(\\mathcal{O}(10)\\,{\\rm GeV})^2$\\\\ \\hline\n1-loop with neutrino &  $(\\mathcal{O}(10^{-3})\\,{\\rm GeV})^2$ & $(\\mathcal{O}(10^{-3})\\,{\\rm GeV})^2$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Typical orders of magnitude of the quantum corrections to the SM-like Higgs doublet mass.}\n\\label{table:QC}\n\\end{table}\n\n\n\\section{Numerical results} \\label{sec:result}\n\nBefore presenting our numerical results,\n   we first discuss constraints on the model parameters from the perturbativity  \n   and the stability of the electroweak vacuum in the renormalization group evolutions. \nIn our analysis, all values of couplings are given at $\\mu=v_\\Phi$.\nFor $v_\\Phi$ at the TeV scale, \n   we find the constraint $g_{B-L}\\lesssim 0.3$ to avoid the Landau pole of the gauge coupling \n   below the Planck scale, while a more severe constraint $g_{B-L} \\lesssim 0.2$ is obtained \n   to avoid a blowup of  the quartic coupling $\\lambda_2$ below the Planck scale. \nFrom $g_{B-L} \\lesssim 0.2$ and\n the experimental bound $M_{Z'} > 2.9$ TeV on the $Z'$ boson mass \\cite{Aad:2014cka,Khachatryan:2014fba},\n we find $v_\\Phi > 7.25$\\,TeV.\nThe electroweak vacuum stability, in other words, $\\lambda_H(\\mu) > 0$ for any scales between  \n  the electroweak scale and the Planck scale, \n  can be realized by sufficiently large $\\lambda_3$ and\/or $\\lambda_4$\n  as $\\lambda_3 =\\lambda_4 \\gtrsim 0.15$.\nTo keep their perturbativity below the Planck scale, \n $\\lambda_3 =\\lambda_4 \\lesssim 0.48$ must be satisfied,\n while we will find that the naturalness of the electroweak scale leads to a more severe upper bound.\n\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[clip, height=6cm]{bound.eps} \\hspace{0.5cm}\n\t\\includegraphics[clip, height=6cm]{Heavy_mass.eps}\n\\end{center}\n\\caption{\nThe relation between $v_\\Phi$ and $\\lambda_{H2\\Phi}$ through  Eq.~(\\ref{Higgs}) (left panel), \n  and the corresponding extra heavy Higgs boson mass spectrum (right panel).  \nThe red and blue lines correspond to  $\\lambda_{\\rm mix}=0$ and\n $\\lambda_{\\rm mix} = 0.1\\times \\lambda_{H2\\Phi}$, respectively. \nThe shaded region shows the perturbativity bound for $g_{B-L}=0.2$.\nThe vertical lines show the upper bound of $v_\\Phi$,\n at which Higgs mass corrections from the two loop diagrams with the $U(1)_{B-L}$ gauge boson \n become $(10\\,{\\rm GeV})^2$ for $g_{B-L}=0.1$ (left) and $g_{B-L}=0.01$ (right), respectively.\n}\n\\label{bound}\n\\end{figure}\n\nTo realize the hierarchy $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$,\n  we take $\\lambda_{H1\\Phi}=0$, for simplicity. \nWhen we consider $\\lambda_{\\rm mix}$ in the range of $0< \\lambda_{\\rm mix} < 0.1\\times \\lambda_{H2\\Phi}$,\n   the relation between $v_\\Phi$ and $\\lambda_{H2\\Phi}$ obtained by Eq.~(\\ref{Higgs}) \n   is shown in the left panel of Fig.~\\ref{bound}. \nHere, we have fixed $\\lambda_3 =\\lambda_4 = 0.15$ as an example.\nThe red and blue lines correspond to  the lowest value $\\lambda_{\\rm mix}=0$ and\n   the highest value $\\lambda_{\\rm mix} = 0.1\\times \\lambda_{H2\\Phi}$, respectively.\\footnote{\nAlthough the bosonic seesaw mechanism does not work for $\\lambda_{\\rm mix}=0$, \n   one may consider the electroweak symmetry breaking through the scalar loop correction shown in Fig.~\\ref{one-loop}.\n}\nThe shaded region shows the perturbativity bound $v_\\Phi > 7.25$\\,TeV.\nThe vertical lines show the upper bound of $v_\\Phi$,\n at which two-loop corrections with the $Z'$ boson to the Higgs mass become $(10\\,{\\rm GeV})^2$\n for $g_{B-L}=0.1$ (left) and $g_{B-L}=0.01$ (right), respectively.\nNote that $\\lambda_{H2\\Phi} v_\\Phi^2$ is almost constant.\nSince all heavy Higgs boson masses are approximately determined by $\\lambda_{H2\\Phi} v_\\Phi^2$,\n  they are almost independent of $v_\\Phi$,\n  as is shown in the right panel of Fig.~\\ref{bound}.\nThe heavy Higgs boson masses lie in the range between 1\\,TeV and 1.7\\,TeV,\n  which can be tested at the LHC in the near future. \n\n\nIn Eq.~(\\ref{Higgs}), it may be natural for the first term from the tree-level couplings \n   dominates over the second term from the 1-loop correction. \nThis naturalness leads to the constraint of  $\\lambda_3 = \\lambda_4 < 0.26$,\n   which is more severe than the perturbativity bound $\\lambda_3 =\\lambda_4 \\lesssim 0.48$ discussed above.\nThis condition is equivalent to the fact that the origin of the negative mass term mainly comes from\n  the diagonalization of the scalar mass squared matrix in Eq.~(\\ref{matrix}),\n  namely, the bosonic seesaw mechanism.\n\n\\begin{figure}[t]\n\\begin{center}\n          \\includegraphics[clip, height=6cm]{10TeV.eps} \\hspace{0.5cm}\n          \\includegraphics[clip, height=6cm]{100TeV.eps} \n\\end{center}\n\\caption{\nRenormalization group evolutions of the quartic couplings for $v_\\Phi=10$\\,TeV (left) and 100\\,TeV (right).\nThe red, green, and blue lines correspond to\n $\\lambda_{H1\\Phi}$, $\\lambda_{H2\\Phi}$ and $\\lambda_{\\rm mix}$, respectively.\nThe rightmost vertical line shows the reduced Planck scale.\n}\n\\label{running}\n\\end{figure}\n\nNow we present the results of our numerical analysis. \nIn Fig.~\\ref{running}, we show the renormalization group evolutions of the quartic couplings.\nHere, we have taken $\\lambda_{H1\\Phi} = 0$,\n and $\\lambda_{H2\\Phi} = 10^{-2}$ and $10^{-4}$\n for $v_\\Phi=10$\\,TeV (left panel) and 100\\,TeV (right panel), respectively.\nThe red, green, and blue lines correspond to the running of\n $\\lambda_{H1\\Phi}$, $\\lambda_{H2\\Phi}$ and $\\lambda_{\\rm mix}$, respectively.\nThe rightmost vertical line denotes the reduced Planck scale $M_{Pl} = 2.4 \\times 10^{18}$ GeV.\nIn this plot, the other input parameters have been set\n  as $g_{B-L}  = 0.17$ and $\\lambda_3  = \\lambda_4  = 0.17$\n  to realize the electroweak vacuum stability without the Landau pole, and $\\lambda_\\Phi  = 10^{-3}$.\nThe value of $\\lambda_1=\\lambda_2=\\lambda_H$ at $\\mu=v_\\Phi$ has been evaluated by \n  extrapolating the SM Higgs quartic coupling with $M_h=125$ GeV from the electroweak scale to $v_\\Phi$. \nFor this parameter choice,\n the $Z'$ boson and the right-handed neutrinos have the masses of the same order of magnitude as\n $M_{Z'}=3.4$ $(34)$ TeV and $M_N=2.0$  $(20)$ TeV for $v_\\Phi=10$ $(100)$ TeV,\n while the $B-L$ Higgs boson mass is calculated as $M_\\phi=0.23$ $(2.3)$ TeV. \nAs is well-known, $M_\\phi \\ll M_{Z'}$ is a typical prediction of the CW mechanism.\nThe masses of the heavy Higgs bosons are roughly 1\\,TeV for both $v_\\Phi=10$\\,TeV and 100\\,TeV.\n\n\nIn order for the bosonic seesaw mechanism to work,\n we have assumed the hierarchy among the quartic couplings as \n $\\lambda_{H1\\Phi} \\ll \\lambda_{\\rm mix} \\ll \\lambda_{H2\\Phi}$ at the scale $\\mu=v_\\Phi$.\nOne may think it unnatural to introduce this large hierarchy by hand.\nHowever, we find from Fig.~\\ref{running} that the large hierarchy \n  between $\\lambda_{H1\\Phi}$ and $\\lambda_{H2\\Phi}$ tends to disappear toward high energies.\nThis is because the beta functions of the small couplings $\\beta_{\\lambda_{H1\\Phi}}$ and $\\beta_{\\lambda_{H2\\Phi}}$ \n  are not simply proportional to themselves, but include terms given by other sizable couplings \n  (see Appendix for the explicit formulas of their beta functions).  \nThis behavior of reducing the large hierarchy in the renormalization group evolutions  \n   is independent of the choice of the boundary conditions for $g_{B-L}$, $\\lambda_3$, $\\lambda_4$  and $\\lambda_\\Phi$. \nTherefore, Fig.~\\ref{running} indicates that once our model is defined at some high energy, say, the Planck scale, \n   the large hierarchy among the quartic couplings, which is crucial for the bosonic seesaw mechanism to work, \n   is naturally achieved from a mild hierarchy at the high energy. \n\n\n\\begin{table}[t]\n\\begin{center}\n\\begin{tabular}{|c|cc|}\\hline\n & $SU(3)_c \\otimes SU(2)_L \\otimes U(1)_Y$ & $U(1)_{B-L}$ \\\\\n\\hline \\hline\n$S_{L,R}$ & (1, 1, 0) & $x$\\\\\n$S'_{L,R}$ & (1, 1, 0) & $x-2$\\\\\n$D_{L,R}$ & (1, 2, 1\/2) & $x$\\\\\n$D'_{L,R}$ & (1, 2, 1\/2) & $x+2$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Additional vector-like fermions. $x$ is a real number.}\n\\label{table:3}\n\\end{table}\n\n\nWe see in Fig.~\\ref{running} that $\\lambda_{\\rm mix}$ is almost unchanged.\nThis is because $\\beta_{\\lambda_{\\rm mix}}$ is proportional to $\\lambda_{\\rm mix}$, \n  which is very small (see Appendix).\nHence, the hierarchy between $\\lambda_{\\rm mix}$ and the other couplings gets enlarged at high energies.\nTo avoid this situation and make our model more natural, \n   one may introduce additional vector-like fermions listed in Table \\ref{table:3}, for example.\\footnote{\nAs another possibility, one may think that some symmetry forbids the $\\lambda_{\\rm mix}$ term\n  and it is generated via a small breaking.}\nAlthough $x$ is an arbitrary real number, we assume $x\\neq 1$ to distinguish the new fermions from the SM leptons.\nThese fermions have Yukawa couplings as\n\\begin{eqnarray}\n\t-{\\mathcal L}_V &=&  Y_{SS} \\overline{S_L} \\Phi S'_R + Y_{SD} \\overline{S'_R} H_2^\\dagger D'_L\n\t\t\t\t\t+ Y_{DD} \\overline{D'_L} \\Phi D_R + Y_{DS} \\overline{D_R} H_1 S_L \\nonumber \\\\\n\t\t\t\t\t&& + Y'_{SS} \\overline{S_R} \\Phi S'_L + Y'_{SD} \\overline{S'_L} H_2^\\dagger D'_R\n\t\t\t\t\t+ Y'_{DD} \\overline{D'_R} \\Phi D_L + Y'_{DS} \\overline{D_L} H_1 S_R\n\t\t\t\t\t+ h.c.,\n\\end{eqnarray}\n so that $\\beta_{\\lambda_{\\rm mix}}$ includes\n terms of $Y_{SS} Y_{SD} Y_{DD} Y_{DS}$ and $Y'_{SS} Y'_{SD} Y'_{DD} Y'_{DS}$,\n which are not proportional to $\\lambda_{\\rm mix}$.\nThese terms originate the diagram shown in Fig.\\,\\ref{mix}.\n\\begin{figure}[t]\n\\begin{center}\n\t\\includegraphics[scale=0.7,clip]{mix_corr.eps}\n\\end{center}\n\\caption{One-loop diagram due to the additional fermions, which is relevant to $\\beta_{\\lambda_{\\rm mix}}$.}\n\\label{mix}\n\\end{figure}\nAccordingly, the minimization condition of $V_\\Phi$ is modified to\n\\begin{eqnarray}\n\t\\lambda_\\Phi \\simeq \\frac{11}{6 \\pi^2}\n\t\t\\left[ 6 g_{B-L}^4 - {\\rm tr} Y_M^4\n\t\t- \\frac{1}{8}\\left( Y_{SS}^4 + Y_{SS}^{\\prime 4} + 2Y_{DD}^4 + 2Y_{DD}^{\\prime 4} \\right) \\right].\n\\label{CW_relation2}\n\\end{eqnarray}\nFrom the conditions $\\lambda_\\Phi>0$ and $g_{B-L}<0.2$,\n the additional Yukawa contribution should satisfy\n $Y_{SS}^4 + Y_{SS}^{\\prime 4} + 2Y_{DD}^4 + 2Y_{DD}^{\\prime 4} \\lesssim 3\\times(0.4)^4$.\nNote that vector-like fermions masses are dominantly generated by $v_\\Phi$,\n and they are sufficiently heavy to avoid the current experimental bounds.\n\n\\begin{figure}[t]\n\\begin{center}\n          \\includegraphics[clip, height=7cm]{100TeV_f.eps}\n\\end{center}\n\\caption{Runnings of quartic couplings for $v_\\Phi=100$\\,TeV\n with additional vector-like fermions.\nThe input parameters are the same as before.}\n\\label{running2}\n\\end{figure}\n\n\nFig.~\\ref{running2} shows the runnings of the quartic couplings for $v_\\Phi=100$ TeV\n  with the additional vector-like fermions. \nThe input parameters are the same as before,\n  while we have taken the Yukawa couplings as\n  $Y_{SS}=Y_{SD}=Y_{DD}=Y_{DS}=0.2$ and $Y'_{SS}=Y'_{SD}=Y'_{DD}=Y'_{DS}=0.1$ at $\\mu=v_\\Phi$, \n   for simplicity. \nToward high energies, $|\\lambda_{\\rm mix}|$ becomes larger,\n   and the hierarchy with the other couplings becomes mild. \nWe can see that $\\lambda_{H1\\Phi}$ is negative below $\\mu\\simeq10^8$ GeV,\n  because the contributions of additional Yukawa couplings to $\\beta_{\\lambda_{H1\\Phi}}$ are effective\n  below $\\mu\\simeq10^8$ GeV.\nAbove the scale, the contribution of $U(1)_{B-L}$ couplings becomes effective,\n  and then $\\lambda_{H1\\Phi}$ becomes positive.\nAs a result, the large hierarchy at the $U(1)_{B-L}$ symmetry breaking scale can be realized \n   with a mild hierarchy at some high energy.  \nWe expect that a ultraviolet complete theory, which provides the origin of \n  the classical conformal invariance, takes place at the high energy.  \n  \n\n\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe have investigated a classically conformal $U(1)_{B-L}$ extension of the Standard Model \n   with two electroweak Higgs doublet fields. \nThrough the Coleman-Weinberg mechanism, the $U(1)_{B-L}$ symmetry is radiatively broken, \n   and a mass scale is generated via the dimensional transmutation. \nThis symmetry breaking is the sole origin of all dimensionful parameters in the model, \n   and the mass terms of the two Higgs doublet fields are generated \n   through their quartic couplings with the $B-L$ Higgs field. \nAll generated masses are set to be positive but, nevertheless, \n   the electroweak symmetry breaking is realized by the bosonic seesaw mechanism.    \nIn order for the bosonic seesaw mechanism to work,\n  we need a large hierarchy among the two Higgs doublet masses, \n   which originates from a large hierarchy among the quartic couplings. \nAlthough it seems unnatural to introduce the large hierarchy by hand at the $U(1)_{B-L}$ symmetry breaking scale, \n   we have found through analysis of the renormalization group evolutions of the quartic couplings \n   that this hierarchy is dramatically reduced towards high energies. \nTherefore,  once our model is defined at some high energy, for example, the Planck scale, \n  in other words, the origin of the classically conformal invariance is provided \n  by some ultraviolet complete theory at the Planck scale, \n  the bosonic seesaw mechanism is naturally realized with a mild hierarchy among the quartic couplings. \nThe requirements for the perturbativity of the running couplings and the electroweak vacuum stability \n  in the renormalization group analysis as well as for the naturalness of the electroweak scale, \n  we have identified the regions of model parameters such as \n  $g_{B-L}(v_\\Phi)\\lesssim0.2$, $0.15 \\lesssim \\lambda_3(v_\\Phi) = \\lambda_4(v_\\Phi) \\lesssim 0.23$, \n  and $v_\\Phi \\lesssim 100$ TeV.  \nWe have also found that all heavy Higgs boson masses are almost independent of $v_\\Phi$,\n  and lie in the range between $1$ TeV and $1.7$ TeV, which can be tested at the LHC in the near future. \n  \n  \n\n\n\\subsection*{\\centering Acknowledgment} \\label{Acknowledgement}\nN.O. would like to thank the Particle Physics Theory Group of Shimane University\n  for hospitality during his visit.\nThis work is partially supported by Scientific Grants\n  by the Ministry of Education, Culture, Sports, Science and Technology (Nos. 24540272, 26247038, and 15H01037)\n  and the United States Department of Energy (DE-SC 0013680).\nThe work of Y.Y. is supported\n  by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists\n  (Grant No. 26$\\cdot$2428).\n\n\n\n\\section*{Appendix}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\nLet $\\Omega$ be a bounded domain in $\\rn$, with $C^{1,\\alpha}$ boundary. We consider the following quasilinear elliptic problem involving the $A$-laplacian operator\n\\begin{equation}\\label{problem}\n\\begin{cases}\n- \\Delta_A u =f(x,u, \\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu  =0 & {\\rm on}\\,\\,\\,\n\\partial \\Omega \\,,\n\\end{cases}\n\\end{equation}\nwhere $A:[0,\\infty)\\to [0,\\infty)$ is a convex function, vanishing at $0$, $A\\in C^2((0,+\\infty))$, \nand $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function. The $A$-laplacian operator is defined by $\\Delta_A u={\\rm div}\\left(A'(|\\nabla u|)\\frac{\\nabla u}{|\\nabla u|}\\right)$. The properties of the function $A$ guarantee that $\\Delta_A u$ makes sense also when $|\\nabla u|=0$. \\\\\nA wide class of operators can be incorporated in \\eqref{problem}. The $p$-Laplacian and the $(p,q)$-Laplacian, for which $A(t)=t^p$ and $A(t)=t^p+t^q$, $t\\geq 0$, respectively, are the most known, but we can also consider functions like $A(t)=(\\sqrt{1+t^2}-1)^\\gamma$, for $t\\geq 0$ and $\\gamma> 1$ or $A(t)=t^p\\lg(1+t)$, for $t\\geq 0$ and $p>1$. All the $\\Delta_A$ corresponding to the functions $A$ considered above appear in many physical contests, like nonlinear elasticity and plasticity theory.\\\\ \nThe presence of the gradient in the nonlinear term, called convection term, makes variational methods not applicable. Among the techniques used to study problems with a convection term, we cite: topological degree method (\\cite{BalFil, Ruiz}), theory of pseudomonotone operators (\\cite{GW}), fixed point theorems (\\cite{BNV, Zou}), sub and super solution methods (\\cite{FMMT, FM, G, NgSch}), approximation methods (\\cite{Tanaka}), or a combination of the techniques above (\\cite{BaTo1, FMP, MotWin}).\\\\\nWe deal with existence, regularity and sign of the solutions to \\eqref{problem}. Results in this direction can be found in \\cite{BalFil, FMMT, FMP, MotWin, NgSch, Ruiz, Tanaka, Zou}. In the papers above there are various growth conditions on $f$, with respect to each variable, which make it necessary to use different methods to approach the problem, depending on the behavior of the convective term.\\\\\nIn all the papers cited above, the abstract framework is the classical Sobolev space $W_0^{1,p}(\\Omega)$ and the growth conditions with respect $(s,\\xi)\\in \\r\\times \\rn$ are of polynomial type. By contrast, we work in Orlicz spaces and take into account a class of operators which, although they depend only on the gradient, cannot be treated in the Sobolev spaces. Furthermore, this allows for $f$ a wider choice than that seen above. Roughly speaking, for a problem with the $p$-Laplacian, a function $f(x,s,\\xi)=-c+\\frac{|s|^{p^*-1}}{\\lg(1+|s|)}+a(|s|)|\\xi|^p$ (see Theorem \\ref{main2}) is allowed. This does not happen if we consider standard growths.\\\\\nIn \\cite{BalFil, FMP, Ruiz, Zou} the authors establish the existence and the regularity of positive solutions for a problem involving the $p$-Laplacian. In \\cite{BalFil, Ruiz} \nthe convection term is a continuous, nonnegative function with subcritical growth with respect to $u$ and growth less then $p$ with respect to $\\nabla u$. In \\cite{FMP} the growth of the convection term is at most $p-1$ with respect to $u$ and $\\nabla u$, while in\n \\cite{Zou} the convection term is superlinear for $(s,\\xi)\\to (0,0)$ and its growth is at most $p$ with respect to $u$ and strictly less than $p$ with respect to $\\nabla u$. \nAn existence and regularity result for the $p$-Laplacian with a convection term that can be singular at $0$ can be found in \\cite{NgSch}. The existence of a suitable pair of sub and supersolutions plays a crucial role in their proof. In general, sub and supersolution methods allow to study also the case of a singular convection term, provided the interval of sub and supersolutions does not contain the singular point.\\\\\nIn \\cite{MotWin} the authors\ngive also sign information on the solutions. They use sub and supersolution methods, in combination with variational techniques, for an operator that can be treated as the $(p,q)$-laplacian. \\\\ \nExistence and regularity results, for a general operator $A(x,\\nabla u)$ can be found in \\cite{NgSch, Tanaka}, and in \\cite{FMMT} for $A(x,u,\\nabla u)$. In \\cite{Tanaka} the convection term is a continuous function with growth less than $p-1$ with respect to $u$ and $|\\nabla u|$, while in \\cite{FMMT, NgSch} the growth is at most $p$ with respect to $|\\nabla u|$.\\\\\nLet's make some more detailed comments on our new existence and regularity results to \\eqref{problem} (Theorems \\ref{main}, \\ref{main1} and \\ref{main2}). \nIn Theorem \\ref{main} we assume the existence of an ordered pair of sub and supersolutions $\\underline u,\\,\\overline u \\in W^{1,\\infty}(\\Omega)$ and use Theorem 3.6 in \\cite{BaTo1} to prove the existence of a regular solution to \\eqref{problem}. The growth condition on $f$, in Theorem \\ref{main}, is weaker than that used the Theorem 3.6 in \\cite{BaTo1}.\nA limit in the use of the method of sub and supersolutions is due to the fact that establishing their existence may not be easy. So we give two existence results, Theorems \\ref{main1} and \\ref{main2}, where a unified hypothesis on $f$ guarantees the existence of a suitable pair of sub and supersolutions and enable us to apply Theorem \\ref{main} to obtain the existence of a regular constant sign solution.\\\\\nThe paper is arranged a follows: in Section \\ref{sec2} we give the basic definitions and collect some auxiliary results. Our main theorems are proved in Section \\ref{sec3}. Finally, in Section \\ref{sec4}, we present some examples in which it is easy to verify the existence of constant sub and supersolutions. \n\n\\section{Preliminaries}\\label{sec2}\n\nIn this Section we give the main definitions on Young functions and define the Orlicz Sobolev spaces that we use in the sequel. For a comprehensive treatment of Young functions and Orlicz spaces we refer the reader to  \\cite{Chl, KrRu, RR1, RR2}. We also collect some auxiliary results for the proof of the main theorems.\n\\begin{definition} A function $A: [0, \\infty ) \\to [0, \\infty]$ is called a Young function if it is convex, vanishes at $0$, and is neither identically equal to $0$, nor to infinity (in $(0,+\\infty)$).\n\\end{definition}\nFor Young functions\n\\begin{equation}\\label{convexity}\nA(\\lambda t) \\leq \\lambda  A(t)\\quad  \\hbox{for  $\\lambda \\leq 1$ and $t \\geq 0$.}\n\\end{equation}\n\\begin{definition}\\label{conjygate} The Young conjugate of a Young function $A$ is the Young function $\\widetilde A$ defined as\n$$\\widetilde A (s) = \\sup\\{ st - A(t):\\ t \\geq 0\\} \\quad \\hbox{for $s \\geq 0$.}$$\n\\end{definition}\n\n\\begin{definition} A  Young function $A$ is said to satisfy the $\\Delta_2$-condition near infinity (briefly $A\\in \\Delta_2$ near infinity) if it is finite valued and there exist two constants $K\\geq 2$ and $M\\geq 0$ such that\n\\begin{equation}\\label{delta2young}\nA(2t)\\leq KA(t)\\quad \\hbox{for }\\ t\\geq M\\,.\n\\end{equation}\n\\end{definition}\n\\begin{definition} The function $A$ is said to satisfy the $\\nabla_2$-condition near infinity (briefly $A\\in\\nabla_2$ near infinity) if there exist two constants $K>2$ and  $M\\geq 0$ such that\n\\begin{equation}\\label{nabla2young}\nA(2t)\\geq KA(t)\\quad \\hbox{for }\\ t\\geq M\\,.\n\\end{equation}\n\\end{definition}\nIf \\eqref{delta2young} or \\eqref{nabla2young} holds with $M=0$, then $A$ is said to satisfy the $\\Delta_2$-condition (globally), or the $\\nabla_2$-condition (globally), respectively. \nGiven a Young function $A \\in C^1([0,+\\infty))$, define the quantities \n\\begin{equation}\\label{i}\n\tp_A=inf_{t>0}\\frac {t\\cdot A'(t)}{A(t)},\\ \\hbox{and}\\quad q_A=sup_{t>0}\\frac {t\\cdot A'(t)}{A(t)}\\,.\n\\end{equation}\nThe conditions\n\\begin{equation*}\\label{ndg}\n\tp_A>1\\ \\hbox{and}\\quad q_A<+\\infty \n\\end{equation*}\nare equivalent to the fact that $A\\in\\nabla_2\\cap\\Delta_2$ (globally). \n\n\\par\nWe give basic definitions and the main properties on the Orlicz spaces.\nLet $\\Omega$ be a measurable set in $\\rn$, with $n\\geq 1$. Given a Young function $A$, the Orlicz space $L^A(\\Omega)$ is the set of all measurable functions $u:\\Omega\\to\\r$ such that the Luxemburg norm\n$$\\|u\\|_{L^A(\\Omega)}=\\inf\\bigg\\{\\lambda>0:\\int_\\Omega A\\big(\\tfrac 1\\lambda|u|\\big)\\,dx\\leq 1 \\bigg\\}$$\nis finite. The functional $\\|\\cdot\\|_{L^A(\\Omega)}$ is a norm on $L^A(\\Omega)$, and the latter is a Banach space (see \\cite{Adams}).\\\\\nIf $A$ is a Young function, then a generalized  H\\\"older inequality\n\\begin{equation}\\label{holderyoung}\n\\int _\\Omega |u v|\\,dx \\leq 2\\|u\\|_{L^A (\\Omega)} \\|v\\|_{L^{\\widetilde A}(\\Omega)}\n\\end{equation}\nholds for every $u\\in L^A (\\Omega)$ and $v\\in L^{\\widetilde A}(\\Omega)$.\\\\\n\nIf $A\\in \\Delta_2$ globally (or $A\\in \\Delta_2$ near infinity and $\\Omega$ has finite measure) then \n\\begin{equation}\\label{intfinito}\n \\int_{\\Omega} A(k|u|)dx <+\\infty \\ \\hbox{for all}\\ u\\in L^A(\\Omega),\\ \\hbox{all}\\ k\\geq 0\\,. \n\\end{equation}\nLet $\\Omega$ be an open set in $\\rn$ with $|\\Omega|<\\infty$. The isotropic Orlicz-Sobolev spaces \n$W^{1,A}_0(\\Omega)$ is defined as\n\\begin{align*}\nW^{1,A}_0(\\Omega)=\\{u:\\Omega \\to \\r: &\\, \\hbox{the continuation of $u$ by $0$ outside $\\Omega$} \\\\ & \\hbox{is weakly differentiable in $\\rn$, \\, |u|,\\, $|\\nabla u| \\in L^A (\\Omega)$}\\}.\n\\end{align*}\nThe space $W_0^{1,A}(\\Omega)$ equipped with the norm \n$$\\|u\\|_{W^{1,A}_0(\\Omega)} = \\| |\\nabla u| \\|_{L^A (\\Omega)}.$$\nis a Banach space. This norm is equivalent to the standard one\n$$\\|u\\|_{W_0^{1,A}(\\Omega)} = \\|u\\|_{L^A(\\Omega)}+\\||\\nabla u|\\|_{L^A (\\Omega)}\\,.$$\n\nFor the Young function $A$ in \\eqref{problem}, we assume:\n\\begin{itemize}\n\\item [[ A1]] \\ $A\\in C^2(]0,+\\infty[)$ (this  implies $A'\\in C^1([0,+\\infty[)$);\n \\item [[ A2]] there exist two positive constants $\\delta$, $g_0>0$ such that\n\\begin{equation}\\label{dg}\n    \\delta\\leq \\frac{tA''(t)}{A'(t)}\\leq g_0\\quad \\hbox{for}\\ t>0\\,.\n\\end{equation}\n\\end{itemize}\n\nWe point out that, \\eqref{dg} guarantees that $A'(0)=0$ and $A\\in \\nabla_2\\cap\\Delta_2$ globally. In fact integrating \\eqref{dg},\n\\begin{equation*}\n\\left(\\frac{t}{t_0}\\right)^{\\delta}\\leq \\frac{A'(t)}{A'(t_0)}\\leq \\left(\\frac{t}{t_0}\\right)^{g_0}\\quad \\hbox{for}\\ t>t_0>0\\,.\n\\end{equation*}\nChoosing $t=2t_0$\n\\begin{equation*}\n\t2^\\delta A'(t_0)\\leq A'(2t_0) \\leq 2^{g_0}A'(t_0)\\quad \\hbox{for}\\ t>t_0>0\\,.\n\\end{equation*}\nThus\n\\begin{equation*}\nA(2t)=\\int_0^{2t}A'(\\tau)d\\tau=2\\int_0^{t}A'(2s)ds\\leq 2^{g_0+1}\\int_0^{t}A'(s)ds=2^{g_0+1}A(t)\\quad \\hbox{for all}\\; t>0,\n\\end{equation*}\nand\n\\begin{equation*}\nA(2t)=\\int_0^{2t}A'(\\tau)d\\tau=2\\int_0^{t}A'(2s)ds\\geq 2^{\\delta+1} \\int_0^{t}A'(s)ds=2^{\\delta+1} A(t)\\quad \\hbox{for all}\\; t>0.\n\\end{equation*}\n\\par\nWe investigate the existence and the regularity of the solutions to problem \\eqref{problem}. The proof of the existence is based on sub and supersolution methods, while the main tool for the regularity is Theorem 1.7 of \\cite{Li1} and the remark immediately after the statement (see also \\cite[Theorem 1]{Li}), that we recall below.\n\\begin{proposition}(see \\cite[Theorem 1.7]{Li1})\\label{Lib}\nLet $\\Omega$ be a bounded domain in $\\r^n$ with $C^{1,\\alpha}$ boundary, for some $0<\\alpha\\leq 1$. Let $g:[0,+\\infty[\\to [0,+\\infty[$ be a $C^1$, increasing function, satisfying $0<\\delta\\leq \\frac{tg'(t)}{g(t)}\\leq g_0$, for $t>0$, and let $G(t)=\\int_0^t g(\\tau)d\\tau$. Consider the problem\n$$div({\\cal A}(x,u,\\nabla u))+B(x,u,\\nabla u)=0\\ \\hbox{in}\\ \\Omega\\,.$$\nSuppose ${\\cal A}$ and $B$ satisfy the structure conditions (here $a_{ij}(x,z,\\eta)=\\frac{\\partial {\\cal A}^i}{\\partial\\eta_j}$) \n\\begin{itemize}\n\t\\item[$(a)$] $\\sum_{i,j=1}^na_{ij}(x,z,\\eta)\\xi_i\\xi_j\\geq  \\frac{g(|\\eta|)}{|\\eta|}|\\xi|^2$\n\\item[$(b)$] $\\sum_{i,j=1}^n|a_{ij}(x,z,\\xi)|\\leq  \\Lambda\\frac{g(|\\xi|)}{|\\xi|}$\n\\item[$(c)$] $|{\\cal A}(x,z,\\xi)-{\\cal A}(y,w,\\xi)|\\leq \\Lambda_1(1+g(|\\xi|)(|x-y|^\\alpha+|z-w|^\\alpha)$\n\\item[$(d)$] $|B(x,z,\\xi)|\\leq \\Lambda_1(1+g(|\\xi|)|\\xi|),$\n\\end{itemize}\nfor some positive constants $\\Lambda$, $\\Lambda_1$, $M_0$, for all $x$ and $y\\in \\Omega$, for all $z,w\\in [-M_0,M_0]$ and for all $\\xi \\in \\mathbb{R}^n$.\t\nThen, any solution $u\\in W^{1,G}(\\Omega)$, with $|u|\\leq M_0$ in $\\Omega$, is  $C^{1,\\beta}(\\overline{\\Omega})$ for some positive $\\beta$. Moreover \n\\begin{align}\\label{boundc1}\n\t\\|u\\|_{C^{1,\\beta}(\\overline{\\Omega})}\\leq C(\\alpha, \\Lambda, \\delta, g_0, n, \\Lambda_1, g(1), \\Omega, M_0)\n\\end{align}\n\\end{proposition}\n\n\\begin{lemma}\\label{LemLib} Let $A$ be a Young function satisfying $[A1]$ and $[A2]$. Put $\\Phi(\\xi)=A(|\\xi|)$. Then \n\\begin{eqnarray}\\label{1.10a''}\n\t\t\\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\n\t\t\\geq\\min\\{\\delta, 1\\}\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,,\n\\end{eqnarray}\n\\begin{eqnarray}\\label{1.10b}\n\\sum_{i,j=1}^n\\left|\\partial_{ij}\\Phi(\\eta)\\right|\n\t\\leq [2\\max\\{|\\delta-1|,|g_0-1|\\}+n]\\frac{A'(|\\eta|)}{|\\eta|}\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,,\n\\end{eqnarray} \t\n\\end{lemma}\nand $\\nabla \\Phi=\\cal A$ satisfies conditions $a--c$ in  Proposition \\ref{Lib}, with $g(t)=\\min\\{\\delta,\\,1\\}A'(t)$, $\\Lambda= \\frac{\\lambda}{\\min\\{\\delta,\\,1\\}}$, where $\\lambda=2\\max\\{|\\delta-1|,|g_0-1|\\}+n$.\\\\\n{\\bf Proof}\nFrom \\eqref{dg} \n\\begin{equation*}\n(\\delta -1) \\frac{A'(|\\eta|)}{|\\eta|}\\leq  A''(|\\eta|)-\\frac{A'(|\\eta|)}{|\\eta|} \\leq (g_0-1)\\frac{A'(|\\eta|)}{|\\eta|}\n\\ \\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\nAlso, $\\partial_i\\Phi(\\eta)=A'(|\\eta|)\\frac{\\eta_i}{|\\eta|}$, and\n\\begin{equation*}\n\\partial_{ij}\\Phi(\\eta)= A''(|\\eta|)\\frac{\\eta_i\\eta_j}{|\\eta|^2} +A'(|\\eta|)\\left(\\frac{\\delta_{ij}}{|\\eta|} -\\frac{\\eta_i\\eta_j}{|\\eta|^3}\\right)\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\nThus\n\\begin{align*}\n\\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j= &\\sum_{i,j=1}^n \\left(A''(|\\eta|)\\frac{\\xi_i\\eta_i\\xi_j\\eta_j}{|\\eta|^2} +A'(|\\eta|)\\frac{\\delta_{ij}\\xi_i\\xi_j}{|\\eta|}- A'(|\\eta|)\\frac{\\xi_i\\eta_i\\xi_i\\eta_j}{|\\eta|^3}\\right)\\\\\n&=\\left(\\frac{A''(|\\eta|)}{|\\eta|^2}- \\frac{A'(|\\eta|)}{|\\eta|^3}\\right)(\\xi,\\eta)^2+\n\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\\\\n &\\geq (\\delta -1)\\frac{A'(|\\eta|)}{|\\eta|^3}(\\xi,\\eta)^2+\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\\qquad\\qquad\\qquad\\qquad\n\\end{align*}\nIf $\\delta\\geq 1$ \n\\begin{equation}\\label{1.10a}\n    \\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\\geq \\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation}\nIf $\\delta<1$ \n\\begin{eqnarray}\\label{1.10a'}\n    \\sum_{i,j=1}^n\\partial_{ij}\\Phi(\\eta)\\xi_i\\xi_j\\geq (\\delta -1)\\frac{A'(|\\eta|)}{|\\eta|^3}|\\xi|^2|\\eta|^2+\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\\\\n    =\\delta\\frac{A'(|\\eta|)}{|\\eta|}|\\xi|^2\\ \\ \\hbox{for all}\\ \\xi \\in \\rn,\\eta\\in \\rn\\setminus\\{0\\}\\,.\\nonumber\n\\end{eqnarray}\nPutting together \\eqref{1.10a} and \\eqref{1.10a'}, we get \\eqref{1.10a''}.\\\\\nConsider\n\\begin{equation*}\n|\\partial_{ij}\\Phi(\\eta)|\\leq\\frac{|\\eta_i||\\eta_j|}{{|\\eta|^2}} \\left|A''(|\\eta|) -\\frac{A'(|\\eta|)}{|\\eta|}\\right|+\n\t\\delta_{ij}\\frac{A'(|\\eta|)}{|\\eta|} \\quad \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{equation*}\t\n\nThus\n\\begin{eqnarray*}\n\\sum_{i,j=1}^n\\left|\\partial_{ij}\\Phi(\\eta)\\right|\\leq\\frac{\\left(\\sum_{i=1}^n|\\eta_i|\\right)^2}{{|\\eta|^2}} \\left|A''(|\\eta|) -\\frac{A'(|\\eta|)}{|\\eta|}\\right|+\nn\\frac{A'(|\\eta|)}{|\\eta|}\\nonumber \\\\\n\\leq\\left[2\\max\\{|\\delta-1|,|g_0-1|\\}+n\\right]\\frac{A'(|\\eta|)}{|\\eta|}\\ \\ \\hbox{for all}\\ \\eta\\in \\rn\\setminus\\{0\\}\\,.\n\\end{eqnarray*} \nSo \\eqref{1.10b} holds with $\\lambda=2\\max\\{|\\delta-1|,|g_0-1|\\}+n$.\n\\qed\n\\section{Main results}\\label{sec3}\nIn this section first we give two existence and regularity results (Theorems \\ref{main1} and \\ref{main2}), in which we assume a global growth condition on $f$, unilateral with respect to $s\\in \\r$. In Theorem \\ref{main1} we require that $f$ satisfies some conditions for $(x,s,\\xi)\\in \\Omega \\times  [0,+\\infty)\\times \\rn$ and obtain the existence of a nonnegative solution. Similarly, in Theorem \\ref{main2}, $f$ satisfies some conditions for $(x,s,\\xi)\\in \\Omega \\times  (-\\infty, 0]\\times \\rn$ that guarantee the existence of a non-positive solution.\\\\\n\\noindent Here is the definition of weak solution to \\eqref{problem}. \n\\begin{definition}\nA function  $u\\in \\w0$ is a weak solution to problem \\eqref{problem} if\n\\begin{equation*}\n\\int_{\\Omega}A'(|\\nabla u|)\\cdot\\frac{\\nabla u}{|\\nabla u|}\\cdot\\nabla v dx= \\int_{\\Omega}f(x,u,\\nabla u)vdx\n\\end{equation*}\nfor all $v\\in \\w0$. \n\\end{definition}\nFor the first two Theorems, we assume\n$$({\\cal{H}}):\n\\left\\{\n\\begin{array}{l}\n\ta:[0,+\\infty[\\to [0,+\\infty[ \\ \\hbox{is a locally essentially bounded function};\\\\\n\t\\rho_1,\\rho_2:\\Omega\\to [0,+\\infty[\\ \\hbox{are two measurable functions},\\ \\rho_1,\\,\\rho_2\\in L^\\infty(\\Omega)\\ \\hbox{and}\\\\ \\rho_2 (x)>0\\ \\hbox{on a set of positive measure};\\\\\n\tg_1,g_2:[0,+\\infty[\\to [0,+\\infty[\\ \\hbox{are two non-decreasing functions such that}\\ g_1(0)=g_2(0)=0\\\\\n\t\\hbox{and there exist}\\ s_0>0,\\,k_1\\in \\left ]0,\\omega_n^{\\frac{1}{n}}|\\Omega|^{-\\frac{1}{n}}\\right[,\\ \\hbox{such that}\\ \n\tg_1(|s|)|s|\\leq A(k_1|s|)\\ \\hbox{for all}\\ |s|\\geq s_0\\,.\n\\end{array}\n\\right.\n$$\nHere $\\omega_n$ is the measure of the unit ball in $\\rn$.\n\\begin{theorem}\\label{main1}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary. \nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function, satisfying $[A1]$ and $[A2]$. Let $f:\\Omega\\times \\mathbb{R}\\times \\mathbb{R}^n\\to \\mathbb{R}$ be a Carath\\'{e}odory function fulfilling\n\t\\begin{equation}\\label{growth f2+}\n\t\t\\rho_2(x)-g_2(s)-a(s)A'(|\\xi|)|\\xi|\\leq f(x,s,\\xi)\\leq \\rho_1(x)+g_1(s)\\ \\hbox{ for a.e.}\\ x\\in \\Omega,\\ \\hbox{all}\\ s\\geq 0,\\ \\hbox{all}\\ \\xi\\in\\rn\\,.\n\t\n\t\\end{equation}\nThe functions $a,\\,\\rho_1,\\,\\rho_2,\\,g_1,\\,g_2$ are as in $(\\cal{H})$.\n\tThen problem \\eqref{problem} has a nontrivial, nonnegative solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$.\\\\\n\n\tIf, in addition, there exist $\\overline{\\delta} >0$ and $k_3>0$ such that\n\t$g_2(s)s\\leq A(k_3s)$ for every $s\\in (0,\\overline{\\delta})$, then $u>0$ in $\\Omega$.\n\\end{theorem}\n\\begin{theorem}\\label{main2}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary.\nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function, satisfying $[A1]$ and $[A2]$. Let $f:\\Omega\\times \\mathbb{R}\\times \\mathbb{R}^n\\to \\mathbb{R}$ be a Carath\\'{e}odory function fulfilling\n\\begin{equation}\\label{growth f2}\n-\\rho_1(x)-g_1(|s|)\t\\leq f(x,s,\\xi)\\leq -\\rho_2(x)+g_2(|s|)+a(s)A'(|\\xi|)|\\xi|\\ \\hbox{for a.e.}\\, x\\in \\Omega,\\ \\hbox{all}\\ s\\leq 0,\\ \\hbox{all}\\ \\xi\\in\\rn\\,,\n\n\t\\end{equation}\nwhere the functions $a,\\,\\rho_1,\\,\\rho_2,\\,g_1,\\,g_2$ are as in $(\\cal{H})$.\n\tThen problem \\eqref{problem} has a nontrivial, non-positive solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$.\\\\\n\n\tIf, in addition, there exist $\\overline{\\delta} >0$ and $k_3>0$ such that\n\t$g_2(s)s\\leq A(k_3s)$ for every $s\\in (0,\\overline{\\delta})$, then $u<0$ in $\\Omega$.\n\\end{theorem}\n\\begin{remark}\nIn \\cite{BalFil}, Theorem 1, the authors prove the existence of a positive solution for a problem with the $p$-Laplacian, and a convection term $f$ satisfying the hypotheses of Theorem \\ref{main2}.\n\\end{remark}\nFor the proof of the Theorems above, we need an abstract existence result, where sub and supersolutions come into play.\\\\\nThe definition of sub and supersolution in general domains, for which a trace theory may not hold, can be found in \\cite{BaTo1}. Our hypotheses on $\\Omega$ allow to adopt the classical definition (see \\cite[Theorem 3.1]{Ctrace}).\\\\\nWe say that $\\overline u\\in W^{1,A}(\\Omega)$ is a supersolution to \\eqref{problem} if $\\overline u_{|\\partial \\Omega}\\geq 0$ (in the sense of traces) and\n\\begin{equation*}\\label{supersolution}\n\t\\int_{\\Omega}A'(|\\nabla\\overline u|)\\cdot\\frac{\\nabla\\overline u}{|\\nabla\\overline u|}\\cdot\\nabla v dx\\geq \\int_{\\Omega}f(x,\\overline u,\\nabla \\overline u)vdx\n\\end{equation*}\nfor all $v\\in W_0^{1,A}(\\Omega)$, $v\\geq 0$ a.e. in $\\Omega$.\\\\\nWe say that $\\underline u\\in W^{1,A}(\\Omega)$ is a subsolution to \\eqref{problem} if $\\underline u_{|\\partial \\Omega}\\leq 0$ (in the sense of traces) and\n\\begin{equation*}\\label{subsolution}\n\t\\int_{\\Omega}A'(|\\nabla\\underline u|)\\cdot\\frac{\\nabla \\underline u}{|\\nabla \\underline u|}\\cdot \\nabla v dx\\leq \\int_{\\Omega}f(x,\\underline u,\\nabla \\underline u)vdx\n\\end{equation*}\nfor all $v\\in W_0^{1,A}(\\Omega)$, $v\\geq 0$ a.e. in $\\Omega$.\\\\\nFor the next Theorem we assume that problem \\eqref{problem} has a subsolution and supersolution, $\\underline{u}$, $\\overline{u}\\in W^{1,\\infty}(\\Omega)$, with $\\underline{u}(x)<\\overline{u}(x)$ for all $x\\in \\Omega$.\nAlso, $f:\\Omega\\times \\r\\times \\rn\\to \\r$ is a Carath\\'{e}odory function satisfying the following growth condition:\n\\begin{itemize}\n\\item [(H)] there exists a function $\\sigma\\in L^{\\infty}(\\Omega)$ and a constant $a>0$, such that\n$$\n|f(x,s,\\xi)|\\leq \\sigma (x)+a A'(|\\xi|)|\\xi|\\quad \\hbox{for a.e.}\\ x\\in \\Omega,\\ \\hbox{all}\\ s\\in [\\underline u(x),\\overline u(x)],\\ \\hbox{all}\\ \\xi\\in \\rn\\, .\n$$\n\\end{itemize}\n\\noindent The local condition on $f$, with respect to $s$, is sufficient for our purposes. The use of the method of sub and supersolutions requires an a priori analysis of the problem. Only once the existence of sub and supersolutions has been established does one proceed to search for the existence of a solution.\n\n\n\n\\begin{theorem}\\label{main}\nLet $\\Omega$ be a bounded domain in $\\rn$ with $C^{1,\\alpha}$ boundary.\nLet $A: [0,+\\infty[\\to [0,+\\infty[$ be a Young function satisfying $[A1]$ and $[A2]$. Let $\\underline{u}$, $\\overline{u}\\in W^{1,\\infty}(\\Omega)$ be as above and assume that $f$ satisfies hypothesis (H). Then problem \\eqref{problem} admits at least a solution  $u\\in C_0^{1,\\beta}(\\overline \\Omega)$. Moreover $\\underline u(x)\\leq u(x)\\leq \\overline u(x)$ a.e in $\\Omega$.\n\\end{theorem}\n{\\bf Proof}. Let $M=\\max\\{\\| \\overline u\\|_\\infty,\\| \\underline u\\|_\\infty\\}$ and $R>\\max\\{\\|\\nabla \\overline u\\|_\\infty,\\|\\nabla \\underline u\\|_\\infty\\}$. Consider the truncated function $f_R$ defined by\n\\begin{equation*}\\label{fR}\n    f_R(x,s,\\xi)=\\left\\{\\begin{array}{cc}\n                          f(x,s,\\xi) & \\hbox{if}\\ |\\xi|\\leq R,\\\\\n                          f(x,s,\\xi)\\cdot\\frac{A'(R)R}{A'(|\\xi|)|\\xi|} & \\hbox{if}\\ |\\xi|> R \\,,\n                        \\end{array}\n    \\right.\n\\end{equation*}\nand the problem\n\\begin{equation}\\label{problem1}\n\\begin{cases}\n-\\Delta_A(u)=f_R(x,u,\\nabla u) & {\\rm in}\\,\\,\\,\\Omega \\\\\nu  =0 & {\\rm on}\\,\\,\\,\\partial \\Omega \\,.\n\\end{cases}\n\\end{equation}\nIn view of the choice of $R$, $\\underline u$ and $\\overline u$ are a subsolution and a supersolution to \\eqref{problem1} respectively. Using the monotonicity of $A'$ we deduce that $|f_R(x,s,\\xi)|\\leq \\sigma (x)+a A'(R)R$, for a.e. $x\\in \\Omega$, all $s\\in [\\underline u(x),\\overline u(x)]$, all $\\xi\\in \\rn$. From Theorem 3.6 in \\cite{BaTo1} problem \\eqref{problem1} admits a solution $u\\in W^{1,A}_0(\\Omega)$ with $\\underline u(x)\\leq u(x)\\leq \\overline u(x)$ a.e in $\\Omega$. Thus $u\\in L^\\infty(\\Omega)$.\\\\ \n\\noindent The functions $A$ and $f$ satisfy the hypotheses of Proposition \\ref{Lib}, with $\\Lambda_1=\\max\\{\\|\\sigma\\|_\\infty, \\min\\{\\delta,1\\}^{-1}a\\}$ (see also Lemma \\ref{LemLib}). Since $|f_R|\\leq |f|$ the same holds for $f_R$ whatever $R$ is.\nDue to Proposition \\ref{Lib} there exist two positive constants $0<\\beta\\leq 1$ and $C$,  independent from $R$, such that any solution to \\eqref{problem1} belongs in $C_0^{1,\\beta}(\\overline \\Omega)$ and $\\|u\\|_{C_0^{1,\\beta}(\\overline \\Omega)}\\leq C$. Choosing $R>C$ we deduce that $u$ is a solution to \\eqref{problem}.\n\\qed\n\\begin{remark}\nWhen the solution $u$ has constant sign, it should be interest to verify if it is positive (or negative) in $\\Omega$. The maximum principle by Pucci-Serrin (see \\cite [Theorem 3.5] {PS}) is a powerful tool, as it ensures that (under some proper conditions on $A$ and $f$) \nany nonnegative solution to \\eqref{problem} is positive in $\\Omega$. \nA quite standard situation occurs when $f$ is bounded below by suitable monotone functions and $A\\in\\Delta_2$ near $0$, as the following corollary shows. \\end{remark}\n\\begin{corollary}\\label{cor1}\nUnder the hypotheses of Theorem \\ref{main}, assume that $f$ satisfies\n\\begin{equation}\\label{growthPS}\n\tf(x,s,\\xi)\\geq -a A'(|\\xi|)-b(s)\\ \\hbox{for all} \\; x\\in \\Omega,\\ s>0,\\ \\xi\\in \\mathbb{R}^n,\\; |\\xi|\\leq 1,\n\\end{equation}\nwhere $a>0$, $b:[0,+\\infty[ \\to [0,+\\infty[$ is a function increasing in $(0,\\overline{\\delta})$ (for some $\\overline{\\delta}>0$), $b(0)=0$, and $b(s)=\\frac{A(ks)}{s}$ for $s\\in (0,\\overline\\delta)$ and some $k>0$.\nThen any nonnegative, nontrivial solution to \\eqref{problem} is positive.\n\\end{corollary}\n{\\bf Proof.}\nLet $u\\in W^{1,A}_0(\\Omega)$ be a nonnegative, nontrivial solution to \\eqref{problem}. Theorem \\ref{main} ensures that $u\\in C_0^{1,\\beta}(\\overline{\\Omega})$. In order to prove that $u>0$ in $\\Omega$ we use Theorem 5.3.1 of \\cite{PS}.\\\\\nConditions $(A1)'$ and $(A2)$ of the Theorem cited above hold, because $A\\in C^2((0,+\\infty))$, $A'(0)=0$, and $s\\mapsto A'(s)$ is strictly increasing. Conditions $(F2)$ and $(B1)$ are satisfied too.\\\\\nIt remains to verify condition $(1.1.5)$ of Theorem 5.3.1 of \\cite{PS}. \nPut $B(s)=\\int_0^s b(t)dt$. Due to the monotonicity of $\\frac{A(t)}{t}$, for $s\\in(0,\\overline\\delta)$, it holds \n$$B(s)=\\int_0^s \\frac{A(kt)}{t}dt\\leq \\int_0^s \\frac{A(ks)}{s}dt=A(ks)\\,.$$\nIf $h\\in \\mathbb{N}$ is such that $k<2^h$ then, in view of \\eqref{delta2young}\n$$A(ks)\\leq K^h A(s)\\quad \\hbox{for all}\\; s\\geq 0\\,.$$ \nLet $b_1=\\max\\{p_A-1,K^h\\}$. \nThen, for $s\\in (0,\\overline{\\delta})$, using \\eqref{convexity} and the inequality above\n\\begin{align*}\n\tH(s)=sA'(s)-A(s)&\\geq (p_A-1)A(s)=\\frac{p_A-1}{b_1}b_1A(s)\\nonumber\\\\\n\t\\geq  b_1 A\\left(\\frac{(p_A-1)s}{b_1}\\right)\\geq  K^hA\\left(\\frac{(p_A-1)s}{b_1}\\right)&\\geq  A\\left(\\frac{k(p_A-1)s}{b_1}\\right) \\geq B\\left(\\frac{(p_A-1)s}{b_1}\\right)\\,, \n\\end{align*}\nor equivalently\n\\begin{equation*}\nH\\left(\\frac{b_1s}{p_A-1}\\right)\\geq B(s)\\quad \\hbox{for}\\ 0<s<\\frac{b_1\\overline{\\delta}}{p_A-1}\\,.\n\\end{equation*}\n$H$ is increasing, so\n\\begin{equation*}\n\t\\frac{b_1s}{p_A-1}\\geq H^{-1}(B(s))\\quad \\hbox{for}\\ 0<s<\\frac{b_1\\overline{\\delta}}{p_A-1}\\,.\n\\end{equation*}\nFinally \n\\begin{equation}\\label{minor per strong}\n\\frac{1}{ H^{-1}(B(s))}\\geq \\frac{p_A-1}{b_1s}\\quad \\hbox{for}\\ 0<s<\\frac{b_1\\overline{\\delta}}{p_A-1}\\,.\n\\end{equation}\nIntegrating \\eqref{minor per strong} from $\\ep$ to $s<\\frac{b_1\\overline{\\delta}}{p_A-1}$ and passing to the limit as $\\ep\\to 0^+$ we obtain condition $(1.1.5)$ of Theorem 5.3.1 of \\cite{PS}. Thus $u>0$ in $\\Omega$.\\qed\n\nNow we accomplish with the proof of Theorems \\ref{main1} and \\ref{main2}.\\\\\n{\\bf Proof of Theorem \\ref{main1}}. \nFrom the proof of Theorem 3.3 of \\cite{BaTo1} we know that there exists a nontrivial solution $\\overline{u}\\geq 0$, to the problem\n\t$$\n\t- \\Delta_A(u) =\\rho_1(x)+g_1(|u|)\\,.\n\t$$\nTheorem 3 of \\cite{Cbound} guarantees that $\\overline{u}$ is bounded. Finally, from Proposition \\ref{Lib}, we have that $\\overline{u}\\in C_0^{1,\\beta}(\\overline{\\Omega})$.\nThe inequalities in \\eqref{growth f2+} show that $\\overline{u}$ is a supersolution to problem \\eqref{problem} and $\\underline{u}=0$ is a subsolution to problem \\eqref{problem}. The assumptions on $\\rho_2$ guarantee that $\\underline{u}=0$ is not a solution. If we put $\\sigma(x)=\\max\\{\\rho_1(x)+g_1(\\overline{u}(x)) ,\\; \\rho_2(x)+g_2(\\overline{u}(x))\\}$ for a.e. $x\\in \\Omega$, then \\eqref{growth f2+} leads to\n$$\n|f(x,s,\\xi)|\\leq \\sigma(x)+a(s)A'(|\\xi|)|\\xi|\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; \\hbox{all}\\ s\\in [0,\\overline{u}(x)],\\; \\hbox{all}\\ \\xi\\in \\mathbb{R}^n\\,.$$\nLet $I=[0,\\sup_\\Omega \\overline u]$ and $a=\\|a\\|_{L^\\infty(I)}$. Then $a<+\\infty$ and $a(s)\\leq a$ for a.e. $s\\in I$. Let $I_0\\subset I$ be a set of null measure, such that $a(s)>a$ for all $s\\in I_0$. For $s\\in I_0$ it holds\n\\begin{align}\n|f(x,s,\\xi)|=\\lim_{t\\to s}|f(x,t,\\xi)|=\\liminf_{t\\to s}|f(x,t,\\xi)|& \\leq \\sigma(x)+\\liminf_{t\\to s}a(t)A'(|\\xi|)|\\xi|\\nonumber\\\\\n\t\\leq \\sigma(x)+aA'(|\\xi|)|\\xi| &\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; \\xi\\in \\mathbb{R}^n\\,.\n\\end{align}\nThus  \n\\begin{align}\n|f(x,s,\\xi)|\\leq \\sigma(x)+aA'(|\\xi|)|\\xi|&\\quad \\hbox{for a.e.} \\; x\\in \\Omega, \\; s\\in [0,\\overline u(x)],\\; \\xi\\in \\mathbb{R}^n\\,.\n\\end{align}\nSo, from Theorem \\ref{main}, problem \\eqref{problem} admits at least a nontrivial solution $u\\in C_0^{1,\\beta}(\\overline{\\Omega})$ such that $0\\leq u\\leq \\overline{u}$.\\\\\nNow we prove that, under the additional condition on $g_2$, $u>0$ in $\\Omega$. We set $b(s)=\\max \\{g_2(s),\\frac{A(k_3s)}{s}\\}$, for $s\\geq 0$ and observe that the left inequality in \\eqref{growth f2+} guarantees that we can apply Corollary \\ref{cor1}.\n\\qed\n\\noindent {\\bf Proof of Theorem \\ref{main2}}. It is enough to put $f_1(x,s,\\xi)=-f(x,-s,-\\xi)$ and to use Theorem \\ref{main1} for $f_1$.\\qed\n\n\\section{Examples}\\label{sec4}\nThis section is devoted to some examples with different Young functions and various nonlinearities.\\\\\n\\noindent In the first two examples we consider the problem\n\\begin{equation}\\label{pqf}\n\\ \\begin{cases}\n- {\\rm div}\\left((|\\nabla u|^{p-2}+|\\nabla u|^{q-2})\\nabla u\\right) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu  =0 & {\\rm on}\\,\\,\\,\\partial \\Omega \\, ,\n\\end{cases}\n\\end{equation}\nwhere $1<q<p<\\infty$. Here $A(t)=\\frac{t^p}{p}+\\frac{t^q}{q}$ for all $t\\geq 0$, and \n\\eqref{dg} holds with $\\delta=q-1$, $g_0=p-1$.\n\\begin{example}\nLet $a:\\r\\to\\r$ and $g:\\r\\to\\r$ be two continuous functions and let $h:\\Omega\\to \\r$ be an essentially bounded function.\nAssume that there exist $s_1,\\, s_2\\in\\r$, such that $g(s_1)=g(s_2)=0$ for some $s_1<0<s_2$, $g(s)\\neq 0$ for all $s\\in ]s_1,s_2[$, and $|\\{x\\in\\Omega:h>0\\}|>0$, $|\\{x\\in\\Omega:h<0\\}|>0$.\\\\\nSet\n\\begin{equation*}\nf(x,s,\\xi)= g(s)h(x)+a(s)A'(|\\xi|)(|\\xi|)\\ \\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn\\,.\n\\end{equation*}\nThen $u_1=s_1$ and $u_2=s_2$ are a subsolution and a supersolution to \\eqref{pqf}, respectively. Also $u\\equiv 0$ is not a solution nor a sub or a supersolution and $f$ satisfies condition (H) \n with $\\sigma (x)=|h(x)|\\max_{[s_1,s_2]}|g(s)|$ and $a =\\max_{[s_1,s_2]}|a(s)|$. By Theorem \\ref{main}, problem \\eqref{pqf} has a nontrivial solution $u\\in C^{1,\\beta}_0(\\overline\\Omega)$ with $s_1\\leq u\\leq  s_2$ a.e. in $\\Omega$.\n\n\\end{example}\n\n\\begin{example}\nLet $a:\\r\\to\\r$ and $g:\\r\\to\\r$ be two continuous functions and let $h:\\Omega\\to \\r$ be an essentially bounded function. Assume that $h(x)\\geq 0$ (or $h(x)\\leq 0$) in $\\Omega$, $h(x)>0$ on a set of positive measure, $g(s_1)=0$ for some $s_1>0$, $g(s)\\neq 0$ for all $s\\in [0,s_1[$, and $g(0)h(x)\\geq 0$ in $\\Omega$.\\\\\nSet\n\\begin{equation*}\n\tf(x,s,\\xi)= g(s)h(x)+a(s)A'(|\\xi|)(|\\xi|)\\ \\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn\\,.\n\\end{equation*}\nThen $u_1\\equiv0$ and $u_2\\equiv s_1$ are a subsolution and a supersolution to \\eqref{pqf}, respectively. Also $u\\equiv 0$ is not a solution and $f$ satisfies condition (H) with $\\sigma (x)=|h(x)|\\max_{[0,s_1]}|g(s)|$ and $a =\\max_{[0,s_1]}|a(s)|$. By Theorem \\ref{main}, problem \\eqref{pqf} has a nontrivial solution $u\\in [0,s_1]$, $u\\in C^{1,\\beta}_0(\\overline\\Omega)$. From Theorem 5.3.1 in \\cite{PS}, $u>0$ in $\\Omega$ (note that $g(s)h(x)\\geq 0$ in $\\Omega\\times[0,s_1]$).\n\\end{example}\n\\begin{example} \nConsider the problem \n\\begin{equation}\\label{problempq}\n\\begin{cases}\n- {\\rm div}(\\lg(1+|\\nabla u|^{q})|\\nabla u|^{p-2}\\nabla u) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\nu  =0 & {\\rm on}\\,\\,\\,\n\\partial \\Omega \\,,\n\\end{cases}\n\\end{equation}\nwith $q>0$, $p>1$.\\\\ \nLet $r\\geq q$, $\\overline \\delta >0$ and define $b:[0,+\\infty[\\to [0,+\\infty[$ as\n\\begin{equation*}\n    b(s)=\\left\\{\\begin{array}{cc}\n         s^{p+q-1}&\\hbox{if}\\ s\\in [0,\\overline \\delta],  \\\\\n          s^{p+r-1}& \\hbox{if}\\ s> \\overline \\delta\\,.\n    \\end{array}\\right.\n\\end{equation*}\nAssume that $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function satisfying\n\\begin{itemize}\n\\item [$(f_0)$] there exists $\\sigma >0$ such that $f(x,\\sigma,0)\\leq 0$ a.e. in $\\Omega$;\n\\item [$(f_1)$] $f(x,0,0)\\geq 0$ a.e. in $\\Omega$, with strict inequality on a set of positive measure;\n\\item [$(f_2)$] there exist $k>0$ such that\n \\begin{equation*}\\label{strong1}\nf(x,s,\\xi)\\geq-k(|\\xi|^{p-1}\\lg(1+|\\xi|^q)+b(s))\\ \\hbox{for}\\ x\\in\\Omega,\\ \\hbox{all}\\ s\\geq 0\\ \\hbox{and all}\\ \\xi\\in \\rn,\\ \\hbox{with}\\ |\\xi|\\leq 1\\,.\n  \\end{equation*}\n\\item [$(f_3)$]  there exists $c >0$ such that \n$|f(x,s,\\xi)|\\leq c(1+|\\xi|^p\\lg(1+|\\xi|^q))\\ \\hbox{for all}\\ x\\in\\Omega,\\ s\\in \\r, \\ \\xi\\in\\rn$.\n\\end{itemize}\nFor the function $A'$ in problem \\eqref{problempq} condition \\eqref{dg} holds with $\\delta=p-1$, $g_0=p-1+q$ and \n$A(t)\\approx  t^{p+q}$ for  $t$ small.\nWe can apply Theorem \\ref{main} and Corollary \\ref{cor1} to obtain the existence of a positive solution $u\\in C_0^{1,\\beta}(\\overline \\Omega)$, and $u\\leq \\sigma$ in $\\Omega$.\n\\end{example}\nThe example above extends in two directions Theorem 6 of \\cite{FMMT}: it allows an higher growth for $f$ and permit also the choice $r=q$ in the lower bound for $f$.\n\n\\begin{example} \nConsider the problem \\begin{equation}\\label{Apq}\n\t\\begin{cases}\n\t\t- {\\rm div}\\left(\\frac{|\\nabla u|^{p-2}\\nabla u}{\\lg^q(1+|\\nabla u|)}\\right) =f(x,u,\\nabla u) & {\\rm in}\\,\\,\\, \\Omega \\\\\n\t\tu  =0 & {\\rm on}\\,\\,\\,\n\t\t\\partial \\Omega \\,,\n\t\\end{cases}\n\\end{equation}\nwith $p>1$, $p-q-1>0$.\nLet $\\rho\\in L^{\\infty}(\\Omega)$ and $g_1,g_2:[0,+\\infty[\\to [0,+\\infty[$ be two unbounded, nondecreasing functions, such that $g_1(0)=g_2(0)=0$. Also, let $a_1,a_2: \\r \\to [0,+\\infty[$ be two locally essentially bounded functions and let $c_1,c_2>0$.\\\\\nAssume that  $f:\\Omega\\times\\r\\times\\rn\\to\\r$ is a Carath\\'{e}odory function satisfying\n\\begin{eqnarray*}\n-c_1+g_1(|s|)-a_1(s)\\frac{|\\xi|^{p}}{\\lg^{q}(1+|\\xi|)}\\leq f(x,s,\\xi)\\leq -c_2+g_2(|s|)\\rho(x)+a_2(s)\\frac{|\\xi|^{p}}{\\lg^{q}(1+|\\xi|)}\\\n\\end{eqnarray*}\n$\\hbox{for}\\ (x,s,\\xi)\\in \\Omega \\times \\r\\times \\rn \\,.$ \\\\\nWe show that problem \\eqref{Apq} has a nontrivial solution $u\\leq 0$ in $\\Omega$.\\\\\nFor the function $A'$ in problem \\eqref{Apq} condition \\eqref{dg} holds with $\\delta =p-1-q$, $g_0=p-1$.\nIf $k:=\\inf\\{s>0\\,:\\, g_1(s)\\geq c_1\\}$, then $\\underline u\\equiv -k$ is a subsolution to \\eqref{Apq}, and $\\overline u\\equiv 0$ is a supersolution but not a solution to \\eqref{Apq}. Let $a=max \\{\\|a_1\\|_{L^\\infty ([-k,0])},\\ \\|a_2\\|_{L^\\infty ([-k,0])}\\}$, $\\sigma(x)=max\\{c_1, -c_2+g_2(k)\\rho(x)\\}$. Then \n\\begin{equation*}\n|f(x,s,\\xi)|\\leq \\sigma(x)|+a A'(|\\xi|)|\\xi|\\quad \\hbox{for}\\ x\\in \\Omega,\\ s\\in\\,[-k,0],\\ \\xi\\in\\rn\\,.\n\\end{equation*}\nBy Theorem \\ref{main}, problem \\eqref{Apq} has a nontrivial solution $u\\in [-k,0]$.\n\\end{example}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nIn the last few decades the detrended fluctuation analysis (DFA) method has become a\none of the standard techniques for the computation of the fractal scaling properties\n\\cite{PenBul1985}. It also serves as a great tool for the detection of long-range correlations \nnonstationary time series. Another very celebrated method is\nbased on the wavelet transformation of the signals \\cite{MuzBac1993,PhiLim2011} which together\nconstitutes widely accepted methods of the fluctuation analysis.\nIn 1941 Kolmogorov introduced multifractal formalism \\cite{Kolmogorov1941} in the context of \nanalysis of the turbulent data. It was extensively developed in the last\ndecade of the last century \\cite{Mandelbrot1999,StaMae1988} and still attracts considerable\nattention which includes such a distant fields like the biological systems \n\\cite{IvaAma1998}, financial markets\n\\cite{OswKwa2005,OhEom2012}, econophysics \\cite{Takayasu2004}, turbulence\n\\cite{Sreenivasan1991}, space data analysis \\cite{MacWaw2011}, physiology\n\\cite{StaAma1988} or medicine \\cite{MakRyn2011,Geras2013} to mention but a few.\n\\begin{figure}[ht!]\n\\centering\n\\subfigure[Signals obtained from muscle at complex work. The task was to tie the\nstandard surgery knots using the laparoscopic tools.]{\n\\includegraphics[width=.48\\linewidth]{fig1_1a}\n\\includegraphics[width=.48\\linewidth]{fig1_1b}\n\\label{fig11ab}}\n\\subfigure[Signals obtained from muscle in the relaxed state.]{\n\\includegraphics[width=.48\\linewidth]{fig1_2a}\n\\includegraphics[width=.48\\linewidth]{fig1_2b}\n\\label{fig12ab}}\n\\subfigure[Signals obtained from muscle at maximum contraction.]{\n\\includegraphics[width=.48\\linewidth]{fig1_3a}\n\\includegraphics[width=.48\\linewidth]{fig1_3b}\n\\label{fig13ab}}\n\\caption{Data collected from the channel 1 (in the vicinity of the trapezius ridge) for\namateur (left column, black) and professional (right column, red) at three\ndifferent states.}\n\\label{fig1}\n\\end{figure}\n\nThe electromyography (EMG) together with the electrocardiography, electroretinography and\nelectroencephalography are nowadays not only the diagnostic instruments in medicine but\nconstitute the proper and powerful scientific tool based on the electro-physiological\nactivities of our body \\cite{Aminoff1986}. In particular the surface EMG (sEMG) has\nbecome a promising apparatus for the non-invasive analysis of muscles\n\\cite{MerPar2004,BotCor2006}. The standard analysis of the sEMG signal covers usually three\naspects: the activation level of the muscle membrane potential, impact of the forces exerted\non the muscle and the degree of muscle fatigue. In this work on the contrary we will\nemploy the multifractal analysis of the electromyogram in order to extend the typical\nanalysis of the sEMG time series. The classical (mono-) fractal aspects \n\\cite{NaiKum2011,PhiLim2009,PhiPhu2012,XiaZhi2005} has also been \nextensively analysed for example in the context of force of contraction of different muscles \n\\cite{ArjKum2014}. \n\nElectromyography itself implies several challenges. It concerns an inappropriate location \nof the electrodes over the group of muscles \\cite{MesMer2009}, variation of the distance \nbetween the electrodes during the measurement and finally modification of the source position \nin relation to each electrodes. These in turn will influence the morphology of the series and\ncan manifest as a change of the shape, but most of all can affect the signal's amplitude.\nAnother important aspect is the cross talk, which is defined as the influence from the \nactivity of the neighbouring muscles. The last mentioned problem which can have quite a large\ncontribution to the systematic error is the inter--individual variability. This has a particular\nmeaning in the situation when we want to compare two or more individuals. This in \ngeneral is caused by the different tissue characteristics, which include\nthe thickness of adipose tissue, skin electrical resistance, sweating, hydration and \nthe thickness of epidermis.\n\nThe paper is organized as follows. In the next section we will describe the experimental\nsetup. Next we will report the details of the multifractal analysis. We will end with\nthe conclusions and summary.\n\n\\section{Set-up}\n\\label{setup}\nWe have performed the measurements on two individuals - a highly skilled one and\na complete beginner. They were given the complex\ntask which should be performed on the laparoscopic trainer. The task was to tie the\nstandard surgery knots using the laparoscopic tools. Usual recording time took around\n60 minutes and the signal was collected from four groups of muscless,\ntrapezius ridge (channels 1 and 5), deltoids (ch. 2 and 6), long palmar muscle and\nulnar wrist flexor (ch. 3 and 7) and abductor muscle of thumb and flexor brevis (ch. 4 and 8).\nChannels 1--4 and 5--8 were linked to the left and the right upper extremity, respectively. \nThe measurements were conducted with the portable 8 channel surface EMG recorder \n(OT Bioelettronica, Torino, IT) with the bipolar surface circular AgCl electrodes of size 15x15 mm. \nThe inter-electrode distance was set to 10 mm.\n\nThe system automatically records the maximum value of the signal from each of the active channels. \nThe Amplitude Rectified Value (ARV) measured in $\\mu$V was collected on MVC (maximum voluntary \ncontraction) recording mode. The ARV is mean value of the rectified EMG over a time interval \nT (125ms). The range of bandwidth was from 34 to 340 Hz. \nThe signal amplitudes were in the range of a few $\\mu$V to\neven more than 1000 $\\mu$V, see figure \\ref{fig11ab} for details. Additionally, we recorded the\nsignals from the relaxed muscles -- the recorded person was asked to sit down and reduce the \nmobility for a few minutes. This record consists of around 3500 data points (about seven minutes), \nsee figure \\ref{fig12ab}. The third measurement corresponds to the muscle in the strong contraction --\nboth subjects were supposed to keep 1 kg with arms bent and kept in parallel to the ground.\nAs it is quite hard to maintain such a posture for a long time, this data are the shortest \nand consists of only few hundreds measurement points (a little over a minute), see figure \\ref{fig13ab}.\n\nThere are visible differences in the electric potential generated by the muscle cells for all three different \nstates presented in figure \\ref{fig1}: the complex task (a, top row), the rest state (b, middle row) \nand the strong constant contraction (c, bottom row). \nThere is also a significant difference in the level of the muscle activity between a skilled person \nand an amateur. The amateur has about five times greater amplitude of the signal. The most probable\ncause for this lies in the different characteristics of the tissue, like the thickness of the skin \nand the adipose tissue which in turn leads to the different skin impedance for both cases.\n\n\\section{Multifractal analysis}\n\\label{mfdfa}\nIn the following we will present the typical Multifractal Detrended Fluctuation Analysis\n(MFDFA) as presented in \\cite{KanSte2002} and \\cite{Ihlen2012}. \nIn short, the analysis requires\nthe following stages. Suppose that we have time series with $N$ data points\n$\\{x_i\\}$, we perform than four consecutive steps\n\\begin{itemize}\n\\item[(i)] Calculate the profile $y_i$ as the cumulative sum from the data\nwith the subtracted mean\n\\begin{equation}\\label{eq1}\ny_i = \\sum_{k=1}^i [x_i - \\langle x \\rangle].\n\\end{equation}\n\n\\item[(ii)] The cumulative signal is split in $N_s$ equal non-overlapping segments of\nsize $s$. Here, for the width of the segments we use the power of two, $s = 2^r$,\nwhere $r = 4,\\dots,\\left \\lfloor \\log_2(N\/10) \\right \\rfloor$. Larger segment sizes \nwill result with rather weak statistics.\nTypically the length of the data will not be accordant\nwith the power of two and some data would have to be dropped from the analysis.\nTherefore the same procedure should be performed starting from the last index, and\nin turn the $2 N_s$ segments will be taken into account.\n\n\\item[(iii)] Calculate the local trend $y^m_{v,i}$ for $v^{th}$ segment by means of the\nleast--square fit of order $m$. Then determine the variance\n\\begin{equation} \nF^2(s,v) \\equiv {1 \\over s} \\sum_{i=1}^{s} \\left( y^m_{v,i} - y_{v,i} \\right)^2 \\label{fsdef} \n\\end{equation}\nfor each segment $v = 1, \\ldots, N_s$. The same procedure has to be \nrepeated in the reversed order (starting from the last index).\nNext determine the fluctuation function being\nthe $q^{th}$ statistical moment of the calculated variance.\n\\begin{eqnarray}\n\\label{eq2}\nF_q(s) &=& \\left(\\frac{1}{2N_s}\\sum_{v=1}^{2N_s} [F^{2}(s,v)] \\right)^\\frac{1}{q},\n\\quad q \\ne 0,\\\\\nF_0(s) &=& \\exp \\left\\{ {1 \\over 4 N_s}\n\\sum_{\\nu=1}^{2 N_s} \\ln \\left[F^2(s,\\nu)\\right] \\right\\},\n\\quad q = 0.\n\\end{eqnarray}\nThe above function needs to be calculated for all segment sizes $s=2^r$.\nWe have exploited several different orders of the fitted polynomials and end up with no\nstatistical difference between the results. Here we will present the analysis with\nthe quadratic fit.\n\n\\item[(iv)] In the last step the determination of the scaling\nlaw of the fluctuation function (\\ref{eq2}) is performed by means of the log--log\nplots of $F_q(s)$ versus segment sizes $s$ for all values of $q$. The function\n$F_q(s) \\sim s^{h(q)}$ is naturally smaller for the smaller fluctuations, which\nresults in the increasing function with the increasing segment size.\nFrom the calculated Hurst exponent $h(q)$ we are able to determine several quantifiers.\nFirstly, we work out the mass exponent using the formula\n\\begin{equation}\\label{mass}\n\\tau(q) = q h(q) - 1.\n\\end{equation}\nSecondly we can obtain the singularity exponent $\\alpha(q)$ by applying the Legendre\ntransform. The last quantifier and the main result of the MFDFA method is the singularity\nspectrum, given by\n\\begin{equation}\\label{spectrum}\nD[\\alpha(q)] = q \\alpha(q) - \\tau(q).\n\\end{equation}\n\\end{itemize}\nThe detailed information on how to read the singularity spectrum can be found in\nthe literature \\cite{MakRyn2011,Kantelhardt2008,MakFul2010}.\nMultifractality is an indication of the complex dynamics where the single exponent (like\nthe fractal dimension) will not be enough to describe the phenomenon. In the case\nwhen the data exhibits not just one individual exponent the continuous spectrum of exponents\nshould be taken into account.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.98\\linewidth]{fig2a}\n\\includegraphics[width=.98\\linewidth]{fig2aa}\\\\\n\\centering\n\\includegraphics[width=.985\\linewidth]{fig2b}\n\\includegraphics[width=.985\\linewidth]{fig2bb}\n\\caption{(color online) The multifractal spectrum for channel 1 (in the vicinity of the trapezius ridge) for\nprofessional (a,b) and amateur (c,d). On each panel three curves depict the multifractal\nspectrum corresponding to the three states of the performance of the the muscle group:\nexecuting complex task (red), at rest (green) and at the maximum contraction (blue).\nThe panels (a) and (c) present the spectra for the original data, while the panels\n(c) and (d) show the spectra for the shuffled data.\nAll figures have the same width for the sake of comparison.}\n\\label{fig2}\n\\end{figure}\n\n\\section{Multifractal spectra at complex work}\n\nThe central result of this work is presented in figure \\ref{fig2}. The plot presents the \nmultifractal spectrum. This relation is defined as the singularity spectrum \nas the function of the singularity exponent $D = D(\\alpha(q))$. \nThe multifractal spectrum describes how often the irregularity of certain degree occurs in the signal. \n$D(\\alpha)$ represent $q$-order singularity dimension and \n$\\alpha(q)$ stands for the $q$-order singularity exponent (\\ref{spectrum}). \nIt illustrates the variability in \nthe fractal structure of the time series. The monofractal time series has dense mf--spectrum \naround the single point $(H,1)$, where $H$ is the global Hurst exponent \n\\cite{Hur1951}.\nTo investigate the scaling properties of our data the analysis of the the position of the \nmaximum value of the mf--spectrum, the global Hurst exponent itself, the half--width and the \nextremes of the mf--spectrum have to be taken into consideration \\cite{MakRyn2011}. \nThe whole analysis was performed for all 8 channels. We have calculated the $q$-th order fluctuation\nfunction $F_q(s)$ for 100 values of the invert power $q \\in [-5,5]$ with the step equal to $0.1$\nas suggested in the literature \\cite{Ihlen2012}.\n\nThere are two general sources of multifractality which can affect the shape of the mf-spectrum:\n(i) one is due to the broad probability density function which lies behind the data (or its\nfluctuations); (ii) second is driven by the different behaviour of the (auto)correlation function for\nlarge and small fluctuations; (iii) both situations simultaneously.\nSimple data shuffling can test the possible source of multifractality. In the case (i)\nshuffling will not change the mf-spectrum, for (ii) will destroy the effect completely\nas the shuffling will destroyed the possible correlations; in the last case (iii) the spectrum\nwill differ from the original one-shuffled series will exhibit somehow weaker multifractality. \nFor the all analysed cases the correlation for large and small fluctuations seem to be\nthe main factor which causes the strong multifractality \\cite{Horvatic2011}\n-- please compare pairwise panels (a) -- (b) and (c) -- (d) in the figure \\ref{fig2}. \nFor the presented analysis this is the usual effect for all of the spectra except\nfor the working state for the professional and the maximal contraction state for the amateur.\nThe Hurst $H$ exponent for all of the cases behaves however in a similar way for the \nshuffled data. In the working state after shuffling the estimated values are very close\nto $0.5$ which suggest that in this very case we deal with the white uncorrelated white noise. \nThis means that the correlations are the only source of multifractality (case ii).\nThe other states show a little bit higher (contraction) and lower (rest) values of $H$ than\nin the work state, which may suggest some sort of the monofractal behaviour (again after\nshuffling) -- see panels (b) and (d).\n\nThe shape and the width of the multifractal spectrum provide the information about the local changes \nof the Hurst exponent. We can see that the value of the spectral width is different for different states \nof muscular tension for both individuals. \nA large difference between periods when small and large fluctuations takes place increases in turn the width of the spectrum.\nThe analysis of the signal where neither weak nor strong local fluctuations dominate will result in the symmetric \nshape of the mf-spectrum. This aspect is visible for the sequence of the nonprofessional performing full task \n-- see the red curve in figure \\ref{fig2}(c) for details. On the contrary the study of the corresponding signal \nbut for the professional exhibits the dominance of the low local fluctuations. This feature is also clearly visible \nin the raw signal, see figure \\ref{fig11ab} on the r.h.s. This situation is a manifestation of the influence\nof the training for the resulting electrical potential generated by group of the examined muscle cells.\nThe trained person will use his locomotor system very effectively, allowing only simple and necessary\nmovements. Therefore the resultant spectrum will in general express close similarity to the spectrum\nfor activity system at rest -- compare green curve in figure \\ref{fig2}(a). Both just described states\nwill show the dominance of the low fluctuations, which is an indicator of the weak excitations in muscle\ncell membranes. \nOn the contrary the spectrum for the untrained person at actual work will show rather broad and symmetric\nmultifractal spectrum. The rare events are as distant from the maximum value as low fluctuations and as \nresult neither weak nor strong fluctuations dominate in the signal. Even at the rest state the\nspectrum is rather wide and symmetric, which indicate constant excitations from the electrical \nactivity of the muscle cell membrane -- i.e. even at rest the amateur will unnecessarily exploit\nthe energy.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=.9\\linewidth]{fig3}\n\\caption{$q$-order Hurst exponent $h(q)$ for the series with the dominance of the \nsmall (green, solid) and large (blue, dashed) local fluctuation for professional.}\n\\label{fig3}\n\\end{figure}\n\nThe time series, or rather its fluctuations, exhibit similarity for both individuals only in the state of \nthe full contraction. The natural\ntendency to strong excitation of the cell membranes will result in the predominance of the high \nfluctuations. This is visible as the high variability in the signal (figure \\ref{fig13ab}), \nas a mf--spectrum with the left truncation or as the leveling of the $q$-order Hurst exponents positive $q$'s,\nsee blue dashed curve in figure \\ref{fig3} for details.\n\nThe difference of the  multifractal structure in the local fluctuation with small and large variation is visible \nalso on $h(q)$ and $\\alpha(q)$ dependence, which respectively presents Hurst and singularity exponent as a function of $q$.\nFor the $h(q)$ dependence see figure \\ref{fig3} for details. The $\\alpha(q)$ is calculated as the tangent slope of the mass exponent \n$\\tau(q)$ (\\ref{mass}). For monofractal signals (or similarly for the white noise) \nthe $h(q)$ and $\\alpha(q)$ would be independent of $q$. On the other hand, for the time series which exhibit \nmultifractality, the discussed quantifiers will typically be monotonically decreasing functions.\nNontheless, for certain behaviors of the series, for instance if only large (small) local fluctuations\nprevail the $h(q)$ dependence will show plateau for positive (negative) values of $q$. This feature\ncause the truncation of the left (right) branch of the corresponding mf--spectrum shown in \nthe figure \\ref{fig2}.\n\n\\begin{table}[htbp]\n\\caption{The parameters of the mf--spectrum, calculated for three states of muscle activity respectively for the raw and integrated data.}\n\\begin{center}\n\\begin{tabular}[H]{|c|c|c|c|} \\hline\nObject & maximal contraction & task & relaxation \\\\\n\\hline\n\\multicolumn{4}{|c|}{Hurst exponent $H^{int} =h^{int}(2)$}\\\\\n\\hline\nProfessional & $0.714$ & $0.31$ & $0.664$ \\\\\nAmateur & $1.01$ & $0.67$ & $0.772$\\\\ \\hline\n\\multicolumn{4}{|c|}{Hurst exponent $H = h(2)$}\\\\ \\hline\nProfessional & $0.062$ & $-0.147$ & $0.031$ \\\\\nAmateur & $0.094$ & $-0.017$ & $0.025$\\\\ \\hline\n\\multicolumn{4}{|c|}{$h_{max}^{int} =h^{int}(0)$ }\\\\ \\hline\nProfessional & $0.759$ & $1.06$ &$0.755$ \\\\\nAmateur & $0.981$ & $1.14$ & $1.05$\\\\ \\hline\n\\multicolumn{4}{|c|}{$h_{max} = h(0)$ }\\\\ \\hline\nProfessional & $0.111$ & $0.166$ &$0.027$ \\\\\nAmateur & $0.149$ & $0.239$ & $0.175$\\\\ \\hline\n\\multicolumn{4}{|c|}{$\\Delta_{1\/2}^{int}$ }\\\\ \\hline\nProfessional & $0.045$ & $0.756$ &$0.092$ \\\\\nAmateur & $0.031$ & $0.467$ & $0.282$\\\\ \\hline\n\\multicolumn{4}{|c|}{$\\Delta_{1\/2}$ }\\\\ \\hline\nProfessional & $0.0485$ & $0.312$ &$0.0044$ \\\\\nAmateur & $0.0556$ & $0.256$ & $0.150$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\label{tab1}\n\\end{table}\n\nThere are several parameters we can use to effectively describe mf--spectrum and, consequently, the signal which lies behind it.\nIn the table \\ref{tab1} we have collected the typical quantifiers -- the values of the typical Hurst exponent $H=h(2)$, \nsingularity exponent located at the maximum of the spectrum $h_{max}$, and\nspectrum half--width $\\Delta_{1\/2}$ defined as the absolute value of the difference between Hurst exponent and $h_{max}$, \nall calculated for both row and integrated data. \nThe integrated data are calculated as cumulative sums of the original (raw) signal. \nIn the case of the monofractal signal the spectrum of the integrated signal is usually the same as for the raw \none, but shifted by 1 to the right. Lower values of this difference suggest that the obtained \nmf--spectra have other than monofractal scaling.\nAgain, for almost all factors there is a significant difference between amateur \nand professional for two states -- at the relaxation and during the assumed task. For maximum contraction the only factor\nwhich distinguishes the professional and amateur is the Hurst exponent calculated for integrated data. \n\n\\section{Summary and conclusions}\nThe comparison of the kinesiological electromyographic signal between a professional (highly trained) and an amateur \nwas presented for the three typical states of work of the human musculo--skeletal system. Based on the multifractal \ndetrended analysis we have shown the differences and similarities for the data fluctuations. The main message\nwhich can be drawn from the analysis is that the locomotor aparatus for the trained person would require\nmuch less energy to perform tasks as it's work would produce much smaller local fluctuations. The multifractal spectrum\nwould in turn look more similar to the one at the rest state. On the other hand there is much less chance to distinguish \nthe depth of training between two persons if one would look at the sEMG data assembled at the strong muscle tension. \nThe muscle cell membranes will in this case tend to react with much higher voltage of the electrical potential as these \ncells will be much stronger activated neurologically.\n\nIn conclusions we would like to suggest the possible application for automatic verification of abilities for \nperforming complex tasks based on the fluctuation analysis. If the person's multifractal spectrum would be wide with\nno truncation on the right side (at higher singularity exponents $\\alpha(q)$) than the low and high local fluctuation\nwould be equally probable. This means that the examined person still operates with too much stress and struggles with\nthe task, therefore some more training would still be needed.\n\n\\section*{Acknowledgements}\nAuthors would like to thank Danuta Makowiec and Jan \u017bebrowski for valuable discussions. \nThis work was partially supported by the Polish Ministry of Science and Higher Education\n(Grant K\/ZDS\/003962).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThis sequel of our paper on the Onsager-Machlup theory \\cite{Jurisch1} is inspired by a serious mistake we noticed in a paper by Taniguchi and Cohen \\cite{Taniguchi}. As it turned out, this mistake has already been made by Onsager and Machlup in their seminal paper \\cite{Machlup}, and has been applied ever since. In the attempt to include inertial effects into the Langevin-equation Onsager and Machlup proposed a second order variational-principle based on a generalization of the Onsager-Machlup Lagrangian. This but violates Ostrogradsky's theorem \\cite{Ostrogradsky}. In the following we elucidate the source of the mistake and suggest a solution of the problem.\n\nThe equation of motion of interest is given by\n\\begin{equation}\nm\\,\\ddot{q}(t)\\,=\\,F[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\quad,\n\\label{Introduction1}\\end{equation}\nwhere $F[q(t)]$ is a conservative force-field, the second term describes Stokes-friction, and the last term is a Gaussian stochastic process. Thus, it is assumed that already the acceleration is distorted by noise. In \\cite{Taniguchi} as in \\cite{Machlup} friction has been introduced by purely phenomenological arguments.\n\nBy the assumption that the motion takes place in the over-damped regime, $\\ddot{q}(t)\\,=\\,0$, Eq. (\\ref{Introduction1}) can be cast into\n\\begin{equation}\n\\dot{q}(t)\\,=\\,-\\,\\mu_{1}^{-1}\\,\\left(F[q(t)]\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\right)\\quad,\n\\label{Introduction2}\\end{equation}\nwhich may be identified as a Langevin-equation, from which the corresponding Onsager-Machlup Lagrangian can be derived. We remark that such a reinterpretation already becomes impossible when Newtonian friction $\\sim\\,\\dot{q}^{2}$ is present, see \\cite{Jurisch1}.\n\nIn a next step, following Onsager and Machlup \\cite{Machlup}, the acceleration is included as an additional drift-field\n\\begin{equation}\n\\dot{q}(t)\\,=\\,\\frac{m}{\\mu_{1}}\\,\\ddot{q}(t)\\,-\\,\\mu_{1}^{-1}\\,\\left(F[q(t)]\\,+\\,\\sqrt{\\sigma_{2}}\\,\\eta(t)\\right)\\quad,\n\\label{Introduction3}\\end{equation}\nwhich is just a rearrangement of Eq. (\\ref{Introduction1}). Along with \\cite{Machlup} Eq. (\\ref{Introduction3}) is still interpreted as a Langevin-equation, which is a highly doubtful interpretation. Furthermore, again following Onsager and Machlup \\cite{Machlup}, the Onsager-Machlup Lagrangian is set up by\n\\begin{equation}\n\\mathcal{L}(\\ddot{q},\\,\\dot{q},\\,q)\\,=\\,\\frac{\\mu_{1}^{2}}{2\\,\\sigma_{2}}\\,\\left(\\dot{q}(t)\\,-\\,\\frac{m}{\\mu_{1}}\\,\\ddot{q}(t)\\,+\\,\\mu_{1}^{-1}\\,F[q(t)]\\right)^{2}\\quad.\n\\label{Introduction4}\\end{equation}\nHowever, this Lagrangian is completely meaningless. The fundamental, but unfortunately scarcely known theorem of Ostrogradsky \\cite{Ostrogradsky} proves that Lagrangians, which depend on higher than first order derivatives in a non-degenerate way do not describe any meaningful dynamics. Non-degeneracy here means that $\\partial\\mathcal{L}\/\\partial\\ddot{q}$ is a function of $\\ddot{q}$, to which the Lagrangian Eq. (\\ref{Introduction4}) applies for. Ostrogradsky's theorem holds in general, it is one of the two hard criteria every Lagrangian, which claims physical content must obey. The second hard criterion are the Helmholtz-conditions \\cite{Helmholtz}, see also Nigam and Banerjee \\cite{Nigam},  Nucci and Leach \\cite{Nucci}, and references therein. A violation of one of these criteria is sufficient to render a Lagrangian meaningless. Consequently, the Lagrangian under consideration here does not exist. This clarifies that the interpretation of Eq. (\\ref{Introduction3}) as a Langevin-equation is ineligible. Furthermore, this rules out the second order variational-principle given by Onsager and Machlup in \\cite{Machlup}. As a consequence, inertial effects cannot be included in a standard way.  We have not checked if the Lagrangian Eq. (\\ref{Introduction4}) also violates the Helmholtz-conditions, since the violation of Ostrogradsky's theorem is sufficient to rule this ansatz out.\n\nSystems described by Lagrangians with higher than first order derivatives suffer from the Ostrogradsky-instability. In short, the phase-space of such systems either explodes or decays almost instantly. This comes from pathological excitations and states of unbounded positive and negative energy. We emphasize that this must not be confused with the Dirac-theory. The Dirac-Lagrangian is sound with respect to it's derivatives, the spinors are relativistic fields and the unbounded scale of energy has a precise physical meaning.\n\nTo illustrate the origin of the Ostrogradsky-instability, we calculate the Hamiltonian of the Lagrangian Eq. (\\ref{Introduction4}). The canonic coordinates are given by\n\\begin{equation}\nQ_{1}\\,=\\,q,\\quad Q_{2}\\,=\\,\\dot{q},\\quad P_{1}\\,=\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\dot{q}}\\,-\\,\\frac{d}{d\\,t}\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\ddot{q}},\\quad P_{2}\\,=\\,\\frac{\\partial\\,\\mathcal{L}}{\\partial\\,\\ddot{q}}\\quad.\n\\label{Introduction5}\\end{equation}\nIt is easy to see that this choice is the natural generalization of the first order case. For our present system, this gives\n\\begin{equation}\nP_{2}\\,=\\,\\frac{m}{\\sigma_{2}}\\,\\left(m\\,\\ddot{q}(t)\\,-\\,\\mu_{1}\\,\\dot{q}(t)\\,-\\,F[q(t)]\\right)\\,\\rightarrow\\,\\ddot{q}(t)\\,=\\,\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2})\\,=\\,\\frac{\\sigma_{2}}{m^{2}}\\,P_{2}\\,+\\,\\frac{\\mu_{1}}{m}\\,Q_{2}\\,+\\,m^{-1}\\,F[Q_{1}]\\quad.\n\\end{equation}\nThe Hamiltonian follows on the standard route\n\\begin{eqnarray}\n\\mathcal{H}(P_{2},\\,P_{1},\\,Q_{2},\\,Q_{1})&=&P_{1}\\,Q_{2}\\,+\\,P_{2}\\,\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2})\\,-\\,\\mathcal{L}\\left(\\mathcal{Q}(P_{2},\\,Q_{1},\\,Q_{2}),\\,Q_{2},\\,Q_{1}\\right)\\nonumber\\\\\n&=&P_{1}\\,Q_{2}\\,+\\,\\frac{\\sigma_{2}}{2\\,m^{2}}\\,P_{2}^{2}\\,+\\,P_{2}\\,\\left(\\frac{\\mu_{1}}{m}\\,Q_{2}\\,+\\,m^{-1}\\,F[Q_{1}]\\right)\\quad.\n\\label{Introduction6}\\end{eqnarray}\nThe term linear in $P_{2}$ goes without problems, it can be removed by quadratic completion and acts like a gyroscopic potential. The term linear in $P_{1}$ but is non-trivial, since it does not resolve by Legendre-transform. The momentum $P_{1}$ is not subject to any constraints, it is completely unbounded and can take any value. By coupling to $Q_{2}$ this lack of boundary tends to disrupt the whole system. It is now easy to see that still higher order derivatives in the Lagrangian make things only worse. This is the origin of the Ostrogradsky-instability. By calculating the canonic equations and going back to the Euler-Lagrange equations one can convince oneself that Ostrogradsky's choice for the canonic coordinates indeed is correct.\n\nFor the whole impact of Ostrogradsky's theorem, please see e.g. Motomashi \\cite{Motomashi}, especially Woodard \\cite{Woodard}, and references therein. For completeness we add that Eq. (\\ref{Introduction4}) must not be confused with the Gaussian action-principle, see e.g. Lanczos \\cite{Lanczos}. The Gaussian action is no Lagrangian, but an application of the method of least squares.\n\\newline\n\n\\emph{Short historical note}: Because Ostrogradsky's theorem is unbeknownst to a broader audience, we think some words about it's history are in order. Ostrogradsky discovered his theorem in 1850, but it got almost unnoticed. At this time the understanding of energy and stability was still in it's childhood, and nobody could grasp what Ostrogradsky's theorem really says. Obviously even Landau and Lifshitz have not known about Ostrogradsky's theorem, nothing else could explain why it is not to be found in their textbook about mechanics. Thus, there is no wonder about that also Onsager and Machlup have not been aware of Ostrogradsky's theorem.\n\nOstrogradsky's theorem was rediscovered in particle physics, general relativity and cosmology. As far as we know, the first who proposed higher order Lagrangians were Pais and Uhlenbeck \\cite{Pais}, ironically in 1950. Skeptics asked if this is allowed. On this route Ostrogradsky's theorem was rediscovered and is a nuisance in particle physics and cosmology ever since.\n\n\n\\section{Suggested solution of the problem}\nIn this section we shall elucidate our suggestion for a solution of the problem posed by Eqs. (\\ref{Introduction1}). Since the Newtonian equation of motion with noise is not tractable by the Onsager-Machlup theory, we need two first-order equations with independent variables to introduce Langevin-equations. Thus, we employ the canonical formalism.\n\nThe equation of motion with Stokes-friction\n\\begin{equation}\n\\ddot{q}(t)\\,=\\,-\\,\\frac{1}{m}\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\quad,\n\\label{Solution7}\\end{equation}\nis a consequence of the Onsager-Machlup Lagrangian, see \\cite{Jurisch1},\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,b(t)^{-2}\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\quad.\n\\label{Solution8}\\end{equation}\nThe function $b(t)=\\exp[\\mu_{1}\/2\\,t]$ is a Helmholtz-factor or, likewise, a Jacobi-multiplier. The Hamiltonian thus reads\n\\begin{equation}\n\\mathcal{H}(p,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p^{2}(t)}{2\\,m}\\,+\\,\\exp[-\\,\\mu_{1}\\,t]\\,U_{\\rm{I}}[q(t)]\\quad,\n\\label{Solution9}\\end{equation}\nwhich also is known as the Caldirola-Kanai Hamiltonian \\cite{Caldirola, Kanai}.\nThe canonic equations follow by\n\\begin{eqnarray}\ndq(t)&=&\\partial_{p}\\mathcal{H}(p,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)}{m}\\,dt\\quad,\\nonumber\\\\\ndp(t)&=&-\\,\\partial_{q}\\mathcal{H}(p,\\,q)\\,=\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,dt\\,+\\,\\sqrt{\\sigma_{2}}\\,dW(t)\\quad,\n\\label{Solution10}\\end{eqnarray}\nwhere we already have introduced the stochastic momentum-process with variance $\\sigma_{2}$. The Onsager-Machlup Lagrangian of the momentum-process then is\n\\begin{equation}\n\\mathcal{L}(\\dot{p},\\,q)\\,=\\,\\frac{1}{2\\,\\sigma_{2}}\\,\\left(\\dot{p}(t)\\,+\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\right)^{2}\\quad.\n\\label{Solution11}\\end{equation}\nThe new Hamiltonian of the system can now safely be written by\n\\begin{equation}\n\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)^{2}}{2\\,m}\\,+\\,\\frac{\\sigma_{2}}{2}\\,P(t)^{2}\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,P(t)\\quad,\n\\label{Solution13}\\end{equation}\nwhere the second and third terms follow by a Legendre-transform of the Lagrangian Eq. (\\ref{Solution11}. The unified canonic equations finally yield\n\\begin{eqnarray}\n\\dot{q}(t)&=&\\partial_{p}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[\\mu_{1}\\,t]\\,\\frac{p(t)}{m}\\quad,\\nonumber\\\\\n\\dot{p}(t)&=&\\partial_{P}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\sigma_{2}\\,P(t)\\,-\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\quad,\\nonumber\\\\\n\\dot{P}(t)&=&-\\,\\partial_{q}\\mathcal{H}(p,\\,P,\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(t)]\\,P(t)\\quad.\n\\label{Solution14}\\end{eqnarray}\nFor $\\sigma_{2}=0$ the canonic equations decouple, and we recover Eq. (\\ref{Solution10}), as it must. As a last step, we eliminate the momenta and calculate the Newtonian equation of motion for the the most probable path $q(t)$. The calculation is easily done by differentiating the first line in Eq. (\\ref{Solution14}) and insert the other equations. This yields\n\\begin{equation}\n\\ddot{q}(t)\\,=\\,-\\,\\frac{1}{m}\\,\\partial_{q}U_{\\rm{I}}[q(t)]\\,+\\,\\mu_{1}\\,\\dot{q}(t)\\,+\\,\\frac{\\sigma_{2}}{m}\\,\\exp[\\mu_{1}\\,t]\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\exp[-\\,\\mu_{1}\\,\\tau]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\right]\\quad.\n\\label{Solution15}\\end{equation}\nThis Newtonian equation of motion elucidates that the noisy distortion influences the trajectory of the most probable path as additional force, which but depends on the interaction $U_{\\rm{I}}[q(t)]$. Note that this force still remains finite for $\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\,=\\,0$. Furthermore, the additional force grows or decays as time evoles. If the force grows, this means that the stochastic noise tends to destroy the action of the potential and the friction, while a decay leaves the action of the potential and the friction sound or even enhances it.\n\nWe can integrate Eq. (\\ref{Solution15}) and obtain the Onsager-Machlup Lagrangian for the combined system, reading\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\left(\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{I}}[q(t)]\\right)\\,+\\,\\sigma_{2}\\,q(t)\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\exp[-\\,\\mu_{1}\\,\\tau]\\,\\partial_{q}^{2}U_{\\rm{I}}[q(\\tau)]\\right]\\quad.\n\\label{Solution16}\\end{equation}\nIn principal, it is always possible to derive the Newtonian equation of motion from the canonic equations, but a reconstruction of the primal Lagrangian of the system may not always be possible. We chose the word \\emph{primal} here to emphasize that this Lagrangian combines the environmental effects all in one, and thus is the Lagrangian to start with for all further examinations. We remark that the time-dependent exponential, last term in Eq. (\\ref{Solution16}), shows the general structure of a Helmholtz-factor.\n\n\n\\section{Notes on the hierarchy of immersions and actions of environments}\nFollowing our classification of interactions in \\cite{Jurisch1}, we see that the additional potential in Eq. (\\ref{Solution16}) acts like an external potential $U_{\\rm{E}}[q(t),\\,t]$. The first term in the Lagrangian Eq. (\\ref{Solution16}) describes an ideal harmonic oscillator, that is immersed in environment (I), which creates Stokes-friction. The stochastic process disturbs the environment (I) from the outside, and consequently may be understood as environment (II), that acts upon environment (I). This leads to a hierarchy, that can be written by sets\n\\begin{equation}\n\\{\\rm{harmonic\\,oscillator}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{environment (II)}\\quad.\n\\label{Solution17}\\end{equation}\nWe chose to write $\\subset$ instead of $\\in$, since we understand the ideal system not as to be an element of environment (I), but as a system immersed into environment (I). Thus, it is an additional subset of environment (I), which can be taken out again without reducing environment (I). If the ideal system would be an element, then a take-out would reduce environment (I).\n\nSome more remarks have to be made about the system described in \\cite{Taniguchi}. There, the potential $U[q(t)]$ is used as an external harmonic potential, that traps Brownian particles. To our regards, the correct initial ansatz for this system must be written by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\quad,\n\\label{Solution18}\\end{equation}\nsince the harmonic trap acts from the outside. The Lagrangian Eq. (\\ref{Solution18}) generates a Newtonian equation of motion with Stokes-friction. The system of Brownian particles itself is thus solely described by the kinetic energy plus the noise, while the harmonic trap surrounds this system, and consequently acts from the outside. According to our results in \\cite{Jurisch1}, this modifies the equation of motion, and thus also leads to a different primal Lagrangian. In this interpretation, the hierarchy of systems as described in \\cite{Taniguchi} is given by\n\\begin{equation}\n\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{environment (II)}\\right\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\quad.\n\\label{Solution19}\\end{equation}\nThis also means that the external harmonic potential $U_{\\rm{E}}[q(t)]$ must only be added to the primal Onsager-Machlup Lagrangian as an external potential. This hierarchy then leads to\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}\\,+\\,\\sigma_{2}\\,q(t)\\right\\}\\,-\\,U_{\\rm{E}}[q(t)]\\quad.\n\\label{Solution20}\\end{equation}\n\nIf, and only if environment (II) shall also disturb the external harmonic trap, then the initial Lagrangian can be taken by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\,-\\,U_{\\rm{E}}[q(t)]\\quad.\n\\label{Solution21}\\end{equation}\nThe set-relation in this case yields\n\\begin{equation}\n\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\right\\}\\,\\leftarrow\\,\\rm{environment (II)}\\quad,\n\\label{Solution22}\\end{equation}\nand the primal Lagrangian follows by\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)\\,=\\,\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\,\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}\\,-\\,U_{\\rm{E}}[q(t)]\\right\\}\\,+\\,\\sigma_{2}\\,q(t)\\,\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\,\\partial_{q}^{2}U_{\\rm{E}}[q(\\tau)]\\right]\\quad.\n\\label{Solution23}\\end{equation}\n\nThe primal Lagrangians Eqs. (\\ref{Solution16}, \\ref{Solution20}, \\ref{Solution23}) elucidate how sensitive the dynamics of a system depends on the hierarchy of immersions and actions of several environments. Just for completeness, we illustrate the case, where the system Eq. (\\ref{Solution23}) is immersed into an environment (III), described by the function $b[q(t),\\,t]$. The set-relation then reads\n\\begin{equation}\n\\left\\{\\left\\{\\{\\rm{free\\,particles}\\,\\subset\\,\\rm{environment (I)}\\}\\,\\leftarrow\\,\\rm{harmonic\\,trap}\\right\\}\\,\\leftarrow\\,\\rm{environment (II)}\\right\\}\\,\\subset\\,\\rm{environment\\,(III)}\\quad,\n\\label{Solution24}\\end{equation}\nand the primal Lagrangian becomes\n\\begin{equation}\n\\mathcal{L}(\\dot{q},\\,q)=b^{-2}[q(t),\\,t]\\left(\\left\\{\\left\\{\\left\\{\\exp[-\\,\\mu_{1}\\,t]\\frac{m}{2}\\,\\dot{q}(t)^{2}\\right\\}-U_{\\rm{E}}[q(t)]\\right\\}+\\sigma_{2}\\,q(t)\\exp\\left[\\int_{t_{0}}^{t}d\\tau\\,\\partial_{q}^{2}U_{\\rm{E}}[q(\\tau)]\\right]\\right\\}\\right)-\\mathcal{V}[q(t),\\,t]\\,,\n\\label{Solution25}\\end{equation}\nwhere the potential $\\mathcal{V}[q(t),\\,t]$ subsumes all additional potentials, that depend on $b[q(t),\\,t]$, see \\cite{Jurisch1}.\n\nBy our discussion we understand that immersions and actions of environments mostly can be implemented directly into the Lagrangian without complications. Only the action of an additional stochastic process on the Newtonian equation of motion requires to go through the formalism we have developed above.\n\n\n\\section{Remark}\nAs a last point, we shall remark that in exotic field-theory methods have been developed, which, under certain circumstances, can remove the Ostrogradsky-instability from the Hamiltonian, see e.g. Chen et. al. \\cite{Chen}. The method of choice is the Dirac-constraint \\cite{Dirac}. If the Ostrogradsky-instability is removable, this leads to a reduction of the phase-space, where the system then evolves on a restricted hyper-surface only. Chen et. al. \\cite{Chen} successfully probe the method of Dirac-constraints on a variety of elementary problems, including the Pais-Uhlenbeck oscillator \\cite{Pais}. However, if a reduction of the phase-space is possible, then this comes at the cost that the resulting Lagrangian is completely remote from the initial problem. Furthermore, in order to be successful, artificial auxiliary terms must be added to the initial ill-defined Lagrangian by Lagrange-multipliers, which are constructed in a way that the Dirac-constraints can act as wished. From a sane bottom-up view, however, such approaches are more than questionable. To enforce something by tinkering, which is just not there ignores a distinct caveat.\n\nOur suggestions is built upon the contrary, in that we enlarge the dimension of the phase-space. This allows us not only to resolve the second order derivative of the Newtonian equation of motion, but also the inclusion of the stochastic process by applying the Onsager-Machlup theory straight.\n\n\n\\section{Conclusion}\nBy using Ostrogradsky's theorem, we ruled out the possibility to reinterpret Newton's equation of motion with Stokes-friction and stochastic noise as a Langevin-equation, since the corresponding Onsager-Machlup Lagrangian contains second order derivatives in a non-degenerate way. Such a Lagrangian is ill-defined and there is no physics in it. This shows that Newton's equation of motion and the Langevin-equation are something completely different. \n\nFurthermore, we suggested a method of how to treat a system where Newton's equation of motion is distorted by noise. Our ansatz is built upon the canonical formalism, and the results we achieved seem to be sound. We were able to derive an Onsager-Machlup Lagrangian, that describes the effects of the stochastic distortion by a force, which additionally influences of the most probable path a particle takes through a disordered environment.\n\nLast, we gave arguments of how to understand the hierarchy of environments, that prove useful for the analysis of systems of higher complexity. This has shown that the structure of the primal Lagrangian sensitively depends on the hierarchy of immersions and actions of the environments.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{A Deep Unsolved Problem}\n\nThis article is proposing a simplistic model to account for the behavior of the viscosity of liquids as a function of temperature. The best illustration of this behavior is the data from Angell's review paper \\cite{Ang95}, that can be found below in Fig.~\\ref{visc.fig-ang}. It shows on one graph the behavior of the logarithm of the viscosity $\\eta$ as a function of inverse temperature, in order to reveal whether or not an Arrhenius law  is valid $\\eta \\sim e^{W\/\\kB T}$. The excitation energy $W$ is represented by the slope of the curve in such coordinate axis. The graph actually shows that the Arrhenius law is valid for a category of materials called {\\em ``strong liquids''}. For most other materials, called {\\em ``fragile''}, the slope changes as the temperature approach the glass transition.\n\n\\begin{figure}[ht]\n   \\centering\n\\includegraphics[width=9cm]{angell.jpg}\n\\caption{\\label{visc.fig-ang} Viscosity of the liquid phase of various glassy materials. The logarithm of the viscosity is maps as a function of the normalized inverse temperature, to exhibit Arrhenius law. The normalization is provided by the glass transition temperature $T_g$. (Taken from \\cite{Ang95}).}\n\\end{figure}\n\nThe glassy and liquid states of a material share many features. It is understood that the difference between them cannot be account for by using equilibrium Thermodynamics: the glass transition is not a phase transition, it is a dynamical process in which the viscosity becomes so large that the typical time scales involved in the material become macroscopic. The viscosity changes by 15 orders of magnitude over a very small temperature range near the transition. The reason why viscosity changes so rapidly, while the atomic structure does not show signs of this change, has been a mystery for a long time. It has been considered as one of the deepest unsolved problems in science \\cite{An95,La07} by the most prominent experts.\n\n The present model is based upon several decades of experimental observations, numerical simulations and attempts to built a theory liable to explain the origin of viscosity. The author is especially indebted to T.~Egami, from Oak Ridge National Laboratory and to J.~Langer from UC Santa Barbara, for having patiently explained their understanding and knowledge of this difficult subject.\n\n The work presented here is a small brick added to the knowledge, hoping that it will open the door to a more accurate theory liable to explain this mystery. However, several evidences point toward this model providing some universal results: (a) it introduces a parameter that permits to distinguish between strong and fragile glasses, (b) it exhibits a {\\em ``cross-over temperature''} $T_{co}$, below which the viscosity varies very quickly {\\em w.r.t.} the temperature. In addition, while being first motivated by the study of bulk metallic glasses, it seems to be relevant in other areas of Physics and Material Sciences, such as colloids \\cite{Ega17}, usual glassy materials in their liquid phase, or even quantum fluids \\cite{Hel17}.\n\n\\subsection{Anankeons}\n \\label{visc.ssect-ananke}\n\nThe fundamental concept on which this work is based will be called {\\em anankeon}. The name came from a long discussion between the present author and T.~Egami, during which the later formulated his view on what were the most fundamental degrees of freedom in the liquid phase \\cite{EB12}. More precisely, in his view, it is the cutting and forming of the atomic bond which matters, an action changing the local topology of atomic connectivity network. To explain quickly what the term anankeon means, it suffices to understand that a liquid, like a solid, is a condensed material. Namely the atoms are jam packed together in a local structure made of well ordered clusters, with not much empty places available around. The local stress felt by atoms leads them to jump or to locally reorder themselves  to find a more comfortable position, trying to minimize their energy. These events are fast. They correspond to crossing a saddle point in the energy landscape, namely a passage through an unstable equilibrium positions \\cite{Ega14}. In practice they happen at unpredictable times, so that their dynamic is more efficiently described in terms of a Markov process. Hence the corresponding stress tensor felt by each atom is constantly varying under the {\\em stress of circumstances}, a concept that can be translated by the word $\\alpha\\nu\\alpha \\gamma\\kappa\\epsilon\\iota\\alpha$ (anagkeia) in Greek. Associated with this concept was the Greek goddess {\\em Ananke}, expressing the fate or destiny, due to uncontrollable forces, constraints, necessity. Hence the name anankeon associated with these stress induced unpredictable atomic moves.\n\n \\subsection{Local Structure, Local Dynamics}\n \\label{visc.ssect-loc}\n\n If there is only one atomic species in a condensed material, the local clusters are likely to have the shape of a regular icosahedron. Since regular icosahedra cannot tile the $3D$-space, there is either a geometrical frustration or the local cluster deform in the solid phase to lead to a crystal. The most likely symmetry of such a crystal is {\\em fcc}. To get a glass,  frustration is necessary. In the case of metallic glasses, frustration is obtained by mixing several species with very different atomic radius in order to prevent the systems to crystallize in a periodic order. However, Egami insisted for a long time that this geometrical description of the atomic distribution, while useful, is insufficient at describing the material. He insisted that an atom is not a hard sphere, but exhibits a bit of elasticity. Consequently, an atom can be seen as having a local stress tensor, in a way similar with particle having spins. This representation is powerful enough to describe the liquid phase in the high temperature regime \\cite{ES82}. In this view, a liquid appears as a free gas of objects represented by non interacting atomic stress tensors. Numerical simulations \\cite{ES82} suggest that this atomic stress tensor is a random Gaussian variable with zero average and with a covariance matrix given by a Maxwell-Bolztman factor using the expression of the elastic energy. It is even quantitatively better to use an effective elastic energy by taking into account the long distance effect of stress propagation using the mean field theory developed by Eshelby \\cite{Esh57}, in analogy with the Clausius-Mossoty theory of effective electric permittivity in dielectric. The equilibrium state of such a free gas can then be explicitly computed using Statistical Mechanics leading to a law of Dulong and Petit for the heat capacity, an observation made for a long time, which found an explanation with this theory.\n\n \\subsection{A Toy Model}\n \\label{visc.ssect-toy}\n\n What happens at lower temperature~? The atomic vibrations, leading to acoustic waves, which will be called abusively {\\em phonons} here, whereas quantized or classical, are still existing in liquids as in solids. However the acoustic waves with short wave length, namely comparable to the inter-atomic distance, are strongly damped in the liquid phase at high temperature. Whenever the typical time scale characterizing the jump dynamics is comparable to the phonon period, the phonon has no time to oscillate and it is damped. The present model investigates this damping more quantitatively. To proceed, the same intuition will be followed as in the original paper of Drude on electronic transport in metals \\cite{Dr1900}. In order to make the model computable, it will be assumed that it describes the mechanical fate of a given atom: (i) as long as no jump occurs, this atom is located near the minimum of a local potential well, and it will oscillate, harmonically, with a frequency given by the local curvature of the well, (ii) then at random times (Poissonian distribution) the atom jumps, finding itself, after the jump in a new potential well, with a new local curvature and a new initial  location relative to the new equilibrium position and a new initial velocity (phase-space position). Hence, the curvature of potential well (expressed in terms of the frequency of oscillation) will be considered as a random variable with a fixed average and a given covariance $\\sigma$ which will be shown to be the most important parameter. Similarly, the new relative initial phase-space position, after the jump, will also be chosen randomly according to a Maxwell-Boltzmann distribution. To make the model even more computable, it will be assumed that only one degree of freedom of oscillation is allowed. It will be seen that, even though this model is so simplistic, it exhibits a transition between two time-scales as $\\sigma$ decreases. The computation of correlation functions, like the viscosity, shows that this change of scale may affect the viscosity in an essential way.\n\n \\subsection{Anankeons in Glass: the STZ-Theory}\n \\label{visc.ssect-stz}\n\n For a very long times, engineers investigating the mechanical properties of materials measured the strain as the material is submitted to an external stress. The strain-stress curves shows usually a line at small stress, corresponding to the elastic response. This response is usually reversible: increasing then decreasing the stress follows the same curve. If the stress applied becomes large most materials just break. But some materials, like bulk metallic glasses, exhibit a plastic region in which the sample can be strained irreversibly, without much of additional stress. In a series of seminal papers \\cite{Fa98,FL98,LaPe03,Lan04,Lan06}, Langer and his collaborators developed a theory of what they called {\\em shear transformation zone} (STZ), to account for the plasticity region in bulk metallic glasses. The idea is that in the plastic region, the local clusters exhibit droplets in which the solid really behaves like a liquid, called an STZ. Considering the size of such droplets as a random variable, acting as a local lubricant, Langer {\\em et al.} proposed a system of effective equation at the macroscopic scale that are modeling the behavior of the material under stress with an amazing accuracy \\cite{RB12}. At the atomic scale though, numerical simulation show that these STZ's are the result of a cascade or avalanche of atomic swaps, namely of anankeons, \\cite{DA05}, starting at a random site, called the germ. This initial anankeon induces other in its neighborhood, because of the disruption created by the propagation of the stress induced by such swaps. The cascade looks like a local catastrophic event, occurring during a very short time, leading to the building of the local liquid-like droplets. The time scale representing the time necessary to create an STZ is however long in comparison with the typical time scale of the anankeon-swap. A model for such a cascade of event was proposed in \\cite{SDM01,FIE15}.\n\n\\section{Liquids at Large Temperature as a Perfect Anankeon Gas}\n\\label{visc.sect-perfect}\n\n The numerical simulation produced by Egami and Srolovitz \\cite{ES82} can be interpreted as a description of the liquid phase, far from the liquid-solid transition, in terms of a free gas of anankeon. The degree of freedom associated with each anankeon will be given in terms of an atomic stress tensor. Their Gaussian distribution observed in the numerical results can be interpreted in terms of a Gibbs state describing the thermal equilibrium in a statistical mechanical approach. This Section is dedicated to describe this in detail.\n\n \\subsection{The Delone Hypothesis and the Ergodic Paradox}\n \\label{visc.ssect-DelHyp}\n\n An instantaneous snapshot of the atomic arrangement can be described as a set of points ${\\mathcal L}$, representing the position of atomic nuclei. With a very good approximation ${\\mathcal L}$ is efficiently described as a {\\em Delone set}. Namely it is characterized by two length scales $0<r\\leq R<\\infty$ defined as follows: (i) any open Euclidean ball of radius $r$ contains at most one point of ${\\mathcal L}$, (ii) any closed Euclidean ball of radius $R$ contains at least one point of ${\\mathcal L}$. The first condition is equivalent to having a minimum distance $2r$ between any pair of distinct atoms. The second condition is equivalent to forbidding holes a diameter larger than $2R$. \n\n\\vspace{.1cm}\n\n However, the principles of Equilibrium Statistical Mechanics and the ergodicity with respect to translations contradict such an hypothesis.  This is the content of the so-called {\\em Ergodicity Paradox} \\cite{Bel15}. This is because in the infinite volume limit, the ergodic theorem implies that with probability one, given any $\\epsilon >0$, somewhere in the atomic configuration two atoms are at a distance shorter than $\\epsilon$. Similarly, with probability one, given any $\\epsilon >0$, somewhere in the atomic configuration there is a hole of size larger than $1\/\\epsilon$. However, such local arrangement are rare, so that dynamically their lifetime is so short that they are unobservable. Hence restricting the atomic configurations to make up a Delone set is a reasonable assumption called the {\\em Delone Hypothesis}\n\n \\subsection{Voronoi Tiling, Delone Graphs and Local Topology}\n \\label{visc.ssect-VDLT}\n\n Given a Delone set ${\\mathcal L}$ and a point $x\\in{\\mathcal L}$, the Voronoi cell $V(x)$ is defined as the set of points in space closer to $x$ than to any other point in ${\\mathcal L}$. $V(x)$ can be shown to be the interior of a convex polyhedron containing the open ball centered at $x$ of radius $r$ and contained in the ball centered at $x$ of radius $R$ \\cite{AK00,AKL13}. The closure $T(x)$ of the Voronoi cell is called a Voronoi tile. Two such tiles can only intersect along a common face. In particular two Voronoi cells centered at two distinct points of ${\\mathcal L}$ do not intersect. The entire space is covered by such tiles, so that the family of Voronoi tiles constitute a tiling, the {\\em Voronoi tiling}. The vertices of the Voronoi tiles will be called {\\em Voronoi points}. Their set ${\\mathcal L}^\\ast$ is usual called the {\\em dual lattice}. For a generic atomic configuration, a Voronoi point has not more than $d+1$ atomic neighbors in space-dimension $d$ ($3$ in the plane and $4$ in the space). In particular, the Voronoi points are the center of a ball with $d+1$ atoms on its boundary, a property called by Delone (who signed Delaunay) the {\\em ``empty sphere property''} \\cite{Del34}. It means that it is at the intersection of $d+1$ Voronoi tiles. The $d+1$ atoms located at the center of those tiles are the vertices of a simplex (triangle for $d=2$, tetrahedron for $d=3$) that generates the so-called {\\em Delaunay  triangulation}.\n\n\\vspace{.1cm}\n\n Two points $x,y\\in{\\mathcal L}$ will be called nearest neighbors whenever their respective Voronoi tiles are intersecting along a facet, namely a face of codimension 1. The pair $\\{x,y\\}$ will be called an {\\em edge}. In practice, though, using physical criteria instead of a geometrical one, an edge might be replaced by the concept of {\\em bond}, to account for the fact that the atoms associates with $x$ and $y$ are strongly bound \\cite{Be59,Be60,Be64}. The term ``strongly'' is defined through  a convention about the binding energy, so as to neglect bonds between atoms weakly bound to each other. Whatever the definition of an edge, geometrical or physical (bonds), this gives the Delone graph ${\\mathscr G}=({\\mathcal L}, {\\mathscr E})$, where ${\\mathcal L}$ plays the role of the vertices and ${\\mathscr E}$ denotes the set of edges or of bonds. At this point the Delone graph needs not being oriented. \n\n\\vspace{.1cm}\n\n Along a graph, a path is an ordered finite set of edges, each sharing a vertex with its successor or with its predecessor. The number of such edges is the length of the path. The graph-distance between two vertices is the length of the shortest paths joining them. A {\\em graph-ball} centered at $x$ of radius $n$ will be the set of all vertices at graph distance at most $n$ from $x$. Such graph balls correspond to local clusters. Two graph balls are called {\\em isomorphic} if there is a one-to-one surjective map between their set of vertices, that preserve the graph distance. As a result, if the atomic positions is slightly changed locally in space, the graph balls might still be isomorphic. Therefore graph ball, modulo isomorphism are encoding what physicists called the {\\em local topology} of the atomic configuration. \n\n\\vspace{.1cm}\n\n The transition between two graph balls can be described precisely through the concept of {\\em Pachner moves} \\cite{Pa91,AK00,AKL13}. This concept was created by experts of computational geometry, to describe the deformation of manifolds via a computer. A Pachner moves can only occur if at least one Voronoi point becomes degenerate, namely it admits at least $d+2$ atomic neighbor at some point during the move. Such moves correspond to a local change in a graph ball, namely an edge disappear another may reappear. A Pachner move corresponds exactly to the concept of anankeon. It can be described through using the Delaunay triangulation as explained in Fig.~\\ref{visc.fig-pachner}.\n\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{pachner2dD.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{pachner3d4V.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{figure}[ht]\n   \\centering\n\\caption{\\label{visc.fig-pachner} Left: a Pachner move in the plane. \nRight: a Pachner move in the $3$-space}\n\\end{figure}\n\n In $3D$-space such a move involves at least five atoms moving quickly to deform the Delaunay triangulation, changing the nature of the Delone graph, namely the local topology. This number five has been seen in numerical simulations \\cite{Yue14} as can be seen in Fig.~\\ref{visc.fig-yue}.\n\n\\begin{figure}[ht]\n   \\centering\n\\includegraphics[width=9cm]{STZstat.jpg}\n\\caption{\\label{visc.fig-yue} Number of atoms involved in a Pachner move \\cite{Yue14}}\n\\end{figure}\n\n \\subsection{Atomic Stress Tensor}\n \\label{visc.ssect-atomstress}\n\n The definition of stress tensor in solids \\cite{BH54,LPKL}, using the hypothesis of Continuum Mechanics, gives an intuition upon how to build a stress tensor at the atomic level \\cite{EMV80}. The present presentation will use a different route, but leads to a similar result. It will be assumed that interatomic forces are described by a two-body isotropic potential $W(r)$, where $r$ denotes the distance between the two atoms. In such a case, given a pair $e=(x,y)$ of atoms bound by an edge, let $n_e$ denotes the unit vector along the line generated by $x,y$ oriented from $x$ to $y$. By isotropy, the force acting upon the atom located at $x$ coming from $y$ is $F_y(x)= W'(r) n_e$. The total force acting on $x$ is therefore given by the sum $F(x)=\\sum_{y} n_e W'(r)$, where the sum includes the nearest neighbors of the atom at $x$. In Continuum Mechanics, the stress tensor is defined so as to satisfy  $F=\\int_{\\partial \\Lambda}\\sigma(s)\\cdot n(s)ds$ for any volume $\\Lambda$ with boundary $\\partial \\Lambda$. At the atomic level though, the integral will be replaced by the previous sum. The role of the volume $\\Lambda$ is played by the Voronoi tile $T(x)$. Each nearest neighbor of $x$ corresponds to a unique facet of $T(x)$, and then the unit vector $n_e$ is nothing but the normalized unit vector perpendicular to the facet oriented outwardly {\\em w.r.t.} the tile. This suggests the following formula for the stress tensor: each bond $e$ gives it a value $\\sigma_e$ so that\n\n$$\\sigma_e=|n_e\\rangle\\langle n_e| \\;\\frac{W'(r_e)}{A_e}\\,,\n$$\n\n where $r_e$ is the Euclidean length of the bond and $A_e$ is the Euclidean area of the facet associated with $e$. It ought to be remarked that the ratio $\\tr(\\sigma_e)=p_e= W'(r)\/A_e$ represents the pressure felt by the atom coming from its neighbor in the direction of the edge $e$. The deviatory stress is the traceless part $\\sigma_e-p_e\/d$ where $d$ is the dimension of the space in which this material lies. The atomic stress at $x$ is just the sum of these contributions, namely for an atom located at the vertex $x\\in{\\mathcal L}$, \n\n\\begin{equation}\n\\label{visc.eq-atomstress}\n\\sigma(x)= \\sum_{\\partial_0e=x} \n |n_e\\rangle\\langle n_e| \\;\\frac{W'(r_e)}{A_e}\\,.\n\\end{equation}\n\n In this expression, $\\partial_0e=x$ means the edges with origin at $x$. The formula above is very similar to the one obtained in \\cite{EMV80}, using an argument from Continuum Mechanics, apart from a possible numerical constant close to $1$.\n\n\\vspace{.1cm}\n\n The numerical simulation in \\cite{ES82} suggests that in the liquid phase, far from the liquid-solid transition, the statistical distribution of the atomic stress tensor follows a Maxwell-Boltzmann-Gibbs law of the form\n\n\\begin{equation}\n\\label{visc.eq-ES}\n\\Pro\\{\\sigma\\in {\\widehat{\\Lambda}}\\}=\\frac{1}{Z_1}\n   \\int_{{\\widehat{\\Lambda}}} e^{-\\beta(p^2\/2B+\\tau^2\/2G)}\\; d\\sigma\\,.\n\\hspace{2cm}\n  \\mbox{\\rm\\bf (Egami-Srolovitz Principle)}\n\\end{equation}\n\n In this expression, $\\beta=1\/\\kB T$, ${\\widehat{\\Lambda}}$ represents a volume in the space of stress tensors, $Z_1$ is a normalization factor, $p$ is the pressure given by the normalized trace of the stress tensor, and $\\tau$ is called the {\\em von Mises stress}. The latter is defined such that $\\tau^2$ is the trace of the square of the deviatory stress (traceless part of $\\sigma$). The coefficients $B,G$ are respectively the {\\em bulk modulus} and the {\\em shear modulus}. The expression in the exponential represents the elastic energy associated with the atomic stress.\n\n\\begin{figure}[ht]\n   \\centering\n\\includegraphics[width=11cm]{heatCap.jpg}\n\\caption{\\label{visc.fig-heatCap} Specific heat as a function of temperature for an alloy made of Gold, Germanium and Silicon signaling a glass-liquid transition \\cite{CT68}}\n\\end{figure}\n\n\\vspace{.1cm}\n\n \\subsection{Equilibrium for Liquid at High Temperature}\n \\label{visc.ssect-liqeq}\n\n Under the Egami-Srolovitz principle, if the atomic stress are uncorrelated, it becomes possible to compute the partition function $Z(T,N)$ of a volume of liquid containing $N$ atoms. Namely $Z(T,N)=Z_1(T)^N$, where\n\n$$Z_1(T)= \\int e^{-\\beta(p^2\/2B+\\tau^2\/2G)}\\; d\\sigma\\,,\n   \\hspace{2cm}\n    \\beta=\\frac{1}{\\kB T}\\,.\n$$\n\n The domain of integration is the space of all real symmetric $d\\times d$ matrices, representing a possible stress tensor. This Gaussian integral is easy to compute giving $Z_1(T)= {\\mathscr Z}\\, T^{d(d+1)\/2}$ where ${\\mathscr Z}$ is a constant depending on $B,G$, but independent of the temperature. The Clausius entropy is given by $S=\\kB \\ln(Z(N,T))$ leading to the following expression of the heat capacity $C_v= c_d N\\kB$ where $c_d=d(d+1)\/2$, which is $6$ in $3D$. In particular, the system follows a {\\em Law of Dulong-Petit} like the contribution of phonons in crystal. However, the physical origin is different in liquid as the main degree of freedom associated with atoms are the atomic stress. Hence a liquid can be seen as a {\\em perfect gas of anankeons}. This prediction gave a satisfactory answer to the experimental observation \\cite{CT68} of the saturation of the heat capacity at high temperature (see Fig.~\\ref{visc.fig-heatCap}).\n\n\n\\section{Phonon-Anankeon Interaction}\n\\label{visc.sect-phan}\n\n What happens at lower temperature, near the solid-liquid transition~? The following model will show that as the temperature decreases, the vibration modes are constraining the shape of local clusters, so as to change the viscosity in a non trivial way.\n\n \\subsection{Time Scales}\n \\label{visc.ssect-tsc}\n\n Since the dynamics plays such an important role in explaining the difference between solids and liquids, the various relevant time scales ought to be discussed. The most elementary is $\\tau_{\\mbox{\\em \\tiny LC}}$, where $LC$ stands for {\\em local configuration}, representing the time required between two consecutive Pachner moves. It corresponds to  the anankeon relaxation time. The next time scale is called {\\em Maxwell's relation time} $\\tau_{\\mbox{\\em \\tiny M}}$ \\cite{HMcD06} is defined by the ratio\n\n\\begin{equation}\n\\label{visc.eq-maxwell}\n\\tau_{\\mbox{\\em \\tiny M}}=\\frac{\\eta}{G_\\infty}\\,,\n\\end{equation}\n\n where $\\eta$ is the viscosity and $G_\\infty$ is the high frequency shear modulus. The Maxwell relaxation time represents the time scale below which the system behaves like a solid and beyond which it can be considered as a liquid. At high temperature $\\tau_M=\\tlc$. Besides these two time scales, there is the $\\alpha$-relaxation time $\\tau_\\alpha$ \\cite{FKST}, defined by the scattering function at the peak of the structure function $S(Q)$. Finally there is the {\\em bond lifetime} $\\tau_B$ \\cite{YO98}. The value of these time scales has been discussed thoroughly in \\cite{IE12,INE13} (see Fig.~\\ref{visc.fig-relTime}). This was done through classical as well as {\\em ab initio} molecular dynamics simulation on various metallic liquids like liquid iron, $Cu_{56}Zr_{44}$ or  $Zr_{50}Cu_{40}Al_{10}$. \n\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{relaxTimes.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{minipage}[b]{7.5cm}\n\\begin{center}\n\\includegraphics[width=7cm]{timeRatio.jpg}\n\\end{center}\n\\end{minipage}\n\\begin{figure}[ht]\n   \\centering\n\\caption{\\label{visc.fig-relTime} Various relaxation times in the liquid phase as function of the temperature. Left: $T_g$ is the temperature at the liquid-solid transition. The Arrhenius law is visible on this graph: the slope of the lines give an estimate of $W$. Right: $T_A$ is the crossover temperature below which the Maxwell time differs from the $\\tau_{\\mbox{\\em \\tiny LC}}$. (\\cite{INE13})}\n\\end{figure}\n\n All these time scales are proportional in the liquid phase, far enough from the liquid-solid transition as shown in Fig.~\\ref{visc.fig-relTime} and are given by an Arrhenius law $\\tau \\sim e^{W\/\\kB T}$.\n\n However, near the liquid-glass transition point, there is a significant discrepancy between the Maxwell time and the local configuration time as can be seen on the right of  Fig.~\\ref{visc.fig-relTime}. This is precisely the discrepancy that the present model is going to describe. It seems natural, then, to choose the following basic time scale for the purpose of the modeling:\n\n\\begin{itemize}\n  \\item\\label{visc.Time-scale} {\\bf Time-scale assumption: } In the toy model $\\tau=\\tau_{\\mbox{\\em \\tiny LC}}$.\n\\end{itemize}\n\n \\subsection{Phonon Dynamic: the main approximations}\n \\label{visc.ssect-phdynapp}\n\n As explained in Section~\\ref{visc.ssect-toy}, the model will describe the situation of one atom in the liquid. Most of the time, this atom is located near the bottom of a potential well, created by the interaction with other atoms around. Since the potential is smooth near its minimum, there is no loss of generality in assuming that it is well approximated by a local paraboloid, at least if the atom does not move too far from the minimum. Hence the atomic motion is well approximated by an harmonic oscillator. To simplify further, it will be assumed that {\\em only one of the three dimension really matter}. This looks drastic, but the method of solving the model is actually the same if all the dimensions are taken into account. Hence, the curvature of the well near the minimum defines the oscillator pulsation $\\omega=2\\pi\\nu$ where $\\nu$ is the frequency. Let then $q=q(t)$ denotes the $1D$ distance of the atom from the potential minimum at any given time $t$. It is convenient to introduce the variable $u=\\omega q$, which has the same dimension as the velocity $v=\\stackrel{\\cdot}{q}$. Hence the $2D$-vector $X$ with coordinate $u,v$ represents the phase-space position of the atom at any given time. The usual equation of motion is given by \n\n$$m\\frac{d^2q}{dt^2}+ kq^2=0\\,,\n   \\hspace{2cm}\n    \\omega= \\sqrt{\\frac{k}{m}}\\,,\n$$\n\n  where $m$ is the mass of the atom and $k$ is the spring constant, expressible in terms of the second derivative of the potential energy. This equation can equivalently be written as\n\n\\begin{equation}\n\\label{visc.eq-harmosc}\n\\frac{dX}{dt}=\\omega J X\\,,\n  \\hspace{2cm}\n   J=\\left[\n\\begin{array}{cc}\n~0 & 1\\\\\n-1 & 0\n\\end{array}\n\\right]\\,.\n\\end{equation}\n\n The solution of this equation, with initial position $u(0)$ and initial velocity $v(0)$, is given by\n\n\\begin{equation}\n\\label{visc.eq-solharm}\nX(t)=e^{\\omega t J} X(0)= \n\\left[\n\\begin{array}{cc}\n\\cos{\\omega t} & \\sin{\\omega t}\\\\\n-\\sin{\\omega t} & \\cos{\\omega t}\n\\end{array}\n\\right]\\;\\left[\n\\begin{array}{c}\nu(0)\\\\\nv(0)\n\\end{array}\n\\right]\\,.\n\\end{equation}\n\n The mechanical energy of this oscillator is given by \n\n$$E= \\frac{m\\stackrel{\\cdot}{q}^2}{2}+\\frac{k q^2}{2}=\n   \\frac{m|X|^2}{2}\\,.\n$$\n\n It ought to be stressed that the spring constant $k$ depends in an essential way upon the local environment seen by the test-atom. In particular, if the local configuration changes due to a Pachner move, or an anankeon, this constant will change as well. This will be described mathematically in the next paragraph.\n\n \\subsection{The toy model for anankeon-phonon interaction}\n \\label{visc.ssect-anandyn}\n\n When an anankeon strikes, the atom is ejected from its position to jump on a position nearby, in a new harmonic potential. To describe such a situation, following the strategy described in Section~\\ref{visc.ssect-toy}, it will be assumed that the anankeons strikes at {\\em random times} $\\{ \\cdots <\\tau_n <\\tau_{n+1}<\\cdots\\}$. Here the integer index $n$ varies from $-\\infty$ to $+\\infty$. To be more precise, the randomness of those times will be assumed to be {\\em Poissonian}, namely the random variables $\\tau_{n+1}-\\tau_n$ are {\\em i.i.d.}, with average\n\n$${\\mathbb E}\\{\\tau_{n+1}-\\tau_n\\}\\stackrel{def}{=}\n   \\langle \\tau_{n+1}-\\tau_n\\rangle= \\tau =\\tlc\\,,\n$$\n\n and an exponential distribution\n\n$$\\Pro\\{(\\tau_{n+1}-\\tau_n)\\in A\\subset {\\mathbb R}\\}= \n   \\int_A e^{-t\/\\tau} \\; \\frac{dt}{\\tau}\\,.\n$$\n\n The phase-space position $X(t)$ after the jump will also be relative to the minimum of the new potential well. In particular, there is a need to update this position right after the jump in terms of the position right before, namely\n\n$$X(\\tau_n+0)=X(\\tau_n-0)+\\xi_n\\,.\n$$\n\n The vector $\\xi_n$ represents the result of the anankeon kick \non the initial phase-space position, relative to the potential minimum. It is reasonable to assume that $\\xi_n$ is also a random variable and that the family $(\\xi_n)_{n\\in{\\mathbb Z}}$ is made of {\\em i.i.d.}'s. In addition it will be assumed that the common distribution is provided by the Gibbs state at the temperature $T$. Hence\n\n\\begin{equation}\n\\label{visc.eq-gibbs}\n\\Pro\\{\\xi_n \\in B\\subset {\\mathbb R}^2\\}= \n   \\frac{\\kB T}{2\\pi m}\n    \\int_B e^{-m|\\xi|^2\/\\kB T} \\; d^2\\xi\\,.\n\\end{equation}\n\n At last, the pulsation $\\omega$ will depend upon the potential well in which the atom fell. Hence between the times $\\tau_n$ and $\\tau_{n+1}$ it will be given by $\\omega_n$. Since the atom this model is describing should be typical, the $\\omega_n$'s should be considered as random variables, and again, the family $(\\omega_n)_{n\\in{\\mathbb Z}}$ will be made of {\\em i.i.d.}'s. At this point, it is unwise to fix the common distribution. It enough to fix the average $\\omega$ and the variance $\\sigma$\n\n\\begin{equation}\n\\label{visc.eq-omega}\n\\omega = {\\mathbb E}\\{\\omega_n\\}\\,,\n \\hspace{2cm}\n  \\sigma=\\Var(\\omega_n)= {\\mathbb E}\\{(\\omega_n-\\omega)^2\\}^{1\/2}\\,.\n\\end{equation}\n\n At last, during the time interval $(\\tau_n,\\tau_{n+1})$, the evolution of $X(t)$ will satisfy eq.~(\\ref{visc.eq-harmosc}) and eq.~(\\ref{visc.eq-solharm}).\n\n \n  \\subsection{Viscosity and correlation function}\n  \\label{visc.ssect-visc}\n\n The viscosity is given by the fluctuation-dissipation theorem as\n\n\\begin{equation}\n\\label{visc.eq-visc}\n\\eta = \\frac{V}{\\kB T}\n   \\int_0^\\infty C_f(t)\\, dt\\,,\n\\end{equation}\n\n where $V$ is the volume of the fluid, $T$ the temperature, and $C_f$ is a correlation functions defined by\n\n\\begin{equation}\n\\label{visc.eq-corr2}\nC_f(t) =\n   {\\mathbb E}\\left\\{ f(X(t))\\;\\overline{f(X(0)} \\right\\}\\,,\n\\end{equation}\n\n where $f$ is a function on the  phase space giving the expression of the stress tensor. For the purpose of this calculation, it is sufficient to consider $f$ as a scalar valued function, because it does not change the conclusion of the computation. The goal will be to show that, as $t\\to\\infty$, the correlation $C_f(t)\\sim e^{-t\/\\tau_M}$ defining the Maxwell time $\\tau_M$. And indeed the time integral in eq.~(\\ref{visc.eq-visc}) will be proportional to $\\tau_M$ as anticipated by eq.~(\\ref{visc.eq-maxwell}).\n\n\\vspace{.1cm}\n\n This correlation function can also be computed using the {\\em dissipative evolution} defined by the operator $P_t$ as the following conditional expectation\n\n\\begin{equation}\n\\label{visc.eq-sg}\nP_tf(x) = {\\mathbb E}\\left\\{ f(X(t))|X(0)=x\\right\\}\\,.\n\\end{equation}\n\n Then, the correlation function is given by the phase-space integral\n\n\\begin{equation}\n\\label{visc.eq-corrHilb}\nC_f(t) =\\int_{{\\mathbb R}^2} \\overline{f(x)}\\,P_tf(x)\\;d^2x\\,.\n\\end{equation}\n\n In order to compute $\\tau_M$ precisely, it is convenient to compute the Laplace transform of $C_f$ instead, namely, given a {\\em complex variable} $\\zeta$\n\n\\begin{equation}\n\\label{visc.eq-lapcorr}\n{\\mathscr L} C_f(\\zeta)=\n   \\int_0^\\infty e^{-t\\zeta}\\, C_f(t)\\, dt\\,.\n\\end{equation}\n\n It turns out that the asymptotic behavior of the correlation at $t\\to \\infty$ can be interpreted as the position of the nearest singularity of ${\\mathscr L} C_f$ in the complex plane. Namely the Laplace transform is analytic in the half-plane $\\Re\\zeta >-1\/\\tau_M$. Hence, in order to compute the correlation function, it is sufficient to compute the value $f(X(t))$ taken by the function $f$ along the random phase-space trajectory of the atom under scrutiny.\n\n\n\n\\section{Computing the Laplace transform of a correlation function}\n\\label{visc.sect-corrcomp}\n\n\\noindent The formula~(\\ref{visc.eq-corrHilb}) suggests to compute the Laplace transform of the dissipative evolution operator $P_t$ instead, acting on some function space describing the relevant functions on the phase-space. In order to do so, it will be more efficient to compute the actions of the various part of the dynamic on functions rather than on the phase-space trajectory. As will be seen this calculus is very similar to the operator calculus used in Quantum Mechanics.\n\n \\subsection{Operator Calculus}\n \\label{visc.ssect-opcal}\n\n\\noindent The first part of the evolution is provided by the phase-space mouvement of the harmonic oscillator, as described in Section~\\ref{visc.ssect-phdynapp}. Namely\n\n\\begin{equation}\n\\label{visc.eq-angmom}\nf\\left(e^{\\omega t J}x\\right)=\n   \\left(e^{-\\omega t{\\mathbb J}}f\\right)(x)\\,,\n    \\hspace{2cm}\n     -{\\mathbb J}=v\\partial_u-u\\partial_v\\,,\n\\end{equation}\n\n\\noindent where $x=(u,v)$ is the phase-space position and ${\\mathbb J}$ is very similar to the {\\em phase-space angular momentum} but for the missing $\\imath$ in front (namely ${\\mathbb J}$ is anti-Hermitian instead) and its $2D$ character in phase-space.\n\n\\vspace{.1cm}\n\n\\noindent Similarly, the impact of the anankeon on the atom is to translate the phase-space position of the atom after a kick, by a vector $\\xi=(a,b)$, leading to the {\\em translation operator}\n\n\\begin{equation}\n\\label{visc.eq-trans}\nf(x+\\xi)= \\left(e^{\\xi\\cdot \\nabla}f\\right)(x)\\,,\n    \\hspace{2cm}\n     \\xi\\cdot \\nabla= a\\partial_u+b\\partial_v\\,.\n\\end{equation}\n\n\\noindent Hence $\\nabla$ plays the role of a phase-space momentum operator apart for the missing $\\imath$ in front (namely $\\nabla$ is anti-Hermitian instead).\n\n\\vspace{.1cm}\n\n\\noindent Let $X(t)$ denote the phase-space trajectory of the atom at time $t$ with initial condition $X(0)=x$. From the description of the model, an harmonic motion takes place between time $\\tau_0=0$ to $\\tau_1$ at pulsation $\\omega_1$, then a phase-space translation by $\\xi_1$, then another harmonic motion between time $\\tau_1$ to $\\tau_2$ at pulsation $\\omega_2$, then a new phase-space translation by $\\xi_2$, and so on, until the time $t$ is reached. It becomes possible to describe it from the functional point of view as follows\n\n\\begin{equation}\n\\label{visc.eq-evol}\nP_tf(x)= {\\mathbb E}\\left(f(X(t))|X(0)=x\\right) =\n{\\mathbb E}\\left\\{\n \\prod_{j=1}^n\n  \\left(e^{-(\\tau_j-\\tau_{j-1})\\omega_{j-1}{\\mathbb J}}\\, \n        e^{\\xi_j\\cdot\\nabla}\n  \\right)\n    e^{-(t-\\tau_n)\\omega_n{\\mathbb J}}\\,f\n        \\right\\}(x)\\,,\n\\end{equation}\n\n\\noindent where the product is ordered from left to right, with $j=1$ on the left and $j=n$ on the right, and where $n$ is the integer such that $\\tau_n\\leq t<\\tau_{n+1}$. From the assumptions made on the model, the $(\\tau_j-\\tau_{j-1})$ are {\\em i.i.d.}, as well as the $\\omega_j$'s and the $\\xi_j$'s. It follows that $n$ is also a random variable. Computing the distribution of $n$ though, is hard enough to try to avoid it. The Laplace transform will allow to avoid such a calculation.\n\n \\subsection{Laplace transformation}\n \\label{visc.ssect-laping}\n\n\\noindent After averaging and taking the Laplace transform the result is\n\n$${\\mathscr L} P_\\zeta f(x)=\n   \\int_0^\\infty e^{-t\\zeta}\\; P_tf(x)\\;dt=\n    {\\mathbb E}\\left\\{\\int_0^\\infty e^{-t\\zeta}\\;f(X(t)\\;dt \\Bigg| X(0)=x\\right\\}\\,,\n     \\hspace{1cm}\n      \\zeta\\in {\\mathbb C}\\,.\n$$\n\n\\noindent Thanks to the linearity of the averaging process, this integral over time can be decomposed, under the averaging, into the time intervals $[\\tau_n,\\tau_{n+1}]$, to give\n\n$${\\mathscr L} P_\\zeta f(x)=\\sum_{n=0}^\\infty\n    {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n     e^{-t\\zeta}\\;f(X(t)\\;dt\\Bigg| X(0)=x\\right\\}\\,.\n$$\n\n\\noindent  Using eq.~(\\ref{visc.eq-evol}), this gives\n\n\\begin{equation}\n\\label{visc.eq-ev}\n{\\mathscr L} P_\\zeta f(x)= \\sum_{n=0}^\\infty\n    {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n     e^{-t\\zeta}\\; \\prod_{j=1}^n\n      \\left(e^{-(\\tau_j-\\tau_{j-1})\\omega_{j-1}{\\mathbb J}}\\, \n        e^{\\xi_j\\cdot\\nabla}\n      \\right)\n       e^{-(t-\\tau_n)\\omega_n{\\mathbb J}}\\,f(x)\\;dt\n        \\right\\}\\,.\n\\end{equation}\n\n\\noindent Since $\\tau_0=0$, a telescopic sum give $t=(t-\\tau_n)+(\\tau_n-\\tau_{n-1})+\\cdots+ (\\tau_1-\\tau_0)$. This leads to distribute the first term $e^{-t\\zeta}$ inside each terms of the product leading to replace each $\\omega_{j-1}{\\mathbb J}$ by $\\omega_{j-1}{\\mathbb J} + \\zeta$, namely\n\n\\begin{equation}\n\\label{visc.eq-ev2}\n{\\mathscr L} P_\\zeta f(x)= \\sum_{n=0}^\\infty\n    {\\mathbb E}\\left\\{\\int_{\\tau_n}^{\\tau_{n+1}} \n     \\prod_{j=1}^n\n      \\left(e^{-(\\tau_j-\\tau_{j-1})(\\omega_{j-1}{\\mathbb J}+\\zeta)}\\, \n        e^{\\xi_j\\cdot\\nabla}\n      \\right)\n       e^{-(t-\\tau_n)(\\omega_n{\\mathbb J}+\\zeta)}\\,f(x)\\;dt\n        \\right\\}\\,.\n\\end{equation}\n\n\\noindent The first operation consists in evaluating the integral over $t$. The change of variable $s=t-\\tau_n$ gives the following operator valued integral\n\n\\begin{equation}\n\\label{visc.eq-sint}\n\\int_0^{\\tau_{n+1}-\\tau_n}\n    e^{-s(\\omega_n{\\mathbb J}+\\zeta)}\\;ds= \n     \\frac{1-e^{-(\\tau_{n+1}-\\tau_n)(\\omega_n{\\mathbb J}+\\zeta)}}\n       {\\zeta+ \\omega_n{\\mathbb J}}\\,.\n\\end{equation}\n\n\\noindent The next step consists in recognizing that the averaging ${\\mathbb E}$ can be decomposed into averaging over the time Poisson process, before averaging over the random phase-space positions $\\xi_j$'s or the random pulsations $\\omega_{j-1}$'s. In addition, the stochastic independence of the $(\\tau_j-\\tau_{j-1})$'s permits to reduce the averaging to each factor \nin the product in eq.~(\\ref{visc.eq-ev2}). Using the the {\\em time-scale assumption} in Section~\\ref{visc.Time-scale} on page~\\pageref{visc.Time-scale}, this averaging reduces to the following integral, where $A$ is an operator\n\n\\begin{equation}\n\\label{visc.eq-pint}\n{\\mathbb E}_\\tau\\left\\{e^{-(\\tau_j-\\tau_{j-1})A}\\right\\}=\n   \\int_0^\\infty e^{-s\/\\tau-sA}\\;\\frac{ds}{\\tau} =\n    \\frac{1}{1+\\tau A}\\,,\n\\end{equation}\n\n\\noindent where ${\\mathbb E}_\\tau$ denotes the average over the Poissonian times. The stochastic independence of each factor in the product, leads to a factorization of the averaging, since, whenever two random variable $X,Y$ are stochastically independent the average of their product is given by the product of their average, that is ${\\mathbb E}(XY)={\\mathbb E}(X){\\mathbb E}(Y)$. After a little algebra, using eq.~(\\ref{visc.eq-sint}, \\ref{visc.eq-pint}), the eq.~(\\ref{visc.eq-ev2}) leads to\n\n\\begin{equation}\n\\label{visc.eq-ev3}\n{\\mathscr L} P_\\zeta f(x)= \\tau \\sum_{n=0}^\\infty\n    {\\mathbb E}\\left\\{ \n     \\prod_{j=1}^n\n      \\left(\n        \\frac{1}{1+\\tau(\\zeta+\\omega_{j-1}{\\mathbb J})}\\, \n        e^{\\xi_j\\cdot\\nabla}\n      \\right)\n       \\frac{1}{1+\\tau(\\zeta+\\omega_n{\\mathbb J})}\\,f(x)\n        \\right\\}\\,.\n\\end{equation}\n\n\\noindent Similarly, since the $\\xi_j$'s and the $\\omega_{j-1}$'s are stochastically independent, it is enough to compute the following averages\n\n\\begin{equation}\n\\label{visc.eq-xiaver}\n{\\mathbb E}_\\xi\\left\\{e^{\\xi_j\\cdot\\nabla}\\right\\}=\n  e^{\\kB T\\,\\Delta\/2m}\\,,\n   \\hspace{2cm}\n    \\Delta=\\nabla\\cdot\\nabla= \\partial_u^2+\\partial_v^2\\,,\n\\end{equation}\n\n\\noindent where ${\\mathbb E}_\\xi$ denotes the average over the phase-space position, using the Gaussian integral in eq.~(\\ref{visc.eq-gibbs}). Similarly, the average over the $\\omega_{j-1}$'s leads to the operator-valued function\n\n\\begin{equation}\n\\label{visc.eq-omaver}\n{\\mathbb A}(\\zeta)=\n {\\mathbb E}_\\omega\\left\\{\n  \\frac{1}{1+\\tau(\\zeta+\\omega_{j-1}{\\mathbb J})}\n            \\right\\}\\,.\n\\end{equation}\n\n\\noindent Evaluating this integral requires to know the distribution of the $\\omega_{j-1}$'s. At this point, no further assumption will be made yet. Using eq.~(\\ref{visc.eq-xiaver}, \\ref{visc.eq-omaver}), the eq.~(\\ref{visc.eq-ev3}) leads to\n\n\\begin{equation}\n\\label{visc.eq-ev4}\n{\\mathscr L} P_\\zeta f(x)= \\tau \\sum_{n=0}^\\infty\n    \\left\\{ \n     {\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}\n    \\right\\}^n\n      {\\mathbb A}(\\zeta)f(x)=\\tau\\,\n       \\frac{1}{1-{\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}}\\,{\\mathbb A}(\\zeta)f(x)\\,.\n\\end{equation}\n\n\\noindent The last series converges only if $\\|{\\mathbb A}(\\zeta)\\,e^{\\kB T\\,\\Delta\/2m}\\|<1$ where $\\|\\cdot\\|$ denotes the operator norm. At this point though, no accurate description of the space on which these operators act has been done, so that the norm condition is still meaningless. It will be the purpose of the discussion of the next Section. With this cautionary statement in mind, the problem is reduced to analyzing the properties of one operator at the end, giving the Laplace transform as a function of $\\zeta$.\n\n \\subsection{Digression: choosing the function space}\n \\label{visc.ssect-fspace}\n\n\\noindent In order to make sense of the calculation made in the previous Section~\\ref{visc.ssect-laping}, as was remarked at the end, the function space on which these operators act must be made more precise. In order to do so, the first question is to know which type of function $f$ are physically relevant. In the calculation of viscosity, the ideal function should be the atomic deviatory stress tensor as given in eq.~(\\ref{visc.eq-atomstress}). Using the finite sum decomposition over the edges linking the atom at $x$ with its neighbors, it boils down to find a space containing the functions of the form\n\n\\begin{equation}\n\\label{visc.eq-function}\ng(x) = \\frac{|y-x\\rangle\\langle y-x|}{|y-x|^2}\\, W'(|y-x|)\\,,\n\\end{equation}\n\n\\noindent where $y$ is a fixed point representing the position of a neighboring atom. First this expression is matrix valued instead of being a scalar. This matrix is $d\\times d$ if the liquid is lying in a $d$-dimensional space. It is real symmetric by construction. Moreover, {\\em only the traceless part} of this matrix matters for the viscosity. In $d$-dimension the space of $d\\times d$ real symmetric matrices is invariant by the rotation group $SO(d)$, and it decomposes into the direct sum of two invariant subspaces, one corresponding to the trace (trivial representation of $SO(d)$) and the traceless part (unit representation of $SO(d)$). For $d=3$ it is well known that the unit representation of $SO(3)$, acting on this space of traceless real symmetric matrices, corresponds to the spin $2$ representation of $SU(2)$.\n\n\\vspace{.1cm}\n\n\\noindent On the other hand, the effective potential energy $W$ decays fast enough at infinity. The decay is mostly exponential due to screening effect of negatively charges electrons over the Coulomb potential of the positively charged nucleus. Moreover, it derivative is likely to share the same property. Hence, the function $g$ decays fast enough as the position $x$ of the atom diverges from its equilibrium position.\n\n\\vspace{.1cm}\n\n\\noindent The previous considerations motivated by the physical problem, have to be translated into the langage of the phase-space for the $1D$ harmonic oscillator replacing the atomic motion. In the previous Sections, the letter $x$ has been confusingly used also to describe the phase-space position $x=(u,v)$, where $u$ represents the $1D$ position of the harmonic oscillator relative to the origin chosen at the minimum of the potential well, while $v$ describes the corresponding velocity. The translation between the language used in $d$ dimensions and the phase-space language will be done in the following way:\n\n\\vspace{.1cm}\n\n(i) The space ${\\mathcal H}$ of relevant functions will be chosen so that any of them decay fast enough at infinity;\n\n(ii) the space ${\\mathcal H}$ needs to be mathematically convenient, in particular, choosing it to be a Hilbert space is a very convenient choice;\n\n(iii) the symmetry properties of the function $f$ used in the calculation of the viscosity must reflect as much as possible the ones of the realistic stress tensor given in eq.~(\\ref{visc.eq-atomstress}).\n\n\\vspace{.1cm}\n\n\\noindent A reasonable choice is ${\\mathcal H}=L^2({\\mathbb R}^2)$, which denotes the space of complex valued square integrable measurable functions over the phase-space ${\\mathbb R}^2$, namely which satisfy \n\n$$\\|f\\|_{\\mathcal H}^2= \\int_{{\\mathbb R}^2} |f(x)|^2\\,d^2x <\\infty\\,.\n$$\n\n\\noindent The symmetry properties might be reflected by the presence of the operator ${\\mathbb J}$, generating the phase-space rotations, namely the group $U(1)$ of rotations in the plane. It is well known that ${\\mathbb J}$ can be defined as an anti-selfadjoint operator on ${\\mathcal H}$ with eigenvalues $\\imath \\ell$ where $\\ell \\in{\\mathbb Z}$ takes on any nonnegative of nonpositive integer values. Moreover the eigenvectors are given by functions of the form (using polar coordinates)\n\n\\begin{equation}\n\\label{visc.eq-evell}\ng_\\ell(x) =e^{\\imath \\ell \\theta} g(r)\\,,\n   \\hspace{2cm}\n    x=(r\\cos{\\ell\\theta},r\\sin{\\ell\\theta})\\in{\\mathbb R}^2\\,,\n\\end{equation}\n\n\\noindent where\n\n\\begin{equation}\n\\label{visc.eq-L2int}\n\\int_0^\\infty |g(r)|^2\\,rdr <\\infty\\,.\n\\end{equation}\n\n\\noindent In particular this eigenspace is infinite dimensional. Let $\\Pi_\\ell$ denote the orthogonal projection from ${\\mathcal H}$ onto this space. Moreover, these spaces are mutually orthogonal and then span the entire Hilbert space ${\\mathcal H}$. \n\n\\vspace{.1cm}\n\n\\noindent The analogy between the realistic situation and the phase-space modeling, suggests to choose the eigenspace of ${\\mathbb J}$ with angular momenta $\\ell=\\pm 2$. It will be seen though, that this restriction does not matter in terms of the qualitative features of the model.\n\n\\vspace{.1cm}\n\n\\noindent Similarly the operator $\\Delta$ becomes self-adjoint with nonpositive generalized eigenvalues. In particular, the operator $\\exp\\{\\kB T \\Delta\/2m\\}$ is selfadjoint with its spectrum given by the interval $[0,1]$, so that $0\\leq \\exp\\{\\kB T \\Delta\/2m\\}\\leq 1$ implying $\\|\\exp\\{\\kB T \\Delta\/2m\\}\\|\\leq 1$.\n\n\\vspace{.1cm}\n\n\\noindent The previous discussion shows that, while a choice has to be made for ${\\mathcal H}$, it will always be partly arbitrary. In particular, the discussion of symmetries might not be totally realistic. But the choice of ${\\mathcal H}$ is not innocent. The properties of the unbounded operators ${\\mathbb J}$ and $\\Delta$ depend crucially upon which choice is made. Hence, the main criterion is whether or not the model is effective at describing the physical properties of the system under study.\n\n \\subsection{Analyticity Domain}\n \\label{visc.ssect-holom}\n\n\\noindent Using the assumptions made in Section~\\ref{visc.ssect-fspace}, the operator $\\{1+\\tau(\\zeta+ \\omega_j {\\mathbb J})\\}^{-1}$ has a spectrum of isolated eigenvalues (each which infinite multiplicity) given by $\\{1+\\tau(\\zeta + \\imath\\omega_j\\ell)\\}^{-1}$ with $\\ell\\in{\\mathbb Z}$. In particular, it has poles at\n\n$$\\zeta_\\ell= -\\left(\\frac{1}{\\tau} + \\imath \\ell \\omega_j\\right)\\,.\n$$\n\n\\noindent As the real valued random variable $\\omega_j$ varies, though, the pole position describes a subset of the vertical line $\\Re{\\zeta}=-1\/\\tau$, the extend of which depends upon the support of the probability distribution of $\\omega_j$. Hence after taking the average over $\\omega_j$, this line becomes a cut in the $\\zeta$-complex plane. It is convenient to introduce the following functions\n\n\\begin{equation}\n\\label{visc.eq-aell}\na_\\ell(\\zeta) ={\\mathbb E}\n \\left\\{\n   \\frac{1}{1+\\tau(\\zeta +\\imath\\omega_j\\ell)}\n \\right\\}\\stackrel{\\Re(\\zeta)\\uparrow +\\infty}{\\simeq}\n  \\frac{1}{1+\\tau\\zeta}+ \n   O\\left(\\frac{1}{(1+\\tau\\zeta)^2}\\right)\\,,\n  \\hspace{1.5cm}\n   \\ell\\in{\\mathbb Z}\\,.\n\\end{equation}\n\n\\noindent In view of the definition of the Laplace transform (see eq.~(\\ref{visc.eq-lapcorr})), it implies that only the part $\\Re(\\zeta)>-1\/\\tau$ is relevant for this analysis. Hence\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 1: } {\\em ${\\mathbb A}(\\zeta)$ and the $a_\\ell(\\zeta)$'s are complex analytic in the $\\zeta$-complex plane in the domain $\\Re(\\zeta)>-1\/\\tau$.}\n\n\\vspace{.1cm}\n\n\\noindent Since ${\\mathbb J}$ is anti-selfadjoint, the operator ${\\mathbb A}(\\zeta)$ is {\\em normal}, namely it commutes with its adjoint. It can actually be expressed in terms of the eigenprojections $\\Pi_\\ell$ using its {\\em spectral decomposition}\n\n\\begin{equation}\n\\label{visc.eq-spdecomp}\n{\\mathbb A}(\\zeta)= \\sum_{\\ell\\in{\\mathbb Z}} a_\\ell(\\zeta)\\;\\Pi_\\ell\\,.\n\\end{equation}\n\n\\noindent Consequently its norm is given by the maximum of the modulus of its eigenvalues, namely, using $|{\\mathbb E}(X)|\\leq {\\mathbb E}(|X|)$, \n\n\\begin{equation}\n\\label{visc.eq-normA}\n\\|{\\mathbb A}(\\zeta)\\|=\n   \\sup_{\\ell\\in{\\mathbb Z}}\n    \\left|\n     {\\mathbb E}_\\omega\\left\\{\n      \\frac{1}{1+\\tau(\\Re{\\zeta}+\\imath (\\Im{\\zeta}+\\omega_j\\ell))}\n              \\right\\}\n    \\right|\\leq \n     \\frac{1}{|1+\\tau\\Re(\\zeta)|}\\,.\n\\end{equation}\n\n\\noindent Going back to eq.~(\\ref{visc.eq-ev4}), the Laplace transform of the stochastic evolution shows a singularity at values of $\\zeta$ for which the spectrum of ${\\mathbb A}(\\zeta)\\exp\\{\\kB T \\Delta\/2m\\}$ contains the number $1$. Since the product of two operators might not commute, the computation of the spectrum might be difficult. However, he Laplacian is always invariant by rotation in any dimension, so that\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 2: } {\\em the operators ${\\mathbb J}$ and $\\Delta$ commute.}\n\n\\vspace{.1cm}\n\n\\noindent More precisely, using the polar coordinates in $2D$ gives\n\n$$\\Delta= \\frac{1}{r}\\,\\frac{\\partial}{\\partial r}\n    \\left(r\\frac{\\partial}{\\partial r}\\right)+ \\frac{{\\mathbb J}^2}{r^2}=\n     -\\left({\\widehat{p}}_r^2+\\frac{-{\\mathbb J}^2}{r^2}\\right))\\,,\n$$\n\n\\noindent where ${\\widehat{p}}_r$ is the analog of the (selfadjoint) {\\em radial momentum}. On the eigensubspace $\\Pi_\\ell{\\mathcal H}$, the operator $-{\\mathbb J}^2$ takes on the value $\\ell^2$. So that the eq.~(\\ref{visc.eq-ev4}) becomes\n\n\\begin{equation}\n\\label{visc.eq-ev5}\n{\\mathscr L} P_\\zeta = \\tau\\sum_{\\ell\\in{\\mathbb Z}}\n \\frac{1}{1-a_\\ell(\\zeta) e^{-({\\widehat{p}}_r^2+\\ell^2\/r^2)}}\\,\n  a_\\ell(\\zeta) \\Pi_\\ell\\,.\n\\end{equation}\n\n\\noindent Since the operator ${\\widehat{p}}_r^2+\\ell^2\/r^2$ is known to admit $[0,\\infty)$ as its spectrum, the {\\em l.h.s.} is analytic in $\\zeta$, in the subdomain of $\\Re(\\zeta)>-1\/\\tau$ made of values for which $a_\\ell(\\zeta)\\notin [1,+\\infty)$ for all the $\\ell$'s that are relevant for the viscosity formula. Without knowing more about the distribution of the $\\omega_n$'s it is not possible to go further.\n\n \\subsection{An explicit case}\n \\label{visc.ssect-expl}\n\n\\noindent There is a distribution of the pulsation that permits do get a closed formula for the $a_\\ell$'s. Namely we assume that the pulsation is uniformly distributed on some finite interval $I$. Since ${\\mathbb E}(\\omega_n)=\\omega$ and $\\Var(\\omega_n)=\\sigma$ (see eq.~(\\ref{visc.eq-omega})), it follows that $I=[\\omega-\\sigma\\sqrt{3}, \\omega+\\sigma\\sqrt{3}]$. This gives\n\n\\begin{equation}\n\\label{visc.eq-aell1}\na_\\ell(\\zeta)= \\frac{1}{2\\imath\\sqrt{3} \\ell \\sigma\\tau}\\,\n \\ln\\left\\{\n   \\frac{1+\\tau(\\zeta+\\imath\\ell\\omega)+\\imath\\sqrt{3}\\ell\\sigma\\tau}\n        {1+\\tau(\\zeta+\\imath\\ell\\omega)-\\imath\\sqrt{3}\\ell\\sigma\\tau}\n    \\right\\}\\,.\n\\end{equation}\n\n\\noindent It follows that $a_0=0$. Changing $\\ell$ into $-\\ell$ and $\\omega$ into $-\\omega$ do not change $a_\\ell$. Hence there is no loss of generality in assuming that $\\ell \\geq 1$. As long as $\\Im(\\zeta)\\neq -\\ell \\omega$, $\\Im\\left(a_\\ell(\\zeta)\\right)\\neq 0$ so that $1-a_\\ell(\\zeta)e^{-p}\\neq 0$ for $p\\geq 0$, so the Laplace transform of $P_t$ is well defined. To find the positions of the singularities it is therefore necessary to assume that $(\\zeta+\\imath\\ell\\omega)=\\Re(\\zeta)\\in {\\mathbb R}$.  From the definition of $a_\\ell$ in eq.~(\\ref{visc.eq-aell}), the branch of the logarithm in eq.~(\\ref{visc.eq-aell1}) is such that as $\\Re(\\zeta)\\to +\\infty$, it satisfies the asymptotics given in eq.~(\\ref{visc.eq-aell}). Hence, using polar coordinates in the complex plane, this gives\n\n$$1+\\tau\\Re(\\zeta)+\\imath\\sqrt{3}\\ell\\sigma\\tau=\n   \\rho e^{\\imath \\theta}\\,,\n    \\hspace{2cm}\n     \\tan{\\theta}= \\frac{\\sqrt{3}\\ell\\sigma\\tau}{1+\\tau\\Re(\\zeta)}\\,.\n$$\n\n\\noindent It ought to be remarked that, in the domain of analyticity, $1+\\tau\\Re(\\zeta)>0$. With the asymptotics for $\\Re(\\zeta)$ large, since $\\ell\\geq 1$ the definition of $\\theta$ implies $0<\\theta<\\frac{\\pi}{2}$. To summarize,\n\n\\begin{equation}\n\\label{visc.eq-aell2}\na_\\ell(\\zeta)= \\frac{\\theta}{\\sqrt{3}\\ell\\sigma\\tau}\\,,\n \\hspace{2cm}\n  \\tan{\\theta}= \\frac{\\sqrt{3}\\ell\\sigma\\tau}{1+\\tau\\Re(\\zeta)}\\,,\n   \\hspace{2cm}\n    0<\\theta<\\frac{\\pi}{2}\\,.\n\\end{equation}\n\n\\noindent The analyticity condition $a_\\ell\\neq [1,+\\infty)$ imposed on the Laplace transform implies, whenever $\\zeta+\\imath\\ell\\omega=\\Re(\\zeta)$, that \n\n(i) if $\\sqrt{3}\\ell\\sigma\\tau\\geq \\pi\/2$, then $a_\\ell(\\zeta)\\neq [1,+\\infty)$ for $\\Re(\\zeta)>-1\/\\tau$;\n\n(ii) If $\\sqrt{3}\\ell\\sigma\\tau< \\pi\/2$, then in order that $a_\\ell(\\zeta)\\neq [1,+\\infty)$, it is necessary that $\\theta<\\sqrt{3}\\ell\\tau\\sigma<\\pi\/2$; since the function $\\theta\\mapsto \\tan{\\theta}$ is increasing on $[0,\\pi\/2)$ then\n\n\\begin{equation}\n\\label{visc.eq-maxtime}\n\\Re(\\zeta)>-\\frac{1}{\\tau_M}\\,,\n   \\hspace{2cm}\n    \\frac{\\tau_M}{\\tau} =\n     \\left(\n       1-\\frac{\\sqrt{3}\\ell\\tau\\sigma}{\\tan{\\sqrt{3}\\ell\\tau\\sigma}}\n     \\right)^{-1}\n\\end{equation}\n\n\\noindent In other words\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 3: } {\\em The Maxwell time is given, in terms of $\\tau=\\tlc$ by the following formulae}\n\n\\begin{equation}\n\\label{visc.eq-maxtime2}\n\\frac{\\tau_M}{\\tlc} =\n\\left\\{\n\\begin{array}{ll}\n1 & \\mbox{\\rm if}\\;\\;\\sqrt{3}\\ell\\tau\\sigma\\geq \\pi\/2\\\\\n\\left(\n       1-\\frac{\\sqrt{3}\\ell\\tau\\sigma}{\\tan{\\sqrt{3}\\ell\\tau\\sigma}}\n     \\right)^{-1}>1 & \\mbox{\\rm otherwise}\n\\end{array}\\right.\n\\end{equation}\n\n\\vspace{.5cm}\n\n\\section{Interpreting the Results}\n\\label{visc.sect-results}\n\n\\noindent From the mathematical computations made in the previous Section~\\ref{visc.ssect-expl}, it follows that there are two regimes concerning the Maxwell time, both being controlled by the dimensionless parameter\n\n$$K = \\sqrt{3}\\tlc\\sigma\\,,\n$$\n\n\\noindent which is proportional to the variance of the distribution of the pulsations $\\omega_n$, representing the curvature of the local potential energy, near its mechanical equilibrium, where the atom is located. In order to draw practical predictions for a liquid material, some physical hypothesis are required. In particular, {\\em it is highly expected that the parameters $\\tlc$ and $\\sigma$ depend on the temperature of the liquid}. If so, the following conclusion can be made\n\n\\vspace{.1cm}\n\n\\noindent {\\bf Result 4: } {\\em Depending upon the material, there may or may not exist a cross-over temperature $T_{co}$, such that (a) for $T\\geq T_{co}$, the anankeons dominate and the Maxwell time coincides with the local configuration time, while, (b) if $T<T_{co}$, the phonons restore some order, survive for a longer while, so that the Maxwell time becomes larger than the local configuration time.}\n\n\\vspace{.1cm}\n\n\\noindent In view of the results showed in Fig.~\\ref{visc.fig-relTime} \\cite{INE13}, it seems natural to assume that the time dependence of $\\tlc$ be given by an {\\em Arrhenius Law}, namely\n\n$$\\tlc=\\tau_\\infty e^{W\/\\kB T}\\,,\n   \\hspace{1cm}\\mbox{\\rm where}\\hspace{1cm}\n    \\tau_\\infty= \\lim_{T\\uparrow \\infty} \\tlc(T)\\,.\n$$\n\n\\noindent In this expression, the energy $W$ can be interpreted as the height of the potential barrier to be passed over as the anankeon strikes. \nHow is the variance $\\sigma$ depends on the temperature~? So far no clues have been given on that. So we are reduced to speculate. By analogy and {\\em without further justification, though}, the same hypothesis can be made on $\\sigma$, \n\n$$\\sigma =\\sigma_\\infty e^{-W_v\/\\kB T}\\,,\n   \\hspace{1cm}\\mbox{\\rm where}\\hspace{1cm}\n    \\sigma_\\infty= \\lim_{T\\uparrow \\infty} \\sigma(T)\\,.\n$$\n\n\\noindent If so, it is legitimate to ask the following question: \n\n\\vspace{.1cm}\n\n\\noindent {\\bf Question: } {\\em what is the meaning of the energy scale $W_v$~?}\n\n\\vspace{.1cm}\n\n\\noindent It is only possible to see that if $W_v$ is small, then the variance of the frequency distribution is weakly dependent of the temperature. If, on the other hand, $W_v$ is large enough, then this variance is strongly depressed at low temperature. It is important to remember that such an Arrhenius Law, while not justified by physical arguments, needs to be valid only in a temperature range of moderate size above the glass transition. In such a case, it is only a convenient way to parametrize the temperature behavior of the variance using only one parameter, the energy scale $W_v$. \n\n\\vspace{.1cm}\n\n\\noindent Once these assumptions are made, it follows that the parameter $K$ behaves like\n\n\\begin{equation}\n\\label{visc.eq-KT}\nK(T)= K_\\infty e^{-(W_v-W)\/\\kB T}\\,,\n   \\hspace{1cm}\\mbox{\\rm where}\\hspace{1cm}\n    K_\\infty= \\sqrt{3}\\,\\tau_\\infty\\, \\sigma_\\infty\\,.\n\\end{equation}\n\n\\noindent The condition required for the existence of a cross-over temperature $T_{co}$ is that, $K(T)$ decreases as the temperature decreases and that $K(T_{co})\\ell=\\pi\/2$, leading to\n\n\\begin{equation}\n\\label{visc.eq-Tco}\nW_v>W\\,, \n \\hspace{1cm}\n  K_\\infty >\\frac{\\pi}{2}\\,,\n   \\hspace{1cm}\\Rightarrow\\hspace{1cm}\n    \\kB T_{co}= \\frac{W_v-W}{\\ln(2 K_\\infty\/\\pi)}\n\\end{equation}\n\n\\noindent The first consequence of these hypothesis is that the difference between strong and fragile glassy materials \\cite{Ang95} can be expressed by the conditions\n\n\\begin{equation}\n\\label{visc.eq-strfra}\n\\left\\{\n\\begin{array}{ll}\nW_v\\leq W & \\mbox{\\rm for strong liquids}\\\\\nW_v>W & \\mbox{\\rm for fragile liquids}\n\\end{array}\\right.\n\\end{equation}\n\n\\noindent The second consequence is that {\\em the ratio between the Maxwell and the local configuration times diverges exponentially fast below the cross-over temperature} (see eq.~(\\ref{visc.eq-maxtime2}))\n\n\\begin{equation}\n\\label{visc.eq-ratio}\n\\frac{\\tau_M}{\\tlc}\\;\\;\n \\stackrel{T\\ll T_{co}}{\\sim}\\;\\;\n  \\frac{1}{\\ell^2K(T)^2} =\n   \\frac{e^{2(W_v-W)\/\\kB T}}{\\ell^2 K_\\infty^2}\\,.\n\\end{equation}\n\n\\noindent Such a law would explain why the viscosity diverges so fast near the glass transition temperature.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nThe practical utility of widely used methods in electronic structure\ntheory is in large part determined by the optimization algorithms\nthey rely on.\nThis basic theme has been repeated throughout the history of\nquantum chemistry, with methods as fundamental as Hartree-Fock\ntheory becoming dramatically more useful with the development\nof superior solution methods such as the direct inversion of\nthe iterative subspace. \\cite{pulay1982diis}\nSimilar transformations have been seen in configuration interaction\n(CI) theory thanks to Davidson's method, \\cite{davidson1975}\nin the density matrix renormalization group (DMRG) approach\nthanks to (among other innovations) the noise algorithm, \\cite{white2005noise}\nand in many other methods besides.\nAs in the case of DMRG, it is usually not so simple as a single innovation\nin the numerical methods that transforms a theory from a promising\nproof of concept into a robust computational tool.\nInstead, such tools often arise as the result of a series of innovations,\nthat, once combined, fit together in a way that makes them more than\nthe sum of their parts.\n\nIn the context of quantum Monte Carlo (QMC), and more specifically\nin its variational (VMC) formulation, the introduction of the\nlinear method (LM) for trial function optimization marked a large\nstep forward along the path to practical utility and reliability.\n\\cite{Umrigar2007}\nHowever, recent research has revealed multiple options for\nbypassing the LM's memory bottleneck, making clear that there is\nstill a great deal of distance to cover in the maturation of\nVMC numerical methods.\nSome of these approaches\n\\cite{Neuscamman2012,Zhao2017}\ndepend, like the LM itself, on knowing\nat least some information about energy second derivatives, but by\navoiding the construction of full Hessian-sized matrices they\nachieve dramatically lower memory footprints compared to the LM.\nOther even more recent approaches, most of which can be classified as\naccelerated descent (AD) methods,\n\\cite{Schwarz2017,Sabzevari2018,Mahajan2019}\navoid second derivative information entirely and are thus even\nmore memory efficient, relying instead on a\nlimited knowledge of the optimization's history of energy\nfirst derivatives or in one case just the signs of\nthese derivatives. \\cite{Luo2018}\nIn the present study, we explore the relative advantages of these\nfirst and second derivative approaches and find that,\nwhen combined, they offer a highly complementary optimization\nstrategy that appears both more robust and more efficient\nthan either class of methods is on its own.\n\nThe ability to optimize larger and more complicated wave function\nforms is becoming increasingly relevant due to rapid\nprogress in other areas of VMC methodology.\nThe introduction of the table method \\cite{Clark2011,Morales2012}\nhas increased the size of CI expansions that can be\nhandled by more than an order of magnitude, and expansion\nlengths beyond 10,000 determinants are no longer unusual.\nA recent improvement to the table method \\cite{Filippi2016,Assaraf2017}\nnow allows the molecular orbital basis to be optimized\nefficiently in the presence of these large expansions,\nwhile the resurgence of interest in selected CI methods\n\\cite{evangelista2016adaptive,Holmes2016,tubman2016asci,\n      Sharma2017,loos2018scipt,zimmerman2018sshci}\nhas provided a convenient route to their construction.\nIn addition to these CI-based advances, other wave function\ninnovations have also led to growing demands on\nVMC optimization methods.\nIncreasingly sophisticated correlation factors,\nsuch as those used in Hilbert space approaches\n\\cite{changlani2009cps,mezzacapo2009eps,neuscamman2012magnet,\n      Neuscamman2012,Schwarz2017,Sabzevari2018,Mahajan2019}\nas well as a steady stream of developments in real space\n\\cite{Casula2004,Sorella2007,LopezRios2012,Luchow2015,Goetz2017,Goetz2018}\nhave also raised the demand for optimization approaches\nthat can deal with large numbers of highly nonlinear parameters.\nAlthough less thoroughly explored, the treatment of correlation effects\nvia back flow transformations also continues to receive attention\nand create new optimization challenges. \\cite{Holzmann2015iterbackflow,Luo2018}\nFinally, in addition to these increases in ansatz sophistication,\nrecent interest in using excited state variational principles\nto expand QMC's excited state capabilities has led to its own\ncollection of optimization difficulties.\n\\cite{Zhao2016,neuscamman2016varqmc,blunt2017ct,Shea2017,\n      robinson2017vm,blunt2018excited,Flores2019}\n\nBy supporting these various advances in QMC methodology, improved\nVMC optimization methods have the potential for large impacts\nin diverse areas of chemistry and solid state physics.\nWork on lattice models, for example, continues to push the\nboundaries on how approximate wave functions are defined.\n\\cite{troyer2017rbm,Kochkov2018}\nIn the area of molecular excited states, QMC methods offer\npromising new routes to high-accuracy treatments of both\ndouble excitations \\cite{Zhao2016,Neuscamman2016}\nand charge transfer excitations, \\cite{blunt2017ct,Flores2019}\nboth of which continue to challenge conventional\nquantum chemistry methods.\nIn QMC's traditional area of simulating real solids, applications\nof both VMC and projector Monte Carlo would benefit immediately\nfrom the ability to prepare more sophisticated trial\nwave functions.\n\\cite{Foulkes2001,morales2018qfqmcnio,zhao2018gaps}\nDiffusion Monte Carlo (DMC) in particular would achieve higher accuracy using \nthe better nodal surfaces determined by well-optimized ansatzes from VMC.\nMore generally, the ability of QMC to combine treatments\nof weak and strong electron correlation effects within a robust\nvariational framework that operates near the basis set limit\nmakes it a powerful general-purpose\napproach for difficult molecular and materials problems where\nhigh accuracy is necessary.\nBy increasing the size and complexity of systems that fall into\nits purview, improvements in QMC wave function optimization methods\ntherefore have the potential to move electronic structure simulation\nforward on a number of fronts.\n\nThe present study seeks to aid in this endeavor by focusing on\nthe relative advantages of recently developed low-memory\nfirst and second derivative methods in VMC and in\nparticular on how they can be used to complement each other.\nUnlike deterministic optimizations, in which second derivative\nmethods are typically preferred so long as they are affordable,\nthe situation is less straightforward when the objective function\nand its derivatives are statistically uncertain.\nOne major concern is that, in practice, it can be more difficult\nto achieve low-uncertainty estimates of the second derivative\nterms that appear in the LM and its descendants.\nWhile this issue can be mitigated by the use of alternative\napproaches to importance sampling,\nthese can increase uncertainty in the energy due to the loss\nof the zero-variance principle.\nThus, as we will demonstrate, statistical precision tends to\nbe higher when using AD methods, which is an advantage on top\nof their ability to converge to the minimum without the bias\nthat arises from the LM's highly nonlinear matrix diagonalization.\nHowever, we will also see that in order to enjoy the advantages\nof a tighter and less biased final convergence,\nAD methods must first reach the vicinity of the minimum.\nFor this task, we find that the LM and its low-memory variants\noutperform all of the first derivative methods that we tested,\nespecially for optimizations in which the wave function contains\ndifferent classes of parameters that vary greatly in their\nnonlinear character and how they couple to each other.\nHappily, we will see that a hybrid approach --- in which\nAD and low-memory LM optimization steps are interwoven ---\nexcels both at reaching the vicinity of the minimum and\nproducing unbiased final energies while simultaneously\nmaintaining a high degree of statistical efficiency.\n\n\\section{Theory}\n\n\\subsection{Variational Monte Carlo}\n\nVMC combines the variational principle of quantum mechanics with Monte Carlo evaluation of high dimensional integrals.\\cite{Umrigar2015}\nTo study the ground state of a system, we pick a trial wave function $\\Psi$ of some particular form and seek to minimize its energy expectation value.\n\\begin{equation}\n    E(\\Psi) = \\frac{\\Braket{\\Psi | H | \\Psi}}{\\Braket{\\Psi | \\Psi}}\n\\end{equation}\n\nIn the language of mathematical optimization, $E(\\Psi)$ is an example of an objective function or cost function.\nFor a typical system with $N$ electrons, this expression contains integrals over $3N$ position space coordinates which for some wave functions can only be evaluated efficiently through Monte Carlo sampling rather than quadrature methods.\nWe rewrite the energy as\n\\begin{align}\n\\notag\n   E &= \\frac{\\int d \\mathbf{R}\\Psi (\\mathbf{R}) H \\Psi(\\mathbf{R})}{\\int d \\mathbf{R}\\Psi (\\mathbf{R})^2}\n      = \\frac{\\int d \\mathbf{R}\\Psi (\\mathbf{R})^2 E_L (\\mathbf{R})}{\\int d \\mathbf{R}\\Psi (\\mathbf{R})^2} \\\\\n     &= \\int d \\mathbf{R} \\rho (\\mathbf{R})E_L (\\mathbf{R})\n\\label{eqn:energy}\n\\end{align}\nwhere $E_L (\\mathbf{R}) = \\frac{H \\Psi (\\mathbf{R})}{\\Psi (\\mathbf{R})}$ is the local energy and $\\rho (\\mathbf{R}) = \\frac{\\Psi(\\mathbf{R})^2}{\\int d \\mathbf{R} \\Psi (\\mathbf{R})^2}$ is the probability density.\nThe zero-variance principle\\cite{Assaraf1999} makes $\\rho (\\mathbf{R})$ the most common choice of probability distribution for obtaining samples, but it is not the only option.\nFor effective estimation of quantities beside the energy, such as the LM matrix elements, other\nimportance sampling functions are often preferred.\n\\cite{trail2008a,trail2008b,robinson2017vm}\nIn our LM and blocked LM calculations in this study,\nwe employ the importance sampling function\n(and the appropriately modified statistical estimate formulas\n\\cite{Flores2019})\n\\begin{align}\n\\label{eqn:is}\n    |\\Phi|^2 \\equiv\n    |\\Psi|^2 + \\frac{\\epsilon}{M}\n    \\hspace{0.5mm} \\sum_{I} |D_I|^2\n\\end{align}\nin which the $D_I$ are the $M$ different $S_z$-conserving single\nexcitations relative to the closed shell reference determinant.\nThe logic behind this choice is that it puts some weight on\nconfigurations that are highly relevant for the orbital rotation\nparameters' wave function derivatives, as small orbital\nrotations can be approximated via the addition of singles.\nWe find that this importance sampling function substantially\nreduces the uncertainty of the LM matrix elements corresponding\nto orbital rotations, which in turn helps reduce the update\nstep uncertainty.\nFor AD, we simply use traditional $|\\Psi|^2$ importance sampling\nas in equation \\ref{eqn:energy}.\n\nBy the variational principle, we are guaranteed that $E$ is an upper bound on the true ground state energy. \nGiven some set of adjustable parameters in the functional form of $\\Psi$, we expect that values of those parameters that yield a lower value of $E$ to correspond to a wave function that is closer to the ground state. \nOne could then imagine the abstract space produced by the possible values of all variational parameters.\nThe set of optimal parameter values that specify the wave function expression which minimizes $E$ can be taken as a point in this space labeled by the vector $\\mathbf{p^*}$.\nIn general, the initial choice for parameters will not be at this energy minimum point, but at some other point $\\mathbf{p_0}$.\nThe problem of determining the best wave function in VMC calculations then relies on an optimization algorithm for finding $\\mathbf{p^*}$ after starting from $\\mathbf{p_0}$.\n\nWithin this framework, one of the most important considerations is that the optimization is inherently stochastic due to the introduction of noise through the Monte Carlo evaluation of the integral in equation \\ref{eqn:energy}.\nThis forms a contrast with many other methods in electronic structure theory including Hartree-Fock, CI, and coupled cluster where various deterministic optimization schemes predominate.\\cite{Helgaker2000}\nMany of the algorithms commonly encountered in a deterministic quantum chemistry context such as steepest descent and the Newton method, have been adapted for use in VMC.\\cite{Harju1997,Lin2000,Lee2005,Sorella2005,Umrigar2005}\nHowever, there is now a need to be robust to the presence of noise.\nHistorically, errors due to finite sampling led to numerical instabilities that prompted interest in minimizing variance\\cite{Kent1999,Umrigar1988} instead of energy, but later optimization developments have sought to mitigate this issue and in this paper we only consider energy minimization.\nAs we will now discuss in their respective sections, both the LM and gradient descent approaches possess features that enable them to operate stably in a stochastic setting.\n\n\\subsection{The Linear Method}\nThe LM\\cite{Nightingale2001,Umrigar2007} begins with a first order Taylor expansion of the wave function.\nFor a set of variational parameters given by vector $\\mathbf{p}$, we have\n\\begin{equation}\n\\label{eqn:lmTaylor}\n    \\Psi (\\mathbf{p}) = \\Psi_0 + \\sum_i \\Delta p_i \\Psi_i\n\\end{equation}\nwhere $\\Psi_i = \\frac{\\partial \\Psi(\\mathbf{p})}{\\partial p_i}$ and $\\Psi_0$ is the wave function at the current parameter values.\n\nFinding the optimal changes to the parameters amounts to solving the generalized eigenvalue problem \n\\begin{equation}\n\\label{eqn:lmEigen}\n    H\\Vec{c} = E S \\Vec{c}\n\\end{equation}\n in the basis of the wave function and its first order parameter derivatives $\\{\\Psi_0,\\Psi_1,\\Psi_2,...\\}$.\n$H$ and $S$ are the Hamiltonian and overlap matrices in this basis with elements \n\\begin{equation}\n\\label{eqn:lmH}\n    H_{ij} = \\Braket{\\Psi_i | H | \\Psi_j} \n\\end{equation}\n\\begin{equation}\n\\label{eqn:lmS}\n    S_{ij} = \\Braket{\\Psi_i  | \\Psi_j}\n\\end{equation}\nThe matrix diagonalization to solve this eigenproblem for eigenvector $\\Vec{c} = (1,\\Delta \\mathbf{p})$ then yields the updated parameter values $\\mathbf{p_1} = \\mathbf{p_0} + \\Delta \\mathbf{p}$.\nAs the matrices $\\bm{H}$ and $\\bm{S}$ both contain a subset\nof the second derivative terms that would be present in a \nNewton-Raphson approach, \\cite{Toulouse2007}\nthe LM is most naturally categorized as a second-derivative\nmethod, and it certainly shares Newton-Raphson's difficulties\nwith regards to dealing with matrices whose dimension\ngrows as the number of variables.\n\nFor practical use with finite sampling, the LM must be stabilized to prevent unwisely large steps in parameter space.\nThis is accomplished by adding shift values\\cite{Umrigar2007} to the matrix diagonal that effectively act as a trust radius scheme similar to those used with the Newton method.\nIn our implementation, the Hamiltonian is modified with two shift values meant to address distinct potential problems in the optimization.\\cite{Kim2018}\n\\begin{equation}\n\\label{eqn:lmShift}\n    \\mathbf{H} \\xrightarrow[]{} \\mathbf{H} + c_I \\mathbf{A} + c_S \\mathbf{B}\n\\end{equation}\n\nThe matrix elements of $\\mathbf{A}$ are given by $A_{ij} = \\delta_{ij}(1-\\delta_{i0})$ so that the shift $c_I$ effectively gives an energy penalty to directions of change from the current wave function.\\cite{Umrigar2007}\nThe second shift is intended to address problems that may arise if some wave function derivatives have norms that differ by orders of magnitude. \nIn this situation, the single shift value $c_I$ is insufficient to preserve a quick yet stable optimization.\nFor a parameter with a large derivative norm, a sufficiently high value of $c_I$ might prevent an excessively large change in its value.\nHowever, all other parameter directions with smaller derivative norms will be so heavily penalized by the large value of $c_I$ that those parameters become effectively fixed.\nThe purpose of the second $c_S \\mathbf{B}$ term is to retain important flexibility in other parameter directions. \nWe can write the matrix $\\mathbf{B}$ as\n\\begin{equation}\n\\label{eqn:bMat}\n\\mathbf{B} = (\\mathbf{Q^T})^{-1} \\mathbf{T} \\mathbf{Q}^{-1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\label{eqn:qMat}\n    Q_{ij} = \\delta_{ij} -\\delta_{i0}(1-\\delta_{j0})S_{0j}\n\\end{equation}\nand \n\\begin{equation}\n\\label{eqn:tMat}\n    T_{ij} = (1-\\delta_{i0}\\delta_{j0})[\\mathbf{Q^T}\\mathbf{S}\\mathbf{Q}]_{ij}\n\\end{equation}\nThe matrix $\\mathbf{Q}$ provides a transformation to a basis where all update directions are orthogonal to the current wave function and the matrix $\\mathbf{T}$ is the overlap matrix in this basis. \nThe optimal choice of shift parameters $c_I$ and $c_S$ may depend on the particular optimization problem.\nIn our implementation, an adaptive scheme adjusts the shifts on each iteration by comparing the energies calculated through correlated sampling on three different sets of shift values and choosing whichever shifts produced the lowest energy.\n\nThe LM has been successfully applied to a\nvariety of systems to prepare good trial wave functions\nfor DMC.\n\\cite{Umrigar2007,Toulouse2007,Toulouse2008,Brown2007,Petruzielo2012,Goetz2017,Goetz2018}\nIt has also been used in the variational optimization of a recent functional for targeting excited states.\\cite{Zhao2016,Shea2017,Flores2019}\nHowever, it possesses a number of limitations, most notably a memory cost that scales with the square of the number of optimizable parameters due to the matrices it builds. \nThis memory cost currently confines routine use of the LM to less than roughly 10,000 parameters though exceptional calculations with up to about 16,000 have been made.\\cite{Clark2011}\nAnother shortcoming is the nonlinear bias of the LM.\nWe are evaluating the elements of the Hamiltonian and overlap matrices stochastically and have a nonlinear relationship between them and our energy through the generally high order characteristic polynomial of the eigenvalue problem of equation \\ref{eqn:lmEigen}.\nAs a result, we in general expect the LM to converge to a point in parameter space slightly offset from the true minimum.\nThis nonlinear bias has been studied for the LM in Hilbert space\\cite{Zhao2016a} and a similar issue arises in the context of Full Configuration Interaction QMC.\\cite{Blunt2018}\nBoth the memory constraint and the nonlinear bias of the LM become more severe for ansatzes with larger numbers of variational parameters, which spurs the search for potential alternatives.\nOne approach suggested for memory reduction is to employ Krylov subspace methods for Eq. 4 to avoid building matrices, but it requires a drastically higher sampling effort due to the need for many matrix-vector multiplications and so we do not pursue the approach here.\\cite{Neuscamman2012}\n\n\\subsection{Blocked Linear Method}\nOne recent approach to bypassing the memory bottleneck is known as the blocked linear method (BLM).\\cite{Zhao2017}\nThe first step of the algorithm is to divide the full set of parameters into $N_b$ blocks. \nNext, a LM-style matrix diagonalization is carried out within each block and some number $N_k$ of the resulting eigenvectors from the blocks are retained as good directions for constructing an approximation for the overall best update direction in the full parameter space.\nFor a particular block of variables, the wave function expansion in the LM is given by\n\\begin{equation}\n\\label{eqn:blmBlock}\n    \\ket{\\Psi_b} = \\ket{\\Psi_0} + \\sum_{i=1}^{M_b} c_i \\ket{\\Psi^i} \n\\end{equation}\nwhere $\\ket{\\Psi^i}$ is the wave function derivative with respect to the $i$th variable in the block, $M_b$ is the number of variables in the block, and $\\ket{\\Psi_0}$ the current wave function as in the normal LM.\nWe can perform the same matrix diagonalization done in the LM, only with parameters outside the block fixed.\nThis yields a set of eigenvectors that we can use to construct another approximate expansion of the original wave function.\nWe can construct a matrix $\\mathbf{B}$ using the $N_k$ eigenvectors with the lowest eigenvalues from each block and write a new expansion\n\\begin{equation}\n\\label{eqn:blmNewBlock}\n    \\ket{\\Tilde{\\Psi}} = \\alpha \\ket{\\Psi_0} + \\sum_{k=1}^{N_b} \\sum_{j=1}^{N_k} A_{kj} \\sum_{i=1}^{M_b} B_{ji}^{(b)} \\ket{\\Psi^{i,b}}\n\\end{equation}\nHaving now pre-identified important directions within each block,\nthe idea is that a subsequent LM-style diagonalization\nin the basis of these good directions (which yields\nthe coefficients $A_{kj}$) should still provide a good\nupdate direction when re-expressed in the full parameter space.\n\nIn order to help retain most of the accuracy of the traditional LM, the first stage of the BLM computation includes $N_o$ other good directions that are used to supply the current block's diagonalization with information about how its variables are likely to couple to those in other blocks.\nIn practice, important out-of-block directions are obtained by\nkeeping a history of previous iterations' updates\nas the optimization progresses.\nWe can rewrite the one block expansion introduced in equation \\ref{eqn:blmBlock} as\n\\begin{equation}\n\\label{eqn:blmHistory}\n    \\ket{\\Psi_b} = \\ket{\\Psi_0} + \\sum_{i=1}^{M_b} c_i \\ket{\\Psi^i} + \\sum_{j=1}^{N_o} \\sum_{k=1,k\\neq b}^{N_b} d_{jk} \\ket{\\Theta_{jk}}\n\\end{equation}\nwhere we take the\n\\begin{equation}\n\\label{eqn:blmTheta}\n    \\ket{\\Theta_{jk}} = \\sum_{l=1}^{M_k} C_{jkl} \\ket{\\Psi^{l,k}}\n\\end{equation}\nas the linear combinations of wave function derivatives from other blocks that were identified as important based on previous iterations' updates.\nThe additional term in the expansion allows us to account for couplings between variables in different blocks and enable the construction of a better space for the second diagonalization.\nWe assemble the matrix $\\mathbf{B}$ and $\\ket{\\Tilde{\\Psi}}$ and then seek to minimize $\\frac{\\Braket{\\Tilde{\\Psi} | H | \\Tilde{\\Psi}}}{\\Braket{\\Tilde{\\Psi}|\\Tilde{\\Psi}}}$ with respect to variational parameters $\\alpha$ and $A_{kj}$ in our BLM wave function expansion in equation \\ref{eqn:blmNewBlock}.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{BLM_flowchart.eps}\n    \\caption{Flowchart depicting steps in the BLM algorithm to arrive at a parameter update.}\n    \\label{fig:blmchart}\n\\end{figure}\n\nFigure \\ref{fig:blmchart} portrays the algorithmic steps described above. Some number of parameters too large to be handled by the standard LM is divided among different blocks whose diagonalizations produce the vectors $\\Vec{b_i}$ for the construction of the space of the second diagonalization that produces the parameter update.\nThe BLM can be thought of as achieving memory savings in the use of smaller matrices at the cost of having to run over the sample twice when the traditional LM must run over it just once.\nA more extensive description of the BLM and its precise memory usage can be found in its original paper.\\cite{Zhao2017}\n\nWe divide parameters evenly among blocks, but one could implement the use of tailored blocks of varying sizes. \nIt is advisable to choose the block size to be large enough to keep important parameters of the same type, such as all of those for a Jastrow factor, within the same block.\nThis enables the expected strong coupling between them to be handled more accurately by the LM-style diagonalization within that block.\nWhile the BLM has been successfully applied up to about 25,000 parameters and found to closely reproduce the results of the standard LM, \\cite{Zhao2017}\nit remains a relatively new method, and the present study will provide additional data on its efficacy.\n\n\\subsection{Gradient Descent Methods}\nIn the last few years, increasing attention\\cite{Schwarz2017,Schwarz2017a,Sabzevari2018,Luo2018,Mahajan2019} has been paid to optimization methods that use only first derivatives to minimize a Lagrangian of the form \n\\begin{equation}\n\\label{eqn:lagrangian}\n    \\mathcal{L}( \\Psi(\\mathbf{p})) = \\Braket{\\Psi | H | \\Psi} - \\mu(\\Braket{\\Psi | \\Psi} - 1) \n\\end{equation}\nwhere $\\mu$ is a Lagrange multiplier and, in practice, a moving average of the local energy.\nThere is no need to solve an eigenvalue problem as in the LM and the memory cost of these approaches scales linearly with the number of parameters.\nWe also note that the stochastic evaluation of derivatives of this Lagrangian will lead to a smaller nonlinear bias compared to what is encountered in the LM.\nWhile there is some nonlinearity present in the product $\\mu \\Braket{\\Psi | \\Psi}$, it is mild compared to the high order polynomials encountered in the solution of the LM eigenvalue problem and can be avoided entirely if desired through modest amounts of extra sampling.\nMinimization of this Lagrangian targets the ground state, but excited states can similarly be targeted with these optimization algorithms merely by using derivatives of one of the excited state functionals that have been developed.\\cite{Choi1970,Ye2017,Zhao2016}\n\nThe simplest method in this category is the steepest descent algorithm.\n\\begin{equation}\n\\label{eqn:steepest}\n    p_i^{k+1} = p_i^k - \\eta_k \\frac{\\partial \\mathcal{L}(\\mathbf{p})}{\\partial p_i}\n\\end{equation}\nIn this case, the value of each parameter on the $k+1$'th step is found simply by subtracting the statistically uncertain parameter derivative times a step size $\\eta_k$.\nThe step size can be taken as constant over all steps in the simplest case, but rigorous proofs on the convergence of stochastic gradient descent (SGD) rely on decaying step sizes satisfying $\\sum_{k} \\eta_k = \\infty$ and $\\sum_{k} \\eta_k^2 < \\infty$.\\cite{Bottou2012}\n\nIt may be worth briefly commenting that the typical formulation of stochastic gradient descent as seen in the machine learning and mathematical optimization literature is slightly different from what we use here within VMC.\nIn a common machine learning scenario,\\cite{Bottou2012} one has a training set of input data $\\{x_1,x_2,...,x_n\\}$ and corresponding outputs $\\{y_1,y_2,...,y_n\\}$ and wishes to minimize a loss function $Q(x,y;w)$ that measures the error produced by a model $f_w(x)$, which predicts $\\widetilde{y}_i$ given $x_i$ and is parameterized by variables $w$.\nFor this setting, the SGD algorithm refers to evaluating the gradient of $Q$ with a randomly chosen pair $(x_j,y_j)$ from the given data set and then computing the parameter update according to $w_{k+1} = w_k - \\eta_k \\nabla_w Q(x_j,y_j)$.\nFor our VMC optimization, we are dealing with a noisy gradient similar to what occurs in this machine learning problem, but the source of our noise is somewhat different and lies in our means of evaluating the underlying 3N dimensional integrals within our Lagrangian derivatives.\nAnother important distinction is that in machine learning applications, complete convergence to the minimum is in fact undesirable because it will overfit the model to the training data and degrade its performance on new sets of test inputs.\nMuch as SGD provides a computational speed up for machine learning problems, we are also able to operate gradient descent methods at a cheap per-iteration cost because we need only a modest number of samples to evaluate sufficiently precise Lagrangian derivatives compared to the Hamiltonian and overlap matrices in the LM.\nHowever, unlike the machine learning case, we do want to come as close as possible to the true minimum, and we will see that even reaching the vicinity of the minimum can be difficult for descent methods when typical VMC initial guesses are employed.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{descent_intuition.eps}\n    \\caption{Illustration of the difficulty faced by steepest descent in red on the lower left with its slow approach to the minimum. Accelerated descent in green on the upper right is able to progress more rapidly to the minimum with its memory of previous gradients.}\n    \\label{fig:narrowvalley}\n\\end{figure}\n\nWhile steepest descent can be guaranteed to eventually reach the minimum of the Lagrangian even in a stochastic setting, its asymptotic convergence is very slow.\nFor some intuition, one could imagine the landscape of the Lagrangian's values forming a very narrow valley near the true minimum.\nIn this situation, steepest descent would produce parameter updates mostly back and forth along the sides of the valley with little improvement of parameter values in the direction directly toward the minimum.\nDue to the limitations of steepest descent, a number of other flavors of accelerated gradient descent (AD) have been developed that include a momentum term with information on previous values of the gradient.\nAs illustrated in Figure \\ref{fig:narrowvalley}, the general intuition is that this additional term provides some memory of the progression along narrow valleys that steepest descent lacks and thereby achieves swifter convergence.\nIn addition, there are multiple schemes for adaptively varying the step sizes used in a manner that draws on the particular derivative values for each individual parameter as the optimization progresses. \nThese methods have recently been applied successfully to Hilbert space QMC. In this study, we work in real space and investigate a combination of Nesterov momentum with RMSprop as presented by the Booth group \\cite{Schwarz2017,Schwarz2017a}, a method using random step sizes from the Clark group\\cite{Luo2018}, AMSGrad, recently used by the Sharma group\\cite{Sabzevari2018,Mahajan2019}, as well as the ADAM optimizer\\cite{Kingma}.\n\nWe now lay out the precise expressions for each of these methods in turn. The RMSprop algorithm used by Booth and co-workers is given by the following recurrence relations.\\cite{Schwarz2017}\n\\begin{equation}\n\\label{eqn:rmspropMomentum}\n    p_i^{k+1} = (1-\\gamma_k)q_i^{k+1} - \\gamma_k q_i^k\n\\end{equation}\n\\begin{equation}\n\\label{eqn:rmspropUpdate}\n   q_i^{k+1} = p_i^k - \\tau_k \\frac{\\partial \\mathcal{L}(\\mathbf{p})}{\\partial p_i}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:rmspropRecur}\n\\lambda_0=0 \\hspace{7mm}\n\\lambda_k = \\frac{1}{2} + \\frac{1}{2}\\sqrt{1+4\\lambda_{k-1}^2} \\hspace{7mm}\n\\gamma_k = \\frac{1-\\lambda_k}{\\lambda_{k+1}}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:rmspropStep}\n    \\tau_k = \\frac{\\eta}{\\sqrt{E[(\\frac{\\partial\\mathcal{L}}{\\partial p_i})^2]^{(k)}} + \\epsilon}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:rmspropAvg}\n    E[(\\partial \\mathcal{L})^2]^{(k)} = \\rho E\\left[\\left(\\frac{\\partial\\mathcal{L}}{\\partial p_i}\\right)^2\\right]^{(k-1)} + (1-\\rho)\\left(\\frac{\\partial\\mathcal{L}}{\\partial p_i}\\right)^2\n\\end{equation}\nAbove, $p_i^k$ denotes the value of the $i$th parameter on the $k$th step of the optimization, $\\tau_k$ is a step size that is adaptively adjusted according to the RMSprop algorithm in equations \\ref{eqn:rmspropStep} and \\ref{eqn:rmspropAvg}.\nThe running average of the square of parameter derivatives in the denominator of $\\tau_k$ allows for the step size to decrease when the derivative is large, which should hedge against the possibility of taking excessively large steps.\nConversely, a smaller denominator when the derivative is small allows for larger steps to be taken.\nThe weighting in the running average is controlled by a factor $\\rho$ that can be thought of as the amount of memory retained of past gradients for adjusting $\\tau_k$, and $\\eta$ again denotes the chosen initial step size.\nIn order to avoid possible singularities when the gradient is very close to zero, a small positive number $\\epsilon$ is included in the denominator of $\\tau_k$.\nEquation \\ref{eqn:rmspropMomentum} shows the momentum effect in which the update for the parameter on the $k+1$ step depends on the update from the previous step as well as the current gradient.\nWe also follow the Booth group in applying a damping factor to the momentum by replacing $\\gamma_k$ with $\\gamma_k e^{-(\\frac{1}{d})(k-1)}$.\nThe quantity $d$ effectively controls how quickly the momentum is turned off, which eventually turns the algorithm into SGD.\nThe values of $d$,$\\eta$, $\\rho$, and $\\epsilon$ may all be chosen by the user of the algorithm and are known as hyperparameters in the machine learning literature.\nIn the results we present using this method, we have used $d=100$, $\\rho = .9$ and $\\epsilon = 10^{-8}$.\nWe have found adjusting these hyperparameters has relatively little influence on optimization performance compared to choices for step size $\\eta$, but their influence could be explored more systematically.\n\nThe Clark group's algorithm takes a far simpler form\n\\begin{equation}\n\\label{eqn:randomStep}\n    p_i^{k+1} = p_i^k - \\alpha \\eta \\frac{\\left|\n    \\frac{\\partial \\mathcal{L}}{\\partial p_i^k}\n    \\right|}{\\frac{\\partial \\mathcal{L}}{\\partial p_i^k}}\n\\end{equation}\nand has been recently used with neural network wave functions in the context of the Hubbard model.\\cite{Luo2018}\nHere $\\alpha$ is a random number in the interval $(0,1)$ and $\\eta$ sets the overall scale of the random step size.\nThe motivation for allowing the step size to be random is that it may help the optimization escape local minima that it encounters.\nWithin VMC, this algorithm can be run with fewer samples per iteration even compared to other gradient descent based algorithms as only the sign of the derivative needs to be known, but it typically requires many more iterations to converge.\n\nADAM and AMSGrad are popular methods within the machine learning community\\cite{Ruder2016,Reddi2018,Kingma} and have similar forms. ADAM is given by:\n\\begin{equation}\n\\label{eqn:adamUpdate}\n    p_i^{k+1} = p_i^{k} - \\eta \\frac{m_i^k}{\\sqrt{n_i^k}}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:adamMomentum}\n    m_i^k = (1-\\beta_1)m_i^{k-1} + \\beta_1 \\frac{\\partial \\mathcal{L}}{\\partial p_i^k}\n\\end{equation}\n\\begin{equation}\n\\label{eqn:adamStep}\n    n_i^k = \\beta_2 \\hspace{0.5mm} n_i^{k-1}\n            + (1-\\beta_2)\\bigg(\\frac{\\partial \\mathcal{L}}{\\partial p_i^k}\\bigg)^2\n\\end{equation}\nAMSGrad is a recent adaptive step size scheme developed in response to the limitations of ADAM \\cite{Reddi2018} and has almost the same form except for a slightly different denominator.\n\\begin{equation}\n\\label{eqn:amsgradStep}\n    n_i^k = \\mathrm{max}\\Bigg(n_i^{k-1}, \\hspace{2mm}\n                (1-\\beta_2) \\hspace{0.5mm} n_i^{k-1} +\n                \\beta_2 \\bigg(\\frac{\\partial \\mathcal{L}}{\\partial p_i^k}\\bigg)^2\\Bigg)\n\\end{equation}\n\nIn our calculations, we have used $\\beta_1 = 0.1$ and $\\beta_2 = 0.01$ for both AMSGrad and ADAM in line with the choice made by the Sharma group.\\cite{Sabzevari2018,Mahajan2019}\nIt may be worth noting that a different convention appears in machine learning literature using $1-\\beta_1$ and $1-\\beta_2$ for what we and the Sharma group call $\\beta_1$ and $\\beta_2$. \\cite{Ruder2016,Reddi2018}\n\nCompared to the LM, these first derivative descent methods have some significant advantages.\nTheir low memory usage and reduced nonlinear bias make them a natural fit for the large parameter sets that the LM struggles to handle.\nThey are remarkably robust in the presence of noise and do not need special safeguards against statistical instabilities such as the LM's shifts.\nAt a basic practical level, the descent methods are also far simpler to implement than the LM and especially its blocked variant.\nHowever, as we will see in our results, they often struggle\nto reach the vicinity of the minimum using a comparable sampling effort.\n\n\\subsection{A Hybrid Optimization Method}\nIn an attempt to retain the benefits of both the LM and the AD\ntechniques, we have developed a hybrid optimization scheme that can be applied to large numbers of parameters.\nOur approach alternates between periods of optimization using AD and sections using the BLM. \nAmong other advantages, this allows us to use gradient descent to identify the $N_o$ previous important directions in parameter space that are used in the BLM via equation \\ref{eqn:blmHistory}. \nThe precise mixture of both methods can be flexibly altered, but a concrete example would be to first optimize for 100 iterations using RMSprop.\nBy storing a vector of parameter value differences every 20 iterations, we would produce 5 vectors that can be used for equation \\ref{eqn:blmHistory} in some number (say three) steps of the BLM. \nAfter the execution of these BLM steps, the algorithm would return to another 100 iterations of descent and the process repeats until the minimum is reached.\nFigure \\ref{fig:hybridschematic} shows a generic depiction of how the ground state energy optimization may behave over the course of the hybrid method.\nThere are extended sections of computationally cheap optimization using gradient descent interwoven with substantial energy improvement over a few BLM steps.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{hybrid_opt.eps}\n    \\caption{Schematic depiction of a typical energy optimization using the hybrid method. The dashed box around a section of descent in green and BLM in red defines a macro-iteration of the method.}\n    \\label{fig:hybridschematic}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{Hybrid_contour.eps}\n    \\caption{Schematic representation of gradient descent corrections in green to the red BLM steps, which we have observed to produce a greater degree of uncertainty about the location of the final minimum.}\n    \\label{fig:hybridcontour}\n\\end{figure}\n\nThe use of AD and the BLM should naturally allow parameter sets beyond the traditional LM limit of about 10,000 variables to be addressed, a limit we will surpass in the present study in the diflurodiazene system.\nFor now, Table \\ref{tab:memCost} lays out how the memory cost of the methods we are considering scales with number of parameters $N$.\nBoth the hybrid method and the BLM steps it contains have a memory scaling that is intermediate between that of the standard LM and the descent methods.\nThe cost is given only approximately because while it is normally dominated by the cost of the $N_b$ blocks in the BLM, there are additional contributions related to how many directions are retained from the first BLM diagonalization and how many old directions are used.\\cite{Zhao2017}\n\n\\begin{table}[htbp]\n  \\centering\n  \\small\n  \\caption{Rough memory cost scaling for the optimization methods we examine, with $N$ the number of optimized parameters and $N_b$ the number of blocks}\n    \\begin{tabular}{llll}\n    Method Type & \\multicolumn{1}{l}{Memory Cost}        \\\\ \\hline\n                & \\\\\n    Standard Linear Method & $O(N^2)$ \\\\ \n               \n                & \\\\\n    Blocked Linear Method & $\\thicksim O\\left(\\frac{N^2}{N_b}\\right)$ \\\\ \n            \n             & \\\\\n    Hybrid Method & $\\thicksim O\\left(\\frac{N^2}{N_b}\\right)$ \\\\ \n             \n              & \\\\\n    Descent Methods & $O(N)$\n      \n    \\end{tabular}%\n  \\label{tab:memCost}%\n\\end{table}%\n\nOne key motivation for including sections of AD, especially when the method is near convergence, is to counteract the noise we observe in LM updates.\nWhile the LM tends to converge in a relatively small number of steps, we find the individual energies still fluctuate from iteration to iteration by multiple m$E_h$, particularly when we are working with wave functions that possess many highly nonlinear parameters.\nFigure \\ref{fig:hybridcontour} shows a cartoon of this behavior near the minimum\nthat prevents tight convergence.\nUnless the shifts are large enough to constrain it to very small steps, the LM will tend to bounce around near the true minimum due to substantial (and biased) statistical uncertainties in its step direction.\nThe resulting energy fluctuations lead to ambiguity in what to report as the definitive LM energy.\nOne could take the absolute lowest energy reached on any iteration, but this is fairly unsatisfactory as it feels too dependent on a \"lucky\" step landing right on the minimum.\nOur practice has been to take an average over multiple steps at the end of the optimization when parameter values should be converged.\nHowever, this will generally include iterations with upward energy deviations due to the step uncertainties.\nThe use of AD offers a way out of this dilemma because it can correct the errors in the LM steps by moving towards the minimum more smoothly and with less bias.\nAs we shall demonstrate in our results, these considerations seem to give the hybrid method a statistical advantage over the LM by achieving lower error bars for the same computational cost.\nThey are also the basis of our recommendation for finishing optimizations with a long section of pure AD, which we shall show tends to improve the energy and greatly diminish the final statistical uncertainty.\n\n\n\n\n\\subsection{Wave Functions}\n\nAn assessment of optimization methods' effectiveness requires consideration of the form of the wave function that they are applied to. \nMulti-Slater determinant wave functions have been a common choice of ansatz in QMC and are typically combined with Jastrow factors that help recover some electron correlation and describe particle cusps.\\cite{Foulkes2001} \nWe specify our Multi-Slater Jastrow (MSJ) wave function with the following set of equations.\n\n\\begin{equation}\n\\label{eqn:psi}\n    \\Psi = \\psi_{MS} \\psi_J \\psi_C\n\\end{equation}\n\\begin{equation}\n\\label{eqn:psiMS}\n    \\psi_{MS} = \\sum_{i=0}^{N_D} c_i D_i\n\\end{equation}\n\\begin{equation}\n\\label{eqn:psiJ}\n    \\psi_J = \\exp{\\sum_i \\sum_j \\chi_k(|r_i - R_j|) + \\sum_k \\sum_{l>k} u_{kl} (|r_k - r_l|)}\n\\end{equation}\n\n\\begin{equation}\n\\label{eqn:psiNCJF}\n    \\psi_C = \\exp(\\sum_{IJ} F_{IJ} N_I N_J + \\sum_K G_K N_K) \n\\end{equation}\n\nIn equation $29$ above, $\\psi_{MS}$ consists of $N_D$ Slater determinants $D_i$ with coefficients $c_i$.\nIt can be generated by some other quantum chemistry calculation such as complete active space self-consistent field (CASSCF) or a selective CI method prior to the VMC optimization.\nIn the one- and two-body Jastrow factor $\\psi_J$, we have\nfunctions $\\chi_k$ and $u_{kl}$, which are constructed from\noptimizable splines whose form is constrained so as to enforce\nany relevant electron-electron and electron-nuclear cusp conditions.\n\\cite{Kim2018}\n\nWhile MSJ wave functions with these types of traditional Jastrow\nfactors (TJFs) have been successfully used in many contexts,\n\\cite{Foulkes2001,Umrigar2007,Clark2011,Assaraf2017,Flores2019}\nmore involved correlation factors can be considered.\nTypically, this involves the construction of many-body Jastrows\nfactors, \\cite{Umrigar1988,Huang1997,Casula2003}\nwhich may involve various polynomials of interparticle distances \n\\cite{Huang1997,Luchow2015,LopezRios2012}\nor an expansion in an atomic orbital basis\n\\cite{Casula2003,Casula2004,Beaudet2008,Sterpone2008,\n      Marchi2009,Barborini2012,Zen2015}\nor a set of local counting functions. \\cite{Goetz2017,Goetz2018}\nThe latter case of many-body Jastrows, known as\nreal space number-counting Jastrow factors (NCJF),\nis employed here as an example many-body Jastrow factor.\nIn real space, Jastrow factors have historically been effective\nat encoding small changes to the wave function associated\nwith weak correlation effects, \\cite{Foulkes2001}\nbut work in Hilbert space and lattice model VMC\nreminds us that they can also be used to aid in the\nrecovery of strong correlations.\n\\cite{gutzwiller_gf,Neuscamman2013,Neuscamman2016}\nOne way to view NCJFs is as an attempt to develop a real space\nmany-body Jastrow factor that can aid in recovering both\nstrong and weak electron correlations.\n\\cite{Goetz2018}\n\nThe form of our NCJFs in equation \\ref{eqn:psiNCJF} has the same structure as previously proposed four-body Jastrow factors,\\cite{Marchi2009} where $N_I$ denotes the population of a region and the $F_{IJ}$ and $G_K$ are linear coefficients.\nThe region populations are computed by summing the values of counting functions at each electron coordinate.\n\\begin{equation}\n\\label{eqn:ncjfPop}\n    N_I = \\sum_i C_I (\\mathbf{r_i})\n\\end{equation}\nIn this work, we use a recently introduced \\cite{Goetz2018} form for the counting functions consisting of normalized Gaussians.\n\\begin{equation}\n\\label{eqn:ncjfCount}\n    C_I = \\frac{g_I(\\mathbf{r})}{\\sum_j g_j (\\mathbf{r})}\n\\end{equation}\nwhere \n\\begin{equation}\n\\label{eqn:ncjfGauss}\n    g_j(\\mathbf{r}) = \\exp ((\\mathbf{r} - \\mu)^T \\mathbf{A} (\\mathbf{r} - \\mu) + K)\n\\end{equation}\ndescribes a Gaussian about a center $\\mu$. \nBy placing these normalized Gaussians at various centers, we can divide up space with a Voronoi tessellation.\nSchemes have been developed to generate partitions that either consist of regions centered on atoms or of finer grained divisions of space that can capture correlation within an atomic shell.\nWe make use of both types of partitioning methods for different wave functions in our study.\nFor simplicity, we only consider optimization of the parameters $F_{IJ}$ in the $F$-matrix of our NCJFs (the coefficients $G_K$ can be eliminated with a basis transformation of the region populations $N_I$),\n\\cite{Goetz2018}\nbut in principle the parameters defining the Gaussians $g_j$ could also be optimized.\nWe provide details of the Gaussians used in our ansatzes in Appendix D.\n\nWe also consider the problem of optimizing the molecular orbital\nshapes alongside the other variational parameters. \nThe ability to relax orbitals is important for successful study of many systems, particularly those involving excited state phenomena.\\cite{Flores2019}\nWe make use of considerable theoretical and computational machinery based on the table method enhancements developed by Filippi and coworkers \\cite{Filippi2016,Assaraf2017} that enables efficient evaluation of orbital rotation derivatives in large MSJ wave functions.\nA rotation of molecular orbitals can be described with a unitary transformation with matrix $\\mathbf{U}$ parameterized as the exponential of an antisymmetric matrix $\\mathbf{X} = - \\mathbf{X^T}$\n\\begin{equation}\n\\label{eqn:unitaryMat}\n    \\mathbf{U} = \\exp (\\mathbf{X})\n\\end{equation}\nImpressively, one can obtain all wave function derivatives with\nrespect to the elements of $\\mathbf{X}$ for a large multi-Slater\ndeterminant ansatz for a cost that is only slightly higher than\nthat of the local energy evaluation.\nFor the details of how this is accomplished, we refer the reader\nto the original publications. \\cite{Filippi2016,Assaraf2017}\nFrom the standpoint of parameter optimization, the main significance\nof the orbitals (and the NCJFs) lies in both their nonlinearity\nand  their strong coupling to other optimizable parameters.\nIn practice, we find that turning on the optimization of orbitals\nand NCJFs greatly enhances the difficulty of the optimization problem\ncompared to MSJ optimizations in which only the CI \ncoefficients and one- and two-body Jastrow parameters are varied.\n\n\\section{Results}\n\n\\subsection{Multi-Slater Jastrow N$_2$}\n\\label{sec:n2simple}\n\nFor a small initial test system, we consider the nitrogen dimer $\\text{N}_2$ at the near-equilibrium and stretched bond lengths of 1.1 and 1.8 \\r{A}. \nThe nitrogen dimer is a known example of a strongly correlated system and a common testing ground for quantum chemistry methods.\\cite{Langhoff1974,Rossi1999,Chan2004,Braida2011,Mazziotti2004,Neuscamman2013,Neuscamman2016}\nThe initial wave function ansatz consists of a modest number of Slater determinants (67 for the equilibrium geometry and 169 for the stretched, the result of a 0.01 cutoff limit on determinant coefficients) with traditional one-body and two-body Jastrow factors.\nThe Jastrow splines provide 30 additional optimizable parameters via 10 point cubic b-splines with cutoff distances of 10 bohr for the electron-nuclear and same-spin and opposite-spin electron-electron components.\nThe Slater determinant expansion is the result of a (10e,12o) CASSCF calculation in GAMESS\\cite{Baldridge1993} using BFD pseudopotentials and the corresponding VTZ basis set.\\cite{Burkatzki2007}\nDue to the simplicity of the variable space in this case, we have\nemployed the $|\\Psi|^2$ guiding function for all optimization\nmethods, including the LM and BLM.\nSee Appendix B for further computational details.\n\nThe first and simplest study we can make is to optimize our ansatzes with our multiple optimization techniques until convergence and compare final energies.\nNote that all of our VMC optimizations with different methods in this study were performed using our implementations within a development version of the QMCPACK software package.\\cite{Kim2018}\nAs N$_2$ is a small enough system that the traditional LM can be easily \nemployed, we take the approach of first obtaining a traditional LM\noptimization result and then using it as a reference against which\nto compare the performance of other methods.\nFor the gradient descent methods, multiple optimizations were attempted \nwith the initial step sizes tweaked from run to run based on a rough \nexamination of how parameter values compared to the LM's results.\nWe find that the chosen values for the step sizes and other\nhyperparameters in the gradient descent algorithms often\nleads to apparent convergence at different energies.\nIt is therefore essential to make effective choices for these\nparameters, which in part seems to rely on one's experience\nwith a given system.\n\nFigures \\ref{fig:n2citjfeq} and \\ref{fig:n2citjfst} show energy differences relative to the LM result when optimizing the equilibrium and stretched nitrogen dimer wave functions respectively.\nTables providing the precise energies and statistical uncertainties as well as the step sizes used are shown in the appendices.\nFirst, we see the choice of step sizes can have a substantial influence on the quality of gradient descent results.\nIn some cases, the same method can appear to converge to energies more\nthan 20 m$E_h$ apart when run with different initial step sizes.\nWhile many of the gradient descent optimizations clearly did not reach\nthe minimum, the energy differences from the LM are only about 5 m$E_h$\nor less when looking at the runs that used what turned out to be the\nbest choices for the hyperparameters.\nWith further tweaking of the hyperparameters, we would guess that at least\nsome of these descent methods could match the performance of the LM\nin this simple test case.\nFinally, we observe that the hybrid method performs about as well\nas the best descent optimizations, typically reaching energies that\nagree with the LM within error bars.\n\n\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=8.3cm]{n2_tjfci_equilibrium.eps}\n    \\caption{Different methods' optimized energies relative to that of the\n         LM for equilibrium $\\text{N}_2$ when optimizing CI coefficients\n         and the TJF.\n         }\n    \\label{fig:n2citjfeq}\n\\end{figure}\n\n\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=8.3cm]{n2_tjfci_stretched.eps}\n    \\caption{Different methods' optimized energies relative to that of the\n         LM for stretched $\\text{N}_2$ when optimizing CI coefficients\n         and the TJF.\n         }\n    \\label{fig:n2citjfst}\n\\end{figure}\n\n\n\\subsection{All parameter N$_2$}\n\\label{sec:n2all}\n\nWe now add a NCJF and enable orbital optimization in order\nto extend the comparison in a setting with a larger number\nand variety of nonlinear parameters.\nWe will consider the relative merits of the optimization methods\nin much greater detail in this setting as it offers\na clearer view of their differences.\nFor the number-counting Jastrow factor, we generated a set of 16 counting regions with 8 octants per atom after dividing space in half with a plane bisecting the bond axis.\nThe details are given in Appendix D, but we will note here that\nthis adds 135  $F$-matrix parameters to the optimization.\nAllowing for orbital optimization adds another 663 and 618 parameters for the equilibrium and stretched cases, respectively.\nNote that our implementation of orbital optimization in QMCPACK removes rotation parameters for orbitals that are not occupied in any determinant and also between orbitals occupied in all determinants, and so the precise number of rotation parameters is a function of the determinant expansion.\nWith orbital optimization enabled, the choice of importance sampling\nfunction becomes an issue, and we now employ $|\\Phi|^2$ \nfor all LM and BLM steps with the $\\epsilon$ weight set to 0.001.\n\nFigures \\ref{fig:n2alleq} and \\ref{fig:n2allst} show converged ground state energies relative to that of the LM.\nFor this more difficult version of the nitrogen dimer, we find that the gradient descent methods are less effective.\nThey now often yield energies that can be 10 m$E_h$ or more above the LM's answer though we again find that choice of step size plays a significant role.\nThe worst results for AMSGrad and ADAM were the result of choosing inappropriately large step sizes and simple reductions in the initial step size produced improvements in energy of tens of m$E_h$ though the final result still remained well above the LM's.\nWhen we examined the optimizations over the course of their iterations, the gradient methods typically displayed some ability to quickly improve the wave function and energy initially, but they would then plateau and only very slowly improve the energy thereafter.\nExtrapolating from our results indicates that even if these gradient descent methods eventually converge to the minimum, they will only do so after thousands more iterations and at a computational cost well beyond that of the LM.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{n2_all_param_equilibrium.eps}\n\\caption{Different methods' optimized energies relative to that of the\n         traditional LM for equilibrium $\\text{N}_2$ when all parameters\n         are optimized simultaneously.\n         See also Table \\ref{tab:n2alleqData}.}\n    \\label{fig:n2alleq}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{n2_all_param_stretched.eps}\n    \\caption{Different methods' optimized energies relative to that of the\n         traditional LM for stretched $\\text{N}_2$ when all parameters\n         are optimized simultaneously.\n         See also Table \\ref{tab:n2allstData}.}\n    \\label{fig:n2allst}\n\\end{figure}\n\n\n\n\\begin{table}[htbp]\n  \\centering\n  \\footnotesize\n \n  \\caption{Energies, uncertainties, and sample numbers for optimization of all parameters in equilibrium N$_2$.}\n    \\begin{tabular}{llll}\n    Method & \\multicolumn{1}{l}{Energy (a.u.)}        & \\multicolumn{1}{l}{Uncertainty (a.u.)} & \\multicolumn{1}{l}{Samples} \\\\ \\hline\n \n    Hybrid  1& -19.9263        & 0.0004      &5,400,000 \\\\ \n      Hybrid  2& -19.9266        & 0.0004        &10,000,000\\\\\n         Hybrid  3& -19.9272        & 0.0004     &42,000,000\\\\ \n          &             &       &           \\\\\\hline\n    RMSprop 1& -19.8974        & 0.0005   & 20,000,000\\\\\n    RMSprop 2& -19.9242        & 0.0004 & 20,000,000\\\\\n          &       &       &  \\\\\\hline\n    AMSGrad 1& -19.9115      & 0.0006 & 20,000,000\\\\\n        AMSGrad 2& -19.9221        & 0.0004 & 20,000,000\\\\\n          &       &       &  \\\\\\hline\n    ADAM   1& -19.8973        & 0.0009 & 20,000,000\\\\\n          ADAM   2& -19.9165        & 0.0005     & 20,000,000 \\\\\n          &       &       &  \\\\\\hline\n    Random  1& -19.9019        & 0.0006 & 20,000,000\\\\\n          Random  2& -19.9157     & 0.0005 & 20,000,000 \\\\\n          &             &     & \\\\\\hline\n    LM    & -19.9280        & 0.0008     & 40,000,000\\\\\n     &             &     & \\\\\\hline\n     BLM    & -19.9297        & 0.0008    & 80,000,000\\\\\n      &             &     & \\\\\\hline\n      DF-BLM & -19.9293  &0.0001    & 90,000,000\\\\\n      DF-Hybrid  1& -19.9290  &0.0001  & 15,400,000\\\\\n      DF-Hybrid  2& -19.9290  &0.0001  & 20,000,000\\\\\n      DF-Hybrid  3& -19.9293  &0.0001  & 52,000,000\\\\\n      \n    \\end{tabular}%\n  \\label{tab:n2alleqData}%\n\\end{table}%\n\n\n\\begin{table}[htbp]\n  \\centering\n \n  \\footnotesize\n  \\caption{Energies, uncertainties, and sample numbers for optimization of all parameters in stretched N$_2$.}\n    \\begin{tabular}{llll}\n    Method & \\multicolumn{1}{l}{Energy (a.u.)}        & \\multicolumn{1}{l}{Uncertainty (a.u.)} & \\multicolumn{1}{l}{Samples}\\\\ \\hline\n       \n    Hybrid  1 & -19.6316       & 0.0004    & 5,400,000\\\\\n    Hybrid  2      & -19.6321        & 0.0004 & 8,400,000\\\\\n     Hybrid  3     & -19.6340       & 0.0004 & 49,200,000\\\\\n          &             &       &           \\\\ \\hline\n    RMSprop 1& -19.6277        & 0.0005      & 20,000,000\\\\\n      RMSprop 2    & -19.6313        & 0.0005 & 20,000,000\\\\\n          &       &       &  \\\\ \\hline\n    AMSGrad 1& -19.5571        & 0.0010 & 20,000,000\\\\\n     AMSGrad 2    & -19.6064        & 0.0008 & 20,000,000\\\\\n     AMSGrad 3    & -19.6268        & 0.0008 & 20,000,000\\\\\n          &         &           &           \\\\ \\hline\n    ADAM 1& -19.5889        & 0.0009 & 20,000,000\\\\\n     ADAM 2     & -19.6179        & 0.0006 & 20,000,000\\\\\n      ADAM 3    & -19.6192        & 0.0005 & 20,000,000\\\\\n          &         &           &           \\\\ \\hline\n    Random 1& -19.6112        & 0.0006 & 20,000,000\\\\ \n     Random 2       &-19.6196        & 0.0006 & 20,000,000\\\\\n          &              &       & \\\\ \\hline\n    LM    & -19.6356        & 0.0009 & 40,000,000\\\\\n     &              &       & \\\\ \\hline\n     BLM    & -19.6354        &0.0008 & 80,000,000\\\\\n     &              &       & \\\\ \\hline\n     DF-BLM & -19.6356  &0.0001 & 90,000,000\\\\\n      DF-Hybrid 1& -19.6352  &0.0001 & 15,400,000\\\\\n      DF-Hybrid 2& -19.6354  &0.0001 & 18,400,000\\\\\n      DF-Hybrid 3& -19.6346  &0.0001 & 59,200,000\\\\\n      \n    \\end{tabular}%\n  \\label{tab:n2allstData}%\n\\end{table}%\n\nA more careful comparison of the different methods can be made by referring to Tables \\ref{tab:n2alleqData} and \\ref{tab:n2allstData}, which list the precise converged energies and their error bars.\nWe also report the total number of samples used in each optimization as a proxy for computational effort, noting that for the BLM and the BLM portion of the hybrid method we double counted samples out of fairness as the BLM steps require running over their samples twice.\nIn assessing cost, one must also consider the statistical uncertainty achieved, where we see that the LM and BLM are at a disadvantage.\nTo help illustrate the update uncertainty contribution to this error, which we first discussed in the theoretical section above, we show the energy versus LM iteration for equilibrium N$_2$ in Figure \\ref{fig:n2lm}.\nThe fluctuations in energy from step to step, sometimes by as much as 2 m$E_h$, demonstrate the difficulty the LM faces from the uncertainty in its steps near the minimum.\nIn this case, we see that the LM's final energy uncertainty is driven by the update step uncertainty rather than the uncertainty in evaluating the energy for a given set of parameter values at a particular iteration.\nWe have observed similar behavior in the BLM and include the result of a BLM calculation in the tables.\nNote that the tabulated energies come from an average over the last ten optimization steps in the case of the standard LM and BLM and from an average over the last 50 descent steps in the case of the hybrid and pure descent methods.\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{LM_opt.eps}\n    \\caption{The standard LM optimization for all parameter equilibrium N$_2$.}\n    \\label{fig:n2lm}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.8cm]{n2_params.eps}\n    \\caption{\n    Values for the first off-diagonal $F$-matrix element ($F_{01}$),\n    the first electron-nuclear TJF spline parameter ($u_0$),\n    and the second orbital rotation variable ($X_{02}$) at each\n    micro iteration of the ``Hybrid 1'' optimization for\n    all parameter equilibrium N$_2$.\n    }\n    \\label{fig:n2param}\n\\end{figure}\n\nFrom the tabulated data, we see that the hybrid optimization can achieve\nlower energies than the gradient descent methods using fewer samples,\nand that its results are typically within a few m$E_h$ of the traditional LM.\nWhile the accelerated descent sections of the hybrid method provide\nsome swift energy reductions early on when the wave function is still far\nfrom the minimum in parameter space, the BLM steps in the algorithm\ngreatly accelerate the process of bringing the parameters near to the\nminimum, as can be seen in Figure \\ref{fig:n2param}.\nLooking at the electron-nuclear spline parameter and the orbital\nrotation parameter, we see typical\ncases in which rapid initial parameter movement during the early\npart of the first RMSprop stage transitions to much slower movement\nlater in that stage, followed by very little movement at all\nin later RMSprop stages.\nNote that the latter can be explained largely by the need to\nkeep initial step sizes small in later stages to avoid\nsignificant upward deviations in the energy as the RMSprop method\nrebuilds its momentum history.\nIn between these AD stages, the BLM updates move the parameter\nvalues in much larger steps, greatly accelerating convergence.\nThis behavior makes the hybrid approach somewhat more black box\nas compared to the pure descent approaches, as the ability to\nget near the minimum with a modest sampling effort is much\nless dependent on the choice of the initial step sizes than for\nthe AD methods.\nThis conclusion is supported by the fact that the hybrid\noptimizations in Tables \\ref{tab:n2alleqData} and \\ref{tab:n2allstData}\nused various initial step size settings (as discussed in Appendix B)\nand nonetheless  produced lower energies than the pure descent methods\nin every case.\n\nAs discussed in our introduction of the hybrid method, another advantage\nis its ability to obtain a lower error bar at convergence\nthan the LM for the same overall computational cost.\nThis is a natural consequence of spending part of its sampling effort\non gradient descent steps that correct for the BLM steps' uncertainty\nand bias (as illustrated earlier in Figure \\ref{fig:hybridcontour})\nand that hew closer to\nthe zero-variance principle by importance sampling with $|\\Psi|^2$.\nTo demonstrate this advantage explicitly, we ran additional sets\nof LM and hybrid optimizations adjusted to have essentially the same\ntotal number of samples.\nWe then compare the standard error for the last ten LM steps\nand the last ten hybrid macro iterations in \nFigures \\ref{fig:n2eqstderr} and \\ref{fig:n2ststderr},\nwhere we find that the hybrid has a substantially lower\nstatistical uncertainty in every case.\nAssuming the usual $N^{-1\/2}$ decay of uncertainty\nwith sample size, the LM would require a factor of roughly\nfour times more samples to reach the hybrid's uncertainty,\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{N2_eq_error.eps}\n    \\caption{Standard Errors for the hybrid method and LM on all parameter\n    equilibrium $\\text{N}_2$ vs different optimizations'\n    total sampling costs.}\n    \\label{fig:n2eqstderr}\n\\end{figure}\n\\begin{figure}\n    \\centering\n   \\includegraphics[width=8.3cm]{N2_st_error.eps}\n    \\caption{Standard Errors for the hybrid method and LM on\n    all parameter stretched $\\text{N}_2$ plotted against different optimizations'\n    total sampling costs.}\n    \\label{fig:n2ststderr}\n\\end{figure}\n\nThese statistical advantages in the final energy can be improved even further if we finish an optimization with a long section of pure descent.\nTo demonstrate this, we have taken the final wave functions produced by the hybrid and BLM optimizations in Tables \\ref{tab:n2alleqData} and \\ref{tab:n2allstData} and applied a further period of optimization using RMSprop with initial step sizes of 0.001 for all parameters.\nThis ``descent finishing'' (DF) adds only a modest additional cost compared to the preceding optimization and yields a large improvement in statistical uncertainty and, in many cases, an improvement in the final energy value as well.\nThese advantages can be seen clearly in Figures \\ref{fig:n2dfeq} and \\ref{fig:n2dfst}, as well as in Tables \\ref{tab:n2alleqData} and \\ref{tab:n2allstData},\nwhere we observe final error bars that are a factor of\neight smaller than those of the LM.\nIn terms of cost, this implies that the traditional LM would\nhave required 64 times the original number of samples\nto achieve the DF-BLM or DF-hybrid precision.\nPut another way, we find that the DF-hybrid\napproach gives an equivalent or lower energy, with a much\nsmaller error bar, at a substantially lower cost.\nNote that, in contrast, we find that this DF approach is not very effective\nwhen used in conjunction with the pure descent methods, where\nit essentially amounts to restarting the methods at the\nparameter values found after the first run of their optimization.\nWhile we do find that this restarting of the accumulation\nof momentum can improve the energy, the wave function\nparameters still do not reach their optimal values and the\nenergy lowering vs total sampling cost is not competitive\nwith the DF-hybrid.\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{eq_N2_descent_finish.eps}\n    \\caption{Converged energies in equilibrium $\\text{N}_2$ before and after a final descent optimization.}\n    \\label{fig:n2dfeq}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=8.3cm]{st_N2_descent_finish.eps}\n    \\caption{Converged energies in stretched $\\text{N}_2$ before and after a final descent optimization.}\n    \\label{fig:n2dfst}\n\\end{figure}\n\n\nOur study of the nitrogen dimer provides some clarity on the relative strengths of the LM and gradient descent, while also pointing the way to a more effective synthesis of the two.\nGradient descent methods struggle in the presence of a variety of different highly nonlinear parameters, although they did perform better when we were only optimizing TJFs and CI coefficients.\nAmong the descent methods, we found that the RMSprop approach came the\nclosest to achieving the LM minimum energy.\nIt is of course difficult to rule out the possibility that this\nand other AD methods could reach the LM energy\nwith additional sampling and more experimentation with\nthe hyperparameters.\nHowever, it is far from obvious that this would be\nbe cost-competitive, and the need to make careful and possibly\nsystem-specific choices for hyperparameters\nis somewhat antithetical to the general aspiration that an\noptimizer be as black-box as possible.\nFor its part, the LM is more effective at moving parameters\ninto the vicinity of the minimum, but tight convergence is then stymied by\nan unsatisfactory level of biased statistical uncertainty.\nAs a side note, this behavior --- in which the first derivative methods give\nbetter convergence once near the minimum but are at a relative disadvantage\nfar from the minimum --- is somewhat the reverse of what one would expect\nin deterministic optimization, where second derivative methods are at their\nstrongest relative to first derivative methods during the final tight\nconvergence in the vicinity of the minimum.\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=8.3cm]{eq_styrene_geometry.eps}\n    \\caption{Equilibrium geometry of styrene. See Appendix C for structure coordinates.}\n    \\label{fig:styrene_geom}\n\\end{figure}\nAlthough things are reversed in the stochastic VMC case, we stress that the\ntwo classes of methodology are strongly complementary, as they compensate\nfor each other's weaknesses.\nBy using a low-memory version (BLM or hybrid) of the LM to get near\nto the minimum and then handing off to an accelerated descent method\nto achieve tight convergence, we find better overall performance than when\nworking with either class of method on its own.\nThese insights in hand, we will now apply this combined approach in a pair of larger and more challenging VMC optimization examples.\n\n\\subsection{Styrene}\nWe first turn to styrene at its equilibrium geometry \n(Figure \\ref{fig:styrene_geom}) which offers an optimization with both\nmore electrons and more variables, but in which the traditional LM\nis still quite achievable for comparison.\nAs in N$_2$, we construct a multi-Slater wave function modified by both TJFs and a NCJF.\nTo generate our Slater determinants, we have employed the\nheatbath selective CI (HCI) method as implemented in the Dice\ncode by Sharma and coworkers. \\cite{Holmes2016,Sharma2017}\nThe orbital basis for the HCI calculation was produced via a (14e,14o) CASSCF calculation in Molpro \\cite{MOLPRO_brief} using a recently developed set of pseudopotentials and their corresponding double zeta basis.\\cite{Bennett2017}\nIn this CASSCF basis, HCI then correlated 32 electrons (out of a total of 40 electrons left over after applying pseudopotentials) in 64 orbitals.\nFor our NCJF, we defined one counting region per atom, giving our $F$-matrix 135 optimizable parameters (see Appendix D for further NCJF details).\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=8.3cm]{addOrbs_F_styrene_descent_finish.eps}\n    \\caption{Converged energies in equilibrium styrene before and after a final descent optimization.}\n    \\label{fig:styrene_energies}\n\\end{figure}\n\nWe optimized our wave function in a staged fashion using the standard LM, the BLM, and the hybrid method.\nFirst, we conducted a partial optimization of the TJFs and the 100 most important CI coefficients.\nWe then turned on the optimization of the orbitals and the NCJF's $F$-matrix, reaching a total of 4,570 parameters, most of them highly nonlinear.\nIn the hybrid and BLM optimizations, the parameters were divided into 5 blocks and we used $N_k = 50$ and $N_o = 5$ for our numbers of kept directions and previous important directions, respectively.\nWe used a value of 0.001 for $\\epsilon$ in the $|\\Phi|^2$ distribution for the LM and BLM sampling.\nThese optimizations were then followed by 1,000 steps of RMSprop.\nAs shown in Figure \\ref{fig:styrene_energies}, we find that our hybrid method reaches a converged energy as low or better than that of the standard and blocked LM, and finishing our optimizations with descent provides a substantial improvement in the statistical uncertainty even in this more challenging case.\n\n\n\n\\begingroup\n\\setlength{\\tabcolsep}{1.5pt}\n\\renewcommand{\\arraystretch}{1.25}\n\\begin{table}\n  \\centering\n  \\small\n  \\caption{A summary of the VMC optimization stages in FNNF showing\n           the number of determinants\n           $N_d$ included from HCI, which parameters are optimized,\n           and the total number $N_p$ of optimized parameters.\n           Note that CI coefficients are optimized at every stage.\n           Stages 2, 3, and 4 start from the parameter values from\n           the previous stage, with newly added determinants' coefficients\n           initialized to zero.\n           We also report the number of iterations performed in each stage,\n           which for stage 4 is simply the number of RMSprop steps.\n           A hybrid iteration, on the other hand, consists of 100 RMSprop steps\n           followed by three BLM steps.\n           All RMSprop steps use 20,000 samples drawn from $|\\Psi|^2$,\n           while the BLM steps each use 1 million samples drawn from the\n           $|\\Phi|^2$ guiding function with $\\epsilon$ set to 0.01.\n          }\n    \\begin{tabular}{cclcccrr}\n    Stage & Method & $N_d$ & TJF & $F$-matrix & Orbitals & $N_p$ & Iterations \\\\\n    \\hline\n    1 & Hybrid & $10^2$ & \\checkmark & & & 139 & 9 \\\\\n    2 & Hybrid & $10^3$ & \\checkmark & \\checkmark & & 1048 & 4 \\\\\n    3 & Hybrid & $10^4$ & \\checkmark & \\checkmark & \\checkmark & 15,573 & 6 \\\\\n    4 & AD     & $10^4$ & \\checkmark & \\checkmark & \\checkmark & 15,573 & 1,000 \\\\\n    \\end{tabular}%\n  \\label{tab:fnnf_stages}%\n\\end{table}%\n\\endgroup\n\n\n\\subsection{FNNF}\n\\label{sec:fnnf}\n\nWe now turn our attention to a strongly correlated transition\nstate of the the diflurodiazene (FNNF) \\textit{cis-trans}\nisomerization, where we test the hybrid optimization approach\non a much larger determinant expansion.\nThe FNNF isomerization  can be thought of as a toy model molecule\nfor larger systems such as photoswitches,\nwhich have potential uses in molecular machines \\cite{Russew2010,Kinbara2005}\nand high-density memory storage. \\cite{tian2004switches}\nIn addition, FNNF itself is of interest as part of the synthesis\nof high energy polynitrogen compounds and has been the subject of\nmultiple electronic structure studies.\\cite{Christe1991,Christe2010,Lee1989}\nHere we focus on its strongly correlated transition state, which is\nthe direct analogue of the out-of-plane TS1 transition state \nin diazene. \\cite{Sand2012}\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=18.5cm]{fnnf_plot.eps}\n    \\caption{Left panel: energies during stages 3 and 4 of the FNNF\n             optimization.  The descent energies are reported as the\n             average over the last 50 RMSprop steps within each block\n             of 100 RMSprop steps, whereas the BLM energies are\n             the energy estimates on the random samples used for the\n             BLM update steps.\n             Right panel:  change in the value of the $F$-matrix\n             parameter that couples the two nitrogen atoms' counting\n             regions over the first three macro iterations\n             of stage 3, with each micro iteration corresponding to\n             one RMSprop or BLM step.\n             The nine BLM points on the right panel correspond to\n             the first nine BLM points on the left panel.\n            }\n    \\label{fig:fnnf_plot}\n\\end{figure*}\n\nOur treatment of this transition state began by locating its\ngeometry via an (8e,8o) CASSCF optimization in the cc-pVTZ basis\nusing Molpro.\\cite{MOLPRO_brief}\nAt this geometry (given in Appendix C)\nwe then switch over to using BFD pseudopotentials\nand their corresponding triple zeta basis,\n\\cite{Burkatzki2007}\nin which we use the Dice code\n\\cite{Holmes2016,Sharma2017}\nto iterate an HCI calculation\nwith 24 electrons\ndistributed in the lowest 50 (8e,8o) CASSCF\norbitals until its variational wave function\nhas reached almost 2 million determinants.\nWe then import the first 10,000 of these determinants\ninto our VMC optimization and combine them with\nTJFs, atom-centered NCJFs, and orbital optimization,\nwhich produces an ansatz with over 15,000 variational parameters.\n\nOur VMC optimization proceeds in stages as summarized in\nTable \\ref{tab:fnnf_stages}.\nThis begins with TJFs and a 100-determinant ansatz from HCI,\nwith later optimization stages adding\nmore determinants and turning on the optimization of\nthe NCJF and orbital rotation variables.\nAs in styrene, the strategy is to bring the parameters\nnear to their optimal values with the help of the LM\nand then to perform a final\nunbiased relaxation via a long run of RMSprop AD.\nDue to the large number of variational parameters,\nwe incorporate the LM via the hybrid scheme, with the BLM\nsteps employing 2, 2, and 10 blocks\nduring stages 1, 2, and 3, respectively.\nIn stages 1 and 2, we used an initial RMSprop step size of 0.01 for TJFs and CI coefficients\nbefore setting it to 0.005 at the beginning of stage 3.\nFor the $F$-matrix parameters, we began by setting the initial\nstep size to 0.001,\nbut after observing a significant rise and fall of the energy\nduring the RMSprop section of the first hybrid macro iteration\nin stage 3, we reduced this to 0.0001 and also lowered the TJF and CI step size to 0.0005\nfor the last 4 macro iterations in that stage.\nFor all steps in stage 4, we maintained the 0.0001 step size for the $F$-matrix parameters and lowered the initial step size for TJFs and CI coefficients to 0.0002.\nAn initial step size of 0.0001 was used for orbital parameters throughout both stages 3 and 4.\nThe BLM steps used the $|\\Phi|^2$ guiding function with a value of 0.01 for $\\epsilon$.\n\n\\begin{table}\n  \\centering\n  \\small\n  \\caption{Energies of the transition state of FNNF.}\n    \\begin{tabular}{llll}\n    Method & \\multicolumn{1}{l}{Energy (a.u.)} &       & \\multicolumn{1}{l}{Uncertainty (a.u.)} \\\\\n    \\hline\n   Hartree-Fock &  -67.112730 & &  \\vphantom{\\big(\\big)} \\\\\n    CASSCF & -67.359100 & &  \\vphantom{\\big(\\big)} \\\\\n      \n       VMC Stage 1 &  -68.1017 & & 0.0011 \\vphantom{\\big(\\big)} \\\\\n      \n       VMC Stage 2 & -68.1213 &  &0.0009 \\vphantom{\\big(\\big)} \\\\\n   \n    VMC Stage 3 & -68.1698 &       & 0.0006 \\vphantom{\\big(\\big)} \\\\\n     \n      VMC Stage 4 \\hspace{2mm} & -68.1750 & &0.0002 \\vphantom{\\big(\\big)} \\\\\n    \\end{tabular}%\n  \\label{tab:fnnf_energies}%\n\\end{table}%\n\nThe energies resulting from this staged optimization\nare shown in Table \\ref{tab:fnnf_energies} and Figure \\ref{fig:fnnf_plot}.\nUnsurprisingly, stage 3 proved to be the most challenging and expensive\nstage, as it is where we hope to move all parameters near to their final\nvalues in a setting where the traditional LM would face severe memory\nbottlenecks.\nAs seen in Figure \\ref{fig:fnnf_plot}, both the AD and BLM steps\nclearly work to lower the energy during the first two macro iterations of\nstage 3.\nIn the last four macro iterations of stage 3, however, the energy decreases\nmore slowly and it is less clear, at least when looking at the energetics,\nwhether the BLM steps are still necessary.\nInstead, their importance is revealed by inspecting the movement of\nthe $F$-matrix values within the NCJF, an example of which is shown\nin the right-hand panel of Figure \\ref{fig:fnnf_plot}.\nAs in N$_2$, these parameters prove to be the most resistant to optimization\nvia AD, and we clearly see that although AD does gradually move their values\nin the same direction as the BLM, the BLM steps dramatically accelerate\ntheir optimization.\nThis effect is seen throughout all six macro iterations of stage 3, and\nso although the BLM energies are not obviously improving at the end of this\nstage, the inclusion of these steps is clearly still beneficial.\nNote that relaxing the NCJF after moving from a 1,000-determinant to a\n10,000-determinant expansion is important, because the larger determinant\nexpansion is better able to capture some of the correlation effects\nthat the NCJF is encoding, and so we expect (and indeed see) that this\ndiminishing of its role leads to smaller $F$-matrix values being optimal.\n\nAlthough we have again found that it would be difficult for AD alone\nto provide a successful optimization of our ansatz, the statistical\nadvantages of its incorporation are still quite clear.\nA close inspection of the sample sizes used in the optimization\nreveals that each of the AD and BLM points in the left panel of\nFigure \\ref{fig:fnnf_plot} corresponds to averaging over 1 million\nrandom samples.\nDespite this equal sampling effort, the uncertainties for the\nAD energy estimates are about one third the size of those for the\nBLM, implying that a pure BLM approach would require an order of\nmagnitude more sampling effort to produce similar results.\nTo understand this statistical advantage, we need to remember\ntwo important differences between the AD and BLM steps.\nFirst, the nonlinearity of the LM and BLM eigenvalue problem\nleads to biases in the update steps that can both increase the\nstep-to-step energy uncertainty and cause the method to optimize\noff-center from the true minimum.\nSecond, the use of an alternative guiding function for the\nBLM samples in order to mitigate this step uncertainty moves\nus away from the zero-variance regime enjoyed by traditional\n$|\\Psi|^2$ sampling.\nIf we were to instead employ traditional sampling, our energy\nestimates for a specific wave function would improve,\nbut the BLM step uncertainty would increase sharply.\nAs the AD methods do not suffer from these issues, they help us to\nfurther mitigate the BLM step uncertainty and to perform\na final, high-precision relaxation during stage 4.\nIn total, incorporating the AD steps in this case roughly\ndoubles the number of samples required, but is well worthwhile given\nthat it improves statistical efficiency by almost an order of magnitude.\n\nWhile it is possible that the NCJF parameters are not quite converged\nin this particular optimization\nand that increasing iteration counts in stages 1 through 3 could\nfurther improve the energy, the lessons learned from investigating\na large MSJ optimization for the FNNF transition state are already clear.\nWhile both the BLM and the AD methods can be used in this 10,000+ parameter\nregime, they bring highly complementary advantages to the optimization\nand so would appear to work better together than apart.\nIn particular, the BLM helps optimize the parameters that change only\nvery slowly during AD, whereas the statistical advantages of AD\ngreatly increase precision at a given sample size and work to\neliminate the statistical biases suffered by the BLM.\n\n\\section{Conclusions and Outlook}\n\\label{sec:conclusion}\n\nWe have found that a combination of first and second derivative\noptimization methods appears to work better than using either\nclass of method on its own when minimizing the energies of\nwave functions in variational Monte Carlo.\nThis is particularly true for wave functions with a wide\nvariety of different types of nonlinear parameters, as for\nexample when dealing simultaneously with traditional\none- and two-body Jastrow factors, many-body Jastrow factors,\nand orbital relaxations.\nWhile the linear method and its low-memory variants show a\nsuperior ability to move these nonlinear parameters into\nthe vicinity of their optimal values, accelerated descent\nmethods prove much more capable of converging them tightly\naround the minimum.\nThis situation stands as an interesting reverse of what\nis typically encountered in deterministic optimization,\nwhere second derivative methods are usually superior\nfor tight final convergence and first derivative methods\nperform relatively better in the early stages of an\noptimization.\nThe realities of working with statistically uncertain\nenergies and energy derivatives turns this expectation\non its head, both because of the need to stabilize the\nstatistics of the linear method's second derivative\nelements through zero-variance-violating importance\nsampling schemes and due to the nonlinear biases that\nare induced when solving the linear method's eigenvalue\nproblem.\nThe linear method's ability to quickly move the parameters\nnear the minimum, however, makes it appear that employing\nit as part of a hybrid approach is well worthwhile.\nIndeed, in our testing, hybridizing low-memory\nlinear method variants with accelerated descent methods\nprovides better energies with smaller\nstatistical uncertainties at a lower computational cost\nwhen compared to the stand-alone use of either the linear method\nor accelerated descent methods.\n\nLooking forward, there are many questions still to be answered\nabout the interplay between first and second derivative methods.\nFor example, although the blocked linear method greatly reduces\nmemory cost vs the traditional linear method, it is not clear that\nit can be applied effectively beyond 100,000 parameters in its\ncurrent form.\nOne thus wonders whether it is necessary to optimize all of the\nparameters during the linear method steps of the hybrid\napproach, or whether it may be possible to identify\n(perhaps during an ongoing optimization?) which parameters\nwould benefit from linear method treatment and which would not.\nWere such a sorting possible, accelerated descent methods with\ntheir even lower memory footprint could be left to deal with\nmost of the parameters, with only a relatively small subset\ntreated by the linear method steps.\nAnother important issue is making the hybrid approach as black\nbox and user friendly as possible.\nAlthough we have tested it here with many different descent\nstep size settings for the different parameter types, this has\nnot been a systematic survey.\nMore extensive testing may allow clear defaults to be settled\nupon so that users can reasonably expect a successful\noptimization without resorting to careful step size control.\nWe look forward to investigating these exciting possibilities\nin future.\n\n\n\\section{Acknowledgements}\nThis work was supported by the Office of Science,\nOffice of Basic Energy Sciences, the US Department of Energy,\nContract No.\\ {DE-AC02-05CH11231}.\nCalculations were performed using the Berkeley Research Computing\nSavio cluster and\nthe National Energy Research Scientific Computing Center,\na DOE Office of Science User Facility supported by the\nOffice of Science of the U.S. Department of Energy\nunder Contract No.\\ {DE-AC02-05CH11231}.\n\n\\section{Appendix A: Additional Energies}\n\n\\begin{table}[H]\n  \\centering\n  \\footnotesize\n  \\caption{Precise Values for optimizing CI coefficients and traditional Jastrow factors in equilibrium N$_2$.}\n    \\begin{tabular}{llll}\n    Method & \\multicolumn{1}{l}{Energy (a.u.)}        & \\multicolumn{1}{l}{Uncertainty (a.u.)} & \\multicolumn{1}{l}{Samples}\\\\ \\hline\n    \n    Hybrid 1& -19.9083        & 0.0005    & 3,900,000\\\\\n     Hybrid 2     & -19.9095        & 0.0006  & 5,400,000\\\\\n     Hybrid 3     & -19.9101        & 0.0005      & 29,000,000\\\\\n          &         &           &           \\\\ \\hline\n    RMSprop 1& -19.8936        & 0.0006   & 20,000,000\\\\\n    RMSprop 2& -19.9091        & 0.0005 & 20,000,000\\\\\n          &       &       &  \\\\\\hline\n    AMSGrad 1& -19.8998        & 0.0007 & 20,000,000\\\\\n    AMSGrad 2      & -19.8926        & 0.0008 & 20,000,000\\\\\n    AMSGrad 3      & -19.9048        & 0.0006 & 20,000,000\\\\\n     AMSGrad 4     & -19.9071        & 0.0005 & 20,000,000\\\\\n          &             &       & \\\\ \\hline\n    ADAM 1& -19.9009        & 0.0005 & 20,000,000\\\\\n     ADAM 2     & -19.9056        & 0.0006 & 20,000,000\\\\\n     ADAM 3     & -19.9078        & 0.0005 & 20,000,000\\\\\n          &         &           &           \\\\ \\hline\n    Random 1& -19.8926        & 0.0006 & 20,000,000\\\\\n     Random 2   &-19.9041        &0.0005 & 20,000,000 \\\\\n          &       &        & \\\\\\hline\n    Linear Method    & -19.9105        & 0.0010 & 40,000,000 \\\\\n    \\end{tabular}%\n  \\label{tab:n2tjfcieqData}%\n\\end{table}%\n\n\\begin{table}[H]\n\\footnotesize\n  \\centering\n  \\caption{Precise Values for optimizing CI coefficients and traditional Jastrow factors in stretched N$_2$.}\n    \\begin{tabular}{llll}\n    Method & \\multicolumn{1}{l}{Energy (a.u.)}        & \\multicolumn{1}{l}{Uncertainty (a.u.)} & \\multicolumn{1}{l}{Samples}\\\\ \\hline\n   \n    Hybrid 1& -19.6141        & 0.0005    & 27,000,0000\\\\ \n          &       &       &  \\\\ \\hline\n    RMSprop 1& -19.6010        & 0.0007 & 20,000,000\\\\\n     RMSprop 2& -19.6137        & 0.0005 & 20,000,000\\\\\n          &       &       &  \\\\ \\hline\n    AMSGrad 1& -19.6044        & 0.0006 & 20,000,000\\\\\n    AMSGrad 2      & -19.5858        & 0.0007 & 20,000,000\\\\\n     AMSGrad 3     & -19.6096        & 0.0006 & 20,000,000\\\\\n          &              &      &  \\\\\\hline\n    ADAM 1& -19.6028        & 0.0006 & 20,000,000\\\\\n    ADAM 2      & -19.6105        & 0.0005 & 20,000,000\\\\\n          &       &       &  \\\\ \\hline\n    Random 1& -19.5937        & 0.0006 & 20,000,000\\\\\n    Random 2        &-19.6147        &0.0006 & 20,000,000 \\\\\n          &             &        &  \\\\ \\hline\n    Linear Method    & -19.6155        & 0.0010 & 40,000,000 \\\\\n    \\end{tabular}%\n  \\label{tab:n2tjfcistData}%\n\\end{table}%\n\n\\section{Appendix B: Details for N$_2$ Optimizations}\n\nThe details for the optimizations behind the final energies in the main paper are presented below.\nIn every case of N$_2$, all flavors of pure gradient descent based optimization were run for 2000 iterations at 10,000 samples per iteration except for the random step size method, which was run for 10,000 iterations at 2000 samples per iteration.\nThe total sampling cost was then 20 million samples, half of the standard linear method's cost of 40 million over 40 steps.\nGiven the descent methods' tendency to plateau after a few hundred iterations and then lower the energy by only a few m$E_h$ afterward, we expect that running them longer to fully match or exceed the linear method's sampling effort would not yield a much better result in most cases.\nWe found that it was often advantageous to allow for different types of parameters to be given different initial step sizes.\nTables \\ref{tab:n2tjfcieqStep} through \\ref{tab:n2allstStep} list the step sizes for different descent optimizations in all cases of N$_2$.\nThe values can be cross-referenced with the energy results in earlier tables to see which choices were most effective.\nSome amount of experimentation was necessary to build up intuition for what choices are effective, but we generally expect more nonlinear parameters such as those in the $F$-matrix and the orbitals to require smaller step sizes.\nWe also found that RMSprop benefited from using larger step sizes compared to other descent algorithms.\nThe energy may be significantly raised on early iterations, but tends to be quickly lowered and eventually brought to an improved result once enough gradient history has built up over more steps. \n\nThe details of the different hybrid method optimizations are slightly more involved and are discussed separately here.\nIn all cases, the blocked linear method steps of the hybrid optimization used 5 blocks with 5 directions from sections of RMSprop to provide coupling to variables outside a block and retained 30 directions from each block to construct the final space for determining the parameter update.\nThese were also the settings given to the blocked linear method optimizations of N$_2$ that appear in the main text.\nAll hybrid optimizations used the RMSprop method for their AD sections with the hyperparameters $d=100$, $\\rho = .9$ and $\\epsilon = 10^{-8}$.\nThe step sizes used in the AD sections varied over the course of the hybrid optimizations.\nWe typically chose larger step sizes for the AD portion of the first macro-iteration in order to obtain more energy and parameter improvement at a low sampling cost before any use of the BLM and these are tabulated separately as \"Hybrid-Initial\".\nThe smaller step sizes reported for the rest of hybrid optimization were used in the later macro-iterations to avoid rises in the energy that might occur before a sufficient gradient history was accumulated.\nWe also list the step sizes in the long RMSprop optimization used to achieve the descent finalized energies.\nThese were sometimes larger than those for the AD sections of the initial hybrid optimizations because the descent finalization was long enough for any early transient rises in the energy to recover.\n\nWe now specify how the hybrid method sampling costs reported in Tables \\ref{tab:n2alleqData} and \\ref{tab:n2allstData} of the main paper were divided between AD and the BLM. In all parameter equilibrium N$_2$, Hybrid 1 consisted of 500 AD steps costing 3 million samples interwoven with 12 BLM steps that cost 2.4 million samples.\nHybrid 2 consisted of the same sequence of steps, but had an increased sampling effort of 6 million samples on descent and 2.4 million on BLM.\nHybrid 3 had a greatly increased sampling cost and consisted of 1400 AD steps for 11.2 million samples interwoven with 19 BLM steps costing 38 million.\nFor all parameter stretched N$_2$, Hybrid 1 used the same sequence of steps and sampling cost breakdown as Hybrid 1 for the equilibrium case.\nHybrid 2 consisted of 600 AD steps that cost 7 million samples and 15 BLM steps that cost 3 million.\nHybrid 3 also had 600 AD steps, now using 12 million samples, and 15 BLM steps, now using 30 million samples.\nFor all descent finalizations in N$_2$, we used 1000 steps of RMSprop at an additional cost of 10 million samples and took an average over the last 500 steps to obtain our reported energies and error bars.\n\nFinally, we give the breakdown of the hybrid sampling costs in Tables \\ref{tab:n2tjfcieqData} and \\ref{tab:n2tjfcistData} of Appendix A.\nFor the equilibrium case, Hybrid 1 had 500 AD steps costing 1.5 million samples and 12 BLM steps costing 2.4 million samples.\nHybrid 2 had the same combination of steps and BLM cost as Hybrid 1 while the AD steps used 3 million samples.\nHybrid 3 used the same sequence of steps, but increased the AD and BLM sampling costs to 5 million and 24 million, respectively.\nIn the stretched case, the hybrid optimization used 500 AD steps with 3 million samples and 12 BLM steps costing 24 million samples.\n\n\\begin{table}[H]\n\\footnotesize\n    \\caption{Step sizes for TJFCI equilibrium N$_2$ optimizations.}\n    \\centering\n    \\begin{tabular}{lccc}\n         Method & 2-Body TJF & 1-Body TJF & CI \n   \\\\ \\hline\n  RMSprop 1 & 0.01  & 0.01  & 0.005 \\\\\n   RMSprop 2 & 0.05  & 0.05  & 0.01 \\\\\n        \n            &       &       &        \\\\ \\hline\n    AMSGrad 1 & 0.05  & 0.05  & 0.01 \\\\\n    AMSGrad 2 & 0.05  & 0.05  & 0.05 \\\\\n    AMSGrad 3 & 0.01 & 0.01 & 0.01\\\\\n    AMSGrad 4 & 0.001 & 0.001 & 0.001\\\\\n        \n            &       &       &        \\\\ \\hline\n    ADAM 1 & 0.01  & 0.01  & 0.01\\\\\n    ADAM 2 & 0.005  & 0.005  & 0.005  \\\\\n      ADAM 3 & 0.001  & 0.001  & 0.001  \\\\\n        \n            &       &       &         \\\\ \\hline\n    Random 1 & 0.01 & 0.01 & 0.01\\\\\n    Random 2 & 0.0005 & 0.0005 & 0.0005 \\\\\n    &       &       &         \\\\ \\hline\n    Hybrid-Initial 1 & 0.1  & 0.1  & 0.01\\\\\n    Hybrid-Initial 2 & 0.1  & 0.1  & 0.01  \\\\\n      Hybrid-Initial 3 & 0.1 & 0.1  & 0.01  \\\\\n        &       &       &         \\\\ \\hline\n     Hybrid 1 & 0.001  & 0.001  & 0.001\\\\\n    Hybrid 2 & 0.005  & 0.005  & 0.005  \\\\\n      Hybrid 3 & 0.005  & 0.005  & 0.005  \\\\\n    \\end{tabular}\n    \\label{tab:n2tjfcieqStep}\n\\end{table}\n\n\\begin{table}[H]\n\\footnotesize\n    \\caption{Step sizes for TJFCI stretched N$_2$ optimizations.}\n    \\centering\n    \\begin{tabular}{lccc}\n         Method & 2-Body TJF & 1-Body TJF & CI \n      \\\\ \\hline\n        RMSprop 1 & 0.01  & 0.01  & 0.005 \\\\\n      RMSprop 2 & 0.05  & 0.05  & 0.01 \\\\\n        \n            &       &       &        \\\\ \\hline\n    AMSGrad 1 & 0.05  & 0.05  & 0.01 \\\\\n    AMSGrad 2 & 0.05  & 0.05  & 0.05 \\\\\n    AMSGrad 3 & 0.01 & 0.01 & 0.01\\\\\n       \n            &       &       &        \\\\ \\hline\n    ADAM 1 & 0.01  & 0.01  & 0.01\\\\\n    ADAM 2 & 0.005  & 0.005  & 0.005  \\\\\n       \n            &       &       &         \\\\ \\hline\n    Random 1 & 0.01 & 0.01 & 0.01\\\\\n    Random 2 & 0.001 & 0.001 & 0.001 \\\\\n    &       &       &         \\\\ \\hline\n    Hybrid-Initial 1 & 0.1 & 0.1 & 0.1\\\\\n    Hybrid 1 & 0.005 & 0.005 & 0.005\\\\\n    \\end{tabular}\n    \\label{tab:n2tjfcistStep}\n\\end{table}\n\\begin{table}[H]\n\\scriptsize\n    \\caption{Step sizes for all parameter equilibrium N$_2$ optimizations.}\n    \\centering\n    \\begin{tabular}{lccccc}\n         Method & 2-Body TJF & 1-Body TJF & $F$-Matrix & CI & Orbitals\n  \\\\   \\hline\n    RMSprop 1& 0.005  & 0.005  & 0.001  & 0.001  & 0.001 \\\\\n    RMSprop 2& 0.05  & 0.05  & 0.01  & 0.01  & 0.01 \\\\\n       \n          &       &       &       &       &  \\\\ \\hline\n    AMSGrad 1 & 0.05  & 0.05  & 0.005 & 0.01  & 0.001 \\\\\n    AMSGrad 2 & 0.005 & 0.005 & 0.001 & 0.001 & 0.001 \\\\\n       \n          &       &       &       &       &  \\\\ \\hline\n    ADAM 1 & 0.05  & 0.05  & 0.005 & 0.01  & 0.001 \\\\\n    ADAM 2 & 0.005 & 0.005 & 0.001 & 0.001 & 0.001 \\\\\n       \n          &       &       &       &       &  \\\\ \\hline\n    Random 1 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    Random 2 & 0.001 & 0.001 & 0.0005 & 0.001 & 0.0005 \\\\\n    &       &       &       &       &  \\\\ \\hline\n    Hybrid-Initial 1 & 0.1 & 0.1 & 0.0001 & 0.01 & 0.01 \\\\\n    Hybrid-Initial 2 & 0.1 & 0.1 & 0.0005 & 0.01 & 0.01 \\\\\n    Hybrid-Initial 3 & 0.01 & 0.01 & 0.001 & 0.01 & 0.001 \\\\\n    &       &       &       &       &  \\\\ \\hline\n    Hybrid 1 & 0.0001 & 0.0001 & 0.0001 & 0.0001 & 0.0001 \\\\\n    Hybrid 2 & 0.001 & 0.001 & 0.0005 & 0.0005 & 0.0005 \\\\\n    Hybrid 3 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    &       &       &       &       &  \\\\ \\hline\n    DF-Hybrid 1 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    DF-Hybrid 2 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    DF-Hybrid 3 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    \\end{tabular}\n    \\label{tab:n2alleqStep}\n\\end{table}\n\n\\begin{table}[H]\n\\scriptsize\n    \\caption{Step sizes for all parameter stretched N$_2$ optimizations.}\n    \\centering\n    \\begin{tabular}{lccccc}\n         Method & 2-Body TJF & 1-Body TJF & $F$-Matrix & CI & Orbitals\n  \\\\   \\hline\n      RMSprop 1 & 0.05  & 0.05  & 0.05  & 0.01  & 0.01 \\\\\n    RMSprop 2 & 0.1   & 0.1   & 0.01  & 0.01  & 0.005 \\\\\n       \n            &       &       &       &       &  \\\\ \\hline\n    AMSGrad 1 & 0.05  & 0.05  & 0.05  & 0.01  & 0.01 \\\\\n    AMSGrad 2 & 0.05  & 0.05  & 0.01  & 0.02  & 0.001 \\\\\n    AMSGrad 3 & 0.005 & 0.005 & 0.001 & 0.002 & 0.001 \\\\\n       \n            &       &       &       &       &  \\\\ \\hline\n    ADAM 1 & 0.05  & 0.05  & 0.05  & 0.01  & 0.01 \\\\\n    ADAM 2 & 0.05  & 0.05  & 0.01  & 0.02  & 0.001 \\\\\n    ADAM 3 & 0.005 & 0.005 & 0.001 & 0.002 & 0.001 \\\\\n       \n            &       &       &       &       &  \\\\ \\hline\n    Random 1 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    Random 2 & 0.001 & 0.001 & 0.0005 & 0.001 & 0.0005 \\\\\n     &       &       &       &       &  \\\\ \\hline\n     Hybrid-Initial 1 & 0.1 & 0.1 & 0.01 & 0.01 & 0.001 \\\\\n    Hybrid-Initial 2 & 0.1 & 0.1 & 0.01 & 0.01 & 0.001 \\\\\n    Hybrid-Initial 3 & 0.1 & 0.1 & 0.01 & 0.01 & 0.001 \\\\\n    &       &       &       &       &  \\\\ \\hline\n    Hybrid 1 & 0.0001 & 0.0001 & 0.0001 & 0.0001 & 0.0001 \\\\\n    Hybrid 2 & 0.001 & 0.001 & 0.0005 & 0.0005 & 0.0005 \\\\\n    Hybrid 3 & 0.001 & 0.001 & 0.0005 & 0.0005 & 0.0005 \\\\\n    &       &       &       &       &  \\\\ \\hline\n    DF-Hybrid 1 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    DF-Hybrid 2 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    DF-Hybrid 3 & 0.001 & 0.001 & 0.001 & 0.001 & 0.001 \\\\\n    \\end{tabular}\n    \\label{tab:n2allstStep}\n\\end{table}\n\n\n\\section{Appendix C: Molecular Geometries}\n\n\\begin{table}[H]\n     \\caption{Structure of equilibrium styrene. Coordinates in \\r{A}.}\n     \\centering\n     \\begin{tabular}{lrrr}\n       \\multicolumn{1}{l}{} &  &  & \\\\ \\hline\n C     &    1.39295   &  0.00000    &  0.00000 \\\\\nC      &    2.16042   & -1.19258    &  0.01801 \\\\\nC      &    2.09421 &    1.23178    & -0.01914 \\\\\nC       &   3.56585  &  -1.15969    &  0.05286 \\\\\nC        &  3.50142   &  1.27211    &  0.01795 \\\\\nC        &  4.23686   &  0.07503    &  0.06081 \\\\\nC         & 0.00000   &  0.00000    &  0.00000 \\\\\nC   &      -0.79515   & -0.93087    &  0.54406 \\\\\nH    &      1.71222   & -2.11161    & -0.00239 \\\\\nH     &     1.59237   &  2.12471    & -0.04753 \\\\\nH      &    4.09818   & -2.03273    &  0.07153 \\\\\nH       &   3.99086   &  2.16987    &  0.01692 \\\\\nH   &       5.25794   &  0.10043    &  0.09503 \\\\\nH    &     -0.46324   &  0.77112    & -0.42775 \\\\\nH     &    -0.43431   & -1.72147    &  1.02278 \\\\\nH      &   -1.78240   & -0.84577    &  0.49296 \\\\\n       \\hline\n     \\end{tabular}\n     \\label{tab:styreneGeo}\n   \\end{table}\n\n\\begin{table}[H]\n     \\caption{Structure of FNNF transition state. Coordinates in \\r{A}.}\n     \\centering\n     \\begin{tabular}{lrrr}\n       \\multicolumn{1}{l}{} &  &  & \\\\ \\hline\n       N &  0.49939 & -0.44656 & -0.59377\\\\\n       N & 0.57066 & 0.41224 &  0.5639 \\\\\n       F &  -0.39084 & 0.14563 & -1.36959 \\\\\n       F & -0.39807 & -0.12032 &  1.39157\\\\\n       \\hline\n     \\end{tabular}\n     \\label{tab:fnnfGeo}\n   \\end{table}\n   \n \\section{Appendix D: NCJF Gaussian Basis}  \n   \n  The form for the three dimensional Gaussian basis functions of NCJFs in the main text can equivalently be written as $g_j(\\mathbf{r}) = \\exp (\\mathbf{r}^T \\mathbf{A} \\mathbf{r} - 2 \\mathbf{B}^T\\mathbf{r} +C)$ where $\\mathbf{A}$ is a symmetric matrix defined by 6 parameters, $\\mathbf{B}$ is a three-component vector, and $C$ is a single dimensionless number.\n  We used 16 basis functions for the all parameter cases of N$_2$, 16 in styrene, and 4 in FNNF.\n  Their complete specifications are presented in Tables \\ref{tab:gaussianA}-\\ref{tab:fnnfGaussian}.\n  The components $A_{xx},A_{xy},A_{xz},A_{yy},A_{yz},A_{zz}$ with units of inverse square bohr are the same for each basis function within a particular system and are therefore listed separately in Table \\ref{tab:gaussianA}.\n  Tables \\ref{tab:n2alleqGaussian}-\\ref{tab:fnnfGaussian} contain components $B_x, B_y, B_z$, with units of inverse bohr and $C$ for each system's basis functions.\n \n \n    \\begin{table}[H]\n\\footnotesize\n    \\caption{Components of the matrix $\\mathbf{A}$ for our systems.}\n    \\centering\n    \\begin{tabular}{lcccccc}\n         System & $A_{xx}$ & $A_{xy}$ & $A_{xz}$ & $A_{yy}$ & $A_{yz}$ & $A_{zz}$\n        \\\\   \\hline\n        Equilibrium N$_2$ & -6.9282 & 0.0 & 0.0 &  -6.9282 & 0.0  & -6.9282  \\\\\n        Stretched N$_2$ & -6.9282 & 0.0 & 0.0 &  -6.9282 & 0.0  & -6.9282  \\\\\n       Styrene &-0.1 & 0.0 &0.0 &-0.1 &0.0 &-0.1 \\\\\n       FNNF  &-0.1 & 0.0 &0.0 &-0.1 &0.0 &-0.1 \\\\\n    \\end{tabular}\n    \\label{tab:gaussianA}\n\\end{table}\n \n   \\begin{table}[H]\n\\footnotesize\n    \\caption{Gaussian components for all parameter equilibrium N$_2$.}\n    \\centering\n    \\begin{tabular}{ccccc}\n         Basis Function & $B_x$ & $B_y$ & $B_z$ & $C$\n      \\\\   \\hline\n        $g_0$ &  -0.8 &-0.8 &-0.8& -0.2771 \\\\\n        $g_1$ & 0.8 &-0.8 &-0.8 & -0.2771\\\\\n        $g_2$ &-0.8 &0.8 &-0.8 & -0.2771\\\\\n        $g_3$ & 0.8 &0.8& -0.8 & -0.2771\\\\\n        $g_4$ & -0.8 &-0.8& 0.8 & -0.2771\\\\\n        $g_5$ &0.8 &-0.8& 0.8  & -0.2771\\\\\n        $g_6$ &-0.8 &0.8& 0.8  &-0.2771 \\\\\n        $g_7$ &0.8& 0.8& 0.8 & -0.2771\\\\\n        $g_8$ &-4.4787& -0.8& -0.8  &-11.2500 \\\\\n        $g_9$ &-2.8787& -0.8 &-0.8  & -4.5981\\\\\n        $g_{10}$ &-4.4787 &0.8& -0.8  & -11.2500\\\\\n        $g_{11}$ &-2.8787& 0.8& -0.8  & -4.5981\\\\\n        $g_{12}$ &-4.4787 & -0.8 & 0.8  & -11.2500 \\\\\n        $g_{13}$ &-2.8787 & -0.8 & 0.8  & -4.5981\\\\\n        $g_{14}$ &-4.4787& 0.8& 0.8  & -11.2500\\\\\n        $g_{15}$ &-2.8787& 0.8& 0.8  & -4.5981\\\\\n    \\end{tabular}\n    \\label{tab:n2alleqGaussian}\n\\end{table}\n\n  \\begin{table}[H]\n\\footnotesize\n    \\caption{Gaussian basis functions for all parameter stretched N$_2$.}\n    \\centering\n    \\begin{tabular}{ccccc}\n         Basis Function & $B_x$ & $B_y$ & $B_z$ & $C$\n \\\\   \\hline\n        $g_0$ &  -0.8 &-0.8 &-0.8& -0.2771 \\\\\n        $g_1$ & 0.8 &-0.8 &-0.8 & -0.2771 \\\\\n        $g_2$ &0.8& 0.8& -0.8  & -0.2771 \\\\\n        $g_3$ &0.8 &0.8& -0.8 & -0.2771\\\\\n        $g_4$ &-0.8 &-0.8 &0.8 & -0.2771\\\\\n        $g_5$ &0.8 &-0.8& 0.8  & -0.2771\\\\\n        $g_6$ &-0.8 &0.8& 0.8  & -0.2771\\\\\n        $g_7$ &0.8 &0.8 &0.8 & -0.2771\\\\\n        $g_8$ &-5.8015 & -0.8& -0.8 & -22.7322 \\\\\n        $g_9$ &-4.2015 &-0.8 & -0.8  & -11.8474 \\\\\n        $g_{10}$ &-5.8015 &0.8 &-0.8  & -22.7322\\\\\n        $g_{11}$ & -4.2015& 0.8& -0.8 & -11.8474\\\\\n        $g_{12}$ &-5.8015 & -0.8 & 0.8  & -22.7322\\\\\n        $g_{13}$ &-4.2015 & -0.8 & 0.8  & -11.8474 \\\\\n        $g_{14}$ &-5.8015 & 0.8 & 0.8  & -22.7322\\\\\n        $g_{15}$ &-4.2015 & 0.8 & 0.8  & -11.8474\\\\\n    \\end{tabular}\n    \\label{tab:n2allstGaussian}\n\\end{table}\n\n\n\n\\begin{table}[H]\n\\footnotesize\n    \\caption{Gaussian basis functions for equilibrium styrene.}\n    \\centering\n    \\begin{tabular}{ccccc}\n         Basis Function & $B_x$ & $B_y$ & $B_z$ & $C$\n       \\\\       \\hline\n        $g_0$ & -0.2632 & 0.0 & 0.0 & -0.6929 \\\\\n        $g_1$ & -0.4083 & 0.2254 & -0.003403& -2.1748\\\\\n        $g_2$ &-0.3957 & -0.2328 &  0.003617  & -2.1081\\\\\n        $g_3$ &-0.6738 & 0.2191 & -0.009989  & -5.0220\\\\\n        $g_4$ &-0.6617 & -0.2404 & -0.003392  & -4.9561\\\\\n        $g_5$ &-0.8007 & -0.01418 & -0.01149  & -6.4137\\\\\n        $g_6$ & 0.0 & 0.0 & 0.0  & 0.0\\\\\n        $g_7$ &0.1503 & 0.1759 & -0.1028  & -0.6409\\\\\n        $g_8$ &-0.3236 & 0.3990 & 0.0004516  & -2.6392\\\\\n        $g_9$ &-0.3009 & -0.4015 & 0.008982  & -2.5184\\\\\n        $g_{10}$ &-0.7744 & 0.3841 & -0.01352  & -7.4750\\\\\n        $g_{11}$ & -0.7542 & -0.41005 & -0.003197  & -7.3691 \\\\\n        $g_{12}$ & -0.9936 & -0.01898  & -0.01796  & -9.8794 \\\\\n        $g_{13}$ & 0.08754 & -0.1457 & 0.08083  & -0.3543\\\\\n        $g_{14}$ & 0.08207 &  0.3253 &  -0.19328  & -1.4992 \\\\\n        $g_{15}$ & 0.3368 & 0.1598 & -0.09316  & -1.4767 \\\\\n    \\end{tabular}\n    \\label{tab:styreneGaussian}\n\\end{table}\n\n \\begin{table}[H]\n\\footnotesize\n    \\caption{Gaussian basis functions for FNNF.}\n    \\centering\n    \\begin{tabular}{ccccc}\n         Basis Function & $B_x$ & $B_y$ & $B_z$ & $C$\n   \\\\         \\hline\n        $g_0$ &  -0.09437 &0.08439 &0.1122& -0.2862 \\\\\n        $g_1$ & -0.1078 & -0.07790 & -0.1066 & -0.2905 \\\\\n        $g_2$ &0.07386& -0.02752& 0.2588  & -0.7320\\\\\n        $g_3$ &0.07522 &0.02274& -0.2630  &-0.7533 \\\\\n    \\end{tabular}\n    \\label{tab:fnnfGaussian}\n\\end{table}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nMuon spin rotation and relaxation spectroscopy ($\\mu$SR\\xspace) and density functional theory (DFT) were first theorized, and later implemented, roughly in the same years.\\cite{PhysRev.136.B864,PhysRev.140.A1133,PhysRev.105.1415} Indeed, during the development of $\\mu$SR\\xspace as an experimental technique for studying magnetism in solid state physics, the analysis of the experimental data has greatly benefited from the theoretical insights provided by the first embryonic density functional based simulations.\\cite{PhysRevLett.40.264}\n\nTo the best of our knowledge, the first example of this kind dates back in 1975.\\cite{meier1975electron} Dr. P. Meier provided the first results from simulations aiming at identifying the effect of a positively charged interstitial muon in elemental metals.\nFrom then on, many works\\cite{PhysRevLett.40.264,Jena_1979} presented and discussed \\emph{ab initio} methods to tackle some common sources of uncertainty that stem from complicated $\\mu$SR\\xspace experiments. \nWe summarize them with the following three questions: {\\em i)} where is the muon? {\\em ii)} Can we estimate the parameters of the $\\mu-e^{-}$ interaction Hamiltonian? {\\em iii)} Is the muon a passive probe?\n\nA renewed effort in trying to answer the above questions with first principles simulations has begun a few years ago.\nIndeed, what has essentially changed from the 70s is our capabilities in simulating the electronic properties of complex materials, strongly reducing the impact of the approximations that must be adopted to solve the many body electronic (and in some cases also nuclear) problem on the parameters under investigation.\nIt is known, for example, that DFT is very accurate in determining the bond distances and unit cell sizes even when adopting the Local Density Approximation (LDA) for the exchange and correlation potential.\nThis is the rougher approximation, that disregards the non-local effects of the exchange interaction, and there are many cases where it is not sufficient to obtain a realistic description of the material.\n\nStarting from the LDA, the Generalized Gradient Approximation generally improves the description of the lattice positions. From there, in recent years, many rungs have been added to the ``Jacob's ladder of density functional approximations for the exchange-correlation energy'' \\cite{vxcladder}. Moreover, many body approaches different from DFT and \\emph{ad hoc} corrections to the Kohn-Sham Hamiltonian have greatly improved the capabilities of the density functional method in describing the electronic degrees of freedom of crystals and molecules. The reader is referred to one of the many review articles that discusses this topic.\\cite{QUA:QUA24521,PhysRevB.67.153106}\n\nAt the same time, the astonishing increase in the computational power obtained during the past 40 years has provided a tangible change in the amount of predictions that simulations can provide.\n\nFor what concerns $\\mu$SR\\xspace, this has also led to the possibility of addressing the three fundamental questions discussed above with fully \\emph{ab initio} methods. \n\n\nIn this short review article we will briefly address each question, respectively in Sect.~\\ref{sec:where}, \\ref{sec:interactions} and \\ref{sec:passive}, surveying the importance of the quantum nature of the muon in Sect.~\\ref{sec:quantum}. The discussion is illustrated by old and recent examples of DFT aided $\\mu$SR\\xspace data analysis. We will limit our attention to the simulations involving positive muons since they constitute the vast majority of the experiments performed with $\\mu$SR\\xspace nowadays. Since the topic is still rather vast, particular attention will be devoted to the analysis of the simulations performed in crystalline materials, even though we will also touch a few aspects of the analysis of the muon\/sample interactions in molecular compounds. \n\n\n\\section{Where is the muon?}\n\\label{sec:where}\n\nPart of the first experiments performed with $\\mu$SR\\xspace where devoted to the validation of the then new technique.\nFor this reason the study of elemental crystals and text-book cases were predominant and represented an important development toward a more precise understanding of the muon\/sample interactions. Positive muon sites identifications were mainly performed with direct experimental approaches like, for example, following the evolution of the muon frequency shift in a transverse field experiment as a function of the applied stress in a single crystal \\cite{PhysRevB.32.293}, notably with the stimulus of theoretical insight \\cite{PhysRevB.29.4170}, or by the symmetries of the same shifts as a function of the applied field direction\\cite{PhysRevB.30.186}, or again obtaining geometrical constraints on the relative position of the muon and the interacting nuclei from avoided level crossing measurements \\cite{PhysRevLett.60.224}.\nAt the same time, computational methods were used to model the electronic density surrounding the muon and provide, in some cases, information regarding the interstitial position of the muon and the interaction parameters between the muon and its surrounding electronic environment.\n\n\n\\begin{table}\n    \n\\begin{tabular}{ccc}\n    \\hline\n    Compound & \\multicolumn{2}{c} {muon site}\\\\\n             & (experiment) & ({\\em ab initio}) \\\\\n    \\hline\n &&\\\\\n    Fe$_3$O$_4$ & \\ \\cite{fe3o4,PhysRevB.77.045115} & -- \\\\\n    $R$FeO$_{3}$ ($R$ =  Sm, Eu, Dy, Ho, Y, Er) & \\ \\cite{PhysRevB.27.5294} & -- \\\\\n    UCoGe & \\ \\cite{PhysRevLett.102.167003} & -- \\\\\n    YBCO & \\ \\cite{Brewer_1991,Weber} & -- \\\\\n    UNi$_2$Al$_3$ &  \\ \\cite{Amato_2000} & -- \\\\\n    GdNi$_5$ & \\ \\cite{Mulders_2000} & -- \\\\\n    PrNi$_5$  & \\ \\cite{Feyerherm_1995} & -- \\\\\n    PrIn$_3$ & \\ \\cite{Tashma_1997} & -- \\\\\n    LiV$_2$O$_4$ &  \\ \\cite{Koda_2004} & -- \\\\\n    TmNi$_2$B$_2$C &  \\ \\cite{Gygax_2003} & -- \\\\\n    YMnO$_3$ &  \\ \\cite{Lancaster_2007} & -- \\\\\n    HoB$_2$C$_2$ &  \\ \\cite{De_Lorenzi_2003} & -- \\\\\n    UN & \\ \\cite{M_nch_1993} & -- \\\\\n    UAl$_2$ & \\ \\cite{Kratzer_1986} & -- \\\\\n    CeAl$_{3}$ & \\ \\cite{PhysRevB.39.11695} & -- \\\\\n    ZnO &\\ \\cite{PhysRevB.85.165211} &\\  \\cite{PhysRevLett.85.1012}\\\\\n    Y$_2$O$_3$ &\\ \\cite{PhysRevB.85.165211} &\\ \\cite{PhysRevB.85.165211} \\\\  \n    CoF$_2$, MnF$_2$ & \\ \\cite{PhysRevB.30.186} & \\ \\cite{PhysRevB.87.121108} \\\\\n    LiF &  \\ \\cite{PhysRevB.33.7813} & \\ \\cite{PhysRevB.87.121108,PhysRevB.87.115148} \\\\\n    YF$_3$ & \\ \\cite{Noakes1993785} & \\ \\cite{PhysRevB.87.115148} \\\\\n    MnSi & \\ \\cite{PhysRevB.89.184425} & \\ \\cite{jp5125876}\\\\\n    La$_2$CuO$_4$ T & \\ \\cite{PhysRevLett.59.1045,Hitti1990} & \\ \\cite {Suter2003329}\\\\\n    La$_2$CuO$_4$ T$^\\prime$ & \\ \\cite{gwenthesis} & \\ \\cite{pbphd} \\\\\n    $R$CoPO ($R$ = La, Pr)& \\ \\cite{PhysRevB.87.064401} & \\ \\cite{PhysRevB.87.064401} \\\\\n    PrB$_2$O$_7$ (B = Sn, Zr, Hf) & \\ \\cite{PhysRevLett.114.017602} &  \\ \\cite{PhysRevLett.114.017602} \\\\\n\\end{tabular}\n\\caption{Tentative list of the non elemental crystalline compounds where the muon site is known (or a few candidates are proposed) from the experiment. Although elemental crystal have been largely studied and the muon site is known in most of them, they have been omitted in this table for simplicity and readability.}\\label{tab:sites}\n\\end{table}\n\n\nIn this context, one instructive problem that has been considered with \\emph{ab initio} approaches is the diffusion of the muon in copper. This topic has been widely studied both experimentally \\cite{PhysRevB.43.3284,PhysRevLett.39.836} and theoretically \\cite{PhysRevB.29.5382}.\nFrom the field and orientation dependence of the decay in single crystalline samples\\cite{PhysRevLett.39.832} and from \\emph{ab initio} calculations it was soon realized that the occupied muon site is the octahedral interstitial.\nFirst principles simulations showed that this is due to the large Zero Point Motion Energy (ZPME) that exceed the self trapping energy gain and makes the tetrahedral interstitial site unstable.\\cite{PhysRevB.29.5382} \n\nStarting from the mid 80s, the increasing computational power allowed to tackle supercells of crystalline materials. Computational efforts were reported mainly for paramagnetic muon states, i.e. muonium, in carbon based and semiconducting materials. Within these simpler crystalline structures \\emph{ab initio} methods allowed the identification of the muon sites \\cite{0022-3719-17-14-009} and the determination important characteristic of the interaction between the muon and the investigated sample. \n\nMore recently, the advent of large computational facilities \nand the development of effective methods for the solution of the Kohn-Sham equations \\cite{PhysRevB.59.1758,PhysRevB.41.7892} \nhas led to a rebirth of the DFT approach for providing a description of the muon implantation process in the very end of its deceleration path.\nIndeed, since it is rarely possible to determine the muon site(s) with experimental methods, the computational approaches are becoming a precious supporting tool for $\\mu$SR\\xspace data analysis.\n\nAs of today  the best compromise between accuracy and efficiency to identify muon sites in molecules and crystalline materials is based on a sampling of the total energy hyper-surface which is obtained for the embedding of the charged particle in the host system. \n\nThe starting point is to treat the muon as a hydrogen atom within the standard Born Oppenheimer (BO) approximation used in DFT. \nDepending on the crystalline or molecular nature of the material under study, different approaches are used. In crystalline solids periodic boundary conditions are normally adopted and the final crystal relaxation around the muon is reproduced by choosing a suitably large supercell, in order to limit the interactions between the periodic muon replicas in the simulated system. It is very important to check the convergence of the results against the supercell size. To this aim two requirements are generally inspected:\n\\begin{enumerate}\n    \\item the interaction between the muon and its replicas must not influence the results,\n    \\item the displacement of atoms must progressively decay as the distance from the muon increases. \n\\end{enumerate}\n\nThis approach has been diffusely used for neutral and charged impurity calculations and has produced accurate results also for $\\mu$SR\\xspace  in Si, as well as in other elemental semiconductors \\cite{PhysRevLett.58.1547,PhysRevLett.60.2761,PhysRevLett.64.669,Jones351,EPL.7.145}.\n\nFor molecular systems the calculation can be less computational expensive since supercells are no longer required. However, in the case of large low symmetry molecules, one must still consider all the possible muon additions to the molecule at inequivalent positions, guiding an educated guess by chemical insight.\\cite{doi:10.1021\/jp107824p,doi:10.1021\/jp305610g}\n\nTable \\ref{tab:sites} is a partial list of crystalline compounds where the muon site is assigned, with certainty or tentatively, by experiment. The last rows list the cases where {\\em ab initio} confirmed the assignment. \n\nThe stability and the formation energy barrier of the various muon embedding configurations can be estimated, within the same BO approximation, with the nudged elastic-band (NEB) approach. This method provides an efficient way to identify saddle points and minimum energy paths between known initial and final ionic configurations.\nInitially, a series of equally separated configurations (called images) along the reaction path is guessed.\nThe images are kept equally separated from each other along the reaction path by adding an elastic band force acting on the images. A constrained optimization of the total energy for all the images, obtained by projecting out the perpendicular and the parallel components to the path of the spring force and of the true force respectively, provides an iterative method to identify the minimum energy path between the initial and final configurations of the muon and its neighborhood.\n\n\\begin{figure}\n\\center\n    \\includegraphics{81411Fig1}\n    \\caption{Formation energy for various types of hydrogen impurities as a function of the Fermi Energy level. The plot shows that H$^+$ is the stable form in ZnO, for Fermi Energy values (bottom scales) within the calculated energy gap (top scale), a fact that is experimentally\\cite{PhysRevLett.86.2601} verified in ZnO.\n    Reprinted figure with permission from Ref.\\onlinecite{PhysRevLett.85.1012}. Copyright (2000) by the American Physical Society.}\\label{fig:zno}\n\\end{figure}\n\nOne of the first success stories was the discovery of a shallow donor state for hydrogen, hence for the  muon as well, in Zinc Oxide. It was somehow surprising since hydrogen in p-type (n-type) semiconductors typically acts as a compensating impurity, in a stable  H$^{+}$(H$^{-}$) charge state at the bond-center. \\cite{PhysRevLett.85.1012}\nA 96 atoms supercell affords the calculation of the formation energies of the various charge states of the hydrogen atom and molecule in ZnO. The large H$^{+}$-oxygen bond strength leads to the formation of shallow donor states with a low electron density at the hydrogen (or the muon) site, as it is shown in Fig.~\\ref{fig:zno}.  The formation energy depends on three critical parameters: the total energy associated to the impurity in the selected charge state (top axis), the Fermi energy (bottom axis)  and the hydrogen\/muon ZPME, discussed in Sect.~\\ref{sec:quantum}. The interplay between these quantities governs the stability of paramagnetic and diamagnetic species, although metastable states must also be taken into account in the case of epithermal implanted muons.\n\nThe experimental demonstration of the unexpected shallow donor came just one year later with the observation of its characteristic precession frequencies by Cox and co-workers\\cite{PhysRevLett.86.2601}, confirming the \\emph{ab initio} predictions, and followed two years later by detailed single crystal studies. \\cite{PhysRevLett.89.255505}\n\nOne common feature of these early examples is that they forcibly concern the simplest crystal structures, that both require more manageable computational efforts, and offer few rather obvious starting guess sites for the muon.\nThe extension to non elemental compounds with more complex structures also implies a strategy for the exploration of the candidate sites. \n\nAn example of exploration strategy along the same line of ZnO is provided by Vil\\~ao  {\\em et al.} \\cite{PhysRevB.84.045201} who compared  experiments on paratelluride ($\\alpha$TeO$_2$) with DFT within GGA on a 3x3x3 supercell (96 atoms) including NEB calculations to discuss diffusion by means of classical energy barriers. They are thus able to identify the experimental results as a donor and a deep trap configuration.\n\n\\begin{figure}\n\\center\n    \\includegraphics[width=\\columnwidth]{81411Fig2}\n    \\caption{(Color online) Muon sites in Y$_{2}$O$_{3}$. Different positions are identified for the various charge states of the hydrogen impurity used to simulate the muon.\n    Reprinted figure with permission from Ref.\\onlinecite{PhysRevB.85.165211}. Copyright (2012) by the American Physical Society.}\\label{fig:yttria}\n\\end{figure}\n\n\nMore recently Silva {\\em et al.}\\cite{PhysRevB.85.165211} assigned the muon species observed in yttria. Figure~\\ref{fig:yttria} shows the various diamagnetic and paramagnetic configurations obtained for the different charge states considered for the impurity in the simulations. The results highlight the striking difference between the embedding sites obtained with the different charge states and underline the importance of considering various electronic configurations when exploring the muon embedding sites. These examples are still guided in some degree by the chemical insight allowed by the dominant covalent nature of the bonds. \n\nAdditional strategies are devised for dominantly ionic compounds. Whenever the candidate muon sites are not easily assigned by educated guess it is unavoidable to explore all the possible interstitial sites. To this end it is useful to set up a grid of positions and to optimize the impurity site with the use of the Hellman-Feynman theorem, that provides the intermolecular forces and allows the identification of equilibrium geometries. This procedure produces candidate interstitial positions for the muon.\nThe number of points in the grid may usually be reduced with the help of symmetry considerations. This can substantially shrink the computational cost of the simulation.\n\nIn recent years, DFT based muon site assignment has been used in several materials.\nGraphene has been investigated \\cite{doi:10.1021\/nl202866q} to try and clarify whether diluted hydrogen decorated defects, mimicked by muons, can give rise to magnetism. The authors support their interpretation of the experimental results with DFT predictions of hydrogen decorated carbon vacancies on 6x6 nanoribbon cluster.  By contrast adsorption of positive muon on perfect graphene predicts \\cite{Pant_2014}  a ground state energy of 0.22 eV directly on top of the carbon atom. \n\nRecently, $\\mu$SR\\xspace has played a prominent role in characterizing the newly discovered iron based high temperature superconductors. As a consequence, the muon site in these materials has been widely studied.\nMost of the initial estimations were based on a naive approach which relies on the analysis of the electrostatic potential of the bulk material obtained from DFT simulations \\cite{PhysRevB.80.094524,0953-2048-25-8-084009,PhysRevB.91.144423,PhysRevB.85.064517}.\nThis method is the direct evolution of the point-charge model widely used in literature.\\cite{PhysRevB.84.054430,PhysRevB.85.054111,PhysRevLett.103.147601,PhysRevB.84.064433}\nIncidentally we mention that this approach has also been used for materials with diverse electronic properties.\\cite{1742-6596-551-1-012052,1742-6596-551-1-012053}\nThe results obtained from the analysis of the unperturbed electrostatic potential (UEP) are generally validated by comparison with the experiment. \nThis is the case of the 11 11 pnictide superconductors\\cite{PhysRevB.80.094524}.A full \\emph{ab-initio} confirmation of the UEP predictions was provided for the isostructural compound LaCoPO with relaxed supercell calculations.\\cite{PhysRevB.87.064401}\n\nThe remarkable success of simple electrostatic potential predictions in simpler three dimensional metals may perhaps be justified by heuristic arguments on the screening of point charges. A recent confirmation is the case of MnSi where accurate  supercell calculations\\cite{jp5125876} introduce only tiny improvements on the UEP predictions. However the accuracy of the UEP method in a layered material like LaFeAsO or LaCoPO is more surprising, and it should probably be considered just a fortunate case. The method cannot be expected to produce reliable results in insulators or in two dimensional materials alternating metallic and charge reservoir layers. \n\nFluorides represent a notable example of insulator where the UEP method fails.\\cite{PhysRevB.87.115148} These materials deserve a special mention. Brewer and coworkers identified a striking effect, characteristic of several of them, where the distortion of the lattice produced by the muon on F atoms is very pronounced\\cite{PhysRevB.33.7813}. The muon-fluorine distance for the most distorted nearest neighbor ions is characterized in details by the experimental measurement, thanks to the quantum entanglement of the muon and fluorine nuclear spins.\n\nThis has been used as an ideal test case by M\\\"oller {\\em et al.} \\cite{PhysRevB.87.121108} and by Bernardini {\\em et al.} \\cite{PhysRevB.87.115148} to verify the accuracy of DFT in reproducing the crystalline distortions introduce by the muon. The first two columns of Table~\\ref{tab:Moeller}  show the experimental and calculated muon-fluorine distances, whose very good agreement is an important validation of the \\emph{ab initio} approach to the muon site identification problem. \n\n\\begin{table}\n    \n\\begin{tabular}{cccc}\n\\hline\\hline\n& $2r_{\\rm DFT}$ (\\AA) & $2r_{\\rm exp}$ (\\AA) & ZPE (eV)\\\\\n\\hline\n(FHF)$^-$ & 2.36 & 2.28 & 0.30\\\\\n(F$\\mu$F)$^-$ & 2.36 & & 0.80 \\\\\nLiF & 2.34 & 2.36(2)\\cite{PhysRevB.33.7813} & 0.76\\\\\nNaF & 2.35 & 2.38(1)\\cite{PhysRevB.33.7813} & 0.76\\\\\nCaF$_2$ & 2.31 & 2.34(2)\\cite{PhysRevB.33.7813} & 0.83\\\\\nBaF$_2$ & 2.33& 2.37(2)\\cite{PhysRevB.33.7813}& 0.79\\\\\nCoF$_2$ & 2.36 & 2.43(2)& 0.73\\\\\n\\hline\n\\hline    \n\\end{tabular}\n\\caption{Calculated (DFT) and experimental (exp) properties of\nthe diamagnetic and molecular ion  fluorine-muon (F-$\\mu$-F) states in solid and vacuum. $r$(\\AA) is the muon-fluoride bond length.\nReprinted table with minor edits with permission from Ref.~\\onlinecite{PhysRevB.87.121108}. Copyright (2013) by the American Physical Society.}\\label{tab:Moeller}\n\\end{table}\n\nAs it has been shown from the very beginning by studying the interstitial muon site in elemental crystals, the ZPME plays a crucial role in this procedure.\\cite{teichler1978microscopic,0305-4608-9-7-013,rath1979effect}\nIndeed the large ZPME of the muon, discussed in Sect.~\\ref{sec:quantum}, may yield classical hopping or quantum tunneling among various interstitial positions. \nThe case of fcc copper represent an instructive example in this perspective. \\cite{PhysRevB.43.3284} The barrier between the tetrahedral and the octahedral interstitial is too small to bind the muon. Thus no localized muon wave-function is found in that positions. At the same time the barrier between the octahedral sites is small enough to permit both quantum tunneling and classical diffusion for relatively small temperatures.\\cite{jp5125876}\n\nIn several instances a successful identification of the muon sites based on a first principle approach relies on the additional evaluation of the ZPME, not just on the solution of a purely electronic problem, and the number of cases where this inclusion proved to be essential is steadily increasing.\\cite{Suter2003329,PhysRevB.84.045201,PhysRevB.85.165211,Pant_2014,PhysRevLett.107.207207,1402-4896-88-6-068510}\n\n\n\\section{Interaction parameters}\n\\label{sec:interactions}\n\nAnother point that has been largely discussed in the literature is the estimation, from first principles, of the interaction parameters between the muon and the electrons (or possibly the nuclei) of the host material.\nSuccessful results have been reported from early studies of the hyperfine coupling for diamagnetic muon sites in metallic magnetic materials and for paramagnetic muon sites in semiconductors.\\cite{PhysRevLett.64.669}\nIn the vast majority of these preliminary studies the electron density and its magnetic polarization at the muon position were used to estimate the hyperfine parameters according to\\cite{PhysRevLett.64.669}\n\n\\begin{align}\n\\label{eq:hyperfine}\n    \\mathcal{H} & = \\sum_i\\left\\{\\frac{2 \\mu_{0}}{3}  \\mathbf{m}^{\\mu} \\cdot \\mathbf{m}^{e} \\delta(\\mathbf{r}_i)\\right. \\nonumber\\\\ &\\left.\\quad + \\frac{\\mu_{0}}{4 \\pi} \\frac{1}{r_i^{3}} \\left[ 3 (\\mathbf{m}_{\\mu} \\cdot \\mathbf{\\hat{r}_i})(\\mathbf{m}_{e} \\cdot \\mathbf{\\hat{r}_i}) - \\mathbf{m}_{\\mu} \\cdot \\mathbf{m}_{e} \\right]\\right\\}\n\\end{align}\n\n\nwhere the first term is the Fermi contact term and the second is the dipolar interaction, $\\delta$ is the Dirac delta function, $\\mathbf{m}^{e} = -g_{e} \\mu_{\\mathrm{B}} \\mathbf{S}^{e}$ and $\\mathbf{m}^{\\mu} = \\gamma_{\\mu} \\hbar \\mathbf{I}^{\\mu}$  are the electronic and the muon magnetic moment operators, $\\mathbf{r}_i$ is the coordinate of the i-th electron in a reference frame centered at the muon and $\\mu_{0}$ is the vacuum permeability.\n\nThe parent compound La$_2$CuO$_4$ in its standard crystal group 64 has been first addressed by a 9 Cu + Mu cluster in the Generalized Gradient Approximation (GGA), without quantum treatment of the muon. \\cite{Suter2003329} The authors extract spin densities on Cu, nearest neighbor O ions and the muon. They remark that spin density on oxygen, although definitely smaller than that on copper, provides a non negligible contribution through the second term of Eq.~(\\ref{eq:hyperfine}), in view of the much shorter distance to the bonded muon, thanks to the $r^{-3}$ dependence of the dipolar term. This remark might be of more general relevance.\n\n\nFor chemically diamagnetic muon species the contact hyperfine coupling at the muon site is generally the result of a rather small imbalance of the spin density, and the accuracy of its determination strongly depends on that of the DFT description of the many body electronic problem. Even assuming small uncertainties of the muon site coordinates and on the electron magnetic moments a large relative numerical error on  the reduced electronic spin polarization at the muon site, produces a considerable relative inaccuracy, that is usually an order of magnitude larger than that typically obtained for the dipolar coupling.\n\nThis remark applies e.g.~to the one-dimensional quantum antiferromagnet AF Cu(pyz)(NO$_3$)$_2$, where a 2x2x1 supercell GGA calculation \\cite{PhysRevB.91.144417} including the muon in both positive and neutral charge configurations predict large contact couplings for the latter and much smaller ones for the former. The rather low experimental zero field muon precession frequencies rule out the large hyperfine, neutral muon site. The authors implicitly acknowledge the inaccuracy of low spin density DFT estimation by establishing a comparison between the experimental local field and the dipolar contribution alone. This comparison justifies the qualitative statement that the Cu moment must be reduced by a large factor, of order 7-8 with respect to the nominal 1 $\\mu_B$, as expected from the low dimensional nature  of the magnetic order. \n\\begin{figure}\n\\center\n    \\includegraphics[width=0.8\\columnwidth]{81411Fig3}\n    \\caption{(Color online) The adiabatic muon energy isosurface in MnSi, \\cite{PhysRevB.89.184425,jp5125876} obtained joining points where the potential acting on the muon is 1.6 times the corresponding muon ground state energy. The dots represent the points used for the potential interpolation, within DAA (see text).} \\label{fig:mnsias}\n\\end{figure}\n\n\nThus, whenever Eq.~(\\ref{eq:hyperfine}) provides a reasonably good account of the muon coupling, $\\mu$SR\\xspace data together with the muon site assignment allow quantitative determination of magnetic moments and, in fortunate cases, even of magnetic structures. In this perspective a promising approach based on the Bayesian analysis,\\cite{Blundell2012113,PhysRevB.89.140413} has been proposed in order to estimate the magnetic moment size and\/or the long range magnetic structure of magnetic materials.\nIndeed by feeding the probabilistic analysis with the DFT results obtained for the identified muon site(s) it is possible to compare quantitatively the expectations for various spin arrangements and determine the most probable long range magnetic order for the sample under investigation.\n\nEquation (\\ref{eq:hyperfine}) represents only a first approximation, since it neglects the quantum nature of the muon. Although the approximation often provides at least the correct order of magnitude, if not the exact experimental value for the hyperfine couplings at the muon site \\cite{Katayama1979431,Jepsen1980575}, there are known cases in which this approach is not sufficiently accurate \\cite{PhysRevB.25.297, PhysRevB.27.53, PhysRevB.15.1560}.\n\nFinally, we notice that estimating {\\em ab initio} the value of the hyperfine field at the muon site may be differently challenging, depending on the material. For strongly correlated electron material it involves an accurate description of their electronic properties that is not always available with conventional DFT approaches. The reliability of the typical approximations, e.g. LDA+U, must therefore be seriously taken in consideration case by case.\n\nAt the opposite end rather accurate results can be obtained for the hyperfine coupling parameters of paramagnetic muonated radicals.\\cite{ct500027z,PhysRevE.87.012504,C4CP00618F,C4CP04899G,Peck} This is mainly due to the fact that the contact hyperfine coupling constants are proportional, in this case, to the electron density at the muon site, which is much larger for paramagnetic species.\nOnce again, the key point in this procedure is an accurate electronic description of the molecule, usually obtained with hybrid exchange and correlation functionals.\\cite{Peck}\n\n\\section{The quantum muon}\n\\label{sec:quantum}\n\n\nIn the case of both diamagnetic and paramagnetic muon centers, improved results are obtained if the quantum nature of the muon, usually far from being negligible, is taken into account. This was already considered in most of the first reports on \\emph{ab initio} studies of the muon \\cite{rath1979effect,PhysRevB.25.297}. Since within DFT the muon is treated as a charged classical particle, the simulations must be extended in order to provide a description of the muon wave-function.  \n\nThe task is fulfilled with a number of different approaches\\cite{1.1356441,Gross1998291,1.3556661,C4CP05192K,PhysRevA.90.042507,PhysRevLett.101.153001,RevModPhys.85.693}. We mention here the estimation of the ground state energy of the muon with the analysis of the phonon-modes, within the BO approximation for the electrons, \\cite{PhysRevB.87.121108} and the double adiabatic approximation (DAA)\\cite{jp5125876}, discussed in details by Soudackov and Hammes-Schiffer\\cite{Soudackov1999503} and Porter {\\em et al}\\cite {PhysRevB.60.13534} in which an adiabatic energy surface for the solution of the Schr\\\"oedinger equation of the muon is obtained.\n\nMore accurate approaches may be provided by the Nuclear-Electronic Orbitals (NEO)\\cite{C4CP06006G}, using Hartree-Fock methods\\cite{Kerridge_2004} on an orthogonal basis for the muon and the electrons. Finally, path integral molecular dynamics (PIMD)\\cite{1.471221, PhysRevLett.81.1873, Yoshikawa2014135,PhysRevLett.99.205504,PhysRevB.60.14197} in principle can yield the most accurate calculations. However, since these techniques are also listed in order of increasing computational costs, their use grows increasingly impractical for materials, such as mixed valence, strongly correlated electron systems, that are already intrinsically complex from an {\\em  ab initio} point of view.\n\nThe first two methods treat the muon as a point-like charged particle.\nThe DAA method and the linear response evaluation of the muon phonon modes in the crystal represent the simplest and less computer intensive approaches. The DAA method approximates the potential of the muon Schr\\\"odinger equation with the DFT total electronic energy recalculated as a function of fixed muon position in a suitable grid. The phonon based mechanism yields the standard harmonic approximation to the mode frequency, which directly provides the ZPME by a projection method. It has the advantage of treating the muon and the nuclei on the same footing, but the harmonic approximation is usually not very accurate in the muon case. This is indirectly shown for example in Fig.~\\ref{fig:mnsias} by the highly non ellipsoidal shape of the muon potential energy isosurface obtained by DAA in the case of MnSi \\cite{jp5125876}. The actual shape of the muon potential energy may be mapped within DAA, to avoid the harmonic approximation, but only inasmuch as the energy scales of the nuclei of the embedding system are well separated from those of the muon. Hence DAA may fail for systems with close-lying muon and hydrogen ions. \n\nThe NEO approach introduces a muon wave-function that is optimized together with electronic wave-functions and overcomes the BO approximation, and PIMD treats the nuclei from a quantum perspective by mapping them onto an isomorphic classical polymer of replicas of each nucleus (called bead). In \\emph{ab initio} PIMD, each bead requires a self consistent calculation, thus the computational cost scales linearly with number of beads.\n\n\\begin{figure}\n\\center\n    \\includegraphics[width=0.8\\columnwidth]{81411Fig4}\n    \\caption{The hopping frequency for a muon in diamond as obtained from PIMD simulations.  The solid line is derived from $\\mu$SR\\xspace measurements. Reprinted figure with permission from Ref.~\\onlinecite{PhysRevLett.99.205504}. Copyright (2007) by the American Physical Society.} \\label{fig:diamond}\n\\end{figure}\n\nAn instance where the drastic approximation of the first two methods (and maybe also of the NEO approach) may fail is the metastability of the neutral H$_0$ charge state in Si, that was first tentatively assigned \\cite{Cox1986516} to the tetrahedral (T) site  Mu$^0_T$. Specifically, the existence of an absolute energy minimum at the bond-center (BC) Mu$^0_{BC}$ site \\cite{Cox1986516} is confirmed, in Si and diamond, by Hartree-Fock cluster calculations that identify the tetrahedral site as a local minimum.\nHowever, contrasting results regarding the height of the barrier between the two sites and the role of the ground state ZPME of Mu$^0_T$ are reported. \\cite{PhysRevLett.71.557, PhysRevLett.58.1547,PhysRevB.36.9122} \n\nRecalling that thermodynamic equilibrium may not be achieved during a muon lifetime, this finding would agree with the low temperature experimental observation of both species in all elemental semiconductors.\nThis fact is partially confirmed by more accurate PIMD calculations in diamond.\\cite{PhysRevLett.99.205504}\nFigure \\ref{fig:diamond} displays both the experimental (solid line) and the theoretical PIMD (dashed line) jump rates for muons in diamond. The jump rate from the T site to the BC site is several orders of magnitude larger than that from the BC to the T site for all experimentally accessible temperatures. Therefore the simulation predicts, in qualitative agreement with experiment, that Mu$^0_T$ must disappears from observation at high temperatures, when its jumping rate time becomes shorter than a few nanoseconds, whereas in the same temperature range the more stable Mu$^0_{BC}$ does not delocalise during a few muon lifetimes. \n\nAs of today there is no universal solution to obtain a detailed description of the quantum nature of the muon. While for small molecules PIMD gives the most accurate results, this approach is usually prohibitively time consuming for muons embedded in crystalline materials.\nIndeed there are two aspects specific to $\\mu$SR\\xspace experiments that makes PIMD very computationally demanding: the small mass of the muon and, in many cases, the (low) temperature that is used in experiments. Both these conditions contribute to the growth of the required number of beads.\nFor this reason, when performing PIMD, the \\emph{ab initio} methods for the electronic structure evaluation are usually abandoned in favor of less demanding approaches like tight-binding Hamiltonians or empirical potentials which may degrade the quality of the description of the electron density.\n\n\n\\section{Is the muon a passive probe?}\n\\label{sec:passive}\n\nThe muon is a positively charged particle\\cite{footnote} and, when it comes to rest in the lattice, it gives rise to a charged defect that can introduce appreciable local modification to its immediate electronic and ionic environment. The key question in this case is whether the muon can alter the properties of the system that it is supposed to probe in the experiment.\nThe answer to this question can be rarely provided just by $\\mu$SR\\xspace experiments and it depends largely on the goals of the measurement. Since a large fraction of the muon studies are directed at magnetic materials the question can be often rephrased into ``is the muon distortion capable of altering the apparent magnetic behavior of the investigated sample with respect to that of the crystal without the muon?''\n\n\n\nThe appreciable crystalline distortions are however generally not a source of concern for magnetic measurements. Their scarce influence may be explained by considering that, firstly, the muon usually forms bonds with the most electronegative atoms of the hosting system, and, in many cases, these atoms do not provide the leading contribution to the exchange integral. Secondly, when the muon does modify the exchange integral, the perturbation is usually confined to the nearest neighbor magnetic atoms. Thus the global magnetic properties are not affected by the perturbation, although the local magnetic field at the muon site may be modified. As long as $\\mu$SR\\xspace experiments regard the relative temperature dependence of the local field, the influence is not relevant. \n\n\n\n\\begin{figure}\n\\center\n    \\includegraphics{81411Fig5}\n    \\caption{(Color online) Perturbation introduced by the muon on its neighboring Pr atoms in Pr$_2$Ir$_2$O$_7$. The positive electric charge of the muon modifies the crystal field levels producing a non negligible hyperfine coupling at the muon site as a consequence of the ground state doublet splitting. \n    Reprinted figure with permission from Ref.~\\onlinecite{PhysRevLett.114.017602}. Copyright (2015) by the American Physical Society.}\\label{fig:pr}\n\\end{figure}\n\nDFT can be directly employed to check the variation of the magnetic moment of equivalent ions as a function of their distance from the muon.  Although most often it is indeed found that the charged particle does not modify significantly the magnetic properties  of its neighbors,\na notably different instance is represented by the one dimensional AF Cu(pyz)(NO$_3$)$_2$ \\cite{PhysRevB.91.144417} discussed earlier.\nFor neutral supercell simulations there are a couple of muon embedding sites which donate an electron to the $3d$ orbitals of their nearest Cu, turning it into the diamagnetic Cu$^{+}$ configuration. Such a substantial local perturbation is, however, still hard to detect.\nIndeed, even though the modification does affect the local value of the magnetic field via the dipolar coupling, it has practically no effect on the collective magnetic properties of the system. Therefore the temperature dependence of the magnetic order parameter or that of its slow fluctuations, that induce muon spin relaxation, remain the same as in an unperturbed environment. \n\n\nBy contrast, a very interesting case in which the magnetic response of the system is altered by the muon has been recently discussed by Foronda and co-authors.\\cite{PhysRevLett.114.017602} \nAn unexpected quasi-static local field at the muon site was observed in geometrically frustrated pyrochlore iridate Pr$_2$Ir$_2$O$_7$. This effect was argued to be related to a muon induced perturbation of the crystal field levels of Pr which leads to the lifting of the non-Kramers degeneracy of the ground state.\\cite{1742-6596-225-1-012031}\n\nThis hypothesis has been nicely demonstrated by the results obtained with DFT simulations which provides the deformation of the oxygen tetrahedra surrounding the Pr atoms.\nThe lifting of the degeneracy, shown in Fig. \\ref{fig:pr}, is reported for the three Pr atoms closer to the muon. The hyperfine interaction between the Pr nuclei and $f$ orbitals is enhanced by the lifted degeneracy and the resulting magnetic moments of the three Pr atoms surrounding the muon are revealed by the $\\mu$SR\\xspace experiment, thus masking the properties of the nonmagnetic unperturbed system.\\cite{PhysRevLett.114.017602}\n\nWhenever $\\mu$SR\\xspace results deviate drastically from expectations, one should critically consider whether they are due to specific local alteration induced by the muon on its surrounding. It also happens that unconventional muon related phenomena are invoked to justify $\\mu$SR\\xspace observations. These cases  too may profit from a comparison with DFT predictions. For example, it has been suggested that spin polarons may interact strongly with muons, in particular in the noncentrosymmetric magnetic metal MnSi \\cite{PhysRevB.83.140404}. In this material the spin polaron would consist of an electron coupled to four Mn neighbors, to form a single large spin entity. Binding of the muon to the spin polaron was claimed to justify two precessions observed by Storchack {\\em et al.} in high transverse magnetic field. \\cite{PhysRevB.83.140404} This explanation is alternative to the conventional hypothesis of a muon site made inequivalent by the application of the field, to justify the appearance of more than one frequency.  For MnSi the site identification by DFT, supporting accurate transverse field experiments and a careful data analysis, proved that the observed frequencies correspond to the latter case. \\cite{PhysRevB.89.184425,jp5125876}\n\nA similar instance is that of deconfined magnetic monopoles that are predicted by theory in spin ices, such as in some rare earth pyrochlores. A recent experiment claimed to have detected by $\\mu$SR\\xspace in Dy$_2$Ti$_2$O$_7$ a second Wien effect, also referred to as magnetricity, i.e.~the dissociation of magnetic charges by an applied field. \\cite{nature461.956} A subsequent work \\cite{PhysRevLett.107.207207} demonstrated this not to be the case, by comparing observed and simulated spectra on the same material, based on DFT site assignment. Incidentally this is a case where the error in interpretation of the earlier experiment turned out to be a trivial one, and its recognition did not actually rule out the existence of deconfined magnetic charges in Dy$_2$Ti$_2$O$_7$. Rather, a more recent work suggested \\cite{PhysRevLett.108.147601} that the observations of Bramwell {\\em et al.} \\cite{nature461.956} are probably still due to magnetricity, although the observation in that work was rather indirect, through a large fraction of muons implanted in close contact to the sample, but outside it, in its cryostat holder.\n\nIn both the MnSi and the Dy$_2$Ti$_2$O$_7$ cases qualitative analysis could suffice to produce plausibility arguments towards the correct conclusion, but the DFT offered a precious quantitative support to the discussion of the experimental findings. This and other examples,\\cite{PhysRevB.87.121108} show the effectiveness of the computational approaches in confirming or rejecting the generally accepted belief that the muon behaves as a passive probe.\n\n\\section{Limits and Perspectives}\nOne can confidently say that DFT calculations nowadays offer methods for predicting muon candidate sites in many crystalline materials, suitable to be directly employed in the design of $\\mu$SR\\xspace experiments and to provide complementary information for the data analysis. We refer here to muon {\\em candidate} sites because it is presently beyond the scope of DFT to actually predict the branching ratios among such sites during muon implantation, which is effectively an epithermal process.   \n\nThe literature on DFT based analysis of the effect of charged impurity is vast and often provides precious guidance for the validation of the muon results obtained by numerical simulations. Three intrinsic limiting factors arise when considering DFT as a tool for complementing muon experimental observations.\n\nOpen challenges are still represented by critical compositions of solid solutions, such as certain intermetallics, or intermediate valence oxides. It is for instance still very hard to accurately simulate by DFT a specific composition like YBa$_2$Cu$_3$O$_{6.35}$, at the onset of high T$_c$ superconductivity, since a very large supercell would be required. But enough insight can be often gathered by considering end members and simple intermediate compositions.   \n\nAnother difficulty is sometimes caused by the mean field approach of the Kohn-Sham method, not always sufficient to describe the electronic properties of the materials in all their relevant details. This is notoriously true already for semiconductors, e.g. when excited states are involved, as for the energy gap, although in this case the shortcomings  for the muon are easily circumvented. Failures may be more difficult to overcome in the case of strongly correlated systems. \n\nA third problem is related to the BO approximation that is commonly adopted when simulating the muon with \\emph{ab initio} approaches. This latter issue may become rather severe when dealing with the interaction between light atoms and the muon. \n\n\n\nSince the purpose of the present review is to concentrate on methods that may be routinely available to assist the analysis of $\\mu$SR\\xspace experiments, not all the known theoretical tools qualify. In this sense a universal viable way to tackle the quantum nature of the muon is still missing. Although from the theoretical point of view, many approaches have been developed,\\cite{gwfornonbo,1742-6596-225-1-012031,1.471221}  most of them become computationally increasingly expensive with the number of atoms, and the number of electrons per atom that are included in the calculation. A compromise between accuracy and speed must be found. \n\nPromising results have been obtained with the nuclear-electronic orbital (NEO) method \\cite{jp053552i,C4CP06006G} which, by treating only a small subset of the atoms with non-BO approaches, improves the description of the muon and of other light nuclei with smaller computational costs with respect to other approaches like, for example, PIMD.\nFor the cases where the computational cost of performing PIMD is sustainable, this approach has demonstrated high accuracy\\cite{ct500027z}.\nNonetheless, as of today, its applicability is limited to simpler systems such as molecular materials, where the single DFT self consistent field simulation is computationally not expensive. \n\nWhenever the muon coupling to its environment is dominated by dipolar interactions the site assignment is already sufficient to obtain a fully quantitative muon data analysis, and the influence of the quantum muon treatment may be much less important. By contrast, when contact hyperfine couplings are required for a chemically diamagnetic site, state of the art DFT techniques may not be sufficiently accurate, although the situation will probably change during the next years owing to the advances of both computational methods' efficiency and computational power availability.\n\nFinally, we have shown that the methods we have described above are already a very valuable tool when critically analyzing the possibility of a muon induced effect. \n\n\\begin{acknowledgment}\n\n\n\nThe seed for both our recent DFT work and the effort to collect the systematics which is the basis of the present review was generated by the MUON JRA of EU FP7 NMI3, under grant agreement 226507. We would like to thank Franz Lang, Johannes M\\\"oller, Stephen Blundell and Fabio Bernardini for useful discussions. We also acknowledge partial funding from PRIN project 2012X3YFZ2 and from the European Union's Horizon 2020 research and innovation programme under grant agreement No 654000. \n\n\\end{acknowledgment}\n\n\\bibliographystyle{jpsj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe effect of quenched random impurities which couple to the local energy\ndensity on the critical behaviour of a system whose pure version undergoes \na continuous transition\nis well understood in terms of the Harris criterion \\cite{Harris}: if\nthe specific heat exponent $\\alpha$ of the pure system is negative,\nsuch impurities do not change the\nqualitative nature of the transition or its universality class, while in\nthe opposite case there is expected to be different behaviour described\nby a new random fixed point of the renormalisation group (RG). Given\nthe volume of literature devoted to this subject, it is rather\nsurprising to find relatively few studies of the effects of such\nimpurities on systems whose pure versions undergo a first-order\ntransition, despite the ubiquity of such systems in nature. A naive\nextension of the Harris criterion, based on the observation that the\neffective value of $\\alpha$ is unity at a first-order transition, might suggest\nthat randomness is always strongly relevant. On the other hand,\nfirst-order behaviour is accompanied by a finite correlation length and\nis commonly assumed to be stable under small perturbations. Following\nearly work by Imry and Wortis \\cite{IW}, Aizenman and Wehr \\cite{AW}\nand Hui and Berker \\cite{HB} showed that the role of dimensionality $d$\nis crucial. In fact the first authors proved a rigorous theorem which\nstates that for $d\\leq 2$ the Gibbs state is always unique, for\narbitrarily small but non-zero concentration of impurities. This means\nthat there can be no phase coexistence at the transition and hence no\nlatent heat. \n\nThis result raises a number of important questions: (a) what is its\nphysical mechanism; (b) is the continuous transition accompanied by a\ndivergent correlation length; (c) if so, what are the (presumably)\nuniversal critical exponents characterising the transition; and (d) what\nhappens for $d>2$? Initial Monte Carlo studies of (c) \\cite{CFL,Dom}\nsuggested that the critical behaviour of all such systems might be strongly\nuniversal, exhibiting Ising-like exponents independent of the\nunderlying symmetry of the model. This received some theoretical\nsupport \\cite{Kardar}. However, more recent numerical studies (to be\ndescribed below) support earlier theoretical results \\cite{Ludwig} in\nsuggesting that the critical behaviour of, for example, the random bond\n$q$-state Potts model depends continuously on $q$, both in the region\nwhere the pure model has a continuous transition, and where it is\nfirst-order. The aim of this talk is to summarise some of these \nrecent developments, and to provide at least partial answers to the\nabove questions.\n\n\\section{Mapping to the random field Ising model}\nFirst, I want to describe a mapping \\cite{CJ} of this problem to the \n\\em random field \\em Ising model (RFIM), which is asymptotically valid for very\nstrong first-order transitions, and which provides a physical\nexplanation of the Aizenman-Wehr result \\cite{AW}, as well as a\nprediction for what happens when $d>2$. \n\nConsider a pure system at a thermal first-order transition point. There\nwill be coexistence between a (generally unique) disordered phase and\nthe (generally non-unique) ordered phases. The internal energies \n$U_1$ and $U_2$ of\nthese two kinds of phase will differ by the latent heat, $L$.\nNow consider a interface between the disordered phase and one of the \nordered phases, with surface tension $\\sigma$ (measured in units of $kT_c$).\nIf $\\sigma$ is large, there will be very few isolated bubbles of the\nopposite phase above or below the main interface. Its equilibrium\nstatistics may therefore be\ndescribed by a free energy functional equal to its area multiplied by\n$\\sigma$. Let us compare this with the interface between the two\n\\em ordered \\em phases of an Ising model, with spins taking the values\n$\\pm1$, at low temperatures. Once\nagain, there will be few bubbles of the opposite phase, and the bare\ninterfacial tension will be $\\sim 2J$, where $J$ is the reduced exchange\ncoupling. In the limit when $\\sigma\\sim 2J$ is large, these\ninterfacial models are therefore identical\\footnote{This is not the case when\nbubbles of the opposite phase are included. For example, regions of\nordered phase appearing in the disordered phase are counted with\nthe degeneracy factor of the ordered phases, as compared with a factor\nof unity for the Ising case.}\n\nNow consider the effects of randomness in these two models, in\nthe first case random bonds, coupling to the local energy density, and\nin the second random fields, coupling to the local magnetisation.\nIn the RFIM, these are accounted for by adding a term $\\Sigma_{r>}h(r)-\n\\Sigma_{r<}h(r)$, where $h(r)$ is the local random field, and in the two\nterms $r$ is summed respectively above and below the instantaneous\nposition of the interface. For random impurities of local concentration\n$\\delta x(r)$ we have, similarly,\n$U_1\\Sigma_{r>}\\delta x(r)+U_2\\Sigma_{r<}\\delta x(r)$. Apart from a constant\nindependent of the position of the interface, this may be written as\n$L$ times $\\Sigma_{r>}\\delta x(r)-\\Sigma_{r<}\\delta x(r)$, exactly the\nsame form as for the RFIM. \n\nWe may therefore set up a dictionary between these two cases, in which\nthe thermal variables of the random bond system are related to the\nmagnetic variables of the RFIM:\n\\begin{eqnarray*}\n\\sigma\/kT_c&\\longleftrightarrow&J\/kT\\\\\n(L\/kT_c)\\,x&\\longleftrightarrow& h_{RF}\/kT\\\\\n(T-T_c)\\,L&\\longleftrightarrow&H\\cdot M,\n\\end{eqnarray*}\nwhere the last relation is between the fields $(T-T_c)$ and a \\em\nuniform \\em magnetic field $H$ which respectively distinguish between\nthe two phases. Although the above mapping may seem ill-defined in its\nuse of the local energy density as a kind of order parameter, it may be\nmade completely explicit, for example, for the $q$-state Potts model\nthrough the mapping to the random cluster model, where $\\sigma\\sim\nL\\sim\\ln q$ for large $q$ \\cite{CJ}.\n\nThe interfacial model of the RFIM has been studied extensively\n\\cite{RFIMinterface} and, in particular, RG equations have been derived\nwhich are asymptotically exact in the limit of low temperatures and\nweak randomness, just where our mapping is valid. For $d=2$ the variable\n$h_{RF}\/J\\sim xL\/\\sigma$ is (marginally) relevant, and, just as this\ndestroys the spontaneous magnetisation for the RFIM, the latent heat\nvanishes for the random bond case, in accord with the Aizenman-Wehr\ntheorem \\cite{AW}. For the RFIM, the RG\ntrajectories flow towards a paramagnetic fixed point at which the\ncorrelation length $\\xi\\sim\\exp({\\rm const}(J\/h_{RF})^3)$ is finite, but\nnote that this is outside the region where the mapping to the random\nbond problem is valid. We cannot conclude, therefore, anything about the\nnature of the latter's true critical behaviour from this argument. In fact,\nnumerical and other analytic studies indicate that the actual\ncorrelation length is divergent, and hence $\\xi$ is merely a crossover \nlength in this case.\n\nThe predictions are more interesting for $d>2$, when the RFIM exhibits a\ncritical fixed point at $kT\/J=0$ and a finite value of $h_{RF}\/J$.\n\\begin{figure}\n\\centerline{\n\\epsfxsize=3in\n\\epsfbox{fig1.ps}}\n\\caption{Critical surface for $d>2$, constructed by analogy with the\nphase diagram of the RFIM. The axes are $(1\/\\sigma,x)$ in the\nnotation of this talk. The shaded region is first-order,\nbounded by a line of tricritical points, along which the RG flows go\ntowards the fixed point at $R$. The upper boundary is the percolation\nlimit, and the partially dashed curve is a conjectured line of fixed\npoints describing the continuous random critical behaviour.}\n\\label{fig1}\n\\end{figure}\nAs shown in Fig.~\\ref{fig1}, there is now a region in which the\nspontaneous magnetisation (latent heat) is non-vanishing as the sign of\nthe uniform field ($T-T_c$) is changed. This region is bounded, in the\ncase of the RFIM by the critical curve, close to which the critical\nbehaviour is determined by the zero-temperature fixed point. Similarly,\nthe first-order region of the random bond system will be bounded by a\nline of \\em tricritical \\em points, above which (presumably) the transition\nbecomes continuous. Since the fixed point occurs in the region where the\nmapping between the two systems is valid, we may infer some of the\ntricritical exponents from those of the RFIM \\cite{CJ}. For example, the latent\nheat should vanish as $(x_c-x)^\\beta$, where $\\beta$ is the usual\nmagnetisation exponent of the RFIM. Similarly, the correlation length on\nthe critical surface should diverge as $x\\to x_c$\nwith the usual exponent $\\nu$ of the RFIM. But the behaviour for \n$T\\not=T_c$ is related to the \\em magnetic \\em properties of the RFIM, and is\ncomplicated by\nthe fact that the temperature at the RFIM fixed point is dangerously\nirrelevant, with an RG eigenvalue $-\\theta$ which is responsible for\nthe violation of hyperscaling $\\alpha=2-(d-\\theta)\\nu$ in that model. \nAs a result, for example, the correlation length in the random bond\nmodel at $x=x_c$ diverges as $(T-T_c)^{1\/y}$, with \n$y=d-\\theta-\\beta\/\\nu$.\n\n\\section{Results for the Potts model.}\nAs is well-known, the pure $q$-state Potts model undergoes a first-order\ntransition for $q>4$ in $d=2$, and it is a relatively simple system to\nstudy numerically in the random bond case. By choosing a suitable\ndistribution of randomness, one can fix the model to be self-dual so that\nthe critical point is determined exactly. One method, which is very\neffective for pure two-dimensional critical system, is to study the\nfinite-size scaling behaviour of the eigenvalues $\\Lambda_i$ of the\ntransfer matrix in a strip of width $N$. By conformal invariance\n\\cite{JCconf}, these\nare related to the scaling dimensions $x_i$ by \n$2\\pi x_i\/N\\sim\\ln(\\Lambda_0\/\\Lambda_i)$ as $N\\to\\infty$. For the Potts\nmodel, it is possible \\cite{JC} to write the transfer matrix in the so-called\nconnectivity basis \\cite{BN}, allowing $q$ to appear as an easily tunable\nvariable.  In the random case, however, the transfer matrices for\ndifferent rows do not commute, and the role of the $\\Lambda_i$ is\ntaken by the Lyapunov exponents which govern the average growth rate\nof the norm of vectors under the action of the transfer matrices.\nThe asymptotic behaviour of the correlation functions is related to\nthese exponents as for the pure case. \nIn order to use conformal invariance, it is necessary to assume\ntranslational invariance which is only recovered after quenched\naveraging. However, the Lyapunov exponents themselves are not\nself-averaging, only their logarithms. This is related to the phenomenon\nof multi-scaling, whereby the average of the $p$th power of the\ncorrelation function is governed by an exponent $x^{(p)}$ which is not\nlinear in $p$. Since this occurs in most random systems, however, I shall\nnot treat the problem in detail here. It still proves possible, effectively\nby measuring the\nwhole distribution of the $\\ln \\Lambda_i$, to extract the decay of the\naverage correlation function, and hence, for example, the magnetic\nexponent $x_1\\equiv\\beta\/\\nu$. The results from our study \\cite {JC}\nare shown in \nFig.~\\ref{fig2}. We see that the value of $x_1$ appears to increase\n\\begin{figure}\n\\centerline{\n\\epsfxsize=4in\n\\epsfbox{fig2.ps}}\n\\caption{Measured values for the magnetic exponent $\\beta\/\\nu$ of the\nrandom-bond $q$-state Potts model for $d=2$, from Refs.~(9,12). The solid\ncurve is the exact value for the pure model for $q\\leq4$, and the\nother is the extrapolated $(q-2)$-expansion of Dotsenko et al. [14].}\n\\label{fig2}\n\\end{figure}\nsteadily with $q$. For $q<3$ it agrees with the analytic\n$(q-2)$-expansion \\cite{Ludwig,Dot}. \nThere appears to be no break at $q=4$ where the pure\ntransition becomes first-order. Our results disagree with earlier Monte\nCarlo work \\cite{CFL} for $q=8$ where a number close to the Ising value of\n$\\frac18$ was reported. However, more recent Monte Carlo results by\nPicco \\cite{Picco} find $x_1=0.150-0.155$ for $q=8$ (and $0.185\\pm0.005$\nfor $q=64$, while Chatelain and Berche \\cite{CB} \nreport $x_1=0.153\\pm0.003$ for $q=8$.\nThe small discrepancy with our results may be explained by a\ncareful study of the crossover behaviour which shows that stronger\nrandomness (beyond the reach of our methods) must be considered\nas $q$ increases \\cite{Picco}.\n\nAll studies report a thermal exponent $\\nu\\approx1$. This approximately\nsaturates the lower bound proved by Chayes et al \\cite {Chayes}. It\nhas been suggested \\cite{Davis} that, in general, this might be a result of the\naveraging procedure and that in some random systems \nthe `true' value of $\\nu$ might be less than unity. However, it can\nshown that this is not the case here.\n\n\\section{Analytic results for weak first-order transitions.}\nAlthough the $q$-state Potts model is relatively simple to analyse\nnumerically, like most other first-order transitions it is difficult to\nstudy in any kind of perturbative RG approach, since the randomness is\nstrongly relevant. However, this is\nnot always the case for systems which exhibit weak `fluctuation-driven'\nfirst-order transitions. An example is afforded by $N$ Ising models\nwith spins $s_i(r)$,\ncoupled through their energy densities, with a hamiltonian\n\\begin{equation}\n{\\cal H}=-\\sum_{r,r'}J(r,r')\\sum_is_i(r)s_i(r')+\ng\\sum_{r,r'}\\sum_{i\\not=j}s_i(r)s_i(r')s_j(r)s_j(r')\n\\end{equation}\nFor uniform couplings $J$, this exhibits a first-order transition when\n$N>2$, and in fact, on the critical surface is equivalent to the\nGross-Neveu model, which may be analysed nonperturbatively to show that\nthe correlation length is $\\xi\\sim\\exp({\\rm const}\/(N-2)g)$ \\cite{GN}. \nEven with\nrandom bonds $J(r,r')$ of strength $\\Delta$\nthe one-loop RG equations may be derived \nBy very simple combinatorial methods \\cite{JCbook}, to give \\cite{RFD}\n\\begin{eqnarray}\ndg\/d\\ell&=&4(N-2)g^2-8g\\Delta+\\cdots\\label{flows1}\\\\\nd\\Delta\/d\\ell&=& -8\\Delta^2+8(N-1)g\\Delta+\\cdots\\label{flows2}\n\\end{eqnarray}\nThese flows are illustrated in Fig.~\\ref{fig3}.\n\\begin{figure}\n\\centerline{\n\\epsfxsize=5in\n\\epsfbox{fig3.eps}}\n\\caption{RG flows in the critical surface for $N$ coupled Ising models,\nfrom Ref.~(21).\nFor non-zero randomness the trajectories curve back towards the\ndecoupled pure fixed point.}\n\\label{fig3}\n\\end{figure}\nWhen $\\Delta=0$, $g$ runs away to infinity, in a manner consistent with\nthe non-perturbative result for $\\xi$. When randomness is added,\nhowever, it is marginally relevant, and in fact the flows are quite\nsimilar to those found for the RFIM quoted earlier if we identify \nthe interfacial tension $\\sim\\xi$.\nHowever, in this case, we also see where the flows\nend: in this case at the critical\nfixed point corresponding to $N$ decoupled pure Ising models. (This is\nthe only example I know of where the infrared and ultraviolet fixed\npoints of a set of RG flows are the same.) There have been various\ngeneralisations of this calculation to coupled Potts models \\cite{gen}. In most\ncases the addition of randomness induces flow towards a critical fixed\npoint which is perturbatively accessible. In higher dimensions, however,\ndifferent outcomes are possible. Adding randomness to models in\n$4-\\epsilon$ dimensions with cubic anisotropy appears not to change\ntheir fluctuation-driven first-order character \\cite{RFD}. However, for impure \n$n$-component superconductors in $4-\\epsilon$ dimensions there is a\ncritical concentration above which the transition becomes continuous\n\\cite{CBoy}.\n\n\\section{Outlook}\nThere are still many open questions in this relatively little explored\narea. A lot more needs to be learned about the nature of the universal\ncritical behaviour (if indeed it is universal), both in $d=2$ and for\n$d>2$ when the randomness is sufficiently strong. In the former case, it\nmay be possible to solve the problem by conformal field theory methods.\nEven the limit of large $q$ appears non-trivial, however. Similar ideas\napply to quantum phase transitions and may have relevance for the\nquantum Hall effect. Most importantly, it should be possible to find\nexperimental systems which realise some of the predictions discussed\nabove. There are, after all, many three-dimensional examples of\nfirst-order transitions. However, it should be stressed that the\nrandomness should couple only to the local energy density, not to the\norder parameter, otherwise this becomes the random field problem. \nIt is also important to ensure that the tricritical\nbehaviour is driven by the effects discussed and not by some simple\nmean-field mechanism (which would give rise to mean-field tricritical\nexponents in $d=3$, part from logs.) Finally it is important to\nunderstand the dynamics of these systems: it may be that, like the\nrandom field problem, they are plagued by logarithmically slow time\nscales close to the critical point \\cite{RFdyn}.\n\nI am grateful to Jesper Jacobsen for his continuing collaboration on\nthis problem. This work was supported in part by EPSRC Grant GR\/J78327.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{Section_1}\n\nIn this paper, we consider the optimal control problem of \n\\begin{align}\t\n\\label{eq_intro_1_1}\n\\textbf{(P)}~~J(x_0,u(\\cdot)) = \\int_0^T l(r,x(r),u(r) \\dd r + h(x_0,x(T)),\n\\end{align}\nsubject to the following state equation with $\\alpha \\in (0,1)$,\n\\begin{align}\n\\label{eq_intro_1_2}\nx(t) = x_0 + \\int_0^t \\frac{f(t,s,x(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align}\nand the state constraints\n\\begin{align}\n\\label{eq_intro_1_3}\n\\begin{cases}\n(x_0, x(T)) \\in F, & \\textrm{(terminal state constraint)},\\\\\n\tG^i(t,x(t)) \\leq 0,~\\forall t \\in [0,T],~ i=1,\\ldots,m, & \\textrm{(inequality state constraint).} \n\\end{cases}\n\\end{align}\nThe precise problem statement of \\textbf{(P)} including the space of admissible controls and the standing assumptions for (\\ref{eq_intro_1_1})-(\\ref{eq_intro_1_3}) is given in Section \\ref{Section_2_2}. We mention that the optimal control problems with state constraints capture various practical aspects of systems in science, biology, engineering, and economics \\cite{Hartl_SICON_1995, Arutyunov_JOTA_2020, Evans_2010, Vinter_book}.\n\nThe state equation in (\\ref{eq_intro_1_2}) is known as a class of Volterra integral equations. The main feature of Volterra integral equations is the effect of memories, which does not appear in ordinary (state) differential equations. In fact, Volterra integral equations of various kinds have been playing an important role in modeling and analyzing of practical physical, biological, engineering, and other phenomena that are governed by memory effects \\cite{Burton_book}. We note that one major distinction between (\\ref{eq_intro_1_2}) and other existing Volterra integral equations is that (\\ref{eq_intro_1_2}) has two different kernels $\\frac{f(t,s,x,u)}{(t-s)^{1-\\alpha}}$ and $g(t,s,x,u)$, in which the first kernel $\\frac{f(t,s,x,u)}{(t-s)^{1-\\alpha}}$ becomes singular at $s=t$, while the second kernel $g(t,s,x,u)$ is nonsingular.  In fact, $\\alpha \\in (0,1)$ in the singular kernel of (\\ref{eq_intro_1_2}) determines the amount of the singularity, in which the large singular behavior occurs with small $\\alpha \\in (0,1)$. \n\nOptimal control problems for various kinds of Volterra integral equations via the maximum principle have been studied extensively in the literature; see \\cite{Vinokurov_SICON_1969, Angell_JOTA_1976, Kamien_RES_1976, Medhin_JMAA_1988, Carlson_JOTA_1987, Burnap_IMA_Control_1999, Vega_JOTA_2006, Bonnans_Vega_JOTA_2013, Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Bonnans_SVA_2010} and the references therein. Specifically, the first study on optimal control for Volterra integral equations (using the maximum principle) can be traced back to \\cite{Vinokurov_SICON_1969}. Several different formulations (with\/without state constraints, with\/without delay, with\/without additional equality and\/or inequality constraints) of optimal control for Volterra integral equations and their generalizations are reported in  \\cite{Angell_JOTA_1976, Kamien_RES_1976, Medhin_JMAA_1988, Carlson_JOTA_1987, Burnap_IMA_Control_1999, Vega_JOTA_2006, Belbas_AMC_2007, Bonnans_SVA_2010}. Some recent progress in different directions including the stochastic framework can be found in \\cite{Bonnans_Vega_JOTA_2013, Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Wang_ESAIM_2018, Hamaguchi_Arvix_2021}. We note that the above-mentioned existing works considered the situation with nonsingular kernels only in Volterra integral equations, which corresponds to $f \\equiv 0$ in (\\ref{eq_intro_1_2}). Hence, the problem settings in the earlier works can be viewed as a special case of \\textbf{(P)}.\n\nRecently, the optimal control problem for Volterra integral equations having singular kernels only (equivalently, $g \\equiv 0$ in (\\ref{eq_intro_1_2})) was studied in \\cite{Lin_Yong_SICON_2020}. Due to the presence of the singular kernel, the technical analysis including the maximum principle (without state constraints) in \\cite{Lin_Yong_SICON_2020} should be different from that of the existing works mentioned above. In particular, the proof for the well-posedness and estimates of Volterra integral equations in \\cite[Theorem 3.1]{Lin_Yong_SICON_2020} require a new type of the Gronwall's inequality. Furthermore, the maximum principle (without state constraints) in \\cite[Theorem 4.3]{Lin_Yong_SICON_2020} needs a different duality analysis for variational and adjoint integral equations, induced by the variational approach. More recently, linear-quadratic optimal control problem (without state constraints) for linear Volterra integral equations with singular kernels only was studied in \\cite{Han_Arxiv_2021}.\n\n\nWe note that Volterra integral equations having singular and nonsingular kernels are strongly related to classical state equations and fractional order differential equations in the sense of Riemann-Liouville or Caputo \\cite{Kilbas_book}. For the case with singular kernels only, a similar argument is given in \\cite[Section 3.2]{Lin_Yong_SICON_2020}. In particular, let $\\mathcal{D}_{\\alpha} ^C [x(\\cdot)]$ be the fractional derivative operator of order $\\alpha \\in (0,1)$ in the sense of Caputo \\cite[Chapter 2.4]{Kilbas_book}. Then applying \\cite[Theorem 3.24 and Corollary 3.23]{Kilbas_book} to (\\ref{eq_intro_1_2}) yields\n\\begin{subequations}\n\\begin{align}\n\\label{eq_intro_1_4}\n\\mathcal{D}_{\\alpha}^C [x(\\cdot)](t) = f(t,x(t),u(t)) & ~\\Leftrightarrow~ x(t) = x_0 + \\frac{1}{\\Gamma(\\alpha)} \\int_0^t \\frac{f(s,x(t),u(s))}{(t-s)^{1-\\alpha}} \\dd s,~ \\textrm{a.e.}~ t \\in [0,T], \\\\\n\\label{eq_intro_1_5}\n\\frac{ \\dd x(t)}{\\dd t} = g(t,x(t),u(t)) & ~\\Leftrightarrow~ x(t) = x_0 + \\int_0^t g(s,x(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align}\n\\end{subequations}\nwhere $\\Gamma(\\cdot)$ is the gamma function. Note that while (\\ref{eq_intro_1_4}) is a class of fractional differential equations in the sense of Caputo, (\\ref{eq_intro_1_5}) is a classical ordinary differential equation.\nInstead of $\\mathcal{D}_{\\alpha}^C [x(\\cdot)]$ in (\\ref{eq_intro_1_4}), we may use the fractional derivative of order $\\alpha \\in (0,1)$ in the sense of Riemann-Liouville \\cite[Chapter 2.1 and Theorem 3.1]{Kilbas_book}. Hence, we observe that (\\ref{eq_intro_1_4}) and (\\ref{eq_intro_1_5}) are special cases of our state equation in (\\ref{eq_intro_1_2}). This implies that the state equation in (\\ref{eq_intro_1_2}) is able to describe various types of differential equations including combinations of fractional (in Riemann-Liouville- or Caputo-type) and ordinary differential state equations. We also mention that there are several different results on optimal control for fractional differential equations; see \\cite{Agrawal_ND_2004, Bourdin_arvix_2012, Kamocki_AMC_2014,  Gomoyunov_SICON_2020} and the references therein.\n\nThe aim of this paper is to study the optimal control problem stated in \\textbf{(P)}. As noted above, since (\\ref{eq_intro_1_2}) has both singular and nonsingular kernels, when $f \\equiv 0$, (\\ref{eq_intro_1_2}) is reduced to the Volterra integral equation with singular kernels only studied in \\cite{Lin_Yong_SICON_2020}. Since \\cite{Lin_Yong_SICON_2020} did not consider the state-constrained control problem, \\textbf{(P)} can be viewed as a generalization of \\cite{Lin_Yong_SICON_2020} to the state-constrained control problem for Volterra integral equations having singular and nonsingular kernels. Moreover, with $g \\equiv 0$, (\\ref{eq_intro_1_2}) is reduced to the classical Volterra integral equation with nonsingular kernels only (e.g. \\cite{Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Burnap_IMA_Control_1999, Medhin_JMAA_1988, Kamien_RES_1976, Bonnans_SVA_2010}). Hence, \\textbf{(P)} also covers the optimal control problems for Volterra integral equations with nonsingular kernels only.\n\n\nUnder mild assumptions on $f$ and $g$, we first obtain the well-posedness (in $L^p$ and $C$ spaces) and precise estimates for generalized Volterra integral equations of (\\ref{eq_intro_1_2}) when the initial condition of (\\ref{eq_intro_1_2}) also depends on $t$ (see Lemma \\ref{Lemma_2_1} and Appendix \\ref{Appendix_B}). This requires the extensive use of the generalized Gronwall's inequality with singular and nonsingular kernels, together with the several different regularities of integrals having singular and nonsingular integrands, where their results (including the generalized Gronwall's inequality) are obtained in Appendix \\ref{Appendix_A}. Note that the main technical analysis for the well-posedness and estimates of (\\ref{eq_intro_1_2}) (see Lemma \\ref{Lemma_2_1} and Appendix \\ref{Appendix_B}) should be different from those for the case with singular kernels only in \\cite{Lin_Yong_SICON_2020}, as the presence of the singular and nonsingular kernels in (\\ref{eq_intro_1_2}) causes various cross coupling characteristics.\n\nNext, we obtain the maximum principle for \\textbf{(P)} (see Theorem \\ref{Theorem_3_1}). Due the presence of the state constraints in (\\ref{eq_intro_1_3}) and the control space being only a separable metric space (that does not necessarily have any algebraic structure), the derivation of the maximum principle in this paper must be different from that for the unconstrained case with singular kernels only studied in \\cite[Theorem 4.3]{Lin_Yong_SICON_2020}. Specifically, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance functions and the generalized Gronwall's inequality (see Appendix \\ref{Appendix_A}), to establish the duality analysis for Volterra-type variational and adjoint equations, which leads to the desired necessary conditions for optimality. \nFurthermore, as (\\ref{eq_intro_1_2}) has both singular and nonsingular kernels, the proof for the maximum principle of this paper should be more involved than that for the classical state-constrained maximum principle without singular kernels studied in the existing literature (e.g. \\cite[Theorem 1]{Bonnans_SVA_2010} and  \\cite{Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Burnap_IMA_Control_1999, Medhin_JMAA_1988, Kamien_RES_1976}). In fact, the analysis of the maximum principle for state-constrained optimal control problems is entirely different from that of the problems without state constraints \\cite{Hartl_SICON_1995, Bourdin_arxiv_2016}. We also note that different from existing works for classical optimal control of Volterra integral equations (e.g. \\cite{Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Burnap_IMA_Control_1999, Medhin_JMAA_1988, Kamien_RES_1976, Bonnans_SVA_2010}), our paper does not assume the differentiability of (singular and nonsingular) kernels in $(t,s,u)$ (time and control variables) and the convexity of the control space.\n\nThe rest of this paper is organized as follows. The notation and the problem statement of \\textbf{(P)} are given in Section \\ref{Section_2}. The statement of the maximum principle for \\textbf{(P)} is provided in Section \\ref{Section_3}. Some examples of \\textbf{(P)} are studied in Section \\ref{Section_5}. The proof of the maximum principle for \\textbf{(P)} is given in Section \\ref{Section_4}. Appendices \\ref{Appendix_A}-\\ref{Appendix_D} give some preliminary results and lemmas including the well-posedness and estimates of (\\ref{eq_intro_1_2}).\n\n\n\\section{Notation and Problem Formulation}\\label{Section_2}\n\n\n\\subsection{Notation}\\label{Section_2_1}\n\nLet $\\mathbb{R}_+$ and $\\mathbb{R}_-$ be the sets of nonnegative and nonpositive numbers, respectively. Let $\\mathbb{R}^n$ be the $n$-dimensional Euclidean space, where $\\langle x,y \\rangle_{\\mathbb{R}^n \\times \\mathbb{R}^n} := x^\\top y$ is the inner product and $|x|_{\\mathbb{R}^n} := \\langle x,x \\rangle^{1\/2}_{\\mathbb{R}^n \\times \\mathbb{R}^n}$ is the norm for $x,y \\in \\mathbb{R}^n$. We sometimes write $\\langle \\cdot,\\cdot \\rangle$ and $|\\cdot|$ when there is no confusion. For $A \\in \\mathbb{R}^{m \\times n}$, $A^\\top$ denotes the transpose of $A$. Let $I_{n}$ be an $n \\times n$ identity matrix.\nLet $\\Delta := \\{(t,s) \\in [0,T] \\times [0,T]~|~ 0 \\leq s \\leq t \\leq T \\}$ with $T > 0$ being a fixed horizon. Define $\\mathds{1}_{A}(\\cdot)$ by the indicator function of any set $A$. A modulus of continuity is any increasing real-valued function $\\omega :[0,\\infty) \\rightarrow [0,\\infty)$, vanishing at $0$, i.e., $\\lim_{t \\downarrow 0} \\omega(t) = 0$, and continuous at $0$. In this paper, the constant $C$ denotes the generic constant, whose value is different from line to line.\n\nFor any differentiable function $f:\\mathbb{R}^n \\rightarrow \\mathbb{R}^l$, let $f_x : \\mathbb{R}^n \\rightarrow \\mathbb{R}^{l \\times n}$ be the partial derivative of $f$ with respect to $x \\in \\mathbb{R}^n$. Note that $f_x =  \\begin{bmatrix}\n\tf_{1,x}^\\top & \\cdots & f_{l,x}^\\top \\end{bmatrix}^\\top$ with $f_{j,x} \\in  \\mathbb{R}^{1 \\times n}$, and when $l=1$, $f_x \\in \\mathbb{R}^{1 \\times n}$. For any differentiable function $f : \\mathbb{R}^n \\times \\mathbb{R}^l \\rightarrow \\mathbb{R}^l$, $f_{x} : \\mathbb{R}^n \\times \\mathbb{R}^l \\rightarrow \\mathbb{R}^{l \\times n}$ for $x \\in \\mathbb{R}^n$, and $f_y: \\mathbb{R}^n \\times \\mathbb{R}^l \\rightarrow \\mathbb{R}^{l \\times l}$ for $y \\in \\mathbb{R}^l$. \n\t\n\t\n\nFor $1 \\leq p < \\infty$, we define the following spaces:\n\\begin{itemize}\n\\item $L^p([0,T];\\mathbb{R}^n)$: the space of functions $\\psi:[0,T] \\rightarrow \\mathbb{R}^n$ such that $\\psi$ is measurable and satisfies $\\|\\psi(\\cdot)\\|_{L^p([0,T];\\mathbb{R}^n)} := \\Bigl ( \\int _0^T |\\psi(t)|^p_{\\mathbb{R}^n} \\dd t \\Bigr )^{1\/p}$;\n\\item $L^{\\infty}([0,T];\\mathbb{R}^n)$: the space of functions $\\psi:[0,T] \\rightarrow \\mathbb{R}^n$ such that $\\psi$ is measurable and satisfies $\\|\\psi(\\cdot)\\|_{L^{\\infty}([0,T];\\mathbb{R}^n)} :=  \\esssup_{t \\in [0,T]} |\\psi(t)|_{\\mathbb{R}^n} < \\infty$;\n\\item $C([0,T];\\mathbb{R}^n)$: the space of functions $\\psi:[0,T] \\rightarrow \\mathbb{R}^n$ such that $\\psi$ is continuous and satisfies $\\|\\psi(\\cdot)\\|_{\\infty} := \\sup_{t \\in [0,T]} |\\psi(t)|_{\\mathbb{R}^n} < \\infty $;\n\\item $\\textsc{BV}([0,T];\\mathbb{R}^n)$: the space of functions $\\psi:[0,T] \\rightarrow \\mathbb{R}^n$ such that $\\psi$ is a function with bounded variation on $[0,T]$.\n\\end{itemize}\nThe norm on $\\textsc{BV}([0,T];\\mathbb{R}^n)$ is defined by $\\|\\psi(\\cdot)\\|_{\\textsc{BV}([0,T];\\mathbb{R}^n)} := \\psi(0) + \\textsc{TV}(\\psi)$, where $\\textsc{TV}(\\psi) := \\sup_{(t_k)_k} \\bigl \\{ \\sum_{k} |\\psi(t_{k+1}) - \\psi(t_k)|_{\\mathbb{R}^n} \\bigr \\} < \\infty$ with the supremum being taken by all partitions of $[0,T]$. Let $\\textsc{NBV}([0,T];\\mathbb{R}^n)$ be the space of functions $\\psi(\\cdot) \\in  \\textsc{BV}([0,T];\\mathbb{R}^n)$ such that $\\psi(\\cdot) \\in  \\textsc{BV}([0,T];\\mathbb{R}^n)$ is normalized, i.e., $\\psi(0) = 0$ and $\\psi$ is left continuous. The norm on $\\textsc{NBV}([0,T];\\mathbb{R}^n)$ is defined by $\\|\\psi(\\cdot)\\|_{\\textsc{NBV}([0,T];\\mathbb{R}^n)} := \\textsc{TV}(\\psi)$. When $\\psi(\\cdot) \\in \\textsc{NBV}([0,T];\\mathbb{R})$ is monotonically nondecreasing, we have $\\|\\psi(\\cdot)\\|_{\\textsc{NBV}([0,T];\\mathbb{R}} = \\psi(T)$. Note that both $(\\textsc{BV}([0,T];\\mathbb{R}^n), \\|\\cdot\\|_{\\textsc{BV}([0,T];\\mathbb{R}^n)})$ and $(\\textsc{NBV}([0,T];\\mathbb{R}^n), \\|\\cdot\\|_{\\textsc{NBV}([0,T];\\mathbb{R}^n)})$ are Banach spaces. \n\n\\subsection{Problem Formulation}\\label{Section_2_2}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{state1.eps}~~~\n\\includegraphics[scale=0.35]{state2.eps}\t\n\\caption{State trajectories when $x_0 = 1$, $f(t,s,x,u) = -0.4 \\sin(2\\pi x)$, and $g(t,s,x,u) = -x$. Note that the state trajectory shows more singular behavior with small $\\alpha \\in (0,1)$.}\n\\label{Fig_1_1_1_1}\n\\end{figure}\n\n\nConsider the following Volterra integral equation:\n\\begin{align}\n\\label{eq_1}\nx(t) = x_0 + \\int_0^t \\frac{f(t,s,x(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align}\nwhere $\\alpha \\in (0,1)$ is the parameter of singularity, $x(\\cdot) \\in \\mathbb{R}^n$ is the state with the initial condition $x_0 \\in \\mathbb{R}^n$, and $u(\\cdot) \\in U \\subset \\mathbb{R}^d$ is the control with $U$ being the control space. In (\\ref{eq_1}), $\\frac{f(t,s,x,u)}{(t-s)^{1-\\alpha}}$ is the singular kernel (with the singularity appearing at $s=t$) and $g(t,s,x,u)$ is the nonsingular kernel, where $f,g:\\Delta \\times \\mathbb{R}^n \\times U \\rightarrow \\mathbb{R}^n$ are generators. We note that $\\alpha \\in (0,1)$ determines the level of singularity of (\\ref{eq_1}); see Figure \\ref{Fig_1_1_1_1}. Notice also that $f$ and $g$ are dependent on two time parameters, $t$ and $s$, where their roles are different. While $t$ is the outer time variable to determine the current time, $s$ is the inner time variable describing the path or memory of the state equation from $0$ to $t$. We sometimes use the notation $x(\\cdot;x_0,u) := x(\\cdot)$ to emphasize the dependence on the initial state and the control.\n\n\\begin{assumption}\\label{Assumption_2_1}\n\\begin{enumerate}[(i)]\n\\item $(U,\\rho)$ is a separable metric space, where $U \\subset \\mathbb{R}^d$ and $\\rho$ is the metric induced by the standard Euclidean norm $|\\cdot|_{\\mathbb{R}^d}$;\n\\item There is a constant $K \\geq 0$ such that for some modulus of continuity $\\omega$,\n\\begin{align*}\n\\begin{cases}\n\t|f(t,s,x,u) -  f(t^\\prime,s,x,u)| + |g(t,s,x,u) -  g(t^\\prime,s,x,u)| \\leq K \\omega(|t-t^\\prime|)(1+|x|), \\\\\n\t\t~~~~~~~~~~ \\forall (t,s),(t^\\prime,s) \\in \\Delta, ~x \\in \\mathbb{R}^n,~ u \\in U; \n\\end{cases}\t\n\\end{align*}\n\\item For $p > \\frac{1}{\\alpha}$, there are nonnegative functions $K_0(\\cdot) \\in L^{ \\frac{1}{\\alpha} +}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{ \\frac{p}{\\alpha p - 1} +}([0,T];\\mathbb{R})$, where $L^{p+}([0,T];\\mathbb{R}^n) := \\cup_{r > p} L^{r}([0,T];\\mathbb{R}^n)$ for $1 \\leq p < \\infty$, such that\n\\begin{align*}\n\\begin{cases}\n\t|f(t,s,x,u) - f(t,s,x^\\prime,u^\\prime)| + |g(t,s,x,u) - g(t,s,x^\\prime,u^\\prime)| \\leq K(s) (|x-x^\\prime| + \\rho(u,u^\\prime)) , \\\\\n\t~~~~~~~~~~ \\forall (t,s) \\in \\Delta,~ x,x^\\prime \\in \\mathbb{R}^n,~ u,u^\\prime \\in U, \\\\\n\t|f(t,s,0,u)| + |g(t,s,0,u)| \\leq K_0(s),~ \\forall (t,s) \\in \\Delta,~ u \\in U;\n\\end{cases}\t\n\\end{align*}\n\\item $f$ and $g$ are of class $C^1$ (continuously differentiable) in $x$, which are bounded and continuous in $(x,u) \\in \\mathbb{R}^n \\times U$.\n\\end{enumerate}\t\t\n\\end{assumption}\n\nFor $p \\geq 1$ and $u_0 \\in U$, the space of admissible controls for (\\ref{eq_1}) is defined by\n\\begin{align*}\n\\mathcal{U}^p[0,T] = \\Bigl \\{u:[0,T] \\rightarrow U~|~ \\textrm{$u$ is measurable in $t \\in [0,T]$} ~\\&~ \\rho(u(\\cdot),u_0) \\in L^p([0,T];\\mathbb{R}_+) \\Bigr \\}\n\\end{align*}\nWe state the following lemma; the proof is provided in Appendix \\ref{Appendix_B} (see Lemmas \\ref{Lemma_B_1} and \\ref{Lemma_B_2}).\n\n\\begin{lemma}\\label{Lemma_2_1}\nLet (i)-(iii) of Assumption \\ref{Assumption_2_1} hold. Then the following results hold:\n\\begin{enumerate}[(i)]\n\\item For any $(x_0,u(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$, (\\ref{eq_1}) admits a unique solution in $C([0,T];\\mathbb{R}^n)$, i.e., $x(\\cdot;x_0,u) \\in C([0,T];\\mathbb{R}^n)$, and there is a constant $C \\geq 0$ such that \n\\begin{align*}\n\\Bigl \\|x (\\cdot;x_0,u) \\Bigr \\|_{L^p([0,T];\\mathbb{R}^n)} \\leq C \\Bigl (1 + |x_0|_{\\mathbb{R}^n} + \\Bigl \\|\\rho(u(\\cdot),u_0) \\Bigr \\|_{L^p([0,T];\\mathbb{R}_+)} \\Bigr )\t;\n\\end{align*}\t\n\\item For any $x_0,x_0^\\prime \\in \\mathbb{R}^n$ and $u(\\cdot),u^\\prime (\\cdot) \\in \\mathcal{U}^p[0,T]$, there is a constant $C \\geq 0$ such that\n\\begin{align*}\n& \\Bigl \\|x(\\cdot;x_0,u) - \tx(\\cdot;x_0^\\prime,u^\\prime) \\Bigr \\|_{L^p([0,T];\\mathbb{R}^n)}  \\leq C  |x_0 - x_0^\\prime |_{\\mathbb{R}^n} \\\\\n&~~~~~ + C \\Biggl [ \\int_0^T \\Bigl ( \\int_0^t \\frac{|f(t,s,x(s;x_0,u),u(s)) - f(t,s,x(s;x_0,u),u^\\prime(s))|}{(t-s)^{1-\\alpha}} \\dd s \\Bigr )^p \\dd t \\Biggr]^{\\frac{1}{p}} \\\\\n&~~~~~ + C \\Biggl [ \\int_0^T \\Bigl ( \\int_0^t | g(t,s,x(s;x_0,u),u(s)) - g(t,s,x(s;x_0,u),u^\\prime(s)) | \\dd s \\Bigr)^p  \\dd t \\Biggr ]^{\\frac{1}{p}}.\n\\end{align*}\n\\end{enumerate}\n\\end{lemma}\n\n\nWe introduce the following objective functional:\n\\begin{align}\n\\label{eq_2}\nJ(x_0,u(\\cdot)) = \\int_0^T l(r,x(r),u(r) \\dd r + h(x_0,x(T)).\n\\end{align}\nThen the main objective of this paper is to solve the following optimal control problem:\n\\begin{align*}\n\\textbf{(P)}~ \\inf_{u(\\cdot) \\in \\mathcal{U}^p[0,T]} J(x_0,u(\\cdot)),~\\textrm{subject to (\\ref{eq_1}),}\n\\end{align*}\nand the state constraints given by\n\\begin{align}\n\\label{eq_3}\n\\begin{cases}\n(x_0, x(T;x_0,u)) \\in F, & \\textrm{(terminal state constraint)},\\\\\n\tG^i(t,x(t;x_0,u)) \\leq 0,~\\forall t \\in [0,T],~ i=1,\\ldots,m, & \\textrm{(inequality state constraint).} \n\\end{cases}\n\\end{align}\n\n\n\\begin{assumption}\\label{Assumption_2_2}\n\t\\begin{enumerate}[(i)]\n\t\\item $l:[0,T] \\times \\mathbb{R}^n \\times U \\rightarrow \\mathbb{R}$ is continuous in $t \\in [0,T]$, and is of class $C^1$ in $x$, which is bounded and continuous in $(x,u) \\in \\mathbb{R}^n \\times U$. Moreover, there is a constant $K \\geq 0$ such that\n\t\\begin{align*}\n\t\\begin{cases}\n\t|l(s,x,u) - l(s,x^\\prime,u^\\prime)| \\leq K (|x-x^\\prime| + \\rho(u,u^\\prime)) ,~ \\forall s \\in [0,T],~ x,x^\\prime \\in \\mathbb{R}^n,~ u,u^\\prime \\in U, \\\\\n\t|l(s,0,u)| \\leq K,~ \\forall s \\in [0,T],~ u \\in U;\n\t\\end{cases}\t\n\t\\end{align*}\n\t\\item $h : \\mathbb{R}^n \\times \\mathbb{R}^n \\rightarrow \\mathbb{R}$ is of class $C^1$ in both variables, which are bounded. Let $h_{x}$ and $h_{x_0}$ be partial derivatives of $h$ with respect to $x$ and $x_0$, respectively. Moreover, there is a constant $K \\geq 0$ such that\n\t\\begin{align*}\n\t|h(x_0,x) - h(x_0^\\prime,x^\\prime)| \\leq K ( |x_0 - x_0^\\prime |\t+ |x^\\prime - x^\\prime | ),~ \\forall (x_0,x),(x_0^\\prime,x^\\prime) \\in \\mathbb{R}^n \\times \\mathbb{R}^n;\n\t\\end{align*}\n\t\\item $F$ is a nonempty closed convex subset of $\\mathbb{R}^{2n}$;\n\t\\item For $i=1,\\ldots,m$, $G^i:[0,T] \\times \\mathbb{R}^n \\rightarrow \\mathbb{R}$ is continuous in $t \\in [0,T]$ and is of class $C^1$ in $x$, which is bounded in both variables.\n\t\\end{enumerate}\n\\end{assumption}\n \n\nUnder Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2}, the main objective of this paper is to derive the Pontryagin-type maximum principle for \\textbf{(P)}, which constitutes the necessary conditions for optimality. Note that Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2} are crucial for the well-posedness of the state equation in (\\ref{eq_1}) by Lemma \\ref{Lemma_2_1} (see also Appendix \\ref{Appendix_B}) as well as the maximum principle of \\textbf{(P)}. Assumptions similar to Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2} have been used in various optimal control problems and their maximum principles; see \\cite{Yong_book, Li_Yong_book, Lin_Yong_SICON_2020, Bettiol_CVOC_2021, Bourdin_arxiv_2016, Bonnans_SVA_2010, Carlson_JOTA_1987, Vega_JOTA_2006, Moon_Automatica_2020_1, Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Bonnans_Vega_JOTA_2013, Vinokurov_SICON_1969, Bourdin_MP_2020, Hamaguchi_Arvix_2021} and the references therein.\n\n\\section{Statement of the Maximum Principle}\\label{Section_3}\n\nWe provide the statement of the maximum principles for \\textbf{(P)}. The proof is given in Section \\ref{Section_4}. \n\n\\begin{theorem}\\label{Theorem_3_1}\n\tLet Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2} hold. Suppose that $(\\overline{u}(\\cdot), \\overline{x}(\\cdot)) \\in \\mathcal{U}^p[0,T] \\times C([0,T];\\mathbb{R}^n)$ is the optimal pair for \\textbf{(P)}, i.e., $\\overline{u}(\\cdot) \\in \\mathcal{U}^p[0,T]$ and the optimal solution to \\textbf{(P)}, where $\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u}) := \\overline{x}(\\cdot) \\in C([0,T];\\mathbb{R}^n)$ is the corresponding optimal state trajectory of (\\ref{eq_1}). Then there exists the tuple $(\\lambda,\\xi,\\theta_1,\\ldots,\\theta_m)$, where $\\lambda \\in \\mathbb{R}$, $\\xi \\in \\mathbb{R}^{2n}$ with $(\\xi_1,\\xi_2) \\in \\mathbb{R}^n \\times \\mathbb{R}^n$, and $\\theta(\\cdot) := (\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\in \\textsc{NBV}([0,T];\\mathbb{R}^m)$ with $\\theta_i(\\cdot) \\in  \\textsc{NBV}([0,T];\\mathbb{R})$ for $i=1,\\ldots,m$, such that the following conditions are satisfied:\n\\begin{itemize}\n\t\\item Nontriviality condition: the tuple $(\\lambda,\\xi,\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot))$ is not trivial, i.e., it holds that \\\\ $(\\lambda,\\xi,\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\neq 0$, where\n\t\\begin{align*}\n\t\\begin{cases}\n\t\\lambda \\geq  0, \\\\\n\t\\xi = \\begin{bmatrix}\n \t\\xi_1 \\\\\n \t\\xi_2\n \\end{bmatrix}  \\in N_F \\Bigl (\\begin{bmatrix}\n\t\\overline{x}_0 \\\\\n\t\\overline{x}(T)\n\\end{bmatrix} \\Bigr ), \\\\\n\t\\theta_i(\\cdot)\t\\in \\textsc{NBV}([0,T];\\mathbb{R})~ \\textrm{with}~  \\|\\theta_i(\\cdot)\\|_{\\textsc{NBV}([0,T];\\mathbb{R})} = \\theta_i(T) \\geq 0,~\\forall i=1,\\ldots,m,\n\t\\end{cases}\n\t\\end{align*}\n\twith $N_F(x)$ being the normal cone to the convex set $F$ defined in (\\ref{eq_4_1}), and $\\theta_i(\\cdot) \\in  \\textsc{NBV}([0,T];\\mathbb{R})$, $i=1,\\ldots,m$, being finite, nonnegative, and monotonically nondecreasing on $[0,T]$;\n\\item Nonnegativity condition:\n\t\\begin{align*}\n\t\\begin{cases}\n\t\\lambda \\geq 0, \\\\\n\t\\dd \\theta_i(s) \\geq 0,~ \\forall s \\in [0,T],~i=1, \\ldots, m,\n\t\\end{cases}\n\t\\end{align*}\n\twhere $\\dd \\theta_i$ denotes the Lebesgue-Stieltjes measure on $[0,T]$ corresponding to $\\theta_i$, $i=1,\\ldots,m$;\n\t\\item Adjoint equation: there exists a nontrivial $p(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ such that $p$ is the unique solution to the following backward Volterra integral equation having singular and nonsingular kernels:\n\\begin{align*}\n\tp(t) & = \\int_t^T \\frac{f_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(r-t)^{1-\\alpha}} p(r) \\dd r - \\mathds{1}_{[0,T)} (t) \\frac{f_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(T-t)^{1-\\alpha}} \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )^\\top \\\\\n&~~~ + \\int_t^T g_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top p(r) \\dd r\t - g_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )^\\top \\\\\n&~~~ - \\lambda l_x(t,\\overline{x}(t),\\overline{u}(t))^\\top  - \\sum_{i=1}^m G_x^{i}(t,\\overline{x}(t))^\\top \\frac{\\dd \\theta_i(t)}{\\dd t},~  \\textrm{a.e.}~ t \\in [0,T];\n\\end{align*}\n\t\\item Transversality condition:\n\t\\begin{align*}\t\n\t0 & \\leq \t\\Bigl \\langle \\xi_1, \\overline{x}_0 - y_1 \\Bigr \\rangle_{\\mathbb{R}^n \\times \\mathbb{R}^n} + \\Bigl \\langle \\xi_2, \\overline{x}(T) - y_2 \\Bigr \\rangle_{\\mathbb{R}^n \\times \\mathbb{R}^n},~ \\forall y = \\begin{bmatrix}\n\t\t\t y_1 \\\\\n\t\t\t y_2\n\t\t\\end{bmatrix} \\in F, \\\\\n\t\\int_0^T p(t) \\dd t &= \\xi_1 + \\xi_2 + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T))^\\top  +\\lambda h_x(\\overline{x}_0,\\overline{x}(T))^\\top;\n\t\\end{align*}\n\t\\item Complementary slackness condition:\n\t\\begin{align*}\t\n\t\\int_0^T G^i(t,\\overline{x}(t;\\overline{x}_0,\\overline{u})) \\dd \\theta_i(t) = 0,~ \\forall i=1,\\ldots,m,\n\t\\end{align*}\n\twhich is equivalent to\n\t\\begin{align*}\n\t\\textsc{supp}(\\dd \\theta_i(\\cdot)) \\subset \\{ t \\in [0,T]~|~ G^i(t,\\overline{x}(t;\\overline{x}_0,\\overline{u})= 0\\},~ \\forall i=1,\\ldots,m,\t\n\t\\end{align*}\n\twhere $\\textsc{supp}(\\dd \\theta_i(\\cdot))$ denotes the support of the measure $\\dd \\theta_i$, $i=1,\\ldots,m$;\n\t\\item Hamiltonian-like maximum condition: \n\t\\begin{align*}\n&\\int_t^T p(r)^\\top  \\frac{f(r,t,\\overline{x}(t),\\overline{u}(t))}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t) \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\frac{f(T,t,\\overline{x}(t),\\overline{u}(t))}{(T-t)^{1-\\alpha}} \\\\\n&~~~ + \\int_t^T p(r)^\\top  g(r,t,\\overline{x}(t),\\overline{u}(t)) \\dd r - \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )  g(T,t,\\overline{x}(t),\\overline{u}(t)) \\\\\n&~~~ - \\lambda l(t,\\overline{x}(t),\\overline{u}(t)) \\\\\n& = \\max_{u \\in U} \\Biggl \\{ \\int_t^T p(r)^\\top  \\frac{f(r,t,\\overline{x}(t),u)}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t) \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\frac{f(T,t,\\overline{x}(t),u)}{(T-t)^{1-\\alpha}} \\\\\n&~~~ + \\int_t^T p(r)^\\top  g(r,t,\\overline{x}(t),u) \\dd r - \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )  g(T,t,\\overline{x}(t),u)  \\\\\n&~~~ - \\lambda l(t,\\overline{x}(t),u) \\Biggr \\},~\\textrm{a.e. $ t \\in [0,T]$.}\n\\end{align*}\n\t\\end{itemize}\n\\end{theorem}\n\nSeveral important remarks are given below.\n\n\n\\begin{remark}\\label{Remark_3_3}\nThe adjoint equation $p$ in Theorem \\ref{Theorem_3_1} includes the (strong or distributional (or weak)) derivative of $\\theta$, which is expressed as $\\frac{\\dd \\theta_i(t)}{\\dd t}$, $i=1,\\ldots,m$. Notice that $\\theta_i$, $i=1,\\ldots,m$, are finite and monotonically nondecreasing by Theorem \\ref{Theorem_3_1}, where their corresponding Lebesgue-Stieltjes measures, denoted by $\\dd \\theta_i$, $i=1,\\ldots,m$, are nonnegative, i.e., $\\dd \\theta_i(s) \\geq 0$, for $s \\in [0,T]$ and $i=1,\\ldots,m$. In fact, $([0,T],\\mathcal{B}([0,T]))$, where $\\mathcal{B}$ is the Borel $\\sigma$-algebra generated by subintervals of $[0,T]$, is a measurable space on which the two nonnegative measures $\\dd \\theta_i$ and $\\dd t $ are defined. Then we can easily see that $\\dd \\theta_i \\ll \\dd t$, i.e., $\\dd \\theta_i$ is absolutely continuous with respect to $\\dd t$. That is, $\\dd \\theta_i(B) = 0$ whenever $\\dd t(B) =  0$ for $B \\in \\mathcal{B}([0,T])$ and $i=1,\\ldots,m$ \\cite[Appendix C]{Conway_2000_book}. By the Radon-Nikodym theorem (see \\cite[Appendix C]{Conway_2000_book}), this implies that there is a unique Radon-Nikodym derivative $\\Theta_i(\\cdot) \\in L^1([0,T];\\mathbb{R})$, $i=1,\\ldots,m$, such that \n\\begin{align*}\n\\frac{\\dd \\theta_i(t)}{\\dd t} = \\Theta_i(t)~\\Leftrightarrow~  \t\\theta_i(t) = \\int_0^t \\Theta_i(s) \\dd s,~ \\forall i=1,\\ldots,m,~  \\textrm{a.e.}~ t\\in [0,T].\n\\end{align*}\nHence, with the Radon-Nikodym derivative $\\Theta_i(\\cdot)$, $i=1,\\ldots,m$, the adjoint equation $p$ in Theorem \\ref{Theorem_3_1} can be written as\n\\begin{align}\n\\label{eq_3_1}\n\tp(t) & = \\int_t^T \\frac{f_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(r-t)^{1-\\alpha}} p(r) \\dd r - \\mathds{1}_{[0,T)} (t) \\frac{f_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(T-t)^{1-\\alpha}} \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )^\\top  \\\\\n&~~~ + \\int_t^T g_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top p(r) \\dd r\t - g_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )^\\top \\nonumber \\\\\n&~~~ - \\lambda l_x(t,\\overline{x}(t),\\overline{u}(t))^\\top  - \\sum_{i=1}^m G_x^{i}(t,\\overline{x}(t))^\\top \\Theta_i(t),~  \\textrm{a.e.}~ t \\in [0,T]. \\nonumber\n\\end{align}\nNote that the well-posedness (existence and uniqueness of the solution) of the adjoint equation in (\\ref{eq_3_1}) follows from Theorem \\ref{Theorem_3_1} (see also Lemma \\ref{Lemma_B_5} in Appendix \\ref{Appendix_B}). \n\\end{remark}\n\n\\begin{remark}\nThe strategy of the proof for Theorem \\ref{Theorem_3_1} is based on the Ekeland variational principle. Moreover, as $U$ is only the (separable) metric space and does not have any algebraic structure, the spike variation technique has to be employed. In contrast to other classical approaches, our proof needs to deal with the Volterra-type variational and adjoint equations having singular and nonsingular kernels in the variational analysis. \n\\end{remark}\n\n\\begin{remark}\\label{Remark_3_2}\nThe nontrivial tuple $(\\lambda, \\xi, \\dd \\theta_1,\\ldots,\\dd \\theta_m, p)$ is a Lagrange multiplier, which is said to be normal when $\\lambda > 0$ and abnormal when $\\lambda = 0$. In the normal case, we may assume the Lagrange multiplier to have been normalized so that $\\lambda =1$.\n\\end{remark}\n\n\n\\begin{remark}\nThe necessary conditions in Theorem \\ref{Theorem_3_1} are of interest only when the terminal state constraint is nondegenerate in the sense that $G_x^i(t,\\overline{x}(t))^\\top \\neq 0$ whenever $G^i(t,\\overline{x}(t)) = 0$ for all $t \\in [0,T]$ and $i=1,\\ldots,m$. \nA similar remark is given in \\cite[page 330, Remarks (b)]{Vinter_book} for the classical state-constrained optimal control problem for ordinary state equations. \n\\end{remark}\n\n\\begin{remark}\\label{Remark_3_4}\nWithout the state constraints in (\\ref{eq_3}), Theorem \\ref{Theorem_3_1} holds with $\\lambda = 1$, $\\xi = 0$, and $\\theta = 0$. This is equivalent to the following statement (see also \\cite[Theorem 4.3]{Lin_Yong_SICON_2020} for the case with singular kernels only): If $(\\overline{u}(\\cdot), \\overline{x}(\\cdot)) \\in \\mathcal{U}^p[0,T] \\times C([0,T];\\mathbb{R}^n)$ is the optimal pair for \\textbf{(P)}, then the following conditions hold:\n\\begin{itemize}\n\\item Adjoint equation:\t$p(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ is the unique solution of the following backward Volterra integral equation having singular and nonsingular kernels:\n\\begin{align*}\n\tp(t) & = \\int_t^T \\frac{f_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(r-t)^{1-\\alpha}} p(r) \\dd r - \\mathds{1}_{[0,T)} (t) \\frac{f_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top}{(T-t)^{1-\\alpha}}  h_x(\\overline{x}_0,\\overline{x}(T))^\\top \\\\\n&~~~ + \\int_t^T g_x(r,t,\\overline{x}(t),\\overline{u}(t))^\\top p(r) \\dd r\t - g_x(T,t,\\overline{x}(t),\\overline{u}(t))^\\top  h_x(\\overline{x}_0,\\overline{x}(T))^\\top  \\\\\n&~~~ - l_x(t,\\overline{x}(t),\\overline{u}(t))^\\top,~  \\textrm{a.e.}~ t \\in [0,T];\n\\end{align*}\n\\item Hamiltonian-like maximum condition: \n\t\\begin{align*}\n&\\int_t^T p(r)^\\top  \\frac{f(r,t,\\overline{x}(t),\\overline{u}(t))}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t)  h_x(\\overline{x}_0,\\overline{x}(T))  \\frac{f(T,t,\\overline{x}(t),\\overline{u}(t))}{(T-t)^{1-\\alpha}} \\\\\n&+ \\int_t^T p(r)^\\top  g(r,t,\\overline{x}(t),\\overline{u}(t)) \\dd r - h_x(\\overline{x}_0,\\overline{x}(T))   g(T,t,\\overline{x}(t),\\overline{u}(t))   -  l(t,\\overline{x}(t),\\overline{u}(t)) \\\\\n& = \\max_{u \\in U} \\Biggl \\{ \\int_t^T p(r)^\\top  \\frac{f(r,t,\\overline{x}(t),u)}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t) h_x(\\overline{x}_0,\\overline{x}(T))  \\frac{f(T,t,\\overline{x}(t),u)}{(T-t)^{1-\\alpha}} \\\\\n&~~~ + \\int_t^T p(r)^\\top  g(r,t,\\overline{x}(t),u) \\dd r - h_x(\\overline{x}_0,\\overline{x}(T))  g(T,t,\\overline{x}(t),u)   -  l(t,\\overline{x}(t),u) \\Biggr \\},~\\textrm{a.e. $ t \\in [0,T]$.}\n\\end{align*}\n\\end{itemize}\n\\end{remark}\n\n\\begin{remark}\nBy taking  $f \\equiv 0$ in Theorem \\ref{Theorem_3_1}, we can obtain the maximum principle for classical Volterra integral equations with nonsingular kernels only. Note that Theorem \\ref{Theorem_3_1} is different from the classical maximum principles for Volterra integral equations with nonsingular kernels only studied in the existing literature   (e.g. \\cite[Theorem 1]{Bonnans_SVA_2010} and \\cite{Dmitruk_MCRF_2017, Dmitruk_SICON_2014, Medhin_JMAA_1988}),  where Theorem \\ref{Theorem_3_1} does not need differentiability of kernels with respect to time variables and the adjoint equation in Theorem \\ref{Theorem_3_1} is expressed by the integral form. \n\\end{remark}\n\n\n\\section{Examples}\\label{Section_5}  \n\nIn this section, we provide two examples of \\textbf{(P)}.\n\n\\begin{example}\t\\label{Example_1}\n\\normalfont\nConsider the minimization of the following objective functional\n\\begin{align*}\nJ(x_0,u(\\cdot)) = \\int_0^3 [ x(s) + \\frac{1}{2} u(s)^2 ] \\dd s + (x_0 + x(3)),\t\n\\end{align*}\nsubject to the Volterra integral equation with singular and nonsingular kernels given by\n\\begin{align}\n\\label{eq_s_4_1}\nx(t) = x_0 + \\int_0^t \\frac{u(s)}{(t-s)^{1-\\alpha}} \\dd s\t+ \\int_0^t u(s) \\dd s,~\\textrm{a.e. $ t \\in [0,3]$,}\n\\end{align}\nand the state constraints\n\\begin{align}\n\\label{eq_s_4_2}\n\\begin{cases}\n\t(x_0,x(3)) \\in F=\\{10\\} \\times \\{-16\\}, & \\textrm{(terminal state constraint)}, \\\\\nG(t,x(t)) = - x(t) - \\Bigl (\\frac{t^2}{5} + 20 \\Bigr ) \\leq 0,~ \\forall t \\in [0,3], & \\textrm{(inequality state constraint)}.\n\\end{cases}\n\\end{align}\nWe assume that the control space $U$ is an appropriate sufficiently large compact subset of $\\mathbb{R}^d$ to satisfy Assumption \\ref{Assumption_2_2}.\n\n\nNote that $F$ is singleton, which is closed and convex. Hence, by (\\ref{eq_s_4_2}), we can choose $\\xi = 0$. This implies that the (candidate) optimal state trajectory holds $\\overline{x}_0 = 10$ and $\\overline{x}(3) = -10$. In addition, the transversality condition leads to $\\int_0^3 p(t) \\dd t = 2 \\lambda$.  Assume by contradiction that $\\lambda = 0$. Then the adjoint equation holds that $p(t) = \\frac{\\dd \\theta(t)}{\\dd t}$. This implies $\\int_0^3 p(t) \\dd t = \\int_0^3 \\dd \\theta(t) =  \\theta(3)  - \\theta(0) =  \\theta(3) = 0$, which, by the fact that $\\theta(0) = 0$ and $\\theta$ is monotonically nondecreasing, contradicts the nontriviality condition of $\\theta$ as well as the adjoint equation $p$ in Theorem \\ref{Theorem_3_1}. Therefore, $\\lambda \\neq 0$, and we may take the normalized case with $\\lambda = 1$. Based on the preceding discussion and by Theorem \\ref{Theorem_3_1}, the following conditions hold:\n\\begin{itemize}\n\\item Nontriviality and nonnegativity conditions:\n\\begin{itemize}\n\\item $\\lambda = 1$ and $\\theta(\\cdot)\t\\in \\textsc{NBV}([0,3];\\mathbb{R})$ with $\\|\\theta(\\cdot)\\|_{\\textsc{NBV}([0,3];\\mathbb{R})} = \\theta(3) \\geq 0$, $\\theta$ being finite and monotonically nondecreasing on $[0,3]$, and $\\dd \\theta(t) \\geq 0$ for $t \\in [0,3]$;\n\\end{itemize}\n\\item Adjoint equation:\n\\begin{align}\n\\label{eq_s_4_3}\np(t) = -  1 + \\frac{\\dd \\theta(t)}{\\dd t},~  \\textrm{a.e.}~ t \\in [0,3];\n\\end{align}\n\\item Transversality condition: \n\\begin{align}\n\\label{eq_s_4_4}\n\\int_0^3 p(t) \\dd t = \\int_0^3 \\Bigl [ -1 + \\frac{ \\dd \\theta(t)}{ \\dd t} \\Bigr ] \\dd t = -3 + \\theta(3) = 2~ \\Rightarrow~ \\theta(3) = 5 > 0;\n\\end{align}\n\\item Complementary slackness condition:\n\\begin{align}\t\n\\label{eq_s_4_5}\n\t\\int_0^3 \\Bigl [ - \\overline{x}(t) - \\Bigl (\\frac{t^2}{5} + 20 \\Bigr ) \\Bigr ] \\dd \\theta(t) = 0\t;\n\\end{align}\n\\item Hamiltonian-like maximum condition: the first-order optimality condition implies\n\\begin{align}\n\\label{eq_s_4_6}\n\\overline{u}(t)\t= - 1 + \\int_t^3 p(r) \\dd r - \\frac{\\mathds{1}_{[0,3)}(t)}{(3-t)^{1-\\alpha}} + \\int_t^3  \\frac{p(r)}{(r-t)^{1-\\alpha}} \\dd r,~\\textrm{a.e. $ t \\in [0,3]$.}\n\\end{align}\n\\end{itemize}\n\nThe numerical simulation results of Example \\ref{Example_1} with $\\alpha = 0.8$  and $\\alpha = 0.5$ are given in Figures \\ref{Fig_1} and \\ref{Fig_11111}. One can easily observe that for each case, the optimal state trajectory holds the terminal condition as well as the inequality constraint in (\\ref{eq_s_4_2}). In addition, $\\theta(\\cdot) \\in \\textsc{NBV}([0,3];\\mathbb{R})$, where $\\theta$ is finite and monotonically nondecreasing on $[0,3]$ and $\\dd \\theta(t) \\geq 0$ for $t \\in [0,3]$, and the adjoint equation holds $p(\\cdot) \\in L^p([0,3];\\mathbb{R})$. The (candidate) optimal solution is obtained from the Hamiltonian-like maximum condition in (\\ref{eq_s_4_6}). Note that the numerical approach that we adopt is as follows:\n\\begin{enumerate}\n\\setlength{\\itemindent}{0.2in}\n\\item[(s.1)] Given $\\theta(0) = 0$ and $\\theta(3) > 0$, provide a guess of the measure $\\dd \\theta$ and then construct $\\theta$;\n\\item[(s.2)] Compute the adjoint equation in (\\ref{eq_s_4_3});\n\\item[(s.3)] Compute the optimal solution in (\\ref{eq_s_4_6});\n\\item[(s.4)] Compute the controlled state equation in (\\ref{eq_s_4_1}) under the optimal solution (\\ref{eq_s_4_6}), which needs to satisfy the terminal and inequality constraints in (\\ref{eq_s_4_2}); \n\\item[(s.5)] Check the complementary slackness condition in (\\ref{eq_s_4_5}) and the transversality condition in (\\ref{eq_s_4_4});\n\\item[(s.6)] If the constraints and conditions in (s.4) and (s.5) hold, stop the algorithm. Otherwise, we iterate (s.1)-(s.5).\n\\end{enumerate}\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.28]{xsolution08.eps}\n\\includegraphics[scale=0.28]{theta08.eps}\n\\includegraphics[scale=0.28]{adjoint08.eps}\n\\includegraphics[scale=0.28]{control08.eps}\n\\includegraphics[scale=0.28]{inequality08.eps}\n\\caption{Simulation results of Example \\ref{Example_1} with $\\alpha = 0.8$.}\n\\label{Fig_1}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.28]{xsolution05.eps}\n\\includegraphics[scale=0.28]{theta05.eps}\n\\includegraphics[scale=0.28]{adjoint05.eps}\n\\includegraphics[scale=0.28]{control05.eps}\n\\includegraphics[scale=0.28]{inequality05.eps}\n\\caption{Simulation results of Example \\ref{Example_1} with $\\alpha = 0.5$.}\n\\label{Fig_11111}\n\\end{figure}\n\\end{example}\n\n\n\\begin{example}\t\n\\label{Example_2}\n\\normalfont\nWe consider the linear-quadratic problem of \\textbf{(P)} without state constraints. The state equation and the objective functional are given by\n\\begin{align*}\nx(t) & = x_0 + \\int_0^t \\frac{A_1(t,s) x(s) + B_1 u(s)}{(t-s)^{1-\\alpha}} \\dd s\t+ \\int_0^t \\Bigl [ A_2(t,s) x(s) + B_2(t,s) u(s) \\Bigr ] \\dd s, \\\\\nJ(x_0,u(\\cdot)) & = \\frac{1}{2} \\int_0^T \\Bigl [ \\langle x(s), Q(s) x(s) \\rangle + \\langle u(s), R(s) u(s) \\rangle \\Bigr ] \\dd s + \\frac{1}{2} \\langle x(T), M x(T) \\rangle,\n\\end{align*}\nwhere (ii) of Assumption \\ref{Assumption_2_1} holds and (see Lemmas \\ref{Lemma_B_1}-\\ref{Lemma_B_5_2323232} in Appendix \\ref{Appendix_B})\n\\begin{align*}\n\\begin{cases}\nA_1(\\cdot,\\cdot),A_2(\\cdot,\\cdot) \\in L^{\\infty}(\\Delta;\\mathbb{R}^{n \\times n}),~ B_1(\\cdot,\\cdot),B_2(\\cdot,\\cdot) \\in L^{\\infty}(\\Delta;\\mathbb{R}^{n \\times d}), \\\\\nQ(\\cdot) \\in L^{\\infty}([0,T];\\mathbb{R}^{n \\times n}),~ R(\\cdot) \\in L^{\\infty}([0,T];\\mathbb{R}^{d \\times d}),~ M \\in \\mathbb{R}^{n \\times n}, \\\\\nQ(t) = Q(t)^\\top  \\geq 0,~ R(t) = R(t)^\\top > c I_d~(c > 0),~ M=M^\\top,~ \\forall t \\in [0,T].\n\\end{cases}\t\n\\end{align*}\nWe further assume that the state space $X$ and the control space $U$ are appropriate sufficiently large compact subsets of $\\mathbb{R}^n$ and $\\mathbb{R}^d$, respectively, to satisfy Assumption \\ref{Assumption_2_2}.\n\nBy Remark \\ref{Remark_3_4} and the first-order optimality condition, the corresponding optimal solution is as follows:\n\\begin{align*}\t\n\\overline{u}(t) & = R(t)^{-1} \\Biggl [ \\int_t^T \\frac{B_1(r,t)^\\top p(r)}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t) \\frac{B_1(T,t)^\\top M \\overline{x}(T)}{(T-t)^{1-\\alpha}} \\\\\n&~~~~~ + \\int_t^T B_2(r,t)^\\top p(r) \\dd r - B_2(T,t)^\\top M \\overline{x}(T) \\Biggr ],~\\textrm{a.e. $ t \\in [0,T]$,}\n\\end{align*}\nwhere $p$ is the adjoint equation given by\n\\begin{align*}\np(t) & = \\int_t^T \\frac{A_1(r,t)^\\top p(r)}{(r-t)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(t)\t\\frac{A_1(T,t)^\\top M \\overline{x}(T)}{(T-t)^{1-\\alpha}} \\\\\n&~~~ + \\int_t^T A_2(r,t)^\\top p(r) \\dd r - A_2(T,t)^\\top M \\overline{x}(T) - Q(t) \\overline{x}(t),~\\textrm{a.e. $ t \\in [0,T]$.}\n\\end{align*}\nAssume that $T=2$, $x_0 = 1$, $A_1 = -1$, $A_2 = 0.2$, $B_1 = 2$, $B_2 = 0.1$, $M=1$, $R=1$, and $Q=0$. By applying the shooting method \\cite{Bourdin_MP_2020}, the numerical simulation results are obtained in Figures \\ref{Fig_3} and \\ref{Fig_323423423}.\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.28]{xsolution_linear.eps}\n\\includegraphics[scale=0.28]{adjoint_linear.eps}\n\\includegraphics[scale=0.28]{control_linear.eps}\t\n\\caption{Simulation results of Example \\ref{Example_2} with $\\alpha = 0.5$.}\n\\label{Fig_3}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.28]{xsolution_linear_001.eps}\n\\includegraphics[scale=0.28]{adjoint_linear_001.eps}\n\\includegraphics[scale=0.28]{control_linear_001.eps}\t\n\\caption{Simulation results of Example \\ref{Example_2} with $\\alpha = 0.01$.}\n\\label{Fig_323423423}\n\\end{figure}\n\\end{example}\n\n\n\\section{Proof of the Maximum Principle}\\label{Section_4}\nThis section is devoted to prove Theorem \\ref{Theorem_3_1}.\n\n\\subsection{Preliminaries on Distance Functions}\\label{Section_4_1}\n\nLet $(X,\\|\\cdot\\|_{X})$ be a Banach space. We denote $(X^*,\\|\\cdot\\|_{X^*})$ by the dual space of $(X,\\|\\cdot\\|_{X})$, where $X^*$ is the space of bounded linear functionals on $X$ with the norm given by $\\|\\psi\\|_{X^*} := \\sup_{x \\in X,~ \\|x\\|_{X} \\leq 1} \\langle \\psi,x \\rangle_{X^* \\times X}$. Here, $\\langle \\cdot,\\cdot \\rangle_{X^* \\times X}$ denotes the usual duality paring between $X$ and $X^*$, i.e., $\\langle \\psi,x \\rangle_{X^* \\times X} := \\psi(x)$. Recall that $(X^*,\\|\\cdot\\|_{X^*})$ is also a Banach space. \n\nWe first deal with the terminal state constraints in (\\ref{eq_3}). Recall that $F$ is a nonempty closed convex subsets of $\\mathbb{R}^{2n}$. Let $d_{F} : \\mathbb{R}^{2n} \\rightarrow \\mathbb{R}_{+}$ be the standard Euclidean distance function to $F$ defined by $d_{F}(x) := \\inf_{y \\in F} |x-y|_{\\mathbb{R}^{2n}}$ for $x \\in \\mathbb{R}^{2n}$. Note that $d_{F}(x) = 0$ when $x \\in F$. Then it follows from the projection theorem \\cite[Theorem 2.10]{Ruszczynski_book} that there is a unique $P_{F}(x) \\in F$ with $P_{F}(x) : \\mathbb{R}^{2n} \\rightarrow F \\subset \\mathbb{R}^{2n}$, the projection of $x \\in \\mathbb{R}^{2n}$ onto $F$, such that $d_{F}(x) = \\inf_{y \\in F} |x-y|_{\\mathbb{R}^{2n}} = |x - P_{F}(x)|_{\\mathbb{R}^{2n}}$. By \\cite[Lemma 2.11]{Ruszczynski_book}, $P_{F}(x) \\in F$ is the corresponding projection if and only if $\\langle x - P_{F}(x),y- P_{F}(x) \\rangle_{\\mathbb{R}^{2n} \\times \\mathbb{R}^{2n}} \\leq 0$ for all $y \\in F$, which leads to the characterization of $P_{F}(x)$. In view of \\cite[Definition 2.37]{Ruszczynski_book}, we have $x - P_{F}(x) \\in N_{F}(P_{F}(x))$ for $x \\in \\mathbb{R}^{2n}$, where $N_{F}(x)$ is the normal cone to the convex set $F$ at a point $x \\in \\mathbb{R}^{2n}$ defined by\n\\begin{align}\n\\label{eq_4_1}\nN_{F}(x) := \\{ y \\in \\mathbb{R}^{2n}~|~ \\langle y,y^\\prime - x \\rangle_{\\mathbb{R}^{2n} \\times \\mathbb{R}^{2n}} \\leq 0,~ \\forall y^\\prime \\in F \\}.\t\n\\end{align}\nBased on the distance function $d_F$, the terminal state constraint in (\\ref{eq_3}) can be written as\n\\begin{align*}\nd_F(\\overline{x}_0, \\overline{x}(T;\\overline{x}_0,\\overline{u})) = 0 ~\\Leftrightarrow~ \t\\begin{bmatrix}\n\t\t\\overline{x}_0 \\\\\n\t\t\\overline{x}(T;\\overline{x}_0,\\overline{u}))\t\t \t\t\n \t\\end{bmatrix} \\in F.\n\\end{align*}\n\n\n\\begin{lemma}\\label{Lemma_4_1}\nThe function $d_F(x)^2$ is Fr\\'echet differentiable on $\\mathbb{R}^{2n}$ with the Fr\\'echet differentiation of $d_F(x)^2$ at $x$ given by $D d_F(x)^2(h) = 2 \\langle x-P_F(x),h \\rangle_{\\mathbb{R}^{2n} \\times \\mathbb{R}^{2n}}$ for $h \\in \\mathbb{R}^{2n}$.\n\\end{lemma}\n\\begin{proof}\nNote that\n\\begin{align*}\nd_F(x+h)^2 - d_F(x)^2  & \\leq |x+h - P_F(x)|_{\\mathbb{R}^{2n}}^2 - |x - P_F(x)|_{\\mathbb{R}^{2n}}^2  = 2 \\langle x - P_F(x),h \\rangle + |h|_{\\mathbb{R}^{2n}}^2,\n\\end{align*}\nand by the fact that the projection operator is nonexpansive, i.e., $|P_{F}(x) - P_{F}(x^\\prime) |_{\\mathbb{R}^{2n}} \\leq |x-x^\\prime|_{\\mathbb{R}^{2n}}$, for all $x,x^\\prime \\in \\mathbb{R}^{2n}$ (see \\cite[Theorem 2.13]{Ruszczynski_book}), \n\\begin{align*}\nd_F(x)^2 - d_F(x+h)^2 & \\leq|x - P_F(x+h)|_{\\mathbb{R}^{2n}}^2 - |x+h - P_F(x+h)|_{\\mathbb{R}^{2n}}^2 \\\\\n& = - 2 \\langle x - P_F(x), h \\rangle +  2 \\langle P_F(x+h) - P_F(x), h \\rangle +  |h|_{\\mathbb{R}^{2n}}^2 \\\\\n& \\leq - 2 \\langle x - P_F(x), h \\rangle + 3|h|_{\\mathbb{R}^{2n}}^2.\n\\end{align*}\nSince $-3 |h|_{\\mathbb{R}^{2n}} \\leq \\frac{|d_F(x+h)^2 - d_F(x)^2 - 2 \\langle x - P_F(x), h \\rangle |}{|h|_{\\mathbb{R}^{2n}}}\t \\leq |h|_{\\mathbb{R}^{2n}}$, this completes the proof.\n\\end{proof}\n\n\n\nNow, we consider the inequality state constraint given in (\\ref{eq_3}). Let $\\gamma: C([0,T];\\mathbb{R}^n) \\rightarrow C([0,T];\\mathbb{R}^m)$ be defined by $\\gamma(x(\\cdot;x_0,u)) := (\\gamma_1(x(\\cdot;x_0,u)),\\ldots, \\gamma_m(x(\\cdot;x_0,u))):= G(\\cdot,x(\\cdot;x_0,u)) = \\begin{bmatrix}\n \tG^1(\\cdot,x(\\cdot;x_0,u)) & \\cdots & G^m(\\cdot,x(\\cdot;x_0,u))\n \\end{bmatrix}$. Moreover, we let $S \\subset C([0,T];\\mathbb{R}^m)$ be the nonempty closed convex cone of $C([0,T];\\mathbb{R}^m)$ defined by $S := C([0,T];\\mathbb{R}_{-}^m)$, where $\\mathbb{R}_{-}^m := \\mathbb{R}_{-} \\times \\cdots \\times \\mathbb{R}_{-}$. Note that $S$ has a nonempty interior. Then the inequality state constraint in (\\ref{eq_3}) can be expressed as follows:\n\\begin{align}\n\\label{eq_4_2}\n\\gamma(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\in S~ \\Leftrightarrow~ G^i(t,\\overline{x}(t;\\overline{x}_0,\\overline{u})) \\leq 0,~\\forall t \\in [0,T],~ i=1,\\ldots,m.\n\\end{align}\n\nRecall that $G^i$, $i=1,\\ldots,m$, are continuously differentiable in $x$. Then $\\gamma$ is Fr\\'echet differentiable with its Fr\\'echet differentiation at $\\gamma(x(\\cdot)) \\in S$ given by $D\\gamma(x(\\cdot))(w) =  G_x(\\cdot,x(\\cdot)) w $ for all $w \\in C([0,T];\\mathbb{R}^n)$ \\cite[page 167]{Flett_book}. The normal cone to $S$ at $x \\in S$ is defined by\n\\begin{align}\n\\label{eq_4_3}\nN_S(x) : = \\{ \\kappa \\in C([0,T]; \\mathbb{R}^m)^*~|~\\langle \\kappa,\\kappa^\\prime - x \\rangle_{C_m^* \\times C_m} \\leq 0,~ \\forall \\kappa^\\prime \\in S \\},\n\\end{align}\nwhere $\\langle \\cdot,\\cdot \\rangle_{C_m^* \\times C_m} := \\langle \\cdot,\\cdot \\rangle_{C([0,T]; \\mathbb{R}^m)^* \\times C([0,T]; \\mathbb{R}^m)}$ stands for the duality paring between $C([0,T]; \\mathbb{R}^m)$ and $C([0,T]; \\mathbb{R}^m)^*$ with $C([0,T]; \\mathbb{R}^m)^*$ being the dual space of $C([0,T]; \\mathbb{R}^m)$.\n\n\\begin{remark}\\label{Remark_4_2_3_5_2_3_2_1}\nNote that $(C([0,T]; \\mathbb{R}^m),\\|\\cdot\\|_{\\infty})$ is a separable Banach space \\cite[Theorem 6.6, page 140]{Conway_2000_book}. Then by \\cite[Theorem 2.18, page 42]{Li_Yong_book}, there exists a norm $\\|\\cdot \\|_{C([0,T]; \\mathbb{R}^m)}$ on $C([0,T]; \\mathbb{R}^m)$, which is equivalent to $\\|\\cdot\\|_{\\infty}$ \\cite[Definition 2.17, page 42]{Li_Yong_book}, such that $(C([0,T]; \\mathbb{R}^m)^*, \\|\\cdot \\|_{C([0,T]; \\mathbb{R}^m)^*})$ is strictly convex, i.e., $\\|x \\|_{C([0,T]; \\mathbb{R}^m)^*} = \\|y \\|_{C([0,T]; \\mathbb{R}^m)^*} = 1$ and $\\|x+y \\|_{C([0,T]; \\mathbb{R}^m)^*} =2 $ imply $x=y$ for $x,y \\in C([0,T]; \\mathbb{R}^m)^*$ \\cite[Definition 2.12, page 41]{Li_Yong_book}.\n\\end{remark}\n\nLet $d_{S}: C([0,T]; \\mathbb{R}^m) \\rightarrow \\mathbb{R}_{+}$ be the distance function to $S$ defined by\n\\begin{align*}\nd_S(x) := \\inf_{y \\in S} \\|x-y\\|_{C([0,T]; \\mathbb{R}^m)}~ \\textrm{for}~x \\in C([0,T]; \\mathbb{R}^m).\n\\end{align*}\nBy definition of $d_S$, (\\ref{eq_4_2}) is equivalent to\n\\begin{align*}\nd_S \\Bigl ( \\gamma(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr ) = 0 ~ \\Leftrightarrow ~ \\gamma \\Bigl (\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u}) \\Bigr ) \\in S.\t\n\\end{align*}\n\n\\begin{lemma}\\label{Lemma_4_2}\nThe distance function $d_S$ is nonexpansive,  continuous, and convex. \n\\end{lemma}\n\\begin{proof}\nTo simplify the notation, let $\\|\\cdot\\| := \\|\\cdot\\|_{C([0,T]; \\mathbb{R}^m)}$. We fix $x,y \\in S$. Let $\\epsilon > 0$ be given. By definition, there is $\\pi \\in S$ such that $d_S(y) \\geq \\|\\pi-y\\| - \\epsilon$. We then have\n\\begin{align*}\nd_S(x) \\leq \\|x-\\pi\\| \\leq \\|x-y\\| +\\|y-\\pi\\| \\leq \t\\|x-y\\| + d_S(y) + \\epsilon.\n\\end{align*}\nSimilarly, we have $d_S(x) \\geq |\\pi-x\\| - \\epsilon$, and\n\\begin{align*}\t\nd_S(y) \\leq \\|y-\\pi\\| \\leq \\|x-y\\| + \\|x-\\pi\\| \\leq \\|x-y\\| + d_S(x) + \\epsilon.\n\\end{align*}\nSince $\\epsilon$ is arbitrary, $|d_S(x) - d_S(y)| \\leq \\|x-y\\|$ holds, which also implies the continuity of $d_S$.\n\nAs $S$ is convex, we have $(1-\\eta) x + \\eta y \\in S$ for $\\eta \\in [0,1]$. By definition of $d_S$, there are $\\pi_x, \\pi_y \\in S$ such that $\\|\\pi_x - x\\| \\leq d_S(x) + \\epsilon$ and $\\|\\pi_y - y\\| \\leq d_S(y) + \\epsilon$. \nDefine $\\pi := (1-\\eta) \\pi_x +  \\eta \\pi_y \\in S$ for $\\eta \\in [0,1]$. It then follows that\n\\begin{align*}\nd_S ((1-\\eta) x + \\eta y) & \\leq \\|\\pi - ((1-\\eta) x + \\eta y) \\|  \n\\leq (1-\\eta) d_S(x) + \\eta d_S(y) + \\epsilon.\n\\end{align*}\nSince $\\epsilon$ is arbitrary, $d_S$ is convex. We complete the proof.\n\\end{proof}\n\nWe define the subdifferential of $d_S$ at $x \\in C([0,T];\\mathbb{R}^m)$ by \\cite[page 214]{Rockafellar_book}\n\\begin{align}\n\\label{eq_4_5_6_5_43_3_5_7_3_3}\n\\partial d_S(x) := \\{ y^\\prime \\in C([0,T];\\mathbb{R}^m)^*~|~\\langle y^\\prime , y -x \\rangle_{C_m^* \\times C_m}\t\\leq d_S(y) - d_S(x),~ \\forall y \\in C([0,T];\\mathbb{R}^m) \\}.\n\\end{align}\nBy \\cite[page 27]{Clarke_book} and Lemma \\ref{Lemma_4_2}, since $d_S$ is continuous, $\\partial d_S(x)$ is a nonempty ($\\partial d_S(x) \\neq \\emptyset$), convex, and weak--$^*$ compact subset of $C([0,T];\\mathbb{R}^m)^*$. Moreover, from \\cite[Proposition 2.1.2]{Clarke_book}, it holds that $\\|y^\\prime\\|_{C([0,T];\\mathbb{R}^m)^*} \\leq 1$ for all $y^\\prime \\in \\partial d_S(x)$. \n\nAn important consequence of Remark \\ref{Remark_4_2_3_5_2_3_2_1} and Lemma \\ref{Lemma_4_2} is as follows:\n\\begin{remark}\\label{Remark_4_1}\nSince $\\partial d_S(x)$ is convex, $\\eta y^\\prime + (1-\\eta) y^{\\prime \\prime} \\in \\partial d_S(x)$ for any $y^\\prime,y^{\\prime \\prime} \\in \\partial d_S(x)$ and $\\eta \\in [0,1]$. Consider $\\|\\eta y^\\prime + (1-\\eta) y^{\\prime \\prime}  \\|_{C([0,T]; \\mathbb{R}^m)^*} = 1$ for $\\eta \\in [0,1]$. Then $\\|y^\\prime\\|_{C([0,T];\\mathbb{R}^m)^*} = 1$ and $ \\|y^{\\prime \\prime} \\|_{C([0,T];\\mathbb{R}^m)^*} = 1$ when $\\eta = 1$ and $\\eta = 0$, respectively. Moreover, when $\\eta = \\frac{1}{2}$, $\\|y^\\prime + y^{\\prime \\prime}  \\|_{C([0,T]; \\mathbb{R}^m)^*} = 2$. Since $(C([0,T]; \\mathbb{R}^m)^*, \\|\\cdot \\|_{C([0,T]; \\mathbb{R}^m)^*})$ is strictly convex, we must have $y^\\prime = y^{\\prime \\prime} \\in C([0,T];\\mathbb{R}^m)^*$, which implies $\\partial d_S(x) =\\{y^\\prime\\}$, i.e., $\\partial d_S(x)$ is a singleton, and $\\|y^\\prime\\|_{C([0,T];\\mathbb{R}^m)^*} = 1$. \n\\end{remark} \n\n\n\\begin{lemma}\\label{Lemma_4_3_1_2_2_1}\nThe distance function $d_S$ is strictly Hadamard differentiable on $C([0,T];\\mathbb{R}^m) \\setminus S$ with the Hadamard differential $D d_S$ satisfying $\\|D d_S(x)\\|_{C([0,T]; \\mathbb{R}^m)^*} = \\|y^\\prime\\|_{C([0,T]; \\mathbb{R}^m)^*} = 1$ for all $x \\in C([0,T];\\mathbb{R}^m) \\setminus S$. Consequently, $d_S(x)^2$ is strictly Hadamard differentiable on $C([0,T];\\mathbb{R}^m) \\setminus S$ with the Hadamard differential given by $D d_S(x)^2 = 2 d_S(x) D d_S(x)$ for $x \\in C([0,T];\\mathbb{R}^m) \\setminus S$. Moreover, $d_S(x)^2$ is Fr\\'echet differentiable on $S$ with the Fr\\'echet differential being $D d_S(x)^2 = 0 \\in C([0,T]; \\mathbb{R}^m)^*$ for all $x \\in S$.\n\\end{lemma}\n\n\\begin{proof}\nThe strictly Hadamard differentiability of $d_S(x)$ and $d_S(x)^2$ on $C([0,T];\\mathbb{R}^m) \\setminus S$ follows from on Lemma \\ref{Lemma_4_2} and Remark \\ref{Remark_4_1}, together with \\cite[Theorem 3.54]{Mordukhovich_book}. The Fr\\'echet differentiability of $d_S(x)^2$ on $S$ with $D d_S(x) = 0 \\in C([0,T]; \\mathbb{R}^m)^*$ for $x \\in S$ follows by the fact that $d_S(x) = 0$ for $x \\in S$ and that $d_S$ is nonexpansive shown in Lemma \\ref{Lemma_4_2}. This completes the proof.\n\\end{proof}\n\n\\subsection{Ekeland Variational Principle}\\label{Section_4_2}\n\nRecall that the pair $(\\overline{x}(\\cdot),\\overline{u}(\\cdot)) \\in C[0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$ is the optimal pair of \\textbf{(P)}. We also write $\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u}) := \\overline{x}(\\cdot)$ to emphasize the dependence of the state equation $\\overline{x}(\\cdot)$ on the optimal initial condition and control $(\\overline{x}_0, \\overline{u}(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$. Note that the pair $(\\overline{x}_0,\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u}))$ holds the state constraints in (\\ref{eq_3}). The optimal cost of \\textbf{(P)} under $(\\overline{x}(\\cdot),\\overline{u}(\\cdot))$ can be written by $J(\\overline{x}_0, \\overline{u}(\\cdot))$.\n\nRecall the distance functions $d_F$ and $d_S$ in Section \\ref{Section_4_1}. For $\\epsilon > 0$, we define the penalized objective functional as follows:\n\\begin{align}\n\\label{eq_4_5}\nJ_{\\epsilon}(x_0,u(\\cdot)) & = \\Biggl ( \\Bigl ( \\bigl [ J(x_0,u(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\Big)^2 + d_F \\Bigl ( \\begin{bmatrix}\nx_0 \\\\\nx(T) \t\n \\end{bmatrix} \\Bigr )^2 + d_S \\Bigl ( \\gamma(x(\\cdot)) \\Bigr )^2 \\Biggr )^{\\frac{1}{2}}.\n\\end{align}\nWe can easily observe that $J_{\\epsilon}(\\overline{x}_0, \\overline{u}(\\cdot)) = \\epsilon > 0$, i.e., $(\\overline{x}_0, \\overline{u}(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$ is the $\\epsilon$-optimal solution of (\\ref{eq_4_5}). Define the Ekeland metric $\\widehat{d}:(\\mathbb{R}^n \\times \\mathcal{U}^p[0,T]) \\times (\\mathbb{R}^n \\times \\mathcal{U}^p[0,T]) \\rightarrow \\mathbb{R}_{+}$ as follows:\n\\begin{align}\t\n\\label{eq_4_6}\n\\widehat{d} \\Bigl ((x_0,u(\\cdot)),(\\tilde{x}_0,\\tilde{u}(\\cdot)) \\Bigr ) := |x_0 - \\tilde{x}_0 | + \\overline{d}(u(\\cdot),\\tilde{u}(\\cdot)),\n\\end{align}\nwhere\n\\begin{align}\n\\label{eq_4_7}\n\\overline{d}(u(\\cdot),\\tilde{u}(\\cdot)) := |\\{t \\in [0,T]~|~ u(t) \\neq \\tilde{u}(t) \\} |,~ \\forall u(\\cdot),\\tilde{u}(\\cdot) \\in \\mathcal{U}^p[0,T].\n\\end{align}\nIt is easy to see that $(\\mathbb{R}^n \\times \\mathcal{U}^p[0,T],\\widehat{d})$ is a complete metric space \\cite[Lemma 7.2]{Ekeland_JMAA_1974}. By Assumption \\ref{Assumption_2_2}, together with Lemmas \\ref{Lemma_4_1} and \\ref{Lemma_4_2}, $J_{\\epsilon}(x_0,u)$ in (\\ref{eq_4_5}) is a continuous functional on $(\\mathbb{R}^n \\times \\mathcal{U}^p[0,T],\\widehat{d})$. \n\n\nIn view of (\\ref{eq_4_5})-(\\ref{eq_4_7}), we have\n\\begin{align}\n\\label{eq_4_8}\n\\begin{cases}\nJ_{\\epsilon}(x_0,u(\\cdot)) > 0,~ \\forall (x_0,u(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T], \\\\\nJ_{\\epsilon}(\\overline{x}_0, \\overline{u}(\\cdot))  = \\epsilon \\leq \\inf_{(x_0,u(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]} J_{\\epsilon}(x_0,u(\\cdot)) + \\epsilon.\n\\end{cases}\n\\end{align}\nBy the Ekeland variational principle \\cite{Ekeland_JMAA_1974}, there exists a pair $(x_0^\\epsilon,u^\\epsilon) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$ such that\n\\begin{align}\n\\label{eq_4_9_1_1_1}\n\\widehat{d}\\Bigl ( (x_0^\\epsilon,u^\\epsilon(\\cdot)),(\\overline{x}_0,\\overline{u}(\\cdot)) \\Bigr ) \\leq \\sqrt{\\epsilon},\t\t\n\\end{align}\nand\n\\begin{align}\n\\label{eq_4_9}\n\\begin{cases}\nJ_{\\epsilon}(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\leq J_{\\epsilon}(\\overline{x}_0, \\overline{u}(\\cdot))  = \\epsilon, \\\\\nJ_{\\epsilon}(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\leq J_{\\epsilon}(x_0,u(\\cdot)) + \\sqrt{\\epsilon} \\widehat{d} \\Bigl ((x_0^\\epsilon,u^\\epsilon(\\cdot)),(x_0,u(\\cdot)) \\Bigr ),~ \\forall (x_0,u(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T].\n\\end{cases}\n\\end{align}\nAs $\\widehat{d} ((x_0^\\epsilon,u^\\epsilon(\\cdot)),(x_0^\\epsilon,u^\\epsilon(\\cdot))) = 0$, the above condition implies that the pair $(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$ is the minimizing solution of the following Ekeland objective functional over $\\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$:\n\\begin{align}\n\\label{eq_4_10}\nJ_{\\epsilon}(x_0,u(\\cdot)) + \t\\sqrt{\\epsilon} \\widehat{d} \\Bigl ((x_0^\\epsilon,u^\\epsilon(\\cdot)),(x_0,u(\\cdot)) \\Bigr ).\n\\end{align}\nWe observe that (\\ref{eq_4_10}) is the unconstrained control problem. By notation, we write $(x^\\epsilon(\\cdot),u^\\epsilon(\\cdot)) := (x^{\\epsilon}(\\cdot;x_0^\\epsilon,u^\\epsilon),u^\\epsilon(\\cdot))  \\in C([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$, where $x^{\\epsilon}(\\cdot;x_0^\\epsilon,u^\\epsilon)$ is the state trajectory of (\\ref{eq_1}) under $(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$.\n\n\\subsection{Spike Variations and First Variational Equation}\\label{Section_4_3}\n\nIn the previous subsection, we have obtained the $\\epsilon$-optimal solution to \\textbf{(P}), which is also the optimal solution to the Ekeland objective functional in (\\ref{eq_4_10}). The next step is to derive the necessary condition for $(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$. We employ the spike variation technique, as $U$ does not have any algebraic structure (hence, it is impossible to use standard (convex) variations).\n\nFor $\\delta \\in (0,1)$, define\n\\begin{align*}\n\\mathcal{E}_{\\delta} := \\{ E \\in [0,T]~|~ |E| = \\delta T\\}\t,\n\\end{align*}\nwhere $|E|$ denotes the Lebesgue measure of $E$. For $E_{\\delta} \\in \\mathcal{E}_{\\delta}$, we introduce the spike variation associated with $u^\\epsilon$, i.e., the optimal solution of (\\ref{eq_4_10}): \n\\begin{align*}\nu^{\\epsilon,\\delta}(s) := \\begin{cases}\n \tu^\\epsilon(s), & s \\in [0,T] \\setminus E_{\\delta}, \\\\\n\tu(s), & s \\in  E_{\\delta},\n \\end{cases}\t\n\\end{align*}\nwhere $u(\\cdot) \\in \\mathcal{U}^p[0,T]$. Clearly, $u^{\\epsilon,\\delta}(\\cdot) \\in \\mathcal{U}^p[0,T]$. Moreover, by definition of $\\overline{d}$ in (\\ref{eq_4_7}),\n\\begin{align}\n\\label{eq_4_12_345234231}\n\t\\overline{d}(u^{\\epsilon,\\delta}(\\cdot),u^\\epsilon(\\cdot)) \\leq | E_{\\delta}| = \\delta T.\n\\end{align}\nConsider also the variation of the initial state given by $x_0 + \\delta a $, where $a \\in \\mathbb{R}^n$. By notation, let us define the perturbed state equation by\n\\begin{align}\n\\label{eq_5_234235457634545245234523}\nx^{\\epsilon,\\delta}(\\cdot) := x^{\\epsilon,\\delta}(\\cdot; x_0^\\epsilon + \\delta a,u^{\\epsilon,\\delta}) \\in C([0,T];\\mathbb{R}^n).\n\\end{align}\nIn fact, $x^{\\epsilon,\\delta}(\\cdot)$ is the state trajectory of (\\ref{eq_1}) under $(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$. We also recall $(x^\\epsilon(\\cdot),u^\\epsilon(\\cdot)) := (x^{\\epsilon}(\\cdot;x_0^\\epsilon,u^\\epsilon),u^\\epsilon(\\cdot)) \\in C([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$, where $x^{\\epsilon}$ is the state trajectory of (\\ref{eq_1}) under $(x_0^\\epsilon,u^\\epsilon(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$. Then by (\\ref{eq_4_9}) and (\\ref{eq_4_12_345234231}), we have\n\\begin{align}\n\\label{eq_4_13_1_1_1_2}\n- \\sqrt{\\epsilon} (|a| +  T) \\leq \\frac{1}{\\delta} \\Bigl ( J_{\\epsilon}(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - \tJ_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))  \\Bigr ).\n\\end{align}\n\n\\begin{lemma}\\label{Lemma_4_4_2342341234234}\nThe following result holds:\n\\begin{align*}\n\\sup_{t \\in [0,T]} \\Bigl | x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t) - \\delta  Z^{\\epsilon}(t) \\Bigr | = o(\\delta),\n\\end{align*}\nwhere $Z^\\epsilon$ is the solution to the first variational equation related to the optimal pair $(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$ given by\n\\begin{align*}\nZ^{\\epsilon}(t) & = a + \\int_0^t \\Bigl [ \\frac{f_x(t,s,x^\\epsilon(s),u^\\epsilon(s))}{(t-s)^{1-\\alpha}} Z^{\\epsilon}(s) \\dd s \t + \\frac{\\widehat{f}(t,s) }{(t-s)^{1-\\alpha}}  \\Bigr ] \\dd s \\\\\n&~~~ + \\int_0^t \\Bigl [ g_x(t,s,x^\\epsilon(s),u^\\epsilon(s)) Z^{\\epsilon}(s) + \\widehat{g}(t,s) \\Bigr ] \\dd s,~  \\textrm{a.e.}~ t \\in [0,T], \\nonumber\n\\end{align*}\nwith for any $u(\\cdot) \\in \\mathcal{U}^p[0,T]$, \n\\begin{align*}\n\\begin{cases}\n\t\\widehat{f}(t,s) := f(t,s,x^\\epsilon(s),u(s)) - f(t,s,x^\\epsilon(s),u^\\epsilon(s)), \\\\\n\t\\widehat{g}(t,s) := g(t,s,x^\\epsilon(s),u(s)) - g(t,s,x^\\epsilon(s),u^\\epsilon(s)). \n\t\\\\\n\\end{cases}\t\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nBy definition and (\\ref{eq_5_234235457634545245234523}), \n\\begin{align*}\n\tx^{\\epsilon,\\delta}(t) = x(t;x_0^\\epsilon + \\delta a,u^{\\epsilon,\\delta}) &= (x_0^{\\epsilon} + \\delta a) + \\int_0^t \\frac{f(t,s,x^{\\epsilon,\\delta}(s),u^{\\epsilon,\\delta}(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x^{\\epsilon,\\delta}(s),u^{\\epsilon,\\delta}(s)) \\dd s, \\\\\n\tx^{\\epsilon}(t)  = x(t;x_0^\\epsilon,u^{\\epsilon})  & = x_0^{\\epsilon} + \\int_0^t \\frac{f(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s)) \\dd s.\n\\end{align*}\nFor $\\delta \\in (0,1)$, let\n\\begin{align}\n\\label{eq_e_1}\nZ^{\\epsilon,\\delta}(t) := \\frac{x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t)}{\\delta},\t~ t \\in [0,T],\n\\end{align}\nwhere based on the Taylor expansion, $Z^{\\epsilon,\\delta}$ holds\n\\begin{align*}\nZ^{\\epsilon,\\delta}(t) & = a + \\int_0^t  \\Bigl [ \\frac{f_{x}^{\\epsilon,\\delta}(t,s)}{(t-s)^{1-\\alpha}} Z^{\\epsilon,\\delta}(s) + \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} \\frac{\\widehat{f}(t,s) }{(t-s)^{1-\\alpha}}  \\Bigr ] \\dd s \\\\\n&~~~ + \\int_0^t  \\Bigl [ g_{x}^{\\epsilon,\\delta}(t,s) Z^{\\epsilon,\\delta}(s) + \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} \\widehat{g}(t,s) \\Bigr ] \\dd s,~ t \\in [0,T]\n\\end{align*}\nwith $f_{x}^{\\epsilon,\\delta}$ and $g_{x}^{\\epsilon,\\delta}$ defined by\n\\begin{align*}\nf_{x}^{\\epsilon,\\delta}(t,s) & := \\int_0^1 f_x(t,s,x^{\\epsilon}(s) + r (x^{\\epsilon,\\delta}(s) - x^{\\epsilon}(s)),u^{\\epsilon,\\delta}(s)) \\dd r, \\\\\ng_{x}^{\\epsilon,\\delta}(t,s) & := \\int_0^1 g_x(t,s,x^{\\epsilon}(s) + r (x^{\\epsilon,\\delta}(s) - x^{\\epsilon}(s)),u^{\\epsilon,\\delta}(s)) \\dd r.\n\\end{align*}\n\nBy Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2}, we have\n\\begin{align}\n\\label{eq_4_16_4534345345}\n\\begin{cases}\n|\\widehat{f}(t,s)| + |\\widehat{g}(t,s)|  \\leq 4 K_0(s) + 4 K(s)|x^\\epsilon(s)| + K(s)(\\rho(u(s),u_0) + \\rho(u^\\epsilon(s),u_0)) =: \\widetilde{\\psi}(s), \\\\\n|f_x^{\\epsilon,\\delta}(t,s)| + |g_x^{\\epsilon,\\delta}(s)|  \\leq K(s).\n\\end{cases}\n\\end{align}\nLet $q > \\frac{1}{\\alpha}$,\nand we replace $p$ by $q$ in Lemma \\ref{Lemma_A_5}. Recall $L^{l+}([0,T];\\mathbb{R}^n) := \\cup_{r > l} L^{r}([0,T];\\mathbb{R}^n)$ for $1 \\leq l < \\infty$, and the $L^p$-spaces of this paper are induced by the finite measure on $([0,T],\\mathcal{B}([0,T]))$. Since $p > \\alpha p > 1$, it holds that $K(\\cdot) \\in L^{\\frac{p}{\\alpha p - 1} + } ([0,T];\\mathbb{R}) \\subset   L^{  \\frac{p}{p-1} +}([0,T];\\mathbb{R})$. Hence, we can choose $q$ so that $K(\\cdot) \\in L^q([0,T];\\mathbb{R}) \\subset L^{\\frac{p}{p-1}}([0,T];\\mathbb{R}) $ and $x^{\\epsilon}(\\cdot) \\in L^p([0,T];\\mathbb{R}^n) \\subset L^{\\frac{pq}{q-p}}([0,T];\\mathbb{R}^n)$. It then follows from the H\\\"older's inequality that\n\\begin{align*}\n\\Bigl ( \\int_0^T |K(s)^p x^{\\epsilon}(s)^p| \\dd s \\Bigr)^{\\frac{1}{p}}  \\leq \\Bigl ( \\int_0^T |K(s)|^q \\dd s \\Bigr )^{\\frac{1}{q}} \\Bigl ( \\int_0^T |x^{\\epsilon}(s)|^{\\frac{pq}{q-p}} \\dd s \\Bigr )^{\\frac{q-p}{qp}} & < \\infty, \\\\\n\\Bigl ( \\int_0^T |K(s)^p (\\rho(u(s),u_0) + \\rho(u^\\epsilon(s),u_0))^p| \\dd s \\Bigr)^{\\frac{1}{p}} &< \\infty.\n\\end{align*}\nTherefore, as $K_0(\\cdot) \\in L^{\\frac{1}{\\alpha} + } ([0,T];\\mathbb{R}) \\subset L^{p+} ([0,T];\\mathbb{R})$, $\\widetilde{\\psi}(\\cdot) \\in L^p([0,T];\\mathbb{R}) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R})$. \n\nBased on Assumption \\ref{Assumption_2_1} and (\\ref{eq_4_16_4534345345}), we can show that\n\\begin{align}\t\n\\label{eq_e_2}\n|x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t) | \n& \\leq  b(t) + \\int_0^t \\frac{K(s)}{(t-s)^{1-\\alpha}} |x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t) |\\dd s + \\int_0^t K(s) |x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t) |\\dd s, \n\\end{align}\nwhere\n\\begin{align*}\nb(t) = |\\delta a  |_{\\mathbb{R}^n} + \\int_0^t \\mathds{1}_{E_{\\delta}}(s) \\frac{\\widetilde{\\psi}(s)} {(t-s)^{1-\\alpha}} \\dd s + \\int_0^t \\mathds{1}_{E_{\\delta}}(s) \\widetilde{\\psi}(s) \\dd s.   \n\\end{align*}\nWe let $\\widetilde{\\psi}(t,\\cdot) := \\widetilde{\\psi}(\\cdot)$ in (\\ref{eq_4_16_4534345345}). As $\\widetilde{\\psi}(0,\\cdot) \\in L^p([0,T];\\mathbb{R}) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R})$, by Lemmas \\ref{Lemma_A_3} and \\ref{Lemma_A_4} (and using Assumption \\ref{Assumption_2_1}), we have $b(\\cdot) \\in L^p([0,T];\\mathbb{R}^n) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R}^n)$. Note also that we can choose $q$ so that $x^{\\epsilon}(\\cdot) \\in L^p([0,T];\\mathbb{R}^n) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R}^n)$ and $|x^{\\epsilon,\\delta}(\\cdot) - x^{\\epsilon}(\\cdot) |_{\\mathbb{R}^n} \\in L^p([0,T];\\mathbb{R}) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R})$. \nIn addition, from Lemmas \\ref{Lemma_A_3} and \\ref{Lemma_A_4}, there is a constant $C \\geq 0$ such that\n\\begin{align*}\t\n\\Biggl | \\int_0^t \\mathds{1}_{E_{\\delta}}(s) \\widetilde{\\psi}(s) \\dd s \\Biggr | + \\Biggl | \\int_0^t \\mathds{1}_{E_{\\delta}}(s) \\frac{\\widetilde{\\psi}(s)} {(t-s)^{1-\\alpha}} \\dd s \\Biggr | \\leq C |E_{\\delta}|^{\\frac{1}{q}}.\n\\end{align*}\nThen applying Lemma \\ref{Lemma_A_5} to (\\ref{eq_e_2}) yields\n\\begin{align}\n\\label{eq_5_4564563452342342342342342}\n|x^{\\epsilon,\\delta}(t) - x^{\\epsilon}(t) |_{\\mathbb{R}^n} & \\leq  b(t) + C \\int_0^t \\frac{K(s)}{(t-s)^{1-\\alpha}} b(s) \\dd s + C \\int_0^t K(s) b(s) \\dd s \\\\\n& \\leq C \\Bigl ( |\\delta a|_{\\mathbb{R}^n} + |E_{\\delta}|^{\\frac{1}{q}} \\Bigr )~ \\rightarrow 0,~ \\textrm{as $\\delta \\downarrow 0$ for all $ t\\in [0,T]$.} \\nonumber\n\\end{align}\n\nOn the other hand, since $Z^{\\epsilon}$ is linear, by Lemma \\ref{Lemma_2_1} (see also the results in Appendix \\ref{Appendix_B}), it admits a unique solution in $C([0,T];\\mathbb{R}^n)$. Hence, as\n\\begin{align*}\n|Z^{\\epsilon}(t)| \\leq |a| + \t\\int_0^t \\frac {K(s)}{(t-s)^{1-\\alpha}}  |Z^{\\epsilon}(s)| \\dd s + \\int_0^t \\frac{\\widetilde{\\psi}(s)}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t K(s) |Z^{\\epsilon}(s)| \\dd s + \\int_0^t \\widetilde{\\psi}(s) \\dd s,\n\\end{align*}\nwe use Lemmas \\ref{Lemma_A_3}-\\ref{Lemma_A_5} to get\n\\begin{align}\n\\label{eq_e_3}\n|Z^{\\epsilon}(t)| & \\leq  \\hat{b}(t) + C \\int_0^t\t\\frac{K(s) }{(t-s)^{1-\\alpha}}  \\hat{b}(s) \\dd s + C \\int_0^t K(s) \\hat{b}(s) \\dd s  \\leq C \\Bigl ( |a|_{\\mathbb{R}^n} + \\|\\widetilde{\\psi}(\\cdot)\\|_{L^p([0,T];\\mathbb{R})} \\Bigr ),\n\\end{align}\nwhere $\\hat{b}(t) := \t|a|_{\\mathbb{R}^n} + \\int_0^t \\frac{\\widetilde{\\psi}(s)}{(t-s)^{1-\\alpha}} \\dd s  + \\int_0^t \\widetilde{\\psi}(s) \\dd s$.\n\nWe obtain\n\\begin{align}\n\\label{eq_4_17_2343234112}\nZ^{\\epsilon,\\delta}(t)\t- Z^{\\epsilon}(t) & = \\int_0^t \\frac{f_x^{\\epsilon,\\delta}(t,s)}{(t-s)^{1-\\alpha}} \\Bigl [ Z^{\\epsilon,\\delta}(s) - Z^{\\epsilon}(s) \\Bigr ] \\dd s + \\int_0^t \\Bigl ( \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} - 1 \\Bigr ) \\frac{\\widehat{f}(t,s)}{(t-s)^{1-\\alpha}} \\dd s \\\\\n&~~~ + \\int_0^t \\frac{f_x^{\\epsilon,\\delta}(t,s) - f_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))}{(t-s)^{1-\\alpha}} Z^{\\epsilon}(s) \\dd s   \\nonumber\\\\\n&~~~ + \\int_0^t g_x^{\\epsilon,\\delta}(t,s) \\Bigl [ Z^{\\epsilon,\\delta}(s) - Z^{\\epsilon}(s) \\Bigr ] \\dd s + \\int_0^t \\Bigl ( \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} - 1 \\Bigr ) \\widehat{g}(t,s) \\dd s \\nonumber\\\\\n&~~~ + \\int_0^t \\Bigl [ g_x^{\\epsilon,\\delta}(t,s) - g_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s)) \\Bigr ] Z^{\\epsilon}(s) \\dd s,~ t \\in [0,T]. \\nonumber\n\\end{align}\nNotice that\n\\begin{align*}\n\\Biggl | \\frac{f_x^{\\epsilon,\\delta}(t,s) - f_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))}{(t-s)^{1-\\alpha}} Z^{\\epsilon}(s) \\Biggr |_{\\mathbb{R}^n} & \\leq \\frac{4 K(s)}{(t-s)^{1-\\alpha}} |Z^{\\epsilon}(s)|,~ \\forall s \\in [0,t), \\\\\n\\Biggl | \\Bigl [ g_x^{\\epsilon,\\delta}(t,s) - g_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s)) \\Bigr ] Z^{\\epsilon}(s) \\Biggr |_{\\mathbb{R}^n} & \\leq 4 K(s) |Z^{\\epsilon}(s)|,~ \\forall s \\in [0,t].\n\\end{align*}\nwhere $\\lim_{\\delta \\downarrow 0} |f_x^{\\epsilon,\\delta}(t,s) - f_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))| = 0$ and $\\lim_{\\delta \\downarrow 0} |g_x^{\\epsilon,\\delta}(t,s) - g_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))| = 0$. In addition, using (\\ref{eq_e_3}), we get\n\\begin{align*}\n\\int_0^T\t\\frac{4 K(s)}{(t-s)^{1-\\alpha}} |Z^{\\epsilon}(s)| \\dd s < \\infty,~ \\int_0^T 4 K(s) |Z^{\\epsilon}(s)| \\dd s < \\infty.\n\\end{align*}\n\nFor convenience, define\n\\begin{align*}\t\nb^{(1,1)}(t) & := \\int_0^t \\frac{f_x^{\\epsilon,\\delta}(t,s) - f_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s))}{(t-s)^{1-\\alpha}} Z^{\\epsilon}(s) \\dd s \\\\\nb^{(2,1)}(t) & := \\int_0^t \\Bigl [ g_x^{\\epsilon,\\delta}(t,s) - g_x(t,s,x^{\\epsilon}(s),u^{\\epsilon}(s)) \\Bigr ] Z^{\\epsilon}(s) \\dd s \\\\\nb^{(1,2)}(t) & := \\int_0^t \\Bigl ( \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} - 1 \\Bigr ) \\frac{\\widehat{f}(t,s)}{(t-s)^{1-\\alpha}} \\dd s \\\\\nb^{(2,2)}(t) & := \\int_0^t \\Bigl ( \\frac{\\mathds{1}_{E_{\\delta}}(s)}{\\delta} - 1 \\Bigr ) \\widehat{g}(t,s)  \\dd s.\n\\end{align*}\nBy the dominated convergence theorem, it follows that\n\\begin{align*}\t\n\\lim_{\\delta \\downarrow 0} b^{(1,1)}(t)  = 0, ~ \\lim_{\\delta \\downarrow 0} b^{(2,1)}(t) = 0,~ \\forall t \\in [0,T].\n\\end{align*}\nIn addition, by letting\n\\begin{align*}\n\\phi(t,s) = \\widehat{g}(t,s),~~~~ \\psi(t,s) = \\widehat{f}(t,s),\n\\end{align*} \nand then invoking Lemmas \\ref{Lemma_D_1} and \\ref{Lemma_D_2} in Appendix \\ref{Appendix_D} (by Assumptions \\ref{Assumption_2_1} and \\ref{Assumption_2_2}, together with Remark \\ref{Remark_B_1}, $\\psi$ holds (\\ref{eq_d_1}) in Appendix \\ref{Appendix_D}), for any $\\delta \\in (0,1)$, there exists an $E_{\\delta} \\in \\mathcal{E}_{\\delta}$ such that\n\\begin{align*}\n|b^{(1,2)}(t)|_{\\mathbb{R}^n}  \\leq \\delta,~ |b^{(2,2)}(t)|_{\\mathbb{R}^n} &  \\leq \\delta,~ \\forall t \\in [0,T].\n\\end{align*}\n\nWith $b^{(1)}(\\cdot) := b^{(1,1)}(\\cdot) + b^{(2,1)}(\\cdot)$ and $b^{(2)}(\\cdot) := b^{(1,2)}(\\cdot) + b^{(2,2)}(\\cdot)$ in (\\ref{eq_4_17_2343234112}), we then have\n\\begin{align*}\n|Z^{\\epsilon,\\delta}(t)\t- Z^{\\epsilon}(t)| & \\leq b^{(1)}(t) + b^{(2)}(t)  + \\int_0^t \\frac{K(s)}{(t-s)^{1-\\alpha}} \\Bigl [ Z^{\\epsilon,\\delta}(s) - Z^{\\epsilon}(s) \\Bigr ] \\dd s \\\\\n&~~~ +  \\int_0^t K(s) \\Bigl [ Z^{\\epsilon,\\delta}(s) - Z^{\\epsilon}(s) \\Bigr ] \\dd s,~ t \\in [0,T],\n\\end{align*}\nand by applying the same technique as above and using Lemma \\ref{Lemma_A_5}, \n\\begin{align*}\n|Z^{\\epsilon,\\delta}(t)\t- Z^{\\epsilon}(t)| & \\leq  b^{(1)}(t) + b^{(2)}(t) + C \\int_0^t \\frac{K(s)}{(t-s)^{1-\\alpha}} \\Bigl [ b^{(1)}(s) + b^{(2)}(s) \\Bigr ] \\dd s \\\\\n&~~~ + C \\int_0^t K(s) \\Bigl [ b^{(1)}(s) + b^{(2)}(s) \\Bigr ] \\dd s,~ t \\in [0,T].\n\\end{align*}\nHence, the dominated convergence theorem implies that\n\\begin{align*}\t\n\\lim_{\\delta \\downarrow 0} |Z^{\\epsilon,\\delta}(t)\t- Z^{\\epsilon}(t)|_{\\mathbb{R}^n} = 0,~ \\forall t \\in [0,T].\n\\end{align*}\nBy definition of $Z^{\\epsilon,\\delta}$ in (\\ref{eq_e_1}), we have the desired result. This completes the proof.\t\n\\end{proof}\n\n\n\n\\subsection{Crucial Facts from Ekeland Variational Principle, together with Passing Limit and Second Variational Equation}\\label{Section_4_4}\n\nWe recall $Z^{\\epsilon,\\delta}(\\cdot) := \\frac{x^{\\epsilon,\\delta}(\\cdot) - x^{\\epsilon}(\\cdot)}{\\delta}$ defined in (\\ref{eq_e_1}). Based on the Taylor expansion, \n\\begin{align*}\n& \\frac{1}{\\delta} \\Bigl ( J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - \tJ(x_0^\\epsilon, u^{\\epsilon}(\\cdot))  \\Bigr ) \\\\\n& = \\frac{1}{\\delta} \\Biggl ( \\int_0^T l(s,x^{\\epsilon,\\delta}(s),u^{\\epsilon,\\delta}(s)) \\dd s + h(x_0^\\epsilon + \\delta a, x^{\\epsilon,\\delta}(T))  - \\int_0^T l(s,x^{\\epsilon}(s),u^{\\epsilon}(s)) \\dd s - h(x_0^\\epsilon, x^{\\epsilon}(T))  \\Biggr ) \\\\\n& =  \\int_0^T l_x^{\\epsilon,\\delta}(s) Z^{\\epsilon,\\delta}(s) \\dd s + \\int_0^T \\frac{\\mathds{1}_{E_{\\delta}}}{\\delta} \\widehat{l}(s) \\dd s  + h_{x_0}^{\\epsilon,\\delta}(T) a + h_{x}^{\\epsilon,\\delta}(T)  Z^{\\epsilon,\\delta}(T),\n\\end{align*}\nwhere \n\\begin{align*}\n\\widehat{l}(s) & := l(s,x^\\epsilon(s),u(s)) - l(s,x^\\epsilon(s),u^\\epsilon(s)), \\\\\nl_x^{\\epsilon,\\delta}(s) & := \\int_0^1 l_x(s,x^{\\epsilon}(s) + r (x^{\\epsilon,\\delta}(s) - x^{\\epsilon}(s)),u^{\\epsilon,\\delta}(s)) \\dd r,\n\\\\\nh_{x_0}^{\\epsilon,\\delta}(T) & := \\int_0^1 \n \th_{x_0}(x_0^\\epsilon + r \\delta a, x^{\\epsilon}(T) + r (x^{\\epsilon,\\delta}(T)-x^{\\epsilon}(T)))   \\dd r \\\\\nh_{x}^{\\epsilon,\\delta}(T) & := \\int_0^1 \n h_x(x_0^\\epsilon + r \\delta a, x^{\\epsilon}(T) + r (x^{\\epsilon,\\delta}(T)-x^{\\epsilon}(T))) \\dd r.\n\\end{align*}\n\nLet us define\n\\begin{align*}\n\\widehat{Z}^{\\epsilon}(T) & = \\int_0^T l_x(s,x^{\\epsilon} (s),u^{\\epsilon} (s)) Z^{\\epsilon}(s) \\dd s  + \\int_0^T \\widehat{l}(s) \\dd s + \n \th_{x_0}(x_0^{\\epsilon},x^{\\epsilon}(T))a + h_x(x_0^{\\epsilon},x^{\\epsilon}(T)) \tZ^{\\epsilon}(T).\n\\end{align*}\nBy definition of $J$ in (\\ref{eq_2}),\n\\begin{align*}\n& \\frac{1}{\\delta} \\Bigl ( J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - \tJ(x_0^\\epsilon, u^{\\epsilon}(\\cdot))  \\Bigr ) - \t\\widehat{Z}^{\\epsilon}(T) \\\\\n& = \\int_0^T l_x^{\\epsilon,\\delta}(s) \\Bigl [ Z^{\\epsilon,\\delta}(s) - Z^{\\epsilon}(s) \\Bigr ] \\dd s + \\int_0^T \\Bigl [ l_x^{\\epsilon,\\delta}(s)  - l_x(s,x^{\\epsilon} (s),u^{\\epsilon} (s)) \\Bigr ] Z^{\\epsilon}(s) \\dd s \\\\\n&~~~ + \\int_0^T \\Bigl ( \\frac{\\mathds{1}_{E_{\\delta}}}{\\delta} - 1 \\Bigr ) \\widehat{l}(s) \\dd s + \\Bigl [ h_{x_0}^{\\epsilon,\\delta}(T) - h_{x_0}(x_0^\\epsilon, x^\\epsilon(T)) \\Bigr ] a \\\\\n&~~~ + h_x^{\\epsilon,\\delta}(T)\\Bigl [ Z^{\\epsilon,\\delta}(T) - Z^{\\epsilon}(T) \\Bigr ] + \\Bigl [ h_x^{\\epsilon,\\delta}(T) - h_x(x^{\\epsilon}(T))  \\Bigr ] Z^{\\epsilon}(T).\n\\end{align*}\nNotice that $\\lim_{\\delta \\downarrow 0} |Z^{\\epsilon,\\delta}(t)\t- Z^{\\epsilon}(t)| = 0$ for all $t \\in [0,T]$ by Lemma \\ref{Lemma_4_4_2342341234234}. Moreover, with $\\phi(t,s) = \\widehat{l}(s)$ in Lemma \\ref{Lemma_D_1} of Appendix \\ref{Appendix_D}, for any $\\delta \\in (0,1)$, there exists an $E_{\\delta} \\in \\mathcal{E}_{\\delta}$ such that \n\\begin{align*}\t\n\\Biggl | \\int_0^t \\Bigl ( \\frac{1}{\\delta} \\mathds{1}_{E_{\\delta}}(s) - 1 \\Bigr ) \\widehat{l}(s) \\dd s \\Biggr | & \\leq \\delta,~ \\forall t \\in [0,T].\n\\end{align*}\nHence, by using a similar technique of Lemma \\ref{Lemma_4_4_2342341234234}, we can show that\n\\begin{align}\n\\label{eq_4_15}\n\\lim_{\\delta \\downarrow 0} \\Biggl | \t\\frac{1}{\\delta} \\Bigl ( J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}) - \tJ(x_0^\\epsilon, u^{\\epsilon})  \\Bigr ) - \t\\widehat{Z}^{\\epsilon}(T) \\Biggr | = 0,\n\\end{align}\nwhich is equivalent to \n\\begin{align}\n\\label{eq_4_16}\n\\Bigl | J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - \tJ(x_0^\\epsilon, u^{\\epsilon}(\\cdot))  - \t\\delta \\widehat{Z}^{\\epsilon}(T) \\Bigr |\t= o(\\delta).\n\\end{align}\n\n\nNow, from (\\ref{eq_4_13_1_1_1_2}), \n\\begin{align}\n\\label{eq_4_17_1_1_1_1}\n-\\sqrt{\\epsilon}(|a| + T) & \\leq \\frac{1}{\\delta} \\Bigl ( J_{\\epsilon}(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - \tJ_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))  \\Bigr ) \\\\\n& = \\frac{1}{J_{\\epsilon}(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) + \tJ_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))} \\nonumber \\\\\n&~~~ \\times \\frac{1}{\\delta} \\Biggl ( \\Bigl ( \\bigl [ J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\Bigr  )^2  - \\Bigl ( \\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\Bigr  )^2 \\nonumber \\\\\n&~~~~~~~ + d_F \\Bigl ( \\begin{bmatrix}\nx_0^\\epsilon + \\delta a \\\\\nx^{\\epsilon,\\delta}(T)\t\n\\end{bmatrix} \\Bigr )^2 - d_F \\Bigl ( \\begin{bmatrix}\nx_0^\\epsilon \\\\\nx^{\\epsilon}(T)\t\n\\end{bmatrix} \\Bigr )^2  + d_S \\Bigl ( \\gamma(x^{\\epsilon,\\delta}(\\cdot)) \\Bigr )^2 - d_S \\Bigl ( \\gamma(x^{\\epsilon}(\\cdot)) \\Bigr )^2 \\Biggr ). \\nonumber\n\\end{align}\n\nBy continuity of $J_{\\epsilon}$ on $(\\mathbb{R}^n \\times \\mathcal{U}^p[0,T],\\widehat{d})$ and (\\ref{eq_4_16}), it follows that  $\\lim_{\\delta \\downarrow 0}  J_{\\epsilon}(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) = J_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))$, which leads to\n\\begin{align*}\n\\lim_{\\delta \\downarrow 0} \\Bigl \\{ J_{\\epsilon}(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) + \tJ_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))\t\\Bigr \\} = 2 J_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot)).\n\\end{align*}\nIn view of (\\ref{eq_4_15}) and Lemma \\ref{Lemma_4_4_2342341234234},\n\\begin{align*}\n& \\frac{1}{\\delta} \\Bigl ( \\bigl [ J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\Bigr  )^2  - \\Bigl ( \\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\Bigr  )^2 \\\\\n& = \\Biggl (  \\bigl [ J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+   +  \\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+  \\Biggr ) \\\\\n&~~~~~~~ \\times \\frac{1}{\\delta}  \\Biggl ( \\bigl [ J(x_0^\\epsilon + \\delta a, u^{\\epsilon,\\delta}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+   -  \\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+  \\Biggr )\n\\\\\n& \\rightarrow  2 \\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+ \\widehat{Z}^{\\epsilon}(T),~ \\textrm{as $\\delta \\downarrow 0$.}\n\\end{align*}\nLet us define (since $J_{\\epsilon}(x_0^\\epsilon,u^\\epsilon(\\cdot)) > 0$ by (\\ref{eq_4_8}))\n\\begin{align}\n\\label{eq_4_17}\n\\lambda^{\\epsilon} := \t\\frac{\\bigl [ J(x_0^\\epsilon, u^{\\epsilon}(\\cdot)) - J(\\overline{x}_0, \\overline{u}(\\cdot)) + \\epsilon \\bigr ]^+}{J_{\\epsilon}(x_0^\\epsilon, u^{\\epsilon}(\\cdot))} \\geq 0.\n\\end{align}\n\nBy Lemmas \\ref{Lemma_4_1} and \\ref{Lemma_4_4_2342341234234}, and the definition of Fr\\'echet differentiability, as $\\delta \\downarrow 0$,\n\\begin{align*}\t\n& \\frac{1}{\\delta} \\Biggl ( d_F \\Bigl ( \\begin{bmatrix}\n\tx_0^\\epsilon + \\delta a \\\\\n\tx^{\\epsilon,\\delta}(T)\n\\end{bmatrix} \\Bigr )^2 - d_F \\Bigl ( \\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr )^2 \n  \\Biggr )  \\rightarrow 2 \\Biggl \\langle \\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} - P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix}  \\Bigr ), \\begin{bmatrix}\na \\\\\nZ^{\\epsilon}(T)\n \\end{bmatrix} \\Biggr \\rangle_{\\mathbb{R}^{2n} \\times \\mathbb{R}^{2n}},\n\\end{align*}\nwhere $P_F: \\mathbb{R}^{2n} \\rightarrow F \\subset \\mathbb{R}^{2n}$ is the projection operator defined in Section \\ref{Section_4_1}. Notice that by the statement in Section \\ref{Section_4_1}, \n\\begin{align*}\nd_F\t\\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr ) = \\Biggl | \\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} - P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix}  \\Bigr ) \\Biggr |_{\\mathbb{R}^{2n}},\n\\end{align*}\nand by (\\ref{eq_4_1}), \n\\begin{align*}\t\n\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} - P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr ) \\in N_F \\Bigl ( P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr ) \\Bigr ).\n\\end{align*}\nWe define (note that $J_{\\epsilon}(x_0^\\epsilon,u^\\epsilon (\\cdot)) > 0$ by (\\ref{eq_4_8}) and $\\xi_1^{\\epsilon},\\xi_2 ^{\\epsilon} \\in \\mathbb{R}^n$)\n\\begin{align}\n\\label{eq_4_18}\n\\xi^{\\epsilon} := \\begin{bmatrix}\n \\xi^{\\epsilon}_1 \\\\\n \\xi^{\\epsilon}_2\t\n \\end{bmatrix}\n := \\frac{\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} - P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr ) }{J_{\\epsilon}(x_0^\\epsilon,u^\\epsilon (\\cdot) )}\t\\in N_F \\Bigl ( P_F \\Bigl (\\begin{bmatrix}\n\tx_0^\\epsilon \\\\\n\tx^{\\epsilon}(T)\n\\end{bmatrix} \\Bigr ) \\Bigr ).\n\\end{align}\n\nBy Lemma \\ref{Lemma_2_1} (see also Lemmas \\ref{Lemma_B_1} and \\ref{Lemma_B_2} in Appendix \\ref{Appendix_B}), it holds that $Z^{\\epsilon}(\\cdot) \\in C([0,T];\\mathbb{R}^n)$. Then using Lemmas \\ref{Lemma_4_3_1_2_2_1} and \\ref{Lemma_4_4_2342341234234}, as $\\delta \\downarrow 0$, we get\n\\begin{align*}\n& \\frac{1}{\\delta} \\Biggl (  d_S \\Bigl (\\gamma(x^{\\epsilon,\\delta}(\\cdot)) \\Bigr )^2 - d_S \\Bigl (\\gamma(x^{\\epsilon}(\\cdot)) \\Bigr )^2 \\Biggr )  \\\\\n&\\rightarrow \\begin{cases}\n2 \\Biggl \\langle d_S \\Bigl ( \\gamma(x^{\\epsilon}(\\cdot)) \\Bigr ) D d_S \\Bigl ( ( \\gamma(x^{\\epsilon}(\\cdot)) \\Bigr ),   G_x(\\cdot,x^{\\epsilon}(\\cdot)) Z^{\\epsilon}(\\cdot) \\Biggr \\rangle_{C_m^* \\times C_m}, & \t \\gamma(x^{\\epsilon}(\\cdot)) \\notin S, \\\\\n0 \\in \\mathbb{R}, & \\gamma(x^{\\epsilon}(\\cdot)) \\in S .\n \\end{cases}\n\\end{align*}\nWe define (since $J_{\\epsilon}(x_0^\\epsilon,u^\\epsilon (\\cdot)) > 0$ by (\\ref{eq_4_8}))\n\\begin{align}\n\\label{eq_4_19}\n\\mu^{\\epsilon} := \\begin{dcases}\n\\frac{d_S \\Bigl (\\gamma(x^{\\epsilon}(\\cdot)) \\Bigr ) D d_S \\Bigl (\\gamma(x^{\\epsilon}(\\cdot)) \\Bigr )}{J_{\\epsilon}(x_0^\\epsilon,u^\\epsilon (\\cdot) )} \\in C([0,T]; \\mathbb{R}^m)^*, &  \\gamma(x^{\\epsilon}(\\cdot)) \\notin S,  \\\\\n0 \\in C([0,T]; \\mathbb{R}^m)^*, & \\gamma(x^{\\epsilon}(\\cdot)) \\in S,\n \\end{dcases} \n\\end{align}\nand since $D d_S \\Bigl ( \\gamma(x^{\\epsilon}(\\cdot)) \\Bigr )$ is the subdifferential of $d_S \\Bigl ( \\gamma(x^{\\epsilon}(\\cdot)) \\Bigr )$ at $\\gamma(x^{\\epsilon}(\\cdot)) \\in S$ (see Lemma \\ref{Lemma_4_3_1_2_2_1}), by (\\ref{eq_4_3}) and (\\ref{eq_4_5_6_5_43_3_5_7_3_3}), we have\n\\begin{align}\n\\label{eq_4_26_23423425345}\n\\mu^{\\epsilon} \\in N_S \\Bigl (\\gamma(x^{\\epsilon}(\\cdot)) \\Bigr ). \t\n\\end{align}\n\n\nIn view of Lemma \\ref{Lemma_4_3_1_2_2_1} and the definitions of $J_\\epsilon$, $d_F$ and $d_S$, this leads to\n\\begin{align}\n\\label{eq_4_20_1_1_1_1_2}\n& |\\lambda^\\epsilon|^2 + \t|\\xi^\\epsilon|^2_{\\mathbb{R}^{2n}} + \\|\\mu^\\epsilon\\|^2_{C([0,T]; \\mathbb{R}^m)^*} = 1.\n\\end{align}\nHence, as $\\delta \\downarrow 0$, applying (\\ref{eq_4_17})-(\\ref{eq_4_26_23423425345}) to (\\ref{eq_4_17_1_1_1_1}) yields\n\\begin{align}\t\n\\label{eq_4_20}\n-\\sqrt{\\epsilon}(|a| + T) & \\leq  \\lambda^{\\epsilon} \\widehat{Z}^{\\epsilon}(T) + \\Bigl \\langle \\xi_1^{\\epsilon}, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2^{\\epsilon}, Z^{\\epsilon}(T) \\Bigr \\rangle + \\Bigl \\langle \\mu^{\\epsilon},  G_x(\\cdot,x^{\\epsilon}(\\cdot)) Z^{\\epsilon}(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m}.\n\\end{align}\n\nThe following lemma shows the estimate between the first and second variational equations, where the first variational equation is given in Lemma \\ref{Lemma_4_4_2342341234234}.\n\n\\begin{lemma}\\label{Lemma_4_5}\nFor any $(a,u(\\cdot)) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$, the following results hold:\n\\begin{align*}\n& \\textrm{(i)}~ \\lim_{\\epsilon \\downarrow 0} \\Bigl \\{ |x_0^\\epsilon - \\overline{x}_0 |_{\\mathbb{R}^n} + \\overline{d}(u^\\epsilon(\\cdot),\\overline{u}(\\cdot)) \\Bigr \\} = 0, \\\\\n& \\textrm{(ii)}~  \\sup_{t \\in [0,T]} \\Bigl | Z^{\\epsilon}(t;a,u) - Z(t;a,u) \\Bigr | = o(\\epsilon),~ \\Bigl | \\widehat{Z}^{\\epsilon}(T;a,u) - \\widehat{Z}(T;a,u) \\Bigr | = o(\\epsilon),\n\\end{align*}\nwhere $Z(\\cdot) := Z(\\cdot;a,u)$ is the solution to the second variational equation related to $(\\overline{x},\\overline{u}(\\cdot))$ and $\\widehat{Z}(\\cdot) := \\widehat{Z}(\\cdot;a,u)$ is the variational equation of $J$, both of which are given below\n\\begin{align*}\n& Z(t)  = a + \\int_0^t \\Bigl [ \\frac{f_x(t,s,\\overline{x}(s),\\overline{u}(s))}{(t-s)^{1-\\alpha}} Z(s) \\dd s \t + \\frac{f(t,s,\\overline{x}(s),u(s)) - f(t,s,\\overline{x}(s),\\overline{u}(s))}{(t-s)^{1-\\alpha}}  \\Bigr ] \\dd s \\\\\n&~~~ + \\int_0^t \\Bigl [ g_x(t,s,\\overline{x}(s),\\overline{u}(s)) Z(s) + \\bigl ( g(t,s,\\overline{x}(s),u(s)) - g(t,s,\\overline{x}(s),\\overline{u}(s)) \\bigr ) \\Bigr ] \\dd s,~  \\textrm{a.e.}~ t \\in [0,T], \\\\\n& \\widehat{Z}(T)  = \\int_0^T l_x(s,\\overline{x}(s),\\overline{u}(s)) Z(s) \\dd s + \\int_0^T \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s \\\\\n&~~~  + h_{x_0}(\\overline{x}_0,\\overline{x}(T))a + h_x(\\overline{x}_0,\\overline{x}(T)) Z(T),~  \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\n\\end{lemma}\n\n\\begin{remark}\\label{Remark_4_8_4534535}\nNote that (i) of Lemma \\ref{Lemma_4_5} follows from the definition of the Ekeland metric in (\\ref{eq_4_6}) (see also (\\ref{eq_4_9_1_1_1})). The proof for (ii) of Lemma \\ref{Lemma_4_5} is similar to that for Lemma \\ref{Lemma_4_4_2342341234234}.\t\n\\end{remark}\n\n\nWe now consider the limit of $\\epsilon \\downarrow 0$.  Instead of taking the limit with respect to $\\epsilon \\downarrow 0$, let $\\{\\epsilon_k\\}$ be the sequence of  $\\epsilon$ such that $\\epsilon_k \\geq 0$ and $\\epsilon_k \\downarrow 0$ as $k \\rightarrow \\infty$. We replace $\\epsilon$ by $\\epsilon_k$. Then by (\\ref{eq_4_20_1_1_1_1_2}),  the sequences $(\\{\\lambda^{\\epsilon_k} \\}, \\{\\xi^{\\epsilon_k} \\},\\{\\mu^{\\epsilon_k}\\}) $ are bounded for $k \\geq 0$. Note also from (\\ref{eq_4_20_1_1_1_1_2}) that the ball generated by $\\|\\mu^{\\epsilon_k}\\|^2_{C([0,T];\\mathbb{R}^m)^*} \\leq 1$ is a closed unit ball in $C([0,T];\\mathbb{R}^m)^*$, which is weak--$*$ compact by the Banach-Alaoglu theorem \\cite[page 130]{Conway_2000_book}. Then by the standard compactness argument, we may extract a subsequence of $\\{\\epsilon_k\\}$, still denoted by $\\{\\epsilon_k\\}$, such that \n\\begin{align}\n\\label{eq_4_28_34_34234343234}\n(\\{\\lambda^{\\epsilon_k} \\}, \\{\\xi^{\\epsilon_k} \\},\\{\\mu^{\\epsilon_k}\\}) \\rightarrow (\\lambda^0, \\xi^0,\\mu^0) = :(\\lambda, \\xi,\\mu),~ \\textrm{as $k \\rightarrow \\infty$,}\n\\end{align}\nwhere $\\{\\mu^{\\epsilon_k}\\} \\rightarrow \\mu$ (as $k \\rightarrow \\infty$) is understood in the weak--$*$ sense \\cite{Conway_2000_book}. \n\nWe claim that from (\\ref{eq_4_17})-(\\ref{eq_4_26_23423425345}), the tuple $(\\lambda,\\xi,\\mu)$ holds\n\\begin{subequations}\n\\label{eq_4_22}\n\\begin{align}\n\\label{eq_4_22_234234234234}\n\t\\lambda &\\geq 0, \\\\\n\\label{eq_4_22_234234234234_wdsdf}\n\t\\xi &\\in N_F \\Bigl (  P_F \\Bigl (\\begin{bmatrix}\n\t\\overline{x}_0, \\\\\n\t\t\\overline{x}(T)\n\\end{bmatrix} \\Bigr ) \\Bigr ), \\\\\n\\label{eq_4_22_234234234234_wdsdfdsfsdf}\n\\mu &\\in N_S \\Bigl ( \\gamma(\\overline{x}(\\cdot)) \\Bigr ).\n\\end{align}\n\\end{subequations}\nIndeed, (\\ref{eq_4_22_234234234234}) holds due to (\\ref{eq_4_17}). Furthermore, (\\ref{eq_4_22_234234234234_wdsdf}) follows from (\\ref{eq_4_18}) and the property of limiting normal cones \\cite[page 43]{Vinter_book}. To prove (\\ref{eq_4_22_234234234234_wdsdfdsfsdf}), we note that (\\ref{eq_4_26_23423425345}) and (\\ref{eq_4_3}) mean that $ \\langle \\mu^{\\epsilon_k}, z - \\gamma(x^{\\epsilon_k}(\\cdot)) \\rangle_{C_m^* \\times C_m} \\leq 0$ for any $z \\in S$. Then (\\ref{eq_4_22_234234234234_wdsdfdsfsdf}) holds, since by (\\ref{eq_4_3}), (\\ref{eq_4_20_1_1_1_1_2}) and (\\ref{eq_4_28_34_34234343234}), together with the boundedness of $\\{\\mu^{\\epsilon_k}\\}$, Lemma \\ref{Lemma_4_5}, and the weak--$*$ convergence property of $\\{\\mu^{\\epsilon_k}\\}$ to $\\mu$, it holds that\n\\begin{align*}\n0 \\geq \\Bigl \\langle \\mu^{\\epsilon_k}, z - \\gamma(x^{\\epsilon_k}(\\cdot) ) \\Bigr \\rangle_{C_m^* \\times C_m}\t& \\geq  \\Bigl \\langle \\mu, z - \\gamma(\\overline{x}(\\cdot) ) \\Bigr \\rangle_{C_m^* \\times C_m} - \\Bigl \\|\\gamma(x^{\\epsilon_k}(\\cdot)) - \\gamma(\\overline{x}(\\cdot)) \\Bigr \\|_{\\infty} \\\\\n&~~~ + \\Bigl \\langle \\mu^{\\epsilon_k}, z - \\gamma(\\overline{x}(\\cdot)) \\Bigr \\rangle_{C_m^* \\times C_m} - \\Bigl \\langle \\mu, z - \\gamma(\\overline{x}(\\cdot)) \\Bigr \\rangle_{C_m^* \\times C_m} \\\\\n& \\rightarrow ~\\Bigl \\langle \\mu, z - \\gamma(\\overline{x}(\\cdot) ) \\Bigr \\rangle_{C_m^* \\times C_m},~ \\textrm{as $k \\rightarrow \\infty$.}\n\\end{align*}\n\nBy (\\ref{eq_4_28_34_34234343234}) and (\\ref{eq_4_20_1_1_1_1_2}), together with Lemma \\ref{Lemma_4_5}, it follows that\n\\begin{align*}\n\\lambda^{\\epsilon_k} \\widehat{Z}^{\\epsilon_k}(T)  & \\leq \\lambda \\widehat{Z}(T) +  |\\widehat{Z}^{\\epsilon_k}(T) - \\widehat{Z}(T)| + |\\lambda^{\\epsilon_k} - \\lambda | \\widehat{Z}(T)~\\rightarrow ~ \\lambda \\widehat{Z}(T),~ \\textrm{as $k \\rightarrow \\infty$,}\t \\\\\n\\Bigl \\langle \\xi_1^{\\epsilon_k}, a \\Bigr \\rangle  &= \t  \\Bigl \\langle \\xi_1, a \\Bigr \\rangle + \\Bigl \\langle \\xi_1^{\\epsilon_k}, a \\Bigr \\rangle - \\Bigl \\langle \\xi_1, a \\Bigr \\rangle~ \\rightarrow~ \\Bigl \\langle \\xi_1, a \\Bigr \\rangle,~ \\textrm{as $k \\rightarrow \\infty$,} \\\\\n\\Bigl \\langle \\xi_2^{\\epsilon_k}, Z^{\\epsilon_k}(T) \\Bigr \\rangle  & \\leq \\Bigl \\langle \\xi_2, Z(T) \\Bigr \\rangle + |Z^{\\epsilon_k}(T) - Z(T)| + |\\xi_2^{\\epsilon_k} - \\xi_2||Z(T)|~\\rightarrow \\Bigl \\langle \\xi_2, Z(T) \\Bigr \\rangle,~ \\textrm{as $k \\rightarrow \\infty$,}\n\\end{align*}\nand similarly, together with the definition of the weak--$*$ convergence,\n\\begin{align*}\n\\Bigl \\langle \\mu^{\\epsilon_k},  G_x(\\cdot,x^{\\epsilon_k}(\\cdot)) Z^{\\epsilon_k}(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m} & \\leq \\Bigl \\langle \\mu,  G_x(\\cdot,x(\\cdot)) Z(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m} + \\| Z^{\\epsilon_k}(\\cdot) - Z(\\cdot)\\|_{\\infty} \\\\\n&~~~ + \\Bigl \\langle \\mu^{\\epsilon_k},  G_x(\\cdot,x(\\cdot)) Z(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m} - \\Bigl \\langle \\mu,  G_x(\\cdot,x(\\cdot)) Z(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m}\t\\\\\n& \\rightarrow ~ \\Bigl \\langle \\mu,  G_x(\\cdot,x(\\cdot)) Z(\\cdot) \\Bigr \\rangle_{C_m^* \\times C_m},~ \\textrm{as $k \\rightarrow \\infty$.}\n\\end{align*}\nTherefore, as $k \\rightarrow \\infty$, (\\ref{eq_4_20}) becomes for any $(a,u) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$,\n\\begin{align}\t\n\\label{eq_4_23}\n0 & \\leq  \\lambda \\widehat{Z}(T) + \\Bigl \\langle \\xi_1, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2, Z(T) \\Bigr \\rangle + \\Bigl \\langle \\mu,  G_x(\\cdot, \\overline{x}(\\cdot)) Z(\\cdot;a,u) \\Bigr \\rangle_{C_m^* \\times C_m}.\n\\end{align}\nNote that (\\ref{eq_4_23}) is the crucial inequality obtained from the Ekeland variational principle as well as the estimates of the variational equations in Lemmas \\ref{Lemma_4_4_2342341234234} and \\ref{Lemma_4_5}. \n\n\\subsection{Proof of Theorem \\ref{Theorem_3_1}: Complementary Slackness Condition}\\label{Section_4_5}\n\nWe prove the complementary slackness condition in Theorem \\ref{Theorem_3_1}. Let $\\mu = (\\mu_1,\\ldots, \\mu_m) \\in C([0,T];\\mathbb{R}^m)^*$, where $\\mu_i \\in C([0,T];\\mathbb{R})^*$, $i=1,\\ldots,m$. Then it holds that\n\\begin{align}\t\n\\label{Eq_4_31_3434534234234123}\n\\Bigl \\langle \\mu, z \\Bigr \\rangle_{C_m^* \\times C_m} = \\sum_{i=1}^m \\Bigl \\langle \\mu_i, z_i \\Bigr \\rangle_{C_1^* \\times C_1},~ \\forall z = (z_1,\\ldots,z_m) \\in C([0,T];\\mathbb{R}^m)\n\\end{align}\nwhere $\\langle \\cdot,\\cdot \\rangle_{C_1^* \\times C_1} := \\langle \\cdot,\\cdot \\rangle_{C([0,T];\\mathbb{R})^* \\times C([0,T];\\mathbb{R})}$ denotes the duality paring between $C([0,T];\\mathbb{R})$ and $C([0,T];\\mathbb{R})^*$. \n\nRecall $\\gamma(\\overline{x}(\\cdot)) = (\\gamma_1(\\overline{x}(\\cdot)),\\ldots, \\gamma_m(\\overline{x}(\\cdot))) = G(\\cdot,\\overline{x}(\\cdot)) = \\begin{bmatrix}\n \tG^1(\\cdot,\\overline{x}(\\cdot)) & \\cdots & G^m(\\cdot,\\overline{x}(\\cdot))\n \\end{bmatrix} \\in S$ and $\\mu \\in N_S \\Bigl (\\gamma(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr )$ by (\\ref{eq_4_22_234234234234_wdsdfdsfsdf}). Based on (\\ref{Eq_4_31_3434534234234123}) and (\\ref{eq_4_3}), this implies that for any $z \\in S$,\n\\begin{align}\n\\label{eq_4_24}\t\n\\Bigl \\langle \\mu, z - \\gamma(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u}))  \\Bigr \\rangle_{C_m^* \\times C_m} = \\sum_{i=1}^m \\Bigl \\langle \\mu_i, z_i - \\gamma_i (\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr \\rangle_{C_1^* \\times C_1} \\leq 0.\n\\end{align}\nTaking $z$ in (\\ref{eq_4_24}) as follows:\n\\begin{align*}\nz &= \\begin{bmatrix}\n G^1(\\cdot,\\overline{x}(\\cdot)) & \\cdots & G^{i-1}(\\cdot,\\overline{x}(\\cdot)) & 2G^{i}(\\cdot,\\overline{x}(\\cdot)) & G^{i+1}(\\cdot,\\overline{x}(\\cdot)) & \\cdots & G^m(\\cdot,\\overline{x}(\\cdot))\t\n \\end{bmatrix} \\in S, \\\\\n z^{(-i)} &= \n \\begin{bmatrix}\n \tG^1(\\cdot,\\overline{x}(\\cdot)) & \\cdots & G^{i-1}(\\cdot,\\overline{x}(\\cdot)) & 0_{\\in C([0,T];\\mathbb{R})} & G^{i+1}(\\cdot,\\overline{x}(\\cdot)) & \\cdots &  G^m(\\cdot,\\overline{x}(\\cdot)) \n \\end{bmatrix} \\in S.\n\\end{align*}\nThen (\\ref{eq_4_24}) is equivalent to\n\\begin{align}\n\\label{eq_4_25}\n\\Bigl \\langle \\mu_i, G^i(\\cdot,\\overline{x}(\\cdot;\\overline{x}_0;\\overline{u})) \\Bigr \\rangle_{C_1^* \\times C_1} & = 0,~ \\forall i=1,\\ldots,m ,\n\\\\\n\\label{eq_4_26}\n\\Bigl \\langle \\mu_i, z_i  \\Bigr \\rangle_{C_1^* \\times C_1} & \\geq 0,~ \\forall z_i \\in C([0,T];\\mathbb{R}_{+}),~ i=1,\\ldots,m.\n\\end{align}\n\nFor (\\ref{eq_4_25}) and (\\ref{eq_4_26}), by the Riesz representation theorem (see \\cite[page 75 and page 382]{Conway_2000_book} and \\cite[Theorem 14.5]{Limaye_book}), there is a unique $\\theta(\\cdot) = (\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\in \\textsc{NBV}([0,T];\\mathbb{R}^m)$ with $\\theta_i(\\cdot) \\in \\textsc{NBV}([0,T];\\mathbb{R})$, i.e., $\\theta_i$, $i=1,\\ldots,m$, being the normalized functions of bounded variation on $[0,T]$, such that every $\\theta_i$ is finite, nonnegative, and monotonically nondecreasing on $[0,T]$ with $\\theta_i(0) = 0$. Moreover, the Riesz representation theorem leads to the following representation:\n\\begin{align}\t\n\\Bigl \\langle \\mu_i, \\gamma_i(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr \\rangle_{C_1^* \\times C_1} &= \\int_0^T G^i(s,\\overline{x}(s;\\overline{x}_0,\\overline{u})) \\dd \\theta_i(s) = 0,~ \\forall i=1,\\ldots,m,   \\nonumber \\\\\n\\label{eq_4_27}\n\\langle \\mu_i, z_i  \\Bigr \\rangle_{C_1^* \\times C_1} &= \\int_0^T z_i(s) \\dd \\theta_i(s) \\geq 0,~ \\forall z_i \\in C([0,T];\\mathbb{R}_{+}),~ i=1,\\ldots,m.\n\\end{align}\n\nNotice that (\\ref{eq_4_27}) always holds as $\\theta_i$ is monotonically nondecreasing on $[0,T]$ with $\\theta(0) = 0$ (equivalently, $\\dd \\theta_i$ is nonnegative) and $z_i \\in C([0,T];\\mathbb{R}_{+})$. Hence,  (\\ref{eq_4_25}) and (\\ref{eq_4_26}) are reduced to\n\\begin{align}\t\n\\label{eq_4_27_4_23_3_4_2_}\n& \\Bigl \\langle \\mu_i, \\gamma_i(\\overline{x}(\\cdot;\\overline{x}_0;\\overline{u})) \\Bigr \\rangle_{C_1^* \\times C_1} = \\int_0^T G^i(s,\\overline{x}(s;\\overline{x}_0,\\overline{u})) \\dd \\theta_i(s) = 0,~ \\forall i=1,\\ldots,m, \\\\\n& \\Leftrightarrow~ \t\\textsc{supp}(\\dd \\theta_i(\\cdot)) \\subset \\{ t \\in [0,T]~|~ G^i(t,\\overline{x}(t;\\overline{x}_0,\\overline{u}))= 0\\},~ \\forall i=1,\\ldots,m, \\nonumber\n\\end{align}\nwhere the equivalence follows from the fact that $G^i(t,\\overline{x}(t;\\overline{x}_0,\\overline{u})) \\leq 0$, $i=1,\\ldots,m$, and $\\dd \\theta_i$, $i=1,\\ldots,m$, are finite nonnegative measures on $([0,T],\\mathcal{B}([0,T]))$. The relation in (\\ref{eq_4_27_4_23_3_4_2_}) proves the complementary slackness condition in Theorem \\ref{Theorem_3_1}.\n\n\\subsection{Proof of Theorem \\ref{Theorem_3_1}: Nontriviality and Nonnegativity Conditions}\\label{Section_4_6}\n\nWe prove the nontriviality and nonnegativity conditions in Theorem \\ref{Theorem_3_1}. Recall (\\ref{eq_4_24}), i.e., for any $z \\in S$, \n\\begin{align}\n\\label{eq_4_24_2345234234234223423}\t\n\\Bigl \\langle \\mu, z - \\gamma(\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr \\rangle_{C_m^* \\times C_m} = \\sum_{i=1}^m \\Bigl \\langle \\mu_i, z_i - G^i(\\cdot,\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) \\Bigr \\rangle_{C_1^* \\times C_1} \\leq 0.\n\\end{align}\nThen by the Riesz representation theorem (see \\cite[page 75 and page 382]{Conway_2000_book} and \\cite[Theorem 14.5]{Limaye_book}) and the fact that $\\theta_i$, $i=1,\\ldots,m$, is finite, nonnegative, and monotonically nondecreasing on $[0,T]$ with $\\theta_i(0) = 0$ (see Section \\ref{Section_4_5}), it follows that $\\|\\mu_i \\|_{C([0,T];\\mathbb{R}^m)^*} = \\|\\theta_i(\\cdot)\\|_{\\textsc{NBV}([0,T];\\mathbb{R})}  = \\theta_i(T) \\geq 0$ for $i=1,\\ldots,m$. In addition, as $\\theta_i$ is monotonically nondecreasing, we have $\\dd \\theta_i(s) \\geq 0$ for $s \\in [0,T]$, where $\\dd \\theta_i$ denotes the Lebesgue-Stieltjes measure corresponding to $\\theta_i$, $i=1,\\ldots,m$. \n\nBy (\\ref{eq_4_22_234234234234_wdsdf}) and the fact that $\\begin{bmatrix}\n\\overline{x}_0 \\\\\\overline{x}(T)) \\end{bmatrix} \\in F$ implies $P_F \\Bigl (\\begin{bmatrix}\n\\overline{x}_0 \\\\\\overline{x}(T)) \\end{bmatrix} \\Bigr ) = \\begin{bmatrix}\\overline{x}_0 \\\\\\overline{x}(T) \\end{bmatrix}$ (see Section \\ref{Section_4_1}), we have $\\xi = \\begin{bmatrix}\n \\xi_1 \\\\\n \\xi_2\t\n \\end{bmatrix} \\in N_F \\Bigl ( \\begin{bmatrix}\\overline{x}_0 \\\\\\overline{x}(T) \\end{bmatrix} \\Bigr )$. \nIn addition, from the fact that $S = C([0,T];\\mathbb{R}_{-}^m)$ has an nonempty interior, there are $z^\\prime \\in S$ and $\\sigma > 0$ such that $z^\\prime + \\sigma z \\in S$ for all $z \\in \\overline{B}_{(C([0,T];\\mathbb{R}^n),\\|\\cdot\\|_{C([0,T];\\mathbb{R}^n)})}(0,1)$ (the closure of the unit ball in $C([0,T];\\mathbb{R}^n)$). Then by (\\ref{eq_4_24_2345234234234223423}), it follows that\n\\begin{align*}\t\n\\sigma \\Bigl \\langle \\mu,z \\Bigr \\rangle_{C_m^* \\times C_m} \\leq \\Bigl \\langle \\mu,  \\gamma(\\overline{x}(\\cdot)) - z^\\prime\t \\Bigr \\rangle_{C_m^* \\times C_m},~ \\forall z \\in \\overline{B}_{(C([0,T];\\mathbb{R}^n),\\|\\cdot\\|_{C([0,T];\\mathbb{R}^n)})}(0,1).\n\\end{align*}\nBy (\\ref{eq_4_20_1_1_1_1_2}) \nand the definition of the norm of the dual space (the norm of linear functionals on $C([0,T];\\mathbb{R}^m)$ (see Section \\ref{Section_4_1})), we get\n\\begin{align*}\n\\sigma \\|\t\\mu \\|_{C([0,T];\\mathbb{R}^m)^*}  = \\sigma \\sqrt{1-|\\lambda|^2 - \t|\\xi|^2_{\\mathbb{R}^{2n}}} \\leq \\Bigl \\langle \\mu,  \\gamma(x^{\\epsilon}(\\cdot)) - z^\\prime\t \\Bigr \\rangle_{C_m^* \\times C_m},~ z^\\prime \\in S.\n\\end{align*}\nNotice that $\\sigma > 0$. When $\\mu = 0 \\in C([0,T];\\mathbb{R}^m)^*$ and $\\xi = 0$, we must have $\\lambda = 1$. When $\\lambda = 0$ and $\\mu = 0 \\in C([0,T];\\mathbb{R}^m)^*$, we must have $|\\xi|_{\\mathbb{R}^{2n}} = 1$. When $\\lambda = 0$ and $\\xi = 0$, it holds that $\\mu \\neq 0 \\in C([0,T];\\mathbb{R}^m)^*$. This implies that the tuple $(\\lambda,\\xi,\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot))$ cannot be trivial, i.e., $(\\lambda,\\xi,\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\neq 0$ (they cannot be zero simultaneously).\n\nIn summary, based on the above discussion, it follows that the following tuple\n\\begin{align*}\n\\begin{cases}\n\t\\lambda  \\geq 0, \\\\\n\t\\xi \\in N_F \\Bigl ( \\begin{bmatrix}\\overline{x}_0 \\\\\\overline{x}(T) \\end{bmatrix} \\Bigr ), \\\\\n\t\\|\\mu_i \\|_{ C([0,T];\\mathbb{R}^m)^*}  = \\|\\theta_i(\\cdot)\\|_{\\textsc{NBV}([0,T];\\mathbb{R})} = \\theta_i(T) \\geq 0,~\\forall i=1,\\ldots,m\n\\end{cases}\n\\end{align*}\ncannot be trivial, i.e., it holds that $(\\lambda,\\xi,\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\neq 0$, and\n\\begin{align*}\n\t& \\begin{cases}\n\t\\lambda \\geq 0, \\\\\n\t\\dd \\theta_i(s) \\geq 0,~ \\forall s \\in [0,T],~i=1, \\ldots, m.\n\t\\end{cases}\n\\end{align*}\nThis shows the nontriviality and nonnegativity conditions in Theorem \\ref{Theorem_3_1}.\n\n\n\n\\subsection{Proof of Theorem \\ref{Theorem_3_1}: Adjoint Equation and Duality Analysis}\\label{Section_4_7}\nRecall the variational inequality in (\\ref{eq_4_23}), i.e., for any $(a,u) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$,\n\\begin{align}\n\\label{eq_5_40_23423402302020202}\n0 & \\leq  \\lambda \\widehat{Z}(T;a,u) + \\Bigl \\langle \\xi_1, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2, Z(T;a,u) \\Bigr \\rangle + \\Bigl \\langle \\mu,  G_x(\\cdot, \\overline{x}(\\cdot)) Z(\\cdot;a,u) \\Bigr \\rangle_{C_m^* \\times C_m}.\n\\end{align}\nSimilar to (\\ref{eq_4_27_4_23_3_4_2_}), by the Riesz representation theorem, it holds that\n\\begin{align*}\t\n\\Bigl \\langle \\mu,  G_x(\\cdot,\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) Z(\\cdot;a,u) \\Bigr \\rangle_{C_m^* \\times C_m} & = \\sum_{i=1}^m \\Bigl \\langle \\mu_i,  G_x^i(\\cdot,\\overline{x}(\\cdot;\\overline{x}_0,\\overline{u})) Z(\\cdot;a,u)  \\Bigr \\rangle_{C_1^* \\times C_1} \\\\\n& = \\sum_{i=1}^m \\int_0^T G_x^{i} (s,\\overline{x}(s)) Z(s;a,u) ) \\dd \\theta_i(s),\n\\end{align*}\nwhere as shown in Section \\ref{Section_4_5}, we have $\\theta(\\cdot) = (\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\in \\textsc{NBV}([0,T];\\mathbb{R}^m)$ with $\\theta_i (\\cdot) \\in \\textsc{NBV}([0,T];\\mathbb{R})$ being finite and monotonically nondecreasing on $[0,T]$. \n\nThen by using the variational equations in Lemma \\ref{Lemma_4_5}, (\\ref{eq_5_40_23423402302020202}) becomes\n\\begin{align}\n\\label{eq_4_30_4_2323234_2}\t\n0 & \\leq  \n\\Bigl \\langle \\xi_1 + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2 + \\lambda h_x(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle  + \\sum_{i=1}^m \\int_0^T G_x^{i} (s,\\overline{x}(s)) Z(s) \\dd \\theta_i(s)  \\\\\n&~~~ + \\int_0^T \\Bigl [ \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f_x(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}}   \\nonumber \\\\\n&~~~~~~~~~~ + \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )  g_x(T,s,\\overline{x}(s),\\overline{u}(s))  + \\lambda l_x(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] Z(s) \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}} \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\Bigl ( g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\Bigr ) \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\lambda  \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s .\\nonumber\n\\end{align}\nBased on Lemma \\ref{Lemma_B_5} and Remark \\ref{Remark_3_3}, let $p(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ be the unique solution to the adjoint equation in Theorem \\ref{Theorem_3_1}. Applying it to (\\ref{eq_4_30_4_2323234_2}) yields\n\\begin{align}\n\\label{eq_4_31_32342323423423423}\n0 & \\leq \t\n\\Bigl \\langle \\xi_1 + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2 + \\lambda h_x(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle \\\\\n&~~~ + \\sum_{i=1}^m \\int_0^T G_x^{i} (s,\\overline{x}(s)) Z(s;a,u) \\dd \\theta_i(s)  + \\int_0^T \\Biggl [ - p(s) - \\sum_{i=1}^m G_x^{i} (s,\\overline{x}(s))^\\top \\frac{\\dd \\theta_i(s)}{\\dd s} \\nonumber \\\\\n&~~~~~~~~~~ + \\int_s^T \\frac{f_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top}{(r-s)^{1-\\alpha}} p(r) \\dd r + \\int_s^T g_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top p(r) \\dd r \\Biggr ]^\\top Z(s;a,u) \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}} \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\Bigl [ g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\Bigr ] \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\lambda  \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s  \\nonumber \\\\\n& = \\Bigl \\langle \\xi_1 + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle + \\Bigl \\langle \\xi_2 + \\lambda h_x(\\overline{x}_0,\\overline{x}(T))^\\top, a \\Bigr \\rangle \\nonumber \\\\\n&~~~ + \\int_0^T \\Biggl [ - p(s) + \\int_s^T \\Bigl [ \\frac{f_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top}{(r-s)^{1-\\alpha}} + g_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top \\Bigr ] p(r) \\dd r \\Biggr ]^\\top Z(s;a,u) \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}} \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\Bigl [ g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\Bigr ] \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\lambda  \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s. \\nonumber\n\\end{align}\n\nIn (\\ref{eq_4_31_32342323423423423}), the standard Fubini's formula and Lemma \\ref{Lemma_4_5} lead to\n\\begin{align*}\n& \\int_0^T \\Biggl [ - p(s) + \\int_s^T \\Bigl [ \\frac{f_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top}{(r-s)^{1-\\alpha}} + g_x(r,s,\\overline{x}(s),\\overline{u}(s))^\\top \\Bigr ] p(r) \\dd r \\Biggr ]^\\top Z(s) \\dd s \\\\\n& = \\int_0^T - p(s)^\\top Z(s) \\dd s\t+ \\int_0^T \\int_0^s p(s)^\\top \\Bigl [ \\frac{f_x(s,r,\\overline{x}(r),\\overline{u}(r)}{(s-r)^{1-\\alpha}} + g_x(s,r,\\overline{x}(r),\\overline{u}(r)) \\Bigr ] Z(r) \\dd r  \\dd s \\\\\n& = \\int_0^T - p(s)^\\top \\Biggl [ Z(s) - \\int_0^s  \\Bigl [ \\frac{f_x(s,r,\\overline{x}(r),\\overline{u}(r)}{(s-r)^{1-\\alpha}} + g_x(s,r,\\overline{x}(r),\\overline{u}(r)) \\Bigr ]Z(r) \\dd r \\Biggr ] \\dd s \\\\\n& = \\int_0^T - p(s)^\\top \\Biggl [a + \\int_0^s \\frac{f(s,r,\\overline{x}(r),u(r)) - f(s,r,\\overline{x}(r),\\overline{u}(r))}{(s-r)^{1-\\alpha}} \\dd r \\\\\n&~~~~~~~ + \\int_0^s \\bigl [ g(s,r,\\overline{x}(r),u(r)) - g(s,r,\\overline{x}(r),\\overline{u}(r)) \\bigr ]   \\dd r \\Biggr ] \\dd s.\n\\end{align*}\nMoreover, by definition of $N_F$ in (\\ref{eq_4_1}), it follows that \n\\begin{align*}\n\\Bigl \\langle \\xi_1, a \\Bigr \\rangle + \\Bigl  \\langle \\xi_2, a \\Bigr  \\rangle & \\leq \\Bigl \\langle \\xi_1, \\overline{x}_0 - y_1 + a \\Bigr \\rangle + \\Bigl \\langle \\xi_2, \\overline{x}(T;\\overline{x}_0,\\overline{u}) - y_2 + a \\Bigr \\rangle,~ \\forall y = \\begin{bmatrix}\n y_1 \\\\ \n y_2\n \\end{bmatrix} \\in F.\n\\end{align*}\nHence, (\\ref{eq_4_31_32342323423423423}) becomes for any $(a,u) \\in \\mathbb{R}^n \\times \\mathcal{U}^p[0,T]$ and $y \\in F$, \n\\begin{align}\t\n\\label{eq_4_32}\n0 & \\leq \\Bigl \\langle \\xi_1, \\overline{x}_0 - y_1 + a \\Bigr \\rangle_{\\mathbb{R}^n \\times \\mathbb{R}^n} + \\Bigl \\langle \\xi_2, \\overline{x}(T;\\overline{x}_0,\\overline{u}) - y_2 + a \\Bigr \\rangle_{\\mathbb{R}^n \\times \\mathbb{R}^n} \\\\\n&~~~ + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T)) a + \\lambda  h_x(\\overline{x}_0,\\overline{x}(T)) a - \\Bigl \\langle \\int_0^T p(s)  \\dd s , a \\Bigr \\rangle \\nonumber \\\\\n&~~~ + \\int_0^T - p(s)^\\top \\Biggl [\\int_0^s \\frac{f(s,r,\\overline{x}(r),u(r)) - f(s,r,\\overline{x}(r),\\overline{u}(r))}{(s-r)^{1-\\alpha}} \\dd r \\nonumber \\\\\n&~~~~~~~ + \\int_0^s \\bigl [ g(s,r,\\overline{x}(r),u(r)) - g(s,r,\\overline{x}(r),\\overline{u}(r)) \\bigr ]   \\dd r \\Biggr ] \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}} \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\Bigl [ g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\Bigr ] \\dd s \\nonumber \\\\\n&~~~ + \\int_0^T \\lambda  \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s. \\nonumber\n\\end{align}\nBelow, we use (\\ref{eq_4_32}) to prove the transversality condition, the nontriviality of the adjoint equation, and the Hamiltonian-like maximum condition in Theorem \\ref{Theorem_3_1}.\n\n\n\\subsection{Proof of Theorem \\ref{Theorem_3_1}: Transversality Condition and Nontriviality of Adjoint Equation}\\label{Section_4_8}\n\nIn (\\ref{eq_4_32}), when $u=\\overline{u}$, we have\n\\begin{align}\n\\label{eq_5_456345345345}\t\n0 & \\leq \\Bigl \\langle \\xi_1, \\overline{x}_0 - y_1 + a \\Bigr \\rangle + \\Bigl \\langle \\xi_2, \\overline{x}(T;\\overline{x}_0,\\overline{u}) - y_2 + a \\Bigr \\rangle \\\\\n&~~~ + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T)) a + \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) a - \\Bigl \\langle \\int_0^T p(s)  \\dd s , a \\Bigr \\rangle,~ \\forall y = \\begin{bmatrix}\n y_1 \\\\ \n y_2\n \\end{bmatrix} \\in F. \\nonumber\n\\end{align}\nWhen $y_1 = \\overline{x}_0$ and $y_2 = \\overline{x}(T;\\overline{x}_0,\\overline{u})$, the above inequality holds for any $a,-a \\in \\mathbb{R}^n$, which implies\n\\begin{align}\n\\label{eq_4_38_1_2_1_2_3_2_1}\n\\int_0^T p(s) \\dd s = \\xi_1 + \\xi_2 + \\lambda h_{x_0}(\\overline{x}_0,\\overline{x}(T))^\\top  +\\lambda h_x(\\overline{x}_0,\\overline{x}(T))^\\top.\n\\end{align}\nUnder this condition, (\\ref{eq_5_456345345345}) becomes\n\\begin{align*}\t\n0 \\leq \\Bigl \\langle \\xi_1, \\overline{x}_0 - y_1 \\Bigr \\rangle + \\Bigl \\langle \\xi_2, \\overline{x}(T;\\overline{x}_0,\\overline{u}) - y_2 \\Bigr \\rangle,~ \\forall y  \\in F.\t\n\\end{align*}\nThis proves the transversality condition in Theorem \\ref{Theorem_3_1}. In addition, as $p (\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ by Lemma \\ref{Lemma_B_5},  (\\ref{eq_4_38_1_2_1_2_3_2_1}), together with the nontriviality condition, shows the nontriviality of the adjoint equation in Theorem \\ref{Theorem_3_1}.\n\n\\subsection{Proof of Theorem \\ref{Theorem_3_1}: Hamiltonian-like Maximum Condition}\\label{Section_4_9}\n\nWe finally prove the Hamiltonian-like maximum condition in Theorem \\ref{Theorem_3_1}. When $y_1 = \\overline{x}_0$, $y_2 = \\overline{x}(T;\\overline{x}_0,\\overline{u})$ and $a=0$ in (\\ref{eq_4_32}), by the standard Fubini's formula, (\\ref{eq_4_32}) can be written as\n\\begin{align}\n\\label{eq_4_33}\n0 & \\leq \\int_0^T -p(s)^\\top \\Biggl [\\int_0^s \\frac{f(s,r,\\overline{x}(r),u(r)) - f(s,r,\\overline{x}(r),\\overline{u}(r))}{(s-r)^{1-\\alpha}} \\dd r \\\\\n&~~~~~~~ + \\int_0^s \\bigl [ g(s,r,\\overline{x}(r),u(r)) - g(s,r,\\overline{x}(r),\\overline{u}(r)) \\bigr ]   \\dd r \\Biggr ] \\dd s  \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}} \\dd s  \\nonumber \\\\\n&~~~ + \\int_0^T \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\Bigl [ g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\Bigr ] \\dd s  \\nonumber \\\\\n&~~~ + \\int_0^T \\lambda  \\Bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\Bigr ] \\dd s\t \\nonumber \\\\\n& = \\int_0^T \\Biggl [ \\int_s^T - p(r)^\\top   \\frac{f(r,s,\\overline{x}(s),u(s)) - f(r,s,\\overline{x}(s),\\overline{u}(s))}{(r-s)^{1-\\alpha}} \\dd r  \\nonumber \\\\\n&~~~~~~~~~~ + \\int_s^T - p(r)^\\top \\bigl [ g(r,s,\\overline{x}(s),u(s)) - g(r,s,\\overline{x}(s),\\overline{u}(s)) \\bigr ] \\dd r    \\nonumber \\\\\n&~~~~~~~~~~ + \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\mathds{1}_{[0,T)}(s) \\frac{f(T,s,\\overline{x}(s),u(s)) - f(T,s,\\overline{x}(s),\\overline{u}(s))}{(T-s)^{1-\\alpha}}  \\nonumber \\\\\n&~~~~~~~~~~ + \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\bigl [ g(T,s,\\overline{x}(s),u(s)) - g(T,s,\\overline{x}(s),\\overline{u}(s))  \\bigr ]  \\nonumber \\\\\n&~~~~~~~~~~ + \\lambda  \\bigl [ l(s,\\overline{x}(s),u(s)) - l(s,\\overline{x}(s),\\overline{u}(s)) \\bigr ]  \\Biggr ] \\dd s. \\nonumber\n\\end{align}\n\nLet us define for $s \\in [0,T]$,\n\\begin{align*}\n& \\Lambda(s,\\overline{x}(s),u) := \\int_s^T p(r)^\\top \\frac{f(r,s,\\overline{x}(s),u)}{(s-r)^{1-\\alpha}} \\dd r - \\mathds{1}_{[0,T)}(s)  \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr ) \\frac{f(T,s,\\overline{x}(s),u)}{(T-s)^{1-\\alpha}} \\\\\n&~~~~~ + \\int_s^T p(r)^\\top g(r,s,\\overline{x}(s),u) \\dd r - \\Bigl ( \\lambda h_x(\\overline{x}_0,\\overline{x}(T)) + \\xi_2^\\top \\Bigr )  g(T,s,\\overline{x}(s),u) - \\lambda l(s,\\overline{x}(s),u).\n\\end{align*}\nThen we observe that (\\ref{eq_4_33}) becomes\n\\begin{align*}\n\\int_0^T \\Lambda(s,\\overline{x}(s),u(s)) \\dd s \\leq \t\\int_0^T \\Lambda(s,\\overline{x}(s),\\overline{u}(s)) \\dd s.\n\\end{align*}\n\nAs $U$ is separable, there exists a countable dense set $U_i = \\{u_i,~ i \\geq 1\\} \\subset U$. Moreover, there exists a measurable set $S_i \\subset [0,T]$ such that $|S_i| = T$ and any $t \\in S_i$ is the Lebesgue point of $\\Lambda(t,\\overline{x}(t),u(t))$, i.e., $\\lim_{\\tau \\downarrow 0} \\frac{1}{2\\tau} \\int_{t-\\tau}^{t+\\tau} \\Lambda(s,\\overline{x}(s),u(s)) \\dd s = \\Lambda(t,\\overline{x}(t),u(t))$ \\cite[Theorem 5.6.2]{Bogachev_book}. We fix $u_i \\in U_i$. For any $ t \\in S_i$, define\n\\begin{align*}\nu(s) : = \\begin{cases}\n \t\\overline{u}(s), & s \\in [0,T] \\setminus (t-\\tau,t+\\tau),\\\\\n \tu_i, & s \\in (t-\\tau,t+\\tau).\n \\end{cases}\t\n\\end{align*}\nIt then follows that\n\\begin{align*}\n0 \\leq \\lim_{\\tau \\downarrow 0} \\frac{1}{2\\tau} \\int_{t - \\tau}^{t + \\tau} \\bigl [ \t\\Lambda(s,\\overline{x}(s),\\overline{u}(s)) - \\Lambda(s,\\overline{x}(s),u_i) \\bigr ] \\dd s  = \\Lambda  (t,\\overline{x}(t),\\overline{u}(t)) - \\Lambda  (t,\\overline{x}(t),u_i).\n\\end{align*}\nSince $\\cap_{i \\geq 1} S_i = [0,T]$, $\\Lambda$ is continuous in $u \\in U$, and $U$ is separable, we must have\n\\begin{align*}\n\\Lambda(t,\\overline{x}(t),u) \\leq \\Lambda(t,\\overline{x}(t),\\overline{u}(t)),~ \\forall u \\in U,~  \\textrm{a.e.}~ t \\in [0,T],\n\\end{align*}\nwhich proves the Hamiltonian-like maximum condition in Theorem \\ref{Theorem_3_1}. This is the end of the proof for Theorem \\ref{Theorem_3_1}.\n\n\n\n\n\n\n\\begin{appendices}\n\t\n\\section*{Appendices}\n\nWe provide some preliminary results, and obtain the well-posedness and estimates of general Volterra integral equations having singular and nonsingular kernels. To simplify the notation, we use $\\|\\cdot\\|_{q} := \\|\\cdot\\|_{L^q([0,T];\\mathbb{R}^n)}$ and $\\|\\cdot\\|_{p} := \\|\\cdot\\|_{L^p([0,T];\\mathbb{R}^n)}$. \n\n\n\n\n\\section{Preliminaries}\\label{Appendix_A}\n\n\n\n\\begin{lemma}\n\\label{Lemma_A_2}\nLet $\\alpha \\in (0,1)$. Suppose that $\\frac{1}{q} + 1 = \\frac{1}{p} + \\frac{1}{r}$ with $p,q \\geq 1$ and $ r \\in [1, \\frac{1}{1-\\alpha})$. Then for any $a<b$, $ \\tau \\in (0,b-a]$, and $\\psi(\\cdot) \\in L^p([a,b];\\mathbb{R})$,\n\\begin{align*}\n\\Bigl ( \\int_{a}^{a+\\tau} \\Bigl | \\int_a^{t} \\frac{\\psi(s)}{(t-s)^{1-\\alpha}} \\dd s \\Big |^q \\dd t \\Bigr )^{\\frac{1}{q}} &\\leq \\Bigl ( \\frac{\\tau^{1-r(1-\\alpha)}}{1-r(1-\\alpha)} \\Bigr)^{\\frac{1}{r}} \\|\\psi(\\cdot) \\|_{L^p([a,b];\\mathbb{R})}, \\\\\n\\Bigl ( \\int_a^{a+\\tau} \\Bigl | \\int_a^t \\psi(s) \\dd s \\Bigr |^q \\dd t \\Bigr )^{\\frac{1}{q}} & \\leq  \\tau^{\\frac{1}{r}} \\|\\psi(\\cdot) \\|_{L^p([a,b];\\mathbb{R})}.\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nLet $\\zeta_{\\tau}(t) := \\frac{1}{t^{1-\\alpha}}\t\\mathds{1}_{(0,\\tau]}(t)$. Note that $\\mathds{1}_{(0,\\tau]}(t-s) = 1$ for $t-s \\in (0,\\tau]$, otherwise $\\mathds{1}_{(0,\\tau]}(t-s) = 0$. It follows that (note that $t \\vee s := \\max \\{t,s\\}$ for $t,s \\in [0,T]$)\n\\begin{align*}\n(\\psi(\\cdot)\\mathds{1}_{[a,b]} * \\zeta_{\\tau}(\\cdot) )(t) & = \\int_{a}^{b} \\frac{\\psi(s)}{(t-s)^{1-\\alpha}} \\mathds{1}_{(0,\\tau]}(t-s) \\dd s \n= \\begin{cases}\n\\int_{a \\vee t-\\tau}^{t} \\frac{\\psi(s)}{(t-s)^{1-\\alpha}} \\dd s, & t \\in [a,b],  \\\\\n0 & t \\notin [a,b].  \n \\end{cases}\n\\end{align*}\nThis leads to\n\\begin{align*}\n& (\\psi(\\cdot)\\mathds{1}_{[a,b]} * \\zeta_{\\tau}(\\cdot) )(t) = \\begin{cases}\n\\int_{a}^{t} \\frac{\\psi(s)}{(t-s)^{1-\\alpha}} \\dd s, & t \\in [a,a+\\tau],  \\\\\n0 & t \\notin [a,a+\\tau].  \n \\end{cases}\n\\end{align*}\nHence, by Young's Inequality (see \\cite[Theorem 3.9.4]{Bogachev_book}), for $\\frac{1}{q} + 1 = \\frac{1}{p} + \\frac{1}{r}$ with $r \\in [1,\\frac{1}{1-\\alpha})$,\n\\begin{align*}\n\\Bigl ( \\int_a^{a+\\tau} \\Bigl | \\int_a^t \\frac{\\psi(s)}{(t-s)^{1-\\alpha}} \\dd s \\Bigr|^q \\dd t \\Bigr )^{\\frac{1}{q}} \n& \\leq \\|\\psi(\\cdot)\\|_{L^p([a,b];\\mathbb{R})} \\|\\zeta_{\\tau}(\\cdot) \\|_{L^r([0,\\tau];\\mathbb{R})}.\n\\end{align*}\nNote that as $r(1-\\alpha) \\in [1-\\alpha,1)$ with $1-\\alpha > 0$,\n\\begin{align*}\n\t\\|\\zeta_{\\tau}(\\cdot) \\|_{L^r([0,\\tau];\\mathbb{R})}^r = \\int_0^{\\tau}  t^{-r(1-\\alpha)} \\dd t = \\frac{1}{1 - r(1-\\alpha)} \\tau^{1 - r(1-\\alpha)}.\n\\end{align*}\nThis proves the first inequality. The second inequality can be shown in a similar way by letting $\\zeta_{\\tau}(t) :=\t\\mathds{1}_{(0,\\tau]}(t)$. We complete the proof.\n\\end{proof}\n\n\\begin{lemma}[Lemma 2.3 of \\cite{Lin_Yong_SICON_2020}]\\label{Lemma_A_3}\nSuppose that $\\alpha \\in (0,1)$ and $p \\geq 1$. Assume that $\\psi :\\Delta \\rightarrow \\mathbb{R}^n$ is measurable with $\\psi(0,\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ satisfying $|\\psi(t,s) - \\psi(t^\\prime,s)|_{\\mathbb{R}^n} \\leq \\omega(|t-t^\\prime|) \\psi^\\prime(s)$ for $t,t^\\prime \\in [0,T]$ and $s \\in [0,T]$, where $\\psi^\\prime (\\cdot) \\in L^p([0,T];\\mathbb{R})$ and $\\omega$ is some modulus of continuity. Let\n\\begin{align*}\n\\varphi(t) &:= \\int_0^t \\frac{\\psi(t,s)}{(t-s)^{1-\\alpha}} \\dd s,~ \\textrm{a.e.}~ t \\in [0,T]. \n\\end{align*}\nThen $\\varphi(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ and $\\|\\varphi(\\cdot)\\|_{p} \\leq \\frac{T^\\alpha}{\\alpha} \\Bigl ( \\|\\psi(0,\\cdot)\\|_{p} + \\omega(T) \\|\\psi^\\prime (\\cdot) \\|_{p} \\Bigr )$.\nFurthermore, if $p > \\frac{1}{\\alpha}$, then  $\\varphi$ is continuous on $[0,T]$ and there is a constant $C$, independent from choice of $\\varphi$, such that \n\\begin{align*}\n|\\varphi(t)|_{\\mathbb{R}^n} \\leq C \\Bigl ( \t\\|\\psi(0,\\cdot)\\|_{p} + \\|\\psi^\\prime (\\cdot) \\|_{p} \\Bigr ),~ \\forall t \\in [0,T].\n\\end{align*}\n\\end{lemma}\n\n\n\\begin{lemma}\\label{Lemma_A_4}\nSuppose that $\\alpha \\in (0,1)$ and $q \\geq 1$. Assume that $\\psi :\\Delta \\rightarrow \\mathbb{R}^n$ is measurable with $\\psi(0,\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ satisfying $|\\psi(t,s) - \\psi(t^\\prime,s)|_{\\mathbb{R}^n} \\leq \\omega(|t-t^\\prime|) \\psi^\\prime(s)$ for $t,t^\\prime \\in [0,T]$ and $s \\in [0,T]$, where $\\psi^\\prime (\\cdot)  \\in L^p([0,T];\\mathbb{R})$ and $\\omega$ is some modulus of continuity. Let\n\\begin{align*}\n\\hat{\\varphi}(t) &:= \\int_0^t \\psi(t,s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\nThen $\\hat{\\varphi} (\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ and $\\|\\hat{\\varphi} (\\cdot) \\|_{p} \\leq T \\Bigl ( \\|\\psi(0,\\cdot)\\|_{p} + \\omega(T) \\|\\psi^\\prime (\\cdot) \\|_{p} \\Bigr )$.\nFurthermore, if $p > \\frac{1}{\\alpha}$, then  $\\hat{\\varphi}$ is continuous on $[0,T]$ and there is a constant $C$, independent from choice of $\\hat{\\varphi}$, such that \n\\begin{align*}\n|\\hat{\\varphi}(t)|_{\\mathbb{R}^n} \\leq C \\Bigl ( \t\\|\\psi(0,\\cdot)\\|_{p} + \\|\\psi^\\prime (\\cdot) \\|_{p} \\Bigr ),~ \\forall t \\in [0,T].\t\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe proof is analogous to that for Lemma \\ref{Lemma_A_3}. Indeed, to prove the continuity, note that\n\\begin{align*}\n|\\psi(t,s)|_{\\mathbb{R}^n} \\leq |\\psi(0,s)|_{\\mathbb{R}^n} + \\omega(t)\\psi^\\prime(s) =: \\overline{\\psi}(s),~ (t,s) \\in \\Delta,\n\\end{align*}\nwhere $\\overline{\\psi} \\in L^p([0,T];\\mathbb{R})$, since $\\psi(0,\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ and $\\psi^\\prime \\in L^p([0,T];\\mathbb{R})$. It holds that\n\\begin{align*}\n& |\\hat{\\varphi}(t) - \\hat{\\varphi}(t^\\prime) |_{\\mathbb{R}^n}\t\n\\leq \\omega(|t-t^\\prime|) T^{\\frac{q-1}{q}} \\|\\psi^\\prime  (\\cdot) \\|_{p} + \\|\\overline{\\psi} (\\cdot)\\|_{p} \\tau^{\\frac{p-1}{p}} + \\|\\overline{\\psi} (\\cdot) \\|_{p}(t^\\prime-t+\\tau)^{\\frac{p-1}{p}}.\n\\end{align*}\nThen the rest of the proof is similar to that of Lemma \\ref{Lemma_A_3}; thus completing the proof.\n\\end{proof}\n\n\\begin{lemma}[Gronwall-type inequality with the presence of singular and nonsingular kernels]\n\\label{Lemma_A_5}\nAssume that $\\alpha \\in (0,1)$ and $p > \\frac{1}{\\alpha}$.\nLet $P(\\cdot) \\in L^p([0,T];\\mathbb{R})$ and $b(\\cdot),z(\\cdot) \\in L^{\\frac{p}{p-1}}([0,T];\\mathbb{R})$, where $b$, $z$, and $P$ are nonnegative functions. Suppose that the following holds:\n\\begin{align}\n\\label{eq_a_1}\nz(t) \\leq b(t) + \\int_0^t  \\frac{P(s)z(s)}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t \tP(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align}\nThen there exists a constant $C \\geq 0$ such that\n\\begin{align*}\nz(t) \\leq b(t) + C \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha}} \\dd s + C \\int_0^t P(s) b(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\t\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nBy the H\\\"older's inequality, we have $P(\\cdot)z(\\cdot) \\in L^1([0,T];\\mathbb{R})$, which implies that the two integrals on the right-hand side of (\\ref{eq_a_1}) are well-defined in the $L^1$ sense. Below, there are several generic constants, whose values vary from line to line.\n\nWe remove the singularity of the right-hand side of (\\ref{eq_a_1}). As (\\ref{eq_a_1}) is linear in $z$, consider,\n\\begin{align*}\n\tz(t) \\leq b(t) + \\int_0^t \\frac{\\hat{P}(s;\\alpha) z(s)}{(t-s)^{1-\\alpha}} \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align*}\nwhere $\\hat{P}(s;\\alpha) := P(s)  + (t-s)^{1-\\alpha} P(s)$ with $(t,s) \\in \\Delta$. Note that $\\hat{P}(\\cdot;\\alpha) \\in L^p([0,T];\\mathbb{R})$ is nonnegative, and since $1-\\alpha > 0$, we have $\\|\\hat{P}(\\cdot;\\alpha)\\|_{p} \\leq  \\|P(\\cdot)\\|_{p} + T^{1-\\alpha} \\|P(\\cdot)\\|_{p} \\leq C \\|P(\\cdot)\\|_{p}$.\n\nIt follows that \n\\begin{align}\n\\label{eq_A_2_23234234}\nz(t) \n& \\leq b(t) + \\int_0^t \\frac{P(s) b(s)}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t P(s) b(s) \\dd s + \\int_0^t \t\\frac{\\hat{P}(s;\\alpha)}{(t-s)^{1-\\alpha}} \\int_0^s \\frac{\\hat{P}(\\tau;\\alpha)}{(s-\\tau)^{1-\\alpha}} z(\\tau) \\dd \\tau \\dd s,\n\\end{align}\nNotice that the double integral above represents the integration over the triangular region with base and height of $t$, where the integration is performed vertically and then horizontally with respect to $\\tau$ and $s$, respectively. Alternatively, we may reverse the order of the double integration above, i.e.,\n\\begin{align}\n\\label{eq_a_3_345345}\n \\int_0^t \\frac{\\hat{P}(s;\\alpha) }{(t-s)^{1-\\alpha}} \\int_0^s \\frac{\\hat{P}(\\tau;\\alpha)}{(s-\\tau)^{1-\\alpha}} z(\\tau) \\dd \\tau \\dd s \n& = \\int_0^t \\int_{s}^t \\frac{\\hat{P}(\\tau;\\alpha) \\hat{P}(s;\\alpha)}{(t-\\tau)^{1-\\alpha} (\\tau-s)^{1-\\alpha}}   z(s) \\dd \\tau \\dd s.\n\\end{align}\n\nLet $v := \\frac{\\tau-s}{t-s}$. Note that $\\tau$ varies from $s$ to $t$, which implies $v$ varies from $0$ to $1$. Moreover, $\\tau = s+ (t-s) v $ and $\\dd \\tau = (t-s) \\dd v$.  Then using the H\\\"older's inequality and changing the integration variable, the integration in (\\ref{eq_a_3_345345}) can be evaluated by\n\\begin{align*}\t\n& \\int_0^t \\int_{s}^t \\frac{\\hat{P}(\\tau;\\alpha) \\hat{P}(s;\\alpha)}{(t-\\tau)^{1-\\alpha} (\\tau-s)^{1-\\alpha}}  \\dd \\tau z(s) \\dd s \\\\\n& \\leq C \\Biggl ( \\|P(\\cdot)\\|_{p} \\int_0^t P(s) \\Bigl ( \\int_s^t \\frac{1}{(t-\\tau)^{ \\frac{(1-\\alpha)p}{p-1}} (\\tau-s)^{ \\frac{(1-\\alpha)p}{p-1}}} \\dd \\tau \\Bigr)^{\\frac{p-1}{p}} z(s) \\dd s + \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}} \\int_0^t P(s) z(s) \\dd s \\Biggr ) \\\\\n& = C^{(1)} \\Biggl ( \\|P(\\cdot)\\|_{p} \\int_0^t \\frac{P(s) z(s) }{(t-s)^{2(1-\\alpha) - \\frac{p-1}{p}}}\n\\Bigl (\\int_0^1 \\frac{1}{(1-v)^{(1-\\alpha) \\frac{p}{p-1}} v^{(1-\\alpha) \\frac{p}{p-1}}  }  \\dd v \\Bigr )^{\\frac{p-1}{p}} \\dd s \\\\\n&~~~~~~~~~~ + \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}} \\int_0^t P(s) z(s) \\dd s \\Biggr ).\n\\end{align*}\nLet $\\alpha[\\alpha]  := 1 - (1-\\alpha) \\frac{p}{p-1} = \\frac{\\alpha p - 1}{p-1} \\in (0,1)$ and $\\alpha^{(1)} := 1 - \\Bigl ( 2(1-\\alpha) - \\frac{p-1}{p} \\Bigr )  = 2 \\alpha - \\frac{1}{p} = \\alpha + \\Bigl (\\alpha - \\frac{1}{p} \\Bigr ) > \\alpha$ (note that $p > \\frac{1}{\\alpha}$). We can show that\n\\begin{align*}\n& C^{(1)} \\|P(\\cdot)\\|_{p} \\int_0^t \\frac{P(s) z(s) }{(t-s)^{2(1-\\alpha) - \\frac{p-1}{p}}}\n\\Bigl (\\int_0^1 \\frac{1}{(1-v)^{1-\\alpha[\\alpha]} v^{1-\\alpha[\\alpha]}  }  \\dd v \\Bigr )^{\\frac{p-1}{p}} \\dd s\t\\\\\n& = C^{(1)} \\|P(\\cdot)\\|_{p} B(\\alpha[\\alpha], \\alpha[\\alpha])^{\\frac{p-1}{p}}  \\int_0^t \\frac{P(s) z(s)}{(t-s)^{1-\\alpha^{(1)}}}  \\dd s =: \\bar{c}^{(1)} \\int_0^t \\frac{P(s) z(s)}{(t-s)^{1-\\alpha^{(1)}}}  \\dd s,\n\\end{align*}\nwhere $B$ is the beta function defined by $B(x,y) := \\int_0^1  t^{x-1} (1-t)^{y-1} \\dd t$ for $x,y > 0$. We also have $C^{(1)} \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}}\t \\int_0^t P(s) z(s) \\dd s =: \\hat{c}^{(1)} \\int_0^t P(s) z(s) \\dd s$. Then with $c^{(1)} := \\max \\{ \\bar{c}^{(1)}, \\hat{c}^{(1)} \\}$, (\\ref{eq_a_3_345345}) is bounded above by\n\\begin{align*}\t\n\\int_0^t \\frac{\\hat{P}(s;\\alpha) }{(t-s)^{1-\\alpha}} \\int_0^s \\frac{\\hat{P}(\\tau;\\alpha)}{(s-\\tau)^{1-\\alpha}} z(\\tau) \\dd \\tau \\dd s  \\leq c^{(1)} \\Bigl (  \\int_0^t \\frac{P(s) z(s)}{(t-s)^{1-\\alpha^{(1)}}}  \\dd s + \\int_0^t P(s) z(s) \\dd s \\Bigr ).\n\\end{align*}\nHence, by letting $c^{(0)} := 1$  and $\\alpha^{(0)} := \\alpha$, together with (\\ref{eq_a_1}), (\\ref{eq_A_2_23234234}) can be evaluated by\n\\begin{align}\n\\label{eq_a_2}\t\nz(t) \n& \\leq b(t) + \\int_0^t \\frac{P(s) b(s)}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t P(s) b(s) \\dd s + \\int_0^t \t\\frac{\\hat{P}(t,s)}{(t-s)^{1-\\alpha}} \\int_0^s \\frac{\\hat{P}(s,\\tau)}{(s-\\tau)^{1-\\alpha}} z(\\tau) \\dd \\tau \\dd s \\\\\n& \\leq b(t) + \\sum_{i=0}^{1} c^{(i)} \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha^{(i)}}} \\dd s + \\sum_{i=0}^{1} c^{(i)} \\int_0^t P(s) b(s) \\dd s  \\nonumber \\\\\n&~~~  + c^{(1)} \\int_0^t \\frac{\\hat{P}(s;\\alpha^{(1)})}{(t-s)^{1-\\alpha^{(1)}}} \\int_0^s \\frac{\\hat{P}(\\tau;\\alpha)}{(t-s)^{1-\\alpha}} z(\\tau) \\dd \\tau  \\dd s,~ \\textrm{a.e.}~ t \\in [0,T]. \\nonumber\n\\end{align}\n\nNote that by using the same technique as above, we can show that\n\\begin{align*}\n& c^{(1)} \\int_0^t \\frac{\\hat{P}(s;\\alpha^{(1)})}{(t-s)^{\\alpha^{(1)}}} \\int_0^s \\frac{\\hat{P}(\\tau;\\alpha)}{(s-\\tau)^{1-\\alpha}} z(\\tau) \\dd \\tau  \\dd s  = c^{(1)} \\int_0^t \\int_s^t \\frac{\\hat{P}(\\tau;\\alpha^{(1)})}{(t-\\tau)^{1-\\alpha^{(1)}}} \\frac{\\hat{P}(s;\\alpha)}{(\\tau-s)^{1-\\alpha}} z(s) \\dd \\tau  \\dd s \\\\\n& \\leq C^{(2)} c^{(1)} \\Biggl ( \\|P(\\cdot)\\|_{p} \\int_0^t  \\frac{P(s) z(s)}{(t-s)^{2-\\alpha-\\alpha^{(1)}  - \\frac{p-1}{p} }} \\Bigl ( \\int_0^1 \\frac{1}{(1-v)^{(1-\\alpha^{(1)}) \\frac{p}{p-1} } v^{ (1-\\alpha) \\frac{p}{p-1}  }} \\dd v \\Bigr )^{\\frac{p-1}{p}} \\dd s  \\\\\n&~~~~~~~~~~ + \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}} \\int_0^t P(s) z(s) \\dd s \\Biggr ).\n\\end{align*}\nLet $\\alpha[\\alpha^{(1)}] := 1 - (1-\\alpha^{(1)}) \\frac{p}{p-1} = \\frac{\\alpha^{(1)} p - 1}{p-1} \\in (0,1)$ and $\\alpha^{(2)} := 1- \\Bigl (2-\\alpha-\\alpha^{(1)} - \\frac{p-1}{p} \\Bigr )  = \\alpha + 2 \\Bigl ( \\alpha - \\frac{1}{p} \\Bigr ) > \\alpha^{(1)} > \\alpha$. We can show that\n\\begin{align*}\n& C^{(2)} c^{(1)} \\|P(\\cdot)\\|_{p} \\int_0^t  \\frac{P(s) z(s)}{(t-s)^{2-\\alpha-\\alpha^{(1)}  - \\frac{p-1}{p} }} \\Bigl ( \\int_0^1 \\frac{1}{(1-v)^{1-\\alpha[\\alpha^{(1)}]} v^{1- \\alpha[\\alpha]}} \\dd v \\Bigr )^{\\frac{p-1}{p}} \\dd s  \t \\\\\n& = C^{(2)} c^{(1)} \\|P(\\cdot)\\|_{p} B(\\alpha[\\alpha],\\alpha[\\alpha^{(1)}])^{\\frac{p-1}{p}} \\int_0^t \\frac{P(s)z(s)}{(t-s)^{1-\\alpha^{(2)}}} \\dd s =: \\bar{c}^{(2)} \\int_0^t \\frac{P(s)z(s)}{(t-s)^{1-\\alpha^{(2)}}} \\dd s,\n\\end{align*}\nand we have $C^{(2)} c^{(1)} \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}} \\int_0^t P(s) z(s) \\dd s =: \\hat{c}^{(2)} \\int_0^t P(s) z(s) \\dd s$.\nLet $c^{(2)} := \\max \\{ \\bar{c}^{(2)}, \\hat{c}^{(2)} \\}$. Then using a similar approach, (\\ref{eq_a_2}) can be evaluated by\n\\begin{align}\n\\label{eq_a_5_3423234234}\nz(t) \n& \\leq b(t) + \\sum_{i=0}^{1} c^{(i)} \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha^{(i)}}} \\dd s + \\sum_{i=0}^{1} c^{(i)} \\int_0^t P(s) b(s) \\dd s     \\\\\n&~~~ + c^{(2)} \\int_0^t \\frac{P(s)z(s)}{(t-s)^{1-\\alpha^{(2)}}}\\dd s + c^{(2)}  \\int_0^t P(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T]. \\nonumber\n\\end{align}\nProceeding similarly, using (\\ref{eq_a_1}), (\\ref{eq_a_5_3423234234}) is evaluated by\n\\begin{align*}\nz(t) & \\leq b(t) + \\sum_{i=0}^{2} c^{(i)} \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha^{(i)}}} \\dd s + \\sum_{i=0}^{2} c^{(i)} \\int_0^t P(s) b(s) \\dd s   \\\\\n&~~~ + c^{(3)} \\int_0^t \\frac{P(s)z(s)}{(t-s)^{1-\\alpha^{(3)}}}\\dd s + c^{(3)}  \\int_0^t P(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align*}\nwhere $\\alpha[\\alpha^{(2)}] := 1 - \\frac{(1- \\alpha^{(2)} )p}{p-1}  = \\frac{\\alpha^{(2)} p - 1}{p-1} \\in (0,1)$, $\\alpha^{(3)} := \\alpha + 3 \\Bigl ( \\alpha - \\frac{1}{p} \\Bigr ) > \\alpha$, $c^{(3)} := \\max \\{ \\bar{c}^{(3)}, \\hat{c}^{(3)} \\}$, $\\bar{c}^{(3)} := C^{(3)} c^{(2)} \\|P(\\cdot)\\|_{p} B (\\alpha[\\alpha], \\alpha[\\alpha^{(2)}])$, and $\\hat{c}^{(3)} := C^{(3)} c^{(2)} \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}}$.\n\nTherefore, by induction, we are able to get\n\\begin{align*}\nz(t) & \\leq b(t) + \\sum_{i=0}^{k-1} c^{(i)} \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha^{(i)}}} \\dd s + \\sum_{i=0}^{k-1} c^{(i)} \\int_0^t P(s) b(s) \\dd s   \\\\\n&~~~ + c^{(k)} \\int_0^t \\frac{P(s)z(s)}{(t-s)^{1-\\alpha^{(k)}}}\\dd s + c^{(k)}  \\int_0^t P(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align*}\nwhere $c^{(0)} = 1$, $\\alpha^{(0)} = \\alpha$, and for $i=1,\\ldots,k$,\n\\begin{align*}\n\\alpha[\\alpha] & := \\frac{\\alpha p - 1}{p-1}  \\in (0,1),~ \\alpha^{(i)} := \t \\alpha + i \\Bigl (\\alpha - \\frac{1}{p} \\Bigr ) > \\alpha,~ c^{(i)} := \\max \\{ \\bar{c}^{(i)}, \\hat{c}^{(i)} \\},\t \\\\\n\\bar{c}^{(i)} &:= C^{(i)} c^{(i-1)} \\|P(\\cdot)\\|_{p} B (\\alpha[\\alpha], \\alpha[\\alpha^{(i-1)}]),~ \\hat{c}^{(i)} := C^{(i)} c^{(i-1)} \\|P(\\cdot)\\|_{p} T^{\\frac{p-1}{p}}.\n\\end{align*}\n\nWe observe that there is $k^\\prime \\geq 1$ such that $\\alpha^{(k)} \\geq 1$ for any $k \\geq k^\\prime$. Hence, with a fixed $k \\geq k^\\prime$, using the H\\\"older's inequality, it follows that\n\\begin{align*}\nz(t) & \\leq b(t) + \\sum_{i=0}^{k-1} c^{(i)} \\int_0^t \\frac{P(s)b(s)}{(t-s)^{1-\\alpha^{(i)}}} \\dd s + \\sum_{i=0}^{k-1} c^{(i)} \\int_0^t P(s) b(s) \\dd s   \\\\\n&~~~ + c^{(k)} T^{\\alpha^{(k)}-1} \\int_0^t P(s)z(s) \\dd s + c^{(k)} T^{\\alpha^{(k)}-1}  \\int_0^t P(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\nNotice that the integrals above do not have the singularity. Hence, there is a constant $C$ such that $\\frac{c^{(i)}}{(t-s)^{1-\\alpha^{(i)}}} \\leq \n\\frac{C}{(t-s)^{1-\\alpha}}$ for all $i$ with $0 \\leq i \\leq k$ and $0 \\leq s < t \\leq T$. \nThis implies that\n\\begin{align*}\t\nz(t) \n& \\leq b(t) + C \\int_0^t \\frac{P(s) b(s)}{(t-s)^{1-\\alpha}} \\dd s + C \\int_0^t P(s) b(s) \\dd s\\\\\n&~~~ + c^{(k)} T^{\\alpha^{(k)}-1} \\int_0^t P(s)z(s) \\dd s + c^{(k)} T^{\\alpha^{(k)}-1}  \\int_0^t P(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T]. \n\\end{align*}\nThen we apply the standard Gronwall's inequality (see \\cite[page 14]{Walter_book}) to obtain the desired result. This completes the proof of the lemma.\n\\end{proof}\n\n\n\\section{Well-posedness and Estimates of Volterra Integral Equations}\\label{Appendix_B}\n\nWe prove Lemma \\ref{Lemma_2_1} in a more general setting when the initial condition of (\\ref{eq_1}) is also dependent on the outer time variable. Consider the following Volterra integral equation:\n\\begin{align}\n\\label{eq_b_1}\nx(t) = x_0(t) + \\int_0^t \\frac{f(t,s,x(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align}\nLet $x(\\cdot;x_0,u) := x(\\cdot)$ be the solution of (\\ref{eq_b_1}) under $(x_0(\\cdot),u(\\cdot)) \\in L^p([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$, where we recall\n\\begin{align*}\n\\mathcal{U}^p[0,T] = \\Bigl \\{u:[0,T] \\rightarrow U~|~ \\textrm{$u$ is measurable in $t \\in [0,T]$} ~\\&~ \\rho(u(\\cdot),u_0) \\in L^p([0,T];\\mathbb{R}_+) \\Bigr \\}\n\\end{align*}\nHere, $(U,\\rho)$ is a separable metric space, where $U \\subset \\mathbb{R}^d$ and $\\rho$ is the metric induced by the standard Euclidean norm $|\\cdot|_{\\mathbb{R}^d}$\n\n\\begin{assumption}\\label{Assumption_B_1}\nFor $p \\geq 1$ and $\\alpha \\in (0,1)$, there are nonnegative $K_0(\\cdot) \\in L^{ (\\frac{p}{1 + \\alpha p} \\vee 1)+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{ (\\frac{1}{\\alpha} \\vee \\frac{p}{p-1})+}([0,T];\\mathbb{R})$, where $L^{p+}([0,T];\\mathbb{R}^n) := \\cup_{r > p} L^{r}([0,T];\\mathbb{R}^n)$ for $1 \\leq p < \\infty$ and $t \\vee s := \\max \\{t,s\\}$ for $t,s \\in [0,T]$, such that\n\\begin{align*}\n\\begin{cases}\n\t|f(t,s,x,u) - f(t,s,x^\\prime,u^\\prime)| + |g(t,s,x,u) - g(t,s,x^\\prime,u^\\prime)| \\leq K(s) (|x-x^\\prime| + \\rho(u,u^\\prime)) , \\\\\n\t~~~~~~~~~~ \\forall (t,s) \\in \\Delta,~ x,x^\\prime \\in \\mathbb{R}^n,~ u,u^\\prime \\in U, \\\\\n\t|f(t,s,0,u)| + |g(t,s,0,u)| \\leq K_0(s),~ \\forall (t,s) \\in \\Delta,~ u \\in U.\n\\end{cases}\n\\end{align*}\n\\end{assumption}\n\n\\begin{lemma}\\label{Lemma_B_1}\nLet Assumption \\ref{Assumption_B_1} hold. Assume that $p \\geq 1$ and $\\alpha \\in (0,1)$. Then for any $(x_0(\\cdot),u(\\cdot)) \\in L^p([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$, (\\ref{eq_b_1}) admits a unique solution in $L^p([0,T];\\mathbb{R}^n)$. In addition, there is a constant $C \\geq 0$ such that (\\ref{eq_b_1}) holds the following estimate:\n\\begin{align}\n\\label{eq_b_1_1_1_2}\n\\Bigl \\| x(\\cdot;x_0,u) \\Bigr \\|_{p} \\leq C \\Bigl (1 + \\|x_0(\\cdot)\\|_{p} + \\|\\rho(u(\\cdot),u_0)\\|_{L^p([0,T];\\mathbb{R}_+)} \\Bigr ).\t\n\\end{align}\nFurthermore, for any $x_0(\\cdot),x_0^\\prime(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$ and $u(\\cdot),u^\\prime(\\cdot) \\in \\mathcal{U}^p[0,T]$, there is a constant $C \\geq 0$ such that\n\\begin{align}\n\\label{eq_b_1_1_1_3}\n& \\|x(\\cdot;x_0,u) - \tx(\\cdot;x_0^\\prime,u^\\prime) \\|_{p}  \\leq C  \\|x_0(\\cdot) - x_0^\\prime(\\cdot) \\|_{p} \\\\\n&~~~~~ + C\\Biggl [ \\int_0^T \\Bigl ( \\int_0^t \\frac{|f(t,s,x(s;x_0,u),u(s)) - f(t,s,x(s;x_0,u),u^\\prime(s))|}{(t-s)^{1-\\alpha}} \\dd s \\Bigr )^p \\dd t \\Biggr]^{\\frac{1}{p}}  \\nonumber \\\\\n&~~~~~ + C\\Biggl [ \\int_0^T \\Bigl ( \\int_0^t | g(t,s,x(s;x_0,u),u(s)) - g(t,s,x(s;x_0,u),u^\\prime(s)) | \\dd s \\Bigr)^p  \\dd t \\Biggr ]^{\\frac{1}{p}}. \\nonumber\n\\end{align}\n\\end{lemma}\n\n\\begin{remark}\\label{Remark_B_1}\n\\begin{enumerate}[(i)]\n\\item By Assumption \\ref{Assumption_B_1}, we have\n\\begin{align*}\n|f(t,s,x,u)| + |g(t,s,x,u)| & \\leq K_0(s) + K(s)(|x| + \\rho(u,u_0)),~ \\forall (t,s) \\in \\Delta,~ (x,u) \\in \\mathbb{R}^n \\times U.\n\\end{align*}\n\\item Unlike Assumption \\ref{Assumption_2_1}, we do not assume $p > \\frac{1}{\\alpha}$ in Assumption \\ref{Assumption_B_1}. In addition, the conditions of $K_0$ and $K$ in Assumption \\ref{Assumption_B_1} are weaker than those in Assumption \\ref{Assumption_2_1} when $p > \\frac{1}{\\alpha}$. Indeed, with $p > \\frac{1}{\\alpha}$, we can show that $K_0(\\cdot) \\in L^{\\frac{1}{\\alpha} + } ([0,T];\\mathbb{R}) \\subset L^{ (\\frac{p}{1 + \\alpha p} \\vee 1)+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{p}{\\alpha p - 1} + } ([0,T];\\mathbb{R}) \\subset   L^{ (\\frac{1}{\\alpha} \\vee \\frac{p}{p-1})+}([0,T];\\mathbb{R})$. This means that Lemma \\ref{Lemma_2_1} can be shown under the weaker assumption than Assumption \\ref{Assumption_2_1}\n\\end{enumerate}\n\\end{remark}\n\n\n\\begin{proof}[Proof of Lemma \\ref{Lemma_B_1}]\nThe main idea of the proof is the extension of \\cite[Theorem 3.1]{Lin_Yong_SICON_2020}, where unlike \\cite{Lin_Yong_SICON_2020} we have to consider the cross coupling characteristics between the singular and nonsingular kernels in (\\ref{eq_b_1}). Furthermore, our proof provides a more detailed statement, which can be viewed as a refinement of \\cite{Lin_Yong_SICON_2020}. \n\nWe first use the contraction mapping argument to show the existence and uniqueness of the solution to (\\ref{eq_b_1}). For $\\tau \\in [0,T]$, where $\\tau$ will be determined below, let us define\n\\begin{align*}\t\n\\mathcal{F}[x(\\cdot)](t) := x_0(t) + \\int_0^t \\frac{f(t,s,x(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t g(t,s,x(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [0,\\tau]. \n\\end{align*}\nFor $q,p \\geq 1$ and $r \\in  [1,\\frac{1}{1-\\alpha})$, set $r= 1+\\beta$, where $\\beta \\geq 0$ (equivalently, $\\beta \\in [0,\\frac{\\alpha}{1-\\alpha})$) and $\\frac{1}{p} + 1 = \\frac{1}{q} + \\frac{1}{1+\\beta}$. By Lemma \\ref{Lemma_A_2} and Remark \\ref{Remark_B_1}, it follows that\n\\begin{align}\n\\label{eq_b_2}\n& \\|\\mathcal{F}[x(\\cdot)] (\\cdot) \\|_{L^p([0,\\tau];\\mathbb{R}^n)} \\\\ \n& \\leq \\|x_0(\\cdot)\\|_{p} + \\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\Bigl \\|K_0(\\cdot) + K(\\cdot)(|x(\\cdot)| + \\rho(u(\\cdot),u_0) ) \\Bigr \\|_{L^q([0,\\tau];\\mathbb{R})}. \\nonumber\n\\end{align}\nBelow, we consider the three different cases.\n\n\\subsubsection*{Case I: $p > \\frac{1}{1-\\alpha}$}\nNote that $\\frac{1}{p} < 1 - \\alpha$.  Moreover, $\\frac{1}{\\alpha} > \\frac{p}{p-1}$, $\\frac{p}{1+\\alpha p} > 1$, and $ 1+\\beta < \\frac{1}{1-\\alpha} = 1+ \\frac{\\alpha}{1-\\alpha}$ (equivalently, $\\beta < \\frac{\\alpha}{1-\\alpha}$). In this case, we have $K_0(\\cdot) \\in L^{\\frac{p}{1+\\alpha p}+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{1}{\\alpha}+}([0,T];\\mathbb{R})$. Observe that $\\frac{1}{q} = \\frac{1}{p} + 1 - \\frac{1}{1+\\beta} < \t\\frac{1}{p} + 1 - \\frac{1}{1 + \\frac{\\alpha}{1-\\alpha}} = \\frac{1}{p} + \\alpha  < 1$ and $\\frac{1}{q} - \\frac{1}{p} = \\frac{p-q}{pq} = 1 - \\frac{1}{1+\\beta} < 1 - \\frac{1}{1+\\frac{\\alpha}{1-\\alpha}} = \\alpha < 1$, \nwhich implies $q \\searrow \\frac{p}{1+\\alpha p} < 1$ and $\\frac{pq}{p-q} \\searrow \\frac{1}{\\alpha}$ as $\\beta \\nearrow \\frac{\\alpha}{1-\\alpha}$. Hence, since $K_0(\\cdot) \\in L^{\\frac{p}{1+\\alpha p}+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{1}{\\alpha}+}([0,T];\\mathbb{R})$, we may choose $\\beta$ close enough to $\\frac{\\alpha}{1-\\alpha}$ so that $K_0 (\\cdot)\\in L^{q}([0,T];\\mathbb{R})$ and $K (\\cdot) \\in L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})$. Therefore, as $\\frac{p-q}{p} + \\frac{q}{p} = 1$, it follows that\n\\begin{align*}\n& \\|K_0(\\cdot) + K(\\cdot)(|x(\\cdot)| + u(\\cdot))\\|_{L^q([0,\\tau];\\mathbb{R})} \\\\\n& \\leq \\|K_0(\\cdot)\\|_{L^q([0,T];\\mathbb{R})} + \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} (\\| x(\\cdot)\\|_{L^p([0,T];\\mathbb{R}^n)} + \\|\\rho(u(\\cdot),u_0) \\|_{L^p([0,T];\\mathbb{R})}).\n\\end{align*}\nThis, together with (\\ref{eq_b_2}), implies\n\\begin{align}\n\\label{eq_b_3}\n\\bigl \\|\\mathcal{F}[x(\\cdot)] (\\cdot) \\bigr \\|_{L^p([0,\\tau];\\mathbb{R}^n)} & \\leq \\|x_0(\\cdot)\\|_{p} + \\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\Bigl [  \\|K_0(\\cdot)\\|_{L^q([0,T];\\mathbb{R})}\t  \\\\\n&~~~  + \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} \\Bigl ( \\| x (\\cdot) \\|_{L^p([0,\\tau];\\mathbb{R}^n)} + \\|\\rho(u(\\cdot),u_0) \\|_{L^p([0,\\tau];\\mathbb{R})} \\Bigr ) \\Bigr ]. \\nonumber\n\\end{align}\nThis shows $\\mathcal{F}[x(\\cdot)]: L^p([0,\\tau];\\mathbb{R}^n) \\rightarrow L^p([0,\\tau];\\mathbb{R}^n)$ for $\\tau \\in [0,T]$. For $x(\\cdot),x^\\prime (\\cdot) \\in L^p([0,\\tau];\\mathbb{R}^n)$, by Lemma \\ref{Lemma_A_2} and Assumption \\ref{Assumption_2_1}, and using the same technique as (\\ref{eq_b_2}) and (\\ref{eq_b_3}), it follows that\n\\begin{align*}\n& \\Bigl \\|( \\mathcal{F}[x(\\cdot)] - \\mathcal{F}[x^\\prime(\\cdot)])(\\cdot) \\Bigr \\|_{L^p([0,\\tau];\\mathbb{R}^n)} \\\\\n& \\leq \\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} \\|x(\\cdot) - x^\\prime(\\cdot) \\|_{L^p([0,\\tau];\\mathbb{R}^n)}.\n\\end{align*}\nTake $\\tau \\in (0,T]$, independent of $x_0$, such that $\\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} < 1$. Then the mapping $\\mathcal{F}[x(\\cdot)]: L^p([0,\\tau];\\mathbb{R}^n) \\rightarrow L^p([0,\\tau];\\mathbb{R}^n)$ is contraction. Hence, in view of the contraction mapping theorem, (\\ref{eq_b_1}) admits a unique solution on $[0,\\tau]$ in $L^p([0,\\tau];\\mathbb{R}^n)$.\n\nFor $[\\tau,2\\tau]$, consider,\n\\begin{align*}\t\n\\mathcal{F}[y(\\cdot)](t) & := x_0(t) + \\int_0^{\\tau} \\frac{f(t,s,x(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^{\\tau} g(t,s,x(s),u(s)) \\dd s \\\\\n&~~~ + \\int_{\\tau}^{t} \\frac{f(t,s,y(s),u(s))}{(t-s)^{1-\\alpha}} \\dd s + \\int_{\\tau}^{t} g(t,s,y(s),u(s)) \\dd s,~ \\textrm{a.e.}~ t \\in [\\tau,2\\tau].\n\\end{align*}\nNote that by Lemma \\ref{Lemma_A_2} and (\\ref{eq_b_3}), we have\n\\begin{align*}\n& \\bigl \\|\\mathcal{F}[y(\\cdot)] (\\cdot) \\bigr \\|_{L^p([\\tau,2\\tau];\\mathbb{R}^n)}  \\\\\n& \\leq \\|x_0(\\cdot) \\|_{p} + C \\Bigl [  \\|K_0 (\\cdot) \\|_{L^q([0,T];\\mathbb{R})}\t + \\|K (\\cdot) \\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} \\Bigl ( \\| x(\\cdot) \\|_{L^p([0,\\tau];\\mathbb{R}^n)}  \\\\\n&~~~ + \\|\\rho(u(\\cdot),u_0) \\|_{L^p([0, \\tau];\\mathbb{R})} \\Bigr ) + \\|K (\\cdot) \\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} \\Bigl ( \\| y(\\cdot) \\|_{L^p([\\tau, 2 \\tau];\\mathbb{R}^n)} + \\|\\rho(u(\\cdot),u_0) \\|_{L^p([\\tau, 2 \\tau];\\mathbb{R})} \\Bigr )  \\Bigr ],\n\\end{align*}\nwhich shows that $\\mathcal{F}[y(\\cdot)]:   L^p([\\tau,2\\tau];\\mathbb{R}^n) \\rightarrow L^p([\\tau,2\\tau];\\mathbb{R}^n)$. Moreover, by a similar argument, it follows that for any $y(\\cdot),y^\\prime (\\cdot) \\in L^p([\\tau,2\\tau];\\mathbb{R}^n)$,\n\\begin{align*}\n& \\bigl \\| (\\mathcal{F}[y(\\cdot)] - \\mathcal{F}[y^\\prime(\\cdot)] )(\\cdot)\\bigr \\|_{L^p([\\tau,2\\tau];\\mathbb{R}^n)} \\\\\n& \\leq \\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} \\|y(\\cdot) - y^\\prime(\\cdot) \\|_{L^p([\\tau, 2 \\tau];\\mathbb{R}^n)}.\t\n\\end{align*}\nAs before, we have $\\Bigl ( \\Bigl ( \\frac{\\tau^{1-(1+\\beta)(1-\\alpha)}}{1-(1+\\beta)(1-\\alpha)} \\Bigr )^{\\frac{1}{1+\\beta}} + \\tau^{\\frac{1}{1+\\beta}} \\Bigr ) \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})} < 1$. Hence, (\\ref{eq_b_1}) admits a unique solution on $[\\tau,2 \\tau]$ in $L^p([\\tau,2 \\tau];\\mathbb{R}^n)$. By induction, we are able to prove the existence and uniqueness of the solution for (\\ref{eq_b_1}) on $[0, \\tau], [\\tau,2 \\tau], \\ldots, \n[\\lfloor \\frac{T}{\\tau} \\rfloor \\tau ,T]$. This shows the existence and uniqueness of the solution for (\\ref{eq_b_1}) on $[0,T]$ in  $L^p([0,T];\\mathbb{R}^n)$. \n\nWe now prove the estimates in (\\ref{eq_b_1_1_1_2}) and (\\ref{eq_b_1_1_1_3}). Let $z(\\cdot) := |x(\\cdot)\t- x^\\prime(\\cdot) |_{\\mathbb{R}^n}$, where $x(\\cdot) := x(\\cdot;x_0,u)$ and $x^\\prime(\\cdot) := x(\\cdot;x_0^\\prime,u^\\prime)$. Then\n\\begin{align*}\nz(t)  \n& \\leq b(t) +  \\int_0^t \\frac{ K(s) z(s) }{(t-s)^{1-\\alpha}} \\dd s   + \\int_0^t K(s) z(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T],\n\\end{align*}\nwhere\n\\begin{align*}\nb(t) & := |x_0(t) - x_0^\\prime(t)|_{\\mathbb{R}^n} + \\int_0^t \\frac{|f(t,s,x(s),u^\\prime(s))-f(t,s,x(s),u(s))|}{(t-s)^{1-\\alpha}} \\dd s \\\\\n&~~~ + \\int_0^t |g(t,s,x(s),u^\\prime(s))-g(t,s,x(s),u(s))| \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\nNote that $z(\\cdot),b(\\cdot) \\in L^p([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{1}{\\alpha}+}([0,T];\\mathbb{R})$. We replace $p$ by $q$ in Lemma \\ref{Lemma_A_5}. Recall that the $L^p$-spaces of this paper are induced by the finite measure on $([0,T],\\mathcal{B}([0,T]))$. Then as $\\frac{1}{q} < \\frac{1}{p} + \\alpha < 1$, we may increase $q$ enough to get $K (\\cdot) \\in L^{q}([0,T];\\mathbb{R}) \\subset L^{\\frac{1}{\\alpha}+}([0,T];\\mathbb{R})$ and $z(\\cdot),b(\\cdot) \\in L^p([0,T];\\mathbb{R}) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R})$. By Lemma \\ref{Lemma_A_5}, it follows that\n\\begin{align*}\t\nz(t) = |x(t)\t- x^\\prime(t) |_{\\mathbb{R}^n} \\leq  b(t) + C \\int_0^t \\frac{K(s) b(s)}{(t-s)^{1-\\alpha}} \\dd s + C \\int_0^t K(s) b(s) \\dd s,~ \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\nHence, similar to (\\ref{eq_b_2}), \n\\begin{align*}\n& \\|x(\\cdot;x_0,u)\t- x(\\cdot;x_0^\\prime,u^\\prime)\t\\|_{p} \n \\leq C \\Bigl ( \\|b(\\cdot)\\|_{p} + \\|K(\\cdot)\\|_{L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})}\\|b(\\cdot)\\|_{p} \\Bigr ) \\leq C \\|b(\\cdot)\\|_{p}.\n\\end{align*}\nThis shows the estimate in (\\ref{eq_b_1_1_1_3}). The estimate in (\\ref{eq_b_1_1_1_2}) can be shown in a similar way.\n\n\n\\subsubsection*{Case II: $1 < p \\leq \\frac{1}{1-\\alpha}$} \n\nThis case implies $1-\\alpha \\leq \\frac{1}{p} < 1$, $\\frac{1}{\\alpha} \\leq \\frac{p}{p-1}$, and $\\frac{p}{1+\\alpha p} \\leq 1$. Moreover, for $\\beta \\in (0,p-1)$ (equivalent to $1 + \\beta  \\in (1,p)$),  we have $1-\\alpha \\leq \\frac{1}{p} < \\frac{1}{1+\\beta}$. Hence,  $K_0 (\\cdot) \\in L^{1+}([0,T];\\mathbb{R})$ and $K (\\cdot)  \\in L^{\\frac{p}{p-1}+}([0,T];\\mathbb{R})$. Then since $\\frac{1}{p} + 1 = \\frac{1}{q} + \\frac{1}{1+\\beta}$, we observe that $\\frac{1}{p}\t< \\frac{1}{q} = \\frac{1}{p} + 1 - \\frac{1}{1+\\beta} \\nearrow 1$ and $\\frac{p-q}{pq} = \\frac{1}{q} - \\frac{1}{p} = 1 - \\frac{1}{1+\\beta} \\nearrow  \\frac{p-1}{p}$ as $\\beta \\nearrow p-1$.\n\nAs $K_0(\\cdot) \\in L^{1+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{p}{p-1}+}([0,T];\\mathbb{R})$, we are able to choose $\\beta$ close enough to $p-1$ to get $q > 1$ and $\\frac{p-q}{pq} < \\frac{p-1}{p}$, which implies $K_0(\\cdot) \\in L^{q}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{pq}{p-q}}([0,T];\\mathbb{R})$. We replace $p$ by $q$ in Lemma \\ref{Lemma_A_5}. Since $p \\in [1,\\frac{1}{1-\\alpha}]$, choose $q$ to get $q > \\frac{p}{p-1} \\geq \\frac{1}{\\alpha}$, which implies $K(\\cdot) \\in L^{q}([0,T];\\mathbb{R}) \\subset L^{\\frac{p}{p-1}+}([0,T];\\mathbb{R})$ and $z(\\cdot),b(\\cdot) \\in L^p([0,T];\\mathbb{R}) \\subset L^{\\frac{q}{q-1}}([0,T];\\mathbb{R})$. Then the technique for Case I can be applied to prove Case II.\n \n\n\\subsubsection*{Case III: $p=1$} \n\nWe have $K_0(\\cdot) \\in L^{1+}([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\infty}([0,T];\\mathbb{R})$. Choose $\\beta = 0$ and use (\\ref{eq_b_2}) to get \n\\begin{align*}\n\\bigl \\|\\mathcal{F}[x(\\cdot)](\\cdot) \\bigr \\|_{L^1([0,\\tau];\\mathbb{R}^n)} & \\leq \\|x_0(\\cdot)\\|_{p} +  \\frac{\\tau^{\\alpha}}{\\alpha} \\Bigl [  \\|K_0(\\cdot)\\|_{L^1([0,T];\\mathbb{R})}\t \\\\\n&~~~~~~~ + \\|K(\\cdot)\\|_{L^{\\infty}([0,T];\\mathbb{R})}( \\| x(\\cdot) \\|_{L^1([0,\\tau];\\mathbb{R}^n)} + \\|\\rho(u(\\cdot),u_0)\\|_{L^1([0,\\tau];\\mathbb{R})}) \\Bigr ].\n\\end{align*}\nThen the rest of the proof is similar to that for Case I. This completes the proof of the theorem.\n\\end{proof}\n\n\n\\begin{remark}\\label{Remark_B_2}\n\tThe integrability of $K_0$ and $K$ in Assumption \\ref{Assumption_B_1} is crucial in the proof of Lemma \\ref{Lemma_B_1}. Comparing between Cases I and II, we see that $(x_0(\\cdot),u(\\cdot)) \\in L^p([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$ has weaker integrability in Case II. Hence, we need stronger integrability of $K$ from $K(\\cdot) \\in L^{\\frac{1}{\\alpha}+}([0,T];\\mathbb{R})$ to $K(\\cdot) \\in L^{\\frac{p}{p-1}+}([0,T];\\mathbb{R})$, and $K_0$ from $K_0(\\cdot) \\in L^{\\frac{p}{1+\\alpha p} + } ([0,T];\\mathbb{R})$ to $K_0(\\cdot) \\in L^{1 + } ([0,T];\\mathbb{R})$  (note that in Case II, $\\frac{1}{\\alpha} \\leq \\frac{p}{p-1}$ and $\\frac{p}{1+\\alpha p} \\leq 1$). Notice that for Case III, by the weakest integrability of $(x_0(\\cdot),u(\\cdot)) \\in L^p([0,T];\\mathbb{R}^n) \\times \\mathcal{U}^p[0,T]$, we need the essential boundedness of $K$, i.e., the strongest integrability condition for $K$. Finally, as the proof relies on the contraction mapping argument, the solution of (\\ref{eq_b_1}) can be constructed via the standard Picard iteration algorithm, which is applied to Examples \\ref{Example_1} and \\ref{Example_2} in Section \\ref{Section_5}.\n\\end{remark}\n\n\nWe state the continuity of the solution under the stronger assumption (see Remark \\ref{Remark_B_1}).\n\\begin{lemma}\\label{Lemma_B_2}\nLet Assumption \\ref{Assumption_2_1} hold and $x_0(\\cdot) \\in C([0,T];\\mathbb{R}^n)$. Then (\\ref{eq_b_1}) admits a unique solution in $C([0,T];\\mathbb{R}^n)$. \n\\end{lemma}\n\n\\begin{proof}\nBased on Lemma \\ref{Lemma_B_1} and Remark \\ref{Remark_B_1}, (\\ref{eq_b_1}) admits a unique solution in $L^p([0,T];\\mathbb{R}^n)$.  Notice that under Assumption \\ref{Assumption_2_1},  $K_0(\\cdot) \\in L^{\\frac{1}{\\alpha} + } ([0,T];\\mathbb{R})$ and $K(\\cdot) \\in L^{\\frac{p}{\\alpha p - 1} + } ([0,T];\\mathbb{R})$. Let $q = \\frac{sp}{s+p}$, where $s > \\frac{p}{\\alpha p - 1}$. We observe $K(\\cdot) \\in L^s([0,T];\\mathbb{R}^n)$. Since $s = \\frac{pq}{p-q}$, we have $\\frac{1}{q} = \\frac{s+p}{sp} = \\frac{1}{p} + \\frac{1}{s} > \\frac{1}{p}$ and $\\frac{pq}{p-q} > \\frac{p}{\\alpha p - 1}\t\\Rightarrow \\frac{p-q}{pq} < \\frac{\\alpha p - 1}{p} \\Rightarrow  \\alpha > \\frac{1}{q}$. This implies $ \\frac{1}{p} < \\frac{1}{q} < \\alpha$, i.e., $p > q > \\frac{1}{\\alpha}$. \n\nNote that $K_0(\\cdot) \\in L^q([0,T];\\mathbb{R})$ and \n\\begin{align}\n\\label{eq_b_6}\n|f(t,s,x(s),u(s))| + |g(t,s,x(s),u(s))| & \\leq K_0(s) + K(s)(|x(s)|_{\\mathbb{R}^n} + |\\rho(u(s),u_0)|) =: \\overline{\\psi}(s).\n\\end{align}\nConsequently, using the H\\\"older's inequality, we get\n\\begin{align*}\n& \\Bigl ( \t\\int_0^T |K(s)|^q(|x(s)|_{\\mathbb{R}^n} + \\rho(u(s),u_0))^q \\dd s   \\Bigr)^{\\frac{1}{q}}  \\\\\n& \\leq \\|K(\\cdot)\\|_{L^{ \\frac{pq}{p-q} }([0,T];\\mathbb{R})}  ( \\|x (\\cdot) \\|_{p} + \\| \\rho(u(\\cdot),u_0)\\|_{L^p([0,T];\\mathbb{R})}  ) < \\infty.\n\\end{align*}\nThis implies $\\overline{\\psi}$ defined in (\\ref{eq_b_6}) holds $\\overline{\\psi}(\\cdot) \\in L^q([0,T];\\mathbb{R})$. As $q > \\frac{1}{\\alpha}$, the continuity of (\\ref{eq_b_1}) follows from Lemmas \\ref{Lemma_A_3} and \\ref{Lemma_A_4}, together with Assumption \\ref{Assumption_2_1}. This completes the proof.\n\\end{proof}\n\n\nWe study linear Volterra integral equations having singular and nonsingular kernels. For $\\alpha \\in (0,1)$ and $x_0(\\cdot) \\in L^p([0,T];\\mathbb{R}^n)$, consider\n\\begin{align}\n\\label{eq_b_34543534234234}\nx(t) = x_0(t) + \\int_{0}^t \\frac{F(t,s)x(s)}{(t-s)^{1-\\alpha}} \\dd s + \\int_0^t H(t,s) x(s) \\dd s,~  \\textrm{a.e.}~ t \\in [0,T],\n\\end{align}\nwhere $F,G:\\Delta \\rightarrow \\mathbb{R}^{n \\times n}$ satisfy $F(\\cdot,\\cdot), H(\\cdot,\\cdot)  \\in L^{\\infty}(\\Delta;\\mathbb{R}^{n \\times n})$.\n\n\n\\begin{lemma}\\label{Lemma_B_5_2323232}\nThe solution of (\\ref{eq_b_34543534234234}) can be written as\n\\begin{align}\n\\label{eq_b_34543534234234_1_1_1}\nx(t) = x_0(t) + \\int_0^t \\Psi(t,s) x_0(s) \\dd s,~  \\textrm{a.e.}~ t \\in [0,T],\n\\end{align}\nwhere $\\Psi$ is the state transition equation defined by\n\\begin{align*}\n\\Psi(t,s) = \\frac{F(t,s)}{(t-s)^{1-\\alpha}} + \\int_s^t \\frac{F(t,\\tau) \\Psi(\\tau,s) }{(t-\\tau)^{1-\\alpha}} \\dd \\tau  + H(t,s) + \\int_s^t H(t,\\tau) \\Psi(\\tau,s) \\dd \\tau,~  \\textrm{a.e.}~ t \\in (s,T].\n\\end{align*}\n\\end{lemma}\n\\begin{proof}\nThe well-posedness of (\\ref{eq_b_34543534234234}) follows from Lemma \\ref{Lemma_B_1}. From (\\ref{eq_b_34543534234234_1_1_1}), it follows that\n\\begin{align*}\n& \\int_{0}^t \\frac{F(t,s)}{(t-s)^{1-\\alpha}} x(s) \\dd s + \\int_0^t H(t,s) x(s) \\dd s \\\\\n& = \\int_{0}^t \\frac{F(t,s)}{(t-s)^{1-\\alpha}}  \\Bigl [x_0(s) + \\int_0^s \\Psi(s,\\tau) x_0(\\tau) \\dd \\tau \\Bigr ] \\dd s  + \\int_0^t H(t,s) \\Bigl [x_0(s) + \\int_0^s \\Psi(s,\\tau) x_0(\\tau) \\dd \\tau \\Bigr ] \\dd s \\\\\n& = \\int_0^t \\Bigl [ \\frac{F(t,s) }{(t-s)^{1-\\alpha}} + \\int_s^t \\frac{F(t,\\tau) \\Psi(\\tau,s) }{(t-\\tau)^{1-\\alpha} } \\dd \\tau + H(t,s) + \\int_s^t H(t,\\tau) \\Psi(\\tau,s) \\dd \\tau \\Bigr ] x_0(s) \\dd s \\\\\n& = \\int_0^t \\Psi(t,s) x_0(s) \\dd s = x(t) - x_0(t),\n\\end{align*}\nwhich completes the proof.\n\\end{proof}\n\n\nConsider the following $\\mathbb{R}^n$-valued backward Volterra integral equation having singular and nonsingular kernels, which covers the adjoint equation in Theorem \\ref{Theorem_3_1}:\n\\begin{align}\n\\label{eq_b_7}\nz(t) & = z_0(t) + \\int_t^T \\frac{F(r,t)^\\top}{(r-t)^{1-\\alpha}} z(r) \\dd r + \\int_t^T H(r,t)^\\top z(r) \\dd r \\\\\n&~~~ + \\sum_{i=1}^m C_i(t)^\\top \\frac{\\dd \\theta_i(t)}{\\dd t} + D(t)^\\top,~  \\textrm{a.e.}~ t \\in [0,T]. \\nonumber\n\\end{align} \n\n\\begin{assumption}\\label{Assumption_6}\n\\begin{enumerate}[(i)]\n\t\t\\item $z_0(\\cdot) \\in C([0,T];\\mathbb{R}^n)$, $\\theta (\\cdot)  = (\\theta_1(\\cdot),\\ldots,\\theta_m(\\cdot)) \\in \\textsc{NBV}([0,T];\\mathbb{R}^m)$, and $\\dd \\theta_i \\ll \\dd t$, i.e., $\\dd \\theta_i$ is absolutely continuous with respect to $\\dd t$ for $i=1,\\ldots,m$;\n\t\t\\item $F,H:\\Delta \\rightarrow \\mathbb{R}^{n \\times n}$ and $C_i,D:[0,T] \\rightarrow \\mathbb{R}^n$, $i=1,\\ldots,m$, satisfy $F(\\cdot,\\cdot),H(\\cdot,\\cdot) \\in L^{\\infty}(\\Delta;\\mathbb{R}^{n \\times n})$ and $C_i(\\cdot),D(\\cdot) \\in L^{\\infty}([0,T];\\mathbb{R}^{n})$, $i=1,\\ldots,m$. \n\\end{enumerate}\n\\end{assumption}\n\n\n\\begin{lemma}\\label{Lemma_B_5}\nLet Assumption \\ref{Assumption_6} hold. Assume that $p \\geq 1$ and $\\alpha \\in (0,1)$. Then for any $z_0(\\cdot) \\in C([0,T];\\mathbb{R}^n)$, (\\ref{eq_b_7}) admits a unique solution in $L^p([0,T];\\mathbb{R}^n)$.\n\\end{lemma}\n\n\\begin{proof}\nNote that by Remark \\ref{Remark_3_3} and the Radon-Nikodym theorem, there is a unique $\\Theta_i(\\cdot) \\in L^1([0,T];\\mathbb{R})$, $i=1,\\ldots,m$, such that   $\\frac{\\dd \\theta_i(t)}{\\dd t} = \\Theta_i(t)$ for $i=1,\\ldots,m$. Hence, we may replace $\\frac{\\dd \\theta_i(t)}{\\dd t}$ by $\\Theta_i(t)$ in (\\ref{eq_b_7}). Let us define\n\\begin{align*}\n\\mathcal{G}[z(\\cdot)](t) & := z_0(t) + \\int_t^T \\frac{F(r,t)^\\top}{(r-t)^{1-\\alpha}} z(r) \\dd r + \\int_t^T H(r,t)^\\top z(r) \\dd r \\\\\n&~~~ + \\sum_{i=1}^m C_i(t)^\\top \\Theta_i(t) + D(t)^\\top,~  \\textrm{a.e.}~ t \\in [0,T].\n\\end{align*}\nClearly, $\\mathcal{G}[z(\\cdot)]: L^p([0,T];\\mathbb{R}^n) \\rightarrow L^p([0,T];\\mathbb{R}^n)$. In addition, for $\\tau > 0$, we apply a similar technique of Lemma \\ref{Lemma_B_1} (with Lemma \\ref{Lemma_A_2} for $p=q$ and $r=1$) to show that\n\\begin{align*}\n\\|(\\mathcal{G}[z(\\cdot)]- \\mathcal{G}[z^\\prime(\\cdot)])(\\cdot)\\|_{L^p([T-\\tau,T];\\mathbb{R}^n)} \\leq \t\\Bigl ( \\frac{\\tau K}{\\alpha} + \\tau K \\Bigr ) \\|z(\\cdot) - z^\\prime(\\cdot)\\|_{L^p([T-\\tau,T];\\mathbb{R}^n)}.\n\\end{align*}\nWe may choose $\\tau$, independent of $z_0$, such that $\\Bigl ( \\frac{\\tau K}{\\alpha} + \\tau K \\Bigr ) < 1$. Then by the contraction mapping theorem, (\\ref{eq_b_7}) admits a unique solution on $[T-\\tau,T]$ in $L^p([T-\\tau,T];\\mathbb{R}^n)$. By induction, we are able to show that (\\ref{eq_b_7}) admits a unique solution on $[0,T]$ in $L^p([0,T];\\mathbb{R}^n)$. \nWe complete the proof.\n\\end{proof}\n\n\n\\section{Auxiliary Lemmas}\\label{Appendix_D}\n\n\\begin{lemma}[Corollary 3.9 and page 144 of \\cite{Li_Yong_book} or Lemma 3 of \\cite{Bourdin_arxiv_2016}]\\label{Lemma_D_1}\nAssume that $(X,\\|\\cdot\\|_{X})$ is a Banach space. For $\\delta \\in (0,1)$, define $\\mathcal{E}_{\\delta} := \\{ E \\in [0,T]~|~ |E| = \\delta T\\}$, where $|E|$ denotes the Lebesgue measure of $E$. Suppose that $\\phi:\\Delta \\rightarrow X$ satisfies the properties such that (i) $\\|\\phi(t,s)\\|_{X} \\leq \\overline{\\phi}(s)$ for all $(t,s) \\in \\Delta$, where $\\overline{\\phi}(\\cdot) \\in L^1([0,T];\\mathbb{R})$, and (ii) for almost all $s \\in [0,T]$, $\\phi(\\cdot,s):[s,T] \\rightarrow X$ is continuous. Then there is an $E_{\\delta} \\in \\mathcal{E}_{\\delta}$ such that\n\\begin{align*}\n\\sup_{t \\in [0,T]} \\Biggl | \\int_0^t \\Bigl ( \\frac{1}{\\delta} \\mathds{1}_{E_{\\delta}}(s) - 1 \\Bigr ) \\phi(t,s) \\dd s \\Biggr | \\leq \\delta.\n\\end{align*}\n\\end{lemma}\n\n\n\\begin{lemma}[Lemma 4.2 of \\cite{Lin_Yong_SICON_2020}]\\label{Lemma_D_2}\nLet $\\mathcal{E}_{\\delta}$ be the set in Lemma \\ref{Lemma_D_1}. Assume $\\psi:\\Delta \\rightarrow \\mathbb{R}$ holds  the following property:\n\\begin{align}\n\\label{eq_d_1}\n\\begin{cases}\n\t|\\psi(0,s)| \\leq \\overline{\\psi}(s),~ s \\in [0,T], \\\\\n\t|\\psi(t,s) - \\psi(t^\\prime,s)| \\leq \\omega(|t-t^\\prime|) \\overline{\\psi}(s),~ (t,s),(t^\\prime,s) \\in \\Delta,\n\\end{cases}\t\n\\end{align}\nwhere $\\overline{\\psi} \\in L^p([0,T];\\mathbb{R})$ with $p > \\frac{1}{\\alpha}$ and $\\omega:[0,\\infty) \\rightarrow [0,\\infty)$ is some modulus of continuity. Then there is an $E_{\\delta} \\in \\mathcal{E}_{\\delta}$ such that\n\\begin{align*}\n\t\\sup_{t \\in [0,T]} \\Biggl | \\int_0^t \\Bigl (\\frac{1}{\\delta} \\mathds{1}_{E}(s) - 1 \\Bigr ) \\frac{\\psi(t,s)}{(t-s)^{1-\\alpha}} \\dd s \\Biggr | \\leq \\delta.\n\\end{align*}\n\\end{lemma}\n\n\n\\end{appendices}\n\n\n\\bibliographystyle{IEEEtranS}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nWith the popularity of GPS-equipped smart devices and wireless\nmobile networks \\cite{deng2013maximizing,kazemi2012geocrowd}, nowadays people can easily identify and participate\nin some location-based tasks that are close to their current\npositions, such as taking photos\/videos, repairing houses, and\/or\npreparing for parties at some spatial locations. Recently, a new\nframework, namely \\textit{spatial crowdsourcing}\n\\cite{kazemi2012geocrowd}, for employing workers to conduct spatial\ntasks, has emerged in both academia (e.g., the database community\n\\cite{chen2014gmission}) and industry (e.g., TaskRabbit\n\\cite{taskrabbit}). A typical spatial crowdsourcing platform (e.g.,\ngMission \\cite{chen2014gmission} and MediaQ \\cite{kim2014mediaq})\nassigns a number of moving \\textit{workers} to do \\textit{spatial\n\ttasks} nearby, which requires workers to physically move to some\nspecified locations and accomplish these tasks.\n\n\n\nNote that, not all spatial tasks are as simple as taking a photo or\nvideo clip (e.g., street view of Google Maps\n\\cite{GoogleMapStreetView}), monitoring traffic conditions (e.g.,\nWaze \\cite{waze}), or reporting local hot spots (e.g., Foursquare\n\\cite{foursquare}), which can be easily completed by providing\nanswers via camera, sensing devices in smart phones, or naked eyes,\nrespectively. In contrast, some spatial tasks can be rather complex,\nsuch as repairing a house, preparing for a party, and performing\nentertainment shows for a ceremony, which may consist of several\nsteps\/phases\/aspects, and require demanding professional skills from\nworkers. In other words, these complex tasks cannot be simply\naccomplished by normal workers, but require the skilled workers with\nspecific expertise (e.g., fixing roofs or setting up the stage).\n\n\n\n\\begin{figure}[t]\\centering\n\t\\scalebox{0.8}[0.8]{\\includegraphics{example_locations_C.eps}}\\vspace{-1ex}\n\t\\caption{\\small An Example of Repairing a House in the Multi-Skill Spatial Crowdsourcing System.}\\vspace{-5ex}\n\t\\label{fig:bbq_example}\n\\end{figure}\n\n\\begin{table}\\vspace{-3ex}\n\t\\parbox[b]{.44\\linewidth}{\n\t\t\\caption{\\small Worker\/Task Skills}\\label{tab:marker_skill}\\vspace{-2ex}\n\t\t{\\small\\scriptsize\n\t\t\t\\begin{tabular}{l|l}\n\t\t\t\t{\\bf worker\/task} & {\\bf  skill key set}\\\\\n\t\t\t\t\\hline \\hline\n\t\t\t\t$w_1$, $w_4$ $w_8$& \\{$a_1$, $a_4$, $a_6$\\}\\\\\n\t\t\t\t$w_2$& \\{$a_5$\\}\\\\\n\t\t\t\t$w_3$, $w_7$& \\{$a_2$,  $a_3$\\}\\\\\n\t\t\t\t$w_5$, $w_6$& \\{$a_1$,  $a_5$\\}\\\\\n\t\t\t\t\\hline\n\t\t\t\t$t_1$, $t_2$, $t_3$& \\{$a_1$ $\\sim$ $a_6$\\}\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\vspace{-1ex}\n\t}\n\t\\hfill\n\t\\parbox[b]{.52\\linewidth}{\n\t\t\\caption{\\small Descriptions of Skills}\\label{tab:skill_description}\\vspace{-2ex}\n\t\t{\\small\\scriptsize\n\t\t\t\\begin{tabular}{c|l}\n\t\t\t\t{\\bf skill key} & {\\bf \\quad  skill description}\\\\\n\t\t\t\t\\hline \\hline\n\t\t\t\t$a_1$& painting walls\\\\\n\t\t\t\t$a_2$& repairing roofs\\\\\n\t\t\t\t$a_3$& repairing floors\\\\\n\t\t\t\t$a_4$& installing pipe systems\\\\\n\t\t\t\t$a_5$& installing electronic components\\\\\n\t\t\t\t$a_6$& cleaning\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\t\\vspace{-1ex}\n\t}\n\t\\vspace{-4ex}\n\\end{table}\n\n\nInspired by the phenomenon of complex spatial tasks, in this paper,\nwe will consider an important problem in the\nspatial crowdsourcing system, namely \\textit{multi-skill spatial\n\tcrowdsourcing} (MS-SC), which assigns multi-skilled workers to those\ncomplex tasks, with the matching skill sets and high scores of the\nworker-and-task assignments.\n\nIn the sequel, we will illustrate  the\nMS-SC problem by a motivation example of repairing a house.\n\n\n\\nop{\n\t\n\tThe spatial tasks in the spatial crowdsourcing system can be\n\tchecking the display shelves at neighborhood stores, taking a\n\tphoto\/video (e.g., street view of Google Maps\n\t\\cite{GoogleMapStreetView}), monitoring traffic conditions (e.g.,\n\tWaze \\cite{waze}), or reporting local hot spots (e.g., Foursquare\n\t\\cite{foursquare}). Since simple tasks are easy for workers to\n\taccomplish (i.e., workers only need to move to some locations, and\n\tprovide the answer via naked eyes, camera, or sensing devices in\n\tsmart phones), they do not require expert skills from workers. Thus,\n\tany worker can be assigned to do simple tasks.\n\t\n\t\n\tOn the other hand, however, spatial tasks can be as complex as\n\trepairing a house, decorating a room, performing entertainment shows\n\tfor a ceremony, or reporting an ad-hoc news, which consist of\n\tseveral steps\/phases\/aspects, and require demanding professional\n\tskills from workers. In this case, the spatial crowdsourcing\n\tplatform has to assign multi-skilled workers only to those complex\n\ttasks that exactly require their skills, which is more challenging\n\tthan the assignment with simple tasks.\n\t\n\t\n}\n\n\n\n\n\n\\vspace{0.5ex}\\noindent {\\bf Example 1 (Repairing a House).} {\\it\n\tConsider a scenario of the spatial crowdsourcing in Figure\n\t\\ref{fig:bbq_example}, where a user wants to repair a house he\/she\n\tjust bought, in order to have a good living environment for his\/her\n\tfamily. However, it is not an easy task to repair the house, which\n\trequires many challenging works (skills), such as repairing\n\troofs\/floors, replacing\/installing pipe systems and\n\telectronic components, painting walls, and finally cleaning rooms.\n\tThere are many skilled workers that can accomplish one or some of\n\tthese skill types. In this case, the user can post a spatial task\n\t$t_1$, as shown in Figure \\ref{fig:bbq_example}, in the spatial\n\tcrowdsourcing system, which specifies a set of required skills\n\t(given in Tables \\ref{tab:marker_skill} and \\ref{tab:skill_description}) for the\n\thouse-repairing task, a valid time period to repair, and the maximum\n\tbudget that he\/she would like to pay.\n\t\n\tIn Figure \\ref{fig:bbq_example}, around the spatial location of\n\ttask $t_1$, there are 8 workers, $w_1 \\sim w_8$, each of whom has a\n\tdifferent set of skills as given in Table \\ref{tab:marker_skill}.\n\tFor example, worker $w_1$ has the skill set $\\{\\text{painting walls},$\n\t$\\text{installing pipe systems},$ $\\text{cleaning}\\}$.\n\t\n\t\n\tTo accomplish the spatial task $t_1$ (i.e., repair the house), the\n\tspatial crowdsourcing platform needs to select a best subset of\n\tworkers $w_i$ ($1\\leq i\\leq 8$), such that the union of their skill\n\tsets can cover the required skill set of task $t_1$, and, moreover,\n\tworkers can travel to the location of $t_1$ with the maximum net\n\tpayment under the constraints of arrival times, workers' moving ranges,  and budgets. For\n\texample, we can assign task $t_1$ with 3 workers $w_2$, $w_7$, and\n\t$w_8$, who are close to $t_1$, and whose skills can cover all the\n\trequired skills of $t_1$.\n\t\n}\n\n\n\n\n\n\\nop{\n\t\n\t\\vspace{0.5ex}\\noindent {\\bf Example 2 (Performing Ad Hoc News\n\t\tReporting).} {\\it A task requester wants to report a breaking news\n\t\tsuddenly occurring at a remote location. The news reporting needs\n\t\tdifferent skills such as taking photos, recording audio\/videos,\n\t\tinterviewing people, news background analysis, and writing the news.\n\t\tIn this situation, the task requester can post a spatial task of\n\t\tnews reporting with the specified location to the spatial\n\t\tcrowdsourcing system, which includes the information such as the\n\t\tbudget, a required skill set, and the valid period. The system can\n\t\tthen assign those workers, who are near the specified location and\n\t\thave the required news reporting skills, to the news reporting task.}\n\t\n}\n\n\n\n\n\nMotivated by the example above, in this paper, we will\nformalize the MS-SC problem, which aims to efficiently assign\nworkers to complex spatial tasks, under the task constraints of\nvalid time periods and maximum budgets, such that the required skill\nsets of tasks are fully covered by those assigned workers, and the\ntotal score of the assignment (defined as the total profit of\nworkers) is maximized.\n\n\nNote that, existing works on spatial crowdsourcing focused on\nassigning workers to tasks to maximize the total number of completed\ntasks \\cite{kazemi2012geocrowd}, the number of performed tasks for a\nworker with an optimal schedule \\cite{deng2013maximizing}, or the\nreliability-and-diversity score of assignments\n\\cite{cheng2014reliable}. However, they did not take into account\nmulti-skill covering of complex spatial tasks, time\/distance constraints, and the assignment\nscore with respect to task budgets and workers' salaries (excluding\nthe traveling cost). Thus, we cannot directly apply prior solutions to\nsolve our MS-SC problem.\n\n\nIn this paper, we first prove that our MS-SC problem in the spatial\ncrowdsourcing system is NP-hard, by reducing it from the \\textit{Set\n\tCover Problem} (SCP) \\cite{karp1972reducibility}. As a result, the\nMS-SC problem is not tractable, and thus very challenging to achieve\nthe optimal solution. Therefore, in this paper, we will tackle the\nMS-SC problem by proposing three effective approximation approaches,\ngreedy, $g$-\\textit{divide-and-conquer} ($g$-D\\&C), and\ncost-model-based adaptive algorithms, which can efficiently compute\nworker-and-task assignment pairs with the constraints\/goals of\nskills, time, distance, and budgets.\n\n\n\n\nSpecifically, we make the following contributions.\n\\begin{itemize}[leftmargin=*]\n\t\n\t\\item We formally define the \\textit{multi-skill spatial\n\t\tcrowdsourcing} (MS-SC) problem in Section \\ref{sec:problem_def}, under the constraints of multi-skill covering, time, distance, and budget for spatial workers\/tasks in the spatial crowdsourcing system.\n\t\n\t\\item We prove that the MS-SC problem is NP-hard, and thus\n\tintractable in Section \\ref{sec:reduction}.\n\t\n\t\\item We propose efficient approximation approaches, namely greedy,\n\t$g$-divide-and-conquer, and cost-model-based adaptive\n\talgorithms to tackle the MS-SC problem in Sections \\ref{sec:greedy}, \\ref{sec:g_D&C}, and \\ref{sec:adpative}, respectively.\n\t\n\t\\item We conduct extensive experiments on real and synthetic data sets,\n\tand show the efficiency and effectiveness of our MS-SC approaches in Section \\ref{sec:exper}.\n\\end{itemize}\n\n\nSection \\ref{sec:framework} introduces a general framework for our MS-SC problem in spatial crowdsourcing systems. Section \\ref{sec:related} reviews previous works on spatial\ncrowdsourcing. Finally, Section \\ref{sec:conclusion} concludes this\npaper.\n\n\n\\section{Problem Definition}\n\\label{sec:problem_def}\n\n\nIn this section, we present the formal definition of the multi-skill\nspatial crowdsourcing, in which we assign multi-skilled\nworkers with time-constrained complex spatial tasks.\n\n\n\\subsection{Multi-Skilled Workers}\n\nWe first define the multi-skilled workers in spatial crowdsourcing\napplications. Assume that $\\Psi=\\{a_1, a_2, ..., a_k\\}$ is a\nuniverse of $k$ abilities\/skills. Each worker has one or multiple\nskills in $\\Psi$, and can provide services for spatial tasks that\nrequire some skills in $\\Psi$.\n\n\n\n\n\\begin{definition} $($Multi-Skilled Workers$)$ Let\n\t$W_p$ $=\\{w_1,$ $w_2,$ $...,$ $w_n\\}$ be a set of $n$ multi-skilled\n\tworkers at timestamp $p$. Each worker $w_i$ ($1\\leq i\\leq n$) has a\n\tset, $X_i$ $(\\subseteq \\Psi)$, of skills, is located at position\n\t$l_i(p)$ at timestamp $p$, can move with velocity $v_i$, and has a maximum moving distance $d_i$.\n\t\\qquad $\\blacksquare$ \\label{definition:worker}\n\\end{definition}\n\nIn Definition \\ref{definition:worker}, the multi-skilled workers\n$w_i$ can move dynamically with speed $v_i$ in any direction, and at\neach timestamp $p$, they are located at spatial places $l_i(p)$, and\nprefer to move at most $d_i$ distance from $l_i(p)$. They can freely\njoin or leave the spatial crowdsourcing system. Moreover, each\nworker $w_i$ is associated with a set, $X_i$, of skills, such as\ntaking photos, cooking, and decorating rooms.\n\n\\subsection{Time-Constrained Complex Spatial Tasks}\n\nNext, we define complex spatial tasks in the spatial crowdsourcing\nsystem, which are constrained by deadlines of arriving at task\nlocations and budgets.\n\n\n\\begin{definition}\n\t$($Time-Constrained Complex Spatial Tasks$)$ Let $T_p=\\{t_1, t_2,\n\t..., t_m\\}$ be a set of time-constrained complex spatial tasks at\n\ttimestamp $p$. Each task $t_j$ ($1\\leq j\\leq m$) is located\n\tat a specific location $l_j$, and workers are expected to reach the\n\tlocation of task $t_j$ before the arrival deadline $e_j$. Moreover, to complete the\n\ttask $t_j$, a set, $Y_j$ $(\\subseteq \\Psi)$, of skills is\n\trequired for those assigned workers. Furthermore, each task $t_j$ is\n\tassociated with a budget, $B_j$, of salaries for workers. \\qquad\n\t$\\blacksquare$ \\label{definition:task}\n\\end{definition}\n\nAs given in Definition \\ref{definition:task}, usually, a task\nrequester creates a time-constrained spatial task $t_j$, which\nrequires workers physically moving to a specific location $l_j$, and\narriving at $l_j$ before the arrival deadline $e_j$. Meanwhile, the task\nrequester also specifies a budget, $B_j$, of salaries, that is, the\nmaximum allowance that he\/she is willing to pay for workers. This\nbudget, $B_j$, can be either the reward cash or bonus points in the\nspatial crowdsourcing system.\n\n\nMoreover, the spatial task $t_j$ is often complex, in the sense that\nit might require several distinct skills (in $Y_j$) to be conducted.\nFor example, a spatial task of repairing a house may require\nseveral skills, such as repairing floors, painting walls and cleaning.\n\n\n\n\\subsection{The Multi-Skill Spatial Crowdsourcing Problem}\n\nIn this subsection, we will formally define the\nmulti-skill spatial crowdsourcing (MS-SC) problem, which assigns spatial\ntasks to workers such that workers can cover the skills required by\ntasks and the assignment strategy can achieve high scores.\n\n\\vspace{0.5ex}\\noindent {\\bf Task Assignment Instance Set.} Before\nwe present the MS-SC problem, we first introduce the concept of task\nassignment instance set.\n\n\n\\begin{definition}\n\t$($Task Assignment Instance Set$)$ At timestamp $p$, given a worker\n\tset $W_p$ and a task set $T_p$, a \\textit{task assignment instance\n\t\tset}, denoted by $I_p$, is a set of worker-and-task assignment pairs\n\tin the form $\\langle w_i, t_j\\rangle$, where each worker $w_i \\in\n\tW_p$ is assigned to at most one spatial task $t_j\\in T_p$.\n\t\n\tMoreover, we denote $CT_p$ as the set of completed tasks $t_j$ that can be reached\n\tbefore the arrival deadlines $e_j$, and accomplished by those assigned workers in $I_p$.\\qquad\n\t$\\blacksquare$ \\label{definition:instance}\n\\end{definition}\n\nIntuitively, the task assignment instance set $I_p$ is one possible\n(valid) worker-and-task assignment between worker set $W_p$ and task\nset $T_p$. Each pair $\\langle w_i, t_j\\rangle$ is in $I_p$, if and\nonly if this assignment satisfies the constraints of task $t_j$,\nwith respect to distance (i.e., $d_i$), time (i.e., $e_j$), budget\n(i.e., $B_j$), and skills (i.e., $Y_j$).\n\n\nIn particular, for each pair $\\langle w_i, t_j\\rangle$, worker $w_i$\nmust arrive at location $l_j$ of the assigned task $t_j$ before its\narrival deadline $e_j$, and can support the skills required by task $t_j$,\nthat is, $X_i \\bigcap Y_j \\neq \\emptyset$. The distance between\n$l_i(p)$ and $l_j$ should be less than $d_i$. Moreover, for all\npairs in $I_p$ that contain task $t_j$, the required skills of task\n$t_j$ should be fully covered by skills of its assigned workers,\nthat is, $Y_j \\subseteq \\cup_{\\forall \\langle w_i, t_j\\rangle \\in\n\tI_p}X_i$.\n\n\nTo assign a worker $w_i$ to a task $t_j$, we need to pay him\/her\nsalary, $c_{ij}$, which is related to the traveling cost from the\nlocation, $l_i(p)$, of worker $w_i$ to that, $l_j$, of task $t_j$.\nThe traveling cost, $c_{ij}$, for vehicles can be calculated by the\nunit gas price per gallon times the number of gallons needed for the\ntraveling. For the public transportation, the cost $c_{ij}$ can be\ncomputed by the fees per mile times the traveling distance. For\nwalking, we can also provide the compensation fee for the worker\nwith the cost $c_{ij}$ proportional to his\/her traveling distance.\n\nWithout loss of generality, we assume that the cost, $c_{ij}$, is\nproportional to the traveling distance, $dist(l_i(p), l_j)$, between\n$l_i(p)$ and $l_j$, where $dist(x, y)$ is a distance function\nbetween locations $x$ and $y$. Formally, we have: $c_{ij} = C_i\\cdot\ndist(l_i(p), l_j)$, where $C_i$ is a constant (e.g.,\ngas\/transportation fee per mile).\n\n\nNote that, for simplicity, in this paper, we use Euclidean distance\nas our distance function (i.e., $dist(x, y)$). We can easily extend\nour proposed approaches in this paper by considering other distance\nfunction (e.g., road-network distance), under the framework of the\nspatial crowdsourcing system, and would like to leave the topics of\nusing other distance metics as our future work.\n\n\n\\vspace{0.5ex}\\noindent {\\bf The MS-SC Problem.} In the sequel, we\ngive the definition of our \\textit{multi-skill spatial\n\tcrowdsourcing} (MS-SC) problem.\n\n\\begin{definition}\n\t(Multi-Skill Spatial Crowdsourcing Problem) Given a time interval $P$, the problem of\n\t\\textit{multi-skill spatial crowdsourcing} (MS-SC) is to assign the\n\tavailable workers $w_i\\in W_p$ to spatial tasks $t_j\\in T_p$, and to obtain a\n\ttask assignment instance set, $I_p$, at each timestamp $p \\in P$, such that:\n\t\n\t\\begin{enumerate}\n\t\t\\item any worker $w_i\\in W_p$ is assigned to only one spatial\n\t\ttask $t_j\\in T_p$ such that his\/her arrival time at location $l_j$\n\t\tbefore the arrival deadline $e_j$, the moving distance is\n\t\tless than the worker's maximum moving distance $d_i$, and \n\t\tall workers assigned to $t_j$ have skill sets fully covering $Y_j$;\n\t\t\\item the total traveling cost of all the assigned workers to task $t_j$ does not exceed the budget of\n\t\tthe task, that is, $\\sum_{\\forall \\langle w_i, t_j\\rangle \\in\n\t\t\tI_p}c_{ij}$ $\\leq B_j$; and\n\t\t\\item the total score, $\\sum_{p \\in P}S_p$, of the task assignment\n\t\tinstance sets $I_p$ within the time interval $P$ is maximized,\n\t\\end{enumerate}\n\t\n\twhere it holds that:\n\t\n\t{\\small\\vspace{-4ex}\n\t\t\\begin{eqnarray}\n\t\t\tS_p &=& \\sum_{t_j \\in CT_p}B'_j, and \\label{eq:assignment_score}\\\\\n\t\t\tB'_j &=& B_j - \\sum_{\\langle w_i, t_j\\rangle \\in I_p}c_{ij}.\\label{eq:flexible_budget}\n\t\t\\end{eqnarray}\n\t\t\\vspace{-2ex}\n\t\t\\label{definition:MS_SC}}\n\\end{definition}\n\n\nIn Definition \\ref{definition:MS_SC}, our MS-SC problem aims to\nassign workers $w_i$ to tasks $t_j$ such that: (1) workers $w_i$ are\nable to reach locations, $l_j$, of tasks $t_j$ on time and\ncover the required skill set $Y_j$, and the moving distance is less than $d_i$; (2) the total traveling cost\nof all the assigned workers should not exceed budget $B_j$; and\n(3) the total score, $\\sum_{p \\in P}S_p$, of the task-and-worker assignment within time interval $P$ should be maximized.\n\nAfter the server-side assignment at a timestamp $p$, those assigned\nworkers would change their status to unavailable, and move to the\nlocations of spatial tasks. Next, these workers will become\navailable again, only if they finish\/reject the assigned tasks.\n\n\\vspace{0.5ex}\\noindent {\\bf Discussions on the Score $S_p$.}\nEq.~(\\ref{eq:assignment_score}) calculates the score, $S_p$, of a\ntask-and-worker assignment by summing up \\textit{flexible budgets},\n$B_j'$ (given by Eq.~(\\ref{eq:flexible_budget})), of all the\ncompleted tasks $t_j\\in CT_p$, where the \\textit{flexible budget} of\ntask $t_j$ is the remaining budget of task $t_j$ after paying\nworkers' traveling costs. Maximizing scores means maximizing the \nnumber of accomplished tasks while minimizing the traveling cost of workers.\n\nIntuitively, each task $t_j$ has a maximum budget $B_j$, which\nconsists of two parts, the traveling cost of the assigned workers\nand the flexible budget. The former cost is related to the\ntotal traveling distance of workers, whereas the latter one can be\nfreely and flexibly used for rewarding workers for their\ncontributions to the task. Here, the distribution of the flexible\nbudget among workers can follow existing incentive mechanisms in\ncrowdsourcing \\cite{rula2014no, yang2012crowdsourcing}, which\nstimulate workers who did the task better (i.e., with more\nrewards).\n\nNote that, in Eq.~(\\ref{eq:assignment_score}), the score, $S_p$, of\nthe task assignment instance set $I_p$ only takes into account those\ntasks that can be completed by the assigned workers (i.e., tasks in\nset $CT_p$). Here, a task can be completed, if the assigned workers\ncan reach the task location before the deadline and finish the task\nwith the required skills.\n\n\nSince the spatial crowdsourcing system is quite dynamic, new\ntasks\/workers may arrive at next timestamps. Thus, if we cannot find\nenough\/proper workers to do the task at the current timestamp $p$,\nthe task is still expected to be successfully assigned with workers\nand completed in future timestamps. Meanwhile, the task requester\ncan be also informed by the spatial crowdsourcing system to increase\nthe budget (i.e., with higher budget $B_j$, we can find more skilled\ncandidate workers that satisfy the budget constraint). Therefore, in\nour definition of score $S_p$, we would only consider those tasks in\n$CT_p$ that can be completed by the assigned workers at timestamp\n$p$, and maximize this score $S_p$.\n\n\n\\subsection{Hardness of Multi-Skill Spatial Crowdsourcing Problem}\n\\label{sec:reduction}\n\nWith $m$ time-constrained complex spatial tasks and $n$\nmulti-skilled workers, in the worst case, there are an exponential\nnumber of possible task-and-worker assignment strategies, which\nleads to high time complexity, $O((m + 1)^n)$. In this subsection,\nwe prove that our MS-SC problem is NP-hard, by reducing a well-known\nNP-hard problem, \\textit{set cover problem} (SCP)\n\\cite{vazirani2013approximation}, to the MS-SC problem.\n\n\n\n\\begin{lemma} (Hardness of the MS-SC Problem)\n\tThe problem of the Multi-Skill Spatial Crowdsourcing (MS-SC) is\n\tNP-hard. \\label{lemma:np}\\vspace{-1ex}\n\\end{lemma}\n\\begin{proof}\n\tPlease refer to Appendix A.\n\\end{proof}\n\n\nSince the MS-SC problem involves multiple spatial tasks whose skill\nsets should be covered, we thus cannot directly use existing\napproximation algorithms for SCP (or its variants) to solve the\nMS-SC problem. What is more, we also need to find an assignment\nstrategy such that workers and tasks match with each other (in terms\nof traveling time\/cost, and budge constraints), which is more\nchallenging.\n\n\nThus, due to the NP-hardness of our MS-SC problem,  in subsequent\nsections, we will present a general framework for MS-SC processing and design 3 heuristic algorithms, namely greedy,\n$k$-divide-and-conquer, and cost-model-based adaptive approaches, to\nefficiently retrieve MS-SC answers.\n\n\n\n\n\n\n\\begin{table}\n\t\\begin{center}\\vspace{-5ex}\n\t\t\\caption{\\small Symbols and Descriptions.} \\label{table0}\n\t\t{\\small\\scriptsize\n\t\t\t\\begin{tabular}{l|l}\n\t\t\t\t{\\bf Symbol} & {\\bf \\qquad \\qquad \\qquad\\qquad\\qquad Description} \\\\ \\hline \\hline\n\t\t\t\t$T_p$   & a set of $m$ time-constrained spatial tasks $t_j$ at timestamp $p$\\\\\n\t\t\t\t$W_p$   & a set of $n$ dynamically moving workers $w_i$ at timestamp $p$\\\\\n\t\t\t\t$e_j$   & the deadline of arriving at the location of task $t_j$\\\\\n\t\t\t\t$l_i(p)$  & the position of worker $w_i$ at timestamp $p$\\\\\n\t\t\t\t$l_j$   & the position of task $t_j$\\\\\n\t\t\t\t$X_i$   & a set of skills that worker $w_i$ has\\\\\n\t\t\t\t$Y_j$   & a set of the required skills for task $t_j$ \\\\\n\t\t\t\t$d_i$   & the maximum moving distance of worker $w_i$ \\\\\n\t\t\t\t$B_j$   & the maximum budget of task $t_j$ \\\\\n\t\t\t\t$I_p$   & the task assignment instance set at timestamp $p$ \\\\\n\t\t\t\t$CT_p$   & a set of tasks that are assigned with workers at timestamp $p$ and\\\\\n\t\t\t\t& \\qquad can be completed by these assigned workers\\\\\n\t\t\t\t$C_i$ & the unit price of the traveling cost of worker $w_i$\\\\\n\t\t\t\t$c_{ij}$ & the traveling cost from the location of worker $w_i$ to that of task $t_j$\\\\\n\t\t\t\t$S_p$   & the score of the task assignment instance set $I_p$\\\\\n\t\t\t\t$\\Delta S_p$   & the score increase when changing the pair assignment\\\\ \\hline\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\\vspace{-6ex}\n\\end{table}\n\nTable \\ref{table0} summarizes the commonly used symbols.\n\n\\section{Framework of Solving MS-SC Problems}\n\\label{sec:framework}\n\n\nIn this section, we present a general framework,  \nnamely {\\sf MS-SC\\_Framework}, in Figure \\ref{alg:framework} for solving the\nMS-SC problem, which greedily assigns workers with spatial tasks for\nmultiple rounds. For each round, at timestamp $p$, we first retrieve\na set, $T_p$, of all the available spatial tasks, and a set, $W_p$,\nof available workers (lines 2-3). Here, the available task set $T_p$\ncontains existing spatial tasks that have not been assigned with\nworkers in the last round, and the ones that newly arrive at the\nsystem after the last round. Moreover, set $W_p$ includes those\nworkers who have accomplished (or rejected) the previously assigned\ntasks, and thus are available to receive new tasks in the current\nround.\n\nIn our spatial crowdsourcing system, we organize both sets $T_p$ and\n$W_p$ in a cost-model-based grid index. For the sake of space\nlimitations, details about the index construction can be found in\nAppendix E. Due to\ndynamic changes of sets $T_p$ and $W_p$, we also update the grid\nindex accordingly (line 4). Next, we utilize the grid index to\nefficiently retrieve a set, $S$, of valid worker-and-task candidate\npairs (line 5). That is, we obtain those pairs of workers and tasks,\n$\\langle w_i, t_j\\rangle$, such that workers $w_i$ can reach the\nlocations of tasks $t_j$ and satisfy the constraints of skill\nmatching, time, and budgets for tasks $t_j$. With valid pairs in set\n$S$, we can apply our proposed algorithms, that is, \\textit{greedy},\n\\textit{$g$-divide-and-conquer}, or \\textit{adaptive\n\tcost-model-based} approach, over set $S$, and obtain a good\nworker-and-task assignment strategy in an assignment instance set\n$I_p$, which is a subset of $S$ (line 6).\n\n\nFinally, for each pair $\\langle w_i, t_j \\rangle$ in the selected\nworker-and-task assignment set $I_p$, we will notify worker $w_i$ to\ndo task $t_j$ (lines 7-8).\n\n\n\\begin{figure}[ht]\n\t\\begin{center}\\vspace{-2ex}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_Framework}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} a time interval $P$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} a worker-and-task assignment strategy within the time interval $P$\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> for each timestamp $p$ in $P$\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> \\> retrieve all the available spatial tasks to $T_p$\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> \\> retrieve all the available workers to $W_p$\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> \\> update the grid index for current $T_p$ and $W_p$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> obtain a set, $S$, of valid worker-and-task pairs from the index\\\\\n\t\t\t\t\t\t\\> (6) \\> \\> \\> use our \\textit{greedy}, \\textit{$g$-divide-and-conquer} or \\textit{adaptive cost-model-based} approach\\\\\n\t\t\t\t\t\t\\> \\> \\> \\>  \\> to obtain a good assignment instance set, $I_p \\subseteq S$\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\> for each pair $\\langle w_i, t_j \\rangle$ in $I_p$\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> \\> inform worker $w_i$ to conduct task $t_j$\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\\vspace{-3ex}\n\t\\caption{\\small Framework for Solving the MS-SC Problem.}\n\t\\label{alg:framework}\\vspace{-5ex}\n\\end{figure}\n\n\\nop{\n\t\n\tThe available tasks include the ones arriving between the last and\n\tthe current rounds, and those that are unfinished after last round.\n\tFor any task $t_j$ of the latter type, we need to maintain the set\n\tof workers, $W_j$, who have accepted the assigned task $t_j$, and\n\tupdate the information of the task $t_j$. When we update the\n\tinformation of unfinished task $t_j$ after last round, we reduce its\n\tbudget $B_j$ to $B'_j=B_j - \\sum_{w_i \\in W_j} c_{ij}$ and change\n\tits required skill set $Y_j$ to $Y'_j = Y_j \/ \\cup_{w_i \\in\n\t\tW_j}X_i$.\n\t\n\t\n\t\n\t\n\tMoreover, we allow workers to accomplish multiple tasks within a\n\tperiod of time. However,\n\teach worker can only be available to accept more tasks after the\n\tcurrent assigned task has been rejected or finished.\n\tWhen we assign more than one task to\n\ta worker in a round, if he\/she uses too\n\tmuch time to finish one task,\n\the\/she may miss the deadlines of the other tasks,\n\twhich could harm the overall performance of the platform.\n\t\n\tIn order to facilitate the processing of the MS-SC problem,\n\twe present an efficient cost-model-based indexing\n\tmechanism, which can maintain workers and tasks and help the\n\tretrieval of MS-SC answers. Different from the grid-index in\n\tour previous work \\cite{cheng2014reliable}, the grid-index in\n\tthis work utilize bitmap synopses to time- and space- efficiently\n\torganize\/manipulate the sets of skills for workers and tasks.\n\tFor the details of our grid-index, please refer to\n\tAppendix E of our technical report \\cite{arxivReport}.\n\t\n}\n\n\n\n\\section{The Greedy Approach}\n\\label{sec:greedy}\n\nIn this section, we will propose a greedy algorithm, which greedily\nselects one worker-and-task assignment, $\\langle w_i, t_j \\rangle$,\nat a time that can maximize the increase of the assignment score\n(i.e., $\\sum_{\\forall p\\in P} S_p$ as given in Definition\n\\ref{definition:MS_SC}). This greedy algorithm can be applied in\nline 6 of the framework, {\\sf MS-SC\\_Framework}, in Fig.\n\\ref{alg:framework}.\n\n\n\\subsection{The Score Increase}\n\\label{subsec:score_increase}\n\nBefore we present the greedy algorithm, we first define the\nincrease, $\\Delta S_p$, of score $S_p$ (given in\nEq.~(\\ref{eq:assignment_score})), in the case where we assign a\nnewly available worker $w_i$ to task $t_j$. Specifically, from\nEqs.~(\\ref{eq:assignment_score}) and (\\ref{eq:flexible_budget}), we\ndefine the score increase after assigning worker $w_i$ to task $t_j$\nas follows:\\vspace{-2ex}\n\n{\\small\n\t\\begin{eqnarray}\n\t\t\\Delta S_p = S_p-S_{p-1} = \\Delta B_j' = \\frac{|X_i \\cap (Y_j -\n\t\t\t\\widetilde{Y_j})|}{|Y_j|} \\cdot B_j -\n\t\tc_{ij},\\label{eq:score_increase}\n\t\\end{eqnarray}\\vspace{-2ex}}\n\n\\noindent where $\\widetilde{Y_j}$ is the set of skills that have\nbeen covered by those assigned workers (excluding the new worker\n$w_i$) for task $t_j$.\n\nIn Eq.~(\\ref{eq:score_increase}), $\\frac{|X_i \\cap (Y_j -\n\t\\widetilde{Y_j})|}{|Y_j|}$ is the ratio of skills for task $t_j$\nthat have not been covered by (existing) assigned workers, but can\nbe covered by the new worker $w_i$. Intuitively, the first term in\nEq.~(\\ref{eq:score_increase}) is the pre-allocated maximum budget\nbased on the number of covered skills by the new worker $w_i$,\nwhereas the second term, $c_{ij}$, is the traveling cost from\nlocation of $w_i$ to that of $t_j$. Thus, the score increase,\n$\\Delta S_p$, in Eq.~(\\ref{eq:score_increase}) is to measure the\nchange of score (i.e., flexible budget) $S_p$, due to the assignment\nof worker $w_i$ to task $t_j$.\n\n\n\\subsection{Pruning Strategies}\n\\label{sub:pruning}\n\nThe score increase can be used as a measure to evaluate and decide\nwhich worker-and-task assignment pair should be added to the task\nassignment instance set $I_p$. That is, each time our greedy\nalgorithm aims to choose one worker-and-task assignment pair in $S$\nwith the highest score increase, which will be added to $I_p$ (i.e.,\nline 6 of {\\sf MS-SC\\_Framework} in Fig. \\ref{alg:framework}).\nHowever, it is not efficient to enumerate all valid worker-and-task\nassignment pairs in $S$, and compute score increases. That is, in\nthe worst case, the time complexity is as high as $O(m\\cdot n)$,\nwhere $m$ is the number of tasks and $n$ is the number of workers.\nTherefore, in this subsection, we present three effective pruning\nmethods (two for pruning workers and one for pruning tasks) to quickly filter out false alarms of worker-and-task pairs\nin set $S$.\n\n\n\\noindent\\textbf{The Worker-Pruning Strategy.} When assigning available workers\nto spatial tasks, we can rule out those valid worker-and-task pairs\nin $S$, which contain either \\textit{dominated} or \\textit{high-wage\n\tworkers}, as given in Lemmas \\ref{lemma:dominate_worker} and\n\\ref{lemma:expensive_worker}, respectively, below.\n\nWe say that a worker $w_a$ is \\textit{dominated by} a worker $w_b$\nw.r.t. task $t_j$ (denoted as $w_a \\succ_{t_j} w_b$), if it holds\nthat $X_a \\subseteq X_b$ and $c_{aj}\\geq c_{bj}$, where $X_a$ and\n$X_b$ are skill sets of workers $w_a$ and $w_b$, and $c_{aj}$ and\n$c_{bj}$ are the traveling costs from locations of workers $w_a$ and\n$w_b$ to task $t_j$, respectively.\n\n\\begin{lemma} (Pruning Dominated Workers) Given two worker-and-task\n\tpairs $\\langle w_a, t_j\\rangle$ and $\\langle w_b, t_j\\rangle$ in\n\tvalid pair set $S$, if it holds that $w_a \\succ_{t_j} w_b$, then we\n\tcan safely prune the worker-and-task pair $\\langle w_a,\n\tt_j\\rangle$. \\label{lemma:dominate_worker}\n\\end{lemma}\n\n\\begin{proof}\n\tPlease refer to Appendix B.\n\\end{proof}\n\n\nLemma \\ref{lemma:dominate_worker} indicates that if there exists a\nbetter worker $w_b$ than worker $w_a$ to do task $t_j$ (in terms of\nboth the skill set and the traveling cost), then we can safely\nfilter out the assignment of worker $w_a$ to task $t_j$.\n\n\\begin{lemma} (Pruning High-Wage Workers) Let $\\widetilde{c_{\\cdot j}}$\n\tbe the total traveling cost for those workers that have already been\n\tassigned to task $t_j$. If the traveling cost $c_{ij}$ of assigning a worker\n\t$w_i$ to task $t_j$ is greater than the remaining budget\n\t$(B_j-\\widetilde{c_{\\cdot j}})$ of task $t_j$, then we will not\n\tassign worker $w_i$ to task $t_j$. \\label{lemma:expensive_worker}\n\\end{lemma}\n\\begin{proof}\n\tPlease refer to Appendix C.\n\\end{proof}\n\n\n\nIntuitively, Lemma \\ref{lemma:expensive_worker} shows that, if the\nwage of a worker $w_i$ (including the traveling cost $c_{ij}$)\nexceeds the maximum budget $B_j$ of task $t_j$ (i.e., $c_{ij} >\nB_j-\\widetilde{c_{\\cdot j}}$), then we can safely prune the\nworker-and-task assignment pair $\\langle w_i, t_j\\rangle$.\n\n\n\\noindent\\textbf{The Task-Pruning Strategy.} Let $W(t_j)$ be a set of valid\nworkers that can be assigned to task $t_j$, and $\\widetilde{W(t_j)}$\nbe a set of valid workers that have already been assigned to task\n$t_j$. We give the lemma of pruning those tasks with insufficient\nbudgets below.\n\n\\begin{lemma} (Pruning Tasks with Insufficient Budgets) If an unassigned worker $w_i\\in\n\t(W(t_j)-\\widetilde{W(t_j)})$ has the highest value of $\\frac{\\Delta\n\t\tS_p}{|X_i \\cap (Y_j - \\widetilde{Y_j})|}$, and the traveling cost,\n\t$c_{ij}$, of worker $w_i$ exceeds the remaining budget\n\t$(B_j-\\widetilde{c_{\\cdot j}})$ of task $t_j$, then we can safely\n\tprune task $t_j$. \\label{lemma:prune_insufficient_task}\n\t\n\\end{lemma}\n\\begin{proof}\n\tPlease refer to Appendix D.\n\\end{proof}\n\nIntuitively, Lemma \\ref{lemma:prune_insufficient_task} provides the\nconditions of pruning tasks. That is, if any unassigned worker\nsubset of $(W(t_j)-\\widetilde{W(t_j)})$ either cannot fully cover\nthe required skill set $Y_j$, or exceeds the remaining budget of\ntask $t_j$, then we can directly prune all assignment pairs that\ncontain task $t_j$.\n\nTo summarize, by utilizing Lemmas \\ref{lemma:dominate_worker},\n\\ref{lemma:expensive_worker} and\n\\ref{lemma:prune_insufficient_task}, we do not have to check all\nworker-and-task assignments iteratively in our greedy algorithm.\nInstead, we can now apply our proposed three pruning methods, and\neffectively filter out those false alarms of assignment pairs, which\ncan significantly reduce the number of times to compute the score\nincreases.\n\n\n\\subsection{The Greedy Algorithm}\n\\label{sub:greedy_algorithm}\n\nAccording to the definition of the score increase $\\Delta S_p$ (as\nmentioned in Section \\ref{subsec:score_increase}), we propose a\ngreedy algorithm, which iteratively assigns a worker to a spatial\ntask that can always achieve the highest score increase.\n\n\n\n\n\\begin{figure}[ht]\\vspace{-3ex}\n\t\\begin{center}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_Greedy}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} $n$ workers in $W_p$ and $m$ time-constrained spatial tasks in $T_p$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} a worker-and-task assignment instance set, $I_p$\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> $I_p = \\emptyset$;\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> compute all valid worker-and-task pairs $\\langle w_i, t_j \\rangle$ from $W_p$ and $T_p$\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> while $W_p \\neq \\emptyset$ and $T_p \\neq \\emptyset$\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> \\> $S_{cand} = \\emptyset;$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> for each task $t_j\\in T_p$\\\\\n\t\t\t\t\t\t\\> (6) \\> \\> \\> \\> for each worker $w_i$ in the valid pair $\\langle w_i, t_j \\rangle$\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\> \\> \\> if we cannot prune dominated worker $w_i$ by Lemma \\ref{lemma:dominate_worker}\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> \\> \\> \\> if we cannot prune high-wage worker $w_i$ by Lemma \\ref{lemma:expensive_worker}\\\\\n\t\t\t\t\t\t\\> (9) \\> \\> \\> \\> \\> \\> \\> add $\\langle w_i, t_j \\rangle$ to $S_{cand}$\\\\\n\t\t\t\t\t\t\\> (10)\\> \\> \\> \\> if we cannot prune task $t_j$ w.r.t. workers in $S_{cand}$ by Lemma \\ref{lemma:prune_insufficient_task}\\\\\n\t\t\t\t\t\t\\> (11)\\> \\> \\> \\> \\> for each pair $\\langle w_i, t_j \\rangle$ w.r.t. task $t_j$ in $S_{cand}$\\\\\n\t\t\t\t\t\t\\> (12)\\> \\> \\> \\> \\> \\> compute the score increase, $\\Delta S_p(w_i, t_j)$\\\\\n\t\t\t\t\t\t\\> (13)\\> \\> \\> \\> else\\\\\n\t\t\t\t\t\t\\> (14)\\> \\> \\> \\> \\> $T_p=T_p - \\{t_j\\}$\\\\\n\t\t\t\t\t\t\\> (15)\\> \\> \\> obtain a pair, $\\langle w_r, t_j \\rangle \\in S_{cand}$, with the highest score increase, \\\\\n\t\t\t\t\t\t\\> \\> \\> \\> \\> \\> $\\Delta S_p(w_r, t_j)$, and add this pair to $I_p$\\\\\n\t\t\t\t\t\t\\> (16)\\> \\> \\> $W_p=W_p - \\{w_r\\}$\\\\\n\t\t\t\t\t\t\\> (17)\\> \\> return $I_p$\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\\vspace{-5ex}\n\t\\caption{\\small The MS-SC Greedy Algorithm.}\\vspace{-4ex}\n\t\\label{alg:greedy}\n\\end{figure}\n\nFigure \\ref{alg:greedy} shows the pseudo code of our MS-SC greedy\nalgorithm, namely {\\sf MS-SC\\_Greedy}, which obtains one\nworker-and-task pair with the highest score increase each\ntime, and returns a task assignment instance set $I_p$ with\nhigh score.\n\nInitially, we set $I_p$ to be empty, since no workers are assigned\nto any tasks (line 1). Next, we find out all valid worker-and-task\npairs $\\langle w_i, t_j \\rangle$ in the crowdsourcing system at\ntimestamp $p$ (line 2). Here, the validity of pair $\\langle w_i, t_j\n\\rangle$ satisfies 4 conditions: (1) the distance between the\ncurrent location, $l_i(p)$, of worker $w_i$ and the location, $l_j$\nof task $t_j$ is less than the maximum moving distance, $d_i$ of\nworker $w_i$, that is, $dist(l_i(p), l_j) \\leq d_i$; (2) worker\n$w_i$ can arrive at the location, $l_j$, of task $t_j$ before the\narrival deadline $e_j$; (3) worker $w_i$ have skills that task $t_j$\nrequires; and (4) the traveling cost, $c_{ij}$, of worker $w_i$\nshould not exceed the budget $B_j$ of task $t_j$.\n\n\nThen, for each round, we would select one valid worker-and-task\nassignment pair with the highest score increase, and add it to set\n$I_p$ (lines 3-16). Specifically, in each round, we check every task\n$t_j$ that is involved in valid pairs $\\langle w_i, t_j \\rangle$,\nand then prune those dominated and high-wage workers $w_i$, via\nLemmas \\ref{lemma:dominate_worker} and \\ref{lemma:expensive_worker},\nrespectively (lines 7-8). If worker $w_i$ cannot be pruned by both\npruning methods, then we add it to a candidate set $S_{cand}$ for\nfurther checking (line 9). After obtaining all workers that match\nwith task $t_j$, we apply Lemma \\ref{lemma:prune_insufficient_task}\nto filter out task $t_j$ (if workers cannot be successfully assigned\nto $t_j$). If task $t_j$ cannot be pruned, we will calculate the\nscore increase, $\\Delta S_p(w_i, t_j)$, for each pair $\\langle w_i,\nt_j \\rangle$ in $S_{cand}$; otherwise, we remove task $t_j$ from\ntask set $T_p$ (lines 10-14).\n\nAfter we scan all tasks in $T_p$, we can retrieve one\nworker-and-task assignment pair, $\\langle w_r, t_j \\rangle$, from\nthe candidate set $S_{cand}$, which has the highest score increase,\nand insert this pair to $I_p$ (line 15). Since worker $w_r$ has been\nassigned, we remove it from the worker set $W_p$ (line 16). The\nprocess above repeats, until all workers have been assigned (i.e.,\n$W_p = \\emptyset$) or there are no tasks left (i.e.,\n$T_p=\\emptyset$) (line 3).\n\n\nFigure \\ref{subfig:validpairs} illustrates an example of valid\npairs, where $n$ available workers and $m$ spatial tasks are denoted\nby triangular and circular nodes, respectively, and valid\nworker-and-task pairs are represented by dashed lines. Figure\n\\ref{subfig:assignment} depicts the result of one assignment with\nhigh score, where the bold lines indicate assignment pairs in $I_p$.\n\n\n\n\\begin{figure}[ht]\\vspace{-2ex}\\centering\n\t\\subfigure[][{\\scriptsize Valid Pairs}]{\n\t\t\\scalebox{0.47}[0.47]{\\includegraphics{valid_pairs.eps}}\n\t\t\\label{subfig:validpairs}}\\hspace{8ex}\n\t\\subfigure[][{\\scriptsize Assignment Instance}]{\n\t\t\\scalebox{0.45}[0.45]{\\includegraphics{assignment_example.eps}}\n\t\t\\label{subfig:assignment}}\\vspace{-2ex}\n\t\\caption{\\small Illustration of the Worker-and-Task\n\t\tAssignment.}\\vspace{-4ex}\n\t\\label{fig:assignment}\n\\end{figure}\n\n\n\n\\vspace{0.5ex}\\noindent {\\bf The Time Complexity.} We next present\nthe time complexity of the greedy algorithm, {\\sf MS-SC\\_Greedy} (in\nFigure \\ref{alg:greedy}). Specifically, the time cost of computing\nvalid worker-and-task assignment pairs (line 2) is given by $O(m\n\\cdot n)$ in the worst case, where any of $n$ workers can be\nassigned to any of $m$ tasks (i.e., $m\\cdot n$ valid worker-and-task\npairs). Then, for each round (lines 3-16), we apply pruning methods\nto $m\\cdot n$ pairs, and select the pair with the highest score\nincrease. In the worst case, pairs cannot be pruned, and thus the\ntime complexity of computing score increases for these pairs is\ngiven by $O(m \\cdot n)$. Moreover, since each of $n$ workers can\nonly be assigned to one spatial task, the number of iterations is at\nmost $n$ times. Therefore, the total time complexity of our greedy\nalgorithm can be given by $O(m \\cdot n^2)$.\n\n\\section{The $g$-Divide-and-Conquer Approach}\n\\label{sec:g_D&C}\n\nAlthough the greedy algorithm incrementally finds one\nworker-and-task assignment (with the highest score increase) at a\ntime, it may incur the problem of only achieving local optimality.\nTherefore, in this section, we propose an efficient\n\\textit{$g$-divide-and-conquer algorithm} ($g$-D\\&C), which first\ndivides the entire MS-SC problem into $g$ subproblems, such that\neach subproblem involves a smaller subgroup of $\\lceil m\/g\\rceil$\nspatial tasks, and then conquers the subproblems recursively (until\nthe final group size becomes 1). Since different numbers, $g$, of\nthe divided subproblems may incur different time costs, in this\npaper, we will propose a novel cost-model-based method to estimate\nthe best $g$ value to divide the problem.\n\nSpecifically, for each subproblem\/subgroup (containing $\\lceil\nm\/g\\rceil$ tasks), we will tackle the worker-and-task assignment\nproblem via recursion (note: the base case with the group size equal\nto 1 can be solved by the greedy algorithm\n\\cite{vazirani2013approximation}, which has an approximation ratio\nof $\\ln(N)$, where $N$ is the total number of skills). During the\nrecursive process, we will combine\/merge assignment results from\nsubgroups, and obtain the assignment strategy for the merged groups,\nby resolving the assignment conflict among subgroups. Finally, we\ncan return the task assignment instance set $I_p$, with respect to\nthe entire worker and tasks sets.\n\n\n\nIn the sequel, we first discuss how to decompose the MS-SC problem\ninto subproblems in Section \\ref{subsec:MS_SC_decomposition}. Then,\nwe will illustrate our $g$-divide-and-conquer approach in Section\n\\ref{subsec:D&C}, which utilizes the decomposition and merge (as\nwill be discussed in Section \\ref{subsec:merge}) algorithms.\nFinally, we will provide a cost model in Section\n\\ref{subsec:DC_cost_model} to determine the best number $g$ of\nsubproblems during the $g$-D\\&C process.\n\n\n\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Original MS-SC Problem}]{\n\t\t\\scalebox{1}[1]{\\includegraphics{beforeDecompose.eps}}\n\t\t\\label{subfig:beforedecomposing}}\\hspace{4ex}\n\t\\subfigure[][{\\scriptsize Decomposed Subproblems}]{\n\t\t\\scalebox{0.9}[0.9]{\\includegraphics{decomposed.eps}}\n\t\t\\label{subfig:decomposed}}\\vspace{-2ex}\n\t\\caption{\\small Illustration of Decomposing the MS-SC Problem.}\\vspace{-5ex}\n\t\\label{fig:decomposing}\n\\end{figure}\n\n\\subsection{MS-SC Problem Decompositions}\n\\label{subsec:MS_SC_decomposition}\n\nIn this subsection, we discuss how to decompose a MS-SC problem into\nsubproblems. In order to illustrate the decomposition, we first\nconvert our original MS-SC problem into a representation of a\nbipartite graph.\n\n\\vspace{0.5ex}\\noindent {\\bf Bipartite Graph Representation of the\n\tMS-SC Problem.} Specifically, given a worker set $W_p$ and a spatial\ntask set $T_p$, we denote each worker\/task (i.e., $w_i$ or $t_j$) as\na vertex in the bipartite graph, where worker and task vertices have\ndistinct vertex types. There exists an edge between a worker vertex\n$w_i$ and a task vertex $t_j$, if and only if worker $w_i$ can reach\nspatial task $t_j$ under the constraints of skills (i.e., $X_i \\cap\nY_j \\ne \\emptyset$), time (i.e., arrival time is before deadline\n$e_j$ of arrival), distance (i.e., the traveling distance is below $d_i$), and\nbudget (i.e., the traveling cost is below task budget $B_j$). We say\nthat the worker-and-task assignment pair $\\langle w_i, t_j\\rangle$\nis \\textit{valid}, if there is an edge between vertices $w_i$ and\n$t_j$ in the graph.\n\n\nAs an example in Figure \\ref{subfig:beforedecomposing}, we have a\nworker set $W_p = \\{w_i | 1\\leq i\\leq 5\\}$, and a spatial task set\n$T_p = \\{t_j | 1\\leq j\\leq 3\\}$, which are denoted by two types of\nvertices (i.e., represented by triangle and circle shapes,\nrespectively) in a bipartite graph. Any edge connects two types of\nvertices $w_i$ and $t_j$, if worker $w_i$ can reach the location of\ntask $t_j$ and do tasks with the required skills from $t_j$. For\nexample, there exists an edge between $w_1$ and $t_1$, which\nindicates that worker $w_1$ can move to the location of $t_1$ before\nthe arrival deadline $e_1$, with the traveling distance under $d_1$,\nwith the traveling cost below budget $B_1$, and moreover with some\nskill(s) in the required skill set $Y_1$ of task $t_1$.\n\nNote that, one or multiple worker vertices (e.g., $w_1$, $w_3$, and\n$w_4$) may be connected to the same task vertex (e.g., $t_1$).\nFurthermore, multiple task vertices, say $t_1$ and $t_2$, may also\nshare some conflicting workers (e.g., $w_3$ or $w_4$), where the\nconflicting worker $w_3$ (or $w_4$) can be assigned to either task\n$t_1$ or task $t_2$ mutual exclusively.\n\n\n\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize} \\vspace{-4ex}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_Decomposition}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} $n$ workers in $W_p$, $m$ time-constrained spatial tasks in $T_p$, and the number\\\\\n\t\t\t\t\t\t\\> \\> \\> \\> of groups $g$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} decomposed MS-SC subproblems, $P_s$ ($1\\leq s\\leq g$)\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> for $s$ = 1 to $g$\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> \\> $P_s = \\emptyset$\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> compute all valid worker-and-task pairs $\\langle w_i, t_j\\rangle$ from $W_p$ and $T_p$, \\\\\n\t\t\t\t\t\t\\> \\> \\> \\> \\> and obtain a bipartite graph $G$\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> for $s = 1$ to $g$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> let set $T_p^{(j)}$ contain the next anchor task $t_j$ and its top-$(\\lceil m\/g \\rceil -1)$\\\\\n\t\t\t\t\t\t\\>  \\> \\>  \\> \\> \\> nearest tasks \/\/ the task, $t_j$, whose longitude is the smallest\\\\\n\t\t\t\t\t\t\\> (6) \\> \\> \\> for each task vertex $t_j \\in T_p^{(j)}$ in graph $G$\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\> \\> obtain all worker vertices $w_i$ that connect with task vertex $t_j$\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> \\> add all pairs $\\langle w_i, t_j\\rangle$ to $P_s$\\\\\n\t\t\t\t\t\t\\> (9) \\> \\>  return $P_s$ (for $1\\leq s\\leq g$)\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\vspace{-5ex}\n\t\\caption{\\small The MS-SC Problem Decomposition Algorithm.}\n\t\\vspace{-4ex}\n\t\\label{alg:decomposing}\n\\end{figure}\n\n\n\n\\vspace{0.5ex}\\noindent {\\bf Decomposing the MS-SC Problem.} Next,\nwe will illustrate how to decompose the MS-SC problem, with respect\nto task vertices in the bipartite graph. Figure\n\\ref{fig:decomposing} shows an example of decomposing the MS-SC\nproblem (as shown in Figure \\ref{subfig:beforedecomposing}) into 3\nsubproblems (as depicted in Figure \\ref{subfig:decomposed}), where\neach subproblem contains a subgroup of one single spatial task\n(i.e., group size = 1), associated with its connected worker\nvertices. For example, the first subgroup in Figure\n\\ref{subfig:decomposed}) contains task vertex $t_1$, as well as its\nconnecting worker vertices $w_1$, $w_3$, and $w_4$. Different task\nvertices may have conflicting workers, for example, tasks $t_1$ and\n$t_2$ share the same (conflicting) worker vertices $w_3$ and $w_4$.\n\n\nIn a general case, given $n$ workers and $m$ spatial tasks, we\npartition the bipartite graph into $g$ subgroups,\neach of which contains $\\lceil m\/g \\rceil$ spatial tasks, as well as their\nconnecting workers. Figure \\ref{alg:decomposing} presents the pseudo\ncode of our MS-SC problem decomposition algorithm, namely {\\sf\n\tMS-SC\\_Decomposition}, which returns $g$ MS-SC\nsubproblems (each corresponding to a subgroup with $\\lceil m\/g \\rceil$ tasks),\n$P_s$, after decomposing the original MS-SC problem.\n\n\n\nSpecifically, we first initialize $g$ empty\nsubproblems, $P_s$, where $1\\leq s\\leq g$ (lines\n1-2). Then, we find out all valid worker-and-task pairs $\\langle\nw_i, t_j\\rangle$ in the crowdsourcing system at timestamp $p$, which\ncan can form a bipartite graph $G$, where valid pairs satisfy the\nconstraints of skills, times, distances, and budgets (line 3).\n\nNext, we want to obtain one subproblem $P_s$ at a time (lines 4-8).\nIn particular, for each round, we retrieve an anchor task $t_j$ and\nits top-$(\\lceil m\/g \\rceil -1)$ nearest tasks, which form a task\nset $T_p^{(j)}$ of size $\\lceil m\/g \\rceil$ (line 5). Here, we\nchoose anchor tasks with a sweeping style, that is, we always choose\nthe task whose longitude is smallest (in the case where multiple\ntasks have the same longitude, we choose the one with smallest\nlatitude). Then, for each task $t_j\\in T_p^{(j)}$, we obtain its\ncorresponding vertex in $G$ and all of its connecting worker\nvertices $w_i$, and add pairs $\\langle w_i, t_j\\rangle$ to\nsubproblem $P_s$ (lines 6-8). Finally, we return all the $g$\ndecomposed subproblems $P_s$.\n\n\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize} \\vspace{-4ex}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_$g$D\\&C}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} $n$ workers in $W_p$, and $m$ time-constrained spatial tasks in $T_p$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} a worker-and-task assignment instance set, $I_p$\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> $I_p = \\emptyset$\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> estimate the best number of groups, $g$, for $W_p$ and $T_p$\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> invoke {\\sf MS-SC\\_Decomposition}$(W_p, T_p, g)$, and obtain subproblems $P_s$\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> for $s=1$ to $g$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> if the number of tasks in subproblem $P_s$ (group size) is greater than 1\\\\\n\t\t\t\t\t\t\\> (6) \\> \\> \\> \\> $I_p^{(s)}$ = {\\sf MS-SC\\_$g$D\\&C}($W_p (P_s)$, $T_p(P_s)$)\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\> else\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> \\> invoke classical greedy set cover algorithm to solve subproblem $P_s$,\\\\\n\t\t\t\t\t\t\\> \\> \\> \\> \\> \\> \\>  and obtain assignment results $I_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (9) \\> \\> for $i=1$ to $g$\\\\\n\t\t\t\t\t\t\\> (10) \\> \\>  \\> find the next subproblem, $P_s$\\\\\n\t\t\t\t\t\t\\> (11) \\> \\>  \\> $I_p$ = {\\sf MS-SC\\_Conflict\\_Reconcile} ($I_p$, $I_p^{(s)}$) \\\\\n\t\t\t\t\t\t\\> (12) \\> \\>  return $I_p$\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\\vspace{-5ex}\n\t\\caption{\\small The $g$-Divide-and-Conquer Algorithm.}\n\t\\vspace{-5ex}\n\t\\label{alg:incrementalFrame}\n\\end{figure}\n\n\n\n\\subsection{The $g$-D\\&C Algorithm}\n\\label{subsec:D&C}\n\nIn this subsection, we propose an efficient\n\\textit{$g$-divide-and-conquer} ($g$-D\\&C) algorithm, namely {\\sf\n\tMS-SC\\_$g$D\\&C}, which recursively partitions the original MS-SC\nproblem into subproblems, solves each subproblem (via recursion),\nand merges assignment results of subproblems by resolving the\nconflicts.\n\n\nSpecifically, in Algorithm {\\sf MS-SC\\_$g$D\\&C}, we first estimate\nthe best number of groups, $g$, to partition, with respect to $W_p$ and\n$T_p$, which is based on the cost model proposed later in Section\n\\ref{subsec:DC_cost_model} (line 2). Then, we will call the {\\sf\n\tMS-SC\\_Decomposition} algorithm (as mentioned in Figure\n\\ref{alg:decomposing}) to obtain subproblems $P_s$ (line 3). For\neach subproblem $P_s$, if $P_s$ involves more than 1 task, then we\ncan recursively call Algorithm {\\sf MS-SC\\_$g$D\\&C} itself, by\nfurther dividing the subproblem $P_s$ (lines 5-6). Otherwise, when\nsubproblem $P_s$ contains only one single task, we apply the greedy\nalgorithm of the classical set cover problem for task set $T_p(P_s)$ and worker set\n$W_p(P_s)$ (lines 7-8).\n\nAfter that, we can obtain an assignment instance set $I_p^{(s)}$ for\neach subproblem $P_s$, and merge them into one single\nworker-and-task assignment instance set $I_p$, by reconciling the\nconflict (lines 9-11). In particular, $I_p$ is initially empty (line\n1), and each time merged with an assignment set $I_p^{(s)}$ from\nsubproblem $P_s$ (lines 10-11). Due to the confliction among\nsubproblems, we call function {\\sf MS-SC\\_Conflict\\_Reconcile}\n$(\\cdot, \\cdot)$ (discussed later in Section \\ref{subsec:merge}) to\nresolve the confliction issue during the merging process. Finally,\nwe can return the merged assignment instance set $I_p$ (line 12).\n\\vspace{-1ex}\n\n\\subsection{Merging Conflict Reconciliation}\n\\label{subsec:merge}\n\nIn this subsection, we introduce the merging conflict reconciliation\nprocedure, which resolves the conflicts while merging assignment\nresults of subproblems (i.e., line 11 of Procedure {\\sf\n\tMS-SC\\_$g$D\\&C}). Assume that $I_p$ is the current assignment\ninstance set we have merged so far. Given a new subproblem $P_s$\nwith assignment set $I_p^{(s)}$, Figure \\ref{alg:conflict_reconcile}\nshows the merging algorithm, namely {\\sf MS-SC\\_Conflict\\_Reconcile},\nwhich combines two assignment sets $I_p$ and $I_p^{(s)}$ by\nresolving conflicts.\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize} \\vspace{-2ex}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_Conflict\\_Reconcile}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} the current assignment instance set, $I_p$, of subproblem $P$ we have merged, \\\\\n\t\t\t\t\t\t\\> \\> \\> \\> and the assignment instance set, $I_p^{(s)}$, of subproblem $P_s$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} a merged worker-and-task assignment instance set, $I_p$\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> let $W_c$ be a set of all conflicting workers between $I_p$ and $I_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> while $W_c \\neq \\emptyset$\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> \\> choose a worker $w_i\\in W_c$ with the highest traveling cost in $I_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> \\> if we substitute $w_i$ with $w_i'$ in $P_s$ having the highest score $S_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> \\> compute the reduction of the assignment score, $\\Delta S_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (6) \\> \\> \\> if we substitute $w_i$ with $w_i''$ in $P$ having the highest score $S_p$\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\> \\> compute the reduction of the assignment score, $\\Delta S_p$\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> if $\\Delta S_p > \\Delta S_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (9) \\> \\> \\> \\> substitute worker $w_i$ with $w_i'$ in $I_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (10)\\> \\> \\> else\\\\\n\t\t\t\t\t\t\\> (11)\\> \\> \\> \\> substitute worker $w_i$ with $w_i''$ in $I_p$\\\\\n\t\t\t\t\t\t\\> (12)\\> \\> \\> $W_c = W_c - \\{w_i\\}$\\\\\n\t\t\t\t\t\t\\> (13)\\> \\> $I_p = I_p \\cup I_p^{(s)}$\\\\\n\t\t\t\t\t\t\\> (14) \\> \\>  return $I_p$\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\\vspace{-5ex}\n\t\\caption{\\small The Merging Conflict Reconciliation Algorithm.}\n\t\\vspace{-5ex}\n\t\\label{alg:conflict_reconcile}\n\\end{figure}\n\nIn particular, two distinct tasks from two subproblems may be\nassigned with the same (conflicting) worker $w_i$. Since each worker\ncan only be assigned to one spatial task at a time, we thus need to\navoid such a scenario when merging assignment instance sets of two\nsubproblems (e.g., $I_p$ and $I_p^{(s)}$). Our algorithm in Figure\n\\ref{alg:conflict_reconcile} first obtain a set, $W_c$, of all\nconflicting workers between $I_p$ and $I_p^{(s)}$ (line 1). Then,\neach time we greedily solve the conflicts for workers $w_i$ in an\nnon-decreasing order of the traveling cost (i.e., $c_{ij}$) in\n$I^{(s)}_p$ (line 3). Next, in order to resolve the conflicts, we\ntry to replace worker $w_i$ with another worker $w_i'$ (or $w_i''$)\nin $P_{s}$ (or $P$) with the highest score $S_p^{(s)}$ (or\n$S_p$), and compute possible reduction of the assignment score,\n$\\Delta S_p^{(s)}$ (or $\\Delta S_p$) (lines 4-7). Note that, here we replace worker $w_i$ with other available workers. If no other workers are available for replacing $w_i$, we may need to sacrifice task $t_j$ that worker $w_i$ is assigned to. For example, when we cannot find another worker to replace\n$w_i$ in $P_s$, the substitute of $w_i$ will be set as an empty worker, which means the\nassigned task $t_j$ for $w_i$ in $I_p^{(s)}$ will be sacrificed\nand $\\Delta S_p^{(s)}=B'_j$ (as calculated in Equation \\ref{eq:flexible_budget}). In the case that\n$\\Delta S_p > \\Delta S_p^{(s)}$, we substitute worker $w_i$ with\n$w_i'$ in $I_p^{(s)}$ (since the replacement of $w_i$ in subproblem\n$S_p^{(s)}$ leads to lower score reduction); otherwise, we resolve\nconflicts by replacing $w_i$ with $w_i''$ in $I_p$ (lines 8-12). After\nresolving all conflicts, we merge assignment instance set $I_p$ with\n$I_p^{(s)}$ (line 13), and return the merged result $I_p$.\n\n\\subsection{Cost-Model-Based Estimation of the Best Number of Groups}\n\\label{subsec:DC_cost_model}\n\n\nIn this subsection, we discuss how to estimate the best number of\ngroups, $g$, such that the total cost of solving the MS-SC problem\nin $g$-divide-and-conquer approach is minimized. Specifically, the\ncost of the $g$-divide-and-conquer approach consists of 3 parts: the\ncost, $F_D$, of decomposing subproblems, that, $F_C$, of conquering\nsubproblems recursively, and that, $F_M$, of merging subproblems by\nresolving conflicts.\n\nWithout loss of generality, as illustrated in Figure\n\\ref{fig:gDC_cost_model}, during the $g$-divide-and-conquer process,\non level $k$, we recursively divide the original MS-SC problem into\n$g^k$ subproblems, $P_1^{(k)}$, $P_2^{(k)}$, ..., and\n$P_{g^k}^{(k)}$, where each subproblem involves $m\/g^k$ spatial\ntasks.\n\\begin{figure}[ht]\\vspace{-3ex}\n\t\\centering\n\t\\scalebox{0.4}[0.4]{\\includegraphics{gDC_cost_model.eps}}\\vspace{-1ex}\n\t\\caption{\\small Illustration of the Cost Model Estimation.}\\vspace{-4ex}\n\t\\label{fig:gDC_cost_model}\n\\end{figure}\n\n\n\n\\noindent {\\bf The Cost, $F_D$, of Decomposing Subproblems.} From\nAlgorithm {\\sf MS-SC\\_Decomposition} (in Figure\n\\ref{alg:decomposing}), we first need to retrieve all valid\nworker-and-task assignment pairs (line 3), whose cost is $O(m\\cdot\nn)$. Then, we will divide each problem into $g$ subproblems, whose\ncost is given by $O(m\\cdot g+m)$ on each level. For level $k$, we\nhave $m\/g^k$ tasks in each subproblem $P^{(k)}_i$. We will further\ndivide it into $g$ more subproblems, $P^{(k+1)}_j$, and each one\nwill have $m\/g^{k+1}$ tasks. To obtain $m\/g^{k+1}$ tasks in each\nsubproblem $P^{(k+1)}_j$, we first need to find the anchor task, which\nneeds $O(m\/g^{k})$ cost, and further retrieve the rest tasks,\nwhich needs $O(m\/g^{k+1})$ cost. Moreover, since we will have $g^{k+1}$\nsubproblems on level $k+1$, the cost of decomposing tasks on level $k$\nis given by $O(m\\cdot g+m)$.\n\n\nSince there are totally $log_g(m)$ levels, the total cost of\ndecomposing the MS-SC problem is given by:\n{\\small\n\t$$F_D=  m\\cdot n+(m \\cdot g+m)\\cdot log_g(m).$$\n\t\\vspace{-2ex}}\n\n\n\n\n\\noindent {\\bf The Cost, $F_C$, of Recursively Conquering\n\tSubproblems.} Let function $F_C(x)$ be the total cost of conquering\na subproblem which contains $x$ spatial tasks. Then, we have the\nfollowing recursive function: \n{\\small\\vspace{-3ex}\n\t$$F_C(m) = g\\cdot F_C(\\left\\lceil \\frac{m}{g}\\right\\rceil).$$\n\t\\vspace{-2ex}}\n\nAssume that $deg_t$ is the average degree of task nodes in the\nbipartite group $G$. Then, the base case of function $F_C(x)$ is the\ncase that $x=1$, in which we apply the greedy algorithm on just one\nsingle task and $deg_t$ workers. Thus, by the analysis of the time\ncomplexity in Section \\ref{sub:greedy_algorithm}, we have:\n{\\small\\vspace{-1ex}\n\t$$F_C(1) = cost_{greedy}(deg_t, 1) = deg_t^2.$$\n\t\\vspace{-3ex}\n}\n\nFrom the recursive function $F_C(x)$ and its base case, we can\nobtain the total cost of the recursive invocation on levels from 1\nto $log_g(m)$ below:\n{\\small\\vspace{-1.5ex}\n\t$$\\sum_{k = 1}^{log_g(m)}F_c(m\/g^k) =\n\t\\frac{1-m}{1-g}deg_t^2$$\n\t\n}\n\n\\noindent {\\bf The Cost, $F_M$, of Merging Subproblems.} Next, we\nprovide the cost, $F_M$, of merging subproblems by resolving\nconflicts. Assume that we have $n_s$ workers who could be assigned\nto more than one spatial task (i.e., conflicting workers). Moreover,\neach worker node has an average degree $deg_w$ in the bipartite\ngraph. During the subproblem merging processing, we can estimate the\nworst-case cost of resolving conflicts for these $n_s$ workers, and\nwe may resolve conflicts for each worker at most $(deg_w-1)$ times.\n\nTherefore, the worst-case cost of merging subproblems can be given\nby: \\vspace{-1ex}$$F_M = n_s\\cdot (deg_w-1).$$\n\n\n\\noindent {\\bf The Total Cost of the $g$-D\\&C Approach.} The total\ncost, $cost_{gD\\&C}$, of the $g$-D\\&C algorithm can be given by\nsumming up three costs, $F_D$, $F_C$, and $F_M$. That is, we have\n\n{\\small\n\t\\vspace{-3ex}\n\t\\begin{eqnarray}\n\t\tcost_{gD\\&C} &=& F_D + \\sum_{k = 1}^{log_g(m)}F_c(m\/g^k) + F_M \\label{eq:D&C_cost}\\\\\n\t\t&=& (mg+m)\\log_g (m) + \\frac{1-m}{1-g}deg_t^2 + n_s (deg_w -\n\t\t1)\\notag \\vspace{-3ex}\n\t\\end{eqnarray}\n\t\\vspace{-3ex}\n}\n\nWe take the derivation of $cost_{gD\\&C}$ (given in\nEq.~(\\ref{eq:D&C_cost})) over $g$, and let it be 0. In particular,\nwe have:\n\n{\\small\n\t\\vspace{-3ex}\n\t\\begin{eqnarray}\n\t\t&&\\frac{\\partial cost_{gD\\&C}}{\\partial g} \\notag\\\\\n\t\t&=&\\frac{m\\log(m)(g\\log(g) - g - 1)}{g\\log(2g)} +\n\t\t\\frac{1-m}{(1-g)^2}deg_t^2 = 0\n\t\\end{eqnarray}\n\t\\vspace{-1ex}\n}\n\nWe notice that when $g=2$, $\\frac{\\partial cost_{gD\\&C}}{\\partial g}$ is much\nsmaller than 0 but increases quickly when $g$ grows. In addition, $g$ can only\nbe an integer. Then we can try the integers, (2, 3, 4... ), until\n$\\frac{\\partial cost_{gD\\&C}}{\\partial g}$ is above 0.\n\n\n\\section{The Cost-Model-Based Adaptive Algorithm}\n\\label{sec:adpative}\n\nIn this section, we introduce a \\textit{cost-model-based adaptive\n\tapproach}, which adaptively decides the strategies to apply our\nproposed greedy and $g$-divide-and-conquer ($g$-D\\&C) algorithms.\nThe basic idea is as follows. Unlike the $g$-D\\&C algorithm, we do\nnot divide the MS-SC problem into subproblems recursively until task\ngroup sizes become $1$ (which can be solved by the greedy algorithm\nof set cover problems). Instead, based on our proposed cost model,\nwe will partition the problem into subproblems, and adaptively\ndetermine when to stop in some partitioning round (i.e., the total\ncost of solving subproblems with the greedy algorithm is smaller\nthan that of continuing dividing subproblems).\n\n\\begin{figure}[ht]\n\t\\begin{center}\n\t\t\\begin{tabular}{l}\n\t\t\t\\parbox{3.1in}{\n\t\t\t\t\\begin{scriptsize} \\vspace{-4ex}\n\t\t\t\t\t\\begin{tabbing}\n\t\t\t\t\t\t12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=12\\=\\kill\n\t\t\t\t\t\t{\\bf Procedure {\\sf MS-SC\\_Adaptive}} \\{ \\\\\n\t\t\t\t\t\t\\> {\\bf Input:} $n$ workers in $W_p$, and $m$ time-constrained spatial tasks in $T_p$\\\\\n\t\t\t\t\t\t\\> {\\bf Output:} a worker-and-task assignment instance set, $I_p$\\\\\n\t\t\t\t\t\t\\> (1) \\> \\> $I_p = \\emptyset$\\\\\n\t\t\t\t\t\t\\> (2) \\> \\> estimate the cost, $cost_{greedy}$, of the greedy algorithm\\\\\n\t\t\t\t\t\t\\> (3) \\> \\> estimate the best number of groups, $g$, and obtain the cost, $cost_{gdc}$, \\\\\n\t\t\t\t\t\t\\> \\> \\>  of the $g$-D\\&C approach\\\\\n\t\t\t\t\t\t\\> (4) \\> \\> if $cost_{greedy} < cost_{gdc}$\\\\\n\t\t\t\t\t\t\\> (5) \\> \\> \\> $I_p$ = {\\sf MS-SC\\_Greedy}($W_p$, $T_p$) \\\\\n\t\t\t\t\t\t\\> (6) \\> \\> else \\qquad \\textit{\/\/ $g$-D\\&C algorithm}\\\\\n\t\t\t\t\t\t\\> (7) \\> \\> \\>  invoke {\\sf MS-SC\\_Decomposition}$(W_p, T_p, g)$, and obtain subproblems $P_s$\\\\\n\t\t\t\t\t\t\\> (8) \\> \\> \\> for each subproblem, $P_s$, \\\\\n\t\t\t\t\t\t\\> (9) \\> \\> \\> \\> $I_p^{(s)}$ = {\\sf MS-SC\\_Adaptive}($W_p(P_s)$, $T_p(P_s)$)\\\\\n\t\t\t\t\t\t\\> (10)\\> \\> \\> for $i=1$ to $g$\\\\\n\t\t\t\t\t\t\\> (11)\\> \\>  \\> \\> find the next subproblem, $P_s$\\\\\n\t\t\t\t\t\t\\> (12)\\> \\> \\> \\> $I_p$ = {\\sf MS-SC\\_Conflict\\_Reconcile} ($I_p$, $I_p^{(s)}$) \\\\\n\t\t\t\t\t\t\\> (13)\\> \\>  return $I_p$\\\\\n\t\t\t\t\t\t\\}\n\t\t\t\t\t\\end{tabbing}\n\t\t\t\t\\end{scriptsize}\n\t\t\t}\n\t\t\\end{tabular}\n\t\\end{center}\\vspace{-5ex}\n\t\\caption{\\small The MS-SC Cost-Model-Based Adaptive Algorithm.}\n\t\\label{alg:hybrid}\\vspace{-5ex}\n\\end{figure}\n\n\n\n\\subsection{Algorithm of the Cost-Model-Based Adaptive Approach}\n\n\n\nFigure \\ref{alg:hybrid} shows the pseudo-code of our\ncost-model-based adaptive algorithm, namely {\\sf MS-SC\\_Adaptive}.\nInitially, we estimate the cost, $cost_{greedy}$, of applying the\ngreedy approach over worker\/task sets $W_p$ and $T_p$ (line 2).\nSimilarly, we also estimate the best group size, $g$, and compute\nthe cost, $cost_{gd\\&c}$ of using the $g$-D\\&C algorithm (line 3).\nIf it holds that the cost of the greedy algorithm is smaller than\nthat of the $g$-D\\&C approach (i.e., $cost_{greedy} < cost_{gdc}$),\nthen we will use the greedy algorithm by invoking function {\\sf\n\tMS-SC\\_Greedy}($\\cdot, \\cdot$) (due to its lower cost; lines 4-5).\nOtherwise, we will apply the $g$-D\\&C algorithm, and further\npartition the problem into subproblems $P_s$ (lines 6-7). Then, for\neach subproblem $P_s$, we recursively call the cost-model-based\nadaptive algorithm, and retrieve the assignment instance set\n$I_p^{(s)}$ (line 9). After that, we merge all the assignment\ninstance sets from subproblems by invoking function {\\sf\n\tMS-SC\\_Conflict\\_Reconcile}($\\cdot, \\cdot$) (lines 10-12). Finally,\nwe return the worker-and-task assignment instance set $I_p$ (line\n13).\n\n\n\n\n\\subsection{Cost Model for the Stoping Condition}\n\n\n\nNext, we discuss how to determine the stopping level, when using our\ncost-model-based adaptive approach to recursively solve the MS-SC\nproblem. Intuitively, at the current level $k$, we need to estimate\nthe costs, $cost_{greedy}$ and $cost_{gdc}$, of using greedy and\n$g$-D\\&C algorithms, respectively, to solve the remaining MS-SC\nproblem. If the greedy algorithm has lower cost, then we will stop\nthe divide-and-conquer, and apply the greedy algorithm for each\nsubproblems.\n\nIn the sequel, we discuss how to obtain the formulae of costs\n$cost_{greedy}$ and $cost_{gdc}$.\n\n\\underline{\\it The Cost, $cost_{greedy}$, of the Greedy Algorithm.}\nGiven a set, $W_p$, of $n$ workers and a set, $T_p$, of $m$ tasks,\nthe cost, $cost_{greedy}$, of our greedy approach (as given in\nFigure \\ref{alg:greedy}) has been discussed in Section\n\\ref{sub:greedy_algorithm}.\n\nIn the bipartite graph of valid worker-and-task pairs, denote\nthe average degree of workers as $deg_w$, and that of tasks as\n$deg_t$. In Figure \\ref{alg:greedy}, the computation of valid\nworker-and-task pairs in line 2 needs\n$O(m\\cdot n)$ cost. Since there are at most $n$ iterations, for each\nround (lines 3-16), we apply two worker-pruning methods to at most\n$(2m \\cdot deg_t)$ pairs, and select pairs with the highest score\nincreases, which need $O(3m \\cdot n\\cdot deg_t)$ cost in total.\nFor the cost of task-pruning, there are totally $n$ rounds (lines 3-16; \ni.e., removing one out of $n$ workers in each round in line 16). \nIn each round, there are at most $deg_w$ out of $m$ tasks (line 5) that may be \npotentially pruned by Lemma \\ref{lemma:prune_insufficient_task} (line10). To check each of $deg_w$ tasks, we need $O(deg_t)$ cost. \nTherefore, the total cost of task-pruning is given by $O(n\\cdot deg_t \\cdot deg_w)$. \nIf we cannot prune a task that was assigned with a worker in the last round (lines 3-16), then we need to update score increases of $deg_t$ workers for that task. Each task will be assigned with workers for $deg_t$  times. Thus, the total update cost for one task is given by $O(deg^2_t)$ (line 12). Therefore, $cost_{greedy}(n, m)$ can be given by:\\vspace{-3ex}\n\n\n\n\n{\\small\n\t\\begin{eqnarray}\n\t\t&& cost_{greedy}(n, m) \\notag\\\\\n\t\t&=&C_{greedy}\\cdot(m\\cdot n + n\\cdot deg_t \\cdot (3m + deg_w) + m\\cdot deg^2_t), \\qquad \\label{eq:greedy_approach_cost}\n\t\\end{eqnarray}}\n\t\\vspace{-3ex}\n\t\n\t\\noindent where parameter $C_{greedy}$ is a constant factor, which\n\tcan be inferred from cost statistics of the greedy algorithm.\n\t\n\t\n\t\n\t\n\t\\underline{\\it The Cost, $cost_{gdc}$, of the $g$-D\\&C Algorithm.}\n\tAssume that the current $g$-divide-and-conquer level is $k$. We can\n\tmodify the cost analysis in Section \\ref{subsec:DC_cost_model}, by\n\tconsidering the cost, $cost_{gdc}$, of the remaining\n\tdivide-and-conquer levels. Specifically, we have the cost, $F_D'$,\n\tof the decomposition algorithm, that is:\\vspace{-2ex}\n\t\n\t{\\small\n\t\t$$F_D'=  m\\cdot n+(m\\cdot g+m)\\cdot k.$$\n\t\t\\vspace{-3ex}}\n\t\n\t\n\tMoreover, when the current level is $k$, the cost of conquering the\n\tremaining subproblems is given by:\\vspace{-2ex}\n\t\n\t{\\small\n\t\t$$\\sum_{i= k}^{log_g(m)}F_c(m\/g^i).$$\n\t\t\\vspace{-3ex}}\n\t\n\tFinally, the cost of merging subproblems is given by $F_M$.\n\t\n\tAs a result, the total cost, $cost_{gdc}$, of solving the MS-SC\n\tproblem with our $g$-D\\&C approach for the remaining partitioning\n\tlevels (from level $k$ to $log_g(m)$) can be given by:\\vspace{-3ex}\n\t\n\t{\\small\n\t\t$$cost_{gdc}= C_{gdc}\\cdot(F_D' + \\sum_{i = k}^{log_g(m)}F_c(m\/g^i)+ F_M),$$\n\t\t\\vspace{-3ex}}\n\t\n\t\\noindent where parameter $C_{gdc}$ is a constant factor, which can\n\tbe inferred from time cost statistics of the $g$-D\\&C algorithm.\n\t\n\tThis way, we compare $cost_{greedy}$ with $cost_{gdc}$ (as mentioned\n\tin line 4 of {\\sf MS-SC\\_Adaptive} Algorithm). If $cost_{greedy}$ is\n\tsmaller than $cost_{gdc}$, we stop at the current level $k$, and apply the greedy algorithm to tackle\n\tthe MS-SC problem directly; otherwise, we keep dividing the original\n\tMS-SC problem into subproblems (i.e., $g$-D\\&C).\n\n\\section{Experimental Study}\n\\label{sec:exper}\n\n\n\\subsection{Experimental Methodology}\n\n\\noindent \\textbf{Data Sets.} We use both real and synthetic data to\ntest our proposed MS-SC approaches. Specifically, for real data, we\nuse Meetup data set from \\cite{liu2012event}, which was crawled from\n\\textit{meetup.com} between Oct. 2011 and Jan. 2012. There are 5,153,886\nusers, 5,183,840 events, and 97,587 groups in Meetup, where each\nuser is associated with a location and a set of tags, each group is\nassociated with a set of tags, and each event is associated with a\nlocation and a group who created the event. For an event, we use the\ntags of the group who creates the event as its tags. To conduct the\nexperiments on our approaches, we use the locations and tags of\nusers in Meetup to initialize the locations and the practiced skills\nof workers in our MS-SC problem. In addition, we utilize the\nlocations and tags of events to initialize the locations and the\nrequired skills of tasks in our experiments. Since workers are\nunlikely to move between two distant cities to conduct one spatial\ntask, and the constraints of time (i.e., $e_j$), budget (i.e.,\n$B_j$) and distance (i.e., $d_i$) also prevent workers from moving\ntoo far, we only consider those user-and-event pairs located in the\nsame city. Specifically, we select one famous and popular city, Hong\nKong, and extract Meetup records from the area of Hong Kong (with\nlatitude from \\ang{22.209} to \\ang{113.843} and longitude from\n\\ang{22.609} to \\ang{114.283}), in which we obtain 1,282 tasks and\n3,525 workers.\n\n\n\n\nFor synthetic data, we generate locations of workers and tasks in a\n2D data space $[0, 1]^2$, following either Uniform (UNIFORM) or\nSkewed (SKEWED) distribution. For Uniform distribution, \nwe uniformly generate the locations of tasks\/workers in the 2D data space.\nSimilarly, we also generate\ntasks\/workers with the Skewed distribution by locating 90\\% of them into\na Gaussian cluster (centered at (0.5, 0.5) with variance = $0.2^2$),\nand distribute the rest workers\/tasks uniformly.  Then, for\nskills of each worker, we randomly associate one user in Meetup data\nset to this worker, and use tags of the user as his\/her skills in\nour MS-SC system. For the required skills of each task, we randomly \nselect an event, and use its tags as the required skills of the task.\n\n\nFor both real and synthetic data sets, we simulate the velocity of\neach worker with Gaussian distribution within range $[v^-, v^+]$,\nfor $v^-, v^+ \\in (0, 1)$. For the unit price, $C_i$, w.r.t. the\ntraveling distance of each worker, we generate it following the Uniform\ndistribution within the range $[C^-, C^+]$. Furthermore, we produce\nthe maximum moving distance of each worker, following the Uniform\ndistribution within the range $[d^-, d^+]$ (for $d^-, d^+ \\in\n(0,1)$). For temporal constraints of tasks, we also generate the\narrival deadlines of tasks, $e$, within range $[rt^-, rt^+]$ with\nGaussian distribution. Finally, we set the budgets of tasks with\nGaussian distribution within the range $[B^-, B^+]$. \nHere, for Gaussian distributions, we linearly map data \nsamples within $[-1, 1]$ of a Gaussian distribution $\\mathcal{N}(0, 0.2^2)$ \nto the target ranges.\n\n\n\\noindent\\textbf{MS-SC Approaches and Measures.} We conduct\nexperiments to compare our three approaches, GREEDY, $g$-D\\&C and\nADAPTIVE, with a random method, namely RANDOM, which randomly\nassigns workers to tasks.\n\nIn particular, GREEDY selects a ``best'' worker-and-task assignment\nwith the highest score increase each time, which is a local optimal\napproach. The $g$-D\\&C algorithm keeps dividing the problem into $g$\nsubproblems on each level, until finally the number of tasks in each\nsubproblem is 1 (which can be solved by the greedy algorithm on each\none-task subproblem). Here, the parameter $g$ can be estimated by a\ncost model to minimize the computing cost. The cost-model-base\nadaptive algorithm (ADAPTIVE) makes the trade-off between GREEDY and\n$g$-D\\&C, in terms of efficiency and accuracy, which adaptively\ndecides the stopping level of the divide-and-conquer. To evaluate\nour three proposed approaches, we need to compare the results with\nground truth. However, as proved in Section \\ref{sec:reduction}, the\nMS-SC problem is NP-hard, and thus infeasible to calculate the real\noptimal result as the ground truth. Alternatively, we will compare\nthe effectiveness of our three approaches with that of a random\n(RANDOM) method, which randomly chooses a task then randomly assigns\nworker to the task. For each MS-SC instance, we run RANDOM for 10\ntimes, and report the result with the highest score.\n\nTable \\ref{table2} depicts our experimental settings, where the\ndefault values of parameters are in bold font. In each set of\nexperiments, we vary one parameter, while setting other parameters\nto their default values. For each experiment, we report the running\ntime and the assignment score of our tested approaches. All our\nexperiments were run on an Intel Xeon X5675 CPU @3.07 GHZ with 32 GB\nRAM in Java.\n\n\n\\begin{table}[t]\n\t\\begin{center}\\vspace{-6ex}\n\t\t\\caption{\\small Experiments Settings.} \\label{table2}\n\t\t{\\small\\scriptsize\n\t\t\t\\begin{tabular}{l|l}\n\t\t\t\t{\\bf \\qquad \\qquad \\quad Parameters} & {\\bf \\qquad \\qquad \\qquad Values} \\\\ \\hline \\hline\n\t\t\t\tthe number of tasks $m$ & 1K, 2K, \\textbf{5K}, 8K, 10K \\\\\n\t\t\t\tthe number of workers $n$ & 1K, 2K, \\textbf{5K}, 8K, 10K \\\\\n\t\t\t\tthe task budget range $[B^-, B^+]$  & [1, 5], \\textbf{[5, 10]}, [10, 15], [15, 20], [20, 25]\\\\\n\t\t\t\tthe velocity range $[v^-, v^+]$   & [0.1, 0.2], \\textbf{[0.2, 0.3]}, [0.3, 0.4], [0.4, 0.5]\\\\\n\t\t\t\tthe unit price w.r.t. distance $[C^-, C^+]$ & [10, 20], \\textbf{[20, 30]}, [30, 40], [40, 50]\\\\\n\t\t\t\tthe moving distance range $[d^-, d^+]$ & [0.1, 0.2], [0.2, 0.3], \\textbf{[0.3, 0.4]}, [0.4, 0.5]\\\\\n\t\t\t\tthe expiration time range $[rt^-, rt^+]$  & [0.25, 0.5], [0.5, 1], \\textbf{[1, 2]}, [2, 3], [3, 4]\\\\\n\t\t\t\t\\hline\n\t\t\t\\end{tabular}\n\t\t}\n\t\\end{center}\\vspace{-7ex}\n\\end{table}\n\n\n\\subsection{Experiments on Real Data}\n\n\n\n\nIn this subsection, we show the effects of the range of task budgets\n$[B^-, B^+]$, the range of workers' velocities $[v^-, v^+]$, and\nthe range of unit prices w.r.t. distance $[C^-, C^+]$.\n\n\n\\noindent\\textbf{Effect of the Range of Task Budgets $[B^-, B^+]$.}\nFigure \\ref{fig:budget_b} illustrates the experimental results on\ndifferent ranges, $[B^-, B^+]$, of task budgets $B_j$ from $[1,5]$\nto $[20,25]$. In Figure \\ref{subfig:b_score}, the assignment scores\nof all the four approaches increase, when the value range of task\nbudgets gets larger. When the average budgets of tasks increase, the\nflexible budget $B'$ of each task will also increase. $g$-D\\&C and\nADAPTIVE can achieve higher score than GREEDY. In contrast, RANDOM\nhas the lowest score, which shows that our proposed three approaches\nare more effective. As shown in Figure \\ref{subfig:b_cpu}, the\nrunning times of our three approaches increase, when the range of\ntask budgets becomes larger. The reason is that, when $B_j \\in [B^-,\nB^+]$ increases, each task has more valid workers, which thus leads\nto higher complexity of the MS-SC problem and the increase of the\nrunning time. The RANDOM approach is the fastest (however, with the\nlowest assignment score), since it does not even need to find local\noptimal assignment. The ADAPTIVE algorithm achieves much lower\nrunning time than $g$-D\\&C (a bit higher time cost than GREEDY), but\nhas comparable score with $g$-D\\&C (much higher score than GREEDY),\nwhich shows the good performance of ADAPTIVE, compared with GREEDY\nand $g$-D\\&C.\n\n\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Scores of Assignment}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{budget_score.eps}}\n\t\t\\label{subfig:b_score}}\n\t\\subfigure[][{\\scriptsize Running Times}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{budget_cpu.eps}}\n\t\t\\label{subfig:b_cpu}}\\vspace{-2ex}\n\t\\caption{\\small Effect of the Range of Task Budgets $[B^-, B^+]$ (Real Data).}\\vspace{-4ex}\n\t\\label{fig:budget_b}\n\\end{figure}\n\n\n\\noindent\\textbf{Effect of the Workers' Velocity Range $[v^-,\n\tv^+]$.} Figure \\ref{fig:velocity_v} reports the effect of the range\nof velocities, $[v^-, v^+]$, of workers over real data. As shown in\nFigure \\ref{subfig:v_score}, when the range of velocities increases\nfrom $[0.1,0.2]$ to $[0.2,0.3]$, the scores of all the approaches\nfirst increase; then, they stop growing for the velocity range \nvarying from [0.2, 0.3] to [0.4, 0.5]. The reason is that, at the\nbeginning, with the increase of velocities, workers can reach more\ntasks before their arrival deadlines. Nevertheless, workers are also\nconstrained by their maximum moving distances, which prevents them\nfrom reaching more tasks. ADAPTIVE can achieve a bit higher scores\nthan $g$-D\\&C, and much better assignment scores than GREEDY.\n\nIn Figure \\ref{subfig:v_cpu}, when the range of velocity $[v^-,\nv^+]$ increases, the running times of our tested approaches also\nincrease, due to the cost of more valid worker-and-task pairs to be\nhandled. Similar to previous results, RANDOM is the fastest, and\n$g$-D\\&C is the slowest. ADAPTIVE requires about 0.5-1.5 seconds,\nand has lower time cost than $g$-D\\&C, which shows the efficiency of\nour proposed approaches.\n\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Scores of Assignment}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{v_score.eps}}\n\t\t\\label{subfig:v_score}}\n\t\\subfigure[][{\\scriptsize Running Times}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{v_cpu.eps}}\n\t\t\\label{subfig:v_cpu}}\\vspace{-2ex}\n\t\\caption{\\small Effect of the Range of Velocities $[v^-, v^+]$ (Real Data).}\\vspace{-4ex}\n\t\\label{fig:velocity_v}\n\\end{figure}\n\n\n\n\\noindent\\textbf{Effect of the Range of Unit Prices w.r.t. Traveling Distance\n\t$[C^-, C^+]$.} In Figure \\ref{subfig:c_score}, when the unit prices w.r.t.\ntraveling distance $C_i \\in[C^-, C^+]$ increase, the scores of all the approaches\ndecrease. The reason is that, when the range of unit prices $[C^-, C^+]$ increases, we need to pay\nmore wages containing the traveling costs of workers (to do spatial\ntasks), which in turn decreases the flexible budget of each task.\nHowever, ADAPTIVE can still achieve the highest score among all four\napproaches; scores of $g$-D\\&C are close to the scores of ADAPTIVE\nand higher than GREEDY.\n\nIn Figure \\ref{subfig:c_cpu}, when the range of unit prices, $[C^-, C^+]$, of the\ntraveling cost increases, the number of valid worker-and-task pairs\ndecreases, and thus the running time of all the approaches also\ndecreases. Our ADAPTIVE algorithm is faster than $g$-D\\&C, and\nslower than GREEDY.\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Scores of Assignment}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{C_score.eps}}\n\t\t\\label{subfig:c_score}}\n\t\\subfigure[][{\\scriptsize Running Times}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{C_cpu.eps}}\n\t\t\\label{subfig:c_cpu}}\\vspace{-2ex}\n\t\\caption{\\small Effect of the Range of Unit Prices w.r.t. Traveling Distance $[C^-, C^+]$ (Real Data).} \\vspace{-1ex}\n\t\\label{fig:constanct_c}\\vspace{-3ex}\n\\end{figure}\n\nIn addition, we also tested the effects of the range, $[d^-, d^+]$,\nof maximum moving distances for workers, and the expiration time\nrange, $[rt^-, rt^+]$, of tasks over the real data set, Meetup. Due to\nspace limitations, please refer to experimental results with respect\nto other parameters (e.g., $[d^-, d^+]$ and $[rt^-, rt^+]$) in\nAppendix F.\n\nFrom experimental results on the real data above, ADAPTIVE can\nachieve higher scores than Greedy and $g$-D\\&C, and it is faster\nthan $g$-D\\&C and slower than GREEDY. Although $g$-D\\&C can achieve\ngood scores close to ADAPTIVE, it is the slowest among all the 4\napproaches.\n\n\\subsection{Experiments on Synthetic Data}\n\nIn this subsection, we test the effectiveness and robustness of our\nthree MS-SC approaches, GREEDY, $g$-D\\&C, and ADAPTIVE, compared\nwith RANDOM, by varying parameters (e.g., the number of tasks $m$\nand the number of workers $n$) on synthetic data sets. Due to space limitations, we present the experimental results for tasks\/workers with Uniform distributions. For similar results with tasks\/workers following skewed distributions, please refer to Appendix G.\n\n\\noindent\\textbf{Effect of the Number of Tasks $m$.} Figure\n\\ref{fig:tasks_m} illustrates the effect of the number, $m$, of\nspatial tasks, by varying $m$ from $1K$ to $10K$, over synthetic\ndata sets, where other parameters are set to default values. For\nassignment scores in Figure \\ref{subfig:m_score}, $g$-D\\&C obtains\nresults with the highest scores among all the four approaches.\nADAPTIVE performs similar to $g$-D\\&C, and achieves good results\nsimilar to $g$-D\\&C. GREEDY is not as good as $g$-D\\&C and ADAPTIVE,\nbut is still much better than RANDOM. When the number, $m$, of\nspatial tasks becomes larger, all our approaches can achieve higher\nscores.\n\nIn Figure \\ref{subfig:m_cpu}, when $m$ increases, the running time\nalso increases. This is because, we need to deal with more\nworker-and-task assignment pairs for large $m$. The ADAPTIVE\nalgorithm is slower than GREEDY, and faster than $g$-D\\&C. In\naddition, we find that the running time of GREEDY performs, with the\nsame trend as that estimated in our cost model (as given in\nEq.~(\\ref{eq:greedy_approach_cost})).\n\n\n\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Scores of Assignment}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{m_score.eps}}\n\t\t\\label{subfig:m_score}}\n\t\\subfigure[][{\\scriptsize Running Times}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{m_cpu.eps}}\n\t\t\\label{subfig:m_cpu}}\\vspace{-2ex}\n\t\\caption{\\small Effect of the Number of Tasks $m$ (Synthetic Data).}\\vspace{-4ex}\n\t\\label{fig:tasks_m}\n\\end{figure}\n\n\\noindent\\textbf{Effect of the Number of Workers $n$.} Figure\n\\ref{fig:workers_n} shows the experimental results with different\nnumbers of workers, $n$, from $1K$ to $10K$ over synthetic data,\nwhere other parameters are set to their default values. Similar to\nprevious results about the effect of $m$,  in Figure \\ref{subfig:n_score}, our proposed three\napproaches can obtain good results with high assignment scores,\ncompared with RANDOM. Moreover, when the number, $n$, of workers\nincreases, the scores of all our approaches also increase. The\nreason is that, when $n$ increases, we have more potential workers,\nwho can be assigned to nearby tasks, which may lead to even larger\nscores.\n\nIn Figure \\ref{subfig:n_cpu}, the running time of our approaches\nincreases, with the increase of the number of workers . This is due\nto higher cost to process more workers (i.e., larger $n$).\nSimilarly,  ADAPTIVE has higher time cost than GREEDY, and lower\ntime cost than $g$-D\\&C.\n\n\\begin{figure}[ht]\\vspace{-2ex}\n\t\\centering\n\t\\subfigure[][{\\scriptsize Scores of Assignment}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{n_score.eps}}\n\t\t\\label{subfig:n_score}}\n\t\\subfigure[][{\\scriptsize Running Times}]{\n\t\t\\scalebox{0.2}[0.2]{\\includegraphics{n_cpu.eps}}\n\t\t\\label{subfig:n_cpu}}\\vspace{-2ex}\n\t\\caption{\\small Effect of the Number of Workers $n$ (Synthetic Data).}\\vspace{-1ex}\n\t\\label{fig:workers_n}\\vspace{-3ex}\n\\end{figure}\n\n\n\nIn summary, over synthetic data sets, our ADAPTIVE algorithm trades\nthe accuracy for efficiency, and thus has the trade-off of\nscores\/times between GREEDY and $g$-D\\&C.\n\n\n\n\n\n\\section{Related Work}\n\\label{sec:related}\n\n\nRecently, with the popularity of GPS-equipped smart devices and\nwireless networks (e.g., Wi-Fi and 4G), the spatial crowdsourcing\n\\cite{deng2013maximizing, kazemi2012geocrowd} that performs\nlocation-based tasks has emerged and become increasingly important\nin both academia and industry. In this section, we review the\nrelated work on spatial crowdsourcing, as well as the set cover\nproblem (and its variants).\n\n\n\n\\vspace{0.5ex}\\noindent {\\bf Spatial Crowdsourcing.} Prior works on\ncrowdsourcing \\cite{alt2010location, bulut2011crowdsourcing} usually\nstudied crowdsourcing problems, which  treat the location\ninformation as a parameter and distribute tasks to workers. In these\nproblems, workers are not required to accomplish tasks on sites.\n\nIn contrast, the spatial crowdsourcing platform\n\\cite{kazemi2012geocrowd} requires workers to physically move to\nsome specific locations of tasks, and perform the requested\nservices, such as taking photos\/videos, waiting in line at shopping\nmalls, and decorating a room. As an example, some previous works\n\\cite{cornelius2008anonysense, kanhere2011participatory} studied the\nsmall-scale or specified campaigns benefiting from\n\\textit{participatory sensing} techniques, which utilize smart\ndevices (equipped by workers) to sense\/collect data for real\napplications.\n\n\nKazemi and Shahabi \\cite{kazemi2012geocrowd} classified the spatial\ncrowdsourcing systems from two perspectives: people's motivation and\npublishing models. From the perspective of people's motivation, the\nspatial crowdsourcing can be categorized into two groups:\nreward-based, in which workers can receive rewards according to the\nservices they supplied, and self-incentivised, in which workers\nconduct tasks voluntarily. In our work, we study our MS-SC problem\nbased on the reward-based model, where workers are paid for doing\ntasks. However, with a different goal, our MS-SC problem targets at\nassigning workers to tasks by using our proposed algorithms, such\nthat the required skills of tasks can be covered, and the total\nreward budget (i.e., flexible budget $B_j'$ in\nEq.~(\\ref{eq:flexible_budget})) can be maximized. Note that, we can\nembed incentive mechanisms from existing works \\cite{rula2014no,\n\tyang2012crowdsourcing} into our MS-SC framework to distribute\nrewards (flexible budgets) among workers, which is however not the\nfocus of our problem.\n\n\nAccording to the publishing modes of spatial tasks, the spatial\ncrowdsourcing can be also classified into two categories:\n\\textit{worker selected tasks} (WST) and \\textit{server assigned\n\ttasks} (SAT) \\cite{kazemi2012geocrowd}. In particular, for the WST\nmode, spatial tasks are broadcast to all workers, and workers can\nselect any tasks by themselves. In contrast, for the SAT mode, the\nspatial crowdsourcing server will directly assign tasks to workers,\nbased on location information of tasks\/workers.\n\n\nSome prior works \\cite{alt2010location, deng2013maximizing} on the\nWST mode allowed workers to select available tasks, based on their\npersonal preferences. However, for the SAT mode, previous works\nfocused on assigning available workers to tasks in the system, such\nthat the number of assigned tasks on the server side\n\\cite{kazemi2012geocrowd}, the number of worker's self-selected\ntasks on the client side \\cite{deng2013maximizing}, or the\nreliability-and-diversity score of assignments\n\\cite{cheng2014reliable} is maximized. For example, Peng et al.\n\\cite{cheng2014reliable} aims to obtain a worker-and-task assignment\nstrategy such that the assignment score (w.r.t. spatial\/temporal\ndiversity and reliability of tasks) is maximized.\n\nIn contrast, our MS-SC problem has a different, yet more general,\ngoal, which maximizes the total assignment score (i.e., flexible\nbudget, given by the total budget of the completed tasks minus the\ntotal traveling cost of workers). Most importantly, in our MS-SC\nproblem, we need to consider several constraints, such as\nskill-covering, budget, time, and distance constraints. That is, the\nrequired skill sets of spatial tasks should be fully covered by\nskills of those assigned workers, which is NP-hard and intractable.\nThus, previous techniques\n\\cite{cheng2014reliable,deng2013maximizing,kazemi2012geocrowd} on\ndifferent spatial crowdsourcing problems cannot be directly applied\nto our MS-SC problem.\n\nSome research communities studied the theory of SAT problems and\ndeveloped some SAT (Satisfiability) solvers. However, standard SAT\nsolvers can only solve decision problems (i.e., NP-complete\nproblems), but not optimization problems (i.e., NP-hard problems,\nlike our MS-SC problem). Thus, we need to design specific heuristic\nalgorithms for tackling the MS-SC problem.\n\nMoreover, some previous works \\cite{pournajaf2014spatial,\n\tto2014framework} utilized \\textit{differential privacy} techniques\n\\cite{dwork2008differential} to protect the location information,\nwhich is used to do the assignment, but may release some sensitive\nlocation\/trajectory data (leading to malicious attacks).\nNevertheless, this privacy issue is out of the scope of this paper.\n\n\n\\vspace{0.5ex}\\noindent {\\bf Set Cover Problem.} As mentioned in\nLemma \\ref{lemma:np}, the \\textit{set cover problem} (SCP) is a\nclassical NP-hard problem, which targets at choosing a set of\nsubsets to cover a universe set, such that the number of the\nselected subsets is minimized. SCP is actually a special case of our\nMS-SC problem, in which there exists only one spatial task. However,\nin most situations, we have more than one spatial task in the\nspatial crowdsourcing system, which is more complex, and thus more\nchallenging, to tackle.\n\nA direct variant of SCP is the \\textit{weighted set cover problem},\nwhich associates each subset with a weight. The well-known greedy\nalgorithm \\cite{vazirani2013approximation} can achieve an\napproximation ratio of $\\ln(N) ( \\approx H(N)$ here $H(N) =\n\\sum_{i=1}^{N}1\/i)$, where $N$ is the size of the universe set.\nOther SCP variants, such as the \\textit{set multicover problem}\n(SMC) and \\textit{multiset multicover problem} (MSMC), focused on\ncovering each element of the universe set for at least specified\ntimes using sets (in SMC, each element in subsets has just one copy)\nor multisets (in MSMC, each element in subsets has a specified\nnumber of copies). Sun and Li \\cite{sun2005mechanism} studied\n\\textit{set cover games problem}, which covers multiple sets. However, they focused on designing a good mechanism to enable each single task to obtain a local optimal result. In contrast, our work aims to obtain a global optimal solution to maximize the score of assignment.\n\n\n\nDifferent from SCP and its variants that cover only one universe\nset, our MS-SC problem is targeting to cover multiple sets, such\nthat the assignment score is maximized. Furthermore, our MS-SC\nproblem is also constrained by budget, time, and distance, which is\nmuch more challenging than SCP. To the best of our knowledge, no\nprior works on SCP (and its variants) have studied the MS-SC\nproblem, and existing techniques cannot be used directly to tackle\nthe MS-SC problem.\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nIn this paper, we propose the problem of the \\textit{multi-skill\n\toriented spatial crowdsourcing} (MS-SC), which assigns the\ntime-constrained and multi-skill-required spatial tasks with\ndynamically moving workers, such that the required skills of tasks\ncan be covered by skills of workers and the assignment score is\nmaximized. We prove that the processing of the MS-SC problem is\nNP-hard, and thus we propose three approximation approaches (i.e.,\ngreedy, $g$-D\\&C, and cost-model-based adaptive algorithms), which\ncan efficiently retrieve MS-SC answers. Extensive experiments have\nshown the efficiency and effectiveness of our proposed MS-SC\napproaches on both real and synthetic data sets.\n\n\\balance\n\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Some recent development}\n\\subsection{Background}\n\nThe object of kinetic theory is the modeling of particles by a distribution function in the phase space, which is denoted by $F(t,x,v)$ for $(t,x,v) \\in [0, \\infty) \\times  {\\Omega} \\times \\mathbb{R}^{3}$ where $\\Omega$ is an open bounded subset of $\\mathbb{R}^{3}$. Dynamics and collision processes of dilute charged particles with an electric field $E$ can be modeled by the (two-species) Vlasov-Poisson-Boltzmann equation\n\\begin{equation}  \\label{2FVPB}\n\\begin{split}\n\\partial_t F_+ + v \\cdot \\nabla_x F_+ +E \\cdot \\nabla_v F_+ = Q(F_+,F_+) + Q(F_+,F_- ),\n\\\\ \\partial_t F_- + v \\cdot \\nabla_x F_- -E \\cdot \\nabla_v F_- = Q(F_-,F_+) + Q(F_-,F_- ).\n\\end{split} \\end{equation}\nHere $F_\\pm(t,x,v) \\ge 0 $ are the density functions for the ions $(+)$ and electrons $(-)$ respectively.\n\n\nThe collision operator measures ``the change rate'' in binary hard sphere collisions and takes the form of\n\\begin{equation} \\begin{split}\\label{Q}\nQ(F_{1},F_{2}) (v)&: = Q_\\mathrm{gain}(F_1,F_2)-Q_\\mathrm{loss}(F_1,F_2)\\\\\n&: =\\int_{\\mathbb{R}^3} \\int_{\\S^2} \n|(v-u) \\cdot \\omega| [F_1 (u^\\prime) F_2 (v^\\prime) - F_1 (u) F_2 (v)]\n \\mathrm{d} \\omega \\mathrm{d} u,\n\\end{split}\\end{equation}   \nwhere $u^\\prime = u - [(u-v) \\cdot \\omega] \\omega$ and $v^\\prime = v + [(u-v) \\cdot \\omega] \\omega$. The collision operator enjoys a collision invariance: for any measurable $G$,  $\n\\int_{\\mathbb{R}^{3}} \\begin{bmatrix}1 & v & \\frac{|v|^{2}-3}{2}\\end{bmatrix} Q(G,G) \\mathrm{d} v = \\begin{bmatrix}0 & 0 & 0 \\end{bmatrix}.$ It is well-known that a global Maxwellian $\\mu$\nsatisfies $Q(\\cdot,\\cdot)=0$, where\n\\begin{equation} \\label{Maxwellian}\n\\mu(v):= \\frac{1}{(2\\pi)^{3\/2}} \\exp\\bigg(\n - \\frac{|v |^{2}}{2 }\n \\bigg).\n\\end{equation}\n\n\nThe electric field $E$ is given by\n\\begin{equation} \\label{Field}\nE(t,x): =  - \\nabla_{x} \\phi(t,x)\n\\end{equation}\nwhere an electrostatic potential is determined by the Poisson equation: \n\\begin{equation}   \\label{Poisson2}\n-\\Delta_x \\phi(t,x) = \\int_{\\mathbb R^3 } (F_+(t,x,v) - F_-(t,x,v))  \\, dv\n\\ \\  \\text{in} \\ \\Omega .\n\\end{equation}\n\nA simplified one-species Vlasov-Poisson-Boltzmann equation is often considered to reduce the complexity. Where we let $F(t,x,v)$ takes the role of $F_+(t,x,v)$, and assume $F_- = \\rho_0 \\mu $ where the constant $\\rho_0 = \\int_{\\Omega \\times \\mathbb R^3} F_+(0,x,v)  \\,dv dx$. Then we get the system\n\\begin{equation}  \\label{Boltzmann_E}\n\\partial_{t} F + v\\cdot \\nabla_{x} F + E\\cdot \\nabla_{v} F = Q(F,F),\n\\end{equation} \n\\begin{equation}   \\label{Poisson}\n-\\Delta_x \\phi(t,x) = \\int_{\\mathbb R^3 } F(t,x,v)  \\, dv -\\rho_0\n\\ \\  \\text{in} \\ \\Omega .\n\\end{equation}\nHere the background charge density $\\rho_0$ is assumed to be a constant.\n\nThroughout this paper, we use the notation\n\\begin{equation}  \\label{iota} \\begin{split}\n \\iota = +  \\text{ or } -, \\text{ and }\n -\\iota = \\begin{cases}\n- &, \\text{ if } \\iota = + \\\\ +&, \\text{ if } \\iota = -.\n\\end{cases}\n\\end{split} \\end{equation}\nAnd for the one-species case, $F_\\iota = F$.\n\n\n\n\n\n\n\nIn many physical applications, e.g. semiconductor and tokamak, the charged dilute\ngas is confined within a container, and its interaction with the boundary, which can be described by suitable boundary conditions, often plays a crucial role in global dynamics. In this paper we consider one of the physical conditions, a so-called diffuse boundary condition:\n\\begin{equation} \\label{diffuse_BC}\nF_\\iota(t,x,v)= \\sqrt{2\\pi} \\mu(v) \\int_{n(x ) \\cdot u>0} F_\\iota(t,x,u) \\{n(x) \\cdot u\\}\\mathrm{d} u \\ \\ \\text{for} \\ (x,v) \\in \\gamma_-.\n\\end{equation}\nHere $\\gamma_-: = \\{(x,v) \\in \\partial\\Omega \\times \\mathbb{R}^3: n(x) \\cdot v<0\\}$, and $n(x)$ is the outward unit normal at a boundary point $x$. \n\n\nDue to its importance, there have been many research activities in mathematical study of the Boltzmann equation. In \\cite{Guo_P}, global strong solution of Boltzmann equation coupled with the Poisson equation has been established through the nonlinear energy method, when the initial data are close to the Maxwellian $\\mu$. In the large-amplitude regime, an almost exponential decay for Boltzmann solutions is established in \\cite{DV}, provided certain a priori strong Sobolev estimates can be verified. Such high regularity insures an $L^\\infty$-control of solutions which is crucial to handle the quadratic nonlinearity. Even though these estimates can be verified in periodic domains, their validity in general bounded domains have been doubted. \n\nDespite its importance, mathematical theory on boundary problems of VPB, especially for strong solutions, hasn't been developed up to satisfactory (cf. renormalized solutions of VPB were constructed in \\cite{Michler}). One of the fundamental difficulties for the system in bounded domains is the lack of higher regularity, which originates from the characteristic nature of boundary conditions in the kinetic theory, and the nonlocal property of the collision term $Q$. This nonlocal term indicates that the local behavior of the solution could be affected globally by $x$ and $v$, and thus prevents the localization of the solution. From that a seemingly inevitable singularity of the spatial normal derivative at the boundary $x \\in \\partial \\Omega$ arises\n$\\partial_n F_\\iota(t,x,v) \\sim \\frac{1 }{ n(x) \\cdot v  }  \\notin L^1_{loc}.$\nSuch singularity towards the grazing set $\\gamma_0 : = \\{(x,v) \\in \\partial\\Omega \\times \\mathbb{R}^3: n(x) \\cdot v=0\\}$ has been studied thoroughly in \\cite{GKTT1} for the Boltzmann equation in convex domain. Here we clarify that a $C^{\\alpha}$ domain means that for any ${p} \\in \\partial{\\Omega}$, there exists sufficiently small $\\delta_{1}>0, \\delta_{2}>0$, and an one-to-one and onto $C^{\\alpha}$-map, $\n\t\\eta_p: \\{( x_{\\|,1}, x_{\\|,2} , x_n ) \\in \\mathbb R^3 : x_n > 0 \\} \\cap B(0; \\delta_1 ) \\to \\Omega \\cap B(p;\\delta_2 )$ with $\\eta_p ( x_{\\|,1}, x_{\\|,2} , x_n ) = \\eta_p( x_{\\|,1}, x_{\\|,2} , 0 ) + x_n [-n (\\eta_p( x_{\\|,1}, x_{\\|,2} , 0))],$\n\tsuch that $\\eta_p (\\cdot, \\cdot, 0) \\in \\partial\\Omega$. \n\tA \\textit{convex} domain means that there exists $C_\\Omega>0$ such that for all $p \\in \\partial\\Omega$ and  $\\eta_p$ and for all $x_\\parallel$,\n\\begin{equation}\\label{convexity_eta}\n\\begin{split}\n\\sum_{i,j=1}^{2} \\zeta_{i} \\zeta_{j}\\partial_{i} \\partial_{j} \\eta _{{p}}   ( x_{\\parallel }  )\\cdot \nn ( x_{\\parallel } )\n  \\leq    - C_{\\Omega} |\\zeta|^{2}  \\ \n  \\text{ for all}   \\ \\zeta \\in \\mathbb{R}^{2}.\n\\end{split}\n\\end{equation}\n\n\n\n\nConstruction of a unique global solution and proving its asymptotic stability of VPB in general domains has been a challenging open problem for any boundary condition. In \\cite{VPB} the authors give the \\textit{first} construction of a unique global \\textit{strong} solution of the one-species VPB system with the diffuse boundary condition when the domain is $C^3$ and \\textit{convex.} Moreover an asymptotic stability of the global Maxwellian $\\mu$ is studied. The result was then extended to the two-species case in \\cite{2SVPB}.\n\n\n\n\n\n\n\n\\subsection{Global strong solution of VPB}\nIn \\cite{VPB, {2SVPB}}, the authors take the first step toward comprehensive understanding of VPB in bounded domains. They consider the zero Neumann boundary condition for the potential $\\phi$: $n \\cdot E \\vert_{\\partial \\Omega }= \\frac{ \\partial \\phi }{\\partial n } \\vert_{\\partial \\Omega } = 0$, which corresponds to a so-called insulator boundary condition. In such setting $(F_\\iota, E) = (\\mu, 0)$ is a stationary solution. \n\n\n\nThe characteristics (trajectory) is determined by the Hamilton ODEs for $f_+$ and $f_-$ separately\n\\begin{equation} \\label{hamilton_ODE1}\n\\frac{d}{ds} \\left[ \\begin{matrix}X_\\iota^f(s;t,x,v)\\\\ V_\\iota^f(s;t,x,v)\\end{matrix} \\right] = \\left[ \\begin{matrix}V_\\iota^f(s;t,x,v)\\\\ \n{-\\iota} \\nabla_x \\phi_f\n(s, X_\\iota^f(s;t,x,v))\\end{matrix} \\right]  \\ \\ \\text{for}   - \\infty< s ,  t < \\infty  ,\n\\end{equation}\nwith $(X_\\iota^f(t;t,x,v), V_\\iota^f(t;t,x,v)) =  (x,v)$. Where the potential is extended to negative time as $\\phi_f(t,x)= e^{-|t|} \\phi_{f_0}(x)$ for $t\\leq 0$.\n\n\nFor $(t,x,v) \\in \\mathbb{R}  \\times  \\Omega \\times \\mathbb{R}^3$, define \\textit{the backward exit time} $t_{\\mathbf{b},\\iota}^f(t,x,v)$ as   \n\\begin{equation} \\label{tb}\nt_{\\mathbf{b},\\iota}^f (t,x,v) := \\sup \\{s \\geq 0 : X_\\iota^f(\\tau;t,x,v) \\in \\Omega \\ \\ \\text{for all } \\tau \\in (t-s,t) \\}.\n\\end{equation}\nFurthermore, define $x_{\\mathbf{b},\\iota}^f (t,x,v) := X_\\iota^f(t-t_{\\mathbf{b},\\iota}(t,x,v);t,x,v)$ and $v_{\\mathbf{b},\\iota}^f (t,x,v) := V_\\iota^f(t-t_{\\mathbf{b},\\iota}(t,x,v);t,x,v)$.\n\n\n\n\n\n\nIn order to handle the boundary singularity, they introduce the following notion\n\\begin{definition}[Kinetic Weight] \\label{kweight} For $\\e>0$\n\\begin{equation} \\label{alphaweight}\\begin{split}\n\\alpha_{f,\\e,\\iota}(t,x,v) : =& \\ \n\\chi \\Big(\\frac{t-t_{\\mathbf{b},\\iota}^{f}(t,x,v)+\\e}{\\e}\\Big)\n|n(x_{\\mathbf{b},\\iota}^{f}(t,x,v)) \\cdot v_{\\mathbf{b},\\iota}^{f}(t,x,v)| \\\\\n&+ \\Big[1- \\chi \\Big(\\frac{t-t_{\\mathbf{b},\\iota}^{f}(t,x,v) +\\e}{\\e}\\Big)\\Big].\n\\end{split}\\end{equation}\nHere they use a smooth function $\\chi: \\mathbb{R} \\rightarrow [0,1]$ satisfying\n\\begin{equation} \\label{chi}\n\\begin{split}\n\\chi(\\tau)  =0,  \\     \\tau\\leq 0, \\ \\text{and} \\  \\ \n\\chi(\\tau)  = 1    ,  \\  \\tau\\geq 1. \n \\ \\ \\frac{d}{d\\tau}\\chi(\\tau)  \\in [0,4] \\ \\   \\text{for all }   \\tau \\in \\mathbb{R}.\n\\end{split}\n\\end{equation}\n\\end{definition}\nAlso, denote\n\\begin{equation}  \\label{matrixalpha}\n\\alpha_{f,\\e}(t,x,v) := \\begin{bmatrix} \\alpha_{f, \\e, +}(t,x,v) & 0 \\\\ 0 & \\alpha_{f, \\e, -}(t,x,v) \\end{bmatrix}.\n\\end{equation}\n\nNote that $\\alpha_{f,\\e,\\iota}(0,x,v)\\equiv \\alpha_{{f_0},\\e,\\iota}(0,x,v)$ is determined by $f_0$. For the sake of simplicity,  the superscription $^f$ in $X_\\iota^f, V_\\iota^f, t_{\\mathbf{b},\\iota}^f, x_{\\mathbf{b},\\iota}^f, v_{\\mathbf{b},\\iota}^f$ is dropped unless they could cause any confusion.\n\n\nOne of the crucial properties of the kinetic weight in (\\ref{alphaweight}) is an invariance under the Vlasov operator: $\n\\big[\\partial_t + v\\cdot \\nabla_x - \\nabla_x \\phi_f \\cdot \\nabla_v \\big] \\alpha_{f,\\e,\\iota}(t,x,v) =0.$ This is due to the fact that the characteristics solves a deterministic system (\\ref{hamilton_ODE1}). This crucial invariant property under the Vlasov operator is one of the key points in their approach in \\cite{VPB, 2SVPB}. \n\n\nDenote $\n\\label{weight}\nw_\\vartheta(v) =  e^{\\vartheta|v|^2}.$\n\n\\begin{theorem}[\\cite{VPB, 2SVPB}] \n\\label{main_existence}\nAssume a bounded open $C^3$ domain $\\Omega \\subset\\mathbb{R}^3$ is convex (\\ref{convexity_eta}). Let $0< \\tilde{\\vartheta}< \\vartheta\\ll1$. Assume the compatibility condition: (\\ref{diffuse_BC}) holds at $t=0$. \nThere exists a small constant $0< \\e_0 \\ll 1$ such that for all $0< \\e \\leq \\e_0$ if an initial datum $F_{0,\\iota} =\\mu + \\sqrt \\mu f_{0,\\iota}$ satisfies\n\\begin{equation} \\label{small_initial_stronger}\n \\|w_\\vartheta f_{0,\\iota} \\|_{L^\\infty(\\bar{\\Omega} \\times \\mathbb{R}^3)}< \\e,  \\|   w_{\\tilde{\\vartheta}}   \\nabla_{v } f_{0,\\iota} \\|_{ {L}^{3 } ( {\\Omega} \\times \\mathbb{R}^3)}< \\infty,\n \\end{equation}\n\\begin{equation} \\label{W1p_initial}\n\\begin{split}\n \\| w_{\\tilde{\\vartheta}} \\alpha_{f_{0,\\iota}, \\e }^\\beta \\nabla_{x,v } f_{0,\\iota} \\|_{ {L}^{p } ( {\\Omega} \\times \\mathbb{R}^3)}\n <\\e\n\\ \\\n\\text{for}  \\  \\ 3< p < 6, \\ \\ \n1-\\frac{2}{p }\n < \\beta<\n\\frac{2}{3}\n \n,\\end{split}\n\\end{equation}\nthen there exists a unique global-in-time solution $F_\\iota(t)= \\mu+ \\sqrt{\\mu} f_\\iota(t) \\geq 0$ to (\\ref{2FVPB}), (\\ref{Field}), (\\ref{Poisson2}), \\eqref{diffuse_BC}. Moreover there exists $\\lambda_{\\infty} > 0$ such that \n\\begin{equation} \\begin{split}\\label{main_Linfty}\n \\sup_{ t \\geq0}e^{\\lambda_{\\infty} t} \\| w_\\vartheta f_\\iota(t)\\|_{L^\\infty(\\bar{\\Omega} \\times \\mathbb{R}^3)}+ \n \\sup_{ t \\geq0}e^{\\lambda_{\\infty} t} \\| \\phi_f(t)  \\|_{C^{2}(\\Omega)}  \\lesssim 1,\n\n\\end{split}\\end{equation}\nand, for some $C>0$, and, for $0< \\delta= \\delta(p,\\beta) $,\n\\begin{equation} \\label{W1p_main}\n \\| w_{\\tilde{\\vartheta}} \\alpha_{f, \\e ,\\iota }^{\\beta } \\nabla_{x,v} f_\\iota(t)  \\|_{L^{ p} ( {\\Omega} \\times \\mathbb{R}^3)} \n \\lesssim e^{Ct} \\ \\ \\text{for all } t \\geq 0\n,\n\\end{equation}\n\n\\begin{equation} \\label{nabla_v f_31}\n\\| \\nabla_v f_\\iota (t) \\|_{L^3_x (\\Omega) L^{1+\\delta }_v (\\mathbb{R}^3)} \\lesssim_t 1  \\ \\ \\text{for all } \\  t\\geq 0.\n\\end{equation}\n \n\n\nFurthermore, if $F_\\iota$ nad $G_\\iota$ are both solutions to (\\ref{2FVPB}), (\\ref{Field}), (\\ref{Poisson2}), \\eqref{diffuse_BC},\nthen \n\\begin{equation} \\label{stability_1+}\n\\| f_\\iota(t) - g_\\iota(t) \\|_{L^{1+\\delta} (\\Omega \\times \\mathbb{R}^3)} \\lesssim_t \\| f_\\iota(0) - g_\\iota(0) \\|_{L^{1+\\delta} (\\Omega \\times \\mathbb{R}^3)} \\ \\ \\text{for all } \\  t\\geq 0.\n\\end{equation} \n\n\\end{theorem}\n\\begin{remark}The second author and his collaborators constructs a local-in-time solution for given general large datum in \\cite{CKL} for the generalized diffuse reflection boundary condition. By introducing a scattering kernel $R(u \\rightarrow v;x,t)$, representing the probability of a molecule striking in the boundary at $x\\in\\partial\\Omega$ with velocity $u$ to be bounced back to the domain with velocity $v$, they consider  \\begin{equation}\\begin{split}\\label{eqn:BC}\n&F(t,x,v) |n(x) \\cdot v|= \\int_{\\gamma_+(x)}\nR(u \\rightarrow v;x,t) F(t,x,u)\n\\{n(x) \\cdot u\\} d u, \\quad \\text{ on }\\gamma_-\n.\n\\end{split}\n\\end{equation}\n In \\cite{CKL} they study a model proposed by Cercignani and Lampis in~\\cite{CIP,CL}. With two accommodation coefficients $  0<r_\\perp\\leq 1,\\quad 0<r_\\parallel<2 ,$\nthe Cercignani-Lampis boundary condition (C-L boundary condition) can be written as\n\\begin{equation}\\label{eqn: Formula for R}\\begin{split}\n &R(u \\rightarrow v;x,t)\\\\\n:=&  \\frac{1}{r_\\perp r_\\parallel (2- r_\\parallel)\\pi\/2} \\frac{|n(x) \\cdot v|}{(2T_w(x))^2}\nI_0 \\left(\n \\frac{1}{2T_w(x)}\\frac{2 (1-r_\\perp)^{1\/2} v_\\perp u_\\perp}{r_\\perp}\n\\right)\n\\\\\n& \\times \n\\exp\\left(- \\frac{1}{2T_w(x)}\\left[\n\\frac{|v_\\perp|^2 + (1- r_\\perp) |u_\\perp|^2}{r_\\perp}\n+ \\frac{|v_\\parallel - (1- r_\\parallel ) u_\\parallel|^2}{r_\\parallel (2- r_\\parallel)}\n\\right]\\right).\n\\end{split}\n\\end{equation}\nHere $T_w(x)$ is a wall temperature on the boundary and $I_0 (y) := \\pi^{-1} \\int^{\\pi}_0e^{y \\cos \\phi } d \\phi$.\nIn this formula, $v_\\perp$ and $v_\\parallel$ denote the normal and tangential components of the velocity respectively: $   v_\\perp= v\\cdot n(x) ,  v_\\parallel = v- v_\\perp n(x)\\,$.\n\\end{remark}\n\nIn \\cite{VPB, 2SVPB} a global $L^\\infty$-bound is proven by $L^2-L^\\infty$ framework. The idea is to use Duhamel's principle to estimate the solution $f$ along the characteristics \\eqref{hamilton_ODE1} to reach\n\\begin{equation}  \\begin{split}\\notag\n   \\| f_\\iota(t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)}\n \\sim  \\| e^{-  t} f_{0,\\iota} \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} + \\int_0^t e^{- (t-s ) } \\| f_\\iota(s) \\|_{L^2(\\Omega \\times \\mathbb R^3) } ds.\n\\end{split} \\end{equation}\nAnd then use the decay of $f_\\iota$ in $L^2$ norm to conclude the decay in $L^\\infty$.\nThe key of this process is to verify\n\\begin{equation} \\begin{split}\n\\frac{ \\partial X_\\iota(s;t,x,v) }{ \\partial v } &\\sim -(t-s)\\text{Id}_{3 \\times 3 } + \\int_s^t \\int_{s'}^t \\nabla_x^2 \\phi(s'') \\frac{ \\partial X_\\iota (s'';t,x,v) }{\\partial v } ds' ds'' \\\\\n&\\sim O(|t-s|) \\text{Id}_{3\\times3}.\n\\end{split}\\end{equation}\nFor which the $C^2$-bound of $\\phi$ seems necessary. Unfortunately such $C^2$ estimate for $\\phi$ falls short of the boarder line case of the Schauder elliptic regularity theory when the source term of the Poisson equation $\\int_{\\mathbb R^3 } (F_+ - F_- )  dv $ in \\eqref{Poisson2} is merely continuous or bounded. They overcome such difficulty by interpolating the $C^2$ norm into a sum of a $C^{2,0+}$ norm and a $C^{1,1-}$ norm:\n\\begin{lemma}\\label{lemma_interpolation}Assume $\\Omega \\subset \\mathbb{R}^3$ with a $C^2$ boundary $\\partial\\Omega$. For $0< D_1<1$, $0< D_2<1$, and $\\Lambda_0>0$,\n\t\t\\begin{equation} \\begin{split}\\label{phi_interpolation}\n\t\t\n\t\t\t\\|\\nabla^2_x \\phi(t )\\|_{L^\\infty (\\Omega)\n\t\t\n\t\t\t\\lesssim_{\\Omega, D_1, D_2}\n\t\t\n\t\t\te^{D_1 \\Lambda_0t}\\|  \\phi(t)\\|_{C^{1,1-D_1}(\\Omega)}\n\t\t\t+ e^{- D_2 \\Lambda_0t}\\|  \\phi(t)\\|_{C^{2, D_2}(\\Omega)}.\n\t\t\\end{split}\\end{equation}\n\t\\end{lemma} \n While an exponential decay of the weaker $C^{1,1-}$ norm can be derived from the exponential decay of $f_\\iota$ in $L^\\infty$, the $C^{2,0+}$ norm is controlled by Morrey's inequality\n\\begin{equation}  \\label{phic2pbd}\n\\| \\phi \\|_{C_x^{2,0 +}} \\lesssim \\sum_{\\iota = \\pm} \\| \\int_{\\mathbb R^3 } f\\iota \\sqrt \\mu dv \\|_{C_x^{0,0+} } \\lesssim  \\sum_{\\iota = \\pm} \\| \\int_{\\mathbb R^3} \\nabla_x f_\\iota \\sqrt \\mu dv \\|_{L^p_x} , \\text{ for } p>3.\n\\end{equation}\n\nNow the spatial derivative of $f_\\iota$ needs to be controlled.\nThey develop an $ \\alpha_\\iota$-weighted $W^{1,p}$ estimate by energy-type estimate of $ \\alpha_\\iota \\nabla_{x,v} f_\\iota$, where the $ \\alpha_\\iota$-multiplication compensates the boundary singularity. This allows us to bound \\eqref{phic2pbd} for $\\frac{p-2}{p} < \\beta < \\frac{p-1}{p}$,\n\\[\n \\| \\int_{\\mathbb R^3} \\nabla_x f_\\iota \\sqrt \\mu dv \\|_{L^p_x} \\lesssim \\|  \\alpha_\\iota^{-\\beta} \\|_{L^{\\frac{p}{p-1}}} \\|  \\alpha_\\iota^\\beta \\nabla_x f_\\iota \\sqrt \\mu \\|_{L^p_{x,v} } \\lesssim \\|  \\alpha_\\iota^\\beta \\nabla_x f_\\iota \\sqrt \\mu \\|_{L^p_{x,v} },\n\\]\nas long as\n\\begin{equation}  \\label{int1overalpha}\n \\alpha_\\iota^{-\\frac{\\beta p}{p-1} } \\sim \\frac{1}{ \\alpha_\\iota(t,x,v)^{1-}} \\in L^1_v \\text{ uniformly for all } x.\n\\end{equation}\n\nA difficulty of the proof of \\eqref{int1overalpha} arises form lack of local representation of $ \\alpha_\\iota (t, x, v)$. $\\alpha_\\iota$ is only defined at some boundary point along (possibly very complicated) characteristics. They employ a geometric change of variables $v \\mapsto (x_{\\mathbf{b},\\iota}(t,x,v),t_{\\mathbf{b},\\iota}(t,x,v) )$ to exam \\eqref{int1overalpha}. By computing the Jacobian there is an extra $ \\alpha$-factor from $dv \\sim \\frac{  \\alpha_\\iota}{|t_{\\mathbf{b},\\iota}|^3 } dt_{\\mathbf{b},\\iota} d x_{\\mathbf{b},\\iota}$, which cancels the singularity of \\eqref{int1overalpha}. Then they use a lower bound of $t_{\\mathbf{b},\\iota} \\gtrsim \\frac{|x_{\\mathbf{b},\\iota}^f-x|}{\\max |V|}$ and a bound $ \\alpha \\lesssim \\frac{|(x-x_{\\mathbf{b},\\iota}^f) \\cdot n(x_{\\mathbf{b},\\iota}^f)|}{t_{\\mathbf{b},\\iota}^f}$ to have\n\\begin{equation} \\label{alpha_bounded_intro}\n\\int_{|v| \\lesssim 1} { \\alpha_\\iota}^{-  \\frac{\\beta p}{p-1}} \\mathrm{d} v \\lesssim \\int_{\\text{boundary}} \\frac{|(x- x_{\\mathbf{b},\\iota}) \\cdot n(x_{\\mathbf{b},\\iota})|^{1-  \\frac{\\beta p}{p-1}}}{|x-x_{\\mathbf{b},\\iota}|^{3-  \\frac{\\beta p}{p-1}}} \\mathrm{d} x_{\\mathbf{b},\\iota}\n+ \\text{good terms}< \\infty, \n\\end{equation}\nwhich turns to be bounded as long as $ \\frac{\\beta p}{p-1}<1$. \n\n\n\nFrom the above estimates and the interpolation, they derive \\textit{an exponential decay} of $\\phi(t)$ in $C_x^2$ as long as $\\|  \\alpha_\\iota^\\beta \\nabla_x f(t) \\|_{L_{x,v}^p} $ grows at most exponentially. With the $C_x^2$-bound of $\\phi$ in hand, they control $\\|  \\alpha_\\iota^\\beta \\nabla_x f(t) \\|_{L_{x,v}^p} $via Gronwall's inequality and close the estimate by proving its (at most) exponential growth.\n\n\nFor the uniqueness and stability of approximating sequence they prove $L^1$-stability. The key observation is that $v$-derivatives of the diffuse BC \\eqref{diffuse_BC} has no boundary singularity, thus is bounded. The equation of $\\nabla_v f_\\iota$ has a singular forcing term $\\nabla_x f_\\iota$.\nFor which they control $\\| \\nabla_x f_\\iota \\|_{L_x^3 L_v^1 } $ as $\\|  \\alpha_\\iota^{-\\beta} \\|_{L_v^{\\frac{p}{p-1}} } \\|  \\alpha_\\iota^\\beta \\nabla_x f_\\iota\\|_{L^p_{x,v} } $, and this term is bounded from \\eqref{int1overalpha}.\n\n\\hide\nFor the two-species VPB system in \\cite{2SVPB}, when performing the expansion for $\\begin{bmatrix} F_+ \\\\ F_- \\end{bmatrix} = \\begin{bmatrix} \\mu + \\sqrt \\mu f_+ \\\\ \\mu + \\sqrt \\mu f_- \\end{bmatrix}$, the vector linearized Boltzmann operator $L$ is defined as\n\\begin{equation}  \\label{L_decomposition}\nL \\begin{bmatrix} g_1 \\\\ g_2 \\end{bmatrix} :=  -\\frac{1}{\\sqrt \\mu } \\begin{bmatrix}  2 Q(\\sqrt \\mu g_1, \\mu ) + Q (\\mu, \\sqrt \\mu ( g_1 + g_2 ) )\n\\\\   2 Q(\\sqrt \\mu g_2, \\mu ) + Q (\\mu, \\sqrt \\mu ( g_1 + g_2 ) )\n \\end{bmatrix}.\n \\end{equation}\nThe null space of $L$ is a six-dimensional subspace of $L^2_v(\\mathbb R^3; \\mathbb R^2 )$ spanned by orthonormal vectors\n\\begin{equation}  \n\\left\\{ \\begin{bmatrix} \\sqrt \\mu \\\\ 0  \\end{bmatrix}, \\begin{bmatrix} 0  \\\\ \\sqrt \\mu \\end{bmatrix}, \\begin{bmatrix} \\frac{v_i}{\\sqrt 2 } \\sqrt \\mu \\\\ \\frac{v_i}{\\sqrt 2 } \\sqrt \\mu   \\end{bmatrix}, \\begin{bmatrix} \\frac{|v|^2 - 3}{2\\sqrt 2} \\sqrt \\mu \\\\ \\frac{|v|^2 - 3}{2\\sqrt 2} \\sqrt \\mu   \\end{bmatrix}\n \\right\\}, \\, i = 1,2,3.\n\\end{equation}\nAnd the projection of $\\mathbf f = \\begin{bmatrix} f_+ \\\\ f_- \\end{bmatrix}$ onto the null space $N(L)$ can be denoted by\n\\begin{equation}  \\begin{split}\n& \\mathbf P \\mathbf f (t,x,v)\n\\\\ & := \\left\\{ a_+(t,x) \\begin{bmatrix} \\sqrt \\mu \\\\ 0  \\end{bmatrix} + a_-(t,x) \\begin{bmatrix} 0  \\\\ \\sqrt \\mu \\end{bmatrix} + b(t,x)  \\cdot \\frac{v}{\\sqrt 2 } \\begin{bmatrix} \\sqrt \\mu \\\\ \\sqrt \\mu  \\end{bmatrix} + c(t,x)  \\frac{|v|^2 - 3}{2\\sqrt 2}\\begin{bmatrix} \\sqrt \\mu \\\\ \\sqrt \\mu  \\end{bmatrix}\n\\right\\}.\n\\end{split} \\end{equation}\nUsing the standard $L^2$ energy estimate of the equation, it is well-known that $L$ is degenerate: $\\left \\langle L \\mathbf f , \\mathbf f \\right \\rangle  \\gtrsim \\left \\|  (I - \\mathbf P) \\mathbf f \\right \\|_{L^2_{x,v }}$. Thus it's clear that in order to control the $L^2$ norm of $f_\\iota(t)$, a way to bound the missing $ \\left \\| \\mathbf P\\mathbf f (t) \\right \\|_{L^2} $ term is needed. Fortunately, the technique of test function method developed in \\cite{EGKM} can be applied to the two-species settings. By the weak formulation of the equation and a set of properly choosing test functions, the $a_\\pm(t,x), b(t,x), c(t,x)$ can be controlled. And the estimate of $\\mathbf P\\mathbf f $  is achieved as\n\\[\n\\left \\| \\mathbf P\\mathbf f (s) \\right \\|_{L_{x,v}^2 } \\lesssim \\left \\| ( I - \\mathbf P ) \\mathbf f \\right \\|_{L_{x,v}^2 } + \\text{ \"good terms\"}.\n\\]\n\\unhide\n\n\n\n\\subsection{Improved regularity under the sign condition}\nOne interesting question is to improve the regularity estimate beyond a weighted $W^{1,p}$ for $p< 6$ of $f_\\iota$ in \\cite{VPB, 2SVPB}. Some work in this direction has been done in \\cite{VPBEP}.\n\nIn \\cite{VPBEP} the author consider the one-species VPB system \\eqref{Field}, \\eqref{Boltzmann_E}, where the potential consists of a self-generated electrostatic potential and an external potential. That is $E = \\nabla \\phi$, where\n\\begin{equation} \\label{VPB2}\n\\phi(t,x) = \\phi_F(t,x) + \\phi_E(t,x), \\text{ with } \\frac{\\partial \\phi_E}{\\partial n } > C_E > 0 \\text{ on } \\partial \\Omega,\n\\end{equation}\nand $\\phi_F$ satisfies \\eqref{Poisson} and the zero Neumann boundary condition $\\frac{ \\partial \\phi_F}{\\partial n } = 0$ on $\\partial \\Omega$. Under such setting, the field $E$ satifies a crucial sign condition on the boundary\n\\begin{equation}  \\label{signEonbdry}\nE(t,x) \\cdot n(x) > C_E > 0 \\text{ for all } t \\text{ and all } x \\in \\partial \\Omega.\n\\end{equation}\n\n\nWith the help of the external potential $\\phi_E$ with the crucial sign condition \\eqref{VPB2}, they construct a short time weighted $W^{1,\\infty}$ solution to the VPB system, which improves the regularity estimate of such system in Theorem \\ref{main_existence}. The key idea of the result is to incorporate a different distance function $\\tilde \\alpha$:\n\\begin{equation}  \\label{alphatilde}\n\\tilde \\alpha \\sim \\bigg[ |v \\cdot \\nabla \\xi (x)| ^2 + \\xi (x)^2 - 2 (v \\cdot \\nabla^2 \\xi(x) \\cdot v ) \\xi(x) - 2(E(t,\\overline x ) \\cdot \\nabla \\xi (\\overline x ) )\\xi(x) \\bigg]^{1\/2},\n\\end{equation}\nwhere $\\xi:\\mathbb R^3 \\to \\mathbb R$ is a smooth function such that $ \\Omega = \\{ x \\in \\mathbb R^3: \\xi(x) < 0 \\}$, and the closest boundary point $\\overline x := \\{ \\bar x \\in \\partial \\Omega :  d(x,\\bar x ) = d(x, \\partial \\Omega) \\}$ is uniquely defined for $x$ closed to the boundary. Note that $\\tilde \\alpha \\vert_{\\gamma_-} \\sim | n(x) \\cdot v |$. A version of a distance function without the potential was used in \\cite{GKTT1}. One of the key contribution in \\cite{VPBEP} is to incorporate this different distance function \\eqref{alphatilde} in the presence of an external field.\n\n\\begin{theorem}[\\cite{VPBEP}]\\label{WlinftyVPBthm} Let $\\phi_E (t,x)$ be a given external potential with $\\nabla_x \\phi_E$ satisfying (\\ref{signEonbdry}), and\n$ \\| \\nabla_x \\phi_E(t,x) \\|_{C^1_{t,x}(\\mathbb{R}_+ \\times\\bar \\Omega )}\n < \\infty.$\n Assume that, for some $0< \\vartheta < \\frac{1}{4}$,\n$\\|  w_\\vartheta \\tilde{\\alpha} \\nabla_{x,v} f_0 \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} + \\| w_\\vartheta f_0 \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)}\n< \\infty.$\nThen there exists a unique solution $F(t,x,v) = \\sqrt \\mu f(t,x,v) $ to (\\ref{Boltzmann_E}), \\eqref{Field},  (\\ref{diffuse_BC}), (\\ref{VPB2}) for $t \\in [0,T]$ with $0 < T \\ll 1$, such that for some $0< \\vartheta' < \\vartheta $, $\\varpi \\gg 1$,\n$\\sup_{0\\le t \\le T} \\| w_{\\vartheta'} f(t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} < \\infty,$ and\n\\begin{equation\n\\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'} e^{-\\varpi  \\langle v \\rangle t } \\tilde{\\alpha}  \\nabla_{x,v}  f^{} (t,x,v) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} < \\infty. \n\\end{equation}\n\\end{theorem}\n\n\nOne of the crucial property $\\tilde \\alpha$ enjoys, under the assumption of the sign condition \\eqref{signEonbdry}, is the invariance along the characteristics:\n\n\\begin{lemma}[Velocity lemma near boundary] \\label{velocitylemma} \nSuppose $E(t,x)$ satisfies the sign condition (\\ref{signEonbdry}).\nThen for any $0 \\le s<t $ and trajectory $X(\\tau), V(\\tau)$ solving (\\ref{hamilton_ODE1}), if $X(\\tau ) \\in \\Omega $ for all $s \\le \\tau \\le t $, then\n\\begin{equation}\n\\begin{split}  \\label{vlemma}\ne^{ - C \\int_s ^ t ( |V(\\tau')| + 1 ) d \\tau'} \\tilde \\alpha ( s,X(s),V(s) ) &\\le \\tilde \\alpha (t,X(t),V(t)) \\\\&\\le  e^{C \\int_s ^ t ( |V(\\tau')| + 1 ) d \\tau'} \\tilde \\alpha (s,X(s),V(s)),\n\\end{split}\\end{equation}\nfor any $C \\gtrsim  (   \\| \\nabla_x \\phi_E(t,x) \\|_{C^1_{t,x}(\\mathbb{R}_+ \\times\\bar \\Omega )} + 1  )\/{C_E} $.\n\\end{lemma}\n\n \n\nThe key ingredient in the $\\tilde \\alpha$-weighted regularity estimate is a dynamical non-local to local estimate which can be stated as\n\\begin{lemma}\\label{keylemma}\nLet $(t,x,v) \\in [0,T] \\times  \\Omega \\times \\mathbb R^3$, $ 1 < \\beta < 3$, $0 < \\kappa \\le 1 $. Suppose $E$ satisfies the sign condition (\\ref{signEonbdry}). Then for $\\varpi \\gg 1$ large enough, and for any $0< C_\\vartheta < \\frac{1}{4}$, $0 < \\delta \\ll 1$,\n\\begin{equation} \\label{nonlocaltolocal}\n \\begin{split}\n & \\int_{\\max\\{ 0, t - t_{\\mathbf{b}} \\}} ^t  \\int_{\\mathbb R^3} e^{ -\\int_s^t\\frac{\\varpi}{2} \\langle V(\\tau;t,x,v) \\rangle d\\tau }  \\frac{e^{-\\frac{C_\\vartheta}{2} |V(s)-u|^2 }}{|V(s) -u |^{2 - \\kappa}  }  \\frac{1}{(\\tilde \\alpha(s,X(s) ,u))^\\beta} du  ds\n\\\\  \\lesssim &   e^{ 2C_\\Omega \\frac{\\| \\nabla E\\|_\\infty  + \\| E \\|_{L^\\infty_{t,x}}^2 + \\| E \\|_{L^\\infty_{t,x}}}{C_E}}  \\frac{  \\delta ^{\\frac{3 -\\beta}{2} }}{\\langle v\\rangle^2(C_E+1)^ {\\frac{\\beta-1}{2}} (\\tilde \\alpha(t,x,v))^{\\beta -2 }  (   \\| E \\|_{L^\\infty_{t,x}}^2 + 1 )^{\\frac{3 -\\beta}{2}}}  \n\\\\ & +  \\frac{ ( \\| E \\|_{L^\\infty_{t,x}}^2+1)^{\\beta -1} } { C_E^{\\beta-1} \\delta ^{\\beta -1 } (\\tilde \\alpha ( t,x,v) )^{\\beta - 1 } } \\frac{2}{ \\varpi },\n\\end{split} \n\\end{equation}\nwhere $(X(s), V(s)) = (X(s;t,x,v), V(s;t,x,v)) $ as in \\eqref{hamilton_ODE1}. \\end{lemma}\nThe same estimate without the external field had been established by the second author and his collaborators in \\cite{GKTT1}. The proof of \\eqref{nonlocaltolocal} is obtained by first making use of a series of change of variables to get the precise estimate of the velocity integration, which is bounded by,\n\\begin{equation}  \\label{kernelbddfor1overalpha}\n\\int_{\\mathbb R^3} \\frac{ e^{-\\vartheta |V(s)- u|^2}}{ |V(s) - u | ^{2 - \\kappa} [\\tilde \\alpha(s, X(s), u ) ]^\\beta } du \\lesssim \\frac{ 1}{( |V(s)|^2 \\xi(X(s)) - C_E  \\xi (X(s)) )^{\\frac{\\beta - 1}{2} } } , \n\\end{equation}\nthen followed by relating the time integration back to ${\\tilde \\alpha}^{-1}$. For the later part of the proof, the velocity lemma \\eqref{vlemma} and the boundedness of the external field to ensure the monotonicity of $|\\xi(X(s))|$ near the boundary, where the change of variable\n$\ndt \\simeq \\frac{d\\xi}{|v\\cdot \\nabla \\xi |},$\ncan be performed and recovers a power of $\\tilde \\alpha$ in the $\\xi$-integration. On the other hand, the sign condition (\\ref{VPB2}) is crucially used to establish a lower bound for $|\\xi(X(s))|$ when it's away from the boundary, which helps to recover a power of $\\tilde \\alpha$ as wanted.\n\n\n\n\\hide\n\\subsection{Generalized Diffuse Boundary Condition}\nIn \\cite{CKL} the authors study the one-species VPB system with generalized diffuse boundary condition.\nThe diffuse boundary condition \\eqref{diffuse_BC} can be generalized by introducing a scattering kernel $R(u \\rightarrow v;x,t)$ through a general balance law of\n\\begin{equation}\\begin{split}\\label{eqn:BC}\n&F(t,x,v) |n(x) \\cdot v|= \\int_{\\gamma_+(x)}\nR(u \\rightarrow v;x,t) F(t,x,u)\n\\{n(x) \\cdot u\\} d u, \\quad \\text{ on }\\gamma_-\n.\n\\end{split}\n\\end{equation}\nPhysically, $R(u\\to v;x,t)$ represents the probability of a molecule striking in the boundary at $x\\in\\partial\\Omega$ with velocity $u$ to be bounced back to the domain with velocity $v$ at the same location $x$ and time $t$. In \\cite{CKL} they study a model proposed by Cercignani and Lampis in~\\cite{CIP,CL}. With two accommodation coefficients $  0<r_\\perp\\leq 1,\\quad 0<r_\\parallel<2 ,$\nthe Cercignani-Lampis boundary condition (C-L boundary condition) can be written as\n\\begin{equation}\\label{eqn: Formula for R}\\begin{split}\n&R(u \\rightarrow v;x,t)\\\\\n:=& \\frac{1}{r_\\perp r_\\parallel (2- r_\\parallel)\\pi\/2} \\frac{|n(x) \\cdot v|}{(2T_w(x))^2}\n\\exp\\left(- \\frac{1}{2T_w(x)}\\left[\n\\frac{|v_\\perp|^2 + (1- r_\\perp) |u_\\perp|^2}{r_\\perp}\n+ \\frac{|v_\\parallel - (1- r_\\parallel ) u_\\parallel|^2}{r_\\parallel (2- r_\\parallel)}\n\\right]\\right)\\\\\n& \\times  I_0 \\left(\n \\frac{1}{2T_w(x)}\\frac{2 (1-r_\\perp)^{1\/2} v_\\perp u_\\perp}{r_\\perp}\n\\right).\n\\end{split}\n\\end{equation}\nHere $T_w(x)$ is a wall temperature on the boundary and $I_0 (y) := \\pi^{-1} \\int^{\\pi}_0e^{y \\cos \\phi } d \\phi$.\nIn this formula, $v_\\perp$ and $v_\\parallel$ denote the normal and tangential components of the velocity respectively: $   v_\\perp= v\\cdot n(x) ,\\quad v_\\parallel = v- v_\\perp n(x)\\,$.\nSimilarly $u_\\perp= u\\cdot n(x)$ and $u_\\parallel = u- u_\\perp n(x)$.\n\nThis model can be considered as a generalization of fundamental boundary conditions. For instance by setting $r_\\perp=1$ and $r_\\parallel=1$, the scattering kernel equals $R(u\\to v;x,t)=\\frac{2}{\\pi (2T_w(x))^2}e^{-\\frac{|v|^2}{2T_w(x)}} |n(x)\\cdot v|.$\nThis corresponds to the diffuse boundary condition \\eqref{diffuse_BC}. With $r_\\perp=0,r_\\parallel=0$, the scattering kernel is given by $R(u\\to v;x,t)=\\delta(u-\\mathfrak{R}_xv)$,\n  with $\\mathfrak{R}_xv=v-2n(x)(n(x)\\cdot v)$. This corresponds the specular reflection boundary condition $F(t,x,v)=F(t,x,\\mathfrak{R}_xv)$. With $r_\\perp=0,r_\\parallel=2$, the scattering kernel is given by $R(u\\to v;x,t)=\\delta(u+v)$, which corresponds the bounce-back reflection reflection boundary condition $F(t,x,v)=F(t,x,-v)$.\n   \nIt is important to note that the C-L boundary condition satisfies the reciprocity property\n\\begin{equation} \\label{eqn: reciprocity}\n    R(u\\to v;x,t)=R(-v\\to -u;x,t) \\frac{e^{-|v|^2\/(2T_w(x))}}{e^{-|u|^2\/(2T_w(x))}}\\frac{|n(x)\\cdot v|}{|n(x)\\cdot u|}\\,,\n\\end{equation}\nand the normalization property $\\int_{\\gamma_-(x)} R(u\\to v;x,t) dv=1$.\n\n\n\n\n\n\n\n\n\n\nDefine the global Maxwellian using the maximum wall temperature as\n\\begin{equation}\\label{eqn: def for weight}\n\\mu:=e^{-\\frac{|v|^2}{2T_M}}\\,,\\ \\text{ with }T_M:=\\max_{x\\in \\partial \\Omega}\\{T_w(x)\\}. \\  \\text{ And set } F = \\sqrt \\mu f.\n\\end{equation}\nThe main result in \\cite{CKL} is:\n\\begin{theorem*}\\label{local_existence}\nAssume $\\Omega \\subset \\mathbb{R}^3$ is open bounded, and convex $C^3$ domain. A wall temperature $T_w(x)>0$ is defined on $x\\in \\partial \\Omega$ and smooth. Assume that two accommodation coefficients satisfy\n\n\\begin{equation}\\label{eqn: Constrain on T}\n\\frac{\\min_{x\\in \\partial \\Omega}\\{T_w(x)\\}}{\\max_{x\\in \\partial \\Omega}\\{T_w(x)\\}}>\\max\\Big(\\frac{1-r_\\parallel}{2-r_\\parallel},\\frac{\\sqrt{1-r_\\perp}-(1-r_\\perp)}{r_\\perp}\\Big)\\,.\n\\end{equation}\nLet $0< \\tilde{\\theta}< \\theta <\\frac{1}{4\\max_{x\\in \\partial \\Omega}\\{T_w(x)\\}}.$\nAssume\n\\begin{equation}\n\\| w_\\theta f_0 \\|_\\infty < \\infty, \\, \\label{eqn: w f_0} \\| w_{\\tilde{\\theta}} \\nabla_v f_0 \\|_{L^{3}_{x,v}}<\\infty,\n\\end{equation}\n\\begin{equation}\n\\| w_{\\tilde{\\theta}} \\alpha_{f_0, \\epsilon }^\\beta \\nabla_{x,v } f_0 \\|_{ {L}^{p } ( {\\Omega} \\times \\mathbb{R}^3)} <\\infty\\quad \\text{for} \\quad 3< p < 6\\,,\\, 1-\\frac{2}{p }< \\beta< \\frac{2}{3}.\\label{eqn: alpha f0}\n\\end{equation}\nThen there is a unique solution $F(t,x,v) = \\sqrt \\mu f(t,x,v)$ to~\\eqref{Boltzmann_E}, \\eqref{eqn:BC}, \\eqref{eqn: Formula for R} in a time interval of $t \\in [0,\\bar{t}]$.\nMoreover, there are $\\mathfrak{C}>0$ and $\\lambda>0$, so that $f$ satisfies\n\\begin{equation}\\label{infty_local_bound}\n\\sup_{0 \\leq t \\leq \\bar{t}}\\| w_{\\theta}e^{-\\mathfrak{C} \\langle v\\rangle^2 t} f  (t) \\|_{\\infty}\\lesssim \\| w_\\theta f_0 \\|_\\infty  , \\ \\ \\sup_{0 \\leq t \\leq \\bar{t}}\\| \\nabla_v f (t) \\|_{L^3_xL^{1+ \\delta}_v}< \\infty,\n\\end{equation}\n\\begin{equation}\\label{W1p_local_bound}\n\\sup_{0 \\leq t \\leq \\bar{t}}\\Big\\{ \\| w_{\\tilde{\\theta}}e^{-\\lambda t\\langle v\\rangle }\\alpha_{f,\\epsilon }^\\beta \\nabla_{x,v} f (t) \\|_{p} ^p+ \\int^t_0  |w_{\\tilde{\\theta}} e^{-\\lambda s\\langle v\\rangle } \\alpha_{f,\\epsilon}^\\beta \\nabla_{x,v} f (t) |_{p,+}^p\\Big\\}< \\infty .\n\\end{equation}\n\\end{theorem*}\n\n\n\\unhide\n \n\n\\section{On the Vlasov-Poisson-Boltzmann system surrounded by Conductor boundary}\n\nIn the second part of the paper, we consider the one-species VPB system surrounded by conductor boundary. More specifically, we consider the system \\eqref{Boltzmann_E}, \\eqref{Field}, where the electrostatic potential $\\phi$ is obtained by\n\\begin{equation}  \\label{phiC}\n-\\Delta_x \\phi(t,x) = \\int_{\\mathbb R^3} F(t,x,v) dv, \\, x \\in \\Omega, \\ \\  \\ \n\\phi = 0, \\, x \\in \\partial \\Omega.\n\\end{equation}\nAn important benefit in the conductor boundary setting \\eqref{phiC} is that $E = - \\nabla_x \\phi$ enjoys the sign condition \\eqref{signEonbdry} from a quantitative Hopf lemma, without the need of an external potential.\n\\begin{lemma}[Lemma $3.2$ in \\cite{BC}]\\label{Hopf}\nSuppose $ h \\ge 0 $, and $h \\in L^\\infty(\\Omega)$. Let $v$ be the solution of\n\\begin{equation}   \\begin{split}\n- \\Delta v = h   \\text{ in } \\Omega,\n \\ \\ v = 0   \\text{ on } \\partial \\Omega.\n\\end{split} \\end{equation}\nThen for any $ x \\in \\partial \\Omega$,\n\\begin{equation}  \\label{hopfquan}\n\\frac{ \\partial v(x) }{ \\partial n} \\ge c \\int_{\\Omega} h(x)  d (x, \\partial \\Omega)dx,\n\\end{equation}\nfor some $c > 0$ depending only on $\\Omega$. Here $d (x, \\partial \\Omega)$ is the distance from $x$ to the boundary $\\partial \\Omega$.\n\\end{lemma}\n\\hide\n\\begin{proof}\nSee . \n\n\n\n\\end{proof}\\unhide\n\nOur goal is to prove a local existence and regularity theorem for the system \\eqref{Boltzmann_E}, \\eqref{Field}, \\eqref{diffuse_BC}, \\eqref{phiC}. Let's first define our distance function $\\tilde \\alpha$.\n\n\nLet $d(x,\\partial \\Omega) := \\inf_{y \\in \\partial \\Omega} \\| x - y \\| $. For any $\\delta > 0$, let $ \\Omega ^\\delta : = \\{ x \\in \\Omega : d(x, \\partial \\Omega ) < \\delta \\}$. For $\\delta \\ll 1$ is small enough, we have for any $x \\in \\Omega ^\\delta$ there exists a unique $\\bar x \\in \\partial \\Omega$ such that $d(x,\\bar x ) = d(x, \\partial \\Omega)$ (cf. (2.44) in \\cite{VPBEP}). \n\n\n\\begin{definition}\nFirst we define for all $(x, v ) \\in  \\Omega ^{\\delta} \\times \\mathbb R^3 $,\n\\[\n\\beta(t,x,v) = \\bigg[ |v \\cdot \\nabla \\xi (x)| ^2 + \\xi (x)^2 - 2 (v \\cdot \\nabla^2 \\xi(x) \\cdot v ) \\xi(x) +2(\\nabla \\phi(t,\\overline x ) \\cdot \\nabla \\xi (\\overline x ) )\\xi(x) \\bigg]^{1\/2}.\n\\]\nFor any $\\epsilon >0$, let $\\chi_\\epsilon: [0,\\infty) \\to [0,\\infty) $ be a smooth function satisfying $  \\chi_\\epsilon(x) =x$ for $0 \\le x \\le \\frac{\\epsilon}{4}$, $\\chi_\\epsilon(x) = C_\\epsilon$ for  $x \\ge \\frac{\\epsilon}{2}$,  $\\chi_\\epsilon(x)$ is increasing for $ \\frac{\\epsilon}{4} < x < \\frac{\\epsilon}{2}$, and $ \\chi_\\epsilon'(x) \\le 1$. Let $\\delta': = \\min \\{| \\xi (x)| : x\\in \\Omega, d(x, \\partial \\Omega) = \\delta \\} $, then we define our weight function to be:\n\n\\begin{equation} \\label{alphadef}\n\\tilde \\alpha(t,x,v) : = \n\\begin{cases}\n (\\chi_{\\delta'}  ( \\beta(t,x,v) ) )^{} & x \\in  \\Omega^\\delta, \\\\\n C_{\\delta'} ^{} & x \\in \\Omega \\setminus  \\Omega^\\delta.\n\\end{cases} \\end{equation}\n\\end{definition}\n\n\n\n\\begin{theorem}[Weighted $W^{1,\\infty}$ estimate for the VPB surrounded by conductor]  \\label{WlinftyVPBthm} \nAssume $F_0 = \\sqrt \\mu f_0$ satisfies\n\\begin{equation}  \\label{VPBf0assumption}\n\\|  w_\\vartheta \\tilde \\alpha \\nabla_{x,v} f_0 \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} + \\| w_\\vartheta f_0 \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} + \\| w_\\vartheta \\nabla_v f_0 \\|_{L^3(\\bar \\Omega \\times \\mathbb R^3)} < \\infty,\n\\end{equation}\nfor some $0< \\vartheta < \\frac{1}{4}$.Then there exists a unique solution $F(t,x,v) = \\sqrt \\mu f(t,x,v) $ to (\\ref{Boltzmann_E}), \\eqref{Field}, \\eqref{diffuse_BC}, (\\ref{phiC}) for $t \\in [0,T]$ with $0 < T \\ll 1$, such that for some $0< \\vartheta' < \\vartheta $, $\\varpi \\gg 1$,\n\\begin{equation}  \\label{linfinitybddsolution}\n\\sup_{0\\le t \\le T} \\| w_{\\vartheta'} f(t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} < \\infty,\n\\end{equation} \n\\begin{equation}\\label{weightedC1bddsolution}\n\\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'} e^{-\\varpi  \\langle v \\rangle t } \\tilde \\alpha  \\nabla_{x,v}  f^{} (t,x,v) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} < \\infty, \n\\end{equation}\n\\begin{equation} \\label{L3L1plusbddsolution}\n \\sup_{0 \\le t \\le T}\\| e^{-\\varpi \\langle v \\rangle t } \\nabla_v f(t) \\|_{L^3_x(\\Omega)L_v^{1+\\delta}(\\mathbb R^3 ) } < \\infty \\text{ for } 0< \\delta \\ll 1.\n\\end{equation}\n\\end{theorem}\n\n\n\n\nThe corresponding equation for $f = \\frac{F}{\\sqrt \\mu } $ is\n\\begin{equation} \\label{VPBsq1}\n(\\partial_t + v \\cdot \\nabla_x - \\nabla \\phi \\cdot \\nabla_v  + \\frac{v}{2} \\cdot \\nabla \\phi + \\nu( \\sqrt \\mu f) ) f^{} = \\Gamma_{\\text{gain}} (f,f),\n\\end{equation}\n\\begin{equation} \\label{VPBsq3}\n-\\Delta_x \\phi(t,x) = \\int_{\\mathbb R^3 } \\sqrt \\mu f dv, \\,\\,   \\phi = 0 \\text{  on } \\partial \\Omega,\n\\end{equation}\n\\begin{equation}  \\label{fbdry}\nf^{}(t,x,v) = c_\\mu \\sqrt{\\mu(v)} \\int_{n\\cdot u >0 } f(t,x,v) \\sqrt{\\mu(u)} (n(x) \\cdot u ) du.\n\\end{equation}\nHere $\n \\nu (\\sqrt \\mu f )(v) : = \\int_{\\mathbb R^3 } \\int_{\\mathbb S^2} |v - u|^{\\kappa} q_0 (\\frac{ v -u}{ |v -u| } \\cdot w ) \\sqrt{\\mu(u) } f(u) d\\omega du$, and $\n \\Gamma_{\\text{gain}} (f_1,f_2) (v): = \\int_{\\mathbb R^3 } \\int_{\\mathbb S^2} |v - u|^{\\kappa} q_0 (\\frac{ v -u}{ |v -u| } \\cdot w ) \\sqrt{\\mu(u) } f_1(u')f_2(v') d\\omega du$.\n\n\n\nLet $\\partial \\in \\{ \\nabla_x, \\nabla_v \\}$. Let $E = - \\nabla_x \\phi $. Denote \n\\begin{equation}  \\label{nuvarpi}\n\\nu_{\\varpi} = \\nu(\\sqrt \\mu f )+ \\frac{v}{2} \\cdot E  + \\varpi \\langle v \\rangle  + t \\varpi \\frac{v}{\\langle v \\rangle }\\cdot E    -  {\\tilde \\alpha}^{-1} ( \\partial_t {\\tilde \\alpha} + v \\cdot \\nabla_x {\\tilde \\alpha} + E \\cdot \\nabla_v {\\tilde \\alpha} ).\n\\end{equation}\nThen by direct computation we get \n \\begin{equation} \\label{seqforc1fixedpotential} \\begin{split}\n \\bigg \\{ & \\partial_t + v\\cdot \\nabla_x  + E \\cdot \\nabla_v + \\nu_{\\varpi} \\bigg\\} ( e^{-\\varpi \\langle v \\rangle t } {\\tilde \\alpha} \\partial f^{})\n\\\\  = & e^{-\\varpi \\langle v \\rangle t }  {\\tilde \\alpha} \\left(  \\partial \\Gamma_{gain} (f,f) - \\partial v \\cdot \\nabla_x f^{} - \\partial E \\cdot \\nabla_v f^{} - \\partial ( \\frac{v}{2} \\cdot E ) f^{} - \\partial (\\nu (\\sqrt \\mu f ) ) f^{}  \\right)\n\\\\  := & \\mathcal{N}(t,x,v).\n\\end{split} \\end{equation}\n\n\n\nIn order to deal with the diffuse boundary condition \\eqref{diffuse_BC}, we define the stochastic (diffuse) cycles as $(t^0,x^0,v^0) = (t,x,v)$,\n\\begin{equation} \\label{diffusecycles} \\begin{split}\n& t^1 = t - t_{\\mathbf{b}}(t,x,v), \\, x^1 = x_{\\mathbf{b}}(t,x,v) = X(t - t_{\\mathbf{b}}(t,x,v);t,x,v), \n\\\\ & v_b^0 = V(t - t_{\\mathbf{b}}(t,x,v);t,x,v) = v_{\\mathbf{b}}(t,x,v),\n\\end{split} \\end{equation}\n and $v^1 \\in \\mathbb R^3$ with $n(x^1) \\cdot v^1 > 0$. \nFor $l\\ge1$, define\n\\[ \\begin{split}\n& t^{l+1} = t^l - t_{\\mathbf{b}}(t^l,x^l,v^l), x^{l + 1 } = x_{\\mathbf{b}}(t^l,x^l,v^l), \n\\\\ & v_b^l = v_{\\mathbf{b}}(t^l,x^l,v^l), \n\\end{split} \\]\nand $v^{l+1} \\in \\mathbb R^3 \\text{ with } n(x^{l+1}) \\cdot v^{l+1} > 0$.\nAlso, define \n\\[\nX^l(s) = X(s;t^l,x^l,v^l), \\, V^l(s) = V(s;t^l,x^l,v^l),\n\\]\n so $X(s) = X^0(s), V(s) = V^0(s)$. We have the following lemma.\n\n\n\n\\begin{lemma}[Lemma $12$ in \\cite{VPBEP}] \\label{expandtrajl}\n\n\n\nIf $t^1 < 0$, then\n\\begin{equation}  \\label{C1trajectoryt1less0}\n\\begin{split}\n&  e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha | \\partial  f (t,x,v) |\n\\\\ &  \\lesssim \\tilde \\alpha(0,X^0(0), V^0(0) ) \\partial f (0, X^0(0) , V^0(0) ) + \\int_0^t \\mathcal N(s,X^0(s), V^0(s) ) ds.\n\\end{split} \\end{equation}\n\n\nIf $t^1 > 0$, then\n\\begin{equation} \\label{C1trajectory}\n\\begin{split}\n& e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha | \\partial  f (t,x,v) |\n\\\\ \\lesssim & e^{-\\frac{\\vartheta}{2}  |v_{\\mathbf{b}}^0| ^2} P(\\| w_\\vartheta f_0 \\|_\\infty) +  \\int_{t^1}^t \\mathcal N(s, X^0(s), V^0(s) ) ds\n\\\\ & +  \\sqrt {\\mu (v_{\\mathbf{b}}^0) } \\langle v_{\\mathbf{b}}^0 \\rangle^2 \\int_{\\prod_{j=1}^{l-1} \\mathcal V_j} \\sum_{i=1}^{l-1} \\textbf{1}_{\\{t^{i+1} < 0 < t^i \\}}  |\\tilde \\alpha  \\partial f  (0,X^{i}(0), V^{i}(0)) |    \\, d \\Sigma_{i}^{l-1}\n\\\\ & +  \\sqrt {\\mu (v_{\\mathbf{b}}^0) } \\langle v_{\\mathbf{b}}^0 \\rangle^2 \\int_{\\prod_{j=1}^{l-1} \\mathcal V_j} \\sum_{i=1}^{l-1} \\textbf{1}_{\\{t^{i+1} < 0 < t^i \\}}  \\int_0^{t^i} \\mathcal N(s, X^i(s), V^i(s) ) ds  \\, d \\Sigma_{i}^{l-1}\n\\\\ & +  \\sqrt {\\mu (v_{\\mathbf{b}}^0) } \\langle v_{\\mathbf{b}}^0 \\rangle^2 \\int_{\\prod_{j=1}^{l-1} \\mathcal V_j} \\sum_{i=1}^{l-1} \\textbf{1}_{\\{t^{i+1} > 0 \\}}  \\int_{t^{i+1}}^{t^i} \\mathcal N(s, X^i(s), V^i(s) ) ds  \\, d \\Sigma_{i}^{l-1}\n\\\\ & +  \\sqrt {\\mu (v_{\\mathbf{b}}^0) } \\langle v_{\\mathbf{b}}^0 \\rangle^2 \\int_{\\prod_{j=1}^{l-1} \\mathcal V_j} \\sum_{i=2}^{l-1} \\textbf{1}_{\\{t^{i} > 0 \\}}   e^{-\\frac{\\vartheta}{2}  |v_{\\mathbf{b}}^{i-1}| ^2} P(\\| w_\\vartheta f_0 \\|_\\infty)  \\, d \\Sigma_{i-1}^{l-1}\n\\\\ & +  \\sqrt {\\mu (v_{\\mathbf{b}}^0) } \\langle v_{\\mathbf{b}}^0 \\rangle^2  \\int_{\\prod_{j=1}^{l-1} \\mathcal V_j}  \\textbf{1}_{\\{t^{l} > 0 \\}}  e^{-\\varpi \\langle v_{\\mathbf{b}}^{l-1} \\rangle t^l } \\tilde \\alpha (t^l,x^l, v_{\\mathbf{b}}^{l-1}) | \\partial  f (t^l,x^l,v_{\\mathbf{b}}^{l-1}) | d \\Sigma_{l-1}^{l-1},\n\\end{split} \\end{equation}\nwhere $\\mathcal V_j = \\{ v^j \\in \\mathbb R^3: n(x^j ) \\cdot v^j > 0 \\}$,\nand \n\\[ \\begin{split}\nd \\Sigma_i^{l-1} = & \\{\\prod_{j=i+1}^{l-1}  \\mu(v^j) c_\\mu |n(x^j ) \\cdot v^j | dv^j \\} \n\\{ e^{\\varpi \\langle v^i \\rangle t^i } \\mu^{1\/4}(v^i) \\langle v^i \\rangle d v^i \\}\n\\\\ & \\{\\prod_{j=1}^{i-1} \\sqrt{\\mu(v_{\\mathbf{b}}^j ) } \\langle v_{\\mathbf{b}}^j \\rangle \\mu^{1\/4}(v^j ) \\langle v^j \\rangle e^{\\varpi \\langle v^j \\rangle t^j } d v^j\\},\n\\end{split} \\]\nwhere $c_\\mu$ is the constant that $ \\int_{\\mathbb R^3 }  \\mu(v^j) c_\\mu |n(x^j ) \\cdot v^j | dv^j = 1$.\n\\end{lemma} \n\nThe following lemma is necessary for us to establish Theorem \\ref{WlinftyVPBthm}.\n\n\\begin{lemma}\nIf $(F, \\phi)$ solves (\\ref{phiC}), write $f = \\frac{F}{\\sqrt \\mu}$, then\n\\begin{equation} \\label{linfinitybdofpotential}\n\\| \\phi_F(t) \\|_{C^{1,1-\\delta} (\\Omega) } \\lesssim_{\\delta,\\Omega} \\| w_\\vartheta f(t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)}, \\text{ for any } 0< \\delta <1,\n\\end{equation}\nand\n\\begin{equation} \\label{c2bdofpotential}\n\\| \\nabla ^2 \\phi_F(t) \\|_{L^\\infty(\\Omega)} \\lesssim \\| w_\\vartheta f(t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)} + \\| e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_{x} f (t) \\|_{L^\\infty(\\bar \\Omega \\times \\mathbb R^3)}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}\nIt is obvious to have (\\ref{linfinitybdofpotential}) from the Morrey inequality and elliptic estimate.\nNext we show (\\ref{c2bdofpotential}).\nBy Schauder estimate, we have, for $p >3$ and $\\Omega \\subset \\mathbb R^3$,\n\\[\n\\| \\nabla^2 \\phi_F(t) \\|_{L^\\infty(\\Omega)} \\le \\| \\phi_F \\|_{C^{2, 1 - \\frac{3}{p}}(\\Omega)} \\lesssim_{p,\\Omega} \\| \\int_{\\mathbb R^3} f (t) \\sqrt \\mu dv \\|_{C^{0, 1 - \\frac{3}{p}}(\\Omega)}.\n\\]\nThen by Morrey inequality, $W^{1,p} \\subset C^{0, 1 -\\frac{3}{p}} $ with $p >3$ for a domain $\\Omega \\subset \\mathbb R^3$ with a smooth boundary $\\partial \\Omega$, we derive\n\\[ \\begin{split}\n \\| & \\int_{\\mathbb R^3} f (t) \\sqrt \\mu dv \\|_{C^{0, 1 - \\frac{3}{p}}}   \\lesssim  \\| \\int_{\\mathbb R^3} f (t) \\sqrt \\mu dv \\|_{W^{1,p} }\n \\\\ \n  & \\lesssim  \\| w_\\vartheta f(t) \\|_\\infty +  \\| e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_{x} f (t) \\|_\\infty \\| \\int_{\\mathbb R^3 } e^{ \\varpi \\langle v \\rangle t } \\sqrt \\mu \\frac{1}{\\tilde \\alpha} dv \\|_{L^p(\\Omega ) }.\n\\end{split} \\]\n\n\nIt suffices to show that for some $\\beta > 1$, \n\\begin{equation}  \\label{int1alphaLpbdd}\n\\| \\int_{\\mathbb R^3 } e^{-\\frac{1}{8} |v|^2 }  \\frac{1}{\\tilde \\alpha^\\beta} dv \\|_{L^p(\\Omega ) } < \\infty.\n\\end{equation}\n Since $\\tilde \\alpha$ is bounded from below when $x$ is away from the boundary of $\\Omega$, it suffices to only consider the case when $x$ is close enough to $\\partial \\Omega$. From direct computation (see \\cite{VPBEP}), we get\n\\begin{equation} \\label{int1overalphadv}\n \\int_{\\mathbb R^3 } e^{-\\frac{1}{8} |v|^2 }  \\frac{1}{\\tilde \\alpha^\\beta} dv \\lesssim \\frac{1}{(\\xi(x)^2 - 2 E(t,\\bar x ) \\cdot \\nabla \\xi(\\bar x) \\xi(x) )^{\\frac{\\beta -1 }{2}}} \\lesssim \\frac{1}{ |\\xi(x) | ^{\\frac{\\beta -1 }{2}}}.\n\\end{equation}\nAnd since $\\xi$ is $C^2$, we have \n\\[\n\\int_{d(x,\\partial \\Omega) \\ll 1 } \\frac{1}{ |\\xi(x) | ^{\\frac{(\\beta -1 )p}{2}}} dx \\lesssim \\int_{d(x,\\partial \\Omega) \\ll 1 } \\frac{1}{ |x - \\bar x | ^{\\frac{(\\beta -1 )p}{2}}} dx.\n\\]\nNow from (\\ref{convexity_eta}), \n\\[\n\\int_{\\Omega \\cap B(p;\\delta_2 ) }  \\frac{1}{|x - \\bar x  | ^{\\frac{(\\beta -1 )p}{2}}} dx \\lesssim \\int_{ |x_n | < \\delta_1 } \\frac{1}{|x_n|^{\\frac{(\\beta -1 )p}{2}}} d_{x_n} < \\infty,\n\\]\nif we pick $\\beta < \\frac{2}{p}+1 $. And since $\\partial \\Omega$ is compact, we can cover $\\partial \\Omega$ with finitely many such balls, and therefore we get (\\ref{int1alphaLpbdd}).\n\n\n\\end{proof}\n\n\n\\begin{proof} [Proof of Theorem \\ref{WlinftyVPBthm}]\nFor the sake of simplicity we only show the a priori estimate. See \\cite{CKL} for the construction of the sequences of solutions and passing a limit.\n\n\n\n\nThe proof of \\eqref{linfinitybddsolution} for $f$ satisfying \\eqref{VPBsq1}, \\eqref{VPBsq3}, and \\eqref{fbdry} is standard. We refer to Theorem $4$ in \\cite{VPBEP}.\n\n\n\nFirst from \\eqref{VPBsq3} and the fact that $ \\int_{\\mathbb R^3 } \\sqrt \\mu f dv \\ge 0$, we apply Lemma \\ref{Hopf} to get\n\\begin{equation}  \\label{hopfm}\n - \\frac{  \\partial \\phi (t,x)}{\\partial n} \\ge c \\iint_{\\Omega \\times \\mathbb R^3} \\sqrt \\mu  f(t,x,v)  \\delta(x) dv dx,\n\\end{equation}\nfor some $c$ depending only on $\\Omega$. \n\nDenote\n\\[\n \\iint_{\\Omega \\times \\mathbb R^3}  F_0(x,v)  \\delta(x) dv dx = c_{E_0}.\n\\]\nThen $\\int_0^T \\iint_{\\Omega \\times \\mathbb R^3 }\\delta(x) \\times \\eqref{Boltzmann_E} \\ dv dx dt$ gives\n\\[ \\begin{split}\n&  \\iint_{\\Omega \\times \\mathbb R^3}F(T,x,v)  \\delta(x) dv dx  \n \\\\ &  =   \\iint_{\\Omega \\times \\mathbb R^3}  F_0(x,v)  \\delta(x) dv dx + \\int_0^T \\iint_{\\Omega \\times \\mathbb R^3 } F v \\cdot \\nabla_x \\delta(x)  dv dx dt.\n\\end{split} \\]\nTogether with \\eqref{linfinitybddsolution} and \\eqref{hopfm} we deduce\n\\begin{equation}  \\label{signphim}\n - \\frac{  \\partial \\phi (t,x)}{\\partial n} \\ge c \\iint_{\\Omega \\times \\mathbb R^3}F(t,x,v)  \\delta(x) dv dx >  \\frac{c c_{E_0}}{2},\n\\end{equation}\nas long as $ T \\lesssim \\frac{ c_{E_0}}{2 M}$.\n\n\n\n\n\nNext, we investigate \\eqref{seqforc1fixedpotential}. Since\n\\[ \\begin{split}\n w_{\\vartheta } \\Gamma_{gain}(\\partial f,f )  \n \\lesssim \\| e^{2\\vartheta' |v|^2 } f \\|_\\infty \\int_{\\mathbb R^3 } \\frac{ e^{-C_{\\vartheta'} |u -v | ^2 } } {|u -v |^{2 -\\kappa} }  |e^{\\vartheta ' |u|^2 } \\partial f(t,x,u ) | du,\n\\end{split} \\]\nand\n\\[ \\begin{split}\nw_{\\vartheta } \\nu(\\sqrt{\\mu} \\partial f ) f \n\\lesssim  \\| e^{2 \\vartheta' |v|^2 } f \\|_\\infty  \\int_{\\mathbb R^3 } \\frac{ e^{-C_{\\vartheta'} |u -v | ^2 } } {|u -v |^{2 -\\kappa} } | \\partial f(t,x,u ) | du.\n\\end{split} \\]\nThus from (\\ref{linfinitybddsolution}) we have the following bound for $\\mathcal N$:\n\n\\begin{equation} \\label{Nbdd}\\begin{split}\n|  \\mathcal   N(t,x,v) | \n  \\lesssim &   (1 + \\| \\nabla^2 \\phi \\|_\\infty)  [ P(\\| w_\\vartheta f_0 \\|_\\infty )  + |w_{\\vartheta'}  e^{-\\varpi \\langle v \\rangle t  }  \\tilde \\alpha  \\partial f(t,x,v) | ]\n\\\\ & +  \\| w_\\vartheta f_0 \\|_\\infty  e^{-\\varpi \\langle v \\rangle t }  \\tilde \\alpha(t,x,v) \\int_{\\mathbb R^3 }  \\frac{ e^{-C_\\vartheta |u -v | ^2 } } {|u -v |^{2 -\\kappa} } | e^{\\vartheta ' |u|^2 }\\partial f(t,x,u ) | du.\n\\end{split} \\end{equation}\n\nRecall the definition of $\\nu_\\varpi$ in \\eqref{nuvarpi}, note that from the velocity lemma (\\ref{vlemma}), and (\\ref{signphim}) we have\n\\[ \\begin{split}\n& \\tilde \\alpha^{-1}  ( \\partial_t \\tilde \\alpha + v \\cdot \\nabla_x \\tilde \\alpha -\\nabla \\phi \\cdot \\nabla_v \\tilde \\alpha ) \n \\\\ \\lesssim & ( \\| \\nabla \\phi \\|_\\infty + \\| \\nabla^2 \\phi \\|_\\infty ) \\langle v \\rangle\n \\\\ \\lesssim  & (\\| w_{\\vartheta'} f(t) \\|_\\infty + \\| e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_{x} f (t) \\|_\\infty)\\langle v \\rangle\n\\\\  \\lesssim & (P(\\| w_{\\vartheta} f_0\\|_\\infty ) + \\| \\tilde \\alpha \\partial f_0 \\|_\\infty)\\langle v \\rangle.\n   \\end{split} \\]\nTherefore we have\n\\begin{equation}  \\label{largeomegabar}\n\\nu_\\varpi \\ge  \\frac{\\varpi}{2} \\langle v \\rangle,\n\\end{equation}\nonce we choose $\\varpi \\gg 1$ large enough.\n   \nFor $t^1 < 0$, using the Duhamel's formulation we have from (\\ref{seqforc1fixedpotential})\n\n\\begin{equation}  \\label{}\n\\begin{split}\n w_{\\vartheta'} & e ^{-\\varpi \\langle v \\rangle t  }  \\tilde \\alpha | \\partial  f (t,x,v) |\n\\\\   \\le &  e^{ -\\int_s^t \\nu_{\\varpi} (\\tau, X(\\tau), V(\\tau) d\\tau}  e^{\\vartheta' |V(0)|^2 } \\tilde \\alpha \\partial f (0, X(0) , V(0) ) \n\\\\ & + \\int_0^t e^{ -\\int_s^t \\nu_{\\varpi} (\\tau, X(\\tau), V(\\tau) d\\tau } \\mathcal N(s,X(s), V(s) ) ds.\n\\end{split} \n\\end{equation}\nThus by (\\ref{Nbdd}) we have\n\\[ \\begin{split}\n \\sup_{0 \\le t \\le T} & \\| \\textbf{1}_{ \\{ t^1 < 0 \\}} e^{-\\varpi \\langle v \\rangle t } w_{\\vartheta'} \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty\n\\\\ \\le & \\sup_{0 \\le t \\le T}  \\| e^{ -\\int_0^t \\nu_{\\varpi} (\\tau, X(\\tau), V(\\tau) d\\tau} e^{\\vartheta' |V(0)|^2 } \\tilde \\alpha \\partial f (0, X(0) , V(0) ) \n\\\\ & + \\int_0^t e^{ -\\int_s^t \\nu_{\\varpi} (\\tau, X(\\tau), V(\\tau) d\\tau } \\mathcal N(s,X(s), V(s) ) ds \\|_\\infty\n \\\\ \\le & \\| w_{\\vartheta'} \\tilde \\alpha \\partial f_0 \\|_\\infty + P(\\| w_{\\vartheta} f_0 \\|_\\infty)   \\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t  } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty\n \\\\ & +T   (1 + \\| \\nabla^2 \\phi \\|_\\infty) [ P(\\| w_{\\vartheta} f_0 \\|_\\infty) +   \\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'}  e^{-\\varpi \\langle v \\rangle t  } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty] \n \\\\ & \\times \\int_0^t \\int_{\\mathbb R^3} e^{ -\\int_s^t\\frac{\\varpi}{2} \\langle V(\\tau;t,x,v) \\rangle d\\tau }  \\frac{e^{-\\varpi  \\langle (s;t,x,v) \\rangle  s }}{e^{-\\varpi  \\langle u \\rangle s}} \\frac{e^{-C_\\vartheta |V(s)-u|^2 }}{|V(s) -u |^{2 - \\kappa}  }  \\frac{\\tilde \\alpha(s,X(s) ,V(s))}{\\tilde \\alpha(s,X(s) ,u)} du  ds.\n\\end{split} \\]\nNow since\n$\n \\langle u \\rangle -  \\langle V(s;t,x,v) \\rangle \\le 2 \\langle u - V(s;t,x,v) \\rangle,\n$\nwe have\n$\n \\frac{e^{-\\varpi  \\langle (s;t,x,v) \\rangle  s }}{e^{-\\varpi  \\langle u \\rangle s}} e^{-C_\\vartheta |V(s)-u|^2 }\n\\lesssim   e^{ -\\frac{ C_\\vartheta |V(s) - u | ^2}{2}}.\n$\nThus\n\\begin{equation} \\label{genlemma1toget}\n\\begin{split}\n\\int_0^t & \\int_{\\mathbb R^3} e^{ -\\int_s^t\\frac{\\varpi}{2} \\langle V(\\tau;t,x,v) \\rangle d\\tau }  \\frac{e^{-\\varpi  \\langle (s;t,x,v) \\rangle  s }}{e^{-\\varpi  \\langle u \\rangle s}}  \\frac{e^{-C_\\vartheta |V(s)-u|^2 }}{|V(s) -u |^{2 - \\kappa}  }  \\frac{\\tilde \\alpha(s,X(s) ,V(s))}{\\tilde \\alpha(s,X(s) ,u)} du  ds\n\\\\ \\lesssim &    \\int_0^t \\int_{\\mathbb R^3} e^{ -\\int_s^t\\frac{\\varpi}{2} \\langle V(\\tau;t,x,v) \\rangle d\\tau }  \\frac{e^{-\\frac{C_\\vartheta}{2} |V(s)-u|^2 }}{|V(s) -u |^{2 - \\kappa}  }  \\frac{\\tilde \\alpha(s,X(s) ,V(s))}{\\tilde \\alpha(s,X(s) ,u)} du  ds.\n\\end{split} \n\\end{equation}\nNote that, for any $\\beta > 1$,\n$\n\\frac{1}{\\tilde \\alpha(x, X(s) ,u ) } \\lesssim \\frac{1}{ (\\tilde \\alpha (x, X(s) , u ) )^\\beta } + 1.\n$\nSo from (\\ref{signphim}) we can let $1 < \\beta \\le 2$, and apply the nonlocal-to-local estimate (\\ref{nonlocaltolocal}) to (\\ref{genlemma1toget}) to have\n\\begin{equation}\n\\begin{split}\n \\int_0^t  & \\int_{\\mathbb R^3} e^{ -\\int_s^t\\frac{\\varpi}{2} \\langle V(\\tau;t,x,v) \\rangle d\\tau }  \\frac{e^{-\\varpi  \\langle (s;t,x,v) \\rangle  s }}{e^{-\\varpi  \\langle u \\rangle s}} \\frac{e^{-C_\\vartheta |V(s)-u|^2 }}{|V(s) -u |^{2 - \\kappa}  }  \\frac{\\tilde \\alpha(s,X(s) ,V(s))}{\\tilde \\alpha(s,X(s) ,u)} du  ds\n\\\\ \\lesssim &  e^{ C ( \\| \\nabla \\phi \\|_\\infty^2 + \\| \\nabla^2 \\phi \\|_\\infty )}   \\left( \\frac{ \\delta^{\\frac{3 - \\beta}{2}} (\\tilde \\alpha(t,x,v) )^{3 - \\beta } }{ (|v|^2 + 1 )^{\\frac{3 -\\beta}{2}} } + \\frac{ (|v| + 1 )^{\\beta - 1} (\\tilde \\alpha(t,x,v) )^{2 - \\beta }}{ \\delta ^{\\beta - 1} \\varpi \\langle v \\rangle  }   \\right)\n\\\\ \\lesssim &  e^{ C ( \\| \\nabla \\phi \\|_\\infty^2 + \\| \\nabla^2 \\phi \\|_\\infty  ) }  \\left( \\delta^{\\frac{3 - \\beta}{2}} + \\frac{1}{\\delta ^{\\beta - 1} \\varpi} \\right),\n\\end{split}\n\\end{equation}\nwhere we used $\\tilde \\alpha(s,X(s) ,V(s)) \\lesssim e^{ C ( \\| \\nabla \\phi \\|_\\infty^2 + \\| \\nabla^2 \\phi \\|_\\infty ) }  \\tilde \\alpha(t,x,v)$.\n\nSimilarly, for $t^1(t,x,v) \\ge 0 $, we again apply the nonlocal-to-local estimate (\\ref{nonlocaltolocal}) to get\n\\[\n\\begin{split}\n| &  \\textbf{1}_{ \\{ t^1 > 0 \\}}  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha  \\partial  f (t,x,v) |\n\\\\ \\lesssim & C_l e^{Cl t^2 } \\left( \\delta^{\\frac{3 - \\beta}{2}} + \\frac{1}{\\delta ^{\\beta - 1} \\varpi}  \\right)  P(\\| w_{\\vartheta} f_0 \\|_\\infty) \\max_{ 0 \\le i \\le l-1 } e^{ C ( \\| \\nabla \\phi \\|_\\infty^2 + \\| \\nabla^2 \\phi \\|_\\infty  )}   \n\\\\ & \\times  \\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty\n\\\\ & +  T(1 + \\|\\nabla^2 \\phi \\|_\\infty) \\sup_{0 \\le t \\le T}  \\| w_{\\vartheta'} e^{-\\varpi  \\langle v \\rangle t } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty \n\\\\  + &  T  l (Ce^{Ct^2 } )^l  (1 + \\|\\nabla^2 \\phi \\|_\\infty)  \\sup_{0 \\le t \\le T} \\| w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty  \n\\\\ + & Tl (Ce^{Ct^2 } )^l    (1 + \\|\\nabla^2 \\phi  \\|_\\infty )P(\\| w_{\\vartheta} f_0 \\|_\\infty) \n+  l (Ce^{Ct^2 } )^l  \\| \\tilde \\alpha \\partial f_0  \\|_\\infty +   P(\\| w_{\\vartheta} f_0 \\|_\\infty) \n\\\\ & + C \\left( \\frac{1}{2} \\right) ^l   \\sup_{0 \\le t \\le T} \\|w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t} \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty.\n\\end{split}\n\\]\n\n\nFinally from \\eqref{c2bdofpotential}, we can choose a large $l$ then large $C$ then small $\\delta$ then large $\\varpi$ and finally small $T$ to conclude \n\\[ \\begin{split}\n \\sup_{0 \\le t \\le T}   \\| e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha  \\partial  f (t,x,v) \\|_\\infty\n\\le  \\frac{C_1}{2} \\left(  \\| w_{\\vartheta } \\tilde \\alpha \\partial f_0 \\|_\\infty + P ( \\| w_\\vartheta f_0 \\|_\\infty ) \\right)\n\\end{split} \\]\nThis proves (\\ref{weightedC1bddsolution}).\n\nNext we prove \\eqref{L3L1plusbddsolution}. Consider taking $\\nabla_v$ derivative of (\\ref{VPBsq1}) and adding the weight function $e^{-\\varpi \\langle v \\rangle t}$, we get\n\\begin{equation} \\begin{split}\\label{eqtn_g_v}\n\t\t\t&[\\partial_t + v\\cdot \\nabla_x - \\nabla_x \\phi \\cdot \\nabla_v + \\frac{v}{2} \\cdot \\nabla_x \\phi + \\varpi \\langle v \\rangle - \\frac{v}{ \\langle v \\rangle } \\varpi t \\cdot \\nabla_x \\phi + \\nu(\\sqrt \\mu f ) ]  (e^{-\\varpi \\langle v \\rangle t} \\nabla_v f )\n\t\t\t\\\\ =& e^{-\\varpi \\langle v \\rangle t} \\left( - \\nabla_v \\nu(\\sqrt \\mu f ) f - \\nabla_x f - \\frac{1}{2} \\nabla_x\\phi f + \\nabla_v \\Gamma_{\\text{gain}}(f,f) \\right),\n\t\t\\end{split}\\end{equation}\nwith the boundary bound for $(x,v) \\in\\gamma_-$\n\t\t\\begin{equation} \\label{bdry_g_v}\n\t\t\\big|\\nabla_v f  \\big| \\lesssim   |v| \\sqrt{\\mu} \\int_{n \\cdot u>0} |f| \\sqrt{\\mu} \\{n \\cdot u \\} \\mathrm{d} u \\ \\ \\text{on } \\ \\gamma_-.\n\t\t\\end{equation}\nAnd \n\\[\n \\frac{v}{2} \\cdot \\nabla_x \\phi + \\varpi \\langle v \\rangle - \\frac{v}{ \\langle v \\rangle } \\varpi t \\cdot \\nabla_x \\phi + \\nu(\\sqrt \\mu f ) > \\frac{\\varpi }{2} \\langle v \\rangle,\n \\]\n for $\\varpi \\gg 1$.\n\n\t\tUsing the Duhamel's formulation, from (\\ref{eqtn_g_v}) we obtain the following bound along the characteristics\n\t\t\\begin{eqnarray}\n\t\t& \\,\\,\\, & |e^{-\\varpi \\langle v \\rangle t }  \\nabla_v  f(t,x,v)|  \\nonumber\n\t\t\\\\\n\t\t& \\le &   \\mathbf{1}_{ \\{ t_{\\mathbf{b}}(t,x,v)> t \\}}  \n\t\t e^{ -\\int_0^t  - \\frac{C}{2}\\langle V(\\tau) \\rangle d\\tau } |\\nabla_v f(0,X(0;t,x,v), V(0;t,x,v))|\\label{g_initial}\\\\\n\t\t& + &   \\ \\mathbf{1}_{ \\{ t_{\\mathbf{b}}(t,x,v)<t \\} \n\t\n\t\t e^{-\\varpi \\langle v_{\\mathbf{b}} \\rangle t_{\\mathbf{b}}} \\mu(v_{\\mathbf{b}})^{\\frac{1}{4}}  \\int_{n(x_{\\mathbf{b}}) \\cdot u>0} \n\t\t| f(t-t_{\\mathbf{b}}, x_{\\mathbf{b}}, u) |\\sqrt{\\mu} \\{n(x_{\\mathbf{b}}) \\cdot u\\} \\mathrm{d} u\\label{g_bdry}\\\\\n\t\t&  +&   \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}} \n\t\t e^{ -\\int_s^t  - \\frac{\\varpi}{2}\\langle V(\\tau) \\rangle d\\tau } e^{-\\varpi \\langle V(s) \\rangle s }  \n\t\t  |\\nabla_x f(s, X(s),V(s))|\n\t\t\\mathrm{d} s\\label{g_x}\\\\\n\t\t&   + &\\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}} \\label{g_Gamma}\n\t\t(1+ \\| w_{\\vartheta'} f \\|_\\infty  \n\t\n\t\n\t\te^{ -\\int_s^t  - \\frac{\\varpi}{2}\\langle V(\\tau) \\rangle d\\tau }  e^{-\\varpi \\langle V(s) \\rangle s } \n\t\t\\\\ \\notag && \\,\\,\\,  \\times \\int_{\\mathbb{R}^3} \\frac{e^{-C_{\\vartheta'} |V(s) - u |^2 }}{ |V(s) - u | ^{2 -\\kappa } } \\nabla_v f(s,X(s),u)| \\mathrm{d} u \n\t\t\\mathrm{d} s\\label{g_K}\\\\\n\t\t& +  & \\label{nablaphigint}\n\t\t\\| w_{\\vartheta'} f\\|_\\infty \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}} \n\t\t  e^{ -\\int_s^t  - \\frac{\\varpi}{2}\\langle V(\\tau) \\rangle d\\tau } e^{-\\varpi \\langle V(s) \\rangle s } e^{-\\vartheta' |V(s) |^2 }\n\t\t  \\\\&& \\notag \\times |\\nabla_x \\phi (s, X(s;t,x,v))| \n\t\t\\mathrm{d} s. \\label{g_phi}\n\t\t\\end{eqnarray}\nWe first have\n\t\t\\begin{equation} \\begin{split}\\label{est_g_initial}\n\t\t\t&\\| (\\ref{g_initial})\\|_{L^3_x L^{1+ \\delta}_v}\\\\\n\t\t\t\\lesssim & \\left(\n\t\t\t\\int_{\\Omega}\n\t\t\t\\left(\\int_{\\mathbb{R}^3} |e^{\\vartheta' |V(0) | ^2 } \\nabla_v f(0,X(0 ), V(0 ))|^3 \n\t\t\t\\right)\n\t\t\t\\left(\n\t\t\t\\int_{\\mathbb{R}^3} e^{-(1+ \\delta) \\frac{3}{2-\\delta} \\vartheta' |V(0) | ^2 } \\mathrm{d} v \\right)^{\\frac{2-\\delta}{1+ \\delta}}\n\t\t\t\\right)^{1\/3} \\\\\n\t\t\t\\lesssim & \\\n\t\t\t\\left(\\iint_{\\Omega \\times \\mathbb{R}^3} |e^{\\vartheta' |V(0) | ^2 } \\nabla_v f(0,X(0;t,x,v), V(0;t,x,v))|^3 \\mathrm{d} v \\mathrm{d} x\\right)^{1\/3}\n\t\t\t\\\\\n\t\t \\lesssim & \\ \\| w_{\\vartheta'} \\nabla_v f (0) \\|_{L^3_{x,v}},\n\t\t\\end{split}\n\t\t\\end{equation}\n\t\twhere we have used a change of variables $(x,v) \\mapsto (X(0;t,x,v), V(0;t,x,v))$.\n\t\t\n\t\t\n\t\t\n\t\t\\hide\n\t\tAlso we use $|V(0;t,x,v)| \\gtrsim |v|$ for $|v|\\gg 1$, from (\\ref{decay_phi}), and hence $\\tilde{w}(V(0;t,x,v))^{- (1+ \\delta) \\frac{3}{2-\\delta}} \\in L_v^1 (\\mathbb{R}^3)$.\\unhide\n\t\t\n\t\tClearly \n\t\t\\begin{equation} \\label{est_g_bdry}\n\t\t\\|(\\ref{g_bdry})\\|_{L^3_x L^{1+ \\delta}_v}  \\lesssim \\sup_{0 \\leq s \\leq t} \\| w_{\\vartheta'} f (s) \\|_\\infty.\n\t\t\\end{equation}\n\t\t\n\t\tFrom $ \\| \\nabla_x \\phi \\|_{L^3} \\lesssim \\| \\phi \\|_{W^{2,2}_x}$ for a bounded $\\Omega \\subset \\mathbb{R}^3$, and the change of variables $(x,v) \\mapsto (X(s;t,x,v), V(s;t,x,v))$ for fixed $s\\in(\\max\\{t-t_{\\mathbf{b}},0\\},t)$,\n\t\t\\begin{equation} \n\t\t\\begin{split}\\label{est_g_phi}\n\t\t\t\\|  (\\ref{nablaphigint})\\|_{L^3_x L^{1+ \\delta}_v} \n\t\t\t \\lesssim  &  \\| w_{\\vartheta'} f\\|_\\infty \\int^t_{\\max\\{t-t_{\\mathbf{b}},0\\}} \\| \\phi (s) \\|_{W^{2,2}_{x} } \n\t\t\t\\\\\n\t\t\t\\lesssim & \\ \\| w_{\\vartheta'} f\\|_\\infty \\int^t_{\\max\\{t-t_{\\mathbf{b}},0\\}} \\| \\int_{\\mathbb R^3} \\sqrt \\mu f(s) dv  \\|_{2}.\n\t\t\t \\lesssim  t  \\| w_{\\vartheta'} f\\|_\\infty \\| w_{\\vartheta'} f \\|_\\infty .\n\t\t\\end{split}\n\t\t\\end{equation}\n\n\t\t\n\n\n\n\n\n\n\t\t\n\t\n\t\tNext we have from (\\ref{int1overalphadv}), for $\\frac{3\\delta}{ 2 (1+\\delta) } < 1$, equivalently $0 < \\delta < 2 $,\n\t\t{\n\t\t\t\t\\begin{equation} \\label{init_p_xf}\n\t\t\\begin{split}\n\t\t\t \\|(\\ref{g_x})\\|_{L^3_x L^{1+ \\delta}_v}  \\le &\\left\\|\\left\\| \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}\n\t\t\t\\nabla_x f(s,X(s),V(s)) \\mathrm{d} s\n\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x}\\\\\n\t\t\t= & \\ \\left\\|\\left\\| \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}}    \\frac{ e^{\\vartheta' |V(s) |^2 } e^{-\\varpi \\langle V(s) \\rangle s} \\tilde \\alpha \\nabla_x f(s,X(s),V(s))}{e^{\\vartheta' |V(s) |^2 } e^{-\\varpi \\langle V(s) \\rangle s} \\tilde \\alpha}\n\t\t\t\\mathrm{d} s\n\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x}\n\t\t\t\\\\\n\t\t\t\\le & \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t} \\tilde \\alpha \\nabla_x f \\right\\|_\\infty \n\t\t\t\\\\\n\t\t\t& \\times \\left\\|\n\t\t\t \\left\\| \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}}  \\frac{e^{- \\vartheta' |V(s) |^2 } e^{\\varpi \\langle V(s) \\rangle s }}{\\tilde \\alpha(s, X(s), V(s) )}\t\t\\mathrm{d} s\n\\right\\|_{L_{v}^{1+\\delta}( \\mathbb{R}^3)}\\right\\|_{L^{3}_x}\\\\\n\t\t\t\\lesssim &   e^{C ( \\| \\nabla \\phi \\|_\\infty + \\| \\nabla \\phi \\|_\\infty^2 +\\| \\nabla ^2 \\phi \\|_\\infty) }\n \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty  \n\t\t\t\t\t\\\\ & \\times  t  \\int_{\\Omega }  \\left( \\int_{\\mathbb R^3 }   \\frac{e^{- \\frac{\\vartheta'}{2} |v |^2 }}{(\\tilde \\alpha (t, x, v ))^{1+\\delta}}  \\mathrm{d} v \\right)^{\\frac{3}{1+\\delta}}  \\mathrm{d} x\t\t\t\n\t\t\\\\ \\lesssim & t e^{C (  \\| \\nabla \\phi \\|_\\infty^2 +\\| \\nabla ^2 \\phi \\|_\\infty) }  \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty.\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\\end{split}\\end{equation}\n\t\t}\n\t\t \t\t\nNext, we consider (\\ref{g_Gamma}). From the computations in (\\ref{int1overalphadv}), and using the fact that $\\frac{1}{\\tilde \\alpha} \\lesssim \\frac{1}{\\tilde \\alpha^\\beta}$, we have\n{\n\\begin{equation} \\label{nonlocall3l1est} \\begin{split}\n & \\|(\\ref{g_Gamma})\\|_{L^3_x L^{1+ \\delta}_v}  \n \\\\ \\le & \\left\\|\\left\\| \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}}e^{ -\\int_s^t  - \\frac{\\varpi}{2}\\langle V(\\tau) \\rangle d\\tau }  e^{-\\varpi \\langle V(s) \\rangle s } \\right. \\right. \n\\\\ & \\left. \\left.  \\quad \\quad \\quad \\quad   \\quad \\quad \\quad \\quad \\times  \\int_{\\mathbb{R}^3} \\frac{e^{-C_{\\vartheta'} |V(s) - u |^2 }}{ |V(s) - u | ^{2 -\\kappa } } \\nabla_v f(s,X(s),u)| \\mathrm{d} u \\mathrm{d} s\n\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x}\n\\\\ \\lesssim &e^{C \\| \\nabla \\phi \\|_\\infty }   \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty \n\\\\ & \\times \\left\\|\\left\\| \\int^t_{\\max\\{t-t_{\\mathbf{b}}, 0\\}}e^{ -\\int_s^t  - \\frac{\\varpi}{2}\\langle V(\\tau) \\rangle d\\tau } \n \\int_{\\mathbb{R}^3} \\frac{e^{-C_{\\vartheta'} |V(s) - u |^2 }}{ |(s) - u | ^{2 -\\kappa } } \\frac{e^{- \\frac{\\vartheta'}{2} |u|^2 }  }{(\\tilde \\alpha(s,X(s),u))^\\beta} \\mathrm{d} u \\mathrm{d} s\n\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x}.\n\\end{split}\n\\end{equation}\nAnd then applying the nonlocal-to-local estimate \\eqref{nonlocaltolocal} to \\eqref{nonlocall3l1est} , we conclude\n\n\\begin{equation} \n\\begin{split}\n & \\|(\\ref{g_Gamma})\\|_{L^3_x L^{1+ \\delta}_v}   \n \\\\ \\lesssim &e^{C (\\| \\nabla \\phi \\|_\\infty   +  \\| \\nabla^2 \\phi \\|_\\infty)}  \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{- \\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty \n\\\\ & \\times \\left\\|\\left\\|\n\\frac{  \\delta ^{\\frac{3 -\\beta}{2} }}{(\\tilde \\alpha(t,x,v))^{\\beta -2 }  (|v|^2 + 1 )^{\\frac{3 -\\beta}{2}}}  + \\frac{ (|v| +1)^{\\beta -1} } { \\delta ^{\\beta -1 } \\varpi \\langle v \\rangle  ( \\tilde \\alpha ( t,x,v) )^{\\beta - 1 } }\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x}\n\\\\ \\lesssim &e^{C (\\| \\nabla \\phi \\|_\\infty   +  \\| \\nabla^2 \\phi \\|_\\infty)}  \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty \n\\\\ & \\times \\left( O(\\delta^{\\frac{3-\\beta}{2}} )   + \\frac{1}{ \\delta ^{\\beta -1 } \\varpi}  \\left\\|\\left\\|\n   \\frac{ 1} { \\langle v \\rangle^{2 -\\beta}  ( \\tilde \\alpha ( t,x,v) )^{\\beta - 1 } }\t\t\t\\right\\|_{L_{v}^{1+ \\delta}(  \\mathbb{R}^3)}\\right\\|_{L^{3}_x} \\right)\n\\\\ \\lesssim & C(\\delta^{\\frac{3-\\beta}{2}}  +  \\frac{1}{ \\delta ^{\\beta -1 } \\varpi}  )e^{C (\\| \\nabla \\phi \\|_\\infty  +  \\| \\nabla^2 \\phi \\|_\\infty)}  \\sup_{0 \\le t \\le T} \\ \\left\\|  w_{\\vartheta'} e^{-\\varpi \\langle v \\rangle t } \\tilde \\alpha \\nabla_x f \\right\\|_\\infty,\n\\end{split} \\end{equation}\n}\nfor $\\beta$ satisfies $\\frac{  (\\beta-1)(1+\\delta) -1}{2} \\frac{3}{1+\\delta} < 1$, which is equivalent to $\\beta< \\frac{5}{3} + \\frac{1}{1+\\delta}$. Therefore any $1 < \\beta < \\frac{5}{3}$ would work.\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\tCollecting terms from (\\ref{g_initial})-(\\ref{nablaphigint}), and (\\ref{est_g_initial}), (\\ref{est_g_bdry}), (\\ref{est_g_phi}), (\\ref{init_p_xf}), (\\ref{nonlocall3l1est}), we derive\n\t\t\\begin{equation} \\begin{split}\\label{bound_nabla_v_g}\n\t\t\t& \\sup_{0 \\leq s \\leq t}\\| e^{-\\varpi \\langle v \\rangle t }  \\nabla_vf(s) \\|_{L^3_xL^{1+ \\delta}_v} \\\\\n\t\\lesssim & \\| w_{\\vartheta'} \\nabla_v f (0) \\|_{L^3_{x,v}} +  \\| w_{\\vartheta'} f \\|_\\infty )^2 +  \\| w_{\\vartheta'} f \\|_\\infty\n\t\\\\ < & \\infty.\n\t\t\t\\end{split} \\end{equation}\nThis proves (\\ref{L3L1plusbddsolution}) and conclude Theorem \\ref{WlinftyVPBthm}.\n\n\n\n\n\n\\end{proof}\n\n\\section{Acknowledgements}This work was supported in part by National Science Foundation under Grant No. 1501031, Grant No. 1900923, and the Wisconsin Alumni Research Foundation.\n\n\n\\input{references}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn \\cite{[Euler]} (see \\cite{[Heine]} for the translation of \\cite{[Euler]} in\nEnglish) Euler proved that there does not exist a perfect map from the sphere\n$S^{2}$ or from a part of $S^{2}$ to the Euclidean plane $E^{2}.$ Recall that\na smooth map $f$ from $S^{2}$ (or from a part of $S^{2})$ to $E^{2}$ is called\n\\textit{perfect} if for each $p\\in S^{2}$\\ there is a neighborhood $U(p)$\\ of\n$p$ in $S^{2}$ such that the restriction of $f$\\ on $U(p)$\\ preserves\ndistances infinitesimally along the meridians and the parallels of $S^{2}$ and\n$f$ preserves also angles between meridians and parallels \\cite{[CP]}. In\nmodern geometric language a perfect map is a local isometry from $S^{2}$ to\n$E^{2}$ and thus Euler's theorem results as a corollary of the Gauss Egregium\nTheorem which was proven many years later. However, Euler's method of proof is\nvery fruitful and can be applied in similar problems, see for instance\nProposition 5 in \\cite{[CP]}. Very briefly, Euler's basic idea for the\nnon-existence of a perfect map from $S^{2}$ to $E^{2},$ is to translate\ngeometrical conditions to a system of differential equations and prove that\nthis system does not have a solution. Using Euler's method, the non-existence\nof a smooth map from a neighbourhood $U$ of $S^{2}$ to $E^{2}$ which preserves\ndistances infinitesimally along the meridians and the parallels of $S^{2}$ and\nwhich sends the meridional arcs of $U\\cap S^{2}$ to straight lines of $E^{2},$\ncan also be proven \\cite{[CP]}.\n\nThe origin of all these problems lies in the ancient problem of cartography,\nthat is, the problem of constructing geographical maps from $S^{2}$ (or from a\nsubset of $S^{2})$ to $E^{2}$ which satisfy certain specific requirements.\nThis problem can also be considered as part of a more general subject\nwhich explores the existence of coordinate transformations that preserves some\ngeometrical properties from one coordinate system to another.\nSeveral prominent mathematicians have been studying this problem from\nantiquity to our days and in the course of this study, $S^{2}$ was replaced\ngradually by surfaces of revolution or by surfaces in $E^{3}$ generally, see\n\\cite{[Papadop]} for an excellent historical recursion on this subject.\n\nThe goal of this work is to determine the class of surfaces of revolution $S$\nfor which there exists a smooth map $\\Phi$ from an open neighbourhood $U$ of\n$S$ to $E^{2}$ preserving distances infinitesimally along the meridians and\nthe parallels of $S$ and sending meridional arcs of $U\\cap S$ to straight\nlines of $E^{2}.$ Furthermore the map\n$\\Phi$ is computed explicitly. For the computation of $\\Phi$ we follow Euler's\nideas, that is, we convert geometrical conditions to differential equations\nwhose solutions allow us to find $\\Phi.$\n\nAs a corollary of the above result we deduce that if $p$ is a point of $S$ and\nif the Gaussian curvature at $p$ is positive, then a map $\\Phi$ as above does\nnot exist in a neighbourhood $U$ of $p.$ We also deduce that if $S_{0}$ is an\nabstract surface of constant negative curvature, such maps $\\Phi$ do not exist\nfrom an open subset $U$ of $S_{0}$ to $E^{2}.$\n\n\\section{Statement of Results}\n\nLet $S$ be  a surface of revolution  in  $E^{3}$.  \nIn the following we assume that all maps are of class $C^s$, $s\\geq2$.   \n We consider a\nparametrization of $S$ given by\n\\begin{equation}\nr(t,u)=(f(u)\\cos t,f(u)\\sin t,g(u)), \\tag{r}\\label{r}%\n\\end{equation}\nwhere $f(u)>0,$ $a<u<b,$ $t\\in \\lbrack0,2\\pi].$ In what follows we will always\nassume that the curve $\\gamma(u)=(f(u),g(u))$ is parametrized by arc-length\ni.e. $(f^{\\prime})^{2}(u)+(g^{\\prime})^{2}(u)=1.$ Therefore the Riemannian\nmetric on $S$ takes the form\n\\[\nds^{2}=du^{2}+f^{2}(u)dt^{2}.\n\\]\nFor $t$ fixed, the $u$-curves $r(t,\\cdot)$ are called \\textit{meridians} of\n$S$ and are geodesics for the metric $ds,$ while for $u$ fixed, the $t$-curves\n$r(\\cdot,u)$ are called \\textit{parallels} and are not geodesics in general.\n\nAt each point $p\\in S$ we may assign a unique pair of coordinates\n$(t,u)\\in(a,b)\\times \\lbrack0,2\\pi)$ since $p=r(t,u).$ Thus, $p$ will be\nidentified with this pair $(t,u).$\n\n\n\n\n\\begin{theorem}\n\\label{main theorem} Assume that $f^{\\prime}(u)\\neq0$ and\n$f^{\\prime \\prime}(u)\\neq 0$ for each $u\\in(a,b)$. Let\n$U$ be an open connected\nsubset of $S$. Then, there\nexists a map $\\Phi:U\\rightarrow\nE^{2}$ having the properties \\\\\n$(C1):$ $\\Phi$ sends the meridional arcs of $S\\cap U$ to straight lines\nof $E^{2};$\\\\\n$(C2):$ $\\Phi$ preserves distances infinitesimally along the meridians\nand the parallels of $S;$\\\\\nif and only if $f^{2}=cu^{2}+du+k$, where $c,d,k$ are constants with $k>0$ and $c>0$.\nFurthermore, assuming that $\\Phi(t,u)=(x(t,u),y(t,u))$, we have that\n\\begin{align*}\nx(t,u)  &  =u\\cos(b(t))+\\int \\sqrt{k}\\cos(\\theta_{0}-b(t))dt,\\\\\ny(t,u)  &  =-u\\sin(b(t))+\\int \\sqrt{k}\\sin(\\theta_{0}-b(t))dt,\n\\end{align*}%\nwhere\n\\[\nb(t)=-\\sqrt{c}t+c_{0}.\n\\]\n\\end{theorem}\n\n\n\\begin{figure}[h] \n\\centering{\\includegraphics[scale = 0.3]{SURFACE-PETR2.eps}\n\\caption{\\small Surface with $f^{2}=u^{2}+1$}}\n\\end{figure}\n\n\nIn Figure 1, a surface of revolution $S$ is drawn which satisfies all\nhypothesis of Theorem \\ref{main theorem}. In this example, taking $f^{2}\n=u^{2}+1$ it results that $g(u)=\\ln(\\sqrt{u^{2}+1}+u),$ $u>0.$ The picture\nconfirms that the Gaussian curvature of each point of $S$ is negative.\n\n\n\\begin{corollary}\n(1) If $p\\in S$ and the Gaussian curvature at $p$ is positive then for each\nneighborhood $U$ of $p$ in $S$ there does not exist a map $\\Phi:U\\rightarrow\nE^{2}$ satisfying the conditions $(C1)$ and $(C2).$\n\n(2) If $S_{0}$ is a Riemannian surface of constant negative curvature then for\neach open neighbourhood $U\\subset S_{0}$ there does not exist a map\n$\\Phi:U\\rightarrow E^{2}$ satisfying the conditions $(C1)$ and $(C2).$\n\\end{corollary}\n\n\n\nCondition $(C1)$ is a natural requirement since meridians are geodesics of $S$\nand thus it is required to be sent to geodesics of $E^{2}$ via $\\Phi.$\n\nCondition $(C2)$ appears in Euler's writings and means that the elementary\nlength between two points $p,$ $q$ on a meridian (resp. two points $p,r$ on a\nparallel) of $S$ is equal to the elementary length of points $P=\\Phi(p),$\n$Q=\\Phi(q)$ (resp. of points $P,$ $R=\\Phi(r)).$ In other words, the `elements'\n$pq$ and $pr$ are equal to the `elements' $PQ$ and $PR$ respectively\n(following the terminology of \\cite{[Euler]}). In order to express $(C2)$\nrigorously, let $p=(t,u),$ $q=(t,u+du),$ $r=(t+dt,u),$ $P=\\Phi(t,u),$\n$Q=\\Phi(t,u+du),$ $R=\\Phi(t+du,u).$ If we denote by $|p_{1}-p_{2}|$ the\ndistance between the points $p_{1},$ $p_{2}\\in S$ and by $||P_{1}-P_{2}||$ the\nEuclidean distance between the points $P_{1},$ $P_{2}\\in E^{2},$ then\ncondition $(C2)$ means that:%\n\\begin{equation}\n\\underset{du\\rightarrow0}{\\lim}\\frac{|q-p|}{|du|}=\\underset{du\\rightarrow\n0}{\\lim}\\frac{||P-Q||}{|du|} \\tag{i}\\label{i}%\n\\end{equation}\n\n\\begin{equation}\n\\underset{dt\\rightarrow0}{\\lim}\\frac{|r-p|}{|du|}=\\underset{dt\\rightarrow\n0}{\\lim}\\frac{||R-Q||}{|du|}. \\tag{ii}\\label{ii}%\n\\end{equation}\n\n\nUsing the coordinate functions $x(t,u)$ and $y(t,u)$ the equalities (\\ref{i})\nand (\\ref{ii}) take respectively the following form:%\n\n\\begin{equation}\n\\sqrt{(\\frac{\\partial x}{\\partial u})^{2}+(\\frac{\\partial y}{\\partial u})^{2}%\n}=1, \\tag{iii}\\label{iii}%\n\\end{equation}\n\n\\begin{equation}\n\\text{\\qquad}\\sqrt{(\\frac{\\partial x}{\\partial t})^{2}+(\\frac{\\partial\ny}{\\partial t})^{2}}=f(u). \\tag{iv}\\label{iv}%\n\\end{equation}\nIndeed, relations (\\ref{iii}) and (\\ref{iv}) correspond to the relations (I)\nand (II) of (\\cite{[Heine]}, p. 5). In \\cite{[CP]}, these relations are\nreproved using a modern mathematical language and they are labelled as\nrelations (6) and (7) respectively. In the present work, relations (\\ref{iii})\nand (\\ref{iv}) are obtained by replacing $\\cos u$ in the parametrization\n$(\\cos u\\cos t,\\cos u\\sin t,\\sin u)$ of $S^{2}$ by the function $f(u)$ in the\nparametrization $(f(u)\\cos t,f(u)\\sin t,g(u))$ of $S$ and then repeating the\nsame steps. As a result, using Euler's method, condition $(C2)$ is translated\nin a system of differential equations consisting of the relations (\\ref{iii})\nand (\\ref{iv}).\n\nCombining $(C1)$ and $(C2)$ we deduce that $\\Phi$ restricted to a meridian of\n$S$ is an isometry onto its image.\n\n\\begin{remark} {\\rm\nIf $f^{\\prime}(u)=0$ for each $u\\in(a,b)$, then, the curve $\\gamma\n(u)=(f(u),g(u))$ is a straight line in the $(x,z)$-plane and so, the surface\n$S$ obtained by revolving $\\gamma$ about the $z$-axis is Euclidean i.e.\nlocally isometric to the Euclidean plane $E^{2}.$\nFurthermore, if $f^{\\prime\n\\prime}(u)=0$ for each $u,$ we deduce that $f^{\\prime}$ and $g^{\\prime}$ are\nconstant functions since by assumption the curve $\\gamma(u)=(f(u),g(u))$ is\nparametrized by arc-length. Therefore $\\gamma(u)$ is a line segment and thus\n$S$ is a locally isometric to $E^{2}.$ }\n\\end{remark}\n\n\n\n\n\\section{Auxiliary Lemmas}\n\nIn this section we give some results that we will use for the proof of Theorem 1.\n\n\\begin{lemma}\n Assume that $f^{\\prime}(u)\\neq 0$ and\n$f^{\\prime \\prime}(u)\\neq 0$, for each $u\\in(a,b)$. Let\n$U$ be an open connected\nsubset of $S$ and  a map $\\Phi:U\\rightarrow\nE^2$ having the properties $(C_1)$ and $(C_2)$. Then, \n there are\nvariables $\\phi$ and $\\omega$ which are functions of $t,$ $u$ satisfying\n\\begin{equation*}\n\\left(\\frac{\\partial x}{\\partial u},\\frac{\\partial y}{\\partial u}\\right)=\n(\\cos \\phi\n,\\sin \\phi), \n\\ \\ \\ \\\n\\left(\\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t}\\right)=(f(u)\\cos\n\\omega,f(u)\\sin \\omega)\n\\end{equation*}\nand\n\\[\nf^{\\prime \\prime}(u)=\\left(  \\frac{\\partial \\omega}{\\partial u}(t,u)\\right)\n^{2}f(u).  \n\\]\n\\end{lemma}\n\\begin{proof}\nProceeding as in the proof of Proposition 5 of \\cite{[CP]} \nwe have that there are\nvariables $\\phi$ and $\\omega$ which are functions of $t,$ $u$ such that\n\\begin{equation*}\n(\\frac{\\partial x}{\\partial u},\\frac{\\partial y}{\\partial u})=(\\cos \\phi\n,\\sin \\phi),  \n\\end{equation*}\n\\begin{equation*}\n(\\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t})=(f(u)\\cos\n\\omega,f(u)\\sin \\omega).  \n\\end{equation*}\n\nSince\n\\[\n\\frac{\\partial^{2}}{\\partial u\\partial t}=\\frac{\\partial^{2}}{\\partial\nt\\partial u},\n\\]\nwe have%\n\\begin{equation*}\n-\\sin \\phi \\cdot \\frac{\\partial \\phi}{\\partial t}=f^{\\prime}\\cdot \\cos \\omega\n-f\\cdot \\sin \\omega \\cdot \\frac{\\partial \\omega}{\\partial u}. \n\\end{equation*}\n\n\\begin{equation*}\n\\cos \\phi \\cdot \\frac{\\partial \\phi}{\\partial t}=f^{\\prime}\\cdot \\sin \\omega\n+f\\cdot \\cos \\omega \\cdot \\frac{\\partial \\omega}{\\partial u}. \n\\end{equation*}\nMultiplying the first of the above equality by \n $\\cos \\omega$, the second by $\\sin \\omega$ and\nadding, we deduce that\n\\begin{eqnarray*}\n\\lefteqn{\n(-\\sin \\phi \\cdot \\cos \\omega+\\cos \\phi \\cdot \\sin \\omega)\\frac{\\partial \\phi}{\\partial\nt}=}\\\\\n& & f^{\\prime}\\cdot \\cos^{2}\\omega-f\\cdot \\sin \\omega \\cdot \\cos \\omega \\cdot\n\\frac{\\partial \\omega}{\\partial u}+f^{\\prime}\\cdot \\sin^{2}\\omega+f\\cdot\n\\cos \\omega \\cdot \\sin \\omega \\cdot \\frac{\\partial \\omega}{\\partial u}%\n\\end{eqnarray*}\nif and only if \n\\begin{equation*}\n\\sin(\\phi-\\omega)\\cdot \\frac{\\partial \\phi}{\\partial t}=f^{\\prime}.\n\\end{equation*}\nSimilarly, multiplying the first equality by\n $\\cos \\phi$, the second by $\\sin \\phi$\nand adding, we obtain%\n\\begin{eqnarray*}\n\\lefteqn{\\sin u\\cdot \\cos \\omega \\cdot \\cos \\phi-}\\\\\n& & \\cos u\\cdot \\sin \\omega \\cdot \\frac{\\partial \\omega}{\\partial u}\\cdot \\cos \\phi-\\sin u\\cdot \\sin \\omega \\cdot \\sin\\phi+\n\\cos u\\cdot \\cos \\omega \\cdot \\frac{\\partial \\omega}{\\partial u}\\cdot \\sin \\phi = 0\n\\end{eqnarray*}\nwhich implies that%\n\\begin{equation*}\nf\\cdot \\sin(\\phi-\\omega)\\frac{\\partial \\omega}{\\partial u}=-f^{\\prime}\\cdot\n\\cos(\\phi-\\omega). \n\\end{equation*}\n\nOn the other hand, the condition $(C1)$ implies that the meridians are\n mapped  to straight lines, and so, we have \n\\begin{equation*}\n\\frac{\\partial \\phi}{\\partial u}=0.\n\\end{equation*}\nThus, \ndifferentiating \n\\begin{equation*}\n\\sin(\\phi-\\omega)\\cdot \\frac{\\partial \\phi}{\\partial t}=f^{\\prime}\n\\end{equation*}\n with respect to $u$ and using the previous equality, we obtain%\n\\begin{equation*}\n-\\cos(\\phi-\\omega)\\cdot \\frac{\\partial \\omega}{\\partial u}\\cdot \\frac\n{\\partial \\phi}{\\partial t}=f^{\\prime \\prime}.\n\\end{equation*}\n\n\nMultiplying \n\\begin{equation*}\n\\sin(\\phi-\\omega)\\cdot \\frac{\\partial \\phi}{\\partial t}=f^{\\prime}\n\\end{equation*}\nby%\n\\[\n\\frac{\\partial \\omega}{\\partial u}\\cdot \\frac{\\partial \\phi}{\\partial t}%\n\\]\nwe have%\n\\[\nf\\cdot \\sin(\\phi-\\omega)\\cdot\\left(\\frac{\\partial \\omega}{\\partial u}\\right)^{2}\\cdot\n\\frac{\\partial \\phi}{\\partial t}=-f^{\\prime}\\cdot \\cos(\\phi-\\omega)\\cdot\n\\frac{\\partial \\omega}{\\partial u}\\cdot \\frac{\\partial \\phi}{\\partial t}.\n\\]\nHence, combining the above equalities,  we deduce%\n\\[\nf\\cdot f^{\\prime}\\cdot(\\frac{\\partial \\omega}{\\partial u})^{2}=f^{\\prime}\\cdot\nf^{\\prime \\prime}%\n\\]\nwhich implies that%\n\\begin{equation*}\nf^{\\prime}(u)(f^{\\prime \\prime}(u)-\\left(  \\frac{\\partial \\omega}{\\partial\nu}\\right)  ^{2}f(u))=0.\n\\end{equation*}\nSince we have $f^{\\prime}(u)\\neq 0$, we obtain the result.\n\\end{proof}\n\n\n\\begin{lemma}\nAssume that $f^{\\prime}(u)\\neq 0$ and\n$f^{\\prime \\prime}(u)\\neq 0$, for each $u\\in(a,b)$. Assume that\n\\begin{equation*}\n2f^{\\prime}a^{\\prime}+fa^{\\prime \\prime}=0.\n\\end{equation*}\nThen, we have $f^{2}=cu^{2}+du+k$ and\n\\[\na=a(u)=\\arctan \\left(  \\frac{2c}{\\sqrt{-\\Delta}}\\left(  u+\\frac{d}{2c}\\right)\n\\right),\n\\]\nwhere $\\Delta=d^{2}-4ck$.\n\\end{lemma}\n\\begin{proof} Putting $y=a^{\\prime}$, we have the differential equation\n\\[\ny^{\\prime}+\\frac{2f^{\\prime}}{f}y = 0.\n\\]\nIts  solution  is\n\\begin{equation*}\ny=a^{\\prime}=Ce^{-2\\int \\frac{f^{\\prime}}{f}du}.\n\\end{equation*}\nIt follows that\n$$(a^{\\prime})^{2}   = C^2  e^{-4\\int \\frac{f^{\\prime}}{f}du},$$\nwhence\n $$\\frac{f^{\\prime \\prime}}{f}=C^2 e^{-4\\int \\frac{f^{\\prime}}{f}du},$$\nand so, we obtain\n$$\\ln f^{\\prime \\prime}-\\ln f    =K-4\\int \\frac{f^{\\prime}}{f}du.$$\nDifferentiating the above equality, we get\n$$\\frac{f^{(3)}}{f^{\\prime \\prime}}-\\frac{f^{\\prime}}{f}   =-4\\frac{f^{\\prime}%\n}{f},$$\nand therefore we deduce\n\\[\nf^{(3)}f+3f^{\\prime \\prime}f^{\\prime}=0.\n\\]\n\nOn the other hand, we have\n\\[\n(ff^{\\prime})^{\\prime \\prime}=((ff^{\\prime})^{\\prime})^{\\prime}=(f^{\\prime\n}f^{\\prime}+ff^{\\prime \\prime})^{\\prime}=2f^{\\prime \\prime}f+f^{\\prime \\prime\n}f+ff^{(3)}=f^{(3)}f+3f^{\\prime \\prime}f^{\\prime},\n\\]\nwhence we get\n\\begin{equation*}\n(ff^{\\prime})^{\\prime \\prime}=0.\n\\end{equation*}\nIt follows that $(ff^{\\prime})^{\\prime}=c$, whence we have\n$ff^{\\prime}=c_{1}u+d_{1}$, and so, we get $(f^{2})^{\\prime}=cu+d$. \nThus, we obtain\n\\[\nf^{2}=cu^{2}+du+k.\n\\]\n\n\nTaking the first and the second derivative, we have\n\\[\nf^{\\prime}=\\frac{1}{2}\\frac{2cu+d}{\\sqrt{cu^{2}+du+k}}\\  \\  \\  \\text{and}%\n\\  \\  \\ f^{\\prime \\prime}=\\frac{4ck-d^{2}}{4(cu^{2}+du+k)^{3\/2}}.\n\\]\nThus\n\\[\n\\frac{f^{\\prime \\prime}}{f}=\\frac{4ck-d^{2}}{4(cu^{2}+du+k)^{2}}%\n\\  \\  \\  \\text{and}\\  \\  \\  \\frac{f^{\\prime}}{f}=\\frac{2cu+d}{2(cu^{2}+du+k)}.\n\\]\nSince\n\\[\n\\frac{f^{\\prime \\prime}}{f}=(a^{\\prime})^{2}= C^2 \ne^{-4\\int \\frac{f^{\\prime}}%\n{f}du},\n\\]\nwe have\n\\[\n\\frac{f^{\\prime \\prime}}{f}=\\frac{C^2}{f^{4}}.\n\\]\nThus, we obtain\n\\[\n\\frac{4ck-d^{2}}{4(cu^{2}+du+k)^{2}}=\\frac{C^2}{(cu^{2}+du+k)^{2}},\n\\]\nand therefore\n\\[\nC^2 =\\frac{4ck-d^{2}}{4}.\n\\]\n\n\nLet $\\Delta=d^{2}-4ck$ be the discriminant of $cu^{2}+du+k$. Thus, we\nget \n\\[\nC=\\frac{\\sqrt{-\\Delta}}{2}.\n\\]\nFurthermore, we have\n\\[\na^{\\prime}=Ce^{-2\\int \\frac{f^{\\prime}}{f}du}=\\frac{\\sqrt{-\\Delta}\/2}{f^{2}%\n}=\\frac{\\sqrt{-\\Delta}\/2}{cu^{2}+du+k}%\n\\]\nand thus\n\\[\na=\\int a^{\\prime}du=\\int \\frac{\\sqrt{-\\Delta}\/2}{cu^{2}+du+k}du=\\int \\frac\n{\\sqrt{-\\Delta}\/2}{c(u+\\frac{d}{2c})^{2}+(\\frac{\\sqrt{-\\Delta}}{2c})^{2}}du.\n\\]\nHence, we obtain \n\\begin{equation*}\na(u)=\\arctan \\left(  \\frac{2c}{\\sqrt{-\\Delta}}\\left(  u+\\frac{d}{2c}\\right)\n\\right).   \n\\end{equation*} \\end{proof}\n\n\n\n\\section{Proof of Theorem 1}\nSuppose that  there exists a map $\\Phi:U\\rightarrow\nE^{2}$ having the properties $(C1)$ and $(C2)$. \nBy Lemma 1,  there are\nvariables $\\phi$ and $\\omega$ which are functions of $t,$ $u$ satisfying\n\\begin{equation}\n\\left(\\frac{\\partial x}{\\partial u},\\frac{\\partial y}{\\partial u}\\right)=(\\cos \\phi\n,\\sin \\phi), \\tag{1}\\label{1}%\n\\end{equation}\n\\begin{equation}\n\\left(\\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t}\\right)=(f(u)\\cos\n\\omega,f(u)\\sin \\omega). \\tag{2}\\label{2}%\n\\end{equation}\nand\n\\[\nf^{\\prime \\prime}(u)=\\left(  \\frac{\\partial \\omega}{\\partial u}(t,u)\\right)\n^{2}f(u). \\tag{3}\\label{3}%\n\\]\nBy $(C1)$, the meridians are mapped  to straight lines, and so, we have \n\\begin{equation*}\n\\frac{\\partial \\phi}{\\partial u}=0. \n\\end{equation*}\nThus (\\ref{1}) yields\n\\begin{eqnarray*}\n\\lefteqn{\\frac{\\partial}{\\partial u}\\left(  \\frac{\\partial x}{\\partial u}%\n,\\frac{\\partial y}{\\partial u}\\right)  =}\\\\\n& &  \\left(  \\frac{\\partial^{2}x}{\\partial\nu^{2}},\\frac{\\partial^{2}y}{\\partial u^{2}}\\right)= \\frac{\\partial}{\\partial\nu}(\\cos \\phi,\\sin \\phi)=\n\\left(-\\sin \\phi \\frac{\\partial \\phi}{\\partial u},\\cos \\phi\n\\frac{\\partial \\phi}{\\partial u}\\right)=(0,0).\n\\end{eqnarray*}\nIt follows\n\\begin{equation}\nx(t,u)=ug_{1}(t)+g_{2}(t),\\  \\  \\  \\ y(t,u)=uh_{1}(t)+h_{2}(t). \\tag{4}\\label{4}%\n\\end{equation}\nTherefore, the function $(\\partial \\omega\/\\partial u)$ is a function depending\nonly on the variable $u$, and hence there exist functions $a(u)$ and $b(t)$\nsuch that\n\\begin{equation}\n\\omega(t,u)=a(u)+b(t). \\tag{5}\\label{5}%\n\\end{equation}\nCombining (\\ref{2}) and (\\ref{5}), we deduce\n\\begin{equation}\n\\left(  \\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t}\\right)\n=(f\\cos(a(u)+b(t)),f\\sin(a(u)+b(t))), \\tag{6}\\label{6}%\n\\end{equation}\nand using that\n\\[\n\\frac{\\partial^{2}x}{\\partial u\\partial t}=\\frac{\\partial^{2}x}{\\partial\nt\\partial u},\n\\]\n(\\ref{4}) and (\\ref{6}) implies that\n\\[\n\\frac{\\partial}{\\partial u}f\\cos(a(u)+b(t))=\\frac{\\partial}{\\partial t}%\ng_{1}(t).\n\\]\nTherefore, for each $t$ and $u$, we deduce\n\\begin{equation}\nf^{\\prime}(u)\\cos((a(u)+b(t))-f(u)\\sin(a(u)+b(t))a^{\\prime}(u)=g_{1}^{\\prime\n}(t). \\tag{7}\\label{7}%\n\\end{equation}\nSimilarly, from\n\\[\n\\frac{\\partial^{2}y}{\\partial u\\partial t}=\\frac{\\partial^{2}y}{\\partial\nt\\partial u}%\n\\]\nwe get\n\\begin{equation}\nf^{\\prime}(u)\\sin((a(u)+b(t))+f(u)\\cos(a(u)+b(t))a^{\\prime}(u)=h_{1}^{\\prime\n}(t), \\tag{8}\\label{8}%\n\\end{equation}\nfor each $t$ and $u$.\n\nBy taking the derivative of (\\ref{7}) with respect to $u$ we have\n\\[\nf^{\\prime \\prime}\\cos \\omega-2f^{\\prime}a^{\\prime}\\sin \\omega-f(a^{\\prime}%\n)^{2}\\cos \\omega-fa^{\\prime \\prime}\\sin \\omega=0\n\\]\nand so, we get\n\\begin{equation*}\n\\sin \\omega(2f^{\\prime}a^{\\prime}+fa^{\\prime \\prime})-(f^{\\prime \\prime\n}-f(a^{\\prime})^{2})\\cos \\omega=0  \n\\end{equation*}\nAssuming that $\\sin \\omega \\neq0$,   we obtain \n\\begin{equation}\n2f^{\\prime}a^{\\prime}+fa^{\\prime \\prime}=0. \\tag{9}\\label{9}%\n\\end{equation}\nIf $\\sin \\omega=0,$ then $\\cos \\omega \\neq0$.\nThus, by taking the derivative of (\\ref{8}) we can derive the same\ndifferential equation (\\ref{9}), restricting if necessary the domain where\nthe functions $f$ and $a$ are defined.\nLemma 2 implies that\n$$f^{2}=cu^{2}+du+k$$\nand\n\\[\na=a(u)=\\arctan \\left(\\frac{2c}{\\sqrt{-\\Delta}}\\left(u+\\frac{d}{2c}\\right)\n\\right), \\tag{10}\\label{10}\n\\]\nwhere $\\Delta=d^{2}-4ck$.\n\n\n Using (\\ref{9}) and (\\ref{10}) we get\n\\[\n\\frac{f^{\\prime}}{f}=-\\frac{a^{\\prime \\prime}}{2a^{\\prime}}=a^{\\prime}\\tan a,\n\\]\nwhich is equivalent to\n \\begin{equation}\n f^{\\prime}\\cos a-fa^{\\prime}\\sin a=0.\n \\tag{11}\\label{11}\n \\end{equation}\n\nIf $f \\cos a =0$, then the above equality implies that \n$fa^{\\prime}\\sin a=0$. Since $f(u) a^{\\prime}(u) \\neq 0$, for every $u$,\nwe have $\\sin a=0$ which is a contradiction. Thus, dividing the above\nequality by  $f \\cos a$, we obtain  $\\frac{f^{\\prime}}{f}\\tan\na+a^{\\prime}=0$. Substituting $f^{\\prime}\/f$ by $a^{\\prime}\\tan a$ we deduce\n $a^{\\prime}(\\tan a)^2+a^{\\prime}=0$, whence $(\\tan a)^2 = -1$\n which is a contradiction. Hence we have \n$$f^{\\prime}\\sin a+fa^{\\prime}\\cos a\\neq 0.$$ \n\n\nOn the other hand, by taking the derivative of\n$f^{\\prime}\\sin a+fa^{\\prime}\\cos a$, we have\n\\[\n(f^{\\prime}\\sin a+fa^{\\prime}\\cos a)^{\\prime}=f^{\\prime \\prime}\\sin\na+f^{\\prime}a^{\\prime}\\cos a+f^{\\prime}a^{\\prime}\\cos a+fa^{\\prime \\prime}\\cos\na-f(a^{\\prime})^{2}\\sin a.\n\\]\nIn order to prove that this expression is zero, it suffices to show that\n\\[\n\\frac{f^{\\prime \\prime}}{f}\\tan a+2\\frac{f^{\\prime}}{f}a^{\\prime}+a^{\\prime \\prime\n}-(a^{\\prime})^{2}\\tan a=0\n\\]\nand one can verify, by a simple replacement, that this relation holds. Furthermore, we have\n\\[\na^{\\prime}(0)=\\frac{\\sqrt{-\\Delta}}{2k},\\text{ }f^{\\prime}(0)=\\frac{d}%\n{2\\sqrt{k}},\\text{ }f(0)=\\sqrt{k},\\text{ }\\tan a(0)=\\frac{d}{\\sqrt{-\\Delta}}.\n\\]\nThen, we obtain\n\\begin{equation}\nf^{\\prime}\\sin a+fa^{\\prime}\\cos a=\\sqrt{c}. \\tag{12}\\label{12}\n\\end{equation}\n\n\n By expanding relation (\\ref{7}), we obtain\n\\begin{eqnarray*}\n\\lefteqn{f^{\\prime}(\\cos a(u)\\cos b(t)-\\sin a(u)\\sin b(t))-}\\\\\n& & fa^{\\prime}(\\sin a(u)\\cos\nb(t)+\\sin b(t)\\cos a(u))=g_1^{\\prime}(t),\n\\end{eqnarray*}\nand from (\\ref{11}), (\\ref{12}) the relation $g_{1}^{\\prime}(t)=\\sqrt{c}\\sin b(t)$ follows.\n\nSimilarly, from (\\ref{9}) we have:\n\\[\nf^{\\prime}(\\sin a\\cos b+\\sin b\\cos a)+fa^{\\prime}(\\cos a\\cos b-\\sin a\\sin\nb)=h_{1}^{\\prime}(t).\n\\]\nwhence\n\\[\n\\cos b(f^{\\prime}\\sin a+fa^{\\prime}\\cos a)+\\sin b(f^{\\prime}\\cos a-fa^{\\prime\n}\\sin a)=h_{1}^{\\prime}(t)\n\\]\nand so, we obtain  $h_{1}^{\\prime}(t)=\\sqrt{c}\\, \\cos b(t)$.\n Therefore, we get\n$$g_{1}^{\\prime}(t)=\\sqrt{c}\\sin b(t) \\ \\ \\ \\text{and}\\ \\ \\  h_{1}^{\\prime\n}(t)=\\sqrt{c}\\, \\cos b(t).$$\n\nWe will proceed now with the computation of the projection $\\Phi.\\medskip$\nBy hypothesis, we have, that $(\\partial \\phi\/\\partial u)=0.$ \n Hence $\\phi$ is a\nfunction only of $t.$ From (\\ref{1}), (\\ref{2}) and (\\ref{4}) we have that\n\\begin{equation}\n\\left(  \\frac{\\partial x}{\\partial u},\\frac{\\partial y}{\\partial u}\\right)\n=(\\cos \\phi(t),\\sin \\phi(t))=(g_{1},h_{1})\\text{ }\\tag{13}\\label{13}%\n\\end{equation}\nand%\n\\begin{equation}\n\\left(  \\frac{\\partial x}{\\partial t},\\frac{\\partial y}{\\partial t}\\right)\n=(f(u)\\cos(a(u)+b(t)),f(u)\\sin(a(u)+b(t)))=(ug_{1}^{\\prime}+g_{2}^{\\prime\n},uh_{1}^{\\prime}+h_{2}^{\\prime}).\\tag{14}\\label{14}%\n\\end{equation}\nConsequently, (\\ref{13}) and (\\ref{14}) we get respectively that\n\\[\n(g_{1})^{2}+(h_{1})^{2}=1\n\\]\nand\n\\[\nu^{2}((g_{1}^{\\prime})^{2}+(h_{1}^{\\prime})^{2})+2u(g_{1}^{\\prime}%\ng_{2}^{\\prime}+h_{1}^{\\prime}h_{2}^{\\prime})+(g_{2}^{\\prime})^{2}%\n+(h_{2}^{\\prime})^{2}=cu^{2}+du+k.\n\\]\nTherefore, we have:\n\\begin{align*}\n(g_{2}^{\\prime})^{2}+(h_{2}^{\\prime})^{2} &  =k\\\\\n2(g_{1}^{\\prime}g_{2}^{\\prime}+h_{1}^{\\prime}h_{2}^{\\prime}) &  =d.\n\\end{align*}\nFrom the first of the previous relations we deduce that there exists a\nfunction $r(t)$ such that\n\\begin{equation}\n(g_{2}^{\\prime},h_{2}^{\\prime})=(\\sqrt{k}\\cos r(t),\\sqrt{k}\\sin r(t))\\tag{15}%\n\\label{15}%\n\\end{equation}\nwhile from the second, in combination with (\\ref{13}) and (\\ref{12}), we deduce\nthat $2\\sqrt{ck}\\sin(b+r)=d$ and thus\n\\[\n\\sin(b+r)=\\frac{d}{2\\sqrt{ck}}.\n\\]\nTherefore, there exists real number $\\theta_{0}$ such that\n$$r(t)=\\theta_{0}-b(t). $$\nFurthermore, from (\\ref{13}) we have that\n\\[\n(g_{1}^{\\prime},h_{1}^{\\prime})=(-\\phi^{\\prime}\\sin \\phi,\\phi^{\\prime}\\cos\n\\phi)=(\\sqrt{c}\\, \\sin b,\\text{ }\\sqrt{c}\\, \\cos b),\n\\]\nand so, we have the following  two cases:%\n\na) $\\phi^{\\prime}=\\sqrt{c}$ and  $\\phi(t)=-b(t)$. \nThus, we have \n$$b^{\\prime}(t)=-\\phi^{\\prime}=-\\sqrt{c},$$\nwhence  \n$$b(t)=-\\sqrt{c}t+c_{0}. $$\nThen, we get\n\\begin{equation}\n(g_{1},h_{1})=(\\cos(-b(t)),\\sin(-b(t)))=(\\cos b(t),-\\sin b(t)).\\tag{16}%\n\\label{16}%\n\\end{equation}\nThus, combining (\\ref{4}), (\\ref{15}) and (\\ref{16}) we\ndeduce\n\\begin{align*}\nx(t,u) &  =u\\cos(b(t))+\\int \\sqrt{k}\\cos(\\theta_{0}-b(t))dt\\\\\ny(t,u) &  =-u\\sin(b(t))+\\int \\sqrt{k}\\sin(\\theta_{0}-b(t))dt.\n\\end{align*}\n\n\nb) $\\phi^{\\prime}=-\\sqrt{c}$ and $\\phi(t)=\\pi-b(t)$. Proceeding\nas above, we deduce the result.\n \nFurthermore, substituting $b(t)$\nin the integrals above we may calculate them and thus we may find explicit\nformulas for the map $\\Phi.$\n\n\nConversely, by substituting the above expressions of $x(t,u)$ and $y(t,u)$ into (iii) and (iv), and supposing that $f^{2}=cu^{2}+du+k$, we see that condition $(C1)$ is easily verified. Also, condition $(C2)$  is  satisfied, since  \n$\\frac{\\partial x}{\\partial u}=\\cos \\phi$\nimplies that $$\\phi=\\arccos(\\frac{\\partial x}{\\partial u})$$ and so, by taking the derivative with respect to $u$, we obtain that \n$\\frac{\\partial \\phi}{\\partial u}=0.$\nHence, Theorem \\ref{main theorem} is proven.\n\n\n\n\\section{Proof of Corollary 1}\n\\\n(1) The Gaussian curvature of each point of $S$ is given by the formula\n\\[\nK=-\\frac{f^{\\prime \\prime}}{f}%\n\\]\n(see Formula (9), p. 162, in the Example 4 of \\cite{[DoCarmo]}). On the other\nhand, in the proof of Lemma 2\n we have shown that $f^{\\prime \\prime\n}\/f>0.$ Therefore, $K<0$ at every point of $S$ and thus statement (1) is proven.\n\n(2) The  surfaces of revolution of constant negative curvature are well known.\nA description of them can be found for example in (\\cite{[Gray]}, Theorem\n15.22, p. 477). Obviously these surfaces of revolution $R$ does not have the\nform of the surface $S$ given in Theorem \\ref{main theorem}. Therefore, for\nany point $p\\in R$ and for any neighborhood $U\\subset R$ of $p$ there does not\nexist a map $\\Phi:U\\rightarrow E^{2}$ satisfying the conditions $(C1)$ and\n$(C2).$ On the other hand, if $S_{0}$ is a Riemannian surface of constant\nnegative curvature $k<0,$ it is well known that $S_{0}$ is locally isometric\nto surface of revolution $R$ of constant curvature $k.$   Therefore our\nstatement follows.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and Preliminaries}\n\n\n\nThe pointwise a.e. convergence of sequences of operators is of paramount importance   and has been widely studied in several areas of analysis, such as harmonic analysis, PDE, and ergodic theory.  This area boasts challenging  problems,  \nindicatively see     \\cite{CRV, C66, DGL, JMZ}, \nand is intimately connected with the boundedness of the associated maximal operators; on this see \\cite{Stein}. \nMoreover, techniques and tools employed to study a.e. convergence have led to important developments in harmonic analysis. \n\nMultilinear harmonic analysis has made significant advances in recent years. The founders of this area are  Coifman and Meyer \\cite{CM2} who   realized the \napplicability  of multilinear operators  and  introduced their study  in analysis  in the \nmid 1970s. \nFocusing on  operators that commute with translations, a\nfundamental difference between the   multilinear  and the  linear  theory  is the existence of a straightforward characterization of boundedness at an initial point, usually $L^2\\to L^2$. The lack of an easy characterization of boundedness at an initial point in the multilinear theory creates difficulties in their study. \nCriteria that get very close to characterization of boundedness have   \nrecently been obtained by  the first two authors and Slav\\'ikov\\'a \\cite{Gr_He_Sl}\nand also  by Kato, Miyachi, and Tomita \\cite{Katoetal}  in the  bilinear case.  \nThese criteria were extended to the general $m$-linear case for $m\\ge 2$ by the authors of this article in \\cite{Paper1}. This reference also contains initial \n$L^2\\times\\cdots\\times L^2\\to L^{2\/m}$ estimates for  \nrough homogeneous multilinear singular integrals associated with \n$L^q$ functions on the sphere and multilinear multipliers of H\\\"ormander type.  \n\n The purpose of this work is to obtain the pointwise a.e. convergence of truncated multilinear homogeneous singular integrals and lacunary multilinear multipliers by establishing   boundedness for their associated maximal operators.\n\n\n\n\nWe first   introduce multilinear truncated singular integral operators.\nLet $\\Omega$ be a integrable function, defined on the sphere $\\mathbb{S}^{mn-1}$, satisfying the mean value zero property \n\\begin{equation}\\label{vanishingmtcondition}\n\\int_{\\mathbb{S}^{mn-1}}\\Omega~ d\\sigma_{mn-1}=0.\n\\end{equation} Then we define\n\\begin{equation*}\nK(\\vec{{y}}\\,):=\\frac{\\Omega(\\vec{{y}}\\,')}{|\\vec{{y}}\\,|^{mn}}, \\qquad \\vec y \\neq 0, \n\\end{equation*} \nwhere $\\vec{{y}}\\,':=\\vec{{y}}\\,\/|\\vec{{y}}\\,|\\in \\mathbb{S}^{mn-1}$, and the \ncorresponding truncated multilinear operator $\\mathcal{L}_{\\Omega}^{(\\epsilon)}$ by\n$$\n\\mathcal L^{(\\epsilon)}_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x):=\\int_{(\\mathbb R^n)^m\\setminus B(0,\\epsilon)}{K(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\,\n$$\n acting on Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$, where $x\\in\\mathbb R^n$ and $\\vec{{y}}\\,:=(y_1,\\dots,y_m)\\in (\\mathbb R^n)^m$.\nMoreover, by taking $\\epsilon\\searrow 0$, we obtain the multilinear homogeneous Calder\\'on-Zygmund singular integral operator \n\\begin{align}\n\\mathcal L_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x)&:=\\lim_{\\epsilon\\searrow 0}\\mathcal L^{(\\epsilon)}_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x) \\label{epsilonto0}\\\\\n&=p.v. \\int_{(\\mathbb R^n)^m}{K(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\, . \\nonumber\n\\end{align}\nThis is still well-defined for any Schwartz functions $f_1, \\dots,f_m$ on $\\mathbb R^n$. Here,   $B(0,\\epsilon)$ is  the ball centered at zero with radius $\\epsilon>0$ in $(\\mathbb R^n)^m$.  \n   In \\cite{Paper1} we   showed that \n   if $\\Omega$ lies in $ L^q\n(\\mathbb{S}^{mn-1}) $ with $q>\\frac{2m}{m+1}$, then the multilinear singular integral operator $\\mathcal{L}_{\\Omega}$ admits a  bounded \nextension from $L^2(\\mathbb R^n)\\times \\cdots \\times L^2(\\mathbb R^n)$ to $L^{2\/m}(\\mathbb R^n)$. In order words, given $f_j\\in L^2(\\mathbb R^n)$, $\\mathcal{L}_{\\Omega}(f_1,\\dots,f_m)$ is well-defined and is in $L^{2\/m}(\\mathbb R^n)$. It is natural to expect that, similar to the linear case, the truncated operator $\\mathcal{L}_{\\Omega}^{(\\epsilon)}(f_1,\\dots,f_m)$ converges a.e. to $\\mathcal{L}_{\\Omega}(f_1,\\dots,f_m)$ as $\\epsilon\\to 0$.\n\n\n\n\n\n\n\n\n\n\n\nOur first main result is as follows.\n\\begin{thm}\\label{CCC1'}\nLet $m\\ge 2$, $\\frac{2m}{m+1}<q\\le \\infty$ and $\\Omega \\in L^q (\\mathbb S^{mn-1})$     satisfy \\eqref{vanishingmtcondition}.\nThen  the  truncated singular integral  \n$\\mathcal{L}_{\\Omega}^{(\\epsilon)}(f_1,\\dots, f_m)$ converges a.e. as $\\epsilon\\to 0$ when $f_j\\in L^2(\\mathbb{R}^n)$, $j=1,\\dots , m$.\nThat is, the multilinear singular integral $\\mathcal{L}_{\\Omega}(f_1,\\dots,f_m)$ is well-defined a.e. when $f_j\\in L^2(\\mathbb{R}^n)$, $j=1,\\dots , m$.\n\\end{thm}\n\n\n\nTo achieve this goal, we need the following result from  \\cite{GDOS}.\n\\begin{prop}\\label{prop00'}  \nLet $0<p_j \\le\\infty$, $1\\le j\\le m$, $0<q<\\infty$ and let $D_j$ be a dense subclass of $L^{p_j}(\\mathbb R^n)$. \nLet $\\{T_t\\}_{t>0}$  be a \nfamily of $m$-linear operators while $T_\\ast$   is the associated maximal operator, defined by\n$$T_{\\ast}(f_1,\\dots,f_m):=\\sup_{t>0} \\big|T_t(f_1,\\dots,f_m)\\big| $$\nfor $f_j\\in D_j$, $1\\le j\\le m$.  Suppose that there is a constant $B$ such that \n\\begin{equation}\\label{100'}\n\\|T_{\\ast} (f_1,\\dots,f_m)\\|_{L^{q,\\infty}(\\mathbb R^n)} \\leq B \\prod_{j=1}^m \\|f_j\\|_{L^{p_j}(\\mathbb R^n)}\n \\end{equation}\n for all $f_j\\in D_j(\\mathbb{R}^n)$. Also suppose  that for all $f_j $ in $D_j$ we have \n\\begin{equation}\\label{tto1'}\n \\lim_{t\\to 0} T_{t} (f_1,\\dots, f_m) = T(f_1,\\dots,f_m)\n \\end{equation}\nexists and is finite. \nThen for all functions $f_j\\in L^{p_j}(\\mathbb{R}^n) $ the limit in \\eqref{tto1'} exists and is finite  a.e., and defines an $m$-linear operator   which uniquely extends $T$ defined on $D_1\\times \\cdots \\times D_m$ and\nwhich is bounded from $L^{p_1}(\\mathbb R^n)\\times \\cdots \\times L^{p_m}(\\mathbb R^n)$ to $L^{q,\\infty}(\\mathbb{R}^n)$. \n\\end{prop}\n\n\n\n\n\n\n \n\n \n \n With the help of this result, we reduce Theorem~\\ref{CCC1'} to the boundedness of \nthe  associated maximal singular integral operator \n\\begin{equation*}\n\\mathcal{L}_{\\Omega}^*\\big(f_1,\\dots,f_m\\big)(x):=\\sup_{\\epsilon>0}\\Big| \\mathcal{L}_{\\Omega}^{(\\epsilon)}\\big(f_1,\\dots,f_m\\big)(x)\\Big|.\n\\end{equation*}\n\\begin{thm}\\label{MAXSINGINT}\nLet $m\\ge 2$, $\\frac{2m}{m+1}<q\\le \\infty$ and $\\Omega \\in L^q (\\mathbb S^{mn-1})$     satisfy \\eqref{vanishingmtcondition}. Then there exists a constant $C>0$ such that\n\\begin{equation}\\label{maxinequal}\n\\big\\Vert \\mathcal{L}_{\\Omega}^*(f_1,\\dots,f_m)\\big\\Vert_{L^{2\/m}(\\mathbb R^n)}\\le C\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2(\\mathbb R^n)}\n\\end{equation}\nfor Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$.\n\\end{thm}\nThis extends and improves a result obtained in  \\cite{BH}  when $m=2$ and $q=\\infty$.\\\\ \n\n\n The essential contribution of this article is to suitably   combine  \nLittlewood-Paley techniques and   wavelet decompositions to reduce the boundedness of $\\mathcal{L}_\\Omega^*$ to   decaying estimates for norms of  maximal operators associated with lattice bumps; see \\eqref{2mmaingoal} for the exact formulation. This   result is actually   proved in terms of Plancherel type inequalities, developed in \\cite{Paper1} and stated in Proposition~\\ref{keyapplication1}. \n\n\n \n\n\n\n\n\\hfill\n\n\n\n\n\n\nThe tools used to establish Theorem~\\ref{CCC1'} turn out to be useful in the study of pointwise convergence problem of several related operators. \nAs an example\nlet us take multilinear multipliers with limited decay  to demonstrate our  idea. \n\nFor a smooth function $\\sigma\\in\\mathscr{C}^{\\infty}((\\mathbb R^n)^m)$ and $\\nu\\in\\mathbb{Z}$\nlet\n\\begin{equation}\\label{defSsinu}\nS_{\\sigma}^{\\nu}\\big(f_1,\\dots,f_m\\big)(x):=\\int_{(\\mathbb R^n)^m}{\\sigma(2^{\\nu}\\vec{{\\xi}}\\,)\\Big( \\prod_{j=1}^{m}\\widehat{f_j}(\\xi_j)\\Big)e^{2\\pi i\\langle x,\\sum_{j=1}^{m}\\xi_j \\rangle}} ~d\\,\\vec{{\\xi}}\\,\n\\end{equation}\nfor Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$, where $\\vec{{\\xi}}\\,:=(\\xi_1,\\dots,\\xi_m) \\in (\\mathbb R^n)^m$.\nWe are interested in the poinwise convergence of $S^\\nu_\\sigma$ when $\\nu\\to-\\infty$.\nWe pay particular attention to $\\sigma$ satisfying   the limited \ndecay property (for some fixed $a$)\n$$\n\\big|\\partial^{\\beta}\\sigma(\\vec{{\\xi}}\\,) \\big|\\lesssim_{\\beta}|\\vec{{\\xi}}\\,|^{-a}\n$$\nfor sufficiently many $\\beta$. Examples of multipliers of this type include $\\widehat \\mu$, the Fourier transform of the spherical measure $\\mu$; see  \\cite{Ca79, CW78,  Ru} for the corresponding linear results.  \n\n\nThe second contribution of this work  is the following result.\n\\begin{thm}\\label{CCC2'}\nLet $m\\ge 2$ and $a>\\frac{(m-1)n}{2}$.\nLet \n$\\sigma\\in\\mathscr{C}^{\\infty}((\\mathbb R^n)^m)$ satisfy\n\\begin{equation}\\label{givenassumption}\n\\big|\\partial^{\\beta}\\sigma(\\vec{{\\xi}}\\,) \\big|\\lesssim_{\\beta}|\\vec{{\\xi}}\\,|^{-a}\n\\end{equation}\nfor all $|\\beta|\\le  \\big[ \\frac{(m-1)n}{2} \\big]+1$, where $ \\left[ r\\right]$ denotes the integer part of $r$.\nThen for \n$f_j$ in $L^2(\\mathbb{R}^n)$, $j=1,\\dots , m$, the   functions \n$\nS_\\sigma^\\nu(f_1,\\dots, f_m) \n$\nconverge a.e. to $\\sigma(0) f_1 \\cdots f_m$ as $\\nu \\to -\\infty$. Additionally, if $\\lim_{y\\to \\infty} \n\\sigma(y)$ exists and and equals $L$, then the functions \n$S_\\sigma^\\nu(f_1,\\dots, f_m) $ converge a.e. to \n$L f_1 \\cdots f_m$ as $\\nu \\to  \\infty$. \n\\end{thm}\n\n\n\n\n\n\n\n\n\nThis problem is also reduced to the boundedness of the associated\n $m$-(sub)linear lacunary maximal multiplier operator  defined by: \n\\begin{equation*}\n\\mathscr{M}_{\\sigma}\\big(f_1,\\dots,f_m\\big)(x):=\\sup_{\\nu \\in\\mathbb{Z}}{\\big|S_{\\sigma}^{\\nu}\\big(f_1,\\dots,f_m\\big)(x)\\big|}.\n\\end{equation*}\n$\\mathscr{M}_{\\sigma}$ is the so-called multilinear spherical maximal function  when $\\sigma=\\widehat\\mu$, which was studied extensively recently by \\cite{AP, Barrionuevo2017, DG, HHY, JL}. In particular a bilinear version of the following theorem was previously obtained in \\cite{Gr_He_Ho2020}.\n\n\n\n\n\n\n\n\n\\begin{thm}\\label{application4}\nLet $m\\ge 2$ and $a>\\frac{(m-1)n}{2}$.\nLet \n $\\sigma\\in\\mathscr{C}^{\\infty}((\\mathbb R^n)^m)$\nbe as in Theorem~\\ref{CCC2'} \nThen there exists a constant $C>0$ such that\n\\begin{equation*}\n\\big\\Vert \\mathscr{M}_{\\sigma}(f_1,\\dots,f_m)\\big\\Vert_{L^{2\/m}(\\mathbb R^n)}\\le C \\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2(\\mathbb R^n)}\n\\end{equation*}\nfor Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$.\n\\end{thm}\n\n\n\n\n\nIt follows from Theorems~\\ref{MAXSINGINT} and ~\\ref{application4} that\n$\\mathcal{L}_{\\Omega}^*$ and $\\mathscr{M}_{\\sigma}$ have  unique bounded extensions from $L^2\\times \\cdots \\times L^2$ to $L^{2\/m}$ by density.\n\n\n\nLet us now sketch the proof of Theorem~\\ref{CCC2'}, taking Theorem \\ref{application4} temporarily for granted.\nWe notice that the claimed \nconvergence  holds pointwise everywhere  for smooth functions $f_j$ with compact support\nby the Lebesgue dominated convergence theorem.\nThen the assertions are immediate consequences of Proposition~\\ref{prop00'}.\\\\\n\n\\begin{comment}\n\\begin{proof}[Proof of Theorem~\\ref{CCC1'} and Theorem~\\ref{CCC2'}]\nTo verify these results, we notice that the claimed \nconvergence   holds pointwise everywhere  for smooth functions $f_j$ with compact support\nby the Lebesgue dominated convergence theorem.  In the case of the Theorem~\\ref{CCC2'} this assertion is straightforward, while in the case of Theorem~\\ref{CCC1'} it is a consequence \nof the smoothness of the function $f_1(x)\\cdots f_m(x)$ at the origin combined with the cancellation of $\\Omega$. The assertions in both theorems then follow by \napplying  Proposition~\\ref{prop00'}.\n\\end{proof}\n\\end{comment}\n\n As Theorems \\ref{CCC1'} and \\ref{CCC2'} follow from Theorems~\\ref{MAXSINGINT} and ~\\ref{application4}, respectively, \nwe   actually focus on the proof of Theorems~\\ref{MAXSINGINT} and ~\\ref{application4} in the remaining sections.\n\n\n\n\n\n\\section{Preliminary material}\\label{11171}\n\n\n We adapt some notations and key estimates  from \\cite{Paper1}.\n For the sake of independent reading we review the main tools and notation. \nWe begin with  certain orthonormal bases of $L^2$ due to Triebel \\cite{Tr2010},  that will be of great use in our work. \nThe idea is as follows.  \nFor any fixed $L\\in\\mathbb{N}$ one can construct real-valued compactly supported functions $\\psi_F, \\psi_M$ in  $\\mathscr{C}^L(\\mathbb{R})$ satisfying the following properties:\n  $\\Vert \\psi_F\\Vert_{L^2(\\mathbb{R})}=\\Vert \\psi_M\\Vert_{L^2(\\mathbb{R})}=1$, \n  $\\int_{\\mathbb{R}}{x^{\\alpha}\\psi_M(x)}dx=0$ for all $0\\le \\alpha \\le L$, and moreover, \nif     $\\Psi_{\\vec{{G}}}$ is a function on $\\mathbb{R}^{mn}$, defined by\n$$\\Psi_{\\vec{{G}}}(\\vec{{x}}\\,):=\\psi_{g_1}(x_1)\\cdots \\psi_{g_{mn}}(x_{mn})$$\nfor $\\vec{{x}}\\,:=(x_1,\\dots,x_{mn})\\in \\mathbb{R}^{mn}$ and $\\vec{{G}}:=(g_1,\\dots,g_{mn})$ in the set     \n $$\\mathcal{I}:=\\big\\{\\vec{{G}}:=(g_1,\\dots,g_{mn}):g_i\\in\\{F,M\\} \\big\\},$$\nthen the family of functions\n\\begin{equation*}\n\\bigcup_{\\lambda\\in\\mathbb{N}_0}\\bigcup_{\\vec{{k}}\\,\\in \\mathbb{Z}^{mn}}\\big\\{ 2^{\\lambda{mn\/2}}\\Psi_{\\vec{{G}}}(2^{\\lambda}\\vec{{x}}\\,-\\vec{{k}}\\,):\\vec{{G}}\\in \\mathcal{I}^{\\lambda}\\big\\}\n\\end{equation*}\nforms an orthonormal basis of $L^2(\\mathbb{R}^{mn})$,\nwhere $\\mathcal{I}^0:=\\mathcal{I}$ and  for $\\lambda \\ge 1$, we set  $\\mathcal{I}^{\\lambda}:=\\mathcal{I}\\setminus \\{(F,\\dots,F)\\}$.\n \n\n\nWe   consistently use   the notation $\\vec{{\\xi}}\\,:=(\\xi_1,\\dots,\\xi_m) $ for elements of $ (\\mathbb R^n)^m$, $\\vec{{G}}:=(G_1,\\dots,G_m)\\in (\\{F,M\\}^n)^m$, and  \n$\n\\Psi_{\\vec{{G}}}(\\vec{{\\xi}}\\,)=\\Psi_{G_1}(\\xi_1)\\cdots \\Psi_{G_m}(\\xi_m).\n$\nFor each $\\vec{{k}}\\,:=(k_1,\\dots,k_m)\\in (\\mathbb Z^n)^m$ and $\\lambda\\in \\mathbb{N}_0$, let \n$$\n\\Psi_{G_i,k_i}^{\\lambda}(\\xi_i):=2^{\\lambda n\/2}\\Psi_{G_i}(2^{\\lambda}\\xi_i-k_i), \\qquad 1\\le i\\le m\n$$\nand\n$$\n\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}(\\vec{{\\xi}}\\,\\,):=\\Psi_{G_1,k_1}^{\\lambda}(\\xi_1)\\cdots \\Psi_{G_m,k_m}^{\\lambda}(\\xi_m).\n$$\nWe also assume that the  support of $\\psi_{g_i}$ is contained in $\\{\\xi\\in \\mathbb{R}: |\\xi|\\le C_0 \\}$ for some $C_0>1$,\nwhich implies that\n\\begin{equation*}\n\\textup{Supp} (\\Psi_{G_i,k_i}^\\lambda)\\subset \\big\\{\\xi_i\\in\\mathbb R^n: |2^{\\lambda}\\xi_i-k_i|\\le C_0\\sqrt{n}\\big\\}.\n\\end{equation*}\nIn other words, the support of $\\Psi_{G_i,k_i}^\\lambda$ is contained in the ball centered at $2^{-\\lambda}k_i$ and radius $C_0\\sqrt{n}2^{-\\lambda}$.\nThen we note that for a fixed $\\lambda\\in\\mathbb{N}_0$, elements of $\\big\\{\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}\\big\\}_{\\vec{{k}}\\,}$ have (almost) disjoint compact supports.\n\n\n\n\n\n\n\n\n\n\n \n\n\nIt is also known in \\cite{Tr2006} that\nif $L$ is sufficiently large, then every tempered distribution $H$ on $\\mathbb{R}^{mn}$ can be represented as\n\\begin{equation}\\label{daubechewavelet}\nH(\\vec{{x}}\\,)=\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}}\\sum_{\\vec{{k}}\\,\\in \\mathbb{Z}^{mn}}b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}2^{\\lambda mn\/2}\\Psi_{\\vec{{G}}}(2^{\\lambda} \\vec{{x}}\\, -\\vec{{k}}\\,)\n\\end{equation}\nand for $1<q<\\infty$ and $s\\ge 0$,\n$$\n\\Big\\Vert \\Big(\\sum_{\\vec G,\\ \\vec k} \\big|b_{\\vec G,\\vec k}^\\lambda \\Psi^\\lambda_{\\vec G,\\vec k}\\big|^2\\Big)^{1\/2} \\Big\\Vert_{L^q(\\mathbb{R}^{mn})} \\le C2^{-s\\lambda}\\|H\\|_{L^q_s(\\mathbb{R}^{mn})} \n$$\nwhere \n\\begin{equation*}\nb_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}:=\\int_{\\mathbb{R}^{mn}}{H(\\vec{{x}}\\,)\\Psi^\\lambda_{\\vec G,\\vec k}(\\vec{{x}}\\,)}~d\\vec{{x}}\\, \n\\end{equation*}\nand $L^q_s$ is the  Sobolev space of functions $H$ such that \n$(I-\\Delta)^{s\/2} H \\in L^q(\\mathbb R^{mn})$. \nMoreover, it follows from the last estimate and \nfrom the (almost) disjoint support property  of the $\\Psi^\\lambda_{\\vec G,\\vec k}$'s that \n\\begin{align}\n\\big\\Vert \\big\\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}\\big\\}_{\\vec{{k}}\\,\\in \\mathbb{Z}^{mn}}\\big\\Vert_{\\ell^{q}}\n\\approx&\\Big(2^{\\lambda mn(1-q\/2)}\\int_{\\mathbb{R}^{mn}}\\Big( \\sum_{\\vec{{k}}\\,} \\big|b^\\lambda_{\\vec G,\\vec k}\\Psi^\\lambda_{\\vec G,\\vec k}(\\vec x)\\big|^2\\Big)^{q\/2}~ d\\vec{{x}}\\,\\Big)^{1\/q}\\notag \\\\\n\\lesssim& \\; 2^{-\\lambda (s-mn\/q+mn\/2)}\\Vert H\\Vert_{L^q_s(\\mathbb{R}^{mn})}. \\label{lqestimate}\n\\end{align} \n\n\n\n\nNow we study an essential estimate in \\cite{Paper1} which will play a significant role in the proof of both Theorems \\ref{MAXSINGINT} and \\ref{application4}.\nWe define the operator $L_{G_i,k_i}^{\\lambda,\\gamma}$ by\n\\begin{equation}\\label{lgklg}\nL_{G_i,k_i}^{\\lambda,\\gamma}f:=\\big( \\Psi_{G_i,k_i}^{\\lambda}(\\cdot\/2^{\\gamma})\\widehat{f}\\big)^{\\vee}, \\qquad \\gamma\\in\\mathbb{Z}.\n\\end{equation} \nFor $\\mu\\in\\mathbb{Z}$ let \n\\begin{equation}\\label{uset}\n\\mathcal{U}^{\\mu}:=\\big\\{ \\vec{{k}}\\,\\in (\\mathbb Z^n)^m: 2^{\\mu-2}\\le |\\vec{{k}}\\,|\\le 2^{\\mu+2},~  |k_1|\\ge\\cdots \\ge |k_m| \\big\\}\n\\end{equation}\nand split the set into $m$ disjoint subsets $\\mathcal{U}_l^{\\mu}$ as below:\n\\begin{align*}\n\\mathcal{U}_1^{{\\mu}}&:=\\big\\{\\vec{{k}}\\,\\in \\mathcal{U}^{{\\mu}}:|k_1|\\ge 2C_0\\sqrt{n}>|k_2|\\ge\\cdots \\ge |k_m| \\big\\}\\\\\n\\mathcal{U}_2^{{\\mu}}&:=\\big\\{\\vec{{k}}\\,\\in \\mathcal{U}^{{\\mu}}:|k_1|\\ge |k_2|\\ge 2C_0\\sqrt{n}>|k_3|\\ge\\cdots \\ge |k_m| \\big\\}\\\\\n&\\vdots\\\\\n\\mathcal{U}_m^{{\\mu}}&:=\\big\\{\\vec{{k}}\\,\\in \\mathcal{U}^{{\\mu}}:|k_1|\\ge \\cdots\\ge |k_m|\\ge 2C_0\\sqrt{n} \\big\\}.\n\\end{align*}\nThen we have the following two observations that appear in \\cite{Paper1}.\n\\begin{itemize}\n\\item For $\\vec{{k}}\\,\\in\\mathcal{U}_l^{\\lambda+\\mu}$,\n\\begin{equation}\\label{lgkest}\nL_{G_j,k_j}^{\\lambda,\\gamma}f=L_{G_j,k_j}^{\\lambda,\\gamma}f^{\\lambda,\\gamma,{\\mu}}  \\quad \\text{ for }~  1\\le j\\le l \n\\end{equation}\ndue to the support of $\\Psi_{G_j,k_j}^{\\lambda}$, where $\\widehat{f^{\\lambda,\\gamma,\\mu}}(\\xi):=\\widehat{f}(\\xi)\\chi_{C_0\\sqrt{n}2^{\\gamma-\\lambda}\\le |\\xi|\\le 2^{\\gamma+\\mu+3}}$.\n\\item For $\\mu\\ge 1$ and $\\lambda\\in\\mathbb{N}_0$, \n\\begin{equation}\\label{L2}\n\\Big( \\sum_{\\gamma\\in\\mathbb{Z}}{\\big\\Vert f^{\\lambda,\\gamma,{\\mu}}\\big\\Vert_{L^2}^2}\\Big)^{1\/2}\\lesssim ({\\mu}+\\lambda)^{1\/2}\\Vert f\\Vert_{L^2} \\lesssim \\mu^{1\/2}(\\lambda+1)^{1\/2}\\Vert f\\Vert_{L^2} \n\\end{equation} \nwhere Plancherel's identity is applied in the first inequality.\n\\end{itemize}\n\n \n\\begin{prop}[{\\cite[Proposition 2.4]{Paper1}}]\\label{keyapplication1}\nLet $m$ be a positive integer with $m\\ge 2$ and $0<q<\\frac{2m}{m-1}$. Fix $\\lambda\\in\\mathbb{N}_0$ and $\\vec{{G}}\\in \\mathcal{I}^{\\lambda}$.\nSuppose that $\\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\}_{\\vec{{G}}\\in \\mathcal{I}^{\\lambda}, \\gamma,\\mu\\in \\mathbb{Z}, \\vec{{k}}\\,\\in (\\mathbb Z^n)^m}$ is a sequence of complex numbers satisfying \n\\begin{equation*}\n\\sup_{\\gamma \\in \\mathbb{Z}}\\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\}_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m} \\big\\Vert_{\\ell^{\\infty}}\\le A_{\\vec{{G}},\\lambda,\\mu}\n\\end{equation*}\n and \n \\begin{equation*}\n \\sup_{\\gamma \\in\\mathbb{Z}}\\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\}_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m} \\big\\Vert_{\\ell^{q}}\\le B_{\\vec{{G}},\\lambda,\\mu,q}.\n \\end{equation*}\nThen the following statements hold:\n \\begin{enumerate}\n \\item For $1\\le r\\le 2$,  there exists a constant $C>0$, independent of $,\\vec{{G}}, \\lambda, \\mu$, such that\n\\begin{align*}\n&\\Big\\Vert \\Big(\\sum_{\\gamma\\in\\mathbb{Z}}\\Big| \\sum_{\\vec{{k}}\\,\\in \\mathcal{U}_1^{\\lambda+\\mu}} b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}L_{G_1,k_1}^{\\lambda,\\gamma}f_1^{\\lambda,\\gamma,\\mu}\\prod\n_{j=2}^{m}L_{G_j,k_j}^{\\lambda,\\gamma}f_j\\Big|^r\\Big)^{1\/r}\\Big\\Vert_{L^{2\/m}}\\\\\n&\\le C A_{\\vec{{G}},\\lambda,\\mu} 2^{\\lambda mn\/2}\\Big(\\sum_{\\gamma\\in\\mathbb{Z}}{\\Vert f_1^{\\lambda,\\gamma,\\mu}\\Vert_{L^2}^r} \\Big)^{1\/r} \\prod_{j=2}^{m}\\Vert f_{j}\\Vert_{L^2}\n\\end{align*}  \nfor Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$.\n \\item For $2\\le l\\le m$ there exists a constant $C>0$, independent of $\\vec{{G}}, \\lambda, \\mu$, such that\n\\begin{align*}\n&\\Big\\Vert \\sum_{\\gamma\\in\\mathbb{Z}} \\Big| \\sum_{\\vec{{k}}\\,\\in \\mathcal{U}_l^{\\lambda+\\mu}} b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\Big( \\prod_{j=1}^{l}L_{G_j,k_j}^{\\lambda,\\gamma}f_j^{\\lambda,\\gamma,\\mu}\\Big)\\Big( \\prod_{j=l+1}^{m}L_{G_j,k_{j}}^{\\lambda,\\gamma}f_{j}\\Big) \\Big| \\Big\\Vert_{L^{2\/m}}\\\\\n&\\le C A_{\\vec{{G}},\\lambda,\\mu}^{1-\\frac{(l-1)q}{2l}}B_{\\vec{{G}},\\lambda,\\mu,q}^{\\frac{(l-1)q}{2l}} 2^{\\lambda mn\/2}\\Big[ \\prod_{j=1}^{l}\\Big(\\sum_{\\gamma\\in\\mathbb{Z}}{\\Vert f_j^{\\lambda,\\gamma,\\mu}\\Vert_{L^2}^2} \\Big)^{1\/2}\\Big] \\Big[\\prod_{j=l+1}^{m}\\Vert f_{j}\\Vert_{L^2}\\Big]\n\\end{align*} \nfor Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$,\n where $\\prod_{m+1}^m$ is understood as empty.\n \\end{enumerate}\n\\end{prop}\n\nIn view of (\\ref{lgkest}), (\\ref{L2}) and Proposition \\ref{keyapplication1}, we actually obtain\n\\begin{align}\\label{1mainprop}\n&\\Big\\Vert \\Big( \\sum_{\\gamma\\in\\mathbb{Z}}\\Big| \\sum_{\\vec{{k}}\\,\\in\\mathcal{U}_{1}^{\\lambda+\\mu}}b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\prod_{j=1}^{m}L_{G_j,k_j}^{\\lambda,\\gamma}f_j\\Big|^2\\Big)^{1\/2}\\Big\\Vert_{L^{2\/m}}\\nonumber \\\\&\\lesssim A_{\\vec{{G}},\\lambda,\\mu}\\mu^{1\/2}2^{\\lambda mn\/2}(\\lambda+1)^{1\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{align}   \nand for $2\\le l\\le m$\n\\begin{align}\\label{2mainprop}\n&\\Big\\Vert  \\sum_{\\gamma\\in\\mathbb{Z}}\\Big| \\sum_{\\vec{{k}}\\,\\in\\mathcal{U}_{l}^{\\lambda+\\mu}}b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\gamma,\\mu}\\prod_{j=1}^{m}L_{G_j,k_j}^{\\lambda,\\gamma}f_j\\Big| \\Big\\Vert_{L^{2\/m}}\\nonumber\\\\\n&\\lesssim A_{\\vec{{G}},\\lambda,\\mu}^{1-\\frac{(l-1)q}{2l}}B_{\\vec{{G}},\\lambda,\\mu,q}^{\\frac{(l-1)q}{2l}} \\mu^{l\/2}2^{\\lambda mn\/2}(\\lambda+1)^{l\/2}    \\prod_{j=1}^{m}\\Vert f_{j}\\Vert_{L^2}.\n\\end{align}    \n    \n    \n\n\n\n\\section{An auxiliary  lemma} \n\n\nWe have the following extension of Lemma~5 in \\cite{BH}.\n\n\\begin{lm}\\label{AUX}\nLet $1<q\\le \\infty$ and $\\Omega \\in L^q(\\mathbb S^{mn-1})$. Suppose $1<p_1,\\dots , p_m<\\infty$ and $1\/m<p<\\infty$ satisfies $1\/p=1\/p_1+\\cdots +1\/p_m$ and \n\\begin{equation}\\label{AUXassume}\n\\frac 1p< \\frac{1}{q}+\\frac{m}{q' }.\n\\end{equation}\n Then the maximal operator\n\\[\n\\mathcal M_\\Omega(f_1,\\dots, f_m)(x)=\n\\sup_{R>0}  \\frac{1}{R^{mn} } \\idotsint\\limits_{|\\vec y|\\le R}   |\\Omega (\\vec{{y}}\\,' ) | \n\\prod_{j=1}^m \\big|  f_j(x-y_j) \\big|  ~ d\\vec{{y}}\\,\n\\]\nmaps $L^{p_1}(\\mathbb R^n) \\times \\cdots \\times L^{p_m}(\\mathbb R^n) $ to \n$L^{p }(\\mathbb R^n) $ with norm  bounded by a constant multiple of  $\\|\\Omega\\|_{L^q(\\mathbb{S}^{mn-1})}$.\n\\end{lm}\n\n\\begin{proof} \nSince $\\Vert \\Omega\\Vert_{L^r(\\mathbb{S}^{mn-1})}\\lesssim \\Vert \\Omega\\Vert_{L^{\\infty}(\\mathbb{S}^{mn-1})}$ for all $1<r<\\infty$ and there exists $1<q<\\infty$ such that $1\/p<1\/q+m\/q'<m~ (=1\/\\infty+m\/1)$,\nwe may assume $1<q<\\infty$. Without loss of generality, we may also assume that $\\|\\Omega\\|_{L^q(\\mathbb{S}^{mn-1})}=1$. \n\nWe split \n\\[\n\\Omega= \\Omega_0+\\sum_{l=1}^\\infty \\Omega_l ,\n\\]\nwhere $\\Omega_0=\\Omega \\chi_{|\\Omega|\\le 2}$ and $\\Omega_l=\\Omega \\chi_{2^l<|\\Omega|\\le 2^{l+1} }$ for $l\\ge 1$. \nThen H\\\"older's inequality and Chebyshev's inequality give\n\\[\n\\|\\Omega_l\\|_{L^1}  \\le |\\textup{Supp} \\, \\Omega_l|^{\\frac{1}{q'}} \\le \\|\\Omega\\|_{L^q}^{\\frac{q}{q'}} 2^{-l \\frac{q}{q'}} =2^{-l \\frac{q}{q'}} \n\\]\nand obviously \n\\begin{equation}\\label{Omegainfty}\n\\|\\Omega_l \\|_{L^\\infty} \\le 2^{l+1}.\n\\end{equation}\n\nWe first claim that for $1<r,r_1,\\dots,r_m<\\infty$ with $1\/r=1\/r_1+\\cdots+1\/r_m$,\n\\begin{equation}\\label{664455-1}\n\\big\\|\\mathcal M_{\\Omega_l} \\big\\|_{L^{r_1}\\times \\cdots \\times L^{r_m}\\to L^{r}} \n\\lesssim \\|\\Omega_l\\|_{L^1(\\mathbb{S}^{mn-1})}  \\lesssim 2^{-l \\frac{q}{q'}}. \n\\end{equation}\n\nTo verify this estimate,  \nwe choose indices $\\mu_1,\\dots,\\mu_m$ satisfying $$1\/\\mu_1+\\cdots+1\/\\mu_m=1$$ and\n$$1<\\mu_j<r_j \\qquad \\text{ for each }~ 1\\le j\\le m. $$\nThen a direct computation using H\\\"older's inequality yields \n\\[\n\\mathcal M_{\\Omega_l}\\big(f_1,\\dots, f_m\\big)(x) \\le  \\int_{\\mathbb S^{mn-1}}   \\big|\\Omega_l (\\vec \\theta\\,  )\\big| \\prod_{j=1}^{m}\\mathcal{M}_{\\mu_j}^{\\theta_j}f_j(x)    ~ d\\vec \\theta\\, ,\n\\] \nwhere the directional maximal operator $\\mathcal{M}_{\\mu_j}^{\\theta_j}$ is defined by\n\\[\n\\mathcal M_{\\mu_j}^{\\theta_j} f_j(x)  := \\sup_{R>0}\\Big(  \\frac{1}{R  }\\int_0^R \\big|  f (x-t\\theta_j) \\big|^{\\mu_j} ~dt\\Big)^{1\/\\mu_j}. \n\\]\nIt follows from this that \n\\[\n \\big\\| \\mathcal M_{\\Omega_l}(f_1,\\dots, f_m)\\big\\|_{L^{r}} \\le  \\int_{\\mathbb S^{mn-1}}   |\\Omega (\\vec \\theta\\,  ) |  \\prod_{j=1}^m  \n\\big\\| \\mathcal M_{\\mu_j}^{\\theta_j}f_j  \\big\\|_{L^{r_j}}  ~ d\\vec \\theta,\n\\]\nwhere Minkowski's inequality and H\\\"older's inequality are applied.\nUsing the $L^{r_j}$ boundedness of $\\mathcal M_{\\mu_j}^{\\theta_j} $ for $0<\\mu_j<r_j$ with constants independent of $\\theta_j$\n(by the method of rotations), we obtain \\eqref{664455-1}.\n\n\nThen the case $p>1$ (for which $q>1$ implies the assumption (\\ref{AUXassume})) in the assertion follows from summing the estimates (\\ref{664455-1}) over $l\\ge 0$.\n\n\nThe other case $1\/m<p\\le 1$ can be proved by interpolation with the $L^1\\times\\cdots\\times L^1\\to L^{1\/m,\\infty}$ estimate.\nLet $\\mathcal{M}$ be the Hardy-Littlewood maximal operator.\nThen it is easy to see the pointwise estimate\n$$\\mathcal M_{\\Omega_l}\\big(f_1,\\dots , f_m\\big)(x) \\le 2^{l+1} \\prod_{j=1}^m \\mathcal M f_j(x).$$\nand this proves that\n\\begin{equation}\\label{664455}\n\\big\\|\\mathcal M_{\\Omega_l} \\big\\|_{L^1\\times \\cdots \\times L^1\\to L^{1\/m,\\infty}} \\lesssim 2^l, \n\\end{equation}\n using H\\\"older's inequality for weak type spaces (\\cite[p 16]{CFA}), the estimate (\\ref{Omegainfty}), and the weak $(1,1)$ boundedness of $\\mathcal{M}$.\n Now we fix $0<p_1,\\dots,p_m<\\infty$ and $1\/m<p\\le 1$, and choose $r>1$ such that \n\\begin{equation*}\n\\frac{1}{p}<\\frac{1}{rq}+\\frac{m}{q'} \\,\\,\\,\\Big( \\!\\!<\\frac{1}{q}+\\frac{m}{q'} \\Big),\n\\end{equation*}\nor,  equivalently,\n$$\\frac{q(m-1\/p)}{q'(m-1\/r)}-\\frac{1\/p-1\/r}{m-1\/r}>0.$$\nThen the interpolation between (\\ref{664455}) and (\\ref{664455-1}) with appropriate $(r_1,\\dots, r_m)$ satisfying $1\/r=1\/r_1+\\dots+1\/r_m$ (using Theorem  7.2.2 in \\cite{MFA}) yields\n\\begin{equation*}\n\\big\\|\\mathcal M_{\\Omega_l} \\big\\|_{L^{p_1}\\times \\cdots \\times L^{p_m}\\to L^{p }}   \\lesssim 2^{l \\frac{1\/p-1\/r}{m-1\/r}} 2^{-l \\frac{q}{q'} \\frac{m-1\/p}{m-1\/r} }=2^{-l(\\frac{q(m-1\/p)}{q'(m-1\/r)}-\\frac{1\/p-1\/r}{m-1\/r})}\n\\end{equation*}\nFinally, the exponential decay in $l$ together \nwith the fact that $\\|\\cdot \\|_{L^p}^p$ is a subadditive quantity for $0<p\\le 1$ implies the claimed conclusion. \n\\end{proof}\n\n \n\n\\section{Proof of Theorem \\ref{MAXSINGINT}}\n\nLet $\\frac{2m}{m+1}<q<2$ and $\\Omega$ in $L^q(\\mathbb S^{mn-1})$.\nWe  use a dyadic decomposition introduced by Duoandikoetxea and Rubio de Francia \\cite{Du_Ru}.\nWe choose a Schwartz function $\\Phi^{(m)}$ on $(\\mathbb R^n)^m$ such that its Fourier transform $\\widehat{\\Phi^{(m)}}$ is supported in the annulus $\\{\\vec{{\\xi}}\\,\\in (\\mathbb R^n)^m: 1\/2\\le |\\vec{{\\xi}}\\,|\\le 2\\}$ and satisfies\n$\\sum_{j\\in\\mathbb{Z}}\\widehat{\\Phi^{(m)}_j}(\\vec{{\\xi}}\\,)=1$ for $\\vec{{\\xi}}\\,\\not= \\vec{\\boldsymbol{0}}$ where $\\widehat{\\Phi_j^{(m)}}(\\vec{{\\xi}}\\,):=\\widehat{\\Phi^{(m)}}(\\vec{{\\xi}}\\,\/2^j)$.\nFor $\\gamma\\in\\mathbb{Z}$  let\n $$ K^{\\gamma}(\\vec{{y}}\\,):=\\widehat{\\Phi^{(m)}}(2^{\\gamma}\\vec{{y}}\\,)K(\\vec{{y}}\\,), \\quad \\vec{{y}}\\,\\in (\\mathbb R^n)^m$$\n and then we observe that $K^\\gamma(\\vec{{y}}\\,)=2^{\\gamma mn} K^0(2^\\gamma \\vec{{y}}\\,)$.\n For $\\mu\\in\\mathbb{Z}$ we define\n \\begin{equation}\\label{kernelcharacter}\nK_{\\mu}^{\\gamma}(\\vec{{y}}\\,):=\\Phi_{{\\mu}+\\gamma}^{(m)}\\ast K^{\\gamma}(\\vec{{y}}\\,)=2^{\\gamma mn}[\\Phi_{{\\mu}}^{(m)}\\ast K^{0}](2^\\gamma \\vec{{y}}\\,).\n\\end{equation}\nIt follows from this definition that \n$$\n\\widehat{K^\\gamma_\\mu}(\\vec{{\\xi}}\\,)= \\widehat{\\Phi^{(m)}}(2^{-(\\mu+\\gamma)}\\vec{{\\xi}}\\,)\\widehat{K^0}(2^{-\\gamma}\\vec{{\\xi}}\\,)=\\widehat{K^0_\\mu}(2^{-\\gamma} \\vec{{\\xi}}\\,),\n$$\nwhich implies that $\\widehat{K^\\gamma_\\mu}$ is bounded uniformly in $\\gamma$ while they have almost disjoint supports, so it is natural to add them together as follows:\n $$K_{\\mu}(\\vec{{y}}\\,):=\\sum_{\\gamma\\in \\mathbb{Z}}{K_{\\mu}^{\\gamma}(\\vec{{y}}\\,)}.$$\n\n\n\\subsection{Reduction}\nWe introduce the maximal  operator\n\\begin{equation*}\n\\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big)(x) := \n\\sup_{\\tau\\in \\mathbb Z} \\bigg| \\sum_{\\gamma<\\tau} \n\\int_{(\\mathbb R^n)^m }{K^{\\gamma}(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\, \n\\bigg| \n\\end{equation*}\nfor $x\\in \\mathbb R^n$. \nThen we claim that\n\\begin{equation}\\label{Est77}\n\\mathcal{L}_{\\Omega}^*\\big(f_1,\\dots,f_m\\big)\n\\le \\mathcal{M}_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x)+  \\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big) \n\\end{equation}\nTo prove \\eqref{Est77} we  introduce the notation \n$$K^{(\\epsilon)} ( \\vec y\\,) := K ( \\vec y\\,)  \\chi_{|\\vec y\\,| \\ge \\epsilon}, \\qquad \\widetilde{K}^{(\\epsilon)} ( \\vec y\\,) := K ( \\vec y\\,) \\big( 1-  \\widehat{\\Theta^{(m)}} (\\vec y\/\\epsilon) \\big),$$\nsetting $\\widehat{\\Theta^{(m)}}(\\vec{{y}}\\,):=1-\\sum_{\\gamma\\in\\mathbb{N}}{\\widehat{\\Phi^{(m)}}(\\vec{{y}}\\,\/2^{\\gamma})}$ so that \\[\n\\textup{Supp}(\\widehat{\\Theta^{(m)}})\\subset \\{\\vec{{y}}\\,\\in (\\mathbb R^n)^m:|\\vec{{y}}\\,|\\le 2\\}\n\\]\n and $\\widehat{\\Theta^{(m)}}(\\vec{{y}}\\,)=1$ for $|\\vec{{y}}\\,|\\le 1$.\n\n\nGiven $\\epsilon>0$ choose $\\rho\\in \\mathbb Z$ such that $2^{\\rho}\\le \\epsilon<2^{\\rho+1} $. \nThen we write\n\\begin{align}\n&\\bigg| \\int_{(\\mathbb R^n)^m\\setminus B(0,\\epsilon)}{K(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\, \\bigg| \n\\notag\\\\\n&\\qquad\\qquad \\le \n\\bigg| \\int_{(\\mathbb R^n)^m }{\\big( K^{(\\epsilon)}(\\vec{{y}}\\,) - \\widetilde{K}^{(2^{\\rho})}(\\vec{{y}}\\,) \\big) \\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\, \\bigg| \\label{651} \\\\\n&\\qquad \\qquad\\qquad \\qquad+ \\bigg| \\int_{(\\mathbb R^n)^m}   \\widetilde{K}^{(2^{\\rho})}(\\vec{{y}}\\,) \\prod_{j=1}^{m}f_j(x-y_j) ~d\\,\\vec{{y}}\\,\\bigg| \\label{652}.\n\\end{align}\nTerm \\eqref{652} is clearly less than\n\\begin{equation*}\n\\bigg|\\sum_{\\gamma\\in\\mathbb{Z}: \\gamma<-\\rho}\\int_{(\\mathbb R^n)^m}K^{\\gamma}(\\vec{{y}}\\,) \\prod_{j=1}^{m}f_j(x-y_j) ~d\\,\\vec{{y}}\\,   \\bigg|\\le \\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big)(x),\n\\end{equation*}\nwhile \\eqref{651} is controlled by\n$\n \\mathcal{M}_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x)\n$\nas\n\\begin{equation*}\n\\big|K^{(\\epsilon)}(\\vec{{y}}\\,) - \\widetilde K^{(2^{\\rho})}(\\vec{{y}}\\,) \\big|\\lesssim |K(\\vec{{y}}\\,)| \\chi_{|\\vec{{y}}\\,|\\approx 2^{\\rho}}\\lesssim \\frac{|\\Omega(\\vec{{y}}\\,')|}{2^{\\rho mn}}\\chi_{|\\vec{{y}}\\,|\\lesssim 2^{\\rho}}.\n\\end{equation*}\nThus \\eqref{Est77} follows after taking the supremum over all $\\epsilon>0$. \n\nSince  the boundedness of $\\mathcal{M}_{\\Omega}$ follows from Lemma~\\ref{AUX} with the fact that $q>\\frac{2m}{m+1}$ implies $\\frac{m}{2} <\\frac{1}{q}+\\frac{m}{q'}$, \nmatters reduce to the boundedness of $\\mathcal{L}_{\\Omega}^{\\sharp}$. \n\nFor each $\\gamma\\in\\mathbb{Z}$ let $$K_{\\mu}:=\\sum_{\\gamma\\in\\mathbb{Z}}K_{\\mu}^{\\gamma}.$$\nIn  the study of multilinear rough singular integral operators $\\mathcal{L}_{\\Omega}$ in \\cite{Paper1} whose kernel is $\\sum_{\\gamma\\in\\mathbb{Z}}K^{\\gamma}=\\sum_{\\mu\\in\\mathbb{Z}}\\sum_{\\gamma\\in\\mathbb{Z}}K_{\\mu}^{\\gamma}=\\sum_{\\mu\\in\\mathbb{Z}}K_{\\mu}$,\nthe part where $\\mu$ is less than a constant is relatively simple because the Fourier transform of $K_{\\mu}$ satisfies the estimate \n\\begin{equation}\\label{symbolest}\n\\big|\\partial^{\\alpha}\\widehat{K_{\\mu}}(\\vec{{\\xi}}\\,)\\big|\\lesssim \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}|\\vec{{\\xi}}\\,|^{-|\\alpha|}Q(\\mu),\\qquad 1<q\\le \\infty\n\\end{equation}\nfor all multiindices $\\alpha$ and $\\vec{{\\xi}}\\,\\in\\mathbb{R}^{mn}\\setminus \\{0\\}$, where  $Q(\\mu)=2^{(mn-\\delta')\\mu}$ if $\\mu\\ge 0$ and $Q(\\mu)=2^{\\mu(1-\\delta')}$ if $\\mu<0$ for some $0<\\delta'<1\/q'$, which is the condition of the Coifman-Meyer multiplier theorem \\cite{CM}, \\cite[Theorem 7.5.3]{MFA} with constant $\\Vert \\Omega\\Vert_{L^q(\\mathbb S^{mn-1})}Q(\\mu)$.\nThe remaining case when $\\mu$ is large enough was handled by using product-type wavelet decompositions.\nWe expect a similar strategy would work in handling $\\mathcal{L}_{\\Omega}^{\\sharp}$.\n\nTo argue strictly, we write\n\\begin{equation*}\n  \\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big)  \n  \\le\n \\widetilde   \\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big)  + \\sum_{\\mu\\in\\mathbb{Z}:2^{\\mu-10}>C_0\\sqrt{mn}} \n  \\mathcal{L}_{\\Omega,\\mu}^{\\sharp}\\big(f_1,\\dots,f_m\\big) ,\n  \\end{equation*}\nwhere we set\n$$\n \\widetilde   \\mathcal{L}_{\\Omega}^{\\sharp}\\big(f_1,\\dots,f_m\\big) (x) \n :=\n  \\sup_{\\tau \\in \\mathbb Z} \\bigg| \n  \\int_{(\\mathbb R^n)^m }{  \\sum_{ \\gamma<\\tau} \\sum_{\\mu\\in\\mathbb{Z}:2^{\\mu-10}\\le C_0\\sqrt{mn}} K^{\\gamma}_{\\mu}(\\vec{{y}}\\,)   \\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\,  \n  \\bigg| \n$$\nand\n$$\n  \\mathcal{L}_{\\Omega,\\mu}^{\\sharp}\\big(f_1,\\dots,f_m\\big)(x) \n  := \\sup_{\\tau\\in \\mathbb Z} \\bigg| \n  \\sum_{\\gamma<\\tau} \n  \\int_{(\\mathbb R^n)^m }{  K^{\\gamma}_{\\mu}(\\vec{{y}}\\,)   \\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\,  \n  \\bigg| .\n  $$\n  Then Theorem \\ref{MAXSINGINT} follows from the following two propositions:\n  \\begin{prop}\\label{propo1}\n  Let $1<p_1,\\dots,p_m\\le \\infty$ and $1\/p=1\/p_1+\\cdots+1\/p_m$. Suppose that  $1<q<\\infty$ and $\\Omega\\in L^q(\\mathbb{S}^{mn-1})$ with $\\int_{\\mathbb{S}^{mn-1}}\\Omega d\\sigma =0$. Then there exists a constant $C>0$ such that\n  \\begin{equation*}\n  \\big\\Vert \\widetilde{\\mathcal{L}}_{\\Omega}^{\\sharp}(f_1,\\dots,f_m)\\big\\Vert_{L^p}\\le C\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^{p_j}}\n  \\end{equation*}\n  for Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$.\n  \\end{prop}\n  \n  \n  \\begin{prop}\\label{propo2}\n  Let $\\frac{2m}{m+1}<q\\le \\infty$ and $\\Omega\\in L^q(\\mathbb{S}^{mn-1})$ with $\\int_{\\mathbb{S}^{mn-1}}\\Omega d\\sigma =0$. \n  Suppose that $\\mu\\in\\mathbb{Z}$ satisfies $2^{\\mu-10}>C_0\\sqrt{mn}$. \n  Then there exist $C, \\epsilon_0>0$ such that  \n\\begin{equation*}\n\\big\\|  \\mathcal{L}_{\\Omega,\\mu}^{\\sharp}(f_1,\\dots,f_m)\\big\\|_{L^{2\/m} } \\lesssim  2^{-\\epsilon_0 \\mu} \\| \\Omega\\|_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{equation*}\n  for Schwartz functions $f_1,\\dots,f_m$ on $\\mathbb R^n$.\n \n  \\end{prop}\n\n\n\n\n\n\n\n\n\\subsection{Proof of Proposition \\ref{propo1}}\n\nWe  decompose $\\widetilde{\\mathcal{L}}_{\\Omega}^{\\sharp}$ further so that the Coifman-Meyer multiplier theorem is involved:\nSetting\n$$\\widetilde{K}(\\vec{{y}}\\,):=\\sum_{\\mu\\in\\mathbb{Z}: 2^{\\mu-10}\\le C_0\\sqrt{mn}}K_{\\mu}(\\vec{{y}}\\,)=\\sum_{\\mu\\in\\mathbb{Z}: 2^{\\mu-10}\\le C_0\\sqrt{mn}}\\;\\sum_{\\gamma\\in\\mathbb{Z}}K_{\\mu}^{\\gamma}(\\vec{{y}}\\,),$$\n$\\widetilde{\\mathcal{L}}^{\\sharp}_{\\Omega}\\big(f_1,\\dots,f_m\\big)(x)$ is controlled by the sum of \n\\begin{equation*}\nT_{\\widetilde{K}}^*\\big(f_1,\\dots,f_m\\big)(x):= \\sup_{\\tau\\in\\mathbb{Z}}\\Big|\\int_{|y|> 2^{-\\tau}}\\widetilde{K}(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)~d\\,\\vec{{y}}\\, \\Big|\n\\end{equation*}\nand\n\\begin{equation*}\n\\mathfrak{T}_{K}^{**}\\big(f_1,\\dots,f_m\\big)(x):= \\sup_{\\tau\\in\\mathbb{Z}}\\Big| \\int_{(\\mathbb R^n)^m}  {K}^{**}_{\\tau}(\\vec{{y}}\\,)    \\prod_{j=1}^{m}f_j(x-y_j)  ~   d\\,\\vec{{y}}\\,   \\Big|, \n\\end{equation*}\nwhere \n\\begin{equation*}\n{K}^{**}_{\\tau}(\\vec{{y}}\\,):=\\Big(\\sum_{\\mu\\in\\mathbb{Z}: 2^{\\mu-10}\\le C_0\\sqrt{mn}}\\; \\sum_{\\gamma<\\tau}K_{\\mu}^{\\gamma}(\\vec{{y}}\\,)\\Big) -\\widetilde{K}(\\vec{{y}}\\,)\\chi_{|\\vec{{y}}\\,|> 2^{-\\tau}}.  \n\\end{equation*}\n\n\nTo obtain the boundedness of $T_{\\widetilde{K}}^*$, \nwe claim that $\\widetilde{K}$ is an $m$-linear Calder\\'on-Zygmund kernel with constant $C\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}$ for $1<q< \\infty$.\nIndeed, it follows from (\\ref{symbolest}) that\n\\begin{equation*}\n\\big| \\partial^{\\alpha} \\widehat{\\widetilde{K}}(\\vec{{\\xi}}\\,)\\big|\\le \\sum_{\\mu\\in\\mathbb{Z}: 2^{\\mu-10}\\le C_0\\sqrt{mn}}\\big|\\partial^{\\alpha}\\widehat{K_{\\mu}}(\\vec{{\\xi}}\\,)\\big|\\lesssim \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}|\\vec{{\\xi}}\\,|^{-|\\alpha|}\n\\end{equation*} as the sum of $Q(\\mu)$ over $\\mu$ satisfying $2^{\\mu-10}\\le C_0\\sqrt{mn}$ converges.\nThen $\\widetilde{K}$ satisfies the size and smoothness conditions for $m$-linear Calder\\'on-Zygmund kernel with constant $C\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}$, as mentioned in the proof of \\cite[Proposition 6]{Gr_To}.\nSince $\\widetilde{K}$ is a Calder\\'on-Zygmund kernel, \nCotlar's inequality in \\cite[Theorem 1]{GT-Indiana} yields that $T_{\\widetilde{K}}^*$ is bounded on the full range of exponents with constant $C\\Vert \\Omega\\Vert_{L^{q}(\\mathbb{S}^{mn-1})}$.\n\n\nTo handle the boundedness of the operator $\\mathfrak{T}_K^{**}$, we observe that the kernel ${K}_{\\tau}^{**}$ can be written as\n\\begin{equation*}\n{K}_{\\tau}^{**}(\\vec{{y}}\\,)=\\sum_{\\mu\\in\\mathbb{Z}:2^{\\mu-10}\\le C_0\\sqrt{mn} }\\; \\Big( \\sum_{\\gamma<\\tau}K_{\\mu}^{\\gamma}(\\vec{{y}}\\,)\\chi_{|\\vec{{y}}\\,|\\le 2^{-\\tau}}-  \\sum_{\\gamma\\ge \\tau}K_{\\mu}^{\\gamma}(\\vec{{y}}\\,)\\chi_{|\\vec{{y}}\\,|>2^{-\\tau}}\\Big)\n\\end{equation*}\nand thus \n\\begin{equation*}\n\\mathfrak{T}_{K}^{**}\\big(f_1,\\dots,f_m\\big)(x) \\le \\sup_{\\tau\\in\\mathbb{Z}}\\; \\sum_{\\mu\\in\\mathbb{Z}:2^{\\mu-10}\\le C_0\\sqrt{mn} } \\mathcal{I}_{\\mu,\\tau}(x)+\\mathcal{J}_{\\mu,\\tau}(x)\n\\end{equation*} where\n\\begin{equation*}\n\\mathcal{I}_{\\mu,\\tau}(x):=  \\sum_{\\gamma<\\tau}\\Big|\\int_{|\\vec{{y}}\\,|<2^{-\\tau}} K_{\\mu}^{\\gamma}(\\vec{{y}}\\,) \\prod_{j=1}^{m}f_j(x-y_j) \\;d\\,\\vec{{y}}\\, \\Big|,\n\\end{equation*}\n\\begin{equation*}\n\\mathcal{J}_{\\mu,\\tau}(x):= \\sum_{\\gamma \\ge \\tau}\\Big|\\int_{|\\vec{{y}}\\,|\\ge 2^{-\\tau}} K_{\\mu}^{\\gamma}(\\vec{{y}}\\,) \\prod_{j=1}^{m}f_j(x-y_j) \\;d\\,\\vec{{y}}\\, \\Big|.\n\\end{equation*}\nWe claim that there exists $\\epsilon>0$ such that\n\\begin{equation}\\label{IJest}\n\\mathcal{I}_{\\mu,\\tau}+ \\mathcal{J}_{\\mu,\\tau}\\lesssim_{C_0,m,n} 2^{\\epsilon \\mu}\\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\mathcal{M}f_j  ~\\text{ uniformly in ~} \\tau\\in\\mathbb{Z}\n\\end{equation}\nfor $\\mu$ satisfying $2^{\\mu-10}\\le C_0\\sqrt{mn}$, where we recall $\\mathcal M$ is the Hardy-Littlewood maximal operator.\nThen, using H\\\"older's inequality and the boundedness of $\\mathcal{M}$, we obtain\n$$\n\\big\\Vert  \\mathfrak{T}_{K}^{**}\\big(f_1,\\dots,f_m\\big) \\big\\Vert_{L^p}\\lesssim \\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})} \\Big\\Vert \\prod_{j=1}^{m}\\mathcal{M}f_j  \\Big\\Vert_{L^p}\\lesssim \\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^{p_j}}\n$$\n for $1<p_1,\\dots,p_m\\le \\infty$ and $0<p\\le \\infty$ satisfying $1\/p=1\/p_1+\\dots+1\/p_m$ as $\\sum_{\\mu: 2^{\\mu-10}\\le C_0\\sqrt{mn}}2^{\\epsilon \\mu}$ converges.\nTherefore, let us prove (\\ref{IJest}).\n\n\n\nUsing (\\ref{kernelcharacter}), we have\n\\begin{align*}\n\\mathcal{I}_{\\mu,\\tau}(x)&\\lesssim \\sum_{\\gamma<\\tau}\\int_{|\\vec{{y}}\\,|<2^{-\\tau}}\\int_{|\\vec{{z}}\\,|\\approx 1}2^{\\gamma mn}2^{\\mu mn}|\\Omega(\\vec{{z}}\\,')|d\\vec{{z}}\\, \\prod_{j=1}^{m}|f_j(x-y_j)|~d\\,\\vec{{y}}\\,\\\\\n &\\lesssim 2^{\\mu mn}\\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\frac{1}{2^{-\\tau mn}}\\int_{|\\vec{{y}}\\,|<2^{-\\tau}} \\prod_{j=1}^{m}|f_j(x-y_j)| ~d\\,\\vec{{y}}\\,\\\\\n &\\lesssim 2^{ \\mu mn}\\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\mathcal{M}f_j(x)\n\\end{align*} as desired.\n\n In addition,\n \\begin{align*}\n \\mathcal{J}_{\\mu,\\tau}(x)\\le \\sum_{\\gamma\\ge \\tau}\\int_{|\\vec{{y}}\\,|\\ge 2^{-\\tau}}2^{\\gamma mn}\\Big|\\int_{|\\vec{{z}}\\,|\\approx 1}{\\Phi_{\\mu}(2^{\\gamma}\\vec{{y}}\\,-\\vec{{z}}\\,)\\Omega(\\vec{{z}}\\,')}d\\,\\vec{{z}}\\, \\Big|\\prod_{j=1}^{m}|f_j(x-y_j)| ~ d\\,\\vec{{y}}\\,.\n \\end{align*}\n   Since $\\Omega$ has vanishing mean, we have\n\\begin{align*}\n&\\Big|\\int_{|\\vec{{z}}\\,|\\approx 1}{\\Phi_{\\mu}(2^{\\gamma}\\vec{{y}}\\,-\\vec{{z}}\\,)\\,\\Omega(\\vec{{z}}\\,')}\\,d\\,\\vec{{z}}\\, \\Big| \\\\\n&\\lesssim 2^{\\mu (mn+1)}\\int_{|\\vec{{z}}\\,|\\approx 1}\\int_0^1\\big| \\nabla \\Phi (2^{\\mu+\\gamma}\\vec{{y}}\\,-2^{\\mu}t\\vec{{z}}\\,)\\big|\\;dt\\; |\\Omega(\\vec{{z}}\\,')|~ d\\, \\vec{{z}}\\,. \n\\end{align*}\nNow we choose a constant $M$ such that $mn<M<mn+1$ and see that\n\\begin{align*}\n\\big| \\nabla \\Phi (2^{\\mu+\\gamma}\\vec{{y}}\\,-2^{\\mu}t\\vec{{z}}\\,)\\big|&\\lesssim_M \\frac{1}{(1+|2^{\\mu+\\gamma}\\vec{{y}}\\,-2^{\\mu}t\\vec{{z}}\\,|)^M}\\\\\n&\\lesssim_{C_0,m,n,M} \\frac{1}{(1+2^{\\mu+\\gamma}|\\vec{{y}}\\,|)^M}\\le \\frac{1}{2^{M(\\mu+\\gamma)}}\\frac{1}{|\\vec{{y}}\\,|^{M}}\n\\end{align*}\nas $|\\vec{{z}}\\,|\\approx 1$, $0<t<1$, and $2^{\\mu-10}\\le C_0\\sqrt{mn}$.\nThis yields that\n\\begin{align*}\n \\mathcal{J}_{\\mu,\\tau}(x)&\\lesssim 2^{\\mu(mn+1-M)}\\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\Big(\\sum_{\\gamma\\ge \\tau}2^{-\\gamma(M- mn)}\\Big)\\\\\n  &\\qquad \\qquad \\qquad \\qquad \\times \\int_{|\\vec{{y}}\\,|\\ge 2^{-\\tau}}{\\frac{1}{|\\vec{{y}}\\,|^M}\\prod_{j=1}^{m}|f_j(x-y_j)|}~  d\\, \\vec{{y}}\\,.\n\\end{align*}\nSince $M>mn$,  the sum over $\\gamma\\ge \\tau$ converges to $2^{-\\tau(M-mn)}$ and the integral over $|\\vec{{y}}\\,|\\ge 2^{-\\tau}$ is estimated by\n\\begin{align*}\n&\\sum_{l=0}^{\\infty}\\int_{2^{-\\tau+l}\\le |\\vec{{y}}\\,|<2^{-\\tau+l+1}}{\\frac{1}{|\\vec{{y}}\\,|^M}\\prod_{j=1}^{m}|f_j(x-y_j)|}~ d\\,\\vec{{y}}\\,\\\\\n&\\lesssim 2^{\\tau(M-mn)}\\sum_{l=0}^{\\infty}2^{-l(M-mn)} \\Big(\\frac{1}{2^{(-\\tau+l+1)mn}}\\int_{|\\vec{{y}}\\,|\\le 2^{-\\tau+l+1}}\\prod_{j=1}^{m}|f_j(x-y_j)| ~ d\\,\\vec{{y}}\\,\\Big)\\\\\n&\\lesssim 2^{\\tau(M-mn)}\\prod_{j=1}^{m}\\mathcal{M}f_j(x).\n\\end{align*}\nFinally, we have\n\\begin{equation*}\n \\mathcal{J}_{\\mu,\\tau} \\lesssim 2^{\\mu(mn+1-M)}\\Vert \\Omega\\Vert_{L^1(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\mathcal{M}f_j ,\n\\end{equation*} which completes the proof of (\\ref{IJest}).\n\n \n\\subsection{Proof of Proposition \\ref{propo2}}\nThe proof is based on the wavelet decomposition and the recent developments in  \\cite{Paper1}.\nRecalling that $\\widehat{K_{\\mu}^0}\\in L^{q'}$, we apply the wavelet decomposition (\\ref{daubechewavelet})  to write \n\\begin{equation*}\n\\widehat{K_{\\mu}^0}(\\vec{{\\xi}}\\,)=\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}}\\sum_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m}b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\Psi_{G_1,k_1}^{\\lambda}(\\xi_1)\\cdots \\Psi_{G_m,k_m}^{\\lambda}(\\xi_m)\n\\end{equation*}\nwhere \n\\begin{equation*}\nb_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}:=\\int_{(\\mathbb R^n)^m}{\\widehat{K_{\\mu}^0}(\\vec{{\\xi}}\\,)\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}(\\vec{{\\xi}}\\,)}~ d\\,\\vec{{\\xi}}\\,.\n\\end{equation*}\nIt is known in \\cite{Paper1} that for any $0<\\delta<1\/q'$,\n\\begin{equation}\\label{maininftyest}\n\\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\}_{\\vec{{k}}\\,}\\big\\Vert_{\\ell^{\\infty}}\\lesssim      2^{-\\delta {\\mu}}2^{-\\lambda (M+1+mn)} \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\n\\end{equation} where $M$ is the number of vanishing moments of $\\Psi_{\\vec{{G}}}$. \nMoreover, it follows from the inequality (\\ref{lqestimate}), the Hausdorff-Young inequality, and Young's inequality that\n\\begin{align}\\label{mainlqest}\n\\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\}_{\\vec{{k}}\\,}\\big\\Vert_{\\ell^{q'}}&\\lesssim 2^{-\\lambda mn (1\/2-1\/q')}\\Vert \\widehat{K_{\\mu}^0}\\Vert_{L^{q'}}\\lesssim 2^{-\\lambda mn(1\/q-1\/2)}\\Vert \\Omega \\Vert_{L^q(\\mathbb{S}^{mn-1})}.\n\\end{align} \n\n\n\n\nNow we may assume that $2^{\\lambda+\\mu-2}\\le |\\vec k|\\le 2^{\\lambda+\\mu+2}$ due to the compact supports of $\\widehat{K_{\\mu}^0}$ and $\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}$. \nIn addition, by symmetry, it suffices to focus on the case $|k_1|\\ge\\cdots\\ge |k_m|$.\nSince $\\widehat{K_{\\mu}^\\gamma}(\\vec{{\\xi}}\\,)=\\widehat{K_{\\mu}^0}(\\vec{{\\xi}}\\,\/2^\\gamma)$, the boundedness of $ \\mathcal{L}_{\\Omega,\\mu}^{\\sharp}$ is reduced to the inequality\n\\begin{align}\\label{2mmainest}\n&\\bigg\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\Big| \\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}} \\sum_{\\gamma\\in\\mathbb{Z}: \\gamma<\\tau}\\sum_{\\vec{{k}}\\,\\in\\mathcal{U}^{\\lambda+\\mu}} b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\prod_{j=1}^{m} L_{G_j,k_j}^{\\lambda,\\gamma}f_j\\Big|\\bigg\\Vert_{L^{2\/m}}\\nonumber\\\\\n&\\lesssim 2^{-\\epsilon_0 \\mu}\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{align} \nwhere the operators $L_{G_j,k_j}^{\\lambda,\\gamma}$ and the set $\\mathcal{U}^{\\lambda+\\mu}$ are defined as in (\\ref{lgkest}) and (\\ref{uset}). \nWe split $\\mathcal{U}^{\\lambda+{\\mu}}$ into $m$ disjoint subsets $\\mathcal{U}_l^{\\lambda+{\\mu}}$ ($1\\le l\\le m$) as before such that\nfor $k\\in \\mathcal{U}^{\\lambda+\\mu}_l$ we have \n$$|k_1|\\ge\\cdots \\ge |k_l|\\ge 2C_0\\sqrt n\\ge |k_{l+1}|\\ge\\cdots\\ge |k_m|.$$\nThen the left-hand side of (\\ref{2mmainest}) is estimated by\n\\begin{equation*}\n\\bigg( \\sum_{l=1}^{m}\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}} \\bigg\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\Big| \\sum_{\\gamma\\in\\mathbb{Z}:\\gamma<\\tau} \\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m) \\Big|     \\bigg\\Vert_{L^{2\/m}}^{2\/m}\\bigg)^{m\/2}\n\\end{equation*} \nwhere $\\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}$ is defined by\n\\begin{equation*}\n\\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}\\big(f_1,\\dots,f_m\\big):=\\sum_{\\vec{{k}}\\,\\in\\mathcal{U}_l^{\\lambda+{\\mu}}}b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu} \\Big(\\prod_{j=1}^{m}L_{G_j,k_j}^{\\lambda,\\gamma}f_j \\Big).\n\\end{equation*}\n\nWe claim that for each $1\\le l\\le m$ there exists $\\epsilon_0, M_0>0$ such that\n\\begin{align}\\begin{split}\\label{2mmaingoal}\n \\bigg\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\Big| &\\sum_{\\gamma\\in\\mathbb{Z}:\\gamma<\\tau} \\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m) \\Big|     \\bigg\\Vert_{L^{2\/m}} \\\\\n&\\lesssim 2^{-\\epsilon_0 \\mu_0}2^{-\\lambda M_0}\\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2},\n\\end{split}\\end{align} \n which concludes (\\ref{2mmainest}). Therefore it remains to prove \\eqref{2mmaingoal}. \n\n\n\\subsubsection*{Proof of (\\ref{2mmaingoal})}\nWhen $2\\le l\\le m$, we apply (\\ref{2mainprop}) with $2<q'<\\frac{2m}{m-1}$, \nalong with  (\\ref{maininftyest}), and (\\ref{mainlqest}) to obtain  \n\\begin{align}\n&\\bigg\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\Big| \\sum_{\\gamma\\in\\mathbb{Z}:\\gamma<\\tau}\\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}\\big(f_1,\\dots,f_m\\big) \\bigg\\Vert_{L^{2\/m}}\\le \\bigg\\Vert  \\sum_{\\gamma\\in\\mathbb{Z}} \\big| \\mathcal{T}_{\\vec{{G}},l}^{\\lambda,\\gamma,\\mu}\\big(f_1,\\dots,f_m\\big) \\big| \\bigg\\Vert_{L^{2\/m}}\\notag\\\\\n&\\lesssim \\big\\Vert \\big\\{ b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\big\\}_{\\vec{{k}}\\,}\\big\\Vert_{\\ell^{\\infty}}^{1-\\frac{(m-1)q'}{2m}} \\big\\Vert \\big\\{ b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\big\\}_{\\vec{{k}}\\,}\\big\\Vert_{\\ell^{q'}}^{\\frac{(m-1)q'}{2m}} 2^{\\lambda mn\/2}(\\lambda+1)^{l\/2}{\\mu}^{l\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\\notag\\\\\n&\\lesssim \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}2^{-\\delta {\\mu}(1-\\frac{(m-1)q'}{2m})}{\\mu}^{m\/2}2^{-\\lambda C_{M,m,n,q}}(\\lambda+1)^{m\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2} ,\\notag \n\\end{align} \nwhere \n$$C_{M,m,n,q}:=(M+1+mn)(1-\\frac{(m-1)q'}{2m})+mn(1\/q-1\/2)\\frac{(m-1)q'}{2m}-\\frac{mn}{2}.$$\nHere we used  the fact that $\\frac{l-1}{2l}\\le \\frac{m-1}{2m}$ for $l\\le m$.\n  Then \\eqref{2mmaingoal} follows from choosing $M$ sufficiently large so that $C_{M,m,n,q}>0$ since $1-\\frac{(m-1)q'}{2m}>0$. \n \n\nNow let us prove (\\ref{2mmaingoal}) for $l=1$.\nIn this case, we first see the estimate\n\\begin{equation}\\label{keykeyest}\n\\Big\\Vert \\Big(  \\sum_{\\gamma\\in\\mathbb{Z}}\\big|     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)   \\big|^2\\Big)^{1\/2}\\Big\\Vert_{L^{2\/m}}  \\lesssim 2^{-\\epsilon_0\\mu} 2^{-M_0\\lambda} \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{equation}\nfor some $\\epsilon_0, M_0>0$, which can be proved, as in \\cite[Section 6]{Paper1}, by using (\\ref{1mainprop}) and (\\ref{maininftyest}).\n\nChoose a Schwartz function $\\Gamma$ on $\\mathbb R^n$ whose Fourier transform is supported in the ball $ \\{\\xi\\in\\mathbb R^n: |\\xi|\\le 2\\}$ and is equal to $1$ for $|\\xi|\\le 1$, and define $\\Gamma_k:=2^{kn}\\Gamma(2^k\\cdot)$ so that\n$\\textup{Supp}(\\widehat{\\Gamma_k})\\subset \\{\\xi\\in\\mathbb R^n: |\\xi|\\le 2^{k+1}\\}$ and $\\widehat{\\Gamma_k}(\\xi)=1$ for $|\\xi|\\le 2^k$.\n\n\nSince the Fourier transform of $\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)$ is supported in the set \n$\\big\\{  \\xi\\in\\mathbb R^n: 2^{\\gamma+\\mu-5}\\le |\\xi|\\le 2^{\\gamma+\\mu+4}   \\big\\}$, we can write\n\\begin{equation*}\n\\sum_{\\gamma\\in\\mathbb{Z}: \\gamma<\\tau}\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)=\\Gamma_{\\tau+\\mu+3}\\ast \\Big( \\sum_{\\gamma\\in\\mathbb{Z}: \\gamma<\\tau}\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)\\Big)\n\\end{equation*}\nand then split the right-hand side into\n \\begin{align*}\n& \\Gamma_{\\tau+\\mu+3}\\ast  \\Big( \\sum_{\\gamma\\in\\mathbb{Z}}\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)\\Big)-\\Gamma_{\\tau+\\mu+3}\\ast  \\Big( \\sum_{ \\gamma\\in\\mathbb{Z} :\\gamma\\ge \\tau}\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)\\Big).\n\\end{align*}\nDue to the Fourier support conditions of $\\Gamma_{\\tau+\\mu+3}$ and $\\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)$, the sum in the second term can be actually taken over $\\tau\\le \\gamma\\le \\tau+9$.\nTherefore, the left-hand side of (\\ref{2mmaingoal}) is controlled by the sum of\n\\begin{equation}\\label{DefI}\nI:=\\bigg\\Vert \\sup_{\\nu\\in\\mathbb{Z}}\\Big| \\Gamma_{\\nu}\\ast \\Big(\\sum_{\\gamma\\in\\mathbb{Z}}{  \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)   } \\Big)\\Big|      \\bigg\\Vert_{L^{2\/m}}\n\\end{equation}\nand\n\\begin{equation}\\label{DefII}\nII:=\\sum_{\\gamma=0}^{9}\\Big\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\big| \\Gamma_{\\tau+\\mu+3}\\ast T_{\\vec{{G}},1}^{\\lambda,\\tau+\\gamma,\\mu}(f_1,\\dots,f_m)\\big|\\Big\\Vert_{L^{2\/m}}.\n\\end{equation}\n\n\nFirst of all, when $0\\le \\gamma\\le 9$,\nthe Fourier supports of both $\\Gamma_{\\tau+\\mu+3}$ and $T_{\\vec{{G}},1}^{\\lambda,\\tau+\\gamma,\\mu}(f_!,\\dots,f_m)$ are $\\{\\xi\\in \\mathbb R^n: |\\xi|\\sim 2^{\\tau+\\mu}\\}$. This implies that\nfor any $0<r<1$,\n\\begin{align*}\n&\\big| \\Gamma_{\\tau+\\mu+3}\\ast T_{\\vec{{G}},1}^{\\lambda,\\tau+\\gamma,\\mu}(f_1,\\dots,f_m)(x)\\big|\\\\\n&\\lesssim 2^{(\\tau+\\mu)(n\/r-n)}\\Big( \\int_{\\mathbb R^n}  \\big| \\Gamma_{\\tau+\\mu+3}(x-y)\\big|^r \\big| T_{\\vec{{G}},1}^{\\lambda,\\tau+\\gamma,\\mu}(f_1,\\dots,f_m)(y)\\big|^r  ~dy\\Big)^{1\/r}\\\\\n&\\lesssim \\Big(\\mathcal{M}\\big(|T_{\\vec{{G}},1}^{\\lambda,\\tau+\\gamma,\\mu}(f_1,\\dots,f_m)|^r\\big)(x)\\Big)^{1\/r}\n\\end{align*}\nwhere the Nikolskii inequality (see \\cite[Proposition 1.3.2]{Tr1983}) is applied in the first inequality.\nSetting $0<r<2\/m$, and using the maximal inequality for $\\mathcal{M}$ and the embedding $\\ell^2\\hookrightarrow \\ell^{\\infty}$\nwe obtain\n\\begin{align}\\label{peetreargument}\nII&\\lesssim \\big\\Vert \\sup_{\\tau\\in\\mathbb{Z}}\\big|  T_{\\vec{{G}},1}^{\\lambda,\\tau,\\mu}(f_1,\\dots,f_m)   \\big| \\big\\Vert_{L^{2\/m}}\\\\\n &\\le \\Big\\Vert \\Big( \\sum_{\\gamma\\in\\mathbb{Z}}\\big|     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)   \\big|^2\\Big)^{1\/2}\\Big\\Vert_{L^{2\/m}}. \\nonumber\n\\end{align}\nThen the $L^{2\/m}$ norm is bounded by the right-hand side of (\\ref{2mmaingoal}), thanks to (\\ref{keykeyest}). This completes the estimate for $II$ defined in \\eqref{DefII} and \nwe turn our attention to $I$ defined in \\eqref{DefI}. \n\n\n In the sequel we will make use of the following inequality: \n if $\\widehat{g_{\\gamma}}$ is supported on $\\{\\xi\\in \\mathbb{R}^n : C^{-1} 2^{\\gamma+\\mu}\\leq |\\xi|\\leq C2^{\\gamma+\\mu}\\}$ for some $C>1$ and $\\mu\\in \\mathbb{Z}$, then\n\\begin{equation}\\label{marshallest}\n\\Big\\Vert \\Big\\{ \\Phi^{(1)}_j\\!\\ast\\! \\Big(\\sum_{\\gamma\\in\\mathbb{Z}}{g_{\\gamma}}\\!\\Big)\\!\\Big\\}_{\\! j\\in\\mathbb{Z}}\\Big\\Vert_{L^p(\\ell^q)}\\lesssim_{C} \\big\\Vert \\big\\{ g_j\\big\\}_{j\\in\\mathbb{Z}}\\big\\Vert_{L^p(\\ell^q)} \\quad \\text{uniformly in }~\\mu\n\\end{equation}  for $0<p<\\infty$. The proof of (\\ref{marshallest}) is elementary and standard, so it is omitted here; see \\cite[(13)]{Gr_He_Ho} and  \\cite[Theorem 3.6]{Ya}  for  related arguments.\n \n\n\nTo obtain the bound of $I$, we note that\n\\begin{equation*}\nI \\approx \\Big\\Vert \\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }\\Big\\Vert_{H^{2\/m}}\n\\end{equation*} \n where $H^{2\/m}$ is the Hardy space. We refer to \\cite[Corollary 2.1.8]{MFA} for the above estimate.\nThen, using the Littlewood-Paley theory for Hardy space (see for instance \\cite[Theorem 2.2.9]{MFA}) and (\\ref{marshallest}), there exists a unique polynomial $Q^{\\lambda,\\mu,\\vec{{G}}}(x)$ such that\n\\begin{align}\\label{3mainest}\n \\Big\\Vert \\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }       -Q^{\\lambda,\\mu,\\vec{{G}}} \\Big\\Vert_{H^{2\/m}}\\notag &\\lesssim  \\Big\\Vert \\Big( \\sum_{\\gamma\\in\\mathbb{Z}}\\big|     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)   \\big|^2\\Big)^{1\/2}\\Big\\Vert_{L^{2\/m}}\\nonumber\\\\\n & \\lesssim 2^{-\\epsilon_0\\mu} 2^{-M_0\\lambda} \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n \\end{align} where (\\ref{keykeyest}) is applied.\nFurthermore, \n\\begin{align*}\n&\\Big\\Vert    \\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }  \\Big\\Vert_{H^{2\/m}}\\\\\n&\\approx \\bigg\\Vert  \\sup_{\\nu\\in\\mathbb{Z}}\\Big|\\Gamma_{\\nu}\\ast\\Big(  \\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }  \\Big)\\Big|\\bigg\\Vert_{L^{2\/m}}\\\\\n&=\\bigg\\Vert  \\sup_{\\nu\\in\\mathbb{Z}}\\Big|\\Gamma_{\\nu}\\ast\\Big(  \\sum_{\\gamma\\in\\mathbb{Z}: \\gamma\\le \\nu-\\mu+5}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }  \\Big)\\Big|\\bigg\\Vert_{L^{2\/m}}\\\\\n&\\lesssim \\bigg\\Vert \\sup_{\\nu\\in\\mathbb{Z}} \\Big| \\sum_{\\gamma\\in\\mathbb{Z}: \\gamma\\le \\nu-\\mu+5}{      \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)     }\\Big|\\bigg\\Vert_{L^{2\/m}}\\\\\n&\\le \\Big\\Vert \\sum_{\\gamma\\in\\mathbb{Z}}{  \\big|    \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)  \\big|   }\\Big\\Vert_{L^{2\/m}}\n\\end{align*} \nwhere the argument that led to (\\ref{peetreargument}) is applied in the first inequality.\nAs we have discussed in \\cite[Section 6.1]{Paper1},   this quantity is finite for all Schwartz functions $f_1,\\dots,f_m$.\nAccordingly, we have \n$$\\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }       -Q^{\\lambda,\\mu,\\vec{{G}}}\\in H^{2\/m}$$\nand \n$$\\sum_{\\gamma\\in\\mathbb{Z}}{     \\mathcal{T}_{\\vec{{G}},1}^{\\lambda,\\gamma,\\mu}(f_1,\\dots,f_m)       }      \\in H^{2\/m},$$\nand thus $Q^{\\lambda,\\mu,\\vec{{G}}}=0$.\nNow it follows from (\\ref{3mainest}) that \n\\begin{equation*}\nI \\lesssim 2^{-\\epsilon_0\\mu} 2^{-M_0\\lambda} \\Vert \\Omega\\Vert_{L^q(\\mathbb{S}^{mn-1})}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2},\n\\end{equation*}\nas expected. This completes the proof of (\\ref{2mmaingoal}).\n\n\n\n\n\n\n\n\n\\section{Proof of Theorem \\ref{application4}}\\label{pfapp4}\nLet $\\mu_0$ be the smallest integer satisfying $2^{{\\mu}_0-3}>C_0\\sqrt{mn}$ and\n $$\\widehat{\\Theta^{(m)}_{{\\mu}_0-1}}(\\vec{{\\xi}}\\,):=1-\\sum_{{\\mu}={\\mu}_0}^{\\infty}{\\widehat{\\Phi_{\\mu}^{(m)}}(\\vec{{\\xi}}\\,)}.$$\nClearly, \n\\begin{equation*}\n\\widehat{\\Theta^{(m)}_{{\\mu}_0-1}}(\\vec{{\\xi}}\\,)+\\sum_{{\\mu}={\\mu}_0}^{\\infty}{\\widehat{\\Phi_{\\mu}^{(m)}}(\\vec{{\\xi}}\\,)}=1\n\\end{equation*}\nand thus we can write\n\\begin{equation*}\n\\sigma(\\vec{{\\xi}}\\,)=\\widehat{\\Theta_{{\\mu}_0-1}^{(m)}}(\\vec{{\\xi}}\\,)\\sigma(\\vec{{\\xi}}\\,)+\\sum_{{\\mu}={\\mu}_0}^{\\infty}{\\widehat{\\Phi_{\\mu}^{(m)}}(\\vec{{\\xi}}\\,)\\sigma(\\vec{{\\xi}}\\,)}=:\\sigma_{{\\mu}_0-1}(\\vec{{\\xi}}\\,)+\\sum_{{\\mu}={\\mu}_0}^{\\infty}{\\sigma_{\\mu}(\\vec{{\\xi}}\\,)}.\n\\end{equation*}\nNote that $\\sigma_{{\\mu}_0-1}$ is a compactly supported smooth function and thus the corresponding maximal multiplier operator $\\mathscr{M}_{\\sigma_{{\\mu}_0-1}}$, defined by\n\\begin{align*}\n&\\mathscr{M}_{\\sigma_{{\\mu}_0-1}}\\big(f_1,\\dots,f_m\\big)(x)\\\\\n&:=\\sup_{\\nu \\in\\mathbb{Z}}\\Big|\\int_{(\\mathbb R^n)^m}{\\sigma_{{\\mu}_0-1}(2^{\\nu} \\vec{{\\xi}}\\,)\\Big( \\prod_{j=1}^{m}\\widehat{f_j}(\\xi_j)\\Big) e^{2\\pi i\\langle x,\\sum_{j=1}^{m}\\xi_j \\rangle}}d\\vec{{\\xi}}\\, \\Big|,\n\\end{align*}\nis bounded by a constant multiple of \n$\n\\mathcal{M}f_1(x)\\cdots\\mathcal{M}f_m(x)\n $ \n where $\\mathcal{M}$ is the Hardy-Littlewood maximal operator on $\\mathbb R^n$ as before.\nUsing H\\\"older's inequality and the $L^2$-boundedness of $\\mathcal{M}$, we can prove\n\\begin{equation*}\n\\big\\Vert \\mathscr{M}_{\\sigma_{{\\mu}_0-1}}(f_1,\\dots,f_m) \\big\\Vert_{L^{2\/m}}\\lesssim \\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}.\n\\end{equation*}\n\nIt remains to show that\n\\begin{equation}\\label{lefttoshow}\n\\Big\\Vert \\sum_{{\\mu}={\\mu}_0}^{\\infty}\\mathscr{M}_{\\sigma_{\\mu}}(f_1,\\dots,f_m)\\Big\\Vert_{L^{2\/m}}\\lesssim \\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}.\n\\end{equation}\nUsing the decomposition (\\ref{daubechewavelet}), write\n\\begin{equation}\\label{sigmajdef}\n\\sigma_{\\mu}(\\vec{{\\xi}}\\,)=\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}}\\sum_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m}{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\Psi_{G_1,k_1}^{\\lambda}(\\xi_1)\\cdots\\Psi_{G_m,k_m}^{\\lambda}(\\xi_m)}\n\\end{equation}\n where \n \\begin{equation*}\n b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}:=\\int_{(\\mathbb R^n)^m}{\\sigma_{\\mu}(\\vec{{\\xi}}\\,)\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}(\\vec{{\\xi}}\\,)}d\\vec{{\\xi}}\\,.\n \\end{equation*} \n \n Let $M:=\\Big[ \\frac{(m-1)n}{2}\\Big]+1$ and choose $1<q<\\frac{2m}{m-1}$ such that\n \\begin{equation}\\label{choiceq}\n \\frac{(m-1)n}{2}<\\frac{mn}{q}<\\min{(a,M)}.\n \\end{equation}\nIn view of (\\ref{lqestimate}), we have\n \\begin{align}\\label{bqpointbound}\n \\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\}_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m}\\big\\Vert_{\\ell^q} &\\lesssim 2^{-\\lambda (M-mn\/q+mn\/2)}\\Vert \\sigma_{\\mu}\\Vert_{L_M^q((\\mathbb R^n)^m)}\\nonumber\\\\\n  & \\lesssim 2^{-\\lambda (M-mn\/q+mn\/2)} 2^{-{\\mu}(a-mn\/q)}\n \\end{align} where the assumption (\\ref{givenassumption}) is applied in the last inequality.\n\n\nWe observe that if ${\\mu}\\ge {\\mu}_0$, then $b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}$ vanishes unless $2^{\\lambda+{\\mu}-2}\\le |\\vec{{k}}\\,|\\le 2^{\\lambda+{\\mu}+2}$\n due to the compact supports of $\\sigma_{\\mu}$ and $\\Psi_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda}$, which allows us to replace the sum over $\\vec{{k}}\\,\\in (\\mathbb Z^n)^m$ in (\\ref{sigmajdef}) by the sum over $2^{\\lambda+{\\mu}-1}\\le |\\vec{{k}}\\,|\\le 2^{\\lambda+{\\mu}+1}$. \nMoreover, we may consider only the case $|k_1|\\ge \\cdots \\ge |k_m|$ as in the previous section.\nTherefore, in the rest of the section, we assume\n\\begin{align*}\n\\sigma_{\\mu}(\\vec{{\\xi}}\\,)&=\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in \\mathcal{I}^{\\lambda}}\\sum_{\\vec{{k}}\\,\\in \\mathcal{U}^{\\lambda+{\\mu}}}{    b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\Psi_{G_1,k_1}^{\\lambda}(\\xi_1)\\cdots\\Psi_{G_m,k_m}^{\\lambda}(\\xi_m) }\\\\\n&=\\sum_{l=1}^{m}\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in \\mathcal{I}^{\\lambda}}\\sum_{\\vec{{k}}\\,\\in \\mathcal{U}_l^{\\lambda+{\\mu}}}{    b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\Psi_{G_1,k_1}^{\\lambda}(\\xi_1)\\cdots\\Psi_{G_m,k_m}^{\\lambda}(\\xi_m) }\\\\\n&=:\\sum_{l=1}^{m}\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}}\\sigma_{\\mu,l}^{\\lambda,\\vec{{G}}}(\\vec{{\\xi}}\\,)\n\\end{align*}\nwhere the sets $\\mathcal{U}^{\\lambda+\\mu}$, $\\mathcal{U}_l^{\\lambda+\\mu}$ are defined as before.\nThen the left-hand side of (\\ref{lefttoshow}) can be controlled by\n\\begin{equation}\\label{boundgoal}\n\\Big(\\sum_{l=1}^{m}\\sum_{{\\mu}={\\mu}_0}^{\\infty}\\sum_{\\lambda\\in\\mathbb{N}_0}\\sum_{\\vec{{G}}\\in\\mathcal{I}^{\\lambda}}\\big\\Vert \\mathscr{M}_{\\sigma_{{\\mu},l}^{\\lambda,\\vec{{G}}}}(f_1,\\dots,f_m)\\big\\Vert_{L^{2\/m}}^{2\/m} \\Big)^{m\/2}.\n\\end{equation}\n\n\n\n\n\nNow we claim that \n\\begin{align}\\label{app4claim}\n&\\big\\Vert \\mathscr{M}_{\\sigma_{{\\mu},l}^{\\lambda,\\vec{{G}}}}(f_1,\\dots,f_m)\\big\\Vert_{L^{2\/m}}\\nonumber\\\\\n&\\lesssim 2^{-\\lambda (M-mn\/q)}(\\lambda+1)^{l\/2}2^{-{\\mu}(a-mn\/q)}{\\mu}^{l\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}.\n\\end{align}\nThen (\\ref{boundgoal}) is less than a constant multiple of \n$\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}$ as desired, due to the choice of $q$ in (\\ref{choiceq}).\n\n\nIn order to prove (\\ref{app4claim}), we use the estimates (\\ref{1mainprop}) and (\\ref{2mainprop}). \nWe first rewrite\n\\begin{equation*}\n\\mathscr{M}_{\\sigma_{{\\mu},l}^{\\lambda,\\vec{{G}}}}\\big(f_1,\\dots,f_m\\big)(x)=\\sup_{\\gamma\\in\\mathbb{Z}}\\Big| \\sum_{\\vec{{k}}\\,\\in \\mathcal{U}_l^{\\lambda+{\\mu}}}{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu} \\Big( \\prod_{j=1}^{m}L_{G_j,k_j}^{\\lambda,\\gamma}f_i(x)\\Big) }\\Big|\n\\end{equation*}\nwhere $L_{G,k}^{\\lambda,\\gamma}$ is defined as in (\\ref{lgklg}).\n\n \n\nWhen $l=1$, applying the embeddings $\\ell^2\\hookrightarrow \\ell^{\\infty}$, $\\ell^q\\hookrightarrow \\ell^{\\infty}$, and (\\ref{1mainprop}), the left-hand side of (\\ref{app4claim}) is less than\n\\begin{align*}\n&\\bigg\\Vert \\bigg(\\sum_{\\gamma\\in\\mathbb{Z}} \\Big| \\sum_{\\vec{{k}}\\,\\in\\mathcal{U}_{1}^{\\lambda+{\\mu}}}{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}   \\prod_{j=1}^{m}{L_{G_j,k_j}^{\\lambda,\\gamma}f_j}             }    \\Big|^2 \\bigg)^{1\/2}\\bigg\\Vert_{L^{2\/m}}\\\\\n&\\lesssim \\big\\Vert \\{  b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}  \\}_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m}\\big\\Vert_{\\ell^{q}} \\mu^{1\/2}2^{\\lambda mn\/2}(\\lambda+1)^{1\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\\\\\n&\\lesssim 2^{-\\lambda(M-mn\/q)}(\\lambda+1)^{1\/2} 2^{-{\\mu}(a-mn\/q)} {\\mu}^{1\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{align*}\nwhere (\\ref{bqpointbound}) is applied in the last inequality.\n\n\nFor the case $2\\le l\\le m$, we can bound the left-hand side of (\\ref{app4claim}) by\n\\begin{align*}\n&\\bigg\\Vert \\sum_{\\gamma\\in\\mathbb{Z}}\\Big| \\sum_{\\vec{{k}}\\,\\in\\mathcal{U}_{l}^{\\lambda+{\\mu}}} b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}  \\prod_{j=11}^{m} L_{G_j,k_j}^{\\lambda,\\gamma}f_j   \\Big| \\bigg\\Vert_{L^{2\/m}}\\\\\n&\\lesssim \\big\\Vert \\{b_{\\vec{{G}},\\vec{{k}}\\,}^{\\lambda,\\mu}\\}_{\\vec{{k}}\\,\\in (\\mathbb Z^n)^m}\\big\\Vert_{\\ell^{q}} {\\mu}^{l\/2}2^{\\lambda mn\/2}(\\lambda+1)^{l\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}.\n\\end{align*}\nHere, we used the inequality (\\ref{2mainprop}) and the embedding $\\ell^q\\hookrightarrow \\ell^{\\infty}$.\nThen the preceding expression is estimated by\n\\begin{equation*}\n2^{-\\lambda (M-mn\/q)}(\\lambda+1)^{l\/2}2^{-{\\mu} (a-mn\/q)}{\\mu}^{l\/2}\\prod_{j=1}^{m}\\Vert f_j\\Vert_{L^2}\n\\end{equation*}\nin view of (\\ref{bqpointbound}). This completes the proof of (\\ref{app4claim}).\n\n\n\n\n \\section{Concluding remarks}\n\n\\begin{comment}\n{\\color{red}\nIn this section we apply our main results to obtain the a.e. convergence of \nmultilinear singular integrals. The following proposition will be useful for this purpose. Its proof \ncan be found in \\cite{GDOS}. \n\n\n\\begin{prop}\\label{prop00}  Let $0<p_i \\le\\infty$, $1\\le i\\le m$, $0<q<\\infty$ and let $T_t$, $t>0$ be a \nfamily of $m$-linear operators while $T_\\ast=\\sup_{t>0} |T_t| $ is the associated maximal operator.  Suppose that there is a constant $B$ such that \n\\begin{equation}\\label{100}\n\\|T_{\\ast} (f_1,\\dots,f_m)\\|_{L^{q,\\infty}} \\leq B \\prod_{j=1}^m \\|f_j\\|_{L^{p_j}}\n \\end{equation}\n for all $f_j\\in L^{p_j}(\\mathbb{R}^n)$. Also suppose  that for all $\\varphi_j $ in a dense subclass $D_j $ of \n $L^{p_j}(\\mathbb{R}^n)$ we have \n\\begin{equation}\\label{tto1}\n \\lim_{t\\to 0} T_{t} (\\varphi_1,\\dots, \\varphi_m) = T(\\varphi_1,\\dots,\\varphi_m)\n \\end{equation}\nexists and is finite. Then for all functions $f_j\\in L^{p_j}(\\mathbb{R}^n) $ the limit in \\eqref{tto1} exists and is finite  a.e., and defines an $m$-linear operator   which uniquely extends $T$ defined on $D_1\\times \\cdots \\times D_m$ and\nwhich is bounded from $L^{p_1}\\times \\cdots \\times L^{p_m}$ to $L^{q,\\infty}(\\mathbb{R}^n)$. \n\\end{prop}\n\nWe now state the two main corollaries of our work. \n\n\\begin{cor}\\label{CCC1}\nLet $\\Omega$ be as in Theorem~\\ref{MAXSINGINT} and let\n\\begin{equation*}\n\\mathcal{L}_{\\Omega}^*\\big(f_1,\\dots,f_m\\big)(x):=\\sup_{\\epsilon>0} \n\\big| \\mathcal{L}_{\\Omega}^\\epsilon(f_1,\\dots, f_m)(x) \\big|, \n\\end{equation*}\nwhere \n\\[\n\\mathcal{L}_{\\Omega}^\\epsilon(f_1,\\dots, f_m)(x)=\n \\int_{(\\mathbb R^n)^m\\setminus B(0,\\epsilon)}{K(\\vec{{y}}\\,)\\prod_{j=1}^{m}f_j(x-y_j)}~d\\,\\vec{{y}}\\,. \n\\]\nThen for $f_j\\in L^2(\\mathbb{R}^n)$, $j=1,\\dots , m$, the  truncated singular integrals  \n$\\mathcal{L}_{\\Omega}^\\epsilon(f_1,\\dots, f_m)$ converge a.e. as $\\epsilon\\to 0$ to  \n$ {\\mathcal{L}_{\\Omega}}(f_1,\\dots, f_m)$, where  $ {\\mathcal{L}_{\\Omega}}$ denotes here the bounded  extension of $\\mathcal{L}_{\\Omega}$ on $L^2\\times \\cdots\\times L^2.$\n\\end{cor}\n\n\n\n\\begin{cor}\\label{CCC2}\nLet $\\sigma$ be as in Theorem~\\ref{application4} \nand for $\\nu\\in \\mathbb Z$ let $S_\\sigma^\\nu$ be as in \\eqref{defSsinu}. Then for \n$f_j\\in L^2(\\mathbb{R}^n)$, $j=1,\\dots , m$, the $L^{2\/m}(\\mathbb{R}^n)$ functions \n$\nS_\\sigma^\\nu(f_1,\\dots, f_m) \n$\nconverge a.e. to $\\sigma(0) f_1 \\cdots f_m$ as $\\nu \\to -\\infty$. Additionally, if $\\lim_{y\\to \\infty} \n\\sigma(y)$ exists and and equals $L$, then the functions \n$S_\\sigma^\\nu(f_1,\\dots, f_m) $ converge a.e. to \n$L f_1 \\cdots f_m$ as $\\nu \\to  \\infty$. \n\\end{cor}\n\nTo verify these corollaries, we notice that the claimed \nconvergence   holds pointwise everywhere  for smooth funcitons with compact support $f_j$ \nby the Lebesgue dominated convergence theorem.  In the case of the Corollary~\\ref{CCC2} this assertion is straightforward, while in the case of Corollary~\\ref{CCC1} it is a consequence \nof the smoothness of the function $f_1(x)\\cdots f_m(x)$ at the origin combined with the cancellation of $\\Omega$. The assertions in both corollaries then follow by \napplying  Proposition~\\ref{prop00}.\n}\n\\end{comment}\n\nAs of this writing, we are uncertain how to extend  Theorem~\\ref{application4} \nin the non-lacunary case. A new ingredient may be necessary to accomplish this. \\\\\n\nWe have addressed the boundedness of several multilinear and maximal multilinear\noperators at the initial point $L^2\\times \\cdots \\times L^2\\to L^{2\/m}$. Our future \ninvestigation  related to this project has two main directions: \n(a) to extend this initial estimate to many other operators, such as the   \ngeneral maximal multipliers considered in \\cite{Gr_He_Ho2020, Ru}, and (b) to obtain \n  $L^{p_1}\\times \\cdots \\times L^{p_m}\\to L^{p}$ bounds for all of these operators in the \n  largest possible range of exponents possible. Additionally, one could consider the \n  study of related endpoint estimates. We hope to achieve this goal in future \n  publications. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\\label{sec:intro} \n\nAccurately counting the number of cells in microscopic images is important for medical diagnoses and biological applications~\\cite{venkatalakshmi2013automatic}.\nHowever, manual cell counting is a very time-consuming, tedious, and error-prone task. An automatic and efficient solution to improve the counting accuracy is highly desirable, but automatic cell counting is challenged by the low image contrast and significant inter-cell occlusions in 2D microscopic images~\\cite{matas2004robust,barinova2012,arteta2012,xing2014automatic}.\n\nRecently, density regression-based counting methods\nhave been employed to address these challenges~\\cite{lempitsky2010,xie2018}.\nThese methods employ machine-learning tools to learn a density regression model (DRM) that estimates the cell density distribution from the characteristics\/features of a given image.\nThe number of cells can be subsequently counted by integrating the estimated density map.\nHowever, in these methods, a large amount of annotated data are required, which can be difficult to obtain in practice.\nUsing annotated synthetic images to train a DRM can mitigate this problem~\\cite{xie2018}.\nHowever, due to the intrinsic domain shift between the experimental and synthetic cell images, a DRM trained with synthetic images may generalize poorly to experimental images~\\cite{lempitsky2010,xie2018}.\nCurrently available DRM methods~\\cite{lempitsky2010,xie2018} cannot address this critical issue.\n\nDomain adaptation methods based on unsupervised adversarial learning have been proposed \nto mitigate the harmful effects of domain shift in computer vision tasks, such as classification~\\cite{tzeng2017adversarial} and segmentation~\\cite{dou2018unsupervised}. \nThese methods aim at learning transformations that map two shifted image domains to a common feature space, so that the learned model that works for one domain can be applied to another.\nThe methods have demonstrated very promising performance for the target tasks~\\cite{tzeng2017adversarial,dou2018unsupervised}.\n\nCombining the advantages of both supervised learning-based density regression methods and unsupervised learning-based domain adaptation methods, \nThis study proposes a novel, manual-annotating-free automatic cell counting method. The proposed method is evaluated on experimental immunofluorescent microscopic images of human embryonic stem cells (hESC).\n\n\\section{Methodology}\n\\label{sec:methodology}\n\n\\subsection{Background: Density Regression-Based Automatic Cell Counting}\n\\label{ssec:density}\n\nThe goal of density regression-based cell counting methods is to learn a density function $F$ \nthat can be employed to estimate the cell density map for a given image~\\cite{lempitsky2010,xie2018}.\nFor a given image $X\\in \\mathbb{R}^{M\\times N}$ that includes $N_c$ cells, \nthe corresponding density map $Y\\in \\mathbb{R}^{M\\times N}$ can be considered as the superposition of a set of $N_c$ normalized 2D discrete Gaussian kernels that are placed at the centroids of the $N_c$ cells. \nLet $S = \\{ ({s_{k_x}},{s_{k_y}})\\in \\mathbb{N}^2: k = 1,2, ..., N_c\\}$ represent the cell centroid positions in $X$. \nEach pixel $Y_{i,j}$ on the density map $Y$ can be expressed as\n\\begin{equation}\nY_{i,j} = \\sum_{k=1}^{N_c} G_\\sigma (i-{s_{k_x}},j-{s_{k_y}}), \\quad \\forall \\quad i \\in M, \\quad j \\in N, \\\\\n\\label{eq:map}\n\\end{equation}\n\n\\noindent where $G_\\sigma(n_x,n_y) = C \\cdot e^{-\\frac{n_x^2+n_y^2}{2\\sigma^2}}\\in \\mathbb{R}^{(2K_G+1)\\times (2K_G+1)}, n_x, n_y = -K_G, -K_G+1,..., K_G$, is a normalized 2D Gaussian kernel that satisfies $\\sum_{n_x=-K_G}^{K_G}\\sum_{n_y=-K_G}^{K_G} G_\\sigma(n_x,n_y) =1$. Here, $\\sigma^2$ is the isotropic covariance, $(2K_G+1)\\times (2K_G+1)$ is the kernel size, and C is a normalization constant\n\nThe density regression-based cell counting process includes three steps: \n(1) map an image into a feature map,\n(2) estimate a cell density map from the feature map, and \n(3) integrate the estimated density map for cell counting.\nIn the first step, each pixel $X_{i,j}$ can be assumed to be associated with a real-valued feature vector $\\phi(X_{i,j})\\in \\mathbb{R}^Z$. \nThe feature map $P\\in \\mathbb{R}^{M\\times N \\times Z}$ of $X$ can be generated using specific feature extraction methods, such as the dense scale invariant feature transform (SIFT) descriptor~\\cite{vedaldi2010vlfeat}, ordinary filter banks~\\cite{fiaschi2012}, \nor codebook learning~\\cite{sommer2011ilastik}.\nIn the second step, the estimated density $\\hat{Y}_{i,j}$ of each pixel $X_{i,j}$ can be obtained by applying a pre-trained density regression function $F$ on the given $\\phi(X_{i,j})$:\n\\begin{equation}\n\t\\hat{Y}_{i,j} = F(\\phi(X_{i,j});\\Theta),\n\t\\label{eq:estimatemap}\n\\end{equation}\n\n\\noindent where $\\Theta$ is a parameter vector that determines the function $F$.\nFinally, in the third step, the number of cells in $X$, $N_c$, can be counted by integrating the estimated density map $\\hat{Y}$ over the image region:\n\\begin{equation}\n\tN_c \\approx \\hat{N_c} = \\sum_{i=1}^{M}\\sum_{j=1}^{N} \\hat{Y}_{i,j}.\n\t\\label{eq:counting}\n\\end{equation}\n\nA key task in density regression-based cell counting methods is learning the function $F$ by use of training datasets.\nThe learning of $F$ and the related cell counting method proposed in this study are described below.\n\n\\subsection{The Proposed Automatic Cell Counting Framework}\n\\label{ssec:method}\n\nThe proposed density regression-based automatic cell counting method is implemented by use of both supervised and unsupervised learning, and employs both annotated synthetic images and unannotated experimental images.\nCombining the advantages of both supervised learning-based density regression methods and unsupervised learning-based domain adaptation methods, \nthe proposed method can learn a DRM by use of annotated synthetic images without the need of manually-annotated experimental images. A domain adaptation model (DAM) will be learned by use of both unannotated synthetic and experimental images.\n\n\\subsubsection{Overview of the Proposed Method}\n\\label{sssec:overview}\n\nThe proposed method, shown in Figure~\\ref{fig:framework}, has three phases: 1) Source DRM training (Section~\\ref{sssec:drm}), 2) DAM training (Section~\\ref{sssec:dam}), and 3) Density map estimation for cell counting based on the target DRM (Section~\\ref{sssec:counting}).\nHere, a source DRM represents the DRM trained with synthetic images (the source domain), while a target DRM is the domain-adapted DRM that can be employed for estimating the density map of a given experimental image (the target domain). \nIn the source domain training phase, a source DRM consisting of an encoder CNN (ECNN) and decoder CNN (DCNN) is trained by use of supervised learning with a set of annotated images in the source domain. The trained ECNN maps the images in the source domain to a feature space of the source domain, while the DCNN maps the feature space to the corresponding density maps. The trained ECNN will be employed for training a DAM, while the trained DCNN is employed as part of the target DRM for density estimation and cell counting. In the DAM training phase, a DAM is trained in an unsupervised manner by use of the trained ECNN and a set of unannotated synthetic images and experimental images. The trained DAM will become part of the target DRM and will be employed to map the images in the target domain to a feature space that has minimum domain shift with that of the source domain. In the cell counting phase, the combination of the trained DAM and DCNN forms the target DRM to estimate the density map for a given experimental image.\n\n\\begin{figure}\n\t\\begin{center}\n\t    \\includegraphics[width=\\textwidth]{framework.pdf}\n\t\\end{center}\n\\caption{Overview of the proposed automatic cell counting framework.}\n\\label{fig:framework}\n\\end{figure}\n\n\\subsubsection{Source DRM Training}\n\\label{sssec:drm}\n\nThe first phase of the proposed method is to train a source DRM with annotated synthetic images, as shown in the left block of Figure~\\ref{fig:drm}.\nThe trained source DRM then determines a density function $F$.\nThe source DRM is designed as a fully convolutional neural network (FCNN) that includes an encoder CNN (ECNN) and a decoder CNN (DCNN). This design is motivated by a network architecture described in the literature~\\cite{xie2018}.\nThe ECNN encodes synthetic images to a low-dimensional and highly-representative feature space, while the DCNN decodes the feature space to estimate the corresponding density map. \nThe architectures of the ECNN and DCNN are shown in Figure~\\ref{fig:drm}.\nEach block in the ECNN or DCNN includes a chain of layers.\nHere, CONV represents a convolutional layer, Pool represents a max-pooling layer, and \\lq\\lq{}Up\\rq\\rq{} represents an up-sampling layer. \nThere are a total of eight CONV layers in the ECNN and DCNN. \nThe numbers of kernels in these eight CONV layers are set to 32, 64, 128, 512, 128, 64, 32, and 1, respectively.\nThe size of each kernel in the first seven CONV layers and that of the kernel in the last CONV layer are set to $3\\times 3$ and $1\\times 1$, respectively.\n\n\nLet the source DRM be denoted as $F(X^s; \\Theta)$, \nwhere $X^s$ represents a synthetic image in the source domain, \nand $\\Theta = (\\Theta^e, \\Theta^d)$ represents the parameter vectors in the ECNN and the DCNN.\nTraining the source DRM is equivalent to learning a function $F$ with a parameter vectors $(\\Theta^e, \\Theta^d)$ that maps $X^s$ to an estimated density map, $\\hat{Y}$, such that\n\\begin{equation}\nY \\approx \\hat{Y} = F(X^s; \\Theta^e, \\Theta^d).\n\\end{equation}\n\nLet the source ECNN and DCNN be denoted as $\\mathcal{A}^s(X^s;\\Theta^{e})$ and $\\mathcal{D}^s(\\mathcal{A}^s(X^s;\\Theta^{e});\\Theta^{d})$, respectively.\nTherefore, $F(X^s; \\Theta) = \\mathcal{D}^s(\\mathcal{A}^s(X^s; \\Theta^{e}); \\Theta^{d})$.\nThe ECNN and the DCNN are jointly trained with a given set of $B$ training data \n$G^s = \\{(X^s_i,Y^s_i)\\}_{i = 1, ..., B}$, \nwhere $X^s_i$ is the $i$-th synthetic image and $Y^s_i$ is its ground truth density map. \nThe training process is implemented through the minimization of a loss function $L(\\Theta)$ that is defined as\n\\begin{equation}\nL(\\Theta) = \\frac{1}{B}\\sum_{i=1}^{B}\\left\\lVert Y_i^s - F(X^s_i;\\Theta)\\right\\rVert ^2.\n\\label{eq:loss1}\n\\end{equation}\n\nThe numerical minimization of $L(\\Theta)$ is performed via a momentum stochastic gradient descent (SGD) method~\\cite{bottou2010large}.\nA trained $F$ with the optimized parameters $\\Theta^*= (\\Theta^{e*}, \\Theta^{d*})$ is obtained in this step.\nThe trained ECNN, $\\mathcal{A}^s(X^s;\\Theta^{e*})$, will be employed for training the DAM as described next.\n\n\\subsubsection{DAM Training}\n\\label{sssec:dam}\n\nThe second phase of the proposed method is to train a DAM by use of images from both the source and target domains and the trained ECNN, $\\mathcal{A}^s(X^s;\\Theta^{e*})$.\nThe trained DAM is employed to map images in the target domain to a feature space that has minimum domain shift with the feature space of images in the source domain.\n\n\\begin{figure} [!ht]\n\\begin{center}\n\t\\begin{subfigure}{0.7\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{architecture.pdf}\n\t\t\\caption{The architecture of the source DRM}\n\t\t\\label{fig:drm}\n\t\\end{subfigure}\n   \\break\n\t\\begin{subfigure}{0.4\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{DAM.pdf}\n    \\caption{The architecture of the DAM}\n    \\label{fig:dam}\n  \\end{subfigure}\n\t\\begin{subfigure}{0.55\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{DCM.pdf}\n    \\caption{The architecture of the DCM}\n    \\label{fig:dcm}\n  \\end{subfigure}\n\\end{center}\n\\caption{\nThe network architectures of the source DRM (ECNN+DCNN), the DAM, and the DCM.\n}\n\\label{fig:dam_training}\n\\end{figure}\n\nThe DAM training process is shown in the middle block of Figure~\\ref{fig:framework}.\nThe network architecture of the DAM is shown in Figure~\\ref{fig:dam}.\nLet the DAM be denoted as $\\mathcal{A}^t(X^t,\\Theta^t)$.\nThe DAM is trained with unannotated source and target images through the minimization of domain shifts between the feature spaces of both the source and target domains. \nThe Wasserstein distance is identified as a metric measuring this domain shift in this study. \nDue to the difficulty of directly computing the Wasserstein distance, as discussed in the literature~\\cite{arjovsky2017wasserstein},\na domain critic model (DCM) is set up as a surrogate to estimate the Wasserstein distance for differentiating the feature distributions of the two domains.\nThe architecture of the DCM employed in this study is shown in Figure~\\ref{fig:dcm}.\nThe DCM includes two CONV layers.\nThe numbers of kernels in these two CONV layers are set to 128 and 256, respectively. \nThe size of each kernel is set to $3\\times 3$. \nAVE represents an average pooling layer, and FC represents a fully connected layer. \nDropout rate is set to 0.5, and the number of neurons in the FC layer is set to 256. \nThe DAM and DCM are optimized iteratively via an adversarial loss function~\\cite{arjovsky2017wasserstein} in an unsupervised manner by use of unannotated synthetic and experimental images.\n\n\n\\subsubsection{Density Estimation for Cell Counting}\n\\label{sssec:counting}\nThe final phase of the proposed method is density estimation for automatic cell counting on a given image\nas shown in the right block of Figure~\\ref{fig:framework}. \nThe combination of the trained DAM and the trained DCNN forms a target DRM, \n$\\mathcal{D}^s(\\mathcal{A}^t(X^t;\\Theta^{t*});\\Theta^{d*})$.\nThe trained DAM maps an experimental image to a feature map in the feature space of the source domain,\nand the trained DCNN estimates the density map of the image with this feature map. \nFinally, the number of cells is subsequently counted by integrating the estimated density map.\n\n\\section{Experimental Results}\n\\label{sec:experiment}\n\n\\subsection{Datasets}\n\\label{ssec:dataset}\n\nThe datasets used in this study are described in Table~\\ref{tab:dataset}.\nIn this experiment, $200$ synthetic bacterial fluorescent microscopic cell images of $256\\times 256$ pixels each were generated by use of methods described in the literature~\\cite{lehmussola2007}.\nThe synthetic images were annotated automatically when generated. The ground truth density map of each synthetic image was generated by placing normalized Gaussian kernels at each annotated cell centroid in the image, according to the methods introduced in Section~\\ref{ssec:density}. The values of $\\sigma$ and $K_G$ were set to $3$ pixels and $10$ pixels, respectively.\n\nIn addition, $122$ experimental immunofluorescent microscopic hESC images of $512\\times 512$ pixels each were employed.\nIn 10 of these $122$ images, the centroids of each cell were manually annotated.\nThe density map for each of the $10$ images was generated with $\\sigma$ and $K_G$ being $3$ pixels and $10$ pixels, respectively.\nThese density maps were employed as the ground truth to evaluate the cell counting performance.\n\\begin{table}[ht]\n\\caption{Datasets employed in this study}\n\\begin{center}\n\\begin{tabular}{c c c c}\n\\hline\\hline\n\\textbf{Images}\t\t& \\textbf{Annotated synthetic}\t& \\textbf{Unannotated hESC}& \\textbf{Annotated hESC}  \\\\\n\\textbf{Dataset size}\t& 200 \t\t\t\t\t\t& 112 \t\t\t\t\t\t& 10 \\\\\n\\textbf{Image size (pixels)} & $256\\times 256 $ \t\t& $512 \\times 512$ \t\t\t&  $512\\times 512$\\\\\n\\textbf{Purpose}\t\t& DRM, DAM training\t\t\t& DAM training       \t\t\t& Cell counting evaluation \\\\\n\\hline\\hline \n\\end{tabular}\n\\end{center}\n\\label{tab:dataset}\n\\end{table}\n\nEach pair of a synthetic image and a ground truth density map was employed for the supervised learning of the source DRM.\nOut of the $200$ samples, $160$ were employed for training the source DRM, and the remaining $40$ were employed as the validation set.\nDifferent from the source DRM training, unannotated synthetic and experimental data were employed in the DAM training phase.\nThe same $200$ synthetic cell images (without annotation) and $112$ unannotated experimental hESC images\nwere employed as the source and target image sets respectively, \nand were used for the adversarial learning of the DAM and DCM. \nThe DAM learning process was described in Section~\\ref{sssec:dam} above. Ten manually-annotated images and their density maps were employed as the ground truths to evaluate the cell counting performance of the proposed method.\n\n\\subsection{Method Implementation}\n\\label{ssec:implement}\n\nThe training and evaluation of the proposed method were performed on a NVIDIA Titan X GPU with 12GB of VRAM. \nSoftware packages employed in our experiments included Python 3.5, Keras 2.0, and Tensorflow 1.0.\nIn the source DRM training, the learning rate was set to $0.001$ and the batch size was set to $100$. The learning rate is a hyper-parameter that controls the stride of updating the values of the parameters in a to-be-trained model in each iteration.\nA mean square error (MSE) loss function was employed for model training.\nAfter $3000$ epochs, the model that resulted in the lowest MSE in the validation set was stored as the to-be-employed source DRM.\n\nIn the DAM training, the DAM was initialized with the weights of the trained ECNN.\nThe learning rates for training the DAM and DCM were both set to $10^{-8}$. \nIn each iteration, $100$ images were randomly selected from the source image set, \nand another $100$ images were randomly selected from the target image set. \nEach selected target image was randomly cropped into an image of $256 \\times 256$ pixels,\nso that the inputs of the ECNN and those of the DAM had the same sizes.\nThe DAM and DCM were iteratively optimized via an adversarial loss~\\cite{arjovsky2017wasserstein}.\n\n\n\\subsection{Results}\n\\label{ssec:result}\n\nWe compared the results obtained from the proposed method (denoted as Adaptation) with those from two other methods.\nThe first alternative method applied the source DRM directly to the experimental cell images (denoted as Source-only).\nThe second method was a fully convolutional regression network (FCRN)-based DRM~\\cite{xie2018} that was trained with annotated experimental images (denoted as Annotated-train).\nFor the Annotated-train method, $5$ out of the $10$ annotated images were employed \nto train the FCRN-based DRM, and the remaining $5$ were employed for the model validation.\n\nThe performances of the three cell counting methods were measured by the mean of absolute errors (MAE) and standard deviation of absolute errors (SAE).\nMAE measures the mean of absolute errors between the estimated cell counts and their ground truths for all 10 annotated hESC images, while SAE measures the standard deviation of the absolute errors.\nThe results of evaluating the three methods on all 10 images are shown in Table~\\ref{tab:mae}.\nIn terms of MAE and SAE, the proposed method demonstrates superior cell counting performance to the Source-only and Annotated-train methods.\n\\begin{table}[ht]\n\\caption{Performance of the proposed cell counting method} \n\\begin{center}       \n\\begin{tabular}{c c c c} \n\\hline\\hline\nPerformance  \t& Adaptation \t\t& Source-only \t\t\t& Annotated-train\\\\\nMAE $\\pm$ SAE \t&  \\textbf{35.82$\\pm$24.98}\t& 170.06 $\\pm$ 50.58 \t& 39.84 $\\pm$ 25.24\\\\\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\label{tab:mae}\n\\end{table}\n\\begin{figure}\n\t\\begin{center}\n\t    \\includegraphics[width=\\textwidth]{prediction_examples.png}\n\t\\end{center}\n\\caption{Density map estimation for 3 out of the 10 annotated hESC image examples.}\n\\label{fig:density_results}\n\\end{figure}\n\nAdditionally, the density maps of three testing hESC images estimated by use of the three methods are shown in Figure~\\ref{fig:density_results}.\nThe figures in each row display the results corresponding to different images.\nIn each row,  the sub-figures from left to right show the experimental image, the ground truth density map, \nand the density maps estimated with the proposed Adaptation method, the Source-only method, \nand the Annotated-train method, respectively.\nThe ground truth number of cells and the number counted from the estimated density maps are indicated at the bottom of each sub-figure.\nAs shown in Figure~\\ref{fig:density_results}, the proposed method can estimate a density map that is visually similar to the ground truth density map. The density maps estimated by use of the Source-only method is much denser than the ground truth, while those estimated by the Annotated-train method are blurred.\n\n\n\n\\section{Discussion}\n\\label{sec:discussion}\nIn this work, a domain adaptation-based density regression framework is proposed for automatic microscopic cell counting.\nInstead of training a DRM using the experimental images that require manual annotating, in this study, we propose to 1) train a source DRM with annotated synthetic images and 2) train a DAM \nwith both unannotated synthetic and experimental images to minimize the domain shift between the source and target domains.\nThe trained source DRM and DAM are jointly employed as a target DRM for automatic cell counting in experimental images. \nThe proposed method reduces or even eliminates the need to manually annotate experimental images and can greatly improve the generalization of the DRM trained with annotated synthetic images. \nTo the best of our knowledge, this is the first study in which a domain adaptation method is employed to support the task of automatic cell counting in microscopic images.\n\nIn addition, the proposed framework allows many flexible modifications. \nFor example, the FCNN network employed in the training of the source DRM can be \nreplaced with other FCNN networks when different tasks need to be addressed.\nThe choice of the FCNN in the current study is motivated by the success of deep neural networks in other computer vision tasks, including image classification~\\cite{he2016}, segmentation~\\cite{ronneberger2015, he2018}, and object detection~\\cite{ren2015}.\nAlthough the network architecture of the FCNN is fixed in this study, tuning of the network architecture is still an open question in deep learning-based researches, but it is not the focus of this work.\n\n\n\n\\section{Conclusion}\n\\label{sec:conclusion}\n\nA novel, manual-annotating-free density regression framework is proposed for automatic microscopic cell counting. \nThe proposed method integrates a supervised learned DRM and an unsupervised learned DAM for automatic cell counting, and achieves promising cell counting performance.\n\n\\acknowledgments\n \nThis work was supported in part by award NIH R01EB020604, R01EB023045, R01NS102213, and R21CA223799.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec0}\n\n\\par\n\nIn the paper, we consider pseudo-differential operators, where\nthe symbols are of infinite orders and possess suitable Gevrey regularities\nand which are allowed to grow sub-exponentially together with all\ntheir derivatives. Our main purpose is to extend boundedness results,\nin \\cite{To14}, of the\npseudo-differential operators when acting on modulation spaces.\n\n\\par\n\nMore specific, the symbols should satisfy conditions of the form\n\\begin{equation}\\label{Eq:SymbCondIntr}\n|\\partial _x^\\alpha \\partial _\\xi ^\\beta a(x,\\xi )|\n\\lesssim h^{|\\alpha +\\beta |}\\alpha !^\\sigma \\beta !^s\\omega _0(x,\\xi ),\n\\end{equation}\nwhere $\\omega _0$ should be a moderate weight on $\\rr {2d}$\nand satisfy boundedness conditions like\n\\begin{equation}\\label{Eq:WeightCondIntr}\n\\omega _0(x,\\xi )\\lesssim e^{r(|x|^{\\frac 1s}+|\\xi |^{\\frac 1\\sigma})}.\n\\end{equation}\nFor such symbols $a$ we prove that corresponding pseudo-differential\noperators $\\operatorname{Op} (a)$ is continuous from the modulation space\n$M(\\omega _0\\omega ,\\mathscr B )$ to $M(\\omega ,\\mathscr B )$.\n(See Section \\ref{sec1} for notations.)\n\n\\par\n\nSimilar investigations were performed in \\cite{To25} in the case\n$s=\\sigma$ (i.{\\,}e. the isotropic case).\nTherefore, the results in the current paper are more general\nin the sense of the anisotropicity of the considered symbol classes.\nMoreover, we use different techniques compared to \\cite{To25}.\n\n\\par\n\nWe also remark that several ideas arise in \\cite{To14}, where similar\ninvestigations were performed after the conditions \\eqref{Eq:SymbCondIntr}\nand \\eqref{Eq:WeightCondIntr} are replaced by\n$$\n|\\partial _x^\\alpha \\partial _\\xi ^\\beta a(x,\\xi )| \\lesssim \\omega _0(x,\\xi )\n$$\nand\n$$\n\\omega _0(x,\\xi )\\lesssim (1+|x|+|\\xi |)^N,\n$$\nrespectively, for some $N\\ge 0$.\n\n\\par\n\nIn \\cite{Fe4}, H. Feichtinger introduced the modulation spaces\nto measure the\ntime-frequency concentration of a function or distribution\non the time-frequency\nspace or the phase space $\\rr {2d}$.\nNowadays they become popular among mathematicians\nand engineers since their numerous applications in\nsignal processing \\cite{Fe8,Fe9}, pseudo-differential and Fourier integral operators\n\\cite{CoGaTo,CoTo,CoGrNiRo1,PT2,PT3,Ta,Te1,Te2,To14,To18,To20,To22,To24,To25}\nand quantum mechanics \\cite{CoGrNiRo,dGo}.\n\n\\par\n\n\n\\par\n\nThe paper is organized as follows. In Section \\ref{sec1}\nwe give the main definition and properties of Gelfand-Shilov and\nmodulation spaces and we recall some essential results.\nIn Section \\ref{sec2} we state our main results on\nthe continuity with anisotropic settings.\n\n\\par\n\n\\section{Preliminaries}\\label{sec1}\n\n\\par\n\nIn the current section we review basic properties for modulation\nspaces and other related spaces. More details and proofs can be found in\n\\cite{Fe2,Fe3,Fe4,FG1,FG2,FG4,GaSa,Gc2,To20} .\n\n\\par\n\n\\subsection{Weight functions}\\label{subsec1.1}\n\n\\par\n\nA function $\\omega$ on $\\rr d$ is called a \\emph{weight} or\n\\emph{weight function},\nif $\\omega ,1\/\\omega \\in L^\\infty _{\\operatorname{loc}} (\\rr d)$\nare positive everywhere.\nThe weight $\\omega$ on $\\rr d$ is called $v$-moderate\nfor some weight $v$ on $\\rr d$, if\n\\begin{equation}\\label{vModerate}\n\\omega (x+y)\\lesssim \\omega (x)v(y),\\quad x,y\\in \\rr d.\n\\end{equation}\nIf $v$ is even and satisfies \\eqref{vModerate} with $\\omega =v$,\nthen $v$ is called submultiplicative. \n\n\\par\n\nLet $s,\\sigma>0$. Then we let $\\mathscr P_E(\\rr d)$ be the\nset of all moderate weights on $\\rr d$, $\\mathscr P_s(\\rr  d)$\n($\\mathscr P_s^0(\\rr  d)$) be the set of all $\\omega\\in\n\\mathscr P_E(\\rr d)$ such that \n$$\n\\omega (x+y)\\lesssim \\omega (x)e^{r|y|^{\\frac 1s}}, \\quad x,y\\in \\rr d,\n$$\nfor some $r>0$ (for every $r>0$), and  $\\mathscr P_{s,\\sigma}(\\rr {2d})$\n($\\mathscr P_{s,\\sigma}^0(\\rr {2d})$) be the set of all\n$\\omega\\in \\mathscr P_E(\\rr {2d})$ such that\n\\begin{equation} \\label{estomega}\n\\omega (x+y,\\xi+\\eta)\\lesssim \\omega (x,\\xi)\ne^{r(|y|^{\\frac 1s}+|\\eta|^{\\frac 1\\sigma})}, \\quad x,y,\\xi,\\eta\\in \\rr d,\n\\end{equation}\nfor some $r>0$ (for every $r>0$).\n\n\\par\n\n\nThe following result shows that for any weight in $\\mathscr P _E$,\nthere are equivalent weights that \nsatisfy strong Gevrey regularity.\n\n\\par\n\n\\begin{prop}\\label{Prop:EquivWeights}\nLet $\\omega \\in \\mathscr P _E(\\rr {2d})$ and $s,\\sigma> 0$.\nThen there exists a weight\n$\\omega _0\\in \\mathscr P _E(\\rr {2d})\\cap C^\\infty (\\rr {2d})$\nsuch that the following is true:\t\n\\begin{enumerate}\t\n\\item $\\omega _0\\asymp \\omega $;\n\n\\vspace{0.1cm}\n\n\\item for every $h>0$,\n\\begin{equation*}\n|\\partial _x ^{\\alpha}\\partial _\\xi ^\\beta\n\\omega _0(x, \\xi)|\n\\lesssim \nh^{|\\alpha +\\beta|}\\alpha !^\\sigma\n\\beta!^s \\omega _0(x,\\xi) \n\\asymp \nh^{|\\alpha +\\beta|}\\alpha !^\\sigma\n\\beta!^s\\omega (x,\\xi).\n\\end{equation*} \n\\end{enumerate}\n\\end{prop}\n\n\\par\n\nProposition \\ref{Prop:EquivWeights} is equivalent to \\cite[Proposition 1.6]{AbCoTo}.\nIn fact, by Proposition \\cite[Proposition 1.6]{AbCoTo} we have that\nProposition \\ref{Prop:EquivWeights} holds with $s=\\sigma$.\nHence, Proposition \\ref{Prop:EquivWeights} implies \\cite[Proposition 1.6]{AbCoTo}.\nOn the other hand, let $s_0=\\min (s,\\sigma)$. Then\n\\cite[Proposition 1.6]{AbCoTo} implies that there is a weight function\n$\\omega_0\\asymp \\omega$ satisfying\n\\begin{align*}\n|\\partial _x ^{\\alpha}\\partial _\\xi ^\\beta\n\\omega _0(x, \\xi)|\n&\\lesssim \nh^{|\\alpha +\\beta|}(\\alpha !\n\\beta!)^{s_0} \\omega _0(x,\\xi) \n\\\\[1ex]\n&\\lesssim\nh^{|\\alpha +\\beta|}\\alpha !^\\sigma\n\\beta!^s \\omega _0(x,\\xi),\n\\end{align*}\ngiving Proposition \\ref{Prop:EquivWeights}.\n\n\n\\par\n\n\n\\subsection{Gelfand-Shilov spaces}\\label{subsec1.2}\n\n\\par\n\nLet $0<h,s,\\sigma \\in \\mathbf R$ be fixed. Then\n$\\mathcal S _{s;h}^{\\sigma}(\\rr d)$ is\nthe Banach space of all $f\\in C^\\infty (\\rr d)$ such that\n\\begin{equation}\\label{gfseminorm}\n\\nm f{\\mathcal S _{s;h}^{\\sigma}}\\equiv \\sup_{\\alpha ,\\beta \\in\n\\mathbf N^d} \\sup_{x \\in \\rr d} \\frac {|x^\\alpha \\partial ^\\beta\nf(x)|}{h^{|\\alpha | + |\\beta |}\\alpha !^s\\, \\beta !^\\sigma}<\\infty,\n\\end{equation}\nendowed with the norm \\eqref{gfseminorm}.\n\n\\par\n\nThe \\emph{Gelfand-Shilov spaces} $\\mathcal S _{s}^{\\sigma}(\\rr d)$ and\n$\\Sigma _{s}^{\\sigma}(\\rr d)$ are defined as the inductive and projective \nlimits respectively of $\\mathcal S _{s;h}^{\\sigma}(\\rr d)$. This implies that\n\\begin{equation}\\label{GSspacecond1}\n\\mathcal S _{s}^{\\sigma}(\\rr d) = \\bigcup _{h>0}\\mathcal S _{s;h}^{\\sigma}(\\rr d)\n\\quad \\text{and}\\quad \\Sigma _{s}^{\\sigma}(\\rr d) =\\bigcap _{h>0}\n\\mathcal S _{s;h}^{\\sigma}(\\rr d),\n\\end{equation}\nand that the topology for $\\mathcal S _{s}^{\\sigma}(\\rr d)$ is the strongest\npossible one such that the inclusion map from $\\mathcal S _{s;h}^{\\sigma}\n(\\rr d)$ to $\\mathcal S _{s}^{\\sigma}(\\rr d)$ is continuous, for every choice \nof $h>0$. The space $\\Sigma _{s}^{\\sigma}(\\rr d)$ is a Fr{\\'e}chet space\nwith seminorms $\\nm \\, \\cdot \\,  {\\mathcal S _{s;h}^{\\sigma}}$, $h>0$. Moreover,\n$\\Sigma _{s}^{\\sigma}(\\rr d)\\neq \\{ 0\\}$, if and only\nif $s+\\sigma \\ge 1$ and $(s,\\sigma )\\neq\n(\\frac 12,\\frac 12)$, and\n$\\mathcal S _{s}^{\\sigma}(\\rr d)\\neq \\{ 0\\}$, if and only\nif $s+\\sigma \\ge 1$.\n\n\n\\medspace\n\nThe \\emph{Gelfand-Shilov distribution spaces} $(\\mathcal S _{s}^{\\sigma})'(\\rr d)$\nand $(\\Sigma _{s}^{\\sigma})'(\\rr d)$\nare the projective and inductive limit\nrespectively of $(\\mathcal S _{s;h}^{\\sigma})'(\\rr d)$.\nIn \\cite{Pi2} it is proved that $(\\mathcal S _{s}^{\\sigma})'(\\rr d)$\nis the dual of $\\mathcal S _s^\\sigma(\\rr d)$, and $(\\Sigma _{s}^{\\sigma})'(\\rr d)$\nis the dual of $\\Sigma _{s}^{\\sigma}(\\rr d)$ (also in topological sense).\n\n\n\\par\n\nThe Fourier transform $\\mathscr F$ is the linear and continuous\nmap on $\\mathscr S (\\rr d)$,\ngiven by the formula\n$$\n(\\mathscr Ff)(\\xi )= \\widehat f(\\xi ) \\equiv (2\\pi )^{-\\frac d2}\\int _{\\rr\n\t{d}} f(x)e^{-i\\scal  x\\xi }\\, dx\n$$\nwhen $f\\in \\mathscr S (\\rr d)$. Here $\\scal \\, \\cdot \\,  \\, \\cdot \\, $ denotes the usual\nscalar product on $\\rr d$. \nThe Fourier transform extends  uniquely to homeomorphisms\nfrom $(\\mathcal S _{s}^{\\sigma})'(\\rr d)$ to $(\\mathcal S _{\\sigma}^{s})'(\\rr d)$,\nand from  $(\\Sigma _{s}^{\\sigma})'(\\rr d)$ to $(\\Sigma _{\\sigma}^{s})'(\\rr d)$.\nFurthermore, it restricts to homeomorphisms from\n$\\mathcal S _{s}^{\\sigma}(\\rr d)$ to $\\mathcal S _{\\sigma}^{s}(\\rr d)$,\nand from  $\\Sigma _{s}^{\\sigma}(\\rr d)$ to $\\Sigma _{\\sigma}^{s}(\\rr d)$.\n\n\\par\n\n\\medspace\n\nSome considerations later on involve a broader family of\nGelfand-Shilov spaces. More precisely, for $s_j,\\sigma _j\\in \\mathbf R_+$,\n$j=1,2$, the Gelfand-Shilov spaces $\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$ and\n$\\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$ consist of all functions\n$F\\in C^\\infty (\\rr {d_1+d_2})$ such that\n\\begin{equation}\\label{GSExtCond}\n|x_1^{\\alpha _1}x_2^{\\alpha _2}\\partial _{x_1}^{\\beta _1}\n\\partial _{x_2}^{\\beta _2}F(x_1,x_2)| \\lesssim\nh^{|\\alpha _1+\\alpha _2+\\beta _1+\\beta _2|}\n\\alpha _1!^{s_1}\\alpha _2!^{s_2}\\beta _1!^{\\sigma _1}\\beta _2!^{\\sigma _2}\n\\end{equation}\nfor some $h>0$ respective for every $h>0$. The  topologies, and the duals\n\\begin{alignat*}{3}\n&(\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2}) &\n&\\quad \\text{and} \\quad &\n&(\\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2})\n\\intertext{of}\n&\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}) &\n&\\quad \\text{and} \\quad &\n&\\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}),\n\\end{alignat*}\nrespectively, and their topologies\nare defined in analogous ways as for the spaces $\\mathcal S _s^\\sigma (\\rr d)$\nand $\\Sigma _s^\\sigma (\\rr d)$ above.\n\n\\par\n\nThe following proposition explains mapping properties of partial\nFourier transforms on Gelfand-Shilov spaces, and follows by similar\narguments as in analogous situations in\n\\cite{GS}. The proof is therefore omitted. Here, $\\mathscr F _1F$\nand $\\mathscr F _2F$ are the partial\nFourier transforms of $F(x_1,x_2)$ with respect to\n$x_1\\in \\rr {d_1}$ and $x_2\\in \\rr {d_2}$,\nrespectively.\n\n\\par\n\n\\begin{prop}\\label{propBroadGSSpaceChar}\nLet $s_j,\\sigma _j >0$, $j=1,2$.\nThen the following is true:\n\\begin{enumerate}\t\n\\item the mappings $\\mathscr F _1$ and $\\mathscr F _2$ on $\\mathscr S (\\rr {d_1+d_2})$\nrestrict to homeomorphisms\n\\begin{align*}\n\\mathscr F _1 \\, &: \\, \\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}) \\to\n\\mathcal S _{\\sigma _1,s_2}^{s_1,\\sigma _2}(\\rr {d_1+d_2})\n\\intertext{and}\n\\mathscr F _2 \\, &: \\, \\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}) \\to\n\\mathcal S _{s _1,\\sigma _2}^{\\sigma _1,s_2}(\\rr {d_1+d_2})\n\\text ;\n\\end{align*}\n\n\\vspace{0.1cm}\n\n\\item the mappings $\\mathscr F _1$ and $\\mathscr F _2$ on\n$\\mathscr S (\\rr {d_1+d_2})$ are uniquely extendable to\nhomeomorphisms\n\\begin{align*}\n\\mathscr F _1 \\, &: \\, (\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2}) \\to\n(\\mathcal S _{\\sigma _1,s_2}^{s_1,\\sigma _2})'(\\rr {d_1+d_2})\n\\intertext{and}\n\\mathscr F _2 \\, &: \\, (\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2}) \\to\n(\\mathcal S _{s _1,\\sigma _2}^{\\sigma _1,s_2})'(\\rr {d_1+d_2}).\n\\end{align*}\n\\end{enumerate}\n\t\n\\par\n\t\nThe same holds true if the $\\mathcal S  _{s _1,s_2}^{\\sigma _1,\\sigma _2}$-spaces and\ntheir duals are replaced by\ncorresponding $\\Sigma  _{s _1,s_2}^{\\sigma _1,\\sigma _2}$-spaces and their duals.\n\\end{prop}\n\n\\par\n\nThe next two results follow from \\cite{ChuChuKim}. The proofs are therefore omitted.\n\n\\begin{prop}\nLet $s_j,\\sigma _j> 0$, $j=1,2$. Then the following\nconditions are equivalent.\n\\begin{enumerate}\n\\item $F\\in \\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$\\quad\n($F\\in \\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$);\n\n\\vspace{0.1cm}\n\n\\item for some $h>0$ (for every $h>0$) it holds\n\\begin{equation*}\n\\displaystyle{|F(x_1,x_2)|\\lesssim e^{-h(|x_1|^{\\frac 1{s_1}} + |x_2|^{\\frac 1{s_2}} )}}\n\\quad \\text{and}\\quad \n\\displaystyle{|\\widehat F(\\xi _1,\\xi _2)|\\lesssim\n\te^{-h(|\\xi _1|^{\\frac 1{\\sigma _1}} + |\\xi _2|^{\\frac 1{\\sigma _2}} )}}.\n\\end{equation*}\n\\end{enumerate}\n\\end{prop}\n\n\\par\n\nWe notice that if\n$s_j+\\sigma _j<1$ for some $j=1,2$, then\n$\\mathcal S _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$\nand $\\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$\nare equal to the trivial space $\\{ 0\\}$.\nLikewise, if $s_j=\\sigma _j=\\frac 12$ for some $j=1,2$, then\n$\\Sigma _{s _1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}) = \\{ 0\\}$.\n\n\\par\n\n\\subsection{Short time Fourier transform and Gelfand-Shilov spaces}\n\n\\par\n\nWe recall here some basic facts about\nthe short-time Fourier transform and weights.\n\n\\par\n\nLet $\\phi \\in \\mathcal S _s^\\sigma (\\rr d)\\setminus 0$ be fixed. Then the short-time\nFourier transform of $f\\in (\\mathcal S _s^\\sigma )'(\\rr d)$ is given by\n$$\n(V_\\phi f)(x,\\xi ) = (2\\pi )^{-\\frac d2}(f,\\phi (\\, \\cdot \\,  -x)\ne^{i\\scal \\, \\cdot \\,  \\xi})_{L^2}.\n$$\nHere $(\\, \\cdot \\,  ,\\, \\cdot \\,  )_{L^2}$ is the unique extension of the $L^2$-form on\n$\\mathcal S _s^\\sigma (\\rr d)$ to a continuous sesqui-linear form on $(\\mathcal S\n_s^\\sigma )'(\\rr d)\\times \\mathcal S _s^\\sigma (\\rr d)$. In the case\n$f\\in L^p(\\rr d)$, for some $p\\in [1,\\infty]$, then $V_\\phi f$ is given by\n$$\nV_\\phi f(x,\\xi ) \\equiv (2\\pi )^{-\\frac d2}\\int _{\\rr d}f(y)\\overline{\\phi (y-x)}\ne^{-i\\scal y\\xi}\\, dy .\n$$\n\n\\par\n\nThe following characterizations of the\n$\\mathcal S _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$,\n$\\Sigma _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$\nand their duals\nfollow by similar arguments as in the proofs of\nPropositions 2.1 and 2.2 in \\cite{To22}. The details are left\nfor the reader.\n\n\\par\n\n\\begin{prop}\\label{Prop:STFTGelfand2}\nLet $s_j,\\sigma _j>0$ be such that $s_j+\\sigma _j\\ge 1$, $j=1,2$, \n$s_0\\le s$ and $\\sigma_0\\le \\sigma$. Also let\n$\\phi \\in \\mathcal S_{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}\n\\setminus 0$)\n($\\phi \\in \\Sigma _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2}\n\\setminus 0$) and let $f$ be a Gelfand-Shilov distribution on\n$\\rr {d_1+d_2}$.\nThen $f\\in  \\mathcal S _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$\n($f\\in  \\Sigma _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})$),\nif and only if\n\\begin{equation}\\label{stftexpest2}\n|V_\\phi f(x_1,x_2,\\xi _1,\\xi _2)|\n\\lesssim\ne^{-r (|x_1|^{\\frac 1{s_1}} + |x_2|^{\\frac 1{s_2}}\n\t+|\\xi _1|^{\\frac 1{\\sigma _1}} +|\\xi _2|^{\\frac 1{\\sigma _2}} )},\n\\end{equation}\nholds for some $r > 0$ (holds for every $r > 0$).\n\\end{prop}\n\n\\par\n\nA proof of Proposition  \\ref{Prop:STFTGelfand2} can be found in\ne.{\\,}g. \\cite{GZ} (cf. \\cite[Theorem 2.7]{GZ}). The\ncorresponding result for Gelfand-Shilov distributions\nis the following improvement of \\cite[Theorem 2.5]{To18}.\n\n\\par\n\n\\begin{prop}\\label{Prop:STFTGelfand2Dist}\nLet $s_j,\\sigma _j>0$ be such that $s_j+\\sigma _j\\ge 1$, $j=1,2$, \n$s_0\\le s$ and $t_0\\le t$. Also let\n$\\phi \\in \\mathcal S_{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})\n\\setminus 0$ and let $f$ be a Gelfand-Shilov distribution on\n$\\rr {d_1+d_2}$.\nThen the following is true:\n\t\\begin{enumerate}\n\\item $f\\in  (\\mathcal S _{s_1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2})$,\nif and only if\n\\begin{equation}\\label{stftexpest2Dist}\n|V_\\phi f(x_1,x_2,\\xi _1,\\xi _2)|\n\\lesssim\ne^{r (|x_1|^{\\frac 1{s_1}} + |x_2|^{\\frac 1{s_2}}\n\t+|\\xi _1|^{\\frac 1{\\sigma _1}} +|\\xi _2|^{\\frac 1{\\sigma _2}} )}\n\\end{equation}\nholds for every $r > 0$;\n\n\\vspace{0.1cm}\n\n\\item if in addition\n$\\phi \\in \\Sigma _{s_1,s_2}^{\\sigma _1,\\sigma _2}(\\rr {d_1+d_2})\n\\setminus 0$, then \n$f\\in  (\\Sigma _{s_1,s_2}^{\\sigma _1,\\sigma _2})'(\\rr {d_1+d_2})$,\nif and only if\n\\begin{equation}\\label{stftexpest2DistA}\n|V_\\phi f(x_1,x_2,\\xi _1,\\xi _2)|\n\\lesssim\ne^{r (|x_1|^{\\frac 1{s_1}} + |x_2|^{\\frac 1{s_2}}\n\t+|\\xi _1|^{\\frac 1{\\sigma _1}} +|\\xi _2|^{\\frac 1{\\sigma _2}} )}\n\\end{equation}\nholds for some $r > 0$.\n\\end{enumerate}\n\\end{prop}\n\n\\par\n\n\\subsection{Broader family of modulation spaces}\\label{subsec1.3}\n\n\\par\n\n\\par\n\n\n\\begin{defn}\\label{bfspaces1}\nLet $\\mathscr B $ be a Banach space of measurable functions on $\\rr d$,\nand let $v \\in\\mathscr P _E(\\rr d)$.\nThen $\\mathscr B$ is called a \\emph{translation invariant\nBanach Function space on $\\rr d$} (with respect to $v$), or \\emph{invariant\nBF space on $\\rr d$}, if there is a constant $C$ such\nthat the following conditions are fulfilled:\n\\begin{enumerate}\n\\item if $x\\in \\rr d$ and $f\\in \\mathscr B$, then $f(\\, \\cdot \\,  -x)\\in\n\\mathscr B$, and \n\\begin{equation}\\label{translmultprop1}\n\\nm {f(\\, \\cdot \\,  -x)}{\\mathscr B}\\le Cv(x)\\nm {f}{\\mathscr B}\\text ;\n\\end{equation}\n\n\\vspace{0.1cm}\n\n\\item if  $f,g\\in L^1_{loc}(\\rr d)$ satisfy $g\\in \\mathscr B$ and $|f|\n\\le |g|$, then $f\\in \\mathscr B$ and\n$$\n\\nm f{\\mathscr B}\\le C\\nm g{\\mathscr B}\\text ;\n$$\n\\vspace{0.1cm}\n\n\\item Minkowski's inequality holds true, i.{\\,}e.\n\\begin{equation}\\label{Eq:MinkIneq}\n\\nm {f*\\varphi}{\\mathscr B}\\lesssim \\nm {f}{\\mathscr B}\\nm \\varphi{L^1_{(v)}},\n\\qquad f\\in \\mathscr B ,\\ \\varphi \\in L^1_{(v)}(\\rr d).\n\\end{equation}\n\\end{enumerate}\n\\end{defn}\n\n\\par\n\nIf $v$ belongs to $\\mathscr P _{E,s}(\\rr d)$\n($\\mathscr P _{E,s}^0(\\rr d)$), then $\\mathscr B$ in Definition \\ref{bfspaces1}\nis called an invariant BF-space of Roumieu type (Beurling type) of order $s$.\n\n\\par\n\nIt follows from (2) in Definition \\ref{bfspaces1} that if $f\\in\n\\mathscr B$ and $h\\in L^\\infty$, then $f\\cdot h\\in \\mathscr B$, and\n\\begin{equation}\\label{multprop}\n\\nm {f\\cdot h}{\\mathscr B}\\le C\\nm f{\\mathscr B}\\nm h{L^\\infty}.\n\\end{equation}\nIn Definition \\ref{bfspaces1}, condition (2)  means that a\ntranslation invariant BF-space is a solid BF-space in the sense of\n(A.3) in \\cite{Fe6}. \n\n\\par\n\n\\begin{example}\\label{Lpqbfspaces}\nAssume that $p,q\\in [1,\\infty ]$, and let $L^{p,q}_1(\\rr {2d})$ be the\nset of all $f\\in L^1_{loc}(\\rr {2d})$ such that\n$$\n\\nm  f{L^{p,q}_1} \\equiv \\Big ( \\int \\Big ( \\int |f(x,\\xi )|^p\\, dx\\Big\n)^{q\/p}\\, d\\xi \\Big )^{1\/q}\n$$\nif finite.\nThen it follows that $L^{p,q}_1$\nis translation invariant BF-spaces with respect to $v=1$.\n\\end{example}\n\n\\par\n\nWe refer to \\cite {Fe4,FG1,FG2,FG4,GaSa,Gc2,RSTT,To20}\nfor more facts about modulation spaces.\nNext we consider the extended class of modulation spaces which we are interested\nin. \n\n\n\\par\n\n\\begin{defn}\\label{bfspaces2}\nAssume that $\\mathscr B$ is a translation\ninvariant QBF-space on $\\rr {2d}$, $\\omega \\in\\mathscr P _E(\\rr {2d})$,\nand that $\\phi \\in\n\\Sigma _1(\\rr d)\\setminus 0$. Then the set $M(\\omega ,\\mathscr B )$\nconsists of all $f\\in \\Sigma _1'(\\rr d)$ such that\n$$\n\\nm f{M(\\omega ,\\mathscr B )}\n\\equiv \\nm {V_\\phi f\\, \\omega }{\\mathscr B}\n$$\nis finite.\n\\end{defn}\n\n\\par\n\nObviously, we have\n$\nM^{p,q}_{(\\omega )}(\\rr d)=M(\\omega ,\\mathscr B )$\nwhen  $\\mathscr B =L^{p,q}_1(\\rr {2d})$ (cf. Example \\ref{Lpqbfspaces}).\nIt follows that many properties which are valid for the classical modulation\nspaces also hold for the spaces of the form $M(\\omega ,\\mathscr B )$.\n\n\\par\n\nWe notice that $M(\\omega ,\\mathscr B )$ is independent of the choice\nof $\\phi$ in Definition \\ref{bfspaces2} cf.\\, \\cite{To25}.\nFurthermore, $M(\\omega ,\\mathscr B )$ is a Banach space in view of \\cite{PfTo}.\n\n\n\\par\n\n\\subsection{Pseudo-differential operators}\n\n\\par\n\nNext we recall some facts on pseudo-differential operators. Let\n$A\\in \\mathbf{M} (d,\\mathbf R)$ be fixed and\nlet $a\\in \\Sigma _1(\\rr {2d})$. Then the pseudo-differential\noperator $\\operatorname{Op} _A(a)$ is the linear and continuous operator on $\\Sigma _1(\\rr d)$,\ndefined by the formula\n\\begin{multline}\\label{e0.5}\n(\\operatorname{Op} _A(a)f)(x)\n\\\\[1ex]\n=\n(2\\pi  ) ^{-d}\\iint a(x-A(x-y),\\xi )f(y)e^{i\\scal {x-y}\\xi }\\,\ndyd\\xi .\n\\end{multline}\nThe definition of $\\operatorname{Op} _A(a)$ extends to\nany $a\\in \\Sigma _1'(\\rr {2d})$, and then $\\operatorname{Op} _A(a)$ is continuous from\n$\\Sigma _1(\\rr d)$ to $\\Sigma _1'(\\rr d)$. Moreover, for every fixed\n$A\\in \\mathbf{M} (d,\\mathbf R)$, it follows that there is a one to\none correspondence between such operators and pseudo-differential\noperators of the form $\\operatorname{Op} _A(a)$. (See e.{\\,}g. \\cite {Ho1}.)\nIf $A=2^{-1}I$, where $I\\in \\mathbf{M} (d,\\mathbf R)$ is the identity matrix, then\n$\\operatorname{Op} _A(a)$ is equal to the Weyl operator $\\operatorname{Op} ^w(a)$\nof $a$. If instead $A=0$, then the standard (Kohn-Nirenberg)\nrepresentation $\\operatorname{Op} (a)$ is obtained.\n\n\\par\n\nIf $a_1,a_2\\in \\Sigma _1'(\\rr {2d})$ and $A_1,A_2\\in\n \\mathbf{M} (d,\\mathbf R)$, then\n\\begin{equation}\\label{pseudorelation}\n\\operatorname{Op} _{A_1}(a_1)=\\operatorname{Op} _{A_2}(a_2) \\quad \\Leftrightarrow \\quad a_2(x,\\xi\n)=e^{i\\scal {(A_1-A_2)D_\\xi}{D_x}}a(x,\\xi ).\n\\end{equation}\n(Cf. \\cite{Ho1}.)\n\n\\par\n\n\n\\par\n\n\\subsection{Symbol classes}\n\n\\par\n\nNext we introduce function spaces related to symbol classes\nof the pseudo-differential operators. These functions should obey various\nconditions of the form\n\\begin{align}\n|\\partial _x^\\alpha \\partial _\\xi ^\\beta a(x,\\xi )|\n&\\lesssim\nh ^{|\\alpha +\\beta |}\\alpha !^\\sigma \\beta !^s  \\omega (x,\\xi ),\n\\label{Eq:symbols2}\n\\end{align}\nfor functions on $\\rr {d_1+d_2}$. For this reason we consider\nsemi-norms of the form\n\\begin{equation}\\label{Eq:GammaomegaNorm}\n\\nm a{\\Gamma _{(\\omega )}^{\\sigma ,s;h}}\n\\equiv \\sup _{(\\alpha ,\\beta) \\in \\nn {d_1+d_2}}\n\\left (\n\\sup _{(x,\\xi) \\in \\rr {d_1+d_2}} \\left (\n\\frac {|\\partial _x^\\alpha \\partial _\\xi ^\\beta a(x,\\xi )|}{\n\th ^{|\\alpha +\\beta |}\\alpha !^\\sigma \\beta !^s \\omega (x,\\xi )}\n\\right ) \\right ) ,\n\\end{equation}\nindexed by $h>0$, \n\n\\par\n\n\\begin{defn}\\label{Def:GammaSymb2}\nLet $s$, $\\sigma$ and $h$ be positive constants,\nlet $\\omega$ be a weight on $\\rr {d_1+d_2}$, and let\n$$\n\\omega _r(x,\\xi )\\equiv e^{r(|x|^{\\frac 1s} + |\\xi |^{\\frac 1\\sigma })}.\n$$ \n\\begin{enumerate}\n\\item The set $\\Gamma _{(\\omega )}^{\\sigma ,s;h}  (\\rr {d_1+d_2})$\nconsists of\nall $a \\in C^\\infty(\\rr {d_1+d_2})$ such that\n$\\nm a{\\Gamma _{(\\omega )}^{\\sigma ,s;h}}$ in\n\\eqref{Eq:GammaomegaNorm} is finite.\nThe set $\\Gamma _0^{\\sigma ,s;h}  (\\rr {d_1+d_2})$ consists of\nall $a \\in C^\\infty(\\rr {d_1+d_2})$ such that\n$\\nm a{\\Gamma _{(\\omega_r )}^{\\sigma ,s;h}}$ is finite\nfor every $r>0$, and the topology is the projective limit topology of\n$\\Gamma _{(\\omega _r)}^{\\sigma ,s;h}  (\\rr {d_1+d_2})$ with respect to $r>0$;\n\n\\vspace{0.1cm}\n\n\\item The sets $\\Gamma _{(\\omega )}^{\\sigma ,s}  (\\rr {d_1+d_2})$ and\n$\\Gamma _{(\\omega )}^{\\sigma ,s;0}  (\\rr {d_1+d_2})$ are given by\n\\begin{align*}\n\\Gamma _{(\\omega )}^{\\sigma ,s}  (\\rr {d_1+d_2})\n&\\equiv\n\\bigcup _{h>0}\\Gamma _{(\\omega )}^{\\sigma ,s;h}  (\\rr {d_1+d_2})\n\\intertext{and}\n\\Gamma _{(\\omega )}^{\\sigma ,s;0}  (\\rr {d_1+d_2})\n&\\equiv\n\\bigcap _{h>0}\\Gamma _{(\\omega )}^{\\sigma ,s;h}  (\\rr {d_1+d_2}),\n\\end{align*}\nand their topologies are the inductive respective the projective topologies\nof $\\Gamma _{(\\omega )}^{\\sigma ,s;h}  (\\rr {d_1+d_2})$ with respect to $h>0$.\t\n\\end{enumerate}\t\n\\end{defn}\n\n\\par\n\n\\par\n\nThe following result is a straight-forward consequence of \n\\cite[Proposition 2.4]{AbCaTo} and the definitions.\n\n\\par\n\n\\begin{prop}\\label{SymbClassModSpace}\nLet $R>0$, $q\\in (0,\\infty ]$, $s,\\sigma >0$ be such that $s+\\sigma \\ge 1$\nand $(s,\\sigma )\\neq (\\frac 12,\\frac 12)$, $\\phi \\in \\Sigma _{s,\\sigma}^{\\sigma ,s}\n(\\rr {2d})\\setminus 0$,\n$\\omega \\in \\mathscr P _{s,\\sigma}(\\mathbf{R}^{2d}),$ and let\n$$\n\\omega _R(x,\\xi, \\eta, y) = \\omega(x,\\xi) e^{-R(|y|^{\\frac 1s} + |\\eta|^{\\frac 1\\sigma})}.\n$$\nThen\n\\begin{align}\\label{iden}\n\\begin{split}\n\\Gamma^{\\sigma ,s}_{(\\omega)}(\\rr {2d})\n&=\n\\bigcup _{R>0}\\sets {a\\in (\\Sigma _{s,\\sigma}^{\\sigma ,s})'\n\t(\\rr {2d})}{\\nm {\\omega\n\t\t_R^{-1}V_\\phi a}{L^{\\infty ,q}} <\\infty },\n\\\\[1ex]\n\\Gamma^{\\sigma ,s;0}_{(\\omega)}(\\rr {2d})\n&=\n\\bigcap _{R>0}\\sets {a\\in (\\Sigma _{s,\\sigma}^{\\sigma ,s})'\n\t(\\rr {2d})}{\\nm {\\omega\n\t\t_R^{-1}V_\\phi a}{L^{\\infty ,q}} <\\infty }.\n\\end{split}\n\\end{align}\n\\end{prop}\n\n\\par\n\nThe following lemma is a consequence of \\cite[Theorem 3.6]{AbCaTo}.\n \n\\par\n\n\\begin{lemma}\\label{Somega}\nLet $s,\\sigma>0$ such that $s+\\sigma\\ge 1$ $\\omega \\in \\mathscr P _{s,\\sigma}(\\rr {2d})$,\n$A_1,A_2\\in \\mathbf{M} (d,\\mathbf R )$, and\nthat $a_1,a_2\\in (\\Sigma _{s,\\sigma}^{\\sigma,s})'(\\rr {2d})$ are such that\n$\\operatorname{Op} _{A_1}(a_1)=\\operatorname{Op} _{A_2}(a_2)$. Then\n\\begin{alignat*}{3}\na_1 &\\in \\Gamma _{(\\omega )}^{\\sigma,s;0}(\\rr {2d})&\\qquad &\\Leftrightarrow &\\qquad a_2\n&\\in \\Gamma_{(\\omega )}^{\\sigma,s;0}(\\rr {2d}) \n\\end{alignat*}\nand similarly for $\\Gamma _{(\\omega )}^{\\sigma,s}(\\rr {2d})$ in\nplace of $\\Gamma _{(\\omega )}^{\\sigma,s;0}(\\rr {2d})$.\n\\end{lemma}\n\n\\par\n\\section{Continuity for pseudo-differential operators\nwith symbols of infinite order}\\label{sec2}\n\n\\par\n\nIn this section we discuss continuity for operators in\n$\\operatorname{Op} (\\Gamma ^{\\sigma,s}_{(\\omega_0)})$ and\n$\\operatorname{Op} (\\Gamma ^{\\sigma,s;0}_{(\\omega _0)})$\nwhen acting on a general class of modulation spaces.\nIn Theorem \\ref{p3.2} continuity is treated where the symbols belong to\n$\\Gamma ^{\\sigma,s}_{(\\omega _0)}$ and in Theorem \\ref{p3.2B}\n continuity is treated where the symbols belong to\n$\\Gamma ^{\\sigma,s;0}_{(\\omega _0)}$.\nThis gives an analogy to \\cite[Theorem 3.2]{To14}\nin the framework of operator theory and Gelfand-Shilov classes.\n\n\\par\n\nOur main result is stated as follows.\n\n\\par\n\n\\begin{thm}\\label{p3.2}\nLet $A\\in \\mathbf{M} (d,\\mathbf R)$, $s,\\sigma\\ge 1$,\n$\\omega ,\\omega _0\\in\\mathscr P _{s,\\sigma}^0(\\rr {2d})$,\n$a\\in \\Gamma _{(\\omega _0)}^{\\sigma,s}(\\rr {2d})$, and that $\\mathscr B$\nis an invariant BF-space on $\\rr {2d}$. Then\n$\\operatorname{Op} _A(a)$ is continuous from $M(\\omega _0\\omega ,\\mathscr B )$\nto $M(\\omega ,\\mathscr B )$.\n\\end{thm}\n\n\\par\n\nWe need some preparations for the proof, and start with the following remark.\n\n\n\\par\n\n\\begin{rem}\\label{rem:adjUniq}\nLet $s,\\sigma>0$ such that $s+\\sigma \\geq 1$.\nIf $a\\in (\\Sigma _{s,\\sigma}^{\\sigma,s})'(\\rr {2d})$,\nthen there is a unique $b\\in (\\Sigma _{s,\\sigma}^{\\sigma,s})'(\\rr {2d})$ such that\n$\\operatorname{Op} (a)^*=\\operatorname{Op} (b)$, where $b(x,\\xi )= e^{i\\scal {D_\\xi}{D_x}}\\overline {a(x,\\xi )}$\nin view of \\cite[Theorem 18.1.7]{Ho1}. Furthermore, by the latter equality and\n\\cite[Theorem 4.1]{CaTo} it follows that\n$$\na\\in \\Gamma _{(\\omega )}^{\\sigma,s}(\\rr {2d})\n\\quad \\Leftrightarrow \\quad\nb\\in \\Gamma _{(\\omega )}^{\\sigma,s}(\\rr {2d}).\n$$\n\\end{rem}\n\n\\par\n\n\\par\n\n\\begin{lemma}\\label{lem:equivfun}\nSuppose $s,\\sigma\\geq 1$,\n$\\omega \\in \\mathscr P _E(\\rr {d_0})$ and that $f\\in C^\\infty\n(\\rr {d+d_0})$ satisfies\n\\begin{equation}\\label{eq:anGeShiEst}\n|\\partial ^\\alpha f(x,y)|\\lesssim h^{|\\alpha |}\\alpha !^\\sigma \ne^{-r|x|^{\\frac 1s}}\\omega (y),\t\\alpha \\in \\nn {d+d_0}\n\\end{equation}\nfor some $h>0$ and $r>0$. Then there are $f_0\\in C^\\infty (\\rr {d+d_0})$\nand $\\psi \\in \\mathcal S _s^\\sigma(\\rr d)$ such that \\eqref{eq:anGeShiEst} holds\nwith $f_0$ in place of $f$ for some for some $h>0$ and $r>0$, and\n$f(x,y)= f_0(x,y)\\psi (x)$.\t\t\n\\end{lemma}\n\n\\par\n\n\\begin{proof}\nBy Proposition \\ref{Prop:EquivWeights}, there is a submultiplicative weight\n$v_0\\in \\mathscr P _{E,s}(\\rr d)\\cap C^\\infty (\\rr d)$ such that\n\\begin{align}\nv_0(x)&\\asymp e^{\\frac r2|x |^{\\frac 1s}}\\label{eq:v0Est}\n\\intertext{and}\n|\\partial ^\\alpha v_0(x)| &\\lesssim h^{|\\alpha |}\\alpha !^\\sigma \nv_0(x),\\qquad \\alpha \\in \\nn d\n\\label{eq:vEst}\n\\intertext{for some $h,r>0$.\t\nSince $s,\\sigma\\ge 1$,\na straight-forward application of Fa{\\`a} di Bruno's formula,\nfor the composed function $\\psi(x)=g(v_0(x))$, where $g(t)=\\frac 1t$,\non \\eqref{eq:vEst} gives}\n\\left | \\partial ^\\alpha \\left (\\frac 1{v_0(x)}\\right ) \\right | &\\lesssim\nh^{|\\alpha |}\\alpha !^\\sigma\\cdot \\frac 1{v_0(x)},\\qquad \\alpha \\in \\nn d\n\\tag*{(\\ref{eq:vEst})$'$}\n\\end{align}\nfor some $h>0$. It follows from \\eqref{eq:v0Est} and \\eqref{eq:vEst}$'$\nthat if $\\psi =1\/{v_0}$, then $\\psi \\in \\mathcal S _s^\\sigma(\\rr d)$.\nFurthermore, if $f_0(x,y)=f(x,y)v_0(x)$,\nthen an application of Leibnitz formula we get\n\\begin{multline*}\n|\\partial ^{\\alpha}_x\\partial ^{\\alpha _0}_yf_0(x,y)|\\lesssim\n\\sum _{\\gamma \\le \\alpha }\\binom{\\alpha}{\\gamma} |\\partial ^\\delta _x\n\\partial ^{\\alpha _0}_yf(x,y)|\n\\, |\\partial ^{\\alpha -\\delta }v_0(x)|\n\\\\[1ex]\n\\lesssim\nh^{|\\alpha |+|\\alpha _0|} \\sum _{\\gamma \\le \\alpha }\n\\binom{\\alpha}{\\gamma}\n(\\gamma !\\alpha _0!)^\\sigma e^{-r|x|^{\\frac 1s}}\n\\omega (y)(\\alpha -\\gamma )!^\\sigma v_0(x)\n\\\\[1ex]\n\\lesssim\n(2h)^{|\\alpha |+|\\alpha _0|}(\\alpha !\\alpha _0!)^\\sigma\ne^{-r|x|^{\\frac 1s}}v_0(x)\\omega (y)\n\\\\[1ex]\n\\lesssim\n(2h)^{|\\alpha |+|\\alpha _0|}(\\alpha !\\alpha _0!)^\\sigma\ne^{-\\frac r2|x|^{\\frac 1s}}\\omega (y)\n\\end{multline*}\nfor some $h>0$, which gives the desired estimate on $f_0$,\nsince it is clear that $f(x,y)= f_0(x,y)\\psi (x)$.\n\\end{proof}\n\n\\par\n\n\\begin{lemma}\\label{Lemma:PrepReThm3.2}\nLet $s,\\sigma \\ge 1$, $\\omega \\in\n\\mathscr P _{s,\\sigma}^0(\\rr {2d})$, $v_1\\in\n\\mathscr P _{s}^0(\\rr {d})$ and $v_2\\in \\mathscr P _{\\sigma}^0(\\rr d)$ be such that\n$v_1$ and $v_2$ are submultiplicative, $\\omega \\in \\Gamma _{(\\omega )}\n^{\\sigma,s}(\\rr {2d})$ is $v_1\\otimes v_2$-moderate.\nAlso let $a\\in \\Gamma _{(\\omega )}^{\\sigma,s}(\\rr {2d})$, \n$f\\in \\mathcal S _s^\\sigma(\\rr d)$, $\\phi \\in \\Sigma _s^\\sigma(\\rr d)$,\n$\\phi_2=\\phi v_1$,\nIf \n\\begin{align}\n\\Phi (x,\\xi ,z,\\zeta ) &= \\frac {a(x+z,\\xi +\\zeta )}\n{\\omega (x,\\xi)v_1(z)v_2(\\zeta )}\\label{eq:Phidef}\n\\intertext{and}\nH(x,\\xi ,y) &= \\iint \\Phi (x,\\xi ,z,\\zeta )\\phi _2(z)v_2(\\zeta )\ne^{i\\scal {y-x-z}{\\zeta}}\\, dzd\\zeta .\\label{eq:Hidentity}\n\\end{align}\nThen\n\\begin{equation}\\label{eq:stftpseudoform}\nV_\\phi (\\operatorname{Op} (a)f)(x,\\xi ) = (2\\pi )^{-d} (f,e^{i\\scal \\, \\cdot \\,  \\xi\n}H(x,\\xi ,\\, \\cdot \\,  ))\\omega (x,\\xi\n).\n\\end{equation}\nFurthermore the following is true:\n\\begin{enumerate}\n\\item $H\\in C^\\infty (\\rr {3d})$ and satisfies\n\\begin{equation}\\label{eq:DerHEst}\n|\\partial _y^\\alpha H(x,\\xi ,y)| \\lesssim h_0^{|\\alpha |}\n\\alpha!^\\sigma e^{-r_0|x-y|^{\\frac 1s}},\n\\end{equation}\nfor every $\\alpha \\in \\rr d$ and some $h_0,r_0>0$;\n\\vspace{0.1cm}\n\n\\item there are functions $H_0\\in C^\\infty (\\rr {3d})$\nand $\\phi _0\\in \\mathcal S _s^\\sigma(\\rr d)$\nsuch that\n\\begin{equation}\\label{eq:HProd}\nH(x,\\xi ,y) = H_0(x,\\xi ,y)\\phi _0(y-x),\n\\end{equation}\nand such that \\eqref{eq:DerHEst} holds for some $h_0,r_0>0$,\nwith $H_0$ in place of $H$.\t\t\n\\end{enumerate}\t\n\\end{lemma}\n\n\\par\n\nLemma \\ref{Lemma:PrepReThm3.2} follows by similar arguments as in\n\\cite{To25}. In order to be self contained we give a different proof.\n\n\\par\n\n\\begin{proof}\nBy straight-forward computations we get\n\\begin{equation}\\label{eq:storformel}\nV_\\phi (\\operatorname{Op} (a)f)(x,\\xi ) \n= (2\\pi )^{-d} (f,e^{i\\scal \\, \\cdot \\,  \\xi }H_1(x,\\xi ,\\, \\cdot \\, \n))\\omega (x,\\xi ),\n\\end{equation}\nwhere\n\\begin{multline*}\nH_1(x,\\xi ,y) = (2\\pi )^{d}e^{-i\\scal y\\xi}(\\operatorname{Op} (a)^*(\\phi (\\, \\cdot \\,  -x)\n\\, e^{i\\scal \\cdot \\xi}))(y)\/\\omega (x,\\xi )\n\\\\[1ex]\n= \\iint \\frac {a(z,\\zeta )}{\\omega (x,\\xi )}\\phi (z-x)\ne^{i\\scal {y-z}{\\zeta -\\xi}}\\, dzd\\zeta \n\\\\[1ex]\n= \\iint \\Phi (x,\\xi ,z-x,\\zeta -\\xi )\\phi _2(z-x)v_2(\\zeta\n-\\xi )e^{i\\scal {y-z}{\\zeta -\\xi}}\\, dzd\\zeta .\n\\end{multline*}\nIf $z-x$ and $\\zeta -\\xi$ are taken as new variables of integrations,\nit follows that the right-hand side is the same as \\eqref{eq:Hidentity}.\nHence \\eqref{eq:stftpseudoform} holds. This\ngives the first part of the lemma.\n\n\\par\n\nThe smoothness of  $H$ is a consequence of the uniqueness of the adjoint\n(cf. Remark \\ref{rem:adjUniq}) and \\cite[Lemma 2.7]{To25}.\n\n\\par\n\nTo show that \\eqref{eq:DerHEst} holds, let\n$$\n\\Phi _0(x,\\xi ,z,\\zeta ) = \\Phi (x,\\xi ,z,\\zeta )\\phi _2(z),\n$$\nwhere $\\Phi$ defined as in \\eqref{eq:Phidef},\nand let $\\Psi =\\mathscr F _3\\Phi_0$, where $\\mathscr F _3\\Phi_0$ is the partial\nFourier transform of $\\Phi _0(x,\\xi ,z,\\zeta )$ with respect to the $z$ variable.\nThen it follows from the assumptions and \\eqref{eq:vEst}$'$ that\n\\begin{multline*}\n|\\partial _z^\\alpha \\Phi _0(x,\\xi ,z,\\zeta )|\n\\lesssim \\sum_{\\gamma\\leq \\alpha}\\binom{\\alpha}{\\gamma}\n\\sum _{\\lambda\\leq \\gamma}\\binom{\\gamma}{\\lambda}\n\\frac{\\left|\\partial_z^{\\gamma-\\lambda}a(x+z,\\xi+\\zeta)\\right|}{\\omega(x,\\xi)v_2(\\zeta)}\n\\\\\n\\times \\partial^{\\lambda}\\prn{\\frac{1}{v_1(z)}}\nh^{|\\alpha-\\gamma|}(\\alpha-\\gamma)!^\\sigma e^{-r|z|^{\\frac 1s}}\n\\\\[1ex]\n\\lesssim \\sum_{\\gamma\\leq \\alpha}\\binom{\\alpha}{\\gamma}\n\\sum _{\\lambda\\leq \\gamma}\\binom{\\gamma}{\\lambda}\nh^{|\\alpha|}(\\alpha-\\gamma)!^\\sigma(\\gamma-\\lambda)!^\\sigma \\lambda!^\\sigma\ne^{-r_0|z|^{\\frac 1s}}\n\\\\[1ex]\n\\lesssim  h^{|\\alpha|} \\alpha!^\\sigma\n\\sum_{\\gamma\\leq \\alpha}\\binom{\\alpha}{\\gamma}\n\\sum _{\\lambda\\leq \\gamma}\\binom{\\gamma}{\\lambda}\n\\prn{\\frac{(\\alpha-\\gamma)!\\gamma!}{\\alpha!}}^\\sigma\n\\prn{\\frac{(\\gamma-\\lambda)!\\lambda!}{\\gamma!}}^\\sigma\ne^{-r_0|z|^{\\frac 1s}}\n\\\\[1ex]\n\\lesssim  (4h)^{|\\alpha|} \\alpha!^\\sigma e^{-r|z|^{\\frac 1s}}\n\\sum_{\\gamma\\leq \\alpha}1\\, \\cdot \\,  \\sum _{\\lambda\\leq \\gamma}1.\n\\end{multline*}\nSince $\\sum _{\\lambda\\leq \\gamma}1 \\lesssim 2^{|\\gamma|}$,\nwe get\n\\begin{equation}\\label{eq:Phi_0Est}\n|\\partial _z^\\alpha \\Phi _0(x,\\xi ,z,\\zeta )|\n\\leq C (16h)^{|\\alpha|}\\alpha!^\\sigma e^{-r_0|z|^{\\frac 1s}}\n\\leq C h_0^{|\\alpha|}\\alpha!^\\sigma e^{-r_0|z|^{\\frac 1s}}\n\\end{equation}\nfor some $C,h_0,r_0>0$. Then $z\\mapsto \\Phi _0(x,\\xi ,z,\\zeta )$\nis an element in $\\mathcal S _s^\\sigma(\\rr d)$.\nMoreover, $\\set{\\Phi _0(x,\\xi ,z,\\zeta )}_{z\\in\\rr d}$ is a bounded set in\n$\\Gamma _{(1)}^{\\sigma,s}(\\rr d\\times\\rr {2d})$.\nIndeed, for a fixed $z_0\\in \\rr{d}$, then an application of Leibnitz formula,\nFa{\\`a} di Bruno's formula, Proposition \\ref{Prop:EquivWeights} and \\eqref{eq:vEst}$'$, give\n\\begin{multline*}\n\\abs{\\partial_x^\\alpha\\partial_\\xi^\\beta\\partial_\\zeta^\\gamma\\Phi _0(x,\\xi ,z_0,\\zeta )}\n\\leq \\sum\n\\binom \\alpha{\\alpha_1}\\binom \\beta{\\beta_1}\\binom \\gamma{\\gamma_1}\n\\partial_x^{\\alpha_1}\\partial_\\xi^{\\beta_1}\\prn{\\frac 1{\\omega(x,\\xi)}}\n\\\\%[1ex]\n\\times\\partial_\\zeta^{\\gamma_1}\\prn{\\frac 1{v_2(\\zeta)}}\\abs{\\partial_x^{\\alpha-\\alpha_1}\n\t\\partial_\\xi^{\\beta-\\beta_1}\\partial_\\zeta^{\\gamma-\\gamma_1}\n\ta(x+z_0,\\xi+\\zeta)}\\, \\cdot \\,  \\frac{|\\phi(z_0)|}{v_1(z_0)}\n\\\\[1ex]\n\\lesssim\n\\sum\n\\binom \\alpha{\\alpha_1}\\binom \\beta{\\beta_1}\\binom \\gamma{\\gamma_1}\nh^{|\\alpha_1+\\beta_1+\\gamma_1|}\\alpha_1!^\\sigma(\\beta_1!\\gamma_1!)^s\n\\\\%[1ex]\n\\times\\prn{\\frac 1{\\omega(x,\\xi)v_1(z_0)v_2(\\zeta)}}\n\\abs{\\partial_x^{\\alpha-\\alpha_1}\n\t\\partial_\\xi^{\\beta-\\beta_1}\\partial_\\zeta^{\\gamma-\\gamma_1}\n\ta(x+z_0,\\xi+\\zeta)}\n\\\\[1ex]\n\\lesssim\nh^{|\\alpha+\\beta+\\gamma|}\\sum\n\\binom \\alpha{\\alpha_1}\\binom \\beta{\\beta_1}\\binom \\gamma{\\gamma_1}\n\\prn{(\\alpha-\\alpha_1)!\\alpha_1!}^\\sigma\n\\prn{(\\beta-\\beta_1)!\\beta_1!}^s\\prn{(\\gamma-\\gamma_1)!\\gamma_1!}^s\n\\\\[1ex]\n\\lesssim\n(4h)^{|\\alpha+\\beta+\\gamma|}\\alpha!^\\sigma(\\beta!\\gamma!)^s,   \t\n\\end{multline*}\nwhere all the summations above are taken over all \n$\\alpha_1\\leq \\alpha, \\beta_1\\leq\\beta$ and $\\gamma_1\\leq\\gamma$.\nIn view of Proposition \\ref{propBroadGSSpaceChar}\nand \\eqref{eq:Phi_0Est} we have \n$$\n|\\partial _\\eta^\\alpha \\Psi (x,\\xi ,\\eta ,\\zeta )|\n\\lesssim h_0^{|\\alpha |}\\alpha !^se^{-r_0|\\eta |^{\\frac 1\\sigma}},\n$$\nfor some $h_0,r_0>0$. Hence\n$$\n|\\partial _\\eta^\\alpha (\\Psi (x,\\xi ,\\zeta ,\\zeta )v_2(\\zeta ))|\n\\lesssim h_0^{|\\alpha |}\\alpha !^se^{-r_0|\\zeta |^{\\frac 1\\sigma}}\n$$\nfor some $h_0,r_0>0$.\n\n\\par\n\nBy letting $H_2(x,\\xi ,\\, \\cdot \\,  )$ be the inverse partial Fourier transform of\n$\\Psi (x,\\xi ,\\zeta ,\\zeta )v_2(\\zeta )$ with respect to the $\\zeta$ variable,\nit follows that\n\\begin{equation}\\label{eq:H2Est}\n|\\partial _y^\\alpha H_2(x,\\xi ,y)|\n\\lesssim h_0^{|\\alpha |}\\alpha !^\\sigma e^{-r_0|y|^{\\frac 1s}}\n\\end{equation}\nfor some $h_0,r_0>0$. The assertion (1) now follows from the latter estimate\nand the fact that $H(x,\\xi ,y)= H_2(x,\\xi ,x-y)$.\n\n\\par\n\nIn order to prove (2) we notice that \\eqref{eq:H2Est} shows that\n$y\\mapsto H_2(x,\\xi ,y)$ is an element in $\\mathcal S _s^\\sigma(\\rr d)$ with values\nin $\\Gamma ^{(1)}_{s,\\sigma}(\\rr {2d})$. It follows by Lemma \\ref{lem:equivfun}\nthat there exist $H_3\\in C^\\infty (\\rr {3d})$ and $\\phi _0\\in \\mathcal S _s^\\sigma(\\rr d)$\nsuch that \\eqref{eq:H2Est} holds for some $h_0,r_0>0$ with $H_3$ in place of $H_2$,\nand\n$$\nH_2(x,\\xi ,y)= H_3(x,\\xi ,y)\\phi _0(-y).\n$$\nThis is the same as (2), and the result follows.\t\n\\end{proof}\n\n\\par\n\n\\begin{proof}[Proof of Theorem \\ref{p3.2}]\nThere is no restriction if we assume that $A=0$.\nLet $G=\\operatorname{Op} (a)f$. In view of Lemma \\ref{Lemma:PrepReThm3.2} we have\n\\begin{multline*}\nV_\\phi G(x,\\xi )\n=\n(2\\pi )^{-\\frac d2}\\mathscr F ((f\\cdot \\overline {\\phi _0(\\, \\cdot \\,  -x)}) \\cdot H_0(x,\\xi ,\\, \\cdot \\,  ))(\\xi )\n\\omega (x,\\xi )\n\\\\[1ex]\n=\n(2\\pi )^{-d} (V_{\\phi _0}f)(x,\\, \\cdot \\,  ) * (\\mathscr F  (H_0(x,\\xi ,\\, \\cdot \\,  )))(\\xi )\n\\omega (x,\\xi ).\n\\end{multline*}\nSince $\\omega$ and $\\omega _0$ belong to $\\mathscr P _{s,\\sigma}^0(\\rr {2d})$,\nthen for every $r_0>0$ and $x,\\xi,\\eta\\in \\rr d$ we have\n$$\n\\omega (x,\\xi )\\omega _0(x,\\xi )\n\\lesssim\n\\omega (x,\\eta )\\omega _0(x,\\eta ) e^{\\frac {r_0}2|\\xi-\\eta|^{\\frac 1\\sigma}},\n$$\n\nthis inequality and (2) in Lemma \\ref{Lemma:PrepReThm3.2} give\n\\begin{equation*}\n|V_\\phi G(x,\\xi )\\omega _0(x,\\xi )|\n\\lesssim\n\\left(\n|(V_{\\phi _0}f)(x,\\, \\cdot \\,  )\\omega (x,\\, \\cdot \\,  )\n\\omega _0(x,\\, \\cdot \\,  )| * e^{-\\frac {r_0}2|\\, \\cdot \\,  |^{\\frac 1\\sigma}}\n\\right)\n(\\xi).\n\\end{equation*}\n\n\\par\n\nIn view of Definition \\ref{bfspaces1},\nwe get for some $v\\in \\mathscr P _{\\sigma}^0(\\rr d)$,\n\\begin{multline*}\n\\nm G{M(\\omega _0,\\mathscr B )}\\lesssim \\nm {|(V_{\\phi _0}f) \n\\cdot \\omega \\cdot \\omega _0|\n* \\delta _0 \\otimes e^{-r_0|\\, \\cdot \\,  |^{\\frac 1\\sigma}}}{\\mathscr B}\n\\\\[1ex]\n\\le\n\\nm {(V_{\\phi _0}f) \\cdot \\omega \\cdot \\omega _0}{\\mathscr B}\n\\nm {e^{-r_0|\\, \\cdot \\,  |^{\\frac 1\\sigma}}v}{L^1}\n\\asymp \\nm f{M(\\omega \\cdot \\omega _0,\\mathscr B )}.\n\\end{multline*}\nThis gives the result.\n\\end{proof}\n\n\\par\n\nBy similar arguments as in the proof of Theorem \\ref{p3.2}\nand Lemma \\ref{Lemma:PrepReThm3.2}  we get the following.\nThe details are left for the reader.\n\n\\par\n\n\n\n\\begin{thm}\\label{p3.2B}\nLet $A\\in \\mathbf{M} (d,\\mathbf R)$, $s,\\sigma\\ge 1$,\n$\\omega ,\\omega _0\\in\\mathscr P _{s,\\sigma}(\\rr {2d})$,\n$a\\in \\Gamma _{(\\omega _0)}^{\\sigma,s;0}(\\rr {2d})$, and that $\\mathscr B$\nis an invariant BF-space on $\\rr {2d}$. Then\n$\\operatorname{Op} _A(a)$ is continuous from $M(\\omega _0\\omega ,\\mathscr B )$\nto $M(\\omega ,\\mathscr B )$.\n\\end{thm}\n\n\\par\n\n\\begin{lemma}\\label{Lemma:PrepReThm3.2B}\nLet $s,\\sigma \\ge 1$, $\\omega \\in\n\\mathscr P _{s,\\sigma}(\\rr {2d})$, $v_1\\in\n\\mathscr P _{s}(\\rr {d})$ and $v_2\\in \\mathscr P _{\\sigma}(\\rr d)$ be such that\n$v_1$ and $v_2$ are submultiplicative, $\\omega \\in \\Gamma _{(\\omega )}\n^{\\sigma,s;0}(\\rr {2d})$ is $v_1\\otimes v_2$-moderate.\nAlso let $a\\in \\Gamma _{(\\omega )}^{\\sigma,s;0}(\\rr {2d})$, \n$f, \\phi \\in \\Sigma _s^\\sigma(\\rr d)$,\n$\\phi_2=\\phi v_1$, and let $\\Phi$ and $H$ be as in Lemma\n\\ref{Lemma:PrepReThm3.2}.\nThen \\eqref{eq:stftpseudoform} and the following hold true:\n\\begin{enumerate}\n\\item $H\\in C^\\infty (\\rr {3d})$ and satisfies \\eqref{eq:DerHEst}\nfor every $h_0,r_0>0$;\n\n\\vspace{0.1cm}\n\n\\item there are functions $H_0\\in C^\\infty (\\rr {3d})$\nand $\\phi _0\\in \\Sigma _s(\\rr d)$\nsuch that \\eqref{eq:HProd} holds,\nand such that \\eqref{eq:DerHEst} holds for every $h_0,r_0>0$,\nwith $H_0$ in place of $H$.\n\\end{enumerate}\n\\end{lemma}\n\n\\par\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec1}\nAccording to Moore's law \\cite{ref31}, traditional computer architectures will reach their physical limits in the near future. Quantum computers \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19, ref20, ref21, ref22,ref23} provide a tool to solve problems more efficiently than ever would be possible with traditional computers \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11}. The power of quantum computing is based on the fundamentals of quantum mechanics. In a quantum computer, information is represented by quantum information, and information processing is achieved by quantum gates that realize quantum operations \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11,p1,p2,p3}. These quantum operations are performed on the quantum states, which are then outputted and measured in a measurement phase. The measurement process is applied to each quantum state where the quantum information conveyed by the quantum states is converted into classical bits. Quantum computers have been demonstrated in practice \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref8, ref9}, and several implementations are currently in progress \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11, ref16, ref17, ref18, ref19}. \n\nIn the physical layer of a gate-model quantum computer, the device contains quantum gates, quantum ports (of quantum gates), and quantum wires for the quantum circuit\\footnote{The term ``quantum circuit'', in general, refers to software, not hardware; it is a description or prescription for what quantum operations should be applied when and does not refer to a physically implemented circuit analogous to a printed electronic circuit. In our setting, it refers to the hardware layer.}. In contrast to traditional automated circuit design \\cite{ref24, ref25, ref26, ref27, ref28, ref29, ref30}, a quantum system cannot participate in more than one quantum gate simultaneously. As a corollary, the quantum gates of a quantum circuit are applied in several rounds in the physical layer of the quantum circuit \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11, ref16, ref17, ref18, ref19}.\n\nThe physical layout design and optimization of quantum circuits have different requirements with several open questions and currently represent an active area of study \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11, ref16, ref17, ref18, ref19}. Assuming that the goal is to construct a reduced quantum circuit that can simulate the original system, the reduction process should be taken on the number of input quantum states, gate operations of the quantum circuit, and the number of output measurements. Another important question is the maximization of objective function associated with an arbitrary computational problem that is fed into the quantum computer. These parallel requirements must be satisfied simultaneously, which makes the optimization procedure difficult and is an emerging issue in present and future quantum computer developments. \n\nIn the proposed QTAM method, the goal is to determine a topology for the quantum circuits of quantum computer architectures that can solve arbitrary computational problems such that the quantum circuit is minimized in the physical layer, and the objective function of an arbitrary selected computational problem is maximized. The physical layer minimization covers the simultaneous minimization of the quantum circuit area (quantum circuit height and depth of the quantum gate structure, where the depth refers to the number of time steps required for the quantum operations making up the circuit to be run on quantum hardware), the total area of the quantum wires of the quantum circuit, the maximization of the objective function, and the minimization of the required number of input quantum systems and output measurements. An important aim of the physical layout minimization is that the resulting quantum circuit should be identical to a high complexity reference quantum circuit (i.e., the reduced quantum circuit should be able to simulate a nonreduced quantum circuit). \n\nThe minimization of the total quantum wire length in the physical layout is also an objective in QTAM. It serves to improve the processing in the topology of the quantum circuit. However, besides the minimization of the physical layout of the quantum circuit, the quantum computer also has to solve difficult computational problems very efficiently (such as the maximization of an arbitrary combinatorial optimization objective function \\cite{ref16, ref17, ref18, ref19}. To achieve this goal in the QTAM method, we also defined an objective function that provides the maximization of objective functions of arbitrary computational problems. The optimization method can be further tuned by specific input constraints on the topology of the quantum circuit (paths in the quantum circuit, organization of quantum gates, required number of rounds of quantum gates, required number of measurement operators, Hamiltonian minimization, entanglement between quantum states, etc.) or other hardware restrictions of quantum computers, such as the well-known \\textit{no-cloning theorem} \\cite{ref22}. The various restrictions on quantum hardware, such as the number of rounds required to be integrated into the quantum gate structure, or entanglement generation between the quantum states are included in the scheme. These constraints and design attributes can be handled in the scheme through the definition of arbitrary constraints on the topology of the quantum circuit, or by constraints on the computational paths.  \n\nThe combinatorial objective function is measured on a computational basis, and an objective function value is determined from the measurement result to quantify the current state of the quantum computer. Quantum computers can be used for combinatorial optimization problems. These procedures aim to use the quantum computer to produce a quantum system that is dominated by computational basis states such that a particular objective function is maximized. \n\nRecent experimental realizations of quantum computers are qubit architectures \\cite{ref1, ref2, ref3, ref4, ref5, ref6, ref7,ref8, ref9, ref10, ref11, ref12, ref13, ref14, ref15, ref16, ref17, ref18, ref19}, and the current quantum hardware approaches focus on qubit systems (i.e., the dimension $d$ of the quantum system is two, $d=2$). However, while the qubit layout is straightforwardly inspirable by ongoing experiments, the method is developed for arbitrary dimensions to make it applicable for future implementations. Motivated by these assumptions, we therefore would avoid the term `qubit' in our scheme to address the quantum states and instead use the generalized term, `quantum states' throughout, which refers to an arbitrary dimensional quantum system. We also illustrate the results through superconducting quantum circuits \\cite{ref1, ref2, ref3, ref4, ref5}; however, the framework is general and flexible, allowing a realization for near term gate-model quantum computer implementations.\n\nThe novel contributions of this paper are as follows:\n\\begin{itemize}\n\\item \\textit{We define a method for designing quantum circuits for gate-model quantum computers.} \n\\item \\textit{We conceive the QTAM algorithm, which provides a quantum circuit minimization on the physical layout (circuit depth and area), quantum wire length minimization, objective function maximization, input size and measurement size minimization for quantum circuits.} \n\\item \\textit{We define a multilayer structure for quantum computations using the hardware restrictions on the topology of gate-model quantum computers.} \n\\end{itemize}\n\nThis paper is organized as follows. In \\sref{relw} the related works are summarized. \\sref{sec2} proposes the system model. In \\sref{sec4} the details of the optimization method are discussed, while \\sref{sec5} studies the performance of the model. Finally, \\sref{sec6} concludes the paper. Supplemental information is inlucded in the Appendix.\n\n\\section{Related Works}\n\\label{relw}\nThe related works are summarized as follows. \n\nA strong theoretical background on the logical model of gate-model quantum computers can be found in \\cite{ref17,ref16,ref18}. In \\cite{ref7}, the model of a gate-model quantum neural network model is defined.\n\nIn \\cite{refa1}, the authors defined a hierarchical approach to computer-aided design of quantum circuits. The proposed model was designed for the synthesis of permutation class of quantum logic circuits. The method integrates evolutionary and genetic approaches to evolve arbitrary quantum circuit specified by a target unitary matrix. Instead of circuit optimization, the work focuses on circuit synthesis.\n\nIn \\cite{refa2}, the authors propose a simulation of quantum circuits by low-rank stabilizer decompositions. The work focuses on the problem of simulation of quantum circuits containing a few non-Clifford gates. The framework focuses on the theoretical description of the stabilizer rank. The authors also derived the simulation cost.\n\nA method for the designing of a T-count optimized quantum circuit for integer multiplication with $4n+1$ qubits was defined in \\cite{int}. The T-count \\cite{tc} measures the number of T-gates, and has a relevance because of the implementation cost of a T gate is high. The aim of the T-count optimization is to reduce the number of T-gates without substantially increasing the number of qubits. The method also applied for quantum circuit designs of integer division \\cite{int2}. In the optimization takes into consideration both the T-count and T-depth, since T-depth is also an important performance measure to reduce the implementation costs. Another method for designing of reversible floating point divider units was proposed in \\cite{div}.\n\nIn \\cite{logic}, a methodology for quantum logic gate construction was defined. The main purpose of the scheme was to construct fault-tolerant quantum logic gates with a simple technique. The method is based on the quantum teleportation method \\cite{tel}.\n\nA method for the synthesis of depth-optimal quantum circuits was defined in \\cite{depth}. The aim of the proposed algorithm is to compute the depth-optimal decompositions of logical operations via an application of the so-called meet-in-the-middle technique. The authors also applied their scheme for the factorizations of some quantum logical operations into elementary gates in the in the Clifford+T set.\n\nA framework to the study the compilation and description of fault-tolerant, high level quantum circuits is proposed in \\cite{ft}. The authors defined a method to convert high level quantum circuits consisting of commonly used gates into a form employing all decompositions and ancillary protocols needed for fault-tolerant error correction. The method also represents a useful tool for quantum hardware architectures with topological quantum codes.\n\nThe Quantum Approximate Optimization Algorithm (QAOA) optimization algorithm is defined in \\cite{ref16}. The QAOA has been defined to evalute approximate solutions for combinatorial optimization problems fed into the quantum computer. \n\nRelevant attributes of the QAOA algorithm are studied in \\cite{refa3}.\n\nIn \\cite{refa4}, the authors analyzed the performance of the QAOA algorithm on near-term gate-model quantum devices. \n\nThe implementation of QAOA with parallelizable gates is studied in \\cite{refa5}.\n\nIn \\cite{refa6} the performance of QAOA is studied on different problems. The analysis covers the MaxCut combinatorial optimization problem, and the problem of quantum circuit optimizations on a classical computer using automatic differentiation and stochastic gradient descent. The work also revealed that QAOA can exceed the performance of a classical polynomial time algorithm (Goemans-Williamson algorithm \\cite{refgw}) with modest circuit depth. The work also concluded that the performance of QAOA with fixed circuit depth is insensitive to problem size.\n\nIn \\cite{refa7}, the authors studied the problem of ultrafast state preparation via the QAOA with long range interactions. The works provides an application for the QAOA in near-term gate-model quantum devices. As the authors concluded, the QAOA-based approach leads to an extremely efficient state preparation, for example the method allows us to prepare Greene-Horne-Zeilinger (GHZ) states with $\\mathcal{O}\\left( 1 \\right)$ circuit depth. The results were also demonstrated by several other examples. \n\nAnother experimental approach for the implementation of qubit entanglement and parallel logic operations with a superconducting circuit was presented in \\cite{song}. In this work, the authors generated entangled GHZ states with up to 10 qubits connecting to a bus resonator in a superconducting circuit. In the proposed implementation, the resonator-mediated qubit-qubit interactions are used to control the entanglement between the qubits and to operate on different pairs in parallel.\n\nA review on the noisy intermediate-scale quantum (NISQ) era can be found in \\cite{refpr}. \n\nThe subject of quantum computational supremacy is discussed in \\cite{refha, aar}.\n\nFor a survey on the attributes of quantum channels, see \\cite{ref11}, a survey on quantum computing technology is included in \\cite{refsur}.\n\n\\section{System Model}\n\\label{sec2}\nThe simultaneous physical-layer minimization and the maximization of the objective function are achieved by the Quantum Triple Annealing Minimization (QTAM) algorithm. The QTAM algorithm utilizes the framework of simulated annealing (SA)  \\cite{ref24, ref25, ref26, ref27, ref28, ref29, ref30}, which is a stochastic point-to-point search method. \n\nThe procedure of the QTAM algorithm with the objective functions are depicted in \\fref{fig1}. The detailed descriptions of the methods and procedures are included in the next sections. \n\n \\begin{center}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle = 0,width=1\\linewidth]{fig1.pdf}\n\\caption{The QTAM method for quantum computers. The quantum gate ($QG$) circuit computation model consists of an input array of $n$ quantum states (depicted by the green box), layers of quantum gates integrated into a quantum circuit (depicted by the purple box), and a measurement phase (depicted by the orange box). The quantum gates that act on the quantum states formulate a quantum circuit with a given circuit height and depth. The area of the quantum circuit is minimized by objective function $F_{{\\rm 1}} $, while the total quantum wire area of the quantum circuit is minimized by $F_{{\\rm 2}} $ ($F_{{\\rm 1}} \\wedge F_{{\\rm 2}} $ is referred via the quantum circuit minimization). The result of the minimization is a quantum circuit of quantum gates with minimized quantum circuit area, minimized total quantum wire length, and a minimized total Hamiltonian operator. The maximization of a corresponding objective function of arbitrary selected computational problems for the quantum computer is achieved by $F_{{\\rm 3}} $ (referred via the objective function maximization). Objective functions $F_{{\\rm 4}} $ and $F_{{\\rm 5}} $ are defined for the minimization of the number of quantum states (minimization of input size), and the total number of measurements (minimization of measurements).} \n \\label{fig1}\n \\end{center}\n\\end{figure}\n\\end{center}\n\n \n\\subsection{Computational Model}\n By theory, in an SA-based procedure a current solution $s_{A} $ is moved to a neighbor $s_{B} $, which yields an acceptance probability \\cite{ref24, ref25, ref26, ref27, ref28, ref29, ref30} \n\\begin{equation} \\label{eq1} \n{\\Pr }\\left(f\\left(s_{A} \\right),f\\left(s_{B} \\right)\\right)=\\frac{{\\rm 1}}{{\\rm 1}+e^{\\left(\\frac{f\\left(s_{A} \\right)-f\\left(s_{B} \\right)}{Tf\\left(s_{A} \\right)} \\right)} } , \n\\end{equation} \n where $f\\left(s_{A} \\right)$ and $f\\left(s_{B} \\right)$ represent the relative performances of the current and neighbor solutions, while $T$ is a control parameter, $T\\left(t\\right)=T_{\\max } {\\rm exp}\\left(-R\\left(t\/k\\right)\\right)$, where $R$ is the temperature decreasing rate, $t$ is the iteration counter, $k$ is a scaling factor, while $T_{\\max } $ is an initial temperature.\n\nSince SA is a probabilistic procedure it is important to minimize the acceptance probability of unfavorable solutions and avoid getting stuck in a local minima.\n\nWithout loss of generality, if $T$ is low, \\eqref{eq1} can be rewritten in function of $f\\left(s_{A} \\right)$ and $f\\left(s_{B} \\right)$ as \n\\begin{equation} \\label{eq2} \n{\\Pr }\\left(f\\left(s_{A} \\right),f\\left(s_{B} \\right)\\right)=\\left\\{\\begin{split} {{\\rm 1,if\\text{ }}f\\left(s_{A} \\right)>f\\left(s_{B} \\right)} \\\\ {{\\rm 0,if\\text{ }}f\\left(s_{A} \\right)\\le f\\left(s_{B} \\right)} \\end{split}\\right. . \n\\end{equation} \n In the QTAM algorithm, we take into consideration that the objectives, constraints, and other functions of the method, by some fundamental theory, are characterized by different magnitude ranges \\cite{ref24, ref25, ref26, ref27, ref28, ref29, ref30}. To avoid issues from these differences in the QTAM algorithm we define three annealing temperatures, $T_{f} \\left(t\\right)$ for objectives, $T_{g} \\left(t\\right)$ for constraints and $T_{c} \\left(t\\right)$ for the probability distribution closeness (distance of the output distributions of the reference quantum circuit and the reduced quantum circuit).\n\nIn the QTAM algorithm, the acceptance probability of a new solution $s_{B} $ at a current solution $s_{A} $ is as \n\\begin{equation} \\label{eq3} \n{\\Pr }\\left(s_{A} ,s_{B} \\right)=\\frac{{\\rm 1}}{{\\rm 1}+e^{\\tilde{d}\\left(f\\right)T_{f} \\left(t\\right)} e^{\\tilde{d}\\left(g\\right)T_{g} \\left(t\\right)} e^{\\tilde{d}\\left(c\\right)T_{c} \\left(t\\right)} } , \n\\end{equation} \n where $\\tilde{d}\\left(f\\right)$, $\\tilde{d}\\left(g\\right)$ and $\\tilde{d}\\left(c\\right)$ are the average values of objective, constraint and distribution closeness domination, see Algorithm 1.\n\nTo aim of the QTAM algorithm is to minimize the cost function \n\\begin{equation} \\label{eq4} \n\\min f\\left({\\rm x}\\right)=\\alpha _{{\\rm 1}} F_{{\\rm 1}} \\left({\\rm x}\\right)+\\ldots +\\alpha _{N_{obj} } F_{N_{obj} } \\left({\\rm x}\\right)+F_{s} , \n\\end{equation} \n where ${\\rm x}$ is the vector of design variables, while $\\alpha $ is the vector of weights, while $N_{obj} $ is the number of primarily objectives. Other $i$ secondary objectives (aspect ratio of the quantum circuit, overlaps, total net length, etc.) are minimized simultaneously via the single-objective function $F_{s} $ in \\eqref{eq4} as \n\\begin{equation} \\label{eq5} \nF_{s} =\\sum _{i} \\alpha _{i} F_{i} \\left(x\\right). \n\\end{equation} \n\n\\subsection{Objective Functions}\nWe defined $N_{obj} =5$ objective functions for the QTAM algorithm. Objective functions $F_{{\\rm 1}} $ and $F_{{\\rm 2}} $ are defined for minimization of $QG$ quantum circuit in the physical layer. The aim of objective function $F_{{\\rm 1}} $ is the minimization of the $A_{QG} $ quantum circuit area of the $QG$ quantum gate structure, \n\\begin{equation} \\label{eq6} \nF_{{\\rm 1}} :\\min \\left(A_{QG} \\right)=\\min \\left(H'_{QG} \\cdot D'_{QG} \\right), \n\\end{equation} \n where $H'_{QG} $ is the optimal circuit height of $QG$, while $D'_{QG} $ is the optimal depth of $QG$.\n\nFocusing on superconducting quantum circuits \\cite{ref1, ref2, ref3, ref4, ref5}, the aim of $F_{{\\rm 2}} $ is the physical layout minimization of the $w_{QG} $ total quantum wire area of $QG$, as \n\\begin{equation} \\label{eq7} \nF_{{\\rm 2}} :w_{QG} =\\min \\sum _{k=1}^{h} \\left(\\sum _{i=1}^{p} \\sum _{j=1}^{q} \\ell _{ij} \\cdot \\delta _{ij} \\left(\\psi _{ij} \\right)\\right), \n\\end{equation} \n where $h$ is the number of nets of the $QG$ circuit, $p$ is the number of quantum ports of the $QG$ quantum circuit considered as sources of a condensate wave function amplitude \\cite{ref1, ref2, ref3, ref4, ref5}, and $q$ the number of quantum ports considered as sinks of a condensate wave function amplitude, $\\ell _{ij} $ is the length of the quantum wire $ij$, $\\delta _{ij} $ is the effective width of the quantum wire $ij$, while $\\psi _{ij} $ is the (root mean square) condensate wave function amplitude \\cite{ref1, ref2, ref3, ref4, ref5} associated to the quantum wire $ij$. \n\nObjective function $F_{{\\rm 3}} $ is defined for the maximization of the expected value of an objective function $C_{L} (\\vec{\\Phi })$ as  \n\\begin{equation} \\label{eq8} \nF_{3} :\\max {C_{L} (\\vec{\\Phi })}=\\max {\\langle \\vec{\\Phi }|C|\\vec{\\Phi }\\rangle }, \n\\end{equation} \n where $C$ is an objective function, $\\vec{\\Phi }$ is a collection of $L$ parameters \n\\begin{equation} \\label{eq9} \n\\vec{\\Phi }=\\Phi _{{\\rm 1}} ,\\ldots ,\\Phi _{L}  \n\\end{equation} \n such that with $L$ unitary operations, state $|\\vec{\\Phi }\\rangle $ is evaluated as \n\\begin{equation} \\label{eq10} \n|\\vec{\\Phi }\\rangle =U_{L} \\left(\\Phi _{L} \\right),\\ldots ,U_{{\\rm 1}} \\left(\\Phi _{{\\rm 1}} \\right)\\left| \\varphi \\right\\rangle  , \n\\end{equation} \n where $U_{i} $ is an $i$-th unitary that depends on a set of parameters $\\Phi _{i} $, while $\\left| \\varphi \\right\\rangle  $ is an initial state. Thus the goal of $F_{{\\rm 3}} $ is to determine the $L$ parameters of $\\vec{\\Phi }$ (see \\eqref{eq9}) such that $\\langle \\vec{\\Phi }|C|\\vec{\\Phi }\\rangle $ is maximized.\n\nObjective functions $F_{{\\rm 4}} $ and $F_{{\\rm 5}} $ are defined for the minimization of the number of input quantum states and the number of required measurements. The aim of objective function $F_{{\\rm 4}} $ is the minimization of the number of quantum systems on the input of the $QG$ circuit, \n\\begin{equation} \\label{eq11} \nF_{{\\rm 4}} :\\min \\left(n\\right). \n\\end{equation} \n The aim of objective function $F_{{\\rm 5}} $ is the minimization of the total number of measurements in the $M$ measurement block, \n\\begin{equation} \\label{eq12} \nF_{{\\rm 5}} :\\min \\left(m\\right)=\\min {\\left(N_{M} \\left|M\\right|\\right)}, \n\\end{equation} \n where $m=N_{M} \\left|M\\right|$, where $N_{M} $ is the number of measurement rounds, $\\left|M\\right|$ is the number of measurement gates in the $M$ measurement block. \n\n\\subsection{Constraint Violations}\n The optimization at several different objective functions results in different Pareto fronts \\cite{ref24, ref25, ref26, ref27} of placements of quantum gates in the physical layout. These Pareto fronts allow us to find feasible tradeoffs between the optimization objectives of the QTAM method. The optimization process includes diverse objective functions, constraints, and optimization criteria to improve the performance of the quantum circuit and to take into consideration the hardware restrictions of quantum computers. In the proposed QTAM algorithm the constraints are endorsed by the modification of the Pareto dominance \\cite{ref24, ref25, ref26, ref27} values by the different sums of constraint violation values. We defined three different constraint violation values.\n \n\\subsubsection{Distribution Closeness Dominance}\n In the QTAM algorithm, the Pareto dominance is first modified with the sum of distribution closeness violation values, denoted by $c_{s} \\left(\\cdot \\right)$. The aim of this iteration is to support the closeness of output distributions of the reduced quantum circuit $QG$ to the output distribution of the reference quantum circuit $QG_{R} $.\n\nLet $P_{QG_{R} } $ the output distribution after the $M$ measurement phase of the reference (original) quantum circuit $QG_{R} $ to be simulated by $QG$, and let $Q_{QG} $ be the output distribution of the actual, reduced quantum circuit $QG$. The distance between the quantum circuit output distributions $P_{QG_{R} } $ and $Q_{QG} $ (distribution closeness) is straightforwardly yielded by the relative entropy function, as \n\\begin{equation} \\label{eq13} \nD\\left(\\left. P_{QG_{R} } \\right\\| Q_{QG} \\right)=\\sum _{i} P_{QG_{R} } \\left(i\\right)\\log _{2} \\frac{P_{QG_{R} } \\left(i\\right)}{Q_{QG} \\left(i\\right)} . \n\\end{equation} \n For two solutions $x$ and $y$, the $d_{x,y} \\left(c\\right)$ distribution closeness dominance function is defined as \n\\begin{equation} \\label{eq14} \nd_{x,y} \\left(c\\right)=c_{s} \\left(x\\right)-c_{s} \\left(y\\right), \n\\end{equation} \n where $c_{s} \\left(\\cdot \\right)$ is evaluated for a given solution $z$ as \n\\begin{equation} \\label{eq15} \nc_{s} \\left(z\\right)=\\sum _{i=1}^{N_{v} } v_{i}^{c} , \n\\end{equation} \n where $v_{i}^{c} $ is an $i$-th distribution closeness violation value, $N_{v} $ is the number of distribution closeness violation values for a solution $z$.\n\nIn terms of distribution closeness dominance, $x$ dominates $y$ if the following relation holds: \n\\begin{equation} \\label{eq16} \n\\begin{split} {\\left(\\left(c_{s} \\left(x\\right)<0\\right)\\wedge \\left(c_{s} \\left(y\\right)<0\\right)\\wedge \\left(c_{s} \\left(x\\right)>c_{s} \\left(y\\right)\\right)\\right)} \\\\ \\vee{\\left(\\left(c_{s} \\left(x\\right)=0\\right)\\wedge \\left(c_{s} \\left(y\\right)<0\\right)\\right),} \\end{split} \n\\end{equation} \n thus \\eqref{eq16} states that $x$ dominates $y$ if both $x$ and $y$ are unfeasible, and $x$ is closer to feasibility than $y$, or $x$ is feasible and $y$ is unfeasible.\n\nBy similar assumptions, $y$ dominates $x$ if \n\\begin{equation} \\label{eq17} \n\\begin{split} {\\left(\\left(c_{s} \\left(x\\right)<0\\right)\\wedge \\left(c_{s} \\left(y\\right)<0\\right)\\wedge \\left(c_{s} \\left(x\\right)<c_{s} \\left(y\\right)\\right)\\right)} \\\\ \\vee{\\left(\\left(c_{s} \\left(x\\right)<0\\right)\\wedge \\left(c_{s} \\left(y\\right)=0\\right)\\right).} \\end{split} \n\\end{equation} \n  \n\\subsubsection{Constraint Dominance}\n The second modification of the Pareto dominance is by the sum of constraint violation values, \n\\begin{equation} \\label{eq18} \nd_{x,y} \\left(g\\right)=g_{s} \\left(x\\right)-g_{s} \\left(y\\right), \n\\end{equation} \n where $g_{s} \\left(\\cdot \\right)$ is the sum of all constraint violation values, evaluated for a given solution $z$ as \n\\begin{equation} \\label{eq19} \ng_{s} \\left(z\\right)=\\sum _{i=1}^{N_{g} } v_{i}^{g} , \n\\end{equation} \n where $v_{i}^{g} $ is an $i$-th constraint violation value, $N_{g} $ is the number of constraint violation values for a solution $z$.\n\nSimilar to \\eqref{eq16} and \\eqref{eq17}, in terms of constraint dominance, $x$ dominates $y$ if the following relation holds: \n\\begin{equation} \\label{eq20} \n\\begin{split} {\\left(\\left(g_{s} \\left(x\\right)<0\\right)\\wedge \\left(g_{s} \\left(y\\right)<0\\right)\\wedge \\left(g_{s} \\left(x\\right)>g_{s} \\left(y\\right)\\right)\\right)} \\\\ \\vee{\\left(\\left(g_{s} \\left(x\\right)=0\\right)\\wedge \\left(g_{s} \\left(y\\right)<0\\right)\\right),} \\end{split} \n\\end{equation} \n thus \\eqref{eq16} states that $x$ dominates $y$ if both $x$ and $y$ are unfeasible, and $x$ is closer to feasibility than $y$, or $x$ is feasible and $y$ is unfeasible.\n\nBy similar assumptions, $y$ dominates $x$ with respect to $g_{s} \\left(\\cdot \\right)$ if \n\\begin{equation} \\label{eq21} \n\\begin{split} {\\left(\\left(g_{s} \\left(x\\right)<0\\right)\\wedge \\left(g_{s} \\left(y\\right)<0\\right)\\wedge \\left(g_{s} \\left(x\\right)<g_{s} \\left(y\\right)\\right)\\right)} \\\\ \\vee{\\left(\\left(g_{s} \\left(x\\right)<0\\right)\\wedge \\left(g_{s} \\left(y\\right)=0\\right)\\right).} \\end{split} \n\\end{equation} \n \n\\subsubsection{Objective Dominance}\nLet $x$ and $y$ refer to two solutions, then, by theory, the $d_{x,y} \\left(f\\right)$ objective dominance function is defined as \n\\begin{equation} \\label{eq22} \nd_{x,y} \\left(f\\right)=\\prod _{i=1,f_{{\\rm 1}} \\left(x\\right)\\ne f_{{\\rm 1}} \\left(y\\right)}^{N_{obj} } \\frac{\\left|f_{i} \\left(x\\right)-f_{i} \\left(y\\right)\\right|}{R_{i} } , \n\\end{equation} \n where $N_{obj} $ is the number of objectives (in our setting $N_{obj} =5$), $R_{i} $ is the range of objective $i$, while $x$ dominates $y$ if $f_{i} \\left(x\\right)\\le f_{i} \\left(y\\right)$ for $\\forall _{i} =1,\\ldots ,N_{obj} $, and for at least one $i$ the relation $f_{i} \\left(x\\right)<f_{i} \\left(y\\right)$ holds. \n \n\\subsection{Objective Function Maximization}\n\\label{A1}\nThe quantum circuit $QG$ executes operations in the ${\\rm {\\mathcal{H}}}$ Hilbert space. The dimension of the ${\\rm {\\mathcal{H}}}$ space is \n\\begin{equation} \\label{eq64} \n{\\rm dim}\\left({\\rm {\\mathcal{H}}}\\right)=d^{n} , \n\\end{equation} \n where $d$ is the dimension of the quantum system ($d=2$ for a qubit system), while $n$ is the number of quantum states.\n\nUsing the formalism of \\cite{ref16, ref17, ref18}, let assume that the computational problem fed into the quantum circuit $QG$ is specified by $n$ bits and $m$ constraints. Then, the objective function is defined as \n\\begin{equation} \\label{eq65} \nC\\left(z\\right)=\\sum _{\\alpha =1}^{m} C_{\\alpha } \\left(z\\right), \n\\end{equation} \n where \n\\begin{equation} \\label{eq66} \nz=z_{{\\rm 1}} \\ldots z_{n}  \n\\end{equation} \n is an $n$-length bitstring, and $C_{\\alpha } \\left(z\\right)=1$ if $z$ satisfies constraint $\\alpha $, and $C_{\\alpha } \\left(z\\right)=0$ otherwise \\cite{ref16, ref17, ref18}.\n\nAssuming a Hilbert space of $n$ qubits, ${\\rm dim}\\left({\\rm {\\mathcal{H}}}\\right)={\\rm 2}^{n} $, using the computational basis vectors $\\left| z\\right\\rangle  $, operator $C\\left(z\\right)$ in \\eqref{eq65} is a diagonal operator in the computational basis \\cite{ref16, ref17, ref18}. Then, at a particular angle $\\gamma $, $\\gamma \\in \\left[0, \\pi \\right]$, unitary $U\\left(C,\\gamma \\right)$ is evaluated as \n\\begin{equation} \\label{eq67} \nU\\left(C,\\gamma \\right)=e^{-i\\gamma C} =\\prod _{\\alpha =1}^{m} e^{-i\\gamma C_{\\alpha } } , \n\\end{equation} \n such that all terms in the product are diagonal in the computational basis.\n\nThen, for the $\\mu $ dependent product of commuting operators, $\\mu \\in \\left[0,\\pi \\right]$ \\cite{ref16, ref17, ref18}, a unitary $U\\left(B,\\mu \\right)$ is defined as \n\\begin{equation} \\label{eq68} \nU\\left(B,\\mu \\right)=e^{-i\\mu B} =\\prod _{j=1}^{n} e^{-i\\mu \\sigma _{x}^{j} } , \n\\end{equation} \n where $B=\\sum _{i} X_{i} $, $X_{i} =\\sigma _{x}^{i} $, $\\sigma _{x} $ is the Pauli $X$-operator, while $\\mu $ is a control parameter \\cite{ref16, ref17, ref18, ref19}, $\\mu \\in \\left[0,\\pi \\right]$ . For a qubit setting, the $\\left| s\\right\\rangle  $ initial state of the quantum computer is the uniform superposition over computational basis states, \n\\begin{equation} \\label{eq69} \n\\left| s\\right\\rangle  =\\frac{{\\rm 1}}{\\sqrt{{\\rm 2}^{n} } } \\sum _{z} \\left| z\\right\\rangle  . \n\\end{equation} \n Let assume that the $G_{QG}^{k,r} $ multilayer structure of the $QG$ quantum circuit contains $n$ quantum ports of several quantum gates, and edge set \n\\begin{equation} \\label{eq70} \n{\\rm {\\mathcal{S}}}_{E} =\\left\\{\\left\\langle jk\\right\\rangle \\right\\} \n\\end{equation} \n of size $m$. Then, the aim of the optimization is to indentify a string $z$ \\eqref{eq66} that the maximizes the objective function \n\\begin{equation} \\label{eq71} \nC=\\sum _{\\left\\langle jk\\right\\rangle } C_{\\left\\langle jk\\right\\rangle } , \n\\end{equation} \n where \n\\begin{equation} \\label{eq72} \nC_{\\left\\langle jk\\right\\rangle } =\\frac{{\\rm 1}}{{\\rm 2}} \\left({\\rm 1}-z_{i} z_{j} \\right), \n\\end{equation} \n where $z_{i} =\\pm 1$.\n\nIn $G_{QG}^{k,r} $ different unitary operations can be defined for the single quantum ports (qubits) and the connected quantum ports, as follows.\n\nLet $U_{q_{s} } \\left(\\mu _{j} \\right)$ be a unitary operator on a $q_{s} $ single port (qubits) in $G_{QG}^{k,r} $, be defined such that for each quantum ports a $\\mu _{j} $ parameter is associated as \n\\begin{equation} \\label{eq73} \nU_{q_{s} } \\left(\\mu _{j} \\right)=e^{-i\\mu _{j} X_{j} } . \n\\end{equation} \n For the collection \n\\begin{equation} \\label{eq74} \n\\vec{\\mu }=\\left(\\mu _{{\\rm 1}} ,\\ldots ,\\mu _{n} \\right), \n\\end{equation} \n the resulting unitary is \n\\begin{equation} \\label{eq75} \nU_{q_{s} } \\left(\\vec{\\mu }\\right)=\\prod _{j} U_{q_{s} } \\left(\\mu _{j} \\right). \n\\end{equation} \n\n\nThe unitary $U_{q_{s} } \\left(\\vec{\\mu }\\right)$ is therefore represents the applications of the unitary operations at once in the quantum ports of the $QG$ quantum circuit.\n\nThen, let unitary $U_{q_{jk} } \\left(\\gamma _{jk} \\right)$ be defined for connected quantum ports $q_{jk} $ in $G_{QG}^{k,r} $, as \n\\begin{equation} \\label{eq76} \nU_{q_{jk} } \\left(\\gamma _{jk} \\right)=e^{i\\gamma _{jk} Z_{j} Z_{k} } , \n\\end{equation} \n where $Z_{i} =\\sigma _{z}^{i} $, where $\\sigma _{z} $ is the Pauli $Z$-operator. Since the eigenvalues of $X_{i} $ and $Z_{j} Z_{k} $ are $\\pm 1$, it allows us to restrict the values \\cite{ref16, ref17, ref18} of parameters $\\gamma $ and $\\mu $ to the range of $\\left[0,\\pi \\right]$.\n\nThen, defining collection \n\\begin{equation} \\label{eq77} \n\\vec{\\gamma }=(\\gamma _{jk}^{{\\rm 1}} ,\\ldots ,\\gamma _{jk}^{h} ), \n\\end{equation} \n where $h$ is the number of individual $\\gamma _{jk} $ parameters, the unitary $U_{q_{jk} } \\left(\\vec{\\gamma }\\right)$ is yielded as \n\\begin{equation} \\label{eq78} \nU_{q_{jk} } \\left(\\vec{\\gamma }\\right)=\\prod _{\\left\\langle jk\\right\\rangle \\in G_{QG}^{k,r} } U_{q_{jk} } \\left(\\gamma _{jk} \\right). \n\\end{equation} \n Assuming that there exists a set ${\\rm {\\mathcal{S}}}_{\\vec{\\mu }}^{u} $ of $u$ collections of $\\vec{\\mu }$'s \n\\begin{equation} \\label{eq79} \n{\\rm {\\mathcal{S}}}_{\\vec{\\mu }}^{u} :\\vec{\\mu }^{\\left({\\rm 1}\\right)} ,\\ldots ,\\vec{\\mu }^{\\left(u\\right)}  \n\\end{equation} \n and a set ${\\rm {\\mathcal{S}}}_{\\vec{\\gamma }}^{u} $ of $u$ collections of $\\vec{\\gamma }$'s, \n\\begin{equation} \\label{eq80} \n{\\rm {\\mathcal{S}}}_{\\vec{\\gamma }}^{u} :\\vec{\\gamma }^{\\left({\\rm 1}\\right)} ,\\ldots ,\\vec{\\gamma }^{\\left(u\\right)} , \n\\end{equation} \n a $\\left| \\phi \\right\\rangle  $ system state of the $QG$ quantum circuit is evaluated as \n\\begin{equation} \\label{eq81} \n\\begin{split}\n \\left| \\phi  \\right\\rangle  &=\\left| \\mathcal{S}_{{\\vec{\\mu }}}^{u},\\mathcal{S}_{{\\vec{\\gamma }}}^{u},C \\right\\rangle  \\\\ \n & ={{U}_{{{q}_{s}}}}\\left( {{{\\vec{\\mu }}}^{\\left( u \\right)}} \\right){{U}_{{{q}_{jk}}}}\\left( {{{\\vec{\\gamma }}}^{\\left( u \\right)}} \\right)\\ldots {{U}_{{{q}_{s}}}}\\left( {{{\\vec{\\mu }}}^{\\left( 1 \\right)}} \\right){{U}_{{{q}_{jk}}}}\\left( {{{\\vec{\\gamma }}}^{\\left( 1 \\right)}} \\right)\\left| s \\right\\rangle ,  \n\\end{split}\n\\end{equation} \n where $\\left| s\\right\\rangle  $ is given in \\eqref{eq69}.\n\nThe maximization of objective function \\eqref{eq65} in the multilayer $G_{QG}^{k,r} $ structure is therefore analogous to the problem of finding the parameters of sets ${\\rm {\\mathcal{S}}}_{\\vec{\\mu }}^{u} $ \\eqref{eq79} and ${\\rm {\\mathcal{S}}}_{\\vec{\\gamma }}^{u} $ \\eqref{eq80} in the system state $\\left| \\phi \\right\\rangle  $ \\eqref{eq81} of the $QG$ quantum circuit. \n\n \n\\subsection{The QTAM Algorithm}\n\\label{sec3}\n\\begin{theorem} The QTAM algorithm utilizes annealing temperatures $T_{f} \\left(t\\right)$, $T_{g} \\left(t\\right)$ and $T_{c} \\left(t\\right)$ to evaluate the acceptance probabilities, where $T_{f} \\left(t\\right)$ is the annealing temperature for the objectives, $T_{g} \\left(t\\right)$ is the annealing temperature for the constraints and $T_{c} \\left(t\\right)$ is the annealing temperature for the distribution closeness.\n\\end{theorem}\n\\begin{proof}\nThe detailed description of the QTAM procedure is given in Algorithm 1. \n\n\\setcounter{algocf}{0}\n\\begin{algo}\n  \\DontPrintSemicolon\n\\caption{\\textit{Quantum Triple Annealing Minimization (QTAM)}}\n\\textbf{Step 1}. Define an archive ${\\rm {\\mathcal{A}}}$ with random solutions, and select a $\\xi $ random solution from ${\\rm {\\mathcal{A}}}$.\n\n\\textbf{Step 2}. Define $\\nu $ as $\\nu =\\Xi \\left(\\xi \\right)$, where $\\Xi \\left(\\cdot \\right)$ is a moving operator. Determine the dominance relation between $\\xi $ and $\\nu $ via ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)$, where function ${\\rm {\\mathcal{D}}}_{P} \\left(\\cdot \\right)$ is the constrained Pareto dominance checking function.\n\n\\textbf{Step 3}. Evaluate acceptance probabilities based on ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)$.\n\n\\textbf{(a)}: If ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\nu \\angle \\xi $ ($\\xi $ dominates $\\nu $, where $\\angle $ is the Pareto dominance operator), then $\\xi =\\nu $, with probability \n\\begin{equation} \\label{eq23} \n{\\Pr }\\left(\\left. \\xi =\\nu \\right|\\nu \\angle \\xi \\right)=\\frac{{\\rm 1}}{{\\rm 1}+e^{\\tilde{d}\\left(f\\right)T_{f} \\left(t\\right)} e^{\\tilde{d}\\left(g\\right)T_{g} \\left(t\\right)} e^{\\tilde{d}\\left(c\\right)T_{c} \\left(t\\right)} } , \n\\end{equation} \n where $\\tilde{d}\\left(f\\right)$, is the average objective dominance, evaluated as \n\\begin{equation} \\label{eq24} \n\\tilde{d}\\left(f\\right)=\\frac{\\left(\\sum _{i=1}^{k} d_{i,\\nu } \\left(f\\right)\\right)+d_{\\xi ,\\nu } \\left(f\\right)}{k+{\\rm 1}} , \n\\end{equation} \n where $d_{x,y} \\left(f\\right)$ is the objective dominance function as given in \\eqref{eq22}, while $\\tilde{d}\\left(g\\right)$ average constraint dominance as \n\\begin{equation} \\label{eq25} \n\\tilde{d}\\left(g\\right)=\\frac{-\\left(\\sum _{i=1}^{k} d_{\\nu ,i} \\left(g\\right)\\right)-d_{\\nu ,\\xi } \\left(g\\right)}{k+{\\rm 1}} , \n\\end{equation} \n where $d_{x,y} \\left(g\\right)$ is the constraint dominance function as given in \\eqref{eq18}, and $\\tilde{d}\\left(c\\right)$ is average distribution closeness dominance as \n\\begin{equation} \\label{eq26} \n\\tilde{d}\\left(c\\right)=\\frac{-\\left(\\sum _{i=1}^{k} d_{\\nu ,i} \\left(c\\right)\\right)-d_{\\nu ,\\xi } \\left(c\\right)}{k+{\\rm 1}} , \n\\end{equation} \n where $d_{x,y} \\left(c\\right)$ is the distribution closeness dominance function as given in \\eqref{eq14}, while $T_{f} \\left(t\\right)$ is the annealing temperature for the objectives \n\\begin{equation} \\label{eq27} \nT_{f} \\left(t\\right)=T_{f_{\\max } } e^{-R\\left(\\frac{t}{k} \\right)} , \n\\end{equation} \n where $R$ is the temperature decreasing rate, $T_{f_{\\max } } $ is a maximum (initial) value for annealing the objectives factor, $T_{g} \\left(t\\right)$ is the annealing temperature for the constraints \n\\begin{equation} \\label{eq28} \nT_{g} \\left(t\\right)=T_{g_{\\max } } e^{-R\\left(\\frac{t}{k} \\right)} , \n\\end{equation} \n where $T_{g_{\\max } } $ is a maximum (initial) value for annealing the constraint factor, and $T_{c} \\left(t\\right)$ is the annealing temperature for the distribution closeness \n\\begin{equation} \\label{eq29} \nT_{c} \\left(t\\right)=T_{c_{\\max } } e^{-R\\left(\\frac{t}{k} \\right)} , \n\\end{equation} \n where $T_{c_{\\max } } $ is a maximum (initial) value for annealing the distribution closeness factor, respectively.  \n\\end{algo}\n\n\\setcounter{algocf}{0}\n\\begin{algo}\n  \\DontPrintSemicolon\n\\caption{\\textit{Quantum Triple Annealing Minimization (QTAM), cont.}}\n\\textbf{(b)}. If ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\left(\\nu \\neg \\angle \\xi \\right)\\wedge \\left(\\xi \\neg \\angle \\nu \\right)$ ($\\nu $ and $\\xi $ are non-dominating to each other) such that $\\nu $ is dominated by $k\\ge {\\rm 1}$ points in ${\\rm {\\mathcal{A}}}$, ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\nu \\angle \\left({\\rm {\\mathcal{A}}}\\right)_{k} $, then $\\xi =\\nu $, with probability \n\\begin{equation} \\label{eq30} \n\\begin{split} {{\\Pr}\\left(\\left. \\xi =\\nu \\right|\\left(\\nu \\neg \\angle \\xi \\right)\\wedge \\left(\\xi \\neg \\angle \\nu \\right),\\nu \\angle \\left({\\rm {\\mathcal{A}}}\\right)_{k} \\right)} \\\\ {=\\frac{{\\rm 1}}{{\\rm 1}+e^{\\tilde{d}\\left(f\\right)_{k} T_{f} \\left(t\\right)} e^{\\tilde{d}\\left(g\\right)_{k} T_{g} \\left(t\\right)} e^{\\tilde{d}\\left(c\\right)_{k} T_{c} \\left(t\\right)} } ,} \\end{split} \n\\end{equation} \n where \n\\begin{equation} \\label{eq31} \n\\tilde{d}\\left(f\\right)_{k} =\\tilde{d}\\left(f\\right)-d_{\\xi ,\\nu } \\left(f\\right), \n\\end{equation} \n where $\\tilde{d}\\left(f\\right)$ is as in \\eqref{eq24}, \n\\begin{equation} \\label{eq32} \n\\tilde{d}\\left(g\\right)_{k} =\\tilde{d}\\left(g\\right)+d_{\\nu ,\\xi } \\left(g\\right), \n\\end{equation} \n where $\\tilde{d}\\left(g\\right)$ is as in \\eqref{eq25}, while \n\\begin{equation} \\label{eq33} \n\\tilde{d}\\left(c\\right)_{k} =\\tilde{d}\\left(c\\right)+d_{\\nu ,\\xi } \\left(c\\right), \n\\end{equation} \n where $\\tilde{d}\\left(c\\right)$ is as in \\eqref{eq26}.\n\n\\textbf{(c)}: If ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\left(\\nu \\neg \\angle \\xi \\right)\\wedge \\left(\\xi \\neg \\angle \\nu \\right)$, and ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,{\\rm {\\mathcal{A}}}\\right)=\\left({\\rm {\\mathcal{A}}}\\neg \\angle \\nu \\right)$, thus $\\nu $ is non-dominating with respect to ${\\rm {\\mathcal{A}}}$, then apply Sub-procedure 1.\n\n\\textbf{(d)}: If ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\left(\\nu \\neg \\angle \\xi \\right)\\wedge \\left(\\xi \\neg \\angle \\nu \\right)$, and ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,{\\rm {\\mathcal{A}}}\\right)=\\left(\\left({\\rm {\\mathcal{A}}}\\right)_{k} \\angle \\nu \\right)$, thus $\\nu $ dominates $k\\ge {\\rm 1}$ points in ${\\rm {\\mathcal{A}}}$, then apply Sub-procedure 2.\n\n\\textbf{(e)}: If ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\xi \\angle \\nu $ such that ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,{\\rm {\\mathcal{A}}}\\right)=\\left(\\nu \\angle \\left({\\rm {\\mathcal{A}}}\\right)_{k} \\right)$, thus $\\nu $ is dominated by $k\\ge {\\rm 1}$ points in ${\\rm {\\mathcal{A}}}$, then set $\\xi =\\nu $, with probability \n\\begin{equation} \\label{eq34} \n{\\Pr }\\left(\\left. \\xi =\\nu \\right|\\xi \\angle \\nu ,\\nu \\angle \\left({\\rm {\\mathcal{A}}}\\right)_{k} \\right)=\\frac{{\\rm 1}}{{\\rm 1}+e^{-\\tilde{d}\\left(\\min \\right)} } , \n\\end{equation} \n where $\\tilde{d}\\left(\\min \\right)$ is evaluated as \n\\begin{equation} \\label{eq35} \n\\tilde{d}\\left(\\min \\right)=\\mathop{\\min }\\limits_{\\forall k} {\\left(d_{\\nu ,k} \\left(f\\right)-\\left(d_{k,\\nu } \\left(g\\right)+d_{k,\\nu } \\left(c\\right)\\right)\\right)}. \n\\end{equation} \n Using \\eqref{eq35}, apply Sub-procedure 3. To evaluate the ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,{\\rm {\\mathcal{A}}}\\right)$ relations between $\\nu $ and the elements of ${\\rm {\\mathcal{A}}}$ at ${\\rm {\\mathcal{D}}}_{P} \\left(\\xi ,\\nu \\right)=\\xi \\angle \\nu $, apply Sub-procedure 4.\n\nStep 4. Apply Steps 2-3, until $i<N_{it} $, where $i$ is the actual iteration, $N_{it} $ is the total number of iterations.  \n\\end{algo}\n\nThe related steps are detailed in Sub-procedures 1-4. In Step 3 of Sub-procedure 1, the best ${{\\mathcal{A}}_{s}}$ solutions refer to those solutions from $\\mathcal{A}$ that have the largest values of the crowding distance \\cite{ref27}. Particularly, in this step, the solutions are also sorted and compared by a crowded comparison operator to find the best solution. \n\n\\setcounter{algocf}{0}\n\\begin{subproc}\n  \\DontPrintSemicolon\n\\caption{\\textit{}} \n\\textbf{Step 1}. Set $\\xi =\\nu $, and add $\\nu $ to ${\\rm {\\mathcal{A}}}$.\n\n\\textbf{Step 2}. If $\\left|{\\rm {\\mathcal{A}}}\\right|>A_{s} $, where $\\left|{\\rm {\\mathcal{A}}}\\right|$ is the number of elements in ${\\rm {\\mathcal{A}}}$, $A_{s} $ is the maximal archive size, then assign $\\Delta _{cr} \\left({\\rm {\\mathcal{A}}}\\right)$ to ${\\rm {\\mathcal{A}}}$, where $\\Delta _{cr} \\left(\\cdot \\right)$ is the crowding distance.\n\n\\textbf{Step 3}. Select the best $A_{s} $ elements.  \n\\end{subproc}\n\n\\begin{subproc}\n  \\DontPrintSemicolon\n\\caption{\\textit{}} \n\\textbf{Step 1}. Set $\\xi =\\nu $, and add $\\nu $ to ${\\rm {\\mathcal{A}}}$.\n\n\\textbf{Step 2}. Remove all the $k$ dominated points from ${\\rm {\\mathcal{A}}}$.  \n\\end{subproc}\n\n\\begin{subproc}\n  \\DontPrintSemicolon\n\\caption{\\textit{}} \n\\textbf{Step 1}. Set $\\xi =k_{\\tilde{d}\\left(\\min \\right)} $, where $k_{\\tilde{d}\\left(\\min \\right)} $ is a point of ${\\rm {\\mathcal{A}}}$ that corresponds to $\\tilde{d}\\left(\\min \\right)$ (see \\eqref{eq35}) with probability ${\\Pr }\\left(\\left. \\xi =\\nu \\right|\\xi \\angle \\nu ,\\nu \\angle \\left({\\rm {\\mathcal{A}}}\\right)_{k} \\right)$ (see \\eqref{eq34}).\n\n\\textbf{Step 2}. Otherwise set $\\xi =\\nu $.  \n\\end{subproc}\n\n\\begin{subproc}\n  \\DontPrintSemicolon\n\\caption{\\textit{}} \n\\textbf{Step 1}. If ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,{\\rm {\\mathcal{A}}}\\right)={\\rm {\\mathcal{A}}}\\neg \\angle \\nu $, i.e., $\\nu $ is non-dominating with respect to ${\\rm {\\mathcal{A}}}$, then set $\\xi =\\nu $, and add $\\nu $ to ${\\rm {\\mathcal{A}}}$. If $\\left|{\\rm {\\mathcal{A}}}\\right|>A_{s} $, then assign $\\Delta _{cr} \\left({\\rm {\\mathcal{A}}}\\right)$ to ${\\rm {\\mathcal{A}}}$, and select the best $A_{s} $ elements.\n\n\\textbf{Step 2}. If ${\\rm {\\mathcal{D}}}_{P} \\left(\\nu ,\\left({\\rm {\\mathcal{A}}}\\right)_{k} \\right)=\\left({\\rm {\\mathcal{A}}}\\right)_{k} \\angle \\nu $, i.e., $\\nu $ dominates $k$ points in ${\\rm {\\mathcal{A}}}$, then set $\\xi =\\nu $, and add $\\nu $ to ${\\rm {\\mathcal{A}}}$. Remove the $k$ points from ${\\rm {\\mathcal{A}}}$.  \n\\end{subproc}\n\n\\end{proof} \n\n\\subsubsection{Computational Complexity of QTAM}\nFollowing the complexity analysis of \\cite{ref24, ref25, ref26, ref27}, the computational complexity of QTAM is evaluated as \n\\begin{equation} \\label{eq36} \n{\\rm \\mathcal{O}}\\left(N_{d} N_{it} \\left|{\\rm {\\mathcal{P}}}\\right|\\left(N_{obj} +\\log _{2} \\left(\\left|{\\rm {\\mathcal{P}}}\\right|\\right)\\right)\\right), \n\\end{equation} \n where $N_{d} $ is the number of dominance measures, $N_{it} $ is the number of total iterations, $\\left|{\\rm {\\mathcal{P}}}\\right|$ is the population size, while $N_{obj} $ is the number of objectives.\n\n\\section{Wiring Optimization and Objective Function Maximization}\n\\label{sec4}\n\\subsection{Multilayer Quantum Circuit Grid}\n An $i$-th quantum gate of $QG$ is denoted by $g_{i} $, a $k$-th port of the quantum gate $g_{i} $ is referred to as $g_{i,k} $. Due to the hardware restrictions of gate-model quantum computer implementations \\cite{ref16, ref17, ref18, ref19}, the quantum gates are applied in several rounds. Thus, a multilayer, $k$-dimensional (for simplicity we assume $k=2$), $n$-sized finite square-lattice grid $G_{QG}^{k,r} $ can be constructed for $QG$, where $r$ is the number of layers, $l_{z} $, $z=1,\\ldots ,r$ . A quantum gate $g_{i} $ in the $z$-th layer $l_{z} $ is referred to as $g_{i}^{l_{z} } $, while a $k$-th port of $g_{i}^{l_{z} } $ is referred to as $g_{i,k}^{l_{z} } $.\n\n\\subsection{Method}\n\n\\begin{theorem}\nThere exists a method for the parallel optimization of quantum wiring in physical-layout of the quantum circuit and for the maximization of an objective function $C_{\\alpha } \\left(z\\right)$.\n\\end{theorem}\n\\begin{proof}\nThe aim of this procedure (Method 1) is to provide a simultaneous physical-layer optimization and Hamiltonian minimization via the minimization of the wiring lengths in the multilayer structure of $QG$ and the maximization of the objective function (see also \\sref{A1}). Formally, the aim of Method 1 is the $F_{{\\rm 2}} \\wedge F_{{\\rm 3}} $ simultaneous realization of the objective functions $F_{{\\rm 2}} $ and $F_{{\\rm 3}} $.\n\nUsing the $G_{QG}^{k,r} $ multilayer grid of the $QG$ quantum circuit determined via $F_{{\\rm 1}} $ and $F_{{\\rm 2}} $, the aim of $F_{{\\rm 3}} $ maximization of the objective function $C\\left(z\\right)$, where $z=z_{{\\rm 1}} \\ldots z_{n} $ in an $n$-length input string, where each $z_{i} $ is associated to an edge of $G_{QG}^{k,r} $ connecting two quantum ports. The objective function $C\\left(z\\right)$ associated to an arbitrary computational problem is defined as \n\\begin{equation} \\label{eq38} \nC\\left(z\\right)=\\sum _{\\left\\langle i,j\\right\\rangle \\in G_{QG}^{k,r} } C_{\\left\\langle i,j\\right\\rangle } \\left(z\\right), \n\\end{equation} \n where $C_{\\left\\langle i,j\\right\\rangle } $ is the objective function for an edge of $G_{QG}^{k,r} $ that connects quantum ports $i$ and $j$.\n\nThe $C^{{\\rm *}} \\left(z\\right)$ maximization of objective function \\eqref{eq38} yields a system state $\\Psi $ for the quantum computer \\cite{ref16, ref17, ref18, ref19} as \n\\begin{equation} \\label{eq39} \n\\Psi =\\left\\langle \\left. \\gamma ,\\mu ,C^{{\\rm *}} \\left(z\\right)\\right|\\right. C^{{\\rm *}} \\left(z\\right)\\left| \\gamma ,\\mu ,C^{{\\rm *}} \\left(z\\right)\\right\\rangle  , \n\\end{equation} \n where \n\\begin{equation} \\label{eq40} \n{\\left| \\gamma ,\\mu ,C^{*} \\left(z\\right) \\right\\rangle} =U\\left(B,\\mu \\right)U\\left(C^{*} \\left(z\\right),\\gamma \\right){\\left| s \\right\\rangle} , \n\\end{equation} \nwhile \n\\begin{equation} \\label{eq43} \nU\\left(C^{{\\rm *}} \\left(z\\right),\\gamma \\right)\\left| z\\right\\rangle  =e^{-i\\gamma C^{{\\rm *}} \\left(z\\right)} \\left| z\\right\\rangle  , \n\\end{equation} \n where $\\gamma $ is a single parameter \\cite{ref16, ref17, ref18, ref19}.\n\nThe objective function \\eqref{eq38} without loss of generality can be rewritten as \n\\begin{equation} \\label{eq44} \nC\\left(z\\right)=\\sum _{\\alpha } C_{\\alpha } \\left(z\\right), \n\\end{equation} \n where $C_{\\alpha } $ each act on a subset of bits, such that $C_{\\alpha } \\in \\left\\{{\\rm 0,1}\\right\\}$. Therefore, there exists a selection of parameters of $\\vec{\\Phi }$ in \\eqref{eq9} such that \\eqref{eq44} picks up a maximized value $C^{{\\rm *}} \\left(z\\right)$, which yields system state $\\Upsilon $ as \n\\begin{equation} \\label{eq45} \n\\Upsilon =\\langle \\vec{\\Phi }|C^{{\\rm *}} (z)|\\vec{\\Phi }\\rangle . \n\\end{equation} \n Therefore, the resulting Hamiltonian $H$ associated to the system state \\eqref{eq45} is minimized via $F_{{\\rm 2}} $ (see \\eqref{eq57}) as \n\\begin{equation} \\label{eq46} \nE_{L} (\\vec{\\Phi })=\\min {\\langle \\vec{\\Phi }|H|\\vec{\\Phi }\\rangle }, \n\\end{equation} \n since the physical-layer optimization minimizes the $\\ell _{ij} $ physical distance between the quantum ports, therefore the energy $E_{L} (\\vec{\\Phi })$ of the Hamiltonian associated to $\\vec{\\Phi }$ is reduced to a minima. \n\nThe steps of the method $F_{{\\rm 2}} \\wedge F_{{\\rm 3}} $ are given in Method 1. The method minimizes the number of quantum wires in the physical-layout of $QG$, and also achieves the desired system state $\\Psi $ of \\eqref{eq39}. \n\n\\setcounter{algocf}{0}\n\\begin{proced}\n  \\DontPrintSemicolon\n\\caption{\\textit{Quantum Wiring Optimization and Objective Function Maximization}}\n\\textbf{Step 1}. Construct the $G_{QG}^{k,r} $ multilayer grid of the $QG$ quantum circuit, with $r$ layers $l_{{\\rm 1}} ,\\ldots ,l_{r} $. Determine the \n\\[C\\left(z\\right)=\\sum _{\\left\\langle i,j\\right\\rangle \\in G_{QG}^{k,r} } C_{\\left\\langle i,j\\right\\rangle } \\left(z\\right)\\] \nobjective function, where each $C_{\\left\\langle i,j\\right\\rangle } $ refers to the objective function for an edge in $G_{QG}^{k,r} $ connecting quantum ports $i$ and $j$, defined as \n\\[C_{\\left\\langle i,j\\right\\rangle } \\left(z\\right)=\\frac{{\\rm 1}}{{\\rm 2}} \\left({\\rm 1}-z_{i} z_{j} \\right),\\] \nwhere $z_{i} =\\pm 1$.\n\n\\textbf{Step 2}. Find the optimal assignment of separation point $\\Delta $ in $G_{QG}^{k,r} =\\left(V,E,f\\right)$ at a physical-layer blockage $\\beta $ via a minimum-cost tree in $G_{QG}^{k,r} $ containing at least one port from each quantum gate $g_{i} $, $i=1,\\ldots ,\\left|V\\right|$. For all pairs of quantum gates $g_{i} $, $g_{j} $, minimize the $f_{p,c} $ path cost (${\\rm L1}$ distance) between a source quantum gate $g_{i} $ and destination quantum gate $g_{j} $ and then maximize the overlapped ${\\rm L1}$ distance between $g_{i} $ and $\\Delta $.\n\n\\textbf{Step 3}. For the $s$ found assignments of $\\Delta $ in Step 2, evaluate the objective functions $C_{\\alpha _{i} } $, $k=1,\\ldots ,s$, where $C_{\\alpha _{0} } $ is the initial value. Let the two paths ${\\rm {\\mathcal{P}}}_{{\\rm 1}} $ and ${\\rm {\\mathcal{P}}}_{{\\rm 2}} $ between quantum ports $g_{i{\\rm ,1}} $, $g_{j{\\rm ,1}} $, $g_{j{\\rm ,2}} $ be given as ${\\rm {\\mathcal{P}}}_{{\\rm 1}} :g_{i{\\rm ,1}} \\to \\Delta \\to g_{j{\\rm ,1}} $, and ${\\rm {\\mathcal{P}}}_{{\\rm 2}} :g_{i{\\rm ,1}} \\to \\Delta \\to g_{j{\\rm ,2}} $. Evaluate objective functions $C_{\\left\\langle g_{i{\\rm ,1}} ,\\Delta \\right\\rangle } \\left(z\\right)$, $C_{\\left\\langle \\Delta ,g_{j{\\rm ,1}} \\right\\rangle } \\left(z\\right)$ and $C_{\\left\\langle \\Delta ,g_{j{\\rm ,2}} \\right\\rangle } \\left(z\\right)$.\n\n\\textbf{Step 4}. Select that $k$-th solution, for which \n\\[{{C}_{{{\\alpha }_{k}}}}\\left( z \\right)=C_{\\left\\langle {{g}_{i,1}},\\Delta  \\right\\rangle }^{\\left( k \\right)}\\left( z \\right)+C_{\\left\\langle \\Delta ,{{g}_{j,1}} \\right\\rangle }^{\\left( k \\right)}\\left( z \\right)+C_{\\left\\langle \\Delta ,{{g}_{j,2}} \\right\\rangle }^{\\left( k \\right)}\\left( z \\right)\\]\nis maximal, where $C_{\\left\\langle i,j\\right\\rangle }^{\\left(k\\right)} $ is the objective function associated to a $k$-th solution between quantum ports $g_{i{\\rm ,1}} $, $g_{j{\\rm ,1}} $, and $g_{i{\\rm ,1}} $, $g_{j{\\rm ,2}} $ in $G_{QG}^{k,r} $. The resulting $C_{\\alpha }^{{\\rm *}} \\left(z\\right)$ for ${\\rm {\\mathcal{P}}}_{{\\rm 1}} $ and ${\\rm {\\mathcal{P}}}_{{\\rm 2}} $ is as \n\\[C_{\\alpha }^{*} \\left(z\\right)=\\mathop{\\mathop{\\max }}\\limits_{k} {\\kern 1pt} \\left(C_{\\alpha _{k} } \\left(z\\right)\\right).\\] \n\n\\textbf{Step 5}. Repeat steps 2-4 for all paths of $G_{QG}^{k,r} $.\n\\end{proced}\n\n\\end{proof}\n\nThe steps of Method 1 are illustrated in \\fref{fig2}, using the $G_{QG}^{k,r} $ multilayer topology of the $QG$ quantum gate structure, $l_{i} $ refers to the $i$-th layer of $G_{QG}^{k,r} $.\n\n \\begin{center}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[angle = 0,width=0.8\\linewidth]{fig2.pdf}\n\\caption{The aim is to find the optimal wiring in $G_{QG}^{k,r} $ for the $QG$ quantum circuit (minimal path length with maximal overlapped path between $g_{i{\\rm ,1}} $ and $g_{j{\\rm ,1}} $,$g_{j{\\rm ,2}} $) such that the $C_{\\alpha } $ objective function associated to the paths ${\\rm {\\mathcal{P}}}_{{\\rm 1}} :g_{i{\\rm ,1}} \\to g_{j{\\rm ,1}} $, and ${\\rm {\\mathcal{P}}}_{{\\rm 2}} :g_{i{\\rm ,1}} \\to g_{j{\\rm ,2}} $ is maximal. (a): The initial objective function value is $C_{\\alpha _{0} } $. A physical-layer blockage $\\beta $ in the quantum circuit allows no to use paths ${\\rm {\\mathcal{P}}}_{{\\rm 1}} $ and ${\\rm {\\mathcal{P}}}_{{\\rm 2}} $. (b): The wire length is optimized via the selection point $\\Delta $. The path cost is $f_{p,c} =11+3f_{l} $, where $f_{l} $ is the cost function of the path between the layers $l_{{\\rm 1}} $ and $l_{{\\rm 2}} $ (depicted by the blue vertical line), the path overlap from $g_{i{\\rm ,1}} $ to $\\Delta $ is $\\tau _{o} =5+f_{l} $. The objective function value is $C_{\\alpha _{{\\rm 1}} } $. (c): The path cost is $f_{p,c} =10$, the path overlap from $g_{i{\\rm ,1}} $ to $\\Delta $ is $\\tau _{o} =4$. The objective function value is $C_{\\alpha _{{\\rm 2}} } $. (d): The path cost is $f_{p,c} =12$, the path overlap from $g_{i{\\rm ,1}} $ to $\\Delta $ is $\\tau _{o} =6$. The objective function value is $C_{\\alpha _{{\\rm 3}} } $. The selected connection topology from (b), (c), and (d) is that which yields the maximized objective function $C_{\\alpha }^{{\\rm *}} $.} \n \\label{fig2}\n \\end{center}\n\\end{figure}\n\\end{center}\n\n\\subsection{Quantum Circuit Minimization}\n For objective function $F_{{\\rm 1}} $, the area minimization of the $QG$ quantum circuit requires the following constraints. Let $S_{v} \\left(P_{i} \\right)$ be the vertical symmetry axis of a proximity group $P_{i} $ \\cite{ref24, ref25, ref26} on $QG$, and let $x_{S_{v} \\left(P_{i} \\right)} $ refer to the $x$-coordinate of $S_{v} \\left(P_{i} \\right)$. Then, by some symmetry considerations for $x_{S_{v} \\left(P_{i} \\right)} $, \n\\begin{equation} \\label{eq47} \nx_{S_{v} \\left(P_{i} \\right)} =\\frac{{\\rm 1}}{{\\rm 2}} \\left(x_{i}^{{\\rm 1}} +x_{i}^{{\\rm 2}} +\\kappa _{i} \\right), \n\\end{equation} \n where $x_{i} $ is the bottom-left $x$ coordinate of a cell $\\sigma _{i} $, $\\kappa _{i} $ is the width of $\\sigma _{i} $, and \n\\begin{equation} \\label{eq48} \ny_{i}^{{\\rm 1}} +\\frac{h_{i} }{{\\rm 2}} =y_{i}^{{\\rm 2}} +\\frac{h_{i} }{{\\rm 2}} , \n\\end{equation} \n where $y_{i} $ is the bottom-left $y$ coordinate of a cell $\\sigma _{i} $, $h_{i} $ is the height of $\\sigma _{i} $.\n\nLet $\\left(\\sigma ^{{\\rm 1}} ,\\sigma ^{{\\rm 2}} \\right)$ be a symmetry pair \\cite{ref24, ref25, ref26} that refers to two matched cells placed symmetrically in relation to $S_{v} \\left(P_{i} \\right)$, with bottom-left coordinates $\\left(\\sigma ^{{\\rm 1}} ,\\sigma ^{{\\rm 2}} \\right)=\\left(\\left(x_{i}^{{\\rm 1}} ,y_{i}^{{\\rm 1}} \\right),\\left(x_{i}^{{\\rm 2}} ,y_{i}^{{\\rm 2}} \\right)\\right)$. Then, $x_{S_{v} \\left(P_{i} \\right)} $ can be rewritten as \n\\begin{equation} \\label{eq49} \nx_{S_{v} \\left(P_{i} \\right)} =x_{i}^{{\\rm 1}} -x_{i} =x_{i}^{{\\rm 2}} +x_{i} +\\kappa _{i} , \n\\end{equation} \n with the relation $y_{i}^{{\\rm 1}} =y_{i}^{{\\rm 2}} =y_{i} $.\n\nLet $\\sigma ^{S} =\\left(x_{i}^{S} ,y_{i}^{S} \\right)$ be a cell which is placed centered \\cite{ref24, ref25, ref26} with respect to $S_{v} \\left(P_{i} \\right)$. Then, $x_{S_{v} \\left(P_{i} \\right)} $ can be evaluated as \n\\begin{equation} \\label{eq50} \nx_{S_{v} \\left(P_{i} \\right)} =x_{i}^{S} +\\frac{\\kappa _{i} }{{\\rm 2}} , \n\\end{equation} \n along with $y_{i}^{S} =y_{i} $. Note that it is also possible that for some cells in $QG$ there is no symmetry requirements, these cells are denoted by $\\sigma ^{0} $.\n\nAs can be concluded, using objective function $F_{{\\rm 1}} $ for the physical-layer minimization of $QG$, a $d$-dimensional constraint vector ${\\rm \\mathbf{x}}_{F_{{\\rm 1}} }^{d} $ can be formulated with the symmetry considerations as follows: \n\\begin{equation} \\label{eq51} \n{\\rm \\mathbf{x}}_{F_{{\\rm 1}} }^{d} =\\sum _{N_{\\left(\\sigma ^{{\\rm 1}} ,\\sigma ^{{\\rm 2}} \\right)} } \\left(x_{i} ,y_{i} ,r_{i} \\right)+\\sum _{N_{\\sigma ^{S} } } \\left(y_{i} ,r_{i} \\right)+\\sum _{N_{\\sigma ^{0} } } \\left(x_{i} ,y_{i} ,r_{i} \\right), \n\\end{equation} \n where $N_{\\left(\\sigma ^{{\\rm 1}} ,\\sigma ^{{\\rm 2}} \\right)} $ is the number of $\\left(\\sigma ^{{\\rm 1}} ,\\sigma ^{{\\rm 2}} \\right)$ symmetry pairs, $N_{\\sigma ^{S} } $ is the number of $\\sigma ^{S} $-type cells, while $N_{\\sigma ^{0} } $ is the number of $\\sigma ^{0} $-type cells, while $r_{i} $ is the rotation angle of an $i$-th cell $\\sigma _{i} $, respectively.\n\n\\subsubsection{Quantum Wire Area Minimization}\nObjective function $F_{{\\rm 2}} $ provides a minimization of the total quantum wire length of the $QG$ circuit. To achieve it we define a procedure that yields the minimized total quantum wire area, $w_{QG} $, of $QG$ as given by \\eqref{eq7}. Let $\\delta _{ij} $ be the effective width of the quantum wire $ij$ in the $QG$ circuit, defined as \n\\begin{equation} \\label{eq52} \n\\delta _{ij} =\\frac{\\psi _{ij} }{J_{\\max } \\left(T_{ref} \\right)h_{nom} } , \n\\end{equation} \n where $\\psi _{ij} $ is the (root mean square) condensate wave function amplitude, $J_{\\max } \\left(T_{ref} \\right)$ is the maximum allowed current density at a given reference temperature $T_{ref} $, while $h_{nom} $ is the nominal layer height. Since drops in the condensate wave function phase $\\varphi _{ij} $ are also could present in the $QG$ circuit environment, the $\\delta '_{ij} $ effective width of the quantum wire $ij$ can be rewritten as \n\\begin{equation} \\label{eq53} \n\\delta '_{ij} =\\frac{\\psi _{ij} \\ell _{eff} r_{0} \\left(T_{ref} \\right)}{\\chi _{\\varphi _{ij} } } , \n\\end{equation} \n where $\\chi _{\\varphi _{ij} } $ is a maximally allowed value for the phase drops, $\\ell _{eff} $ is the effective length of the quantum wire, $\\ell _{eff} \\le \\left(\\chi _{\\varphi _{ij} } \\delta _{ij} \\right)\/\\psi _{ij} r_{0} \\left(T_{ref} \\right),$ while $r_{0} \\left(T_{ref} \\right)$ is a conductor sheet resistance \\cite{ref1, ref2, ref3, ref4, ref5}.\n\nIn a $G_{QG}^{k,r} $ multilayer topological representation of $QG$, the $\\ell _{ij} $ distance between the quantum ports is as \n\\begin{equation} \\label{eq54} \n\\ell _{ij} =\\left|x_{i} -x_{j} \\right|+\\left|y_{i} -y_{j} \\right|+\\left|z_{i} -z_{j} \\right|f_{l} , \n\\end{equation} \n where $f_{l} $ is a cost function between the layers of the multilayer structure of $QG$.\n\nDuring the evaluation, let $w_{QG} \\left(k\\right)$ be the total quantum wire area of a particular net $k$ of the $QG$ circuit, \n\\begin{equation} \\label{eq55} \nw_{QG} \\left(k\\right)=\\sum _{i=1}^{p} \\sum _{j=1}^{q} \\ell _{ij} \\cdot \\delta _{ij} \\left(\\psi _{ij} \\right), \n\\end{equation} \n where $q$ quantum ports are considered as sources of condensate wave function amplitudes, while $p$ of $QG$ are sinks, thus \\eqref{eq7} can be rewritten as \n\\begin{equation} \\label{eq56} \nF_{{\\rm 2}} :w_{QG} =\\min {\\sum _{k=1}^{h} w_{QG} \\left(k\\right)}. \n\\end{equation} \n Since $\\psi _{ij} $ is proportional to $\\delta _{ij} \\left(\\psi _{ij} \\right)$, \\eqref{eq56} can be simplified as \n\\begin{equation} \\label{eq57} \nF_{{\\rm 2}} :w'_{QG} =\\min {\\sum _{k=1}^{h} w'_{QG} \\left(k\\right)}, \n\\end{equation} \n where \n\\begin{equation} \\label{eq58} \nw'_{QG} \\left(k\\right)=\\sum _{i=1}^{p} \\sum _{j=1}^{q} \\ell _{ij} \\cdot \\psi _{ij} , \n\\end{equation} \n where $\\ell _{ij} $ is given in \\eqref{eq54}.\n\nIn all quantum ports of a particular net $k$ of $QG$, the source quantum ports are denoted by positive sign \\cite{ref24, ref25, ref26} in the condensate wave function amplitude, $\\psi _{ij} $ assigned to quantum wire $ij$ between quantum ports $i$ and $j$ , while the sink ports are depicted by negative sign in the condensate wave function amplitude, $-\\psi _{ij} $ with respect to a quantum wire $ij$ between quantum ports $i$ and $j$.\n\nThus the aim of $w_{QG} \\left(k\\right)$ in \\eqref{eq55} is to determine a set of port-to-port connections in the $QG$ quantum circuit, such that the number of long connections is reduced in a particular net $k$ of $QG$ as much as possible. The result in \\eqref{eq56} is therefore extends these requirements for all nets of $QG$.\n\n\\paragraph{Wave Function Amplitudes}\nWith respect to a particular quantum wire $ij$ between quantum ports $i$ and $j$ of $QG$, let $\\psi _{i\\to j} $ refer to the condensate wave function amplitude in direction $i\\to j$, and let $\\psi _{j\\to i} $ refer to the condensate wave function amplitude in direction $j\\to i$ in the quantum circuit. Then, the let be $\\phi _{ij} $ defined for the condensate wave function amplitudes of quantum wire $ij$ as \n\\begin{equation} \\label{eq59} \n\\phi _{ij} =\\min {\\left(\\left|\\psi _{i\\to j} \\right|,\\left|\\psi _{j\\to i} \\right|\\right)}, \n\\end{equation} \n with a residual condensate wave function amplitude \n\\begin{equation} \\label{eq60} \n\\xi _{i\\to j} =\\phi _{ij} -\\psi _{i\\to j} , \n\\end{equation} \n where $\\psi _{i\\to j} $ is an actual amplitude in the forward direction $i\\to j$. Thus, the maximum amount of condensate wave function amplitude injectable to of quantum wire $ij$ in the forward direction $i\\to j$ at the presence of $\\psi _{i\\to j} $ is $\\xi _{i\\to j} $ (see \\eqref{eq60}). The following relations holds for a backward direction, $j\\to i$, for the decrement of a current wave function amplitude $\\psi _{i\\to j} $ as \n\\begin{equation} \\label{eq61} \n\\bar{\\xi }_{j\\to i} =-\\psi _{i\\to j} , \n\\end{equation} \n with residual quantum wire length \n\\begin{equation} \\label{eq62} \n\\Gamma _{j\\to i} =-\\delta _{ij} , \n\\end{equation} \n where $\\delta _{ij} $ is given in \\eqref{eq52}.\n\nBy some fundamental assumptions, the ${\\rm {\\mathcal{N}}}_{R} $ residual network of $QG$ is therefore a network of the quantum circuit with forward edges for the increment of the wave function amplitude $\\psi $, and backward edges for the decrement of $\\psi $. To avoid the problem of negative wire lengths the Bellman-Ford algorithm \\cite{ref24, ref25, ref26} can be utilized in an iterative manner in the residual directed graph of the $QG$ topology.\n\nTo find a path between all pairs of quantum gates in the directed graph of the $QG$ quantum circuit, the directed graph has to be strongly connected. The strong-connectivity of the $h$ nets with the parallel minimization of the connections of the $QG$ topology can be achieved by a minimum spanning tree method such as Kruskal's algorithm \\cite{ref24, ref25, ref26}.\n \n\\begin{lemma}\nThe objective function $F_{{\\rm 2}} $ is feasible in a multilayer $QG$ quantum circuit structure.\n\\end{lemma}\n\\begin{proof}\n The procedure defined for the realization of objective function $F_{{\\rm 2}} $ on a $QG$ quantum circuit is summarized in Method 2. The proof assumes a superconducting architecture.\n \n\\setcounter{algocf}{1}\n\\begin{proced}\n  \\DontPrintSemicolon\n\\caption{\\textit{Implementation of Objective Function $F_2$}}\n\\textbf{Step 1}. Assign the $\\psi _{ij} $ condensate wave function amplitudes for all $ij$ quantum wires of $QG$ via Sub-method 2.1.\n\n\\textbf{Step 2}. Determine the residual network of $QG$ via Sub-method 2.2.\n\n\\textbf{Step 3}. Achieve the strong connectivity of $QG$ via Sub-method 2.3.\n\n\\textbf{Step 4}. Output the $QG$ quantum circuit topology such that $w_{QG} $ \\eqref{eq7} is minimized.  \n\\end{proced}\n\n\nThe sub-procedures of Method 2 are detailed in Sub-methods 2.1, 2.2 and 2.3.  \n\n\\setcounter{algocf}{0}\n\\begin{subproc2}\n  \\DontPrintSemicolon\n\\caption{}\n\\textbf{Step 1}. Create a ${\\rm M}_{QG} $ multilayer topological map of the network ${\\rm {\\mathcal{N}}}$ of $QG$ with the quantum gates and ports.\n\n\\textbf{Step 2}. From ${\\rm M}_{QG} $ determine the $L_{c} $ connection list of ${\\rm {\\mathcal{N}}}$ in $QG$.\n\n\\textbf{Step 3}. Determine the $\\delta _{ij} $ the effective width of the quantum wire $ij$ via \\eqref{eq52}, for $\\forall ij$ wires.\n\n\\textbf{Step 4}. Determine $\\phi _{ij} $ via \\eqref{eq59} for all quantum wires $ij$ of the $QG$ circuit.\n\n\\textbf{Step 5}. For a $k$-th net of $QG$, assign the wave function amplitude values $\\psi _{ij} $ to $\\forall ij$ quantum wires such that $w_{QG} \\left(k\\right)$ in \\eqref{eq55} is minimized, with quantum wire length $\\ell _{ij} $ \\eqref{eq54}.  \n\\end{subproc2}\n\n\\begin{subproc2}\n  \\DontPrintSemicolon\n\\caption{}\n\\textbf{Step 1}. Create a $\\bar{{\\rm M}}_{QG} $ multilayer topological map of the ${\\rm {\\mathcal{N}}}_{R} $ residual network of $QG$.\n\n\\textbf{Step 2}. From $\\bar{{\\rm M}}_{QG} $ determine the $\\bar{L}_{c} $ connection list of the ${\\rm {\\mathcal{N}}}_{R} $ residual network of $QG$.\n\n\\textbf{Step 3}. For $\\forall i\\to j$ forward edges of $\\bar{{\\rm M}}_{QG} $ of ${\\rm {\\mathcal{N}}}_{R} $, compute the $\\xi _{i\\to j} $ residual condensate wave function amplitude \\eqref{eq60}, and for $\\forall j\\to i$ backward edges of $\\bar{{\\rm M}}_{QG} $, compute the quantity $\\bar{\\xi }_{j\\to i} $ via \\eqref{eq61}.\n\n\\textbf{Step 4}. Compute the residual negative quantum wire length $\\Gamma _{j\\to i} $ via \\eqref{eq62}, using $\\delta _{ij} $ from \\eqref{eq52}.\n\n\\textbf{Step 5}. Determine the $\\bar{C}$ negative cycles in the ${\\bar{{\\rm M}}_{QG}} $ of the ${\\rm {\\mathcal{N}}}_{R} $ residual network of $QG$ via the ${\\rm {\\mathcal{A}}}_{BF} $ Bellman-Ford algorithm \\cite{ref24, ref25, ref26}.\n\n\\textbf{Step 6}. If $N_{\\bar{C}} >0$, where $N_{\\bar{C}} $ is the number of $\\bar{C}$ negative cycles in $\\bar{{\\rm M}}_{QG} $, then update the $\\psi _{ij} $ wave function amplitudes of the quantum wires $ij$ in the to cancel out the negative cycles.\n\n\\textbf{Step 7}. Re-calculate the values of \\eqref{eq60}, \\eqref{eq61} and \\eqref{eq62} for the residual edges of ${\\rm {\\mathcal{N}}}_{R} $.\n\n\\textbf{Step 8}. Repeat steps 5-7, until $N_{\\bar{C}} >0$.  \n\\end{subproc2}\n\n\\begin{subproc2}\n  \\DontPrintSemicolon\n\\caption{}\n\\textbf{Step 1}. For an $i$-th $sn_{k,i} $ subnet of a net $k$ of the $QG$ quantum circuit, set the quantum wire length to zero, $\\delta _{ij} =0$ between quantum ports $i$ and $j$, for all $\\forall i$.\n\n\\textbf{Step 2}. Determine the $L{\\rm 2}$ (Euclidean) distance between the quantum ports of the subnets $sn_{k,i} $ (from each quantum port of a subnet to each other quantum port of all remaining subnets \\cite{ref24}).\n\n\\textbf{Step 3}. Weight the $\\delta _{ij} >0$ non-zero quantum wire lengths by the calculated $L{\\rm 2}$ distance between the connections of the subnets of the $QG$ quantum circuit \\cite{ref1, ref2, ref3, ref4, ref5}, \\cite{ref24, ref25, ref26}.\n\n\\textbf{Step 4}. Determine the minimum spanning tree ${\\rm {\\mathcal{T}}}_{QG} $ via the ${\\rm {\\mathcal{A}}}_{K} $ Kruskal algorithm \\cite{ref24}.\n\n\\textbf{Step 5}. Determine the set $S_{{\\rm {\\mathcal{T}}}_{QG} } $ of quantum wires with $\\delta _{ij} >0$ from ${\\rm {\\mathcal{T}}}_{QG} $. Calculate $\\delta _{S_{{\\rm {\\mathcal{T}}}_{QG} } } =\\max \\left(\\delta _{ij} ,\\delta '_{ij} ,\\delta _{0} \\right)$, where $\\delta _{0} $ is the minimum width can be manufactured, while $\\delta _{ij} $ and $\\delta '_{ij} $ are given in \\eqref{eq52} and \\eqref{eq53}.\n\n\\textbf{Step 6}. Add the quantum wires of $S_{{\\rm {\\mathcal{T}}}_{QG} } $ to the ${\\rm M}_{QG} $ multilayer topological map of the network ${\\rm {\\mathcal{N}}}$ of $QG$.\n\n\\textbf{Step 7}. Repeat steps 4-6 for $\\forall k$ nets of the $QG$ quantum circuit, until ${\\rm M}_{QG} $ is not strongly connected.  \n\\end{subproc2}\n\nThese conclude the proof.\n\\end{proof}\n\n\\subsubsection{Processing in the Multilayer Structure}\n\nThe $G_{QG}^{k,z} $ grid consists of all $g_{i} $ quantum gates of $QG$ in a multilayer structure, such that the $g_{i,k}^{l_{z} } $ appropriate ports of the quantum gates are associated via an directed graph ${\\rm {\\rm G}}=\\left(V,E,f_{c} \\right)$, where $V$ is the set of ports, $g_{i,k}^{l_{z} } \\subseteq V$, $E$ is the set of edges, and $f_{c} $ is a cost function, to achieve the gate-to-gate connectivity.\n\nAs a hardware restriction we use a constraint on the quantum gate structure, it is assumed in the model that a given quantum system cannot participate in more than one quantum gate at a particular time.\n\nThe distance in the rectilinear grid $G_{QG}^{k,z} $ of $QG$ is measured by the $d_{{\\rm L1}} \\left(\\cdot \\right)$ ${\\rm L1}$-distance function. Between two network ports $x,y\\in V$, $x=\\left(j,k\\right)$, $y=\\left(m,o\\right)$, $d_{{\\rm L1}} \\left(\\cdot \\right)$ is as \n\\begin{equation} \\label{eq63} \nd_{{\\rm L1}} \\left(x,y\\right)=d_{{\\rm L1}} \\left(\\left(j,k\\right),\\left(m,o\\right)\\right)=\\left|m-j\\right|+\\left|o-k\\right|. \n\\end{equation} \n The quantum port selection in the $G_{QG}^{k,r} $ multilayer structure of $QG$, with $r$ layers $l_{z} $, $z=1,\\ldots ,r$, and $k=2$ dimension in each layers is illustrated in \\fref{figA1}.\n\n\\begin{center}\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[angle = 0,width=1\\linewidth]{figA1.pdf}\n\\caption{The method of port allocation of the quantum gates in the $G_{QG}^{k,r} $ multilayer structure, with $r$ layers $l_{z} $, $z=1,\\ldots ,r$, and $k=2$ dimension in each layers. The aim of the multiport selection is to find the shortest path between ports of quantum gates $g_{i} $ (blue rectangle) and $g_{j} $ (green rectangle) in the $G_{QG}^{{\\rm 2,}r} $ multilayer structure. (a): The quantum ports needed to be connected in $QG$ are port $g_{i{\\rm ,1}} $ in quantum gate $g_{i} $ in layer $l_{{\\rm 1}} $, and ports $g_{j{\\rm ,1}} $ and $g_{j{\\rm ,2}} $ of quantum gate $g_{j} $ in layer $l_{{\\rm 3}} $. (b): Due to a hardware restriction on quantum computers, the quantum gates are applied in several rounds in the different layers of the quantum circuit $QG$. Quantum gate $g_{j} $ is applied in two rounds in two different layers that is depicted $g_{j}^{l_{{\\rm 3}} } $ and $g_{j}^{l_{{\\rm 2}} } $. For the layer-$l_{{\\rm 3}} $ quantum gate $g_{j}^{l_{{\\rm 3}} } $, the active port is $g_{j{\\rm ,1}}^{l_{{\\rm 3}} } $ (red), while the other port is not accessible (gray) in $l_{{\\rm 3}} $. The $g_{j{\\rm ,1}}^{l_{{\\rm 3}} } $ port, due to a physical-layer blockage $\\beta $ in the quantum circuit of the above layer $l_{{\\rm 2}} $ does not allow to minimize the path cost between ports $g_{i{\\rm ,1}} $ and $g_{j{\\rm ,1}}^{l_{{\\rm 3}} } $. The target port $g_{j{\\rm ,1}}^{l_{{\\rm 3}} } $ is therefore referred to as a blocked port (depicted by pink), and a new port of is $g_{j}^{l_{{\\rm 3}} } $ selected for $g_{j{\\rm ,1}}^{l_{{\\rm 3}} } $ (new port depicted by red). (c): For the layer-$l_{{\\rm 2}} $ quantum gate $g_{j}^{l_{{\\rm 2}} } $, the active port is $g_{j{\\rm ,2}}^{l_{{\\rm 2}} } $ (red), while the remaining port is not available (gray) in $l_{{\\rm 2}} $. The white dots (vertices) represent auxiliary ports in the grid structure of the quantum circuit. In $G_{QG}^{{\\rm 2,}r} $, each vertices could have a maximum of 8 neighbors, thus for a given port $g_{j,k} $ of a quantum gate $g_{j} $, ${\\rm deg}\\left(g_{j,k} \\right)\\le {\\rm 8}$.} \n \\label{figA1}\n \\end{center}\n\\end{figure*}\n\\end{center}\n\n\\paragraph{Algorithm}\n\\begin{theorem}\nThe Quantum Shortest Path Algorithm finds shortest paths in a multilayer $QG$ quantum circuit structure.\n\\end{theorem}\n\\begin{proof}\nThe steps of the shortest path determination between the ports of the quantum gates in a multilayer structure are included in Algorithm 2. \n\n\\setcounter{algocf}{1}\n\\begin{algo}\n  \\DontPrintSemicolon\n\\caption{\\textit{Quantum Shortest Path Algorithm (QSPA)}}\n\\textbf{Step 1}. Create the $G_{QG}^{k,r} $ multilayer structure of $QG$, with $r$ layers $l_{z} $, $z=1,\\ldots ,r$, and $k$ dimension in each layers. From $G_{QG}^{k,r} $ generate a list $L_{{\\rm {\\mathcal{P}}}\\in {\\rm {\\rm Q}{\\rm G}}} $ of the paths between each start quantum gate port to each end quantum gate port in the $G_{QG}^{k,r} $ structure of $QG$ quantum circuit.\n\n\\textbf{Step 2}. Due to the hardware restrictions of quantum computers, add the decomposed quantum gate port information and its layer information to $L_{{\\rm {\\mathcal{P}}}\\in {\\rm {\\rm Q}{\\rm G}}} $. Add the $\\beta $ physical-layer blockage information to $L_{{\\rm {\\mathcal{P}}}\\in {\\rm {\\rm Q}{\\rm G}}} $.\n\n\\textbf{Step 3}. For a quantum port pair $\\left(x,y\\right)\\in G_{QG}^{k,r} $ define the $f_{c} \\left(x,y\\right)$ cost function, as \n\\[f_{c} \\left(x,y\\right)=\\gamma \\left(x,y\\right)+d_{{\\rm L1}} \\left(x,y\\right),\\] \nwhere $\\gamma \\left(x,y\\right)$ is the real path size from $x$ to $y$ in the multilayer grid structure $G_{QG}^{k,r} $ of $QG$, while $d_{{\\rm L1}} \\left(x,y\\right)$ is the ${\\rm L1}$ distance in the grid structure as given by \\eqref{eq63}.\n\n\\textbf{Step 4}. Using $L_{{\\rm {\\mathcal{P}}}\\in {\\rm {\\rm Q}{\\rm G}}} $ and cost function $f_{c} \\left(x,y\\right)$, apply the $A^{{\\rm *}} $ parallel search \\cite{ref24, ref25, ref26} to determine the lowest cost path ${\\rm {\\mathcal{P}}}^{{\\rm *}} \\left(x,y\\right)$.  \n\\end{algo}\n\n\\end{proof}\n\n\\paragraph{Complexity Analysis}\nThe complexity analysis of Algorithm 2 is as follows. Since the QSPA algorithm (Algorithm 2) is based on the $A^{{\\rm *}} $ search method \\cite{ref24, ref25, ref26}, the complexity is trivially yielded by the complexity of the $A^{{\\rm *}} $ search algorithm.\n\n\\section{Performance Evaluation}\n\\label{sec5}\nIn this section, we compare the performance of the proposed QTAM method with a multiobjective evolutionary algorithm called NSGA-II \\cite{com1}. We selected this multiobjective evolutionary algorithm for the comparison, since the method can be adjusted for circuit designing. \n\nThe computational complexity of NSGA-II is proven to be ${\\rm {\\mathcal O}}\\left(N_{it} N_{obj} \\left|{\\rm {\\mathcal P}}\\right|^{2} \\right)$ in general, while at an optimized nondominated procedure, the complexity can be reduced to ${\\rm {\\mathcal O}}\\left(N_{it} N_{obj} \\left|{\\rm {\\mathcal P}}\\right|\\log _{2} \\left|{\\rm {\\mathcal P}}\\right|\\right)$. We take into consideration both situations for a comparison. The complexity of QTAM is given in \\eqref{eq36}.\n\nThe complexity of the methods in terms of the number of iterations, $N_{O}$, is compared in \\fref{figA2}. The performance of QTAM is depicted in \\fref{figA2}(a), while \\fref{figA2}(b) and \\fref{figA2}(c) illustrate the performances of the NSGA-II and optimized NSGA-II, respectively.\n\nFor the comparison, the $N_{obj} $ parameter is set to $N_{obj} =5$, while for the QTAM method, $N_{d} $ is set to $N_{d} =3$. \n\n\\begin{center}\n\\begin{figure*}[htbp]\n\\begin{center}\n\\includegraphics[angle = 0,width=1\\linewidth]{figP.pdf}\n\\caption{(a): The computational complexity ($N_{O} $: number of operations) of QTAM in function of $N_{it} $ and $\\left|{\\rm {\\mathcal P}}\\right|$,  $N_{it} \\in \\left[1,100\\right]$, $\\left|{\\rm {\\mathcal P}}\\right|\\in \\left[1,500\\right]$. (b): The computational complexity of the NSGA-II method in function of $N_{it} $ and $\\left|{\\rm {\\mathcal P}}\\right|$,  $N_{it} \\in \\left[1,100\\right]$, $\\left|{\\rm {\\mathcal P}}\\right|\\in \\left[1,500\\right]$. (c): The computational complexity of the optimized NSGA-II in function of $N_{it} $ and $\\left|{\\rm {\\mathcal P}}\\right|$,  $N_{it} \\in \\left[1,100\\right]$, $\\left|{\\rm {\\mathcal P}}\\right|\\in \\left[1,500\\right]$.} \n \\label{figA2}\n \\end{center}\n\\end{figure*}\n\\end{center}\n\nIn the analyzed range, the maximized values of $N_{O} $ are $N_{O} \\left({\\rm QTAM}\\right)\\approx 2\\cdot 10^{6} $, $N_{O} (\\text{NSGA-II})\\approx 1.25\\cdot 10^{8} $, and for the optimized NSGA-II scenario, $N'_{O} \\left({\\text{NSGA-II}}\\right)\\approx 2.25\\cdot 10^{6} $, respectively. In comparison to NSGA-II, the complexity of QTAM is significantly lower. Note, while the performance of QTAM and the optimized NSGA-II is closer, QTAM requires no any optimization of the complexity of the nondominated procedure. \n\n\n\\section{Conclusions}\n\\label{sec6}\nThe algorithms and methods presented here provide a framework for quantum circuit designs for near term gate-model quantum computers. Since our aim was to define a scheme for present and future quantum computers, the developed algorithms and methods were tailored for arbitrary-dimensional quantum systems and arbitrary quantum hardware restrictions. We demonstrated the results through gate-model quantum computer architectures; however, due to the flexibility of the scheme, arbitrary implementations and input constraints can be integrated into the quantum circuit minimization. The objective function that is the subject of the maximization in the method can also be selected arbitrarily. This allows a flexible implementation to solve any computational problem for experimental quantum computers with arbitrary hardware restrictions and development constraints. \n\n\n\\section*{Acknowledgements}\nThis work was partially supported by the European Research Council through the Advanced Fellow Grant, in part by the Royal Society's Wolfson Research Merit Award, in part by the Engineering and Physical Sciences Research Council under Grant EP\/L018659\/1, by the Hungarian Scientific Research Fund - OTKA K-112125 and in part by the Engineering and Physical Sciences Research Council under Grant EP\/L018659\/1.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThis paper combines four lines of research: (a) studying variations of domination problems, here the Roman domination~\\cite{Cocetal2004,Dre2000a,HHS98}; (b) input-sensitive enumeration of minimal solutions, a topic that has drawn attention in particular from people also interested in domination problems \\cite{AbuHeg2016,CouHHK2013,CouLetLie2015,GolHKKV2016,GolHegKra2016}; (c) related to (and motivated by) enumeration, extension problems have been introduced and studied in particular in the context of domination problems\\footnote{Historically, a logical extension problem~\\cite{BorGurHam98} should be mentioned, as it has led to \\cite[Th\\'eor\\`eme 2.16]{Mar2013a}, dealing with an extension variant of 3-\\textsc{Hitting Set}; also see \\cite[Proposition 3.39]{Mar2013a} concerning implications for \\textsc{Extension Dominating Set}.} in \\cite{Bazetal2018,BonDHR2019,CasFKMS2019a,CasFGMS2021,KanLMNU2015,KanLMNU1516,Mar2013a}: is a given set a subset of any minimal dominating set?; \n(d) the \\textsc{Hitting Set Transversal Problem} is the question if all minimal hitting sets of a hypergraph can be enumerated with polynomial delay (or even output-polynomial) only: this question is open for four decades by now and is equivalent to several enumeration problems in logic, database theory and also to enumerating minimal dominating sets in graphs, see \\cite{CreKPSV2019,EitGot95,GaiVer2017,KanLMN2014}. By way of contrast, we show that enumerating all minimal Roman domination functions is possible with polynomial delay, a result which is  quite surprising in view of the general similarities between the complexities of domination and Roman domination problems.\n\n\\begin{figure}[bth]\n\\begin{center}\n\\includegraphics[width=.65\\textwidth]{Roman-mymap.pdf}\n\\end{center}\n\\caption{\\label{fig-Roman-map}The Roman Empire in the times of Constantine}\n\\end{figure}\n\n\\textsc{Roman Domination}\\ comes with a nice (hi)story: \n namely, it should reflect the\nidea of how to secure the Roman Empire by positioning the armies (legions)\non the various parts of the Empire in a way that either\n(1)  a specific region $r$ is also the location of at least one army or \n(2) one region $r'$  neighboring $r$\nhas two armies, so that $r'$ can afford sending off one army to the region $r$ \n (in case of an attack) without diminishing self-defense capabilities.\n More specifically, Emperor \nConstantine had a look at a map of his empire (as discussed in~\\cite{Ste99}, also see Fig.~\\ref{fig-Roman-map}).\\footnote{\nThe historical background is also nicely described\nin the online Johns Hopkins Magazine,  visit \n\\url{http:\/\/www.jhu.edu\/~jhumag\/0497web\/locate3.html} to pre-view~\\cite{ReVRos2000}.} \nRelated is the island hopping strategy pursued by General MacArthur in World War II in the Pacific theater to gradually increase the \nUS-secured areas. \n\n\\textsc{Roman Domination}\\ has received a lot of attention from the algorithmic community in the past 15 years~\\cite{Ben2004,ChaCCKLP2013,Dre2000a,Fer08,Lie2007,Lieetal2008,LiuCha2013,Pagetal2002,PenTsa2007,ShaWanHu2010}.\nRelevant to our paper is the development of exact algorithms for \\textsc{Roman Domination}: combining ideas from \\cite{Lie2007,Roo2011}, an $\\mathcal{O}(1.5014^n)$ exponential-time and -space algorithm (making use of known \\textsc{Set Cover} algorithms via a transformation to \\textsc{Partial Dominating Set})  was presented in~\\cite{ShiKoh2014}. \n In \\cite{Chaetal2009,CheHHHM2016,Favetal2009,HedRSW2013,KraPavTap2012,LiuCha2012,LiuCha2012a,MobShe2008,XinCheChe2006,XueYuaBao2009,YerRod2013a}, more combinatorial studies can be found. This culminated in a chapter on Roman domination, stretching over nearly 50 pages in the monograph~\\cite{HayHedHen2020}. There is also an interesting link to the notion of a \\emph{differential} of a graph, introduced in~\\cite{Masetal2006}, see \\cite{BerFerSig2014}, also adding further algorithmic thoughts, as expressed in \\cite{AbuBCF2016,BerFer2014,BerFer2015}. For instance, in~\\cite{BerFer2014} an  exponential-time algorithm was published, based on a direct Measure-and-Conquer approach.\n \nOne of the ideas leading to the development of the area of \\emph{extension problems} (as described in~\\cite{CasFGMS2021}) was  to cut branches of search trees as early as possible, in the following sense:\nto each node of the search tree, a so-called pre-solution~$U$ can be associated,  and it is asked if it is possible\nto extend $U$ to a meaningful solution~$S$. In the case of \\textsc{Dominating Set}, this means that $U$ is   a set of vertices and a `meaningful solution' is an inclusion-wise minimal dominating set. Notice that such a strategy would work not only for computing smallest dominating sets, but also for computing largest minimal dominating set, or for counting minimal solutions, or for enumerating them. \nAlas, as it has been shown by many examples, extension problems turn out to be quite hard problems.\nEven for combinatorial problems whose standard decision version is solvable in polynomial time (for instance, \\textsc{Edge Cover}), its extension variation is \\textsf{NP}-hard. In such a case, the approach might still be viable, as possibly parameterized algorithms exist with respect to the parameter `pre-solution size'.\nThis would be interesting, as this parameter is small when a big gain can be expected in terms of an early abort of a search tree branch.\nIn particular for \\textsc{Extension Dominating Set}, this hope is not fulfilled. To the contrary, with this parameterization $|U|$, \\textsc{Extension Dominating Set}\\ is one of the few problems known to be complete for the parameterized complexity class \\textsf{W}[3], as shown in~\\cite{BlaFLMS2019}.\n\nWith an appropriate definition of the notion of minimality, \\textsc{Roman Domination}\\ becomes one of the few examples where the hope seeing extension variants being efficiently solvable turns out to be true, as we will show in this paper. This is quite a surprising result, as in nearly any other way, \\textsc{Roman Domination}\\ behaves most similar to \\textsc{Dominating Set}.\nTogether with its combinatorial foundations (a characterization of minimal Roman domination functions), this constitutes the first main result of this paper. \nThe main algorithmic exploit of this result is a non-trivial polynomial-space enumeration algorithm for minimal Roman domination functions that guarantees polynomial delay only, which is the second main result of the paper. As mentioned above, the corresponding question\nfor enumerating minimal dominating sets is open since decades, and we are not aware of any other modification of the concept of domination that seems to preserve any other of the difficulties of \\textsc{Dominating Set}, like classical or parameterized or approximation complexities, apart from the complexity of extension and enumeration.\nOur enumeration algorithm is a branching algorithm that we analyzed with a simple Measure \\& Conquer approach, yielding a running time  of $\\mathcal{O}(1.9332^n)$, which also gives an upper bound on the number of minimal Roman dominating functions of an $n$-vertex graph. This result is complemented by a simple example that proves a lower bound of $\\Omega(1.7441^n)$ for the number of minimal Roman dominating functions on graphs of order~$n$.\n\n\n\n\\section{Definitions}\n\nLet $\\mathbb{N}=\\{1,2,3,\\dots\\}$ be the set of positive integers. For $n\\in\\mathbb{N}$, let $[n]=\\{m\\in\\mathbb{N}\\mid m\\leq n\\}$.\nWe only consider undirected simple graphs. \nLet $G=\\left(V,E\\right)$ be a graph. For $U\\subseteq V$, $G[U]$ denotes the graph induced by~$U$. \nFor $v\\in V$, $N_G(v)\\coloneqq\\{u\\in V\\mid \\lbrace u,v\\rbrace\\in E\\}$ denotes the \\emph{open neighborhood} of~$v$, while $N_G[v]\\coloneqq N_G(v)\\cup\\{v\\}$ is the \\emph{closed neighborhood}  of~$v$. We extend such set-valued functions $X:V\\to 2^V$ to $X:2^V\\to 2^V$ by setting $X(U)=\\bigcup_{u\\in U}X(u)$. Subset $D\\subseteq V$ is a  \\emph{dominating set}, or ds for short, if $N_G[D]=V$. \nFor $D\\subseteq V$ and $v\\in D$, define the \\emph{private neighborhood} of $v\\in V$ with respect to~$D$ as $P_{G,D}\\left( v\\right)\\coloneqq N_G\\left[ v\\right] \\setminus N_G\\left[D\\setminus \\lbrace v\\rbrace\\right]$.\nA function $f\\colon V \\to \\lbrace 0,1,2 \\rbrace$ is called a \\emph{Roman dominating function}, or rdf for short, if for each $v\\in V$ with $f\\left(v\\right) = 0$, there exists a $u\\in N_G\\left( v \\right)$ with $f\\left(u\\right)=2$. \nTo simplify the notation, we define $V_i\\left(f\\right)\\coloneqq \\lbrace v\\in V\\mid f\\left( v\\right)=i\\rbrace$ for $i\\in\\lbrace0,1,2\\rbrace$. The \\emph{weight} $w_f$ of a function $f\\colon V \\to \\lbrace 0,1,2 \\rbrace$ equals $|V_1\\left(f\\right)|+2|V_2\\left(f\\right)|$. The classical \\textsc{Roman Domination}\\ problem asks, given $G$ and an integer $k$, if there exists an rdf for~$G$  of weight at most~$k$. Connecting to the original motivation, $G$ models a map of regions, and if the region vertex~$v$ belongs to~$V_i$, then we place $i$ armies on~$v$.\n\nFor the definition of the problem \\textsc{Extension Roman Domination}, we need to define the order $\\leq$ on $\\lbrace 0,1,2\\rbrace^{V}$ first:\nfor $f,g \\in \\lbrace 0,1,2\\rbrace^{V}$, let $f\\leq g$ if and only if $f\\left(v\\right)\\leq g\\left(v\\right)$ for all $v\\in V$. In other words, we extend the usual linear ordering $\\leq$ on $\\{0,1,2\\}$ to functions mapping to $\\{0,1,2\\}$ in a pointwise manner. \n We call a function $f\\in \\lbrace 0,1,2\\rbrace^{V}$ a \\emph{minimal Roman dominating function} if and only if $f$ is a rdf and there exists no rdf $g$, $g\\neq f$, with $g\\leq f$.\\footnote{According to \\cite{HayHedHen2020}, this notion of minimality for rdf was coined by Cockayne but then dismissed, as it does not give a proper notion of \\emph{upper Roman domination} number. However, in our context, this definition seems to be the most natural one, as it also perfectly fits the extension framework proposed in \\cite{CasFGMS2022}. We will propose in \\autoref{sec:alternative-notion} yet another notion of minimal rdf that also  fits the mentioned extension framework.} The weights of minimal rdf can vary considerably. Consider for example a star $K_{1,n}$ with center~$c$. Then, $f_1(c)=2$, $f_1(v)=0$ otherwise; $f_2(v)=1$ for all vertices~$v$; $f_3(c)=0$, $f_3(u)=2$ for one $u\\neq c$, $f_3(v)=1$ otherwise, define three minimal rdf with weights $w_{f_1}=2$,  and $w_{f_2}=w_{f_3}=n+1$. \n\n\\vspace{5pt}\n\n\\noindent\n\\centerline{\\fbox{\\begin{minipage}{.99\\textwidth}\n\\textbf{Problem name: }\\textsc{Extension Roman Domination}, or \\textsc{ExtRD} for short\\\\\n\\textbf{Given: } A graph $G=\\left( V,E\\right)$ and a function $f\\in \\lbrace 0,1,2 \\rbrace^V.$\\\\\n\\textbf{Question: } Is there a minimal rdf $\\widetilde{f} \\in \\lbrace 0,1,2\\rbrace^V$ with $f\\leq \\widetilde{f}$?\n\\end{minipage}\n}}\n\n\\vspace{5pt}\n\nAs our first main result, we are going to show that \\textsc{ExtRD} can be solved in polynomial time in \\autoref{sec:poly-time-ExtRD}.\nTo this end, we need some understanding of the combinatorial nature of this problem, which we provide in \\autoref{sec:properties-minimal-rdf}.\n\nThe second problem that we consider is that of enumeration, both from an output-sensitive and from an input-sensitive perspective.\n\n\\vspace{5pt}\n\n\\noindent\n\\centerline{\\fbox{\\begin{minipage}{.99\\textwidth}\n\\textbf{Problem name: }\\textsc{Roman Domination Enumeration}, or  \\textsc{RDEnum} for short\\\\\n\\textbf{Given: } A graph $G=\\left( V,E\\right)$.\\\\\n\\textbf{Task: } Enumerate all minimal rdf ${f} \\in \\lbrace 0,1,2\\rbrace^V$ of~$G$!\n\\end{minipage}\n}}\n\nFrom an output-sensitive perspective, it is interesting to perform this enumeration without repetitions and with polynomial delay, which means that there is a polynomial $p$ such that between the consecutive outputs of any two minimal rdf of a graph of order~$n$ that are enumerated, no more than $p(n)$ time elapses, including the corner-cases at the beginning and at the end of the algorithm. From an input-sensitive perspective, we want to upper-bound the running time of the algorithm, measured against the order of the input graph. The obtained run-time bound should not be too different from known lower bounds, given by graph families where one can prove that a certain number of minimal rdf must exist. \nOur algorithm will be analyzed from both perspectives and achieves both goals.\nThis is explained in \\autoref{sec:enum-minimal-rdf-simple} and in \\autoref{sec:enum-minimal-rdf-refined}.\n\n\n\\section{Properties of Minimal Roman Dominating Functions}\n\\label{sec:properties-minimal-rdf}\n\n\\begin{theorem}\\label{t_1_2_neigborhood}\nLet $G=\\left(V,E\\right)$ be a graph and $f: \\: V \\to \\lbrace 0,1,2\\rbrace$ be a minimal rdf. Then $N_G\\left[V_2\\left(f\\right)\\right]\\cap V_1\\left(f\\right)=\\emptyset$ holds.\n\\end{theorem}\n\\begin{pf}\nAssume that there exists a $\\lbrace u,v\\rbrace\\in E$ with $f\\left(v\\right) = 2$ and $f\\left(u\\right)=1$.\nLet\n$$ \\widetilde{f}:V\\to \\lbrace 0,1,2\\rbrace,\\: w\\mapsto\\begin{cases}\nf\\left(w\\right), &w\\neq u\\\\\n0,& w=u\n\\end{cases}$$\nWe show that $\\widetilde{f}$ is a rdf, which contradicts  the minimality of $f$, as $\\widetilde{f}\\leq f$ and $\\widetilde{f}\\left( u \\right) < f\\left( u\\right)$ are given by construction.\nConsider $w \\in V_0\\left(\\widetilde{f}\\right)$. If $w= u$, $w$ is dominated by $v$, as $\\lbrace u,v\\rbrace\\in E$. Consider $w\\neq u$. Since $f$ is a rdf and $V_0\\left(f\\right)\\cup\\lbrace u\\rbrace = V_0\\left(\\widetilde{f}\\right)$, there exists a $t\\in N_G\\left[ w\\right]\\cap V_2\\left(f\\right)$. By construction of $\\widetilde{f}$, $V_2\\left(f\\right)=V_2\\left(\\widetilde{f}\\right)$ holds. This implies $N_G\\left[ w\\right]\\cap V_2\\left(\\widetilde{f}\\right)\\neq \\emptyset$.  Hence, $\\widetilde{f}$ is a rdf. \n\\end{pf}\n\n\n\n\n\\begin{theorem}\\label{t_private_neighborhood}\nLet $G=\\left(V,E\\right)$ be a graph and $f: \\: V \\to \\lbrace 0,1,2\\rbrace$ be a minimal rdf. Then for all $v\\in V_2\\left( f \\right)$,  $P_{G\\left[V_0\\left(f\\right) \\cup  V_2\\left(f\\right)\\right], V_2\\left(f\\right)}\\left(v\\right) \\nsubseteq \\lbrace v \\rbrace$ holds.\n\\end{theorem}\n\\begin{pf}\nDefine $G'\\coloneqq G\\left[V\\setminus V_1\\left( f \\right)\\right] = G\\left[V_0\\left(f\\right) \\cup  V_2\\left(f\\right)\\right]$. In contrast to the claim, assume that there exists a $v\\in V_2\\left( f \\right)$ with $P_{G', V_2\\left(f\\right)}(v) \\subseteq \\lbrace v \\rbrace$. Define \n$$ \\widetilde{f}:V\\to \\lbrace 0,1,2\\rbrace,\\: w\\mapsto\\begin{cases}\nf\\left(w\\right), &w\\neq v\\\\\n1,& w=v\n\\end{cases} $$\nWe show that $\\widetilde{f}$ is a rdf, which contradicts  the minimality of $f$, as $\\widetilde{f}\\leq f$ and $\\widetilde{f}\\left( v \\right) < f\\left( v\\right)$ are given by construction.\nLet $u\\in V_0\\left(\\widetilde{f}\\right)=V_0\\left({f}\\right)$. We must show that some neighbor of $u$ belongs to $V_2\\left(\\widetilde{f}\\right)=V_2\\left(f\\right) \\setminus \\lbrace v\\rbrace$.\nThen, $\\widetilde f$ is a rdf.\n\nFirst, assume that $u$ is a neighbor of~$v$. By the choice of~$v$, $u$ is not a private neighbor of~$v$. Hence, there exists a $w\\in N_{G}\\left[u\\right] \\cap\\left( V_2\\left(f\\right) \\setminus \\lbrace v\\rbrace\\right) = N_{G}\\left[u\\right]\\cap V_2\\left(\\widetilde{f}\\right)$. Secondly, \nif $u\\in V_0\\left(\\widetilde{f}\\right)$ is not a neighbor of $v$, then there exists a $w\\in V_2\\left(f\\right)\\setminus\\{v\\}$ that dominates~$u$, i.e.,  $w\\in N_{G}\\left[u\\right]\\cap \\left(V_2\\left(f\\right)\\setminus \\lbrace v\\rbrace\\right) =  N_{G}\\left[u\\right]\\cap V_2\\left(\\widetilde{f}\\right)$.\n\\end{pf}\n\n\n\\noindent\nAs each $v\\in V_0\\left(f\\right)$ has to be dominated by a $w\\in V_2\\left(f\\right)$, the next claim follows.\n\n\\begin{corollary}\\label{c_min_dom}\nLet $G=\\left(V,E\\right)$ be a graph and $f\\in \\lbrace 0,1,2\\rbrace^V$ be a minimal rdf. Then, $V_2\\coloneqq V_2\\left(f\\right)$ is a minimal ds of $G\\left[ N_G[V_2]\\right]$, with \n$N_G[V_2]=V_0\\left(f\\right)\\cup V_2$.\n\\end{corollary}\n\n\\begin{remark}\nWe can generalize the last statement as follows: Let $G=\\left(V,E\\right)$ be a graph and $f: \\: V \\to \\lbrace 0,1,2\\rbrace$ be a minimal rdf. Let $I\\subseteq V_1(f)$ be an independent set in $G$. Then, $V_2\\left(f\\right)\\cup I$ is a minimal ds of $G\\left[ V_0\\left(f\\right)\\cup V_2\\left(f\\right)\\cup I\\right]$. If $I$ is a maximal independent set in $G[V_1(f)]$, then $V_2\\left(f\\right)\\cup I$ is a minimal ds of $G\\left[ V_0\\left(f\\right)\\cup V_2\\left(f\\right)\\cup V_1(f)\\right]$.\n\\end{remark}\n\n\\noindent\nThis allows us to deduce the following characterization result.\n\n\\begin{theorem}\\label{t_porperty_min_rdf}\nLet $G=\\left(V,E\\right)$ be a graph, $f: \\: V \\to \\lbrace 0,1,2\\rbrace$ and abbreviate\n$G'\\coloneqq G\\left[ V_0\\left(f\\right)\\cup V_2\\left(f\\right)\\right]$. Then, $f$ is a minimal rdf if and only if the following conditions hold:\n\\begin{enumerate}\n\\item$N_G\\left[V_2\\left(f\\right)\\right]\\cap V_1\\left(f\\right)=\\emptyset$,\\label{con_1_2}\n\\item $\\forall v\\in V_2\\left(f\\right) :\\: P_{G',V_2\\left(f\\right)}\\left( v \\right) \\nsubseteq \\lbrace v\\rbrace$, also called \\emph{privacy condition}, and \\label{con_private}\n\\item $V_2\\left(f\\right)$ is a minimal dominating set of $G'$.\\label{con_min_dom}\n\\end{enumerate}\n\\end{theorem}\n\\begin{pf}\nThe ``only if'' follows by  \\autoref{t_1_2_neigborhood}, \\autoref{t_private_neighborhood} and  \\autoref{c_min_dom}. \n\nLet $f$ be a function that fulfills the three conditions. Since $V_2\\left(f\\right)$ is a dominating set on $G'$, for each $u\\in V_0\\left( f\\right)$, there exists a $v\\in V_2\\left(f\\right)\\cap N_{G}\\left[u\\right]$. Therefore, $f$ is a rdf. \nLet $\\widetilde{f}:V \\to \\lbrace 0,1,2 \\rbrace$ be a minimal rdf with $\\widetilde{f}\\leq f$. Therefore, $\\widetilde{f}$ (also) satisfies the three conditions by  \\autoref{t_1_2_neigborhood}, \\autoref{t_private_neighborhood} and \\autoref{c_min_dom}. \nAssume that there exists a $v\\in V$ with $\\widetilde{f}\\left( v\\right) < f\\left( v \\right)$. Hence,  $V_2\\left(\\widetilde{f}\\right)\\subseteq V_2\\left(f\\right)\\setminus \\lbrace v\\rbrace$.  \\\\\n\\textbf{Case 1:} $\\widetilde{f}\\left( v\\right)=0, f\\left( v \\right) =1$. Therefore, there exists a $u\\in N_G\\left(v\\right)$ with $f\\left(u\\right)\\geq \\widetilde{f}\\left(u\\right)=2$. This contradicts Condition~\\ref{con_1_2}.\\\\\n\\textbf{Case 2:} $\\widetilde{f}\\left( v\\right)\\in\\lbrace 0, 1\\rbrace, f\\left( v \\right) =2$.  Let  $u\\in N_G\\left(v\\right)$ with $f(u)=0$. This implies $\\widetilde{f}(u)=0$ and\n$$\\emptyset \\neq  N_G\\left[u\\right] \\cap V_2\\left(\\widetilde{f}\\right)\\subseteq N_G\\left[u\\right] \\cap V_2\\left(f\\right)\\setminus \\lbrace v \\rbrace$$ holds. Therefore, $N_G\\left( v \\right) \\subseteq N_G\\left[ V_2\\left(f\\right) \\setminus \\lbrace v\\rbrace\\right]$. This contradicts Condition~\\ref{con_private}.\n\nThus, $\\widetilde{f}=f$ holds and $f$ is minimal. \n\\end{pf}\n\n\\noindent\nWe conclude this section with an upper bound on the size of $V_2(f)$.\n\n\\begin{lemma}\\label{lem:V2-bound}\nLet $G=\\left(V,E\\right)$ be a graph and $f: V \\to \\lbrace0,1,2\\rbrace$ be a minimal rdf. Then $2 \\: \\vert V_2\\left(f\\right)\\vert \\leq \\vert V \\vert  $ holds.\n\\end{lemma}\n\n\\begin{pf}\nConsider a graph $G=\\left(V,E\\right)$ and a minimal rdf $f:V\\to\\lbrace 0,1,2\\rbrace$. For each $v\\in V_2\\left(f\\right)$, let $P_f\\left(v\\right)= P_{G\\left[V_0\\left(f\\right) \\cup  V_2\\left(f\\right)\\right], V_2\\left(f\\right)}\\left(v\\right) \\setminus \\lbrace v \\rbrace\\subseteq V\\setminus V_2\\left(f\\right) $. By \\autoref{t_private_neighborhood}, these sets are not empty and, by definition, they do not intersect.  Hence, we get:\n$$ \\vert V \\vert =  \\vert V_2\\left( f\\right) \\vert +\\vert V\\setminus V_2\\left( f\\right) \\vert \\geq \\vert V_2\\left( f\\right) \\vert +\\left\\vert\\bigcup_{v\\in V_2\\left(f\\right)}P_f\\left(v\\right) \\right\\vert\\geq 2\\: \\vert V_2\\left( f\\right) \\vert\\,. $$ \nTherefore, the claim is true. \n\\end{pf}\n\n\\section{A Polynomial-time Algorithm for \\textsc{ExtRD}}\n\\label{sec:poly-time-ExtRD}\n\nWith \\autoref{t_porperty_min_rdf}, we can construct an algorithm that solves the problem \\textsc{Extension Roman domination} in polynomial time. \n\\begin{algorithm}\n\\caption{Solving instances of \\textsc{ExtRD}}\\label{alg}\n\\begin{algorithmic}[1]\n\\Procedure{ExtRD Solver}{$G,f$}\\newline\n \\textbf{Input:} A graph $G=\\left(V,E\\right)$ and a function $f\\colon V\\to \\lbrace0,1,2\\rbrace$.\\newline\n \\textbf{Output:} Is there a minimal Roman dominating function $\\widetilde{f}$ with $f\\leq \\widetilde{f}$?\n\\State $\\widetilde{f}\\coloneqq f$. \\label{alg_init}\n\\State $M_2\\coloneqq V_2\\left(f\\right)$. \\{ Invariant: $M_2=V_2(\\widetilde{f})$ \\} \\label{alg_invariant}\n\\State $ M \\coloneqq M_2$. \\{ All $v\\in V_2(\\widetilde{f})$ are considered below; invariant: $M\\subseteq M_2$. \\}\n\\label{alg_before_while}\n\\While{$M\\neq \\emptyset$}\n\\State Choose $v\\in M$. \\{ Hence, $\\widetilde{f}(v)=2$. \\}\n\\For {$u\\in N\\left(v\\right)$}\n\\If {$\\widetilde{f}\\left(u\\right)=1$}\\label{alg_if_no}\n\\State $\\widetilde{f}\\left(u\\right)\\coloneqq 2$.\n\\State Add $u$ to $M$ and to $M_2$.\n\\EndIf\n\\EndFor\n\\State Delete $v$ from $M$. \n\\EndWhile\n\\For{$v\\in M_2$}\\label{alg_for_no}\n\\If{$N_G\\left(v\\right)\\subseteq N_G\\left[M_2\\setminus \\lbrace v\\rbrace\\right]$}\\label{alg_private_test}\n\\State \\textbf{Return No}. \\label{alg_no}\n\\EndIf\n\\EndFor\n\\For{$v\\in V\\setminus N_G\\left[ M_2\\right]$}\\label{alg_fil_for}\n\\State $\\widetilde{f}\\left(v\\right)\\coloneqq 1$.\n\\EndFor \n\\State\\textbf{Return Yes}.\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{theorem}\\label{theorem:correctness_alg}\nLet $G=\\left(V,E\\right)$ be a graph and $f\\colon V\\to\\lbrace0,1,2\\rbrace$. \nFor the inputs $G,f$, Algorithm~\\ref{alg} returns yes if and only if $\\left( G,f\\right)$ is a yes-instance of \\textsc{ExtRD}. In this case, the function~$\\widetilde f$ computed by Algorithm~\\ref{alg} is a minimal rdf.\n\\end{theorem}\n\\begin{pf}First observe that the invariants stated in Lines~\\ref{alg_invariant} and~\\ref{alg_before_while} of Algorithm~\\ref{alg} are true whenever entering or leaving the while-loop.\n\nLet the answer of the algorithm be \\emph{yes} and $\\widetilde{f}$ be the function computed by the algorithm. We will show that~$\\widetilde{f}$\nsatisfies the \nconditions formulated in \\autoref{t_porperty_min_rdf}.\n\nObserving the if-condition in Line~\\ref{alg_if_no}, clearly after the while-loop, no neighbor~$u$ of $v\\in V_2\\left(\\widetilde{f}\\right)$ fulfills $\\widetilde{f}\\left( u\\right)=1$. Hence, $\\widetilde{f}$ satisfies Condition~\\ref{con_1_2}.\nIf the function $\\widetilde{f}$ would contradict Condition~\\ref{con_private} of \\autoref{t_porperty_min_rdf}, then we would get to Line~\\ref{alg_no} and the algorithm would answer \\emph{no}. As we are considering a \\emph{yes}-answer of our algorithm, we can assume that this privacy condition holds after the for-loop of Line~\\ref{alg_for_no}. \nWe also can assume that $M_2=V_2\\left(\\widetilde{f}\\right)$ is a minimal ds of the graph $G\\left[N_G\\left[M_2\\right]\\right]$. Otherwise, for such a $v\\in M_2$ and each $u\\in N_G\\left(v\\right)$, there would exist a $w\\in N_G\\left[u\\right]\\cap\\left( M_2\\setminus\\lbrace v\\rbrace\\right)$. In this case, the algorithm would return \\emph{no} in Line~\\ref{alg_no}.\nIn the for-loop of Line~\\ref{alg_fil_for}, we update for all $v\\in V\\setminus N_G\\left[ M_2\\right]$ the value $\\widetilde{f} \\left( v\\right)$ to~$1$. With the while-loop, this implies $N_G\\left[M_2\\right] = V_0\\left(\\widetilde{f}\\right)\\cup V_2\\left(\\widetilde{f}\\right)$. Therefore, $V_2\\left(\\widetilde{f}\\right)$ is a minimal ds of $G\\left[V_0\\left(\\widetilde{f}\\right)\\cup V_2\\left(\\widetilde{f}\\right)\\right]$. \nSince we do not update the values of $\\widetilde{f}$ to two in this last for-loop, Condition~\\ref{con_private} from  \\autoref{t_porperty_min_rdf} holds. By the while-loop and the for-loop starting in Line~\\ref{alg_fil_for}, it is trivial to see that Condition~\\ref{con_1_2} also holds for the final $\\widetilde{f}$.\nWe can now use \\autoref{t_porperty_min_rdf} to see that $\\widetilde{f}$ is a minimal rdf. \n\nSince we never decrease $\\widetilde{f}$ in this algorithm, starting with $\\widetilde f=f$ in Line~\\ref{alg_init}, we get $f\\leq \\widetilde{f}$. Therefore, $\\left(G,f\\right)$ is a \\emph{yes}-instance of \\textsc{ExtRD}.\n\n\nNow we assume that $\\left(G,f\\right)$ is a \\emph{yes}-instance, but the algorithm returns \\emph{no}. Therefore, there exists a minimal rdf $\\overline{f}$ with $f\\leq \\overline{f}$. Since $N_G\\left[V_2\\left(\\overline{f}\\right)\\right]\\cap V_1\\left(\\overline{f}\\right)=\\emptyset$, $\\widetilde{f}\\leq \\overline{f}$ holds for the function~$\\widetilde{f}$ in Line~\\ref{alg_for_no}. This implies $M_2=V_2\\left( \\widetilde{f}\\right)\\subseteq V_2\\left(\\overline{f}\\right)$. \nThe algorithm returns \\emph{no} if and only if there exists a $v\\in M_2$ with \n$$ N_G\\left(v\\right)\\subseteq N_G\\left[ M_2\\setminus \\lbrace v \\rbrace\\right]\\subseteq N_G\\left[V_2\\left(\\overline{f}\\right)\\setminus \\lbrace v\\rbrace\\right].$$\nApplying again Theorem~\\ref{t_porperty_min_rdf}, we see that $\\overline{f}$ cannot be a minimal rdf, contradicting our assumption.\n\\end{pf}\n\n\\noindent\nIn \\autoref{propos:runtime}, we prove that our algorithm needs polynomial time only.\n\n\\begin{proposition}\\label{propos:runtime}\nAlgorithm~\\ref{alg} runs in time cubic in the order of the input graph.\n\\end{proposition}\n\\begin{pf} Let $G=(V,E)$ be the input graph. \nDefine $n=\\vert V\\vert$. Up to Line~\\ref{alg_before_while}, the algorithm can run in linear time. As each vertex can only be once in $M$ and we look at the neighbors of each element in $M$, the while-loop runs in time $\\mathcal{O}\\left(n^2\\right)$. In the for-loop starting in Line~\\ref{alg_for_no},  we build for all $v\\in M_2$ the set $N_G\\left[M_2\\setminus \\lbrace v\\rbrace\\right]$. This needs $\\mathcal{O}\\left( n^3\\right)$ time. The other steps of this loop run in time $\\mathcal{O}\\left(n^2\\right)$.\nThe last for-loop requires linear time. Hence, the algorithm runs in time $\\mathcal{O}\\left(n^3\\right)$.   \n\\end{pf}\n\n\\section{Enumerating  Minimal RDF  for General  Graphs}\n\\label{sec:enum-minimal-rdf-simple}\n \nFor general graphs, our general combinatorial observations allow us to strengthen the (trivial) $\\mathcal{O}^*(3^n)$-algorithm for enumerating all minimal rdf  for graphs of order~$n$\ndown to $\\mathcal{O}^*(2^n)$, as displayed in Algorithm~\\ref{alg:enum}.\nTo understand the correctness of this enumeration algorithm, the following lemma is crucial.\n\n\\begin{algorithm}\n\\caption{A simple enumeration algorithm for minimal rdf}\\label{alg:enum}\n\\begin{algorithmic}[1]\n\\Procedure{RD Enumeration}{$G$}\\newline\n \\textbf{Input:} A graph $G=\\left(V,E\\right)$.\\newline\n \\textbf{Output:} Enumeration of all  minimal rdf $f:V\\to\\{0,1,2\\}$.\n\\For {all functions $f:V\\to\\{1,2\\}$}\n\\For {all $v\\in V$ with $f(v)=1$}\n\\If {$\\exists u\\in N_G(v): f(u)=2$}\n\\State $f(v)\\coloneqq 0$.\n\\EndIf\n\\EndFor\n\\State Build graph $G'$ induced by $f^{-1}(\\{0,2\\})=V_0(f)\\cup V_2(f)$.\n\\State $\\text{private-test}\\coloneqq 1$.\n\\For {all $v\\in V$ with $f(v)=2$}\n\\If {$P_{G',V_2(F)}(v)\\subseteq\\{v\\}$}\n\\State $\\text{private-test}\\coloneqq 0$.\n\\EndIf\n\\EndFor\n\\If{$\\text{private-test}=1$ and if $f^{-1}(2)=V_2(f)$ is a minimal ds of $G'$}\n\\State Output the current function  $f:V\\to\\{0,1,2\\}$.\n\\EndIf\n\\EndFor\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{lemma}\\label{lem:extend2}\nLet $G=(V,E)$ be a graph with $V_2\\subseteq V$ such that $P_{G,V_2}\\left( v \\right) \\nsubseteq \\lbrace v\\rbrace$ for each $v\\in V_2$ holds. Then there exists exactly one minimal rdf $f\\in \\lbrace 0,1,2 \\rbrace ^ V$ with $V_2=V_2\\left(f\\right)$. Algorithm~\\ref{alg} can calculate $f$.\n\\end{lemma}\n\\begin{pf}\nDefine \\\\[-4.5ex]\n$$f:V\\to \\lbrace 0, 1, 2\\rbrace, v\\mapsto \\begin{cases}\n2, & v \\in V_2\\\\\n1, & v \\notin N\\left[ V_2\\right]\\\\\n0, & \\text{otherwise}\n\\end{cases}$$\nHence, $N_G\\left[ V_2 \\right]=V_2\\cup V_0\\left(f\\right)$. With the assumption $P_{G,V_2}\\left( v \\right) \\nsubseteq \\lbrace v\\rbrace$, $V_2$ is a minimal ds of $G[V_2\\cup V_0\\left(f\\right)]$. Furthermore,  $N_G\\left[V_2\\right]\\cap V_1\\left(f\\right)=\\emptyset$. As $V_2=V_2\\left(f\\right)$, all conditions of \\autoref{t_porperty_min_rdf} hold and $f$ is a minimal rdf.\n\nLet $\\widetilde{f}\\in\\lbrace 0,1,2\\rbrace^V$ be a minimal rdf with $V_2= V_2\\left(\\widetilde{f}\\right)$. If there exists some $v\\in V_0\\left( f \\right) \\cap V_1\\left(\\widetilde{f}\\right)$, this contradicts Condition~\\ref{con_1_2}, as $v\\in N_G\\left[V_2\\right] = N_G\\left[V_2\\left(\\widetilde{f}\\right)\\right]$. Therefore, $ V_0\\left( f\\right)\\subseteq V_0\\left( \\widetilde{f}\\right) $ holds. By the assumption that $\\widetilde{f}$ is a rdf, for each $v\\in V_0\\left( \\widetilde{f} \\right)$ there exists a $u\\in V_2\\left(\\widetilde{f}\\right)\\cap N\\left[v\\right] = V_2\\cap N\\left[v\\right]$. This implies $v\\in N_G\\left[V_2\\right]\\setminus V_2= V_0\\left( f\\right)$. Therefore, $ V_0\\left( f\\right) = V_0\\left( \\widetilde{f}\\right)$ holds. This implies $f=\\widetilde{f}$.\n\n\\vspace{5pt}\nDefine: \n$$\\widehat{f}:V\\to \\lbrace 0, 1, 2\\rbrace, v\\mapsto \\begin{cases}\n2, & v\\in V_2\\\\\n0, & v\\notin V_2\n\\end{cases}.$$\nIt is trivial to see that $\\widehat{f}\\leq f$. By \\autoref{theorem:correctness_alg}, Algorithm~\\ref{alg} returns \\emph{yes} for the input $\\widehat{f}$. Let $\\overline{f}$ be the the minimal rdf produced by Algorithm~\\ref{alg}, given~$\\widehat{f}$. We want to show that $V_2=V_2\\left( \\overline{f} \\right)$. We do this by looking at the steps of the algorithm.\nSince $V_1\\left( \\widehat{f} \\right) = \\emptyset$, the algorithm never gets into the If-clause in Line~\\ref{alg_if_no}. This is the only way to update a vertex to the value 2. Therefore, $V_2= V_2\\left(\\overline{f}\\right)$. \n\\end{pf}\n\n\\begin{proposition} Let $G=\\left(V,E\\right)$ be a graph. For minimal rdf $f,g \\in \\lbrace0,1,2\\rbrace ^ V$ with $V_2\\left(f\\right)=V_2\\left(g\\right)$, it holds $f=g$.\n\\end{proposition}\n\n\\begin{pf}\nBy \\autoref{t_private_neighborhood}, $V_2\\left(f\\right)$ fulfills the conditions of \\autoref{lem:extend2}. Therefore, there exists a unique minimal rdf $h\\in\\lbrace0,1,2\\rbrace ^ V$ with $V_2\\left( h \\right)=V_2\\left( f \\right) = V_2\\left( g \\right)$. Thus. $f=g=h$ holds.\n\\end{pf}\n\nHence, there is a bijection between the minimal rdf of a graph $G=(V,E)$ and subsets $V_2\\subseteq V$ that satisfy the condition of \\autoref{lem:extend2}.\n\n\\begin{proposition}\\label{prop:RomanEnum}\nAll minimal rdf of a graph of order~$n$ can be enumerated in time\n$\\mathcal{O}^*(2^n)$.\n\\end{proposition}\n\n\\begin{pf}\n Consider Algorithm~\\ref{alg:enum}. The running time claim is obvious. The correctness of the algorithm is clear due to \\autoref{t_porperty_min_rdf} and \\autoref{lem:extend2}.  \n\\end{pf}\n\n\nThe presented algorithm clearly needs polynomial space only, but it is less clear if it has polynomial delay.\nBelow, we will present a branching algorithm that has both of these desirable properties, and moreover, its running time is below $2^n$. How good or bad such an enumeration is, clearly also depends on examples that provide a lower bound on the number of objects that are enumerated. The next lemma explains why the upper bounds for enumerating minimal rdf must be bigger than those for enumerating minimal dominating sets.\n\n\\begin{lemma}\nA disjoint collection of $c$ cycles on five vertices yields a graph of order $n=5c$ that has $(16)^c$ many minimal rdf. \n\\end{lemma}\n\n\\begin{pf}\nLet $C_5$ be a cycle of length 5 with $V\\left(C_5\\right)=\\lbrace v_1,\\ldots,v_5 \\rbrace$ and $E\\left( C_5 \\right) = \\lbrace\\lbrace v_i,v_{i+1}\\rbrace\\mid i\\in [4] \\rbrace \\cup \\lbrace \\lbrace v_1,v_5 \\rbrace \\rbrace$. For a $f\\in\\lbrace0,1,2\\rbrace^{V\\left( C_5\\right)}$ there are at least the following sixteen possibilities for $\\left( f\\left( v_1\\right),\\ldots,f\\left( v_5\\right)\\right)$:\n\\begin{itemize}\n    \\item zero occurrences of 2: $(1,1,1,1,1)$;\n    \\item one occurrence of 2: $(2,0,1,1,0)$ and four more cyclic shifts;\n    \\item two adjacent occurrences of 2:  $(2,2,0,1,0)$  and four more cyclic shifts;\n    \\item two non-adjacent occurrences of 2: $(2,0,2,0,0)$   and four more cyclic shifts.\n\\end{itemize}\n\nTherefore, there are at least 16 minimal rdf on $C_5$. To prove that these are all the minimal rdf, we use Lemma~\\ref{lem:V2-bound}, which implies $\\vert V_2\\left(f\\right)\\vert\\leq \\frac{\\vert V\\left(C_5\\right)\\vert}{2} <3$. Hence, the number of minimal rdf on $C_5$ is at most $\\binom{5}{0}+\\binom{5}{1}+\\binom{5}{2}=16$. \n\\end{pf}\n\n\n\\begin{corollary}\nThere are graphs of order $n$ that have at least ${\\sqrt[5]{16}\\,}^n\\in\\Omega(1.7441^n)$ many minimal rdf.\n\\end{corollary}\n\nWe checked with the help of a computer program that there are no other connected graphs of order at most eight that yield (by taking disjoint unions) a bigger lower bound.\n\n\\section{A Refined Enumeration Algorithm}\n \\label{sec:enum-minimal-rdf-refined}\n\nIn this section, we are going to prove the following result, which can be considered as the second main result of this paper.\n\\begin{theorem}\\label{thm:minimal-rdf-enumeration}\nThere is a polynomial-space algorithm that enumerates all minimal rdf of a given graph of order $n$ with polynomial delay and in time $\\mathcal{O}^*(1.9332^n)$.\n\\end{theorem}\n\nNotice that this is in stark contrast to what is known about the enumeration of minimal dominating sets, or, equivalently, of minimal hitting sets in hypergraphs. Here, it is a long-standing open problem if \nminimal hitting sets in hypergraphs can be enumerated with polynomial delay.\n\nThe remainder of this section is dedicated to describing the proof of this theorem.\n\n\\subsection{A bird's eye view on the algorithm}\n\nAs all along the search tree, from inner nodes we branch into the two cases if a certain vertex is assigned $2$ or not, it is clear that (with some care concerning the final processing in leaf nodes) no minimal rdf is output twice. Hence, there is no need for the branching algorithm to store intermediate results to test (in a final step) if any solution was generated twice. Therefore, our algorithm needs only polynomial space, as detailed in \\autoref{prop:poly-space} and \\autoref{cor:poly-space}.\n\nBecause we have a polynomial-time procedure that can test if a certain given pre-solution can be extended to a minimal rdf, we can build (a slightly modified version of) this test into an enumeration procedure, hence avoiding unnecessary branchings. \nTherefore, whenever we start with our binary branching, we know that at least one of the search tree branches will return at least one new minimal rdf. Hence, we will not move to more than $N$ nodes in the search tree before outputting a new minimal rdf, where $N$ is upper-bounded by twice the order of the input graph. This is the basic explanation for the claimed polynomial delay, as detailed in \\autoref{prop:poly-delay}.\n\nLet $G=(V,E)$ be a graph.\nLet us call a(ny partial) function \n\n\\vspace{-15pt}\n\\begin{equation*}\n\\begin{split}\n f: V \\longrightarrow \\{0,1,2,\\overline{1}, \\overline{2}\\}\n\\end{split}\n\\end{equation*}\n a \\emph{generalized Roman domination function}, or grdf for short. \nExtending previously introduced notation, let $\\overline{V_1}(f) = \\{x\\in V\\mid  f(x) = \\overline{1}\\}$, and $\\overline{V_2}(f) = \\{x\\in V\\mid  f(x) = \\overline{2}\\}$. A vertex is said to be \\emph{active} if it has not been assigned a value (yet) under~$f$; these vertices are collected in the set $A(f)$. Hence, for any grdf $f$, we have the partition $V=A(f)\\cup V_0(f)\\cup V_1(f)\\cup V_2(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f)$. \n\nAfter performing a branching step, followed by an exhaustive application of the reduction rules, any grdf~$f$ considered in our algorithm always satisfies the following \\textbf{(grdf) invariants}:\n\\begin{enumerate}\n    \\item $\\forall x\\in \\overline{V_1}(f)\\cup V_0(f)\\,\\exists y\\in N_G(x):y\\in V_2(f)$,\n    \\item $\\forall x\\in V_2(f):N_G(x)\\subseteq \\overline{V_1}(f) \\cup V_0(f) \\cup V_2(f)$,    \n    \\item $\\forall x\\in V_1(f):N_G(x)\\subseteq \\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f)$,\n    \\item if $\\overline{V_2}(f)\\neq\\emptyset$, then $A(f)\\cup \\overline{V_1}(f)\\neq \\emptyset$.\\footnote{This condition assumes that our graphs have non-empty vertex sets.}\n\\end{enumerate}\n\nFor the extension test, we will therefore consider the function $\\hat f:V\\to\\{0,1,2\\}$ that is derived from a grdf~$f$ as follows: \n$$\\hat f(v)=\\begin{cases}0, & \\text{if }v\\in  A(f)\\cup V_0(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f)\\\\\n1, & \\text{if }v\\in V_1(f)\\\\\n2, & \\text{if }v\\in V_2(f)\n\\end{cases}$$\n\n\nThe enumeration algorithm uses a combination of reduction and branching rules, starting with the nowhere defined function $f_\\bot$, so that $A(f_\\bot)=V$. The schematics of the algorithm is shown in Algorithm~\\ref{alg:refined-enum}. \nTo understand the algorithm, call an rdf $g$ as \\emph{consistent} with a grdf $f$ if $g(v)=2$ implies $v\\in A(f)\\cup V_2(f)\\cup \\overline{V_1}(f)$ and $g(v)=1$ implies $v\\in A(f)\\cup V_1(f)\\cup \\overline{V_2}(f)$ and $g(v)=0$ implies $v\\in A(f)\\cup V_0(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f)$.\nBelow, we start with presenting some reduction rules, which also serve as (automatically applied) actions at each branching step, whenever applicable. The branching itself always considers a most attractive vertex $v$ and either gets assigned~2 or not. \nThe running time analysis will be performed with a measure-and-conquer approach. Our simple measure is defined by  $\\mu(G,f)=|A(f)|+\\omega_1 |\\overline{V_1}(f)| + \\omega_2 |\\overline{V_2}(f)|\\leq |V| $\nfor some constants $\\omega_1$ and $\\omega_2$ that have to be specified later.\n\nThe measure never increases when applying a  reduction rule.\n\n\n\\begin{algorithm}\n\\caption{A refined enumeration algorithm for minimal rdf}\\label{alg:refined-enum}\n\\begin{algorithmic}[1]\n\\Procedure{Refined RD Enumeration}{$G,f$}\\newline\n \\textbf{Input:} A graph $G=\\left(V,E\\right)$, a grdf $f:V\\to\\{0,1,2,\\overline{1},\\overline{2}\\}$.\\newline\n \\emph{Assumption:} There exists at least one   minimal rdf  consistent with $f$.\\newline\n \\textbf{Output:} Enumeration of all  minimal rdf  consistent with $f$.\n\\If {$f$ is everywhere defined and $f(V)\\subseteq \\{0,1,2\\}$}\n\\State Output $f$ and return.\n\\EndIf\n\\State \\{\\,We know that $A(f)\\cup \\overline{V_1}(f)\\neq\\emptyset$.\\,\\}\n\\State Pick a vertex $v\\in A(f)\\cup \\overline{V_1}(f)$ of highest priority for branching.\n\\State $f_2\\coloneqq f $; $f_2(v)\\coloneqq 2$.\n\\State Exhaustively apply reduction rules to $f_2$. \\{\\,Invariants are valid for $f_2$.\\,\\}\n\\If{$\\textsc{GenExtRD Solver}\\left(G,\\widehat{f_2},\\overline{V_2}(f_2)\\right)$}\n\\State $\\textsc{Refined RD Enumeration}\\left(G,{f_2}\\right)$.\n\\EndIf\n\\State $f_{\\overline{2}}\\coloneqq f $; \\textbf{if} $v\\in A(f)$ \\textbf{then} $f_{\\overline{2}}(v)\\coloneqq {\\overline{2}}$ \\textbf{else} $f_{\\overline{2}}(v)\\coloneqq 0$.\n\\State Exhaustively apply reduction rules to $f_{\\overline{2}}$. \\{\\,Invariants are valid for $f_{\\overline{2}}$.\\,\\}\n\\If{$\\textsc{GenExtRD Solver}\\left(G,\\widehat{f_{\\overline{2}}},\\overline{V_2}(f_{\\overline{2}})\\right)$}\n\\State $\\textsc{Refined RD Enumeration}\\left(G,{f_{\\overline{2}}}\\right)$.\n\\EndIf\n\\EndProcedure\n\\end{algorithmic}\n\\end{algorithm}\n\n\\noindent\nWe are now presenting details of the algorithm and its analysis.\n\n\\subsection{How to achieve polynomial delay and polynomial space}\nIn this section, we need a slight modification of the problem \\textsc{ExtRD} in order to cope with pre-solutions. In this version, we add to an instance, usually specified by $G=\\left( V, E\\right)$ and $f:V\\to \\lbrace 0, 1, 2\\rbrace$, a set $\\overline{V_2}\\subseteq V $ with $V_2\\left(f\\right)\\cap \\overline{V_2}=\\emptyset$. The question is if there exists a minimal RDF $\\widetilde{f}$ with $f \\leq \\widetilde{f}$ and $V_2\\left(\\widetilde{f}\\right)\\cap \\overline{V_2}=\\emptyset$. We call this problem a \\emph{generalized} rdf extension problem, or  \n\\textsc{GenExtRD} for short.\nIn order to solve this problem, we  modify Algorithm \\ref{alg} to cope with \\textsc{GenExtRD} by adding an if-clause after Line~\\ref{alg_if_no} that asks if $u\\in\\overline{V_2}$. If this is true, then the algorithm returns \\emph{no}, because it is prohibited that $\\tilde{f}(u)$ is set to~2, while this is necessary for minimal rdf, as there is a vertex $v$ in the neighborhood of $u$ such that $\\tilde{f}(v)$ has been set to~1. We call this algorithm  \\textsc{GenExtRD Solver}.\n\n\\begin{lemma}\\label{lem:GenExtRD}\nLet $G=\\left(V,E\\right)$ be a graph, $f : V \\to \\lbrace0,1,2\\rbrace$ be a  function and $\\overline{V_2} \\subseteq V$ be a set with $V_2\\left({f}\\right)\\cap \\overline{V_2}=\\emptyset$. \\textsc{GenExtRD Solver} gives the correct answer when given the \\textsc{GenExtRD} instance $(G,f,\\overline{V_2})$.\n\\end{lemma}\n\n\n\\begin{pf}\nIn Algorithm~\\ref{alg}, the only statement where we give a vertex the value $2$ is in the if-clause of Line~\\ref{alg_if_no}. The modified version would first check if the vertex is in $\\overline{V_2}$. If this is true, there will be no minimal RDF solving this problem. Namely, if we give the vertex the value~$2$, this would contradict $V_2 \\left( \\widetilde{f} \\right) \\cap \\overline{V_2}=\\emptyset$. If the value stays $1$, this would contradict Condition~\\ref{con_1_2}. By \\autoref{theorem:correctness_alg}, $\\widetilde{f}$ will be a minimal rdf with $V_2\\left(\\widetilde{f}\\right) \\cap \\overline{V_2}=\\emptyset$ if the algorithm returns \\emph{yes}.\n\nAssume there exists a minimal RDF $\\overline{f}$ with $V_2\\left(\\overline{f}\\right) \\cap \\overline{V_2}=\\emptyset$ but the algorithm returns  \\emph{no}. First we assume that  \\emph{no} is returned by the new if-clause. This implies that a vertex $u\\in V_1\\left( f \\right)$ is in the neighborhood of a vertex $v\\in V$ that has to have the value~2 in any minimal rdf that is bigger than $f$ (because \\autoref{theorem:correctness_alg}). But this would lead to a similar contradiction as above.\n\nTherefore, the answer \\emph{no} has to be returned in Line \\ref{alg_no}. That would contradict Condition~\\ref{con_private} or Condition~\\ref{con_min_dom}. Thus the algorithm would correctly return \\emph{yes}.\n\\end{pf}\n\nLet $f$ be a generalized rdf at any moment of the branching algorithm. The next goal is to show that \\textsc{GenExtRD Solver} could tell us in polynomial time if there exists a minimal rdf that could be enumerated by the branching algorithm from this point on.   \n\n\\begin{proposition}\nLet $G=\\left(V,E\\right)$ be a graph, $f : V \\to \\lbrace0,1,2,\\overline{1},\\overline{2}\\rbrace$ be a partial function. Then,  \\textsc{GenExtRD Solver} correctly answers if there exists some minimal rdf $g: V \\to \\lbrace0,1,2\\rbrace$ that is consistent with $f$ when \\textsc{GenExtRD  Solver} is given the instance $(G,\\hat f,\\overline{V_2}(f))$.\n\\end{proposition}\n\nThe following proof makes use of the grdf invariants presented above, which are only formally proved to hold in the next subsection, in \\autoref{prop:invariants}.\n\n\\begin{pf} We have to show two assertions: (1) If \\textsc{GenExtRD  Solver} answers \\emph{yes} on the instance $(G,\\hat f,\\overline{V_2}(f))$, then there exists a minimal rdf $g: V \\to \\lbrace0,1,2\\rbrace$ that is consistent with $f$. (2) If there exists a minimal rdf $g: V \\to \\lbrace0,1,2\\rbrace$ that is consistent with $f$, then \\textsc{GenExtRD  Solver} answers \\emph{yes} on the instance $(G,\\hat f,\\overline{V_2}(f))$. \n\n\\smallskip\\noindent\n\\underline{ad (1)}: Assume  \\textsc{GenExtRD  Solver} found a minimal rdf $g$ such that $\\hat f\\leq g$ and $V_2(g)\\cap \\overline{V_2}(f)=\\emptyset$. Let $v\\notin A(f)$. First assume that $g(v)=2$. Clearly, vertices in $V_2(f)=V_2(\\hat f)$ do not get changed, as they cannot be made bigger. Hence, assume $v\\notin V_2(f)$ exists with $g(v)=2$. As \\textsc{GenExtRD Solver} will only explicitly set the value~2 for vertices originally set to~1 (by their $f$-assignment) that are in the neighborhood of vertices already set to value~2 and that do not belong to $\\overline{V_2}(f)$, we have to reason about a possible $v\\in V_1(f)=V_1(\\hat f)$. \nBy the third grdf invariant, the neighborhood of $v$ contains no vertex from $V_2(f)=V_2(\\hat f)$, so that the case of some  $v\\notin V_2(f)$  with $g(v)=2$ can be excluded.\n\nSecondly, assume that $g(v)=1$. The case $v\\in V_1(f)$ is not critical, and $v\\in V_2(f)$ is not possible, as reasoned above. Notice that $g(v)=1$ was set in the last lines of the algorithm. In particular, $N_G(v)\\cap V_2(g)=\\emptyset$. As $V_2(f)\\subseteq V_2(g)$, also $N_G(v)\\cap V_2(f)=\\emptyset$. By the first grdf invariant, \n$v\\notin \\overline{V_1}(f)\\cup V_0(f)$. Hence, only $v\\in \\overline{V_2}(f)$ remains as a possibility.\n\nThirdly, assume that $g(v)=0$. As $f(v)\\in\\{1,2\\}$ is clearly impossible, $v\\in V_0(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f)$ must follow.\nHence, $g$ is consistent with $f$.\n\n\\smallskip\\noindent\n\\underline{ad (2)}: Assume that there exists a minimal rdf $g: V \\to \\lbrace0,1,2\\rbrace$ that is consistent with $f$. We have to prove that $\\hat f\\leq g$ and that $V_2(g)\\cap \\overline{V_2}(f)=\\emptyset$, because then \\textsc{GenExtRD Solver} will correctly answer \\emph{yes} by \\autoref{lem:GenExtRD}.\nFor $g(v)=2$, then consistency implies $f(v)\\neq \\overline{2}$, and trivially $\\hat f(v)\\leq g(v)$. \nFor $g(v)=1$, $v\\in A(f)\\cup V_1(f)\\cup \\overline{V_2}(f)$, and hence $\\hat f(v)\\in\\{0,1\\}$, so that $\\hat f(v)\\leq g(v)$. \nIf $g(v)=0$, then $v\\in A(f)\\cup V_0(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f)=V_0(\\hat f)$, so that again  $\\hat f(v)\\leq g(v)$. \n\\end{pf}\n\n\nAn important consequence of the previous proposition is stated next.\nNotice that our algorithm behaves quite differently from what is known about algorithms the enumerate minimal ds.\n\n\\begin{proposition}\\label{prop:poly-delay}\nProcedure \\textsc{Refined RD Enumeration}, on input $G=(V,E)$, outputs functions $f:V\\to\\{0,1,2\\}$ with polynomial delay.\n\\end{proposition}\n\n\\begin{pf} Although the reduction rules are only stated in the next subsection, it is not hard to see by quickly browsing through them that they can be implemented to run in polynomial time. Moreover, \\textsc{GenExtRD Solver} runs in polynomial time.\nHence, all work done in an inner node of the search tree needs polynomial time only.\nBy the properties of \\textsc{GenExtRD Solver}, the search tree will never continue branching if no outputs are to be expected that are consistent with the current grdf (that is associated to that inner node). Hence, a run of the procedure \\textsc{Refined RD Enumeration} dives straight through setting more and more values of a grdf, until it is everywhere defined with values from $\\{0,1,2\\}$, and then it returns from the recursion and dives down the next promising branch. Clearly, the length of any search tree branch is bounded by $|V|$, so that at most $2|V|$ many inner nodes are visited between any two outputs. This also holds at the very beginning (i.e., only polynomial time will elapse until the first function is output) and at the very end (i.e., only polynomial time will be spent after outputting the last function). \nThis proves the claimed  polynomial delay.\n\\end{pf}\n\n\\begin{proposition}\nProcedure \\textsc{Refined RD Enumeration} correctly enumerates all minimal rdf that are consistent with the input grdf, assuming that at least one consistent rdf exists.\n\\end{proposition}\n\n\\begin{pf}\nAs there exists a consistent rdf, outputting the input function is correct if the input function is already an rdf, which is checked, as we test if the given grdf is everywhere defined at has only images in $\\{0,1,2\\}$. Also, before \\textsc{Refined RD Enumeration} is called recursively, we  explicitly check if  at least one consistent rdf exists.\n\nIf the input grdf~$f$ is not everywhere defined or if $\\overline{V_2}(f)\\cup \\overline{V_1}(f)\\neq\\emptyset$, then $A(f)\\cup\\overline{V_1}(f)\\neq\\emptyset$ by the fourth grdf invariant. Hence, whenever \\textsc{Refined RD Enumeration} is called recursively, $A(f)\\cup\\overline{V_1}(f)\\neq\\emptyset$ holds, as these calls are immediately after applying all reduction rules exhaustively.\n\nHence, by induction and based on the previous propositions, \\textsc{Refined RD Enumeration} correctly enumerates all minimal rdf that are consistent with the input grdf.\n\\end{pf}\n\n\\begin{corollary}\\label{cor:correct-enumeration}\nProcedure \\textsc{Refined RD Enumeration} correctly enumerates all minimal rdf of a given graph $G=(V,E)$ when provided with the nowhere defined grdf $f_\\bot$.\n\\end{corollary}\n\n\\begin{pf}\nDue to the previous proposition, it is sufficient to notice that all minimal rdf are consistent with  $f_\\bot$ and that the function that is constant~1 is a minimal rdf consistent with  $f_\\bot$.\n\\end{pf}\n\n\\begin{proposition}\\label{prop:poly-space}\nProcedure \\textsc{Refined RD Enumeration} never enumerates any minimal rdf consistent with the given grdf on the input graph $G=(V,E)$ twice.\n\\end{proposition}\n\n\\begin{pf}\nNotice that the enumeration algorithm always branches by deciding for a vertex~$v$ from $A(f)\\cup \\overline{V_1}(f)$, where $f$ is the current grdf, if $f(v)$ is updated to~$2$ or not, which means that either $v\\in A(f)$ is set to $\\overline{2}$, or $v\\in \\overline{V_1}$ is set to~0. Then, reduction rules may apply, but they never change the decision if, in a certain branch of the search tree,  $f(v)=2$ is either true or false.\nMoreover, they never set any vertex to~$2$. \nAs any  minimal rdf that is ever output in a certain branch will be consistent with the grdf~$f$ associated to an inner node of the search tree, Procedure \\textsc{Refined RD Enumeration} never enumerates any minimal rdf twice.\n\\end{pf}\n\nAn important consequence of the last claim is that there is no need to store all output \nfunctions in order to finally parse them to see into enumerating any of them only once.\n\n\\begin{corollary}\\label{cor:poly-space}\nAlgorithm  \\textsc{Refined RD Enumeration} lists all minimal rdf consistent with the given grdf on the input graph $G=(V,E)$ without repetitions and in polynomial space.\n\\end{corollary}\n\n\\subsection{Details on reductions and branchings}\n\nFor the presentation of the following rules, we assume that $G=(V,E)$ and a grdf $f$ is given. We also assume that the rules are executed exhaustively in the given order.\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule LPN (Last Potential Private Neighbor).} If $v\\in V_2(f)$ satisfies $|N_G(v)\\cap  (\\overline{V_2}(f)\\cup A(f))|=1$, then set $f(x) = 0$ for $\\{x\\}=N_G(v)\\cap  (\\overline{V_2}(f)\\cup A(f))$.\n\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule $V_0$.} Let  $v \\in V_0(f)$. Assume there exists a unique $u\\in V_2(f)\\cap N_G(v)$. \nMoreover, assume that for all $x\\in N_G(u)\\cap (V_0(f)\\cup \\overline{V_1}(f)\\cup \\overline{V_2}(f))$, $|N_G(x)\\cap V_2(f)|\\geq 2$ if $x\\neq v$. Then, for any  $w \\in N_G(v)\\cap A(f)$, set $f(w) = \\overline{2}$ and  for any  $w \\in N_G(v)\\cap \\overline{V_1}(f)$, set $f(w) =0$.\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule $V_1$.} Let  $v \\in V_1(f)$. For any  $w \\in N_G(v)\\cap A(f)$, set $f(w) = \\overline{2}$.  For any  $w \\in N_G(v)\\cap \\overline{V_1}(f)$, set $f(w) =0$.\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule $V_2$.} Let  $v \\in V_2(f)$. For any  $w \\in N_G(v)\\cap A(f)$, set $f(w) = \\overline{1}$.  For any  $w \\in N_G(v)\\cap \\overline{V_2}(f)$, set $f(w) =0$.\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule NPD (No Potential Domination).} If $v\\in \\overline{V_2}(f)$ satisfies $N_G(v)\\subseteq \\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f)$, then set $f(v) = 1$ (this also applies to isolated vertices in $\\overline{V_2}(f)$).\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule NPN (No Private Neighbor).} If $v\\in A(f)$ satisfies $N_G(v)\\subseteq V_0 \\cup\\overline{V_1}(f)$, then set $f(v) = \\overline{2}$ (this also applies to isolated vertices in $A(f)$).\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule Isolate.} If $A(f)=\\emptyset$ and if $v\\in \\overline{V_1}(f)$ satisfies $N_G(v)\\cap\\overline{V_2}(f)= \\emptyset$, then set $f(v) = 0$.\n\n\\vspace{5pt}\n\\noindent\n{\\bf Reduction Rule Edges.} If $u,v\\in \\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f)$ and $e=uv\\in E$, then remove the edge~$e$ from $G$.\n\n\\vspace{5pt}\n\\noindent\nIn the following, we first take care of the claimed grdf invariants. \n\n\\begin{proposition}\\label{prop:invariants}\nAfter exhaustively executing the proposed reduction rules, as indicated in Algorithm~\\ref{alg:refined-enum}, the claimed grdf invariants are maintained.\n\\end{proposition}\n\n\\begin{pf}We argue for the correctness of the grdf invariants  by induction one by one. Notice that (trivially) all invariants hold if we start the algorithm with the nowhere defined grdf.\n\\begin{enumerate}\n    \\item $\\forall x\\in \\overline{V_1}(f)\\cup V_0(f)\\,\\exists y\\in N_G(x):y\\in V_2(f)$.\n    \n    We need to show  that $N_G(x) \\cap V_2(f) \\neq \\emptyset$ holds for each $x\\in V_0(f) \\cup \\overline{V_1}(f)$. For the inductive step, we only have to look at the reduction rules, since the branching rules only change the value to $0$ if the vertex was already in $\\overline{V_1}(f)$. For each reduction rule where we set a value of a vertex to~$0$ or to~$\\overline{1}$, there exists a vertex in the neighborhood with value~$2$, which is seen as follows.\n\n\\begin{tabular}{rl}\n    LPN:& We explicitly consider $x\\in N_G(V_2(f))$ only to be set by $f(x)=0$.\\\\\n    $V_0$\\,\\&\\,$V_1$:& We only set $w$ to $0$ if it has been in $\\overline{V_1}$. By induction  hypothesis,  \\\\\n    &$w$ has a neighbor in $V_2(f)$.\\\\\n    $V_2$:& We explicitly consider $w\\in N_G(V_2(f))$ only to be set to~$0$ or to~$\\overline{1}$.\\\\\n    Isolate:& Only vertices from $\\overline{V_1}(f)$ are set to~0; apply induction hypothesis. \n\\end{tabular}\n    \n    \\item $\\forall x\\in V_2(f):N_G(x)\\subseteq \\overline{V_1}(f)\\cup V_0(f) \\cup V_2(f)$.\n    \n    This property can only be invalidated if new vertices get the value~2 or if vertices from  $\\overline{V_1}(f)\\cup V_0(f) \\cup V_2(f)$ are changed to a value other than this or if edges are deleted (as vertices are never deleted). The only way in which a vertex gets~$v$ the value~2 is by branching. Immediately afterwards, the reduction rules are executed: LPN and $V_2$ will install the invariant for the neighborhood of~$v$. No reduction rule ever changes the value of a vertex from $V_0(f) \\cup V_2(f)$, while vertices from $\\overline{V_1}(f)$ might be set to~$0$ or~$2 $. The Reduction Rule Edges deletes no edges incident to vertices from $V_2(f)$. \n    \\item $\\forall x\\in V_1(f):N_G(x)\\subseteq \\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f)$.\n    \n    The invariant is equivalent to the following  three conditions: (a) $N(V_1(f))\\cap A(f)=\\emptyset$, (b) $N(V_1(f))\\cap \\overline{V_1}(f)=\\emptyset$ and (c) $N(V_1(f))\\cap V_2(f) =\\emptyset$. Conditions (a) and (b) are taken care of by Reduction Rule $V_1$.\n    Condition (c) immediately follows by the already proven second invariant.\n    \\item If $\\overline{V_2}(f)\\neq\\emptyset$, then $A(f)\\cup \\overline{V_1}(f)\\neq \\emptyset$.\n    \n    Consider some $x\\in \\overline{V_2}(f)$. By the second invariant, $N_G(x)\\cap V_2(f)=\\emptyset$. By the Reduction Rule Edges, $N_G(x)\\cap \\left(\\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f)\\right)=\\emptyset$. As the Reduction Rule NPD did not apply, the only possible neighbors of $x$ are in $A(f)\\cup \\overline{V_1}(f)$.\\qed\n\\end{enumerate}\n\\renewcommand{\\qed}{}\n\\end{pf}\n\nWe have now to show the \\emph{soundness} of the proposed reduction rules. In the context of enumerating minimal rdf, this means the following: if $f,f'$ are grdf of $G=(V,E)$ before or after applying any of the reduction rules, then $g$ is a minimal rdf that is consistent with $f$ if and only if it is consistent with $f'$.\n\n\\begin{proposition}\\label{prop:rdf-reductionrules-soundness}\nAll proposed reduction rules are sound.\n\\end{proposition}\n\n\\begin{pf}\nFor the soundness of the reduction rules, we also need the invariants proven to be correct in \\autoref{prop:invariants}.\nWe now prove the soundness of each reduction rule, one at a time.\n\nIf possible, we apply Reduction Rule LPN first. Consider $v\\in V_2(f)$ with $\\{x\\}=N_G(v)\\cap  (\\overline{V_2}(f)\\cup A(f))$. Before the branching step, due to the second invariant, neighbors of $V_2(f)$-vertices are either in $V_0(f)$, $V_2(f)$ or in $\\overline{V_1}(f)$. As no reduction rule adds a vertex to $V_2(f)$, $v$ must have been put into $V_2(f)$ by the last branching step.\nBy the first invariant, we know that all $y\\in N_G(v)\\cap\\left( \\overline{V_1}(f)\\cup V_0(f)\\right)$ are dominated by vertices different from~$v$. As  $v\\in V_2(f)$, it still needs a private neighbor to dominate. As $N_G(v)\\cap  (\\overline{V_1}(f)\\cup A(f))$ contains one element~$x$ only, setting $f(x)=0$ is enforced for any minimal rdf (see \\autoref{t_private_neighborhood}).\n\nNext, we prove Reduction Rule $V_0$. We consider $v\\in V_0(f)$ and $u\\in V$ with $\\{u\\}=V_2(f)\\cap N_G(v)$. We can use the rule, since $u$ needs a private neighbor which can only be $v$, by the assumption that every other neighbor of~$u$ is dominated at least twice. To maintain the property that $v$ is a private neighbor of~$u$, each $A(f)$-neighbor of~$v$ is set to~$\\overline{2}$ and each $\\overline{V_1}(f)$-neighbor of $v$ is set to~$0$. This annotates the fact that any minimal rdf~$g$ compatible with~$f$ will satisfy $(g(x)=2\\implies x=u)$ for each $x\\in N_G(v)$.\n\nThe soundness of Reduction Rule $V_1$ and Reduction Rule $V_2$ mainly follows from \\autoref{t_1_2_neigborhood}. \n\nComing to the Reduction Rule NPD, notice that setting $f(v)=0$ would necessitate $f(u)=2$ for some neighbor $u$ of $v$, which is impossible.\n\nFor the Reduction Rule NPN, we use the fact that $N_G(v) \\cap V_2(f) \\neq \\emptyset$ holds for each $v\\in V_0(f) \\cup \\overline{V_1}(f)$, which is the first invariant.\\footnote{More precisely, we also have to check that the possibly newly introduced vertices in $V_0(f)$ or $\\overline{V_1}(f)$ by the branching or by the reduction rules up to this point do maintain the invariant, but this is nothing else then re-checking the induction step of the correctness proof of this invariant, see the proof of \\autoref{prop:invariants}.}  This implies that  there is no element left for $v$ to dominate (therefore it has no private neighbor except itself). Thus, if $v$ has the value $2$, then it would contradict with \\autoref{t_private_neighborhood}.\n\nFor the soundness of Reduction Rule Isolate, we note that, since $A(f)$ is empty, $v\\in \\overline{V_1}(f)$ can only have neighbors in $V_1(f)\\cup V_2(f)\\cup \\overline{V_1}(f)$, as $\\overline{V_2}(f)$-neighbors are prohibited. As  Reduction Rule $V_1$ was (if possible) executed before, $N_G(v)\\cap V_1(f)= \\emptyset$ holds. Therefore, $v\\in \\overline{V_1}(f)$ would not have a private neighbor if $f(v)=2$, cf. the first invariant.\\footnote{Again, one has to partially follow the induction step of the proof of \\autoref{prop:invariants}.}\n\nFinally, the soundness of Reduction Rule Edges follows trivially from the fact that an element of $\\overline{V_2}(f) \\cup V_0(f) \\cup V_1(f)$ cannot dominate any vertex in $\\overline{V_2}(f) \\cup V_0(f) \\cup V_1(f)$. Hence, a minimal rdf $g$ is consistent with $f$ if and only it is consistent with $f'$, obtained by applying the Reduction Rule Edges to~$f$.\n\\end{pf}\n\n\nIn order to fully understand Algorithm~\\ref{alg:refined-enum}, we need to describe priorities for branching.\nWe describe these priorities in the following in decreasing order for a vertex $v\\in A(f)\\cup \\overline{V_1}(f)$.\n\\begin{enumerate}\n    \\item $v\\in A(f)$ and $|N_{G}(v)\\cap (A(f)\\cup \\overline{V_2}(f))|\\geq 2$;\n    \\item any $v\\in A(f)$;\n    \\item any $v\\in \\overline{V_1}(f)$, preferably if $|N_{G}(v)\\cap \\overline{V_2}(f)|\\neq 2$.\n\\end{enumerate}\n\nThese priorities also split the run of our algorithm into phases, as whenever the algorithm was once forced to pick a vertex according to some lower priority, there will be never again the chance to pick a vertex of higher priority thereafter.\nIt is useful to collect some \\textbf{phase properties} that instances must satisfy after leaving Phase~$i$, determined by applying the $i^\\text{th}$ branching priority.\n\n\\begin{itemize}\n\\item Before entering any phase, there are no edges between vertices $u,v$ if $u,v\\in V_0(f)\\cup V_1(f)\\cup \\overline{V_2}(f)$ or if $u\\in V_2(f)$ and $v\\in \\overline{V_2}(f)\\cup A(f)$ or if $u\\in V_1(f)$ and  $v\\in \\overline{V_1}(f)\\cup A(f)$, as we can assume that the reduction rules have been exhaustively applied.\n    \\item After leaving the first phase, any active vertex with an active neighbor is either pendant or has only further neighbors from $\\overline{V_1}(f)\\cup V_0(f)$.\n    \n    \\item After leaving the second phase, $A(f)=\\emptyset$ and $N_G(\\overline{V_2}(f))\\subseteq \\overline{V_1}(f)$. Moreover, any vertex $x\\in \\overline{V_2}(f)$ has neighbors in $\\overline{V_1}(f)$.\n    \\item After leaving the third phase, $A(f)=\\overline{V_2}(f)=\\overline{V_1}(f)=\\emptyset$, so that $f$ is a Roman dominating function. \n\\end{itemize}\n\n\\begin{proposition}\nThe phase properties hold.\n\\end{proposition}\n\n\\begin{proof} We are considering the items on the list separately.\n\\begin{itemize}\\item Reduction Rule Edges shows the first claim. Reduction Rules $V_2$ and $V_1$ show the other two claims.\n\\item \nBy the branching condition, we know that after leaving the first phase, $|N_{G}(v)\\cap (A(f)\\cup \\overline{V_2}(f))|< 2$ for any active vertex~$v$. Since $v$ has a neighbor in $A(f)$ (say $u$) this implies that there cannot be any other neighbor in $A(f)\\cup \\overline{V_2}(f)$. Moreover, by the Reduction Rule $V_1$, $N_G(v)\\cap V_1(f)=\\emptyset$, and by the Reduction Rule $V_2$, $N_G(v)\\cap V_2(f)=\\emptyset$. Hence, $N_G(v) \\setminus \\lbrace u\\rbrace \\subseteq \\overline{V_1}(f)\\cup V_0(f)$.\n\\item The second phase branches on each $v\\in A(f)$. Therefore, it ends if $A(f)=\\emptyset$.  Let $v\\in \\overline{V_2}(f)$. By Reduction Rule Edge, we get $N_G(v) \\cap \\left( \\overline{V_2}(f) \\cup V_0 \\cup V_1 \\right)=\\emptyset$. Reduction Rule $V_2$ implies that $v$ does not have a neighbor in $V_2(f)$. Therefore we get $N_G(v) \\subseteq \\overline{V_1}(f)$. If $N_G(v)$ is empty, Reduction Rule NPD will be triggered. Therefore, $N_G(v)$ has at least one element.\n\\item The third phase runs on the vertices in $\\overline{V_1}(f)$. Thus, $\\overline{V_1}(f)=\\emptyset$ holds at the end of this phase. Since we never put a vertex into $A(f)$ again, $A(f)$ is empty. To get $\\overline{V_2}(f) = \\emptyset$, we can use the same argumentation as in the property before, since a vertex goes only from $A(f)$ to $\\overline{V_2}(f)$.\\qed \n\\end{itemize}\n\\end{proof}\n\n\\subsection{A Measure \\& Conquer Approach}\n\nWe now present the branching analysis, classified by the described branching priorities.\nWe summarize a list of all resulting branching vectors in \\autoref{tab:branching-vectors}.\n\n\\begin{table}[tb]\n    \\centering\n    \\begin{tabular}{|l|l|}\\hline\n         Phase \\#&  Branching vector\\\\\\hline \n         1.1 & $(1-\\omega_2,3-2\\omega_1)$\\\\\n         1.2 & $(1-\\omega_2,1+2\\omega_2)$\\\\\n         1.3 & $(1-\\omega_2,2+\\omega_2-\\omega_1)$\\\\\\hline\n         2.1 \\& 2.2.b & $(1-\\omega_2,2)$\\\\\n         2.2.a & $(1-\\omega_2,1+\\omega_2+\\omega_1)$\\\\\n         2.2.c & $(1+\\omega_2,1)$\\\\\\hline\n         3.1 & $(\\omega_1,\\omega_1+3\\omega_2)$\\\\\n         3.2.a & $(\\omega_1,2\\omega_1+\\omega_2)$\\\\\n         3.2.b \\& 3.3.a & $(\\omega_1+\\omega_2,\\omega_1+\\omega_2)$\\\\3.3.b & $(2\\omega_1+2\\omega_2,2\\omega_1+2\\omega_2,2\\omega_1+2\\omega_2,2\\omega_1+2\\omega_2)$\\\\\\hline\n    \\end{tabular}\n    \\caption{The branching vectors of different branching scenarios of the enumeration algorithm for listing all minimal Roman domination functions of a given graph}\n    \\label{tab:branching-vectors}\n\\end{table}\n\n\\subsubsection{Branching in Phase~1.}\nWe are always branching on an active vertex $v$. In the first branch, we set $f(v) = 2$.\nIn the second branch, we set $f(v) = \\overline{2}$.\nIn the first branch, in addition the Reduction Rule $V_2$ triggers at least twice. In order to determine a lower bound on the branching vector, we describe three worst-case scenarios; all other reductions of the measure can be only better.\n\\begin{enumerate}\n    \\item $|N_G(v)\\cap A(f)|=2$, i.e., $v$ has two active neighbors $x$ and $y$.  The corresponding recurrence is: $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+2(1-\\omega_1)))$, as either $v$ moves from $A(f)$ to $V_2(f)$ and $x,y$ move from $A(f)$ to $\\overline{V_1}(f)$, or $v$ itself moves from $A(f)$ to $\\overline{V_2}(f)$. The branching vector is hence: $(1-\\omega_2,3-2\\omega_1)$, as noted in the first row of \\autoref{tab:branching-vectors}.\n    \\item $|N_G(v)\\cap \\overline{V_2}(f)|=2$. The corresponding recurrence is:  $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+2\\omega_2))$, see the second row of \\autoref{tab:branching-vectors}.\n    \\item $|N_G(v)\\cap A(f)|=1$ and $|N_G(v)\\cap \\overline{V_2}(f)|=1$, leading to  $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+(1-\\omega_1)+\\omega_2))$, see \\autoref{tab:branching-vectors}, third row.\n\\end{enumerate}\n\n\\subsubsection{Branching in Phase~2.}\nWe are again branching on an active vertex $v$. By Reduction Rule NPN, we can assume that $N_G(v)\\neq\\emptyset$. In the first branch, we set $f(v) = 2$.\nIn the second branch, we set $f(v) = \\overline{2}$.\n\n\\begin{enumerate}\n    \\item\nIf $N_G(v)\\cap A(f)=\\{x\\}$, then $N_G(v)\\cap \\overline{V_2}(f)=\\emptyset$ in this phase. Therefore, in the first branch, $f(x)=0$ is enforced by Reduction Rule LPN. Notice that this might further trigger Reduction Rule $V_0$ if $N_G(x)\\cap (A(f)\\cup \\overline{V_1}(f))$ contains vertices other than~$v$. The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+1))$, see \\autoref{tab:branching-vectors}, fourth row.\n    \\item\nIf $N_G(v)\\cap  \\overline{V_2}(f)=\\{x\\}$, then $N_G(v)\\cap A(f)=\\emptyset$ in this phase. Therefore, in the first branch, $f(x)=0$ is enforced by Reduction Rule LPN. We consider several sub-cases now.\n\\begin{enumerate}\n    \\item $N_G(x)\\cap \\overline{V_1}(f)\\neq\\emptyset$. Reduction Rule $V_0$ will put all these vertices into $V_0(f)$. The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+\\omega_2+\\omega_1))$, see \\autoref{tab:branching-vectors}, fifth row.\n    \\item $|N_G(x)\\cap A(f)|\\geq 2$. Reduction Rule $V_0$ will put all these vertices into $\\overline{V_2}(f)$  (except for $v$). The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-(1-\\omega_2))+T(\\mu-(1+\\omega_2+(1-\\omega_2)))$, see \\autoref{tab:branching-vectors}, fourth row.\n    \\item Recall that by Reduction Rule Edges, $N_G(x)\\cap (\\overline{V_2}(f)\\cup V_0(f)\\cup V_1(f))=\\emptyset$, so that (if the first two cases do not apply) now we have $N_G(x)\\setminus\\{v\\}\\subseteq V_2(f)$. By the properties listed above, also $N_G(x)\\cap V_2(f)=\\emptyset$ is clear, so that now $|N_G(x)|=1$, i.e., $x$ is a pendant vertex. In this situation, we do not gain anymore from the first branch, but when $f(v)=\\overline{2}$ is set, Reduction Rule NPD triggers and sets $f(x)=1$. The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-(1+\\omega_2))+T(\\mu-(1-\\omega_2+\\omega_2))$, see \\autoref{tab:branching-vectors}, sixth row.\n\\end{enumerate}\n\\end{enumerate}\n\n\n\\subsubsection{Branching in Phase~3.}\nAs $A(f)=\\emptyset$, we are now branching on a vertex $v\\in \\overline{V_1}(f)$. Due to Reduction Rule Isolate, we know that $N_G(v)\\cap\\overline{V_2}(f)\\neq \\emptyset$.\nIn the first branch, we consider setting $f(v)=2$, while in the second branch, we set $f(v)=0$.\nAgain, we discuss several scenarios in the following.\n\n\\begin{enumerate}\n    \\item Assume that $|N_G(v)\\cap \\overline{V_2}(f)|\\geq 3$. If we set $f(v)=2$, then Reduction Rule $V_2$ triggers at least thrice. The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-\\omega_1)+T(\\mu-(\\omega_1+3\\omega_2))$, with a branching vector of $(\\omega_1,\\omega_1+3\\omega_2)$, see \\autoref{tab:branching-vectors}, seventh row.  \n    \\item Assume that $|N_G(v)\\cap \\overline{V_2}(f)|=1$, i.e., there is some (unique) $u\\in \\overline{V_2}(f)$ such that $N_G(v)\\cap \\overline{V_2}(f)=\\{u\\}$. We consider two sub-cases:\n    \\begin{enumerate}\n        \\item If  $|N_G(u)\\cap \\overline{V_1}(f)|\\geq 2$, then if we set $f(v)=2$, then first Reduction Rule LPN triggers $f(u)=0$, which in turn sets $f(w)=0$ for all $w\\in N_G(u)\\cap \\overline{V_1}(f)$, $w\\neq u$, by Reduction Rule $V_0$. The corresponding worst-case recurrence is: $T(\\mu) = T(\\mu-\\omega_1)+T(\\mu-(\\omega_2+2\\omega_1))$, see \\autoref{tab:branching-vectors}, eighth row.  \n        \\item If  $|N_G(u)\\cap \\overline{V_1}(f)|=1$, then $u$ is a pendant vertex. Hence, in the first branch, we have (as above) $f(v)=2$ and $f(u)=0$, while in the second branch, we have $f(v)=0$ and $f(u)=1$ by Reduction Rule NPD. This decreases the measure by $\\omega_1+\\omega_2$ in both branches, see \\autoref{tab:branching-vectors}, nineth row. This scenario happens in particular if the graph $G'$ induced by $\\overline{V_1}(f)\\cup \\overline{V_2}(f)$ contains a connected component which is a $P_2$. Therefore, we refer to this (also) as a \\emph{$P_2$-branching} below.\n    \\end{enumerate}\n    \\item Assume that $|N_G(v)\\cap \\overline{V_2}(f)|=2$, i.e., there are some $u_1,u_2\\in \\overline{V_2}(f)$ such that $N_G(v)\\cap \\overline{V_2}(f)=\\{u_1,u_2\\}$. Notice that in the first branch, when $f(v)=2$, Reduction Rule $V_2$ triggers twice, already reducing the measure by  $\\omega_1+2\\omega_2$.\n    We consider further sub-cases:\n    \\begin{enumerate}\n        \\item If  $|N_G(u_1)\\cap \\overline{V_1}(f)|=1$, then $u_1$ is a pendant vertex. As in the previous sub-case, this helps us reduce the measure in the second branch   by $\\omega_1+\\omega_2$ due to Reduction Rule NPD, which obviously puts us in a better branching than \\autoref{tab:branching-vectors}, nineth row. Similarly, we can discuss the case  $|N_G(u_2)\\cap \\overline{V_1}(f)|=1$.\n        \n        \\item If  $|N_G(u_1)\\cap \\overline{V_1}(f)|=2$, then we know now that the graph $G'$ induced by $\\overline{V_1}(f)\\cup \\overline{V_2}(f)$ is  bipartite after removing edges between vertices from $\\overline{V_1}(f)$,  and vertices from $\\overline{V_1}(f)$ all have degree two and vertices from $\\overline{V_2}(f)$ all have degree at least two. The worst case for the following branching is hence given by a $K_{2,2}$ as a connected component in $G'$: Testing now all possibilities of setting the $\\overline{V_2}(f)$-vertices to $0$ or to $1$ will determine all values of the $\\overline{V_1}(f)$-vertices by reduction rules. Hence, we have in particular for the  $K_{2,2}$ a scenario with four branches, and in each branch, the measure is reduced by $2\\omega_2+2\\omega_1$ \\emph{($K_{2,2}$-branching)}.\n      \n    \\end{enumerate}\n\\end{enumerate}\n\n\n\\begin{proposition}\\label{prop:run-time}On input graphs of order~$n$, \nAlgorithm \\textsc{Refined RD Enumeration} runs in time $\\mathcal{O}^*(1.9332^n)$.\n\\end{proposition}\n\n\\begin{pf}\nWe follow the run-time analysis that led us to the branching vectors listed in \\autoref{tab:branching-vectors}. The claim follows by choosing as weights $\\omega_1= \\frac{2}{3}$ and $\\omega_2=0.38488$.\n\\end{pf}\n\nThe two worst-case branchings (with the chosen weights $\\omega_1= \\frac{2}{3}$ and $\\omega_2=0.38488$) are 1.1, 3.2.b and 3.3.\nIf we want to further improve on our figures, we would have to work on a deeper analysis in these cases.\nFor the $P_2$-branching, it might be an idea to combine it with the branchings where it could ever originate from. Notice that adjacent $\\overline{V_1}$-$\\overline{V_2}$-vertices can be only produced in the first branching phase. But we would then have to improve also on Phase 3.3, the worst case being a $K_{2,2}$-branching in Case 3.3 (b).\n\n\\smallskip\\noindent\nLet us finally summarize the corner-stones of our reasoning.\n\\begin{proof}[\\autoref{thm:minimal-rdf-enumeration}]\nSeveral important properties have been claimed and proved about Algorithm~\\ref{alg:refined-enum} that show the claim of our second main theorem.\n\\begin{itemize}\n    \\item The algorithm correctly enumerates all minimal rdf; see \\autoref{cor:correct-enumeration}.\n    \\item The algorithm needs polynomial space only; see \\autoref{cor:poly-space}.\n    \\item The algorithm achieves polynomial delay; see \\autoref{prop:poly-delay}.\n    \\item The algorithm runs in time $\\mathcal{O}^*(1.9332^n)$ on input graphs of order~$n$; see \\autoref{prop:run-time}.\\qed \n\\end{itemize}\n\\end{proof}\n\n\\section{An Alternative Notion of Minimal RDF}\n\\label{sec:alternative-notion}\n\nSo far, we focused on an ordering of the functions $V\\to\\{0,1,2\\}$ that was derived from the linear ordering $0<1<2$. Due to the different functionalities, it might be not that clear if 2 should be bigger than 1.\nIf we rather choose as a basic partial ordering $0<1,2$, with $1,2$ being incomparable, this yields another ordering for the functions $V\\to\\{0,1,2\\}$, again lifted pointwise. Being reminiscent of partial orderings, let us call the resulting notion of minimality PO-minimal rdf.\nRecall that the notion of minimality for Roman dominating functions that we considered so far and that we also\nview as the most natural interpretation of this notion has been refuted in the literature, because it leads to a trivial notion of \\textsc{Upper Roman Domination}, because the minimal rdf $f:V\\to\\{0,1,2\\}$ with biggest\nsum $\\sum_{v\\in V}f(v)$ is achieved by the constant function $f=1$. This is no longer true for the (new) problem \\textsc{Upper PO-Roman Domination}. \n\nAlso, this can be seen as a natural pointwise lifting of the inclusion ordering, keeping in mind that $f\\leq_{PO}g$ iff $V_1(f)\\subseteq V_1(g)$ and $V_2(f)\\subseteq V_2(g)$.\n\nMore interesting for the storyline of this paper are the following results:\n\n\\begin{theorem}\\label{t_porperty_min_rdf}\nLet $G=\\left(V,E\\right)$ be a graph, $f: \\: V \\to \\lbrace 0,1,2\\rbrace$  \nand abbreviate\n$G'\\coloneqq G\\left[ V_0\\left(f\\right)\\cup V_2\\left(f\\right)\\right]$. Then, $f$ is a PO-minimal rdf if and only if the following conditions hold:\n\\begin{enumerate}\n\\item$N_G\\left[V_2\\left(f\\right)\\right]\\cap V_1\\left(f\\right)=\\emptyset$,\\label{con_1_2_PO}\n\\item $V_2\\left(f\\right)$ is a minimal dominating set of $G'$.\\label{con_min_dom_PO}\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{pf}\nFirst we look into the ``only if''-part. The first condition follows analogously from \\autoref{t_1_2_neigborhood}. For the other condition, we assume that there exists a graph $G=\\left(V,E\\right)$ and a PO-minimal-rdf $f: V \\to \\lbrace 0,1,2\\rbrace$ such that $V_2(f)$ is not a minimal dominating set in $G'$. Since $f$ is a rdf, $V_2(f)$ is a dominating set in~$G'$. Thus, $V_2(f)$ is not irredundant in $G'$. Hence, there exists a $v\\in V_2(f)$ such that $N[v] \\subseteq N_G[V_2(f)\\setminus \\lbrace v\\rbrace]$. Define \n$$ \\widetilde{f}:V\\to \\lbrace 0,1,2\\rbrace,\\: w\\mapsto\\begin{cases}\nf\\left(w\\right), &w\\neq v\\\\\n0,& w=v\n\\end{cases}. $$\nClearly, vertices $w\\in \\left( V_0(f) \\cup V_1(f)\\right)\\setminus N_G[v]$ are dominated by $\\widetilde{f}$. But $N_G[v]$ is also dominated, since $N_G[v] \\subseteq N_G[V_2(f)\\setminus \\lbrace v\\rbrace]$ holds. This would contradict the PO-minimality of $f$. \n\nLet $f$ be a function that fulfills the two conditions. Since $V_2\\left(f\\right)$ is a dominating set in $G'$, for each $u\\in V_0\\left( f\\right)$, there exists a $v\\in V_2\\left(f\\right)\\cap N_{G}\\left[u\\right]$. Therefore, $f$ is a rdf. \nLet $\\widetilde{f}:V \\to \\lbrace 0,1,2 \\rbrace$ be a PO-minimal rdf such that $\\widetilde{f}$ is smaller than $f$ with respect to the partial ordering. Therefore, $\\widetilde{f}$ (also) satisfies the two conditions. \nAssume that there exists a $v\\in V$ with $\\widetilde{f}\\left( v\\right) < f\\left( v \\right)$. Hence,  $V_2\\left(\\widetilde{f}\\right)\\subseteq V_2\\left(f\\right)\\setminus \\lbrace v\\rbrace$.  \n\n\\noindent\n\\textbf{Case 1:} $\\widetilde{f}\\left( v\\right)=0 ~and~ f\\left( v \\right) =1$. Therefore, there exists a vertex $u\\in N_G\\left(v\\right)$ with $f\\left(u\\right)\\geq \\widetilde{f}\\left(u\\right)=2$. This contradicts Condition~\\ref{con_1_2}.\n\n\\noindent\n\\textbf{Case 2:} $\\widetilde{f}\\left( v\\right)=0 ~and~ f\\left( v \\right) =2$. Thus, for each $u\\in V_0(f) \\cap N_G[v]\\subseteq V_0(\\widetilde{f})\\cap N_G[v]$ there exists a $w\\in V_2(\\widetilde{f})\\cap N_G(u)\\subseteq V_2(f)\\cap N_G(u)$. This implies, that $V_2(f)$ is not irredundant in $G'$, which contradicts the second condition.\n\nTherefore, $\\widetilde{f} = f$ holds and $f$ is PO-minimal.\n\\end{pf}\n\nBased on this characterization of PO-minimality, we can again derive a positive algorithmic results for the corresponding extension problem.\n\n\\begin{theorem}\\label{theorem:P_ExtPoRDF}\nThe extension problem \\textsc{ExtPO-RDF} can be solved in polynomial time.\n\\end{theorem}\n\n\\begin{pf}\nFor this problem, we have to modify Algorithm \\ref{alg} again. This time, we have to modify Line \\ref{alg_private_test} to \\textbf{if} $N_G[v]\\subseteq N[M_2\\setminus \\lbrace v\\rbrace]$ \\textbf{do}. The rest of the proof is analogous to the proof of  \\autoref{theorem:correctness_alg}.\n\\end{pf}\n\nFurthermore, we can show that for PO-minimal rdf, the simple enumeration algorithm is already provably optimal.\n\n\\begin{theorem}\nThere is a polynomial-space algorithm that enumerates all PO-minimal rdf of a given graph of order $n$ in time $\\mathcal{O}^*(2^n)$ with polynomial delay. Moreover, there is a family of graphs $G_n$, with $G_n$ being of order $n$, such that $G_n$ has $2^n$ many PO-minimal rdf.\n\\end{theorem}\n\n\\begin{pf}\nThe algorithm itself works similar to Algorithm~\\ref{alg:enum}, but we have to integrate the extension tests as in Algorithm~\\ref{alg:refined-enum}. Therefore, we need to combine our two modifications for Algorithm~\\ref{alg}. This new version would solve the \\textsc{GenExtPO-RDF}, where a graph $G=(V,E)$, a function $f:V\\to\\lbrace0,1,2\\rbrace$ and a set $\\overline{V_2}$ are given and we need to find a PO-minimal rdf $\\widetilde{f}$ with $f\\leq \\widetilde{f}$ and $\\overline{V_2}\\cap V_2\\left( f\\right)=\\emptyset$ (to prove this, combine the proofs of \\autoref{lem:GenExtRD} and \\autoref{theorem:P_ExtPoRDF}). \nTo see optimality of the enumeration algorithm, notice that the null graph (edge-less graph) of order~$n$ has any mapping $f:V\\to\\{1,2\\}$ as a PO-minimal rdf. \n\\end{pf}\n\nIt follows that the (relatively simple) enumeration algorithm is optimal for PO-minimal rdf.\nIf one dislikes the fact that our graph family is disconnected, consider the star $K_{1,n}$ that has $2^{n}+1$ many different PO-minimal rdf: If $V(K_{1,n})=\\{0,1,\\dots,n\\}$, with $0$ being the center and $i\\in\\{1,\\dots,n\\}$ being the `ray vertices' of this star, then either put $f(0)=2$ and $f(i)=0$ for $i\\in\\{1,\\dots,n\\}$, or $f(j)=1$ for $j\\in \\{0,1,\\dots,n\\}$, or $f(0)=0$ and $f(i)\\in\\{1,2\\}$ is arbitrary for  $i\\in\\{1,\\dots,n\\}$ (except for $f(j)=1$ for $j\\in \\{1,\\dots,n\\}$).\nThis example proves that there cannot be any general enumeration algorithm running in time  $\\mathcal{O}((2-\\varepsilon)^n)$ for any $\\varepsilon>0$, even for connected graphs of order~$n$.\n\n\\section{Conclusions}\n\nWhile the combinatorial concept of Roman domination leads to a number of complexity results that are completely analogous to what is known about the combinatorial concept of  domination, the two concepts lead to distinctively different results when it comes to enumeration and extension problems. These are the main messages and results of the present paper.\n\nWe are currently working on improved enumeration and also on counting of minimal rdf in special graph classes.\nOur first results are very promising; for instance, there are good chances to completely close the gap between lower and upper bounds for enumerating minimal rdf for some graph classes.\n\nAnother line of research is looking into problems that are similar to Roman domination, in order to better understand the specialties of Roman domination in contrast to the classical domination problem. What makes Roman domination behave different from classical domination when it comes to finding extensions or to enumeration?\n\nFinally, let us mention that our main branching algorithm also gives an input-sensitive enumeration algorithm for minimal Roman dominating functions in the sense of Chellali \\emph{et al.}~\\cite{CheHHHM2016}. However, we do not know of a polynomial-delay enumeration algorithm in that case. This is another interesting line of research.\nHere, the best lower bound we could find was a repetition of a $C_4$, leading to $\\sqrt[4]{8}\\geq 1.68179$ as the basis.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe study the entropy dissipation for the Boltzmann collision operator without cutoff for a wide range of power law potentials.\nThe main result of the present work gives a bound on the entropy dissipation from below by a weighted Lebesgue-Norm. \n\nThe Boltzmann equation is a nonlinear integro-differential equation which describes the dynamics of a diluted gas. \nIt is of the form\n\\begin{equation}\\label{eq:inhomogboltzmann}\n \\partial_t f(t,x,v)  + v\\cdot\\nabla_x f(t,x,v) = Q(f,f)(t,x,v), \\quad f(0,x,v) = f_0(x,v),\n \\end{equation}\nwhere $t\\geq 0$ and $x,v\\in\\mathds{R}^d$ for $d\\geq 2$. \nThe solution $f$ describes the density of particles at time $t\\geq 0$ with position $x\\in\\mathds{R}^d$ having velocity $v\\in\\mathds{R}^d$. \nWhile the left-hand side of \\eqref{eq:inhomogboltzmann} describes the transport of particles, the right-hand side takes interactions between particles into account. \n\nIn the special case of spacial homogeneity, the Boltzmann equation simplifies to\n\\begin{equation}\\label{eq:homogboltzmann}\n \\partial_t f(t,v)  = Q(f,f)(t,v), \\quad f(0,v) = f_0(v)\n \\end{equation}\n\nThe operator $Q$ on the right-hand side of the Boltzmann equation denotes the so-called Boltzmann collision operator, which acts on the function $f(t,x,\\cdot)$ for fixed values of $t,x$ and is given by\\footnote{Throughout the paper, we will drop the dependence on $t$ and $x$ of a function, whenever they are not directly involved.}\n\\[ Q(g,f)(v) = \\int_{\\mathds{R}^{d}}\\int_{\\mathds{S}^{d-1}} (g(v_{\\ast}')f(v') - g(v_{\\ast})f(v))B(|v-v_{\\ast}|,\\cos\\Theta)\\, \\d\\sigma \\d v_{\\ast},  \\]\nwhere for $v,v_{\\ast}\\in\\mathds{R}^d$, $\\sigma\\in\\mathds{S}^{d-1}$\n\\begin{center}\n\\begin{tabular}{ l l }\n$\\displaystyle v' = \\frac{v+v_{\\ast}}{2} + \\frac{|v_{\\ast}-v|}{2}\\sigma$ \\hspace*{2.5cm}  & $\\displaystyle\\cos\\Theta= \\sigma\\cdot\\frac{v_{\\ast}-v}{|v_{\\ast}-v|}$, \\\\\n$\\displaystyle v_{\\ast}'= \\frac{v+v_{\\ast}}{2} - \\frac{|v_{\\ast}-v|}{2}\\sigma.$ & \\, \n\\end{tabular}\n\\end{center}\nThe type of interactions of the particles is determined by the so-called collision kernel $B(|v-v_{\\ast}|,\\cos\\Theta)$. A classical assumption on $B$ is the integrability of the collision kernel, referred to as Grad's cutoff assumption. In this paper, we study collision kernels that do not satisfy Grad's cutoff assumption.\nTo be more precise, we consider collision kernels of the form $B(|v-v_{\\ast}|,\\cos\\Theta) = \\Phi(|v-v_{\\ast}|)b(\\cos\\Theta)$.\nWe assume\n\\begin{equation}\\label{def:Phi}\n\\Phi(|v-v_{\\ast}|) = c_{\\Phi}|v-v_{\\ast}|^{\\gamma}\n\\end{equation} \nfor some $c_{\\Phi}>0$ and $b$ satisfies\n\\begin{equation}\\label{def:b}\n c_b^{-1} \\Theta^{-1-2s} \\leq \\sin(\\Theta)^{d-2}b(\\cos\\Theta) \\leq c_b \\Theta^{-1-2s} \\quad \\text{for all } \\Theta\\in\\left(0,\\frac{\\pi}{2}\\right]\n \\end{equation}\nfor some $c_b>0$, where $\\gamma>-d$ and $s\\in(0,1)$. The case $\\gamma \\geq 0$ is referred to as the hard potential case and $\\gamma < 0$ as soft potential case. In particular, the sub-case $\\gamma+2s<0$ is known as the very-soft potential case. \n\nMany questions about the regularity of solutions are open in the very soft potential range. For a recent review and open problems, see \\cite{ImbSilReg20}. In that case, the reaction term in the collision operator (we write it $Q_2$ in \\eqref{def:Q1Q2}) is more singular. It is difficult to control it with the diffusion part of the operator. In particular, there is no known method that leads to $L^{\\infty}$ estimates in the very soft potential case, even for space homogeneous solutions. A similar difficulty arises for the Landau equation in the very soft potential range, and in particular for Coulomb potentials. Our main objective in this paper is to derive an entropy dissipation estimate that applies in the very soft potential range, similar to the well known result by L. Desvillettes \\cite{DesvillettesColoumb} for the Landau equation with Coulomb potentials. Our main estimate is in a weighted $L^p$ space, with hopefully sharp asymptotics for large velocities.\n\nThe entropy dissipation for the Boltzmann collision operator is given by the following formula\n\\begin{equation}\\label{eq:genentropydiss}\n\\begin{aligned}\nD(f) &=-\\left\\langle Q(f,f),\\ln f\\right\\rangle_{L^2(\\mathds{R}^d)}\\\\\n& = \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} f(v_{\\ast})f(v) \\left[\\ln(f(v))-\\ln(f(v'))\\right] B(|v-v_{\\ast}|,\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v.\n\\end{aligned}\n\\end{equation}\n\nThe expression for $D(f)$ applies to $f$ as a function of $v$, for each \\emph{frozen} values of $t$ and $x$. The following entropy dissipation formula applies to solutions of the Boltzmann equation.\n\\begin{equation} \\label{eqn:dissipationintegratedintime}\n\\partial_t \\iint_{\\mathds{R}^d \\times \\mathds{R}^d} f \\log f \\d v \\d x = -\\int_{\\mathds{R}^d} D(f) \\d x.\n\\end{equation}\nIn the case of space-homogeneous solutions, the formula is simpler (and more powerful) since it does not involve integration with respect to $x$.\n\nThe expression for $D(f)$ is nonnegative. Due to the difficulty to obtain any other coercive quantities associated to the Boltzmann equation, it is interesting to study lower bounds for $D(f)$ that lead to a priori estimates in standard function spaces.\n\nWith the aim of sudying estimates for $D(f)$, we consider a nonnegative function $f = f(v) : \\mathds{R}^d \\to [0,\\infty)$. There is no point in keeping track of the dependence of $f$ with respect to $t$ and $x$ since $D(f)$ applies to $f$ as a function of $v$ only. Our estimate will depend on the mass, energy and entropy of $f$. To be more precise, it depends on upper bounds for the energy and entropy, and upper and lower bounds for the mass of $f$ of the following form.\n\\begin{align}\n& 0 < m_0 \\leq \\int_{\\mathds{R}^d} f(v) \\d v \\leq M_0, \\label{ass:mass} \\\\\n& \\int_{\\mathds{R}^d} f(v) |v|^2 \\d v \\leq E_0, \\label{ass:energy} \\\\\n& \\int_{\\mathds{R}^d} f(v) \\log f(v) \\d v \\leq H_0. \\label{ass:entropy}\n\\end{align}\nFor the spatially homogeneous Boltzmann equation, due to the conservation of mass and energy, and the monotonicity of the entropy, it suffices that these inequalities hold initially for them to hold for positive time. For the space-inhomogeneous Boltzmann equation, the estimates in this paper would apply provided that the inequalities above hold for every value of $t$ and $x$.\n\nBefore, we formulate the main result of the present paper, let us recall the weighted Lebesgue Norm. For $p\\geq 1$ and $\\ell\\in\\mathds{R}$, we define\n\\[ \\|g\\|_{L^p_{\\ell}} =  \\left(\\int_{\\mathds{R}^d} \\langle v \\rangle^{\\ell p}|g(v)|^p\\, \\d v \\right)^{1\/p}, \\]\nwhere $\\langle v \\rangle = (1+|v|^2)^{1\/2}$.\n The main result of the present paper is the following entropy dissipation estimate:\n\\begin{theorem}\\label{thm:entropydissipation}\nLet $-d<\\gamma \\leq 2$ and $s\\in(0,1)$. Let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}. \nAssuming $\\gamma \\leq 0$, there is a finite constant $c>0$, depending on $d$, $b$, $\\gamma$, $s$ and the macroscopic bounds $m_0,M_0,E_0$ and $H_0$, and $C>0$ depending on $d$, $b$, $\\gamma$ and $s$ only, such that\n\\begin{equation}\\label{eq:mainresult}\n  D(f) \\geq c\\|f\\|_{L^p_{-q}} - C M_0^2,\n\\end{equation} \nwhere $1\/p = 1-2s\/d$ and $q=2s\/d - \\gamma-2s$.\n\nWhen $\\gamma > 0$, a similar estimate follows but depending on a higher moment instead of $M_0$.\n\\begin{equation}\\label{eq:mainresult-hardpotential}\n  D(f) \\geq c\\|f\\|_{L^p_{-q}} - C \\left( \\int_{\\mathds{R}^d} \\langle v \\rangle^\\gamma f \\d v \\right)^2.\n\\end{equation} \n\\end{theorem}\nThe proof of \\autoref{thm:entropydissipation} takes advantage of the simple idea of using the non-negativity of the integrand in the entropy dissipation and replace the kinetic factor $\\Phi$ by a smaller bounded function $\\psi$ without the singularity on $v=v_{\\ast}$ for $\\gamma<0$. The estimate in a weighted $L^p$ space, with a precise exponent, follows from an explicit formula for integral quadratic forms. This estimate, given in Proposition \\ref{prop:quadratictoLebesgue}, is one of the main novelties of this paper.\n\nAs a corollary of \\autoref{thm:entropydissipation}, we see that H-solutions to the Cauchy problem \\eqref{eq:homogboltzmann} are in a weighted Lebesgue space, that is \n\\[ f\\in L^{1}\\left([0,T], L^p_{-q}(\\mathds{R}^d)\\right),\\]\n where $p$ and $q$ are as in \\autoref{thm:entropydissipation}. This implies in particular that H-solutions to \\eqref{eq:homogboltzmann} are weak solutions in the usual sense. For details, see Section \\ref{sec:weaksol}.\n\nThe Landau equation can be derived as the grazing collision limit of the Boltzmann equation, see e.g. \\cite{desvillettes1992asymptotics, goudon1997boltzmann, Villaninewclass, AlexandreVillani2002boltzmann, alexandre2004landau} and the references therein. Precisely, the Boltzmann collision operator $Q(f,f)$, properly normalized, converges to the Landau operator as $s \\to 1$. In \\cite{DesvillettesColoumb}, Desvillettes proves an entropy dissipation estimate for the Landau equation with Coulomb interaction and presents applications to weak solutions. The estimate we obtain in \\autoref{thm:entropydissipation} is an analogous result but for the Boltzmann collision operator. If we take $d=3$ and $\\gamma=-3$, and take $s \\to 1$ in Theorem \\ref{thm:entropydissipation}, we see that $p\\to 3$ and $q\\to 5\/3$. While, the exponent $p$ coincides with the one in \\cite{DesvillettesColoumb}, our exponent in the weight $q$ is improved ($L^3_{-5\/3}$ as opposed to $L^3_{-3}$), suggesting that the weight exponent in \\cite{DesvillettesColoumb} may not be optimal. It is worth noting that the proof given in \\cite{DesvillettesColoumb} cannot be applied to the Boltzmann collision operator. The method in this paper is related to a simpler proof presented in \\cite{golse2019partial}.\n\nThe entropy dissipation is an important quantity in the analysis of the Boltzmann equation. It has various applications such as the construction of renormalized solutions to the Cauchy problem for the Boltzmann equation, see \\cite{DiPernaLionsCauchy}, or the introduction of H-solutions for the Boltzmann equation and Landau equations, see \\cite{Villaninewclass}. There are various lower bounds for the entropy dissipation in the literature. However, they have serious limitations when $\\gamma<0$ that we seek to overcome with the present result.\n\nOne of the first and best known entropy dissipation estimates appeared in \\cite{AlexDesvVilWenn}. Their analysis applies when $\\Phi$ is bounded below which, strictly speaking, is only the case when $\\gamma = 0$. For $\\gamma < 0$, some further analysis in \\cite{AlexDesvVilWenn} leads to \\emph{local} estimates restricted to bounded values of $v$. For other works on entropy dissipation estimates and their applications, see for instance \\cite{VillaniReg, AlexandreVillani2002boltzmann, Des03Fourier, desvillettes2005trend, mouhot2006explicit} and the references therein.  \nIn \\cite{Gressmann-Strain-Global2010, GressmanStrainGlobal}, Gressmann and Strain introduce a metric which captures the anisotropic structure of the Boltzmann operator. Using this metric and the associated spaces, the same authors obtain entropy dissipation estimates in \\cite{Gressmann-Strain-2011} with sharp asymptotics for large velocities. The estimates in \\cite{Gressmann-Strain-2011} depend on a quantity (that the authors call $C_g$) that is only controlled by moments of $f$ when $\\gamma \\geq 0$ (the hard potentials case). \n\nIn addition of our main result, we present a refinement of the entropy dissipation result of \\cite[Theorem 3]{Gressmann-Strain-2011} so that it applies to the soft potential range without resorting to higher integrability assumptions on $f$. We recall the anisotropic fractional Sobolev norm of Gressman and Strain:\n\\begin{equation}\\label{def:weightedanisoSobolev}\n |f|_{{\\dot{N}^{s,\\gamma}}}^2 = \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\frac{(f(v')-f(v))^2}{d_{GS}(v,v')^{d+2s}} \\left( \\langle v \\rangle \\langle v' \\rangle \\right)^{(\\gamma+2s+1)\/2} \\mathds{1}_{\\{d_{GS}(v,v')\\leq 1\\}} \\d v' \\d v,\n\\end{equation}\nwhere  \n\\begin{equation}\\label{eq:GSdist}\n d_{GS}(v,v') = \\sqrt{|v-v'|^2+\\frac{1}{4}(|v|^2-|v'|^2)^2}\n \\end{equation}\nmeasures the distance in the lifted paraboloid $\\{v\\in\\mathds{R}^{d+1}\\colon v_{d+1} = \\frac12|(v_1,\\dots,v_d)|^2 \\}$.\n\nWe derive the following entropy dissipation estimate for soft potentials.\n\n\\begin{proposition}\\label{prop:entropydissipation}\nLet $-d<\\gamma \\leq 0$ and $s\\in(0,1)$. Let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}.\nThere is a constant $c>0$ depending on $d$, $s$, $\\gamma$ and the macroscopic bounds $m_0,M_0,E_0$ and $H_0$, and a constant $C$ depending only on $d$, $s$ and $\\gamma$ only, such that\n\\[ D(f) \\geq c |\\sqrt{f}|_{\\dot{N}^{s,\\gamma}}^2 - C M_0^2. \\]\n\\end{proposition}\n\nThe novelty of Proposition \\ref{prop:entropydissipation} compared with \\cite[Theorem 3]{Gressmann-Strain-2011} is that our negative error term is in terms of the mass of $f$ only. The result in the cited paper has a negative error term depending on higher integrability assumptions on the function. Roughly, they require $f \\ast |v|^\\gamma$ to be locally bounded in \\cite[Assumption U]{Gressmann-Strain-2011}. There is no apparent upper bound in terms of the hydrodynamic quantities for their parameter $C_g$ when $\\gamma<0$. Proposition \\ref{prop:entropydissipation} implies that weak solutions to the space-homogeneous Boltzmann equation belong to $L^1([0,T], N^{s,\\gamma})$, which was not available from earlier results in the literature.\n\nWhile it is conceivable that one could potentially derive our main result in Theorem \\ref{thm:entropydissipation} from Proposition \\ref{prop:entropydissipation} combined with some sharp form of a weighted fractional Sobolev inequality (not readily available in the literature), we chose to prove Theorem \\ref{thm:entropydissipation} directly, and then present an independent proof of Proposition \\ref{prop:entropydissipation}. The direct proof of Theorem \\ref{thm:entropydissipation} is relatively short and elegant. So, we think it is worth presenting Theorem \\ref{thm:entropydissipation} as an independent result.\n\n\\subsection*{Notation}We write $a\\lesssim b$ if there is a universal constant $c>0$ such that $a\\leq cb$. The notation $a\\gtrsim b$ means that $b\\lesssim a$ and $a \\approx b$ that $a\\lesssim b$ and $a\\gtrsim b$.\n\\section{Entropy dissipation estimates}\nThe Boltzmann collision operator clearly plays a central role in our analysis, since the entropy dissipation is defined through it. In the following, we will briefly discuss the decomposition of the operator and some selected properties. \n\nThe Boltzmann collision operator $Q(f,g)$ can be decomposed into the sum of an integro-differential operator $Q_1(f,g)$ and a lower order term $Q_2(f,g)$ (see \\cite{SilvestreBoltzmann}):\n\\begin{equation}\\label{def:Q1Q2}\nQ_1(f,g)(v) := (\\mathcal{L}_{K_f}g)(v) \\quad \\text{and} \\quad Q_2(f,g)(v) = g(v)\\int_{\\mathds{R}^d}f(v-w)\\widetilde{B}(|w|)\\, \\d w.\n\\end{equation}\nThe integro-differential operator $\\mathcal{L}_{K_f}$ is defined by\n\\[ \\mathcal{L}_{K_f}g(t,v) = pv\\int_{\\mathds{R}^d} (g(t,v')-g(t,v))K_{f}(t,v,v')\\, \\d v',\\]\nwhere pv denotes the Cauchy principal value around $v\\in\\mathds{R}^d$. The kernel $K_f(t,v,v')$ depends on the \nfunction $f$ and is given by the formula\n\\begin{equation}\\label{def:kernel}\n\\begin{aligned}\n K_f(t,v,v') &= \\frac{2^{d-1}}{|v'-v|}\\int_{w\\perp (v'-v)} f(t,v+w) \\, B(r,\\cos\\Theta) r^{-d+2}\\, \\d w,\n \\end{aligned}\n \\end{equation}\nwhere\n \\begin{equation}\\label{def:rvwreps}\n\\begin{aligned}\nr&=\\sqrt{|v'-v|^2+|w|^2}, \\qquad & \\cos(\\Theta\/2)= \\frac{|w|}{r}, \\\\\nv_{\\ast}'&= v+w, & v_{\\ast}= v'+w.\n\\end{aligned}\n\\end{equation}\nWhile $Q_1$ represents the singular part of the collision operator, the part $Q_2$ is of lower order. The lower order term can be handled using the cancellation lemma \\cite[Lemma 1]{AlexDesvVilWenn}.\nThe function $\\widetilde{B}$ in \\eqref{def:Q1Q2} is given by\n\\begin{align*}\n\\widetilde{B}(z)& =\\int_{\\mathds{S}^{d-1}}(2^{d\/2}(1-\\sigma\\cdot e)^{-d\/2}\\left(B\\left(\\sqrt{2} z\/(1-\\sigma\\cdot e),\\cos\\Theta\\right) -B(z,\\cos\\Theta)\\right) \\, \\d \\sigma\\\\\n& =C_b \\Phi(z)=C_bc_{\\Phi}|z|^{\\gamma},\n\\end{align*} \nwhere $C_b$ is a positive constant depending on the angular function $b$. For details on the decomposition of the Boltzmann collision operator, see \\cite{SilvestreBoltzmann}.\n\nThe idea of controlling the singularity of the operator near $v=v_{\\ast}$ for $\\gamma<0$ is to introduce an auxiliary collision kernel, where we replace the kinetic factor $\\Phi$ by a smaller bounded function $\\psi$ in which we cut out this singularity. \nSince the integrand of the entropy dissipation is non-negative, it can be bounded from below by the same expression with $B$ replaced by a smaller collision kernel.\nFor a given function $\\psi$ on $\\mathds{R}^d$, we define the generalized collision kernel $B^{\\psi}$ by\n\\[ B^{\\psi}(|v-v_{\\ast}|,\\cos\\Theta) = \\psi(|v-v_{\\ast}|)b(\\cos\\Theta). \\]\nThe auxiliary collision kernel $B^{\\psi}$ and the collision kernel $B$ differ only in the fact that we replace the kinetic factor $\\Phi$ with the function $\\psi$. \n\nBy simply replacing $\\Phi$ by $\\psi$,  we can define the generalized kernel $K_f^{\\psi}$ by\n\\begin{equation}\\label{def:genkernel}\n \\mathcal{K}_f^{\\psi}(t,v,v') = \\frac{2^{d-1}}{|v'-v|}\\int_{w\\perp (v'-v)} f(t,v'+w) \\, \\psi(r) b(\\cos\\Theta) r^{-d+2 }\\, \\d w,\n \\end{equation}\nwhere $r$ and $\\cos\\Theta$ are defined in \\eqref{def:rvwreps} and $\\widetilde{B}^{\\psi}$ by\n\\begin{equation}\\label{def:genB}\n\\widetilde{B}^{\\psi}(z) = C_b \\psi(z).\n \\end{equation}\nThis leads to the decomposition of the auxiliary collision operator $Q^{\\psi}(f,g)$ (with the collision kernel $B$ replaced by $B^{\\psi}$) into the sum of an integro-differential operator $Q^{\\psi}_1(f,g)$ and a lower order term $Q^{\\psi}_2(f,g)$, where the operators $Q_1^\\psi$ and $Q_2^\\psi$ \nare defined as in \\eqref{def:Q1Q2} with $K_f$ and $\\widetilde{B}$ replaced by $K_f^{\\psi}$ resp. $\\widetilde{B}^{\\psi}$.\n\nWe define the auxiliary function $\\psi:\\mathds{R}^d\\to\\mathds{R}$ to be non-negative function satisfying $\\psi\\leq \\Phi$ and:\n\\begin{equation}\\label{def:psi}\n\\begin{aligned}\n\\text{if } \\gamma<0:  \\quad & \\begin{cases}1\\leq \\psi(|z|) \\leq 2 \\quad &\\text{if } |z|\\leq 1, \\\\ \\psi(|z|)=\\Phi(|z|) &  \\text{if } |z|> 1,\n\\end{cases} \\\\\n\\text{if } \\gamma \\geq  0:  \\quad & \\psi(|z|) =\\Phi(|z|) \\quad \\text{for all } z\\in\\mathds{R}^d.\n\\end{aligned}\n\\end{equation}\nNote that by this choice, for any value of $\\gamma$,\n\\[ \\int_{\\mathds{R}^d} \\int_{\\mathds{R}^d} f(v) f(v_{\\ast}) \\psi(|v-v_{\\ast}|) \\, \\d v_{\\ast}\\, \\d v \\lesssim \\begin{cases} \nM_0^2 & \\text{ if } \\gamma \\leq 0, \\\\\n\\left( \\int f \\langle v \\rangle^\\gamma \\d v \\right)^2 & \\text{ if } \\gamma > 0.\n\\end{cases} \\]\nThe entropy dissipation is naturally connected to a quadratic form, coming from the singular part of the collision operator and a lower order term.\n\\begin{lemma}\\label{lem:entropuquadratic}\nLet $-d<\\gamma\\leq 2$ and $s\\in(0,1)$. Let $\\psi$ be a non-negative function satisfying $\\psi\\leq \\Phi$ and \\eqref{def:psi} and let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}. \nThere is an universal constant $C>0$ such that\n\\begin{equation}\nD(f) \\geq \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left(\\sqrt{f(v')} - \\sqrt{f(v)}\\right)^2 K^{\\psi}_f(v,v') \\, \\d v' \\, \\d v  - C \\left( \\int_{\\mathds{R}^d} f(v) \\langle v \\rangle^{\\gamma_+} \\d v \\right)^2.\n\\end{equation}\n\\end{lemma}\n\\begin{proof} \nUsing the non-negativity of the entropy dissipation $D(f)$ and $\\psi\\leq \\Phi$, we get\n\\begin{align*}\nD(f) & = \\frac 12 \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} (ff_\\ast - f' f'_\\ast) \\ln\\left(\\frac {ff_\\ast}{f'f'_\\ast} \\right) B(|v-v_{\\ast}|,\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& \\geq \\frac 12 \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} (ff_\\ast - f' f'_\\ast) \\ln\\left(\\frac {ff_\\ast}{f'f'_\\ast} \\right) \\psi(|v-v_{\\ast}|) b(\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& = \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} f(v_{\\ast})f(v) \\left[\\ln(f(v))-\\ln(f(v'))\\right] \\psi(|v-v_{\\ast}|) b(\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& \\geq \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} f(v_{\\ast}) \\left(\\sqrt{f(v')} - \\sqrt{f(v)}\\right)^2\\psi(|v-v_{\\ast}|) b(\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& \\quad -\\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} f(v_{\\ast}) \\left(f(v') - f(v)\\right)\\psi(|v-v_{\\ast}|) b(\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& = I_1 - I_2.\n\\end{align*}\nIn the second estimate, we used the inequality $x(\\ln x - \\ln y) \\geq \\left(\\sqrt{y} - \\sqrt{x}\\right)^2 - (y-x)$ for all $x,y\\geq 0$ (See \\cite{AlexDesvVilWenn}).\nBy the definition of the kernel $K_f^{\\psi}(v,v')$, the term $I_1$ can be written as\n\\[ I_1 = \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left(\\sqrt{f(v')} - \\sqrt{f(v)}\\right)^2 K_f^{\\psi}(v,v') \\, \\d v' \\, \\d v. \\]\nFor the term $I_2$  we use the cancellation lemma (See \\cite{AlexDesvVilWenn} or \\cite{SilvestreBoltzmann} for details), which leads to\n\\begin{align*}\nI_2  &= C \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} f(v)f(v_{\\ast}) \\psi(|v-v_{\\ast}|) \\d v_{\\ast} \\d v \\\\ \n&\\leq C \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} f(v)f(v_{\\ast}) \\langle v-v_{\\ast} \\rangle^\\gamma \\d v_{\\ast} \\d v \\\\ \n &\\leq C \\begin{cases} \nM_0^2 & \\text{ if } \\gamma \\leq 0, \\\\\n\\left( \\int f \\langle v \\rangle^\\gamma \\d v \\right)^2 & \\text{ if } \\gamma > 0.\n\\end{cases}\n\\end{align*}\nHere $C>0$ is a finite universal constant.\n\\end{proof}\n\nOur main result \\autoref{thm:entropydissipation} follows immediately from \\autoref{lem:entropuquadratic} and an estimate of the quadratic form from below by a weighted Lebesgue norm.\n\n\\begin{proposition}\\label{prop:quadratictoLebesgue}\nLet $\\gamma>-d$ and $s\\in(0,1)$. Let $\\psi$ be a non-negative function satisfying \\eqref{def:psi}, let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy} and $g \\in L_2^1(\\mathds{R}^d)$. \nThere is a constant $c>0$, depending on the dimension $d$ and the macroscopic bounds $m_0,M_0,E_0$ and $H_0$, such that\n\\[  \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left(\\sqrt{g(v')} - \\sqrt{g(v)}\\right)^2 K^{\\psi}_f(v,v') \\, \\d v' \\, \\d v \\geq c \\|g\\|_{L^p_{-q}}, \\]\nwhere $1\/p = 1-2s\/d$ and $q=2s\/d - \\gamma-2s$. In particular, $g \\in L^p_{-q}(\\mathds{R}^d)$ when the left hand side is bounded.\n\\end{proposition}\nThe proof of this proposition needs some auxiliary results and is given at the end of this section.\n\nSince the estimate in \\autoref{prop:quadratictoLebesgue} has no restrictions on $\\gamma>-d$ and $s\\in(0,1)$, \nit covers soft as well as hard potentials for the Boltzmann collision operator. It is perhaps most interesting that it works in the case of very-soft potentials $\\gamma+2s<0$. Note that outside of that range, if $\\gamma+2s>2s\/d$, the exponent $q$ changes its sign.\n\nAn essential tool for the proof of \\autoref{prop:quadratictoLebesgue} are cones of nondegeneracy introduced in \\cite{SilvestreBoltzmann}.\nBefore we recall the cones of nondegeneracy and some important properties, we first give a lower bound on the generalized kernel.\n\\begin{lemma}\\label{lemma:kernelpsi}\nLet $-d<\\gamma<0$ and $s\\in(0,1)$. Let $\\psi$ be a non-negative function satisfying \\eqref{def:psi}.\nThen \n\\begin{equation}\\label{Kernelpsi}\n\\begin{aligned}\nK^{\\psi}_f(v,v') \\gtrsim |v-v'|^{-d-2s}\\int_{w\\perp (v'-v)} f(v'+w)  \\min\\Big( |w|^{\\gamma+2s+1}, |w|^{2s+1}\\Big) \\, \\d w.\n \\end{aligned}\n\\end{equation}\n\\end{lemma}\nNote that in the hard potential case $\\gamma\\geq 0$, the auxiliary kernel $K_f^{\\psi}$ coincides with the kernel $K_f$ and therefore, there is nothing to do in this case. The respective result is given in \\cite[Corollary 4.2]{SilvestreBoltzmann}, namely\n\\begin{equation}\\label{eq:lowerboundKf}\nK_f(v,v') \\gtrsim |v-v'|^{-d-2s}\\int_{w\\perp (v'-v)} f(v'+w)   |w|^{\\gamma+2s+1} \\, \\d w,\n\\end{equation} \nwhich provides a better lower bound. Nevertheless, the estimate \\eqref{Kernelpsi} is sufficient for our applications in the case of soft potentials.\n\\begin{proof}[Proof of \\autoref{lemma:kernelpsi}]\nAs in the proof of \\cite[Corollary 4.2]{SilvestreBoltzmann}, we study the two  cases $\\cos\\Theta\\geq 0$ and $\\cos\\Theta< 0$. \n\\begin{enumerate}\n\\item[(i)] If $\\cos\\Theta\\geq 0$, then $|w|\\approx r$ and $b(\\cos\\Theta)\\approx |v-v'|^{-d+1-2s}r^{d-1+2s}$. Hence, \n\\begin{align*} \n\\psi(r) b(\\cos\\Theta) r^{-d+2 } &\\approx |v-v'|^{-d-2s+1}\\left( |w|^{1+2s}\\mathds{1}_{\\{r\\leq 1\\}}(r) + |w|^{1+2s+\\gamma}\\mathds{1}_{\\{r> 1\\}}(r)\\right) \\\\\n&\\geq  |v-v'|^{-d-2s+1}\\min\\Big( |w|^{\\gamma+2s+1}, |w|^{2s+1}\\Big).\n\\end{align*}\n\\item[(ii)] In the case $\\cos\\Theta<0$, we have $|v'-v|\\approx r$ and $|w|=r\\cos(\\Theta\/2)$. \nTherefore, $b(\\cos\\Theta) = \\cos(\\Theta\/2)^{\\gamma+2s+1}$. If $r\\leq 1$, then \n\\begin{align*}\n\\psi(r) b(\\cos\\Theta) r^{-d+2} &\\approx r^{-d-2s+1} |w|^{\\gamma+2s+1} \\\\\n&\\approx  |v'-v|^{-d-2s+1} |w|^{\\gamma+2s+1} \\\\\n&\\geq  |v-v'|^{-d-2s+1}\\min\\Big( |w|^{\\gamma+2s+1}, |w|^{2s+1}\\Big).\n\\end{align*}\nOn the other hand, if $r\\geq 1$, we have\n\\begin{align*}\n\\psi(r) b(\\cos\\Theta) r^{-d+2} &\\approx r^{-d+2} \\cos(\\Theta\/2)^{\\gamma+2s+1}r^{\\gamma} \\\\\n&\\approx |v-v'|^{-d-2s+1}|w|^{2s+1}\\cos(\\Theta\/2)^{\\gamma}r^{\\gamma}  \\\\\n&\\approx |v-v'|^{-d-2s+1}|w|^{\\gamma+2s+1} \\\\\n& \\geq |v-v'|^{-d-2s+1}\\min\\Big( |w|^{\\gamma+2s+1}, |w|^{2s+1}\\Big).\n\\end{align*}\n\\end{enumerate}\nThis finishes the proof of \\autoref{lemma:kernelpsi}.\n\\end{proof}\nNote that by the bound on the mass and energy, a certain amount of the mass of the function $f$ lies inside a ball centered around zero with radius depending on $m_0$ and $E_0$. The bound on the entropy $H_0$ provides that this mass in not concentrated on a set of measure zero.\nThese observations lead to cones of nondegeneracy constructed by sets of the form $\\{f\\geq \\ell\\}$. To be more precise, for any point $v\\in\\mathds{R}^d$, there is a symmetric cone of directions $A(v)$ such that its perpendicular planes intersect the set $\\{f\\geq \\ell\\}$ on a set with $\\mathcal{H}^{d-1}$ positive Hausdorff measure. \nAs a consequence of \\autoref{lemma:kernelphi} resp. \\eqref{eq:lowerboundKf}, \nwe get the existence of a cone of non-degeneracy for the kernel $K^{\\psi}_f$. \nHere, one can simply follow the lines of the proof of \\cite[Lemma 4.8 and Lemma 7.1]{SilvestreBoltzmann} and use the bound on the kernel $K_f^{\\psi}$ given in Lemma \\ref{lemma:kernelpsi}\n\\begin{lemma}{\\cite[Lemma 7.1]{SilvestreBoltzmann}}\\label{lemma:kernelphi}\nLet $\\gamma>-d$ and $s\\in(0,1)$.  Let $\\psi$ be a non-negative function satisfying \\eqref{def:psi}\nand let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}.\nFor any $v\\in\\mathds{R}^d$, there exists a symmetric subset $A(v)\\subset\\mathds{S}^{d-1}$ such that \n\\begin{enumerate}\n\\item[(i)] $\\displaystyle |A(v)|>\\mu\\langle v \\rangle^{-1}$, where $|A(v)|$ denotes the $d-1$-Hausdorff measure of $A(v)$,\n\\item[(ii)]  $\\displaystyle  K^{\\psi}_f(v,v')\\geq \\lambda \\langle v \\rangle^{1+2s+\\gamma}|v-v'|^{-d-2s}$ whenever $(v'-v)\/(|v'-v|)\\in A(v)$.\n\\item[(iii)] For every $\\sigma\\in A(v)$, $|\\sigma\\cdot v| \\leq C$.\n\\end{enumerate}\nThe constants $\\mu$, $\\lambda$ and $C$ depend on $d$ and on the hydrodynamic bounds $m_0, M_0, E_0$ and $H_0$.\n\\end{lemma}\nThe set $A(v)$ in the previous lemma describes a set of directions $A(v)$ along which the kernel $K^{\\psi}_f$ has a lower bounds given in property (ii). We denote the corresponding cone of nondegeneracy by $\\Xi(v)$ that is\n\\[ \\Xi(v) := \\left\\{  v' \\in\\mathds{R}^d\\colon \\frac{(v'-v)}{|v'-v|}\\in A(v)\\right\\}. \\]\nFurthermore, the cone of nondegeneracy degenerates as $|v|\\to\\infty$ and satisfies\n\\[ |B_R(v) \\cap \\Xi(v)| \\approx R^d \\langle v \\rangle^{-1} . \\]\n\nUsing these cones of nondegeneracy, we can finally prove \\autoref{prop:quadratictoLebesgue}.\n\\begin{proof}[Proof of \\autoref{prop:quadratictoLebesgue}]\nWe initially assume that $g \\in L^p_{-q}(\\mathds{R}^d)$ and prove the estimate. A posteriori, we prove that the quadratic form must be infinite if $g \\notin L^p_{-q}(\\mathds{R}^d)$ by an approximation procedure using that $g \\in L^1_2(\\mathds{R}^d)$.\n\nRecall that by \\autoref{lemma:kernelphi} for every $v\\in\\mathds{R}^d$, we know that $K^{\\psi}_f(v,v') \\geq \\lambda \\langle v \\rangle^{1+2s+\\gamma}|v-v'|^{-d-2s}$, whenever $v'\\in \\Xi(v)$. \nFurthermore, for every $v\\in\\mathds{R}^d$, let us choose $R=R(v)>0$ so that for some large $C>0$,\n\\[ g(v)^p R^d \\langle v\\rangle^{-qp-1} \\approx C\\|g\\|_{L^p_{-q}}^p. \\]\nWith this choice,\n\\[\\{v'\\in (B_R(v)\\cap \\Xi(v))\\colon g(v') \\leq g(v)\/2\\}| \\gtrsim  |B_R(v)\\cap \\Xi(v)| \\approx R^d \\langle v \\rangle^{-1}.\t\\]\nHence, using this information, we get\n\\begin{align*}\n\\int_{\\mathds{R}^d} \\left( \\sqrt{g(v')} - \\sqrt{g(v)} \\right)^2K_f^{\\psi}(v,v')\\, \\d v'  & \\gtrsim R^d \\langle v \\rangle^{-1} g(v) \\left(\\lambda \\langle v\\rangle^{1+2s+\\gamma}R^{-d-2s} \\right) \\\\\n& \\geq c_1 R^{-2s} g(v) \\langle v \\rangle^{\\gamma+2s}  \\\\\n& = c_2 \\|g\\|_{L^p_{-q}}^{-2sp\/d} g(v)^{1+2sp\/d}\\langle v \\rangle^{\\gamma+2s-2s(qp+1)\/d},\n\\end{align*}\nwhere $c_1,c_2$ are constants depending on $m_0, M_0, E_0, H_0$.\nOur choice of $p$ and $q$ was made so that $p=1+2sp\/d$ and $-qp =\\gamma+2s-2s(qp+1)\/d$. \nIntegrating over $v\\in\\mathds{R}^d$ finally gives us\n\\[ \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left( \\sqrt{g(v')} - \\sqrt{g(v)} \\right)^2K_f^{\\psi}(v,v')\\, \\d v' \\d v   \\geq c\\|g\\|_{L^p_{-q}}^{-2sp\/d} \\|g\\|_{L^p_{-q}}^p = c\\|g\\|_{L^p_{-q}}. \n\\]\nThis concludes the proof provided that $g \\in L^p_{-q}$. For a function $g \\notin L^p_{-q}$, we consider $g_m(v) := \\max(\\min(g(v),m),-m)$. Since we assumed that $g \\in L^1_2(\\mathds{R}^d)$, we have $g_m \\in L^p_{-q}$ and the inequality holds. Moreover, $\\|g_m\\|_{L^p_{-q}} \\to \\infty$ and therefore, applying the monotone convergence theorem,\n\\begin{align*}\n\\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} &\\left( \\sqrt{g(v')} - \\sqrt{g(v)} \\right)^2K_f^{\\psi}(v,v')\\, \\d v' \\d v \\geq \\\\\n& \\geq \\lim_{m \\to \\infty} \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left( \\sqrt{g_m(v')} - \\sqrt{g_m(v)} \\right)^2K_f^{\\psi}(v,v')\\, \\d v' \\d v \\\\\n& \\geq \\lim_{m \\to \\infty} c \\|g_m\\|_{L^p_{-q}} = +\\infty. \n\\end{align*}\n\\end{proof}\nAs already mentioned, the main result \\autoref{thm:entropydissipation} now follows from a combination of \\autoref{lem:entropuquadratic} and \\autoref{prop:quadratictoLebesgue}.\n\n\\begin{remark}\nEven though we asumme $g \\in L^1_2$ in Proposition \\ref{prop:quadratictoLebesgue}, the estimate does not depend on $\\|g\\|_{L^1_2}$. There are various possible ways to relax this assumption, but it cannot be removed altogether. It is there to rule out the obvious counterexample of $g$ being a nonzero constant function for which the quadratic form vanishes. It can be replaced by any condition that allows some form of the final approximation procedure for the case $g \\notin L^p_{-q}$.\n\\end{remark}\n\n\\subsection{Entropy dissipation estimate involving an anisotropic fractional Sobolev space}\nIn this subsection, we present a second entropy dissipation estimate for the Boltzmann collision operator. \nThis estimate involves the anisotropic distance by Gressmann and Strain \\cite{Gressmann-Strain-2011}.\n\nLet us first briefly recall the sharp anisotropic coercivity estimate for the Boltzmann collision operator from \\cite{Gressmann-Strain-2011}.\nNote that $\\left\\langle Q(g,f),f\\right\\rangle$ can be rewritten as\n\\begin{equation}\\label{eq:NgKg}\n\\begin{aligned}\n\\left\\langle Q(g,f),f\\right\\rangle &= \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} g(v_{\\ast})f(v) \\left[f(v')-f(v)\\right] B(r,\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& = \\frac12 \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} g(v_{\\ast})\\left[f(v')^2 -f(v)^2\\right] B(r,\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v \\\\\n& \\quad \\qquad -\\frac12 \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d}\\int_{\\mathds{S}^{d-1}} g(v_{\\ast})\\left[f(v')-f(v)\\right]^2 B(r,\\cos\\Theta)\\d\\sigma \\d v_{\\ast} \\d v\\\\\n& =:K_g(f) - N_g(f).\n\\end{aligned}\n\\end{equation}\n\nIn \\cite[Theorem 1]{Gressmann-Strain-2011}, the authors prove that under mild assumptions on the function $g$, the term $N_g(f)$ can be bounded from below the weighted anisotropic Sobolev semi-norm\n$|f|_{{\\dot{N}^{s,\\gamma}}}^2$ defined in \\eqref{def:weightedanisoSobolev}. This estimate provides an entropy dissipation estimate in terms of the anisotropic fractional Sobolev space. However, the result depends on certain parameter $C_q$ that would be difficult to bound when $\\gamma<0$. In this section we prove Proposition \\ref{prop:entropydissipation}, which is effectively a refinement of \\cite[Theorem 1]{Gressmann-Strain-2011} in the soft potential range.\n\n\nIn \\cite{imbertsilvestreglobal2022}, Imbert and Silvestre introduce a change of variables, which is used to turn local H\u00f6lder and Schauder estimates into global ones. For $v_0\\in\\mathds{R}^d$ let \n\\[ \\overline{v} :=  \\begin{cases} v_0 + v  \\quad & \\text{if } |v_0|<2, \\\\ v_0+T_0v &  \\text{if } |v_0|\\geq 2,\\end{cases} \\]\n where\n \\begin{equation}\\label{def:T0}\n  T_0(av_0+w) = \\frac{a}{|v_0|}v_0 + w, \\qquad \\text{for all } w\\perp v_0, a\\in\\mathds{R}. \n \\end{equation}\n The function $T_0:\\mathds{R}^d\\to\\mathds{R}^d$, introduced in \\cite{imbertsilvestreglobal2022}, has a strong connection to the anisotropic distance \n\\eqref{eq:GSdist}.\nIn \\cite[Lemma A.1]{imbertsilvestreglobal2022} it is shown that for any given $v_0\\in\\mathds{R}^d$ with $|v_0|\\geq 2$, we have \n\\begin{equation}\\label{eq:comparabilitymetric}\nd_{GS}(v_1,v_2) \\asymp |T_0^{-1}(v_1-v_2)|\n\\end{equation}\nfor all $v_1,v_2\\in E_1(v_0) := v_0+T_0(B_1)$. \nWe define\n\\begin{equation}\\label{def:Kbar}\n\\overline{K}^{\\psi}_f(v,v') = |v_0|^{-1-\\gamma-2s}K^{\\psi}_f(\\overline{v}, v_0+T_0v'),\n\\end{equation}\nwhere $\\psi$ is the auxiliary function $\\psi:\\mathds{R}^d\\to\\mathds{R}$ defined in\n\\eqref{def:psi}. \nIn \\cite{imbertsilvestreglobal2022}, the authors derive the global coercivity estimate by Gressmann and Strain by using the above mentioned transformation and a local coercivity estimate. \nBy using similar methods, we are able to prove \\autoref{prop:entropydissipation}.\nBefore we draw our attention to the proof of \\autoref{prop:entropydissipation}, we need some auxiliary results.\nLet us first state a local coercivity estimate.\n\\begin{lemma}\\label{lemma:localcoercivity}\nLet $\\gamma>-d$ and $s\\in(0,1)$. Let $\\psi$ be a non-negative function satisfying $\\psi\\leq \\Phi$ and \\eqref{def:psi} and let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}. \nThere is a constant $\\lambda>0$, depending on the macroscopic bounds $m_0, M_0, E_0$ and $H_0$, such that for every $g:\\mathds{R}^d\\to\\mathds{R}$\n\\[ \\int_{B_1}\\int_{B_1} \\left(g(v') - g(v)\\right)^2 \\overline{K}^{\\psi}_f(v,v') \\d v' \\d v \\geq \\lambda \\int_{B_{1\/2}}\\int_{B_{1\/2}} \\frac{\\left(g(v') - g(v)\\right)^2}{|v-v'|^{d+2s}} \\d v' \\d v. \\]\n\\end{lemma}\n\\begin{proof} \nBy \\autoref{lemma:kernelphi}, there is a cone of non-degeneracy for the kernel $K^{\\psi}_f$. Hence, following the lines of the proof of \\cite[Lemma 5.6]{imbertsilvestreglobal2022}, there is also a cone of non-degeneracy for $\\overline{K}^{\\psi}_f$. \nNow the result follows from the coercivity condition \\cite[Theorem 1.3]{chakersilvestrecoerc}.\n\\end{proof}\nLet $v_0\\in\\mathds{R}^d\\setminus B_2$ be given.\nFor $v\\in\\mathds{R}^d$, let $\\overline{v}$ be such that $v=v_0+T_0(\\overline{v})$, where $T_0$ is defined in \\eqref{def:T0}. Furthermore, let $\\overline{g}(v)= g(\\overline{v})$.\n\\begin{lemma}\\label{lemma:coerc1}\nLet $\\gamma>-d$ and $s\\in(0,1)$. Let $\\psi$ be a non-negative function satisfying $\\psi\\leq \\Phi$ and \\eqref{def:psi} and let $f$ be a non-negative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}. \nThere are $c>0$, $R\\in (2,\\infty)$ and $\\rho\\in (0,1]$, depending on the macroscopic bounds $m_0, M_0, E_0$ and $H_0$, such that for all $g:\\mathds{R}^d\\to\\mathds{R}$\n\\begin{align*} \n\\iint_{d_{GS}(v,v')<R} &(g(v)- g(v'))^2 K_f^{\\psi}(v,v')\\,\\, \\d v' \\, \\d v   \\\\\n& \\geq c\\iint_{d_{GS}(v,v')<\\rho}  (g(v)- g(v'))^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^{(\\gamma+2s+1)\/2}}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v . \n\\end{align*}\n\\end{lemma}\n\\begin{proof} The proof follows as in \\cite[Lemma A.2]{imbertsilvestreglobal2022}. \nIt uses the transformation \\eqref{def:T0}, the comparability \\eqref{eq:comparabilitymetric} and the local coercivity estimate \\autoref{lemma:localcoercivity}.\nHaving these results on hand, one can proceed in the exact same way as it is done in \\cite[Lemma A.2]{imbertsilvestreglobal2022}. \n\\end{proof}\n\nThe integral of Lemma \\ref{lemma:coerc1} is over the points $v,v' \\in \\mathds{R}^d$ so that $d_{GS}(v,v') < \\rho$. This value of $\\rho$ is not important. The next lemma shows that we can enlarge it at will without altering the result.\n\n\\begin{lemma} \\label{lem:rho32rho}\nLet $g : \\mathds{R}^d \\to \\mathds{R}$ be any measurable function, $q \\in \\mathds{R}$ and $\\rho < 2$. The following inequality holds for some $c>0$ depending only on dimension, $s$ and $q$.\n\\begin{align*}\n\\iint_{d_{GS}(v,v')<\\rho}  (g(v)- g(v'))^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v &\\geq \\\\\nc \\iint_{d_{GS}(v,v')< \\frac 32 \\rho}  (g(v)- g(v'))^2\\, &\\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v\n\\end{align*}\n\\end{lemma}\n\n\\begin{proof}\nGiven $v,v' \\in \\mathds{R}^d$ so that $d_{GS}(v,v') < \\frac 32 \\rho$, define\n\\[ N(v,v') := \\left\\{ w \\in \\mathds{R}^d : d_{GS}(v,w) < \\frac 23 d_{GS}(v,v') \\text{ and } d_{GS}(v',w) < \\frac 23 d_{GS}(v,v') \\right\\}.\\]\n\nFrom the triangle inequality, we observe that $d_{GS}(v,w) > \\frac 13 d_{GS}(v,v')$ for all $w \\in N(v,v')$. Moreover, since $d_{GS}(v,v') < \\frac 32 \\rho < 3$, we also see that $\\langle v \\rangle \\approx \\langle v' \\rangle \\approx \\langle w \\rangle$ for all $w \\in N(v,v')$.\n\nLet us also define $M(v,w) := \\{ v' \\in \\mathds{R}^d: w \\in N(v,v')\\}$. The sets $N(v,v')$ and $M(v,w)$ are substantial portions of a ball with respect to the distance $d_{GS}$. It is not hard to estimate their volumes $|N(v,v')| \\approx |M(v,w)| \\approx \\langle v \\rangle^{-1} d_{GS}(v,v')^d$.\n\nWith this notation, we proceed with the computation\n\\begin{align*}\n\\iint_{d_{GS}(v,v')< \\frac 32 \\rho}  &(g(v)- g(v'))^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v = \\\\\n&= \\iint_{d_{GS}(v,v')< \\frac 32 \\rho}  (g(v)- g(v'))^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, |N(v,v')|^{-1} \\int_{N(v,v')} \\d w \\d v' \\, \\d v \\\\\n\\intertext{Using that $(g(v)- g(v'))^2 \\leq 2(g(v)- g(w))^2 + 2(g(w)- g(v'))^2$ and $|N(v,v')| \\approx \\langle v \\rangle^{-1} d_{GS}(v,v')^d$}\n&\\lesssim \\iint_{d_{GS}(v,v')< \\frac 32 \\rho}\\int_{N(v,v')}  \\left( (g(v)- g(w))^2 + (g(w)- g(v'))^2 \\right) \\, \\frac{ (\\langle v \\rangle \\langle v' \\rangle)^{q+1\/2}}{d_{GS}(v,v')^{2d+2s}}\\, \\d w \\d v' \\, \\d v \\\\\n\\intertext{Note the symmetry respect to $v$ and $v'$ and that $\\langle v \\rangle \\approx \\langle v' \\rangle \\approx \\langle w \\rangle$.}\n& \\approx \\iint_{d_{GS}(v,w)<\\rho} (g(v)- g(w))^2 \\frac{ \\langle v \\rangle^{2q+1}}{d_{GS}(v,w)^{2d+2s}} \\left( \\int_{v' \\in M(v,w)} \\d v' \\right) \\d w \\d v \\\\\n& \\approx \\iint_{d_{GS}(v,w)<\\rho} (g(v)- g(w))^2 \\frac{ \\langle v \\rangle^{q} \\langle w \\rangle^{q}}{d_{GS}(v,w)^{d+2s}} \\d w \\d v \\\\\n\\end{align*}\n\\end{proof}\n\n\\begin{corollary} \\label{cor:rhoisirrelevant}\nLet $g : \\mathds{R}^d \\to \\mathds{R}$ be any measurable function, $q \\in \\mathds{R}$ and $\\rho < 1$. The following inequality holds for some $c>0$ depending only on dimension, $s$, $q$ and $\\rho$.\n\\begin{align*}\n\\iint_{d_{GS}(v,v')<\\rho}  (g(v)- g(v'))^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v &\\geq \\\\\nc \\iint_{d_{GS}(v,v')< 1}  (g(v)- g(v'))^2\\, &\\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^q}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v\n\\end{align*}\n\\end{corollary}\n\n\\begin{proof} We iterate Lemma \\ref{lem:rho32rho} $m$ times so that $(3\/2)^m \\rho \\geq 1$.\n\\end{proof}\n\nWe finally have all tools to proof our main result of this section, that is \\autoref{prop:entropydissipation}.\n\\begin{proof}[Proof of \\autoref{prop:entropydissipation}]\nLet $\\psi$ be a non-negative satisfying $\\psi\\leq \\Phi$ and \\eqref{def:psi}. \nProceeding as in the proof of \\autoref{lem:entropuquadratic},\n\\[ D(f) \\geq  N_f^{\\varphi}(\\sqrt{f}) - C M_0^2\\]\nwhere $C>0$ is a finite universal constant and\n\\[ N_f^{\\psi}(\\sqrt{f}) = \\int_{\\mathds{R}^d}\\int_{\\mathds{R}^d} \\left(\\sqrt{f(v')} - \\sqrt{f(v)}\\right)^2 K_f^{\\psi}(v,v') \\d v' \\d v.\\]\n\nBy \\autoref{lemma:coerc1}, there are $c_1>0$ and $\\rho\\in(0,1)$, depending on $m_0, M_0, E_0$ and $H_0$, such that\n\\[ N_f^{\\psi}(\\sqrt{f}) \\geq c_1 \\iint_{d_{GS}(v,v')<\\rho}  (\\sqrt{f(v)}- \\sqrt{f(v')})^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^{(\\gamma+2s+1)\/2}}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v.\\]\n\nBecause of Corollary \\ref{cor:rhoisirrelevant}, we can replace $\\rho$ in the formula above by $1$ by adjusting the constant $c_1$. We get\n\\[ N_f^{\\psi}(\\sqrt{f}) \\geq c_2 \\iint_{d_{GS}(v,v')<1}  (\\sqrt{f(v)}- \\sqrt{f(v')})^2\\, \\frac{\\left( \\langle v \\rangle \\langle v' \\rangle \\right)^{(\\gamma+2s+1)\/2}}{d_{GS}(v,v')^{d+2s}}\\, \\d v' \\, \\d v.\\]\nTherefore, we conclude that\n\\[ D(f) \\geq c |\\sqrt{f}|_{\\dot{N}^{s,\\gamma}}^2 - C M_0^2. \\]\n\n\\end{proof}\n\\section{Applications to weak solutions}\\label{sec:weaksol}\nIn this section, we discuss an application of the main result \\autoref{thm:entropydissipation} for weak solutions to the spatially homogeneous Boltzmann equation. Let us have a look at the weak formulation of the Boltzmann collision operator. \n In the following, let $f\\in L^\\infty([0,\\infty); L_2^1(\\mathds{R}^d)) \\cap C([0,\\infty); \\mathcal{D}'(\\mathds{R}^d))$ be a nonnegative function satisfying \\eqref{ass:mass}, \\eqref{ass:energy} and \\eqref{ass:entropy}.\n\nIn \\cite{Villaninewclass}, Villani introduces a class of weak solutions, called H-solutions, to the spatially homogeneous Boltzmann equation with bounded entropy dissipation. A solution in this class might not be a weak solution in the usual sense. As explained in \\cite[Section 7, Application 2]{AlexDesvVilWenn} and \\cite{Villaninewclass}, this problem appears because of the lack of an a priori estimate in the very soft potential case of the form\n\\begin{equation}\\label{eq:apropri}\n \\int_{0}^T \\int_{B_R} \\int_{B_R} f(t,v) f(t,v_{\\ast})|v-v_{\\ast}|^{\\gamma+2} \\, \\d v \\, \\d v_{\\ast} \\, \\d t < \\infty. \n\\end{equation}\nIt is mentioned in \\cite{AlexDesvVilWenn} that the (local) entropy dissipation estimate shows that H-solutions are weak solutions in the usual sense when $\\gamma+2s \\geq d-2$. The computation is sketched without explicit details. Using our estimate in Theorem \\ref{thm:entropydissipation}, we show that \\eqref{eq:apropri} holds whenever $\\gamma+2s > -2$, covering the whole physical range of exponents. There is no fundamental difference between the computation presented here and the one proposed in \\cite{DesvillettesColoumb}. It is not clear why they stated a suboptimal range in \\cite{AlexDesvVilWenn}, suggesting that it is possibly a typo in the paper. We explain the computation explicitly below.\n\nOne important ingredient in the proof of \\eqref{eq:apropri} is that H-solutions are in a weighted Lebesgue space. It follows immediately from \\autoref{thm:entropydissipation} combined with the entropy dissipation formula \\eqref{eqn:dissipationintegratedintime} (without integrating in space).\n\n\\begin{corollary}\\label{corollary:weightedlebesgue}\nLet $T>0$, $-d<\\gamma\\leq 2$ and $s\\in(0,1)$. \nLet $f$ be a non-negative H-solution to the Cauchy problem \\eqref{eq:homogboltzmann} with initial datum $f_0$. Assume $f_0$ satisfies \\eqref{ass:entropy}. \\\\\nThen $f\\in L^{1}\\left([0,T], L^p_{-q}(\\mathds{R}^d)\\right)$, where $1\/p = 1-2s\/d$ and $q=2s\/d - \\gamma-2s$.\n\\end{corollary}\n\\begin{proof}\nBy definition, H-solutions satisfy the space-homogeneous form of \\eqref{eqn:dissipationintegratedintime}. We get that $\\int_0^T D(t) \\d t \\leq \\int f_0 \\log f_0 \\d v$. The corollary follows applying Theorem \\ref{thm:entropydissipation}.\n\\end{proof}\n\nWe use \\autoref{corollary:weightedlebesgue} to show that H-solutions are weak solutions in the usual sense by proving \\eqref{eq:apropri}.\n\n\\begin{corollary}\\label{cor:aprioriestimate}\nLet $T>0$, $-d<\\gamma\\leq 0$ and $s\\in(0,1)$, so that $\\gamma+2s > -2$. \nLet $f$ be a non-negative H-solution to the Cauchy problem \\eqref{eq:homogboltzmann} with initial datum $f_0$. Assume $f_0$ satisfies \\eqref{ass:entropy}. Then $f$ satisfies \\eqref{eq:apropri}.\n\\end{corollary}\n\n\\begin{proof}\n\\begin{align*}\n \\int_{0}^T & \\int_{B_R}  \\int_{B_R} f(t,v) f(t,v_{\\ast}) |v-v_{\\ast}|^{\\gamma+2}\\mathds{1}_{\\{|v-v_{\\ast}| \\leq 1\\}}) \\, \\d v \\, \\d v_{\\ast} \\, \\d t \\\\\n& \\leq \\int_{0}^T \\|f\\|_{L^1} \\sup_{v \\in B_R} \\int_{B_R} f(t,v_{\\ast}) |v-v_{\\ast}|^{\\gamma+2} \\, \\d v_{\\ast} \\, \\d t \\\\\n\\intertext{We apply H\\\"older's inequality with $1\/p = 1 - 2s\/d$ as in Corollary \\ref{corollary:weightedlebesgue}.}\n& \\leq \\int_{0}^T \\|f(t,\\cdot)\\|_{L^1} \\|f(t,\\cdot)\\|_{L^p(B_R)} \\sup_{v \\in B_R} \\||v-\\cdot|^{\\gamma+2} \\|_{L_{p'}(B_R)} \\d t \\\\\n&\\leq \\|f\\|_{L^\\infty([0,T],L^1(B_R))} \\|f\\|_{L^1([0,T],L^p(B_R))} \\sup_{v \\in B_R} \\||v-\\cdot|^{\\gamma+2} \\|_{L_{p'}(B_R)}\n\\end{align*}\nIt only remains to check whether the last factor is finite. We have\n\\begin{align*}\n\\||v-\\cdot|^{\\gamma+2} \\|_{L_{p'}(B_R)} =\\left( \\int_{B_R} |v-v_\\ast|^{(\\gamma+2)\\frac{d}{2s}} \\, \\d v_\\ast \\right)^{\\frac{2s}d} \n\\end{align*}\nThe integral is finite provided that $(\\gamma+2)d\/(2s) > -d$. This is clearly the case when $\\gamma+2s > -2$.\n\\end{proof}\n\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n    \\begin{table}[t]\n    \\begin{center}\n    \\begin{tabular}{|c|c|c| }\n    \\hline\n    Additive Stretch & Size & Time \\\\\n    \\hline \\hline\n         +2 &  $O(n^{3\/2})$ & $O(mn^{1\/2})$ \\cite{aingworth1999fast}  \\\\\n         \\hline\n         +4 & $\\widetilde{O}(n^{7\/5})$ & $O(mn)$  \\cite{chechik2013new} \\\\\n         \\hline\n         \\textbf{+4} & $\\bm{\\widetilde{O}(n^{7\/5})}$ & $\\bm{\\widetilde{O}(mn^{3\/5})}$ \\textbf{(this paper)}\\\\\n         \\hline\n         +6 & $\\widetilde{O}(n^{4\/3})$ &  $\\widetilde{O}(n^2)$ (\\cite{Baswana2010} and \\cite{woodruff2010additive})\\\\\n         \\hline\n         +8 & $O(n^{4\/3})$ & $O(n^2)$ \\cite{knudsen2017additive}\\\\\n         \\hline\n    \\end{tabular}\n    \\caption{Notable Additive Spanner Constructions}\n    \\end{center}\n    \\end{table}\n\nA graph on $n$ nodes can have on the order of $m = O(n^2)$ edges. For very large values of $n$, this amount of edges can be prohibitively expensive, both to store in space and to run graph algorithms on. Thus it may be prudent to operate instead on a smaller approximation of the graph. A \\textit{spanner} is a type of subgraph which preserves distances between nodes up to some error, which we call the stretch. Spanners were introduced in \\cite{peleg1989graph} and \\textit{additive} spanners were first studied in \\cite{liestman1993additive}.\n\n\\begin{definition}[Additive Spanners]\n\nA $k-$additive spanner (or ``$+k$ spanner\") of a graph $G$ is a subgraph $H$ that satisfies $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t) + k$ for each pair of nodes $s,t \\in V(G)$. $k$ is called the (additive) \\textit{stretch} of the spanner.\n\n\\end{definition}\n\n\nNote that since $H$ is a subgraph of $G$, the lower bound $\\texttt{dist}_G(s,t) \\leq \\texttt{dist}_H(s,t)$ is immediate (the error is one-sided). Spanners have found applications in distance oracles \\cite{baswana2006faster}, parallel and distributed algorithms for computing almost shortest paths \\cite{cohen1998fast,elkin2005approximating}, synchronizers \\cite{peleg1987optimal}, and more. \n\nSpanners are a tradeoff between the size of the subgraph and the stretch; spanner size can be decreased at the cost of a greater stretch and vice versa. The +2 spanner construction due to Aingworth et al. produces spanners of size $O(n^{3\/2})$, and this size is optimal \\cite{aingworth1999fast}. The +4 spanner construction is due to Chechik, which produces smaller spanners of size $\\widetilde{O}(n^{7\/5})$ (though this bound is not known to be tight; it is conceivable that further improvements may reduce size up to $O(n^{4\/3})$)\\cite{chechik2013new}. The +6 construction due to Baswana et al. \\cite{Baswana2010} and +8 construction due to Knudsen \\cite{knudsen2017additive} achieve an $O(n^{4\/3})$ spanner. It is known that any $+k$ spanner construction has a lower bound of $n^{4\/3 - o(1)}$ edges on the spanners it produces, so error values greater than +6 are not existentially of interest; the +8 spanner exchanges error for a polylog improvement in construction speed. \n\nIn addition to finding spanner constructions that produce the smallest spanners possible, it is also in our interest that these construction algorithms be fast. Because spanners are meant to make graphs more compact, they are mainly of interest for very large graphs. Thus, for very large $n$, a polynomial time speedup to an algorithm for producing $+k$ spanners is highly desirable. There is a long line of work done in the interest of speeding up spanner constructions, including \\cite{ roditty2004dynamic, baswana2007simple,woodruff2010additive,knudsen2017additive, alstrup2019constructing}.\nFor a comprehensive survey, see \\cite{ahmed2020graph}. \n\nSome additive spanner size and efficiency results are summarized in Table 1 above. As mentioned above, the $+8$ spanner due to Knudsen \\cite{knudsen2017additive} exchanges error over the $+6$ spanner for a polylog improvement in construction time. In this paper, we present a polynomial speedup to Shiri Chechik's 4-additive spanner construction presented in \\cite{chechik2013new}, at no cost in error.\n\n\n\n\n\n\n\n\n\n\n\\begin{theorem} [Main Result]\nThere is an algorithm that constructs (with high probability) a $+4$ spanner on $\\widetilde{O}(n^{7\/5})$ edges in $\\widetilde{O}(mn^{3\/5})$ time.\n\\end{theorem}\n\nFor comparison, the bottleneck to Chechik's original construction is solving the All-Pairs-Shortest-Paths (APSP) problem; with combinatorial methods, this has an $O(mn)$ runtime, and with matrix multiplication methods, $O(n^\\omega)$ \\cite{seidel1995all} \\footnote{We note that when $m> n^{\\omega-0.4}$, the bottleneck in the algebraic case is instead the second stage of the algorithm described in section 1.2 ($\\widetilde{O}(mn^{2\/5})$).}. Currently, $\\omega <2.373$ \\cite{alman2021refined}. See Section 1.2 for a full runtime analysis of the original construction.\n\nOur speedup relies on avoiding the APSP problem. We do this by realizing that we can weaken the path finding methods in Chechik's original construction without compromising error. In particular, we introduce a new problem in Section 2.2, which we call the ``Weak Constrained Single Source Shortest Paths\" (weak CSSSP) problem. We give a Dijkstra-time solution to this problem and apply it to create our new +4 spanner construction in Section 2.3.\n\n\n\nWith matrix multiplication methods for APSP, Chechik's algorithm has an $\\widetilde{O}(n^\\omega)$ implementation \\cite{seidel1995all}\nHowever, we note that there is a range of $m$ values where our combinatorial algorithm potentially outperforms algebraic methods; for example if $\\omega \\geq 2.3$, then when $m < n^{1.699}$, the complexity of our algorithm is polynomially faster\n\n\nFinally, we extend our weak CSSSP method to the construction of spanners in the weighted setting. While the unweighted setting is predominant in the study of additive spanners, the generalization was made to the weighted setting by Elkin et al. in \\cite{elkin2019almost}, and further work was done in \\cite{ahmed2020weighted,elkin2020improved,ahmed2021additive}. There are two different weighted generalizations of $+k$ spanners in the literature; the weaker generalization (introduced in \\cite{ahmed2020weighted}) requires the spanner $H$ to satisfy $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t) + kW$ for each $s,t \\in V(G)$, where $W = \\max_{e \\in E(G)} w(e)$ is the maximum edge weight of the edges of $G$. These are called $+kW$ spanners. The stronger generalization (studied in \\cite{elkin2020improved, ahmed2021additive}) defined below restricts the edge weight stretch factor to the maximal edge weight over shortest $s \\leadsto t$ shortest paths, denoted $W(s,t)$. For this reason this generalization is known as ``local weighted error\".\n\n\n\\begin{definition}[Weighted Additive Spanner (Global Error)]\nA $+kW$ spanner of a graph $G$ is a subgraph $H$ that satisfies $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+kW$ for all $s,t \\in V(G)$, where $W = \\max_{e \\in E}w(e)$\n\\end{definition}\n\n\\begin{definition}[Weighted Additive Spanner (Local Error)]\nA $+kW(\\cdot,\\cdot)$ spanner of a graph $G$ is a subgraph $H$ that satisfies $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+kW(s,t)$ for all $s,t \\in V(G)$, where $W(s,t)$ is the maximum edge weight along a shortest path $\\pi_G(s,t)$ in $G$.\n\\end{definition}\n\n\nIn \\cite{ahmed2021additive}, Ahmed et al. generalized the 4-additive spanner construction presented in \\cite{chechik2013new} to the (strong) weighted setting. Their algorithm constructs a $+4W(\\cdot,\\cdot)$ spanner on $\\widetilde{O}(n^{7\/5})$ edges, and can be implemented in $\\widetilde{O}(mn^{4\/5})$ time by using AliAbdi et al's Bi-SPP algorithm \\cite{AliAbdi2019} for constrained shortest path finding. Our contribution in section 3 will be a $+4W(\\cdot,\\cdot) + \\epsilon W$ spanner in $\\widetilde{O}(mn^{3\/5})$ time, on $\\widetilde{O}_\\epsilon(n^{7\/5} \\epsilon^{-1})$ edges with high probability, for any error $1>\\epsilon >0$.\n\n\n\\begin{theorem}\nFor any weighted graph $G = (V,E)$ and $\\epsilon >0$, there is a $+4W(\\cdot,\\cdot)+\\epsilon W$ spanner on $\\widetilde{O}_\\epsilon(n^{7\/5})$ edges and computable in $\\widetilde{O}(mn^{3\/5})$ time, with high probability.\n\\end{theorem}\n\\noindent\nWe note that while Ahmed et. al's construction in  \\cite{ahmed2021additive} doesn't have the extra $+\\epsilon W$ stretch, our construction comes with a polynomial speedup. \n\n\\subsection{Notations}\n\nWe will use $\\pi_G(s,t)$ to refer to a canonical shortest path between two nodes $s,t \\in V(G)$. $P(s,t)$ is a variable we use in some of our algorithms that describes some computed $s \\leadsto t$ path. For a node $v \\in V(G)$, $\\Gamma_G(v)$ denotes the neighborhood of $v$ (the set containing $v$ and its neighbors) in $G$. When $S$ is a set, $\\Gamma_G(S):= \\bigcup_{v \\in S} \\Gamma_G(v)$. $\\mathcal{P}(u,v)$ denotes the set of all paths between nodes $u,v$. If $A,B$ are subsets of $V$, then $\\mathcal{P}(A,B) = \\bigcup_{u\\in A, v\\in B}\\mathcal{P}(u,v)$\n\n\\subsection{Current Runtime of the +4 Spanner Construction}\n\n\nIn \\cite{chechik2013new}, Shiri Chechik presents a spanner construction that produces a +4 spanner on $\\widetilde{O}(n^{7\/5})$ edges on average with probability $>1-1\/n$. The runtime complexity, however, was not analyzed. In this section, we will describe Chechik's algorithm and then give a runtime analysis. Chechik's construction of a +4 spanner $H$ of an input graph $G$ can be split into three stages:\n\n\\begin{enumerate}[(i)]\n    \\item All edges adjacent to ``light nodes\" (nodes with degree $< \\mu = \\lceil n^{2\/5}\\log^{1\/5}n\\rceil$) are added to $H$.\n    \n    \\item Nodes are sampled for inclusion into a set $S_1$ with probability $9\\mu\/n$. BFS (Breadth-First Search) trees for these nodes are computed, and the edges for these trees are added to $H$.\n    \n    \\item Nodes are sampled for inclusion into a set $S_2$ with probability $1\/\\mu$. For each ``heavy\" node (nodes with degree $\\geq \\mu$ in the original graph) $v$ that is not in $S_2$, but is adjacent to some node of $S_2$, we arbitrarily choose a neighbor $x \\in S_2$ and add the edge $(v,x)$ to $H$. These choices also define the ``clusters\" of the graph: for each $x \\in S_2$, $C(x)$ is the set containing $x$ and its adjacent heavy nodes that were paired with $x$ in the previous step. We now find, for each pair $x_1,x_2 \\in S_2$, the shortest path $P(s,t)$ subject to the constraint that $s \\in C(x_1)$, $t \\in C(x_2)$, and $P(s,t)$ has $\\leq \\mu^3\/n$ heavy nodes. We use $\\texttt{heavy\\_dist}_G(P(s,t))$ to refer to the number of heavy nodes on $P(s,t)$ in $G$.\n\n\\end{enumerate}\n\nAlgorithm \\ref{alg:chechik} gives the full details. Stage (i) takes $O(n)$ time and stage (ii) takes $\\widetilde{O}(mn^{2\/5})$ time with high probability. The computationally dominant step of this algorithm is the task of finding these shortest paths between the clusters in (iii) (unless algebraic methods are used, in which case stage (ii) dominates). For worst case inputs, the expected number of clustered nodes (nodes in some cluster) is $\\Omega(n)$. Thus, this algorithm's runtime will be bottlenecked by the all-pairs-shortest-paths problem. We now show that the heavy-node constraint on the paths does not increase the runtime.\nTo see this, we note that it's enough to search over paths of the form\n$$P(x_1, x_2) = (x_1,s)\\circ \\pi_G(s,t) \\circ (t,x_2) \\footnote{Paths of this form will be important in our own construction (see Definition 5).}$$\n($\\circ$ denotes path concatenation), where $x_1,x_2$ range over $S_2$ and $s \\in C(x_1)$, $t \\in  C(x_2)$. Specifically, we want the shortest path of this form for each pair of clusters $C(x_1),C(x_2)$, where $\\pi_G(s,t)$ is a constraint-satisfying ($\\leq\\mu^3\/n$ heavy nodes) path that is also a shortest path in $G$.\n\nTo find such paths, we first solve APSP ($O(mn)$ time combinatorially, $O(n^{\\omega})$ time algebraically) to get shortest paths $\\pi_G(s,t)$ for each $s,t \\in V$. Then for all pairs of clustered nodes $s,t$, with cluster centers $x_1,x_2$ respectively: if $\\texttt{heavy\\_dist}_G(\\pi_G(s,t)) \\leq \\mu^3\/n$, set $P(x_1,x_2) = (x_1,s) \\circ \\pi_G(s,t) \\circ (t,x_2)$ as the current best path for the cluster pair $(C(x_1),C(x_2))$ if one hasn't yet been selected, otherwise replace the current path iff $P(x_1,x_2)$ is shorter. This is an APSP time process for finding the shortest valid canonical shortest path connecting each cluster pair; at the end of the process, we add the edges of these best paths. We note that because we're not searching over \\textit{all} paths, but only one set of canonical shortest paths, it's possible we fail to find valid (constraint satisfying) paths between some cluster pairs. This does not impede correctness, as we only require these paths in the cases that they exist. \n\n\n\\begin{figure}[h]\n\\begin{algorithm}[H]\n    \\label{alg:chechik}\\caption{Chechik's 4-Additive Spanner Construction \\cite{chechik2013new}}\n    \\KwIn{$n$-node graph $G=(V,E)$}\\vspace{1em}\n    $E'=$ All edges incident to light nodes\\\\\n    Sample a set of nodes $S_1$ at random, every node with probability $9\\mu\/n$\\\\\n    \\ForEach{node $x\\in S_1$}{\n        Construct a BFS tree $T(x)$ rooted at $x$ spanning all vertices in $V$\\\\\n        $E'=E'\\cup E(T(x))$\\\\\n    }\\vspace{1em}\n    %\n    Sample a set of nodes $S_2$ at random, every node with probability $1\/\\mu$\\\\\n    \\ForEach{heavy node $x$ so that $(\\{x\\}\\cup\\Gamma_G(x))\\cap S_2 = \\varnothing$}{\n        Add all incident edges of $x$ to $E'$\\\\\n    }\n    \\ForEach{node $x\\in S_2$}{\n        C(x) = \\{x\\}\n    }\n    \\ForEach{heavy node $v$ so that $v\\not\\in S_2$ and $\\Gamma(v)\\cap S_2\\neq\\varnothing$}{\n        Arbitrarily choose one node $x\\in\\Gamma_G(v)\\cap S_2$\\\\\n        $C(x) = C(x) \\cup \\{x\\}$\\\\\n        $E' = E' \\cup \\{(u,v)\\}$\\\\\n    }\n    \\ForEach{pair of nodes $(x_1, x_2)$ in $S_2$}{\n        Let $\\hat{\\mathcal{P}}=\\{P\\in\\mathcal{P}(C(x_1),C(x_2))\\ |\\ \\texttt{heavy\\_dist}_G(P)\\leq\\mu^3\/n\\}$\\\\\n        Let $P(\\hat y_1, \\hat y_2)$ be the path in $\\hat{\\mathcal{P}}$ with minimal $\\left|P(\\hat y_1, \\hat y_2)\\right|$\n        $E'=E'\\cup E(P(\\hat y_1, \\hat y_2))$\n    }\n    \\Return $H=(V,E')$\n\\end{algorithm}\n\\end{figure}\n\n\\FloatBarrier \n\n\n\n\n\\section{Fast Construction of the +4 Spanner}\n\n\nIn this section, we present our main result; a modification of Chechik's +4 spanner construction that has $\\widetilde{O}(mn^{3\/5})$ runtime with high probability, with no compromise to size or error.\n\n\\subsection{Constrained Shortest Paths}\n\nChechik's original algorithm required the computation of shortest paths subject to a constraint on the number of heavy nodes in the paths. This was a proxy for constraining the number of edges that had not yet been added to the spanner at that point in the construction. We will call CSSSP the ``Constrained Single Source Shortest Path Problem\". This is similar to the GB-SPP (``Gray-Vertices Bounded Shortest Path Problem\") presented by AliAbdi et al. in \\cite{AliAbdi2019}, but our constraint is on the edges instead of on the nodes.\n\n\\begin{definition}[CSSSP] The constrained single-source shortest paths problem is defined by the following algorithm contract:\n\\begin{itemize}\n    \\item \\textbf{Input:} An (unweighted, undirected) graph $G=(V,E)$, a set of ``gray\" edges $E_g \\subset E$, a source vertex $s \\in V$, and a positive integer $g$.\n    \n    \\item \\textbf{Output:} For every $t \\in V$, a path $P(s,t)$ on $\\leq g$ gray edges, where $|P(s,t)| \\leq |P'(s,t)|$ for all $s \\leadsto t$ paths $P'(s,t)$ on $\\leq g$ gray edges.\n\\end{itemize}\n\\end{definition}\n\nOur modification to Chechik's construction will also make use of constrained shortest path finding, but the CSSSP problem is stronger than necessary for our purposes, and we can get away with a better runtime by solving a weaker problem. In this section, we define and give an efficient algorithm for a weaker variation on CSSSP, which we'll call weak CSSSP. In particular, we will only need to find constrained shortest paths from $s$ to $t$ in situations where a certain type of $s \\leadsto t$ constrained path already exists. We define these paths and call them \\textbf{g-short paths}\n\n\n\\begin{definition}[g-short path]\nFor two nodes $s,t$, an $s \\leadsto t$ path is called ``g-short\" if it has $<g$ gray edges and is of the form $(s,s') \\circ \\pi_G(s',t') \\circ (t,t')$, where $\\pi_G(s',t')$ is a shortest path.\n\\end{definition}\n\n\\noindent\nThese g-short paths naturally arise in the analysis of our 4-additive spanner construction later in the paper. Note that g-short paths are not necessarily \\textit{shortest} paths, but they are at most $+2$ edges longer than a shortest path (in the present unweighted case). Now we are ready to define weak CSSSP:\n\n\\begin{definition}[Weak CSSSP] The weak constrained single-source shortest paths problem is defined by the following algorithm contract:\n\n\\begin{itemize}\n    \\item \\textbf{Input:} An (unweighted, undirected) graph $G=(V,E)$, a set of ``gray\" edges $E_g \\subset E$, a source vertex $s \\in V$, and a positive integer $g$.\n    \n    \\item \\textbf{Output:} For every $t \\in V$, a path $P(s,t)$ on $\\leq 5g$ gray edges, satisfying the following:\n    \\begin{itemize}\n        \\item If there exists an $s\\leadsto t$ \\textbf{g-short path}, then the \\textbf{g-optimality condition} holds: $|P(s,t)| \\leq |P_g(s,t)|$ for any $s \\leadsto t$ path $P_g(s,t)$ on $<g$ gray edges.\n\t\\item If no such path exists, then $P(s, t)$ can be anything.\n        \n       \n    \\end{itemize}\n\n\\end{itemize}\n\\end{definition}\n\n\nInformally, if there is a ``short enough\" path from $s$ to $t$ with $<g$ gray edges, then the outputted path $P(s,t)$ has $\\leq 5g$ gray edges and is shortest among all $s \\leadsto t$ paths with $<g$ gray edges. Besides the constant factor on the gray-edge bound, the only difference between weak CSSSP and CSSSP is that we put a precondition on a g-short $s \\leadsto t$ path existing. When we need weak CSSSP later in the paper, it will only be in situations where this g-short path exists.\n\n\nOne can solve the weak CSSSP problem by simply solving the general CSSSP problem. AliAbdi et al. in \\cite{AliAbdi2019} present a label-setting algorithm ``Bi-SPP\" (Bi-Colored Shortest Path Problem) for solving GB-SPP. This is the problem of finding shortest paths from a source node $s \\in V$ to every other node $t \\in V$ subject to the constraint that the paths have $\\leq g$ gray \\textbf{nodes}.\n\n\\begin{theorem}[Implicit in Prior Work]\nThere is an $\\widetilde{O}(gm)$ solution to the weak CSSSP problem\n\\end{theorem}\n\\begin{proof}\nGiven input to an instance of the weak CSSSP problem: we designate a \\textit{node} to be gray if it is adjacent to a gray edge. It is clear that if a path has $< g$ gray edges, it must then have $< 2g$ gray nodes. Thus if we solve GB-SPP on this graph with parameter $g'=2g$, the resulting paths $P(s,t)$ satisfy $|P(s,t)| \\leq |P_g(s,t)|$ for any path $P_g(s,t)$ on $<g$ gray edges. Furthermore, the paths $P(s,t)$ have $< 2g \\leq 5g$ gray edges. Therefore these resulting paths satisfy the weak CSSSP requirements. Using the Bi-SPP algorithm, this can be done in $O(g(m+n \\log n)) = \\widetilde{O}(gm)$ time \\cite{AliAbdi2019}.\n\n\\end{proof}\n\n\n\nWe now present a new algorithm for solving weak CSSSP with the same runtime as plain SSSP in the weighted setting - using Dijkstra's algorithm with Fibonacci heaps, a runtime of $O(m + n\\log n) = \\widetilde{O}(m)$. We give each non-gray edge weight 1, and each gray edge weight $1+g^{-1}$. We run Dijkstra's algorithm with these weights and report the paths it computes. The intuition behind this approach is that we ``punish\" gray edges by a value that ensures both the g-optimality condition and $\\leq 5g$ gray edges in the paths Dijkstra reports. The punishment is big enough so that it is impossible for the reported path to have too many gray edges without violating the fact that a g-short $s\\leadsto t$ path exists, and small enough so that the total punishment on the $P_g(s,t)$ paths described in the algorithm contract is less than $1$, effectively acting as a tiebreaker between such paths.\n\n\n\\begin{figure}[h]\n\\begin{algorithm}[H]\n    \\label{alg:weak}\\caption{Weak CSSSP via Edge Weighting}\n    \\KwIn{Undirected, unweighted, graph $G=(V,E)$\\\\\n    Source vertex $s\\in V$\\\\\n    Set of gray edges $E_g \\subseteq E$\\\\\n    Positive integer $g$\n    }\\vspace{1em}\n   \n  \n   \n    \\ForEach{edge $(u,v) \\in E$}{\n        $w(u,v)\\leftarrow 1+g^{-1}$ if $(u,v)$ is gray, and $w(u,v)\\leftarrow 1$ otherwise.\n    }\n   \n    \n    \n    Run Dijkstra on $s$ with weight function $w$ to get paths $P(s,t)$ for each other $t \\in V$\\\\\n    \\Return these $P(s,t)$ paths\n   \n\n\\end{algorithm}\n\\end{figure}\n\n\n\\begin{theorem}\nAlgorithm \\ref{alg:weak} solves weak CSSSP in $O(m + n\\log n) = \\widetilde{O}(m)$ time.\n\\end{theorem}\n\\begin{proof}\nThe time complexity follows immediately from the complexity of Dijkstra's algorithm, which is the dominant stage of the algorithm. We now prove correctness.\\\\\n\n\n\\noindent\nLet $t \\in V$ and suppose a g-short $s\\leadsto t$ path $P'(s,t)=(s,s') \\circ \\pi_G(s',t') \\circ (t',t)$ exists. We will first show that $P(s,t)$ has $\\leq 5g$ gray edges. Suppose to show a contradiction that it has $>5g$ gray edges. Note by construction that the weight of a path is its length plus $g^{-1}$ times the number of gray edges. Thus  $w(P(s,t)) > |P(s,t)| + g^{-1}\\cdot 5g = |P(s,t)| + 5$. Furthermore, $w(P'(s,t)) < |P'(s,t)| + g^{-1} \\cdot g = |P'(s,t)| + 1$. But we also have, by the fact that $P(s,t)$ is the lowest-weight $s \\leadsto t$ path, that $w(P(s,t)) \\leq w(P'(s,t))$, and thus \n\n\\begin{align*}\n    |P(s,t)|+5 &< |P'(s,t)| + 1\\\\\n    &\\leq |\\pi_G(s',t')| + 2 + 1\\\\\n    & \\leq |\\pi_G(s,t)| +2+ 2 + 1\\\\\n    &= \\texttt{dist}_G(s,t)+5\n\\end{align*}\n\\noindent\nThis implies that $|P(s,t)| < \\texttt{dist}_G(s,t)$, which is a contradiction. Thus the computed path $P(s,t)$ has $\\leq 5g$ gray edges. We now show the g-optimality condition to complete the proof: let $P_g(s,t)$ be an arbitrary $s\\leadsto t$ path on $<g$ gray edges. We have that \n\n\\begin{align*}\n    |P(s,t)| &\\leq w(P(s,t))\\\\\n    &\\leq w(P_g(s,t))\\\\\n    & < |P_g(s,t)| + g\\cdot g^{-1}\\\\\n    &=|P_g(s,t)|+1\n\\end{align*}\n\n\\noindent\nThus $|P(s,t)| \\leq |P_g(s,t)|$ as required.\n\n\\end{proof}\n\n\\subsection{Application to 4-Additive Spanner Construction}\n\nWe are now ready to state our modification to Chechik's spanner construction. Two insights allow us to improve the efficiency: (i) instead of finding the constrained shortest paths between the clusters, it is sufficient to only do this for paths between $S_2$ nodes. Furthermore, (ii) it is sufficient to compute the weak CSSSP paths for this task. The constraint we place on the paths we find will be a constraint on ``heavy edges\", which are edges adjacent to heavy nodes\n\n\\begin{definition}[heavy edge]\nAn edge $(u,v)$ is called \\textbf{heavy} if either $u$ or $v$ are heavy nodes (having degree $\\geq \\mu$).\n\\end{definition}\n\nBesides these changes, the construction is the same as Chechik's original construction. We note that (i) alone was used by Ahmed et al. in their spanner construction for \\textit{weighted} graphs \\cite{ahmed2021additive}; combining (i) with AliAbdi et al.'s algorithm for CSSSP \\cite{AliAbdi2019} yields an $\\widetilde{O}(mn^{4\/5})$ run time, though this was not known to the authors at the time [Personal Communication]. (ii) is a novel method and is responsible for the next polynomial step down to $\\widetilde{O}(mn^{3\/5})$. We now prove our main result through the following series of lemmas:\n\n\\begin{figure}[h]\n\\begin{algorithm}[H]\n    \\label{alg:best}\\caption{Faster 4-Additive Spanner Construction}\n    \\KwIn{$n$-node graph $G=(V,E)$}\\vspace{1em}\n    $E'=$ All edges incident to light nodes\\\\\n    Sample a set of nodes $S_1$ at random, every node with probability $9\\mu\/n$\\\\\n    \\ForEach{node $x\\in S_1$}{\n        Construct a BFS tree $T(x)$ rooted at $x$ spanning all vertices in $V$\\\\\n        $E'=E'\\cup E(T(x))$\\\\\n    }\\vspace{1em}\n    %\n    Sample a set of nodes $S_2$ at random, every node with probability $1\/\\mu$\\\\\n    \\ForEach{heavy node $x$ so that $(\\{x\\}\\cup\\Gamma_G(x))\\cap S_2 = \\varnothing$}{\n        Add all incident edges of $x$ to $E'$\\\\\n    }\n   \n   \n   \n    \\ForEach{heavy node $v$ so that $v\\not\\in S_2$ and $\\Gamma_G(v)\\cap S_2\\neq\\varnothing$}{\n        Arbitrarily choose one node $x\\in\\Gamma_G(v)\\cap S_2$\\\\\n       \n        $E' = E' \\cup \\{(x,v)\\}$\\\\\n    }\n   \n   \n   \n    \n    \\ForEach{node $x_1 \\in S_2$}{\n        Compute weak CSSSP on $G$, with $g = \\mu^3\/n+2$, $x$ as the source vertex, and $E_g$ as the set of heavy edges, to get paths $P(x_1,x_2)$ for each $x_2 \\in V$.\\\\\n        Add $E(P(x_1,x_2))$ to $H$ for each $x_2 \\in S_2$\n    }\n    \n\n    \\Return $H=(V,E')$\n\\end{algorithm}\n\\end{figure}\n\n\n\n\\noindent\nThe first lemma is standard:\n\n\\begin{lemma}[\\cite{chechik2013new, Baswana2010}]\nFor a shortest path $\\pi_G(s,t)$ in a graph $G$ and for any vertex $v \\in V(G)$, $v$ has $\\leq 3$ neighbors in $\\pi_G(s,t)$.\n\\end{lemma}\n\\begin{proof}\nSuppose to show a contradiction that $v$ has four neighbors $u_1,u_2,u_3,u_4 \\in \\pi_G(s,t)$, and assume WLOG $u_1 \\prec u_2 \\prec u_3 \\prec u_4$ in $\\pi_G(s,t)$. This implies $\\texttt{dist}_G(u_1,u_4) \\geq 3$. But the path $(u_1,v) \\circ (v,u_4)$ has length $2$, which is a contradiction.\n\\end{proof}\n\n\nWe now show that for any two nodes $s,t$ of $H$, we have with very high probability that $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4$. Note that it's sufficient to prove this result when $s,t$ don't have all of their edges included in $H$. We call such nodes ``uncovered\". This is because when $s,t$ are not both uncovered, it's enough to demonstrate this stretch for the subpath of $\\pi(s,t)$ beginning and ending at the first and last uncovered nodes respectively.\n\n\\begin{definition}\nA node $v \\in V(G)$ is said to be ``covered\" in $H$ if all of its edges are included in $H$.\n\\end{definition}\n\nThis is because if $s,t$ were not both covered, we could let $s'$ be the first uncovered node on $P(s,t)$ and $t'$ the last uncovered node, and then $\\texttt{dist}_H(s',t') \\leq \\texttt{dist}_G(s',t')+4 \\implies \\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4$. This is because all the edges of $P(s,t)$ occurring before $s'$ or after $t'$ are already in $H$, so any stretch must occur between $s'$ and $t'$. This allows us to assume that $s,t$ are in $\\Gamma_G(S_2)$, as all other nodes are covered by our algorithm. The proof of the following lemma is identical to the first part of the proof of Lemma 2.2 in \\cite{chechik2013new}, but we repeat it here for completeness.\n\n\n\n\\begin{lemma}\nFor any two uncovered nodes $s,t\\in V(G)$ such that the canonical shortest path $\\pi_G(s,t)$ has $>\\mu^3\/n$ heavy nodes, we have $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4$ with probability $\\geq 1-\\frac{1}{n^3}$\n\\end{lemma}\n\\begin{proof}\nIn this case, we claim that there is a $\\geq 1-1\/n^3$ probability that $\\pi_G(s,t)$ is adjacent to a BFS tree in $H$. $\\pi_G(s,t)$ has $>\\mu^3\/n$ heavy nodes, each of degree $\\geq \\mu$. Thus the sum of the degrees of nodes on $\\pi_G(s,t)$ is $> \\mu^4\/n$. By Lemma 5, this implies there are at least $\\mu^4\/3n$ nodes adjacent to $\\pi_G(s,t)$. Each node $v$ has probability $9\\mu\/n$ of being included in $S_1$, and thus having a shortest-path tree rooted at $v$ in $H$. Therefore the probability that \\textit{none} of these nodes adjacent to $\\pi_G(s,t)$ have such a tree rooted at them is\n   \\begin{align*}\n       &\\leq (1-9\\mu\/n)^{\\mu^4\/3n}\\\\\n       &=(1-9\\log^{1\/5}n\/n^{3\/5})^{n^{3\/5}\\log^{4\/5}n\/3}\\\\\n       &=(1-9\\log^{1\/5}n\/n^{3\/5})^{(n^{3\/5}\/9\\log^{1\/5}n)\\cdot 3\\log n}\\\\\n       &\\leq \\left(\\frac{1}{e}\\right)^{3 \\log n}\\\\\n       &\\leq 1\/n^3\n   \\end{align*}\n   where we used the fact that $(1-\\frac{1}{x})^x < 1\/e$ for $x\\geq 1$. Thus, we have a $> 1-1\/n^3$ probability of the existence of a node $r$ neighboring some $u \\in \\pi_G(s,t)$ such that a BFS tree rooted at $r$ is in $H$. When this is the case, we can simply take the $s\\leadsto r$ followed by the $r \\leadsto t$ shortest paths provided by the BFS tree, which has a stretch factor of 2 as shown below:\n \\begin{align*}\n    \\texttt{dist}_H(s,t) &\\leq \\texttt{dist}_H(s,r) + \\texttt{dist}_H(r,t)\\\\\n    &= \\texttt{dist}_G(s,r) + \\texttt{dist}_G (r,t)\\\\\n    &\\leq \\texttt{dist}_G(s,u) + 1 + \\texttt{dist}_G (u,t)+1\\\\\n    &= \\texttt{dist}_G(s,t)+2\n   \\end{align*}\n\n\\end{proof}\n\n\\begin{lemma}\nFor any two uncovered nodes $s,t\\in V(G)$ such that the canonical shortest path $\\pi_G(s,t)$ has $\\leq \\mu^3\/n$ heavy nodes, we have $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4$ with probability $\\geq 1-\\frac{1}{n^3}$.\n\\end{lemma}\n\\begin{proof}\nBoth $s,t$ are uncovered and thus in $\\Gamma_G(S_2)$. Let $x_1,x_2 \\in S_2$ such that $s \\in \\Gamma_H(x_1)$ and $t \\in \\Gamma_H(x_2)$. We assume $x_1 \\neq x_2$ as this case is trivial.\\\\\n\nCall an edge $(u,v)$ ``heavy\" if both $u$ and $v$ are heavy nodes. Since $\\pi_G(s,t)$ is a (shortest) path on $\\mu^3\/n$ heavy nodes, it has $<\\mu^3\/n = g-2$ heavy edges, meaning $P'(x_1,x_2) := (x_1,s) \\circ \\pi_G(s,t) \\circ \\pi_G(t,x_2)$ has $<g$ gray edges and is thus a g-short path\\footnote{If $x_1=s$ or $x_2=t$, $(x_1,s)$ (resp. $(t,x_2)$) are by convention empty paths.}. Thus when we compute the weak CSSSP path $P(x_1,x_2)$, we have that $|P(x_1,x_2)| \\leq |P'(x_1,x_2)|$ . Furthermore, $|P'(x_1,x_2)| \\leq \\pi_G(s,t)+2 \\leq \\texttt{dist}_G(x_1,x_2)+4$, since $\\pi_G(s,t)$ is a shortest path in $G$. Let $P(s,t) = (s,x_1) \\circ P(x_1,x_2) \\circ (x_2,t)$, and note that $P(s,t)$ is a path in $H$. This path witnesses that\n\n\\begin{align*}\n    \\texttt{dist}_H(s,t) &\\leq 2 + |P(x_1,x_2)|\\\\\n    &\\leq 2 + |P'(x_1,x_2)|\\\\\n    &\\leq 2 + 2 + |\\pi_G(s,t)|\\\\\n    &= \\texttt{dist}_G(s,t)+4\n\\end{align*}\n\nThus we deterministically have the required stretch for such node pairs.\n\n\\end{proof}\n\nCorrectness follows by the above two lemmas and the union bound, which gives us that with probability $>1-1\/n$, $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t) + 4$ holds for all $s,t \\in V(G)$.\n\n\\begin{lemma}\nThe subgraph $H$ produced by Algorithm \\ref{alg:best} has $O(n\\mu) = \\widetilde{O}(n^{7\/5})$ edges with high probability.\n\\end{lemma}\n\\begin{proof}\nWe can separate the addition of edges to $H$ into 4 types:\n\n\\begin{enumerate}\n    \\item The edges incident to light nodes are added. Each light node is incident to $\\leq \\mu$ edges by definition, so $O(n\\mu) = \\widetilde{O}(n^{7\/5})$ edges are added.\n    \n    \\item The BFS tree of each node in $S_1$ is added. Each such tree contributes $O(n)$ edges. The probability of a node being added to $S_1$ is $9\\mu\/n$, so $|S_1|=\\Theta(\\mu)$ with high probability, and thus $O(\\mu \\cdot n) = \\widetilde{O}(n^{7\/5})$ edges are added with high probability\n    \n    \\item Edges adjacent to heavy nodes $v$ that are  $\\notin \\Gamma_G(S_2)$ are added. Nodes are added to $S_2$ with probability $1\/\\mu$, thus the probability of $v$ being neither in $S_2$ nor adjacent to a node in $S_2$ is $\\leq (1-1\/\\mu)^{\\deg(v)+1}$. If $\\deg(v) = \\Omega(\\mu \\log n)$, then it is adjacent to a node in $S_2$ with high probability. Thus the number of edges added for $v$ is at most $1+\\deg(v)(1-1\/\\mu)^{\\deg(v)} < \\mu$ with high probability. Unioning over all $v$, this adds $O(n\\mu)=\\widetilde{O}(n^{7\/5})$ edges with high probability.\n    \n    \\item Edges on paths between $S_2$ nodes with $\\leq 5\\mu^3\/n$ heavy edges are added. $|S_2| = \\Theta(n\/\\mu)$ with high probability, yielding $\\Theta(n^2\/\\mu^2)$ pairs of $S_2$. All the light edges (edges adjacent to light nodes) have already been added to $H$, so each path between these $S_2$ pairs adds at most $\\Theta(\\mu^3\/n)$ edges. Unioning over the number of pairs, this adds $O(\\mu n)=\\widetilde{O}(n^{7\/5})$ edges with high probability. \\qedhere\n\\end{enumerate}\n\n\n\n\n\n\n\n\n\n\n\\end{proof}\n\n\n\\begin{lemma}\nOn an $n-$node $m-$edge input graph, Algorithm \\ref{alg:best} runs in $\\widetilde{O}(mn^{3\/5})$ with high probability.\n\\end{lemma}\n\\begin{proof}\n\nThe only two superlinear stages of the algorithm are (a) the generation of the $S_1$ Breadth-First Search trees, and (b) solving weak CSSSP for each node of $S_2$. For (a): nodes are sampled to be in $S_1$ with probability $9\\mu\/n$, so $|S_1| = O(\\mu) = \\widetilde{O}(n^{2\/5})$ with high probability. BFS has worst-case runtime $O(m)$. Thus this stage is $O(m\\mu)=\\widetilde{O}(mn^{2\/5})$ time. For (b): we showed in section 2.1 an algorithm that solves weak CSSSP in $\\widetilde{O}(m)$ time, which we run for each node of $S_2$. Multiplying this over the size of $S_2$ (which has size $\\widetilde{O}(n^{3\/5})$ with high probability), we get $\\widetilde{O}(mn^{3\/5})$ time with high probability.\n\n\n\\end{proof}\n\n\n\\noindent\nTheorem 1 now follows from Lemmas 6-9.\n\n\n\n\\section{Weighted +4 Spanner}\n\n\nIn this section, we prove Theorem 2 by first synthesizing a weighted analogue of the weak CSSSP problem from section 2.1, then we apply this in a similar fashion to create our $+4W(s,t)+\\epsilon W$ construction. We note that a $+4W$, $\\widetilde{O}(n^{7\/5})$ edge, $\\widetilde{O}(mn^{3\/5})$ time spanner construction is a very straightforward generalization of the methods presented in the unweighted part of this paper. However, $W(s,t)$ error is preferable (except in the case when all the edge weights are equal). \n\n\n\n\n\\subsection{Weighted Constrained Shortest Paths}\n\nIn our first step towards our weighted spanner, we will generalize weak CSSSP from section 2.1 to the weighted setting, and give a Dijkstra-time algorithm for solving it. This is also where the $\\epsilon$ error of the construction will be incurred, so our generalization will incorporate an error parameter. There are many ways to generalize weak CSSSP to the weighted setting, but we choose this one as it is what we've been able to find a use for in the weighted spanner construction.\n\n\\begin{definition}[Weighted Weak CSSSP (with error)] The weighted weak constrained single-source shortest paths problem with error is defined by the following algorithm contract:\n\n\\begin{itemize}\n    \\item \\textbf{Input:} An (undirected) graph $G=(V,E)$, a set of ``gray\" edges $E_g \\subset E$, a source vertex $s \\in V$, a weight function $w:V\\to \\mathbb{R}^+$, an error parameter $1>\\epsilon>0$, and a positive integer $g$.\n    \n    \\item \\textbf{Output:} For every $t \\in V$, a path $P(s,t)$ on $\\leq 5g\/\\epsilon$ gray edges, satisfying the following:\n    \\begin{itemize}\n        \\item If an $s\\leadsto t$ g-short \\footnote{We reuse the same definition from section 2.1.} path exists, then the \\textbf{near g-optimality condition} holds: $w(P(s,t)) < w(P_g(s,t)) + \\epsilon W$ for any $s\\leadsto t$ path $P_g(s,t)$ on $<g$ gray edges, where $W = \\max_{e \\in E}w(E)$.\n        \n        \\item If no such paths exist, $P'(s,t)$ can be anything.\n        \n       \n    \\end{itemize}\n\n\\end{itemize}\n\\end{definition}\n\nThe only difference between this version of the problem and the unweighted version is that it is in terms of path weight instead of path length, and it allows an $\\epsilon$ error in terms of the heaviest edge of the graph. As before, one can use AliAbdi et al's Bi-SPP \\cite{AliAbdi2019} to solve this, but it is stronger than necessary. A modification to Algorithm \\ref{alg:weak} gives us the fastest solution. It is the same as before, except that we now punish each gray edge by $+\\epsilon W g^{-1}$ instead of $+g^{-1}$.\n\n\\begin{figure}[h]\n\\begin{algorithm}[H]\n    \\label{alg:deltaweighted}\\caption{Weighted Weak CSSSP with Error by Edge Punishing}\n    \\KwIn{Undirected graph $G=(V,E)$\\\\\n    Source vertex $s\\in V$\\\\\n    Set of gray edges $E_g \\subseteq E$\\\\\n    Weight function $w:E \\to \\mathbb{R}^{+}$\\\\\n    Error $1>\\epsilon>0$\\\\\n    Positive integer $g$\n    }\\vspace{1em}\n   \n   Let $W = \\max_{e \\in E} w(e)$\n   \n   \n    \\ForEach{edge $(u,v) \\in E$}{\n        $w'(u,v)\\leftarrow w(u,v)+\\epsilon W g^{-1}$ if $(u,v)$ is gray, and $w'(u,v)\\leftarrow w(u,v)$ otherwise.\n    } \n    \n    \n    Run Dijkstra's algorithm on $G$ with source node $s$ and weight function $w'$ to get paths $P(s,t)$ for each other $t \\in V$\\\\\n    \\Return these $P(s,t)$ paths \n\n\\end{algorithm}\n\\end{figure}\n\n\n\\begin{theorem}\nAlgorithm \\ref{alg:deltaweighted} solves Weighted Weak CSSSP  with error in $O(m + n \\log n) = \\widetilde{O}(m)$ time.\n\\end{theorem}\n\\begin{proof}\n\nThe time complexity follows immediately from the complexity of Dijkstra's algorithm, which is the dominant stage of the algorithm. We now prove correctness.\n\nLet $t \\in V$ and suppose a g-short $s\\leadsto t$ path $P'(s,t)=(s,s') \\circ \\pi_G(s',t') \\circ (t',t)$ exists. We will first show that $P(s,t)$ has $\\leq 5g\/ \\epsilon$ gray edges. Suppose to show a contradiction that it has $>5g\/\\epsilon$ gray edges. Thus\n\n\n\\begin{align*}\n    w(P(s,t)) + 5W &< w'(P(s,t))\\\\\n    &\\leq w'(P'(s,t))\\\\\n    &< w(P'(s,t)) + g\\cdot (\\epsilon W g^{-1}) = w(P'(s,t)) + \\epsilon W\\\\\n    &\\leq w(P'(s,t)) + W\\\\\n    &\\leq w(\\pi_G(s',t'))+2W+W\n\\end{align*}\n\nTherefore $w(P(s,t)) + 4W < w(\\pi_G(s',t')) + 2W$. Since $\\pi_G(s',t')$ is a shortest path, $w(\\pi_G(s',t')) \\leq w(P(s,t)) + 2W$. This implies $w(P(s,t)) +4W < w(P(s,t)) + 4W$, a contradiction.\\\\\n\n\nNow we prove near g-optimality. Let $P_g(s,t)$ be an arbitrary $s\\leadsto t$ path on $<g$ gray edges. Then \n\n\\begin{align*}\nw(P(s,t)) &\\leq w'(P(s,t))\\\\\n&\\leq w'(P_g(s,t))\\\\ \n&< w(P_g(s,t)) + g \\cdot (\\epsilon W g^{-1})\\\\\n&= w(P_g(s,t)) + \\epsilon W\n\\end{align*}\n\nas required.\n\n\\end{proof}\n\n\n\n\\subsection{$+4W(\\cdot,\\cdot)+\\epsilon W$ Spanner}\n\nIn this section we modify our Section 2 construction to construct $+4W(\\cdot,\\cdot) + \\epsilon W$ spanners, for any $\\epsilon > 0$, on $\\widetilde{O}_\\epsilon(n^{7\/5}e^{-1})$ edges in $\\widetilde{O}(mn^{3\/5})$ time. This will make use of our weak CSSSP generalization to the weighted setting with error discussed in the previous subsection, which is where the $+\\epsilon W$ stretch is incurred\n\nThe construction differs from Algorithm \\ref{alg:best} in three important ways: (i): instead of adding all the edges of light nodes to the spanner $H$, we perform a \\textbf{$\\mu$-lightweight initialization} - that is, we add the $\\mu$ lightest edges of each node to $H$ (breaking ties arbitrarily), a technique introduced in \\cite{ahmed2021additive}.\n\n\\begin{definition}[$d-$lightweight initialization \\cite{ahmed2021additive}]\nA $d-$lightweight initialization $H = (V,E')$ of a weighted graph $G = (V,E)$ is a subgraph created by selecting the $d$ lightest edges of every node of $G$, breaking ties arbitrarily.\n\\end{definition}\n\n(ii): Instead of computing standard weak CSSSP (as seen in Section 2.2) on the $S_2$ nodes, we compute our new weighted version with error. (iii): We now omit the step of connecting ``heavy nodes\" with nodes of $S_2$. Instead we rely on these connections happening ``naturally\" due to our $\\mu$-lightweight initialization, which allows us to make use of the properties the lightweight initialization gives us.\n\n\\begin{figure}[h]\n\\begin{algorithm}[H]\n    \\label{alg:weightedspan}\\caption{$+4W(\\cdot,\\cdot) + \\epsilon W$ Spanner}\n    \\KwIn{$n$-node graph $G=(V,E)$\\\\\n\tWeight function $w:E \\mapsto \\mathbb{R}^+$\\\\\n\tError parameter $1>\\epsilon>0$\\\\\n\t}\\vspace{1em}\n    Let $H_0=(V,E')$ be a $\\mu$-lightweight initialization of $G$.\\\\\n    \n    Sample a set of nodes $S_2$ at random, every node with probability $1\/\\mu$\\\\\n    \\ForEach{node $x$ so that $(\\{x\\}\\cup\\Gamma_{H_0}(x))\\cap S_2 = \\varnothing$}{\n        Add all incident edges of $x$ to $E'$\\\\\n    }\n    Sample a set of nodes $S_1$ at random, every node with probability $9\\mu\/n$\\\\\n    \\ForEach{node $x\\in S_1$}{\n        Construct a shortest-path tree $T(x)$ rooted at $x$ spanning all vertices in $V$\\\\\n        $E'=E'\\cup E(T(x))$\\\\\n    }\\vspace{1em}\n    %\n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n   \n    \n    \\ForEach{node $x_1 \\in S_2$}{\n        Compute weighted weak CSSSP with error on $G$, with $g = \\mu^3\/n+2$, $x_1$ as the source vertex, $E_g = E \\setminus E'$, and $\\epsilon$ as the error parameter, to get paths $P(x_1,x_2)$ for each $x_2 \\in V$.\\\\\n        Add $E(P(x_1,x_2))$ to $E'$ for each $t \\in S_2$\n    }\n    \n\n    \\Return $H=(V,E')$\n\\end{algorithm}\n\\end{figure}\n\n\\noindent\nNote that $H_0$ in Algorithm 5 denotes the $\\mu-$initialization of $G$, before any additional edges are added to $E'$. We will refer to $H_0$ in our proofs as it will allow us to make use of the fact that the edges added are only the ones form the lightweight initialization. We now prove correctness by the following series of lemmas. We will make use of the following theorem due to Ahmed et al.\n\n\n\\begin{theorem}[\\cite{ahmed2020weighted}]\nLet $H$ be a $d-$lightweight initialization of an undirected, weighted graph $G$. Then if a shortest path $\\pi_G(s,t)$ is missing $l$ edges in $H$, there are $\\Omega(dl)$ different nodes adjacent to $\\pi_G(s,t)$ in $H$.\n\\end{theorem}\n\n\n\\begin{lemma}\nFor any two nodes $s,t\\in V(G)$ such that the canonical shortest path $\\pi_G(s,t)$ is missing $>\\mu^3\/n$ edges in $H_0$, we have $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4W(s,t)$ with probability $\\geq 1-\\frac{1}{n^3}$\n\\end{lemma}\n\n\\begin{proof}\n\nThis implies that there are $>\\mu^3\/n$ \\textit{nodes} along $\\pi_G(s,t)$ with missing incident edges in $H_0$. Call the set of these nodes $S$. Since $H_0$ is a $\\mu-$lightweight initialization, it follows that nodes missing an edge in $H_0$ must have degree $>\\mu$ in $G$. We utilize the above theorem from \\cite{ahmed2020weighted}, which allows us to conclude that there are $\\Omega(\\mu^4\/n)$ different nodes adjacent to $\\pi_G(s,t)$ in $H_0$. \n\n\n\nAs shown in Lemma 4 (the details of which we won't repeat), we have a $> 1-1\/n^3$ probability of one of these neighbors being a member of $S_1$. In this case, let $r\\in S_1$ be adjacent to a node $q$ of $\\pi_G(s,t)$ in $H_0$. Since $(q,r) \\in H_0$ but $q$ is disconnected from $\\pi_G(s,t)$ in $H_0$, it follows from the fact that $H_0$ is a $\\mu-$lightweight initialization that $w(q,r)$ is lighter than the missing edge incident to $q$ on this path, i.e. $w(q,r) \\leq W(s,t)$.\n\nSince $r \\in S_1$, it is the root of a shortest path tree of $G$ included in $H$. Thus, since $\\pi_G(s,r)$ is a shortest path, $w(\\pi_G(s,r)) \\leq  w(\\pi_G(s,q)) + w(q,r) \\leq w(\\pi_G(s,q))+W(s,t)$. Likewise, $w(\\pi_G(r,t)) \\leq w(\\pi_G(q,t)) + W(s,t)$. Thus, the path $\\pi_G(s,r) \\circ \\pi_G(r,t)$, which belongs to $H$, witnesses that\n\n\\begin{align*}\n\\texttt{dist}_H(s,t) &\\leq w(\\pi_G(s,q))+W(s,t) + w(\\pi_G(q,t)) + W(s,t)\\\\\n&= w(\\pi_G(s,t)) + 2W(s,t)\\\\\n&= \\texttt{dist}_G(s,t) + 2W(s,t)\n\\end{align*} \n\nThus with probability $>1-1\/n^3$, we achieve the required stretch.\n\n\\end{proof}\n\n\\begin{lemma}\nFor any two nodes $s,t\\in V(G)$ such that the canonical shortest path $\\pi_G(s,t)$ is missing $< \\mu^3\/n$ edges in $H_0$, we have $\\texttt{dist}_H(s,t) \\leq \\texttt{dist}_G(s,t)+4W(s,t) + \\epsilon W$.\n\\end{lemma}\n\\begin{proof}\nJust as in the unweighted case, we can assume WLOG that $s$ and $t$ are uncovered, and thus are each in the neighborhood of an $S_2$ node (all nodes $\\notin \\gamma(S_2)$ are covered by the algorithm). Let $x_1,x_2 \\in S_2$ such that $s \\in \\Gamma(x_1)$ and $t \\in \\Gamma(x_2)$. We can furthermore assume that the first and last edges of $\\pi_G(s,t)$ are missing from $H_0$, otherwise we could simply push our analysis to the first\/last nodes on $\\pi_G(s,t)$ to be severed from the path in $H_0$.\n\nSince $\\pi_G(s,t)$ is a shortest path with $< g-2 = \\mu^3\/n$ missing (gray) edges, $P'(x_1,x_2):=(x_1,s')\\circ \\pi_G(s',t') \\circ (t',x_2)$ is a g-short path. Thus by weighted weak CSSSP with error, our construction yields a path $P(x_1,x_2)$ with $w(P(x_1,x_2)) \\leq w(P'(x_1,x_2)) + \\epsilon W$. Furthermore, by the fact that $H_0$ is a $\\mu$-lightweight initialization, the edge $(x_1,s)$ must be lighter than the first edge of $\\pi_G(s,t)$, and $(t,x_2)$ must be lighter than the last edge of $\\pi_G(s,t)$. Thus $w(x_1,s),w(t,x_2) \\leq W(s,t)$.\\\\\n\nTherefore, $w(P(x_1,x_2)) \\leq w(\\pi_G(s,t))+ 2W(s,t) + \\epsilon W$. Thus, the path $(s,x_1) \\circ P(x_1,x_2) \\circ (x_2,t)$ in $H$ witnesses that\n\n\\begin{align*}\n\\texttt{dist}_H(s,t) &\\leq w(s,x_1) + w(P(x_1,x_2)) + w(x_2,t)\\\\\n&\\leq 2W(s,t) + w(P(x_1,x_2))\\\\\n&\\leq 2W(s,t) + 2W(s,t) + \\epsilon W + w(\\pi_G(s,t))\\\\\n&= \\texttt{dist}_G(s,t) + 4W(s,t) + \\epsilon W\n\\end{align*}\n\\noindent\nThus we deterministically have the required stretch for such node pairs.\n\n\\end{proof}\n\n\\noindent\nCorrectness now follows by the above two lemmas and the union bound.\n\n\\noindent\nWe now show that $H$ has the desired edge bound. We note that the details of this proof are mostly the same as the corresponding proof for our unweighted construction.\n\n\\begin{lemma}\n$H$ has (with high probability) $\\widetilde{O}_\\epsilon(n^{7\/5}\\epsilon^{-1})$ edges.\n\\end{lemma}\n\\begin{proof}\nAlgorithm \\ref{alg:weightedspan} adds edges to $H$ in 4 stages:\n\\begin{enumerate}[(i)]\n    \\item We begin with a $\\mu-$lightweight initialization of $G$, which adds $\\widetilde{O}(n^{2\/5})$ edges per node, giving a total of $\\widetilde{O}(n^{7\/5})$ edges. This covers all light nodes (of degree $<\\mu$).\n    \n    \n   \n    \\item We add the edges of a shortest path tree for each node of $S_1$. The probability of a node's inclusion into $S_1$ is $9\\mu\/n$, thus $|S_1| = \\widetilde{O}(n^{2\/5})$ with high probability. Adding $O(n)$ edges for each of the $S_1$ nodes thus yields $\\widetilde{O}(n^{7\/5})$ edges added with high probability.\n    \n    \n   \n    \\item We add all the edges of heavy nodes $v$ (of degree $>\\mu$) not in the neighborhood of any $S_2$ node. Nodes are added to $S_2$ with probability $1\/\\mu$, thus the probability of $v$ being neither in $S_2$ nor adjacent to a node in $S_2$ is $\\leq (1-1\/\\mu)^{\\deg(v)+1}$. If $\\deg(v) = \\Omega(\\mu \\log n)$, then it is adjacent to a node in $S_2$ with high probability. Thus the number of edges added for $v$ is at most $1+\\deg(v)(1-1\/\\mu)^{\\deg(v)} < \\mu$ with high probability. Unioning over all $v$, this adds $O(n\\mu)=\\widetilde{O}(n^{7\/5})$ edges with high probability.\n    \n    \\item Edges on paths between $S_2$ nodes with $\\leq 5\\epsilon^{-1}\\mu^3\/n$ additional edges are added to $H$. $|S_2| = \\Theta(n\/\\mu)$ with high probability, yielding $\\Theta(n^2\/\\mu^2)$ pairs of $S_2$. Since each path between these pairs incurs $\\leq 5\\epsilon^{-1}\\mu^3\/n$ extra edges, this adds $O_\\epsilon(\\mu n \\epsilon^{-1})=\\widetilde{O}_\\epsilon(n^{7\/5}\\epsilon^{-1})$ edges with high probability.\n\\end{enumerate}\n\\end{proof}\n\n\nFinally, we show that Algorithm \\ref{alg:weightedspan} has the desired runtime.\n\n\\begin{lemma}\nOn an $n-$node $m-$edge input graph, Algorithm \\ref{alg:weigtedspan} runs in $\\widetilde{O}(mn^{3\/5})$ with high probability.\n\\end{lemma}\n\\begin{proof}\n\nThe only two superlinear stages of the algorithm are (a) the generation of the $S_1$ shortest-path trees, and (b) solving weak weighted CSSSP with error for each node of $S_2$. For (a): nodes are sampled to be in $S_1$ with probability $9\\mu\/n$, so $|S_1| = O(\\mu) = \\widetilde{O}(n^{2\/5})$ with high probability. Dijkstra has worst-case runtime $\\widetilde{O}(m)$. Thus this stage is $\\widetilde{O}(mn^{2\/5})$ time. For (b): we showed in section 3.1 an algorithm that solves weak CSSSP in $\\widetilde{O}(m)$ time, which we run for each node of $S_2$. Multiplying this over the size of $S_2$ (which has size $\\widetilde{O}(n^{3\/5})$ with high probability), we get $\\widetilde{O}(mn^{3\/5})$ time.\n\n\n\\end{proof}\n\n\\noindent\nBy Lemmas 12-15, we have now proven the main result of this chapter (Theorem 2).\n\n\n\n\n\\section{Conclusion}\n\nIn this paper we have presented a new state-of-the-art $\\widetilde{O}(mn^{3\/5})$ complexity result for constructing the +4 spanner, doing so by solving a novel pathfinding problem (weak CSSSP). This fills in a literature gap that has existed between +2,+6, and +8 spanners, as this is the first paper studying the efficiency of the +4 spanner construction. We also extended our methods to the weighted setting, where we were able to to derive a construction for $+4W(s,t)+\\epsilon W$ spanners, with the same runtime and an $\\epsilon^{-1}$ stretch to spanner size.\n\nWe believe that to find further polynomial time improvements to our construction would require a polynomial reduction in the number of $S_2$ nodes to compute shortest path trees on. The next bottleneck to the algorithm is the time needed to build the BFS trees rooted at the $S_1$ nodes, which is $\\widetilde{O}(mn^{2\/5})$ with high probability. For the weighted spanner construction, we hope to see a reduction in error to $(4+\\epsilon)W(s,t)$ or $+4W(s,t)$ without a compromise to runtime, eliminating global error entirely.\n\n\n\n\n\n\n\n\n\\section*{Acknowledgements}\n\nMany thanks to Greg Bodwin, without whose guidance this paper would not be possible. I also thank fellow students Eric Chen and Cheng Jiang, with whom I discussed an earlier version of the paper.\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe concept of fuzzy transform ($F$-transform) was firstly introduced by Perfilieva \\cite{per}, a theory that attracted the interest of {many researchers}. It has now been greatly expanded upon, and a new chapter in the theory of semi-linear spaces has been opened. The main idea of the $F$-transform is to factorize (or fuzzify) the precise values of independent variables by using a proximity relation, and to average the precise values of dependent variables to an approximation value. The theory of $F$-transform has already been developed and used to real-valued to lattice-valued functions (cf., \\cite{{per},{irin1}}), from fuzzy sets to parametrized fuzzy sets \\cite{st} and from the single variable to the two (or more variables) (cf., \\cite{{mar2},{mar}, {mar1}, {step}}). Recently, several studies have begun to look into $F$-transforms based on any $L$-fuzzy partition of an arbitrary universe (cf., \\cite{kh1,jir,mo, mocko, anan,spt1,rus,spt}), where $L$ is a complete residuated lattice. Among these researches, the relationships between $F$-transforms and {semimodule homomorphisms} were introduced in \\cite{jir}; a categorical approach of $L$-fuzzy partitions was studied in \\cite{mo}; while, the relationships between $F$-transforms and similarity { relations were} discussed in \\cite{mocko}. Further, in \\cite{anan}, an interesting link among $F$-transforms, $L$-fuzzy topologies\/co-topologies and $L$-fuzzy approximation operators (which are concepts used in the study of an operator-oriented view of fuzzy rough set theory) was established, while in \\cite{spt1}, the relationship between fuzzy pretopological spaces and spaces with $L$-fuzzy partition was shown. Also, in a different direction, a generalization of $F$-transforms was presented in \\cite{rus} by considering the so-called $Q$-module transforms, where $Q$ stands for an unital quantale, while $F$-transforms based on a generalized residuated lattice {were studied} in \\cite{spt}. Further, classes of $F$-transforms taking into account the well-known classes of implicators, namely {$R-,S-, QL-$implicators} were discussed in \\cite{tri}. The several researches carried out in the application fields of $F$-transforms, e.g., trend-cycle estimation \\cite{holc}, {data compression \\cite{hut}}, numerical solution of partial differential equations \\cite{kh}, scheduling \\cite{li}, time series \\cite{vil}, data analysis \\cite{no}, denoising \\cite{Ir}, face recognition \\cite{roh}, neural network {approaches \\cite{ste} and trading} \\cite{to}.\\\\\\\\\n{It is easy to see how the dynamics of the species would be impacted when they interact. Many studies have been conducted on the problem of the food chain. If the growth rate of one species increases while that of the other decreases during their interaction, we say they are in a predator-prey situation. Such a situation arises when one species (predator) feeds on another species (prey). The Lotka-Volterra model is the first fundamental system representing the interaction between prey and predator species. During the first world war, the Lotka-Volterra model was developed to explain the oscillatory levels of certain fish in the Adriatic sea in \\cite{elet,murray}. Several features of predator-prey models have been the subject of many mathematical and ecological studies. There are many factors such as functional response \\cite{liu}, competition \\cite{cush,gak}, cooperation \\cite{elet} affecting dynamics of predator-prey model. The stability and other dynamical behavior of predator-prey models could be found in \\cite{aziz,bell,liou,hsu,gakk,brau,oat,past,jp}. The parameters in all the above-cited models are crisp in nature. However, in real-world ecosystems, many parameters may oscillate simultaneously with periodically varying environments. They also change due to natural and human-caused events, such as fire, earthquakes, climate warming, financial crisis, etc. As a result, environmental variables significantly impact the interaction process between the species and its dynamics. Thus the fuzzy mathematical model is more effective than the crisp model. Therefore we have considered the fuzzy set theory to create the prey-predator model. Specifically, the imprecise parameters are replaced by fuzzy numbers in the fuzzy approach. The analysis of the behavior of most phenomena is often based on mathematical models in the form of differential equations. Fuzzy differential equations are equations in which uncertainties are modeled by fuzzy sets (possibility). In recent years, analyzing the dynamical behavior of prey-predator systems whose mathematical models have been considered fuzzy differential equations. Obtaining the solution of the fuzzy differential equation has been investigated under the concept of H-derivative, SGH-derivative \\cite{bede}, gH-derivative and g-derivative \\cite{bede1}, H$_2$-differentiability \\cite{maz}, and gr-derivative \\cite{mazan}, it has been researched how to solve fuzzy differential equations. Also, in \\cite{hoa,hoa1,long,long1,long2}, fractional calculus has been used to discuss fuzzy differentiable equations. In \\cite{sun1}, it has been investigated how stable fuzzy differentiable equations using the second kind of Hukuhara derivative are in the application.\\\\\\\\\nAs long as the underlying fuzzy functions are highly generalized Hukuhara differentiable, the approach described in \\cite{sun} could be used for the stability analysis. Thus the limitation of the method as mentioned above relates to the existence of a highly generalized Hukuhara derivative. As the method presented in \\cite{sun} is based on what is known as Fuzzy Standard Interval Arithmetic (FSIA), then it has a flaw known as the UBM phenomenon (see \\cite{mazan} for more details). A novel fuzzy derivative idea termed granular derivative terms of relative-distance-measure fuzzy interval arithmetic (RDM-FIA) was presented in \\cite{mazan} to overcome the drawbacks of the FSIA-based approach. A new idea of the conventional membership functions called the horizontal membership functions (HMFs) proposed in \\cite{piegat} was used to construct RDM-FIA. Based on the findings in \\cite{land,piegat1,piegat2,son}, it has been established that the RDM-FIA is a more helpful application tool than the FSIA.} \\\\\\\\ \nDifferential equations cannot always be solved analytically, requiring numerical methods. Therefore in scientific research, numerical methods for solving differential equations have been elaborated frequently. In this connection, differential equations are successfully solved using fuzzy techniques. The fuzzy transform ($F$-transform) introduced by Perfilieva \\cite{per} is one of the fuzzy techniques that has been introduced in the literature. An approximation method based on $F$-transform for second order differential equation was introduced in \\cite{chen}. In \\cite{ali,kh,stev,p}, numerical methods based on $F$-transform to solve initial value problem and boundary value problem were introduced. Also, a numerical method based on $F$-transform to solve a class of delay differential equations by $F$-transform was introduced in \\cite{tom}.\\\\\\\\\nIt is to be pointed out here that the numerical solution of a differential equation or fuzzy differential  based on $F$-transform (by using the concept of level sets) was studied, but the numerical solution based on granular $F$-transform of a fuzzy mathematical model, which is represented by fuzzy differential equations under granular differentiability, is yet to be done. In which all parameters and initial conditions can be uncertain. Specifically,\n\\begin{itemize}\n\\item[$\\bullet$] we introduce the concepts of granular $F$-transform and granular inverse $F$-transform associated with the fuzzy function and discuss some basic results by using the concept of granular metric;\n\\item[$\\bullet$] we formulate a fuzzy prey-predator model and investigate the equilibrium points and their stability in terms of fuzzy numbers;\n\\item[$\\bullet$] we establish a numerical method based on granular $F$-transform for the fuzzy prey-predator model; and\n\\item[$\\bullet$] we present a comparison between two numerical solutions with the exact solution.\n\\end{itemize}\n\\section{Preliminaries} Herein, the ideas associated with fuzzy number, horizontal membership function, granular differentiability and fuzzy partition (cf., \\cite{maz,mazan,naj,naja,per,kh}, for details). Throughout this chapter, $E^1$ denotes the collection of fuzzy numbers defined on the real number $\\mathbb{R}$ and $E^m=E^1\\times E^1\\times...\\times E^1$. The $\\alpha$-level sets of $\\widehat{a}\\in E^1$ is $\\widehat{a}^\\alpha=[\\underline{a}^\\alpha,\\overline{a}^\\alpha]$, where $\\underline{a}^\\alpha $ and $ \\overline{a}^\\alpha$ are the left and right end points of $\\widehat{a}$.\n\\begin{def1}\\label{grhor}\nFor a fuzzy number $\\widehat{a}: [a_1,a_2] \\subseteq \\mathbb{R} \\rightarrow [0, 1]$ and $\\widehat{a}^\\alpha=[\\underline{a}^\\alpha,\\overline{a}^\\alpha]$, the\n{\\bf horizontal membership function} is a function $ a^{gr} : [0, 1] \\times [0, 1] \\rightarrow [a, b]$ such that $a^{gr}(\\alpha, \\mu_a)=\\underline{a}^\\alpha+(\\overline{a}^\\alpha-\n\\underline{a}^\\alpha)\\mu_a$, where $\\mu_a\\in [0, 1]$ is called the relative-distance-measure (RDM) variable.\n\\end{def1}\n\\begin{rem}\\label{grr1} (i) The horizontal membership function of $\\widehat{a}\\in E^1$, i.e., $a^{gr}(\\alpha, \\mu_{a})$ is also denoted by $\\mathbb{K}(\\widehat{a})$. Also, the $\\alpha$-level set of $\\widehat{a}$ can be given by \\[\\mathbb{K}^{-1}(a^{gr}(\\alpha, \\mu_{a}))=\\widehat{a}^\\alpha=[\\inf\\limits_{\\beta\\geq\\alpha}\\min\\limits_{\\mu_{a}}a^{gr}(\\beta, \\mu_{a}),\\sup\\limits_{\\beta\\geq\\alpha}\\max\\limits_{\\mu_{a}}a^{gr}(\\beta, \\mu_{a})].\\]\n\\begin{itemize}\n \\item[(ii)] For the fuzzy numbers $\\widehat{a_1},\\widehat{a_2}\\in E^1$, $\\widehat{a_1}=\\widehat{a_2}$ iff $\\mathbb{K}(\\widehat{a_1})=\n \\mathbb{K}(\\widehat{a_2})$ and $\\widehat{a_1}\\geq \\widehat{a_2}$ if $\\mathbb{K}(\\widehat{a_1})\\geq\n \\mathbb{K}(\\widehat{a_2}),\\,\n \\forall\\,\\mu_{a_1}=\\mu_{a_2}\\in [0,1]$. \n \\item[(iii)] Let each of addition, subtraction, multiplication and division operations between fuzzy numbers $\\widehat{a_1},\\widehat{a_2}\\in E^1$ be represented by $\\odot$. Therefore $\\widehat{a_1}\\odot\\widehat{a_2}=\\widehat{m}\\in E^1$ iff $\\mathbb{K}(\\widehat{m})=\n \\mathbb{K}(\\widehat{a_1})\\odot\\mathbb{K}(\\widehat{a_2})$.\n \\item[(iv)] Let $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3}\\in E^1$. Then we have\n \\item[$\\bullet$] $\\widehat{a_1}-\\widehat{a_2}=-(\\widehat{a_1}+\\widehat{a_2})$,\n \\item[$\\bullet$] $\\widehat{a_1}-\\widehat{a_1}=0$,\n \\item[$\\bullet$] $\\widehat{a_1}\\div\\widehat{a_1}=1$, and\n \\item[$\\bullet$] $(\\widehat{a_1}+\\widehat{a_2})\\widehat{a_3}=\\widehat{a_1}\\widehat{a_3}+\\widehat{a_2}\\widehat{a_3}$.\n \\item[(v)] A fuzzy function is a generalization of a classical function in which the domain or range, or both, is a subset of the fuzzy numbers set.\n \\end{itemize}\n \\end{rem}\n\\begin{def1}\n The horizontal membership function of $\\widehat{g}(\\widehat{p_{1}}(u),\\widehat{p_{2}}(u),...,\\widehat{p}_m(u))$ is defined by $\\mathbb{K}(\\widehat{g}(\\mathbb{K}(\\widehat{p}_1(u)),\\mathbb{K}(\\widehat{p_{2}}(u)),...,\\mathbb{K}(\\widehat{p}_m(u))))$, where $\\widehat{g}:E^m\\rightarrow E^1$ and $\\widehat{p_i}:[a_1,a_2]\\subseteq\\mathbb{R}\\rightarrow E^1,\\,i=1,2,...,m$, are fuzzy functions.\n\\end{def1}\n\\begin{def1}\\label{grmetric}\nA fuzzy function $d^{gr}:E^1\\times E^1\\rightarrow \\mathbb{R}^+\\cup \\{0\\}$ \nis called the {\\bf granular metric} if \\[d^{gr}(\\widehat{a_1},\\widehat{a_2})=\\sup\\limits_{\\alpha\\in[0,1]}\\max\\limits_{\\mu_{a_1},\\mu_{a_2}\\in[0,1]}|a_1^{gr}(\\alpha,\\mu_{a_1})-a_2^{gr}(\\alpha,\\mu_{a_2})|,\\,\\forall\\,\\widehat{a_1},\\widehat{a_2}\\in E^1.\\]\n\\end{def1}\n\\begin{def1}\\label{grcont}\nA fuzzy function $\\widehat{g}: [a_1,a_2] \\subseteq \\mathbb{R} \\rightarrow E^1$ is called the {\\bf granular continuous} (gr-continuous) if for all $u_0\\in[a_1,a_2],\\epsilon>0$ there is a $\\delta>0$ such that $$d^{gr}(\\widehat{g}(u),\\widehat{g}(u_0))<\\epsilon, \\,\\,\\mbox{whenever}\\, |u-u_0|<\\delta.$$\n\\end{def1}\n\\begin{def1} \\label{grgrdif}\nA fuzzy function $\\widehat{g}: [a_1,a_2] \\subseteq \\mathbb{R} \\rightarrow E^1$ is called the {\\bf granular differentiable} (gr-differentiable) at the point\n$u\\in [a_1,a_2]$ if there is a fuzzy number $\\mathcal{D}^{gr} \\widehat{g} (u)\\in E^1$ such that the following limit exists:\n\\[\\mbox{lim}_{h\\to 0}\\frac{\\widehat{g}(u + h)-\\widehat{g}(u)}{h}=\\mathcal{D}^{gr} \\widehat{g} (u).\\]\n\\end{def1}\n\\begin{pro}\\label{grgrdif1}\nAt any point $u\\in [a_1,a_2]$, a fuzzy function $\\widehat{g} : [a_1,a_2] \\subseteq \\mathbb{R} \\rightarrow E^1$ is gr-differentiable iff $\\mathbb{K}(\\widehat{g})$ is differentiable w.r.t. $u$ at that point. In addition, \n$$\\mathbb{K}(\\mathcal{D}^{gr} \\widehat{g} (u))=\\frac{\\partial}{\\partial t}\\mathbb{K}(\\widehat{g} (u)).$$\n\\end{pro}\n\\begin{pro}\\label{grgrpdif}\nThe fuzzy function $\\widehat{g}(\\widehat{p}_1(u),\\widehat{p_{2}}(u),...,\\widehat{p}_m(u))$ is gr-partial differentiable w.r.t. $\\widehat{p}_i(u)$ iff its horizontal membership function is differentiable w.r.t. the horizontal membership function of $\\widehat{p}_i(u)$, where $\\widehat{g} : E^m \\rightarrow E^1$ and $\\widehat{p}_i : [a_1,a_2] \\subseteq \\mathbb{R} \\rightarrow E^1,\\,i = 1, 2, . . . ,m$, are fuzzy functions.. Moreover,\n\\[\\mathbb{K}(\\frac{\\partial^{gr}}{\\partial \\widehat{p}_i} \\widehat{g}(\\widehat{p}_1(u),\\widehat{p_{2}}(u),...,\\widehat{p}_m(u)))=\\frac{\\partial}{\n\\partial \\mathbb{K}(\\widehat{p}_i)}\\mathbb{K}(\\widehat{g}(\\mathbb{K}(\\widehat{p}_1(u)),\\mathbb{K}(\\widehat{p_{2}}(u)),...,\\mathbb{K}(\\widehat{p}_m(u))))\n.\\]\n\\end{pro}\n\\begin{def1}\\label{grgrint}\nLet $g^{gr}(u, \\alpha, \\mu_g )$ is integrable on $u\\in [a_1,a_2]$, where $g^{gr}(u, \\alpha, \\mu_g )$ is a horizontal membership\nfunction of a gr-continuous fuzzy function $\\widehat{g} : [a_1,a_2]\\subseteq \\mathbb{R}\\rightarrow E^1$. In addition, let integral of $\\widehat{g} $ on $[a_1,a_2]$ be represented by $\\oint^{a_2}_{a_1} \\widehat{g} (u)du$. If there exists a fuzzy number $\\widehat{m} = \\oint^{a_2}_{a_1} \\widehat{g} (u)du$ such that $\\mathbb{K}(\\widehat{m})=\\int^{a_2}_{a_1} \\mathbb{K}(\\widehat{g} (u))du$, then the fuzzy function $\\widehat{g}$ is called the {\\bf granular fuzzy integrable} on $[a_1,a_2]$.\n\\end{def1}\n\\begin{def1}\\label{grgrpol}\nA {\\bf granular fuzzy polynomials} is an expression consisting of fuzzy variables and fuzzy coefficients that involves only the granular operations of addition, subtraction, multiplication and nom-negative with integer exponents of fuzzy variables. \n\\end{def1}\nFor example- $\\widehat{g}(\\widehat{p})=\\widehat{a}_m\\widehat{p}^n+\\widehat{a}_{m-1}\\widehat{p}^{m-1}+...+\\widehat{a}_1\\widehat{p}+\\widehat{a}_0,\\,\\forall\\,\\widehat{a}_i\\in E^1,\\,i=1,...,m$.\n\\begin{def1}\\label{grroot}\nA {\\bf fuzzy root} of a granular fuzzy polynomial $\\widehat{g}(\\widehat{p})$ is a fuzzy number $\\widehat{p}_i$ such that $\\widehat{g}(\\widehat{p}_i)=0$.\n\\end{def1}\n\\begin{rem}\\label{grr2}\nIt is easy to check that if $\\widehat{p}_k$ is a fuzzy root of $\\widehat{g}(\\widehat{p})$, then $\\mathbb{K}(\\widehat{p}_i)$ is a root of $\\mathbb{K}(\\widehat{g}(\\mathbb{K}(\\widehat{p})))$, i.e., $\\widehat{g}(\\widehat{p}_i)=0\\Rightarrow \\mathbb{K}(\\widehat{g}(\\mathbb{K}(\\widehat{p_i}) ))=0$.\n\\end{rem}\nNext, the concepts of fuzzy partition and $F$-transform introduced by \\cite{per} are recalled. \n\\begin{def1}\\label{grFP} Let $u_1 <...<u_m$ be fixed nodes within $[a_1,a_2]$, where $u_1 = a_1, u_m = a_2$ and $m \\geq 2$. Then fuzzy sets $P_1, . . . , P_m$ (are called {\\bf basic functions}) identified with their membership functions $P_1(u), . . . , P_m(u)$ defined on $[a_1,a_2]$, form a {\\bf fuzzy partition} of $[a_1,a_2]$ if they satisfy the following properties for $i = 1, . . . ,m$,\n\\begin{itemize}\n  \\item[(i)] $P_i :[a_1,a_2]\\rightarrow [0,1],\\, P_i(u_i)=1$,\n   \\item[(ii)] $P_i(u) = 0$ if $t\\notin (u_{i-1},u_{i+1})$, where for the uniformity of the notation, we put $x_0 = a_1$ and $u_{m+1} = a_2$,\n   \\item[(iii)] $P_i(u)$ is continuous,\n   \\item[(iv)] $P_i(u), i= 2, . . . ,m$, strictly increases on $[u_{i-1}, u_{i}]$ and $P_i(u), i = 1, . . . ,m-1$, strictly decreases on $[u_i, u_{i+1}]$, and\n \\item[(v)] $\\sum_{i=1}^{m}P_i(u)=1,\\,\\forall\\,u\\in[a_1,a_2],\\,$.\n   \\end{itemize}\n\\end{def1}\nIf $u_1, . . . , u_m,\\,m \\geq 2$,\nare equidistant, then the fuzzy partition of $[a_1,a_2]$ is $h$-uniform. That is to say that $u_i= a_1+ h (i -1) , i = 1, . . . ,m$, where $h = \\frac{a_2-a_1}{m-1 },m\\geq2$ and the two\nadditional properties are satisfied:\n\\begin{itemize}\n  \\item[(i)] $P_i(u_i -u) = P_i(u_i +u ), i= 2, . . . ,m -1, \\forall\\, u\\in [0, h]$, and\n  \\item[(ii)] $P_i(u) = P_{i-1}(u -h)$ and $P_{i+1}(u) = P_i(u -h), i= 2, . . . ,m -1,\\,\\forall u\\in [u_k, u_{i+1}]$.\n\\end{itemize} \n\\begin{lem}\\label{grlem}\nLet $P_1,...,P_m,\\,m\\geq 3$, be basic functions that form the uniform fuzzy partition of $[a_1,a_2]$. Then\n\\[\\int_{u_1}^{u_2}P_1(u)du=\\int_{u_{m-1}}^{u_{m}}P_m(u)du=\\frac{h}{2},\\,\\,and\\]\n\\[\\int_{u_{i-1}}^{u_{i+1}}P_i(u)du=h,\\, i=2,...,m-1\\]\nwhere $h$ is the distance between two adjacent nodes.\n\\end{lem}\n\\begin{def1}\\label{grft} Let $P_1, ...,P_m$ be basic functions that form a fuzzy partition of $[a_1,a_2]$ and ${g}$ be the continuous function on $[a_1,a_2]$. Then the {\\bf $F$-transform} of ${g}$ w.r.t. $P_1,...,P_m$ is the $m$-tuple\nof real numbers (components) $F[{g}] = (F_1[{g}], ..., F_m[{g}])$, where\n\\[F_i[{g}]=\\dfrac{\\int^{a_2}_{a_1} {g}(u)P_i(u)du}{\\int^{a_2}_{a_1} P_i(u)du},\\,i=1,2,...,m,\\]\nwhere $F_i[{g}]$ (or simply $F_i)$) is $i$-th component of the $F$-transform.\n\\end{def1}\n\\begin{def1}\\label{grift} Let $F[\\widehat{g}] = (F_1, ..., F_m)$ be the $F$-transform of ${g}$ w.r.t. $P_1, ...,P_m$. Then the function \n$$\\hat{g}_m(u)=\\sum_{i=1}^{m}F_i[\\widehat{g}]P_i(u),\\,\\forall\\,u\\in[a_1,a_2],$$\nis called the {\\bf inverse $F$-transform}.\n\\end{def1}\n\\section{Granular fuzzy transform} \nThis section introduces the concept of the granular fuzzy transform ($F$-transform), which creates a relationship between a set of gr-continuous fuzzy functions from $[a_1,a_2]$ to $E^1$ and the collection of fuzzy numbers. The formula, which will refer to as a granular inverse $F$-transform (inversion formula), converts a fuzzy number into another gr-continuous fuzzy function that approximates the original one. Further, we investigate their basic properties. Now, we initiate with the following.\n\\begin{def1}\\label{grgrft} Let $P_1, ...,P_m$ be basic functions that form a fuzzy partition of $[a_1,a_2]$ and $\\widehat{g}: [a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Further, let $\\widehat{g}(u)P_i(u)$ be the granular fuzzy integrable on $[a_1,a_2]$. Then the {\\bf granular $F$-transform} of $\\widehat{g}$ w.r.t. $P_1,...,P_m$ is the $m$-tuple of fuzzy numbers (components) $F^{gr}[\\widehat{g}] = (F^{gr}_1[\\widehat{g}], ..., F^{gr}_m[\\widehat{g}])$, where\n\\[F^{gr}_i[\\widehat{g}]=\\dfrac{\\oint^{a_2}_{a_1}\\widehat{g}(u)P_i(u)du}{\\oint^{a_2}_{a_1} P_i(u)du},\\,i=1,2,...,m,\\]\nwhere $F^{gr}_i[\\widehat{g}]$ (or simply $F^{gr}_i)$) is $i$-th component of the granular $F$-transform.\n\\end{def1}\nFurther, let $P_1, . . . ,P_m$ be basic functions that form a uniform fuzzy partition of $[a_1,a_2]$. Then from Lemma \\ref{grlem}, the components of granular $F$-transform can be written as \n\\begin{eqnarray*}\n\\nonumber F^{gr}_1[\\widehat{g}]&=&\\dfrac{2}{h}\\oint^{u_2}_{u_1}\\widehat{g}(u)P_1(u)du,\\\\\n\\nonumber F^{gr}_m[\\widehat{g}]&=&\\dfrac{2}{h}\\oint^{u_m}_{u_{m-1}}\\widehat{g}(u)P_m(u)du,\\\\\n\\nonumber F^{gr}_i[\\widehat{g}]&=&\\dfrac{1}{h}\\oint^{u_{i+1}}_{u_{i-1}}\\widehat{g}(u)P_i(u)du,\\,i=2,...,m-1.\\\\\n\\end{eqnarray*}\n\\begin{rem}\\label{grr3}\n(i) From Definition \\ref{grgrint}, it is clear that $\\mathbb{K}(F^{gr}[\\widehat{g}])=F[\\mathbb{K}(\\widehat{g})]$, where $F[.]$ is $F$-transform in \\cite{per}; and\\\\\\\\\n(ii) For any crisp function $g:[a_1,a_2]\\rightarrow \\mathbb{R},\\,F^{gr}[g]=F[g]$, i.e., granular $F$-transform becomes $F$-transform in \\cite{per}.\n\\end{rem}\nThe following are towards the some fundamental properties of the granular $F$-transform associated with the fuzzy function. \n\\begin{pro}\\label{grp1}\nLet $\\widehat{g_1},\\widehat{g_2}:[a_1,a_2]\\rightarrow E^1$ be gr-continuous fuzzy functions. Then the granular $F$-transform is linear, i.e.,\n\\[F^{gr}[\\widehat{a_1}\\widehat{g_1}+\\widehat{a_2}\\widehat{g_2}]=\\widehat{a_1}F^{gr}[\\widehat{g_1}]+\\widehat{a_2}F^{gr}[\\widehat{g_2}],\\,\\forall\\,\\widehat{a_1},\\widehat{a_2}\\in E^1.\\]\n\\end{pro}\n\\textbf{Proof:} Let $\\widehat{a_1},\\widehat{a_2}\\in E^1$. Then from Remarks \\ref{grr1} and \\ref{grr3}, we have\n\\begin{eqnarray*}\n\\mathbb{K}(F^{gr}[\\widehat{a_1}\\widehat{g_1}+\\widehat{a_2}\\widehat{g_2}])&=&F[\\mathbb{K}(\\widehat{a_1}\\widehat{g_1}+\\widehat{a_2}\\widehat{g_2})]\\\\\n&=&F[\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{g_1})+\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{g_2})]\\\\\n&=&\\mathbb{K}(\\widehat{a_1})F[\\mathbb{K}(\\widehat{g_1})]+\\mathbb{K}(\\widehat{a_2})F[\\mathbb{K}(\\widehat{g_2})]\\\\\n&=&\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(F^{gr}[\\widehat{g_1}])+\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(F^{gr}[\\widehat{g_2}])\\\\\n&=&\\mathbb{K}(\\widehat{a_1}F^{gr}[\\widehat{g_1}]+\\widehat{a_2}F^{gr}[\\widehat{g_2}]).\n\\end{eqnarray*}\nThus $F^{gr}[\\widehat{a_1}\\widehat{g_1}+\\widehat{a_2}\\widehat{g_2}]=\\widehat{a_1}F^{gr}[\\widehat{g_1}]+\\widehat{a_2}F^{gr}[\\widehat{g_2}]$.\n\\begin{pro}\\label{grp2} Let $P_1, . . . ,P_m,m\\geq3$, be basic functions that form a uniform fuzzy partition of $[a_1,a_2]$ and $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-differentiable fuzzy function. Moreover, $\\mathcal{D}^{gr}\\widehat{g}$ is the gr-continuous fuzzy function on $[a_1,a_2]$. Then $F^{gr}[\\widehat{g}]$ gives minimum to the fuzzy function defined on $S\\subseteq E^1$ such that\n\\[\\widehat{\\phi}(\\widehat{a})={\\oint^{a_2}_{a_1}(\\widehat{g}(u)-\\widehat{a})^2P_i(u)du},\\,i=1,2,...,m,\\]\nwhere $S=\\{\\widehat{a}\\in E^1:\\widehat{g}(a_1)\\leq \\widehat{a}\\leq\\widehat{g}(a_2)\\}.$\n\\end{pro}\n\\textbf{Proof:} Since the fuzzy function $(\\widehat{g}(u)-\\widehat{a})^2P_i(u)$ is the gr-continuously differentiable w.r.t. $\\widehat{a}$ on $S$. Therefore we can write \n\\begin{eqnarray}\\label{grmin}\n\\nonumber\\mathcal{D}^{gr}\\widehat{\\phi}(\\widehat{a})&=&\\mathcal{D}^{gr}\\left(\\oint^{a_2}_{a_1}(\\widehat{g}(u)-\\widehat{a})^2 P_i(u)du\\right).\n\\end{eqnarray}\nNow, from Remark \\ref{grr1} and Proposition \\ref{grgrdif1}, we have\n\\begin{eqnarray}\\label{grmin1}\n\\nonumber\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{\\phi}(\\widehat{a}))&=&\\mathbb{K}\\left(\\mathcal{D}^{gr}\\left(\\oint^{a_2}_{a}(\\widehat{g}(u)-\\widehat{a})^2P_i(u)du\\right)\\right)\\\\\n\\frac{\\partial }{\\partial \\nonumber \\mathbb{K}(\\widehat{a})}\\mathbb{K}(\\widehat{\\phi}(\\mathbb{K}(\\widehat{a})))&=&-2\\int^{a_2}_{a_1}(\\mathbb{K}(\\widehat{g}(u))-\\mathbb{K}(\\widehat{a}))P_i(u)du.\n\\end{eqnarray}\nAlso, it can be easily check that $\\mathbb{K}(\\widehat{\\phi}(\\mathbb{K}(\\widehat{a})))$ reaches its minimum at that time, resulting in a solution to the equation $\\frac{\\partial }{\\partial \\mathbb{K}(\\widehat{q})}\\mathbb{K}(\\widehat{\\phi}(\\mathbb{K}(\\widehat{a})))=0$, i.e.\n\\[ \\mathbb{K}(\\widehat{a})=\\dfrac{\\oint^{a_2}_{a_1}\\mathbb{K}(\\widehat{g}(u))P_i(u)du}{\\oint^{a_2}_{a_1} P_i(u)du}=\\mathbb{K}(F^{gr}_i[\\widehat{g}]).\\]\nThus we have $\\widehat{a}=F^{gr}_i[\\widehat{g}]$.\\\\\\\\\nIn the following, we demonstrate that several assumptions about the smoothness of $\\widehat{g}$ can be used to estimate each granular $F$-transform component $F^{gr}_i,\\,i = 1,...,m$.\n\\begin{pro}\\label{grp3}\nLet $P_1, . . . ,P_m,\\,m\\geq3$, be basic functions that form a uniform fuzzy partition of $[a_1,a_2]$ and $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. In addition, let $F^{gr}_i[\\widehat{g}], i= 1, . . . ,m$, be the granular $F$-transform components of $\\widehat{g}$ w.r.t. $P_1, . . . ,P_m$. Then\n\\begin{eqnarray*}\n d^{gr}(\\widehat{g}(u),F^{gr}_i[\\widehat{g}])&\\leq& \\omega(2h,\\widehat{g}),\\\\\nd^{gr}(\\widehat{g}(u),F^{gr}_{i+1}[\\widehat{g}])&\\leq& \\omega(2h,\\widehat{g}),\\,\\forall\\,i= 1, . . . ,m-1,\\, u\\in [u_i, u_{i+1}],\n\\end{eqnarray*}\nwhere $h=\\frac{a_2-a_1}{m-1}$ and $\\omega(2h,\\widehat{g})=\\max\\limits_{|\\delta|\\leq 2h}\\max\\limits_{t'\\in[a_1,a_2-\\delta]}d^{gr}(\\widehat{g}(u'+\\delta),\\widehat{g}(u'))$ is the modulus of gr-continuity of $\\widehat{g}$ on $[a_1,a_2]$.\n\\end{pro}\n\\textbf{Proof:} Let $u\\in[u_i,u_{i+1}]$ and $1\\leq i\\leq m-1$. Then \n\\begin{eqnarray*}\n d^{gr}(\\widehat{g}(u),F^{gr}_i[\\widehat{g}])&=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-F_i[\\mathbb{K}(\\widehat{g})]|\\\\\n &=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-\\frac{1}{h}{\\int^{u_{i+1}}_{u_{i-1}}g^{gr}(u',\\alpha,\\mu_g)P_i(u')du'}|\\\\\n &=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|\\frac{1}{h}{\\int^{u_{i+1}}_{u_{i-1}}(g^{gr}(u,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g))P_i(u')du'}|\\\\\n &\\leq&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}\\frac{1}{h}{\\int^{u_{i+1}}_{u_{i-1}}|g^{gr}(u,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g)|P_i(u')du'}\\\\\n &\\leq&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}\\max\\limits_{|\\delta|\\leq 2h}\\max\\limits_{t'\\in[a_1,a_2-\\delta]}|g^{gr}(u'+\\delta,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g)|\\\\\n &=&\\max\\limits_{|\\delta|\\leq 2h}\\max\\limits_{t'\\in[a_1,a_2-\\delta]}d^{gr}(\\widehat{g}(u'+\\delta),\\widehat{g}(u'))\\\\\n &=&\\omega(2h,\\widehat{g}).\n\\end{eqnarray*}\nSimilarly, we can show that $d^{gr}(\\widehat{g}(u),F^{gr}_{i+1}[\\widehat{g}])\\leq \\omega(2h,\\widehat{g})$.\n\\begin{pro}\\label{grp4}\nLet $P_1, . . . ,P_m,\\,m\\geq3$, be basic functions which form a uniform fuzzy partition of $[a_1,a_2]$ and $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Then for $c_{i1} \\in [u_{i-1}, u_{i}]$ and $c_{i2} \\in [u_{i}, u_{i+1}]$, the components of granular $F$-transform satisfy the following condition:\n\\[F^{gr}_i[\\widehat{g}]=\\frac{1}{h}{\\oint^{c_{i2}}_{c_{i1}}\\widehat{g}(u)du},\\,i= 2, . . . ,m-1.\\]\nMoreover, if $i=1$ and $i=m$, then there exists $c\\in [u_{1},u_{2}]$ \\mbox{and} $c\\in [u_{m-1}, u_{m}]$ such that\n\\[F^{gr}_1[\\widehat{g}]=\\frac{2}{h}{\\oint^{c}_{u_1}}\\widehat{g}(u)du,\\]\n\\[F^{gr}_m[\\widehat{g}]=\\frac{2}{h}{\\oint^{u_{m}}_{c}}\\widehat{g}(u)du,\\,\\,\\mbox{respectively}.\\]\n\\end{pro}\n\\textbf{Proof:} Let $2\\leq i \\leq m-1$. Then from Definition \\ref{grFP} , $P_i(u)$ increases monotonically on $[u_{i-1}, u_{i}]$ and decreases monotonically on $[u_{i}, u_{i+1}]$,\nwe find\n\\[F^{gr}_i[\\widehat{g}]=\\frac{1}{h}{\\oint^{u_{i+1}}_{u_{i-1}}\\widehat{g}(u)P_i(u)du}.\\]\nNow, from Remark \\ref{grr1}, we can write\n\\begin{eqnarray}\n\\nonumber\\mathbb{K}(F^{gr}_i[\\widehat{g}])&=&\\mathbb{K}\\left(\\frac{1}{h}{\\oint^{u_{i+1}}_{u_{i-1}}\\widehat{g}(u)P_i(u)du}\\right)\\\\\n\\nonumber&=& \\frac{1}{h}\\int^{u_{i+1}}_{u_{i-1}}\\mathbb{K}(\\widehat{g}(u))P_i(u)du\\\\\n\\nonumber&=& \\frac{1}{h}\\int^{u_{i}}_{u_{i-1}}\\mathbb{K}(\\widehat{g}(u))P_i(u)du+\\frac{1}{h}\\int^{u_{i+1}}_{u_{i}}\\mathbb{K}(\\widehat{g}(u))P_i(u)du\\\\\n\\nonumber&=&\\frac{1}{h}{\\int^{u_{i}}_{c_{i1}}}\\mathbb{K}(\\widehat{g}(u))du+\\frac{1}{h}\\int^{c_{i2}}_{u_{i}}\\mathbb{K}(\\widehat{g}(u))du\\\\\n\\nonumber&=&\\frac{1}{h}{\\int^{c_{i2}}_{c_{i1}}}\\mathbb{K}(\\widehat{g}(u))du\\\\\n\\nonumber&=& \\mathbb{K}\\left(\\frac{1}{h}{\\oint^{c_{i2}}_{c_{i1}}}\\widehat{g}(u)du\\right).\n\\end{eqnarray}\n{Thus $F^{gr}_i[\\widehat{g}]=\\frac{1}{h}{\\oint^{c_{i2}}_{c_{i1}}}\\widehat{g}(u)du$. Similarly, we can obtain} $${F^{gr}_1[\\widehat{g}]=\\frac{2}{h}{\\oint^{c}_{u_1}}\\widehat{g}(u)du\\left(F^{gr}_m[\\widehat{g}]=\\frac{2}{h}{\\oint^{u_{m}}_{c}}\\widehat{g}(u)du\\right).}$$\nFrom the above result, $F^{gr}_i$ can be defined as an integral mean value of $\\widehat{g}$ within the interval $[c_{i1}, c_{i2}]$ that accumulates information about the fuzzy function $\\widehat{g}$. However, for the given fuzzy function and nodes of the partition, this interval cannot be specified precisely. We can demonstrate the closeness between the components of the granular $F$-transform and fuzzy function at the corresponding nodes. The estimation of closeness is given below.\n\\begin{pro}\\label{grp5}\nLet the conditions of Proposition \\ref{grp4} be hold and the fuzzy function $\\widehat{g}$ be twice gr-continuously differentiable in $(a_1,a_2)$. Then for all $i= 1, . . . ,m$\n\\[F^{gr}_i[\\widehat{g}] = \\widehat{g}(u_k) + O(h^2).\\]\n\\end{pro}\n\\textbf{Proof:} Let $2\\leq i\\leq m-1$. Then\n\\[\\vspace{-1mm}{F^{gr}_i[\\widehat{g}]=\\frac{1}{h}{\\oint^{u_{i+1}}_{u_{i-1}}\\widehat{g}(u)P_i(u)du}.}\\]\nFrom Remark \\ref{grr1}, the above expression can be written as\n\\begin{eqnarray}\n\\nonumber\\mathbb{K}(F^{gr}_i[\\widehat{g}])&=&\\mathbb{K}\\left(\\frac{1}{h}{\\oint^{u_{i+1}}_{u_{i-1}}\\widehat{g}(u)P_i(u)du}\\right)\\\\\n\\nonumber&=& \\frac{1}{h}\\int^{u_{i+1}}_{u_{i-1}}\\mathbb{K}(\\widehat{g}(u))P_i(u)du.\n\\end{eqnarray}\nNow, by using trapezoidal rule with nodes $u_{i-1},u_i,u_{i+1}$, we have \n\\begin{eqnarray}\n\\nonumber \\mathbb{K}(F^{gr}_i[\\widehat{g}]) &=&\\frac{1}{h}\\int^{u_{i+1}}_{u_{i-1}}\\mathbb{K}(\\widehat{g}(u))P_i(u)du\\\\\n\\nonumber&=&\\frac{1}{h}\\frac{h}{2}\\left[\\mathbb{K}(\\widehat{g}(u_{i-1}))P_i(u_{i-1})+\\mathbb{K}(\\widehat{g}(u_{i+1}))P_i(u_{i+1})+ 2\\mathbb{K}(\\widehat{g}(u_{i}))P_i(u_{k})\\right]+O(h^2)\\\\\n\\nonumber&=&\\mathbb{K}(\\widehat{g}(u_{i}))+ O(h^2)\\\\\n\\nonumber&=&\\mathbb{K}(\\widehat{g}(u_{i})+ O(h^2)).\n\\end{eqnarray}\nThus $F^{gr}_i[\\widehat{g}] = \\widehat{g} (u_i) + O(h^2)$. Similarly, for $i=1\\,(i=m)$, $F^{gr}_1[\\widehat{g}] = \\widehat{g} (u_1) + O(h^2)\\, (F^{gr}_m[\\widehat{g}] = \\widehat{g} (u_m) + O(h^2))$.\\\\\\\\\nThe following is towards the notion of the granular inverse $F$-transform associated with the fuzzy function. Now, we initiate with the following.\n\\begin{def1}\\label{grgrift} Let $F^{gr}[\\widehat{g}] = (F^{gr}_1, ..., F^{gr}_m)$ be the granular $F$-transform of $\\widehat{g}$ w.r.t. $P_1, ...,P_m$. Then the function \n$$\\widehat{g}^{gr}_m(u)=\\sum_{i=1}^{m}F^{gr}_i[\\widehat{g}]P_i(u),\\,\\forall\\,u\\in[a_1,a_2],$$\nis called the {\\bf granular inverse $F$-transform}.\n\\end{def1}\n\\begin{rem}\n(i) From Definition \\ref{grgrift}, it is easy to say that $\\mathbb{K}(\\widehat{g}^{gr}_m(u))=\\sum_{i=1}^{m}\\mathbb{K}(F^{gr}_i[\\widehat{g}])P_i(u)=\\sum_{i=1}^{m}F_i[\\mathbb{K}(\\widehat{g})]P_i(u)=\\hat{g}_m(u)$, where $F[.],\\hat{g}_m(u)$ are $F$-transform and inverse $F$-transform in \\cite{per}, respectively; and\\\\\\\\\n(ii) For any crisp function $g:[a_1,a_2]\\rightarrow \\mathbb{R},\\widehat{g}^{gr}_m(u)=\\hat{g}_m(u)$, i.e., granular $F$-transform and granular inverse $F$-transform become $F$-transform and inverse $F$-transform in \\cite{per}, respectively.\n\\end{rem}\nBelow, we demonstrate that the granular inverse $F$-transform $\\widehat{g}^{gr}_m$ can approximate the original gr-continuous fuzzy function $\\widehat{g}$ with an arbitrary precision. To do this, we need the following remark.\n\\begin{rem}\\label{grremark} (i) Let $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Then for all $\\epsilon > 0$ there is a $\\delta>0$ such that for all $u\\in[a_1,a_2]$, $d^{gr}(\\widehat{g}(u),\\widehat{g}(u))<\\epsilon$, whenever $|t-t'|<\\delta$. As $d^{gr}(\\widehat{g}(u),\\widehat{g}(u'))=\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g)|$. So, by using the fact that a function continuous on $[a_1,a_2]$, is uniformly continuous, we have $$\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g)|<\\epsilon.$$\n(ii) \\vspace{-1mm}{By using (i), we can show that for all $u\\in [a_1,a_2]$ and $i=1,...,m-1,$ $$d^{gr}(\\widehat{g}(u),F_i^{gr}[\\widehat{g}])\\leq\\epsilon,\\,d^{gr}(\\widehat{g}(u),F_{i+1}^{gr}[\\widehat{g}])\\leq\\epsilon.$$\nNow, from Proposition \\ref{grp3}, we have}\n\\begin{eqnarray}\n\\nonumber d^{gr}(\\widehat{g}(u),F_i^{gr}[\\widehat{g}])&\\leq&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}\\frac{1}{h}{\\int^{u_{i+1}}_{u_{i-1}}|g^{gr}(u,\\alpha,\\mu_g)-g^{gr}(u',\\alpha,\\mu_g)|P_i(u')du'}\\\\\n\\nonumber&<&\\frac{\\epsilon}{h}{\\int^{u_{i+1}}_{u_{i-1}}P_i(u')du'}\\\\\n\\nonumber&=&\\epsilon.\n\\end{eqnarray}\n{Similarly, we can show that $d^{gr}(\\widehat{g}(u),F_{i+1}^{gr}[\\widehat{g}])\\leq\\epsilon$.}\n\\end{rem}\n\\begin{pro}\\label{grp6}\nLet $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Then for all $\\epsilon > 0$ there is $m_\\epsilon$ and the fuzzy partition\n$P_1, . . . ,P_{m_\\epsilon}$ of $[a_1,a_2]$ such that for all $u\\in[a_1,a_2]$,\n\\[d^{gr}(\\widehat{{g}}_m(u) , \\widehat{g}^{gr}_{m_\\epsilon}(u))\\leq\\epsilon,\\]\nwhere $\\widehat{g}^{gr}_{m_\\epsilon}$ is the granular inverse $F$-transform of $\\widehat{g}$.\n\\end{pro}\n\\textbf{Proof:} Let $u\\in[a_1,a_2]$ and $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Then from Definition \\ref{grmetric} and Remark \\ref{grremark} (ii),\n\\begin{eqnarray*}\n\\nonumber d^{gr}(\\widehat{{g}}(u), \\widehat{g}^{gr}_m(u))&=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-{g}^{gr}_m(u,\\alpha,\\mu_g)|\\\\\n\\nonumber&=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)-\\sum_{i=1}^{m}F_i[\\mathbb{K}(\\widehat{g})]P_i(u)|\\\\\n\\nonumber&=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|g^{gr}(u,\\alpha,\\mu_g)\\sum_{i=1}^{m}P_i(u)-\\sum_{i=1}^{m}F_i[\\mathbb{K}(\\widehat{g})]P_i(u)|\\\\\n\\nonumber&=&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}|\\sum_{i=1}^{m}(g^{gr}(u,\\alpha,\\mu_g)-F_i[\\mathbb{K}(\\widehat{g})])P_i(u)|\\\\\n\\nonumber&\\leq&\\sup\\limits_{\\alpha\\in [0,1]}\\max\\limits_{\\mu_g\\in [0,1]}\\sum_{i=1}^{m}|g^{gr}(u,\\alpha,\\mu_g)-F_i[\\mathbb{K}(\\widehat{g})]|P_i(u)\\\\\n\\nonumber&\\leq&\\epsilon\\sum_{i=1}^{m}P_i(u)\\\\\n\\nonumber&=&\\epsilon.\n\\end{eqnarray*}\nProposition \\ref{grp6} can be formulated for the uniform fuzzy partitions of $[a_1,a_2]$.\n\\begin{cor}\\label{grco1}\nLet $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function and $\\{(P^{(m)}_1, . . . ,P^{(m)}_{m})_m\\}$ be a sequence of uniform fuzzy partitions of $[a_1,a_2]$ for all $m$ . In addition, let $\\{\\widehat{g}^{gr}_m(u)\\}$ be the sequence of granular inverse $F$-transforms w.r.t. $\\{(P^{(m)}_1, . . . ,P^{(m)}_{m})_m\\}$, respectively. Then for all $\\epsilon > 0$ there exist $m_\\epsilon$ such that $m>m_\\epsilon$ and\n\\[d^{gr}(\\widehat{{g}}_m(u) , \\widehat{g}^{gr}_{m}(u))\\leq\\epsilon,\\,\\forall\\,u\\in [a_1,a_2].\\]\n\\end{cor}\n\\textbf{Proof:} Follows from Proposition \\ref{grp6}.\n\\begin{cor}\nLet the assumptions of Corollary \\ref{grco1} be hold. Then the sequence of granular inverse $F$-transforms $\\{\\widehat{g}^{gr}_m(u)\\}$ uniformly\nconverges to fuzzy function $\\widehat{{g}}$ .\n\\end{cor}\n\\textbf{Proof:} Follows from Corollary \\ref{grco1}.\\\\\\\\\nThe following result describes how to estimate the difference between any two approximations of a given fuzzy function by the granular inverse $F$-transforms based on different sets of basic functions. As can be observed, it depends on the original fuzzy function's smoothness behavior, as defined by its modulus of continuity.\n\\begin{pro}\\label{grp7} \nLet $P_1,...,P_m$, $P'_1,...,P'_m,\\,m\\geq 3$, be basic functions that form different uniform fuzzy partitions of $[a_1,a_2]$ and $\\widehat{g}:[a_1,a_2]\\rightarrow E^1$ be the gr-continuous fuzzy function. Further, let $\\widehat{g}^{gr}_m,\\widehat{g}^{gr'}_m$ be the granular inverse $F$-transforms of\n$\\widehat{g}$ w.r.t. different sets of basic functions. Then\n\\[d^{gr}(\\widehat{g}^{gr}_m(u),\\widehat{g}^{gr'}_m(u))\\leq 2 \\omega(2h,\\widehat{g}), \\,\\forall\\,u\\in [a_1,a_2],\\]\nwhere $h=\\frac{a_2-a_1}{m-1}$ and $2 \\omega(2h,\\widehat{g})$ is the modulus of continuity of $\\widehat{g}$ on $[a_1,a_2]$.\n\\end{pro} \n\\textbf{Proof:} Follows from Proposition \\ref{grp3} and Definition \\ref{grgrift}.\\\\\\\\\nThe following is towards the graphical representation of horizontal membership functions and level sets of the granular $F$-transform and the granular inverse $F$-transform associated with a fuzzy function are presented.\n\\begin{exa}\\label{grexa}\n\\end{exa}\nLet $P_1 = (0, 0, 1), P_2= (0, 1, 2), P_3 = (1, 2, 3), P_4 = (2, 3, 3)$ be the triangular fuzzy numbers that form a fuzzy partition of $[0,3]$. To calculate the components of the granular $F$-transform and the granular inverse $F$-transform, we assume a fuzzy function $\\widehat{g}(u)=(\\frac{u^3}{3},\\frac{u^3}{3}+u+3,\\frac{2u^3}{3}+4)$ and its horizontal membership function is given by \n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{g}(u))&=&g^{gr}(u,\\alpha,\\mu_g)\\\\\n&=&\\frac{u^3}{3}+\\alpha(u+3)+\\mu_g(1-\\alpha)(\\frac{u^3}{3}+4),\\,u\\in[0,3],\\alpha,\\mu_g\\in\\{0,0.5,1\\}.\n\\end{eqnarray*}\nNext, the horizontal membership functions of the components of the granular $F$-transform and the granular inverse $F$-transform associated with the fuzzy function $\\widehat{g}(u)$ corresponding to different values of $\\alpha,\\mu_g$, are presented in Figure \\ref{grfig:0}. In which, at $\\alpha=1$, the fuzzy function $\\widehat{g}(u)$ shows crisp behavior and the granular $F$-transform and the granular inverse $F$-transform become $F$-transform and inverse $F$-transform as given in \\cite{per}, respectively. Also, the $\\alpha(=0)$-level sets of the fuzzy function $\\widehat{g}(u)$, granular $F$-transform and granular inverse $F$-transform are given in Figure \\ref{grfig:00}.\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.5]{figure\/1fex.eps} \n    \\includegraphics[scale=0.5]{figure\/4fex.eps}\n      \\includegraphics[scale=0.5]{figure\/2fex.eps}\n        \\includegraphics[scale=0.5]{figure\/5fex.eps}\n       \\includegraphics[scale=0.5]{figure\/3fex.eps}\n         \\includegraphics[scale=0.5]{figure\/6fex.eps}\n     \\includegraphics[scale=0.5]{figure\/7fex.eps}\n     \\caption{Red, blue, green curves show horizontal membership function (HMF) of fuzzy function $\\widehat{f}(t)$, granular $F$-transform and granular inverse $F$-transform presented in Example \\ref{exa} corresponding to different values of $\\alpha,\\mu$, respectively.}\n    \\label{fig:0}\n\\end{figure}\n\\begin{figure}\n    \\centering\n    \\includegraphics[scale=0.55]{figure\/level3.eps} \n   \n    \\caption{Red, blue, green curves show $\\alpha(=0)$-level sets of of fuzzy functions $\\widehat{f}(t)$, granular $F$-transform and granular inverse $F$-transform presented in Example \\ref{exa}, respectively. }\n    \\label{fig:00}\n\\end{figure}\n\\section{Fuzzy prey-predator model}\nIn this section, we formulate a fuzzy prey-predator model and study the dynamic behavior of the same. This section is divided into two subsections; the first is towards the formulation of the model, and the latter is towards its dynamical behavior.\n\\subsection{Formulation of fuzzy prey-predator model}\nIn this subsection, we present a fuzzy prey-predator model in which one predator team interacts with two teams of prey. In the presence of a predator, prey groups assist one another, but in the absence of a predator, they compete. We consider two teams of prey with densities $\\widehat{p}(u)$ and $\\widehat{q}(u)$, interacting with one team of predators with density $\\widehat{r}(u)$ in a fuzzy environment, respectively, which is chiefly motivated from a crisp model given in \\cite{elet}. The proposed fuzzy prey-predator model is as follows:\n\\begin{eqnarray}\\label{gr1}\n\\begin{array}{ll}\n{\\mathcal{D}^{gr}\\widehat{p}(u)}=\\widehat{a_1}\\widehat{p}(u)(1-\\widehat{p}(u))-\\widehat{p}(u)\\widehat{r}(u)+\\widehat{p}(u)\\widehat{q}(u)\\widehat{r}(u)=\\widehat{g_1}(u,\\widehat{p},\\widehat{q},\\widehat{r}),\\\\ \n{\\mathcal{D}^{gr}\\widehat{q}(u)}=\\widehat{a_2}\\widehat{q}(u)(1-\\widehat{q}(u))-\\widehat{q}(u)\\widehat{r}(u)+\\widehat{p}(u)\\widehat{q}(u)\\widehat{r}(u)=\\widehat{g_2}(u,\\widehat{p},\\widehat{q},\\widehat{r}),\\\\ \n{\\mathcal{D}^{gr}\\widehat{r}(u)}= -\\widehat{a_3}\\widehat{r}^2(u)+\\widehat{a_4}\\widehat{p}(u)\\widehat{r}(u)+\\widehat{a_5}\\widehat{q}(u)\\widehat{r}(u)=\\widehat{g_3}(u,\\widehat{p},\\widehat{q},\\widehat{r}),\\\\\n\\widehat{p}(0)=\\widehat{p}_0,\\,\\widehat{q}(0)=\\widehat{q}_0 ,\\,\\widehat{r}(0)=\\widehat{r}_0,\n\\end{array}\n\\end{eqnarray}\nwhere $\\widehat{p},\\widehat{q},\\widehat{r}:[0,1]\\subseteq \\mathbb{R}\\rightarrow E^1$ are gr-differentiable fuzzy functions and $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$, $\\widehat{p}_0,\\widehat{q}_0,\\widehat{r}_0$ are positive fuzzy numbers. Based on\nRemark \\ref{grr1} and Proposition \\ref{grgrdif1}, the system (\\ref{gr1}) can be written as\n\\begin{eqnarray}\\label{gr2}\n\\nonumber\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{p}(u))&=&{\\frac{\\partial}{\\partial t}\\mathbb{K}(\\widehat{p}(u))}=\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{p}(u))({1}-\\mathbb{K}(\\widehat{p}))-\\mathbb{K}(\\widehat{p}(u))\\mathbb{K}(\\widehat{r}(u))+\\\\\n\\nonumber&&\\mathbb{K}(\\widehat{p}(u))\\mathbb{K}(\\widehat{q}(u))\\mathbb{K}(\\widehat{r}(u))=\\mathbb{K}(\\widehat{g_1}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),\\\\ \n\\nonumber\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{q}(u))&=&{\\frac{\\partial}{\\partial t}\\mathbb{K}(\\widehat{q}(u))}=\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{q}(u))({1}-\\mathbb{K}(\\widehat{q}(u)))-\\mathbb{K}(\\widehat{q}(u))\\mathbb{K}(\\widehat{r}(u))+\\\\\n\\nonumber&&\\mathbb{K}(\\widehat{p}(u))\\mathbb{K}(\\widehat{q}(u))\\mathbb{K}(\\widehat{r}(u))=\\mathbb{K}(\\widehat{g_2}(\\mathbb{K}(u,\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),\\\\ \n\\nonumber\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{r}(u))&=&{\\frac{\\partial}{\\partial t}\\mathbb{K}(\\widehat{r}(u))}= -\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r}(u))^2+\\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p}(u))\\mathbb{K}(\\widehat{r}(u))+\\\\\n\\nonumber&&\\mathbb{K}(\\widehat{a_5})\\mathbb{K}(\\widehat{q}(u))\\mathbb{K}(\\widehat{r}(u))=\\mathbb{K}(\\widehat{g_3}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),\\\\\n\\mathbb{K}(\\widehat{p}(0))&=&\\mathbb{K}(\\widehat{p}_0),\\,\\mathbb{K}(\\widehat{q}(0))=\\mathbb{K}(\\widehat{q}_0)\n ,\\,\\mathbb{K}(\\widehat{r}(0))=\\mathbb{K}(\\widehat{r}_0).\n\\end{eqnarray}\nThe system (\\ref{gr2}) can be given as\n\\begin{eqnarray}\\label{gr2a}\n\\nonumber\\frac{\\partial}{\\partial t}p^{gr}(u,\\alpha,\\mu_p)&=&a_1^{gr}(\\alpha,\\mu_{a_1})p^{gr}(u,\\alpha,\\mu_p)({1}-p^{gr}(u,\\alpha,\\mu_p))-p^{gr}(u,\\alpha,\\mu_p)r^{gr}(u,\\alpha,\\mu_r)\\\\\n\\nonumber&&+p^{gr}(u,\\alpha,\\mu_p)q^{gr}(u,\\alpha,\\mu_q)r^{gr}(u,\\alpha,\\mu_r),\\\\\n\\nonumber\\frac{\\partial}{\\partial t}q^{gr}(u,\\alpha,\\mu_q)&=&a_2^{gr}(\\alpha,\\mu_{a_2})q^{gr}(u,\\alpha,\\mu_q)({1}-q^{gr}(u,\\alpha,\\mu_q))-q^{gr}(u,\\alpha,\\mu_q)r^{gr}(u,\\alpha,\\mu_r)\\\\\n\\nonumber&&+p^{gr}(u,\\alpha,\\mu_p)q^{gr}(u,\\alpha,\\mu_q)r^{gr}(u,\\alpha,\\mu_r),\\\\\n\\nonumber\\frac{\\partial}{\\partial t}r^{gr}(u,\\alpha,\\mu_r)&=& -a_2^{gr}(\\alpha,\\mu_{a_3})r^{2,gr}(u,\\alpha,\\mu_r)+d^{gr}(\\alpha,\\mu_{a_4})p^{gr}(u,\\alpha,\\mu_p)r^{gr}(u,\\alpha,\\mu_r)+\\\\\n&&a_5^{gr}(\\alpha,\\mu_{a_5})q^{gr}(u,\\alpha,\\mu_q)r^{gr}(u,\\alpha,\\mu_r),\n\\end{eqnarray}\n$p^{gr}(0,\\alpha,\\mu_p)=p_0^{gr}(\\alpha,\\mu_{p_0}),q^{gr}(0,\\alpha,\\mu_q)=q_0^{gr}(\\alpha,\\mu_{q_0}),r^{gr}(0,\\alpha,\\mu_r)=r_0^{gr}(\\alpha,\\mu_{r_0}).$\n\\subsection{Dynamical behavior of the fuzzy prey-predator model}\nIn this subsection, we investigate the equilibrium points and their stability of the proposed fuzzy prey-predator model. Now, we initiate with the following.\n\\begin{def1}\\label{greq}\n A point $(\\widehat{p_e},\\widehat{q}_e,\\widehat{r}_e)$ is called the {\\bf fuzzy equilibrium point} of the system (\\ref{gr1}) if\n \\begin{eqnarray*}\n \\mathcal{D}^{gr}\\widehat{p}(u)&=&\\widehat{g_1}(\\widehat{p_e},\\widehat{q}_e,\\widehat{r}_e)=0,\\\\\n \\mathcal{D}^{gr}\\widehat{q}(u)&=&\\widehat{g_2}(\\widehat{p_e},\\widehat{q}_e,\\widehat{r}_e)=0,\\\\\n \\mathcal{D}^{gr}\\widehat{r}(u)&=&\\widehat{g_3}(\\widehat{p_e},\\widehat{q}_e,\\widehat{r}_e)=0.\n \\end{eqnarray*}\n\\end{def1}\n\\begin{rem}\\label{grremeq}\nIt can be easily seen that $(\\widehat{p_e},\\widehat{q}_e,\\widehat{r}_e)$ is a fuzzy equilibrium point of the system (\\ref{gr1}) iff $(\\mathbb{K}(\\widehat{p_e}),\\mathbb{K}(\\widehat{q}_e),\\mathbb{K}(\\widehat{r}_e))$ is an equilibrium point of the system (\\ref{gr2}).\n\\end{rem}\n After algebraic calculation for the system (\\ref{gr1}), we get several fuzzy equilibrium points $(\\widehat{p},\\widehat{q},\\widehat{r})$, e.g.\n\\begin{itemize}\n\\item[(i)] $\\widehat{E}_0({0},{0},{0})$,\n\\item[(ii)] $\\widehat{E}_1({1},{0},{0})$,\n\\item[(iii)] $\\widehat{E}_2({0},{1},{0})$,\n\\item[(iv)] $\\widehat{E}_3({1},{1},{0})$,\n\\item[(v)] $\\widehat{E}_4({0},\\widehat{q_{e_4}},\\widehat{r_{e_4}})$, where\n\\begin{eqnarray*}\\widehat{q_{e_4}}&=&\\frac{\\widehat{a_2}\\widehat{a_3}}{\\widehat{a_2}\\widehat{a_3}+\\widehat{a_5}},\\\\\n\\widehat{r_{e_4}}&=&\\frac{\\widehat{a_2}\\widehat{a_5}}{\\widehat{a_2}\\widehat{a_3}+\\widehat{a_5}},\n\\end{eqnarray*}\n\\item[(vi)] $\\widehat{E}_5(\\widehat{p_{e_5}},{0},\\widehat{r_{e_5}})$, where \n\\begin{eqnarray*}\n\\widehat{p_{e_5}}&=&\\frac{\\widehat{a_1}\\widehat{a_3}}{\\widehat{a_1}\\widehat{a_3}+\\widehat{a_4}},\\\\\n\\widehat{r_{e_5}}&=&\\frac{\\widehat{a_1}\\widehat{a_4}}{\\widehat{a_1}\\widehat{a_3}+\\widehat{a_4}},\n\\end{eqnarray*}\n\\item[(vii)] $\\widehat{E}_6({1},{1},\\widehat{r_{e_6}})$, where \n\\[\\widehat{r_{e_6}}=\\frac{\\widehat{a_4}+\\widehat{a_5}}{\\widehat{a_3}},\\,\\mbox{and}\\]\n\\item[(viii)] $\\widehat{E}_7(\\widehat{p_{e_7}},\\widehat{q_{e_7}},\\widehat{r_{e_7}})$, where\n\\begin{eqnarray*}\n\\widehat{p_{e_7}}&=&\\frac{\\widehat{a_2}\\widehat{a_3}+\\widehat{a_5}\\left({1}-\\sqrt{\\frac{\\widehat{a_2}}{\\widehat{a_1}}}\\right)}{\\widehat{a_5}+\\widehat{a_4}\\sqrt{\\frac{\\widehat{a_2}}{\\widehat{a_1}}}},\\\\\n\\widehat{q_{e_7}}&=&\\frac{\\widehat{a_3}\\sqrt{\\widehat{a_1}\\widehat{a_2}}-\\widehat{a_4}\\left({1}-\\sqrt{\\frac{\\widehat{a_2}}{\\widehat{a_1}}}\\right)}{\\widehat{a_5}+\\widehat{a_4}\\sqrt{\\frac{\\widehat{a_2}}{\\widehat{a_1}}}},\\\\\n\\widehat{r_{e_7}}&=&\\sqrt{\\widehat{a_1}\\widehat{a_2}}.\n\\end{eqnarray*}\n\\end{itemize}\nThe existence of ${E}_0,{E}_1, {E}_2,{E}_3,{E}_4,{E}_5$ and ${E}_6$ are obvious. Now, we are concentrating on the existence of fuzzy equilibrium point $\\widehat{E}_7$. Therefore the fuzzy equilibrium point ${E}_7$ exists if \n\\begin{eqnarray*}\n&&\\widehat{a_3}\\sqrt{\\widehat{a_1}\\widehat{a_2}}\\leq\\widehat{a_4}+\\widehat{a_5},\\\\\n&&\\widehat{a_1}\\widehat{a_3}+\\widehat{a_4}>\\widehat{a_4}\\sqrt{\\frac{\\widehat{a_2}}{\\widehat{a_1}}},\\\\\n&&\\widehat{a_2}\\widehat{a_3}+\\widehat{a_5}>\\widehat{a_5}\\sqrt{\\frac{\\widehat{a_1}}{\\widehat{a_2}}}.\n\\end{eqnarray*}\nFrom Remark \\ref{grremeq}, $(\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))$ is an equilibrium point of the system (\\ref{gr2}), e.g. \n\\begin{itemize}\n\\item[(i)] $E_0({0},{0},{0})$,\n\\item[(ii)] $E_1({1},{0},{0})$,\n\\item[(iii)] $E_2({0},{1},{0})$,\n\\item[(iv)] $E_3({1},{1},{0})$,\n\\item[(v)] $E_4({0},\\mathbb{K}(\\widehat{q_{e_4}}),\\mathbb{K}(\\widehat{r_{e_4}}))$, where\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{q_{e_4}})&=&\\frac{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_3})}{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_5})},\\\\\n\\mathbb{K}(\\widehat{r_{e_4}})&=&\\frac{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_5})}{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_5})},\n\\end{eqnarray*}\n\\item[(vi)]$E_5(\\mathbb{K}(\\widehat{p_{e_5}}),{0},\\mathbb{K}(\\widehat{r_{e_5}}))$, where\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{p_{e_5}})&=&\\frac{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_3})}{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_4})},\\\\\n\\mathbb{K}(\\widehat{r_{e_5}})&=&\\frac{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_4})}{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_4})}\n\\end{eqnarray*}\n\\item[(vii)] $E_6({1},{1},\\mathbb{K}(\\widehat{r_{e_6}}))$, where \n\\[\\mathbb{K}(\\widehat{r_{e_6}}=\\frac{\\mathbb{K}(\\widehat{a_4})+\\mathbb{K}(\\widehat{a_5})}{\\mathbb{K}(\\widehat{a_3})},\\,\\mbox{and}\\]\n\\item[(viii)] $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$, where \n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{p_{e_7}})&=&\\frac{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_5})\\left({1}-\\sqrt{\\frac{\\mathbb{K}(\\widehat{a_2})}{\\mathbb{K}(\\widehat{a_1})}}\\right)}{\\mathbb{K}(\\widehat{a_5})+\\mathbb{K}(\\widehat{a_4})\\sqrt{\\frac{\\mathbb{K}(\\widehat{a_2})}{\\mathbb{K}(\\widehat{a_1})}}},\\\\\n\\mathbb{K}(\\widehat{q_{e_7}})&=&\\frac{\\mathbb{K}(\\widehat{a_3})\\sqrt{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_2})}-\\mathbb{K}(\\widehat{a_4})\\left({1}-\\sqrt{\\frac{\\mathbb{K}(\\widehat{a_2})}{\\mathbb{K}(\\widehat{a_1})}}\\right)}{\\mathbb{K}(\\widehat{a_5})+\\mathbb{K}(\\widehat{a_4})\\sqrt{\\frac{\\mathbb{K}(\\widehat{a_2})}{\\mathbb{K}(\\widehat{a_1})}}},\\\\\n\\mathbb{K}(\\widehat{r_{e_7}})&=&\\sqrt{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_2})},\n\\end{eqnarray*}\n\\end{itemize}\nare the equilibrium points of the system (\\ref{gr2}).\\\\\\\\ \nBelow, we discuss the local and global fuzzy stability of the fuzzy equilibrium points corresponding to the fuzzy eigenvalues, respectively. Before stating the next, we introduce the following.\n\\begin{def1}\\label{grstable} Let $\\widehat{\\lambda}_i,\\,i=1,2,...,m$, be a fuzzy eigenvalue of the fuzzy matrix $\\widehat{M}=[\\widehat{a_1}_{ij}]_{m\\times m}$.\nThen the fuzzy equilibrium points of the system of $\\mathcal{D}^{gr}\\widehat{p}(u)=\\widehat{M}\\widehat{p}(u)$ are called {\\bf fuzzy stable} and {\\bf fuzzy unstable} iff $Re(\\widehat{\\lambda}_i)<0$ and $Re(\\widehat{\\lambda}_i)>0$, respectively.\n\\end{def1}\n\\begin{def1}\\label{grlocal}\nA fuzzy equilibrium point $\\widehat{p_e}$ of $\\mathcal{D}^{gr}\\widehat{p}(u)=\\widehat{M}\\widehat{p}(u)$ is called {\\bf locally asymptotically fuzzy stable} if all eigen values of the corresponding variational fuzzy matrix have negative real parts.\n\\end{def1}\n\\begin{def1}\\label{grglobal} A fuzzy equilibrium point $\\widehat{p_e}$ of $\\mathcal{D}^{gr}\\widehat{p}(u)=\\widehat{M}\\widehat{p}(u)$ is called {\\bf globally asymptotically fuzzy stable} if there exists a $gr$-continuous and differentiable function $\\widehat{U}:D\\subseteq E^m\\rightarrow E^1$ such that \n\\begin{itemize}\n  \\item[(i)] $\\widehat{U}(\\widehat{p_e})=0$;\n    \\item[(ii)] $\\widehat{U}(\\widehat{p})>0,\\,\\forall\\,\\widehat{p}\\in D-\\{\\widehat{p_e}\\}$;\n      \\item[(iii)] $\\widehat{U}(\\widehat{p})$ is radially unbounded; and\n        \\item[(iv)] $\\mathcal{D}^{gr}\\widehat{U}(\\widehat{p})<0,\\,\\forall\\,\\widehat{p}\\in D-\\{\\widehat{p_e}\\}$.\n\\end{itemize}\nAlso, the fuzzy function $\\widehat{U}$ is called {\\bf fuzzy Lyapunov function}.\n\\end{def1}\n\\begin{thm}\\label{grlogo}\nA fuzzy equilibrium point $\\widehat{p_e}$ of $\\mathcal{D}^{gr}\\widehat{p}(u)=\\widehat{M}\\widehat{p}(u)$ is locally or globally asymptotically fuzzy stable iff the equilibrium point $\\mathbb{K}(\\widehat{p_e})$ of $\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{p}(u))=\\mathbb{K}(\\widehat{M})\\mathbb{K}(\\widehat{p}(u))$ is locally or globally asymptotically stable, respectively.\n\\end{thm}\n\\textbf{Proof:} Let $\\widehat{p_e}$ be a fuzzy equilibrium point of $\\mathcal{D}^{gr}\\widehat{p}(u)=\\widehat{M}\\widehat{p}(u)$. Then from Remark \\ref{grremeq}, $\\mathbb{K}(\\widehat{p_e})$ is an equilibrium point of $\\mathbb{K}(\\mathcal{D}^{gr}\\widehat{p}(u))=\\mathbb{K}(\\widehat{M})\\mathbb{K}(\\widehat{p}(u))$. Also, let $\\mathbb{K}(\\widehat{\\lambda}_i),\\,i=1,2,...,m$ be an eigen value of the corresponding variational matrix. Then the equilibrium point $\\mathbb{K}(\\widehat{p_e})$ is locally asymptotically stable iff $Re(\\mathbb{K}(\\widehat{\\lambda}_i))<0$, i.e., $\\mathbb{K}(Re(\\widehat{\\lambda}_i))<0$. Therefore from Remark \\ref{grr1}, we have $Re(\\widehat{\\lambda}_i)<0$. Thus the fuzzy equilibrium point $\\widehat{p_e}$ is locally asymptotically\nfuzzy stable. Also, the equilibrium point $\\mathbb{K}(\\widehat{p_e})$ is globally asymptotically\nstable if there exists a suitable Lyapunov function $\\mathbb{K}({\\widehat{U}})$ such that $\\mathbb{K}({\\widehat{U}}(\\mathbb{K}(\\widehat{p_e})))=0$, $\\mathbb{K}({\\widehat{U}}(\\mathbb{K}(\\widehat{p})))>0,\\frac{\\partial\\mathbb{K}({\\widehat{U}})}{\\partial t}<0,\\forall \\,\\mathbb{K}(\\widehat{p})$ (except for the equilibrium point $\\mathbb{K}(\\widehat{p_e})$ and $\\mathbb{K}({\\widehat{U}}(\\mathbb{K}(\\widehat{p})))$ is radially unbounded. Therefore from Remark \\ref{grr1}, the fuzzy function $\\widehat{U}$ is the fuzzy Lyapunov function, i.e., satisfies all conditions given in Definition \\ref{grglobal}. Thus the fuzzy equilibrium point $\\widehat{p_e}$ is globally asymptotically\nfuzzy stable and conversely. \\\\\\\\\nThe variational fuzzy matrix of the system (\\ref{gr1}) is defined as\n\\begin{equation}\\label{grm}\n\\widehat{M_1} = \n\\begin{bmatrix}\n\\widehat{a_1}-2\\widehat{a_1}\\widehat{p}-\\widehat{r}+\\widehat{q}\\widehat{r} & \\widehat{p}\\widehat{r} & -\\widehat{p}+\\widehat{p}\\widehat{q} \\\\\n\\widehat{q}\\widehat{r} & \\widehat{a_2}-2\\widehat{a_2}\\widehat{q}-\\widehat{r}+\\widehat{p}\\widehat{r} & -\\widehat{q} +\\widehat{p}\\widehat{q}\\\\\n\\widehat{a_4}\\widehat{r} & \\widehat{a_5}\\widehat{r} & -2\\widehat{a_3}\\widehat{r}+\\widehat{a_4}\\widehat{p}+\\widehat{a_5}\\widehat{q} \n\\end{bmatrix}.\n\\end{equation}\nNow, the variational matrix of the system (\\ref{gr2}) is given by\n\\begin{equation}\\label{grm1}\n\\mathbb{K}(\\widehat{M_1}) = \n\\begin{bmatrix}\n\\mathbb{K}(\\widehat{a}_{11})& \\mathbb{K}(\\widehat{p})\\mathbb{K}(\\widehat{r}) & -\\mathbb{K}(\\widehat{p})+\\mathbb{K}(\\widehat{p})\\mathbb{K}(\\widehat{q}) \\\\\n\\mathbb{K}(\\widehat{q})\\mathbb{K}(\\widehat{r} )& \\mathbb{K}(\\widehat{a}_{22}) & -\\mathbb{K}(\\widehat{q})+\\mathbb{K}(\\widehat{p})\\mathbb{K}(\\widehat{q}) \\\\\n\\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{r}) & \\mathbb{K}(\\widehat{a_5})\\mathbb{K}(\\widehat{r}) & \\mathbb{K}(\\widehat{a}_{33})\n\\end{bmatrix},\n\\end{equation}\nwhere\n\\[\\mathbb{K}(\\widehat{a}_{11})=\\mathbb{K}(\\widehat{a_1})-2\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{r})+\\mathbb{K}(\\widehat{q})\\mathbb{K}(\\widehat{r}), \\]\n\\[\\mathbb{K}(\\widehat{a}_{22})=\\mathbb{K}(\\widehat{a_2})-2\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{r})+\\mathbb{K}(\\widehat{p})\\mathbb{K}(\\widehat{r} ),\\]\n\\[\\mathbb{K}(\\widehat{a}_{33})=-2\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r})+\\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p})+\\mathbb{K}(\\widehat{a_5})\\mathbb{K}(\\widehat{q}).\n\\] \nTo check the stability of the fuzzy equilibrium points of the systems (\\ref{gr1}), we define the characteristic equation of the matrix (\\ref{grm1}) by $det(\\mathbb{K}(\\widehat{\\lambda})I-\\mathbb{K}(\\widehat{M_1}))=0$.\n\\begin{itemize}\n\\item[(i)] The variational matrix (\\ref{grm1}) corresponding to the equilibrium point $E_0(0,0,0)$ has eigenvalues\n$\\mathbb{K}(\\widehat{\\lambda})=0,\\mathbb{K}(\\widehat{a_1}),\\mathbb{K}(\\widehat{a_2})$. In which two eigenvalues are positive. Thus $E_0(0,0,0)$ is an unstable equilibrium point. From Remark \\ref{grr1}, the variational fuzzy matrix (\\ref{grm}) corresponding to the fuzzy equilibrium point $\\widehat{E}_0(0,0,0)$ has fuzzy eigenvalues $\\widehat{\\lambda}=\\widehat{0},\\widehat{a_1},\\widehat{a_2}$ and $\\widehat{E}_0(0,0,0)$ is an unstable fuzzy equilibrium point.\n\\item[(ii)] The variational matrix (\\ref{grm1}) corresponding to the equilibrium point $E_1(1,0,0)$ has\n$\\mathbb{K}(\\widehat{\\lambda})=-\\mathbb{K}(\\widehat{a_1}),\\mathbb{K}(\\widehat{a_2}),\\mathbb{K}(\\widehat{a_4})$. In which two eigenvalues are positive. Thus $E_1(1,0,0)$ is an unstable equilibrium point. From Remark \\ref{grr1}, the variational fuzzy matrix (\\ref{grm}) corresponding to the fuzzy equilibrium point $\\widehat{E}_1(1,0,0)$ has fuzzy eigenvalues $\\widehat{\\lambda}=-\\widehat{a_1},\\widehat{a_2},\\widehat{a_4}$ and $\\widehat{E}_1(1,0,0)$ is an unstable fuzzy equilibrium point.\n\\item[(iii)] The variational matrix (\\ref{grm1}) corresponding to the equilibrium point $E_2(0,1,0)$ has eigenvalues\n$\\mathbb{K}(\\widehat{\\lambda})=\\mathbb{K}(\\widehat{a_1}),-\\mathbb{K}(\\widehat{a_2}),\\mathbb{K}(\\widehat{a_5})$. In which two eigenvalues are positive. Thus $E_2(0,1,0)$ is an unstable equilibrium point. From Remark \\ref{grr1}, the variational fuzzy matrix (\\ref{grm}) corresponding to the fuzzy equilibrium point $\\widehat{E}_2({0},{1},{0})$ has fuzzy eigenvalues $\\widehat{\\lambda}=\\widehat{a_1},-\\widehat{a_2},\\widehat{a_5}$ and $\\widehat{E}_2({0},{1},{0})$ is an unstable fuzzy equilibrium point.\n\\item[(iv)] The variational matrix (\\ref{grm1}) corresponding to the equilibrium point $E_3(1,1,0)$ has eigenvalues\n$\\mathbb{K}(\\widehat{\\lambda})=-\\mathbb{K}(\\widehat{a_1}),-\\mathbb{K}(\\widehat{a_2}),\\mathbb{K}(\\widehat{a_4})+\\mathbb{K}(\\widehat{a_5})$. In which one eigenvalues are positive. Thus $E_3(1,1,0)$ is an unstable equilibrium point. From Remark \\ref{grr1}, the variational fuzzy matrix (\\ref{grm}) corresponding to the fuzzy equilibrium point $\\widehat{E}_3(1,1,0)$ has fuzzy eigenvalues $\\widehat{\\lambda}=-\\widehat{a_1},-\\widehat{a_2},\\widehat{a_4}+\\widehat{a_5}$ and $\\widehat{E}_3(1,1,0)$ is an unstable fuzzy equilibrium point.\n\\item[(v)] The equilibrium point $E_4(0,\\mathbb{K}(\\widehat{q_{e_4}}),\\mathbb{K}(\\widehat{r_{e_4}}))$ is locally asymptotically stable if \n\\[\\mathbb{K}(\\widehat{a_1})<\\frac{\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_5})^2}{(\\mathbb{K}(\\widehat{a_2})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_5}))^2}.\\]\nTherefore $\\widehat{E}_4({0},\\widehat{q_{e_4}},\\widehat{r_{e_4}})$ is locally asymptotically fuzzy stable if \\[\\widehat{a_1}<\\frac{\\widehat{a_2}\\widehat{a_5}^2}{(\\widehat{a_2}\\widehat{a_3}+\\widehat{a_5})^2}.\\]\n\\begin{exa}\\label{grex1}\n\\vspace{-5mm}\n\\end{exa} Let $\\widehat{a_1}=(0.01,0.02,0.03),\\widehat{a_2}=(2,4,6),\\widehat{a_3}=(0.1,0.2,0.3),\\widehat{a_4}=(3,4,5),\\widehat{a_5}=(1,2,3),\\widehat{p}_0=0.1,\\widehat{q}_0=0.2,\\widehat{r}_0=0.3$, where $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$ are triangular fuzzy numbers. Now, the horizontal membership functions of the given triangular fuzzy numbers are given by\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_1})&=&a_1^{gr}(\\alpha,\\mu_{a_1})=0.01+0.01\\alpha+\\mu_{a_1}(0.02-0.02\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_2})&=&a_2^{gr}(\\alpha,\\mu_{a_2})=2+2\\alpha+\\mu_{a_2}(4-4\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_3})&=&a_3^{gr}(\\alpha,\\mu_{a_3})=0.1+0.1\\alpha+\\mu_{a_3}(0.2-0.2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_4})&=&a_4^{gr}(\\alpha,\\mu_{a_4})=3+\\alpha+\\mu_{a_4}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_5})&=&a_5^{gr}(\\alpha,\\mu_{a_5})=1+\\alpha+\\mu_{a_5}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{p}_0)&=&0.1,\\mathbb{K}(\\widehat{q}_0)=0.2,\\mathbb{K}(\\widehat{r}_0)=0.3.\n\\end{eqnarray*}\nFurther, we assume $\\mu_p,\\mu_q,\\mu_r,\\mu_{a_1},\\mu_{a_2},\\mu_{a_3},\\mu_{a_4},\\mu_{a_5}=\\mu\\in\\{0,0.4,0.6,1\\}$ and $\\alpha\\in\\{0,0.5,1\\}$.\n\\begin{table}[ht!]\n \\begin{center}\n    \\begin{adjustbox}{max width=0.9\\linewidth}\n  \\begin{tabular}{|p{.4cm}|p{3.1cm}| p{3.1cm}|p{3.1cm}|}\n  \\hline\n   $\\mu$ & $\\alpha=0$&  $\\alpha=0.5$& $\\alpha=1$\\\\\n   \\hline\n   0&(0,0.1667,1.6667)&(0,0.2308,2.3077)&(0,0.2857,2.8571) \\\\\n   \\hline\n    0.4&(0,0.2647,2.6471)&(0,0.2754,2.7536)&(0,0.2857,2.8571) \\\\\n   \\hline\n    0.6&(0,0.3056,3.0556)&(0,0.2958,2.9578)&(0,0.2857,2.8571)\\\\\n   \\hline\n    1&(0,0.375,3.75)&(0,0.3333,3.3333)&(0,0.2857,2.8571)\\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n   \\caption{Equilibrium point $E_4(0,\\mathbb{K}(\\widehat{q_{e_4}}),\\mathbb{K}(\\widehat{r_{e_4}}))$ for different values of $\\alpha,\\mu$}\n  \\label{grtab:table1}\n \\end{center}\n\\end{table}\nFor the given data set, we realize that the stability condition of $E_4$ is well satisfied and the system (\\ref{gr2a}) have different equilibrium points corresponding to different values of $\\alpha,\\mu$, as shown in Table \\ref{grtab:table1}. From Table \\ref{grtab:table1}, we realize that for a fixed value of $\\alpha$ the equilibrium point increases with increasing value of $\\mu$ and at $\\alpha=1$ remain same because the left and right interval of triangular fuzzy number coincides with each other. For $\\mu=0,0.4$ the equilibrium point increases while for $\\mu=0.6,1$ the equilibrium point decreases with increasing value of $\\alpha$. Figure \\ref{grfig:1} shows that for the given data in Example \\ref{grex1}, the population density $\\mathbb{K}(\\widehat{p}(u))$ of prey eventually extinct while the population densities $\\mathbb{K}(\\widehat{q}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$ of prey and predator persist, respectively and eventually get their steady states given in Table \\ref{grtab:table1} corresponding to different values of $\\alpha,\\mu$ and become asymptotically stable. At $\\alpha=1$, it shows crisp behavior.\n\\begin{figure}\n \\centering\n   \\includegraphics[scale=0.75]{figure\/1.eps} \n   \\includegraphics[scale=0.75]{figure\/5.eps}\n  \\includegraphics[scale= 0.75]{figure\/9.eps}\n \\includegraphics[scale=0.75]{figure\/2.eps}\n   \\includegraphics[scale=0.75]{figure\/6.eps}\n   \\includegraphics[scale=0.75]{figure\/10.eps}\n  \\includegraphics[scale=0.75]{figure\/3.eps}\n    \\includegraphics[scale=0.75]{figure\/7.eps}\n      \\end{figure}\n    \\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.75]{figure\/11.eps} \n     \\includegraphics[scale=0.75]{figure\/4.eps}\n     \\includegraphics[scale=0.75]{figure\/8.eps}\n \\includegraphics[scale=0.75]{figure\/12.eps}        \\caption{Variation of preys and predator populations against the time for the system (\\ref{gr2a}) for different values of $\\alpha,\\mu$. Blue, green and red curves show the horizontal membership function of the population densities $\\mathbb{K}(\\widehat{p}(u)), \\mathbb{K}(\\widehat{q}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$, respectively.}\n  \\label{grfig:F}\n\\end{figure}\n\\item[(vi)] The equilibrium point $E_5(\\mathbb{K}(\\widehat{p_{e_5}}),0,\\mathbb{K}(\\widehat{r_{e_5}}))$ is locally asymptotically stable if \n\\[\\mathbb{K}(\\widehat{a_2})<\\frac{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_4})^2}{(\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_3})+\\mathbb{K}(\\widehat{a_4}))^2}.\\]\nTherefore the fuzzy equilibrium point $\\widehat{E}_5(\\widehat{p_{e_5}},{0},\\widehat{r_{e_5}})$ is locally asymptotically fuzzy stable if \n\\[\\widehat{a_2}<\\frac{\\widehat{a_1}\\widehat{a_4}^2}{(\\widehat{a_1}\\widehat{a_3}+\\widehat{a_4})^2}.\\]\n\\begin{exa}\\label{grex2}\n\\end{exa}\n Let $\\widehat{a_1}=(2,4,6),\\widehat{a_2}=(0.01,0.02,0.03),\\widehat{a_3}=(0.1,0.2,0.3),\\widehat{a_4}=(1,2,3),\\widehat{a_5}=(3,4,5),\\widehat{p}_0=0.1,\\widehat{q}_0=0.2,\\widehat{r}_0=0.3$, where $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$ are triangular fuzzy numbers. Now, the horizontal membership functions of the given triangular fuzzy numbers are given by\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_1})&=&a_1^{gr}(\\alpha,\\mu_{a_1})=2+2\\alpha+\\mu_{a_1}(4-4\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_2})&=&a_2^{gr}(\\alpha,\\mu_{a_2})=0.01+0.01\\alpha+\\mu_{a_2}(0.02-0.02\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_3})&=&a_3^{gr}(\\alpha,\\mu_{a_3})=0.1+0.1\\alpha+\\mu_{a_3}(0.2-0.2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_4})&=&a_4^{gr}(\\alpha,\\mu_{a_4})=1+\\alpha+\\mu_{a_4}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_5})&=&a_5^{gr}(\\alpha,\\mu_{a_5})=3+\\alpha+\\mu_{a_5}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{p}_0)&=&0.1,\\mathbb{K}(\\widehat{q}_0)=0.2,\\mathbb{K}(\\widehat{r}_0)=0.3.\n\\end{eqnarray*}\nFurther, we assume $\\mu_p,\\mu_q,\\mu_r,\\mu_{a_1},\\mu_{a_2},\\mu_{a_3},\\mu_{a_4},\\mu_{a_5}=\\mu\\in\\{0,0.4,0.6,1\\}$ and $\\alpha\\in\\{0,0.5,1\\}$.\n\\begin{table}[ht!]\n \\begin{center}\n   \\begin{adjustbox}{max width=0.9\\linewidth}\n  \\begin{tabular}{|p{.4cm}|p{3.1cm}| p{3.1cm}|p{3.1cm}|}\n  \\hline\n   $\\mu$ & $\\alpha=0$&  $\\alpha=0.5$& $\\alpha=1$\\\\\n   \\hline\n   0&(0.1667,0,1.6667)&(0.2308,0,2.3077)&(0.2857,0,2.8571) \\\\\n   \\hline\n    0.4&(0.2647,0,2.6471)&(0.2754,0,2.7536)&(0.2857,0,2.8571) \\\\\n   \\hline\n    0.6&(0.3056,0,3.0556)&(0.2958,0,2.9577)&(0.2857,0,2.8571)\\\\\n   \\hline\n    1&(0.375,0,3.75)&(0.3333,0,3.3333)&(0.28571,0,2.8571)\\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n   \\caption{Equilibrium point $E_5(\\mathbb{K}(\\widehat{p_{e_5}}),0,\\mathbb{K}(\\widehat{r_{e_5}}))$ for different values of $\\alpha,\\mu$}\n  \\label{grtab:table2}\n \\end{center}\n\\end{table}\nFor the given data set, we observe that the stability condition of $E_5$ is well satisfied and the system (\\ref{gr2a}) have different equilibrium points corresponding to different values of $\\alpha,\\mu$, as shown in Table \\ref{grtab:table2}. From Table \\ref{grtab:table2}, we realize that for a fixed value of $\\alpha$ the equilibrium point increases with increasing value of $\\mu$ and at $\\alpha=1$ remain same because the left and right interval of triangular fuzzy number coincides with each other. For $\\mu=0,0.4$ the equilibrium point increases while for $\\mu=0.6,1$ the equilibrium point decreases with increasing value of $\\alpha$. Figure \\ref{grfig:2} shows that for the given data in Example \\ref{grex2}, the population density $\\mathbb{K}(\\widehat{q}(u))$ of prey eventually extinct while the population densities $\\mathbb{K}(\\widehat{p}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$ of prey and predator persist, respectively and eventually get their steady states given in Table \\ref{grtab:table2} corresponding to different values of $\\alpha,\\mu$ and become asymptotically stable. At $\\alpha=1$, it shows crisp behavior.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.75]{figure\/13.eps}  \n  \\includegraphics[scale=0.75]{figure\/17.eps}\n\\includegraphics[scale= 0.75]{figure\/21.eps}\n  \\includegraphics[scale=0.75]{figure\/14.eps}  \n  \\includegraphics[scale=0.75]{figure\/18.eps}\n   \\includegraphics[scale=0.75]{figure\/22.eps}\n    \\includegraphics[scale=0.75]{figure\/15.eps}\n    \\includegraphics[scale=0.75]{figure\/19.eps}\n     \\end{figure}\n     \\begin{figure}\n  \\centering\n    \\includegraphics[scale=0.75]{figure\/23.eps}\n     \\includegraphics[scale=0.75]{figure\/16.eps}\n      \\includegraphics[scale=0.75]{figure\/20.eps}\n     \\includegraphics[scale=.75]{figure\/24.eps}\n     \\caption{Variation of preys and predator populations against the time for the system (\\ref{gr2a}) for different values of $\\alpha,\\mu$. Blue, green and red curves show the horizontal membership function of the population densities $\\mathbb{K}(\\widehat{p}(u)), \\mathbb{K}(\\widehat{q}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$, respectively.}\n  \\label{grfig:2}\n\\end{figure}\n\\item[(vii)] The equilibrium point $E_6(1,1,\\mathbb{K}(\\widehat{r_{e_6}}))$ is locally asymptotically stable if \n\\[\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_2})>\\frac{(\\mathbb{K}(\\widehat{a_4})+\\mathbb{K}(\\widehat{a_5}))^2}{\\mathbb{K}(\\widehat{a_3})^2}.\\]\nTherefore the fuzzy equilibrium point $\\widehat{E}_6({1},{1},\\widehat{r_{e_6}})$ is locally asymptotically fuzzy stable if \n\\[\\widehat{a_1}\\widehat{a_2}>\\frac{(\\widehat{a_4}+\\widehat{a_5})^2}{\\widehat{a_3}^2}.\\]\n\\begin{exa}\\label{grex3}\n\\end{exa} Let $\\widehat{a_1}=(3,4,5),\\widehat{a_2}=(2,4,6),\\widehat{a_3}=(3,4,5),\\widehat{a_4}=(1,2,3),\\widehat{a_5}=(0.01,0.02,0.03),\\widehat{p}_0=0.1,\\widehat{q}_0=0.2,\\widehat{r}_0=0.3$, where $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$ are triangular fuzzy numbers. Now, the horizontal membership functions of the given triangular fuzzy numbers are given by\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_1})&=&a_1^{gr}(\\alpha,\\mu_{a_1})=3+\\alpha+\\mu_{a_1}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_2})&=&a_2^{gr}(\\alpha,\\mu_{a_2})=2+2\\alpha+\\mu_{a_2}(4-4\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_3})&=&a_3^{gr}(\\alpha,\\mu_{a_3})=3+\\alpha+\\mu_{a_3}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_4})&=&a_4^{gr}(\\alpha,\\mu_{a_4})=1+\\alpha+\\mu_{a_4}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_5})&=&a_5^{gr}(\\alpha,\\mu_{a_5})=0.01+0.01\\alpha+\\mu_{a_5}(0.02-0.02\\alpha),\\\\\n\\mathbb{K}(\\widehat{p}_0)&=&0.1,\\mathbb{K}(\\widehat{q}_0)=0.2,\\mathbb{K}(\\widehat{r}_0)=0.3.\n\\end{eqnarray*}\nFurther, we assume $\\mu_p,\\mu_q,\\mu_r,\\mu_{a_1},\\mu_{a_2},\\mu_{a_3},\\mu_{a_4},\\mu_{a_5}=\\mu\\in\\{0,0.4,0.6,1\\}$ and $\\alpha\\in\\{0,0.5,1\\}$.\n\\begin{table}[ht!]\n \\begin{center}\n  \\begin{adjustbox}{max width=0.9\\linewidth}\n  \\begin{tabular}{|p{.4cm}|p{2.0cm}| p{2.0cm}|p{2.0cm}|}\n  \\hline\n   $\\mu$ & $\\alpha=0$&  $\\alpha=0.5$& $\\alpha=1$\\\\\n   \\hline\n   0&(1,1,0.3367)&(1,1,0.4329)&(1,1,0.5050) \\\\\n   \\hline\n    0.4&(1,1,0.4784)&(1,1,0.4921)&(1,1,0.5050) \\\\\n   \\hline\n    0.6&(1,1,0.5290)&(1,1,0.5173)&(1,1,0.5050)\\\\\n   \\hline\n    1&(1,1,0.6060)&(1,1,0.5611)&(1,1,0.5050)\\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n   \\caption{Equilibrium point $E_6({1},{1},\\mathbb{K}(\\widehat{r_{e_6}}))$ for different values of $\\alpha,\\mu$}\n  \\label{grtab:table3}\n \\end{center}\n \\end{table}\nFor the given data set, we observe that the stability condition of $E_6$ is well satisfied and the system (\\ref{gr2a}) have different equilibrium points corresponding to different values of $\\alpha,\\mu$, as shown in Table \\ref{grtab:table3}. From Table \\ref{grtab:table3}, we realize that for a fixed value of $\\alpha$ the equilibrium point increases with increasing value of $\\mu$ and at $\\alpha=1$ remain same because the left and right interval of triangular fuzzy number coincides with each other. For $\\mu=0,0.4$ the equilibrium point increases while for $\\mu=0.6,1$ the equilibrium point decreases with increasing value of $\\alpha$. Figure \\ref{grfig:3} shows that for the given data in Example \\ref{grex3}, initially the population density $\\mathbb{K}(\\widehat{r}(u))$ of predator decreases while the population densities $\\mathbb{K}(\\widehat{p}(u))$ and $\\mathbb{K}(\\widehat{q}(u))$ of preys increases, respectively and eventually get their steady states given in Table \\ref{grtab:table3} corresponding to different values of $\\alpha,\\mu$ and become asymptotically stable. At $\\alpha=1$, it shows crisp behavior.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.75]{figure\/25.eps}\n  \\includegraphics[scale=0.75]{figure\/29.eps}\n  \\includegraphics[scale= 0.75]{figure\/33.eps}\n   \\includegraphics[scale=0.75]{figure\/26.eps}\n   \\includegraphics[scale=0.75]{figure\/30.eps}\n   \\includegraphics[scale= 0.65]{figure\/34.eps}\n\\includegraphics[scale=0.75]{figure\/27.eps}\n\\includegraphics[scale=0.75]{figure\/31.eps}\n\\end{figure}\n\\begin{figure}\n  \\centering\n\\includegraphics[scale= 0.65]{figure\/35.eps}\n\\includegraphics[scale=0.8]{figure\/28.eps}\n\\includegraphics[scale=0.8]{figure\/32.eps}\n\\includegraphics[scale= 0.65]{figure\/36.eps} \\caption{Variation of preys and predator populations against the time for the system (\\ref{gr2a}) for different values of $\\alpha,\\mu$. Blue, green and red curves show the horizontal membership function of the population densities $\\mathbb{K}(\\widehat{p}(u)), \\mathbb{K}(\\widehat{q}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$, respectively.}\n \\label{grfig:3}\n\\end{figure}\n\\item[(viii)] The equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$ is locally asymptotically stable if \n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_4})+\\mathbb{K}(\\widehat{a_5})&>&\\mathbb{K}(\\widehat{a_3})\\sqrt{\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_2})},\\\\\n\\mathbb{K}(\\widehat{a_2})&\\geq& \\mathbb{K}(\\widehat{a_1}),\\\\\n\\mathbb{K}(\\widehat{a_1})\\mathbb{K}(\\widehat{a_3})&\\geq& 1.\n\\end{eqnarray*}\nTherefore the fuzzy equilibrium point $\\widehat{E}_7(\\widehat{p_{e_7}},\\widehat{q_{e_7}},\\widehat{r_{e_7}})$ is locally asymptotically fuzzy stable if \n\\begin{eqnarray*}\n\\widehat{a_4}+\\widehat{a_5}&>&\\widehat{a_3}\\sqrt{\\widehat{a_1}\\widehat{a_2}},\\\\\n\\widehat{a_2}&\\geq&\\widehat{a_1},\\\\\n\\widehat{a_1}\\widehat{a_3}&\\geq& 1.\n\\end{eqnarray*}\n\\begin{exa}\\label{grex4}\n\\end{exa} Let $\\widehat{a_1}=(1,2,3),\\widehat{a_2}=(2,4,6),\\widehat{a_3}=(1,2,3),\\widehat{a_4}=(3,4,5),\\widehat{a_5}=(1,2,3),\\widehat{p}_0=0.1,\\widehat{q}_0=0.2,\\widehat{r}_0=0.3$, where $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$ are triangular fuzzy numbers. Now, the horizontal membership functions of the given triangular fuzzy numbers are given by\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_1})&=&a_1^{gr}(\\alpha,\\mu_{a_1})=1+\\alpha+\\mu_{a_1}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_2})&=&a_2^{gr}(\\alpha,\\mu_{a_2})=2+2\\alpha+\\mu_{a_2}(4-4\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_3})&=&a_3^{gr}(\\alpha,\\mu_{a_3})=1+\\alpha+\\mu_{a_3}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_4})&=&a_4^{gr}(\\alpha,\\mu_{a_4})=3+\\alpha+\\mu_{a_4}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_5})&=&a_5^{gr}(\\alpha,\\mu_{a_5})=1+\\alpha+\\mu_{a_5}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{p}_0)&=&0.1,\\mathbb{K}(\\widehat{q}_0)=0.2,\\mathbb{K}(\\widehat{r}_0)=0.3.\n\\end{eqnarray*}\nFurther, we assume $\\mu_p,\\mu_q,\\mu_r,\\mu_{a_1},\\mu_{a_2},\\mu_{a_3},\\mu_{a_4},\\mu_{a_5}=\\mu\\in\\{0,0.4,0.6,1\\}$ and $\\alpha\\in\\{0,0.5,1\\}$.\n\\begin{table}[ht!]\n \\begin{center}\n  \\begin{adjustbox}{max width=0.9\\linewidth}\n  \\begin{tabular}{|p{0.5cm}|p{4cm}| p{4cm}|p{4cm}|}\n  \\hline\n   $\\mu$ & $\\alpha=0$&  $\\alpha=0.5$& $\\alpha=1$\\\\\n   \\hline\n   0&(0.3025,0.5068,1.414)&(0.6014,0.7181,2.1213)&(0.9366,0.9552,2.8284) \\\\\n   \\hline\n    0.4&(0.7993,0.8581,2.5456)&(0.8675,0.9063,2.6870)&\n    (0.9366,0.9552,2.8284) \\\\\n   \\hline\n    0.6&(1.0773,1.0546,3.1113)&(1.0066,1.0046,2.9698)&\n    (0.9366,0.9552,2.8284)\\\\\n   \\hline\n   1&(1.6639,1.4695,4.2426)&\n   (1.2934,1.2075,3.5355)&\n    (0.9366,0.9552,2.8284)\\\\\n    \\hline\n  \\end{tabular}\n  \\end{adjustbox}\n \\caption{Equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$ for different values of $\\alpha,\\mu$}\n  \\label{grtab:table4}\n \\end{center}\n \\end{table}\nFor the given data set, we observe that the stability condition of $E_7(\\widehat{p_{e_7}},\\widehat{q_{e_7}},\\widehat{r_{e_7}})$ is well satisfied and the system (\\ref{gr2a}) have different equilibrium points corresponding to different values of $\\alpha,\\mu$, as shown in Table \\ref{grtab:table4}. From Table \\ref{grtab:table4}, we realize that for a fixed value of $\\alpha$ the equilibrium point increases with increasing value of $\\mu$ and at $\\alpha=1$ remain same because the left and right interval of triangular fuzzy number coincides with each other. For $\\mu=0,0.4$ the equilibrium point increases while for $\\mu=0.6,1$ the equilibrium point decreases with increasing value of $\\alpha$. Figure \\ref{grfig:4} shows that for the given data in Example \\ref{grex4}, initially the population densities $\\mathbb{K}(\\widehat{p}(u)),\\mathbb{K}(\\widehat{q}(u))$ and $\\mathbb{K}(\\widehat{r}(u))$ of preys and predator, respectively increases and eventually get their steady states given in Table \\ref{grtab:table4} corresponding to different values of $\\alpha,\\mu$ and become asymptotically stable. At $\\alpha=1$, it shows crisp behavior.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.75]{figure\/37.eps}\n   \\includegraphics[scale=0.75]{figure\/41.eps}\n     \\includegraphics[scale= 0.75]{figure\/45.eps}\n   \\includegraphics[scale=0.75]{figure\/38.eps}\n    \\includegraphics[scale=0.75]{figure\/42.eps}\n  \\includegraphics[scale= 0.65]{figure\/46.eps}\n    \\includegraphics[scale=0.75]{figure\/39.eps} \n    \\includegraphics[scale=0.75]{figure\/43.eps}\n    \\end{figure}\n  \\begin{figure}\n  \\centering\n    \\includegraphics[scale= 0.65]{figure\/47.eps}\n     \\includegraphics[scale=0.75]{figure\/40.eps} \n     \\includegraphics[scale=0.75]{figure\/44.eps}  \\includegraphics[scale=0.65]{figure\/48.eps}\n  \\caption{Variation of preys and predator populations against the time for the system (\\ref{gr2a}) for different values of $\\alpha,\\mu$. Blue, green and red curves show the horizontal membership function of the population densities $\\widehat{p}(u), \\widehat{q}(u)$ and $\\widehat{r}(u)$, respectively.}\n  \\label{grfig:4}\n\\end{figure}\n\\end{itemize}\n\\begin{thm}\nThe nontrivial fuzzy equilibrium point $\\widehat{E}_7({\\widehat{p_{e_7}}},{\\widehat{q_{e_7}}},{\\widehat{r_{e_7}}})$ is globally asymptotically fuzzy stable.\n\\end{thm}\n\\textbf{Proof:} To study the global fuzzy stability of the system (\\ref{gr1}), it is enough to check that the global stability of the system (\\ref{gr2}). For which, we construct a suitable Lyapunov function $\\mathbb{K}(\\widehat{U})=\\left(\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}})-\\mathbb{K}(\\widehat{p_{e_7}})\\log\\left(\\frac{\\mathbb{K}(\\widehat{p})}{\\mathbb{K}(\\widehat{p_{e_7}})}\\right)\\right)+\\left(\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}})-\\mathbb{K}(\\widehat{q_{e_7}})\\log\\left(\\frac{\\mathbb{K}(\\widehat{q})}{\\mathbb{K}(\\widehat{q_{e_7}})}\\right)\\right)+\\linebreak \\left(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}})-\\mathbb{K}(\\widehat{r_{e_7}})\\log\\left(\\frac{\\mathbb{K}(\\widehat{r})}{\\mathbb{K}(\\widehat{r_{e_7}})}\\right)\\right)$. It is simple to verify that the function $\\mathbb{K}(\\widehat{U})$ is zero at the equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$ and positive for other values of\\linebreak $(\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))$. Then\n\\begin{eqnarray}\n    \\begin{array}{ll}\\label{grlya}\n       \\frac{\\partial \\mathbb{K}(\\widehat{U})}{\\partial t}=\\frac{(\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}}))}{\\mathbb{K}(\\widehat{p})}\\frac{\\partial \\mathbb{K}(\\widehat{p})}{\\partial t}+\\frac{(\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}}))}{\\mathbb{K}(\\widehat{q})}\\frac{\\partial \\mathbb{K}(\\widehat{q})}{\\partial t}+\\frac{(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}}))}{\\mathbb{K}(\\widehat{r})}\\frac{\\partial \\mathbb{K}(\\widehat{r})}{\\partial t}\\\\\n    \\hspace{1cm}= (\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}}))(\\mathbb{K}(\\widehat{a_1})(1-\\mathbb{K}(\\widehat{p}))-\n    \\mathbb{K}(\\widehat{r})+    \\mathbb{K}(\\widehat{q})    \\mathbb{K}(\\widehat{r}))\\\\\n    \\hspace{1cm}+ (\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}}))(\\mathbb{K}(\\widehat{a_2})(1-\\mathbb{K}(\\widehat{q}))-\n    \\mathbb{K}(\\widehat{r})+    \\mathbb{K}(\\widehat{p})    \\mathbb{K}(\\widehat{r}))\\\\\n    \\hspace{1cm}+ (\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}}))(-\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r})+\n    \\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p})+    \\mathbb{K}(\\widehat{a_5})\\mathbb{K}(\\widehat{q})).\n    \\end{array}\n\\end{eqnarray}\nFor the equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$, we have the set of equilibrium equations\n\\begin{eqnarray}\\label{grseteq}\n    \\begin{array}{ll}\n      \\mathbb{K}(\\widehat{a_1})(1-\\mathbb{K}(\\widehat{p_{e_7}}))-\n    \\mathbb{K}(\\widehat{r_{e_7}})+    \\mathbb{K}(\\widehat{q_{e_7}})    \\mathbb{K}(\\widehat{r_{e_7}})=0\\\\\n    \\mathbb{K}(\\widehat{a_2})(1-\\mathbb{K}(\\widehat{q_{e_7}}))-\n    \\mathbb{K}(\\widehat{r_{e_7}})+    \\mathbb{K}(\\widehat{p_{e_7}})    \\mathbb{K}(\\widehat{r_{e_7}})=0\\\\\n-\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r_{e_7}})+\n    \\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p_{e_7}})+    \\mathbb{K}(\\widehat{a_5})\\mathbb{K}(\\widehat{q_{e_7}})=0.\n    \\end{array}\n\\end{eqnarray} \nNow, we simplify the Equations (\\ref{grlya}) and (\\ref{grseteq}) in the form\n\\begin{eqnarray}\n    \\begin{array}{ll}\n       \\frac{\\partial \\mathbb{K}(\\widehat{U})}{\\partial t}= -\\mathbb{K}(\\widehat{a_1})(\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}}))^2-\\mathbb{K}(\\widehat{a_2})(\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}}))^2\n       \\\\\n      \\hspace{1cm} -\\mathbb{K}(\\widehat{a_3})(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}}))^2+(\\mathbb{K}(\\widehat{a_4})-1)(\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}}))(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}}))\\\\\n      \\hspace{1cm} +\\left(\\frac{\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r_{e_7}})}{\\mathbb{K}(\\widehat{q_{e_7}})}-\\frac{\\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p_{e_7}})}{\\mathbb{K}(\\widehat{q_{e_7}})}-1\\right)(\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}}))(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}})).\n      \\end{array}\n\\end{eqnarray}\nNext, we assume that $\\mathbb{K}(\\widehat{a_4})<1,\\,\\frac{\\mathbb{K}(\\widehat{a_3})\\mathbb{K}(\\widehat{r_{e_7}})}{\\mathbb{K}(\\widehat{q_{e_7}})}<\\frac{\\mathbb{K}(\\widehat{a_4})\\mathbb{K}(\\widehat{p_{e_7}})}{\\mathbb{K}(\\widehat{q_{e_7}})}+1$ and $(\\mathbb{K}(\\widehat{p})-\\mathbb{K}(\\widehat{p_{e_7}})),\\,(\\mathbb{K}(\\widehat{q})-\\mathbb{K}(\\widehat{q_{e_7}})),\\,(\\mathbb{K}(\\widehat{r})-\\mathbb{K}(\\widehat{r_{e_7}}))$ are of same sign. Then $ \\frac{\\partial \\mathbb{K}(\\widehat{U})}{\\partial t}<0$ except for the equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$. Thus the equilibrium point $E_7(\\mathbb{K}(\\widehat{p_{e_7}}),\\mathbb{K}(\\widehat{q_{e_7}}),\\mathbb{K}(\\widehat{r_{e_7}}))$ is globally asymptotically stable. Now, from Remark \\ref{grr1} and Theorem \\ref{grlogo}, we have $\\mathcal{D}^{gr}\\widehat{U}<0$, except for the fuzzy equilibrium point $\\widehat{E}_7(\\widehat{p_{e_7}},\\widehat{q_{e_7}},\\widehat{r_{e_7}})$. Thus the equilibrium point $\\widehat{E}_7(\\widehat{p_{e_7}},\\widehat{q_{e_7}},\\widehat{r_{e_7}})$ is globally asymptotically fuzzy stable.\n\n\\section{Numerical solution of the fuzzy prey-predator model by using granular $F$-transform}\nThis section presents a numerical method by using the concept of granular $F$-transform to solve the system (\\ref{gr1}). To solve the system (\\ref{gr1}), we first need to solve the system (\\ref{gr2}) by using the concept of $F$-transform. The method of $F$-transform consists in applying the $F$-transform to both sides of the system (\\ref{gr2}). It leads to a system of algebraic equations, whose solution is a discrete representation of an analytical solution of the system (\\ref{gr2}). Accordingly, we apply granular $F$-transform to both sides of the system (\\ref{gr1}), whereby we get the following equations:\n\\begin{eqnarray}\\label{gr5}\n\\nonumber{F^{gr}[\\mathcal{D}^{gr}\\widehat{p}}]&=&F^{gr}[\\widehat{g_1}(u,\\widehat{p},\\widehat{q},\\widehat{r})],\\\\ \n\\nonumber{G^{gr}[\\mathcal{D}^{gr}\\widehat{q}}]&=&G^{gr}[\\widehat{g_2}(u,\\widehat{p},\\widehat{q},\\widehat{r})],\\\\ \n{H^{gr}[\\mathcal{D}^{gr}\\widehat{r}}]&=& H^{gr}[\\widehat{g_3}(u,\\widehat{p},\\widehat{q},\\widehat{r})].\n\\end{eqnarray}\nFrom Remark \\ref{grr1}, the system (\\ref{gr5}) can be written as\n\\begin{eqnarray}\\label{gr6}\n\\nonumber{\\mathbb{K}(F^{gr}[\\mathcal{D}^{gr}\\widehat{p}}])&=&\\mathbb{K}(F^{gr}[\\widehat{g_1}(u,\\widehat{p},\\widehat{q},\\widehat{r})]),\\\\ \n\\nonumber{\\mathbb{K}(G^{gr}[\\mathcal{D}^{gr}\\widehat{q}])}&=&\\mathbb{K}(G^{gr}[\\widehat{g_2}(u,\\widehat{p},\\widehat{q},\\widehat{r})]),\\\\ \n\\nonumber{\\mathbb{K}(H^{gr}[\\mathcal{D}^{gr}\\widehat{r}]})&=& \\mathbb{K}(H^{gr}[\\widehat{g_3}(u,\\widehat{p},\\widehat{q},\\widehat{r})]).\n\\end{eqnarray}\nFrom Proposition \\ref{grgrdif1}, the above system can be rewritten as\n\\begin{eqnarray}\\label{gr7}\n\\nonumber{F[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{p})]}&=&F[\\mathbb{K}(\\widehat{g_1}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r})))]=F[g_1],\\\\\n\\nonumber{G[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{p})]}&=&G[\\mathbb{K}(\\widehat{g_2}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r})))]=G[g_2],\\\\ \n{H[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{p})]}&=&H[\\mathbb{K}(\\widehat{g_3}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r})))]=H[g_3],\n\\end{eqnarray}\nwhere \n\\[g_1=\\mathbb{K}(\\widehat{g_1}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),\\]\n\\[g_2=\\mathbb{K}(\\widehat{g_2}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),\\]\n\\[g_3=\\mathbb{K}(\\widehat{g_3}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))).\\]\n Let us choose $n > 2$ and create an $h$-uniform fuzzy partition $P_1,... P_m$ of $[a_1,a_2]$, where $h =\\dfrac{a_2-a_1}{m-1}$.\n We denote the $F$-transforms of the functions on left and right sides of the system (\\ref{gr7}) by \n \\[F[\\mathbb{K}(\\widehat{p})] = (X_1,...,X_m)^T,\\]\n \\[G[\\mathbb{K}(\\widehat{q})] = (Y_1,...,Y_m)^T ,\\]\n \\[H[\\mathbb{K}(\\widehat{r})] = (Z_1,...,Z_m)^T,\\]\n \\[F[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{p})] = (X'_1,...,X'_m)^T,\\]\n \\[G[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{q})] = (Y'_1,...,Y'_m)^T ,\\]\n \\[H[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{r})] = (Z'_1,...,Z'_m)^T ,\\]\n \\[F[g_1] = (F_1,...,F_m)^T,\\]\n \\[G[g_2] = (G_1,...,G_m)^T,\\]\n \\[H[g_3] = (H_1,...,H_m)^T.\\]\n Also, we set \\[X_1=\\mathbb{K}(\\widehat{p}_a),\\,Y_1=\\mathbb{K}(\\widehat{q}_a),\\,Z_1=\\mathbb{K}(\\widehat{r}_a),\\] respectively. Thus from the system (\\ref{gr7}), we obtain the following system of algebraic equations:\n\\begin{eqnarray}\\label{gr8}\n\\nonumber&&X'_i=F_i,\\,Y'_i=G_i,\\,Z'_i=H_i,\\,i=2,....m-1,\\\\\n&&X_1=\\mathbb{K}(\\widehat{p}_a),\\,Y_1=\\mathbb{K}(\\widehat{q}_a),Z_1=\\mathbb{K}(\\widehat{r}_a),\n\\end{eqnarray}\nwhere \\[X'_i=F_i[\\frac{\\partial}{\\partial t}\\mathbb{K}(\\widehat{p})],\\,Y'_i=G_i[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{q})],\\,Z'_i=H_i[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{r})],\\]\n\\[X_i=F_i[\\mathbb{K}(\\widehat{p})],\\,Y_i=G_i[\\mathbb{K}(\\widehat{q})],\\,Z_i=H_i[\\mathbb{K}(\\widehat{r})],\\] \\[F_i=F_i[g_1],\\,G_i=G_i[g_2],\\,H_i=H_i[g_3]\\]\nare $F$-transform components. Also, by using schemes of the method of central differences, we replace \n\\begin{eqnarray*}\nX'_i&=&\\dfrac{X_{i+1}-X_{i-1}}{2h},\\\\\nY'_i&=&\\dfrac{Y_{i+1}-Y_{i-1}}{2h},\\\\\nZ'_i&=&\\dfrac{Z_{i+1}-Z_{i-1}}{2h},\\,i=2,...,m-1\n\\end{eqnarray*}\nand assume $X_2=X_1+hF_1,Y_2=Y_1+hG_1,Z_2=Z_1+hH_1$. Thus we can introduce $(m-2)\\times m$ matrix\n\\begin{equation}\\label{grm2}\n{D} = \\frac{1}{2h}\n\\begin{bmatrix}\n-1 & 0 &1&0&0&0&...&0\\\\\n0 & -1 &0&1&0&0&...&0\\\\\n0 & 0&-1&0&1&0&...&0\\\\\n .& &&&.&& &.\\\\[-.2cm]\n . & &&&&.& &.\\\\[-.2cm]\n . & &&&&&. &.\\\\\n  0 & 0&0&0&...&-1&0 &1\\\\\n\\end{bmatrix}.\n\\end{equation}\nNow, we can rewrite the system (\\ref{gr8}) to the following system of linear equations:\n\\[D F[\\mathbb{K}(\\widehat{p})]=F[g_1],\\,D G[\\mathbb{K}(\\widehat{q})]=G[g_2],\\,D H[\\mathbb{K}(\\widehat{r})]=H[g_3],\\]\nwhere \\[\nF[\\mathbb{K}(\\widehat{p})] = (X_1,...,X_{m-1})^T,\\]\n\\[G[\\mathbb{K}(\\widehat{q})] = (Y_1,...,Y_{m-1})^T,\\] \\[H[\\mathbb{K}(\\widehat{r})] = (Z_1,...,Z_{m-1})^T,\\]\n\\[F[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{p})] = (X'_2,...,X'_{m-1})^T,\\]\\[G[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{q})] = (Y'_2,...,Y'_{m-1})^T,\\]\\[H[\\frac{\\partial }{\\partial t}\\mathbb{K}(\\widehat{r})] = (Z'_2,...,Z'_{m-1})^T,\\]\n\\[F[g_1] = (F_2,...,F_{m-1})^T,\\]\\[ G[g_2] = (G_2,...,G_{m-1})^T,\\]\\[H[g_3] = (H_2,...,H_{m-1})^T.\\] Next, the matrix $D$ is completed by adding the first and second row as initial values, as seen below:\n\\begin{equation*}\\label{grm3}\n{D'} = \\frac{1}{2h}\n\\begin{bmatrix}\n1 & 0 &0&0&0&0&...&0\\\\\n0& 1 &0&0&0&0&...&0\\\\\n-1 & 0 &1&0&0&0&...&0\\\\\n0 & -1 &0&1&0&0&...&0\\\\\n0 & 0&-1&0&1&0&...&0\\\\\n .& &&&.&& &.\\\\[-.2cm]\n . & &&&&.& &.\\\\[-.2cm]\n . & &&&&&. &.\\\\\n  0 & 0&0&0&...&-1&0 &1\\\\\n\\end{bmatrix},\n\\end{equation*}\nso that the matrix $D'$ is nom-singular. Now, we can expand $F[ g_1 ],G[g_2]$ and $G[g_3]$ by adding the first and second elements using initial conditions and the matrix $D'$, as follows:\n\\[F[g_1] = \\left(\\frac{\\mathbb{K}(\\widehat{p}_a)}{2h},\\frac{\\mathbb{K}(\\widehat{p_{2}})}{2h},g_2,...,F_{m-1}\\right)^T,\\]\n\\[G[g_2] = \\left(\\frac{\\mathbb{K}(\\widehat{q}_a)}{2h},\\frac{\\mathbb{K}(\\widehat{q}_2)}{2h},G_2,...,G_{m-1}\\right)^T, \\] \n\\[H[g_3] = \\left(\\frac{\\mathbb{K}(\\widehat{r}_a)}{2h},\\frac{\\mathbb{K}(\\widehat{r}_2)}{2h},H_2,...,H_{m-1}\\right)^T,\\]\nwhere $\\mathbb{K}(\\widehat{p_{2}})=X_2$, $\\mathbb{K}(\\widehat{q}_2)=Y_2$ and $\\mathbb{K}(\\widehat{r}_2)=Z_2$. Thus we have \n\\begin{eqnarray}\\label{grmat}\nD' F[\\mathbb{K}(\\widehat{p})]=F[g_1],\\,D' G[\\mathbb{K}(\\widehat{q})]=G[g_2],\\,D' H[\\mathbb{K}(\\widehat{r})]=H[g_3].\n\\end{eqnarray}\nThe system (\\ref{grmat}) can be written as\n\\begin{eqnarray}\\label{grmat1}\n F[\\mathbb{K}(\\widehat{p})]=(D')^{-1}F[g_1],\\, G[\\mathbb{K}(\\widehat{q})]=(D')^{-1}G[g_2],\\, H[\\mathbb{K}(\\widehat{r})]=(D')^{-1}H[g_3].\n\\end{eqnarray}\nNow, to solve the system (\\ref{grmat1}), we compute the inverse matrix $(D')^{-1}$. If $m$ is even then inverse matrix is given as\n\\begin{equation*}\\label{grm4}\n({D'})^{-1} = {2h}\n\\begin{bmatrix}\n1 & 0 &0&0&0&0&...&0\\\\\n0& 1 &0&0&0&0&...&0\\\\\n1 & 0 &1&0&0&0&...&0\\\\\n0 & 1 &0&1&0&0&...&0\\\\\n1 & 0&1&0&1&0&...&0\\\\\n .& &&&.&& &.\\\\[-.2cm]\n . & &&&&.& &.\\\\[-.2cm]\n . & &&&&&. &.\\\\\n  0 & ..&0&1&0&1&0 &1\\\\\n\\end{bmatrix},\n\\end{equation*}\nand when $m$ is odd, we obtain\n\\begin{equation*}\\label{grm5}\n({D'})^{-1} = {2h}\n\\begin{bmatrix}\n1 & 0 &0&0&0&0&...&0\\\\\n0& 1 &0&0&0&0&...&0\\\\\n1 & 0 &1&0&0&0&...&0\\\\\n0 & 1 &0&1&0&0&...&0\\\\\n .& &&&.&& &.\\\\[-.2cm]\n . & &&&&.& &.\\\\[-.2cm]\n . & &&&&&. &.\\\\\n  0 & ..&0&1&0&1&0 &1\\\\\n   1 & 0&1&0&...&1&0 &1\\\\\n\\end{bmatrix}.\n\\end{equation*}\nThus from the system (\\ref{grmat1}), we have the following \\begin{eqnarray}\\label{gr9}\n\\nonumber X_{i+1}&=&X_{i-1}+2hF_i,\\\\\n\\nonumber Y_{i+1}&=&Y_{i-1}+2hG_i,\\\\\n\\nonumber Z_{i+1}&=&Z_{i-1}+2hH_i,\\\\\n\\nonumber X_2&=&\\mathbb{K}(\\widehat{p_{2}}),\\,Y_2=\\mathbb{K}(\\widehat{q}_2),\\,\nZ_2=\\mathbb{K}(\\widehat{r}_2),\\\\\nX_1&=&\\mathbb{K}(\\widehat{p}_a),\\,Y_1=\\mathbb{K}(\\widehat{q}_a),Z_1=\\mathbb{K}(\\widehat{r}_a),\\,i=2,...,m-1.\n\\end{eqnarray}\nThe system (\\ref{gr9}) can be used to the computation of $X_3, . . . , X_m,Y_3, . . . , Y_m$ and $Z_3, . . . , Z_m$. However, it can not be used directly by using the functions $\\mathbb{K}(\\widehat{g_1}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))),$ $\\mathbb{K}(\\widehat{g_2}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r})))$ and $\\mathbb{K}(\\widehat{g_3}(u,\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r})))$, because these functions use unknown functions $\\mathbb{K}(\\widehat{p}), \\mathbb{K}(\\widehat{q})$ and $\\mathbb{K}(\\widehat{r})$. Therefore we can use the same technique as in \\cite{p} and substitute functions by their $F$-transform components:\n\\begin{eqnarray}\n\\nonumber\\hat{F}_i[g_1]&=&\\dfrac{\\int_{a}^{b} \\mathbb{K}(\\widehat{g_1}(u,X_i,Y_i,Z_i))P_i(u)du}{\\int_{a}^{b}P_i(u)du },\\\\\n\\nonumber\\hat{G}_i[g_2]&=&\\dfrac{\\int_{a}^{b} \\mathbb{K}(\\widehat{g_2}(u,X_i,Y_i,Z_i))P_i(u)du}{\\int_{a}^{b}P_i(u)du },\\\\\n\\hat{H}_i[g_3]&=&\\dfrac{\\int_{a}^{b} \\mathbb{K}(\\widehat{g_3}(u,X_i,Y_i,Z_i))P_i(u)du}{\\int_{a}^{b}P_i(u)du },i=1,...,m-1.\n\\end{eqnarray}\nThus the system (\\ref{gr9}) can be written as\n\\begin{eqnarray}\\label{gr10}\n\\nonumber X_{i+1}&=&X_{i-1}+2h\\hat{F}_i[g_1],\\\\\n\\nonumber Y_{i+1}&=&Y_{i-1}+2h\\hat{G}_i[g_2],\\\\\n\\nonumber Z_{i+1}&=&Z_{i-1}+2h\\hat{H}_i[g_3],\\\\\n\\nonumber X_{2}&=&X_{1}+h\\hat{F}_1[g_1],\\\\\n\\nonumber Y_{2}&=&Y_{1}+h\\hat{G}_1[g_2],\\\\\n\\nonumber Z_{2}&=&Z_{1}+h\\hat{H}_1[g_3],\\,i=2,...,m-1,\\\\\nX_1&=&\\mathbb{K}(\\widehat{p}_a),\\,Y_1=\\mathbb{K}(\\widehat{q}_a),Z_1=\\mathbb{K}(\\widehat{r}_a).\n\\end{eqnarray}\nThe proposed method is based on the same assumptions as the well-known Euler mid-point method. It has the same degree of accuracy. The solution vector $(F_m[\\mathbb{K}(\\widehat{p})],G_m[\\mathbb{K}(\\widehat{q})],H_m[\\mathbb{K}(\\widehat{r})])$ approximates the (unique) solution $(\\mathbb{K}(\\widehat{p}),\\mathbb{K}(\\widehat{q}),\\mathbb{K}(\\widehat{r}))$ of the system (\\ref{gr2}) in the sense that $(\\mathbb{K}(\\widehat{p}_i),\\mathbb{K}(\\widehat{q}_i),\\mathbb{K}(\\widehat{r}_i))\\approx (X_i,Y_i,Z_i), i= 1,...,m$, where nodes\n$u_1,...,u_m$ are determined by fuzzy partition $P_1,...,P_m$.\\\\\\\\\nThe system (\\ref{gr10}) can be written as in terms of granular $F$-transform, i.e.,\n\\begin{eqnarray}\\label{gr11}\n\\nonumber\\mathbb{K}(F^{gr}_{i+1}[\\widehat{p}])&=&\\mathbb{K}(F^{gr}_{i-1}[\\widehat{p}])+2h\\mathbb{K}(\\hat{F}^{gr}_i[\\widehat{g_1}]),\\\\\n\\nonumber\\mathbb{K}(G^{gr}_{i+1}[\\widehat{q}])&=&\\mathbb{K}(G^{gr}_{i-1}[\\widehat{q}])+2h\\mathbb{K}(\\hat{G}^{gr}_i[\\widehat{g_2}]),\\\\\n\\nonumber\\mathbb{K}(H^{gr}_{i+1}[\\widehat{r}])&=&\\mathbb{K}(H^{gr}_{i-1}[\\widehat{r}])+2h\\mathbb{K}(\\hat{H}^{gr}_i[\\widehat{g_3}]),\\\\\n\\nonumber\\mathbb{K}(F^{gr}_{2}[\\widehat{p}])&=&\\mathbb{K}(F^{gr}_{1}[\\widehat{p}])+h\\mathbb{K}(\\hat{F}^{gr}_{1}[{\\widehat{g_1}}]),\\\\\n\\nonumber\\mathbb{K}(G^{gr}_{2}[\\widehat{q}])&=&\\mathbb{K}(G^{gr}_{1}[\\widehat{q}])+h\\mathbb{K}(\\hat{G}^{gr}_{1}[{\\widehat{g_2}}]),\\\\\n\\nonumber\\mathbb{K}(H^{gr}_{2}[\\widehat{r}])&=&\\mathbb{K}(H^{gr}_{1}[\\widehat{r}])+h\\mathbb{K}(\\hat{H}^{gr}_{1}[{\\widehat{g_3}}]),\\\\\n\\mathbb{K}(F^{gr}_{1}[\\widehat{p}])&=&\\mathbb{K}(\\widehat{p}_a),\\mathbb{K}(G^{gr}_{1}[\\widehat{q}])=\\mathbb{K}(\\widehat{q}_a),\\mathbb{K}(H^{gr}_{1}[\\widehat{r}])=\\mathbb{K}(\\widehat{r}_a),\n\\end{eqnarray}\nwhere \n\\begin{eqnarray*}\n\\widehat{g_1}&=&\\widehat{g_1}(F^{gr}_i[\\widehat{p}],G^{gr}_i[\\widehat{q}],H^{gr}_i[\\widehat{r}]),\\\\ \\widehat{g_2}&=&\\widehat{g_2}(F^{gr}_i[\\widehat{p}],G^{gr}_i[\\widehat{q}],H^{gr}_i[\\widehat{r}]),\\\\ \\widehat{g_3}&=&\\widehat{g_3}(F^{gr}_i[\\widehat{p}],G^{gr}_i[\\widehat{q}],H^{gr}_i[\\widehat{r}]),\\,i=2,...,m-1. \n\\end{eqnarray*}\nFrom Reamrk \\ref{grr1}, the system (\\ref{gr11}) becomes\n\\begin{eqnarray}\\label{gr12}\n\\nonumber F^{gr}_{i+1}[\\widehat{p}]&=&F^{gr}_{i-1}[\\widehat{p}] +2h \\hat{F}^{gr}_i[\\widehat{g_1}],\\\\\n\\nonumber G^{gr}_{i+1}[\\widehat{q}]&=&G^{gr}_{i-1}[\\widehat{q}] +2h \\hat{G}^{gr}_i[\\widehat{g_2}],\\\\\n\\nonumber H^{gr}_{i+1}[\\widehat{r}]&=&H^{gr}_{i-1}[\\widehat{r}] +2h \\hat{H}^{gr}_i[\\widehat{g_3}],\\\\\n\\nonumber F^{gr}_{2}[\\widehat{p}]&=&F^{gr}_{1}[\\widehat{p}])+h\\hat{F}^{gr}_{1}[{\\widehat{g_1}}],\\\\\n\\nonumber G^{gr}_{2}[\\widehat{q}]&=&G^{gr}_{1}[\\widehat{q}])+h\\hat{G}^{gr}_{1}[{\\widehat{g_2}}],\\\\\n H^{gr}_{2}[\\widehat{r}]&=&H^{gr}_{1}[\\widehat{r}]+h\\hat{H}^{gr}_{1}[{\\widehat{g_3}}],\\,i=2,...,m-1\n\\end{eqnarray}\nwhere $F^{gr}_{1}[\\widehat{p}]=\\widehat{p}_a,G^{gr}_{1}[\\widehat{q}]=\\widehat{q}_a,H^{gr}_{1}[\\widehat{r}]=\\widehat{r}_a$. The solution vector $(F^{gr}[\\widehat{p}],G^{gr}[\\widehat{q}],H^{gr}[\\widehat{r}])$ approximates the (unique) solution $(\\widehat{p},\\widehat{q},\\widehat{r})$ of the system (\\ref{gr1}) in the sense that $(\\widehat{p}_i,\\widehat{q}_i,\\widehat{r}_i)\\approx (F^{gr}_i,G^{gr}_i,H^{gr}_i), i= 1,...,m$, where nodes\n$u_1,...,u_m$ are determined by fuzzy partition $P_1,...,P_m$. Applying the granular inverse $F$-transform to the extended vector $(F^{gr}[\\widehat{p}],G^{gr}[\\widehat{q}],H^{gr}[\\widehat{r}])$,\nwe obtain an approximate solution $(\\widehat{p},\\widehat{q},\\widehat{r})$ of the system (\\ref{gr1}) in the form of fuzzy function function:\n\\begin{eqnarray*}\n\\widehat{{p}}^{gr}_m(u)&=&\\sum_{n=1}^{m} F^{gr}_{i}[\\widehat{p}]P_i(u),\\\\\n\\widehat{{q}}^{gr}_m(u)&=&\\sum_{n=1}^{m} G^{gr}_{i}[\\widehat{q}]P_i(u),\\\\ \n\\widehat{{r}}^{gr}_m(u)&=&\\sum_{n=1}^{m} H^{gr}_{i}[\\widehat{r}]P_i(u),\n\\end{eqnarray*}\nwhere $P_i(u), i= 1, . . . ,m$ are given basic functions. \n\\begin{exa}\\label{grex5}\n\\end{exa} Let $\\widehat{a_1}=(0.01,0.02,0.03),\\widehat{a_2}=(2,4,6),\\widehat{a_3}=(0.1,0.2,0.3),\\widehat{a_4}=(3,4,5),\\widehat{a_5}=(1,2,3),\\widehat{p}_0=0.1,\\widehat{q}_0=0.2,\\widehat{r}_0=0.3$, where $\\widehat{a_1},\\widehat{a_2},\\widehat{a_3},\\widehat{a_4},\\widehat{a_5}$ are triangular fuzzy numbers and assume that fuzzy partition is a triangular fuzzy partition. Now, the horizontal membership functions of the given triangular fuzzy numbers are given by\n\\begin{eqnarray*}\n\\mathbb{K}(\\widehat{a_1})&=&a_1^{gr}(\\alpha,\\mu_{a_1})=0.01+0.01\\alpha+\\mu_{a_1}(0.02-0.02\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_2})&=&a_2^{gr}(\\alpha,\\mu_{a_2})=2+2\\alpha+\\mu_{a_2}(4-4\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_3})&=&a_3^{gr}(\\alpha,\\mu_{a_3})=0.1+0.1\\alpha+\\mu_{a_3}(0.2-0.2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_4})&=&a_4^{gr}(\\alpha,\\mu_{a_4})=3+\\alpha+\\mu_{a_4}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{a_5})&=&a_5^{gr}(\\alpha,\\mu_{a_5})=1+\\alpha+\\mu_{a_5}(2-2\\alpha),\\\\\n\\mathbb{K}(\\widehat{p}_0)&=&0.1,\\mathbb{K}(\\widehat{q}_0)=0.2,\\mathbb{K}(\\widehat{r}_0)=0.3.\n\\end{eqnarray*}\nFurther, we assume $\\mu_p,\\mu_q,\\mu_r,\\mu_{a_1},\\mu_{a_2},\\mu_{a_3},\\mu_{a_4},\\mu_{a_5}=\\mu\\in\\{0,0.4,0.6,1\\}$, and $\\alpha\\in\\{0,0.5,1\\}$. A comparison of the numerical solutions obtained by F-transform (FT-Euler mid-point method) and Euler method with exact solution of the system (\\ref{gr2}) corresponding to different values of $\\alpha,\\mu$ and $h=0.01$ . For the accuracy of numerical solutions over exact solution, root mean square error (RMS error) is used, which is defined by formula\n\\[\\mbox{RMS error}=\\sqrt{\\frac{1}{m}\\sum_{i=1}^{m}({\\Delta}y_k)^2},\\] where ${\\Delta}y_i=y_k^{exact}-y_k^{numerical}$ and $y_k^{exact},y_k^{numerical}$ are exact and numerical solution, respectively. Tables \\ref{grtab:5}, \\ref{grtab:6}, \\ref{grtab:7} and \\ref{grtab:8} show comparison among the exact solution, Euler method and the FT-Euler mid-point method for the system (\\ref{gr2}). The graphical representation of this system shows that exact solution and the FT-Euler mid-point method are very close to each other in comparison to Euler method. Therefore from above comparison, we can say that FT-Euler mid-point method is a trustworthy numerical method and the above figures show that the population density $\\mathbb{K}(\\widehat{p}(u))$ of prey decreases whereas the population densities $\\mathbb{K}(\\widehat{q}(u)),\\mathbb{K}(\\widehat{r}(u))$ of prey and predator, respectively increase corresponding to different values of $\\alpha,\\mu$.\n\\begin{figure}\n  \\centering\n  \\includegraphics[scale=0.5]{figure\/eumid1.eps} \\includegraphics[scale=0.5]{figure\/eumid2.eps}\n   \\includegraphics[scale=0.5]{figure\/eumid3.eps}\n   \\includegraphics[scale=0.5]{figure\/eumid4.eps}\n   \\includegraphics[scale=0.5]{figure\/eumid5.eps}\n  \\includegraphics[scale=0.5]{figure\/eumid6.eps}\n  \\includegraphics[scale=0.5]{figure\/eumid7.eps}\n\\includegraphics[scale=0.5]{figure\/eumid8.eps}\n\\end{figure}\n\\begin{figure}\n  \\centering\n\\includegraphics[scale=0.5]{figure\/eumid9.eps}\n  \\caption{Variation of preys and predator populations against the time for the system (\\ref{gr2a}) for different $\\alpha,\\mu$. Blue, green, red curves show population densities corresponding to exact, FT-Euler mid-point and Euler method, respectively.}\n  \\label{grfig:1}\n\\end{figure}\n\\begin{table}[ht]\n\\centering\n \\begin{adjustbox}{max width=0.90\\linewidth}\n\\begin{tabular}{|c|c |ccc|ccc|ccc|}\n\\hlin\n $\\mu$ &$u_i$ &\\multicolumn{3}{|c|}{Exact} &\\multicolumn{3}{|c|}{ Euler}\n &\\multicolumn{3}{|c|}{FT-Euler mid-point}\n\\\\ \n & &$\\mathbb{K}(\\widehat{p}(u))$&$\\mathbb{K}(\\widehat{q}(u))$&$\\mathbb{K}(\\widehat{r}(u))$ &$\\mathbb{K}(\\widehat{p}(u))$& $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ & $\\mathbb{K}(\\widehat{p}(u))$ & $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ \\\\\n\\hline\n & 0&0.1000&0.2000&0.3000&0.1000&0.2000&0.3000 & 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0954&\t0.2573&\t0.3309 &0.0954&\t0.2571&\t0.3307& 0.0954&\t0.2573&\t0.3309\\\\\n{0} & 0.4& 0.0910&\t0.3210&\t0.3681&0.0910&\t0.3207&\t0.3677 & 0.0910&\t0.3210&\t0.3681 \\\\\n &0.6&0.0867&\t0.3871&0.4135 &0.0867&0.3868&0.4128 &0.0867&\t0.3872&0.4135 \\\\\n &0.8 &0.0825&\t0.4505&\t0.4689 &0.0825&0.4504&\t0.4679 & 0.0825&\t0.4506&\t0.4690\\\\\n  & 1&0.0785&0.5060&0.5362&0.0785&\t0.5062&\t0.5347&0.0784&0.5060&0.5363\\\\\n  \\hline\n  & 0&0.1000&0.2000&0.3000&0.1000&\t0.2000&\t0.3000 & 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0956&0.3218&0.3505 &0.0956&0.3209&0.3499& 0.0956&0.3218&0.3505 \\\\\n{0.4} & 0.4& 0.0916\t&0.4646&0.4275&0.0916\t&0.4635&0.4257 &0.0916\t&0.4646&0.4275 \\\\\n & 0.6&0.0878&0.5958&0.5449&0.0878&0.5958&0.5412&0.0878&0.5958&0.5449 \\\\\n & 0.8 &0.0843&\t0.6861&\t0.7174& 0.0844&\t0.6876&\t0.7111&0.0843&\t0.6861&\t0.7174\\\\\n  & 1&0.0806 &0.7258&0.9572 &0.0807\t&0.7285&0.9478 & 0.0806 &0.7258&0.9572 \\\\\n  \\hline\n  & 0&0.1000&\t0.2000&\t0.3000 &0.1000&\t0.2000&\t0.3000 & 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0957&\t0.3569\t&0.3636 & 0.0957&\t0.3555\t&0.3612& 0.0957&\t0.3570\t&0.3622\\\\\n{0.6} & 0.4&0.0919&0.5383&\t0.4723&\t0.0919&0.5373&\t0.4656&\t0.0919&0.5385&\t0.4687\\\\\n &0.6 &0.0884&\t0.6823&\t0.6541 &0.0884&\t0.6823&\t0.6404 & 0.0885&\t0.6830&\t0.6472\\\\\n &0. 8&0.0850&\t0.7534&0.9379 &0.0852&\t0.7577&0.9138 & 0.0851&\t0.7548&0.9256\\\\\n  & 1&0.0809&\t0.7565&\t1.3412 & 0.0813&\t0.7630&\t1.3042 & 0.0811&\t0.7590&\t1.3214\\\\\n  \\hline\n  & 0&0.1000&\t0.2000&\t0.3000 &0.1000&\t0.2000&\t0.3000& 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0960&0.4313&0.3901 &0.0960&0.4289&0.3879 & 0.0960&0.4313&0.3901\\\\\n{1} & 0.4& 0.0926&\t0.6726&\t0.5823 &0.0926&\t0.6726&\t0.5744& 0.0926&\t0.6727&\t0.5822 \\\\\n & 0.6&0.0897&\t0.7961&\t0.9541&0.0898&\t0.7997&\t0.9361 & 0.0897&\t0.7963&\t0.9538 \\\\\n & 0.8&0.0860&\t0.8010&\t1.5690 &0.0862&\t0.8062&\t1.5379&0.0860&\t0.8012&\t1.5685\\\\\n  & 1&0.0789&\t0.7335&\t2.4055 &0.0796&\t0.7395&\t2.3667& 0.0789&\t0.7342&\t2.4046 \\\\\n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{Comparison of numerical results of Example \\ref{grex5} for $\\alpha=0$}\n\\label{grtab:5}\n\\end{table}\n\\begin{table}[ht]\n\\centering\n \\begin{adjustbox}{max width=0.90\\linewidth}\n\\begin{tabular}{|c|c |ccc|ccc| ccc|}\n\\hlin\n $\\mu$ &$u_i$ &\\multicolumn{3}{|c|}{Exact} &\\multicolumn{3}{|c|}{ Euler}\n &\\multicolumn{3}{|c|}{FT-Euler mid-point}\n\\\\ \n & &$\\mathbb{K}(\\widehat{p}(u))$&$\\mathbb{K}(\\widehat{q}(u))$&$\\mathbb{K}(\\widehat{r}(u))$ &$\\mathbb{K}(\\widehat{p}(u))$& $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ & $\\mathbb{K}(\\widehat{p}(u))$ & $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ \\\\\n\\hline\n & 0&0.1000&0.2000&0.3000&0.1000&\t0.2000&\t0.3000 & 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0956&\t0.2967&\t0.3426&0.0956&\t0.2961&\t0.3422 & 0.0956&\t0.2967&\t0.3426 \\\\\n{0} & 0.4& 0.0913&\t0.4092&\t0.4020&0.0913&\t0.4083&\t0.4009&0.0913&\t0.4092&\t0.4020\\\\\n & 0.6&0.0874&\t0.5202&\t0.4857&0.0874&\t0.5198&\t0.4835 & 0.0874&\t0.5203&\t0.4857 \\\\\n & 0.8&0.0836&\t0.6107&\t0.6015&0.0836&\t0.6113&\t0.5979 & 0.0836&\t0.6108&\t0.6015 \\\\\n  & 1&0.0799&\t0.6685&\t0.7569&\t0.0800&\t0.6702&\t0.7515\t&0.0799&\t0.6685&\t0.7569 \\\\\n  \\hline\n  & 0&0.1000&0.2000&0.3000&0.1000&\t0.2000&\t0.3000&0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0957&\t0.3305&0.3533 &0.0957&0.3294&\t0.3526 & 0.0957&\t0.3304&0.3533\\\\\n{0.4} & 0.4& 0.0916&\t0.4832&\t0.4370&0.0916&\t0.4821&\t0.4349& 0.0916&\t0.4832&\t0.4370 \\\\\n & 0.6&0.0880\t&0.6193&\t0.5678 &0.0880&\t0.6194&\t0.5634&0.0880\t&0.6193&\t0.5677\\\\\n & 0.8&0.0845&\t0.7067&\t0.7633&0.0846&\t0.7086&\t0.7559& 0.0845&\t0.7068&\t0.7632\\\\\n  & 1&0.0808&\t0.7382&\t1.0374 &0.0809&\t0.7414&\t1.02627& 0.0808&\t0.7384&\t1.0372 \\\\\n  \\hline\n  & 0&0.1000&\t0.2000&\t0.3000 &0.1000&\t0.2000&\t0.3000 & 0.1000&0.2000&0.3000 \\\\\n &0. 2&0.0957&\t0.3480&\t0.3592&0.0957&\t0.3467&\t0.3583& 0.0957&\t0.3480&\t0.3591\\\\\n{0.6} &0.4&0.0918&\t0.5202&\t0.4576&\t0.0918&\t0.5190&\t0.4549& 0.0918&\t0.5202&\t0.4576 \\\\\n & 0.6&0.0883&\t0.6629&\t0.6189&0.0883&\t0.6636&\t0.6130 & 0.0883&\t0.6630&\t0.6188\\\\\n & 0.8&0.0850&\t0.7409&\t0.8672&0.0850&\t0.7435&\t0.8570 & 0.0850&\t0.7410&\t0.8672\\\\\n  & 1&0.0810&\t0.7545&\t1.2193&0.0812&\t0.7583&\t1.2042 & 0.0810&\t0.7546&\t1.2192\\\\\n  \\hline\n & 0&0.1000&\t0.2000&\t0.3000 &0.1000&\t0.2000&\t0.3000 & 0.1000&0.2000&0.3000 \\\\\n & 0.2&0.0958&\t0.3843&\t0.3719&0.0958&\t0.3825&\t0.3706& 0.0958&\t0.3843&\t0.3719 \\\\\n{1} & 0.4&0.0922&\t0.5918&\t0.5061 &0.0921\t&0.5908&\t0.5015& 0.0922&\t0.5918&\t0.5060 \\\\\n & 0.6&0.0890&\t0.7351&\t0.7447&0.0890&\t0.7371&\t0.7346& 0.0890&\t0.7352&\t0.7446 \\\\\n & 0.8&0.0856&\t0.7841&\t1.1288 &0.0858&0.7880&\t1.1111 & 0.0856&\t0.7842&\t1.1286 \\\\\n  & 1&0.0808&\t0.7598&\t1.6740 &0.0812&\t0.7647&\t1.6489& 0.0808&\t0.7600&\t1.6738 \\\\\n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{Comparison of numerical results of Example \\ref{grex5} for $\\alpha=0.5$} \n\\label{grtab:6}\n\\end{table}\n\\begin{landscape}\n\n\\begin{table}[ht]\n\n\\centering\n \\begin{adjustbox}{max width=0.65\\linewidth}\n \\begin{tabular}{|c |rrr|rrr| rrr|}\n\\hlin\n $u_i$ &\\multicolumn{3}{|c|}{Exact} &\\multicolumn{3}{|c|}{ Euler}\n &\\multicolumn{3}{|c|}{FT-Euler mid-point}\n\\\\ \n&$\\mathbb{K}(\\widehat{p}(u))$&$\\mathbb{K}(\\widehat{q}(u))$&$\\mathbb{K}(\\widehat{r}(u))$ &$\\mathbb{K}(\\widehat{p}(u))$& $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ & $\\mathbb{K}(\\widehat{p}(u))$ & $\\mathbb{K}(\\widehat{q}(u))$ & $\\mathbb{K}(\\widehat{r}(u))$ \\\\\n\\hline\n 0&0.1000&0.2000&0.3000&0.1000&0.2000&0.3000 & 0.1000&0.2000&0.3000 \\\\\n  0.2&0.0957&0.3392&\t0.3562 &0.0957&\t0.3380&\t0.3554& 0.0957&\t0.3380&\t0.3554 \\\\\n 0.4& 0.0917&\t0.5018&\t0.4470&0.0917&\t0.5006&\t0.4446& 0.0917&\t0.5018&\t0.4470\\\\\n 0.6&0.0882&\t0.6417&\t0.5924 &0.0882&\t0.6421&\t0.5873 & 0.0882&\t0.6418&\t0.5924\\\\\n  0.8&0.0848&\t0.7250&\t0.8132&0.0848&0.7272&\t0.8044 & 0.0848&\t0.7250&\t0.8132\\\\\n  1&0.0809&\t0.7477&\t1.1247&0.0811&\t0.7512&\t1.1117 & 0.0809&\t0.7478&\t1.1246\\\\\n \n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{Comparison of numerical results of Example \\ref{grex5} for $\\alpha=1,\\mu=0,0.4,0.6,1$}\n\\label{grtab:7}\n\\end{table}\n\\begin{table}[ht]\n\\centering\n \\begin{adjustbox}{max width=1.0\\linewidth}\n\\begin{tabular}{|c|ccc|ccc|ccc|ccc|ccc|ccc|}\n\\hlin\n $\\mu$&\\multicolumn{9}{|c|}{ Euler}\n &\\multicolumn{9}{|c|}{FT-Euler Mid-point}\n\\\\\n& & $\\alpha=0$& && $\\alpha=0.5$ & & & $\\alpha=1$ & &&$\\alpha=0$& && $\\alpha=0.5$ & & & $\\alpha=1$ & \\\\\n\\hline\n& &\t& & & & & & &\t& & & & & & & & & \\\\\n{0} & 4.0000e-3&\t2.1213e-4&\t8.1035e-4&0.0000&5.7773e-5&5.7773e-5&4.0000e-3&8.7369e-4&\t2.8381e-3& 0.0000& 5.7732e-5& 0.0000&8.1650e-5 &1.8317e-3 &6.7928e-3 & 0.0000&4.9497e-4 & 3.2914e-4\\\\\n & &\t& & & & & & &\t& & & & & & & & & \\\\\n  \\hline\n \n  & &\t& & & & & & &\t& & & & & & & & & \\\\\n {0.4} & 5.7732e-5 & 1.3880e-3&4.7097e-3 & 0.0000& 0.0000&0.0000 &5.7732e-05&\t1.6467e-3&5.8375e-3&0.0000&\t1.0000e-4&\t1.0000e-4& 8.1650e-5 &1.8317e-3 &6.7928e-3 & 0.0000&4.9497e-4 & 3.2914e-4 \\\\\n & &\t& & & & & & &\t& & & & & & & & & \\\\\n \\hline\n \n & &\t& & & & & & &\t& & & & & & & & & \\\\\n{0.6} &1.8257e-4&\t3.2583e-3&1.6367e-2&1.0000e-4&\t1.2076e-3&\t3.1054e-3 & 8.1652e-5\t&2.0339e-3 &7.9053e-3\t& 0.0000&1.0000e-4\t&1.2910e-4\t & 8.1650e-5 &1.8317e-3 &6.7928e-3 & 0.0000&4.9497e-4 & 3.2914e-4 \\\\\n& &\t& & & & & & &\t& & & & & & & & & \\\\\n  \\hline\n \n & &\t& & & & & & &\t& & & & & & & & & \\\\\n{1} & 3.0000e-4&3.6914e-03&\t2.1848e-2&\t0.0000 &3.2145e-4&\t4.3970e-4&1.8708e-4\t & 2.7404e-3&1.3332e-2\t& 0.0000&1.2247e-4 &1.2247e-4\t&\t8.1650e-5 &1.8317e-3 &6.7928e-3 & 0.0000&4.9497e-4 & 3.2914e-4 \\\\\n& &\t& & & & & & &\t& & & & & & & & & \\\\\n\\hline\n\\end{tabular}\n\\end{adjustbox}\n\\caption{RMS error for Example \\ref{grex5} for different values of $\\alpha,\\mu$} \n\\label{grtab:8}\n\\end{table}\n\\end{landscape}\n\\section{Conclusion}\nIn this contribution, we have introduced and studied the concepts of granular $F$-transforms to enrich the theory of $F$-transforms and explore new applications. Accordingly, we have initiated the said theory, formulated a fuzzy prey-predator model consisting of two prey and one predator team, and discussed the equilibrium points and their stability for this model. Moreover, we have established a numerical method based on the granular $F$-transform to find the numerical solution to the proposed model. Finally, a comparison between two numerical solutions with the exact solution is discussed. In the future, it will be interesting to use the proposed numerical method based on granular $F$-transforms to solve fuzzy fractional differential equations and analyze error estimation. \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2\u00a9 Springer Science+Business Media Dordrecht 2014\n\nVolume 3\n\nDao Companions to Chinese Philosophy\n\nSeries EditorYong Huang\n\nWhile \"philosophy\" is a Western term, philosophy is not something exclusively Western. In this increasingly globalized world, the importance of non-Western philosophy is becoming more and more obvious. Among all the non-Western traditions, Chinese philosophy is certainly one of the richest. In a history of more than 2500 years, many extremely important classics, philosophers, and schools have emerged. As China is becoming an economic power today, it is only natural that more and more people are interested in learning about the cultural traditions, including the philosophical tradition, of China.\n\nThe Dao Companions to Chinese Philosophy series aims to provide the most comprehensive and most up-to-date introduction to various aspects of Chinese philosophy as well as philosophical traditions heavily influenced by it. Each volume in this series focuses on an individual school, text, or person.\n\nFor further volumes: http:\/\/\u200bwww.\u200bspringer.\u200bcom\/\u200bseries\/\u200b8596\n\nEditor\n\nVincent Shen\n\nDao Companion to Classical Confucian Philosophy\n\nEditor\n\nVincent Shen\n\nDepartment of Philosophy and Department of East Asian Studies, University of Toronto, Toronto, Ontario, Canada\n\nISBN 978-90-481-2935-5e-ISBN 978-90-481-2936-2\n\nSpringer Dordrecht Heidelberg New York London\n\nLibrary of Congress Control Number: 2013950617\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\nThis work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher's location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law.\n\nThe use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.\n\nWhile the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.\n\nPrinted on acid-free paper\n\nSpringer is part of Springer Science+Business Media (www.springer.com)\n\nContents\n\n1 Introduction:\u200b Classical Confucianism in Historical and Comparative Context 1\n\nVincent Shen\n\nPart I Historical Development\n\n2 The Fading of Political Theology and the Rise of Creative Humanism 23\n\nVincent Shen\n\n3 The Philosophy of Confucius 53\n\nPeimin Ni\n\n4 The Philosophy of Confucius' Disciples 81\n\nYuet Keung Lo\n\n5 Zisi and the Thought of Zisi and Mencius School 119\n\nChen-Feng Tsai\n\n 6 The Daxue ( Great Learning ) and the Zhongyong ( Doctrine of the Mean )  139\n\nAndrew H. Plaks\n\n7 Philosophical Thought of Mencius 153\n\nWing-cheuk Chan\n\n8 Xunzi as a Systematic Philosopher:\u200b Toward Organic Unity of Nature, Mind, and Reason 179\n\nChung-ying Cheng\n\nPart II Philosophical Issues\n\n9 Early Confucian Perspectives on Emotions 203\n\nCurie Vir\u00e1g\n\n10 Art and Aesthetics of Music in Classical Confucianism 227\n\nJohanna Liu\n\n11 Wisdom and Hermeneutics of Poetry in Classical Confucianism 245\n\nVincent Shen\n\n12 Early Confucian Moral Psychology 263\n\nKwong-loi Shun\n\n 13 Early Confucian Virtue Ethics: The Virtues of Junzi  291\n\nAntonio S. Cua\n\n14 Early Confucian Political Philosophy and Its Contemporary Relevance 335\n\nTongdong Bai\n\n15 Ultimate Reality and Self-Cultivation in Early Confucianism:\u200b A Conceptual\/\u200bExistential Approach 363\n\nZhonghu Yan\n\n16 Confucian Harmony:\u200b A Philosophical Analysis 379\n\nChenyang Li\n\nIntroduction to Contributors395\n\nIndex397\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_1\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 1. Introduction: Classical Confucianism in Historical and Comparative Context\n\nVincent Shen1\n\n(1)\n\nDepartment of Philosophy and Department of East Asian Studies, University of Toronto, Toronto, Ontario, Canada\n\nVincent Shen\n\nEmail: vincent.shen@utoronto.ca\n\nAbstract\n\nWe use the term \"classical Confucianism,\" or \"early Confucianism,\" or \"pre-Qin Confucianism,\" to cover the thoughts, doctrines and practical wisdoms developed in the first stage of Confucianism. Confucianism began with the founder Confucius (551\u2013479BCE), and developed through Confucius' disciples and his grandson Zisi\u5b50\u601d (483\u2013402BCE) to Mencius (372\u2013289BCE), and finally Xunzi\u8340\u5b50 (325\u2013238BCE). Not long after the death of Xunzi, the warring states were all conquered and unified by the Qin Empire (221\u2013206 BCE), in the process of which Confucian books were infamously burned and scholars were buried, most of them Confucian, by the first emperor of the Qin-Empire, and so ended this first period of Confucianism. It is thus distinguished from the later Confucianism of the Han Dynasty (202BCE\u2013220CE), which became a state ideology with the canonization of the early Confucian founding texts, and the Neo-Confucianism that developed from the eleventh to eighteenth Century in Song, Ming and Qing Dynasties, as the revival of Confucianism after its silence for almost eight centuries under the challenges of Neo-Daoism and Buddhism.\n\n## 1 Introduction\n\nWe use the term \"classical Confucianism,\" or \"early Confucianism,\" or \"pre-Qin Confucianism,\" to cover the thoughts, doctrines and practical wisdoms developed in the first stage of Confucianism. Confucianism began with the founder Confucius (551\u2013479BCE), and developed through Confucius' disciples and his grandson Zisi \u5b50\u601d (483\u2013402BCE) to Mencius (372\u2013289BCE), and finally Xunzi \u8340\u5b50 (325\u2013238BCE). Not long after the death of Xunzi, the warring states were all conquered and unified by the Qin Empire (221\u2013206 BCE), in the process of which Confucian books were infamously burned and scholars were buried, most of them Confucian, by the first emperor of the Qin-Empire, and so ended this first period of Confucianism. It is thus distinguished from the later Confucianism of the Han Dynasty (202BCE\u2013220CE), which became a state ideology with the canonization of the early Confucian founding texts, and the Neo-Confucianism that developed from the eleventh to eighteenth Century in Song, Ming and Qing Dynasties, as the revival of Confucianism after its silence for almost eight centuries under the challenges of Neo-Daoism and Buddhism.\n\nTherefore, \"early Confucianism\" is thus called because of the fact that it is the initial period of Confucianism in Chinese intellectual history. It is also called \"pre-Qin Confucianism\" because this period of Confucianism ended by the time that the Qin Empire unified China. However, philosophically more important, it is called \"classical Confucianism\" because all the founding texts, the so-called \"Six Classics\" (liujing \u516d\u7d93),1 were produced in this period, when its early founders lived and developed their thoughts. These texts had been collected, edited and used as teaching materials since Confucius, even if their canonization came later in the Han Dynasty.2\n\nIn Chinese intellectual history, classical Confucianism has many dimensions: religious, philosophical, cultural, ideological, political, etc. In this volume, we will focus mainly on its philosophical dimension, putting it in the context of ancient Chinese intellectual history, and sometimes comparing it with Western philosophy, in order to make it more understandable for today's readers living in a time of cultural exchange and dialogue between East and West.\n\nHistorically, \"Confucius,\" the name of the founder of this school, was the latinization of Kongfuzi \u5b54\u592b\u5b50 (Teacher Kong) when his works were introduced to Europe by the earliest Jesuits in China from the late sixteenth Century onward. The term \"Confucianism\" was used to name the school of thought founded by Confucius. However, in China, scholars and common people usually call him simply Kongzi \u5b54\u5b50 (Master Kong), while the school founded by Confucius is called rujia \u5112\u5bb6 (school of ru \u5112), instead of \"Confucianism.\"\n\nIt should be noted that, historically, the profession of ru existed much earlier than Confucius himself and thus before the rise of Confucianism. Ru was a profession or class of people who served as coordinators at public and private ritual occasions and as teachers of rites and related knowledge. The majority came from the surviving Yin \u6bb7 people of Shang \u5546 Dynasty (1766\u20131122BCE), a very religious people, which was conquered and replaced by the Zhou \u5468 Dynasty (1122\u2013221BCE). Since Yin people practiced rituals in their everyday public and private life, the rus were very well versed in all details of the knowledge and practice of sacrificial, funeral, ceremonial, and other kinds of ritual. After the Yin people lost their power to the Zhou Dynasty, the rus should have been invited by the Duke of Zhou to design and establish the Zhouli \u5468\u79ae (the Rites of Zhou Dynasty) because of their expertise in the rites from their Shang legacy. Because of their participation in the ritual reform or redesigning of the Zhouli, which was a most important political and cultural project of the Duke of Zhou in the early days of Zhou Dynasty, the social status of ru from Yin Dynasty was thereby relatively enhanced in Zhou Dynasty; some of them held the office of rites and served as teachers of rites. This explains why the Duke of Zhou was always an ideal and honorable person for the ru people. Confucius himself would take his frequent dreaming of the Duke of Zhou as a sign of his younger days' ambition for such positions. However, during the Spring and Autumn period (722\u2013481BCE), most of the rus lost their office and earned their living by serving as private teachers of rites and ritual coordinators. What they taught were the liuyi \u516d\u85dd (Six Arts): ritual, music, archery, driving, writing, and calculating.\n\nThe Chinese family name of Confucius is Kong \u5b54, and his first name is Qiu \u4e18, also known as Zhongni \u4ef2\u5c3c. His ancestors were from the Yin \u6bb7 people who lived in the Song\u5b8b State and moved from there to the Lu \u9b6f State, today's Shandong \u5c71\u6771 area. This family background, together with Yin people's traditional familiarity with rites, explains his early interest in practicing li and the fact that, in his earlier career, he served more or less the same function as one of the rus. He was indeed the most famous and influential among the rus, because he had the largest number of students (said to number 3,000), and systematically organized his teaching materials into six parts. These later became the liujing \u516d\u7d93 (Six Classics), which were the founding scriptures of Confucianism. Most importantly, Confucius had laid the philosophical foundation of his teaching of the rituals by a transcendental deduction from ren\u4ec1 (humaneness) to yi \u7fa9 (righteousness) to li \u79ae (rituality, propriety).\n\nAmong the Six Classics, the legendary Yuejing \u6a02\u7d93 (Classic of Music) was believed to be lost under the reign of the First Qin Emperor (221\u2013207BCE) who threw Confucian books into the fire.3 Thus, more realistically we have only wujing \u4e94\u7d93 (Five Classics), that is, the Shijing \u8a69\u7d93 (Classic of Poetry), the Shangshu \u5c1a\u66f8 or Shujing \u66f8\u7d93 (Book of Documents or Classic of Documents), the Liji \u79ae\u8a18 (Book of Rites), the Yijing \u6613\u7d93 (Classic of Changes), and the Chunqiu \u6625\u79cb (Spring and Autumn Annals).4\n\nThe ru profession existed long before Confucius came on the historical scene of ancient China. Hence it existed long before Confucius formed a community with his disciples and students of his disciples that had a strong sense of belonging to a school, a school that was later divided into several sects. The original Chinese term of Confucianism or rujia \u5112\u5bb6 (Confucian School) appeared only later in the Shiji \u53f2\u8a18 (Record of the Grand Historian) of Sima Qian \u53f8\u99ac\u9077 (145\u201386BC) in 100BC. This means that the term rujia appeared only after the end of classical Confucianism.\n\nConcerning the lineage of classical Confucianism in Chinese intellectual history, we should say simply that, in its first phase, Confucius (551\u2013479BC), the founder of classical Confucianism, after his teaching to his disciples, was followed by his grandson Zisi \u5b50\u601d (493\u2013406BC), and then by Mencius (371\u2013289BC) in the second phase, and by Xunzi \u8340\u5b50 (298\u2013238BC) in the third phase. These could be seen as the three phases in the historical development of classical Confucianism. This volume will focus on the philosophical thoughts developed in these three phases of classical Confucianism.\n\nHowever, it is worthwhile to mention briefly the later development of Confucianism after the classical period, to be dealt with more pertinently in other volumes of this series. Confucianism in the Han Dynasty was somehow related to the last phase of classical Confucianism because of the fact that it was based on Xunzi's philosophy. Indeed, Xunzi's idea of Heaven as Nature and his combination of li \u79ae (ritual) with fa \u6cd5 (law) were followed by most Confucians in the Han Dynasty to serve the Han emperors and to reinforce the political stability of the state. In particular, Dong Zhongshu \u8463\u4ef2\u8212 (c179-c104BCE) was responsible for making Confucianism the state ideology of the Han Dynasty. For him, Heaven was nature, but, to counter-balance the Emperor's power, he developed the idea of Mandate of Heaven in the form of laws of nature. These laws were constituted by yin \u9670 \/yang \u967d and five phases to form an organic whole, in which everything was well connected and things felt and responded to the same class of things. There is precise correspondence between Man and Nature. For Dong Zhongshu, human nature was factual\/empirical, not transcendental. History was determined by dynamic change of the five phases. (Chan 1963: 271\u2013288) Another Han Confucian, Wang Chong \u738b\u5145 (27\u2013100CE) continued the line of naturalistic Confucianism. His criticism of all superstitions of his time, including those of Confucianism, Daoism and popular religions, manifested a rationalistic attitude. Unfortunately, since the end of the Han Dynasty, Confucianism became dormant and less influential for intellectuals, who were led away first by Neo-Daoism, and then by Chinese Mahayana Buddhism.\n\nThen, after almost eight centuries of silence, in the mid eleventh Century, Confucianism began to revive in the North Song \u5b8b Dynasty as \"Neo-Confucianism.\" This developed itself through three lines of thought. First, from the five masters of North Song Dynasty, Zhou Dunyi \u5468\u6566\u9824 (1017\u20131073), Zhang Zai \u5f35\u8f09 (1020\u20131077), Shao Yung \u90b5\u5eb8 (1011\u20131077), Cheng Hao \u7a0b\u705d (also known as Cheng Mingdao \u7a0b\u660e\u9053 1032\u20131085) and Cheng Yi \u7a0b\u9824 (also known as Cheng Yichuan \u7a0b\u4f0a\u5ddd1033\u20131107), to ZHU Xi \u6731\u71b9 (1130\u20131200) in the South Song Dynasty: this line could be called Neo-Confucianism of the Realist Type. Second, from Lu Xiangsan \u9678\u8c61\u5c71 (1139\u20131193) to Wang Yangming \u738b\u967d\u660e (1472\u20131529): this line could be called Neo-Confucianism of the Idealist type. Third, thinkers from late Ming Dynasty to mid-Qing Dynasty, such as Wang Fuzhi\u738b\u592b\u4e4b (1619\u20131692), Yan Yuan \u984f\u5143 (1635\u20131704), Li Gong \u674e\u5868 (1659\u20131733), Dai Zheng \u6234\u9707 (1723\u20131777), etc., constituted Neo-Confucianism of the Naturalist type. These three orientations, the Realist, the Idealist and the Naturalist, constituted the diversity of philosophical discussions of the Neo-Confucianism.5\n\n## 2 Five Classics and Their Meaning in Comparative Context\n\nThe task of a philosopher could be said to be to make explicit the universalizable meaning of existence implicit in particular beings, events, actions, or stories. Here \"universalizable\" means being capable of being shared by many different others. Since we do not presume the factual existence of universality pure and simple in this human historical world, to think philosophically for us is not to develop a purely abstract argument in defense of the presumed universality, but rather to reveal the meaningfulness of existence in the historical-local yet universalizable context. It is in this sense that comparison of different philosophical traditions is both doable and inspiring for us finite human beings longing for truth and ultimate reality. Thus we consider it helpful for understanding the meaning of the Five Classics of classical Confucianism to compare them, in this introductory chapter, with the Scriptures in another tradition of the world, that is, the Bible of Judeo-Christian tradition.\n\nIt is intriguing for scholars of comparative religion\/philosophy to see whether there is something comparable between the mode of revealing of meaningfulness of existence in the Confucian Classics and that in other religious traditions, such as the Judeo-Christian Holy Scriptures. For example, in Confucianism, the dominant school of thought in Chinese culture, can we find in the wujing \u4e94\u7d93 (Five Classics) something similar to the mode of revealing in the Bible? The discussion of this question is interesting here for us in order to give a comparative context to make explicit the Confucian sense of the meaningfulness of existence for readers now existing in a world of globalization when China is meeting with the West. Short of space, I will take up only the Biblical tradition for the purpose of comparison, while a comparison with the classics of other traditions is also urgently needed and I regret not to be able to do it here. Hopefully the limited comparison here is helpful for extending the understanding of the philosophical depth of classical Confucianism in regard to its mode of revealing the meaning of existence.\n\nIn fact, as we will see in the next chapter, there is a movement of thought in ancient China from the revealing of God's Will in the form of political theology to the rising of a creative humanism, changing thereby the mode of revealing of the meaningfulness of existence. In this context, I shall first put the Five Classics into a broader and comparative context by referring to Paul Ric\u0153ur's analysis of the concept of \"revelation\" in the Bible. For Paul Ric\u0153ur, the Bible, conceived as revealing the Words of God, contains various forms of discourse by which revelation is to be expressed: prophetic discourse, narrative discourse, prescriptive discourse, sapiential discourse, and poietic discourse (Ricoeur 1977: 15\u201354).\n\nFirst, in regard to the prophetic discourse, there has been a dominant view among Chinese scholars that the Confucians do not play the role of prophet in the Confucian Classics comparable to the Jewish tradition. Among these scholars, Tu Wei-ming is one of the most influential. According to Tu, the Confucian intellectual\/officer is different from the Greek philosopher and the Jewish prophet.6 However, this kind of differentiation could lead to mutual misunderstanding between these cultural traditions. Therefore it needs more sophisticated discernment in order that, while comparing their differences, some common concern could become manifest to help in their mutual understanding.\n\nIn the Biblical tradition, the prophetic discourse is normally characterized by the predictive visions or unveiling of the future7 and the structure of double authorship.8 In comparison, we find there was indeed no prophetic discourse in the sense of double authorship in classical Confucian texts. It is in this sense that Tu Weiming is right when he says that Confucians do not play the role of a prophet. This means simply that there was no Confucian prophet speaking the Words of God. However, since the diviners in ancient China interpreted the crackled sign on the bone or tortoiseshell, or the hexagram obtained in operating with the yarrow stalks and its lines and related texts, as good or bad omens, and thereby predicted the future consequence or destiny of a specific human action, they were somehow playing a role like that of a prophet as revealing the will of High God or that of Heaven. This is most clear in the Fragments of Shang Divination where we often read texts such as, \"The King divined as such...With the approval of di\" (di ruo \u5e1d\u82e5), or \"di does not approve\" (di bu ruo \u5e1d\u4e0d\u82e5), or \"The King makes the divination, and says: the result is auspicious, with the approval of di.\" (Guo 1978: 745, 1173) Note here shangdi \u4e0a\u5e1d (High God) passively approved the request or demand of kings and diviners. However, He never took the initiative to reveal Himself, as was the case of God in the Bible.\n\nIn the Yijing (Classics of Changes), there seems to be an affirmation of a kind of pre-knowledge, foresight, prediction of the direction of how things were going to happen in the future. This was also affirmed in the Chap. 24 of the Zhongyong \u4e2d\u5eb8 (Doctrine of the Mean): \"It is characteristic of the most entire sincerity to be able to foreknow. When a nation or family is about to flourish, there are sure to be happy omens; and when it is about to perish, there are sure to be unlucky omens.\" (Legge 1991: 417) However, we cannot find any apocalyptic literature in Confucian Scriptures, although it is true that we can find some cyclical eschatological idea of catastrophe (jie\u52ab) and prediction in Religious Daoism, worthy of further comparative study elsewhere as to the Chinese and Christian concepts of eschatology, but not in this volume.\n\nSecond, in comparison with the Biblical narrative discourse in which God is revealed through the telling of stories, especially the stories of history-making events,9 we can find this kind of revealing narrative in the Classical Confucian historical texts; those \u00e9v\u00e9nements fondateurs of Chinese people recorded in the Classics of Documents, the Spring and Autumn Annals, and the three Commentaries of the Spring and Autumn Annals (the Zuo's, the Gongyang's and the Guliang's) that were also promoted to the status of jing \u7d93 (Classics) in Tang Dynasty. Chinese people in particular are fond of looking into historical founding events for the revealing or manifestation of dao (the Way) and the meaningfulness of individual and collective existence. Also, it is very interesting to see that, in the three Commentaries of the Spring and Autumn Annals, telling a story was itself providing interpretative commentary on the very succinct texts of the Annals. To tell a story is to interpret or to show the hermeneutic function of human reason through the narrative discourse of telling stories which integrate human actions, events, and decisions.\n\nPrescriptive discourse is the third form of expressing God's revelation in the Judeo-Christian Scriptures, such as in the Ten Commandments. We find also in the Confucian Classics, both the ideas of cosmic regulation based on the concept of Heavenly dao and the codes of human behavior based on the concept of li \u79ae (ritual) are revealed in a prescriptive sense to show the way to a meaningful life. Even if the authority of the Ten Commandments in Judeo-Christian tradition is based on God, whereas the Confucian li is based on ethical requirement, for both traditions there is always the necessary mediation of regulations for the fulfillment of human existence.\n\nHowever, if viewed from the perspective of Kant's moral philosophy of autonomy, the revealed laws such as the Ten Commandments are seen as a kind of heteronomy. The understanding of the biblical prescriptive discourse as purely external obligation and therefore heteronomy has created a misunderstanding on the part of modern new Confucians vis-\u00e0-vis Christian ethics. For example, Mou Zongshan (1909\u20131995), a modern new Confucian, in following Kant's view, criticizes Christian ethics as a kind of heteronomy, \"depending totally on the decision of God\" (Mou 1985: 156). On the contrary, Mou and his followers presume Confucian ethics, \"like Kant, totally depends on the morality of autonomy\" (Mou 1985: 136), because of its foundation in the full unfolding of human subjectivity's self-awareness.\n\nHowever, we should understand that both Christian and Confucian ethics are relational ethics. Biblical laws are to be understood with the idea of the Covenant that designates a whole complex of relations, the core of which is love and justice. For Jesus, in the Kingdom of love, the Law would be fulfilled to its last iota. For Him, the Law and the Prophets were summed up in the Golden Rule from Deuteronomy: \"So always treat others as you would like them to treat you; that is the meaning of the Law and the Prophets\" (Matt. 7:12). All laws could be summarized in this: \"Love God and love others as yourself.\" In other words, all laws are completed and subsumed under God's infinite generosity and unconditional love.\n\nIn Confucian ethics, even if virtue is prior to obligation, still the meaningfulness of life cannot go without li. Reciprocity constitutes the essence of the golden rule implied in li, either negatively as \"Do not do to others what you do not want them to do to you\" (Analects 15:24, Chan 1963: 44.), or more positively as \"A man of humanity, wishing to establish his own character, also establishes others; wishing to be prominent himself, he also helps others to be prominent\"(Analects. 6: 28, Chan 1963: 30). However, even if an initial generosity was not absent in Confucianism, and the inner dynamism of humaneness there could be extended to all human beings and all things, still in Confucianism we do not find infinite generosity and unconditional love. Unfortunately, the initial generosity in classical Confucianism was constrained in the relation of reciprocity; and when later in Han Dynasty this was implemented in the framework of hierarchical institutions, Confucianism would have lost its original generosity and love.\n\nFourth, the Biblical revelation is expressed also in the form of sapiential discourse or wisdom, in the Wisdom Books such as Job, the Proverbs, the Ecclesiates\/Qoheleth, the Book of Wisdom, the Ecclesiaticus\/Ben Sira, etc. Wisdom fulfills one of religion's fundamental functions in binding together ethos and cosmos, the sphere of human action and the sphere of the world, in the very point of their discordance: suffering, and more precisely, in unjust suffering. Wisdom teaches us how to endure, how to suffer. For Paul Ric\u0153ur, this is the most profound meaning of the book of Job, the best example of wisdom.\n\nIn comparison, we can say that Confucian Classics like the Analects, the Mencius, the Zhongyong (Doctrine of the Mean), and the Daxue (Great Leaning) are books of wisdom. However, the true nature of their wisdom consists in offering very rich ideas to inspire human beings in their self-cultivation and harmonization of relationships, dealing with the essential problem of how to lead a harmonious life in society and within the universe. There we find fewer words touching on the problem of suffering. We must wait till Buddhist Scriptures come into China to tell about suffering in the doctrine of Four Noble truths. However, the early Buddhist emphasis on the doctrine of suffering later changed to the focus on Enlightenment and the One Mind in Chinese Mahayana Buddhism. The Yijing (Classics of Changes), especially the Yizhuan (Interpretations of the Yijing), puts its emphasis not at all on suffering, but rather on the harmony and creativity of the universe, and human work as assisting cosmic creativity.\n\nFinally, the biblical revelation also takes the form of hymn or poetic discourse, like the Psalms and the Song of Songs. For Paul Ric\u0153ur, the Psalter may be said to be revealed in the sense that the sentiments or affections expressed in praise, supplication, thanksgiving, and celebration are all engendered by what the human heart allows to exist and become manifest in surpassing pathos and suffering discerned in wisdom when it transforms suffering. As Ric\u0153ur says, \"The revelation is this very formation of feelings that transcends the everyday modalities of human feeling\" (Ricoeur 1977: 30). In the Confucian Classics, it is in the Shijing (Classic of Poetry) and the lost Classic of Music or its residue in the Yueji (Notes on Music) included in the Liji (the Book of Rites) that we find similar emotional expressions. The Shijing, like the Psalms and Song of Songs, expresses and thereby interprets through its poetic language human affectivity as the original mode of human existence, thereby revealing the truth or the authenticity of life itself. The recently unearthed manuscripts Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry) start by saying \"Poetry could not be without willing; music could not be without feeling; literature could not be without wording\" (Confucius 2001: 123). Also, in the recently unearthed Guodian Bamboo Slips, a text entitled Xing Zhi Ming Chu \u6027\u81ea\u547d\u51fa (Human Nature Comes from Mandate), now attributed to the so-called Zisi-Mencius school, says,\n\n> Dao begins with human feeling,\n> \n> Human feeling is born from human nature,\n> \n> Those who begin with human feeling,\n> \n> Will end up with righteousness. (Jinmen Museum 1998: 179)\n\nThus, according to this recently unearthed text, Dao starts to reveal itself through human feeling and accomplishes itself in the ethical relation of righteousness. Also, a careful reading of the shijing shows that affective relations between man and woman, subjects and kings, human beings and Heaven, sometimes with love, sometimes with joy, sometimes with anxiety, sometimes with bitterness, sometimes even with hateful blame, depending on the situations by which they are affected and the ways they are treated, these are all expressed through poetic language. Confucius' comments on the function of poetry seem to have well grasped this web of existence constituted of affective relations, as we can find in the Shijing. Here, classical Confucianism is not a stringent political and ethical philosophy, but an affective poetic discourse revealing the authenticity of existence, the primacy of feeling and affectivity over reason and rational discourse, and the primacy of existence over pure thinking.\n\nIn summary, we can say that the Confucian Five Classics are somehow comparable to the Judeo-Christian Scriptures in the sense that they reveal also the meaningfulness of existence in assuming a role of prediction or foresight over what is going to happen in the future, the role of unfolding meaning through historical narratives and human actions, and the role of revealing the normative dimension of human existence in compliance with the basic norms of human action. They also play the role of Wisdom Books, such as the Analects, the Mencius, the Zhongyong and the Daxue, etc., in which wisdom is proposed to guide the junzi's self-cultivation and harmonization of relationships. Finally, they also reveal the affective dimension of human existence through the poetic expressions of the Shijing.\n\nThus, the Shijing (Classic of Poetry) singles out affectivity as essential to human existence, exploring thereby the imaginative function of human reason and the manifestation of human affectivity in poetic verses. The Shangshu (Classic of Documents), the Chunqiu (Annals of Spring and Autumn) and its three Commentaries show us human historicity and the function of practical reason in judging historical events. There is also a certain hermeneutic function in the Confucian interpretation of the meaning of texts and historical events either by giving comments or by narrating stories in detail. Also, with the Chapter Hongfan \u6d2a\u7bc4 (Great Model) of the Shangshu and the Yijing, there emerges a speculative function of Chinese philosophy, related somehow to the archaic religions and cosmology of ancient China. Many of these texts or their contents existed long before the birth of Confucius and Laozi.\n\nConfucius himself taught the Shijing, the Shangshu, together with li (rite) and yue (music). He studied the Yijing and composed\/edited the Chunqiu in the later years of his life. However, it was after Confucius that his disciples and the Confucian school as a whole started to take the Shijing, the Shangshu, the Liji, the Yijing, and the Chunqiu as essential to their teaching and even had them canonized.\n\n## 3 Characteristics of Chinese Philosophy in Comparative Context\n\nSince this volume focuses on the philosophy of classical Confucianism in the context of Chinese intellectual history, we may ask: what does \"philosophy\" mean in China? In fact, there is no exact term in Chinese Classics for \"philosophy\" as in Western civilization. Instead, terms such as daoshu \u9053\u8853 (Dao and art of its realization), daoxue \u9053\u5b78 (Learning of Dao), lixue \u7406\u5b78 (Learning of Principles), etc., were used by Chinese scholars. The term philosophia, when first introduced to China by M. Ricci and his colleagues G. Aleni and F. Furdato, was translated either phonetically as feilusuofeiya \u6590\u9304\u6240\u8cbb\u4e9e or as aizhi zhixue \u611b\u77e5\u4e4b\u5b78 (the science of loving wisdom). It was a Japanese scholar, Nishi Amane\u897f\u5468 (1829\u20131892), who translated \"philosophy\" into \"tetsugaku\", which, in Chinese, reads as zhexue \u54f2\u5b78. This neologism was first appropriated by Liang Qichao \u6881\u555f\u8d85 who used it to introduce philosophy in his Newspaper published in Yokohama, Japan.\n\nIn the Chinese language, zhe \u54f2means \"wise,\" while xue \u5b78 means \"to learn\" or \"learning\" and \"science.\" Together they mean \"leaning to be wise,\" or \"science of wisdom.\" The concept is thus different from the meaning of its Western counterpart \"philosophy\" as \"love of wisdom.\" In Western philosophy, according to Socrates, only gods have wisdom, whereas men are only lovers of wisdom, and, if one loves wisdom, it means one does not have yet wisdom in him\/herself. By contrast, Chinese philosophers would think that if one does not have wisdom, how could one teach others about it? Only the wise can teach wisdom. Even if no one is totally wise, at least one should be relatively wise to be able to teach others about wisdom. Also, all people should have the potential to become wise, otherwise nobody can receive the teaching of wisdom, no matter how differently the concept of wisdom is conceived by different schools of philosophy.\n\nEven if Chinese Philosophy is very different from other philosophical traditions in the world, however, different philosophical traditions, as expressions of human reason, could also share some complementary features. Therefore, we may, for example, compare Chinese philosophy and Western philosophy in the following ways. First, Western philosophy uses languages based on the alphabetical system and therefore is more abstract, while Chinese philosophy uses pictograms and ideograms, which express imaged ideas such as ren \u4eba (human beings), tian \u5929 (Heaven), ren \u4ec1 (humanness), dao \u9053 (the Way), xin \u5fc3 (Mind\/heart), etc. These ideas are indeed linguistically linked with image, and therefore they are produced by a balanced use of the right side and the left side of human brain in featuring both language and image.\n\nProbably because of its linguistic use of pictograms and ideograms, Chinese philosophy expresses itself by Image-Ideas; therefore it is different from Western philosophy that refers to Pure Ideas. Chinese philosophy prefers metaphors and narratives to the concepts and argumentations used by Western philosophy. Generally speaking, Chinese philosophy, when grasping the Reality Itself in an enlightening insight by human speculative reason, tends to form a kind of Image-Idea, something between a pure Idea and an iconic\/sonoric image, and it thereby keeps the holistic character of the manifestation or the intuitive reception of the Reality Itself. The Idea-Image is seen as expressive and evocative of, though never exhaustive of, the richness of Reality Itself and therefore enjoys merely the status of a metaphor.\n\nIn other words, Chinese philosophers, when using their speculative reason, grasp intuitively the Ultimate Reality and call it the shangdi \u4e0a\u5e1d (High God), or the Taiji \u592a\u6975 (Great Ultimate), or dao \u9053, ren \u4ec1 (humaneness), xin\u5fc3 (mind\/heart), cheng \u8aa0 (sincerity\/true reality), kong \u7a7a (emptiness), or yixin \u4e00\u5fc3 (One Mind), etc. All these are but metaphorical interpretations of the Ultimate Reality thus grasped intuitively by Chinese philosophers of various traditions, such as Confucianism, Daoism and Chinese Mahayana Buddhism.\n\nIn Chinese artistic creativity, by the imaginative function of reason and its poetic transformation, artists would render the Idea-Image into a sort of concrete iconic\/sonoric image and thereby materialize it in Chinese painting, music, poetry, etc. This is the essence of Chinese philosophy of art. In moral and ethical actions, the practical function of reason would bring the Idea-Image into the judgment of events and the intervention of one's own action into the course of events and thereby take responsibility of it. This is crucial to Chinese moral philosophy. In their function of historical reason, Chinese historians would take the Ultimate Reality to be manifested through human actions that constitute events, and also through events that constitute historical stories via emplotment. This is fundamental to the Chinese philosophy of history. Stories bring us hope because, somehow or other, the meaningfulness of existence is to be revealed or manifested through stories, although always in a metaphorical way. Through stories of our own and those of many others, we get a kind of access to the Ultimate Reality.\n\nComparatively speaking, in Western philosophy, the pre-Socratic philosophers such as Thales, Anaximander, Anaximenes, Heraclitus, etc., still kept a very intimate relation with the original Idea-Image, in relating, for example, the idea of Arche and Physis to water, to the unlimited, to air, to fire etc. However, the main stream of Western philosophy from Parmenides and Plato onward pushed the Idea-Image into pure idea, and then, with logical definition, conceptualized it and related one concept with other concepts logically in propositions. Concepts were detached deliberately from images, things, and events, and defined and related one to another logically in descriptive and argumentative discourses. By this detachment, concept and argumentation can help the human mind to develop the critical function of reason, in not limiting itself to the particularity of images, things, and events, but paying attention to the abstract universalizability of concepts and the rigor of their logical relation. Although the validity of concepts and argumentation might be absolutized in a way so as to claim for pure universality and rational structure, in fact, they allow us to see Reality and its structure only in an abstract way. By contrast, metaphors, mostly related to one another by poetic phrases and stories, are different from abstract concepts and well-structured argumentation; yet they still keep an intimate relation with images and events.\n\nWe may also compare Chinese philosophy with Western philosophy by tracing the respective perceptions of their own origin. When exploring the meaning of existence, ancient Greek philosophy emphasized a disinterested theoretical examination of being and truth, while Confucian philosophy engaged in practical thinking on human nature and destiny. Philosophy, according to Aristotle, had its origin in the Greek notion of theoria, the disinterested pursuit of truth and sheer intellectual curiosity. In comparison, Confucian philosophy seemed to be more pragmatically motivated. For Aristotle, the episteme began as a result of the attitude of wonder, which led to the theoretical construction of scientific and philosophical knowledge. In comparison, Confucian philosophy began with the attitude of concern, which led finally to a practical wisdom for guiding human destiny, both individual and collective.\n\nAristotle pointed out in the Metaphysics that the way of life in which knowledge began was constituted of leisure (rastone) and recreation (diagoge), as it was in the case of Egyptian priests who invented geometry. Aristotle believed that in leisure and recreation, the human being needed not care about the daily necessities of life and could thereby wonder about the causes of things and search for knowledge for knowledge's own sake. The result of wonder was theoria.10 According to Aristotle, the philosophical meaning of theoria was defined against praxis, or, as Aristotle put it, \"not in virtue of being able to act, but of having the theory for themselves and knowing the cause.\" (Aristotle 1984: 1553, 981b 6\u20137) On the other hand, it was defined with respect to a universal object, which was seen by Aristotle as the first characteristic of episteme, thus leading itself to philosophy and ending up with ontology (Aristotle 1984: 1554, 1584: 982a 20- 982b 10; 1003a 20\u201330).\n\nThus, Western philosophy was historically grounded in the Greek heritage of theoria, which regarded human existence no longer as determined by diverse practical interests, but as submitted henceforward to a universalizing and objective norm of truth. Theoria and philosophy, in Aristotle's Metaphysics, culminated in the science of ontology, which according to Aristotle investigated \"being qua being\" as the most general and comprehensible aspect of all beings.\n\nBy contrast, Confucian philosophy originated with the attitude of concern, which led not to universalizable theories but to universalizable praxis. It was because of the concern with the individual as well as the collective destiny that a Confucian mind started to philosophize. The Great Appendix to the Book of Changes, arguably attributed to Confucius, started to give an explanation of the beginning of the Book of Changes and saw its author as in a situation of great care and sorrow with compassionate concern. There we read:\n\n> Those who composed the Yijing should have great care and sorrow... The study of Yi rose at the ending period of Yin Dynasty, the time when the power of Zhou is flourishing, the time when happens the affair of King Wen fighting against the tyrant Zhou. This is why its words are warning against danger. Being warned against danger, one can create peace; on the contrary, he who takes things lightly tends to fall down. The Dao of this book is great, and none of the hundred things is omitted. Keeping oneself always vigilant from the start to the end, then in principle there will be nothing to blame. This is the Dao of the Changes. (Zhu Xi 1999: 261, 264)\n\nThis text shows that in the eyes of its author, the philosophy of Yi (Changes) as a serious intellectual activity begins with the attitude of concern in the situation of great care and sorrow, not at all in the situation of leisure and recreation, as Aristotle suggests. It emerges with the concern for both personal and collective destiny. The proposition that \"the Dao of this book is great, and none of the hundred things is omitted\" suggests that Confucian philosophy intends to be a practical wisdom capable of guiding a universalizable praxis.\n\nSince whether or not there is universality pure and simple is still a question open to debate, I prefer to use the term \"universalizability,\" a common concern of which may show us a convergence between ancient Greek philosophy and Confucian philosophy. Even if ancient Greek philosophy concerns itself more with the theoretical universalizability, whereas Confucian philosophy concerns itself more with the practical universalizability, however, both of them try to go beyond particular interest and to transcend the limit of particularity in view of a universalizable value. In a certain sense, both of them target the ideal of universalizability. In this light, theoria and praxis are different yet complementary in the search of universalizability.\n\n## 4 The Contents and Movement of Thought in the Following Chapters\n\nThis volume offers a handbook on the philosophy of classical Confucianism with both a historical and a systematic approach. Taking into account the newly unearthed materials and most recent scholarship, the first part will present the historical development of classical Confucianism by introducing its rise upon the fading of ancient political theology and the arrival of a creative humanism, and the development of the philosophical ideas of the major philosophers, such as Confucius, Confucius' disciples, Confucius' grandson Zisi and the Zisi-Mencius School, Mencius, and finally Xunzi. Together with the historical development, the philosophy in the Confucian Classics and other major works of these philosophers are analytically and critically made explicit and assessed.\n\nSince there are important philosophical ideas and systematic issues comparable to today's philosophical concerns yet not so much developed by the historical part, we will present them in the second section that deals with feeling and emotion, aesthetic appreciation of music, wisdom in poetry, moral psychology, virtue ethics, political thoughts, relation with the Ultimate Reality, and finally the concept of harmony in Confucianism.\n\nTherefore, following this introductory chapter, I will proceed to discuss the rise of classical Confucianism in the context of early Chinese intellectual history that moved from the political theology of the Shangshu to the emergence of creative humanism in the Yijing. After having discussed why and how the political theology faded in ancient China, I will focus on the Yijing, which evolved from divination to ethical interpretation, and then to philosophical construction, to see human beings as participating in the process of cosmic creativity and leading a meaningful life. Thus the Yijing's emphasis was put upon a human agent's creative interpretation of the cosmic laws and logical structures in the unfolding of human historicity. That is why there is always an ethical core in classical Confucianism, as is well exposed in the Analects and the Spring and Autumn Annals. The fading of political theology and the rise of creative humanism led to the founding of classical Confucianism by Confucius and its development through Confucius' disciples, Zisi-Mencius school and Xunzi, to be discussed in the following chapters.\n\nIn Chap.\u200b 3, Ni Peimin uses historical, exegetical, and comparative methods to tell the story of Confucius' life and his most important philosophical concepts such as tian\u5929 (Heaven), ren \u4ec1 (humaneness or human-heartedness), yi \u7fa9and li \u79ae (appropriateness and ritual propriety), zheng \u653f\/\u6b63 (political philosophy), and xue \u5b78 (learning to be human). Ni does this with the support of his textual analysis of the essential quotations from the Analects, sometimes in comparison with Western philosophy, sometimes with the interpretation of today's human experience, thus leading us to a fuller understanding of Confucius's philosophy in a comparative and updated context.\n\nAfter Confucius, we will move to Confucius' disciples. In Chap.\u200b 4, Lo Yuet Keung gives us a vivid description of Confucius' legacy to his disciples. He focuses on the disciples' relations with their Master and among themselves, how Confucius taught them, and how his teaching was received by them. Although not many historical records are available about individual disciples, Lo is able to give us a convincing account of the most important figures among Confucius' disciples such as Yan Hui \u984f\u56de, Zengzi \u66fe\u5b50, Mi Zijian \u5b93\u5b50\u8ce4, and other disciples from whom it became clear how Confucius' philosophy was passed on to future generations.\n\nAfter Confucius' disciples, it was Confucius' grandson Zisi who transmitted classical Confucianism to the further major development by Mencius. Recent unearthed bamboo slip texts indicate that there existed a Si-Meng Xuepai \u601d\u5b5f\u5b78\u6d3e(Zisi-Mencius School). In Chap.\u200b 5, Tsai Zhenfeng deals with the philosophy of Zisi, and the historical questions related to his masters and his works, and he attempts to answer these questions from recently unearthed texts such as the Wuxing \u4e94\u884c (Five Actions) in order to highlight the specificity of Zisi's philosophy. Also, he explores the characteristics of the continuous development of Zisi's philosophy till Mencius.\n\nIn the past, all scholarship on classical Confucianism has jumped from Confucius immediately to Mencius, without explaining what happened in the gap of some 120 years between them. However, in this book, Chaps.\u200b 4 and  offer our readers a lineage of continuity from Confucius to his disciples, then from Confucius' disciples to Zisi, and then from Zisi to Mencius, as supported by recent unearthed materials.\n\nChapter 6 by Andrew Plaks deals with the Daxue and the Zhongyong, which, together with the Analects and the Mencius, are most familiar to all Confucians of all ages, and have been put together in the South Song Dynasty by Zhu Xi \u6731\u71b9 (1130\u20131200) as the Four Books. As is well demonstrated by Andrew Plaks, the Daxue and Zhongyong are most significant works in classical Confucianism because of their development of the Confucian philosophical discourse with fundamental or core concepts essential to Confucianism in particular and Chinese thought in general: the basic Confucian concepts like zhong \u4e2d (centrality), he \u548c (harmony), cheng \u8aa0 (sincerity), the relation between the concepts of tian\u5929 (heaven), xing\u6027 (nature) dao \u9053 (the way), jiao \u6559 (education), and the process of formation of a virtuous life from chengyi \u8aa0\u610f (sincere intention), zhenxin \u6b63\u5fc3 (upright heart), xiusheng \u4fee\u8eab (self-cultivation), qijia \u9f4a\u5bb6 (family in harmony), zhiguo \u6cbb\u570b (kingdom well-governed), to ping tianxia \u5e73\u5929\u4e0b (all under heaven in peace), which have become so classically familiar for all Confucians and all educated people in China. Also, they provide the core ideas and issues for the great revival of Confucianism in the Northern and Southern Song periods, in response to the cosmological and spiritual challenges of neo-Daoism and Buddhism since the third Century AD. Plaks discusses the textual history, the meaning of the titles, the structure of the text, the integral arguments, and the interpretations of these two important texts, which are indeed very valuable for understanding their textual, historical and philosophical meanings.\n\nThe development from Zisi to Mencius having thus been made clear, Chap.\u200b 7 by Chan Wing-cheuk details further logical and philosophical developments of classical Confucianism by Mencius. He puts Mencius in dialogue with his contemporary philosophers. Chan starts with the four types of analogical arguments, that is, exemplification (pi \u8f9f), parallel (mou \u4f94), imitation (yuan \u63f4), and extension (tui \u63a8), that Mencius uses in his critical conversation with Gaozi in regard to the subject of human nature. Also, Mencius tries to justify his thesis phenomenologically: the feelings of commiseration, of shame and dislike, of respect and reverence, of right and wrong, etc., are pure, transcendental, and ontological feelings. Chan understands these four feelings, in Heidegger's term, as Seinsk\u00f6nnen (capacity to be) of man, well illustrated by the fact that a man has a feeling of alarm and distress and tries to rescue a child about to fall into a well, as he explains in Chap.\u200b 7. In this way, Chan attempts to show that, with Mencius, Chinese philosophy makes progress toward philosophical argumentation and phenomenological justification.\n\nAfter Mencius, we move to Xunzi, the last great classical Confucian. In Chap.\u200b 8, titled \"Xunzi as a Systematic Philosopher: Toward Organic Unity of Nature, Mind, and Reason,\" Cheng Chung-ying attempts to bring out, with an intellectual effort of rethinking and re-evaluation, the theoretical system of Xunzi. This is necessary for understanding the fundamental issues of nature at large, human nature, mind, language, human society, and human government. Starting with Xunzi's distinction and relation between Heaven and Human, Cheng argues that Xunzi sees human beings as the most dynamic and creative part of nature, and takes the creation of human community and human culture as an absolute must for fulfilling the nature of both the Human and Heaven. Thus, Xunzi offers a systematic framework in which a human person can identify the end and meaning in his or her life, and in which one can relate human rationality and morality to the full development of humanity. In this light, Cheng develops Xunzi's systematic philosophy that targets the full-fledged development of individual, society, and politics in the universe.\n\nIn these last three chapters, it is most interesting to see that Chinese philosophical argumentation and its systematic character come to maturity with Daxue and Zhongyong's core conceptual network, Mencius' philosophical argumentation and justification, and Xunzi's philosophical system, with which this historical part of classical Confucianism comes to an end. This brings us to the second part dealing with the systematic issues of classical Confucianism.\n\nThis part of the book consists of a series of discussions related to some crucial issues in the philosophical system of classical Confucianism. Given that the recently unearthed Confucian text Xing Zi Ming Chu \u6027\u81ea\u547d\u51fa (Human Nature Comes from Mandate) affirms that \"Dao started with qing \u60c5 (feeling and emotions),\" and that \"he who started with feeling would end up with righteousness\" (Jinmen Museum 1998: 179), we start with the topic of human feeling. Thus, Chap.\u200b 9 by Curie Virag explores early Confucians' thoughts on feeling and emotion in discussing Confucius, Mencius, Xunzi and the recently unearthed Xing Zi Ming Chu's positive attitude toward feeling, which is quite different from the negative attitude of Mohists and Daoists. Virag has convincingly argued that, because of this positive attitude, early Confucians affirm human feelings as the foundation and starting point of human life itself. They see the human self as in constant interaction with the real things of the world, and thus look at feelings as human response to the external reality.\n\nAfter the discussion on human feeling and emotion, we move on to their cultivation through music, ritual, and poetry. Chapter 10 by Johanna Liu deals with music, which in ancient China was closely related to li (rituals). For Liu, the Chinese character \u6a02 is endowed with double pronunciations yue\/le and double meanings music\/pleasure. Since the Yueji \u6a02\u8a18 (Notes on Music) in the Liji \u79ae\u8a18 (Book of Rites) is now proved to be a work produced in Han Dynasty, i.e. later than the classical Confucianism, it is not to be considered a main source for discussing the classical Confucian philosophy of music. Fortunately, the recently unearthed Xing Zi Ming Chu \u6027\u81ea\u547d\u51faprovides us with a new insight into the aesthetic meaning of yue in classical Confucianism. Thus, Johanna Liu explores in detail the thoughts on yue\/le \u6a02 (music and pleasure) in the Xing Zi Ming Chu both textually and intertextually (in reference to the Zuozhuan, the Xunzi and the Zhouli) in order to re-define and re-understand the classical Confucian issues of music, such as the ideas of music pursued by Confucian scholars, the place of music in the self-cultivation of junzi, the music of Zheng and Wei as new sounds and melodies criticized by the Confucians, etc. She also will discuss the aesthetic meaning of music by referring to the Confucian theory of qing \u60c5 (sentiment\/feelings\/emotions), and deal with the in-depth relation between yue (music) and qing.\n\nAfter music, we will focus on poetry, which occupies a crucial position in the classical Confucian educational program. Besides the traditional discussion on the words related to poetry in the Analects, the Zuozhuan and the Mencius, the recently unearthed bamboo slips of Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry) add much to our knowledge of Confucius' teaching on poetry. In Chap.\u200b 11, I will focus on the hermeneutics of poetry and the wisdom in poetry, which has more philosophical implications and is essential to Confucius' teaching. Confucius' poetic wisdom is to be found in his comments on different kinds of poetry and individual poems, in the way he organizes his teaching on the Shijing, especially in his vision of affectivity as a primordial mode of human existence and the function of poetry in self-cultivation and public life. We will also discuss Confucius' hermeneutics of poetry and its development by Mencius.\n\nAfter the Confucian visions of feeling and its cultivation through music, ritual and poetry, we will move to morality and ethics. In Chap.\u200b 12, entitled \"Early Confucian Moral Psychology,\" Shun Kwong-loi discusses the issue of self-transformation, which means one's own reflective efforts at moral self-improvement. Shun focuses on the psychology of this process within the individual person. For Shun, the early Confucians, in reflecting on concrete situations that involve specific individuals with whom they are in interaction, see clearly that ethical problems generally have their source in the depths of human psychology. This is expressed clearly in all early Confucian texts. Shun discusses first the notion of self, then the three aspects of the early Confucian ethical ideal: ren \u4ec1 (humanity, benevolence), li \u79ae (propriety, observance of the rites), and yi \u7fa9 (dutifulness, righteousness). He shows that each of these three aspects of the Confucian ideal involves one's moving away from a certain kind of self-centeredness. Then, he discusses the nature of the process involved in such a transformation and, finally, ways in which one's ethical attention may be properly or improperly directed in such a process.\n\nChapter 13, by Antonio Cua, will discuss classical Confucian virtue ethics. Here, Antonio Cua's focus is on what he calls \"virtues of junzi \u541b\u5b50 (superior person)\" in the Lunyu \u8ad6\u8a9e, with occasional employment of insights from the Mencius, the Xunzi, and the Liji \u79ae\u8a18. Cua starts with a critical appreciation of various interpretations of junzi before he presents a map of the virtues of junzi. These excellences or personal merits of the junzi depict some salient characteristics of junzi, with emphasis on the embodiment of the fundamental, interdependent, ethical virtues, and the selective, dependent virtues. Then, Cua proceeds to discuss two ways for dealing with conflict: tiaohe \u8abf\u548c (reconciliation and harmonizing of differences) and the art of zhongyong \u4e2d\u5eb8 (keeping to the centrality). Finally, he discusses the flexibility of junzi in the exercise of quan \u6b0a (moral discretion) in exigent circumstances.11\n\nChapter 14 by Bai Tongdong will focus on early Confucian political philosophy and discuss its contemporary relevance. This chapter, based on its analysis of the Four Books, argues that the political dimension of human life should be taken as the key concern of early Confucians, and, seen from their problematic, their political philosophy has a very strong relevance for today. Bai even claims that it is more justified to call the early Confucians \"modern\" rather than \"classical\" thinkers in the Western sense. This chapter shows how the early Confucians answer some of the key problems of their times, especially the search for a new social glue, and tries to make explicit the relevance of their thoughts for today. This chapter shows also how the early Confucians answer other key issues of their time, especially the selection of the ruling class. It argues also that the regime proposed by the early Confucians and their philosophical considerations can serve as a critique of contemporary liberal democratic regimes, and that could inspire us to think on ways to a more constructive political system.\n\nIn Chap.\u200b 15, entitled On Confucian Self-cultivation and Ultimate Reality, Yan Zhonghu discusses, with an approach that is both conceptual and existential, Confucius' concept of tian \u5929 (Heaven) as the Ultimate Reality. Based on Confucius' own words, he develops the idea of tian as the force behind all natural processes, as a conscious being and as the assigner of a person's earthly mission. Then he discusses how this concept of Ultimate Reality is involved in the self-cultivation of the three major Confucian virtues of wisdom (zhi \u77e5\/\u667a), humanity (ren \u4ec1), and courage (yong \u52c7) that make a person fully human. Yan's emphasis on yong or courage, a virtue that enables one to face life's challenges, is most interesting. He claims that Confucius himself demonstrated great courage in the face of danger or death because of his firm belief in the Way of Heaven in much the same way that Christians hold belief in God in the face of existential despair.\n\nIn the last chapter of this book, serving as its conclusion, Li Chenyang discusses classical Confucian philosophy of harmony, as well as its program for realizing harmony in the world. He explores the meaning and the ideal of world harmony as illustrated in Confucian classics, and the guiding principles toward world harmony from the Confucian perspective. This chapter is both reconstructive and constructive. It aims to draw on various ideas espoused by ancient Confucian philosophers and to connect the dots in order to present a coherent picture of the Confucian ideal of world harmony. Li claims that, for today, the three classical Confucian principles, that is, the primacy of moral force, empathetic understanding of others, and harmony with differences, together constitute the foundation of Confucian philosophy for world harmony. They are the valuable contributions that Confucians make to the contemporary world.\n\nReferences\n\nAristotle. 1984. Metaphysics (trans: Ross, W.D.). In The complete works of Aristotle, ed. Jonathan Barnes, Princeton: Princeton University Press.\n\nChan, Wing-tsit. 1963. A source book in Chinese philosophy, 4th ed. Princeton: Princeton University Press.\n\nConfucius, Konzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on the Book of Odes). 2001. In The Shanghai Museum's Chu bamboo books of the warring states, ed. Ma Chenyuan \u99ac\u627f\u6e90, Vol. 1. Shanghai: Shanghai Museum (the most precious unearthed materials to update us with Confucius' teaching on poetry).\n\nFang, Thom\u00e9 H. 1981. Chinese philosophy: Its spirit and development. Taipei: Linking Publishing. (A comprehensive history of Chinese philosophy written in English by one of the most important contemporary Chinese philosophers).\n\nGuo, Moruo, ed. 1978\u20131983. Jiaguwen heji \u7532\u9aa8\u6587\u5408\u96c6 (Complete collection of oral bone scripts). Beijing: Zhonghua Shuju.\n\nJinmen Museum. 1998. Guodian Chu Mu Zhujian \u90ed\u5e97\u695a\u5893\u7af9\u7c21 (Bamboo Slips of Guodian Chu Tomb). Beijing: Wenwu Press. (The most significant unearthed materials to update us with Chinese intellectual history and Chinese philosophy, especially for the classical Confucianism and classical Daoism).\n\nLegge, James, trans. 1991 Doctrine of the mean. In The Chinese classics, Vol. 1, Reprint. Taipei: SMC Publishing.\n\nMou, Zongsan. 1985. Yuanshan lun \u5713\u5584\u8ad6 (On perfect goodness). Taipei: Student Bookstore.\n\nRicoeur, Paul. 1977. Herm\u00e9neutique de l'id\u00e9e de R\u00e9v\u00e9lation. In La R\u00e9v\u00e9lation, ed. par P. Ricoeur, E. Levinas, E. Haulotte, E. Corn\u00e9lis, Cl. Geffr\u00e9, Bruxelles: Publications des Facult\u00e9s Universitaires Saint-Louis.\n\nRicoeur, Paul. 2004. Learning to be human: Spiritual exercises from Zhu Xi and Wang Yangming to LiuZhongzhou. In Confucian spirituality, ed. Tu Weiming and Mary Evelyn Tucker, New York: The Crossroad Publishing Company.\n\nTu, Weiming. 2002. In Tu Weiming Wenji (Collected works of Tu Weiming), ed. Guo Qiyong and Zheng Wenlong, Vol. 5. Wuhan: Wuhan Press.\n\nZhu, Xi. 1999. Zhouyi Benyi \u5468\u6613\u672c\u7fa9 (The original meaning of Zhou Yi), Reprint. Taipei: Da'an Press (very convenient version for checking the texts of the Book of Changes with philosophical and religious meanings).\n\nFootnotes\n\n1\n\nAmong the Six Classics, the Yuejing \u6a02\u7d93 (Classic of Music) was said to be lost during the reign of the First Qin Emperor, therefore in fact we have now only Wujing \u4e94\u7d93 (Five Classics).\n\n2\n\nIn the early Han Dynasty, Emperor Wen\u6587 institutionalized the Doctor of the Shujing \u66f8\u7d93 (The Classic of Documents) and the Doctor of Shijing \u8a69\u7d93 (The Classic of Poetry), while Emperor Jing \u666f institutionalized that of the Chunqiu \u6625\u79cb (The Spring and Autumn Annals). It was in 136 BC that Emperor Wu \u6b66 institutionalized also Doctors for the Yijing \u6613\u7d93 (The Classic of Changes) and the Li \u79ae (The Book of Rites), altogether Five Classics, and deinstitutionalized all other doctors of various other schools.\n\n3\n\nThe Yueji \u6a02\u8a18 (Notes on Music) in the Liji (Book of Rites) was arguably, as thus considered by many scholars, a residual part of the Yuejing \u6a02\u7d93 (Classic of Music).\n\n4\n\nUnlike the conventional English translation of the Shijing \u8a69\u7d93 as \"Book of Odes,\" the Yijing\u6613\u7d93 as \"Book of Changes,\" we translate those classical Confucian scriptures titled with the word jing \u7d93 consistently as \"Classic,\" thus the Yijing \u6613\u7d93 is translated as Classic of Changes, the Shijing \u8a69\u7d93 as Classics of Poetry, and the Shangshu \u5c1a\u66f8, when called as Shujing \u66f8\u7d93, we translate it as Classic of Documents. For the rest, titled without the term jing, I will follow the conventional translation of Liji \u79ae\u8a18 as Book of Rites, and the Chunqiu \u6625\u79cb as Spring and Autumn Annals.\n\n5\n\nHere I follow the distinction of these three types of Neo-Confucianism made by Thom\u00e9 H. Fang (see Fang 1981: 8\u201311).\n\n6\n\nTu differentiates Confucians as intellectual\/officer from Jewish prophets in several places of his writings, for example, in Tu 2002: 14 and Tu 2004: 152.\n\n7\n\nThe predictive visions of the future includes that of the \"last days,\" therefore some apocalyptic predictions as revealed by God in the Revelation to John.\n\n8\n\nThe structure of double authorship was shown more specifically in the Prophets, such as \"Listen, you heaven, earth, attend, for Yahweh is speaking\" (Isaiah 1:2), or, \"The words of Yahweh were addressed to me\" \"So, the Lord Yaweh says this\" (Ezekiel, everywhere). The double authorship here in question means that, behind the prophet's mouth, there is God's Word revealed through it.\n\n9\n\nSuch as the narrative genre of discourse in the Pentateuch, the Deuteronomic History, the synoptic Gospels and the Book of Acts, etc. God's revealing here is done through those \"history-making events,\" or in Jacques Ellul and Paul Ric\u0153ur's terms, the \"founding events\" (\u00e9v\u00e9nements fondateurs), like the election of Abraham, the Exodus, the anointing of David, etc., in the Old Testament, and the birth, teaching, death, and resurrection of Christ for the early Christian church etc. The idea of revelation then appears as connected to the very character of these events and the plots that connect many events into the unity of a story. The faith of Israel and that of the early Christian Church are tied up here in the confession of the transcendent character of such nuclear founding and instituting events, which are seen as the imprint, mark, or trace of God's act.\n\n10\n\nAristotle wrote in Metaphysics: \"For it is owing to their wonder that men both now begin and at first began to philosophize; they wondered originally at the obvious difficulties, then advanced little by little and stated difficulties about the greater matters,... therefore since they philosophized in order to escape from ignorance, evidently they were pursuing science in order to know, and not for any utilitarian end\" (Aristotle 1984: 1554, 982b 12\u201322).\n\n11\n\nI should mention here that Antonio Cua was the first to submit his chapter to me, before he sadly passed away, which was a great loss of the academic world of Chinese and comparative philosophy. Indeed, Antonio Cua himself was a junzi. Were his spirit still around somewhere, I would like to use this occasion to thank him for this excellent contribution.\n\n# Part 1  \nHistorical Development\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_2\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 2. The Fading of Political Theology and the Rise of Creative Humanism\n\nVincent Shen1\n\n(1)\n\nDepartment of Philosophy and Department of East Asian Studies, University of Toronto, Toronto, Ontario, Canada\n\nVincent Shen\n\nEmail: vincent.shen@utoronto.ca\n\nAbstract\n\nTo understand the origin of classical Confucianism in the historical context of ancient China, we need to know first how ancient Chinese thought moved from the religious experience with High God's or Heaven's revealing to the humanist construction of the meaningfulness of human existence. This concerns the way early Chinese people conceived the Ultimate Reality in terms of shangdi\u4e0a\u5e1d (High God) or tian \u5929 (Heaven) and His\/Its revelation. In particular, this was shown through the evolution of their religious experience and the rise of a creative humanism in the process of ancient Chinese history. It is evidenced by such major texts as the Shangshu \u5c1a\u66f8 (Book of Documents) and the Yijing\u6613\u7d93 (Classics of Changes), and the development of this legacy till the time of Confucius. In order to examine this development, we will also need to touch upon, albeit very briefly, the Lunyu\u8ad6\u8a9e (Analects) and the Chunqiu\u6625\u79cb (Spring and Autumn Annals). Thus this chapter will serve the purpose of introducing the basic philosophical ideas in these Confucian Classics and their historical changes.\n\nTo understand the origin of classical Confucianism in the historical context of ancient China, we need to know first how ancient Chinese thought moved from the religious experience with High God's or Heaven's revealing to the humanist construction of the meaningfulness of human existence. This concerns the way early Chinese people conceived the Ultimate Reality in terms of shangdi \u4e0a\u5e1d (High God) or tian \u5929 (Heaven) and His\/Its revelation. In particular, this was shown through the evolution of their religious experience and the rise of a creative humanism in the process of ancient Chinese history. It is evidenced by such major texts as the Shangshu \u5c1a\u66f8 (Book of Documents) and the Yijing \u6613\u7d93 (Classics of Changes),1 and the development of this legacy till the time of Confucius. In order to examine this development, we will also need to touch upon, albeit very briefly, the Lunyu \u8ad6\u8a9e (Analects) and the Chunqiu \u6625\u79cb (Spring and Autumn Annals). Thus this chapter will serve the purpose of introducing the basic philosophical ideas in these Confucian Classics and their historical changes.\n\n## 1 High God or Heaven's Revealing in Ancient China\n\nLike many civilizations in the world, Chinese culture arose in deep relation with its religious experiences. Etymologically, the term \u795e shen (God, divinities, or spirit) and all things related to God or divinities, such as \u79ae li (ritual), \u796d ji (religious sacrifice), were all composed of the idea of revealing: the root shi \u793a meant \"the revealing of signs from Heaven manifesting good fortune or misfortune\" (tian chuixiang yi jian jixiong \u5929\u5782\u8c61\u4ee5\u898b\u5409\u51f6). Now, one of the implicit philosophical questions in this kind of religious experience is: How did ancient Chinese people conceive the Ultimate Reality and what was the relation between human beings and the Ultimate Reality? This is indeed a religious question with deep philosophical implications.\n\nIn ancient China the answer to this question was conceived, in particular in the Shang Dynasty (1751\u20131112BCE), in terms of the High God (shangdi \u4e0a\u5e1d) and His revelation to individuals and various ethnic groups. Historically speaking, in ancient China, there was a very strong faith in a High God, named as di \u5e1d or shangdi in the Shang Dynasty, or tian \u5929 (Heaven) in the Zhou Dynasty (1111\u2013249BCE), though always within a polytheistic context. In Shang Dynasty, di or shangdi, arguably evolved from the Shang people's cult of ancestry, dominated over a pantheon of divinities composed of natural powers (like the sun, Yellow River, mountains, etc.), former lords, and pre-dynastic ancestors.\n\nIn the Zhou dynasty, tian \u5929 evolved from di into a more universal power symbolized by the physical sky covering universally all things under it. It was no longer limited to the ancestral divinity, but rather was the supreme ruler of both human society and the whole universe, which was still not independent of the surrounding multiple divinities, such as those of mountains and rivers, ancestral spirits, and even those of the stove and the household shrine.2 In the period of transition from Shang to Zhou, the concepts of di and tian were sometimes used interchangeably, although the direction was moving toward the more universal idea of tian.\n\nTo the earliest Chinese mind, the Ultimate Reality, either as shangdi or as tian, was accessible to human beings through prayer or sacrifice; also it revealed itself through natural\/supernatural phenomena and historical events produced by human action. In ancient China, the earliest form of text that recorded both natural\/supernatural phenomena and historical events was from the wu \u5deb. Their functions were very similar to a shaman, supposed to be capable of communicating with the High God, spirits and ghosts and they served as professional diviners, using objects, dream contents, and astronomical phenomena as revealing messages from di or shangdi or other forms of spirits. Therefore we may call the Chinese earliest historians wu-historians (wushi \u5deb\u53f2): wu conducted and recorded the results of divination, itself an important part of the state ritual, for what was asked in each divination involved important events for the state and public life. This is to say that Chinese history started with divinatory records, or records of God's revealing, usually on the shoulder bones of bigger animals such as cows and sheep, and on the tortoise shell, as well as records on the bronze stencils, such as inscriptions on the bronze tripods. These records showed, first, the time and place of the divination and the name of the diviner(s); second, the intended question for the divination; third, the divinatory explanation concerning the revealing of the good fortune or misfortune of the act in question, based on the interpretation of the crackled signs on the bones or tortoise shell; and fourth, words of verification that recorded the facts that verified the revealed divine will.3 These indeed were the first historical records in China. When we come to the transition from late Shang to early Zhou in the twelfth century BCE, there the earliest part of the Yijing (the Classics of Changes), such as the guaci \u5366\u8fad (hexagram dictum), and yaoci \u723b\u8fad (stroke dictum), somehow still kept the record more or less this way. These texts in the Yijing could be seen as records of divinatory actions, though taken to be and interpreted as the exemplar texts by later generations because they were conducted by Sage-Kings.\n\nAs mentioned previously, ancient Chinese people had faith in a High God, the di \u5e1d or shangdi \u4e0a\u5e1d in the Shang Dynasty, or tian \u5929 (Heaven) in the Zhou Dynasty, and His revealing to human beings their fortune\/misfortune when taking a particular course of action. In Shang dynasty, the idea of Mandate from divinities justified the political leaders' power. However, during the time of transition, Zhou people, in order to justify its revolution against the tyranny of the King Zhou \u7d02 of Shang, though still keeping the idea of Mandate (of Heaven now for Zhou people), would believe that the Mandate was always assured by the kings' virtues. However, even if the Mandate of Heaven was primarily assured by political leaders' ethical virtues, there was still a strong feeling that it was hard to maintain the Mandate, and therefore the practice of asking the advice of Heaven's will via divination was always a part of state rituals. Many poems in the Shijing \u8a69\u7d93 (The Classics of Poetry) showed deep admiration of King Wen's virtue, though still with a sense of uncertainty because the Mandate was hard to maintain:\n\n> The Mandate of Heaven,\n> \n> How beautiful and unceasing!\n> \n> Oh, how glorious\n> \n> Was the purity of King Wen's virtue!\n> \n> With blessings he overwhelms us.\n> \n> We will receive the blessings.\n> \n> They are a great favor from our King Wen.\n> \n> May his descendants hold fast to them. (Chan 1963: 6)\n\nOr again,\n\n> Heaven's Mandate is not constant,\n> \n> The officers of Yin were fine and alert,\n> \n> They assist at the libation in our Capital,\n> \n>.......\n> \n> Cultivate your virtue,\n> \n> Always strive to be in harmony with Heaven's Mandate.\n> \n> Seek for yourself the many blessings.\n> \n> Before Yin lost its army,\n> \n> Its kings were able to be counterparts to the Lord on High.\n> \n> In Yin you should see as in a mirror\n> \n> That the Mandate is not easy to keep. (Chan 1963: 7)\n\nIn the first poem we read that the Mandate of Heaven was related to King Wen's virtue and there was an overwhelming admiration for him; however, in the second there was a warning of the difficulty in keeping Heaven's Mandate, which was seen as \"not constant.\" Since the Mandate of Heaven was hard to keep, even by means of one's virtue, there was always need of divine revelation to see God's will, for human virtue was not an absolute assurance of Heaven's Mandate and would never be able to replace God's revealing.\n\nIt was arguably certain, as evidenced by all historic records and archeological findings, that Shang people were very religious and always conducted divination to the di or shangdi. Their kings and princes always practiced divination before any major action was taken, such as an act of war, marriage, diplomacy, travelling, hunting, paying an important visit. Special officers were in charge of divination, named buren \u535c\u4ebaor zhenren \u8c9e\u4eba (diviners). The results of divination were seen as revealing the will of di or shangdi, as shown in the Fragments of Shang Divination. There we read such texts as: \"The King divined as such...With the approval of di (\u5e1d\u82e5)\"; \"di does not approve (\u5e1d\u4e0d\u82e5)\"; \"The King makes the divination, and says: the result is auspicious, with the approval of di\" (\u738b\u5360\u66f0:\u5409,\u5e1d\u82e5) (Guo 1978: 745, 1173).\n\nThis was also the legacy of Shang as perceived and understood by the Zhou people. Thus we read in the Chapter Hongfan \u6d2a\u7bc4 (Great Model) of the Shangshu (Book of Documents), supposedly exposing a legacy consisting of nine categories coming down from Shang Dynasty and told by Shang's Viscount of Ji \u7b95 to King Wu of Zhou. In the seventh Category we read,\n\n> The Seven Category is the Examination of doubts. Select and appoint officers for divination by tortoise shells and by stalks, and command them thus to divine.... The calculation of the passage of events is the function of experts whose duty is to perform the divination. If you have any doubt about important matters, consult with your own heart, with your ministers and officers, with the common people, and the tortoise shells and stalks. (Chan 1963: 10)\n\nIt is clear from this text that, according to Shang legacy, divination was used in the process of decision making for any important action to be taken. Divination (with tortoise shells and stalks) played a major role in achieving consensus, together with other components of decision making such as the political leader's judgment (consult with your own heart), consultation with experts (with your ministers and officers), and semi-democratic participation (with the common people). The function of divination in achieving consensus is supported by a later text, the Shenzi \u614e\u5b50, in which we read, \"Therefore the purpose of using divinatory stalks and tortoise shell is to establish consensus\" (Shen 1972: 2, my translation).\n\nAs to the procedure of divination, the king first had to make clear his intention or purpose about what was to be asked before the divination. Then he proceeded to the act of divination by burning a tortoise shell or a piece of the shoulder bone of a cow, and after it showed a pattern of crackling, he obtained the interpretation of it from the diviners. Usually the same divination was performed by three different persons, of which the agreed results (lucky or unlucky) of two were taken as revealing the will of High God or Heaven. The decision was therefore made on the rule of majority. This was also confirmed by the Great Appendix II of the Yijing where it was said, \"When three people perform together, their number decreases by one. When one person performs alone, he finds a companion. This is to say their words agree with each other's\" (Zhu 1999: 258).4 As shown in the Fragments of Shang Divination, as well as in the Book of Documents, there were cases in which divination was practiced by two persons, the so-called \"er ren gongzhen\" (\u4e8c\u4eba\u5171\u8c9e co-divination by two persons), in which the agreement of the two diviners was essential for making a decision. That is why in the Great Appendix of the Yijing, it was said, \"Two people with the same heart. This is in favor of achieving a decisive resolution like cutting a metal\" (Zhu 1999: 241, my translation).\n\nIn the early period of Zhou, the tortoise shell was still quite often used in the divination to make decisions, although the final decision regarding a major political act always had to be made by the king himself. For example, in his choice of Hao as Capital of Zhou, King Wu performed the divination by tortoise shell and made the decision, as the Shijing reads,\n\n> He examined and divined, did the King.\n> \n> About settling in the capital of Hao.\n> \n> The tortoise shell decided the site,\n> \n> And King Wu completed it.\n> \n> A sovereign true was King Woo! (Legge 1949a: 463)\n\nIt should be noted that there were always factors that rendered the result of divination uncertain. For example, sometimes the results were contradictory and therefore no consensus was obtained; or, sometimes a king, following his own desire, did not listen to the result of the divinatory, or he listened only to the result in favor of his own desire. We will come back to these issues a little later when we discuss the uncertainty of God's revealing via divination.\n\n## 2 Political Theology in the Revealed World Image\n\nIn fact, the legacy of Shang's divination was retold by the Zhou historian in the Hongfan \u6d2a\u7bc4 (Grand Model) chapter of the Book of Documents. In this document, we find a historical narrative representing a political theology. There was reported the Nine Categories constituting a worldview with an emphasis on the ideal of politics and human affairs as supported by the intelligible structure of the world. It was said there that King Wu of Zhou, in 1121BCE, the 13th year of his reign after having conquered Shang, went to ask Shang's Viscount of Ji about the principle of achieving good relations among people. Though the Viscount of Ji refused to serve Zhou by reason of his fidelity to Shang, nevertheless he told King Wu the wisdom of the Grand Model, as a legacy from the Emperor Yu of Xia dynasty. He said,\n\n> I have heard that of old Gun (Great Yu's father) dammed up the flood and thereby created a chaos among the Five Agents. The di was aroused to anger and did not give him the Great Nine Categories. The various virtues and their relations declined in due course, and Gun was executed. Yu thereupon rose to continue the heritage. Heaven gave him the Great Norm with its nine categories. And the various virtues and their relations were regulated. (Chan 1963: 9)\n\nThus, the Hongfan was seen, or at least thus interpreted in the text, as a revelation of the High God to Yu \u79b9, in giving him a somewhat structural vision of the world in nine categories constituting the earliest Chinese vision and concepts of Nature, human activities and their excellence, politics and good governance according to my classification:\n\n1.\n\nConcerning the vision of Nature, we find in the Hongfan several categories related to the elements and process of Nature. The first category concerned the five agents or five dynamic elements, namely Water, Fire, Wood, Metal, and Earth that constituted the basic elements of the natural world. They should not be seen as five functions of life as some would presume or as five phases of history as Dong Zhongshu \u8463\u4ef2\u8212 (179\u2013104BCE), the Confucian scholars of Western Han Dynasty, would interpret. Also, the fourth category concerned the five arrangements or divisions of time, namely, the year, the month, the day, the stars, and zodiacal signs, and the calendrical calculation. Also there was the eighth category of confirmation from the weather, namely, rain, sunshine, heat, cold, wind, and seasonableness.\n\n2.\n\nConcerning human activities and their excellence, there was the second category of five activities, which concerned moral and intellectual conduct and their virtues: \"The second category is the Five Activities, namely appearance, speech, seeing, hearing, and thinking. The virtue of appearance is respectfulness; that of speech is accord [with reason]. That of seeing is clarity; that of hearing is distinctness; and that of thinking is penetration and profundity\" (Chan 1963: 9). About the happiness and misfortune of human life, we see in the ninth category of five blessings of happiness (longevity, wealth, physical and mental health, cultivation of excellent virtues, and an end crowning a good life) and six misfortunes (premature death, sickness, sorrow, poverty, wickedness, and weakness).\n\n3.\n\nConcerning politics and governance, there we find the third category of eight governmental offices, which concerned branches of administration, such as \"food, commodities, sacrifices, public works, education, justice, the reception of guests, and the army.\" The fifth category of the Grand (Royal) Ultimate will be discussed below. The sixth category of three virtues governed in a way responding to different people and different times: \"correctness and uprightness, strong government and weak government. In times of peace and tranquility, apply correctness and uprightness; in times of violent disorder, apply strong government; in times of harmony and order, apply weak government. Apply strong government to the reserved and retiring, and apply weak government to the lofty and intelligent\" (Ibid.). Also, there is the seventh category of the examination of doubts, which we have already mentioned above.\n\nThe most important, central to all these nine categories, is the fifth, which concerned the category of the Royal (Grand) Ultimate. We read,\n\n> Fifth, of Royal Ultimate, the highest, having established his highest standard of excellence, accumulates the five happiness and diffuses them to bestow to the people. Then the people will keep with you the ultimate standard. Without deflection, without unevenness, pursue royal righteousness; without any selfish likings, pursue the royal way. Without any selfish likings, pursue the royal path.\n> \n> Without partiality, without deflection, the royal path is level and ease. Without perversity, without one-sidedness, the royal path is right and straight. Seeing this perfect excellence, turn to this perfect excellence. (Legge 1991: 331\u2013332)\n\nAs central to all nine categories, the Royal Ultimate was theological in the sense that it was bestowed and revealed by di \u5e1d, as was said, \"these words stated in the Royal Ultimate were instruction not from the kings but from the di\" to become the ultimate principle of the royal power, which was to become impartial and universalizable, without deflection, without selfishness. Told to the conqueror by a Viscount belonging to the conquered ethnic group, it might be a request of fairness and impartiality of the conqueror to the conquered. But, beyond that, the idea of centrality or middleness interpreted as impartiality or fairness had its universalizable meaning for all individuals and all social groups toward many others. With it, royal power or political leaders would be able to bring the five blessings, as indicated in the ninth category, to their people as mentioned previously, that is, longevity, wealth, physical and mental health, cultivation of excellent virtues, and a good end crowning a good life, which have long been the core values of Chinese people.\n\nBesides its social political meaning, the idea of centrality as fairness and impartiality expressed in the category of the Royal Ultimate had also its religious meaning. Similar to Mircea Eliade's claim that religious man experienced the world as having a sacred centre and sought to live there, the centrality as affirmed in the fifth category was to be seen as the axis mundi, the vertical centrality as the centre of the world and as linking together all three cosmic levels: Heaven, earth and the underworld. This justified their belief that their world was holy because it was closest to the center of the universe. Eliade noted that temples might be seen as equivalents of sacred mountains. Religious man might understand his world as being at the center of the world on three scales: country, city, and sanctuary. In this view, in Shang Dynasty, An Yang \u5b89\u967d was seen by the Yin people as the center of the world; however, for the Zhou Dynasty, Hao Jing \u93ac\u4eac was seen by the Zhou people as the center of the world. There might be other centers of the world according to different ethnic groups.\n\nThe words of Viscount Ji marked also the transition from Shang to Zhou. In the text, the term di \u5e1d (High God) was first used, but later changed to the term tian \u5929 (heaven). That is why we should see the Hongfan as already a perception of Shang legacy by the people of Zhou. In Zhou dynasty, the concept of di evolved into that of tian, a more universal divine power no longer limited to the ancestral divinity, but rather the supreme ruler of both human society and the whole universe. While in the period of transition these two concepts were sometimes used interchangeably, the basic direction was moving toward the more universalizable idea of tian.\n\n## 3 The Decline of Political Theology\n\nThe dependence on the revealing of God had declined as the concept of Ultimate Reality underwent a process of change, first from di or shandi to tian, and then the idea of tian itself also changed to a more humanistic interpretation. It was not only that institutionally, in the Zhou dynasty, the division of work made diviner and historian two different jobs, which were, again, differentiated into various functions. Also, the divination itself became more technical and was full of uncertainty of God's revealing. These factors explain the fading of political theology and the move toward more humanistic intellectual concerns.\n\nFirst, in the Zhou dynasty, the division of work became more differentiated and complicated. The Zhouli \u5468\u79ae, a book arguably produced in the Warring States, when discussing Zhou institutions, talked about the distinction between the diviner (bu\u535c), the priest (zhu \u795d), the shaman dancers (wu \u5deb, again divided into male wu and female wu), and the historian (shi \u53f2). The shi was again divided into great historian (dashi \u5927\u53f2) and small historian (xiaoshi \u5c0f\u53f2). It mentioned also the internal historians (neishi \u5167\u53f2), and external historians (waishi \u5916\u53f2), and royal historians (yushi \u5fa1\u53f2) (see Zhouli: 37\u201341). All these institutional distinctions and functions might be an idealistic plan-making rather than a factual description of the Zhou Institution. However, we can tell at least that from the Zhou dynasty onward, the historians were separated from the diviners. When divested of their religious function of divination, historians in the Zhou dynasty were keepers of royal books and records, and they gave advice to kings and political leaders, based on the experience of the past as shown in these textual records. This was the cradle of all intellectuals, thinkers, and therefore philosophers, before they could rise from among common people in later ages like those after the Spring and Autumn period. The revealing of God's will therefore changed to the revealing of meaning through the interpretation of events, plots and their textual records. The narrative itself revealed meaning by way of interpretation. Indeed, historians preceded the rise of philosophers, and some philosophers had once been historians, like Laozi, who was said to have served as a keeper of the Royal library in Zhou's court.\n\nSecond, the decline of political theology was also caused by the uncertainty of the results of divination as to God's revelation. Sometimes the results were contradictory among themselves, and sometimes they depended on the king's choice, as when a king, following his own desire, did not listen to the result of the divinatory, or listened only to the result in favor of his own desire, such as in the story narrated by the Zuo's Commentary to the Spring and Autumn Annals. In the 4th year of Duke Xi's \u50d6\u526c reign (655BCE), Duke Xian of Jin State \u6649\u737b\u526c desired to marry Liji \u5b4b\u59ec, a woman famous for her beauty. The tortoise shell predicted that this would be unlucky. However, the milfoil predicted it lucky. The duke said, \"I will follow the milfoil.\" The diviner of tortoise shell said, \"The milfoil is inferior to the tortoise shell. Better follow the latter.\" The diviner added that if the duke married Liji, there would be catastrophic consequences. However the duke did not listen to this advice, and married Liji (Legge 1949b: 141\u2013142, translation modified and pinyin used). This was a case of a king who listened only to the result in favor of his own desire, which meant, despite the necessity of practicing the ritual of divination, the king wanted to hold his fate in his own hands.\n\nIn other cases, the results of divination might be contradictory, which would make the revealing of the Divine will uncertain. For example, in the 7th month of the 6th year of Ai Duke's \u54c0\u526c reign (489BCE), the King Zhao of Chu \u695a\u662d\u738b, while settled in Chengfu \u57ce\u7236, intended to succor Cheng \u9673 State; therefore he consulted the tortoise shell about fighting, and got an unfavorable answer. Then he consulted it about retreating, and got another unfavorable answer. Facing these contradictory results, King Zhao said, \"If it's to die all the same, it's better to die rather than defeating the Chu army. It's better to die than to turn back to our ally and evade the enemy. If I have to die in both cases, I'll die at the hands of my enemy.\" Then he made a decision to attack (Legge 1949b: 810).\n\nThe contradiction of divinatory results showed itself the uncertainty of God's revealing, and pushed political leaders, as holders of the collective destiny, to appeal to either more humane values, as in the aforementioned case, or more rational explanation. Here is an example of the rational explanation: on the 5th month of the 24th year of Duke Zhao's \u662d\u526creign (518BCE), there was an eclipse of the sun, regarding which Zisheng \u6893\u614e said, \"There will be floods.\" However, Zhaozi \u662d\u5b50said, \"There will be drought.\" The reason given by Zhaozi was that \"The sun has passed the equinox, and the yang influence has not predominated. When it does so, it will be in a very great degree, and we must have drought. The yang influence, not getting vent, will be accumulated\" (Legge 1949b: 701). This is a much more rational, meteorological and astronomical explanation, based on the movement of stars, such as that of the sun over the equinox, and the interchange of the forces of yin and yang, against the divinatory prediction.\n\nSometimes the reaction was more humanistic and against any blind belief in God's intervention. Here is a case of Zichan's \u5b50\u7522 humanistic response to the divination: According to the Zuo's commentary on Spring and Autumn Annals, in the month of May of the18th year of Duke Zhao's reign (524BCE), it was narrated that,\n\n> The Star Huo made its first appearance at dusk, On Binzi there was wind, and Zisheng predicted fire would break out there in seven days.... In a few days, messengers from Song state, Wei state, Chen state and Zheng State all reported cases of fire. Bei Zhao said, \"If you don't do as I said, Zheng will suffer from fire again.\" The people begged that his advice be taken.... Zichan replied, \"The Way of heaven is distant, while the Way of Man near. We cannot reach to the former, what means have we of knowing it? How should Zhao know the Way of Heaven? He is a great talker, and we need not wonder if his words sometime come true. (Legge 1949b: 671)\n\nZichan's comment that the diviner was a great talker and thereby his words sometimes came true showed a certain probabilistic view on the fulfillment of divinatory prediction. This was also related to the feeling of the uncertainty of divine revelation and the contingency by which a diviner's prediction might just happen to come true. Also, Zichan's thought that the Way of Heaven was far and the Way of Man near, emphasizing a humanistic concern with the Way of man, would lead also to the reasonableness of human words and deeds rather than God's will. It was in this spirit that Zhao Wenzi \u8d99\u6587\u5b50, after listening to one of Zichan's discourses, said, \"His speeches are reasonable. To go against reasonable speeches is inauspicious\" (Legge 1949b: 516). Here the good or misfortune seemed to depend on human reasonableness rather than divine will.\n\nNote that all these recorded events happened during Confucius' life time. Confucius (551\u2013479BCE), who in his late 50s was troubled by the difficulties of his exile and therefore focused on the study of the Yijing, giving philosophical comments on it and practicing divination using milfoil rather than tortoise shell, knew all these that evoked a more humanistic attitude towards divination.\n\nThird, the successive changes in the concept of Ultimate Reality also laid a metaphysical\/religious foundation that did not require reference to divine revelation and led to a stronger sense of its uncertainty. In Chinese religious and philosophical history, not only had the Shang people's concept of di or shangdi changed to the Zhou people's concept of tian, the idea of tian itself had also undergone changes after the rise of Confucianism and Daoism. It had changed from its religious transcendental meaning to more human-centered meaning. As said previously, the earlier concept of tian, in the Classics of Poetry and the Book of Documents, as a personal God capable of revealing Himself had changed at the time of Confucius (551\u2013479BCE), who, whilst keeping its religious meaning, focused more on a humanistic concern. Then, through the mediation of Zisi \u5b50\u601d (493\u2013406BCE) to Mencius (371\u2013289BCE), tian was seen more like the highest principle of morality immanent in and therefore accessible to human nature. In the Zhongyong (Doctrine of the Mean), usually attributed to Zisi, the concept of cheng \u8aa0, which meant sincerity on the psychological and ethical level, meant also True Reality on the metaphysical level. For Mencius, if one could unfold fully one's heart\/mind, one should be able to understand one's own nature, and when one understood one's own nature, one understood tian. This means there was a humanistic tendency in classical Confucianism that turned the transcendent tian into an immanent principle accessible to human subjectivity, in the process of which political theology would fade away.\n\nAll these reasons explain why political theology was replaced by humanism, though still open to divine revelation and the cosmic dynamism. The focus now was on human beings as agents, and on events and stories as revealing their destiny. Certainly something true and meaningful must be revealed in the existential time as constituted of events and human actions, not in the abstract time as form of succession and continuity. The meaning of human life consisted not in revealing a substantial spiritual Being imposing His will on human beings, but rather in human creativity in accord with Dao manifesting itself in the process of the universe.\n\n## 4 The Rise of the Classics of Changes\n\nThe Yijing \u6613\u7d93 (Classics of Changes) is one of the oldest classics of Chinese philosophy. In fact, it was originally a book of divination of the Zhou people. The act of divination concerned the consultation of the oracles of good or bad omen in order to know the good fortune or misfortune of the action to be taken or in people's future destiny. However, even in this act of asking, somehow with a sense of passive waiting for the divine response in terms of revelation, we find already in the Yijing the emergence of active participation of human agents, which led eventually to the philosophical construction of a meaningful existence. In the act of divination of the Yijing, there was a search for the ethical meaning of the divine revelation and then a deeper concern with the manifestation of the direction of changes. From this concern with the vector of becoming in the universe, there emerged a philosophical interpretation and conceptual construction of creative humanism.\n\nIn the Shang dynasty, mostly the tortoise shell or the shoulder bones of cows were used for divination, because they were used in the religious ceremonies and therefore considered to be imbued with mystic powers. Divination was a way of deciphering what came out naturally as signs resulting from the burning and crackling of the tortoise shell or bones, and the results were seen as representing the will of di or shangdi. When we come to the Zhou Dynasty, there was a change from the use of tortoise shell to that of the milfoil, although in its early period the tortoise shell was still quite often used to make decisions. The divinatory method of Zhouyi, when created by King Wen, founder of the Zhou dynasty, still used tortoise shells, as indicated by some of the guaci \u5366\u8fad (hexagram dictums) and yaoci \u723b\u8fad (stroke dictums). For example, the guaci of Hexagram 27 Yi \u9824 said, \"Second Sixth. Someone gives you a precious tortoise, costing about 100 shells, don't refuse the gift.\" Or, the guaci of the Hexagram Yi said, \"First Ninth. Give up your own spirit tortoise, and observe my mouth eating. Bad omen.\"5\n\nThus the change might have been caused by the fact that tortoise shell was precious and not commonly available. Milfoil or yellow stalks were, on the other hand, more easily accessible. Therefore, at the times of Spring and Autumn (722\u2013481BCE) and Warring States (403\u2013222BCE), divination in consultation with the Zhouyi changed to the use of milfoil or yellow stalks. The first historical record of the use of this method is found in the Zuozhuan \u5de6\u50b3 (Zuo's Commentary of the Spring and Autumn Annals). In the 22nd year of Duke Zhuang's \u838a\u526c reign (672BCE), it was recorded that, \"In the younger days of Jingzhong \u656c\u4ef2, the Marquis of Chen had met a Zhou historian who showed him the Zhouyi. The Marquis of Chen asked him to perform the divination by stalks [on the future of the boy]. When he found the hexagram guan \u89c0 going toward pi \u5426 hexagram, he said, \"here is the deliverance: we behold the light of the State. This is auspicious for one to be the King's guest'''' (Legge 1949b: 102\u2013103; translation modified).\n\nThis historical record showed for the first time that the divination according to the Zhouyi was practiced in Chen State, in particular in performing \"the divination by stalks.\" It is reasonable to presume that, after this method was invented by King Wen, its use was limited only to the Zhou court before it spread into other states. It was around the early years of the Spring and Autumn that other states, such as here the State of Chen, began to practice the Zhouyi divination by stalks and took its texts such as guaci and yaoci as objects of interpretation in order to reveal the meaning and future development of an event, a person or an issue at stake in the divination. Also, it was first recorded here that, each time the diviners got a hexagram in divination, they used the method of biangua \u8b8a\u5366 (in the previous quotation, \"the hexagram guan going toward pi hexagram\") to see how things were going to evolve. They then consulted the gua dictum and the yao dictums in order to know the manifestation of the divine will and its evolution for the decision of the good or bad omen of things and persons in question.\n\nIn comparison with the use of tortoise shell, the use of milfoil\/yellow stalks was technically more manageable and thereby gave a sense of human participation in knowing the revealing of God's will and in controlling one's own destiny.6 Since the time of late Spring and Autumn and the Warring States, to consult the Yijing, people used milfoil, a divining plant, to obtain a hexagram and its biangua, and to consult their guaci and yaoci and interpret their meaning by him\/herself or with the diviner. This depended much on the interpretative function of human reason and the choice made, according to the rules of divination, by human beings in referring also to their reasoning and judgment. In the divination by tortoise shell there was less space for the intervention of human subjectivity, whereas in the divination by yarrow stalks there was more possibility for such an intervention, thus leading to a more humanist construction of a meaningful world. Divination, even though still in use, became more a technique for making decisions and attaining consensus, rather than for revealing God's will as such. Confucius, who in his later years practiced the divinatory method using milfoil rather than tortoise shell, took more a humanistic and philosophical attitude towards divination, as he said, \"I look into the Yijing only for its ethical and philosophical meaning\" (in Zhao: 269).\n\n## 5 Divination, Ethical Interpretation, and Philosophical Construction\n\nWe have to put the revelation of divine will in the Yijing in the context of ancient Chinese people's effort to construct a meaningful world on both an individual and collective level. I consider the world of meaningfulness in the Yijing as moving from divination to ethical interpretation, and then from ethical interpretation to a philosophical, in particular cosmological, construction. In this process, the uncertainty of God's revealing and the inclination to rely on human factors played a great role in pushing the development of human thought towards rational and philosophical considerations of people's destiny.\n\nThis can be seen first from the textual composition of the Yijing, which is constituted of two parts: the older part is called the Zhouyi \u5468\u6613 (Zhou's Book of Changes), and the newer part is called the Yizhuan \u6613\u50b3 (Interpretations of the Zhouyi). The earliest version of the Zhouyi was merely constituted of 64 hexagrams with gua (hexagram) dictums and yao (lines) dictums. Historically these were the record of divinations in the late Shang and early Zhou periods. The results of these divinations were usually obtained by burning the tortoise shell or ox shoulder bone. However, the textual record of divination done in the past with regard to historical events became the sample texts to be consulted by later diviners. These, like the Shang oracles, could be seen as historical records of early Zhou people's divination, and since these divinations were related to major actions taken by their founding political leaders, they were taken by later diviners as exemplar texts. Therefore, as historical texts, they could be read together with other texts such as Classics of Odes and Book of Documents to get a more complete picture of China's ancient history. As divinatory texts, these recorded the divinations of Zhou Dynasty's founders, while in later days they began to serve as exemplary divinatory texts susceptible of various interpretations by subsequent generations.\n\nThis is to say that, since the early Spring and Autumn period, the Zhouyi served as exemplar divinatory texts, seemingly because of the fact that they were done by King Wen, King Wu, and the Duke of Zhou, honored as Sage Kings or Sages by later generations. Thus their status changed from being the textual records of the original manifestation of divine will to that of model texts susceptible to various interpretations, sometimes even conflicting interpretations, by many diviners on different occasions in later times. This was a change from manifestation-revealing to hermeneutic-revealing, though always targeting the will of the divine. In the hermeneutic-revealing, a diviner, with a due process of using divinatory stalks, got one hexagram, which, in the case of biangua, changed to another hexagram, and then he consulted the guaci (hexagram dictum) to interpret it properly in order to answer the inquirer's question by message of the revealed divine will. Then, he consulted the yaoci, i.e. the stroke dictums, and through his interpretation, sought to figure the unfolding of the event by interpreting them one after another.\n\nOn the other hand, the Yizhuan (Interpretations of the Zhouyi) appeared mostly in the late Spring and Autumn and Warring States. Historically they were attributed to Confucius; however, there were some texts produced by later hands, even as late as the time of the Han dynasty, which sought to give a systematic interpretation to the Zhouyi. Among these interpretations, the Xiangzhuan (\u8c61\u50b3 Symbolism Interpretations) almost always gave ethical interpretations of the guaci and yaoci, while the Tuanzhuan (\u5f56\u50b3 Judgment Interpretations), though quite often delivering ethical interpretation, referred from time to time to the cosmological background of ethics. For example, in the Judgment Interpretation of Hexagram 1 Qian \u4e7e, it was said,\n\n> Great indeed is Qian the Original, by which myriad things start to exist. It dominates over the process of the whole universe. It moves the cloud and gives the rain, and all kinds and categories of things form themselves and change with it. From beginning to end, the sun drives six dragons timely to run over the sky. With the Way Qian changes and transforms, each and everything fulfill their genuine nature and destiny, while altogether they achieve an optimal harmony. Qian is that which gives birth to myriad things, and that which renders myriad countries peaceful. (Zhu 1999: 30)\n\nWe should say that the texts in the First and Second Xici \u7e6b\u8a5e (Great Appendixes), together with the Xugua \u5e8f\u5366 (Sequence of the Hexagrams) and the beginning chapters of the Shuogua \u8aaa\u5366 (Remarks on the Trigrams) were the most philosophical texts in the Yijing. They established a systematic interpretation of the dao of all things and the principles of human affairs contained implicitly in the Zhouyi. It is mostly in these texts that the Yijing became a book of philosophy, because here it extended to all realms of existence, not limiting itself to the function of divination and telling the good\/bad fortune of human affairs. As to its comprehensive functions, the Great Appendix says,\n\n> The Yijing contains a fourfold dao of sages. As to speech, one should be guided by its words; as to action, one should be guided by its changes; as to making utensil, one should be guided by its images; as to divination, one should be guided by its oracle. Therefore the superior man, whenever he has to take a certain action, or to depart for somewhere, consults the Yijing, and he does so in articulating his intent in words. The Yi responds to his query like an echo; no matter it is far or near, dark or deep, he learns of the things of the future. (Zhu 1999: 246)\n\nAs we can see, here divination is only one of the four functions of the Yijing. Besides divination, it has also the functions of guiding discourse, taking action, and technological invention, therefore touching upon various aspects of human existence and diverse things in the universe, to which the principles elaborated by the Yijing all apply. Thus, the Xici \u7e6b\u8fad says, \"The Yi \u6613 is that which has enabled the sages to reach all depths and to grasp the initial moment of all things. Only through knowing its depth could one penetrate all intents of people under Heaven. Only through knowing the initial moment can one complete all affairs under Heaven. Since it is so spiritual, one can achieve all rapidly without being in haste and reach the goal without further ado\" (ibid).\n\nIn short, the development from the 64 hexagrams, the gua dictums, and the yao dictums, which constituted the Zhouyi, to the Xiangzhuan (\u8c61\u50b3 Symbolism Interpretations), which were attached to these original texts, was a development from divination to ethical interpretation; then, through the mediation of the Tuanzhuan (\u5f56\u50b3 Judgment Interpretations), the Yijing turned to the cosmological and comprehensive visions developed in the Xici, Xuguazhuan, and Shuoguazhuan. This was a development from ethical interpretation to philosophical construction.\n\n## 6 Natural Regularity and Logical Structure\n\nDivination was an act of asking for the manifestation of God's will with regard to a particular course of action. Unlike God in the Bible, the shangdi or tian never showed Himself, not even through the mouths of the prophets, but only as interpreted by the diviners over the signs on bones\/tortoise shell, dream contents, natural phenomena, or the hexagrams and their attached texts of explanation, which were always susceptible of human interpretation. In the system of Yijing, unlike Shang's di or shangdi who was said to manifest His will through the crackling of shoulder bones and tortoise shell, the divine will was manifested through hexagrams, their strokes, and texts attached to them in terms of metaphors related to the nature and regularity of things. Therefore, there was an implicit move from the direct manifestation of divine will to the manifestation of the regularity of things and the logic of reasoning with suggestive ethical implications.\n\nWithout explicit reference to divine will, the manifestation of the good fortune or misfortune of a certain action when one consulted the gua dictums and yao dictums was expressed noticeably through two interesting kinds of language. The first is the revelation of change of human affairs through the change of natural phenomena; the second is the reasoning of good fortune or misfortune of human affairs based on the logical structure of the hexagram in question. The first is an analogical understanding of human affairs in comparison to natural phenomena. The second concerns the logical construction of 64 hexagrams through the combination of logical possibilities of undivided stroke and divided stroke, or yin and yang.\n\nFirst, let us take the self-understanding of human affairs through an analogical understanding to the becoming of natural phenomena: For example, in the yao dictum of the second stroke, undivided, of the Daguo \u5927\u904e Hexagram, it says, \"Second Ninth. Dry poplar sprouts new buds. Old man marries with a young wife. There is nothing disadvantageous\" (Zhu 1999: 122, without specific mentioning, the following translations are all mine). And the yao diction of the fifth stroke, undivided, of the same hexagram says, \"Fifth Ninth. Dry poplar gives birth to new flowers. Old lady marries with a strong man. No blame no praise\" (Zhu 1999: 123). The production of shoots or flowers on an old decayed willow tree means the regeneration of life in nature; when this is shown in answering the question of an old bachelor or an old widow who consulted the Yijing by way of divination, it means a favorable time for him or her to marry again.\n\nSecond, the constitution of hexagrams, which serves as the logical foundation of manifestation, implies a logic and mathematic of two values, exemplified by an unbroken stroke, \u2500, and a broken stroke, \u2013. It is well known that Leibniz in his article Explication de l'arithm\u00e9tique binaire (Leibniz 1703) claimed that in the hexagrams of the Yijing he had found the universality of the binary system based on 0 and 1.\n\nAlthough the record of a hexagram and its texts was generated historically by the act of divination on a particular occasion, it is amazing that the totality of 64 hexagrams shows a rigorous logical internal structure. This fact has led to many theoretical speculations by scholars. For example, Thom\u00e9 Fang in his Logical Problems of the Yijing, shows that there is a process of logical derivation of the 64 hexagrams (Fang 1987: 1\u201329). The two constituent strokes, \u2500 and \u2013, taken alternatively, generate the two cardinal trigrams, qian \u4e7e and kun \u5764, through trivergence. In turn, these two congruent cardinal trigrams give rise to their six descendants through selective cross-linkage. On the one hand, the two cardinal trigrams, qian and kun, continue to generate hexagrams by alternative superposition. On the other, the other six rudimentary trigrams, by way of super-adding, generate another group of hexagrams. In this way, after 18 steps of logical derivation, we have a system of 64 hexagrams. Fang's explanation, though but one among many tentative theories to show the logic of the Yijing, goes well with what is explained in the Xuogua \u8aaa\u5366 (Remarks on the Trigrams),\n\n> Each trigram embraced those three categories, and, being repeated, its full form consists of six lines. A distinction was made of the positions assigned to the yin and yang lines, which were variously occupied, now by the strong, now by the soft forms, and thus the figure of each hexagram was completed. (Zhu 1999: 268; my corrected translation based on that of Legge 1963: 424)\n\nIt suffices to say now that there is a certain logical structure in the Yijing, and, by way of logical derivation, we have an open system of 64 hexagrams, in which any two co-ordinate hexagrams are put into a state of perfect congruence explicable in terms of logical correspondence. No matter how scholars may interpret the nature and structure of the structural intelligibility implied in the logic of the Book of Changes, there is undeniably a certain kind of logical structure linking the 64 hexagrams.\n\nThus, in interpreting the results of divination, either the manifestations of divine will or the dao of becoming, the language is expressed, first, in terms of metaphors coming from natural phenomena and their regularities, and second, in a systematic logical-mathematical structuration. Thus, the manifestation is constituted of a cosmological meaning and a formal meaning. In both meanings, the major aim of divination is to follow the dao of becoming in taking a particular course of action. It is true that the divination targets only the action to be taken and its good or bad fortune. However, the gua dictum and the yao dictum add also some lessons to human action concerning the cultivation of virtue and the conduct for a meaningful life.\n\n## 7 Making Explicit the Ethical Interpretation: Human Intervention into Structure\n\nThe structural aspect of the Yijing is constructed by the logical-mathematical system of hexagrams. Nevertheless, human subjectivity and its inner energy toward a meaningful life could always intervene into the structure, even to the point of reorganizing it. This means the human agent and its search for meaningful existence could give interpretation to the structure, in the sense not only of identifying specific cases that fit the structure, but also serving as an organizing-principle of the structure, or better, structuration. This is shown in the Xiangzhuan (\u8c61\u50b3 Symbolism Interpretations) and Tuanzhuan (\u5f56\u50b3 Judgment Interpretations). Xiangzhuan gives almost completely ethical interpretations of the guaci and yaoci, whereas the Tuanzhuan (\u5f56\u50b3 Judgment Interpretations), though quite often delivering ethical interpretation, refers from time to time to the cosmological background of ethics. In the Xiangzhuan and Tuanzhuan, we find several basic concepts, such as dang \u7576 (properness), ying \u61c9 (responding), zhong\u4e2d (centrality), and shi \u6642 (timing). Based on these concepts, several theories were developed for deciding what is good fortune or misfortune, as shown in the interpretation of Xiangzhuan and Tuanzhuan.\n\nThe first is the dangwei shuo \u7576\u4f4d\u8aaa (Theory of Proper Position), as proposed in the Xiangzhuan and the Tuanzhuan: An action is of good fortune when it happens in a proper position, of misfortune when the position is improper. Although in the beginning, this interpretation was more concerned with the proper position in the structural hierarchy of a hexagram, later it introduced also the Confucian ethical concept and Daoist cosmic concept of yin and yang. The main idea of this theory is that the yang stroke should be in the yang position, that is, on the odd lines; and the yin stroke should be in the yin position, that is, on the even lines. A hexagram is constituted of six strokes (lines), in which the first, the third, and the fifth lines, counting up from the bottom of each hexagram, are the yang position; whereas the second, the fourth, and the sixth lines, counting also from the bottom of each hexagram, are the yin positions. The unbroken lines, which are thus called yang yao, should take yang position as its proper position; otherwise it will be improper. The broken lines, which are thus called yin yao, should take yin position as their proper position; otherwise it will be improper. To be in a proper position is to have good fortune, whereas to be in an improper position is to have misfortune.\n\nFor example, in the 63rd hexagram, the Jiji \u65e2\u6fdf Hexagram, all yang and yin strokes are in their proper positions. That is why the Tuanzhuan says, \"Advantage for the divination. The strong and soft lines are rightly arranged, each in its appropriate position\" (Ibid: 226). On the other hand, in the 54th hexagram, the Guimei \u6b78\u59b9, where all other lines except the first and the sixth are not in their proper positions, the gua dictum says that, \"It is dangerous to advance in taking action. Not leading to any advantage\" (Ibid: 200). And its Tuanzhuan says, \"It's dangerous to advance in taking action, because the positions of the lines are not appropriate to them. Not leading to any advantage, because the soft rides upon the strong\" (Ibid.).\n\nThe second is the yingwei shuo \u61c9\u4f4d\u8aaa (Theory of Respondent Position). The theory of proper position does not suffice to explain all fortunate and unfortunate cases. Thus the Tuanzhuan introduces the Theory of Respondent Position. This theory says that, when in the groups of the first and the fourth lines, the second and the fifth lines, and the third and the sixth lines, there is a yin line responding to a yang line, or a yang line responding to a yin line, then that line with respondent will be a case of good fortune. When, in case of improper position, there is no such a respondent, it will be a case of misfortune.\n\nFor example, in the dayou \u5927\u6709 Hexagram, the fifth line is not in its proper position. Nevertheless, the yao dictum of its fifth line says it is of good fortune: \"Fifth Sixth. The sign is friendly good, honorable, and is of good fortune\" (Ibid: 83). The justification is offered by the Theory of Respondent Position in the Tuanzhuan that reads, \"In dayou the soft (yin) line has the honorable position of grand centrality, and is responded to by the strong lines above and below\" (Ibid: 82). So, the fifth line, a yin stroke, becomes the sign of good fortune because of its being in central position and is responded to by yang lines. This leads to the following considerations.\n\nThe third is the zhongwei shuo \u4e2d\u4f4d\u8aaa (Theory of Central Position). In the case that there is neither proper position nor respondent position, it would not necessarily go to the misfortunate side, because it could still be remedied by the fact that a stroke occupies the central place in the upper or lower trigram. The Theory of Central Position says that the stroke that appears in the divination in the second line, which is central to the lower trigram, or in the fifth line, which is central to the upper trigram, will be a sign of good fortune. The previously discussed yao dictum of the fifth line of the dayu \u5927\u6709 Hexagram is an example. Take another example, in the weiji \u672a\u6fdf Hexagram, the fifth stroke, a yin stroke, is not in its proper position, but its yao dictum tells of good fortune. To explain this, the Tuanzhuan says, \"Weiji is a successful divination because the soft (yin) line is in a central position\" (Ibid: 229). Hence, the Central Position can remedy the case of improper position, while in the case of proper position, it can offer positive reinforcement.\n\nFinally, there is qushi shuo \u8da3\u6642\u8aaa (Theory of Acting in Proper Time), which says that the good fortune or misfortune of a hexagram depends on the proper (right) or improper character of the time in which it appears. If it appears at a proper time, it is good fortune; otherwise, it is misfortune. The fact that a stroke is in the central position does not necessarily make it good fortune. It is good fortune when it is at a proper time, and misfortune when at an improper time. For example, in the Jie \u7bc0 Hexagram, the yao dictum of the second line says, \"If you are not going out of your house door and courtyard, it will be bad fortune\" (Ibid: 218). However, both the second and the fifth lines are in central position, where the Tuanzhuan says, \"Jie, successful, because the strong (yang) and the soft (yin) lines are well distributed and the strong line occupies the central position\" (Ibid: 217). Nevertheless, the xiangzhuan says, \"If you are not going out of your house door and courtyard, it will be bad fortune, because he loses extremely his proper time\" (Ibid.). Time is therefore the ultimate criterion for judging good or bad fortune. This goes well with what the Xici II (The Great Appendix II) says, \"The strong and soft lines are the fundamental principle according to which the Changes were founded, while their change and success depend on the properness of time\" (Ibid: 252).\n\nIt becomes clear now that the Yijing, even when it contains an essential structural aspect, will never neglect the role of human agent and human historicity. When human beings attempt to know the good fortune or misfortune of their action, they must pay attention to the logical, numeric, and orderly structures, to the point of even implying something like Leibniz's Mathesis Universalis or Universal Grammar. However, these structures must submit themselves to the interpretation of human beings and their historicity in order to render themselves meaningful. It is based upon this vision that, later, the Xici I (Great Appendix I) says about the structural factors,\n\n> The Yi changes through the cross-linkage of lines, and unfold its numeric structure by the reversal and alternative superposition of trigrams. When one could penetrate all cases of change, one may obtain the logical structure of the universe. When one can exhaust the numeric structure, one may determine the meaning of all images under Heaven. If not with this supreme change of all under Heaven, how could it achieve this? (Ibid: 246)\n\nThis important text shows the marvelous effect of the combination of structural factors. On the other hand, the Shuogua \u8aaa\u5366 (Remarks on the Trigrams) also says,\n\n> In ancient time, when the sages composed the Yi, in order to give assistance to the Spiritual Intelligence, they created the rule for the use of the divining plant. The number three was assigned to heaven, number two to earth, and from these came the other numbers. They contemplate the changes through the yin (broken) and yang (unbroken) lines and formed the trigrams. From the movements taking place in the strong (yang) and soft (yin) lines, they created the separate lines or yao. There ensued a harmonious conformity to the Way of Dao and virtue to the discernment of what is just and right. They made an exhaustive investigation into the principle of all things to understand the mandate of Heaven. (Ibid: 267; Legge 1963: 423 with my corrections)\n\nThis equally important text shows us that there is a dynamic interconnection between the formal and numeric structures and the human agent's self-realization. In following the numeric and formal structures and in giving an interpretation to it, there is a way of discerning what is just and right, to the extent of following the Way and realizing one's own mandate of Heaven.\n\n## 8 From Ethical Interpretation to Philosophical Construction\n\nThe Zhouyi represents the process from divinatory manifestation to ethical interpretation, while the Yizhuan represents the process from ethical interpretation to philosophical and conceptual construction. It is in the Yizhuan that the Yijing becomes a book of philosophy, because here it is extended to all realms of existence, not limiting itself to the function of divination and that of telling the good or bad fortune of human affairs. The Xici I says,\n\n> The Yi is that which has enabled the sages to reach all depths and to grasp the initial moment of all things. Only through knowing its depth could one penetrate all intents of people under Heaven. Only through knowing the initial moment can one complete all affairs under Heaven. Since it's so spiritual, one can achieve all in a fast way without being haste and reach the goal without further ado. (Zhu 1999: 246)\n\nHere, we may reconstruct the steps by which the meaningfulness of existence was constructed in the Yizhuan. Following the Shuogua and the Xici, I will analyze them into the following constructive and progressive stages.\n\n1.\n\nConstruction of elementary representations: According to the Xici I, \"The sages were able to observe all the traces on phenomena under heaven, and therefore were able to grasp their forms and contents, and represent things properly. These were then called xiang \u8c61 (manifesting representations)\" (Ibid: 241). What were to be represented were the greatest natural phenomena in our environmental world: Heaven and earth, mountain and lake, thunder and wind, water and fire etc. In addition, they represent also various things in the environment, such as plants and animals. For example, qian \u4e7e represents heaven, jade, metal, ice, horse, etc; li \u96e2 represents fire, sun, lightning, turtle, crab, tortoise, etc.\n\n2.\n\nConstruction of directions of action: Actions are taken in the nexus of space and time. Since all actions are undertaken in time, therefore the vector of time is the most important direction. In the Yijing, the direction of time is conceived in two ways: Toward the past, it is a natural process; toward the future, it is an anticipatory process. The Shuogua \u8aaa\u5366 says, \"The numbering of the past is a natural process; the knowledge of the coming is anticipation. Therefore in the Yi we have both anticipation and the natural process\" (Zhu 1999: 268; Legge 1963: 424) As to spatial directions of action, which is always taken in time, eight principal spatial directions are possible for any action, as indicated in the Shuogua \u8aaa\u5366: East, South East, South, South West, West, North West, North, North East (Zhu 1999: 269, Legge 1963: 425\u2013426). More detailed spatial directions are elaborated with the complex interaction of hexagrams. In the Yijing, spatial direction is crucial for the good or bad fortune of an action, and that is why it is often indicated in the gua dictums. For example, it is said in the gua dictum of the kun\u5764 hexagram, \"Kun. Great success. In favor of the divination regarding the mare (female horse). If you take the long trip, you will get lost first before you find a host. You will gain some friends in the south west, you will lose friends in the north east. It's good for the diviner to stay\" (Zhu 1999: 39).\n\n3.\n\nRepresentations of the human body: It is the human body that takes action and moves toward different directions. Therefore the human body is represented and situated in the realm of reality. It is said in the Shuogua that, \"Qian suggests the idea of a head; kun, that of the belly; zheng, that of the feet; shun, that of the thighs; kan, that of the ears; li, that of the eyes; gen, that of the hands; and dui, that of the mouth\" (Zhu 1999: 271, Legge 1963: 429).\n\n4.\n\nConstruction of human relationships, especially ethical relationships: The Shuogua says, \"Qian is the symbol of heaven, and hence has the appellation of father. Kun is the symbol of earth, and hence has the appellation of mother. Zheng shows the first application of qian to kun, resulting in the first of its male, and hence is called its eldest son. Shun shows a first application of qian to kun, resulting in the first of its female, and hence is called its eldest daughter. Kan shows a second application of kun to qian, resulting in the second of its male, and hence is called its second son. Li shows a second application of qian to kun, resulting in the second of its female, thus is called its second daughter. Gen shows a third application of kun to qian, resulting in the third of its male, and hence is called the youngest son. Dui shows a third application of qian to kun, resulting in the third of its female, and hence is called its youngest daughter\" (Zhu 1999: 271; Legge 1963: 429\u2013430). Thus there is an ethical network with cosmo-ontological implications. Human beings always act and live in the ethical relationship, they never act and live as isolated individuals, because they exist in the ontology of dynamic relation, in which ethical relations could combine with other dimensions of construction, such as the spatial-temporal. This is illustrated by the fact that, in a traditional Chinese house, the father lives in the room of the North-West corner, the mother that of the South-East, the eldest son that of the East, the eldest daughter that of the South-East, the second son that of the North, etc.\n\n5.\n\nConstruction of the system of hexagrams: In order that the emblematic representations could cover the whole universe and human existence, there should be an original creativity giving birth to myriad things, as well as the logical derivation of all hexagrams. On the ontological level, Taiji \u592a\u6975 (the Great Ultimate) represents the Original Creative Power that gives birth incessantly to all beings. This is an ontology of creativity in which \"To be is to be creative.\" On the logical level, this is also the order by which all hexagrams are produced by a process of logical derivation. According to the Xici, \"Therefore in Yi there is the Taiji that generates the two primary models. The two primary models generate the four xiang \u8c61 (manifesting representations). The four xiang generate the eight trigrams. The eight trigrams determine good fortune and bad fortune. Good fortune and bad fortune create the great field of action\" (Zhu 1999: 48). Or, again, \"The eight trigrams are arranged in order, and they contain images in them. Thereupon they are doubled, thus to contain each the [six] lines. The strong and the soft exchange and extend mutually, and there the change is contained\" (Zhu 1999: 252). \"The Yi changes through the cross-linkage of lines, and unfold its numeric structure by the reversal and alternative superposition of trigrams. When one could penetrate all cases of change, one may obtain the logical structure of the universe. When one can exhaust the numeric structure, one may determine the meaning of all images under Heaven\" (Zhu 1999: 246). Although the system of hexagrams was elaborated in the Yijing only to the 64th hexagram, it is possible to continue further; thus it is an open system.\n\n6.\n\nThe Construction of universalizable norms of action: Universalizable standard of action must be established to guide human actions and life praxis. As the Xici says, \"The Dao of this book is great, and none of the hundred things is omitted. Keeping oneself always vigilant from the start to the end, then in principle there will be nothing to blame. This is the Dao of the Changes\" (Zhu 1999: 264). Here, the terms \"great\" and \"none of the hundred things is omitted\" mean the universalizability on the pragmatic level. The wisdom in the Yijing therefore results from a deep concern with individual and collective destiny based on practical universalizability, somewhat different from the search for theoretical universalizability in Western philosophy. Both are interested in universalizability, but while the Chinese one is practical, the Western one is theoretical.\n\nThus the Yijing embraces a pragmatism that concerns itself with human action and its good or bad fortune. Since the going-to-the-better or going-to-the-worse is most crucial in human affairs, the Yijing's concern with the fortune or misfortune of human action touches the heart of human destiny. However, even in this humanistic concern, the Yijing still remains open to a sense of religiosity in its faith in the Great Ultimate, the Origin of all things, despite the uncertainty of God's revealing. Yijing's philosophy is a philosophy of action, and a philosophy that brings action to the betterment of human beings in the process of history. According to Yijing's philosophy, with its faith in the Great Ultimate and its optimistic trust in cosmic creativity, human beings are invited to participate in the process of cosmic advancement and to lead a meaningful life in their history, all in emphasizing human responsibility, in the sense of the ability to respond, and their creative interpretation of laws and structures in nature and history.\n\n## 9 Philosophical Breakthrough: Confucius\n\nSince the ethical interpretation and philosophical meaning in Yijing's later development were somehow related to Confucius, we have to say something about the emergence of classical Confucianism at the time of Confucius, its founder. The time in which Confucius appeared could be properly characterized, in Karl Japers' term, as the first \"Axial Age,\" or in Talcott Parsons' term, as an epoch of \"philosophical breakthrough.\" In the late Spring and Autumn period, Confucius' philosophical thought evolved from a philosophy of li (rituals) to the philosophy of ren (humaneness), then from the philosophy of ren to the philosophy of yi (Changes). This reading of Confucius' life is confirmed by his own words that indicate the evolution of his thought:\n\n> The Master said, \"At fifteen, my heart and mind was set upon learning; at thirty I established myself; at forty I had no more perplexities; at fifty I knew my Mandate of Heaven. At sixty I was at ease with whatever I heard. At seventy I could follow my heart's desire without transgressing moral principles.\" (Analects 2.4. Chan 1963: 22; with my corrections)\n\nAccording to my understanding, what Confucius learnt at 15 was li (ritual). Since he said, \"one should establish himself on li\" (Analects 8:8), he should have established himself at 30 by becoming a ru \u5112 teaching li and serving on the occasions of public and private li to earn his living. At 40, he had laid the philosophical foundation of li on ren\u4ec1, and therefore had no more perplexity. In his 50s, that is, between 50 and 60, in particular after he left Lu state when he was 54 years old, there was a period of time in which he and his disciples endured unstable, troublesome, and even dangerous years of exile, when Confucius studied the Yijing closely and consulted it regularly.7 This understanding of the self and the nature of things led him to a deeper understanding of his own destiny (Mandate of Heaven), and what other people were saying and feeling while he was in his 60s, and his spiritual freedom in the last few years of his life. From this interpretation of Confucius' short bio-data, the evolution of his thought could be divided into three stages:\n\n  * First, from age 15 to his 30s, Confucius' main concern was first learning li, and, at the age of 30, earning his living in developing a career of a ru by teaching and practicing li.\n\n  * Second, from his 30s to his 40s, his major concern was ren, taking this as the transcendental foundation of li so as to revitalize Zhouli. At this stage, his intellectual development focused on the philosophy of ren and how it gave foundation to li.\n\n  * Third, from his mid-50s onward, Confucius focused on the Yijing and History. He ordered, prefaced and gave commentaries to the Yijing. Arguably most of the Yizhuan could be from the hand of Confucius, either in his writing, or in his teaching as recorded and developed by his disciples. Sima Qian (c.145\u201390BCE) in his Shiji \u53f2\u8a18 (Record of the Grand Historian) said, \"Confucius in the later years of his life was fond of the Yijing. He had ordered and prefaced the Xianzhuan, the Xicis, the Tuanzhuan, the Shuogua, and the Wenyan\"(Sima 1976: 609). Here Sima Qian confirmed Confucius' contributions to the Yizhuan. Not to indulge in the debates over the question of authenticity, we may say that most of the texts in the First and Second Xici \u7e6b\u8a5e (Great Appendixes), the Xugua \u5e8f\u5366 (Sequence of the Hexagrams), the beginning chapters of the Shuogua \u8aaa\u5366 (Remarks on the Trigrams) and the qian and kun hexagrams' Wenyan \u6587\u8a00, were by Confucius.\n\nAs to Confucius' study of history, Sima Qian also told in his Shiji that Confucius in the later years of his life said of himself, \"A junzi would not pass away without a dignified name to remain in the world. Since my Way could not spread in the world, how could I show myself to the future generations?\" (Sima 1976: 609, my translation). Upon that, he proceeded to compose the Spring and Autumn Annals, in which he put down chronologically and gave his judgment on historical events.\n\nWe can say that Confucius, in the Yizhuan, focused more on the relation between ethics, cosmic creativity and the Ultimate Reality; his ideas of human affairs, ethics and his laying the foundation of li on ren, were to be found in the Analects; his vision of history was developed in the Spring and Autumn Annals, which provides us with his practical and axiological judgment on human actions and historical events. However, these refer always to his basic ideas of li \u79ae, yi \u7fa9, and ren \u4ec1, and that is why a philosophical understanding of these concepts is necessary here.\n\nEtymologically, the Chinese character li \u79ae is composed of both \u793a and \u8c4a, in which the sign \u793a signifies the enlightening or a signal coming from shangdi or tian (Heaven) to reveal a good or bad omen of individual and collective destiny, while the sign \u8c4a represented two wine cups used in the ceremony of libation, mostly religious. Religious rituals were rife in Shang Dynasty. Notice that, from Shang to Zhou, different people in the social hierarchy performed different religious ceremonies: for example, the King or Son of Heaven performed sacrifice to Heaven, princes and marquises to mountains and rivers, and common people only to their ancestry, each having their codes of behavior in their daily life.\n\nThus, in its actual meaning, li has three essential aspects: first, as the sacrificial ceremonies; second, as the social and political institutions; and third, as codes of daily behavior. Sacrificial ceremonies were ways to communicate with the divinities or deities; social and political institutions were organizational structures to set up a social order out of human beings' tendency to chaos and conflict, and codes of behavior regulated human action and attributed value to a specific action in question. All together, we can say that the functions of li were to communicate, to establish order and to render values. In these senses we can relate these three meanings to the ideal dimension of li. In its ideal meaning, li in Zhou Dynasty, the so-called Zhouli \u5468\u79ae, represented an ideal of Zhou civilization established by the Duke of Zhou, in the process of communication with the divine and many other human beings, according to which a civilized society should develop in an order imbued with a sense of beauty or harmony.\n\nIt is obvious that li was not created by Confucius and it never came to his mind to create a new system of li. He himself admired the rich meaning and content of Zhouli, saying that \"The Zhou dynasty looked back to the precedent two [Xia and Shang] dynasties. Such a wealth of culture! I follow the Zhou\" (Analects 3.14, Ames and Rosemont 1998: 85). What Confucius endeavored to do was to revitalize Zhouli by making it meaningful again in basing it on the self-awareness of each and every person's inherent and transcendental ability. Here the term \"transcendental\" means that which was within human ability prior to experience and that rendered experience possible.\n\nConfucius tried to revitalize Zhouli by tracing back to its transcendental origin and basing it on ren, which signified the sensitive interconnectedness between a human being and other human beings, nature, and Heaven. Ren manifested human beings' inner self and responsibility, in the original sense of the ability to respond in and through his\/her sincere moral awareness. Also, it meant the intersubjectivity giving support to all social and ethical life. Thus, we may understand that, with ren, human being has an inner dynamism of generously going outside of one's self to many others, in the meanwhile he will not lose his\/her own self. That is why Confucius said that ren is not remote from or difficult for any human being; as soon as an individual willed for it, ren was already there within him\/herself (Analects: 7.30). By this, Confucius laid a transcendental foundation to human beings' interaction with nature, with society, and with Heaven.\n\nAnd then, from ren, Confucius derived yi, righteousness or dutifulness, which represented for him respect for many others and the proper actions toward each of them. Not much was said by Confucius about yi, though what was said was very essential to Confucianism: \"A superior man makes righteousness (yi) the substance of everything; he practices it according to ritual order (li). He brings it forth in modesty. And he carries it to its conclusion with faithfulness. He is indeed a superior man!\" (Analects 15.18; Chan 1963: 43 with my corrections). Notice here that li was that which a wise and good man used to carry out or practice yi, which was seen by him as the substance of everything. For Confucius, yi was also the criterion by which good men and base guys were discerned (Analects 4.16). Upon rightness was based all moral norms, moral obligations, our consciousness of them, and even the virtue of always acting according to them.\n\nNow, from yi, Confucius derived li, the ritual order or proprieties, which represented the ideal meaning of harmony with a sense of beauty, and the actual meaning of codes of behavior, social institutions, and religious ceremonies. Youzi \u6709\u5b50, a disciple of Confucius, once said, \"Among the functions of propriety (li) the most valuable is that it establishes harmony. The beauty of the way of ancient kings consists of this. It is the guiding principle of all things great and small\" (Analects 1.12, Chan 1963: 21). We could presume therefore that Confucius understood li as an overall concept of a cultural ideal, as harmony with a sense of beauty, or a graceful order leading to beauty and harmony. With it, human life in the past is worthy of keeping in memory; in the future, worthy of expectation; and in the present, full of meaningfulness.\n\n## 10 Meaningfulness of Historical Events: The Spring and Autumn Annals\n\nThe Spring and Autumn Annals, authored by Confucius in the last years of his life, shows us the meaningfulness in human actions, events, and stories and the role human agent plays in the historical events. There the dao is to manifest through human actions that take place as events, and the emplotment that combines different actions and events into a story as an understandable narrative unity. Implicitly, there is a historical ontology that shows the dao in history, already made explicit in the Yijing.\n\nIn Confucius' time, historians of each state recorded their own historical events and established their own Annals. Confucius seemed to have consulted all the records of the past, especially those of Lu \u9b6fState. He said of himself, \"To transmit but do not create. I am trustful and devoted to antiquity\" (Analects 7.1). In his Annals, Confucius not only recorded historical facts, but more importantly he gave his judgments from his moral standards based on ren, yi, and li. That is why it was said that the rebellious ministers and stealers of power all dreaded his Annals. Therefore, by \"transmit\" he meant not only to organize historical materials to transmit factual accounts, but also to transmit them with value judgment. This seemed very laborious work, and he put a lot of energy into the composition of the Annals. That's why he said that \"in the future generations, it is by the Annals that people know my merit; it is also by the Annals that people would blame me\" (Sima 1976: 609).\n\nSince the wording of Confucius' Annals was very compact, there were three interpretations of Confucius' Annals: The Zuo's Commentary, the Gongyang's Commentary, and the Guliang's Commentary, all done in the Warring States period, most probably by developing the teaching of Confucius in the Annals transmitted by Zixia, one of Confucius' disciples. These three commentaries were so classical that they became themselves part of the Confucian Thirteen Classics. In giving commentaries on Confucius' Annals they made clear a reading of history with moral judgment. For example, in November of the 22nd year of Duke Xi's reign (637BCE), Confucius wrote,\n\n\"In winter, in the 11th month, on the jishi \u5df1\u5df3 day, the first day of the moon, the Duke Xiang of Song fought with an army of Chu near the Hong River, when the army of Song was disgracefully defeated.\"\n\n> To this record, Zuo's Commentary reads as follows:\n> \n> An army of Chu invaded Song in order to relieve Zheng. The Duke of Song being minded to fight, his Minister of War remonstrated strongly with him, saying, \"Heaven has long abandoned the House of Shang. Your Grace may wish to raise it again, but such opposition to Heaven will be unpardonnable.\" The Duke, however, would not listen to advice, and in winter, in the 11th month, on the jishi day, the first day of the month, he fought with the army of Chu near the Hong River.\n> \n> The men of Song were all drawn up for battle, before those of Chu had all crossed the river, and the Minister of War said to the Duke, \"They are many, and we are few. Pray let us attack them, before they have all crossed over.\" The duke refused; and again, when the minister asked leave to attack them after they had crossed, but when they were not yet drawn up, he refused, waiting they were properly marshalled before he commanded the attack.\n> \n> The army of Song was shamefully defeated; one of the duke's thighs was hurt and the warders of the gates were all slain. The people of the State all blamed the duke, but he said, \"The superior man does not inflict a second wound, and does not take prisoner of any of the gray hairs. When the ancient had their army in the field, they would not attack an enemy when he was in a defile. And though I am but a poor representative of a fallen dynasty, I would not sound my drums to attack an unformed host.\"(Legge 1949b: 183)\n\nNote that Zuo's commentary had the special hermeneutic method that by narrating the story without giving any explicit argument using abstract terms, it gave its interpretation of the meaning of Confucius' succinct text in the Annals. This means that in Zuo's commentary, telling a story itself is already doing a hermeneutic work, that is, giving interpretation or rendering meaning to a text difficult to understand. However, the reader, through reading the events put together in a plot narrating a story, such as this one of the Duke of Song's war against Chu army, could feel strongly his moral standard out of his humanness and rightness as expressed in emphasizing the ritual of doing war.\n\nNow, we can compare this with the Gongyang's Commentary, which says about the same text of Confucius:\n\n> It sufficed to date an appointed war. Why does it here mention the first day of the moon? When the Spring and Autumn Annals used more words generously not to save words, it means the action itself is upright. In what sense is it upright? The Duke of Song had an appointment of war with the Chu army to fight at Hong River. When the Chu men were crossing the river to come to him, his officer of war reported to him, \"Pray to attack them before they all cross the river.\" The duke of Song said, \"No. I heard that a superior man does not put others in danger. Even if I am only a representative of a fallen dynasty, I cannot bear to do it.\" When the Chu men all crossed the river but not yet drew themselves up, the officer of war again said, \"Pray to attack them while they are not drawn up well.\" The Duke of Song says, \"No. I heard it, the superior man does not attack a not well drawn up army.\" Then the Chu army was well drawn up, and the duke of Song called the drum for attack, thereupon the Song army was disgracefully defeated. Therefore, a superior man puts the emphasis on not attacking the unlined-up. He will not forget the li in facing a crucial event. He has the attire of a king but without good ministers and subjects under him. We can say that even a war done by King Wen is not more moral than this one. (Gongyangzhuan 1965: 28)\n\nThis commentary of the Gongyang was composed of two parts, though well mixed into one story: a narrative part and a moralizing part in the form of textual explanation. The story telling part was similar to the Zuo's commentary; however, the explanatory part rendered explicit the moral meaning of it, as to Confucius' admiration of the uprightness of the Duke of Song by allowing more words making precise the time in which the event happened, and by comparing Duke Xiang of Song to King Wen of Zhou.\n\nIt is a general style of the Gongyang to mix up moralizing discourse with the story-telling of historical discourse. A sensitive reader, in reading the story told by the Zuo's commentary, could already understand or make explicit the moral implication of it. In comparison, the Gonyang would be more helpful for those who are not sensitive enough to read out the moral in the story. Also, it puts down a clearer and stronger Confucian moral philosophy of history. The text of the Gongyang's Commentary started with the idea of Dayitong \u5927\u4e00\u7d71 (Great Unification) under the leadership of King Wen. That's why in the above text the same idea was repeated by saying that \"even a war done by the King Wen is not more moral than this one.\"\n\nShort of space, I shall not delve into the depth of Guliang's Commentary, which is constituted also of a narrative part and a moralizing or explanatory part. Just let me take the same story as an example, the Guliang in its moralizing part gave the comment that, \"Human beings as humans depend on his language.... Language as language depends on its trustfulness.... Trustfulness as trustfulness depends on the dao. The most precious in dao is its timeliness, and its implementation depends on the situational power\" (Guliangzhuan 1965: 24). This comment seemed to praise Duke Xiang of Song as trustworthy in his words for the appointment of war. Nevertheless, it also blamed him for not knowing the situational power for implementing the dao.\n\nLater, from Gongyang's Commentary was developed a kind of reformative ideology. In particular, Gongyang's idea of dayitong \u5927\u4e00\u7d71 (Great Unification) was developed in Han Dynasty, during the time when Dong Zhongshu \u8463\u4ef2\u8212 (179\u2013104BCE) was making Confucianism the state ideology of Han, excluding all other schools of thought, under the reign of Emperor Wu \u6b66 (158\u201387BCE), to become a progressivist vision of history developing to a \"Great Unification.\" Dong Zhongshu interpreted Confucius' Spring and Autumn Annals as divided into three Ages, the transmitted-heard age (of the past), the heard-about age (of the present), and the envisioned age (in the future), thus a progressively evolutional vision of history. Also, he saw Confucius' Annals as promulgating some historical laws for future dynasties as well as embodying the natural laws. This interpretation of history as evolving through Three Ages was developed in the later Han Dynasty by He Xiu \u4f55\u4f11 (129\u2013182AD) in his Chunqiu Gongyang Zhuan Jiegu \u6625\u79cb\u526c\u7f8a\u50b3\u89e3\u8a41 (Notes and Explanations of Gongyang Commentary). According to He, Confucius, in the Spring and Autumn Annals, proposed three stages of historical development. First, the Age of Disorder (juluan shi \u64da\u4e82\u4e16), in which one was familiar only with one's own country, and taking other Chinese states as foreign. Second, the Age of Rising Peace (shengping shi \u5347\u5e73\u4e16), in which all highly civilized Chinese states become one, taking the uncivilized countries as foreign. Finally, the Age of Great Peace (taiping shi \u592a\u5e73\u4e16), in which all Under Heaven, including the close and the far away, the small and the great, are as if forming one country.\n\nThis vision of the Great Unification was very inspiring both philosophically and politically for the Chinese people. This ideology based on its philosophy of history was revitalized in the late Qing period by Kang Yuwei \u5eb7\u6709\u70ba (1858\u20131927), and keeps on inspiring Confucian political movement even till very recently in China.\n\n## 11 Conclusion\n\nIn ancient Chinese intellectual history, the revelation of meaningfulness of human existence moved from political theology to creative humanism. The ultimate reality in the form of political theology was di or shangdi (High God), which moved, when the creative humanism emerged, to Taiji (Great Ultimate), the Original source of creativity to be realized in human actions, events, and stories in the cosmic process. In other words, it moved from the Book of Documents to the Yijing, which played a crucial role in Chinese religious and philosophical history. Turning itself away from the passive awaiting of and dependence on the High God's revealing, always uncertain and sometimes contradictory, the Yijing, evolving from divination to ethical interpretation, then from ethical interpretation to philosophical construction, had directed its thoughts to a meaningful world constructed by human creativity, even if still referring to the Great Ultimate. The Yijing appealed to cosmic laws and intelligible structures to understand human destiny, always with the possibility of human interpretation and intervention. In the philosophy of Yijing, there was an optimistic trust in the Great Ultimate and its creativity in the natural and human world. Human beings were invited to participate in the process of cosmic creativity and lead a meaningful life, emphasizing human responsibility and creative interpretation in the unfolding of human historicity.\n\nWith this emphasis on human moral effort, decision, and responsibility, there was always an ethical core in classical Confucianism, so well exposed in the ethical insights of the Analects and the ethical historicism of the Spring and Autumn Annals. Thus, for the Analects, the fundamental ethical values such as ren, yi, li and other virtues constituted an onto-ethical meaningfulness of existence, to be realized in human self-cultivation, society and governance. For the Spring and Autumn Annals, the meaning of existence was to be manifested through major human decisions, actions, historical events and stories, and judged in reference to some basic Confucian moral values. The fading of political theology and the rise of creative humanism led to the founding of classical Confucianism by Confucius and its further development through Confucius' disciples, Zisi-Mencius School and Xunzi, to be discussed in the following chapters.\n\nReferences\n\nAmes, R., and Henry Rosement. 1998. The analects of confucius, a philosophical translation. New York: Ballantine Books (English translation of the Analects with philosophical sensitivity to Western philosophical assumptions).\n\nChan, Wing-tsit. 1963. A source book in Chinese philosophy. Princeton: Princeton University Press (A comprehensive selection of Chinese philosophical works covering its entire historical development with helpful introductions and explanations).\n\nde Bary, Wm Theodore, and Irene Bloom, eds. 1999. Source of Chinese traditions. 2nd ed., Vol. I. From earliest times to 1600. New York: Columbia University Press (A comprehensive selection of Chinese works most significant in the intellectual history of China with good introductions).\n\nFang, Thom\u00e9 H. 1981. Chinese philosophy: Its spirit and development. Taipei: Linking Publishing. (A comprehensive history of Chinese philosophy written in English by one of the most important contemporary Chinese philosophers).\n\nFang, Thom\u00e9 H. 1987. Logical problem of the book of changes. In Creative creativity. Taipei: Liming Press (an insightful study of the logical system of the Yijing by a famous contemporary Chinese philosopher).\n\nGongyangzhuan (Gongyang's Commentary on the Spring and Autumn Annals). 1965. In Duanju Shisanjing Jingwen (Punctuated texts of thirteen scriptures). Taipei: Kaiming Bookstore (punctuated original texts in a convenient portable Confucian thirteen classics).\n\nGraham, Angus C. 1989. The disputers of the dao: Philosophical argument in ancient China. La Salle: Open Court.\n\nGuliangzhuan (Guliang's Commentary on the Annals of Spring and Autumn). 1965. In Duanju Shisanjing Jingwen (Punctuated texts of thirteen classics), Taipei: Kaiming Bookstore (punctuated original texts in a conveniently portable Confucian thirteen classics).\n\nGuo, Moruo, ed. 1978\u20131983. Jiaguwen heji \u7532\u9aa8\u6587\u5408\u96c6 (Complete collection of oral bone scripts). Beijing: Zhonghua Shuju.\n\nGuodian Chu Mu Zhujian \u90ed\u5e97\u695a\u5893\u7af9\u7c21 (Guodian Chu Tomb Bamboo Slips). 1998. ed. Jingmen Shi Museum. Beijing: Wenwu Chubanshe (containing pre-Qin Daoist and Confucian texts unearthed at Guodian Chu Tomb in 1993).\n\nLaozi, Laozi Sizhong \u8001\u5b50\u56db\u7a2e (Four versions of the Laozi). 1999. Taipei: Da'an Press (most convenient collection of the four essential versions of Laozi).\n\nLegge, James. 1991. The shoo king, or the book of historical documents. Reprint, Taipei: SMC Publishing.\n\nLegge, James. trans. 1949a. The She King. Reprint, SMC Publishing.\n\nLegge, James. trans. 1949b. The Ch'un Tsew, with the Tso Chuen. Reprint, SMC Publishing Inc.\n\nLegge, James. trans. 1963. The I Ching, 2nd ed. New York: Dover.\n\nLeibniz, Gottfried. 1703. Explication de l'arithm\u00e9tique binaire. Memoires de l'Acad\u00e9mie Royale des Sciences. 3(1703): 85\u201393.\n\nLi, Ling. 2002. Shangbo Chujian Sanpian Jiaodu Ji (Records of Proofreading on Three Pieces of Chu Bamboo Slips Published by Shanghai Museum). Taipei: Wanjuanlou Chubanche.\n\nLiao, Mingchun. \u5ed6\u540d\u6625 1999. Jing Men Guodian Chu Tomb Bamboo Slips and Pre-Qin Confucianism. In Guodian Chu jian yanjiu. Shenyang: Liaoning Jiaoyu Chubanshe.\n\nSchwartz, Benjamin. 1985. The world of thought in ancient China. Cambridge: Belknap Press (A recommendable intellectual history of ancient China with insightful comparisons to Western philosophy).\n\nShen, Dao \u614e\u5230. 1972. The Shenzi \u614e\u5b50. Reprint, Taipei: World Books.\n\nShima, Qian. 1976. Shiji \u53f2\u8a18. Reprint, Taipei: Donghua Publishing House.\n\nWilhelm, Richard, and Carr F. Baynes, trans. 1977. The I Ching: book of changes. Princeton: Princeton University Press.\n\nWilhelm, Hellmut, Richard, and Carr F. Baynes, trans. 1979. Understanding the I Ching: The Wilhelm lectures on the book of changes. Princeton: Princeton University Press.\n\nWu, Haoshen \u5433\u6d69\u5764, and Pan Yu \u6f58\u60a0. 1985. Zhongguo Giaguxue Shi \u4e2d\u570b\u7532\u9aa8\u5b78\u53f2 (A history of Chinese studies on turtle shells and bones). Shanghai: Shanghai Renmin Press.\n\nZhao, Jianwei \u8d99\u5efa\u5049. 2000. Chutu Jianbo Zhouyi Shuzheng \u51fa\u571f\u7c21\u5e1b\u300a\u5468\u6613\u300b\u758f\u8b49 (Exegesis of the unearthed silk text Zhouyi). Taipei: Wanjuanlou Publishing House.\n\nZhouli \u5468\u79ae. 1965. Duanju Shisanjing Jingwen \u65b7\u53e5\u5341\u4e09\u7d93\u7d93\u6587 (Punctuated texts of thirteen scriptures). Taipei: Kaiming Bookstore.\n\nZhu, Xi. 1999. Zhouyi Benyi \u5468\u6613\u672c\u7fa9 (The original meaning of Zhou Yi). Reprint, Taipei: Da'an Press (A convenient reprint of the philosophical and religious study of the Book of Changes by Zhu Xi with original texts).\n\nFootnotes\n\n1\n\nThe Chapter Hong Fan \u6d2a\u7bc4 (Grand Model) in the Shangshu \u5c1a\u66f8 (Book of Documents), and the Yijing \u6613\u7d93 (Classics of Changes) were seen by Thom\u00e9 H. Fang as the two major sources of Classical Confucianism, or Primordial Confucianism as he calls it, at its origin (Fang 1981: 2, 38, 539). The Yijing, though usually translated by scholars as Book of Changes in the past, is translated here as Classics of Changes, to transmit the meaning of the term jing as \"Classics.\"\n\n2\n\nAs evidenced by Wan-shun Jia's conversation with Confucius: \"It is better to pay homage to the spirit of the stove than to the spirits of the household shrine. What does this mean?\" The Master replied: \"It is not so. A person who offends against tian has no where to pray\" (Analects 3.13; Ames and Rosemont 1998: 85).\n\n3\n\nThese records can be seen in the early divinatory texts written on turtle shells and bones, see for example, Wu Haoshen and Pan Yu 1985: 86\u201387.\n\n4\n\nThere are many available Chinese editions of the Yijing. However, for the convenience of reference, I will use Zhu 1999.\n\n5\n\nFor the Chinese text of the Yijing, I use the convenient version of Zhu 1999. However, this version, like many other traditional versions, has the fault of anachronism or at least historical confusion by inserting the tuanci \u5f56\u8fad (Judgment Text) under the guaci \u5366\u8fad (hexagram dictum), and xiangci \u8c61\u8fad ( Symbolism Text) under yaoci \u723b\u8fad (line dictum). Now we know there must be at least more than seven hundred years' difference between them. In the following, all the English translations are of mine. I will not specify their pages.\n\n6\n\nThus, this method was used first only in the court of Zhou, before it became more popular in other states, and was last used by Qin state in the West and Chu State in the south.\n\n7\n\nThe silk text manuscript Yao \u8981 discovered at Mawangdui, Hunan Province, China in 1973 said, \"Kongzi is fond of Yijing when getting old. When he stays at home, the book Yijing is always on his mat; when he travels, it is in his hand bag. Zigong the disciple says to him, 'In the past you taught us that \"if one loses virtue whereupon ghost and spirit grow, or if one is far from wisdom and deep considerations, he practice divination more often.\" I thought this was correct, and made effort to live up to it. Why are you getting old to love it?'......\"Do you believe in its divination?\" Confucius says, \"Out of my hundred divinations seventy were fulfilled.... As to Yijing, I look behind the divination and regard more to its virtue and meaning\" (Zhao 2000: 276).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_3\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 3. The Philosophy of Confucius\n\nPeimin Ni1\n\n(1)\n\nGrand Valley State University, Michigan, USA\n\nPeimin Ni\n\nEmail: nip@gvsu.edu\n\nAbstract\n\nIf any single person has exerted more influence on the entire Chinese philosophical tradition than anyone else, it must be Kong Fuzi, better known in the West by the Latinized term, Confucius (551\u2013479 B.C.E.). While \"Kong\" is his family name, \"Fuzi,\" close in meaning to \"master,\" is a respectful way of addressing the man. His given name, Qiu, is rarely applied in references to him.\n\n## 1 Introduction\n\nIf any single person has exerted more influence on the entire Chinese philosophical tradition than anyone else, it must be Kong Fuzi, better known in the West by the Latinized term, Confucius (551\u2013479 B.C.E.). While \"Kong\" is his family name, \"Fuzi,\" close in meaning to \"master,\" is a respectful way of addressing the man. His given name, Qiu, is rarely applied in references to him.\n\nThough the time in which Confucius was born was a chaotic period, it was the golden age of the Chinese philosophical tradition. The glory of the early Zhou \u5468 Dynasty was declining, but still fresh in the minds of the people. The founders of the Zhou, King Wen and his son Duke Zhou, laid the foundation for a humanitarian government in emulation of the ancient sage-kings and refined the feudal ritual system. By the Spring and Autumn period (770\u2013476 B.C.E.), however, the social order of the Zhou was crumbling. It was against such a historical background that China had its most glorious period in philosophy. Not only were the founders of the two most influential philosophies in Chinese history, Confucianism and Daoism, born during the period, many other brilliant minds brought about the so-called \"hundred schools of thought,\" making the time comparable to ancient Greece in terms of the importance to their respective civilizations.\n\nConfucius was born near Qufu \u66f2\u961c, a town in the state of Lu \u9b6f in China known for its preservation of the early Zhou rituals and music. Little about his parents is known for certain. It is commonly believed that his father was a low ranking military officer, who died when Confucius was only 3 years old. His mother, from whom he received his primary education, raised him in a relatively humble situation. Confucius set his heart upon learning at the age of 15, and since that time he was a determined learner. He is believed to be the first person in the history of China to set up a school that offered, in today's term, \"liberal education\" in an institutional way. According to a likely exaggerated account, he had over 3,000 students during his life time, and 72 of them became conversant with the \"six Arts\" that he taught \u2013 ritual, music, archery, charioteering, writing, and arithmetic.\n\nConfucius considered himself a \"transmitter\" rather than a creator. According to him, the wisdom that he taught was already entailed in the ancient traditional rituals, the history, music, poetry, and the limited written works, which were, though corrupted over the ages, still largely available. He is believed to have edited some of the most basic Chinese classic books, including the Book of Rites, the Book of History, the Classics of Odes, the Classics of Music, and the Classics of Changes. Though his teachings are inseparable from these traditional works, Confucius is actually an innovative and even revolutionary thinker of the time. With deep understanding and insights about what was most valuable in the tradition, he creatively reconstructed the tradition and through him, the works became a set of canons that the later ages would abide by. But above all these other works, his own major teachings, recorded and collected by his students and their students into the book known as Lunyu \u8ad6\u8a9e (the Analects), later became the most sacred canon of all in traditional China.\n\nWith a strong sense of mission and ambition to bring the world into a harmonious order, Confucius spent a considerable amount of his life traveling around, often with his close disciples, trying to implement his humanistic ideas in political affairs. However, he was deeply disappointed with the rulers. Having survived some life-threatening situations, he finally returned to his home state Lu at the age of 68, and died 5 years later with no anticipation of his subsequent fame as China's first and foremost teacher, a supreme sage, and a \"king without a crown.\"\n\nHis teachings were carried on and developed by the persistent effort of his followers, especially Meng Zi \u5b5f\u5b50 (Latinized as \"Mencius,\" 390\u2013305 B.C.E.) and Xun Zi \u8340\u5b50 (325\u2013238 B.C.E.). Tested against the rival schools of thought and having endured sweeping attacks from the First Emperor of Qin Dynasty (reigned from 221\u2013209 B.C.E.), Confucianism was finally recognized as the official state ideology during the Han Dynasty around 100 B.C.E. Since then, though with ups and downs, Confucianism enjoyed the status of being China's principles of morality, of law, of government, of education, and of life in general, which everyone was supposed to follow, from the emperor down to the ordinary people. During the Song and Ming Dynasties (roughly from the tenth \u2013 thirteenth century and the fourteenth \u2013 seventeenth century respectively), Confucian scholars brought another upsurge of Confucianism by their creative interpretations of it in response to the challenges from its strong rivals, Daoism and Buddhism, and further consolidated its dominant position in China. The status of being an official state ideology, however, was not simply a blessing to Confucianism, for once endorsed as an official doctrine it could hardly avoid being dogmatized and became alienated as a means for political advantages. Most of the repressive practices of the feudalistic China were conducted under the name of Confucianism, though in most cases they were contrary to the real spirit of the Master's own teachings.\n\nAt the beginning of the twentieth century, with the arrival of Western material powers and ideas in China, Confucianism was criticized and considered to be responsible for all that was backward and benighted in China. In the recent decades, however, a rival voice has been growing stronger in scholarly circles and public media. Accompanied by the success of the four \"small dragons\" in Asia \u2013 Hong Kong, Singapore, South Korea, and Taiwan, where Confucianism retained its strong hold, the interest in Confucianism revived. It is characterized by Mou Zongshan and other Confucian scholars such as Tu Weiming as the \"Third Epoch of Confucianism,\" succeeding the first one, represented by Confucius himself and his immediate followers such as Meng Zi and Xun Zi, and the second, represented by the Song and Ming Confucians. With a critical re-appropriation and transformation, many contemporary Confucian scholars believe that Confucianism can provide valuable philosophical resources for addressing disturbing situations in the post-modern world.\n\nSince the philosophical ideas in the Analects, the \"Bible\" of Confucianism, are contained in excerpts put together with no apparent logical order and with little articulation, they need to be unpacked and re-organized so that contemporary readers who are accustomed to the discourse style of writing in philosophy have less difficulty in assessing and appreciating them. Doing this, however, involves the risk of cannibalizing the dynamic system, in which the passages are related like different sides of a crystal that reflect the lights of each other. Readers of this article are therefore cautioned to pay special attention to the mutual entailment of the ideas, and not to treat them simply as distinct parts of a system.1\n\n## 2  Tian \u5929\u2013 Heaven\n\nThe word \"Confucianism\" is unknown to most Chinese, because in China the school is referred to as \"ru jia \u5112\u5bb6\" \u2013 the school of ru, where \"ru\" refers not to Confucius, but to the practices and the way of life most distinctively represented by him. There is neither dependency on deity worship nor priesthood in Confucianism, and Confucius himself is deemed as a model human being rather than a god. There are indeed Confucian temples all over China, but they are more like memorials than monasteries. This significant fact shows that Confucianism is different from most religions and is less likely to develop into fundamentalism.\n\nConfucius' own attitude toward issues regarding deities and life after death is partly skeptical and partly pragmatic. He says, \"To say you know when you know, and to say you do not when you do not. That is knowledge\/wisdom\" (2\/17). It was said of him, \"The Master did not talk about strange phenomena... or supernatural beings\" (7\/21). His advice is to \"Keep a distance from supernatural beings while showing them due reverence\" (6\/22). \"If you are not yet able to serve other people, how can you serve supernatural beings?\" \"If you do not yet understand life, how can you understand death?\" (11\/12) It is evident that the Master did not conceal his lack of knowledge about matters related to supernatural beings or to life after death. He refrained from speculating or conjecturing about things that he had no knowledge about. At the same time, he remained open to the possibility that there might be deities. When Confucius offered sacrifice to his ancestors or to other spirits, he did it as if the spirits were actually present. He said, \"If I am not fully present in the sacrifice, it is no different from not having done the sacrifice\" (3\/12).\n\nConfucius' focus is always on this life and this world. When his disciple Zigong asked whether those who were dead had consciousness, the Master is reported to have responded with these remarks:\n\n> If I were to say that they do have awareness, I am afraid that those who are filial to their parents and grandparents would send off the dead ones as if they were alive [and hence have lavish burials]. If I were to say that they don't have awareness, I am afraid that those who are not filial would discard the dead ones unburied. Ci [Zigong], do you want to know whether people have awareness after they die? When you die, you will eventually know. It will not be too late to know by then. (Sun and Guo 1998: 21)\n\nThe response interestingly contains no direct answer to Zigong's question about the afterlife. All that concerns him is how those who are alive would behave. It seems that for the Master, as long as one lives a decent human life in this world, one will have nothing to regret, whether there is an afterlife or not. This attitude is also reflected in another passage in the Analects. When his disciple Zai Wo inquired, \"The 3 year mourning period on the death of one's parents is already too long,... surely a year is enough,\" the Master replied, \"Would you then feel at ease [an \u5b89] eating fine rice and wearing colorful brocade?\" (17\/21) Clearly the Master's concern is not so much about pleasing the spirits, much less about knowledge pertaining to the existence of the afterlife. His concern is about the appropriateness of one's own feelings and dispositions.\n\nConfucius' this-worldly attitude does not mean that there is no spiritual dimension to his thought. He firmly believes that his mission is bestowed by tian \u5929, usually translated as \"heaven\" (9\/5). For this mission, one should be determined to travel a journey that is not supposed to end before one's death (8\/7), and the aim is even more important than life itself (15\/9). Scholars have varying interpretations of the Confucian notion of tian with regard to whether it is personal or impersonal, transcendent or immanent. While some have gone so far as to claim that it resembles the Christian notion of a personal and transcendental God, others claim that it is entirely impersonal and immanent, no other than the natural order of the universe displayed through the change of the seasons and dynasties and the like. Still others hold opinions somewhere in between, maintaining that the Confucian tian is \"immanently transcendent.\"2 While these conflicting interpretations reflect the ambiguity in Confucius' notion of tian, scholars generally agree that it is important to look at the notion's historical emergence. Tian is a notion that Confucius' early Zhou predecessors used to replace and to depersonalize the Shang Dynasty notion of Shang Di \u4e0a\u5e1d, \"Lord-on-High.\" Even though the notion of heaven found in the Zhou literature such as Shu Jing \u66f8\u7d93 (the Book of History) and in the Analects still carries with it the sense of a being that governs worldly affairs, it already showed itself in the realm of this world rather than being entirely transcendent. The will of heaven was no longer considered so much as the will of an anthropomorphic deity that issues orders and gives blessings and sanctions from above; it immanently exhibited itself in popular consensus and in regular patterns of discernible social and natural events, and it could be affected by the moral undertakings of the people. From a passage in the Shu Jing that says \"Heaven sees through the eyes of the people, heaven listens through the ears of the people,\" we can see that what appears to be anthropomorphic here is rather more anthropogenic. In the Analects, Confucius is quoted as saying \"Does heaven speak? And yet the four seasons turn and the myriad things are born and grow within it\" (17\/19). Most scholars take this passage as evidence that for Confucius, heaven is the principle according to which natural events take place. The same philosophical implication is contained in the Classics of Changes. Divination based on reading the pattern on tortoise shells after they are heated, or yellow stalks after they are scattered, entails the belief that everything in the universe is governed by the same principle, whether in an intentional or in a purely naturalistic way. The Classics of Changes also entails the view that humans can affect their destiny through their own activities. It tells people not only what situation they are facing, but also, given the specific situation, what kind of action should be taken. Under such a notion, rulers were considered sacred only so long as they were able to continue to be \"entrusted\" with tian ming \u5929\u547d, the mandate of heaven.\n\nBecause heaven displays itself through worldly phenomena, it is possible for humans to know its mandate. Confucius says: \"At the age of 50, I knew tian ming\" (2\/4). He did not explain specifically how he came to know it, but he shows a strong confidence that heaven has bestowed virtue\/virtuosity upon him (7\/23). Since the word for virtue\/virtuosity, de \u5fb7, also means power, his confidence might be derived from his faith in the power given to him by heaven. Even though otherwise heaven often appeared to be at odds with him, the Master believed that \"It is human who are able to make the Way great, not the Way that can make human great\" (15\/29). This statement also sheds light on the notion of \"the Way\" (dao \u9053), because it suggests that the Way is a trajectory, a mode of acting, which is itself road building, rather than a metaphysical entity that is purely objective and external to human conduct.\n\nThis feature of the Confucian spirituality is naturally accompanied by a strong sense of anxiety and responsibility, a realization of a close interdependence between people's fortunes and their own conduct. Not only do people have to rely on themselves for their own fortune, they also have to take the responsibility of affecting the fortune of other people, depending on how great their influences can be. The more power one has, the more one is accountable. As contemporary Confucian scholar Xu Fuguan \u5f90\u5fa9\u89c0 says, in other religions piety is a state of the mind when one dissolves one's own subjectivity and throws oneself entirely before God, yet in the Confucian spirituality, as the embodiment of heaven, human subjectivity becomes highly concentrated and piety becomes staying sincere to one's own responsibility (Xu 1984: 22). The Analects records that Yao, an ancient sage King, said to his successor: \"On \u2013 you Shun! The line of succession conferred by heaven rests on your person. Grasp it sincerely and without deviation. If all within the four seas sink into dire straits, the honors bestowed on you by heaven will be terminated forever.\" King Tang of the Shang Dynasty said in a sacrificial ceremony, \"If I personally do wrong, let not the 10,000 states be implicated; if the 10,000 states do wrong, the guilt lies with me alone!\" (20\/1) This sense of subjectivity is well displayed in the well-known passage by the Song Dynasty Confucian Zhang Zai \u5f35\u8f09:\n\n> To establish heart-mind for heaven and the Earth,\n> \n> To shape destiny (ming \u547d) for the common people;\n> \n> To revive the lost scholarship for ancient sages,\n> \n> To generate peace for ten-thousand generations to come.\n\nThe word ming \u547d, a term usually translated as \"destiny\" or \"fate,\" also appears many times in the Analects. Since the word also means decree or mandate, and is sometimes used as an abbreviation for tian ming, it is easy to confuse the two. Ming is different from the mandate of heaven in that the latter is more of a moral imperative, and the former is more of a definite order or sequence of certain phenomena. For instance, when a natural event happens beyond a person's control, it would be considered ming (see, for example, 6\/10 and 12\/5). Similarly, whether the Way eventually prevails or not, given the human efforts involved, is determined by ming (14\/36). Since heaven is more like nature that is beyond personal control, so what is by ming is also by tian, heaven.\n\nConfucius fully recognizes the fact that, even though humans do not have full control of everything, we can make great differences to our life within certain limits. On the one hand, as Mencius puts it, \"he who understands ming does not stand under a wall on the verge of collapse\" (Mencius: 7A\/2). On the other hand, one can actively change one's ming by various means. For example, the Analects tells us that when Confucius' disciple Sima Niu \u53f8\u99ac\u725b lamented, \"Everyone has brothers except me,\" another disciple Zixia \u5b50\u590f said to him: \"Life and death are a matter of ming (destiny); Wealth and honor lie with heaven. The exemplary person is deferential and faultless, respectful of others and refined, and everyone in the world is his brother. Why would the exemplary person worry about not having brothers?\" (12\/5) By becoming a morally exemplary person and redefining what it means to have brothers, Zixia showed Sima Niu how he could gain control of his ming, or minging (ordering) his ming (destiny)!\n\nEven those who take tian to be entirely immanent would not take Confucianism as simply a secular humanism that is complicit with the status quo, and hence would not reject the view that Confucius advocates transcendence in the sense of going beyond the status quo or the surface appearance. The spiritual aim of Confucius is often characterized in the phrase \"the unity of heaven and human.\" Compared to the Christian aim of going to Heaven and being united with God, the transcendental creator, but not becoming God oneself, the Confucian unity is to become sacred oneself through the unity and it is achieved through one's relatedness with other people in this world. Confucius never felt that a lack of personal immortality would lead to the lack of meaning for life, for a life may go infinitely beyond its narrow, personal biological span. A common application of this broad notion of immortality is the continuation of one's family line, which Confucians also honor. Among the things that are considered bad to do to one's parents, the worst is to have no heir, says Mencius (Mencius: 4A\/26). Another bad thing to do to one's parents is to behave immorally and to make the parents feel ashamed. Both of these \"bad things\" done to one's parents need to be understood as an extension of the parents' own immediate personal state of existence to the way they \"exist\" in others. That Chinese people take how others regard them very seriously, and that they expect their children to bring honor to their family and community, are clear indications of their conception of themselves. Their existence extends to the lives of other people. Confucius' saying that \"If for three years [of mourning] one does not change from the way of his father, he may be called filial\" (1\/11) can also be understood in this light, for it is a way in which the father continues to be \"alive.\" Confucius may very well be aware of the notion of \"Three Immortalities\" that existed in his home state, Lu, according to which one can become immortal by establishing words for others to keep in mind, achievements for others to benefit from, and virtue\/virtuosity for others to follow (see Chan 1963: 13). All these three immortalities are achieved relationally through continued existence and manifestation of one's efficacy in the community and in history rather than personal survival after death. Confucius' endorsement of this idea is indicated in the Analects. He says, for example, \"Exemplary persons despise the thought of ending their days without having their names properly established\" (15\/20). In the context of his overall teachings, one's \"name\" clearly relates to how one continues to \"live\" in the lives of others, particularly through the three ways mentioned above (see, e.g. 16\/12).\n\nObviously this kind of immortality, or the unity between heaven and human, is not guaranteed or dependent on the mercy of a deity and is not attained in another world. It is dependent on oneself in this life and this world. It is therefore, as Herbert Fingarette puts it as the title of his influential book on Confucianism, \"secular as sacred\" (see Fingarette 1972). But dependence on oneself does not mean solitary effort. One needs to pursue it in the course of one's life within the community. To be more specific, this is to be ren and practice li.\n\n## 3  Ren \u4ec1 \u2013 Human-Heartedness\n\nCentral to Confucius' philosophy, the word \"ren \u4ec1\" appears frequently in the Analects. Yet nowhere can one find a precise definition of it, nor have translators reached agreement on any satisfying English translation. Whether one takes it as \"benevolence,\" \"human-heartedness,\" \"authoritative person\/conduct,\" \"altruism,\" \"humanity,\" or \"goodness,\" there always seems to be something that is left out, or something not quite fit that would creep in. Ren seems to be an ideal that even the Master himself claimed never to have fully reached (7\/34), and yet it is so close to everyone that the Master says, \"Is ren far away? No sooner do I seek it than it has arrived\" (7\/30).\n\nSome observations, however, can help us to get a good grasp of it. First, as \"ren\" is occasionally used in Confucian texts interchangeably with \"human\" or \"person,\" \u2013 ren \u4eba \u2013 which is, in Chinese, homophonic to it (see Mencius: 1B\/15 and Zhongyong: Chap. 20), we have reason to believe that \"the distinction between the two terms [\u4eba and \u4ec1] must be qualitative: two distinguishable degrees of what it means to be a person\" (Hall and Ames 1987: 114). As in English, \"human\" is sometimes used in Chinese to carry the moral expectations for being a human, and thereby making the expression \"a human should be like a human\" more than a simple tautology. For this reason \"ren \u4ec1\" can be interpreted as a quality that makes a person an authentic human being, which every biological human should strive toward.\n\nSecond, the Chinese character \"ren \u4ec1\" consists of two parts, \"person \u4eba\" and the number \"two \u4e8c.\" This etymological analysis, say Ames and Rosemont, \"underscores the Confucian assumption that one cannot become a person by oneself \u2013 we are, from our inchoate beginnings, irreducibly social\" (Ames and Rosemont 1998: 48). Indeed, many descriptions of ren in the Analects are about interpersonal relations. \"Ren\" is to \"love people\" (12\/22), says the Master, and the method to be ren is \"shu \u6055\" \u2013 comparing one's own heart with other hearts with compassion (6\/30). The interrelatedness of a person is so important to Confucius that some contemporary Confucian scholars believe that for Confucius, one does not play the roles of being a father, a friend, a teacher, etc.; one is these roles (see, for instance, Rosemont 1991: 72). They argue that the Confucian concept of a person is different from those who take humans as atomic, autonomous individuals.3 Ren is essentially relational, and the ren person defers to his or her relationship and interaction with \"the other\" for the completion of oneself. No one can become fully human in isolation, nor can one say that what happens to others has nothing to do with him or herself. This relational dimension in the concept of the individual serves as an important philosophical foundation not only for the social and political philosophy of Confucius, but also for his ideas of religiosity or spirituality.\n\nThird, we observe that, while Confucius never offered any definition of ren, he gave different answers to different disciples when they asked about it. This would indeed be puzzling and confusing if ren were understood as a concept to be grasped by the intellect. However, it would be totally understandable if it is more like an art or a disposition that needs to be mastered, embodied, and displayed in one's life, including in one's gestures and manners. The Master's different answers should be seen as practical instructions offered in accordance to each of the disciple's particular needs for their attainment of ren.\n\nThese three observations lead us to the conclusion that ren is a quality pertaining to one's caring disposition toward others that has to be developed and fully embodied before a biological person can become an authentic human being. Ren is more associated with one's heart and the whole bodily disposition than with rational knowledge or decisions. It must be embodied, and not merely understood and followed as a universal principle or imperative. For this reason, it is a matter of cultivation rather than gaining propositional knowledge, and by its nature unsuitable for conceptual formulation.\n\nTwice when asked about ren, Confucius answers \"Do not impose upon others what you yourself do not want\" (12\/2, 15\/24). It is known as a negative version of the \"Golden Rule,\" \u2013 it is \"golden\" because it seems to capture what is moral at a substantial level, and can be considered a general rule of conduct after which other rules would follow; it is \"negative\" because it tells people what not to do. Actually Confucius provides a positive version of it also, for he says: \"If you want to establish yourself, establish others. If you want to promote yourself, promote others\" (6\/30). However, we must be clear that Confucius never took it as a rule, much less \"Golden.\" He \"rejects... inflexibility and rigidity\" (9\/4, see also 15\/37), and he states clearly that a morally exemplary person, jun zi \u541b\u5b50, \"is never for or against anything invariably. He is always on the side of the appropriate\" (4\/10). Confucius himself was characterized as a sage who acted according to circumstances rather than rules (Mencius: 5B\/1). The art of flexibility is deemed by the Master so highly that he says, to find a partner good enough in the exercise of quan \u6b0a (discretion) is more difficult than finding a partner good enough in taking a stand or in the pursuit of the Way (9\/30). The word \"quan\" originally means \"scale,\" and thus the action of \"weighing\" or \"making discretion\" as well. According to Gong Yang Zhuan \u526c\u7f8a\u50b3, a Chinese classic dated probably to the Warring State period (476\u2013221 BCE), \"quan means moral goodness resulting from transgressing well-established canons\" (\"The 11th Year of the Duke of Heng\"). Indeed, to take the teachings as the \"Golden Rule\" would result in the problem of basing rights and wrongs on personal unqualified likes and dislikes, and thereby subject to counterexamples that are contrary to our moral intuitions. For a person who likes to be bribed, the Golden Rule (positive version) would not only permit him to bribe others; it would obligate him to do so. For a judge who does not like to be put in jail, the Golden Rule (negative version) would make the criminal justified in disputing such a punishment. Confucius is not at all uncritical about one's desires and wants. In fact one of Confucius' major descriptions of ren is \"To restrain the self\" (12\/1). For Confucius, cultivating one's heart-mind is almost a life-long journey. He says that he himself did not reach the state where he could follow his heart's desire without overstepping the line until the age of 70 (2\/4).\n\nHow should we take the Confucian \"Golden Rule,\" if it is not meant to be a rule at all? Let us look at the contexts where the statements are found. When a disciple asked, \"Is there one expression that can be acted upon throughout one's entire life?\" the Master replied, \"There is shu \u6055. Do not impose on others what you yourself do not want\" (15\/24). In another passage, right after the statement of his positive version of the \"Golden Rule,\" the Master says: \"Taking an analogy near at hand is the method of becoming ren\" (6\/30). The word \"shu\" consists of two parts, the upper part \"ru \u5982\" means \"like,\" \"as if,\" \"resemble,\" and the lower part, \"xin \u5fc3,\" means \"heart-mind.\" This etymological analysis helps us to understand that for Confucius, the application of the \"Golden Rule\" is to take one's own heart as an analogy near at hand, and to extend one's considerations to the wants and needs of others empathically.\n\nAccording to such a reading, the \"Golden Rule\" statements in the Analects should be read as no different from \"shu,\" a method to be ren. Even though there is no guarantee that the method will lead to right actions all the time, it helps a person to become sensitive to the interests of others. Unlike a rule which allows no exceptions and is typically imposed upon the agent as an obligation and proscribing certain acts, a method is mastered by the agent, enables the agent to perform the right action, and is certainly not to be used when its application is unwarranted.\n\nThis leads us back to a notion we mentioned earlier \u2013 de \u5fb7, or virtue\/virtuosity. Often, ren is considered as a virtue in Confucianism, and due to its central place, Confucianism is taken as a version of virtue ethics comparable to Aristotle's, since both of them focus on building the moral agent, and not on formulating rules of conduct. Both Aristotle's aret\u00e9 (virtue) and Confucian de are dispositions or abilities required for living an excellent life, and both need to be embodied through constant practice so that they become almost like our second nature. Both philosophers agree that the virtuous person possesses the ability to discern particularities in individual situations that are not subject to formulations of rules. However, there are important differences between the two. Based on a teleological metaphysics, the Aristotelian virtue is a matter of moral obligation for people to develop for the sake of fulfilling the pre-established telos (aim) of a human being. The Confucian de, on the other hand, is more a power or an art that enables a person to develop their human potential creatively as authors of their own life.\n\nAnother important difference between Aristotle and Confucius is that Aristotle places intellectual virtue at the center of his theory, taking rationality to be a defining feature of being a human and contemplation to be the most distinctive human activity. The Confucian de, on the other hand, centers on the affective aspect of caring and loving, and Confucius takes the cultivation and manifestation of proper emotions and attitudes to be more important for a human than the use of the intellect. Confucius characterizes ren as to \"love the people\" (12\/22). Even though this love is similar to moral duty in that, unlike the way a mouse loves rice, it extends beyond personal interests, it differs from moral duty in that love must be out of the heart, and not merely out of the rational faculty of the mind. Whether in daily life or in governmental affairs, a ren person is always considerate and has others' interests at heart. In running a government, the ren ruler \"is frugal in his expenditures and loves his subordinates, and puts the common people to work only at the proper season of the year\" (1\/5). In daily life, a ren person \"loves the multitude broadly\" (1\/6). She \"does not exploit others' fondness of her, nor does she exhaust others' devotion to her\" (Book of Rites, Chap. 1. 1987: 11). She \"does not intimidate others by showing off her own talent, nor belittle others by revealing their shortcomings\" (Book of Rites, Chap. 32. 1987: 293).\n\nJust like the \"Golden Rule,\" love by itself does not guarantee that what it dictates is always morally right. One of the Confucian qualifications for a proper love is to love with distinction. The Confucian understanding of relatedness is reciprocal but not symmetrical. It starts locally and concretely from family love and extends outward without limit. The abstract idea of everyone being equal is therefore not only foreign to Confucius, but would also be considered misleading. Confucius differentiates according to relationships and social roles. \"When his stables caught fire, the Master hurried back from court and asked, 'Was anyone hurt?' He did not inquire after the horses\" (10\/17). It does not mean that he cared nothing about animals. \"The Master fished with a line, but did not use a net; he used an arrow and line, but did not shoot at roosting birds\" (7\/27). It only means that in comparison, we are closer in relation to our fellow human beings. Among humans, he also believes that one should start with loving one's own parents and gradually extend the love to others according to the degrees of closeness in relations. Not only is it more natural for us to care more about those who we consider to be one of \"us,\" it is also the starting point for extended love. The Analects states clearly that filial piety (xiao \u5b5d) is the very root of a proper social order. \"The morally exemplary person concentrates his efforts on the root; for the root having taken hold, the Way will grow therefrom. Aren't filial piety and fraternal deference the roots of becoming human-hearted indeed?\" (1\/2) Here we find the love to be both a characteristic of the ren person and a method of becoming ren. By practicing ren, ren grows. If we do not start our love from the immediate context of our life and with those who we immediately encounter, it will not start at all.\n\nThis methodological function of filial piety shows that it should not be simply taken as a moral imperative or principle. Confucius' endorsement of his fellow villagers' way of dealing with their own family members' misconduct by mutual concealment (13\/18) becomes a defense of injustice if we take it to be an ethical principle, but it becomes sacrificing a branch for the sake of saving the root when we treat it as a method of becoming ren.4 The Confucian strategy is, as Mencius puts it beautifully, to \"Treat the aged of your own family in a manner befitting their venerable age and extend this treatment to the aged of other families; treat your own young in a manner befitting their tender age and extend this to the young of other families, and you can roll the Empire on your palm\" (Mencius: 1A\/7).\n\nConfucius also differentiates love according to circumstances. He would rather help the needy than make the rich richer (6\/4). Unlike \"an eye for an eye and a tooth for a tooth\" or Jesus Christ's exhortation to turn the other cheek, Confucius repays ill will with zhi \u76f4, uprightness or straightforwardness, or, taking the word zhi as a verb, \"to straighten,\" \"to correct,\" or \"to help grow.\" If you repay ill will with kindness, says the Master, \"then how would you repay kindness?\" (14\/34. See also 19\/3). The proactive attitude of helping the wrong-doer to correct the wrong provides us with a thought-provoking alternative to merely sticking on social justice or to \"love your enemy\" by letting them continue to do harm.\n\nRen is also what makes a person worthy of respect and the basis for respecting others. Unlike the Kantian who conceives humans as ends in themselves because, as rational beings, humans can make free choices and hence are the source of all values, or the Christian who believes that humans have dignity because we are made in the image of God, for Confucius, human dignity is more an achievement than a natural quality given by nature or by God. For Confucius, one earns respect from others by being respectful oneself. \"If one is respectful, one will not suffer insult\" (17\/6, 1\/13). Confucius says, \"The exemplary person does not speak more than what he can accomplish, and does not behave across the line of proper conduct, people revere him without being forced to\" (Book of Rites, Chap. 27. 1987: 277). The respect one deserves is therefore in proportion to the wellness of one's cultivation. It does not mean that for a well cultivated person, no one will initiate insult, but rather that the insult will only display the insulter's own lack of humanity. When someone spoke disparagingly of Confucius, Zigong made the following remarks:\n\n> This is in vain. Confucius would not be hurt. The superior character of other people is like a mount or a hill which can still be stepped on, but Confucius is the sun and moon which no one can climb beyond. When people cut themselves off from the sun and moon, what damage does this do to the sun and moon? It would only demonstrate that such people do not know their own limits. (19\/24)\n\nThis teaching reminds everyone to cultivate themselves and not to disrespect others, even those who are not well cultivated. Repeatedly Confucius reminds his students to set strict standards for themselves and to be lenient to others (see 15\/15, 4\/14, 14\/30, 15\/19, and 15\/21). Viewed from this perspective, one's respect for others is more a requirement for one's own humanity than a moral act based on judging others.\n\nNo passage in the Analects embarrasses contemporary Confucians more than 17\/25, where Confucius says, \"It is only n\u00fc zi \u5973\u5b50 (typically taken to mean 'women') and petty persons who are difficult to provide for. Drawing them close, they are immodest, and keeping them at a distance, they complain.\" Modern advocates of Confucianism, like their opponents, take the saying to be descriptive, and denounce it as sexist. Putting aside whether the term \"n\u00fc zi\" refers to women in general or only to female servants, as some scholars have speculated (since at the time \"fu ren \u5a66\u4eba\" was a more common term for women in general),5 few have noticed the instructive message beneath the surface that can be found by relating it to other passages in the Analects: Can one be like the Master, as described by his disciples, \"respectful and yet at ease\" and \"commanding but not ferocious\" (7\/38), so that even the most difficult to provide for are \"pleased, if close, and attracted, if at a distance\" (13\/16)? Reading the passage as a reminder to cultivate oneself rather than a description of fact, this passage looks more consistent with the rest of the book and the overall spirit of Confucianism.\n\n## 4  Yi \u7fa9 and Li \u79ae \u2013 Appropriateness and Ritual Propriety\n\nWhile ren is the internal quality or disposition that makes a person an authentic person, yi \u7fa9 is the appropriateness of actions that typically originates from ren. For being appropriate in one's action, however, one also needs the guidance of li \u79ae, ritual propriety, for otherwise a person of ren can still go astray. \"Not being mediated by the observance of the ritual propriety, in being respectful a person will wear himself out, in being cautious he will be timid, in being bold he will be unruly, and in being forthright he will be rude\" (8\/2).\n\nThe word li originally meant holy ritual or sacrificial ceremony, and it is used by Confucius to mean more broadly behavior patterns established and accepted as appropriate through the history by a community, including what we call manners, etiquette, ceremonies, customs, rules of propriety, etc. The metaphor of holy ritual serves as a reminder that the most ordinary activities in our life can also be ritualistic or ceremonial, and it is the ceremonial that sets human activities apart from those of animals. The way we greet each other, a handshake, for instance, is ritualistic, for it is not a mere physical touching of hands. We stand up to greet our guests, and walk them to the door as they leave. These are rituals because, from the point of view of efficiency, they can be spared in most cases. We address people in a certain manner, though practically, calling someone's attention can be done in many other ways. We even look at each other according to some implicit rules \u2013 at a very young age we start to learn that it is impolite to stare at people, especially at certain parts of their body. When a disciple asked about filial conduct, the Master replied: \"Those who are called filial today are considered so because they are able to provide for their parents. But even dogs and horses are given that much care. If you do not respect your parents, what is the difference?\" (2\/7) By serving food and dining with respect and appreciation in a proper setting, mere physical nourishment becomes a ceremony, and thereby becomes human.\n\nIndeed the learning is basic but not at all primitive. As any \"zhi \u8cea\" (basic stuff, substance, inside principles, or characters) needs certain \"wen \u6587\" (refined pattern, form, style, and outside appearance) to exist and unfold, appropriateness (yi) needs ritual propriety (li) to be its form of expression. \"An exemplary person takes appropriateness as the zhi [substance of his conduct], and carries it out in the form of ritual propriety\" (15\/18). \"When there is a preponderance of zhi over wen, the result will be churlishness; when there is a preponderance of wen over zhi, the result will be pedantry. Only a well-balanced admixture of these two will result in an exemplary person\" (6\/18). When someone said, \"Exemplary persons should focus on the substance, what do they need refined form for?\" Confucius' disciple Zigong replied, \"Refined form is no different from substance; substance is no different from refined form. The skin of a tiger or a leopard, shorn of hair, is no different from that of a dog or a sheep\" (12\/8). According to this interpretation, the refined form and the substance of an exemplary person are so closely connected to each other that not only should they match each other, without one the other cannot exist! An appropriate conduct must have a refined ritual form, otherwise it would not be appropriate; and a ritual form cannot be considered refined unless it is appropriate.\n\nAs the Master says, \"In referring time and again to observing ritual propriety, how could I just be talking about the presence of jade and silk?\" (17\/11) Learning rituals is no different from learning to be a human. Humans are like raw materials \u2013 they need to be carved, chiseled, grounded, and polished to become authentic persons (1\/15). Learning ritual propriety is such a process. By practicing ritual propriety, a person can be transformed and established (8\/8). Most people learn their basic moral lessons at a young age, not by studying Kantian formulations of categorical imperative or utilitarian calculations, but by repeated use of rituals such as saying \"thank you,\" which increases one's sense of appreciation, and \"I am sorry,\" which increases one's sensitivity to others' pain. Not only does the skill of dealing with sophisticated human relationships have to be learned from actual life, and in this sense it is not merely a matter of remembering what is right and wrong in the brain, the habit of rituals is not something extrinsic to the person, but the result of transformation of the person. Through the process not only does one know what is expressive of humanity but one also becomes an excellent human. Only from this perspective can we understand and appreciate the passages in the Analects that give detailed accounts of how the Master greeted his guests, dressed himself, ate, sat, etc. The subtlety and complexity of the coordinated ritual acts are certainly beyond what can be capsulated in any abstract principle. This matter of ethical importance, nicely captured by Joel Kupperman as the \"style\" of life, is unfortunately neglected by most Western ethical theories (see Kupperman 1999: 26\u201335).\n\nIn his influential work, Confucius \u2013 the Secular as Sacred, Herbert Fingarette articulates the significance of the Confucian teachings about ritual propriety by referring to J. L. Austin's work in the early 1960s on a class of linguistic actions called \"performative utterances.\"6 Fingarette says that the lesson of Austin's work is not so much about language as it is about ceremony, for all the performative uses of language are ceremonies or rituals or they are nothing. They cannot be done out of ritual context. \"No purely physical motion is a promise; no word alone, independent of ceremonial context, circumstances and roles can be a promise.\" \"In short, the peculiarly moral yet binding power of ceremonial gesture and word cannot be abstracted from or used in isolation from ceremony. It is not a distinctive power we happen to use in ceremony; it is the power of ceremony\" (Fingarette 1972: 12, 14). In an apparently puzzling conversation, Confucius compared his disciple Zigong to a sacrificial vase of jade (5\/4). According to Fingarette, the passage reveals the profound significance of ritual propriety, for the vessel's sacredness does not reside in the preciousness of its beauty; it is sacred \"because it is a constitutive element in the ceremony.... By analogy, Confucius may be taken to imply that the individual human being, too, has ultimate dignity, sacred dignity by virtue of his role in rite, in ceremony, in li\" (Fingarette 1972: 75). Outside of a certain ritual setting, a vessel would not be sacred, no matter how beautiful it is. Of course the analogy should not be taken so far as to mean that one can be sacred by merely being in a ritual setting without personal cultivation and active participation in the ritual. It is typical that before a holy ritual takes place, the participant has to take a shower, fast for a certain period of time, and meditate for a while to calm the mind and make the will sincere. If placement alone was enough to confer the holiness, all those processes would be simply meaningless.\n\nAs all ritual practices involve encountering others, the passage also implies that no one can become sacred in solitude. It is through ritual propriety that social activities and human relations are coordinated in a civilized, and hence, sacred way. The Master says, \"Achieving harmony is the most valuable function of observing ritual propriety\" (1\/12). Harmony (he \u548c) is different from conformity (tong \u540c). \"The exemplary person pursues harmony rather than conformity; the petty minded is the opposite\" (13\/23). Just like simply adding water to water would not make a delicious broth, or playing the same note would not make beautiful music, diversity is a precondition for harmony. While the parts forced into conformity are in agreement with each other at the cost of their individuality, the parts of a harmonious whole mutually enhance each other without sacrificing their uniqueness. People in harmonious relations participate in social activities and construction. They are not merely constituents of them.\n\nOf course, harmony also presupposes something fundamental that the participants will have to agree upon. Confucius made it clear that \"People who have chosen different Ways (dao) cannot make plans together\" (15\/40). The harmony of a broth is dependent on the condition that no ingredient is sharply at odds with other ingredients. This requirement would be very difficult to meet and problematic if the universal agreement is expected to be in the form of principles stated in words. The principles often tend to be either too rigid to allow for differences and creativity or too thin (abstract) to be practically meaningful. The practice of ritual propriety, however, is ambiguous and leaves maximum space for uniqueness and creativity. A handshake in itself does not specify what is agreed upon, and yet a certain trust and mutual recognition can be established through it. Not only can the meaning carried by a handshake be richer than any agreement on a principle, it will not lose mutuality for the sake of having an agreement, nor will it lack emotional content for the sake of retaining rationality.\n\nThe ritual order envisioned by Confucius implies social hierarchy, and for this reason it is often criticized as elitist and opposed to human equality. But it is supposed to be one of reciprocity and deference to excellence rather than a rigid order of aristocracy. The Master is himself a role model: \"At court, when speaking with lower officials, he was congenial, and when speaking with higher officials, straightforward yet respectful. In the presence of his lord, he was reverent though composed\" (10\/2). In these rituals the social roles and relationships are confirmed and communicated, the responsibilities that come with them are accepted, and human-heartedness is displayed. Those who are in superordinate positions enjoy more status of authority while bearing more responsibility, and those who are in subordinate positions enjoy more protection and less responsibility but are expected to return the former with respect and wholehearted devotion (zhong \u5fe0). This does not mean that the subordinates will always have to obey their superiors. Confucius says, \"In political matters, if no one says anything different from what their ruler says, it is fine when what the ruler says is good, but it would ruin a state if what the ruler says is not good\" (13\/15). Even though he did not have a concept of modern democracy, he believes that:\n\n> when a state has subjects who dare to stand up and appeal to the King, it will not be in danger. When a father has a son who dares to speak up, he will not deviate from ritual propriety. When a person has friends who dare to speak up, he will not take inappropriate action. Hence how can we say that the son is filial, if he obeys whatever his father says? How can a subject be considered whole-heartedly devoted if he follows whatever his King orders? Those who examine what they are expected to follow, that is what filiation and whole-hearted devotion means. (Xun Zi 1967: Hsun Tzu, Chap. 29)\n\nConfucius himself even endorsed King Wen and Duke Zhou, who overthrew their corrupted king and became rulers themselves.\n\nThough Confucius values traditional ritual proprieties so highly that he says, \"If for a single day one were able to return to the observance of ritual propriety, the whole empire would defer to ren\" (12\/1), nowhere did he say that they must be unchangeable. When asked about the root of observing ritual propriety, the Master replied, \"What an important question! In observing ritual propriety, it is better to be modest than extravagant; in mourning, it is better to express real grief than to worry over formal details\" (3\/4). The principle behind this is the humanitarian spirit, ren, not mere traditional formality (see also 9\/3, 11\/1).\n\nTo Confucius, ritual propriety is no less aesthetic than it is educational or social-political. He often puts the word li, ritual propriety, together with yue or le \u4e50, a word that, when pronounced as yue, means aesthetic activities such as music, dance and poetry, and when pronounced as le, means happiness or joy (see 11\/1, 16\/2, 16\/5, 17\/11). Ritual ceremonies were traditionally composed of dance, song, and music. The beauty of the music, the dance, and the songs which constitute the rituals reinforces the ethical and social meanings of the rituals by giving them an aura of sacredness. At a more fundamental level, the persons who are refined by rituals and the social order resulting from and exemplified by ritual proprieties can themselves be considered artistic. Refined by ritual propriety, a person will have the grace that enhances the natural beauty of the body profoundly. To the contrary, lacking proper manner, the natural beauty of a person will diminish dramatically, and in extreme cases, to nothing but what is of the flesh. An unsightly behavior is always opposed to ritual propriety, and a conduct in accord with ritual propriety is always elegant and aesthetically pleasing.\n\nThe beauty of the social order resulting from ritual propriety is compatible to the beauty of the natural world, in which objects are different but mutually dependent, and they rise and fall rhythmically. The beauty of each object is dependent upon its place within the whole, in relation to its environment. The surrounding can enhance or reduce its beauty, depending upon how it is placed within the environment and in relation to everything else. By ritual propriety, humans can correspond and interact with each other artistically, like performers in a well-conducted orchestra, in which the artistic performance of each is aesthetically dependent on and enhanced by her cooperation with and coordination within the whole.7\n\n## 5  Zheng \u6b63 \u2013 Political Philosophy\n\nConfucius' humanitarian spirit is well exemplified in many teachings in the Analects. He clearly says that one should \"love the multitude at large\" (1\/6). He teaches his disciples that an exemplary person should be gracious in deporting himself, deferential in serving his superiors, generous in attending to needs of the common people, and appropriate in employing their services (5\/16). This is the Confucian ideal society:\n\n> When the great Way prevails, a public and common spirit is everywhere under the sky. People of talents and virtue are chosen, trustworthiness advocated and harmony cultivated. People love not merely their own parents, nor treat as children only their own children. The aged are provided till their death, the able-bodied all have places to utilize their ability, and the young have the means for growing up. Widows, widowers, orphans, childless, disabled, and ill, all sufficiently maintained. Men get their share and women have their homes. People hate to throw goods of value away upon the ground, but see no reason to keep them for their own gratification. People dislike not putting their strengths into use, but see no reason to use them only to their own advantage. Therefore schemings diminish and find no development; robberies, thefts, rebellions, and treason do not happen. Hence the outer doors need not be shut. This is called the Grand Union. (Book of Rites, Chap. 9. 1987: 120)\n\nWhat is needed for achieving such a Grand Union is basically ren (human-heartedness) and li (ritual propriety). This section will serve as a further elaboration and clarification of how the principles are applied in the social-political realm.\n\nConfucius explicitly placed the use of administrative and law enforcement as inferior to the way of ritual propriety and moral excellence for securing social order. First of all, during that particular historical era, it was unlikely that the rulers could sanction anything except through arbitrary and power driven orders from above, whereas ritual propriety, a repository of past insights into morality, was already there as a tradition, actualized in custom in the society. More importantly, \"Lead the people with administrative injunctions and keep them orderly with penal law, and they will avoid punishments but will be without a sense of shame. Lead them with virtue\/virtuosity (de \u5fb7) and keep them orderly through observing ritual propriety, and they will develop a sense of shame, and moreover, will order themselves\" (see 2\/3). Compulsion and punishment can only ensure outward conformity, at best. People will stay out of trouble not because they are ashamed of doing wrong, but because they fear the punishment, and in places where legal enforcement cannot reach or no one else is around to see, they may still do wrong. However, if the social order is secured by virtue\/virtuosity and ritual proprieties, an internal supervision will develop, which is much more effective in its penetration into people's lives, saturated as a way of life itself. The Master said, \"In hearing litigation, I am no different from anyone else. But if you insist on a difference, it is that I try to get people to have no need to resort to litigation in the first place\" (12\/13). He did not mean to be just idealistic and optimistic. Litigation is often needed when things are not going well, but it tends to inflict ill feeling and animosity, a side-effect hard to avoid even in the most fair court rulings.\n\nFamilies are small societies on the bases of which the larger society is structured. If the person can be a good member of a family, that person can be a good member of a larger community; if the person can regulate the family well, that person can rule a country well. When someone asked Confucius, \"Why do you not actively search for a career in governing?\" the Master replied,\n\n> the Book of Documents says, \"It is all in filial conduct (xiao \u5b5d)! Just being filial to your parents and befriending your brothers is carrying out the work of government.\" In so doing a person is also taking part in government. How can there be any question of my having actively taking part in governing? (2\/21)\n\nExtending the family model to the art of ruling, Confucius believed that the way to conduct zheng \u653f (to govern) requires one to be zheng \u6b63, a homophone that means \"being proper,\" \"straight,\" \"orderly,\" or \"to correct,\" \"to make straight.\" When the Master was asked about the order of importance among the following three things for an effective government \u2013 sufficient food to eat, sufficient arms for defense, and that the common people have confidence in their leaders, he put confidence in leaders at the top. \"Death has been with us from ancient times, but if common people do not have confidence in their leaders, community will not endure\" (12\/7). The words zheng \\--to govern and zheng -being correct are not merely homophonous; they have an intrinsic affinity. He believes that \"the excellence of the exemplary person is the wind, while that of the petty person is the grass. As the wind blows, the grass is sure to bend\" (12\/19). In public affairs, if those who are in superior positions are fond of ritual propriety, the common people will be easy to command (14\/41), so easy that they will even follow without any command (13\/6). \"If there was a ruler who achieved order without taking any action (wu wei \u7121\u7232), it was surely Shun. What did he do? He simply assumed an air of respectfulness and faced due south. That was all\" (15\/5). 8 \"Being proper in their own position, what difficulty would the rulers have in governing? But if not able to set themselves proper, how can they set others proper?\" (13\/13. See also 12\/19 and 12\/17).\n\nThis is the Confucian notion of wu wei \u2013 action by non-action, a notion more well known for its Daoist affiliation. It is non-action because the agent does not seem to be making efforts or exerting any force at all. While the Daoist wu wei is to do things naturally and spontaneously, the Confucian wu wei is to accomplish the intended results by ritual proprieties enlivened by their de \u2013 virtue\/virtuosity. Here we may take up a troubling passage in the Analects, \"the common people can be made to follow a path, but not to know \u6c11\u53ef\u4f7f\u7531\u4e4b,\u4e0d\u53ef\u4f7f\u77e5\u4e4b\" (8.9), to illustrate the importance of reading Confucius holistically. A careful reading reveals that the key word \"ke\u53ef\" in this passage may be taken either as \"be permitted to\" or \"can.\" The first reading would make the passage read \"the common people are permitted to follow a path, but not permitted to know.\" Based on this reading, critics of Confucianism argue that Confucianism is authoritarian, elitist, and opposed to human freedom and democracy. Though no government can afford to have total transparency (if a government were to reveal all threats to the public as soon as they were reported, it would cause the public to panic and lead the society into chaos), it is still wrong to keep the common people entirely in the dark. Confucius does think that common people are in the dark and do not know what is good for them (see 15\/25); but it is precisely because of this that he thinks they should be educated (see 2\/20, 13\/9, 13\/29, 13\/30). His principle of education is to provide teachings for everyone who shows a sincere willingness to learn and to do this without discrimination (15\/39). It would be totally inconsistent to interpret the saying as wanting to have the common people kept in the dark.\n\nTaking \"ke\" to mean \"can\" and reading the passage as saying that common people can be made to follow a path but cannot be made to know the path still leaves a number of different ways to explain the reason for it. One is that common people are low in intelligence and lack the ability to know it. Another, held by Cheng Yi \u7a0b\u9824, argues that the reason that common people cannot be made to know the path is that this kind of knowledge is knowledge of\/as virtue. Unlike knowledge gained through hearing and seeing, which can be understood by one's mind, knowledge of\/as virtue has to be experienced in one's heart so that the person is willing to act accordingly. By its very nature, knowledge of\/as virtue cannot be taught but has to be attained by oneself (zide \u81ea\u5f97). Still another explanation, offered by Zhu Xi \u6731\u71b9, maintains that the sage cannot go door to door to explain the reason to everyone. It is simply impractical.9 While all these readings have some plausibility, a most convincing reading, maintained by a number of Confucian scholars such as Dai Xi \u6234\u6eaa, He Yan \u4f55\u664f, and Zhang Ping \u5f35\u6191, is that the saying is about the sage's way of transforming people. The sages can make the common people follow them like having an invisible spiritual force that excites and motivates people. When the common people plow the land to get their food, or dig their well to get their water, they do not think that the power of their Emperor has anything to do with them. If there were something for the common people to know, it means that the ruler's way is still visible and not very effective (see Dai 1999: 9). Here the whole passage is understood as advice, consistent with Confucius' teachings about zheng: it is better when people are made to follow the path naturally by the magical forces of examples of excellence and by ritual proprieties than by visible forces.\n\nSince ritual propriety is dependent on clarification of roles, Confucius takes zhengming \u6b63\u540d, ordering or rectifying names, seriously, so that each person will know what is expected of him or her in a given web of particular relationships. When asked what would be the priority in bringing order to a state, Confucius replied, \"Without question it would be to order names properly\" (13\/3). Using names, for Confucius, is like implementing strategies or devices that stipulate expectations and norms of conduct. \"The ruler must rule, the minister minister, the father father, and the son son\" (12\/11). Each \"name\" carries with it a certain norm that whoever bears the name is expected to follow, and others are also expected to treat them accordingly. This is why Confucius advises people to \"speak cautiously\" (2\/18). It is said that when Confucius was editing the Spring and Autumn Annals, he paid special attention to the use of words, since words carry the force that can affect reality. An important feature of this Confucian pragmatic orientation toward language is that it focuses on the acceptability of names, or, as Chad Hansen puts it, \"social-psychological techniques for shaping inclinations and feelings that direct behavior in accord with a moral way\" (Hansen 1985: 495). It does not aim at trying to get to the truth in a correspondence sense, nor to invoke proofs or propositional knowledge. In this orientation, reality is supposed to match words, or term-beliefs, and the aim of the rectification of names is to ensure that, as pegs of role expectations, the names are acceptable, and people will have proper dispositions to respond to them. In contrast, in the referential use of language familiar to the Western philosophies preoccupied with obtaining truth and knowledge, sentential beliefs are supposed to match reality, and the aim of rectification is to ensure that they capture what is true.\n\nIt would be inadequate to take Confucius' teachings as merely advocating a social and political order, because they also aim at achieving personal freedom and aesthetic creativity. It is true that in the works of Confucius, there is no word close to the Western idea of \"freedom.\" Meanwhile, ritual propriety seems to serve as intangible rope that even limits what one can and cannot think or will. Furthermore, Confucius' emphasis on ritual propriety and the importance of acting in accordance with one's roles in the web of social relationships seem to place limitations on individual freedoms. However, actually Confucius has a different understanding of freedom. In his famous short autobiography, Confucius says, \"At the age of 70, I was able to follow my heart-mind's (xin \u5fc3) will (yu \u6b32) without overstepping the line\" (2\/4). The statement entails that for Confucius, freedom does not mean the lack of any \"lines\" of conduct that constrain what one can and cannot do, nor the ability to deliberate between alternatives and act upon what one chooses. For him, freedom is a cultivated spontaneity that frees one even from making choices because one would know what the proper lines are and would not have any intention of overstepping them! In this state of existence the \"lines\" to the person would be no more than \"No smoking\" signs to non-smokers. Just as any decent human being would not deliberate on whether or not to kick an innocent child for fun, a well cultivated person will have no need for deliberation in most cases (see Kupperman 1999: 102\u2013114). Indeed, an uncultivated person is like a novice chess player who does not know what to do when faced with all available alternatives, and hence would not be considered by Confucius as having any real freedom. One who is totally indifferent to alternatives would be like Buridan's ass, which starved to death between two equally good piles of hay because it could not find a reason to go to one pile and not the other. Of course, the freedom of cultivated spontaneity is not a natural state that people are born into and can simply enjoy without having to earn it, nor is it a matter of knowing all the relevant facts and making deliberate choices. One has to develop proper inclinations or dispositions, which is to learn proper rituals and cultivate ren. \"Those who know are not perplexed,\" says Confucius; but that is not all. He also says that one must be ren to be free from anxiety, and courageous to be free from fear (9\/29).\n\nTaking one's own right as an autonomous choice maker for granted and making demands on external conditions is not Confucius' way of assuring freedom. Though he clearly does not think that people should submit to whatever is imposed on them, he believes that \"If one sets strict standards for oneself and makes allowances for others when making demands on them, one will stay clear of ill will\" (15\/15; see also 1\/16, 4\/17, 15\/21 and 20\/1). Confucius even describes ren, the central quality of an exemplary person, in part as \"reforming the self\" (12\/1). Not having the person cultivated to a certain level of maturity, the availability of alternatives could even endanger the person.\n\nOf course for Confucius freedom is not a personal matter. Just as water is a necessary condition for one to swim, and adjusting bodily movement according to the nature of water increases one's freedom in the water, specific relationships with others are a necessary condition for an individual to be free within the given social environment, and adjusting the relationships according to ritual propriety increases one's freedom in it. A person is so inseparable from others that her domain of choice is itself defined and transformed by her interaction with others. It is with this understanding that Confucians take the selection of one's own residence to be a serious matter \u2013 not about the material wealth of the neighborhood, for \"were an exemplary person to live in it, what crudeness could there be?\" (9\/14); the most important feature of a residence is the presence of human-hearted (ren) persons in the neighborhood (see 4\/1).\n\nConfucianism is often criticized for ignoring, if not opposing, individual rights and for the lack of any idea of democracy. However, some contemporary Confucian scholars have argued that not only are the criticisms unfair to classic Confucianism in the way that Confucius himself represented it, but also that, in exactly these areas, Confucianism has particularly valuable insights to offer. Henry Rosemont, for instance, argues that the modern Western notion of the autonomous right-bearing individual is fundamentally flawed. \"99 % of the time I can fully respect your first generation civil and political rights [the rights of speech, of religion, of a fair trial, etc.] simply by ignoring you. You certainly have a right to speak, but no right to make me listen\" (Rosemont 1998: 59). Without a communitarian notion of self, i.e. the self as a nexus of social relations, the practice of human rights could result in \"excessive individualism, competitiveness, and vicious litigiousness\" that \"is not only endangering the well-being of others but also detrimental to our own wholesomeness\" (Tu 1998: 305). The Confucian notion of the embodied, relational, duty-bearing person allows each member of a society to have a clear sense of mutual dependence on other people, and to develop a sense of caring for the interest of others. On this basis we can construct our notions of democracy and human rights as the right and duty of every member of the community to participate in public affairs and take the public welfare of all as one's own. This Confucian notion of democracy is fully compatible with the spirit of democracy \u2013 a way of social and political life that promotes the interest of each member of the community as a vital part of the society of, by, and for the people. The key Confucian constituent in this theory is that there can be a good life over and above individual preferences, a life in which \"the desired would not be equated with the desirable, and democratic political participation \u2013 being a citizen \u2013 would involve engaging in collective dialogue about the appropriate means for achieving agreed-upon ends\" (Rosemont 1991: 93). It is worth noting that when Carsun Chang, the Chinese delegate to the U.N. and a distinguished Confucian scholar, added the clause that all men \"should act towards one another in a spirit of brotherhood\" (which is derived from the Confucian saying that \"all within the four seas are brothers,\" 12\/5 of the Analects), to the Declaration of Human Rights, he could not have expected to merely establish a procedure that would guarantee all the \"brothers\" the basic rights to speak, to be fed, etc. The spirit of brotherhood requires a sense of caring, love, and respect beyond just not legally violating the rights of a sibling. When a person acts toward his brother in a legal but otherwise unkindly manner, there is nothing a legal authority can do.\n\n## 6  Xue \u5b78 \u2013 Learning to Be Human\n\nAs a system of teachings that revolves around the theme of personal cultivation and manifestation of ren and li, naturally the topics of learning, thinking, and attaining knowledge are important to Confucianism.\n\nOne distinctive feature of the Confucian way of dealing with learning, thinking, and knowledge is that they all contain the heart part of xin \u5fc3, heart-mind. The heart is engaged in the process of reaching a deeper understanding, of critical evaluation, and of appropriating what is learned so that one is able to apply it creatively and artistically. Responding to a disciple's question about shortening the period of mourning for his parents, Confucius asks \"Would you feel at ease?\" (17\/21) This question forces the disciple to bring whatever feelings and ideas he has on the matter in front of his moral subjectivity and examine whether or not he can accept them at ease. When another disciple asked Confucius about the exemplary person, the Master said, \"The exemplary person is free from worry and apprehension.... If there is nothing to be ashamed of upon self-reflection, what can the person be worried about and afraid of?\" (12\/4) It is no coincidence that the Chinese language contains lexicons that can be illustrative of the bodily characteristic of the Confucian way of thinking and knowing, such as \"ti yan \u9ad4\u9a57,\" bodily experience, \"ti hui \u9ad4\u6703,\" bodily understanding, \"ti cha \u9ad4\u5bdf,\" bodily examination, \"ti zhi \u9ad4\u77e5,\" bodily knowing, and \"ti ren \u9ad4\u8a8d,\" bodily recognition. We do not passively receive impressions, nor do we merely reason intellectually. We experience with the body engaged, understand with our heart in empathy, examine with the sensitivity of the body, and know with the body disposed to act upon what is known.\n\nSince the process of learning and reflection involves dispositioning the body and transformation of the entire person, it requires practice to perfect. Biologically, human beings are similar in nature. It is by xi \u7fd2 (practice) that they diverge (17\/2). The outcome is not nearly as much an accumulated stock of propositional knowledge as a set of abilities, which Song and Ming dynasty Confucians call \"gongfu \u529f\u592b\" (kung fu), obtained through receiving training from masters and through one's own diligent practice. The Analects shows that when asked about ren by the disciples, Confucius never tried to describe ren per se. He talked about what a ren person would be like, how they would act, and he gave instructions according to each disciple's particular condition, letting them know on what level and in which respect they should start or continue their practice. The teaching method is indeed more typical of gongfu masters than of philosophy teachers in the ordinary sense of the term.\n\nThe Confucian gongfu culminates at zhongyong (see 6\/29). \"Zhong \u4e2d\" means \"centrality,\" \"not to be one-sided.\" \"Yong \u5eb8\" means \"ordinary\" or \"commonality,\" \"practicality,\" and \"constancy.\" When \"zhong\" and \"yong\" are used together as one term, it can be translated as \"centering the commonality.\" There is a considerable overlap between the Confucian doctrine of zhongyong and the Aristotelian Golden Mean. Both mean the virtue (not necessarily moral virtue), or excellence, of avoiding two extreme vices \u2013 deficiency and excess (see 11\/16, 11\/22, and 20\/1), and not a state of being mediocre. By associating \"zhong\" with \"yong,\" Confucius advises people to constantly practice zhong in the ordinary or everyday life that makes our heart-mind always at ease. Since everyday life situations are dynamic and hence there is no rigid rule to follow, the person has to embody the gongfu to respond to differing situations in a consistent way, and be creative as a co-creator of the universe. After all, the unity between heaven and human is not a combination of two entities or a human's ascendance to another world; it is rather one's becoming truly human in serving parents, taking care of children, respecting teachers, helping friends, and finding enjoyment in these activities (see 7\/19).\n\nThe kind of knowledge obtained through the Confucian sense of learning is closer to what is now commonly referred to as \"knowing how [to do something],\" as opposed to \"knowing what [is the case].\" The former is dispositional, and the latter is propositional. The teachings of Confucius are like road signs, or directives, that guide people's life and their actions, leading them to their own new discovery and their own unique life. An obvious omission in the Confucian learning program is natural sciences. Confucius felt close to nature, but he never displayed any interest in the dispassionate, objective analysis of nature, as scientists do. His remarks about nature are without exception the objectification of his moral and aesthetic sentiments and virtues. As Xu Fuguan points out, the names of plants and animals in the 300 Odes are sentiments and virtues of the poets, not botany or zoology. Western science interprets human as part of nature; Confucius interprets nature in terms of human.\n\nBut certain caution is needed in taking the Confucian learning as \"knowing how.\" First of all, while it is true that Confucius never really delved into natural sciences, the holistic and correlative way of thinking that was prevalent in Chinese philosophy, including Confucius' teachings, has led to remarkable achievements and insights about how the universe (including our own bodies) functions. It is best exemplified in traditional Chinese medicine. Confucius' own observation about the connection between human-heartedness and longevity (6\/23) and his followers' contribution to Chinese medicine and the Chinese theories of health are indications that Confucianism may have more profound understanding of how the natural world functions than modern medical science does, though the latter is indisputably much more advanced in detailed local areas (that is, if we put aside the fact that the two systems may be incommensurable scientific \"paradigms\"). The most remarkable feature of the Confucian outlook on the natural world is that it helps us to understand the inseparableness between the body and the mind, between moral cultivation and the overall wellness of a person, and between the state of an individual and her interpersonal relationships, and that the universe is a continuum, which must be viewed holistically (see Ni 1996, 1999).\n\nSecond, it is important to stress that the Confucian learning is not merely a matter of acquiring motor skills, as \"knowing how\" is commonly understood. It is more a transformation of the person. I may know how to overcome procrastination, but not disposed to do it or have the ability to do it. On the other hand I may be able to do it, but not know specifically how. The aim of the Confucian learning is to achieve the full capacity, including the strong inclination, to achieve excellence.\n\nAn interesting observation here is that, in this philosophy that aims primarily at prescribing a good way of life and offering instructions for people to grow mature and live well, the notion of time may also be different. \"The Master was standing on the riverbank, and observed, 'Isn't life's passing just like this, never ceasing day or night!'\" (9\/17) On the surface, this appears to be just a lament on how fast time goes. But the Master always draws moral lessons from observations of nature. The other passages succeeding this quote indicate that he is more likely reminding people that one should be like the river, making constant efforts to improve oneself. Time for a learner is not a passage that always passes evenly. \"As in pilling up earth to erect a mountain, if, only one basketful short of completion, I stop, I have stopped. As in filling a ditch to level the ground, if, having dumped in only one basketful, I continue, I am progressing\" (9\/19). \"If there was anyone who was never tired of practicing what he was taught, it was surely Yan Hui\" (9\/20), for he is a person that \"I only saw his progress; I never saw him stop\" (9\/21). \"There are indeed seedlings that do not flower, and there are flowers that do not fruit\" (9\/22). Though no one can be perfect, one should aim at constantly perfecting oneself.\n\nConfucius himself is a model learner. He says, \"In strolling in the company of just two other persons, I am bound to find a teacher. Identifying their strengths, I follow them, and identifying their weaknesses, I reform myself accordingly\" (7\/22). One must not hesitate in correcting oneself when in error (1\/8). When he was told that he misjudged someone, the Master said, \"I am fortunate. If I make a mistake, others are sure to inform me\" (7\/31). \"When you erred and yet not to correct yourself, that is to err indeed\" (15\/30). Often the problem is that, when something goes wrong, people tend to blame others or bad fortune rather than to search for the answer in themselves. Yet the exemplary person is like an archer who first searches for the fault within when he fails to hit the mark (Zhongyong 2001: Chap. 14; see also the Analects: 14\/35).\n\nWhile everyone should strive to become a junzi \u541b\u5b50 (exemplary person), the highest perfection of learning is to become a shengren \u8056\u4eba, or sage. A Confucian sage is not one who has eliminated all natural human desires. The ideal is actually to regulate and transform natural desires to the human level. There is no question that for Confucius, as for all the great thinkers of his time, humans need to regulate their desires. But Confucius never advocated the elimination of human desires. He himself is one who likes fine food, not merely whatever can fill the stomach (see 10\/8). With regard to sex, he only cautions young people not to overindulge in it (16\/7). The Songs (Odes) that he repeatedly quotes and advises his disciples to study are full of love themes. He is also fond of having fame (15\/20), though he also says that one should not be frustrated if one is not recognized by others (1\/1). He is very frank in saying that \"Wealth and honor are what people want, but if they are the consequence of deviating from the way, I would have no part in them\" (4\/5). \"If wealth were an acceptable goal, even though I would have to serve as a groom holding a whip in the marketplace, I would gladly do it. But if it is not an acceptable goal, I will follow my own devices\" (7\/12). The point is \"to desire but not to be greedy\" (20\/2), and to obtain what one desires without deviating from the proper way.\n\nPerhaps no passage in the Analects states Confucius' own orientation toward personal enjoyment more clearly than section 11\/26. The passage records a detailed conversation between Confucius and four of his close disciples, Zilu, Zeng Xi, Ran You, and Zihua. The Master said, with an obvious intention to create a relaxed atmosphere and to encourage the students to speak their minds, \"Just because I am a bit older than you do not hesitate on my account. You keep saying, 'No one recognizes my worth!' but if someone did recognize your worth, how would you be of use to them?\" The four students all expressed their wishes. Zilu said that his wish was to govern a state that was in trouble and to bring it back to a sure direction. Ran You's wish was to rule a small territory and to bring prosperity to the people in 3 years. Zihua stated his modest wish that he wanted to serve diligently as a minor officer and do his job well. Only Zeng Xi (also known as Zeng Dian), after plucking a final note on his zither to bring the music to an end and setting the instrument aside, said that he would choose something different from the rest. Here is how Zeng Xi expressed his wish:\n\n> At the end of spring, with the spring clothes having already been finished, I would like, in the company of five or six young men and six or seven children, to cleanse ourselves in the Yi River, to revel in the cool breezes at the Altar for Rain, and then return home singing.\n\nTypically people would expect the Master to endorse the first three disciples' wishes, for they all expressed moral and political ambitions. However, after hearing all, the Master heaved a deep sigh, and said that he would be with Zeng Xi, whose wish was nothing but enjoyment of a beautiful environment and pleasant company. The detailed description of Zeng Xi's relaxed way of expressing his wish is indeed part of the message: rather than anxiously waiting for his turn, he kept playing his music to the end even after the Master asked him \"What about you, Dian?\" Instead of being a nervous moralist who always worries about staying within the moral boundary, a real sage is one whose heart-mind is at ease, who is able to enjoy life artistically by spontaneous participation in the revival of everything, and celebrate the transformation of heaven and earth.10\n\nThis brings us to the question of the Confucian approach to art and aesthetics. If we say that today's conventional notion of art associates artworks with studios and galleries, the Confucian art is the artistic way of life itself. While the conventional artist dissolves the opposition between the mind and the \"hands,\" the Confucian sage achieves unity with heaven, and is able to participate in heaven's creation. Aesthetic enjoyment is actually the culmination of the Confucian learning. Reading the passage 7\/6 as a process of cultivation, we see that Confucius puts \"sojourn in the arts\" as the highest ideal, above other stages of practice, such as \"Set your will on the way. Have a firm grasp on virtue. Rely on humanity.\" Similarly, passage 8\/8 describes \"finding fulfillment or consummation in music\/enjoyment\" as a result of getting inspiration from the Odes and taking a stand from observing ritual proprieties. For Confucius, \"knowing that it cannot be done and yet doing it\" (a remark someone made about Confucius, 14\/38) is at its best when one \"takes pleasure\" in doing it (6\/20). When the Duke of She asked Zilu about Confucius, and Zilu did not answer, Confucius said: \"Why didn't you say that I am a person who forgets his food when engaged in vigorous pursuit of something, is so happy as to forget his worries, and is not aware that old age is coming on?\" (7\/19). The word \"forget\" is a strong indication of the pure non-utilitarian aesthetic ideal. These passages reflect that the ideal of Confucian learning is not in morality in the Kantian sense \u2013 not moral for the sake of being moral. It is the opposite: Morality is valuable because of its utilitarian function of leading to the aesthetic ideal (which is itself non-utilitarian)11 or that what is moral is itself beautiful and enjoyable. Once the ideal is achieved, there is no need to worry about morality. One simply follows the will of the heart-mind without overstepping the line and enjoys the freedom this entails, just like \"When the Dao prevails in the world, the common people do not debate affairs of state\" (16\/2).\n\nThe Confucian sage embodies zhi, knowledge or wisdom, and is therefore not perplexed; she embodies ren, human-heartedness, and is therefore not worrisome; she is courageous, and is therefore not timid (9\/29). But that is not all. The person is profoundly joyful. She enjoys water, for wisdom is like water, dynamic and creative; she enjoys mountains, for human-heartedness is like the mountains, enduring and full of dignity (6\/23). The Confucian artistic creation is displayed in one's entire life, and not merely in those \"big moments\" like jumping into a burning building to save a child, or taking up a dangerous mission to protect one's own country. After all, it is more difficult to be aesthetic consistently in daily activities, providing all sorts of social services, and building strong social relationships (this is where the \"Zeng Dian spirit\" differs from the Daoist and Buddhist ideals). Such a person may not be considered a prominent artist in the conventional sense of the term, yet for Confucius, she will necessarily be truly prominent (see 12\/20), because she cannot become such an artist without turning her own relatedness with others into what is harmonious and pleasant. Indeed, as Roger Ames and David Hall point out, the social order brought forth by the Confucian sage will be an aesthetic one. In contrast with logical or rational order, which enforces some external or transcendental rule or principle from without, the aesthetic order emerges as embodied rules and principles through self-cultivation and mutual coordination. In a logical\/rational order, there is no creativity, but only consistency and continuity, whereas in an aesthetic order, individuals and communities are able to creatively interpret and re-interpret the rules and principles. In a logical\/rational order, everyone is equal, because everyone is conceived abstractly as an agent, and hence they are substitutable with any other agent, whereas in an aesthetic order, individuals are concrete particulars, un-substitutable, and the inequality is a matter of deference to excellence (see Hall and Ames 1987: 131\u20138).\n\nReferences\n\nAmes, Roger T., and Henry Rosemont, Jr. 1998. The analects of Confucius: A philosophical translation. New York: Ballantine Books. (\"The Bible of Confucianism\" translated into English with philosophical sensitivity to Western philosophical assumptions.)\n\nBook of Rites (Li Ji \u79ae\u8a18). 1987. Shanghai: Shanghai Guji Chubanshe. (One of the \"Five Chinese Classics,\" composed around 200 B.C.E., this book records and describes in detail traditional rituals of ancient China during and prior to Confucius' time.)\n\nChan, Wing-tsit. 1963. A source book in Chinese philosophy. Princeton: Princeton University Press. (A comprehensive, though selective, collection of Chinese philosophical works covering its entire historical development, with helpful explanatory aids.)\n\nCheng, Shude \u7a0b\u6a39\u5fb7. 1990. Collected interpretations of the analects \u8ad6\u8a9e\u96c6\u91cb. Beijing: Zhonghua Shuju. (A most recent comprehensive collection of traditional commentaries of the Analects with the author's own brief analysis.)\n\nDai, Xi \u6234\u6eaa. 1999. Shigu questions and answers of the analects \u77f3\u9f13\u8ad6\u8a9e\u7b54\u554f, Vol. 2, in Siku Quanshu \u56db\u5eab\u5168\u66f8, Wenyuange copy, Classics Part, Four Books Section. Hong Kong: Dizhi Wenhua Chuban Co. Limited & Chinese University of Hong Kong. (A commentary of the Analects written during the Southern Song dynasty as a textbook for Shigu Shuyuan \u77f3\u9f13\u66f8\u9662, or \"Stone Drums College.\")\n\nDao. 2008. Symposium: Filial piety as the root of morality or the source of corruption (Part 1). A Journal of Comparative Philosophy 7(1): 1\u201355.\n\nFingarette, Herbert. 1972. Confucius \u2013 The secular as sacred. New York: Harper & Row. (A landmark work on Confucianism in English language which places great emphasis on ritual propriety and relates it to contemporary Western philosophical theories about language and mind.)\n\nHall, David L., and Roger T. Ames. 1987. Thinking through Confucius. Albany: State University of New York Press. (An in-depth discussion of Confucianism through comparison with Western philosophical theories.)\n\nHansen, Chad. 1985. Chinese language, Chinese philosophy, and 'truth'. Journal of Asian Studies XLIV(3): 491\u2013519. (A thought provoking essay that claims Western concept of truth to be alien to Chinese language and philosophy.)\n\nHuang, Yong. 2007. Confucian theology: Three models. Religion Compass 1(4): 455\u2013478. (Comprehensive guide to various interpretations of the religiosity of Confucianism.)\n\nHuang, Yong. 2008. Neo-Confucian hermeneutics at work: Cheng Yi's philosophical interpretation of Analects 8.9 and 17.3. Harvard Theological Review 101.1: 169\u2013201. (A thorough and clear analysis of different interpretations of the two difficult passages in the Analects.)\n\nIvanhoe, P.J. 2007. The shade of Confucius: Social roles, ethical theory, and the self. In Polishing the Chinese mirror: Essays in honor of Henry Rosemont, Jr. ed. Marthe Chandler and Ronnie Littlejohn, 34\u201349. New York: Global Scholarly Publications. (A critique of Rosemont's view that the Confucian conception of persons as essentially role-bearing, and questions the adequacy of such a view as the basis for moral theory.)\n\nKupperman, Joel. 1999. Learning from Asian philosophy. New York: Oxford University Press. (Illuminating illustrations on how Asian philosophy can enrich current philosophical practice and offer unique insights about life.)\n\nLi, Chenyang. 2000. The sage and the second sex, Confucianism, ethics, and gender. Chicago\/La Salle: Open Court. (An informative and inspiring anthology on the role of women in ancient China and the Confucian position about it.)\n\nMencius. 1970. Trans. D. C. Lau. New York: Penguin Books. (Attributed to Meng Zi, the \"Second Sage\" of the Confucian tradition, the book was considered one of the \"Four Canons\" of Confucianism.)\n\nNeville, Robert Cummings. 2008. Ritual and deference, extending Chinese philosophy in a comparative context. New York: State University of New York Press. (A lucid articulation of the relevance of Confucian ideas of ritual and the works of Xun Zi in contemporary life.)\n\nNi, Peimin. 1996. A qigong interpretation of Confucianism. The Journal of Chinese Philosophy 23(1): 79\u201397. (Articulation of basic Confucian values from what the author later calls a \"gongfu\" perspective.)\n\nNi, Peimin. 1999. Confucian virtues and personal health. In Confucian bioethics, ed. Ruiping Fan. Dordrecht\/Boston: Kluwer Academic. (From the Confucian holistic vision, the essay reveals how Confucian cultivation may be understood far beyond the scope of morality.)\n\nRosemont, Henry, Jr. 1991. A Chinese mirror: Moral reflections on political economy and society. La Salle: Open Court. (A sharp critique of contemporary industrial capitalism through reflections about the Chinese philosophical tradition and the modern China.)\n\nRosemont, Henry, Jr. 1998. Human rights: A bill of worries. In Confucianism and human rights, ed. Wm. Theodore de Bary and Tu Weiming. New York: Columbia University Press. (Critique of liberal autonomous individual as the conceptual basis for human rights.)\n\nSun, Xing Yan \u5b6b\u661f\u884d & Guo Yi \u90ed\u6c82. 1998. Collected sayings of Confucius, proofread with new additions \u5b54\u5b50\u96c6\u8a9e\u6821\u88dc. Shangdong: Qilu Shushe. (A collection of Confucius' sayings from books other than the Analects.)\n\nTu, Weiming. 1998. Epilogue: Human rights as a Confucian moral discourse. In Confucianism and human rights, ed. Wm. Theodore de Bary and Tu Weiming. New York: Columbia University Press. (Placed within the context of global human rights discourse, the essay argues that Confucian core values are not only compatible with, but can also enhance the universal appeal of human rights.)\n\nXu, Fuguan \u5f90\u5fa9\u89c0. 1984. The history of Chinese theories on human nature \u2013 Pre-Qin period \u4e2d\u570b\u4eba\u6027\u8ad6\u53f2: \u5148\u79e6\u7bc7, 7th ed. Taiwan: Commercial Press. (Through a survey of pre-Qin Chinese theories of human nature, the author tries to answer the question of what are the main characteristics of Chinese culture.)\n\nXun Zi. 1967. In basic writings of Mo Tzu, Hsun Tzu, and Han Fei Tzu. Trans. Burton Watson. New York: Columbia University Press. (A pre-Qin Confucian classic and a rival of the Mencius, known for its emphasis on ritual propriety and its claim that human nature is bad.)\n\nZhongyong. 2001. Focusing the familiar, a translation and philosophical interpretation of the Zhongyong. Trans. Roger Ames and David Hall. Honolulu: University of Hawaii Press. (Originally a chapter in the Book of Rites and later regarded as one of the four canons, it is arguably the most metaphysical work in Confucianism.)\n\nFootnotes\n\n1\n\nThe translation of the Chinese texts in this chapter is mostly based on the English books listed in the bibliography at the end, often with some modifications, or, in the case of direct quotes from Chinese texts, my own. Citations from the Analects will be given simply in parentheses with the chapter number and the section number. For example, \"(2\/1)\" means Chap.\u200b 2, Sect. 1 from the book of the Analects.\n\n2\n\nSee Huang (2007) for a comprehensive summary of the controversy.\n\n3\n\nOf course this raises both the philosophical question about whether such a notion can be consistent with human subjectivity, namely human beings as decision makers, as initiators of our actions, etc., which Confucius certainly acknowledges, and the interpretive question about whether Confucius does not at the same time also think that humans are individual entities. P.J. Ivanhoe, for instance, raises a number of objections to this interpretation of the Confucian notion of self (see Ivanhoe 2007 and Rosemont's response in the same book).\n\n4\n\nThere is a heated debate on the issue in the recent years, initiated by Liu Qingping's criticism of Confucian morality as a basis for favoritism toward one's own family, and hence is opposed to the principle of justice. See Dao: A Journal of Comparative Philosophy (2008) for a collection of essays on this controversy.\n\n5\n\nRefer to Li's The Sage and the Second Sex (Li 2000) for the issue about Confucianism and gender, and pages 3\u20134 of the book for the specific issue about interpreting \"n\u00fczi.\"\n\n6\n\nThe utterances are not acts of descriptions about certain facts or acts of instructions to induce some other action; they are the very execution of the acts itself. A promise is a typical example \u2013 the utterance of \"I promise\" is the very act of promising.\n\n7\n\nReaders may refer to Neville (2008) for elaboration of the modern significance of the Confucian idea of ritual.\n\n8\n\nShun was an ancient sage king that Confucius revered greatly. According to traditional Chinese ritual, south is the direction to which the superior's seats face.\n\n9\n\nSee Huang (2008) for a detailed analysis of these interpretations.\n\n10\n\nHere I am basically following Cheng Yi and Zhu Xi's reading of the passage. There are some other interpretations. For instance, Cheng Shude takes the Master to be lamenting that given his age and the fact that there was no sage king to employ him, he had no chance to implement his visions in his life. His agreement with Dian was simply saying that he had no choice but to be like Dian (see Cheng 1990 : 812). Still another interpretation, held by Zhang L\u0215xiang, holds that the four disciples' aspirations show an ascending order. Zilu's was to bring peace to the land, which is necessary for implementing Ran You's wish \u2013 to let people have sufficient material supplies. Ran You's, in turn, is a necessary condition for implementing Zihua's wish, i.e. using ritual propriety to teach people and to transform the society. Finally, only when all the three wishes mentioned above become reality can people truly enjoy the kind of pleasure that Zeng Dian was talking about. The Master's \"with Dian\" is then actually consenting to this highest ideal (see Cheng 1990: 816).\n\n11\n\nThough this does not mean that arts cannot have utilitarian functions. For instance the Master says that the Songs \"can arose your sensibilities, strengthen your powers of observation, enhance your ability to get on with others, and give expression to complaints\" (17\/9). It only means that an aesthetic ideal does not need any utilitarian function for its justification.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_4\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 4. The Philosophy of Confucius' Disciples\n\nYuet Keung Lo1\n\n(1)\n\nDepartment of Chinese Studies, National University of Singapore, 5 Arts Link, Singapore, 117570, Singapore\n\nYuet Keung Lo\n\nEmail: chsloyk@nus.edu.sg\n\nAbstract\n\nAs the Zhou feudal order began to crumble in his times, one of Confucius' (551\u2013479 B.C.E.) ambitions was to train a new social class of people of learning and with moral aspirations and commitment and he called these people noble scholars (shi \u58eb). When they were properly and sufficiently trained in the various arts and classical learning, the noble scholars often tried to seek political office to restore the society that was in ritual disorder, and in this sense they were also called gentleman (junzi \u541b\u5b50). To Confucius, learning was not confined to books and must be translated into actual practices in everyday life and the political arena. Indeed, his own academy itself constituted a living environment where his teachings were transmitted to his disciples, who were expected to internalize and manifest them in their day-to-day interactions with one another in a fellowship that formed and shaped the earliest Confucian community. In the study of early Confucian thought, the physical environment where the real-life relationships and exchanges that took place and the teachings of the master were first practiced and eventually modified and transformed has been largely ignored. Thus it is worthwhile to examine how the earliest Confucian community was formed and how the master's teachings were practiced and tested before we discuss the philosophy of Confucius' disciples.\n\n## 1 Introduction\n\nAs the Zhou feudal order began to crumble in his times, one of Confucius' (551\u2013479 B.C.E.) ambitions was to train a new social class of people of learning and with moral aspirations and commitment and he called these people noble scholars (shi \u58eb). When they were properly and sufficiently trained in the various arts and classical learning, the noble scholars often tried to seek political office to restore the society that was in ritual disorder, and in this sense they were also called gentleman (junzi \u541b\u5b50). To Confucius, learning was not confined to books and must be translated into actual practices in everyday life and the political arena. Indeed, his own academy itself constituted a living environment where his teachings were transmitted to his disciples, who were expected to internalize and manifest them in their day-to-day interactions with one another in a fellowship that formed and shaped the earliest Confucian community. In the study of early Confucian thought, the physical environment where the real-life relationships and exchanges that took place and the teachings of the master were first practiced and eventually modified and transformed has been largely ignored. Thus it is worthwhile to examine how the earliest Confucian community was formed and how the master's teachings were practiced and tested before we discuss the philosophy of Confucius' disciples.\n\n## 2 Gate to Education\n\nWe know that Confucius was one of the first teachers who offered education to the common folk on a regular basis. Disciples clustered around Confucius but we do not know exactly how they actually spent their time together. In Chinese we have the term kongmen \u5b54\u9580, which literally means \"the gate of Kong,\" Kong being the family name of Confucius. Kongmen refers to the community or fellowship which comprises Confucius and his numerous disciples. Yet, this term is not original to the Confucian community it describes; rather, it was probably coined around the late fifth century1 and gained wide currency since the Tang dynasty (618\u2013906). Even though kongmen evokes a strong sense of physical community, in actuality, it pinpoints the cultural, moral, and social identity of Confucius and his followers, as the family name of Confucius unmistakably indicates. This is indeed the way the term was used especially in the early days when it was coined.2 Thus kongmen does not seem to tell us how the earliest Confucian community was constructed and how it functioned. At least, the family of Confucius does not get us very far in this regard.\n\nBut what about Confucius' \"gate\"? The Han exegete Bao Xian's \u5305\u54b8 (6 B.C.E.\u221265 C.E.) offered us an extremely valuable clue.3 As we are all taught, the first chapter of the Analects talks about friendship and self-cultivation, among other things. The line in question reads, \"When friends come from afar, isn't it joyful?\" Virtually all translators of the Analects render the term peng as \"friends\" (see, for instance, Slingerland 2003: 1). In glossing \"peng\" in the line, however, Bao says, \"Peng refers to people who share the same gate.\" This seems to be shocking, as far as our customary understanding is concerned. Indeed, for people in the Han period there was nothing unusual about Bao's gloss. For them, tongmen \u540c\u9580 can only mean \"people who share the same gate,\" namely, people who studied under the same master.4 It does not mean \"friends,\" as we have learned. In plain English, tongmen means \"fellow students\" or \"classmates.\"5 This peculiar meaning of tongmen continues in use to this day.\n\nWhat interests us here is the gate in question. Whose gate is it? Given that tongmen refers to fellow students who study under the same teacher, it would seem that the gate must refer to the teacher's gate. This can be supported by two considerations. First, since the Confucian community is denoted by the term kongmen, the gate evidently belongs to Confucius the teacher. The term is marked, as it were, with the family name of Confucius. Second, the term menren \u9580\u4eba (people of the same gate) appears five times in the Analects,6 and it clearly refers to Confucius' disciples in four occurrences. If his disciples could be referred to as \"people of the same gate,\" it seems that the gate must belong to Confucius. It does not seem to make sense that while the disciples studied with Confucius, their identity would be symbolized by a gate of someone other than their master.\n\nSeen in this light, Huang Kan's \u7687\u4f83 (485\u2013545) elaboration of Bao Xian's interpretation is particularly informative in his Lunyu jijieyishu \u8ad6\u8a9e\u96c6\u89e3\u7fa9\u758f (Subcommentaries on the Analects). He says,\n\n> [People who] share the same teacher are called peng whereas [people who] hold fast to the same ambition are called friends. Peng is synonymous with \"comrade;\" peng are people who form a comradeship at the gate of a [common] teacher.7\n\nHuang makes it explicitly clear that the gate belonged to the teacher (shimen \u5e2b\u9580).8 However, the Qing scholar Mao Qiling \u6bdb\u5947\u9f61 (1623\u20131716) disagrees and argues that peng was originally the appellation for \"gate\" and this \"gate\" actually referred to that which guarded the dormitories of the students rather than the gate that belonged to the teacher. According to Mao, Confucius' disciples lived together in the same quarters, and the \"gate\" in question, then, referred to the gate of their dormitories. Mao substantiates his interpretation with a variety of early classical sources, but a commonsensical reflection would seem sufficient to support his view. Given that many, if not most, of Confucius' disciples came from places other than his native state of Lu \u9b6f, they had to find a place to stay while studying under his tutelage. Even for disciples coming from Lu, their homes might be too far away from where Confucius was for them to commute, and they needed temporary lodging as well. It is, then, plausible that the disciples cohabitated in a place provided by Confucius. Since the dormitories were probably provided by Confucius, it also seems to make sense to call their front doors the teacher's gates. In fact, Confucius himself did make such a reference (Analects 11.15, 11.2). He called his academy, metonymically and metaphorically, \"Qiu's gate\" (Qiu zhi men \u4e18\u4e4b\u9580), Qiu being Confucius' personal name. Hence, the seemingly different interpretations of Huang Kan and Mao Qiling are indeed compatible, if not actually identical.\n\nThe sheer number of the disciples makes it unlikely that they all lived with Confucius in his own house. Once, a disciple named Chen Kang \u9673\u4ea2 asked Confucius' son Boyu \u4f2f\u9b5a if he had received any special teaching from the master given their unique relation. In reply, Boyu mentioned what he had learned from his father while running into him in the courtyard on two separate occasions (Analects 16.13).9 The courtyard in question cannot be just any courtyard, for it was one mutually understood by Chen Kang and Boyu. Given the explicit reference to private access to Confucius, the courtyard then must be inside Confucius' residence, and this means that the disciples probably did not live there with their master as Boyu appeared to be the only person who had such a privilege.\n\nIn the Analects, there are passages that seem to hint at the dormitory setting. For instance, we all know that Yan Hui \u984f\u56d8 (521\u2013481 B.C.E.) was the most beloved student of Confucius and the master considered him to be the only student who was truly fond of learning (Analects 6.3, 11.7). Yet, Yan Hui almost always kept his mouth shut in the master's presence; he seldom responded to the master's instruction and appeared, even in the observant eyes of Confucius, to be stupid. However, Confucius eventually came to the conclusion that Yan Hui was in fact not stupid at all. This is because the master had observed Yan Hui in private (si \u79c1) when the deceptively stupid student was either by himself or with his classmates, and he realized that Yan Hui's behavior actually could elaborate or even illuminate what had been taught to him (Analects 2.9).10 Where did Confucius observe his students? The key word si rules out the \"classroom.\" One viable possibility is the place where the students lived.11\n\nFrom Confucius' visit to Yan Hui's living quarters, it is plausible that the student dormitories were not far away from the master's residence. In fact, an episode concerning Ran Qiu \u5189\u6c42 (522-? B.C.E.) evidently supports our conjecture. One day when Ran Qiu returned from court, he ran into Confucius, and the master asked why he came back so late (Analects 13.14). How would Ran Qiu run into Confucius when he returned home? The most reasonable explanation seems to be that his dwelling was in the vicinity of the master's place. Still, there is another anecdote that may further substantiate our claim. One day when Confucius was out at court, a fire broke out in the stables on his residence. When the master returned, he asked if anybody was injured without a care about the horses (Analects 10.17). We do not know whom the master asked about possible human casualties, but it is plausible that the disciples were at the scene helping to put out the fire. If so, their dormitories could not be far away from the master's residence.\n\nIt should be noted that while the students lived in their own quarters, instruction might take place in Confucius' residence. Once, Ru Bei \u5b7a\u60b2 sent a messenger to seek audience with Confucius, but the master declined on the pretext of illness. As the messenger was on his way out, the master picked up his zither and sang, and he made sure it was heard (Analects 17.20).12 The incident suggests that some of the students were around Confucius in his home at that time, presumably for the sake of receiving instruction. That is why this incident was recorded. Book 10 of the Analects records the conduct of Confucius in private, including his preference and insistence on various quotidian details of everyday life such as eating and sleeping. These records all indicate that the students studied with the master in his residence, even if occasionally. In fact, at least some of the disciples had to practice their learning in the master's house (Analects 14.44, 6.5).\n\nBased on the Analects, then, we can conclude that the disciples' dormitories were not far away from Confucius' own residence and that Confucius and his disciples probably lived in close proximity to each other. As luck would have it, we still have a valuable fragment from an encyclopaedic work of the third century by the title of Huanglan \u7687\u89bd (For the Eyes of the Emperor), compiled under the order of Emperor Wen of Wei \u9b4f\u6587\u5e1d (reigned 220\u2013225). This fragment can confirm the finding of our investigation. In it we are told that \"the houses of the disciples with wells and water pots still remain\" (Quoted in Kong 1983: 446.86). And according to Kong Chuan \u5b54\u50b3 (fl. mid-twelfth century), the 47th-generation descendant of Confucius, the dormitories of the disciples were actually only a mere five li \u91cc (less than two miles) east of Confucius' residence, sandwiched by River Si \u6cd7\u6c34 to the north behind and River Zhu \u6d19\u6c34to the south in the front. Kong further specified that there was actually a lecture hall near the dormitories where Confucius would instruct his disciples, yet it had become dilapidated even though its site survived into the twelfth century (Quoted in Kong 1983: 446.86). The attachment of the lecture hall to the dormitories and its geographical proximity to Confucius' residence must have contributed significantly to the palpable sense of camaraderie and solidarity that characterized the earliest Confucian community. It is well known that Confucius tailor-made his pedagogy to suit the various learning habits and personalities of his students (Analects 11.22). Although this tailor-made approach does not presuppose a dormitory setting, it does require the teacher to get to know his students personally and understand their personalities, character flaws, and learning habits as well as individual strengths and weaknesses. Without such thorough understanding of his students, it is impossible for the teacher to educate them in accordance with the unique character and talents of each student. To get to know a student thoroughly, close contact and regular communication would definitely help.13 The physical proximity between the students' dormitories and their master's residence was crucial as it provided them ample opportunities to interact constantly with each other, and in a real sense, we can consider them to be living together in the same community. Finally, it is noteworthy that the Confucian community inaugurated by Confucius was apparently carried on through various lines of disciples even after the master died, for the term menren and mendizi \u9580\u5f1f\u5b50 continued in use to refer to the second generation disciples of Confucius as well (Analects 8.3, 19.12).\n\n## 3 The Disciples and the Earliest Confucian Community\n\nConfucius began his teaching career when he was in his early 30s, and over the course of his life, it is well known that he taught more than 3,000 students and some 70 of them were particularly outstanding in their well-rounded excellences. While Confucius was not the only or earliest teacher who made education accessible to the commoners, his disciples turned out to be the most influential presence in the social, political, and intellectual landscapes during the Warring States Period (404\u2013221 B.C.E.). In Sima Qian's \u53f8\u99ac\u9077 (145?\u201387? B.C.E.) magnum opus, Shiji \u53f2\u8a18 (Records of the Grand Historian), a special niche was created for them in a collective biography entitled \"Zhongni dizi liezhuan\" \u4ef2\u5c3c\u5f1f\u5b50\u5217\u50b3 (Arrayed biographies of Zhongni's [Confucius'] disciples). It was not a mere matter of paying tribute to Confucius14; rather, it was, to a large extent, a factual account of the widespread and lasting influence of the master's disciples in their own time and thereafter. The Grand Historian was just being truthful.\n\nWhile the identity of the vast majority of the three thousand disciples had long fallen into oblivion, the names of 77 were recorded in Sima Qian's collective biography, and they were, according to the Grand Historian, recognized by Confucius himself as \"noble scholars of extraordinary abilities\" who \"thoroughly embodied the training they had received.\" 15 According to the biography of Confucius \u5b54\u5b50\u4e16\u5bb6 (\"Kongzi shijia\") in the Shiji, the training in question was on the Odes, the Documents, the rites, and music (See Takigawa 1977: juan 67). Another earlier source, Kongzi jiayu \u5b54\u5b50\u5bb6\u8a9e (Family Sayings of Confucius) mentions the names of 76 disciples, which do not completely overlap with those in the Shiji, although the order in which the common names appear is basically identical.16 And the Kongzi jiayu says the 76 disciples had \"ascended to the master's hall and entered into his private chamber\" (\u5347\u5802\u5165\u5ba4). The two lists from the Shiji and Kongzi jiayu together yield 96 names.17 While the expression \"ascending to the master's hall and entering into his private chamber\" is taken metaphorically today, it probably was meant in the literal sense originally and indeed depicted the Confucian community graphically as Confucius himself used it to measure the progress and achievements of his disciples (Analects 11.15).\n\nApparently, Confucius did not demand any tuition; so long as a prospective student presented himself with a bundle of dried meat (Analects 7.7), he would welcome him. According to Xunzi \u8340\u5b50, (340\u2013245 B.C.E.) Confucius' disciple Zigong \u5b50\u8ca2 (520-? B.C.E.) even said that the master would not turn anyone away who came to seek education; he compared his teacher to a skilled physician, who had many patients at his door (Wang Xianqian 1997: 2.536). The sheer number of his disciples seems to attest to what Zigong said. They came from virtually all over China. From what we can still identify today, 43 came from the state of Lu, Confucius' homeland; six from the state of Wei, three from Qin, two each from Chen and Qi; and one each from Song, Chu and Cai (Zhang 2007: 24). As the master himself said, \"In teaching there is no difference in kind\" (Analects 15.39). What he meant here was that education was open to all in his community. A student's personal background was not relevant. Once, a young boy from a village notorious for its recalcitrant residents came to request admission from the master to the Confucian community, and when he was accepted, some of the disciples were nonplussed. Confucius then explained, \"When a person purifies his self (jieji \u7d5c\u5df1) before he announces his presence, I would approve his purity of mind even though I cannot guarantee what will become of him\" (Analects 7.29). From this example we can see that Confucius basically adopted an open-door admission policy; the bundle of dried meat was only meant to be a ritualistic token of the prospective student's sincerity. The fact that the young boy purified himself before visiting Confucius suggests that the master's insistence on sincerity was well known.18 Indeed Confucius' students would pay for their education a sum commensurate with their financial assets, and this was also considered a ritual gesture to their teacher as well. With this income, Confucius no longer needed to rely on an official career to make a living (Qian 1991: 16).\n\nThere does not seem to be any hard-and-fast restriction on the age of the student. We know that young boys (Analects 7.29, 11.26, 14.44) as well as adults only a few years the master's junior were admitted to the Confucian community.19 The disciples came from a variety of social backgrounds; some of them were noblemen20 while the majority were simply underprivileged folks. Yan Hui, for example, lived in wretched conditions that nobody \"was able to bear\" (Analects 6.11). Similarly, Ran Yong \u5189\u96cd (style Zhonggong \u4ef2\u5f13, 522-? B.C.E.) came from so humble a family that some suspected that he might not be offered an official post on that account (Analects 6.6), even though Confucius expressly recommended that he was \"capable of being a lord\" (Analects 6.1). Of the disciples of humble origin, some, like Yan Zhuoju \u984f\u6dbf\u805a, were even former robbers21 while Gongye Chang \u526c\u51b6\u9577 had been a prisoner even though, in Confucius' opinion, he was actually not guilty of any crime (Analects 5.1).\n\nThe Confucian community comprised an intricate mosaic of human relationships. Some disciples were father and son to each other; some were son and nephew to the master himself. And many others were probably friends who decided to come to study with Confucius together. Between Confucius and his disciples, we also witness a variety of relationships of different degrees of intimacy. For instance, Confucius publicly professed that he treated Yan Hui as if he were his son, and his love and care for his beloved disciple were mutual for it was no secret that Yan Hui also respected the Master like he would his father (Analects 11.11). In reality, most, if not all, of the disciples treated Confucius as their father. It is quite certain that Confucius was well aware of their love and respect. Little wonder that he would rather die in their hands than in the hands of retainers (Analects 9.12). The master preferred to die under the care of his \"family.\" When Confucius passed away, the disciples unanimously agreed to observe the 3-year mourning ritual at the master's gravesite, and Zigong, in particular, even continued to stay on for three more years (Takigawa 1977: juan 47, 746). Since the 3-year mourning period was reserved only for parents, the disciples' observance of the ritual in effect represented the most unequivocal statement of their profound respect for their mentor. The fictive relationship can actually become real. Confucius appreciated the personal qualities of some of his disciples so much that he, for example, gave his own daughter to Gongye Chang (Analects 5.1) in marriage and the daughter of his brother to another disciple Nan Rong \u5357\u5bb9 (Analects 5.3, 11.6). All in all, the Confucian community in effect functioned much like a close-knit family. The only missing members seem to be girls and women.22\n\nThe majority of the disciples naturally came and left; the length of their stay with Confucius varied according to their aspirations, ambitions, and of course, the availability of an official post. There does not seem to be a set length of time for the Confucian program of learning. Some students wanted to learn as much as necessary within a relatively short period of time. One young student, for example, was asked to be the master's messenger; it was an opportunity for him to learn the proper rituals of sending and receiving guests. When someone asked if the teenager had made progress in his learning, Confucius replied that he was actually only trying to wrap up his learning as soon as possible (Analects 14.44). Perhaps for the same reason, the master also warned a more accomplished disciple Zixia \u5b50\u590f (507-? B.C.E.), even after the latter had an official career as Governor of Jufu \u8392\u7236. Confucius told Zixia not to hurry things for petty gains if he wanted to succeed in larger enterprises (Analects 13.17). On the other hand, Confucius would also send his disciples out to start an official career if he deemed them capable and ready (Analects 5.6). Even though the disciples could decide on the pace and length of their study, it appears that a serious disciple would spend at least 3 years under the tutelage of Confucius before he looked for or accepted a job (Analects 8.12). Ultimately, Zixia summed it up most appropriately: \"When one has the leisure from studying, one should pursue an official career\" (Analects 19.13). Every student could have his own timetable.\n\nWhy did people come to study with Confucius? If we look at Confucius' curriculum, we may get a telling clue. Among many other things taught in the Confucian community, the so-called Six Arts were well known. They were rituals, music, archery, charioteering, writing, and mathematics. These were practical skills required for employment in the service of feudal vassals. In other words, people came to Confucius to learn the skills for a respectable and, hopefully, profitable career under the patronage of feudal vassals, who in many cases were then trying to recruit fresh talents to help them scramble for political clout and power. The prospective students came to find a way to improve their material life; that is why some of them would even ask about the art of agronomy and horticulture (Analects 13.4). Students such as Zizhang \u5b50\u5f35 (503-? B.C.E.), who had higher aspirations, would seek instructions on how to pursue an official career (Analects 2.18). All in all, the prospective student certainly did not come to find out what his self was and how to cultivate it.23 Once, Confucius recommended Qidiao Kai \u6f06\u96d5\u958b (540-? B.C.E.) for an official post, and when the disciple declined because he did not yet feel confident, the master was visibly pleased (Analects 5.6). Confucius was pleased because Qidiao Kai's modesty reflected his seriousness in pursuing the Way, which was evidently somewhat uncommon in the Confucian community. Hence, we can almost hear Confucius' inaudible sigh when he bemoaned that \"It is not easy to find a man who, after studying for 3 years, does not think about a salary\" (Analects 8.12).\n\nWhat distinguishes Confucius from other teachers in or before his time is that he discouraged learning strictly for material gains. When Zizhang came from the far away state of Chen \u9673, to express his wish to study with the master in hopes of an official career, Confucius admonished the young man, who was 48 years his junior, that he should study for the sake of learning and for his own good. If he could avoid making mistakes in his words and deeds, the master emphasized, an official career would only be a matter of course (Analects 2.18). To another student Zixia, who was 44 years younger than himself, the master also exhorted him to be a \"gentlemanly scholar\" (junzi ru \u541b\u5b50\u5112), not a \"petty\" one (xiaoren ru \u5c0f\u4eba\u5112) [Analects 6.13]. After all, the master said, \"The gentleman seeks the Way; he does not seek to be fed. Plough the fields and you may still go hungry. Apply yourself to learning and you may yet make a career. Therefore, a gentleman worries that he may not find the Way, he does not worry that he may remain poor\" (Analects 15.32). On numerous other occasions, Confucius reminded his students repeatedly that the true noble scholar (shi \u58eb) should aspire for the Way rather than merely worrying over his material provisions. Hence, he said, \"A noble scholar sets his heart on the Way, and if he feels ashamed of shabby clothes and coarse food, he is not worth any consideration\" (Analects 4.9).\n\nIn the Analects, numerous chapters deal with the subject of wealth and poverty and they should be read in light of the master's emphasis on the pursuit of the Way even if at the expense of acquiring wealth. Confucius himself confesses, \"Even though I may have coarse grain for food, water for drink and my bent arm for a pillow, I'll find joy therein. Wealth and honor inappropriately acquired are nothing but fleeting clouds to me\" (Analects 7.16). Being his true self and being joyful are what Confucius seeks for himself; this is one of the crucial implications of what he calls \"learning for one's own self\" (weiji \u70ba\u5df1) [Analects 14.24]. For this reason, he praises Yan Hui for being able to enjoy himself with simple food and drink even in a miserable dwelling (Analects 6.11). By the same token, he is proud of Zilu \u5b50\u8def (542\u2013480 B.C.E.) for his being able to stand in his tattered gown without feeling ashamed alongside people who wear fine furs (Analects 9.27).\n\nMaterial provisions pertain to personal, petty considerations while the pursuit of the Way demands lifelong dedication to moral and cultural obligations (Analects 8.7). As Confucius said, \"You can make the gentleman understand an issue in light of what is right, whereas you can make the petty man understand an issue in light of the gains involved\" (Analects 4.11). Material provisions can be acquired by offering one's skills and expertise to a feudal lord while the self-appointed mission of carrying on with the Way makes one a morally fulfilled person and authenticates one's human existence. That is why Confucius reminded his students, \"People in the past learned for their own selves (weiji \u70ba\u5df1) whereas people today study for others (weiren \u7232\u4eba)\" [Analects 14.24].\n\nNevertheless, material provisions and the pursuit of the Way are not necessarily incompatible goals. A sensible man as he was, Confucius acknowledged that \"wealth and rank are what people desire\" (Analects 4.5) and he himself claimed that if he could acquire riches by driving a chariot, he would do it (Analects 7.12). On the other hand, one of Confucius' goals in education was to train political talents, and this was particularly emphasized in the early phase of his teaching career. This is what he meant by \"learning for the sake of others,\" namely, to serve in the political arena with one's acquired learning and skills. That was the reality of his time and he acutely realized it. Therefore, Confucius did intend his disciples to pursue an official career, and perhaps to become wealthy. While Yan Hui was well known for his moral accomplishments, it is erroneous to assume that he was averse to a career in political governance,24 and he was indeed a disciple from the early phase of Confucius' teaching career when the master was primarily intent on training political talents. As mentioned earlier, he would urge his disciples to pursue an official career when they were ready. Contrary to common assumptions, Confucius never meant to emphasize learning only for learning's sake. For him, one of the main goals of the study of the Odes, for instance, was to provide training in politics and diplomacy (Analects 13.5, 17.9). Indeed, Confucius' professed purpose of education was so well known that \"scouts\" from other states would come to find out if any of his disciples was suitable and ready for employment (Analects 5.8, 6.8, 11.24, 11.7).25\n\n## 4 The Confucian Program of Learning\n\nIn spite of the evolving spiritual progress of Confucius and the myriad differing motives of his disciples, the distinctive calling to study for one's own self was no doubt the trademark of the Confucian community. The calling was understandably challenging and it might be a clarion call to the disciples to recognize and confront their inner selves that were deep in the core of their persons. For instance, when Zaiwo \u5bb0\u6211 suggested that the typical 3-year mourning period for one's deceased parents be reduced to 1 year, Confucius asked him if he would be able to feel at ease eating white rice and wearing fine silk after only 1 year of mourning. The master was not trying to persuade the seemingly unfeeling disciple on rational grounds; rather, he was compelling him to wake up to his genuine feelings for his parents and become morally alive. And when Zaiwo answered in the affirmative, the master could only say in distress that Zaiwo was not humane (buren \u4e0d\u4ec1) [Analects 17.21].\n\nThis episode concerning Confucius and Zaiwo captures the spirit of the master's teaching most graphically. First of all, Confucius respects Zaiwo as an independent person who can will and feel on his own. This is the baseline for the master's teaching. Therefore, as much as he wants to see a change of heart in his disciple, Confucius does not appeal to the \"universal\" tradition of mourning practice; neither does he abuse his own authority as a teacher. Confucius once asked Yan Hui rhetorically, \"In the practice of humaneness does a person follow [the feelings and will of] his own or [those of] others?\" (Analects 12.1) We can thus appreciate that when the master criticizes Zaiwo for being inhumane he is saddened that his disciple has failed to establish himself on his own. The self of Zaiwo is not sensitive enough to erect itself; it appears to stoop to material comfort and selfish enjoyment.\n\nConfucius firmly believes that an individual should and must discover for himself his inner core of being where the sprout of humaneness lies dormant. The dormant humaneness would spring to life upon its host interacting with his fellow beings, the most important of whom are his parents and family members. Only then will the personal self be gradually nurtured into moral fruition through a lifelong process of becoming in the art of interpersonal negotiation. The nurturing of the personal self entails determination to sustain its growth and development vis-\u00e0-vis an expanding circle of human relationships.\n\nSecond, while we all recognize that Confucius was particular about rituals, we may overlook the fact that he was indeed unrelentingly critical of the rituals of his time. On most occasions in the Analects where he discussed rituals, he was actually showing his dissatisfaction over or even disgust at the ritual abuses of the day.26 For him, the spirit of rituals did not lie in their exterior forms which might very well have become fossilized and outmoded (Analects 17.11). Confucius was keenly interested in revitalizing and refashioning many of the obsolete rituals prevailing in his time (Analects 9.3). However, for him, the 3-year mourning ritual was not ready to change. He did not look at the 3-year mourning as a rigid custom; rather, he considered it a most appropriate ritual gesture to express our genuine feelings for our deceased parents. In other words, the ritual gesture simultaneously embodied and revealed our inarticulate feelings. But more importantly, for Confucius, the ritual form was not only integral to our love and respect that were amorphous, but it indeed consummated and embodied them and authenticated us as bona fide human beings.\n\nThe 3-year mourning ritual, then, was still a very much vibrant legacy of the traditional culture which Confucius inherited and admired. The observance of the ritual would make us not only morally but authentically indebted to a time-honoured cultural tradition whose revival and continual transformation was Confucius' lifelong dream. When we are thus incumbent, we can be called an authentic shi, or a noble scholar. Zeng Can \u66fe\u53c3 (honorifically, Zengzi \u66fe\u5b50, 505\u2013435 B.C.E.) describes it best when he says, \"To be a noble scholar, one cannot afford not to be strong and resolute, for one's burden is heavy and one's road is long. When a person takes up humaneness as his burden \u4ec1\u4ee5\u7232\u5df1\u4efb (lit., the burden for his self), is that not heavy? When the road ends only with his death, is that not long?\" (Analects 8.7) On the one hand, the intimate relationship between one's personal self (ji \u5df1) and humaneness is highlighted here. The analogy of a lifelong journey demanded of the humane self, on the other hand, suggests that the individual must carry on a cultural tradition much larger than his personal self. Given such an unyielding demand on the aspiring noble scholar, we can appreciate why some disciples felt intimidated and were too feeble-hearted to take up the noble challenge. Ran Qiu, for instance, told Confucius that even though he was fond of pursuing the Way as the master taught, he did not have the moral stamina to sustain his efforts (Analects 6.12). Perhaps for this very reason, Confucius often exhorted his disciples to apply themselves to follow the dictates of their humane self, even if only for one single day (Analects 4.6).\n\n## 5 Harmony and Tension\n\nAs has been analyzed, the Confucian community consisted of an intricate mesh of human relationships. Existentially speaking, then, the cultivation of the humane self was inevitably implicated in the crisscrossing of human interactions on a regular, if not daily, basis. In fact, Confucius himself offered an emphatic assurance that \"a virtuous person will not stand alone; he is bound to have neighbors\" (Analects 4.25). Thus a virtuous person should expect to cultivate himself in a neighborhood of like-minded people. It was only with this communal understanding that Confucius would praise the living in a neighborhood of humane residents as \"beautiful\" (Analects 4.1).\n\nFor a community to function properly there must be some guiding principles to which its members will adhere in negotiating their daily transactions with one another. Even though Confucius expressly pronounced that he found it difficult to educate people who could spend a whole day together showing off their shallow wits without ever saying anything appropriate (Analects 15.17), as far as we know, he did not formally stipulate any rules to regulate the Confucian community. Yet, it should have been clear to all his disciples that the master exhorted them to become true gentlemen.27 After all, the disciples understood that the purpose of the collective life in the Confucian community was to cultivate humaneness with each other's assistance (Analects 12.24).28 In this connection, we can imagine that what Confucius had said on numerous occasions about the self-definition of the noble scholar must have guided all members in the Confucian community in their everyday lives. Confucius, for instance, said, \"The gentleman should be proud without being contentious, sociable without being partisan\" (Analects 15.22). Apparently, the only occasion on which the gentleman would engage in competition is archery (Analects 3.7). Clearly, the master asked his disciples to respect themselves as honorable denizens in his community; this self-respect must have come from their determination to become gentlemen. And with a strong will of which even an army cannot deprive (Analects 9.26), every aspiring gentleman should have a sense of lofty pride. But as a member of the Confucian community, everyone should also learn to be sociable and act according to what is right and appropriate. Since partisanship and competition for personal advantages tend to undermine the solidarity and camaraderie of the community, they are discouraged. In fact, Confucius made it crystal clear that \"The gentleman seeks harmony, but not conformity\" (Analects 13.23). No doubt, the master would like to see harmony among his disciples who lived together in a close-knit family that was the Confucian community. Indeed, he evidently enjoyed the company of his disciples in such a harmonious environment (Analects 11.13). Yet, at the same time, Confucius also respected the differences of his disciples, and he no doubt would try his best to foster mutual respect among them.\n\nThe camaraderie of the disciples clearly reveals itself in the various interactions among themselves. One of the most visible expressions of such amity is mutual sharing among the disciples. For instance, when Zihua \u5b50\u83ef was sent on a mission to the state of Qi \u9f4a, Ran Qiu requested an allowance of grain for Zihua's mother from Confucius (Analects 6.4). Similarly, when Yan Hui died prematurely, we can understand why his father Yan Lu \u984f\u8def would ask Confucius to spare his own carriage in order to provide for an outer coffin for his son (Analects 11.8). Yet, many other disciples also pleaded to give their brilliant classmate a lavish burial. And in spite of the emphatic objection of the master, they proceeded with their plan (Analects 11.11). Their love and respect for their fellow student is unmistakable. Once, Confucius had a conversation with Yan Hui and Zilu in which they each shared their personal aspirations. For Zilu, he wished that he could share his carriages, horses, clothes, and furs with his classmates and friends without being upset when they damaged his property (Analects 5.26). This seems to be a less than lofty aspiration, but in light of the noble spirit of mutual assistance in the Confucian community, Zilu's aspiration to unconditional sharing was indeed noble enough.29\n\nThe shared destiny of the Confucian community perhaps was most readily seen when the master and disciples encountered dangers on the road. The best evidence for the solidarity in the Confucian community is revealed in the fact that several times when Confucius and company were trapped on their journey and their lives were threatened, everybody stayed together in one common destiny with unswerving faith in the moral values their master had taught them. Even though their master was marooned like a \"dog without a home\" (sangjia zhi gou \u55aa\u5bb6\u4e4b\u72d7), none of the disciples deserted him or the Way they pursued with him. In fact, they fought hard to defend their master and their Heaven-ordained destiny.30 In the case of Yan Hui, when Confucius and his disciples were stranded in Kuang \u5321, Yan Hui and Confucius became separated during the chaos. When the two were eventually reunited, the master said, \"I thought you were dead.\" In reply, Yan Hui said, \"While the master is still alive, how would I dare to die?\" (Analects 11.23) It is interesting to note that Yan Hui's father was still alive and well when this narrow escape from death took place; yet, the reason Yan Hui did not dare to die was because he believed Confucius had survived the life-threatening episode. This suggests that the spiritual bond between Yan Hui and Confucius was even greater than the blood relation between the devoted disciple and his father Yan Lu.\n\nThe camaraderie of the disciples can certainly be considered as an expression of harmony in the Confucian community. It should be noted that in the Confucian doctrine while harmony recognizes moral egalitarianism, it thrives on social distinctions. Thus, even though Yuoruo \u6709\u82e5, who was 30 years younger than Confucius, concluded that the application of the rites prized harmony (Analects 1.12), we should not overlook the fact that the spirit of rituals actually lies in making and affirming social distinctions. Confucius certainly recognized social distinctions and in fact took them seriously. Allegedly, he advocated the doctrine of the rectification of names (zhengming \u6b63\u540d) [Analects 13.3], and he taught that we should respect our elders and superiors. Yet, it is not clear whether seniority did carry any actual weight and was respected in the Confucian community. In fact, there appears to be signs that indicate that seniority was not always given due respect.\n\nWhile we expect that most of the disciples would learn the art of music (one of the Six Arts), not all of them were equally skilled in it. Zilu was one of the students who came to study under Confucius in the master's early teaching career. Perhaps not insignificantly, Zilu was only 9 years the master's junior. As far as we know today, he was the second oldest student in the Confucian community. The Analects certainly portrays him as an intimate companion to Confucius, and therefore we would expect that he commanded respect among his classmates. Apparently he did until 1 day when Confucius seemed to complain about Zilu's musical skills, and on that account some of the disciples ceased to respect this senior student. As a result, Confucius attempted to resolve this insidious discord by saying that \"Zilu had actually ascended to the hall even though he had yet to enter the chamber\" (Analects 11.15). In other words, Confucius personally endorsed Zilu's accomplishments as his student. At the very least, this seems to be an unusual act of intervention by the master. Why would Confucius feel compelled to speak in favor of Zilu? We do not know how effective the master's endorsement of Zilu's accomplishments turned out to be, but apparently, a disciple's status in the Confucian community had something to do with the appraisal he received from the master. In fact, the master's approval seems to matter a lot more than a disciple's own accomplishments. More importantly, the respect a disciple commanded among his fellow students might not be commensurate with his seniority in the community, and this suggests that while Confucius insisted on the junior showing respect to his senior, this doctrine was not necessarily practiced even in the very community where it was taught.\n\nIn the Analects, we sometimes find the disciples soliciting their master to comment on their own abilities (Analects 5.4), and they sometimes seem to ask their master bluntly to compare their talents to those of their classmates. These queries seem innocuous enough, but if we examine them in the context of an emergent Confucian community whose denizens supposedly took it upon themselves to live in strict accordance with what they had personally learned from their master and realized their shared commitment to a noble ideal of gentlemanly living, then these innocent inquiries might reveal traces of tension and rivalry among members of the community.\n\nAs mentioned earlier, Confucius acknowledged that Yan Hui was very much like a son to him. The master also made it known that Yan Hui was the only student who was fond of learning as much as he was, even after his premature death. We can imagine that the master would publicly show his special affection and care for his beloved student as well. No doubt Yan Hui was a gifted and hardworking student, but the slighting of Zilu by his classmates suggests that talent and industry might not be sufficient to command respect in the Confucian community. And in Yan Hui's case, he lived in unusual poverty31 and even the master admitted that he appeared to be dumb in his presence (Analects 2.9). He simply did not seem to stand out in any particular way. All things considered, Yan Hui's esteemed status must have owed a great deal to the master's generous accolades to him.\n\nWith this in mind, we can understand why other disciples sometimes ventured to ask Confucius to compare themselves with Yan Hui. And we must emphasize that the comparisons seem to focus on Yan Hui, Zigong, and Zilu, all of them being senior and advanced students in the Confucian community. To be sure, it is not clear whether all of these recorded comparisons were sought by the concerned students, yet given their senior status and plausibly high profiles in the Confucian community, it is possible that some of the concerned students might have a following even though that might very well go against their own wishes. It is probable that the \"secret\" followers of these esteemed students would try to settle the rank order of their heroes among themselves by seeking the master's assessment on the matter. For instance, Confucius was reported to have said, \"Yan Hui came close to perfection, I suppose, and yet he often suffered poverty.32 Zigong did not accept his lot and went into trading; his speculation is often right\" (Analects 11.19). Granted that we do not know why Confucius would make such a comparison, it seems clear that it was made for the curiosity of someone who was interested in ranking two of the most gifted students in the Confucian community.\n\nIt is also noteworthy that while we appreciate Confucius' candor, his answer was purposefully unequivocal as if he wanted to settle the matter once and for all. This apparent intention is even more obvious when the master initiated the comparison between Yan Hui and Zigong on another occasion. We do not know why Confucius would bring up such an issue; perhaps Zigong himself had said something to trigger the comparison in the first place. In any case, here is what happened.\n\n> Confucius asked Zigong, \"Which is better, you or Yan Hui?\"\n> \n> \"How do I dare to compare with Hui? Hui can deduce ten things from one thing he learns but I can only deduce two from one thing I learn,\" replied Zigong.\n> \n> The master said, \"Indeed, you are not his equal; and neither am I\" (Analects 5.9).\n\nIf Confucius indeed wanted to put a stop to the kind of potentially insidious comparison he was asked to make, his affirmative answer to Zigong could not have been more effective. If the master himself was not Yan Hui's equal, then who could be?\n\nIf it is unfair to second-guess Zigong's intention in the comparison between him and Yan Hui, Zilu could hardly extricate himself from his jealousy of Yan Hui. Zilu evidently would have liked to win the favor of Confucius. Once, Confucius regretfully wished that he could take a raft and put out to sea since he was disappointed that the Way did not prevail in his time. And he added that perhaps Zilu was the only person to accompany him on the trip. Upon hearing this, the senior student was overjoyed (Analects 5.7). It is irrelevant that Confucius never did take the trip, but Zilu's reaction to Confucius' suggestion clearly indicates that he cared very much about what kind of place he might occupy in the heart of the master.\n\nThus it is understandable that when Confucius gave unconditional and exclusive approval to another disciple, Zilu might squirm in jealousy. Once, Confucius had a conversation with Yan Hui and Zilu, and the master said, \"To come out when needed and to hide when dismissed \u2013 only you and I can do this.\" Zilu was evidently displeased and he ventured to challenge the master. Since he was known for his courage and martial skills, Zilu confronted the master by asking, \"If you had command of the Three Armies, whom would you take as your lieutenant?\" (Analects 7.11) Obviously, Zilu was trying to compel Confucius to ascertain his own niche in the master's heart. We cannot but sense a subtle rivalry between Yan Hui and Zilu.\n\nPerhaps Zilu's jealousy was not unique to him. When Yan Hui died, Confucius was greatly saddened and he wailed wildly. The disciples in the funeral procession then reminded him, \"Master, is such grief not excessive?\" (Analects 11.10) We should recall that Confucius normally advocated the doctrine of the Mean, so it is easy to understand why his disciples found him a little dramatic in showing his grief in public.33 Perhaps more importantly, Confucius apparently did not show the same grief when his own son Boyu died.34 Yet, in the final analysis, it is more likely that the disciples felt that Confucius might be too visibly partial to his beloved disciple.35\n\nIt should be pointed out that even after the death of Confucius, the rivalry between some disciples apparently continued to persist. It did not involve Yan Hui as he had already predeceased the master. The most notable rivalry, and perhaps the only one that existed, was between Zizhang, Zixia and Ziyou \u5b50\u6e38 (506-? B.C.E.), the latter two being the exemplary students of literature in the Confucian community (Analects 11.3). According to the Analects, the rivalry between these second-generation masters in literature, in fact, might very well have been purely academic (Analects 19.3, 19.12, 19.15) and did not involve any personal competition or rivalry for status or for reputation. Their differences concerned approaches to self-cultivation but their common objective that had been passed down from Confucius had not changed. They all strove for the consummation of humaneness (Analects 19.15, 19.16). In the end, they remained faithful to the fundamental doctrine of their master: \"The humane person wishes to help others achieve what he wishes to achieve for himself; he wishes to help others to obtain what he wishes to obtain for himself\" (Analects 6.30).\n\nHowever real the jealousy and rivalry were among some disciples in the Confucian community, evidently they did not turn out to be destructive or divisive. The disciples actually lived in communal peace. Even if some disciples did feel jealous of Yan Hui, Yan Hui was indeed a positive influence that pervaded in the Confucian community. The master himself admitted that since he had admitted Yan Hui to the Confucian community, his disciples became much more cohesive. Confucius said that in teaching he never hid anything from his disciples (Analects 7.24) and even his own son did not have any privilege in this regard (Analects 16.13). In other words, he treated everyone in the Confucian community with equal respect and fairness; he did not play favoritism. This spirit of fair-mindedness inspired all members of the Confucian community and overrode petty jealousy and rivalry before and after the death of Confucius. We learn that many disciples chose to settle around the gravesite of the deceased master and thus the Confucian community continued to exist both physically and spiritually and throve in the absence of the master (Takigawa 1977: juan 47, 746\u2013747). Yet, none of the senior disciples came forward to assume the leadership of their community. When someone suggested that Zigong was actually more worthy and competent than the master himself, the senior disciple readily and emphatically dismissed the insinuation that he could and should now assume the moral leadership of the Confucian community (Analects 19.25). Zigong's refutation not only was meant to defend the unrivalled greatness of the master, but it also declared that all disciples of Confucius were truly equal and respected one another. There might be occasional innocuous rivalry but it was perhaps only meant to win the favor or get the attention of the master whom they dearly adored. In the final analysis, it can be argued that the disciples by and large followed their master's teaching; they acknowledged both that they were all different in their abilities and talents and that they might have a different yet unique niche in their master's heart. They could live together in harmony while respecting one another's differences (Analects 13.23). Thus, they did not need a moral leader. In spite of Confucius' death, the earliest Confucian community did not disintegrate because of factional strife, as every member sincerely upheld what the master had taught.\n\n## 6 Confucian \"Family\"\n\nThe Confucian community virtually constituted a family in terms of its living environment, fictive kinship, emotional bonding, and philosophical outlook. Confucius was revered as not only as a surrogate father of the family but also the \"father\" of the \"school\" (jia \u5bb6, literally, family) of thought called Confucianism (rujia \u5112\u5bb6). When the Han Grand Historian Sima Tan \u53f8\u99ac\u8ac7 (?-110 B.C.E.) classified the six major philosophical doctrines of the pre-Qin period, he considered each of them to be forming a \"family\" of its own. The idea of \"family\" was not merely a convenient metaphor but actually signified the historical reality that a philosophical doctrine actually resulted from the collective wisdom of numerous thinkers over generations who shared a common core of fundamental concerns as well as approaches and solutions to them. 36 No doubt the metaphor is most apt and realistic for Confucianism as the doctrine developed over time along then recognizable lineages even after the demise of Confucius. To express his extraordinary veneration for Confucius, Sima Qian violated his own organizational principles of his magnum opus Shiji and accorded the master a special biography in the category of hereditary houses which was reserved only for feudal lords and nobles of hereditary enfeoffments, who of course established their biological families through actual kinship. The analogy hinges upon the fictive kinship relations between the master and his disciples, which not only signified their emotional bonding but their intellectual affiliation as well. Graphically, the portraits of Confucius and his 72 disciples were often painted together in the Imperial Academy and tombs, and the portraits of the disciples rarely appeared by themselves.\n\nAs Confucius' teaching career spanned over four decades or so, students came to follow him at different times. In the master's own words, they can be classified into the early (xianjin \u5148\u9032) disciples and later (houjin \u5f8c\u9032) disciples (Analects 7.7), who began to study with Confucius when he returned to his home state at the age of 68 after a 14-year-long peripatetic search for a feudal lord who would implement his political vision. Thus roughly speaking, there were two generations of disciples from the earliest Confucian community. Owing partly to the personal transformation of Confucius himself as a philosopher and partly to the change of the ethos in late Chunqiu times, the learning of the early and later disciples differed in character. The early disciples such as Zilu, Ran Qiu, and Zigong aspired to employ themselves in politics while Ziyou, Zixia and Zengzi were keen on studying culture, which includes rites and music (Qian 2002: 94\u201395). Of course, there are exceptions, and some disciples such as Yan Hui and Zigong actually straddled both generations. In any case, the later disciples actually classified the specialties of both generations into four main categories: virtue, oratory, government, and culture (Analects 11.3). On the other hand, Confucius in his teaching referred to literature and real-life situations and instructed on how to do one's best and act in good faith (Analects 7.25). Evidently, the disciples' classification and the master's pedagogy coincide.\n\nThe disciples were not trained as philosophers; they received training in the Odes, the Documents, the rites, and music; these were all practical arts for political service and governance. Much worse, with the exception of Zengzi, virtually none of them left us with any writing of their own. The bibliographical section of the Han shu (Han History) lists only three works by Confucius' disciples and four by his second-generation disciples.37 Except for a redaction of the work of Zengzi in ten scrolls, none survives today. To make things even more complicated, the disciples' works all belonged to a philosophic family and thus similarities might overshadow individual differences. Toward the end of the second century B.C.E., Han Fei \u97d3\u975e reported that Confucianism had developed into eight branches, each of which was identified with the family name (shi \u6c0f) of its purported founder. Granted that Han Fei's characterization is ambiguous and puzzling, it nonetheless implies that Confucianism was perceived as comprising an extended family over generations. Only three of these branches \u2013 Yan shi \u9854\u6c0f (Yan Hui),38 Zizhang shi \u5b50\u5f35\u6c0f (Zizhang), and Qidiao shi \u6f06\u96d5\u6c0f (Qidiao Kai) \u2013 can be attributed to the immediate disciples of Confucius. Unfortunately, Han Fei mentioned nothing about their doctrines. What is certain is that only the later disciples and their own followers began to distinguish themselves with their doctrines. The Confucian texts from the pre-Qin period archived in the Han imperial library confirm this phenomenon. As we all know, Confucius had little to say about human nature but according to Wang Chong \u738b\u5145 (27\u2013?97), the issue of human nature became a preoccupation among his later disciples and their own followers. Wang said,\n\n> Shi Shuo \u4e16\u78a9 (a second-generation disciple of Confucius)39 believed that human nature contains both good and evil. Focus on the good in human nature and nurture it to its full development, good will grow. [Focus on] the evil and nurture it to its full development, evil will grow. Thus human nature contains yin and yang as well as good and evil; what matters is nurture. Thereupon Shizi composed the Book on Nurturing in one chapter. Mi Zijian \u5b93\u5b50\u8ce4 (521\/502-? B.C.E.), Qidiao Kai and Gongsun Nizi \u526c\u5b6b\u5c3c\u5b5040 also debated about feelings and human nature and [their views] corresponded to that of Shizi in that they all held that human nature contains both good and evil.\n\nConfucius reportedly only said, \"By nature humans are similar to one another and by practice they become far apart\" (Analects 17.2). And from everything else he said, he seemed to entertain a much broader view of human nature and did not exclusively confine it to the dualistic moral categories of good and evil.41 In this sense, the later disciples did divert from their master in their understanding of human nature. Insofar as the anonymous authors of the recently unearthed bamboo texts related to the Confucian doctrine from around the fourth century B.C.E. can be considered to be later-generation disciples of Confucius, they also provide evidence to this proliferation and diversification of the Confucian doctrine on human nature. For instance, in the Guodian bamboo text now entitled Xing zi ming chu \u6027\u81ea\u547d\u51fa (Human nature comes from destiny) it explicitly says, \"It is human nature that there is good and not good\" (\u5584\u4e0d\u5584,\u6027\u4e5f).42 Still later, even though Mencius and Xunzi held seemingly diametrically opposite views on human nature, they inherited the same dualistic framework of good and evil from Confucius' later disciples.\n\n## 7 Yan Hui\n\nGiven the various difficulties as mentioned, I will focus on the philosophy of the immediate disciples of Confucius only, and in particular, Yan Hui, Zengzi, and some other later disciples. This is because Yan Hui, an early disciple, was unanimously respected as the most accomplished in the Confucian community and enjoys the same prestige even until today. Beginning with the Han dynasty he was worshipped along with Confucius himself and in 1,330 Emperor Wenzong \u6587\u5b97of the Yuan dynasty bestowed upon him the title of Lord of Realized Sagehood (fusheng gong \u5fa9\u8056\u526c). The Fusheng Temple (\u5fa9\u8056) in his honor still stands in Qufu, Shangdong today. Zengzi was probably the most respected among Confucius' later disciples and we are fortunate to have a partial redaction of his sayings. Some of the later disciples are also discussed because they had new philosophical insights that perhaps might have set them apart from the majority of their cohort.\n\nInformation about Yan Hui in the Analects is scanty but it should be emphasized that not only did he figure quite prominently in many other pre-Qin and early Han accounts, but they are also consistent with his basic character in the Analects. In a sense, we can say that Yan Hui enjoyed an afterlife that was richer in detail than his physical one. Inasmuch as the two lives are consistent, coherent, and unified, they can be viewed as one, philosophically if not strictly historically. Based on the Analects, Yan Hui pursued both learning for one's sake and learning for others' sake but he was more inclined to embrace the former. Contrary to common perception, he appears to be fully capable of political leadership (Analects 15.11; Hightower 1952: 49\u201351). Personal temperament and perhaps the unconducive political circumstances contributed to his decision.43 He was determined to attain self-sufficiency and spiritual autonomy, and his self-cultivation was particularly focused on the heart-mind. He was able to unify himself with his environment with his personal self in a blissful oblivion; he was also able to manage his own feelings and maintain them in harmony. He evidently was able to emanate a unique charm that could draw people together and help them cohere. This was the remarkable achievement of Yan Hui's learning for his own sake.\n\nConfucius once compared his disciples in terms of their spiritual and moral accomplishments and he said, \"[Yan] Hui does not violate humaneness in his heart-mind for 3 months without lapse while the others attain it only now and then\" (Analects 6.7).44 This perhaps is the highest virtue Yan Hui attained in his short life and should be recognized as the bodily expression of his philosophy. For Confucius, humaneness was without doubt the highest virtue, which he seldom volunteered to discuss (Analects 9.1). This is a rare occasion and he pinpointed the locus of the virtue in one's heart-mind.45 It cannot be overemphasized that Confucius said that he himself \"followed the wishes of his heart-mind without transgression\" (Analects 2.4) when describing the apex of his lifelong self-cultivation. It can be assumed that the master's achievement was not limited to any length of time. This sustained effortless expression of the wishes of the heart-mind always in accordance with what is appropriate in a given circumstance is a succinct description of what humaneness is as an embodied experience. It is a unity of the inner and the outer as well as self and other. Yan Hui was not the most talkative disciple and in fact he quietly observed Confucius' teaching without so much as uttering a word so that he even appeared stupid to the master, but it is important to note that Confucius realized that Yan Hui's practices could actually illuminate his own teachings (Analects 2.9). Yan Hui's embodiment of humaneness should thus be considered as his personal understanding and practice of the master's doctrine on that virtue. According to a dialogue between Confucius and Yan Hui reported in the Xunzi, when Yan Hui was asked to describe the wise person and the humane person, he replied, \"The wise person knows himself and the humane person loves himself\" (\u77e5\u8005\u81ea\u77e5,\u4ec1\u8005\u81ea\u611b) [Wang 1997: 2:533]. Insofar as the report is reliable, and it is certainly consistent with what the Analects tells us, Yan Hui's cultivation of humaneness centres on the self and specifically, on self-understanding and the management of one's own feelings.46\n\nConfucius said, \"People of the past learned for themselves (weiji \u70ba\u5df1); people today learned for the service of others (weiren \u7232\u4eba)\" (Analects 14.24). It is often misunderstood that learning for the sake of others is morally undesirable as one is perceived to be learning just to impress others rather than for one's own good.47 In fact, what Confucius meant by weiren is that one learns in order to be employable in the government so that one can serve one's lord as well as the people under his care. One of the main goals of the master's popular education was to train people in the practical skills of governance, and he would urge his disciples to take up a post in the government when he deemed them ready (Analects 5.5). Indeed, many if not most of his disciples came to receive training from the master precisely for a political career. Thus the master could not but sigh that \"it is not easy to find a man who does not think of earning a salary after 3 years of learning\" (Analects 8.12). The purpose of a political career no doubt is to serve others \u2013 it is the ultimate goal of learning for others' sake. Yet if a person's goal of serving others is to earn a salary, he does not live up to the master's expectation of a noble scholar (shi \u58eb), who sets his heart on the Way and would not feel ashamed of having poor food and poor clothes (Analects 4.9; cf. Analects 15.32). As mentioned above, Confucius' disciples classified themselves into four main categories of virtue, oratory, government, and culture. Of these, only virtue, which is not a skill in an ordinary sense, is the goal of learning for one's own sake; the rest are all technical expertise that could be used to serve others.48 Confucius reminded his disciples that \"the gentleman is not a utensil\" (junzi buqi \u541b\u5b50\u4e0d\u5668) [Analects 2.12]. One would be a mere utensil if one learns with the only goal to serve others.\n\nNeedless to say, the two kinds of learning are not incompatible with each other. That Yan Hui was versed in the practical skills of governance and well-rounded in the abilities of political leadership, and that he excelled in the cultivation of virtue suggests that he was a true gentleman, not a utensil. On one occasion, he revealed to Confucius his dual ambition that he should like never to boast of his own goodness (fashan \u4f10\u5584) and show off his meritorious achievements (shilao \u65bd\u52de) [Analects 5.26]. Evidently, Yan Hui aimed at both learning for his own sake and learning for the sake of others. Little wonder that Confucius proudly acknowledged that only he and Yan Hui had the ability to practice what they had learned when employed and to hide themselves when dismissed (Analects 7.11). Confucius in effect announced that Yan Hui was the only disciple who could actually master both learning for one's own sake and learning for the sake of others.\n\nWhile learning to serve others is a perfectly legitimate endeavor, learning for one's own sake clearly takes precedence and assumes fundamental importance in self-cultivation. Confucius praised Yan Hui for not taking his anger out on others (bu qiannu \u4e0d\u8cb3\u904e) and not committing the same mistake twice (bu erguo \u4e0d\u4e8c\u904e) [Analects 6.3]. Both achievements are grounded solidly in the cultivation of the self; they are the outcomes of learning for one's own sake. Evidently, the ability to manage one's anger is a sign of cultivation of the heart-mind. Yan Hui was not just able to contain his anger within himself; rather, he could deal with it on its own without unburdening it on others. He could master his anger without being enslaved and driven by it as this would mean he had lost control of his self. He could express his anger as he wished yet without violating the proper confines of personal anger. He is the master of his anger, not the other way around. In the words of the Zhongyong \u4e2d\u5eb8 (Doctrine of the Mean), Yan Hui's anger probably can attain due measure and degree (zhongjie \u4e2d\u7bc0) when aroused and his heart-mind would be in a state of harmony (he \u548c). Confucius once lamented that the virtue of zhongyong (mean) was rare among the common people in his time (Analects 6.29), but we may wonder if Yan Hui was not one of the rare exceptions.\n\nYan Hui's other outstanding accomplishment is to not repeat the same mistake. This may seem insignificant to us but it was not the case with Confucius or Yan Hui himself. Notably, Confucius in the Analects often talked about the issue of making mistakes.49 He was keen in differentiating mistakes in light of people's various temperaments and levels of cultivation (Analects 4.7). He emphasized learning from one's mistakes and regretted that he had never seen anyone who could see his own mistakes and interrogate himself in his heart (Analects 5.27). Confucius was visibly impressed when Qu Boyu's \u8627\u4f2f\u7389 messenger told him that his lord wanted to minimize his errors but had not been successful (Analects 14.25). Thus it is clear that Confucius was deeply concerned about doing things right. Even after he had attained the mind that was free of doubts at the age of 40 (Analects 2.4), he wished that he could live on to the age of 50 in a few more years so that he could study the Book of Changes and free himself from major mistakes (daguo \u5927\u904e) [Analects 7.17].50 We should keep in mind that his self-proclaimed spiritual accomplishment at the age of 70 is all about doing things right in a spontaneous seamless harmony of inner self and outer dictates.\n\nIn order to understand precisely the merit of Yan Hui's ability to avoid repeating the same mistake, it is important to know what kind of mistakes Confucius' might have in mind in summing up his beloved disciple's moral accomplishments. When Yan Hui asked Confucius about humaneness, he was told to cultivate it by returning to the observance of the rites through overcoming the self (keji fuli weiren \u514b\u5df1\u5fa9\u79ae\u7232\u4ec1) [Analects 12.1]. And Confucius reminded him that the cultivation of humaneness comes from one's self (weiren you ji \u7232\u4ec1\u7531\u5df1). When Yan Hui ventured to ask about the method of cultivation, the master told him not to see, listen, speak, or act when it is not in accordance with the rites. In the Analects, seven disciples had asked their master about humaneness on different occasions; all except Yan Hui focused on its political dimension.51 Naturally Confucius gave them different answers and it was only to Yan Hui that he referred to specific practices of self-cultivation.52\n\nIn the entire Analects, Yan Hui had asked Confucius only two questions: one on humaneness and one on governance (Analects 15.11). No doubt they were of utmost concern to him. The four negative imperatives Confucius specified are noteworthy as they were intended to help Yan Hui navigate the minefield of ritual transgressions. In this light, it seems probable that Confucius was referring to ritual violation when he praised his disciple for being able not to repeat the same mistake. That he was so persistent all his life in trying to restore the ritual institutions of the Zhou dynasty should then seem an inevitable destiny to him. Interestingly, the master himself committed a factual error in trying to avoid making one regarding the rites. He discreetly acknowledged it when his disciple Wuma Qi \u5deb\u99ac\u671f (521 B.C.E.-?) brought it to his attention but apparently did not regret it (Analects 7.31). He did not cover it up like a petty person would (Analects 19.8). Wuma Qi was 31 years Confucius' junior, and if we assume that he became the master's student at the age of 20, then Confucius was already 51 years old when he made that factual error. It was the time when he hoped he could avoid making \"major mistakes.\" As the rites were considered an indispensable means of political governance and social interaction among humans as well as between the human world and the natural and spiritual realms,53 perhaps for Confucius, \"major mistakes\" were in fact about ritual transgressions. That is why the four imperatives he stipulated for Yan Hui all concern the rites and they were articulated as prohibitions. Little wonder then that Yan Hui said, \"The master broadened me with matters of culture and tie them together for me with the rites\" (Analects 9.11).54\n\nIt seems that Yan Hui's self-cultivation revolved around the practices of the rites and he would regulate his conduct with them. A later disciple Youruo once said, \"In the application of the rites, harmony is prized...It does not always work if one aims at harmony for the sake of harmony without regulating it with the rites\" (Analects 1.12).55 Clearly, this sort of strenuous pursuit of harmony is far from Confucius' spontaneous realization of his heart-mind's wishes without overstepping the boundaries. In this light, Yan Hui's cultivated abilities of managing his anger in a harmonious way and not repeating the same mistake are perhaps two sides of the same coin, and for this reason they were mentioned in tandem by the master. These accomplishments \u2013 inner harmony and overcoming the self \u2013 stem from the cultivation of the heart-mind and no doubt constitute learning for one's own sake. Once, Confucius asked his disciples about their aspirations, and Yan Hui replied, \"I wish I would never boast of my good qualities or call attention to my good deeds\" (Analects 5.26). In contrast, Zilu wished he would never be bothered if his friends damaged the material goods he had loaned them. Clearly Yan Hui was keen on the cultivation of his own self. Confucius described what would be highly regarded as Yan Hui's paradigmatic attainment of self-cultivation as follows:\n\n> How remarkable Hui is! A bowl of rice to eat, a gourd of water to drink, dwelling in a humble alley \u2013 no one can bear such hardship, yet Hui does not allow this to affect his joy. How remarkable Hui is! (Analects 6.11)\n\nIn spite of the unusual misery he lived in, Yan Hui was oblivious to it. His heart-mind was autonomous and his inner joy was impervious to external contingencies. In Zengzi's characterization, \"he has yet appears to lack; he is full yet appears to be empty\" (\u6709\u82e5\u7121, \u5be6\u82e5\u865b) [Analects 8.5]. Indeed Yan Hui aimed at self-sufficiency in his heart-mind. Insofar as joy was concerned, he was able to follow his heart-mind's wishes much like Confucius at age 70. It is perhaps in this connection that the master commended his ability of sustaining humaneness in his heart-mind for 3 months without lapse. Interestingly, Confucius was known to have forgotten the taste of meat also for 3 months when he was totally immersed in the Shao \u97f6 music of sage-king Shun \u821c (Analects 7.14), which he lauded as \"perfectly beautiful and perfectly good\" (Analects 3.25). Both master and disciple had the capacity to be lost in the oblivion of joy.56 It was a unique experience to which no other disciple apparently was spiritually adequate enough to have access.\n\n## 8 The Later Disciples\n\nIn Mencius's time it was well known that Confucius' later disciples Zixia, Ziyou, and Zizhang all acquired the features of the sage whereas his early disciples Ran Niu \u5189\u725b, Min Ziqian, and Yan Hui embodied his totality in epitome, and Mencius himself consented to this view (Mencius 2A2). As far as we can tell today, this Warring-States opinion is basically sound, and in light of our analysis above, the case of Yan Hui at least can be confirmed. With regard to the various features the later disciples acquired from their master, the trio mentioned by Mencius were all employed under different political administrations. Zixia stood out in his expertise in music and in the study of the Documents 57 and Ziyou specialized in the study of rites with particular regard to governance. Little is known about Zizhang except his expertise in governance. In the Analects, virtually every question he asked Confucius pertains to governance. In fact, that seems to be his sole concern in studying with Confucius; he was the only disciple who asked the master about how to secure a position in the government (Analects 2.18).58 The trio, it would seem, could be considered \"utensils\" in Confucian terms.59\n\nBesides the trio, Zengzi was probably one of most respected of Confucius' later disciples. He was known for his daily self-introspection on three counts: doing his best (zhong \u5fe0) when he is asked for counsel; being trustworthy in dealing with his classmates and friends; and personal practice before giving instructions to others (Analects 1.4). Even though it is not clear how Zengzi conceived of the practice in a broader philosophical context, he highlighted the importance of being watchful when one is alone (shendu \u614e\u7368),60 a doctrine which would receive greater attention at a later time in the Warring States period.61 Zengzi also identified zhong \u5fe0 (doing one's best, loyalty) and shu \u6055 (empathy) as the one thread that ties the philosophy of Confucius together (Analects 4.15). Like Zixia, he was also known for a method of cultivating courage. According to Mencius, it is similar to that of Meng Shishe \u5b5f\u65bd\u820d in that he would treat defeat as victory. Even though he cannot be certain of victory, he can rid himself of fear (Mencius 2A2). Zengzi's writings indeed focus almost exclusively on personal ethics \u2013 filial devotion and the moral integrity of the gentleman. Unlike many of the later disciples, Zengzi did not gain much, if any, experience in actual governance, so he had little to say about politics or the relation between ruler and minister. In fact, except for the parent\u2013child relation, he rarely addressed other human relations.62 Moreover, most of his discussions are geared toward concrete behavior and lack a theoretical bent and he did not offer a view on human nature.\n\nPerhaps Zengzi was best known for being a filial son not only in the Warring States period but throughout Chinese history. As tradition has it, the Xiaojing \u5b5d\u7d93 (Classic on Filial Devotion) we have today is actually the record of a lecture on the virtue given to him by Confucius.63 In the Analects, Confucius himself did not volunteer to speak about filial devotion; he only counselled his various later disciples on specific filial conduct.64 His early disciples apparently did not ask him about the virtue either. There seems to be a growing interest among the later disciples in the nature of filial devotion. Curiously, Zengzi had virtually nothing to say about filial devotion in the Analects even though he indeed developed an elaborate doctrine on the virtue. In the Analects 8.3 we are told that when Zengzi was seriously ill and approached the end of his life, he summoned his disciples and told them, \"Take a look at my hands. Take a look at my feet. The Odes say, 'In fear and trembling\/as if standing at the edge of a deep abyss\/As if treading on thin ice.' Only now am I certain that I am spared, my young friends.\" Evidently, Zengzi's filial devotion consists in taking good care of his person and physical body so that its integrity would not be compromised either morally or physically. It was his personal practice for a lifetime rather than a mere philosophical doctrine. Perhaps from his personal experience, Zengzi differentiates three types of filial devotion: the highest venerates the parents (zunqin \u5c0a\u89aa); the second highest does not subject them to humiliation (buru \u4e0d\u8fb1), and the lowest can only feed them (neng yang \u80fd\u990a).65 While Confucius' advice to his disciple on filial devotion primarily focuses on the son's genuine love for his parents and its proper expressions, Zengzi tends to magnify the virtue and expands it to cover every small behavior in the son's daily living and indeed his entire life. Filial devotion is elevated to the foundation of all virtues. For Zengzi, the virtue of filial devotion is the main cord that binds all under heaven (\u5929\u4e0b\u4e4b\u5927\u7d93).66 It fills up the space between heaven and earth and covers the four seas. It can be practiced day and night and extend to posterity. It is a universal standard wherever it is applied (Wang 1983: 84).\n\nNot only does filial devotion become the guiding power and moral inspiration in monitoring the minutiae of the son's everyday life, but it is virtually the teleology of his entire being. Thus if a son injures his leg by accident at home, he could be considered unfilial. As Zengzi puts it meticulously, \"The gentleman does not forget about his parents when he lifts his foot, nor does he forget them when he utters a word\" (\u541b\u5b50\u4e00\u8209\u8db3\u4e0d\u6562\u5fd8\u7236\u6bcd, \u4e00\u51fa\u8a00\u4e0d\u6562\u5fd8\u7236\u6bcd) [Wang 1983: 85].67 This is because one's body is a gift from one's parents; therefore, in conducting it, how can one not be reverential? (Wang 1983: 82\u201383) \"At birth, the parents give the son a body in whole, and if the son can return it intact [at death], he can be called filial. A body without injury is considered intact\" (\u7236\u6bcd\u5168\u800c\u751f\u4e4b,\u5b50\u5168\u800c\u6b78\u4e4b, \u53ef\u8b02\u5b5d\u77e3\u3002\u4e0d\u8667\u5176\u9ad4, \u53ef\u8b02\u5168\u77e3) [Wang 1983: 85].68 Compared to Confucius who treats the son as the moral agent and pinpoints directly his affective feelings and thoughtfulness in interacting with his parents, Zengzi begins to shift the focus of filial love to its recipients such that they become the determinant power in guiding the son's conduct. Confucius' idea of filial devotion is son-centred (Analects 1.11); it is an intricate art of reinforcing a special emotive bonding between parent and son. In contrast, Zengzi's idea of filial devotion is parent-centred and it borders on being a moral discipline to avoid making mistakes in any possible way.69 It is perhaps no coincidence that the three counts on which Zengzi examined daily all concern doing things right for others and that he took zhong and shu to be the single thread that unifies Confucius' teachings as the two virtues \u2013 doing one's best and being sympathetic \u2013 are crucial to filial devotion that aims at serving the persons of the parents rather than fulfilling one's humanity as a son. Of course, Zengzi's moral practice, filial service, and self-fulfilment are fully integrated.\n\nThere is another critical development in Zengzi's doctrine on filial devotion. When Confucius was once asked why he did not take part in government, he replied that \"if a man is filial to his parents and friendly to his siblings, he is making a contribution to the government\" (Analects 2.21). The connection between filial virtue and government is indirect at best and Confucius was not philosophizing on filial devotion or fraternal love. However, the later disciple Youruo actually theorized on the virtue and he considered filial devotion and fraternal obedience to be the roots of humaneness (ren zhi ben \u4ec1\u4e4b\u672c). For him, if a person is filial to his parents and obedient to his siblings, he would never offend his superiors, much less to stage a revolt. Therefore, the gentleman should cultivate the roots because once the roots are established, the Way will develop (Analects 1.2). Familial virtues now are directly linked to political worthiness; furthermore, the former will develop into the latter. While Zengzi did not articulate it as succinctly, he no doubt endorsed the same view, and in his characteristically detailed fashion, he pronounced:\n\n> When one is not dignified at home, one is not filial; when one does not serve one's lord to the best of one's abilities, one is not filial; when one is not reverential in conducting one's official duties, one is not filial; when one is not trustworthy to one's friends, one is not filial; when one is not courageous in the battlefield, one is not filial. (Wang 1983: 83)\n\nIt is clear that filial devotion can be translated into political loyalty, professional dedication, personal trustworthiness, and even military courage. Needless to say, the beneficiaries of filial devotion are no longer confined to the parents. In this sense, Zengzi's idea of filial devotion is more beneficiary-oriented than son-centred. In Zengzi's discourse on filial devotion, he particularly highlights its intimate relation to political loyalty.70 He says, \"If one knows how to serve one's parents, one can serve one's lord. If one knows how to serve one's elder brothers, one can serve one's teachers and superiors. The way one treats one's son is just the same as the way [a lord] treats his subjects\" (Wang 1983: 79) because \"a filial son is good at serving his lord and a fraternally obedient brother is good at serving his elders\" (Wang 1983: 82). This peculiar symbiosis of filial devotion and political loyalty soon would find its way into the Classic of Filial Devotion toward the end of the third century B.C.E.71 even as Zengzi became a legendary filial exemplar of all ages.\n\n## 9 Mi Zijian \u5b50\u8ce4 and Confucian Political Philosophy\n\nWhile law and punishments have their place in governance, Confucius thought people should be guided by virtue and shaped uniformly by the rites so that not only will they develop a sense of shame, but they will strive to reform themselves (Analects 2.3). The crux of his doctrine lies in the fact that people have the innate ability to seek the good on their own when they are properly educated and inspired. Confucius was interested in transforming people from within. Thus he would prefer to make lawsuits unnecessary rather than merely adjudicate them (Analects 12.13). This perhaps should be in line with the spirit of ruling by wuwei \u7121\u7232 (noninterference) that Confucius admired in the sage-king Shun \u821c (Analects 15.5).\n\nFundamentally keen on self-cultivation for his own sake as he was, Yan Hui was also interested and apparently competent in political governance. He asked Confucius about how to govern a state (rather than a specialist position that Zilu or Zigong held as a \"utensil\"), and the master told him, \"Use the calendar of the Xia, ride in the carriage of the Yin, and wear the ceremonial cap of the Zhou. As for music, employ the Shao (of Shun) and Wu (of King Wu of Zhou). Abandon the tunes of Zheng and keep the clever talkers at bay. The tunes of Zheng are wanton and clever talkers are dangerous\" (Analects 15.11). The master offered models of political institutions; he did not counsel on practical matters and his only realistic advice was to avoid clever talkers. There is no record of Yan Hui's view on governance in the Analects, but if we can rely on an account in the early Han dynasty source Hanshi waizhuan \u97d3\u8a69\u5916\u50b3 (Outer Commentary on the Han Version of the Odes),72 which contains numerous episodes in which Confucius discussed various issues with his disciples, we shall gain valuable insight into his political philosophy. In a conversation between Confucius and his disciples on their political aspirations, after Zilu and Zigong spoke of the desire to employ their talents in the military and diplomacy respectively, Yan Hui said,\n\n> I wish I might be minister in a small state. The ruler would govern by the True Way, and his subjects would be reformed by his transforming virtue. Prince and subjects would be of one mind, and those inside and those outside [the court] would respond to one another. Of the various states and the feudal lords, none but would fall in line with what is appropriate and be subject to [my] influence. The able-bodied would rush to come forward, and the old would come leaning on their staves. My teachings would pass to the four barbarians. Everyone would give up his weapons and assemble inside the four gates [of my capital]. In the world everywhere enduring peace would prevail. Flying or crawling, each [creature] would rejoice in his own nature. I would advance the worthy and employ the able, each to be in charge of the office suited to himself. Then the prince above would be tranquil and his subjects below would be in harmony. I would let [my robes] fall, fold my hands, and practice noninterference (\u5782\u62f1\u7121\u7232). What was done would coincide with the True Way, naturally and easily adhering to the rites. Those who spoke of humaneness and righteousness I would reward, and those who spoke of war and strife I would put to death.73\n\nThe scenario Yan Hui presented may not be Daoist in substance, but the natural, harmonious dynamics between the prince and his subjects, among the subjects themselves, as well as between humans and other creatures is characteristically Daoistic. It is perhaps no coincidence that Yan Hui was only interested in governing a small state. In Laozi 80, we are given the Daoist utopia, which is also \"a small country with a sparse population\" where weapons would be of no use. What is most important, however, is the way Yan Hui's utopia is governed and achieved and it is explicitly called wuwei.74 How exactly wuwei worked remains unclear in this portrayal but Yan Hui emphasized that the need for military action or diplomacy would be obviated in a utopia governed by it. And this in itself is indeed a form of preemptive wuwei. As Confucius exclaimed in approval, \"In a government with Hui, how would you, You [Zilu] and Ci [Zigong], have a chance to show your abilities?\" And he called Yan Hui a \"sage\" (sheng \u8056).\n\nIt is not always easy to see how a philosopher is at work in real life. Yan Hui never did serve in any government and Confucius left us no record of his actual governance. Fortunately, Confucius' later disciple Mi Zijian provides us with some clues as to the actual work of a Confucian governor. Mi Zijian enjoyed a legendary reputation in his political career as the local governor of Shanfu \u55ae\u7236 in the state of Lu \u9b6f. We are told that he played the zither all the time and never came down from the hall where he was supposed to work, yet Shanfu was in perfect order. On the other hand, Wuma Qi \u5deb\u99ac\u671f, another disciple of Confucius who was probably his predecessor, also governed Shanfu well but in a different way. He went out to his duties while the stars were still out and did not return until they again came out at night. He gave himself no rest, taking care of everything in person. When Wuma Qi asked Mi Zijian for the reason, he said he himself governed by \"relying on people\" (ren ren \u4efb\u4eba) whereas his predecessor governed by \"relying on strength\" (ren li \u4efb\u529b). For this reason, people called Mi Zijian a gentleman.75 They said, \"While he rested his four limbs, preserved his sight and hearing, kept his mind and spirit quiet, the various officers would be in order. All he did was to make use of their natural capacities\" (\u9038\u56db\u80a2,\u5168\u8033\u76ee, \u5e73\u5fc3\u6c23\u800c\u767e\u5b98\u7406\u3002\u4efb\u5176\u6578\u800c\u5df2\u77e3).76 This is perfectly in line with the Daoist idea of ruling by wuwei. For instance, Laozi 63 says, \"Practice noninterference, engage in nonengagement\" (\u70ba\u7121\u70ba, \u4e8b\u7121\u4e8b). And Laozi 48 says, \"Obtain all under heaven always by nonengagement. With engagement, it is not sufficient to obtain all under heaven\" (\u53d6\u5929\u4e0b\u5e38\u4ee5\u7121\u4e8b, \u53ca\u5176\u6709\u4e8b, \u4e0d\u8db3\u4ee5\u53d6\u5929\u4e0b). Clearly, Mi Zijian was practicing noninterference and nonengagement in governing Shanfu.\n\nIn actual practice, Mi Zijian's wuwei can be illustrated in a situation where his noninterference would seem counterintuitive but epitomized the true spirit of the Daoist value. Once during the time of wheat harvest, the state of Qi planned to attack the state of Lu and on its way it passed through Shanfu. The local elders then asked their governor if the citizens were allowed to make their harvests before the Qi army arrived, since this could give them more food instead of it going to their enemies. They made their requests repeatedly but Mi Zijian refused to listen. Soon, all the wheat was taken by the Qi troops. Jisun (Mi Zijian's superior) was infuriated when he heard the news, and he sent someone to reprimand Mi Zijian. But he explained, \"We lost our wheat this year but we can grow it again next year. Yet if I allowed those who did not work in the field to take the harvest, then people would love to have raiding robbers here. Moreover, 1 year of wheat harvest would not make Lu more powerful; neither would Lu become less powerful with its loss. But if we allow the people to think that they can take things from others as they wish, the damage will definitely remain for several years to come.\" 77\n\nIn this episode Mi Zijian literally did not do anything to make sure that the citizens of Shanfu would not get used to the idea of taking other people's property willfully. He wanted them to respect others as well as themselves but he did it without interfering in the unfolding of events. This is wuwei par excellence. Not only did his superior agree with him, but the people of Shanfu were transformed during his tenure such that they could not bring themselves to deceive their governor.78 Mi Zijian's actual governance substantiates his master's claim that when guided by virtue and shaped by the rites, people will have a sense of shame. The Grand Historian Sima Qian actually made a special mention of the transformative influence of Mi Zijian's wuwei rulership in the Shiji.79 Most significantly, it was also noted in the Daoist text Huai'nanzi \u6dee\u5357\u5b50.80 And if we compare Mi Zijian's practice with the Daoist rulership described in the text, the credit he earned was well deserved. The Huai'nanzi says,\n\n> The Perfect Man rules by hiding his intelligence and eradicating his embellishments. He follows the Way and abandons his cleverness, abides by what is proper with the people, shuns his temptations, discards his desires, and dismisses his deliberations. To minimize that which to observe, one can be clear about things. To curtail that which to obtain, one can be successful (He 1998: 60\u201361).\n\nAs the Daoist Perfect Man \"rules by hiding his intelligence and eradicating his embellishments,\" Mi Zijian \"rested his four limbs, preserved his sight and hearing.\" The Perfect Man \"follows the Way and abandons his cleverness\" while Mi Zijian \"kept his mind and spirit quiet.\" They both keep things simple and let them run their course as they see fit. They abide by this fundamental spirit of noninterference. The Huai'nanzi says, \"He who embodies the Way is relaxed and cannot be exhausted\" (\u9ad4\u9053\u8005\u9038\u800c\u4e0d\u7aae) [He 1998: 32]. Mi Zijian aptly fits the description.\n\nTo be sure, Mi Zijian was not a Daoist exemplar. The Huai'nanzi also depicts the Daoist sage as follows:\n\n> Sageliness lies not in governing people but in attaining the Way; joy lies not in wealth and high station but in harmony in virtue; if one knows the self is great and all under heaven is small, one is close to the Way. (He 1998: 65\u201366)\n\nWe have no evidence that Mi Zijian was interested in attaining the Way or cultivating harmony in virtue even though he appeared to prize himself over the city he governed. Nevertheless, the case of Mi Zijian perhaps suggests that Confucian political philosophy began to accommodate Daoistic elements in the middle of the fifth century B.C.E. as Confucius was coming to the end of his life. In fact, the spiritual accomplishment of the master at age 70 undoubtedly took on a bit of Daoistic character. He could follow the wishes of his heart-mind without overstepping the boundaries.\n\nFurthermore, the character of Yan Hui, as analyzed above, seems to match the Daoist sage described in the Huai'nanzi as well. While it is difficult to take the accounts of Yan Hui in the Zhuangzi literally, the fact that he is usually treated as a Daoist adept in the text suggests that the followers of the Yan \"school\" in Han Fei's account of late Warring States Confucianism may indeed espouse a doctrine that at least was then perceived to be consistent with Zhuangzi's philosophy. Yan Hui might not be a mere flat caricature at the mercy of Zhuangzi's Daoist manipulation. At least some of the anecdotes about Confucius' disciples also appear in Confucian texts from early Han times. Like Yan Hui, Yuan Xian \u539f\u61b2 (515-?), for example, refused to serve in government in spite of his extreme poverty, apparently because he felt ashamed to do so when the Way had fallen into disuse.81 His story was recorded in the Zhuangzi where he reputedly said he could never bring himself to \"learn for other people's sake and to teach for one's own sake.\"82 The same story was repeated in two early Han texts.83 It should not be a mere coincidence that only Confucius' disciples such as Yan Hui and Yuan Xian were cast in a favorable light in the Zhuangzi.\n\nAs mentioned above, both Mi Zijian and Qidiao Kai developed a theory of human nature and shared the view that human nature contains both good and evil. Even though Qidiao Kai was probably related to one of the eight \"schools\" of Confucianism identified by Han Fei, little is known about him except that he did not seem to enjoy serving in government (Analects 5.6) and that he might have been meted some form of corporal punishment even though he did not actually commit any wrongdoing.84 It is no coincidence that Zhuangzi was averse to serving in government as was Qidiao Kai. Interestingly, it is well known that quite a few handicapped figures were featured in the Zhuangzi and they invariably represented accomplished men of authentic Daoist virtues.\n\nThe Warring States period was an unusually turbulent time; new ideas were conceived and new experiments were introduced in virtually every aspect of life. Innovation and adaptation carried the day. The fact that the teachings transmitted by Confucius eventually branched off into eight different Confucian doctrines says it all. Bamboo texts on Confucian teachings recently unearthed amply show that the master's unified doctrine indeed proliferated itself into a kaleidoscopic variety. We know that the Zhongyong and the Xici Commentary of the Yijing, both products of the late Warring States period, are hybrids of Confucian and Daoist metaphysics and ethics. Between Confucius and these two composite texts, there must be a trajectory of adaptation and accommodation. New archaeological finds may help us chart it in more definite outlines. Meanwhile, a meticulous study of Warring States and early Han texts may still prove to be most rewarding to the open-minded reader who is interested in tracing the history of evolving doctrines that would later be formally labeled as Confucianism and Daoism.\n\nReferences\n\nBan Gu \u73ed\u56fa. 2002. Hanshu \u6f22\u66f8, Vol. 12. Beijing: Zhonghua. The standard history of the Han dynasty from premodern times.\n\nBoltz, William G. 1993. Hsiao ching. In Early Chinese texts: A bibliographical guide, ed. Michael Loewe, 141\u2013153. Berkeley: Society for the Study of Early China: Institute of East Asian Studies, University of California. A standard and authoritative introduction in English to the textual history of and studies on the Xiaojing.\n\nChen Li \u9673\u7acb. 1994. Baihu tong shuzheng \u767d\u864e\u901a\u758f\u8b49, Vol. 2. Beijing: Zhonghua. The best commentary on the Baihu tong.\n\nChen, Qiyou \u9673\u5947\u7337. 1974. _Hanfeizi jishi_ \u97d3\u975e\u5b50\u96c6\u91cb. Shanghai: Shanghai remin (The best commentary on the Legalist work _Hanfeizi_).\n\nChen, Qiyou \u9673\u5947\u7337. 1984. L\u00fcshi chunqiu jiaoshi \u5442\u6c0f\u6625\u79cb\u6821\u91cb, Vol. 4. Shanghai: Xuelin. The best commentary on the L\u00fcshi chunqiu.\n\nCsikszentmihalyi, Mark. 2004. Material virtue: Ethics and the body in early China. Leiden: Brill. A detailed and careful study of the excavated text \"Wuxing\" and its relation to the development of early Confucian thought.\n\nDuan Yucai \u6bb5\u7389\u88c1. 1974. Shuowen jiezi zhu \u8aaa\u6587\u89e3\u5b57\u6ce8. Taipei: Lantai. An authoritative annotation on the Han lexicon Shuowen jiezi.\n\nGuo Qingfan \u90ed\u6176\u85e9. 1985. Zhuangzi jishi \u838a\u5b50\u96c6\u91cb, Vol. 4. Beijing: Zhonghua. This work contains the earliest extant commentary on the Zhuangzi by Guo Xiang (d. 312) and a subcommentary by Cheng Xuanying (fl. 631\u2013650) with additional commentaries from Qing-dynasty scholars supplemented by the compiler.\n\nHe Ning \u4f55\u5be7. 1998. _Huai'nanzi jishi _ \u6dee\u5357\u5b50\u96c6\u91cb. 3 Vols. Beijing: Zhonghua. A good modern commentary on the _Huai'nanzi_.\n\nHightower, James R. trans. 1952. Han Shih Wai Chuan: Han Ying's Illustrations of the Didactic Application of the Classic of Songs. Cambridge: Harvard University Press. An excellent translation of the Hanshi waizhuan.\n\nHightower, James R. trans. 1993. Han shi wai chuan. In Early Chinese texts: A bibliographical guide, ed. Michael Loewe, 125\u2013128. Berkeley: Society for the Study of Early China: Institute of East Asian Studies, University of California. A standard and authoritative introduction in English to the textual history of and studies on the Hanshi waizhuan.\n\nHuang Kan \u7687\u4f83. 1991. Lunyu jijie yishu \u8ad6\u8a9e\u96c6\u89e3\u7fa9\u758f, Vol. 2. Taipei: Guangwen. Collected commentaries and subcommentaries on the Analects compiled in the fifth century.\n\nIkeda Tomohisa \u6c60\u7530\u77e5\u4e45. 2005. Mawangdui Hanmu boshu Wuxing yanjiu \u99ac\u738b\u5806\u5e1b\u66f8\u4e94\u884c\u7814\u7a76. Beijing: Zhongguo shehui kexue, Xianzhuang. A careful and detailed study of the \"Wuxing\" from Mawangdui that discusses its dating and authorship as well as philosophical concepts; the \"Wuxing\" is fully annotated with a modern Chinese translation.\n\nKong Chuan \u5b54\u50b3. 1983. Dongjia zaji \u6771\u5bb6\u96dc\u8a18. In Yingyin Wenyuange Siku quanshu \u666f\u5370\u6587\u6df5\u95a3\u56db\u5eab\u5168\u66f8. Taipei: Shangwu. An important work on Confucius by his descendant of the 47th generation in the twelfth century; it includes topics on Confucius' ancestral genealogy, the dates of his birth and death, his mother and wife, his posthumous honorary titles as well as the temple dedicated to him and the steles therein, among others.\n\nKramers, R.P. 1993. K'ung tzu chia y\u00fc. In Early Chinese texts: A bibliographical guide, ed. Michael Loewe, 258\u2013262. Berkeley: N.P. The Society for the Study of Early China and the Institute of East Asian Studies, University of California. A standard and authoritative introduction in English to the textual history of and studies on the Kongzi jiayu.\n\nLao, Yueqiang (Lo Yuet Keung) \u52de\u60a6\u5f37. 2007. Cong Lunyu wei n\u00fczi yu xiaoren wei nanyang ye zhang lun Zhuxi de quanshixue \u5f9e\u300a\u8ad6\u8a9e\u300b\u3008\u552f\u5973\u5b50\u8207\u5c0f\u4eba\u70ba\u96e3\u990a\u7ae0\u3009\u8ad6\u6731\u5b50\u7684\u8a6e\u91cb\u5b78. In _Chinese Studies_ \u6f22\u5b78\u7814\u7a76 25.2 (Dec. 2007), 131\u2013163.\n\nLao, Yueqiang (Lo Yuet Keung) \u52de\u60a6\u5f37. 2009. Shan e guan yiwai de Kongzi xing lun \u2013 yige sixiangshi de kaocha \u5584\u60e1\u89c0\u4ee5\u5916\u7684\u5b54\u5b50\u6027\u8ad6--\u4e00\u500b\u601d\u60f3\u53f2\u7684\u8003\u5bdf. In Guannianzi jiedu yu sixiangshi tan suo \u89c0\u5ff5\u5b57\u89e3\u8b80\u8207\u601d\u60f3\u53f2\u63a2\u7d22, ed. Zheng Jixiong \u912d\u5409\u96c4, 73\u2013124. Taipei: Xuesheng. An original study of Confucius' concept of xing (human nature) that argues that the master did not understand xing in terms of good and evil.\n\nLau, D.C., and Chen, Fong Ching. 1995. A Concordance to the Lun Yu. In The ICS ancient Chinese texts concordance series, Classical Works No.14. Hong Kong: The Commercial Press. A reliable concordance to the Analects in a critical edition.\n\nLi, Ling \u674e\u96f6. 2007. Shangbo Chujian san pian jiaodu ji \u4e0a\u535a\u695a\u7c21\u4e09\u7bc7\u6821\u8b80\u8a18. Beijing: Zhongguo renmin daxue. The work contains critical reading by one of the authorities on three excavated Chu texts in the Shanghai Museum with their counterparts from the Guodian tomb.\n\nLi, Qiqian, and Wang Shilun complied. 1991. Kongzi dizi ziliao huibian \u5b54\u5b50\u5f1f\u5b50\u8cc7\u6599\u532f\u7de8. Ji'nan: Shangdong youyi shushe. A useful and convenient collection of historical sources on the disciples of Confucius.\n\nLiu, Pansui \u5289\u76fc\u9042. 1958. Lunheng jijie \u8ad6\u8861\u96c6\u89e3, Vol. 2. Taipei: Shijie. A useful modern commentary on the Lunheng.\n\nLo, Yuet Keung. 2008. Teachers-Disciples, or Friends? \u2013 An Historico-Exegetical Approach to the Analects. In Confucian Ethics in Retrospect and Prospect, eds. Vincent Shen and Kwong-loi Shun, 27\u201360. Washington DC: The Council for Research in Values and Philosophy. An intellectual-historical study of the term peng in Analects 1.1 that demonstrates that commentaries on the Analects were significantly informed by the commentator's philosophical concerns engendered by his peculiar historical milieu.\n\nMeng, Wentong \u8499\u6587\u901a. 2006. Qidiao zhi ru kao \u6f06\u96d5\u4e4b\u5112\u8003. In Chuanda shixue Meng Wentong juan \u5ddd\u5927\u53f2\u5b78\u8499\u6587\u901a\u5377, ed. Meng Mo \u8499\u9ed8, 152\u2013156. Chengdu: Sichuan daxue. One of the earliest and solid studies of Confucius's disciple Qidiao Kai and its philosophical legacy in pre-Qin times.\n\nOuyang, Xun \u6b50\u967d\u8a62. 1982. Yiwen leiju \u85dd\u6587\u985e\u805a, Vol. 2. Beijing: Zhonghua. A useful encyclopaedic work from the seventh century on a wide variety of classified topics.\n\nQian Mu \u9322\u7a46. 1991. Kongzi zhuan \u5b54\u5b50\u50b3. Taipei: Dongda. A good study of Confucius' life with solid historical evidence.\n\nQian Mu \u9322\u7a46. 2002. Xian-Qin zhuzi xinian \u5148\u79e6\u8af8\u5b50\u7e6b\u5e74. Beijing: Shangwu. An authoritative study that attempts to ascertain the chronology of pre-Qin philosophers and their interconnections.\n\nRosemont, Jr., Henry, and Ames, Roger T. trans. 2009. The Chinese Classic of Family Reverence: A Philosophical Translation of the Xiaojing. Honolulu: University of Hawaii Press. The latest translation of the Xiaojing with a lengthy introduction that discusses the historical, philosophical, and religious dimensions of the Xiaojing with a focus on articulating a Confucian notion of role-ethics rooted in a relational conception of the person.\n\nSlingerland, Edward. trans. 2003. Confucius Analects \u2013 With Selections from Traditional Commentaries. Indianapolis\/Cambridge: Hackett Publishing. One of the recent translations of the Analects that includes selected commentaries on the classic from premodern China.\n\nTakigawa Kametar\u014d \u7027\u5ddd\u9f9c\u592a\u90ce. 1977. Shiki kaich\u00fck\u014dsh\u014d \u53f2\u8a18\u6703\u6ce8\u8003\u8b49. Taipei: Hongye. Arguably the best collected Chinese and Japanese annotations from premodern times on the Shiji.\n\nWang, Pingzhen \u738b\u8058\u73cd. 1983. Da Dai Liji jiegu \u5927\u6234\u79ae\u8a18\u89e3\u8a41. Beijing: Zhonghua. The standard commentary on the Da Dai Liji from premodern times.\n\nWang, Xianqian \u738b\u5148\u8b19. 1997. Xunzi jijie \u8340\u5b50\u96c6\u89e3. Beijing: Zhonghua. The standard commentary on the Xunzi from premodern times.\n\nWei, Shou \u9b4f\u6536. 1974. Weishu \u9b4f\u66f8, Vol. 8. Beijing: Zhonghua. The standard history of the Wei dynasty from premodern times.\n\nXiao, Zixian \u856d\u5b50\u986f. 1972. Nan Qi shu \u5357\u9f4a\u66f8, Vol. 3. Beijing: Zhonghua. The standard history of the Southern Qi dynasty from premodern times.\n\nYan, Zhenyi \u95bb\u632f\u76ca, and Zhong Xia \u9418\u590f. 2000. annot. Xin shu jiaozhu \u65b0\u66f8\u6821\u6ce8. Beijing: Zhonghua. A critical, annotated edition of Xin shu, the work of Han-dynasty thinker Jia Yi \u8cc8\u8abc.\n\nYang, Zhaoming \u694a\u671d\u660e. 2005. Kongzi jiayu tongjie \u5b54\u5b50\u5bb6\u8a9e\u901a\u89e3 \u2013 \u9644\u51fa\u571f\u8cc7\u6599\u8207\u76f8\u95dc\u7814\u7a76. Taipei: Wanjuanlou. A good annotated modern edition of the Kongzi jiayu with modern Chinese translation; relevant excavated texts from the Warring States period and a critical study of the transmission of the Kongzi jiayu are also included.\n\nZhang, Rujiao \u5f35\u8339\u5b0c. 2007. Kongmen xueshu dili kao \u5b54\u9580\u5b78\u8853\u5730\u7406\u8003. In Luming youyou \u2013 Xinjiapo guoli daxue zhongwenxi yanjiusheng lun rujia wenhua \u9e7f\u9cf4\u5466\u5466 \u2013 \u65b0\u52a0\u5761\u570b\u7acb\u5927\u5b78\u4e2d\u6587\u7cfb\u7814\u7a76\u751f\u8ad6\u5112\u5bb6\u6587\u5316. ed. Lao Yueqiang (Lo Yuet Keung) \u52de\u6085\u5f37, 21\u201372. Singapore: Qingnian. A collection of studies on various aspects of Confucian culture and thought.\n\nZhu Xi \u6731\u71b9. 2001. _Sishu zhangju jizhu_. Beijing: Zhonghua. One of the most authoritative and influential commentaries on the Fouar Books from premodern times.\n\nFootnotes\n\n1\n\nThe term kongmen first appeared in Xiao Zixian 1972: juan 39, 2:686. Xiao lived in South China, but around the same time when he was composing his historical work, another historian in North China Wei Shou \u9b4f\u6536 (506\u2013572) was completing his own, in which he also used the term kongmen. See Wei Shou 1974 juan 84, 5:1850. It is possible that the term was coined to mark the uniqueness of the Confucian tradition vis-\u00e0-vis Buddhism and Daoism, which had been gaining considerable currency since the third century. At the same time, the term rumen \u5112\u9580 (literally, \"Confucian gate\") was also in wide circulation even though its first use dates back to the first century during the Eastern Han dynasty. See Liu Pansui 1958: juan 13, 1:281, and juan 30, 2:590.\n\n2\n\nSince the Song (960\u20131276) period, the term was often used in contradistinction to Daoism and Buddhism, and thus carried a ring of ideology with it.\n\n3\n\nBao Xian was invited by Emperor Guangwu of Eastern Han to tutor the heir-apparent on the Analects, and while in this capacity Bao wrote a zhangju \u7ae0\u53e5 commentary on the work. The commentary was sanctioned by the government and accepted as one of the official commentaries on the Analects. It was later incorporated into He Yan's \u4f55\u664f (190\u2013249) Lunyu jijie \u8ad6\u8a9e\u96c6\u89e3 (Collected Commentaries on the Analects) in the early third century.\n\n4\n\nFor a detailed study of the interpretations of the term peng from the Han to the Qing periods, see Lo 2008.\n\n5\n\nFor most China scholars, the most common term for \"classmates\" actually is \"tongxue\" (\u540c\u5b78, literally, people who study together), yet it was not coined until around the first century. See Ban Gu \u73ed\u56fa 2002: juan 78, 10:3271.\n\n6\n\nThe term menren appears five times in three different chapters in the Analects. See Analects 4.15, 7.29, 11.11 (twice) and 11.15. Sometimes, the term mendizi \u9580\u5f1f\u5b50 was also used to refer to the disciples (Analects 8.3, 9.2). Interestingly, both menren and mendizi were used to refer to the second-generation disciples of Confucius as well (Analects 19.3, 8.3). References to the Analects are based on D.C. Lau and Chen Fong Ching 1995.\n\n7\n\nHuang Kan 1991: 1.4. Huang Kan's gloss of peng as dang actually was based on at least two Han sources: Xu Shen's \u8a31\u614e (fl. second century) Shuowen jiezi \u8aaa\u6587\u89e3\u5b57 (Preface dated 121) and Ban Gu's \u73ed\u56fa (32\u201392) Baihu tong \u767d\u864e\u901a. For the citation on Shuowen, see Duan Yucai 1974: 150. For Baihu tong, see Chen Li \u9673\u7acb 1994: juan 8, 1:376.\n\n8\n\nThe term shimen began to appear for the first time in the first century during the early years of the Eastern Han \u2013 exactly the same time when Bao Xian wrote his commentary on the Analects.\n\n9\n\nThe Analects records that Confucius queried his son on the Odes on one occasion. See Analects 17.10.\n\n10\n\nConfucius expected his students to tell him about the other three corners of a table when they were shown one. See Analects 7.8. Yan Hui evidently had more than lived up to such expectation.\n\n11\n\nIn the case of Yan Hui, being a native of Lu, it is also possible that he lived in his own place, which would not be far away from the home of Confucius, since the Analects mentions, in particular, the miserable condition of Yan Hui's dwelling. As Yan Hui's misery was unique, the Analects could not be referring to the communal setting of the Confucian dormitory.\n\n12\n\nOn another occasion, a friend of Confucius named Qu Boyu \u8627\u4f2f\u7389 also sent a messenger to deliver his greetings to the master, and the conversation between the host and the guest was recorded in Analects 14.25. It seems evident that some of the disciples were, again, around when the conversation took place. In fact, in Analects 14.44, a young disciple of Confucius did serve as an attendant to receive guests in the master's household.\n\n13\n\nConfucius appears to comment, at least occasionally, on the individual strengths and weaknesses of his disciples. See, for instance, Analects 11.18. More detailed discussion follows below.\n\n14\n\nSima Qian also placed the biography of Confucius among those of the Hereditary Houses, which were reserved for feudal vassals and no other pre-Qin thinker or teacher was treated with the same honor.\n\n15\n\nWhether Sima Qian's figure was accurate, it seems that the total number of students who had studied with Confucius was significantly large.\n\n16\n\nSee the \"Qishier dizi jie\" \u4e03\u5341\u4e8c\u5f1f\u5b50\u89e3 chapter. Even though the chapter and its title explicitly say 72 disciples, 76 names are actually mentioned in the text.\n\n17\n\nMore disciples could still be identified and so far at least 102 names are confirmed. See Zhang Rujiao 2007: 21\u201372. It should be noted that only 27 or so of these disciples were mentioned in the Analects.\n\n18\n\nConfucius also purified himself before he sought audience with the Duke of Lu. See Analects 14.21.\n\n19\n\nYan Hui's father Yan Lu \u984f\u8def was only 6 years younger than Confucius and Zilu was 9 years his master's junior. See Takigawa, Shiki kaich\u016b k\u014dsh\u014d, juan 67, p. 863 and p. 856.\n\n20\n\nNangong Jingshu \u5357\u5bae\u656c\u53d4 (Analects 14.5) and Sima Niu \u53f8\u99ac\u725b (Analects 12.3) were noblemen.\n\n21\n\nSee Chen Qiyou 1984: juan 4, 1:205. Yan Zhuoju came from Liangfu \u6881\u7236 in the state of Lu and eventually became an official in the state of Qi \u9f4a. He was also known as Yan Zhuozou \u984f\u6dbf\u9112 and was mentioned in Sima Qian's biography of Confucius' disciples. See Shiji, juan 17, p. 743.\n\n22\n\nIt is probable that there were young maids working in the household of Confucius. See Analects 17.25 and Lao 2007: 1310163.\n\n23\n\nPerhaps Yan Hui and Min Ziqian \u9594\u5b50\u9a2b (536-? B.C.E.) were exceptions but their unusual aspirations simply proved the rule. We learn that Min Ziqian was even compelled to flee when an official post was repeatedly offered to him (Analects 6.9). Yan Hui and Min Ziqian were both exemplary students in terms of moral cultivation, and the latter was particularly noted for his filial devotion. Yet, evidently, Min Ziqian was also cut for a political career. See Analects 11.3, and 11.5.\n\n24\n\nSee Hanshi waizhuan \u97d3\u8a69\u5916\u50b3, juan 9. For an English translation, see Hightower 1952: 303\u2013304. Yan Hui was known to be of a quiet type and he hardly spoke in the Analects, and the only question he had ever asked Confucius was about governance. See Analects 15.11.\n\n25\n\nAfter Confucius died, feudal vassals continued to come to seek advice from his disciples before they made an offer to a potential candidate for an official post. See Analects 19.19 (in this case it was a second-generation disciple).\n\n26\n\nFor instance, see Analects 3.1, 3.2, 3.10.\n\n27\n\nConfucius admitted that he had never had the privilege to see a sage and that he would be glad as long as he could see a true gentleman. See Analects 7.26.\n\n28\n\nIt seems common for the disciples to help each other out when they had difficulty in learning. See, for instance, Analects 12.22, where Fan Chi \u6a0a\u9072 (505\/515-?), who being puzzled about Confucius' answer on the meaning of humaneness, asked his classmate Zixia specifically on the very issue. See also Analects 7.5.\n\n29\n\nZilu once recommended his classmate Zigao \u5b50\u7f94 to be the Governor of Bi \u8cbb. Even though Confucius did not think Zigao was ready for the appointment and thus found Zilu's recommendation ill-advised, nevertheless, this does not detract from Zilu's kind-heartedness and caring for his classmate. See Analects 11.25.\n\n30\n\nShiji, juan 47, pp. 734\u2013736, pp. 739\u2013740.\n\n31\n\nZigong was good at business speculation, and apparently he was wealthy in the Confucian community. He himself seemed to be quite aware of his wealthy status (Analects 1.15). He also seemed to be recognized as one of the esteemed seniors among the disciples. His opinion was often consulted when doubts and concerns arose (Analects 1.10, 19.25).\n\n32\n\nThe fact Confucius referred to Yan Hui's poverty suggests that his beloved disciple might fail to inspire respect owing to his financial straits.\n\n33\n\nWhen Yan Hui died, Confucius was so saddened that he exclaimed, \"Alas! Heaven has made me bereft! Heaven has made me bereft!\" The master indeed was weeping and was not aware of his unusual outpour of sorrow, and when his disciples reminded him that he was showing undue sorrow, he simply replied, \"Am I?\" See Analects 11.9, 11.10.\n\n34\n\nWhen another disciple Sima Niu was sick, Confucius visited him and, holding his disciple's hand, he said, \"We are going to lose him. It must be destiny. Alas! Such a man is afflicted with such a disease! Such a man is afflicted with such a disease!\" Like Yan Hui, Sima Niu was also known for his moral virtues but Confucius' reaction to his serious illness, albeit heart-broken, was within reasonable limits, at least in the eyes of his disciples. See Analects 6.10.\n\n35\n\nIt should be pointed out that Confucius also preferred grief to formality in funerals. See Analects 3.4.\n\n36\n\nAs mentioned above, Confucius referred to his academy as \"Qiu's gate\" and he measured the achievements of his disciples by pinpointing their relative positions in his household (Zigong, for instance, had ascended to the hall but yet to enter into the chamber), in this sense his disciples were considered members of his physical family and were indeed addressed as \"sons and younger brothers\" (dizi \u5f1f\u5b50). In light of this, Confucius' teaching was a family doctrine and a work entitled Kongzi jiayu (Family Sayings of Confucius) that consists of dialogues between Confucius and his disciples very much similar to the Analects is still extant today. For a discussion of the Kongzi jiayu, see R. P. Kramers, \"K'ung tzu chia y\u00fc\" in Michael Loewe 1993: 258\u2013262.\n\n37\n\nZengzi \u66fe\u5b50 in 18 chapters (pian \u7bc7); Qidiaozi \u6f06\u96d5\u5b50 in 13 chapters; Mizi \u5b93\u5b50in 16 chapters; (Jingzi \u666f\u5b50 in 13 chapters; Ban Gu's note says \"it contains Mizi's sayings and Jingzi appears to be his disciple.\" 6.1724); Shizi \u4e16\u5b50 in 21 chapters (Ban Gu's note says \"Shizi's name was Shuo \u78a9, and he was a second-generation disciple and came from the state of Chen; Li Ke \u674e\u514b in seven chapters (Zixia's disciples, minister to Marquis Wen of Wei \u9b4f\u6587\u4faf); and Gongsunnizi \u526c\u5b6b\u5c3c\u5b50 in 28 chapters (second-generation disciple).\n\n38\n\nIf Yan shi indeed refers to Yan Hui, it must be a later attribution by his disciples to acknowledge the origin of their new doctrine because Yan Hui died before Confucius did, so he probably would not have started his own lineage of teaching while the master was still alive. Given that Yan Hui tried to emulate the master in every possible way, it is not likely that he would start his own line of teaching in the first place.\n\n39\n\nBan Gu in his own annotation to the work of Shizi says, \"[Shizi], named Shuo, he was a native of Chen and a disciple of [one of] the 70 disciples of Confucius.\"\n\n40\n\nAccording to Shen Yue \u6c88\u7d04 (441\u2013513), the \"Yueji\" \u6a02\u8a18 chapter of the Liji \u79ae\u8a18 actually came from the hands of Gongsun Nizi. The \"Yueji\" chapter does contain explicit references to human nature.\n\n41\n\nLao Yueqiang, \"Shan e guan yiwai de Kongzi xing lun\u2014yige sixiangshi de kaocha\" in Zheng 2009: 73\u2013124.\n\n42\n\nLi 2007: 102. The same quotation also appears in a similar yet different bamboo text called \"Xing qing\" \u6027\u60c5 (Human nature and emotions), published by the Shanghai Museum. See Li Ling 2007: 155.\n\n43\n\nConfucius enthusiastically commended Yan Hui as the only disciple who, like himself, would come out when employed and would hide when denied to serve (Analects 7.11). The wisdom to decide when to come out and when to hide no doubt was implied. Hence, it was a matter of choice that Yan Hui lived in poverty. Confucius also said on a separate occasion, \"Poverty and humble station are what people hate, but if one gets them the right way, one will not leave them. If the gentleman abandons humaneness, how could he be considered one of such name? The gentleman never abandons humaneness for the moment it takes to finish a meal...\" (Analects 4.5). Yan Hui no doubt fits the description of such a gentleman.\n\n44\n\nRan Yong (style Zhonggong), along with Yan Hui, was also regarded one of the most accomplished in moral cultivation (Analects 11.3), but Confucius did not think he had achieved humaneness (Analects 5.5; cf. Analects 5.8).\n\n45\n\nLater, Mencius would make the same identification. See Mencius 6A11.\n\n46\n\nIn contrast, Zilu's replies to the same questions are: The wise person causes other people to know him and the humane person causes other people to love him. Similarly, Zigong's replies are: The wise person knows other people and the humane person loves other people. Both disciples focus on others rather than one's own self.\n\n47\n\nIn the early commentary on Analects 14.24 that is extant today, the Han-dynasty exegete Kong Anguo \u5b54\u5b89\u570b said, \"For the sake of one's own sake means one embodies the Way and practices what one learns while for the sake of others means one can only talk about what one learns.\" Huang 1991: 2.511. Clearly, the distinction between the two types of learning is amoral. The moralistic reading begins with Cheng Yi \u7a0b\u9824 (1033\u20131107). See Zhu 2001: 155.\n\n48\n\nIt should be noted that Confucius did not classify his teaching into learning for one's sake and learning for others' sake. Nor did he separate the four areas of learning into formal disciplinary categories. It is rather the learner's ambition that could make such a distinction.\n\n49\n\nAnalects 1.8, 4.7, 5.27, 6.3, 7.17, 7.31, 14.25, 15.30, and 19.8.\n\n50\n\nWhether Confucius actually studied the Book of Changes may be still controversial to some, but this does not alter the fact that even as Confucius was approaching to age 50 when he would \"know heaven's mandate\" \u77e5\u5929\u547d (Analects 2.4), he was still very much concerned with avoiding mistakes.\n\n51\n\nAnalects 5.19, 6.26, 6.30, 12.1, 12.2, 12.3, 12.22, 15.10, and 17.6. While it is not clear if Yan Hui also asked specifically about the political implications of humaneness, Confucius did refer to it explicitly in his reply. The difference between Yan Hui and these other disciples is that he was the only one who went on to seek advice on how to cultivate humaneness personally after being given an answer on the political significance of the virtue.\n\n52\n\nTo another accomplished disciple Zigong, who was just interested in seeking Confucius' confirmation of his understanding of humaneness, the master told him only a general principle of cultivating humaneness after pointing out his mistake. See Analects 6.30.\n\n53\n\nYouruo said, \"The beauty of the way of the ancient kings lies in the rites. All matters, great and small, were observed in accordance with them\" (Analects 1.12).\n\n54\n\nConfucius himself proclaimed this pedagogy in Analects 6.27 (repeated in 12.15).\n\n55\n\nThis is a theorization of the rites and it is no coincidence that it came from a later disciple who excelled in literary culture. It is quite possible that Yan Hui would agree to it.\n\n56\n\nAccording to the Zhuangzi, when Confucius asked him to serve in the government, Yan Hui replied that \"the Way that I am learning from the master is sufficient to give me joy.\" See \"Rang wang\" \u8b93\u738b Chapter, in Guo 1985: 4:978.\n\n57\n\nZixia was known for his cultivation of courage. According to Mencius, his method was similar to Beigong You \u5317\u5bae\u9edd in that they never showed submission on their face or let anyone outstare them. See Mencius 2A2. Zixia himself describes his kind of courage in the Hanshi waizhuan. See Hightower 1952: 213. It was also known that Zixia had a battle in his heart between what to pursue: the joy of wealth and high station and the joy of the Way taught by Confucius. See Sima Qian, \"Li shu\" in Li and Wang 1991: 499.\n\n58\n\nBut Zizhang was not interested in politics merely for personal gains; he actually warned against pursing profit at the expense of what is right (Analects 19.1). Ultimately, it appears that he was concerned about larger issues of cultural and political order. (Analects 2.23).\n\n59\n\nZixia was actually warned by Confucius not to be a petty scholar (xiaoren ru \u5c0f\u4eba\u5112) and to aspire to be a noble one (junzi ru \u541b\u5b50\u5112) [Analects 6.13].\n\n60\n\nIn the Great Learning (Daxue \u5927\u5b78), in a discussion of chengyi \u8aa0\u610f (sincerity of intention) where the idea of being watchful when one is alone, Zengzi was quoted to say, \"With ten eyes looking at you and ten hands pointing at [you], isn't fearsome?\"\n\n61\n\nMost notably in the bamboo text called \"Wuxing\" \u4e94\u884c. Studies on this important text are too numerous to name, but see Csikszentmihalyi 2004) and Ikeda 2005.\n\n62\n\nIn Analects 12.24, Zengzi said, \"The gentleman makes friends by being cultivated in literature and cultivates humaneness with the support of his friends.\" He addressed the nature and significance of friendship.\n\n63\n\nFor a discussion of the text, see William G. Boltz, \"Hsiao ching,\" in Loewe 1993: 141\u2013153. For a new translation of the Xiaojing, see Rosemont and Ames 2009.\n\n64\n\nAnalects 2.5, 2.6, 2.7, and 2.8.\n\n65\n\nWang 1983: 82. Confucius, in answering the later disciple Ziyou on filial devotion, said that if one does not feed one's parents with reverence, one is treating them much like one would feed dogs and horses (Analects 2.7).\n\n66\n\nChapter 7 of the Classic of Filial Devotion says, 'Filial devotion is the cord [that binds] heaven, the proper standard for earth, and the conduct that people observe\" (\u5b5d, \u5929\u4e4b\u7d93\u4e5f, \u5730\u4e4b\u7fa9\u4e5f, \u6c11\u4e4b\u884c\u4e5f).\n\n67\n\nOther unfilial mundane behaviors include climbing the mountains, treading on dangerous areas, laughing casually, getting angry wilfully. See Wang 1983: 79.\n\n68\n\nZengzi said this was what he had heard from Confucius. In the opening chapter of the Xiaojing, Confucius was reported to have said to Zengzi, \"Our physical body and limbs as well as our hair and skin are given by our parents; we dare not subject them to any injury.\"\n\n69\n\nOnce, Zengzi cut off the roots of the gourds by accident while he was weeding in the field, and his father Zeng Xi, himself also a disciple of Confucius, was infuriated. In a fit of anger he struck Zengzi with a huge staff and the son fell down and lost consciousness. After a while, Zengzi came to and he asked his father if he hurt himself since he struck him so hard. He then went back to his room and began to play the zither so that his father could hear him play and thought he was fine. When Confucius heard about this, he was angry and refused to see Zengzi. See Yang 2005: 188\u2013189. It should also be pointed out that after Zeng Xi died when Zengzi was 31 years of age, he could not help but weep every time he read the book on mourning rites. See Shizi \u5c38\u5b50 (a work of the Warring States period), quoted in Ouyang 1982: 623. Zengzi also could not bring himself to eat jujubes as it was his father's favorite food. See Mencius 7B36.\n\n70\n\nIt should be noted that Zengzi sometimes uses the term zhong \u5fe0 in the sense of reverence (jing \u656c) and considers it to be the root of filial devotion (xiao zhi ben \u5b5d\u4e4b\u672c). See Wang 1983: 79.\n\n71\n\nIn Chap.\u200b 14 of the Classic of Filial Devotion, it says, \"The gentleman in being filial to his parents can thus transfer his loyalty to his lord and in being obedient to his elder sibling can thus transfer his deference to his superiors.\" A similar statement can also be found in Chap.\u200b 2.\n\n72\n\nFor a discussion of the text, see James R. Hightower, \"Han shi wai chuan\" in Loewe ed., Early Chinese Texts: A Bibliographical Guide, 1993: 125\u2013128.\n\n73\n\nHanshi waizhuan, in Li and Wang 1991: 23. The translation is modified from Hightower 1952: 249.\n\n74\n\nConfucius himself also applauded the sage-king's rulership, which he also characterized as ruling by non-interference (wuwei). See Analects 15.5.\n\n75\n\nIn fact, Confucius himself praised Mi Zijian as a gentleman as well (Analects 5.3).\n\n76\n\nHanshi wai zhuan, in Li and Wang 1991: 688. It is possible that Mi Zijian was intending a double meaning with the pun word guan, which can mean both \"organ\" and \"officer\" here. When he rested his four limbs, preserved his sight and hearing, kept his mind and spirit quiet, his various organs would be in order accordingly. Due to its charismatic and moral influence, his subordinates and officers would also be functioning as they should.\n\n77\n\nSee \"Shenwei\" Chapter, in Yan and Zhong 2000: 75. As mentioned earlier, Mi Zijian entertained the view that human nature contains both good and evil. One may wonder if his analysis of the inclination of the Lu people to reap other people's harvest when an imminent battle made it convenient and possible does not actually reflect his view of human nature.\n\n78\n\nSee L\u00fcshi chunqiu, \"Jubei\" Chapter, in Li and Wang 1991: 687.\n\n79\n\nSee \"Guji liezhuan\" in Li and Wang 1991: 690.\n\n80\n\nSee \"Daoyin xun\" Chapter, in Li and Wang 1991: 689.\n\n81\n\nWhen Yuan Xian asked about shame, Confucius told him, \"When the Way prevails in the state, serve. When the Way falls into disuse in the state, it is shameful to serve.\" See Analects 14.1.\n\n82\n\n\"Rang wang\" Chapter in Li and Wang 1991: 700.\n\n83\n\nHanshi waizhuan and Liu Xiang's \u5289\u5411 Xin xu \u65b0\u5e8f, in Li and Wang 1991: 701 and 704, respectively.\n\n84\n\nKongcongzi \u5b54\u53e2\u5b50, \"Jie Mo\" \u8a70\u58a8 Chapter, in Li and Wang 1991: 769. According to Han Fei, the followers of Qidiao \"never show submission on their face or let anyone outstare them. When their behavior is not straight, even if they face a captive, they would avoid him. When their behavior is straight, even if they face a feudal lord, they would reprimand him.\" See Chen 1974: 2:1085. For study of Qidiao \"school,\" see Meng Wentong, \"Qidiao zhi ru kao\" in Meng 2006: 152\u2013156.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_5\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 5. Zisi and the Thought of Zisi and Mencius School\n\nChen-Feng Tsai1\n\n(1)\n\nDepartment of Chinese Literature, National Taiwan University, Taiwan, China\n\nChen-Feng Tsai\n\nEmail: tsaichen@ntu.edu.tw\n\nAbstract\n\nZisi \u5b50\u601d was a grandson of Confucius (552\u2013479 BCE). According to QIAN Mu\u9322\u7a46 (Qian 1935: 161), he was born in the thirty-seventh year of the reign of the Zhou Dynasty's King Jing and lived to the ripe old age of 82 sui\u6b72 (483\u2013402 BCE). Despite Zisi's exalted ancestry, gaps in the historical record have made it extremely difficult for scholars to obtain an understanding of his thought with any certainty. The following two sections will briefly review some of the issues surrounding Zisi's works and the school of thought to which he belonged as a preface to further discussion of him in the context of Chinese intellectual history.\n\n## 1 The Zisi of Intellectual History\n\nZisi \u5b50\u601d was a grandson of Confucius (552\u2013479 BCE). According to QIAN Mu \u9322\u7a46 (Qian 1935: 161), he was born in the thirty-seventh year of the reign of the Zhou Dynasty's King Jing and lived to the ripe old age of 82 sui \u6b72 (483\u2013402 BCE). Despite Zisi's exalted ancestry, gaps in the historical record have made it extremely difficult for scholars to obtain an understanding of his thought with any certainty. The following two sections will briefly review some of the issues surrounding Zisi's works and the school of thought to which he belonged as a preface to further discussion of him in the context of Chinese intellectual history.\n\n### 1.1 Zisi's Works\n\nA number of works have been attributed to Zisi throughout the centuries. Ostensibly, the least problematic works should have been eponymous works: the bibliographical treatise of the Han shu (\u6f22\u66f8) mentions a work entitled Zisi in 23 juan, while the corresponding section of the Sui shu (\u968b\u66f8) lists a similarly titled work, the Zisizi (\u5b50\u601d\u5b50), in seven juan. Unfortunately, though, both works were lost sometime during the period of the Northern and Southern dynasties. Still, it is possible that portions of these works could have survived in other forms, for the musicology section of the Sui shu also quotes the Southern Dynasty scholar Shen Yue \u6c88\u7d04 (441\u2013513) as saying that portions of the Zisizi are still to be seen in the Book of Rites (\u79ae\u8a18), namely the Doctrine of the Mean (Zhong yong \u4e2d\u5eb8), Biao ji \u8868\u8a18, Fang ji \u574a\u8a18, and Zi yi \u7dc7\u8863 chapters. The biography of Confucius in the Shiji (\u53f2\u8a18) also names Zisi as the author of the Doctrine of the Mean.\n\nBased on the Shiji and the Sui shu, we can see that the Doctrine of the Mean has traditionally been attributed to Zisi. And while that traditional ascription was not challenged for centuries, some questions raised during that time as to the original organization of the 41 sections in the transmitted version recorded in the Book of Rites are relevant to our discussion. The Shiji does not give any information about any subdivisions of this work, but the bibliographical treatise of the Han shu lists a Commentary on the Doctrine of the Mean (Zhong yong shuo \u4e2d\u5eb8\u8aaa) in two chapters. This led the Song dynasty scholar Wang Bo \u738b\u67cf (1197\u20131274) to speculate that the transmitted version should be divided into two chapters, one containing the first 20 sections, the other of the final 11 sections. Despite his doubts about the arrangement of this work, Wang never went so far as to doubt that any of the 31 sections were not written by Zisi. That was left to the Qing dynasty scholar Cui Shu \u5d14\u8ff0 (1740\u20131816). Cui based his argument that Zisi was not the author of the Doctrine of the Mean on textual evidence. For instance, he argued that the language of the Doctrine of the Mean was elaborate and obscure, especially when compared to the simplicity and clarity characteristic of the Analects and the Mencius. The difference in style of expression was taken as evidence for a later date of composition for the Doctrine of the Mean. More important, perhaps, was Cui's observation that a phrase in the 20th section of the work, \"when those in an inferior position do not have the confidence of their superiors\" was virtually identical to a passage in the Mencius. Cui pointed out that when quoting Confucius or Zisi in other contexts, the Mencius always began with the formulation \"Confucius said\" or \"Zisi said.\" The lack of such an attribution, Cui argued, meant that the passage in question was to be understood as being spoken by Mencius (372\u2013289 BCE) himself. In other words, the Doctrine of the Mean should be understood as quoting the Mencius, rather than the other way around, and therefore could not have been the work of Zisi.\n\nSome modern scholars have tried to combine the insights of Wang and Cui by suggesting multiple authorship of the Doctrine of the Mean. Specifically, it has been suggested that either the entire first half or the second through to the nineteenth sections thereof were penned by Zisi, with the remaining sections being added later by students of either Mencius or Zisi.1 Other evidence has been put forth to support the view that the Doctrine of the Mean is made up of materials written by Zisi and other unknown author or authors during the transition from the Qin dynasty to the Han dynasty. For example, it has been suggested that the marking of quotations of Confucius with the phrases \"Confucius said\" (\u5b50\u66f0) and \"Zhongni said\" (\u4ef2\u5c3c\u66f0) indicates different relations on the part of the author with Confucius. Likewise, the use of elements from two independent genres, namely the discursive essay and collected sayings, can perhaps be taken as indicative of different authors writing at different points in time.2\n\nIn addition to the Doctrine of the Mean, Zisi is also attributed with putting forth a theory of wuxing \u4e94\u884c (five actions) in the \"Contra Twelve Philosopher\" chapter of the Xunzi \u8340\u5b50. However, what exactly this theory entailed baffled scholars until the recent discovery of two versions of a text known as the Wuxing: the first, a silk manuscript, was discovered at the Mawangdui site in Changsha, Hunan Province in 1973, while the second, a bamboo manuscript, was discovered at the tomb of a Chu king at Guodian in Jingmen, Hubei Province in 1993. The silk manuscript is divided into two sections, the Wuxing text, which is for the most part identical to the bamboo manuscript, and a commentary on that text.3 The fact that the Guodian grave was sealed up sometime before 300 BCE means that Zisi would have been alive when this text was composed. That happy coincidence, combined with the connection between Zisi and something called the \"wuxing\" in the Xunzi has led most scholars to agree that the two manuscripts represent a single text that was originally written by Zisi or his students. On the strength of these finds, then, we can also see that the term wuxing in the Xunzi does not refer to the five cosmological elements \u2013 metal, wood, water, fire, and earth \u2013 that flow from yin and yang. Instead, it refers to a set of five kinds of actions: benevolence, righteousness, ritual propriety, wisdom, and sagacity (Pang 2000).\n\nTwo other bamboo manuscripts with connections to Zisi were also uncovered in the Guodian find, the Lumugong wen zisi \u9b6f\u7a46\u526c\u554f\u5b50\u601d (Duke Mu of Lu Asks Zisi) and the Zi yi mentioned previously. The first of the two is generally seen as being written by Zisi or his students. The basis for this attribution, besides the fact that the text presents itself as a dialogue with Zisi, is the appearance of the notion that one should \"follow the Way and not the ruler.\" This formulation is consistent with the description of Zisi's ideas given in the Mencius. As for the Zi yi, the evidence in favor of authorship by Zisi includes Shen Yue's attribution mentioned earlier and its appearance together with other works attributable to Zisi at Guodian. However, in his Jingdian shiwen \u7d93\u5178\u91cb\u6587, Lu Mingde \u9678\u5fb7\u660e (circa 550\u2013630) quotes a scholar as saying that the Zi yi was authored by Gongsun Nizi \u526c\u5b6b\u5c3c\u5b50. For this reason, a conservative conclusion would be to see this work as a collection of sayings by Confucius that was passed on by both Zisi and Gongsun Nizi (Li 2002: 70\u201371).\n\nFrom the above, we can see that our most reliable sources of information on the thought of Zisi are the Wuxing and the Doctrine of the Mean (from the second to the nineteenth sections). The bamboo manuscript of the Lumugong wen zisi can also be seen as a secondary reference.\n\n### 1.2 Zisi's Scholarly Affiliation\n\nThe question of Zisi's scholarly genealogy has given rise to considerable speculation throughout the centuries. Given the year of his birth, it would have been impossible for Zisi to have studied under Confucius. There is also no evidence suggesting that he was a student of either Zengzi \u66fe\u5b50 (505\u2013436 BCE) or Ziyou \u5b50\u6e38 (506\u2013443 BCE). During the Tang dynasty, though, some scholars, including Han Yu \u97d3\u6108 (768\u2013824) suggested that Zisi was indebted to Zengzi (Han Changli ji \u97d3\u660c\u9ece\u96c6 20:14b\u201315a). This claim was accepted by prominent Neo-Confucians in the Northern Song dynasty and expanded on by Zhuzi (Zhu Xi\u6731\u71b9 1130\u20131200), who included Zengzi in his genealogy of the transmission of the Way.4 Some modern scholars have also suggested that Zisi's thought can be traced to Ziyou (Meng 2008: 22\u201323, and Jiang 2000), but the argument for this connection has come under sharp criticism (Ye 2000).\n\nThe question of Zisi's intellectual descendants, or more particularly Zhuzi's claim that the Way was transmitted from Zisi to Mencius, has also produced much discussion. But whereas the proposed connection between Zengzi and Zisi might have been mere speculation on the part of scholars in the Tang and Song dynasties,5 there is actual evidence that some of Zisi's ideas were transmitted by Mencius. For instance, the Shiji \u53f2\u8a18 states that, \"Meng Ke was from Zou and studied under a student of Zisi,\" while the \"Contra Twelve Philosophers\" chapter of the Xunzi claims that \"Zisi sang the tune (of the Wuxing) and Mencius provided the harmony.\" However, evidence to the contrary can certainly be found as well. The Han feizi (\u97d3\u975e\u5b50) states that among the \"eight schools of ru\" there were the \"ru of Zisi\" (\u5b50\u601d\u4e4b\u5112) and the \"ru of Meng\" (\u5b5f\u6c0f\u4e4b\u5112), although it is possible that this was not a reference to Mencius but rather to someone else with the same surname.\n\nGiven the contradictory evidence, it is not surprising that the relation between Zisi and Mencius has continued to be a source of controversy.6 Two reasons are usually put forth for doubting Zhuzi's genealogy and whether the two men belonged to the same school. To begin with, Zisi was already dead when Mencius was born, and there is no indication in the Mencius that Mencius either studied under Zisi's students or considered himself to have been championing Zisi's ideas. Furthermore, the chapter of the Xunzi mentioned above is generally seen as being written by followers of Xunzi interested in promoting the thought of their teacher at the expense of Mencius.7 Thus, its claims should be seen as being polemical in nature and not necessarily reflecting historical reality.\n\nThe discovery of the silk and bamboo copies of the Wuxing has resulted in a reconsideration of the relationship between Zisi and Mencius. The appearance of the term \"wuxing\" in the Xunzi (as being characteristic of their thought) and in the newly discovered texts has led to the conclusion that what has been found is a lost text from the school of thought led by Zisi and Mencius. Some more adventurous scholars have even gone so far as to suggest that the jing \u7d93 (classics) portion of the silk manuscript was written by Zisi, with the commentary being authored by Mencius (See, for example, Chen 2008a: 1\u201311). This, however, seems to be based on an uncritical acceptance and mechanical application of the claim in the Xunzi that \"Zisi sang the tune and Mencius provided the harmony,\" a claim we have suggested cannot be taken at face value. A more nuanced understanding of the question is possible if we understand this phrase as indicating a similarity or continuity between the thought of these two men, rather than an actual teacher-student relation. This would allow us to tentatively agree that the \"classic\" portion of the Wuxing was written by Zisi or his students and that the commentary was written either by Mencius or his students.\n\n## 2 Philosophical Issues for the ru Tradition Posed by the Wuxing Manuscript\n\nWhile we cannot say with certainty that texts such as the Doctrine of the Mean, the Wuxing, and the Lumugong wen zisi were written by Zisi, we can be confident that they are at least closely associated with his school of thought. By the same token, even though these texts may not lend themselves to a discussion of Zisi's thought per se, they do provide us with a window through which we can see what sort of issues Zisi was interested in discussing. This is the same reading strategy that allows us, for instance, to take a statement by Zigong \u5b50\u8ca2 in Book 5 of the Analects that, \"The Master's discourse on nature and the Way of Heaven cannot be heard,\" as indicating that the topics of nature and the Way of Heaven were of much interest to students of Confucius even though he himself did not spend much time discussing them.\n\nCan Zisi be seen as providing a new twist to questions of the sort that Zigong was interested in? While we would hesitate to claim that the statement, \"What Heaven confers is called nature\" in the opening passage of the Doctrine of the Mean represents Zisi's view of these topics, we can say on the strength of the Wuxing that these topics held a place of importance in Zisi's school of thought. We can go further and add that given the way in which the Way of Heaven and the Way of Man are theorized in the Wuxing, at the very least the distinction is being made between autonomous and heteronymous morality, the moral and the natural good, the sage and sagacity, and between tacit and explicit knowledge. These distinctions will be explored in detail below.\n\n### 2.1 Internal Formation of the Five Actions\n\nThe Wuxing discovered at the Guodian grave opens with the following: 8\n\n1.\n\nWhen benevolence forms internally, it is called \"virtuous action\"; when not formed internally, it is called \"action.\" When righteousness forms internally, it is called \"virtuous action\"; when not formed internally, it is called \"action.\" When ritual propriety forms internally, it is called \"virtuous action\"; when not formed internally, it is called \"action.\" When wisdom forms internally, it is called \"virtuous action\"; when not formed internally, it is called \"action.\" When sagacity forms internally, it is called \"virtuous action\"; when not formed internally, it is called \"virtuous action.\"\n\nThe five kinds of actions in the Wuxing \u2013 benevolence, righteousness, ritual propriety, wisdom, and sagacity \u2013 can be understood as corresponding to the five virtues that are articulated when these actions take form. However, a distinction is made with respect to the first four virtues between instances in which they are formulated internally and instances in which they are not. In the former case, the result is \"virtuous action,\" while in the latter case, there is simply \"action.\" These two terms refer to two types of moral conduct that are predicated on different principles. \"Virtuous action\" is action based on an internal, independent will and is seen as moral conduct stemming from an autonomous principle. By contrast, \"action\" is an external realization based on other considerations (such as social codes, custom, or considerations of self-interest) and is seen as moral conduct stemming from a heteronymous principle. This distinction does not apply for the fifth action or virtue, sagacity. Regardless of whether it is realized internally, the Wuxing claims that sagacity is always to be considered \"virtuous action.\" The gist of this last point would seem to be that all conduct that is a realization of sagacity is based on an autonomous principle.\n\n### 2.2 Virtue and Good\n\nThe second section of the Wuxing states:\n\n2.\n\nWhen the five \"virtuous actions\" are harmonized, it is called \"virtue\"; when four actions are harmonized, it is called \"good.\" Good is the Way of Man; virtue is the Way of Heaven.\n\nHere, the harmonization of the five \"virtuous actions\" is itself called \"virtue\" and associated with the Way of Heaven. By contrast, the harmonization of the four types of action is identified as \"good\" and associated with the Way of Man. The term \"harmonized\" here should be understood as referring to something that transcends the categories of the analytic and the synthetic. Thus, the \"virtue\" resulting from the harmonization of the five actions indicates an active state of moral good. By contrast, the \"good\" stemming from the harmonization of the four actions refers to a completed state of natural good reached by bringing together benevolence, righteousness, ritual propriety, and wisdom.9 Here \"natural good\" refers to a relativistic conception of good and bad where a value is assigned to something depending on whether it is suited as a means or an instrument for achieving a desired end. As causal relations prevail here, this state is understood as belonging to the Way of Man. \"Moral good,\" on the other hand, is based on the principle of reason devoid of all a posteriori elements. Being as it is completely outside the realm of causal relations or considerations of personal benefit, this state is understood as belonging to the Way of Heaven.\n\nThe terms \"virtue\" and \"good\" are discussed further in sections 3 and 10 through 11 of the Wuxing:\n\n3.\n\nIf the five actions all form internally and are acted out in a timely manner, the (one who does so) is called a \"Gentleman.\" A candidate official who aspires to the Way of the Gentleman is called an \"aspiring candidate official.\" Without being enacted, good could never be approached. Without being willed, virtue could never be brought to completion. Without being reflected upon, wisdom could never be obtained.\n\n  * 10\u201311 When a Gentleman performs good acts, there is a beginning and an end; when a Gentleman performs virtuous acts, there is a beginning but no ending. \"A metal bell sounding and a jade stone vibrating it\" is a person of virtue. \"A metal bell sounding\" is good; a \"jade tone\" is sagely. Good is the Way of Man; virtue is the Way of Heaven. Only with a person of virtue may there be \"a metal bell sounding and a jade stone vibrating it.\"\n\nSection 3 implies that moral good has nothing to do with the relation that obtains between a means or instrument chosen and the desired end. Instead, it is completely dependent on the independent volition of the will. We can see from the statement \"without being willed, virtue could never be brought to completion\" that virtue is always to be found in the active state of the will. Based on sections 10 and 11, the performance of a good act is predicated on the causal connection between a means and an end, that is, between a beginning and an ending. By contrast, the performance of virtuous acts stems from an autonomous moral nature without any concern for instrumental means. Thus there is a beginning (the moral nature) but no ending. Two metaphors, those of the \"metal bell sounding\" and a \"jade stone vibrating,\" are also used to explain, respectively, the performance of good acts and virtuous acts. These metaphors can be used in this way because the phrase \"a metal bell sounding\" carries the meaning of a beginning, while \"a jade stone vibrating\" implies an ending.10 Taken together, the phrase \"a metal bell sounding and a jade stone vibrating\" thus describes a virtuous sagacity that is able to combine the natural good of the Way of Man with the moral good of the Way of Heaven. Put another way, this virtuous sagacity is able to partake in the function of the natural good without violating the moral good of the latter.\n\n### 2.3 The Sage and Sagacity\n\nIn the above discussion of good and virtue, virtue is shown to be a kind of good that stems from an independent will. The question, then, is how that virtue comes about. This question is treated in sections 15, 17 and 26:\n\n  * 15: Not having heard the Way of the Gentleman is called \"not sharp-eared.\" Not having seen a worthy is called \"not clear-sighted.\" Hearing the Way of the Gentleman and not knowing that it is the Way of the Gentleman is called \"un-sagely.\" Seeing a worthy and not knowing he has virtue is called \"un-wise.\"\n\n  * 17: Hearing the Way of the Gentleman is being \"sharp-eared.\" Hearing and knowing it is sagacity. The sage knows the Way of Heaven. To know and act on it is righteous. To act on it in a timely way is virtue. Seeing a worthy is being \"clear-sighted.\" Seeing and knowing it is wisdom. Knowing and being content therewith is benevolence. Being content and respectful is ritual propriety. Sagacity gives birth to wisdom, ritual and music,11 and is the harmonization of the five kinds of action.\n\n  * 26 Seeing to know it is called approaching it. Drawing on metaphors to know it is called approaching it. Drawing on comparisons to know it is called approaching it. Knowing by intuition, that is Heaven. \"Shangdi \u4e0a\u5e1d is with you; be not of two minds.\" That is what is meant.\n\nWe have seen above how the harmonization of the five \"virtuous actions\" is related to the realization of virtue. But how is that harmonization achieved? The answer to this question cannot be found by making appeal to the idea of the sage, for by definition the conduct of the sage is an expression of virtue. It would appear, instead, that this harmonization is closely linked with the idea of sagacity.\n\nTo better understand what sagacity refers to, we need first to recognize the distinction between the sage and sagacity. A close reading of the Wuxing text shows that the two terms are used differently. For example, in section 17, the term \"sage\" seems to refer to a particular type of person (e.g., a ruler such as King Wen) and not to the average person. Sagacity, on the other hand, refers to having aural knowledge of the Way of the Gentleman. The difference between a sage and an average person, then, stems from the former's ability to know Heaven. But even though the average person does not, by definition, have this knowledge, there is no reason to assume that the innate value of sagacity is also denied to him. To put it another way, the Wuxing presents each of the five types of virtue as corresponding to a different reflective and cognitive capability. Wisdom corresponds to distinguishing the worthies, while benevolence, righteousness, and ritual propriety correspond, respectively, to whether the mind can be content with, implement, and respect the Way of the Gentleman. Sagacity, for its part, is paired with having knowledge of the Way of the Gentleman. Thus in some fashion knowledge of the Way of the Gentlemen clears the way for the harmonization of the five virtues.\n\nUp until now we have taken it for granted that the Way of the Gentleman is \"out there\" to be heard and recognized. But how is it that this information is available to be heard in the first place? To answer that question, appeal will ultimately have to be made to the sage and his knowledge of the Way of Heaven, for the sage not only has this knowledge, but acts on it and does so in a timely manner. In other words, it seems that the Way of the Gentleman can be known by others precisely because the sage embodies and articulates it through his conduct. It is significant to note that this articulation is achieved through actions instead of through language, a point that brings us to an implicit distinction between two types of knowledge in the Wuxing.\n\n### 2.4 Tacit and Explicit Knowledge\n\nThe Wuxing argues that people, by means of sagacity, can \"hear\" and thus know the Way of the Gentleman as articulated by the sage. However, in section 26 we can see that the type of knowledge associated with sagacity, what has up until now been referred to as knowledge based on hearing, is fundamentally different from other types of knowledge discussed in the Wuxing. In this section, we are presented with four different ways of knowing. The first three, knowledge based on \"seeing,\" \"metaphors\" and \"examples,\" are associated with the category of \"man\" and can be taken to refer to knowledge and deductions associated with the sensual facilities. The fourth, that associated with sagacity and the category of \"Heaven\" is knowledge based on intuition (ji \u5e7e). This type of knowledge should be understood as tacit knowledge, that is to say, knowledge of an innate value or metaphysical truth obtained without the aid of conscious thought. It is exactly this distinction between two types of thought, or \"two minds,\" that allows the author of the Wuxing to use the phrase \"Shangdi is with you; be not of two minds\" to sum up section 26. The term shangdi makes the connection with the \"Heaven\" that is known intuitively, while the admonition \"be not of two minds\" points to the superiority of tacit knowledge. It is also significant that the notion of knowing intuitively is associated, as is \"hearing and knowing,\" with sagacity. The connection between these two kinds of knowing allows us to visualize the following theoretical procession: First there is a sage with knowledge of the Way of Heaven, who then articulates this knowledge as the Way of the Gentleman. This then allows others to gain knowledge (intuitive knowledge, also referred to as \"hearing and knowing\") of the Way of the Gentleman. This in turn ultimately results in the manifestation of wisdom as well as rites and music (i.e., benevolence, righteousness, and ritual propriety, respectively).\n\nIn light of the above analysis, we can better understand another part of section 2:\n\n> A Gentleman without the anxiety of the inner mind will be without the wisdom of the inner mind. Without the wisdom of the inner heart, he will be without the joy of the inner mind. Without the joy of the inner mind, he will be discontented. Discontented, he will be unhappy. Unhappy, he will be without virtue.\n\nThe term \"anxiety of the inner mind\" refers to a fearful dread on the part of the Gentleman that the Way of the Gentleman will not be put into action. It is only because of this dread that the mind, based on moral sentiments and intellectual consideration, arranges for various actions to be undertaken. Thus all subsequent thought and action is an opportunity for the mind to take the initiative of putting into action the Way of the Gentleman that has been \"heard.\" This in turn will produce feelings of joy, contentment, and happiness and will ultimately result in virtue.12\n\nIn the Wuxing, Heaven is explained by the phrase \"Shangdi is with you\". This could be interpreted as suggesting a tendency towards theonomous ethics or divine command theory whereby the morality of an act is determined by reference to a divine will. We would suggest, however, that that is not the case. Instead, ethical thought in the Wuxing is closer to a rational or autonomous ethical position. To begin with, it should be noted that there is no obvious notion of supreme deity in the text. It is true that there is a passage that states: \"'Shangdi is with you; be not of two minds.' That is what is meant.\" It is also true that the term shangdi can be taken to mean a supreme deity. However, the fact that the phrase, \"that is what is meant\" is added suggests that shangdi should be understood as an analogy instead of factually. Furthermore, while there is a Heaven\/man dichotomy in the Wuxing, there is no sense of Heaven as a transcendental deity. Instead, the emphasis seems to be on showing that the category of Heaven is incompatible with man's instrumental thought. The Wuxing does stress that the Way of Heaven is superior to the Way of Man, just as the virtuous action is superior to mere action and virtue is superior to good. However, this is done simply to indicate that the highest ethical principle has a transcendental nature unrelated to the calculated achievement of self-interested goals. But this transcendental nature does not entail a religious fear of a deity. Instead what is called for is an understanding of transcendental moral principles and choices in conduct based thereon. This is what is meant in section 3 when it is stated:\n\n> Without being enacted, goodness could never be approached. Without being willed, virtue could never be brought to completion. Without being reflected upon, wisdom could never be obtained. Reflection that is not clear cannot examine (its object); reflection that is not thorough cannot give form (to its object). Being unable to give form (to its object) one will be discontent. Being discontent, one will be unhappy. Being unhappy, one will be without virtue.\n\nIn other words, the overall position of the Wuxing is that anyone aspiring to the Way of the Gentleman can use his innate sagacity to obtain tacit knowledge of moral principles and demands. Once those moral demands are grasped, that person can go on to perform good actions externally and cultivate virtue internally.\n\n## 3 From Zisi to Mencius\n\nGiven that the bamboo manuscript of the Wuxing was composed earlier than the Mencius, a comparison of the two texts can also help us understand developments in Confucian thought during this time. Likewise, by contrasting the above discussion of some of the theoretical aspects of the Wuxing with the Analects and the Mencius, we can get a better idea of the place this text has in Chinese intellectual history.\n\n### 3.1 Inner Virtue and Moral Autonomy\n\nThe Wuxing makes a distinction between autonomous morality and heteronomous morality by contrasting the formation of the five actions internally with the failure to do so. This is a distinction that is largely lacking in the Analects. While it is true that the Analects stresses that authentic rituals and music are predicated on benevolence, Confucius is only seen exhorting his students to approach or to complete benevolence. He does not delve into the theoretical nuances of the difference between ordinary \"ritual and music\" and music and ritual that stem from benevolence, nor does he delineate a hierarchical ranking or explicate an order of importance in terms of practice. As a result, students such as Ziyou and Zixia \u5b50\u590f formed divergent views on these issues, a disagreement that is aired out in Book 19 of the Analects. There, Ziyou criticizes the emphasis students of Zixia place on \"sprinkling and sweeping,\" \"answering and replying,\" \"advancing and retiring,\" as \"not knowing the essential.\" Zixia, for his part, insists that there should be a sequence in one's studies, a \"beginning and an end.\" This kind of dispute indicates that Confucius did not leave behind a definitive answer to questions of this type.\n\nThe distinction in the Wuxing between \"virtuous action\" and mere \"action\" is at the very least a much more detailed explanation of the difference between autonomous morality and heteronomous morality than Confucius ever gave. In that sense, the Wuxing can be seen as making a clear theoretical distinction between ordinary ritual and music (sans internal formation) and ritual and music stemming from benevolence (with internal formation). Mencius seems to be making a similar point when he criticizes Gaozi \u544a\u5b50 for having never understood righteousness because he treated it as something external to himself. Mencius, of course, argues that both benevolence and righteousness are internal. Likewise, Mencius' idea of \"acting from benevolence and righteousness rather than enacting benevolence and righteousness\" seems to be inspired by the same line of thought that led the Wuxing to contrast \"virtuous action\" with ordinary action. In both cases, the argument hinges on the internal origin of a kind of conduct.\n\nThe Mencius builds further on the distinction between \"internal\" and \"external\" to espouse an autonomous morality. For instance, Mencius states:\n\n> The mind of compassion pertains to benevolence, the mind of shame to righteousness, the mind of respect to ritual propriety, and the mind of right and wrong to wisdom. Benevolence, righteousness, ritual propriety and wisdom do not coat me in a luster from the outside. They are inside me to begin with, only I didn't think of it. 6A6\n\nHere, the assertion about benevolence, righteousness, ritual propriety and wisdom being internal is a restatement of Mencius' idea that the liangzhi \u826f\u77e5, or the mind, is the final arbiter of moral principles in his system. In other words, the fact that both virtues and moral principles are internal leads naturally to the notion of an autonomous morality. This move is further strengthened by the distinction Mencius makes between more noble \"honors bestowed by Heaven\" (tianjue \u5929\u7235) and ordinary \"honors bestowed by man\" (renjue \u4eba\u7235). The ascription of \"nobility\" to the former serves to underscore the moral nature of men and suggests that they are not to be used instrumentally, both key features of an autonomous morality.\n\n### 3.2 An Emphasis on Inner Reflection\n\nAs we have seen, the Wuxing distinguishes between explicit knowledge (knowing through seeing, analogies, and comparison) and tacit knowledge (knowing intuitively). But in terms of tacit knowledge, there is an emphasis on knowledge of external objects, whether the intuiting of the Way of Heaven, hearing the Way of the Gentleman, or the seeing of worthies.\n\nThe Mencius also discusses tacit knowledge in a number of places:\n\n> What one can do without having to learn is one's inborn ability; what one knows without having to reflect is one's inborn knowledge. There is no one who does not know to love their parents when they are young children and respect their elder brothers when they grow up. 7A15\n\nHere it should be noted that the fact that one can know without reflection shows that the verb \"know\" must be taken as referring to tacit knowledge. The knowledge (zhi \u77e5) part of the liangzhi \u826f\u77e5 is completely independent from explicit knowledge, which is precisely what makes it liang \u826f (noble). If we compare the Mencius with the Wuxing, however, we find that the kind of tacit knowledge emphasized in the Mencius is qualitatively different from that discussed in the Wuxing. For where the Wuxing emphasizes tacit knowledge of external objects, in the Mencius it comes from an awareness of one's own inner liangzhi.\n\nThe liangzhi, besides being discussed in terms of its appearance as compassion, shame, respect, and a sense of right and wrong, can also be seen as stemming from a sense of filial piety. This can be seen when Mencius, answering questions about the sage Shun, does not begin his discussion of the sage with a consideration of the Way of Heaven or the definition of a Gentleman or a worthy (see Mencius, 5A) Instead of starting from such external objects, Mencius takes filial piety as his starting point.13\n\nThe prominence of ideas like being filial to one's parents and being sincere to oneself (chengshen \u8aa0\u8eab) in the Mencius suggests that tacit knowledge is treated primarily as inner moral knowledge, and not the type of external knowledge of Heaven or the Way of the Gentleman seen in the Wuxing. As a result, the moral principles (or moral thought) discussed in the Mencius do not require the apprehension of a metaphysical principle or the Way of the Gentleman. Instead, it revolves around inner reflection and the finding of an inherent, inner moral mind-nature (xinxing \u5fc3\u6027) that starts with consideration of the subtle outward manifestations of that mind-nature.\n\nThe tacit knowledge described by Mencius is also related to his concept of thought (si \u601d). Thus we see statements like:\n\n> The desire to be exalted is common to the minds of all men. Everyone has something to be exalted inside them, they just haven't thought of it. 6A17\n> \n> Benevolence, righteousness, ritual propriety and wisdom do not coat me in a luster from the outside. They are inside me to begin with, only I didn't think of it. 6A6\n> \n> The function of the mind is to think: think and it will be obtained: don't think and it will not be obtained. This has been conferred upon me by Heaven. If one establishes what is of greater importance to begin with, things of less importance will be unable to dislodge it. 6A15\n\nFrom this we can see that for the most part Mencius' discussion of tacit knowledge emphasizes thought that is directed towards what is common to the minds of all men. This \"thought\" is not just any thought but rather should be understood as the reflection on and inspection of the mind. And insofar as reflection and inspection play a significant role in Confucian practice, \"thought\" in Mencius' hands comes to have important implications for the theory and practice of cultivation. By contrast, statements in the Wuxing such as \"Lacking benevolence, thought cannot be precise,\" \"lacking wisdom, thought cannot be thorough\" and \"lacking sagacity, thought cannot be swift,\" show correct thought and knowledge to be dependent on the virtues of benevolence, wisdom, and sagacity. Here, clearly, the object and content of thought is of secondary importance. From this we can see that Mencius is not concerned simply with thought in a formal sense, as is the case in the Wuxing, but also with its object and content.\n\n### 3.3 A De-emphasizing of Sagacity\n\nIn the Mencius, we can see statements that approximate claims in the Wuxing about sages knowing the Way of Heaven, having aural knowledge of the Way of the Gentleman, and visual knowledge of the worthies. For example:\n\n> From Yao and Shun to Tang, it was over five hundred years. Men like Yu and Gao Yao saw and knew them, while those like Tang heard and knew of them. From Tang to King Wen, it was over five hundred years. Men like Yi Yin and Lai Zhu saw and knew of them, while those like King Wen heard and knew of them. From King Wen to Confucius it was over five hundred years. Men such as Tai Gongwang and Sanyi Sheng saw and knew of them, while those like Confucius heard and knew of them. From Confucius to the present it has been over a hundred years. As such we are not far removed in time from the time of the sage, just as we are quite close to the home of the sage, and yet no one has anything of them, no one has anything of them. 7B84\n\nAccording to this, included among the sages of old were those with aural knowledge and those with visual knowledge. But in doing so, Mencius effectively does away with the hierarchy put forth in the Wuxing whereby aural knowledge is associated with the sage, who is superior to the worthy with his visual knowledge. Menicus' disregard for the previously established ordering could be taken as an indication that he felt no need to differentiate between the sage and the worthy. To put it another way, and perhaps more precisely, Mencius believed that sagacity could be subsumed under the category of wisdom.\n\nThis subsuming of sagacity to wisdom should be viewed in conjunction with the following passage:\n\n> Mencius said, \"The way the mouth is disposed to tastes, the eyes to colors, the ear to sounds, the nose to smells, and the four limbs to ease is human nature. It is also conferred (by Heaven), and so the Gentleman does not call it 'nature.' The way benevolence pertains to the relation between a father and a son, righteousness to the relations between a ruler and the subject, ritual propriety to the relation between a host and a guest, wisdom to the worthy and the sage to the Way of Heaven, that is conferred (by Heaven). But it is also human nature, and so the Gentleman does not say it is conferred (by Heaven).\" 7B70\n\nIn this passage we see Mencius distinguishing between a biological nature consisting of the ears, eyes, mouth, nose, and four limbs and a moral nature consisting of benevolence, righteousness, ritual propriety, wisdom and the relation of the sage to the Way of Heaven. It is important to note that in contrast to the Wuxing, sagacity and aural knowledge of the Way of the Gentleman are missing in the Mencius. This could be taken as an indication that Mencius believes that the autonomous moral principle is achieved directly from tacit knowledge produced by one's own liangzhi rather than indirectly from aural knowledge of the Way of the Gentleman.\n\nThe listing of the sage and the Way of Heaven together with the virtues of benevolence, righteousness, ritual propriety, and wisdom would seem to indicate that the five are of equal standing. However, we arrive at a different understanding of their relative status once we consider the idea of the \"four germs\" (siduan \u56db\u7aef, see 2A6). From a theoretical standpoint, the statement that \"the mind of compassion is the germ of benevolence; the mind of shame is the germ of righteousness; the mind of courtesy is the germ of ritual propriety; the mind of right and wrong is the germ of wisdom\" should be seen as being equivalent to the \"hint\" (ji \u5e7e) of the Way of Heaven that makes intuition of it possible. In his commentary on the Mencius, Zhuzi glosses the term translated here as \"germ\" (duan \u7aef) as \"tip of a thread\" (xu \u7dd2), adding \"it is as if there is something inside and its tip is showing.\" In other words, he understands the \"germs\" to have a meaning similar to the external \"subtle signs\" (jiwei zhizao \u5e7e\u5fae\u4e4b\u5146) of the Book of Changes.\n\nThis similarity is significant because it suggests a parallel between the \"four germs\" and the sage and his knowledge of the Way of Heaven. As we have seen, the Wuxing argues that the sage has virtue and access to the Way of the Gentleman because he is able to gain (intuitive) knowledge of the Way of Heaven from subtle \"hints.\" It is possible that Mencius lists the sage together with the four virtues because he views them as representing two different paths for gaining access to the Way of the Gentleman. One path, the one outlined in the Wuxing, was the one taken by the sages Yao and Shun: they knew the Way of Heaven directly. Mencius' innovation is to suggest that there is an alternative path, and one that is available to ordinary people, that involves \"thought\" and \"examination\" directed at one's own liangxin \u826f\u5fc3. So while acknowledging that the sage knows Heaven, the Mencius also argues that it is possible to have knowledge of internal moral principles by way of one's liangxin. This opens up the possibility that ordinary people can obtain the same results as Yao and Shun simply by developing their \"four germs.\"\n\nWhy does Mencius open up this alternative path? The answer to that question lies in his reluctance to claim that he was a sage even as he hoped to establish his own moral system. When asked if he himself is a sage, Mencius invokes Confucius, and, in addition to his strong denial, adds the comment: \"A sage is something Confucius did not claim title to\" (2A2). So, while Confucius is referred to as a sage several times in the Mencius, here we see Mencius implying that he could no more recklessly claim sagehood for himself than Confucius could. But given the context of the Wuxing and the exclusive status of the sage, it is difficult to see how Mencius could hope to legitimately develop his own moral system without claiming to be a sage and having knowledge of the Way of Heaven. In that light, Mencius may have felt the need to show that even though he was not the type of sage that knew Heaven, he still had access to an alternate method by which he could walk the path of the sage.\n\nAccording to Mencius, there is a universal morality that originates in a moral mind common to everyone. Thus, he states:\n\n> All palates have the same preference in taste; all ears in sound; all eyes in beauty. Should the mind prove to be the only exception in possessing nothing in common? What is it, then, that the mind has in common? Principles and correctness. The sage is simply the first to find what is common to my mind. It is for this reason that principles and correctness please my mind in the same way that meat pleases my palate. 6A7\n\nThis \"mind\" is the moral mind, which in other places is referred to by Mencius as the \"original mind\" (benxin \u672c\u5fc3) or liangzhi, while \"principles and correctness\" refer to the content of moral principles. Mencius here argues that the similarity (universality) of moral principles stems from the commonality of the original mind. For this reason he can say that the sage is, in this respect, no different from the average person. If there is any difference, it is only in terms of whether or not they have found what is common to everyone's mind. In this way, Mencius allows for the possibility that the average person can grasp moral principles by finding them in his own mind.\n\n### 3.4 A New way of Knowing Heaven\n\nEven though he does not emphasize the sages' knowledge of the Way of Heaven, Mencius does not overlook this issue. In fact, there are two passages in the Mencius that discuss knowledge of Heaven:\n\n> To give full realization to one's mind, one must know one's nature. To know one's nature, one must know Heaven. 7A1\n> \n> If a person in an inferior position does not have the confidence of his superiors, he cannot govern the people. There is a way to win the confidence of your superiors: if your friends do not trust you, you will not win the confidence of his superiors. There is a way to win the trust of friends: if you don't please your parents when serving them, you will not win the trust of friends. There is a way to please your parents: if reflecting on yourself you find yourself to be insincere, you will not please your parents. There is a way to be sincere to yourself: if you do not understand goodness, you will not be sincere to yourself. For this reason, sincerity is the Way of Heaven. Pondering sincerity is the Way of Man. 4A12\n\nAccording to the first passage, fully realizing one's mind allows knowledge of one's nature, which in turns allows knowledge of Heaven. This is how Mencius can claim that \"everybody can be a Yao or a Shun.\" Mind, nature, and Heaven are seen by him as forming a kind of continuum that allows for upward progression based on one's level of personal cultivation. Given a universal moral mind, everyone has the same potential for achieving knowledge of Heaven on par with the great sages. This lets Mencius deemphasize the virtue of sagacity and its aural knowledge of the Way of the Gentleman and develop his moral system directly from the idea of realizing one's mind. It also takes him way beyond anything seen in the Analects and the Wuxing in terms of the universality of the moral subject.\n\nZhuzi believed that realizing the mind was the ultimate result of \"exhausting principles\" (qiongli \u7aae\u7406), glossing it as \"meaning that 'knowledge is complete'.\" Elsewhere he explained the terms as \"meaning without exception that all of the principles of object are known.\"14 However, Zhuzi's interpretation is most likely inconsistent with Mencius' original meaning. Mencius argues that even though the completion of cultivation results in the people being governed, the actual realization of moral principles comes from a series of increasingly inner-oriented steps, from winning the trust of superiors, winning the trust of friends, pleasing the parents, and finally being sincere to oneself (Mencius 4A12). This on-going process of reflection begins with being sincere to oneself. But the term \"being sincere to oneself,\" insofar as it implies \"being faithful to the mind\" or \"being faithful to the four 'germs' of the mind,\" basically means the same as \"realizing the mind.\" This is also what Mencius meant by pondering sincerity (sicheng \u601d\u8aa0): he wanted his students to seek out emanations of a mind that cannot be deceived. Thus, when Mencius says, \"Sincerity is the Way of Heaven. Pondering sincerity is the Way of Man,\" the term \"Way of Heaven\" should be understood as something that man is incapable of fully comprehending but which is nevertheless imposed on him externally as with laws and regulations. By contrast, the \"Way of Man\" is to reflect on and realize those same inscrutable laws and regulations.\n\nFrom the above, we see that Mencius takes the Way of Heaven to be the source of the Way of Man. This is quite different from the approach taken in the Wuxing, where the two are seen as involving separate moral principles and the Way of Heaven is not only given a higher status but is held up as a goal towards which man should strive.15\n\n## 4 Conclusion: Another Look at the Zisi\/Mencius School of Thought\n\nBefore the discovery of the bamboo Wuxing manuscript, any discussion of Zisi' s thought inevitably began with the Doctrine of the Mean, the main feature of which is the conception of the Way of Heaven. The opening section boldly makes the claim that, \"What Heaven confers is called nature; according with nature is called the Way; the cultivation of the Way is called teaching.\" Section 20 states, \"Sincerity is the Way of Heaven. The attainment of sincerity is the Way of Man,\" and Section 21 goes on to say, \"Clarity resulting from sincerity is called nature. Sincerity resulting from clarity is called teaching.\" Here we see intentional, and thus meaningful, associations being made between Heaven and man on the one hand, and nature, Way, and teaching on the other.\n\nMou Zongsan \u725f\u5b97\u4e09 suggested that the Doctrine of the Mean took the idea of an objective and transcendent Decree of Heaven (tianming \u5929\u547d) as a starting point for its system of thought. As such, he argued that given our understanding of the development of intellectual thought, the Doctrine of the Mean should be of a later date than the Mencius with its notion that a person who fully realizes his mind knows his nature and a person who knows his nature knows Heaven (See Mou 1969: 46\u201347). This kind of argument from philosophical analysis can certainly serve to bolster arguments, such as Cui Shu's, based on textual studies that the Doctrine of the Mean was composed later than the Mencius.16 At the same time, it also suggests that some parts of the Doctrine of the Mean, and specifically its ideas about the Way of Heaven, cannot be used to compare Zisi and Mencius.\n\nHappily, there is no question about the dating of the Wuxing. Even if we cannot say with certainty whether it was penned by Zisi, we do know that it predated the Mencius. It is also consistent with the comment in the Xunzi about Zisi and Mencius' relation to this theory. At the very least, then, we can say that the Wuxing is from the period of time between the death of Zisi and the birth of Mencius. From the above discussion, we can see that the distinction between autonomous and heteronymous morality and explicit and tacit knowledge is featured more prominently in the Wuxing than in the Analects. These distinctions were also picked up and expanded by Mencius. However, the virtue of sagacity seen in the Wuxing was effectively dropped by Mencius. In the Wuxing, the distinction between sagacity and wisdom parallels the distinction between hearing the Way of the Gentleman and seeing worthies. But despite being seen as kinds of tacit knowledge, aural and visual knowledge have to be obtained indirectly from contact with the outside world. Mencius moves his emphasis away from the external world and places it firmly in the inner world. For this reason he emphasizes that benevolence, righteousness, ritual propriety and wisdom are rooted in the moral mind and argues that knowledge of moral principles can be gained from the four \"germs\" of their minds.\n\nThis idea can be understood on the one hand as indicating the expression of the moral mind as the four different moral facets. On the other hand, it can also be understood as the expression of the moral mind in human relationships. With respect to the latter, the personal aspect (benevolence in the relation between father and son) and the social aspect (righteousness in the relation between ruler and official) are felt much more directly than the other two dimensions of ritual propriety in the relation between host and guest and wisdom in relation to the worthies. By virtue of this felt priority, \"benevolence and righteousness\" can be used to stand in for the full expression \"benevolence, righteousness, ritual propriety and wisdom.\" This can be seen, for example, when Confucius is quoted in the \"Man in the World\" chapter of the Zhuangzi (\u838a\u5b50\u2027\u4eba\u9593\u4e16) as saying:\n\n> In the world, these are the two great decrees: one is fate, and the other is duty. That a son should love his parents is fate \u2013 you cannot erase this from his mind. That a subject should serve a ruler is duty \u2013 there is no place he can go and be without a ruler, no place he can escape to between Heaven and earth.\n\nThis passage would also be appropriate in describing how Mencius used the term \"benevolence and righteousness\" as a synonym for the moral mind or to express the sum of all moral principles. One further step simplifies \"benevolence and righteousness\" to the single term \"benevolence,\" a universal term with its base in the sentiments of filial piety and brotherly love and the mind sensitive to the suffering of others. From this kind of argumentation we can see that the mature thought of Mencius draws directly from Confucius' idea of benevolence. It was for this reason that Zisi is not referred to as a teacher in the Mencius and why Mencius is portrayed as stressing that \"I was never a student of Confucius, but I have learned of him from others\" (4B50).\n\nZigong \u5b50\u8ca2 says: \"The Master's discourse on nature and the Way of Heaven cannot be heard.\" While we can't tell what Zisi's view of nature was from the Wuxing, there is a discussion of the difference between the Way of Heaven and the Way of Man in the text. However, on this point, Mencius is not completely in agreement with the Wuxing. That is primarily because in the Wuxing the sage and only the sage is able to know the Way of Heaven and have his virtue harmonize with Heaven. The common people, and even the worthies, are left out of the equation. For Mencius, on the other hand, realizing the mind is followed by knowing nature and knowing Heaven. Likewise, he saw benevolence and righteousness, virtues that are internal to the mind and common to all, as the essence of realizing the mind, thereby providing an avenue by which ordinary people could have access to transcendent moral principles. By claiming that nature and Heaven could be known in this manner, Mencius took a step further in making (human) nature a metaphysical concept. As such, Mencius can be seen as attempting to resolve the important, and unanswered, question left behind by Confucius regarding the relation between nature and the Way of Heaven.\n\nReturning to the question of a school of thought associated with Zisi and Mencius, we can see from the above discussion that while Mencius was certainly influenced by Zisi, the extent of this influence was not greater than that of Confucius and his students (such as Zengzi). Therefore, Mencius would most likely not have considered himself to have belonged to such a school of thought. We would therefore suggest that those who spoke of such a school of thought, such as Xunzi and his students did not have a correct understanding of Mencius' thought.\n\nReferences\n\nChen, Jing \u9673\u975c. 2008. The historical construction of the Zisi\/Mencius school \u601d \u5b5f\u5b78\u6d3e\u7684\u6b77\u53f2\u5efa\u69cb. In Philosophy, texts, and history: A new discussion of the Zisi\/Mencius school \u601d\u60f3.\u6587\u737b.\u6b77\u53f2:\u601d\u5b5f\u5b78\u6d3e\u65b0\u63a2. Beijing \u5317\u4eac: Beijing daxue chubanshe \u5317\u4eac\u5927\u5b78\u51fa\u7248\u793e.\n\nChen, Lai \u9673\u4f86. 2008a. Zisi or Mencius? A review of theories of authorship of the five virtues text and a discussion of the historical significance of the discovery of the Guodian Chu bamboo slip manuscript \u300a\u4e94\u884c\u300b\u7d93\u8aaa\u5206\u5225\u70ba\u5b50\u601d\u3001\u5b5f\u5b50\u6240\u4f5c\u8ad6:\u517c\u8ad6\u90ed\u5e97\u695a\u7c21\u300a\u4e94\u884c\u300b\u7bc7\u51fa\u571f\u7684\u6b77\u53f2\u610f\u7fa9\u3009\u4e00. In Philosophy, texts, and history: A new discussion of the Zisi\/Mencius school \u601d\u60f3.\u6587\u737b.\u6b77\u53f2:\u601d\u5b5f\u5b78\u6d3e\u65b0\u63a2. Beijing \u5317\u4eac: Beijing daxue chubanshe \u5317\u4eac\u5927\u5b78\u51fa\u7248\u793e.\n\nChen, Lai \u9673\u4f86. 2008b. A Study of the bamboo slip manuscript of the five virtues and Zisi's thought \u7af9\u7c21\u300a\u4e94\u884c\u300b\u7bc7\u8207\u5b50\u601d\u601d\u60f3\u7814\u7a76. In Philosophy, texts, and history: A new discussion of the Zisi\/Mencius school \u601d\u60f3.\u6587\u737b.\u6b77\u53f2:\u601d\u5b5f\u5b78\u6d3e\u65b0\u63a2. Beijing \u5317\u4eac: Beijing daxue chubanshe \u5317\u4eac\u5927\u5b78\u51fa\u7248\u793e.\n\nCsikszentmihalyi, Mark. 2004. Material virtues: Ethics and the body in early China. Leiden: Brill. (An extensive study of Wuxing text as it relates to the Zisi-Mengzi school and contemporary trends in medical, musical, and moral thought).\n\nFeng, Youlan \u99ae\u53cb\u862d. 1961. A history of Chinese philosophy \u4e2d\u570b\u54f2\u5b78\u53f2. Beijing \u5317\u4eac: Zhonghua shuju \u4e2d\u83ef\u66f8\u5c40. (An early review of Chinese philosophy heavily informed by contemporary trends in Western materialistic philosophy).\n\nFeng, Youlan \u99ae\u53cb\u862d. 1982. The question of the dating of the Doctrine of the Mean \u4e2d\u5eb8\u7684\u5e74\u4ee3\u554f\u984c. In Critique of ancient history \u53e4\u53f2\u8fa8, Vol. 4. Shanghai \u4e0a\u6d77: Guji chubanshe \u53e4\u7c4d\u51fa\u7248\u793e.\n\nJiang, Boqian \u8523\u4f2f\u6f5b. 1984. A comprehensive study of Pre-Qin philosophers \u8af8\u5b50 \u901a\u8003. Zhejiang \u6d59\u6c5f: Guji chubanshe \u53e4\u7c4d\u51fa\u7248\u793e.\n\nJiang, Guanghui\u59dc\u5ee3\u8f1d. 2000. Reflections on the Guodian Chu bamboo slips \u90ed\u5e97 \u695a\u7c21\u8207\u9053\u7d71\u6538\u95dc. Chinese Philosophy \u4e2d\u570b\u54f2\u5b78 21: 13\u201340.\n\nJiao, Xun \u7126\u5faa. 1992. The correct reading of Mencius (\u5b5f\u5b50\u6b63\u7fa9). Taipei \u53f0\u5317: Shijie shuju \u4e16\u754c\u66f8\u5c40.\n\nLi, Minghui \u674e\u660e\u8f1d. 1994. Kant's idea of inherent evil. \u5eb7\u5fb7\u7684\u6839\u672c\u60e1\u8aaa. In A reconstruction of Kantian Ethics and Mencian moral thought \u5eb7\u5fb7\u502b\u7406\u5b78\u8207\u5b5f\u5b50\u9053\u5fb7\u601d\u8003\u4e4b\u91cd\u5efa. Taipei \u53f0\u5317: Zhongyang yuanjiuyuan zhongguo wenzhe yanjiusuo \u4e2d\u592e\u784f\u7a76\u9662\u4e2d\u570b\u6587\u54f2\u784f\u7a76\u6240.\n\nLi, Ling \u674e\u96f6. 2002. Notes on annotating the Guodian Chu bamboo slips \u90ed\u5e97\u695a\u7c21\u6821\u8b80\u8a18. Beijing \u5317\u4eac: Beijing daxue chubanshe \u5317\u4eac\u5927\u5b78\u51fa\u7248\u793e.\n\nMeng, Peiy\u00fcan \u8499\u57f9\u5143. 2008. The philosophical characteristics of 'Nature Comes From Fate' and the relation to the Zisi\/Mencius school \u3008\u6027\u81ea\u547d\u51fa\u3009\u7684\u601d\u60f3\u7279\u5fb5\u53ca\u5176\u8207\u601d\u5b5f\u5b78\u6d3e\u7684\u95dc\u806f. In Collected papers from the international conference on the Zisi\/Mencius school of confucianism \u5112\u5bb6\u601d\u5b5f\u5b78\u6d3e\u570b\u969b\u5b78\u8853\u7814\u8a0e\u6703\u8ad6\u6587\u532f\u7de8. Jinan \u6fdf\u5357: Qilu shushe \u9f4a\u9b6f\u66f8\u793e.\n\nMou, Zongsan \u725f\u5b97\u4e09. 1969. Substance of mind and substance of human nature \u5fc3\u9ad4\u8207\u6027\u9ad4, Vol. 3 Taipei \u53f0\u5317: Zhongzheng shuju \u4e2d\u6b63\u66f8\u5c40. (An influential treatment of Neo-Confucian thought heavily influenced by Kantian philosophy).\n\nPang, Pu \u9f90\u6a38. 2000. Annotations and research on the silk manuscript of the five virtues \u7af9\u5e1b\u4e94\u884c\u7bc7\u6821\u6ce8\u53ca\u7814\u7a76. Taipei \u53f0\u5317: Wanjuanlou tushu gongsi \u842c\u5377\u6a13\u5716\u66f8\u526c\u53f8.\n\nPuett, Michael J. 2002.To become a God: Cosmology, sacrifice, and self-divizination in early China. Cambridge: Harvard University Asia Center (A thought-provoking study of conceptualizations of relationships between man and gods in early China with comparisons to ancient Greece).\n\nQian, Mu \u9322\u7a46.1935. A study of the chronology of Pre-Qin philosophers \u5148\u79e6\u8af8\u5b50\u7e6b \u5e74\u8003\u8fa8. Shanghai \u4e0a\u6d77: Shangwu yinshuguan \u5546\u52d9\u5370\u66f8\u9928.\n\nRen, Jiy\u00fc \u4efb\u7e7c\u6108. 1983. A history of the development of Chinese philosophy: Pre-Qin \u4e2d\u570b\u54f2\u5b78\u767c\u5c55\u53f2.\u5148\u79e6\u5377. Beijing \u5317\u4eac: Renmin chubanshe \u4eba\u6c11\u51fa\u7248\u793e.\n\nTakeuchi, Yoshio \u6b66\u5167\u7fa9\u96c4. 1931. A study of Pre-Qin writings \u5148\u79e6\u7d93\u7c4d\u8003, Vol. 2. Trans. Jiang Xia'an \u6c5f\u4fe0\u5eb5. Beijing \u5317\u4eac: Shangwu yinshuguan \u5546\u52d9\u5370\u66f8\u9928.\n\nXu, Fuguan \u5f90\u5fa9\u89c0. 1969. A history of Chinese discourse on human nature: Pre-Qin \u4e2d\u570b\u4eba\u6027\u8ad6\u53f2.\u5148\u79e6\u7bc7. Taipei \u53f0\u5317: Shangwu yinshuguan \u5546\u52d9\u5370\u66f8\u9928. (A comprehensive review of debates on human nature in classic Chinese thought).\n\nYe, Guoliang \u8449\u570b\u826f. 2000. The question of the philosophical genealogy of confucian writings in the Guodian collection \u90ed\u5e97\u5112\u5bb6\u8457\u4f5c\u7684\u5b78\u8853\u7cfb\u8b5c\u554f\u984c. In The bulletin of the department of Chinese literature, Vol. 2, 1\u201326. National Taiwan University \u81fa\u5927\u4e2d\u6587\u5b78\u5831.\n\nZhu, Xi \u6731\u71b9. 1997. The complete works of Zhu Zi arranged by topic, Vol. 8 \u6731\u5b50\u6587\u96c6\u5927\u5168\u985e\u7de8. compiled by Zhu Y\u00fc \u6731\u7389. Taipei \u53f0\u5317: Zhuangyan wenhua shiye youxian gongsi \u838a\u56b4\u6587\u5316\u4e8b\u696d\u6709\u9650\u526c\u53f8.\n\nFootnotes\n\n1\n\nFeng Youlan \u99ae\u53cb\u862d argues that the second section through the first part of the nineteenth section were written by Zisi, while the opening section and everything after the phrase \"when those in inferior positions do not obtain the confidence of their superiors\" (dao qian ding ze bu qiong \u9053\u524d\u5b9a\u5247\u4e0d\u7aae) was written by Mencius. See Feng (1961: 447\u20138). Xu Fuguan \u5f90\u5fa9\u89c0 takes a slightly different position, arguing that Zisi wrote the first section and that the second half was written by either Zisi or students of his writing prior to Mencius. See Xu (1969: 105\u20136). TAKEUICHI Yoshio held that the second through nineteenth sections were penned by Zisi, with the first section and everything after the nineteenth section being written by students of Zisi. See Takeuchi (1931: 121\u201323).\n\n2\n\nTakeuchi thought that the first half of the Doctrine of the Mean contained portions of the Zisizi and was written during the early portion of the Warring States period, while the second half was composed towards the end of the Qin dynasty. The basis for the later dating of the second half came from the reference in Sect. 26 \"it carries the Hua and Yue mountains without feeling their weight\" (Takeuchi argued that a person from Lu would not have considered those two mountains to be large) and another reference in Sect. 28 to \"Today under the Heavens the same wheels are used for carriages, the same characters are used for writing, and the same rules are used for conduct\".\n\n3\n\nThe silk manuscript of the Wuxing can be divided into two parts, the first part containing a discussion of the five virtues, and the second part containing a running commentary on the first part. Pang Pu refers to the first part as the \"classic\" (jing \u7d93) and the second part as the \"commentary\" (shuo \u8aaa). See Pang (2000: 100).\n\n4\n\nAccording to Zhuzi, the Way was passed from Confucius to Zengzi to Zisi and finally to Mencius. See Zhu Xi's Zhongyung jijie xu.\n\n5\n\nBased on the results of recent research, it would appear that the Tang dynasty view that Zisi's thought can be traced to Zengzi should be understood not as saying that Zisi was a student of Zengzi, but rather that their thought was similar in a number of respects.\n\n6\n\nScholars who have viewed Zisi and Mencius as belonging to the same school include Zhang Taiyan \u7ae0\u592a\u708e,Guo Muoruo \u90ed\u6cab\u82e5, and Hou Wailu \u4faf\u5916\u5eec. Ren Jiyu \u4efb\u7e7c\u6108 argues that there was no such school of thought during the pre-Qin period (Ren 1983: 290\u20139).\n\n7\n\nThe Southern Song dynasty scholar Wang Yinglin \u738b\u61c9\u9e9f argued that this criticism was falsely attributed to Xunzi by students of his like Han Fei and Li Si interested in defaming the sages and worthies. More recently, Chen Jing argues that the criticism was added by Liu Xiang and Liu Xin (Chen 2008: 159\u201380).\n\n8\n\nTranslations of the Guodian Wuxing manuscript are based on the text arranged and edited by Li Ling (2002).\n\n9\n\nThe concepts of the natural good and the moral good have been adopted from Kant. For a relevant discussion, see Lin (1994).\n\n10\n\nFor an explanation of these metaphors in terms of a beginning and ending, see Jiao Xun \u7126\u5faa, Mengzi zhengyi \u5b5f\u5b50\u6b63\u7fa9 10.397.\n\n11\n\nChen Lai \u9673\u4f86 reads this sentence as \"Sagacity and wisdom gives birth to ritual and music.\" This reading is apparently motivated by a desire to emphasize that sagacity and wisdom are critical for the harmonization of the five actions and that benevolence is critical for the harmonization of the four kinds of action. Our reading differs from Chen's, but seems to be equally acceptable. Rites and music can include benevolence, righteousness and ritual propriety, in which case the entire sentence can be read as meaning that the four actions are born from sagacity. Sects. 5 and 6 of the Wuxing discuss three kinds of thought: that of benevolence, that of wisdom, and that of sagacity. Chen takes this to mean that the five actions are grouped into two sets: one of sagacity and wisdom and the other of the remaining three. However, the introduction of these three kinds of thought does not seem to necessitate grouping the actions into two sets. It is our view that sagacity is the highest of the five kinds of action and that for this reason it is a virtuous action regardless of whether it has been formed internally. Wisdom and benevolence form a second tier of concepts, with wisdom being related to intellectual capacities and benevolence standing in for benevolence, righteousness and ritual propriety and being related to moral capabilities. See Chen (2008b).\n\n12\n\nThis part of Sect. 2 can further be seen in light of the discussion of the caution of the Gentleman when in solitude in Sects. 8 and 9.\n\n13\n\nThis strategy is reminiscent of Youzi's \u6709\u5b50 claim in Chap. 1 of the Analects that \"filial piety and brotherly respect are the essence of benevolence.\"\n\n14\n\nThis passage appears in the Zhuzi yulei \u6731\u5b50\u8a9e\u985e, juan 60.\n\n15\n\nPang Pu \u9f90\u6a38 understands \"Virtue is the Way of Heaven\" to mean that virtue is conferred by Heaven. Chen Lai offers a different explanation, arguing that this sentence should be understood as meaning: \"The nature of the Way of Heaven is that is has a beginning and no end; the formation of the five actions internally and their harmonization is an articulation of the harmony of the Way of Heaven; sages know the Way of Heaven and for this reason their actions harmonize with it.\" Therefore, the sentence \"Virtue is the Way of Heaven\" here indicates a harmonization of virtue with the Way of Heaven and not that virtue is received from Heaven.\" We are in agreement with Chen's interpretation. See Pang (2000: 159) and Chen (2008b: 17).\n\n16\n\nFeng Youlan \u99ae\u53cb\u862d (Feng 1982: 184) divides the Doctrine of the Mean into three sections and argues that the first and final sections were later additions. Jiang Boqian (Jiang 1984: 337\u2013338) sees a connection between the Doctrine of the Mean and the Xunzi, while Qian Mu (Qian 1935: 459) claims that the text of the Zi yi is very similar to the Xunzi.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_6\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 6. The Daxue (Great Learning) and the Zhongyong (Doctrine of the Mean)\n\nAndrew H. Plaks1\n\n(1)\n\nPrinceton University, Princeton, NJ, USA\n\nAndrew H. Plaks\n\nEmail: aplaks@princeton.edu\n\nAbstract\n\nIn the historical development of classical Confucian philosophy, the two brief treatises known as Daxue \u5927\u5b78 (Great Learning) and Zhongyong \u4e2d\u5eb8 (Doctrine of the Mean) are among the most important milestones marking the transition of Confucian thought, from the kernels of ancient wisdom encapsulated in the early canonic writings: The Shujing \u66f8\u7d93 (Classic of Documents), The Yijing\u6613\u7d93 (Classic of Changes) and The Shijing \u8a69\u7d93 (Classic of Poetry) together with the diffuse ethical teachings set forth in the name of Confucius in the Lunyu \u8ad6\u8a9e (The Analects), toward the gradual emergence of Confucianism as a spiritually profound and intellectually complex philosophical system. In this light, their position as foundational documents in the history of classical Chinese thought parallels that of the Platonic Dialogues in transforming the diffuse concepts of the pre-Socratics into a unified mode of philosophical inquiry. Alongside the related integral arguments presented in the pre-Qin writings attributed to Mencius and Xunzi, now supplemented by a handful of independent treatises outside of the received tradition that have been rediscovered among the archaeological treasures unearthed in China in recent years (including such texts as those known under the titles: Taiyi sheng shui\u592a\u4e00\u751f\u6c34 (The Great One Gave Birth to Water), Xing zi ming chu\u6027\u81ea\u547d\u51fa (Nature comes from Mandate), Hengxian\u6046\u5148 (The Constant Precedes), and Wuxingpian\u4e94\u884c\u7bc7 (Five Actions), these works set the terms of discourse and the modes of argumentation that gradually crystallized through the centuries from Han through Tang to form the primary discourse of the Confucian strain of early Chinese thought. Ultimately, these works came to provide the core ideas and issues of the great revival of Confucian thought, in response to the profound spiritual challenge of Buddhist philosophy, that took shape during the Northern and Southern Song periods (a movement conventionally known in contemporary Sinological writings as \"Neo-Confucianism\"), and continued to dominate intellectual life in China through the Yuan, Ming and Qing dynasties, down to the end of imperial period and beyond.\n\n## 1 Introduction\n\nIn the historical development of classical Confucian philosophy, the two brief treatises known as Daxue \u5927\u5b78 (Great Learning) and Zhongyong \u4e2d\u5eb8 (Doctrine of the Mean) are among the most important milestones marking the transition of Confucian thought, from the kernels of ancient wisdom encapsulated in the early canonic writings: The Shujing \u66f8\u7d93 (Classic of Documents), The Yijing \u6613\u7d93 (Classic of Changes) and The Shijing \u8a69\u7d93 (Classic of Poetry) together with the diffuse ethical teachings set forth in the name of Confucius in the Lunyu \u8ad6\u8a9e (The Analects), toward the gradual emergence of Confucianism as a spiritually profound and intellectually complex philosophical system. In this light, their position as foundational documents in the history of classical Chinese thought parallels that of the Platonic Dialogues in transforming the diffuse concepts of the pre-Socratics into a unified mode of philosophical inquiry. Alongside the related integral arguments presented in the pre-Qin writings attributed to Mencius and Xunzi, now supplemented by a handful of independent treatises outside of the received tradition that have been rediscovered among the archaeological treasures unearthed in China in recent years (including such texts as those known under the titles: Taiyi sheng shui \u592a\u4e00\u751f\u6c34 (The Great One Gave Birth to Water), Xing zi ming chu \u6027\u81ea\u547d\u51fa (Nature comes from Mandate), Hengxian \u6046\u5148 (The Constant Precedes), and Wuxingpian \u4e94\u884c\u7bc7 (Five Actions), these works set the terms of discourse and the modes of argumentation that gradually crystallized through the centuries from Han through Tang to form the primary discourse of the Confucian strain of early Chinese thought. Ultimately, these works came to provide the core ideas and issues of the great revival of Confucian thought, in response to the profound spiritual challenge of Buddhist philosophy, that took shape during the Northern and Southern Song periods (a movement conventionally known in contemporary Sinological writings as \"Neo-Confucianism\"), and continued to dominate intellectual life in China through the Yuan, Ming and Qing dynasties, down to the end of imperial period and beyond.\n\nDespite the common impression of these works \u2013 in the eyes of those whose familiarity with them is based upon casual or compulsory study \u2013 as straightforward, unambiguous presentations of ethical and cosmological platitudes, it can be shown upon thoughtful reading and analysis that they both constitute carefully-wrought expositions of some of the most essential tenets of Confucian thought. Whether read as separate texts or taken together as a mutually complementary pair of writings, they provide a crucial elucidation of the central issues of mature Confucianism, regarding the process of self-cultivation, the ideal of perfect self-realization (or, \"sagehood\"), the anthropocentric conception of the cosmic order, the attainment of \"integral wholeness\" (cheng \u8aa0) and the outward extension from inner selfhood to the fulfillment of the intrinsic potential of all human beings and even all \"things\" in the objective world.\n\n## 2  Daxue (The Great Learning)\n\n### 2.1 Textual History\n\nThe Daxue stands among the seminal texts of the Confucian canon under two separate bibliographical headings: first, as an important chapter in the canonic compendium of the Han ritual corpus, the Book of Rites (Liji \u79ae\u8a18), and then, alongside the Analects, Mengzi, and the Doctrine of the Mean, as one of the so-called \"Four Books\" (Sishu \u56db\u66f8) elevated to scriptural status by Zhu Xi and other leading figures among his predecessors and followers in the reconstitution of Confucian thought in Song times. From that point on, these works took on immense cultural significance far beyond their physical size, forming the core curriculum of traditional Confucian education, the central focus of intellectual discourse and the textual basis of the imperial examination system.\n\nThe dating of the original composition of this brief text, prior to its inclusion in the Liji, remains uncertain. Its traditional attribution to Zengzi \u66fe\u5b50(505\u2013436 BCE?, personal name: Zeng Can \u66fe\u53c3, alt. reading: Zeng Shen), one of the leading disciples of Confucius whose name is mentioned prominently in the Analects and to whom a number of miscellaneous early writings are assigned, would place it as early as the fifth century BCE. But a variety of alternative theories of authorship, stylistic indicators, and, above all, inter-textual links to a number of other important writings of the Warring States period (475\u2013221 BCE), make it far more likely that it was composed in its present form as late as the Western Han period (206 BCE \u2013 24 CE), at a point much closer in time to its eventual selection for inclusion in the Liji compendium. The dating of the work is further confused by certain inter-textual echoes linking the Daxue to the parallel literature of early Confucianism. Chief among these are a number of strikingly similar passages in the Mengzi and the writings of Xunzi roughly predating or postdating the probable formation of the original layers of the text. The connections to the text of the Mengzi, in particular, led earlier generations of Confucian thinkers to view the Daxue as a crucial link in the orthodox chain of the \"filiation of the Way\" (daotong \u9053\u7d71), from the Master of the Analects down through the Song masters, and then on to latter-day exponents of Confucian doctrine. These same textual echoes, however, have been taken by a greater number of pre-modern and modern scholars as evidence that the text could only have been composed by a later follower of Mengzi \u5b5f\u5b50 (372\u2013289 BCE) during the later Warring States, sometimes pushing it as far forward as the Qin or early Han periods. Other intellectual historians deny or deemphasize the relation between the Daxue and Mengzi's strain of early Confucian thought, reading it instead as a development of the arguments of Xunzi \u8340\u5b50 (312\u2013230 BCE) at the very end of the Warring States period, or even as a reflection of the syncretic writings of early Han thinkers such as Dong Zhongshu \u8463\u4ef2\u8212 (179\u2013104 BCE).\n\nThe earliest extant text of the Daxue, that transmitted in the Liji, is but the first in a chain of alternate recensions. These include one version engraved on a stone tablet unearthed during the Ming period (1368\u20131644), alleged to have been inscribed during the Northern Wei \u9b4f dynasty (386\u2013532) but later dismissed as a forgery, and a whole series of variant texts, as many as 20 or more in number, representing the revisions of later commentators purporting to restore the \"original\" form of the canon. While these competing recensions amount in nearly all cases to little more than a reshuffling of the constituent passages or larger sections of the text, the resulting changes in the structure of the argument are heatedly defended by their respective editors and commentators as vital to the correct understanding of the meaning of the work. This textual controversy is particularly significant in the case of Zhu Xi's (1130\u20131200) reconstruction of the order of the text in accordance with the teachings of Cheng Yi \u7a0b\u9824 (1033\u20131107), his intellectual forebear and predecessor in the orthodox lineage, that which was to become the officially sanctioned version of the Daxue that gained new scriptural status as one of the Four Books. Still, doubts continued to be raised regarding the definitive form of the work, particular among later polemical opponents of Zhu Xi, notably Wang Shouren \u738b\u5b88\u4ec1 (1472\u20131529, alt. name Wang Yangming \u738b\u967d\u660e) in the mid-Ming period, many of whom championed competing versions provocatively claimed to represent the true \"ancient text\" (guben \u53e4\u672c) of the classic.\n\n### 2.2 Explication of Title\n\nThe traditional title of this treatise \u2013 taken simply from the first two words of the text \u2013 is almost always given in Western translations as one or another variant of the expression \"great learning.\" This conventional rendering is misleading, both on general philological grounds and because it fails to describe the specific contents of the book. Even a cursory reading of the text makes it immediately apparent that the educational process envisioned in the second term of the title: xue \u5b78 refers here to a paradigm of moral fulfillment in every phase of human capacity, a conception far broader than what is expressed in the narrow sense of the English \"learning.\" Rather, the word xue in Confucian discourse covers the full spectrum of personal accomplishment from the active to the contemplative spheres. It begins at the more advanced levels of actual instruction, but then extends to the widest realization of human potential. The idea of the perfection of individual character at the heart of these spheres of Confucian attainment precisely matches the scope and meaning of the central ideal of \"self-cultivation.\"\n\nRegarding the first term in the title, the difficulty with the standard translations lies not in its literal meaning (\"large\" or \"great\"), but in the particular conception of greatness to which it refers. In one very influential gloss on this word, Zhu Xi explains the whole phrase as an abbreviation for daren zhi xue \u5927\u4eba\u4e4b\u5b78 (the learning of the great person). He then goes on to define the \"great one\" in ways that clarify the higher order of learning he envisions: either the higher status conferred by age, specifically the age (15 years) at which a young student moves up to a more advanced level in his intellectual and moral training, or, more compellingly, the greatness of the highest degree of cultivation, that of the Confucian \"sage\" (shengren \u8056\u4eba). At first glance, Zhu Xi's reading seems to clash with that of Zheng Xuan \u912d\u7384 (127\u2013200) and Kong Yingda \u5b54\u7a4e\u9054 (574\u2013648), the leading earlier commentators on the text of the Daxue preserved in the canonic Liji. Their gloss of the two words of the title as boxue \u535a\u5b78 (broad cultivation) seems to emphasize simple breadth of learning \u2013 until they go on to qualify their understanding of this capaciousness of mind as applying specifically to the capacity to govern, a sense not far removed from Zhu Xi's explication of the \"great one\" as the sage-ruler. Once Zhu's dominant interpretation was in place, subsequent commentators, even those at pains to revise or even overturn some of the basic assumptions of Song orthodoxy, continued to incorporate this essential reading into their interpretations. To give only the most conspicuous example, Wang Yangming, the famous- exponent of the so-called \"philosophy of mind\" (xinxue \u5fc3\u5b78) in the sixteenth century, explained the \"greatness\" of the title as an allusion to the man who embraces the \"ultimate unity of all things\" (wanwu yiti \u842c\u7269\u4e00\u9ad4).\n\n### 2.3 Structure of the Text\n\nRegardless of which version of the text one uses, its structure can in all cases be divided very neatly into a brief opening section, setting forth the core arguments of the treatise in concise outline form, followed by a series of ten explanatory, or \"expansion\" chapters, taking up and explicating one by one the key terms introduced in the initial section. In orthodox Confucian learning, these two textual divisions came to be designated as the \"canonic core\" (jing \u7d93) of the work, traditionally taken as the direct teachings of Zengzi himself, and the attached \"commentarial traditions\" (zhuan \u50b3) loosely ascribed to various subsequent generations of Confucian thinkers. The opening chapter itself is further analyzed into two primary components: an initial tripartite statement known as the \"three general principles\" (san gangling \u4e09\u7db1\u9818), and the central passage known as the \"eight specific points\" (ba tiaomu \u516b\u689d\u76ee) setting forth, in the rhetorical form of a chain syllogism, a series of consecutive stages, or complementary phases, that define the different spheres of inner conception and outward action through which the perfect cultivation of the Confucian sage is to be realized. This core section of the opening chapter is both preceded and followed by important transitional passages, in which primary stress is placed upon the importance of the proper sequential and conceptual ordering of these essential phases of self-cultivation.\n\nThe \"expansion chapters\" that make up the remainder of the book explicate, one by one, each of the main ideas set forth in the opening chapter. The bulk of these sections is composed of sets of illustrative \"proof texts\" drawn primarily from the received \"canonic corpus\", especially the \"Classic of Documents\" (Shujing \u66f8\u7d93) and the \"Classic of Poetry\" (Shijing \u8a69\u7d93), followed by statements of an expository or interpretive nature developing the meaning of the specific aspect of Confucian fulfillment under discussion in that chapter. Of these ten chapters, Chaps.\u200b 1, , and  (following the numbering in Zhu Xi's recension) take up the loaded terms of the \"three general principles\": \"causing the light of one's inner moral force to shine forth\" (ming mingde \u660e\u660e\u5fb7), \"bringing the people to a state of renewal\" (xin min \u65b0\u6c11), and \"coming to rest in the highest good\" (zhi yu zhishan \u6b62\u65bc\u81f3\u5584). At the other end of the treatise, Chaps.\u200b 6, , , , and  explicate the first five of the \"eight specific points\", in reverse order from their rhetorical sequence in the opening chapter: those expounding \"achieving integral wholeness in one's inner consciousness\" (cheng qi yi \u8aa0\u5176\u610f), \"setting straight one's emotional and cognitive faculties\" (zheng qi xin \u6b63\u5176\u5fc3), \"perfecting one's individual character\" (xiu qi shen \u4fee\u5176\u8eab), \"putting one's family into proper balance\" (qi qi jia \u9f4a\u5176\u5bb6), and \"establishing orderly rule in one's kingdom\" (zhi qi guo \u6cbb\u5176\u570b). This leaves Chaps.\u200b 4 and  to deal with what appear as the last two phases in the standard order of the initial eight-point chain: \"expanding to the utmost one's range of comprehension\" (zhi zhi \u81f4\u77e5), and \"extending to all things the correct conceptual grid\" (gewu \u683c\u7269). Both of these sections present formidable problems of interpretation, since Chap.\u200b 4, vaguely designated as \"the roots and the branches\" (benmo \u672c\u672b), touches only tangentially upon the expansion of one's range of comprehension, while Chap.\u200b 5 is, in all the different extant versions, reduced to nothing more than a cryptic heading regarding \"understanding the fundamental core,\" with no actual explication of the subject. This glaring gap is filled for us in the canonic text established in the Four Books, where Zhu Xi adds a long explanatory note to take the place of the ostensibly \"lost\" fifth chapter. This supplementary note takes the form of a disquisition on the meaning of the final term in the eight-point sequence, gewu, in an unabashedly Neo-Confucian vein.\n\n### 2.4 Integral Argument\n\nThe central theme of the Daxue concerns the substance and the ordering of the Confucian process of self-cultivation. This is immediately clear in the prevailing reading of the title of the treatise by some of the major traditional commentators as: \"the learning of the great one,\" or \"learning to be great.\" The essential ideas set forth in the \"three principles\" and the \"eight points,\" however, immediately raise a number of difficult questions of interpretation regarding the nature of this process. These questions begin with the initial problem of defining one's inner moral force (de) and what it means to cause it to \"shine forth\" (ming mingde), how one is to go about \"renewing the people\" (xinmin, frequently read according to an alternate recension as qinmin \u89aa\u6c11, \"loving the people\"), and why the point of ultimate attainment is here described with the term \"supreme good\" (zhi shan \u81f3\u5584), a glowing ideal that may sound quite familiar to Western ears, but is rarely invoked in early Chinese writings. In the passage that follows, the affirmation of the absolute importance of the temporal ordering of the process of Confucian cultivation is cast into further doubt as to its essential logical thrust: are these stages to be conceived as sequential steps or as simultaneous spheres of self-perfection? As the text moves from the broader to the narrower spheres of human endeavor, we learn that each stage of outer realization must be preceded by prior conditions of more personal, and ultimately, internal cultivation. But upon arriving at the deepest layers of consciousness, the order of progression seems to shift direction, and the core of inner attainment is redefined in terms of an outward-directed engagement with the entire phenomenal world, as expressed in the elusive term gewu (\"extending to all things the correct conceptual grid\"). The answer to some of these questions is adumbrated in a common thread of meaning running through many of the passages in the \"expansion chapters.\" Here the recurrent theme is one of finding within one's own inner self, in both the \"heart\" or \"mind\" (xin \u5fc3) that interacts with the outside world and the \"inner consciousness\" (yi \u610f), closer to the core of one's being, a reflection of the abiding patterns of meaning intrinsic in the entire cosmic order. Once one has succeeded in apprehending these inherent principles, one can then extrapolate from them to build a solid ground for moral judgment and for effective engagement with others at every level of human interaction.\n\nThe difficulty of defining the terms of discourse used in the Daxue, and the profound implications of its essential message regarding the inner core of selfhood, made it a central focus of philosophical debate in the Confucian school, at least from Tang times on. Even after it was recanonized as one of the Four Books in the Song period, and thus became one of the obligatory texts of formal education for all Confucian scholars for the next 1,000 years, its essential meaning remained a matter of great controversy, particularly between followers of the orthodox \"learning of principle\" (lixue \u7406\u5b78), and thinkers of the so-called \"school of the mind\" (xinxue \u5fc3\u5b78) associated with many leading thinkers of the late-Imperial period, from Lu Xiangshan \u9678\u8c61\u5c71 (1139\u20131192) to Wang Yangming.\n\n### 2.5 Interpretation\n\nWhen one re-examines the overall flow of ideas in this treatise from the perspective of the primary assertions in each of its component chapters, a fairly well focused core argument begins to come into view. The first three expansion chapters, corresponding to the opening tripartite sequence of the first chapter, tell us in the most succinct manner that moral cultivation must be self-generated (ziming \u81ea\u660e), it must be extended to others (tui yi ji ren \u63a8\u4ee5\u6975\u4eba), and it must be fulfilled by reaching a point of dynamic equilibrium in the fulfillment of cardinal human roles. Chapters 6, , , , and  add to this the teaching that true cultivation is predicated upon paying heed to one's innate moral predisposition, avoiding the destabilizing pull of emotional impulses and affective inclinations and seeking the ground of moral interaction within oneself \u2013 measuring one's responses to others by one's own internal yardstick, yet never cutting oneself apart from the universal fabric of the human community. What this all adds up to is a compelling statement of the solid ground of cultivated selfhood upon which Confucian commitment to the various spheres of external human engagement must rest. This is a view of human capacity frequently paraphrased in the well-known Chinese ideal: \"sagely qualities in one's inner self and kingly virtues toward the outside world\" (neisheng waiwang \u5167\u8056\u5916\u738b), or, in words that will find their most eloquent expression in the Zhongyong: \"bringing oneself to completion, and thereby bringing to completion one's fellow man and all existing things\" (chengji ... chengwu \u8aa0\u5df1...\u8aa0\u7269).\n\nThis understanding of the overall trajectory of cultivation traced through the five final phases of the paradigm helps us to explicate the two crucial ideas that are left conspicuously unexplained in the initial sequence of the \"expansion chapters\". The first of these is the uncertain expression gewu \u683c\u7269. This most difficult and controversial of Confucian terms is composed of two separate semantic elements: a verb and a noun of vague and uncertain meaning. The first character: ge is glossed in the standard commentaries on various occurrences in the Classic of Poetry and the Classic of Documents as an archaic verb meaning \"to come\" or \"to arrive\" (one such citation appears in Chap.\u200b 16 of the Zhongyong). On this basis, the compound phrase may be understood as referring to some idea of \"arriving at\" or \"reaching out\" to all things (wu \u7269) in the world. In this context it is clearly the reach of the mind, rather than any material grasp, that is intended. Drawing upon an alternate use of the same word to indicate a grid-like window frame, the expression can be understood to mean something like reaching out to all things in the world through a mental process of extrapolation by analogy, i.e. what Sinologists refer to as \"correlative thinking\". Thus the term suggests understanding the place and meaning of all things in the world in accordance with their positions in the universal grid of interrelated categories and individual correspondences.\n\nThe phrase gewu \u683c\u7269 used to define the innermost core of selfhood in the eight-point chain (that would have been elucidated, one assumes, in the missing fifth chapter), can finally be construed in terms of the Confucian (and more particularly, Mencian) faith that, at the point of maximum introversion, what the true seeker of cultivation finds within himself is his own inalienable integration into the universal patterns that govern all men and things. Zhu Xi describes this moment of self-discovery in terms strikingly reminiscent of Buddhist notions of sudden enlightenment. The process of individual attainment then becomes a matter of extrapolating outward from this inner core, extending the grid of conception to define and realize one's place in the objective world of Confucian fulfillment.\n\nThis redefinition of the locus of selfhood may also help us to reinterpret the puzzling citation of the passage from the Analects (12:13) that makes up most of Chap.\u200b 4. The insertion of Confucius' words regarding his personal experience as a magistrate is presented in the heading as an illustration of the idea of \"comprehending the root\" (zhiben \u77e5\u672c) of the whole system. Given the general lack of respect accorded in early Chinese writings to the entire sphere of legal disputation \u2013 in sharp contrast to the more honored position of principled advocates and wise judges in, say, Platonic or Mishnaic contexts \u2013 it is initially quite difficult to see why the author of these lines chose this as an elucidation of the \"fundamental core.\" The solution to this riddle may lie in recognizing that this proof-text is cited not for what Confucius reportedly said about his own experience as a judge, but rather as a canonic attestation of one particular formulation: the words \"one should put oneself in the place of others.\" By recasting the original source expressed in the first person: \"I am like other men\" (wo you ren ye \u6211\u7336\u4eba\u4e5f), the application of this citation to the argument of the Daxue raises it to the force of a general proposition on the proper ordering of Self and Other.\n\n## 3  Zhongyong (The Doctrine of the Mean)\n\n### 3.1 Textual History\n\nThe Zhongyong has consistently been paired with the Daxue through its entire history as a canonical text: from its inclusion as a chapter in the Book of Rites (Liji) in the Western Han period to its incorporation into the Four Books (Sishu) in the restructuring of the canon in the time of Zhu Xi and others in the Song period. It, too, is of uncertain provenance. Its traditional attribution to Confucius' grandson, the second generation disciple Zisi (also known by his personal name Kong Ji \u5b54\u4f0b), is generally discounted by modern scholars, but in this case a body of other writings in the Liji compendium ascribed to this same figure, plus certain items in early bibliographical sources, lend a greater degree of credibility to the claim. On the strength of this attribution, Zisi is also revered as a transitional key figure in the orthodox chain of transmission (daotong \u9053\u7d71), linking the first-generation disciples of the Master to the first full exposition of Confucian thought in the writings of Mengzi 150 years later. Even more than the Daxue, however, the Zhongyong is tied by a dense network of inter-textual borrowings and allusions to a variety of philosophical writings dating from both before and after its first attested recension in the early Han. Of particular interest to scholars is the presence in the text of strong echoes, and in some cases even direct paraphrases, of important passages in the Mengzi, on the one hand, and Xunzi, on the other \u2013 in a manner reminiscent of certain inter-textual allusions in the Daxue. These have provided the spark for ongoing controversies regarding which of these two strains of early Confucianism best reflects the intellectual orientation of the original author. Despite the uncertainties regarding its date and authorship, on the other hand, the text of the Zhongyong has remained remarkably stable over the course of its development, with no major variants to speak of in its canonic recension.\n\n### 3.2 Explication of Title\n\nAlong with the many other points on which the Zhongyong exhibits a shared textual history with the Daxue, it, too, has been subject to misleading renderings of the name of the work in Western translations. In this case, the title is not simply drawn from the opening words of the text, as in the Daxue, but represents an expression appearing prominently at a later point. It is this phrase, whose literal translation gives us the conventional \"Doctrine of the Mean,\" by which the book is almost universally known in Western writings about China. The problem with this formulation is not its implied analogy to the concept of the \"mean\" in Aristotelian thinking. The notion of dynamic equilibrium developed in this text is, in fact, sufficiently close to that of Aristotle, both in its basic sense and in its more complex application, to warrant what might otherwise be considered a questionable transfer of a Greek term to the Chinese context. Even popular etymologies based upon the early graphic form of the character zhong \u4e2d (a vertical line passing through the center of a circle) \u2013 suggesting the image of a kind of well-balanced spinning top \u2013 do not automatically invalidate the use of the \"mean\" as an equivalent expression. The principal problem with the standard Doctrine of the Mean is that it simply ignores the second term of the equation. The word yong \u5eb8 is typically glossed in the traditional commentaries either by its near homophone meaning \"to use,\" or by another character (chang \u5e38) meaning \"common,\" readings that can be easily combined to yield the sense of \"common practice.\" Among those Western translators who have paid proper attention to this philological point, most have still misrepresented the relationship between the two words in the title, either treating them as coordinate terms: \"A and B\" (for example, Tu Wei-ming's Centrality and Commonality), or taking the second as a modifier of the first (as in many early Latin and French versions).\n\nAll of these readings of the title tend to lose sight of two central facts about this book. First, while it opens with significant reference to the concept of the \"mean,\" this is clearly not the primary theme of the work as a whole, and the word drops almost entirely out of the text after the first quarter of its length. Instead, the Zhongyong is deeply concerned with the philosophical implications of transposing the unattainable ideal of perfect balance \u2013 that attributed to the cosmic order, to the praxis of finite human existence, putting it into practice in modes of behavior that trace from the lowest to the highest the entire range of human capacity.\n\n### 3.3 Structure\n\nAn initial perusal of the internal structural arrangement of the Zhongyong shows it to be strikingly similar to that found in the Daxue. It also begins with an opening section (here numbered Chap.\u200b 1) that sets forth a programmatic overview of the basic message of the text (similarly termed the \"canonic core,\" or jing), followed by a series of \"expansion chapters,\" also traditionally referred to as the \"commentarial traditions\" (zhuan), that develop in detail the ideas outlined at the outset. In this case, however, the expansion chapters are not keyed, as in the Daxue, to specific terms or lines in the opening section. Rather, they take up the central threads of the argument and probe their meaning through a patchwork of proof texts, interpretive comments and philosophical argumentation. These 32 chapters fall neatly into three distinct sections. The first (Chaps.\u200b 2, , , , , , , , , and ) consists of a series of actual or fabricated quotations from the Master expressing the supreme difficulty of realizing the ideal of the mean in common practice. The following section (Chap.\u200b 12 through most of Chap. 20) explores the conceptual ground of successful Confucian cultivation, notably in cardinal human relations, especially filial piety and other ritual obligations, and expanding outward to the broader human context in the exercise of benevolent rulership. The final section (from the end of Chaps. 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, and 33) introduces a new definition of the ideal of perfect cultivation, the concept of \"self completion,\" or \"integral wholeness\" (cheng), and, with this as its central focus, proceeds to explore the substance and the significance of the highest degree of self-cultivation, that of Confucian \"sagehood,\" with its metaphysical implications for Man as a central pillar uniting Heaven and Earth.\n\nThe sharp division between these three sections in their terms of discourse and their intellectual focus: beginning with the notions of perfect equilibrium in the cosmic sphere and finite harmony in the human realm (zhonghe \u4e2d\u548c), and culminating in the ideal of \"integral wholeness\" (cheng) extending to metaphysical dimensions, has led many scholars, from as early as the Song period down to modern times, to speculate about the possibility that the Zhongyong may in fact be a composite text rather than a single unitary work, perhaps even reflecting formal disputation between competing philosophical schools, or at least divergent views regarding some of the central issues of Confucian thought. A close reading and analysis of the text, however, reveals a coherent central thread of conception running through the different phases of the argument. In order to follow this argument, one must immediately note that the term constituting the traditional title of the treatise: zhongyong cannot be understood here in the simple sense of \"moderation,\" or \"mediocrity,\" as it is used in a well-known passage in the Analects (and from there passed into common use). Nor can the text be read as a mere restatement of the \"doctrine of the mean,\" because, after introducing the conception of cosmic equilibrium in the opening chapter, the remainder of the text then turns its entire attention to other issues.\n\nOne is therefore forced to reinterpret the words zhongyong here, in line with the major commentaries, as referring to the limited application of the cosmic model of perfect equilibrium in the concrete context of Confucian moral practice. The exploration of this conception in the text takes us, after an initial exposition of the major themes and concepts, from a series of statements on the virtual impossibility (or all but perfect sages) of fully attaining this ideal, through a discussion of varying forms and degrees of putting it into practice, and finally to a lofty contemplation of the human and cosmic implications of its highest conceivable realization.\n\n### 3.4 Central Argument\n\nThe central thread of the argument of the Zhongyong can be grasped more clearly when one appreciates the rhetorical method of the text, whereby certain statements are used mainly to lay the conceptual groundwork, or to lead through a chain of assertions to its primary propositions. This is seen immediately in the potentially misleading opening line of the work, where two terminologically overloaded clauses on the \"nature of things\" (xing \u6027) ordained by Heaven and on the fundamental relation between this \"nature\" and \"the Way\" are used to set the ontological grounding for the following clause, in which the primary focus of the text is turned to the \"cultivation of the Way\" in the concrete human realm. The same method of rhetorical analysis must be applied in the best-known passage in the opening section, where the notion of a hypothetical state of perfect equilibrium (zhong), posited as temporally or logically prior to the emergence of concrete existence \u2013 as perceived through the emotional markers of human experience (xinuaile zhi weifa \u559c\u6012\u54c0\u6a02\u4e4b\u672a\u767c), is introduced to provide the logical underpinning for the main point at issue in this text: that is, the process of seeking a degree of harmonious balance (he \u548c) in the real world. As in the context of music (to which the Chinese term he \u548c, like its Greek counterpart is linked), \"harmony\" signifies not a celestial state of perfect balance, but rather a compensatory process of re-balancing the disharmonies of human experience, subject to the limiting conditions imposed by the parameters of concrete existence (yifa \u5df2\u767c). When the text is read in this way, the crucial passage in Chap.\u200b 12: \"The Way is not far from humanity\" takes on its full meaning, to assert that sagely fulfillment is rooted in the essential ground of biological and social reality. The same sense emerges in the key passage in Chap. 20 marking the transition to the final section of the text, where the \"Way of Man\" (ren zhi dao \u4eba\u4e4b\u9053), in contradistinction to the unmediated wholeness of the \"Way of Heaven'\" (tian zhi dao \u5929\u4e4b\u9053), is distinguished by the need for unceasing, concerted effort in order to strive toward a less cosmic level of moral self-completeness. Finally, this human ideal of wholeness is linked, by way of the common ground of being shared by the individual with all other people and all existing things in the finite world, to a notion of the perfect cultivation of the sage that, at its highest conceivable level, enables such a person to participate, alongside Heaven and Earth, in the creative processes of the entire cosmic scheme.\n\nThe profound metaphysical implications attached to the process of human cultivation in the Zhongyong, especially in the opening chapter and in the final expansion chapters, help to explain why it became a primary focus of Confucian philosophical debate. Not only were the expressions weifa \u672a\u767c (prior to concrete manifestation) and yifa \u5df2\u767c (in the world of concrete manifestation) later extracted from the text and used as shorthand indicators for the metaphysical and existential realms, respectively, but the deeper meanings of such terms as \"nature\" (xing \u6027) \"equilibrium and harmony\" (zhong he \u4e2d\u548c) and \"self-completeness\" (cheng \u81ea\u6210) as developed here came to provide the core concepts for a large portion of the intellectual discourse of the Neo-Confucian period.\n\nIn the space of just a few dozen pages, the Zhongyong deals with a very wide range of philosophical issues. It opens by positing a tripartite relation between human cultivation (jiao \u6559), the Way (dao \u9053), and the intrinsic nature of things (xing \u6027), then considers the cosmic condition of the perfectly balanced mean with respect to the human ideal of ceaseless rebalancing (shizhong \u6642\u4e2d). From this point, it surveys the entire range of possibilities for translating the abstract ideal of the mean into concrete practice; it asserts the grounding of all the spheres of human relations in the roots of individual experience and the anchoring of all external accomplishment in the perfection of individual character; and it explores the significance of the paradigmatic Confucian acts associated with ritual propriety and benevolent rulership. Finally, it contemplates the achievement of integral wholeness on both the cosmic and the individual levels, affirming the necessity of extending individual self-realization to the realization of all other beings; and it envisions the metaphysical implications of the highest imaginable attainment of these ideal states of being. In the eyes of a number of traditional and modern readers, this breadth of enquiry into so many separate topics has supported the suspicion that the treatise as we have it may be a composite compilation rather than a single integral text.\n\nIn view of the sequence of arguments in the respective chapters and the overall structural movements that make up this sweeping scope of discussion, however, the separate topics can be seen to fall into a coherent unified argument. The key to integrating the disparate elements of the text lies in grasping and consistently applying the logical method implied at the very outset in Chap.\u200b 1, and then reinforced in each of the subsequent contexts of discussion. This is the insistence on the essential distinction between the cosmic and the sub-lunar planes of being, according to which primary significance is focused on the concerted acts of human striving required for the ordering and perfecting of the finite human realm. In this light, each of the textual divisions outlined above takes its place in the gradually rising scaffold of this compact but elegant intellectual edifice. After positing the logical limits of human self-perfection in the definitions of moral instruction (with its mirror image: cultivation) and in the ideal of harmony in the opening section, the text goes on to explore the full range of attainment to which the active and contemplative human faculties are directed. The first of the major structural divisions traces this range of possible attainment from the low end of the scale, observing the prevailing condition of zero-degree accomplishment of the mean in practice, this set off even more starkly by glimpses of sagely cultivation reserved for the most perfect of human exemplars alone (the Sage-Emperor Shun \u821c, and the paradigmatic disciple of Confucius Yan Hui \u984f\u56de). We then turn in the second movement to a more detailed examination of finite degrees of Confucian cultivation in varying circumstances. After first consolidating the conceptual foundation for such activities by asserting the proposition that the roots from which all such accomplishment springs must lie close to, indeed are intrinsic within, the fundamental parameters of individual existence, the text then goes on to reaffirm this through the classic Confucian paradigms of moral action in ritual observance and benevolent rulership. This culminates, at the start of the final textual movement, in the redefinition of the highest state to which man can aspire in terms of the central concept of \"integral wholeness\" (cheng \u8aa0). Here, again, the recurrent logic of the treatise is invoked to reassert the fundamental distinction between a hypothetical perfect state of cosmic wholeness, and that degree of proximate wholeness approachable only through concerted human striving. This conception is further deepened in subsequent sections by the crucial corollary that this pursuit requires both external acts of cultivation and an internal state of conscious understanding, and by the even bolder assertion that this so-called \"wholeness\" remains necessarily incomplete until it is extended beyond the self to effect the full \"realization\" of other men and things. In the final chapters, the author lifts his eyes to the highest conceivable plane of individual fulfillment, envisioning consequences of human perfection that bring us to the upper limit of the trajectory of attainment traced in the text. Even as we contemplate these glowing images at the height of the Confucian spiritual vision, we are reminded in the final chapter that none of this perfection can be separated from the fundamental ground of moral consciousness actualized only in the sphere of human interaction.\n\nReferences\n\nChan, Wing-tsit. 1963. A sourcebook in Chinese philosophy, The great learning, 84\u201394. Princeton: Princeton University Press.\n\nGardner, Daniel K. 1986. Chu Hsi and the Ta Hsueh: Neo-Confucian reflection on the Confucian canon. Cambridge: Harvard University Press. Close reading of Daxue text based on textual commentaries of Zhu Xi.\n\nGraham, A.C. 1989. Disputers of the Tao: Philosophical argument in ancient China. La Salle: Open Court. General survey of early Chinese thought from the perspective of polemical discourse among different \"schools\".\n\nHenderson, John B. 1991. Scripture, canon, and commentary: A comparison of Confucian and western exegesis. Princeton: Princeton University Press. First comprehensive study of Confucian textual commentary.\n\nHu, Zhikui \u80e1\u5fd7\u594e. 1984. Evidential explication of Daxue and Zhongyong \u5b78\u5eb8\u8fa8\u8b49. Taipei: Linking Publishing House. (Study of primary textual issues regarding Daxue and Zhongyong).\n\nHughes, E.R. 1943. The great learning and the mean-in action. New York: E. P. Dutton. Translation of Daxue and Zhongyong with running commentary by author.\n\nLau, D.C. 1967. A note on Ke-Wu. Bulletin of the School of Oriental and African Studies 30: 353\u2013357. New philological approach to the meaning of the key term gewu.CrossRef\n\nLegge, James. 1923. The four books. Shanghai. (Reprint, New York: Paragon Books, 1966). (\"classic\" English translations of basic Confucian canons).\n\nLoewe, Michael. 1993. Early Chinese texts: A bibliographical guide. Berkeley: Institute of East Asian Studies.\n\nNivison, David S. 1996. The ways of Confucianism. La Salle: Open Court. Good introduction to the different streams of early confucian thought.\n\nNylan, Michael. 2001. The five Confucian classics. New Haven: Yale University Press. Broad introduction to textual history and intellectual significance of the \"Five Classics\".\n\nRiegel, Jeffrey. 1978. The Four Tzu Ssu chapters of the Li Chi: An analysis and translation. Stanford: Stanford University dissertation. (argument for interpretation of key chapters in Liji based on their presumed common authorship)\n\nSchwartz, Benjamin. 1989. The world of thought in ancient China. Cambridge: Harvard University Press. General survey of early Chinese philosophy.\n\nTang, Junyi \u5510\u541b\u6bc5. 1973. Treatise on Chinese philosophy \u4e2d\u570b\u54f2\u5b78\u539f\u8ad6, Hong Kong: New Asia College. (Penetrating discussions of central issues in traditional Chinese thought).\n\nTu, Wei-ming. 1989. Centrality and commonality: An essay on Chung Yung. New York: State University Press. Translation of Zhongyong with embedded commentary by author.\n\nTwichett, Denis, and Michael Loewe. 1986. Cambridge history of China \u2013 Ch'in and Han. Cambridge: Cambridge University Press.CrossRef\n\nXu, Fuguan \u5f90\u5fa9\u89c0. 1963. History of Chinese theories of human nature \u4e2d\u570b\u4eba\u6027\u8ad6\u53f2. Taizhong: Donghai University Press. (Comprehensive review of discussions on topic of essential human nature in classic Chinese thought).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_7\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 7. Philosophical Thought of Mencius\n\nWing-cheuk Chan1\n\n(1)\n\nDepartment of Philosophy, Brock University, St. Catharines, Canada\n\nWing-cheuk Chan\n\nEmail: wchan@brocku.ca\n\nAbstract\n\nTraditionally, Mencius (Mengzi\u5b5f\u5b50 371\u2013289 B.C.) has been identified as the second major founder of Confucianism. His family was originally noble, though his parents were quite poor. It has been a famous legend that for the sake of a better education environment for him, his mother moved 3 times. Like Confucius (Kongzi\u5b54\u5b50), during his earlier life Mencius travelled from one state to another to convince the lords to accept his political ideas. After several failed attempts, Mencius retreated to his home state of Lu \u9b6f and started to teach students. With the help of his disciples, he was able to complete a major work entitled the Mencius. This book, consisting of seven chapters, has been identified as the second cornerstone for the founding of Confucianism after the Analects. In Mencius' time, both Mohism (Mojia \u58a8\u5bb6) and Yangism (Yangzhu xuepai\u694a\u6731\u5b78\u6d3e) were very popular and hence posed a threat to Confucianism. Mencius accordingly was determined to overcome these two opponents. Given his important contribution to the development of Confucianism, Mencius has been named \"the second sage\" (yasheng\u4e9e\u8056).\n\n## 1 Introduction\n\nTraditionally, Mencius (Mengzi \u5b5f\u5b50 371\u2013289 B.C.) has been identified as the second major founder of Confucianism. His family was originally noble, though his parents were quite poor. It has been a famous legend that for the sake of a better education environment for him, his mother moved 3 times. Like Confucius (Kongzi \u5b54\u5b50), during his earlier life Mencius travelled from one state to another to convince the lords to accept his political ideas. After several failed attempts, Mencius retreated to his home state of Lu \u9b6f and started to teach students. With the help of his disciples, he was able to complete a major work entitled the Mencius. This book, consisting of seven chapters, has been identified as the second cornerstone for the founding of Confucianism after the Analects. In Mencius' time, both Mohism (Mojia \u58a8\u5bb6) and Yangism (Yangzhu xuepai \u694a\u6731\u5b78\u6d3e) were very popular and hence posed a threat to Confucianism. Mencius accordingly was determined to overcome these two opponents. Given his important contribution to the development of Confucianism, Mencius has been named \"the second sage\" (yasheng \u4e9e\u8056).\n\nIn fact, Confucius, Mencius, and Xunzi in Chinese philosophy have been compared to Socrates, Plato, and Aristotle in Western philosophy. Just as Whitehead says that the whole of Western philosophy is a footnote to Plato's thought, one might say that the whole of Confucianism is a footnote to Mencius' thought.1 More specifically, Mencius' major contributions are shown in developing a philosophical anthropology, an ethics, and a political philosophy from a Confucian standpoint. Mencius' philosophical anthropology starts with the thesis that human nature is originally good. He thereby criticized Gaozi's \u544a\u5b50 thesis of the neutrality of human nature. Mencius not only tried to justify his thesis argumentatively, but also experientially. His ethics and political philosophy are founded upon his philosophical anthropology. Mind (xin \u5fc3) is arguably the central concept in Mencius' thought.\n\nFirst, this chapter aims to show that pure feeling constitutes the essence of mind in Mencius. This enables us to develop a non-naturalistic picture of Mencius. On the methodological level, as we will see, analogical thinking plays a key role in Mencius' philosophy. From a historical standpoint, the rise of Mencius' thought signifies that Chinese philosophy no longer only consists of dogmatic assertions. In addition, in terms of a settlement of the debate between Shun Kwong-loi \u4fe1\u5ee3\u4f86 and Liu Xiusheng \u5289\u79c0\u751f in modern scholarship on Mencius, we will try to overcome the paradigm of radical rationalism which is founded by Mou Zongsan \u725f\u5b97\u4e09 in the East and the paradigm of moral psychology which is introduced by David Nivision in the West. As a result, one has to shift to the new paradigm of a phenomenology of pure feeling. Finally, starting with Mencius' \"politics of the sentiment,\" we will explore its possible contribution to overcome the controversy between modernity and post-modernity. This indicates that Mencius' philosophy is not past, but is significantly relevant to our age. From a standpoint of intellectual history, all this will help us to discover that it was neither Zhu Xi \u6731\u71b9 (1130\u20131200), nor Lu Xiangshan \u9678\u8c61\u5c71 (1139\u20131193) and Wang Yangming \u738b\u967d\u660e (1472\u20131529), but rather Liu Zongzhou \u5289\u5b97\u5468 (1578\u20131645) who was the faithful follower of Mencius.\n\n## 2 Mencius' View of Human Nature in Light of His Debate with Gaozi\n\nLogic in traditional Chinese philosophy is essentially different from Aristotelian syllogism. In contrast to the deductive logic of the West, Chinese logic is mainly analogical. Accordingly, most of the arguments developed by Chinese philosophers are primarily analogical. From a historical standpoint, only with the rise of the later Mohists did Chinese logic receive a first systematic articulation. But this does not preclude the possibility that even philosophers of other schools mainly argued by means of the Mohist logic. Thus it is helpful for us to start by highlighting the Mohist logic as a theory of analogical reasoning, in order to appreciate the arguments raised by Gaozi and Mencius.2\n\nIn the Mohist logic, there are four basic types of analogical reasoning: (1) exemplification (pi \u8f9f) (2) parallel (mou \u4f94); (3) imitation (yuan \u63f4); (4) extension (tui \u63a8). First, exemplification (pi) is attributive analogical reasoning. According to the Mohists, \"exemplification is to put forth another thing in order to illuminate this thing\" (Graham 1978: 522; I mainly follow Lau 1970: 261). This is a type of reasoning in terms of comparing properties. Its validity is mainly determined by the existence of the similarity of attributes.\n\nSecond, parallel (mou) is relational analogical reasoning. According to the Mohists, \"parallel is to set [two] propositions [expressing relations] side by side and show that they will both do\" (Graham 1978: 522; I mainly follow Lau 1970: 261). This is a type of reasoning in terms of comparing relations. The existence of the proportionality between the relations is the major criterion for the validity.\n\nThird, imitation (yuan) is copying analogical reasoning. According to the Mohists, \"imitation is to say: 'You can do this, why am I alone not allowed to do that?'\" (Graham 1978: 522). This is a type of reasoning in terms of mimics. When the similarity between the proponent's and the opponent's position is found, the argument is valid. Pragmatically, imitation aims at defending one's own position.\n\nFinally, extension (tui) is destructive analogical reasoning. According to the Mohists, \"Extension is to assimilate what has not been accepted to what has been accepted, so as to refute the opponent's thesis\" (Graham 1978: 522). This is a type of reasoning which targets the opponent's thesis in terms of assimilation. If there is a self-inconsistency in the opponent's position, then the argument is valid. Extension can be divided into two forms: (1) direct refutation; (2) indirect refutation. Pragmatically, the goal of extension is destructive with regard to the opponent's position by uncovering its self-inconsistency. Such an outline of the Mohist theory of analogical reasoning enables us to develop a typological analysis of the arguments in the Gaozi-Mencius debate.\n\nFirst of all, in justifying his claim that human nature is neutral, Gaozi said:\n\n> G I: (i) Human nature is like the willow tree, and righteousness is like a cup or a bowl;\n> \n> (ii) To turn human nature into humanity and righteousness is like turning the willow tree into cups and bowls. (Mencius 6A1; Graham 1978: 522\u2013523; here I mainly follow Lau 1970: 262)\n\nGI (i) is an exemplification (pi). For Gaozi, both human nature and the willow tree are natural givenness, while righteousness and a cup are artificial products. The point of this argument is to show that righteousness is nothing natural. GI (ii) is a parallel (mou). According to Gaozi, turning the willow tree into cups and bowls is against its nature; likewise, to turn human nature into humanity and righteousness is against its nature. The point of this argument is to show that humanity and righteousness do not originally belong to human nature.\n\nAs a rebuttal, Mencius pointed out,\n\n> M I: Sir, can you follow the nature of the willow tree and make the cups and bowls, or must you violate the nature of the willow tree before you can make the cups and bowls? If you are going to violate the nature of the willow tree in order to make cups and bowls, then must you also violate human nature in order to make it into humanity and righteousness? Your words, alas! would lead all people in the world to consider humanity and righteousness as calamity. (Mencius 6A1)\n\nImplicit in this passage is an extension (tui). Mencius' point is that if Gaozi's argument is valid, then it would give rise to the following undesirable consequence: All people in the world \u2013 including Gaozi himself \u2013 would consider humanity and righteousness as calamity. For Gaozi's position implies that to be moral is against human nature. In terms of Mencius' counter-argument, one can see that the controversy is intensified on whether to be moral is devastating to the original human nature.\n\nTo this, Gaozi responded with a new argument which is a parallel (mou):\n\n> G II: Man's nature is like whirling water.\n> \n> If a breach in the pool made to the east it will flow to the east.\n> \n> Man's nature is indifferent to good and evil, just as water is indifferent to east and west. (Mencius 6A2)\n\nIn maintaining that there is still a similarity between turning water flow to east or west and turning human nature to good or evil, Gaozi no longer appealed to the notion of intrinsic nature, but rather argued that the direction of turning is determined in an accidental manner. He aimed to show that just as water's flowing to east or west is determined by external factors, human nature's turning to good or evil is determined by external factors. For him, to be good or evil has no intrinsic ground in human nature. In this way, he tried to escape the challenge brought up by Mencius that to be moral is to be against human nature.\n\nLikewise, Mencius employed a parallel (mou) to respond:\n\n> M II: Man's nature is naturally good just as water naturally flows downwards. Man can be made to do evil as water is forced to flow uphill. (Mencius 6A2)\n\nFor Mencius, it is in accordance with the nature of water that it flows downward, but it is against its nature and due to the external force that it flows upward; likewise, it is in accordance with his nature that man is good, but it is against his nature and due to external influence that man is made evil. In his eyes, it is nothing accidental for human beings to be moral, for it is grounded in human nature. That is to say, there is something a priori in human nature that ensures the capacity of becoming moral.\n\nIn denying such an a priori concept of human nature, Gaozi started with his own definition of \"nature\": \"What is inborn is called nature\" (Mencius 6A3). For him, \"nature\" is an empirical, or, more precisely, a biological, concept. Accordingly, it is only a contingent truth that human nature is good.\n\nFor the sake of defending the necessity of the thesis that human nature is good, Mencius rejoined, \"When you say what is inborn is called nature, is that like saying white is white?\" Given Gaozi's positive answer, Mencius said:\n\n> M III: If you agree with saying that \"the whiteness of a white feather is the same as the whiteness of snow,\" and that \"the whiteness of snow is the same as the whiteness of white jade,\" then you have to accept that \"the nature of a dog is the same as the nature of an ox, and the nature of an ox is the same as the nature of a man.\" (Mencius 6A3)\n\nThis is, in reality, an extension (tui). In Mencius' eyes, if Gaozi does not accept the thesis that \"the nature of an ox is the same as the nature of a man,\" then he has to abolish the thesis that to say what is inborn is called nature is like saying white is white. The latter is necessarily true, whereas the former is not.\n\nIn defending his definition of \"nature,\" Gaozi said: \"By nature we desire food and sex. Humanity is internal and not external, whereas righteousness is external and not internal\" (Mencius 6A4). As a further support for this thesis, Gaozi introduced the following parallel (mou):\n\n> G III: When I see an old man and respect him for his age, it is not that the oldness is within me, just as, when something is white and I call it white, I am merely observing its external appearance. I therefore say righteousness is external. (Mencius 6A4)\n\nGaozi's point is that by observing the external appearance of a white thing, we call it white; likewise, when we see the external appearance of an old man, we respect him for his age. At first glimpse, this might raise a challenge to the understanding of Gaozi's view of human nature as neutral, for he now clearly claims that humanity is internal. However, what Gaozi actually means by saying that humanity is internal is rather that human nature is inclined to be good. This indicates that Gaozi merely understands humanity to be de facto internal. In other words, for Gaozi, \"humanity\" is still only an \"ontic,\" rather than an \"ontological\" concept. Accordingly, Gaozi's internalization of humanity is in reality similar to attributing desiring food and sex to human nature. That is, humanity is \"internal\" only in the sense of being \"instinctual.\" So this will not change the fact of Gaozi's identification of human nature as neither good nor evil.\n\nIn rejecting such a \"naturalization\" of humanity, Mencius employed the same analogy but in a different way:\n\n> M IV: There is no difference between our considering a white horse to be white and a white man to be white. But is there no difference between acknowledging the age of an old horse and the age of an old man? And what is it that we call righteousness, the fact that a man is old or the fact that we honor his old age? (Mencius 6A4)\n\nFrom a logical standpoint, Mencius' above argument is an extension (tui). Although Mencius accepted that there is no discrepancy between the white of the white horse and that of the white man, he doubted whether Gaozi would agree that there is no difference between the oldness of the old horse and that of the old man; for Gaozi had to accept that the oldness of the old man has an ethical implication, whereas the white of the white horse does not. More importantly, this shows that righteousness is shown in the respect for the old man, rather than in being the old man. As a result, the similarity between calling a white thing a white thing in seeing its whiteness and respecting an old man in seeing his oldness is destroyed.\n\nHowever, Gaozi insisted that the respect for the old man exactly indicates that righteousness is external. He hence introduced another extension (tui) which can be reformulated as follows:\n\n> G IV: (1) You agree that since my love of my own younger brother but not the younger brother of a man from the state of Qin is determined by my pleasant feeling, it is called \"internal.\"\n> \n> (2) Now in the case of my respect for the old man from the state of Chu as well as my own elders, what determines my pleasant feeling is age itself. Therefore, it is called \"external.\"\n> \n> (3) To be determined by my subjective feeling is different from being determined by the other's oldness. While the former is \"internal,\" the latter is \"external.\"\n> \n> (4) In maintaining the internality of humanity, how can you oppose the externality of righteousness?\n\nIn order to escape such a charge, Mencius continued his effort in denying the externality of righteousness. He rejoined with another extension (tui):\n\n> M V: We love the roast meat of Qin as much as we love our own. This is even so with respect to material things. Then are we going to say that our love of the roast meat is also external? (Mencius 6A4)\n\nMencius' point is that if Gaozi identifies our love of the roast meat as external, then it would give rise to a self-inconsistency in G IV.\n\nAlthough Gaozi himself did not respond to Mencius' counter-attack, his disciple Meng Jizi \u5b5f\u5b63\u5b50 raised a question to Mencius' disciple Gongduzi \u526c\u90fd\u5b50, \"What does it mean to say that righteousness is internal?\"\n\n> Gongduzi said, \"We practice reverence, and therefore it is called internal.\"\n> \n> [MENG Jizi asked:] \"Suppose a fellow villager is 1 year older than your older brother. Whom are you going to serve with reverence?\"\n> \n> [Gongduzi replied:] \"I shall serve my brother with reverence.\"\n> \n> [MENG Jizi asked:] \"In offering wine at a feast, to whom will you offer it first?\"\n> \n> [Gongduzi answered:] \"I shall offer wine to the villager first.\"\n> \n> MENG Jizi concluded: \"Now you show reverence to one but honor the age of the other. What determines your actions certainly lies without and not within.\" (Mencius 6A5)\n\nMeng Jizi's argument can be reformulated as follows:\n\n> G V: You accept that when a fellow villager is one year older than your older brother, you will offer wine to the villager first.\n> \n> However, you show reverence to your older brother in your heart while honoring the age of the villager.\n> \n> Therefore, there is a self-inconsistency in your position.\n\nThis is also an argument in the form of an extension (tui). Meng Jizi's point is that in such a ceremony actually only the external appearance is the key factor. This implies that righteousness is external.\n\nSince Gongduzi himself was not in a position to respond, he turned to Mencius for help. Mencius told him,\n\n> If you ask him whether he will serve with reverence his uncle or his younger brother, he will say he will serve with reverence his uncle. Then you ask him, in case his younger brother is acting at a sacrifice as the representative of the deceased, then to whom is he going to serve with reverence? He will say he will serve the younger brother with reverence. Then you ask him \"Where is your reverence for your uncle?\" He will then say, \"[I show reverence to my younger brother] because he represents the ancestral sprit in an official capacity.\" You can likewise say, \"[I show reverence to the villager] because of his position.\" Ordinarily, the reverence is due to the elder brother but on special occasions it is due to the villager. (Mencius 6A5)\n\nIn reality, Mencius introduces the following argument:\n\n> M VI: Ordinarily, the reverence is due to the elder brother, but on special occasions it is due to the villager.\n> \n> Just as ordinarily the reverence is due to your uncle, but when your younger brother represents the ancestral sprit in an official capacity it is due to him.\n\nThis is an argument of parallel (mou). For Mencius, these two cases are similar in changing the order of reverence due to the special occasions. So what matters is the difference between the standard and the secondary, rather than that between the internal and the external.\n\nWhen Meng Jizi learnt this, he said:\n\n> G VI: We show reverence to the uncle when reverence is due to him, and we show reverence to the younger brother when reverence is due to him. Certainly what determines it lies without and does not come from within. (Mencius 6A5)\n\nThis is also an argument of parallel (mou). As a follower of Gaozi, Meng Jizi wanted to stress that it is the role of the person that determines the order of reverence, and that this is the evidence that righteousness is external.\n\nFollowing Mencius' instruction, Gongduzi rejoined:\n\n> M VII: In the winter we drink things hot. In the summer, we drink things cold. Does it mean that what determines eating and drinking also lies outside? (6A5)\n\nTo this argument Mou Zongsan remarks:\n\n> Gongduzi's argument does not work. It is messy and muddle-minded. On the surface, it looks like what Mencius said before: \"We love the roast meat of Qin as much as we love our own. Then are we going to say that our love of the roast meat is also external?\" In reality, they are different. While Mencius's argument is valid, Gongduzi's argument is not. To hold that \"In the winter we drink things hot, and in the summer, we drink things cold\" can only lend support to Gaozi's position. Namely, righteousness should be external. This indicates that Gongduzi is illogical. In saying that \"We love the roast meat of Qin as much as we love our own,\" Mencius aims to stress love is internal. (Mou Zongsan 2004: 9)\n\nHowever, Mou Zongsan's critique can only be justified from a behaviourist standpoint. That is to say, only when the behaviour is entirely reducible to the conditioned reflex, then Mou Zongsan's criticism is valid. Gongduzi's original point is rather that just as what determines the love of roast meat (as is shown in Mencius' argument) comes from within, what determines the love of drinking comes from within. Like Mencius, he is able to emphasize the spontaneity of the subjectivity. Structurally, his argument is a parallel (mou). Our difference from Mou Zongsan's position reinforces that the validity of an analogical argument is context-dependent or intention-dependent.\n\nOne can now sum up the difference between Mencius and Goazi as follows: on the one hand, for Mencius, apart from human nature no humanity and righteousness would be possible; on the other hand, for Goazi, humanity is internal to human nature, whereas righteousness is external to it. More importantly, while Mencius' concept of human nature is ontological, Goazi's concept of human nature is ontic. In other words, what Mencius means by \"nature\" is Being in the sense of the \"way to be.\" In contrast, what Gaozi has in mind is basically a naturalistic, or more precisely a biological concept of nature.\n\nBut a set of questions remains: Is it possible to conclude that Mencius wins the argumentation? If this is the case, then in what sense does Mencius win the debate? Does Gaozi commit any logical fallacy?\n\nIn order to answer these questions, let us start with an exposition of the Mohist theory of logical fallacy. As the Mohists write,\n\n> Things may have similarities, but it does not follow that therefore they are completely similar. When propositions are parallel, there is a limit beyond which this cannot be pushed. For each thesis, there is a ground. Despite the similarity of [the two] theses, their grounds can well be different. For any choice of a position, there is always a criterion. Despite the similarity in the [two] chosen positions, their criteria may be different. Therefore, as far as the arguments in the form of exemplification, parallel, imitation, and extension are concerned, they may go too far, become different in validity, turn to be dangerous, and lose the foundation. So it is necessary to be careful in employing them, and to avoid seeing them as a rule. Given the possibility of ambiguity in speech, distinction in kind, and difference in grounding, one has to go beyond seeing things from any partial perspective. (Graham 1978: 522\u2013523; here I mainly follow Lau 1970: 262)\n\nThis passage also includes an analysis of the causes of the rise of fallacies in the four types of analogical reasoning. First, in the case of exemplification (pi), the validity of an inference is grounded in the similarity between the properties of two things. However, \"Things may have similarities, but it does not follow that therefore they are completely similar.\" That is, the similarity might be just local. If one falsely generalizes a local affinity to be a global one, then it might give rise to a fallacy. This is the fallacy of \"going too far.\" Secondly, in the case of parallel (mou), the validity is grounded in the similarity between two relations. Nonetheless, \"When propositions are parallel, there is a limit beyond which this cannot be pushed.\" If one oversteps the limit, then one commits to a fallacy. This is the fallacy of \"difference in validity.\" Thirdly, in the case of imitation (yuan), the validity of the reasoning is grounded in the similarity between two theses. However, \"For each thesis, there is a ground. Despite the similarity of [the two] theses, their grounds can well be different.\" If one fails to recognize such a possibility, then one might commit a fallacy of \"turning to be dangerous.\" Finally, in the case of extension (tui), the validity of an inference is grounded in the inconsistency shown in the opponent's two similar positions. Nevertheless, \"For the choice of a position, there is a criterion. Despite the similarity in the [two] chosen positions, their criteria may be different.\" Overlooking such a difference can give rise to the fallacy of \"losing the foundation.\"\n\nArmed with such a Mohist theory of fallacy, we can start to examine the arguments involved in the debate between Gaozi and Mencius. First of all, the validity of GI is grounded in the similarity between making man moral and making the willow tree into cups. However, in MI Mencius not only tries to show that making man moral and making the willow tree into cups are different, but also aims to argue that GI can lead to disastrous consequences for being moral. For we have to do violence to the nature of the willow tree in order to make it into cups, whereas to be moral does not imply doing any violence to human nature. Furthermore, the claim that we have to do violence to our nature in order to be moral would undermine the authority and positive status of morality. Therefore, GI commits the fallacy of \"going too far and that of difference in validity.\" To this extent, one can conclude that MI succeeds in overriding GI.\n\nThe validity of GII as a parallel is grounded in the similarity between the indifference water shows in turning east or west and the equal possibility of human nature in becoming good or evil. MII rather aims to show that as far as the similarity between human nature and water is concerned, human nature's being good should be compared to water's flowing downward. From a logical point of view, GII and MII are different parallels. Unless Gaozi claims that GII starts with the premise that water and human nature are completely alike, one cannot charge him for committing the fallacy of \"becoming different in validity.\"\n\nAs an extension MIII takes issue with Gaozi's identification of human nature as what is inborn. Given Gaozi's refusal to see the nature of an ox as the nature of a man, he would give up the claim that \"Nature is what is inborn\" would be a necessary truth like \"White is white.\"\n\nNonetheless, in insisting on the externality of righteousness, Gaozi tries to compare an old man to the color white. The validity of GIII as an exemplification is grounded in the similarity between their appearances. In his counter-attack, Mencius points out that if Gaozi agrees that there is a distinction between an old man and an old horse, then he should give up the attempt to prove the externality of righteousness in terms of a comparison between the quality of being old and the quality of being white, for righteousness is only shown in the respect for the elder. MIV as an extension aims to uncover that if GIII holds, then we should also show respect for an old horse. But given that Gaozi does not accept such an undesirable consequence, he would withdraw GIII; otherwise, he would commit the fallacy of \"going too far.\"\n\nSince Gaozi maintains that the respect for an old man exactly shows the externality of righteousness, he contrasts the distinction shown in loving my brother and loving another's brother to the identity shown in the respect for an old man from my country and the respect for an old man from another country. GIV as an extension uncovers that it is based on my subjective preference that humanity is said to be \"internal,\" while it is based on the appearance of the old man that righteousness is said to be \"external.\" But even if Mencius denies this contrast, he would not commit the fallacy of \"losing the foundation.\" For Mencius maintains that it is not due to the \"outer\" appearance of an old man, but rather the \"internal\" feeling that we show respect to him.\n\nMV as an extension aims to shows that there is a self-inconsistency in GIV. For Mencius, it is rather Gaozi who commits the fallacy of \"losing the foundation.\" As Gaozi said before, to say that humanity is \"internal\" is due to the fact that \"I am the one to determine that pleasant feeling\" (Mencius 6A4). But although our love of the roast meat is also determined by our pleasant feeling, he identifies it as \"external.\"\n\nAs a counter-attack, GV in form of an extension aims to demonstrate that it is rather MV which commits the fallacy of \"losing the foundation.\" For, while showing respect to your brother whole-heartedly, you are forced to honor the elder villager. This implies that without admitting the externality of righteousness, Gongduzi would be inconsistent with himself. If Gongduzi agrees that age is the most decisive factor, then he will lose his position.\n\nAt first glance, MVI seems to confirm that the external factor is most decisive. For this indicates that to honor the younger brother is like honoring the elder villager. As is shown in GVI, similar to the case of honoring the younger brother, the criterion for honoring the elder villager is also external.\n\nIn order to block such a possibility, MVII aims to show that it is self-consistent to see the determining factor for whether to drink soup in winter or to drink water in summer as internal. One might wonder to what extent such a parallel can become valid. Clearly, if the disciples of Gaozi also assume that the subjective feeling of comfort, rather than the variation of external conditions, plays the key role in deciding whether to drink soup or water, then MVII is able to attain its goal.\n\nAs a whole, Gaozi finally gives up defending his position. In the first place, this is because, without the concept of moral competence, he cannot make sense of his own thesis of the internality of humanity. On the other round, given his thesis of the internality of humanity, he cannot reduce morality to the level of performance only. From a logical standpoint, in either case, it would give rise to the fallacy of self-inconsistency. In the second place, as Chong Kim-chong \u838a\u9326\u7ae0 points out, \"Mengzi [Mencius] is questioning Gaozi's application of 'internal' and 'external' to both ren \u4ec1 and yi \u7fa9 and showing that it has absurd consequences\" (Chong 2002: 104). To this extent, one can conclude that Mencius wins the debate. But Chong Kim-chong sees the origin of this debate in the conflict between a moral psychology and a psychology of desire. In contrast, for us, it is rather due to the more fundamental fact that Mencius and Gaozi have different views in philosophical anthropology. More precisely, in developing their respective views of human nature, Mencius' starting point is \"pure feeling,\" whereas Gaozi's is one of \"sensibility.\" As will be seen, Mencius' position transcends a moral psychological approach, whereas Gaozi's doctrine is imprisoned in moral psychology.\n\n## 3 Mencius' View of the Four Beginnings as Pure Feelings\n\nBesides the argumentative approach, Mencius tried to justify his thesis phenomenologically. For him, the feelings of commiseration, of shame and dislike, of respect and reverence, of right and wrong, are found in all men. These feelings can be, respectively, cultivated into the virtues of humanity, righteousness, propriety (li \u79ae), and wisdom. Together these feelings define what a man is as a man. That is, they are intrinsic to human nature (Mencius 2A6). As phenomenological evidence, Mencius pointed out that when a man suddenly sees a child about to fall into a well, he has a feeling of alarm and distress, and then tries to rescue the child. His motivation is not to favor the child's parents, nor to gain the praise of his neighbors and friends, nor to avoid the criticism from the other if he did not rescue the child (Mencius 2A6). All this shows that human nature is originally good and the moral mind is essentially affective. In terms of pure affectivity, for Mencius, human nature is nothing abstract.\n\nThe feelings of commiseration, of shame and dislike, of respect and reverence, of right and wrong are, however, the four beginnings. It is only when they are fully developed that one can become virtuous. This explains why, although all men have the same starting point, only a few might eventually become sages. Since Mencius maintained the original goodness of human nature, it is also necessary for him to account for the origin of evil. In his explanation, Mencius introduced the analogy of barley seeds. As he said in 6A7,\n\n> Sow the seeds and cover them with soil. The place is the same and the time of sowing is also the same. The plants shoot up and by the summer solstice they all ripen. If there is any unevenness, it is because the soil varies in richness and there is no uniformity in the fall of rain and dew and the amount of human effort devoted to tending it.\n\nThis shows that it is mainly due to the lack of effort in fully putting our moral capacity into the reality that evil arises. The feelings of commiseration, of shame and dislike, of respect and reverence, of right and wrong, are only the seeds. It is only when they are fully developed that virtues, instead of evils, can be attained. That is to say, evil arises when man fails to fully concretize his moral competence. Therefore a good beginning does not sufficiently guarantee a happy ending. Another necessary condition is self-cultivation.\n\nConcerning the know-how in fully developing the four beginnings, Mencius likewise employed an analogy in explanation. He pointed out, \"Even if you had the keen eyes of Lilou and the skill of Gong Shuzi, you could not draw squares or circles without a carpenter's square or a pair of compasses\" (Mencius 6A1). In the case of moral cultivation, the former sages such as Yao and Shun can show us how to use the carpenter's square and compass. Modeling on these sages' behaviour, it is possible for us to rectify our behaviour. To this extent analogical thinking also plays a key role in moral cultivation.\n\nAt this juncture, one might ask, how can the sages achieve their goal. Mencius' answer runs as follows:\n\n> The sages, having taxed their eyes to their utmost capacity, went on to the compass and the square, the level and plumb-line, which can be used endlessly for the production of squares and circles, planes and straight lines; having taxed their ears to their utmost capacity, they went on to the six pipes which can be used endlessly for setting the pitch of the fives notes (Mencius 4A1).\n\nIn the realm of morality, what functions as the square, the compass, and the five notes is li (principles). As Mencius further wrote, \"What is common to all hearts? Principles and rightness. The sage is simply the man first to this common element in my heart\" (Mencius 4A7).\n\nDespite the thesis that morality is primarily based on feelings, Mencius is a rationalist in stressing the universality of the moral principles. For him, moral principles are first of all universalized feelings. More precisely, the feeling of commiseration corresponds to the principle (li \u7406) of humanity, the feeling of shame and dislike corresponds to the principle of righteousness, the feeling of deference and compliance corresponds to the principle of propriety, the feeling of right and wrong corresponds to the principle of wisdom. Without these feelings, moral principles would become empty. In this way, there is no contrast between feeling and reason.\n\nBut what is the ontological status of the four beginnings as feelings? In dealing with \"feeling\" in Mencius' sense, many scholars tend to identify these feelings as instinctual. That is, they are understood as psychological emotions. For example, in arguing that \"Mencius held a picture of the role of emotion in moral motivation that militates against a general separation of reason from emotion,\" David Wong only approaches Mencius' feeling from the standpoint of moral psychology (Wong 1991: 31). Likewise, although traditional Confucians such as Zhu Xi insist on the difference between the \"four beginnings\" as feelings, i.e., pleasure, anger, sorrow and joy, on the one hand, and the \"seven emotions,\" i.e., happiness, bitterness, worry, delight, love, hatred and desire, on the other hand, they try to account for the difference in terms of the distinction between the principle (li \u7406) and the material force (qi \u6c23). As Zhu Xi said, \"The four beginnings are issued from the principle (li\u7406), whereas the seven emotions are issued from the material force (qi \u6c23)\" (Zhu Xi 1986: vol. 4, 1297) This nonetheless does not help change the fact that for Zhu Xi, the four beginnings, like the seven emotions, are still sensible. So the uniqueness of the moral feeling in Mencius' sense is only shown in its being aroused by reason alone. All this indicates that Mencius' feeling is identified either as rational (Mou Zongsan) or as sensible (Zhu Xi and David Wong). As a consequence, the autonomy of \"feeling\" in Mencius' sense has been eliminated; for \"feeling\" in Mencius' sense is pure, rather than sensible. In particular, to say that pure feeling is issued from reason overlooks the fact that the four beginnings can play the role of the Bestimmungsgrund (determining ground), additional to that of the Bewegungsgrund (motivating ground) of morality. As Mencius pointed out, \"Children carried in the arms all know to love their parents. As they grow, they all know to respect their elder brothers\" (Mencius 7A15). That is to say, all of them are not results from learning, but rather arise spontaneously. However, such \"innate\" ability and knowledge should not be understood in a biological or naturalistic sense. If this is the case, then one would ask Mencius for statistical grounds. This would also signify that for Mencius, moral truth is probable, rather than necessary.\n\nThe thesis that Mencius' four beginnings are pure feelings is also confirmed by his assertion that \"Therefore moral principles please my mind as beef and mutton and pork please our mouths\" (Mencius 6A7). It is because moral principles are non-sensible, and the pleasure aroused by them must be likewise non-sensible. Furthermore, as Chong Kim-chong observes, \"it would be wrong to construe Mengzi [Mencius] as suggesting that the heart-mind is a sensory organ in the way the palate is\" (Chong 2002: 114) But one cannot hence agree with Chong Kim-chong's claim that here Mencius \"is making a naturalistic assumption that just as there is something that pleases my palate, there is also something that pleases my heart-mind\" (Chong 2002: 114). As Chong Kim-chong himself acknowledges, \"It is clear, that it is li and yi that pleases the heart-mind, not the sensation of taste\" (Chong 2002: 114). But apart from the concept of pure feeling, one can hardly make sense of such an account.\n\nMencius also spoke of the \"strong, moving force\" (haoran ji qi \u6d69\u7136\u4e4b\u6c23). Such a force is immaterial. Furthermore, \"As force, it is accompanied by righteousness and the Way\" (Mencius 2A2). While pure feeling goes hand-in-hand with the immaterial force, sensible emotion belongs to the dimension of the material force.\n\nMore importantly, the reason why Mencius' moral feeling must not be identified as \"instinctual,\" \"sensible,\" \"biological,\" or \"psychological,\" is shown in his concept of human nature. As seen before, Mencius strongly opposes Gaozi's understanding of human nature. For, in his eyes, Gaozi's concept of human nature is biological or naturalistic. Given Mencius' identification of human nature with the four beginnings, the reduction of feeling to sensible emotion would blur the essential distinction between Mencius' and Gaozi's doctrines of human nature. Indeed, for Gaozi, desiring food and sex is \"instinctual.\" Therefore, to identify the four beginnings as \"instinctual\" would collapse Mencius' concept of human nature into that of Gaozi.\n\nInsofar as Mencius clarifies human nature in terms of the four feelings, it would be more accurate to say that he grants a primacy to feeling over the distinction between reason and sensibility. Ontologically, the four feelings constitute the Being of human beings. But to characterize the four feelings as \"instinctual\" implies that they are merely \"ontic\" features of human beings. That is, these feelings merely belong to the human being as a being only, and hence have nothing to do with the ontological structure of the human being. Accordingly, this kind of understanding of the four beginnings can hardly make sense of Mencius' thesis:\n\n> a man without the feeling of commiseration is not a man; a man without the feeling of shame and dislike is not a man; a man without the feeling of deference and compliance is a not a man; and a man without the feeling of right and wrong is not a man. (Mencius 2A6)\n\nAll this indicates that the four feelings constitute the ontological structure of man. This implies that for Mencius, man is first of all a feeling subject. That is, moral feeling is constitutive of xin\u5fc3 (mind). Although we are sympathetic with Irene Bloom's claim that Mencius' \"conception directly entails the idea of a universal human nature,\" we disagree with her identification of \"four beginnings\" as \"moral dispositions\" (Bloom 1994: 24; 44). In reality, these are pure, rather than psychological, feelings. Pace Heidegger, one might characterize the Being of pure feeling in Mencius' sense as follows:\n\n> Ontologically pure feeling is a primordial kind of Being for man, in which man is disclosed to himself priori to all cognition and volition, and beyond their range of disclosure. And we are never free of pure feelings. (see Heidegger 1962a: 175)\n\nIn short, Mencius' four feelings should be understood as man's Seinsk\u00f6nnen (capacity of to be). There is accordingly an essential distinction between pure feeling and sensible emotion. First, pure feeling is ontological, while sensible emotion is ontic. Second, pure feeling arises spontaneously, whereas sensible emotion is aroused by the external factor. Third, there is always an identity between pure feeling and ethical principle; in contrast, there is usually a discrepancy or even a conflict between sensible emotion and ethical principle. It should be noted that Craig Ihara's claim, \"Mencius does not himself use a word that can translate directly as 'emotion'\" (Ihara1991: 48), is not justified if the term \"emotion\" refers to pure feeling. On the other hand, one might agree with David Wong's thesis that, \"It seems straightforward to classify innate sensitivity to others as an emotion, or at least the primitive beginnings of an emotion such as compassion,\" only when the term \"emotion\" is changed into \"pure feeling\" (Wong 1991: 31\u201344).\n\nSeparating \"pure feeling\" from \"psychological emotion\" is also crucial for a correct understanding of the example of the child falling into the well. Ihara claims, \"Certainly there are no cognitive emotions in the example of the child in the well. That is exclusively an illustration of instinctive sympathetic responses\" (Ihara 1991: 50). If the feeling of commiseration is merely an instinctive sympathetic response, then its rise is exclusively elicited by certain features of a situation. This implies that the feeling of commiseration would not arise out of spontaneity of the mind. Such a consequence cannot be accepted by Mencius. For example, as he said, \"Shun understood the way of things and had a keen insight into human relationship. He followed the path of morality. He did not just put morality into practice\" (Mencius 4B19; my italic). This shows that the four beginnings as pure feelings are nothing instinctual, but rather self-conscious. That is also the reason why Mencius is able to claim, \"if one finds oneself in the right, one goes forward even against men in the thousands\" (Mencius 2A2). Obviously, for him, morality is never a \"blind\" action.\n\nOn the other hand, following Ronald de Sousa, David Wong argues that there is no principle of practical reason but only emotion which is capable of framing \"what features of a situation appear as salient features\" (Wong 1991: 33). In reality, it is pure feeling, rather than sensible emotion, that can address the \"frame problem.\" For the rise of sensible emotion is conditioned by the affecting factor external to the subjectivity. Nevertheless for Mencius, the feeling of commiseration, for example, is constitutive of human nature. It is due to its existence prior to seeing the child falling into the well that the feeling of commiseration can exercise the function of framing the salient features of the situation.\n\nAll this shows that the four beginnings as human feelings should be understood from a transcendental, rather than a moral psychological, perspective. However, Liu Xiusheng argues that Mencius is a moral realist. For him, this is the only way for Mencius to account for the objectivity of moral judgment. According to moral realism, as Liu Xiusheng understood, \"a moral judgment ascribes, truly or falsely, a moral quality to a person, action, or object\" (Liu 2003: 162). In justifying this moral realist picture of Mencius, Liu Xiusheng turns to Mencius' frequent comparison of \"the mind\/heart's enjoyment of moral qualities to the eye's enjoyment of certain colours, the ear's enjoyment of certain sounds, the mouth's enjoyment of certain flavours\" for help (Liu 2003: 162). Seen from the textual perspective, the evidence enumerated by Liu Xiusheng might be too thin in supporting his claim. As a matter of fact, Mencius never spoke of any \"moral quality,\" not to mention that he never granted any \"objective reality\" to such \"moral qualities,\" which can be compared to the \"secondary qualities\" in the empiricist sense. More importantly, there seems to be a vicious circle in Liu Xiusheng's position. In explaining why respecting an old horse does not involve moral approval while respecting an old man does, he says that it is \"because the former does not evoke characteristically human feelings, but the latter does\" (Liu 2003: 164; my italic). This indicates that those characteristically human feelings are only the \"passive receivers\" of moral facts. But at the same time he writes:\n\n> For Mencius, feelings and so on contribute to cognition in the following two ways. (a) They have a capacity to establish what properties or features of a situation appear as salient. (b) Certain feelings indicate the presence of a moral quality. From (a) and (b) we can say, as some prefer, that certain feelings and sentiments are the equipment by which we identify moral facts. (Liu 2003: 163)\n\nTo this extent, those human feelings are more than just \"passive receivers,\" and have a kind of function in determining what can be counted to be \"moral qualities\" or \"moral facts.\" Therefore, it remains unclear in regard to the \"ultimate\" authority in judging what can be counted as \"moral.\" At this juncture, Liu Xiusheng might find himself faced with the following dilemma. If he grants the ready-made status to moral facts, then he has to fall back to the metaphysical moral realism. On the other hand, if he lets human feelings be the sole determining grounds of \"being moral,\" then he has to side with emotivism. Since Liu Xiusheng explicitly refuses to identify Mencius as either a metaphysical realist or an emotivist, he has to face an insurmountable difficulty.\n\nAll in all, it is rather due to Liu Xiusheng's overlooking the distinction between pure feeling and sensible emotion that he identifies Mencius as a moral realist. In comparing Mencius to Hume, Liu Xiusheng merely understands the four beginnings as sensible emotions. As he writes:\n\n> For Hume, \"feeing\" or \"sentiment\" generally means (a) a tendency to be moved by various situations (as such it is also called a \"sensibility\" or \"propensity\") and (b) in general an affective mental state and in particular an attitude that issues from the tendency described in (a). Hume's use of \"feeling\" or \"sentiment\" is very similar to Mencius' use of \"xin\" (\u5fc3)... as in \"ce yin zhi xin\" (\u60fb\u96b1\u4e4b\u5fc3)... which is often translated as \"feeing\" or \"sense.\" (Liu 2003: 98)\n\nGiven his opposition of an emotivist picture of Mencius, Liu Xiusheng sees a realist interpretation of Mencius as the only way out. As he explains,\n\n> If Yi [yi] \u7fa9 and the performance of Yi [yi] \u7fa9 (to respect an elder) do not essentially involve any objective fact but depend entirely upon the projection of the agent's natural feelings or desire or attitudes, then a \"judgment\" of Yi [yi] \u7fa9will be entirely arbitrary and radically relativistic. (Liu 2003: 162)\n\nNonetheless, if one, from the very beginning, rejects such a \"naturalization\" or \"psychologization\" of the four feelings, then there is no need to (falsely) turn Mencius into a moral realist, in order to account for the possibility of the \"objectivity\" of moral judgment. In other words, the cause for Liu Xiusheng's error lies in his blindness to the fact that for Mencius, the \"rationality\" (or \"objectivity\") of moral judgment is grounded in what is common in the heart. As is seen before, Mencius explicitly identified principles and righteousness (yi \u7fa9) as what is common in the heart. That is, it is in terms of what is common in the heart, rather than any objective moral qualities, that Mencius attempts to account for the possibility of the \"objectivity\" of moral judgment. That is the reason why Mencius had to point out, \"A gentleman differs from other men in that he retains his heart\" (Mencius 4B28). Thus in account of the objectivity of moral judgment, there is an alternative to identifying Mencius as a moral realist. When Liu Xiusheng goes back to Mencius 6A4 and 6A7 to find evidence for his claim, he might also suffer from failing to differentiate Mencius' position from Gaozi's naturalism. In Mencius 6A4 Mencius tries to show that the quality \"being elder\" is not the major factor in determining morality; and what counts is only the \"respect.\" Such a respect is nothing but an \"internal\" pure feeling. From a textual standpoint, Liu Xiusheng has, besides, to face the challenge that his moral realist interpretation can hardly make sense of Mencius' doubt: \"And what is it that we call righteousness, the fact that a man is old or the fact that we honor his old age?\" Obviously, for Mencius, \"being old\" is not any moral quality which can determine the \"objectivity\" of the moral judgment that \"I ought to respect the elder.\" Finally, it is only with the interpretation of Mencius as a \"sensibility theorist\" that Liu Xiusheng can ignore any conflict between his realist and internalist view of Mencius. For him, Mencius has merely developed a sensibility theory of feeling. Accordingly, Mencius, as an internalist, only conceives of feeling as \"subjective.\" To be sure, with such a kind of feeling, Mencius can hardly explain the objectivity of moral judgment. But in saying that, \"We have all kinds of feelings, which establish the salience of different kinds of fact,\" Liu Xiusheng should have recognized the \"transcendental\" function of Mencius' moral feeling (Liu 2003: 164). Indeed, only pure feelings, rather than psychological emotions (or sensibility), are able to exercise such a function.\n\nAll this confirms that moral feelings in Mencius' sense are primarily \"pure.\" To this extent, Mencius is closer to Kant than Hume \u2013 contrary to the major thesis raised by Liu Xiusheng. It is no wonder that scholars such as Mou Zongsan and Li Minghui have compared Mencius to Kant (see Li 1990). In fact, as Heidegger remarks, when Kant said that \"respect for the law is not the incentive to morality; it is morality itself,\" he hinted at the possibility of \"pure\" moral feeling (Heidegger 1962b: 163). For Kant, in contrast to the pathological feeling, respect as moral feeling is \"produced solely by reason\" (Kant 1956: 79). But since Kant himself maintained that \"all feeling is sensuous,\" it had to wait for Heidegger to reveal the reality of \"respect as a pure feeling\" (Heidegger 1962b: 164). More importantly, even what Kant originally meant by \"pure moral feelings\" refer to those aroused by moral laws, it does not change the fact that they are still sensuous. As a result, unlike Mencius, Kant also failed to recognize the \"identity\" of the pure feeling and the moral law. Kant insisted that \"no kind of feeling, [even] under the name of a practical or moral feeling, may be assumed as prior to the moral law and as its basis\" (Kant 1956: 77). In identifying the moral law as the subjective cause of the feeling of respect, Kant did not allow moral feeling to serve \"as a basis of the objective moral law itself\" (Kant 1956: 79). To this extent, Michel Henry's following observation might more faithfully reflect Kant's limitation: \"The determination of the specific affective reality of respect, namely, of respect itself, nevertheless presupposes a pure concept of affectivity which is totally lacking in Kant\" (Henry 1973: 524). In contrast, Mencius is able to take care of the \"identity\" of the moral principle and the pure feeling. For him, the moral principle is nothing but the \"universalized\" pure feeling. In terms of Mencius' concept of pure moral feeling, one can also appreciate Henry's thesis: \"far from being opposed to morality, affectivity is its condition\" (Henry 1973: 529). Thus to reduce the four beginnings to psychological emotions would conceal their \"transcendental\" status. For Henry, what is wrong with Kant is the \"absence of a transcendental philosophy of affectivity\" (Henry 1973: 530). Furthermore, according to Kant, the submission to moral laws brings comfort. He however insisted that, \"This comfort is not happiness, not even the smallest part of happiness\" (Kant 1956: 91). Due to his lack of an explicit identification of moral feeling as pure, he could not produce any positive characterization of such a kind of comfort. He could at best say that, \"This inner satisfaction is therefore merely negative with reference to everything which might make life pleasant\" (Kant 1956: 91). On the other hand, when Mencius declared that, \"There is no greater joy than to examine oneself and be sincere\" (Mencius 7A4), what he had in mind is joy as pure feeling, rather than as sensible emotion. At this juncture, Mencius' position is superior to that of Kant. More importantly, according to Mou Zongsan, Kant's failure to recognize that morality can please our mind indicates his insufficiency in articulating the concept of moral motive (see Mou 2005: 11). Mou Zongsan also points out that the introduction of an \"original feeling\" which is non-sensible and immaterial is necessary for fixing this problem. But regrettably he seems to ignore that in Mencius the four beginnings are originally pure feelings.\n\nMencius' doctrine of pure feelings also shows that to simply explicate his concept of \"human nature\" in terms of the Aristotelian \"essence\" is not adequate. For Mencius' approach is primarily \"ontological,\" rather than \"logical.\" So we agree with Roger Ames that Mencius is not an essentialist. But for us xing is primarily an ontological, rather than a cultural, concept (see Ames 1991). That is the reason why Mencius explicitly declared, \"He who exerts his mind to the utmost knows his nature. He who knows his nature knows Heaven\" (Mencius 7A1). In this context, while \"nature\" means \"Being of man,\" \"Heaven\" means \"Being itself.\" With such a thesis, as Mou Zongsan points out, Mencius paves the way for the rise of Confucian \"moral metaphysics\" (moralische Metaphysik) (see Mou 1968: vol. 1). That is, moral praxis not only reveals the Being of man, but also Being itself. Therefore, apart from pure feeling, Mencius would not be able to pinpoint the essential difference between man and other beings, not to mention the possibility of demonstrating the ontological implications of moral praxis.\n\nUndeniably, on average most of us are not perfect in moral praxis. But this does not mean that we should no longer try to improve ourselves in moral praxis. Neither does it imply that human nature is not originally good. In illustrating this point Mencius introduces the famous analogy of the trees of the Niu Mountain: The trees of this mountain were once beautiful. However, due to the hewing down by humans and being pastured by the animals, the mountain became bold. When people see its boldness, they tend to think that it never had any timber. But as this is not the nature of a mountain, it is impossible for man to be originally lacking of humanity and righteousness. People seeing that a man behaves like an animal will tend to think that he never has the original endowment for goodness. Nevertheless, this is contrary to the human feeling (Mencius 6A8).\n\nAll this shows that for Mencius, possibility is higher than actuality. Such a standpoint lends support to our thesis that the four beginnings as feelings cannot be understood in a \"naturalistic\" way. For naturalism grants a priority to actuality over possibility. But to say that Mencius grants a primacy to the four beginnings as pure feelings by no means implies that he thereby undermines the importance of sensible feelings. As he said in Mencius 7A20, \"A gentleman delights in three things.\" They are: (1) \"His parents are alive and his brothers are well\"; (2) \"Above, he is not ashamed to face Heaven; below, he is not ashamed to face man\"; (3) \"He has the good fortune of having the most talented pupils in the Empire.\" In these cases, the rise of delight results from the existence of a certain state of affairs in the world. Moreover, the difference between pure feeing and sensible emotion does not exclude the possibility of their co-existence. As Mencius pointed out in Mencius 3A2, sensible emotions might be the immediate expressions of pure feelings. For example, during the mourning the future Duke Wen of Teng wept so bitterly that his people were greatly delighted. It is because the prince was able to mourn out of his heart. At this juncture, one might agree with Xiao Yang's \u856d\u967dthesis that for Mencius, \"our virtuous actions are always a natural expression of what lies deep within our hearts\" (Xiao Yang 2006: 270). The only thing we have to add is that the latter primarily refers to pure feeling, rather than psychological emotion.\n\nOn the whole, our \"ontological\" interpretation of human nature and \"transcendental\" understanding of pure feeling in Mencius' sense might shed new light on settling a significant controversy in modern scholarship. Concerning a correct understanding of Mencius' thesis of the \"internality\" of humanity (ren \u4ec1) and righteousness (yi \u7fa9), Shun Kwong-loi considers three interpretations:\n\n1.\n\nThe motivation interpretation \"takes the internality of yi \u7fa9 to be the claim that an act is yi \u7fa9 only if it is performed not just because it is proper, but because the agent is fully inclined to so act.\" (Shun 1997: 98)\n\n2.\n\nThe nature interpretation \"regards the internality of yi \u7fa9as the claim that yi \u7fa9 is a part of hsing [xing] \u6027, in the sense that human beings already share yi \u7fa9 as one of the four desirable attributes or are already disposed to yi \u7fa9 behaviour.\" (Shun 1997: 98)\n\n3.\n\nThe knowledge\/recognition interpretation \"regards it [the internal thesis] as the claim that one's knowledge of yi \u7fa9 derives from certain features of the heart\/mind [xin] \u5fc3.\" (Shun 1997: 99)\n\nWhile admitting that these three interpretations \"are related and Mencius may have subscribed to all three,\" Shun Kwong-loi rejects the first two in favour of the third (Shun 1997: 99). In challenging such a position, Liu Xiusheng argues that the rejection of the first two interpretations is groundless, and \"the third interpretation alone is, at best incomplete in explicating Mencius' internalist thesis\" (Liu 2003: 158). Accordingly, it is necessary to incorporate all these three interpretations in order to develop a defensible understanding of Mencius' position. As an alternative, he integrates these three interpretations into his \"Mencian internalism\" which claims:\n\n1.\n\nMencius is an internalist in terms of the relationship between moral judgment and motivation.\n\n2.\n\nMencius, however, is not an irrealist internalist and in particular not an emotivist; he is a moral realist.\n\n3.\n\nMencius is not a metaphysical moral realist who claims evidence-independent moral reality. He claims that moral reality is conceptually tied to certain human sensibilities. That is, Mencius is a realist of the sensibility theory kind. (Liu 2003: 158)\n\nAs a settlement of this debate, first of all, our understanding of the four beginnings as pure feelings can justify the claim that \"internal\" means \"necessarily involves motivation.\" For the point of Mencius' analogy is that as the desire to eat the roast meat necessarily motivates me, in showing respect to an elder, the motivating factor lies in me. In Kantian terms, this means that feeling functions as the Bewegungsgrund of morality. As Kant said, \"Respect for the moral law is therefore the sole and undoubted moral incentive\" (Kant 1956: 81). However, for us, in distinction from Hume, Kant, Shun Kwong-loi, and Liu Xiusheng, \"feeling\" here refers to \"pure feeling.\" Since yi \u7fa9 is non-sensible, it can only be related to \"pure feeling.\" Secondly, our \"ontological\" understanding of Mencius' xing \u6027 as Being enables us to make sense of the nature interpretation. For obviously the Being of the human being is different from that of the horse. More precisely, there is an ontological difference between human beings and horses. So in terms of such an ontological difference, one can understand why treating an old man as old is different from treating an old horse as old. Finally, our clarification of the \"transcendental\" status of pure feeling helps us to understand that for Mencius, like for Kant, moral actions are \"done wholly out of respect for duty and not from aroused feelings.\" (Kant 1956: 88) This also explains the possibility of the point rightly emphasized by Liu Xiusheng that \"It is not the old person who is Yi [yi] \u7fa9 (as an adjective in Chinese); rather it is the person who respects the old person who is Yi [yi] \u7fa9\" (Liu 2003: 158). Mencius would definitely welcome Kant's following thesis as expounded by Heidegger: \"respect constitutes the essence of the person as the moral self\" (Heidegger 1962b: 163). On the other hand, there is no need for us to commit to the so-called \"Mencius internalism.\" Nevertheless, we agree with Liu Xiusheng's observation: \"Though not an emotivist, Mencius does believe that characteristically human feelings play a role in the moral enterprise\" (Liu 2003: 15). So, when these feelings are understood as pure, it is possible for us to account for both the subjectivity and objectivity of ethical values. In sum, the major error common to Liu Xiusheng and Shun Kwong-loi is the missing of the possibility of Mencius' four beginnings as \"pure feelings.\"\n\nFinally, Mencius' thesis that everybody can become Yao or Shun demonstrates the egalitarian spirit of Confucian ethics. At this point, even Xunzi has to follow Mencius. Despite his route on the way to becoming a sage being different from that of Mencius, Xunzi also stresses that everyone can become a sage (Chan 1963: 54, 133). But unlike Mencius, Xunzi fails to provide a necessary guarantee for his claim. For Xunzi's ground remains empirical and naturalistic. The concept of pure feeling is clearly absent in Xunzi.\n\nFrom a critical standpoint, one might wonder why Mencius has to stress that moral activity aims to reveal xing \u6027 and Heaven. Indeed, the rise of Confucianism signifies a humanistic turn in the development of Chinese philosophy. Like Socrates, Confucius shifts the focus from nature to society. The major concern of Confucius is human affairs. As Zigong \u5b50\u8ca2 remarked on the master, \"one cannot get to hear his views on xing \u6027 (Being) and the Dao of Heaven\" (Lau 1970: 41; with modification). At this juncture, one might recall what Leo Strauss said about Socrates' turn in Western philosophy: \"Contrary to appearance, Socrates' turn to the study of human things was based, not upon disregard of the divine or natural things, but upon a new approach to the understanding of all things\" (Strauss 1953: 122). If Strauss' remark can be equally applied to Confucius, then one might understand why Mencius' thesis is not deviant. Indeed, it is nothing accidental for Mencius to quote \"Great Declaration\" in saying that \"Heaven sees as my people see; Heaven hears as my people hear\" (Mencius 5A5). In Mencius' eyes, the Dao of Heaven is reflected in the hearts of the people. In ancient China, there was a lack of the concept of natural rights. In order to set a rational control on the ruler, Mencius could at best assert: \"It was Heaven that gave the empire to him\" (Mencius 5A5). As the Dao of Heaven is reflected in the hearts of the people, it is important for a ruler to be accepted by the people. Therefore, a ruler must keep in mind the principle that \"To preserve one's mind and to nourish one's nature is the way to serve Heaven\" (Mencius 7A1). Unlike the purely metaphysical concern of Song-Ming Neo-Confucians, Mencius might be more interested in the political implications of the concept of the Dao of Heaven. Granted the primacy of pure feeling, Mencius' thesis of revealing Heaven by moral praxis can also give rise to a concept of cosmic feeling. It is in terms of such a cosmic feeling that Mencius is able to claim that \"All the ten thousand things are there in me\" (Mencius 7A4). The former Kings can function as the models, for they were able to reveal Heaven with such a cosmic feeling. This means that they well understood the hearts of the people.\n\n## 4 Mencius' Political Philosophy and Its Implications\n\nMencius' political philosophy basically results from an extension of his doctrine of moral feelings. As he told the rulers, \"If you let people follow their feelings (original nature), they will be able to do good\" (Mencius 6A6). When Xunzi later attacked Mencius, one of his arguments was based on the concept of natural feelings. Xunzi's point is that \"to follow man's nature and his feelings will inevitably result in strife and rapacity, combine with rebellion and disorder, and end in violence\" (Chan 1963: 128). Mencius would blame Xunzi for overlooking the possibility of pure feelings. What Xunzi understands by \"feeling\" only refers to natural feeling implicit in Gaozi's conception of human nature as what is inborn. That is to say, Xunzi is still imprisoned in the naturalist concept of feeling (this reinforces our thesis that it is misleading to characterize Mencius' doctrine of feeling as a moral psychology). As a result, Xunzi fails to recognize the possibility that when people keep to their ontological nature, i.e., pure feeling, they would love excellent virtues and thereby establish a harmonious society. Like Gaozi, Xunzi reduces feelings to desires. However, when Mencius speaks of feelings, they are \"feelings proper to the originally good nature of man\" (Chan 1963: 54). It is due to Xunzi's overlooking of the difference between pure and sensible feelings that he identifies feelings as the source of evil desires. Nevertheless, in order to account for the possibility of the first sage, it is necessary to start with the pure feelings intrinsic to the originally good human nature.\n\nFor Mencius, it is the original goodness of human nature that makes possible a \"politics of sentiment.\" That is the reason why he said,\n\n> When a government that cannot bear to see the suffering of the people is conducted from a mind that cannot bear to see the suffering of others, the government of the empire will be as easy as making something go round in the palm. (Mencius 2A6)\n\nIn a concrete case, Mencius reassured King Xuan that given his feeling for the ox, the king can definitely feel for his people. In accounting for this story, Ihara maintains that King Xuan's \"sympathetic response was itself psychologically based upon the ox's resemblance to an innocent man being taken away for execution\" (Ihara 1991: 51; my italic). However, if Ihara's account is justified, then Mencius' effort would be in vain. For this would rather confirm Gaozi's thesis of the externality of righteousness. When Mencius concludes that the starting point of benevolent governance is the governance that cannot tolerate the suffering of the people, the sympathetic response is rather issued from the four beginnings as pure feelings. This also implies that the concept of \"the hearts of the people\" occupies a central position in politics. As is shown in the past, \"Jie \u6840 and Zhou \u7d02 lost their empires because they lost the people and they lost the people because they lost the hearts of the people\" (Mencius 4A9). Therefore, to win the hearts of the people is the fundamental principle of governing. But how can one win the hearts of the people? Mencius' answer runs as follows: \"It is to collect for them what they like and do not do to them what they do not like\" (Mencius 4A9). Clearly, this is an extension of Confucius' golden rule to the political realm. For Mencius, this fundamental principle works because \"The people turn to the humanity [of the ruler] as water flows down and as beasts run to the wilderness\" (Mencius 4A9). Such an analogical representation reminds us that the original goodness of human nature is the ultimate foundation for Mencius' political philosophy. This also shows that apart from pure feelings, Mencius' benevolent government would become impossible.\n\nNowadays, Mencius' thesis of the priority of humanity and righteousness over profit in governing seems to be naive and impractical. But it would be mistaken to charge Mencius with ignoring the importance of profit. In his eyes, there is nothing wrong for a ruler to love wealth and sex, if he can let his people enjoy the same (Mencius 1B5). What is not acceptable is to override humanity and righteousness through profit-seeking.\n\nIn promoting Confucius' idea of a harmonious society, Mencius not only argued for its existence in the time of Yao, Shun, and Yu, but also tried to reassure its possibility in the original goodness of human nature. For him, \"Yao and Shun practiced humanity and righteousness because of their nature\" (Mencius 7A30). Thus when the ruler is able to imitate what Yao and Shun did before, the idea of government by humanity and righteousness could return to reality. Yao, Shun, and Yu are therefore set by Mencius as the role models in governing the state. Indeed, he reassured, \"No one ever erred through following the example of the Former Kings\" (Mencius 4A1). Nevertheless this does not imply that Mencius urges us to go back to the past. For what a ruler should learn from Yao and Shun is rather their humanity to all people. Methodologically, analogical thinking is also fundamental for Mencius' political philosophy. But this does not imply that Mencius is a political conservativist. His point is rather that one should learn from the history. As he noted, \"The Three Dynasties won the Empire through benevolence and lost it through cruelty\" (Mencius 4A3).\n\nFrom a structural perspective, a harmonious society is an identity of difference: the aspect of identity is represented by humanity, whereas that of difference is represented by righteousness. Mencius rejected the Mohist doctrine of universal love for its ignorance of the distinctions in love implied an elimination of the aspect of difference. He also challenged the Yangists' egoism. For this would undermine the aspect of identity and hence make the state impossible. In reality, both the Mohist universal love and the Yangist selfish love do not go beyond Gaozi's concept of human nature as what is inborn. This also indicates that like Gaozi, they lack the concept of pure feeling.\n\nThe fundamental principle of Mencius' political philosophy can be summed up as follows: \"[In a state] the people are the most important; the spirits of the land and grain (guardians of territory) are the next; the ruler is of slight importance\" (Mencius 7B14). Following this principle Mencius even grants legitimacy to the people's overthrow of the tyrant. This constitutes the most innovative and provocative step in Chinese political philosophy. However, one might not thereby agree with some modern scholars such as Chan Wing-tsit in claiming that \"it also made him the greatest advocate of political democracy in Chinese history\" (Chan 1963: 50). Undeniably, for Mencius, without the support of the people no rulership can last long. \"Therefore,\" as he said, \"to gain [the hearts of] the peasantry is the way to become an emperor\" (Mencius 7B14). Mencius might be able to introduce the principle: \"for the people.\" But he never proclaimed the idea: \"of the people\"; he rather granted the ownership of the state to the ruler. Following the Book of Odes, he believed: \"There is no territory under Heaven which is not the king's; there is no man the borders of the land who is not his subject\" (Mencius 5A4). It would be also too modern to claim that he conceived of revolution as a \"right.\" More importantly, he missed the idea: \"by the people.\" The goal of revolution in Mencius' sense is to replace a tyrant with a humane ruler, rather than with an elected government from the people. Therefore, as Hsiao Kung-chuan \u856d\u526c\u6b0a observed, \"Mencius' theory of the importance of the people differed from modern democracy\" (Hsiao Kung-chuan 1971: vol. 1, 161).\n\nOn the other hand, it is interesting to explore Mencius' legacy in overcoming the problem of modernity. Generally, thinkers such as Strauss claim that modernity gives rise to nihilism. For these opponents of modernity, the crisis is basically shown in the elimination of morality. In the debate on modernity, there seems to be a tension between good and basic rights. In granting a priority to rights over good, liberalism has tried to relegate morality to the private realm. As a result, there is a tendency to exclude morality from the political dimension. For the anti-modernists, this signifies the disappearance of the distinction between good and evil. To be sure, Mencius' political philosophy remains pre-modern. But given his primary concern with the feelings of the people, he would welcome modern democracy. Certainly, he has to give up the idea of a sage-king. However, this does not necessarily imply any retreat of morality from the political dimension. Instead of seeing good and rights in competition, Mencius would grant an equal status to them. Moral good in the Confucian sense is universalistic. That is to say, it is not specific to Chinese culture. The thesis that human nature is originally good is not limited to the Chinese.\n\nFrom Mencius' standpoint, what is wrong with modernity is the forgetfulness of the subject of pure feelings. For modernity only sees the essence of subjectivity in rationality. As is shown in John Rawls' theory of justice, the subject in the \"original position\" is merely capable of making rational choice (see Rawls 1971). On the other hand, in granting a primacy to communicative rationality, Habermas founds an ethics only upon the normative presuppositions of a rational discourse. For the discourse ethics, subjects are primarily rational speakers. But Habermas seems to ignore that the argumentative approach is only one form of justification. Indeed, moral feeling can play the role analogical to perception in the non-discursive form of justification (see Chan Wing-cheuk 1987).\n\nTo the extent that modernity grants a primacy to the distinction of reason and sensibility, the subject is understood as the rational subject. For the post-modernists, the subject is understood as the subject of desire. As Deleuze and Guattari pointed out, \"Desire...is, rather, the subject that is missing in desire\" (Deleuze and Guattari 1983: 26). This indicates that the post-modernists are not yet entirely free from the premise of modernity. What is overlooked by both modernity and post-modernity is the possibility of pure feeling and its priority to the division between reason and sensibility. They accordingly fail to recognize the subject of pure feelings. At this juncture, Mencius' concept of human nature as pure feelings might provide an alternative in overcoming the problem of modernity.\n\nAccording to Strauss, the symbol of modernity is \"the Beast Man as opposed to the God Man: it understands man in the light of the sub-human rather than of the superman\" (Strauss 1958: 296\u2013297). Even in a pre-modern period, Mencius is able to anticipate such a possibility. He said, \"This is the way the common people: once they have a full belly and warm clothes on their back they degenerate to the level of beasts if they are allowed to lead idle lives, without education\" (Mencius 3A4). What Mencius meant by education is, in the first place, moral cultivation. The aim of such education is to reactivate our moral feelings. Now we can understand why Mencius said, \"Slight is the difference between man and the brutes. The common man loses this distinguishing feature, while the gentleman retains it\" (Mencius 4B19). But Mencius would also agree with Strauss in stressing that, \"Human nature is one thing, virtue or the perfection of human nature is another\" (Strauss 1953: 145). The problem of how to fully develop human nature so as to assist the overcoming of the problem of modernity gives rise to a pressing task for contemporary Confucians.\n\nIn his interpretation of Mencius, Mou Zongsan opposes Zhu Xi's identification of the four beginnings as feelings. For Mou Zongsan, the four beginnings are rational (see Mou 2004: 6). As a result, he refuses to identify them as feelings. Like Zhu Xi, he tends to understand feeling only as psychological. Particularly, he fails to appreciate that in stating: \"When one does not please one's parents, one cannot be a man\" (Mencius 4A28), what Mencius has in mind is primary an ethical feeling. For Mencius, such an ethical feeling has a political implication. That is the reason why he said, \"Once the Blind Man was pleased, the Empire was transformed\" (Mencius 4A28). More importantly, if one ignores the significance of pure feelings, then the potentiality of Mencius' thought in overcoming the crisis of modernity would be overlooked.\n\nAs a conclusion one can say that Mencius was highly skilful in argumentation, though he confessed: \"I am not fond of disputation. I have no alternative\" (Mencius 3B9). Despite Xunzi's challenge, his thesis of the original good of human nature has become the orthodox in the later development of Confucianism. In the field of Confucian ethics, he emphasizes the role played by the conscience. To this extent, he shifts from Confucius' stress on the primacy of the community to that of immanence. But this does not imply that Mencius commits the fallacy of solipsism. Neither does it imply that he hence reduces man to an apolitical being. On the contrary, with his doctrine of human nature as pure feeling, he is not only able to do justice to the \"transcendental\" status of affectivity, but also able to develop a politics of sentiment. To this extent, he promotes the development of Confucian political philosophy. One might not necessarily agree with Mou Zongsan's thesis that Mencius' doctrine of nobility of Heaven gives rise to a Confucian perfect teaching (see Mou 1985: xii). But Mencius' slogan that everyone can become a sage shows his affinity with the ek\u0101y\u0101na Buddhism. His doctrine of moral praxis particularly plays a key role for the rise of the Southern School of Chan Buddhism. Both emphasize the importance of the idea of self-power. Although his ideal of government by humanity and righteousness remains utopian, his emphasis on the importance of sentiment of the people is a legacy for the promotion of democracy in China as well as for a possible overcoming of the global crisis of modernity. Historically, Mencius was identified as the pioneer of the School of Xin \u5fc3 (Mind) founded by Lu Xiangshan and Wang Yangming in Song-Ming Neo-Confucianism. However, this school seems to ignore the key role played by pure feelings. In contrast, Liu Zongzhou was able to link feeling to Being (xing \u6027). At this juncture, unlike Zhu Xi, he understood feeling as pure, rather than sensible (see Tang Chun-i 1975: 305\u2013331). Given the central position of pure feelings in Mencius' thought, Liu Zongzhou might be the most faithful disciple of Mencius. As far as the modern research on Mencius is concerned, there are two major paradigms. In the Chinese circle, there is a paradigm founded by Mou Zongsan. According to this paradigm, the four beginnings are understood as rational. That is, Mencius' pure feeling is reduced to reason. In Western scholarship, David Nivison initiates a paradigm of moral psychology (see Nivision 1996). In this paradigm the four beginnings are interpreted as sensible emotions. Nonetheless, a proper understanding of the Mencian objection to Gaozi's conception of xing as what is inborn and Xunzi's naturalism shows the limitation of the paradigm of moral psychology. On the other hand, even Mou Zongsan finally recognizes that commiseration is a sort of feeling. This brings forth a challenge to the paradigm of strict rationalism. Our above exposition should have shown that these two paradigms suffer from undermining the autonomy of pure feeling. Positively, it points to a paradigm shift towards the phenomenology of pure feeling. This new paradigm might have already been anticipated by Liu Zongzhou.\n\nReferences\n\nAmes, R. 1991. 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Mencian hermeneutics: A history of interpretations in China. New Brunswick\/London: Transaction Publishers. (A historical survey of the Chinese interpretations of Mencius).\n\nIhara, Craig. 1991. David Wong on emotions in Mencius. Philosophy East and West 41(1): 45\u201353.CrossRef\n\nIvanhoe, Philip J. 2002. Ethics in the Confucian tradition: The thought of Mengzi and Wang Yangming, 2nd ed. Indianapolis: Hackett Publishing Company. A sinologist approach.\n\nKant. 1956. Critique of practical reason. Trans. Lewis White Beck. Indianapolis: Bobbs-Merill.\n\nLau, D.C. \u5289 \u6bbf\u7235. 1970. Mencius. Harmondsworth: Penguin. (A philosophical translation).\n\nLi, Minghui \u674e\u660e\u8f1d. 1990. Confucianism and Kant \u5112\u5bb6\u8207\u5eb7\u5fb7. Taipei: Lianjing. (A Kantian approach).\n\nLiu, Xiusheng, and P.J. Ivanhoe (eds.). 2002. Essays on the moral philosophy of Mencius. Indianapolis: Hackett Publishing Company. A moral-psychological approach.\n\nLiu, Xiusheng \u5289\u79c0\u751f. 2003. Mencius, Hume, and the foundation of ethics. London: Ashgate. (A Humean approach).\n\nMou, Zongsan \u725f\u5b97\u4e09. 1968. Mind as substance and nature as substance \u5fc3\u9ad4\u8207\u6027\u9ad4, Vol. 1. Taipei: Zhengzhong.\n\nMou, Zongsan \u725f\u5b97\u4e09. 1985. Theory of perfect good \u5713\u5584\u8ad6. Taipei: Xuesheng.\n\nMou, Zongsan \u725f\u5b97\u4e09. 2004a. Lectures on Mencius \u5b5f\u5b50\u6f14\u8b1b\u9304, IV. Legein 351: 3\u201310.\n\nMou, Zongsan \u725f\u5b97\u4e09. 2004b. Lectures on Mencius \u5b5f\u5b50\u6f14\u8b1b\u9304, VI. Legein 353: 2\u20138.\n\nMou, Zongsan \u725f\u5b97\u4e09. 2005. Lectures on Mencius \u5b5f\u5b50\u6f14\u8b1b\u9304, VIII. Legein 355: 43\u201349.\n\nNivision, David. 1996. Ways of Confucianism: Investigations in Chinese philosophy, ed. Bryan van Norden. Chicago: Open Court.\n\nRawls, John. 1971. A theory of justice. Cambridge: Harvard University Press.\n\nShun, Kwong-loi \u4fe1\u5ee3\u4f86. 1997. Mencius and early Chinese thought. Stanford: Stanford University Press. (An analytical approach).\n\nShun, Kwong-loi, and David Wong (eds.). 2004. Confucian ethics: A comparative study of self, autonomy and community. New York: Cambridge University Press.\n\nStrauss, Leo. 1953. Natural right and history. Chicago: The University of Chicago Press.\n\nStrauss, Leo. 1958. Thoughts on Machiavelli. Glencoe: The Free Press.\n\nTang, Chun-i \u5510\u541b\u6bc5. 1975. Liu Tsung-chou's doctrine of moral mind and practice and His Critique of Wang Yang-ming. In The unfolding of Neo-Confucianism, ed. W. Theodore de Bary, et. al. New York: Columbia University Press.\n\nVan Norden, Bryan. 2008. Mengzi: With selection from traditional commentaries. Indianapolis: Hackett Publishing Company. (A philological approach).\n\nWong, David. 1991. Is there a distinction between reason and emotion in Mencius? Philosophy East and West 41(1): 31\u201344.CrossRef\n\nXiao, Yang \u856d\u967d. 2006. When political philosophy meets moral psychology: Expressivism in the Mencius. Dao: A Journal of Comparative Philosophy 5(2): 257\u2013271.\n\nZhu, Xi \u6731\u71b9. 1986. Collected works of Zhu Xi \u6731\u5b50\u8a9e\u985e. Taipei: Wenjin.\n\nFootnotes\n\n1\n\nFor more recent investigations on Mencius' philosophy, please see: Shun Kwong-loi 1997; Huang Chun-chieh 2001; Alan K. L. Chan 2002; Liu Xiusheng and P.J. Ivanhoe 2002; Liu Xiusheng 2003. I would like to express my gratitude to Dr. Chang Chan-yuan and Miss Chen Yi's assistance. Regarding the quotations from Mencius' original text, I have followed either the English translation in Chan Wing-tsit 1963 or in Lau 1970. Occasionally, I may make an amendment of their translations.\n\n2\n\nSuch a connection was noted by D. C. Lau (Cf.: Lau 1970: 235\u2013263). But no one has thus far systematically analyzed these arguments from the standpoint of the Mohist logic in a complete manner.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_8\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 8. Xunzi as a Systematic Philosopher: Toward Organic Unity of Nature, Mind, and Reason\n\nChung-ying Cheng1\n\n(1)\n\nDepartment of Philosophy, University of Hawaii at Manoa, Honolulu, Hawaii, USA\n\nChung-ying Cheng\n\nEmail: ccheng@hawaii.edu\n\nAbstract\n\nTo understand Xunzi \u8340\u5b50 (298\u2013238 BCE) requires us to read through and think through all his essays (32 chapters in his collected works) in totality and to recognize the major theses he has developed through a train of thinking which relates one thesis to another. It is to be recognized that Xunzi is a systematic philosopher who follows Confucius in trying to reach \"a thread of penetrating unity of my way\" (wudao yiyi guanzhi\u543e\u9053\u4e00\u4ee5\u8cab\u4e4b). But he is also like Mencius in responding to issues of his times in order to reach a better solution of these issues in light of his political ideal for a well-ordered society. It is therefore only when we see his time and his life as contexts of understanding that his approach and his views can be better understood and their significance can be better appreciated. For example, why does he assert that human nature is bad in opposition to Mencius' position that human nature is good? Why does he advocate following both later kings and ancient sages? Why does he offer a new and detailed reconstruction of language and names for developing political governance and social control? Of course I am not here to suggest a merely intellectual historical approach to Xunzi. On the contrary, I wish to bring out the theoretical and systematic side of Xunzi which is necessary for understanding fundamental issues of nature at large, human nature, mind, language, human society, and human government. These are the philosophical issues which came to the forefront in Xunzi's time and demanded a rethinking and re-evaluation.\n\n## 1 Introduction\n\nTo understand Xunzi \u8340\u5b50 (298\u2013238 BCE) requires us to read through and think through all his essays (32 chapters in his collected works) in totality and to recognize the major theses he has developed through a train of thinking which relates one thesis to another. It is to be recognized that Xunzi is a systematic philosopher who follows Confucius in trying to reach \"a thread of penetrating unity of my way\" (wudao yiyi guanzhi \u543e\u9053\u4e00\u4ee5\u8cab\u4e4b). But he is also like Mencius in responding to issues of his times in order to reach a better solution of these issues in light of his political ideal for a well-ordered society. It is therefore only when we see his time and his life as contexts of understanding that his approach and his views can be better understood and their significance can be better appreciated. For example, why does he assert that human nature is bad in opposition to Mencius' position that human nature is good? Why does he advocate following both later kings and ancient sages? Why does he offer a new and detailed reconstruction of language and names for developing political governance and social control? Of course I am not here to suggest a merely intellectual historical approach to Xunzi. On the contrary, I wish to bring out the theoretical and systematic side of Xunzi which is necessary for understanding fundamental issues of nature at large, human nature, mind, language, human society, and human government. These are the philosophical issues which came to the forefront in Xunzi's time and demanded a rethinking and re-evaluation.\n\nIt must be pointed out that in recent Western (American) studies of Xunzi it is Antonia Cua who has paid most serious attention to Xunzi by both articulating and analyzing Xunzi in the modern language of ethical thinking and using Xunzi as the basis for understanding new dimensions of modern ethics. In this sense Cua can be said to provide a good example and a good key for studying Xunzi as a systematic philosopher. This is also the way we are going to treat Xunzi: we see Xunzi as a systematic philosopher. Given this initial basis for understanding Xunzi and for that matter for any philosopher in the classical period, we can now assert that Xunzi has developed his system of philosophy over a period of time starting with concern with the distinction and relation between heaven and man, learning and knowledge, moving to issues of political governance, reflecting his understanding of the Confucian project as reconstruction of li \u79ae and yi \u7fa9. But this project requires three basic considerations of humanity for development. The first is an understanding of human nature and human mind. It is important to see how mind and nature are to be distinguished in Xunzi whereby human nature can be said to be bad whereas human mind can become perceptive, reflective, and develop critical powers of understanding and evaluation, providing moral norms of relevant and correct action. The second is an understanding of nature at large as cosmic resources to be explored and learned for the benefit of developing human facilities. Xunzi comes to a more positivist account of heaven based on our rational ability to know and to valuate. The third is an understanding of names in relation to reality. Xunzi develops a philosophy of language whereby we can see how we come to share common norms of values and action so that we can act in unison and achieve communal order of harmonious and productive forms of living.\n\nThe development of language is so essential that Xunzi believes that our proper understanding and use of language is the medium and foundation for achieving efficient political governance and social order. It is true that we need adequate knowledge of nature at large and proper understanding and cultivation of human nature and human mind for reconstructing social order and political authority which give rise to unity of the world and peace under heaven. But this cultivation is possible and is to be only ensured on the basis of the proper understanding and use of language as a tool and as a vehicle for establishing functionality of social order and legitimacy of political governance. This is a new attempt to reconstruct, but not to restore, the li \u79ae (ritual and social ethics), which implies a new sense of li and a new political philosophy based on deeper recognition of the human mind and human nature. It is on this basis that Xunzi is capable of criticizing Mencius on one hand, and on the other, gives inspiration to his disciples (such as Han Fei \u97d3 and Li Si \u674e\u65af) who unfortunately went to the extreme of legalism which instrumentally conduced to the unification of China, but which also generally undermined people's trust in law.\n\nWith this understanding of Xunzi we may see how Xunzi could fit into the classical philosophy of Confucianism by seeing how it can be said to have fruitfully developed the social philosophy of Confucius, and to a great extent, to have replaced or replenished the Mencian model for establishing a political government of the wangdao \u738b\u9053 (the king's way) with wangzhi \u738b\u5236 (system of the king). Yet it must be also pointed out that he could integrate Mencius as part of his system without having to reject Mencius as he appears to do In this sense his philosophy of logic, language and political governance could be still enriched and enhanced for a more comprehensive theory of nature, mind and politics. Given this new perspective, we are able to find a high level of interpretive relevance and practical applicability of Xunzi's philosophy in today's China and the contemporary globalized world.\n\n## 2 Starting with Antonia Cua\n\nAlthough Xunzi's philosophy has been variously described and examined in histories of Chinese philosophy and introductions to English translations of the texts of Xunzi, no full systematic study of Xunzi has been attempted in modern times. We must however credit Antonia Cua (1933\u20132007) with being the first contemporary scholar to undertake a study of Xunzi's moral philosophy from an analytical point of view. Although Cua's study cannot be said to cover all the aspects of Xunzi, he has made Xunzi an outstanding center of attraction for contemporary Chinese philosophy and in some important sense secured a central and unique place of Xunzi in the classical Confucian tradition. Cua begins his study of Xunzi with his work Ethical Argumentation: A Study in Hsun Tzu's Moral Epistemology in 1965. From then on Cua has derived many insights from his study of Xunzi in his work Moral Vision and Tradition: Essays in Chinese Ethics in 1998. In his 2005 book Human Nature, Ritual and History Cua eventually came to formulate a philosophy of human nature of Xunzi and dealt with many other topics in connection with Xunzi's ethical and religious philosophy. He comes to endorse a position similar to Tang Chun-yi in considering the religious aspect of Xunzi as an extension of the ethical philosophy of Xunzi and speaks of spiritual beings as our own creations (wei \u507d) or as \"supervenient qualities of our reflective ethical experience, the resultant attributes of the expression of our ethical emotions or thought in the context of religious observances\" (Cua 2005: 189), relating to W. D. Ross's book The Right and the Good. Following the threads of his thinking on Xunzi, one may reasonably come to see how a fuller systematic study of the philosophy of Xunzi is necessary for the purpose of covering all aspects of Xunzi's position as rooted in some sort of unity of virtue and ritual.1\n\nWith this understanding, it is fair to say that Cua has opened the study of Xunzi in the contemporary contexts of analytical philosophy. His communication with me in the early 1970s on Confucian ethics provides some basis for this new departure and adventure, clearly a result of attraction by Xunzi's own analytical and discursive approach. What Cua does is to further conceptually analyze the key notions of Xunzi and reach for a formulation of the main theses of Xunzi boosted with both logical and linguistic arguments. These are important methods Cua has brought to bear on the study of Xunzi which nevertheless conformed with my earlier proposal of making a rational reconstruction of the Chinese philosophy for modern and Western readers. Cua is quite successful in doing this and thus demonstrates the effectiveness of conceptual analysis as a method of study of Chinese philosophy texts. But one needs also point out that it is due to Cua's own concern with contemporary ethics that he was able to bring contemporary concepts of ethics to bear on the Chinese texts.\n\nIn a sense Cua has made a philosophical hermeneutical study of Xunzi as it is a \"from-interpretation\" in contrast with a \"to-interpretation,\" which requires comprehensive systematic understanding.2 Not quite touching on the ontological and cosmological issues of heaven in Xunzi, Cua has come to see the human being as primarily an ethical animal which requires community as the basis for its fuller development. As we shall see in the following discussion, it is important to see how Xunzi could be differently understood in his conception of human development in a systematic framework of his distinction and relation between heaven and man and in contrast with Mencius. Yet we can see how the conceptual analysis by Cua could be complemented with a holistic understanding of man in Xunzi from a systematic philosophy point of view as advocated by me.\n\n### 2.1 Distinction and Relation Between Heaven and Human\n\nIt was said that there were 322 essays by Xunzi in circulation at the time of 32 BCE when Liu Xiang came to edit and emend Xunzi. Finally, 32 essays were selected and authenticated as genuine representations of Xunzi, which is our current reader for Xunzi.3 On reading through Xunzi and in light of our understanding of his biographical background there is little doubt that Xunzi is a systematic philosopher, who has thought through almost all the philosophical major issues of his time and has written and argued for a well-conceived idea and end of human society and community, which consists in the development of society and community according to ritual and reason (to be called liyi \u79ae\u7fa9), principle and law (to be called lifa \u7406\u6cd5), in which not only no strife and conflict would arise to disturb the productive relationships of human persons but human potential as human beings will be fully realized as a consequence.4 In this sense Xunzi is both a social idealist and a social realist, and his social idealism of lifa society is based on his social realism of learning (xue \u5b78), education (jiao \u6559) and institutional and policy control (wangzhi \u738b\u5236) with a rational reflection and logical persuasion toward understanding the way (zhidao \u77e5\u9053) as both a method and an end.\n\nGiven this brief sketch of the framework and content of his philosophical thinking, we may raise the question, what are the core and most basic concerns of Xunzi's philosophy as a systematic philosophy in which every thesis of his philosophy could be well-related and integrated. To raise this question is to ask for an insight on the basic source idea and starting points of Xunzi's philosophy. This question is important because we could come to know many important concerns and themes of Xunzi's and yet be in need of an understanding of how to premise his philosophy. We need some central concern which throws light on other concerns and leads us to a better grasp of issues and problems of Xunzi in his philosophical thinking in various areas. For example, scholars have been concerned with Xunzi's theory of human nature as bad. Is this a starting point for his philosophy of nature and man? No, we need to look elsewhere to identify a theme that enables us to explain why Xunzi holds human nature to be bad and in what sense of badness.\n\nI suggest that the core and key notion for understanding Xunzi correctly and systematically is his notion of the distinction and relation made between heaven (tian \u5929) and humanity (ren\u4eba). This distinction and relation has been referred as \"tianren xiangfen \u5929\u4eba\u76f8\u5206\" (the mutual distinction of heaven and human).5 Why and how this distinction and separation are possible remains a difficult question. Yet it is quite clear that it is due to this distinction and separation that humans have both freedom and responsibility to develop their intelligence and sense of control over their own destiny, which means their own self-development and self-realization. This distinction lies in the fact that, whereas heaven has its stable regularities which normally do not change in the course of time, humans have the intelligence and cognitive power to explore things of nature for their survival and development. Whereas heaven has a law-like existence, the human being is a teleological being that seeks its own end of well-being and prosperity in nature. But if humans do not pay attention to the regularities of nature and adapt to them, they would not be able to survive or prosper. On the other hand, if human beings develop their cognitive and rational power of knowledge and evaluation, they would then be in a superior position to fulfill their end and potentiality of being. In this sense there is an internal relation of dependence of human on nature or more strictly the relation of dependence of human knowledge on nature. This relationship is well-expressed in the first paragraph of Xunzi's Tianlun \u5929\u8ad6 (Essay on Heaven):\n\n> The natural course of heaven has its regularities, it does not exist for Yao \u582f (the sage king), and it does not perish for Jie \u6840 (the bad tyrant). If a ruler responds [to heaven] with good governance, heaven will produce good fortune; if a ruler responds [to heaven] with disorderly conduct, heaven will produce calamity. If a ruler strengthen his national resources and be thrift in use, heaven would not impoverish his state; if a ruler takes precaution in preparation and act according to season, heaven would not make his people ill; if a ruler follows the law of nature and does not divert, heaven would not cause any harm. (Xunzi 17\u20131) (translation mine)\n\nIn this essay Xunzi may have addressed his view on the distinction between heaven and human to rulers of states and hence his talk of Yao and Jie. But this distinction no doubt applies to the individual person. An individual person must adjust and adapt to nature for his survival and his prosperity. In fact, the whole human species must pay attention to natural laws of cause and effect in human action. If one does not work hard and yet spends without restraint, one cannot count on heaven to become rich. If one does not save in resources of life and act out of order with time, one cannot count on heaven to become secure. If one betrays the way of nature and arbitrarily pursue one's desires, one cannot count on heaven to succeed. In this sense humans are masters of their own lives and cannot depend on heaven to become prosperous without making their own efforts.\n\nAs one can see, it is due to the law of natural causation that one's efforts are rewarded with success, while one's laziness is rewarded with loss. One may even say that it is nature or heaven that makes the hard working person succeed and the lazy person lose out. After all, humans are not separated from heaven; even heaven has its own stable regularities, for as human beings we depend on heaven for making our efforts in developing ourselves a matter of gain and our lack of such development a matter of loss. This view of the power of heaven is fully compatible with Xunzi's idea of the distinction and relation between human and heaven. Heaven's stable regularities have potentialities to make a difference to the well-being of the human person and the human community. It can be seen that community formation and societal development, political organization and moral-institutional rule by li and yi are themselves matters of the human use of human intelligence and reason that are conducive to the fulfillment of the ends of humanity. It is in this sense that heaven or nature has provided a basis for human development and human self-fulfillment only if human beings are to recognize this and have the wisdom to take advantage of this recognition. Hence Xunzi says near the end of his Tianlun essay:\n\n> Rather than to make heaven glorified and think of it, would it be better to take heaven as resource and control it? Rather than to follow heaven by praising it, would it be better to take charge of the heaven's order and make use of it? Rather than to look up time and to wait for it, would it be better to respond to time and enable it to serve us? Rather than to let things accumulate by themselves, would it be better to use our capabilities to transform things? Rather than to ponder how to use things, would it be better to organize things and do not let it be wasted? Rather than contemplate on how things arise, would it be better to consider how things grow? Hence to forget the human ability and merely think of heaven, one will lose the reality of ten thousand things. (Xunzi 17\u201344) (translation mine)\n\nXunzi recognizes that heaven has its autonomy of harmonious functioning which is exhibited in the movements of the sun and the moon and the four seasons. Heaven for Xunzi is largely cosmic nature in the sense of having natural powers to produce life and to sustain the growth of things. It lets all things \"have their own harmonies for living and have their own nourishment for becoming\" (ibid). He even conceives of heaven as a power that no one knows regarding its form and everyone knows regarding what it has accomplished. His view on heaven reflects the apparent influence of Daoists like Laozi and Zhuangzi, for his view of heaven eventually makes heaven a matter of the dao, which one cannot identify in concrete form but which one can experience in the concrete things dao produces and sustains. In this powerful announcement, Xunzi makes it clear that heaven has the resources to be utilized by humans and humans' ability to utilize heaven is a way to approach heaven and a way of development of heaven for the development of humans. In this sense one sees that there is also a kind of unity between heaven and humans, not in the direction of emotional attachment to an object of worship, nor in the direction of mere cosmological understanding or metaphysical contemplation, but in an effort to engage heaven for the actualization of heaven's power in order to fulfill the well-being of the human as an individual, as a community and as a state.\n\n## 3 Developing Humanity from Knowledge of Heaven\n\nIn this sense Xunzi shows that heaven is not only the origin and source of human life but also the resource for development of human life, so that not only does human being depend on heaven for its well-being but also heaven depends on human being for making its power and potential felt and effective. If one substitutes the term \"dao\" for \"tian,\" one can easily recognize an element of the Daoist philosophy, whereby dao makes itself available for things to be born and grow without self-assertion. But there is still a major difference between Xunzi and Laozi or Zhuangzi: Whereas Laozi and Zhuangzi want humans to imitate and follow the dao so that they do not exert themselves to accomplish their well-being, Xunzi makes it absolutely clear that a human being needs to undertake utmost development and action like dao in achieving his or her own well-being and accomplishing his or her own life potential. In this regard Xunzi is no doubt more Confucian than Daoist. We may bring out this point more explicitly in later discussions.\n\nHumans need to develop themselves, because, according to Xunzi, the human is a part of nature and in fact the most dynamic and creative part of nature. That is why Xunzi says: \"Water and fire have vital force and does not have life. Grass and wood have life and do not have knowledge, birds and beasts have knowledge and yet have no righteousness. It is human being who has vital force, life, knowledge and also righteousness. Hence man is the most valuable part of the world\" (Xunzi 9.70, translation mine). Hence humans could by nature rise above all things as the best representatives of the potential of heaven. Yet human potential has to be developed as part of nature, and this should enable humans to be truly representative of heaven. It is obvious that Xunzi has developed the insight of Confucius: \"It is not the way that glories humans; it is humans who are capable of glorifying the dao\" (Analects 15.29). What Confucius considers to be dao is dao of heaven, for it is heaven that created this world and humans and there is a way in such creativity. Being conscious of their own creative potential, it is the humans' role to continue this cosmic creativity and to make creation of human culture and civilization possible. To do this can be said to be a way of developing and glorifying heaven that lies in the actual fulfillment of human potential at the same time.\n\nIn this sense Xunzi has followed the Confucian insight and hence takes the creation of human community and human culture as an absolute must for fulfilling the nature of both humans and heaven. We see how Xunzi has reached his version of the unity of heaven and human in a significantly different way from Mencius. In Mencius one sees a quest for understanding the identity of human in heaven and hence the identity of human in its intrinsic goodness which gives rise to natural morality. For Mencius morality is a matter of \"return to oneself to reach integrity\" (fanshen er cheng \u53cd\u8eab\u800c\u8aa0). Whereas we may think that Mencius, following Zisi, has argued for the identity of human with heaven at the source creativity of heaven, Xunzi argues for the development of human potential for fulfillment of heaven as the end of humans. In this sense his argument for the distinction and relation between humans and heaven is a form of identity of heaven and human with heaven as a means for the end of human development. If we can see Mencius' view as an \"enlightenment theory of human nature,\" we can no doubt see Xunzi's view as an \"achievement theory of human nature.\" We need not see absolute opposition between the two views. Both are initiated by Confucian concern with achieving humanity. Together they form a complementary pair for spelling out the full implication of the Confucian vision of humanity as arising from heaven to develop heaven.\n\nHowever, I wish to make two more important observations regarding Xunzi's view on heaven. First, Xunzi recognizes the natural functions of heaven and the natural functions of humans. What is being natural (tian\u5929) is observed to be by the things themselves without human intervention. It is not to ascribe pre-conceived qualities to things or to heaven. It has nothing to do with essentialism, neither is it a matter of modern anti-realism.6 Thus the movement of seasons and change of climates are part of the natural functions of nature. Humans need not take charge of these functions. Xunzi says that \"achieving by not acting, acquiring by not seeking, this is called the function of heaven (tianzhi \u5929\u8077)\" (Xunzi 17.5). He asserts that humans should not compete with the natural functions of heaven (buyu tian zhengzhi \u4e0d\u8207\u5929\u722d\u8077). In this he makes a point of repudiating Zisi's view on participating in the works of heaven and earth in the Zhongyong. He points out that \"heaven has its own time, earth its own riches, and human has its own ordering power\" (Xunzi 17.5). To recognize these distinctions and do one's own job is called nengcan \u80fd\u53c3 (able to coordinate with heaven and earth). If one goes beyond this, it is only a mere confusion. Here lies the distinction of functions in nature and their proper coordination through man.\n\nClearly, Xunzi regards Zisi and Mengzi's theory of unity of heaven and human as a matter of humans' competition with and usurpation of heaven and earth. Xunzi's theory is significant in light of today's environmental ethical concerns, for according to Xunzi, the human attitude to nature is to use heaven by coordinating with heaven, although there is also an ambiguity in Xunzi's own explanation. He does want humans to explore heaven's powers in order to use them to human advantage. How far can humans go in using heaven without being said to harass nature and cause harm to both heaven and human? Apparently, for Xunzi, there must be a natural limit of such exploration and use, and it is also up to humans to discover this limit.7\n\nBy the same token, human beings have their natural functions, which however need to be developed and transformed by way of knowledgeable adjustment to nature. In other words, a human being needs to know his end of life by knowing the heaven or the dao of heaven which is exhibited in the natural functions of heaven. To know this is to adjust oneself to them instead of trying to change them. In this sense, Xunzi wants to redefine the notion of \"zhitian \u77e5\u5929,\" which Mencius regarded as a matter based on knowing one's own nature (zhixing \u77e5\u6027). In making this redefinition, Xunzi speaks of the natural functions of sense organs and the natural function of the mind as the ruler of the five senses: He says:\n\n> Ears, eyes, nose, mouth and body all have their abilities to contact outside things, and yet they cannot substitute for one another. These are called the natural organs of humans. It is heart which sits in the void of the chest and governs all the five senses and hence is called the natural ruler (tianjun \u5929\u541b). (Xunzi 17.5; translation mine)\n\nFor Xunzi the natural functions of natural organs of humans should be maintained in order as well as developed for better use, and this is how a sage could be distinguished from ordinary people. The latter's natural organs are often darkened, confounded, abandoned and violated and therefore could not fulfill their natural functions to the full. This is a matter of what he latter called bi \u853d (obscurations, see his essay 21, Jiebi Pian \u89e3\u853d\u7bc7 [On Removing Obscurations]). On the other hand, a sage is capable of removing obscurations and gives clarity to his \"natural ruler\" (tianjun \u5929\u541b: heart\/mind), regulates his senses (tianguan \u5929\u5b98), cultivates his natural capacities (tianyang \u5929\u990a), complies with what is required for good governance (tianzheng \u5929\u653f), and preserves his natural feelings (tianqing \u5929\u60c5), so that he can accomplish his natural ends (tiangong \u5929\u529f).\n\nIn order to do this, a sage must begin with learning and reflect on his own power of thinking, cognition, and understanding so that he may become enlightened of what is required of him for developing his natural functions and how this is to be done. He needs to become self-conscious of his powers, his needs and his ends. He needs to know what type of being he belongs to (zhilei \u77e5\u985e) so that he can properly develop himself within the capacities of humans. He needs to be transformed by his own self-cultivation in natural and social contexts so that he may make use of heaven and help others develop by constructing and developing cultural forms and institutional norms and rituals which he calls li and yi. In this sense Xunzi speaks of the distinction between junzi \u541b\u5b50and xiaoren \u5c0f\u4eba. The former is one who respects what he has and does not long for what is in heaven, whereas the latter gives up what he has and longs for what belongs to heaven. The junzi, by developing what he has could eventually become the sage (shengren \u8056\u4eba) who is in a position to develop principles and norms for social and cultural establishment without obscurations in his mind. It is interesting to note that for Xunzi any excess of desires and partial views deviating from a norm for total human development are considered forms of obscurations (for this see his essay 21 Jiebi Pian \u89e3\u853d\u7bc7).\n\n## 4 End of Humans and How to Learn\n\nGiven this clarification of Xunzi's understanding of the distinction and relation between humans and nature, all major propositions of Xunzi's philosophy would fall into order. This understanding in fact provides a framework for orientating humanity and evaluating human worth. It is a framework in which a human person could identify the end of his life and the direction of his efforts and thus find meaning in his life. It is a framework in which one can speak of human rationality and morality according to which humanity can be said to be developed and achieved. One can see that if one wishes to develop oneself, one needs to learn and acquire knowledge of nature so that one can use natural knowledge for avoiding mistakes and achieving understanding and hence making correct judgments for action. One also needs to be sure that one is capable of development of oneself and makes an effort to go beyond oneself in order to master principles, norms, and disciplines so that one may form responsible and reliable judgments of action. One needs also to be awakened to a vision of human community and society on which one's own self-realization depends. One must therefore come to know the status and position (lei \u985e) one's own person so that one can properly and persistently pursue the long-term good of life. A human person will have to learn to know ultimately the ideal end of humanity and strive for its realization.\n\nFor Xunzi, one has to learn from both former sage-kings and latter-day kings and teachers. For it is in the conduct of the sages that one sees the example of developing one's character and transforming one's desires so that one's character would not become obscured or beclouded and thus not fall into narrow channels of selfish egoism and bigotry. It is Xunzi's observation that without efforts to learn and without being engaged in learning from history and experience one simply could not properly use one's organs and form a correct understanding of things outside. One would not be able to have clear judgments as to what is the best for human well-being and thus fail to achieve a better state of development, which makes human life worthwhile, meaningful, and even happy.\n\nThe concern with learning leads Xunzi to deal with basic issues of understanding. One issue is how to come to know the dao or dao of heaven. One reply is that one must remove obscurations one has encountered in one's life. Xunzi has listed ten forms of such obscurations, bi \u853d: likes, dislikes, stressing only the beginnings, stressing only endings, stressing distant things, stressing near-by things, seeing things in the broad, seeing things superficially, stressing the ancient or stressing the present (see Xunzi 21.6). As a matter of fact, since there are all different things and different perspectives, all these differences could becloud the whole comprehensive understanding. Hence in order to know the dao one needs to reflect whether one has been obsessed with one aspect of things or seen things from one point of view or from a presupposed state of feeling. It is natural that one could easily fall into this narrowness of mind and heart. Being an empiricist may not prevent one from taking views narrowly. The question is whether the human mind\/heart could remain open and reflective on positions taken so that one may see other points of view and transform one's view in light of a larger vision of things and a wider horizon of understanding. Without going into details, we can see that Xunzi has suggested three basic approaches for opening one's mind\/heart in order to transcend bi and reach a clear and comprehensive understanding.\n\nOne is to learn from historical examples. One can see how different failure and success stories of rulers and ministers have been useful in overcoming bi in the whole context of history. We can see those stories as simply a matter of trying struggle between bi and ming \u660e (clarity, illumination) which is the opposite of bi.8 That is why to learn from history is so important for humans, for humans must learn from experience, and historical experience is a matter of experience. With learning from history and experience one probably can become enlightened regarding the nature of things and one's ability of judgment will develop so that one could come to know norms among differences of things. Xunzi calls a person who comes to know or form norms among differences of things a sage (\u8056\u4eba shengren), for such a person has demonstrated a great achievement of mind and reason in seeing things clearly, in grasping the ends of things and in forming norms for correct action. With these a sage would be able to hold differences of things without being subject to their mutual beclouding.\n\nAs to how these achievements could happen to a person, Xunzi simply appeals to the enlightening function of mind that would become the basis for knowing the norm of things. We may consider this the second point of learning the dao, which consists in self-cultivation of the human mind-heart. In fact Xunzi holds that the basic function of the mind-heart is to know the dao and it can know the dao because one could cultivate oneself into a state of void, oneness, and tranquility (xuyi er jing \u865b\u4e00\u800c\u975c). He has this to say:\n\n> A human being is born to have perception; with perception one has memory. Memory is a matter of storing and yet it has void, which consists in receiving new things without harming what it has received. Mind is born with knowledge, and to know is to know the difference. The so-called difference is a matter which mind can hold at the same time. Mind has oneness as it does not let one harm the other which is oneness. Mind is such that it dreams in sleep, moves of its own, and plans when motivated. Hence mind cannot be said not moving, yet it has its tranquility, which consists in not letting dreams to disturb its cognitive power. Before acquiring the dao and yet seeking the dao, one must have void, oneness, and tranquility.... Being void, one, and tranquil, this is called great pure illumination (daqingming \u5927\u6e05\u660e). (Xunzi 21.36\u201339; translation mine)\n\nIt is with this state of daqingming that one can come to see things in their comprehensive totality without being trapped into any one partiality. One would be able to come \"to see all things within four seas in one room\" and \"to control great principles (dali \u5927\u7406) and put the whole universe in order\" (Ibid). In other words, one would rise above all different things and acquire a total view of things together with a norm for ordering them and guiding their further development. It is a state where \"all things are displayed and amidst them there is the norm (zhong xuan heng yan \u4e2d\u61f8\u8861\u7109)\" (Xunzi 21.29).9 Xunzi would consider Confucius as actually achieving this state of mind and hence becoming a sage. It is important to note that the so-called heng is what Xunzi has designated as the dao. In this sense dao has two basic functions of organizing and ordering which is more than what a Daoist would regard as the dao. Perhaps we could regard the dao as more epistemological and moral than ontological and cosmological, even though it may originate from a cosmological source as one sees in the natural functions of heaven. This dao therefore is more Confucian than Daoist again: it is the result of comprehensive understanding and penetrating insight into how things could be organized and developed toward a better state from a teleological point of view. In this sense Xunzi speaks of \"mind knows the dao\" (xinzhidao \u5fc3\u77e5\u9053) and hence speaks of \"prescribing according to the dao\" (kedao \u53ef\u9053) because of this knowledge. This knowledge is both descriptive and prescriptive, both empirical and axiological understanding of the dao on the part of the human and his mind.\n\nThere is also the third way of removing one's bi and avoiding future bi and hence knowing the dao is to learn from former and latter kings and sages. Xunzi stresses this point in the last paragraph of his essay on Jiebi. He considers that there are sages who have comprehended all the basic natures of things while there are kings who have comprehended all the basic principles of governance. Here Xunzi speaks from the point of view of accomplished development, not from the viewpoint of seeking development or in the middle of development. From this point of view, the norms are already established and hence an ultimate norm of rightness (longzheng \u9686\u6b63) resulting from these must be followed in determining what is right and what is wrong, what is bad and what is good.10 This could become a point of view of governance by law but it could also become a point of view of authority, which would inevitably lead to legalism and legalistic domination. With this said, it is obvious that the first two approaches are more relevant for understanding how human beings could develop and how an ideal society and government could be still maintained or sustained in an open process of open reflection and open learning.\n\n## 5 Communication and Institutionalization\n\nFrom these major premises of the distinction and relation between heaven and human one could also see how rectification of names and institutionalization of li and yi are important and consequent measures for maintaining and sustaining a developed human community, society, and government. In order to have a sustainable community, society, and government, communication must be maintained with correct use of names and language. This is because, if a human person wishes to fulfill his own potentiality and achieve central values of humanity for survival and prosperity, he needs to achieve solidarity and trust with other people through good communication and thus through proper use of language. Xunzi's strong interest in the issue of \"rectifying names\" (zhengming \u6b63\u540d) consists in recognizing the realist and empiricist foundations of the names and their logical consistency (see Cheng 1969). He wishes to see that names and language be preserved in this consistency and yet can be at any time used to serve the social function of communication and solidarity of human feelings and minds. This matter is therefore dealt with in his essay 22 on rectifying names. Again this shows how Xunzi has conscientiously followed Confucius' suggestion on this issue and therefore can be said to explicate and expound the Confucian doctrine of rectifying names in a more sophisticated and comprehensive manner (see Xunzi 8 and 24).\n\nBased on the theory of rectifying names, Xunzi is able to go one step further in the development of human potentiality for the achievement and flourishing of humanity. This is to explore various ways in which an ideal society and an ideal state can be established. This is actually the main focus of Xunzi's efforts: he wishes to establish an ordered society by a sagely king who is a paragon of human development and hence the center of emulation for all people under his rule. For this reason Xunzi can be seen to have devoted more space and time in arguing for and formulating such an ideal rule. In the present collection of his works, there are 11 essays which can be said to be on this topic of ideal rule. Among the 11 essays, essay 9 on Wangzhi (\u738b\u5236 King's Institutions) stands out as the most comprehensive discussion for the topic of weizheng (\u70ba\u653f doing government or conducting government) in which all aspects of political governance together with organizational issues and administrative-managerial issues are examined, some with illustrations from historical cases. This essay deserves independent analysis and exposition particularly in light of contemporary theories of public administration, political governance, and governmental ethics (see Cheng 2006). It suffices to say that with this essay the political philosophy and social idealism of Xunzi has reached it acme and apex.\n\n## 6 Is Human Nature Really Bad?\n\nOne main purpose for identifying the basic framework of Xunzi's philosophy in terms of the distinction and relation between heaven and human is to show that his theory of human nature needs to be re-evaluated and given a new interpretation. It has been traditionally understood that when Xunzi speaks of human nature as being bad, he has shown a pessimistic attitude toward the quality of human existence and therefore a low esteem for humanity in the world. If human nature is bad, how could we redeem it or trust it in the first place? Perhaps it is for this reason that Xunzi was excluded unfairly from the main stream of Confucianism since the later Tang Period. In contrast, Mencius was conceived to show great confidence in human nature because he regards it as good and as such as capable of achieving goodness in human society. But as matter of fact, we must note that even when Xunzi argues in opposition to Mencius that human nature is bad (renxing e \u4eba\u6027\u60e1), he is more focused on his observation of common desires exhibited in common people for the purpose of arguing why the behavior of the common people needs to be improved. There are three terms in need of logical clarification in his observation: namely, the human person (ren \u4eba), the nature (xing \u6027) of human beings and the bad (e \u60e1). Let us first proceed with the notion of the human person. As an observation, we do see most common people exhibiting the three selfish traits of behavior which Xunzi identified in his essay on Human Nature Being Bad: love of profit, emotions of jealousy and hatred, and desire for sensual pleasures. Hence Xunzi's statement is a generalization on most people who we may regard as xiaoren\u5c0f\u4eba(petite person)11 in contrast with junzi who have overcome their natural desires of such. If such desires can be seen as deeply rooted in human nature, they must not be changed or removed from human nature. The only way to deal with them is to let them go or to control them by means of other human powers.\n\nIt is obvious from Xunzi that we can adopt the latter method of improving the human condition, namely by introducing reason or knowledge as a means of restraining selfish desires. It is obvious that we do have reason and reflective knowledge that can help us overcome our desires. But then, does this ability of reason and reflection belong to our nature as well? Is human nature only confined to average observation? Or maybe human nature does have a depth that is to be revealed in proper moments under the right circumstances? We shall say more about this point on \"nature\" a little later. Finally, what is \"bad\" about human desires and nature? What do we mean by human nature being bad? What is bad, perhaps, is that the first sort of desire, if not checked, will lead one into a situation of conflict and clash. Similarly for the second sort of desire, if not checked, in following it through, one will encounter loss of virtues and morality in an individual and thus cause instability in society. The third sort of emotion, love for sensual pleasures, if unchecked, could create both conflict and strife. In all these three cases, to follow one's desires and emotions (called qingxing \u60c5\u6027 by Xunzi) will not lead and conduce to larger social gain and welfare but instead will result in social disasters and possible political suppression. Hence to say that human nature is bad is equivalent to saying that it can bring bad results or consequences, and that that is undesirable from a political-social-moral point of view. In light of this fact, moral words like \"good\" and \"bad\" in their use have been intended to prescribe what to do and what not to do in order to avoid undesirable results and to achieve desirable ends.\n\nWhat is good is to be accepted and to be done, what is bad is not to be accepted and not to be done. From this point of view, it is possible to see that for Xunzi to say that human nature is bad is to say that certain desires need to be checked in order to avoid undesirable consequences. Or to put it in a different way, to say that a desire when unchecked or restrained is bad is to anticipate some consequences such as disorder and suppression, which are not acceptable and not to be done. Hence we can indeed speak of consequential sense of good and bad as indicated by Tony Cua in his paper on Xunzi's philosophy of human nature (see Cua 2005: 8\u20139). As these consequences have not yet happened in a given situation, we may speak even more precisely of the good and the bad in an anticipatory sense. It is not those feelings and desires that are intrinsically bad (in fact they can be instrumentally good for some purpose under certain conceivable social and economic conditions),12 it is the anticipated results of pursuing them that are considered bad. Hence they are to be regarded as bad as a matter of anticipation of their results. Once we see this anticipatory aspect of the use of the good and bad, we can also come to see another anticipatory aspect of the use of these two terms, namely to say that human nature is bad is to say that it needs to be improved, guided, and refined. If we focus on this latter sense of anticipation for the social-political-moral statement that human nature is bad, there is not much condemnation or pessimism as one may view them in a consequentialist light.\n\nGiven these basic analyses of the three terms \"ren \u4eba\/xing \u6027 \/e \u60e1,\" we can see Xunzi's statement or observation \"renxing e \u4eba\u6027\u60e1\" as a forewarning for potential harm for not developing one's nature and as a reminder of the necessity of development of human nature in light of consequential good and benefit. With this understanding, what would be wrong with Xunzi's statement or his observation? For those who wish to defend Mencius' view that human nature is good, must we say that Mencius is obligated to deny this statement and its practical implication? It is quite conceivable that Mencius would not deny that human nature could be developed and improved, even though he may regard human nature as originally good (good at origin) or intrinsically good. He may consent to Xunzi's statement as an observation and agree with Xunzi in the need for improvement of human nature in a broadened notion of human nature which includes desires as commonly seen.\n\nOne may see Xunzi's retort of Mencius' thesis that human nature is good as an ambiguous reading of the word \"xing \u6027.\" For Mencius human nature or xing is rooted in heaven and earth and will naturally present itself when there are no contravening factors such as desires for certain concrete things. It is true that people in general could desire profit and sensual pleasures and evince hatred and jealousy in concrete circumstances, but it is also true that people in general could desire love and praise for noble acts, spiritual peace and virtue and even show sympathetic benevolence and tolerance under other concrete circumstances. People could naturally hate acts of hatred and dislike jealous outbursts. The passionate-impulsive nature of the human person can be very complex and toward various objects they may react with various emotions and desires, some of which we may regard as anticipatorily bad, and some of which we may regard as anticipatorily good. Hence Mencius and Xunzi need not controvert each other on this matter of evincing of different desires and emotions regarding their passionate-impulsive natures of the human being. In terms of the dispute between Xunzi and Mencius, Mencius could accept what Xunzi says without giving up his observation and theoretical understanding that human nature is good at the origin of its existence. So our next question is whether Xunzi could accept the Mencian view if we may clarify Mencius' position in light of the distinction and relation between heaven and human in Xunzi.\n\nMy response is that we can indeed reformulate Mencius in a way that would suit Xunzi's understanding of human nature in an enlarged sense. What Mencius has argued is that given optimum circumstances all virtuous emotions could be identified in the four beginnings of the virtues (siduan \u56db\u7aef). Take the example of the feeling of sympathy for an innocent about to fall into the well. Do we experience an impulse to run to help? As an empirical statement, the saying that people generally experience such an impulse may carry no weight for proving the beginning of the virtue of benevolence, because the virtue may have to contain both the rational element of justification and the feeling element one experiences and not just the feeling element alone. But one may argue that the feeling element serves as an occasion for discovery of something deeper in human nature so that one may come to rationally adopt the feeling and urge for action as a moral rule and a moral command. With this one can see how Xunzi may regard feelings and desires which are directed to selfishness as also an occasion for rational reflection and thus as a basis for rational development of virtues such as li and yi. As a matter of fact, Xunzi has argued and brought to light the fact that the human mind has the capacity of knowing the way of heaven which is to see balance, harmony, restraint, and wholeness of relations as the normative basis for judgment of moral evaluation and appraisal and hence as the basis for actualizing or developing of a deeper level of human nature.\n\nIn this sense there is no reason why Xunzi may not come to see Mencius as providing a level of humanity which would naturally accommodate his theory of human knowledge of the dao as he indicated in the Jiebi \u89e3\u853d essay. There is no reason why his theory of selfish desires and emotions may not be reconciled with Mencius' theory of pure emotions and will which is also reflected and embodied in his theory of reflective reason and normative dao. Of course there is still a basic difference between the status of reason and dao in Xunzi and the status of emotion and will in Mencius. But this difference only points to the fact that each has come across an aspect of the human mind-heart, which consists in requiring conscious understanding as well as emotional response as factors of determinative power for action. The human mind-heart is an inseparable unit of reason\/emotion and will which enables the human person to know and to act on knowledge and to know via action. It is the way that human nature is structured and composed if we can speak of a reflective understanding of human nature as we wish not only to explain our actions by our desires and emotions and knowledge but we wish to also plan our life along this line of knowledge-emotion and action interaction in a process of defining and growing ourselves as human beings.\n\nWhat is significant in Xunzi is that he sees that human nature is a dynamic process which needs and requires development as it is seen in both historical and theoretical contexts. Humans' relation to nature and heaven is such that we need to move beyond our selfish desires and emotions so that we may transform ourselves into a higher and larger being by awakening our rationality and sensibility to the dao that is embodied in the regularities of heaven. Xunzi's essay on heaven has pinpointed this potential aspect of human nature and it is on this basis that the whole philosophy of social li and yi is to be developed. It is a realistic theory because it is based on observation of the reality of heaven and its way. In point of fact, as we see above in the quotations from Jiebi Pian and Tianlun Pian, Xunzi has presupposed or identified a level of human nature which contains potential good for human development. We may see this potential good as the capacity for checking selfish desires and as the capacity to form divisions of labor and to organize communities and groups for collective productivity and mutual enhancement. They are implicit in the human nature that Xunzi has generally described as yi \u7fa9 as he says that \"Human beings have vital force, life and perception and righteousness\" in the Wangzhi essay. In a sense he has recognized this implicit and deep level of human nature. But unlike Mencius, he may think that one can only get them by way of learning and exposure to historical lessons from the past and the teachings of sage kings. But that one can learn from these resources does not rule out that one can feel the relevance of those levels in one's direct encounter with life. They have to be realized in one's life and experienced in order to be the basis for one's action. As history and moral lessons can be effective, they have a basis in the human feelings that can be said to be a part of the enlarged human nature.\n\nWhat I have said above makes it clear that for Xunzi human nature is not bad in all possible senses. In his use of the bad, bad may come with the good. His idea that because of the lack of good one may desire bad is an instructive one. But the fact that one may aspire for the good on the basis of feeling bad could be said to prove that human nature is indeed good after all for in this good bad is a simply a sign of privation.\n\nWe must further observe that there is indeed a sense of bad which is morally bad not as an anticipatory consequence for the social community, namely the bad in the sense of consciously and intentionally seeking selfish ends at the expense of others with knowledge used for achieving this intention. Xunzi has touched on this sense of intentional badness in his historical examples but he has not discussed it in distinction from the non-intentional badness which is anticipatorily attributed to the natural desires and feelings of most people in their life courses.\n\nAs one final question concerning this clarified theory of human nature in Xunzi, we may ask whether ordinary people can become sages. The answer is that if a sage can become a sage, an ordinary person can also become a sage, for there is no ontological basic difference between a sage and an ordinary person in their basic and extended natures. Xunzi has developed his theory of the human mind that would allow sagely wisdom to develop so that one can speak of everyone knowing the dao. There is no reason why anyone may not learn the dao as an ultimate norm and dao as an ultimate end of life. Xunzi has also provided three major ways of knowing the dao as I have exposed in the above. Hence there is no reason that any person is not capable of becoming a sage. As both a social realist and a social idealist, Xunzi sees his philosophy of human nature as part of his theory of human development in his fundamental framework of the distinction and relation between man and heaven and his consequent views on the human faculty of using language and developing human minds. Hence in the last resort there is a convergence of Xunzi and Mencius in the comprehensive system of reflections of Confucius.\n\n## 7 Conclusion\n\nGiven the above new understanding of Xunzi, which combines conceptual analysis with a holistic systemic integration, one can have a full and balanced view of Xunzi both with regard to his theoretical content and with regard to his position in the classical Confucian philosophy. There are many unique features of Xunzi which reflect Xunzi's experience and reflection on the problem of human self and human person, human nature, human morality, human society, and government. One great contribution lies in his discursive analytic and systematic approach to various topics of humanity, which are essential for understanding individual, society, and government. He takes his observation of life and society seriously and sees history as providing examples and case studies in a theoretical framework. He has thus provided Confucianism with a new perspective which is required for integrating history, theory, and experience. He is also careful in working out a full philosophy of social motivation and political administration based on a moral reflection on the potential and purposefulness of humanity, which has been only programmatically discussed by Mencius and other philosophers. In this sense he has integrated both the tradition of Daxue and Zhongyong and founded his philosophy on both experience and reflection.\n\nApart from methodology and political philosophy, Xunzi has come to present an implicitly full theory of human nature based on experience and reflection and in reference to the relation and distinction between heaven and human. It is he who has critically examined the Zisi and Mencian theory of humanity and its role and disclosed a dimension of reality which needs to be dealt with by ethics and education. In this manner he has provided a justification of human history and culture as resources of morality that is required for the full development of humans, which in turn is essential for human survival and prosperity. In a sense he is able to pull all threads of thinking of Confucius together and re-establish the Confucian vision both as a social-political ideal and as an individually committed realizable project. In this sense he has comprehensively covered both the externalist and internalist issues in Confucianism and gives it a form of completion and organic interrelation. Although he was often understood as a mere naturalist philosopher, he is in fact not. His ideas of dao and heaven, and human mind knowing the dao, provide a basis for reconnecting human and heaven in a richer context than his Confucian predecessors and perhaps even the Daoists.\n\nAlthough Xunzi appears to advocate a philosophy of human nature which many people dislike, it is precisely by focusing on a realistic aspect of human existence that the concept of human nature becomes more dynamic, open and even creative . It is fair to conclude that given our holistic and systematic understanding of Xunzi, there cannot be a complete and integrated understanding of the classical Confucianism without this understanding of Xunzi.13\n\nReferences\n\nCheng, Chung-ying. 1969. Rectifying names (Cheng-Ming) in classical confucianism. In Proceedings of the XXV international congress on orientalists: Essays in Asian studies. ed. Harry Lambly. Honolulu: University of Hawaii Asian Studies Publications. (The first study of the zheng-ming theory in Confucius and Xunzi from logical and semantic perspectives).\n\nCheng, Chung-ying. 1996. On Chinese and Western spirits of philosophy \u8ad6\u4e2d\u897f\u54f2\u5b78\u7cbe\u795e, 3rd ed. Shanghai: Dongfang Chubanshe. (This book discusses aspects of logico-analytic, ontological and pragmatic thinking in contemporary Western philosophy in contrast with those of onto-cosmological, onto-hermeneutic and onto-ethical thinking in Chinese philosophical tradition.).\n\nCheng, Chung-ying. 2005. A confucian-Kantian reflection on mutuality and complementarity: Virtue and law. In Structuren der Macht \u2013 Studien zum politischen Denken Chinas, ed. Harald Holz and Konrad Wegmann, Vol. 13. Muenster: Lit Verlag Muenster. (In this essay the development of ideas of virtue and law in Kant and classical Confucianism shows how Kantian and Confucian philosophies can be said to be complementary in development of a moral and political community of men.).\n\nCheng, Chung-ying. 2006. The C theory: Chinese philosophy of management C \u7406\u8ad6:\u4e2d\u570b\u7ba1\u7406\u54f2\u5b78. Beijing: Renmin Daxue Chubanshe. (This is a philosophical study of administration and governance based on interpretation of five powers theory, to show how one could learn from nature by reflection.).\n\nCook, Scott. 1997. Xun zi on ritual and music. Monumenta Serica 45: 1\u201338. A useful discussion of Xunzi's theory of li \u793c and le \u4e50.\n\nCua, Antonio. 1977. The conceptual aspect of Hs\u00fcn Tzu's philosophy of human nature. Philosophy East and West 27(4): 373\u2013390 (A penetrating analysis of Xunzi's notion of xing \u6027 in regard to both its moral and non-moral implications).CrossRef\n\nCua, Antonio. 1985. Ethical argumentation: A study of Hs\u00fcn Tzu's moral epistemology. Honolulu: University of Hawaii Press. A conceptual and logical analysis and integration of the moral psychological ideas such as xin \u5fc3 and zhi \u77e5.\n\nCua, Antonio. 2005. Human nature, ritual and history. Washington, DC: The Catholic University of America Press. Again a collection of papers on Chinese views on xing \u6027, li \u793c and historical lessons.\n\nGoldin, Paul Rakita. 1999. Rituals of the way: The philosophy of Xunzi. LaSalle: Open Court. This work relates Xunzi's theory of li \u793c to fulfillment of the dao \u9053 or comprehensive order of humanity and society.).\n\nHutton, Eric. 2002. Moral reasoning in Aristotle and Xunzi. Journal of Chinese Philosophy 29(3): 355\u2013384. A good comparison on moral reasoning between Aristotle and Xuanzi in both similarity and difference.CrossRef\n\nIvanhoe, Philip J. 1991. A happy symmetry: Xunzi's ethical thought. Journal of the American Academy of Religion 59(2): 309\u2013322. A useful statement of Xunzi's ethics).CrossRef\n\nKline III, T.C., and Philip J. Ivanhoe (eds.). 2000. Virtue, nature, and moral agency in the Xunzi. Indianapolis: Hackett. A collection of papers on various topics to do with Xunzi.\n\nKnoblock, John. 1988\u20131994. Xunzi: A translation and study of the complete works, Vol. 3. Stanford: Stanford University Press. (A scholarly translation of Xunzi with detailed annotations).\n\nLee, Janghee. 2005. Xunzi and early Chinese naturalism. Albany: State University of New York Press. A treatise on how Xunzi is related to empirical observation of and reflection on nature in early Chinese texts.\n\nMachle, Edward J. 1993. Nature and heaven in the Xunzi. New York: State University of New York. Again a naturalistic account of Xunzi's philosophy of cosmic reality.\n\nQi, Sihe \u9f4a\u601d\u548c. 1986. A concordance of Xunzi \u8340\u5b50\u5f15\u5f97. Shanghai: Guji Chubanshe. (A useful locator of text items in Xunzi).\n\nRobins, Dan. 2001\u20132002. The development of Xunzi's theory of Xing. Early China 26\u201327: 99\u2013158. (Another account of how Xunzi comes to his theory of xing \u6027).\n\nSato, Masayuki. 2003. The Confucian quest for order: The origin and formation of the political thought of Xun Zi. Leiden: Brill. A fairly thorough discussion of Xunzi's philosophy of government by li \u793c and lifa \u793c\u6cd5.\n\nWang, Zhong \u6c6a\u4e2d. 1977. A general introduction to Xunzi \u8340\u537f\u5b50\u901a\u8ad6. Teipei: Chengwen Chubanshe. (Again another study of Xunzi by a Qing poet-scholar).\n\nWang, Xianqian \u738b\u5148\u8b19. 2010. Collected interpretations of Xunzi \u8340\u5b50\u96c6\u89e3. Beijing: Zhonghua Shuju. (A useful traditional commentary on Xunzi by one learned and famous historian-textual scholar in late Qing \u6e05 Dynasty).\n\nWatson, Burton. 1963. Hs\u00fcn Tzu: Basic writings. New York: Columbia University. A useful basic reader for Xunzi.\n\nYearley, Lee. 1980. Hsun Tzu on the mind: His attempted synthesis of Confucianism and Taoism. Journal of Asian Studies 39(3): 465\u2013480. A relative early study of Xunzi's idea of mind in English language.\n\nFootnotes\n\n1\n\nIt is regrettable that Cua was not given time to do this. On the other hand, one can appreciate how Cua was influenced by his study of Xunzi or by Xunzi in his formulation of his own ethical philosophy. His ideas of reasonable action and paradigmatic individuals are notable examples as indicated in his writings.\n\n2\n\nIn my onto-hermeneutics as a methodology I distinguish between reading and interpretation with pre-understanding which forms the \"from-interpretation\" and reading and interpretation based on objective and rational and conceptual analysis and synthesis of the given texts which forms the \"to-interpretation\" (see Cheng 1996: 55ff).\n\n3\n\nThere are many versions of texts of Xunzi. In this article I used the traditional well-known Chinese texts as annotated by Qing scholar Wang Xianqian \u738b\u5148\u8b19, titled Xunzi Jijie \u8340\u5b50\u96c6\u89e3. I also consulted Qing scholar Wang Zhong \u6c6a\u4e2d's well-known introduction to Xunzi titled Xunqingzi Tong Lun \u8340\u537f\u5b50\u901a\u8ad6. For identification of passages in Xunzi, I used Xunzi Yinde \u8340\u5b50\u5f15\u5f97 as compiled by Qi Sihe \u9f4a\u601d\u548c,Shanghai: Guji Publishing 1986.\n\n4\n\nI have devoted a long paper in integrating law and virtue in light of Xunzi's moral and political philosophy in order to show how yili and lifa could be unified as a system of Confucian political governance. See Cheng 2005: 291\u2013333.\n\n5\n\nThis term has been also understood as \"separation of heaven and human,\" which is, however, mistaken, for it is precisely in the link between heaven and human that human is able to develop itself. But this link lies in the observation that heaven is independent of human will and has its own regularities of movement and change. It is up to the individual person to make adjustment to changes in nature and of nature, whereas nature will release its potential power in ways beneficial to genuine, well \u2013 cultivated individuals.\n\n6\n\nThere is currently some phobia against essentialism in both Western Philosophy and Chinese philosophy, which can be well-understood as a result of doing comparison of Chinese philosophy with Western or Greek philosophy. Whereas Aristotle wishes to make a definition of things in essences to be conceptualized in his ultimate system of metaphysics, Chinese philosophers always appeal to observation and experiences of things as ultimate ground of understanding of reality. Hence there is no Greek metaphysical ascription of essences in a close system of thinking in Chinese philosophy. On the other hand, due to the observational approach of the Yijing, nothing in the cosmos remains unchanged or unchangeable, but their natural qualities as experienced and observed by humans could be understood in relation to humans and in relations among themselves. But this is not to say that these qualities are constructed by humans nor that they are artificial or simply conventional as products of human caprice. On the contrary, qualities of things are real qualities of real things as reality is an experienced fact, and their names are conventional in the sense that they reflect common understanding among humans who adopt common names for these things and their qualities. In this sense names of things represent both human experiences of things and social conventions for naming those experiences of things. For this read Xunzi's essay on Zhengming (Essay 22, \"Rectifying Names\"). Hence one should make a distinction between metaphysical, essentialistic, and Aristotelian realism versus empirical, naturalistic, non-Aristotelian realism.\n\n7\n\nIt is to be pointed out that neither Xunzi's achievemental view nor Mengzi's enlightenmental view constitutes a violation of nature in the modern sense of industrial overdevelopment, which creates pollution and imbalance in nature and endangers the survival of human species and other living beings.\n\n8\n\nXunzi uses the term \"ming \u660e\" (illumination) to describe the state of mind in freedom from bi (obscuration). It is a state of mind that suggests both being illuminated and illuminating. That mind illuminates implies that mind has the light of reason which could shine light rays on things and thus opens itself to clear understanding of experience and objects. It is not quite clear that the state of xuyierjing \u865b\u4e00\u800c\u975c must be a state of illumination. But one may liken it to light in space which may not been detected but which when shining on an object will make the object known.\n\n9\n\nThis state of understanding conforms to what I have described as \"chaorong \u8d85\u878d\" (transcendental integration). But Xnnzi seems also to require that one sees a norm for judging proper places of different things and for guiding or prescribing proper development of things. Perhaps I could describe such a state as \"chaorong er zhuzai \u8d85\u878d\u800c\u4e3b\u5bb0\" ( transcendental integration with normative control).\n\n10\n\nSee how language develops as a help or as a block? See how language develops to an extent that we come to have common framework of reference and meaning and yet there is the possibility of confusion and confounding (luan). This is because there is no way to prevent people from seeing the reference of language as having its own validity \u2013 the issue of reification. This ends up in the problem of yi ming luan sh i\u4ee5\u540d\u4e82\u5be6. There is also the issue of lack of abstraction as in the case of yi shi luan ming \u4ee5\u5be6\u4e82\u540d. There is the problem of yi ming luan ming \u4ee5\u540d\u4e82\u540d: forgetting that each name has its own use context and purpose to serve, and one cannot mix them together and draw conclusions. If killing a robber is not killing a human, it is because killing a robber is to kill a human for his or her crime of robbery where to kill a human is to kill a human for no crime of his or her own but for some external reason of someone else and hence possibly a crime. Our language serves the function of identifying the purpose of using the language: to kill a human is to identify the killing as simply killing without justification, whereas to kill a robber is to kill a person for his robbery. Hence the two can be related as having the same reference: a robber is a human, so to kill a robber is to kill a human but it is to kill the human for his robbery and there is no need to worry about their separate purposes and identities.\n\n11\n\nThe term \"xiaoren\" need not be taken as a morally condemning term. It is originally used to refer to common people who are subjects to be ruled by the junzi, the rulers.\n\n12\n\nSuch as competition in an economic market. As Adam Smith sees it, it is the private interests and selfish desires which motivate competition and innovation in an open market which, when combined with some minimum regulating government, could generate profits for the public through an \"invisible hand,\" which represents a rational institutionalization of the economy.\n\n13\n\nXunzi's notion of human nature being bad becomes a source of the notion of \"qizhizhixing \u6c23\u8cea\u4e4b\u6027\" (nature of temperament) in the Song-Ming Neo-Confucianism. It is largely conceived by Zhang Zai, the Cheng Brothers and Zhu Xi that human nature has both the nature of temperament and the nature of righteousness and reason (yilizhixing \u7fa9\u7406\u4e4b\u6027), which of course requires adequate integration for both understanding the whole person and the self-cultivation project of a Confucian junzi.\n\n# Part 2  \nPhilosophical Issues\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_9\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 9. Early Confucian Perspectives on Emotions\n\nCurie Vir\u00e1g1\n\n(1)\n\nDepartment of East Asian Studies, University of Toronto, Toronto, Canada\n\nCurie Vir\u00e1g\n\nEmail: curie.virag@utoronto.ca\n\nAbstract\n\nThe emotions were a central and constant feature of early ethical debates from antiquity onwards. Early Chinese thinkers diverged greatly in their attitudes toward the emotions, and in how they imagined their place in moral life. While for Confucius (Kongzi \u5b54\u5b50, 551\u2013479 BCE) the morally cultivated person was one whose virtue was completed by joy and pleasure, likes and dislikes, and the fulfillment of his own desires, the Daodejing \u9053\u5fb7\u7d93 called for the eradication of these very emotions and desires as a prerequisite to the achievement of oneness with the Way. While Mencius (Mengzi \u5b5f\u5b50, 372\u2013289 BCE) reasoned that certain innate feelings constituted the natural beginnings, or \"sprouts\" (duan \u7aef) of virtue and moral action, Mozi \u58a8\u5b50(c. 480\u2013390 BCE) regarded spontaneous feelings as inherently partial and thus the cause of injustice, corruption and extravagance on the part of rulers. Underlying such radically diverging perspectives on the role of emotions in moral and political were, more fundamentally, competing visions of what it meant to be human and how to fully realize our moral potential: do we achieve our proper human state through stillness, oneness, and solitude, or through movement, change, and engagement in the world of people and things? Can we rely on our \"natural\" and spontaneous feelings to lead us to right action or must we constrain and order these feelings through our intelligence and the use of institutions and cultural forms?\n\n## 1 Introduction\n\nThe emotions were a central and constant feature of early ethical debates from antiquity onwards. Early Chinese thinkers diverged greatly in their attitudes toward the emotions, and in how they imagined their place in moral life. While for Confucius (Kongzi \u5b54\u5b50, 551\u2013479 BCE) the morally cultivated person was one whose virtue was completed by joy and pleasure, likes and dislikes, and the fulfillment of his own desires, the Daodejing \u9053\u5fb7\u7d93 called for the eradication of these very emotions and desires as a prerequisite to the achievement of oneness with the Way. While Mencius (Mengzi \u5b5f\u5b50, 372\u2013289 BCE) reasoned that certain innate feelings constituted the natural beginnings, or \"sprouts\" (duan \u7aef) of virtue and moral action, Mozi \u58a8\u5b50(c. 480\u2013390 BCE) regarded spontaneous feelings as inherently partial and thus the cause of injustice, corruption and extravagance on the part of rulers. Underlying such radically diverging perspectives on the role of emotions in moral and political were, more fundamentally, competing visions of what it meant to be human and how to fully realize our moral potential: do we achieve our proper human state through stillness, oneness, and solitude, or through movement, change, and engagement in the world of people and things? Can we rely on our \"natural\" and spontaneous feelings to lead us to right action or must we constrain and order these feelings through our intelligence and the use of institutions and cultural forms?\n\nIn recent years, scholars of early China have paid increasing attention to the early discourse of emotions, recognizing the need to take this discourse into account if we are to have a fuller picture not only of the thought and moral outlook of the period, but also of the forces that drove political, social, and cultural change. This trend reflects a seismic shift affecting all academic disciplines, from the humanities to the social sciences to the neurosciences, in which emotions \u2013 and thinking about emotions \u2013 have become central topics of investigation in their own right. This development goes hand in hand with a growing awareness that emotions cannot be fully excised from those \"rational\" and intelligent human processes that were one assumed to be distinct from them (see esp. Solomon 2004; de Sousa 1987; Nussbaum 2001). Such hospitable circumstances for the investigation of emotions have reinvigorated the study of early thought in China. As a number of scholars have pointed out, emotions were central in the ethical and political discourse of pre- and early imperial China, reflecting the fact that \"emotion, intuition, and moral judgment were thought to work in tandem, rather than in opposition to one another\" (Nylan 2001a: 99).1\n\nStudies of emotions in much of the existing scholarly literature have tended to focus on qing \u60c5 \u2013 the Chinese term most commonly associated with the western concept of \"emotions.\" In their discussions of this concept, commentators have invariably made note of its underlying duality of meaning: (1) \"situation,\" \"circumstance,\" or \"essential reality,\" on the one hand and (2) \"passion,\" \"emotion,\" or \"feeling\" on the other. While scholars of early Chinese thought have approached this duality historically and have been concerned with identifying the temporal moment in which the second sense of qing as passion or emotion might have first come into being, studies of the post-Han period have tended to focus on the simultaneity and correspondence of these two basic meanings of qing, and thus on the implied connection between inner and outer realities.2 This connection might emphasize how the external world is mirrored in the inner realm as feelings, or how the inner world is fully revealed in the outer through expression \u2013 primarily in the form of poetry and music.3\n\nIn this chapter, I will go beyond qing to consider a broader range of ideas and concepts that are at least as important for understanding how early thinkers thought about emotions. These include accounts of the fully realized individual, such as the junzi \u541b\u5b50 (the worthy), the sage, or the person of ren \u4ec1 (benevolence); concepts referring to the constituent parts of the human person, such as xing \u6027 (inborn nature) and xin \u5fc3 (heart-mind); terms referring to particular feelings such as xi \u559c (joy), nu \u6012 (anger), ai \u54c0 (sorrow), and le \u6a02 (delight\/pleasure); and a myriad other emotive terms such as yu \u6b32 (desire) and zhi \u5fd7 (direction, intention, will, etc). By considering the history of thinking about emotions more holistically, this study seeks to understand the diverse ways in which the self, in this critical historical juncture, was configured ethically and ontologically.\n\nI will focus here on key texts of the early Confucian tradition, namely, the Analects of Confucius, the Mencius, the Xunzi \u8340\u5b50, and the recently discovered Guodian text, the Xing Zi Ming Chu \u6027\u81ea\u547d\u51fa. My goal is not only to show what these early texts might represent within the larger evolution of thinking about emotions, but also to reveal how their various approaches to the emotions point to the special problems and priorities driving the development of moral thought in this period. Historicizing the history of thinking about the emotions in this way shows us that what was at stake in the debates over the nature and moral status of the emotions was not simply moral theory pertaining to the life of the individual. Accounts of emotions had implications for thinking about community and state, culture and power. It is no wonder, then, that early Chinese thinkers, whose ideas were by and large put forth to gain the ear of rulers and offer a blueprint for political reform (Lloyd 1996: 20\u201346), rarely failed to discourse on the emotions. Or that by the end of the Warring States period, when the rising imperial powers began to create forms of knowledge integrating theories of the self with theories of the body politic and of the cosmos, the emotions \u2013 far from being suppressed or defined out of the picture \u2013 would play a pivotal role in uniting these previously disparate realms. Theories of the human self were the battleground upon which values and ideas about culture and statecraft were waged, and competing visions of the emotions offered different interpretations of how our goals as a society, both individually and collectively, should be realized.\n\n## 2 The Pleasure of Learning and the Joy of Self-Realization in the Analects of Confucius\n\nThe Analects of Confucius (Lunyu \u8ad6\u8a9e) reveals what, from a modern perspective, might appear as a striking mingling of affective and moral discourses.4 In its opening passage it relates the joys and pleasures arising from learning and friendship:\n\n> The Master said, \"Is it not a pleasure (yue \u8aaa), having learned something, to try it out at due intervals? Is it not a delight (le \u6a02) to have friends come from afar? Is it not worthy (jun \u541b\u5b50) not to take offence when others fail to appreciate your abilities?\" (Analects 1.1; ICS Lunyu1.1\/1\/3\u20134.)5\n\nHere, Confucius juxtaposes pleasure (yue \u8aaa, \u6085), delight (le \u6a02), and being a noble person (junzi \u541b\u5b50), as though they were parallel virtues, suggesting that finding pleasure and joy in the important things of life are moral ends at the level of being a worthy person. Other passages in the text go further than this to assert joy and pleasure as signs of moral attainment. In the case of learning, Confucius notes that the ultimate level of achievement is one in which a person has found delight (le \u6a02) in what one has learned: \"The Master said, 'To be fond of something is better than merely to know it, and to delight in it is better than merely to be fond of it'\" (Analects 6.20). And in his reflections on his own life trajectory, Confucius signposts the various stages of achievement as they lead from his youthful embarkation upon a life of learning to his realization of its ultimate end:\n\n> The Master said, \"At fifteen I set my heart on learning; at thirty I took my stand; at forty I came to be free from doubts; at fifty I understood the Decree of Heaven; at sixty my ear was attuned; at seventy I followed my heart's desires (cong xin suo yu \u5f9e\u5fc3\u6240\u6b32) without overstepping the line.\" (Analects 2.4; ICS Lunyu 2.4\/3\/1\u20132.)\n\nFor one who aspires to realize himself, the moment of self-fulfillment comes when what one spontaneously desires to do fully coincides with what one ought to do.\n\nIn addition to emphasizing the feelings of joy and pleasure, and the fulfillment of one's desires in learning and self-cultivation, the Analects also stresses emotional engagement. The moral ideal is not one of detachment and impartiality, as Mozi would advocate (see below), but the very capacity to have preferences grounded in one's genuine convictions. A morally cultivated person \u2013 one who fully embodies the virtue of ren \u4ec1 (humaneness or benevolence) \u2013 is someone who possesses clear likes (hao \u597d) and dislikes (e \u60e1):\n\n> The Master said, \"It is only the benevolent person who is capable of liking and disliking others.\" (Analects 4.3; ICS Lunyu 4.3\/7\/9.)\n\nZigong said, \"Does even the gentleman (junzi) have dislikes?\"\n\n> The Master said, \"Yes. The gentleman has his dislikes. He dislikes those who proclaim the evil in others. He dislikes those who, being in inferior positions, slander their superiors. He dislikes those who, while possessing courage, lack the spirit of the rites. He dislikes those whose resoluteness is not tempered by understanding (Analects 17.24; ICS Lunyu 17.24\/50\/19\u201324.)\n\nFor a worthy, there is no contradiction between the spontaneous, emotional responses of liking and disliking, on the one hand, and making proper moral judgments, on the other. Indeed, having strong feelings and preferences about something is to fully realize one's relationship with it.6 Confucius thus compares the lover of benevolence with the lover of beautiful women: \"The Master said, 'I have yet to meet the man who is as fond of virtue as he is of beauty in women.'\" (Analects 9.18; ICS Lunyu 9.18\/21\/20. Cf. Analects 15.13). In other passages, Confucius notes that one's actions must be accompanied by emotional involvement if they are to be meaningful. Thus, for example, just as the feelings of pleasure and joy must accompany true learning and understanding, proper feeling must also be present in one's moral conduct for it to be genuine. In the case of filial piety (xiao \u5b5d):\n\n> The Master said, \"What is difficult to manage is the expression on one's face. As for the young taking on the burden when there is work to be done or letting the old enjoy the wine and the food when these are available, that hardly deserves to be called filial.\" (Analects 2.8; ICS Lunyu 2.8\/3\/14\u201315. see also Analects 2.7 and 8.2)\n\nThis passage notes how, although we may be able to perform filial actions, we might not be able to achieve the more difficult task of genuinely feeling the devotion and respect owed to one's parents. It is easy to go through the motions of being filial to one's parents. Far more difficult is to perform these acts sincerely \u2013 with authentic feeling. Unaccompanied by such feeling, our actions \u2013 however well-intentioned \u2013 become merely external performance, devoid of moral significance (Cf. Analects 3.12).\n\nFor Confucius, then, to be truly benevolent means not only doing what is right and appropriate in any given situation, but also being fully engaged emotionally. Thus, in a fully realized individual, inner and outer are part of a unified whole. This has tremendous implications for Confucius' understanding of ritual (li \u79ae), which he regarded as the beginning and end of moral self-cultivation. If Confucius stressed the need to constantly abide by ritual, exhorting others not to \"look,\" \"speak,\" or \"move\" without doing so in accordance with ritual propriety (Analects 12.1), this need for constant vigilance was as much about awareness of one's emotional commitment as it was about outer performance.7 Right emotions, then, were essential to the Confucian conception of virtue, which was none other than the art of living properly and well.\n\nWhat are the ethical and historical implications of Confucius' stress on emotional presence as a necessary feature of the perfected individual? Historically, the Confucian discourse of emotions emerged as part of the search for a new and universal basis of values in a time of moral flux. It signifies a broader ethical shift, embraced by certain members of the newly emergent scholar-literati class, that the true locus of authority was moral, and that this site of moral authority lay within the individual.8 Confucius proposed that neither secular authority, nor established social norms, nor even the ritual practices of the Zhou state as codified in the transmitted texts and passed down through convention, could provide reliable models by which we should order our lives individually and collectively. The true guides to living lay within the self. Being present in the very structure of our feeling and desiring, they were not simply transferred from the outside. To the extent that they could be subjectively experienced by all individuals, they could be said to be universal.9 This general theme of the self-reliance and autonomy of the cultivated individual can be seen in a number of the Master's pronouncements about ren:\n\n> The Master said, \"Is ren really so far away? No sooner do I desire it than it is here.\" (Analects 7.30; ICS Lunyu 7.30\/17\/12.)\n> \n> The Master said, \"What the gentleman seeks, he seeks within himself; what the petty person seeks, he seeks in others.\" (Analects 15.21; ICS Lunyu 15.21\/43\/17.)\n> \n> \"When it comes to being ren defer to no one, not even your teacher.\" (Analects 15.36; ICS Lunyu 15.36\/44\/20.)\n\nThe importance of emotions in Confucius' account of the perfected individual is a symptom of his attempt to shift the locus of morality authority inwards.10 The ultimate ethical attainment depended upon our individual capacities, which meant that what we needed for living life well and meaningfully was inside us, not \"out there\" somewhere. It was we who broadened the Way, and not the other way around (Analects 15.29; cf. 7.10, 7.30, 15.21).\n\nConfucius' claim that right action had to be accompanied by spontaneous emotions points to an inner basis of moral authority, suggesting that living properly in the world is as much a personal attitude and dispositional state as much as it is about following accepted norms. But on the theoretical questions of what it was that made human emotional responses normative, and how generally the human moral character was to be understood and refined through cultivation, Confucius offered no explanation. Indeed, he was famously reticent about the subject of human nature and the \"Way of Heaven\" (Analects 5:13). His humanistic and this-worldly orientation, with its emphasis on the total engagement of the person and on the very transparency of the self, stressed the way in which individuals actualized their moral potential in relation to other people and things, and in specific contexts. The self was thus a practical entity, coming into perspective as individuals made themselves visible through their conduct \u2013 above all, through engaging in proper ritual, which ideally emerged out of emotions that were aligned with one's virtues.11\n\n## 3 Mencius and the Emotive Grounds of Morality\n\nSome of Confucius' most prominent followers in the fourth and third centuries BCE grappled with the issue of how the emotional disposition of human beings, as contained within the \"inborn nature\" (xing \u6027), was related to virtue and moral action. They took as their point of departure the basic human virtues as elaborated by Confucius, as well as his assumptions about bodily integrity, self-cultivation, and sagehood. But in an age of intellectual ferment and broadened horizons in philosophical investigations, moral inquiry called for greater awareness of the workings of the human mind and body, as well as of the larger forces and patterns that governed the world (Harper 1999: 813\u2013852). These explorations of the outer world shaped, in turn, knowledge of the human self, helping to give rise to what Erica Brindley has characterized as \"a new intellectual preoccupation with the body and psyche\" (Brindley 2006b: 3. See also Sivin 1995 and Lewis 2006). It was thus during the fourth century that the concept of xin \u5fc3 \u2013 the heart-mind \u2013 emerged as a major subject of philosophical discussion.\n\nDuring this time, xin was broadly conceived in the context of a more or less unified cosmological vision \u2013 one based on the processes and movements of qi \u6c23, or \"configurative energy\" (Porkert 1974: 167).12 Conceived as the site not only of thought (si \u601d, l\u00fc \u616e) and will (zhi \u5fd7), but also of feelings and desires, the recurring usage of the term xin belies a more general concern with the emotional and affective dimensions of life (Brindley 2006a: 247). Within this context, the emotions became a central concern among later followers of Confucius. Among the most prominent of these was Mencius, who emphasized the inherent, natural origins of morality in human beings.13 This was in stark contrast to Xunzi (further discussed below), who held that human nature possessed no innate moral direction, and therefore required the guidance of extrinsic and artificial devices (wei \u507d) in order to be made good. Thus, while Mencius identified in human emotions the \"sprouts\" (duan \u7aef) of virtue, Xunzi saw them as generally steering people towards immoral conduct and therefore needing to be channeled and controlled through ritual, music, and other institutions arising from a well-ordered state. For both thinkers, however, morality depended on understanding the universal emotional tendencies of human beings, and knowing how to channel them properly.\n\nLike Confucius, Mencius conceived of the perfected individual as one for whom inner and outer were integrated. He went beyond Confucius, however, in his insistence that, through people's eyes and words, it was possible to know their inner character.14 Moreover, he took further the idea of \"knowing\" the self: subjecting the inner realm to deeper scrutiny, he traced the origins of virtue to certain basic emotional inclinations. The central moral claim for which Mencius is known \u2013 that human nature (xing \u6027) is originally good (shan \u5584) \u2013 assumes that all human beings share the same emotional constitution, which furnishes the \"sprouts\" of virtue within ourselves. The famous example of the child about to fall into the well in Mencius 2A6 exemplifies the workings of one of four emotional dispositions that represent the origin of moral awareness \u2013 the feeling of compassion (ce yin zhi xin \u60fb\u96b1\u4e4b\u5fc3). \"No one,\" Mencius claims, \"is devoid of a heart sensitive to the sufferings of others,\" and his proof of this is that every single human being would feel a twinge of compassion when faced with the scenario of a child about to fall into a well. Not everyone will necessarily act upon this feeling, but everyone would feel it, and this fact is sufficient, according to Mencius, to make his case. This feeling of compassion is accompanied by three other innate human feelings \u2013 shame (xiu wu zhi xin \u7f9e\u60e1\u4e4b\u5fc3), respect (gong jing zhi xin \u606d\u656c\u4e4b\u5fc3), and approval and disapproval (shi fei zhi xin \u662f\u975e\u4e4b\u5fc3) 15 \u2013 all of which represent the \"sprouts\" of moral action:\n\n> The feeling of compassion is the sprout of humaneness; the feeling of shame, of rightness; the feeling of respect, of ritual propriety; the feeling of approval and disapproval, of wisdom. Man has these four sprouts just as he has four limbs. For a man possessing these four sprouts to deny his own potentialities is for him to cripple himself. (Mencius 2A6; ICS Mengzi 3.6\/18\/8\u201310)\n\nOur moral sentiments, as signified by the proper functioning of our emotional disposition, were as much a part of our humanness as our sense faculties \u2013 our ears, eyes, mouths and noses \u2013 and our four limbs. As such, to deny their existence was tantamount to \"crippling\" ourselves.\n\nFor Mencius, then, the potentiality for moral consciousness and action was as characteristic to the essential and underlying disposition of human beings (qing \u60c5) as our sense faculties were intrinsic to our underlying physical constitution (Mencius 6A6).16 The affective, moral and physical realms, however, were not simply parallel, analogous aspects of the human experience; they were in fact inseparable and fully integrated with one another. Conceived in this way, morality was none other than nurturing our underlying tendencies, both physical and mental: for just as our emotions were the starting point for our moral awareness, so was the functioning of our bodies part involved in our embodiment of the moral order:\n\n> Mencius said, \"The way the mouth is disposed towards tastes, the eye towards colours, the ear towards sounds, the nose towards smells, and the four limbs towards ease is human nature, yet therein also lies the Decree (ming \u547d). That is why the gentleman does not describe it as nature.\" For a defense of the idea that Mencius is offering a biological argument for morality, see Bloom (1997). (Mencius 7B24; ICS Mengzi 14.24\/75\/22\u201324.)\n\nFor Mencius, the workings of our sense faculties were inseparable from the norms that should govern our lives. That is, morality is not about a higher set of guidelines that should be imposed over and above our senses and feelings but was integrated with them. The two realms were thus part of a unified whole pervaded by the shared substance of qi \u2013 that \"vital force\" or \"configurative energy\" that explained the workings of the physical body, as well as the inner life of the self. As Alan K. Chan has pointed out, Mencius was but one of many thinkers of his time who understood the human person as \"constituted by qi\" and who conceived the process of moral self-cultivation as \"a process of taming, forcing, or channeling the powerful energy that informs the heart\" (Chan 2002: 52).\n\nThe fact that there existed some inner force that could tame and channel the qi suggests that human beings possessed a capacity within themselves that was not determined by extrinsic physical circumstances. This capacity lay within the heart, and Mencius was careful to point out a crucial difference in the cognitive functioning of the heart and the sense faculties:\n\n> The organs of hearing and sight are unable to think (si \u601d) and can be misled by external things (wu \u7269). When one thing acts on another, all it does is to attract it (yin zhi er yi \u5f15\u4e4b\u800c\u5df2). The organ of the heart can think. But it will find the answer only if it does think; otherwise, it will not find the answer. This is what Heaven has given me. (Mencius 6A15; ICS Mengzi 11.15\/60\/27\u201311.15\/61\/1.)\n\nThus, while the senses were vulnerable to the push and pull of external things (wu \u7269), the heart possessed the capacity to think or reflect (si \u601d), which allowed it to resist the onslaught of things and pursue something other than what one might be immediately attracted to at a given moment. This seems to suggest that si \u2013 the process of thinking or reflecting that the xin is uniquely capable of \u2013 is fundamentally distinct from feeling, which is physically responsive to one's circumstances. Does that imply, then, that Mencius recognizes a distinction between feeling and thinking? Most scholars who have addressed this issue directly have argued against such a reading, either by insisting that Mencius regarded feelings as legitimate and proper sources of motivation for moral action (Van Norden, Behuniak) or that what Mencius imputes to feelings in fact involve significant levels of cognition and reasoned judgment (Wong 2000; Nivison 1980).\n\nIn 6A6, Mencius seems to reiterate his argument in 2A6 about the emotive basis for humaneness, rightness, ritual propriety and wisdom, but instead of referring to the various feelings of compassion, shame, and so on, as the \"sprouts\" of these virtues, he simply equates them:\n\n> The feeling of compassion is benevolence. The feeling of shame is rightness. The feeling of respect is propriety. The feeling of approval and disapproval is wisdom. Benevolence, rightness, propriety, and wisdom are not welded onto us from the outside. We inherently have them. It is simply that we do not reflect upon them. (ICS Mengzi 11.6\/58\/22\u201324.)\n\nThe four basic virtues, then, are not \"welded\" onto us from without, but are inherent to our constitution. The reason they are not always actualized is because people fail to si them \u2013 to direct, concentrate and focus them.17 Recognizing the way in which si integrates the realms of feeling, judgment and cognition, 6A15 does not seem necessarily to warrant a thought\/feeling dichotomy. That is, morality for Mencius does not depend on a categorical distinction between our biologically rooted desires and emotional inclinations, on the one hand, and our capacity for reasoned thought and behavior, on the other \u2013 with the latter guiding the heart to the proper objects of its interest: normative principle\/pattern (li \u7406) and propriety (yi \u7fa9). Instead, our heart seek out, and experience satisfaction in, the pursuit and realization of virtue, just as our sense faculties desire and experience satisfaction in things they crave, such as delicious food and beautiful objects (Mencius 6A7). What is necessary for the nurturing of our hearts is therefore not the eradication of desire but the cultivation of desire for the right things.18 The heart and the senses thus remain in a state of dynamic tension with one another, with the possibility of mutual influence. In the case of a junzi, who had fully mastered both realms, the operations of his \"floodlike qi\" (hao ran zhi qi \u6d69\u7136\u4e4b\u6c23) lead him spontaneously to do the right thing, and to be entirely one with his actions. Mencius' moral vision, premised on the ideal of a unity between feelings and right judgment, points to the claim that human beings ultimately contain their own moral compass, and therefore do not need to have virtue instilled from them from without. The true directives for how to live in this life reside within the self:\n\n> Mencius said, \"A junzi steeps himself in the Way because he wishes to find it in himself. When he finds it in himself, he will be at ease in it; when he is at ease in it, he can draw deeply upon it; when he can draw deeply upon it, he finds its source wherever he turns. This is why a junzi wishes to find the Way in himself.\" (Mencius 4B14)\n\nIn emphasizing the inner grounds of morality, Mencius openly disputed thinkers like Mozi \u58a8\u5b50(473 BCE \u2013 ca. 391 BCE) and his own contemporary Gaozi \u544a\u5b50 (ca. 420\u2013350 BCE), who were not so confident in the innate human disposition to find the proper course.19 Mozi and his followers had regarded the feelings as the source of social injustice and wastefulness. Recognizing a deep contradiction between our natural inclinations and desires, which led to material self-indulgence and partiality in our relationships with others, Mozi advocated that we strive for impartial concern (jian ai \u517c\u611b) for the wellbeing of all people so as to enable our sense of propriety or correctness to hold sway (Mozi (2003) Ch. 16). Gaozi, for his part, argued that moral propriety was simply about external directives that were internalized within the self (Mencius 6A1-4). In the midst of these debates about the true sources of morality, Mencius supported his assertions by claiming that all human beings possessed in common emotional sensibilities that \u2013 if properly nurtured \u2013 would lead us to virtue. Morality, in short, was the proper actualization of feelings that we all possessed by virtue of being human.\n\n## 4 Emotions and the Beginning of Culture in the Xing Zi Ming Chu\n\nEmotions figure prominently in the Xing Zi Ming Chu (\u6027\u81ea\u547d\u51fa), a text that forms part of the now well-known cache of philosophical writings discovered in the Guodian tombs. Some of these writings are believed to date as far back as the fourth century BCE.20 Containing lengthy discussions on such topics as human nature (xing \u6027), the heart-mind21 (xin \u5fc3) and feelings (qing \u60c5), the Xing Zi Ming Chu offers a particular ethical theory of the self and then sets about to explain how this self \u2013 given its attributes \u2013 is to achieve proper realization in the world of things. The text has received particular scrutiny among scholars working in the field of Warring States thought \u2013 particularly those exploring the early history of qing. 22 It is clear, however, that while it shares the Mencian premise that feelings and desires are defining elements of being human, and are therefore of vital relevance to thinking about moral and political order, it arrives at vastly different conclusions about what our emotions entail for the proper shaping of human life.\n\nThe Xing Zi Ming Chu opens with a pronouncement about the nature of human beings, which is the starting point for the ethical theory that follows:\n\n> Generally speaking, although human beings possess an inborn nature (xing \u6027), their heart-minds (xin \u5fc3) lack a fixed orientation (dian zhi \u5960\u5fd7). It depends on things (wu \u7269) and only then does it act; it depends on pleasure and only then does it move; it depends on practice and only then does it become fixed. (Guodian chumu zhujian 179)23\n\nThere is a twofold claim being made here about human nature and the heart-mind, which potentially creates a tension within the individual. On the one hand, all human beings are said to possess a common nature (xing) that \"arises from the Decree (ming \u547d)\" sent down by Heaven itself. On the other hand, they also possess a heart-mind (xin) that has no fixed orientation and thus assumes shape and direction according to \"things\" (wu), our emotional responses to them, and the habits we form from our circumstances. In contrast to the Mencius, the text asserts that our heart-minds lack an inherent direction and are therefore vulnerable to external conditions, and to the unstable responses of our emotions.\n\nAfter establishing the heart-mind's dependence upon \"things\" to achieve a sense of direction, the text directly proceeds to the various emotions \u2013 joy and anger, sorrow and sadness \u2013 that arise within the nature:\n\n> The qi of joy and anger, sorrow and sadness (xi nu ai bei zhi qi \u559c\u6012\u54c0\u60b2\u4e4b\u6c23) are the realm of the inborn nature. As for their becoming manifest outside, it is due to things grabbing hold of them (wu qu zhi ye \u7269\u53d6\u4e4b\u4e5f). The nature emerges from the Decree, and the Decree descends from Heaven. (Guodian chumu zhujian 179)\n\nAlthough these emotions are said to be part of the inborn nature, it is only when they encounter things in the world that they become manifest. That is, emotions represent the locus of encounter between our inner nature and the world outside. The text then introduces the term qing: \"The Dao begins in feeling\/qing (Dao shi yu qing \u9053\u59cb\u65bc\u60c5). And feelings\/qing are born from the nature (Qing sheng yu xing \u60c5\u751f\u65bc\u6027)\" (Guodian chumu zhujian 179). If in the Mencius, qing refers to the \"essential nature\" or \"what is genuine,\" here it refers to something closer to the realm of feelings and desires. Qing represents the aspect of human feelings that arise from contact with the things and events of the external world. But at the same time, as that which is \"born from the nature,\" qing is inborn and therefore universal to all human beings. As such, it is the starting place of the Dao, and thus the very foundation of the proper life itself.\n\nThe realm of qing \u2013 properly conceived here as emotions or, as Michael Puett has put it more precisely, \"dispositional responsiveness\" \u2013 opens up a problem that needs to be resolved. If all human beings, though sharing a common emotional disposition, are subject to the determinations of things in the external world, how does it become possible to achieve a unified sense of self with a \"fixed purpose\"? How is a virtuous and moral existence possible? If early Daoist thinkers resolved this dilemma by advocating that we free ourselves from our dependence upon the external world, and from the allures of the senses, the author of the Xing Zi Ming Chu insisted that the inborn nature could not be properly realized unless one engaged with the realm of things. And thus, if our emotional disposition was something inborn and gave us the capacity to live as we ought \u2013 if, indeed, qing was the starting place of the Dao \u2013 the proper course lay in successfully channeling one's emotions in a manner that conformed with propriety (yi \u7fa9): \"At the beginning one is close to qing, and at the end one is close to propriety\" (Guodian chumu zhujian 179).\n\nAsserting qing as the starting place of the Dao, this text focuses on inborn, universally shared tendencies as the ground for proper action; but it also identifies propriety as the end point \u2013 as something that can only be achieved through education (jiao \u6559). Education refers to external forces that shape human nature. The text, accordingly, engages in an extended discussion of how human nature requires the presence of things (wu \u7269) and circumstances (shi \u52e2) for its proper flourishing and realization:\n\n> Generally speaking, regarding the nature, some things move it, some things entice it, some things interact with it, some things discipline it, some things bring it forth, some things nurture it, some things make it grow. Generally speaking, what moves the nature are things; what delights the mind is pleasure; what joins with the nature are intentions; what disciplines the nature is propriety; what brings the nature forth is circumstance; what nurtures the nature is practice; what makes the nature grow is the Dao. (Guodian chumu zhujian 179)\n\nThis is a theory of human nature and the heart-mind that explains how to lead the innate emotional disposition, in the face of its encounter with things and circumstances, to its proper realization in accordance with propriety. The various forces that come to shape the person \u2013 circumstances, propriety, repeated study, the Way \u2013 are not described abstractly and generally, but concretely: they are particular experiences that are bound up with tangible things (wu) that can be seen and experienced:\n\n> Generally speaking, what is seen (jian zhe \u898b\u8005) are called things; what brings delight to the self is called pleasure; the circumstances of things are called circumstance; having intentional activity is called purpose. Propriety is the accumulation of a myriad goodnesses. Habitual cultivation (xi \u7fd2) means having that by which to habitually cultivate one's nature. Dao is the Dao of the myriad things. (Guodian chumu zhujian 179)\n\nIn the midst of a world filled with things good and bad that literally take hold of us, what brings order and direction is the fact that there is a normative human way for people to follow:\n\n> As for the four techniques of the way, only the human way is able to serve as the guide (ke dao \u53ef\u9053). As for the remaining three, one guides them (dao zhi \u9053\u4e4b) and that is all. (Guodian chumu zhujian 179)\n\nThis human way is the way consisting of Poetry, Documents, Rites and Music, and these \u2013 having emerged out of human feeling \u2013 furnish the proper guidelines for living. The text explains the emergence of these achievements of civilization historically: they represent the primordial sages' expression and documentation of actual human actions and events, and they arose from their qing. The work of the sages was to compile, order and embody them, and also to use them to pattern their own qing:\n\n> As for Poetry, Documents, Rites and Music, when they first appeared they arose from human beings. Poetry came about when there were events, and people enacted them. Documents came about when there were events and people spoke of them. Rites and music came about when there were events and people raised them up. The sages, having compared their variety, arranged and assembled them; having perceived their priorities, they added admonishments; having given substance to their propriety, they put them in order; having patterned their qing, they manifested and internalized them. Afterwards, they again used them for instruction (jiao \u6559). Instruction is that by which to give birth to virtue within the self. The Rites arose from qing. (Guodian chumu zhujian 179)\n\nThe final sentence in this passage adds that the Rites arose from qing, thus emphasizing the way in which self-cultivation was a matter of according with one's emotions, rather than opposing them, even while it involved the need for external forces to shape the moral nature.24\n\nThis positing of a normative cultural tradition that was arranged and embodied by the sages introduces into the moral discourse a model of the self as political subject in the literal sense: a being that is to be analyzed and shaped by those empowered to rule. The \"psychological\" perspective that is characteristic of this and other texts from this period thus corresponds to an interest in defining the human moral nature as an entity that is in constant interaction with \"things\" and that requires outer sources of completion. Although the text's recognition of the emotions as a starting point for human morality gives it an important point of commonality with earlier Confucian texts, its portrayal of the self as passive, directionless, and requiring external sources of direction represents an important departure: cultural institutions have become, for some, the carriers of moral order and value.25\n\n## 5 Conscious Activity and the Fulfillment of Natural Feelings in Xunzi\n\nIn many ways, the perspectives introduced in the Xing Zi Ming Chu prefigures the ethical stance of Xunzi \u8340\u5b50 (ca. 312\u2013230 B.C.). Xunzi, too, elaborated at length not only on the importance of the emotions and their proper expression, but also on the necessity of cultural institutions such as rites and music to shape human nature. But whereas in the Xing Zi Ming Chu, emotions (qing) are cited to show how the proper guide to living (Dao) and cultural institutions actually originate from the human emotional disposition, in the Xunzi, the theory of emotions describes a more complex relation between natural human feelings and morality. On the one hand, Xunzi asserts that the emotions are innate and universal, and as such, possess normative content. In the \"Tian lun \u5929\u8ad6\" chapter, he explains how the form and spirit, body and mind, the sense faculties and emotions, all came to be produced by Nature (tian \u5929):\n\n> When Nature's work has been established, and its achievements have been perfected, the physical form becomes complete and the spirit is born. Love and hate, happiness and anger, sadness and joy are all contained within, and these are called the natural feelings (tian qing \u5929\u60c5). The eyes, ears, nose, mouth and physical form are all able to make contact [with things], but are not able to act on behalf of one another. These are called the natural faculties (tian guan \u5929\u5b98). (Xunzi 17.3; ICS Xunzi 17\/80\/9\u201310.)26\n\nLater in this section Xunzi explains that the human possession of these emotions and faculties imply a kind of moral imperative to abide by and preserve these natural endowments:\n\n> To obscure one's natural ruler (tian jun \u5929\u541b), confuse one's natural faculties, reject one's natural nourishment, rebel against the natural order of things and to turn away from the natural feelings so as to destroy one's natural virtues \u2013 this is indeed a great calamity. The sage purifies his natural ruler, rectifies his natural faculties, completes his natural nourishment, accords with the natural order of things, and nurtures his natural feelings so as to complete one's natural virtues. If one is able to do this, then he knows what to do and what not to do. And Heaven and Earth will preside and the myriad things will serve (Xunzi 17.3; ICS Xunzi 17\/80\/11\u201314.)\n\nOur possession of emotions and sense faculties, therefore opens the path to a directive that we must follow in order to live life well: we must nurture (yang \u990a) these aspects of ourselves if we are to fully realize our human potential. In this way, we not only fulfill our individual human lives but also the larger destiny of Heaven and Earth. Thus it is that the sage, having successfully cultivated and refined his own nature, brings it about that \"Heaven and Earth will preside and the myriad things will serve.\"\n\nThis nurturing of what is \"natural,\" however, requires going beyond what is nature-given. If the forces of nature, or Heaven and Earth are able to bring the myriad things into being, it is incapable of ordering them: \"Heaven is able to bring things into being, but it is not able to differentiate them. Earth is able to sustain people, but it is not able to govern them.\" (Xunzi 19.6; ICS Xunzi 19\/95\/3.) One's natural state, therefore, does not provide us with a moral compass by which to live: as Xunzi famously declared, \"Human nature is bad.\" (Xing'e \u6027 \u60e1). Human nature, for Xunzi, refers to the most basic, instinctive, nature that all human beings possess and that accounts for people's affective responses to their condition: they desire food when hungry, warmth when cold, rest when tired, and so on. It also refers to the distinctive features of the five senses \u2013 sight, hearing, taste, smell and touch (Xunzi 4\/64). Part and parcel of this natural human tendency towards the satisfaction of personal desire is the realm of \"human feelings\" (renqing \u4eba\u60c5), which in their raw state lead one to pursue what is immediately gratifying to the self, rather than what is proper:\n\n> Yao asked Shun: \"What are human feelings like?\"\n> \n> Shun replied: \"Human feelings are truly unlovely things (bu mei \u4e0d\u7f8e). What need is there to ask about them? When a man's life is complete with wife and child, his filial devotion towards his parents diminishes. When he attains what he desires, his sincerity towards his friends diminishes. When he has fulfilled his ambition for high office, his loyalty towards his lord diminishes. People's feelings! People's feelings! How unlovely they are! Only in the case of the worthy is this not so.\" (Xunzi 23.6a\/ICS Xunzi 23\/116\/25\u201323\/117\/1.)27\n\nSince our qing tends towards the satisfaction of our immediate desires, allowing them to operate unimpeded would lead to strife, mutual destruction and disorder.\n\nHow, then, are we to go beyond the impulses and desires of our inborn human nature so as to realize ourselves properly? For this we must look beyond the inherent qualities of human nature to external standards of virtue such as ritual and propriety. These things, Xunzi claims, do not come about because of any inherent disposition of the inborn nature, but are the result of artifice, or conscious activity (wei \u507d). Morality, in fact, depends upon both nature and artifice: If the inborn nature (xing \u6027) is the \"beginning, the root, the raw material and the basic substance\" of the self, \"conscious activity\" is its \"pattern, the ordering principle, the extension and the flourishing.\" (Xunzi 19.6\/ICS Xunzi 19\/95\/1.) But it is not simply that natural emotional dispositions need to be molded and shaped by artificial, external devices. The latter, in fact, \"nurture\" human desires and provide the means to satisfy them (Xunzi 19\/417). Unlike the Xing Zi Ming Chu, which had explained ritual and the Way as arising out of the natural and spontaneous realm of qing, Xunzi makes no claims as to the natural origins of ritual and morality. These instruments of culture, moreover, served to order society into a clear hierarchy based on differences in status and age, as well as intellectual, moral, and physical capability (Xunzi 4\/69). In this way, the ancients managed to preserve the harmony of society.\n\nXunzi's apparent ambivalence towards the emotions is philosophically significant.28 It enables Xunzi to put forth a more nuanced vision of the self than what has been put forth by previous thinkers \u2013 a vision that would also have profound consequences for the later development of thought and political ideology. At the most general level, what Xunzi lays out is a conception of self in which previously recognized polarities are brought together under a single vision. At its foundation is a kind of two-self theory: while Xunzi claims to speak of a single, universal human nature, rather than of different natures belonging to different types of people, he nevertheless seems to have two different types of people in mind. On the one hand, there is the ruler, who \"fully realizes\/enables the full realization of human qing (essential nature)\" (Hutton 2000: 228), and whose earliest representatives \u2013 the ancient sage kings \u2013 were morally refined enough to come up with rituals and cultural models that could be used to assist others. On the other hand there is the common person, whose originally \"bad\" nature needs to be shaped and molded through artifice in order to be made good. Moral self-cultivation is signified by a passage or movement in which one's nature of an ordinary person is transformed into that of a sage.\n\nXunzi proposes, then, a model of self that is capable of change. The trajectory of change is represented by the mediation of a set of inter-related polarities:\n\n1.\n\nNature versus artifice\n\n2.\n\nInner versus outer\n\n3.\n\nPassiveness versus activeness\n\n4.\n\nEmptiness versus fullness\n\nOne aspect of this change is a movement from the inborn nature to a nature that has been refined through artificial conscious activity:\n\n> Without an inborn nature, there would be nothing for conscious activity to contribute to; if there were no conscious activity, the nature would not be able to better itself. When the nature and conscious activity are joined, then one lives up to the name of 'Sage' and the virtue of uniting all under Heaven is realized. (Xunzi 19.6; ICS Xunzi 19\/95\/1\u20132.)\n\nThe sage is the figure in whom the natural and acquired nature are brought together. Thus, the progression is not a movement from nature to artifice, but rather, from a raw state of nature (in which emotions and desires are spontaneously realized) to a refined state of nature, in which emotions and desires have been channeled in the proper direction.29 The model is one of a balance of oppositions, and this balance characterizes not only human goals, but also the very workings of the cosmos, explaining the vorigin and proper ordering of all things:\n\n> Thus it is said that when Heaven and Earth join together, the myriad things are produced; when yin and yang converge, the changes and transformations are set into motion; and when the inborn nature and conscious activity are joined together, all under Heaven is put into order. Heaven is able to produce things, but it is not able to differentiate them; Earth is able to sustain people, but it is not able to order them; Those who count themselves among the creatures and living people under Heaven depend on sages to assign them to their proper station. (Xunzi 19.6; ICS Xunzi 19\/95\/2\u20134.)30\n\nA similar logic penetrates the dialectic between inner and outer realms of the self. According to Xunzi, human nature is fundamentally characterized by desire \u2013 a longing for things that makes the inborn nature liable to depart from the path of goodness:\n\n> People's natural inclinations (qing \u60c5 ) are such that for food they desire meat of grass- and grain-fed animals; for clothing they desire patterned and embroidered fabrics; and for traveling they desire horse and carriage. In addition to this they desire the wealth of extra funds and stored-up provisions so that even through extended periods of meager earnings they will not experience shortages.(Xunzi 4.11\/ICS Xunzi 4\/16\/5\u20136.)\n\nHuman beings are invariably caught up in a mechanism of interdependence with the world of things and circumstances. This is what makes us, to a certain extent, passive and vulnerable to things. But this also makes us capable of positive change through our encounter with things. The solution, then, is not to reject this world of things, but to bring about a situation in which we are caught up in a positive relationship with it.\n\n> The junzi does not differ from others by birth; he is just good at borrowing (jia \u5047) from things. (Xunzi1.3\/ICS Xunzi 1\/1\/14\u201315.)\n> \n> The junzi arranges (she \u8a2d) things; the petty man is arranged by things. (Xunzi 2.5; ICS Xunzi 2\/6\/12\u201313.)31\n\nAlthough the goal is to cultivate ourselves in such a way that we remain open to things, we must do so without becoming completely transformed by them. And ultimately, just as sagehood is the ultimate end of self-cultivation, so it is that being active \u2013 rather than passive \u2013 remains the model of proper behavior. Making use of a political analogy, Xunzi explains that the senses are like the many government bureaus, each with its distinct function (guan \u5b98), and within this system, \"the heart-mind (xin \u5fc3) is the lord that controls the body just as the lord of men controls and governs (jun \u541b) society.\" (Xunzi 17.3a\/ICS Xunzi 17\/80\/10.)\n\nHere, the heart-mind, as the ruler of the body, the interface between inner and outer, and the access point of the Way, represents the juncture of apparent oppositions. Its substance is characterized by both emptiness and fullness, which gives it autonomy and yet opens it to the possibility of activity and learning:\n\n> What do people use to know the Way? I say that it is the heart-mind. How does the heart-mind know? I say, through emptiness, oneness, and stillness. The heart-mind has never stopped storing, but nevertheless it has what is called emptiness. The heart-mind has never stopped being two, but nevertheless it has what is called oneness. The heart-mind has never stopped moving, but nevertheless it has what is called stillness. (Xunzi 21.5d\/ICS Xunzi 21\/103\/25\u201326.)\n\nXunzi's theory of mind thus affirms positive learning \u2013 learning that takes place through a consistent and ongoing process of accumulation (ji \u7a4d) that focuses on the knowledge and experience of external things achieved through one's sense faculties. But because the mind is empty and unified, it is capable of a lifetime's worth of learning without ever losing the upper hand in its relationship with things. And ultimately, one's fate as an individual \u2013 whether one becomes a robber or an artisan or a farmer or a sage \u2013 depends on \"the accumulated effect of circumstances.\" Indeed, Xunzi claimed that \"As for Yao and Yu, it was not that they were fully realized by birth; rather, they rose up by transforming themselves, completed themselves through self-cultivation and conscious activity, and only having exerted themselves did they become perfected\" (Xunzi 4.10\/ICS Xunzi 4\/15\/13\u201314.).\n\nWhat Xunzi offers is a model of how moral change becomes possible through an intelligible process of practical self-cultivation. Xunzi's view of emotions is an essential feature of his ethical vision of self, for the very moral ambivalence of the emotions allows him both to lay claims as to how people actually are, and to provide a more dynamic picture of how they are capable of change. But the moral significance of the emotions goes beyond this for Xunzi. At another level, positive emotions in the forms of pleasure and joy are also essential to his conception of the fully realized life. And it is in this aspect that Xunzi's thought bears a strong resemblance to the ideas contained in the Analects of Confucius. If the emotional nature of ordinary people leads them away from the proper path of living, that of the sages is in perfect harmony with the Way, which is why \"the sage follows his desires and is one with his feelings, but what regulates them is moral principle (li \u7406).\" This is also why, in contrast to the person of humane principles, whose thought is \"reverent\" the thought of the sage is \"joyous\" (le \u6a02) (Xunzi 21\/493-4). Cultivating one's emotional nature in accordance with moral principles does not mean that one's desires will be moderated or suppressed. Quite the contrary: it makes the very satisfaction of these desires possible. Thus, as for the junzi, who through learning has come to perfect himself: \"His eyes come to love it more than the five colors, his ears come to love it more than the five sounds, his mouth comes to love it more than the five flavors and his heart-mind benefits from everything in the world\" (Xunzi 1.14\/ICS Xunzi 1\/4\/18\u201319.).32\n\n## 6 Conclusions\n\nAs I have endeavored to trace here, a central thread uniting the early tradition of Confucian thought was a shared recognition of human emotions and desires as a defining aspect of the human character, and an attempt to conceptualize morality as developing out of these emotions. How thinkers saw the link between emotions and morality varied in fundamental ways: if for Confucius and Mencius, a certain emphasis on genuineness of feeling and on the unity of inner and outer meant that the affective life offered a direct channel to moral understanding and engagement with the world, for Xunzi and the author of the Xing Zi Ming Chu the emotions implied, or necessitated, the creation of cultural forms that helped to guide the self along the proper path to sagehood. And by their very character, the emotions pointed to the existence of multiple oppositions within us all \u2013 between nature and artifice, inner and outer, passiveness and activeness, emptiness and fullness \u2013 which we traversed through our endeavor to cultivate and realize our human moral potential.\n\nBased on our reading of all of these thinkers, we see that the early Confucian tradition \u2013 far from suppressing or rejecting the emotions \u2013 affirmed their importance as the very foundation and starting point of the proper life itself. We see, moreover, that this tradition was also united by a shared vision of the self as an entity that is in constant contact with the real things of the world, and that takes its direction from its encounters with these things. On both these points, early Confucian thinkers were in fundamental disagreement with the assumptions of its major critics, including the Mohists and, more significantly, the Daoists. A recurring theme in the Daodejing \u9053\u5fb7\u7d93 is that emotions \u2013 particularly the desires (yu \u6b32) arising from the physical senses \u2013undermine the ultimate human goal of achieving a state of total perceptual awareness of, and unity with, the Way. Thus the text repeatedly warns against the dangers of the senses:\n\n> The five colors make man's eyes blind;\n> \n> The five notes make his ears deaf;\n> \n> The five tastes injure his palate. (Daodejing Ch. 12, Lau 1963: 68)\n\nIt also calls for the need to lesson one's desires: for \"no crime is greater than having excessive sensual desire (ke yu \u53ef\u6b32).\" (Daodejing Chap. 46, Lau 1963: 128)\n\nMore ambiguously, the \"Inner Cultivation\" (Nei ye \u5167\u696d) chapter of the Guanzi describes sageliness as a state in which the natural completeness of the mind is allowed actualize its \"inner reality\" (qing \u60c5). Here, the presence of emotions and desires is understood as disrupting our ideal state of quiescence (jing \u9759), setting our lives in disorder (luan \u4e82) and undermining our ability to find contentedness (huan \u6b61) in life. In this case, however, the text does not simply denounce the emotions and the disturbances caused by them, but offers a way to join the realm of emotions with the higher faculty of perception by opening the possibility of managing the emotions constructively:\n\n> If joy and anger (xi nu \u559c\u6012) are excessive,\n> \n> Deal with them in a planned manner.\n> \n> Moderate the five desires and get rid of the two violent emotions.\n> \n> Be neither joyous nor angry, then equanimity and good judgment will fill your breast. (Guanzi Ch. 36; Rickett 1998:12)\n\nThe goal is not simply to transcend the body and to eliminate all traces of emotions and desires, but to achieve a unity of body and mind so that one achieves a state of contentedness and happiness (fu \u798f) arising from a sense of ease and oneness within one's self. Accordingly, the sage is not simply one who possesses no desires, but is capable of cultivating himself in such a way that self-destructive emotions and desires can be eliminated.\n\nWhat emerges from these competing accounts of the place of emotions and desires in moral life is an ongoing debate over what it means to be human. This controversy was deeply enmeshed in a host of other contemporary disputes over such enduring questions as the sources of morality and the proper norms by which to live; the nature of our relationship to things and to others; the meaning of kingship and statecraft; and the origins and function of culture. That early thinkers turned to the realm of emotions and desires in their search for answers reveals a deep fissure between two possible visions of self: one in which the individual possessed agency, control, and the power of self-determination versus one in which the self was subject to external influence and vulnerable to the motions of the external world. The implications of these alternatives for power are only too clear, and it was thus hardly a coincidence that during this time, when the knowledge of the body, moral psyche and the cosmos were being mapped onto a unified world picture, the reality of empire was just looming over the horizon.\n\nReferences\n\nAllan, Sarah. 1997. The way of water and sprouts of virtue. Albany: State University of New York Press (Analyzes early Chinese philosophy by way of two basic and pervasive metaphors \u2013 water and plants).\n\nAllan, Sarah, and Crispin, Williams, eds. 2000. 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In ed Kline III, and Ivanhoe.\n\nFootnotes\n\n1\n\nThe question of whether early Chinese thinkers themselves saw thinking and feeling (or reason and emotion) as distinct has been an issue of deep controversy. See, e.g., the debate over the cognitive content of Mencius' view of emotions in Wong 1991 and Ihara 1991. Recently, Liu Qingping has challenged the traditional enterprise of analyzing the history of Chinese philosophy in terms of the presence or non-presence of \"rational principles\" by proposing that Chinese thought is rooted in \"emotional principles\" (see Liu 2011).\n\n2\n\nFor positions on this debate over early meanings of qing, see, among others, Graham 1990: 59; Hansen 1995: 194\u2013203; Allan 1997: 85; and Puett 2004: 37\u201343. Christoph Harbsmeier has written about the wide semantic range of the term qing, differentiating among seven basic groups of meanings. See Harbsmeier 2004. Eifring 2004 offers the most focused and historically comprehensive survey of qing and its evolving conceptions from antiquity through imperial times.\n\n3\n\nFor an extensive discussion of how this correspondence plays out in traditional Chinese thinking about literature, see Owen 1985. Among later thinkers who opposed this merging of self and world (as conceived in terms of the pairing, qing \u60c5 and jing \u666f) and stressed the importance of recognizing the subjective and objective as distinct realms, was Wang Fuzhi, who believed that their confounding resulted in a failure to see things in their true aspect. On this subject see Wong Siu-kit 1978.\n\n4\n\nLike many early texts, there is a messy textual history behind the Analects and we cannot, properly speaking assume that the received text represents the collected sayings of a historical figure Kong Qiu of Lu. I will refer to the Master's voice in the text as \"Confucius\" out of a convenience justified by its general unity of perspective.\n\n5\n\nThe translations given in this chapter are modified from Lau 1982. The given source passages from the Lunyu follow the format in Lau, Wah and Ching (1995a), which is given as ICS Lunyu. I have also actively consulted Edward Slingerland's 2003 translation.\n\n6\n\nShun Kwong-loi has thus argued for a broadened conception of ren as a \"cluster of emotional dispositions and attitudes\" that are embodied in li. In this context, Shun notes, li should likewise be considered more broadly as various norms that govern human conduct,\" rather than as \"ritual propriety,\" which is the conventional translation. Shun 2002: 67.\n\n7\n\nHerbert Fingarette has likewise argued for an understanding of Confucian ritual in a way that emphasizes the aspect of commitment \u2013 focusing, however, on the way in which ritual enables individuals to actualize their virtue and create a sense of the sacred through the forging of community. In this way, ritual enables one's moral convictions to become outwardly visible, transmuting moral identity into socially-embedded conduct (Fingarette 1998: 1\u201317).\n\n8\n\nSee Yuri Pines' important study of the intellectual and historical background to the emergence of Confucian thought (Pines 2002).\n\n9\n\nThe \"we\" here needs to be qualified, for Confucius does not necessarily assume a universal \"we.\" It is, after all, only those who have embarked upon an arduous path of self-cultivation who are able to \"fully realize their desires without crossing the line.\" Moreover, as Erica Brindley has shown, the very possibility of this self-cultivation is strictly delimited by class and gender (Brindley 2009).\n\n10\n\nThis increasing emphasis on realms of human life that are within human control can be seen in the evolving history of core concepts that would become central to Confucius' ethical vision: tian \u5929 (Heaven), de \u5fb7 (virtue), ming \u547d (mandate) ren \u4ec1 (humaneness, etc). See Pines 2002: 164\u2013204.\n\n11\n\nHerbert Fingarette has noted the \"absence of an elaborated doctrine of an 'inner psychic life'\" in the Analects, arguing, however, that the vision of self found in this text is distinctive in Asian philosophical traditions in its assertion of the self and its aspirations as both positive and meaningful. Based on his examination of vocabulary connected to self and to willing (ji \u5df1, shen \u8eab, y\u00fc \u6b32, and zhi \u5fd7), Fingarette concludes, \"Confucius is affirmative, rather than negative, about the role of personal will and of the self it expresses. Specifically, Confucius appeals to us to activate our will \u2013 something only we as individuals can do \u2013 as a prime means of realizing the ideal life\" (Fingarette 1979: 134).\n\n12\n\nSee Behuniak 2005: 1\u201322 for a thorough account of the conceptual history of qi and its connection to moral thought in Mencius' thought and in Warring States philosophy in general. On the central role of qi in Mencius' conception of the heart and of self-cultivation, see Chan 2002b.\n\n13\n\nFor a comprehensive discussion of the moral and conceptual background of the issues discussed here, see Shun 1997.\n\n14\n\nMencius 4A15; ICS Mengzi 7.15\/38\/14\u201315. \"Mencius said, 'There is in man nothing more ingenuous than the pupils of his eyes. They cannot conceal his wickedness. When he is upright within his breast, a man's pupils are clear and bright; when he is not, they are clouded and murky. How can a man conceal his true character if you listen to his words and observe the pupils of his eyes?\" Translations from the Mencius are adapted from Lau 1970. Original passage citations follow the format of Lau, Wah and Ching (1995b), which is given as ICS Mengzi. I have also consulted Van Norden's 2008 translation and commentary.\n\n15\n\nI follow James Behuniak (who in turn follows Dobson, Legge, and Chan) in reading xin \u5fc3 as \"feeling\" rather than \"mind\" or \"heart-mind\" here since it is clear that that Mencius is referring to specific feelings and not a particular physical faculty or organ. There are, however, passages in which Mencius has in mind a particular faculty and not simply the operations of this faculty, as in Mencius 6A15, discussed below. For a discussion of the problem of translating xin, see Behuniak 2005: 26.\n\n16\n\nBoth Lau 1970: 163 and Shun: 214\u2013215 follow this line of reading qing and translate the term as \"what is genuine.\"\n\n17\n\nVan Norden points out this key distinction between 2A6 and 6A6, and proposes taking si as \"reflection\" in the following sense: \"'Reflection'\" is focusing one's attention upon and thinking about one's feelings and the situations that elicit them. It is an activity that involves feelings, thoughts, and perception. (Van Norden 2008: 149).\n\n18\n\nIn 7B35 Mencius also emphasizes the importance of reducing desires, which also suggests that what is problematic is not desire per se, but misdirected and immoderate desire.\n\n19\n\nOn Mencius' disputes with the views of other philosophical schools including the Yangists, Mohists, and Legalists, see Graham 1990, Bloom 1994, and Chen 2002.\n\n20\n\nIndeed, a major re-evaluation of early Chinese thought has been underway since the Guodian findings, and scholars like Tang Yijie have suggested that one major impact that the discovery of these texts has had on our understanding of Warring States philosophy is that it makes clear the central role of emotions in early thought. These findings have likewise prompted scholars to speculate that there existed a \"Si-Meng\" school (Si Meng Xue Pai) of thought that included Mencius, the author of the Guodian texts, and authors of critically significant chapters of the Li Ji, including the Yue Ji and the Zhong Yong. See Li Xueqin 1999: 75\u201379, Liao Mingchun 1999: 36\u201374, and Behuniak 2005: xviii-xx. On the Guodian materials and their intellectual content and contexts, see esp. Allan 2000, Ding 2000, GCBM 1999, SGCBM 1999, and PISGCBM.\n\n21\n\nthe term xin assumes a more cognitive dimension in this text and its meanings cannot be effectively rendered as \"heart,\" as in the case of Mencius.\n\n22\n\nOther Guodian texts containing discussions of human nature (xing \u6027) and feelings (qing \u60c5) are the Cheng zhi wen zhi \u6210\u4e4b\u805e\u4e4b and the Yucong \u8a9e\u53e2 nos. 1, 2, and 3. See Puett 2004 and Tang 2003.\n\n23\n\nI follow the published version of this text in Jingmen Shi bowuguan 1998. I have been greatly aided in my translation of this and subsequent passages by Michael Puett's readings in Puett 2004.\n\n24\n\nJohanna Liu theorizes about the larger philosophical meaning of qing in her assertion that, in the Xing Zi Ming Chu, qing is \"understood as the beginning of openness to the other in terms of 'all things' and 'yi [propriety]' as the ending, the final fulfillment towards which human feeling tends.\" Liu 2008: 71.\n\n25\n\nAs Michael Puett has argued, the ethical significance of qing in this text lies in its invocation as an explanation and justification of cultural institutions. Puett 2004: 43\u201350.\n\n26\n\nIn English the only complete translation of Xunzi's writings is Knoblock 1988\u20131994. Notable partial translations include Watson 1963 and Hutton 2001. I have actively consulted and borrowed from Knoblock's translation in his renderings of certain passages. The passage numbering follows Knoblock and original source citations follow the format in Lau, Wah and Ching (1996), which will be given as ICS Xunzi.\n\n27\n\nFollowing Knoblock in translating bu mei \u4e0d\u7f8e.\n\n28\n\nA major concern in Puett's scholarship on qing (as well as on the issue of creation, zuo \u4f5c) is the issue of ambivalence, which points to a deeper agenda to justify the creation and use of culture. See Puett 2004. On this see Puett 2001.\n\n29\n\nOn Xunzi's account of self-cultivation as a process of shaping and habituation new dispositions and desires, see Kline 2006.\n\n30\n\nBorrowing Knoblock's phrasing for fen \u5206.\n\n31\n\nCompare with similar statements in Guanzi 49, \"Neiye,\" 16.3a and Zhuangzi 20, 7.9a.\n\n32\n\nThis passage has been significantly altered from Knoblock, who reads it as \"His eye comes to love the five colors\" etc., which is implausible given that Xunzi regards sensual pleasure as part of the inborn nature. It is much more likely that Xunzi is referring here to the new-found appreciation for learning in those who have fulfilled their human potential through self-cultivation. Michael Nylan elaborates at length upon Xunzi's emphasis on pleasure as an end-product of self-cultivation. She notes how this concern was part of a larger discourse in Warring States\/early Han thought in which the most prominent thinkers focused on the achievement of pleasure as \"one of the best possible tools of persuasion to induce members of the ruling class to right action\" as they conceived of it (Nylan 2001b: 73).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_10\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 10. Art and Aesthetics of Music in Classical Confucianism\n\nJohanna Liu1\n\n(1)\n\nDepartment of East Asian Studies, University of Toronto, Toronto, Ontario, Canada\n\nJohanna Liu\n\nEmail: chienm.liu@gmail.com\n\nAbstract\n\nWhen we look into the classical Confucianism for its interest in art and aesthetics, what immediately comes to our attention is its emphasis on music and poetry, as shown in the fact that both the lost Yuejing \u6a02\u7d93 (Classics of Music) and the Shijing \u8a69\u7d93 (Classics of Poetry) were regarded as belonging to the six fundamental Confucian Classics, liujing\u516d\u7d93. It is also confirmed in the recently unearthed Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry) and the Xing zi ming chu\u6027\u81ea\u547d\u51fa (Nature comes from Mandate). This chapter will focus on the philosophical issues of music (yue) as discussed in the recently unearthed Xing zi ming chu, in reference to the Yueji\u6a02\u8a18 (\"Record of Music\") chapter in the Liji\u79ae\u8a18 (Book of Rites).\n\n## 1 Introduction\n\nWhen we look into the classical Confucianism for its interest in art and aesthetics, what immediately comes to our attention is its emphasis on music and poetry, as shown in the fact that both the lost Yuejing \u6a02\u7d93 (Classics of Music) and the Shijing \u8a69\u7d93 (Classics of Poetry) were regarded as belonging to the six fundamental Confucian Classics, liujing \u516d\u7d93. It is also confirmed in the recently unearthed Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry) and the Xing zi ming chu \u6027\u81ea\u547d\u51fa (Nature comes from Mandate). This chapter will focus on the philosophical issues of music (yue) as discussed in the recently unearthed Xing zi ming chu, in reference to the Yueji \u6a02\u8a18 (\"Record of Music\") chapter in the Liji \u79ae\u8a18 (Book of Rites).1\n\nYue \u6a02, as one of the six arts (liu yi \u516d\u85dd) pertaining to the classical training of Confucian scholars, has a rather ambiguous feature as to its theoretical status, especially when compared with shi (poetry). The difficulty consists not only in the historical fact that the Classics of Music was lost after the burning of books in Qin Dynasty, but also due to the complicated relationship between yue and li \u79ae (ritual). Besides, on the linguistic and semantic level, the Chinese character yue \u6a02, which represents music, is endowed with the double pronunciations yue\/le and the double meanings music\/pleasure. The recently unearthed Xing zi ming chu, among other bamboo slips unearthed at Guodian \u90ed\u5e97, with a major treatise on yue, provides us with a new clue to re-think the aesthetic meaning of yue in classical Confucianism. This is the main purpose of this chapter. By the strategy of intertextuality, which is applicable to the reading and interpretation of text, the first part of this chapter will contrast the texts on yue in the Xing zi ming chu with other ancient texts in the Liji \u79ae\u8a18, the Zuozhuan \u5de6\u50b3 (Zuo's Commentary on Spring and Autumn Annals), the Xunzi \u8340\u5b50, the Shiji \u53f2\u8a18 (Record of the Grand Historian), etc., to identify the problems to be re-defined and re-understood, such as the ideas of music pursued by Confucian scholars, the crisis produced by Zheng \u912d and Wei \u885b music as new sounds and melodies criticized by the Confucians, the place of music in the self-cultivation of junzi \u541b\u5b50, etc. The second part of this chapter will focus on the aesthetic meaning of music by referring to the Confucian theory of qing \u60c5 (sentiment, affection, situation), also based on the Xing zi ming chu, which has dealt with the in-depth relation between yue and qing.\n\n## 2 Intertextuality and the Interpretation of Xing zi ming chu\n\nThe text entitled Xing zi ming chu, written on bamboo slips discovered at Guodian in 1993, has been considered by contemporary scholars to be one of the most important unearthed documents pertaining to the theory of music in pre-Qin Confucianism. Researchers could find, in its transcribed version established by the Jingmen \u834a\u9580 Museum and published by Wenwu \u6587\u7269 Publisher in 1998, that one third of the text, distributed among 67 pieces of bamboo slips, is devoted to the discussion of music. It would be probably too rash to claim that a new theoretical understanding of classical Confucian music could be built upon this newly unearthed text, because of the fragmentary character of the text itself and the uncertainty of its authorship. Nevertheless, it undoubtedly provides us with at least a new view and a critical reflection on the insufficiency of the received theories, which have been based on other conventional texts that consider Confucian music mostly from its cultural ideological function in keeping peace and harmony in the society, rather than as an art with which people can enjoy more or less purely aesthetic value.\n\nA comparative study of the similarity between the text of Xing zi ming chu and other known texts in the Liji and Yueji, the Zhongyong \u4e2d\u5eb8, the Xunzi, etc., has led many scholars to infer that the Xing zi ming chu could be attributed to the so called Si-Meng Xuepai \u601d\u5b5f\u5b78\u6d3e (Zisi \u5b50\u601d and Mencius' \u5b5f\u5b50 School). One of the contributions of this line of research consists in having traced some texts in the Liji, especially that of the Yueji chapter, back to the period of Warring States. It concerns also some problems involved in the debates between scholars of jinwen \u4eca\u6587 (New Text) and guwen \u53e4\u6587 (Old Text) about the authorship of the Yueji. 2\n\nThe main focus of this chapter is not to get involved in the debate about the authenticity and authorship of these unearthed texts. Instead, the problem with which we are concerned is how to achieve an in-depth understanding of the aesthetic\/artistic meaning of music (yue) in classical Confucianism, through the application of the reading strategy of intertextuality or intertextual analysis to the Xing zi ming chu. The term \"intertextuality\"3 was coined by Julia Kristeva in 1967 and later developed by Roland Barthes. According to Julia Kristeva, every text is \"constructed as a mosaic of quotations,\" and \"absorption and transformation of another.\" (Kristeva 1986: 37) Kristeva claims that reading is an on-going dialogue between the writing subject, the addressee (or ideal reader), and other exterior texts, and she suggests that the text be viewed by both horizontal and vertical axes, since \"the word's status is thus defined horizontally (the word in the text belongs to both writing subject and addressee) as well as vertically (the word in the text is oriented towards an anterior or synchronic literary corpus)\" (Kristeva 1986: 36\u201337). Roland Barthes develops this idea of textual intersection and considers every text as the outcome of interconnection of cultural artifacts. He says,\n\n> One of the paths of this deconstruction-reconstruction is to permute texts, scraps of texts that have existed or exist around and finally within the text being considered: any text is an intertext; other texts are present in it, at varying levels, in more or less recognizable forms: the texts of the previous and surrounding culture. (Barthes 1981: 74)\n\nThat is to say, a text is never a solitary or isolated work done by an isolated writer, but a network of writings by quoting one text from another, or by alluding one text to another, through and by which a continual deferment of an idea or a meaning in a particular culture would be able to continue.\n\nIn view of the literary texts in Chinese classics, this type of intertextuality could be found almost everywhere since the time of Confucius, who claims that: \"I transmit but do not innovate\" (Analects 7.1, my translation). In this chapter, the study of Xing zi ming chu could be seen as a good example for decoding Chinese textual meaning by intertextual analysis, which takes the Xing zi ming chu as an interconnected body of cultural texts from both synchronic (horizontal) and diachronic (vertical) views. According to the results of scientific examination of all excavated relics in the Guodian Chu tomb, it is supposed that those bamboo slips and their writings were transcribed presumably no later than 300 BC, that is, in the middle-late period of Warring States. The owner of these scripts was supposed to be a Confucian scholar of Chu \u695a,4 arguably a teacher of the crown prince Heng \u6a6b, skilled at reading both Confucian Classics and Daoist texts, as evidenced by the co-existence of fragments related to both Zisi and Laozi \u8001\u5b50. Some parts of the unearthed texts, including the Xing zi ming chu, are apparently related to other Confucian Classics. Some scholars assume that the author was a follower of Zisi and Mencius. The text, supposed to have been used by the owner as teaching materials, could be viewed horizontally, as having a dialectic relation between addresser (compilers\/teachers), addressee (readers\/students), and vertically, as interacting with previous texts and various forms of its contemporary cultures.\n\nBy contrasting the text on music in the Xing zi ming chu with other texts quoted from other Confucian Classics, or when alluded to in other texts, or otherwise in connection with the cultural form of the day, two main questions remain to be asked and examined: (1) What is the artistic meaning of music in classical Confucianism? (2) How is the aesthetic foundation of Confucian music related to the concept of qing \u60c5 (sentiment)? Some other issues relevant to the Chinese aesthetic of music will also be discussed, such as the ideal of music that Confucian scholars were pursuing; the symbolic meaning of ritual music; the crisis of Zheng \u912d and Wei \u885b music as new sounds and melodies criticized by early Confucians, and the place of music in the four ways of self-cultivating of a junzi \u541b\u5b50.\n\n## 3 The Artistic Meaning of Music in Classical Confucianism\n\nGenerally speaking, the Xing zi ming chu, as one of the Confucian teaching materials, is an article in which we find major discussion of the means of self-cultivation to become a junzi by way of music, given that music is supposed to contain spiritual power that may have influence on the formation and transformation of human nature. In it, there is a particular paragraph that elucidates the educational role of music as one of the three arts by which the Sages teach the way of realization of the Dao in humans that allows them to get along harmoniously with all things. It reads,\n\n> The Dao is a way of getting along with all beings. The major concern of Dao consists in the art of mind. Among the four arts\/ways to the Dao, only the Art\/Way of being human is the way through which Dao could manifest itself. The other three arts\/ways (e.g. the art of poetry, the art of history, and the art of ritual music [liyue]) are human ways of expressing the Dao. Poetry, history and ritual music, all these three are originally produced by human beings. Poetry is versed by capable persons, history is narrated by capable persons, ritual music is performed by capable persons. (Guodian [Guodian Chu mu zhu jian] 1998: 179)\n\nUnder my textual analysis, there are three points implied in this paragraph that deserve our attention:\n\n1.\n\nDao means the human Dao by which one is supposed to get along well with all beings, including those from Heaven, from Earth, and among people.\n\n2.\n\nThe ways of Dao contain two levels: human Dao and three arts (san shu \u4e09\u8853), including poetry (shi \u8a69), history (shu \u66f8) and ritual music (liyue \u79ae\u6a02).5\n\n3.\n\nThe three arts (san shu) are originally produced by those who are capable of carrying out the human Dao.\n\nOur further question now is how to understand the formational meaning of liyue in the context of Confucian culture. How should we understand the meaning of liyue: li of yue, or yue of li? What kind of music does the liyue refer to? Why do Confucian scholars emphasize the self-cultivating function of liyue? What have they learned from liyue? Furthermore, what is the artistic meaning of liyue from the viewpoint of Confucian aesthetics?\n\n## 4 Religious Function of Ritual Music in Zhou Dynasty\n\nIt was an old tradition in Confucian culture to consider music as having a transforming power on the individual heart\/mind and on social customs. Since Zhou Dynasty, music had been considered as one important topic in the educational curriculum including four disciplines for cultivating the sons of the royal family and eminent people selected from the State to be trained as prominent future leaders. In the Liji it was said in the chapter \"On Royal Regulation\" (Wangzhi \u738b\u5236) that,\n\n> The (board for) the direction of Music gave all honour to its four subjects of instruction, and arranged the lessons in them, following closely the poems, histories, ceremonies, and music of the former kings, in order to complete its scholars.... The eldest son of the king and his other sons, the eldest son of all the feudal princes, the sons, by their wives proper, of high ministers, Great offices, and officers of the highest grade, and the eminent and select scholars from (all) the states, all repaired (to their instruction), entering the schools according to their years. (Legge 1967a: 232\u2013233)\n\nAlso in Chap. VII, \"King Wen as Son and Heir\" (Wenwang shizi \u6587\u738b\u4e16\u5b50), it was said that, \"In the education of the crown princes adopted by the founders of the three dynasties, the subjects were the rules of propriety and music\" (Legge 1967a: 349).\n\nAccording to the chapter \"Spring Ministry\" (Chunguan \u6625\u5b98) in the book of Zhouli \u5468\u79ae (the Rituals of Zhou), it was the director of music (dasiyue \u5927\u53f8\u6a02) who took charge of the school of grand studies (chengjun \u6210\u5747), and taught the heir-sons and the young generations the six ways of music performance with ethical values,6 that is, centrality, harmony, respect, moderation, piety, and friendship, and taught them the six artistic forms of musical language, that is, figurativeness (xing \u8208), discourse (dao \u9053), ironic (feng\u8af7), narrative (song \u980c), speech (yang \u8a00), and wording (yu \u8a9e). After they became capable of performing music with ethical values and expressing music in various forms of musical language, the heir-sons and eminent young scholars were taught the six pieces of ritual dance inherited from previous dynasties: Cloud Gate (Yunmen \u96f2\u9580) and Grand Scroll (Dajuan\u5927\u5377), Grand Concord (Daxian \u5927\u54b8), Grand Shao (Dashao \u5927\u97f6), Grand Majesty (Daxia \u5927\u590f), Grand Exaltation (Dahuo \u5927\u6fe9), and Grand Warrior (Dawu \u5927\u6b66) (see Zhouli).7 The objective of teaching the heir-sons to play short flute and string music and to perform the various kinds of ritual dance consisted in cultivating their capacity to conduct ceremonies with ritual music, rather than to become professional musicians such as vocalists, instrumentalists, or composers, all these roles often being played by the so-called gu meng \u77bd\u77c7 (the blind).\n\nAs to the value of music, what has been stressed in the Zhouli and Liji was its religious function in the rituals of sacrificial offering, such as the sacrifice to Heaven and Earth offered by Son of Heaven, that to the spirits of the land and grains by princes of the states, and the five sacrifices of the house offered by great officers.8 All ceremonies of offerings were accompanied by performances of different kinds of ritual music, songs, and dances.9 The six pieces of ritual dance are accompanied by the ritual music of the six dynasties in ancient China. As with other ancient civilizations in the world, the complete repertoires of their music performance were lost, but in China some textual descriptions of the titles, the performances, and the religious and social-political functions of its ancient ritual music still remained and could be read in some texts in the Zuozhuan, the Analects, the Liji, the Zhouli, the Guoyu \u570b\u8a9e, the L\u00fc shi chun qiu \u5442\u6c0f\u6625\u79cb, etc., which could still serve as textual evidence revealing to us a certain idea about the function of music in pre-Qin China.\n\nApparently, in referring to the above texts, the Xing zi ming chu's mention of watching the ritual dances of Lai \u8cda and Wu \u6b66, Shao \u97f6 and Xia \u590f, could be understood as dealing with and manifesting the religious value of ritual music from an aesthetics point of view. As it reads,\n\n> In watching the dance of Lai and that of Wu, there arises a feeling of being well arranged in order. In watching the dance of Shao and that of Xia, there arises a sense of beauty of simplicity. (Guodian: 180)10\n\nNow we can be sure that, in classical Confucian education, learning ritual music, for the heir-sons, is different from learning music for self-entertainment and for passing leisure time. For them, the purpose of learning music is to cultivate their spiritual sensibility, with heavy aesthetic and ethical implications, to the revealing of Heaven, Earth, and ancestors, through their training in the art of sounds. This means the religious function of ritual music has its aesthetic foundation in the human mind, as expressed by the word \"qing\" in the Xing zi ming chu. Before we discuss in more detail the relation between music and qing, we have to review briefly the shift of musical value in Confucian thought from religious function to more humanistic concerns.\n\n## 5 The Confucian Idea of Music as a Way of Self-Cultivation to be an Integral Person\n\nAlong with the collapse of Zhou aristocracy and the rise of various schools of thought in the periods of late Spring and Autumn and early Warring States, the right to receive education was no longer the privilege of royal family members. In this process, the value of music in the cultivation of the human heart\/mind degenerated. For example, the Laozi emphasized the quietness and silence of Nature, and claimed that too many sounds (five tones) would make people deaf. In the Mozi \u58a8\u5b50we find a chapter that criticizes music and there we read the claim that indulgence in the pleasure of music was a cause of corruption. Among various intellectual schools, classical Confucianism was the only school that kept the traditional idea of education and put the emphasis on the important cultural meaning of music. Confucius himself was a man of music; he used to sing, play musical instruments such as qing \u78ec, qin \u7434and se \u745f, and he even knew how to compose a piece of musical work.11 He had put to right order the repertoires of music for the Odes, and corrected their tones after his trip from Wei \u885b back to Lu \u9b6f.12 He had discussed issues related to the performance of music with the Grand Music Master of Lu saying, \"How to play music may be known. At the commencement of the piece, all the parts should sound together. As it proceeds, they should be in harmony while severally distinct and flowing without break, and thus on to the conclusion\" (Analects 3.23).\n\nConfucius taught his disciples music as one of the six arts, and considered music an essential element in the completion of cultivation of a junzi or a condition sine qua non of a complete (integral) person.13\n\nMusic, considered as essential to the formation of a complete (integral) person, did not consist merely in musical performance such as playing an instrument, but in the realization, through music, of the human Dao, i.e.. the virtue of humanity (ren), without which music, as an art of sound, would become meaningless. Confucius said: \"If a man be without the virtues proper to humanity, what has he to do with music?\" (Analects 3.3) Only with the human Dao of ren, would music become properly a human art of playing with sound, and thereby the following question, proposed by Confucius himself, would have the possibility of finding an answer: \"Ritual, ritual, does it mean no more than gems and silk? Music, music, does it mean no more than bells and drums?\" (Analects 17.11)\n\nBasically, this question touched upon a crucial problem in Chinese aesthetics of music, and would prompt a series of questions on the essence and existence of music as an art. How could the sounds of bells and drums be musical and be considered as belonging to the art of music? If the answer is that their sounds are produced merely by the performance of a musician, then, what kind of music player could be considered a musician-artist? If the answer is that those who know how to play bells and drums and perform in a way that is proper to music, then the question will turn back to the original question: What is music? Who is a musician?\n\nIn contemporary Western philosophy, Martin Heidegger has taken \"the Coming of Being\/Truth in things\" as the starting point for answering questions on the origin of a work of art (Heidegger 1971: 17\u201376). In comparison, classical Confucians would take a different approach than the ontological one taken by Martin Heidegger in answering these questions. They took an ethico-aesthetic approach to consider the artistic value of \"music\" and \"musician.\" For example, the passage on the origin of music in the \"Record of Music,\"14 emphasized the moral relation between music and the human mind, rather than the technique of composing the sounds in music. Thus only junzi can understand the profound meaning of music as art. It read,\n\n> All modulations of sound take their rise from the mind of man; and music is the intercommunication of them in their relations and differences. Hence, even if beasts know sound, but they know not its modulations; and masses of the common people know the modulations, but they do not know music. It is only the superior man who can (really) know music. (Legge 1967b: 95)\n\nIn Confucian thought, music should always go along with the practice of li (ritual, propriety). Anyone who is good at musical sounds, but not familiar with li, won't deserve the name of a good musician, that is, a musician as a complete (integral) person. This is illustrated by a story that was told about Kui \u5914, who was reputed at the practice of musical sounds, but there was a rumor that he had only one leg (yizu \u4e00\u8db3). Duke Ai of Lu \u9b6f\u54c0\u526c doubted it and went to ask Confucius. Confucius explained that Kui was not a person with one leg, but a man who was capable only of playing sounds, which was insufficient, or one-legged (zu \u8db3) in metaphor, for a good musician. That is to say Kui is merely sufficient (zu \u8db3) as a musician, not a \"good\" musician.15\n\nIn a dialogue on the virtue of li, during Confucius' leisure time at home, Zi Gong\u5b50\u8ca2 asked a similar question about whether Kui was a good musician. Confucius explained, \"To be versed in the ceremonial usages, and not versed in music, we call being poorly furnished. To be versed in music, and not versed in the ceremonial usages, we call being one-sided. Now Khuei [Kui] was noted for his acquaintance with music, and not for his acquaintance with ceremonies, and therefore his name has been transmitted with the account of him (which your question implies)\" (Legge 1967b: 275\u2013276).\n\nIt is clear then, for Confucius, Kui was a man who knew enough musical sounds and performed music well, but his one-sided knowledge was not enough for him to become a good musician in the sense of having a real knowledge of music as completing his human personality.\n\n## 6  Qing as the Aesthetic Foundation of Confucian Music\n\nThe purpose of learning music was not merely to know musical sounds, but, more than that, to cultivate the capacity of realizing human Dao in its completeness. The humanistic meaning of music is therefore based on the Confucian theory of self-cultivation, which now gradually takes on a more significant role than its religious function.\n\nIn the Xing zi ming chu, the cultivation of music as an art should go along with the cultivation of li \u79ae (rule of propriety), shi \u8a69 (poetry), and shu \u66f8 (History), considered as the san shu \u4e09\u8853 (three arts), constituting thereby an integral way of orientating toward the human Dao. By learning shi, shu, liyue \u79ae\u6a02, the ability of junzi would gradually develop under the teaching of the Sages, which consisted in the formation of a complete (integral) human ability of unifying all things by analogy, learning lessons from observing the sequence of things, measuring human activities by examining the righteousness of will, and ordering human feelings in receiving them in and in expressing them out (Guodian: 178).16 The ability was obtained from the training of shi, shu, liyue functions as a whole, without neglecting one or the other, no matter by way of san shu (three arts) or liu yi (six arts). It makes sense that in the Xing zi ming chu, the emphasis on music's value in self-cultivation in no way neglected its relation with poetry (ability of using language), with ritual propriety, and with history.\n\nAccording to the Xing zi ming chu, the realization of human Dao should start from cultivating the ability of feeling (qing \u60c5). The Xing zi ming chu said: \"The Dao begins with qing (dao shi yu qing \u9053\u59cb\u65bc\u60c5)\" (Guodian: 179). As Tang Yijie \u6e6f\u4e00\u4ecb has well pointed out, \"It makes sense to say 'dao begins with qing' rather than 'dao arises from qing,' because dao exists from the start on account of human qing rather than emerging out of qing\" (Tang 2003: 271). Tang Yijie explains in his notes that, \"This is not to say that it cannot emerge at all, for it can also emerge out of rationality or study\" (Tang 2003: 279).\n\nMost of scholars' discussions on qing in the Confucian classics focus on the status of qing, referring generally to the various psychological forms of emotion, such as the seven qings (joy, anger, sadness, fear, love, disliking, liking) in the Liji,17 or the six qings (likes and dislikes, delights and angers, grieves and joys) in the Xunzi, 18 in the context of their ethical discussions about the relation between the human mind (xin \u5fc3) and human nature. The explanation of the moral function of xi \u559c (pleasure), nu \u6012 (anger), ai \u54c0 (sorrow), le \u6a02 (joy) in the Zhongyong,19 and the annotations on the concept of zhonghe \u4e2d\u548c (equilibrium and harmony) in Zhu Xi's \u6731\u71b9 Zhongyong Zhangju \u4e2d\u5eb8\u7ae0\u53e5 (Commentaries on Chapters and Sentences in the Zhongyong) have well provided us with the ethical model of interpreting the meaning of human affectivity (qing), but unfortunately all of them have left the aesthetic dimension of feeling untouched. However, this aesthetic dimension is always there in the creativity of poetry and music, and also, we should say, it exists vividly in the daily life of all people.\n\nThe interpretation of qing in the Xing zi ming chu, following the Zhongyong, has also laid the foundation of feeling (qing) on human nature (xing \u6027),20 but its emphasis was put on the aesthetic function of qing in its expression through yue\/le (music\/pleasure) and li (ritual\/propriety). In the Xing zi ming chu, the term \"qing\" is understood as the beginning of openness to the other in terms of \"all things,\" and \"yi\" as the ending, the final fulfillment, toward which human feeling tends; and \"those who understand feeling can express it properly, and those who understand yi can realize it in oneself properly\" (Guodian: 179).\n\nIn short, according to the Confucian tradition, the learning of music and ritual propriety is to cultivate the capacity of a complete person as to his\/her aesthetic feeling, which is rooted in human affectivity (qing), to be integrated with his\/her moral feeling and religious sentiment, which are expressed through yi \u7fa9 and li \u79ae.\n\n## 7 The Aesthetic Dimension of yue\/le in the Xin zi ming chu\n\nApart from its ethical function in Confucian culture, music, together with the ritual propriety with which it operates, has an aesthetic dimension as well. This consists in the pleasure (le) obtained by a sympathetic feeling that is able to share the world of other people by apprehending various affections communicated through sounds produced by other people.\n\n### 7.1 Sounds and Music, Music\/Pleasure and Ritual Propriety\n\nThe \"Record of Music\" said that \"music produces pleasure;\u2013 what the nature of man cannot be without\" (Legge 1967b: 127). Enjoying the art of music by singing songs, playing musical instruments, or simply listening to a beautiful melody, is the common aesthetic experience of music among people. A famous story about Confucius studying the Chinese lute from Shi Xiangzi \u5e2b\u8944\u5b50 tells us that for Confucius, the aesthetic pleasure of music as art does not consist only in the rhythm and melody, or only in playing with the mathematically intelligible structure of sounds, but, more so, in the existential meaningfulness conveyed through the sounds of the music, understood in a humanistic way.21 This does not mean that the Confucian theory of music has neglected the embodiment of music in sounds. On the contrary, it claims that only those who know sounds are able to talk about music and understand it. The \"Record of Music\" said, \"Hence with him who does not know the sounds we cannot speak about the airs, and with him who does not know the airs we cannot speak about the music\" (Legge 1967b: 95).\n\nThrough the aesthetic feeling produced by musical experience, a human being is given access to various kinds of pleasure in sounds as well as in music, and enjoys the experience of values revealed through them. The Xing zi ming chu has vividly described the variety of pleasures in the aesthetic experience of listening, such as listening to the sound of laughter that makes one feel lively and happy; in hearing the ballad, which makes one feel contented and excited; in listening to the melody of qin and se, when a profound feeling of praise is inspired; in watching the dance of Lai and the dance of Wu, when there arises a feeling of being well arranged in order; in watching the dance of Shao and the dance of Xia, when there arises a sense of beauty of simplicity.22\n\nThe pleasure obtained from sounds can stay no longer than the echo of laughter in the air; whereas the pleasure obtained from a musical melody will last as long as it resounds in one's own mind\/heart. The experience of Confucius in hearing the music of Shao in Qi State makes him ignore the taste of meat for 3 months (Analects 7:14). This is a typical aesthetic experience of musical art.\n\nConcerning the relation between the aesthetic pleasure of music and the self-cultivation of a junzi, the Xing zi ming chu pointed out that learning spiritual pleasure through music would be the faster way to reform one's heart.23 The longer the mind keeps the spiritual pleasure of music, the more serious it would be in returning to its own original good nature and its original qing, and the more smoothly it would be in expressing outward and in receiving inward. This is the way of realizing one's virtue (see Guodian: 180).\n\nOne of the meanings of connecting li with yue consists in the fact that the practice of ritual propriety should be realized with spiritual pleasure in heart\/mind, given that the true meaning of li is based on the feeling of respect. In daily life, sincere smiling is enough to display the pleasure of heart\/mind in the friendly exchange of agreeable words. As to the diplomatic meeting among states or nations, a concert in the state or national banquet represents the magnificence of the diplomatic rituals. We need not mention again the pious feeling in the performance of ritual music during the sacred offerings in a temple. Therefore, it makes sense for the Xing zi ming chu to claim that \"smiling is the superficial side of ritual propriety, whereas music\/spiritual joy is the deep side of ritual propriety\" (guodian: 180).\n\nConfucius once described the presentation of music in diplomatic courtesy and explained the symbolic function of music performed in the diplomatic ceremony during the visit of a ruler. This happened in Confucius' leisure time at home, after he had talked to his disciples Zizhang \u5b50\u5f35, Zigong \u5b50\u8ca2 and Yan You (\u8a00\u6e38 or Ziyou \u5b50\u6e38), on the value of li.\n\n> When one ruler is visiting another ruler, they bow to each other, each courteously declining to take the precedence, and then enter the gate. As soon as they have done so, the instruments of music, suspended from their frames, strike up. They then bow and give place to each other again, and ascend to the hall, and when they have gone up, the music stops. In the court below, the dances Hsiang and Wu are performed to the music of the flute, and that of Hsia proceeds in due order with (the brandishing of feathers and) fifes. (After this), the stands with their offerings are set out, the various ceremonies and musical performances go on in regular order, and the array of officers provided discharge their functions. In this way the superior man perceives the loving regard (which directs the entertainment). They move forward in perfect circles; they return and form again the square. The bells of the equipages are tuned to the Khai-khi [\u91c7\u9f4a]; when the guest goes out they sing the Yung [\u96cd]; when the things are being taken away, they sing the Khan-yu; and thus the superior man (sees that) there is not a single thing for which there is not its proper ceremonial usage. (Legge 1967b: 274\u2013275)\n\nIn reading Confucius' detailed description of the diplomatic courtesy and music performance in the court today, we still can feel the magnificence of li and yue in ancient China. The focus of Confucius was the symbolic function of music in showing cultivated good feeling, virtue and historical knowledge, as the text goes on to say,\n\n> The striking up of the instruments of metal, when they enter the gate, serves to indicate their good feeling; the singing of the Khing Miao [\u6e05\u5edf], when they have gone up to the hall, shows the virtue (they should cultivate); the performance of the Hsiang to the flute in the court below, reminds them of events (of history). Thus the superior men of antiquity did not need to set forth their views to one another in words; it was enough for them to show them in their music and ceremonies. (Legge 1967b: 274)\n\n### 7.2 Music, Qing, and the Sentiment of Grief\n\nIt is by the aesthetic feeling, i.e. the sense of beauty, and the moral feeling, i.e. the sentiment of respect, that superior men of antiquity could set forth their views and communicate with each other without the necessity of using verbal language. That is why the Xing zi ming chu says, \"Being in trust without saying a word are those who have the sense of beauty\" (Guodian: 181).\n\nThe temporary pleasure brought about by the musical sounds is not enough to carry on the formation of an individual's virtues and a group of people's ethos. There is no need to say it is not good enough for the good governance of a country. Confucius' criticism of the songs of Zheng and Wei was in the context of his reply to Yan Yuan's question on the government of a State. For the purpose of serving as Music of a State, Confucius recommended the dance of Shao and he alerted rulers to keep away from the sounds of Zheng, due to the latter's excessive indulgence in the pleasures of sounds, which was unqualified to serve in the ritual ceremony in a temple or in the court (Analects 15.11). It seems that Confucius didn't deny the cognitive value of the sounds of Zheng that revealed a local people's ethos. What made Confucius discontented was the mixture of the court music of ya with the popular music of Zheng (Analects 17.18).\n\nAlthough the purely melodic aspect of music is not enough to be qualified for being performed in the sacred ritual, as music of ya is thus qualified, it is still quite practical for the training of musical skill of an instrumentalist or vocalist. That is why the Xing zi ming chu says, \"The ancient music is good for mind\/heart, and the new sounds are good for the fingers, both are for the cultivation of the people\" (guodian: 180).\n\nAlong with its affirmation of the aesthetic value of music, Xing zi ming chu did not ignore the aesthetic quality of the feeling of grief. In this sense it is quite different from Zi Zhang who took grief and joy as belonging to two separate categories of crying and music: \"to grief, there belong crying and tears; to joy, songs and dancing\" (Zuozhuan: 708). By contrast, the Xing zi ming chu considered pleasure and grief as a pair of feelings that produce each other: \"The extreme development of music\/pleasure accompanies itself certainly with grief. Crying, with grief, too. All of them touch human feelings\" (Guodian: 180).\n\nThe aesthetic pleasure produces a sense of openness to other people's joy, whereas the feeling of grief produces a sympathetic feeling for other people's sorrow. The sound of crying expresses the feeling of grief as well as that of pleasure. It is only in the highest form of music that brings the highest pleasure that one would be able to convey a comprehensive feeling of sympathy capable of discerning various states of mind from sounds produced by other people. The human mind tends to play with various kinds of sounds, in which crying is but one kind of decipherable sound among others. Moreover, the great music conveys the sentiment of sadness, similar to the sound of crying. And the great pleasure in music will not be separated from the sentiment of sadness. That is to say, the sentiment of grief, as the deepest feeling (qing) of the human heart\/mind, plays an important role with a primary value in the aesthetics of Chinese music since the pre-Qin period.24\n\nHowever, the extreme happiness in the heart\/mind of a junzi caused by a melody with the sentiment of grief has nothing to do with the psychological emotions, which are changing all the time with the transient process of myriad things. It refers rather to the qing, the ability of human dao, inherent to the heart\/mind of a junzi. The following passage in the Wuxing Pian \u4e94\u884c\u7bc7 (The Five Actions) reveals the deep concern of the grief embodied in junzi's mind. \"While I do not see junzi, my grief heart cannot but agitated. Now that I have seen junzi, my heart cannot be but happy\" (Guodian: 149). This passage is followed by the verses quoted from the Shijing, \"While I do not see junzi, my grieved heart is agitated. Now that I have seen this junzi, my heart cannot be happy [like others]\" (Legge 1960: IV. 23\u201324). This song is an air of the Zhaonan \u662d\u5357 chapter in the Shijing, singing on the ascent of the southern hill for the gathering of ferns, when one is surrounded by the sounds of grass-insects and the leaping of grass-hoppers. The song should be happy, evoking delightful images of the excursion and a sense of love and satisfaction. However, paradoxically, the song also transmits a mood of grief to the audience. According to the commentary of Confucian tradition, for example in the Hanshi waizhuan \u97d3\u8a69\u5916\u50b3 (Outer Commentary to the Classics of Poetry by Han Ying),25 the grief conveyed in this poem is an allegory of the sorrowful feeling of a junzi who is worried about the lack of discontentment in his daily life full of pleasurable satisfaction. In the Hanshi waizhuan, Confucius says that a junzi has three kinds of grief: the lack of knowing, the lack of learning caused by self-contented knowledge, and the lack of action after learning. 26 His saying \"not seeing a junzi\" does not only mean the missing of someone who is away and absent, but the missing of someone who lacks the sentiment of grief, which is necessary for being a junzi. In short, the didactic function of music has its ground in its cultivation of the qing of the human heart\/mind, which operates with the dialectic interaction between yue\/le (happiness) and bei (grief).\n\n## 8 The Ethico-aesthetic Theory of Chinese Music and the \"Record of Music\"\n\nThe problems of authorship and time of composition have been issues of long debate among scholars. Nevertheless, most contemporary scholars have now agreed that the Yueji (Record of Music) chapter in the Liji was compiled in Western Han by Liu De \u5289\u5fb7, Mao Chang \u6bdb\u8407 and other Han Confucian scholars, therefore much later than the period of classical Confucianism and should not be our main concern in this chapter. 27 However, since it has been claimed by scholars to be an important Confucian work on music, we will say a few words about it here at the end of this essay. In continuity with the Xing zi ming chu, the Yueji emphasized the educational, political, and moral function of music as based on the relation between qing and yue. The statements such as \"the yue\/le \u6a02 (music\/pleasure) is something unchangeable of qing \u60c5\" (Legge 1967b: 114; translation modified); 28 \"to go to the very root of [the things] and know the changes [which they undergo] is the qing (sentiment\/essence) of pleasure in music (le\/yue)\" (Legge 1967b: 114; translation modified); all indicate the essential role of qing for understanding the meaning of music. Today, it is arguable to state that the Yueji, as a representative work of Han Confucian scholars, has integrated various pre-Qin schools' thoughts on music, including Daoism, Legalism, Mohism, Yin-yang School and Miscellaneous School, into Confucian thoughts to establish its own theory of music (Cai Zhongde 2003: 252). This syncretistic discourse on the origin of music, on the relation between rites and music, on the cultural function of music, has been an undeniable contribution to the ethico-aesthetic theory of Chinese music in the Confucian tradition.\n\nOne the one hand, the Yueji follows Confucius' thought in taking music as an ideal way to self-cultivation, to improving quality of state governance, and rectifying the folklore customs; indeed an ideal way of pursing a happy life that embodies the aesthetic value of music\/ritual music in basing it on moral righteousness. On the other hand, it follows Xunzi's thought on music in epitomizing his theory and in quoting a lengthy text directly from the Yuelun \u6a02\u8ad6 (Treatise on Music) chapter from the Xunzi, to explain how musical sounds penetrate into the human mind with deep moral affections and political consequences. From Mohism, it adopts the criticism of the decadent music of Zheng and Wei in order to propose a theory on the mutual communication between musical sounds and quality of governance. For example, it says, \"The airs of Zheng and Wei were those of an age of disorder, showing that those states were near such an abandoned condition. The airs near the river Pu, at the mulberry forest, were those of a state going to ruin. The government (of Wei) was in a state of dissipation, and the people were unsettled, calumniating their superiors, and pursuing their private aims beyond the possibility of restraint\" (Legge 1967b: 94). Since the airs of Zheng and Wei allude to a kind of decadent life, therefore they are not suitable to serve as State Music. That's why when Yan Yuan asked how the government of a country should be administered, Confucius advised, \"Let the music be the Shao with its pantomimes. Banish the songs of Zheng, and keep far from specious talkers. The songs of Zheng are licentious; specious talkers are dangerous\" (Analects 17.11).\n\nThus the Confucian concept of music does not confine itself to the perception of fluency of melody, but sees in it the pursuit of the ideal state of harmony, including the cosmological harmony between Heaven and Earth, moral harmony between heaven and the human heart\/mind, and political harmony between individuals and society. In this sense, the role of qing playing on the level of emotions should be transformed and harmonized through music in order to purify the desiring heart. Thus the Yueji says, \"Hence the superior man returns to the (good) affections (proper to his nature) in order to bring his will into harmony with them, and compares the different qualities (of actions) in order to perfect his conduct\" (Legge 1967b: 112). Thus the emphasis is put on the aspect of social harmony rather than on the aspect of art. The Yueji says, \"Therefore, when the music has full course, the different relations are clearly defined by it; the perceptions of the ears and eyes become sharp and distinct; the action of the blood and physical energies is harmonious and calm; (bad) influences are removed, and manners changed; and all under heaven there is entire repose\" (Legge 1967b:112).\n\n## 9 Conclusion\n\nIt is a common sense to say that, in general, Chinese culture as defined by classical Confucianism and reputed as a liyue culture, has always put its emphasis on the educational function of music in forming individual moral characters and people's ethos. Most of the discussions on music in ancient Confucian documents have been focused on the educational and ethical effect of music, especially in emphasizing the grandiose music of ya. In this historical and ideological context, the newly unearthed Xing zi ming chu shows us a very interesting example, in which we find a continuity of the same Confucian stereotypical idea that considers music as one of the three arts (san shu), as an important way of realizing human Dao. This represents the mainstream ideas of Confucius' legacy. On the other hand, Xing zi ming chu also provides us with something new, that is, the aesthetic value of qing, the artistic value of musical sounds, and the dialectical relation between the feeling of pleasure and that of grief in music. It considers the aesthetic quality of musical sounds, as different from other kinds of sounds, both physical and human, and relates them to the irreducible moral and affective dimensions of human existence, all in promoting them with ethical and religious values. In this sense, even if the Xing zi ming chu may not be taken as great as other Confucian classical texts, nevertheless, it can inspire us with a remarkable aesthetic of music, and convey to us the richness of Confucian music culture in the pre-Qin Era.\n\nReferences\n\nBarthes, Roland. 1981. Theory of the text. In Untying the text, a post-structuralist reader, ed. Young Robert. Boston: Routledge & Kegan Paul.\n\nBrindley, Erica. 2006a. Music and 'Seeking One's Heart-Mind' in the 'Xing Zi Ming Chu.' Dao: A Journal of Comparative Philosophy 5.2: 247\u2013255.\n\nBrindley, Erica. 2006b. Music, cosmos, and the development of psychology in early China. T'oung Pao 92: 1\u201349.CrossRef\n\nCai, Yong \u8521\u9095. 2002. Qin cao \u7434\u64cd. Xu xiu si ku quan shu, vol. 1092. Reprint. Shanghai: Shanghai Guji chubanshe.\n\nCai, Zhongde \u8521\u4ef2\u5fb7. 2003. Zhong guo yin yue mei xue shi \u4e2d\u56fd\u97f3\u4e50\u7f8e\u5b66\u53f2 (History of Aesthetics of Chinese Music). Beijing: Renmin yinyue chubanshe.\n\nCook, Scott. 1995. Yue Ji \u6a02\u8a18 \u2013 Record of music: Introduction, translation, notes, and commentary. Asian Music 26(2) (University of Texas Press): 1\u201396 (English translation of the Yue Ji with embedded commentary by the author).\n\nEgan, Ronald. 1997. The controversy over music and 'Sadness' and changing conceptions of the Qin in middle period China. Harvard Journal of Asiatic Studies 57(1): 5\u201366 (An aesthetic study on the concept of music and it's relation with qin \u7434 in Classical Chinese culture).\n\nGuodian Chu mu zhu jian \u90ed\u5e97\u695a\u5893\u7af9\u7c21 (Guodian Chu Tomb Bamboo Slips). 1998. ed., Jingmen Shi Museum. Beijing: Wenwu Chubanshe (Unearthed documents for both early Confucianism and Daoism).\n\nHeidegger, Martin. 1971. The origin of the work of art. In Poetry, Language, Thought (trans: Hofstadter, A). New York: Harper & Row.\n\nHightower, James Robert, trans. 1952. \u97d3\u8a69\u5916\u50b3 Han Shih Wai Chuan. Han Ying's illustrations of the didactic application of the classic of songs. An annotated translation. Cambridge: Harvard University Press.\n\nKristeva, Julia.1969. Le mot, le dialogue et le roman. In Recherches pour une s\u00e9manalyse. Paris: \u00c9ditions du Seuil.\n\nKristeva, Julia. 1986. Word, Dialogue and Novel. In The Kristeva reader, ed. Toril Moi. Oxford: Blackwell.\n\nLegge, James. 1967a. Li Chi: Book of rites. Part 1, Reprinted. New Hyde Park: University Books.\n\nLegge, James. 1967b. Li Chi: Book of rites. Part 2, Reprinted. New Hyde Park: University Books.\n\nLegge, James, trans. 1960. The Chinese classics, Vols. 5, Reprinted. Hong Kong: Hong Kong University Press.\n\nLi, Ling \u674e\u96f6. 2002. Shang bo Chu jian san pian jiao du ji\u4e0a\u535a\u695a\u7c21\u4e09\u7bc7\u6821\u8b80 (Records of Proofreading on Three Pieces of Chu Bamboo Slips Published by Shanghai Museum). Taipei: Wan juan lou Publisher.\n\nLi, Xue Qin \u674e\u5b66\u52e4. 1987. Zhu jian jiayu yu Han Wei Kongshi jiaxue \u7af9\u7b80\u3008\u5bb6\u8bed\u3009\u4e0e\u6c49\u9b4f\u5b54\u6c0f\u5bb6\u5b66. Confucius Studies \u5b54\u5b50\u7814\u7a76 2:60\u201364.\n\nPuett, Michael. 2004. The ethics of responding properly: The notion of Qing \u60c5 in early Chinese thought. In Love and emotions in traditional Chinese literature, ed. Eifring, Halvor. Leiden: Brill (A general survey on the concept of qing \u60c5 in Chinese thought).\n\nSima, Qian \u53f8\u99ac\u9077. 1967. Shiji, Reprint. Taipei: Donghua Shuju.\n\nTang, Yijie \u6c64\u4e00\u4ecb. 2003. Emotion in pre-Qin Ruist moral theory: An explanation of 'Dao Begins in Qing' (trans: Brian Bruya and Hai-ming Wen.) Philosophy east and west 53(2). Hawai'i: University of Hawai'i Press (One of the first philosophical interpretations on the concept of \"Dao shi yu qing\" in the Xing zhi ming chu).\n\nWang Xiangshen \u738b\u5148\u614e, ed. 1896. Han Feizi Jijie \u97d3\u975e\u5b50\u96c6\u89e3, Reprint. Taipei: Yiwen Yinshuguan, 1983.\n\nWatson, Burton, trans. 1963. Hsun Tzu. New York: Columbia University Press.\n\nXu Fuguan \u5f90\u5fa9\u89c0. 1966. The spirit of Chinese art \u4e2d\u570b\u85dd\u8853\u7cbe\u795e. Taipei: Xuesheng Shuju (Comprehensive study on philosophy of Chinese art).\n\nZhouli \u5468\u79ae. 1815. In Shisanjin zhushu \u5341\u4e09\u7d93\u6ce8\u758f. Comp. Ruan Yuan \u962e\u5143 (1764\u20131849), Vol. 3, Reprint. Taipei: Yiwen yinshuguan, 1963.\n\nFootnotes\n\n1\n\nThe Liji has been translated by James Legge, titled as Li Chi: Book of Rites (Legge 1967b).\n\n2\n\nIn the compilation of the Five Classics of Confucianism, the theory of music was arranged in the Liji (Book of Rites). According to the explanation of guwen scholars, it was due to the disappearance of the Yuejing (Classics of Music) after Qin's fire. Nevertheless, according to jinwen scholars' understanding, a book on the theory of music never existed before. What have really existed were the documents on the rules of music sound. Since there is not enough documentary evidence to certify the original source of the texts in the Yueji, some scholars claimed that the Yueji was created by Han scholars and falsely attributed to Pre-Qin Confucians. Others claimed that the writer of Yueji was named Gongsun Ni \u526c\u5b6b\u5c3c, a Confucian scholar in the Spring and Autumn period. For detail see Cook 1995: 3\u201310.\n\n3\n\nThe word intertextuality was used by Julia Kristeva to explain the transposition in textual system. Cf. Kristeva, Julia. 1969. \"Le mot, le dialogue et le roman.\" Recherches pour une s\u00e9manalyse. Paris: \u00c9ditions du Seuil. P. 146. English translation as \"Word, Dialogue and Novel.\" The Kristeva Reader, edited by Toril Moi. Oxford: Blackwell. 1986.\n\n4\n\nThe owner of the tomb was presumably related to Chen Liang \u9673\u826f, a Confucian scholar, recorded in the Mencius.\n\n5\n\nAccording to the annotation by Li Ling, here \"dao si shu \u9053\u56db\u8853\" should be understood as consisting in four arts, say, art of mind, art of poetry, art of history and art of ritual music; whereas \"san shu\" (three arts) means, respectively, shi (poetry), shu (history), and liyue (ritual music). For the coherence of meanings, the two characters \"liyue \u79ae\u6a02\" should be read together as one way\/art, instead of being read separately as two different arts: art of li and art of yue (Li Ling 2002: 70).\n\n6\n\nAccording to the commentary of Dong Zhongshu \u8463\u4ef2\u8212 (179\u2013104 BC), quoted by Zheng Xuan \u912d\u7384 (127\u2013200) in his annotations on Liji, the meaning of dezhe \u5fb7\u8005 could be understood as the person capable to perform. Here I follow Zheng Xuan's commentary that understands the yuede \u6a02\u5fb7 as a way of music performance.\n\n7\n\nCf. \"Spring Ministry with the Overseer of Ritual Affairs\" (Chunguan Zongbo \u6625\u5b98\u5b97\u4fdd) in the Zhouli(1815).\n\n8\n\nCf. \"Royal Regulation\" in the Book of Rites, \"The son of Heaven sacrificed to Heaven and Earth; the princes of the states, to the (spirits of the) land and grain; Great officers offered the five sacrifices (of the house)\" (Legge 1967a: 225).\n\n9\n\nCf. \"Spring Ministry\" (Chunguan) in the Zhouli (1815).\n\n10\n\nI translate this passage in reference to Confucius' words about Shao and Wu in the Analects translated by James Legge: \"the Master said of the Shao that it was perfectly beautiful and also perfectly good. He said of the Wu that it was perfectly beautiful but not perfectly good.\" (Analects 3.25; Legge 1960: 164).\n\n11\n\nIn the Qin Cao \u7434\u64cd (a collection of ancient tunes of qin), Cai Yong \u8521\u9095 (132\u2013192 CE) noted that Confucius composed the Yilan Cao \u7317\u862d\u64cd (Tune of Elegant Orchid) to convey his poetic mood, on the returning road from Wei to his native state Lu, when he passed a hidden vale and observed a fragrant orchid flourishing alone (see Cai Yong 2002: 147). In the Analects, Confucius played the qing \u78ec (a sounding stone) while traveling in Wei. It is read, \"The Master was playing, 1 day, on a musical stone in Wei when a man, carrying a straw basket, passed door of the house where Confucius was, and said, \"His heart is full who so beats the musical stone\" (Analects 14. 40).\n\n12\n\nConfucius said, \"I returned from Wei to Lu, and then the music was reformed, and the pieces in the royal songs and praise songs all found their proper places\" (Analects 9.15).\n\n13\n\nConfucius said, \"It is by the odes that the mind is aroused. It is by the rules of propriety that the character is established. It is from music that the finish is received\" (Analects 8.8). In answering Zilu's question about a complete person, Confucius said, \"suppose a man with the knowledge of Zang Wu-Zhong, the freedom from covetousness of Gong Chuo, the bravery of Bian Zhuang Zi, and the varied talents of Ran Qiu; add to these the accomplishments of the rules of propriety and music; such a one might be reckoned a complete man\" (Analects 14.12).\n\n14\n\nBefore the discovery of the unearthed bamboo slips, \"The Record of Music,\" complied in Western-Han as a document of Confucian lineage, has been considered as the unique clue to the understanding of Confucian thought on music. For example, Xu Fuguan, in his Zhongguo yishu jingshen (On Spirit of Chinese Art), contributed a chapter to investigate the spirit of Confucian thought on art through music, where he claimed that the theory of music in the \"Record of Music\" transmitted the legacy of Confucian thought on music that highly valued the relation between morality (Xu Fuguan 1966: 12).\n\n15\n\nThis story can be found in the Han Feizi jijie \u97d3\u975e\u5b50\u96c6\u89e3 (The Collected Annotations of Han Feizi) juan12: 33 \"Waichu Shuo Zuo Xia \u5916\u5132\u8aaa\u5de6\u4e0b\" (Wang Xiangshen 1896: 465).\n\n16\n\nWith different translation on the text of this passage, Michael Puett makes a comment with pedagogical meaning of the learning the san shu: \"The sages took the worthy traditions from the past, organized them, patterned (li) their qing, and thereby made them available to educate the latter-born\" (Puett 2004: 50).\n\n17\n\n\"What are the feelings of men? They are joy, anger, sadness, fear, love, disliking, liking. These seven feelings belong to men without their learning them\" (Legge 1967a: 379).\n\n18\n\nIt is read in the \"Rectifying Names\"(Zhengming \u6b63\u540d) chapter in the Xunzi: \"The likes and dislikes, delights and angers, grieves and joys of the nature are called emotions\" (Watson 1963: 139).\n\n19\n\n\"While there are no stirrings of pleasure, anger, sorrow, or joy, the mind may be said to be in the state of Equilibrium. When those feelings have been stirred, and they act in their due degree, there ensues what may be called the state of Harmony\" (Legge 1960: 384).\n\n20\n\n\"Qing arises from xing (qing sheng yu xing \u60c5\u751f\u65bc\u6027)\" (Guodian: 179).\n\n21\n\nCf. Sima Qian 1967: Book 47: \"Confucius studied [the tunes of] qin from Shi Xiangzi. For 10 days [Confucius] did not advance. Shi Xiangzi said, you may go on studying more. Confucius said, I have learned the qu \u66f2 (melody) [of this tune] but I haven't get the shu\u6578 (mathematic structure). Later [Shi Xiang Zi] said, you have learned the structure you may go on. But Confucius said, I haven't learned the zhi \u5fd7 (ideal\/poetic meaning). Later [Shi Xiangzi] said, you have learned the ideal\/poetic meaning, you may go on. But Confucius said, I have not yet got the demeanour of the author. Later, getting a feeling of a person characterized by majesty with profound thoughts, a gentle, venerable person but with lofty ideal, [Confucius] said, I learned the demeanour of the author, it was someone of very dark appearance, large in size and looking far off at the sea, like the king of four countries. If it isn't Wen Wang, who else could it be? Shi Xiangzi stood up, bowed and said, the tune you were talking is exactly the Wen Wang Cao \u6587\u738b\u64cd.\" This narrative can be read also in Hanshi waizhuan, juan 5 and in Kongzi Jiayu \u5b54\u5b50\u5bb6\u8a9e (The school sayings of Confucius) Chap. 35 Bianyue \u8faf\u6a02\u89e3 (Explanation about Music). The Jiayu has been proved as authentic rather than pseudographic after the unearthed documents excavated in China since 1970s. For example, see Li Xueqin (1987).\n\n22\n\n\"When you hear singing and chanting, you will feel jovial. This is excitement. When you listen to the sounds of the lute and zither, you will feel stirred. This is distress. When you watch the Lai and Wu dances, you will feel confrontational. This is being incited. When you watch the Shao and Xia dances you will feel focused. This is frugality\" (Guodian: 180; translation in Brindley 2006b: 25, 28\u201329).\n\n23\n\n\"In general the difficult thing about learning is 'seeking one's heart-mind.' If one follows from what one has done, one is close to obtaining it, but it is not comparable to the speed with which music achieves the same end\" (Guodian: 180; translation in Brindley 2006a: 248).\n\n24\n\nThe discussion on the primary value of sadness in the aesthetics of Chinese music, see Egan 1997: 5\u201366.\n\n25\n\nThe Hanshi waizhuan \u97d3\u8a69\u5916\u50b3, attributed to Han Ying \u97d3\u5b30 (active. 200\u2013130 BC), was one of the four schools (Mao \u6bdb, Lu \u9b6f, Qi \u9f4a, Han \u97d3) of Odes learning in Han dynasty. The Hanshi waizhuan was the longest surviving document of the Odes interpretation outside the tradition of the Maoshi \u6bdb\u8a69.\n\n26\n\nCf. Hanshi waizhuan, juan 1: 18. This paragraph has been literally translated by James Robert Hightower: \"Confucius said, 'The superior man has three worries: That he does not know\u2014can he not but worry? That he knows but does not study [what he knows]\u2014can he not but worry? That he studies but does not practice what he has studied\u2014can he not but worry? The Ode says: When I have not yet seen the superior man,\/My sorrowful heart is very sad\" (Hightower 1952: 26).\n\n27\n\nConcerning the studies on the problem of the authorship and the time of composition of the Yueji, see Scott Cook (1995: 3\u201310).\n\n28\n\nJames Legge translated this passage with a different interpretation: \"In music we have the expression of feelings which do not admit of any change\" (James Legge 1967b: 114).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_11\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 11. Wisdom and Hermeneutics of Poetry in Classical Confucianism\n\nVincent Shen1\n\n(1)\n\nDepartment of Philosophy and Department of East Asian Studies, University of Toronto, Toronto, Ontario, Canada\n\nVincent Shen\n\nEmail: vincent.shen@utoronto.ca\n\nAbstract\n\nPoetry occupied a crucial position in the classical Confucian educational program. Not only did the Shijing \u8a69\u7d93 (Classic of Poetry) constitute one of the fundamental teaching materials in the Six Classics, but poetry also played an important role in the political, diplomatic, and cultural arenas in ancient China. In the cultural and educational context of ancient China, shi\u8a69 (poetry) was closely connected with yue\u6a02 (music) and li \u79ae (propriety, rites), as confirmed by what Confucius said, \"One gets inspired with poetry, established with propriety, and filled with a sense of completeness by music\" (Analects 8.8, my translation and same below). Also, the learning of poetry was necessary for elegant language and public speech, as Confucius was said to have taught Bo Yu\u4f2f\u9b5a, his son, \"One will not know what to say if one does not learn poetry\" (Analects 16.13).\n\n## 1 Introduction\n\nPoetry occupied a crucial position in the classical Confucian educational program. Not only did the Shijing \u8a69\u7d93 (Classic of Poetry) constitute one of the fundamental teaching materials in the Six Classics, but poetry also played an important role in the political, diplomatic, and cultural arenas in ancient China. In the cultural and educational context of ancient China, shi \u8a69 (poetry) was closely connected with yue \u6a02 (music) and li \u79ae (propriety, rites), as confirmed by what Confucius said, \"One gets inspired with poetry, established with propriety, and filled with a sense of completeness by music\" (Analects 8.8, my translation and same below). Also, the learning of poetry was necessary for elegant language and public speech, as Confucius was said to have taught Bo Yu \u4f2f\u9b5a, his son, \"One will not know what to say if one does not learn poetry\" (Analects 16.13).\n\nAs to the practical application of poetry in the political and diplomatic arenas, Confucius said, \"If someone is able to recite the 300 odes, yet cannot achieve anything when entrusted with political power, or cannot respond to other's specific intent when sent to a diplomatic mission, no matter how many poems he has learnt, of what use is it?\" (Analects 13.5). Here Confucius' words confirmed the historical fact that till Confucius' time, poems or odes were still quite often recited to express one's intent on political and diplomatic occasions.\n\nSuch a crucial position occupied by poetry in ancient Chinese culture and classical Confucianism must have its philosophical implications. Therefore we may ask what is the philosophical meaning and wisdom implied in Confucian teaching of poetry, which is the most succinct and essential form of language? What are the Confucian hermeneutic methods related to the interpretation of meaning in poetry? In short, what is the philosophy of Confucius' teaching of poetry?\n\nIn several places in the Analects where Confucius referred explicitly to the Shijing we can find some important messages about Confucius' understanding of poetry and his interpretation of poems.1 However, problems such as how Confucius taught poetry to his students, and what is his philosophy of teaching poetry etc., were not very clearly discussed in the Analects. Fortunately, the recently unearthed bamboo slips of Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry)2 add much to our knowledge of Confucius' contribution to the teaching of poetry in ancient China. These newly unearthed texts, which might have been transcribed by Confucius' later disciples, or even disciples of his disciples, help us clarify and solve some historically debated problems. In this chapter, I will focus only on the philosophy and wisdom that seem very essential in Confucius' teaching of poetry and in his comments on poems. We can find Confucius' poetic wisdom in his comments on different kinds of poetry and individual poems, in the way he organized his teaching on the Shijing, especially in his vision of affectivity as the primordial mode of human existence and the function of poetry in both self-cultivation and public life. We will also touch upon the hermeneutic issues of how to interpret the meaning of poetry.\n\n## 2 Historical Context of Confucius' Teaching on Poetry\n\nBefore Confucius came onto the historical scene, there had been in ancient China collections of Odes by shi \u5e2b (Master), musical masters and professional collectors in each city-state, which then went through the process of royal selection by tai shi \u592a\u5e2b (Great Master), Zhou's musical officers, for the purpose of royal files and public education. The group of bamboo slips now entitled Kongzi Shilun could be read as a syllabus recording Confucius' teaching on the Shijing (Classic of Poetry). It constitutes in fact the first treatise we know till now on the Shijing for the purpose of education in early Confucianism.\n\nThese recently unearthed fragments confirm the historical facts not only that Confucius taught and gave comments on the Shijing, but also that his teaching on the Shijing was in fact quite different from Han scholars', especially that of Mao's commentaries on the Shijing, which had constituted the traditional version of the Shijing.3 First, in regard to the order, the traditional (Mao's) version of Shijing follows the order of Guofeng \u570b\u98a8, Xiaoya \u5c0f\u96c5, Daya \u5927\u96c5, and Songs \u980c, whereas in these newly unearthed fragments, Confucius' comments follow the order of Songs \u980c, Daxia \u5927\u590f (Daya \u5927\u96c5), Xiaoxia \u5c0f\u590f (Xiaoya \u5c0f\u96c5), and Bangfeng \u90a6\u98a8 (instead of Guofen \u570b\u98a8).4 Also, we find in these fragments some different titles for the same odes in the traditional version as well as some new titles that were not there in the past.5 This confirms the fact that poems selected in the Shijing were largely stable, while Confucius used an earlier version with different order of arrangement and some different titles from Mao's version in Han Dynasty.\n\nThe real difference between Confucius' teaching of poetry and that of Mao's commentary consists in their interpretations. According to Mao's commentaries, most poems in the Daya, Xiaoya, and Guofeng were produced either for praising or satiric criticism of King Xuan \u5ba3\u738b and King Li \u53b2\u738b. For example, more than half of the Guofeng poems were presumed by Mao's commentaries to be composed for political purposes. In other words, most of the Odes contained in the Daya, the Xiaoya, and the Guofeng were presumed to be either glorifying praise or satirical criticism of kings, royal families, or political leaders of the time in the form of poems, while those expressing common people's feelings and more universal human affectivity were neglected. However, when we come to the newly unearthed fragments of the Kongzi Shilun, we find that there more attention was paid to human feeling. In these fragments, except in the case of Jie Nan San \u7bc0\u5357\u5c71 and Yu Wu Zheng \u96e8\u7121\u6b63, we do not find any interpretation of verses as satirical criticism of bad kings. In this sense, the Shijing were not necessarily envisaged as poetic expression of political criticism or flattering praise, though they might sometimes have such a function. Their function of expressing human feeling and affectivity in poetic language was put in priority. It was then that a poem could have in addition other functions such as political and pragmatic ones.\n\n## 3 Confucius' Method of Teaching Poetry\n\nThe way in which Confucius commented on the Shijing, as shown in the newly unearthed bamboo slips, seems to be very systematic and pedagogical, which constituted a very intelligent way of teaching poetry on his part. We find some bamboo slips in which Confucius characterized poetry in general and the relation of poetry with other cultural activities such as music and literature. Then, we find some bamboo slip texts in which he proceeded to discuss the general characteristics of those major types of poem such as Guofeng, Xiaoxia, Daxia, and Songs. These comments might be considered as a preface or introduction to poetry in general and its major types. Then, we find some bamboo slips in which Confucius commented on each individual poem in giving their wholesome meaning as well as featuring some of their most remarkable key verses. We can suppose then that the texts of Kongzi Shilun were arranged and structured in the following order: general introduction to poetry, classification of poems and their main characteristics, summary of and comments on individual poems and the key verse(s) in each poem according to their classification. This gives us an impression of a very systematically organized syllabus of Confucius' teaching on the Shijing. Let me explain each aspect of the syllabus in the following order.\n\nFirst, concerning the nature of poetry in general, Confucius said in Fragment 1, \"Poetry could not be without earnest thoughts; music could not be without feeling; literature cannot be without good wording\" (Confucius 2001: 123).6 These words related poetry to music and literature, based upon the fact that in ancient China, odes were sung and performed with music and elegant wording. In the Zuo's Commentary on the Spring and Autumn Annals, for example, the narrative of Jizha's \u5b63\u624e reviewing Zhou's music during his diplomatic visit to the State of Lu \u9b6f, which took place in the 29th year of Duke Xiang's Reign \u8944\u526c\u4e8c\u5341\u4e8c\u5e74 (544BC), gave us typical textual evidence that odes were performed at that time with music (Legge 1994c: 545\u2013551). In Fragment 2 of the bamboo slip texts, Confucius said, \"The music itself is easy and slow. Its song is accompanied harmoniously by xun \u58ce and ci \u7b8e. Its thought is deep and far-reaching. The odes in the Daxia are full of praises for his virtues\" (Confucius 2001: 127). This comment also gave us the textual evidence that confirmed the fact that poetry was at that time performed with music. If poetry was related essentially to music, then its essential function of expressing earnest human thoughts was deeply related to human affectivity or human feeling. This brings us to Confucius' vision of affectivity as an essential mode of human existence, quite different from the traditional emphasis on his ethical and political pragmatism. We will come back to this part of Confucian wisdom after we discuss his views on different types of poems and individual pieces of poem.\n\nSecond, Confucius' comments on different types of odes: after the characterization of poetry in general and its relation with music and literature, Confucius proceeded to give comments on different types of odes such as Songs, Daxia, Xiaoxia, and Bangfeng. This order, very well articulated in the Kongzi Shilun, was in fact in reverse to the order presented in Mao's version.\n\nThe Songs, first commented on by Confucius, were generally performed in ancient China together with music in temples and royal courts, mostly for the purpose of praising the present king or the founding kings' benevolent and brilliant virtues. In the fragments where Confucius mentioned the Songs, he seemed to emphasize those kings who had achieved great merit for the country, especially King Wen. For example, in Fragment 2 of the bamboo slips, we read, \"...is it, thereby King Wen received the mandate. This Song sings his merit of conquering, which is often spoken of by his offspring. The music itself is easy and slow. Its song is accompanied harmoniously by xun \u58ce and ci \u7b8e. Its thought is deep and far-reaching. The odes in the Daxia are full of praises for his virtues, often spoken of...\" (Confucius 2001: 127). Also, in Fragment 5, we read, \"What to do with those who have great merits? To praise them in the sacrificial Songs. The verses of the Qingmiao \u6e05\u5ebf praise how supreme the merit of King [Wen] has been\" (Ibid: 131).\n\nAs shown in Fragment 2, Confucius' comments on the Songs were followed by those on the Daxia. The newly unearthed texts seem to confirm the traditional opinion that Daya poems were composed mostly for the praise of the virtues of Zhou's founding Kings, especially King Wen, for example in Fragment 2 mentioned above. Also, Fragment 7 belongs to the Daxia. It reads,\n\n> What does it mean by saying that \"[God said to King Wen:] 'I think cherishingly of your bright virtue.\" It is to say about King Wen's sincerity. 'The mandate is from Heaven. Giving the throne to King Wen.' It is a mandate by sincerity. Trustfully it is. That's why Confucius says, 'It is a Heavenly Mandate. Is it by his own multiple talents that King Wen gets it? It is his mandate.' (Ibid: 134)\n\nThese comments on the Daxia were followed by those on Xiaoxia, which was constituted of poems composed by the elites or officials of each country\/city state. There, the fragment on which Confucius gave his comment seemed to emphasize the expression of common people and the king's subjects' suffering, their worries and complaints in difficult times. Confucius said in Fragment 3, \"[The odes of Xiaoxia] express people's worries and complaints in difficult times, lamenting that their rulers are declining and being deprived of virtue\" (Ibid: 128\u2013129). These words of Confucius give us a general characterization of the Xiaoxia.\n\nAfter this, in Fragments 8 and 9, we are able to read Confucius' comments on individual poems contained in the Xiaoxia:\n\n> The Shiyue \u5341\u6708 is good at describing slanderous situations. The Yu Wu Zheng \u96e8\u7121\u6b63 and the Jie Nan San \u7bc0\u5357\u5c71 are all expressing the decline of their Lord, and kings and dukes feel ashamed. There are a lot of doubts in the Xiaomin \u5c0f\u65fb, saying that one's willing is not pleased. The Xiaowan \u5c0f\u5b9b gives no hateful words, though there seems to be no calm conscience either. The Xiaobian \u5c0f\u5f01 and the Qiaoyan \u5de7\u8a00 all talk about the harm done by those who slander. (Ibid: 135\u2013136)\n> \n> Most precious is that he blames his own self. The Tianbao \u5929\u4fdd says that he owns his happiness forever. Even when the food offered is tiny, he still keeps on his virtue like in the old days. The Qifu \u7948\u7236 also blames with reason. The Huangming \u9ec3\u9cf4 shows that the author was then stuck in difficulty and desired to return to the old days. Most people would be shamed against its anger. The Jinjin Ze Er \u83c1\u83c1\u8005\u83aa talks about the enrichment made by talented people. (Ibid: 137).\n\nIn comparison with the Xiaoxia, which was constituted of poems composed by the elites or officials of each country\/city state, the Bangfeng \u90a6\u98a8 was constituted rather of popular songs of each individual state. In the newly unearthed bamboo slips of Kongzi Shilun, most of Confucius' comments on the Shijing were on the Bangfeng, or popular songs of different states. The later term \"Guofeng\" was named Bangfeng in Confucius' time, of which the word \"bang\" was later changed to \"guo\" in Mao's version, probably in order to avoid breaking taboo against Liu Bang, name of the first emperor of Han Dynasty. The term \"feng \u98a8\" could mean transformation by way of education, custom, the ethos of the people, and the popular songs, including love songs, of the 13 states existing in the time of Spring and Autumn not including the Zhounan \u5468\u5357 and the Zhaonan \u662d\u5357. This could be seen especially in the narrative of Jizha's review of Zhou Music by which Jizha understood well the ethos of the people and the virtues of political leaders of each country well documented and performed in the court of Lu state. The theory goes that Bangfeng recorded and expressed first of all the ethos of people in each state, whether or not it had the educational function of transforming its people or the political function of criticizing or satirizing its political leaders, such as Mao and other Han scholars emphasized. In Fragments 3 and 4 of the Kongzi Shilun, Confucius says,\n\n> The Odes of Bangfeng contain a lot of things. They look generally into the customs of people by abundantly collecting materials from different countries. Its wording is elegant and its voice kind. (Ibid: 128\u2013129)\n> \n> Odes are like open doors, given to the lower people for expressing themselves. What could it be their intentions? It is to express in the Bangfeng. When people are exhausted and tired, especially when the upper elites and the lower people are not in harmony, what could it be their intentions? (Ibid: 130)\n\nAs we can see clearly from the fragments quoted above and in other comments of Confucius on individual poems, although they are not quoted here, Confucius' method of interpreting poems went much like this: he first summarized his characterization of each poem together with other poems in the same group he was commenting on before he went on to the details of each poem. He used the method of \"key verse\" by picking the most interesting verse(s) in each poem to exemplify his judgment on the poem or on the group of poems in question. In his choice of key verses and his appropriation of the meaning of each poem, there was manifested a kind of poetic phronesis,7 or poetic wisdom as an art of expressing one's value judgment.\n\n## 4 Human Affectivity and the Cultivation of Feeling\n\nThe Shijing, like the Song of Songs in the Old Testaments, expresses and thereby interprets through poetic language human affectivity as the original mode of human existence. Confucius was very clear about this point when, combining poetry, music, and literature, he said in Fragment 1 that \"Poetry could not be without willing; music could not be without feeling; literature could not be without wording.\" This line of thought should have been followed by Zisi, the grandson of Confucius, and become a legacy in the so-called Zisi-Mencius School. That is why in the text entitled Xing zi ming chu (Human Nature Comes from Mandate), a Confucian text, also in the Guodian Bamboo Slips, now attributed to the so-called \"Zisi-Mencius School,\" we read,\n\n> Human nature comes out from mandate,\n> \n> Mandate descends from Heaven,\n> \n> Dao begins with human feeling,\n> \n> Human feeling is born from human nature,\n> \n> Those who begin with human feeling,\n> \n> Will end up with righteousness. (Jinmen Museum 1994: 179. my translation)\n\nWe should notice that human feeling or affectivity is here very much emphasized, even to the point of relating it to the Dao and human nature, in saying that \"Dao begins with human feeling,\" and that \"Human feeling is born from human nature.\" In particular, it puts emphasis on the role of feeling in human self-cultivation and ethical behaviour. \"Those who begin from human feeling will end up with righteousness.\" Therefore, somehow it bases the unfolding of human nature and the fulfilment of human existence on human affectivity.\n\nMao's commentary in the Great Preface to the Shijing, though overemphasizing the political function of poetry in praising or satirizing powerful kings, still followed Confucius' idea of human feeling\/affectivity and saw it as essential to poetry. In that line of thought, Mao's Great Preface said, \"Poetry is the product of earnest thought. Thought cherished in the mind becomes earnest. Exhibited in words, it becomes poetry. The feelings move inwardly, and are embedded in words. When words are insufficient for them, recourse is had to sighs and exclamations. When sighs and exclamations are insufficient, recourse is had to prolonged utterances of song. When those prolonged utterances of song are insufficient for them, unconsciously the hands begin to move and the feet to dance\" (Legge 1994b: 34).\n\nJames Legge, when commenting on the Shijing, followed also this line of thought, though tracing it back only to Mao's Great Preface because of his ignorance of Confucius' own teaching on poetry. He said, \"By poetry, according to the Great Preface and the views generally of Chinese scholars, is denoted the expression, in rhymed words, of thought impregnated with feeling; which so far as it goes, is a good account of the species of composition\" (Legge 1994b: 1).\n\nA careful reading of the Shijing shows that the affective relation between man and woman, subjects and kings, human beings and Heaven, sometimes with love, sometimes with joy, sometimes with anxiety, sometimes with bitterness, sometimes even with hateful blame, depending on the situation they were affected and the way they were treated, were expressed through poetic language: for example, affective relation between man and woman, as in the Guanju \u95dc\u96ce (Cry of Ospreys)8; or affective relation with parents and ancestors, as in the Xiaowan \u5c0f\u5b9b9; or affective relation between subjects and his superiors and king, as in the Qifu \u7948\u7236 that showed a deep sense of complaint to the minister of war.10 The relation between subjects and king was much better in the founding period of Zhou dynasty, as exemplified by those poems concerning King Wen.11 Because of his virtues, King Wen was assured of his Mandate of Heaven as justification of the legitimacy of his political leadership. The Mandate of Heaven bestowed upon those who had virtues, such as King Wen. This presupposed an affective relation between humankind and Heaven, or God on High, in whom there was a strong belief in the period from Shang Dynasty to the early Zhou Dynasty.\n\nConfucius' comment on the function of poetry seems to have well grasped this web of existence constituted of affective relations as we can find in the Kongzi Shilun. It evokes in us an image of a great thinker, not that of a stringent political and ethical philosopher, but first of all a human person in whom we can recognize the authenticity of existence, the primacy of affectivity over discourse, and the primacy of existence over thinking.\n\nConfucius emphasized the cultivation of human affectivity, to the point of thinking innocently upon those poems expressing passionate emotions and sexual insinuations. In fact, there expressed many such kinds of emotion, such as joy, anger, sadness, blaming, hate, and lust. Some poems might appear to be sexually indecent, such as the poems of the Zhengfeng \u912d\u98a8 and the Weifeng \u885b\u98a8, yet Confucius would wrap them up in the basic spirit of the Shijing in saying that, \"All 300 odes can be covered by one of their sentences, and that is, \"Have no depraved thoughts\" (Analects 2.2; Chan 1969: 22).\n\nIn the bamboo slips of Kongzi Shilun, Confucius seemed to be very open to human feelings, especially to the feeling of love such as that expressed in the Cry of Ospreys. He made this clear in his summary characterization of the Cry of Osprey as \"joyful\" in Fragment 10, saying that, \"The Guanju sings of joyfulness.\" Then he proceeded to say, \"The Quechao \u9d72\u5de2 sings of marriage. The Gantang \u7518\u68e0 sings of praise. The Lu Yi \u7da0\u8863 sings of thinking. The Yanyan \u71d5\u71d5 sings of love. Why does it mention propriety? All for the purpose of tracing back to its origin. The Guanju understands li \u79ae by way of beauty\" (Confucius 2001: 139). Again, in Fragment 11, Confucius said, \"...it's about love. The Guanju sings of joyfulness and is enriched by the author's thinking. The Qiumu \u6882\u6728 sings of timeliness because of the author's happiness. The wisdom in the Hanguang \u6f22\u5ee3 is an unattainable wisdom. The marriage in the Quechao \u9d72\u5de2 is paired with...\" (Confucius 2001: 141). Now it is obvious that Confucius, in encouraging the expression of human feeling and the joyfulness of love, would put them back into the ethical relationship and the cultivation of feeling by li \u79ae, understood as the ritual behavior leading to harmonious relation with a sense of beauty. We can see this in Fragment 12 where Confucius said, \"The happiness [of Man and woman] should be traced back and integrated in li; that's why it could be joyful. The Qiumu sings of happiness realized by junzi\" (Confucius 2001: 142).\n\nThus, we can assume that Confucius condones the expression of human feelings through poetic language, and he sees this as a basic function of poetry. This is also corroborated by the Analects, where Confucius says, \"My young friends, why do you not study the Odes? The Odes can arouse your feelings, broaden your observations, enlarge your fellowship, and express your grievances. They help you in your immediate service to your parents and in your remote service to your rulers. They widen your acquaintance with the names of birds, animals and plants\" (Analects 17.9; Chan 1969: 47, with my modification). Confucius seems to take centrality as the principle of cultivating one's affectivity, as in the case of love songs, such as the Cry of Osprey. Confucius says, \"The Cry of Osprey is pleasing without being excessive, is mournful without being injurious\" (Analects 3.20; Ames and Rosemont 1998: 86). Here the so-called \"pleasing without being excessive,\" \"mournful without being injurious\" refer implicitly to a principle of centrality or middleness \u2013 that one should always take the middle way. The principle of centrality\/middle way allows human feelings to express themselves without becoming excessive, all to the end of achieving harmony with a sense of beauty. In short, poetry expresses human affectivity in its authentic mode, while ethics cultivates human affectivity by referring it to the measure of li and to the principle of centrality\/middle way.\n\n## 5 Function of Judgment in Meaning Appropriation: Confucian Hermeneutics of Poetry\n\nConfucius shows his wisdom in appropriating the meaning of a poem and in his judgment on it. Confucius' reading of poems may be seen as a verbal and decisional concretization of pragmatic wisdom. That is what is meant by \"appropriation of meaning by cutting or selecting text\" (duanzhan quyi \u65b7\u7ae0\u53d6\u7fa9) or, in short, \"featuring key verses,\" which in fact is a kind of poetic wisdom implying his judgment in the process of textual selection and interpretation. In the bamboo slip texts of Kongzi Shilun, we find Confucius commenting on poems by highlighting a certain key verse in order to represent the whole poem. For example, Fragment 6 reads,\n\n> [The Qingmiao \u6e05\u5edf says,] \"Great is the number of the officers, assiduous followers of the virtue of King Wen.\" I pay my homage to this. The Liewen \u70c8\u6587 says, \"What is most powerful is being the Man.\" \"What is most distinguished is being virtuous.\" \"Ah, the former kings are not forgotten.\" I am delighted by all these. [The Haotian You Chengming \u660a\u5929\u6709\u6210\u547d says,] \"The Heaven made its determinate mandate, which our two sovereigns received.\" They are indeed highly honored and powerful. The Songs... (Confucius 2001: 133)\n\nHere in this text Confucius puts together a group of key verses of different poems in the Songs to emphasize the idea that the moral admirability and the ethical power of the person of King Wen consist in his virtue, seen as the honorable assurance of the Mandate of Heaven bestowed on him. The theme of \"Mandate of Heaven\" in the case of King Wen seems to be very much cherished by Confucius. We read, for example, in Fragments 7 and 22, that,\n\n> What does it mean by saying that \"[God said to King Wen:] 'I think cherishingly of your bright virtue\"? It is to say about King Wen's sincerity. \"The mandate is from Heaven, giving the throne to King Wen.\" It is a mandate by sincerity. Trustfully it is. That's why Confucius says, \"It is a Heavenly Mandate. Is it by his own multiple talents that King Wen gets it? It is his mandate.\" (Confucius 2001: 134)\n> \n> When the Wanqiu \u5b9b\u4e18 said, \"Full of kindly affection, yet without anything to look up to.\" I say it is good. When the Yijie \u7317\u55df said \"All four arrows again and again return to the same place. One is able to withstand rebellion.\" I like it. When the Sijiu \u9cf2\u9ce9 said that \"His deportment is uniformly coherent, his heart being tight to what is right.\" I do believe it. When the Wen Wang \u6587\u738b said that, \"King Wen is on high. Oh, bright is he in Heaven.\" I say it is excellent. (Confucius 2001: 151)\n\nThe hermeneutic method of \"featuring key verses\" presupposes a certain common familiarity with the odes and a common measure of understanding them, so that it suffices to mention the title and the key verse of a poem to make the audience understand the message in one's discourse. In other words, it is enough to pick one verse of a poem, or sometimes even only the title of that poem, to convey the affective and intellectual thoughts of the user of the poem. In ancient China, this collective familiarity with a more or less common collection of poems was a result of the fact that there were collectors of poems on the level of each city-state, and then, on a higher level, there were royal selectors of poems in the Zhou court. There were also schools where one could learn the poems, which, on the level of Zhou court, were named Piyong \u8f9f\u5ef1, and, on the level of each city-state, Pangong \u6cee\u5bae. In these schools, elites could learn the collections of poems on both central and local levels that served as their common reference. Also, they had the opportunity to learn poems from different states so as to understand the ethos of other states and the virtues of their political leaders.\n\nThe hermeneutic process was not rigid; in fact it was quite flexible. Sometimes the users or interpreters' appropriation of meaning of some words in a particular poem might be different from its original meaning. This was especially the case with Confucius when he quoted a poem to express his assessment of a person or an event. We should say that Confucius, when quoting from the Shujing \u66f8\u7d93 (Classic of Documents), attached himself more often to the original wording and meaning in it. However, when quoting from the Shijing, Confucius would depart quite often from the original meaning of a poem. This was probably because of the fact that what he quoted from the Classic of Documents, which is historical in nature, and history was supposed to be as faithful as possible to what had actually happened; whereas the Classic of Poetry is about human intentions and affectivity, which could suffer from changes in space and time and were susceptible to undergo freer subjective interpretations. This special characteristic of poetry allowed Confucius the flexibility to appropriate meanings by selecting verses, which showed a creative interpretation of his own. For example, in the Analects, we read,\n\n> Zizhang asked about the exaltation of virtue and the resolution of perplexities. The Master said, \"Make it your guiding principle to do your best to others and to be trustworthy in what you say, and move yourself to where rightness is, then you will be exalting virtue. When you love a man you want him to live and when you hate him you want him to die. If, wanting him to live, you also want him to dies, is this not being perplexed? [The Odes says] \"If you did not do so for the sake of riches, You must have done so for the sake of difference\" (Analects12.10; Lau 1983: 113; emphasis added)12\n\nHere Confucius quoted from verse 3 of the Woxing Zhi Ye \u6211\u884c\u4e4b\u91ce in the Xiaoya \u5c0f\u96c5. The original verse sang about someone who changed his mind after a marriage was made, though not necessarily because the new wife was richer, at least because she was different from the previous wife. Confucius used here the term yi \u7570 (difference) to explain the origin or cause of doubt in respect to virtue, that is, if one changed one's mind because of facing something or someone different, one would be in quandary. Confucius re-contextualized the verse by changing the case of marriage to that of virtue. We have to keep in mind that here the relation of marriage served as a metaphor for virtue, in the sense that, just as one should not change one's mind in marriage because of the new wife's being richer or being different, in the case of virtue, one should not change one's respect for virtue because of facing different situations.\n\nIn fact, Confucius' famous saying, that \"All 300 odes can be covered by one of their sentences, and that is, 'Have no depraved thoughts,'\" was itself an exemplary case in which he had appropriated the meaning of poetry by creative selection and interpretation. Originally the verse \"Have no depraved thoughts\" came from the poem entitled Jiong \u99c9(Stallions) in the Lu Songs \u9b6f\u980c, which was sung when someone was pasturing horses, where it was sung \"Ah, how they are without depravity!\"13 In the original Chinese texts, the term \"si \u601d\" was merely an auxiliary ending term similar to \"si \u516e,\" and the whole verse would say only something like, \"Ah, don't go astray,\" or \"Ah, how they are without depravity,\" as James Legge translated it. However, Confucius used the term \"si\" to denote \"thought\" and read the whole verse as \"thought without depravity.\" Thus, it was a creative reading on the part of Confucius.\n\nNow, according to classical Confucianism, what were the hermeneutic criteria by which users of poem could achieve mutual understanding? Here we should point out one principle that says the meaning of a poem should be \"like\" (lei \u985e), which means both \"similarity\" and \"befittingness.\" On the one hand, the true intention of the user of poems should be similar to the literal, figural, or imaginable meaning of words, verses, or odes used in the situation. On the other hand, the poems used should be befitting to the occasion and be used by other users for a similar purpose. Similarity of one's intention to one's wording means sincerity. Similarity of purpose to other users of poems means friendship or alliance for common purpose. Especially on diplomatic occasions, the use of a poem contrary to the principle of \"lei\" might create conflict or even lead to war among states. For example, according to the Confucian Classics Zuzhuan \u5de6\u50b3 (Zuo's Commentary on the Spring and Autumn Annals), in the 16th year of Duke Xiang's Reign \u8944\u526c\u5341\u516d\u5e74 (557BC), during the banquet of Marquis of Jin \u6649 with other princes at Wen \u6eab, the Marquis of Jin said that \"The poems sung in the banquet must be similar and befitting the occasion, now the poem sung by Gao Hou \u9ad8\u539a from Qi \u9f4a State is not so.\" Thereafter a war was waged against Qi. (Legge 1994c: 472). That the misuse of a poem could have a serious consequence, even to the point of provoking a real war, was the most remarkable negative consequence of misused poetry that ever happened in the history of international affairs.\n\nWe should say that cutting verse to appropriate meaning and the criteria of similarity and befittingness concerned mostly the users of poems, yet till the time of Confucius the hermeneutic criterion of those who read or listened to poems was left untouched. It was much later, in the time of Warring State that Mencius proposed to \"trace the expressed intention by understanding\" (yiyi nizhi \u4ee5\u610f\u9006\u5fd7). We read in the upper chapter of Wanzhang \u842c\u7ae0 of the Mencius that, \"Therefore he who interprets the Odes, should not be stuck by words in detriment of a sentence, and he should not be stuck by sentence in detriment of earnest thoughts. If one can trace back to the earnest thought by understanding, he then is said to have caught its meaning\" (Mencius 5A4; my translation).\n\nUsing one's understanding of a poem to trace back to its original and earnest intent could mean, on the level of poetic hermeneutics, something very similar to what Wilhelm Dilthey says about the function of understanding in historiography. For Wilhelm Dilthey's historical hermeneutics, human life is teleological in the sense that it tends to the creation of meaning by expressing its creativity in words, deeds, and works, which, in their turn, could be understood by enacting this process of creative expression in a sympathetic understanding. While the process of creativity goes from the dynamic teleology of life to meaningful expression in words, deeds, and works, the process of understanding goes inversely from the expressed words and deeds, to trace back to the dynamic and creative process of life, via the intelligible structure of words, deeds, and works in question. When added to Confucius' \"hermeneutics from the user's point of view,\" Mencius' \"hermeneutics from the reader's point of view\" could be seen as having completed the circle between the user and the reader in the classical Confucian hermeneutics of poetry.\n\n## 6 The Use of Poetry in Public Affairs\n\nWisdom, conceived as prudence demonstrated in judgment, can be shown not only in the appropriation of meanings of poems, but also in the public use of poems. When using poems on public occasions, especially on a diplomatic occasion, which, as we have mentioned, was a common practice in ancient China, one should be wise to quote the right poem and the right verse for the right occasion. In some sense, Confucius' teaching of poetry was necessitated by his ambition to train disciples capable of taking positions in public life, of becoming officers or diplomats, for which purpose they should be well versed in the Odes. As we have quoted from the Analects, Confucius said, \"If someone is able to recite the 300 odes, yet cannot achieve anything when entrusted with political power, or cannot respond to other's specific intent when sent to a diplomatic mission, no matter how many poems he has learnt, of what use is it?\" (Analects 13.5).\n\nHere the necessity to have a common reference so as to facilitate common understanding and avoid alienation or conflict situations seems to exclude the possibility that Confucius might have edited an original collection of poems for his own use in his teaching of poetry. As I see it, although Confucius had a better and more systematic method of teaching poetry, it would seem to the disadvantage of his disciples if he had a different collection of poems than the one commonly used. In the Zuo's Commentary of the Spring and Autumn Annals, where poems were reported to have been performed 73 times, only in two performances were the quoted poems unrecognized because of the ignorance of the person in question. This shows that there was a very high consensus as to what poems to learn and there was no need for Confucius to break this consensus by producing a different collection of poems for his own teaching and for the good career of his students. On the other hand, according to the Analects, Confucius had reorganized the order and the musical tuning of those collected poems. We read in the Analects that,\n\n> The Master said, \"It was after my return from Wei to Lu that music was put right, with the ya \u96c5 and the song \u980c being assigned their proper places.\" (Lau 1983: 81)\n\nConfucius' teaching of the poem was for the purpose of his disciples' self-cultivation as well as their future career in public service. Confucius said in the Analects, \"Unless you study the Odes, you will be ill-equipped to speak\" (Lau 1983: 167). That is also why he was concerned that his disciples could recite all the poems, and yet, when sent to foreign states, they were unable to quote them properly in responding to specific poems sung by their hosts. All these were about the public use, especially the diplomatic use, of the Odes. As proved historically by Zuo's Commentary on the Spring and Autumn Annals, the diplomatic use of the Odes was a very common practice in ancient China during the period of Spring and Autumn, especially during the period from 637BC to 505BC. In this period of time, the intellectuals, officials, and diplomats learnt poems by heart as part of their intellectual and professional formation. They used, quoted, and performed poems to express their intents, as a rhetoric device and as a token of culture. \"Perform odes to express one's intent\" (fushi yanzhi \u8ce6\u8a69\u8a00\u5fd7) became a diplomatic practice by which one used or quoted poems to express one's own wish, one's worry, and one's solicitation, and one's interlocutor or entertainer would also respond by quoting relevant verses.\n\nFor example, according to the record of the Zuozhuan, in the 23rd year of Duke Xi's Reign \u50d6\u526c\u4e8c\u5341\u4e09\u5e74 (637BC), Jin \u6649 prince Chong Er \u91cd\u8033 in his exile visited Qin State. Chong Er sang the Hesui \u6cb3\u6c34 (River Water),14 the Earl Mu of Qin responded with the Liuyue\u516d\u6708, singing of King Xuan of Zhou's order to Yin Jifu \u5c39\u5409\u752b to conquer the Yanyun \u513c\u72c1, and one could read there a verse like \"The King had ordered the expedition, to help the son of Heaven.\" Chong Er, advised by Zhao Shuai \u8d99\u8870, bowed to express his gratitude to Earl Mu of Qin, in acknowledgement of his encouraging him to retake the control of his own country and become a great assisting Earl to Zhou King. This was a story of properly responding to poems on a diplomatic occasion.\n\nOn the other hand, it would be a shameful matter if a diplomat sent to another state did not understand the words and intents of the poem sung to him. For example, the Zuozhuan told the story that, in the 12th year of Duke Zhao's Reign (530BC), the State of Song sent Huading \u83ef\u5b9a on a diplomatic mission to the State of Lu, where Duke Zhao of Lu entertained him with a banquet and the Luxiao \u84fc\u856d was played for him, which sang of the delight of seeing the diplomat. The four verses sang respectively of the happiness of seeing the noble man with delightful words in banquet, with appreciation of favor and brightness, with joyfulness in being virtuous, and with the shared happiness in their getting together. Unfortunately, Huading did not understand these verses and made no response to them. For that reason Duke Zhao got angry and said, \"He is sure to be driven into exile. He cherished not that 'We feast and talk'; he declared not his sense of that 'They favors me, they brighten me'; He understood not that 'Excellent virtues'; He accepts not that 'Common happiness'; how should he continue to be in that position in Song?\" (Legge 1994c: 639, my correction in bold). This is one of the two examples of poems that were sung yet ignorantly not understood in the 73 performances of poems mentioned in the Zuo's Commentary.\n\n## 7 Wisdom in the Poems\n\nTo be wise or not is also an important criterion by which Confucius judges poems. As can be seen in the newly unearthed bamboo slips of the Kongzi Shilun, Confucius characterizes a poem as wise or not in the Fragments 10, 11, 13, 27, 28, 29. So it seems that Confucius uses the term \"wise\" or \"wisdom\" quite frequently in his characterization of poems. If wisdom concerns a prudent way of delivering judgment, then caution should be paid as to cases in which judgment could be distorted in both ethical and political matters. This is generally called a case of slander, a kind of malicious statement, or an utterance of defamatory statements injurious to the wellbeing of a person. If wisdom concerns judgment that reveals truth, then slander is a dissimulation of truth that causes injury to one's wellbeing or fame. Slander is generally used, in private as well as in public situations, as a tactic to gain worldly advantage by irresponsibly uttering statements that dissimulate truth and thereby cause injury to one's opponents in a competitive or conflict situation. In the newly unearthed bamboo slips of Kongzi Shilun, Confucius has very wisely picked certain poems complaining about slanders and appealing for justice. For example, Fragment 8 reads,\n\n> The Shiyue \u5341\u6708 is good at describing slanderous situations. The Yu Wuzheng \u96e8\u7121\u6b63 and the Jie Nansan \u7bc0\u5357\u5c71 are all expressing the decline of their Lord, and kings and dukes feel ashamed. There are a lot of doubts in the Saomin \u5c11\u65fb, saying that one's willing is not pleased. The Xiaowan \u5c0f\u5b9b gives no hateful words, though there seems to be no calm conscience either. The Xiaobian \u5c0f\u5f01 and the Qiaoyan \u5de7\u8a00 all talk about the harm done by those who slander. (Confucius 2001: 135\u2013136)\n\nIt is evident then that those poems related to slanders imply always certain painful complaint. Negatively speaking, in some cases, what Confucius means by wisdom seems to be related to slanderous situations. For example, in fragment 28, Confucius says, \"...The Qiangyouqi \u7246\u6709\u85ba is with great caution and secrecy and yet knowing nothing about what to say. The Qingying \u9752\u8805 is wise\" (Confucius 2001: 158). In the Book of Odes, the poem Qingying expresses an anxious complaint against slanders that aim at separating someone from his king, using the buzzing of blue flies as a metaphor of slanderous speeches. We read the following verses of the Qingying,\n\n> They buzz about, the blue flies,\n> \n> Lighting on the fences.\n> \n> O happy and courteous sovereign,\n> \n> Do not believe slanderous speeches\n> \n> They buzz about, the blue flies,\n> \n> Lighting on the jujube trees.\n> \n> The slanderous observe no limits,\n> \n> And throw the whole kingdom into confusion. (Legge 1994b: 394\u2013395)\n\nAs shown by the Zuo's Commentary of the Spring and Autumn Annals, this poem was also used by the chieftains of barbarian tribes to complain about slanders during a diplomatic ritual of alliance. Since quoting and performing poems on social, political, or diplomatic occasions constituted a common practice among the nobles, it belonged to a part of the know-how of noble class, especially related to the li or rituals. Under the power of this ancient Chinese poetic culture, even the chieftains of barbarian tribes were familiar with it when they were establishing diplomatic relations with the Han people. For example, according to the Zuo's Commentary of the Spring and Autumn Annals, in the 14th year of Duke Xiang's Reign \u8944\u526c\u5341\u56db\u5e74 (558BC), it was mentioned that Fan Xuanzi \u8303\u5ba3\u5b50 blamed the barbarian chief Ju Zhi \u99d2\u652f for his disloyalty to Jin, and refused him participation in the morning court meeting. Ju Zhi refuted this accusation by saying that Jin was too unfair toward the barbarian tribes, and then he recited the Qingying \u9752\u8805 before he retreated from the court.\n\nWisdom is in contrasting position to slander; yet both are related to the proper or improper use of language. Confucius knows how wording as a vehicle to reveal truth is difficult. The learning of poems will equip a person well in what to say properly, whereas without learning poems, one would be ill-equipped with skill in speaking. This seems to be very much emphasized by Confucius in the newly unearthed Kongzi Shilun. For example, in Fragment 25, Confucius says that, \"The Dangdang \u8569\u8569 talks about the common people. The Youtu \u6709\u5154 expresses being in wrong time. The last chapter of the Datian \u5927\u7530 knows what to say and behaves courteously.\" In Fragment 29, he says that, \"In the Juan Er \u5377\u8033 we find no wise man. The Seqin \u6d89\u79e6 talks about quietude. The Zhu'er \u8457\u800c talks about the coming bridegroom. The Jiaozhen \u89d2\u6795 talks about woman. The Hesui \u6cb3\u6c34 talks about wisdom.\" Because of the fact that the Hesui is lost, we do not have the textual basis to judge what Confucius has in mind when he says that the Hesui is wise. Yet we can always say that wisdom on this level always concerns truth as revealed or distorted by language.\n\n## 8 Openness to the Unknown: Unending Quest of Wisdom\n\nIn conclusion, we may say that the classical Confucian classics, such as the Shijing, the Analects, the Zhuozhuan, and the Mencius, together with the newly unearthed bamboo slips Kongzi Shilun, show us the importance of poetry in ancient Chinese society in general and in Confucian teaching in particular. Especially the Kongzi Shilun conveys to us Confucius' wisdom in his systematic teaching of poetry, and in his seeing affectivity as the original mode of human existence. For Confucius, the human feelings, when expressed, should be cultivated by li leading to a virtuous life of harmony imbued with a sense of beauty. Confucius' poetic wisdom of appropriating the meaning of poetry is itself a creative interpretation and reasonable judgment of the poem's positive relevance to self-cultivation and the public realm.\n\nStill, it seems that wisdom is not totally exhausted in the cultivated expression of human affectivity and judgment, which always keeps to words that reveal truth and prudently avoids words that distort truth. Wisdom seems to be more than all these. It seems that, beyond the revealed and the expressed, wisdom is still open to the unfathomable, the hidden, and the unknown. In Fragment 10, Confucius first characterizes a poem as wise in saying that the Hanguan \u6f22\u5ee3 sings of wisdom (Confucius 2001: 139). Following this, in Fragment 11, he continues to say that \"the wisdom in the Hanguan is unattainable\" (Confucius 2001: 141). Then, in Fragment 13, he says that \"The Hanguan sings for wisdom which is uneasy to attain. Wisdom, though unattainable, is it not eternal?\" (Confucius 2001: 142).\n\nWhen we come to the traditional version of Hanguan in the Shijing, it sings in fact of some unapproachable beautiful lady (ladies) in making comparison to the uncrossable river Han. It reads,\n\n> In the South rise the trees without branches,\n> \n> Affording no shelter.\n> \n> By the Han River are girls rambling about,\n> \n> But it is vain to solicit them.\n> \n> The breath of the Han,\n> \n> Can not be dived across;\n> \n> The length of the Jiang,\n> \n> Can not be navigated with a raft.\n> \n> Many are the bundles of firewood;\n> \n> I would cut down the thorns [to form more]\n> \n> Those girls that are going to their future home,\u2013\n> \n> I would feed their horses.\n> \n> The breath of the Han,\n> \n> Can not be dived across;\n> \n> The length of the Jiang,\n> \n> Can not be navigated with a raft. (Legge 1994b: 15\u201316)\n\nIn Mao's commentary, the praise of ladies in this poem is said to be used as a metaphor to praise King Wen, under whose reign the dissolute manners of people, especially women, in the region south of Zhou, had undergone great transformation. However, contrary to Mao's commentary, the song could be read also simply as a love song, without any ideologizing ethical and political insinuation, which at most serves as a metaphor for wisdom. Thus, Confucius could interpret it as singing of the unattainability of wisdom. This is another case of Confucius appropriating meaning by creative interpretation, using Han River's uncrossability to express metaphorically a sense of the unattainability of the lady, and, furthermore, using the unattainability of the lady to tell of the unattainability of wisdom. The eternal wisdom is therefore comparable to a lady so beautiful, so attractive, yet unattainable. On this point it is quite similar to what the Book of Wisdom says, \"Wisdom I loved and searched for from my youth; I resolved to have her as my bride. I felt in love with her beauty\" (The New Jerusalem Bible 1998: 792).\n\nThe more the eternal wisdom is attractive, the more it is unattainable. Wisdom therefore possesses certain unfathomability. Confucian poetic wisdom is very humanistic in the sense that he puts emphasis on human affectivity and its appropriate expression through self-cultivation and public ritual, and yet he still opens himself to the hidden and the unfathomable dimension of existence. This implies an implicit understanding that human feeling and human desire are always longing for the infinite, while the ultimate truth is always unfathomable. Therefore, wisdom is still to be sought in the process of unending quest.\n\nReferences\n\nAmes, Roger and Henry Rosemont, trans. 1998. The analects of confucius, a philosophical translation. New York: Ballatine Books (English translation of the Analects with philosophical sensitivity to Western philosophical assumptions).\n\nChan Wing-Tsit. 1969. A source book in Chinese philosophy. Princeton: Princeton University Press (A comprehensive selection of Chinese philosophical works covering its entire historical development, with helpful introductions and explanations).\n\nConfucius. 2001. Konzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry). In The Shanghai Museum's Chu Bamboo books of the warring states, ed. Ma Chenyuan, Vol. 1. Shanghai: Shanghai Museum (recently unearthed bamboo slips on Confucius' teaching of the Classic of Poetry published by Shanghai Museum).\n\nHan Ying. 2006. Hanshi waizhuan \u97d3\u8a69\u5916\u50b3. Beijing: Beijing Tushuguan Chubanshe.\n\nHightower, James Robert, trans. 1952. Han Shih Wai Chuan, Han Ying's illustrations of the didactic application of the classic of songs. Cambridge: Harvard University Press (An annotated English translation of the Han Shih Wai Chuan).\n\nJinmen Museum, ed. 1994. Bamboo texts of Guodian Chu Tomb. Beijing: Wenwu Press (containing pre-Qin Daoist and Confucian texts unearthed in 1993 at Guodian Chu Tomb).\n\nKarlgren, Bernhard. 1942\u201346. Glosses on the book of odes. Bulletin of the Museum of Far Eastern Antiquities. Stockholm.\n\nKarlgren, Bernhard. (1950) The book of odes. Bulletin of the Museum of Far Eastern Antiquities, Stockholm (English translation with annotations by a famous Swedish sinologist and linguist).\n\nLau, D.C., trans. 1983. Confucius: The analects. Hong Kong: The Chinese University Press (one of the best English translations of the Analects).\n\nLegge, James, trans. 1994a. Confucian analects. In The Chinese classics, Vol. I. Taipei: SMC Publishing.\n\nLegge, James. 1994b. The she king. In The Chinese Classics, Vol. IV. Taipei: SMC Publishing.\n\nLegge, James. 1994c. The Ch'un Ts'ew with the Tso Chuen. In The Chinese Classics, Vol. V. Taipei: SMC Publishing.\n\nOwen, Stephen. 2006. The making of early Chinese classical poetry. Cambridge: Harvard East Asian Monograph Series (an updated work with new approach to classical Chinese poetry composed between the end of the first century B.C. and the third century A.D.).\n\nPuett, Michael. 2004. The ethics of responding properly: The notion of Qing \u60c5 in early Chinese thought. In Love and emotions in traditional Chinese literature, ed. Eifring, Halvor. Leiden: Brill.\n\nTang Yijie. 2003. Emotion in Pre-Qin Ruist moral theory: An explanation of 'Dao Begins in Qing'. Translated by Brian Bruya and Wen Hai-ming. Philosophy East and West 53.2.\n\nThe New Jerusalem Bible. 1998. Edited by Howard Wansbrough. London: Longman and Todd.\n\nWaley, Arthur. 1937. The book of songs. London\/New York\/Boston: George Allen and Unwin\/Houghton Mifflin.\n\nXu Fuguan \u5f90\u5fa9\u89c0. 1966. Zhongguo yishu jingshen. Taipei: Xusheng Shuju (A philosophical analysis of the essence of Chinese art by one of the most famous modern new Confucians).\n\nYu, Pauline, Peter Bol, Stephen Owen, and Willard Peterson. ed. 2000. Ways with words: Writing about reading texts from early China. Berkeley: University of California Press.\n\nFootnotes\n\n1\n\nSee for example, the Analects, 2.2, 7.18, 8.8, 9.15, 13.5, 16.13, 17.9, 17.10,...etc., in which Confucius talked about either the Classic of Poetry in general or various individual odes in it.\n\n2\n\nThese bamboo slips, unearthed from the Warring States Chu tombs in the area of Jinmen City, Hubei Province, China, where the Jinmen Bamboo Slips were also discovered in 1993, were found a bit later in Hong Kong antiquity market in the spring of 1994. They were then bought back, conserved and edited by the Shanghai Museum, China. The first volume of a series of books entitled Shanghai Museum's Chu Bamboo Books of the Warring States was published in November, 2001. There we find three pieces of works, now named respectively the Kongzi Shilun \u5b54\u5b50\u8a69\u8ad6 (Confucius on Poetry), the Ziyi \u7dc7\u8863 (Colorful Clothes), and the Xingqin Lun\u6027\u60c5\u8ad6 (On Human Nature and Human Affectivity). The texts now entitled Kongzi Shilun consists of 29 bamboo slips, which have been transcribed by philological scholars into 29 fragments. This is the first time we have so many fragments of Confucius' teaching on the Shijing among so many recent archaeological findings.\n\n3\n\nMao's commentaries to the Shijing were traditionally attributed to Mao Heng and Mao Chang, who were active in early Western Han Dynasty (206BC-9BC) without exact dates of birth and death. Their commentaries contained the earliest version of the Shijing.\n\n4\n\nSince the words xia \u590f and ya \u96c5 were very similar both phonetically and morphologically in ancient China, we may presume there was no big difference between the titles Daxia \u5927\u590f and Daya \u5927\u96c5, Xiaoxia \u5c0f\u590f and Xiaoya \u5c0f\u96c5. As to Bangfeng \u90a6\u98a8, it should be the original title, and was later changed to Guofen\u570b\u98a8.\n\n5\n\nAs to different titles, for example, there were Shiyue \u5341\u6708 instead of Shiyue zhijiao\u5341\u6708\u4e4b\u4ea4, Tang zhisui\u6e6f\u4e4b\u6c34 instead of Yang zhisui \u63da\u4e4b\u6c34, Bei baizhou \u5317\u767d\u821f instead of Bozhou \u67cf\u821f...etc. As to new titles, for example, Keshi\u53ef\u65af, Zhongshi \u4e2d\u6c0f, L\u00fc'er\u5f8b\u800c, Heshui \u6cb3\u6c34...etc.\n\n6\n\nAll the English translations of the Kongzi Shilun in this chapter are mine.\n\n7\n\nWe adopt the term phronesis from Aristotle, for whom it is practical wisdom in ethical sense. We use this term to express a meaning not limited to practical wisdom in ethics as for Aristotle. We extend its meaning also to poetics.\n\n8\n\nFor example, in the Guanju \u95dc\u96ce we read, \"Kwan-kwan go the ospreys, on the islet in the river. The modest, retiring, virtuous young lady; after whom a young gentleman loves to look for mate\" (Legge 1994b: 1; with my modification).\n\n9\n\nWe read, in the Xiaowan \u5c0f\u5b9b, \"Small is the cooing dove, but it flies aloft up to Heaven. My heart is wounded with sorrow, and I think of our forefathers. When the dawn is breaking. And I cannot sleep, the thoughts in my breast are of my own parents\" (Legge 1994b: 333\u2013334).\n\n10\n\nWe read, in the Qifu \u7948\u7236, \"Minister of War, We are the claws and teeth of the king. Why have you rolled us into this sorrow? So that we have no abiding place? Minister of War, We are the taloned soldiers of the king. Why have you rolled us into this sorrow? So that there is no end [of our toils]? Minister of War, we have indeed acted without discrimination. Why have you rolled us into this sorrow? So that our mothers have to do all the labor of cooking?\" (Legge 1994b: 298\u2013299).\n\n11\n\nFor example, we read, in the Wei Tian Zhi Ming \u7dad\u5929\u4e4b\u547d (The Mandate of Heaven): \"How solemn and unceasing! Oh, how illustrious was the purity of King Wen's virtue! With blessings he overwhelms us. We all receive the blessings. They are a great favour from our King Wen. May his remote descendents hold fast to them\" (Legge 1994b: 570\u2013571; with my modifications).\n\n12\n\nMy correction of D.C. Lau's translation of \"novelty\" into \"difference,\" this being more faithful to the term yi \u7570 in the Chinese text.\n\n13\n\nThe Jiong \u99c9 (Stallions) reads, \"Fat and large are the stallions, on the plains of the far-distant borders. Of those stallions, fat and large, some are cream-coloured; some, red and white; some, with white hairy legs; some, with fish eyes; All, stout carriage horses, Ah, how they are without depravity; He thinks of his horses, and thus serviceable are they\" (Legge 1994b: 613).\n\n14\n\nThe Hesui is lost now and we find only its title here in this text. However, it is possible that \"Hesui\" might be another title of the Miansui \u6c94\u6c34, in which it is sung, \"In large volume, those flowing waters go to the court of the sea. Rapid is that flying flacon, now soaring, now resting. Alas! Among my brethren, my countrymen, my friends, no one is willing to think of the prevailing disorder. But who has not parents to suffer from it?\" (Legge 1994b: 295).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_12\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 12. Early Confucian Moral Psychology\n\nKwong-loi Shun1\n\n(1)\n\nChinese University of Hong Kong, Hong Kong, China\n\nKwong-loi Shun\n\nEmail: kls.cuhk@gmail.com\n\nAbstract\n\nIn this chapter, I discuss an aspect of early Confucian ethical thought that arguably is one of its more distinctive characteristics, namely, its emphasis on self-transformation. By self-transformation, I refer to the transformation of oneself that comes about as a result of one's own reflective efforts at self-improvement; the process that involves such efforts I will refer to as \"self-cultivation.\" This aspect of Confucian thought is related to its distinctive nature as a kind of ethical thought. Suppose we use the term \"ethics\" broadly in the sense that an ethical concern has to do with a concern with how to live, or how one should live, where the scope of \"one\" is supposed to include most of those whom we nowadays would refer to as \"human beings.\" This way of describing the scope of \"one\" is intended to accommodate the fact that the term ren \u4eba (human beings), which is the term closest to \"human beings\" in early Chinese, is not understood in biological terms in early China and, on some scholarly views, might have a more restricted scope than that of the contemporary term \"human beings.\"\n\nThis chapter is part of my work on a multivolume project on Confucian ethics, and it draws occasionally on ideas in previously published articles cited in the references.\n\n## 1 Introduction\n\nIn this chapter, I discuss an aspect of early Confucian ethical thought that arguably is one of its more distinctive characteristics, namely, its emphasis on self-transformation. By self-transformation, I refer to the transformation of oneself that comes about as a result of one's own reflective efforts at self-improvement; the process that involves such efforts I will refer to as \"self-cultivation.\" This aspect of Confucian thought is related to its distinctive nature as a kind of ethical thought. Suppose we use the term \"ethics\" broadly in the sense that an ethical concern has to do with a concern with how to live, or how one should live, where the scope of \"one\" is supposed to include most of those whom we nowadays would refer to as \"human beings.\" This way of describing the scope of \"one\" is intended to accommodate the fact that the term ren \u4eba (human beings), which is the term closest to \"human beings\" in early Chinese, is not understood in biological terms in early China and, on some scholarly views, might have a more restricted scope than that of the contemporary term \"human beings.\"\n\nWhile early Confucian thinkers did exhibit an ethical concern in this sense, the way they engage in ethical thinking and teaching is very different from the way this is done in a contemporary academic setting. Their primary concern is not with developing or transmitting a systematic general account of the ethical life, but is more immediately practical. Their attention is directly focused on the daily ethical experiences of themselves and of their associates, the concrete ethical challenges they confront, and the way for themselves and their associates to properly navigate the ethical complexities of the world. As we can see from the record of his sayings, Confucius' attention was directed primarily to providing concrete ethical guidance to specific individuals, including rulers and officials as well as his students and other close associates. Confucian thought did subsequently evolve in a more general direction, leading to more general and systematic discussions of such topics as human nature (xing \u6027). Still, even when expounding on the human condition in general terms, most major Confucian thinkers continue to direct attention to concrete situations involving specific individuals they encounter in their daily life, in a way often reflected in their more general discourse.\n\nGiven the orientation of their ethical thinking, their attention is often focused on what we would now describe as the psychology of the individual person. In reflecting on concrete situations involving specific individuals with whom they are in interaction or in discourse, they see clearly that ethical problems generally have their source in the depths of the human psychology, and it is here that the fundamental ethical task resides. And while childhood upbringing is important, the task also involves a continuous self-reflective reshaping of oneself in adult life. \"Self- cultivation\" refers to this process, and \"self-transformation\" to the goal of this process.\n\nIn elaborating on this aspect of early Confucian thought, I will take as my primary sources Chinese texts up to the early Han that are usually classified as Confucian texts, including the Lunyu \u8ad6\u8a9e (Analects) of Confucius, the Mengzi \u5b5f\u5b50 (Mencius), the Xunzi \u8340\u5b50, the Daxue \u5927\u5b78 (Great Learning), and the Zhongyong \u4e2d\u5eb8 (Centrality and Commonality). The term usually translated as \"Confucianism\" or \"Confucian thought\" is rujia \u5112\u5bb6, or the school of ru \u5112, and this term was introduced in early Han as part of a retrospective classification of the pre-Han intellectual scene. Although ru \u5112 (Confucianism) was an identifiable social group of professional ritualists and educators at that time, the school of ru \u5112 (Confucianism) was not a well defined movement before the Han. Still, the texts just referred to do share some commonalities that warrant our grouping them together, and when referring to early Confucian thought, I will be referring primarily to ideas in these texts. For convenience, when this will not lead to confusion, I might sometimes refer to these ideas simply as Confucian ideas, without specifying that they come from the early period.\n\nSo far, I have freely used the notion of self in discussing early Confucian thought. This notion requires explanation, and this will be the task of Sect. 1. In Sects. 2, 3, and 4, I discuss three aspects of the early Confucian ethical ideal related to, respectively, ren \u4ec1 (humanity, benevolence), li \u79ae (propriety, observance of the rites), and yi \u7fa9 (dutifulness, righteousness), with a focus on how the self figures within this ethical ideal. I will show that there is a sense in which each of these three aspects of the Confucian ideal involves one's moving away from a certain kind of self-centeredness, and these three sections together describe the kind of self-transformation advocated by the early Confucians. In Sect. 5, I discuss the nature of the self-cultivation process involved in such a transformation, and in Sect. 6, I discuss ways in which one's ethical attention may be properly or improperly directed in such a process.\n\n## 2 The Self\n\nTo explain the notion of self we will be ascribing to the early Confucians, let us begin with some of the terms that early Chinese thinkers use to talk about various features of a person (Shun 2004: 183\u2013199). They use the term ti \u9ad4 (body), often translated as \"body,\" to talk about the person's body, and there are also ways of referring to parts of the body, such as the four limbs (to which ti \u9ad4 also refers) and the senses. These parts of the body not only have certain capacities, such as the eyes' capacity of sight, but they also exhibit certain characteristic tendencies, such as the way the eyes are drawn toward beautiful colors. These tendencies are referred to as yu \u6b32 (desires), a term often translated as \"desires.\" This term is used not just of parts of the body, but also of the person as a whole to describe how the person is drawn toward things like life and honor. That human beings have such tendencies as part of their basic constitution is regarded as a fact about them that is pervasive and difficult to alter, a fact that is referred to as the qing \u60c5 (facts) of human beings. Later, qing \u60c5 (emotions) comes to refer to what we would describe as emotions, including such thing as joy, sorrow and anger, these also being regarded as part of the basic constitution of a person.\n\nThere is another aspect of the Chinese view of the person for which it is difficult to find a western counterpart. The body of a person is supposed to be filled with qi \u6c23 (the life forces), a kind of energy or force that flows freely in and gives life to the person. Qi \u6c23 (the life forces) is responsible for the operation of the senses, and it can be affected by what happens to the senses. It is linked to the emotions, and what we would describe as a person's physical and psychological well-being is regarded as dependent on a proper balance of qi \u6c23 (the life forces). For example, both illness and such emotional responses as fear are explained in terms of the condition of qi \u6c23 (the life forces).\n\nAmong the different aspects of the person, early Confucians attach special significance to xin \u5fc3 (heart, mind), the organ of the heart which is viewed as the site of what we would describe as cognitive and affective activities. Xin \u5fc3 (heart, mind), a term often translated as \"heart\" or \"mind\", can have desires (yu \u6b32) and emotions (qing \u60c5), and can also deliberate and focus attention on things. It has the ability to set directions that guide one's life and shape one's person as a whole, and these directions of the heart\/mind are referred to as zhi \u5fd7 (directions of the heart\/mind). Zhi \u5fd7 (directions of the heart\/mind) can refer to specific intentions or general goals, and it is something that can be set up, nourished, and attained. It can also be altered by oneself or swayed under others' influence, and lost through insufficient persistence or distraction by other things. Zhi \u5fd7 (directions of the heart\/mind) has to do with the heart\/mind's focusing itself on and constantly bearing in mind certain courses of action or goals in life, in such a way that it will guide one's action or one's life unless it is changed by oneself or under others' influence, or unless one is led to deviate from it by other distractions.\n\nZhi \u5fd7 (directions of the heart\/mind) differs from yu \u6b32 (desires) in that, while zh \u5fd7 (directions of the heart\/mind) pertains specifically to the heart\/mind, yu \u6b32 (desires) can pertain to the heart\/mind or to parts of the body such as the senses. Also, while zhi \u5fd7 (directions of the heart\/mind) involves focusing the heart\/mind in a way that guides one's actions or one's life in general, yu \u6b32 (desires) involves tendencies that one may choose to resist rather than act on. There is another term, yi \u610f (thoughts and inclinations), often translated as \"thought,\" which refers to tendencies that differ from both zhi \u5fd7 (directions of the heart\/mind) and yu \u6b32 (desires). The term can refer to one's thoughts or opinions, or to one's inclinations, which involve one's wanting to see certain things happen, or one's thinking of bringing about certain things. Unlike yu \u6b32 (desires), which can involve tendencies (such as sensory desires) that just happen to obtain without one's having a reflective awareness of them, yi \u610f (thoughts and inclinations) is more reflective in that its object is something one is aware of as part of one's thoughts, which pertain to the heart\/mind. On the other hand, yi \u610f (thoughts and inclinations) is in a less directed state than zhi \u5fd7 (directions of the heart\/mind) in that, while yi \u610f (thoughts and inclinations) can be just a thought in favor of something without one's actually having decided to act in that direction, zhi \u5fd7 (directions of the heart\/mind) involves one's actually forming the intention or aim to so act.\n\nWith this account of the early Chinese view of the person as background, we can now turn to the notion of self that might be ascribed to the early Confucians. In using the notion, I have in mind three phenomena that can be found in early Confucian texts. First, as we just saw, we find a range of terms used in early Chinese texts to talk about the constitution of an individual person, including what we nowadays would describe as physical as well as psychological aspects of a person. Thus, in early China, there is a conception of the individual person as an identifiable entity distinct from other persons. The way some thinkers view what is distinctive of a human person might make reference to the way the individual person relates to other people. For example, both Mozi and Xunzi explicitly state that they view a person (ren \u4eba) in social terms, namely, in terms of their capacity to draw and abide by social distinctions (Mozi 1948: 16\/12\/4\u20135; Xunzi 1965: 3.3b\u20134a, cf. 5.7b\u20138a). Nevertheless, while holding such views, these thinkers still speak in a way that allows them to talk about the relation between the individual person and other people, showing that they still have a conception of the individual person as distinct from, though intimately related to, other people.\n\nSecond, in classical Chinese, there are first personal pronouns, such as wo \u6211 (oneself) and wu \u543e (oneself), that can be used to refer to oneself. In addition to these first personal pronouns, the classical Chinese language also has two characters with the meaning of \"oneself\"; zi \u81eaand ji \u5df1. The two characters differ in that the former emphasizes one's relation to oneself, while the latter emphasizes oneself as contrasted with others (ren \u4eba). These linguistic observations show that the early Chinese not only have a conception of an individual person as distinct from other people, but they also regard each individual person as having a conception of oneself as an identifiable entity distinct from other people, as well as a conception of the way one relates to oneself.\n\nThird, in early Confucian texts, the characters just mentioned are often used to talk about one's examining oneself and making improvements to oneself on the basis of such self-examination. This shows that early Confucian thinkers also work with a conception of one's being related to oneself in a self-reflective manner, with the capacity to reflect on, examine, and bring about changes in oneself. They ascribe this capacity to the heart\/mind, which plays a guiding role in one's life. And, as we will discuss in greater detail later, the heart\/mind also has the capacity to hold on to the directions it sets without being swayed by external forces, as well as the capacity to constantly step back from and reflect on its own activities and to reshape them in accordance with its conception of what is proper.\n\nIn speaking of the Confucian conception of the self, I have in mind these three aspects of the way the early Confucians view the individual person \u2013 that they have a conception of an individual person as distinct from (though also intimately related to) other people, that they see each such person as also having a conception of oneself as distinct from other people and of the way one relates to oneself, and that they regard each such person as having, through the capacity of the heart\/mind, the capacity to constantly reflect on and bring about changes to oneself, including changes to the activities of the heart\/mind itself. Throughout the chapter, I will be using the notion of self in this minimal sense, without being committed to more substantive accounts of what constitutes the self, such as whether it is constituted by a stream of consciousness.\n\nHaving discussed the Confucian conception of the self, I turn now to the content of the Confucian ethical ideal. Throughout the history of Confucian thought, ren \u4ec1 (humanity, benevolence), li \u79ae (propriety, observance of the rites), and yi \u7fa9 (dutifulness, righteousness) continue to be three prominent concepts used in characterizing this ideal. I will consider these three aspects of the Confucian ideal in the next three sections, focusing on how the self figures in each. I will show that, in each case, there is a sense in which what is advocated involves one's moving away from a certain kind of self-centeredness. Together, these three aspects of the Confucian ideal point to a kind of self-transformation that involves properly situating the self in relation to others and in relation to certain ethical standards.\n\n## 3 Concern for Others\n\nRen \u4ec1 (humanity, benevolence) as part of the Confucian ethical ideal has to do with one's concern for the well-being of others. The way such concern is spelt out takes different forms in the history of Confucian thought, and ren \u4ec1 (humanity, benevolence) is also related to what some might call 'metaphysical' views, especially in later Confucian thought. In addition, there are different scholarly views on the earlier connotations of the term; for example, some believe the term carried in earlier times the connotation of fully embodying the distinctive characteristics of a human person, which accounts for the occasional translation of ren \u4ec1 (humanity, benevolence) as \"humanity.\" In my discussion, I will focus on that aspect of ren \u4ec1 (humanity, benevolence) having to do with one's concern for the well-being of others, and set aside these other dimensions of ren \u4ec1 (humanity, benevolence). This aspect of ren \u4ec1 (humanity, benevolence) has been discussed in the English language literature using various contemporary terms such as \"love,\" \"compassion,\" \"empathy,\" \"sympathy,\" or \"benevolence.\" These terms, however, often carry certain specific connotations, especially in contemporary philosophical analysis of the phenomena they refer to, that might not be present in the Confucian understanding of ren \u4ec1 (humanity, benevolence). To avoid inadvertently ascribing these connotations to ren \u4ec1 (humanity, benevolence), I will avoid the use of these terms, and instead speak of concern for the well-being of others, or simply concern for others, when referring to the relation between self and others that Confucian thinkers advocate in their discourses on ren \u4ec1 (humanity, benevolence).\n\nTo be more specific, in speaking of concern for others, I am referring to the ways in which one's attention maybe focused directly on the well-being of others without mediation by other considerations, and in a positive manner in that one seeks to promote the good of others and to alleviate their negative conditions. This way of characterizing concern for others is intended to exclude other ways in which one maybe related to the well-being of others, such as when one's attention is directed to the well-being of others as a way to accomplish other goals, or when one's attention is focused on others in a way that is conducive to their well-being though their well-being is not the object of one's attention. An example of the latter is the posture of jing \u656c (treat respectfully), which I will discuss in the next section. Characterized in this manner, the notion of concern for others is specific enough to capture the aspect of ren \u4ec1 (humanity, benevolence) under consideration, while also broad enough to accommodate the different ways in which this aspect of ren \u4ec1 (humanity, benevolence) is spelt out in the history of Confucian thought. This characterization is also non-committal on questions such as whether such concern involves one's imaginatively placing oneself in the other's position, unlike contemporary terms such as \"empathy.\" All Confucian thinkers regard such concern for others as part of the ethical ideal, and some, such as Mencius and later Confucian thinkers under his influence, believe it to be already to some extent part of the basic human constitution.\n\nOne's concern for others takes at least three distinct forms in the early Confucian discourse on ren \u4ec1 (humanity, benevolence). First, I may have a concern for the well-being of a specific individual by virtue of some special situation she is in, or some situation in which she and I are involved in some special way. A number of such examples are presented in the Mengzi \u5b5f\u5b50 (Mencius). One passage observes how one would react with ce yin \u60fb\u96b1 (commiseration) upon suddenly seeing a baby on the verge of falling into a well, and another observes how King Xuan of Qi (Qi Xuan Wang \u9f4a\u5ba3\u738b) reacted with bu ren \u4e0d\u5fcd (unable to bear the suffering of others) upon seeing an ox being led to be killed as part of a sacrificial rite (Mengzi 1984: 1A:7, 2A:6). The reactions are presented as immediate responses of the heart\/mind to some imminent negative condition of a living thing, responses that are felt with some degree of intensity and that move one to act in certain ways to alleviate or pre-empt the negative condition. These responses are not mediated by any kind of calculative attitude, though some kind of imaginative exercise might be involved \u2013 King Xuan's response to the plight of the ox is described in terms of his viewing the ox as if it were an innocent person being led to the place of execution. Beyond these general observations about the responses, the textual evidence is not sufficient to enable us to draw further conclusions about whether these responses also carry more specific connotations that are highlighted in contemporary philosophical discussions of such phenomena as compassion, empathy, or sympathy.\n\nThere is another kind of response to a specific individual that involves a situation in which I am in a position to possibly treat the individual in a certain manner. The response is conveyed by the notion shu \u6055 (reciprocity), a notion found in most of the early Confucian texts and explained in the Lunyu \u8ad6\u8a9e (Analects) in terms of my not imposing on another what I would not wish to be imposed on myself (Lunyu 1980: 15.24, cf. 12.2). Shu \u6055 (reciprocity) has to do with potentially negative conditions of an individual in that the contemplated treatment from which I should refrain is either unwelcome to the individual or not in her interest. Though shu \u6055 (reciprocity) is presented in the Lunyu \u8ad6\u8a9e (Analects) as a more reflective kind of exercise, it is likely that ideally, for the Confucians, one should cultivate oneself to the point when one would so respond without engaging in such explicit reflection (Zhu 1986: 116, 358, 672; Zhu 1983\u20131986a: 3.4b\u20135a, 1983\u20131986b: 11.30a).\n\nThe second kind of concern for others involves a concern for the well-being of another individual on an ongoing basis, by virtue of some special relation I stand to that individual. This phenomenon is related to the Confucian advocacy of a gradation in one's concern for others depending on how one is related to them. This kind of concern differs from the first in a number of ways, as can be seen from the example of one's relation to one's parents. Although one would still respond to specific situations involving potentially negative conditions of one's parent, one's attention can be directed to her in a way that is not dependent on awareness of any such situations. Instead, one's concern for her is ongoing \u2013 for example, one continues to think of her while traveling afar, hopes that her aging does not impair her health, and is constantly on guard against doing anything that would bring her disgrace. Furthermore, unlike the first kind of concern, one's attention need not be restricted to negative or potentially negative conditions of hers. Instead, one actively seeks to promote her well-being such as by looking after her health, and to bring her joy in various ways. Indeed, the relation to one's parent is even more complex as it also involves another kind of attitude which is conveyed by the term jing \u656c (treat respectfully) and which we will consider in the next section.\n\nThe third kind of concern has to do with individuals who might not be related to one in any special way. It is directed not to a specific individual, but to living things, or human beings, or a sub-class of human beings, in general. Such attitude is portrayed from time to time in early texts in connection with those who have people under their care, and is sometimes put in terms of one's forming one body (ti \u9ad4) with those under one's care. For example, the ideal ruler is described as someone who regards the common people as part of his body or who forms one body with the common people (e.g., Liji 1965: 17.16a; Guanzi 1965: 10.18a). The best illustration of this kind of attitude is probably the story of Yu's (Da Yu \u5927\u79b9) efforts to channel the flood that had caused immense suffering to the people; he was so devoted to alleviating the plight of the people that he three times passed the door of his own household without entering (Mengzi 1984: 4B29). The way a caring ruler or official feels for the people is also sometimes portrayed using the relation between parents and children as analogy \u2013 the ruler or official is like the parents (fu mu \u7236\u6bcd) of the common people, whom they look after as if caring for a new born infant (chi zi \u8d64\u5b50) (e.g., Mengzi 1984: 3A5). This kind of concern differs from the first in that one's concern is directed not to any specific individual but to people generally, and it can be directed not just to alleviating their suffering but also to positively promoting their well-being.\n\nWhile the three kinds of concern just described differ in important ways, they share the common underlying idea that, ideally, one's attention should be directed to, and one should be sensitive to, the well-being of others in certain appropriate ways. Falling short of this ideal involves one's being overly focused on one's own interests and well-being, and as a result being insufficiently attentive and sensitive to the interests and well-being of others. Put differently, self and others should ideally be connected in certain appropriate ways, and it is a kind of self-centeredness that separates one from others. Part of the self-transformation that the early Confucians advocate involves one's moving away from this kind of self-centeredness, properly situating the self in relation to others. This aspect of the early Confucian ideal is later taken up by Sung-Ming Confucians and developed into the idea of 'one body.' Each individual person ideally forms one body with all living things in that one is sensitive and responsive to the conditions of all living things, and it is through self- centeredness (si \u79c1) that one separates oneself from others (Shun 2005: 1\u20139).\n\n## 4 Lowering Oneself and Elevating Others\n\nLet us turn next to a cluster of attitudes related to li \u79ae (propriety, observance of the rites), a term often translated as \"rites.\" Li \u79ae (propriety, observance of the rites) originally referred to rites of sacrifices, but later came to be used of rules of conduct governing ceremonial behavior in various recurring social contexts. Subsequently, its scope broadened even further to include rules governing behavior appropriate to one's social position, though it continued to be used frequently in connection with ceremonial behavior. From a contemporary western perspective, in which there is not one single term whose scope even approximates that of li \u79ae (propriety, observance of the rites), it might seem puzzling how this variety of rules can be subsumed under a single term. What gives unity to the rules of li \u79ae (propriety, observance of the rites) is the spirit behind li \u79ae (propriety, observance of the rites), which is presented in early texts as jing \u656c (treat respectfully), a term often translated as \"respect\". Li \u79ae (propriety, observance of the rites) is also related to other attitudes, such as rang \u8b93 (letting others have what is good or honorable), and Mencius describes gong jing \u606d\u656c (treat respectfully and with specific postures in being respectful) and ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable) as the basis for li \u79ae (propriety, observance of the rites) (Mengzi 1984: 2A:6, 6A:6). While these attitudes are directed toward others, they are unlike the kind of concern for others considered in the previous section, which involves one's attending directly to the well-being of others.\n\nJing \u656c (treat respectfully), though often directed toward deities or persons, can also be directed toward things and affairs, such as de \u5fb7 (virtue) or one's official responsibilities. In early texts, it is often paired with shen \u614e (a cautious and attentive attitude), and with jie \u6212 (an attitude of being on guard). It likely involves devotion, focus of mental attention, caution, and being on guard against things going wrong. It is occasionally used in early texts such as the Yijing \u6613\u7d93 (The Classics of Changes) to refer to a posture that is not directed to any specific object, and this usage is highlighted by later Confucians to refer to a posture of seriousness that is part of the self-cultivation process. But more often, it is used in early texts to refer to an attitude directed toward deities, persons, or affairs to which one should be devoted, such as one's official responsibilities (for further elaboration on jing \u656c, see Shun 1997: 52\u201354).\n\nTo see how jing \u656c (treat respectfully) differs from the kind of concern for others considered in the previous section, let us consider one's relation to one's parents. Imagine, for example, that one is serving a meal to an elderly parent who cannot take care of herself (cf. Lunyu 1980: 2.7; Mengzi 1984: 4A19, 7A37). In being concerned for her well-being, one would ensure that she is well-nourished and that the kind of food served is pleasing to her. But one could have done something similar for an animal one is keeping as a pet. Jing \u656c (treat respectfully t) toward the parent involves an attitude that goes beyond just attending directly to her well-being. It involves a certain posture toward the parent that is displayed in the way one serves the food \u2013 one's demeanor in passing the food, one's attentiveness to her reactions, the words one uses when speaking to her, one's not being distracted by non-urgent business when serving her, etc. The attentiveness and seriousness one displays, though pleasing to her, are not themselves focused directly on her material needs or on what she finds pleasing. Instead, they are focused directly on her as a person whom one should treat seriously and with attention. If we are to find a western equivalent, the posture involved is probably close to that of treating someone respectfully \u2013 while the person one treats in this manner will be pleased by such treatment, one's attention in treating the person respectfully is not directed to seeking ways to please her.\n\nWhen directed to persons, jing \u656c (treat respectfully) may have to do with certain specific qualities that pertain to its objects. For example, it may be directed to superiors in government or to elders; in these cases, it is by virtue of certain qualities that these individuals have \u2013 their superior position in government or their age \u2013 that one treats them with jing \u656c (treat respectfully). And because jing \u656c (treat respectfully) is often translated as \"respect,\" comparison with contemporary western discussions of respect for persons might lead one to draw the inference that, when Confucian thinkers advocate jing \u656c (treat respectfully) toward people in general, jing \u656c (treat respectfully) is also a response to a certain intrinsic quality shared by all human beings. Such a quality maybe labeled variously as \"human worth\" or \"human dignity,\" and treating human beings with jing \u656c (treat respectfully) maybe regarded as a matter of according them 'due regard' in the sense that jing \u656c (treat respectfully) is a response called for by such an intrinsic quality that all human beings share (for an excellent discussion that takes this direction, see Chan 2006: 229\u2013252. Though my interpretation of jing is different, my thoughts on the subject has been stimulated by Chan's discussion).\n\nIt is unclear that there is textual evidence in support of such an understanding of jing \u656c (treat respectfully t). While Confucian thinkers do advocate jing \u656c (treat respectfully), as well as other kinds of attitudes, toward special classes of people, or people in general, whether they regard these attitudes as responses called for by certain intrinsic qualities of their objects will depend on how they explain their advocacy. For example, while early Confucians advocate love toward one's parents, it does not follow that they regard such love as a response to certain intrinsic qualities pertaining to one's parents. Instead, they might explain it in terms of the past relationship between parents and children, such as how parents have cared for their children when the latter were young (e.g. Lunyu 1980: 17.21). Similarly, in urging that we treat people generally with jing \u656c (treat respectfully), it does not follow from such advocacy by itself that Confucian thinkers regard jing \u656c (treat respectfully) as a response to certain intrinsic qualities that all people share.\n\nAs far as I can tell, there is no evidence that early Confucians subscribe to the contemporary idea of intrinsic human worth or view jing \u656c (treat respectfully) in a way close to the contemporary understanding of respect for persons. As mentioned earlier, if we are to find some western equivalent to jing \u656c (treat respectfully), the attitude involved in jing \u656c (treat respectfully) is probably closer to 'treat respectfully' than to 'respect,' and the idea that we should treat people respectfully does not carry the kind of connotations that are often associated with the idea of respect for persons. That the way jing \u656c (treat respectfully) is used in early texts is unlike the way \"respect\" is used nowadays can be seen from other considerations. For example, while we might come to respect someone on learning about some special accomplishment of the person, I am not aware of any instance in which jing \u656c (treat respectfully) is similarly used in early texts. Also, in early texts, attitudes related to jing \u656c (treat respectfully) are at times presented in such a way that it involves one's viewing people as if one were dealing with them in certain contexts that did not actually obtain. For example, in the Lunyu \u8ad6\u8a9e (Analects), officials are urged to deal with people one encounters in one's travels as if one were receiving important guests, and to employ the common people as if one were conducting an important sacrifice (Lunyu 1980: 12.2). A contemporary example of a similar nature would be for senior scholars to be urged to speak with junior colleagues at professional conferences as if they were accomplished scholars. In these examples, what one is invited to do is to engage in an imaginary exercise that makes it easier for one to adopt a posture of a certain kind when dealing with others, rather than to respond to a certain intrinsic quality that exist in those whom one deals with. The translation \"treat respectfully\" better conveys this connotation of jing \u656c (treat respectfully), and helps us avoid the misleading connotations potentially generated by the translation \"respect.\"\n\nTurning to some of the other attitudes associated with li \u79ae (propriety, observance of the rites), gong \u606d (specific postures in being respectful) is often paired with jing \u656c (treat respectfully) though the two differ in their emphases. Whereas jing \u656c (treat respectfully) is an attitude of caution, seriousness, and mental attention that can be directed toward people and affairs, gong \u606d (specific postures in being respectful) is a more specific attitude having to do with attention to one's appearance, posture, manners, and manner when dealing with others. Both gong \u606d (specific postures in being respectful) and jing \u656c (treat respectfully) have to do with the way one's attention is directed, and gong \u606d (specific postures in being respectful) is not a mere matter of outward appearance. Still, in gong \u606d (specific postures in being respectful), one's attention is directed primarily to externals of the kind just described. As for ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable), ci \u8fadinvolves politely declining, and rang \u8b93 letting others have, something good or of honor to oneself. An example of ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable) is to politely declare one's incompetence to address a question put to one by an elder before responding to the question, rather than immediately proffering an answer (Liji 1965: 1.3b). By so doing, one conveys that one sees the opportunity to respond to the question as an honor, something that one does not necessarily deserve (For further elaboration on gong \u606d and cirang \u8fad\u8b93, see Shun 1997: 54\u201355).\n\nAlthough gong jing \u606d\u656c (treat respectfully and with specific postures in being respectful) and ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable) differ in the manner described, they probably refer to different aspects of a more general attitude that Confucians believe underlies li \u79ae (propriety, observance of the rites). Jing \u656c (treat respectfully) involves taking the other person seriously, focusing one's attention on and treating the other person with caution. Gong \u606d (specific postures in being respectful) involves attending to one's outward presentation of oneself in dealing with the other person, including one's appearance, posture, manners, and demeanor. Together, gong jing \u606d\u656c (treat respectfully and with specific postures in being respectful) demonstrates a serious regard for the other person, in a way that would have been appropriate to someone of higher status than oneself, such as deities, superiors, or elders. Ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable), on the other hand, involves a posture that focuses on declaring one's being in some sense 'lower' than the other, such as being less deserving of an honor that has been offered. This does not mean that one literally has a low opinion of oneself or lacks awareness of good qualities that one might have. Rather, it is a matter of not having oneself at the forefront of one's thinking when interacting with others \u2013 one does not display oneself nor seek attention or admiration. Together, gong jing \u606d\u656c (treat respectfully and with specific postures in being respectful) and ci rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable) are two sides of a more general attitude that underlies li \u79ae (propriety, observance of the rites), an attitude that is described in the Liji \u79ae\u8a18 (The Book of Rites) as \"lowering oneself and elevating others\" (zi bei er zun ren \u81ea\u5351\u800c\u5c0a\u4eba) (Liji 1965: 1.3a).\n\nThe nature of this attitude can be brought out further by comparing ren \u4ec1 (humanity, benevolence) with li \u79ae (propriety, observance of the rites). Ren \u4ec1 (humanity, benevolence) has to do with a view of human beings as beings that have material needs, are vulnerable to pleasure and pain, etc. By contrast, li \u79ae (propriety, observance of the rites) has to do with a view of human beings as beings that have a sense of dignity and sensitivity to the way they are treated. This aspect of human beings requires a degree of self-awareness that is not shared by other animals. For example, while non-human animals may accept food given in whatever manner when hungry, a person would have reluctance accepting food given in an abusive manner even when starving (e.g., Mengzi 1984: 6A:10). Li \u79ae (propriety, observance of the rites), as a set of rules governing behavior in recurring social contexts, is a codification of our acknowledgement of this aspect of humans, and helps build and reinforce our conception of human beings as distinct from other animals in the manner just described. And for people standing in special social relations, such as rulers and subordinates, or parents and children, li \u79ae (propriety, observance of the rites) further codifies the acknowledgement of the special status of people in certain social positions. Human beings naturally tend to put more weight than appropriate on their own interests and well-being, and ren \u4ec1 (humanity, benevolence) helps to steer us away from such a tendency so that we also attend in appropriate ways to the interests and well-being of others. Likewise, human beings naturally tend to over-emphasize their own importance, and li \u79ae (propriety, observance of the rites) helps to steer us away from such a tendency so that we also pay appropriate attention to others in our interactions with them. The idea of 'lowering oneself and elevating others' (zi bei er zun ren\u81ea\u5351\u800c\u5c0a\u4eba) used in the Liji \u79ae\u8a18 (The Book of Rites) to characterize the attitude behind li \u79ae (propriety, observance of the rites) is not a matter of our believing ourselves to be literally in a lower position. Rather, it is a matter of our shifting our attention away from ourselves toward others, in a way that is akin to one's attitude when interacting with people in a higher position. Such redirection of attention is particularly important for those actually in a higher social position, as it is particularly tempting for them to treat those in a lower social position in a disrespectful manner. The remark in the Lunyu \u8ad6\u8a9e (Analects) about dealing with people one encounters as if one were receiving important guests, and employing the common people as if one were conducting an important sacrifice, is directed to those in office vulnerable to this tempting tendency (Lunyu 1980: 12.2).\n\nThus, as part of the Confucian ethical ideal, li \u79ae (propriety, observance of the rites) steers one away from another form of self-centeredness, different from that discussed in the previous section in connection with ren \u4ec1 (humanity, benevolence). The latter involves a move away from an undue emphasis on one's own interests and well-being that might result in insufficient attention and sensitivity to the interests and well-being of others. The former, on the other hand, involves a move away from attaching an undue importance to oneself that might result in one's not treating other people with sufficient seriousness and attentiveness. In both cases, the emphasis is on how the self should be properly situated in relation to others. In the next section, we will consider the kind of attitude associated with yi \u7fa9 (dutifulness, righteousness). While this attitude also involves a move away from a kind of self-centeredness, the emphasis in this case is on the submission of the self to certain ethical standards.\n\n## 5 The Self and Ethical Standards\n\nIn early texts, yi \u7fa9 (dutifulness, righteousness) is often related to ru \u8fb1 (disgrace), or disgrace \u2013 to be subject to disgrace is to be lacking in yi \u7fa9 (dutifulness, righteousness). Likely, the earlier meaning of yi \u7fa9 (dutifulness, righteousness) has to do with a sense of honor and an absence of disgrace; it is a matter of self-regard, not allowing oneself to be subject to disgraceful treatment. One possible attitude toward disgrace (ru \u8fb1) is wu \u60e1 (dislike, aversion), and early thinkers regarded the desire for honor (rong \u69ae) and aversion to disgrace (ru \u8fb1) as part of the fundamental constitution of human beings. But wu \u60e1 (dislike) can be directed at anything that one dislikes, such as death, unpleasant sights and sounds, or insecurity. Although all these things relate to oneself \u2013 it is one's own death or insecurity, and it is the unpleasant sights or sounds that one experiences, that one dislikes \u2013 ru \u8fb1 (disgrace) is related to oneself in a more intimate manner. The ru \u8fb1 (disgrace) one suffers is not just something that one dislikes; it reflects adversely on oneself and results in a lowering of one's standing. One's attitude toward ru \u8fb1 (disgrace) can therefore take on a special form, which is referred to as chi \u6065 (shame, regard as shameful), a term often translated as \"shame\" or \"regard as shameful.\" Unlike wu \u60e1 (dislike), chi \u6065 (shame, regard as shameful) focuses on ru \u8fb1 (disgrace) as something that is beneath oneself or lowers one's standing. Thus, chi \u6065 (shame, regard as shameful) involves a more reflective concern with the self, having to do with thoughts about the effect on oneself of certain occurrences. Though often translated as \"shame\" or \"regard as shameful,\" chi \u6065 (shame, regard as shameful) differs from contemporary western notions of shame in important respects. Chi \u6065 (shame, regard as shameful) can be directed toward something contemplated as well as toward what has already come about. It is not associated with the thought of being seen or the urge to hide oneself. Instead, it is associated with the thought of being tainted and the urge to cleanse oneself of what is tainting; the idea of cleansing oneself is conveyed by the expression xue chi \u96ea\u6065 (cleanse the tainted). Chi \u6065 (shame, regard as shameful) is linked to a resolution to either remedy the disgraceful situation if it has already obtained, or to distance oneself from or pre-empt a potentially disgraceful situation if it has not yet come about (for further elaboration on chi \u6065, see Shun 1997: 58\u201361).\n\nNow, in early China, what is regarded as disgraceful is often treatment that is insulting by public standards, such as being beaten in public, being stared in the eyes, or being treated in violation of certain accepted protocols of conduct that include li \u79ae (propriety, observance of the rites). \"Insulting\" is used here as a translation of the character wu \u4fae (insult), a character that refers to the kind of treatment just described, treatment that is inappropriate by certain generally accepted public standards. It differs from ru \u8fb1 (disgrace) in that while wu \u4fae (insult) is a matter of how certain forms of treatment measure against generally accepted public standards, ru \u8fb1 (disgrace) focuses on the viewpoint of someone who is subject to such treatment, involving a perception of the treatment as somehow diminishing oneself. Ru \u8fb1 (disgrace) is closely identified with wu \u4fae (insult) in early China \u2013 that is, what one regards as diminishing oneself is usually treatment that is insulting by generally accepted public standards. As a result, insulting treatment is viewed with chi \u6065 (regard as shameful), which is often associated with anger at a situation and the urge to fight back to avenge the situation (the discussion that follows draws on Shun (forthcoming). See also Shun 1997: 61\u201363).\n\nFighting of this kind was so pervasive in early China that it led one early thinker to propose that, if one stops seeing what is insulting as a disgrace, such fighting would stop (see the presentation of Songzi's position in Xunzi 1965: 12.11a\u201311b). Xunzi took note of this view, but disagreed on the ground that whether people fight depends on what they dislike, and as long as they dislike insulting treatment, the fighting will not stop regardless of whether one regards such treatment as a disgrace. Contrary to Xunzi, though, this thinker has probably made a valid point \u2013 in not regarding the insulting treatment as disgraceful, one no longer sees it as a personal affront even if one still dislikes it, and it is seeing something as a personal affront that leads to the kind of fierce fighting that has become problematic. In any instance, Xunzi's own position shares something in common with that of the other early thinker in that he also advocates a fundamental change in what one regards as truly disgraceful. According to him, what we regard as disgraceful should not be tied to the way others view or treat us, but should be a matter of our own ethical conduct, which also includes the way we respond to others' treatment of ourselves (Xunzi 1965: 12.12b).\n\nThis view is shared by practically all Confucian thinkers. Several passages in the Lunyu \u8ad6\u8a9e (Analects) also make the point that what one regards as shameful, that is, the proper object of chi \u6065 (shame, regard as shameful), should be a matter of one's own qualities and actions rather than the way one is viewed or treated by others. In the Mengzi \u5b5f\u5b50 (Mencius), we find a contrast between the ranks of Heaven (tian \u5929) and the ranks of human beings, and between what is truly worthy of esteem and esteem that is conferred by humans; the contrast is between the ethical attributes and official ranks in government (Mengzi 1984: 6A:16, cf. 6A:17). In addition, Mencius also advocates a higher form of courage over a lower form of courage. The latter has to do with fighting in response to insulting treatment; the former, by contrast, has to do with the resolve to correct situations that one regards as ethically problematic (Mengzi 1984: 1B:3, 2A:2). Thus, while Mencius continues to relate yi \u7fa9 (dutifulness, righteousness) to chi \u6065 (regard as shameful), he also holds the view that the kind of situations to which chi \u6065 (regard as shameful) should be directed are situations that are disgraceful by ethical standards. It follows from the early Confucian way of viewing the proper object of chi \u6065 (shame, regard as shameful) that chi \u6065 (shame, regard as shameful) is no longer linked to the thought of avenging oneself, as its object is no longer the way one is treated by others. Instead, chi \u6065 (shame, regard as shameful) has more to do with the resolve to distance oneself from certain situations that can be ethically tainting on oneself, and to correct such situations should they arise.\n\nWhat is innovative about the early Confucian position is a different way of viewing the proper object of chi \u6065 (shame, regard as shameful), while still retaining the linkage between yi \u7fa9 (dutifulness, righteousness) and chi \u6065 (shame, regard as shameful). It continues to regard a person of yi \u7fa9 (dutifulness, righteousness) as a person with a sense of self-regard, someone with the resolve to distance oneself from disgraceful situations that are tainting on oneself. But it views what is truly disgraceful in ethical terms: what is tainting on one is for one to be ethically inferior or to conduct oneself unethically. One implication of this view is a move away from a certain kind of focus on oneself. Even when I have been treated inappropriately by others, my focus is not on the situation viewed as a personal affront to myself. Instead, I view it as an ethical situation, and my focus is on how I could respond in an ethically appropriate manner to the situation. Thus, though there is still a sense of proper self-regard, this sense of self-regard is focused on my not responding in an ethically problematic manner to the situation, rather than on my not being treated in a certain manner. Having decoupled chi \u6065 (shame, regard as shameful) from the perspective of seeing something as a personal affront, and having linked it to one's own ethical conduct, chi \u6065 (shame, regard as shameful) can now be a response to a situation in which others are treated inappropriately. An example is that of King Wu (Wu Wang \u6b66\u738b), who upon seeing a tyrant heaping miseries on the people, viewed with chi \u6065 (shame, regard as shameful) the prospect of his not doing something to correct the situation (Mengzi 1984: 1B:3). The kind of self-regard highlighted in this Confucian view is different not just from the view that focuses on one's not being treated in an insulting manner, but also from the views of self-respect presented in certain contemporary philosophical discussions. On certain philosophical accounts, self-respect has to do either with a certain assessment of oneself, related to the worth of oneself or excellences that one possesses, or with protecting one's rights to or claims on not being treated in certain ways (Darwall 1977: 47\u201349). By contrast, the kind of self-regard highlighted in the Confucian view focuses on the idea of not falling below certain ethical standards in one's own behavior, involving the viewpoint that it is tainting on oneself to fall below such standards.\n\nIn this way, though the Confucian understanding of yi \u7fa9 (dutifulness, righteousness) is still related to self-regard, the focus is now on properly subordinating the self to certain ethical standards. These standards are regarded as something that the heart\/mind can grasp in a process that is akin to perception. On a number of occasions, Mencius compares the heart\/mind to the senses. Just as the senses are drawn toward and take pleasure in certain sensory objects, the heart\/mind is drawn toward and takes pleasure in liyi \u7406\u7fa9 (morality) (Mengzi 1984: 6A:7). The relation between the heart\/mind and yi \u7fa9 (dutifulness, righteousness) is described in terms of si \u601d (focusing or directing the attention of the heart\/mind); when the heart\/mind si \u601d (focusing or directing the attention of the heart\/mind), it will attain yi \u7fa9 (dutifulness, righteousness) (Mengzi 1984: 6A:15). Si \u601din early Chinese texts has the connotation of focusing or directing the attention of the heart\/mind, just as one focuses or directs the attention of the ears or eyes when one listens or looks (Shun 1997: 149\u2013153). This perceptual metaphor for describing the operation of the heart\/mind is also found in the Xunzi \u8340\u5b50. While Xunzi uses zhi \u77e5 (know, understand), a term often translated as \"know\" or \"understand,\" to describe the relation of the heart\/mind to dao \u9053 (Way) or li \u7406 (pattern), zhi \u77e5 (know, understand) is itself presented in terms of a perceptual metaphor. The zhi \u77e5 (know, understand) of the heart\/mind can be ming \u660e (bright), or bright (e.g., Xunzi 1965: 1.1a), where ming \u660e (bright) is illustrated by the brightness of fire or of the sun and moon (e.g., Xunzi 1965: 11.13a).\n\nThus, for both Mencius and Xunzi, the kind of ethical standards to which one should subordinate oneself are something that the heart\/mind can 'perceive' and to which we should respond. Furthermore, for both, there should be a firm commitment to these standards of such a kind that it can override personal interests of the most pressing kind, including one's own life. That the individual can exhibit such firmness of commitment is because zhi \u5fd7 (directions of the heart\/mind) is independent of external control in that the heart\/mind has the capacity to hold on to the directions it sets without being swayed by external forces. For example, while both the Lunyu \u8ad6\u8a9e (Analects) and the Mengzi \u5b5f\u5b50 (Mencius) emphasize the guiding role of zhi \u5fd7 (directions of the heart\/mind), comparing it to the commander of an army, the Lunyu \u8ad6\u8a9e (Analects) notes one point of dissimilarity \u2013 while an army can be deprived of its commander, even a common person cannot be deprived of the directions (zhi \u5fd7) set by the heart\/mind (Lunyu 1980: 9.26; Mengzi 1984: 2A2). Such directions can, of course, be influenced by outside factors, but the point is that the heart\/mind has the capacity to resist such influences and, for the Confucian thinkers, one should ideally cultivate oneself to attain such a steadfastness of purpose after having set the heart\/mind in the proper directions. This independence of the heart\/mind from external control is also emphasized by Xunzi, who compares the heart\/mind to the position of the ruler and the senses to the offices of government; like the ruler, the heart\/mind issues order but does not take order from anything (Xunzi 1965: 11.10a\u201310b, 15.5b\u20136a). To ensure that zhi \u5fd7 (directions of the heart\/mind) is not swayed by external influences, we need also to cultivate qi \u6c23 (the life forces), which support zhi \u5fd7 (directions of the heart\/mind). The Mengzi \u5b5f\u5b50 (Mencius) talks about nourishing the flood-like qi \u6c23 (the life forces), acknowledging that without adequate support from a cultivated qi \u6c23 (the life forces), zhi \u5fd7 (directions of the heart\/mind) can collapse and the heart\/mind can be moved (Mengzi 1984: 2A:2). Xunzi likewise emphasizes that self-cultivation involves giving order to qi \u6c23 (the life forces) and nourishing the heart\/mind (zhi qi yang xin \u6cbb\u6c23\u990a\u5fc3), again giving recognition to the complementary roles of zhi \u5fd7 (directions of the heart\/mind) and qi \u6c23 (the life forces) in this steadfastness of purpose (e.g., Xunzi 1965: 1.9a).\n\nSo far, we have discussed in connection with the Confucian conception of yi \u7fa9 (dutifulness, righteousness) the shift of attention away from the way we are treated by others to our conforming to certain ethical standards. For the early Confucians, this shift of attention to the ethical should also happen in other areas of life, including adverse circumstances of life of all kinds. This position is conveyed using the term ming \u547d (destiny, fate, decree), a term sometimes paired with yi \u7fa9 (dutifulness, righteousness) in early texts. Ming \u547d (destiny, fate, decree), often translated as \"destiny,\" \"fate,\" or \"decree,\" is used in early Confucian texts to convey a certain attitude toward adverse external conditions of life, which include not just the way we are viewed and treated by others, but also things like sickness or death. These conditions do matter to the Confucians. For example, in the Lunyu \u8ad6\u8a9e (Analects), we see Confucius lamenting the lack of appreciation by others (Lunyu 1980: 14.35), as well as expressing sorrow at the death of his favorite disciple Yan Hui \u9854\u56de (Lunyu 1980: 11.9, 11.10). At the same time, from the Confucian perspective, even though these things do matter, they pale in significance compared to our own ethical qualities. When we do not fare well in relation to the former, at least the latter is something within our control and something we can fall back on and take consolation in. This contrast between what is of true significance and within our control, and what is of comparatively lesser significance and not entirely within our control, is highlighted in a passage in the Mengzi \u5b5f\u5b50 (Mencius), which characterizes the latter in terms of ming \u547d (destiny, fate, decree) (Mengzi 1984: 7A:3).\n\nDespites its usual translation as \"destiny\" or \"fate,\" ming \u547d (destiny, fate, decree) is not used by early Confucians to express beliefs about pre-determination of things or about historical forces at work that cannot be resisted. Instead, it is used to convey an attitude that might be described as one of willing acceptance (Mengzi 1984: 7A:2). This is not a general attitude directed toward external occurrences in general, but a posture one takes up in response to specific adverse conditions of life that one actually encounters. It might be an undesirable condition that is literally not within one's control, such as the death of a beloved one. Or it might be a condition that one can still alter but only by improper means, such as avoiding one's own death by succumbing to evil. One can still be emotionally affected by these adverse conditions of life and wish things could be otherwise \u2013 one would still grieve at the death of beloved ones, be disappointed by the lack of appreciation by others, and lament the corruption that prevails. However, one would not direct one's emotional energy to blaming others or complaining about the outcome, nor become bitter and resentful (Lunyu 1980: 14.35). It does not mean that one is resigned to the situation in the sense that one becomes totally passive, a kind of fatalistic surrender to one's environment. Nor is it a matter of submission to the environment, as when a slave 'accepts' being enslaved, or a matter of inertia, as when one lets things proceed without bothering. Instead, there is an element of activism in the Confucian attitude. One would still await and welcome the possibility of change, and even when such opportunities do not arise, one would redirect one's energy in a positive direction, just as Confucius redirected his energy to teaching after having come to a realization of the futility of his political endeavors. And accompanying this acceptance of the adverse external conditions of life is a positive affirmation of the ethical values that one stands by and in which one takes consolation.\n\nFrom the preceding discussion of yi \u7fa9 (dutifulness, righteousness) and ming \u547d (destiny, fate, decree), we see that early Confucian thinkers advocate a submission of the self to certain ethical standards that the heart\/mind can grasp. They advocate a total submission of the self to such standards, even at the expense of grave consequences for oneself. This firm ethical commitment involves a move away from attaching undue significance to the interests of oneself understood in ordinary terms. Such interests include not just the way one is treated or viewed by others, but also other adverse conditions of life of all kinds. It is one's ethical qualities that are of fundamental importance, and anchoring the self in such ethical standards enables one to respond to all kinds of adverse circumstances of life without being emotionally perturbed (an idea conveyed in Mengzi 1984: 2A:2 in terms of the heart\/mind's being 'unmoved). Thus, this aspect of the Confucian ideal also involves a move away from a kind of self-centeredness that focuses on one's interest understood in more ordinary terms, to a perspective that anchors the self in the ethical standards to which one should conform.\n\n## 6 Self-Cultivation\n\nWe see from the above discussion that early Confucians advocate a fundamental transformation of the self that involves properly situating the self in relation to others and in relation to certain ethical standards. On the one hand, one submits oneself to certain ethical standards and focuses one's attention on living up to such standards rather than on how one is treated by others or on the external conditions of life. On the other hand, one attaches proper significance to the interests and well-being of others and interacts with them with proper attentiveness and seriousness. Such transformation involves a move away from certain tempting kinds of self- centeredness \u2013 an undue focus on one's own interests and well-being, on one's own importance, or on the external conditions of life to which one attaches importance. That the fundamental ethical task has to do with a move away from certain kinds of self-centeredness is hinted at in early Confucian texts, such as the observation in the Lunyu \u8ad6\u8a9e (Analects) about how attaining the ethical ideal involves \"overcoming the self\"(ke ji \u514b\u5df1) (Lunyu 1980: 12.1). This idea is taken up and highlighted in later Confucian thought, which ascribes ethical failure primarily to certain forms of self-centeredness and regards the move away from self-centeredness (si \u79c1) as the fundamental ethical task (for a discussion of Zhu Xi's \u6731\u71b9 elaboration on this idea, see Shun 2005: 1\u20139).\n\nAttaining a proper positioning of the self in one's perspective takes effort, and while Confucian thinkers also emphasize proper childhood upbringing, what is distinctive of Confucian ethical thought is its emphasis on the self-reflective ethical efforts that one undertakes as an adult. I will use \"self-cultivation\" to refer to the self-reflective process that one undertakes to attain such transformation of the self. The idea of a process of this kind is conveyed in the notion xiushen \u4fee\u8eab (self-cultivation), a term that occurs three times in the Mengzi \u5b5f\u5b50 (Mencius), is the title of a chapter in the Xunzi \u8340\u5b50, and is used to refer to one of the eight items of adult learning in the Daxue \u5927\u5b78 (Great Learning).\n\nRegarding the content of this process, it is spelt out in different ways and with different emphases in the history of Confucian thought. Since part of the transformation has to do with submitting the self to ethical standards that the heart\/mind can grasp, learning is needed for us to properly grasp such standards. The Chinese term usually translated as \"learning,\" xue \u5b78 (learning), has the connotation not just of learning in the contemporary sense, but also of drawing moral lessons from and embodying in one's daily life what one has learnt. For the Confucians, its object includes all aspects of the cultural heritage, including such items as poetry, history, rites (li \u79ae), music, and archery. The notion is particularly highlighted in the Lunyu \u8ad6\u8a9e (Analects) and the Xunzi \u8340\u5b50, the latter having a whole chapter devoted to the subject. The importance of learning is emphasized by Confucius in his own autobiographical statement that he set his heart on learning at the age of 15 (Lunyu 1980 : 2.4). And, as presented in the Xunzi \u8340\u5b50, what one has learnt from li \u79ae (rites) and from other items of the cultural heritage will permeate the whole person, and their accumulated effects will totally reshape the person in an ethical direction (e.g., Xunzi 1965: 1.4b).\n\nLearning by itself is not sufficient. Having ensured that the directions of one's heart\/mind, or zhi \u5fd7 (directions of the heart\/mind), is properly directed, one still need to cultivate one's qi \u6c23 (the life forces) to give it support to ensure that one maintains the kind of steadfastness of purpose described earlier. More importantly, there are all kinds of forces at work within oneself that can lead one astray, and so it is also important to work on one's own heart\/mind to ensure its proper ethical orientation. One might work on the heart\/mind to ensure that it performs its guiding and regulatory roles in relation to others forces at work within oneself, or to ensure that its own operations is properly oriented. I will consider these two kinds of exercises in turn.\n\nThe guiding role of the heart\/mind in relation to the senses is highlighted in a variety of early texts. The Mengzi \u5b5f\u5b50 (Mencius) presents the senses as potentially problematic if not regulated by the heart\/mind (Mengzi 1984: 6A:15), and the Huainanzi \u6dee\u5357\u5b50 likewise observes how sensory objects can distort the operation of the senses and how it is only under the heart\/mind's regulation that the senses attain their proper place (Huainanzi 1965: 7.3a, 14.7b). The Guanzi \u7ba1\u5b50 observes how external things can distort the operation of the senses, which in turn can distort the operation of the heart\/mind (Guanzi 1965: 13.6a, 16.3a). The text compares the position of the heart\/mind to that of the ruler; when the heart\/mind is in order, the senses will also be in order (Guanzi 1965: 13.1a, 16.3b). The governing role of the heart\/mind over the senses is also emphasized in the Xunzi \u7b4d\u5b50 (Xunzi 1965: 11.10a\u201310b). The Xunzi \u8340\u5b50 in addition emphasizes the importance of the heart\/mind's role in regulating desires (yu \u6b32); if unregulated, chaos will result from people's pursuing without constraint the fulfillment of their desires (e.g., Xunzi 1965: 13.1a). The \"Yueji\" chapter \u6a02\u8a18 of the Liji \u79ae\u8a18 (The Book of Rites) makes a similar point. It observes how, when human beings come into contact with external things, likes and dislikes arise. Humans are affected by things without limit, and if their likes and dislikes are not regulated, human beings would be moved to exhaust their human desires (ren yu \u4eba\u6b32) and things become problematic (Liji 1965: 11.8b\u20139a). Thus, for early Confucians, an important part of self-cultivation is to train the heart\/mind to properly guide and regulate all kinds of human desires.\n\nEthical problems can also arise from within the heart\/mind. As we have seen, early Confucians advocate self-transformation of a kind that moves us away from certain forms of self-centeredness that are themselves common and tempting human tendencies. The subtle workings of the heart\/mind can reflect such tendencies and pose obstacles. As part of the self-cultivation process, one also needs to work on the subtle activities of the heart\/mind to ensure that they are properly oriented. One illustration of this idea is the idea in the Daxue \u5927\u5b78 (Great Learning) and Zhongyong \u4e2d\u5eb8 (Centrality and Commonality) that the heart\/mind should cautiously watch over its own activities to ensure that all of its activities, however minute or subtle, are completely oriented in an ethical direction. This idea is presented in terms of one's cautiously watching over du \u7368 (what one alone can access), where du \u7368 (what one alone can access) likely refers to the minute and subtle workings of the heart\/mind that are not yet manifested outwardly and to which one alone has access (Zhu 1983\u20131986d Daxue Chap.\u200b 6, 1983\u20131986c Zhongyong Chap.\u200b 1) (for a more elaborate discussion of the idea of watching over du, see Shun 2008: 262\u2013266). In the Daxue \u5927\u5b78 (Great Learning), this idea is related to the idea of cheng yi \u8aa0\u610f (to make one's thoughts fully oriented in an ethical direction), which is a process of making one's yi \u610f (thoughts and inclinations) fully oriented in an ethical direction. Yi \u610f (thoughts and inclinations), as we saw earlier, has to do with the more reflective inclinations of the heart\/mind, unlike yu \u6b32 (desires) which can be pre-reflective. Thus, this aspect of self-cultivation is concerned with the activities of the heart\/mind itself, not just with the pre-reflective desires that need to be guided and regulated by the heart\/mind. This aspect of Confucian thought shows that the Confucians ascribe to the heart\/mind a self-reflexiveness \u2013 for any of its own activities, however minute and subtle, it has the capacity to reflect on and reshape such activities to ensure their orientation in an ethical direction. And this self-reflexiveness is related to the independence of the heart\/mind from external control \u2013 even though its activities can be influenced by external circumstances, the heart\/mind has the capacity to constantly step back and reshape its own activities under the conception of what is proper that it forms on the basis of its own reflections (the discussion of this and the next paragraph draws on Shun 2004: 188\u2013190).\n\nWe have so far focused on the heart\/mind in our discussion of self-cultivation \u2013 how the heart\/mind can grasp certain ethical standards through learning, how it can guide and regulate our feelings and desires, and how it can monitor and reshape its own operations. While the heart\/mind does play a key role in self-cultivation, it is important to note, though, that the effect of self-cultivation does not stop with the heart\/mind. For the early Confucians, the heart\/mind and other aspects of the person are mutually interacting. In early Chinese texts, we see mention of how the life forces (qi \u6c23) that fill the body can be affected by what happens to the body, such as the tastes that the mouth takes in and the sounds that the ear hears; conversely, the life forces can generate speech in the mouth and sight in the eyes. Also, the directions (zhi \u5fd7) of the heart\/mind can guide and shape the life forces (qi \u6c23) while depending on the life forces for their execution; conversely, the directions of the heart\/mind can be swayed if the life forces are not adequately nourished. It follows from the intimate link between the heart\/mind and the life forces, and between the life forces and the body, that the heart\/mind is also intimately linked to the body. For the early Confucians, the condition of the heart\/mind will inevitably be manifested in the body and in one's outward behavior and demeanor, and be perceivable by others. In emphasizing the need to be watchful over du \u7368 (what one alone can access), both the Daxue \u5927\u5b78 (Great Learning) and the Zhongyong \u4e2d\u5eb8 (Centrality and Commonality) also point out that, although the minute and subtle tendencies of the heart\/mind are initially known to oneself alone and not yet perceived by others, they will eventually become manifest (Zhu 1983\u20131986c Zhongyong Chap.\u200b 1, 1983\u20131986d Daxue Chap.\u200b 6). The Mengzi \u5b5f\u5b50 (Mencius) speaks of how the condition of the heart\/mind is also manifested in the body, not just in action and speech, but also in the face, the look of the eyes, the four limbs, and in one's physical bearing in general; indeed, according to Mencius, it is only through self-cultivation that one can give complete fulfillment to the body (Mengzi 1984: 4A:15, 7A:21, 7A:38). Likewise, the Daxue \u5927\u5b78 (Great Learning) speaks of how virtue (de \u5fb7) adorns the body just as riches adorn a house, and how when the heart\/mind is expanded the body is also at ease (Zhu 1983\u20131986d Daxue Chap.\u200b 6). Thus, for the early Confucians, the effect of self-cultivation does not stop with the heart\/mind, but affects the person as a whole.\n\nIndeed, for them, the effect of self-cultivation does not even stop with the individual person, but also extends to others. Early Confucian texts emphasize how self-cultivation has a transformative power on others, a power that the Confucians regard as the ideal basis for government. For example, for both Confucius and Mencius, the goal of government is to transform people's character, and the way to accomplish this is to first cultivate oneself and to let the transformative power of one's cultivated character take effect (Lunyu 1980: 2.1, 2.3, 12.19, 13.4, 13.13, 15.5; Mengzi 1984: 4A:20, 7A:19). This does not mean that governmental policies are not important, but proper policies are themselves a manifestation of the cultivated character of those in power, and properly carrying out policies transmitted from the past also requires a cultivated character (Mengzi 1984: 2A:6, 4A:1; cf. 7B:5). So, the ultimate basis for order in society lies with cultivating oneself, an idea that the Daxue \u5927\u5b78 (Great Learning) expresses by describing self-cultivation as the basis for regulating the family, giving order to the state, and ultimately bring peace to the whole Empire (Zhu 1983\u20131986d Daxue text; Mengzi 1984: 4A:5, 4A:12, 7B:32). A similar point is also presented in terms of cheng \u8aa0 (complete ethical orientation of the heart\/mind), which is used in early Confucian texts to refer to the complete ethical orientation of the heart\/mind. The notion is highlighted in the Daxue \u5927\u5b78 (Great Learning), the Zhongyong \u4e2d\u5eb8 (Centrality and Commonality), and also occurs in parts of the Mengzi \u5b5f\u5b50 (Mencius) and the Xunzi \u8340\u5b50. Cheng \u8aa0 (complete ethical orientation of the heart\/mind) has the connotation of being real and complete, and it refers to a state in which one embodies the ethical attributes to the fullest extent. The most elaborate presentation of this state is in the second half of the Zhongyong \u4e2d\u5eb8 (Centrality and Commonality), in which cheng \u8aa0 (complete ethical orientation of the heart\/mind) is presented as the basis for social and political order. A cheng \u8aa0 (complete ethical orientation of the heart\/mind) person will have a transformative effect on others' character and, when in government, will also ensure that the 10,000 things take their proper places. Since the transformative effect on others is itself a natural outgrowth of one's own self-transformation, it follows that, for the early Confucians, there is no clear line between self-transformation and the transformation of others.\n\n## 7 Attending to the Self\n\nI have in the chapter highlighted self-transformation as a distinctive feature of Confucian thought, and have discussed in some detail the nature of the self-cultivation process. The transformation involved affects the whole person in fundamental ways. The proper situating of the self in relation to others involves a fundamental reshaping of one's outlook on life, including the way one assesses one's own interests and well-being, as well as one's sense of one's own importance. It involves overcoming natural tendencies to place oneself at the center of things, and requires constant vigilant attention to the subtle activities of the heart\/mind. The firm commitment to the ethical is particularly demanding as it involves a steadfastness of purpose that could incur grave personal sacrifices, including giving up one's own life. These dimensions of Confucian thought provide a sense in which it can also be described as a kind of spiritual thought, if the spiritual is understood in a way that is divorced from pietistic and devotional practices. This Confucian concern with self-transformation, however, might itself raise a potential worry, namely, that it might itself exhibit a problematic form of self-centeredness that involves a misdirection of one's ethical attention. In this concluding section, I will discuss some possible forms that this worry can take. I will discuss two kinds of concern that the early Confucian thinkers themselves have raised and cautioned their audience against, and then move on to two other kinds of concern that might potentially be directed against the Confucian thinkers themselves.\n\nAs we have discussed in the previous section, self-transformation is viewed by Confucian thinkers as the basis for the social and political order. This idea is built into the early Chinese notion de \u5fb7 (virtue, power), a term often translated as \"virtue\" and sometimes as \"power.\" In its earliest use, de \u5fb7 (virtue, power) probably carried primarily religious connotations, referring to an attitude of the king that enabled his communion with Heaven (tian \u5929). It eventually came to refer in addition to qualities such as generosity, humility, and receptiveness to instruction, as well as to certain powers associated with these qualities, including a compulsion to respond on the part of the recipients of generous or sacrificial acts and a non-coercive power on others of attraction and transformation. Early Confucian thinkers continued this tradition, and regarded the power associated with de \u5fb7 (virtue, power) as the ideal basis for government \u2013 it is de \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness), that enables one to become a true king (wang \u738b). At the same time, they also noted the associated potential for a misdirection of attention that results from one's aiming at \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) primarily for its perceived political advantages. Mencius draws a number of distinctions in this connection: between those who truly act out of ren yi \u4ec1\u7fa9 (benevolence and righteousness) and those who enact ren yi \u4ec1\u7fa9 (benevolence and righteousness) for political advantages, between those who rely on de \u5fb7 (virtue, power) to practice ren \u4ec1 (humanity, benevolence) and those who rely on force to make use of ren \u4ec1 (humanity, benevolence), and between those who use goodness to nourish the people and those who make use of goodness to gain people's allegiance (Mengzi 1984: 2A:3, 4B:16, 4B:19). The point is to warn rulers against misdirecting their attention to the political advantages of \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) \u2013 someone so motivated would not truly attain \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness), and would therefore not attain the desired goals. A similar misdirection of attention can happen among officials, who cultivate themselves for the purpose of being appreciated by others thereby attaining employment or high rank in government. In this connection, Mencius again draws a distinction between the honors of Heaven (tian \u5929), which have to do with the ethical attributes, and the honors of humans, which have to do with high ranks in government, lamenting the fact that people during his times would aim at the former for the sake of the latter (Mengzi 1984: 6A:16). Confucius makes a similar point by saying that learning should be for the self and not for others; it should be directed at one's own character and ability as such and not at appreciation by others and the resulting employability (Lunyu 1980: 14.24, cf. 1.1, 1.16, 4.14, 14.30, 15.19).\n\nA second kind of misdirection of attention that early Confucian thinkers criticize has to do with a focus on external perception that is guided by a concern with one's reputation rather than by political ambitions. This is the situation of the village worthy, who is concerned primarily with the reputation of being good, without a genuine concern for its substance (Lunyu 1980: 13.21, 17.13, 17.18, cf. 13.24; Mengzi 1984: 7B:37). This individual's way of life is entirely geared to social opinion, and since he adjusts his way of life to seek the approval of others, it is difficult to find fault with him. His way of life appears good, everyone approves of him, and he regards himself as living properly. And yet he has no genuine commitment to goodness, as can be seen from the fact that he would adjust his words and deeds to what he perceives as the expectations of others. Whatever qualities he has would change in response to the changing expectations of others, and so would lack the stability that morally good traits are often expected to have. Thus, he is not truly virtuous though he resembles someone who is and is mistakenly taken by people to be virtuous; in this sense, he 'steals' the name of \u5fb7 (virtuous) and so is a \"thief of de\"(de zhi zei \u5fb7\u4e4b\u8cca).\n\nBoth Confucius and Mencius condemn such an individual, and there is something about the village worthy that is deeply disturbing. He exhibits a high level of self-awareness. He is aware that he is being judged, and wishes himself to be seen to have certain qualities; so he is not just reflecting on his own inner states, but also reflecting on how others would perceive his inner states. In his motives, he is subtly deceptive in a way that those who aim at being good for the resulting political advantages need not be. As can be seen from some of Mencius' dialogues with the rulers of states, the latter can, at least, be somewhat explicit about their true motives. The former, however, would hide his true motive, which is to please others, as part of his pretence; he would do that as long as he is aware that his audience would not think well of his true motives. In this regard, he is like the type of moral hypocrite who has contrary values and qualities that others would not approve of and who, in seeking the approval of others, has to hide these values and qualities. But unlike this other type of moral hypocrite, the village worthy does not even have contrary values and qualities, as his sole preoccupation is to be seen as good. As a result, unlike this other type of moral hypocrite, he is less likely to betray himself in inattentive moments, and thus more likely to succeed in his deception. Through such deception, he is blurring the distinction between genuine goodness and semblances of the kind that he exhibits, thereby undermining his audience's understanding of what genuine goodness is, and hence undermining the very conception of that to which he makes a false claim (see Kittay 1982; McKinnon 1991 for a discussion of the other type of moral hypocrite and for an elaboration on the last point about what is problematic about the pretence involved).\n\nWhat we have considered are two kinds of misdirection of ethical attention that are in a sense too externally directed \u2013 one's ultimate goal is to attain certain political advantages in the first case or to acquire a certain reputation in the second. Ultimately, though, one's attention is also self-directed in a problematic way \u2013 it is after all one's own political advantages or one's own reputation that one aims at. The issue still comes down to problematic forms of self-centeredness, and the early Confucians are vividly aware of such dangers and speak vehemently against them. An interesting question, though, is whether the Confucians are themselves, in their own focus on self-cultivation, vulnerable to a similar criticism that they are themselves overly self-centered. In the remainder of this section, I will consider two forms such criticism can take, and the potential responses that can be given on behalf of the Confucians (the discussion that follows draws on Shun 2001: 229\u2013244).\n\nOne form that such criticism can take is that, even on the Confucians' own position, the concern with cultivating \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) in oneself can still involve an excessive concern with others' opinion of oneself, even if one's attention is not directed explicitly to the opinion of others in the way that the attention of the village worthy is. The reason is that terms like \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) are terms that typically occur in third-person descriptions of the good person rather than in the content of the first personal deliberation of the truly good person \u2013 to use a contemporary parallel, the truly benevolent person would typically not be thinking of herself as benevolent or her actions as benevolent acts. Thus, it appears, the first-personal exercise of cultivating these qualities in oneself involves one's being concerned primarily with the way others would describe oneself. To aim at \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) is to aim at one's being describable by others in a certain way, and this kind of concern seems misdirected in a disturbing way, just like the village worthy's concern with pleasing others.\n\nOn behalf of the Confucians, one might note that even though they talk in general terms about \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness), self-cultivation for them typically would not involve thinking in these terms. Instead, one's attention in self-cultivation is more often directed to something more specific than these general qualities, as can be seen from our discussion of the idea of being watchful over du \u7368 (what one alone can access). Still, this response would not suffice as the Confucians do from time to time advocate in more general terms a concern with the overall quality of one's character, conveyed in terms of \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness). To supplement this initial response, we may add that while \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness) is typically a third person description, someone engaged in self-cultivation may be using it as a description of others by oneself, rather than of oneself by others. That is, in being concerned with \u5fb7 (virtue, power), or ren yi \u4ec1\u7fa9 (benevolence and righteousness), one's concern need not be with being describable by others in these terms, but may instead be with one's becoming like the kind of person that one would oneself describe in this manner. In having this concern, one's attention is directed not to the description itself, but to one's having a certain quality which, as it happens, can be described in this way.\n\nAnother possible criticism of the Confucian emphasis on self-cultivation is that, even if one is not concerned with others' description of oneself, one's attention may still be too self-directed. One may be making one's own character ethically the most important thing, more important than other-regarding considerations. In fact, Mencius was himself occasionally accused of this kind of self-centeredness. The Mengzi \u5b5f\u5b50 (Mencius) contains several examples of his refusing to see a ruler because he had not been summoned or treated in accordance with certain rules of propriety appropriate to his position. His critics made the point that, if only he had been willing to 'bend' himself a little and have audience with the ruler, he might have been able to accomplish desirable political changes and thereby help the people (Mengzi 1984: 3B:1, 5A:7, cf. 4A:17). His self-righteousness, it seems, had come at the expense of missing the opportunity to help others.\n\nPut in general terms, the Confucian response would refer to both the content of the ethical ideal they advocate and the effect of one's attaining this ideal. The content involves proper attention and sensitivity to the well-being of others as well as due acknowledgement of their importance, and the effect of self-cultivation also extends to the transformation of others' character. Accordingly, there cannot be a divergence between one's own self-cultivation on the one hand, and the well-being or the transformation of others' on the other. Mencius' response to his critics draws on this idea. According to him, what he sought to accomplish in the political realm was to 'straighten' those in power, and 'straightening' others depends on one's being 'straight' oneself; there has never been a case of one's 'bending' oneself and yet succeeding in 'straightening' others (Mengzi 1984: 3B:1, 5A:7; cf. Lunyu 12.22, 13.13). So, to the extent that the well-being of the people depends on a reform of the political order, there cannot be a conflict between a concern for one's character and a concern for the well-being of others. And given that the transformative effect on others' character is a natural outgrowth of one's cultivating one's own character, there also cannot be a conflict between a concern for one's character and a concern for others' character.\n\nThis Confucian response draws on an optimistic belief about the transformative power of a cultivated character. While we do see repeated statements of this belief in early Confucian texts, we also get the sense that the early Confucians were not unaware that this belief could be overly optimistic and might not match the practical political realities of the times. Consider, for example, the fact that the attitude of acceptance that we discussed earlier in connection with the notion ming \u547d (destiny, fate, decree) is sometimes also directed to a failure to bring about desired political changes. In Confucius' words, whether the Way prevails or falls into disuse is itself a matter of ming \u547d (destiny, fate, decree) (Lunyu 1980: 14.36). Now, if one's own ethical qualities are within one's control as the Confucians think, and if one's own self-cultivation will lead naturally to the transformation of others, especially those in power, then the prevailing of the Way should have been something that one can bring about. Thus, statements like Confucius' are implicit acknowledgements that perhaps the belief about the transformative power of a cultivated character is itself overly optimistic. So, while there is a Confucian response to the kind of criticism directed against Mencius, perhaps that response is itself potentially undermined by this tension between the early Confucians' belief in the transformative power of a cultivated character and their own political failure and the resulting frustration with the political realities of their times.\n\nReferences\n\nChan, Sin Yee. 2006. The Confucian notion of Jing \u656c (Respect). Philosophy East & West 56: 229\u2013252.CrossRef\n\nDarwall, Stephen L. 1977. Two kinds of respect. Ethics 88: 36\u201349.CrossRef\n\nGuanzi \u7ba1\u5b50. 1965. Sibu beiyao \u56db\u90e8\u5099\u8981 edition. Taibei: Zhonghua shuju.\n\nHuainanzi \u6dee\u5357\u5b50. 1965. Sibu beiyao \u56db\u90e8\u5099\u8981 edition. Taibei: Zhonghua shuju.\n\nKittay, Eva Feder. 1982. On hypocrisy. Metaphilosophy 13: 277\u2013289.CrossRef\n\nLiji \u79ae\u8a18. 1965. Sibu beiyao\u56db\u90e8\u5099\u8981 edition. Taibei: Zhonghua shuju.\n\nLunyu \u8ad6\u8a9e. Yang, Bojun \u694a\u4f2f\u5cfb. 1980. The Analects with translation and commentary \u8ad6\u8a9e\u8b6f\u6ce8. 2nd ed. Beijing: Zhonghua shuju.\n\nMcKinnon, Christine. 1991. Hypocrisy, with a note on integrity. American Philosophical Quarterly 28: 321\u2013330.\n\nMengzi \u5b5f\u5b50. Yang, Bojun \u694a\u4f2f\u5cfb. 1984. Mencius with translation and commentary \u5b5f\u5b50\u8b6f\u6ce8. 2nd ed. Beijing: Zhonghua shuju.\n\nMozi \u58a8\u5b50. 1948. A Concordance to Mo Tzu. Harvard-Yenching Institute Sinological Index Series, Supplement 21. Peking: Harvard-Yenching Institute.\n\nShun, Kwong-loi. 1997. Mencius and early Chinese thought. Stanford: Stanford University Press (A primarily textual study of early Chinese thought with focus on Mencius. It also includes discussions of Confucius, Xunzi, and the intellectual opponents of the early Confucians such as the Yangists and the Mohists).\n\nShun, Kwong-loi. 2001. Self and self-cultivation in early Confucian thought. In Two roads to wisdom? Chinese and analytic philosophical traditions, ed. Mou Bo. Chicago\/La Salle: Open Court. A collection of essays that study traditional Chinese thought, includling Confucianism, Daoism, and the Yijing from a comparative perspective.\n\nShun, Kwong-loi. 2004. Concept of the person in early Confucian thought. In Confucian ethics: a comparative study of self, autonomy and community, ed. Kwong-loi Shun and David B. Wong. New York: Cambridge University Press (A collection of essays that engage in a comparative study of Confucian thought, with focus on key topics such as the notion of the self, rights and community).\n\nShun, Kwong-loi. 2005. Zhu Xi on Gong and Si. Dao 5: 1\u20139.CrossRef\n\nShun, Kwong-loi. 2008. Wholeness in Confucian thought: Zhu Xi on Cheng, Zhong, Xin, and Jing. In The imperative of understanding: Chinese philosophy, comparative philosophy, and onto-hermeneutics, ed. On-cho Ng. New York: Global Scholarly Publications. (A collection of essays on Chinese thought, with focus on issues related to interpretation and hermeneutics).\n\nShun, Kwong-loi. Forthcoming. On anger: an experimental essay in Confucian moral psychology. In Rethinking Zhu Xi: emerging patterns within the supreme polarity, eds. David Jones, and He Jinli. Albany: State University of New York Press.\n\nXunzi \u8340\u5b50. 1965. Sibu beiyao \u56db\u90e8\u5099\u8981edition. Taibei: Zhonghua shuju.\n\nZhu, Xi \u6731\u71b9. 1983\u20131986a. Collected Annotations on the Analects \u8ad6\u8a9e\u96c6\u6ce8. Siku quanshu \u56db\u5eab\u5168\u66f8 edition. Taibei: Shangwu yinshuguan.\n\nZhu, Xi \u6731\u71b9. 1983\u20131986b. Questions and Answers on the Analects \u8ad6\u8a9e\u6216\u554f. Siku quanshu \u56db\u5eab\u5168\u66f8 edition. Taibei: Shangwu yinshuguan.\n\nZhu, Xi \u6731\u71b9 1983\u20131986c. Section and Sentences Commentary on the Doctrine of the Mean \u4e2d\u5eb8\u7ae0\u53e5. Siku quanshu \u56db\u5eab\u5168\u66f8 edition. Taibei: Shangwu yinshuguan.\n\nZhu, Xi \u6731\u71b9 1983\u20131986d. Section and Sentences Commentary on the Great Learning \u5927\u5b78\u7ae0\u53e5. Siku quanshu \u56db\u5eab\u5168\u66f8 edition. Taibei: Shangwu yinshuguan.\n\nZhu, Xi \u6731\u71b9. 1986. Topically Arranged Conversations of Master Zhu \u6731\u5b50\u8a9e\u985e. Beijing: Zhonghua shuju.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_13\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 13. Early Confucian Virtue Ethics: The Virtues of Junzi\n\nAntonio S. Cua1\n\n(1)\n\nSchool of Philosophy, Catholic University of America, Washington, DC, USA\n\nAntonio S. Cua\n\nEmail: vincent.shen@utoronto.ca\n\nAbstract\n\nConfucius' Analects (Lunyu \u8ad6\u8a9e) provides an ample vocabulary for virtues or excellences of ethical character. Throughout the text, we find frequent occurrence of certain terms such as ren \u4ec1 (benevolence, humaneness), li \u79ae (rules of proper conduct, ritual, rites), and yi \u7fa9 (rightness, righteousness. fittingness), indicating Confucius' ongoing concern with the cultivation of fundamental virtues. His remark that there is one thread (yiguan \u4e00\u8cab) that runs through his teachings may be cited as a partial support for ascribing a holistic perspective to his thought. However, we do not find a systematic scheme for conduct in the Analects. Also, throughout the history of Chinese thought and contemporary Chinese and Western writings on Confucianism, we find a great variance of interpretation of fundamental concepts such as ren (Chan 1955, 1975).\n\nA short version of Sect. 2 in this chapter was presented as the keynote speech to the International Conference on Confucianism: Retrospect and Prospect, September 1\u20132, 2005, organized by Vincent Shen at the University of Toronto.\n\n## 1 Introduction\n\nConfucius' Analects (Lunyu \u8ad6\u8a9e) provides an ample vocabulary for virtues or excellences of ethical character.1 Throughout the text, we find frequent occurrence of certain terms such as ren \u4ec1 (benevolence, humaneness), li \u79ae (rules of proper conduct, ritual, rites), and yi \u7fa9 (rightness, righteousness. fittingness), indicating Confucius' ongoing concern with the cultivation of fundamental virtues. His remark that there is one thread (yiguan \u4e00\u8cab) that runs through his teachings may be cited as a partial support for ascribing a holistic perspective to his thought. However, we do not find a systematic scheme for conduct in the Analects. Also, throughout the history of Chinese thought and contemporary Chinese and Western writings on Confucianism, we find a great variance of interpretation of fundamental concepts such as ren (Chan 1955, 1975).\n\nThe unsystematic character of Confucius' ethical thought in part reflects his emphasis on the concrete and the particular. The Analects portrays a mental attitude that emphasizes the particular and the concrete as a way of explaining concepts. Owing to the influence of the Analects, it may even be said that characteristic of the Chinese way of thinking is the preference for \"particular, concrete, and intuitive explanations.\" Hajime Nakamura claims that this preference \"may be seen in their way of explaining ideas and teaching people by the use of particular examples. To most Chinese, therefore, ethics is not understood or taught as part of a universal law, but is grasped on the basis of particular experiences, and is then utilized to realize human truth.\" (Nakamura 1964: 198). While one may raise questions about the adequacy of this attribution to Chinese ethics, the conception is well-supported by the Analects, as it is quite evident, for example, in the frequent occurrence of proper names intended as exemplars of Confucius' ideal. Yao and Shun, regarded by Confucius as paragons of sagely rulership, were frequently commended as models of virtue (Analects 6.28, 14.45, 20.1.). Other purportedly historical personages were also cited as particular exemplars of conduct (Analects 14.12). More telling is Confucius' extensive use of notion of junzi, instead of principles, for explaining ethical virtues and instruction on their practical significance in human conduct. Plausibly, Confucius' notion of junzi encapsulates a concern for flexibility in coping with changing circumstances. An emphatic autobiographical remark indicates this attitude of flexibility: \"I have no preconceptions about the permissible or impermissible (wu ke wu buke \u7121\u53ef\u7121\u4e0d\u53ef)\" (Analects 18.8). This aspect of Confucian ethics is perhaps best approached by way of Confucius' notion of junzi \u541b\u5b50 as a notion of an ideal individual, who functions as a paradigmatic standard for practical morality. In this light, Confucius' ethical thought, unlike that of Mencius (Mengzi \u5b5f\u5b50) or Xunzi \u8340\u5b50, is best characterized as an ethics of junzi or paradigmatic individuals.\n\nAlthough Confucius believed that only a sage (shengren \u8056\u4eba), divinely inspired and intuitively wise, can envision and establish a harmonious social order, he did not regard the ideal of becoming a sage as practically attainable. He once remarked that he cannot hope to meet a sage, but can only hope to meet a junzi (Analects 7.26). In Analects 14.12, however, we do find a characterization of chengren \u6210\u4eba, which may be rendered as \"perfect man.\" When Zilu asked what constituted a chengren, Confucius cited a combination of excellences exemplified by certain historical persons, such as wisdom, courage, absence of greed, variety of talents, and in addition, an enhancement of these qualities by rites and music. But he added: \"nowadays it is not necessary to have all these qualities in order to be a chengren. If a man can think of what is right (yi) when he sees personal advantage, and in the presence of personal danger, is ready to give up; and does not belie the professions of his life \u2013 such a man may also be considered a chengren.\"2 In this saying, chengren may be construed more clearly as chengde zhi ren \u6210\u5fb7\u4e4b\u4eba,3 a person of accomplished virtue, that is, an ethically accomplished person who has moral convictions, and a particular concern with yi or right conduct, in situations that seem to advance personal gain.4\n\nMencius \u5b5f\u5b50 and Xunzi \u8340\u5b50, as well as some major Song and Ming Confucians, aspire to a greater height. For Cheng Hao \u7a0b\u9865 and Wang Yangming \u738b\u967d\u660e, for example, all humans are capable of becoming sages.5 More realistic is Confucius' insight into the practical possibility of attaining human excellence. To him it is possible for ordinary humans to become junzi, presuming that they have a whole-hearted commitment to the ideal of dao (zhi yu dao \u5fd7\u65bc\u9053), to ren, li, and yi; and have exerted their best efforts in self-cultivation even in adversity or in the best of circumstances. Becoming a sage is an object of wish, not a reasonable, realistic objective of ethical pursuit.\n\nIn this chapter, I present a reconstruction of some principal aspects of Confucius' thought as an ethics of junzi.6 The focus is on the virtues of junzi in the Analects. Occasionally I employ insights of Mencius, Xunzi, and the Liji \u79ae\u8a18 for elaboration. Section 2 is a critical appreciation of the varying interpretations of junzi. Section 3 presents a map of the virtues of junzi. These excellences or personal merits of the junzi depict some salient characteristics of junzi, with emphasis on the embodiment of fundamental, interdependent, ethical virtues and selective, dependent virtues. Section 4 discusses two ways for dealing with conflict: (IV.1) reconciliation and harmonizing of differences (tiaohe \u8abf\u548c) with a preference for arbitration or mediation over adjudication or litigation, and (IV.2) the art zhongyong \u4e2d\u5eb8. Section 5 focuses on the flexibility of junzi in the exercise of quan \u6b0a (moral discretion) in exigent circumstances.\n\n## 2 Interpretations of Junzi in the Lunyu\n\nThe difficulty of settling on a definitive or adequate English translation of junzi in the Analects is reflected in some translations and notes since the nineteenth century. Here is a short list of translations of junzi: \"superior man\" (Legge, Chan, Bodde, Dubs), \"gentleman\" (Waley, Lau, Watson), and \"noble man or person\" (Giles, Fingarette, Schwartz, de Bary). In some recent, explicitly interpretive studies, we also find \"paradigmatic individual\" (Cua 1969b, 1971; both incorporated in Cua 1978, Chaps.\u200b 3 and ), \"model of emulation\" (Munro 1969), or \"exemplary person\" (Hall and Ames 1987). In the case of The Doctrine of the Mean (Zhongyong), we also find \"profound person\" (Tu). Since there is no English equivalent, junzi is best left untranslated. This proposal does not deny the necessity of choice in translation at the service of readability and consistency. But whatever English term is chosen, the writer must offer some explanation and justification for this choice. Let us consider some examples.\n\n### 2.1 James Legge\n\nJames Legge, the pioneering translator of the Lunyu in the late nineteenth century employed \"superior man\" for translating most occurrences of the ethical uses of junzi. In two exceptional cases (Analects 1.2 and 1.14), we find \"man of complete virtue.\" The first exception and also the first occurrence in the Analects (Analects 1.1) is this saying of Confucius: \"Is he not a man of complete virtue [junzi], who feels no discomposure though men may take no note of him?\" (Legge 1960: 1). Legge gives a terse comment on junzi:\n\n> Literarily, it is a \"princely man\"... It is a technical term in Chinese moral writers, for which there is no exact correspondency in English, and which cannot be rendered always in the same way. (Legge 1960: 2)\n\nThe second case is Analects 1.14:\n\n> The Master said, \"He who aims to be a man of complete virtue [junzi] in his food does not seek to gratify his appetite, nor in his dwelling place does he seek the appliances of ease; he is earnest in what he is doing, and careful in his speech; he frequents the company of men of principle that he may be rectified: -such a person may be said indeed to love to learn.\" (Legge 1960: 7\u20138)\n\nLegge gives no explanation for using \"man of complete virtue\" in these remarks. The translation also eradicates the distinction between junzi and sage, for \"a man of complete virtue\" conveys the idea of ethically perfected state or highest degree of ethical attainment of being a sage, rather than a junzi who, as we shall see later, is an imperfect being \u2013 a being who has reached a high or superior degree of ethical attainment as compared with ordinary people who have lesser or no distinguishing ethical achievement. A junzi in this sense is a chengren \u6210\u4eba, an ethically accomplished person. This sense of junzi provides an explanatory justification for Legge's preference for \"superior man\" for the other occurrences of junzi in the Analects.\n\n### 2.2 Lionel Giles\n\nWhile agreeing with Legge that junzi literarily means \"princely man,\" Giles thinks that \"superior man\" is a misleading translation. For Analects 1.2, Giles offers this translation: \"The Master said: Is he not a princely man [junzi] \u2013 he who is never vexed that others know him not?\" A long note follows Giles' translation.\n\n> This is the much-discussed ch\u0171n-tzu [junzi], an expression of which the stereotyped English equivalent is \"the superior man.\" But in this there is, unhappily, a tinge of blended superciliousness and irony absolutely foreign to the native phrase, which in my opinion makes it unsuitable. \"Princely man\" is as nearly as possible in literal translation, and sometimes, as we shall see, it actually means \"prince.\" But in the majority of cases the connotation of rank or authority is certainly not explicit, and as a general rendering I have preferred \"the higher type of man,\" \"the nobler sort of man,\" or sometimes more simply as \"the good man.\" Perhaps the nearest approximation in any European language is to be found in the Greek ho kalos kagathos, because that implies high mental and moral qualities combined with all the outward bearing of a gentleman. Compare also with Aristotle's ho spoudaious who is however more abstract and ideal. (Giles 1907: 52n)\n\nIn defense of Legge we may say that Giles' remark actually echoes Legge's reminder that there is no exact English \"correspondency\" or equivalent for junzi. The difference in translation lies in the explanatory addendum, which provides the rationale for translation. Considered as a descriptive term for an ethical ideal today, \"superior man\" does not have the connotation of \"a tinge of blended superciliousness and irony,\" though we would prefer a gender-neutral term \"individual\" or \"person.\" A superior person is one who is above average in excellence, merit, or intelligence. As we pointed out earlier, junzi is \"superior\" to other people because of his greater ethical character and achievement. Giles' preference for \"the higher type of man,\" or \"the nobler sort of man\" is no less misleading, for it may suggest that a junzi, as distinguished from ordinary persons, belongs to a special class of persons with \"higher mental and moral qualities\" \u2013 an idea Confucius would reject. For Confucius, humans are born pretty much alike; it is practice that sets them far apart (Analects 17.2). For major Confucians, Mencius and Xunzi, since all ordinary people have the capacity to become junzi or sages, their achieved ethical status does not mark them as members of a special class of humanity. As Xunzi put it:\n\n> With respect to inborn nature and endowment, intelligence and ability, the junzi and small-minded persons are one and the same. In desiring honor and averting shame, the junzi and small-minded persons are the same. However, if we consider the way they pursue these matters, we would find them to be diametrically opposite. (Li 1979: 60)7\n\nFor purposes of distinguishing the degrees of ethical attainment and assignment of praise and blame, Xunzi would differentiate the ideally great Confucians (da Ru \u5927\u5112) from those who are merely refined or vulgar (Li 1979: 149). However, Giles' allusion to the Greek ideal of kalos kagathos and Aristotle's notion of spoudaious as somewhat analogous to junzi offers valuable hints for comparative ethical study. When we focus on junzi's concern for li (rites), particularly in the observance of wen \u6587 (cultural refinement), it is illuminating to compare junzi with kalos kagathos, especially if we construe junzi as a refined Confucian (ya ru \u96c5\u5112) in Xunzi's sense \u2013 one who exalts li and yi. A similar remark may be made about the now popular translation of junzi as \"gentleman.\" More interesting and significant is Giles' interesting comparison of junzi with Aristotle's notion of spoudaious, for it is plausible to consider junzi and spoudaious as functionally analogous notions of paradigmatic individuals.\n\n### 2.3 Waley, Lau, and de Bary\n\nLet us turn to the familiar translation of junzi as \"gentleman,\" suggested also by Giles. It is interesting to note the difference in rationales for the same translation, by Arthur Waley and D.C. Lau. Says Waley:\n\n> As regards the translation of the term ch\u0171n-tzu [junzi], I see no alternative but to use \"gentleman,\" though the effect is occasionally somewhat absurd in English. One needs a word, which primarily signifies superiority of birth, but also implies moral superiority. Neither Legge's \"superior man,\" nor Soothill's various equivalents (\"man of the higher type,\" \"wise man\") fulfill this condition. (Waley 1938: 37)\n\nContrast this view with Lau's, where \"superiority of birth\" is, rightly, de-emphasized:\n\n> In the Lunyu... ch\u0171n-tzu [junzi] and hsiao-ren [xiaoren] are essentially moral terms. The ch\u0171n-tzu is the man with a cultivated moral character, while the hsiao-ren is the opposite. It is worth noting that the two usages indicating the social and moral status are not exclusive, and, in individual cases, it is difficult to be sure whether, besides their moral connotations, these terms may not also carry their usual social connotations as well. (Lau 1979: 14)\n\nFor distinguishing these two usages of junzi in the Analects, de Bary proposes the distinction to use \"gentleman\" and \"noble man\" respectively. Says Wm Theodore de Bary:\n\n> [T]he great majority of reference to him [junzi] in the Analects can be understood simply as \"gentleman,\" referring to a class of well-bred persons with gentle ways, impeccable manners, and a well-developed moral sense. It is only in the minority of cases \u2013 though I would say a significant minority \u2013 that the ch\u00fcn-tzu [junzi] is cast in a very lofty and self-sacrificing role demanded of the noble man as a leader of others. He is said to be content even in poverty and in humble circumstances that would be beneath the dignity and refined tastes of the gentleman (Analects 6.9, 15.32, 18.7). No less than the sacrifice of one's life may be called for (Analects 15.8). But above and beyond this the noble man is a model for everyone who might play a leadership role in society, a life of higher responsibilities (Analects 19.7). (de Bary 1991b: 28)\n\nSince the main stress of Confucius is the junzi's virtues of character, inclusive of ren \u4ec1, yi \u7fa9, and li \u79ae or observance of the li (rites, rules of proper conduct), one would expect his \"nobility\" to include the main characteristics of the \"gentleman\" without special reference to class membership. As we shall see shortly, a well-articulated notion of gentleman will also include some fundamental ethical concerns expressed in ren, yi, and li (Analects 2.1).\n\nIn a nutshell, for Confucius, as well as for Mencius and Xunzi, junzi expresses an ideal of a cultivated, ethical character. Although more explanation is needed to avoid misleading interpretations, the various translations of junzi may be viewed as valuable attempts to bring forth the translator's own appraisal of the salient features of this ideal of ethical character in a way that will be intelligible to English readers. In this light, we may regard junzi as a sort of emphatic term that, in context, serves to accentuate certain ethically desirable and commendable virtues (meide \u7f8e\u5fb7) or qualities of ideal person, in short, ethical excellences. In Hume's term, these excellences are \"personal merits\" that deserve emphasis especially in moral education. (Hume 1957: 97) In the case of Xunzi, we have insightful essays on learning and self-cultivation (Xunzi, Books 1\u20132). We find forceful statements of the practice of li, yi, and ren; the importance of learning the classics, guidance of exemplary, learned teachers, self-examination and reflection, and accumulation of good deeds. For becoming a junzi, Xunzi insists on the possession of integrity (quan \u5168) and purity (cui \u7cb9) or incorruptibility. Allied with this emphatic function, the junzi also serves as an exemplar of how the fundamental Confucian ethical concerns \u2013 ren, li, and yi \u2013 have concrete significance, practically attainable in varying degrees by ordinary moral agents.\n\n### 2.4 Chan\n\nIn general junzi is a notion of a morally excellent person, a paradigmatic individual who sets the tone and quality of life of ordinary moral agents. A junzi is a person who embodies ren, and yi, and li. Every person may strive to be a junzi in the sense of a guiding paradigmatic individual, rather than a xiaoren (small-minded person). There are of course degrees of personal ethical achievement, depending on the situation, character, ability, and opportunity of moral agents.\n\nLegge's translation of junzi as \"superior man\" brings out the junzi as a person who has a distinguished, superior ethical character and aptitude. In this sense, a junzi may be said to be a chengren \u6210\u4eba or ethically accomplished person (Analects 14.12). The translation of junzi as \"true gentleman\" focuses on junzi's relation to the cultural setting of his actions, his ability to satisfy, so to speak, the stylistic requirements of a form of life. A junzi, in this sense, is an embodiment of a \"cultural life style.\" Some of the qualities of junzi resemble those of the ideal of gentleman as articulated by Douglas McGee.\n\n> The gentleman as defined by tradition aspired to nothing less than becoming a concrete universal. Guided and sustained by his limitation, he took as his moral ideal the cultivation of humanity.... His conduct was judged by its appropriateness, a measure that took account of particular circumstances, but always in conformity to the human ideal. Conduct regulated by this form felt congruous and fitting, hence purposeful. In the widest sense of \"manners,\" the manners of the gentleman were textures of his life, a texture isomorphic with the structure of his class. (McGee 1966: 222; emphasis added)\n\nIf we attend to the words we have stressed, we can appreciate the functional equivalents of the Confucian cardinal notions. For ren may be properly viewed as the ideal of the \"cultivation of humanity,\" especially when this ideal is expressed as \"an affectionate concern for the well-being of the fellow members of the community.\"8 Perhaps for this reason, Wing-tsit Chan often rendered ren as \"humanity.\" \"Appropriateness of conduct\" expresses the concern for yi. In the Doctrine of the Mean (Zhongyong \u4e2d\u5eb8 Sect. 20), we find a concise remark to the effect that yi is \"appropriateness\" (yi zhe yi ye \u7fa9\u8005\u5b9c\u4e5f).9 The focus on the \"manners\" of life recalls the main concern of li as a set of formal prescriptions for proper behavior, which include, among other things, good manners, courtesy and respectfulness. In sum, the ethical ideal of junzi expressed in the Lunyu may be regarded as setting forth the different requirements or qualities for a life of moral excellence. A junzi's serious commitment to dao (Analects 4.9) involves a singleness of purpose and acting in accordance with the requirements of the basic, interdependent virtues of ren, yi, and li. More formally, ren, li, and yi are basic, cardinal and interdependent virtues.\n\n## 3 Basic, Interdependent, and Dependent Virtues\n\n### 3.1 General Remarks\n\nConcern with the basic interdependent virtues of ren, yi, and li also involves particular dependent virtues such as filiality (xiao \u5b5d), magnanimity (kuan \u5bec), trustworthiness (xin \u4fe1), and courage (yong \u52c7). These particular virtues are called dependent virtues in the sense that their ethical significance depends on connection with the basic, interdependent virtues. Dependent virtues are not subordinate or logical derivatives of the basic virtues.10 The ethical significance of the particular dependent virtues is determined by ren and yi, since these are criteria of moral virtues.11 Of course, when li is invested with an ennobling function, it entails the presence of ren and yi.12 Arguably, as Chen Daqi maintains, what Confucius meant by de \u5fb7, in the sense of excellence or virtue (meide \u7f8e\u5fb7), pertains to the product of the intersection of ren and yi. Thus both ren and yi may be said to be the constituent elements of de.13\n\nFor forestalling misunderstanding, let us note that dependent virtues are virtues, as they reflect personal merits, although their ethical significance is determined by their connection with one or more basic interdependent virtues. At issue is their ethical significance, not their value status as deserving of praise in appropriate non-ethical contexts. Also, their value status may be appreciated in the light of their function as specifications of the concrete significance of the cardinals, which are basically abstract general concepts. To borrow Xunzi's distinction, the cardinals ren, yi, and li are gongming \u5171\u540d, or generic terms, and dependent virtues are bieming \u5225\u540d or specific terms, that is, terms that specify the concrete significance of the cardinals in particular contexts of discourse.\n\nFor elaborating interdependent and dependent virtues, we may employ Chen Daqi's distinction between complete virtues (quande \u5168\u5fb7) and partial virtues (piande \u504f\u5fb7). Interdependent virtues, ren, yi, and li are cardinal or fundamental virtues. They may be said to be quan \u5168 or complete in the sense that their ethical value is intrinsic, i.e., it does not depend on anything else. In this sense, quande \u5168\u5fb7 are complete or whole (quan \u5168). Moreover, these cardinals are relevant to all situations of human life as our actions always have effects on others especially in exigent circumstances. Normal, unproblematic situations in human intercourse do not need to invoke the cardinals. On the other hand, piande \u504f\u5fb7 or partial virtues are so-called because their ethical significance is limited, not only in their application to circumstances, but also insofar as their ethical value depends on connection to the cardinals. It is important to note, however, that piande have values. Here we may invoke Xunzi's distinction with dao as a whole and its various pian or partial aspects. Xunzi's is critical of some thinkers, not because they espoused faulty or irrational doctrines, but because they comprehend only partial aspects of the Dao. Mozi, for example, rightly appreciates the importance of uniformity, but he fails to attend to the value of diversity; Songzi rightly appreciates the value of having few desires, but he fails to see the value of having many desires. Says Xunzi, \"Dao embodies constancy, but encompasses all changes. A single corner is insufficient to exhaust its significance\" (Li 1979: 381, 480).\n\n### 3.2 Supportive and Constitutive Virtues\n\nIn the Analects, we do find some of Confucius' remarks that mention both cardinals and dependent virtues in the same contexts, for example, ren, zhi \u667a(knowledge, wisdom), and yong \u52c7 (boldness or courage) in 14.28; gong \u606d (respectfulness), zhong (loyalty), jing \u656c (reverence), and yi \u7fa9 in 16.10; li \u79ae and zhong \u5fe0 in 3.19; li \u79ae, yi \u7fa9, and xin \u4fe1 (trustworthiness) in Analects 13.4 and 15.18. Once it was reported that the Master taught four subjects: wen \u6587 (culture, cultural refinement), xing \u884c (conduct of life), zhong \u5fe0, and xin \u4fe1 (Analects 7.25).\n\nFor heuristic purposes, we may regard dependent virtues as belonging to two different clusters. One cluster consists of those dependent virtues that are closely related to one basic, cardinal virtue rather than another. Another cluster consists of \"overlapping\" dependent virtues in the sense that they seem especially germane to the practice of one or more cardinal virtues (henceforth, cardinals). For convenience, let us introduce the distinction between supportive and constitutive virtues. Supportive virtues are those virtues that are genial or helpful, though not necessary, to the development of the cardinals such as ren, yi, and li. Constitutive virtues, on the other hand, are those that are constitutive of the quality of the cardinals actualized. Differently put, constitutive virtues are the contributory means to the pursuit of a certain end, that is, they are means that become constituents of the end achieved.\"14 In general, virtues can be admired and can also inspire ideal achievement when they are viewed as constitutive features of an achieved state of a person. However, detached from the governing guide of moral ideals, virtues are mere objects of praise that may not possess a transforming significance for moral agents.\n\nConstitutive virtues are those that are integral parts of the state of ren achieved, and thus may be termed \"integral virtues.\" However, it is important to allow that there are some dependent virtues that are both supportive and integral. In the Analects, items of both clusters are sometimes mentioned together. Below I discuss briefly junzi's basic qualities of character as embodying a concern with the Confucian cardinals and some supportive and constitutive virtues as a preliminary to dealing with Confucius' idea of the flexibility or adaptability of junzi.\n\nAgain the distinction between supportive and constitutive dependent virtues is not intended as a dichotomy. Depending on the character and temperament, what is a supportive trait in one person may be a constitutive virtue for another. Kuan \u5bec (magnanimity, generosity of spirit, broadmindedness), for example, may be constitutive for a person of mild temperament, but merely supportive for another who has an inordinate self-confidence in the practice of ren. In the discussion below, although I sometimes propose a specific interpretation, the classificatory question is always an open question. Moreover, the distinction is offered in a tentative spirit. Perhaps, on closer analysis, the distinction may have only a practical, not theoretical value, i.e., helpful to individual agent's reflection on how best to constitute his or her character, on which dispositions are the most congenial for development in the light of temperament and circumstance.\n\n### 3.3  Ren \u4ec1\n\nA sincere and serious commitment to ren is a weighty and momentous burden (Analects 8.7), for it requires constancy. \"It is hard for a man to have constancy who claims to have when he is wanting, to be full when he is empty and to be comfortable when he is in straightened circumstances\" (Analects 7:26).15 If a junzi disavows ren and gives up his commitment to ren, he does not deserve the title of being a junzi. The junzi \"never deserts ren, not even for as long as it takes to eat a meal\" (Analects 4:5). Whether he confronts an urgent or difficult situation, he abides by ren. A person genuinely committed to the practice of ren would even sacrifice his or her life for the sake of ren. Says Confucius, \"A determined scholar or a ren person will not seek to live at the expense of doing harm to ren. He will even sacrifice his life to fulfill ren\" (Analects 15.9).16\n\nRen, in the broad sense, is Confucius' dao \u9053, his vision of the good, an ideal theme of concern for humanity. The term \"ideal theme\" is an appropriation of the notion of theme familiar in various linguistic contexts. We talk of a theme as a topic of discourse or discussion, or of a theme as \"an idea, point of view, or perception embodied or expanded upon in a work of art,\" or as \"a melody forming the basis of variations or other development in a composition.\"17 Unlike ideal norms, ideal themes do not provide precepts, rules, directives or principles for action.18 They are ideal points of orientation that have an import for committed agents. Such terms as development, clarification, and expansion are thus quite at home in discussing ideal themes. Whereas in the case of ideal norms, terms such as application, compliance and extension are more appropriate.\n\nRen is like a theme in literary or musical composition, amenable to polymorphous, creative expressions, depending on the committed person's interpretation of the significance of the ideal for his or her life. Fundamentally, ren is the love of fellow humans (Analects 12.22), or affectionate concern for the well-being of humanity. Commitment to ren involves benevolence, that is, desire to do good to others as well as to \"study the good of others\" (See Hutcheson 1729: 158). Confucius said: \"The junzi helps others to realize their (ethically) praiseworthy qualities (mei \u7f8e); he does not help them to realize their bad qualities (e \u60e1). The small man does the opposite\" (Analects 12.16).19 Contributory to and constitutive of the realization of ren, is the development of particular dependent, constitutive virtues such as zhong \u5fe0 and shu \u6055. Zhong and shu are perhaps the most important constitutive or integral virtues of ren.20\n\n### 3.4  Ren-Dependent Virtue: Zhong \u5fe0\n\nZhong \u5fe0 is often translated as \"loyalty, devotion,\" sometimes, \"doing one's best.\"21\n\nFor constructive interpretation, all these renderings may be used for indicating a unified conception if we adopt, say, Josiah Royce's preliminary definition of \"loyalty\": \"The willing and practical and thoroughgoing devotion of a person to a cause.\"22 \"Thoroughgoing devotion to a cause\" implies constancy (heng \u6052) and doing one's best to realize the cause or object of one's devotion, that is, in doing one's utmost with one's whole heart and mind (jin xin \u76e1\u5fc3) to realize the object of commitment (jinzi \u76e1\u5df1). (Zhu 1980: 23) Alternatively, to be zhong is to give the fullest expression to one's commitment, in short, to be true to oneself.23\n\nAs a self-regarding virtue, zhong implies a commitment to a self-governing standard for conduct. The object of one's devotion may be another person. For example, when Fan Chi \u6a0a\u9072 asked about ren, Confucius replied: \"While at home maintain your respectful attitude (gong \u606d); in handling affairs, be reverend (jing \u656c); in dealing with others, be zhong \u5fe0. These are the qualities that cannot be put aside, even if you go and live among the barbarians\" (Analects 13.19). The object of zhong may be a person in a superior position. Thus, in one sense, to be zhong is to be loyal to someone superior in the social, political hierarchy, especially to a ruler (Analects 2.20, 3.10, 12.14,24), for example, \"The ruler should employ the services of his subjects in accordance with rites (li \u79ae). A subject should serve his ruler by zhong.\" (Analects 3.19).\n\nNotably, zhong also occurs in a non-hierarchical sense (Analects 1.4, 7.23, 13.19, 16.10). The object of zhong may also be an equal. When Zigong \u5b50\u8ca2 asked about friendship, Confucius replied: \"Advise them in the spirit of zhong and tactfully guide them. If that is not possible, do not disgrace yourself\" (Analects 12.23). On another occasion, Confucius said about himself: \"In a hamlet of ten households, there are bound to be those who are my equal regarding zhong and xin \u4fe1 (trustworthiness, being true to one's words), but they are unlikely to be as eager to learn as I am\" (Analects 5.28). It is important to note that the object of zhong is people in general; it is not confined to either one's superior or equal.25\n\nAs a self-regarding virtue, zhong implies a commitment to a self-governing standard for conduct. As a ren-dependent virtue, zhong is not a blind devotion to persons or matters of concern. Even a ruler's conduct is also subject to criticism by subordinates (e.g., Analects 13.15, 13.23). Xunzi expatiates this theme: In the case of an ethically responsible minister, whenever the ruler has departed from the dao \u9053, it is quite proper for the minister to follow dao rather than the ruler. So also, while filial piety enjoins obedience, there are situations where disobedience is morally justified. When, for example, one's compliance with parental wishes will bring them disgrace rather than honor; when one will endanger the lives of the parents rather than bring them peace; and when the parents' wishes are such as to compel one to behave like a dumb creature rather than a man of moral cultivation. In all these cases, one must follow yi \u7fa9 or one's sense of what is right.26 In short, in appropriate situations in the light of yi or right conduct the object of zhong may not be worthy of devotion. The critical committed agent preserves his sense of yi and independent judgment.27 Moreover, while zhong is a ren-dependent virtue, a person may have zhong without ren (Analects 5.19), although it may sometimes be admired as a personal merit, as in the case of a person of blind devotion to an ethically unworthy person or institution.\n\n### 3.5  Ren-Dependent Virtue: Shu \u6055\n\nLet us now turn to shu \u6055, which expresses the idea of consideration of others. Viewed separately or together, zhong and shu involve reflection and judgment. Zhong expresses loyalty to and conscientious regard for the moral standard or the ideal of ren, i.e., an attitude of sincerity and seriousness in one's commitment to ren; shu more especially pertains to other-regarding conduct. A commitment to ren is a commitment to realizing ren in the personal relations between oneself and another. In other words, it is an adoption of an attitude of moral regard. Shu may be said to be \"the golden rule\" that governs the exemplification of the ren attitude. Zigong asked, \"Is there a single word which can serve as a guide to conduct throughout one's life?\" The master said, \"It is perhaps the word 'shu.' Do not impose on others what you yourself do not desire (yu \u6b32)\" (Analects 15:24).\n\nIn other words, to be guided by shu is to use \"oneself as a measure in gauging the desires of others\" \u2013 an idea expressed in Analects 4.30 and 6.15.28 Alternatively expressed: \"What I do not desire (yu \u6b32), I ought not to do it to another\" (Analects 12:2). In both formulations, what is crucial is the notion of yu \u6b32 or desire. It is misleading to say that shu concerns the nature of desire in the ordinary sense, for it has more to do with the manner of satisfaction than with the nature of occurrent desires. A plausible explication of shu thus requires a distinction between occurrent and reflective desires; thus what I desire now may, on reflection, be something I ought not to desire. This distinction seems implicit in a passage in the Xunzi: \"A single desire which one receives from nature (tian) is regulated and directed by the mind in many different ways, and thus it is difficult to identify and classify it with those which we receive from nature.\"29\n\nIn this sense, shu has to do primarily with reflective rather than occurrent desires. Zhong and shu may be said to be a method of reflection on occurrent desires, for assessing their appropriateness in the context of human relations. In this way, the exercise of shu presupposes a capacity of self-reflection and self-evaluation. To pay heed to shu is to deal earnestly with the question: Do I want my present desire to be satisfied as I want other's analogous desires to be satisfied in a way that comports with ren? The wanting here is a reflective desire. Thus a deliberate consideration on the character of occurrent desires has consequences in terms of the moral character of one's acts. Shu as moral regard has a practical import only when the agent has subjected his occurrent desires to reflective evaluation in the light of ren.\n\n### 3.6 Golden Rule\n\nWe may regard shu as functionally equivalent to the Golden Rule, which is negatively expressed as \"Do not do to others as you would not expect them to do to you,\" or positively, \"Do unto others as you would have them do unto you\" (Singer 1977: 122). Our above interpretation of shu as pertaining to reflective desire is compatible with this general formulation of the Golden Rule. For at issue is the question of one's willingness or desire to be treated in a certain way rather than the content of occurrent desires. The issue has to do with the standard governing the satisfaction of one's occurrent desires. What I morally want to do is to subject my present desires to the assessment in terms of the standard I am committed to. For a Confucian, to be a man of ren is to engage in reflection that brings the ideal of ren to bear in actual conduct, and this obviously implies a desire or willingness to make other persons' desires a relevant consideration in light of ren. To pay heed to shu is to have an other-regarding attitude. Coupled with zhong or one's own sincere commitment to ren, such an other-regarding attitude is an aspect of self-regard, i.e., a regard for one's own character and moral condition. Such a commitment entails \"doing one's best\" in appropriate circumstances. According to Confucius, \"a man of ren desiring (yu \u6b32) to establish his own character, also establishes the character of others, and desiring to be prominent himself, also helps others to be prominent. To be able to judge others by what is near to ourselves may be called the method of realizing humanity\" (Analects 6:30).30 In this way, the moral agent's own conduct serves as a measure or standard for others. This is possible because his own character is achieved by way of embracing others as an integral component of his own moral development. Shu as extensive concern for others is thus a component of one's preoccupation with moral attainment or moral condition on the whole. My extensive concern for others is a concern for their moral being or condition on the whole. In establishing or developing my own moral character in light of ren, I am also engaged in establishing or developing others' moral character, not in the sense of directly urging others to do so nor of asserting myself to be a moral paradigm; but in the sense that my own case serves as an embodiment of the possibility and actuating import of ren-realization. In this way I indirectly contribute to the development of others' moral character. When zhong and shu are construed as a practical rule of conduct, it is a procedural rather than a substantive guide to ren-realization.\n\nArguably, when shu, especially its negative formulation, is construed as a version of \"Golden Rule,\" apart from being a useful means for \"the elimination of self-partiality and inward dishonesty,\" as Kant would put it (See Abbott 1948: lii), it may also be viewed as a counsel of moderation and humility in making claims concerning one's knowledge of the good.31 Recall that the vision of ren or the good is an indeterminate ideal theme, and as such it is subject to diverse, concrete specifications within the lives of committed agents. At any given time, a reasonable agent would make such a specification based on a partial knowledge of the significance of the holistic vision. The ideal of impartiality implicit in the notion of shu, as opposed to partiality of the knowledge of the good, serves as a reminder of one's imperfection or incompleteness of ethical knowledge. By construing the negative formulation of shu (\"What I do not desire, I ought not to impose on others\" (Analects 12.12 and 15:23) as a counsel of modesty and humility, we can appreciate its importance by attending to a characteristic of reasonable persons. Modesty pertains to the moderation of one's claims or demands upon others. One ordinary sense of 'reasonable' indicates that a reasonable person will refrain from making excessive or extravagant demands on others. As Kurt Baier remarks, \"We say of demands or requests that they are excessive if, though we are entitled to make them, the party against whom we make them has good reasons for not complying, as when the landlord demands the immediate vacation of the premises in the face of well-supported pleas of hardship by the tenant\" (Baier 1958: 316). More importantly, in the light of the vision of dao or ideal of the good human life, we would expect reasonable, committed persons to be modest in making their demands and requests, because no one possesses the knowledge of all possible, concrete, and appropriate specifications of the significance of the good for individual human life. As a result, one's specification of the good is always made from a limited perspective. In this sense, the negative formulation of the shu is a counsel of modesty, underscoring the importance of being aware of one's limited perspective. As we shall see in Sects. 3 and 4, the salient qualities of junzi are markedly those of the reasonable persons.\n\nHume justly observes that modesty, in the sense \"opposed to imprudence and arrogance\" is one personal quality that is agreeable to others, for it \"expresses a diffidence of our own judgment and a due attention and regard for others\" (Hume 1957: 7). We find a similar idea of modesty in one passage of the Analects: \"The junzi considers yi \u7fa9 as the substance (zhi \u8cea) of his conduct; he puts it into practice by observing the li \u79ae; he gives it expression by being modest (xun \u905c). He completes it with trustworthiness (xin \u4fe1)\" (Analects 15.18). Perhaps, the inner significance of modesty is best appreciated in terms of humility, in the sense that the reasonable persons know their place in relation to others' thoughts and affections, informed by an awareness and estimate of the limited character of their importance in relation to others. This attitude of humility is compatible with self-respect or even pride in one's achievement (Isenberg 1949: 7\u20138). Moreover, humility, in the light of commitment to the good, is an acknowledgement of \"the lack of knowledge about what is good [for oneself and another] and its consequent attitude of not claiming to be or thinking oneself as good\" (Allinson 1985: 306). Deprived of complete knowledge of the specification of the concrete significance of the good for all humans, a reasonable person will be humble. For any claim to knowledge about one's good and another's is at best a plausible presumption but always defeasible. The possibility of negation of such claims furnishes a basis for the adoption of humility. The ethical value of humility also provides a way of appreciating the Confucian stress on the observance of li or rules of propriety, as well as Cicero's on decorum.32\n\nOn the basis of the foregoing, it should be clear that the negative version of the shu imposes a more stringent requirement than the positive version (analogous to that of negative moral rules or precepts). Imagining oneself in \"the dual roles of agent and patient,\" while expressing self-respect and respect for others, does not provide a reasoned justification for imposing one's conception of the good. This is perhaps the force of Confucius' saying: \"To say you know when you know, and to say you do not when you do not, that is knowledge.\"67 One's claim to knowledge about what is good for oneself and another must be proportional to accessible information and experience. While such knowledge may provide grounds for a claim of its significance for future conduct, reasonable persons would avow their sense of fallibility, in expressing modesty and humility. As we shall see in Sect. 4, sagacious or judicious judgments will also be informed by a sense of timeliness (shi \u6642), that is, an adaptation to the current situation in order to achieve equilibrium and an adjustment to varying, changing circumstances.\n\nOur stress on the significance of the negative aspect of shu \u6055 is consistent with practical expectation of mutual regard or consideration. For this reason, shu is often rendered as \"reciprocity,\" conveying the idea of mutual expectation of the agent and the recipient. In the Xunzi, we find a remark attributed to Confucius: \"The junzi has three different ways of practicing shu: When he is incapable of serving his ruler and yet expects lower officials to serve him, this is not shu. When he is incapable of requiting the affections of his parents and yet expects his sons to be filial, this is not shu. When he cannot respect (jing \u656c) his elder brothers and yet expects his younger brothers to listen to him, this is not shu. If a scholar-official (shi \u58eb) clearly understands these three ways of shu, then it is possible that he can rectify himself\" (Li 1979: 661).\n\n### 3.7 Other Ren-dependent Virtues\n\nLet us consider briefly some other ren-dependent virtues. On one occasion, responding to an inquiry about ren, Confucius said that a man of ren practices five things: \"Gong \u606d (respectfulness), kuan \u5bec (magnanimity, generosity, open-mindedness), xin \u4fe1 (trustworthiness, being true to one's words), min \u654f (adroitness), and hui \u60e0 (beneficence)\" (Analects 17.6). I suppose that kuan and hui are dependent, constitutive virtues of ren \u4ec1, for ren is basically expressed in love, or affectionate concern (ai \u7231). Similarly, warm-heartedness (wen \u6eab) is also ren-dependent, constitutive virtue (1.10). Ren as an affectionate concern for others would also be expressed in loving-kindness (ci \u6148) (2.20), in some contexts, would be expressed in kuan \u5bec. Hui \u60e0 or beneficence is also an expression of ren concern. Xin seems to be another constitutive virtue of ren, as indicated in the pairing of zhong \u5fe0 and xin \u4fe1 (Analects 1.8, 1.9, 9.21, 15.19). For instance, when Zizhang \u5b50\u5f35 asked about conduct (xing \u884c), Confucius replied: \"Make zhong and xin your master guides\" (Analects 15.6). As zhong involves doing one's best on behalf of the object of loyalty, min \u654f (agility or adroitness) would be a virtue of resourcefulness in handling affairs on behalf of the object of loyalty. While gong \u606d is a dependent, supportive virtue of li \u79ae, it is also a supportive virtue of ren when the spirit of ren informs its expression according to li. As Confucius remarked: \"If a man has no ren, what has he to do with li\" (Analects 3.3). Moreover, as involving rang \u8b93, gong would be merely supportive as in the case of the agent's refusal to yield (rang \u8b93) the practice of ren to his teacher (Analects 15.36). As we shall see later, jing \u656c (reverence) is a constitutive virtue of both ren and li, since it is an essential attitude required in filial conduct (xiao \u5b5d) \u2013 a foundation for the practice of ren (Analects 1.2; 2.7).\n\n### 3.8 Overlapping Constitutive Virtues of Ren, Li, and Yi: Keji \u514b\u5df1 and Yong \u52c7\n\nAt this point let us interpose by briefly attending to keji \u514b\u5df1 and yong \u52c7 as overlapping, constitutive virtues of ren, li, and yi. When Yan Yuan \u984f\u6df5 asked about ren, Confucius said: \"To return to the observance of the li through self-control (keji \u514b\u5df1) constitutes ren\" (Analects 12.1). Elsewhere, Confucius also remarked, \"If a man has no concern for ren \u4ec1, what has he to do with li \u79ae?\" (Analects 3.3). These two sayings show the interdependence of ren and li. Self-control is constitutive of the practice of ren as it involves overcoming emotions and desires that may well hamper ren-performance, especially in difficult situations of the moral life. Particularly problematic, as Xunzi points out, are those self-serving desires that interfere with the practice of ren. The li, as delimiting the proper boundary for the pursuit of self-satisfaction, are the means for self-control. In the case of yi, self-control is indispensable in observing the constraints of flexibility \u2013 a topic we will take up in Sect. 2.15.\n\nYong \u52c7, as an aretaic or virtue term, is perhaps best rendered as \"courage\" \u2013 the quality of character that shows itself in facing danger undaunted despite fear or lack of confidence.33 Yong is clearly a dependent virtue of ren, for \"the ren person certainly possesses yong, but a yong person does not necessarily possesses ren\" (Analects 14.4). Likewise, yong is a dependent virtue of li; for its ethical significance depends on its connection with li. Thus, the yong person who does not have regard for li is likely to be unruly (Analects 8.2). It is an open question whether yong is a constitutive virtue of li. Arguably a person committed to the observance of li, in some context, may need yong to act in the absence of the detail rituals involved. Here the agent may need yong or a sense of venture, risking embarrassment or humiliation, or even shame. In the case of yi, yong is clearly a dependent, constitutive virtue. For example, when Zili asked : \"Does the junzi cherish yong?\" The Master said: \"For the junzi, it is yi that is considered supreme. Possessed of yong but devoid of yi, a junzi will make trouble, but a small man will be a brigand\" (Analects 17.23). That Yong is constitutive of yi seems evident in this passage: \"To see yi (the right thing to do) and leave it undone shows a lack of yong\" (Analects 2.24). At any rate, yong requires learning (Analects 17.8) and judgment, which inform the exercise of yi. We will deal with this topic in Sect. 2.15.\n\n### 3.9 Confucius Reasoning\n\nBefore we turn to li-dependent virtues, let us note Confucius' reasoning in our passage above (Analects 17.6). His reasoning consists in a series of hypothetical propositions, conveniently numbered below.\n\n1.\n\nIf you are respectful (gong \u606d), you will not be treated with disrespect.\n\n2.\n\nIf you are generous (kuan \u5bec), you will win the people.\n\n3.\n\nIf you are trustworthy (xin \u4fe1), people will entrust responsibility to you.\n\n4.\n\nIf you are agile (min \u654f), you will accomplish much.\n\n5.\n\nIf you are beneficent (hui \u60e0), you will be in an adequate position to employ the services of others.\n\nNote that the consequent in each hypothetical proposition is an appeal to the notion of reflective desirability.34 In (1)\u2013(5), a desirable state of affairs, the consequent, is said to follow if the antecedent occurs. The consequents are desirable states of affairs, which Confucius took to be plausible presumptions, that is, matters of common knowledge. These presumptions are of course defeasible. If this interpretation is correct, particular moral virtues cited, with others such as knowledge and courage (Analects 15.l7), are recognized as having moral values insofar as they advance the realization of a life of ren. They are virtues in the sense that the absence of them would lead to undesirable consequences, which hamper the pursuit of realization of ren.\n\nWe may represent Confucius' appeal to reflective desirability in this form: \"If you reflect seriously on the desirable consequences of adopting and acting in accord with virtues, you would discover where your true interest lies.\" Indeed, the notion of reflective desirability (or undesirability) is implicit in the idea of shu, and it is an important aspect of Confucian argumentation. Notably, our passage clearly implies that possessing the five virtues is beneficial to the individual, as they are wanted and as they serve the true interest of the individual. For our present purpose, it is sufficient to note that a junzi cultivates the particular virtues for the purpose of attaining to a life of ren and this task presupposes knowledge of reflective desirability of actions and human affairs.35\n\n### 3.10 Dependent Virtues of Li \u79ae\n\nLi and yi are also fundamental ethical virtues. Because of the Confucian stress on tradition and civility, and the need for independent ethical judgment of committed individual moral agents, li and yi deserve special attention in Confucian ethics. Our discussion in the preceding section on ren and particular ren-dependent virtues pertains, so to speak, to the internal aspect of ren-morality. The accent is on the ideal and self-cultivation of desirable qualities or dispositions rather than the external or outward form of conduct. Unlike a small-minded person in times of moral failure, a junzi seeks the cause within himself rather than others (Analects 15:21). For a person committed to the Confucian ideal dao or ren, he or she must examine himself for the cause of failure. As an eminent disciple, Zengzi \u66fe\u5b50 reminds his pupils: \"Daily I engage in self-examination on three matters: In what I have undertaken on another's behalf, have I failed to do my best (zhong \u5fe0)? In my dealings with my friends, have I failed to be trustworthy in what I say (xin \u4fe1)? Have I failed to practice repeatedly what has been passed on to me?\" (Analects 1.4). Among other things, one contribution of Mencius to the development of Confucian ethics is his emphasis on the function of xin \u5fc3 (mind\/heart) in connection with moral failure. Xunzi also gives additional insight on the manner in which an ethically informed human mind may be obscured (bi \u853d) by concern with immediate satisfaction of desires without regard to distant consequences.\n\nMore fundamentally, an action conforming to a ritual requirement of li has its ethical significance because such an action is performed in the light of a concern for ren. As Confucius put it, \"If a man has no concern for ren \u4ec1, what has he to do with li \u79ae?\"(Analects 3.3) Without a regard for ren, ritual observances would amount to mere formal gestures vacuous of moral substance. Notably, in addition to imposing restraint on human behavior, as Xunzi points out, the li also support the satisfaction of desires within the defined boundaries of proper conduct. And when a junzi's compliance with li is informed by the spirit of ren, li has also an ennobling function, exemplifying the junzi's respect for li as an ideal, ren embedded tradition.36 This attitude toward li signifies also a respect for the reality of the situation, the background and possibility that furnish the context for successful moral performance. The Confucian emphasis on li is one justification for the Confucian homage to the concrete. Every action, on this view, has a conventional aspect for understanding its normative meaning and import. Whether or not we accept this stress on li, some sort of convention for identifying the normative import of action must be an essential element in any moral theory. Granted the importance of ethical convention or tradition, attention to the aesthetic and religious dimensions of li will also lead us to an appreciation of valuable facets of human life in different cultures and civilizations.\n\n### 3.11  Li-Dependent Virtue: Gong \u606d and Jing \u656c\n\nPerhaps the most important dependent virtues of Li are gong \u606d and jing \u656c. Both terms pertain to expression of respect for others. In vernacular Chinese, gongjing \u606d\u656c is a compound term (jianming \u517c\u540d) for \"respect or respectfulness.\" In the Lunyu, gong and jing occur as single terms (danming \u55ae\u540d),37 and the difference is roughly indicated by different translations, such as \"respect\" for gong and \"reverence\" for jing. However, in some contexts, as we shall see later, jing may also be rendered as \"respect.\" In one passage we cited earlier in connection with zhong \u5fe0 (Sect. 2.3), we have a suggestion that gong and jing are dependent, supportive constitutive virtues of ren.\n\nFor distinguishing gong from jing, we may say that the former pertains primarily to outward appearance, the latter to one's inner attitude. As Zhu Xi put it: \"Gong's principal focus is appearance (rong \u5bb9), jing on human affairs. Gong is seen in outward expression (wai \u5916), jing focuses on what is within (zhong \u4e2d)\" (Zhu 1980: 91). This explanation is supported by Confucius' remark that among the nine things that occupy junzi's thought (jiusi \u4e5d\u601d) is \"to think of appearing respectful (gong \u606d) when it comes to demeanor (mao si gong \u8c8c\u601d\u606d)\" and \"to think of being reverend when attending to human affairs (shi si jing \u4e8b\u601d\u656c)\" (Analects 16.10). Differently put, gong pertains to one's bearing or deportment. Courtesy or politeness, as it has to do with manners of behavior, is an example of gong. Jing, however, pertains to virtuous conduct, more specifically to one's inner attitude. The idea is also present in the Yijing: \"Being straight means correctness, and being square means yi \u7fa9 (righteousness). The junzi \u541b\u5b50 applies jing \u656c to straighten the internal life (nei \u5167) and yi to square the external life (wai \u5916). As jing and yi are established, one's virtue will not be an isolated instance.\"38 Chan's rendering jing as \"seriousness\" brings out the serious attitude; however, it is better to render it as \"reverence\" in the sense of \"deep respect and feeling for something or someone,\" which is a serious, attentive state of mind.39 In so far as li involves gong and jing, compliance with li has thus outer and inner aspects. Unlike gong, however, jing is a constitutive virtue of li \u79ae.\n\nAs we mentioned earlier, jing \u656c can be also rendered as \"respect.\" When Fan Chi asked about wisdom (zhi \u667a), the Master replied: \"To know the duties (yi \u7fa9) due to the people, to respect (jing \u656c) while keeping the gods and spirits (guishen \u9b3c\u795e) at a distance may be called wisdom\" (Analects 6.22). Note that jing as respect here conveys no sense of the depth of feeling of deferential esteem, as in the case jing for one's parents. Our passage seems to indicate Confucius' agnostic attitude toward gods and spirits, while acknowledging that jing or paying them proper respect is important, perhaps as a concession to common practices without seriously endorsing the common belief in their existence.40 Presumably, then, Confucius was simply expressing a formal gesture of respect for a cultural tradition, but redirecting attention of humans toward a concern for ren or serving humanity and understanding humanity rather than gods and spirits. When Ji Lu \u5b63\u8def asked about serving the spirits of the dead and gods, Confucius replied, \"You are not able even to serve men. How can you serve the spirits?\" When asked about death, Confucius said, \"You do not understand even life. How can you understand death?\" (Analects 11.12)\n\nThe important thing to note is that jing can be used to express respect in ways that differ from gong \u606d, which focuses on the formal character of speech or behavior rather than feelings. I can pay respects or obeisance to the dead, to acquaintances, to superiors, men or women of authority without any feeling of esteem, or even liking. As Confucius remarked, \"Only a man of ren is capable of liking and disliking people,\" in the sense of liking their good and disliking their bad conduct. However, \"if a person has the will (or firm determination) to become a man of ren, to that extent, he can be free from dislike on account of his misconduct\" (Analects 4.4). Xunzi gives us an elaboration of the degrees of jing feeling. \"The ren person must respect (jing \u656c) others. However, there is dao for respecting others. If the other is a worthy, then you should try to be close to him, honor him and respect (jing) him as well. If the other is unworthy, be fearful of him but keep him at distance, while showing him respect. The expression of respect is the same, but one's inner feeling is different\" (Li 1979: 298). Xunzi's remark seems to be an echo of Confucius' attitude toward gods and spirits. If you don't pay respects to them, you will probably incur the censure of the populace. Xunzi goes even further by employing the language of shen (gods), perhaps as an expression for the respect for current linguistic practice. He would probably agree with Hobbes that \"Words are wise men's counters; they do but reckon by them; but they are the monies of fools\"(Hobbes 1952: 29 with spelling modernized).\n\n### 3.12  Li-dependent Virtue: Rang \u8b93\n\nAnother important li-dependent virtue is rang \u8b93, which can be rendered in two different ways: \"to decline politely (tuici \u63a8\u8fad),\" and rang, as in Mencius' cirang zhi xin \u8fad\u8b93\u4e4b\u5fc3--the seed of the virtue of li--has to do with \"yielding\" (Mengzi, 2A6). In both cases, rang may be considered as an example of concern with gong \u606d. One should yield to others in some circumstances, say, in dealing with one's parents or elders, as one can respectfully decline their request. In the either case, as we shall see in Sect. 2.14, the exercise of reasonable judgment in accordance with yi \u7fa9 is a crucial determinant.\n\n### 3.13  Li-dependent Virtue: Wen \u6587\n\nPerhaps the most prominent dependent and constitutive li-dependent virtue is wen \u6587 (culture, cultural refinement). Wen is reported to be one of the four subjects of Confucius' teachings. \"The Master instructs under four heads: culture (wen), moral conduct (xing \u884c), doing one's best (zhong \u5fe0), and being trustworthy in what one says (xin \u4fe1)\" (Analects 7.25). For Confucius, the junzi is \"widely versed in culture but brought back to essentials by the li can, I suppose, be relied upon not to turn against what he stood for\" (Analects 6.27). Although the li is fundamentally a code of formal rules of proper conduct, apart from its connection with ren, it has an aesthetic aspect. Learning is for the sake of self-improvement, not for the sake of impressing other people (Analects 14.24). Xunzi would add, \"The junzi uses learning to beautify his own person (mei qi shen \u7f8e\u5176\u8eab). The small-minded man uses learning as a bribe to win attention from others.\"41 Implicit in the idea of wen is the beautification of character in the light of cultural refinement. The idea of wen, in the light of the connection of li with ren, in effect, appertains to the ennobling of the virtuous character of persons. Alternatively, wen expresses the ennobling function of li. In this light, the junzi is a \"beautiful\" person, as his life and conduct exemplify the \"beauty of virtue\" in an eminent way, reminiscent of the popular concern with \"the beauty of virtue and the deformity of vice\" among the British Moralists of the eighteenth century.\n\nAs a dependent virtue of li, a regard for wen, as Xunzi would put it, is \"to honor the roots\" of human existence. (Li 1979: 424) However, exaggerated emphasis on wen without regard to zhi \u8cea has dubious ethical value. Confucius said, \"When there is a preponderance of [native] substance (zhi \u8cea) over acquired refinement (wen \u6587), the result will be churlishness. When there is a preponderance of acquired refinement over substance, the result will be pedantry. Only a well-balanced admixture of the two do we have a junzi\" (Analects 6.18; see also Analects 12.19).\n\n### 3.14  Yi as the Virtue of Flexibility\n\nThe well-balanced admixture of native substance (zhi \u8cea) and cultural refinement (wen \u6587) does not indicate the ideal, for fundamentally yi \u7fa9 is the substance (zhi) of the ethical life (Analects 15.18). Yi is the Confucian virtue of flexibility. According to Confucius, the junzi, in his dealings with the world, \"is not invariably for or against anything. He is on the side of yi \u7fa9 (Analects 4.10). Recall also Confucius' autobiographical remark cited in the Introduction: \"I have no preconceptions about the permissible or impermissible (wu ke wu buke \u7121\u53ef\u7121\u4e0d\u53ef)\" (18.8). This freedom from predilection, prejudgment, inflexibility, and egotism is said to be characteristic of Confucius. \"There were four things the Master refused to have anything to do with: he refused to entertain conjectures or to insist on certainty; he refused to be inflexible or to be egotistical\" (Analects 9.4). These qualities may also be ascribed to the junzi, qualities which are necessary to maintain his freedom of thought and action in advance of encounter with particular problematic situations, although Confucius, perhaps out of modesty, disclaimed being a junzi. Confucius said, \"The dao of the junzi is threefold, none of which I was able to attain: being humane (ren \u4ec1), he is free from worries (you \u6182); wise (zhi \u77e5), from delusions (huo \u60d1); courageous, he is free from fear.\" Zigong said, \"What the Master has just quoted is a description of himself\" (Analects 14.28). On another occasion, Confucius said, \"In the knowledge of letters and the arts, I may perhaps compare myself with other men. But as for the character of a junzi who carries out in his personal conduct what he professes \u2013 that is something to which I have not yet attained\" (Analects 7.32, this is Ku's translation. Ku 1984: 33).\n\nIf yi is \"to square with\" external life of the junzi, then its primary function is to deal with matters external to the individuals, seen as demands or requirements that need to be made compatible with the concern of their inner life. These external demands may appear in the form of duties (yiwu \u7fa9\u52d9) imposed by societal custom or tradition, along with institutional rules and regulations, more generally, demands for compliance with li as a set of formal prescriptions for proper behavior. This sense of yi, which is functionally equivalent to li, is often rendered as \"duty.\" One example is the use of yi in the passage cited in Sect. 2.11. The Liji mentioned ten duties of human relationships (renyi \u4eba\u7fa9), such as \"The father's loving-kindness (fuci \u7236\u6148) the son's filial piety (zixiao \u5b50\u5b5d), gentleness on the part of elder brother (xiongliang \u5144\u826f), and obedience (shun \u9806) of the younger brother\" (Wang 1977 Vol.1: 300). The Li as a corpus of rules of proper conduct can be quite complex and burdensome even for committed persons. The vastness of the rules staggers our imagination. A chapter (Liqi \u79ae\u5668) in the Liji \u79ae\u8a18 mentioned three hundred \"great (da \u5927)\" or important rules (dali \u5927\u79ae) and three thousand \"smaller\" rules (xiaoli \u5c0f\u79ae), but points out that \"they all lead to the same thing.\" \"No one can enter an apartment but by the door.\" The junzi, in his consideration of the rules, finds those which are carried directly to practice; those in which one has to bend and make some modification; those which are regular and the same for all classes.\" (Legge 1966: 405)\n\nYi \u7fa9, in the sense of rightness, appropriateness or fittingness, would be the basis of modification of li. Moreover, the relevance of the li to the present, particularly in exigent situations, is a matter of reasoned judgment based on his sense of appropriateness or yi and appreciation of the regulative, supportive, and ennobling functions of li.42 Wherefore, the li are subject to revision or even elimination. Notably, as Zhu Xi points out, we must reject rules that are unreasonably onerous and superfluous, and retain rules that are practicable and essential to the maintenance of the social order.\n\nBecause of the complexity and the competence necessary to apply the li, ordinary persons do not have the time, the knowledge, nor interest in the mastery of li. The application of the great or important li, such as funeral rites and sacrifices, required ritual experts, who ideally possess a sense of yi. Confucius seemed to be quite aware of the problem of complexity and the rationales underlying the application of li when he said, \"The common people can be made to follow a path, but not to understand (zhi \u77e5) it\" (Analects 8.9).\n\nMoreover, the appropriate and reasonable determination of the application of li in accordance with yi must attend to the time and circumstance of the individuals. It is explicitly stated in the Liji: \"In [observing] the li, what is appropriate (yi \u5b9c) [for the time and circumstances] should be followed\"(Legge 1966 Vol. 1: 62\u201363); Wang 1977 Vol. 1: 3). Given a set of ritual rules, its application cannot be determined a priori. \"Should words be understood only in one way? Each saying has its own appropriate application.\"(Legge 1966 Vol. 2: 214; Wang 1977 Vol. 2: 609) Or to paraphrase Wittgenstein, ritual rules, like all rules, stand there like signposts for the guidance of our will and action.(Wittgenstein 1968, par. 85) There is always a possibility of doubt in actual cases, where one must interpret their concrete significance. In the application of li in particular, in its requirement of compliance with formal procedures, it is all the more important to pay heed to the circumstances of the individual. For example, the proper execution of formal procedures often requires the possession of material instruments that may go beyond the means of ordinary persons. The Liji is quite emphatic on the economical aspect of ritual observances: \"goods and wealth are not to be expected from the poor in their discharge of the rules of propriety (li \u79ae).\"43 It goes without saying that an enlightened knowledge of other factors that affect different individuals in different circumstances or the same individual in different circumstances is essential to the exercise of yi, which entails a judgment of what is right and reasonable in applying the li.\n\nIn sum, concern for yi is generally a concern for right conduct, which is deemed fitting or appropriate to a particular situation.44 However, one problematic area of conduct, to use Xunzi's expression, is our fondness for profit or personal gain (haoli \u597d\u5229). In Confucius' words, \"The junzi understands what constitutes right conduct (yi \u7fa9); the small-minded man understands what is profitable\" (Analects 4.16). Confucius said of himself that \"wealth and rank attained through unrightful means (buyi \u4e0d\u7fa9) have as much to do with me as passing clouds\" (Analects 7.16). In situations when we are tempted to do what promotes personal gain, Confucius would advise that \"when you see something that is conducive to attaining personal gain, you must think of right conduct (jian de siyi \u898b\u5f97\u601d\u7fa9)\" (Analects 14.12; 16.10), that is, whether the contemplated, self-serving act is the right thing to do. This contrast between yi and self-serving benefit suggests the Confucian distinction between morality and egoism. Perhaps it is the assumption of this contrast that accounts for common translation of yi as having to do with morality. For example, in the passages we just cited, Lau rendered yi \u7fa9 as \"what is moral\" (Analects 7.16) and buyi \u4e0d\u7fa9 as \"immoral means\" (Analects 14.12).\n\n### 3.15 Dependent Virtues of Yi \u7fa9: Kuan\n\nLet us consider some of the dependent virtues of yi as a virtue of flexibility. The idea of kuan \u5bec suggests catholicity and neutrality, which may be regarded as the main supportive and constitutive virtues of yi. For Confucius, the junzi is not an implement (Analects 2:12), which is fit for specific, and narrow purpose. \"Instead, he should have broad vision, wide interests, and sufficient ability to do many things\"(Chan 1963: 24). He is broad-minded and non-partisan (Analects 2:14; also 7:30), and aspires to higher and more valuable things in life (Analects 14:23). Alternatively, we may say that yi as a virtue of flexibility has kuan \u5bec as a supportive and constitutive virtue. Earlier, we mentioned that kuan, as a dependent, constitutive virtue of ren \u4ec1, is a virtue of magnanimity (Sect. 2.1). Understood as kuanhong \u5bec\u5b8f, kuan embraces also broadmindedness and generosity (Sect. 2.7), either in the form of liberality in giving or in the form of generous-mindedness, impartiality or fair-minded. Let us consider some of the dependent virtues of yi as a virtue of flexibility. The idea of kuan suggests catholicity and neutrality, which are the main supportive and constitutive virtues of yi. Earlier, we mentioned kuan, as a dependent, constitutive virtue of ren \u4ec1. For elaborating the complex notion of kuan as a constitutive virtue of both ren \u4ec1 and yi \u7fa9, we may appropriate Xunzi's conception of three desirable qualities of participants in argumentation. Xunzi says of the argumentative discourse of the scholars and junzi:\n\n> With a humane mind (renxin \u4ec1\u5fc3) he explains his ideas to others, with a learning mind (xuexin \u5b78\u5fc3) he listens to their words, and with an impartial mind (gongxin \u526c\u5fc3) he makes his judgment. He is not moved by the censure or praise of the mob; he does not try to bewitch the ears and eyes of the observers; he does not cringe before the power and authority of eminent men... He honors what is fair and upright and despises meanness and wrangling. (Watson 1963: 148\u2013149 emended)\n\n### 3.16  Kuan as a Composite Virtue\n\nA different way of indicating the virtue of kuan, in the light of Xunzi's remark and his distinction between generic (gong \u5171\u540d) and specific terms (bieming \u5225\u540d), is to say that kuan, a composite virtue, as a generic term, may be concretely specified in three virtues: humaneness (renxin \u4ec1\u5fc3), learning mind (xuexin \u5b78\u5fc3), and impartiality or fair-mindedness (gongxin \u526c\u5fc3). In the context of the exercise of yi \u7fa9, renxin expresses a concern for others as one's conduct may affect others. More especially in discourse, renxin would council the agent to be vigilant (shen \u614e) in using words that may hurt others' feelings, for the ren person has concern for others' feelings and well-being (Analects 12.22). Says Xunzi, \"Hurtful words engender wounds deeper than those inflicted by spears or halberds\" (Li 1979: 55). Xunzi would elaborate the close connection of ren and yi in the context of deploying arms.\n\n> Chen Xiao \u9673\u56c2 said to Xunzi, \"When you talk about the use of arms, you always speak of ren \u4ec1 and yi \u7fa9 as being the basis for deploying arms or soldiers. A humane person loves others (renzhe airen \u4ec1\u8005\u7231\u4eba), the righteous person acts in accordance with what is right and reasonable (yizhe xunli \u7fa9\u8005\u5faa\u7406). Why, then, do they have recourse to arms in the first place? Xunzi replied, \"This is not something that you would understand. The humane person does indeed love others, and because he loves others, he hates to see men do them harm. The righteous person acts in accordance with what is right and reasonable, and thus he hates to see them do wrong. He takes up arms, in order to put an end to violence and to do away with harm, not in order to contend with others for spoils.\"45\n\n### 3.17  Gongxin \u526c\u5fc3 as a Specific Virtue of Kuan\n\nAs gongxin \u526c\u5fc3, impartiality or fair-mindedness, is a specific virtue of kuan \u5bec, characteristic of the junzi's neutrality and catholicity, we shall attend to xuexin \u5b78\u5fc3, the learning mind (xuexin), which for Xunzi, in discourse, is the virtue of receptivity, the ability to listen to others without prepossession or prejudgment. This virtue is important, as it is an amplification of what it means to be an impartial person that displays kuan as generous-mindedness. To be impartial in discussion, one must have an open mind, a mind that is receptive to contrary opinions, even those that disagree with one's own, prior to assessment of their relevance or correctness. In the Analects, Confucius frequently stresses the importance of extensive study or learning (boxue \u535a\u5b78) and application (Analects 6.27, 1.1). Learning, however, as we stated in Sect. 2.13, is for the sake of self-improvement, not for the sake of impressing others (Analects 14.24). Quite in the spirit of Confucius' teaching, Zixia \u5b50\u590f said: \"As workmen labor in their workshops to learn their trade, so a junzi gives himself to study in order to attain the Way\" (Analects 19.7).\n\nLearning is not merely the passive absorption of textual materials or learning from one's own or other people's experiences, but must involve a certain amount of thinking or reflection (si \u601d). As Confucius said, \"Learning without thinking is labor lost. Thinking without learning is perilous\" (Analects 2.15). Similarly discussion is important in teaching, as indicated in Confucius' self-reflective remark: \"It is these things that cause me concerns: failure to cultivate virtue, failure to discuss (jiang \u8b1b) more deeply into what I have learned, inability, when I am told what is right (yi \u7fa9), to move to where it is, and inability to reform myself when I have defects\" (Analects 7.3). Confucius did not seem to council deep thinking in situations that require effective decision, as indicated in this report: \"Ji Wenzi \u5b63\u6587\u5b50 always thought three times before taking action. When the Master was told of this, he commented, \"Twice is quite enough\" (Analects 5.20). Presumably, \"thinking thrice\" is poor advice for dealing with matters at hand, which call for relatively fast and effective decision. Also, when habitual, \"thinking thrice\" may result in the separation of learning or knowledge. As Wang Yang-ming \u738b\u967d\u660e complained against the scholars of his time on the separation of learning or knowledge and action, pursuing them separately, thereby disjoining knowledge from action. This tendency of separate pursuits is likely to result in a sort of \"abulia,\" an incapacitation of the agent to will effectively or to make decisions leading to action.46 Of course, on occasion when the agent can afford the time and energy, he must deliberate (l\u0171 \u616e) about the distant future. Failure to do so may give rise to \"worries much closer at hand\" (Analects 15.12).47\n\n### 3.18 Supportive and Constitutive Virtue: Caution (Shen \u614e)\n\nIn addition to the learning mind, Confucius also stressed the importance of caution or circumspection (shen \u614e), dignity, seriousness, solemnity of manners (zhuang \u838a), and firmness, resoluteness, and steadfastness (gang \u525b).48 Shen \u614e is a supportive and constitutive virtue of yi \u7fa9, for caution in speech and conduct (Analects 4.24) is essential to the exercise of impartiality. It is also a supportive and constitutive virtue of ren \u4ec1, since it is implicit in the ideal of shu \u6055, which requires impartiality (Sect. 2.6). Gang \u525b or gangyi \u525b\u6bc5 (Analects 13.27) is an important supportive and constitutive virtue of yi for resoluteness in commitment to yi and the decisiveness in judgment according to yi is an indispensable prerequisite for its exercise. So also, gang, like shen, is a supportive and constitutive virtue of ren \u4ec1. Perhaps, the most important supportive and constitutive virtue of ren, li, and yi is yong \u52c7 (courage or valor). Great courage is required in the readiness of sacrificing one's life for the sake of ren. Confucius said, \"The determined scholar and the man of ren will not seek to live at the expense of ren. They will even sacrifice their lives in order to realize ren\" (Analects 15.10). Here we may say that yong is the courage to be a ren person. In the context that requires the exercise of yi \u7fa9, yong is the courage to do the right thing. As Confucius put it, \"Faced with what is right, to leave it undone shows a lack of courage\" (Analects 2.24). In sum, yong is an overlapping dependent, constitutive virtue of both ren and yi.\n\n### 3.19 Summary\n\nThe foregoing Sects. (2.1\u20132.19) present a map of the virtues of junzi, consisting of cardinal, interdependent virtues such as ren, li, and yi, and their dependent supportive and\/or constitutive virtues. The distinction between interdependent and dependent virtues is a heuristic device for sorting out the virtues. Alternatives such as that of Chen Daqi (Sect. 2) and Ye Jingkui \u8449\u7d93\u6842 are available for comparison.49 Note also that I made no claim as to completeness or to a sharp division of dependent virtues as belonging to one cardinal rather than another, for as we have pointed out, there are overlapping dependent virtues of ren and yi such as yong \u52c7 (Sects. 2.8 and 2.18) and kuan \u5bec (Sects. 2.7 and 2.15). There is a complementary way of grouping the dependent virtues suggested by Zhongyong \u4e2d\u5eb8, Sect. 27: honoring moral character (zun dexing \u5c0a\u5fb7\u6027) and following the path of inquiry and learning (dao xuewen \u9053\u554f\u5b78), much reminiscent of Aristotle's distinction of virtues of character, and virtues of intellect in Nicomachean Ethics.\n\nDependent virtues of ren and li are essentially the virtues of character, and those of yi, virtues of the intellect. Notably the virtues of the intellect are complementary to the virtues of character, and they comprise a few virtues not particularly emphasized by Aristotle, but the idea of phronimous or man of practical wisdom seems to be implicit in the idea of the exercise of yi, which may be elaborated by Xunzi's conception of zhil\u0171 \u77e5\u616e or wise and well-informed deliberation. Xunzi would also concur with Aristotle that persons who possess practical wisdom are those who have \"the capacity to do well about what is good and advantageous for oneself\" (Aristotle 1962: 1140a), for a well-informed and wise choice must be the outcome of careful and well-considered weighing of consequences in the light of benefits and harms to the agent.\n\nAlso, for developing an adequate Confucian ethics of virtue, there is the crucial task of elaborating its theoretical or practical significance, and presently, of dealing with difficult problems attendant to our discussion of yi as a virtue of flexibility, e.g., such problems as the role or status of ethical rules and principles, and the possible contribution of Confucian ethics as an ethics of character or junzi to contemporary virtue ethics, as well as deontology and consequentialism. I have dealt with some of the problems in various essays in Moral Vision and Tradition, especially in a reconstructed formulation of a Confucian moral philosophy, comprising the idea of the Confucian tradition, basic conceptual framework, and principles for dealing with intercultural ethical conflict as preconditions of adjudication. In the next two sections, I shall expand somewhat the ways in which the junzi deal with human conflict, and return to the problems of yi as a virtue of flexibility.\n\n## 4 Non-contentious Ways for Dealing with Human Conflict\n\n### 4.1 Human Conflict\n\nBecause of the concern for ren, or well-being of humanity, Confucius was well aware that human beings, as we know them here and now, are prompt to conflict or altercation concerning what they view as legitimate claims, especially regarding inheritance or division of properties within the family. Xunzi would later draw our attention to the problematic character of the original nature of our basic motivational structure as consisting of feelings and desires. Confucius' attitude toward human conflict seems implicit in his remark: \"The junzi is dignified (jin \u77dc), but not contentious (buzheng \u4e0d\u722d); he associates with other people, but does not join parties (budang \u4e0d\u9ee8)\"(Analects 15.22). \"The junzi does not engage in contention, except perhaps in archery. But when he enters the contest, he courteously makes a vow before he advances to take his place among the contestants; he does the same when he steps down after participating in the event; when he loses, he drinks his cup of forfeit. Thus in this single case of contention, he still shows himself as a junzi\"(Analects 3.8, Ku 1984: 10). As we shall see shortly, non-contentiousness is an indispensable precondition for employing the two methods for dealing with conflict of interests, for it contributes to the maintenance of impartiality and fairness (gong \u526c), essential to the exercise of yi \u7fa9 (Sect. 2.15), and later for Xunzi, a desirable quality of participants in ethical argumentation.\n\n### 4.2 The art of he \u548c\n\nThe Lunyu intimates two ways of dealing with human conflict: he \u548c and zhong \u4e2d or zhongyong \u4e2d\u5eb8. Confucius said, \"The junzi is concerned with he \u548c, but not \u540c tong\" (13.23). He means \"harmony.\" Elsewhere, Youzi \u6709\u5b50remarked that the most valuable function of li \u79ae is to promote he (harmony) (Analects 1.12). This emphasis on the harmonizing function of li is instructive in that the prescriptions of li are primarily designed as schemes of mutual accommodation for securing orderly and harmonious conduct. We can read he \u548c in our passage as tiaohe \u8abf\u548c, \"to mediate or to reconcile.\" Confucius' saying may thus be rendered as \"The junzi is conciliatory, but does not identify with others; the inferior man identifies with others, but is not conciliatory\" (Analects 13.23).50 In the sense of tiaohe, the junzi is concerned with reconciling and harmonizing differences in cases of conflicting interests or values among people. This suggests that the junzi would deal with human conflict by the method of mediation or arbitration rather than the method of adjudication or litigation.\n\nPreference for arbitration is reflected in Confucius' attitude toward litigation: \"In hearing litigation, I am no different from any other men. But if you insist on a difference, it is, perhaps, that I try to get the parties not to resort to litigation in the first place\" (Analects 12:13). In the Xunzi, we have presumably an account of Confucius' handling of a case while serving as a justice minister (sifa da chen \u53f8\u6cd5\u5927\u81e3) in Lu \u9b6fstate. The case is a litigation of a father against his son, presumably for lack of filial piety (xiao \u5b5d). Confucius confined the son in prison and for three months did not decide the case. When the father requested permission to stop the proceeding, Confucius released the son from prison. When the head of Ji \u5b63 family heard about the incident, he was piqued, saying, \"The revered old man has deceived me. I was told that in ruling the country the ruler must advocate the way of filial piety. Now he should execute the unfilial son to set an example, instead he released him from prison.\"\n\n> When [the disciple] Ranzi \u5189\u5b50 related this to Confucius, he sighed deeply and exclaimed, \"Alas! When superiors fail to execute subordinates on account of it \u2013 is that proper! Not having instructed the people and yet to decide criminal prosecutions against them is to kill innocent people... When all living things have their seasons, to make exactions without regard to the season constitutes oppression. Not to instruct the people, yet to require from them completion of allotted tasks constitutes cruelty. It is only when these three practices have been ended that punishments may be considered.\" (Li 1979: 642; Knoblock 1990: 2: 28.3)\n\nIt is also alleged that Confucius was \"good at persuading people not to litigate,\" while serving in the same official capacity in the Lu state. \"Mutual rang \u8b93 (yielding, making concessions and compromises) was practiced among the people\" (Chen 2003). It is possible that Confucius appealed to their sense of shame (chi \u6065) as an internal monitor. Recall Confucius' saying: \"If in government you depend upon laws and enforce the laws by meting out punishments, you may keep the people from wrongdoing, but they will lose the sense of shame (chi \u6065). If, on the other hand, in government you depend on virtues and maintain order by encouraging the rites (li \u79ae) the people will have a sense of shame for wrongdoing and, moreover, will emulate what is good\" (Analects 2.3). Presumably the appeal to people's sense of shame is well exemplified in the rule of Zhou Wen Wang \u5468\u6587\u738b.\n\n> He [Wen Wang] ruled so well that the peasants made compromises with one another regarding land boundaries, and the people respected and took care of the elderly. Nobles of feudal domains who had disputes with one another would go to Wen Wang for arbitration. In one case, when the disputing parties arrived at Zhou territory, they were so impressed by the harmony and mutual forbearance of its people that they felt ashamed of themselves, realizing that the Zhou people would consider a shame to quarrel over the kind of matter, which they were now quarrelling. They therefore decided immediately and entered into a compromise. (Chen 2003)\n\nAs I stated elsewhere, at issue in arbitration is an impartial resolution of disputes oriented toward the reconciliation of the contending parties in the light of the concern for harmonious human intercourse. The arbitrator, chosen by the parties in dispute, is concerned with repairing the rupture of human relationship (lun \u502b) rather than with deciding the rights or wrongs of the parties. The task of an arbitrator is not only to interpret the meaning of a current practice, but also to shape the expectations of the contending parties along the line of mutual concern, to get them to appreciate one another as interacting members in a community. Albert Chen points out, \"The Chinese terms for this [Confucian, official] practice is called tingsong \u807d\u8a1f (beseeching the parties to drop litigation) and xisong \u606f\u8a1f (dissolving the litigation). The ultimate aim is the reconciliation of the disputants to each other and hence the restoration of the personal harmony and social solidarity that have been temporarily breached by the conflict\" (Chen 2003). This practice is reminiscent of Xunzi's jianshu (art of accommodation) (see Cua 1985: 11\u201312).\n\nIn sum, the difference between litigation and arbitration is a difference in orientation toward persons in conflict. The former directs to the problem or issue that beset the parties in dispute, and aims at resolution of conflict. Conflict of interest is seen to be a problem or issue to be resolved. The latter directs attention to the persons in conflict and aims at a reconciliation of the persons involved. The art of he \u548c is thus an art of reconciliation, and not a method of conflict resolution between contending parties.\n\n### 4.3 The Art of Zhong \u4e2d: Striking a Balance\n\nComplementary to the art of he or reconciliation is the art of zhong \u4e2d or zhongyong \u4e2d\u5eb8. The idea of zhong as the mean between extremes, reminiscent of Aristotle's doctrine of the mean, is fairly indicated in this dialogue in the Analects (11.16):\n\n> Zigong \u5b50\u8ca2 asked, \"Who is superior, Shi \u5e2b or Shang \u5546?\" The Master said, \"Shi overshoots the mark; Shang falls short.\" \"Does that mean that Shi is in fact better?\" The Master said, \"There is little to choose between overshooting the mark and falling short (guo you buji \u904e\u7336\u4e0d \u53ca).\"\n\nMore explicit on the mean is Confucius' remark: \"Supreme indeed is the Mean (zhong \u4e2d) as a virtue. It has been rare among the common people for quite a long time\" (Analects 6.29). The nominal use of zhong \u4e2dmeans \"the center,\" thus it may be construed as the mean between excess and deficiency as implicit in our first passage (Analects 11.6). The verbal use of zhong suggests, \"hitting the mark.\" It is significant that the binomial shizhong \u6642\u4e2d couples zhong with shi \u6642, implying that hitting the mark is a matter of timing \u2013 a key concept in Yijing \u6613\u7d93 and a prominent theme in Song Confucianism (see Cheng 2003b). Notice that without self-cultivation, learning, and experience, hitting an appropriate target is likely to be a hit-and-miss affair. Whether one has hit the target in an occurrent situation at a particular time and place is, as Mencius points out, a matter of the exercise of quan \u6b0a. Thus, in the last analysis, tiaohe \u8abf\u548c and zhong \u4e2d, as ways of dealing with human conflict, are guidelines, rather than decision procedures. Below we take up the concept quan as a defining characteristic of the virtue of flexibility of the junzi.\n\n## 5 Exercise of Quan \u6b0a in Accord with Yi \u7fa9\n\n### 5.1  Quan and Jing\n\nThis key Confucian concept may be rendered as \"weighing of circumstances,\" \"exigency,\" or \"moral discretion.\" Understanding quan depends on its contrast with jing \u7d93 (the normal or standard).51 This distinction reflects a similar Confucian concern with chang \u5e38 (the constant) and bian \u8b8a (the changing). Indeed, the latter is indispensable to elucidating the former. As Zhu Xi put it, \"Jing pertains to the constant aspect of dao, and quan to the changing aspect of dao)\" (Zhu 1962: 6:1a).\n\nTranslation of quan by \"expediency\" would be misleading if it were to suggest the agent's concern for self-serving purpose rather than a concern for what is appropriate or proper under the circumstances. As we have seen (Sect. 2.14), expediency, in the first sense, is contrary to the Confucian concern with yi \u7fa9 (rightness, righteousness). In the second sense, however, it is functionally equivalent to the other renderings, that is, weighing (the importance or unimportance) of circumstances and discretion. D. C. Lau's \"moral discretion\" is perhaps the best rendering of quan in a perplexing passage in Analects 9:30.\n\nReflection on the nature of moral discretion may provide us with a systematic way of dealing with Zhu Xi's preoccupation with the jing-quan distinction. In fact, as Wei Cheng-t'ung (Wei Zhengtong \u97cb\u6b63\u901a) points out, Zhu Xi's concern was in part motivated by his students' repeated query on the notion of quan in Analects 9:3052:\n\n> The Master said, \"A man good enough as a partner in one's studies need not be good enough as a partner in the pursuit of the Way [dao]; a man good enough as a partner in the pursuit of the Way need not be good enough as a partner in a common stand; a man good enough as a partner in a common stand need not be good enough as a partner in the exercise of moral discretion [quan]\" (Analects 9.30).\n\nThe exercise of quan is quite properly an exercise of discretion in the sense of the power of the individual to act according to his or her judgment in dealing with uncertain, exigent situations, or \"hard cases.\" As contrasted with the \"soft cases\" or normal problems in human life, the hard cases are rule-indeterminate, thus the established standards of conduct (jing) offer no clear guidance. Even when such standards are deemed appropriate, there may be a problem of application, which calls for interpretive judgment and discretion. The problem cannot be resolved by some mechanical or deductive procedure.\n\n### 5.2 Constraints for Discretion\n\nMost moral and legal traditions thus allow for the exercise of discretion, though such an exercise is always subject to constraints. In the Confucian context, the constraints comprise the li, the operative ritual rules or formal prescriptions of proper conduct. Arguably, Confucius has a recurrent interest in discretion as an indispensable means for coping with the hard cases of moral life. For that reason, he stressed the virtue of flexibility of junzi (Sect. 2.14). Recall again his autobiographical remark: \"I have no preconceptions about the permissible or impermissible\"; and the student's description of his character: \"There were four things the Master refused to have anything to do with: he refused to entertain conjectures or insist on certainty, he refused to be inflexible or to be egotistical\" (Analects 18:8, 9:4).\n\nIn two different ways, the focus on yi (rightness) especially brings out the moral aspect of quan or discretion. First, yi is contrasted with personal gain or self-serving interest. Secondly, yi focuses on doing the right thing as determined by a judgment of the relevance of moral rules to particular circumstances (Sect. 2.14). More importantly, the exercise of yi is required in dealing with changing, exigent situations of human life. Xunzi's emphasis on the use of yi in varying one's response to changing circumstances (yiyi bianying \u4ee5\u7fa9\u8b8a\u61c9) echoes the same concern.53 We would expect an adequate account of the jing-quan distinction to give a pivotal role to yi, since the exercise of quan or discretion is fundamentally an exercise of yi.\n\nThe need for discretion in the interpretation of li or ethical rules can in part be accounted for by the open texture of natural languages. That is, there is always a possibility of vagueness in the empirical and practical application of words in relation to the natural and the human world (Waismann 1952). More fundamentally, the need for discretion arises out of two deficiencies of humanity: \"our relative ignorance of facts\" and \"indeterminacy of aim\" (Hart 1961, Chap.\u200b 7, Sect. 1). In formulating and\/or modifying rules of conduct, we rely on our tradition and experience. We cannot always foresee the consequences of the enforcement of established rules nor anticipate without error our future situations, especially those that are exigent, demanding immediate attention and action. In these cases, discretion is necessary, given the human predicament. Thus, Aristotle considers the subject matter of ethics as one that cannot be treated with exactitude, and that \"the truth\" can only be indicated \"with a rough and general sketch.\" In the case of legal justice, there is a need to supplement it by equity or reasonableness (epieikeia) as a corrective (Aristotle 1962: 1094b).\n\n### 5.3  Quan as a Holistic Idea\n\nAs indicated by his preference for voluntary arbitration or mediation over adjudication in settling disputes within the community, and by his recurrent emphasis on the importance of being a junzi or paradigmatic individual (Sect. 3), Confucius would concur with Aristotle on the role of equity or reasonableness in human affairs. The exercise of quan, moral discretion, is also necessary in the light of Confucian dao as a holistic ideal. This ideal of the good human life as a whole is more of a theme than a norm (Sect. 2.3). However, this ideal is a topic of communal discourse, somewhat analogous to a theme in literary or musical composition. Discretion is essential especially in specifying practical objectives to be pursued in the hard cases of the moral life. It is this \"indeterminacy of aims\" in conjunction with human fallibility or \"ignorance of facts\" that renders such moral discretion (quan) unavoidable.\n\nHence, if the Confucian agent is to cope with exigent, changing circumstances in the course of pursuing the dao, he or she must be disposed to exercise quan. This is perhaps the force of Zhu Xi's saying, \"This substance of dao is vast and inexhaustible (zhe daoti hao-hao wu qiongzhong \u9019\u9053\u9ad4\u6d69\u6d69\u7121\u7aae\u7d42)\" (Zhu 1962: 8:1a). Wang Yang-ming more concisely put it: \"Dao cannot be exhausted with finality (dao wu chongqiong \u9053\u7121\u7d42\u7aae)\" (Chan 1963: 46), that is, the concrete significance of dao cannot be specified with any claim to finality. Any attempt to do so must be informed by a sense of timeliness (shih \u6642) to attain the Mean or equilibrium (zhong \u4e2d), aiming at doing the right thing (yi \u7fa9) in accord with the agent's judgment of what a particular situation demands. Perhaps, this is the basis for Mencius' remark that Confucius was \"the sage whose actions were timely\" (Mencius, 5B:1; 2A:2). The specification of the concrete significance of dao must be a reasoned or principled one. The exercise of quan or moral discretion is thus constrained by the exercise of practical reason.\n\nAs in the study of the classics, so in the proper exercise of quan or moral discretion, an open mind is required (Sects. 2.15\u20132.16). While the objective is to achieve timely equilibrium (shizhong), the Confucian tradition must not favor any one doctrine of interpretation, even if it happens to be a moderate position between extremes. As Mencius says:\n\n> Holding on to the middle [zhong \u4e2d] is closer to being right, but to do this without moral discretion [zhizhong wu-quan \u57f7\u4e2d\u7121\u6b0a] is no different from holding to one extreme. The reason for disliking those who hold to one extreme is that they cripple the Way [dao \u9053]. One thing is singled out to the neglect of a hundred other. (Lau 1970, 7A:26, emended)\n\nMoreover, Xunzi would advise the Confucian agent to consider carefully all the salient features of the situation before pronouncing judgment on his current desires and aversions. In his words:\n\n> When one sees something desirable, he must carefully consider (l\u00fc \u616e) whether or not it will lead to detestable consequences. When he sees something beneficial, he must carefully consider (l\u00fc) whether or not it will lead to harmful consequences. All these consequences must be weighed together (jianquan \u517c\u6b0a) in any mature plan before one determines which desire or aversion, choice or rejection, is to be preferred. (Ibid.)\n\nIn sum, quan (1) essentially pertains to assessment of the importance of moral considerations to a current matter of concern. Alternatively put, the exercise of quan consists in a judgment of the comparative importance of competing options answering to a current problematic situation. (2) The situation is such that it presents a hard case, that is to say, a case falling outside the normal scope of the operation of standards of conduct. (3) Quan is an exercise of moral discretion and as such must conform to the requirement of yi \u7fa9 (rightness). (4) The judgment must accord with li (principle, reason), that is, be a principled or reasoned judgment.\n\n### 5.4  Yi as a Virtue of Flexibility\n\nIn the light of the foregoing discussion of quan, we can appreciate better yi as a virtue of flexibility of the junzi. A junzi may thus be said to be a reasonable, rather than a rational, person. Arguably, there is an important conception of reasonableness in Confucian ethics, especially as it is exemplified in Wang Yangming. Some of the characteristics of reasonable persons are already indicated in the dependent virtues of yi, such as gongxin \u526c\u5fc3, impartiality or open-mindedness, extensive concern for others as renxin \u4ec1\u5fc3, and sense of appropriateness implicit in the virtue of yi. And, given its orientation toward ethical practice, as a reasonable person, a junzi would appreciate the actuating import of the cognitive content of learning and inquiry, in Wang Yangming's words, the unity of knowledge and action (zhixing heyi \u77e5\u884c\u5408 \u2013).\n\nSumming up, the notion of junzi is Confucius' ideal of a paradigmatic individual, which functions a guiding standard for practical conduct. In Confucius' view ordinary moral agents may not attain to sagehood (sheng). However, they can look to a junzi for guidance and may become junzi themselves. The notion of junzi, though an exemplar for practical conduct, is not an ideal of a perfect man, but an ideal of an ethically superior person who embodies the various virtues we have discussed. Plausibly, the notion of junzi in Confucius' thought serves as a mediating schema between his ideal of dao or ren. Historical characters such as Yao and Shun serve a similar function. In the classical Daoism of Laozi and Zhuangzi, fictional characters serve the same purpose in mediating the abstract ideal and the actual world.\n\nMore importantly, construed as an ideal of the paradigmatic individual, the junzi provides a standard of inspiration, rather than aspiration, that is, an embodiment of an ideal theme that functions more like a beacon of light than a norm of conduct (Sect. 2.1). For this reason, the ideal of junzi does not imply any given set of principles or rules. His freedom of judgment lies in his exercise of discretion (quan) especially in exigent situations. However, quan must be exercised with caution, circumspection, informed by knowledge, experience, and reflection. The judgment so reached is defeasible and the moral agent, when challenged, must vindicate himself by engaging in ethical argumentation.54 Arguably, Confucius' notion of junzi expresses the idea of paradigmatic individuals as exemplary embodiments of the spirit and vitality of the tradition. In addition to their role in moral education, they also serve as living exemplars of the transformative significance of the ideal of the tradition, thus invigorating the tradition.\n\nBoth Xunzi and Zhu Xi emphasize the role of inspired and insightful persons regarding classical learning. This emphasis somewhat echoes Confucius' notion of junzi or paradigmatic individuals, that is, persons who through their life and conduct embody ren, li, yi and other ethical excellences (Sect. 2). The junzi are the exemplars of the actuating significance of the Confucian tradition, serving as standards of inspiration for committed agents. These paradigmatic individuals are generally ascribed an authority by their contemporary adherents or posterity. The authority, however, is based on the acknowledgement of their superior knowledge and achievement, and especially their exercise of yi in interpretive judgments concerning the relevance of rules in exigent situations, or in Xunzi's words, on dao and de (virtue). They are the chengren \u6210\u4eba, the ethically accomplished individuals (see Introduction). However, while possessed of an authoritative ethical status, as standard bearers and interpreters of culture and tradition, the junzi are not arbiters of moral disputes, for their interpretations of the Confucian tradition represent individual efforts toward \"the repossession or reconstitution of dao\" (de Bary 1991a: 9), thus essentially contestable and subject to the reasoned assessment of their contemporaries and posterity.\n\n## 6 Conclusion\n\nFor concluding this long study of the virtues of junzi, let me briefly remark on some problems that call for further exploration.55 In our map for the virtues of junzi, in the distinction between basic, cardinal, and interdependent virtues and dependent, supportive\/constitutive virtues, it may be said that the unity of virtues is presupposed without argument. This is a difficult issue that deserves extensive discussion. In studying this issue in Xunzi's moral philosophy in two papers in the 1980s, I proposed what I called the completion thesis. Concisely stated, this thesis is that ren, li, and yi are interdependent concepts, for an adequate explication of one must involve the other concept. This thesis pertains to ideal unity of virtues, since ren, in the broad sense, is an ideal theme. Given the interdependence of these cardinals, the ideality of ren will also pervade through the ennobling function of li and the exercise of yi as exemplified in renxin \u4ec1\u5fc3 or humane mind. The actual conflict between dependent virtues such as xiao \u5b5d to parents and zhong \u5fe0 to the state, historically for the people has been a difficult one. Confucius' preference for the former is indicated in a well-known, but highly perplexing and disturbing passage about a son's concealing his father's misconduct and a father concealing the misconduct of his son as constituting uprightness (zhi \u76f4) \u2013 an attitude that markedly differs from that of Euthypro in Plato's dialogue of that name (Analects 13.18).\n\nAt issue is the case of stealing sheep. In the light of yi \u7fa9, a committed Confucian may differ from Confucius, as Xunzi would most likely council the agent to follow yi, rather than the father's wish from avoiding prosecution. Perhaps the agent at issue, were he to follow Confucius, could say that there are grounds for his decision to conceal his father's misconduct. First, it is a reasonable attempt to resolve a difficult and perplexing situation, a decision to do what is right as he sees it. He may even use the language of \"ought,\" as suggested by Fung Yulan, who construed yi as \"oughtness\" of the situation (Fung 1950: 42). However, the \"oughtness\" of the situation, though a characteristic of obligatory actions, has its central focus on the right act as appropriate to the particular situation that a moral agent encounters. Secondly, the decision rests on ren in the light of Confucius' saying: \"The ren person is at home in practicing ren (renzhe anren \u4ec1\u8005\u5b89\u4ec1)\"(Analects 4.2). More clearly, Mencius said, \"The ren person is man's peaceful abode (ren, ren zhi anzai ye \u4ec1, \u4ec1\u4e4b\u5b89\u5b85\u4e5f) and yi its proper path (yi, renlu ye \u7fa9, \u4eba\u8def\u4e5f).\" The question is, Can I live with the decision to inform on my father? Can I feel at home with the decision without self-reproach? Of course, when challenged, the agent must vindicate himself against imputation of wrongness. And thirdly, perhaps more fundamental, harmony (he \u548c) in the family is always a first consideration (Sect. 3.2). In short, to an agent in a hard and\/or exigent circumstance, he may find no pre-existing guidelines. Wang Yangming comments on the case of Shun's \u821c decision to marry the emperor's two daughters without informing his parents:\n\n> As for Shun's marrying without first telling his parents, was there someone before him who did the same thing and served as an example for him, which he could find out by looking into certain records and asking certain people, after which he did as he did? Or did he search into his liangzhi \u826f\u77e5 [his sense of right and wrong] in an instant of thought and weigh all factors as to what was proper, after which he could not help doing what he did? (Chan 1963: 109\u2013110)\n\nFor Shun, there are no previous cases of paradigmatic agents' decisions to guide his decision, nor is there any existing rule, say, a priority rule or principle that determines a right decision over conflicting claims of filiality and marital relations. All he can do is to rely on his liang-chih, his sense of what is right and good, weigh a variety of factors (quan \u6b0a), and arrive at a judgment based on these factors. For the exercise of quan in hard cases of the moral life, there are no fixed rules; although the decision and judgment involved are subject to argumentation. The agent does his best to arrive at a reasonable decision, while fully aware that it may turn out to be a bad one.\n\nIn this connection, our thesis on the exercise of quan in accord with yi may appear problematic even for one sympathetic to Confucian rather than Kantian ethics. While appreciative of my stress on arbitration for dealing with human conflict, David Wong nevertheless thinks that I underemphasize adjudication \"to be found in Confucius, Mencius, and Xunzi.\" He gives no citations in support of this claim. In an accompanying note to his text, he states:\n\n> It would seem that the very concepts of yi (righteousness) and ren (when it connotes the necessity of expressing respect and concern for others) would have to involve a judgment that certain kinds of actions are simply wrong \u2013 that an action done purely from profit and purely to humiliate another person is simply wrong, for instance.(Wong 2004: 48n33)\n\nI very much appreciate Wong's comment as it affords an occasion for further clarification. Consider the first example of an action done purely from profit. Such an action, by itself, is not wrong or objectionable, because the question of yi arises only when it appears, in the language of Lunyu 7.16, to be a case of buyu \u4e0d\u7fa9. Recall again Confucius' saying, \"When you see something that is conducive to attaining personal gain, you must think of right conduct (jian de siyi \u898b\u5f97\u601d\u7fa9)\" (Analects 14.12; 16.10). (See Sect. 2.14.) The question that could be raised is when there is a suspicion that the action made use of unrightful means (buyi) to obtain profit.\n\nAs regards the second case of an action done \"purely to humiliate another person,\" I quite agree with Wong that the action is wrong, except that I would prefer a different way of stating the point. The action quite clearly belongs to the ambit of the practice of ren, which calls for refrain from harm doing. Given the interdependence of ren and yi, certainly, a junzi would refrain from such an action, but for quite different reasons. Consider an act \"done purely to humiliate another,\" a junzi may agree or even say that it is \"simply wrong.\" Recall that shu \u6055 is a dependent, constitutive virtue of ren (Sect. 2.3). Refrain from harm doing is constitutive of the conduct of a ren person who has an affectionate concern for his fellows (Sects. 2.16 and 2.17). However, \"wrong,\" in this example, functions more like a reminder of an act not to be done, not a discriminator, for it provides no information about the nature of the act. It must be admitted, however, that the use of \"simply wrong\" serves only as an emphasis of the kind of action that a junzi would not perform. To use the language of Mencius: the person who humiliates another for its own sake must be buren \u4e0d\u4ec1 (Mencius 4A1), a completely heartless or uncaring person. For that reason the action is ethically unacceptable.\n\nIn the light of the foregoing discussion, we can now see better that converting the ideality of the unity of virtues into the actuality of the practice of the virtues is not a theoretical task. In spirit, our thesis on the interdependence of the cardinals is akin to that of J. L. 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As we shall see later, the idea of yi (rightness, righteousness) as contrasted with profit or personal gain or advantage is a central tenet of Confucian ethics and plays a key role in the exercise of quan \u6b0a (moral discretion).\n\n3\n\nThis is Liu Baonan's \u5289\u5bf6\u6960 explanation cited by Mao Zishui \u6bdb\u5b50\u6c34, see Mao (1977: 223).\n\n4\n\nQian Mu \u9322\u7a46 points out that the text indicates two sets of standards, higher and lower, as exemplified by persons past and present. This is puzzling, since the so-called higher standard exemplified by men of the past did not include ren. Perhaps, as an occasional dialogue, Confucius thought that the answer is a sufficient response to Zilu, who was more impressed with external than internal merits, the focus of the second set. However, the \"lower\" set of standard, construed as prerequisites for the achievement of the higher, focuses more on the internal psychological attributes of the agents or, in Alasdair MacIntyre's words, the internal goods of the ethical practice. Note also that Xunzi gives a more elaborate description of chengren in terms of quan cui \u5168\u7cb9 (integrity and purity) in Quanxue pian \u52f8\u5b78\u7bc7 19. Reference to Xunzi text is based on Li Disheng's annotated edition.\n\n5\n\nFor more discussion, see Cua (1998b), Chaps.\u200b 7 and .\n\n6\n\nThis chapter is an update of previously published materials presented in two entries: Junzi: The Moral Person\" and \"Quan: Moral Discretion\" in Cua 2003b and other earlier writings. The aim is to present a coherent ethics of junzi focusing on the virtues of character and virtues of flexibility along with the supportive and constitutive virtues and the problem of conflict resolution.\n\n7\n\nFor a comparative study of the notion of shame in Xunzi and Aristotle, see Cua (2003); also Cua (2005, Chap.\u200b 8).\n\n8\n\nFor further elaboration of ren as a vision of human community, see Cua (1984); or Cua (2005, Chap.\u200b 11).\n\n9\n\nRecently Kwon-loi Shun proposed the translation of yi as propriety. Legge, however, uses this term for translating li. Both these translations bring out the concern of both yi and li with proper conduct, with what is proper and fitting to the occasion, but they differ in orientation. Yi is situation-oriented, li, on the other hand, is rule-oriented. For further discussion, see Cua (1978: 69). More extensive discussion is given in Cua (1998b, Chap.\u200b 14).\n\n10\n\nThe distinction between basic and dependent virtues is not the distinction between basic and subordinate virtues mistakenly attributed to me by Schoper, citing my earlier paper \"Hs\u0171n Tzu and the Unity of Virtues\" (Cua 1987). See Schoper 2000. For elaboration of the relation between basic, interdependent virtues and dependent virtues, see Cua (1998b, Chap.\u200b 13).\n\n11\n\nIn Cua (1978), I considered ren as an internal criterion of morality and li as the external criterion. Since the application of li as rules of propriety is determined by yi, yi can also be regarded as an internal criterion, as it is an exercise of judgment concerning the applicability of li. Moreover, \"just as jen [ren] cannot be practiced without li, or the cultural setting, ren cannot be realized without i [yi], or the judgment of the relevance of jen and li in concrete situations of moral performance\" (Cua 1978: 51\u201357, 67\u201369). In Cua (1998b), based on a modification of Chen Daqi's work on the Analects (Chen 1977), I discussed the criteria for determining the central or fundamental concepts in the Analects.\n\n12\n\nSee Cua (1989b); incorporated and expanded in Cua (1998b, Chap.\u200b 13).\n\n13\n\nChen Daqi elaborates ren and yi as constituents of de: \"The core of ren is ai \u611b (affectionate concern), thus ai as the main concern of ren. The fundamental nature of yi is appropriateness (yi* \u5b9c), thus appropriateness is the main concern of yi. Consider xin \u4fe1 (trustworthiness or being true to one's words). Because of affectionate concern for people, one will not allow people to be deceived. One's words must be suited to the action, and action must be suited to the words. This is the core of ren. For cogency (zhongken \u4e2d\u80af) and for the sake of obtaining good results, one should adhere to xin only if such adherence is appropriate and should not adhere to xin if such adherence is inappropriate. This is the fundamental nature of yi.\" Chen goes on to distinguish ren and yi from particular virtues by way of the distinction between complete virtues (quande \u5168\u5fb7) from partial or incomplete virtues (\u504f\u5fb7). The former are said to be \"perfect virtues free from any defects whatsoever. If a virtue has the ren element but does not possess the yi element, it can only be called a partial virtue\" (Chen 1977: 230). Chen's distinction is quite different from my distinction between basic interdependent and dependent virtues, for at issue is not completeness or possession of both ren and yi, but the ethical significance of particular virtues. In other words, in the absence of the connection to ren and yi, particular virtues may have non-ethical values and may well be commendable from the prudential point of view, provided of course, they are not exercised contrary to ren and yi. Nevertheless, Chen's distinction, as we shall see shortly, is useful for elaborating my distinction between interdependent and dependent virtues.\n\n14\n\nHere I adopt Lau's felicitous terminology of \"constitutive means\" as contrasted with instrumental means (Lau 1970: 245). In Western ethical theory, the more familiar terminology is \"contributory\" as contrasted with instrumental means. For an insightful discussion, see Lewis (1946, Chap.\u200b 16).\n\n15\n\nMore elaborate is Mencius's stress on constant heart (hengxin \u6052\u5fc3) as one crucial deficiency that accounts for moral failure, owing perhaps to (a) the enticement of personal gain without considering the relevance of yi, or (b) failure in preserving moral integrity. Says Mencius, \"Only a junzi can have a constant heart (hengxin) in spite of a lack of constant means of support. The people, on the other hand, will not have constant hearts if they are without constant means. Lacking constant hearts, they will go astray and fall into excesses, stopping at nothing\" (Mengzi, 1A7).\n\n16\n\nMy translation. Note here the Confucian view that morality involves sacrifice of one's own interest even to the extent of accepting death.\n\n17\n\nAmerican Heritage Dictionary of the English Language (1969).\n\n18\n\nFor the distinction between ideal norm and ideal theme, see Cua (1978, Chap.\u200b 8).\n\n19\n\nIn this translation, I read mei as mei de \u7f8e\u5fb7, ethically admirable qualities or virtues.\n\n20\n\nThe interpretation proposed below is a reconstruction that makes no claim to being faithful to the original text. It draws some materials from Cua (1984) and Cua (1995). Notably I discuss zhong and shu as distinct, supportive and constitutive virtues of ren. The interpretation does not deal with zhong-shu as a pair, thus leaving open the interpretative issues. For a brief critical survey of different interpretations of zhong and shu as a related pair, see Nivison (2003, 1996).\n\n21\n\nSee Lau (1979: 15). Note that Confucius occasionally paired zhong and xin \u4fe1 (trustworthiness). Xin is also an important dependent virtue. For an infor mative, historical survey, see Shun (2003). For an excellent analytical study of shu, see Mou (2004).\n\n22\n\nAdopting this definition implies no commitment to Royce's conception of \"loyalty to loyalty\" as a supreme good. See Royce (1920: 16\u201317).\n\n23\n\nFor translation of jinxin as \"doing the best one can, being true to one's nature;\" occasionally \"being true to oneself,\" see Gardner (2003: 206, 157).\n\n24\n\nPresumably, these passages are partly the basis for Nivison's view that zhong be construed as \"loyalty\" as expressing the standard governing the conduct of an inferior to a superior or to an equal.\n\n25\n\nFor this reason, Chen Daqi endorses Zhu Xi's interpretation of zhong as jinzi \u76e1\u5df1. This interpretation is plausible when we draw attention to its ethical basis in ren. See Chen (1977: 236\u201337).\n\n26\n\nSee Li (1979: 651). Also, cf. Analects, 4.18: The Master said, \"In serving your father and mother you ought to dissuade them from doing wrong in the gentlest way. If you see advice being ignored, you should not become disobedient but remain reverend. You should not complain even if in doing so you wear yourself out.\"\n\n27\n\nAn alternative interpretation of zhong is to focus on li \u79ae. Bo Mou maintains: \"The primary meaning of 'zhong' in the Analects, especially when it is used together with 'shu' to unify Confucius's ideas as a whole, is a moral agent's sincere and devoted commitment to one's responsibilities and duties specified by the li (the ritual rules) or those culturally and historically established social institution; the implementation of such sincere and devoted commitment as a virtue can be towards any involved moral recipients, no matter to whom \u2013 regardless of the social status of the moral recipient involved\" (Mou 2004). Admittedly many passages in the Analects, particularly those involving yan (speech) would support Mou's interpretation, as he ably discussed in his excellent paper. Still, it is a puzzle how this account can explain a couple of passages involving zhong and xin \u4fe1 without any reference to li, e.g., Analects 1.8 (repeated in 9.25) and 12.10, where on a couple of occasions Confucius recommended zhong and xin as the master concerns or main guides to conduct. Even if the li are relevant, one would expect the sense of yi \u7fa9 to be a determining consideration for the appropriate application of particular ritual rules. See Sect. 2.14.\n\n28\n\nThis is Lau's gloss. Lau continues: \"It interesting to note that when Tzu-kung [Zigong] remarked that if he did not wish others to impose on him neither did he wish to impose on others. Confucius' comment was that this was beyond his ability\" (Lau 1979: 135n7).\n\n29\n\nMore formally, we may restate the distinction as one between first-order and second-order desire. \"Someone has a desire of the second-order either when he wants simply to have a certain desire or when he wants a certain desire to be his will\" (Frankfurt 1971). The distinction is implicit in a passage in the Xunzi, which seems puzzling to Dubs and Watson, and corrupt to Knoblock. My reading is consistent with Xunzi's thesis that the transformation (hua \u5316) of original human nature (hua \u5316) results in new characteristics, which cannot be identified and classed with those things we receive from tian or nature. As Xunzi put it, \"The original nature of man is the beginning and material; acquired characteristics are the beautification and glorification of the original nature. Without acquired characteristics, the original nature could not become beautiful of itself. When original nature and acquired characteristics unite in character development, then only the name of Sage becomes inseparable from that of man\" (Li 1979: 439, Dubs 1929: 234\u201335, Watson 1963: 102. My distinction seems implicit in Liang Qixiong's \u6881\u555f\u96c4 distinction between tianxing yu \u5929\u6027\u6b32 (desires as endowed by nature or natural desires) and lixing yu \u7406\u6027\u6b32 (desires guided by reason or reflective desires). See Liang 1978: 323. Cf. Dubs 1929: 294; Watson 1963: 141; and Knoblock 1994 Vol.3: 21.22.5a.\n\n30\n\nThe translation cited is Chan's, except for translating yu \u6b32 as \"desiring\" in lieu of \"wishing.\" This minor modification is made to conform to my earlier discussion of yu as \"reflective desire.\" Elsewhere, Confucius points out that one establishes (li \u7acb) oneself on the li (Analects 8.8, 16.13). In the text, I follow Chan in construing 6.28 as the positive version of shu.\n\n31\n\nThe following remarks on modesty and humility are inspired by Allinson's perceptive essay on the negative version of the Golden Rule or shu in the Lunyu. However, my claim here is stronger than Allinson's. In my view, if the Confucian ren is brought in as the background for interpreting the negative formulation of shu, it is not merely consonant with the Confucian attitudes of modesty and humility, but more fundamentally, constitutive of reasonable commitment to ren as an ideal of the good human life as a whole. See Allinson (1985). For a discussion of reasonable vindication of the adoption of ren, see Cua (1982: 94\u2013100).\n\n32\n\nFor the Confucian notion of li, see Cua 1989b; and Cua 2000. Cf. Fingarette 1972. For Cicero, see Higgenbotham 1967, Chaps. 27 and 28.\n\n33\n\nOther renderings of yong are possible in different contexts, e.g., \"bravery, boldness, being daring, audacity, fearlessness.\" One passage (Analects 14.28) clearly says that a yong person has no fear (yongzhe buju \u52c7\u8005\u4e0d\u61fc). (See also Analects 9.29.) I leave the translation issue open, since my discussion deals only with the relation of yong to ren, li, and yi.\n\n34\n\nAnother example of appeal to reflective desirability is Analects 8.2, which will be discussed later. Perhaps, the most interesting form of logical reasoning is the use of sorites or chain argument in Analects 13.3. For a critical discussion of deductive reasoning in Confucian ethics, see Cua (1998b: 203\u201305).\n\n35\n\nAs I wrote elsewhere: \"A reasonable solution to a practical question is a concretely temporal solution and not necessarily a universalizable decision that claims cogency of application to all similar cases or to all persons in similar circumstances. The universalizable feature, if present in a ruling, is a consequence of acceptance of that ruling as a paradigm for future situations. What is fitting and appropriate to act in a particular situation remains open to ruling. On this Confucian view, the validity of a moral principle depends on assessment in an actual case. Every moral situation has, so to speak, integrity of its own quite apart from its possible subsumption under a pre-established standard. Its function in practical contexts depends on the agent's appeal to some notion of reasonableness as occasionally determined with a view to the reflective desirability of pursuing certain courses of action. In this sense the notions of reasonableness and reflective desirability are internally related within the concern for the ideal of jen [ren]. Thus some sort of consensual background of shared moral attitude is presupposed in discourse. The use of Confucian sorties in the classics, [e.g., The Great Learning] can also be understood in this light as discourses on reflective desirability rather than attempts at determining the logical consequences of moral beliefs.\" (From Cua 1975).\n\n36\n\nFor a discussion of the three functions of li, see Cua (1989b); More extensive discussion of li and its connection with ren and yi is given in Cua (1998b, Chap.\u200b 13).\n\n37\n\nFor the distinction between danming and jianming in Xunzi, see Li (1979: 515).\n\n38\n\nZhou Yi \u5468\u6613, \u5764\u6587\u8a00.\n\n39\n\nGraham points out that the word jing \u656c, as used by the Cheng brothers (Cheng Hao \u7a0b\u9865 and Cheng Yi \u7a0b\u9824) \"cannot be translated by 'reverence\"; and Bruce's \"seriousness\" is utterly inadequate, although accusation can be made against Bruce, it is difficult to find a better alternative. The two aspects of ching are interdependent; to collect oneself, be attentive to the person or thing implies that one respects him or takes it seriously; and to be respectful implies that one is collected and attentive. But there is no English word which covers both, and the only course seems to use \"reverence\" for one and a different word for the other.\" See Graham (1958: 69).\n\n40\n\nAs Mao Zishui's \u6bdb\u5b50\u6c34 remarks: \"It appears that Confucius does not believe in the existence of gods and spirits. His 'jing' attitude toward gods and spirits is entirely because of concession to 'the established teaching on the legend of spirits.\" (This is a free reading of Mao's remark. See Mao 1977: 87.) It is interesting to note that Confucius's remark, which we render as \"respect the gods and spirits\" has a modern usage that is completely devoid any implication of respect, instead it is used to express disrespect. Lin Yutang, for example, offers this translation of the saying \"jing guishen er yuan zhi \u656c\u9b3c\u795e\u800c\u9060\u4e4b\" as \"to act correctly to nasty people and keep them at a distance\" (Lin 1972: 304A).\n\n41\n\nThe second sentence is Watson's translation. See Watson (1963: 20).\n\n42\n\nFor the notion of yi \u7fa9 as appropriateness (yi \u5b9c), see Zhongyong \u4e2d\u5eb8, Sect. 20 in Chan 1963:104. For this notion of yi and its general significance as ruling on the relevance of moral rules to particular circumstances, see Cua (1971: 44\u201346); elaborated in Cua (1978), Chaps.\u200b 5 and . Similar interpretation may be found in Cheng (1972), Lau (1979: 49\u201350), and Chen (1977, Chap.\u200b 3).\n\n43\n\nQuli \u66f2\u79ae, Legge 1966 Vol. 1:78; Wang 1977 Vol. 1: 20. See also, Tangong \u6a80\u5f13, Legge 1966 Vol. 1: 153\u201354; Wang 1977 Vol. 1: 106.\n\n44\n\nFor the notion of yi \u7fa9 as appropriateness (yi \u5b9c), see Zhongyong, Sect. 20 in Chan (1963: 104). For this notion of yi and its general significance as ruling on the relevance of moral rules to particular circumstances, see Cua (1978, Chaps.\u200b 5 and ).\n\n45\n\nLi (1979: 328); Watson's translation emended. Rendering yizhe xunli \u7fa9\u8005\u5faa\u7406 as \"right and reasonable\" is an interpolation of li \u7406 as \"reason\" in the sense of \"what is right and reasonable,\" which brings out an important function of yi \u7fa9. Knoblock's proposal of \"rational order\" for li \u7406 is also acceptable, in the light of Xunzi's overall concern with order (zhi \u6cbb) as opposed to chaos or disorder (luan \u4e82) (Knoblock 1990 Vol. 2: 15.2). Violence and harm doing, especially on a large-scale, are liable to produce chaos or disorder. Dubs's translation of li \u7406 as \"principles\" in this passage is also acceptable (Dubs 1929: 168), if the notion of principles is construed in the sense of principia, the originating and fundamental basis of ethical conduct, which for the Confucians are ren and yi. Moreover, especially in the context of intercultural ethical conflict, the principled interpretation and formulation of ren and yi is quite appropriate, as it emphatically draws attention to the core of Confucian ethics. See Cua 1997.\n\n46\n\nFor further discussion of Wang's doctrine of the unity of knowledge and action, see Cua (1982).\n\n47\n\nThe theme of learning, thinking, and deliberation is more extensively discussed in the Xunzi. For further discussion, see Cua (1985: 3\u20134, 65\u201369), passim; and Cua (1993).\n\n48\n\nFor shen, see 4.24, 7.13, 8.2; for zhuang, see 2.20, 11.21, 15.33; and for gang, see 5.11, 13.27, 16.7, 17.8.\n\n49\n\nYe proposed a fourfold classification of virtues: (1) self-regarding (duiji \u5c0d\u5df1), (2) other-regarding (duiren \u5c0d\u4eba), (3) attitudes toward things in general (duiwu \u5c0d\u7269), and (4) attitudes toward affairs of human life (duishi \u5c0d\u4e8b) (Ye 1977). It seems to me that Confucius drew no such sharp division between (1) and (2), and between (3) and (4). Chen's proposal is more illuminating and, as I have shown in Sect. 2.1, can be adopted for elaboration of my own proposal, but I leave this as an open issue.\n\n50\n\nThis is Chan's translation. Chan refers to 2.14 for comment: \"The superior man (junzi \u541b\u5b50) is broadminded (kuan \u5bec) but not partisan; the inferior man is partisan but not broad-minded.\" See Chan 1963: 41. For our earlier discussion of kuan, see Sect. 2.15 above.\n\n51\n\nThis section draws materials from my entry \"Quan (Ch'\u0171an): Weighing Circumstances\" in Cua (2003), which is a highly abbreviated account based on Cua (1998a: 260\u2013266).\n\n52\n\nWei 1986, \"Chu Hsi on the Standard and the Expedient,\" in Chan (1986: 255).\n\n53\n\nLi (1979: 43, 306). The judgment involved may be challenged and thus subject to argumentation, see Cua (1985, Chap.\u200b 3).\n\n54\n\nSee my Ethical Argumentation.\n\n55\n\nIn my Keynote Address to the 11 Conference of the International Society for Chinese Philosophy held in Taiwan in 1999, I mentioned other problems: \"What is the role of the developing tradition as the background of Confucian ethics? To what extent can the ideal of dao or ren be concretely specified in a conceptual framework comprising ren, li, and yi? How are these fundamental notions to be further shaped to accommodate the evolving normative problems in the ethical life today and tomorrow, problems that are quickly acquiring greater transcultural and global significance? In the context of inter-traditional and\/or intercultural ethical conflict, what degree of success can one expect from the employment of my proposed ground rules or transcultural principles of adjudication, such principles as non-prescriptivity or cultural integrity, mutuality, procedural justice, rectification, and reconsideration? (see Moral Vision and Tradition, Essay 14.). Perhaps additional or other principles will do a better job in conflict resolution.\" See Cua (2000a).\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_14\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 14. Early Confucian Political Philosophy and Its Contemporary Relevance\n\nTongdong Bai1\n\n(1)\n\nFudan University, Shanghai, China\n\nTongdong Bai\n\nEmail: baitongdong@gmail.com\n\nAbstract\n\nIn this chapter, I will discuss the political aspect of early Confucian philosophy. The texts I will focus upon are the so-called \"Four Books,\" i.e., the Analects, the Mencius, the Great Learning, and the Doctrine of the Mean, which can be considered forming a coherent whole. In the first section, I will argue that the political aspect should be taken as the key concern of early Confucians, and from the problems they were facing, that it is more justified to call them \"modern\" than classical thinkers in the Western sense. In the second section I will show how they answer some of the key questions of their times, especially the search for a new form of social glue, and how their answers can still be relevant to contemporary political issues. In the third section, I will show how they tackle other key issues of their time, especially the selection of the ruling class. In the fourth section, I will show how the regime proposed in the third section combined with ideas discussed in the second section can be (to a large extent) reconciled with, critical of, and constructive to contemporary liberal democratic regimes.\n\nI wish to thank Vincent Shen for his invitation to contribute and very helpful comments on an earlier draft. The research for this chapter is supported by the Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, the \"New Century Excellent Talents in University\" grant, Shanghai Educational Development Foundation (Shuguang Project), the \"Pujiang Talents\" grant from the Shanghai government, the Shanghai Philosophy and Social Sciences Projects, and by research grants from Fudan University (the Guanghua program, 985 Project 2011RWXKZD009, 985 Project 2011RWXKZD010 and others). I am grateful for their support.\n\n## 1 Early Confucianism as a \"Modern\" Political Philosophy\n\nIn this chapter, I will discuss the political aspect of early Confucian philosophy. The texts I will focus upon are the so-called \"Four Books,\" i.e., the Analects, the Mencius, the Great Learning, and the Doctrine of the Mean, which can be considered forming a coherent whole. In the first section, I will argue that the political aspect should be taken as the key concern of early Confucians, and from the problems they were facing, that it is more justified to call them \"modern\" than classical thinkers in the Western sense1. In the second section I will show how they answer some of the key questions of their times, especially the search for a new form of social glue, and how their answers can still be relevant to contemporary political issues. In the third section, I will show how they tackle other key issues of their time, especially the selection of the ruling class. In the fourth section, I will show how the regime proposed in the third section combined with ideas discussed in the second section can be (to a large extent) reconciled with, critical of, and constructive to contemporary liberal democratic regimes.\n\nTo understand early Confucianism, in my view, we should first understand the world early Confucians lived in and thus the problems they were faced with. I will give a brief overview without sketching out interesting and important details.2\n\nEarly Confucians lived during what was called the \"Spring and Autumn\" and \"Warring States\" periods (since these two periods will be mentioned together often in this chapter, I will use \"SAWS\" for brevity, and note the fact that they lasted roughly from 770 to 222 B.C.E), which was about the same time as early classical Western thinkers (the Greek thinkers) lived. This might be one of the reasons why the former are also called \"classical.\" The philosopher Karl Jaspers placed both as well as the early Indian thinkers together in the \"Axial Age\" (Jaspers 1953). These labels are fine if we understand them, especially the term \"classical\" as \"foundational.\" That is, texts of both ancient Greek thinkers and early Confucian thinkers laid the foundations for later developments. But there is a different meaning of \"classical,\" and it is used in contrast with \"modern.\" In addition to the meaning of periodization, these two terms also suggest the nature of the political problems classical or modern thinkers are dealing with. In this sense, we will have to investigate the central issues early Confucians were facing with before we label them as \"classical\" or \"modern.\" For example, the dominant regime in ancient Greece was that of the city-state (polis), and political matters were centered upon the political life in this regime, but the dominant political regime in China before early Confucians was a form of feudalism, and this regime was collapsing during their times. The urgent issue for early Confucians was how to restore order in their world. Thus, to call them classical thinkers (vis-\u00e0-vis modern thinkers) can be misleading in the area of politics.\n\nIf we label Confucian thinkers during the SAWS with regard to the nature of their central political problems, I think it is less misleading to call them modern thinkers than classical thinkers. (Thus throughout the rest of this chapter, I will refer to these Confucians as \"pre-Qin Confucians\" or \"early Confucians.\") The political changes China experienced during the SAWS are comparable to those in early Western modernization. Both were transitions from some form of feudalism to \"modern\" states. During the Western Zhou period (the middle of the eleventh century B.C. to 771 B.C.), which was before the SAWS, the political structure was like a pyramid. The Zhou King was at the very top, and he directly ruled over princes of various ranks, in addition to his own fiefdom. The princes then ruled over their ministers, their ministers over lesser lords, and so on down the level of common people. Although the Zhou state as a whole was relatively large, on each level, one ruler ruled over a limited number of people. The noble ranks were hereditary, and li \u79ae, which includes rites, rituals, and codes of conduct were what bonded the noblemen on each level together. The political structure in Europe during the Middle Ages bore many similarities to the Zhou regime. Of course, there was not someone or an office that held the long-standing status of the Zhou King. The Pope and his office were of as long-standing as the Zhou King, but the former may not have enjoyed the same kind of over-lordship as the latter. The Europeans had heritage from the ancient Greeks and Romans, which the Chinese didn't.\n\nDuring the SAWS in China and toward the end of the Middle Ages in Europe, the feudalistic structure was collapsing. On each level, the ruler was losing his or her control over the lesser noblemen. The Zhou King became but a local lord and eventually was eliminated. What emerged were a few strong states in which one ruler had to rule a large and populous state. With the overlord gone, these states were vying for domination. Hereditary nobility disappeared, and so did the old ruling class and ruling structure. A de facto equality emerged, and political and military offices became more and more plebianized. Li that served as the social glue in feudalistic times couldn't hold a large state of strangers together anymore, and a new social bond was desperately needed. Similar changes also happened in Europe during its transition from feudalism to modernity. There were two pressing \"domestic\" issues for these states in transition: what could serve as the new social glue and how could they reconstruct a ruling class from a more egalitarian and mobile (and \"liberal\" in the sense that one could now choose one's career, in contrast to the fixed role one was born into in feudalistic times) society. Pressing \"international\" questions included how to deal with the newly emerging international relations, and whether it was possible to achieve peace among states.3\n\nThese political questions, in my view, were the foci for pre-Qin Confucians (and other pre-Qin Chinese thinkers). Other dimensions, such as moral psychology and moral metaphysics, were secondary or in service to the political dimension. If my thesis \u2013 i.e. that pre-Qin Confucianism centered upon the aforementioned pressing questions which were also shared by European thinkers during the European transition to modernity \u2013 holds, then the alleged influence Chinese thoughts in general and Confucianism in particular had on European thinkers in their modernization efforts was neither accidental nor superficial.4\n\nHowever, the political aspect has been ignored, even vilified, in recent history. For Confucianism has been linked by many with the alleged authoritarianism in traditional China. In the May 4th Movement in 1919, the attack on authoritarianism and the advocacy of democracy was partially expressed through the slogan of \"down to the Confucian store\" (dadao kongjiadian \u6253\u5012\u5b54\u5bb6\u5e97). Even the cultural conservatives, such as the New Confucians, fully embraced Western democracy, limiting the contemporary significance of Confucianism to the ethical and cultural spheres.\n\nIn the next two sections, I will focus on the political dimension of pre-Qin Confucianism. I will show how the apparently ethical and metaphysical teachings of early Confucianism can be interpreted as being centered upon the aforementioned political issues, thus showing that pre-Qin Confucianism is a political philosophy first and foremost. Since, according to my interpretation, early Confucianism was already addressing issues of modernity, we should consider it offering a political paradigm comparable to and competing with other modern political paradigms, liberal democracy included. If there are differences between the Confucian paradigm and Western modern paradigms, they should be considered competing ones and should be judged as such. One shouldn't dismiss the Confucian one as a political proposal for a bygone era. As long as we still live in modernity, and as long as we refuse to accept the belief that history comes to the end with the triumph of liberal democracy, as the political theorist Francis Fukuyama did (1992), we should evaluate the merits and problems of the Confucian political proposal against other modern political proposals so that we can envision a more ideal and yet humanly possible political regime. Although the elaboration of the political dimension of early Confucianism should already show its contemporary relevance, in the last section of this chapter, I will focus on how, following early Confucian ideas, we can develop a regime that is compatible to contemporary liberal democracy and human rights in many aspects, and is different from and superior to the latter in some other aspects.\n\n## 2 Humanity and Compassion as the New Social Glue\n\n### 2.1 The Ideas of Humanity and Compassion\n\nAs argued in the previous section, the world pre-Qin Confucians faced was a chaotic one due to the collapse of the old regime. To restore order, apparently, one can either restore the old regime or construct a new one. It can be argued that pre-Qin Confucians, especially in the case of Confucius (but far less so in the case of Mencius), wished to restore the old regime. Confucius claimed that \"I transmit but do not innovate; and I like the antiquity with a trustful attitude\" (Analects 7.1), and that \"the Zhou [li]5 was based upon those of the two previous dynasties, and was resplendent in culture. I am for the Zhou [li]\" (Analects 3.14). But the great historian Sima Qian \u53f8\u99ac\u9077 keenly observed in the Shiji \u53f2\u8a18 (Records of the Grand Historian) that what Confucius did was to \"carve the historical records Spring and Autumn Annals [into his own version]\" (Vol. 48 of the Shiji; Sima 1981: 228).\n\nNot to deny that Confucius and even some early Confucians can be interpreted as purely conservative, and that there is textual and historical evidence for this belief, I want to maintain that they can also be interpreted as reformers and even revolutionaries with a conservative fa\u00e7ade, and there is also strong evidence to support this argument. Indeed, the latter reading is perhaps philosophically more interesting and more relevant not only to us, but even to Chinese political thinkers and practitioners since the SAWS, because to go back to the old regime was already a lost cause during the SAWS, and much more so in later times in Chinese history.\n\nIn this chapter, then, I will adopt the reading of early Confucianism as trying to address the new issues of their times by reinterpreting the old. In fact, although apparently, Confucius wished to restore Zhou li, first, he acknowledged the need to adjust it to the changing social and political reality (Analects 9.3), and second, he offered a new basis for following it. This new basis is ren \u4ec1. \"Ren\" can be translated as benevolence or kindness, but I will use the translation as \"humanity\" in this chapter; for it is pronounced exactly like the Chinese word for \"human,\" although the Chinese characters for these two words are different. There may be a deliberate play of this identity of pronunciations, and thus \"humanity\" or \"humane\" can catch this play the best.\n\nWhen discussing with his pupil Zai Wo \u5bb0\u6211 whether the old ritual of 3-year mourning \u2013 according to which one is required to stay away from an office, and away from luxuries and entertainments for 3 years after a parent dies \u2013 should be observed, a crucial reason Confucius offered is that one wouldn't and shouldn't feel at ease in having luxuries during the 3 years after the death of a parent (Analects 17.21). If a person enjoys luxuries in this situation, Confucius said that he is \"inhumane,\" and he ruefully asked if this person \"was not loved by his parents for 3 years?!\"6 At another place, Confucius explicitly said, \"What can a man do with li if he is not humane?\" (Analects 3.3), implying that li (and the conservation of it) is not fundamental, but humanity is, which is the ultimate foundation of li.\n\nMencius developed the concept of humanity and introduced the idea of compassion. In a very famous example in the Mencius, he said:\n\n> The reason for me to say that all human beings have the heart that cannot bear [to see the sufferings of] others is this. If men suddenly see a child about to fall into a well, they all have a feeling of alarm and distress, not to gain friendship with the child's parents, nor to seek the praise of their neighbors and friends, nor because they dislike the cry of the child. From this we see that a man without the feeling of compassion is not a man... (Mencius 2A6)\n\nA few things should be noted here. First, the term for \"compassion,\" ce yin zhi xin \u60fb\u96b1\u4e4b\u5fc3, was translated as \"a feeling of alarm and distress\" in its first occurrence in this passage, which is the original meaning of this term. Perhaps thanks to Mencius, this term is later used to refer to compassion. Second, Mencius did not say that all people will then act on their instinctive moral sentiment of compassion. Taking an action will depend upon the cultivation of moral sentiments. Third, he then claimed that those who don't have the heart of compassion are not human, implying that his account is normative rather than descriptive. Of course, most people would probably react in the way Mencius described, and this shows that Mencius' normative account is rather realistic.\n\nConfucius' concept of humanity and Mencius' concept of compassion are often read as accounts of moral psychology or moral metaphysics. But they may be first and foremost accounts of political philosophy. They offered an answer to one of the fundamental problems of their times: the search for a new social bond. In feudalistic society, li can be sufficient to hold a small group of acquaintances together, but a fundamental change in China's early modernity is the emergence of a large society of strangers. Humanity and compassion are precisely moral sentiments for strangers. In Mencius 7A45, Mencius explicitly pointed out that humanity is feeling toward the people (i.e., strangers). In the aforementioned example, he focused on our immediate and instinctual reaction to the falling child, and in this immediacy, we are not able to recognize whose child it is. Thus, this instinctual reaction is also directed toward strangers.\n\nAnother textual evidence to support this political reading of compassion can be found if we look into the context where the discussion of a falling child began. That passage began with the following statements:\n\n> No human being is devoid of a heart that cannot bear [to see the suffering of others]. The Former Kings had it, which is why they had the governance that could not bear [to see the sufferings of its people]. Applying the heart that cannot bear and practicing the governance that cannot bear, the world can be run like being played on one's palm. (Mencius 2A6)\n\nWe can clearly see the focus on politics in this passage.\n\nFrom the above passage, we can also see that although compassion among people is important, Mencius' focus is the compassion a ruler has for his people. One can argue that traditional China since the collapse of the feudalistic Zhou dynasty had still remained a society of acquaintances. Due to the agrarian nature of its economy, this may well be the case, although social mobility was still far greater than it was during the feudalistic era. But now a ruler or an official had to face strangers, unlike in the feudal structure, where a higher nobleman ruled over a limited number of lesser nobles, whom the higher nobleman knew personally. This personal connection between a ruler or a minister and his subjects was gone in China's early modernity, and the search for a new connection was, as I speculated, what early Confucians tried to address.\n\nAccording to this understanding, we can see a difference between China's early modernity and the European one; for the latter came with the industrial revolution, while China stayed an agrarian society for a very long time after its early modernity. In traditional China, officials who moved out of their hometown and lived in big cities still had their economic roots in rural areas. But in industrial society, these roots could be thoroughly cut off. In general, industrial society leads to far stronger social mobility and thus creates a society of strangers to a far higher degree. In this sense, China's early modernity is modernity 1.0, while the European one is modernity 2.0. Nonetheless, these two versions of modernity share some fundamental problems.\n\nThis speculation can further be supported if we look into Western political philosophy and history. As the German philosopher Friedrich Nietzsche pointed out, compassion was not valued as a virtue by ancient Greeks and Romans. It became a virtue with the introduction of Christianity to Europe, and was prevalent as a virtue during Nietzsche's times (c.f., Nietzsche 1994, 2002). His diagnosis of this \"symptom\" was that this change of values was a result of slave revolt, perpetrated by the people of \"black art,\" the Jews. But this account couldn't explain why compassion could play such a central role in early Confucianism, because the Confucians didn't suffer the kind of hopeless oppression that the Jews in Nietzsche's account did, and thus they had no secret plan to change their status by re-inventing the value system. The real reason for compassion becoming a value and even an important value may have been the rise of a society of strangers. The ancient Greek polis and the early Roman Republic were small city-states, and virtues such as justice (which is a contractual form of codes of conduct) and sentiments such as friendship could serve as the effective social bond. In the Roman Empire, societies of strangers emerged, and so did compassion as a virtue. But the progress of the latter emergence was held up when Europe became feudalistic. When feudalism collapsed and modern states (that are large and populous, lacking the pyramid-like structure of feudalism) emerged, compassion became a dominant virtue, which upset Nietzsche, a former classicist and probably a lover of ancient Rome, who correctly saw the change of values, and mourned the lost world of the ancients, but might have failed to understand the nature of modernity and offered a wrong diagnosis of it. Thus, our understanding of the role of compassion is applicable to the history of political thought in Europe as well, which in turn supports and corroborates our understanding.\n\n### 2.2 The Cultivation of Humanity and Compassion\n\nNow that humanity and compassion are introduced as the new social bond and as the ultimate foundation of li, the next issue is how to cultivate them. The general principle is \"to take as analogy what is near at hand\" (Analects 6.30). What is commonly near at hand is one's own self and family. Self and family are often considered, especially in modern Western philosophy, as being in conflict with the other and the public. Early Confucians understood this conflict. But they also saw the complementary aspect between the two sides. To understand the self and to understand the other are inseparable from each other. In Mencius 1B5, a king confesses that he has weaknesses which are a fondness for money and for beautiful women, but Mencius said that these were not necessarily a problem for the king. Indeed, a ruler can be good precisely because he understands, from his own fondness, that his people must share it too, and then act upon this understanding to secure livelihood and family for his people. An interesting contrast is Immanuel Kant's account of moral worth. In one example, Kant said that if a man is overshadowed by his own grief which extinguishes all his sympathy for the fate of others, or if by nature he is cold and indifferent to the sufferings of others, the philanthropic action that he commits then has \"its genuine moral worth\" (Kant 1998: 12\u201313; 4:398\u20134:399)!\n\nBut to understand other's emotions and needs by understanding one's own doesn't mean that one would do something to help others. An extreme counter-example is that a sadist gets his or her pleasure precisely from understanding that his or her object of torture has certain emotional needs, and he or she deprives them of the object. What is needed for this understanding to become compassion is the motivation to help others. Confucians saw that a unique place to cultivate compassion is within the family. For, on the one hand, family is what most people feel akin to and have a loving feeling for, almost as natural as one's self-love. On the other hand, the care for family members is also the first step to stepping out of one's mere self and self-love, and to go toward others. This is perhaps why filial love played such a central role in early Confucianism. In this sense, filial love is not merely an ethical value for personal conduct, but a crucial political value. This again supports the political philosophical reading of early Confucianism.\n\nFrom a passage in the Analects, we can clearly see the political dimension of filial love and early Confucians' concern with it:\n\n> It is a rare thing for someone who is filial to his parents and respectful to his older brothers to defy superiors. And it is unheard of for those who do not defy superiors to be keen on initiating rebellion. Exemplary persons [junzi] concentrate their efforts on the root, for the root having taken hold, the Way will grow therefrom. Being filial to one's parents and being respectful to one's older brothers is the root of humanity! (Analects 1.2)\n\nThus, for Confucians, communal and political relations are analogous to, and should be modeled after, familial relations. To understand family is important for us to understand politics. If we can cultivate filial love and drive the compassion that grows out of this love outward, we will eventually become caring toward everyone and even everything in the world. As Mencius put it,\n\n> Treat the elderly of my own family [as they should be], and extend this treatment to the elderly of other families; treat the young of my own family [as they should be], and extend this to the young of other families... Thus extending one's humanity outward can protect everyone within the Four Seas [the alleged boundaries of the world], and not extending one's humanity cannot even protect one's wife and sons. (Mencius 1A7)\n\nThus, starting from the self and from family, one should cultivate one's care outward, and develop compassion for strangers. But it should be noted that for Confucians, it is also natural and justified for everyone, including a person of universal love (if there is ever such a person), to care about those closest to them more, in that they would try to save their drowning mother before saving anyone (or anything) else. This is the Confucian idea of graded love (ai you chadeng \u611b\u6709\u5dee\u7b49). Thus, the Confucian love is universal and hierarchical or unequal at the same time.\n\n### 2.3 Confucian Ideas of State Identity, Just War, and Peace Through Unity Under Humanity\n\nPre-Qin Confucians introduced humanity and compassion as the new social glue for a large state, especially between rulers and subjects. Compassion, fully cultivated, can be extended to everyone and everything in the world, but it will still be hierarchical. On the issue of international relations, it means that a person will care about his or her compatriots more than foreigners, but at the same time, this person will consider foreigners human and care for them accordingly. This means that one can enjoy being identified with one's home state, but this identification doesn't mean a total disregard for others.\n\nThis understanding offers a Confucian model of international relations that addressed the need of theories of these relations during the SAWS. The Zhou King stopped being a supranational authority, and states became independent entities of interests. This process was similar to what happened in the post-Westphalian Europe. However, the Confucian treatment of statehood and international relations is different from a certain form of nation-state model that emerged in Europe. This form of nation-state was based upon nationality which was in turn based upon (imagined?) blood relations or other exclusive relations that are closed to aliens. Associated with this kind of nation-state is often a sharp boundary between \"us\" and \"aliens,\" which is a nation-state form of the friend-enemy distinction that existed in Greek polis. Its motto is \"my country, right or wrong,\" and its principle for international relations is Realpolitik. The Confucian model offers an alternative that is morally superior to this kind of nation-state model. It is also more realistic than the kind of cosmopolitanism that is based upon universal and equal love.\n\nIn addition to compassion, early Confucians offered another bond for the state, and it is culture. It comes from the classics of the past and the spirit of humanity lying beneath them. Culture as it is understood by Confucians is open to \"aliens.\" A passage from the Mencius beautifully illustrates this openness (Mencius 3A4). In the passage, Mencius argued that, because he managed to grasp the Chinese culture (in the form of the teachings of Confucius) better than the scholars from the Chinese states, Chen Liang, a native of the state of Chu (a state that the Chinese looked down upon as almost barbaric and not as a bona fide \"Xia\" (Chinese) state), should be considered a Chinese. In contrast, because some others, though from a Xia state, failed to follow the Chinese culture, they degraded themselves to the level of non-Chinese or barbarians.\n\nThis culture-based model of national identity also leads to a possibility of peaceful assimilation among different ethnicities, in contrast to the nation-state model that is based upon blood or other exclusive relations. An alien can become a member of a state the identity of which is based upon a \"thin\" kind of culture by adopting this culture, but an alien cannot become a member of a nation-state the identity of which is based upon blood or other exclusive relations. The only conversion for the latter is elimination. One can object to this claim by saying that the crusaders also wished to convert the aliens, but the fundamental conflicts between religions (one can't be both a Muslim and a Christian; indeed, one can't even be a Catholic Protestant!) mean that this conversion is very close to elimination (either by destroying the body of the alien or by destroying his or her spiritual belonging). Here we should note that culture, as understood by Confucians, is or can be rather \"thin.\" It is so thin that one can be a Confucian Muslim, a Confucian Christian (for a contemporary example, think about the so-called Boston Confucians), or a Confucian Jew. A good historical example is that although Jews have been persecuted all over the world in the past, the only known case of peaceful assimilation happened in China, where the Jews who came to China for commerce willingly adopted the Chinese way of life while maintaining their faith for quite a long time.7 The openness brought about by culture as national identity helped explain how, in spite of being defeated or even conquered by \"barbaric\" nomads on a few occasions in history, Chinese people and Chinese culture have lasted, making China a country that has the longest continuation of state identity. For the conquerors in the past either got assimilated into the Chinese culture and thus became Chinese themselves, or they were eventually driven out due to their failure to assimilate themselves into the Chinese culture.\n\nWe should see clearly by now the merits of the bond for a state that Confucians offered. It doesn't demand people to do the impossible, such as caring for everyone equally, as some Christian theories, Kantian philosophy, or some forms of cosmopolitanism demand. It preserves the boundaries of a state and thus preserves competition among states. This competition, like the function of physical exercise for one's health, is good for different states to develop, instead of falling into a non-competitive and stagnant uniformity, the kind of world of \"last men\" that Nietzsche worried about (1954). But this competition is culture-based, and is aimed at peaceful conversion and not bloody extinction, as some forms of nation-state are. Patriotism in the Confucian model doesn't mean a total disregard for aliens' interests. In short, the Confucian form of national identity is like a life full of friendly but serious sport, in comparison to a sedentary lifestyle and life-or-death strife.\n\nBut there are some problems with the Confucian model. On the one hand, the kind of culture that serves as the identity of the state may be still too thick. Part of the national identity of the U.S. is also culture, but it is more inclusive than the Confucian one. For example, a difficult issue for China now, both practically and theoretically, is how to absorb fully Tibetans and Uyghurs into a Chinese state, which wouldn't pose a theoretical problem for the U.S.8 On the other hand, one can complain that the Confucian cultural identity is too thin. The kind of culture as Confucians understand it can be shared by two states, and it won't be sufficient to draw state boundaries.\n\nThe thinness of Confucian cultural identity of a state, together with the concepts of humanity and compassion, offer an answer to another key question in China's transition to its early modernity during the SAWS, that is, how to achieve \"international\" peace. One period of the SAWS is literally called \"Warring States,\" and every state in this period seemed to believe that the only way to eliminate wars was for one state to unify China. The way to unity was through wars and conquests. For Confucians, however, constant strife is bad because it is against humanity, and unity is desirable only if it stops the strife in a humane manner. Unity is a good secondary to humanity, and unity without humanity is not desirable. Thus, it is still desirable if we can live peacefully and prosperously together without a unified state. Similarly, to maintain two distinctive states is not Confucians' concern, either. In reality, however, the fact that all the warring states thought that they had shared culture did help them go down the road of unity, which would achieve international peace. But Confucians thought that this unification had to be achieved willingly.\n\nA few textual supports are in order here. For example, in a conversation (Analects 16.1), two of Confucius' pupils wished to help their lord (Lord Ji \u5b63\u6c0f) attack a nearby fief because its very existence threatened a fief of his lord. Confucius pointed out that the stability of the state or a fief depends on treating its own people well. When this is done, if people from a different state or fief are still un-submissive and even posing a potential threat, we should improve our civilization and moral character to attract them. Finally, Confucius pointedly observed that the threat to the fief of Lord Ji's did not really come from outside, but from within, i.e., from Lord Ji's lack of humanity and from his practice of Realpolitik. In fact, Lord Ji himself was among those who usurped all the power of the throne of the state of Lu through his strength. An interesting twist to this story is that, in the next chapter of the Analects, the magistrate of Lord Ji's fief that was said to be threatened by another lord's fief used the strong fortification to rebel against Lord Ji (Analects 17.5). The fact that this happened is really a small wonder if we consider the moral example Lord Ji set for his subjects, which is what Confucius alluded to when he claimed that the threat to Lord Ji really came from within. To sum up the principle of domestic and international peace and prosperity, Confucius said that good governance is for the people who are near (that is, under the government in question) to be happy and for those who are afar (that is, from other states or under other magistrates) to wish to come (Analects 13.16).\n\nIn the Mencius, there are many passages that discuss wars and unity. For Mencius, the strength of a state lies in the practice of humanity. For example, in a conversation with a king of one of the seven strong states (the state of Wei \u9b4f) during the Warring States period, he stated,\n\n> A territory which is only a hundred li\u91cc by a hundred li [\"li\" is a length unit, which might have been around half a kilometer during Mencius' times] is sufficient for its ruler to become a true King [ruling over all states]. If Your Majesty will practice humane governance to the people\u2014sparing in the use of punishments and fines, and making the taxes and levies light, so that the fields shall be ploughed deep and the weeds shall be removed in time, and that the strong-bodied, during their days of leisure, shall cultivate their filial piety, fraternal respectfulness, loyalty, and trustworthiness, serving thereby, at home, their fathers and elder brothers, and, outside home, their elders and superiors\u2014then you can make these people who are armed with nothing but staves inflict defeat on the armies of Qin \u79e6 and Chu \u695a [two powerful states during the Warring States period] who are armed with the strong armor and sharp weapon. [For] the rulers of those States rob their people of their time [during the farming season], so that they cannot plough and weed their fields to support their parents. Their parents suffer from cold and hunger, and their brothers, wives, and children are separated and scattered. Those rulers push their people into pits and into water. If Your Majesty should go to punish them, who will be there to oppose you? Hence it is said, \"The man of humanity has no match.\" I beg of you not to have any doubts anymore. (Mencius 1A7)\n\nA few clarifications need to be made here. First, the message here may appear to be overly idealistic, because Mencius seemed to be saying that a small but humane state can conquer the world. But we have to remember that he was talking to the king of a powerful state, and he might have been trying to be encouraging with the idealistic rhetoric. When he actually talked to the king of a small state who had a reputation of being a humane ruler, Mencius didn't repeat the claim; rather, he offered two options to the king: to run away and find a safe place, hoping that his state would become strong over generations and that a future king would lead his people back to the homeland, or to defend his state with his people at the risk of being eliminated (Mencius 1B13, 1B14, and 1B15). Second, he described the other states as being tyrannical, but he didn't seem to support this claim with any historical fact. I think that this description was not really meant to be an accurate picture of other states, but was meant to be a normative requirement. That is, Mencius implicitly warned the king that only if another state is an inhumane state can it be attacked.\n\nThus, when it is humane and strong, a state may need to engage in wars of liberation. Mencius offered clear criteria for these kinds of wars. That is, the people of the state to be liberated have to welcome the invader, and the welcome has to be clearly expressed and long-lasting. Talking about historical and legendary events, Mencius said that a justified war of liberation is one in which those who are attacked later in a military campaign would complain about this (being attacked later) (Mencius 3B5 and &7B4). The people thought that when being attacked by a humane state, their sufferings would finally come to an end (Mencius 1B11 and 3B5). When the attack happened, the people would welcome invaders by coming to the streets and offering them good food and drink (Mencius 1B10, 1B11, and 3B5). This welcoming also has to be long-lasting, and the invasion would turn out to be unjustified if the invaders failed to deliver good life to the people (Mencius 1B10).\n\nFrom these criteria we can also see the caution early Confucians urged against wars of liberation. Even if people of another state are actually suffering, it may take time for them to realize it and to be ready for change. Otherwise, the consequence might be unnecessary loss of life and even push the suffering people to flock to the bad rulers. The reason that the suffering people have to be ready themselves is that, for example, the Mencius expresses a firm belief in every human's capacity to be enlightened, and the moral enlightenment has to done by the person willingly. In a tale in the Mencius, a stupid and impatient farmer wishes to help the seedlings to grow by pulling them up, but, as we can imagine, he ends up killing them (Mencius 2A2). In short, Mencius saw both the potential of human beings and their limits.\n\nIn sum, early Confucians thought that they could solve the conflicts among states with their concepts of humanity and compassion, their understanding of state identity that is based upon culture, and their theories of \"just wars.\" When inhumane states are eliminated, a desirable peace will be achieved. There could be a few humane states, although their boundaries are not taken so seriously by Confucians. It could also be the case that one humane state unifies all in the world. These two options make little difference to early Confucians.\n\n### 2.4 Private and Public in Early Confucianism\n\nIn this section, we have seen how early Confucian ideas that are often taken primarily as ideas of moral psychology and moral metaphysics are first and foremost those of political philosophy. They were addressing some of the key issues in China's transition to early modernity. Since there are shared issues between China's early modernity and the modernity we are still facing today, many ideas and theories discussed so far have clear implications for the contemporary world. Most of these implications have already become obvious when we discussed those ideas themselves. In this subsection, I will elaborate on the contemporary relevance of one set of ideas discussed previously.\n\nIn our discussion of the cultivation of humanity and compassion, a key feature is the Confucian view that there is a continuous and complementary aspect between the private and the public, and it is utilized to overcome the conflict between the two. This offers a model for dealing with the private-public issue. In Plato's Republic and in the Chinese classic Han Fei Zi, much of the private is taken as conflicting with the public, and for the public good, the private is largely suppressed. This is often viewed as a characteristic of various forms of oppressive government. Liberal thinkers since John Stuart Mill (for example, in his On Liberty) object to this oppression, trying to protect the private from the public or governmental intrusion. But this thinking shares with the Republic and the Han Fei Zi\u97d3\u975e\u5b50 the idea that there is a clear divide between the private and the public, and their relation is one of conflict. This is the basis for the contemporary belief in privacy and the separation between the public good and private virtues, or the virtues of moderns and the virtues of ancients. This separation may have been rooted in the memory of the unity between the church and the state and religious oppressions in the West. But as we already saw, Confucian cultural values could be thin enough to become the substratum of a society with diverse faiths. For example, the emphasis on family values is equated in the U.S. with conservatism and implies a public intrusion into private choices. But for Confucians, family is a crucial stepping stone for a private person to open himself or herself up to others, to the public. It is neither merely private nor merely public, and its dual role is precisely the reason that it is so crucial to the politically-minded Confucians. Very simply put, for example, it doesn't matter whether you are a Christian, a Muslim, a Jew, or an atheist, and as long as you acknowledge the importance of the stability of families to the stability of the state and the desirability of the latter, you will consider family arrangements important to the public. In short, the Confucian understanding of the relations between the private and the public poses a challenge to the dominant liberal ideology that pushes too many moral values into the private sphere.9\n\nInterestingly, the above Confucian understanding can also support some liberal agenda, for example, gender equality in politics. One issue as regards women's participation in politics is that, barring any medical miracle, women still have to bear the burden of pregnancy, child-birth, and breast-feeding, if not more. This forces women to retreat within the family for a period of time even if they hold some public office. If we take the conflict model between the private and the public, this retreat will put women at a disadvantage even if they have, before the retreat, the same ability and experience as their male counterparts. However, for Confucians, family plays a dual role, and the private can be constructive to the public. After all, they support the ritual of 3-year mourning, according to which an official has to leave office if a parent of his dies. The hidden assumption of this support is that this 3-year absence wouldn't hurt the official's political capacity. Indeed, by retreating from public life and by recalling the loving memories of the deceased parent, this official may become an even better one when he returns to office. One can use a similar argument to support women in politics by saying that the time women spend in child-birth and child-raising may actually help them become better politicians than they otherwise would be.\n\n## 3 Confucian Hybrid Regime\n\n### 3.1 Government for the People\n\nWith the collapse of feudalism during the SAWS, the political structure had to be rebuilt.10 One issue concerns the legitimacy of the sovereign. In the feudalistic structure of Zhou, the King was called the son of Heaven (tian zi \u5929\u5b50), meaning the legitimacy of the king is from the divine (although the idea of Heaven during the Western Zhou (which was pre-SAWS) might have been already humanized). The king then appointed lesser lords, the lesser lords appointed even lesser lords, and so on and so forth. When the \"son of Heaven\" of Zhou was not respected and eventually eliminated, and when previously vessel states became de facto independent states, the question of the legitimacy of the sovereign emerged.\n\nEarly Confucians, especially Mencius, argued that the legitimacy of the sovereign lies in the service to its people, and this echoes early Confucians' central concept of humanity. In 3A4 of the Mencius, by describing what an ideal government in the (legendary) past did, Mencius clearly offered a normative account of government's duty: it is to improve natural, technological, and economic conditions for people to have their material needs met. But this by itself can make people beast-like at best. For them to be truly human, certain human relationships (wu lun \u4e94\u502b) have to be established with the assistance of the government. Thus, the government should be responsible for satisfying both basic material and spiritual needs of its people.\n\nTherefore, similar to the contemporary democratic understanding of government, the Confucian ideal government is also for the people. But the latter is required to address people's basic spiritual or moral needs, which makes it different from the democratic government. It is a common belief of contemporary democratic peoples that the government will be oppressive if it is in charge of people's spiritual lives. But as we saw in the previous section, the Confucian understanding of morality can be a very thin one that can be endorsed by people with different faiths and spiritualities. Moreover, early Confucians like Mencius believed that all human beings have the equal potential to become moral (see, for example, 2A6, 4B19, 6A7, and 6B2), and to impose morality upon them is only counterproductive, as illustrated by the story of the stupid farmer, who thought he could help seedlings to grow by pulling them up (Mencius 2A2). Thus, the Confucian emphasis on the government's moral role doesn't necessarily mean oppression.\n\nAlthough both contemporary democratic governments and the Confucian ideal government pay attention to satisfying people's material needs, early Confucians also recognize the importance of this satisfaction not only to people's material well-being but also to people's spiritual well-being. Mencius pointed out,\n\n> Only an exemplary person (jun zi \u541b\u5b50) can have a constant [or \"stable\"] heart in spite of a lack of constant means of support [or \"stable possessions or properties\"]. The people, on the other hand, will not have constant hearts if they are without constant means. Lacking constant hearts, they will go astray and fall into excesses, stopping at nothing. To punish them after they have fallen foul of the law is to set a trap for the people. (1A7; see also 3A3)\n\nFor Mencius, then, the common people need to have basic material needs met for them to be moral. Governmental failure to satisfy these needs would make the government responsible for crimes and moral failures of the common people. But at the same time, as mentioned, Mencius also believed in people's ability to become good in spite of a hostile environment, and those who fail morally are also responsible for their failures. By emphasizing both the significance of virtue and the significance of material well-being to one's virtues, early Confucians combined both the American liberals' and conservatives' arguments on governmental responsibility.\n\nTherefore, the Confucian ideal government is a government for the people, and should be held accountable for satisfying people's basic material and moral needs. The general principle, as Mencius put it, is \"Heaven sees with the eyes of its people; Heaven hears with the ears of its people\" (Mencius 5A5).11 If the ruler fails to satisfy people's needs and people are not happy, the ruler can be punished even with death. In one conversation with the king of a powerful state, Mencius asked, if a friend miserably fails to do what he promised and he shouldn't be considered a friend anymore, and if an official fails to do his duty and he should be dismissed, what about a king in a similar situation? The king ducked the question, but the answer is plainly obvious (Mencius 1B6). In another conversation with the same king, Mencius argued that the reason for Confucians who are perceived by the king as loyalists to support the removal and even the killing of certain kings in the past is that these kings fail to fulfill their duties and are kings in name only. Confucians are against regicide only if the king in question is truly a king, and a king who brutalizes his people is a \"mere fellow\" or a tyrant, the killing of whom is completely justified (Mencius 1B8).\n\n### 3.2 But Not by the People\n\nEarly Confucians' view that the legitimacy of the sovereign lies in the satisfaction of the people makes them sound rather democratic. In fact, their \"democratic\" aspect doesn't stop here. As mentioned in the first section, the collapse of feudalism led to the emergence of de facto equality, for hereditary nobility was gone. Early Confucians embraced this equality in both practice and theory.\n\nConfucius was said to be the first private teacher in Chinese history, daringly bringing teachings that were reserved for the nobles at that time to all that were willing to learn (Analects 7.7 and 15.39). As mentioned, Mencius believed that every human being has the potential to become a sage (see, for example, Mencius 2A6, 4B19, 6A7, and 6B2). Xunzi, despite having a view of human nature that is apparently the polar opposite to Mencius', seemed to believe the same (Xunzi Chap. 23; for a partial English translation, see Chan 1969: 133).\n\nParadoxically, however, early Confucians also seemed to believe in hierarchy. In many places in the Analects, Confucius warned not to teach common people higher things (Analects 6.21, 8.9, 15.8, 16.9, and 17.3). For example, he said, \"the common people can be made to follow the way, but they cannot be made to understand it\" (Analects 8.9), with the common people defined as those who \"don't study even after having been vexed by difficulties\" (Analects 16.9). Mencius famously or notoriously maintained the distinction between the \"great men\" (da ren \u5927\u4eba) and the \"small men\" (xiao ren \u5c0f\u4eba), where the great men are those who use their minds and rule, and the small men are those who use their muscles and are ruled and support the former (Mencius 3A4).\n\nHow can we solve this apparent contradiction? As mentioned, early Confucians were reformers with a conservative fa\u00e7ade because they tried to conserve the old by reinterpreting it. The hierarchical terminology in early Confucianism often has its root in feudalism. But from the passages quoted above, we can see that the Confucian hierarchy is not based upon birth, but upon one's merits. Thus, although everyone has the same potential in the case of Mencius or should be educated for whatever reasons in the case of Confucius, both of them seemed to think that only a few can actually \"make it.\" We have to keep in mind that they think that the government is responsible for satisfying people's basic material and moral needs. Thus, those who fail to make it have nobody but themselves and fortune to blame, in contrast to some poor people in contemporary societies, whose failure to excel is partially caused by the failure of the government. For Confucians, then, we are equal in potentiality or at the beginning of our formation, and the government should do everything in its power to help people to move up the ladder. But the reality is that only a few can actually make it. We can't change it, and so we'd better take advantage of it for the public good. We acknowledge the superior political status of those who actually excel themselves in learning and virtues (compassion in particular), but in exchange, they have to justify their superior status by serving the masses. Borrowing the term of \"difference principle\" from John Rawls which dictates that economic inequality can be accepted if the least advantaged are benefited (Rawls 1971: 75\u201383), we can call the Confucian reasoning here the \"political difference principle\"; that is, political inequality (in terms of the power in political decisions) can be accepted if the least advantaged are benefited.\n\nTherefore, although potentially we can all become great men, in actuality, only a few of us can have the level of political wisdom and virtue (compassion) to participate in the art of ruling. Leisure has to be provided to these competent and virtuous great men (Mencius 3A4), and all these conditions are necessary for making sound political decisions. Thus although, as argued in the previous sub-section, the legitimacy of the sovereign lies in satisfying people's basic interests (broadly construed), it is not the people who have the sole power to decide on political matters. It is one thing that \"Heaven hears with the ears of its people,\" but what to do about it is another thing. The Confucian ideal government is for the people, and by the competent and virtuous people (with regard to political decisions).\n\nFor example, in the passage that leads to the conclusion of \"Heaven hears with the ears of its people,\" Mencius was talking about how the legendary sage-ruler Yao passed his throne to the sage-ruler Shun. In spite of the apparent democratic message in the conclusion, Shun was not directly elected by the populace. Rather, he was appointed by Yao as his prime minster for 28 years. After watching his performance for so long, people chose to follow him instead of others after Yao died. In another passage, Mencius said that in the promotion, demotion, and capital punishment of an official, the opinions of the ruler's inner circle and his ministers don't count, and only those of the people do. But the ruler shouldn't simply follow people's opinions; rather, he needs to investigate the case and then make a decision (Mencius 1B7).\n\nLet's now put all the considerations on government of early Confucians, especially Mencius, together. People with merits that are selected from a level playing field should be entrusted with ruling. But people's satisfaction should be clearly expressed and serve as the final measurement of the effectiveness of ruling. The Confucian ideal government that is by the competent and virtuous people will then require some channel for the popular will to be expressed and measured, and in this sense the masses still need to participate in politics to a certain degree. If \"government by someone\" means participation by someone, the Confucian ideal government should be understood as \"for the people, and partly by the competent and virtuous people and partly by all the people.\" It is a hybrid regime that combines popular elements with meritocratic elements.\n\n## 4 Confucianism and Liberal Democracy\n\n### 4.1 The Compatibility Between the Confucian Ideal Regime and Liberal Democracy\n\nIn the previous two sections, I discussed how early Confucianism addressed problems of China's early modernity. These problems were also shared by Western modernity. Thus, the contemporary relevance of the Confucian ideas is obvious. But in most cases, I didn't discuss directly and explicitly how they interact with contemporary ideas, especially ideas of liberal democracy, which are dominant in the contemporary world. I will discuss these interactions in this section. Many of the Confucian ideas were already introduced in previous sections, but some of them were not, and these will now be discussed.\n\nOn the surface, Confucianism is fundamentally in conflict with ideas considered essential to liberal democracy. Liberal democracy is often perceived as being built upon individualism, the pursuit of self-interest, equality, rule of law, and market economy (capitalism), whereas Confucianism is said to advocate collectivism, altruism, hierarchy, rule by virtue, and the belittlement of commerce.\n\nSome of the accusations made against Confucianism are, however, plainly false. Early Confucians never condemn commerce. In Mencius 3A4, Mencius emphatically claimed that \"it is a matter of fact that things are unequal,\" and used this claim to challenge a radical form of egalitarianism in the economy. Confucius claimed that \"it is a shame to be poor and humble when the Way prevails in the state,\" and he considered wealth and fortune to be bad only when the state is in bad shape, meaning that the wealth cannot be gathered in a virtuous manner (Analects 8.13). Thus, Confucians are only in conflict with a radical form of capitalism in which greed is celebrated, and the gathering of wealth is justified no matter how it is achieved.\n\nEarly Confucians did emphasize the role of virtue, but this doesn't mean that they allowed no room for the application of laws. For example, if we read carefully Confucius' own words on the relations between virtues and laws (Great Learning Chap.\u200b 4; Analects 2.3, 12.18, 12.19, 13.3, 13.6, and 20.2), the conclusion we should draw is as follows. First, the use of laws alone wouldn't be sufficient for a state to be good, for the people may be law-abiding, but they would always be ready to violate laws whenever they have a chance if they had no basic virtues. Such a state is in constant danger of degenerating into chaos. Second, for a state to be stable for the right reasons, the masses need to have some basic virtues, and more importantly, the leaders have to set examples for their people. Third, ideally, if all people were virtuous, laws wouldn't be necessary. But from the discussion of the previous section, we saw that early Confucians didn't have high expectations of the common people. In particular, they wouldn't expect the masses to be so virtuous as to regulate themselves. Thus, laws can and need to be used as \"fallback mechanisms\" that regulate the common people as the last resort. Still, laws themselves should be the embodiment of some basic moral concerns. Fourth, there are also matters, such as people's manners, that cannot or should not be regulated by laws, and they should be guided by li, that is, social norms that have moral content. In this sense, li and basic moral codes (such as basic family values) also serve as a \"fallback mechanism\" to legal regulations.\n\nIt is true that the Confucian ideal regime contains strong meritocratic elements. But from the discussion in the previous section, we saw that this meritocracy is built upon equality. That is, early Confucians, especially Mencius and Xunzi, believed in the equality of all in terms of their potentials to become wise and virtuous. The door to climbing up the political ladder is open to everyone, and the government is held accountable for offering all necessary conditions it can to people to move up in the political hierarchy.\n\nEarly Confucians did emphasize the effort to cultivate one's compassion toward others. But this doesn't mean that everyone is able to do so. More importantly, this apparently other-oriented requirement is actually for one's own good, and is not built upon the negation of the self. Confucius stated that the ideal person is for himself, and the non-ideal person is for others (Analects 14.24). Early Confucians differ from radical individualists because they understood the self as fundamentally social and intertwined with others. Again, Confucius claimed that the establishment of oneself is inseparable from the establishment of others (Analects 6.30). As mentioned earlier, Mencius stated that, lacking five basic human relations, a well-fed human look-alike shouldn't be called a human being, and is close to beasts (Mencius 3A4). Thus, the Confucian understanding of the self is different from the understanding of an autonomous self that can exist independent of social conditions, but early Confucians nevertheless acknowledged the self as an entity.\n\nTherefore, the conflict between Confucianism and ideas often associated with liberal democracy is not as sharp and fundamental as it appears to be. But one can argue that there is still conflict between the two, especially if we take a radical version of certain democratic ideas. The questions, then, are the following. First, how can we justify these ideas themselves? For example, if we take a radical version of the autonomous individual, believing that individuals do or should exist before or independent of society, how can we justify it? If we take it as a descriptive account, it is in conflict with the known empirical facts.12 If we take it as a normative account (i.e., human beings should be autonomous individuals), a little exposure to the history of philosophy will tell us that the indisputable justification of it is even harder to establish.\n\nLet us try another indirect defense of the importance of these ideas, that is, by taking advantage of the almost sacred, \"end-of-history\" status of liberal democracy. One can argue that in order to embrace liberal democracy, we have to embrace these ideas. That is, to embrace these ideas is a necessary condition for accepting liberal democracy. If so, there is an even bigger problem. Clearly, a significant minority or even a majority in any large state won't embrace these ideas if no oppression is applied, which is dictated by the fact of pluralism. If accepting these ideas is a necessary condition to embrace democracy, the fact of pluralism will mean that a significant minority or a majority won't accept democracy. How, then, can we have a democracy that is both pluralistic and stable for the right reasons (i.e., not achieved through oppression and other non-liberal means)? This is precisely the question Rawls raised in his later philosophy (Rawls 1996: 36\u201338; see also xxvii and 78).\n\nTo solve this problem, let us take a look at Rawls' answer. Very simply put, it is to make liberal democracy a free-standing political concept, freed from metaphysical ground, including those \"democratic ideas.\" What is necessary for democracy to enjoy stability for the right reasons is for different \"comprehensive doctrines\" (such as a moral metaphysics) to endorse this free-standing concept from itself. This concept doesn't have to be a clearly defined concept, and only needs to be the \"overlapping consensus\" among all the doctrines that can endorse liberal democracy as a political concept. Put differently, what Rawls did was to \"thin down\" the concept of liberal democracy, and to allow it to be read differently (within a loosened boundary) by different comprehensive doctrines. Moreover, the concept of liberal democracy doesn't have to be the highest concept in these doctrines, and doesn't even have to be derivable from these doctrines. All that is needed is an endorsement.\n\nOf course, Rawls still had his own idea of the loosened boundary. If we put this aside, but follow his general approach to this problem, we can argue that for Confucianism to be compatible with liberal democracy, it doesn't have to embrace the aforementioned \"democratic ideas,\" and only needs to endorse liberal democracy with its own reading.\n\nFor example, as already argued, Confucians can endorse the rule of law, taking it as a fallback mechanism and an effective way to regulate common people, although it cannot take it as the highest, sacred principle. Certain moral values are. But this doesn't mean that they can be used to interfere with laws arbitrarily. Rather, these values can play their supreme role through laws that are embodiments of these values, and regulate people's behavior where laws are silent. Moreover, these values are very thin, serving the common interests of all reasonable citizens and leaving the matter of faith and comprehensive belief system to each to choose.13\n\nAnother important issue is rights. If rights have to be built upon the view that we human beings are or should be autonomous individuals, and, as such, we have inalienable rights, Confucians can't accept them. But as in the case of the rule of law, Confucians can endorse certain rights as fallback mechanisms. Confucians can also endorse certain other rights by subjecting them to some higher good in Confucianism. For example, Confucians can endorse freedom of speech by arguing that this is essential to good policy-making because, as argued earlier, the people's will has to have a secure channel of expression, and Confucians can endorse the view that open discussions among the informed and concerned citizens are more likely to lead better political decisions.\n\nAnother possible way for Confucians to endorse rights is as follows. Confucians prefer the language of obligations, but this language can lead to practices similar to how the language of rights does. For example, using the language of rights, we can argue that a criminal has a right not to be tortured. For Confucians, if someone commits a heinous crime, feels no remorse whatsoever, and has no possibility to ever repent (although Mencius might think that there is always a chance of rehabilitation), he or she is not even a real human being, and it is ridiculous to talk about his or her \"human\" rights. Even so, the torture of him or her shows that those who inflict torture lack compassion. The criminal should be punished, but no excessive punishment should be allowed. To enjoy carrying out excessive punishment is a sign of lack of humanity. Thus, Confucians would argue that even though the criminal has no human right not to be tortured, we, as human beings who are superior to this criminal, have an obligation not to torture him or her. Generally, the readings of certain practices may be different between Confucians and some rights theorists, but there can be sufficient overlapping consensus on the practical level, if not on a theoretical level as well.\n\nOne can argue that there is a profound distinction between rights and obligations: rights are enforceable or claimable, but obligations are not. But it is not clear why the latter are not. For example, one's obligation not to torture a criminal can be claimed by the criminal and enforced by law. One can further argue that obligations claimed or enforced by law are oppressive. But again, this seems to be a myth. Why is the claim and enforcement of not torturing a criminal, or a father's obligation to pay child support oppressive? Don't we do it all the time in a liberal democracy, though under the language of rights?\n\nAnother objection to the Confucian \"rights\" is that they are fragmented. If we start with autonomous individuals and then find an a priori way to list their rights, it will be far more systematic. But the problem is how we can ever agree upon one a priori way to do it. The distinction between the a priori and the arbitrary can be arbitrary. With the same reasoning, we can answer another objection: Confucian \"rights\" are often subjected to some higher good, and can thus be sacrificed when in profound conflict with this higher good. For example, freedom of speech, in Confucianism, is justified on the grounds of good policy-making. If a speech is obviously useless to discovering good policies and is somehow problematic, it can be suppressed in the Confucian regime. But unless we take freedom of speech as supreme, it is an open question whether and where there should a limit on freedom of speech. Even among contemporary liberal democracies in the West, there are various degrees of limit to this freedom (for example, the censorship of Nazi symbols in Germany and of pornography and racism in the U.S.). The potential restriction Confucians may impose on speech is then not problematic on its own, although, clearly, we can discuss where the proper limit should be, which, I think, is a far more interesting and promising question to discuss.\n\nIn sum, rights can be largely endorsed in Confucianism, although they may be read differently. Confucians may have different \"rights,\" for example, a child's obligation to support his or her old parents, the government's obligation to provide education to its citizens, etc., and may not endorse certain rights or may not endorse them to the same degree as they are in certain liberal democracies. That is, even with a Rawlsian thin requirement, there may still be some remaining real differences between commonly received rights and Confucian \"rights.\" But whose position is better is an open question.\n\nFinally, on democratic participation and election, Confucians can accept democratic participation as a way for people's will to be expressed, and can use election as a way to select the wise and virtuous. The Confucian reading of election may even improve on the mores of political culture. For example, the dominant ideology in American democracy is that election is a way to get rid of the bad politicians, and this can lead to a \"mud-slinging\" culture which breeds cynicism in politics. Under the Confucian reading, a political candidate will argue that \"I am better than my opponent\" instead of \"My opponent is worse than me,\" and this might push politics to higher goals, instead of a race to the bottom.14\n\nBut a fundamental difference remains. Although Confucians can accept popular participation as a way for the popular will to be expressed, and in this way the popular will can be a factor in the process of political decision-making, Confucians must give room to the wise and virtuous in order for them to have a voice in this process. As argued in the previous section, the Confucian ideal regime is a hybrid regime. In the following, I will show how this Confucian regime is actually better than the present liberal democratic regimes in dealing with some fundamental issues of modern states.\n\n### 4.2 Fundamental Problems Within Liberal Democracies\n\nAs we saw, even with a \"thin\" reading of liberal democracy, there are still remaining differences between liberal democracy and the Confucian ideal regime. But if we are not close-minded and don't believe that history ends with liberal democracy, we shouldn't dismiss the Confucian regime because of the remaining differences, but should ask the question: which regime is actually better in facing the contemporary world?\n\nIn my view, there are four problems with today's liberal democracies in facing the contemporary world. First, the ideologies behind some people's readings of liberal democracy are problematic because they lead to problems in contemporary societies, such as a radical version of individualism that denies any sacrifice for the common good or for others, a blind faith in the market economy, the assumption of the equally evil nature of all human beings that serves as the foundation of check and balance, etc. Confucians would be against these ideologies, and thus offer a cure to them. Of course, one can imagine that these ideologies can be challenged and corrected by some liberal theories as well.\n\nThe second problem is that, combined with the ideology of radical individualism and the retreat of moral education, the institution of one person one vote has a built-in defect of not being able to pay sufficient attention to non-voters, such as past and future generations and aliens. For non-voters have no voice in this process, and voters are not guided to consider the interests of non-voters. This is perhaps why issues about budget deficit and environment (interests of future generations), foreign aid (interests of aliens), etc. are hard to address adequately in present democracies.\n\nThird, even among voters, the vocal and powerful voters tend to suppress the opinions of the silent and the silenced. Especially when the rule of law is not well-established, this leads to problems such as the mistreatment of minorities within a state, and policy-making being hijacked by the so-called \"issue-voters\" who are opinionated about certain matters (such as abortion) and let these matters override all other concerns.\n\nFourth, even in terms of interests of the voters, the voters themselves are often not the best judges. There are many studies that support this.15 One can argue that contemporary democracies are not directly participatory, but representative. But if voters can't understand their interests well, how can they understand which representative represents their interests well?\n\nThe causes of the last three problems, in my view, are the following. First, it is argued that the institution of one person one vote itself encourages citizens' interests \"to shrink and center on their private economic concerns to the detriment of the bonds of community\" (Rawls 1999: 73).16\n\nSecond, the size of most contemporary states is much too large, making state affairs too complicated for common people to understand and making political manipulations easy to achieve and difficult to detect. Moreover, the large size of the state makes possible the large concentration of wealth These wealthy individuals and organizations (for example, think of big global companies) then develop interests of their own that are not necessarily in line with the common good of a state, and they have a motive and the ability to manipulate politics in their favor.\n\nThird, to make the matter worse, most people in contemporary societies have to work for their living. In Mencius 3A4, Mencius argued that a ruler needs to have leisure in order to rule. But this necessary condition for someone to participate in politics is lacking in today's societies. Again, contemporary democracies are representative, but the lack of leisure among citizens together with the complexity of state affairs make them disqualified even from selecting the right representatives.\n\nFourth, even if one had some remarkable capacity and went through all the troubles, his or her vote would be equivalent to someone else's who votes in an uninformed or casual way. Indeed, statistically, in a large society, one vote never counts. This means that to vote responsibly or just to vote is not a rational decision one should make (Hardin 2002).\n\nFifth, there are people who have no interest in politics or choose not to be concerned with politics. In a liberal and pluralistic society, their way of apolitical life should be tolerated. But their political voice should also be limited (either through their own choice or, if we don't think we can count on their good will, through some political arrangement). Otherwise, political decision-making will be jeopardized.\n\n### 4.3 Why Is the Confucian Ideal Regime Better?\n\nMany liberal and democratic theorists also see some of the four problems with contemporary liberal democracies listed above, and try to offer their own solutions within the framework of liberal democracy (see, for example, Rawls 1996, 1999; Ackerman and Fishkin 2004, 2005). They also realize that, for participatory democracy to function well, voters need to meet certain criteria, such as getting informed and having some basic moral sense. The state should offer all necessary means for voters to meet these criteria, such as satisfying people's basic needs, offering education, and making it possible for voters to deliberate on political matters (such as having a day called \"deliberation day,\" as Ackerman and Fishkin 2004, 2005 suggested). But as we saw in the previous subsection, the problems are rooted in the nature of modern states and human psychology. As long as modern states remain large and populous, most people in these states have to work for a living, and human willingness and capacity to understand politics is limited; these problems cannot be ultimately solved by the internal tinkerings. These tinkerings offered by democratic thinkers, already radical in today's politics, may improve the situation, but can't fundamentally and adequately address the aforementioned problems.\n\nIn contrast, the Confucian ideal regime may be better equipped to address these problems. As I have argued, Confucians can endorse the rule of law and most rights. They acknowledge governmental responsibilities to the citizens, such as satisfying their basic material and spiritual needs, and hold government accountable for that. The same is also advocated by liberal and democratic thinkers. But Confucians would also emphasize the basic moral and political education of all citizens, thicker than the civil education in contemporary democracies but still thin enough for citizens with a variety of reasonable comprehensive doctrines to endorse.\n\nConfucian moral and political education is rooted in Confucian political philosophy discussed in the second and third sections of this chapter, and is able to address some of the issues discussed in the previous sections. It tells citizens that voting is not a mere expression of one's narrowly-defined material interests or one's article of faith, and it is not a peaceful process for fighting factions to achieve a temporary truce through a non-violent suppression of minority opinions. Rather, voting should be based upon one's interests broadly construed, which include short- and long-term material interests, interests of others, and moral needs, and voting is a process of discovering public good. In order to vote, one has to have these understandings (which is a moral requirement) and also has to be informed about political affairs. Check and balance as well as other political institutions are not only there to punish the bad, but also to select the good. Government is not a necessary evil, but should be an institution that secures for its citizens a basic standard of living and social relations. When the politicians do their job, they should be considered politically superior and be respected accordingly.\n\nThe Confucian \"civil\" education may already be considered too radical for democratic thinkers. But there is something even more radical from a democratic point of view. Although Confucians such as Mencius believe in equality of all human beings in terms of their potential to become wise and virtuous, and that government is responsible for helping people to realize their potentials with any means necessary, they nevertheless believe that only a few human beings can actually \"make it.\" As we saw, Confucians' concern with the masses' lack of intellectual and moral capacities that are necessary to meaningful political participation is supported by the aforementioned conditions of contemporary states. Then, Confucians think that to correct this deficiency of the masses, the competent and virtuous should be given more power in the process of political decision-making. These people are trained and selected on the basis of whether they are compassionate toward others and are able to grasp political matters, and thus they are better equipped for dealing with problems that challenge contemporary liberal democracies.\n\nThis radical reform of liberal democracy is destined to encounter many objections, and it is beyond the scope of this short chapter to deal with them in any adequate manner. Let me just very briefly mention a few. One issue is how to institutionalize this Confucian hybrid regime. There are many ways of doing so.17 One of them is to turn the upper house of a bicameral legislature into a house of the competent and virtuous. Members of this house can be selected through exams (or making exams a requirement for them to be selected) that test their knowledge relevant to political decision-making and through evaluations of their performances in past assignments in order to see if they have the moral character relevant to being a good politician; they can also be selected from high-officials or from large NGOs who have done a good job in their offices. In the past the Chinese have experimented with many ways of selecting the wise and the virtuous, and thus traditional Chinese politics offers a rich resource for thinking about how to select members of this meritocratic branch. The other house can be popularly elected, and clearly, there are different ways to distribute power between these two houses. Clearly, to make all the relevant procedures fair, laws and institutional regulations are necessary.\n\nThe Confucian hybrid regime may remind people, especially those in the West, of the kind of aristocracy in the Middle Ages, or democracies at their earlier stages, in which many people were unfairly disenfranchised and thus held a deep resentment toward the elite. But as we saw, the Confucian meritocracy is built upon equality and upward mobility. The door to meritocracy is open to everyone both in the sense that everyone has the potential to climb up the political ladder and in the sense that the government is responsible for offering all means necessary for this mobility. Since there is no cut-off line after which one cannot participate in politics anymore, one's seclusion from meritocratic institutions such as the upper house is never permanent. All the above may address the problems of disenfranchisement and resentment.\n\nAnother common objection to my arguments here is that contemporary democracies are actually already meritocratic. There are more wealthy and well-educated people in the legislature and other political institutions, proportion-wise, than in the populace. This may well be true. But first, these meritocrats may have the wrong kind of merits from a Confucian point of view. The Confucians want people with compassion and competence to have a greater voice in government, and merits such as wealth and celebrity status by themselves don't count. Second, even if some politicians have the right kind of merits from a Confucian point of view, they are \"closeted\" meritocrats. Due to the pressure of popular elections, sometimes they have to hide their merits in order to appeal to the average voters, and their political decisions are often hijacked by the short-term and oftentimes misguided interests of these voters. That is, they fail to play the role of meritocrats, as expected by Confucians and necessary to right the wrongs of participatory democracy.\n\nTo propose the Confucian hybrid regime as a correction of contemporary liberal democracies is radical. There are many details that cannot be dealt with due to the limited space of this chapter. But even if there were more space, I am sure many readers would still find faults with this proposal. Here I can only beg for charity and open-mindedness. As I argued in this chapter, Confucian ideas may have had real influences on European modernity. In contrast, for whatever reasons, the Chinese didn't learn from the West soon enough to avoid the humiliations in recent history. If we learn from these historical lessons, if we understand early Confucians as attempting to wrestle with issues of modernity and offering competing paradigms, and if we don't consider the end of history has already come, we should open ourselves to the \"old\" and the \"alien,\" in order to usher in a better future for ourselves.\n\nReferences\n\nAckerman, Bruce, and James Fishkin. 2004. Righting the ship of democracy. Legal Affairs, January\/February 2004, 34\u201339.\n\nAckerman, Bruce, and James Fishkin. 2005. Deliberation day. New Haven: Yale University Press.\n\nBai, Tongdong. 2008. A Mencian version of limited democracy. Res Publica 14(1): 19\u201334.CrossRef\n\nBai, Tongdong. 2009a. The price of serving meat \u2013 On Confucius's and Mencius's views of human and animal rights. Asian Philosophy 19(1): 85\u201399.CrossRef\n\nBai, Tongdong. 2009b. 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Beyond good and evil, eds. Rolf-Peter Horstmann, and Judith Norman. Trans. Judith Norman. Cambridge: Cambridge University Press.\n\nPollak, Michael. 1998. Mandarins, Jews, and missionaries: The Jewish experience in the Chinese empire. New York: Weatherhill.\n\nRawls, John. 1971. A theory of justice. Cambridge: Harvard University Press.\n\nRawls, John. 1996. Political liberalism. New York: Columbia University Press.\n\nRawls, John. 1999. The law of peoples with \"The Idea of Public Reason Revisited\". Cambridge: Harvard University Press.\n\nShapiro, Sidney. 1984. Jews in old China, studies by Chinese scholars. Trans. and ed. Shapiro Sidney. New York: Hippocrene Books.\n\nSima, Qian \u53f8\u9a6c\u8fc1. 1981. Records of the grand historian\u53f2\u8bb0. In the twenty-five historical records. Shanghai: Shanghai Guji Press.\n\nXu, Xin. 2003. The Jews of Kaifeng, China. Jersey City: KTAV.\n\nFootnotes\n\n1\n\nAs I will argue in this section, to use \"modern\" to describe a group of philosophers is with regard to the nature of problems they dealing with, and not with regard to the time they live.\n\n2\n\nFor references in English, see Bai 2011, 2012a, b.\n\n3\n\nIt is bold to claim that China's transitions during the SAWS were a form of modernity, comparable to the European transitions to its modernity. But there are people who have made this claim independent of my work, and have conducted their own comparative studies. See, for example, Fukuyama 2011; Hui 2005.\n\n4\n\nFor the Chinese influence on European modernizations, see Hobson 2004.\n\n5\n\nConfucius didn't specify which aspect of Zhou he talked about here, and it is only one possibility that he referred to the Zhou li.\n\n6\n\nThe English translations of Chinese classics in this chapter are all mine. For different translations of the Analects and the Mencius, see Lau 2000, 2003. For different translations of the Great Learning and the Doctrine of the Mean, see Chan 1969.\n\n7\n\nSee, for example, Shapiro 1984, Pollak 1998, and Xu 2003. One on-line account can be found here: http:\/\/\u200ben.\u200bwikipedia.\u200borg\/\u200bwiki\/\u200bKaifeng_\u200bJews (accessed on February 16, 2012).\n\n8\n\nOf course, the (shameful) elimination of most of Native Americans greatly helped solve the unity problem for the U.S.\n\n9\n\nFor a more detailed discussion, see Chap.\u200b 6 of Bai 2009b.\n\n10\n\nA far more detailed discussion of the topics in this and the next sections can be found in Chaps.\u200b 2, , and  of Bai 2009b. For some earlier and partial versions of these chapters, see Bai 2008, 2009a.\n\n11\n\nThis line was from the chapter \"Taishi \u6cf0\u8a93\" (\"Great Declaration\") of Shangshu\u5c1a\u66f8 (The Book of Documents), and this chapter was allegedly a Zhou document. This was why I said earlier than the humanization of divine will might have happened in Zhou already. But whether this chapter was really a document from Zhou was controversial.\n\n12\n\nSee, for example, the primatologist Frans de Waal's criticism of this asocial version of individualism (de Waal 2006).\n\n13\n\nI use the term \"reasonable\" in this chapter as Rawls did in Rawls 1996, 1999.\n\n14\n\nSee Chan 2012 for a more detailed discussion.\n\n15\n\nSee, for example, Ackerman and Fishkin 2004, 2005, and, more recently, Caplan 2008. For a more popular account, see Kristof 2008.\n\n16\n\nTo be clear, this view is not Rawls's. It is what he describes as a \"Hegelian view.\"\n\n17\n\nChapter  of Bai 2009b contains many detailed discussions of this issue and other issues with Confucian hybrid regime. An earlier English version is Bai 2008. See also Bell 2006.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_15\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 15. Ultimate Reality and Self-Cultivation in Early Confucianism: A Conceptual\/Existential Approach\n\nZhonghu Yan1\n\n(1)\n\nHangzhou Normal University, Hangzhou, China\n\nZhonghu Yan\n\nEmail: zhonghuyan@yahoo.ca\n\nAbstract\n\n\"Self-cultivation\" is the most commonly accepted English equivalent for the Chinese term xiushen\u4fee\u8eab, which covers the exercises including physical as well as psychological and spiritual levels. In the Confucian context, this self-cultivation has an undeniably strong moral sense (Ames 1985). However, this morality has also a metaphysical dimension that has frequently been overlooked by scholars. Therefore, in this chapter, I will attempt to illustrate this profound and often neglected dimension of classical Confucianism by a conceptual and existential analysis of its founder Confucius' Analects. I will begin by discussing the Ultimate Reality as conceived in the Analects and proceed to discuss the three consummate virtues that Confucius expounded, namely, wisdom, benevolence, and courage. I will conclude the essay by highlighting the metaphysical basis of Confucian self-cultivation.\n\nI wish to kindly acknowledge the permission from Religion Compass to reuse \"Ultimate Reality in Confucianism\" and from Cambria Press to reuse portions of An Existential Reading of the Confucian Analects.\n\n## 1 Introduction\n\n\"Self-cultivation\" is the most commonly accepted English equivalent for the Chinese term xiushen \u4fee\u8eab, which covers the exercises including physical as well as psychological and spiritual levels. In the Confucian context, this self-cultivation has an undeniably strong moral sense (Ames 1985).1 However, this morality has also a metaphysical dimension that has frequently been overlooked by scholars. Therefore, in this chapter, I will attempt to illustrate this profound and often neglected dimension of classical Confucianism by a conceptual and existential analysis of its founder Confucius' Analects. I will begin by discussing the Ultimate Reality as conceived in the Analects and proceed to discuss the three consummate virtues that Confucius expounded, namely, wisdom, benevolence, and courage. I will conclude the essay by highlighting the metaphysical basis of Confucian self-cultivation.\n\n## 2 Ultimate Reality in Early Confucianism\n\nThe concept of tian \u5929 (Heaven) appears often in early Confucian literature. It does not just refer to the physical sky but also to the Ultimate Reality behind the visible world. When human beings suffer, they invariably call upon Heaven in some way or other. In contemporary colloquial Chinese, such an expression is wo de tian a \u6211\u7684\u5929\u554a, which means \"Oh, My Heaven.\" In English, we have \"Oh. My God!\" or \"Good Heavens!\" Surprise and shock lead us to search for the cause of suffering. Confucius believed that suffering is part of the physical and social structure of human beings. The correct attitude should be to acknowledge the inevitability of this aspect of human existence, as he said to his disciple Zigong, \"I do not complain against Heaven, nor do I blame Man\" (Analects14.35). What can we infer about the conception of Heaven in early Confucianism?\n\n> Firstly, tian is the force behind all natural processes. Thus we read in the Analects,\n> \n> The Master said, \"I am thinking of giving up speech.\" Zigong said, \"If you did not speak, what would there be for us, your disciples, to transmit?\" The Master said, \"What does Heaven ever say? Yet there are four seasons going round and there are the hundred things coming into being. What does Heaven ever say?\" (Analects 17.19)\n\nHeaven causes the seasons to alternate and plants to grow. And it sustains our lives. Because the Chinese did not make clear linguistic distinctions between the created world and the force of creation itself, the debate on the immanent and transcendent nature of Heaven has persisted (see Huang 2007). The debate is due largely to our dualistic thinking: Something immanent must not be transcendent and something transcendent must not be immanent.2 The Chinese mind does not accept this strict dualism.\n\nSecond, tian is also a conscious being. Heaven creates the myriad things in the world, but it does not seem to leave them to their own devices. It seems to be always overseeing human conduct. It is able to perceive the heart of the people. When Confucius felt that nobody understood him or his vision, he lamented, \"I start from below and get through to what is up above. If I am understood at all, it is, perhaps, by Heaven\" (Analects 14.35). As a man with a great vision, Confucius strongly felt that only Heaven understood his heart. Confucius may even have communicated with Heaven through prayers. He held the transcendent source in the greatest respect. We hear Confucius say, \"When you have offended against Heaven, there is nowhere you can turn to in your prayers\" (Analects 3.13).\n\nAs a conscious being, Heaven seems to be watching over the moral conduct of human beings. Once when Confucius' disciples wished to bring honor to him by acting or sending their own disciples to serve as his retainers, Confucius reprimanded them by saying: \"In pretending that I had retainers when I had none, who would we be deceiving? Would we be deceiving Heaven?\" (Analects 9.12)\n\nFinally, Heaven is the assigner of a person's earthly mission. Heaven also assigned a mission to a person of its choosing. In the Chinese tradition, prophets as seen in the Old Testament are not common,3 but once in a long while, the Chinese do see sages emerge, whose mission was believed to come from Heaven. In the Analects, a few such sages are mentioned, such as Shun, Duke of Zhou and King Wen. It is significant that Confucius considered himself to be called by Heaven to carry out the mission to transmit the Zhou culture. This is apparent in the following passage.\n\n> When under siege in Kuang, the Master said, \"With King Wen dead, is not this culture of ours invested here in me? If Heaven intends this culture of ours to be destroyed, those who come after me will not be able to have any part of it. If Heaven does not intend this culture of ours to be destroyed, then what can the men of Kuang do to me?\" (Analects 9.5)\n\nThis sense of mission is also confirmed in Analects 3.24, where a border official said to Confucius' disciples about their Master: \"The world has long been without the Way. Heaven is about to use your Master as a wooden tongue for a bell.\" Confucius' disciple Zigong also expressed the idea that Heaven exercised its will on the Master, when he said, \"Heaven set him on the path of sagehood\" (Analects 9.6).\n\nFrom being a force behind the movement of nature, to a conscious being and a being that assigns mission, Heaven is seen at the same time as far away and near, as immanent and transcendent. This dual nature of tian sheds a new light on the question of theodicy from a Confucian perspective. As Confucius did not believe in an all-knowing, all-powerful, absolutely transcendent being, tian cannot be held fully responsible for the evil that causes human suffering. As human beings have a participatory role in maintaining a good universe, they should not be exempt from the responsibility for causing suffering. What Confucius proposed in times of suffering is not to complain against Heaven nor blame one's fellow men. Instead, one should take suffering as a learning opportunity.\n\nIn his Chinese Thought from Confucius to Mao Tse-Tung, G. H. Creel asked: \"What did Confucius understand by the term 'Heaven'?\" His response is, \"Not anthropomorphic being. Heaven was seldom so connected in his time, and there is explicit reason for rejecting this idea in connection with Confucius.\" He continues, \"If we examine the way in which Confucius refers to Heaven, it appears that this term stood, in his thinking, for a vaguely conceived moral force in the universe. He placed the utmost emphasis on striving by the individual, but he seems to have hoped that Heaven, as we say, 'helps those who help themselves'\" (Creel 1953: 36). Professor Creel's interpretation of Heaven has a strong appeal to those who no longer hold a religious frame of mind. But I beg to differ from him with my analysis of the passages above. Considering Heaven as a conscious being is not only compatible with the general thrust of thought in the Analects, but also compatible with the belief in the Mandate of Heaven in the time of Confucius and earlier. In the popular religious belief, which may be traced back to a very early period, Heaven is also called tiangong \u5929\u526c, the Lord of Heaven, which suggest a personal character of Heaven. Professor Creel might have been influenced by Max Weber in his rationalistic interpretation, as he quoted a passage from the latter with a strong overtone of rationalism:\n\n> In the sense of absence of all metaphysics and almost all residues of religious anchorage, Confucianism is rationalist to such a far-going extreme that it stands at the extreme boundary of what might call a \"religious\" ethics. At the same time, Confucianism is more rationalist and sober, in the sense of the absence and the rejection of all-non-utilitarian yardsticks, than any other ethical system, with the possible exception of J. Bentham. (Creel 1953: 38\u201339)\n\nIt is the dual character of Heaven that can help to explain the reality of Chinese experience. The fact that the emperor was called the son of Heaven and the fact that sacrifice was offered at the place of worship called the Temple of Heaven suggest further that Heaven after all is not an impersonal thing. As the Ultimate Reality, Heaven also provides the religious ground and ultimate purpose of self-cultivation as will be seen below.\n\n## 3  Zhi \u77e5\/\u667a (Wisdom) and Its Relation to Ultimate Reality\n\nOne of the goals of self-cultivation in early Confucianism is the acquisition of wisdom. Much scholarly ink has been spilled on key notions of Confucius' teaching such as ren \u4ec1, benevolence, and li \u79ae, rites. While the former is considered the goal to which the students are taught to aspire and the qualities that the future leaders of the nation should possess, the latter is considered the concrete practice toward this goal. These are surely the two key notions of the Analects, considering their number of occurrences: 104 instances of ren and 74 instances of li. Confucius also emphasized a separate human capacity, without which ren is incomplete. This human capacity is called zhi \u77e5\/\u667a, a term that could perhaps be translated as \"knowledge\" or \"wisdom.\"\n\n### 3.1 Notion of Zhi in the Analects\n\nThere are 116 instances of the word zhi in the Analects. All but five are attributed to Confucius. In 89 cases, it is used as a verb and in 22 cases it is used as a noun or an adjective. When it is used as a verb, it takes either a single noun or a sentence as its object. When it takes the sentence as its object, it expresses a kind of factual statement:\n\n> The Master said, \"I don't see (zhi) how a man can be acceptable who is untrustworthy in word. When a pin is missing in the yoke-bar of a large cart or in the collar bar of a small cart, how can the cart be expected to go?\" (Analects 2.22)\n\nThe factual statement is also seen in Analects 5.8 and 14.1. Invariably, zhi means \"to judge\" when it expresses a factual statement. When it takes a single object the matter becomes much more complicated, for there is a wide range of objects that zhi takes. I will divide them into four categories.\n\nA. Persons. Zhi involves first of all the capacity for appreciation and recognition of one's talent (Analects 1.16 and 13.2). This must have been a very big concern for Confucius' disciples whose implicit purpose in studying with the Master was to gain an official position. Time and again, Confucius asked them not to worry about the position as long as they have the right qualifications (Analects 4.14, 14.30 and 15.19). Instead, they should develop the capacity to know other people. We will discuss what the knowledge of other people means when we come to the usage of zhi as a noun. For now, it is worth bearing in mind that recognizing a whole person in others is never considered an easy job. Even Confucius himself would at times lament that he is not recognized or understood by the people.\n\n> The Master said, \"There is no one who understands (zhi) me.\" Zigong said, \"How is it that there is no one who understands (zhi) you?\" The Master said, \"I do not complain against Heaven, nor do I blame Man. In my studies, I start from below and get through to what is up above. If I am understood (zhi) at all, it is, perhaps, by Heaven !\" (Analects 14.35)\n\nIt is from Heaven's perspective that Confucius is understood. How much less can we human beings understand another person!\n\nB. Li , the rites. Besides zhi ren \u77e5\u4eba, understanding or recognizing a person, Confucius strongly recommended that one understand the rites, as seen in Analects 3.22 and 7.31. A careful reading of the Analects tells us that understanding the rites does not mean theoretical knowledge; instead, one judges a person's practical performance as a basis upon which to judge whether he is zhi li \u77e5\u79ae or not. Therefore, there is a practical dimension to zhi. Without correct performance of the rites, one cannot position oneself well or win recognition in society. This is seen in the following passage:\n\n> Confucius said, \"A man has no way of becoming a exemplar person unless he understands (zhi) Destiny; he has no way of taking his stand unless he understands (zhi) the rites; he has no way of judging men unless he understands (zhi) words.\" (Analects 20.3)4\n\nWithout understanding or performing the rites correctly, one cannot have a smooth interaction with the people around one and one's career suffers.\n\nC. Speech. For Confucius, to know when, how and what to speak to others is very important, for it is through speech that one communicates to others what kind of person one is (Analects 1.14, 4.22 and 16.6). Equally important is to know how to interpret what others have said and identify the motives behind the words. Confucius especially warned against cunning and insincere words (Analects 5.25; 15.27). These words damage the integrity of a person who speaks them. Finally, the Master made it clear that \"he has no way of judging men unless he understand words\" (Analects 20.3), for apart from behavior, comprehending language is the only means to gain access to the minds of other people.\n\nKnowledge of words, then, involves a keen observation of the speaker as he reveals himself. Also, it involves the capacity to distinguish sincerity and insincerity in the spoken words and therefore gain knowledge of the speaker. This skill requires years of training.\n\nKnowing one's fellow human being as a whole person, practical knowledge of the rites and intuitive grasp of the metaphysical realm of existence and speech appropriately uttered or comprehended constitute the main components in deciding whether a person could be considered zhi, or a wise person. In the Analects, there is no distinction in character between the zhi as a verb meaning to know or understand and zhi as an adjective or noun meaning wise or wisdom. At least, we can say that a wise person is one who possesses salient features of wisdom.\n\nD. Wisdom and Metaphysical Reality. Besides knowing a person and the rites, the Analects also mentions the metaphysical level of existence such as ming \u547d, destiny and death. The passage quoted above speaks of how \"understanding destiny\" is essential for being a junzi \u541b\u5b50. The idea of such a relationship is reinforced in the following passage.\n\n> The Master said, \"At fifteen I set my heart on learning... at fifty I understood (zhi) the Decree of Heaven.\" (Analects 2.4)\n\nZhi ming \u77e5\u547d, understanding destiny, or zhi tianming \u77e5\u5929\u547d, understanding the decree of Heaven, is repeated in Analects 16.8 and 12.3. Ming and tianming are basically interchangeable. What does it mean to \"understand the Decree of Heaven?\" These passages suggest to us that zhi ming or zhi tianming does not necessarily mean that one understands the law of the universe. Instead, it means that there are things in the universe that one should be wise enough to be resigned to, or that one knows what purpose Heaven has for one. This kind of zhi is by no means an intellectual one, and involves an action in light of this understanding. This is best exemplified in two cases: one is when Confucius' disciple Bo-niu \u4f2f\u725bcame down with a very bad disease (Analects 6.10); the other is when he was asked about Yan Y\u00fcan \u984f\u6df5 who died early (Analects 6.3). In both cases, the Master mentioned ming, destiny, and accepted it.\n\nE. Who could be counted as zhi zhe \u77e5\u8005 (wise person)? The person who possesses wisdom is considered a wise man. The wise man displays certain characteristics presumably informed by his wisdom. A wise man knows what to do in a conflict (Analects 9.29, 14.28), knows how to address his speech to the right person (Analects 15.8), and takes delight in movement and change (Analects 6.23).\n\nTherefore, in Confucius' thinking, a wise man is a person who is able to cope with life's difficult situations and retain a healthy state of mind. Confucius never equates technical skill with wisdom. Indeed, he does not use the term zhi for technical skill. Instead, he uses neng \u80fd (Analects 9.6). Confucius did not regard technical skills as worthy of pursuit for a junzi.\n\nAs one of many virtues mentioned in the Analects, zhi is a quality that can be cultivated in a person. Ren, a virtue that is more socially oriented, seems to be more highly regarded by Confucius himself. When discussing the wisdom of Confucius, one needs to distinguish between its narrow sense and its broad sense. In its narrow sense, wisdom is a specific quality. In answer to Fan Chi's questions, Confucius used zhi in a narrow sense (Analects 6.22, 12.22). It is the mental capacity to know the people and things in the world and to act accordingly. In its broad sense, wisdom is the capacity to bring order to oneself, to the community and the whole world in such a way that a wise person lives a happy life. It is in the broad sense of the word that we find that wisdom has its metaphysical foundation. The kind of wisdom that Confucius pursued had its source in Ultimate Reality (Analects 9.29, 6.23). In a certain sense, the fear of Heaven is the beginning of wisdom.\n\n## 4  Ren and Ultimate Reality\n\n### 4.1 Where Do the Roots of Ren, Benevolence, Grow?\n\nThe etymology of ren \u4ec1 suggests to us how relationships with other people are indispensable for forming the character of ren. This is a very important suggestion, for the significance of the individual or individuality has been deemphasized. This emphasis on interpersonal relationship is central in Confucius' teaching about ren. For instance, we have a passage relating the words of his disciple Youzi \u6709\u5b50:\n\n> The exemplary person devotes his effort to the roots, for once the roots are established, the Way will grow therefrom. Being good as a son and obedient as a young man is, perhaps, the root of man's character (ren). (Analects 1.2)\n\nFamily is where the virtue of ren is first cultivated. One needs first to be a good son to one's parents and be sensitive to their needs. As a younger brother in the family, one should also be sensitive to the respect due to the elder brother. Failure to perform one's duty in the family, on the other hand, is considered not ren. For instance, when his disciple Zaiwo \u5bb0\u6211 expressed his reluctance to observe 3 years' mourning for his parents, Confucius said:\n\n> How unfeeling Yu is. A child ceases to be nursed by his parents only when he is three years old. Three years' mourning is observed throughout the world. Was Yu not given three years' love by his parents? (Analects 17.21)\n\nIf one is able to be a good son or younger brother, by extension, one will do well when he steps outside the home and meets people in the community. Sensitivity towards others' feelings is an essential quality; indeed, one meaning of ren that has come down to us as an idiom is mamu bu ren \u9ebb\u6728\u4e0d\u4ec1 (too unfeeling to be ren).5 This suggests that sensitivity is an essential quality for a ren person. Without this quality one is as unfeeling as dead wood.6\n\nThere is a scholarly debate, however, over whether ren is an inner psychological state or an external aspect of man (Hall and Ames 1987). While Tu Weiming and Chan Wing-tsit hold to the former point of view, Fingarette emphasizes the latter. If we take it that ren is the result as well as the process of cultivating sensitivity, the tension between the inner and outer interpretation breaks down. On the one hand, sensitivity is the basis for one's voluntary action that may involve attitude and will and is therefore an inner state. On the other hand, sensitivity is directed to an outside object or person and therefore is an external aspect of man. I suggest that sensitivity consists of a deeper core than what the disparate inner or outer meanings might convey. This deeper core is the source from which Confucian benevolence flows. Confucian benevolence is capable of responding to fellow human beings as well as the Being of all existences.\n\n### 4.2 How to Cultivate Ren\n\nThe questions of how much this deep core of human sensitivity is inborn and how much it is cultivated are not explicitly addressed in the Analects. But from the number of occurrences of this term, namely 104, the topic of ren is obviously a great concern of Confucius and his disciples. Its importance and complexity are indicated by the fact that his disciple Fan Chi asked about it 3 times, but was given a different answer on each occasion. And four other disciples asked the same question, but were also given quite different answers. By analyzing Confucius' responses to the question of ren, we should understand what this concept involves. I begin by analyzing Fan Chi's three exchanges with the Master.\n\n> 1.\n> \n> Fan Chi asked about benevolence. The Master said, \"The benevolent man reaps the benefit only after overcoming difficulties. That can be called benevolence.\" (Analects 6.22)\n> \n> 2.\n> \n> Fan Chi asked about benevolence. The Master said, \"Love your fellow men.\" (Analects 12.22)\n> \n> 3.\n> \n> Fan Chi asked about benevolence. The Master said, \"While at home hold yourself in a respectful attitude; when serving in an official capacity be reverent; when dealing with others do your best (be loyal). These are qualities that cannot be put aside, even if you go and live among the barbarians.\" (Analects 13.19)\n\nFan Chi seems to have been relatively dull-witted, and Confucius' answers tried to accommodate Fan's intellectual capacity. In Analects 12.22, Fan Chi's incapacity to understand Confucius' answers is made explicit. Even after Confucius further explained what he meant, Fan Chi was not able to understand. But from these exchanges, we can still get an idea of some complexity in the meaning of ren.\n\nThese exchanges point out that ren is not something easily defined, but there are some signs or actions that lead one to become a man of ren. From Confucius' answers to Fan Chi's questions, we can see that cultivating ren starts from the family and gradually extends to one's circle of concerns. The \"respectful attitude\" at home means filial piety and brotherly respect; this is the root of ren. This respect is translated into reverence when it comes to the discharge of official responsibility. And when it comes to interpersonal relationships in general, zhong \u5fe0, serious commitment, becomes a guiding principle, for this is the way to win respect or trust from other people. Confucius understood that these teachings are easy to remember but hard to practice. Therefore, he emphasized in Analects 6.22 that the practice of ren involves great difficulties. And in Analects 13.19, he emphasized that the practice of ren demands persistent effort. One should not abandon it even if one is in a barbarian state. Finally, the practice of ren requires that one love one's fellow men, ai ren \u611b\u4eba.\n\nThe love of one's fellow men does not mean that one need only love people from the same social class. And love here does not mean an emotional attachment towards others. Instead it means a sensitive concern for the welfare of others. This love differs from Mohist jian ai \u517c\u611b (universal love), which involves love without any distinction. For later followers of Confucius, the Mohist conception of love is too idealistic and against one's natural feelings. One should certainly be concerned for other people, but one must consider that people are different in relation to oneself. Not considering the differences is a sign of insensitivity. Insensitivity may defeat the very purpose of your love. The Confucian love derives its ultimate source from a firm belief in the prevalence of the Way. In this sense, it is similar to Christian love, which is inspired by a firm belief in the Kingdom of God.7\n\nThe meaning of ren is not exhausted by the above quoted exchanges. Other more quick-witted disciples also raised the question about ren. Confucius' answer to Zhong Gong \u4ef2\u5f13 (12:2) carries a similar spirit as in 13:19. Ren is first of all practiced at home. One should act in such a fashion that no one in the family has complaints. This attitude of respect should be maintained when one meets colleagues or people in the community. It should also be maintained when one acts in the capacity of an official. One should also find opportunities to improve oneself and befriend those who possess the quality of ren.\n\nOne sign of ren is appropriateness in speech, and Confucius often emphasized its importance. People tend to speak more than act. For Confucius, this should absolutely be avoided. In his answer to Sima Niu \u53f8\u99ac\u725b, reluctance to speak is considered a distinctive quality of ren (12:3).\n\nFor Confucius, willingness to practice ren depends on oneself alone, but through examples, de \u5fb7, virtue that accumulates through this practice can spread all over the world. This is probably what is meant by \"If for a single day a man could return to observance of the rites through overcoming the self, then the whole world would consider benevolence to be his\" (12:1).\n\n### 4.3 Intrinsic Rewards from the Practice of Ren\n\nThe analysis of the ren-related passages above suggests to us that everyone has the inborn capacity to become a person of ren. However, because human beings are driven by desires to satisfy themselves immediately, the ren-quality lacks opportunities to shine forth. For Confucius, if men are able to restrain desire through the rites, they will become persons of ren.\n\nRestraining one's desire does not necessarily mean asceticism. Following the rites is an aesthetic experience as well. One joins a cosmic dance, so to speak. The practice of ren surely needs discipline, for the whole being is involved, but the practice of ren has its intrinsic rewards. Indeed, only persons of ren can really enjoy a happy life. For example:\n\n> The Master said, \"One who is not benevolent cannot remain long in straitened circumstances, nor can he remain long in easy circumstances. The benevolent person is attracted to benevolence because he feels at home in it. The wise person is attracted to benevolence because its wisdom benefits benevolence's realization.\"(Analects 4.2)\n\nAnd twice it is mentioned that a benevolent person never worries (Analects 9.29; 14.28). Not only in words but in deeds, Confucius show himself to be a person who rarely worries. Once when Huan Tui threatened to kill him, Confucius remained very calm. He said, \"Heaven is author of the virtue that is in me. What can Huan Tui do to me?\" (Analects 7.23)\n\nIn another instance, when under siege in Kuang, the Master said, \"With King Wen dead, is not culture invested here in me? If Heaven intends culture to be destroyed, those who come after me will not be able to have any part of it. If Heaven does not intend this culture to be destroyed, then what can the men of Kuang do to me?\" (Analects 9.5) Just as an ordinary person, a person of ren will encounter difficulties, but a person of ren can face those difficulties with courage. In the words of Confucius, \"A person of ren is sure to possess courage.\" (Analects 14.4)\n\nConfucius' teaching does not promise anything for the life to come as Jesus' does. It is a teaching that maximizes one's happiness, nonetheless. Human existence is such that many things are beyond our control. For an uncultivated person, these things become the objects of worry or concern. Illness and death are the two most salient examples of this. However, for Confucius, existential concerns go beyond illness and death. As we live in the human world, how we interact with other members of the family and by extension with society at large is also a top concern. Confucius believes that people can try to cultivate themselves to such a degree that they can remain unworried under all circumstances. This quality is ren. Because of his sensitivity to his role in human relationships, such a one can always perform well and remain calm. Because of his sensitivity to forces beyond his control, he can accept what comes to him, even illness and death. Indeed, Confucius explicitly says that unless one understands his destiny, he cannot become an exemplary person (Analects 20.3).\n\n### 4.4 Seeming Paradox of Ren\n\nConfucius often commented on the central concept of ren. From these comments, we learn how qualities of ren may be acquired and what the roots of ren and the rewards of acquiring ren are. There are also a few passages that evade immediate understanding and seem paradoxical. On the one hand, Confucius rarely attributed ren to his contemporaries. Even his best and favorite disciple was said to be able not to go against ren for only 3 months, to say nothing of others who could only attain ren once in a long while (Analects 6.7). Perhaps Zengzi captured this spirit well when he said,\n\n> An exemplary man must be strong and resolute, for his burden is heavy and the road is long. He takes benevolence as his burden. Is that not heavy? Only with death does the road come to an end. Is that not long? (Analects 8.7)\n\nOn the other hand, Confucius told Yan Hui that if for a single day a man could return to observance of the rites through overcoming the self, then the whole world would consider benevolence to be his (Analects 12.1). Furthermore, we have:\n\nIs ren really far away? No sooner do I desire it than it is here in me. (Analects 7.30)\n\nFor Confucius, ren is both difficult and easy. It is easy because it is not something external to us. Every one of us has an inner sensitivity capable of responding to the world. As long as we remove impurities from it by cultivating our heart and mind, we are on the way to becoming ren or a responsive person. In a large sense, Confucius emphasized self-reliance, but we must realize that this self-reliance is not absolute, for relationships with other fellow human beings is equally if not more important. Indeed, a person's self-perfection is significant only in his capacity of relating to others. Still, it is in this generally self-reliant nature that we see the strength and weakness of Confucius' teaching. Its strength lies in a positive confidence that people can transcend the limits of self. It is according to this positive attitude that the above quotation from the Analects 7.30 should be understood. The weakness of Confucius' teaching also lies in his over-confidence, for it blinds us to other aspects of human nature, which might have an evil tendency.\n\nThis positive view of human nature and the human capacity to reach the ren state is not shared by all of Confucius' disciples or later Confucians. In Analects 8.7, we already find that Zengzi thought of ren as a heavy burden and therefore hard to achieve. Later, Xunzi went even further and pointed out that human nature is evil, contrary to the view of Mencius. The paradox that we observed in the Analects about ren can be resolved, for such a paradox indicates an evolution of the view of ren and human nature in general in the overall context of the early Confucian tradition. One thing is true for all periods of this evolution: li, the rites, are essential for overcoming the self and thus generating ren. The egoistic self is most difficult to overcome, but if we can merge our stream of self into the great ocean of life through the rites, or better still each of us play our roles in the orchestra of life, our response to others will become spontaneous. This newly gained virtue of power will influence people at home and abroad and, by restoring the rites, will ultimately bring the whole world to ren.\n\n### 4.5  Ren and Human Affiliation\n\nConfucius believes that man is essentially a social being. When he is born, it is into a web of social relations. He is the son to his parents, a younger brother to his elder brother; when he grows up, he needs to assume responsibility according to these roles. When he enters society, he needs to assume more roles. He is a friend to all friends, and inferior to his senior colleagues. As a social being, he belongs to a certain community. Within that community, most of his needs, including the need for recognition, are met, and as a member of the community, he has to assume multiple roles as well. Newly assumed responsibilities pose great challenges to him, but the socialization process starts at home. Confucius believes that if one performs a proper role as a younger brother at home, he will know intuitively how to be a junior friend or colleague, for proper respect due to one's seniors is applicable in all circumstances. On the other hand, piety towards one's parents is also translatable into obedience towards all authority. This sociological analysis helps us better understand the role of ren in human relations. The etymology of ren reveals a person-to-person relationship that shapes lives. And this relationship has the home as its foundation. Cultivating ren at home leads to the life of virtue abroad. In others words, private virtue is inextricably intertwined with public virtue.\n\nTherefore a person of ren is one who is able to shine forth his virtue both in private and public life. In other words, he is able to respond spontaneously on all occasions. Confucius realized that this capacity for natural response is inborn and shared by all people alike, but has been contaminated by egoistic desires. What a person needs to do is to recover this inborn capacity through cultivation (Analects 17.2). This inborn capacity, according to Confucius, is endowed by Heaven, the ultimate source for what a person could be at the highest level.\n\nConfucius' teaching on the ren is therefore not pragmatic in its true sense. Rather, it is idealistic in a sense. It is predicated on the belief that there exists an Ultimate Reality of Heaven, which endows us with the power of virtue, and to which we should be held responsible. The meaning of our existence is to be found ultimately in this reality.\n\n## 5  Yong \u52c7 (Courage)\n\nAmong the three consummate virtues,8 yong \u52c7 (courage), is the least discussed concept among Confucian scholars. The fact that this virtue is listed along with the other virtues such as ren and zhi indicates that it is a highly important concept in Confucius' thinking. For example,\n\n> 1.\n> \n> The Master said, \"The man of wisdom is never in two minds; the man of benevolence never worries; the man of courage is never afraid.\" (Analects 9.29)\n> \n> 2.\n> \n> The Master said, \"The Way of the exemplary man consists of these three, none of which I have succeeded in following: the man of benevolence never worries; the man of wisdom is never in two minds; the man of courage is never afraid\" (Analects14.28).\n\nThese three virtues focus on three different aspects of the human heart\/mind. Zhi is concerned with the human intellect; i.e. to make sound judgments. Ren is concerned with its affective or emotional aspect; i.e. the human capacity to remain calm and joyful under all circumstances. Yong emphasizes its volitional aspect; i.e. the will to combat fear and anxiety. And while zhi and ren can stand alone as positive virtues, yong, courage, usually requires a supporting virtue to go with it.\n\n### 5.1 Following the Rites\n\n> The Master said, \"Unless a man has the spirit of the rites, in being respectful he will wear himself out...in having courage he will become unruly...\" (Analects 8.2)\n\nCourage as a virtue is a mixed blessing. It will become a positive quality only when it is restrained by the rites.\n\n### 5.2 Righteousness\n\n> Zilu said, \"Does the exemplary man consider courage a supreme quality?\" The Master said, \"For the exemplary man it is morality (yi) that is supreme. Possessed of courage but devoid of morality,9 an exemplary man will make trouble while a small man will be a brigand.\" (Analects 17.23)\n\nYi \u7fa9, righteousness, involves proper action based on a sound judgment made on the basis of information given. Without attention to the appropriate situation, displayed courage may lead to chaos.\n\n### 5.3 Learning\n\nTo love courage without loving learning is liable to lead to insubordination. (Analects 17.8)\n\nThe book or intellectual learning is considered conducive to a balanced mind when one pursues the virtue of courage.\n\nThe relationship between courage and benevolence is also observed in the Analects. In it, courage is considered subordinate to the higher virtue of benevolence:\n\n> The Master said, \"...A benevolent man is sure to possess courage, but a courageous man does not necessarily possess benevolence.\" (Analects 14.4)\n\nAs benevolence enables one to be free from worry in all circumstances (Analects 9.29; 14.28), it is presupposed that one is not afraid. This is not true the other way around. Though a courageous man is not afraid (9:29; 14:28), especially in a dangerous situation, nonetheless he will worry in other situations.\n\nOur philological study could only take us thus far. We need to probe into the existential situation of Confucius to see how courage, though subordinate to benevolence, played an important part in Confucius' overall thought system.\n\nIn Confucius' lifetime, he was reportedly threatened with death several times (Analects 7.23; 9.5 and 15.2), and experienced the death of his close associates (Analects 11.9). In the face of death or non-being, Confucius showed the courage to affirm himself or the courage to be. According to Paul Tillich, the self-affirmation or courage to be in the face of death or fate is different from the self-affirmation in the face of sin and guilt in Christianity. While the former leads to the question of renunciation, the latter leads to the question of salvation (Tillich 2000: 17).\n\nWe do find in the Analects instances of renunciation (Analects 6.10, 12.5), but in the face of fate, Confucius encouraged cultivation of moral virtues such as wisdom and benevolence to offset the impact of the contingent circumstances. The following passage is a description of how he cultivated himself in different stages of his life and reached the final stage when he was able to act spontaneously in all circumstances. Though he may still not have been able to take his fate into his own hands, he was able morally to transcend his limitations.\n\n> The Master said, \"At fifteen I set my heart on learning; at thirty I took my stand; at forty I came to be free from doubts; at fifty I understood the Decree of Heaven; at sixty my ear was attuned; at seventy I followed my hearts' desire without overstepping the line.\"(Analects 2.4)\n\nAlthough it is true that Confucius does not teach about the sin, guilt and salvation that Christianity does, there is a salvation of another kind. Confucius believed that paradise should be built in this world, and his aspiration is to make the elder contented, friends trust in him and younger people cherish a good memory of him. (Analects 5.26) To make this world a paradisiacal place to live, human relationships, in all their complexity, should be smooth. Confucius recommended various methods to make this happen. Externally, one follows accepted norms with a certain degree of creativity. Internally, one cultivates the inherent virtues and lets them shine forth. For the world is fundamentally a human world. It is a real bliss to see ourselves capable of interacting with others in a smooth and spontaneous way.\n\nThough Confucius is mostly concerned with this world, he believe that transcendence is the source of the courage to affirm oneself in the face of fate and death. Confucius repeatedly resorted to Heaven when he experienced the threat of death or when he saw the death of his close disciples, suggesting Heaven is his ultimate source of courage. A close reading of some other passages in the Analects shows that the related concept of Dao, the Way, is an equally important source of courage:\n\n> The Master said, \"He has not lived in vain who dies the day he is told about the Way.\" (Analects 4.8)\n\nThis seemingly pragmatic conception of Dao often blinds us to its implicit mystical dimension. Dao permeates Heaven, Earth and the human world. It is through personal and interpersonal cultivation that Dao is grasped, which in turn enables one to move spontaneously in the world. In this, we believe Confucius' teaching did not differ much from early Daoist teaching. The only conspicuous difference is that Daoism derived its inspiration from the natural world and Confucianism from the human world. In the mystical union in Dao, one person can gain strength of courage and conquer the anxiety of fate and death. For Confucius, the ultimate source for the courage to be in the face of death is the Way of Heaven. Perhaps because of its mystic nature, Confucius rarely talks about it (Analects 5.13).\n\nThe unspeakable nature of the ultimate source of courage is also experienced in the Christian tradition. In his analysis of courage and transcendence, Paul Tillich asserts that mysticism and theism should both be transcended. He concludes, \"The courage to be is rooted in the God who appears when God has disappeared in the anxiety of doubt\"(Tillich 2000: 190).\n\nHe implies that the root of courage is beyond our common conception of God. For Confucius, the root of courage to be can be found in Heaven.\n\n## 6 Conclusion\n\nConfucius' Analects details three consummate virtues of wisdom, benevolence and courage. While wisdom emphasizes the intellectual aspect and benevolence the affective aspect of virtue, courage emphasizes the volitional aspects. All these three virtues find their sources in the Ultimate Reality, Heaven. True wisdom in a person is the capacity endowed by Heaven to understand other people and things around that person. True benevolence in a person is the quality to shine forth the virtue endowed by Heaven. True courage in a person is the calm acceptance of the destiny of Heaven. Our understanding of early Confucianism would be impoverished if we ignore its metaphysical dimension.\n\nThe correlation of Ultimate Reality with self-cultivation finds the best embodiment in shengren \u8056\u4eba, the concept of the Confucian sage. In the Confucian tradition, the sage is the moral exemplar par excellence. In his conception of the universe, everything is related to everything else. All work together to constitute a perfect universe. There is no radical break from the source of creation as is assumed in Western religions. In such a universe, harmony among the constituent parts, such as person, society and nature, is considered normal. The most conspicuous human predicament is the demand for harmony versus the impulse of the self with the egoistic tendency to disrupt harmony. A sage is just such a man who is able to transcend the egoistic drive and to take the interest of the world as his personal interest. The harmony that a sage brings to the world becomes a model that a less spiritually advanced person can emulate. If the whole world acts according to the model, harmony becomes the order of the day. The sage in the Confucian tradition has both secular and sacred dimensions of existence. Secularly, he acts as leader in the world and effects a transformation of the world so much so that worldly harmony is achieved. Sacredly, the sage engages in self transformation, so much so that his ego is diminished and his self is merged with the cosmos. In the expression used by Zhuangzi \u838a\u5b50, such a person is \"inwardly sage and outwardly king\"(see Burton Watson 1968). The ideal and reality thus come as one.\n\nReferences\n\nAmes, Roger. 1993. The meaning of body in classical Chinese philosophy. In Self as body in Asian theory and practice, ed. Thomas P. Kasulis, Roger T. Ames, and Wimal Dissanayake. Albany: State University of New York Press.\n\nAmes, Roger. 1985. The common ground of self-cultivation in classical Taoism and Confucianism. Tsing Hua University Journal 65\u201396.\n\nCheng and Cheng. 1984. Er Cheng ji \u4e8c\u7a0b\u96c6 (Collections of Two Chengs). Beijing: Zhonghua Shuju. (A most important primary source for the study of two Cheng brothers)\n\nChing, Julia. 1977. Confucianism and Christianity: A comparative study. Tokyo; New York: Kodansha International. An authority on a broad comparison between Confucian and Christian traditions.\n\nChing, Julia. 1997. Mysticism and kingship in China: The heart of Chinese wisdom. New York: Cambridge University Press. A religious interpretation of Chinese statecraft and its contemporary relevance.CrossRef\n\nCreel, G.H. 1953. Chinese thought from Confucius and Mao Tse-Tung. Chicago: University of Chicago Press. An interpretation of Chinese thought by a foremost scholar in America, a classic.\n\nDe Bary, W.T., and Bloom Irene (eds.). 1999. Sources of Chinese tradition from earliest times to 1960. New York: Columbia University Press. A widely used sourcebook for Chinese thought.\n\nFingarette, Herbert. 1972. Confucius: The secular and sacred. New York: Harper & Row. A highly insightful interpretation of the Confucian thought by a professional philosopher.\n\nGuodian Chumu Zhujian \u90ed\u5e97\u695a\u5893\u7af9\u7c21 (Guodian Chu Tomb Bamboo Slips). 1998, ed. Jinmen Shi Bowuguan (Jinmen Metropolitan Museum). Guodian Chumu Zhujian Beijing: Wenwu Press.\n\nHall, David, and Roger Ames. 1987. Thinking through Confucius. Albany: State University of New York Press. A successful collaborative work by a Sinologist and American philosopher. Insightful interpretations of Chinese and Western cultural sensibilities.\n\nHuang, Yong. 2007. Confucian theology: Three models. Religious Compass 1(4): 455\u2013478.CrossRef\n\nKnoblock, John. 1988. Xunzi: A translation and study of the complete works, vol. 1. Stanford: Stanford University Press. First full translation of Xunzi based on solid scholarship.\n\nKuang, Yaming. \u5321\u4e9e\u660e 1985. \u5b54\u5b50\u8a55\u50b3 (A Critical Biography of Confucius). Jinan: Qilu Shushe (A Study of Confucius' life and thought by a foremost authority in China).\n\nLau, D. C.1979. Confucius: The analects. Harmondsworth; New York: Penguin Books. (A widely used scholarly translation).\n\nLau, D. C. 1970. Mencius. Trans. D.C. Lau. Harmondsworth, Eng: Penguin Books. (A widely used scholarly translation).\n\nLegge, James. 1959. The Tao the king, chapter 10. In The texts of Taoism. Trans. Legge, James. New York: Julian Press.\n\nLegge, James. 1967. Li chi, the book of rites. Trans. Legge, James. New York: University Books.\n\nMunro, Donald J. 1969. The concept of man in early China. Ann Arbor: Center for Chinese Studies, University of Michigan. (An excellent conceptual analysis of early Chinese thought).\n\nPlato. 1969. The republic, in dialogues of Plato. Trans. Benjamin Jowett; ed. introductory notes Justin D. Kaplan. New York: Washington Square Press.\n\nRawley, H.H. 1956. Prophecy and religion in ancient China and Israel. London: Athlone Press, University of London, Harper. A remarkable study comparing Chinese and Hebrew traditions.\n\nRoth, Harold. 1991. Psychology and self-cultivation in early Taoistic thought. Harvard Journal of Asiatic Studies 51(2): 599\u2013650.CrossRef\n\nRoth, Harold. 1999. Original Tao: Inward training (Neiyeh) and the Foundations of Taoist Mysticism. New York: Columbia University Press. A recovery and interpretation of the early Taoist source by a foremost scholar in the study of early Taoism.\n\nShen, Vincent. 2003. Ren (Jen): Humanity. The encyclopedia of Chinese philosophy, ed. Antonio Cua. New York: Routledge.\n\nShun, Kwong-loi. 1997. Mencius and early Chinese thought. Stanford: Stanford University Press. An excellent philosophical interpretation of Mencius.\n\nSmith, Huston. 1991. The world's religions. San Francisco: Harper. A scholarly book with high popular interest.\n\nStace, Walter. 1960. Mysticism and philosophy. 1st ed., reprinted. Los Angeles: Jeremy P. Tarcher, Inc.\n\nTillich, Paul. 2000. The courage to be. New Haven: Yale University Press. A classic and deeply insightful book by an eminent philosopher\/theologian in last century.\n\nTu, Weiming. 1994. Embodying the universe: A note on Confucian self-realization. In Self as person in Asian theory and practice, ed. Ames Roger et al. Albany: State University of New York Press.\n\nWatson, Burton. 1968. Chuang-tzu: The complete works. New York: Columbia University Press. a classic in translation.\n\nXinzhong, Yao. 1997. Confucianism and Christianity: A comparative study of Jen and Agape. Brighton: Sussex Academic Press. An important work comparing a major theme in Confucian and Christian traditions.\n\nYearley, Lee. 1990. Mencius and Aquinas: Theories of virtue and conceptions of courage. Albany: State University of New York Press. A classic in the study of comparative religious ethics.\n\nFootnotes\n\n1\n\nAlso, John Knoblock translated \u4fee\u8eab as \"Self-cultivation\" in Chap.\u200b 2 of the Xunzi.\n\n2\n\nThe strict dichotomy does not even apply to the Christian conception. The Christian conception of God is immanent as well as transcendent. For its transcendence, Christianity inherited the Hebrew conception of God. For its immanence, Christ bridged the gap between the human and divine. See Acts 17: 27, 28. In this sense, the Chinese conception of Heaven is comparable to it.\n\n3\n\nFor a comparative study of Hebrew prophets and the Chinese sages, see Rawley 1956.\n\n4\n\nThe idea that without knowing the rites there is no way one can take a stand in the world is repeated in Analects 8.8 and 16.13.\n\n5\n\nCheng Hao \u7a0b\u705d seems to be the first one to correlate physical insensitivity with moral insensitivity (see Cheng and Cheng 1984: 33).\n\n6\n\nIt is interesting to note that Confucius once compared his lazy disciple to dead wood (Analects 5.10).\n\n7\n\nThough the Confucian community based on blood and geographical ties is formed in a different way from a Christian community based on the same belief in Jesus Christ, the ultimate goal of both communities is to expand themselves indefinitely so that the Way or the Kingdom of God prevails. For a larger comparative study of Christianity and Confucianism, see Ching 1977. For a discussion of the Confucian community, see pp.96\u2013102. For a focused study of Confucian love, ren, and Christian love, agape, see Xinzhong 1997.\n\n8\n\nRen, benevolence, zhi, wisdom, and yong, courage, are considered three consummate virtues in the Doctrine of the Mean.\n\n9\n\nFor justification for translating yi as morality, see Lau 1979: 26\u20137.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2_16\n\n\u00a9 Springer Science+Business Media Dordrecht 2014\n\n# 16. Confucian Harmony: A Philosophical Analysis\n\nChenyang Li1\n\n(1)\n\nSchool of Humanities and Social Sciences, Nanyang Technological University, Singapore, Singapore\n\nChenyang Li\n\nEmail: Cyli@ntu.edu.sg\n\nAbstract\n\nHe\u548c (harmony, harmonization) is the most cherished ideal in Chinese culture, and more specifically, in Confucianism. Focusing on the Confucian ideal of he developed during the classical period, this chapter explores its meaning and provides a philosophical analysis and exposition.\n\nUnder the title of \"The Confucian Ideal of Harmony,\" an earlier version of this chapter was published in Philosophy East & West 56.4 (2006): 583\u2013603. I thank the University of Hawaii Press for permission to use the article.\n\nHe \u548c (harmony, harmonization) is the most cherished ideal in Chinese culture, and more specifically, in Confucianism. Focusing on the Confucian ideal of he developed during the classical period, this chapter explores its meaning and provides a philosophical analysis and exposition.\n\n## 1 The Confucian Ethical, Political, and Metaphysical Ideal of He\n\n\"He\u548c\" is usually translated in English as \"harmony,\" though it may be more appropriately rendered as \"harmonization\" in many contexts. The word predates Confucianism. Its earliest form can be found in inscriptions on bones and tortoise shells of the Shang Dynasty (sixteenth to eleventh century BCE.) and later more frequently in inscriptions on bronze utensils of the Zhou Dynasty (1066?\u2013256 BCE.) (see Guo 2000). In the earliest Confucian texts, we can find numerous occurrences of \"he.\" The meaning of he in these texts mostly has to do with sounds and how sounds interact with one another. It was probably used more frequently as a verb than a noun. The Yijing: Zhong Fu \u6613\u7d93\u00b7\u4e2d\u5b5a states, \"a crane sings in the woods and its baby birds respond (he) to it\" (TTC 1980: 71). The Zuozhuan: Zhuanggong 22nd Year \u5de6\u50b3\u00b7\u838a\u526c\u4e8c\u5341\u4e8c\u5e74 states, \"male and female phoenixes fly together and their sounds mutually respond (he) vigorously\" (TTC 1980: 1775). The Shijing: Zhengfeng \u8a69\u7d93\u00b7\u912d\u98a8 contains the expression \"responding (he) to brothers with songs\" (TTC 1980: 342). In the Analects, we find \"when Confucius sang with others and saw someone did well, he always made the person repeat the song before he responded (he)\" (Analects 7.31; TTC 1980: 2484). Finally, in the Zhouli: Diguan \u5468\u79ae\u00b7\u5730\u5b98, there is the statement, \"to use [the musical instrument] chun to respond (he) to drums\" (TTC 1980: 721). In all these instances, \"he\" evidently is used to describe how various sounds, of animals, of people, and of instruments, respond to one another. This meaning of \"responding\" is preserved in the modern Chinese language when \"he\" is used as a verb (with the fourth tone), as in \"he shi \u548c\u8a69,\" namely composing a poem in response to another poem by someone else. Xu Shen \u8a31\u614e (30\u2013124) in his lexicon Shuowenjiezi \u8aaa\u6587\u89e3\u5b57 defines \"he\" simply as \"mutual responsiveness (of sounds)\" (Shuowenjiezi 1992: 57). As a lexical definition, Xu's is a report of these usages of \"he\" in earlier texts. It summarizes the root meaning of the word.\n\nHowever, mere mutual responsiveness is not yet harmony. Harmony results only when sounds respond to one another in a mutually enhancing way. One of the earliest definitions of he as harmony, several 100 years before Xu Shen, can be found in the Guoyu \u570b\u8a9e, a classical text written during the Spring and Autumn period (770\u2013476 BCE) in close association with the Confucian tradition. In the chapter \"Zhouyu B \u5468\u8a9e\u4e0a,\" it is stated that \"when sounds correspond and mutually bao one another it is called he\" (Lai 2000: 166). \"Bao \u4fdd\" has a number of interrelated meanings such as \"protect\" (bao hu \u4fdd\u8b77), \"nurture\" (yang yu\u990a\u80b2), \"rely on\" (yi kao \u4f9d\u9760), \"stabilize\" (an ding \u5b89\u5b9a), and \"assure\" (bao zheng \u4fdd\u8b49) (Sources of Terms 1995: 117).\n\nUnderstood in this way, he is not just that sounds mutually respond; it is that various sounds respond to one another in a mutually promoting, mutually complementing, and mutually stabilizing way. In this sense, some of the expressions cited above can be interpreted as harmonization. For example, the expression in the Zuozhuan: Zhuanggong 22 can be interpreted as that \"male and female phoenixes fly together and their sounds harmonize with vigor\" (TTC 1980: 1775). The expression in the Zhouli: Diguan can be read as saying, \"to use [the musical instrument] chun to harmonize the [sounds of] drums\" (TTC 1980: 721). As such, expressions like \"the he of the five sounds\"1 in the Zuozhuan: Xigong 24th Year \u5de6\u50b3\u00b7\u50d6\u526c\u4e8c\u5341\u56db\u5e74 do not merely mean the mutual response of sounds, but also the harmonious interplay of these sounds (TTC 1980: 1818). \"He\" is used here as a noun, standing for a (dynamic) state of music rather than simply one sound responding to another. Therefore, we may conclude that the original meaning of he as harmony comes from the rhythmical interplay of various sounds, either in nature or by human beings, that is musical to the human ear, and that the prototype of he is found in music.2 From the notion of he as the harmonious interplay of sounds, it is not difficult to see how it can be expanded, by analogous thinking, to mean harmony in other things and hence harmony in general.\n\nThus, we can say that, philosophically, harmony presupposes the existence of different things and implies a certain favorable relationship among them. Understood this way, harmony is not about conforming to a fixed order in the world. To the contrary, it is about creating orders.\n\nOne of the earliest ideas of he was proposed by a pre-Confucian scholar-minister, Shi Bo \u53f2\u4f2f (551\u2013475 BCE). In the Guoyu: Zhengyu \u570b\u8a9e\u00b7\u912d\u8a9e Shi Bo elaborates on he:\n\n> Harmony (he) is indeed productive of things. But sameness does not advance growth.3 Smoothing one thing with another is called harmony. For this reason things come together and flourish. If one uses the same thing to complement the same thing, it is a dead end and will become wasted (Lai 2000: 746).\n\nThis is so because,\n\n> A single sound is nothing to hear, a single color4 does not make a pattern, a single taste does not satisfy the stomach, and a single item does not harmonize (Lai 2000: 747).5\n\nAccording to Shi Bo, he was the philosophy of ancient sage-kings and it enabled their societies to prosper:\n\n> Therefore, the early kings mixed Earth with Metal, Wood, Water, and Fire, and produced varieties of things. They balanced one's taste with the five flavors, strengthened the four limbs in order to guard the body, harmonized (he) the six measures of sounds to improve the hearing, made the seven parts of the body upright to maintain the heart\/mind, balanced the eight body parts to complete the whole person, established the nine social rules to set up pure virtues, and put together the ten offices to regulate the multitude. Therefore, there came into existence thousands of categories and tens of thousands of methods used in calculating millions of things and evaluating myriads of properties. They maintained constant incomes and managed countless items. Therefore the kings had land of nine provinces and had incomes to raise the multitude. They taught the people adequate lessons and harmonized (he) them as one family. Thus, it was harmony (he) at the highest level (Lai 2000: 746\u2013747).\n\nThese sage-kings also set themselves as examples in promoting he:\n\n> The early kings married their wives from other families, sought wealth in all directions, and chose ministers who could remonstrate with the ruler. This way they reconciled a multitude of things. They were engaged in harmonization (he-tong) (Lai 2000: 747).\n\nFor Shi Bo, a harmonious world must be a diverse world. This is so because a healthy and prosperous world relies on its diverse things to go together. This is why the ancient sage-kings sought diversity. As in good cooking and in good music-making, a healthy family must consist of spouses from different tribes, a prosperous nation must seek wealth from various sources, and a good government must have ministers capable of holding different opinions. Harmony out of diversity produces a lively world; sameness without adequate difference can only lead to a dead end.\n\nThe Zuozhuan: Zhaogong 20th Year \u5de6\u50b3\u00b7\u662d\u526c\u4e8c\u5341\u5e74 records a discussion of he by another scholar-minister, Yan Zi \u664f\u5b50(?-500 BCE):\n\n> Harmony (he) is like making soup. One needs water, fire, vinegar, sauce, salt, and plum to cook fish and meat. One needs to cook them with firewood, mingle (he) them together in order to balance the taste. One needs to compensate for deficiencies and reduce excessiveness. The virtuous person (junzi) eats [such balanced food] in order to purify his heart\/mind (TTC 1980: 2093).\n\nThe cook needs different things to make a balanced soup. She needs ingredients that taste and smell very different. Water and fire are usually seen as diametrically opposed to each other, yet neither is dispensable for cooking. One important aspect of good cooking is to be able to balance one excessive flavor with another. Yan Zi believes that enjoying such harmonized food can purify a virtuous person's heart\/mind (xin). He continues:\n\n> Sounds are like flavors. Different elements complete one other: one breath, two styles, three types, four instruments, five sounds, six measures, seven notes, eight winds, and nine songs. Different sounds complement one another: the pure and the impure, the big and the small, the short and the long, the fast and the slow, the sorrowful and the joyful, the strong and the tender, the late and the quick, the high and the low, the in and the out, and the inclusive and the non-inclusive. Listening to this kind of music, the heart\/mind of the virtuous person (junzi \u541b\u5b50) is purified (TTC 1980: 2093\u20134).\n\nFor Yan Zi, good music (e.g., a symphony) requires a variety of sounds in various modes. A good musician is like a good cook, capable of mingling together various sounds, some in sharp contrast, to make a coherent and harmonious piece. Like a good soup, enjoying such good music can also purify people's heart\/mind.\n\nBased on this understanding of he, Yan Zi argues that he must be distinguished from another notion, tong \u540c (sameness). In Yan Zi's conversation with the duke of Qi, the duke evidently confuses he with tong when he praises how harmonious (he) it is between him and his minister Ju \u64da. Yan Zi points out that the relationship between the duke and his minister Ju is described more appropriately as tong, not he. Yan Zi uses the above examples of cooking and making music to show that he is not to be confused with tong. Yan Zi says that the moral of his examples of he also applies to the relationship between the ruler and ministers. He says:\n\n> When the duke says \"yes,\" Ju also says \"yes;\" when the duke says \"no,\" Ju also says \"no.\" This is like mixing water with water. Who can eat such a soup? This is like using the same kind of instruments to produce music. Who can enjoy such music? This is why it is not all right to be tong (TTC 1980: 2094).\n\nFor Yan Zi, this kind of relationship between the duke and his minister Ju is sameness or conformity, not harmony. A harmonious relationship presupposes that they have different perspectives and different views on various issues. One may say that tong without adequate differences precludes harmony, and such a state is like a soup made of only one ingredient or a symphony composed of only one kind of instrument. A soup made of only one ingredient is tasteless, a symphony composed of only one instrument is boring, and a government consisting of only one voice is stagnant and dangerous.\n\nThe thought of Shi Bo and Yan Zi was later appropriated in the Confucian classics Zuozhuan and Guoyu. 6 He later became a central ideal in Confucianism. In the Analects 13.23 Confucius (551\u2013479 BCE) adopts the ideal of he, making he a criterion for the good person (junzi). He says that \"The junzi harmonizes but does not seek sameness, whereas the petty person seeks sameness but does not harmonize\" (TTC 1980: 2508). For Confucius, a sensible person should be able to respect different opinions and be able to work with different people in a harmonious way. A major function of li \u79ae (rites, rituals of propriety) is precisely to harmonize people of various kinds. The Confucian disciple Youruo \u6709\u82e5 is recorded in the Analects 1.12 as saying that \"of the functions of li harmonization is the [most] precious\" (TTC 1980: 2458). There is little need to emphasize how Confucius valued the use of li.7 Confucius and Confucians see a direct connection between li and he. They take li to be a central aspect of government and believe that, through the good use of li, good government results in a harmonious society. According to the Zhouli: Tianguan \u5468\u79ae\u00b7\u5929\u5b98, another Confucian classic, one of the six primary functions of the state official Greater Minister is \"[to minister] state rituals (li), in order to harmonize the nation\" (TTC 1980: 645).8\n\nMencius (372\u2013289 BCE), too, highly values he. He comments that among the three important things in human affairs, the harmony of people is the most important: \"good timing is not as good as being advantageously situated, and being advantageously situated is not as good as having harmonious people\" (Mencius 2B1; TTC 1980: 2693). In order to achieve a major goal in social affairs, one would need all three: good timing, being advantageously situated, and having harmonious people. The most precious thing, however, is to have people who work harmoniously with one another. Mencius praises Liu Xiahui \u67f3\u4e0b\u60e0 as \"the sage who is able to harmonize\" (Mencius 5B.1; TTC 1980: 2741). Liu is well known for his familiarity with li and for his conviction of harmonious co-existence.9\n\nXunzi \u8340\u5b50 (313?\u2013238 BCE) also emphasizes he. In the Xunzi: Xiushen \u8340\u5b50\u00b7\u4fee\u8eab, he concurs with Confucius on the importance of li to harmony. Xunzi says that \"[only] when following li is one harmonious and regulated\" (TTM 1986: 289). He also echoes Confucius in saying that \"To harmonize with others by goodness is being reasonably accommodating\" and \"to harmonize with others by wickedness is fawning\" (Xunzi: Xiushen; TTM 1986: 289). For Xunzi, harmony is not without principle. This strikes a similar note to Confucius, whose ideal of harmony is \"harmony without mindlessly following others\" (Liji: Zhongyong 10; TTC 1980: 1626).\n\nHe is not just about human relationship. The Jiaotesheng\u90ca\u7279\u7272chapter of the Confucian classic Liji \u79ae\u8a18 states, \"when yin and yang harmonize, the myriad things get their due\" (Liji; TTC 1980: 1446). Xunzi elaborates on the same idea in more detail: \"with the great transformation of yin and yang, the generous supply of the wind and the rain, the myriad things each get harmonized so they can live, and get their nurture so they grow\" (Xunzi: Tianlun; TTM 1986: 327). Here Xunzi concurs with the Yijing, the Tuanzhuan \u5f56\u50b3chapter of which develops the notion of \"grand harmony\" (taihe \u592a\u548c): \"How great is the Qian \u4e7e (Heaven)! From it the myriad things originate under Heaven....With the changes of the Qian way, the myriad things all keep on their own path of life. Thus they preserve the grand harmony\" (Yijing; TTC 1980: 14). \"Grand Harmony\" is the most important ideal in the Yijing. The world is full of different things, yet all these things harmonize as they go through incessant changes. Confucians have faith in this ultimate harmony of the world.\n\nPerhaps the most concentrated articulation and elaboration of this Confucian ideal of harmony is found in the Zhongyong \u4e2d\u5eb8 chapter of the classical text Liji. The Zhongyong states, \"Centrality is the great foundation under Heaven, and harmony is the great way under Heaven. In achieving centrality and harmony, Heaven and Earth maintain their appropriate positions and the myriad things flourish\" (Liji; TTC 1980: 1625). In a separate essay I have argued that, in the Zhongyong, centrality is the way to achieve harmony (Li 2004). Without harmony, Heaven and Earth would be out of their proper places, and nothing in the world would be able to flourish. Therefore, harmony is the highest ideal in the Zhongyong. The Zhongyong lays out the foundation for Confucian metaphysics and promotes harmony as the highest ideal.\n\nHarmony, as understood in Confucianism, can take place at various levels. It can take place within the individual. A person can harmonize various parts of his or her body, the mind-heart, and various pursuits in life into a well-functioning, organic whole (Li 2010). Harmony can take place between individuals at the level of the family, the community, the nation, and the world.10 This may include harmony between societies, harmony within a society with different ethnic groups (or political organizations), harmony within the same ethnic group with different kinships, and harmony within the same kinship. Harmony also can take place between human beings and the natural universe. Confucianism does not exclude intrapersonal harmony, which Daoism emphasizes, but Confucianism puts tremendous weight on interpersonal harmony, such as harmony between ruler and ministers, parents and children, husband and wife, between siblings and between friends. It also places tremendous weight on the harmony between human society and the natural world. Its ultimate goal is to achieve grand harmony throughout the cosmos.\n\nFor Confucians, harmony defines the kind of life a person should live, the kind of society people should construct, the kind of world humanity should maintain. The difference between harmony and disharmony is one between right and wrong, good and bad, and success and failure. The Wuxing \u4e94\u884c text of the Guodian Chu Bamboo Slips, which is generally accepted as a Confucian text, states that \"the harmony of the Five virtuous practices is called Virtue; the harmony of the Four practices is called Goodness. Goodness is the way of humanity. Virtue is the way of Heaven\" (Liu 2003: 69). The Four virtuous practices are Humanity, Appropriateness, Propriety, and Wisdom; the Five virtuous practices also include an additional Sageliness. The harmonious practices of these virtues are the way of humanity and the way of Heaven. They are the right way.11 Dong Zhongshu \u8463\u4ef2\u8212 (179\u2013104 BCE), the influential Han Confucian, declared that \"the greatest virtue is harmony\" and advocated to \"use central harmony in managing society\" (Dong 1986; TTM 1986: 805). For Dong, the ability to harmonize in the world is indeed the most precious.\n\nAs far as the need for harmony is concerned, Confucians tend to see more consistency than distinctions between the \"private\" and the \"public\" (as is seen in Western liberalism), between the political and the non-political, and between human society and the natural world. When persons and things are engaged in a healthy, stable interplay and each gets its due, it is deemed as harmony; the opposite as disharmony. When a plant is harmonized with its surroundings it thrives; when a person is harmonized with his environment, he flourishes. The ideal of an individual is not only to harmonize within one's own person, but also with other individuals. The ideal of a society is not only to harmonize within the society, but also with other societies. The ideal of humanity is not only to harmonize among its members but also with the rest of the cosmos as well.\n\n## 2 A Philosophical Analysis and Exposition of He\n\nBased on the study in Sect. 1, I would like to make the following observations concerning the Confucian notion of harmony. First, harmony is a metaphysical as well as an ethical notion; it describes both how the world at large operates and how human beings ought to act. Harmony is the Way, the Confucian way. Second, harmony is by its very nature relational. It presupposes the co-existence of multiple parties; \"a single item does not harmonize.\" As far as harmony is concerned, these parties possess relatively equal significance. Harmony is always contextual; a mentality of harmony is a contextual mentality. Epistemologically, it calls for a holistic approach. Persons of harmonious mentality see things, and make judgment of things, in relation, in context, not in isolation or separation. In this connection, harmony is not a one-directional \"adjustment\" in conformity to pre-existing cosmic orders as Max Weber has read it (Weber 1951: 152\u201353). The requirement of harmony places a constraint on each party in interaction, and, in the meantime, provides a context for each party to have optimal space to flourish. In the Confucian view, the world is not there just for one item or one kind of thing. It is for the \"myriad things.\" Nothing in the world can claim absolute superiority to all others. Parties in a harmonious relationship are both the condition and the constraint to one another's growth. A harmonious relationship implies mutual complement and mutual support between the parties (Cheng 1991: 187). There is mutual benefit, even though harmony cannot be reduced to mutual benefit. Third, Confucian harmony is by no means \"perfect agreement\" as is often (mis)understood. In harmony, co-existing parties must be in some way different from one another; while harmony does not preclude sameness of all kinds, sameness itself is not harmony. Harmony is different from stagnant concordance in that harmony is sustained by energy generated through the interaction of different elements in creative tension. Although a harmonious relationship in society may involve friendliness and love, friendliness or love is not a necessary condition for harmony. Even unfriendly parties can co-exist in harmony. This point, as I will show shortly, has implications for an appropriate understanding of the relation between harmony and strife.\n\nIn what follows, I will first articulate a Confucian view on the relation between harmony, sameness, and strife, then explore in the context of Confucian philosophy why harmony is so important to Confucians, and finally, through a brief comparison of Confucianism and Christianity, try to answer some questions about Confucian harmony by discussing issues regarding gaps or potential gaps between moral ideals and their implementation.\n\nIt is important that we do not simple-mindedly interpret the Confucian attitude toward sameness. Difference and sameness between things exist at various levels. Confucians do not advocate difference for the sake of difference, nor do they reject sameness altogether. Obviously, even though the sage-kings are said to have sought spouses from other tribes, the couples were all human beings (i.e., a kind of sameness). Even though one would need different instruments in order to make good music, they are still musical instruments (i.e., another kind of sameness). Indeed, some ancient Chinese texts advocate a certain kind of sameness. For instance, in the Jingfa \u7d93\u6cd5: Sidu \u56db\u5ea6of the Silk Texts of the Mawangdui Han Tombs, is the statement that \"Joining the Heaven and Earth, responding to the people's will, establishing both the civilian and military service, this is called the superior tong \u540c (sameness)\" (Chen 2007: 103). Of course, the \"superior tong \u4e0a\u540c\" is not just any kind of sameness; it is sameness at the optimal level. Confucius also advises us \"not to plan together with people of different roads (pursuits)\" (Analects 15.39; TTC 1980: 2518). Conversely, we should plan together with those following the same (tong) roads. Mencius praises the sage-king Shun \u821c \"good at tong with others\" (Mencius 2A8; TTC 1980: 2691). Furthermore, the Zuozhuan: Chenggong 16th Year \u5de6\u50b3\u00b7\u6210\u526c\u5341\u516d\u5e74 states that: \"when people live in abundance, they will be together harmoniously and listen [to the ruler's commands]\" (TTC 1980: 1917). The expression of \"being together harmoniously\" combines both \"he \u548c\" and \"tong \u540c.\" We can say that, at an appropriate level, sameness is an ingredient of harmony and, as such, must be maintained and valued. Therefore, the kind of sameness that ancient and Confucian scholars reject must not be understood as sameness per se, but \"over-presence of sameness,\" like the one between the duke and his minister Ju in the Yan Zi story. An over-presence of sameness can occur in many ways. It can be caused by a lack of diversity when diversity is called for. For example, a cook only has one kind of ingredient for a soup. It can be caused by forced sameness when sameness is sought for the sake of sameness. For example, a powerful person forces the same opinion on everyone else, or a fawning person pretends the same view in conformity to his superior for ulterior purposes; both undercut conditions necessary for harmony.\n\nFor Confucians, over-presence of sameness, whether seeking spouses within the same tribe or making music with the same type of instrument, is not conducive to harmony. Even though members of a stamp collection club have to share common interest at some level (i.e., they are fond of collecting stamps), collecting the same stamp, e.g., the Year of the Monkey stamp issued by the U.S. Post Office in 2004, does not make a good stamp collection club; such a club will probably not last, because this kind of sameness \"does not advance growth,\" as Shi Bo said. In the political arena, even though laws must be made to treat everyone equally (a kind of sameness), making laws that overly demand uniformity is not conducive to good society. What kind of sameness is appropriate to harmony is always a contextual matter and cannot be determined in absolute terms. Because the harmony orientation takes conflict as its primary opponent, there is a tendency to overlook the other side of the danger, namely the over-presence of sameness. Ancient scholars emphasize the difference between harmony and sameness because people tend to confuse them, taking mere sameness as harmony. Strictly speaking, over-presence of sameness is a lack of harmony, and therefore it is a kind of disharmony.\n\nWhile emphasizing the harmonious interplay of different things, Chinese philosophers appear to have not given much attention to strife, or they may have left the impression that strife is to be avoided for the sake of harmony. A certain type of strife, however, is inherent in harmony. This point is implied in the ancient scholars' emphasis on the difference between harmony and sameness (tong).\n\nHarmony presupposes differences and has to be achieved through differences. Difference entails, at least potentially, strife. We may say that there can be two kinds of strife between various things. The first kind I call tension and cooperative opposition. Different things in a relationship have tension between them in that they compete, at a minimum, for space and time. For instance, people at a busy train station heading in various directions may experience tension and opposition (potential conflict); they may step in one another's way. Tension and opposition like this are obviously not harmony; they can jeopardize harmony. This happens, for instance, when the tension and opposition between busy people erupts into a stampede. Opposing parties, however, can also be harmonized without harming one another. For example, this happens when people at a busy train station take turn, make room for one another, and move forward in an orderly way, even though in different directions. The transition from mere tension and opposition to harmony takes coordination or cooperation of the involved parties, either consciously or unconsciously. Taking a panoramic view of such a busy station, one has to marvel how a harmonious scene is being produced by people heading in different, even opposite, directions. This kind of difference as tension and opposition is a pre-condition for harmony; without it there is mere sameness rather than harmony.\n\nThe second type of strife is more severe. It is the kind in which one force aims at destroying or eliminating the other or others. For example, a murderer kills innocent people, and wolves eat sheep. Let us call this second type antagonistic opposition. This type of strife is either itself disharmony or a key element of disharmony. In this case, harmony is achieved through overcoming strife. An overcoming process may take one of two forms. First, it can be the elimination of strife. We remove a serial murderer and the community is restored to harmony. However, for Confucians, elimination is not the prototypal path to harmony. For the most part, harmony can be achieved through the second form by putting strife under control without elimination, namely by turning the second type of strife into the first type. Whereas a large population of sheep tends to increase the population of wolves, the population of wolves has to be lowered when they over-eat sheep to cause a shortage in food supply. Eventually, the wolves and the sheep have to strike a balance through some kind of natural \"negotiation.\"12 When harmony is achieved, the sheep provide food for the wolves while the wolves weed out the unhealthy and keep the sheep population in check. Of course, sometimes violent disharmony is inevitable. On the individual level, the strife between a wolf and a sheep may end in elimination. Where the population of wolves and the population of sheep strike a balance, some sheep and\/or wolves have already been lost in the process. In cases like this, Confucians would say that strife between the two individuals serves as an instrumental step toward harmony in the long run and at a large scale for the world. Dong Zhongshu writes, \"the system of Heaven and Earth integrates harmony and disharmony, centrality and non-centrality; these are employed timely to be most effective\" (Dong 1986; TTM 1986: 805). Since harmony is not only a state, but more importantly a process, disharmony is necessarily present during the process of harmonization.\n\nIn this sense, human beings can exist harmoniously with nature even though we have to consume natural recourses. In order to survive, we have to consume, and therefore to eliminate lives in nature. But we can do it in a balanced way and achieve harmony with nature. Unlike the natural world, human beings have the capacity to play an active role to promote harmony in the world. This capacity enables human beings to avoid unnecessary damage and harm in achieving harmony. For instance, imposing too large a human population burden on nature may eventually destroy the human habitat, and consequently reduce the human population to a level bearable to Earth. But human beings have the power to restrain their behavior and to maintain a balance with nature before such a catastrophe takes place.\n\nConfucian harmony is not pure submissiveness or absolute avoidance of conflict. In Sect. 10 of the Liji: Zhongyong, Confucius's disciple Zilu \u5b50\u8def asks about strength. Confucius identifies two types of strength. One type is the strength of the northerners, who will fight unto death for the right cause; the southerners have a different kind of strength: they are tolerant and flexible, and they do not avenge the unjust. Confucius approves both types of strength, including the strength of \"central standing without leaning to one side\" (Liji; TTC 1980: 1626). This passage suggests that Confucius endorses an integration of both kinds of strength into a harmonious interplay, rather than simply taking the mean between the two. The strength of the northerners may appear extreme (\"to fight unto death\"), but at times it is necessary in order to achieve and maintain harmony in the large picture of human affairs. Therefore, it should not be ruled out. This is why Confucius tells Zilu that \"the junzi harmonizes without following the flow\" (Liji; TTC 1980: 1626). To blindly follow the flow of other people is what Xunzi has called \"fawning\" as opposed to \"reasonably accommodating\" (Xunzi: Xiushen; TTM 1986: 289); it will lead to an over-presence of sameness rather than long-term harmony. A harmony deteriorating into over-presence of sameness loses internal energy; it cannot maintain itself as harmony and needs renewal.\n\nOn this understanding, Confucian harmony is not mere agreement without difference; it is not to preserve peace at any cost. Harmony is harmonization; real harmony is a dynamic process. It does not rule out strife, but uses strife in order to achieve greater harmony. Harmony comes from, and is maintained through, harmonization; it requires action. Having faith in God does not mean that Christians will not fight for their cause. Similarly, Confucians do not just sit there waiting for harmony to present itself. To the contrary, Confucians have a mission in life: to harmonize the world in the process of the formation of the triad with Heaven and Earth. What makes Confucianism a philosophy of harmony rather than a philosophy of strife is that it takes harmony as the ultimate goal and measures success or failure of an action by whether it contributes to greater harmony in the world.\n\nThen, why is harmony so central to Confucianism? Admittedly, Confucianism is not the only tradition that values harmony. After all, who would not think harmony to be a good thing? My point is that, in comparison with other major world traditions, Confucianism gives harmony utmost importance not matched by most major world traditions and it holds firmly that harmony can be realized. Here I would like to show that the absence of a pre-determined fixed order in the Confucian cosmos and the Confucian belief in the goodness of human nature are among the main reasons why harmony is so central to Confucianism.\n\nDavid Hall and Roger Ames have pointed out insightfully that Confucian harmony, or he, marks the difference between Western \"Truth-seekers\" on the one hand and Chinese \"Way-seekers\" on the other (Hall and Ames 1998: 180).13 For Hall and Ames, \"Truth-seeking\" is the prototypal mode of doing philosophy in the West, while \"Way-seeking\" is its counterpart in China. Seeking \"Truth,\" with a capital \"T,\" is to look for something absolute, eternal, and ultimately true, like Plato's Forms. In contrast, the Chinese Way is not pre-set and needs to be generated through human activities. In Hall and Ames' terminology, whereas Westerners typically follow a logical order in their interpretation of the world, the Chinese typically follow an aesthetic order. Logical order is achieved by application to a given situation of an antecedent pattern of relatedness. Aesthetic order is achieved by the creation of novel patterns (Hall and Ames 1998: 16). Aesthetic order requires openness, disclosure, and flexibility. In the Chinese aesthetic order, various things have to be synthesized in order to generate a harmonious whole, like yin and yang, and the five processings (wuxing \u4e94\u884c).\n\nIn support of Hall and Ames' interpretation, I would point out that ancient Chinese cosmology holds the belief that the world has evolved from chaos, the belief that there is a process from no order to the generation of an order, as articulated in such texts as the Huainanzi: tianwenxun \u6dee\u5357\u5b50\u00b7\u5929\u6587\u8bad and later in Zhou Dunyi's \u5468\u6566\u9890 An Explanation of the Diagram of the Great Ultimate (Taiji tushuo \u592a\u6781\u56fe\u8bf4). Based on this belief, the Chinese order is fluid and open-ended; the order of yin and yang, and of the five processings at most provides a general direction rather than a detailed road map for human action.\n\nThis Confucian understanding of world order differs significantly from some other major world traditions (see Li 2008). In Christianity, for example, God created the world with a purpose for each and every part of it. The will of God is carved in nature as natural law. God has set up order in the world and boundaries between all parts of the creation. Accordingly, the right way in this world is to follow God's rules and to obey the order given by God. Unlike most major world traditions, Confucianism does not believe in an anthropomorphic God as creator. Consequently, in the Confucian world there is no order or natural law from God. Without a pre-set fixed order, the world has to generate an order of its own. Although sometimes the Confucian Heaven (tian \u5929) appears to play a role that resembles the Christian God in some way, it is not a transcendent power as the Christian God is.14 The Confucian Heaven is a member of the Triad Heaven-Earth-Humanity (tian-di-ren \u5929\u5730\u4eba), and it does not have the power to impose a pre-determined order on the world from without. Rather, as merely one member of the Triad the Confucian Heaven needs to achieve order through coordination with the other two-thirds, namely Earth and Humanity.\n\nFor Confucians, therefore, order in the world has to be achieved through harmonization. Take again the example of the populations of sheep and wolves. On the Confucian thinking, there is no fixed order from God about their populations. Nature keeps a balance between the two. Ancient Confucians believe that the will of Heaven is revealed through the people. When the ruler loses the mandate of Heaven, it is time for the people to replace him. Perhaps the fact of the matter is that there is no such thing as the mandate of Heaven. Rather it may be that when the ruler is so oppressive and causes so much social disharmony that the people can no longer put up with him, it is time to replace him with another ruler, with one who can harmonize with the people. Confucians see a harmony (not necessarily friendliness) coming out of this continuous interplay of opposing forces. Through such interplay various parts of the world \"negotiate\" with one another in order to strike a balance, not from a pre-destined principle but through some kind of \"compromise,\" some kind of \"give-and-take.\" It is like rocks and water in the river: both can have their way yet both have to yield in some way. In such a world, any order that exists has to be an outcome of harmonization.\n\nElsewhere I have argued that in Chinese social practice, the Chinese understanding of truth plays an important role that differs from the West (Li 1999: Chap.\u200b 2). The Chinese typically do not see truth as correspondence with an objective fact in the world; rather they understand truth more as a way of being, namely being a good person, a good father or a good son. For them there is no objective truth carved in stone and, consequently, there is not an ultimate fixed order in the world for things to operate. The Confucian Dao is the process of generating an actual order in the world rather than a pre-given fixed order itself. Without pre-determined truth, humanity in coordination with Heaven and Earth has to set boundaries for themselves and for other things as they move forward in the world.\n\nBetween being too principled and being too flexible, Christian theology is more likely to risk the former than Confucian theology, and Confucian theology is more likely to risk the latter than Christian theology.15 Acting out of principles (e.g., the Ten Commandments) from God, the Christian stands firm, but she may misunderstand God (as during the Crusades); being principled leaves her little room for flexibility: God is always right and so are God's commands. Acting on the ideal of harmony, Confucians have few specifics to go by before they themselves make the specifics, and being made by human beings, the specifics should never be taken as absolute. Thus Confucians tend to be less rigid, but risk being too flexible in the pursuit of harmony. It would be na\u00efve, however, to think that a happy union of the Christian and the Confucian would solve all the problems: such a union carries the weakness of not being principled enough when needing to be principled and of not being flexible enough when in need of flexibility.\n\nIn addition to the absence of a pre-determined fixed order in the world, the Confucian belief in harmony has been re-enforced by a faith in the goodness of human nature. Early on in history, Mencius articulated and argued emphatically that human nature is good. His doctrine has had a long-lasting influence on the Confucian tradition. This notion sets the orientation for Confucians to look for ways to resolve conflicts in society without elimination. This orientation is fundamentally different from that of looking to identify and eliminate \"evil-doers\" in the world. In order to root out \"evil-doers,\" the primary approach would be to attack and destroy the enemy as effectively as possible; negotiation is merely a waste of time and of opportunity. On the other hand, in order to work out differences between people who are basically good, the primary approach should be looking for ways to negotiate with them; even though confrontation cannot always be ruled out, it should be minimized. Based on the Confucian belief in the goodness of human nature, it is readily conceivable that the world at large is not fundamentally antagonistic to human existence, because the non-human world, which is incapable of consciously harming others, is either benign or neutral toward humanity. Therefore, harmonization, rather than elimination, should be the primary consideration in dealing with problems in the world.\n\nIn practice, the ideal of harmony translates into a kind of pragmatic attitude or mentality. It is this mentality that makes the whole world of difference between the philosophy of harmony and the philosophy of combativeness. The attitude of harmony has a strategic significance. It makes us more willing to engage in negotiation, more willing to compromise, and less willing to resort to confrontation and conquest. It enables us take into consideration the whole picture of an issue and give each party its due. Therefore, it is more conducive to peaceful solutions to problems in the world.\n\nAt this point, it is appropriate to discuss issues related to the Confucian ideal of harmony and its implementation. First, having faith in harmony does not imply that things always harmonize. As a cherished ideal, harmony provides the guideline as well as an account of certain cultural patterns in Confucian society. Having faith in harmony is to have faith in a world that gives everything its due and lets the myriad things flourish. However, things may not always harmonize. In Christianity, God's will is not always followed by human beings; natural law gets violated; and not all Christians love their neighbors as they should. Similarly, in the Confucian world, harmony is not always achieved and maintained; disruptions take place and disharmony ensues; not all Confucians cherish harmony as much as they should. A Confucian who has faith in the harmony of the world is somewhat like a Christian who has faith in God, even though God and harmony are by no means parallel: Sometimes things go terribly wrong; yet a Christian would keep her faith that things eventually will turn around because all is in the hands of God; similarly, a Confucian in trying times would believe that disharmony is temporary and harmony will prevail. It is harmony as the ultimate ideal that makes the Confucian world a meaningful world, and it is harmonization that gives the Confucians a sense of sacred mission in the world.\n\nSecond, embracing an ideal does not imply a consensus on its application. Under the same ideal, people may seek different ways to implement it. Christians may not agree with one another on how they should put God's words into practice, even though they share a common belief in God and a common belief that one should obey God's will. Similarly, although Confucians take harmony as their highest ideal and all strive for its achievement, they may not always agree with one another on how harmony is identified and how it is best achieved. It is possible that what is called \"harmony\" by one person may be disguised oppression to another. Indeed, there have been times when under the name of \"harmony\" oppression persisted, just as oppression under the name of God existed in the history of Christianity. Obviously it is not the case that once we embrace the concept of harmony, all problems will disappear. My aim in this chapter is to elucidate the Confucian concept and ideal of harmony, not to provide a precise conception of harmony on a particular issue. Just as Christians need to figure out among themselves what God's words mean to them, Confucians have to translate the ideal of harmony into specific terms. Nevertheless, for Confucians it is still important to promote the ideal of harmony rather than disharmony, and to prioritize the goal of harmony, just as it is still important to promote justice rather than injustice, even though we may not agree on exactly what justice is in a specific case.\n\nFinally, one may ask, how are you supposed to promote harmony when your enemy is attacking you? In a world of conflicts today, isn't the Confucian ideal of harmony a mere naivet\u00e9? The Confucian would respond as follows: When your enemy is attacking you, you need to protect yourself. If he slaps your right cheek, you should not give him your left cheek. Harmonization requires action and resolve to overcome disharmonious elements in the world. However, you must understand that in the long run, the best life is to be lived in harmony and peace. Therefore, you should avoid doing extreme things that create or perpetuate your enemy, and even when you engage in fighting with your enemy, you should try to turn conflict into harmony. In other words, one should maintain a harmony mentality rather than the combatant mentality. The Confucian classic Zhouli: Dongguan \u5468\u79ae\u00b7\u51ac\u5b98 states that \"harmony results in peace\" (he ze an \u548c\u5247\u5b89) (TTC 1980: 914). Peace cannot be obtained and maintained without harmony. Temporary peace through oppression and suppression is not real peace, and it does not last. In order to achieve real peace and to maintain peace throughout the world, we would do well to learn from the Confucian ideal of harmony.\n\nReferences\n\nAnalects \u8ad6\u8a9e.1980. In Thirteen classics (TTC).\n\nChen, Guying \u9673\u9f13\u61c9. 2007. An annotation, interpretation and commentary on the four classics of the Yellow Emperor \u7687\u5e1d\u56db\u7d93\u4eca\u8a3b\u4eca\u8b6f. Beijing \u5317\u4eac: Shangwu Chubanshe \u5546\u52d9\u51fa\u7248\u793e.\n\nCheng, Chung-ying. 1991. New dimensions of Confucian and Neo-Confucian philosophy. Albany: SUNY Press. A collection of essays on Confucian philosophy.\n\nDong, Zhongshu \u8463\u4ef2\u8212. 1986. Chunqiu Fanlu \u6625\u79cb\u7e41\u9732. In Twenty-two masters (TTM).\n\nGuo, Qi \u90ed\u9f4a. 2000. The formation of the philosophical category He in Chinese history \u4e2d\u570b\u6b77\u53f2\u4e0a\u54f2\u5b78\u7bc4\u7587\u300c\u548c\u300d\u7684\u5f62\u6210. Bulletin of the institute of the Chinese literature and philosophy 16: 451\u2013466.\n\nHall, David, and Roger Ames. 1998. Thinking from the Han. Albany: SUNY Press. A study of Pre-Qin Confucian philosophy.\n\nLai, Kehong \u4f86\u53ef\u5f18. (2000). The Guoyu interpreted\u300a\u570b\u8a9e\u76f4\u89e3\u300b. Shanghai \u4e0a\u6d77: Fudan University Press \u5fa9\u65e6\u5927\u5b78\u51fa\u7248\u793e.\n\nLi, Chenyang. 1999. The Tao encounters the West: Explorations in comparative philosophy. Albany: State of New York University Press. A comparative study of such important issues as being, language, truth, ethics, and justice.\n\nLi, Chenyang. 2004. Zhongyong as grand harmony: An alternative reading to Ames and Hall's focusing the familiar. Dao: A Journal of Comparative Philosophy 3(2): 173\u2013188.\n\nLi, Chenyang. 2007. Li as cultural grammar: The relation between Li and Ren in the Analects. Philosophy East & West 57(3): 311\u2013329.CrossRef\n\nLi, Chenyang. 2008. The ideal of harmony in ancient Chinese and Greek philosophy. Dao: A Journal of Comparative Philosophy VII(1): 81\u201398.\n\nLi, Chenyang. 2010. Confucian moral cultivation, longevity, and public policy. Dao: A Journal of Comparative Philosophy 9(1): 25\u201336.\n\nLi, Chenyang. 2011. Harmony of virtues in the Wuxing Bamboo text \u7af9\u5e1b\u300a\u4e94\u884c\u300b\u95dc\u65bc\u5fb7\u6027\u548c\u8ae7\u7684\u601d\u60f3\u7814\u7a76. Journal of Chinese Studies \u570b\u5b78\u5b78\u520a 12: 59\u201366.\n\nLi, Chenyang. 2013. The Confucian Philosophy of Harmony. Abingdon\/New York: Routledge (A systematic examination of Confucian harmony philosophy, both theoretical and practical).\n\nLiji \u79ae\u8a18. 1980. In Thirteen classics (TTC).\n\nLiu, Zhao \u5289\u91d7. 2003. Annotated Guodian Chu Bamboo texts\u90ed\u5e97\u695a\u7b80\u6821\u91cb. Fuzhou\u798f\u5dde: Fujian People's Press \u798f\u5efa\u4eba\u6c11\u51fa\u7248\u793e.\n\nMencius \u5b5f\u5b50.1980. In Thirteen classics (TTC).\n\nShen, Vincent. 2003. Some thoughts on intercultural philosophy and Chinese philosophy. Journal of Chinese Philosophy 30(3&4): 357\u2013372.CrossRef\n\nShuowenjiezi \u8aaa\u6587\u89e3\u5b57\u6ce8. 1992. Xu Shen and Duan Yucai \u8bb8\u614e\u64b0 \u6bb5\u7389\u88c1\u6ce8. Shanghai \u4e0a\u6d77: Shanghai Shudian \u4e0a\u6d77\u66f8\u5e97.\n\nSources of Terms \u8fad\u6e90. 1995. Beijing \u5317\u4eac: Commercial Publisher \u5546\u52d9\u51fa\u7248\u793e. (A lexicon on sources of Chinese words).\n\nThirteen Classics (TTC). 1980.\u300a\u5341\u4e09\u7d93\u6ce8\u758f\u300b. Beijing \u5317\u4eac: Zhongguo Shudian \u4e2d\u570b\u66f8\u5e97.\n\nTwenty-Two Masters (TTM). 1986.\u300a\u4e8c \u5341 \u4e8c \u5b50\u300b. Shanghai \u4e0a\u6d77 : Shanghai Guji Chubanshe \u4e0a\u6d77\u53e4\u7c4d\u51fa\u7248\u793e.\n\nWeber, Max. 1951. The religion of China. Hans H. Gerth. New York: Free Press. Analyzes major religions of ancient China.\n\nXunzi \u8340\u5b50.1986. In Twenty-two masters (TTM).\n\nYijing \u6613\u7d93. 1980. In Thirteen classics (TTC).\n\nFootnotes\n\n1\n\nAncient Chinese used a five-tone scale in music.\n\n2\n\nAnother source of the early meaning of \"he\" is the word \"\u76c9,\" which means the mixing of wine with water. See Li 2008.\n\n3\n\nI here translate \"ji\u7e7c\" as \"advance growth.\" See the Analects 6.3, \"the superior person helps those in emergency but does not advance the cause of the rich\" (junzi zhouji bu jifu \u541b\u5b50\u5468\u6025\u4e0d\u7e7c\u5bcc) (TTC 1980: 2478).\n\n4\n\nLiterally it reads \"a single thing\" or \"a single item.\" In his commentary of Zhouli:Chunguan:Baozhangshi \u5468\u79ae\u00b7\u6625\u5b98\u00b7\u4fdd\u7ae0\u6c0f, Zheng Xuan writes, \"wu means color\" (wu se ye \u7269,\u8272\u4e5f) (TTC 1980: 819).\n\n5\n\n\"\u8072\u4e00\u7121\u807d, \u7269\u4e00\u7121\u6587, \u5473\u4e00\u7121\u679c, \u7269\u4e00\u4e0d\u8b1b\u3002\" The Shuowenjiezi defines jiang (\u8b1b) as \"harmonizing (\u548c\u89e3\u4e5f\") (Shuowenjiezi 1992: 95).\n\n6\n\nUnlike the Zuozhuan, the Guoyu is not one of the Confucian 13 Classics. But it has been attributed to the same author as the Zuozhuan and has a similar philosophical orientation.\n\n7\n\nFor my discussion of li, readers can see Li 2007.\n\n8\n\nBeside li (\u79ae), he is also closely related to other key Confucian concepts such as li \u7406, ren \u4ec1, and yi \u7fa9. Space does not allow me to expand my study into these concepts in relation to he in the present chapter.\n\n9\n\nWhen the state of Qi attacked Liu Xiahui's native state Lu in 634 B.C.E., Liu sent off people to persuade the ruler of Qi \"not to harm one another\" (Lai 2000: 295).\n\n10\n\nFor an insightful discussion of intercultural harmony from a Confucian perspective and its implications for the contemporary world, see Shen 2003. For a detailed explication of Confucian harmony at these levels, see Li 2013.\n\n11\n\nFor a detailed discussion of the Confucian harmony of virtues in the Wuxing, see Li 2011.\n\n12\n\nObviously human beings can affect the level of the balance between the two populations, so can other species.\n\n13\n\nObviously, not all Westerners are \"Truth-seekers\" and not all Chinese are \"Way-seekers.\" But to the extent that these two patterns have been the predominant tendencies respectively, I believe that Hall and Ames are right.\n\n14\n\nSome people may think otherwise. Here I follow Hall and Ames' interpretation (Hall and Ames 1998). Perhaps we should distinguish popular Chinese beliefs and Confucian theology here. An average person may believe that Heaven is a fixed entity and that Heaven has set a pre-determined order in the world. Confucian theology, as delineated in such texts as the Liji:Zhongyong and the Yijing, clearly offers a different view.\n\n15\n\nThis is not to say that a Confucian may not be as stubborn as a Christian on one's commitment to a particular moral ideal. But the typical Confucian has less faith in an objective Platonic moral order set in the universe than the typical Christian, and therefore the Confucian's fundamental moral principles are less clearly cut. In this regard, the Confucian probably stands at the mid-point between the \"water-like\" Daoist and the \"righteous\" Christian.\nVincent Shen (ed.)Dao Companions to Chinese PhilosophyDao Companion to Classical Confucian Philosophy201410.1007\/978-90-481-2936-2\u00a9 Springer Science+Business Media Dordrecht 2014\n\n## Introduction to Contributors\n\nBai, Tongdong is a professor of philosophy at Fudan University, Shanghai. His research focuses on political philosophy and the contemporary relevance of Confucian political thoughts. He has published China: The Middle Way of the Middle Kingdom . Forthcoming will be his A New Mission of an Old State: the Contemporary and Comparative Significance of Traditional Confucian Political Philosophy .\n\nChan, Wing-cheuk is a professor at the Department of Philosophy, Brock University, Canada. He has been research fellow in Germany, India and Taiwan. His research interests consist in phenomenology, Yog\u0101c\u0101r\u0101 Buddhism, Chinese and comparative philosophy.\n\nCheng, Chung-ying is a professor at the Department of Philosophy, University of Hawaii at Manoa. He has founded both the Journal of Chinese Philosophy and the International Society of Chinese Philosophy. He publishes in the areas of Confucianism, the Yijing , Comparative Philosophy and Onto-Hermeneutics.\n\nCua, Antonio was a professor of philosophy at the School of Philosophy, Catholic University of America. He published Dimensions of Moral Creativity; The Unity of Knowledge and Action; Moral Vision and Tradition ; Human Nature, Ritual, and History: Studies in Xunzi and Chinese Philosophy. He edited the Encyclopedia of Chinese Philosophy .\n\nLi, Chenyang is now an associate professor at the School of Humanities and Social Sciences, Nanyang Technological University, Singapore. He served as Professor and Chair of the Department of Philosophy, Central Washington University, USA, (1999\u20132010). His research interest focuses on Chinese and comparative philosophy.\n\nLiu, Johanna is a professor of Chinese art and aesthetics at the Department of East Asian Studies, University of Toronto, Canada. Her main research interests are: philosophical aesthetics in China and the West, Chinese art theories and literary criticism.\n\nLo, Yuet Keung is an associate professor at the Department of Chinese Studies, National University of Singapore. He specializes in early Chinese thought and cross-fertilization among Buddhism, Daoism, and Confucianism, as well as Buddhist narrative and female ethics in early medieval China.\n\nNi, Peimin is a professor of philosophy at the Grand Valley State University, Michigan, USA. His research interest is in the area of Chinese and comparative philosophy. Ni is a former President of the Association of Chinese Philosophers in America, and former President of the Society of Asian and Comparative Philosophy.\n\nPlaks, Andrew H ., Ph.D. Princeton University 1973, professor of Chinese Literature and Comparative Literature (1973\u20132007); professor of East Asian Studies, Hebrew University of Jerusalem (1992\u20132012). His research interests consist in Chinese philosophy and literature, especially classical Confucianism, Chinese narratology and Chinese novels.\n\nShen, Vincent holds the Lee Chair in Chinese Thought and Culture at the Department of Philosophy and Department of East Asian Studies, University of Toronto. His research interests are in the areas of Chinese and comparative philosophy, phenomenology, and philosophical problems of technology, culture and religion.\n\nShun, Kwong-loi is currently Sin Wai Kin Professor of Chinese Culture and Chair Professor of Philosophy at the Chinese University of Hong Kong. He was formerly Professor of Philosophy at the University of California at Berkeley, and Professor of Philosophy and East Asian Studies at the University of Toronto.\n\nTsai, Cheng-Feng is a professor at the Department of Chinese Literature at National Taiwan University. He specializes on the study of East Asian Confucianism, Chinese intellectual history, and the history of Chinese Buddhism.\n\nVirag, Curie , Ph.D., Harvard University, is an assistant professor at the Department of East Asian Studies, University of Toronto. Her main area of research is pre-modern Chinese intellectual history and thought, especially the ethics of feeling in premodern China.\n\nYan, Zhonghu is a professor at Hangzhou Normal University, China. Prior to this appointment, he taught at Hope College and the University of Saskatchewan. His research interest consists in Chinese and comparative philosophy, especially the religiosity of Confucianism in dialogue with Christianity.\n\nIndex\n\nA\n\nAesthetics\n\nAmane, Nishi\n\nAmes, Roger\n\nAnalects , the ( Lunyu \u8ad6\u8a9e)\n\nAristotle\n\nArt\n\nAutonomous morality\n\nAxial age\n\nB\n\nBa tiaomu \u516b\u689d\u76ee (eight specific points)\n\nBao Xian\u5305\u54b8\n\nBarthes, Roland\n\nBeauty\n\nBenevolence\n\nBerlin, Isaiah\n\nbi \u853d (be obscured)\n\nBible\n\nBo Yu \u4f2f\u9b5a\n\nBook of Documents ( Shangshu \u5c1a\u66f8) or Classics of Documents ( Shujing \u66f8\u7d93)\n\nC\n\nCentrality\n\nC hen Daqi \u9673\u5927\u9f4a\n\nC hen Lai \u9673\u4f86\n\nC hen Li \u9673\u7acb\n\nC hen Liang \u9673\u4eae\n\nC han Wing-tsit \u9673\u69ae\u6377\n\nCheng \u8aa0 (integrity, sincerity)\n\nC heng Chung-ying \u6210\u4e2d\u82f1\n\nC heng Hao \u7a0b\u704f\n\ncheng qi yi \u8aa0\u5176\u610f or cheng yi \u8aa0\u610f\n\nC heng Yi \u7a0b\u9824\n\nChengji chengren \u6210\u5df1\u6210\u4eba\n\nChengren \u6210\u4eba (an ethically accomplished person)\n\nChi \u6065 (shame, regard as shameful)\n\nC hong Kim-chong \u838a\u9326\u7ae0\n\nChristianity\n\nChunqiu , the \u6625\u79cb (Spring and Autumn Annals)\n\nGongyang's Commentary ( Gongyangzhuan \u526c\u7f8a\u50b3)\n\nGuliang's Commentary ( Guliangzhuan \u7a40\u6881\u50b3)\n\nZuo's Commentary ( Zuozhuan \u5de6\u50b3)\n\nCi rang \u8fad\u8b93 (politely declining and letting others have what is good or honorable), 270\n\nConfucian community\n\nConfucianism\n\nclassical Confucianism\n\nearly Confucianism\n\nneo-Confucianism\n\npre-Qin Confucianism\n\nConfucius (Konfuzi \u5b54\u592b\u5b50, Kongzi \u5b54\u5b50, Kong Qiu \u5b54\u4e18)\n\nConsequentialism\n\nCorrelative thinking\n\nCosmology\n\nCourage ( yong \u52c7)\n\nCreel, G. Herrlee\n\nCua, Antonio\n\nC ui Shu \u5d14\u8ff0\n\nCulture\n\nD\n\nD ai Zhen \u6234\u9707\n\ndali \u5927\u7406\n\nDao \u9053 (the Way)\n\nDao De Jing ( Daodejing ) \u9053\u5fb7\u7d93\n\nDaoism\n\nDaotong \u9053\u7d71\n\ndao wenxue \u9053\u554f\u5b78 (following the path of inquiry and learning)\n\nDance\n\nDaxue \u5927\u5b78 ( Great Learning )\n\nDayitong \u5927\u4e00\u7d71 (Great Unification)\n\nde Bary, William\n\nde Sousa, Ronnie\n\nDeliberation\n\nDecree of heaven (\u5929\u547d)\n\nDeleuze, Gill\n\nDemocracy\n\nDesire ( yu \u6b32)\n\nDilthey, Wilhem\n\nDivine command theory\n\nDoctrine of the Mean ( Zhongyong , the)\n\nD ong Zhongshu \u8463\u4ef2\u8212\n\ndu \u7368 (what one alone can access)\n\nduan \u7aef (sprouts of virtue)\n\nduanzhang quyi \u65b7\u7ae0\u53d6\u7fa9 (appropriation of meaning by key text)\n\nDuke of Zhou (Zhougong \u5468\u526c)\n\nE\n\nEducation\n\nEmotion (see also qing \u60c5)\n\nEnlightenment\n\nEnvironment\n\nEquilibrium\n\nEssentialism, essentialist\n\nEthics\n\nEvil\n\nExcellence\n\nExhausting principles ( qiong-li \u7aae\u7406)\n\nExplicit knowledge\n\nF\n\nFaith\n\nFaithfulness\n\nFamily\n\nF an Chi \u6a0a\u9072\n\nFeeling\n\nFilial piety\n\nFingarette, Herbert\n\nFour Books\n\nFounding events ( \u00e9v\u00e9nements fondateurs )\n\nFreedom\n\nFriendship\n\nfu \u798f (happiness)\n\nFukuyama, Francis\n\nG\n\nGaozi \u544a\u5b50\n\nGenerosity\n\nGentleman\n\nGlobalization\n\nGod\n\ngewu \u683c\u7269\n\nGolden rule(s)\n\nGongduzi \u526c\u90fd\u5b50\n\ngong jing zhi xin \u606d\u656c\u4e4b\u5fc3 (feeling of respect)\n\nG ongsun nizi \u526c\u5b6b\u5c3c\u5b50\n\nGood\/goodness\n\nGood will\n\nGovernance\n\nGraham, Angus\n\nGreat Learning\n\nGrief\n\nGuan \u5b98 (office, function)\n\nGuanzi \u7ba1\u5b50\n\nGuattari, Felix\n\nG uo qingfan \u90ed\u6176\u7c53\n\nGuben \u53e4\u672c\n\nGuodian \u90ed\u5e97\n\nG uo Xiang \u90ed\u8c61\n\nG uo Yi \u90ed\u6c82\n\nH\n\nHabermas, J\u00fcrgen\n\nHall, David\n\nH an feizi , or Han Feizi, the \u97d3\u975e\u5b50\n\nH an Ying \u97d3\u5b30\n\nHan Shiwaizhuan \u97d3\u8a69\u5916\u50b3\n\nHan Yu \u97d3\u6108\n\nHansen, Chad\n\nhao ran zhi qi \u6d69\u7136\u4e4b\u6c23 (floodlike qi )\n\nHappiness\n\nhe \u548c (harmony)\n\nH e Xiu \u4f55\u4f11\n\nH e Yan \u4f55\u664f\n\nHeart\/mind ( xin \u5fc3)\n\nHeaven\n\nHeavenly principle\n\nHeidegger, Martin\n\nHenry, Michel\n\nHermeneutics\n\nHeteronomous morality\n\nHexagram\n\nHexagram Kun \u4e7e\u5366\n\nHexagram Qian \u5764\u5366\n\nHongfan \u6d2a\u7bc4 ( Great Model )\n\nH ou Wailu \u4faf\u5916\u76e7\n\nH uang Chun-chieh \u9ec3\u4fca\u5091\n\nH uang Kan \u9ec3\u4f83\n\nH uang Yong \u9ec3\u52c7\n\nHumaneness (humanity, ren \u4ec1)\n\nHumanism, creative\n\nHume, David\n\nI\n\nIdealism\n\nImmanence, immanent\n\nImpartiality\n\nIhara, Craig\n\nIntegral wholeness\n\nInternalism\n\nIntuition\n\nIvanhoe, Philip. J.\n\nJ\n\nJaspers, Karl\n\nji \u7a4d (accumulation)\n\njian ai \u517c\u611b (detached concern, equalitarian love)\n\njiao \u6559 (education, instruction, teaching)\n\nJ iao Xun \u7126\u5faa\n\njing \u9759 (quiescence)\n\njing \u656c (treat respectfully)\n\njing\/zhuan \u7d93\/\u50b3\n\njixion \u5409\u51f6 (fortune or misfortune)\n\njunzi \u541b\u5b50 (worthiness, a person of moral attainment)\n\njustice\n\nJizha \u5b63\u624e\n\nK\n\nK ang Yuwei \u5eb7\u6709\u70ba\n\nKant, Immanuel\n\nKarlgren, Bernhard\n\nke ji \u514b\u5df1 (self-control, overcoming the self)\n\nKing Wen\u6587\u738b\n\nKing Xuan \u5ba3\u738b\n\nKnoblock, John\n\nK ong Yingda \u5b54\u7a4e\u9054\n\nKongzi Shilun \u5b54\u5b50\u8a69\u8ad6 ( Confucius on Poetry )\n\nKristeva, Julia\n\nL\n\nLaozi\n\nLao D.C.\n\nle \u6a02 (joy, joyous)\n\nLearning (learning \u5b78)\n\nlearning for one's own sake\n\nlearning for the sake of others\n\nLegge, James\n\nli \u7406 (normative principle\/pattern)\n\nli \u79ae (ritual, ritual propriety, rites)\n\nliangxin \u826f\u5fc3\n\nliberal democracy\n\nLiji \u79ae\u8a18 (Book of Rites)\n\nliujing \u516d\u7d93 Six Confucian Classics\n\nlixue \u7406\u5b78\n\nliyi \u7406\u7fa9 (morality)\n\nL i Gong \u674e\u5868 (1659\u20131733)\n\nL i Minghui \u674e\u660e\u8f1d\n\nL iu Xiusheng \u5289\u79c0\u751f\n\nL iu Zongzhou \u5289\u5b97\u5468\n\nL o Yuet Keung \u52de\u6085\u5f37\n\nl\u00fc \u616e (thought)\n\nL u Xiangshan \u9678\u8c61\u5c71\n\nluan \u4e82 (disorder)\n\nLunyu \u8ad6\u8a9e ( Analects )\n\nM\n\nMacIntyre, Alaidair\n\nMawangdui \u99ac\u738b\u5806\n\nM ao Qiling \u6bdb\u5947\u974883\n\nMencius, Mengzi\u5b5f\u5b50\n\nMeng Jizi \u5b5f\u5b63\u5b50\n\nMetaphysics\n\nM i Zijian \u5b93\u5b50\u8ce4\n\nMill, J. Stuart\n\nM in Ziqian \u9594\u5b50\u9a2b\n\nming \u547d (decree, mandate)\n\nming \u660e (illumination)\n\nming \u540d (name)\n\ngongming \u5171\u540d (generic terms)\n\nbieming \u5225\u540d (specific terms)\n\njianming \u517c\u540d (compound term)\n\ndanming \u55ae\u540d(single terms)\n\nming mingde \u660e\u660e\u5fb7\n\nModernity\n\nMoral good\n\nMoral psychology\n\nM ou Zongsan \u725f\u5b97\u4e09\n\nMozi \u58a8\u5b50\n\nMohist Logic\n\nexemplification ( pi \u8f9f)\n\nextension ( tui \u63a8)\n\nimitation ( yuan \u63f4)\n\nparallel ( mou \u4f94)\n\nMusic\n\nN\n\nNatural good\n\nNei ye \u5167\u696d (inner cultivation)\n\nneisheng waiwang \u5167\u8056\u5916\u738b\n\nN i Peimin \u502a\u57f9\u6c11\n\nNietzsche, Friedrich\n\nNeo-Confucianism\n\nidealist neo-confucianism\n\nnaturalist neo-confucianism\n\nrealist neo-confucianism\n\nNivision, David\n\nNorden W. Van\n\nnu \u6012 (anger)\n\nO\n\nOriginal mind (\u672c\u5fc3)\n\nOntology\n\nO uyang Xiu \u6b50\u967d\u4fee\n\nO uyang Xun \u6b50\u967d\u8a62\n\nOwen, Stephen\n\nP\n\nP ang Pu \u9f90\u6a38\n\nPassion\n\nPatience\n\nPeace\n\npeng \u670b\n\nPhilosophy\/Philosophia\n\nphronesis (practical wisdom)\n\nPlaks, Andrew\n\nPlato\n\nPoetry\n\nPolitical theology\n\nPondering sincerity (\u601d\u8aa0)\n\nPostmodern, postmodernity\n\nPuett, Michael\n\nQ\n\nqi \u6c23 (life force, vital force, configurative energy)\n\nQ ian Mu \u9322\u7a46\n\nQ idiao Kai \u6f06\u96d5\u958b\n\nqing \u60c5 (dispositions, emotions, feelings, facts...etc)\n\nai \u611b (love)\n\nai \u54c0(sorrow)\n\ne \u60e1 (dislike)\n\nhao \u597d(fondness, to like)\n\nhuan \u6b61 (contentedness)\n\ne \u6a02 (delight\/pleasure)\n\nnu \u6012(anger)\n\nxi \u559c(joy)\n\nxinuaile \u559c\u6012\u54c0\u6a02\n\nyu \u6b32 (desire)\n\nquan \u6b0a (weighing, moral discretion)\n\nR\n\nRang \u8b93\n\nRational ethics\n\nRawls, John\n\nRealizing the mind ( jinxin \u76e1\u5fc3)\n\nReciprocity\n\nren \u4ec1\n\nrenqing \u4eba\u60c5 (human feelings)\n\nRicoeur, Paul\n\nRitual\n\nRosement, Henry\n\nRoss, W.D.\n\nru \u8fb1\n\nru \u5112 (Confucian)\n\nda Ru \u5927\u5112(great Confucian)\n\nya Ru \u96c5\u5112 (refined Confucian)\n\nS\n\nSagacity\n\nSagehood ( sheng \u8056)\n\nsan gangling \u4e09\u7db1\u9818 (three general principles)\n\nshangdi \u4e0a\u5e1d (High God)\n\nSelf\n\nSelf-centeredness\n\nSelf-cultivation\n\nShame\n\nshan \u5584 (good)\n\nshengren \u8056\u4eba(sage)\n\nS hen Tsing-song \u6c88\u6e05\u677e (Vincent Shen)\n\nS hen Yue \u6c88\u7d04\n\nshi \u52e2 (circumstances)\n\nshi \u58eb (scholar-official)\n\nshi \u6642 (timeliness)\n\nshi fei zhi xin \u662f\u975e\u4e4b\u5fc3 (sense of right and wrong)\n\nShi Xiangzi \u5e2b\u8944\u5b50\n\nshizhong \u59cb\u7d42\n\nshu \u6055 (altruistic extension, sympathetic understanding, reciprocity)\n\nS hun Kwong-loi \u4fe1\u5ee3\u4f86\n\nShuo Wen Jie Zi \u8aaa\u6587\u89e3\u5b57\n\nsi \u79c1 (self-centeredness)\n\nsi \u601d (thought)\n\nsiduan \u56db\u7aef (four sprouts)\n\nfeeling of compassion ( ce yin zhi xin \u60fb\u96b1\u4e4b\u5fc3)\n\nrespect ( gong jing zhi xin \u606d\u656c\u4e4b\u5fc3)\n\nsense of right and wrong ( shi fei zhi xin \u662f\u975e\u4e4b\u5fc3)\n\nshame ( xiu wu zhi xin \u7f9e\u60e1\u4e4b\u5fc3)\n\nSi-Meng xue pai \u601d\u5b5f\u5b78\u6d3e (Zisi-Mencius School)\n\nSima Qian \u53f8\u99ac\u9077\n\nSocrates\n\nStrauss, Leo\n\nT\n\nTacit knowledge\n\nTaiji \u592a\u6975 (Great Ultimate)\n\nT ang Junyi \u5510\u541b\u6bc5\n\nT ang Yijie \u6e6f\u4e00\u4ecb\n\nTechnology, technique\n\nTillich, Paul\n\ntianren xiangfen \u5929\u4eba\u76f8\u5206 (mutual distinction of heaven and man)\n\nTranscendence, transcendental\n\nTheonomous ethics\n\ntian \u5929 (Nature, Nature-endowed)\n\nTrust\n\nTrustworthiness\n\nTruth\n\nT sai Zheng-Feng \u8521\u632f\u8c50\n\nU\n\nUltimate reality\n\nUniversalizability\n\nV\n\nValue\n\nVan Norden, Bryan\n\nVir\u00e1g, Curie\n\nVirtues ( de \u5fb7)\n\nbasic virtues\n\ncomplete virtues ( quande \u5168\u5fb7)\n\ndependent virtues\n\nexcellence or virtue ( meide \u7f8e\u5fb7)\n\ninterdependent virtues\n\npartial virtues ( piande \u504f\u5fb7)\n\nVirtuous action\n\nW\n\nWaley, Arthur\n\nWang Bo \u738b\u67cf\n\nWang Chong \u738b\u5145\n\nWang Fuzi \u738b\u592b\u4e4b\n\nWang Nianshun \u738b\u5ff5\u5b6b\n\nWang Yangming \u738b\u967d\u660e\n\nwangdao \u738b\u9053 (the king's way)\n\nwangzhi \u738b\u5236 (system of the king)\n\nWatson, Burton\n\nWay of heaven ( tian zhi dao \u5929\u9053)\n\nWay of man ( ren zhi dao \u4eba\u9053)\n\nwei \u507d (conscious activity, artifice)\n\nWei Zhengtong \u97cb\u6b63\u901a\n\nweifa yifa \u672a\u767c\u5df2\u767c\n\nwen \u6587\n\nWhitehead, Alfred\n\nWisdom\n\nWittgenstein, Ludwig\n\nWong, David\n\nwu (\u5deb) (shaman)\n\nwu \u7269 (things, external things)\n\nWujing \u4e94\u7d93 (Five Classics)\n\nwuwei \u7121\u70ba\n\nwuxing \u4e94\u884c\n\nX\n\nxi \u559c (delight)\n\nxiao \u5b5d (filial piety)\n\nXiao Yang\n\nxin \u5fc3 (heart-mind)\n\nhumaneness ( renxin \u4ec1\u5fc3)\n\nimpartiality or fair-mindedness ( gongxin \u526c\u5fc3)\n\nlearning mind ( xuexin \u5b78\u5fc3)\n\nxing \u6027 (inborn nature)\n\nXing zi ming chu \u6027\u81ea\u547d\u51fa\n\nxinxue \u5fc3\u5b78\n\nxiu wu zhi xin \u7f9e\u60e1\u4e4b\u5fc3 (feeling of shame)\n\nX u Fuguan \u5f90\u5fa9\u89c0\n\nxuyi er jing \u865b\u4e00\u800c\u975c (void, oneness and tranquility)\n\nXunzi, Xunzi , the \u8340\u5b50\n\nY\n\nY an Hui \u984f\u56de\n\nYan Zi \u664f\u5b50\n\nYang \u990a (to nurture)\n\nY ang Bojun \u694a\u4f2f\u5cfb\n\nyi \u6613 (change)\n\nyi \u610f (direction of heart\/mind, intention)\n\nyi \u7fa9 (righteousness)\n\nyiyi nizhi \u4ee5\u610f\u9006\u5fd7 (trace the expressed intention by understanding)\n\nYijing \u6613\u7d93\n\nguaci \u5366\u8fad (hexagram dictum)\n\nyaoci \u723b\u8fad (stroke dictum)\n\nYizhuan \u6613\u50b3 (Interpretations of the Zhouyi )\n\nShuogua \u8aaa\u5366 (Remarks on the Trigrams)\n\nTuanzhuan \u5f56\u50b3 (Judgment Interpretations)\n\nXiangzhuan \u8c61\u50b3(Symbolism Interpretations)\n\nXici \u7e6b\u8a5e (Great Appendixes)\n\nXugua \u5e8f\u5366 (Sequence of the Hexagrams)\n\nYijing's concepts and theories of deciding jixion \u5409\u51f6 (fortune or misfortune)\n\ndang \u7576 (properness)\n\nshi \u6642 (timing)\n\nying \u61c9 (responding)\n\nzhong \u4e2d (centrality)\n\nYijing's concepts and theories of deciding jixion \u5409\u51f6 (fortune or misfortune) ( cont .) dangwei shuo \u7576\u4f4d\u8aaa (Theory of Proper Position)\n\nqushi shuo \u8da3\u6642\u8aaa (Theory of Acting in Proper Time)\n\nyingwei shuo \u61c9\u4f4d\u8aaa (Theory of Respondent Position)\n\nzhongwei shuo \u4e2d\u4f4d\u8aaa (Theory of Central Position)\n\nyin and yang \u9670\u967d\n\nYu, Da Yu \u79b9, \u5927\u79b9\n\nyu \u6b32 (desire)\n\nYu, Pauline\n\nyue \u6a02 (music)\n\nYueji \u6a02\u8a18\n\nZ\n\nZ ai Wo \u5bb0\u6211\n\nZengzi \u66fe\u5b50\n\nZ hang Zai \u5f35\u8f09\n\nZ heng Xuan \u912d\u7384\n\nzhi \u5fd7 (directions of the heart\/mind, intention, will, etc)\n\nzhong \u4e2d (centrality)\n\nzhong \u5fe0 (loyalty)\n\nzhonghe \u4e2d\u548c(centrality and harmony)\n\nZhongyong \u4e2d\u5eb8 ( Centrality and Commonality, or Doctrine of the Mean )\n\nZ hou Dunyi \u5468\u6566\u9824\n\nZhouyi \u5468\u6613\n\nZhouli \u5468\u79ae\n\nZ hu Xi \u6731\u71b9\n\nZhuangzi, Zhuangzi , the \u838a\u5b50\n\nZichan \u5b50\u7522\n\nZigong \u5b50\u8ca2\n\nZilu \u5b50\u8def\n\nZisi \u5b50\u601d\n\nZiyou \u5b50\u6e38\n\nzun dexing \u5c0a\u5fb7\u6027 (honoring moral character) \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n# Copyright\n\nHarperCollins _Publishers_\n\n1 London Bridge Street\n\nLondon SE1 9GF\n\nwww.harpercollins.co.uk\n\nFirst published by HarperCollins _Publishers_ 2018\n\nFIRST EDITION\n\n\u00a9 Chris Salewicz 2018\n\nCover layout design Claire Ward \u00a9 HarperCollins _Publishers_ 2018\n\nCover photograph \u00a9 Gijsbert Hanekroot\/Redferns\/Getty Images\n\nExtract from _I'm With the Band: Confessions of a Groupie_ by Pamela Des Barres published by Omnibus Press \u00a9 Panela Des Barres\n\nA catalogue record of this book is available from the British Library\n\nChris Salewicz asserts the moral right to be identified as the author of this work\n\nWhile every effort has been made to trace the owners of copyright material reproduced herein and secure permissions, the publishers would like to apologise for any omissions and will be pleased to incorporate missing acknowledgements in any future edition of this book.\n\nAll rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the nonexclusive, non-transferable right to access and read the text of this e-book on screen. No part of this text may be reproduced, transmitted, downloaded, decompiled, reverse engineered, or stored in or introduced into any information storage retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nFind out about HarperCollins and the environment at\n\nwww.harpercollins.co.uk\/green\n\nSource ISBN: 9780008149291\n\nEbook Edition \u00a9 July 2018 ISBN: 9780008149307\n\nVersion: 2018-06-21\n\n# Dedication\n\nFor Alex and Cole\n\n# Contents\n\n_Cover_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nPreface\n\nIntroduction\n\n1 SPANISH GUITAR IN SURREY\n\n2 FROM NELSON STORM TO SESSION PLAYER\n\n3 SHE JUST SATISFIES\n\n4 BECK'S BOLERO\n\n5 BLOW-UPS\n\n6 'YOU'RE GOING TO KILL ME FOR A THOUSAND DOLLARS?'\n\n7 'LIKE A LEAD ZEPPELIN'\n\n8 AMERICAN ADVANCES\n\n9 'WHOLE LOTTA LOVE'\n\n10 LED ZEPPELIN II\n\n11 'DO WHAT THOU WILT'\n\n12 THE BEAST 666\n\n13 ALL THAT GLITTERS\n\n14 THE LEGEND OF ZOSO\n\n15 CITY OF ANGELS\n\n16 THE KING AND JIMMY PAGE\n\n17 COCAINE NIGHTS AND HAUNTED HOUSES\n\n18 AN ACCIDENT IN EXILE\n\n19 THE KENNETH ANGER CURSE\n\n20 FACE TO FACE\n\n21 THE RULES OF ENGAGEMENT\n\n22 BONZO'S LAST STAND\n\n23 THE HERMIT\n\n24 MIDDLE-AGED GUITAR GOD\n\n25 THE SORCERER'S APPRENTICE\n\n26 PHOENIX RISING\n\nPicture Section\n\nSources\n\nAcknowledgements\n\nIndex of Searchable Terms\n\n_About the Publisher_\n\n# Preface\n\nOne chilly February evening in 1975, Jimmy Page journeyed in a black Cadillac limousine to David Bowie's rented house on 20th Street in Manhattan. The Led Zeppelin leader and Bowie had known each other since the mid-sixties, Page having played on several of Bowie's early records.\n\nThe pair were also linked through Lori Mattix, Page's Los Angeles-based underage lover and a cause of considerable concern in the Zeppelin camp, thanks to the criminal complications this could create for the Biggest Band in the World. What few knew was that Bowie had taken Mattix's virginity when she was just 14.\n\nWith the superstar pair having been reintroduced by Mick Jagger, Bowie had invited Page over to his place for an evening's entertainment largely comprising lines of cocaine and glasses of red wine, along with Ava Cherry, Bowie's girlfriend.\n\nMired in his _Cracked Actor_ phase, Bowie was known to be living on milk and cocaine, and on the edge of madness. He had been inspired to devour the writings of Aleister Crowley, whose philosophy he had first dabbled in during the late 1960s: Bowie believed that Page's deep knowledge of Crowley had enhanced the guitarist's aura until it was rock hard and ringing with power.\n\nBut despite being intrigued, Bowie was extremely wary of Page. Conversation was somewhat stiff, although there was brief talk about Page's progress, or lack of it, on the soundtrack to filmmaker Kenneth Anger's occult masterpiece _Lucifer Rising_. Attempting to inquire how Page had developed his extreme aura, Bowie found his questions were never answered: Page would simply smile mysteriously.\n\n'It seemed that he did believe he had the power to control the universe,' wrote Tony Zanetta, the head of Bowie's management organisation MainMan, in his book _Stardust_. Besides, Page was only too aware that Bowie was picking his brain, endeavouring to crack a magician's tricks.\n\nAt one point Bowie disappeared out of the room, and Page accidentally spilled red wine on a satin cushion. When the singer returned, Page tried to blame Ava Cherry, who wasn't in the room.\n\nHis guest's inscrutable behaviour had already rankled Bowie, and now he knew that Page was lying. 'I'd like you to leave,' he said.\n\nPage's response was simply to smile at Bowie. The window was open, and Bowie pointed at it, snapping his words out furiously: 'Why don't you leave by the window?'\n\nPage remained sitting there, maintaining his enigmatic rictus smile, gazing through Bowie. Finally, the Led Zeppelin leader stood up silently, stepped towards the front door and left, shutting the door forcefully behind him.\n\nBowie was terrified. Immediately afterwards he ordered that the house be exorcised. A sensitive soul whose perceptions were addled by drugs, Bowie believed 'it had become overrun with satanic demons whom Crowley's disciples had summoned straight from hell'.\n\nWhen he later ran into Page at a party, Bowie straightaway fled the event.\n\n# Introduction\n\nJohn Bindon, Led Zeppelin's security guard, had stagehand Jim Matzorkis pinned to the floor of a backstage trailer at Oakland Coliseum. Bindon, a sometime actor and London gangland heavy who had reputedly once bitten off a man's testicles and would stab another man to death the following year, was viciously pummelling Matzorkis with his fists and feet. But it was only when Bindon started trying to gouge out the stagehand's eyes that Matzorkis fully appreciated the danger he was in.\n\nFor much of this day of Saturday 23 July 1977 the possibility of such a grim outcome had been building. Many of Zeppelin and their crew seemed in a state of permanent rage, as if they had surrendered control to the large quantities of drugs consumed during the course of a 51-date tour that had begun on 1 April.\n\nLater, Jimmy Page would be obliged to deny to me that what happened that day was karmic recompense for his flirtations with the occult. 'I don't think we were doing anything... evil,' he said, two years later.\n\nIt was especially ironic that what happened at Oakland Coliseum that day, which would utterly transform the fortunes and career trajectory of Led Zeppelin, should be on the turf of Bill Graham, whose Fillmore West had been a temple of popularity for the Yardbirds, Page's previous group, and, along with Graham's New York showcase the Fillmore East, the scene of early break-out triumphs for Led Zeppelin. Although the confluence of the interests of Bill Graham and Peter Grant, Led Zeppelin's manager, had proved mutually advantageous in the past, it was always a disaster waiting to happen. Graham was the most powerful music promoter in the United States; Grant had reinvented the relationship between managers and promoters in the United States, often through heavy-handed behaviour. And as much as Grant terrified people, Graham also possessed a fierce temper.\n\nThe previous evening, Graham, who was promoting Day on the Green, as this event was billed, for an audience of 65,000, had been summoned to Led Zeppelin's hotel, the San Francisco Hilton, to honour a sudden demand for a $25,000 cash advance against their fee for the shows. Entering their suite, Graham noticed a cowboy-hat-wearing local dealer of hard drugs; in a flash he realised what the money was needed for.\n\nArriving only 20 minutes before the start of the gig, Page was so evidently befuddled from his drug consumption, by that stage largely heroin, that Graham could only watch as the Led Zeppelin leader set off for the stage in entirely the wrong direction; he was rescued by an aide who stopped him and despatched him on the correct course. Midway through the set Bindon crawled out onto the stage on his hands and knees and licked Page's boots.\n\nAs the wheezing, out-of-shape man-mountain that was Peter Grant lumbered up to the stage, Jim Downey, a member of the stage crew, commiserated with him about the excessively steep climb. For this presumption Bindon, who was accompanying Grant, punched Downey with such force that he slammed him into a concrete pillar and knocked him out.\n\n'What happened? The fuck did I do?' wondered the victim as he came to. Downey was clearly unaware of an extraordinary management edict \u2013 egregiously pathetic in its arrogance \u2013 on what would become the final Led Zeppelin American tour: no one was permitted to speak to any member of the act, or to Grant, unless they were first spoken to. (Flying on the group's plane, journalist Steven Rosen had been made fully aware of this. He was startled when the normally benign bass player, John Paul Jones, had verbally assaulted him, demanding all of his interview tapes \u2013 a response to an apparently unfavourable comparison of Zeppelin to the Jeff Beck Group that Rosen had made years previously. This incident was indicative of the prevailing mood on the tour.)\n\n'What I didn't like about Led Zeppelin was that they came with force,' Graham wrote later. 'I had heard stories from other cities about how they had muscled promoters to get better deals. How they had shaken them down for money... I had heard about the ugliness of their security.'\n\nBut far worse was to come. Midway through Led Zeppelin's show, during Page's acoustic set, Matzorkis noticed a young boy removing wooden plaques from the doors of the dressing-room trailers: plaques that had the names of the artists on them, and which would be required for the repeat show the following day. Also on the bill, as support acts, were Judas Priest and Rick Derringer. Matzorkis told the kid to put them back. The kid was insistent that he would be taking them. So Matzorkis took them from him \u2013 allegedly cuffing him round the back of the head, although Matzorkis has always denied this \u2013 and wandered over to the storage trailer to lock them away. Unbeknown to Matzorkis, the boy was Warren Grant, the 11-year-old son of Peter Grant.\n\nA few minutes later Matzorkis was still in the storage trailer when John Bonham, Led Zeppelin's drummer \u2013 surplus to requirements during Page's acoustic set \u2013 called to him from the bottom of the short flight of steps outside it. Peter Grant was with him.\n\n'You don't talk to _my_ kid that way,' Grant said, as Matzorkis stood in the trailer's doorway. The burly Bonham, an effigy of muscle and blubbery booze-fat, simply stood there, as though he was Grant's security wingman. Then Grant escalated matters, like the barrack-room lawyer and bully he could be: 'You don't talk to _my_ kid that way. Nobody does. I can have your job.' Grant continued in this vein, accusing Matzorkis of 'roughing this kid up. I heard you hit this child.'\n\nStepping up the stairs, Bonham then kicked Matzorkis between the legs, sending the stagehand flying back into the trailer. Bonham followed him in, while a pair of bodyguards endeavoured to restrain the drummer, shouting at Matzorkis to get out of there. He did, through a rear door.\n\nWhen he learned of the incident, Graham went to find Grant in his trailer. For 20 minutes Graham put up a spirited defence of Matzorkis, yet Grant refused to budge from his thinking: 'Your man put his hands on _my_ people. On my _son_. How could you let this happen? How could you hire these people? I'm very disappointed in you.'\n\n'Let me speak to this man,' Grant repeatedly demanded. When Grant finally insisted that all he wanted was to meet again with 'the man' who had caused these problems to 'make my peace with him', Graham somewhat reluctantly agreed to the manager's request.\n\nWalking over to find Matzorkis in another trailer, Graham observed that Grant was now flanked by a pair of other men; one of them was Bindon.\n\nWhen Graham introduced Grant to Matzorkis, Led Zeppelin's manager seized the stagehand, yanked him towards him and smashed him full in the face with a ring-covered fist, knocking him back into his seat. When Graham lunged at Grant, one of the security men picked up the promoter, threw him down the steps and shut the door of the trailer, standing guard in front of it.\n\nInside the trailer Bindon held Matzorkis from behind, while Grant started to work him over, punching him ceaselessly in the face, knocking out a tooth and kicking him in the balls.\n\nSomehow, Matzorkis, who was screaming for help, broke free of Bindon. He managed to manoeuvre himself to the rear of the trailer, but this was when Bindon leapt upon him and went for his eyes. Fuelled by adrenaline, Matzorkis finally twisted away and got to the door.\n\nDespite the security man guarding it, Matzorkis managed to get out of the trailer and run off across the backstage area.\n\nMeanwhile, tour manager Richard Cole, armed with a chunk of metal pipe, had been attempting to enter the trailer. Bob Barsotti, who with his brother served as Bill Graham's creative director, had prevented him, so he then turned on him. Realising that Cole was demented from the drugs he had seen him consuming during the day, Barsotti ran off, leading him on a merry dance down into the car park, where Cole ran out of steam.\n\nBy now several of Graham's security men had gone to retrieve their 'pieces' from where they were stashed in their cars' boots, but a seasoned member of the backstage crew reminded all concerned that the next day there would be another Led Zeppelin show: if the group did not play, 65,000 fans might very well riot. Yet the Graham crew consensus was that somehow the next day they would 'deal' with Zeppelin and their team. The promoter agreed with this thinking. He also made an offer: if they couldn't 'do' Led Zeppelin and their cohorts the next day, then he would personally fly 25 of his men to New Orleans, the next date on the tour, and they could mete out revenge there.\n\nAt Graham's home that evening, where the promoter had taken Matzorkis for protection following his release from hospital, he received a call from Led Zeppelin's US lawyer: he demanded that Matzorkis sign an indemnity waiver, giving Led Zeppelin protection against being sued over what he referred to as a 'minor altercation'. Unless this was received, Led Zeppelin 'would find it difficult to play' the next day.\n\nGraham agreed to sign; his own lawyer told him that as he was acting under duress, it would not be legally binding. Besides, Graham had a plan. Knowing that Led Zeppelin would be staying in San Francisco for a further night following the Sunday show in Oakland, he had arranged with the local district attorney to arrest those culpable on the Monday morning.\n\nAt the Sunday concert the loathing of Graham's entire crew towards Led Zeppelin was palpable: they glowered at the band and anyone connected with them. Page played most of the show seated, and he and Jones both looked bored. Robert Plant, however, sang very well indeed, dropping occasional words of commiseration in the direction of Graham; bootlegs indicate that it was a far better show than the previous day, partly because Led Zeppelin appeared drug-free. Still, it was a tense affair, and many in the audience were drunk and rowdy. Rumours were running round that a murder had been committed the previous day.\n\nThe next day, Bonham, Grant, Bindon and Cole were arrested at the San Francisco Hilton and taken, hands cuffed behind their backs, across the bay to Oakland to be booked, where they were held in a cell for three hours. There was a very real chance that if the case went to a criminal court, all involved would be deported and never be able to work again in the United States, a serious financial worry.\n\nBonham was charged with a single count of battery, as was Grant; Cole and Bindon were each charged with two counts of battery. The news of their arrest and the incident at Oakland Coliseum made the news all around the world. When they were eventually released they were bailed at $250 on each charge.\n\nAs the arrests at the hotel were taking place, Jones was exiting the Hilton through a rear door. He climbed into a camper van with his family and drove out of San Francisco towards Oregon and Washington state, on a planned holiday before Led Zeppelin's next date, in New Orleans, at the city's Louisiana Superdome. He was set to rejoin the band there on 30 July, in time for the show that night.\n\n'As far as I was concerned, every one of those guys in the band was absolutely 100 per cent accountable for that shit. Because they allowed it to go on,' said Bob Barsotti. 'And we weren't the only ones it happened to. We were just the _last_ ones. We were the only ones who stood up and said something. When we started looking into it, there were incidents like that all across the country on that tour. Trashed hotel rooms. Trashed restaurants. Literally like twenty-thousand dollars' worth of damages at some restaurant in Pennsylvania. Really outrageous stuff. Like where they physically abused waiters and people in the restaurant, and then just bought them off.'\n\n'They would do things after the show,' said Peter Barsotti. 'The traditional \"go get chicks out of the audience for the band\". I remember standing by the ramp and seeing these guys get girls to come over. It was like no other feeling I'd ever experienced. It was like these girls were going to be _sacrificed_. I wanted to go out and grab these girls and say, \"Don't do it, honey. Don't do it.\" I'm as hardcore as the next guy. But I was afraid for these girls.'\n\nIf it could be possible, worse was to come. Arriving on 26 July at the Royal Orleans Hotel in New Orleans, Plant received a phone call from his wife in England: she told her husband that Karac, their five-year-old son, was seriously ill and had been taken to hospital.\n\nThen came another call from her. Karac had died.\n\nA devastated Plant flew back to England. All remaining dates on that eleventh Led Zeppelin tour of the United States were cancelled.\n\nAt the funeral of Karac, Plant was joined by Bonham and Cole. But there was no Page, who had flown instead to Cairo, where he was ensconced by the pyramids in the luxurious Mena House Hotel. Jones, for his part, had simply resumed his family holiday. And Grant had also remained in the US. Plant would not forget this.\n\nOn 26 July Graham received a call from the Zeppelin manager. 'I hope you're happy,' Grant muttered down the line.\n\n'What are you talking about?' Graham asked.\n\n'Thanks to you, Robert Plant's kid died today.'\n\nThat one absurd assessment by Grant captured everything that had gone wrong with Jimmy Page's Led Zeppelin project.\n\nJust considering the death of Karac Plant sets off an inescapable collision of images of those nude blond children crawling up the boulders on the _Houses of the Holy_ sleeve, and of the child being held aloft, as though for sacrifice. You can't help but feel that this might have crossed the mind of the bereft Robert Plant on his wretched plane ride home.\n\nThe Oakland incident, and the death of his singer's son, marked an extraordinary, certainly hubristic fall for Jimmy Page, who since the beginning of Led Zeppelin in 1968 had become the greatest archetype of a globally successful rock star that Britain has ever seen.\n\n'Jimmy Page grew up in the hypocrisy of the United Kingdom in the 1950s and found three chords that saved him,' said his friend Michael Des Barres. 'As Led Zeppelin developed, heroin was obviously the fuel of that mad coach ride through the countryside. And inevitable.'\n\nThe mystery of Led Zeppelin had been established almost entirely through the endless enigma that is Page; later, as the apprentice matured, Plant would offer a separate sort of leadership within the group. In tandem with their extraordinarily lyrical atmospherics, Zeppelin's complex beats were the dominant soundtrack for popular culture for nigh on a decade. But the music was only one part of it; without Page's extremely pure comprehension of the intangibles of rock 'n' roll \u2013 the perfect manner in which to exit a limousine, for example \u2013 Led Zeppelin would not have been granted their place in the pantheon of rock 'n' roll gods.\n\nFrom the very start \u2013 those first publicity pictures with his fluttering eyelashes and choirboy's face \u2013 Page displayed a slightly smirking look of utter confidence and haughty control, with a hidden promise of something sinister cloaked beneath it. There is that very early photograph of the four Led Zeppelin musicians in 1968 clustered around the bonnet of a Jaguar Mk 2 3.8, which had a reputation as a bank robber's car. Page is encased in a then fashionable double-breasted overcoat, its collar pulled rakishly up; he stares at the camera from between those curtains of crimped black hair, smouldering with self-assurance and poise. It is an image maintained in the first official promotional shot of the band, issued by Atlantic Records: the utter Capricorn control of Page leaning over the other three members \u2013 his string-pulling hands resting on the shoulders of the two Midlands neophytes, drummer John Bonham and Plant, who resembles a frightened faun caught in the headlights.\n\nTheir look \u2013 especially that of Page \u2013 is like that of Charles I's cavaliers, perhaps especially of _Lord John Stuart and His Brother, Lord Bernard Stuart_ , Anthony van Dyck's 1638 painting of two teenagers who would be killed in the English Civil War.\n\nHalf an inch under six foot tall, permanently clad in sensuous velvet and sexy ruffled shirts, his jawline frequently dusted with five o'clock shadow, and always with that aura of androgynous otherness, Page looked to many women \u2013 and plenty of men too \u2013 like dirty sex on a stick. This image was as integral to his art as the 20-minute guitar solos with which he would blast his audience's eardrums \u2013 the violin bow he would employ when performing 'Dazed and Confused' clearly doubled as a wizard's wand to manipulate concertgoers.\n\nAnd it only gets better: this romantic dandy lives in a castle with a moat. Jimmy Page does very bad drugs seemingly forever and \u2013 unlike Keith Richards, a mere also-ran in the greatest ever UK rock star stakes \u2013 never gets busted... at least until Zeppelin is over. Moreover, he is held responsible for an entire genre of music \u2013 heavy metal! \u2013 with which his group is only tangentially involved, his true focus being a blending of UK and US folk traditions with a garage band sense of hard rock.\n\nIn his renowned isolation he is like a rock 'n' roll version of Howard Hughes. But in many ways, the very idea of Jimmy Page is as much a construct as any of David Bowie's personae. And \u2013 lest we take this too seriously \u2013 it is worth considering that when his own persona is deconstructed, Page is sometimes little more than a high-art version of Screaming Lord Sutch, the plumber rock 'n' roll showman on whose attractively kitsch shock-rock records he played session guitar.\n\n'Everyone I worked with in the 1960s thought that rock 'n' roll was really an aspect of showbiz,' said Dave Ambrose, who played bass in Shotgun Express (with Rod Stewart) and the Brian Auger Trinity, who supported Led Zeppelin in San Francisco in April 1969. Later, as an A&R man, Ambrose signed the Sex Pistols, Duran Duran and the Pet Shop Boys, among others.\n\nMany of Page's expenditures \u2013 the palatial residences, the vintage cars he was unable to drive (he never passed his test), the enormous collection of rare guitars \u2013 seemed designed to garner respect and support among the world's wealthy and influential, to make people aware of him, to elevate his extraordinarily inscrutable profile, and to establish himself as one of the principal men about whichever town he found himself in.\n\nBut at the same time, here was a rebel cocking a snook at the Establishment, having what he knew he wasn't meant to have. With Led Zeppelin there always was that sense of being resolutely 'underground', a card played with perfect panache by the band for most of their career: hardly ever on television, with no singles released in their homeland, Zeppelin existed from the very beginning as their own outsider identity. In a sense the damning review of their first album by John Mendelsohn in _Rolling Stone_ , a magazine Page came to loathe, was perfect for them; it set in motion the 'us against them' agenda from which Led Zeppelin's success soared.\n\nBy 1977, the year their myth savagely unravelled, they would come to be seen as the embodiment of behemoth rock, all that the new punk movement stood against, but when Led Zeppelin started out in 1968 their anti-Establishment stance was about as punk as it could be.\n\n'The big question today is, Why hasn't he done new music?' said Michael Des Barres. 'Well, why does he have to? Jimmy Page is his own art piece, a performance artist, and he's busy curating his legacy. There is nobody else whose roadie was Aleister Crowley. And it worked. Led Zeppelin were not a band; they were a cult.\n\n'Led Zeppelin brought together all those kids who otherwise would just hang around parking lots in two-bit American cities, kids for whom the obvious decadence of the Rolling Stones didn't really connect. Instead, Led Zeppelin were their cult; they became a focus for and brought together all those disaffected, lost souls who would take the fantasy world of the group and its subject matter and project onto it their own interpretation of what they were.'\n\nThe world was ready for just such a package. Around the time the Rolling Stones were writing 1968's 'Sympathy for the Devil', Mick Jagger and Keith Richards had dabbled in a friendship with the Californian film director and occultist Kenneth Anger, but \u2013 as though proof that in such areas they were distinctly lightweight \u2013 fled his company the next year after the debacle of Altamont. Instead it was Led Zeppelin, driven by Page's assiduous academic interest in altered states and realities, that provided the soundtrack to the building public interest in the occult. In 1972 _TIME_ magazine ran a cover story bearing the strapline 'Satan Returns'. Colin Wilson's mammoth groundbreaking study simply titled _The Occult_ had been published in 1971. More populist was the _Man, Myth and Magic_ partwork series, which commenced in 1970, providing highly readable accounts of a secret world that was exciting to the newly stoned with their now-opened third eyes. As was the manner of partworks, _Man, Myth and Magic_ was extensively plugged in television adverts, featuring an image of a demonic figure, painted by Austin Osman Spare. Spare had been close to Aleister Crowley and was sometimes described as 'Britain's greatest unknown artist'; Page would become the world's leading collector of Spare's work.\n\nBy then there was something frightening about the very notion of Led Zeppelin. After I interviewed Page in 1979 in a relatively forthright manner for the _NME_ , a senior editorial member asked me if I wasn't nervous of any potential repercussions. When I told casual acquaintances I was writing this book, I was met with similar responses: 'Jimmy Page? Black magic?'\n\nFor some years \u2013 a decade or so \u2013 this was the prevailing view of Page. But of course time is a healer, so it should be no great surprise that by the end of the first decade of the twenty-first century, and in his own seventh and eighth decades, Page had redeemed himself to become the most loved and revered of all classic rock stars.\n\nThis redemption was fitting, given that this is the man who almost singlehandedly established the notion of the guitar hero as part of contemporary culture. 'What about Eric Clapton?' you may ask. No: Clapton was too diverse in the paths he trod. It was the singularity of Page's work with his vehicle Led Zeppelin, underpinned by his extraordinarily startling and sinisterly attractive appearance, that awarded him the guitar hero crown. Guitar hero? Guitar god, more like.\n\nHis is an extraordinary story that has taken him to the very darkest of areas \u2013 but always driven by the search for his art. You might not approve of the methods employed to unleash and liberate his creativity, and you can't avoid the impression that Page was vain, arrogant, fanatical and power-hungry, and indulged in a scandalous private life \u2013 much, of course, to the adoration of his fans. Yet many of the accusations against him were probably fabricated or at least exaggerated by his numerous enemies \u2013 though many of these, in the timbre of the times, were no more than cosmic spivs.\n\nCertainly, Page was a man of his age \u2013 ambitious, worldly and pleasure-loving \u2013 but the demonic caricature of evil is mostly an elaborate myth. Not that he didn't gladly play it himself, of course. By mentioning in a very matter-of-fact manner how the congregation of the original church at Boleskine House, a home of Crowley, had burned to death, Page was positioning himself as being metaphysically hard, a cosmically tough motherfucker with complex connections to ghoulish gangs of strange spirits. It was, of course, a good way to attract impressionable women, a variant on those college student 'astrologers' who would take girls back to their rooms to read their charts and then shag them.\n\nFor a time Page was fascinated with \u2013 to give it its full title \u2013 the Isis-Urania Temple of the Hermetic Order of the Golden Dawn, a late-19th-century group of occultists whose members had included the Irish poet W. B. Yeats and Crowley, who \u2013 unsurprisingly \u2013 considered his poetry superior to that of Yeats and had a bitter falling out with the Irishman.\n\n'Much of the Golden Dawn magic,' wrote Gary Lachman in his biography of Crowley, 'as well as Crowley's, has to do with what is called the \"assumption of the god form\", when the magician imagines he has become the particular god he wants to invoke by visualising his form enveloping his own.' Except you might feel that the 'particular god' Page wished to invoke was none other than himself: Jimmy Page, rock god.\n\nAnd this stance was carried through to every aspect of his existence, including his appearances on stage.\n\n'On the surface,' writes the American cultural commentator Erik Davis, 'Page's live performances present typical rockist values of spontaneity, virtuosity and sweaty abandon. But Page adds a novel element to the figure of the guitar hero, an element... of mystery. So even as Page bares his cock rock before tens of thousands of fans, the Zoso doodle emblazoned on his clothes, he reminds us that he knows something that we don't. There is a gap between the hero whose performance we consume and the sage behind the curtain, who remains concealed, literally occult. This mystique makes Page far creepier than Ozzy Osbourne, who is hiding nothing, except maybe his debt to _The Munsters_.'\n\nA balanced appreciation of Page's character reveals traits both admirable and detestable, but claims of his ethical failings have sometimes overshadowed an appreciation of his keen creative mind. Besides, his flamboyantly dissolute lifestyle was hardly different from that of many other rock stars of his age \u2013 such as David Bowie, or Mick Jagger, or Rod Stewart.\n\nBut Page had a longstanding relationship with the art of destruction, and had been preparing for a career of hotel-wrecking since early in his life. At the rear of the secondary school he went to, on Epsom's Danetree Road, there was a bomb shelter left over from the war. Although efforts to destroy it had been made on several occasions by his fellow pupils, it was a 14-year-old Jimmy Page who finally succeeded.\n\nIn what seems less an example of urban terrorism and more like a yarn from one of Richmal Crompton's _Just William_ stories, an older boy had combined sodium sulphate, weed killer and icing sugar to make several miniature bombs. A couple of these had exploded in the school grounds, with the blame always attributed to the rough kids from the neighbouring council estate.\n\nBut then this arms race escalated. Another boy constructed a pipe bomb and placed it inside the bomb shelter. Once lit, however, the fuse on the bomb burned interminably slowly. After some 20 minutes without much progress, one of boys offered a solution: he had a fuse taken from a Jetex, a motor for model aircraft that was popular at the time, which they put in the pipe bomb.\n\n'But nobody dared light it,' remembered Page's friend Rod Wyatt. 'So Jimmy said, \"I'll do it.\" So he goes in the entrance of the shelter, and then he comes running out. As he runs out it goes off: _P-F-O-O-F! B-R-A-MMM!_ And the whole corner, which was thick concrete, flies up in the air, bricks following it. And Jimmy is running out, laughing his head off into the playground.\n\n'Reflecting on this, I thought, \"Was that a sign of the times? That he was going to be part of the loudest rock 'n' roll band ever?\" This gentlemanly young guitar player says, \"I'll light the jet engine. I don't mind.\" It fits Led Zeppelin perfectly.'\n\n## 1\n\n# SPANISH GUITAR IN SURREY\n\nBorn at 4 a.m. on 9 January 1944 in Heston, Middlesex, on the far fringes of London's western suburbs, James Patrick Page was the son of an industrial personnel manager, also called James, and Patricia, a doctor's secretary. The future superstar musician's name was a combination of both of his parents', who had been married at Epsom Register Office on 22 April 1941.\n\nAccording to the mythology of his rock-star legend, Jimmy Page was 'born on a full moon', with all the occult, mystical weight that that phrase carries. Yet this is not precisely true. He was in fact born 31 hours before the full moon of 10 January 1944. While the baby boy and his mother might have felt the powerful energy of the rising Cancer full moon as he was born, the earth's only natural satellite was not yet at its peak. In time Page would become a student of astrology; he would learn that in his astrological chart his moon was in moody Cancer, his sun sign was determinedly ambitious Capricorn, and he had Scorpio rising, with its suggestion of powerful sexuality and interest in arcane areas of life.\n\nTo an extent this only child \u2013 until he started school at the age of five he hardly knew any other children \u2013 was always self-educated, manifesting a strong sense of self-fulfilment, even destiny \u2013 though there is often something fixed and inflexible about the self-taught. 'That early isolation probably had a lot to do with the way I turned out,' he said later. 'Isolation doesn't bother me at all. It gives me a sense of security.'\n\nHeston, his birthplace, has a distinct sense of J. G. Ballard-like suburban anonymity, the net curtains firmly drawn on all manner of potential darkness. Heston lies on the direct flight-path into Heathrow Airport, less than three miles away, and today is a place blighted by the ever-present roaring reverse thrust of descending jet airliners. It was the beginnings of this noise pollution that led the Page family to move first to nearby Feltham, a distance of some four miles, where unfortunately the noise from aeroplanes was even more acute, and then the ten miles or so to the south-east, to 34 Miles Road in Epsom, Surrey, in 1952. (In 1965 Page would, with Eric Clapton, record a song entitled 'Miles Road'.)\n\nEight-year-old Jimmy Page was enrolled at Epsom County Pound Lane Primary School, and at the age of eleven he moved on to Ewell County Secondary School, on Danetree Road in adjacent West Ewell. His headmaster, Len Bradbury, who took over in 1958 when Page was in year three, had played football for Manchester United, and Bradbury's arrival at the school was only a few months after his former team had been decimated in the Munich air disaster. (Many years later Bradbury would be a guest of honour at Manchester United's ground, with pictures of him taken with the team's captain, Roy Keane, and Ryan Giggs.) Page was in the proximity of celebrity \u2013 and he could see that such individuals were sort of ordinary people.\n\nAt 34 Miles Road, Page discovered a Spanish guitar that had been left behind, presumably by the previous occupiers. Had it ever been played? Perhaps not. In the 1950s a Spanish guitar as an _objet d'art_ in the home was considered a sign of sophistication. 'Nobody seemed to know why it was there,' he told the _Sunday Times_. 'It was sitting around our living room for weeks and weeks. I wasn't interested. Then I heard a couple of records that really turned me on, the main one being Elvis's \"Baby Let's Play House\", and I wanted to play it. I wanted to know what it was all about. This other guy at school showed me a few chords and I just went on from there.'\n\n'This other guy' was called Rod Wyatt. Although fascinated by the Spanish guitar, Page all the same had been flummoxed by how to play the instrument. During a lunch break at secondary school, he came across Wyatt, who was in a class a couple of years above him. The owner of an acoustic guitar of his own, Wyatt was running through a version of 'Rock Island Line', a chart hit by the revered Lonnie Donegan, when he met Page. In response to the younger boy's query, Wyatt instructed him to bring his Spanish acoustic to school and he would show him how to tune the instrument. From then on the pair became firm friends.\n\n'My mate Pete Calvert and I were always at Jimmy's house bashing out our guitars for a couple of hours on a Saturday afternoon,' said Wyatt. 'Sometimes I'd go round to Jimmy's and his mum would say, \"No, he's practising.\" When he suddenly realised he had it, he spent a lot of time practising. Sometimes six or seven hours a day. He told me he needed to improve his technique. And he eventually became the all-round perfect guitarist. Practice is what it's all about.'\n\nWhat had really turned Page on in Elvis Presley's rockabilly song 'Baby Let's Play House', released in the UK six days before Christmas 1955, was the guitar playing of Scotty Moore, who served as Presley's guitarist from 1954 to 1958. On 5 July 1954 Moore had, with bassist Bill Black and Elvis himself, mutated the original arrangement of 'That's All Right' by bluesman Arthur Crudup into a version that combined blues and country music, creating one of the foundations of rock 'n' roll.\n\n'Scotty Moore had been a major inspiration in my early transitory days from acoustic to electric guitar. His character guitar playing on those early Elvis Sun recordings, and later at RCA, was monumental. It was during the fifties that these types of song-shaping guitar parts helped me see the importance of the electric guitar approach to music,' said Page.\n\nOn 'Baby Let's Play House' Moore played a burnished rockabilly rhythm. Page's love of the tune would not leave him, even after almost 20 years: about nine minutes into the live version of 'Whole Lotta Love' in the Led Zeppelin film _The Song Remains the Same_ he breaks into a close simulacrum of Moore's licks. But that was a long way in the future: for now the 12-year-old Page assiduously studied and copied Moore's parts. There could hardly be a purer, more perfect example of this brand new musical form, an ideal introduction to what his life would become. Every day he would take his guitar to school on Danetree Road.\n\n'When I grew up there weren't many other guitarists,' he told America's National Public Radio in 2003. 'There was one other guitarist in my school who actually showed me the first chords that I learned and I went on from there. I was bored so I taught myself the guitar from listening to records. So obviously it was a very personal thing.'\n\nBut he also had a seemingly separate life as a choirboy. Each Sunday, wearing the appropriate surplice and cassock, he would sing hymns at Epsom's St Barnabas Anglican Church. The first image in his 2010 photographic autobiography is of him in this mode \u2013 clearly he is being ironic. As so often in the life of Jimmy Page, cold realism lay behind the impetus for his choirboy stints.\n\n'In those days it was difficult to access rock 'n' roll music,' he said to the _Sunday Times_ in 2010, 'because after all the riots happened in the cinemas, when people heard \"Rock Around the Clock\" in the film _Blackboard Jungle_ , the authorities tried to lock it all down. So you needed to tune in to the radio or go to places where you could hear it. It just so happened that in youth clubs they would play records and you'd get to hear Elvis, Jerry Lee Lewis and Ricky Nelson \u2013 but you had to either go to church or be a member of the choir to go to the youth club.'\n\nPage had many of the characteristics of the only child, burying himself in books and, almost the ultimate clich\u00e9, collecting overseas postage stamps. But more and more since discovering that Spanish guitar at Miles Road, he immersed himself in becoming adept on the instrument. 'The choirmaster at St Barnabas remembered that I used to take my guitar to choir practice,' he said, 'and ask if I could tune it up to the organ.'\n\nIn Epsom there is a prominent motorcar showroom named Page Motors. It has often been claimed, even by myself, that this business is owned by members of Page's family. But this is not the case at all. His relatives on his father's side came from Grimsbury in Northamptonshire, and his paternal grandfather had been a nurseryman, tending to plants. (An irony that would not be lost on Page, who later had a Plant of his own to deal with, of course.) On his father's side there was Irish blood.\n\nAt 122 Miles Road, at the far end of the street from the Page family home at number 34, lived a boy of similar age to Page called David Williams. According to Williams, Miles Road, which lay to the west of Epsom, was distinctly the wrong side of the high street. To the east lay the plush property in which affluent commuters to London were ensconced. The Page house backed onto the railway line that transported these people to the capital, less than 20 miles away, and was identical to the Williams home, having a downstairs living room and dining room and a pair of bedrooms upstairs. Downstairs, beyond the kitchen, was an outside toilet. Although most of these houses, including the Pages', later had this feature adapted into a full bathroom, there is no getting away from the fact that these were distinctly basic homes.\n\nPage's father worked in nearby Chessington, a personnel manager at a plastic-coating factory, and his mother was the secretary at a local doctor's practice. Despite the impeccable 'BBC English' \u2013 as such received pronunciation was known at the time \u2013 with which rock star Jimmy Page would express himself, his background was no more than lower middle-class, almost jumped-up working-class.\n\nAnother good friend, Peter Neal, lived on Miles Road. Page, Peter Neal and David Williams would hang out at each other's houses. Gradually Page's home \u2013 without any brothers or sisters to get in the way \u2013 became a focal point. He also had the advantage of enjoying both parents \u2013 although Wyatt mentions some growing tension between his mother and father. When Williams was just 13, his own mother had died: 'I am certain that Jim's mother was the initial driving force behind his musical progression. She was a petite, dark-haired woman with a strong personality, a glint in her eye and wicked sense of humour. She liked to tease me in a good-natured way, but let me hang out endlessly in their front room with Jim. I think she must have known my mother and, given the new circumstances I found myself in, I guess she felt sorry for me. Although I didn't realise it at the time, I can now appreciate her kindness and tolerance, for I must have been a fairly constant presence in her house.'\n\nAfter hearing Chuck Berry, the black poet of rock 'n' roll sensibility, on American Forces Network radio broadcasting fuzzily from Germany in 1956, Williams acquired a UK EP that gathered together the songs 'Maybellene', 'Thirty Days (To Come Back Home)', 'Wee Wee Hours' and 'Together (We Will Always Be)'. He and Page played it incessantly, the latter being especially taken with 'Maybellene' and its tale of amorous automobile class struggle, and with 'Thirty Days'.\n\nFrom the equipment Page quickly began to amass, it seems that this only child was a little spoilt \u2013 or, at least, certainly lucky. He was the first of his friends to acquire a reel-to-reel tape recorder, which he soon replaced with a newer model, selling the older one to Williams so he could then pass on the tapes of songs he would diligently record off the radio.\n\nDiscerning in their taste, certainly to their own minds, these boys truly cared for only a handful of artists: Elvis Presley, Gene Vincent, Little Richard, Chuck Berry and Jerry Lee Lewis \u2013 the eccentric and wild Jerry Lee, with his 13-year-old bride, being Page's especial preference. Eddie Cochran would soon emerge to join this pantheon.\n\nThey would visit and revisit their local cinemas to watch such films as _The Girl Can't Help It_ , a minor triumph in 1956 that featured Little Richard, Fats Domino, Eddie Cochran, Julie London and the Platters. Also released that year was the more pedestrian _Rock, Rock, Rock!_ , which had a highlight performance of 'You Can't Catch Me' by Chuck Berry, his patent-leather pompadour and sneering grin permanently lending him the appearance of one of those black pimps whose look Elvis Presley had tried so hard to emulate. One Saturday afternoon in 1960 Page and Williams would hitchhike 50 miles to Bognor Regis to catch Berry performing a solitary tune in the classic film _Jazz on a Summer's Day_.\n\nIn the record department of Rogers, an electrical goods shop on Epsom High Street, the three boys ingratiated themselves with the girl behind the counter. This ally would provide them with glimpses of record-company schedules of forthcoming releases. The boys would search out the most interesting names. 'Frankie Avalon and Bobby Rydell were clearly to be overlooked in favour of the likes of Screamin' Jay Hawkins or Big \"T\" Tyler,' recalled Williams. 'Also, song titles could often be a good indication of something a little stronger. The dreaded \"A White Sport Coat (and a Pink Carnation)\" was hardly going to evoke the sort of enthusiasm and anticipation we would have for titles such as \"Rumble\", \"I Put a Spell on You\" or \"Voodoo Voodoo\", was it?'\n\nSoon appreciating the limitations of his Spanish acoustic instrument, Page worked for some weeks during the school summer holidays on a milk round, until he had saved up enough money to buy a H\u00f6fner President acoustic guitar. 'It was a hollow-bodied acoustic model with a simple pickup,' said Williams, 'but when he attached it to a very small amplifier, it made something like the sound we all admired. I can recall that Saturday morning when I was summoned to his house to first feast eyes on it. Jim was like the cat with the cream. Pete and I were allowed a strum, but by now we realised that any aspirations we might have had in that direction were going to be dwarfed by Jim's talent, desire and progress.'\n\nAfter Page had acquired his H\u00f6fner President, his parents paid for lessons from a guitar tutor. But the teenager, anxious to play the hits of the day, found himself mired in learning to sight-read; soon he abandoned the lessons, preferring to attempt to learn to play by ear. Later he would appreciate that his impatience had been an error, finally picking up the skill of reading music in the mid-1960s.\n\n'Rock Island Line', the tune that Wyatt had been playing when Page approached him, was a Top 10 hit in 1956 for Lonnie Donegan on both sides of the Atlantic \u2013 in the UK alone the record sold over a million copies. The song was an interpretation of the great bluesman Leadbelly's own version, and it became the flagship for skiffle music.\n\nSkiffle, a peculiarly British grassroots companion movement to rock 'n' roll that required no expensive equipment, was played on guitars but also on homemade and 'found' instruments. Donegan, a member of Ken Colyer's Jazzmen, would play with a guitar, washboard and tea-chest bass during intervals at Colyer's traditional jazz sets. In 1957 the BBC launched its first 'youth' programme, _Six-Five Special_ , with a skiffle title song. The craze swept Britain at an astonishing rate: it was estimated that in the UK there was a minimum of 30,000 youngsters \u2013 maybe almost twice that \u2013 playing the musical form. Across the country groups were created: John Lennon's Quarrymen were a skiffle group that would lead to the formation of the Beatles.\n\nIn accord with this spirit of the age, Page formed such a skiffle group, which his parents permitted to rehearse in their home. Really, this 'group' was little more than a set of likeminded friends, sharing their small amounts of knowledge about this new upstart form. Yet they seemed bestowed with a measure of blessing: in 1957, with Page just 13 years old, the James Page Skiffle Group, following an initial audition, won a spot on an early Sunday evening children's BBC television show, _All Your Own_ , hosted by Huw Wheldon, a 41-year-old rising star (11 years later Wheldon would become director of BBC television). The slot in which they were to be featured was one hinged around 'unusual hobbies'. How did they get this television slot? By chance, runs part of their appearance's myth: the show was looking for a skiffle act, and someone working on it was from Epsom and had heard of Page's band. But there is also a suggestion that Page's ever-supportive mother wrote a letter to the programme, suggesting her son's group.\n\nUnfortunately, the membership of the James Page Skiffle Group has largely been lost in time. For the television appearance, a boy named David Hassall, or perhaps Housego, was involved. Not only did his family own a car, but his father possessed a full set of drums, which David endeavoured to play.\n\nOn a day during the school holidays when the show was to be recorded, Page and Williams caught a train up to London to the BBC studio. Page's mother had phoned William's father to ask if his son would accompany him to the recording. 'The electric guitar itself was already heavy enough for him to carry, but the amplifier was like a little lead box and he clearly could not carry both.'\n\nAt around 4 p.m. Huw Wheldon appeared, fresh from a boozy media lunch, and asked: 'Where are these fucking kids then?'\n\nHis hair Brylcreemed into a rock 'n' roller's quiff and his shirt collar crisply fitted in the crew-neck of his sweater, Page \u2013 his H\u00f6fner President guitar almost bigger than himself \u2013 led his musical cohorts through a pair of songs, 'Mama Don't Want to Skiffle Anymore' and 'In Them Old Cottonfields Back Home'. (Page had also prepared an adaptation of Bill Doggett's 'Honky Tonk', but he was \u2013 rightly \u2013 doubtful he would get to play it, as he felt certain they would need him to sing.) After the performance, he was interviewed by Wheldon, and, with an irony now all too evident, Page declared to the avuncular presenter that his intention was to make his career in the field of 'biological research', modestly declaring himself not sufficiently intelligent to become a doctor. His 'biological research' remark certainly was not glib; Page wanted, he told Wheldon, to find a cure for 'cancer, if it isn't discovered by then'. Clearly this was a serious, thoughtful young man.\n\nYou can only imagine the confidence that this TV appearance must have engendered in the boy who had just become a teenager: in 1957 no one knew anyone who had appeared on the magical new medium of television.\n\nHaving been watched by an audience of hundreds of thousands at the age of 13, why not carry on as he had begun? Success might not have been instant, but within four years Jimmy Page would become a professional musician.\n\nIn the meantime, BBC television had finally begun to give limited exposure to rock 'n' roll, and Buddy Holly appeared on its solitary television channel. 'When he was killed in a plane crash in 1959,' said David Williams, 'I recall that Pete, Jim and I put on black ties and went to the local paper shop to buy all the newspapers that carried photos and obituaries of one of our heroes.'\n\nIn his woodwork class Page carved a reasonable simulacrum of a Fender Jazz Bass, modelling it on the instrument used by Jerry Lee Lewis's bassist in the film _Disc Jockey Jamboree_. 'It sounded good enough,' said Williams.\n\n'To say Jim was dedicated would be an understatement. I hardly ever saw him when he wasn't strapped to his guitar trying to figure out some new licks.' Williams noted that Page's principal inspiration was no longer Elvis Presley or the anguished Gene Vincent, but the ostensibly more wholesome Ricky Nelson. This should not be a surprise: Nelson's upbeat rockabilly tunes featured the acclaimed James Burton on guitar, as much an inspiration to Page as Scotty Moore. Ten years later Burton would be leader of Elvis Presley's TCB band, playing with the King until Elvis's death in 1977.\n\n'Those old Nelson records might seem pretty tame now, but back then the guitar solos (including the ones played by Joe Maphis) were cutting-edge stuff and greatly impressed my pal,' said Williams. 'I remember that he struggled for a long time with the instrumental break of \"It's Late\", but eventually someone showed him the fingerings he was after and he happily moved on.'\n\nNow Page set about forming a group that played more than skiffle. He found a boy who played rhythm guitar \u2013 though with little of the feel of rock 'n' roll \u2013 in nearby Banstead, and then he found a pianist.\n\nAlthough lacking either a drummer or a name, the trio were, after a number of practice sessions, deemed sufficiently ready by Page to play their first show at the Comrades Club, a drinking establishment for war veterans in Epsom town centre.\n\nThe gig was not a colossal success. In fact, Williams said it was 'a complete shambles'. Certainly, it didn't help that the three musicians lacked a drummer to propel the tempo; later in his career Page would ensure he played with the very best drummer he could find.\n\n'As rock 'n' roll progressed,' said Wyatt, 'Jimmy and I added pickups to our guitars; we were going electric. Pete Calvert, a left-handed guitarist and friend of Jimmy's and mine, had a small early Watkins amplifier and I had a Selmer. Jimmy had a bigger Selmer, a sign of what was to come? All three of us were always around each other's houses banging rock 'n' roll. Tommy Steele was making headlines as Britain's first rock 'n' roller, and although that was cool we preferred the grittier sound of the American artists such as Eddie Cochran, Gene Vincent and the Blue Caps and, of course, Elvis. And for Jimmy and me, the sound made by Gene Vincent's lead guitarist, Cliff Gallup: that was the style and guitar sound we loved the best in those days.'\n\nPage knew something had to change. At an electronics trade fair at London's Earls Court Exhibition Centre, he watched a young schoolboy called Laurie London stand up to sing on one of the stands. (Soon London would be at the top of the charts, in both the UK and USA, with his interpretation of the gospel song 'He's Got the Whole World in His Hands'.) Page noticed that the guitarist in London's backing group was playing a Fender Telecaster, the solid-body guitar he truly coveted that he had seen Buddy Holly playing on television. After the performance, Page spoke with London's guitarist, took the Telecaster in his hands and played 'Go Go Go (Down The Line)', a Roy Orbison tune covered by Ricky Nelson, with Page's idol James Burton on the guitar parts.\n\nFender Telecasters, made in the United States, were extremely pricey. Far more affordable, and on sale in London's musical-instrument shops, was the Futurama Grazioso, a Fender copy replete with tremolo arm, manufactured in Czechoslovakia. Page acquired a second-hand version of this instrument.\n\nConcert venues across the United Kingdom were responding to the new youth market for rock 'n' roll. By 1958 Epsom's Ebbisham Hall, little more than a church-hall-type building, had been renamed the 'Contemporary Club' for the rock 'n' roll events it put on each Friday night.\n\nBut with another group with whom he briefly played, Page would not even get as far as the Contemporary Club. At around the age of 14, Page briefly became a member of a fledgling local act called Malcolm Austin and the Whirlwinds. On lead vocals was the aforementioned Austin. Tony Busson played bass; Stuart Cockett was on rhythm guitar; there was a drummer named Tom whose surname has evaporated with time; and 'James Page', as he was billed, played lead guitar. It was Wyatt who had introduced the various musicians to each other. In 1958 Malcolm Austin and the Whirlwinds played at Busson's school Christmas concert, a set largely consisting of covers of Chuck Berry and Jerry Lee Lewis tunes; they played no more than another couple of dates.\n\nBusson, who was two years older than the group's guitarist, said that 'James' Page was 'very trendy: Italian jackets and Italian shoes \u2013 very pointed. Very cool in his tight jeans and trousers, but very baby-faced. We would go round to his house with our acoustic guitars and listen to his 45s and albums. His mum was always very receptive. She'd give us soft drinks. All we really talked about were guitars and pop music. When I first met Jimmy he only had a semi-acoustic H\u00f6fner. Then he got a solid electric, a Futurama Grazioso. He was a great fan of Gene Vincent and the Bluecaps, and also of Scotty Moore. I think he liked anything that was a bit complicated and a bit different.'\n\nThe guitarist's home, remembered Busson, was 'very lower middle-class.' But Page struck him as 'very arty: I thought if he didn't have a career as a musician he'd be an artist. He left school at 15. I thought he would make it. But I also wondered, \"How are you going to support yourself in the interim?\"' Soon Busson would receive an answer.\n\nFor Epsom also had larger venues in which more prestigious acts would perform. Wyatt recalled the buzz when a genuine professional rock 'n' roll show came to Epsom \u2013 creating an atmosphere like that of a circus or fair arriving in town. The concert was held at the local swimming baths. 'Top of the bill,' said Wyatt, 'was a singer, one Danny Storm, whose claim to fame was being Cliff Richard's double. He was a dead ringer. The second headliner was the Buddy Britten Trio. Buddy was a Buddy Holly lookalike. Both Jimmy and I went along to the show, which was very exciting at the time. Halfway through, the comp\u00e8re announced an open-mike talent show; Jimmy and I entered. We both got to play a guitar. I did \"Mean Woman Blues\" and Jimmy did an instrumental, either \"Peter Gunn\" or \"Guitar Boogie Shuffle\".'\n\nUndaunted by the experience of his show at the Comrades Club, Page had persevered and found a drummer and come up with a name: the Paramounts. And at the end of the summer of 1959 he had a show booked for the Paramounts at the Contemporary Club, supporting Red E. Lewis and the Redcaps, a London group modelled largely on the act, antics and material of Gene Vincent and His Bluecaps.\n\nAlthough the Paramounts even had a vocalist of sorts, their material that night largely consisted \u2013 in the manner of the time \u2013 of an instrumental set; Page's strident guitar playing on Johnny and the Hurricanes' recent hit 'Red River Rock' was notable, impressing Red E. Lewis. Lewis informed his group's manager, one Chris Tidmarsh, of this guitarist's prowess: at the end of the Red E. Lewis and the Red Caps' set, Page came out onstage, borrowed the solid-body guitar of Red Caps guitarist Bobby Oats and played a few guitar parts, including some Chuck Berry solos.\n\nFrom the rear of the hall Page was watched by his parents. Did they believe he would grow out of this silly interest? He told me: 'No. Actually they were very encouraging. They may not have understood a lot of what I was doing, but nevertheless they had enough confidence that I knew what I was doing: that I wasn't just a nut or something...'\n\nAlso watching the Paramounts that night, from nearer the stage than his parents, was Sally Anne Upwood, Page's girlfriend at school, a relationship that lasted for a couple of years. Older than her boyfriend, Sally Anne was in Wyatt's class and able to observe Page's musical development.\n\nJimmy Page and the Paramounts played further shows at the Contemporary Club; they supported such acts as the Freddie Heath Combo, who would later be known as Johnny Kidd & the Pirates, one of the greatest English rock 'n' roll groups. And when Bobby Oats left Red E. Lewis and the Redcaps at Easter 1959, Chris Tidmarsh invited 15-year-old Page to audition, above a pub in Shoreditch, East London. He got the gig, at \u00a320 a week.\n\nClearly Page's life was expanding \u2013 philosophically, as well as musically. 'My interest in the occult started when I was about 15,' he told me in 1977. At this time in his life, when still at school, he read Aleister Crowley's _Magick in Theory and Practice_ , a lengthy treatise on Crowley's system of Western occult practice; not an easy book to first comprehend, and a clear indication of the full extent of Page's precocious intelligence. The book struck into his core, and he said to himself, 'Yes, that's it. My thing: I've found it.' From that age he was on his course.\n\n## 2\n\n# FROM NELSON STORM TO SESSION PLAYER\n\nAt first Jimmy Page could only play with Red E. Lewis and the Redcaps at weekends; he was, after all, still at school.\n\nIn fact, at first his father had nixed the idea of his playing with the group. Chris Tidmarsh had needed to come down to 34 Miles Road to see him; it was only when he explained that almost all of the Redcaps' dates were at the weekend and would hardly interfere with his son's schooling that Jimmy's dad agreed. 'Oh, okay then,' said the elder James Page.\n\nYet soon Page had a major contretemps with Miss Nicholson, the deputy headmistress. When he informed her that he intended to be a pop star when he left school, this martinet was utterly dismissive of him. The minimum school-leaving age was 15 at the time, so he walked out of the school with his four GCE O levels and never looked back.\n\n'Jimmy's playing was constantly evolving,' recalled Rod Wyatt. 'After he left school he could play lead and pick like Chet Atkins; he was a real prodigy. We still jammed at each other's houses, but not so frequently. The thing about Jimmy was that, unlike most guitarists of those early days, he could play many styles and genres of music.'\n\nChris Farlowe and the Thunderbirds were an emerging act on the R&B circuit in 1960. Page had first seen Farlowe perform three years before, at the British Skiffle Group Championship at Tottenham Royal in north London.\n\nFarlowe's throaty soul vocals fronted the outfit, but it was his guitarist Bobby Taylor who Page would assiduously study. 'He would sit there and watch Bobby playing. Then he'd come backstage and say, \"Oh man, what a great guitar player you are.\" So Bobby influenced him a great deal,' Farlowe told writer Chris Welch. 'Jimmy was very keen to meet him as he thought he was the coolest guitarist he'd ever seen. Bobby Taylor was a very handsome bloke and always dressed in black... Jimmy used to come to our gigs at places like the Flamingo. Then one day he walked up to us at some hall in Epsom, where he lived, and said, \"I'd like to finance an album of you and the band.\"'\n\nClearly the 16-year-old Page, who was the same age as Farlowe, had a lucid eye on his future, as he had saved the money, aware that this creative investment would eventually repay him handsomely. And he also declared that he would be the producer of this album, a pronouncement of almost shocking confidence and self-possession from one so young.\n\nThe album was recorded at R. G. Jones Studios in Morden, Surrey. Page, observed Farlowe, seemed thoroughly _au fait_ with the workings of a recording studio: 'He knew what to do and just plugged the guitar directly into the system without using any amplifiers. He didn't play any guitar himself. He didn't want to, not with Bobby Taylor playing in the studio.'\n\nThe songs included a powerhouse version of Carl Perkins's 'Matchbox' and a hard rendition of Barrett Strong's 'Money', driven by a thundering Bo Diddley beat. But the LP would not be released until 2017, on Page's own label.\n\nNot content with working his way to becoming the greatest rock guitarist, Page's intuition had clearly told him to study the art of record production too. Did he have a glimmer that he would bring all this together in the not-too-distant future?\n\nIn 1960 Red E. Lewis and the Red Caps were introduced to the beat poet Royston Ellis, who was looking for musicians to back him at a series of readings.\n\nEllis was born in 1941, three years before Jimmy Page, in Pinner, north-west London, an outer suburb, like Heston in far west London, where Page first lived. Leaving school at 16, he was determined to become a writer, and at the age of 18 he had his first book published, _Jiving to Gyp_ , a collection of his poetry. Ellis was immediately taken with rock 'n' roll; he would supplement his meagre earnings from poetry by writing biographies of the likes of Cliff Richard and the Shadows and James Dean. In 1961 he published his account of UK pop music, _The Big Beat Scene_.\n\nEllis referred to his live events, mixing beat music and poetry, as 'Rocketry'. At first he had been supported by Cliff Richards's backing group, the Drifters, who, upon changing their name to the Shadows to avoid confusion with the American R&B vocal group, almost immediately had a number one hit with 'Apache' and could no longer fulfil this function for the poet.\n\nDeterminedly bisexual and looking for someone to pick up, in 1960 Ellis had encountered George Harrison in the Jacaranda coffee bar in Liverpool. Although Harrison managed to avoid the poet's advances, the Beetles \u2013 as they were then known \u2013 ended up backing his poetry reading in the city. Ellis always claimed it was he who suggested they substitute the second 'e' in their name for an 'a'. Lennon later said he saw Ellis as 'the converging point of rock 'n' roll and literature'; the song 'Paperback Writer' was said to have been inspired by him.\n\nThrough Red E. Lewis and the Redcaps, Ellis had learned of the stimulant effects of chewing the Benzedrine-covered cardboard strip inside a Vick's inhaler, useful for the increasing array of late-night shows the group needed to play in far-flung parts of Britain. The poet turned the Beatles on to this, staying up talking to them until nine o'clock the next morning.\n\nWhen it came to his backing music, Ellis decided he did not require the entire musical combo of Red E. Lewis and the Red Caps. Instead he settled on only one musician: Jimmy Page.\n\nBetween late 1960 and July 1961 Page played several stints backing Ellis. One of the most significant dates they played was a television broadcast, on ITV's Southern Television, recorded in Southampton with Julian Pettifer. Ellis would later claim that he had secured Page his first TV appearance, though this was manifestly not the case.\n\nPage was still playing his second-hand Futurama Grazioso; soon it would be replaced with a genuine Fender Telecaster. On 4 March 1961 he and Ellis played together at Cambridge University, at the Heretics Society. And on 23 July 1961, having played in assorted coffee bars and small halls, the pair were faced with a bigger challenge. Twenty-year-old Ellis, accompanied by seventeen-year-old Page, was part of the Mermaid Festival at the newly opened experimental Mermaid Theatre, by London's Blackfriars Bridge. Such illustrious names as Louis MacNeice, Ralph Richardson, Flora Robson and William Empson were also giving readings at the festival.\n\n'Jimmy Page was very dedicated to my poetry, understood it, and we worked well together, producing a dramatic presentation that was well received both on TV and stage,' said Ellis.\n\n'Jimmy composed his own music to back my poems \u2013 usually ones from _Jiving to Gyp_ , although I might have been performing the one with the line \"Easy, easy, break me in easy\" from my subsequent book _Rave_. The Mermaid show was the peak \u2013 and possibly the final one \u2013 of our stage performances.'\n\n'Royston had a particularly powerful impact on me,' said the musician of the poet's work. 'It was nothing like I had ever read before and it conjured the essence and energy of its time. He had the same spirit and openness that the beat poets in America had.\n\n'When I was offered the chance to back Royston I jumped at the opportunity, particularly when we appeared at the Mermaid Theatre in London in 1961. It was truly remarkable how we were breaking new ground with each reading.\n\n'We knew that American jazz musicians had been backing poets during their readings. Jack Kerouac was using piano to accompany his readings, Lawrence Ferlinghetti teamed with Stan Getz to bring poetry and jazz together.'\n\nThese arty events with Royston Ellis were, however, rare and unusual for Page. More commonly he simply toured incessantly with Red E. Lewis and the Redcaps \u2013 and then Neil Christian and the Crusaders.\n\nRed E. Lewis and the Redcaps' manager Chris Tidmarsh had decided that he would become the group's singer, renaming himself Neil Christian. In accordance with his own change of identity, Tidmarsh\/Christian gave the group the moniker the Crusaders, and Page became 'Nelson Storm'. Rhythm guitarist John Spicer was henceforth known as 'Jumbo', while drummer Jim Evans was given the sobriquet of 'Tornado'.\n\nPlaying the same circuit, with Screaming Lord Sutch and the Savages, was a guitarist who had also grown up in Heston. His name was Ritchie Blackmore, and he had a bright future as a founding member of Deep Purple and celebrated guitar hero in his own right.\n\n'I met Jimmy Page in 1962. I was 16, 17,' he recalled of their first meeting, at a time when 'Nelson Storm' had acquired a new instrument. 'We played with Neil Christian and the Crusaders. Jimmy Page was playing his Gretsch guitar. I knew he was going to be somebody then. Not only was he a good guitar player, he had that star quality. There was something about him. He was very poised and confident. So I thought, \"He's going to go somewhere, that guy \u2013 he knows what he's doing.\" But he was way ahead of most guitar players. He knew he was good too. He was the type of guy, who... he wasn't arrogant, but he was very comfortable within himself.'\n\nAfter two years of life on the road, Page came down several times with glandular fever, a lingering virus that was a consequence of exhaustion and a bad diet \u2013 and possibly too regular an ingestion of the Vick's Benzedrine strip. In October 1962, when he was only 18, 'Nelson Storm' quit the Neil Christian outfit.\n\nAlmost immediately he enrolled at Sutton Art College in Surrey to study painting, a love almost as great as the guitar. Needless to say, Page's love of music was undimmed, and he had extremely broad taste, eagerly lapping up classical music, both old and new, especially the groundbreaking work of Krzysztof Penderecki, the Polish composer whose 1960 work _Threnody to the Victims of Hiroshima_ conveyed the devastation wrought on 6 August 1945 on the Japanese city. Page's study of Penderecki's work would be reflected much later in his use of the violin bow on his guitar.\n\n'I was travelling around all the time in a bus,' he told Cameron Crowe in _Rolling Stone_ in 1975. 'I did that for two years after I left school, to the point where I was starting to get really good bread. But I was getting ill. So I went back to art college. And that was a total change in direction... As dedicated as I was to playing the guitar, I knew doing it that way was doing me in forever. Every two months I had glandular fever. So for the next eighteen months I was living on ten dollars a week and getting my strength up. But I was still playing.'\n\nOnly days after Jimmy Page left Neil Christian and the Crusaders he experienced something of an epiphany. For the very first time a package tour of American blues artists was scheduled to play in the United Kingdom. Following concerts in Germany, Switzerland, Austria and France, the American Folk Blues Festival had a date scheduled on 22 October 1962 at Manchester's Free Trade Hall, with both afternoon and evening performances. On the bill were Memphis Slim, Sonny Terry and Brownie McGhee, Helen Humes, Shakey Jake Harris, T-Bone Walker and John Lee Hooker.\n\nPage arranged to go with his friend David Williams, but opted to catch the train and meet him in Manchester rather than travel together by road. By now he seemed to have registered that one of the causes of his ill health had been squeezing into an uncomfortable van to travel those long distances across Britain with the Crusaders.\n\nDavid Williams travelled with a trio of companions he had met at Alexis Korner's Ealing Jazz Club, in reality no more than a room in a basement off Ealing Broadway in West London. Fellow aficionados of this music, these companions had recently formed a group. Its name? The Rolling Stones. These new friends were called Mick Jagger, Keith Richards and Brian Jones.\n\nAlthough the first set of the American Folk Blues Festival rather failed to fire, perhaps an expression of the wet Manchester afternoon that it was, the evening house more than lived up to their expectations. Especially when John Lee Hooker closed the show with a brief, three-song set, accompanied only by his guitar. 'It may have been a damp and grey Manchester outside, but we thought we were sweltering down on the Delta,' said Williams. Hooker had been preceded by T-Bone Walker, the 'absolute personification of cool', according to Williams. 'He performed his famous \"Stormy Monday\" on his light-coloured Gibson. His playing seemed effortless, and his set just got better and better as he dropped the guitar between his legs and then swung it up behind his head for a solo. I did not look at Jim, Keith or any of the others while this was all going on, but I can tell you that afterwards they were full of praise and mightily impressed.'\n\nPage, Jagger, Richards, Jones and Williams then drove back to London through the night, Jones nervous about the rate of knots at which they were travelling. In 1962 the M1, Britain's only motorway, extended no further from London than to the outskirts of Birmingham, a hundred miles north of the capital. 'Eventually we made it to the motorway and came across an all-night service station. Again, for most of us this was a real novelty. However, Jim was a seasoned night-traveller by now and he clearly enjoyed talking me through the delights of the fry-up menu. After a feed we resumed our journey, and it was still dark when we reached the outskirts of London.'\n\nEarly on in his time at Sutton Art College, Page encountered a fellow student called Annetta Beck. Annetta had a younger brother called Jeff, who had recently quit his own course at Wimbledon Art College for a job spray-painting cars. Hot-rod-type motors would become an obsession for Beck.\n\nIn a 1985 radio interview on California's KMET, Beck told host Cynthia Foxx: 'My older sister, as I remember it, came home raving about this guy who played electric guitar. I mean she was always the first to say, \"Shut that racket up! Stop playing that horrible noise!\" And then when she went to art school the whole thing changed. The recognition of somebody else doing the same thing must have changed her mind. She comes home screaming back into the house saying, \"I know a guy who does what you do.\" And I was really interested because I thought I was the only mad person around. But she told me where this guy lived and said that it was okay to go around and visit. And to see someone else with these strange-looking electric guitars was great. And I went in there, into Jimmy's front room... and he got his little acoustic guitar out and started playing away \u2013 it was great. He sang Buddy Holly songs. From then on we were just really close. His mum bought him a tape recorder and we used to make home recordings together. I think he sold them for a great sum of money to Immediate Records.'\n\nBeck and Page began to spend afternoons and evenings at Page's parents' home, playing together and bouncing ideas off each other. Page would be playing a Gretsch Country Gentleman, running through songs like Ricky Nelson's 'My Babe' and 'It's Late', inspired by Nelson's guitarist James Burton \u2013 'so great', according to Beck. They would play back and listen to their jams on Page's two-track tape recorder. The microphone would be smothered under one of the sofa's cushions when they played. 'I used to bash it, and it would make the best bass drum sound you ever heard!' said Beck.\n\nBut this extra-curricular musical experimentation was not necessarily in opposition to what Page was doing on his art course at Sutton Art College: rather, these two aspects of himself complemented each other. Many years later, when asking Page about his career with Led Zeppelin, Brad Tolinski suggested that 'the idea of having a grand vision and sticking to it is more characteristic of the fine arts than of rock music: did your having attended art school influence your thinking?'\n\n'No doubt about it,' Page replied. 'One thing I discovered was that most of the abstract painters that I admired were also very good technical draftsmen. Each had spent long periods of time being an apprentice and learning the fundamentals of classical composition and painting before they went off to do their own thing.\n\n'This made an impact on me because I could see I was running on a parallel path with my music. Playing in my early bands, working as a studio musician, producing and going to art school was, in retrospect, my apprenticeship. I was learning and creating a solid foundation of ideas, but I wasn't really playing music. Then I joined the Yardbirds, and suddenly \u2013 bang! \u2013 all that I had learned began to fall into place, and I was off and ready to do something interesting. I had a voracious appetite for this new feeling of confidence.'\n\nDespite starting his studies at Sutton, Page would, from time to time, step in during evening sessions with Cyril Davies's and Alexis Korner's R&B All-Stars at the Marquee Club and other London venues, such as Richmond's Crawdaddy Club or at nearby Eel Pie Island. Soon he was offered a permanent gig as the guitarist with the R&B All-Stars, but he turned it down, worried that his illness might recur.\n\nThere was another guitar player on the scene at the time, a callow youth nicknamed Plimsolls, on account of his footwear. Although at first Plimsolls could hardly play at all, he was known to have a reasonably moneyed background that enabled him to own a new Kay guitar. He had another sobriquet, Eric the Mod, a reflection of his stylish dress sense, and he would shortly enjoy greater success when, rather like his idol Robert Johnson, he seemed to suddenly master his instrument, and he reverted to his full name: Eric Clapton.\n\nPage recalled that one night after he had sat in with the R&B All-Stars, 'Eric came up and said he'd seen some of the sets we'd done and told me, \"You play like Matt Murphy,\" Memphis Slim's guitarist, and I said I really liked Matt Murphy and actually he was one of the ones that I'd followed quite heavily.'\n\nEric Clapton was not the only one to note Page's expertise. Soon came an approach from John Gibb of the Silhouettes, a group from Mitcham, on the furthest extremes of south London. Gibb asked Page to help record some singles for EMI, starting off with a tune called 'The Worryin' Kind'.\n\nThe Silhouettes would later occasionally feature Page's new friend Jeff Beck on guitar. What has always been a fascinating psychogeographical truth is that Jimmy Page, Eric Clapton and Jeff Beck, Britain's greatest and most creative rock guitarists of the 1960s, grew up within a radius of about 12 miles of each other.\n\nHaving witnessed Page's guitar skills with the R&B All-Stars at the Marquee Club, Mike Leander, a young arranger and producer, pulled him into a studio for a stint as rhythm guitarist in late 1962. Page was moonlighting from art school \u2013 something that would become a pattern.\n\nLeander had been alerted to go and check out Page by Glyn Johns, another Epsom boy, a couple of years older than Page who had watched him play years before and was now a tape operator. Of that first meeting, years before, Johns said: 'One evening we had a talent night. I remember a boy in his early teens no one had seen before, who sat with his legs swinging over the front edge of the stage and played an acoustic guitar. He was pretty good, he may have even won, but I don't think anyone in the hall that night had any idea that he was to become such an innovative force in modern music.'\n\n'I was really surprised,' Page told _Beat Instrumental_ magazine in 1965 about Leander asking him to play. 'Before that I thought session work was a \"closed shop\".\n\n'Mike was an independent producer then. And he wanted me to play on \"Your Momma's Out of Town\" by the Carter-Lewis group. The record was released and I believe it helped him considerably in joining Decca full time.'\n\nSoon Mike Leander had another, more prestigious session gig for Page. This studio date was with Shadows expatriates Jet Harris, a bass player, and drummer Tony Meehan. The resulting tune, 'Diamonds', was a number one smash, the first of a run of hits for the duo. Later, Harris and Meehan hired one John Baldwin to accompany their act on the road. Already there were glimmers of some kind of destiny at work; soon John Baldwin would metamorphose into John Paul Jones, so renamed for a solo single by Rolling Stones manager Andrew Loog Oldham; Baldwin's new moniker was derived from the film _John Paul Jones_ , a popular 1959 movie about the famed US naval commander.\n\nEven though Harris and Meehan had departed the Shadows, it was a huge honour for the 18-year-old Page to play with the alumni of the group that, prior to the Beatles, was the biggest in the UK, largely due to the skill and charisma of Hank Marvin, their bespectacled guitarist.\n\n'When did I first discover Hank Marvin? When I was about 14, because in those days it was really skiffle for young kids who wanted to learn three chords and have a good time,' Page told John Sugar on BBC Radio 4 in 2011. 'But going past that, more into the world of the American rockabilly and rock 'n' roll was starting to seduce us all as kids. Then you had Cliff Richard and the Drifters [as the Shadows were first known] at that time, putting forward a really, really damned good rendition of it, but it still had that sort of grit identity to it.\n\n'So really it was a question of seeing Hank playing with Cliff as a kid, looking at Hank on the television. He was good, but he came alive with the Shadows. I mean it was such a really, really good band and Hank was such a stylist... I mean, he was so cool. He was and still is. He had this image... He was such a fluent player... In those early years, all of us \u2013 Jeff Beck, myself, Eric Clapton \u2013 we all played things like \"Apache\", \"Man of Mystery\", \"F.B.I.\", those sort of [Shadows] hits...\n\n'Hank managed to come up with this unique sound, and that sound is just so recognisable. He inspired so many guitarists in those days as kids, kids who had no idea they may even be rock stars themselves one day.'\n\nPlaying guitar with Jet Harris and Tony Meehan on live dates, along with John Baldwin, was one John McLaughlin, who had given Page some guitar lessons when McLaughlin was working in a guitar shop. 'I would say he was the best jazz guitarist in England then, in the traditional mode of Johnny Smith and Tal Farlow,' said Page. 'He certainly taught me a lot about chord progressions and things like that. He was so fluent and so far ahead, way out there, and I learned a hell of a lot.'\n\nThe 'Diamonds' session kick-started a new career for Page, one in which he would play up to ten sessions a week, although he was still officially at Sutton Art College. For the next three and a half years, again while officially still a student, he became one of the UK's top two guitar session players: the other was Big Jim Sullivan, and for a time the boy from Epsom, where he continued to live at his parents' house, became known as 'Little Jim'. Working with all manner of artists and styles, he honed his guitar playing during this period. In his early sessions he largely used a black Les Paul Custom he had played with Neil Christian. Known as 'the fretless wonder', and with a trio of pickups, it gave Page great tonal flexibility. When required, he also used a 1937 Cromwell archtop acoustic guitar and a Burns amplifier, or from time to time a Fender Telecaster.\n\nIn June 1963 Page was interviewed on Channel Television, ITV's smallest franchise, broadcasting only to the 60,000 inhabitants of the Channel Islands. With his hair immaculately swept back, Page certainly looked the rocking part, yet his accent and precise, formal speech constructions suggested someone from a rather more middle-class background than his actually was. Noted for the duty-free alcohol and tobacco on offer in this UK tax haven, Jersey and Guernsey enjoyed a certain hip cachet as a holiday location: was that why Page was there?\n\nThe interview was filmed on an outdoor quayside. The first question was the set-up:\n\n_What is a session guitarist?_\n\n'A guitarist who's brought in to make records, not necessarily doing one-night stands, hoping they'll get into the hit parade \u2013 only getting an ordinary fee.'\n\n_He doesn't work for any particular singer all the time?_\n\n'Not necessarily.'\n\n_I gather there are only a few young session guitarists like yourself: why is that?_\n\n'Well, it seems to be quite a closed shop. The Musicians' Union have their own chaps in and they don't really like to get the young people in because the old boys need the work.'\n\n_So how did you become a session guitarist?_\n\n'I don't know. Perhaps it was because I had the feel for it.'\n\n_How long have you been playing the guitar?_\n\n'Four years.'\n\n_Have you always been a session guitarist?_\n\n'No, no. For the last 18 months.'\n\n_Do you play for a regular group yourself?_\n\n'Yes: Neil Christian and the Crusaders.' [By now Page was no longer playing with them.]\n\n_And what sort of things do you do with this group?_\n\n'Well, we do one-night stands all over England.'\n\n_What are the big names that you have backed on disc?_\n\n'Jet Harris and Tony Meehan, Eden Kane, Duffy Power.'\n\n_What is it like working with some of the really big names of show business?_\n\n'Disappointing.'\n\n_Why is that?_\n\n'Well, they don't come up to how you expect them to be. Rather disappointing on the whole, I would say.'\n\n_That's probably bad news for some record fans. What is your professional ambition? Do you want to be a guitarist all the time? Do you want to make your own records?_\n\n'No, not necessarily. I'm very interested in art. I think I'd like to become an accomplished artist.'\n\n_Rather than a guitarist?_\n\n'Yes, possibly.'\n\n_Is this a means to an end for you? Are you hoping to earn enough money through your guitar playing?_\n\n'Yes. Yes. I'm hoping to finance my art by the guitar.'\n\nThe quality of the acts Page worked with built steadily. Carter-Lewis and the Southerners' 'Your Momma's Out of Town' was an early example. 'He was a fast player, he knew his rock 'n' roll and he added to that,' said John Carter. 'He was also quiet and a bit of an intellectual.'\n\nJohn Carter and Ken Lewis were essentially songwriters, with a sub-career as backing singers: the first hit they wrote was 'Will I What?', the number eighteen follow-up to Mike Sarne's 1962 novelty number one hit 'Come Outside'. They had been persuaded to form Carter-Lewis and the Southerners to promote their material. As a session musician Page played guitar on 'That's What I Want', a Carter-Lewis song that became a Top 40 hit in 1964 for the Marauders, a group from Stoke-on-Trent. Then Page briefly became an actual member of the group. Viv Prince, later drummer with the Pretty Things, played with the group while Page was with them, along with Big Jim Sullivan and drummer Bobby Graham. By 1964 Carter-Lewis and the Southerners would become the Ivy League and then magically transform into the Flower Pot Men, who in 1967 hit the UK number four slot with 'Let's Go to San Francisco'.\n\nPage also played on records as diverse as 'Walk Tall' by Val Doonican, a kind of Irish Perry Como, and \u2013 on 6 November 1964 \u2013 with another Irish singer, Them vocalist Van Morrison, on the Belfast group's 'Baby, Please Don't Go', its B-side 'Gloria', and follow-up single 'Here Comes the Night'.\n\nThe twin pillars of Them were Van Morrison and guitarist Billy Harrison. 'We were brought over,' said Harrison, 'in the middle of 1964 and stuck in Decca's West Hampstead studio to see what we had. We did \"Baby, Please Don't Go\", \"Gloria\" and \"Don't Stop Crying Now\", which was released as the first single and died a death.'\n\nThe sessions were produced by Bert Berns, a streetwise New Yorker who had become a songwriter and record producer of some significance; a crucial figure at Atlantic Records \u2013 he revived the career of the Drifters and brought Solomon Burke to the label \u2013 he would later run Atlantic's BANG label, kickstarting the solo careers of Van Morrison and Neil Diamond. At first, influenced by Jerry Leiber and Mike Stoller, the white songwriting team from Los Angeles who via their cartoon-like wit transformed the subject matter of rhythm and blues, Bert Berns had been a composer of considerable success, subtly lacing his tunes with hypnotic Latin influences, especially mambo. Installed in New York's famous Brill Building, the endlessly and effortlessly enthusiastic Berns co-wrote the Isley Brothers' 'Twist and Shout', Solomon Burke's 'Everybody Needs Somebody to Love', The Exciters' 'Tell Him', Them's 'Here Comes the Night' and the McCoys' 'Hang On Sloopy', among many others. (As befitted the sometimes sleazy, occasionally Mob-affiliated world of New York popular music, Bert Berns, who everyone found a fabulous human being, attractive and glamorous to those with a fondness for boho chic, was allegedly 'connected', and possibly even a 'made man'. From his rarefied perspective he would have given Page interesting instruction about the US music business. At their first sessions Led Zeppelin recorded a song about him, 'Baby Come On Home', subtitled 'Tribute to Bert Berns', an exceptionally beautiful soul tune of precisely the type Berns would have produced for Atlantic, which was not released until 1993. In Page's guitar playing on this 1968 recording you can hear his love for Bert Berns.)\n\nIt was Bert Berns's writing of the song 'Twist and Shout' that had first brought him to London. Covered by the Beatles, with John Lennon's extraordinary, searing performance taking the song to a show-stopping further level, 'Twist and Shout' closed _Please Please Me_ , the Liverpool group's first album, the number one LP in the UK for 30 weeks in 1963. Although the Beatles meant nothing in America at that time, Berns's first royalty cheque for his song on the album was for $90,000. In October 1963 he came over to London to see what was going on, producing a handful of no-hoper acts.\n\nAlready working in the British capital was Shel Talmy. A Los Angeleno who had worked with Capitol Records, he had been hired as staff producer by Dick Rowe, the Decca head of A&R \u2013 the man who famously turned down the Beatles but redeemed himself somewhat by signing the Rolling Stones. Rowe now decided that Bert Berns might fit as producer with the Belfast act he had signed named Them.\n\n'Twist and Shout' had been covered yet again, by Brian Poole and the Tremeloes, who turned it into a Top 10 UK hit for Decca Records. It was something of a revenge release, as the act had been signed by Dick Rowe in preference to the Beatles \u2013 Brian Poole and the Tremeloes were, after all, from Essex, which was far more geographically convenient for a Londoner like Rowe than Liverpool.\n\nIn London, where he and Talmy were the only American producers working, Bert Berns had secured work through Decca Records, taking on 'Little Jimmy' Page as his principal session guitarist, recognising his talent and befriending him. 'With the new breed of British producers such as Mickie Most or Andrew Loog Oldham trying as hard as they could to make records that sounded American,' wrote Berns's biographer Joel Selvin, 'Berns was the first American producer trying to make records that sounded British.'\n\nThe sessions with Them for Decca proved as much, the resulting recorded songs utterly unique in the resounding clarity of their sound. 'Bert Berns was inveigled into producing the session,' said Billy Harrison. 'And he brought in Jimmy Page, and Bobby Graham on drums. There was much grumbling, mostly from me, because I felt we could play without these guys. Jimmy Page played the same riff as the bass, chugging along. I played the lead: I wrote the riff.\n\n'Bert Berns had arguments with us about the sound. I thought we were playing it okay: if someone brought in session men you took it as a bit of a sleight. I was very volatile in those days.\n\n'There were various rows. Jimmy Page didn't really seem to want to talk to anybody. Just a stuck-up prick who thought he was better than the rest of the world. Sat there in silence. No conversation out of the guy. No response.'\n\nPossibly Billy Harrison was misinterpreting the shyness that other musicians felt characterised the quiet Jimmy Page. And he may have been projecting his personal prejudices. 'He seemed above everybody, above these Paddies. That was the days when guest houses would have a sign up: \"No salesmen, no coloured, no Irish\". Page had that sort of sneering attitude, as though he was looking down on everybody. He's a fabulous technician, but there's nothing wrong with a bit of friendliness.'\n\n'Their lead vocalist, Van Morrison, was really hostile as he didn't want session men on his recordings,' said drummer Bobby Graham. 'I remember the MD, Arthur Greenslade, telling him that we were only there to help. He calmed down but he didn't like it.'\n\n'Whatever Morrison's reservations, they worked well together, and Graham's frenzied drumming at the end of \"Gloria\" is one of rock's great moments,' wrote Spencer Leigh in his _Independent_ obituary of Graham.\n\nAnd the opening guitar riff on 'Baby Please Don't Go' is one of the defining moments of popular music in the sixties. This was all Billy Harrison's own work. 'What annoyed me later,' he said, 'was that you would start to see how it was being said that Jimmy Page had played a blinding solo on \"Baby Please Don't Go\". I got narked about that: he never said he did it, but he never denied it.'\n\n'For a long time,' said Jackie McAuley, who joined Them the next year, 'Jimmy Page got credit for Billy Harrison's guitar part. But he's owned up about it.'\n\nBert Berns also pulled Page in for 'Shout', a cover of the Isley Brothers' classic that was the debut hit for Glasgow's Lulu & The Luvvers. And he had him add his guitar parts to her version of 'Here Comes the Night', a majestic version that was released prior to Them's effort, but spent only one week in the UK charts.\n\nShel Talmy, a former classmate of Phil Spector at Fairfax High School in Los Angeles, also loved Page's playing, and the guitarist was equally taken with him: a studio innovator, Shel Talmy would play with separation and recording levels, techniques that Page would assiduously study.\n\nSoon after he had arrived in London and started working for Decca, Talmy came across the guitarist: 'Somebody mentioned that they'd heard this 17-year-old kid who was really terrific, and I went and checked him out and I used him. We got along great and he was fabulous. I thought, \"This kid is really gonna go somewhere,\" and I only regret that he didn't call me when he formed Led Zeppelin. It's a shame! I would like to have done that.\n\n'He got it. I mean, he was original. At that time in London there were very few really current musicians: a lot of good musicians, but kind of mired slightly in the past. There were, like, one or two good rhythm sections and that was it. I originally started using Big Jim Sullivan, who was the only other one, and then I found Jimmy, who I thought was even better because he was more with it. He was doing what I thought should be done and certainly what was being done in the States, so it was a no-brainer.'\n\nFitting Page together with drummer Bobby Graham, and from time to time John Baldwin on bass, the producer had a team that was highly resourceful and fast. Talmy has described Graham as 'the greatest drummer the UK has ever produced'. While playing with Joe Brown and the Bruvvers, Graham had been approached by Brian Epstein at a gig at the Tower Ballroom in New Brighton, in June 1962. Would he care to replace Pete Best in the Beatles? Epstein asked him. Graham turned down the offer, leaving the way clear for Ringo Starr.\n\nGraham first met Page when the guitarist was playing with Neil Christian and the Crusaders; they had supported Joe Brown at a show in Aylesbury, Buckinghamshire. 'I was so impressed. We became very good friends, and when I became a producer I always used Jimmy. We started a publishing company called Jimbo Music, for stuff we wrote. Jimmy wasn't one of the most way-out and weirdest characters I ever met: he was very quiet, very shy. Jimmy had a slightly dirtier sound than Big Jim Sullivan \u2013 they used to alternate a lot. Unless the arranger wanted a certain thing they'd fight it out amongst themselves.'\n\nNeither Page nor Bobby could sight-read music \u2013 though the guitarist would learn how to do so over the next couple of years. 'I had to rely on what felt right,' said Graham, who estimated that he played on 15,000 tunes in the course of his career. 'I was loud. My trick was, if the singer took a breath, fill in. I was one of the first of the new generation coming in. Jim Sullivan was already in. Jimmy Page \u2013 same thing, couldn't read a note but had a great feel.'\n\nPlaying sessions paid good money, \u00a39 a time when the average working man earned little more than that a week. And, as befitted the rules of the Musicians' Union, there would be three sessions a day: 10 a.m. until 1 p.m.; 2 till 5; and 7 until 10 p.m. During each session the musicians were expected to finish four songs, and afterwards they would be handed small brown envelopes containing their fees in cash. If you worked all three sessions, you'd come away with almost \u00a330 for a day's work. At the end of each evening, Page, Big Jim Sullivan and Graham would adjourn to such fashionable _bo\u00eetes_ of the day as the Cromwellian and Annie's Room.\n\n'The weirdest thing I ever did with Jimmy was _Gonks Go Beat_ ,' reflected Graham. 'Charlie Katz had booked us into Decca number three studio, the cathedral where they did all the classical recordings. I wasn't supposed to be at that session \u2013 it was the only time at the wrong place. My part looked like a map of the London Underground. Jimmy came over and said, \"I think we're in the wrong place. I can't read my part.\" The musical director said, \"Are you ready, gentlemen?\" and there was complete silence. He looked vaguely in my direction, and I thought he was talking to somebody behind me. He said, \"Bob, you're in at the start,\" and I struggled. Finally he put the baton down and came over and ran it through with me. During the session I looked across and Jimmy was thundering away. At the end of the session I said, \"You looked all right, Jim.\" He said, \"I turned my amp off.\"'\n\nWith Shel Talmy, the trio of the two Jims and Bobby worked with a seemingly endless list of aspirant acts and tunes, such as the Lancastrians' 'We'll Sing in the Sunshine', Wayne Gibson's 'See You Later Alligator' and the First Gear's 'Leave My Kitten Alone', a cover of a Little Willie John tune and the B-side of 'A Certain Girl'. 'Leave My Kitten Alone' was deemed 'Page's most outstanding solo prior to \"Whole Lotta Love\"' by US rock critic Greg Shaw.\n\nOn 15 January 1965, again for Talmy, Page worked with 17-year-old David Jones, the leader of the Manish Boys, on 'I Pity the Fool', a cover of the Bobby Bland tune, backed with 'Take My Tip'. So as not to be mistaken for Davy Jones of the Monkees, David Jones would soon change his name to David Bowie. (In 1964 Page had been a 'member' of Jones\/Bowie's Society for the Prevention of Cruelty to Long-Haired Men, a clear publicity gimmick that succeeded in getting Jones on television news. And Page had already played with a pair of earlier David Bowie line-ups, Davy Jones' Locker and Davy Jones and the Lower Third, both with Shel Talmy producing. And he worked on David Bowie's first, eponymous album, for Deram Records, produced by Mike Vernon.)\n\n'That \"I Pity the Fool\" session was phenomenal,' said Wayne Bardell, then working in Francis, Day and Hunter, a record shop on London's Charing Cross Road, but soon to become a successful manager. 'I was at the session at IBC as a guest of the not-yet Bowie, with Shel Talmy producing, Glyn Johns the engineer and Jimmy Page on guitar.'\n\n'Well, it's definitely not going to be a hit,' Page said, correctly, of the tune that day \u2013 it sold no more than 500 copies. But during the Manish Boys' sessions he gave David Jones a guitar riff that the young singer didn't yet know how to use: as David Bowie he fitted this riff into two separate songs, first on 1970's 'The Supermen' on his _The Man Who Sold the World_ album, and again on 'Dead Man Walking' in 1997. 'When I was a baby,' said David Bowie later, 'I did a rock session with one of the bands, one of the millions of bands that I had in the sixties \u2013 it was the Manish Boys, that's what it was \u2013 and the session guitar player doing the solo was this young kid who'd just come out of art school and was already a top session man, Jimmy Page.'\n\nAnd 'this young kid' had every right to be very excited about the part he played on 'I Pity the Fool', which, despite his misgivings, was a sensationally great record that should have been a hit; this was thanks in no small part to Jimmy Page adding searing, hard-rock guitar, like something Mick Green could have provided for the Pirates.\n\n'I Pity the Fool' might have flopped, but Talmy produced breakthrough singles from a pair of acts that would become two of the biggest UK groups of the 1960s: 'You Really Got Me', the Kinks' third 45, and the Who's 'I Can't Explain'. Page played rhythm guitar on a version of the latter track.\n\n'Because Shel wasn't sure I could play a solo, he asked his favourite session guitarist, Jimmy Page, to sit in,' wrote Pete Townshend in his autobiography. 'And because our band had rehearsed the song with backing vocals in Beach Boys style, but not very skilfully, Shel arranged for three male session singers, the Ivy League, to chirp away in our place. Shel Talmy got a good sound, tight and commercial, and although there was no guitar feedback, I was willing to compromise to get a hit.'\n\nIn _Guitar Masters: Intimate Portraits_ by Alan DiPerna, Townshend referred to Page as 'a friend of mine'. The guitarists certainly had something in common: a fling with Anya Butler, the beautiful \u2013 and older \u2013 assistant to Who co-manager Chris Stamp. Townshend was initially puzzled by Page's presence at the session: 'I said to Jimmy, \"Well, what are you doing here?\" He said, \"I'm here to give some weight to the rhythm guitar. I'm going to do the guitar on the overdubs.\" And I said, \"Oh, great.\" And he said, \"What are you going to play?\" \"A Rick 12,\" I told him. And he said, \"I'll play a...\" Whatever it was. It was all very friendly. It was all very convivial.'\n\nAnd on 'Bald Headed Woman', the B-side of 'I Can't Explain', it was Page who played the fuzzbox licks. On the liner notes to the Who's _Two's Missing_ compilation album, Who bassist John Entwistle said: 'The fuzz guitar droning throughout is played by Jimmy Page. The reason being, he owned the only fuzzbox in the country at that time.'\n\nEntwistle was not exactly correct. Gibson guitars had put a fuzz-tone pedal into production in 1962, giving it the brand name of 'Maestro Fuzz-Tone FZ-1'. Although in limited supply, the devices, imported from the US, could be found from time to time in London's more select musical equipment stores, and it was from one of these that Page had acquired his gadget.\n\nLike many technological developments, the origins of the fuzzbox and the dirty edge it added to a guitar's sound \u2013 which Page would employ to his maximum advantage and could be heard to its defining fullest when played by Keith Richards on the Rolling Stones' 'Satisfaction' \u2013 were accidental. In 1951 Jackie Brenston and his Delta Cats \u2013 actually Ike Turner's Kings of Rhythm \u2013 had hit the number one slot in the US rhythm and blues chart with 'Rocket 88'. A distinctive feature of 'Rocket 88' was the growling sound of Willie Kizart's guitar. On his way from Clarksdale, Mississippi, to Sam Phillips's Sun Studio in Memphis in 1951 to record the tune, Kizart's amplifier had fallen from his car while a tyre was being replaced. Endeavouring to repair the resulting damage to the speaker cone, the guitarist stuffed it with paper: the marginally distorted sound that resulted became a feature of the 'Rocket 88' single, which is often cited as one of the first rock 'n' roll records. From then on, guitarists sought out the means to deliver a similar grimy sound, the likes of Link Wray \u2013 who would poke holes in his loudspeaker \u2013 and Buddy Guy consciously damaging their amps to replicate such a tone. And in 1961 the great country singer Marty Robbins's 'Don't Worry' single hit number three in the US national charts, largely courtesy of his guitarist Grady Martin's muttering instrument being played through a faulty amplifier. Martin soon put out his own single, 'The Fuzz', thus bestowing the malfunction with a semi-official term.\n\nIn Los Angeles a radio-station technician developed an electronic device to create such an effect for producer Lee Hazelwood, who employed it on Sanford Clark's 'Go On Home' 45 in 1960. And in the same city, super session player Orville 'Red' Rhodes, who would become a member of the celebrated Wrecking Crew and was also an electronics whizz, developed a similar device, which was utilised by fellow Wrecking Crew guitarist Billy Strange on Ann Margret's 'I Just Don't Understand'. In turn this led to Strange employing Rhodes's invention with the instrumental surf band the Ventures, a kind of US version of the Shadows, on their late-1962 release 'The 2,000 Pound Bee'. It was this tune especially that had come to the attention of Page; anxious to replicate its juddering sound, he had purchased his own Maestro Fuzz-Tone.\n\nYet it was not entirely to his satisfaction. Luckily, he already knew someone who could assist him with this. Roger Mayer was a friend from the Epsom music scene. By 1964 he was working for the Admiralty Research Laboratory in Teddington, in the Acoustical Analysis section, having developed into something of an electronics boffin. And their friendship persisted: Page and Mayer would visit each other's homes to listen to American records. 'Jimmy came to me,' said Mayer, 'when he got hold of the Maestro Fuzz and said, \"It's good but it doesn't have enough sustain... it's a bit staccato.\" I said, \"Well, I'm sure we can improve on that.\" That conversation spurred me to design my first fuzzbox.'\n\n'I suggested that Roger should try to make something that would improve upon the distortion heard on \"The 2,000 Pound Bee\" by the Ventures,' said Page. 'He went away and came up with the first real good fuzzbox... the first thing that really generated this wonderful sustain.'\n\nRunning off a 6-volt battery, Mayer's fuzzbox was constructed within a custom-made casing, which contained controls for gain and biasing along with a switch that would modify the tonal output. 'Right from square one,' said Mayer, 'Pagey and I wanted something that sustained a lot, but then didn't start jittering as it went away. One of the things that became very, very apparent early on was that you didn't want nasty artefacts. It's very easy to design a fuzzbox \u2013 anybody can do it \u2013 but to make one sound nice and retain articulation in notes, now that's something else.'\n\nPage's part in the Kinks' career is more cloudy. Although it has often been claimed that he played the iconic solo on 'You Really Got Me', this is not the case. 'Jimmy did play rhythm on the first Kinks LP, and certainly did not play lead on \"You Really Got Me\", which preceded the LP by several weeks, or anything else for that matter. I only brought him in to play rhythm because at the time Ray wanted to concentrate on his singing,' said Shel Talmy. In fact, Page had already played acoustic 12-string guitar on 'I've Been Driving on Bald Mountain' and 'I'm a Lover Not a Fighter', on the Kinks' eponymously titled debut album. (In 1965 Page played the solo on an instrumental version of 'You Really Got Me'; almost identical to Dave Davies's original guitar part, it was included on an instrumental album by the Larry Page Orchestra entitled _Kinky Music_.)\n\n'My presence at their sessions was to enable Ray Davies to wander around and virtually maintain control of everything, without having to be down in the studio all the time,' said Page later. 'Ray was producing those songs as much as Shel Talmy was... more so, actually, because Ray was directing them and everything. At one point, there were even three guitars playing the same riff.'\n\n'I'll tell you something about Jimmy Page,' Ray Davies told _Creem_ magazine. 'Jimmy Page thinks he was the first person in the world to ever put a B string where a G string should be. And for me, that's his only claim to fame. Other than that, I think he's an asshole... Jimmy Page and a lot of other people subsequently came to our sessions when we became hot, and I think he played rhythm 12-string on \"I'm a Lover Not a Fighter\", and he played tambourine on \"Long Tall Shorty\".'\n\nIn fact, Page did not 'put a B string where a G string should be'. He told _Melody Maker_ that he would substitute the B string with a top E. Rather than the conventional E string he would swap it for a banjo octave string, either tuned to G or A: 'You'll get a raving, authentic blues sound that you hear on most pop records with that string-bending sound.'\n\n'I didn't really do that much on the Kinks records,' Page later admitted. 'I know I managed to get a couple of riffs in on their album, but I can't really remember. I know that Ray didn't really approve of my presence. The Kinks just didn't want me around when they were recording. It was Shel Talmy's idea. One aspect of being in the studio while potential hits were being made was the press \u2013 too many writers were making a big fuss about the use of session men. Obviously I wasn't saying anything to the press but it just leaked out... and that sort of thing often led to considerable bad feeling.'\n\nFor most of these sessions Page employed a Gibson Les Paul Custom, with the frets filed down 'to produce a very smooth playing action... it just sounded so pure and fantastic,' he told John Tobler and Stuart Grundy for BBC's Radio 1.\n\nDespite the griping of Ray Davies and Billy Harrison, Page played on a number of records that were significant cornerstones of mid-sixties British pop \u2013 outright classics, some of them. These included Shirley Bassey's theme song for _Goldfinger_ , the third James Bond film, on which he played with Big Jim Sullivan and Vic Flick, another renowned UK session guitarist \u2013 the tune was a Top 10 US hit. Then there was Tom Jones's 'It's Not Unusual', number one in the UK and Top Ten in the US; Petula Clark's 'Downtown', a US number one; Kathy Kirby's 'Secret Love'; Marianne Faithfull's 'As Tears Go By'; P. J. Proby's 'Hold Me'; the Merseys' 'Sorrow', covered by David Bowie on his _Pin Ups_ album; the Nashville Teens' 'Tobacco Road'; Brian Poole and the Tremeloes' 'Candy Man'; Twinkle's 'Terry', a motorcycle-death record in the tradition of the Shangri-Las' 'Leader of the Pack' that was number four in the UK charts at Christmas 1964 and banned by the BBC for being in 'poor taste'; 'Baby What's Wrong' and its B-side 'Be a Sect Maniac', the first single from the Downliners Sect, a wild R&B outfit who made the Pretty Things seem like Cliff Richard.\n\nAs it had been with Bert Berns, much of Page's session work was for the Decca label, at their studios in Broadhurst Gardens, West Hampstead, a plain, nondescript building, built like an office block.\n\nHe worked extensively with Dave Berry, a Decca solo star from Sheffield whose first hit had been a cover of his namesake Chuck Berry's 'Memphis Tennessee'. He was one of British rock 'n' roll's first anti-heroes, a true original. 'I noticed how strippers used to tease the audience in Hamburg,' he said of his time playing the circuit in the German port. So almost an entire Dave Berry set might consist of him singing his songs from behind the stage curtain, with only his microphone and hand tantalisingly visible.\n\nWhen Elvis Presley covered Arthur Crudup's 'My Baby Left Me', Scotty Moore's guitar licks had proved such an inspiration for the teenage Jimmy Page. Now Page took the lead guitar part himself on Dave Berry's sensational version of the song, with \u2013 as was customary \u2013 Big Jim Sullivan on rhythm.\n\nBerry's 'My Baby Left Me' only grazed the Top 40, but his sultry 'The Crying Game' was a Top 5 tune when it was released in July 1964. However, this time it was Big Jim Sullivan who took the lead part, with Page providing rhythm; on drums, as per usual, was Bobby Graham. There was a picture in the music press, recalled Berry, of Page standing next to him, along with the engineer Glyn Johns, listening to a playback of 'The Crying Game'. 'Many of the session musicians would have left as soon as they had done their part,' said Berry. 'But Jimmy Page, being a proper player, would listen to his own part. He would sometimes want to do it again. Mind you, at the time Jimmy was in Carter-Lewis and the Southerners: by 5 p.m. he'd be gone to do a gig.'\n\nThe specific session players he used, said Berry, 'were really into it. I must have done a quarter of my career with Decca with that line-up: 25 to 30 songs. Mike Smith would call me with the studio booked. But if Big Jim and Jimmy Page were not available we'd cancel it and wait.' There were at least four tracks on which Page played harmonica: 'C.C. Rider', for example, and Buster Brown's 'Fannie Mae', which relies on a harmonica riff. Meanwhile, Page played both lead guitar and the harmonica part on 'Don't Gimme No Lip Child', the B-side of 'The Crying Game'.\n\nWas Page, who was only 20 years old, anxious to impose his personality in the studio? Not at all, said Berry: 'He was very quiet. The true professional players don't have any edge to them anyway. The bigger the artist, the less edge they have to them. These two guitarists were really great players. And they didn't stick to how this stuff was written out. Big Jim would be improvising his solo. You could hear him doing a vocal counter-melody. We'd say, \"Leave that in, it's real.\" You could work with these guys and suggest things. In 2010, when I met him again, Jimmy seemed exactly the same \u2013 a normal and quiet person. I was very proud of my output: it had a vast range. So when Jimmy Page was in the biggest band in the world I was very proud of my association with them. When I'd meet up with him I'd feel very proud, like a child.'\n\nOn 27 March 1964 Page played heavy fuzz-tone guitar on Carter and Lewis's 'Skinny Minnie'.\n\nBy now this was becoming customary practice for the guitarist. Again, in early 1964, on a session for Screaming Lord Sutch and the Savages, Page augmented his guitar with his Gibson Maestro Fuzz-Tone on a single that was released in October that year, 'Dracula's Daughter', and its B-side 'Come Back Baby', a studio date engineered by the legendary Joe Meek in his tiny Holloway Road set-up. (David Sutch, as his name was registered at birth, was an eccentric English rocker who appeared onstage in a coffin, sometimes dressed as Jack the Ripper \u2013 also the title of an earlier Decca single on which Page played \u2013 and based his act on the American Screamin' Jay Hawkins, who had written and recorded 'I Put a Spell on You'. Sutch's Savages proved a fertile training ground, employing \u2013 among many others \u2013 guitarists Jeff Beck and Ritchie Blackmore and drummer Carlo Little, who had played briefly with the Rolling Stones prior to Charlie Watts. In 1963 Sutch stood as a candidate in a UK by-election, representing the Monster Raving Loony Party, the beginning of a career as a perennially unsuccessful Parliamentary candidate. Later, in 1964, Sutch founded Radio Sutch, a pirate broadcaster based in a wartime fort near the Thames estuary. Before the decade was out, Lord Sutch would reappear in the life of Jimmy Page.)\n\nIn September 1964 Decca Records paid for the dynamic, soulful American singer Brenda Lee, who was signed to the label, to come to London to record at Broadhurst Gardens. 'She said to me, \"I've come here to make a record with the British sound.\" She felt she wouldn't get the same sound in Nashville because they're only just catching up on the British beat group sound of about six months ago,' said producer Mickie Most to _Rolling Stone_ magazine.\n\nThe tune chosen to acquaint Little Miss Dynamite with the zeitgeist was 'Is It True', another song written by Page's musical allies John Carter and Ken Lewis. The guitarist used an early wah-wah pedal on the record, which hit the same number 17 spot on both sides of the Atlantic.\n\nBy now Pete Calvert, Page and Rod Wyatt's guitar-playing buddy from Epsom, had rented a London flat, 4 Neate House in Pimlico. Page would drop in and sometimes stay over if he had an early gig the next day. Soon Chris Dreja of the Yardbirds moved in to one of the rooms.\n\nA desire to improve upon and expand his natural abilities seemed second nature to Page. Having bought a sitar almost as soon as he learned of the instrument's existence, he became one of its earliest exponents in the UK. 'Let's put it this way,' he said. 'I had a sitar before George Harrison. I wouldn't say I played it as well as he did, though. I think George used it well... I actually went to see a Ravi Shankar concert one time, and to show you how far back this was, there were no young people in the audience at all \u2013 just a lot of older people from the Indian embassy. This girl I knew was a friend of his and she took me to see him after the concert. She introduced him to me and I explained that I had a sitar, but did not know how to tune it. He was very nice to me and wrote down the tunings on a piece of paper.' On 7 May 1966 _Melody Maker_ , the weekly British music paper that considered itself intellectually superior to the rest of the pop press, ran an article entitled 'How About a Tune on the Old Sitar?', with much of its information taken from Page.\n\nThis questing side of him surfaced again in his efforts to improve his abilities on the acoustic guitar. 'Most great guitarists are either great on electric or great on acoustic,' said Alan Callan, who first met Page in 1968 and in 1975 became UK vice president of Swan Song Records, Led Zeppelin's label. 'But Jim is equally great on both, because he is always faithful to the nature of the instrument. He told me that, quite early on, he'd gone to a session and the producer had said, \"Can you do it on acoustic rather than electric?\" And he said he came out of that session thinking he hadn't nailed it, so he went home and practised acoustic for two months.'\n\nThe first half of the 1960s was a boom period for UK folk music, with several emerging virtuosos, revered by young men learning the guitar or \u2013 in Page's case \u2013 always eager to improve. John Renbourn, Davey Graham \u2013 who incorporated Eastern scales into his guitar playing \u2013 and Bert Jansch were the holy triumvirate of these players; Page was especially turned on by Jansch, who introduced him to 'the alternate guitar tunings and finger-style techniques he made his own in future Zeppelin classics such as \"Black Mountain Side\" and \"Bron-Y-Aur Stomp\",' according to Brad Tolinski in his book _Light and Shade_.\n\n'He was, without a doubt, the one who crystallised so many things,' Page said. 'As much as Hendrix had done on the electric, I really think he's done on the acoustic.' Al Stewart, a folk guitarist and singer, and, like Jansch, a Glaswegian, explained to Page that Jansch's guitar was tuned to D-A-G-G-A-D \u2013 open tuning, as it was known. Page started to employ this himself.\n\n## 3\n\n# SHE JUST SATISFIES\n\nWhile much of Jimmy Page's work consisted of bread-and-butter pop sessions, from time to time he would be offered the opportunity to indulge his creative side. On the morning of 28 January 1965, for example, half a dozen of Britain's most accomplished musicians met at IBC Recording Studios at 35 Portland Place in London for the morning session slot. Page was on guitar, Brian Auger on organ, Rick Brown played the bass and Mickey Waller was the drummer, with Joe Harriott and Alan Skidmore on saxophones. They were assembled to record an album with the American blues legend Sonny Boy Williamson. 'We started at 10 a.m. and it was all done by 1 p.m.,' recalled Waller. 'Also, it was done completely live: there were no overdubs. We all sat in a circle and played.' After Williamson grew progressively more drunk, his skewed sense of timing made the session increasingly difficult.\n\nPage later recalled: 'Sonny Boy was living in [Yardbirds' manager] Giorgio Gomelsky's flat. Somebody told me once that they went to the house and they heard Sonny Boy plucking a live chicken. I don't know how true that was. That didn't happen when I was there. Sonny Boy and I rehearsed these numbers in the manager's flat, and by the time we got into the studio a couple of days later Sonny Boy had forgotten all of the arrangements. It was cool. Good music comes out of that.' (During Sonny Boy Williamson's time in Britain, the bluesman performed at Birmingham Town Hall: there, a 16-year-old Robert Plant, stunned almost breathless from watching his performance, didn't permit his awe to prevent him from sneaking backstage and stealing one of the blues master's harmonicas \u2013 revenge, apparently, for the legendarily acerbic Williamson having told Plant to 'fuck off' when the teenager attempted to greet him while standing side by side at a urinal.)\n\nWhen the Sonny Boy Williamson album session took place, Page had just become involved with another American \u2013 one who was blonde and female. Jackie DeShannon, hailing from Kentucky, was a beautiful singer-songwriter and a musical prodigy from an early age. By the time she was 11 she had her own radio show. In her early teens she had become a recording artist, at first singing country music. Her records 'Buddy' and 'Trouble' came to the attention of the great American early rocker Eddie Cochran. With Chuck Berry and Buddy Holly, Cochran was a rock 'n' roll singer-songwriter; by a measure of synchronicity he was also a hero of Page \u2013 Led Zeppelin would sometimes feature covers of some of Cochran's greatest songs: 'C'mon Everybody', 'Summertime Blues', 'Nervous Breakdown' (which effectively was what 'Communication Breakdown' was) and 'Somethin' Else'. 'You know, you look like a California girl,' Eddie said to her. 'I think that you should be in California if you want to have a great career.'\n\nDeShannon moved to Los Angeles, where Cochran was based, and he teamed her up with singer-songwriter Sharon Sheeley, his girlfriend, who wrote 'Poor Little Fool' for Ricky Nelson. The two girls started to write songs together, resulting in 'Dum Dum', a hit for Brenda Lee, and 'I Love Anastasia', which scored for the Fleetwoods. (Along with Gene Vincent, Sharon Sheeley was injured in the car crash in England that took Eddie Cochran's life on 17 April 1960.)\n\nAt the age of 15 DeShannon became a girlfriend of Elvis Presley, which became part of her myth; she also had a relationship with Ricky Nelson. Although she failed to have big chart hits of her own in the United States, her version of Sonny Bono and Jack Nitzsche's 'Needles and Pins' was a number one record in Canada before the Searchers covered it in the UK, where it also topped the charts. The Searchers soon covered DeShannon's own song 'When You Walk in the Room', releasing it in September 1964, when it reached number three in the UK charts. In August and September of 1964 she was also one of the support acts on the Beatles' first tour of the USA \u2013 her backing musicians on those dates included a young Ry Cooder.\n\nSniffing the cultural wind, as Shel Talmy, Bert Berns and Brenda Lee had done, Jackie DeShannon arrived in the UK to record at the EMI Studios on Abbey Road at the end of 1964. 'I was very used to working with people like Glen Campbell and James Burton and Tommy Tedesco \u2013 all these great, great guitar players,' she recalled. 'So when I was there I said, \"Who's an amazing acoustic guitar player that I can have on my sessions?\" and they all said that Jimmy Page was the guy, because he had played on a lot of different hit records at the time and was one of the guys on the \"A list\" of studio musicians to call.\n\n'So I said, \"Great, let's have him,\" and they said, \"Well, you can't get him here because he's in art school.\" I said, \"What?\" He showed up... with paint on his jeans and he was the youngest player in the room. I went over to him to play a few of my piddling chords, and when he played them back to me I was almost knocked out of the room. Even then, he was spectacular. I knew right then that he was an amazing talent, so he played on a song of mine called \"Don't Turn Your Back on Me\" and we did some writing together.'\n\nThere was, however, an attraction beyond his guitar-playing skills and Jackie DeShannon's own musical abilities. 'We got together afterwards,' Page said in an interview in 1977 when he revealed how she enticed him with a very attractive proposition: 'She said, \"I've got a copy of Bob Dylan's new album if you'd like to hear it,\" and I said, \"Would I like to hear it?\"'\n\nFor most of 1965 Page and Jackie DeShannon were an item, and this almost-eminent American songwriter took him under her wing; together they wrote the song 'Dream Boy', a rocking single, like Ronnie Spector handling a surf tune. At the time Marianne Faithfull felt that Page was 'rather dull'. She saw his relationship with DeShannon as his way of 'undulling himself'. Tony Calder, her manager and close associate of Rolling Stones manager Andrew Loog Oldham, recalled that 'One night I couldn't get into our hotel room because Jimmy and Jackie DeShannon were in there shagging, so I yelled, \"When you've finished could you write a song for Marianne?\"'\n\nThe result was Marianne Faithfull's second hit, 'Come and Stay with Me', which reached number four. 'In My Time of Sorrow', an album track for Marianne, also emerged from this partnership.\n\n'We wrote a few songs together, and they ended up getting done by Marianne, P. J. Proby and Esther Phillips or one of those coloured artists... I started receiving royalty statements, which was very unusual for the time, seeing the names of different people who'd covered your songs,' said Page.\n\nBut how must it have felt for Page to be having sex with someone who had been to bed with two of his idols, Elvis Presley and Ricky Nelson? No doubt it considerably boosted his sense of himself and who he could be, given his increasing belief in psychic connections and the powerful, allegedly transcendent energies of Aleister Crowley's 'sex magick'.\n\nPage was always financially astute. Early in 1965 he had set up his own publishing company, and soon, urged on creatively and emotionally by Jackie DeShannon, he was making his first solo record. 'She Just Satisfies' was a Kinks-style rocker, released on the Fontana label, on which he sang; its B-side was another Page\u2013DeShannon tune, 'Keep Moving'. Hearing it now, 'She Just Satisfies' sounds like it could have been a likely chart contender.\n\n'\"She Just Satisfies\" and \"Keep Moving\" were a joke,' Page later said, dismissing the record with an element of false modesty. 'Should anyone hear them now and have a good laugh, the only justification I can offer is that I played all the instruments myself except the drums.'\n\nIn his 1965 interview with _Beat Instrumental_ magazine, Page was asked about the possibility of a follow-up to 'She Just Satisfies'. He rejected the notion saying, 'If the public didn't like my first record, I shouldn't think they'll want another.'\n\nIn March 1965 DeShannon took Page on his first trip to the United States, first to New York and then to Los Angeles. For the guitarist, who later admitted his first impressions of the USA came from the glamour of Chuck Berry's witty, lyrically descriptive songs, life was suddenly opened up almost unimaginably.\n\nIn New York, Page crashed in the spare room of Bert Berns's sumptuous Manhattan penthouse apartment, with the producer, who was in town, and his Great Dane and pair of Siamese cats. While Page was in the Big Apple, the ever-dynamic Berns produced Barbara Lewis's 'Stop That Girl', a song written by Page and DeShannon; this mid-tempo heartbreak ballad was included on the Michigan soul songstress's _Baby, I'm Yours_ LP, released on Atlantic that year. Through Berns, Page met Ahmet Ertegun and Jerry Wexler, the head honchos of Atlantic Records; it was a connection that would prove exceptionally valuable. Berns also 'took him to an Atlantic session, where he strummed along uncredited because of the union and immigration'.\n\nThen Page flew to the West Coast. With its warm whisper of fecund possibility, Los Angeles in the mid-1960s seemed like a mythical setting. Almost all knowledge of the city was informed by its portrayal in movies, with Hollywood almost like the Holy Grail. With many of its inhabitants drawn from across the globe by the hope of stardom, LA housed some of the most beautiful individuals, of either sex, that any urban conurbation could boast.\n\nThe perfect weather of Los Angeles, its motorised modernity, gorgeous landscapes and fascination with alternative, free-thinking lifestyles \u2013 since the 1920s the city had been known for its practitioners of the more arcane, esoteric arts \u2013 made for an attractive package.\n\nBut its air of affluence could be illusory. In August 1965, when the foreign press went looking for the riots in the south Los Angeles district of Watts, many newshounds famously drove straight through the neighbourhood. Searching for a 'black ghetto', they were unable to believe that this place, with its palm trees and neat bungalows, could be the scene of murderous urban discontent.\n\nThe Watts riots were in stark contrast to the received wisdom about Los Angeles and southern California in general. But they were also a metaphor for the darkness that lay at its heart, always ready to erupt, like the city's ever-present threat of earthquake.\n\nIt was already apparent that Page had a nose for the zeitgeist, and here he was ahead of the times: for shortly Los Angeles would become the world's popular-music capital. 'I first came here in 1965 when I was a studio musician,' he told the _Los Angeles Times_ in 2014. 'Bert Berns brought me out. He invited me to stay at his place. I met Jackie DeShannon, I saw the Byrds play at Ciro's [the Byrds debuted at Ciro's on 26 March 1965], which I think is now the Comedy Store. It was a magical time to be here. It was really happening.'\n\nOne morning he walked into the coffee shop at the Hyatt House on Sunset Boulevard. Seated there, having his breakfast, was Kim Fowley, a veteran of the Hollywood music scene who had been involved with the Hollywood Argyles, B. Bumble and the Stingers, and the Rivingtons; in London, where he had met Page, Fowley had worked with P. J. Proby. 'In he comes, Mister boyish, dressed in crushed velvet. He spotted me, and came and sat down. He told me he'd just had the most insane, disturbing experience.\n\n'A well-known singer-songwriter of the time, a pretty blonde, had asked him over to her house. When he got there, she'd detained him. He said she'd used restraints. I asked if he meant handcuffs and he said yes, but also whips \u2013 for three days and nights. He said it was scary but also fun. They say there's always an incident that triggers later behaviour. I contend that this was it for Jimmy Page. Because being in control \u2013 that became his deal.'\n\nAfter a few months this early example of a rock 'n' roll couple went their separate ways. 'He wanted to split from the music world because he was getting disillusioned,' said Jackie DeShannon. 'Jimmy wanted to go to Cornwall or the Channel Islands and sell pottery. He couldn't stand the business, the strain, and I couldn't stand his dream of quietness, so we split, but I guess he's changed a lot since then.'\n\nThe song 'Tangerine' on _Led Zeppelin III_ is said to have been inspired by Jackie DeShannon.\n\nIn May 1965 Bert Berns was back in London, producing tracks for Them's first album, this time at Regent Sounds Studio on Denmark Street. Clearly uninfluenced by the protests of Billy Harrison, Berns again brought in Jimmy Page, who provided a 'vibrato flourish' on an interpretation of the Josh White folk song 'I Gave My Love a Diamond'.\n\nPrior to the arrival in October 1962 of the Beatles with 'Love Me Do', their first Parlophone single, and their subsequent phenomenon, not just as performers but supreme songwriters, British pop music was dominated by material that traditional 'Tin Pan Alley' publishers touted to acts. Despite the Liverpool quartet's success, which led to so many emergent acts writing their own material, by 1965 the Beatles had by no means overthrown this system. The Yardbirds, a group largely from the extremes of south-west London, were an act that demonstrated the severe disparity between their singles, chosen by such a method, and the material in the five-piece group's live sets \u2013 essentially another version of the harmonica-wailing, thunderously paced and mutated rhythm 'n' blues sound that the Rolling Stones and Pretty Things and other lesser UK groups like the Downliners Sect were somewhat histrionically howling.\n\nThe Yardbirds had formed after Chris Dreja, Anthony 'Top' Topham and Eric Clapton met at Surbiton Art College. 'It was through Top Topham's father,' said Chris Dreja, 'who had this amazing collection of 78s from America [that was] not available to anybody. It was black blues music, and that was the initial turn-on, of course. Discovering that music was like the genie coming out of the bottle, really. We had really rather kitsch pop music with no free fall and very little emotion back in the depressing post-war fifties and sixties.\n\n'And poor old Anthony Topham gets left out, doesn't he? He was quite pivotal, actually. The band was made up of two halves originally. One half was Top and me at art college, and Clapton was in the same art stream. In Surbiton, Surrey, of all places.\n\n'At that stage Top Topham was perhaps as agile and skilled a guitar player as Eric Clapton. He was only 16, however, and his career with the Yardbirds was stymied when his parents insisted that he must remain in full-time education.\n\n'Topham is still a great guitar player. He went on to play for Chicken Shack. Out of all us he was actually the most talented artist around. Clapton and I were all into music, but he got dropped at Kingston Art School because his attentions were elsewhere. But Top's parents, when we were getting wages from it, grounded him, unfortunately, and that is when we got Clapton. He was really the only professional player we knew out there who had any background in the music we were doing.'\n\nKeith Relf, the group's singer, knew Eric Clapton better than the pair of students who were at college with him, so he went and 'tracked him down', as Dreja remembered. Clapton had already moved on to Kingston College of Art, but had been dismissed after his first year; it was considered by his instructors that he was focused on music and not on art.\n\nBy the end of 1964 the Yardbirds had a blossoming reputation and were considered one of the coolest UK acts, clearly on the cusp of breaking out from being a cult attraction, not least because of their by now revered guitarist. The Yardbirds \u2013 who otherwise consisted of flaxen-haired vocalist Keith Relf, second guitarist Chris Dreja, bass player Paul Samwell-Smith and drummer Jim McCarty \u2013 were managed by Giorgio Gomelsky, who had had the Rolling Stones stolen from him by Andrew Loog Oldham and Eric Easton.\n\nThe song that would pull the Yardbirds up to full pop stardom was 'For Your Love'. One of the first two tunes written by Manchester's Graham Gouldman, later of 10cc, 'For Your Love' had been intended for his group the Mockingbirds, until they turned it down, as did Herman's Hermits, who in Harvey Lisberg had the same management as Gouldman. Undaunted by this negative response, Lisberg, who was very impressed by 'For Your Love', then offered it to the Beatles when they played a season of Christmas shows at the Hammersmith Odeon in 1964. Unsurprisingly, the Fab Four, who had their own abundant source of material, displayed no interest. But supporting the Beatles on these Hammersmith shows were the Yardbirds, and they recognised its chart potential and recorded it.\n\nA good call: the single, released in March 1965, was a big hit. Yet 'For Your Love' was at considerable odds with the rest of the band's previous material. The Yardbirds had already put out a pair of what might be considered more characteristic tunes: 'I Wish You Would', a version of the 1955 Billy Boy Arnold Chicago blues tune; and 'Good Morning, School Girl', an adaptation of the 1937 Sonny Boy Williamson song, often titled 'Good Morning Little Schoolgirl' \u2013 a title that in later years would have guaranteed zero radio airtime. On the live version of 'Good Morning, School Girl', on the Yardbirds' first live album, _Five Live Yardbirds_ , the vocal duties on the song were taken by bass player Paul Samwell-Smith and Eric Clapton rather than singer Keith Relf.\n\nHe might have been underestimated in his days at the Ealing Jazz Club, but by now Clapton was showing that he was very much his own man, utterly singular in his purist vision of the kind of music he should be playing: he was determined that the next Yardbirds single should be an Otis Redding cover. His stance, and clear supreme abilities on the guitar, were beginning to transform him into a hero for his fans. And in March 1965 the _Melody Maker_ headline told the story: Clapton Quits Yardbirds \u2013 'too commercial'.\n\n'I thought it was a bit silly, really,' said Clapton of 'For Your Love'. 'I thought it would be good for a group like Hedgehoppers Anonymous. It didn't make any sense in terms of what we were supposed to be playing. I thought, \"This is the thin end of the wedge.\"'\n\nIn the story that accompanied the _Melody Maker_ headline, Keith Relf gave his version of what had taken place. 'It's very sad because we are all friends. There is no bad feeling at all, but Eric did not get on well with the business. He does not like commercialisation. He loves the blues so much I suppose he does not like it being played badly by a white shower like us! Eric did not like our new record \"For Your Love\". He should have been featured but did not want to sing or anything, and he only did a boogie bit in the middle. His leaving is bound to be a blow to the group's image at first because Eric was very popular.'\n\nChris Dreja put the problem more succinctly: 'We had this massive record and we had no lead guitar player.'\n\nWithin two weeks Clapton had joined John Mayall's Bluesbreakers. After a couple of months graffiti began to appear around London: 'Clapton is God'. John Mayall, a bohemian Mancunian who rivalled Alexis Korner as the godfather of the UK blues scene, had already offered Page the job with the Bluesbreakers, but he had turned it down, clearing the way for Clapton.\n\nPage was clearly in demand. The Yardbirds and their manager Giorgio Gomelsky, at the suggestion of Eric Clapton himself, first approached Page to be the guitarist's replacement.\n\n'It was thought,' remembered Jim McCarty, 'that maybe we could get Jimmy Page because Jimmy was the hottest session player, and Giorgio knew Jimmy. He asked Jimmy if he'd join the band but at that time Jimmy was so busy playing sessions that he wasn't into joining a live band. He said why don't you try one of my understudies, a guy called Jeff Beck. So we went down to see Jeff and asked him to join the band.'\n\nPage's friend Jeff Beck was playing with the Tridents, a rocking blues group he had joined in August 1964, never missing their weekly residence at Eel Pie Island, which would draw up to 1,500 people. Beck accepted the offer.\n\nPage was still shadowed by the ill health that had dogged him during his time with Neil Christian and the Crusaders, and was also aware of the large amounts of money he continued to earn as a session player. But his main reason for turning down the offer, he said, was because of his growing friendship with Clapton. 'If I hadn't known Eric, or hadn't liked him, I might have joined. As it was, I didn't want any part of it. I liked Eric quite a bit and I didn't want him to think I'd gone behind his back.'\n\n(This was not the only act of generosity that Page displayed towards his friend Jeff Beck. When, in 1962, he announced he would be leaving Neil Christian and the Crusaders, Page had suggested Beck as a replacement for himself.)\n\nJeff Beck played his first show with the Yardbirds on Friday 5 March 1965 at Fairfield Halls in Croydon, south London, only two days after Eric Clapton quit the group. They were second on the bill to the Moody Blues, flying high for the first time with their hit 'Go Now'.\n\nBeck was so grateful that Page had recommended him as Clapton's replacement that he went round to Page's parents' house in Epsom and presented his friend with his 1959 Fender Telecaster. 'A beautiful gesture,' said Page later. But Beck's gratitude was realistic: although distinguished in his blues guitar band the Tridents, it was not until he joined the Yardbirds, whom he enriched with his fuzz-soaked solos, that he found the vessel for his upward flight into the guitar stratosphere. In fact, he soon borrowed a quite specific vehicle from Page: his Roger Mayer fuzzbox, on which Beck worked out the Eastern-flavoured riff for 'Heart Full of Soul', the first Yardbirds' single he was involved in. 'I still remember the time Jeff came over to my house when he was in the Yardbirds and played me \"Shapes of Things\". It was just so good \u2013 so out there and ahead of its time. And I seem to have the same reaction whenever I hear anything he does,' recalled Page.\n\n'The great thing about Jeff,' said Chris Dreja, 'is that his roots were also the blues and rock 'n' roll, but he was also much wider in his musical tastes. And he had a mind and a talent that wanted to go much further than playing rock 'n' roll and blues riffs, which was perfect for us because we were about to enter a phase of all sorts of experimentation. In retrospect we put Jeff under a lot of pressure. We would work on stuff and then bring Jeff in. Like, for example, the sitar sound he got on \"Heart Full of Soul\". We brought in a sitar player... but it sounded thin and weedy. We said to Jeff, \"Can you do it?\" And he came in and created this incredible sound. Jeff Beck became a prototype of late-sixties psychedelia. He got chords from Stax and Motown records. The locked-up sound in the band gave it that sound.'\n\n## 4\n\n# BECK'S BOLERO\n\nIt was in 'the latter years of the first half of the 1960s' that Jeff Dexter first encountered Jimmy Page. Dexter was a Mod scenemaker, the DJ at Tiles who, as a 15-year-old, famously demonstrated the Twist on BBC television; he had hung around the 2i's Coffee Bar in London's Old Compton Street, the seed-ground of British rock 'n' roll, where Cliff Richard, among others, had been discovered, and would be instrumental in founding the Middle Earth venue, out of which sprang UK underground rock. 'Jimmy was running around town at that time. But we really became friends about 1965 or 66. We both had an eye for a nice suit and a nice shirt.'\n\nPage and Dexter would meet up in Soho to rummage through the wares of the area's assorted rag-trade specialists, sometimes seeking cloth for jackets and trousers, more often looking for potential shirt fabric. An especial temple of suitably exotic material was found at Liberty, the Tudor-fronted department store on Great Marlborough Street. Armed with their cloth, they would then make their way to Star Shirtmakers on Wardour Street, two doors from the Whisky A Go Go. Star Shirtmakers was run by a Hungarian husband-and-wife tailoring team; they would knock up the fabric into shirts styled precisely as their customers desired, for \u2013 even then \u2013 the ludicrously cheap price of 11 shillings, the equivalent of 55 pence. (After the Beatles had been to their tailor, Dougie Millings, Dexter took them to nearby Star Shirtmakers, beginning a flood of celebrity shoppers at the place.)\n\nOne day, when they were leaving Liberty, Page and Dexter found themselves strolling down Kingly Street, which ran off the west end of the store. There they discovered an art gallery called 26 Kingly Street, with extraordinary lighting, sheets of Perspex and glittery screens. London's first psychedelic gallery, 26 Kingly Street was run by Keith Albarn (whose son Damon is the singer in Blur). 'We'd just discovered acid,' said Dexter. 'I tripped in Jimmy's house but never tripped with him \u2013 I sought refuge with him a few times. I went to a Yardbirds rehearsal when I'd dropped acid. And he looked after me.'\n\nContrary to Billy Harrison's dismissal of Page, Dexter insists that his friend was highly regarded on the London music scene, not simply for his musical accomplishments but as an empathetic human being. 'He was a lovely chap. One of the boys. You'd see him at record launches, and the odd club \u2013 though he didn't go to the Speakeasy as much as many others \u2013 and stuff like that. When I was running the Implosion shows Jimmy would come along. Ian Knight, my cohort on those events, went from Middle Earth to becoming the Yardbirds' staging and lighting guy, and went on to have the same job with Led Zeppelin. We hung out at some of those crazy happenings, like the 14 Hour Technicolor Dream at Alexandra Palace.'\n\nPage had moved up from Epsom and was living in a flat off Holland Road in west London, a thoroughfare that ran from Shepherd's Bush to Kensington High Street; it was an area in which it seemed every single one of its myriad bedsits contained a hippie hash-dealer. In 1966 Dexter was invited to Dei\u00e0 in Majorca, home to a bohemian community, by Lady June, an artist and _\u00e9minence_ _grise_ of the psychedelic scene. In Dei\u00e0 he encountered an especially louche breed of Portobello Road-style hippie chick, some of whom relocated back to London, becoming habitu\u00e9s of Blaises, a nightclub in South Kensington: 'Jimmy and one of these dodgy birds used to get really stoned and play Buffalo Springfield again and again and again. \"This is the direction I want to go in,\" he would say. \"I want to have a band that does magical things.\"'\n\nThe folk scene, to which Page was always drawn, remained a prominent feature of Swinging London. 'I used to go to Les Cousins,' Dexter said of London's dominant folk venue. 'I was best friends with Beverley and John Martyn. Nick Drake only felt comfortable at their flat in Hampstead.'\n\nDexter was always impressed with Page's phenomenal knowledge of art: 'He was a collector. Of everything. He's kept every piece of clothing he's had since he was a child. His mother was incredibly neat and tidy. And so is he.'\n\nDexter also became friends with another woman who would have a significant impact on Page: a French model called Charlotte Martin. She was 20 when they first met, Dexter 19. 'I first clapped eyes on her in a place called Westaway and Westaway, a fantastic shop near the British Museum that sold Scottish knitwear. All the young birds would gather there or at the Scotch House. She was a fabulous model who did it all: magazine level, and then once people saw how gorgeous she was she was employed all over the place. She did all the modelling with the Fool, for their collection. She was great friends with them because they all hung out at Eric's place in the Pheasantry.' 'Eric', of course, was Eric Clapton, and he and Charlotte Martin were an item.\n\nBy now Page had effectively dropped out of art college. Even though he would later acquire a considerable reputation for financial canniness, it is somewhat cheap to suggest that it was only his considerable earnings from session playing that continued to attract him to the craft. In fact, for him the art of recording, and coming to as full an understanding of it as possible, appears to have held far more attraction than treading the rock 'n' roll boards. And in the most select quarters his skills were being further recognised. In August 1965 came the press announcement about the formation of Immediate Records, an independent label that was the pet project of Andrew Loog Oldham, Rolling Stones manager and wunderkind of UK pop, and his business partner Tony Calder. 'Immediate will operate in the same way as any good, small independent label in America,' said Oldham. 'We will be bringing in new producers, while our main hope lies with the pop session guitarist turned producer Jimmy Page and my two friends, Stones Mick and Keith.'\n\nPage had first worked with Andrew Loog Oldham in 1964, on one of Oldham's versions of the Stones' songs, performed by the Andrew Oldham Orchestra, part of his endeavour to become the UK's Phil Spector. That session was at Kingsway Studios in Holborn, London; the producer was John 'Paul Jones' Baldwin.\n\nPage then went on the road with Marianne Faithfull, who that summer had hit the Top 10 with her first release, 'As Tears Go By', written by Mick Jagger and Keith Richards, and on which Page had played.\n\nPage had been recommended to Oldham by Charlie Katz, who booked musicians for his sessions. 'He said to me one day, \"There's this young lad, Jimmy, we are trying him out. Why don't you give him a go? He doesn't read but Big Jim Sullivan will take him under his wing.\" And so Jimmy started playing on my sessions,' said Loog Oldham. 'One of the first was Marianne Faithfull's \"As Tears Go By\". He was a bright spark. It was nice having him on the floor... All smiles and not much talk.'\n\nSoon Page found himself playing on the Rolling Stones' 'Heart of Stone', though it was a version that would not be released until the Stones' _Metamorphosis_ album, in 1975.\n\n'Jimmy was like a wisp,' said Loog Oldham. 'I don't really know what kind of a person he was, because the great ones keep it hidden and metamorphose on us, so that the room works.'\n\nAndrew Loog Oldham decided to take their relationship up a level, hiring Page as Immediate's producer and A&R man. 'In those days if you got on with people you tried to work with them. It seemed logical and Jimmy liked the idea... I thought he was very good. What he went on to do kind of proves it, doesn't it?'\n\nAs for sessions with the Rolling Stones, Loog Oldham recalled: 'He played on some of the demos Mick, Keith and I did that ended up on the album released in 1975 called _Metamorphosis_. The Stones did not play on that. I think he was on a Bobby Jameson single that Keith and I wrote and produced... I only considered people the way they considered themselves. Jimmy was a player, an occasional writer at that time with me and with Jackie DeShannon. I never considered him as a solo artist and I don't think he did either.'\n\nPage worked on a trio of demos for the Stones themselves: 'Blue Turns to Grey', 'Some Things Just Stick in Your Mind' and the aforementioned 'Heart of Stone'. Although the version of 'Blue Turns to Grey' on which Page played was never released, a later edition of the song was included on the Stones' 1965 US album _December's Children (And Everybody's)_ , and Cliff Richard's cover of the song was a number 15 hit in 1966. 'Some Things Just Stick in Your Mind' and 'Heart of Stone' were included on _Metamorphosis_ , the first song being covered by Vashti Bunyan for an unsuccessful release on Immediate.\n\nWhat had specifically drawn Page to the Immediate production gig was the chance to work with his old mucker Eric Clapton, now with John Mayall and the Bluesbreakers, who had formed an arrangement with the label.\n\nIn June 1965 John Mayall and the Bluesbreakers went into Pye Studios at Great Cumberland Place in London's West End. Page was at the production helm for what would turn out to be a landmark session in the history of contemporary music.\n\n'I'm Your Witchdoctor' and 'Telephone Blues' were the tunes involved. They featured John Mayall on keyboards, Hughie Flint on drums, John McVie on bass and Eric Clapton on guitar \u2013 the John Mayall's Bluesbreakers line-up that had recorded the celebrated _Beano_ -cover album. 'When \"Witchdoctor\" came to be overdubbed, Eric had this idea to put this feedback wail over the top,' said Page. 'I was with him in the studio as he set this up, then I got back into the control room and told the engineer to record the overdub. About two thirds of the way through he pulled the faders down and said: \"This guitarist is impossible to record.\" I guess his technical ethics were compromised by the signal that was putting the meters into the red. I suggested that he got on with his job and leave that decision to me! Eric's solo on \"Telephone Blues\" was just superb.'\n\nIt was Page who intuited how Clapton's solos could be enhanced by pouring reverb onto them, bringing out the flames in his playing, characterised by Clapton's overdriven one-note sustain.\n\nBut \u2013 as Page noted \u2013 Clapton's plangent, lyrical playing on 'Telephone Blues', the B-side, is perhaps even more distinguished, the first time that he gets to really stretch out with a beautiful, mature stream of notes. You are struck by the clarity of the separation \u2013 and simultaneous harmony \u2013 of the instruments. Clearly Page had learned much from his countless hours in recording studios, learning to appreciate how the very best rock 'n' roll records were assiduously constructed, put together piece by piece.\n\nTellingly, for his first go in the control seat for Immediate, the subject matter of the single's title track alluded to the kind of dark material with which Page would later be associated, perhaps even tarnished by. The opening couplet ran:\n\n'I'm your witchdoctor, got the evil eye\n\nGot the power of the devil, I'm the conjurer guy.'\n\nOn one hand this was no more than the stock imagery that peppered blues music; yet, in the bigger picture, it holds an interesting subtext. It was as though Page was toying with \u2013 giving a test run to, really \u2013 the entire mysterious and dark philosophy that would form the aura of Led Zeppelin.\n\n'The significance of this session cannot be emphasised enough, for it represented the birth of the modern guitar sound. And while Clapton did the playing, it was Page who made it possible for his work to be captured properly on tape,' wrote Brad Tolinski.\n\nThat year Page also worked with the distinguished American composer Burt Bacharach on his album _Hit Maker! Burt Bacharach Plays the Burt Bacharach Hits_. 'Page respected Bacharach's meticulous approach to rehearsing and recording,' wrote George Case in _Jimmy Page: Magus, Musician, Man_. Again, it was part of Page's learning curve. 'Bacharach, in turn, admired the young Briton's politeness and polish.'\n\nAs part of his deal with Immediate, Page played guitar with Nico, a German actress, model and singer based in France whom Andrew Loog Oldham had met in London, where she was soaking up the scene. Loog Oldham and Page co-wrote a song for her, 'The Last Mile', and Page arranged, conducted, produced and played on the tune. It was relegated to the role of B-side, however, to the Gordon Lightfoot number 'I'm Not Saying' \u2013 again, Page played guitar on this track.\n\n'Brian Jones brought Nico to my attention,' said Loog Oldham, 'and Jimmy and I wrote a song, which we recorded with her as a B-side. It might have been better than the A-side. It should have been the A-side, because that was fucking awful. It really was stiff as Britain. Then he went on the road with Marianne Faithfull. We were all impressed by this new wave of women who were coming in.'\n\nPage's friendship with Eric Clapton continued to blossom, and soon Slowhand, as Clapton ironically became nicknamed, would often be accompanied by his beautiful French girlfriend, Charlotte 'Charly' Martin, who was friends with Jeff Dexter. Clapton met her in the Speakeasy nightclub in the summer of 1966, while he was forming his next group, Cream.\n\nProblems with Immediate Records, however, almost created a rupture in the camaraderie between the two guitarists. Without informing Clapton, the label released some tunes that he had recorded onto Page's Simon tape recorder when Clapton had stayed at his house \u2013 which led to Clapton mistrusting Page for a time. Yet this suspicion was misplaced. 'I argued that they couldn't put them out, because they were just variations of blues structures, and in the end we dubbed some other instruments over some of them and they came out, with liner notes attributed to me... though I didn't have anything to do with writing them. I didn't get a penny out of it, anyway,' Page said, revealing what was for him generally a key subtext to any endeavour. (The musicians who overdubbed these instruments onto Clapton's basic tracks were Bill Wyman, Charlie Watts and Mick Jagger, on harmonica.)\n\nBorn in 1939 in west London, Simon Napier-Bell was the son of a documentary filmmaker. After attempting to become a jazz musician in the United States, he drifted into music supervision for movies in Canada; eventually he returned to London, where he continued in the same line of work, including on the 1965 screwball comedy _What's New Pussycat?_ He expanded into the production of records and demos, and he would use popular London studios such as Advision on Bond Street and Cine-Tele Sounds Studios, popularly known as CTS, in Kensington Gardens Square, the top film-music studio in London. He would employ session musicians recommended by Dick Katz, who booked all the top players in London.\n\nHighly intelligent and witty, Napier-Bell became something of an archetypal character of Swinging London, a gay man who was known for driving around in an imported Ford Thunderbird, a cigar clenched between his teeth. His best friend was Vicki Wickham, the producer of _Ready Steady Go!_ , the hip pop music show broadcast every Friday night on ITV. Almost as a jape, he and Wickham co-wrote the English lyrics for the Italian ballad 'Io Che Non Vivo (Senza Te)', which had been featured at the 1965 San Remo Festival; their friend Dusty Springfield sang at the event and had been moved to tears by the song's music and melody. Knocking up their set of English lyrics to match the music in an hour so that they could head out to a London nightclub, Wickham and Napier-Bell gave the tune its title, 'You Don't Have to Say You Love Me'. Recorded by Dusty Springfield, the song was a number one hit in the UK and number four in the United States; in subsequent years it would be a hit again many times over, across the globe, with even Elvis Presley doing a version of it in 1970.\n\nBy the time Napier-Bell wrote the 'You Don't Have to Say You Love Me' lyrics, he was manager of the Yardbirds, having replaced Giorgio Gomelsky. That the talented and fascinating Gomelsky had been fired was perhaps not surprising; later he declared, 'I should never have been a manager: I need someone to manage me.' Though there were no suggestions of impropriety, the Yardbirds had dismissed him because of his inability to turn a profit for the group. All the same, Gomelsky had been an inspirational figure for the Yardbirds, under whose auspices they had become a hit recording act. During their first US tour in 1965 he had even secured a recording session at Sun Studio in Memphis with Sam Phillips, who had mentored Elvis Presley early in his career. The tune they recorded? The Tiny Bradshaw 1951 jump-blues classic 'Train Kept A-Rollin'', reworked in a rockabilly style by the Johnny Burnette Trio in 1956, and included on the Yardbirds' US album release _Having a Rave Up_. 'Train Kept A-Rollin'' was a song that would replay significantly in the Yardbirds' career.\n\n'Some time around the end of 1965 or the beginning of 1966,' recalled Napier-Bell, 'Paul Samwell-Smith, who played bass with the Yardbirds, called me. His girlfriend, later his wife, was Vicki Wickham's secretary. I went to a gig the Yardbirds played in Paris. I quickly realised that a manager's job was to keep the group together.'\n\nBehind Napier-Bell's management of the Yardbirds lay a continuous awkward subtext: 'The Yardbirds were blokes in a pub talking about football. I was gay and couldn't really enter into that world.'\n\nDuring his time working in recording studios Napier-Bell had always employed session musicians. 'You never think session players aren't playing well: they know they are in the top league, the best in the world. They can play next to the guys in LA who would play with Sinatra.'\n\nNapier-Bell's first choice for guitarist was 'always Big Jim Sullivan'. Even though, he says, 'these guys were all infuriating. They'd put you through it. Big Jim Sullivan would always have a paperback book with him that he would read as you did a take: it would be balanced on his music stand. He would even read it halfway through the take until it came to his moment \u2013 he would be doing it to show off.'\n\nIf an additional guitarist was required, it would invariably be Page. 'He and Big Jim would work out their parts between them. I talked to Jimmy Page enough to know he was a real session player. I knew he was a brilliant technician and admired by others. We'd also use John Paul Jones, who did all the arrangements for Herman's Hermits. But I never really liked Jimmy Page. He had a sneer about him. At school the people who bullied me had this terrible, frightening sneer and Jimmy Page reminded me of those people. People who sneer have usually had unhappy childhoods.'\n\nOn 16 and 17 May 1966, at IBC Studios in London's West End, Jeff Beck and Page were involved in what in retrospect can be seen as one of the very first super-sessions.\n\nThe tune was 'Beck's Bolero'. Maurice Ravel's _Bol\u00e9ro_ , which was first performed in 1928 at the Paris Opera, provided the basis for 'Beck's Bolero'; the Russian ballet dancer Ida Rubinstein had commissioned Ravel to write the work, an undulating, insistently repetitive piece based around the Spanish music and dance known as bolero.\n\nBy 1965, largely influenced by the tastes of the likes of Paul McCartney, always an assiduous culture vulture, assorted classical composers had become fashionable among fans of what formerly would have been known as 'pop'. These composers included Bach, Sibelius, Stravinsky, Prokofiev, Gershwin, Debussy and Ravel, whose _Bol\u00e9ro_ was relatively well known in 1966. The song's structure is considerably amended in such a way that it could be interpreted as the first blow of the hard-rock sound that Led Zeppelin would very soon develop.\n\n'Beck's Bolero' employed a formidable cast: Beck on lead guitar, Page on acoustic, revered session pianist Nicky Hopkins, the Who's Keith Moon on drums and John Paul Jones on bass. The Who's John Entwistle, Moon's bass-playing rhythm partner, had originally agreed to do the session, but when he failed to turn up John Paul Jones was called in.\n\n'I heard rumours that Jimmy was talking with Keith Moon about joining his supergroup,' said Napier-Bell. 'I don't think the name Led Zeppelin was in the air at that time, though it may have been mentioned between them. Cream was being formed at the same time. Whether that had much influence on Beck, Page and Moon, I don't know. The Who's managers, Kit Lambert and Chris Stamp, were in the same building as Clapton's manager, Robert Stigwood. So when he was putting Cream together, they would have known all about it, as I did too. Keith Moon would have heard from Kit and Chris as to what was going on too. From my point of view, I was thinking only of keeping Jeff in the group [the Yardbirds]. Jimmy, I think, was thinking of a new group, which would be a blend of all their talents.'\n\n'I always try to do things wholeheartedly or not at all,' said Beck, offering a slight rewrite of history, 'so I tried to imagine what my ideal band would be. We had the right producer, Keith Moon on drums, Jimmy on guitar and John Paul Jones on bass. You could feel the excitement in the studio, even though we didn't know what we were going to play. I thought, \"This is it! What a line-up!\" But afterwards nothing really happened because Moony couldn't leave the Who. He arrived at the studio in disguise so no one would know he was playing with another band.'\n\n'Jim Page and I arranged a session with Keith Moon in secret, just to see what would happen,' said Beck. 'But we had to have something to play in the studio because Keith only had a limited time \u2013 he could only give us like three hours before his roadies would start looking for him. I went over to Jim's house a few days before the session and he was strumming away on this 12-string Fender electric that had a really big sound. It was the sound of that Fender 12-string that really inspired the melody. And I don't care what he says, I invented that melody. He hit these Amaj7 chords and the Em7 chords, and I just started playing over the top of it... He was playing the bolero rhythm and I played the melody on top of it, but then I said: \"Jim, you've got to break away from the bolero beat \u2013 you can't go on like that for ever!\" So we stopped it dead in the middle of the song \u2013 like the Yardbirds would do on \"For Your Love\" \u2013 then we stuck that riff into the middle. And I went home and worked out the other bit [the uptempo section].'\n\n'Even though he said he wrote it, I wrote it,' said Page, presenting an argument that would become somewhat familiar.\n\n'Moon did this amazing fill around the kit, and a U47 mic just left its stand and went flying across the room; he just cracked it one,' said John Paul Jones.\n\n'I remember Jimmy at the studio yelling at us and calling us fucking hooligans,' said Beck. 'Everyone had prior commitments. That session that day, it was one day that really started my head turning \u2013 we were almost doing it.'\n\nThat band, claimed Beck, was the original Led Zeppelin \u2013 'not called \"Led Zeppelin\", but that was still the earliest embryo of the band'.\n\n'It was going to be me and Beck on guitars, Moon on drums, maybe Nicky Hopkins on piano. The only one from the session who wasn't going to be in it was Jonesy, who played bass,' said Page. 'It would have been the first of all those bands, sort of like Cream and everything. Instead, it didn't happen \u2013 apart from the \"Bolero\". That's the closest it got... The idea sort of fell apart. We just said, \"Let's forget about the whole thing, quick.\" Instead of being more positive about it and looking for another singer, we just let it slip by. Then the Who began a tour, the Yardbirds began a tour, and that was it.'\n\nIn fact, there had been some efforts by Page and Beck to find an appropriate vocalist to transmogrify the 'Beck's Bolero' studio line-up into a working outfit, as Page told _Guitar World_ 's Steve Rosen: 'Well, it was going to be either Steve Marriott from the Small Faces or Steve Winwood.' Marriott was managed by Don Arden, the self-styled 'Al Capone of pop'. 'In the end, the reply came back from his office: \"How would you like to have a group with no fingers, boys?\" Or words to that effect.' Sufficiently warned off, the pair never even approached the Spencer Davis Group's Steve Winwood.\n\nThere was even controversy over the 'Beck's Bolero' production credit. Mickie Most claimed it, part of a contractual issue between him and Beck, his managerial client. Simon Napier-Bell would insist it was his, and Jimmy Page claimed that he had done the record's production, staying behind in the studio long after Napier-Bell had gone home.\n\n'The track was done and then the producer just disappeared,' Page told Steve Rosen in September 1977. 'He was never seen again: he simply didn't come back. Napier-Bell, he just sort of left me and Jeff to it. Jeff was playing and I was in the box [studio]. And even though he says he wrote it, I'm playing the electric 12-string on it. Beck's doing the slide bits, and I'm basically playing around the chords.'\n\nSimon Napier-Bell has a different point of view. 'Jimmy Page was being demeaning when we were making the record: he was sneering. Later, when Beck and Page were discussing how the mix should be, I went away to leave them to it. The purpose of a producer is so that the record ends up as it should. That's why I went away \u2013 to leave them to it. As for Mickie Most, my agreement with him over the management of the Yardbirds was that all product reverted to him. I just said, \"What the hell, I don't need it.\" I didn't really \u2013 but that track became a rock milestone.'\n\nWhen Pete Townshend discovered that Keith Moon had played on the session, he was furious. He began to refer to Beck and Page as 'flashy little guitarists of very little brain'. Page's response? 'Townshend got into feedback because he couldn't play single notes.' Townshend later commented: 'The thing is, when Keith did \"Beck's Bolero\", that wasn't just a session, it was a political move. It was at a point when the group was very close to breaking up. Keith was very paranoid and going through a heavy pills thing. He wanted to make the group plead for him because he'd joined Beck.'\n\nLater, it was claimed that Moon had declared that if the studio line-up became an actual band, it would go down 'like a lead balloon'. According to Peter Grant, Entwistle then added, 'like a lead zeppelin'. (Entwistle was always adamant that he came up with the Lead Zeppelin name all on his own; and also that he had the idea of a flaming Zeppelin as an album cover.)\n\nWhen writing his Keith Moon biography _Dear Boy_ , Tony Fletcher interviewed Jeff Beck about the 'Beck's Bolero' session. Fletcher asked Beck if Moon had been using him to pressure the Who for his own ends. Beck replied that that wasn't the case at all: the subtext to the 'Beck's Bolero' session was the relationship between Jimmy Page and himself: 'No, it was something to do with Jimmy and me. I had done sessions for Jimmy. He used to get me to do all the shit he didn't want to do. He used to get me to pick him up in my car and pay for the petrol, and I'd find out he was on the session anyway. When he heard what I was doing on the sessions... he started taking an interest in my style, and then we went from there. And then the Yardbirds got in the way \u2013 I can't remember the sequence of events. I remember thinking, \"Why can't I have what I want instead of what I've got?\" That's always the way. To have someone who's so musically aware and talented as Jim alongside me was something I could really do with. But that wasn't to be until later on in the Yardbirds. Meanwhile, I'm watching the Who going from strength to strength with a fantastic powerful drummer and knowing that that was what I really needed to get myself going. So it was a guiding light in one sense and fragmented what I already had. I was never content being in the Yardbirds, and I left Jim to paddle his canoe in the Yardbirds.'\n\nThe Yardbirds' rhythm-guitar player Chris Dreja was yet another denizen of the Surrey Deep South from which Page, Clapton and Beck hailed. Brought up in Surbiton, where he continued to live, he would from time to time run into Page while the guitarist was studying at Sutton Art College. On more than one occasion he came across him in nearby Tolworth, outside the tropical-fish shop: Page was a tropical-fish enthusiast. 'Hello, Chris, I've just bought a nice thermometer for my fish,' he once greeted him.\n\nOn 18 June 1966 Page travelled up to Oxford in the passenger seat of Jeff Beck's maroon Ford Zephyr Six to watch his friend play with Chris Dreja and the other Yardbirds at the May Ball at Queen's College. They were on the same bill as seasoned Manchester hitmakers the Hollies. Not that that 'seasoned hitmaker' rubric couldn't also have been applied to the Yardbirds. Since Beck had joined the band it seemed like they were never out of the UK charts \u2013 and increasingly the US hit lists were also welcoming the group's 45s: 'Heart Full of Soul', 'Evil Hearted You', 'Shapes of Things' and 'Over Under Sideways Down' had all been hit singles. Meanwhile, their critically acclaimed _Yardbirds_ album \u2013 more generally known as _Roger the Engineer_ \u2013 was about to be released in the middle of July 1966.\n\nAlthough events such as the May Ball paid well, they were formal, black-tie occasions, in the main quite uptight and, as defined in the new underground vocabulary, extremely 'straight'. Becoming increasingly drunk as the evening progressed, Keith Relf, the Yardbirds' singer, took exception to this prevalent social posture, and he began to harangue and berate his audience of bright young things. It was a stance that Page, notwithstanding his fondness for tropical fish, always a rebel and in tune with the more esoteric minutiae of pop culture, admired greatly: he thought Relf put on 'a magnificent rock 'n' roll performance'. Paul Samwell-Smith, the Yardbirds' bass player, was so appalled by Relf's stance, however, that as soon as their show was over he quit.\n\nWith further dates coming up, the Yardbirds were concerned about how they could play them without a bass player. On the spot, Page volunteered his services. 'They had a show at the Marquee Club, and Paul was not coming back. So I foolishly said, \"Yeah, I'll play bass.\" Jim McCarty says I was so desperate to get out of the studio that I'd have played drums.'\n\nFor some time Page had harboured doubts about whether he could continue working as a session player. 'My session work was invaluable. At one point I was playing at least three sessions a day, six days a week. And I rarely ever knew in advance what I was going to be playing. But I learned things even on my worst sessions \u2013 and believe me, I played on some horrendous things. I finally called it quits after I started getting calls to do Muzak. I decided I couldn't live that life any more; it was getting too silly.'\n\n'I remember the May Ball,' said Beck. 'Jimmy Page actually came to that gig. He came to see the band and I told him things were not running very smoothly. There were these hello-yah Princess Di types around. Trays with drinks with sticks. And as soon as we started Keith fell over back into Jim's drums. After, I said, \"Oh sorry, Jim, I suppose you're not interested in joining the band?\" He said it was the best thing ever when Keith fell back into the drums. It wasn't going to put him off that easily.'\n\nAnd so, slightly oddly, Jimmy Page joined the Yardbirds. That Marquee show took place three days later, on 21 June. How was Page's performance? Terrible, according to Beck. 'Absolute disaster. He couldn't play the bass for toffee. He was running all over the neck. Four fat strings instead of six thin ones.'\n\nWhatever. As soon as Page became a part of the group, he changed his look entirely: he presented himself as a highly stylised, very chic rock star, someone of ineffable good taste and class. And very quickly, with his characteristic diligence, he became adept on the bass guitar.\n\nBefore Page formally joined the Yardbirds, he went round to Simon Napier-Bell's flat in Bressenden Place, near Buckingham Palace, for a meeting with the group and their manager. 'When he arrived,' said Napier-Bell, 'he had an enormous swollen lip. Nobody knew who'd done it. He said some people had stopped him in the street and hit him. I remember thinking that if you're Jimmy Page that could happen to you because of your sneering. Jimmy's superciliousness was hard to take. When Jimmy Page looked as nice as he does, maybe he thought he could get away with it.\n\n'He came into the group. I said, \"We don't really get on.\" \"You're my manager: I want to see the contract,\" he said. I said, \"You won't. I'll take my percentage off four-fifths of the money, and I won't manage you.\" Because I knew he would want to pull a stunt and say the contract was terrible.\n\n'I always thought Jimmy Page was partially gay. He didn't have a great childhood: because he was such a cunt, you knew he didn't have a great childhood. And later he got into transvestism. Which meant he thought he was straight.\n\n'I said to Jeff Beck, \"Jimmy Page is coming in to the Yardbirds and you will leave.\" He said, \"No, I won't.\"'\n\nAlthough 'Jeffman' had been proprietorially spray-painted onto the rear of the Telecaster Beck had gifted him, Page customised the instrument, giving it a psychedelic colour-wash, adding a silver plate to catch stage lighting and reflect it back at the audience, a simple but extraordinarily effective trick. For some time this Telecaster became identified with Page.\n\nYet before he could graduate to playing this guitar with the Yardbirds, Page remained the group's bass player. It must have been a baptism of fire: vocalist Keith Relf, an asthmatic, would drink all day, the singer's inner turmoil perhaps exacerbated by the fact that, oddly, the Yardbirds' tour manager was his own father. Between the Marquee show and the end of July, the Yardbirds played 24 dates, all over the UK. For Page it must have been like getting back on the gruelling road with Neil Christian and the Crusaders. There was a show with the Small Faces in Paris on 27 June, and a set of dates in Scotland early in July; at one of these Beck and Page were spat at for wearing the German Iron Cross. Couldn't these former art students have explained that they were simply indulging in a spot of street-level conceptual art? (Or perhaps it was an indication of chronic immaturity? A decade later comparable attempts at shock would be made by the likes of punk stars Sid Vicious and Siouxsie Sioux, similarly employing Nazi regalia. As with the Yardbirds' efforts, wouldn't you consider this to be comparable to naughty ten-year-olds drawing such images on their school exercise books?)\n\nAlthough he had his flat off Holland Road in west London, Page was still frequently staying at his parents' house in Epsom. But in the mid-sixties his parents separated and then divorced. This brought great pain to their son. Moreover, shockingly, his father had been living a double life: he had created a separate family with another woman. This would have been devastating news. 'You would never trust anyone again, especially intimate people,' commented Nanette Greenblatt, a renowned London life coach. 'In any relationship you were in, you would be worrying, \"This is okay for now, but how will it turn out?\" Accordingly, you would want to control people. There would be a strong distrust of male figures. And Aleister Crowley would fit in nicely as an unreliable father figure.'\n\nThe kind of trauma to which Page and his mother were subjected by this egregious information about his father must have been almost overwhelming. But, just as a new birth is said to bring good luck, so a figurative death in the shape of divorce can sometimes offer the opportunity for a phoenix-like rise to escape the grief of the event. Something like that happened to Page. Some constraints fell away, to expose a desire for the ultimate promulgation of who he could be, and how far he could take it. You can have this fantasy image of yourself, which, if you work hard enough at it, is what you become. In other words, find your true will: who you are meant to be in this existence and what you are here for \u2013 the meaning of Crowley's 'Do what thou wilt', appropriated almost predictably from a freemasonry text. Once upon a time, 'Jimmy Page' was a construct in Page's own mind. But because he meant it, and, more importantly, needed it, it became him, and he it. With some assistance from his beliefs, of course.\n\n## 5\n\n# BLOW-UPS\n\nJimmy Page finally moved away from his childhood home. First, he took the flat off Holland Road in Kensington. But part of the mood of the age was the need to connect communally with the essence of the earth, a need that would later be expressed by the likes of the Band, who creatively isolated themselves in Woodstock in upstate New York, and the English group Traffic, fronted by Page's friend Steve Winwood, who wanted to 'get it together' in a cottage in rural Berkshire.\n\nAnd it was to that same verdant county of Berkshire that Page now moved. Kenneth Grahame, author of the children's classic _The Wind in the Willows_ , had retired to Pangbourne, situated on the River Pang four miles west of Reading, in later life, and now Page made the village his home, buying a former boathouse on the river, water frequently being close to the homes he would purchase, solace for his Scorpio rising. Later, he would develop a reputation for being to some extent a recluse; it was here that such an existence was first nurtured, one he found creatively beneficial. 'I really enjoyed that bachelor existence \u2013 working and creating music, and going out on my boat at night on my own; switching off the engine and just coasting in the twilight. I liked all that,' he told the _Sunday Times_. His tank of tropical fish survived the journey from west London, although his long absences away on the road eventually obliged him to give up this hobby.\n\nAnd anyway, Page was almost straightaway off on the road again. This time it was to the United States.\n\nThat same month, June 1966, someone else demonstrated an intriguing element of experimental good taste \u2013 unsurprisingly, given that the record was produced by Shel Talmy, Page's longtime studio champion, a master of innovation. 'Making Time', the stunning, sneering tune in question, was released by a Hertfordshire group called the Creation; Eddie Phillips, the guitarist, would at times play his instrument with a violin bow. It has long been alleged that Page took note of this and copied the effect. 'Eddie Phillips deserves to be up there as one of the great rock 'n' roll guitarists of our time, and he's hardly ever mentioned,' said Talmy. 'He was one of the most innovative guitarists I've ever run across. Jimmy Page stole the bowing bit of the guitar from Eddie. Eddie was phenomenal.'\n\nHowever, Page claims another source for the inspiration. At the Burt Bacharach session he played on, David McCallum, Sr., another session player, who played violin with both the London Philharmonic and Royal Philharmonic orchestras and was father of the co-star of the hit television series _The Man from U.N.C.L.E._ , asked the guitarist if he had tried to bow his instrument as though it were a violin. Page borrowed the violinist's bow \u2013 a wand finding its own wizard \u2013 and had a go, in front of McCallum. 'Whatever squeaks I made sort of intrigued me. I didn't really start developing the technique for quite some time later, but he was the guy who turned me on to the idea.'\n\nThe summer months of 1966 were a pivotal time for much of 'Swinging London', as _Time_ magazine had dubbed the city. In May, at the _NME_ Poll Winners' Concert at Wembley's Empire Pool, the Beatles played what would prove to be their last ever live show in Britain. However, as far as viewers of the televised broadcast of the _NME_ poll were concerned, the Yardbirds closed the concert. Acting on a perverse whim, Andrew Loog Oldham decided that the Rolling Stones' segment should not be broadcast. When Beatles manager Brian Epstein learned of this, he also demanded \u2013 for anxious reasons of 'cool' status \u2013 that the Beatles be kept off the TV, thereby depriving viewers of the Beatles' last scheduled live performance in the UK.\n\nOnly a few weeks after Page joined the Yardbirds came the debut performance of an act that would seismically shift the entire music scene. On 29 July 1966 Cream \u2013 which Clapton had formed with drummer Ginger Baker and bassist and vocalist Jack Bruce \u2013 performed their first ever stage show, at Manchester's Twisted Wheel, a venue more accustomed to hosting pill-popping Mod all-nighters, itself an indicator of changing times. Within a year Cream would be touring the United States to enormous acclaim, and, in a new twist, long-haired and stoned American audiences would reverently sit cross-legged on ballroom floors as Clapton tore through epic, frequently self-indulgent, guitar solos.\n\nThe day after the first Cream gig, on 30 July 1966, England won the football World Cup for the first time, on home soil, and national confidence surged. And by the end of the year another power trio, the Jimi Hendrix Experience, were ferociously tearing up and down the UK's venues and charts, with their first hit 'Hey Joe' and sensational live performances.\n\nFollowing that Wembley _NME_ date, the Rolling Stones were on something of a hiatus. Anita Pallenberg, Brian Jones's girlfriend, had bought a flat at 1 Courtfield Road, behind Gloucester Road tube station, and the Rolling Stone had moved in with her. Like several women in this rarefied bohemian milieu, Anita had about her an intriguing high-priestess aspect; she was attracted to the occult and was rarely without a bag containing rolling papers, tarot cards and occasionally the odd bone. Christopher Gibbs, a fashionable Chelsea art and antiques dealer, had insisted to Anita that she must buy the property, which only had one room and a set of stairs leading to a minstrel's gallery that formed a bedroom of sorts.\n\nPage had been friendly with the Rolling Stones, especially Brian Jones, since he first saw them at Ealing Jazz Club four years earlier. Now, in those first months with the Yardbirds, he was an occasional visitor to the Courtfield Road flat, along with Keith Richards and Tara Browne, the Guinness heir who would be dead by the end of the year, crashing his Lotus Elan after leaving the flat, his death celebrated in the Beatles' 'A Day in the Life'. It was at 1 Courtfield Road that Jones and Pallenberg began to regularly ingest LSD, soon introducing Richards to its glimpses of another reality. It is unlikely that Page, who developed a fondness for psychedelic drugs and was no longer confined by the rigidity of session work's time constraints, did not also enter this arcane coterie.\n\nThrough Robert Fraser, a Mayfair art dealer and major player in the Swinging London scene, the trio of Jones, Pallenberg and Richards became friends with the revered independent filmmaker and occultist Kenneth Anger, a disciple of Aleister Crowley. Anger's very beautiful short movies, marinaded in metaphysical matters, were like visual poems. Anger's use of pop music to tell the story in his films would prove to be hugely influential. Martin Scorsese would replicate it in his breakthrough film _Mean Streets_ , and Anger used Bobby Vinton's 'Blue Velvet' in his 1963 movie _Scorpio Rising_ , 23 years before David Lynch's film _Blue Velvet_. Anger considered Pallenberg to be 'a witch' \u2013 in turn she claimed that everything she knew about witchcraft had been learned from the filmmaker \u2013 and Brian Jones too, and that 'the occult unit within the Stones was Keith and Anita and Brian'. Keith had been turned on to such matters by Anita, and the pair would soon become lovers. A principal consequence of such out-on-the-edge thinking was the writing and recording of the Rolling Stones' 'Sympathy for the Devil', a song that \u2013 as Altamont would suggest \u2013 may not have been without its consequences. Page was yet to meet Kenneth Anger. But when he finally did, some years later, it began a relationship also not without penalties.\n\nAt the beginning of August 1966 the Yardbirds went into IBC Studios to record a new single, with Simon Napier-Bell at the production helm. Although 'Happenings Ten Years Time Ago', as the song was titled, came from the germ of an idea that Page and Keith Relf had come up with, the composing credits for the song would be attributed to all five group members. 'Happenings Ten Years Time Ago' would be the most psychedelic of all the Yardbirds' singles. Like a pointer to the future there were dual lead guitars on the record, Page and Jeff Beck, and so Page once more brought in his friend John Paul Jones to play bass. Beck, who had been suffering from ill health, added his own guitar parts later, along with a piece of spoken-word absurdism based on his experiences in a sexual health clinic. Aside from this whimsy, the lyrics themselves had considerable poignancy, relating to experiences of d\u00e9j\u00e0 vu or even of past-life existences \u2013 appropriately complex subject matter as the pop-based first half of the 1960s gave way to the rockier second half.\n\n'It was a compressed pop-art explosion, with a ferocious staccato guitar figure, a massive descending riff and rolling instrumental break and LSD-inspired lyrics that questioned the construction of reality and the nature of time,' wrote Jon Savage in _1966: The Year the Decade Exploded_. But by some it was seen as wilfully clever clogs. Penny Valentine, _Disc and Music Echo_ 's reliable record reviewer, was extremely dismissive: 'I have had enough of this sort of excuse for music. It is not clever, it is not entertaining, it is not informative. It is boring and pretentious. I am tired of people like the Yardbirds thinking this sort of thing is clever when people like the Spoonful and Beach Boys are putting real thought into their music. And if I hear the word psychedelic mentioned again I will go nuts.'\n\nIn fact, in the UK 'Happenings Ten Years Time Ago' only stuttered to the edge of the Top 30. As far as Britain was concerned, the Yardbirds were \u2013 in the jargon of the time \u2013 on their way out.\n\nOn 5 August 1966 Page played bass with the Yardbirds in the eighth-floor Auditorium of Dayton's Department Store in Minneapolis, Minnesota, a regular gig venue. Although he had visited the United States twice before, this was the very first American show that he ever performed. This, the Yardbirds' third US tour, had been scheduled to kick off a week earlier, but Beck was bedridden with tonsillitis, which was why he had not been present at the initial 'Happenings Ten Years Time Ago' session. At Dayton's Department Store there were two concerts, one at 1 p.m., the other at 4. 'The surroundings felt quite surreal,' Page remarked on his website many years later. For the show he was clad in a purple jumpsuit.\n\n'Shapes of Things' had been a US hit when it was released in February that year, reaching the number 11 slot; it was followed by 'Over Under Sideways Down', out in June and two months later still in the US charts, where it would peak at number 10, the group's biggest-ever American hit. The _Roger the Engineer_ album had been released in July in the USA as _Over Under Sideways Down_ , and that esteemed LP would get to number 52, not bad for a group starting its career in the USA.\n\nThe tour trundled through the Midwest \u2013 Chicago, Detroit, dates in Indiana \u2013 and down into Texas and assorted Bible Belt areas, playing the same sort of state fairs that had so irritated the Rolling Stones on their early US tours \u2013 unaware perhaps that these sort of shows were where Elvis Presley first honed his live act. Jeff Beck had not fully recovered from his illness, and would frequently take out his ire on the inadequate amplifiers at the venues. 'If he didn't get his sound right, he'd just kick the amps offstage,' recalled drummer Jim McCarty.\n\nNot that the shows weren't without a modicum of glamour. Due to a national airline strike, Simon Napier-Bell had been obliged to charter a small private plane. After Beck had smashed his own Marshall amp, he insisted that he would not carry on without an identical replacement. According to Napier-Bell, Page used the dilemma to undermine him: '\"Simon is our manager,\" he said, \"so it is on him to find a replacement.\" There were hardly any Marshall amps at all in the USA at that time \u2013 probably no more than 20 \u2013 so when I finally tracked one down we had to send the plane to pick it up, which cost an absolute fortune, far more than the amp cost. But that kept Jeff Beck happy, and allowed Jimmy Page to feel he'd got one over on me.'\n\nOn 23 August 1966 the Yardbirds played Santa Catalina Island, a resort 22 miles from Los Angeles in the Pacific Ocean. The group arrived on a boat from Long Beach filled with competition winners from radio station KFWB, and one of these fans noticed that Beck seemed in a 'difficult' mood. He had been in this 'mood' for the entire tour: his tonsillitis had never fully gone away and now seemed to be on the offensive once again.\n\nFor Simon Napier-Bell, however, this show at the island's Casino Ballroom was the best he had ever seen by the group. Beck played a solo, he said, that seemed to last for ever. During it he interplayed with Page's thundering bass guitar, and their sonic concoction drifted off into the soundscapes of what already was becoming known \u2013 and dreaded by Penny Valentine \u2013 as 'psychedelia'.\n\nBy the time the Yardbirds returned to Long Beach, it was clear that Beck was not in good shape. Although he retired for the night to the arms of Mary Hughes, his Los Angeles girlfriend, he was so sick the next day that that night's show, at Monterey County Fairgrounds, was cancelled.\n\nBeck's health was bad enough for him to have to drop out of the rest of the tour, a cause of considerable controversy within the group, but a decision of huge significance for Page. For the remaining 12 dates, beginning on 25 August at San Francisco's Carousel Ballroom, Chris Dreja switched from rhythm guitar to bass, and Page, wearing the newly fashionable wide-flared trousers, took over as lead guitarist. 'It was really nerve-wracking,' he said, 'because this was the height of the Yardbirds' concert reputation and I wasn't exactly ready to roar off on lead guitar. But it went all right, and after that night we stayed that way. When Jeff recovered, it was two lead guitars from then on.'\n\nDuring the tour Page would hear his own guitar work on the radio, on a single from a recent session he had played in London produced by Mickie Most. Now 'Sunshine Superman' by Donovan, an innovative and definitive sound of the summer of 1966 that heralded a golden period and shift of style for the former folk singer, was rising up and up the US charts, reaching the number one slot for a week. Although not released in the UK until December that year, it would almost emulate its US chart position, reaching number two.\n\nJimmy Page and Donovan Leitch were like-minded musical souls, each with their own interest in metaphysics. Playing on 'Sunshine Superman' with Page was \u2013 yet again \u2013 John Paul Jones on bass. At those same sessions Page played the haunting guitar on Donovan's equally memorable 'Season of the Witch', which was on the _Sunshine Superman_ album. Built around a D ninth chord shown to Donovan by master guitarist John Renbourn, 'Season of the Witch' was ideal for extended versions and would be frequently employed as soundcheck material by Led Zeppelin.\n\nBack in the UK, Jeff Beck's health returned, and he drove over to Page's Pangbourne home, its interior design already beginning to reflect the prevailing rock-star rococo style. The two guitarists worked out a stage routine that would allow each to play lead guitar, intertwining with one other and mutually strengthening their playing and that of the group. Among the songs they developed was a version of Freddie King's 'Goin' Down' \u2013 though this was never recorded.\n\nThey needed to work fast. On 23 September the Yardbirds again hit the road, supporting the Rolling Stones and the Ike and Tina Turner Revue on a 12-date tour of the UK, two shows a night, ending on 9 October 1966.\n\nThe tour kicked off at London's Royal Albert Hall, where the Yardbirds allegedly blew the Rolling Stones off the stage. Yet the review they received from the _NME_ , especially for Jeff Beck, was extremely sniffy; it irritated him that he was described as 'a guitar gymnast'. Simon Napier-Bell utterly disagreed with such an assessment: 'They were really fantastic. What Jeff and Jimmy were doing was playing Jeff Beck's solos, but in harmony. It was astonishing to hear, and to watch.'\n\nDifficulties soon arose, however. According to Chris Welch in _Led Zeppelin: The Book_ : 'One problem was that Jeff couldn't handle the competition and would try to blow Jimmy off the stage. Page was always on the ball, but Jeff's returning fire in guitar exchanges would be unpredictable and relied on volume when accuracy failed.' Napier-Bell was in agreement in an interview he did with Jim Green in the October 1981 edition of _Trouser Press_ : 'Jimmy deliberately upstaged Jeff, Jeff got moody and walked out towards the end, fortunately we finished it.'\n\nQuickly forgiven, Beck was admitted back into the Yardbirds fold. This was a necessity, as they were about to join the winter leg of Dick Clark's Caravan of Stars \u2013 a regular feature of the American popular music calendar organised by the legendary DJ. But before that there was another avenue to explore.\n\nAn exaggerated, dramatised version of the high-end hippiedom found at Anita Pallenberg and Brian Jones's home was expressed in the party scene in the classic \u2013 yet occasionally extremely pretentious \u2013 metaphysical thriller _Blow-Up_ , which Italian director Michelangelo Antonioni started filming in London in April 1966. (The scene was actually shot in Christopher Gibbs's Cheyne Walk house.) In the film a fashion photographer, played by David Hemmings and clearly based on _enfant terrible_ David Bailey, thinks he has witnessed a murder.\n\nAfter completing the London shoot in June, Antonioni decided that to fully represent the capital's glamorous, swinging spirit he should shoot a sequence in a rock 'n' roll club. In September he returned to London and booked a meeting with Kit Lambert, manager of the Who. The day before the meeting, Lambert had lunch with Napier-Bell at the Beachcomber restaurant in London's Mayfair Hotel; the two men were close friends and Lambert wanted to pick Napier-Bell's brain about how to approach Antonioni. Napier-Bell decided to set him up to the advantage of his own group: 'I told him to ask for \u00a310,000 and insist that the Who had final edit on their sequence. Antonioni kicked him out after about a minute.\n\n'Then I went to see Antonioni: \"We don't want money. This is art. Of course I don't want to edit it.\"' (In fact the Yardbirds received \u00a33,000 for their part in _Blow-Up_ : car-freak Jeff Beck immediately spent his share on a second-hand Corvette Stingray.)\n\nNapier-Bell had scored. The Yardbirds' _Blow-Up_ sequence was filmed at Elstree Studios in Borehamwood, north of London, doubling for Windsor's hip Ricky-Tick Club, in the week beginning 6 October 1966. The band played 'Stroll On', as it was called in the film's credits, the lyrics having been rewritten the previous night by Keith Relf for copyright reasons \u2013 in other words, so he could snatch the credit. But it was better known by fans who had experienced the song when rivetingly performed by the Johnny Burnette Trio as 'Train Kept A-Rollin'' \u2013 the same song that the Yardbirds had recorded at Sun Studio in Memphis the previous year. (Howlin' Wolf's 'Smokestack Lighting', a Yardbirds live favourite, had been first choice, but the idea was shelved when Antonioni decided it lacked the relentless pace he needed for the scene.)\n\n'Train Kept A-Rollin' would be the very first number that Led Zeppelin would play in their initial rehearsal; in _Blow-Up_ the Yardbirds essay an angry, explosive version of the song before a consciously static audience, which includes a young Michael Palin and a dancing, silver-coated Janet Street-Porter. Page stands stage-right to Beck's stage-left and, rather in the manner of the Kinks' Dave Davies, his hair is parted in the centre, his mutton-chop sideburns kissing his jawline and peeking out beneath the twin tonsorial curtains waterfalling from his head. He wears an open black jacket, a trio of badges balanced symmetrically on each lapel. In the sequence Beck freaks out over a malfunctioning amp, smashing his guitar to splintered pieces in a manner that only Pete Townshend would actually do in reality. Upon learning of his role, Beck had recoiled: would he have to destroy his new Gibson Les Paul? No fuckin' way!\n\nA bunch of cheap H\u00f6fner replica guitars were brought in. 'Jeff Beck had to be coaxed into smashing the guitar. And then he did it half a dozen times,' recalled Napier-Bell. It is only after the guitar has been destroyed that the audience breaks out into a feverish response.\n\nSmashing the guitar wasn't the only problem Beck had on the film, however. 'Antonioni was a pompous oaf. I didn't like him at all,' he said. 'The film was a bit of a joke. Crap. I thought, \"Oh, that's the end of us.\" Because I saw the premiere in LA. But people loved it. It kept us going.'\n\nThis scene from _Blow-Up_ appears to be the only filmed record of Page and Beck playing together in the Yardbirds. Though he is only briefly glimpsed in this scene, and Beck's thuggish fury unquestionably steals their joint screen time, Page's almost girlishly pretty looks \u2013 which crease momentarily into a smile \u2013 contrast powerfully with Beck's pugnacious posture, like that of a yob awaiting his borstal sentence. You can see why, for the brief time they played together, the pair proved such a potent force in the Yardbirds.\n\nAlong with the Yardbirds on the Dick Clark Caravan of Stars tour were bill-toppers Gary Lewis & the Playboys, whose singer was the son of comedian Jerry Lewis. The group \u2013 safe in a Herman's Hermits\/Gerry and the Pacemakers kind of way \u2013 had seven successive US Top 10 singles. Then there were 'Wooly Bully' hitmakers Sam the Sham and the Pharaohs; the Distant Cousins; Bobby Hebb, high in the US charts that summer with his sophisticated, sexy 'Sunny'; and early sixties vocal star Brian Hyland, purveyor of puppy-love pop. Soon, as album rock became the principal market force, several of those acts would see their careers nosedive forever.\n\nThe tour wound through the south, the Midwest and East Coast before winding up in Huntington, West Virginia, on 27 November. 'Thirty-three dates, I think, and of those twenty-five were doubles, two shows in one night,' said Page. 'You'd think a double would be played in the same town, but it wasn't \u2013 it was two different towns. The show was in two halves. When the first half finished, and there was an interval... the performers would get on the coach driving to the next venue, while the second half carried on. Then, they in turn carried on to the next place, where the others had by then finished. It was the worst tour I'd ever been on, as far as fatigue is concerned. We didn't know where we were or what we were doing.'\n\nTravelling conditions were abysmal, the artists being driven 600 miles or so a day in a pair of converted Greyhound buses to play four songs each at every show. 'The other acts had little or nothing in common with us,' said Jim McCarty. 'Sam the Sham and his Pharoahs, Brian Hyland. I mean, they were just so different, though Sam the Sham had his moments. Anyway, when they let us off the bus, we'd go onstage and they'd shout, \"Turn the guitars down!\" Jimmy was getting through it because he was a professional. Chris and I stood up to it because we were creating humour from all sorts. Keith was drinking his way along. But Jeff...'\n\nJeff Beck was becoming a specialist in crossing the United States in a bad mood: 'The bus was supposed to have air-conditioning, but didn't seem to. And all the American groups on the bus played their guitars non-stop, and were always singing. Could you imagine? Cooped up on a stuffy bus with everyone around you singing Beatles songs in an American accent?'\n\nHis technique honed on the set of _Blow-Up_ , Beck cut a destructive swathe through the tour's initial dates: amps thrown out of windows, instruments smashed. 'Jeff Beck had to be coaxed into smashing his guitar for the _Blow-Up_ scene,' said Napier-Bell. 'And then he fell in love with doing it and with smashing his amp. I'm sure Jimmy Page was counting the nights, because then Jeff Beck left.'\n\nThe frustration within the group began to mount, and, as Beck would admit in hindsight: 'I was quite messed up. At 21 I was really on my last legs. I just couldn't handle it.'\n\n'One time in the dressing room,' recalled Page, 'Beck had his guitar up over his head, about to bring it down on Keith Relf, but instead smashed it to the floor. Relf looked at him with total astonishment, and Beck said, \"Why did you make me do that?\"'\n\nIn the middle of the tour, in Harlington, Texas, Beck caught a taxi to the airport and flew to Los Angeles \u2013 where, of course, Mary Hughes awaited him. Beck's explanation for his departure? A return of his chronic tonsillitis. He was going to have treatment and would soon return, he said.\n\nThe day after Beck disappeared, Napier-Bell was obliged to appear on a local television show to announce that the next Yardbirds concert had to be cancelled. Showing that they were perfectly adaptable under stress, Relf and Page scoured the area for a joke shop. When Napier-Bell took a lengthy pull on the cigar his pair of charges had presented to him, he had to jump back as it exploded in his mouth \u2013 in the middle of the live television broadcast. At the time there was a term for such behaviour among UK groups: looning. Page's part in this moment clearly showed that he was not above indulging in this, although, given the complicated relationship he had with his manager, was there a subconscious maliciousness in the very notion of this deed?\n\nOnce again, Page took over as the sole lead guitarist. 'Jimmy was always a real pro,' said Chris Dreja, 'whereas Jeff was a man of emotion. I think Jeff always found it harder than Jimmy because he was prone to playing according to how he felt, whereas Jimmy's idea was always, \"We're professional entertainers.\"'\n\n'I didn't like my territory being encroached upon, and I wanted to be it, to do all the guitar playing,' admitted Beck. 'And when it got to the point when I was exhausted, we then embarked on a six-week Dick Clark tour. Six hours in that thing was enough for me. To be faced with those kind of travel problems and emotional tear-ups, and you'd get to the end and play a toilet gig with music you didn't feel comfortable with, was a recipe for disaster. Things just got on top of me and I cracked up, basically. I wanted to do something other than travelling... So it's not important whether I was kicked out or I left \u2013 it just happened.'\n\nAfter a period of reflection in Los Angeles, Beck attempted to return to the Yardbirds. He realised that he had essentially been suffering from a minor nervous breakdown. But when word got back to the Yardbirds that their AWOL guitarist had been seen enjoying himself in LA nightclubs, they voted Beck out of the group.\n\n'That was the point at which I gave up on trying to manage the Yardbirds,' said Napier-Bell. 'I thought it was too difficult and the only person I really liked in the group \u2013 apart from Chris Dreja, who is a nice guy, a good person \u2013 was Jeff Beck. So I kept the management of Jeff Beck. I found Jimmy very difficult to deal with. Always narky.'\n\nPage, however, felt it was unsurprising he was considered awkward: 'Bloody right. We did four weeks with the Rolling Stones and then an American tour and all we got was \u00a3112 each!'\n\nOne aspect of Page that Napier-Bell had observed was the guitarist's tendency to be tucked away in a corner reading yet another esoteric volume by Aleister Crowley. 'People would ask him about it, and he would reply something along the lines of, \"Oh, you wouldn't understand it. You're not intelligent enough.\"'\n\n## 6\n\n# 'YOU'RE GOING TO KILL ME FOR A THOUSAND DOLLARS?'\n\nIf Aleister Crowley performed a role as a kind of absent metaphysical father figure to Jimmy Page, the fastidiously loyal Peter Grant would prove to be a very physical manifestation of one.\n\nBorn into considerable poverty on 5 April 1935 in London's South Norwood, Peter James Grant was the illegitimate son of a secretary. He never met his father, who was rumoured to have left his mother because she was Jewish \u2013 even though she was employed as a typist for the Church of England Pensions Board.\n\nSoon Grant and his mother moved to a tiny terraced house in down-at-heel Battersea, on the edge of the Thames. During the Second World War his school was evacuated, and he was sent to Godalming in Surrey, where he was educated at posh public school Charterhouse. Bullied by resentful incumbents, he developed an abiding loathing of the upper classes that would remain with him all his life. Emotionally distraught at being torn from his mother and home, meagre though it was, Grant began to put on the excessive weight that would characterise him.\n\n'This boy will never make anything of his life,' read the final report by the headmaster of the south London school where Grant finished his education.\n\nThe first few weeks of Peter Grant's working life were spent labouring in a sheet-metal factory. At six feet, five inches tall and on his way to being just as wide, he certainly had the physique for physical work, but he quickly changed course, first becoming a messenger delivering photographs on Fleet Street for the Reuters news agency, and then a stagehand at the Croydon Empire Theatre.\n\nAfter National Service in the Royal Army Ordnance Corps, where he attained the rank of corporal, he became doorman \u2013 bouncer, really \u2013 at the 2i's Coffee Bar on Old Compton Street, the birthplace of British rock 'n' roll and the venue from which sprang Cliff Richard, Tommy Steele and Adam Faith, among others. It was during this time that he met Mickie Most, who had recently returned to London after finding success as a pop star in his wife's native South Africa.\n\nThrough connections he made with Most, Grant was swept up into the newly popular world of televised professional wrestling, appearing as both Count Massimo and Count Bruno Alassio of Milan. From this springboard Grant secured bit parts in a number of films \u2013 among them _A Night to Remember_ , _The Guns of Navarone_ and the epic _Cleopatra_ \u2013 and television shows \u2013 _The Saint_ , _Dixon of Dock Green_ and _The Benny Hill Show_ , among others.\n\nGrant pulled together his 2i's connections, bought a minibus and started a business transporting rock 'n' roll musicians to their gigs. He started off with the Shadows, before he was approached by Don Arden, who gave Grant the task of road-managing American acts he brought over for tours \u2013 the Everly Brothers, Chuck Berry, Bo Diddley, Little Richard, Gene Vincent and Brian Hyland \u2013 a role he performed with alacrity, personally ensuring the artists were paid every penny owed, in cash.\n\nArden had a very nasty reputation: he was rumoured to have held Steve Marriott, of the Small Faces, out of a third-floor office window by his ankles when the diminutive singer requested an accurate accounting of his earnings. Grant learned from such behaviour. 'When he was with Led Zeppelin he was a batterer of people,' said Napier-Bell. 'Although that was probably because of the drugs. When he had cleaned up he became a very nice person.'\n\nBut that was all yet to come. Grant also acted as road manager for the Animals and the New Vaudeville Band, and began sharing an office with Mickie Most at 155 Oxford Street, on the border of then sleazy Soho in London. When Most took over the management of the Yardbirds from Napier-Bell, Grant became their de facto manager and Most's partner in RAK Management.\n\n'They needed someone like Peter Grant,' said Napier-Bell. 'He wouldn't stand for Jimmy Page's sneering.'\n\nThe group was losing traction in the UK, but in America they still carried considerable cachet. Grant travelled with the band on the road in the US, and, for the first time, they returned to England with money in their pockets. 'He was a great manager for the time,' said Chris Dreja. 'He was hands-on, nuts and bolts. He travelled with the band. He made sure they didn't get screwed. He loved his artists. He changed the music scene. He was responsible, especially with Zeppelin, of course, because they had such a huge audience, for changing the percentage points around between the record companies and the artists and the promoters. He was just a fantastic manager.'\n\nOn a date in a snowbound northern American state, the bad weather caused the Yardbirds to arrive late, almost missing their call time. Furious at being so put out, the pair of Mafia promoters refused to pay the group's fee, one of them pulling a gun. Peter Grant walked his considerable girth up to the man: 'What? You're going to kill me for a thousand dollars? I don't think so.' He got the Yardbirds their money.\n\nGrant became close to Page, who, he noted, seemed in control of himself and intelligent, far more businesslike than the other Yardbirds and apparently much older than his 22 years. In this odd-couple relationship, Grant expressed his 'utter faith' in the young but extremely seasoned guitarist. 'It was funny how well Jimmy and Peter got on because Jimmy was a very softly spoken, gentle guy and Peter was from a very different background and education,' said Napier-Bell. Among other things, the pair shared an interest in antiques, and they would go shopping for them together on tour.\n\n'\"Peter, there's only one problem with the band,\"' Grant had been told by Napier-Bell. '\"There's a guy there who's a real smart arse, a real wise guy.\" I said, \"Who's that?\" He said, \"Jimmy Page.\" I was a bit puzzled. I thought, he must know I've known Jimmy since 1962\/63. Apart from Neil Christian, when I was in business with Mickie Most, he did all the Herman's Hermits and Donovan. So when I met Jimmy I said, \"I hear you're a bit of troublemaker and I should get rid of you. What have you been up to?\" He said, \"We did a four-week tour of the UK with the Stones and an American tour and we got \u00a3112 each.\" And he was the only one who had the balls or savvy to say something. By then Mickie Most was recording them. Mickie Most is a pop producer, an excellent pop producer. And there was always a bit of friction there. The way I saw the band going, the way they wanted to carry on, was against the pop thing.'\n\nYet Mickie Most appeared unaware of the cultural wind of change. 'The intention,' he said, 'was to try and resuscitate their pop career.'\n\nIn October 1967 Most insisted that a new Yardbirds 45 was released in the United States: 'Ten Little Indians', a song penned by Harry Nilsson and included on his second album _Pandemonium Shadow Show_. A truly dreadful record, it climbed no higher in the American charts than number 96, although Page had attempted to save the song, which featured a cloying brass section, by turning this into a feature after it had been subjected to what became known as 'reverse echo'.\n\nThat 'Ten Little Indians' was only released in America was a testament to how out of touch Mickie Most had become. Both Page and Grant were well aware of the emerging new underground scene in America, the more reflective, less materialistic outlook of the hippie audiences at Bill Graham's Fillmore West auditorium in San Francisco, which became almost a temple to the Yardbirds. The soundtrack to this counter-culture was provided by the advent of FM radio and its new 'progressive rock' stations like San Francisco's KSAN, New York's WNEW and Orlando's WORJ, which were prepared to play an entire album with no commentary from a DJ (in the UK this was mirrored to an extent by John Peel's late-night _The Perfumed Garden_ show on the pirate-ship Radio London).\n\nThe 'Season of the Witch' was upon us. There was a new generation of American music-makers with very strange, surreal and hitherto unimaginable names that suggested copious drug consumption: Strawberry Alarm Clock, Captain Beefheart, Love, the Doors, Iron Butterfly, Jefferson Airplane, Moby Grape, Quicksilver Messenger Service, the Grateful Dead. They were all allied to the burgeoning 'rock' album audience, a development spurred by the arrival in late 1965 of the first relatively cheap stereo systems. Long-haired, free-loving, pot-smoking and acid-dropping, this new market was cemented together by the considerable schism in American society brought about by the ceaselessly expanding war in Vietnam. Crisscrossing the United States with the Yardbirds, Page and Grant witnessed the success of first Cream and then the Jimi Hendrix Experience, seeing how they fitted perfectly into this new world. It was a musical and cultural sea change.\n\nAnother of these novel new acts, the Velvet Underground, championed in New York City by the artist Andy Warhol, supported the Yardbirds on several shows in the winter of 1966, most notably a show at Michigan State Fairgrounds. Soon the Yardbirds started dropping a snatch of the Velvet Underground's 'I'm Waiting for the Man', the group's paean to heroin dealers, into the middle section of an extended version of their own tune 'I'm a Man'. Page had heard the Velvets' first album while touring the USA with the Yardbirds. 'I'm pretty certain we were the first people to cover the Velvet Underground,' he said. At one of those Manhattan parties at which Andy Warhol was ubiquitous, the artist asked the guitarist to take part in a screen test for him for a movie he had in mind.\n\nAs the sole guitarist with the now four-piece Yardbirds, Page spent much of 1967 and the first half of the next year on long, gruelling tours in far-flung places. It was relentless. There were five American tours, a UK tour, a European tour and in January 1967 an Australasian tour with Roy Orbison and the Walker Brothers, playing two shows a night. But it was not without its rewards. 'When Jeff left and we carried on,' he said, 'the pure nature of the band was that they had a lot of numbers you could really stretch out on.'\n\nBack in Britain from Australia during February 1967, Page worked with Brian Jones at IBC Studios on the soundtrack for _A Degree of Murder_. Directed by German New Wave filmmaker Volker Schl\u00f6ndorff and starring Jones's girlfriend Anita Pallenberg, the film was entered for competition at the 1967 Cannes Film Festival. Although both Page and Nicky Hopkins, the celebrated session pianist, played on the soundtrack, along with Small Faces drummer Kenney Jones, with Glyn Johns engineering, there was never an official release for Brian Jones's music. 'Brian knew what he was doing,' said Page to _Rolling Stone_. 'It was quite beautiful. Some of it was made up at the time; some of it was stuff I was augmenting with him. I was definitely playing with the violin bow. Brian had this guitar that had a volume pedal \u2013 he could get gunshots with it. There was a Mellotron there. He was moving forward with ideas.'\n\n'I don't remember much about the sessions other than we got Jimmy Page to come and play some amazing guitar during the murder scene and that the German director was thrilled with the end result,' recalled Glyn Johns.\n\nPage still had time to play the occasional session. For the past couple of years, Johns had produced Johnny Hallyday's records, a Gallic Elvis who was indubitably the biggest music star in France. Hallyday would often record in London, and a distinct attraction for anyone working on his sessions was that he always paid in cash. But on this occasion he decided to work in Paris, with his own band, which included Mick Jones, later of Spooky Tooth and then Foreigner. 'I took Jimmy Page,' recalled Johns. 'He was nothing short of brilliant.'\n\nOn the tune '\u00c0 Tout Casser' Page performed one of his greatest session moments. And on 'Psychedelic' he employed a bluesy, Albert King-like bending riff that would resurface a couple of years later on Led Zeppelin's 'Whole Lotta Love'. The tune has a classic freak-out section as Hallyday repeats the word 'psychedelic' over and over to blasts of Jimmy Page guitar.\n\nDuring this period in early 1967 Page became briefly involved with 19-year-old model Heather Taylor. A friend of New York photographer Linda Eastman, Taylor had run the fan club for Monkee Davy Jones and briefly been his girlfriend; she had also been a lover of Jimi Hendrix \u2013 'Foxy Lady' was allegedly written for her \u2013 and Jeff Beck.\n\nAfter Page met her at Ondine, the fashionable Manhattan nightclub, she had followed him to London. But he quickly told her they were 'seeing too much of each other' \u2013 after only three dates in London. Taylor was later introduced to the Who's Roger Daltrey by her Californian friend Catherine James, who was living in London with former Moody Blue Denny Laine. Taylor would go on to marry Daltrey, while Catherine James would crop up again in Page's future.\n\nFresh from his experience on the _A Degree of Murder_ soundtrack, which was very much part of the 'progressive' new musical order in his eyes, Page was keen on leading the Yardbirds in a similar heavier and more experimental direction. Later in 1967, this would include the group's rendition of a new song, sometimes known as 'I'm Confused', but more reliably as 'Dazed and Confused'.\n\nOn 25 August 1967, the peak of the year's alleged Summer of Love, the Yardbirds played two shows at the Village Theater in downtown Manhattan. They were supported by the Youngbloods and by Jake Holmes, a singer-songwriter who spun a twist on the folk tradition by working with a guitar and two bass instruments but no drums. Watching from the wings, Jim McCarty was impressed by Holmes's song 'Dazed and Confused', with its descending riff, as was Page. The next day they went and bought _The Above Ground Sound of Jake Holmes_ , his debut album, on which 'Dazed and Confused' was featured. Adapting the song, but only to an extent, with Keith Relf altering the lyrics, 'Dazed and Confused' became a stand-out performance in Yardbirds live shows for the last ten months of the group; Page's dramatic and highly effective flourishing of a violin bow on his guitar during the performance was an undoubted highlight. 'Dazed and Confused' became a showcase tune on the first Led Zeppelin album.\n\nIn the UK the Yardbirds were beginning to seem increasingly irrelevant, but in the United States they remained a substantial concert attraction. Yet even there, the hits didn't keep coming, largely because \u2013 as with 'Ten Little Indians', which would be released later that year in the US \u2013 their selection made almost no sense.\n\n_The Yardbirds' Greatest Hits_ was released in March 1967 in America, and made number 28, their biggest-selling US album. But whereas the Yardbirds' three biggest singles had been written by group members, from now on Mickie Most brought in \u2013 as he had done with their first hit 'For Your Love' \u2013 songs by established hit-making teams.\n\nAccordingly, the choice of some of the 45s released seemed baffling. Written by Harold Spiro and Phil Wainman, 'Little Games' was released in March 1967 in the US and a month later in the UK. It only reached number 51 in America and didn't even chart in the UK. As the title track of the next Yardbirds album, released in mid-July only in the United States \u2013 a mark of their dwindling UK status \u2013 the single was intended as a trailer for the LP.\n\nThere were further odd decisions. 'Ha Ha Said the Clown', for example, had been a Top 5 hit in the UK for Manfred Mann, but had failed to gain any chart movement whatsoever in America. Most persuaded the Yardbirds \u2013 or, more accurately, Keith Relf plus session musicians, including John Paul Jones, who arranged much of Most's material \u2013 to record the song and release it in the States. In July 1967 'Ha Ha Said the Clown', not featured on _Little Games_ , staggered up to the number 45 slot in America. The breezy pop tune was an extraordinary choice for the Yardbirds to release, utterly inappropriate for the market they were trying to build in the States. But the B-side was another matter altogether: the explosive 'Tinker, Tailor, Soldier, Sailor', written by Page and Jim McCarty, with a bass-line like a relentless train rolling, featured the first recorded instance of Page using a violin bow on his guitar.\n\nThe _Little Games_ album \u2013 derided by Page as 'horrible' \u2013 did contain some interesting moments: 'Tinker, Tailor, Soldier, Sailor' itself, and 'Glimpses', a suggestion of the way Page's compositional mind was developing. 'It featured the violin bow, and when it was played in concert I had tapes that played all this stuff \u2013 the Staten Island Ferry, locomotives, shock sounds \u2013 with textures from the bow. But we didn't get a chance, with the Yardbirds, to take it far enough,' he told _Rolling Stone_ 's David Fricke; and 'White Summer', a mesmerising instrumental piece that he would perform live, both with the Yardbirds and Led Zeppelin, on his 1961 Danelectro 3021 or his 1967 Vox Phantom XII 12-string guitar.\n\n'Goodnight Sweet Josephine' was the Yardbirds' last single, released in March 1968. Written by Tony Hazzard, who had also composed 'Ha Ha Said the Clown', it was about a man-eating groupie who lived in Clapham in south London \u2013 where Hazzard himself was based. Uncertain whether to commemorate its clear music-hall origins or celebrate its implied cultural comment, it did neither and ended up truly awful \u2013 a shockingly bad record. First recorded at the end of 1967 at Advision Studios in London, with Mickie Most producing, the Yardbirds disliked the final result so much that they insisted on doing a further version, on 6 February, at De Lane Lea Studios. It was no help, however; the single was the worst they ever released, and the group quickly asked for it to be recalled, reissuing a further version in the United States and Australasia. On the other hand, the single's B-side 'Think About It' had much more going for it \u2013 a shredding solo midway through that Page would take in all its glory into Led Zeppelin's version of 'Dazed and Confused'. With his assiduous thoroughness, the guitarist had grabbed the production of the B-side from Mickie Most.\n\nOn 6 March 1968 Page gamely went to plug 'Goodnight Sweet Josephine' on the long-running BBC radio show _Saturday Club_ \u2013 the show had been a Saturday morning feature of the Light Programme before the station was transformed into Radio 1 the previous autumn, a supposed substitute for the now-banned pirate stations. Interviewed by the ever-avuncular Brian Matthew, the self-styled 'old mate' of his audience, Page described the troubles around making the record in his boyish, accentless voice and concluded, 'It's quite a good product now.'\n\nHowever, Jeff Beck's summary was more to the point. 'When I heard \"Goodnight Sweet Josephine\" I thought, Thank God I left the Yardbirds.'\n\nThree days after that _Saturday Club_ performance the Yardbirds were in Paris. They played at the Assas Faculte du Droit, supported by the Brian Auger Trinity, featuring Julie Driscoll, and recorded the _Bouton Rouge_ television show, performing 'Train Kept A-Rollin'', 'Dazed and Confused' and 'Goodnight Sweet Josephine'.\n\nThe next day, 10 March, the Yardbirds played at Paris's legendary Olympia venue. Afterwards the group moved on to a private party thrown by Eddie Barclay, who ran the renowned Barclay Records. Among the celebrated guests was Brigitte Bardot, dressed in leather motorcycle gear. She looked 'hot', noted Page.\n\nBack in London on 15 March, Page played a session for Joe Cocker, for his 'Marjorine' single and its B-side, 'The New Age of the Lily'. (Later in the summer he would play on Cocker's epochal version of Lennon and McCartney's 'With a Little Help from My Friends', the song that made Cocker's career.) The next day he was again in Paris with the Yardbirds for a show.\n\nA week later, following a concert at Retford College in Nottinghamshire on 23 March, the Yardbirds flew to New York for the start of their American tour. The opening date, 160 miles from Manhattan, was at the Aerodrome in Schenectady in New York state, a venue with a famously tremendous sound system, the kind favoured by Page for the more complex sound he was developing on tunes like 'Think About It', 'White Summer' and 'Dazed and Confused'. But such aural developments would not be employed for much longer with the Yardbirds, as the band knew before they even set off that this was to be their final ever tour.\n\nInspired by the prevailing softer sounds of the likes of \u2013 curiously \u2013 the Turtles and Simon & Garfunkel, and by LSD, Keith Relf and Jim McCarty had concluded that they no longer wanted to be part of the Yardbirds and would form a more folk-influenced outfit instead.\n\nTwo nights later, at Manhattan's Anderson Theater, the Yardbirds played a pair of shows that Epic Records recorded for a live album, which would eventually be released in January 1971 but quickly withdrawn after Page protested that it was a cash-in on the success of Led Zeppelin. 'We knew the American tour was going to be the last one, and all the pressure was off,' said the guitarist. 'We played well and had a really good time. We even managed to play consistently good venues; it was almost entirely universities and psychedelic ballrooms. The only low point was the Andersen Theater gig in New York, which was recorded for a live album. The rats at Epic had got wind of the break-up and decided to get the last drop of potential profit out of us. It was pure convenience for them, being based in New York, where we didn't like playing anyway. It should have been done at somewhere like the Shrine in LA, or the Fillmore. The Anderson Theater was a horrible place, very cold and unfriendly, and it didn't help that the Vanilla Fudge, currently local heroes, were playing across town at the Fillmore East. To cap it all, the Epic sound team had no idea how to record us. They were really straight and they just draped a few mikes around. It was pathetic. When they discovered the inadequacies of the recording, they dubbed on all those ridiculous bullfight cheers.'\n\nAs the tour headed into the American heartland, the audiences noted that the group seemed in tune with the prevailing psychedelic currents. 'I was a real Yardbirds fan,' posted John B on Facebook on 18 May 2015 of their show at Chicago's Blue Village, 'and was a bit high at the time. The band, especially Relf and Jimmy, seemed a bit higher than me. I didn't feel they were \"drunk\", but perhaps tripping. All of that aside, they practically melted the place, and most of the audience (me included) weren't really sure how to process the event \u2013 it was jaw-dropping. Page played the Tele with a bow and a wah-wah. I remember seeing broken strands of the bow hair trailing as he sawed away. The fellows were dressed very \"psychedelically\" and Relf was sporting a long moustache. I specifically recall \"Over Under Sideways Down\" as being a highlight and thinking, Oh my god, this guy can actually play this stuff!'\n\n'I was there that night at the Cellar club,' posted Don Dawson on Facebook, 7 March 2015, of the show at the Cellar club in Arlington Heights, Illinois, on 20 April. 'Page was really wasted but managed to pull himself back together after rallying with a brilliant solo rendition of \"White Summer\" played seated. Then the rest of the band got back in gear to finish their set.'\n\nMoving over to California on 10 May, the Yardbirds played the Earl Warren Showgrounds in Santa Barbara, a venue with a growing underground reputation. The British group Dave Dee, Dozy, Beaky, Mick & Tich had been scheduled as support, along with Turquoise. But the UK outfit was not there, replaced by Three Dog Night. In the audience was a Londoner called Paul Reeves, a former dancer on _Ready Steady Go!_ and a fashion designer of some reputation who was visiting California to sniff the wind. 'After the Yardbirds' set I decided to go backstage and introduce myself, as Englishmen did in those days in California because we were a relative rarity,' said Reeves. 'We got on like a house on fire and I travelled with Jimmy and the band for the rest of the tour, even sleeping on his hotel-room floor in New York.'\n\nThe pair, who remain close friends to this day, bonded over their art-school backgrounds and mutual interest in design and fashion: they had both bought clothes at John Stephen's first shop on Carnaby Street; they both loved the garments you could buy at Granny Takes a Trip, Hung on You, and Emmerton & Lambert in the Chelsea Antiques Market \u2013 where they stocked some of the shirts Reeves designed. The pair also shared a love of the Arts and Crafts movement, and of the then very fashionable Pre-Raphaelites \u2013 from whom Page appeared to have gleaned much of his very being.\n\nBack in London that year Paul Reeves would launch Alkasura, a high-end, futuristic clothes shop on the Kings Road frequented by the likes of the Beatles, the Rolling Stones, the Faces, David Bowie and later \u2013 naturally \u2013 Led Zeppelin. Together, Page and Reeves drove down to the famed Costa's in Tooting Bec, south London, a Greek Cypriot shoemaker who made them buckled snakeskin boots, the inspiration for which derived from an image in _East of the Sun and West of the Moon_ , a book by the early 20th-century illustrator Kay Nielson. (Page's pair eventually became part of the Victoria and Albert Museum's collection, and in 2008 Paul Reeves curated an exhibition at the museum called _The Best of British_ , dedicated to 19th- and 20th-century design from these islands. Page contributed several pieces, including the superb Edward Burne-Jones Pre-Raphaelite tapestry _The Quest for the Holy Grail: The Achievement_ , worth at that point around \u00a31 million.)\n\nPaul Reeves was not the only person Page met on this return to southern California. Catherine James had left Denny Laine and moved back to her birthplace of Los Angeles, bringing her baby son home with her.\n\nMoving into a house in Bronson Canyon, on the edge of the Hollywood Hills, she took a job in the ticket office of Thee Experience, a cutting-edge rock venue on the Sunset Strip. After the Yardbirds' date in Santa Barbara, they had a further show in southern California \u2013 on 11 May 1968 they played in Anaheim, south of Los Angeles \u2013 and two days later recorded a television interview that heavily featured Page.\n\nDuring this time Page visited Thee Experience and linked up with Catherine James. 'I had plenty of handsome suitors, but no one really had my heart until I encountered the ethereal Jimmy Page,' Catherine James wrote in her autobiography. 'Heather had given me a pretty accurate description, but there were no real words to describe his seductive allure and painful charm. It wasn't just his lithe, elegant, rock-and-roll mien, or his Pre-Raphaelite angelic face and soft, long curls. He had an indescribable smouldering look in his eyes, like he had a secret he might share with you one day. When we kissed he inhaled my breath like he was savouring my soul. I'd never felt anything quite as dark or sensuous as James Patrick Page. He captivated me with stories of the English countryside and his lone manor in Pangbourne... He always made me feel like we were somewhere off in the mists of Avalon.\n\n'Although he hadn't exactly spoken the words, he did some heavy alluding. I had illusive visions of being the blithe young maiden of the manor; we'd give life to a band of seraphim and be eternally happy. Little did I know several other girls were under the same identical spell.'\n\nWould Catherine James have been more nervous if she had learned what other members of the Yardbirds knew of their guitarist's liaisons? 'Jimmy Page used to show us Polaroids of often blurred close-up parts of female anatomy, often featuring fruit, I seem to remember,' recalled Chris Dreja.\n\nWhat Catherine was aware of was that her lover was making plans for the future: 'Jimmy was still playing guitar with the Yardbirds, but he was also working on putting his own group together in England. He said he was hoping to come back to California soon. If the band did well, then they'd do a West Coast tour.'\n\nAlthough both Page and Peter Grant were thoroughly cognisant of the shifts in the music scene in the United States, other members of the Yardbirds were not as clued up. 'We weren't aware that the market was changing into an album market,' said Jim McCarty. 'We thought that maybe an updated _Roger the Engineer_ would have worked for us. At this time we'd all wear kaftans... The thing about Keith and I was that we were so tired of travelling around and playing \"Smokestack Lightning\". And we couldn't see any future for it. Jimmy had been in the band a year or so. And was fresh.'\n\nIn fact, Page had been in the Yardbirds for almost two years. But his steely determination remained undimmed: 'When the band finally folded and I wanted to try something new, I just wanted to carry on rocking. I had this great stockpile of material in my mind and songs and riffs I had written down on tape at the time. So it was really handy. I knew which way I wanted the group to go if I could get a group together. Thankfully I did.'\n\nHowever, Grant hadn't only discussed the shifting American sands with Page. The guitarist's old sparring partner Jeff Beck \u2013 himself a client of RAK Management, Simon Napier-Bell having finally decided to absent himself from Beck's career \u2013 also benefitted from this information, as he told _Uncut_ magazine: 'Peter was fantastic. What he realised, long before the others, was that there was a sizeable audience for an underground scene away from the Top 40. We'd been playing shitholes in England and then he took us to America...'\n\nThe Yardbirds' last show was at Luton Technical College on 7 July 1968. With the band to all intents dead and buried, Grant went on the road in the US with Beck, who \u2013 having turned down the task of replacing Syd Barrett in Pink Floyd \u2013 had formed the Jeff Beck Group, with former Steampacket and Shotgun Express singer Rod Stewart, bass player Ron Wood and drummer Micky Waller.\n\nThe Jeff Beck Group's US debut was on 14 June 1968 in New York, at Bill Graham's recently opened Fillmore East, where they played four shows, the opening salvo of a six-week tour. Thanks to the headliners, the Grateful Dead, the 3,000-strong venue was sold out. And although Rod Stewart suffered from such extreme stage fright that he at first delivered his raspy, soulful vocals from behind the amps, the act's performance was a colossal success, described by Robert Shelton the next day in the _New York Times_ as 'wild and visionary', the writer declaring that the British group had upstaged the Grateful Dead. The Jeff Beck Group's burgeoning reputation was only enhanced when, four days later, they began a week's residency at Manhattan's Scene Club, during which both Jimi Hendrix and Eric Clapton regularly got up onstage to jam with the four-piece. And, as a consequence of the wildly successful Fillmore East shows, they were booked to play six nights at San Francisco's Fillmore West.\n\nWhat was most extraordinary about this groundbreaking tour was that the Jeff Beck Group did not have an album in the stores in the United States. Mickie Most's hit-singles-factory ethos meant he was indifferent to Beck's blues inklings and to his group as a whole. He was also dismissive of Rod Stewart's singing strengths, making Beck take the lead vocals on 'Hi Ho Silver Lining', a Top 20 UK hit in the early spring of 1967, a song more suited for pub sing-along choruses than as a flagship single for a pioneering British guitarist. 'Hi Ho Silver Lining' did, however, have the redeeming feature of including 'Beck's Bolero' as its B-side.\n\nBolstered by Robert Shelton's _New York Times_ review, Grant contacted the head of Epic Records in New York and Beck's _Truth_ album, already recorded during several long, stretched-out sessions, was rush-released in August 1968, quickly climbing to number 15 in the US charts. Among the varied tracks, which included a reworking of the Yardbirds' 'Shapes of Things', Jerome Kern and Oscar Hammerstein's 'Ol' Man River' and Howlin' Wolf's 'I Ain't Superstitious', was a version of Muddy Waters' 1962 classic 'You Shook Me'. Jimmy Page pricked up his ears.\n\n## 7\n\n# 'LIKE A LEAD ZEPPELIN'\n\nYou know that inner sensation when you can almost touch the several directions your life might take, when you know the decision you are about to make will probably be final and there will be no return to what was there before? It can be troublesome and worrying. Or it can be enlightening and energising and exciting. For Jimmy Page it was certainly the latter.\n\nAlthough the Yardbirds had split up, they were still contracted to undertake a Scandinavian tour in September 1968. Keith Relf and Jim McCarty left the band during the summer and were in the process of putting together Renaissance, their folk-influenced art-rock project, but they licensed Page to use the Yardbirds name on the Scandinavian dates only.\n\nThe idea of forming a very specific type of band had been gestating gradually in Page's mind ever since the 'Beck's Bolero' session in 1966. The singer, he had then hoped, might be Steve Marriott of the Small Faces, or Stevie Winwood of the Spencer Davis Group. In the summer of 1968 he was still searching for that kind of white soul singer.\n\nPage had been working at Immediate on sessions with his old friend Chris Farlowe, whose first, unreleased solo album he had produced and paid for. Farlowe was a possibility for Page's 'the New Yardbirds', the working title for the band, as he possessed a similar vocal style to those blue-eyed soul stalwarts, but unfortunately little of their charisma. 'Jimmy liked me,' Farlowe told _Record Collector_ in 2017. 'But I was an established artist and a bit of a mod.' (Later, after Led Zeppelin, Farlowe would be called back into service by Page, singing on three tracks on his 1988 solo album _Outrider_.)\n\nPage moved on to another emerging singer, like himself another teenage prodigy: Terry Reid, conveniently managed by Mickie Most, and \u2013 even though he was still only 19 \u2013 the handsome former vocalist with UK dancehall favourites Peter Jay and the Jaywalkers. Attempting to reinvent himself as an 'underground' artist, Reid had released a critically acclaimed album, _Bang, Bang You're Terry Reid_. But Reid had just accepted the task of opening for Cream on their final US tour and was obliged to turn down Page's approach. However, he recommended that Page check out another vocalist he had heard in the Midlands, one Robert Plant. Reid had played on the same bill as Plant's outfit Band of Joy, and mentioned that the group also had a remarkable drummer \u2013 John Bonham.\n\nBy now Robert Plant had been somewhat taken under the wing of Alexis Korner, who would let him stay at his Bayswater flat, sleeping on the same couch that had \u2013 something Korner would always point out \u2013 once been used for the same purpose by Muddy Waters.\n\nPlaying blues music with Alexis Korner appealed greatly to Plant, and at first he failed to respond to the phone messages Peter Grant left for him at the Three Men in a Boat pub in Bloxwich, where Plant had rented a room.\n\nOn 20 July 1968 the shockingly named Obs-Tweedle \u2013 the latest group Plant was singing in \u2013 played a show at the West Midlands College of Education. There was only a small crowd, but in it were Page, Grant and Chris Dreja. 'This big guy with a University of Toronto sweatshirt let us in backstage,' said Grant, 'and I remember Jimmy saying, \"Crikey, they've got a big roadie!\" It turned out to be Robert Plant! Jimmy loved Robert straight away.'\n\nAlthough they were singularly unimpressed by Obs-Tweedle, Plant's clear vocal magic shook Page. How on earth, he wondered, had this singer not yet been discovered? Page was utterly unaware that Plant had already had a pair of flop singles released by CBS, or of his work with Alexis Korner, or that Move manager Tony Secunda was fluttering around him. Instead he worried that the vocalist might have some worrying hidden character defect that had so far kept him from the stellar heights he clearly deserved.\n\nAll the same, a few days later Plant received a telegram: 'Priority \u2013 Robert Plant. Tried phoning you several times. Please call if you are interested in joining the Yardbirds. Peter Grant.'\n\nAt the end of July 1968 Page invited Plant to his Pangbourne home. Arriving off the train, Plant, who was still only 19, was extremely impressed by the house, which was stuffed with dazzling antiques and was an audiophile's dream, with Tannoy speakers and Fischer amplifier, on which Page would excitedly play the Beatles' _Magical Mystery Tour_ double EP. Of course, these possessions were thanks to Page's session work rather than from his relatively brief career with the Yardbirds; later Plant would discover that Page was extremely shrewd financially. And it wasn't just the house that impressed the young singer: 'He was so charismatic,' thought Plant.\n\nPage, however, was not so sure about Plant: 'I liked Robert. He obviously had a great voice and a lot of enthusiasm. But I wasn't sure yet how he was going to be on stage: what he'd be like once we actually got together as a band and started playing.' He also told Mick Wall: 'I didn't know what he'd be like yet as a songwriter. When I first saw him he was a singer first and foremost.'\n\nNevertheless, Plant stayed at Page's Pangbourne home for an entire week. The singer was already especially taken by the West Coast sounds of acts like Buffalo Springfield, a particular favourite of Page's, Jefferson Airplane, Moby Grape and Love. However, he also had a thorough grounding in relatively esoteric blues music. And Page's musical library not only included all these favourites, but extended far beyond them.\n\nPlant shared Page's adoration of the early work of Elvis Presley, as well as such blues staples as Sonny Boy Williamson, Solomon Burke, Howlin' Wolf, Muddy Waters \u2013 whose 'You Shook Me' was on the Pangbourne playlist \u2013 and Robert Johnson, as well as rock 'n' roll greats like Jerry Lee Lewis, Chuck Berry and Eddie Cochran. And such arcane curios as Don and Dewey's sensational 'Justine'.\n\nThey listened to the folk-rock sound of Fairport Convention's 'If I Had a Ribbon Bow', and the joy on Plant's face was writ large when he discovered that Page owned a copy of Joan Baez singing 'Babe I'm Gonna Leave You', written in the 1950s by Anne Bredon. Page had already decided, he told the infectiously enthusiastic Midlands Mod-turned-hippie, that the group he was planning would play an electric version of the song. He had played an acoustic 'Babe I'm Gonna Leave You' while touring with Marianne Faithfull.\n\nPlant's position in the group was sealed when Page asked him if he knew any good drummers. Having been unable to prise Keith Moon away from the Who, Page had had his eye out for a drummer of similar stature and talent, but he'd been turned down by the one he liked most: B. J. Wilson, who played with Procol Harum, riding high on their year-old mega-hit 'Whiter Shade of Pale'. Page had also checked out Mitch Mitchell, who was believed to be dissatisfied in the Jimi Hendrix Experience; Aynsley Dunbar, a Liverpudlian who had played with John Mayall's Bluesbreakers and Jeff Beck; and Clem Cattini, a super-session drummer and friend of Page's who refused to be parted from his regular and substantial session pay.\n\nNaturally Plant suggested his old mucker John Bonham, but this was definitely no old-mates act. Plant knew that Bonham was almost certainly the finest drummer he had ever encountered, an opinion shared by most musicians who came across him.\n\nBonham was lengthily touring Britain with Tim Rose, a gruff-sounding American singer-songwriter who had acquired kudos on the 'underground' scene in London. And when Page went to see them play on 31 July 1968 at the Country Club in Belsize Park, he immediately knew who his drummer would be.\n\nThis would prove important for Plant's place within Led Zeppelin, providing him with a certain subconscious strength. Like a neophyte offering sacrifice, he had brought someone into the group. He had helped form Led Zeppelin. Bonham's presence also meant that he immediately had an ally in the group.\n\nNoisy and boisterous, determinedly non-Londoners, Plant and Bonham were the antithesis of the quiet, rather shy but sophisticated Page; in some ways, of course, they permitted him to present a more fully formed variant of himself, round him off as he vicariously lived out fantasies through them.\n\nPage needed a new bass player too, as Chris Dreja had finally decided to bow out and become a photographer. His replacement \u2013 after Page had been unable to get John Entwistle as part of a failed raid on the ever-fractious Who's rhythm section \u2013 was someone with whom Page was already well acquainted.\n\nBorn on 3 January 1946 \u2013 like Page he was a Capricorn \u2013 and brought up in Kent, the boarding-school-educated John Paul Jones came from a family of professional musicians. He first met Page at a Decca session in West Hampstead in 1964.\n\n'He was very passionate about music, which is why I immediately took to him,' John Paul Jones told _Uncut_ magazine. 'Very knowledgeable about music too. About old records. He was always very interested in recording. We were kind of geeks in those days, in a way. At the end of a session, most of the musicians would sit back and read their golf magazines, but we would always go into the control room to listen to playbacks and watch the engineers and producers. We both wanted to know how things were done. He was quiet, and he was reserved...\n\n'He was the youngest session musician until I came along. We were always really glad to see each other on the sessions, because it meant that you had a young, hip rhythm section. The drummer would be older, I suppose, and Big Jim Sullivan was older, but we were relatively young compared with all the other session musicians. In those days, to be a session musician was considered the pinnacle of your professional music career. If you got a foot in, at that early age, you kind of held on to it.'\n\nJones was already aware of Page when they met: 'I remember him having a reputation almost before I turned professional in early 1963, when he was with Neil Christian and the Crusaders. It was always, \"You've got to hear this guy.\" In fact, I never actually heard him before we worked together, but yes, I knew of his reputation.'\n\nWhen the four musicians rehearsed together for the first time in the middle of August 1968 in a tiny rehearsal room on Gerrard Street in London's Chinatown, their amps almost falling on top of them, the tune they immediately worked on was 'Train Kept A-Rollin'', straight out of recent Yardbirds live sets.\n\nJones thought their performance was 'stunning'. And Page was equally as enthusiastic: 'It was there immediately. It was so powerful that I don't remember what we played after that. For me it was just like, \"Crikey!\" I mean, I'd had moments of elation with groups before, but nothing as intense as that. It was like a thunderbolt, a lightning flash \u2013 boosh! Everyone sort of went, \"Wow.\"'\n\n'He had this whole thing about a dynamic rock band... a whole light-and-shade thing,' said Jones, 'which was pivotal, and it informed every musical decision that he made. I mean, there weren't dynamic rock bands in those days. Everything was either a soft, folky-rock type thing, or just blasting all the time. It was very important to him.'\n\nAt the end of that first rehearsal, Page organised baked beans on toast for his three hungry new musical compatriots \u2013 then diligently collected from each of them the few pence that the food cost. With his accountancy training, Plant looked on approvingly.\n\nBefore they could even get out on the road as 'the New Yardbirds', fulfilling dates already booked, the four musicians found themselves in the recording studio. There was another previous booking to honour: Jones was scheduled to record an entire album with P. J. Proby, the American _enfant terrible_ of British pop music. Proby had earned tremendous media approbation for having split his trousers on several dates on a 1965 package tour. But he had a powerful, almost operatic voice, reminiscent of Elvis Presley's deep baritone register; in fact, like Ral Donner, with whom Plant was always taken, Proby had sung demo records of songwriters' tunes for Elvis.\n\nThere were some definite financial upsides to the obligation, which Jones \u2013 listed as 'producer' \u2013 could not get out of: 'I was committed to doing all the arrangements for the album. As we were talking about rehearsing at the time, I thought it would be a handy source of income. I had to book a band anyway, so I thought I'd book everybody I knew.'\n\nPage had already worked with P. J. Proby, on his 'Hold Me' single and his first album, _I Am P. J. Proby_ , in 1964, playing rhythm guitar to Big Jim Sullivan's lead. The two days of sessions for the album that would be known as _Three Week Hero_ began on 25 August 1968, and without vocal duties to perform Plant played harmonica on the LP.\n\nProby was so impressed with the results that he asked the four musicians if they would back him on an American tour. Unable to entirely forget his old session-playing ways, Page half-agreed to do it, though with the proviso that they were planning an exploratory set of dates on their own in the United States. So could he make a decision after those shows?\n\nProby said, 'Look, from what I've heard and the way you boys played tonight, not only are you not going to be my backing band, I'm going to say goodbye right now, because I don't think I'm ever going to see you again. That's how successful you're going to be. You're exactly what they want; you play all that psychedelic stuff and everything. You're going to go over there and go down so great I don't think you're ever going to come home.'\n\nA Scandinavian tour had been booked in for the Yardbirds that had to be honoured, and it offered an opportunity to iron out the new group's set. Out of sight of the UK media, the New Yardbirds played their first-ever show at Denmark's Gladsaxe Teen Club, a gymnasium complex a few miles out of Copenhagen, on 7 September 1968.\n\nThere was a healthy turnout, some 1,200 strong. 'It was only a short while before the concert that we realised it wouldn't be the \"real\" Yardbirds that were going to play,' said Jorgen Angel, who was 17 and photographed the performance. Page wore a white ruffled shirt over white trousers. Both Plant and Bonham chose floral shirts, Plant's unbuttoned to the waist. Jones wore a long satin coat.\n\nThe set included a number of Yardbirds staples \u2013 'For Your Love', 'White Summer', 'You Shook Me', 'Dazed and Confused' and 'Train Kept A-Rollin''. But they also played 'Communication Breakdown', 'I Can't Quit You Baby', 'How Many More Times' and a cover of Garnet Mimms's 'As Long As I Have You'.\n\nA subsequent newsletter issued by the Gladsaxe Teen Club assessed this first Led Zeppelin show in a most positive light: 'Their performance and their music were absolutely flawless, and the music continued to ring nicely in the ears for some time after the curtains were drawn after their show. Let me in particular give my praise to Jimmy Page, who has made a great job with the three new men. They really succeeded and in particular the guitar solo by Page created huge applause... We can therefore conclude that the new Yardbirds are at least as good as the old ones were.'\n\nThe gig was followed by another that evening, at the Brondby Pop-Club in the south of Copenhagen, and the next day the New Yardbirds played three more shows. Then they had three days off before crossing the border into Sweden, where they played five concerts, many at outdoor amusement parks. The Swedish crowd adored them, pounding their feet into the floor, demanding more.\n\nJust over two weeks after the end of that Scandinavian tour, on 27 September 1968, the four musicians went into Olympic Sound Studios in Barnes, which had an eight-track set-up that was becoming popular with other 'underground' acts \u2013 that summer the Rolling Stones had made _Beggars Banquet_ there \u2013 to record the first Led Zeppelin album.\n\nPage was in charge of the sessions, booked on weekend nights' downtime, and he was personally footing the bill. He played the Fender Telecaster that Jeff Beck had given him as thanks for getting him the job in the Yardbirds: by now the guitar had received its psychedelic paint job. As he would be on all Led Zeppelin records, Page was producer, with the in-house engineer Glyn Johns ably assisting him.\n\n'I wanted artistic control in a vice-like grip, because I knew exactly what I wanted to do with the band. That first record sounded so good because I had gotten so much experience in the recording studio. I knew precisely what I was after and how to get it.'\n\nAll his experience and mastery in studio sessions came to the fore. 'Certainly on the first album I had a very good idea of what I wanted to try and get with the band. Because at that stage I was extremely instrumental in the total direction of it. Obviously there was a definite concept of what one was trying to achieve there, and it was done. But we were definitely right out there on a limb, weren't we? Doing what we believed in. And it didn't really follow any sort of trend that was going on at all \u2013 or certainly nothing relative to any other band.'\n\nEmploying natural room ambience, Page enhanced the reverb and recording texture on the record. In addition to placing a microphone in front of the drums and amplifiers, he put an additional microphone some 20 feet from the amplifier, recording the time-lag balance between the two. 'Everything leaked into everything else. Which was part of the sound,' said Jones.\n\n'Backwards echo was a technique that I invented at the time,' Page said. 'The engineer said it couldn't be done but I knew it could because I'd already suggested it earlier on a Yardbirds track. You reverse the tape, record the echo, then put it around the right way again so that the echo precedes the signal. It created a fantastic sound, but it was really only employed in the first album, on the end of \"You Shook Me\".'\n\nIn total there were 30 hours of sessions, costing \u00a31,782. A high point was 'Babe I'm Gonna Leave You', over which Page and Plant had bonded at Pangbourne. And there was a clutch of further great songs: 'Dazed and Confused', 'Communication Breakdown', 'Good Times Bad Times', 'Your Time Is Gonna Come' and a pair of Willie Dixon tunes, 'I Can't Quit You Baby' and 'You Shook Me'.\n\nBy personally financing the first Led Zeppelin album, Page displayed his knowledge of the music business. Almost the only other successful British artist to own the master tapes of his songs was Dave Clark, leader of the phenomenally successful Dave Clark Five, who had rivalled the Beatles in the USA and was famously financially astute. Page knew that Clark's fortune was a great deal larger than the Beatles' simply because he had always owned his own recordings.\n\n'I was up for it,' said Glyn Johns of his engineering work on _Led Zeppelin_ , 'as Jimmy and I grew up in the same town and had been pals since the early sixties, and John had been the number-one session bass player in London for years. When I was an engineer I would see him almost every day, and a nicer guy you could not wish to meet.\n\n'Knowing that anything these two put together was bound to be pretty good, I turned up at Olympic a couple of weeks later, not having any real idea of what I was walking into. I was blown off my feet. The album that we made in the next nine days was a landmark in rock and roll history, taking it to another level altogether.\n\n'The sound they created, the arrangements they came up with, and the standard of musicianship were equally astonishing. Each session seemed to be more exciting than the last, as what they had prepared unravelled in front of me. All I had to do was press record, sit back and try to contain the excitement of being in the same room as what was going on.\n\n'The stereo mix of this record is certainly one of the best sounding that I ever made, but the credit has to go to the band, as all I did was try to faithfully put down on tape what they were giving me, adding a little echo here and there to enhance the mood.'\n\nJohns took the LP's acetate to a production meeting for _The Rolling Stones Rock and Roll Circus_ television film. He played it to Mick Jagger, hoping to persuade the Stones singer to include Led Zeppelin on the show: 'I felt the band was going to be huge and therefore we should have them on the show, as it would be an enormous coup. My suggestion fell on deaf ears, as Mick did not get it at all.'\n\nMick Jagger's was not the only stellar testimonial that Glyn Johns sought: 'A couple of months later I dragged George Harrison to Olympic on the way home from a Beatles session and played him the master tape with the same result \u2013 he didn't get it either. I found this slightly disconcerting, as I could not understand why they did not get what was so exciting to me... Jimmy and John Paul Jones were from the same era with the same influences, and yet Mick and George openly disliked what they had done, seeing no value in it at all. In any event, they were perfectly entitled to their opinion, and fortunately a large proportion of the record-buying population disagreed with them.'\n\nThere were those who did get it, however. A little later, on Christmas Eve, Johns played the album again, to another guitarist, Peter Frampton, his house guest. 'My jaw dropped,' remembered Frampton. Then the phone rang. It was Steve Marriott on the line from Paris, where the Small Faces had just played a show. 'He said, \"I've left: can I join your band?\"' And so was formed Humble Pie, who from time to time would become something of a simulacrum of Led Zeppelin. When the second record, _Led Zeppelin II_ , was released, however, Steve Marriott would find reasons to seriously dislike Jimmy Page's group.\n\nExciting new album or not in the works, there was still the small matter of the Yardbirds' schedule to fulfil. To Page's concern, these shows were often only half-full at best.\n\nThe UK debut of the New Yardbirds came on 4 October 1968, at Newcastle's Mayfair Ballroom, an established venue on the rock circuit \u2013 but only a few dozen people turned up. 'We originally thought that by calling ourselves the New Yardbirds we would be able to keep a sort of continuity from the early days of the old group,' explained Page. Yet on finding a group onstage that lacked Keith Relf, a few audience members asked for their money back. 'We realised we were working under false pretences. The thing had quickly gone beyond where the Yardbirds left off. We all agreed there was no point in retaining the \"New Yardbirds\" tag, so that was when we decided to change the band's name.'\n\nIronically, Terry Reid, who had been considered for the new line-up by Page and had recommended Plant, was supposed to be on the bill and featured on the early promotional posters, but his support-act slot on Cream's farewell tour of the United States meant he'd had to pull out of the Newcastle show. He was replaced, at a week's notice, by New York Public Library. 'We'd met Jimmy Page years before, when he was in Neil Christian's Crusaders,' said Tez Stokes, the New York Public Library's guitarist. 'He was a very well-respected guitar player, but very quiet, reserved and shy. John Paul Jones I met when he was the bass player with the Tony Meehan band.'\n\n'Before the gig, either the band or their roadie asked if they could borrow our organ,' said Charlie Harcourt, guitarist with the Junco Partners, also on the bill. 'They didn't seem very well prepared. We told them no, they should have brought their own.'\n\n'I stood right at the front and watched,' said Tez Stokes. 'There was hardly anyone there. Some people were sitting in the balcony, looking down.'\n\nThe Mayfair had a revolving stage, on which the next act would set up while the group preceding them were playing. 'Zeppelin weren't playing when the stage came round \u2013 they were farting around. Jones was on the left side in front of a silver-fronted Fender Bassman amp,' said Charlie Foskett, who was in the audience that night. 'He hit a note and the cloth on the front of the amp wobbled, which I thought was cool. Page posed about with his Les Paul. They weren't using monitors, all of that came later. The riser, which had the amps, waiting for someone to go: \"One, two, three, four.\" That came pretty quickly.\n\n'A lot of the audience went: \"Where's the Yardbirds? Who's this?\" It wasn't the Yardbirds, it was another band. After five minutes it was: \"This is great!\" Everybody had forgotten about the Yardbirds. Nobody knew the name Led Zeppelin then, of course, so it was: \"Yeah, this is the New Yardbirds!\"\n\n'I thought the band were terrific. The first thing was the energy. Everything was a bunch of riffs. Page's guitar was ripping away. I've heard better soloists, but he was a great showman, a walking skeleton with skin-tight bell-bottom jeans, the hair and tons of posing.\n\n'I was very much into their overall noise and energy, along with the visual thing. They really had their chops together. It was as loud as the gear would allow it to be at that point, and full throttle, including the couple of Yardbirds hits they did. The discerning music lovers were glued to it.'\n\n'I wasn't very impressed with Robert Plant. He was a bit fey,' thought Ray Laidlaw, the drummer with Downtown Faction, also on the bill, who would evolve into Lindisfarne. 'I was more interested in the mechanics of the band. Bonham was pretty damn good. John Paul Jones kept himself in the background, and Jimmy Page was a fantastic player. But the band wasn't finely tuned.\n\n'Page was basing it on the Jeff Beck Group. He knew how successful they'd been in the US, and was using that as his model. I wouldn't be surprised if after each gig he'd go through it with the band: a little more of this, a bit less of that, polish that, drop this number, bring that one in. Like managing a football team.'\n\n'There was a track on _Truth_ called \"You Shook Me\", with Rod Stewart singing,' recalled Bob Sargeant, the keyboard player with the Junco Partners. 'The New Yardbirds did that number at the Mayfair. They also did \"Shapes of Things\", the old Yardbirds hit. And they played \"Communication Breakdown\" and a couple of other songs that turned up on the first Zeppelin album.'\n\nFourteen days later, the New Yardbirds performed their second UK show, at London's Marquee Club. They were observed to play at what would become their characteristic phenomenal volume. Although the club wasn't entirely full, there was a disproportionate percentage of male drummers making up the audience. John Bonham's reputation preceded him.\n\nThe next night they played at Liverpool University. As they entered the building, two students, John Bold and his friend Chris Wallis, offered to carry in Page's guitar and other equipment, thereby guaranteeing free entry to the hall. Meanwhile, Wallis sold Plant a lump of hash. 'The group was sensational,' said Bold. 'Phenomenally loud. And very dramatic as Jimmy Page waved his violin bow about during \"Dazed and Confused\".'\n\nBy the next show, six days later, at the University of Surrey in Guildford, the 'Lead Zeppelin' name had been decided upon, though the group were already billed as the New Yardbirds. Always with a keen eye on things over the Atlantic, Peter Grant was concerned that US record buyers might think the name was pronounced 'Leed' Zeppelin, so he made an executive decision to drop the 'a' in 'Lead', transmogrifying the name to Led Zeppelin.\n\nThe group were paid \u00a3300 for the show, with an audience of no more than 200. During their soundcheck they played a pair of Creation songs, 'Painter Man' and 'Making Time', as they did regularly at this time. Creation's Eddie Phillips was, of course, a primary source for Page's appropriation of the violin bow during 'Dazed and Confused'.\n\nGlen Colson was a drum roadie for the first couple of months of Led Zeppelin's life: 'Guildford was one of the first gigs I did with them. I set the drums up. And I used to sit behind John Bonham and hand him sticks. Why was he so good? Because he was incredibly technically brilliant. Musically, he was the driving force in Led Zeppelin.\n\n'They were only playing the first album, slow and arranged blues songs. They were so heavy and good that it was obvious there was something going on: I was into music but I had no idea what they were up to. They were heavy and positive, louder and flashier than anyone else. Their angle was that they'd found this incredibly heavy drummer, like the heaviest session man you've ever seen. It was all very professional. A foregone conclusion they would be massive.'\n\nYet even this early in their career, the extraordinarily talented drummer seemed to come with a difficult subtext. 'Bonham behaved like someone who might be a gypsy. He nicked everything he could out of the dressing room: not only the rug, which he just rolled up, but also the tannoy speakers. Then he smashed the place up. The others were just sitting there, watching. Robert Plant was the PR guy, saying hello to everyone, very polite. The other two just sat there looking at me. I was embarrassed to be in the dressing room. Page sat there, saying nothing, like a voyeur. John Paul Jones never said a word: you could mistake him for a road-sweeper. Occasionally they would snigger about something to each other.' In fairness, it must be said that often the last thing a performer requires before going onstage is a cosy chat with the staff. They are focusing on delivering a performance, their opening lines or notes ready in their heads.\n\nWhat surprised Colson was when his friend Kenny Pickett told him that the four musicians knew he was _au fait_ with modern music and asked if he could provide a list of records they could use as inspirational reference points. 'Although Page and Jones had worked in studios on sessions for years, they had never really written any music. They weren't sure if they could do it consistently: they were looking for ideas. So I gave them a list of stuff, including Spirit and Moby Grape, I recall \u2013 though Plant may have already been familiar with those.'\n\nIn the British provinces, audiences remained hard to find, while in London it was easier: on 9 November Led Zeppelin topped the bill at the Roundhouse in Chalk Farm, already established as the premier underground venue in the capital. Their fee was \u00a3150; when they played Bath Pavilion five weeks later, they received only \u00a375.\n\nThe morning before the Roundhouse date, at West Bromwich register office, 19-year-old Robert Plant married Maureen Wilson, his girlfriend since 1966. Their daughter Carmen was born shortly before, on 21 October, and being the well-brought-up, socially responsible Midlander, Plant was legitimising Carmen and making an honest woman of Maureen, a qualified nurse whose father, once head of Calcutta's mounted police, owned a steel factory in the West Midlands. During the previous couple of years of struggle, Maureen had often kept them financially afloat.\n\nDriving down to London in time for the Roundhouse soundcheck, Plant and his new wife celebrated their marriage following the show.\n\nAlthough on occasions Plant was considered by some to be trying a little too hard in his live performances, he and John Bonham \u2013 the raw recruits \u2013 were settling in well with the pair of seasoned session players. 'Jimmy was a member of the Yardbirds and he was a session musician, so he was successful,' said the singer. 'Jonesy was much more the backroom boy... So their sort of positions and previous roles weren't really that daunting.\n\n'But how they handled themselves with us was important. Jonesy was a bit... not withdrawn, but he stands back a little and shoots the odd bit of dialogue into the air. It's good stuff but an acquired taste really. And Jimmy's personality initially was... I don't think I'd ever come across a personality like it before. He had a demeanour that you had to adjust to; certainly it wasn't very casual to start with. But then again, the music was so intense that everything was intense.\n\n'The ambition was intense and the delivery was intense, and where we were going was intense. Nobody knew where the fuck it was, but we all knew that this power was ridiculous from the beginning. So it was very hard to relax, sit down and have a beer and be the guys from the Black Country. Bonzo and I were much more basic in every respect, in how to deal with everything \u2013 including Jimmy. Because he had to be dealt with.'\n\n## 8\n\n# AMERICAN ADVANCES\n\nThe first person to whom Peter Grant offered the Led Zeppelin album was Chris Blackwell, who owned Island Records, the premier underground label in the UK. Island's publishing division occupied another floor in the same building as Grant's at 155 Oxford Street. Blackwell almost bit. 'I knew Peter and really liked what I heard. I almost signed them. In the end, after all that happened, I was glad I didn't. I thought they treated women very badly. Jimmy Page was already very rich before he even started Led Zeppelin. But he was always kinda cheap,' said Blackwell.\n\nBut Blackwell did offer Grant a $25,000 per album deal for world rights, excluding the US and Canada. Blackwell later told Robert Greenfield in _The Last Sultan_ : 'It was a handshake deal, but I was dealing with Grant and so it wasn't a deal until it really was a deal.' Accordingly, armed with Blackwell's offer, Grant and Page flew to New York at the end of November 1968. Grant went straight to Atlantic Records, originally formed as a jazz and R&B label in 1948, a beacon for soulful black music of every kind. He had already befriended Ahmet Ertegun and his wife Mica, an established interior designer who restyled Atlantic's London offices, and of course Page had connected with both Ertegun and his partner at Atlantic, Jerry Wexler, when he had visited New York at the behest of Bert Berns.\n\nBack in June 1966, Ertegun had flown up to Los Angeles from Mexico City, where he had watched Tottenham Hotspur defeat the Mexican national football team in a friendly fixture by a single goal, in a 1\u20130 result. Ertegun had been urged to take the trip to LA by Wexler, as there was a group in the city who were attracting plenty of interest from other labels, but Wexler, who was only really interested in black music, needed Ertegun's opinion. The band was Buffalo Springfield, and the Atlantic Records founder was knocked out. 'First of all, the songs they wrote didn't resemble anything that anybody else was doing,' he said in Greenfield's _The Last Sultan_. 'They also had three outstanding lead singers who were also great guitar players \u2013 Neil Young, Stephen Stills, and Richie Furay... The power in Buffalo Springfield was too incredible. They were one of the greatest rock 'n' roll bands I've ever heard in my life.'\n\nSo rapidly did Buffalo Springfield become the sensation of first the Sunset Strip and then the entire American recording industry that you can almost taste their nearly inevitable rapid disintegration less than a year later, after they played their final gig at the Long Beach Arena in California.\n\nShortly before Buffalo Springfield split up, Neil Young mentioned an act called Iron Butterfly to Ertegun that he thought Atlantic should sign. Young's management team \u2013 very soon to be his ex-management team \u2013 went to see them. 'I thought they were shit, actually,' said Brian Stone, one of Young's managers, in _The Last Sultan_. 'Cacophonous. Out of tune. Awful. I called Ahmet and I said we had a new band, and he saw them at the Purple Onion or some crazy place on Sunset Strip and signed them. Iron Butterfly became such a giant act that it made Jerry Wexler go out and sign Led Zeppelin.'\n\nIron Butterfly's first album was almost not even released by Atlantic: Ertegun considered each potential single to be 'shit'. But then there was a twist of fate that was a kind of zeitgeist moment in the history of the rise of underground rock: the new college FM stations began playing Iron Butterfly's album's title track, 'In-A-Gadda-Da-Vida', itself a twist on 'In the Garden of Eden'. As a consequence Iron Butterfly's _In-A-Gadda-Da-Vida_ album became the number two album in the US charts for the next two years.\n\nAnd Atlantic's diversification didn't stop with Iron Butterfly; they had signed several other white rock groups and enjoyed considerable success with them, including Vanilla Fudge and \u2013 the most famous of all, part of Page's template for Led Zeppelin \u2013 Cream. Clapton's act, who had been signed by Ertegun, were at that time on their final tour of the United States, having announced they were to split up.\n\nIn late 1968 Jerry Wexler was producing _Dusty in Memphis_ for another Springfield \u2013 Dusty this time \u2013 an LP that would seal her reputation. 'The story,' recalled Grant, 'is that she was down at Jerry Wexler's house and he told her about this new group that was in the offing with Jimmy Page and John Paul Jones. She said she'd worked with Jonesy on arrangements and such like, and Jerry was knocked out.'\n\nNot only was Wexler acquainted with Page's assiduous approach to his art, he had somehow also learned of John Bonham being 'a hell of a drummer' \u2013 though he had never heard of Robert Plant.\n\nSo when Grant arrived in New York with Page that November with the tapes of Led Zeppelin's first album, the timing was perfect. Wexler was up for signing an act that would keep him at the vanguard of this new 'underground' world, while Grant used the Chris Blackwell offer as a bargaining chip to push up the Atlantic figure. Wexler offered Grant and Page $75,000 per album for American and Canadian rights for five albums over five years.\n\nWexler was then offered Led Zeppelin's remaining world rights for $35,000. After the UK head of Polydor Records turned down Wexler's tentative offer of $20,000 for the UK rights to Led Zeppelin, the Atlantic man bought those remaining world rights for the suggested $35,000. For Wexler and Atlantic Records it would prove to be an extraordinary financial coup.\n\n'To tell you the truth,' said Blackwell, 'I'm glad I didn't get them because it wouldn't have worked for us at Island. Too dark: I couldn't have dealt with it.'\n\nAfter some wrangling, Wexler gave Grant a cheque for over $100,000, the highest advance paid for any act at that time. The deal included a promise \u2013 insisted upon by Page \u2013 that the group would appear on the Atlantic label itself, not on Atco, the subsidiary label on which the white rock acts so far had appeared. Page had also made it clear to Grant what his deal would be with Led Zeppelin: he would receive 50 per cent of all earnings, the rest to be shared among the other three musicians and Grant.\n\nOnce Wexler signed Led Zeppelin, he had almost nothing more to do with them, passing everyday control of the group over to Ertegun. Aware that Led Zeppelin were Page's group, the ever-diplomatic Ertegun focused on building a relationship with the guitarist and his manager, somewhat to the detriment of the perpetually charming Plant.\n\nNotwithstanding the group's on-the-road excesses, Ahmet Ertegun would accompany them on tour, turning up unexpectedly in their dressing rooms at some backwoods venue to wish them well before they went onstage. Although he became good friends with Grant, there were also constant arguments with the manager over album costs or record release dates. The efforts Ertegun made to keep Led Zeppelin happy would pay off a thousand-fold. 'Ahmet was the guy who went to the concerts and dealt with the managers,' said Wexler. 'He plunged into it heart and soul. He became their friend and it was those efforts that made Atlantic a monster company.'\n\nHowever, Led Zeppelin never formally signed to Atlantic itself. Having set up publishing and production companies for his prot\u00e9g\u00e9s, Grant was able to cut a deal from a different angle than normal: 'We didn't sign direct to the label, we had a production company called Superhype. The title came from Jimmy, who was aware of the hype surrounding us at the time. So I did a tongue-in-cheek number and called it Superhype Music Inc. We sold off the publishing company some years later. The whole deal with Atlantic gave us various clauses that we were able to use in our favour.'\n\nThis was a difficult age when it came to expressing personal ambition, artistic, financial or otherwise. In the underground it was 'uncool' to value possessions of any sort \u2013 despite the evident wealth of the Rolling Stones and the Beatles, who were the true figureheads of the new hippie counter-culture. Led Zeppelin's huge advance from Atlantic Records was interpreted as evidence of their moral laxity and greed; you may feel that overlooking the conceptual irony of calling your production company Superhype said it all. Superhype? With its tongue firmly in its cheek, the company's name was utterly of its satirical time, like the Great American Disaster, the fashionable Fulham Road purveyors of genuine US-style hamburgers in London, where the walls were decorated with front pages of the _New York Times_ , reporting such national traumas as the Wall Street Crash and the assassination of JFK; or Mr Freedom, which opened in 1969 at 430 Kings Road, peddling pop-art fashion by Tommy Roberts, Page's good friend. Didn't anyone notice the almost Ionesco-like absurdity of their name, Led Zeppelin?\n\nInterestingly, Grant never had a formal contract with the four musicians in Zeppelin. 'That was a very strange thing,' said John Paul Jones. 'In fact, when Atlantic eventually found out, they nearly went mad. They said, \"You can't be serious.\" But we just had a gentlemen's agreement... We were signed to Atlantic, but we weren't signed to Peter. We never had a management contract. He got the normal management fees and royalties from the records as executive producer.'\n\nGrant himself had an extremely liberal, rather evolved philosophy of what his role as manager should be, which belied his reputation for controversy. 'In the old days everybody thought the artist worked for the manager,' he said. 'In America they'd say: \"Oh, so and so owns those people.\" You don't own artists. They hire you and give you a percentage of their money to do the very best for them.'\n\nAccording to Jones, the four musicians' relationship with Grant was extremely satisfactory: 'All pretty above board and as a result it was a really happy band. We could never believe how other bands got on. They never spoke to each other and travelled in separate cars. Why did they play together if it was that bad? Everybody thought we were the prima donnas, yet there was hardly an ounce of attitude in the whole band. Page and I had seen it all before. We just didn't want to make the obvious mistakes.'\n\nJeff Beck was in New York, and he encountered Page and Grant, fresh from securing their deal. 'Page said, \"Listen to this. Listen to Bonzo, this guy called John Bonham that I've got.\" And so I said I would, and my heart just sank when I heard \"You Shook Me\". I looked at him and said, \"Jim, what?\" and the tears were coming out with anger. I thought, \"This is a piss-take \u2013 it's got to be.\" I mean, there's _Truth_ still spinning on everybody's turntable, and this turkey's come out with another version. Oh boy... then I realised it was serious, and he did have this heavyweight drummer, and I thought, Here we go again \u2013 pipped at the post kind of thing.'\n\nIn the United States Beck was signed to Epic, a division of CBS Records, to whom the Yardbirds had also been signed. However, Page himself had never signed a contract with CBS. All the same, Clive Davis, the legendary head of CBS, assumed that his label would have the first offer on Page's new band. When Grant, accompanied by his attorney Steve Weiss, paid a courtesy call on Davis, the CBS honcho assumed it was to discuss the new Jimmy Page act. When Grant informed him that Led Zeppelin had already signed to Atlantic, Davis became apoplectic with rage. Grant and Weiss soon left the building. Later, Epic would release the live Yardbirds' album recorded in New York, and Page immediately would cancel its release with an injunction. 'I never signed any contracts with Epic \u2013 they didn't want me at the time,' he said.\n\nThe _Led Zeppelin_ LP was set to be released in the USA on 12 January 1969, three days after Jimmy Page's twenty-fifth birthday. By then Led Zeppelin would be into the third week of their first tour of America, promoted by Frank Barsalona's Premier Talent agency, the first to concentrate exclusively on rock music; Premier Talent had already blazed forward with a pair of acts with similar power line-ups to Page's group: the Who and the Jimi Hendrix Experience.\n\nPage, Plant and Bonham flew to Los Angeles on 24 December 1968. At the airport they were met by Richard Cole, their newly appointed tour manager, who, according to Grant's wishes, had the three ensconced in bungalows in the legendary Chateau Marmont on Sunset Boulevard.\n\nCole, a former roofing scaffolder and old Mod, had pulled himself up by his boot straps and transformed himself into a respected road manager known for his dedication to the acts he worked with. On his first job, with popular soul roadshow Herbie Goins and the Night-Timers, he met Jones, who was playing with the act on live dates. After Cole had worked with Unit 4 + 2, an act managed by Don Arden \u2013 useful for all sorts of nefarious tips, one imagines \u2013 in quick succession he was employed by the Who, the Yardbirds (where he became acquainted with Page), Jeff Beck, Vanilla Fudge, the Rascals, the Searchers and the New Vaudeville Band. Most recently he had been working with Terry Reid on Cream's farewell tour of America, an irony, all things considered.\n\nJones was to meet them in Denver, Colorado, on 26 December for their first-ever US date. With his wife Mo accompanying him, he had disgorged himself into New York as their plane from Heathrow landed at JFK, leaving the other three musicians to catch a connecting flight to Los Angeles; spending Christmas at the New Jersey home of Madeline Bell, a largely UK-based black American singer, he then made his own way to Denver for the Boxing Day concert. Jones's act of independence became a characteristic of his behaviour in the group, an early statement of his need to create 'my own space within the band'.\n\nOne assumes that the classy Chateau Marmont had been intelligently chosen for its luxurious essence, a psychological boost for the new troops; this was what they could look forward to paying for themselves in the hopefully not-so-distant future. But it was pricey, and for once Page was happy to go along with this. It was Page, Jones and Grant who bankrolled this first US tour, on which everything else was trimmed to the minimum; this was long before the days of record companies offering 'tour support': in other words underwriting the cost of their acts going out to play live dates. Mickie Most had put up \u00a31,000 for amps and speakers, and in return took 1 per cent of Led Zeppelin's income in perpetuity.\n\n'We didn't do a tour budget,' said Cole. 'In those days, it was like: Okay, the hotels are going to cost us this much, the air fares this. It wasn't even worked out, the job had to be done. We haven't got that much money, we've got to make it as economical as possible without it being too uncomfortable. As there was more money coming in for shows, things were escalating.\n\n'TWA used to have a thing called Discover America, where you'd buy your airline tickets, you would work out the route of an entire tour, and as long as it went in a circle so you flew into New York and ended up back in New York and didn't go back to the same city twice, you'd get 50 per cent off. The only problem with that was if dates come in or fall out, you'd have to redo all the tickets again. They'd hate it when you took the tickets into the Hilton, where TWA had a desk, because everything was written by hand in those days. And each ticket had maybe 20 stops on it, but everything was economically worked out, even the 707 jets. With a jet like that, you didn't need to order room service because it was supplied, so most of the times we ate on the plane.'\n\nAt the Chateau Marmont, which had catering facilities in each bungalow, Bonham cooked Christmas dinner: turkey with all the trimmings. Yet everyone \u2013 especially Plant \u2013 was so nervous they could barely eat a mouthful. Crippled by the jetlag from the eight-hour time difference that hits most people travelling to the West Coast for the first time, both Bonzo and Plant were freaked out about leaving their families behind at Christmas. 'It's a sacrifice,' said Page, 'but there's going to be a pay-off. This band has a lot going for it. Let's make the best of it.' As befitted his position in the band, Page was then able to retire to his own bungalow at the hotel. While the others shared accommodation \u2013 Plant and Bonham together, Jones with Cole \u2013 Page never did.\n\nBright and early on 26 December, Cole roused his three charges at the Chateau Marmont, drove them out to LAX Airport and caught a flight to Denver for Led Zeppelin's debut live performance in the United States.\n\nLinking up with Jones in the mile-high city, the tension among the four musicians was palpable. How would American audiences respond to them? Would they be as lukewarm as so many of the UK crowds had been?\n\nTheir worries were not without foundation. They had been added late to the bill and were not even included on the posters for a show at the Auditorium Arena headlined by Vanilla Fudge and with Zephyr, a local blues-based hard-rock outfit, on before them. To make matters more complex, Led Zeppelin had only snagged the concert after the Jeff Beck Group pulled out of this and several subsequent shows. Vanilla Fudge had toured with Beck's outfit as support, and had already become friendly with Cole and Grant. The Fudge were hardcore: their moody, keyboard-dominated interpretations of classic tunes such as the Beatles' 'Eleanor Rigby' and the Supremes' 'You Keep Me Hangin' On' had won them a substantial audience; as Led Zeppelin hoped they would do with them, Vanilla Fudge had won their status by playing second fiddle to such acts as Cream, the Doors, the Who and Jimi Hendrix, and frequently blowing these headliners off the stage.\n\nAs the four nervously chain-smoked their way through Zephyr's set, the youngest of them all \u2013 Robert Plant \u2013 was undergoing a deep personal crisis. _How on earth would America take to him?_\n\n'Ladies and gentlemen, for their first American appearance, from London, England, please welcome... Led Zeppelin!' Then for Plant, problems were exacerbated even further. The stage on which they were playing was constantly revolving: every time he tried to establish eye contact with someone in the audience, the stage would shift yet again and he'd find himself staring out into space, disconcertingly searching for the last connection he had made. 'Good Times Bad Times', 'Dazed and Confused' with Page on his violin bow, 'Communication Breakdown' were all stormed through... Then Plant had a sudden realisation: matters were proceeding exceedingly well \u2013 the audience loved them.\n\nAnd with that acknowledgement from the crowd's cheers, the four-piece hit their stride: a 40-minute set was extended to just over an hour, concluding with deafening applause.\n\nLed Zeppelin's first show in the United States was a hit. With their music from the distant future and ancient past, and its fragmented and jagged dominance of the instant present, Led Zeppelin punched their audience in their third eye, coring into their startled beings. They would not leave you alone, like primaeval pterodactyls snatching at your clothing and dragging you towards them; entrancing you with the visual double-act of cherubic Jimmy Page and primal Robert Plant \u2013 even as early as this, cock-rocker Plant paradoxically squealing like a bitch on heat, like a 12-year-old who had lived for a thousand years, drawing out the androgynous, screamingly libidinous nature of both frontmen, interweaving with the musical dramatics.\n\nIn the eyes of Atlantic Records, Led Zeppelin were a replacement for Cream. But Page's band were far more precise and measured \u2013 certainly more calculating \u2013 than Cream, whose sprawling self-indulgence had only increased on their final tour. And at first, judging them only on the sound of their first album, Led Zeppelin seemed to stand in stark contrast to the increasingly soft rock emerging from LA's Laurel Canyon.\n\nThere were 25 shows in all, and a week supporting Vanilla Fudge, whom Led Zeppelin successfully went out of their way to upstage on each occasion. There were four more dates in 1968, in Seattle, Vancouver, Portland \u2013 where they were billed as 'Led Zeppilen featuring Jimmy Page' \u2013 and Spokane \u2013 where the poster advertised an appearance by 'Len Zefflin'. Following these shows in the chilly northern regions of the West Coast, Led Zeppelin were set to move down to southern California and the warmer climes of Los Angeles.\n\nHowever, a fierce snowstorm descended on the Spokane region and the airport was closed. But 200 miles away in Seattle planes were still taking off and landing. Despite the ever-thickening snow, Richard Cole made the decision to drive the highway to Seattle, slipping and slithering all the way. Even when state police ordered them off the dangerous route, Cole simply returned to it at the first opportunity. In the rear of the car, Page was wrapped up in every item of clothing he could find, as he sweated out a case of Hong Kong flu, his temperature rising to over 40 degrees. Accordingly, the guitarist 'didn't have much energy to complain about anything', said Cole.\n\nThe Whisky a Go Go on the Sunset Strip, where Led Zeppelin were booked to play four consecutive nights starting on 2 January 1969, was a perfect setting for the group, who were supported by a new act called Alice Cooper. Page already loved LA, and for Plant it \u2013 along with San Francisco \u2013 was a power-point temple, a land where acts beloved to him such as Buffalo Springfield, the Doors and Love had formed.\n\nOriginally Led Zeppelin had been scheduled to play two shows a night at the Whisky, but Page's flu meant he was only capable of playing the first house on each evening. He still managed to be driven down to Hollywood's tiny Mirror Sound, where Bobby Fuller and Ritchie Valens had once recorded: he was deeply impressed with the energy palpable in the very walls of the studio. Efforts to employ the city's Gold Star Studios, from whence Phil Spector had driven his Wall of Sound around the world, were less successful, the building's 'vibes' disappointing.\n\nOnce he had again checked into the Chateau Marmont, however, Page resisted permitting his fever to interfere with his fun. 'Led Zeppelin hit Los Angeles like napalm,' wrote Catherine James. 'After their dazzling debut at the Whisky a Go Go, Led Zeppelin became the new rock-and-roll gods. Jimmy stayed at my house for the first two luminous nights, then the band went up to San Francisco for more accolades at the Fillmore.\n\n'The next thing I knew Jimmy had disappeared. I heard he was holed up at the Hyatt House on Sunset with... Hollywood's chief groupie, Miss Pamela. My romantic teenage heart was shattered. I agonised, longed and generally stirred myself into a state.'\n\nSoon after, Catherine James fell into the arms of a new singer-songwriter called Jackson Browne, and her soul was comforted.\n\nFrom Los Angeles, Page, his health marginally improved, and the rest of Led Zeppelin flew the 350 miles up to San Francisco. Such was the kudos around the city's by now legendary Fillmore West that the three nights they were about to play there promised to be key in breaking the band in the USA.\n\nIn San Francisco, Peter Grant was waiting for them. As was Maureen Plant, permitted to fly out for this stretch of the road primarily because Grant was aware of her husband's insecurities about playing the Fillmore. 'Robert did lack a bit of confidence,' Grant admitted later.\n\nThe Fillmores West and East were the personal suzerainty of Bill Graham, the legendary \u2013 and sometimes legendarily difficult \u2013 promoter who would transform concert promotion in the United States, for much of the time carrying Zeppelin in his wake. Graham had long been a supporter of the Yardbirds, for whom the Fillmore West had always proved a shrine.\n\nFor this first show, on 9 January, Page's 25th birthday, the headliners were the ragtag political act Country Joe and the Fish, with the iconoclastic and innovative Taj Mahal opening the evening's music. Led Zeppelin's set blew both these acts out of everyone's memories. Page said: 'The early interest was partly caused by the fact that I'd been in the Yardbirds and a lot of Americans had liked that group and wanted to see what Jimmy Page had moved on to. But then when they saw what we had to offer... I mean, Led Zeppelin was frightening stuff! The concept of psychedelic music was about roaming and roving but never actually coming together. That's why Zeppelin succeeded: there was a real urgency about how we played. Everyone would be getting laid-back and we'd come on and hit 'em like an express train. Playing the Fillmore West was the moment when I knew we'd broken through. There were other gigs, like the Boston Tea Party and the Kinetic Circus in Chicago... where the response was so incredible we knew we'd made our impression. But after the San Francisco gig it was just \u2013 bang!'\n\nThe subsequent Fillmore shows were as kinetically powerful, astounding the audiences with their light and shade, the extremities of the set's tensions \u2013 Plant again squealing like a bitch on heat as he hit his vocal stride.\n\nOn 12 January the album _Led Zeppelin_ was released in the US \u2013 the day after the last of the group's three nights at the Fillmore. The LP would not be released in the UK until March.\n\nThe record sleeve was created by George Hardie. Page rejected his first effort \u2013 a zeppelin largely shrouded by clouds \u2013 and instead the chosen image was of the _Hindenburg_ disaster in 1937, in which the airship caught fire and 35 of its 97 passengers were killed. Hardie received \u00a360 for his design. The rear picture, of the four musicians, was taken by Chris Dreja \u2013 proof of the continuity of their bond.\n\nWhile the record-buying population voted with their cash, Mick Jagger and George Harrison's inability to comprehend Led Zeppelin's oeuvre was a reflection of how establishment thinking would respond to the act.\n\nThis was the age when CBS Records was running ads in _Rolling Stone_ magazine declaring 'The Man Can't Bust Our Music'. It was as though the company hippies had scaled the ramparts and won the revolution. But that wasn't the case at _Rolling Stone_ itself. They were having no truck whatsoever with these po-faced, pretty-boy Limeys. The media often imagines it is utterly in touch with the zeitgeist when, most often, it is at least six months behind: across the nation, fans were already going gaga for Led Zeppelin at their live shows. Yet here, in _Rolling Stone_ 's dismissive review of Led Zeppelin's first album by LA-based writer John Mendelsohn, was an unalloyed example of someone who needed to get out more, preferably to those Whisky shows he had clearly missed. The review was especially damning in its comparison of Zeppelin to Jeff Beck: 'In their willingness to waste their considerable talent, the Zeppelin has produced an album which is sadly reminiscent of _Truth_.' Page was characterised as 'a very limited producer and a writer of weak, unimaginative songs, and the Zeppelin album suffers from him having produced and written most of it (alone or in combination with his accomplices in the group)'.\n\nMendelsohn's sneering review would signal the start of years of animosity between Led Zeppelin and the highly influential _Rolling Stone_. Page, who could be described as sometimes oversensitive to press criticism, would not forget what Mendelsohn had written.\n\nMind you, Ritchie Yorke, writing in the Toronto _Globe and Mail_ , clearly got it: 'Led Zeppelin will be the next super group in the US... they have merged with a positive, driving, distinctive sound... Page's guitar work skims across the melody with pure joy... Jones's bass and organ rhythms are forceful and invigorating... unlike many groups, Led Zeppelin has managed to maintain simplicity while striving for depth... the best debut album by any group since _Are You Experienced_ by the Jimi Hendrix Experience.'\n\nWhen the LP was released two months later in the UK, there were only a pair of positive notices: in the satirical magazine _Punch_ Chris Welch described the record as 'the definitive recorded rock performance of the year'; and Felix Dennis, writing in the underground _OZ_ magazine, declared it to be 'a turning point in rock'.\n\nFollowing San Francisco, there was a date in San Diego, down on the Mexican border. Then, via a show in Iowa, it was over to the other side of the country for gigs in rapid succession in Baltimore, followed by three nights in rock'n'roll-loving Detroit \u2013 also considered a potential break-out city for Led Zeppelin. In New York City they played the Fillmore East, blowing the headlining Iron Butterfly off the stage. 'The audience was still going \"Zeppelin, Zeppelin\" when Iron Butterfly had started their set!' said Peter Grant. 'Good band, not a bad band... but no match for Zeppelin.'\n\n'Led Zeppelin landed at Fillmore East and in the first of four weekend shows, the British quartet showed it could develop into the next big super group,' wrote Fred Kirby in _Billboard_. 'Page, a former member of the Yardbirds, ranks with the top pop guitarists in the world and his performance substantiated his reputation. Plant is a blues-style screamer and wailer, whose vocalising was wild. Iron Butterfly had a tough assignment in following Led Zeppelin.'\n\nBut after the Fillmore West, the tour's greatest triumph occurred in Boston, Massachusetts. There they played the 400-seater Boston Tea Party for three nights on 24\u201326 January 1969, legendary performances in which their 90-minute set was extended to four-and-a-half hours, the material effectively being performed twice, with mass encores including Beatles and Rolling Stones songs. 'A laughing, sweating Page could only cough into his cigarette and beg the others: \"What songs do you know?\"' said Mick Wall. If ever there was an East Coast breakthrough moment for Led Zeppelin, this was it.\n\nSlipping across the border to Toronto in Canada for another triumphant show, the band met Ritchie Yorke, the Australian writer who had given the _Led Zeppelin_ album such a prescient and positive review. A good man, Ritchie Yorke would prove a stout ally of Zeppelin over the years.\n\nThere then followed shows at Chicago's Kinetic Playground, an archetypal 'head' venue. A guitarist named Joe Wright was at the first gig, on 7 February 1969: he organised the Tuesday-night jam sessions at the Playground and would always be let in for free, with backstage access; having played for the US Olympic ice hockey team, he was possessed of a measure of self-belief. That night at the venue he had with him his 1964 Les Paul Standard, a rare and beautiful guitar. As soon as Page saw the instrument, he made Wright an offer: 'Oh, this is a lovely guitar. I'll give you $800 for it.'\n\n'I'll give you $950,' said Bonham.\n\n'I'll give you $1,300,' butted in Plant.\n\n'I found out later,' said Wright, 'that they did this a lot. They had to be one up on the other. But you always had to let Pagey win. It was always building up to that. He was in charge. For example, they loved that South-west turquoise Native American jewellery. Robert would get a ring, Bonzo would get a little piece, but Pagey would get the cr\u00e8me de la cr\u00e8me. So you could never top Jimmy. But I wouldn't sell him my guitar. And that's when I met Zeppelin, that moment.\n\n'Then they played. My jaw hit the floor. I fell in love with that superpower that they presented with the jamming, with the blues on steroids. You never saw any of that before with white people. I fell in love with them.'\n\nAfter the show, Joe Wright went backstage again. He had a task to fulfil: one of his friends had formed a group and was searching for a name for it. Joe decided to ask Page if he had any ideas. 'Call them the Wankers,' suggested the guitarist. Unaware of English slang, Wright relayed this suggestion to his mate, who acted upon it, touring for the next few months as a member of the Wankers. Until Wright took a job roadying with the Who and discovered what the name meant.\n\nAfter Chicago, Led Zeppelin moved on to Memphis, where the group went on a pilgrimage to Graceland and failed to utilise the facilities at the legendary Sun Studio. 'By the time of our second album, it wasn't the original Sun Studio anymore,' said Page, recalling how _Led Zeppelin II_ was recorded on the hop, during the brief lulls between live work. They played the Thee Image in Miami, Florida, before Led Zeppelin's triumphant and musically revolutionary first tour of the USA concluded at Baltimore's Civic Center on 16 February 1969.\n\nThe day before, _Rolling Stone_ had published its 27th edition, which would become notorious as the 'Groupie' issue, as much of the magazine was given over to the study of a social group most people had never heard of. The lengthy article was credited to a trio of writers: John Burks, Jerry Hopkins and Paul Nelson. The words were accompanied by Baron Wolman's portrait pictures of the likes of the GTOs (Girls Together Outrageously) and the Plaster Casters of Chicago; the Plaster Casters' speciality was to take plaster-of-Paris moulds of rock stars' penises. With exceptions like the Sanchez twins, Judy and Karen, and Catherine James, you were struck by how unattractive, unglamorous and rather sad looking so many of these girls were.\n\n'The phenomenon of the groupie \u2013 it exists because of glamour and power,' commented life coach Nanette Greenblatt. 'It emerges from old paradigm culture originating in the perception of women as virgins, wives, mothers, whores and under the control\/ownership of the bigger primate. The rock and roll lifestyle seemed to make it easy to find willing people, sufficiently glamourised, to seduce.'\n\nThe _Rolling Stone_ review of the _Led Zeppelin_ album did not appear until 15 March. Otherwise, would Page have been so ready to offer the magazine his thoughts on these self-styled rock 'n' roll girls? The guitarist was remarkably forthcoming when he was interviewed \u2013 though he was no doubt aware that by appearing in such an article he would help secure his reputation as a deeply sexual figure, a powerful element of Led Zeppelin's attraction. Asked whether he found more or fewer groupies during this first American tour with his new group than he did during the eight tours of the country he undertook with the Yardbirds, he was characteristically equivocal: 'By now I've got friends I look up, or they call me, in nearly every city. But the other boys in the band who've not been here before \u2013 I kind of prepare mental lists in my mind to try to predict who they'll pair off with. I know which chicks are going to come out and the kind of fellows they go for, and I know the taste of the guys in the band, and so I try to make my predictions, girl by girl. I can come pretty close.'\n\nIn Page's opinion, the most beautiful groupies were in New York. Yet he found the girls in San Francisco more likely to develop 'personal relations in depth with the musicians'. But, he added, 'You take each as they come, though. And they're all over the world. We even found them in Singapore.'\n\nAs they stepped aboard a plane to return to the UK, the _Led Zeppelin_ album had risen to number 90 in the US Billboard album charts. It would leap upwards until, three weeks later, it was in the Top 20, rising to as high as number 10. It would spend literally years in the charts.\n\nPage was reluctant to depart America. He felt they should stay and push on with more dates. They would be heading back very shortly, Peter Grant advised his client. First, they had a set of UK shows to play.\n\n## 9\n\n# 'WHOLE LOTTA LOVE'\n\nLed Zeppelin's homecoming was not a triumphant one, playing dates at the likes of Plymouth's Van Dyke Club or Hornsey's Wood Tavern, often for no more than a flat fee of \u00a3140.\n\nIn the UK, the band were viewed as a capitalist act whose only purpose was to milk the United States for as much money as possible. 'We just couldn't seem to do anything right as far as the critics were concerned,' complained Jimmy Page. 'At which point I think we all just sort of gave up on the idea of ever pleasing them.'\n\nOn another brief Scandinavian jaunt, however, things were a bit more like it for them. In Denmark, Led Zeppelin performed a full set in front of television cameras, the first time they had ever done so. The audience sat cross-legged on the floor in front of them. 'They don't cheer too madly there, you know?' said Page. 'It was sort of an experimental concert to see if we were any good, I guess.'\n\nWhat was on his mind was the need for new material. The group were buzzing \u2013 they couldn't come down from the US tour \u2013 with adrenaline running round and round their jetlagged and exhausted but exhilarated beings.\n\nThat year Led Zeppelin played four live sessions \u2013 concerts more like \u2013 for the BBC, beginning with a show at London's Playhouse Theatre, essentially a BBC venue, on 3 March. This was especially aimed at offering material for John Peel's _Top Gear_ show.\n\nThe radio performances were a reflection of Led Zeppelin's seemingly overnight success in America. Their US fame had washed across the Atlantic, picked up by fans, if not by dismissive doyens of the underground. Led Zeppelin went out of their way in these BBC sessions to emphasise the improvisational skills with which they enhanced their album material during live performances. Really, there was no other way to get this across to the very many who still had not seen Led Zeppelin live. (More than 20 years later these live versions would finally be released, as _BBC Sessions_.)\n\nApart from their Danish experience, and a mimed Swedish version of 'Communication Breakdown', Led Zeppelin hardly ever appeared on television. Like releasing singles, TV was uncool, and, like their later b\u00eates noires, the Clash, Led Zeppelin never once performed on _Top of the Pops_ , a show that almost guaranteed chart success. Like some twist of karmic victory for such a stance, a version of 'Whole Lotta Love' by Alexis Korner's CCS became the show's weekly theme tune. (On 21 March the group did make their UK television debut, performing 'Communication Breakdown' on _How Late It Is_ , a BBC 2 arts show.)\n\nThis was quite a statement. Everyone else was saying singles were lower-status artefacts and that they were now serious album artists. Ever since _Pet Sounds_ and _Sgt. Pepper_ , this had been the thinking.\n\nBut everyone else continued to release them: the Who's 'Pinball Wizard', Cream, Jimi Hendrix seemingly ad infinitum, Traffic, the Beatles and Stones inevitably.\n\nNow the supposedly hyped Led Zeppelin were at the vanguard of insurrectionist thinking as virtually the only act to stand by their principles and not release singles. Of course, doing so only increased exponentially the level of mystique that began to build around the group; it provided them with an image of distant power, as though they were puppet-masters \u2013 which in a way they were. After all, they were telling the man, which now included their record company, that no one could bust their music.\n\nFrom time to time singles leaked out, a consequence of over-enthusiastic record company thinking. 'Good Times Bad Times', coupled with 'Communication Breakdown', was pressed up for DJ copies by Atlantic in the US \u2013 and quickly nixed by Peter Grant. There were efforts to release 'Whole Lotta Love' as a 45 in the UK, another idea shot down in flames by the manager. But the lack of a single made radio promotion awkward, almost hopeless. Hence the BBC sessions.\n\nHistory can make it seem all part of a master plan. Although Page and Grant had a strategy, there was much more reacting to events as they happened on a day-to-day basis. Still, Page's vision remained clear. Interviewed by Nick Logan of the _NME_ , he showed no doubts: 'I can't see the heavy thing going out. Ever since the underground thing happened a couple of years ago, people's tastes have been broadening. You can have a group... who are into a light, folky thing on the one hand, and us on the other. The whole scene is broad enough to take us all in, and I don't see why that situation shouldn't continue.'\n\nThere are a trio of Led Zeppelin songs that have transcended their catalogue of material to become aural calling cards for the group: 'Whole Lotta Love', 'Stairway to Heaven' and 'Kashmir'.\n\nDespite the stratospheric success of 'Stairway to Heaven', it is the insidious five-note riff of 'Whole Lotta Love', the first hard-rock standard and a tune that radically altered expectations of the sound of the electric guitar and rock vocal, that has most insinuated itself into global popular culture.\n\nYet the origins of 'Whole Lotta Love', which would open the second Led Zeppelin LP, are less straightforward. The song is by no means a Page\u2013Plant original, as the composing credits attest. 'Whole Lotta Love' is a reworking of Muddy Waters' 'You Need Love', which had been taken up by the Small Faces, retitled as 'You Need Loving' and featured on their first album, but credited to Small Faces singer Steve Marriott and bass-player Ronnie Lane. The song was a high point of the Small Faces' live sets.\n\nSteve Marriott told Paolo Hewitt in his book _Small Faces: The Young Mods' Forgotten Story_ that Plant was a regular at their shows. '\"Whole Lotta Love\" by Led Zeppelin was nicked off that album. Percy Plant was a big fan. He used to be at all the Small Faces gigs. We did a gig with the Yardbirds which he was at, and Jimmy Page asked me what that number was we did. \"'You Need Loving',\" I said, \"it's a Muddy Waters thing,\" which it really is, so they both knew it, and Percy used to come to the gigs whenever we played in Kidderminster or Stowbridge, where he came from. He was always saying he was going to get this group together. He was another nuisance. He kept coming into the dressing room, just another little Mod kid. We used to say, \"That kid's here again.\" Anyway, we used to play this number and it became a stock opener after that album. After we broke up they took it and revamped it. Good luck to them. It was only old Percy who'd had his eyes on it. He sang it the same, phrased it the same, even the stops at the end were the same; they just put a different rhythm to it.'\n\nPlant's vocals \u2013 the breathing, the intonations, the inflexions \u2013 on 'Whole Lotta Love' were directly modelled on Steve Marriott's performance of 'You Need Loving'. But there is a fundamental flaw in any complaint made by Steve Marriott: it's that Marriott\u2013Lane credit on the first Small Faces album. For it certainly was not written by the duo; Muddy Waters' 'You Need Loving' was written by Willie Dixon, the extraordinarily influential performer, songwriter and producer who had shaped the sound of post-Second World War Chicago blues music.\n\nWhen, many years later, Willie Dixon's daughter Shirley brought 'Whole Lotta Love' to her father's attention, he realised that it was his tune. A lawsuit followed, in 1985, and Dixon ultimately received a substantial out-of-court settlement. 'Some people said later that \"Whole Lotta Love\" was based on Willie Dixon's \"You Need Love\" and the Small Faces' \"You Need Loving\",' Page said in his defence to Marc Myers, author of _Anatomy of a Song_. 'My riff \u2013 the basis for the entire song \u2013 sounds nothing like either of them. Robert had referenced the Dixon lyrics because with my riff, they felt right. This eventually forced us to give Dixon a co-credit on our song. But if you take Robert's vocal out, there's no musical reference to either song.'\n\n(All the same, Dixon also received further substantial financial compensation when a further song on _Led Zeppelin II_ , 'Bring It On Home', which closed the LP, was revealed as being another of his compositions. Despite the settlement, writer Gavin Martin was asked by Dixon's daughter not to mention Led Zeppelin when he went to interview her father, who was still extremely distressed about the entire matter. And the charges of copyright theft did not end there. 'The Lemon Song', with its coyly prurient, schoolboy-like reference to lemon-squeezing, itself a direct lift from Robert Johnson's 'Travelling Riverside Blues', was revealed as the bastard son of Howlin' Wolf's 'Killing Floor'. Sued in the early 1970s by Arc Music, Wolf's songwriting publisher, Zeppelin again settled out of court.)\n\n'Wherever it comes from,' Plant told Ritchie Yorke, 'it was all about that riff. Any tribute that flows in must go to Jimmy and his riffs. They were mostly in the key of E and you could really play around with them. Since I've been playing guitar myself, I've realised more than ever that the whole thing, the whole band really, came straight from the blues. Everything.'\n\n'Just a basic rock tune, using some electronic sounds in the middle section,' Page said as he made his characteristically equivocal and modest case to Yorke. 'It does sound great on headphones. It reminds me of a sort of Rolling Stones feel.'\n\nThe reference to headphones that Page made is significant. With the cheaper stereo systems readily available, especially in the United States, headphones \u2013 'cans', as they were known \u2013 had become integral to underground culture. Some radicals had dispensed altogether with their pair of stereo speakers and only had headphones. The backward psychedelic section of 'Whole Lotta Love' was perfect for stoned listening in such a manner, the song's arrant sexuality bouncing backwards and forwards within your skull. Headphones culture, as useful for listening to the newly popular FM radio as to a record playing in your room, was yet another reason for the growing popularity of Led Zeppelin in the USA.\n\n'John Paul and I knew our way around a recording studio,' explained Page to Marc Myers in _Anatomy of a Song_ , 'so we weren't going to waste studio time or produce something that wasn't cohesive. More important, I wanted to expand our approach to ensure that our album wouldn't be chopped up into singles for AM radio. To make sure that didn't happen, I produced \"Whole Lotta Love\" \u2013 and our entire second album \u2013 as an uneditable expression, a work that had to be aired on stereo FM to make sense.'\n\nPage had first come up with the famous five-note riff during the summer of 1968, while doodling around with his guitar on the deck of his Pangbourne boathouse. 'I suppose my early love for big intros by rockabilly guitarists was an inspiration, but as soon as I developed the riff, I knew it was strong enough to drive the entire song, not just open it. When I played the riff for the band in my living room several weeks later, during rehearsals for our first album, the excitement was immediate and collective. We felt the riff was addictive, like a forbidden thing.'\n\nAt rehearsals for _Led Zeppelin II_ Page decided that of all their material, 'Whole Lotta Love' was so uniquely strong that it should be the album opener. 'So I wanted to record the song first,' he said.\n\nOn 10 April 1969 Led Zeppelin went into Olympic Sound Studios with engineer George Chkiantz, who had worked with Jimi Hendrix in the same studio on _Axis: Bold as Love_. Chkiantz noted that Page had brought a theremin with him to the studio. Its 'eerie sound', as Page described it, encouraged further unorthodox approaches; he would detune his guitar and tug on the strings, searching for sounds. To boost the drums, Chkiantz made a significant alteration to the studio's set-up: a platform was erected so the drums were some 18 inches off the wooden floor, which stopped the rumble transmitting to the other microphones. Page wanted the listener to precisely feel every drum stroke, the song's foundations for what he envisaged as a panoramic audio experience. Above the drums Chkiantz set up a stereo microphone on a boom.\n\n'I was playing a sunburst 1958 Les Paul Standard guitar I had bought from Joe Walsh in San Francisco when we were out there on tour,' said Page. 'The Standard had this tonal versatility, allowing me to get a blistering high pitch.'\n\nEnergised and empowered thanks to Led Zeppelin's success in the US, Plant had fully overcome the nervousness and uncertainty he had felt on those first American dates. Now he was in a new creative phase. First recording his vocals in the studio itself, he soon retired to the cocooned privacy of the vocal booth. 'Robert's vocal was just as extreme,' said Page. 'He kept gaining confidence during the session and gave it everything he had. His vocals, like my solos, were about performance. He was pushing to see what he could get out of himself. We were performing for each other, almost competitively.'\n\nWhen _Led Zeppelin II_ was mixed at Atlantic's studio in Manhattan late that June, engineer Eddie Kramer \u2013 who had first worked with Page in 1964 on the Kinks' debut album and had worked on several of the second Zeppelin album's songs at Olympic \u2013 discovered that the original tape contained 'bleed-through of a previously recorded vocal in the recording of \"Whole Lotta Love\". It was the middle part where Robert screams, \"Woman. You need it.\" Since we couldn't re-record at that point, I just threw some echo on it to see how it would sound and Jimmy said, \"Great! Just leave it.\"'\n\n'I hadn't heard anything like that before, and I loved it,' Page told Marc Myers. 'I was always looking for things like that when I recorded. That's the beauty of old recording equipment. Robert's faraway voice sounded otherworldly, like a spirit anticipating the vocal he was about to deliver.'\n\nKramer also flung Page's guitar parts about between the speakers, increasing the song's sense of sexual frenzy, splashing on reverb. 'Because Jimmy was a studio brat, he really understood how we could push the limits,' said Kramer to Marc Myers of the Zeppelin leader's electronic animism. 'When you have limitations in the studio, you go for it and stretch your imagination.'\n\n'I knew what I wanted, and I knew how to go about it,' Page told _Guitar World_. 'It was just a matter of doing it. I created most of the sounds with a theremin and my guitar. The theremin generates most of the higher pitches and my Les Paul makes the lower sounds.'\n\n'I detuned it radically and just basically pulled on the strings to make an assortment of growling noises,' he said. 'Evil sounds that you're not supposed to hear on commercial radio. I might've detuned it to a chord, but really I'm just pulling on strings and making them howl! And then, during the mix, with the aid of Eddie Kramer, we did all the panning and added the effects, including using low-frequency oscillators on the tape machine to really pull the whole thing down and lift it back up so the sound is moving in rhythm. It was something no one had ever done before in that context, let alone in the middle of a song. That's how forward thinking we were, that's how avant-garde it was, and that's how much fun we were having.'\n\nIn April 1969 Page gave an interview to _Record Mirror_. He was extremely pleased, he remarked, that Led Zeppelin had now become accepted in the United Kingdom: the first album was already high in the charts. Yet he was only too aware where the principal market for the group lay: 'We're working every day here now \u2013 but before we went to the States nobody wanted to know. And it's not just London \u2013 it's all over the country. Very pleasing reaction. I still reckon the States is our main market, though.'\n\nThere was hardly time for more BBC sessions anyway. Another American tour had been rapidly booked by Peter Grant for the end of April. And for this second tour, they were far higher up the bill, co-headlining with Vanilla Fudge on some shows; it would depend on the popularity of each act in the individual city as to who closed the night. They were also earning approximately four times the $1,500 they had earned per concert on the first set of dates.\n\nThey also variously toured with the Brian Auger Trinity, featuring 'face of 68' Julie Driscoll, Three Dog Night, and Delaney and Bonnie. On certain shows Led Zeppelin performed as the only act on the bill, setting a template for the future.\n\nThe tour kicked off at the Fillmore West, switching \u2013 because of the large ticket demand \u2013 to Bill Graham's Winterland Ballroom for two nights, then returning to the Fillmore for a final San Francisco show; on all four of these San Francisco dates they were supported by the Brian Auger Trinity. Then it was straight down to LA for two nights at the Whisky a Go Go.\n\nWhen they arrived in Los Angeles an earthquake erupted in the nearby desert, a 5.6 on the Richter scale. Believing that the safest place to be in an earthquake was in a bathtub \u2013 because the piping would keep the bath in place, so the myth ran \u2013 Page insisted that only he should be able to use the bath in the two-bedroom suite at the Chateau Marmont he was sharing with Richard Cole. The tour manager was obliged to find other facilities for his ablutions.\n\nDuring these dates Page shifted professional tack. He no longer appeared onstage with the psychedelic 1958 Telecaster he had used on early UK dates. Returning to Pangbourne after Zeppelin's first American tour, he discovered that a friend had 'thoughtfully' removed all of his customising, returning the instrument to its original, innocent state. Page was flabbergasted by this misjudged act of kindness; paint had seeped into the guitar itself, even into the pickups, destroying the instrument utterly. From here on it would be the Gibson Les Paul slung low, almost to his knees, in a stance that would be copied by countless other guitar players, a boon to osteopaths and chiropractors everywhere.\n\nA different sort of story began to emerge about Led Zeppelin on this tour, of an act whose libidinous pursuits were elevated sometimes to the level of art form. Not least when Page, covered in offal, was wheeled on a hospital trolley by John Bonham into a Los Angeles hotel room filled with groupies, who proceeded to devour him.\n\nOn 29 April Led Zeppelin played the Whisky a Go Go. At the first show that night, Pamela Miller and Miss Mercy, members of Frank Zappa's Girls Together Outrageously, the GTOs, went along. Pamela, known as Miss Pamela, was 20 years old. 'I got sticky thighs over the very naughty Jimmy Page while I watched him reinvent guitar playing,' she wrote in her book _I'm With the Band: Confessions of a Groupie_. 'He was wearing a pink velvet suit and his long black curls stuck damply to his pink-velvet cheeks. At the end of the set he collapsed to the floor, and was carried up the stairs by two roadies, one of them stopping to retrieve Jimmy's cherry-red patent-leather slipper.'\n\nLater, at the after-party at Thee Charming Experience, Miss Pamela observed Richard Cole carrying a girl upside down, her panties spinning around one of her ankles; Cole's face was buried in her crotch. Another girl was being fucked on a table. She could not help but notice that Page, who personally inclined to a light, gentle manner with women, was sat rather aside from this bacchanalia, 'observing the scene as if he had imagined it: overseer, creator, impossibly gorgeous pop star'.\n\nAt Mystic Studios in Los Angeles the next day, Led Zeppelin recorded 'Moby Dick' and 'The Lemon Song'. Page had discovered the studio the previous month, when assisting his old friend Screaming Lord Sutch. Temporarily based in the city, the indomitable, eccentric David Sutch had persuaded a ragtag assortment of newly stellar names who had performed in his group the Savages to play on what Page had first thought were no more than demo sessions. John Bonham came along for the ride, although drummer stalwart Carlo Little was there already, along with Jeff Beck, Noel Redding and pianist Nicky Hopkins. When the scrappy consequences of these sessions were released the following February as _Lord Sutch and Heavy Friends_ , Page's ire was hard to ignore.\n\nAfter playing the hip Whisky, Led Zeppelin stormed southern California with a series of shows in more conventional venues, notably two nights at the 3,000-seater Santa Monica Civic Center on 4 and 5 May, a university show in Irvine and at the Rose Palace in Pasadena; and then in Santa Barbara, 90 miles up the coast from LA. They concluded this leg of the tour in their favoured US region, joining Lord Sutch's show at Thee Experience on 8 May. For the encore all four members of Led Zeppelin carried Sutch onstage in a coffin.\n\nThen it was time to head north, up over the border into Canada for shows in Edmonton and Vancouver. 'The hottest new rock band from Britain stalked onstage at the Gardens Friday night and let loose an earthquake of sound and frenzy,' wrote Bob Harvey in the _Edmonton Journal_. 'Their music's loud, almost to the point of pain, but they don't use volume to cover up deficiencies. The volume is part of their attack. They don't titillate or tease audiences to share their inspiration. Instead, they blast out with raw, jagged power, enough to bust a new door into your brain. They use their instruments like a brush and palette, creating frenzied visions that tumble through space and time.'\n\nThe group thundered back down the West Coast to Seattle, where they topped the bill with Three Dog Night, to whom the local reviewer awarded the trophy of best act.\n\nAfter a break in the sun of Honolulu for a show at the Civic Auditorium they headed for another city where Zeppelin would always reign supreme: Detroit and its legendary venue the Grande Ballroom, with its reputation for rock 'n' roll rowdiness. On their pair of shows they were supported by the remarkable cosmic jazz artist Sun Ra and Dutch hard-rockers Golden Earring.\n\nIn Detroit they were joined by the cultural journalist Ellen Sander, assigned by the prestigious _Life_ magazine to cover the group.\n\nLater she would assess Page as 'ethereal, effeminate, pale and frail'. When she asked about the already evident on-the-road abuse of women by his group, Page answered, revealingly, with the social grace of a car salesman: 'Girls come around and pose like starlets, teasing and acting haughty. If you humiliate them a bit, they tend to come on all right after that. Everyone knows what they come for.'\n\nOn 25 May, following shows in Athens, Ohio, and two nights at Chicago's Kinetic Playground, the four-piece hit Columbia, Maryland, midway between Washington, DC and Baltimore, where they shared the bill with the Who, the only time the two groups would ever play together; Led Zeppelin accepted playing second on the bill. When they ran over time, Zeppelin had their amps unplugged by the Who's crew, a cause of considerable brouhaha from Cole. 'We played together only once,' said Pete Townshend. 'I think it must have been the last gig they did playing before another group on the bill... and we just about got it together, we just about topped what they'd done... you get this ideology thing happening for the ideal rock guitarist \u2013 like the B.B. King syndrome, the Eric Clapton thing, the Jimmy Page trip \u2013 and I think Jimmy Page has been right in it and probably invented it.'\n\nUp to Massachusetts, Page and his boys played three nights at their stronghold the Boston Tea Party before winding up this second US tour with two nights at Bill Graham's Fillmore East, on 30 and 31 May, supported by Woody Herman and His Orchestra, and Delaney and Bonnie and Friends, an act that had been championed by George Harrison and which frequently included his old friend Eric Clapton. In keeping with Fillmore East policy, there were two shows a night: one at 8, the next at 11.30. Led Zeppelin 'left the audience paralysed', wrote Denise Kelly in _World Countdown_.\n\nEllen Sander stayed with the group until the end of the tour. In the Fillmore East dressing room she claimed she was attacked by a pair of Led Zeppelin members, notably John Bonham: 'Shrieking and grabbing at my clothes, totally over the edge.' Although Peter Grant stepped forward to save her, it was not until Sander's assailants had ripped the rear of her dress.\n\nLater, Page would be asked about Sander's disparaging reaction to his group. 'That's not a false picture,' he admitted. 'But that side of touring isn't the be all and end all. The worst part is the period of waiting before going on. I always get very edgy, not knowing what to do with myself. It's the build-up where you reach a point almost like self-hypnosis. There's a climax at the end of the show and the audience goes away, but you're still buzzing and you don't really come down. That's when you get a sort of restlessness and insomnia, but it doesn't bother you too much if there's a creative stream coming through. Maybe it's necessary to that creative stream. What's bad is that it's not always a release. You build yourself to that pitch and the release doesn't come. There are different ways of releasing that surplus adrenaline. You can smash up hotel rooms \u2013 it can get to that state. I think we've learnt to come to terms with it. I've learnt to enjoy it and achieve something creative from it too.'\n\nFollowing that second show at the Fillmore, Ahmet Ertegun and Atlantic Records threw a reception for their hot act at New York's swish Plaza Hotel. Each member was presented with a gold record for sales of their first album. 'We were touring until the day we were presented with a gold record. I thought, \"My goodness! A gold record!\"' Page said, keenly aware that it was less than a year since the group had formed. At the party, however, it was impressed on him how urgently the next Zeppelin LP was required, to catch the Christmas sales boost. Straight after the party he took Led Zeppelin back into the studio to continue their work on the album.\n\nTheir glittering gold discs were symbolic of the tremendous breakthrough that had taken place. Whereas the first tour had lost money, now their earnings were very different: they would be divvying up somewhere in the region of $150,000 from these shows.\n\n## 10\n\n# LED ZEPPELIN II\n\nIn June 1969 Led Zeppelin went into Morgan Studios in north-west London. 'Ramble On' was concluded; 'Living Loving Maid', a rockabilly song for which Jimmy Page never professed any fondness, was written by himself and Plant and knocked off in an afternoon; as was their version of Ben E. King's 'We're Gonna Groove', a frequent addition to the live set during the encore finale of other artists' songs. But, just like in the US, these studio dates were snatched between further live dates.\n\nOn the quasi-art nouveau poster for the first Bath Festival of Blues, scheduled for 28 June 1969, Led Zeppelin were ranked fourth, beneath headliners Fleetwood Mac, John Mayall's Bluesbreakers and Ten Years After \u2013 all stalwarts, in different ways, of 'British blues'. It is worth recalling that, subsequent to the demise of Cream, the always creative and exciting Fleetwood Mac, led by John Mayall alumnus Peter Green, were deservedly the most credible act in the UK. Before his sad demise, due to psychological problems, Green was one of the most revered guitarists Britain had ever produced.\n\nAs for Ten Years After, in less than two months' time Alvin Lee's machine-gun-like approach to guitar playing at the Woodstock festival \u2013 and therefore in the _Woodstock_ movie released the following year \u2013 would transform him briefly into a serious contender for Jimmy Page's role as king of guitar heroes. In the late 1960s it seemed that everywhere you looked there was a new gunslinger, vying for the title of fastest guitar player in the West.\n\nLed Zeppelin performed at Bath in the middle of the afternoon, and Page's violin-bow routine on 'Dazed and Confused' caused jaws to drop. 'Nobody had coerced the youth of England into becoming Zep fans, but there they were, cheering Page, Jones, Bonham and Plant, as the drums thundered and the guitars roared,' wrote Chris Welch in _Melody Maker_.\n\nThe next night was two shows at the 'Pop Proms' at London's Royal Albert Hall, the most prestigious venue in the British capital at the time, with a capacity of some 4,500. Zeppelin were supported by Blodwyn Pig, founded by former Jethro Tull guitarist Mick Abrahams, and Liverpool Scene, a semi-satirical trio comprised of the poets Adrian Henri, Mike Evans and Andy Roberts. Individuals from each act came onstage to perform with Led Zeppelin on their final encore, 'Long Tall Sally', Little Richard's 12-bar blues. 'When the group returned to the stage, they found the power had been switched off. \"Hey, put the power on,\" demanded singer Robert Plant as the group stood bewildered. Stalemate: Plant took up a harmonica and let fly on that and all the others could do was clap until a few minutes later the flow of juice was resumed. With the first few bars of \"Long Tall Sally\", the audience was on its feet, dancing in the aisles and in the boxes, and there was incredible mayhem happening on and around the stage. The saxists from Blodwyn Pig and Liverpool Scene added their support to the Zeppelin's rock,' ran Nick Logan's review in the _NME_.\n\nThen it was back to America, for their third US tour.\n\nOn the first of these dates, at the first Atlanta International Pop Festival on 5 July 1969, Jimmy Page was in an introverted frame of mind. In London that afternoon, the Rolling Stones had played a free concert in Hyde Park with their new guitarist Mick Taylor, and their set was dedicated to Brian Jones, the group's founder and Page's friend, who had died two days previously.\n\nThe 16th annual Newport Jazz Festival on Rhode Island ran from 3\u20136 July 1969. In contrast to such acts on the bill as Miles Davis, Dave Brubeck and the Sun Ra Space Arkestra, the Friday night of 4 July was given over entirely to rival rockers to Led Zeppelin, such as the Jeff Beck Group and Ten Years After, as well as the more pastoral sounds of Jethro Tull. Led Zeppelin were set to play the closing show on Sunday night. Such had been the excitement generated by the Friday night acts, the local authorities demanded that Led Zeppelin be thrown off the bill, the festival promoter releasing a specious tale of Page being sick and unable to perform.\n\nJoe Wright had driven over to Newport from Chicago with his 'hippy girlfriends'. 'Those were the days when all you needed was a little bit of long hair and a guitar, and you were instantly backstage. So on the Saturday afternoon I go for a little walk, along with my guitar, and I never came back. I go backstage: \"I'm with Led Zeppelin.\" Then here comes Jimmy Page, walking all by himself.\n\n'I walked straight up to him and I said, \"Hey, asshole: remember me? Joe Wright? From Chicago?\" And he takes a couple of steps back: \"Oh, I'm all alone: there's no bodyguards.\" \"You're the guy who told me to tell my buddy to call his band the Wankers.\" Then he burst out laughing.\n\n'So the ice was broken. Then Clive Coulson, the head roadie, came over, who I knew from the Kinetic. He said, \"What are you doing here?\" I said, \"I've just come to give shit to Jimmy Page. Which I've done.\" He said, \"You're kidding?\" But I explained and we all started laughing. Then Clive starts moaning about his assistant roadie: \"I wish I had someone to replace him.\" I said, \"I'm in!\" So he went off to talk to Jimmy and to Peter, and I was hired on the spot.'\n\nDespite the story having been put out that Page was too sick to close the festival, and the alleged ban by the local authorities, the guitarist and his group paid no heed, taking to the stage at 1 a.m. and playing a sensational 90-minute set to close the festival.\n\nBefore taking the boards, Page had been having a conversation with Joe Wright. 'I didn't know about black American music,' said Wright, 'which is where everything Zeppelin did came from. \"We're all very happy to have you here, but why are you not back in Chicago?\" Jimmy Page asked me. Back in Chicago, though, I got into the black blues clubs, where I would be the only white boy. I really got into it, heavy.'\n\nNewport was only the second date of this summer tour. When Joe Wright joined up, there were another 45 to go. 'They paid me really well. Peter Grant paid me the same as Robert Plant and John Bonham: I was getting $500 a week. They were on salary. Bonzo almost got fired, because he was so drunk and so stoned. He ran the risk of being fired several times. I remember the meetings: \"We've got to get rid of him, we've got to get rid of him.\" \"No, no,\" insisted Jimmy. Absolutely. They didn't call him \"the Beast\" for nothing.\n\n'I think Bonzo, like so many people in show business, didn't know how ugly it could be. He was a Gemini, and you can't get more schizo than that. He was a lovely guy but I don't think he knew how much bullshit was coming with that kind of success. The heavy pressure of the fans, of the idiots chasing you, of the death threats, of the marriage proposals, the endless dates. Jimmy knew that was coming because of the Yardbirds, but I don't think Bonzo had a clue. I want to be rich and famous, think all those kids, but they don't realise what they have to do to get it.'\n\nAt this stage Led Zeppelin were short on songs to play live.\n\n'Zeppelin used to do a little bit of cover material,' recalled Joe. 'They'd do \"Fresh Garbage\" by Spirit.\n\n'Zeppelin never played their own material the same way twice; Jimi Hendrix didn't either. They were overwhelmed by Hendrix. But Jimi was a singing guitar player, which Page and Jeff Beck were not. As a pure player they always envied his fluidity.'\n\nJoe Wright quickly discovered that his job had far more to it than merely humping equipment. And Page gifted him with a new name. 'Jimmy would always be late for rehearsals. I was like his understudy and would play the guitar while everyone was waiting for him at the soundchecks. Robert would be on drums and Bonham on bass, and we'd be goofing around jamming.\n\n'But then Pagey says, \"You're a guitar player: why don't you take care of my stuff?\" To me that was when \"guitar tech\" was born.\n\n'In every single one of those towns I would always find the nightclubs where you could go and jam. And I'd be up onstage with the local guys. So whenever Led Zep would come in from their gig they'd see me playing. At first my nickname was an insult from Pagey: \"Oh, Joe the jammer: he's always jamming.\" But then I became \"Joe Jammer\".'\n\nOn 12 July, Led Zeppelin shared the bill at the Philadelphia Summer Pop Festival with Jeff Beck, Johnny Winter, Blood, Sweat and Tears, Buddy Guy's Blues Band, Al Kooper and Jethro Tull, somewhat to the chagrin of Tull's leader Ian Anderson. 'After an unfortunate occasion in Philadelphia at the Spectrum, where we followed them onstage,' said the singer in _Uncut_ magazine in 2015, 'I learned it is much better to go on before Led Zeppelin. But if you are to go on after, you've got to man up and face the music \u2013 or lack of. I always felt pretty good with Zeppelin. The only awkwardness lay between Robert and me. I probably didn't do enough to make him feel that I was a relaxed co-conspirator in the world of rock music. He probably saw me as being a bit remote and aloof and unwilling to chat. But I was intimidated, frankly. Jimmy was more at ease, a natural guy. John Paul Jones never spoke to anybody. Peter Grant was always a real gentleman with me. I used to watch Led Zeppelin and know there were things quite clearly that I could not do. In particular, Robert's performance. You'd have to put a cross in the box saying \"Don't try that\". In the Robert Plant department, there were a lot of crosses in my box. I was very jealous of his vocal abilities and his stage swagger.'\n\nAlthough having played together for less than a year, Led Zeppelin were utterly steaming as a live act, a fact reflected in their loftier status in the bills they played. Vanilla Fudge were now regularly supporting them, while on other dates they would be billed as co-headliners. After their Vancouver show on 26 July, they earned a tremendously positive and perceptive review from J. Hesse in the _Vancouver Sun_ , one that fully acknowledged the part played by the group's founder: 'Led Zeppelin exists on the genius of lead guitarist Jimmy Page, whose baby face belies his musical message, that of jarring and unnerving the listeners with a fortissimo yowl that never lets up. Never allows time for recovery... it is made up of four individual musical artists attuned to each other's whims, capable of ensemble performance as well as separate forays into the jungle of lonely escapades. How perfectly Led Zeppelin has assessed the hang-ups of their listeners is evident when Plant screams, \"Do you feel all right?\" And the seething mob below the platform screams back, \"Yes!\"'\n\nIn the late 1960s the biggest band in the United States was the Doors, led by the charismatic, histrionic Jim Morrison. At the Seattle Pop Festival, the day after the Vancouver show, Zeppelin played straight after the Doors \u2013 and wiped the stage with them. But the English group's notoriety was enhanced not by their live set, but by an episode at their hotel, the Edgewater Inn, situated on Seattle's Puget Sound, where guests were able to fish directly from their windows.\n\nAccording to myth, the band tied a girl to a bed and then stuffed pieces of a mud shark into her vagina and anus, the proceedings filmed by Mark Stein, Vanilla Fudge's keyboard player. The incident was a key passage in Stephen Davis's bestselling _Hammer of the Gods_. Yet the truth is apparently far more prosaic. The principal perpetrator was not any of the group \u2013 although John Bonham was peripherally involved \u2013 but Richard Cole and possibly Carmine Appice, Vanilla Fudge's drummer. Nevertheless, when Davis's book was published it only enhanced Zeppelin's reputation for rock 'n' roll debauchery.\n\nWhatever might have happened had nothing to do with Page. He was elsewhere, looking ahead to a show in four days' time, at Earl Warren Showgrounds in Santa Barbara, California.\n\nOn 31 July Page flew into New York on the red-eye from Salt Lake City in Utah, where he had just played two shows at the Terrace Ballroom. He went directly to A&R Studios in Manhattan. Waiting for him was Ritchie Yorke, who had been brought in by Peter Grant to listen to a mixing session of some of the new material. 'Dressed in Chelsea Kings Road-style Regency splendour \u2013 the buckled burgundy patent-leather boots, the flaming red velvet bell-bottoms, the dusty pink-brushed velvet jacket... Page was calmly slouched behind an extensive mixing console with engineer Eddie Kramer. He looked up, munching on a prune Danish pastry, which he was coaxing down with a plastic cupful of teabag swill,' recalled Yorke in _Led Zeppelin: Led to Gold 1967\u201389_.\n\nThe tune Page was working on was 'Bring It On Home', which would close _Led Zeppelin II_ , a record that would rocket upwards the already seemingly unstoppable career of the group.\n\nAs Page controlled the production with an iron-hard clarity, Yorke watched closely: 'Eddie Kramer, one of the finest in his field, manoeuvred the myriads of dials and faders, translating Page's notions into sounds. Occasionally Kramer tossed up a suggestion and Page listened. More often than not, he had an intuitive projection of what the outcome would be. The sound splintered into muddier and dirtier incarnations as Page pushed for the raw and heavy echo he wanted, a style of recording closer to the classic early Chess milieu.'\n\nEddie Kramer had worked on, among others, the Beatles' 'All You Need Is Love' and Jimi Hendrix's _Electric Ladyland_ , released the previous October, a record Page was especially fond of. He had nothing but admiration for Page's abilities in the studio. 'Jimmy was an excellent producer,' said the engineer, '[who] had a very definite picture of what he wanted to capture on tape. He was very demanding, but at the same time completely open to suggestions. Jimmy's biggest asset was he knew how to draw the best performances out of the band,' Kramer later said.\n\nYorke was given a playback of 'Living Loving Maid', 'Heartbreaker', with its magnificent assembly and architecture, and 'What Is and What Should Never Be'. 'Like myself, Grant had yet to hear these completed mixes. We sat in stunned silence as the speakers burst forth.'\n\n'Page seemed genuinely pleased,' wrote Yorke, 'by our natural enthusiasm for what we were hearing \u2013 it was the first reaction he'd observed outside of the four band members and engineer Eddie Kramer.'\n\nJerry Wexler, Atlantic Records' 'spiritual Godfather of rock and roll', as Yorke called him, felt similarly rapturous about the three tracks when they were played to him. 'I would have to say that this is the best white blues I have ever heard,' he said.\n\nPage defined his extraordinary pace of work to Yorke: 'We've been so busy that we just weren't able to go into one studio and polish the whole album off. It's become ridiculous \u2013 we put a rhythm track down in London, add the vocals in New York, overdub harmonica in Vancouver and then come back to New York to do the mixing here at A&R. We never really expected to be as big as this. We just wanted to be able to come over to America to play some gigs a couple of times a year. It's almost gotten out of hand now.' Is he being disingenuous? Is this false modesty? Unlikely: Page was always strong on understatement.\n\n'There are so many guitarists around who I think are better than me. Everywhere I go I hear some cat who sounds better than I do. That's the trouble: everyone's good these days. I'm a trifle disappointed with some of the guitar playing on the second album. When I'm in the studio I really miss the rapport you get with a live audience. There's only a few people there looking at you through a window: it's all very depressing really. The hardest thing in the world is to get excitement on to a piece of plastic. I really do think that we all play better on stage than we do on record.'\n\nLater, in April 1998, Page insisted, to _Uncut_ 's Nigel Williamson that he never said that.\n\n'I do worry,' Page told Yorke, that the second album is turning out to be so different from the first \u2013 we may have overstepped the mark. But then again, I suppose there are enough Led Zeppelin trademarks in there. It's very hard rock, there's no doubt about that. There aren't many bands into hard rock these days and I think that might account for some of our success. All sorts of people are into folk and country and softer stuff. We just like to play it hard and bluesy. There aren't many exponents of real contemporary blues either. John Mayall doesn't do it any more. But there's always a market for it, I think. Taj Mahal is my idea of contemporary blues. Our stuff I think is a combination of everything.'\n\nAccording to Yorke, Page harboured considerable worry over the worth of _Led Zeppelin II_ : 'The album took such a long time to make... it was all on and off. It was quite insane really. We had no time and we had to write songs in hotel rooms. By the time the album came out, I was really fed up with it. I just heard it so many times in so many different places. I really think I lost confidence in it. Even though people were saying it was great, I wasn't convinced myself.\n\n'There was probably more attack on the second album because it was written while we were on the road and only getting into the studio when we could find an opening. I suppose that feeling of playing all the time is evident in the new album. There wasn't much time to sit back and think about it.'\n\nAfter his day in the A&R studios in New York City, Page got straight on a plane to Los Angeles. In her autobiography _I'm With the Band_ , Miss Pamela listed extracts from her diary entries. Mr Carlos, a member of the BTOs, the male companion group to the GTOs, had run into Page in Paris. Page had declared his intention to bed Miss Pamela. Her diary entry for 31 July 1969 read: 'Jimmy Page is coming to town today. I don't know whether I want to be with him or not, who knows what diseases I'd get? Such a sweet and lovely precious-looking cherub, why is it that he's perverted? Maybe he's not?? Perhaps I'll find out.'\n\nDespite her trepidation, that night Miss Pamela went along to Thee Experience. There, she sat 'sipping red wine through a straw' as she waited for Zeppelin to show. 'I was feeling haughty one minute and petrified the next, trying to get a little tipsy before the demonic darling darkened the seedy doorstep.' Although Robert Plant and Richard Cole arrived, there was no sign of Page. 'Richard Cole stumbled over and handed me a scrap of paper with Jimmy's number at the Continental Hyatt House scribbled on it. He leaned into me and mumbled thickly in my ear, \"He's waiting for you.\"'\n\nAlthough the Hyatt House was only a short walk away, Miss Pamela instead sat back and watched Bo Diddley play his set.\n\nPlaying it cool seemed to work to Miss Pamela's advantage. The next day, Page called her up, asking why on earth she hadn't shown at the Hyatt the previous night. Led Zeppelin were playing that evening at the Earl Warren Showgrounds in Santa Barbara, ninety miles north of Los Angeles, on the Pacific coast; Page asked her to join him on his journey to the venue. 'But I came down by myself to show a little more hard-to-getness... He seems so shy and delicious, grey eyes gazing into mine, sweet sweetness, pale white skin, gentle gentleman with something to hide. What does he want from me?'\n\nPicking up the ticket and backstage pass Page had left for her at the venue, Miss Pamela found herself escorted backstage after the show by Richard Cole to a waiting limousine. 'The long ride from Santa Barbara was one of these dream experiences that leave you glowing in the dark. From the moment Jimmy slid his small velvet-clad ass across the seat of the limo, right next to mine, until the door was thrown open in front of Thee Experience, we cooed and giggled like doves in heat. It was a hundred-mile drive, which gave him plenty of time to come out with \"all the lines\". He told me he had gotten my number the last time he was in town but was too nervous to use it until the last day, and he called and called but the line was constantly busy... He said he wanted to spend time with me more than anything in the world. Tell me more. I kissed and slobbered all over the inside crease of his slim white arm until he rolled his head back against the plush seat, gasping, \"Oh, Pamela, yes, yes, yes.\" He warned me that his previous LA girlfriend would probably be in the club and that I would have to give him the chance to \"explain\" to her about me. Uh-oh.\n\n'I climbed out of the warm, dark backseat womb, full of wet kisses and flaming glazed eyes, and found myself in the precarious position of sharing this splendid divinity with Catherine James, the most gorgeous rock courtesan alive. She and I hissed at each other from a dark distance, and I beat the old hasty retreat back to my cozy pad.'\n\nIn her diary entry dated 2 August 1969, Miss Pamela describes how her friend Michelle told her that Page was astonished she had left Thee Experience the previous evening. 'He was asking everybody if they'd seen me. He looked all over the club after \"explaining\" to Catherine, and left alone.'\n\nLater in the day her phone rang. She picked it up and heard a voice declare: 'Long distance, Mr Page calling.'\n\n'He knew what to say all right; he could have given a Master's course in how to turn a fairly sane girl into a twittering ninny. No one had ever gushed over me, or given me all the lines before, and I could feel myself falling apart and turning into one of those gooey unrecognisable substances. He told me he was going to come to my door, sweep me off my feet, and take me away in his white chariot; he told me he was my knight in shining armour; he told me he didn't know what was coming over him, he had never felt like this before... He acted like he couldn't believe I ever gave him a second glance. When I told him I missed him, he came out with, \"Oh Miss P. Really? Are you telling me the truth?\" My melting heart wasn't ready for this guy. I swallowed it all whole, and it was fucking delicious.'\n\nHe called her from Houston, where Zeppelin had played the Music Hall on 3 August, saying he would be coming round for her the next evening. And he arrived in a white limousine, taking her to the Palomino Club, something of a country music venue, in North Hollywood. The Everly Brothers were performing. 'We got all caught up in those glorious harmonies. Jimmy's eyes misted up and he squeezed my hand on certain meaningful lyrics: \"Mmmmmm: I never knew what I missed until I kissed you...\" He looked hard at me with a tiny smile on his rosebud lips, making me sweat about the long night to come. He put something into my hand, and it turned out to be a silver ring with twenty pieces of turquoise embedded into it, and I wondered if I was going steady with the best guitar player in the world. He always messed with his black curls, poofing and fluffing them around his flawless face; he wore emerald velvet and white chiffon, thin little socks, and the most perfect brooch on his lapel. I couldn't wait to get back to the hotel and take it all off.'\n\nPage, Miss Pamela noticed, was forever checking out how he looked in the mirror, 'putting perfect waves in his long black hair with a little crimping machine. He used Pantene products, and whenever I smelled them... I remembered being buried in his hair.'\n\nWhen they went to bed, she wrote, 'Such a face, so gentle and soft, I'm amazed at his sadistic tendencies; they're such a part of him that I doubt if he'll ever stop. It was really frightening, he changed into another person, but all he did was chew me and slap me a little.'\n\nWhen Miss Pamela noticed the set of whips in his suitcase, he promised, '\"Don't worry Miss P, I'll never use those on you, I'll never hurt you like that.\" Then he sucked on my neck, and when I could feel the bruise being called up out of my bloodstream, he tossed me down on the bed and told me he would throw away the whips to show how much I meant to him.'\n\nMiss Pamela described how Page had with him a test pressing of _Led Zeppelin II_ , which he played again and again, taking notes. For her part, she turned him on to Nudie Cohn, who had tailored sumptuous outfits for such singing cowboy royalty as Gene Autry and Roy Rogers, as well as making Elvis Presley's $10,000 gold lam\u00e9 suit. The entire group, along with Peter Grant, went to Nudie's store in North Hollywood, to be fitted out with the legendary tailor's western wear. 'We went to the Glass Farmhouse, where Jimmy got a long antique coat embroidered with a dragon and a silly velvet hat with a feather in it.'\n\nIn Las Vegas, Miss Pamela was with them to watch Elvis Presley from front-row seats at the International Hotel. Following the show, one of the King's people emerged to ask if Page would care to come backstage for an audience with Elvis. 'No, thank you,' he replied.\n\nWhen Led Zeppelin left the West Coast, Miss Pamela was on her own and somewhat distraught. At the end of the month, however, the group were to play in New York, at the Singer Bowl in Queens. Page sent her a plane ticket, and she was with him in the city for three days, where he introduced her to people as 'Mrs Page'. She then travelled with him to the Texas International Pop Festival in Lewisville for their last date of this tour, where Zeppelin played to 80,000 people, before it really was goodbye, and he and the band returned to England to finish _Led Zeppelin II_.\n\nComing to the UK at the invitation of Page, Joe Jammer found himself with a manager, Peter Grant, who secured a record deal for him with EMI, on the Regal Zonophone label. At the instigation of Page, Joe became yet another of Screaming Lord Sutch's apprentices, playing guitar with him. But the American was aware of one of the downsides to the guitarist's character: 'I learned early that they had a nickname for Jimmy: Led Wallet. Because he would never pay for anything for anybody. And it came to a time when Zep were becoming more famous, but Jimmy would come in from Pangbourne hitchhiking. To save money. They talked him into at least taking a train. But he would only do second-class. And some hippy guy came up to him: \"Jimmy, why are you here in second-class?\" He told that story in the office. So Peter Grant made a decision. He hired a car to bring Jimmy Page into town. And Jimmy refused to pay for it. So the other four had to pay for Jimmy's car. He had the money. But he was very tight. Maybe there was something way back when with mum and dad and the Second World War. He knew he had to be smart, super-smart.\n\n'Peter was hired by Jimmy and John Paul Jones. They paid him out of their own money at first. He was one of the first managers ever who didn't screw his band, the first manager ever who defended his band \u2013 to the point where everybody outside hated him for being Led Zeppelin's manager. Jimmy did call all the shots. Peter made them happen.\n\n'Jimmy was like a musical Hitler. He had a big master plan to take over the world and he certainly did. There was no question about it. No \"we're goofing around and gonna have fun\"; we might have fun along the way but this was serious work.\n\n'It may not have been that premeditated. But \"we're gonna make it superbig\" was certainly the position. \"We're going to be the biggest band ever\" \u2013 that was premeditated.\n\n'The Led Zeppelin sound came through that last Yardbirds album, which Mickie Most produced. And Mickie Most was under the impression that he was going to be allowed to produce Led Zeppelin. But they said, \"No, Jimmy's going to do it.\" And that had a lot to do with Mickie and Peter falling out. Mickie didn't like all the drugs: he was dead straight. Mickie was a drinker: later he went to AA. But Led Zeppelin eventually went really hardcore and Mickie didn't like that.' Joe Jammer miming injecting a syringe into his arm should make clear what he means by hardcore.\n\n'When I was with them, it was just pot and booze. But they noticed that the pot made them hungry and they ate and got fat. So then they discovered the other thing: old Charles. And then Henry kind of came along on the heels of Charles. But not all of them: Jimmy and Bonham \u2013 Bonham was into anything, anytime, anywhere.'\n\nJohn Paul Jones, however, was another kettle of fish altogether: 'He was the most serious of them all. And he was the quietest. And they always say, \"The wise man is the silent man. He who speaks does not know. And he who knows does not speak.\" So Jonesy, while the fucking chaos was swirling around him, as a bass player was absolutely a fucking world-class master, but just wanted to do a fucking good job.\n\n'He was like the Brian Jones of Led Zeppelin, always introducing weird new instruments, playing keyboards with his foot. Super-musical. What he did gave them was a big advantage over any other four-piece rock band.'\n\nJoe Jammer also quickly perceived the importance of Richard Cole to the Zeppelin operation. 'Richard Cole is the one man who knows everything about everybody. He was the key, the axle in the wheel. He knows everything about everybody. Hence his book _Stairway to Heaven_. Richard knew everything about everybody: without him none of it would have happened. He's the key man. He was third man down the chain: Jimmy Page, Peter Grant, Richard Cole. Jonesy and Jimmy were equal. Because they started it together and paid for it together. I think it was 50\/50.\n\n'The hierarchy of Led Zeppelin was like the Cosa Nostra. I was the guy who didn't get shot. I was the guy who didn't get beat up. I was the guy who didn't get abused. You've got to have one. I thought it was very brutal. When the junk started, they started losing their liking of everybody. Jimmy got real mean to me. But it was the junk doing all the talking.'\n\nThe _Led Zeppelin II_ sleeve design was now ready. Page had asked David Juniper, who had studied with him at Sutton Art College, to come up with 'an interesting idea'. Juniper took a First World War photograph of German flying ace the Red Baron and his esteemed Flying Circus, and then airbrushed the faces of Led Zeppelin's four members over them \u2013 an angelically dangerous-looking Page sat in the centre of the group. Further iconic faces were added: Blind Willie Johnson, for example, and what was at first thought to be Neil Armstrong, but was in fact another astronaut, Frank Borman. There was also the actress Glynis Johns, a sly dig at engineer Glyn Johns, with whom there had been a disagreement.\n\nTo coincide with the release of _Led Zeppelin II_ the group returned to America on 17 October for a brief tour. This time there were no shows in Los Angeles; instead they played three sold-out nights at San Francisco's Winterland Ballroom, with its 5,400 capacity, supported by Roland Kirk and Isaac Hayes.\n\nFor security reasons the group decided to stay in Sausalito, in hip Marin County. Miss Pamela flew up from Los Angeles, and, while taking her on a tour of the city, Page found an art gallery where he bought a number of etchings, at $500 a pop, by the Dutch graphic artist M.C. Escher; he had long loved Escher's work. 'He bought me a book of Sulamith W\u00fclfing's ethereal paintings, and I clutched it to my chest,' she wrote.\n\nWhen it came time to leave once again for London, Page said to Miss Pamela at the airport: 'P, you're such a lovely little girl. I don't deserve you. I'm such a bastard, you know.'\n\nThere were problems with the first factory pressing of _Led Zeppelin II_ : due to the sound of the bass on the record being so heavy and overmodulated, the record kept skipping, necessitating Jerry Wexler to throw a hundred thousand dollars' worth of pressed albums in the bin and start all over again.\n\nThe record was finally released on 22 October 1969. Rattling and bustling with the energy of the group's live shows, around which it had been recorded, and kicking off with the very unique 'Whole Lotta Love', the album surpassed expectations. It was a major statement of Led Zeppelin's position \u2013 as a cutting-edge, zeitgeist act, endlessly inventive and exciting at an extremely primal level. It was one of the raunchiest records ever made. 'It was still blues-based but with a much more carnal approach to the music,' said Plant.\n\n'They were the first numbers written with the band in mind,' Page told Mick Wall. 'It was music more tailor-made for the elements you've got. Like knowing that Bonzo's gonna come in hard at some point, and building that in.' Almost US garage-band punk in the simplicity of some of the songs, there is something curiously amateurish about material like 'Ramble On', indubitably part of the appeal.\n\nUtterly contrary to what Page wanted \u2013 although he was often prepared to be pragmatic when it came to the success of his group \u2013 Atlantic, anxious for a trailer to advertise the new LP from their expensive signing, edited 'Whole Lotta Love' and released it as a single in the USA, where it hit the number four spot. Attempts to do the same in the UK had Peter Grant expressing his feelings in no uncertain terms to Phil Carson, who ran Atlantic there.\n\nAt the end of 1969, at London's prestigious Savoy Hotel, Led Zeppelin were presented with gold discs for their first album by Gwynneth Dunwoody, a minister at the Board of Trade in Prime Minister Harold Wilson's government, in honour of the group's contributions to the UK's balance of trade. Afterwards Grant took his client to the swish neighbourhood of Berkeley Square, where he bought Page a brand new Rolls-Royce \u2013 despite the guitarist not having a driving licence.\n\nIn the US there were 400,000 advance orders for _Led Zeppelin II_. It jumped straight into the Billboard chart at number 15. By Christmas it had replaced the Beatles' _Abbey Road_ at the top of the US charts. It was still in that position in February 1970, when the album hit the number one spot in the UK. As it did so, Led Zeppelin were undertaking a UK tour. Called back for a fourth encore at Leeds University on 24 January, the quartet of musicians played a soul-shaking medley of Eddie Cochran songs intertwined with snippets of 'Communication Breakdown'.\n\nOn 9 January 1970, Jimmy Page's 26th birthday, Led Zeppelin had played again at London's Royal Albert Hall. When they had performed there only six months before, they were somewhat nervous. But now, having knocked the Beatles off the top of the US charts, and with no support act, their confidence was supreme. Among those at the show were John Lennon, Eric Clapton and Jeff Beck.\n\nThey kicked off with 'We're Gonna Groove'. Plant said to the audience, 'This is Jimmy Page: lead guitar,' the Led Zeppelin leader's hair dangling to his shoulder blades.\n\n'Albert Hall was a massive gig for us, and we really wanted to do the best we could,' Page told _Guitar World_ in 2003. 'It was a magic venue. It was built in Victorian times, and you are in there thinking about all the musical history that has preceded you. On top of that, it was something of a homecoming for John Paul Jones and me, because we had both grown up around there. So we were all really paying attention to what we were doing.'\n\nSuch scrutiny clearly paid off. The show, filmed by Peter Whitehead, was utterly sensational. There is something extraordinarily sexual about the sliding of Jimmy Page's violin bow during 'Dazed and Confused', and the reviews echoed the frenzied atmosphere.\n\n'For 10 years, rock and roll had been working towards something that would combine the extraordinary capacities of electronic instruments with the anarchic energy of youth, and there in the Albert Hall on 9 January 1970, [we] found it... the sound came up to me with a force that pummelled me breathless. No other band ever managed to make a sound like that. It was certainly loud, but it was also driving, pushing along with incredible energy,' wrote the eminent feminist Germaine Greer in the _Daily Telegraph_. Meanwhile the _NME_ claimed that the show 'completely destroyed the ever-weakening argument about British reserve'.\n\nIf that colossally successful evening wasn't enough celebration for his birthday, Page received an even greater present after the show. Heather Taylor, his former flame who was by now regularly on the arm of Roger Daltrey, introduced Page to her friend, the model Charlotte Martin. In fact, it was something of a re-introduction: Page had already met her several times, when she was with Eric Clapton.\n\nA fanatical vanity and an almost immeasurable self-adoration would appear to be important prerequisites for any aspiring rock star. But although he may have possessed these attributes, what was far more dominant in Page's case was an impulsive drive that derived much more from an almost obsessive self-knowledge, underpinned by what he had learned about himself from his studies of both art and the occult and the consequent confidence it gave him. Frail, sexually androgynous and mentally muscular, he exuded a complex iron toughness; this was someone who would not back down. When he met Charlotte Martin again he knew they had been drawn together by other forces.\n\nPage and Martin were immediately smitten with each other. He asked Richard Cole to give them a ride to her nearby apartment. Soon she moved into his Pangbourne home. 'Charlotte was the type of girl you couldn't look at just once,' said Cole in _Stairway to Heaven_. 'Tall. Thin. Blonde. Perfect features. You had to glance a second time.' All the same, he was not necessarily taken with her. 'At least in her relationship with me, she was aloof, unfriendly and indifferent. It was my feeling that unless she really liked you, she had a \"take it or leave it\" attitude. Frankly, I wasn't impressed.'\n\nLed Zeppelin's next US tour, their fifth, which ran from 21 March to 19 April, was almost entirely performed in arena-sized venues.\n\nThe group's enormous, sudden success had offered Peter Grant a battering ram with which to smash down the barriers erected by the somewhat amateurish, hit-and-miss \u2013 and often corrupt and Mob-affiliated \u2013 nature of much of American concert promotion. Led Zeppelin's earnings had jumped significantly: a pair of shows at Chicago's Kinetic Playground in February 1969 earned the four-piece $7,500. Returning in May for a similar slot, they picked up $12,500 per night.\n\nNow Grant set about renegotiating the very structure of the deals he struck with US promoters, which would make his clients extremely rich. Zeppelin insisted on 90 per cent of the gross take of ticket sales, leaving just 10 per cent profit for the local promoter. Page could hardly disguise his approval. 'The new system is to put groups on a percentage of the gate money and we drew $37,000 from one amazing gig in Los Angeles,' he said.\n\nFollowing this US tour, London's _Financial Times_ declared that Led Zeppelin had now earned over $5 million in the United States. Was the story run, at the instigation of Peter Grant, to cock a snook at those stalwarts of the underground who still considered Led Zeppelin a 'capitalist' group? Quite probably.\n\nMidway through the tour, however, there was a problem. At Winnipeg Airport in Canada a road manager noticed that something was missing: Page's 'Black Beauty' guitar, a Les Paul Keith Richards gave him that he had played on the entire tour. 'Whole Lotta Love' had been recorded with Page playing the Black Beauty. (Most of his Led Zeppelin slide work was on his Danelectro.)\n\nDespite Led Zeppelin standing to gross a further $1 million for this tour, the guitar's loss put Page in a dark mood for much of the rest of their time on the road. 'Jimmy never seemed to fully recover from the loss,' said Richard Cole. 'We got through all 29 performances without any noticeable impact upon his playing. It really didn't matter that every show was a sellout, that it was a guaranteed million-dollar tour before it had even begun. To Jimmy, the loss of the guitar ruined everything.'\n\nIn case Led Zeppelin were beginning to take their sudden, colossal success for granted, it was on this tour that they received a severe wakeup call. On 8 April 1970, playing an arena in Raleigh, North Carolina \u2013 resolutely in the distinctly unliberal 'South' \u2013 a Zeppelin employee overheard a pair of local cops planning to plant narcotics on a member of the group. Immediately Richard Cole went into action, trying to hire security from the local Pinkerton's office. But then he gave up and called Steve Weiss, the Zeppelin lawyer, in New York. Weiss arranged for a pair of local private detectives to stand guard over the group at all times until they left Raleigh.\n\nWere those Raleigh police officers responding to the signs of excess around Zeppelin that were everywhere evident during their spring 1970 tour of the USA? This was the first time that an unalloyed sense of decadence had begun to hover about the group: drink, drugs and excessive sexuality were omnipresent. Whereas once all four members would have been happy with a drink and a spliff, now cocaine had entered their world, as well as something even worse: when they played in Los Angeles, John Bonham was on heroin.\n\nAnd the overindulgence of these dates took its toll. Suffering for much of the latter days of the tour from a bad cold, Robert Plant's vocals became increasingly ragged. Finally, following their show at the Arizona Veterans Memorial Coliseum in Phoenix on 18 April, Page startled the audience with an announcement following the end of 'Whole Lotta Love': 'You've been a fantastic audience, but there's been something happening tonight... Robert's been very ill, and as he came off he's just collapsed and we've just called for a doctor. We'd really like to do more, but obviously it's impossible.'\n\nThe show scheduled for the next day, in Las Vegas, was cancelled. Led Zeppelin flew home to London.\n\n## 11\n\n# 'DO WHAT THOU WILT'\n\nThe day after Led Zeppelin returned to London, Jimmy Page appeared as a solo guest on Julie Felix's folk-lite BBC television show _Once More with Felix_ , which was broadcast three days later, on 26 April. Page appearing alone was unusual, and his choice of songs was curious too. He played his medley of 'White Summer'\/'Black Mountain Side' \u2013 in Phoenix the previous week he had performed the same material \u2013 for what would be the last time until it resurfaced on the group's 1977 tour. Julie Felix introduced him in a manner befitting of someone who had acquired the status of rock nobility: 'My next guest this evening is a member of certainly the most successful group to come out of Britain in the last couple of years. Led Zeppelin LPs top both the British and American charts, and the lead guitarist in that group is definitely a very talented and special musician.'\n\nBack home in the Midlands, however, Robert Plant was not enjoying quite as warm a homecoming. His voice fully recovered, he was now employing it in arguments with his wife Maureen. In his state of semi-permanent jetlag, Plant was enduring a period of angst about his rapidly improved position in life. Returning home from a tour where he had been worshipped as a god had not readily acclimatised him to adjusting to simple domestic life. However, Plant was an intelligent man, and he felt he could come up with a solution: going on holiday with Jimmy Page and their respective 'old ladies'.\n\nPlant had spent a summer holiday with his parents near Bron-Yr-Aur, a broad, two-storey cottage in north Wales close to the mountain Cadair Idris, which is steeped in mythology, legendarily the site of the last battle of King Arthur. And as spring began to bloom, Plant sold Page on the idea of spending time in the area. 'I had already written \"Immigrant Song\",' recalled Page to the _Guardian_ in 2014, 'and came up with \"Friends\" a day or two before we got together at Robert's house in the West Midlands. I was keen to do \"Gallows Pole\" because I had an arrangement I wanted to try, and I had \"Tangerine\" from a number of years beforehand \u2013 before Led Zeppelin.'\n\nWith Clive Coulson and Sandy Macgregor, a pair of Zeppelin roadies, strapped into service by Richard Cole, the party journeyed up to north Wales, invoking the getting-away-from-it-all creative ethos of acts like the Band and Traffic. 'We had done a lot in 1969,' said Page. 'We'd done six months of solid touring in America. We'd performed all over Europe. We'd also done an album on the road. It was exhilarating, but it was nice to be able to stop. We finally had the chance to slow down, and that's reflected in our third album.'\n\n'Whatever our physical state was at the end of _Led Zeppelin II_ , there was now only one place to go: up into the misty mountains with our families,' said Plant.\n\nIt was perfect peace, a long way physically and \u2013 more importantly \u2013 culturally from the Hyatt House. So long as you didn't mind the lack of running water \u2013 it was necessary to adjourn to a nearby pub for a bath \u2013 or electricity, and the constant drizzle of the rain billowing in from nearby Snowdonia's mountains. Old hippie that he was, still regularly to be found immersed in yet another rereading of J. R. R. Tolkien's _The Lord of the Rings_ , Plant loved it, bringing along his daughter Carmen and their spotted Border collie Strider \u2013 named after a _Lord of the Rings_ character \u2013 as well as Maureen. You wonder how the sophisticated model Charlotte Martin took to it all.\n\n'We were just trying to get some ideas down to bring back and work on,' said Plant. 'So there we were, up in the mountains with one of the very first Sony cassette recorders and a bunch of Eveready batteries, just sitting going: \"Well, what shall we do?\" Carmen was two and needed bathing. Strider needed feeding. And we needed cider.'\n\nIt was here that Page and Plant, armed with a pair of acoustic guitars, first began to actively write together. A pair of songs would immediately find their way onto the next album. The ballad 'That's the Way' \u2013 initially titled 'The First Time' \u2013 resulted from a set of chords that Plant unearthed; taking a long walk with Page's guitar and a portable cassette recorder, they hummed and sang the song \u2013 essentially an eco hymn with reflections on the negative reception they sometimes received from rednecks in the US \u2013 into completion before they had returned. '\"That's the Way\" came out of there as a complete song,' said Page.\n\n'Friends' was a drone-like tune cloaked in mystic sounds that Page came up with after he'd had a severe row with Charlotte Martin. Half an hour after finishing the song, Page had make-up sex with Charlotte, conceiving their child Scarlet in the process.\n\n'Bron-Y-Aur Stomp' was a tune that Plant composed in honour of his dog Strider; on the final recorded version Page's acoustic work revealed the influence of the British folk guitarists Bert Jansch and Davy Graham. The song was first tried out as a rocking electric tune entitled 'Jennings Farm Blues', the name of Plant's new home in the West Midlands.\n\nThere were more. 'Over the Hills and Far Away' and 'Down by the Seaside' would appear on later records, as would 'Black Country Woman' and 'The Rover'. 'Another Way to Wales' and 'I Wanna Be Her Man' never appeared on any record by the group.\n\nEqually as important as this new material was the fact that this was the first opportunity for Page and Plant to form a genuine friendship, to come to a measure of mutual understanding. 'I really came to know Robert, actually living together at Bron-Yr-Aur, as opposed to occupying nearby hotel rooms. The songs took us into areas that changed the band. It established a standard of travelling for inspiration, which is the best thing a musician can do,' Page said later, to Cameron Crowe in _Rolling Stone_ magazine.\n\n'Jimmy was much more of a man of the world than I was,' admitted Plant to the _Guardian_ in 2016. 'He'd travelled a lot with the Yardbirds, he'd done those Dick Clark tours... And he was the face, so I was master at arms at that point in the development of our relationship \u2013 it was all developing before me. It was a very easygoing and very unaffected time \u2013 miraculous really.'\n\nThere are suggestions that recording for the album then commenced at Headley Grange, a rotting 18th-century mansion house in Hampshire. But Page firmly nixes this notion.\n\nInstead he insists that _Led Zeppelin III_ was recorded at Olympic Studios and at Island Records' Basing Street Studios in Notting Hill. 'The whole thing,' confirmed Andy Johns, the engineer and younger brother of Glyn, 'was done at Studio 2 at Olympic. I remember for sure, then I got them to go to Island, where we did the mixes. And that went pretty well. They used to work very quickly.'\n\nOn 22 June there was a break from the routine of recording. The British Council asked Led Zeppelin to represent British popular music on a cultural visit to Iceland. With its ancient independent tradition celebrated in the island's sagas, Plant was stirred into writing the lyrics for the number that would kick off the new LP: 'Immigrant Song'.\n\nEqually importantly, the gig in Iceland's capital Reykjavik, before a 5,000-strong audience, served as a warm-up date for what would be the largest UK showcase so far for the group. On 28 June 1970 Led Zeppelin topped the bill at the second Bath Festival of Blues and Progressive Music, promoted by Frederick Bannister; the four musicians had come a long way from their mid-afternoon slot at the festival the previous year. The band cancelled a pair of shows at New York's Madison Square Garden to appear at Bath after Bannister promised Peter Grant a minimum crowd of 200,000 people. Led Zeppelin received \u00a360,000 for their appearance at Bath, less than they would have picked up for Madison Square Garden.\n\nThe exciting festival bill featured a feast of acts, the cream of underground rock in the US \u2013 Canned Heat, the Byrds, Steppenwolf, Santana, Johnny Winter, Country Joe and the Fish, the Mothers of Invention, Jefferson Airplane, Dr John and Flock \u2013 and at home \u2013 John Mayall, Pink Floyd, the Moody Blues, Fairport Convention, Donovan and many more.\n\nPeter Grant insisted that Led Zeppelin's set was to begin at 8.30 p.m. A week on from the summer solstice, this would ensure that the sun set a few minutes after Zeppelin began their show with the brand new 'Immigrant Song', a tune whose ironic title celebrated Viking rampages to other lands but seemed like a metaphor for Led Zeppelin itself. In the line 'Our only goal will be the western shore', Plant's lyrics clearly refer to both the documented Norse journeys to America and Led Zeppelin cutting a broad swathe across the United States to California. Yet in Plant's barked utterance 'We are your overlords' was there not a suggestion of the hubris that would finally undo Zeppelin \u2013 especially in a song where Page's guitar riff was not really as commanding as it needed to be? Really, the galloping 'Immigrant Song' seemed like a subconscious effort to replicate the power and simple complexity of 'Whole Lotta Love' \u2013 and came up wanting.\n\nFurther new songs were tried out: 'Celebration Day' and 'Since I've Been Loving You', the third song of the set and future fan favourite that featured John Paul Jones on Hammond organ. This song had been played in a different form at the Royal Albert Hall on Page's birthday.\n\n'We've been playing America a lot recently and we really thought that coming back here we might have a dodgy time,' pronounced Plant from the stage. 'There's a lot of things going wrong in America at the moment that are getting a bit sticky. It's really nice to come to an open-air festival where there are no such bad things happening and everything's turned out beautiful.'\n\nMind you, there was the by now established atmosphere of 'heaviness' about Zeppelin's performance. Richard Cole and three of the group's roadies stepped onto the stage during Flock's set and aggressively pulled out the plugs so they could play no more.\n\n'Unbeknownst to Freddy Bannister, I went down to the site and found out from the Meteorological Office what time the sun was setting,' said Peter Grant. 'It was going down right behind the stage. By the band going on at sunset, I was able to bring up the stage lights a bit at a time. It was vital we went on stage at the right time. That's why I made sure the previous band, Flock \u2013 or whoever they were \u2013 got off on time.'\n\nEach member of Led Zeppelin was in vogue with a beard \u2013 a hangover of that time in Wales when water itself was at a premium. Page was accoutred in a tight double-breasted overcoat \u2013 the identical one he'd worn in that 1968 first publicity shot with the group posed about a 3.8 Jaguar \u2013 and a bucket-like hat, giving him the appearance of an especially spectral scarecrow. With his flowing, shoulder-length locks complemented by a trimmed beard and moustache, Plant now appeared like a character from the court of Louis XIV. And anyone watching might have felt that the Zeppelin singer had taken more than a hint of stagecraft from the Who's Roger Daltrey, who was similarly tonsorially styled and was the prototype of the microphone lassoist into whom Plant had evolved.\n\nWhat was specifically significant about Led Zeppelin's performance at the Bath Festival, where they would perform four encores, was an alteration to the tone of the set that would remain with the group for most of the rest of their time together. Some 40 minutes in, John Paul Jones put down his bass guitar and picked up a mandolin; meanwhile, Page swapped his Gibson Les Paul for a Martin D-28 acoustic guitar. As the two musicians tuned up, Plant cracked a joke: 'This is a medley of Lonnie Donegan tunes.' Then he announced the next song: 'This is called \"The Boy Next Door\", for want of a better title.' And so 'That's the Way' \u2013 as the song would be renamed on the album \u2013 became the first song the group played acoustically in the UK. Suddenly an entirely new side of Led Zeppelin was revealed. Those who carped at notions of sudden change accused the four musicians of aping the styles and subject-matter of such newly popular West Coast singer-songwriters as Crosby, Stills, Nash and Young, handily omitting the wealth of evidence from their first two LPs of similarly styled material.\n\nThe rigorously rehearsed set was paced to build to a series of climaxes, ending a full three hours later, after the four encores, the final one a rock 'n' roll medley featuring 'Long Tall Sally', 'Say Mama', 'Johnny B. Goode' and, finally, 'That's All Right Mama'. Led Zeppelin came offstage after three hours, to the adoring words of Mike Raven, the Radio 1 DJ: 'Unbelievable! Led Zeppelin! You're fantastic! Led Zeppelin: England adores you!'\n\nThe show was an utter triumph. With the Beatles having announced that they were breaking up two months previously, Led Zeppelin became the kings of British rock music. Yet even so, there were 'freaks' who basked in the snobbish cool of informing you that they had left the festival site before Led Zeppelin had come onstage; among 'intelligent' fans of rock there were many who were still utterly dismissive of Page's group.\n\nEarlier in the day Page noticed that Roy Harper was wandering around the backstage area. Harper was a gnarled, idiosyncratic artist whom the Zeppelin guitarist had first watched on the London folk circuit in the mid-sixties. Although they had never spoken before, Page approached Harper and asked to be shown how to play an instrumental track, 'Blackpool', from Harper's first LP. 'I played it for him and he said, \"Thanks very much.\" We exchanged pleasantries and then he walked away. The only thing I thought as I watched him leave was, \"That guy's pants are too short for him.\"'\n\nOnly when Page went onstage with Led Zeppelin did Harper appreciate who he was. 'During the second song, all the young women in the crowd started to stand up involuntarily, with tears running down their faces. It was like, \"Jesus, what's happening here then?\" In the end, you knew you'd seen something you were never going to forget.'\n\nHarper and Page became close friends, a relationship celebrated in the 'Hats Off to (Roy) Harper' song on _Led Zeppelin III_ , which served as an introduction to the world's record buyers of a performer whose rigorously uncommercial music had so far been understandably sidelined. Later in the year, at Led Zeppelin's office, Page gave Harper a copy of the new LP, urging him to look at it closely. When he discovered the tribute to him the folk singer was deeply moved. He became an integral feature of the Led Zeppelin scene, sometimes opening for the act.\n\nWhen Zeppelin played the 1970 Bath Festival, Joe Jammer was also on the bill. Although he was no longer part of the group's technical crew, his position as a companion to Page was resoundingly confirmed when, one afternoon at Pangbourne, Charlotte Martin drew up his astrological chart: to the astonishment and then fascination of first Martin and then Page, it was revealed that Joe Jammer and Aleister Crowley shared an almost identical planetary set-up. 'Double Libra, moon in Leo,' said Jammer. 'Same natal chart as Aleister Crowley \u2013 that was the first time I had heard that name. And that really flipped Jimmy out. And Charlotte.' When they first met, Page and Jammer had almost immediately clicked; now it seemed as though this friendship had been written, almost literally, in the heavens.\n\n'Jimmy would use me because there weren't multi-track machines then. So you could only play one part and had to imagine the other part. He'd call me into his room on that 1969 US tour, and ask me to play a little riff as he played something opposite against me. And this went on all the while after I came to London. I'd go out to Pangbourne to help him with some songs. There's a Zep song called \"Black Dog\". But I couldn't learn the riff. I mean, this guy is a master guitarist. He plays me this incredibly complicated riff, and says, \"Play that!\" I couldn't do it, so now he's kinda getting a little mad. But he was gracious: \"Let's move on to something else!\" So he said, \"What do you like to play?\" I said, \"Well, I'm into funk.\" I taught Jimmy how to play funk on that session that day. And it emerged in that Led Zeppelin tune \"The Crunge\". Which is exaggerated acid-trip super-funk. Robert goes, \"Where's the bridge? Can't find the bridge?\"'\n\nAgain assisted by Page, Jammer considerably boosted his income by becoming a session player in the UK, playing on over 150 albums during a ten-year period. 'It was good money. I used to get union scale. But by the time he stopped doing it Jimmy had negotiated his pay up to three times union scale. He had his own Jimmy Page Music right from early on. His mum advised him on that. He also wrote an enormous amount of B-sides for singles. He was certainly a millionaire by the start of Led Zeppelin.'\n\nLed Zeppelin returned to the recording studio to record 'Hats Off to (Roy) Harper' \u2013 'a piece of spontaneous combustion initiated by Page late one night inspired by some frenzied slide-guitar channelling of Bukka White's \"Shake 'Em on Down\"', wrote Mick Wall. 'Gallows Pole' was also laid down, an interpretation of the old English folk song 'The Maid Freed from the Gallows'.\n\n'Since I've Been Loving You' could be seen as the ancestor of the power ballads that would emerge during the 1980s. Yet again there would be questions about the inspiration for this song, which would become a _tour de force_ of their live sets. The genesis of 'Since I've Been Loving You' was surely the song 'Never' by Plant favourites Moby Grape. 'Never' begins with the lines 'Working from eleven to seven every night, ought to make life a drag, yeah, and I know that ain't right'. 'Since I've Been Loving You', meanwhile, starts off with the lines 'Working from seven to eleven every night, it really makes life a drag, yeah, and I know that ain't right'. Compare and contrast, pop fans. 'There are also uncanny echoes in the music, both songs being extrapolations from a stately B.B. King-style blues, working it up into a melodramatic musical statement with beginning, middle and end where the Grape working is more content to meander forth in the one-take jam style it was intended,' considered Mick Wall.\n\nThe engineer on 'Since I've Been Loving You' was a trainee called Richard Digby Smith. In 2000 he was interviewed about the session by _Mojo_ magazine: 'I can see Robert at the mike now. He was so passionate. Lived every line. What you got on the record is what happened. His only preparation was a herbal cigarette and a couple of shots of Jack Daniel's... I remember Pagey pushing him, \"Let's try the outro chorus again, improvise a bit more\"... There was a hugeness about everything Zeppelin did. I mean, look behind you and there was Peter Grant sitting on the sofa \u2013 the whole sofa.'\n\nAlthough he had vowed never to repeat the process of the second album, lugging tapes around and ducking into studios while on tour, Page found history repeated itself. A US tour was scheduled to begin on 5 August 1970, although the first week's dates were cancelled when John Paul Jones's father fell seriously ill. For Page, however, the timing was fortuitous: he flew down to Memphis, linking up with his old friend Terry Manning, with whom he had toured in the Yardbirds and who now ran Ardent Studios in the city. Among other acts that Manning had worked with there were Sam and Dave, Isaac Hayes and Booker T. and the MG's \u2013 Stax Records used Ardent as a kind of overspill facility when their own studio was fully booked. Soon the legendary Big Star would make their records here.\n\nPage worked on mixing the sound of the new album at Ardent, and finally accepted that the guide solo, both plangent and searing, that he had laid down when they first began recording 'Since I've Been Loving You' was precisely what the track needed for its conclusion. 'It's my all-time number-one favourite rock guitar solo,' said Terry Manning. 'We took three or four other takes and tried to put takes together and come up with something, and they were all great. But there's something magic about that one take he did, that stream of consciousness.'\n\n'With \"Since I've Been Loving You\" we were setting the scene of something that was yet to come,' Page told Brad Tolinski. 'It was meant to push the envelope. We were playing in the spirit of blues, but trying to take it into new dimensions dictated by the mass consciousness of the four players involved.\n\n'The same thing goes for the folk stuff as well. It's sort of, \"Well, this is how it was done in the past, but it now has to move.\" It's got to keep moving, moving. There's no point in looking back. You've got to keep moving onwards. Another factor was that my playing was also improving, and it was developing around the band. I didn't play any of this stuff when I was doing studio work or even in the Yardbirds. I was just inspired with this energy that we had collectively. I don't think there was any way to look backwards.'\n\nIt was Page's scrupulous attention to the most minute detail that truly stood out for Manning. The apparent roughness of some of the production \u2013 Bonham calling out 'Fuck!' at the beginning of 'Celebration Day', for example \u2013 was absolutely deliberate, asserted Manning. Quickly, he realised that Page was 'a really brilliant producer'. 'Not to demean or cast any aspersions,' Manning told Mick Wall, 'but I think he harmed himself perhaps in a few ways later on. But at that particular time, the very early days, Jimmy was an incredibly insightful, true musical genius, in my opinion, and I've seen a lot of musical people. I would say that very little happened by accident. Yes, there would be the occasional take that you can't repeat so you go with that, but it did take the insight to know that. He studied everything. When it says \"Produced by Jimmy Page\", it seriously was.'\n\nDiscussing the Bonham expletive on 'Celebration Day', which no one seemed to notice, Page explained to Manning: 'That's not why I wanna leave it, not 'cos that's cool. I like the sonic texture of everything. I like the feel that you're really there.'\n\n'With \"Since I've Been Loving You\", we wanted to do a blues track that wasn't a 12-bar, because we'd done that on the first album,' reflected Page in the _Guardian_ in 2016. 'So we came up with a blues in a minor key. It wasn't something that took forever and a day: it came together very quickly and it was perfect, with its sentiment and how it was approached.\n\n'It felt right for the album to have a rocky side and a folky side \u2013 and the rocky side clearly had to start with \"Immigrant Song\". With that hypnotic riff and Robert's bloodcurdling scream, I thought, \"That's the way to open an album.\"'\n\n'\"Immigrant Song\" was written after we had been to Iceland,' said Plant. 'It was a cultural exchange \u2013 although who they sent over here, I can't imagine. The song was a vignette of what it was like being there. Of course, it ended up spawning generations of guys with crossed axes tattooed on their arms. It's slightly funny now.\n\n'From what little I can remember of the subsequent recording sessions at Olympic Studios in London, it was a case of using mike placement and the right engineers to capture the rawness, energy and attitude of each track. Sometimes it was a bit of a struggle, but we did things quickly and were serious about making something pretty as well as powerful. The outro of \"Gallows Pole\" is great, with all the manic singing I'd overcooked horrendously prior to that. It started having some meaning. I was learning how to syncopate. I was flourishing.'\n\n'Gallows Pole', a traditional blues tale that opened side two of _Led Zeppelin III_ , was discovered by Page on a 1962 album titled _Twelve-String Guitar_ by American acoustic performer Fred Gerlach. Originally the song had been recorded in 1939 by legendary bluesman Leadbelly as 'The Gallis Pole'.\n\n'Tangerine', meanwhile, had been recorded at the final Yardbirds recording session in April 1968; at that point it was titled 'Knowing that I'm Losing You', a tribute to Page's ex-girlfriend Jackie DeShannon. As to his present girlfriend Charlotte Martin, her pregnancy was becoming visible, the unborn child who had been conceived during their time together at Bron-Yr-Aur.\n\nFinally at Ardent in Memphis the new LP was ready to be mastered. Manning was aware of the possibility of adding a small amount of handwritten text to the run-out groove at the end of a side of vinyl. He asked Page if there was anything he would care to write. The musician did indeed have something. On side one he chose the words 'So mote be it'; and on side two 'Do what thou wilt', an abbreviation of 'Do what thou wilt shall be the whole of the law. Love is the law, love under will. There is no law beyond do what thou wilt.' Both these aphorisms were central tenets of Aleister Crowley's Thelemic philosophy. Their appearance in such a central position on the first vinyl pressings of the work brought Page's interest in Crowley more firmly into the minds of Led Zeppelin's followers.\n\nFrom now on rumours began to swirl of a supernatural darkness at the heart of the group, immeasurably adding to its sense of mystery and mystique, its place in an apparently impenetrable world. Now there was a cachet of sinister glamour to everything Led Zeppelin touched.\n\n## 12\n\n# THE BEAST 666\n\n'If one were to take the bible seriously one would go mad. But to take the bible seriously, one must be already mad.'\n\nAleister Crowley\n\nAt around 1.40 p.m. on 23 December 2015, a passing driver saw flames raging through Boleskine House, the former abode of both Aleister Crowley, once branded 'the wickedest man in the world', and Jimmy Page. In a 1976 interview in _Sounds_ , Page declared that 'Aleister Crowley is the great misunderstood genius of the twentieth century'. (Others disagree. Gary Lachman, a former member of Blondie and now a successful author specialising in occult matters, refers to Crowley's outlook as 'kitsch-Nietzschean ideas about a super-race'.)\n\nAfter only two hours of the blaze, 60 per cent of the low-built, broad house had been destroyed. At the time of the fire, Boleskine, which is situated on the southern bank of Scotland's Loch Ness, was unoccupied, its owners absent. After a seven-hour battle by over 30 fire crews, the blaze was finally extinguished. But the entire building had been gutted, rendered utterly uninhabitable.\n\nLike many aspects of the lives of both Aleister Crowley and Jimmy Page there seemed an extraordinary poetry about this fire, as though it were a statement from on high. Precisely why should this have occurred on 23 December 2015, less than 36 hours before Christmas Day, Christianity's principal celebration?\n\nPage bought Boleskine in 1970. Not only did the purchase satisfy his fascination \u2013 some might say obsession \u2013 with the work and life of Aleister Crowley \u2013 'I'm attracted by the unknown,' the Zeppelin leader told the late writer Timothy White while discussing Boleskine \u2013 but the 18th-century property was intended to be a retreat where the Led Zeppelin guitarist could muster his forces and write new material.\n\nCharles Pace, an occult artist, was commissioned to paint appropriately atmospheric magic murals in the house. Then Page called Miss Pamela to ask her to search out some rare Crowley works that he had heard were in a Hollywood bookstore; this included a copy of Crowley's sexually explicit _White Stains_ , privately published in an edition of a hundred copies in 1898: almost every book had been confiscated and destroyed by Her Majesty's Customs, alerted to how variations of sodomy, pederasty, bestiality and necrophilia are interwoven into the verse. What Miss Pamela found was 'a killer-diller item: a typed manuscript with notes in the margins written by Al himself. Jimmy wired me $1,700 and I sent this treasure across the ocean, wishing I could go with it.'\n\nAs some sort of exchange for his wounded seemingly ex-girlfriend, Page did send Pamela a Christmas present, a necklace bearing an antique turquoise phoenix, which contained a beautiful pearl. And there was a note: 'For my dear P. With all my love at Christmas, Jimmy. XXXXXXX' Seven kisses, in numerological terms the number of the intellectual explorer of the obscure; Page's interests in other dimensions had brought him a measure of self-knowledge.\n\nIn the end, however, despite owning the house for over 20 years before he finally sold it, Page never spent more than a total of six weeks at Boleskine. Yet he had still been observed driving in the Loch Ness area in a Land Rover, a set of stag's antlers resplendent on the vehicle's bonnet, as though he were a Scottish aristocrat.\n\nCertainly Page's purchase of Crowley's Scottish home confirmed the musician's interest in the Beast 666, as Crowley styled himself, a nickname first bestowed on him by his mother; in consideration of this apparent parental damnation, it should never be overlooked that Crowley's extremely erudite writings reveal a man with an attractively impish sense of humour, someone who clearly didn't take himself as seriously as might first be suspected. Although Crowley had been given that damning 'wickedest man in the world' rubric by the _Daily Express_ columnist John Bull in 1923, his reputation in less orthodox circles had only risen since those days. And in hip mid-sixties London Page was far from the only aficionado of the visionary that he perceived Crowley to be.\n\nNone other than the mighty Beatles had declared their allegiance to Crowley in the most public of ways: by including his image among those of the inspirational figures collated by artist Peter Blake on the cover of their _Sgt. Pepper's Lonely Hearts Club Band_ LP. The record sleeve has Crowley's shaven head and bulging eyes peering from between Indian guru Sri Yukteswar Giri and actress Mae West: aptly, as mysticism and sex formed the heart of Crowley's life, his libidinous appetites apparently undiminished by his extreme heroin intake later in life. Other figures sharing space on the _Sgt. Pepper_ cover with Crowley \u2013 Aldous Huxley, Carl Jung, Sri Mahavatar Babaji and Sri Paramahansa Yogananda \u2013 show that by the time the Beatles came to know about the Beast, magic, mysticism and altered states had become the seminal ideas driving the counter-culture.\n\nShortly before his death John Lennon gave an interview to David Sheff for _Playboy_ magazine. 'The whole Beatles idea was to do what you want... do what thou wilst, as long as it doesn't hurt anybody' \u2013 he misquoted Crowley's famous edict, 'Do what thou wilt', which, in the parlance of the time, transmogrified into the indulgent hippie mantra of 'Do your thing'. In fact, Crowley was speaking of a need to find oneself and one's own path in life, one's purpose. As to the choice of Crowley and others on the cover of _Sgt. Pepper_ , Paul McCartney commented that such individuals were 'our heroes'; Ringo Starr acknowledged that they were people 'we like and admire'. Interestingly, the album sleeve is full of masonic symbolism. There are eleven freemasons depicted: of these eleven, three are master masons \u2013 Karl Marx, H. G. Wells and Crowley. (Is it not worth bearing in mind that EMI, for whom the Beatles recorded, had always been whispered to be a hotbed of freemasonry?)\n\nAmong underground figureheads, LSD champion Timothy Leary was extremely influenced by Crowley's vision. David Bowie referred to Crowley in 'Quicksand' on his 1972 album _Hunky Dory_ \u2013 he was 'immersed in Crowley's uniform', he sang. (According to Angie Bowie, for a time Bowie believed he was in a secret duel with Page to become head warlock and chief Crowley acolyte.) On the rear cover of the Doors' 1970 compilation album _13_ , Jim Morrison and the other members of the group are shown posing with a bust of Crowley. In 1980, on his first solo album, Ozzy Osbourne released a tune titled 'Mr Crowley'. Iron Maiden, of course, recorded 'The Number of the Beast', also giving their 1982 album on which it featured that title. Marilyn Manson has said he is a follower of Crowley. And Manic Street Preachers feature Crowley in the video for their song 'You Love Us'.\n\nAt the peak of his interest, Jimmy Page owned Crowley's 'artefacts \u2013 books, first editions, manuscripts, hats, canes, paintings, even the robes in which Crowley had conducted rituals'. There were apparently mumblings of discontent from less well-heeled Crowleyites that the guitarist was hogging so much Crowley arcana.\n\nAleister Crowley was a man of considerable achievements. Not only was he a prolific writer and poet, he was also \u2013 for a while \u2013 arguably the world's finest mountaineer. The author Somerset Maugham even declared him to be 'the best whist player of his time'. Reputedly responsible for introducing yoga to Western society, Crowley was also a prominent member of the Hermetic Order of the Golden Dawn, the late-19th and early 20th-century society dedicated to the exploration of metaphysics, headed during Crowley's membership by W. B. Yeats, the great Irish poet. Breaking away from the Golden Dawn, Aleister Crowley joined the Ordo Templi Orientis, the O.T.O., as it is more familiarly known. Although it was founded at the beginning of the 20th century by Carl Kellner and Theodor Reuss, Aleister Crowley soon became its leader, reorganising it with the Law of Thelema as its religious core.\n\nIndeed, Crowley was most famous, or infamous, for having founded Thelema, a religion whose beliefs had been, he asserted, dictated to him while he was in Cairo, Egypt, on 8\u201310 April in the year 1904 by Aiwass, a mystic entity. Allegedly 'channelled' to him, the knowledge Crowley received was published as _The Book of the Law_. One of Thelema's principal tenets was a belief in the power of tantric sex, which Crowley defined as 'sex magick'. It was a reputation that lingered among its disciples, and was much of the cause behind rumours about Page.\n\nBoleskine was built on the site of a church that had burned down during the 10th century, along with its congregation, as mass was being celebrated. The house was constructed as a hunting lodge in the late 18th century by Archibald Fraser, the son of Lieutenant General Simon Fraser, Lord Lovat, whose estate surrounded Boleskine's land. The head of an executed man \u2013 believed to be a previous Lord Lovat who fought with the English during the 1745 Jacobite Rising \u2013 allegedly could be heard rolling around the house. 'I certainly used to hear the \"rolling of the head\",' wrote the Beast in his autobiography, _The Confessions of Aleister Crowley_ , 'but when I put in a billiard table, the old gentleman preferred it to the corridor and confined his amusements to the gunroom. Even before that, he had always stopped at the pylon of the corridor which marked off from the rest of the house the wing which was consecrated to Abramelin... We used to listen at the door of the gunroom, and the head would roll merrily up and down the table with untiring energy. The moment we opened the door the noise would stop.'\n\nCrowley's father was a leading figure in the Plymouth Brethren, a fundamentalist Christian sect almost guaranteed to lead to rebellion in his offspring. And his intelligent son certainly did that: while still a teenager, Crowley had come to loathe the notion of organised Christianity, viewing it as a system to control its subjects \u2013 in fact, he took this further, detesting all organised religion. Part of his belief system hinged on what he considered true Christianity, the original Gnostic church, which portrayed Christ as a rebel, entwined in a sexual relationship with Mary Magdalene \u2013 part of the inspiration for Crowley's theory of 'sex magick'.\n\nWhen his father passed away Crowley was 11 years old: he came into a fortune, inheriting the wealth of the successful family brewery. Crowley bought Boleskine in 1899, when he was only 25. He had been searching for an appropriate location to carry out a series of rituals from _The Book of Abramelin_ , a text central to Thelema, that would enable him to make contact with his holy guardian angel.\n\nA secluded property was required with a north-facing door, which Boleskine had. From there Crowley would deliver the requisite words to the demons, and when they appeared on Boleskine's terrace he would banish them.\n\nThis process would take up to six months. During this time Crowley was obliged to remain in residence at Boleskine, tending to his enactments. Midway through the ritual, however, he was obliged to suddenly depart for Paris on urgent business. The consequence was that the demons were left en masse at Boleskine.\n\nOr so the legend runs. In 1970 the house was bought by Page. Especially attractive to him was the notion of finding your true purpose in life, which Crowley's rituals could go a long way to uncovering.\n\nIn his 1975 interview with Timothy White, the guitarist said of Boleskine: 'Strange things have happened in that house which have nothing to do with Crowley. The bad vibes were already there.'\n\nAlthough he bought Boleskine House as a retreat in which he intended to immerse himself in work, its remote location made this hard to achieve. Yet Page's intellectual and emotional relationship with the legacy of Crowley certainly added to the guitarist's mystique. Later, when stories emerged of \u2013 among other things \u2013 the whips he would carry with him on tour, this only intensified the dark sexual aura he already held in his very appearance, with its more than glimmer of androgyny. There were whispered tales, apocryphal no doubt, of Page, when Led Zeppelin kicked off, ensuring that all the business details were in place, with everything set to go: but, just in case, he had also sacrificed a goat. And another, equally preposterous, asserted that, in order to guarantee success, three members of Led Zeppelin each signed their recording contract in blood \u2013 John Paul Jones had demurred. There are those who do not believe this myth to be preposterous.\n\nA suggestion of something extremely dark at the core of Led Zeppelin became part of the group's myth, even adding to its attraction, in a kind of B-movie Hammer-horror or Dennis Wheatley-novel way. But then Page was already very aware of the power of mystique in rock 'n' roll. Later, Ed Bicknell, who became the manager of Dire Straits, would report on Peter Grant telling him how he would always play the guitarist's interests in the dark arts to their advantage: 'It was really useful sometimes, especially when we went into a record company office. Sometimes I'd take Jimmy into Atlantic in New York and everybody would hide in their offices because they thought he was going to put a spell on them. He was very good at intimidating them.'\n\nHow had Crowley inserted himself back into contemporary culture with such an impact? Since his death in 1947, his reputation had certainly been bedraggled. But it was restored after the publication in Paris in 1960 of _Le Matin des Magiciens_ , a kind of compendium of the occult's greatest hits, frequently brimming with inaccuracy. Translated into English and published in 1963 as _The Morning of the Magicians_ , Louis Pauwels and Jacques Bergier's tome became a zeitgeist book, one of those publications found on every thinking person's bookshelf. Among other things, it reinstated Crowley as an avatar of alternative thinking; almost inevitably, he had been a proponent of 'free love'. In the October 1967 edition of _International Times_ , the influential broadsheet of the underground, a lengthy article billed him as 'Aleister Crowley: the Proto-Hippie'. Although he was on tour in the United States with the Yardbirds when the magazine was published, you can be certain that Page would have read that article.\n\nBut you can also be certain that Page would have been aware of the tragic circumstances of Crowley's death: penniless, lodged in a boarding house in the end-of-the-line East Sussex town of Hastings, still chronically addicted to heroin. As Mick Brown wrote: 'An addict to his last breath, his heroin would be despatched by Heppell's the chemists in London. He was said to be taking 11 grains of heroin a day \u2013 approximately 666 milligrams.\n\n'On 1 December 1947, Crowley died of asthma and chronic bronchitis. The nurse at his bedside reported that his penultimate words were \"I'm perplexed.\" And his very last words? \"Sometimes I hate myself...\"'\n\nBut what is Page's actual relationship to the formal aspects of being a Thelemite? He is not a member of the O.T.O. But he had very strong relationships with some members.\n\n'The only extant thing I know of, tying him to Thelemic religion, is the brief shot of him staring at the stele in _Lucifer Rising_ ,' said Steve Parsons, the former singer with the Sharks and himself a Thelemite, of the Kenneth Anger film that would cause Page such controversy. 'I find it odd that after they fell out, Anger left him in the film. Jimmy has put it on record about finding the book _Magick_ when he was 14 or so, and it makes perfect sense. He wouldn't have to be a member of an organisation. His internal prowess would be on another level.'\n\nAs a deeply creative individual Page takes the attitude that he will make it anew and do it himself: he will create his own cosmic theme park, which will be called Led Zeppelin, in which he will banish what he doesn't want or need.\n\nWith Led Zeppelin he built a magical universe from such thinking. And Led Zeppelin has made a serious impact in the real world. Whatever Page does on an esoteric level, exoterically there it equally is. Put on a Led Zeppelin record and the room is full of spirits. The piece of art that he titles 'Led Zeppelin' has a profound power. There is nothing at all of the rock 'n' roll dilettante about Page.\n\nThrough soaking up what their forms have encountered, voodoo and hoodoo mutate every couple of decades. Led Zeppelin had a similar ability to soak up all influences \u2013 blues, rockabilly, soul, world music, reggae, psychedelia \u2013 and throw them back at you in a radical form. It does not always work, but when it does it is with colossal majesty. It is not music to rearrange the room, like an audio piece of soft furnishing; it is too powerful for that. There is nothing cute whatsoever about Led Zeppelin.\n\n## 13\n\n# ALL THAT GLITTERS\n\nLed Zeppelin's sixth tour of the United States of America finally got underway on 15 August 1970 in New Haven, Connecticut, under a full moon. 'Lest one think that Led Zeppelin is a sedate, socially acceptable group, he may perish the thought. Led Zeppelin is the very reincarnation of the apocalyptic bad trip,' wrote P. Gionfriddo somewhat breathlessly in the _New Haven Register_. Turning this comment on its head, the writer concluded: 'With the dust lingering behind the empty bowl, the people were going home stoned on electric Kool-Aid running full tap from Led Zeppelin's keg of true rock professionalism.'\n\nJohn Paul Jones's father's declining health continued to hover over the bass player, and on 26 August the show in Cleveland, Ohio, had its start time of 8.30 moved earlier to 5.30. Jones needed the extra hours to fly home for his father's funeral. Ever the professional \u2013 as you'd expect from a man who had never missed a booked studio session \u2013 Jones played the gig and departed for the local airport and his private plane before the final encores. But the audience was so insistent that the group return to the stage that the rest of the band complied: Robert Plant busking on his harp as Jimmy Page fixed a broken string, while a local girl employed by the promoter picked up Jones's bass and joined in.\n\nThe set on this tour included 'Since I've Been Loving You', 'Out on the Tiles' and 'Bron-Y-Aur Stomp'. As was their wont, it was when Led Zeppelin reached California that they really hit their stride. On 2 September at the Oakland Coliseum, across the bay from San Francisco, the four-piece played an astonishing set. By now 'Whole Lotta Love' had evolved into a rock 'n' roll blues medley, which would be set inside the original structure of the tune. At Oakland the mixture included John Lee Hooker's 'Boogie Chillen'', Carl Perkins's 'Boppin' the Blues', Lloyd Price's 'Lawdy Miss Clawdy', 'For What It's Worth' by Page and Plant's beloved Buffalo Springfield, Don Hinton's rockabilly classic 'Honey Bee', Muddy Waters' 'Long Distance Call', Hank Snow's 'I'm Moving On', Allen Toussaint's 'Fortune Teller' and Arthur Crudup's 'That's All Right Mama'; this choice of cover material gives a sense of how broad were the tastes and influences within Led Zeppelin. Similarly, the structure of 'Communication Breakdown' was broken up so that 'Good Times Bad Times' could be slipped inside it.\n\nMoving down to the US\u2013Mexico border the following night, they played the Sports Arena in San Diego. Suffering from malfunctioning equipment during his acoustic set, Page scrapped his intended performance of 'Bron-Y-Aur', shifting into 'Since I've Been Loving You' instead.\n\nThe next night, Friday 4 September 1970, at the Forum in Los Angeles, Led Zeppelin were at their peak. At least two bootlegged albums of the show have since appeared, most notably _Live on Blueberry Hill_ , named after Fats Domino's huge 1956 hit, which Zeppelin played as an encore. Among other highlights was a rare performance of 'Out on the Tiles', both heavy-metal thunder and Beatle-like pop, and a version of 'Communication Breakdown' that segued into snatches of 'Good Times Bad Times', 'For What It's Worth' and the Beatles' 'I Saw Her Standing There'.\n\nThe group's acoustic set was also rapturously received. Immediately after the show, they headed to the celebrated Troubador on Santa Monica Boulevard in West Hollywood, where they joined Fairport Convention onstage during their second set of the evening.\n\nUK recognition was also officially theirs when they were voted Best Group in the World in the _Melody Maker_ readers' poll, usurping the now defunct Beatles. On 16 September, Zeppelin, minus Jones, attended _Melody Maker_ 's awards ceremony at London's luxurious Savoy Hotel. They were photographed with Fairport Convention's Sandy Denny, who had won the Best Female Vocalist award.\n\nThe next month, at Island Studios, Page played both electric and slide guitar on 'Lady Wonder' for Mike Heron, a solo effort by this member of the extremely influential Incredible String Band; accompanying him were Fairport Convention's bass player Dave Pegg and drummer Dave Mattacks. At Abbey Road's Studio 3 that October he also played acoustic guitar on Roy Harper's 'The Same Old Rock', a song on Harper's _Harvest_ LP, produced by Peter Jenner.\n\nEarlier in the year, Page had been in touch with Richard Drew, who worked under the name of Zacron; Drew had been at Kingston College of Art, where he had become friends with Eric Clapton. Page asked him to work on the sleeve design for the new album, taking as its basic premise a rotating volvelle based on crop-rotation calendars.\n\nSo elaborate was the resulting packaging that Atlantic Records required a formal meeting to figure out where their own logo would sit. In the end it was tucked in amidst the assorted images of this most pop art of record covers. Note that the label is given as 'Atlantic Deluxe', indicating a higher cost than usual for this LP, thanks to its very expensive sleeve.\n\nIn a similar vein to the collage of images on the _Sgt. Pepper_ sleeve, _Led Zeppelin III_ 's was dotted with chunks of quasi-symbolism.\n\nFor example, the image of a walnut at the top of the sleeve front, above the second 'p' of Zeppelin, is 'like a thing cruising through the sky with a brain in it. A boat that can think,' said Drew. There's a vintage car crossing a bridge, with the initials 'JP' on it: the picture was shot on the humpbacked bridge coming into Pangbourne. A man who had been seated in a boat near Guildford was turned, thought Drew, into 'a sort of Mary Poppins'. The image of a screw in almost the centre of the sleeve insert was a reference to the complicated construction of the revolving cover \u2013 and perhaps to something else.\n\nBut in a way the elaborate revolving cover was a mistake: along with the inscriptions from Crowley on each side of the vinyl, it took away from the greatness of the music, almost overwhelming it and becoming the main talking point about _Led Zeppelin III_. The sleeve seemed almost like a gimmick, something Page had always hated in pop music. Wouldn't a single image on the front of a simple card sleeve have been more potent?\n\n_Led Zeppelin III_ was released on 5 October in the US and 18 days later in the UK. The record had advance orders \u2013 which means the number of copies ordered in by stores \u2013 of 750,000 in America and 60,000 in the UK. Accordingly, on its first day on sale in the US the record went gold.\n\nWhereas its predecessor had been a highly successful hotchpotch of songs recorded in the unlikeliest of circumstances, _Led Zeppelin III_ had been conceived in less stressful conditions, allowing time for it to be considered much more as a total work.\n\nYet, mystifyingly, critics were no kinder. Lester Bangs's _Rolling Stone_ review began with a devastating putdown: his relationship with the group, he said, sprang from 'genuine interest and indefensible hopes, in part from the conviction that nobody that crass could be all that bad'. 'Most of the acoustic stuff,' he continued, 'sounds like standard Zep graded down decibel-wise, and the heavy blitzes could've been outtakes from _Zeppelin II_. In fact, when I first heard the album my main impression was the consistent anonymity of most of the songs.'\n\nThe _Los Angeles Times_ went further, in its pseudo-analysis of the nature of the group's fans: 'Their success may be attributable at least in part to the accelerating popularity among the teenage rock 'n' roll audience of barbiturates and amphetamines, drugs that render their users most responsive to crushing volume and ferocious histrionics of the sort Zeppelin has heretofore dealt in exclusively.'\n\nWhy did Led Zeppelin draw out such intellectual snobbery among critics? It wasn't just in the US; many music writers in the UK would also snigger about Page's group. In fact, the record's contents precisely exemplified his initial concept of Led Zeppelin: that it would reflect 'light and shade'. Although I might carp at the weaknesses of 'Immigrant Song', _Led Zeppelin III_ has more than stood the test of time as one of the group's very greatest works and a testament to the guitarist's creative vision.\n\nBy now Page had become a collector of classic cars: a Bentley, a Cord Sportsman, an Austin Champ and an ancient Mercedes, among others. Yet still he never learned to drive; from time to time he would head down to his garage and sit in these motorised pieces of art.\n\nTo possess a skill so prosaic as the ability to drive could have been advantageous. For he had been on the lookout for a new country property, and no such home would be accessible without a motor vehicle.\n\nAs 1970 drew to a close, Jimmy Page was moving house. His boathouse in Pangbourne had become known to fans, some of whom would attempt to pay him a visit, and, besides, with Charlotte Martin pregnant, it was too small a home in which to raise a family.\n\nThe musician had discovered a country estate for sale in East Sussex called Plumpton Place, near Brighton and within easy commuting distance of London. The cost of the house was an indication of Page's earnings in the two years that Led Zeppelin had been running: \u00a3200,000, a colossal price at the time.\n\nA 16th-century manor house with its own moat, Plumpton Place was an archetypal example of rock-star rococo, one that only enhanced Page's increasingly distant image. In 1927 it had been bought by Edward Hudson, the founder of _Country Life_ magazine. At this time Plumpton Place was in serious need of repair, and Hudson brought in his favourite architect Edwin Lutyens and garden designer Gertrude Jekyll; they set about a major restoration of the main house, the mill house and the 60 acres of land and lakes. In 1969 George and Pattie Harrison had come across the property, which was for sale, and tried to buy it. The vendor, however, disliked the notion of one of those longhaired rock 'n' roll musicians living in her delightful abode. Accordingly, she sold it to a local doctor. Who, three years later, sold Plumpton Place to Jimmy Page. In line with what would come to be seen as Page's fondness for such matters, Plumpton Place was also haunted. People interested in occult phenomena, after all, like to hone their skills; being certain of the existence of a ghost or two would be an act of rebellion against his choirboy upbringing; it was also somewhat sensationalist proof of his occult resilience and toughness.\n\nIn November 1970, as though in conscious defiance of the negative reviews picked up by _Led Zeppelin III_ , Page and Plant returned to Bron-Yr-Aur, which was now decidedly autumnal and even more damp. There they came up with further rough song ideas, including a tune called 'Going to California' and another with the working title of 'Stairway to Heaven' \u2013 although nothing became as complete as on the previous Bron-Yr-Aur stay.\n\nIn December the four musicians reconvened in Barnes at Olympic Sound Studios to work up further ideas. But then it was decided that a more creative use of time would be for the group to stay together in one place. The original idea was to use the Rolling Stones' mobile studio while staying at Stargroves, Mick Jagger's Hampshire mansion. However, when Page discovered that in addition to paying for the recording unit, they would also be charged a high fee to stay at Jagger's house, he balked at the idea and looked elsewhere. Led Zeppelin's office was set to work finding an alternative location, and came up with Headley Grange, which had been built in 1795 as a poorhouse, and more recently had been used for rehearsals by Fleetwood Mac, who lived nearby. Headley Grange is in Headley, East Hampshire, near Bordon, on the Hampshire\u2013Surrey border.\n\n'It was a horrible place \u2013 cold and freezing,' said Richard Cole. 'They were all staying there, though later on it'd just be Jimmy and some of the crew that kept rooms. It was too miserable for the rest of us.\n\n'They'd all wander downstairs at different times. Jimmy would be messing around with something. Jonesy would listen in and add a little bit to it. Robert would sit there writing his lyrics. It was as informal as that.'\n\nUnsurprisingly, Headley Grange was rumoured to be haunted. Again, you might question what it was that drew Page to buildings that contained evident spirits. As he was someone who spent so much of his time in an alternative world, might you not feel that this somehow brought him comfort? Except this doesn't seem to always have been the case. 'It was very Charles Dickens... I'm pretty sure it was haunted,' Page told Brad Tolinski in 2002. 'I remember going up the main staircase on the way to my room one night and seeing a grey shape at the top. I double-checked to see if it was just a play of light, and it wasn't \u2013 so I turned around pretty fast, because I didn't really want to have an encounter with something like that.'\n\nNotwithstanding such spectral visions, the miserable winter weather focused Led Zeppelin's collective mind and they worked hard. There were to be six days of rehearsals, followed by six days of recording; the Rolling Stones' mobile was brought in, Andy Johns at the controls.\n\nOften when an artist attempts a new style \u2013 as Led Zeppelin had done with _III_ \u2013 their effort is initially dismissed, until a subsequent significant work allows it to become appreciated as transitional, moving their art on to a further level.\n\nAll the same, Page had been aware of the extent to which his group was perceived as a cock-rock boys' band, one with a largely male audience. Since the spring of 1970 at his Pangbourne home he had consciously been evolving an epic work that female fans of the group would respond to. 'It begins with the concept of trying to have something that would unravel in layers as the song progressed,' he told the _Guardian_ 's Michael Hann. 'You've got the fragile guitar that is going to open the whole thing, you've got the vocal over that fragile guitar, and then it moves into the more sensual wave with the twin 12-strings, and the electric piano as well.'\n\nAlthough none of the group appreciated at the time what an impact the song would have, it was at Headley Grange that the writing and original recording of 'Stairway to Heaven' was completed, the beginning roughed out in Wales.\n\n'The music came first,' said Page to Pete Frame. 'I'd written it over a long period; the intro in Bron-Yr-Aur, in the cottage, and other parts came together piece by piece.'\n\nMost of Robert Plant's words effectively poured out of him in a stream of consciousness as he sat on the floor at Headley Grange, in front of a blazing fire with a notepad. 'When we came to record it, at Headley Grange,' said Page, 'we were so inspired by how the song could come out, with the building passages and all the possibilities, that Robert came out with the lyrics just like that... I'd say that he produced 80 per cent of the lyrics almost immediately.'\n\nAnd the singer's words and the music of 'Stairway to Heaven' intertwined with perfect harmoniousness. 'Bonzo and Jonesy had gone off to the Speakeasy Club in London \u2013 to relax, I think that's a good term for it,' said Plant. 'Jimmy and I stayed in, and we got the theme and the thread of it there and then.'\n\nBy slowly speeding up, 'Stairway to Heaven' defied conventional recording rules. In fact, it was essential for Page that the song's tempo would gradually build. 'You would find that, having been a studio musician,' he observed, 'the one thing any trained musician will tell you that you don't do is speed up or slow down. This, as far as I could see, was in league with the whole concept of classical writing where everything is moving, moving, moving.'\n\nIn time 'Stairway to Heaven' would become the most played song ever on FM radio in the US, with over three million plays totted up by the year 2000 \u2013 an extraordinary achievement by any standard, but given that the song logged in at just over eight minutes, simply remarkable. Despite his ambitions for the song, Page at first saw nothing especially unique in it; but as he worked on it more and more, with John Paul Jones also working on the arrangement, he felt a growing magical quality: 'It would keep unfolding and more layers would be introduced into the equation. Keeping John Bonham on drums to come in for effect was a trick I'd used before, on things like \"Ramble On\". I knew that would be successful, because whenever he came in he made so much difference. And then it's two more verses before you get to the solo, with this sort of fanfare approach to it. Then everything's flying at that point.'\n\n'There's almost a hysterical trill at the end of the solo that leads into the finale... \"And as we wind on down the road...\" It's like an orgasm at the end. It's whatever you want it to be,' said Plant, emphasising the ever-present sexuality in the music of Led Zeppelin.\n\nWith Jones on keyboard and Page on acoustic guitar, they structured the separate sections, working out how they would elide with each other. As the group leader had said, for maximum impact Bonham's hammering drums were to be held back. 'When John Bonham comes into it,' Page told the _Guardian_ in 2014, 'you need to have the confidence that he knows there's a whole passage that's going to go by without him coming in, otherwise he's going to think, \"That's a bit shambolic. I'll just come in at the beginning.\" It needed proper structure and proper discipline right from the beginning. And then Robert's writing his lyrics, and it's almost like he's channelled the damn thing.'\n\nPage has always insisted that, above all else, it is Plant's words \u2013 the only Zeppelin lyrics ever printed on one of their albums, in an Arts and Crafts typeface \u2013 that make 'Stairway to Heaven' the song that it is. 'I contributed to the lyrics on the first three albums, but I was always hoping that Robert would eventually take care of that aspect of the band,' he said.\n\nThe subject of Plant's words is materialism and those who believe that possessions can lead to salvation, personified by the woman who fallaciously believes all that glitters is gold and it will buy her that stairway to heaven. Although cloaked in pastoral Romanticism, the song is a kind of blues lament against selfish gold-digger females, a grasping breed all four members of Led Zeppelin would have encountered during their treks on tour. And as the song comes to its close there is salvation, as we are shown there is another way to lead our lives. In the somewhat impenetrable maze of its mystical narrative there was a universal truth; each listener was able to interpret it according to his or her experience \u2013 which was why it held such resonance for the colossal amount of record buyers of _Led Zeppelin IV_. Like so many similar parables, its meaning was ripe for dissection \u2013 which both Plant and Page always studiously avoided. Plant also cleverly allowed the potential for turning any interpretation of the lyrics on their head with the suggestive, ambiguous line: 'Cause you know sometimes words have two meanings...'\n\nPage would come to consider that 'Stairway to Heaven' 'crystallised the essence of the band. It had everything there and showed us at our best. It was a milestone. Every musician wants to do something of lasting quality, something which will hold up for a long time. We did it with \"Stairway\".'\n\n## 14\n\n# THE LEGEND OF ZOSO\n\nOn a February afternoon in 1971, Bruno Wizard, a denizen of the very rundown Notting Hill, was walking along Portobello Road with a friend. All of a sudden a white Rolls-Royce Silver Cloud pulled up next to them. The rear window rolled down to reveal the faces of Jimmy Page and Robert Plant. 'Hey,' Page said, summoning Wizard over. 'Do you know anywhere we can get some hash?'\n\nBruno Wizard, who clearly looked the kind of character to whom such a question might be posed, was delighted. In his flat around the corner he had eight ounces of such a substance: recently purchased, powerful black Afghani hashish. Answering in the affirmative, he was given an address \u2013 Island Records' Basing Street Studios.\n\nFrom Headley Grange Led Zeppelin had moved back to London, checking in to Island Studio 1. When Bruno and his mate delivered the goods, they were then permitted to hang out in the studio with the superstar musicians. 'What I realised,' said Bruno, 'was that just because people were huge rock stars, they still had to go out and find their drugs. They didn't magically arrive in their lives.'\n\nAs Bruno and his friend lounged on one of the studio's couches, rolling joints, Page turned to Plant: 'So do you want to try out those lyrics for that song?'\n\nNodding, Plant stepped up to the mike. And opened his voice: 'There's a lady who's sure all that glitters is gold...'\n\n'So I was there at an epochal moment,' recalled Bruno. Later, in the punk era, Bruno would form his own band, the Homosexuals, becoming a musical _\u00e9minence grise_ , a true lyrical philosopher.\n\nThe engineer for these sessions was Phil Brown, who had worked with the group at Olympic Sound Studios on their first album. He felt there had been a distinct shift since the innocence of those days.\n\n'It was really rough,' he told author Paul Rees. 'Peter Grant was there pretty much all the time. He used to sit either with me at the mixing desk or on this big settee behind. He unnerved me. He was like an East End hood. He was about 350 pounds then \u2013 big, sweaty and aggressive. Having him sat there with a couple of 220-pound minders, it wasn't the usual way we did things at Island.\n\n'At that point Jimmy Page seemed really messed up. There were obviously a lot of drugs around but he was also into this Aleister Crowley thing. It just put an edge to the session. There was something unpleasant about the whole thing.\n\n'The rest of the band were okay. Robert seemed very polite. He'd make the odd comment but more to Page than anyone else. John Paul Jones was a bit of a sweetheart, very clever musically. Bonham could be full-on and aggressive, though I didn't see him that much because he wasn't needed.'\n\n'Unlike at the Grange,' writes Rees, 'the pace of work was slow and laborious. Time and again Page took a pass at his \"Stairway to Heaven\" solo.' 'The track was still in something of a skeletal state,' recalled Brown. 'We did take after take with Page and for days on end. Robert told me that Jimmy does a lot of experimenting, and out of all the best bits he moulds a solo. Nothing was explained to me at the time, though. I was just left to wonder what the hell was going on.'\n\nAs he had on 'Since I've Been Loving You' on the third LP, Page worked ceaselessly on his Telecaster to establish the absolutely appropriate guitar solo for the song. He had three distinctly separate examples, finally deciding almost arbitrarily late one night which one he would use; it would become the most famous solo in the history of popular music. 'I had the first phrase worked out, and a link phrase here and there, but on the whole that solo was improvised. I think I played it through a Marshall,' he told Brad Tolinski.\n\nThe layering of guitars on 'Stairway to Heaven' posed some difficulties. 'I've employed so many guitars on \"Stairway\" \u2013 there are two lots of 12-strings going on, there's an acoustic, there's a solo \u2013 that how am I gonna do this live? The answer to that was a double-necked guitar, so I could do the opening on the 6-string, go to the 12-string, do the solo on the 6-string, then go back to the 12-string. That seemed to make the most sense.' Page knew what he needed: a 12- and 6-stringed Gibson. He had wanted one ever since he saw bluesman Earl Hooker working with one. Although they were no longer in production, Gibson agreed to come up with the double-neck for Page, a custom-built, cherry-red EDS-1275. It would become an additional element of his onstage mystique. (To maintain the song's profundity when it was performed live, Plant was instructed by Peter Grant not to speak after the band had finished 'Stairway to Heaven'. )\n\n'Page was all over the shop, too,' remembered Phil Brown. 'Out of tune a lot of the time. He didn't communicate with me at all. A lot of it was done in the control room, through an amp in the studio and with him sat right next to me. He would just go: \"Again... Again... Again.\" It was all very aggressive. He seemed a dark character, that's the only way I can explain it.'\n\nOr did Jimmy Page seem 'dark' simply because he was so immersed in his art? For ultimately this album would prove to be Led Zeppelin's defining statement, an enormous success with around 40 million copies sold.\n\nThe record itself? Well, what was it called? The truth is, no one knew. In defiance at the reception given critically to _Led Zeppelin III_ , Page was adamant that what was clearly _Led Zeppelin IV_ should have no title \u2013 or the name of the artist \u2013 on its cover whatsoever. That this would lead to record-company apoplexy at the removal of such a clear and useful marketing device should be no great surprise.\n\nVisiting a second-hand shop in Reading, on the way to Headley Grange with Jimmy Page, Robert Plant unearthed a 19th-century painting of a rustic character bent low by the burden he carried on his back. This figure, Page would immediately have noted, bore a distinct resemblance to 'Old George' Pickingill, who it was believed had first instructed Aleister Crowley in the occult arts.\n\nIs this story true? Or was the image of this fellow created specifically for the album? Furthermore, although this rustic figure is considerably older, there is a distinct resemblance to the character in the tarot card the Ten of Wands. The Ten of Wands can be interpreted as the need to rise to responsibilities and pressure, which Page certainly felt work on this new album was realising.\n\nWhatever its provenance, this was the central image that featured on the front sleeve of Zeppelin's fourth album, affixed to a decaying house wall and overlooked by a Birmingham tower block; almost unnoticed is a poster for Oxfam that reads: 'Someone dies from hunger everyday.' Were Led Zeppelin \u2013 or more likely Page, who was heavily involved with the design of each LP sleeve \u2013 professing to be carrying the weight of the eco world on their back? That seemed the most clear reading, confirmed by the guitarist in an interview with Dave Schulps of _Trouser Press_ in 1977: 'It represented the change in the balance which was going on. There was the old countryman and the blocks of flats being knocked down. It was just a way of saying that we should look after the earth, not rape and pillage it.'\n\n'The cover was supposed to be something that was for other people to savour rather than for me to actually spell everything out, which would make the whole thing rather disappointing on that level of your own personal adventure into the music,' he told James Jackson in _The Times_ in January 2010, indicating how broad the levels of creative thought were that overhung every area of Led Zeppelin, the personal suzerainty of Jimmy Page.\n\nEach member of Led Zeppelin was represented on the sleeve by a sigil, a rune-like symbol, a clear reflection of Page's occult interests, and also of Robert Plant, even if the latter's were less obsessive: the singer's fondness for the Viking oracle method of runes had been solidified on the group's trip to Iceland.\n\nThese were not archetypal symbols but devised by each individual in the band. Both John Paul Jones and John Bonham took their sigils from Rudoph Koch's _The Book of Signs_. Jones's image was appropriately of an individual who possessed both confidence and competence. Bonham's three interlocking rings represented the man, woman and child \u2013 of his marriage, presumably; twisted upside down, much to the delight of the rest of the band, Bonham's image became the logo of Ballantine beer, his Midlands local brew. Robert Plant devised his own symbolic image, a feather within a circle, an icon that spoke very much of Native Americans but which the singer claimed was sourced from the ancient Mu civilisation.\n\nBut what of Jimmy Page's rune, the sigil that became known as Zoso, which _Led Zeppelin IV_ was sometimes called before Page himself adopted it as a kind of sobriquet? (He even named his own photographic autobiography, published much later, _Zoso_.) As might be expected from the ever precise and measured Page, the origins of Zoso were considerably more arcane.\n\nIn recent years it has become known that the world's most prolific collector of paintings by Austin Osman Spare, sometimes described as Britain's greatest unknown artist, is Jimmy Page. The work he has collected includes Spare's 1907 _Portrait of the Artist_. It should not surprise you to learn that Spare was not only a visionary artist, but also a philosopher and occult musician; he was the inspiration for what is now known as chaos magic. One of Spare's specialities were his sidereal paintings, as though you were looking at a cinema screen from the side. Looking at Spare's paintings, you may receive similar impressions as you would when listening to the music of Led Zeppelin: images from the far, far past coupled with those from a distant science-fiction future \u2013 what you imagine and what you see are equally valid and interrelated.\n\nSpare was a Clerkenwell policeman's son who became a teenage painting prodigy, celebrated in exhibitions while still at the Royal College of Art. Turned off by the commercial art world, he rebelled, downgrading himself and selling his work at what were nothing more than inebriated evenings masquerading as exhibition openings in south London pubs. Always taken with notions of mysticism and other worlds, Spare \u2013 like his mentor for a brief time, Aleister Crowley \u2013 claimed to have had direct experience of the existence of extra-terrestrial intelligence. (Around the time of the recording of _Led Zeppelin IV_ , Spare was championed by Kenneth Grant, a prot\u00e9g\u00e9 of Crowley, in the popular _Man, Myth and Magic_ encyclopaedia partwork, which commenced publication in 1970.)\n\nAmong the several methods through which Spare communicated his art was his use of sigils, influenced by Egyptian hieroglyphics. He would elaborate these sigils by condensing letters of the alphabet into what have been described as 'diagrammatic glyphs of desire, which were to be integrated in postural practices' \u2013 yoga, in other words. These sigils would thereby become 'monograms of thought, for the government of energy'. Spare was endeavouring to rediscover the evangelical concept of the 'word' as a magical complex image. 'Spare's \"sentient symbols\" and his \"alphabet of desire\" situate this mediatory magic in a libidinal framework of Tantric \u2013 which is to say cosmological \u2013 proportions.'\n\nAlso a writer, in his grimoire _The Book of Pleasure_ Spare spoke of the Zos Kia Cultus, a philosophy of magic he developed that focuses on one's individual universe and the influence of the magician's will on it; a way of thinking \u2013 influenced by Crowley \u2013 that was very familiar to Page.\n\nAnd Zos, of course, is only one letter away from Zoso. For his part Page has often maintained that Zoso was intended purely as a representative of Saturn, the ruling planet of his Capricorn sun sign, the ruler of hard work, adamantine will and strength, and necessary strengthening restriction.\n\nThe influence of Austin Osman Spare on Page's choice of rune \u2013 or sigil \u2013 seems rather clear.\n\nThe occult elements of the sleeve were only followed through on the central gatefold image, a painting that was a reworking of the Hermit, the ninth card of the Major Arcana in the Rider-Waite tarot pack, which represents Prudence. The staff the Hermit bears is a symbol of his authority. In its archetypal sense the Hermit, a reclusive, solitary figure, shines the light of a lamp on matters, and desires to give solitary time for thought to himself, while simultaneously not permitting others to stand in his way. The Hermit represents a character who has acquired or is seeking to acquire wisdom in order to better guide others using his lantern, very much as you might imagine Page may have perceived himself at that time. As an image it evidently appealed to him: in the 1976 Led Zeppelin film _The Song Remains the Same_ , he would choose such a character as a representation of himself. (Interestingly, here Page appeared to be crossing party lines: Crowley himself allegedly disapproved of the Rider-Waite tarot pack, and set about having his own set devised, the Thoth deck; Waite had been a member of the Golden Dawn at the same time as Crowley and was considered a direct rival by Page's spiritual mentor.)\n\nIt is worth bearing in mind that, as we see things coming to pass in the life of Jimmy Page, the Hermit, despite its positive and success-inspiring energies, is warning against the isolation that could be harmful to him at certain times: it is very important that he always endeavours to achieve the right balance. The Hermit invites us to discover wisdom and the progress that comes with study; the card also indicates that the Hermit is a person of integrity, but that he is scared to trust in others and completely express what he is feeling \u2013 very much as Page was, polite to the point of sometimes being a little boring. The painting of the Hermit on the inner sleeve was by a supposed friend of Page called Barrington Coleby. There is no record whatsoever of any such person, and there were those who believed the real painter was none other than Page himself.\n\nAll the same, there was something very appealing and romantic about the notion of Jimmy Page as a rock 'n' roll recluse, dabbling with potions and spells in his high tower.\n\nThere were certainly magical elements to the design of the _Led Zeppelin IV_ LP sleeve; could these in any way have helped contribute to its astonishing success, both commercially and creatively?\n\nBut above that, there was the stunning music. The record drew you in straightaway with 'Black Dog', supposedly named after a literal black Labrador retriever at Headley Grange, but containing an interesting double-meaning: the 'black dog' was, of course, Winston Churchill's euphemism for the spells of depression that afflicted him from time to time. Both being intelligent men, you imagine Page and Plant were aware of this. This 'Black Dog', however, was a powerful anti-depressant, like musical cocaine. There was nothing downer whatsoever about this pounding, primal tune with one of Page's characteristic low-register fuzz riffs. Although always present, the guitarist is very restrained: holding back, nothing really flashy going on at all, punctuating the tune with perfect panache. 'Rock and Roll', the next tune, maintains the pace, charging relentlessly on, living up to its title, fresh and exciting. The combination of the two tunes is an invigorating way to start an album.\n\nTrack three, 'The Battle of Evermore', which at first seems to be returning us to the light and shade of the previous record, has a great sense of confidence and clarity about it. An old folk song, replete with additional Tolkien references in Plant's lyrics, and a determined blues-like drive.\n\nPage wrote it on the mandolin, an instrument he'd never played before, at Headley Grange one afternoon. In 1977 he told _Guitar Player_ magazine: 'On \"The Battle of Evermore\", a mandolin was lying around. It wasn't mine, it was Jonesey's. I just picked it up, got the chords and it sort of started happening. I did it more or less straight off. But, you see, that's fingerpicking again, going back to the studio days and developing a certain amount of technique \u2013 at least enough to be adapted and used. My fingerpicking is a sort of cross between Pete Seeger, Earl Scruggs and total incompetence.'\n\nWith Plant in the room tossing out lyrics inspired by a book on Scottish history the singer had read, appended by thoughts from Robert Graves's _The White Goddess_ and Lewis Spence's _The Magic Arts in Celtic Britain_ , the song emerged, fully formed, almost straightaway. As is evident from the words, they concern the battle between night and day, good and evil. There is a reference to 'Avalon'.\n\n'The band was sitting next to the chimney in Headley, drinking tea,' recalled sound engineer Andy Johns, 'when Jimmy grabbed a mandolin and started playing. I gave him a microphone and stuck a Gibson echo on his mandolin. Jimmy had brought this stuff before and had asked me to take a look at it. Suddenly Robert started singing and this amazing track was born from nowhere.'\n\nAs on 'Black Dog', Page again reverses into 'The Battle of Evermore' with gently rising volume, but now on the far more delicate mandolin, while the song featured something unique: a guest, Sandy Denny sharing the vocals with Plant. Page thought that the song 'sounded like an old English instrumental first off. Then it became a vocal and Robert did his bit. Finally we figured we'd bring Sandy by and do a question-and-answer-type thing.'\n\nErik Davis, who wrote a book about the album in the 33 1\/3 series, considered that gender also played a role in the song, with Plant's Prince of Peace and Denny's Queen of Light occupying true male\/female roles as opposed to the gender-blurred androgyny of Page and Plant themselves. As a reward for being the only singer to perform on a Led Zeppelin record other than Plant, Denny was given her own three-pyramid rune on the album sleeve.\n\nEmotionally, the album's next song was very different to what Led Zeppelin had up until then offered to their audience. The closing tune on side one of the vinyl edition: 'Stairway to Heaven'.\n\nFollowing 'Stairway' on the album was the extremely funky 'Misty Mountain Hop'. Inspired by Plant's visit to a 'legalise pot' rally in London's Hyde Park in July 1968, just before he joined what would become Led Zeppelin, the title refers to the Misty Mountains mentioned in J. R. R. Tolkien's _The Hobbit_ \u2013 the perpetual, Leonine hippie in Robert Plant imbued Led Zeppelin with a necessary warmth, firing on top of not only Jimmy Page's Capricorn earthiness, but also that of John Paul Jones, another 'Cap'; Bonzo was the communicative Gemini, pounding out a life beat.\n\nAnd with the meter shifting from 5\/8 to 6\/8, 'Four Sticks' \u2013 so titled because Bonham played the song with two sticks in each hand \u2013 was propelling a series of classic Page riffs around its complex time signature. 'It was supposed to be abstract,' the guitarist said of the song.\n\nAlthough 'Going to California', first titled 'Guide to California', is sometimes said to have been inspired by Joni Mitchell's song 'California', this could not have been the case; Mitchell's _Blue_ album, on which 'California' featured, did not hit the stores until June 1971. Instead it is more likely to have been influenced by Mitchell's 1968 tune 'I Had a King'.\n\nEssentially a song of heartbreak, 'Going to California' is an acoustic number: behind Plant's vocals John Paul Jones is featured on mandolin and Page on guitar, with Bonham absent from the session. A hugely popular Led Zeppelin song, tucked in as the penultimate track on the album, it changes the character of the record; by the time it has finished you have forgotten the pounding openers of 'Black Dog' and 'Rock and Roll'.\n\nBut it is the album's closing track, 'When the Levee Breaks', that raises _Led Zeppelin IV_ to even greater heights. The song matches 'Stairway to Heaven' in scope and stateliness: forceful, free, hypnotic and deceptively simple, with its timing manipulated in the studio, it is one of Zeppelin's greatest-ever recordings. The song is a lament over the Great Mississippi Flood of 1927, when the river broke through the protecting levees.\n\nFor Page it had always been the drum sound that was paramount in Led Zeppelin's recordings: refining Bonham's deeply skilled but deceptively simple meat-and-potatoes power-hammering until it became perfection, anything could be built on it. 'Distance equals depth' was Page's adage. And achieving that paradox so masterfully on 'When the Levee Breaks' was one of his finest production achievements. It was done by the simple expedient of putting Bonham's drums at the base of the naturally echoing stairwell at Headley Grange, and exaggerating that echo and slowing everyone down except Plant.\n\nWhen _Led Zeppelin IV_ was finally released, US rock critic Robert Christgau chose the song as the highpoint of a record he lauded: 'I think the triumph here is \"When the Levee Breaks\". As if by sorcery, the quasi-parodic overstatement and oddly cerebral mood of Led Zep's blues recastings is at once transcended (that is, this really sounds like a blues), and apotheosised (that is, it has the grandeur of a symphonic crescendo) while John Bonham, as ham-handed as ever, pounds out a contrapuntal tattoo of heavy rhythm.'\n\n'I call it the heaviest track of all time,' blogged music critic Bob Lefsetz, 'because I remember the force pounding in my ears, like Bonzo was hitting the skins with baseball bats, like I was on DMT; this cut with absolutely no airplay entranced me, made me feel like I bonded with these madmen.'\n\nIn late February 1971 engineer Andy Johns, Jimmy Page and Peter Grant had flown to Los Angeles to mix the new album at Sunset Sound Recorders. Almost as soon as they arrived an earthquake hit LA. Was it an omen? By the time they returned to London and heard what had come out of Sunset Sound, they appreciated that it may well have been: they were bitterly disappointed with the results. 'I still don't know what happened,' said Page. 'Maybe the monitors were giving us a totally false sound picture, because Sunset Sound had these real state-of-the-art monitors that were able to reproduce a big stretch of frequencies. Who knows?'\n\nThey returned to Island Studios and restored the eight tracks to their true greatness, and then, almost immediately, they were off on tour again, this time a set of UK dates played at small venues. As much as anything, this back-to-their-roots tour was a public-relations exercise set up by Peter Grant, an effort to quell British complaints about Zeppelin spending all their time playing large arenas in the US.\n\n'Stairway to Heaven' was first played live at Belfast's Ulster Hall on 5 March 1971, but the song, which Led Zeppelin would become even more associated with than 'Whole Lotta Love', drew little more than a smattering of applause. 'Stairway' was followed by another new song, 'Going to California', and Tony Wilson, reviewing in _Disc_ magazine, deemed the show 'sensational'.\n\nGetting back to their roots wasn't without its pitfalls, of course. Sniffing round the poky dressing room of Newcastle's Mayfair Ballroom, where they played their first-ever UK date, Page complained to Richard Cole: 'Once you've played in the big places, these small clubs are murder. It's nice to be near the audience, but you forget how small the dressing rooms are. At this point in our career, I think we're entitled to more luxury than this. This is really hard to believe.'\n\nIn Manchester, after a show at the university, the group visited Mr Smith's nightclub and returned to their hotel with some girls. They were hanging out in the room of promoter Tony Smith, when Page led one of the girls into the bathroom. There soon came the sound of running water, and after a few minutes smoke began to emerge from under the bathroom door. 'Jimmy had somehow started a fire in the middle of the sink with newspapers and towels,' wrote Cole. Robert Plant rushed to find a fire extinguisher, and he put out the fire \u2013 but filled the bathroom with foam. When the hotel manager arrived he was told that Page had been involved in a 'religious rite'.\n\n'Incidentally,' wrote Cole, 'we never did find out what Jimmy's bathroom conflagration was all about. Of course, there was his growing preoccupation with the occult. Perhaps the fire was somehow related to that. When I asked him about it, all he said was, \"I liked Percy's explanation best. Let's just say that it was an ancient rite that went up in flames.\"'\n\n_Led Zeppelin IV_ was finished and ready for release. But, unsurprisingly, Jimmy Page and Atlantic Records again found themselves at odds over how a Led Zeppelin record should be packaged. The Zeppelin leader was adamant that the new LP's sleeve would contain no indication whatsoever as to the identity of the act within it. Unsurprisingly, Atlantic Records in New York were equally insistent that this would mean commercial suicide. 'I had to go in personally and argue with the record company about it,' Page told Mick Wall. 'That's why the album took so long to come out, because they wanted to put a name on the cover. But I went in there with Peter, and I stayed in there even after Peter left, talking to them about it. Finally, it came down to, \"Well, let's have a symbol on it\"... I said, \"You can't have one symbol, because no one's going to agree on what it should be. Let's have four symbols, and everyone can choose their own.\" And with the four symbols, that also made it _Zeppelin IV_ , so it was a completely organic process.'\n\nBy so imposing the four runes on their audience Page had employed an elemental number through which Led Zeppelin were fused together: in other words, it was the record cover \u2013 and record \u2013 as spell.\n\nFrom the sleeve inwards there is magic all over this record. You begin to wonder \u2013 comparing the success of both records \u2013 if Page had disempowered _Led Zeppelin III_ with that Crowley inscription etched into the run-out groove? It had mystified people, and rather worried them, leaving the album with a strange, almost sinister taste for many.\n\nAt the very beginning of the next month, on 1 April 1971, the band appeared before an audience of 400 fans at the BBC's Paris Cinema on Lower Regent Street in central London. In consideration of the slim possibilities for many fans of getting tickets for Zeppelin's small-venues tour, Peter Grant had arranged for the band to play their set so it could be broadcast to the whole of the UK. In fact, the show had originally been scheduled for 25 March, but Robert Plant had developed laryngitis, a consequence of hanging out in draughty Northern dressing rooms, and they'd had to postpone. Page and Plant went to the BBC's editing facilities in Maida Vale the day after the performance to edit down the rock 'n' roll section of 'Whole Lotta Love'\u2013 losing 90 seconds of Page's guitar solo \u2013 and entirely removing 'Trucking Little Mama', 'For What It's Worth', 'Honey Bee' and 'The Lemon Song' for its broadcast slot on 4 April.\n\nBy now, there had been a significant development in the personal life of Jimmy Page. On 23 March Led Zeppelin played at the Marquee Club in London, the last day of their roots tour. The Marquee could pack in up to 500 people \u2013 in extremely uncomfortable circumstances. When Zeppelin played there the club was so rammed that it was not necessarily a pleasant experience.\n\nThe very next day, Charlotte Martin gave birth to the child conceived at Bron-Yr-Aur on Page and Plant's first songwriting expedition there: a daughter, Scarlet Lilith Eleida Page.\n\nIn the baby girl's names there were strong references to the thinking of Aleister Crowley. For example, Crowley always gave his girlfriends the nickname 'Scarlet Woman'. And Lilith, the name of one of Crowley's own children, in Jewish tradition was a predecessor of Eve, and an alleged wilderness spirit who was considered a figure of uncontrolled sexuality in the Talmud. In more recent times, specifically in esoteric circles during the 1960s and later, she has become a feminist icon, the representation of the 'bad girl'. All in all, a rather strange name for someone to give his daughter, but again, Jimmy Page \u2013 and Charlotte Martin \u2013 were kicking against convention, so often being deliberately bad themselves.\n\nNow 27 years old, Page was on the brink of his first maturity, about to make that transition into adulthood that esoteric sources \u2013 the sort to which he would pay heed \u2013 assert arises from that age for the next four years. How would he manage to transform himself within the Peter Pan-like, perpetually immature and indulged world of rock 'n' roll? Not especially well, time would tell.\n\nHis chosen employment was not necessarily in accord with such life changes. The first few months of a child's life are not the easiest of times for a relationship, not least because of all the sleepless nights. So when Page again found himself on tour just a few weeks later, this time in Europe, it may well have come as a sense of relief for the musician.\n\nThe relief would be short-lived. On 5 July Led Zeppelin performed in Milan, at the Velodromo Vigorelli, before 15,000 fans. In those days Italian shows were notorious for heavy-handed behaviour by the local police. And this show, in a purpose-built cycling track, was no exception. As soon as Zeppelin hit the stage at 8.30, groups of left-wing radicals in the audience stormed the heavily armed riot police, who responded by firing barrages of tear gas back.\n\n'The promoters ran onstage,' said Page. The group was asked 'to tell the crowd to stop lighting fires. So Robert asked them and suddenly there was smoke by the front of the stage and it was actually tear gas. It was just pandemonium, with nowhere immune from this blasted tear gas, including us.'\n\nAlthough they endeavoured to complete their set, even when fans climbed onstage to get away from the police, the band eventually came offstage to take refuge in a backstage room. Later, Page claimed to have been 'terribly upset'. 'I couldn't believe we'd been used as the instrument for a political demonstration,' he said.\n\nLed Zeppelin would never play Italy again.\n\nIn the summer of 1971, Roy Harper's _Stormcock_ LP was released, his fifth album, on EMI's 'alternative' Harvest label. By now, immensely assisted by the 'Hats Off to (Roy) Harper' tune on Zeppelin's third album, the grizzled, eccentric folkie had become an icon of the counterculture. A powerful outsider record, with its roots in the 1960s Soho folk scene, _Stormcock_ featured Jimmy Page on the track 'The Same Old Rock', which clocked in at over 12 minutes. With its heartfelt diatribe against organised religion \u2013 something Aleister Crowley would have approved of \u2013 Harper recorded his most powerful song ever. It was a triumph, only enhanced by Page's stunningly beautiful, flamenco-like blues solo, the flourishing equal of anything he'd ever recorded, with or without Led Zeppelin. On the record, which sold in only small amounts after Harvest hardly promoted it, Page was credited as 'S. Flavius Mercurius', because of his contractual agreement with Atlantic.\n\nIn stark contrast to this, there was another album that deeply offended Page's creative sensibility. The Yardbirds' Anderson Theater show in New York City on their last tour, which Epic Records had recorded, had been rejected by the band for release as a live album because the sound engineer's approach was amateurish. Among the tracks on the LP was the Yardbirds' rendition of 'I'm Confused', which \u2013 with additional lyrics \u2013 Zeppelin had released as 'Dazed and Confused'.\n\nNow, despite the band's previous objections, Epic Records released the album in May 1971, under the title _Live Yardbirds: Featuring Jimmy Page_. Adding insult to injury, there had been attempts by the label to disguise the unsatisfactory sound with noises from, among other sources, bullfight crowds.\n\nIncensed by Epic's perfidy, Page threatened the label with legal action unless the album was removed from the stores. Epic quickly withdrew it. Page hardly wanted such a substandard representation of his sound available in the US: tickets were already on sale for a further Led Zeppelin American tour, 31 dates set to begin on 19 August.\n\nWith no new record in the stores Atlantic were extremely sceptical about the point of their act playing the dates, with some considerable lack of perception deeming it 'professional suicide'. Although the tour had been booked in anticipation of the new record being available, the twist that timing had taken now meant that touring with nothing to promote might well have been the entire point.\n\nThe tour was fast-paced, with few days off, putting a strain on Robert Plant's voice. On 19 August, the singer's 23rd birthday, things kicked off in Vancouver, where Peter Grant famously smashed up what he believed to be a recording device. Later, it was learned that it was the local government's equipment, measuring the volume of the Zeppelin performance.\n\nA couple of days later they were down in their favourite US city, Los Angeles, for a Saturday night show at the Forum in Inglewood, some 10 miles down La Cienega Boulevard from where they were holed up in the Hyatt House on Sunset.\n\nWhen Led Zeppelin played 'Stairway to Heaven' in California for the first time, the capacity crowd responded. 'Not all of the audience stood up. It was about 25 per cent of it, and I thought, Hey, that's pretty good,' said Page. 'They were really moved and I thought, This is wonderful, this is great. This is what we hoped for, that people would be that receptive to our new music.'\n\nClearly invigorated in LA, the band started the following night's set at the same venue with a run-through of 'Walk Don't Run' by the Ventures, the local instrumental group that had once rivalled the Shadows and the inspiration for Page's first fuzzbox.\n\nThe _Cashbox_ review of the first show was laudatory \u2013 hyperbolic, even: 'They're at it again. I feel weightless after the music's over, escaping from the crowd, riding through the quiet; time and I are strangers, atlas shrugging in the city, remembering how much I enjoy the feeling. And these are only things.\n\n'They call it Rock AND Roll, a fragment of the truth.'\n\nWriting in the _Los Angeles Times_ , even the venerated Robert Hilburn was largely positive, if a bit sniffy, about their Forum set: 'I do, strangely, have something nice to say about Led Zeppelin. Not without reservations, mind you, but at least something. Let me explain. There are some rock groups that strike me as so unimportant and uninteresting, both musically and sociologically, that there doesn't seem to be any reason to ever see them again. Deep Purple is an excellent recent example.\n\n'There are other groups, however, that may be depressingly tedious musically, but there is something interesting about their concert \u2013 either in an occasional strong piece of music or an ability to create an unusually high degree of audience enthusiasm \u2013 that makes you feel another look at the group is justified. Grand Funk \u2013 perhaps the clearest current example of low musical quality and high audience response \u2013 is an example of this kind of group. Led Zeppelin is another.'\n\nYet though he was dismissive of their material, Hilburn seemed to get it by the set's end: 'Just when you are ready to dismiss Zeppelin as a group with one lucky song [\"Whole Lotta Love\"], it begins moving through a number of songs once associated with Elvis Presley: \"That's All Right Mama\", \"Got a Lot o' Livin' to Do\" and \"A Mess of Blues\". And it was sensational. On this 20-minute medley, Zeppelin became an interesting, surging, powerfully effective rock 'n' roll band, totally free of the predictable treadmill they normally employ. On those numbers, Page played a marvellously effective Sun-oriented guitar and Plant rivalled anything Rod Stewart (one of my favourite singers at the moment) and Elvis Presley (favourite singer of the past) have done with \"That's All Right Mama\".'\n\nNow firmly interwoven into the set was an acoustic section, which introduced another new song, the appropriately titled 'Going to California', as well as the more familiar 'That's the Way' and 'Bron-Y-Aur Stomp'.\n\nPrior to that Saturday night show at the Forum, Miss Pamela had visited Page at the Hyatt. She only saw her former paramour for a brief while, having a chat about Aleister Crowley, before she went to Plant's room. Although the singer confessed his own interest in her, later, in the dressing room at the Forum, he told her, 'Jimmy isn't too happy at home. I know he wants you to be with him.'\n\nAfter the show, Page climbed into a limousine with Miss Pamela, and they headed for the Whisky, before going back to the Hyatt to spend the night together. 'Jimmy told me tales of Charlotte and little baby Scarlet. They haven't gotten married, and have lots of disagreements,' Miss Pamela wrote in her diary. It was just over four months since the birth of Scarlet Page: was anyone around Page telling him about the inevitable stresses that come with the birth of a child?\n\nDespite Page's pleasant peccadillo in Los Angeles, it appeared that some of the previous spirit of Zeppelin touring America had vanished on this, their seventh tour of the country. 'The incident in Milan had taken a lot out of us,' wrote Richard Cole in _Stairway to Heaven_. 'Once we arrived in America, Zeppelin almost seemed to become reclusive as we moved from city to city, rarely leaving the hotels except for the concerts themselves. Occasionally, Jimmy would venture out to an antique store in New Orleans or Dallas, but those expeditions were more the exception than the rule.'\n\nAlthough the overt hedonism was less in evidence, it had only disappeared behind literally closed curtains, manifesting in darker forms. In Chicago the promoter received a telephone call: 'Jimmy is going to be shot by someone in the audience tonight.' A similar threat was made in Detroit. Extra security was hired and the local police informed. It was only two years since Charles Manson's Family had killed Sharon Tate and her companions in the Hollywood Hills. Underground culture had thrown up its share of spectres: there seemed to be plenty of dysfunctional, rather frightening characters connected to the American rock 'n' roll circus. For Page the worm had turned; the widely misunderstood Aleister Crowley inscriptions on the third album may not have helped.\n\nThe tour ended in Honolulu with a pair of shows on 16 and 17 September and a four-day break on the nearby island of Maui. The wives of the other three members of the group flew out to join them, so Page insisted that the mountaintop mansion rented for the entire group would be for him and himself alone, so as not to cramp his sexual style.\n\nFrom there they flew to Japan for five dates. Playing two nights at Tokyo's Budokan, they were startled by the hush that descended over the audience as soon as they appeared onstage. 'We weren't used to all of this after coming from America where the whole show people were drinking and smoking joints and going nuts... But we could have a laugh in Zeppelin. We did enjoy ourselves,' said Page.\n\nWhen Robert Plant came onstage at the Budokan, however, his lip was partially split. There had been an incident backstage that in another group might have led to its demise. Before the show John Bonham asked the singer to finally settle up over a longstanding debt of \u00a337, a petrol bill the drummer had settled for him. When the singer, whose nickname of 'Percy' was a play on the word 'purse', something he was as reluctant to dip into as his guitarist, yet again tried to sidestep the request, Bonzo lamped him.\n\nAn element of self-discipline was rather lacking on this, their first Japanese tour. So great were Led Zeppelin's expenses that, extremely unusually for such a parsimonious outfit, they actually lost money. And the lack of discipline wasn't confined to financial matters.\n\nDisapproving of the choice of music at Tokyo's Byblos club, Bonham pissed on the DJ from an upstairs balcony. Endeavouring to get the almost comatose drummer back to their accommodation at the Hilton, Richard Cole finally gave up, leaving Bonham asleep on the pavement outside the hotel. The pair purchased samurai swords the next day and staged a sword fight in the hotel, wrecking everything that came within their range. After a sleeping John Paul Jones was found in the hotel corridor, Led Zeppelin were banned from the Hilton chain for life.\n\nBut a far more egregious example of rock-star bad behaviour took place on the bullet train to their show in Osaka; one that demonstrated the degree of decadence that was seeping everywhere around Led Zeppelin. A pretty Japanese girl called Kanuko, whom Page met in Tokyo, was travelling with them. When the couple disappeared to eat in the train's dining car, salt-of-the-earth John Bonham \u2013 always up for a laugh, ho-ho \u2013 discovered the girl's handbag. He then took it to the bathroom. When he made his way back to his seat, Bonham gleefully announced to Cole: 'I just shit in her handbag!'\n\nWhen Kanuko and Page came back from the dining car, she went to the bathroom with her bag. She returned, in tears, ashen.\n\nA furious Page sought in vain for the culprit.\n\nWhen it came time to return home, Page, Plant and Cole flew to Bangkok, where Page bought a life-size gold, wood and glass Pegasus. Later, the trio visited the city's red-light area, about which Page remarked, 'They must have invented the term \"fucking your brains out\" here.'\n\nWhen they moved on to Bombay Page jammed with local musicians, and he, Plant and Cole again visited the city's red-light district. All three of them experienced violent diarrhoea after they asked their driver to take them on a 'locals' experience' \u2013 for a meal somewhere he ate regularly.\n\n## 15\n\n# CITY OF ANGELS\n\nFinally the fourth album \u2013 _Led Zeppelin IV_ , _Zoso_ , _Runes_ or whatever you liked to call it \u2013 had a release date: 7 November in the United States and 18 November in the UK. A mysterious series of ads in the music press, each displaying just one or all of the sigils, had preceded its release.\n\nJimmy Page was spotted wearing a sweater bearing the Zoso sigil, a gift he received backstage before playing Wembley Empire Pool. Zoso would become comparable to the Rolling Stones' somewhat ugly tongue-and-lip logo. Like Nike's newly minted swoosh, it was a statement of powerful marketing. There is no overstating it: the four Led Zeppelin symbols integrated themselves into popular culture, even being worn as body art.\n\n'All the stopgap titles we throw at the thing are lame: _Led Zeppelin IV_ , _[Untitled]_ , _Runes_ , _Zoso_ , _Four Symbols_. In an almost Lovecraftian sense, the album was _nameless_ , a thing from beyond, charged with manna,' wrote Erik Davis in his book _Led Zeppelin's Led Zeppelin IV_.\n\nA UK tour began. After an apparently mediocre performance on the opening night at Newcastle's City Hall, followed by shows at Sunderland, Dundee, Ipswich, Plant and Bonham's homeland of Birmingham, and Sheffield, they performed at Wembley's Empire Pool on 20 and 21 November. This pair of London shows was constructed very much as an event, called 'Electric Magic', featuring support acts Stone the Crows, Bronco and Dundee, as well as circus acts such as dancing pigs, clad in hats and ruffles, a Peter Grant conceit that drew onstage criticism from Robert Plant: 'I expected a bit more from the pigs! Did you? I could have brought some goats.'\n\nThe first night at Wembley brought forth a sensational performance from the group, captured in the ecstatic _Melody Maker_ review by Roy Hollingworth, a long-time supporter of Zeppelin: 'This was an English band playing like crazy, and enjoying every minute they stood there on stage. They played non-stop for the best part of three hours. Enormous. They played about everything they've ever written. Nothing, just nothing, was spared. This was no job, this was no \"gig\". It was an event for all. So they get paid a lot of bread, well, people paid that bread, and I reckon they got every penny's worth. It was a great night.'\n\nA bad strain of influenza arrived in Britain that winter, and when Led Zeppelin played Manchester on 24 November, Plant announced onstage: 'Gosh. I think I got that flu that's going about.'\n\nSo that the potentially afflicted Plant could take a mid-set breather, Page inserted the wah-wah section from the theme of the movie _Shaft_ into 'Dazed and Confused'. Written by Isaac Hayes, the utterly distinctive soundtrack for the hit blaxploitation flick _Shaft_ had been a number one US album that year.\n\nBy the end of the tour, scheduled for Salisbury, close to Stonehenge, on 15 December, Plant's flu had spread through the rest of the group. The Salisbury date was rescheduled for six days later, when Page would hopefully have recovered from the bug.\n\n_Led Zeppelin IV_ went straight to number one in the UK charts. Jimmy Page breathed a sigh of relief: 'It was probably more painful to get this one out than childbirth itself.' But in America, although it was certified a gold album on the day of its release and was destined to become one of the biggest-selling albums of all time, the record never climbed higher than number two, kept off the top of the charts by Carole King's _Tapestry_.\n\nFor once it was _Rolling Stone_ who stuck the flag in the summit, via Lenny Kaye. 'It might seem a bit incongruous to say that Led Zeppelin \u2013 a band never particularly known for its tendency to understate matters \u2013 has produced an album which is remarkable for its low-keyed and tasteful subtlety, but that's just the case here,' he began his finely written review. 'The march of the dinosaurs that broke the ground for their first epic release has apparently vanished.'\n\nAnd Kaye, who was struck by the sheer variety of the record, saved his real applause until his final paragraph: 'The end of the album is saved for \"When the Levee Breaks\", strangely credited to all the members of the band plus Memphis Minnie, and it's a dazzler. Basing themselves around one honey of a chord progression, the group constructs an air of tunnel-long depth, full of stunning resolves and a majesty that sets up as a perfect climax. Led Zep have had a lot of imitators over the past few years, but it takes cuts like this to show that most of them have only picked up the style, lacking any real knowledge of the meat underneath.'\n\nSo what was going on beneath the fa\u00e7ade of the swaggering rock 'n' roller known as Jimmy Page? By now his interest in the occult \u2013 by no means strange in that era, of course \u2013 was widely known. Any mention of his name would often draw up that two-word response: 'Black magic?'\n\nIn Richard Cole's chronicle of his years with Zeppelin, _Stairway to Heaven_ , he considered Page's relationship with his muse: 'I had always felt that, more than the others, Jimmy was much too complex an individual to be living for music alone. I knew that his dabbling in the occult continued, although he still kept that side of his life very private.' Although Page would occasionally mention the name of Aleister Crowley, the road manager knew little about the supposed Great Beast.\n\nOn assorted occasions, he recalled, Page phoned him to declare himself in the mood 'for shopping for some Crowley artefacts'. 'We'd drive from auction house to rare book showrooms, where Jimmy would buy Crowley manuscripts or other belongings (hats, paintings, clothes).'\n\nWhen Cole questioned him for some details about Crowley, Page drove the question off to the touchline: '\"The guy was really remarkable. Someday we'll talk about it, Richard.\"'\n\nNeedless to say, they never did. 'If the public felt there was a certain mystery surrounding Led Zeppelin, they weren't alone,' Cole concluded. 'As close as I was to them, I felt there was something within Jimmy that he never let anyone see. Particularly when it came to Pagey's preoccupation with Crowley, s\u00e9ances, and black magic, I had a lot of unanswered questions.'\n\nPart of Page's mystery was that he was simultaneously a slightly androgynous boy-man and also carried a rather frightening aura: this was assisted by the atmosphere of menace around Led Zeppelin as a whole. His seeming a little scary was part of the attraction for many of his fans.\n\nSome of Page's more unfortunate character aspects disguised the fact that he was really an extremely evolved human being, and also essentially a nice bloke. For example, there was no doubting the personal loyalty that Page could display. His old friend Jeff Dexter managed America, a trio of expat Yanks who lived in the UK. With their lilting soft-rock sound they had enjoyed huge success globally with their 'A Horse with No Name' single, released the same day as _Led Zeppelin IV_.\n\nBut when Dexter was sacked by the band, Page was concerned, as was Peter Grant, who was also well acquainted with the ceaselessly scene-making Dexter. Grant and Dexter had known each other since the days of the celebrated 2i's Coffee Bar on Old Compton Street in Soho. When Dexter became a dancer with Cyril Stapleton's Orchestra, Grant \u2013 having been a wrestler himself \u2013 suggested Dexter take up the same craft under the moniker the Twisting Wrestler. 'It's all choreography, Jeff. Nothing to do with hurting each other,' Grant promised him. The diminutive Dexter demurred.\n\nDexter was now managing Isaac Guillory, a uniquely styled American folk singer based in London. To help out their mate, Page and Grant spoke enthusiastically to Atlantic Records about Guillory, and secured him \u2013 and his manager \u2013 an extremely substantial recording deal with the label.\n\nPerhaps it was a consequence of his shyness in his early years \u2013 as well as his damaged family background \u2013 that meant Page lacked immediate warmth and was suspicious of outsiders, which was ironic, as that was precisely what he was himself, to a rather extreme degree.\n\nCertainly, with the instant success of _Led Zeppelin IV_ he should have felt utterly validated. Even though he always pretended they meant nothing to him, the bad reviews had certainly hurt. Now he even had thorough approval from _Rolling Stone_ , whom he had once perceived as an enemy. But he was not so easily won over.\n\nIn a little over three years his Led Zeppelin project had become the biggest group in the world. They far outsold, both in terms of records and concert tickets, the Rolling Stones, the only other contender for the title. The fact that the Rolling Stones' albums never sold in large quantities was at that time little known. Yet in achieving his ambition for Zeppelin, Page encountered a similar phenomenon to that which Paul McCartney had found with the Beatles: having reached the absolute zenith of success, he discovered that there was nothing there.\n\nPage was getting into his shit. And, like many other successful rock 'n' rollers before him, he was also going slightly mad from the daily pressure. All those arcane books he had read and yet all these prosaic responsibilities with Zeppelin. Attention to detail had always been his thing, to a necessarily obsessive degree: who cared if modern stereo systems were not sufficiently sensitive to pick up the most subtle filigree flourishes he was applying to his recordings? It would linger there in the tune's backstory. But this was all so very wearing, leaving little time for sleep. 'I'm sure people aren't aware of this. I'm sure they think we sit on our arses all day long, but we don't. All I know is I haven't stopped for three years. If it gives you any indication, I haven't had a holiday ever since the group started,' he said the following year.\n\nOf course, the sleeplessness was hardly assisted by the amount of drugs that were now about the group. Industrial quantities of cocaine followed Led Zeppelin wherever they went; one story had a pair of cops turning up backstage at a venue in the southern US and finding Grant seated at a table piled up with what looked like a sizeable chunk of Colombia's annual cocaine production. They began to go through the motions of placing the manager under arrest. Grant glanced at the policemen and picked up a pile of cash, and then proceeded to deal it out: $5,000, $10,000, $15,000... But to no avail, it seemed, as they continued to read him his rights... $35,000, $40,000, $45,000, $50,000. Finally, the policemen looked at each other and shrugged. One of them leaned over, picked up the cash \u2013 and they left. As they stepped out the door, Grant uttered a final line: 'Pleasure doing business with you, gentlemen.'\n\nIn early February 1972 Led Zeppelin boarded an Air India flight at London's Heathrow Airport. They flew to Bombay for a four-day break on their way to their first tour of Australia; Peter Grant had secured round-the-world tickets from Air India for only \u00a3500 for each traveller. Following a reprise of the sexual shenanigans on their previous visit to Bombay, the musicians spent an afternoon enjoying a camel ride along an Indian beach, which left their bodies sore and battered. Both Page and Plant \u2013 especially Robert Plant, in fact \u2013 were taken with the notion of imbuing their own work with music from other cultures.\n\nIn line with Australia's much-trumpeted tall-poppy view of life \u2013 Someone getting above their station? Cut the buggers down to size! \u2013 the country's media and forces of law and order had a reputation for meting out harsh justice to visiting celebrities.\n\nAnd it seemed that an example would be made of Led Zeppelin. They arrived in Perth for their 16 February show at the Subiaco Oval \u2013 an outdoors show, as all the dates on this Australasian tour would be \u2013 before a crowd of 80,000 people. It was a sensational concert \u2013 there was even a stage invasion \u2013 and the crowd size was boosted by 300 or so fans who, to the chagrin of the local police, cut their way through fences and gates to get in.\n\nFollowing the show Led Zeppelin went to a nightclub in the city, where they ended the night banging out classic rock 'n' roll songs together on a tiny stage.\n\nAfter the group returned to the Scarborough Hotel and put their heads down on their pillows, there came a knocking at their door. It was the police, with warrants to search their rooms, which they duly did. As Led Zeppelin had only just arrived in Australia, they had not even made any drug connections yet. Always a professional despite his severe jet lag, Page was shocked at such incompetence: 'Those stupid arseholes. If they had waited a day or two, we might have had something.' Grant slept through the entire episode, and when he learned of it in the morning, he immediately hired a security team of former police detectives to act as a buffer against any further raids.\n\nAlthough a thunderstorm delayed their second date, a show in Adelaide, by two days, when it finally happened the local press were ecstatic about the gig, the loudest there ever: 'From the start, all eyes were on brilliant lead guitarist Jimmy Page. His electric guitar work was extraordinary. At one stage, using a bow, he smashed out a string of piercing notes only to end with a run of delicate sitar-sounding music. Thunderous applause followed all his work.'\n\nIn Sydney, a city famed for its gay population, they hung out with the transsexuals from a cabaret club called Les Girls. Questions were asked of the four members' sexuality.\n\nIn Auckland in New Zealand they played to the largest crowd ever gathered together in the country up to that point. Back in Australia, in Queensland's semi-tropical Brisbane, a version of Dion's 'The Wanderer' was slipped into the 'Whole Lotta Love' medley \u2013 for the only time in their career.\n\nReturning home via Bangkok, with the intention of pursuing further sexual possibilities, the musicians were refused entry to Thailand. Their long hair was a no-no. When Robert Plant pointed out that they had been allowed into the country the previous year, they were simply informed that the rules had now changed. Trouble with the forces of law and order in both Australia and Thailand: was someone keeping an eye on them?\n\nBack in the UK in March, Page immediately took the group into rehearsals for the new material he had been working on since January. The rehearsal space was a remote farmhouse in Puddletown, Dorset, arguably the UK's most beautiful county, on the shores of the River Piddle, which came with its own studio, called Jabberwocky.\n\nEddie Kramer, the engineer from New York's Electric Lady Studios, was flown over to join the group. Despite the success of the fourth album, Andy Johns had been replaced, perhaps as a result of his misguided desire to mix the album in Los Angeles. Soon, despite the high rental fee demanded by Mick Jagger \u2013 \u00a31,000 a week \u2013 they were all staying at Stargroves, the Stones singer's country mansion in Hampshire, with the Rolling Stones mobile studio wired up in the grounds.\n\nOne of the first tracks recorded, 'The Song Remains the Same', originally titled 'The Plumpton and Worcester Races', was laid down on 18 May. Page used a Fender electric 12-string, with Les Paul overdubs. In the tune there are traces of the highly organised arrangements of Yes, with Page sometimes recalling Steve Howe's six-string flourishes.\n\nFollowing the successful completion of this epic track, which would become the album opener, at around 5 a.m., Robert Plant had the idea to emulate his idol Ral Donner, who had recorded so many demos for Elvis Presley; he had in mind something in the vein of Donner's 'You Don't Know What You've Got', a hit in 1961.\n\nWhat transpired was not precisely in Donner's mood, but more like the bluebeat sound Plant had heard at Midlands Mod clubs in the mid-sixties. After Page rapidly constructed a chord sequence, John Paul Jones laid down a ska-style bass line, quickly picked up on drums by John Bonham. Almost immediately this transmogrified into the joyful 'D'yer Mak'er', a clear play on the word 'Jamaica', if the song's rhythm alone did not reveal its origins. Page's guitar playing gives the song an old-fashioned feeling \u2013 as though he is from time to time reprising his old hero Hank Marvin's work with the Shadows.\n\nThe slow, moody 'Rain Song' was consciously written by Page in response to George Harrison taunting Bonham that Zeppelin were incapable of writing a ballad. 'The Crunge', with its throwaway line 'Can you take me to the bridge?', was more a tribute to James Brown than to the oft-mentioned Otis Redding.\n\n'No Quarter', a melange of cosmic mess and obscure world-music references, had leapt straight from John Paul Jones's keyboard at his home studio in Crowborough in East Sussex; he had presented the song to the group almost fully formed. Always the quietest and most self-contained member of Led Zeppelin, Jones was the anchor of the unit, always there to assist Page on song arrangements. Jones was a fellow earth sign Capricorn, and Page was always adamant that this created a solid foundation for the quartet.\n\n'Black Country Woman' was recorded out on the lawn at Stargroves. A song that would not appear until the _Physical Graffiti_ album, it had Robert Plant's vocals almost interrupted by a jetliner passing overhead. At the singer's insistence, the aircraft's sound was kept on the track.\n\nAfter just over a week, Page decided that the acoustics at Stargroves were simply not up to scratch, certainly not on a level with those at Headley Grange. Led Zeppelin relocated back to Olympic Sound Studios in West London.\n\nDuring 1972 a distinct shift occurred in the dynamic of Led Zeppelin, which had up until now been seen as the child of the substantially pedigreed Jimmy Page, the archetypal manifestation of the guitar hero. When the group had started out Robert Plant seemed little more than a nervous, overgrown schoolboy, struggling with all his might to find his place in the group and the world. But in the three and a bit years since, the singer had evolved into the group's dominant presence, both in the eyes of fans and in the media. Invariably bare-chested, his leonine mane kissing his shoulders, rock's golden god \u2013 as he was frequently now described \u2013 had replaced the apparently moody Page as the visual representation of the group. Prone to all manner of rock star eccentricities \u2013 lax timekeeping most certainly one of them \u2013 Page had had that nickname 'Led Wallet' bestowed on him by crew-members shaking their heads at his financial tightness. But his singer was no better. Following their Alexandra Palace shows in London at the end of 1972, Plant would offer the entire road crew a single bottle of whisky for Christmas.\n\nIt was perhaps inevitable that the singer would replace the guitarist as the visual image of Led Zeppelin. Plant was the man who would stand at the front of the stage and address and embrace the audience. An open individual, he was also happy and at ease both with himself and in his relationship with the media, and the media reciprocated this, respectful of how he was usually ready with an intelligent answer to their questions. The old hippie within him was hard to suppress, and Plant was unquestionably a sensitive soul, yet he was no slouch when it came to self-belief, and he could also be seen as being a little too cocky and rather arrogantly pleased with himself.\n\nHe articulated his position to journalist Cliff Jones: 'Page, and Bonham to a degree, were different to me. I reasoned that I could sing about misty mountains and then chip a football into the back of the net on my days off. That felt like the good life to me back then. It wasn't all-consuming for me in the way it was for the others.\n\n'Don't forget that we had no choice but to get caught up in all that hotel-room and drug-binge kind of thing, because it was part of the experience, almost expected. But I always knew there was a time out when I'd get off the bus and come home.'\n\nBack in the US for a summer tour, Page spent time at Electric Lady Studios in New York, tidying up the new record. On Wednesday 14 June 1972, Led Zeppelin played Nassau Coliseum on New York's Long Island. The concert was reviewed by Robert Christgau in _Newsday_ : 'Jimmy Page is a highly proficient electric blues guitarist whose expertise is essential to the group's effect, but the star of the show is vocalist Robert Plant.' Christgau also saw through to the heart of the construction of the group's material: 'At some deep level, Led Zeppelin's music is about technology. Philosophically, the band prefers humanity pure and simple, but in practice it must realise its humanity technologically. That seems truer than most good-time pastoral fantasies.'\n\nOn the Friday, Catherine James, who had not seen Page since their fling in Los Angeles in 1969, received a surprise phone call. By now she was living with her son Damian in Ridgefield, Connecticut, some 50 or so miles from Manhattan. 'No matter how much time went by, the sound of Jimmy's soft, reticent voice always stopped my heart cold,' she said. 'I hadn't seen him in over two years, but his voice still made my insides stir. He was in New York at the Drake Hotel, and asked if he could come up for the weekend. I thought, \"This has to be a dream.\"'\n\nA couple of hours later Page drew up outside her house in a shining black limousine. 'Jimmy was just as dazzling as the last time we met. He's also the only man who has ever left me at a loss for words. I don't know why, but I always felt slightly shy with him... I thought I was a bit more self-assured, but the mere sight of him getting out of his limo and strolling up my walk gave me the familiar jitters. Trying to maintain some sense of coolness, I greeted Jimmy like he was a distant cousin. With a quick kiss, I announced, \"Damian and I are just off to the market; you're welcome to wait here or come along if you like.\"\n\n'He said he wanted to go with us, and I showed him how I made the silver Pontiac fly over the dips in the road. We were having a sweet time, and my coolness quickly turned into mush. We had a late-night candlelit dinner and kissed till our lips were swollen.'\n\nThey spent the weekend together, going for walks in the woods, listening to Joni Mitchell, kissing and cuddling, everything Catherine James could have dreamt of. 'Jimmy finally said the magic words: \"Why don't you come to England?\"'\n\nWouldn't you think Catherine James would have jumped at the idea? But no, she was in a different life now, looking after her son in the country, where she had built a new and satisfying existence for herself. More to the point, on a purely practical level where would Page have lodged her? What would Charlotte Martin's view of this have been, had she learned of it? Was Page reacting to Catherine in this manner because he was in despair over his relationship with Charlotte? Or was this simply a classic example of self-indulgent rock-star philandering?\n\nWhatever it was, Catherine felt she had no choice but to turn down the opportunity. However, she gladly accepted his offer to fly with him to Los Angeles, where Zeppelin were due to play the Forum on 25 June. She arranged child care for Damian and spent a week at the Hyatt House with Page. 'The shows were stunning. The silhouette of Jimmy cloaked in shimmery velvet, moons and stars, and the haunting, abandoned sound of the bow gliding across his guitar were enchantingly sensual. To me, Jimmy was Led Zeppelin, the sorcerer behind the magic; he definitely had it down.\n\n'After my whirlwind week of rock and steamy, romantic sex, I actually looked forward to the sanctuary of my little house, away from the glitz and glamour of Hollywood.'\n\nThat show at the Forum was the band at one of their ultimate peaks. Thirty years later, a live recording of the concert, plus further extracts from their Long Beach Arena gig two days later, was released as the sensational _How the West Was Won_ , a triple live CD, one of Led Zeppelin's greatest-ever and most representative releases. Page was fully aware of the show's magnificence. 'I think,' he told _The Times_ in 2010, 'what we did on... _How the West Was Won_ \u2013 that 1972 gig \u2013 is pretty much a testament of how good it was. It would have been nice to have had a little more visual recordings, but there you go. That's the conundrum of Led Zeppelin!'\n\nJust back from America in July 1972, Jimmy Page once again went into the studio with his friend Roy Harper at Abbey Road. He played electric guitar on two tracks for the _Lifemask_ album, 'Bank of the Dead' and 'The Lord's Prayer'.\n\nPage retreated briefly to Boleskine. There he was joined one afternoon by fellow Heston native Ritchie Blackmore, former member of Lord Sutch's Savages, a fertile musical training ground. Blackmore had by now become the powerhouse guitarist with Deep Purple and was also an aficionado of Aleister Crowley; Purple would tour the US only a couple of notches in popularity below Zeppelin.\n\nBlackmore was accompanied to Boleskine by Dick O'Dell, his group's lighting director. O'Dell found the experience 'very interesting and a little scary. The house was spooky \u2013 and Gothic, of course. It was a classically rainy, misty, dark Loch Ness day. The house had lots of amazing art, sculpture and weird logos from Jimmy Page's various societies. Including, obviously, pentacles.' That afternoon there was, said O'Dell, an unmistakable element of rivalry between the two gunslinger guitarists: 'Cordial but slightly like two matadors... I guess Ritchie wanted to see the house. I reckon he was jealous: probably wanted to buy it himself. I always thought Page was much more serious about Crowley, Druidism, Magick, Celtic folklore etc. Blackmore liked the image!'\n\nThey had afternoon tea and a couple of joints, before making their excuses and departing.\n\nIn October 1972 Page and Plant returned to Bombay in India to investigate the possibility of recording with Indian musicians. The guitarist had been in touch with his friend Ravi Shankar, asking the sitar master to help him in choosing appropriate musicians. The eminent Indian had a think, then he recommended Vijay Raghav Rao, Films Division of India's principal soundtrack composer and arranger. For what would be termed _The Bombay Sessions_ , Rao brought in Sultan Khan, a master of the very difficult sarangi, a bowed Indian instrument.\n\nWith the recording of the new album complete, there was the not-insignificant matter of the record's sleeve to be designed, which had been something of an issue on the previous two LPs. The most highly regarded design company in London at the time were Hipgnosis; since their design for _A Saucerful of Secrets_ , the second album by Pink Floyd, for whom they became designers of choice, Hipgnosis had proven themselves to be on the zeitgeist of contemporary album art. Their work combined the wilfully surreal, sex and satire, playing with the senses in their distortion of convention. After Storm Thorgerson, who founded the company in 1968 with Aubrey Powell, had designed the cover for Douglas Adams's book _The Hitchhiker's Guide to the Galaxy_ , the author described him as 'the best album designer in the world'. As well as working with the Floyd, Hipgnosis delivered sleeves for, among others, Free, the Nice, Emerson, Lake and Palmer, and Yes. The previous year, 1971, they had designed the packaging for T. Rex's _Electric Warrior_ , the biggest-selling LP in the UK during that 12-month period. (A little-known fact was that in 1970, out of friendship, Page had personally tutored Marc Bolan in the art of playing the electric guitar.)\n\nWhen Thorgerson was called in for a meeting with Grant, Page and Plant, he was informed that the new Zeppelin album was to be the first to have a title: _Houses of the Holy_. Through his esoteric studies, Page had concluded that human beings were 'houses of the Holy Spirit'.\n\nDespite plenty of evidence in the tone of Hipgnosis's previous album sleeves, the trio had not really appreciated the humour at the heart of much of the design company's work. Page was more than a little surprised when Thorgerson returned with what he claimed was his first rough attempt: a photograph of a tennis court and a tennis racquet. When the mystified guitarist requested an explanation, Thorgerson had a three-word explanation: 'Racquet. Racket. Geddit?' Page asked Thorgerson to leave, although he did concede that the designer had shown 'some balls'. Surprisingly, the band decided to stick with Hipgnosis, and they set up a meeting with Aubrey Powell.\n\nPowell had a rather different concept to his partner for the fifth Led Zeppelin album sleeve: to create an image of children ascending a rocky hill. The idea came from science-fiction writer Arthur C. Clarke's novel _Childhood's End_ ; the fact that Clarke was later outed as a paedophile lends an awkward subtext to this source material.\n\nIn the end only two children, both naked, were involved in the photographic shoot, siblings Stefan and Samantha Gates, their images repeated several times. Page claimed the image symbolised, 'Innocence... the idea of us being vessels of houses of the holy'. It was only two years since Blind Faith had adorned their first and only album's cover with an image of a nude pre-pubescent girl: how times would change. The inner sleeve of _Houses of the Holy_ contained even greater grounds for controversy, depicting one of the children being held aloft in a sacrificial posture.\n\nThe ten-day shoot for the sleeve took place on the Giant's Causeway on the north coast of Northern Ireland. When Powell mentioned to Peter Grant that the complex logistics of the expedition might prove very expensive, the Falstaffian manager barked back at him: 'Expensive? We're Led Zeppelin \u2013 we don't care about money!'\n\nWhen it was finally released on 28 March 1973, the complex diversity of material on _Houses of the Holy_ again led to baffled reviews. By this time Page knew what he was likely to be reading: 'People still have this preconceived notion of what to expect. But this will be something really new. If you carry on on just one plane, you just repeat yourself.'\n\nOn 4 May 1973 Led Zeppelin were once again on tour in the United States. _Houses of the Holy_ had been released six weeks previously and, although it went to number one almost immediately, it would soon find itself sliding into the slipstream of the slowly building juggernaut sales of _Led Zeppelin IV_.\n\nThe audience size at the first couple of dates was worthy of Zeppelin's new status as the Biggest Band in the World. An eminence reflected in their use of the _Starship_ , a Boeing 720 available to rent for $2,500 per flight hour. They were the first act to use the renovated plane, which came with its own organ as well as bedrooms and bars; the flight attendants were delighted to work aboard, helping themselves to rolled-up $100 bills when the band and their entourage had departed.\n\nSome 50,000 fans came to Atlanta's Fulton County Stadium for the first concert, and 56,800 the next day at Florida's Tampa Stadium, the latter producing gross earnings of $309,000 \u2013 not bad for a night's work. The size of the Tampa crowd unquestionably raised the bar for Led Zeppelin, and Peter Grant ordered full-page ads to be taken out in major newspapers across the US advising that the date was the largest attendance for a single performance in history.\n\n'Hello. It seems between us we've done something nobody's ever done before... and that's fantastic,' uttered Robert Plant to the crowd as the show kicked off. But perhaps the majesty of the occasion trammelled the group's chops: parts of the Tampa set were irredeemably sloppy, something Zeppelin \u2013 like their great rivals the Rolling Stones \u2013 could be more often than they would like to admit.\n\nTwo days later, however, journalist Lisa Robinson was transported by what she considered a magical performance in Jacksonville, Florida. 'There were no intermissions, no waiting, no tuning up, no bullshit,' wrote Robinson, who would become tight with Jimmy Page, in the UK's _Disc_ magazine. 'Just music. Just gorgeous rock 'n' roll music at its most desperate. The performance was so incredibly timed that you were never completely aware of just exactly when one number started and another began, and the acoustic numbers and the ballads blended in perfectly with the rockers. You couldn't help but beg for more. It was impossible to be a part of that experience and not watch, and listen with a total awe. We don't have bands like this, you know. YOU don't have bands like this... but you do have Led Zeppelin. And they know what they have. They know the high they can achieve, and they're here again \u2013 their album is number one in the country and they're going to go and play everywhere and celebrate rock 'n' roll. God bless them.'\n\nFollowing their experience in Sydney, Led Zeppelin seemed to have developed a fondness for transvestite bars. In New Orleans, while hanging out in the French Quarter, they spent their time in gay bars. 'The New Orleans drag queens always seemed to be having much more fun than the people we'd meet in straight clubs,' thought Richard Cole. 'We also loved shocking people, and there was no better place to do that than a bar where it was often only us and a few transvestites.'\n\nBut not everything ran smoothly on this tour. The 30 May show at Led Zep temple the Forum in Los Angeles was moved to 3 June after Page sprained his hand by leaning on a wire fence at LAX while signing autographs. 'Quite honestly, I don't know why we've had such phenomenal success,' Page told the _Los Angeles Times_ during this brief hiatus. 'Perhaps you could relate it to street music and the fact that people feel more of an affinity to Zep's music because it's not constantly hammered down their throats from every direction. All I can say is that whenever we've gone on stage or into the studio, we've always done our best. We've never really been involved in the media, we've never done a TV programme, and air play, of course, is limited because of the fact that we don't record singles.'\n\n'By the end of a tour, everybody is getting pretty much blown out,' Roy Harper, who was the opening act for Led Zeppelin for most of their 1973 US dates, said in 1974. 'I mean you have no idea, you have absolutely no idea, of what the pressures are like. Peter Grant jestfully referred to me as the social worker on the last trip. There's a limit to what human beings can take, and on a cross-country American tour you can watch the gradual disintegration. And when you run into the average hotel jobsworth, you don't actually want to play around with him too much.\n\n'You're going grey, so life gets into a pattern of activity that's inescapable. You reach the ultimate, which is, \"What can we do to keep ourselves together?\" I've seen cars driven into swimming pools, and walls blown out of hotels with dynamite. But that only happens when you've reached the point of total stress, when you've gone beyond all endurance.'\n\nWith its film world and show-business foundations, coupled with its location at the most extreme edge of the Western world, Los Angeles was home to much that was bizarre. This reflected its relatively recent history. During the 1850s the tiny City of Angels had been infamous as one of the most murderous societies in America, where lynching often took the place of legal execution. 'There is no brighter sun... no country where nature is more lavish of her exuberant fullness,' a local wrote in 1853. 'And yet, with all our natural beauties and advantages, there is no country where human life is of so little account. Men hack one another to pieces with pistols and other cutlery as if God's image were of no more worth than the life of one of the two or three thousand ownerless dogs that prowl about our streets and make night hideous.'\n\nOver a hundred years later, nothing seemed to have changed too much. Many of the city's inhabitants had come in search of fame \u2013 though not everybody, by any means \u2013 and it was still a wild place. As the 1960s shifted into the next decade, the thinking \u2013 sometimes exaggerated by drugs \u2013 behind the era's supposed sexual revolution had turned somewhat rancid. Charles Manson and his family might have had something to do with this.\n\n'You never knew what people were capable of,' said B. P. Fallon, Led Zeppelin's new press agent. 'They'd want me to get a message to Jimmy, and you didn't know whether they were going to pull out a knife or what. So instead I introduced Zeppo to people like Les Petits Bon Bons, who were these wonderful raving young queens from California who'd give Bonzo pictures of their cocks with glitter on them. We went to gay clubs because it was more fun, and simply brought the girls with us.'\n\nIn Los Angeles in the first part of the 1970s there was a sense that anything could happen, especially in Hollywood. As well as prime Mexican and Colombian grass, cocaine was readily available. Nestled amidst the head shops peddling pipes, bongs and packets of rolling papers, there was a 'massage parlour' south of Sunset Strip tellingly titled The Institute of Oral Love, a favourite of road crews.\n\nThe Los Angeles baby groupies were a further sign of the times, as though summoned by the spirit of the Rolling Stones' 'Stray Cat Blues', with its challenging I-don't-care-if-you're-15-years-old line. Fitted out with easily obtainable fake IDs, these LA girls \u2013 often barely into their teens \u2013 were often the scions of fractured Beverly Hills showbiz marriages.\n\nRodney Bingenheimer's English Disco \u2013 Rodney's, as it became known \u2013 opened in 1972 on Sunset Boulevard, and served Watney's Red Barrel beer on tap, which was imported from England, as was much of the music. The sounds of the Sweet, in particular, sailed out of its doors, but also Gary Glitter, Mud and Suzi Quatro. Bingenheimer had visited the UK in 1971, where he was taken by the first stirrings of glam rock; urged on by David Bowie, he returned to LA the next year determined to make his mark.\n\n'Once inside, everybody's a star,' wrote Richard Cromelin the next year. 'The social rules are simple but rigid: All you want to hear is how fabulous you look, so you tell them how fabulous they look. You talk about how bored you are, coming here night after night, but that there's no place else to go. If you're not jaded there's something wrong... If you're 18 you're over the hill.'\n\nThere were other groupie watering holes \u2013 the Whisky a Go Go and the Rainbow Bar and Grill, both on Sunset Strip, while the rooftop pool of the nearby Continental Hyatt House on Sunset was a daytime option \u2013 but Rodney's was tailor-made for the needs of visiting rock stars, often British. Although they were wary of their wives and girlfriends seeing photographs of them with the club's female clientele, Led Zeppelin thought Rodney's was a very good idea, which was hardly surprising. Rodney's was a paean to rock-star narcissism, its female assets offered in the pages of _Star_ magazine, a kind of house publication that only lasted for five issues before it closed following an onslaught of criticism from a shocked general public. Edition one included an article entitled 'Those Foxy Hollywood High Girls', which revealed the magazine's demographic. The lead feature in that issue \u2013 'Your Very Own Superfox: How You'll Know It's Him' \u2013 expressed the philosophy of the baby groupies: 'Foxy Lady, it is written in the stars and in the hearts of every worshipper of the Zodiac, from the inner brooding soul of the Capricorn woman to the deep, easy warmth of the Lady of Sagittarius, that there is a time for her to have her very own SUPERFOX. It is written that the time must come for a girl to move forward and up from the ranks of the shy, blushing Teenybopper, and to express herself as a brave new woman in a brave new world.'\n\nThe queen of these brave new women was Sable Starr, who, in the June 1973 edition of _Star_ , published when she was 15, revealed that she was 'close friends' with David Bowie, Marc Bolan and some other big names. Spirit's Randy California had taken her virginity when she was just 12. She would eventually run away from home at the age of 16, moving to New York for an inevitably ill-fated relationship with Johnny Thunders after the New York Dolls played in Los Angeles.\n\nAlthough they liked to think of themselves as muses, most of these girls were little more than trinkets, abused victims. And, as their circumstances dictated, many of them were also hard as nails and capable of being considerably vicious. Jimmy Page would later tell of how one of his girlfriends from that scene had once bitten into a sandwich to discover it contained razor blades. 'I'd be on the road with Hawkwind or whoever, writing for the _NME_ , and we'd check into the Hyatt and Zeppelin would be there,' said Mick Farren. 'And the whole place was full of the stinkiest fucking groupies. There was something very unclean about the whole deal. Rod and the Faces sort of kicked it off, but it went to some kind of zenith with Zeppelin... with Zeppelin it just seemed to be running in semen and beer and unpleasantness and old Tampaxes.'\n\n'Something about Zeppelin's energy really altered the _joie de vivre_ of the LA rock scene,' said Miss Pamela. 'They thought they could get away with anything \u2013 and they could, because everyone wanted to get near them.'\n\nWhat on earth was going on here with Led Zeppelin \u2013 what was the drive behind such behaviour? Clearly it was a power trip: these young girls were not going to tell these guys that they were full of shit. But aren't these guys, our Led Zeppelin rock 'n' roll gods, also chronically immature? Rock 'n' roll often freezes people emotionally at the age they enter it, with their every wish catered for from then on. As Steve Winwood once remarked to me: 'When I left Traffic to become a solo act, I didn't even know how to go to the post office and buy a stamp.' And people who become rock stars are almost always driven by some psychological need, usually deriving from some form of damage that leads them to need the validation of audience acceptance. I asked life coach Nanette Greenblatt about the consequences of this behaviour.\n\n'I agree the young women will not be calling out the behaviour of these guys,' she said. 'The immaturity is linked also to addictions. Finding stuff outside of yourself to prop you up, and make you feel okay. Often inner resources and the creative journey to make the music, write the songs and be on the road, would bring up great highs and proper lows. This takes one psychologically to the edges. Coping with these probably fuelled a lot of dysfunctional behaviours. Those who managed to navigate healthy, connected relationships and friendships sometimes fared better. I think also some people had a calling to express their music. Driven by that urge and unable to contain or understand drives from their conditioning, they could go off the rails. In artists there is a high incidence of seeking and finding religions, occult, sex cults, drugs etc. The creative process can have a devastating effect, especially in the public eye. The public are also fickle. So careers can be unstable. Huge amounts of luxury can feel empty and unfulfilling \u2013 all the things that are supposed to make you happy, and they don't, which is confusing. You are adored but you aren't loved for you. It can feel empty. Re-inventing yourself is a challenge. Inspiration can dry up. A kind of numbness can lead to even more extreme behaviours in an attempt to feel alive: the self-harming model, where hurting yourself makes you feel a touch more alive.'\n\nA close contender for LA's number-one groupie was Lori Mattix, Sable Starr's best friend, a mixed-race girl whose photograph, when shown to him by Led Zeppelin press agent B. P. 'Beep' Fallon, appealed to Page to the point of infatuation. He got her phone number from Fallon.\n\n'I got to photograph Lori because she was in my room,' said Beep Fallon. 'Why would I not photograph her? The thing about groupies that's misunderstood is that it was all consensual. The girls were the predators, not the bands. Lori and Sable were very funny, very bright; they wouldn't take any shit from anybody.'\n\nTall, with a trim frame, her pretty face and large brown eyes framed by cascading chestnut curls, Lori had many admirers, not least David Bowie, to whom she had lost her virginity when she was just 14. But none more so than Jimmy Page. As a Scorpio, she would have made a strong connection with Page's Scorpio rising.\n\nWhile on the road in Texas he called her: 'Hi, this is Jimmy Page and I wanna meet you.'\n\n'Yeah, sure,' said Lori, assuming it was a prank call and hanging up.\n\nBut when Zeppelin hit Los Angeles, Lori and her friends went up to the Hyatt House's rooftop pool, with its views of the Hollywood Hills behind. Page approached her, but she knew that he was still, to an extent, involved with Miss Pamela, who had a reputation for physical intolerance towards anyone trying to get close to him. Addressing him as 'Mr Page', she told him: 'I really can't talk to you.' 'Honey, if you come with me, nobody'll touch you,' he replied.\n\nAt Rodney's the next night, Page arrived with Miss Pamela. But when Lori went to the bathroom, he followed her in, she claims. That night was the start of their relationship.\n\nWith her tumbling, curly locks, Lori Mattix was, to an extent, a mirror image of Jimmy Page, not unlike Mick and Bianca Jagger. No wonder he loved her so much: she looked just like him. Also, as Jagger had done with other girlfriends, Page styled Lori: 'He was always very conventional and conservative. He wanted me to be such a lady. He used to make me wear long dresses and look gypsy-like.'\n\nWhat on earth did Lori Mattix's mother think of her pubescent daughter hooking up with a man almost twice her age? Rather like the parents of Priscilla Beaulieu when a suitor called Elvis Presley came calling on their underage daughter, she seemed to go along with it, to all intents and purposes.\n\n'He was a real gentleman,' Lori Mattix told Stephen Davis. 'And she knew that he was a really respectable guy and that he had money and, I mean, what is she gonna say? You know, she knew that I was doing it anyway, so she figures if I'm gonna be doing it, who better with?\n\n'After that first year, Jimmy took me along to all the shows. Sometimes they would dedicate a show to me! And if I wasn't with him, he would call me every day from wherever he was. Especially at the time he was in his prime, 73 to 75, that was the prime of Zeppelin. After that, he got kinda smacked out.'\n\nOn 20 July 1973, seven days before the first of three dates at Madison Square Garden that would conclude the US tour, Peter Grant called American-born film director Joe Massot. Page and Grant had long discussed a feature film, almost certainly a documentary, about Led Zeppelin. Peter Whitehead's footage from their 1970 Royal Albert Hall show had been intended as the raw material for it, but at the time it was deemed that the film had been badly lit; this footage would emerge later, in the 2003 release _Led Zeppelin DVD_.\n\nIn 1970 Joe Massot, whose film _Wonderwall_ featured George Harrison's first soundtrack, had moved into a house in Pangbourne, and he and his wife first became friends with Charlotte Martin, and then Page himself after Charlotte invited them to the boathouse for dinner.\n\nPage invited him to Zeppelin's groundbreaking performance at the Bath Festival, and Massot decided there and then that he would like to film the act. He had made several offers to film Zeppelin since, but Grant always turned him down.\n\nWhen Page moved to Plumpton he lost touch with Massot. But when the filmmaker saw the scale of the venues the group would be playing in the US he reacquainted himself with Page and pitched an idea for a documentary that would incorporate live performance and more intimate, backstage film. Page liked the notion and referred him to Richard Cole, who also appreciated the concept of the film, although Cole said that Grant would have his own ideas. And he did: Grant wanted a better-known director, one with a first-class pedigree. Yet Massot's persistence paid off, and finally Grant agreed to the director filming his act.\n\nSo celebratory was the mood on tour that Grant and Page decided it was time to move ahead with the project and film the three nights at Madison Square Garden. 'It all started in the Sheraton Hotel, Boston,' said Grant. 'We'd talked about a film for years and Jimmy had known Joe Massot was interested \u2013 so we called them and over they came.' It was all very quickly arranged. Grant declared he would personally finance the movie and Massot would receive a fee, with all footage being the legal domain of the act, in the firm hands of Steve Weiss, their extremely efficient lawyer.\n\nMassot hurriedly assembled a three-camera crew in time for Led Zeppelin's Baltimore date, on 23 July 1973, and the next night in Pittsburgh, as a dry-run; it was only the New York shows that he would actually shoot. These were the last concerts that the group would play that year. But afterwards, despite New York filming costs of $85,000, it became clear that there were plenty of missing elements from the footage. Remarkably, after three separate performances having been filmed, there was not a single complete version of 'Whole Lotta Love', Led Zeppelin's signature song. And there were niggling details that made things difficult: despite all four members being asked to wear the same clothes each night, John Paul Jones claimed not to have received the instructions and wore a different outfit each time, a cause of considerable continuity problems during editing. Massot did film some interesting backstage footage, however, including Peter Grant berating an unofficial-merchandise seller.\n\nIt was during the second Madison Square Garden show that an extraordinary, and still unsolved, mystery took place at the Drake, Led Zeppelin's New York hotel \u2013 glimpses of which ended up in the resulting film's footage.\n\nRichard Cole and lawyer Steve Weiss had placed in the region of $203,000, largely in $100 bills, in a safe-deposit box in the hotel. Why would they have so much cash? The group liked having it on tap, Cole said, so they could always 'buy a guitar in the middle of the night, or a bit of blow'. In fact, Cole would regularly travel with $50,000 in his pocket, prepared for such an occasion. Because they needed to pay the cost of their plane at the end of the tour, they had even more than they usually carried.\n\nThe previous night, Cole had opened the safe to take out $8,000 for Page to buy a classic Les Paul guitar, and all the money was there.\n\nAs the group were about to leave for the venue for the second night, however, Cole again checked the safe to make sure there was sufficient to pay both Massot's film crew and the fee for the _Starship_.\n\nThere was no money there whatsoever. Cole told Grant and Weiss, who were waiting in the lobby, that all their cash had vanished. The frequently irascible Grant simply reacted to the large problem with sangfroid.\n\nCalling the police meant first involving the three men in another task: completely cleaning out each of the group's accommodation of any drug residue, especially cocaine, as it was likely that everyone's rooms would be searched.\n\nCole was initially the chief suspect in the eyes of the police, and he was given a lie-detector test, which he passed with flying colours.\n\nAs the group returned to the hotel from yet another triumphant show, news of the missing money had broken. Grant took it upon himself to grab a camera from the neck of a persistent _New York Post_ photographer and toss it to the ground. For thus validating his reputation as the most dangerous and volatile manager in the music business, he was immediately arrested for assault and taken downtown to the Tombs. After an hour or so of questioning by the FBI about the theft, he was released without charge. He was never handcuffed, though; the police officer who took him to the station had once drummed with a group supporting the Yardbirds and recognised the manager immediately.\n\nNew York photographer Joe Stevens was in the hip Manhattan nightclub Max's Kansas City: 'The joint filled with people from the Zeppelin gig at Madison Square Garden. An hour or so later Page strolls in with their Irish publicist B. P. Fallon. We chatted about the gig and there was general blah-blah. But then someone blurted out to the two about how the newspapers in New York City were making big headlines about Zep's manager Peter Grant arranging to have the previous night's Madison Square Garden take stolen from their hotel's safe. The two laughed. Page said, \"It's probably true.\" Fallon said he was there when the money was deposited, and he saw Grant get a receipt for the money. At that point the conversation changed to an art installation of a crushed auto that had begun to leak transmission fluid onto the nightclub's floor. The Andy Warhol contingent were holding court in the back of the club dining room and neither heard nor experienced any of this.'\n\nIn October, back in the UK, Joe Massot set about filming additional fantasy sequences. Almost inevitably, Robert Plant was filmed riding a steed in the role of a medieval knight, with his wife and children by a babbling country brook; John Paul Jones read stories to his kids; John Bonham drove a hot rod and ploughed fields. And Jimmy Page?\n\nPage insisted on being shot on the full moon, on 12 October 1973, on a rocky outcrop behind Boleskine House. In the filmed sequence he climbs upwards until he reaches the top, where he is met by a hooded hermit figure, of the same ilk as in the tarot pack and the gatefold of _Led Zeppelin IV_. The hermit's face then morphs through time, revealing it to be that of an ancient Jimmy Page, an antique Zoso: Page's self-perception, the ancient man of wisdom, essential for spreading his myth and that of Led Zeppelin.\n\nWith his purchase of Plumpton Place, Page had set himself up as one of Britain's new aristocrats. But a gentleman with a country estate requires a town house in London, so he set about finding the right one.\n\nThe Tower House, at 29 Melbury Road in London's exclusive Holland Park, is a neo-medieval structure built by architect William Burges as a 'palace of art', in the Gothic Revival style. Since his teens, Page had been fascinated by Burges, whose work was interwoven with the Pre-Raphaelite movement.\n\nDominated by a distinctive cylindrical tower and conical roof, the house was built of red brick. The ground floor had a library, drawing room and dining room, and there were only two bedrooms. Each room had its own theme, such as astrological symbols, butterflies, the ocean. A Grade I listed building, the place felt like high art, as though ordained from above.\n\nPerhaps presciently, the bedroom decoration \u2013 exaggerated poppies covering the panels of a bedside cupboard \u2013 suggested another of Burges's abiding interests: opium.\n\nThe poet John Betjeman, who lived there in the early sixties, had this to say of the house: 'A great brain has made this place. I don't see how anyone can fail to be impressed by its weird beauty... awed into silence from the force of this Victorian dream of the Middle Ages.'\n\nThe actor Richard Harris purchased the house from Lady Jane Turnbull in 1969 for \u00a375,000, gazumping Liberace after learning that the flamboyant American entertainer had yet to put down a deposit. 'It was a strange building and had eerie murals painted on the ceiling... I sensed evil,' wrote the artist Danny La Rue in his autobiography of a visit to Tower House. Harris claimed the house was haunted by the ghosts of children who had lived in a demolished orphanage formerly on the site \u2013 the actor placated them by buying them toys.\n\nA haunted mansion? Just the thing for Jimmy Page. In 1973 he bought Tower House's lease for \u00a3300,000, pipping David Bowie, his fellow Capricorn superstar and Crowleyite rival, to the post. 'I was still finding things 20 years after being there \u2013 a little beetle on the wall or something like that. It's Burges's attention to detail that is so fascinating,' said Page, for whom detail had always held its own high place, in a 2012 interview.\n\nSome of the detail Page himself brought to Tower House was equally culturally significant, like the rows of velvet suits, neatly hung according to colour. Page has never thrown away a single item of clothing, preserving it all, a further art statement assiduously cared for by dedicated staff who fight the eternal war against moths.\n\nSome ten minutes walk from the house, at 4 Holland Street, just off Kensington Church Street, Page set up his own occult bookshop, the Equinox Booksellers and Publishers, as it was grandly titled. _The Equinox_ had been the name of Aleister Crowley's own biannual magazine.\n\nDesigned expensively for Page by an architect in the manner of a _fin de si\u00e8cle_ occult lodge, his shop had its share of Art Deco detail and Egyptian motifs, as well as Crowley's birth chart pinned to one wall. He set up the Equinox because he had become frustrated at entering such shops on his travels and never finding what he was looking for. 'There was not one bookshop in London with a good collection of occult books and I was so pissed off not being able to get the books I wanted,' he wrote in his photographic autobiography.\n\nHe intended to republish long-out-of-print occult works. The first and only books published by the Equinox Booksellers and Publishers were _The Book of the Goetia_ , personally translated by Aleister Crowley himself, and _Astrology, A Cosmic Science_ by Isabel Hickey. But the musician would find the world of esoteric publishing more complex than he had perhaps expected. Mind you, the shop itself would become a focus for aspects of life that Crowley himself might have approved: the consumption of heroin was commonplace. 'They were all busy doing smack in the top room at Equinox,' said Jeff Dexter. 'The initial burst of smack in London came in with Eric [Clapton] and Delaney and Bonnie. And John Lennon.'\n\n'I used to go over there to score,' said John Dunbar, the former husband of Marianne Faithfull.\n\n## 16\n\n# THE KING AND JIMMY PAGE\n\nOn 28 October 1973 Led Zeppelin's legendarily substantial contract with Atlantic Records was due for renewal. The band were now responsible for 25 per cent of Atlantic's sales, and Ahmet Ertegun was anxious to retain such a cash cow.\n\nAlthough already wealthy beyond his wildest dreams, Jimmy Page now saw an opportunity to truly secure his pension. As the Beatles, Rolling Stones and more recently Elton John had done before them, he and Peter Grant decided to form their own record label, something to which the other three Led Zeppelin members each agreed. Unlike the Beatles' and Stones' labels, however, they were insistent that this would not turn into a vanity project: they would sign cutting-edge artists and promote them to success. By setting up their own label through Atlantic, Ertegun and Jerry Wexler's company would essentially become Zeppelin's distributors, although Atlantic remained involved in the marketing.\n\nGrant held a meeting with Ertegun and, after an all-night session between these two music-business _\u00e9minences grises_ , a deal was finally struck. The Atlantic boss emerged into the dawn light with the line: 'Peter, you've taken me to the cleaners.'\n\nBut what would the label be called? There were some predictable suggestions: in addition to Led Zeppelin Records, Stairway, Slag, Slut and Eclipse were mooted, but thankfully dismissed. Then came an odd contender, Swan Song, an idea from Page, a reference to the pair of Australian black swans that roamed freely about his Plumpton residence. Page considered the noise these swans emitted in the moments before their death to be 'one of the most beautiful sounds in the world'. He had played around with the name for a tune he was working on, but instead it became the name of Led Zeppelin's own record company.\n\nAs a piece of symbolism, the name 'Swan Song' had considerable resonance, as the ensuing difficult years would prove. The traditional meaning of the phrase is the end of a final act, a last statement before the endgame is arrived at. Was this Page's subconscious means of somehow drawing to an end the subtly ongoing crisis that the entity of Led Zeppelin was creating in the lives of all around it? For if you use Swan Song Records as a chapter marker in the life of the group, it is clear that from then on matters begin to unravel at a frightening pace.\n\nAs you would expect from a group whose leader had such a creative aesthetic, the Swan Song logo seemed like high art. It was based around a work by the 19th-century painter William Rimmer, _Evening: Fall of Day_ , which Page found in Philippe Jullian's book _Dreamers of Decadence_. Rimmer's painting had been 'improved', to an extent, for the logo: now it appeared to show Icarus falling from his celestial splendour after flying too close to the sun. Later, in an interview with _Mojo_ in 2015, Page said: 'If you think about it in relation to the original Led Zeppelin idea of a lead balloon, it's carrying on the original idea of that, isn't it? The idea of the swan song is the dying song of the swan, okay? So there's a parallel.'\n\nThe Swan Song label was an almost immediate success, as early as June of 1974. Although Bad Company were signed to Island Records in the UK, elsewhere they were a Swan Song act. Moreover, they were managed by Peter Grant. Clearly a supergroup-in-waiting, Bad Company had Free's impressive blues-wailing singer Paul Rodgers and drummer Simon Kirke; playing guitar was Mott the Hoople's Mick Ralphs; and on bass was Boz Burrell from King Crimson.\n\nBad Company's eponymous first album went to number one in the United States and number three in the UK. 'Swan Song was initially a great idea,' said Simon Kirke. 'It had that Island Records vibe, in that it was formed by musicians to look after their musical interests and those of other bands. Of course it was a business and was run as such, but it wasn't a shrine to capitalism by any stretch of the imagination.'\n\nBut there were a couple of other acts on Swan Song who did not fare so well. Peter Grant's old friend Maggie Bell, late of club favourites Stone the Crows, was signed up, the intention being to turn her into the UK's Janis Joplin. The first time Bell met Grant, she told him that he reminded her of Orson Welles in _Citizen Kane_. His response? 'Yeah, do you want to be my little Rosebud?'\n\n'He was absolutely enormous but he was a very kind man. I was like a daughter to him, but looking back he didn't really know what to do with me. He managed loads of guys and my career suffered through that.' Maggie Bell's LP _Suicide Sal_ , on which Page played guitar \u2013 on 'If You Don't Know' and 'Comin' On Strong', a couple of hours' work \u2013 was released in 1975. She then moved to New York for a year, recording a pair of albums that Swan Song never released. 'I paid for them,' she said. 'But to this day I don't know what happened to them. I was told they weren't good enough, but believe me they were!' Instead, Maggie Bell ended up working on reception at the Swan Song offices.\n\nThe Pretty Things were also signed to Swan Song. One-time rivals to the Rolling Stones on the London R&B circuit, their singer Phil May was friends with Page, who adored their 1964 single 'Rosalyn'. Their 1968 album _S.F. Sorrow_ , the first ever concept LP, had been a critical success, but there was a feeling that the group were past their sell-by date. This did not prevent their _Silk Torpedo_ album being given a lavish launch in the UK, however. Chislehurst Caves, to the south-east of London, was hired for a Halloween launch party in 1974, one that would go down in rock 'n' roll legend. As Ahmet Ertegun, whose new velvet jacket was sprayed with jelly at the event, once commented: 'Throw a party. And promote the hell out of the party!' With strippers dressed as nuns doing tricks with crosses and candles, Aleister Crowley would have adored the occasion. As everyone grew progressively more drunk and drugged, the party finally wound up with John Bonham and his cronies hurling food at everyone.\n\nJanine Safer, Swan Song's US publicist, had a different point of view to Simon Kirke: 'Swan Song was for and about Led Zeppelin, primarily. To a lesser extent it was about Bad Company. To hardly any extent it was about the Pretty Things. Maggie, though, held a very special place in the pantheon, because Peter loved her and Jimmy loved her. Detective had a brief flurry of activity, but it was a case of Hurry Up and Wait, long months of doing absolutely nothing. No one would hear from Peter and no one would hear from the band.'\n\nPremises were acquired for the label's London headquarters at 484 New Kings Road, in the World's End section of Chelsea, a rundown, scruffy neighbourhood that had always had a criminal subtext. Swan Song's offices were no improvement on the area: beat-up, battered furniture, uncarpeted stairs, more like a squat than the centre of operations for the Biggest Band in the World. Jimmy Page lived in palatial splendour, which he rarely left for 484 New Kings Road: why should they be bothered about what the place resembled? Besides, there was something of an early distressed look about 484 New Kings Road, situated above a branch of the British Legion, which could almost be judged as an anti-aesthetic.\n\nNear Swan Song's HQ was a pricey bar called J Arthur's, favoured by Zeppelin members and Swan Song employees, as well as posh girls and gangsters. The legendary Granny Takes a Trip, from where Page bought some of his most beautiful clothes and which also for a time dispensed heroin to those in the know, was almost next door at number 488 \u2013 though the shop would close the year Swan Song moved in. With what in hindsight seems a poetic significance, Malcolm McLaren and Vivienne Westwood had set up their own small fashion emporium, now named Sex, a few hundred yards to the east along the same block. Very soon the Sex Pistols, who would deliver especial antipathy towards Led Zeppelin, would form in that store.\n\nAs manager of both Led Zeppelin and Bad Company, and as titular head of Swan Song Records, with a 90-minute train ride in to London from his home near Lewes in Sussex, Peter Grant had his plate full. The pressure would soon start to show, with staff becoming genuinely nervous of the manager's furious tantrums \u2013 which increasingly could be diagnosed as cocaine rage.\n\nAs though adding to his own mystique, Jimmy Page had resolutely refused to meet Elvis Presley when he saw the King at Las Vegas's International Hotel in 1969 with Miss Pamela. Could he simply have been overawed by the very notion of meeting this idol, the man who had introduced the world to the liberating force of rock 'n' roll?\n\nAgain, when the entire group went to see Elvis Presley on Saturday 10 June 1972 at Madison Square Garden, they didn't get to meet him.\n\nOn 11 May 1974 this all changed. Page, Robert Plant and John Bonham went to watch Elvis once more, at the venue Led Zeppelin had made their own, the Forum in Los Angeles.\n\nThe group were in town for the launch party of Swan Song Records the previous night, thrown at the sumptuous, distinctly pink Beverly Hills Hotel, and the evening had not been without considerable controversy. Page had taken up with Bebe Buell, the girlfriend of Todd Rundgren, the mercurial musician and producer. Inevitably this was to the considerable chagrin and pain of the now 16-year-old Lori Mattix.\n\nBebe Buell had first met Page the previous year in New York, when she had been introduced to him by her friend Patti D'Arbanville, celebrated in Cat Stevens's 1970 song 'Lady D'Arbanville'. Page had wanted her to come with him to the Plaza Hotel, where he was staying, so they could 'party', with all the euphemism that word contains. But Buell had been preoccupied with returning home to continue an ongoing row she was having with Rundgren \u2013 over D'Arbanville, in fact, whom she had found sitting on her boyfriend's knee when she had arrived home earlier.\n\nLater that year, at an Eric Clapton show in Manhattan, Mick Jagger tried to chat her up. Part of his come-on line revolved around warning her off Page: 'You haven't been out with Jimmy Page? You must never date Jimmy Page.' Wouldn't you think these to be the kind of words that would only increase her fascination with the Zeppelin guitarist? Jagger was clearly showing some nervousness about his rock 'n' roll love rival.\n\nAt the beginning of May 1974 Buell flew with Rundgren to Los Angeles, checking in at the Hyatt House. After yet another row, Rundgren headed up to San Francisco to play a show, leaving Buell at the Hyatt, along with Rundgren's South American raccoon, almost inevitably named Kundalini, after the primal energy of the coiled female serpent in Hindu philosophy.\n\nIn the Hyatt's lobby Buell ran into Kim Fowley and Rodney Bingenheimer, who were there because Led Zeppelin were coming in from the airport 'to host the biggest party of the year for the launch of their Swan Song label. They were the biggest band in the world in 1974.'\n\nExtremely distressed at having been treated badly by Rundgren, she was sobbing to Fowley and Bingenheimer in one of the hotel's elevators when the door opened and Page, Plant and Cole stepped in. They had just arrived at the hotel and were heading for their rooms. 'Jimmy was wearing a pair of dainty black boots, crushed blue-velvet pants, a beautifully ruffled Edwardian shirt, and a velvet jacket worthy of Beau Brummell,' Buell wrote in her book _Rebel Heart_. 'His pale, handsome face was framed by exquisite black ringlets. He looked like Sir Lancelot.'\n\nPage asked Buell if he hadn't met her with Patti D'Arbanville in New York, which she immediately denied, snuffling and running out of the elevator. Page got her room number from Rodney Bingenheimer.\n\nAt five the next morning a sleeping Bebe Buell received a phone call. It was Jimmy Page asking her to come up to his suite and have breakfast with him. Playing the sympathy card, he told her he was extremely upset: he had just returned from a session with Joe Walsh, 'playing guitar on a song about his wife and child, who were recently killed in a car accident. Please come and join me for breakfast. I'm very upset and lonely.'\n\nBuell said she would only go up to see him on condition that she could bring Kundalini with her, feeling that the raccoon gave her some edge in the rock 'n' roll eccentricity stakes. Besides, she couldn't leave her on her own. She went up to his suite in her pyjamas, wearing no make-up.\n\n'Come in, darling,' Page said, welcoming her with a bow. 'Would you like some cocaine? Are you hungry? I've got a great idea. Why don't we give this fruit basket to the raccoon and put them both in the bathroom?'\n\n'We proceeded to hang out,' said Buell, 'drinking mimosas, doing some lines, and comparing notes on our current problems.'\n\nWhen Peter Grant appeared, he questioned Buell as to whether her boyfriend would want to come and shoot Page in the kneecaps. She assured him that Rundgren had better things to do, having effectively dumped her and run off with someone else in San Francisco.\n\nGrant then went to the bathroom, only to discover \u2013 to his great surprise \u2013 the raccoon. Not only that, but Kundalini had unleashed her primal energy: the bathroom was covered in poo, probably a consequence of all that fruit.\n\nPage was moved to another suite, and he and Bebe Buell finally went to bed together, the musician immediately going down on her: 'He told me I would sleep much better if I had an orgasm; otherwise, the cocaine would keep me awake, and I didn't want any of the Valium he was offering me.'\n\nNot long after falling asleep, they were woken by a frantic Lori Mattix attempting to enter the room: 'I just remember the cacophony of some girl screaming \"Jimmy!\" outside the door. Then Richard [Cole] suddenly showed up and dragged her off in hysterics.' A couple of hours later they were woken yet again; Buell's room remained unpaid for, and her luggage was about to be removed. Page immediately handed Cole the outstanding $50 for the room. 'Well, Richard, it's very simple,' he said. 'Get the bags and bring them here. She'll be staying with me.' Buell was 'kind of shocked by his presumption, but I was also very happy.'\n\nThen Page told her about the 'lovely party' they would be throwing for Swan Song at the Beverly Hills Hotel, saying, 'I would love you to be my companion for the launch.'\n\n'I became Jimmy's girlfriend for the week,' said Buell. 'I stayed with him and was never far from his side.' Every night they ate at expensive restaurants; during the day they would visit metaphysical bookstores. 'We spent a lot of time in bed... We had a powerful sexual relationship. It was very beautiful. He never tried to have anything but completely straight sex with me, although he had one weird penchant. When he kissed me, he loved to spew his saliva into my mouth. It was odd. I thought of it as his way of coming in my mouth without coming in my mouth. Otherwise, we had the most wholesome sex imaginable... Jimmy was never violent; he never tried to practise black magic.'\n\nThe group leader may never have resorted to physical aggression, but others in the Zeppelin camp were known to throw their considerable weight around. One night Page and Buell went to the Rainbow Bar and Grill to have dinner. 'I noticed that there was a lot of violence surrounding Zeppelin,' she said. 'Somebody came over to our table, and that irritated everyone. Richard Cole hit the guy with his elbow and he fell to the ground, next to his teeth. That was a very disturbing thing for me to witness, because I never thought that violence was part of rock 'n' roll... But Peter Grant brought the gangster image to the rock 'n' roll scene, and he prospered. Richard and Peter were not to be tangled with. They were rough players.'\n\nThere was a public scene as security guards kept Lori Mattix \u2013 who had arrived off her head on Quaaludes \u2013 away from Page, and therefore Buell, at the Swan Song party. 'And it was made very clear to Lori \u2013 who was probably the most beautiful girl in LA and had given herself exclusively to Jimmy from age 14 to 16 \u2013 and Miss Pamela that they were history.\n\n'I was very attracted to Jimmy Page,' said Buell. 'He was my type \u2013 very dashing, very English, very Renaissance. He had that otherworldly, other-time vibe. If you fantasised about being a princess and having a prince come and sweep you off your feet and take you on horseback to his castle, Jimmy Page was Sir Lancelot. What girl wouldn't enjoy that fantasy, even if it was only for a week?'\n\nThe night following the Swan Song party, Buell did not accompany Page to the Forum to watch Elvis Presley.\n\nTowards the end of the show, Elvis asked for the arena's house lights to be switched on so he could take a look at the audience. Beginning his next song, Willie Nelson's 'Funny How Time Slips Away', he suddenly paused and spoke to his band. 'Wait a minute, wait a minute, hold it,' he said, laughing. 'If we can we start together, fellas, because we've got Led Zeppelin out there, and Jimmy Darren, and, uh, a whole bunch of people, and let's try to look like we know what we're doing, whether we do or not... Now, what were we doing?'\n\nFollowing the concert, Page, Plant, Bonham and Grant were taken to the King's hotel suite, where he was staying with his girlfriend, a recent _Playboy_ centrefold. 'We went up to his suite and his girlfriend Ginger was there with just a few other people,' Page said later. 'I can tell you, we were really nervous. When he came to the door, he started doing his famous twitch. You know, he didn't put that on \u2013 that was something he really did.'\n\nAlthough scheduled to visit Elvis for no more than 20 minutes, they ended up being with him for over two hours. John Bonham broke the ice with a classic-car conversation, and later Elvis admitted to his guests that he had never heard any of Zeppelin's songs, other than the inevitable 'Stairway to Heaven'.\n\nWhat he had heard, however, were scurrilous tales of Zeppelin on the road. Were they true? Elvis wondered.\n\nIt was left to tactful Robert Plant to issue a denial: 'Of course not. We're family men. I get the most pleasure out of walking the hotel corridors, singing your songs.'\n\nPlant recalled that 'we all stood in a circle and discussed this whole phenomenon, this lunacy... Elvis was very focused, very different to what you now read.'\n\nPage told the King that Led Zeppelin rarely soundchecked, and that when they did, all Plant wanted to do was sing Elvis songs. Laughing, Elvis asked Plant which of his tunes the singer selected. 'I told him I liked the ones with all the moods, like that great country song \"Love Me\".' To Elvis's considerable amusement, Plant proceeded to sing his rendition of 'Love Me' to the King of Rock 'n' Roll. Elvis broke up in laughter.\n\nAs they left Elvis's suite, extremely happy to have met their hero, they heard a voice behind them. 'So when we were leaving, after a most illuminating and funny 90 minutes with the guy, I was walking down the corridor,' remembered Plant. 'He swung round the door frame, looking quite pleased with himself, and started singing that song: \"Treat me like a fool...\" I turned around and did Elvis right back at him. We stood there, singing to each other.'\n\nLed Zeppelin then flew to New York, booking the entire first-class cabin of an American Airlines plane so that the raccoon could fly with them. 'We had a party flying across America at 35,000 feet,' said Bebe Buell.\n\nIn Manhattan they checked into the Pierre on Fifth Avenue, and Page had two connecting suites: 'He used one suite for entertaining and the other to live in, because he was fastidious and he didn't like the reek of beer cans and drug paraphernalia, or getting the smell of cigarette smoke on his clothes.'\n\nNot that the lovers confined themselves to their suite all of the time. 'During the days, we went to art galleries and museums, but mostly we went shopping. Jimmy bought great quantities of shirts and jackets; occasionally, he would buy me a dress or a pair of shoes, and he also bought me a wonderful book on the Pre-Raphaelite painter [Edward] Burne-Jones.'\n\nWhen it came time for Page to return to London, Buell drove out to the airport with him in a limousine. 'He gave me the impression that their relationship [with Charlotte Martin] was souring and he wasn't sure what its future was going to be. He left me dangling, but it was very romantic and he told me he would call me.'\n\nIn September 1974 Bebe Buell called Led Zeppelin's London office, asking for Jimmy Page to be informed that she would shortly be arriving in London. Clearly he had not given her his home phone number. What Buell did not mention was that she would be accompanying her boyfriend Todd Rundgren on a promotional tour for his record company, Warner Brothers.\n\nLodged at the prestigious Savoy Hotel, Buell and Rundgren were sitting down to breakfast when a huge bunch of flowers arrived, accompanied by a card from Page that contained his phone number. Rundgren's response? To take his girlfriend's plate of scrambled eggs and throw them against the wall. Then he delivered a diatribe against Page: 'He's bad. He's evil. He's dark. He's the devil. He's got a woman and a kid. Once he gets tired of you, he'll retire you. Soon you'll be too old for him.' Buell swore to Rundgren that she would never see Page again.\n\nThe pair went to Paris and Amsterdam on further promotional work, before returning to London and checking in at the Montcalm Hotel, close by Marble Arch. As soon as Rundgren disappeared for a meeting with publicity guru Derek Taylor, Buell called Page, and he talked her into coming round to see him at Tower House. She had told Rundgren she was meeting some model friends and would link up with him that night at the Speakeasy.\n\nBuell and Page started drinking champagne as soon as she arrived at Tower House, and within 15 minutes they were in bed together. 'In Jimmy's defence,' she wrote, 'he never attempted to use a whip on me, never hurt me sexually, never tried to do anything weird to me... When I hear the legends and myths, I'm always perplexed. I was Guinevere and he was Sir Lancelot, and Todd was King Arthur... And I really thought Jimmy loved me. I thought there was something there.'\n\nShe woke up the next day, having missed the Speakeasy. In considerable anguish, she called Derek Taylor, who told her that they had the police looking for her.\n\nBriefly reconciled with Rundgren at the Montcalm, Buell remained in their room as he went to the Warner Brothers offices. Then she called Page, who told her she could only come to his place if she was going to stay there with him. She said she would come for a couple of days. 'Ten minutes after swearing to him that I wouldn't get in touch with Jimmy, I couldn't stop myself. Maybe it was the music. Maybe it was his Satanic Edwardian quality. Maybe it was the medieval Sir Lancelot vibe. I didn't know and I didn't care. I just wanted to be with Jimmy.'\n\nReluctantly, Rundgren let her stay on in London when he flew back to New York. 'Todd had this theory that Jimmy was evil and he was good. And I was the virgin angel sacrifice in the middle of this duel between the two powers. He felt that Jimmy collected women and that once Jimmy had wooed me away from him and dominated me completely, I would become yesterday's news.'\n\nWhat did Bebe Buell experience when she once again visited Tower House? 'When I went back to Jimmy, he was a vision in a beautiful white suit. He got an erection, which I couldn't help noticing. He looked at me and said, \"See what you do to me? You make me so hard.\"'\n\nThe pair took mescaline and went to bed, where they seemed to spend most of the time. 'When you're tripping, an erection looks even larger than it is; it takes on a whole other dimension,' she said. 'I was afraid to have sex with him for a minute, because I was hallucinating that his penis stretched to the other side of the room. I needed to pull myself together and realise that it was an average-size penis.' That night, still tripping on the mescaline, they went into the Tower House's sizeable garden 'to look for fairies'.\n\n'Jimmy was very sexy. I think that he was having fun, enjoying a very pure exchange with a woman. I was different. Most of the women he knew were wilder and expected the macabre from him. I think if a woman had wanted her ass spanked and her hair pulled, he would gladly have delivered. If a woman had wanted to be beaten with whips, he would have obliged. If a woman wanted a cigarette put out on her chest, he would have done it.'\n\nShe described them going out to eat at a high-end Indian restaurant, and how Page would keep people at arm's length when he was in public. 'It was because people didn't know who he was. I think it was because they were genuinely afraid that he was Satan, that he had some sort of evil allure... Mick Jagger is supposed to be Lucifer, but people never act that way around him. He gets a lot of, \"Hey, it's Mick Jagger. Hi!\" I never saw that happen to Jimmy Page. Furthermore, unless it was a good-looking girl, Jimmy would act oblivious when someone was having a conversation with him. If some testosterone-driven guy came running up, saying, \"Oh my God! You're my hero,\" he would just look straight ahead and not even acknowledge the person.'\n\nPage asked Buell for her date of birth and drew up her astrological chart. But this reading concerned him: something in it seemed inappropriate for him.\n\nAll the same, Page's intentions seemed to be to have Buell live in the house in London while Charlotte and Scarlet remained in Plumpton. 'Of all the men I saw in these years, Jimmy Page was the only one I was seriously considering leaving Todd for.'\n\nIn the November 1974 edition of _Playboy_ , in which Buell was the centrefold, she was identified as Rundgren's girlfriend. And in the editorial copy that accompanied her nude spread, Mick Jagger was mentioned as a friend who 'called regularly from Montauk'.\n\n'That annoyed Jimmy Page no end,' she said.\n\nHowever, the _Playboy_ spread caused problems for her modelling career. It was suggested she move to London and work there until the heat evaporated in the more puritanical USA. Moreover, Page had started calling again, asking her to come and visit him in London, followed by a cable sent by Peter Grant, requesting her presence in London in November. So she went.\n\nBebe was supposed to be staying in the flat above Equinox on Holland Street. But Eric, the store's manager, told her that Charlotte and Scarlet would be staying there instead. For, in a touch of Dionysian symmetry, there had been another scandal.\n\nRon Wood, still a member of the Faces, and his wife Krissy had been to visit Page and Charlotte at Plumpton. After telling her husband she no longer loved him, Krissy Wood became Page's live-in girlfriend for the next 12 or so months. Ron Wood briefly went off with Charlotte, and then Pattie Boyd. When Led Zeppelin played the Nassau Coliseum on 13 and 14 February 1975, Ron Wood joined them each night on guitar for the encore of 'Communication Breakdown'. Earlier, 'Woody', who seemed to live life as though in permanent disbelief and gratitude at the fortunate hand fate had dealt him, had phoned Page. 'How's our bird?' he asked him.\n\n## 17\n\n# COCAINE NIGHTS AND HAUNTED HOUSES\n\nWhen Kenneth Anger met Jimmy Page he would no doubt have mentioned that, in his youth, just prior to the Second World War, he had seen a Nazi zeppelin prowling the air along the Californian Pacific coast.\n\nBrought up in Los Angeles, an alumnus of Beverly Hills High School, Kenneth Anglemeyer \u2013 he altered his name to the simple but telling 'Anger' for reasons of brevity \u2013 was turned on to movies by his actor grandmother, a friend of the director Max Reinhardt, and he set up a film society to show little-known European films. Anger was one of the first openly gay directors in Hollywood, and the homoerotic _Fireworks_ , made in 1947, was his first film. Later he took his friend Dr Alfred Kinsey, the famous sexologist, to Aleister Crowley's abbey in Sicily, where he spent a summer scraping off whitewash that had been applied to Crowley's paintings after the Great Beast had been expelled under Mussolini's orders.\n\nAnger financed his own films during the latter half of the 1970s through his book _Hollywood Babylon_ , a magazine-style tome that recounted many of Hollywood's dirtiest secrets. (When I bought a copy imported from the US in 1975, it was from a small bookshop on Holland Street, only a few yards from Equinox; it seems not unreasonable to assume that Anger himself \u2013 who would have been in residence at Tower House at the time, certainly visiting Equinox \u2013 could have provided the store with its copies of _Hollywood Babylon_.)\n\nAnger first met Page in 1973 at a Sotheby's auction in London, where they were both bidding for a pornographic manuscript by Aleister Crowley. 'He, of course, had more money than I did,' Anger said later.\n\nBonding over their fascination with Crowley, Anger accepted Page's invitation to go with him to Boleskine House and exorcise the building of the headless man who had so thoughtlessly been rolling his head about the property. Then Anger asked Page to write the soundtrack for the film he was working on, _Lucifer Rising_.\n\nThe two had much in common beyond a love of Crowley, not least a rock-hard core belief in a total extension of consciousness to embrace all possibilities. 'I think that's what you have to keep in mind when you are hearing\/looking at Jimmy and Kenneth's work,' Steve Parsons, formerly Snips, the singer with 1970s cult band the Sharks, whom Page would later call on, told me. 'It's an audible\/visible extension of a much more involved and complicated process, which includes mental, spiritual and physical reconstruction of the senses.\n\n'Both men have made contact with and seek to \"bring down\" star beings from the past and the future \u2013 Kenneth works in elegant silent gesture and Jimmy utilises thunder and lightning to power his musical machine.\n\n'Our boys did give it a go but, as is mostly the case, it ended in tears. Not sure what Kenneth thinks about _Lucifer Rising_ now as he was obviously angry (the clue's in the name) for a long time and, as always, Jimmy keeps his thoughts to himself (probably in another dimension on a finely printed page).'\n\nExtremely iconoclastic, _Lucifer Rising_ is based on the theory that the audience will 'get' Anger's redefining the notion of Lucifer \u2013 in his terms, the bringer of light, who had essentially been given a bad rap. Set in Egypt, presenting a series of Thelemite rituals, _Lucifer Rising_ seems to be about Crowley having _The Book of the Law_ dictated to him in Egypt in 1904. Visually stunning, _Lucifer Rising_ is a beautiful film, the dream-like substance of Anger's filmmaking providing a sumptuous and rather profound experience, though non-Thelemites may find the film slightly (but only slightly) difficult.\n\n'For me,' says Snips, ' _Lucifer Rising_ is essentially an optimistic piece concerning renewal through the actions of the primordial mother and father summoning their daughter, Lilith, from the hell worlds below up to the earth plane, where she will greet visitors from space.'\n\nAnger's film had yet to be edited, and with editing equipment installed in Tower House to save the shambles of _The Song Remains the Same_ , Page invited Kenneth Anger to his London home to work on the movie. Meanwhile, he promised, he would work on the soundtrack.\n\nThe Madison Square Garden shows filmed for _The Song Remains the Same_ in July 1973 were the last Led Zeppelin played until 11 January 1975. After the concerts, with all due secrecy, Robert Plant had an operation to remove nodules from his vocal chords \u2013 a perennially worrying medical development for untrained singers like Plant. During the 1973 American tour his vocals could be unpredictable to say the least. When Zeppelin played Chicago on 7 July 1973, his voice had seemed utterly shot right from the beginning of the set, and there was similar evidence at several further shows.\n\nFollowing the operation, Plant was obliged not to speak for three weeks. A further consequence of the procedure was that Led Zeppelin's singer would never again be able to reach the high notes that had been such a feature of his vocal performances. His singing on the soon-to-be-recorded _Physical Graffiti_ is different from previous Zeppelin recordings: he sounds more comfortable, controlled and accomplished, with almost as much raw power still present, but rarely hitting those bitch-on-heat wails of earlier days \u2013 'In My Time of Dying' is an exception.\n\nPlant's need for a vocal remedy was indicative of a broader malaise: the group's lifestyle hardly leant itself to disciplined vocal practice. Beginning with the 1972 Japanese tour, on which John Bonham behaved so egregiously, the levels of excess in and around the act had developed to a worrying extent.\n\nThe 1973 American tour featured enormous amounts of cocaine: before he performed his 'Moby Dick' drum solo, Bonham would scarf up a handful of coke from a sugar bowl kept by his drum-stool; meanwhile, the other group members would step backstage for their own lines of coke, frequently with the addition of a blowjob. Alcohol was ubiquitous: the drummer was almost permanently drunk; and even Page's booze intake was now a cause for concern. Occasionally heroin was present; nothing like it would become, but it was edging in. And Page, a Capricorn, an astrological sign rather partial to hard drink and drugs, was being pulled into its potentially lethal orbit.\n\nBack in Britain in 1973 he spent some time in his personal archives, tracking down tunes that Zeppelin had recorded over the years but never released. He hoped they would furnish him with sufficient material to ensure that the next Led Zeppelin release, the first on their own label Swan Song, could be a double album. The two-LP set had become a mark of status for all the major players in popular music: the Beatles, Rolling Stones, Bob Dylan, Jimi Hendrix and the Who had all released significant double albums. If the biggest band in the world did not follow suit it could be interpreted as diminishing status. A double record would also bring in substantially more cash for Swan Song, a point that would not have been overlooked by the ever financially cautious guitarist.\n\n'I didn't want it to be a double album with any padding on it,' he told _Rolling Stone_. 'It would be a double album with all character pieces, the way that Led Zeppelin did their music with the sort of ethos of it, if you like, that everything sounded different to everything else.'\n\nBrand-new material would be recorded to complement what he curated. _Led Zeppelin IV_ , by far the group's most commercially successful album, had been recorded at Headley Grange. For their sixth LP, Page decided that it should be used once again, and sessions were booked for November 1973.\n\nThen an utterly unexpected crisis arose: John Paul Jones, always the almost unknown face of the group but an essential musical fit, was considering leaving Led Zeppelin. He cited his time away from his family as his principal gripe, and claimed he had been offered the job of choirmaster at Winchester Cathedral. The Headley Grange sessions were abandoned, and Jones was given leave of absence from the group to consider his options. The time booked at Headley Grange was not wasted, however, as Swan Song's premier signing stepped in to record their debut album _Bad Company_.\n\nJones eventually decided against leaving Led Zeppelin, and recordings for what would become _Physical Graffiti_ resumed in mid-January 1974, running for six weeks until the end of February. Ronnie Lane's mobile studio \u2013 a cheaper option than the one owned by the Rolling Stones \u2013 was employed at Headley Grange, with Ron Nevison as recording engineer.\n\nNo one involved with the project particularly wanted to spend their nights at creepy Headley Grange, and everyone instead checked in to the nearby Frensham Pond Hotel \u2013 with one abstainer. Page, seemingly at ease wherever spirits roamed, was happy to spend his nights at Headley Grange alone, able to work on ideas until dawn if necessary. (In his fascination with haunted properties, was there an element of showing off \u2013 of metaphysical machismo? Jimmy Page, the only one who'd sleep with the spectres? Well, he would, wouldn't he? It could only enhance his four-dimensional image.)\n\nPage had already done plenty of work for _Physical Graffiti_ at Plumpton Place. At the top of his Sussex house he had had a multi-track studio installed, where he would work on 'textures'. 'I had the whole of \"Ten Years Gone\", all of the guitar orchestration, prepared in that house. I came up with \"The Wanton Song\" and \"Sick Again\", and I had the whole concept of \"Kashmir\" basically there,' he told _Rolling Stone_.\n\nBut it was the return to Headley Grange that Page found inspiring. 'I knew how we did the drums in the main hall for the fourth album's \"When the Levee Breaks\". And some numbers would come out of thin air, like for example the way \"Rock and Roll\" did on the fourth album, and then on _Physical Graffiti_ , \"Trampled Under Foot\", which came out of thin air like that, just starting out of a riff. I was basically musically salivating on the way there.'\n\nEven before setting foot in Headley Grange, the band had three tracks already recorded there while making _Led Zeppelin IV_ : 'Boogie with Stu', 'Night Flight' and 'Down by the Seaside', which was heavily influenced by Neil Young's 'Down by the River'. 'Bron-Yr-Aur' had been written for the third album. Plus there was 'Houses of the Holy', the title track for the fifth album, which hadn't fitted the mix of that record's other tunes; 'Black Country Woman', in which Robert Plant sang freely about his sexual relationship with his wife's younger sister, had also been recorded for _Houses of the Holy_ in the garden of Stargroves, that plane flying overhead at the beginning of the tune. 'The Rover' too had first been essayed on the _Houses of the Holy_ sessions: 'The whole thing about \"The Rover\" is the whole swagger of it, the whole guitar-attitude swagger. I'm afraid I've got to say it, but it's the sort of thing that is so apparent when you hear \"Rumble\" by Link Wray \u2013 it's just total attitude, isn't it? So that sort of thing, which is sort of probably in my DNA to be honest,' said Page.\n\nThe Headley Grange _Physical Graffiti_ sessions began with just Page and John Bonham there. Page couldn't wait, he said, 'to get the drums in the hall, to get this big drum sound and then play the riff of... \"Kashmir\"'. Originally titled 'Driving to Kashmir', 'Kashmir', a place neither Page nor Plant had ever been, was developed from another piece Page had been working on: 'I had it before going in there [to record]. I had a piece of music that I'd been working on, and just on the tail end of it I had that riff. I thought, \"Uh-oh. This is something I really want to try.\" I couldn't wait to get into Headley Grange with John Bonham and do this.' The song, which he knew had to be built around the drum kit, is 'like a child's riff. Musically, it's a round, like \"Fr\u00e8re Jacques\", where you can lay things on top of it.' Robert Plant told Page he had some lyrics he had worked on when they had visited Morocco together, although this was well after the entire musical structure had been assembled.\n\nWith its epic sweep and shifts of pace, 'In My Time of Dying', which ran to over 11 minutes long, was a close rival to the mighty 'Kashmir' for the greatest track on _Physical Graffiti_. It was a reworking of Blind Willie Johnson's 1927 song 'Jesus Make Up My Dying Bed', itself a version of a traditional gospel song; Bob Dylan had recorded another version of the song. 'There were no edits or drop-ins or overdubs to the version you hear,' said Jimmy Page. 'This is Led Zeppelin just going for it for an 11-minute song with all the changes in it and everything and the musical map that you have to remember when it goes 1-2-3-4, tapes rolling.' At the end of the song, Plant's exultation 'Oh my Jesus' mutates into 'Oh my Gina'. Page expressed his surprise at Plant's daring in so admitting his affair with another rock star's wife.\n\nThere was the occasional absurdity. One morning John Bonham arrived with a bag containing 1,500 pills of Mandrax, the UK equivalent of Quaalude. Bonham intended to hide the pills by taping them to the inside of his drum heads, but his plan was flawed: as his drum roadie pointed out to him, Bonham had a Perspex kit. On another occasion, recording stopped while the band led farm animals up to Headley Grange's first floor \u2013 an example of 1960s-style 'looning' \u2013 while flares were let off. Recording was abandoned for some days after Peppy, one of the roadies, drove Bonham's brand-new BMW 3.0 CSi into a wall. Terrified, Peppy hid in a wardrobe for 36 hours until Bonzo calmed down.\n\nSome of the new material was caught and wrapped on the first take, as both 'Custard Pie', a Zeppelin term for female genitalia, and 'Trampled Under Foot' were, though the latter was subjected to copious overdubs. On the new tune 'Sick Again' Plant seemed at times to be channelling Steve Marriott, as he had on 'Whole Lotta Love'.\n\nSilverhead were an early sleaze-rock outfit, fronted by swaggering vocalist Michael Des Barres, his androgynous look perfectly suited to the dying embers of glam rock. From an eminent British family, Des Barres had been educated at the exclusive Repton School; as a consequence of his love of drama, he was given a part in the 1967 controversial anti-racism film _To Sir, with Love_ , starring Sidney Poitier. In 1977 he would marry Miss Pamela Miller, who became Pamela Des Barres. Before that he was extremely close friends with Jimmy Page, an indubitable drug buddy.\n\nThe pair first met in the summer of 1974, when B. P. Fallon brought all of Led Zeppelin and Peter Grant to a Silverhead show at a Birmingham club, one of the last gigs that Des Barres's band would play \u2013 the group exploding, he said, 'in a cocaine frenzy. Our fan-base was coke-dealers. They got Peter to come and see the band. We were playing this club in Birmingham and there were eleven people in the audience, and four of them were Led Zeppelin.\n\n'I was a really existential bastard and that day I'd eaten so much hash. I was still floating and didn't give a fuck, and loved Jimmy from that one moment when I'd met him.\n\n'At an early age I met Sidney Poitier: I've never met anyone who had that much charisma. So I had the advantage of that. All the same, going back after the gig to Bonzo's farm for a jam in a tiny room, I marvelled at the ease and confidence with which Jimmy would play: what he did was staggering. So that I thoroughly enjoyed.'\n\nHe and Page clicked, said Des Barres, 'because I mentioned Rimbaud at the right time. Jimmy is an academic. He would stroll around New York buying books. Pamela would go with him.'\n\nDes Barres found himself invited down to Plumpton Place. 'The real story is that Jimmy is a great artist and Shel Talmy recognised it first. We'd sit around, off our heads, talking about Shel Talmy, playing with a set of tarot cards that Aleister Crowley had not only designed but also owned, Jimmy wearing a cloak that had been Crowley's.\n\n'Led Zeppelin was not necessarily fun for Jimmy: it was colossally hard work. The hugeness, those dark winters, both on tour in the United States and of the soul. You're left with a visual image that incorporates that violin bow and Jimmy Page's hair.'\n\nIn August 1974 Led Zeppelin adjourned to Shepperton Studios to restage the Madison Square Garden nights. By now Joe Massot had been taken off the project, replaced by Peter Clifton.\n\nAt the very beginning of September, Jimmy Page and Peter Grant flew out to Austin, Texas, for ZZ Top's First Annual Rompin' Stompin' Barn Dance and Bar BQ at the University of Texas Memorial Stadium. Before a 90,000-strong audience, ZZ Top were supported by Santana, Joe Cocker and Bad Company, who were kicking off the event. Bad Company effectively stole the show, Page joining them onstage to play guitar on 'Rock Me Baby'.\n\nPage bonded with the group, especially with singer Paul Rodgers, a pointer to the future. Four days later he joined Bad Company onstage again, on the same number, at the Schaefer Music Festival in Central Park in New York. (With John Paul Jones accompanying him that time, Page would repeat his encore appearance with Bad Company at the end of the year, at their 19 December gig at London's Rainbow.)\n\nOn 14 September 1974, Page, along with John Bonham, played with Crosby, Stills, Nash and Young at the after-party of their Wembley Stadium concert; the tunes to which he added his guitar were 'Vampire Blues' and the title track from Neil Young's magnificent new _On the Beach_ album. The setting was the extremely upmarket restaurant Quaglino's, which had a small stage on which the musicians could play until the sun came up.\n\nOn 16 October 1974 Page recorded a tune he had written entitled 'Scarlet', dedicated to his now two-year-old daughter. Joining him were Keith Richards on guitar and lead vocals, bass player Rick Grech, formerly of Family and Blind Faith, and drummer Bruce Rowland, then with Fairport Convention; original Rolling Stones pianist Ian 'Stu' Stewart also played on the tune. 'It sounded very similar in style and mood to _Blonde on Blonde_ tracks,' the Led Zeppelin leader said in an interview with Nick Kent the next year, referencing Bob Dylan's classic double LP; he also defined the song 'a folk ballad with reggae guitars'. The extra-curricular interests of two of the musicians he worked with suggested they were not the healthiest of companions for Page; both Richards and Grech were regular heroin users. And Page was spending more and more time with Richards, linking up with him in Tramp, the high-end nightclub in London's Jermyn Street, and travelling back with him \u2013 notwithstanding Page's romance with Krissie Wood \u2013 to the Wick, Ron Wood's Richmond house, where the Rolling Stone was hiding out from police raids on his Cheyne Walk home. When Mick Taylor departed the Stones, there were even unlikely rumours that Page might be his replacement. (When Keith Richards was busted for heroin in Canada in 1977, again it was rumoured that, if Richards was imprisoned, Page might deputise for him on tour with the Rolling Stones.)\n\nThe time recording 'Scarlet' with Keith Richards, asserted Page in an interview in 1975, was 'great, really good. We stayed up all night and went down to Island Studios, where Keith put some reggae guitars over one section. I just put some solos on it, but it was eight in the morning of the next day before I did that.' Although Page believed that 'Scarlet' might be put out as the B-side of a Rolling Stones single, it never happened. Talk of the track did, however, trigger rumours that Page had recorded a solo album. He resolutely denied it when asked about it the next year by Cameron Crowe for an article he was writing for _Rolling Stone_. 'Chalk that off to Keith Richards's sense of humour,' said Page. 'He took the tapes to Switzerland, and someone found out about them. Keith told people that it was a track from my album. I don't need to do a solo album, and neither does anybody else in the band. The chemistry is such that there's nobody in the background who's so frustrated that he has to bring out his own LPs.' Was this a shot across the bows in the direction of Robert Plant? The singer had recently expressed to Peter Grant his desire to make his own record. But who would play on it? Well, Plant wanted... Jimmy Page, John Paul Jones and John Bonham. Grant explained the redundancy of such thinking to the vocalist.\n\nFor Christmas, Page sent out cards offering 'Thelemic Greetings' from Jimmy, Charlotte and Scarlet. Close friends may have been surprised, given that he was firmly ensconced with Krissie Wood. But then he always was a stickler for the form of convention.\n\n_Physical Graffiti_ was released on 24 February 1975. It debuted at number one in the album charts in both the US and UK, the first time Led Zeppelin had done this. It stayed there for ten weeks, boosting the sales of the five preceding LPs, which all joined _Physical Graffiti_ at various positions in the US charts.\n\nWe all had a question about the record: where did the clever title come from? 'Graffiti was appearing, and I imagined something, which was like a physical reaction to it. Since we were in a recording studio, if you're doing a recording, and it's going on the tape, even though its magnetic tape, that's like a graffiti in itself. The music was a physical manifestation.'\n\n_Physical Graffiti_ didn't just sound great; it looked great too. There was a suggestion that the sleeve art was like a more successful version of that on _Led Zeppelin III_. Again, there was the repetition of an almost Advent calendar theme: through the die-cut windows of a pair of tenement buildings were Led Zeppelin members in drag, Peter Grant, the Virgin Mary, Lee Harvey Oswald, King Kong, Neil Armstrong, Judy Garland and the cast of _The Wizard of Oz_ , Queen Elizabeth II, Laurel and Hardy, and body-building champion Charles Atlas.\n\nThe five-storey buildings on the sleeve had been scouted by designer Peter Corriston at 96 and 98 St Mark's Place in New York's East Village, a rundown location that would soon become extremely hip. The front cover was a daytime shot, while the rear was taken at night. On the inner sleeve, designed by Mike Doud, the song titles were written on closed window shades.\n\nWhen the record came out Page and Plant were on holiday in the Caribbean, on the island of Dominica, where the local dreads had provided them with ganja and a hallucinogenic jelly fruit. _Physical Graffiti_ had been conceived in a heroin haze, the guitarist seeking ways in which to access further portals to his creativity; the day-to-day scenes Jeff Dexter had witnessed at the Equinox bookstore were only a reflection of the existence of its owner. Concerned about his increasing partiality to smack, Page knew he would be unlikely to be able to obtain heroin on Dominica, which was part of its appeal as a holiday destination.\n\nThe pair were taking a break in the middle of Led Zeppelin's tenth US tour, on which the band employed a slick stage presentation involving powerful lasers and lighting for the first time. Tucked up in the Plaza Hotel in Manhattan for the nights Zeppelin played Madison Square Garden and Nassau Coliseum, Page complained that his suite was pretentious. 'Something comparable to the Versailles Palace,' he said to Richard Cole.\n\nThe roots 'n' culture existence he shared with Plant and the local Dominican dreads was a world he found far more palatable. Page would play _Burnin'_ , the second Wailers album on Island Records, in his suite throughout the tour.\n\nThe tour restarted in Houston, Texas, three days after _Physical Graffiti_ 's release.\n\nAt their second night at Long Beach, Plant \u2013 also clearly influenced by their Caribbean sojourn \u2013 requested that the soundtrack from the classic Jamaican movie _The Harder They Come_ be played over the sound system before they went onstage; he personally transported a copy of the record down to the venue, but bad traffic ensured that the band arrived so late that the idea was nixed.\n\nIn an endeavour to further broaden Led Zeppelin's market, a public-relations company had been taken on. For the first time ever, Led Zeppelin were on the cover of _Rolling Stone_. Sixteen-year-old reporter Cameron Crowe accompanied the band on the _Starship_ to write a lengthy, perceptive article about the band. Plant, now seen very much as the face of Led Zeppelin, was intensely cooperative; Page less so, although he was reasonably forthcoming.\n\nOne of his replies was fascinating. When Crowe asked him, 'How much do you believe in yourself?', Page gave a lengthy response: 'I may not believe in myself, but I believe in what I'm doing. I know where I'm going musically. I can see my pattern and I'm going much slower than I thought I'd be going. I can tell how far I ought to be going. I know how to get there, all I've got to do is keep playing... I'm not a guitarist as far as a technician goes, I just pick it up and play it. Technique doesn't come into it. I deal in emotions. It's the harmonic side that's important. That's the side I expected to be much further along on than I am now. That just means to say that I've got to keep at it.'\n\nDanny Goldberg, Led Zeppelin's young public-relations executive, had a brainwave that would potentially give a boost to the band's intellectual credibility. He would link Page and William Burroughs, the legendary Beat writer. Burroughs would interview him for _Crawdaddy_ magazine, a New York-based publication that had been described by none other than _Rolling Stone_ as 'the first serious publication devoted to rock 'n' roll news and criticism'. Burroughs attended a Zeppelin show at Madison Square Garden, then the pair retired to the author's downtown loft on Franklin Street, where they shared 'two fingers of whiskey'. Burroughs wanted a conversation rather than a more formal question-and-answer session. Both the musician and the writer shared a fondness for heroin, though this was not mentioned in the article, titled 'Led Zeppelin, Jimmy Page and Rock Magic', that Burroughs penned for the June 1975 edition of _Crawdaddy_.\n\n'We started talking over a cup of tea,' wrote Burroughs, 'and found we have friends in common: the real-estate agent who negotiated Jimmy Page's purchase of the Aleister Crowley house on Loch Ness; John Michel, the flying-saucer and pyramid expert; Donald Cammell, who worked on _Performance_ ; Kenneth Anger, and the Jaggers, Mick and Chris. The subject of magic came up in connection with Aleister Crowley and Kenneth Anger's film _Lucifer Rising_ , for which Jimmy Page did the soundtrack.\n\n'Since the word \"magic\" tends to cause confused thinking, I would like to say exactly what I mean by \"magic\" and the magical interpretation of so-called reality. The underlying assumption of magic is the assertion of \"will\" as the primary moving force in this universe \u2013 the deep conviction that nothing happens unless somebody or some being wills it to happen. To me this has always seemed self-evident. A chair does not move unless someone moves it. Neither does your physical body, which is composed of much the same materials, move unless you will it to move. Walking across the room is a magical operation. From the viewpoint of magic, no death, no illness, no misfortune, accident, war or riot is accidental. There are no accidents in the world of magic. And will is another word for animate energy. Rock stars are juggling fissionable material that could blow up at any time... I found Jimmy Page equally aware of the risks involved in handling the fissionable material of the mass unconscious.'\n\nAmong other things, Page mentioned to Burroughs that on their early US tours, the group would ensure they were never in the country for more than six months, as they would have been eligible for the draft for the Vietnam War.\n\nMuch of the interview was less substantial than one might have hoped: interesting yet hardly revelatory. 'We talked about magic and Aleister Crowley,' wrote Burroughs. 'Jimmy said that Crowley has been maligned as a black magician, whereas magic is neither white nor black, good nor bad \u2013 it is simply alive with what it is: the real thing, what people really feel and want and are. I pointed out that this \"either\/or\" straitjacket had been imposed by Christianity when all magic became black magic; that scientists took over from the Church, and Western man has been stifled in a non-magical universe known as \"the way things are\". Rock music can be seen as one attempt to break out of this dead soulless universe and reassert the universe of magic.\n\n'Jimmy told me that Aleister Crowley's house has very good vibes for anyone who is relaxed and receptive.'\n\nBurroughs asked Page his opinion about the 'monster' allegedly lurking in the depths of Loch Ness (Boleskine House sits on the shore of the loch). Page believed that something along those lines did exist. 'I wondered if it could find enough to eat, and thought this unlikely \u2013 it's not the improbability but the upkeep on monsters that worries me. Did Aleister Crowley have opinions on the subject? He apparently had not expressed himself.'\n\n_Circus_ magazine had published a one-shot paperback entitled _Robert Plant_ , and when Led Zeppelin's tour reached San Diego, Page had words with _Circus_ writer Steven Gaines: not to demand a similar tome about himself, but to ensure that there would not be a follow-up book named _Jimmy Page_.\n\nOn that 1975 US tour Jimmy Page wore a black silk stage suit, gorgeously embroidered with Chinese dragons, crescent moons and stars. 'Page's look was cosmic Nudie suit, Zen-style rockabilly,' wrote Stephen Davis, on tour with the group.\n\nThe night of the San Diego show, Davis sat down with Page in the bar of the Hyatt House and threw a handful of questions the guitarist's way.\n\n'What about the magic thing?' Davis bravely asked.\n\n'Magic is a system of will, and of strength,' came Page's reply, before he effectively sidestepped the question. 'That's what interests me about magic. I can't produce magic, real magic, so what we offer is the illusion of magic \u2013 mechanical devices that perform illusions while we play music. And in my own mind, the difference between the illusion and the reality, of the lasers and the theremin and all that, is... hazy. What's a laser beam? Magic, isn't it?'\n\nDavis noted how exhausted Page looked. 'How long are you going to do this?' he asked the musician.\n\nAs Page got up to leave, he replied: 'Nothing lasts forever. I'm going to enjoy it while I can.'\n\nAlthough Page was ostensibly staying at the Hyatt House with Krissie Wood, Lori Mattix had been installed in another suite on a separate floor. And a room had been booked for Bebe Buell, who was expected to arrive imminently. But there were problems with another girl the day before the San Diego show. A woman in her mid-twenties, dressed in a maroon cloak replete with cowl, approached the hotel's front desk in the morning. She was insistent that she needed to speak to Jimmy Page: the guitarist's life was in danger, she said, and she must get a warning to him. Danny Goldberg, who during the course of the tour had been elevated to the position of vice-president of Swan Song, was in his suite, having breakfast with Stephen Davis. He suggested she come up to see him.\n\nOnce in Goldberg's room, the girl declared herself to have arrived as an emissary. 'Someone' \u2013 she wouldn't say who \u2013 had seen some 'bad energy' around Page. And this could manifest the next night in San Diego. Refusing to reveal anything further \u2013 she would only tell all that she knew to Page personally \u2013 she was finally persuaded to pen a note to Page, which she did in a left-handed scrawl, placing her missive in an envelope and sealing it with her tongue. When she left the room, after one more attempt to see Page, she was angry and slammed the door.\n\nGoldberg then took the unopened envelope and burned it, spreading the ashes before an image of Lord Krishna. He informed Peter Grant of the threat to Page's life, and a further layer of security was added to the San Diego show.\n\nSix months later, when President Gerald Ford visited Sacramento, the California state capital, one 'Squeaky' Fromme, a member of Charles Manson's Family, pointed a gun at the President. A Secret Service agent grabbed the weapon before she could squeeze the trigger. Both Goldberg and Davis felt she looked exactly like the girl who had come up to Goldberg's suite. To this day, Davis believes that the girl so anxious to deliver a 'warning' to Page was Squeaky Fromme.\n\n## 18\n\n# AN ACCIDENT IN EXILE\n\nLed Zeppelin returned to Britain on 10 May 1975, and went straight into rehearsals at Shepperton Studios. Significantly they were about to use up 16 of the 60 days that they would be allowed to spend in the UK during the financial year that had commenced at the beginning of April 1975. For the four members, along with Peter Grant, had decided to be non-resident in their native land in order to avoid paying what they considered punitive tax during a golden financial period for Zeppelin. Their last US tour had earned them $40 million \u2013 multiply that by five for a contemporary equivalent figure. And those dates were only the precursors to a set of late-summer stadium shows that would commence in August: dates that would make their take from the late-winter tour they had just completed seem like toy-town money. What they were about to embark on was known as 'tax exile'. Although in the case of Jimmy Page, it could be seen in retrospect as a dangerous exile from his own soul, one that \u2013 perhaps more than anything \u2013 was fuelled by his dependence on heroin.\n\nBut that was to come. For now this was their triumphant homecoming: on 17 May they played the first of five nights at London's Earls Court Exhibition Centre, a cavernous concrete barn that was the largest venue in the capital, holding some 17,000 people. Their American show, involving a vast PA, was airfreighted to Earls Court, along with the equipment to fire their laser beams \u2013 which, in the end, would prove somewhat underwhelming. An enormous video screen hovered over the band at the rear of the stage, the first to be used in such a manner in the UK. Production costs were so high that Page later remarked, 'We were so determined to do the same sort of show and more than what we'd been doing in America that in the end we came out of it with just a few hundred pounds over the five days, but it didn't matter because the vibe was so electrifying.'\n\nAn acoustic set was built into the show, almost acting the part of an intermission, and during each concert Robert Plant delivered ill-considered criticism of Denis Healey, the Chancellor of the Exchequer, whose punitive taxation measures had been the cause of Led Zeppelin's tax exile. On Sunday 25 May, Plant uttered his final jibe in this vein: 'Thank you, Great Britain, for five glorious days. Thanks for being a great audience, and if you see Denis Healey... tell him we're gone!' As barely a single member of the band's audience would have suffered similar financial angst, it was hard to feel sympathy for a millionaire rock star bellyaching about the amount of tax he was obliged to pay. Plant's remarks were later held up as an example of dinosaur rock-star thinking, and a justification for the imminent emergence of punk rock, the genesis of which was continuing to develop at Malcolm McLaren and Vivienne Westwood's fashion boutique, just along the street from Swan Song's London HQ: six months later the Sex Pistols would play their very first gig.\n\nOn that final Sunday night, an after-show party was thrown backstage at Earls Court, in a sizeable downstairs section. The adored R&B act Dr Feelgood performed, specifically for the edification of Ahmet Ertegun, with whom the Feelgoods hoped to sign a US deal \u2013 they eventually went to CBS. The party was a suitably pricey affair, with excellent food and champagne on tap. Plant, John Bonham, John Paul Jones and Peter Grant table-hopped, the quintessence of conviviality. But there was no sign of Page. Finally, at around 4 a.m., the guitarist could be seen gingerly manoeuvring the stairs that led into the party area, accompanied by a slinky Krissie Wood. An etiolated figure, seemingly floating on air, Page hovered down to the event, as though he had little idea of where he was. He seemed utterly out of it.\n\nLed Zeppelin would not perform again in their home country until their Knebworth shows in 1979.\n\nThe next day, 26 May, Robert Plant and his family left for Morocco, heading straight to Agadir, the coastal resort in the south of the country. Meanwhile, Jimmy Page flew to New York to work on the material recorded on their 1973 tour, readying it for the soundtrack album that would accompany their much-postponed movie.\n\nThree weeks later the two Led Zeppelin frontmen linked up again, in Marrakech, to attend the city's folk festival. Page was accompanied by Charlotte Martin and Scarlet, who was now four years old. After the festival, they hired a Range Rover and drove a long way south, heading for the town of Taifa, on the edge of the Sahara desert. But the dismal road conditions defeated them before they got there. Moreover, while they were there the Western Sahara War was breaking out over the territorially disputed Spanish Sahara.\n\nPlant had a top-of-the-range recording device with him, and he was intrigued by the possibility of recording rhythms employed by local tribesmen, notably Berbers. But the political environment and transport conditions stymied his ambitions.\n\nAfter four weeks, they turned the Range Rover around and headed north, catching a boat in Tangier that took them to Gibraltar, from where they drove up through Spain and France to Switzerland. Peter Grant had chosen Montreux, the home of convenient numbered bank accounts, as his tax-exile domicile. It was also the site of Claude Nobs's legendary annual Montreux Jazz Festival. With the rest of the band also in town, they had a meeting about their imminent US tour, due to start at Oakland Coliseum on 23 August; it was agreed they would meet in Paris on 10 August for rehearsals. The four musicians hung out at the jazz festival, which was in full flow.\n\nPlant, however, had developed a taste for sun and sand in Morocco \u2013 and he wanted more of it. The Greek island of Rhodes had been recommended to him as a convivial destination by Phil May, singer with the Pretty Things. May had rented a house in Rhodes from Pink Floyd's Roger Waters, and Plant arranged to link up with him there. The Led Zeppelin singer decided to drive there from Montreux with his family; Page followed behind in another vehicle, accompanied by not only Charlotte and Scarlet, but also Shirley \u2013 Maureen Plant's sister (she of 'What Is and Should Never Be') \u2013 and her husband.\n\nOn 3 August Page flew off from Rhodes to Sicily; he was keen to see Aleister Crowley's Abbey of Thelema \u2013 little more than a farmhouse, really. It was Kenneth Anger who, having spent that summer cleaning up Crowley's desecrated wall art at the house, had recommended the property to him. The guitarist was considering adding the abbey to his already impressive property portfolio.\n\nOn 4 August, beneath the cloudless blue sky of another perfect day on Rhodes, Maureen Plant was driving her husband, their two children Karac and Carmen, as well as Scarlet Page, back from the beach in a rented Austin Mini when she misjudged a bend. The Mini shot off the road and ploughed into a tree, the impact smashing Robert Plant's right ankle and elbow as well as bones in his right leg. As the children in the back of the car screamed in terror, Plant looked over at Maureen: she was bleeding, with a gash in her head. The singer believed his wife to be dead; in fact, she was unconscious, with a broken pelvis and leg, and a fractured skull. Their son Karac had a broken leg, and their daughter Carmen a broken wrist. Only Scarlet, seated between the two Plant children, escaped relatively unscathed, with only a few cuts and bruises. Travelling behind them in another Mini were Charlotte Martin, with Maureen's sister Shirley and her husband. Although they endeavoured to summon help, it took several hours before a fruit truck with a flatbed was commandeered, on which the wounded were transported to the nearest hospital.\n\nMaureen Plant had lost plenty of blood, and hers was a rare blood type. Thank goodness her sister Shirley was with them \u2013 she shared the same type \u2013 but even so, Maureen needed even more fresh blood than Shirley could give her.\n\nIn great distress, that evening Charlotte called Richard Cole in London. Cole immediately went into action. Through the Greek Embassy in London he contacted Dr John Baretta, a Harley Street physician who spoke Greek, and a celebrated orthopaedic surgeon, Dr Mike Lawrence, was also contacted. Both men agreed to fly immediately to Rhodes, but there was a further absurd frustration: Peter Grant was on holiday and uncontactable. And without his say-so, the Swan Song accountants refused to access the funds for a private plane to bring Plant and his family back. Thankfully, Baretta had a vital contact: he was the personal doctor for Sir Robert McAlpine, who owned not only a significant UK construction firm, but also several private jets. One of these, wrote Cole, 'could be turned into a flying ambulance, equipped with special supports for stretchers'.\n\nWhen the plane took off for a night flight to Rhodes, the aircraft's refrigerator contained eight pints of Maureen's blood type. Arriving at the hospital in Rhodes at 6 a.m., the team were shocked by the sight of cockroaches scurrying about. And there was a further potential problem. 'The police are investigating the accident to see if alcohol or drugs were involved,' the hospital administrator informed the English arrivals. 'Your friends can't leave the country until the police have decided whether they're going to press charges against someone.'\n\nHeeding the advice of Baretta, Cole made the decision to immediately fly the wounded Plant family back to the UK. He rented an ambulance and a pair of station wagons and parked them by a side entrance to the hospital. At 2 in the morning Robert and Maureen Plant \u2013 her IV bottle still inserted into her arm and held aloft by Cole \u2013 along with Karac and Carmen were whisked away to the local airport.\n\nAfter refuelling in Rome, the plane was approaching Heathrow Airport at around 11.30 p.m., when all of a sudden it began circling: orders had come in from Peter Grant not to land until after midnight \u2013 to ensure the singer did not use up another precious day of his tax-exile status.\n\nThe family were transported to Guy's Hospital by London Bridge, where Maureen had an operation and Plant's leg was reset. But he was given a chilling warning by his surgeon: 'You probably won't walk again for six months, maybe more. And there's no guarantee that you'll ever recover completely.'\n\nThe US tour booked for late August and September was put on hold, as were planned dates in Europe and the Far East. Everyone involved was aware that Plant's car accident could herald the end of Led Zeppelin.\n\nBebe Buell spent some of that summer hanging out with Mick Jagger at Montauk in East Hampton, at the end of Long Island. Back in New York City, she one day found herself speaking on the phone to Jimmy Page, who was in London: 'At the end of what seemed like a brief, lighthearted conversation, he told me that he would send me a sign that night at midnight.'\n\nBuell then drove up to Woodstock, where she spent the evening with her girlfriend Jeanne Theis. 'We were sitting in the small dining room when, on the stroke of midnight \u2013 I swear \u2013 we heard a crash upstairs. It could only have come from the bathroom. Rushing up there, we discovered that the large antique wood-framed oval mirror that had been carefully hung above the sink had been hurled a good ten feet across the room and lay shattered at the edge of the sunken bathtub.' The two girls immediately left the house. And Buell had the property exorcised.\n\nAfter his operation at Guy's Hospital, with his badly broken leg and ankle reset, Robert Plant was prepared for a lengthy period of convalescence. However, Peter Grant was quick to point out that, if he stayed longer in the UK, his tax-exile status would be in jeopardy, costing him literally millions of pounds.\n\nOnce again thanks to a contact of Richard Cole, Plant was flown to the tax-exile island of Jersey, in the English Channel Islands. Page was extremely keen for some effort at work to begin as rapidly as possible. 'The longer we wait,' he told Cole, 'the harder it's gonna be to come back.' 'This could be the end of Led Zeppelin,' said Grant. For his part, Plant drank plenty of beer and regularly played the piano. Confined to a wheelchair, the singer now had a plaster cast extending from the top of his right hip to his toes, and was in considerable physical and mental discomfort; he was also extremely depressed, fearful he might never walk again.\n\nIt was decided that, in lieu of their planned dates and any subsequent touring, Led Zeppelin would take their fans to the movies to watch the band onstage. _The Song Remains the Same_ would be rush-released, and Led Zeppelin would begin working on a new album as soon as Plant could participate. For now, Page felt they were like 'technological gypsies' in search of a home.\n\nWhere would that home be? The answer was relatively straightforward: their favourite American city, Los Angeles.\n\nA pair of houses on the Pacific Ocean beach were rented at the Malibu Colony, one for Page and Plant, and one for Grant. A gated community, the Colony was beloved of A-list movie stars and musicians: Jascha Heifetz, the classical violinist, Robbie Robertson of the Band and Neil Diamond were currently in residence. A gated community with a mile-long beach, part of its appeal was that it was located some 20 miles up the Pacific Coast Highway from the temptations of Hollywood.\n\nBut who knew what went on behind closed doors at the Malibu Colony. Certainly some of the stellar neighbours might have been shocked by what transpired at the Page abode. 'We called the house in Malibu \"Henry Hall\",' Benji Le Fevre, personal assistant to Robert Plant, told Richard Cole; 'Henry' was a euphemism for heroin. There were also large quantities of that Zeppelin staple cocaine, which Plant avidly ingested, even when advised that it might hinder the time it took to recover from his injuries. 'I was the go-between when they were out there in Malibu,' remembered Michael Des Barres. 'The narcotic indulgences were so intense that they really separated these four wonderful beings.'\n\nAll the same, Page and Plant did begin to write together, notably on a pair of epics that would open and close the new record that was being developed. Partially written in Morocco that summer, the ironically titled 'Achilles Last Stand' \u2013 the Greek hero had suffered a critical leg injury of his own \u2013 and 'Tea for One', a fast tune during the LA rehearsals that had significantly slowed down by the time it was recorded.\n\nNotwithstanding his drug interests, Page was resolutely in charge of the new LP, every bit as much as he had been on the very first Led Zeppelin album. _Presence_ , as it would come to be titled, was his record, almost a solo album, as he was desperately aware that he needed to save his group. Pay heed to the tonal textures he magics on 'Tea for One', layer after layer, a development from Led Zeppelin's not dissimilar 'Since I've Been Loving You'. By the time John Paul Jones and John Bonham arrived in LA, the songs and arrangements were almost set to go.\n\nAnd the technological developments employed on the album had risen several levels. As Erik Davis writes of Page: 'By the mid-1970s, he was using digital delays, guitar synthesizers, and a live set-up that included wah-wah, MXR effects, and what he admitted was \"total flash\": harmoniser, theremin, Echoplex, and the famous violin bow. By pushing the envelope on sound, these gadgets extended the virtuosity associated with the guitar hero into the domain of techno-acoustic experimentation. But these tools also gave Page a way to create the dramatic atmospheres so important to his sense of \"light and shade\". In particular, Page used electronics to explore what music buffs call timbre; the textural quality of a tone, its sheen, or grain, or colour. Page's timbral flavours define his guitar playing as much as his licks or his blend of acoustic and electric styles.'\n\nBut all Robert Plant had was a piece of wood. Employing a walking cane, the singer began to take his first few hesitant steps, finally mustering the nerve to walk up and down the thick sand of the Colony beach every day, gradually strengthening his shattered right leg. To an extent he was also recovering psychologically, heartened by the strength of the new songs and anxious to take them into a recording studio.\n\nThe temptations of the Sunset Strip appeared to have lost their appeal for the members of Led Zeppelin. When Cameron Crowe had interviewed the group at length for _Rolling Stone_ earlier in the year, Plant had recalled first coming to Los Angeles at the end of 1968: 'Nineteen years old and never been kissed. I remember it well. It's been a long time. Nowadays we're more into staying in our rooms and reading Nietzsche. There was good fun to be had, you know, it's just that in those days there were more people to have good fun with than there are now. The States were much more fun. LA was LA. It's not LA now. LA infested with jaded 12-year-olds is not the LA that I really dug... it's a shame to see these young chicks bungle their lives away in a flurry and rush to compete with what was in the old days the good-time relationships we had with the GTOs and people like that. When it came to looning, they could give us as much of a looning as we could give them.'\n\nWhen John Bonham arrived for rehearsals he clearly did not feel that times had shifted, moving into a room at the Hyatt House. There, he moped about being away from his family, and stayed drunk \u2013 except when he was snorting smack.\n\nRousing himself one night to go to the Rainbow Bar and Grill, the drummer pulled up a stool at the bar and ordered 20 black Russians, a powerful cocktail of vodka and Kahl\u00faa. Downing ten of them immediately, he turned around and his eyes met those of Michelle Myer, who worked for Kim Fowley. When she smiled at him \u2013 clearly oblivious of his reputation \u2013 Bonzo went berserk, staggering over to Myer and punching her full in the face. 'Don't ever look at me that way again,' he barked, returning to the bar and knocking back his remaining ten black Russians. When, on another occasion, he took on the Rainbow's bouncer, he was surprised to discover that the slight fellow was a martial-arts expert who put the drummer on his back in moments.\n\nJohn Paul Jones was living apart from the other three, sometimes impossible to contact when it came to rehearsing the material at SIR Studios, way down on Sunset Boulevard in Hollywood. 'If you see John Paul Jones, shoot him on sight,' Page half-joked to Danny Goldberg.\n\nPage couldn't find John Paul Jones? It was more like the other way round, on account of Page's unorthodox sleeping habits and time-keeping. Every night, Jones said, he would arrive at SIR, along with Plant and Bonham, to wait and wait and wait. 'I learned all about baseball during that period, as the World Series was on and there was not much else to do but watch it.'\n\nWhen all four musicians did manage to link up, wrote Stephen Davis in _Hammer of the Gods_ , 'the new music was hot. The effort and energy that would have gone into the fall tour went into the new music instead, and Led Zeppelin burned with a rubbery new funk that was taking the band to unexpected destinations. Moroccan white guitar noise dashed with New Orleans \"second-line\" rhythms. Time signatures were brutal and labyrinthine. Bonzo was as tough as nails, and Robert was singing in his wheelchair, wiggling inside his cast.'\n\nAs is so often the case in southern California, the weather was postcard perfect. Until a storm ravaged the coastline, almost taking away this new temporary home. Always alert to signs and portents, Page interpreted this as an invitation to move elsewhere. Besides, if they remained in the US they would shortly be liable to pay income tax.\n\nAfter a meeting with Peter Grant, it was decided that they would relocate to Munich in Germany: specifically, to Musicland Studios, which Thin Lizzy had just vacated, having recorded their career-changing _Jailbreak_ album there, and which the Rolling Stones were about to enter, to make _Black and Blue_. There was only a short window of 18 days in which to record the new album, and Page appreciated the sense of urgency this would impose on the project.\n\nThe musicians stopped off in New York on the way to Germany. From New York, Page and Danny Goldberg flew down to Washington, DC on a Swan Song A&R mission, having been tipped off about a sensational blues guitarist called Bobby Parker. When they found Parker he was playing with an undistinguished group on a nearby army base. Page was called up to the stage to play \u2013 badly \u2013 on some blues tunes with Parker. Having recorded the set on a cassette player, Page played the inadequate tape to Plant and Bonham. They were both singularly unimpressed. Bobby Parker was not destined to become a Swan Song act.\n\nThe band travelled separately to Munich, and Plant, Page and Jones were no doubt pleased that they did not travel with Bonham. Uncontrollably drunk on free champagne on the flight, Bonzo fell asleep and woke to discover he had pissed himself. His first-class seat was soaked with urine, which soon gave off a distinctly unsavoury aroma. Summoning Mick Hinton, his personal roadie, from his seat in economy, Bonzo changed into a pair of trousers that his assistant always had with him for such an emergency. Then he made Hinton sit in his dripping first-class accommodation, while he moved back to the roadie's seat in economy.\n\nMusicland, which had a strong reputation as one of the best studios in Europe, was situated in the basement of Munich's Arabella Hotel. What especially pleased Page was that it was located in an uninspiring neighbourhood; minimum distractions were required to make that 18-day deadline. There was, however, one principal diversion: heroin, which Page and Bonham were using during the day at the studio, courtesy of Richard Cole, who had located a dealer close to Musicland. Cole was also using smack. 'None of us seemed the worse for it,' considered the road manager.\n\nFor Robert Plant the sessions were not easy. From his wheelchair, he felt his vocals were considerably lacking in power; he worried that his voice sounded fatigued. In fact, Plant's hypnotic, shrill singing acted as a fabulous counterpoint to the balanced rhythmic fury of the record. On one occasion he tripped in the studio, falling on his damaged right leg, and the crew were both astonished and impressed with the way Page raced across the room to pick up the singer, who was taken to hospital for tests. All the same, Plant felt he had been hustled into making this record by Page and Grant; it was, he believed, keeping him from being with his wife and kids. Maureen was only slowly recovering from her serious injuries, and Plant was beginning to have serious doubts about the point of remaining in Led Zeppelin. 'I was furious with Page and Peter Grant. I was just furious that I couldn't get back to the woman and the children that I loved. And I was thinking, is all this rock 'n' roll worth anything at all?'\n\nJimmy Page, for his part, was feeling confident, as though his guitar playing had moved up several levels. As always, he was resolutely in charge of the sessions. At first he would put in 12-hour working days, but by the final week, it was more like 18 hours. And the pressure had its desired effect: both 'Candy Store Rock', a rockabilly tune on which Plant sounded more like his idol Ral Donner than ever, and 'Hots On for Nowhere', were both written in the studio in around an hour.\n\nThe thundering _Presence_ \u2013 almost a forerunner of the punk sounds that would be omnipresent by the end of the year \u2013 emphasised the creative relationship in Musicland between Page and Bonham, the drummer's power driving the album from its free-standing epic opener 'Achilles Last Stand', one of the greatest of all Zeppelin tracks, via the snorting, harmonica-enhanced 'Nobody's Fault but Mine' \u2013 like _Physical Graffiti_ 's 'In My Time of Dying', adapted from another Blind Willie Johnson masterpiece \u2013 to its suitably mysterious closing tune, 'Tea for One', with its extraordinary sense of loss and loneliness.\n\nRunning out of time to overdub and mix the record, Page called Mick Jagger and explained his predicament. Graciously, the Rolling Stone offered him three days of their time in Musicland.\n\nWorking 24-hour days, dropping asleep at the mixing desk for a couple of hours at a time, Page completed the record in this additional window of time, and every single searing guitar overdub was added in just one inspired day. 'I'll tell you about doing all the guitar overdubs to \"Achilles Last Stand\",' Page told Dave Schulps of _Trouser Press_ in 1977. 'There were basically two sections to the song when we rehearsed it. I know John Paul Jones didn't think I could succeed in what I was attempting to do. He said I couldn't do a scale over a certain section, that it just wouldn't work. But it did. What I planned to try and get that epic quality into it, so it wouldn't just sound like two sections repeated, was to give the piece a totally new identity by orchestrating the guitars, which is something I've been into for quite some time. I knew it had to be jolly good, because the number was so long it just couldn't afford to be half-baked. It was all down to me how to do this. I had a lot of it mapped out in my mind, anyway, but to make a long story short, I did all the overdubs in one night... I thought as far as I can value tying up that kind of emotion as a package and trying to convey it through two speakers, it was fairly successful.'\n\nRichard Cole noted how calm Page was at the end of the final session. 'With some of the earlier albums,' he wrote, 'he would leave the studio feeling pangs of insecurity, convinced that there might have been something else he could have done to make the tracks even better. With _Presence_ , however, he seemed perfectly content. Through a frail smile, he told me that not an ounce of energy had gone to waste.' And not an ounce of heroin, either.\n\nLed Zeppelin were still clearing their equipment out of the studio on 2 December 1975 when the Rolling Stones arrived. Page again thanked Mick Jagger for handing him the extra days. When Jagger asked if they had got some songs recorded, the Zeppelin leader informed him that they had completed their album.\n\nThe leader of the Rolling Stones, notoriously tardy in the studio, was amazed: 'You've only been here three weeks.'\n\n'That's all we needed,' replied Page, once again getting one over on Jagger.\n\nCritically undervalued at the time, the riff-heavy _Presence_ \u2013 the first album by the band to feature almost no keyboards \u2013 was one of Led Zeppelin's greatest records. 'I think it was just a reflection of the total anxiety and emotion of that period,' said Page. 'There's a hell of a lot of spontaneity about that album. We went in with virtually nothing and everything just came pouring out.'\n\n'The whole testament of the Munich album,' said the guitarist, 'is that it proved once and for all that there is no reason for the group to split up. I can't think of too many groups that have been around as long as we have and still retain that spontaneity. We started screaming in rehearsals and never stopped.'\n\nBut the fuel to make _Presence_ would exact a considerable cost: both Page and Bonham, not to mention Cole, were now addicted to heroin.\n\nIn Los Angeles that December on business relating to _The Song Remains the Same_ , Peter Grant questioned Cole about Page, who was also in town. 'Something's different about Jimmy. He acts nervous and jumpy. Something's not right,' said the manager. Page was suffering from heroin-withdrawal symptoms: a runny nose and flu-like aching limbs.\n\nShortly after, Cole and Page flew back to London together, as the guitarist was anxious to attend Scarlet's Christmas school play. During the journey he turned to Cole: 'Chrissakes, Richard, don't get into this shit.'\n\nAlthough he well knew what his boss was referring to, Cole asked him what he meant.\n\n'Heroin. I think I'm hooked. It's terrible.'\n\nCole asked him if he'd tried to stop.\n\n'I've tried, but I can't. It's a real bastard.'\n\nWith the recording of the album \u2013 though not the mixing \u2013 having been completed on Thanksgiving Day, Page's first suggestion for the record's title was 'Thanksgiving'.\n\nWhen discussions with Hipgnosis over the sleeve began, designer George Hardie declared: 'When I think of the group, I always think of power and force. There's a definite presence there.' Hardie had a notion for an iconic object being displayed on the sleeve; he thought the record should be titled 'Obelisk'. But Page responded to the energy of Hardie's words and felt that 'Presence' was the true title for their new music. Besides, the ambiguities of the word appealed to the musician: Presence meaning \u2013 as the designer suggested \u2013 charismatic power and force; Presence as 'presents'; Presence as a spectral appearance.\n\nThe sleeve was extremely unexpected: an archetypal nuclear family \u2013 the husband bearing a resemblance to the actor Jack Nicholson \u2013 seated around a table on which was placed a black, obelisk-shaped object. It managed to look both creepy and sinister \u2013 as you would expect with Led Zeppelin.\n\nBack in Jersey after Munich, where he had not had a single night out, Robert Plant began to feel relatively energised. Behan's West Park, a nightspot in St Helier, became a regular haunt. On 10 December, along with resident pianist Norman Hale, a former member of the Tornadoes, all of Led Zeppelin played an impromptu rock 'n' roll set; Plant was tucked away on a high stool, almost behind John Bonham.\n\nLater, Page was suitably enigmatic in his explanation of the album sleeve to _Melody Maker_ : 'It could be either viewed as past or present. If you look at it, it could be the forties and it could be the seventies. It's got to be viewed in its entirety, otherwise the whole point would be lost. I'm sorry to be elusive on it, but I don't think I should say it's this, that and the other, because it's an ambiguous thing. Photographically it's an ambitious statement, so it's not the right thing to lay down an impression because somebody might have a more illuminating one.'\n\nUnder the terms of their tax exile, Led Zeppelin and Peter Grant were permitted to be in the UK for 60 days of that tax year. Returning to their families in England for Christmas had always been factored into their time out of the country, and Plant, Jones and Bonham went back to their relative forms of domestic bliss. Page, who had already been to Los Angeles since the Musicland sessions, flew to New York to do more work on the soundtrack for _The Song Remains the Same_.\n\nIn March 1976 _Presence_ was finally released. It was the fastest-selling Led Zeppelin LP ever. In the US it shipped to the stores as a platinum album, with over a million copies pre-sold. In both the US and the UK _Presence_ was number one within two weeks. However, without a supporting tour, sales fell off rapidly, and ultimately it was the weakest selling of all Zeppelin records, with some 3.5 million sales.\n\n## 19\n\n# THE KENNETH ANGER CURSE\n\nIn 1973 Peter Grant approached Paul Reeves \u2013 the fashion designer and taste-maker who became firm friends with Jimmy Page after meeting him on the Yardbirds' last US tour \u2013 and asked him to manage the renovation and refurbishment of a mews house he had purchased in London's West End. Money, Grant said, was no object. 'I told him I'd only do it if he didn't come near,' said Reeves.\n\nInvolving his pal Jon Wealleans, an architect and artist, as well as friends from the Royal College of Art, Reeves spent almost two years on the project. As Paul Gorman writes in _The Look_ , 'Reeves sourced everything from curtain material to cutlery, discovering a talent for interior design and love of beautifully made British furniture and textiles along the way.'\n\nAware of the frightening reputation of the gargantuan Grant, the pair were extremely nervous when the Zeppelin manager finally arrived in 1975 to view his freshly designed property. As Gorman writes: '\"I opened the door and it may be a clich\u00e9, but he literally blotted out the sun,\" laughs Wealleans. Reeves, meanwhile, had prudently put some champagne on ice. \"He spent around five minutes looking around, not saying a word,\" says Reeves. \"Then he pronounced, 'I gotta say Paul... it's fucking amazing!' We got the champagne out and a couple of grams of coke and everything was all right!\"'\n\nNo doubt as part of Led Zeppelin's desire to expand their market, appeal and creative credibility, the Paul Reeves and Jon Wealleans transformation of Grant's house was featured in both the _Observer_ and _Ideal Home_ magazines.\n\nAt the peak of both Led Zeppelin's and his own success that year, earning enormous acclaim and wealth, Peter Grant \u2013 who the next year would be approached by Colonel Tom Parker about a European tour by Elvis Presley \u2013 did not appreciate that he too was about to experience a devastating personal crisis. While he was living as a tax exile in a rented house on Long Island, his wife Gloria, a petite former dancer, remained in the UK with their two children at Horselunges Manor, an Elizabethan manor house. Then Gloria left him for another man.\n\nThe divorce came close to destroying Grant and, although he eventually \u2013 somehow \u2013 won custody of their children, there was still the stench of failure about the settlement. And failure was something he was utterly unaccustomed to. Plunged into depression and angst, and perhaps influenced by two of his clients, he began to dabble with heroin.\n\nTower House, meanwhile, was the site of a mysterious issue for Page. A couple staying at his London home were impersonating the guitarist and Charlotte Martin, and they were summarily expelled from the property. 'That got very ugly,' he said to Nick Kent in the _NME_.\n\nLed Zeppelin spent January 1976 in a freezing New York City, lodged at the Park Lane Hotel on Central Park South. Page was still in the recording studio, editing and adjusting the soundtrack for the perpetually imminent _The Song Remains the Same_.\n\nWhen he went to Los Angeles to do more work on the film, Page appeared to be in an even worse state than he was in the autumn. Having signed Michael Des Barres's new group Detective to Swan Song for $1 million, Jimmy Page found himself transported from his rental home in Malibu to the Beverly Hills Hotel for what was known in the music business as a 'signing' photograph. Unfortunately, he was rather too relaxed from the smack he had ingested prior to leaving Malibu. So relaxed indeed that it was found impossible to wake him when he arrived at the hotel. The resulting picture showed Page, asleep, surrounded by the perkily awake members of Detective. Afterwards, he blamed his parlous condition on a Valium he had taken.\n\nDetective would become a problem, though largely for themselves. 'They gave us a million dollars: what the fuck do you think is going to happen?' said Des Barres. 'I spent a month getting a snare-drum sound. It was a great band but we didn't have any moral support, sitting around waiting for Jimmy Page to produce us. And when Bonzo owns a fifth of you, there's a problem.'\n\nAs a fellow heroin addict, was he concerned about what he could see happening to Jimmy Page? 'I was concerned: for myself. When you're a drug addict you don't care about anybody else. When you're a junkie you don't want to be a junkie. I was annoyed because I couldn't get him into the studio. No, I wasn't worried for him.'\n\nConsidering how well acquainted Page was with the works and life of Aleister Crowley, who died a heroin addict, wouldn't the guitarist have felt warned off a relationship with smack?\n\n'It's not that simple,' said Des Barres. 'Jimmy Page's affection for Aleister Crowley was in his ideas of self-liberation. When you get to the hierarchical position that Jimmy found himself in by then, you are at the top of your vision. Jimmy's thoughts were, \"I will experiment with life.\" He emulated something Crowley did, but there are so many more complexities in terms of the narcissism and brilliance of Aleister Crowley.'\n\n_The Song Remains the Same_ was finally released on 20 October 1976. Though the movie had been troubled from the start, the group's enormous cult of fans turned out for the film in their droves. On its initial run it grossed $10 million. So that the music in the film replicated as near as feasible the live sound of the band, Cinema I in Manhattan was equipped at great cost with a quadrophonic sound system for its premiere. Yet, much to Page's chagrin, this was not the case when the film premiered on the West Coast, a screening he and the rest of the band also attended \u2013 as they did the London premiere.\n\nInterviewed by Nick Kent in the _NME_ the next month, Page admitted the film's defects: ' _The Song Remains the Same_ is not a great film, but there's no point in making excuses. It's just a reasonably honest statement of where we were at that particular time. It's very difficult for me to watch it now, but I'd like to see it in a year's time just to see how it stands up.'\n\n_The Song Remains the Same_ is flat in tone, shot with little flair, with the sense of a television programme about it. Had no one advised either of the film's directors that the most efficient and efficacious manner in which to convey onstage movement on camera is to shoot from the sides of the stage? The onstage shots are largely unadventurous, and there are often lighting problems. Even Page's hair seems confused: no longer the flowing Pre-Raphaelite locks of yore, but almost a semi-mullet. He even looks a touch jowly. After the godlike perfection in his image from the 1970 Royal Albert Hall gig, he is, at 32 years old, clearly no longer a kid.\n\nIn fact, in their performance Led Zeppelin seem very much like they are just doing a job, getting through another gig, firing on maybe only three cylinders. Page even looks slightly confused as to who he is, although that is alleviated by the dramatics \u2013 his semi-duck walk, for example \u2013 and the magic of the stage and outfits.\n\nThe Madison Square Garden performance \u2013 cleverly recreated at Shepperton Studios \u2013 is intercut with cameos of each of the group, as well as Peter Grant and Richard Cole. It is rather predictable in its self-mythologising; Robert Plant's sword-and-sorcery horseback ride to a castle, followed by a sword fight in which his opponent drops into the moat, makes one think, Yes, of course Plant would see himself like that.\n\nPage's first appearance has him turning to the camera to reveal pink sunglasses, from which pink light pours. That is all the shot is: Page projecting his sinister mystery, an essential part of his package, and a little creepy. There is the scent of pantomime occultist about it; the use of the red eyes in the film is a complete power trip. But it may also suggest considerable anxiety: are his eyes pinned from smack usage? A man fully at ease with himself might not have pulled such a stunt.\n\nThen there is that sequence \u2013 filmed at his insistence on a dank, full-moon night \u2013 in which he climbs a rocky hill to meet a tarot-like hermit atop the cairn, his face morphing until we appreciate that the hermit is indeed an older Jimmy Page \u2013 an interesting self-perception, and commendably accurate where certain aspects of his later life are concerned.\n\nMuch later, in 2008, Page discussed this sequence with _Guitar World_. He was thrown a perceptive line: 'I find it interesting that you were choosing to represent yourself as a hermit at a time when you were really quite a public figure.'\n\n'Well, I was hermetic,' replied Page. 'I was involved in the hermetic arts, but I wasn't a recluse. Or maybe I was... The image of the hermit that was used for the artwork on _Led Zeppelin IV_ and in the movie actually has its origins in a painting of Christ called _The Light of the World_ by the Pre-Raphaelite artist William Holman Hunt. The imagery was later transferred to the Waite tarot deck. My segment was supposed to be the aspirant going to the beacon of truth, which is represented by the hermit and his journey toward it. What I was trying to say through the transformation was that enlightenment can be achieved at any point in time; it just depends on when you want to access it. In other words, you can always see the truth, but do you recognise it when you see it or do you have to reflect back on it later?'\n\nThe interviewer brought up the matter of the guitarist's occult studies: 'It may have been subtle, but you weren't really hiding it.'\n\n'I was living it,' replied Page. 'That's all there is to it. It was my life \u2013 that fusion of magick and music.'\n\nThe _Guitar World_ interviewer pointed out how advanced was the Led Zeppelin leader's employment of symbolism, such as the sigils on _Led Zeppelin IV_ and the embroidery on his stage clothes, used almost as a form of branding.\n\n'You mean talismanic magick?' Page replied. 'Yes, I knew what I was doing. There's no point in saying much about it, because the more you discuss it, the more eccentric you appear to be. But the fact is \u2013 as far as I was concerned \u2013 it was working, so I used it. But it's really no different than people who wear ribbons around their wrists: it's a talismanic approach to something. Well, let me amend that: it's not exactly the same thing, but it is in the same realm. I'll leave this subject by saying the four musical elements of Led Zeppelin making a fifth is magick into itself. That's the alchemical process.'\n\nTowards the end of _The Song Remains the Same_ a kid is beaten backstage for some unknown sin, a shockingly arrogant moment showing the possibility of darkness within the Led Zeppelin operation. Surely this brief moment had been consciously injected into the film, letting us all know the menace behind the band?\n\nPeter Clifton, who succeeded Joe Massot as director, had been subjected to numerous indignities, including having his house broken into by order of the increasingly paranoid Peter Grant, who was suspicious that the director was secreting away outtakes. Following the end of his relationship with the band, Clifton revealed unalloyed contempt for everyone involved with them: 'The group that comprised Led Zeppelin, whether individually or collectively, were the rudest, most arrogant and inhumane people I ever encountered in my 25 years of filming music. They were all dreadful and behaved appallingly. They were allowed to get away with their horrible behaviour due to their instant commercial success. I can say this with some authority after the ordeal of broken promises and daredevil tactics I put myself through because of my ambition to make the world's most successful rock 'n' roll film. That ambition in 1974 revolved around their co-operation and commitment, which guaranteed the necessary funds. There was no question in my mind that Led Zeppelin were the most enigmatic of all rock bands. They never granted interviews or appeared on TV, never advertised a concert, and yet every one was sold out within hours of the release of tickets. Their popularity lay in myriad reasons: their indisputable talent, their sex appeal and sheer power. The nasty seventies fitted them like an iron glove. It was Jimmy's band and what Jimmy said \u2013 or rather what Peter Grant said on Jimmy's behalf \u2013 was the way it was.'\n\nIn Nick Kent's _NME_ interview with Jimmy Page that November 1976, it was revealed that he had returned to Charlotte Martin and their daughter Scarlet from his dalliance with Krissie Wood \u2013 'a more domestically ordered existence', mused Kent. 'Charlotte's been very ill,' said Page. 'But that's something one doesn't need to go into, really, only that... if you've been with someone for a long time and they get ill, then you immediately have that responsibility... I don't really need to say anymore.'\n\n'Page,' assessed Kent, 'seems a changed man from the days that seemed to reach their hiatus during the 75 tour of America. Then, the guitarist, at once unattached, was staying up for days and nights on end in some kind of mortal combat with the forces of Nature, pushing virtually everything to the limits and cultivating some potentially bad habits in the process.'\n\nWhen Kent suggested this, Page informed him that what he witnessed then was nothing compared to the process of making _Presence_ : 'That was the ultimate test of that whole... lifestyle. I mean, that was 18 hours a day at a real intensity every day. You just plunge in and, I mean, you don't start thinking about three meals a day.'\n\nPage told Kent that following the making and release of _Presence_ 'it was a case of sorting out a year's problems in... say, a month, and not finding the whole process as simple as that. I mean, suddenly I had time to look around, and suddenly I became aware of certain people who'd been taking incredible advantage of me in the year I'd been away.'\n\nAs 1976 galloped along, there seemed little respite from the traumas that the last 18 months had brought. And as so often in the esoteric matters to which Jimmy Page was attracted, there was frequently a scintilla of paradox. For example, Aleister Crowley considered cats to be creatures from the underworld, and useful for sacrifices. Yet Page was a cat lover and certainly would not have agreed with Crowley's brutal thinking. Yet in psychological terms, dreams of cats are frequently seen as representative of women, on whom Crowley was even more down than he was towards feline creatures. And this was a sentiment with which Page seemed to be in total agreement: 'Crowley didn't have a very high opinion of women and I don't think he was wrong.'\n\nMeanwhile, there were further, very real problems at Tower House. In the mystical world there is an old adage: two magicians should never meet. And in the second half of 1976 Page found himself embroiled in a battle of wizards. His opponent was Kenneth Anger, and by the end of the year the filmmaker would declare that Page had been 'fired' for failing to complete his work as composer on the soundtrack of _Lucifer Rising_.\n\nAnger decried the guitarist for time-wasting and a lack of dedication to the project, and claimed that Page's personal problems \u2013 code for his heroin habit \u2013 had made him impossible to work with. Page had supposedly been working on the film for the past three years, but at the point Anger spoke out against him had delivered only 28 minutes of completed tape.\n\nSince their first meeting in 1973, the collaboration between Anger and Page had continued intermittently: Anger was commuting between London and New York, overseeing the publication and distribution of _Hollywood Babylon_ , while Page was committed to Led Zeppelin performances and recording.\n\nFor some three months Anger had been using the film-editing facilities in the basement of Tower House, chopping down the 17 hours of film he had in the can.\n\nAn extraordinary sequence of events unfurled, and Anger apparently became the unwitting victim of a domestic fracas. He was ordered to leave the house by Charlotte Martin. (Later, it was suggested that it was Page's housekeeper, not Martin, who was behind this.)\n\nNo reason was given for his eviction, and when Anger returned to the house the next morning to collect his film materials and belongings, he found the door locked and bolted. That afternoon, Anger, unable to reach Page himself, informed Swan Song that the film collaboration was off and that he had fired Page from the project.\n\nThe following day Anger was able to recover some of his belongings and the film from the now empty Tower House. Page, who was in town for a friend's funeral, was unavailable for comment, but a spokesperson from Swan Song claimed to be totally mystified by the news that the guitarist had been fired from the _Lucifer_ project; he even expressed surprise at the information that Anger was in London at all.\n\nAnger returned once again to Tower House the next day. Now he removed the last of his belongings, as well as artefacts from the film \u2013 these included Lucifer's crown, made of paste studded with rhinestones from a dress once worn by Mae West.\n\n'I haven't laid eyes on Jimmy Page since early June,' Anger told me. 'I've been trying to get in contact with him since then: I've fixed meetings through his office and been stood up half-a-dozen times. I've left messages on his Kafka-esque answering machine. All I've had is promises that the soundtrack is on its way, but nothing's materialised. I've got a fucking film to finish.\n\n'The way he's been behaving is totally contradictory to the teachings of Aleister Crowley and totally contradictory to the ethos of the film. Lucifer is the angel of light and beauty. But the vibes that come off Jimmy are totally alien to that \u2013 and to human contact. It's like a bleak lunar landscape.\n\n'By comparison, Lucifer is like a field full of beautiful flowers \u2013 although there may be a few bumblebees waiting to sting you if you are not careful. I'm beginning to think Jimmy's dried up as a musician. He's got no themes, no inspiration, no melodies to offer. I'm sure he doesn't have another \"Stairway to Heaven\", which is his most Luciferian song. _Presence_ was very much a downer album. In the first place his commitment to _Lucifer_ seemed to be totally serious, and he was very enthusiastic about the project. And he's very into enterprise and hard work. But on the other hand he has this problem dragging him down. He's been acting like Jekyll and Hyde, and I have to have someone who's 100 per cent. This film is my life's work.\n\n'I really don't think he has the zing \u2013 the capabilities to do it. If he'd have said he was bored with the project I'd have understood, but he's just strung me along. Now he's no longer on the project; I'm no longer interested in having him.'\n\nThe situation was complex, however. According to Anger, Page was never formally employed to compose the soundtrack: 'There was never any discussion about money; the whole idea was that it should be an offering of love. The idea was to go 50\/50 on the film's profits and that Jimmy should have all the proceeds from any soundtrack album that came out of it. We never put anything down on paper. We had a gentleman's agreement, which to me is more serious than anything written down by lawyers.'\n\nThe subject of _Lucifer Rising_ , Anger's most ambitious project to date, was of course the 'fallen angel' of orthodox Christian mythology, who in Anger's film was restored to his Gnostic status as 'the Bringer of Light', an implicit part of Crowley's teachings. No one with a serious interest in metaphysics would have been surprised that such subject matter might attract controversy.\n\nAnger had spent the best part of the previous nine years attempting to complete _Lucifer Rising_ , and his difficulties with Page were simply the latest in a catalogue of upsets, misfortunes and disruptions that had stymied the progress of the film. However, until this point Anger's principal problem had been finding someone to take the part of Lucifer.\n\nFollowing the accidental death of the five-year-old boy originally chosen to play the part, the role was taken by Bobby Beausoleil, a former guitarist with the group Love. But Beausoleil was fired from the movie after a prolonged altercation with Anger, and he left, taking most of the completed film with him. Two years later, after he fell under the spell of Charles Manson, Beausoleil was sentenced to death \u2013 commuted to life imprisonment \u2013 for the murder of Gary Hinman.\n\nWith the little footage remaining from the Beausoleil episode, Kenneth Anger shaped another film, _Invocation of My Demon Brother_ , with a synthesiser soundtrack by Mick Jagger. The Rolling Stone was evidently taken with Anger's work, and indeed the filmmaker claimed it was their conversations that inspired Jagger to write 'Sympathy for the Devil'. Jagger agreed to take the part of Lucifer, but he backed out before shooting began, apparently fearing that the Satanic aura he had once sought to cultivate was becoming too tangible for comfort. His brother Chris took his place, but an on-set row with Anger led to his dismissal. Eventually, a Middlesbrough steel worker named Leslie Huggins was recruited for the part, and with Marianne Faithfull and Donald Cammell (co-director of the epochal _Performance_ , and son of a biographer and friend of Crowley) also taking principal roles, filming began.\n\nIn 1976, having fired Page, Anger continued editing the film, hoping to meet a Christmas deadline. He was looking for another musician to write the soundtrack. 'I'm seriously questioning whether to use a musician from the rock world,' Anger told me in the _NME_ 's Carnaby Street offices. 'It seems like most of today's rock music is savage, deliberate bad taste. It's not optimistic, constructive or even fun anymore. I'm certainly jaded with the rock-superstar syndrome. They're like renaissance bandits. Who needs those people?' The director would not come anywhere near to meeting his self-imposed Christmas 1976 deadline, though a single shot of Page would make it into the finished film.\n\nAs for the soundtrack, Anger eventually decided to let Bobby Beausoleil do it from prison in the late 1970s. The CD was released in 2004 \u2013 as Beausoleil continued to languish behind bars.\n\nWhen I asked if he felt vindictive towards Page, Anger declared: 'You bet I do. I'm not a Christian, turn the other cheek kind.' He allowed a thin smile. 'In fact, I'm all ready to throw a Kenneth Anger curse...'\n\nPersistent attempts at the time to contact Swan Song for a statement from or on behalf of Page were met with silence. Not even a 'no comment'.\n\nThe 'Kenneth Anger curse', however, was no idle threat. Almost 30 years later, a still-angry Anger confirmed that he had indeed followed through with it. 'He was a multi-millionaire miser,' he said. 'He and Charlotte, they had so many servants, yet they would never offer me a cup of tea or a sandwich. Which is such a mistake on their part because I put the curse of King Midas on them. If you're greedy and just amass gold you'll get an illness. So I turned her and Jimmy Page into statues of gold.'\n\nPage was utterly dismissive of this. He told me later that this 'curse' consisted of newspaper cuttings underlined in red ink that Anger sent to him. 'It was quite pathetic, actually. _Lucifer Rising_ was going to be a masterpiece, but he didn't manage to pull it off.'\n\nAll the same, it cannot be overlooked that from this point on, Led Zeppelin's nosedive was inexorable. Now mired in heroin addiction and alcoholism, Jimmy Page's days of greatest creativity with the group were behind him. What lay in the future for Led Zeppelin was remorseless tragedy and death, followed by an eventual phoenix-like rebirth for its principal player.\n\n## 20\n\n# FACE TO FACE\n\nIn February of 1977, an absolutely pivotal year for Jimmy Page and Led Zeppelin, I secured an interview with him for an American magazine called _Gig_.\n\nIn the UK, due to the sudden emergence of punk, a cultural shift was underway. In December 1976, Led Zeppelin began rehearsals for an American tour scheduled to begin on 27 February 1977. By January 1977 they were ensconced at Emerson, Lake and Palmer's Manticore Studios at the bottom of Fulham's North End Road. Two months previously I had been to Fulham Town Hall, only a few yards away, to see the Clash, one of the inspirational new punk acts that had sprung up in the UK during 1976. The only time I would ever come again to Manticore would be 18 months later, to watch the Clash \u2013 by then risen considerably in status \u2013 play a secret pre-tour warm-up gig at the ELP establishment.\n\nSo there was a cultural and poetic twist to Led Zeppelin rehearsing at the studio of another rock behemoth for a tour that would be the utter antithesis of punk. This was Zeppelin's most massive tour ever of America, 51 shows in 30 cities, largely playing stadiums, and naturally the band stood to earn a colossal fortune. For example, for the date in Pontiac, Michigan, Led Zeppelin would pack in an audience of 76,229 devotees and earn $900,000. 'The big business nature of the band has always been more of a hazard than anything else,' said Page. 'One day you're just playing guitar and the next day there's a knock on the door and you realise you're in the realms of high finance. It's very heavy.'\n\nIf the speed and attitude with which the first Led Zeppelin album had been recorded was thoroughly in the spirit of punk, which it really was, it is unsurprising that \u2013 traduced as the four musicians may have been by Year Zero attacks on them as the very worst examples of irrelevant dinosaur rock \u2013 they found immediate empathy with this iconoclastic new form.\n\nAt the urging of their former publicist B. P. Fallon, who was now working on the new, sort-of-punk Stiff Records releases, Page and Robert Plant went along with Beep to Covent Garden's Roxy club on 13 January 1977 to watch the Damned, who were supporting \u2013 an example of punk's almost perverse egalitarianism: the Damned had released the first UK punk single and already had an album out (on Stiff, which afforded Fallon dual loyalties) \u2013 Eater, famous on the punk scene largely for their 13-year-old drummer. 'I was aware there was a bit of nudging going on from the audience when they saw us in there,' said Page. 'But we felt very comfortable. And when the Damned kicked off... it was absolutely fantastic. You felt this wall of sound almost pressing down on you.'\n\nPage and Plant loved the Damned's high-speed, slightly bonkers set, and on the subsequent US tour Page would irritate the more reactionary elements of the Zeppelin entourage with his insistence on repeatedly playing the Damned's first album, _Damned Damned Damned_. Guests at the Plaza Hotel in Manhattan would complain about the volume at which he was playing the LP in his suite.\n\n'I was standing onstage at the Roxy Club,' said Captain Sensible, the Damned's bass player, 'and I saw two hippies standing at the back. I thought, \"What are they doing here?\" Then I looked a bit closer, and it was Jimmy Page and Robert Plant.'\n\n'I was at the bar talking to Jimmy Page and Robert Plant,' said Glen Matlock, bass player with the Sex Pistols. 'Everybody was winding everybody up. \"What are they doing down here?\"'\n\nFour days later, Plant returned to the Roxy, this time accompanied by John Bonham. Desmond Coy, on the door, told Plant to go to the back of the queue. And charged him full admission. At the bar, he noted, the Zeppelin members were asking for receipts, for considerably larger amounts than they had paid. Again, the Damned were playing, this time supported by Eater and the Boys. After the Damned's brief set, according to Glen Matlock, 'John Bonham was onstage going, \"What is this? We play for four fucking hours! Get the band up there without that Mouse Scabies and I'll play drums!\" He was really pissed.'\n\n'Bonham was carried out,' said the Damned's guitarist Brian James. 'He was out of his head, drinking vodka all night. It was cool that these geezers were willing to chance their arms down there with no security.'\n\nThese visits by three-quarters of Led Zeppelin to punk's Mecca earned them considerable column inches in the next week's music press, with pictures of Page and Plant at the Roxy.\n\nThere was a pretty, rather stylish and classy girl at Swan Song who showed me in to a broad room furnished with a pair of opposing couches for my _Gig_ interview with Jimmy Page. I liked the girl and we smiled at each other as she left. 'She's lovely, isn't she?' said Page. Then, surprisingly, almost sadly, dispensing with any expectation of his acting out the role of _droit de seigneur_ , 'But nobody knows her story.' (Well, you and your cohorts could always engage her in conversation, I thought.)\n\nI had been asked by the Swan Song office, to whom I had pitched the idea of an interview with Page, to come to Manticore; I was to accompany my friend Pennie Smith, the _NME_ photographer who had been commissioned to shoot some snaps of Led Zeppelin in rehearsals. It was a brief connection \u2013 they were not actually playing when we arrived \u2013 and I felt I was being checked out. But there was none of Zeppelin's legendary horribleness to the press: everyone seemed in good spirits and rather nice, even the allegedly terrifying John Bonham. Robert Plant flirted with Pennie; he was immediately tactile with her, almost as if he felt obliged to act this way as part of his job as Sex God. Worryingly, however, he seemed slightly lopsided, as though putting weight on his left leg to alleviate the stress on his damaged right limb. Page was off to one side, smiling but largely silent.\n\nUltimately, the first 12 dates of this US tour would be cancelled thanks to Plant suffering from laryngitis. A singer unable to sing on what were essentially comeback dates? Sounded psychosomatic to me...\n\nBy the way, at this time I had heard only vague rumours of Page's problems with heroin. Anyway, this is what I wrote for my _Gig_ piece in February 1977:\n\nThe overriding first impression that emanates from both Led Zeppelin's music and the legendary self-isolation the band and its entourage maintains is one of _power_. Something akin to a mega-sized armour-plated rhinoceros moving relentlessly \u2013 and often, one suspects, humourlessly \u2013 through contemporary rock music.\n\nIn what seemed initially to be thoroughly in keeping with this assumed tradition, it soon became apparent that endeavouring to be placed in an 'Interview Situation' with Jimmy Page would not prove to be the easiest journalistic task I had ever undertaken. Indeed, there were moments when scoring this interview seemed to be taking on all the elements of a parody of The Quest for the Rap with the Big Name Rock Star.\n\nNegotiations commenced at the end of November last year. They were consummated in the second week of February at Swan Song's offices on London's Kings Road. In the interim, Page had cancelled two scheduled appointments, though we had actually met on one of these occasions. There had been a further meeting at Emerson, Lake and Palmer's converted cinema rehearsal studios, where Zeppelin was rehearsing for their first tour since Robert Plant sustained severe injuries in a car smash on the Greek island of Rhodes in the summer of '75.\n\nAlmost predictably, when the interview did actually take place, six days before the band was due to fly out to Texas (where the first dates would be postponed because of Plant's laryngitis, though that's another story), Page revealed none of the superstar arrogance or aggression one might expect, talking at length of Led Zeppelin with an almost religious fervour.\n\nWe also discussed his fascination with the occult and, in particular, with the self-styled 'Great Beast', Aleister Crowley (whose former Scottish home, Boleskine House, Page now owns), and his related interests in ecological matters. The guitarist seemed more content and at ease when dealing with these subjects than when talking about the band; though by the time they were raised he had warmed to the task of being interviewed.\n\nFor the first 15 minutes of the interview he sat on the edge of a couch, huddled over and shivering into the cup of tea he was holding in both hands. His speech was frequently little more audible than a whisper. Indeed, he seemed so fragile and it appeared to be such an exhausting emotional effort to talk about Led Zeppelin, the impression remained that it might completely upset his thought pattern \u2013 or he might actually call off the interview \u2013 if he were asked to speak up.\n\nLater on in the interview I would look up from my notebook and see Jimmy Page lying back on the couch and looking at me through his legs. Or stretched out with both eyes firmly shut and a hand stuffed down the crotch of his frayed jeans as he delivered his semi-audible soliloquy.\n\nNotwithstanding an acute bronchial cough that punctuated his speech, he chain-smoked throughout the interview...\n\nHis eyes were ringed with the kind of wrinkles that some would describe as laugh lines and others might attribute to the effects of constant nervous tension. In fact, in the autumn of last year Page spent some time as an in-patient at an exclusive health farm near London. This was supposedly to recuperate from the effects of having become dangerously underweight. Now, though, as he told me in his soft accentless Home Counties voice, 'I just needed to get away for a while and see things from a different perspective. There was nothing sinister. I needed to get into a regular pattern. A regular schedule. And it seems to be working.'\n\n'What sort of... uh... line do you want to take on this?' he asked me with what seemed to be a slight edge of suspicion.\n\nI summarised the majority of my questions. After that, he seemed a little more comfortable: 'Okay. Well, fire away. You can always edit out what you don't want.'\n\n_Okay, then. So how have the rehearsals been? Pretty rigorous?_\n\n'The rehearsals started a month before Christmas and with the Christmas period off we've been working consistently ever since. They've been going well. _Really_ well.\n\n'Of course, the first task was to clean off the rust which is obviously going to set in after 18 months without being on stage... Although we recorded _Presence_ some 14 months ago it's not quite the same because a tour is a concentrated series of dates. We'll have three days on and one day off. And we have like a three and a half hour concert to contend with.\n\n'Consequently there was a stamina aspect involved apart from anything else. Plus the constant dilemma that appears from tour to tour about the repertoire as such. What to drop, what not to drop... Which is always a great problem when you've seen everything go down really, _really_ well for its own merit, for its own _vibe_ , so to speak, and the atmosphere that that particular number's portrayed and evoked. As has been the intention.\n\n'As far as our playing goes, the way it ended up was everyone was just a hundred per cent confident and really bursting to go.'\n\n_Have you dropped any numbers that were in the last live set?_\n\n'We've dropped a few things. But nothing of great importance. None of the epic things but...'\n\n_What have you added?_\n\n'Stuff from the new LP. \"Candy Store Rock\", \"Achilles\", \"Nobody's Fault but Mine\"... We've also added \"Ten Years Gone\", a number we never did in the past. It offered such a challenge as far as the guitars went. It's really my baby because I worked it out note for note at home. At one point there were _nine_ guitars going to present all the harmonies so obviously we lack some of that. But nevertheless the overall _feeling_ of the number comes across. And comes across very well.\n\n'And we're doing \"The Battle of Evermore\" which I think [laughs] is a very sort of noble challenge really. We'll probably get applause for the sheer guts of the thing rather than anything else.\n\n'But they're coming off good. All of them.'\n\n_So it's all happening okay, then, as far as live..._\n\n'Yeah.'\n\n_Do you get very apprehensive about going back on the road?_\n\n'Well, obviously when you haven't played a concentrated tour for two years... well, it goes on for months. And obviously you've got to take into account the fatigue aspect and all this sort of thing.\n\n'But I'm pretty confident. Everyone's confident. As far as the playing goes there's no problems at all. We've got such a variety in there \u2013 oh, \"Babe, I'm Gonna Leave You\" is another number that we've been doing... pedal steel guitar instead of the usual. So it sounds pretty different from the original.\n\n'It's very interesting. And yet we've obviously kept in there all the key epics: \"Achilles\", \"Kashmir\", \"Stairway\"... things like that.\n\n'But the most amazing thing, really, is that we started nine years ago putting out albums, albums which have been constantly subject to change as far as content goes, to the point where we stuck our necks out only because that's the material that we've got when it comes to the time of recording. But rather than stick to the previous formula and work stuff around it we've just stuck to our guns. And because of that we've got a wide variety of material...\n\n'But nevertheless it's marvellous to think that after what is basically a two year break \u2013 even though the film's come out \u2013 to hit the top of all these polls [a reference to Zeppelin having swept the boards of various assorted music publication polls throughout the world] is really quite stunning, really quite... _awe_ -inspiring. A confidence boost beyond all measure, you know, to realise you're still thought of as being really contemporary instead of a... [laughs] _nostalgia_ -band, shall we say.\n\n'Not that I ever thought we'd become a nostalgia band because we've got too much up our sleeves to fall into that bracket. Nevertheless, it's nice to be reassured. In the warmest possible way.'\n\n_You obviously feel as strongly about the band now as ever._\n\n' _Yes_ [very decisively]. And I feel very strongly about the _timeless_ quality of the songs too. I think that's where it's at... That's probably why we're so critical about their construction when we start putting them together. So you don't just rest on the obvious sort of clich\u00e9s that were around in '72.\n\n'I mean, it's so easy to sit there and jam a soul riff and to make a song out of a soul riff.'\n\n_The reaction that I have to Led Zeppelin's music \u2013 and I know I'm not unique in this \u2013 is unlike what I get from any other band. On first hearing a new Zeppelin record it often seems to slightly grate. There are usually edges to it which you have to grow to understand. Also, whenever I haven't played any of the records for a couple of months I'll put them on and immediately get a colossal rush._\n\n_There's something there \u2013 something indefinable unless you accept that it's just down to the chemistry of the four members \u2013 that suggests the band contains the true essence of rock 'n' roll._\n\n'Well, _inventive_ rock 'n' roll... it's got the root which is in all rock 'n' roll... the earthiness. But it's also got all the other facets that, shall we say, musicians of today have been able to get. You know, finger style, folk areas and things like that. And traces of jazz. Generally the three strong areas. Which is so important.'\n\n_And also whether it's in the hard rock or the acoustic side there's also this enigmatic sense of power and strength. Yet many people still look on you as just a heavy metal band._\n\n'Well, they did. I think it's dawned on them now that we are a band that's going to be subject to change all the time. Like it or loathe it.\n\n'But the most encouraging thing about that is that when an LP is announced the advance orders are so great that it seems they automatically assume there is going to be a certain overriding quality to it. Which is really reassuring because that's what we've been going for. Which again related to that lasting quality. Which one can only _hope_ for.\n\n'One can say \u2013 and it sounds pretentious \u2013 but it's the test of time which shows it.'\n\n_So was that your idea of the band when you first conceived it after the Yardbirds ended? That it should have incredibly strong quality and inventiveness?_\n\n' _Definitely_. Yeah. I mean, the first album has got the catalyst for so much. You know, the blues and the sort of step off from that. And the working together between Robert and myself and the acoustic work and the way you can stretch an acoustic number... you know, keep the dramatic quality there. Which is, after all, the atmosphere that you're trying to convey. And trying to, you know, develop the mystery. Just really because the whole aspect of what's going to come round the corner as far as writing goes is the dark element, the mysterious element. You just don't know what's coming. So many good things have come out of that that it would be criminal to interrupt a sort of alchemical process like that. And we're aware of that and we wish to play forever.\n\n'And I hope that's still there. I think it is.'\n\n_How do you feel about_ Presence _?_\n\n'I think it's the most important album myself. In many respects. Because of the amount of time it took to do. Working up against a deadline it took three weeks: the tracks took about a week. Then we had a slight break in the middle when Robert fell over and he thought his leg had gone again. Then we started again and continued.\n\n'And it was pretty much up to me because after that the group really left it to me. Which is really an honour after so long to have that sort of trust; that they know you're not going to sort of _mess it up_ , and don't mind if you embellish the thing or whatever... Because some of those tracks changed immensely by the time all the overdubs and effects had gone on.\n\n'Although it may not be the best track on the album \"Royal Orleans\"... the first verse of that is as it was. As a riff. After that you hear all these guitars coming in [hums guitar parts]. And that's the sort of thing which I can do to change the whole mood of how a number comes out.'\n\n_So when you did_ Physical Graffiti _, for example, you'd work in the same way whereby you'd just be left alone to produce the tracks?_\n\n'Yeah. Pretty much. There'll be the tracks and then Robert will come in and do his vocal parts and sometimes Jones will come in and do a little synthesiser work on the tracks. But usually it's down to me and [laughs] I'm quite happy with that.\n\n'As I say, it's an honour to have that sort of relationship with the band.\n\n'The time spent on _Physical Graffiti_ was really because of work that came in between time and there was the problem of studio availability. We just couldn't get in.\n\n'And there was the new material which related to that point and I wanted it to have a sort of chronological touch about it. And that's why there are all those tracks that go way, way back and reflect the development of the band. And then you have the apex of \"Kashmir\".'\n\n_What do you feel in retrospect about_ The Song Remains the Same _film?_\n\n'Well, I think the film's successful in so much as it is a frozen celluloid statement of an evening.\n\n'And the soundtrack as such... [pause] I wouldn't call it a live album because we've got so much live stuff in the bag going back to '69 at the Albert Hall. We've got some _fabulous_ live stuff. And, it wasn't necessarily the _best_ live material we had but it was the live material that went with the footage so it had to be used. So, you know, it wasn't like A Magic Night. But it wasn't a poor night. It was an honest sort of mediocre night.\n\n'You know, I've always thought of the band as being reasonably consistent as far as the concerts go. I think we always start off _shaky_ and it's at the end when the whole thing builds. Which we build up between ourselves. We build up the \u2013 I don't know what you might call it \u2013 the ESP aspects of it where when you do start jamming and entering areas which are open to free form and you start coming across the different rhythms and you might just _stop_ it and start and _stop_. And use some shock tactics.\n\n'A lot of that is just off the cuff, you see. And that's where everybody's really working. You can just anticipate what's coming. And a lot of bands don't manage to be able to do that... a lot of larger bands anyway. They play it safe with everything just about note for note perfect apart from some change in the solo or something.\n\n'But they don't let the solos go on for a long time on purpose so they can really get their teeth into improvising and showing what can really be done.\n\n'And consequently you hear a number one year and so much has changed from a few years before. Because there is that quality.\n\n'Again _Presence_ by the way... We really _needed_ that as a band that has been together for such a long time to prove to ourselves that... You know, we've always spoken of the instant chemistry and how the band get together and start jamming and within those jams, riffs come out. And it doesn't take long before you've got the framework of a song which often gets reviewed but nevertheless it's there from the inception.\n\n'Whereas you do hear stories of big bands that get together for two or three weeks and they can't get anything together. There's like two or three different strong frameworks every rehearsal. And that sort of three week thing proved that it can be done.\n\n'Mind you, there was a helluva lot of emotional anxiety and frustration related to that as well. You know, the uncertainty of Robert's position and one thing and another... And the way that the band really stuck together during this whole thing because of the loyalty between the band and each other. And that was the emotional release of getting it all out.\n\n'That's why it's an important album to me. Because it reflects all the spontaneous aspects.'\n\n_How did you personally react when you heard about Robert's car smash? What were your immediate reactions?_\n\n'Well, I was shattered. [profound concern in his voice] I'd been with Robert the day before and I'd just left to go to Sicily. And I was in Sicily when I heard the news.'\n\n_So did you hear what state he was in or did you just get garbled reports?_\n\n'No, I just heard that he was hurt very seriously. A doctor had to come out from London immediately and put him on a plane because the medical facilities there were so impossible.'\n\n_Was there any time when you thought, 'Well, what happens if he doesn't get better or if he dies? What happens to the future of the band?'_\n\n'Well, I've always felt \u2013 and this has been discussed \u2013 that no matter what happened, provided he could still play and sing, and even if we could only make albums, that we'd go on forever.\n\n'Just really because the whole aspect of what's going to come round the corner as far as writing goes is the dark element, the mysterious element. You just don't know what's coming. So many good things have come out of that that it would be criminal to interrupt a sort of alchemical process like that. And we're aware of that and we wish to play forever.\n\n'There's a lot of important work to be done yet anyway. It's only just started.'\n\n_You're obviously very confident of the future of the band now. Have you always been so?_\n\n'Yeah.'\n\n_Never any doubts at all?_\n\n'No, no, no.'\n\n_But there does seem to be a contradiction in that you're so into the music..._\n\n'It's the all important thing, yeah.'\n\n_But at the same time Led Zeppelin is a colossal business empire._\n\n'Well, that's just one of those things that happened to snowball up behind us really. But nevertheless the music _is_ the most important thing. If we'd been conscious of trying to sustain a particular sort of market then we'd have stuck to a formula. Which is a terribly dangerous thing. When you know you're going through changes it obviously reflects on your music \u2013 lyrics, especially. If you try and suppress that, then you get into trouble. If you suppress it for the sake of a formula.\n\n'But then that's _our_ philosophy of what we're up to and other people have a different way of looking at it.\n\n'But it's only the test of time which can lay down the importance of what you're doing. You see, there's been so much flak directed towards us. I knew that it would take a time before a proper perspective was reached about what we were really doing. The fourth album probably being the first milestone in that respect, even though the third should have been.\n\n'When the third LP came out, Crosby, Stills, Nash and Young had just toured. They'd done an acoustic set followed by an electric set. And, of course, our third album having more predominant acoustic work on it then the reviews related it to Crosby, Stills, Nash and Young and said, 'Well, obviously they've been influenced by them.' And missed the point altogether. They forgot that we'd used acoustic guitars very heavily on the first album. Not quite so much on the second but it was there.\n\n'And, you know, after we'd been on the road after the second album had been released we really wanted to have a rest, and consequently sitting around a log fire in Wales you don't put up a 200-watt Marshall set-up but you get your acoustic guitars. Consequently acoustic numbers came out.\n\n'And if they've got a validity you owe it to yourself to lay them down.'\n\n_Oh, incidentally, I've always felt that John Paul Jones's situation within the band was exceptionally important, especially in the way he seems to cloak a lot of the changes. Do you not feel that he's underrated?_\n\n'Underrated? Well, quite possibly. Yeah, as far as a bass player relative to a rhythm section. Yeah.\n\n'But as far as the writing goes everybody has an equal share of coming forth with the ideas they've got. But it always seems to end up with myself when it comes to most of it. And I think Robert and I are sort of very sympathetic to the sort of loony vibe because we've always been working together.'\n\n_Yeah, there is a sense of you and Robert as the Terrible Duo, the Inseparables..._\n\n'Yeah. Yeah. But it's not to the point of a Jagger\u2013Richards thing where nobody else gets a look in. It isn't a cut-off situation. It's just developed from the early days where, say, Jonesy may come up with one riff for one section and Bonzo the same whereas I'll probably come up with the whole framework. Or piece together all those little bits and pieces.'\n\n_How is Robert standing up to the stagework?_\n\n'Fine. Fine. He's been rehearsing ten hours a day. Ten hours on the trot.'\n\n_I understand that a changed Robert Plant who has taken to reading Nietzsche on plane journeys has emerged since the accident. I know that you were ill for about nine months prior to joining the Yardbirds and I've heard you're supposed to have spent much of that time reading. Was that when your interest in the occult began?_\n\n[Fifteen second pause] 'My interest in the occult started when I was about 15.'\n\n_Do you agree that whereas Western society tends to see occult matters as a very dark \u2013 a very black \u2013 thing, it is, in fact, a very light and enlightening thing?_\n\n'Well, there has been a major revival, a spiritual revival, throughout the world and it reflects all over the place. Not just within the West.\n\n'And there's a great interest in the Celtic mysteries and the Dark Ages and the areas where a lot of these truths were just erased for the sake of the Church, you know. But I'm quite fascinated by these things.'\n\n_So obviously the folkie Traditional English side of Zeppelin all emanates from one logical area of interest, no?_\n\n'Yeah. Well, a man's a product of his environment. It depends how much he wants to educate himself in that framework. You know, in relationship to his craft. There should be no boundaries, so just carry on as far as you can and do it.'\n\nPage, of course, is an ardent aficionado of occultist and magician Aleister Crowley (1875\u20131947). Indeed, the guitarist owns Equinox, an occult bookshop situated off London's Kensington High Street, which has a large section devoted to Crowley's works as well as having his birth chart pinned to one wall. And, as already mentioned, Page spends most of his time on British shores at the home that Crowley once owned, Boleskin House. [This wasn't actually true \u2013 although I didn't appreciate this at the time.]\n\nNot unexpectedly, such matters are beginning to arouse the interests of the more sensational end of the British press. In fact, only a few weeks ago a _National Enquirer_ -like weekly magazine featured an aerial photograph of the house on its cover along with details of collapsing staircases and the appropriate 'Dark Man of Pop' blurb about Page.\n\n'Well,' says the guitarist, 'they should have gone into the history of the house and Crowley would've come out like a shining angel compared with what else went on.\n\n'I mean, it's had a history of suicides and con tricks. Plus the site of the house is on the site of a church and a graveyard, and the church was burnt down by an arsonist with the whole congregation in it. So the actual foundations of the house are built on hallowed ground.\n\n'But I'm not really interested in going on about Crowley in so much as, say, Pete Townshend does about Meher Baba. I'm not interested in trying to turn anybody on in any way whatsoever. You know, there are a thousand paths and they can choose their own.\n\n'All I know is that it's a system that works... [laughs] Although, of course, there's not much point in following a system that _doesn't_ work.'\n\n_But what about the hassles you've had with Kenneth Anger? (Page wrote the score for filmmaker occultist, and author of_ Hollywood Babylon _, Kenneth Anger's imminent film,_ Lucifer Rising _, but was turned down by Anger towards the end of last year and replaced by none other than Mansonite Bobby Beausoleil. Since then Anger has denounced Page on every possible occasion.)_\n\n'I think it's more the problems he's had with himself. All I know is that at the end of the film I promised him \u2013 as I had before \u2013 the loan of a three-speed projector which makes the editing so much easier. I said to him, \"Well, it's just going to be your own time invested.\" And I also told him that he must put the music on after he put the footage together so I was just waiting for him to contact me, really. He had other music that I'd done instead of the stuff that I'd delivered which he said he wanted to use. Nevertheless, I still needed to hear from him. And I never heard anything.'\n\n_Didn't he come down here and stick things onto the door of this record company?_\n\n'Oh, that was his curse. That was _pathetic_. His curse amounted to sending letters to people. Silly letters saying \"Bugger off, Page\" and this sort of thing.\n\n'How can you take that sort of thing seriously? [Sounds quite deeply disappointed]. A man you had thought to be a genuine occultist and it turns out to be just... _theatre_. It's a shame, really.'\n\n_Although it's quite acceptable these days, do you wish your occult interests weren't known about?_\n\n'I just don't want it rammed down people's throats as though I'm saying it's the be-all and end-all and the only way you'll be able to put things together. I'm not saying that at all. You might go off and study the Gurdjieff system and be equally...\n\n'But what I can relate to is Crowley's system of self-liberation. In which repression is the greatest work of sin. It's like being in a job when you want to be doing something else. That's the area where the true will should come forward. And when you've discovered your true will you should just forge ahead like a steam train. If you put all your energies into it there's no doubt you'll succeed. Because that's your true will. It may take a little while to work out what that is, but when you discover it, it's all there.\n\n'You know, when you realise what it is you're supposed to be here for. I mean, everyone's got a talent for something. Not necessarily artistic but whatever you care to say. And it's just a process of self-liberation. I mean, I just find his writings to be _twentieth century_. As a lot of the others weren't.\n\n'And there's really nothing more to say than that. I find him quite a curious, highly enigmatic character. Consequently I enjoy my researches into him. But it doesn't want to be blown out of all proportion, though, because that would be... silly, you know. I'm just another artist, too.'\n\n_Yeah, it's an interest in all things occult and, as you said, all things English or, rather, of Albion. And that's just one area, right?_\n\n'Mmmmm.'\n\n_Uhh... Returning to the music for a moment, do you feel any great responsibility in your position as one of the ruling triumvirate of rock 'n' roll along with the Stones and the Who? Do you feel any great responsibility towards rock 'n' roll?_\n\n'Well, I've always felt a commitment, shall we say? Because I got into it because I was so turned on by the sounds that I heard when I was really young and I just wanted to be involved in it. It was just something thrilling that could send chills up your spine.'\n\n_Presumably your parents told you you'd grow out of it..._\n\n'No, actually they were very encouraging. They may not have understood a lot of what I was doing but nevertheless they had enough confidence that I knew what I was doing; that I wasn't just [laughs] a nut or something...'\n\n_Do you and the other three members of Zeppelin see much of each other socially?_\n\n'Not that much. But we do. We don't live in each other's pockets so it's always a great joy to see each other again.'\n\n_Do you by any chance find the rock 'n' roll lifestyle a strain in any way?_\n\n'What side of it?'\n\n_Well, and, taking into account your stay at the health farm, the irregularities of the hours, for example?_\n\n'Well, that taxes you physically.'\n\n_And inevitably, therefore, it must tax your mental powers..._\n\n'Well, in a way, yes. Except that when I'm very tired I can do my best writing. You know, late at night because there's nothing to distract you and all those day-to-day problems have gone. And I can just start concentrating on the guitar and get lost within it and I find that all these things are coming out.'\n\n_But there are those who bemoan rock 'n' roll as being vastly uneconomic in terms of both financial and human terms, especially human terms..._\n\n'But the willpower gets you through. And the adrenaline and the feedback from the audience at live concerts \u2013 which is maybe what we've been missing \u2013 is the thing that charges you up like a battery.'\n\n_You really enjoy playing on stage, do you?_\n\n'Oh yeah.'\n\n_You don't prefer recording from playing live or..._\n\n'Both. And things have been out of balance in that respect. And one knows something's missing and gets edgy about it. But it's not until you play again \u2013 when you rehearse \u2013 that you know what it is.\n\n'Of course, a lot of it has been done by Robert's recovery. And certainly not even wanting to breathe a word of the subject of a tour until nature had dictated her terms and things became good.\n\n'But then there is that bond between us that gives enough confidence to just wait and see. Not just go off making solo albums or kicking with others, as they do.\n\n'There's incredible dedication in what we're doing. Be that rightly or wrongly. Subjectively, that's just the way it is. That's the way it's got to be. There's nothing complacent about things. The minute you start not criticising what you're doing then you're in trouble.\n\n'And if you start thinking everything you're doing is a masterpiece [laughs] then you're in trouble.'\n\n## 21\n\n# THE RULES OF ENGAGEMENT\n\nTen days before Led Zeppelin's 1977 tour of the US was scheduled to begin, the _Starship_ developed a serious mechanical problem: during a flight one of its engines had come close to falling off, and the plane had been grounded at Long Beach Airport. Aware that the four Zeppelin musicians were always nervous of airborne transport \u2013 none more so than Jimmy Page \u2013 Richard Cole never mentioned this small difficulty and simply lined up another private 720 jetliner, _Caesar's Chariot_ , owned by the Las Vegas casino Caesars Palace. _Caesar's Chariot_ was almost identical to the _Starship_ , although it lacked the Thomas organ on which John Paul Jones had regularly led his fellow passengers in archetypal London pub sing-songs.\n\nThe postponed tour finally kicked off on April Fool's Day 1977, in Texas, at the Dallas Memorial Auditorium, followed two days later by a show in Oklahoma City at the Myriad. But these two dates, at which Led Zeppelin seemed extremely rusty, even lacking in confidence, with Page playing solos in the wrong key, were like warm-up shows for the first residency dates on this, their eleventh tour of the US; as Stephen Davis points out in _Hammer of the Gods_ , 11 was Aleister Crowley's favourite number, the reason he added the letter 'k' to the word magic. At Chicago Stadium \u2013 a misnomer, as it was only arena-sized, holding audiences of around 16,000 \u2013 Led Zeppelin were set to play four shows, with a one-day break in the middle.\n\nWhat was already becoming evident was that Page was in a permanent altered state. And it wasn't bringing out the best in him. 'I showed up on the third date at the start of the tour,' said Jack Kalmes, the head of Showco, the production company for the tour. 'The mood was ugly and there had been a buzz in the PA and Jimmy had come over and thrown a trash can over one of the main techs.'\n\nOn another occasion, in another location, Page spat in the face of a technician midway through the acoustic set, before a crowd of 50,000.\n\nCould Page's diet \u2013 or lack of it \u2013 have had anything to do with his unsociable frame of mind? 'I prefer to eat liquid food,' he claimed. 'Something like a banana daiquiri, which I can put powdered vitamin in. I'm not really into solid foods very much. I know I'll never turn down some alcohol, so a banana daiquiri, with all the food protein, is the answer.'\n\nHow was Robert Plant adapting to stage performance after such a disastrous accident and long lay-off? Not too well. Apart from when he was sitting during the acoustic segment, he would stand for the three-hour sets; sometimes he would literally be carried off the stage at the end of the show. His damaged right foot was considerably inflamed; much of the time the singer was in a state of semi-pain, constantly consuming painkillers administered by the doctor travelling full-time with Led Zeppelin, a Harvard graduate who had been on tour with other rock bands, notably the Rolling Stones on their 1972 US tour.\n\nBut Plant wasn't the only one in the group who needed to be physically assisted around the stage. By the time Led Zeppelin began their six-night stretch at the Forum in Los Angeles, Page would be physically hoisted onto the performing platform by Richard Cole \u2013 and carried off later. The doctor accused Page of stealing Quaaludes from his supply. The guitarist's response? To threaten to fire him: 'Accusing me? Who the fuck does he think is paying his salary?'\n\nThe doctor was not always able to help his patients. On the third Chicago date Page seemed even more out of sorts than usual, playing the intro to 'Since I've Been Loving You' while the other band members were beginning 'Nobody's Fault but Mine'; trying to play on a single-neck guitar what he required his double-neck for; during 'Ten Years Gone', the sixth number, he actually passed out, although nodded out might be the more accurate way to describe it. 'Jimmy has got a bout of gastroenteritis, which isn't helped by firecrackers' \u2013 a reference to the artillery barrage of firecrackers that rained on the stage \u2013 'so we've gotta take a five-minute break,' declared Plant. Page was in no condition to return to the stage and, after some time, Richard Cole came out onstage. 'Jimmy does not want to do a half-hearted show tonight,' he told the audience, advising them to hang on to their tickets, as the concert would be rescheduled.\n\nLed Zeppelin, who once had only Cole with them, now had a large entourage. As well as his own personal roadie, each band member also had his own assistant; Page had Rick Hobbs, his butler and chauffeur from London. And Cole brought in, as special minder to Peter Grant, handsome John Bindon, whose favourite party trick \u2013 which had allegedly impressed Princess Margaret, supposedly a lover \u2013 was to hang a row of pint mugs on his enormous dick. Bindon was also a very dangerous man. But Grant needed someone with him; enormously depressed by the explosion of his marriage and dabbling with the heroin that seemed omnipresent on the tour \u2013 even the crew were using \u2013 Zeppelin's manager was no longer the life and soul of the party.\n\n'I told one of our roadies that to me this seemed like the beginning of the end for the band,' wrote Cole, who was himself using smack. 'The soul that had driven Zeppelin since 1968 just seemed to have weakened, and drugs played too much of a role in everyone's life.'\n\n'There were bodyguards everywhere, and that was a real big sea change from 75 to 77. There was just a cloud that seemed to hang over everybody,' recalled _Creem_ journalist Jaan Uhelszki.\n\nBy 10 April, Led Zeppelin had wrestled themselves into delivering their first fully successful performance of the tour, at their fourth show in Chicago. 'Jimmy was feeling ill last night, but it was only a false pregnancy, so that's all right,' Plant told the audience. Answering a local radio station's suggestion that drugs and drink may have been behind the guitarist's mysterious collapse the previous day, the singer defended him to the audience: 'Mr Page neither smokes, drinks, takes women or does anything like that, so we want an apology tomorrow and a crate of alcohol!'\n\nPlant's argument was somewhat reduced by Page's rather curious appearance that night: he was clothed in a Nazi storm trooper's uniform, a cap upon his head and jackboots on his feet and spindly legs \u2013 he might have been in the Sex Pistols. Similarly strange were the rooms of his suite at Chicago's Ambassador East Hotel: the curtains were perpetually drawn, electric light was forbidden and candles gently fluttered.\n\nLate on the night of 7 April he entertained journalist Lisa Robinson there. When she inquired about the band's feral reputation, he gave a four-word reply: 'We haven't really stopped.' When asked about the myriad rumours that sat about Led Zeppelin, Page simply replied, 'I must have had a good time.' His soft voice, she noted, was 'slurred'. Robinson came to the conclusion that he was 'either very tired, or very stoned'.\n\nPeople were genuinely concerned about his appearance. When _Trouser Press_ writer Dave Schulps, scheduled to conduct a much-delayed interview with Page, first saw him on _Caesar's Chariot_ , he was surprised: 'At first sight I was struck by how extremely frail he appeared, escorted by a bodyguard who seemed almost to be propping him up.'\n\nThere were tales of Page having to be pushed through airports in a wheelchair. When I asked him about this two years later, he vehemently denied the suggestion: 'I may have done that for a laugh \u2013 not seriously. No, no. That wasn't happening at all.'\n\nWhen interviewed by _Guitar Player_ magazine's Steve Rosen, who flew with the band, Page expressed the concern he had felt when the first dates needed to be cancelled: 'We had done our rehearsals, and we were really on top, really in tip-top form. Then Robert caught laryngitis and we had to postpone a lot of dates and reshuffle them, and I didn't touch a guitar for five weeks. I got a bit panicky about that \u2013 after two years off the road that's a lot to think about.'\n\nRosen later revealed the written rules of engagement for journalists permitted on the band's plane:\n\n  1. Never talk to anyone in the band unless they first talk to you.\n  2. Do not make any sort of eye contact with John Bonham. This is for your own safety.\n  3. Do not talk to Peter Grant or Richard Cole \u2013 for any reason.\n  4. Keep your cassette player turned off at all times unless conducting an interview.\n  5. Never ask questions about anything other than music.\n  6. Most importantly, understand this \u2013 the band will read what is written about them. The band does not like the press nor do they trust them.\n\nLater on the tour, John Paul Jones learned that, eight years previously, Rosen had unfavourably compared Led Zeppelin to the Jeff Beck Group. The suddenly, irrationally furious bass player demanded that Rosen hand over all his cassette tapes.\n\nThe chucking of cherry bombs onto the stage had become like the dark antithesis of the jelly beans peppering the Beatles 12 years before. In Louisville, Kentucky, a lobbed bottle hit Page's guitar: he and the other musicians momentarily quit the stage, but soon returned.\n\nBut it was not all angst and moodiness. On 28 April in Cleveland, Ohio, there was a magical performance where everything gelled. And the enormous show at the Pontiac Silverdome was similarly inspiring.\n\nFollowing it Plant, Jones and Bonham returned to their families in the UK for a two-week break in the schedule. And Page? He flew direct to Cairo, lodging himself at the luxurious Mena House Hotel, adjacent to the Great Pyramid. Cairo, of course, was where _The Book of the Law_ , the foundation of Thelemite thinking, had been dictated to Aleister Crowley by an outer-world entity.\n\nOn 12 May 1977 Page was back in London. He and the other members of Led Zeppelin, along with Peter Grant, were at the Grosvenor House Hotel in Park Lane for the annual Ivor Novello Awards: they were presented with the 1977 award for Outstanding Contribution to British Music.\n\nA week later they were on a plane to the United States for the second leg of their American tour. It kicked off in Birmingham, Alabama, at the Coliseum, where Jimmy included a snatch of 'Dixie' in a guitar solo.\n\nWhen the band played in Maryland, adjacent to Washington, DC, Peter Grant invited a number of Soviet diplomats to the show, having already dined with them at the Russian Embassy. Grant had arranged for John Paul Jones to slip a section of Rachmaninoff's Variations into his set-piece solo during 'No Quarter'. The Soviet officials were delighted. As was Peter Grant: his ambition was for Zeppelin to play in Moscow.\n\nBut everywhere there seemed \u2013 appropriately enough for that year of change that was 1977 \u2013 near-anarchy from the fans. All around them on that final American tour, the dystopian, anarchic future promised by the very nature of Led Zeppelin, which had long lain dormant, appeared to have risen out of its primordial slime. On the tour's first leg there had been over a hundred arrests when a thousand fans tried to break into the Cincinnati Riverfront Coliseum; during a second show at the venue a youth fell from high up on a gantry and died.\n\nOn 21 May in Texas, a state that held almost as much empathy and adoration for Zeppelin as southern California, the band played the Houston Summit Arena; forty people were arrested and over half a million dollars' worth of damage was caused in the city after a post-show audience riot.\n\nA fortnight later there was far greater mayhem. In Tampa, Florida, always prone to the eccentric extremities of its sub-tropical climate, there was a flash flood 20 minutes into Led Zeppelin's set.\n\nThis show had already earned a difficult reputation: on the day that tickets had gone on sale, Zeppelin fans, tiring of waiting in line, had gone on a rampage of destruction, stealing food from stands, tearing up offices and ripping out seats. To control this outbreak, a SWAT team had been brought in from Miami.\n\nThe Tampa date was an outdoor event, before an audience of 70,000. Peter Grant, unusually for him but perhaps unsurprising considering his depressed and drugged frame of mind, had neglected to note that the contract with the promoters contained a 'rain or shine' clause; in other words, whatever the weather the show must go on. Since Leslie Harvey, Maggie Bell's guitarist in Stone the Crows, had been electrocuted onstage at Swansea Top Rank after touching an unearthed microphone with wet hands, Grant had been ever alert to the dangers of untrammelled electric force. He was always insistent that the outdoor venues Led Zeppelin played, such as the one at Tampa Stadium, must have the stage covered by a solid metal roof.\n\nAgain, this last edict had been overlooked at Tampa by the promoters; there was only a canvas awning above the stage, already overloaded with water from earlier rainfall. When the tropical storm broke 20 minutes into Zeppelin's set, their manager immediately pulled the band offstage, ending the concert. The 'rain or shine' clause meant the event could not be rescheduled for the following day. On learning this, a major riot broke out among the audience, who were chanting, 'We want Zeppelin! We want Zeppelin!' As the roadies began to dismantle the equipment, bottles rained down on them. Fans even attacked other fans.\n\nWaiting outside the stadium were 40 cops in full riot gear, most of whom were extremely unsympathetic to the ever-controversial notion of Led Zeppelin and their fans. Now they swung into action, quite literally, racing into the venue and bringing their billy clubs down on the fans' heads, who fought back with fists and any weapon they could find, as a full-scale insurrection appeared to be taking place.\n\nIt was a shocking scene: 60 fans were taken to hospital, along with 12 police officers. The next day, at Grant's hectoring insistence, the promoters placed a full-page advertisement in the local newspaper accepting full responsibility for the disastrous show, absolving Led Zeppelin of any blame \u2013 which was theoretically correct.\n\nThe next dates, in the relative civilisation of New York City, were punctuated by further firecracker attacks: when one hit Page on the hand while he was onstage, he returned to the dressing room for five minutes. He returned when it was apparent that he had not suffered any injury. But these six nights at Madison Square Garden also restored a considerable measure of triumph to the tour. Page proved himself to be the guitar god for which he was revered, delivering blistering solos and performances that were of the very highest order. Always aware of the value of prestigious venues, it is almost certainly no coincidence that, after a number of creaky shows in the American provinces, he rose to exhibit the full majesty of his status in New York City.\n\nThe Madison Square Garden shows allowed Led Zeppelin to briefly become social fixtures on the Manhattan scene: Page hung out with Keith Richards and Ron Wood, visiting Trax disco with them; and John Bonham's fondness for throwing television sets and sundry furniture off his balcony led to the band being ejected from the Plaza Hotel \u2013 for ever. Robert Plant purchased a new Lincoln Mark VI, which he immediately had shipped back to the UK.\n\nWhen Led Zeppelin left New York for California, Page was so out of it that he had to be virtually carried onto _Caesar's Chariot_. The party checked into the Beverly Hilton in Los Angeles, once again eschewing the pleasures of the Hyatt House. There was an English woman living in the city called Pennie who was friends with Plant and his personal tour manager Benji LeFevre. Jimmy Page, she was led to believe by the singer, was in a bad state during that tour because of drugs. For the seven nights at the Forum in Los Angeles no one would really see him until he emerged from his hotel room to go to the gig, where his playing would still be sensational. 'I was there all seven nights at the Forum and I can tell you that nobody slept,' said Michael Des Barres. 'Every evening Jimmy was carried onstage by Richard Cole and carried off.'\n\nBenji LeFevre would add 'sound tricks' to Plant's vocals with an electric harmoniser that permitted him to sing with himself. Then Page would perform a virtuoso guitar performance that could last for up to an hour, and for which the fans would go crazy; during such soloing Plant would sigh, 'Jimmy, oh Jimmy!' However, Plant and LeFevre would refer to this as 'Jimmy Page's Comedy Hour'. A sign of insurrection in the camp? An indication perhaps of the 'lack of camaraderie' that Richard Cole claimed was prevalent on Led Zeppelin's eleventh American tour? It was rare, Cole noted, for the individual band members to spend time with each other: 'When we did socialise, streaks of hostility or maliciousness towards other members of the group sometimes surfaced.'\n\nAlthough Page might have appeared to be missing in action, apart from when he was onstage, he was in fact right at the centre of the action. In his suitably murkily lit suite, he gave a few interviews and trawled through tour photographer Neal Preston's concert slides, anxious not to let out into the public domain any pictures in which his now slightly protruding belly was visible.\n\nWhen she finally gained admission to Page's suite, _Creem_ 's Jaan Uhelszki found herself in one of the most extraordinary interviews of her life: 'I'd been on the road with them for over a week and couldn't get him to agree to an interview. Finally, on the last day of the tour, he agreed to an audience on the condition that the publicist had to be there. I agreed, but didn't realise the implication until I began asking my questions. Jimmy stipulated that I must first ask the publicist my question and then she relay the question to him \u2013 even though we all spoke the same language and I was sitting a mere six feet from him. This went on for about an hour, and was so odd, and rather humiliating.'\n\nBy now Miss Pamela had married Michael Des Barres; in a nice twist, the ceremony had taken place in the back garden of Catherine James's house in Laurel Canyon. 'We saw a lot of Zeppelin, and they were not aging gracefully, except for Robert, who still had his shoulders thrown back,' Pamela Des Barres wrote of these Forum dates. 'Jimmy wore a Third Reich costume, made the Heil Hitler! gesture, and had to be propped up by two flunkies at all times. I saw him take 20 minutes to crawl across the room to get to a black bag full of pills. He kept toppling over, and everyone else in the room pretended not to notice. Or maybe they really didn't notice.'\n\nThere was a three-week break after the Los Angeles dates. Again, the rest of the group flew back to England. Again, Page flew to Cairo.\n\nOn 17 July 1977 the third leg of Led Zeppelin's epic tour of the US began in Seattle, before 62,000 people. It was only an average show. 'It was a night of pot, pills and popcorn with the popcorn coming in a close third to the other two,' wrote the _Seattle Post-Intelligencer_. 'But overall, the Led Zeppelin concert at the Kingdome came off without too much trouble. There were several arrests, lots of dope and booze smuggled in \u2013 either under coats or inside bodies \u2013 and some very sick kids from drinking too much. Plant promised that the 1977 tour would be \"blood, thunder and the hammer of the gods\". A squad of paramedics was geared up for the blood and everybody else was geared up for the blood and thunder part.'\n\nIf Seattle had been just average, the next gig, three days later, at Arizona State University Activity Center Arena in Tempe, was claimed to be 'a strong contender for their worst ever show'. After the concert began, an hour late with no explanation given, Page seemed utterly off his head. He fluffed intros, going into 'Black Mountain Side' for ten seconds before shifting into 'Kashmir', leaving the rest of the baffled group to catch up. Utterly static and seemingly uninspired throughout the set, he was thrown onto his back by one of the flash-pot explosions at the climax of 'Achilles Last Stand'. 'Moby Dick' was dropped, the set concluded with 'Stairway to Heaven' and there was no encore.\n\nThree days later Led Zeppelin and their considerable entourage boarded _Caesar's Chariot_ and flew to San Francisco for two dates just across the bay, in Oakland, California. Then they would fly to New Orleans.\n\nThe first Oakland show, of course, was the one at which Bill Graham's employee Jim Matzorkis was attacked and beaten by John Bonham, Peter Grant and John Bindon. Thoroughly egregious, it was the moment when all of Led Zeppelin's legendary inhumane treatment of lesser beings fully came under global exposure in all its shocking dysfunctionalism. 'I wasn't there, but I do know it was nothing really heavy. Certainly nothing heavier than I'd witnessed out front during the concert,' Page said to me in 1979 about the Oakland assaults.\n\nOn receiving the devastating news of the death of his five-year-old son Karac, Robert Plant immediately flew back to London that night, Tuesday 26 July, accompanied by John Bonham and Richard Cole; the last seven dates of Led Zeppelin's tour were summarily cancelled. _Caesar's Chariot_ was not licensed for travel outside the US and Atlantic Records had lent their corporate jet to President Jimmy Carter; Cole booked the three of them onto a flight from New Orleans to Newark, and then first-class seats on a British Airways plane to London. At Heathrow, Cole had arranged for a private jet to immediately whisk them up to Birmingham Airport. Plant never broke down, but tears were frequently visible on his face. 'Karac was the apple of Robert's eye. They idolised one another,' said Plant's father, who met his son at Birmingham Airport. 'All this success and fame,' his father added. 'What is it worth? It doesn't mean very much when you compare it to the love of a family.'\n\nDid Richard Cole have time during that unhappy flight back to London to ponder on what had happened in Oakland? He, John Bonham, Peter Grant and John Bindon had been arrested and charged with assault. They were on $250 bail each, and awaiting trial.\n\nWith the assistance of his old friend Bonham, Plant withdrew into himself, alone with Maureen and Carmen. He gave up all drugs, cancelling them out of his life in an instant. But he did drink so much beer that by the next spring he considered himself to be obese. By then Maureen was pregnant.\n\nWhat still seems utterly extraordinary, a certain sign of the complete collapse of the flimsiest moral structure or humanity within Led Zeppelin, as though they didn't give a fuck, was that neither Jimmy Page, John Paul Jones nor Peter Grant \u2013 who was probably still blaming Bill Graham for Karac Plant's death \u2013 attended the poor boy's funeral, held in the first week of August. What on earth was each of them thinking? Were they so consumed by rock-star narcissism that they were simply unable to view life from Robert Plant's point of view and see that he would certainly need their support? Or \u2013 certainly in the case of Page and Grant \u2013 were they simply so out of it on smack and downers and booze that their thinking was utterly befuddled? A bit of each, one might feel. It was a big mistake, not least in the eyes of Plant.\n\n'Shortly after Karac died,' said Joe Jammer, 'I bumped into Robert and Maureen outside of Swan Song. He was getting into a Jaguar. And he says to me, \"Stay away from Jimmy. Stay away from Jimmy. He's evil, he's evil. It's his fault that Karac is dead.\" And Maureen is crying her eyes out. And I'm standing there going, \"Oh, it's Jimmy's fault?\" And then they were gone.'\n\nWith the UK media more concerned with the seemingly unstoppable upwards surge of punk rock, the summer months of 1977 were a strange time: it felt as though the old order was being utterly overturned \u2013 Jimmy Page's world of Pre-Raphaelite whimsy seemed rather outdated and irrelevant. On 16 August, Elvis Presley, the King of Rock 'n' Roll, who had so inspired both Page and Plant to become musicians, died of a heart attack at the age of 42 while seated on the toilet. In September 1977, hurrying back from the pub in his Jensen Interceptor, John Bonham ran out of road near his home and ended up in a ditch, breaking two ribs in the process; abandoning his vehicle, he struggled home on foot before police arrived.\n\nIn the mythology of the street Led Zeppelin's tragic fall was interpreted as a perhaps deserved response to the group's gargantuan ambitions, aspirations that had been driven by Page's supposed obsession with the occult. Rumours now surfaced of that deal having been done with the Devil to ensure Zeppelin's success when the band first formed; of the blood sacrifice in which Page, Plant and Bonham had participated, but which John Paul Jones had stepped back from. In the subsequent reading of this, following one further large tragedy, the proof was declared to be in Jones emerging from his time with Led Zeppelin apparently unscathed.\n\nZeppelin biographer Mick Wall claimed that the distance Plant felt from the rest of the group, apart from loyal, supportive Bonham, was profound; that Page, Jones and Grant's non-appearance at Karac's funeral created a rift that never truly healed. 'Until then, Robert was still in thrall to Jimmy and what he had created with Zeppelin. After that incident, Jimmy no longer held the same mystique for Robert,' Wall remarked in 2011. 'It was also the beginning of Robert having much more power over what the band did or didn't do next. He truly no longer cared and therefore was ready to walk at any point if they didn't fit in with him. And that's the way it remains to this day.'\n\nFor his part, Plant was considering giving up singing altogether; he enrolled on a course to become a teacher of the Rudolf Steiner method of education.\n\nPage, however, was intent on keeping the Led Zeppelin flag flying. On 5 September 1977 he went with Atlantic Records UK boss Phil Carson to the annual WEA Records convention, held that year at the Metropole Hotel in Brighton, not far from Plumpton Place. As the evening wore on, the guitarist found himself onstage in a loose jam with the soon-to-be celebrated British composer and musician John Altman, who was on piano, along with Carson on bass. They played 'an instrumental blues which went on for at least 30 minutes \u2013 Jimmy kept apologising for losing his place. And there was a more up-tempo number also,' recalled Altman. Towards the end of the set a Warner employee took off all her clothes and writhed on top of Altman's piano \u2013 at which point, as he would do at Rodney's in LA in such circumstances, Page made himself scarce. Altman was at the convention to play sax with a pack of the first wave of British rockers, including Vince Eager, Marty Wilde and Wee Willie Harris, who all witnessed Page's guitar efforts. 'Jimmy was pretty out of it, I have to say,' thought Altman.\n\nAlthough he was only performing in front of an invited audience \u2013 at what would later be defined as a 'corporate event' \u2013 Page was displaying himself in the best way he knew how. Much was being written and spoken about Plant having twice fallen victim to 'the Led Zeppelin curse'; not perhaps that initiated by Kenneth Anger, but a more nebulous business, a product of Page's endless dabblings with darkness. The Led Zeppelin leader was only too aware of such supposition. Mired, however, in addictions to heroin, cocaine, downers and alcohol, he was floating only inches above a black slab of depression generated by his own confusion about the cause of recent occurrences, including the all-consuming madness of the feral audiences on that last US tour. Why had everything become so crazy? Was he responsible for it? Even though he refused ever to acknowledge it in any interview form, he would have to have been very gone indeed not to have considered this. Maybe he was.\n\nAfter Page and Grant had a meeting in which each was adamant that their singer be left to nurse his pain for as long as necessary, the guitarist declared that he would take a holiday, in Guadeloupe in the French Caribbean, with Charlotte Martin and Scarlet. When Page suggested that Richard Cole accompany them, he revealed a double purpose: the holiday would be an endeavour to get 'clean', as heroin would be impossible to score in Guadeloupe.\n\n'It means about two weeks without heroin, but with plenty of white rum,' he told Cole, revealing his intention to mollify heroin withdrawal on the island by getting drunk on powerful overproof rum; it sounded like Page had been taking drug-dependency advice from Keith Richards, who famously substituted cocaine and vodka for heroin.\n\nPage and Cole duly drank their way through the holiday. You wonder what Charlotte Martin and Scarlet thought to this. Despite their essentially successful efforts to kick their addictions, both Page and Cole returned to junk when they came back from the Caribbean. 'I had heard that Pagey got back into heroin before long, but because I didn't see him for a while, I had no way of knowing for sure. When the band re-formed the following year, however, he seemed as immersed in smack as I was,' wrote Cole. In fact, Cole confessed that within two days of his return he was back on the gear. And his level of alcohol consumption had increased, it would seem, exponentially: Cole went to Peter Grant's home straight from the airport, drinking 40 cans of beer as Grant debriefed him over the condition of their star asset.\n\nNo one in Led Zeppelin was doing anything of note, which gave plenty of opportunity for those so inclined to do very bad drugs. Normally Page would have been eager to seize such free time for ceaseless studio liberation, the science lab of his great art, working out new creations. But apart from allegedly going through Led Zeppelin live material from 1969 for a potential live album, which never materialised, he seemed to do very little. There were some press interviews that he really only undertook to deny the persistent reports that Led Zeppelin were over. And there was that other unrelenting rumour: that he would replace Keith Richards in the Rolling Stones if the Stone received the seven-year prison sentence he was facing in Canada for large-scale heroin possession. That one junkie would be replaced by another never seemed to be factored into this increasingly well-lit fantasy.\n\nYes, Jimmy Page is into junk. But it's not that at all. Why is he doing the junk? On the one hand because of the colossal pressures of being the biggest rock star in the world (though Robert Plant is a strong rival for that position), but also because, in his Pre-Raphaelite vision, he really wants to see what life's possibilities are. As Michael Des Barres says, he's in a position to do so. And he wants to see what it feels like.\n\nYou just can't come down: after the shows Page's adrenaline is firing so fast on the buzz of the never-guaranteed achievement \u2013 and its accomplished success \u2013 and the fiery love of the audience that it is impossible to even consider going to bed. Life \u2013 the endless party! \u2013 has now just started. Get high, that's the first edict. Because otherwise you might succumb to the nights of sleep you've missed. Get high. And get going, whatever it is: sexual, narcotic, intellectual or a bit of all of them, which is the best, clearly the most interesting, as Aleister Crowley would have prescribed. But Jimmy Page doesn't have this anymore. He is at home, exhausted. It has all finally caught up with him. He is drained beyond all belief, even without Plant's terrible difficulties. Where can he replenish his energies? Smack is the anaesthetist, but not the route, as the paucity of material for the next album demonstrated. But that was a little in the future.\n\nIn the spring of 1978 Roy Harper gave an interview to a farming magazine. Much of the interview concerned the sheep he kept on his farm. But he also mentioned that he had been working on the words for new Led Zeppelin material with Jimmy Page. Considering himself also to be a farmer, Robert Plant subscribed to the same publication. And he was outraged by what he read. Brimming with anger, the singer phoned Page, who denied the story. Yet it seemed to be a creative connection, and accordingly Page conversed with Peter Grant.\n\nIn May 1978 Grant booked several days at Clearwell Castle, which Bad Company had already used; on the Welsh border, it was not far from Plant's home. As seemed obligatory with properties associated with Page, the neo-Gothic edifice was haunted, this time by a female ghost with a penchant for singing lullabies to her spectral child; for added effect she would employ a music box. For some time the four musicians played and jammed, but there was almost none of the requisite magic. Page was urging the others on, insisting that it was time for Led Zeppelin to resume business. 'Robert, however, wasn't so sure,' wrote Richard Cole. 'He knew that once the band got back into motion, there would be no turning back. And he wasn't convinced that he was ready, that music was as important to him now as it had been before Karac's death.' Several times during those Clearwater Castle sessions Plant declared that he would never again sing 'Stairway to Heaven'.\n\nA couple of months later, the singer got in touch with an outfit local to him fetchingly named the Turd Burglars. In a small church hall in Worcestershire, Robert Plant essayed a handful of songs with them, including the rock 'n' roll classic 'Blue Suede Shoes'. In August, while holidaying in Ibiza, he went to the club Amnesia with Dr Feelgood, a sometimes magnificent R&B outfit who peppered their own material with such classics as 'Riot in Cell Block Number 9' and who had played at Led Zeppelin's after-show party at Earls Court; Plant joined in on their repertoire. A few weeks later he appeared onstage in Birmingham with Dave Edmunds, who had been signed to Swan Song; the Zeppelin singer worked out on material similar to that played with the Feelgoods.\n\nDuring the summer of 1978, Peter Grant, who was only 43 years old, suffered a minor heart attack. He was advised to utterly change his lifestyle: to give up alcohol and drugs, and to change his diet in order to lose some of his considerable weight, which had shot up to 28 stone. The coronary thrombosis perhaps could have been predicted: as well as his main source of income potentially drying up, the manager had been involved in that bitter divorce battle over the custody of his children, which he had won. Moreover, he had spent much of the time since the Oakland debacle involved in the legal fall-out.\n\nAt the court case in Oakland, six months after the event, Grant, John Bindon, Richard Cole and John Bonham had filed a joint plea of _nolo contendere_ (a Californian legal byway that effectively means 'I will not plead guilty'); found guilty all the same, they each were fined and given suspended prison sentences. Grant was heard to remark that 'the biggest mistake I ever made was employing John Bindon'. (Later that year John Bindon would kill Johnny Darke in a knife fight near Putney Bridge, further along the New Kings Road from Swan Song. This stabbing was allegedly a hit, the bankrupt Bindon having been offered \u00a310,000 to murder Darke. Badly wounded himself, Bindon managed to escape to Ireland, where he had an operation. At his trial for manslaughter the next year, he was somehow found not guilty, assisted considerably by the character-witness statement of the actor Bob Hoskins. John Bindon would die in 1993 of an AIDS-related illness.)\n\nThe four Led Zeppelin members only met again in September 1978, at Cole's wedding, a joint ceremony with Simon Kirke, the drummer with Bad Company. Wary of his potential response, no one mentioned to Plant that Led Zeppelin might now come back together.\n\nFinally it was felt that sufficient time had passed so that Led Zeppelin could enter a recording studio. But where? Certainly not in the UK. There were considerable tax advantages to be enjoyed by British musicians who chose not to record in their home country.\n\nTime was booked at Polar Studios in Stockholm, from 14 November to 21 December 1978. It was perhaps not the best time of year to visit Sweden, which was bitterly cold and snowbound when Led Zeppelin arrived the day before the sessions began. Polar, which had only opened in May that year, was owned by Bj\u00f6rn Ulvaeus and Benny Andersson from Abba; keen to popularise their venture with overseas artists, they gave Led Zeppelin free time at their studio. Mr Led Wallet jumped at the idea.\n\nThe four musicians had been rehearsing new material for six weeks at Eazy Hire in north London, much of which had come from John Paul Jones, aware that someone needed to be in the driving seat. During the Eazy Hire sessions, Jimmy Page brought in a Gizmotron, a bowing gadget for the guitar invented by 10cc's Kevin Godley and Lol Creme. Page adored it, and used it extensively on the moody introduction to 'In the Evening'.\n\nFor the Polar Studios sessions, Led Zeppelin would fly to Stockholm each Monday lunchtime, returning on an evening flight to London the next Friday night. Yet Page was clearly not match fit. Frequently he would arrive at the studio \u2013 as Jones put it \u2013 'two days late', as would John Bonham, half the band in thrall to what Kenneth Anger called 'the white lady': Richard Cole had found a connection close to the studio. On the other hand Jones and Robert Plant would turn up at ten o'clock each morning, fresh-faced and eager for work. They would lay down their parts, which Page would take back at the weekend to Plumpton Place, where he would add his own sounds. The tracks would sound utterly different when they returned to Polar. (Down at Plumpton, Page was even known to jam in the village pub, the Half Moon. 'I remember being in JP's house where he got his first home studio,' said Joe Jammer. 'It was for Charlotte's birthday: me, Jimmy, Jeff Beck, Eric with Pattie Boyd, Ronnie Wood and John Paul Jones, and in the kitchen he had a giant picture of the Sex Pistols. He took me up to the top of the house in the country and showed me his first ever studio. Lots of equipment. He wanted knobs made of Bakelite. He played me his first composition in this studio, and it was the chorus from Handel's _Messiah_ , which he did himself. But it ain't funky. I brought a whole load of James Brown albums to play. He warned me of the future: \"There's a new movement coming called digitalisation.\" He said, \"Resist it as much as you can.\" He said it will be good for editing and for mixing, but it will be no good for recording. Always record analogue and always mix and edit digitally.')\n\nAlthough Page would later claim that this last Led Zeppelin album, which would be tellingly titled _In Through the Out Door_ , was somewhat lightweight, he needed to shoulder most of the blame: his smacked-out lack of vision was the major contributory factor. Jones and Plant were only too aware of Page's condition, as well as Bonham's, and simply seized the reins. After all, didn't someone have to?\n\nAlthough Page would still be credited as 'producer', all such work on _In Through the Out Door_ was essentially done by Jones, with Plant also a force. Frozen in his heroin addiction, Page had minimal input. Over the years Jones had accumulated plenty of material that not had been used by Led Zeppelin. And he had lately acquired a Yamaha GX-1 synthesiser \u2013 a harbinger of the electronically dominated sounds of the imminent eighties \u2013 whose sound washes he applied rather too lavishly to the project, although in so doing he invented what would soon become clich\u00e9s. So the new sound that Led Zeppelin displayed on _In Through the Out Door_ could be interpreted as their being ahead of the cutting edge. Or it could be viewed as a somewhat desperate effort to release some new product and shore up the foundations of an increasingly dysfunctional outfit in which, with the exception of Jones, all the other players were in one way or another exceptionally vulnerable \u2013 including their manager, who seemed never to be at Swan Song and was almost uncontactable. Swan Song indeed seemed especially dormant: both the Pretty Things and Detective had broken up, and it was now that Maggie Bell was reduced to manning the reception desk at the company's offices.\n\nThe rockabilly 'Hot Dog' somehow set the tone: the fourth of the seven songs on the record, it was like an Elvis Presley outtake, momentarily appealing but instantly forgettable; in the US it would be released as an unsuccessful single. There was one great track: the opener 'In the Evening', moody, Moroccan and misty with the sound of meditative contemplation. In the tradition of Zeppelin's epic set-piece tunes like 'Kashmir' or 'Achilles Last Stand', it was the album's standout track, running to almost seven minutes.\n\nJones and Plant wrote 'All My Love', the album's other significant song, with no input at all from Page. Clearly a tribute to Karac, Plant's vocals are almost heartbreaking in their intensity. Later Page was dismissive of 'All My Love': 'I could just imagine people doing the wave and all of that... I wouldn't have wanted to pursue that direction in the future.' His cold assessment of the song seemed rather to miss its point.\n\n'I'm Gonna Crawl', which closed the LP, was an archetypal Zeppelin blues lament. 'On this album,' wrote Stephen Davis, 'where he seemed to have run out of ideas, Jimmy Page turned back to the blues for solace, and his solo seemed to cry with abject remorse.'\n\nIt couldn't be denied \u2013 there was something desperate about _In Through the Out Door_. Yet one creative act by the still-grieving Plant and his wife had been successful: on 21 January 1979 Maureen gave birth to a son, Logan Romero Plant.\n\n## 22\n\n# BONZO'S LAST STAND\n\nIn his _Rolling Stone_ review of _In Through the Out Door_ , Charles M. Young, who had written the magazine's first article about UK punk, argued that Jimmy Page's diminished creativity had left Robert Plant with a lack of strong material to work with; somewhat unsympathetically in the circumstances, he also condemned Plant's lyrics as inane. In _Melody Maker_ Chris Bohn asserted that Zeppelin were 'totally out of touch' and 'displaying the first intimations of mortality'. By contrast, Nick Kent, in the much more punk hard-line _NME_ , found the LP in no way an 'epitaph', seeing its 'potential points of departure' as being worthy of further plays.\n\nAll the same, _In Through the Out Door_ was number one after only two weeks in the US, and sold 6 million copies; in the UK it was similarly number one, shifting 300,000 units.\n\nWith a new album about to be released, on 15 August 1979, what would any act do? They would endeavour to promote it with live shows, the umbilical link between 'product' out in the marketplace and the touring circuit.\n\nHaving somewhat recovered from his coronary, Peter Grant started to put out feelers. Rather than announce a tour, he wanted Led Zeppelin to come back \u2013 for a 'comeback', as showbiz parlance would have it, was essentially what this would be, a career salvation operation \u2013 with an enormous staged event, not dissimilar to Bob Dylan's surprise return to the stage at the 1969 Isle of Wight Festival after his motorbike accident.\n\nGrant started to talk to his old friend Freddy Bannister, who had staged the Bath Festival of Blues and Progressive Music, which had been such a memorable triumph for Led Zeppelin in 1970, their true UK breakthrough point. Since 1974 Bannister had held annual one-day festivals at Knebworth House, in Hertfordshire, 30 miles north of London. These always involved one stellar name, with a strong bill of supporting acts. The first event featured the Allman Brothers, with support from \u2013 among others \u2013 Van Morrison, Tim Buckley and the Sensational Alex Harvey Band. Indeed, before Grant settled on Earls Court, it had been mooted that Led Zeppelin would be the headline act at Knebworth in 1975. Part of Bannister's ethos was to give his audiences the best possible value for money by keeping tickets at as low a price as possible.\n\nIn May it was announced that Led Zeppelin would play an outdoor concert at Knebworth House on 4 August 1979. Freddy Bannister had balked at the idea at first: Grant was demanding a cool \u00a31 million for two shows, on successive Saturdays. There was no question that the event would be a success. Yet from the off the atmosphere of excess about everything connected to it seemed utterly contradictory to the prevailing do-it-yourself mood of punk. The paradoxically puritanical United States of America is endlessly attracted to prestige: the excess that Zeppelin purveyed \u2013 money, power, arrogance \u2013 was how megastars were expected to behave. It didn't go down so well in the UK.\n\nOffered the opportunity to interview 'Pagey' to promote the Knebworth dates, I willingly accepted. My article was published in the 4 August 1979 edition of the _NME_ , the day of the first Knebworth show. I had interviewed Page eight days previously, on a late Friday afternoon. It was given the title \u2013 not chosen by me \u2013 of 'Smiling Men with Bad Reputations'. Here it is:\n\nOf all the old superfart bands it is certainly Led Zeppelin who have been and still are the most reviled by the New Wave.\n\nWhatever jerk-off socialite absurdities Jagger may have got himself into, the Rolling Stones have at least always had one of the prime punk archetypes in Keith Richards. The Who, meanwhile, have the ever-perceptive Townshend, a man who appears to have gone through something of a personal rejuvenation that seems to be a direct result of his encounters with Punk.\n\nFor whatever reasons, though, the manner in which Led Zeppelin have consistently presented themselves has made the band's name synonymous with gratuitous excess. Even the almost equally guilty Pink Floyd have at least had the decency and sensitivity not to quit these shores just for the sake of saving money.\n\nDon't sell your soul for silver and gold, as Lee Perry once said. If rock 'n' roll is essentially an all-encompassing roots culture, then obviously any musician who isolates himself away in some anal-retentive tax-exile lifestyle is neither responding to his obligations nor in harmony with those roots. Also, his initial purpose and motivation must be doubted.\n\nThe Clash's Paul Simonon summed up pretty well the total lack of respect that the new bands feel towards Zeppelin. 'Led Zeppelin??? I don't need to hear the music \u2013 all I have to do is look at one of their album covers and I feel like throwing up!'\n\nIn some ways part of the reason for the venomous loathing directed at the band is not just because they've let themselves down, but also because you know damn well that Jimmy Page at least \u2013 like many of the new Punk icons a former art student \u2013 certainly knows better.\n\n'I've read about many records which are supposed to have turned me on to play rock 'n' roll,' the guitarist told _Trouser Press_ in September 1977, 'but it was \"Baby Let's Play House\" by Presley... I heard that record and I wanted to be part of it; I knew something was going on. I heard the acoustic guitar, slap bass and electric guitar \u2013 three instruments and a voice \u2013 and they generated so much energy I had to be part of it. That's when I started.'\n\nYet, in the same way that the death of the original, classic rock 'n' roll punk the previous month to the publication of that article could have been seen as a serious warning of the false paths and box canyons into which Babylon could misroute rock 'n' rollers, it also appeared at the time that perhaps the whole mighty edifice which Led Zeppelin had created itself to be was starting to crumble away as inevitably as the Malibu Beach Colony will one day slide into the Pacific Ocean.\n\nBy the middle of the year when the two sevens clashed, the belief that the whole Led Zeppelin operation had got it all more than a little bit wrong appeared to be backed up by concrete facts. The band appeared to be in a state of crisis. In artistic terms they seemed to have reached an absolute nadir. Following the turgid _Presence_ LP released in the spring of the previous year there'd then been, six months later, the critically lambasted _The Song Remains the Same_ film and double soundtrack album. Even this emphasis on double records \u2013 two out of the three LPs the band had put out since they'd formed their own Swan Song label had been two-record sets when none had been released before \u2013 suggested attempts to milk their market for all it was worth while fighting a rear-guard action to forestall an inevitable end.\n\nPerhaps more to the point, though, a general atmosphere of personal doom and gloom appeared to surround the once invincible Zeppelin. From the outset the lengthy US tour undertaken by the outfit in the spring of '77 seemed ill-fated. It was to have been the band's first live work since 1975 when vocalist Robert Plant had been severely injured in a car smash on the Greek island of Rhodes during a year of British tax exile. It was ominous then that the first dates were cancelled when Plant developed a throat infection.\n\nJimmy Page himself was also believed not to be in a good state, an assumption fuelled by the news that the full-time services of a doctor were being employed to care for the guitar hero. Now Page denies that the medic was there to look after him alone \u2013 'We had a doctor to look after all of us, period. It was a bloody long tour' \u2013 with the same ease that he dismisses reports of his having been wheeled around between gigs in a wheelchair \u2013 'I may have done that for a laugh \u2013 not seriously. No, no. That wasn't happening at all.'\n\nIn addition, manager Peter Grant was said to be severely depressed following a divorce. Matters appeared to reach what seemed to be an inevitably unpleasant culmination when, following a Bill Graham-promoted San Francisco gig, one of the promoter's security men was badly beaten up by Grant, drummer John Bonham and one John Bindon, a Zeppelin employee.\n\nIf some form of near tragedy during the tour had seemed unavoidable, however, it was yet to wreak its worst toll. This happened some two weeks later when Robert Plant's five-year-old son died of a sudden mystery virus infection and the tour was abandoned while the grief-stricken singer flew home.\n\nNow, of course, all these incidents may be seen as random happenings, as the chance intervention of fate. However, if you believe that you create your own fate and that human beings do not exist in isolation from one another and from the universe but are part of a far greater, interacting scheme in which actions and activities of the past create those of the future, then all this begins to look rather different.\n\nCertainly, Jimmy Page's interests in the occult suggest that he should believe in such a cosmic overview. Indeed, there are those who would claim that it is solely down to Page's interests in these matters that such a tragic atmosphere has surrounded Led Zeppelin in its latter years. Personally, though, I don't think that Jimmy Page has inked a pact with Satan. To think like that is mere superstition and that's taking into account certain rumours which have floated about the music business the past 18 months or so that there are even certain members of the Zeppelin entourage themselves who lay blame for these assorted misfortunes on Page's fascination with Aleister Crowley.\n\nWhen it comes down to it, though, I don't really think that there's been some clear-cut metaphysical holy war of good and evil waged on the rock 'n' roll boards the band has been treading the past ten years. In fact, it's probably that outside interest which has kept the guitarist's head relatively together during the most successful years of the band. The occult, after all, is concerned with knowledge and plumbing one's own mystic depths for certain truths that are beneficial to the whole of humanity. Yes, of course, it can be used in a malevolent manner, but to view all occult activity as the work of the Devil is a red herring laid down by Babylon and therefore is the work of the tricky Devil himself.\n\nI think, though, that Jimmy Page is very confused. His confusion doesn't spring from his occult interests but, I feel, from the very nature of Led Zeppelin itself and his position with regard to the band of which he is indubitably the leader. Indeed, when we met in Swan Song's London office on the hottest day of the year, it became glaringly obvious that Jimmy Page was totally comfortable and, at times, positively exhilarated when talking about these extra-curricular activities.\n\nIt was noticeable, however, that when the conversation changed to the subject of his band he appeared frequently to find eye contact exceptionally awkward. Now, it's quite possible to blame that on the fact that in the isolated, self-enclosed existence in which Jimmy Page dwells he probably doesn't have that much verbal interchange with people outside his own sphere. Also, like many musicians who are far more at ease when living out their fantasies onstage, he may well be slightly nervous. Mind you, although a hermetic, fairly newsless lifestyle is part of the whole Led Zeppelin problem anyway, Page's behaviour does suggest that he is not always totally convinced by his arguments \u2013 and Page is adept in the art of being a media salesman for his band while at the same time revealing little about himself; check how many times the word 'Knebworth' gets mentioned in this piece.\n\nIt's the very nature of Led Zeppelin itself that is the problem. Let's not mince words, it's always been regarded as a 'heavy' operation. There has been a slightly odd vibe about it.\n\nNow, of course, part of the nature of rock 'n' roll is the manner in which it allows people involved with it to live out their childhood Cowboy and Indian fantasies. So I don't know whether Peter Grant really is a figure from the fringes of the underworld or whether he just enjoys people thinking that he is. I suppose it doesn't really matter (though in a way it does) because I've no doubt, as Page himself comments later on, that certain of the behind-the-scenes music industry figures with whom he has to deal, particularly in the States, actually are dodgy characters. So maybe it gives him an edge over them. (Unless they're also all just living out their fantasies \u2013 in which case it all gets a bit complicated and self-perpetuating, and a bit pointless too.)\n\n'The whole point of the bit in _The Song Remains the Same_ film,' Page tells me when I ask him about this, 'where Peter plays a gangster, was just to send up all that and show how it was just a joke anyway.'\n\nNevertheless, I counter, there was the slight problem with the security guy in San Francisco. There's certainly concern in his voice when he replies, 'I didn't see it, you know, so I can't say exactly what happened. There were no million-dollar lawsuits put out on me, y'know.\n\n'But,' he continues, 'you must remember that Bill Graham has a very heavy reputation, that all his security people have a reputation for heaviness. As for Peter... Well, he's a very big guy and, if people are coming up to him all the time and calling him a bastard and telling him to piss off to his face, then he's probably going to react accordingly.'\n\nAlright, fair enough. But let's not forget that John Bindon is currently in Brixton either awaiting or serving a sentence for a subsequently committed manslaughter [in fact, John Bindon would soon be acquitted] \u2013 an incident which wasn't connected in any way with Led Zeppelin. Once again, judging from his reply to being reminded of this, there's no doubt that this genuinely troubles Page, much more out of real concern for Bindon, I feel, than for any unhappiness about him being linked with Zeppelin. It's a pity I forgot at the time, but I'd like to have also got his reaction to Nick Kent's claim that John Bonham once threw a drink over the hapless writer for a negative review.\n\nBut that's by the by, I suppose. It seems more important to tell the guitarist that, whether he's aware of this or not, an oft-expressed opinion on Led Zeppelin has been that the problems Robert Plant has faced have been something of a karmic backlash that Plant, as the most accessible and open band member, has had directed towards him.\n\nPage seems very shocked by this. 'I don't think that's so,' he replies slowly, almost as though slightly dazed, 'if what we were doing was really evil then... then I suppose we'd just put out lots of records and try and make loads of money... I hope that's not so.'\n\nSometime about the middle of last Friday morning I'd had a call from the Swan Song press office. Could I arrive maybe an hour before the interview was due to begin? That way I could be given an earful of the new Zep waxing. I can't pretend the idea exactly thrilled me to the bones, especially in the light of the last studio album, _Presence_ , which I find utterly unenjoyable. If I felt the same way about the new, as yet untitled, LP, it could mean a chilly start to an interview.\n\nBy lunchtime, however, this potentially awkward situation had been resolved by Jimmy Page himself. A further phone call passed on the information that the guitarist felt it pointless for me to hear the record as it was, apparently, 'a separate entity' \u2013 from what I'm not certain. Obviously it did cross my mind that maybe he was thinking the same way as myself and saw little gain in the songs being numbered some time before the record was even released.\n\nPerhaps predictably, when the record did come to be mentioned he was full of enthusiasm for it. The titles of the new numbers are: Side 1 \u2013 'In the Evening', 'South Bound Suarez', 'Hot Dog'; Side 2 \u2013 'Carouselambra', 'All My Love', 'I'm Gonna Crawl'. The Knebworth bashes will feature 'at least two songs from the new album plus several numbers from previous LPs that haven't been performed live in the past. What can I say?'\n\nI was also asked for some idea of the sort of questions I'd be asking. As I was at that time deciding on these for myself I couldn't really help out there. Besides, would this not have detracted from the natural spontaneity of the occasion? I was, however, informed that questions about the death of Plant's son and about Aleister Crowley were strictly taboo. This did not augur particularly well, especially when, while waiting for the assistant-editor chap from the _Melody Maker_ to finish his rap with Jim, photographer Adrian Boot emerged from that session to inform Pennie and me that Page was 'doing a Chuck Berry' and ignoring most of Michael Watts's questions. The guitarist was also apparently none too happy about Boot's snapping needs.\n\nIn the event, of course, Page gave Pennie plenty of pix-taking time prior to our encounter. Also, as far as my interview went, Page and I just started talking conversationally (but not before he made a rapid attempt to flog Knebworth) rather than adhering to any strict question and answer form. This situation lasted for much of the interview.\n\nPage was drinking pints of lager from a straight plastic glass and chain-smoking Marlboros. So was I. It was probably down to a combination of the booze and the hot weather, but the conversation quickly became very speedy. Maybe we were also blocked on the carbon-monoxide fumes wafting in through the open window from the early evening rush-hour traffic three floors below on the Kings Road.\n\nWhat with the roar of London Transport Roadmasters stopping just past the offices and the constant rumble of jets on their way to Heathrow overhead, it was often hard for either of us to make out what the other was saying. Though his enunciation is very clear indeed, Jimmy Page's soft Surrey accent \u2013 the family business is Page Motors in Epsom [Not so!] \u2013 makes him perhaps the most quietly spoken interviewee I've ever come across. Even so, I was pleased that he didn't pull the slumped-out whispering wimp number that I'm told is one of his favourite interview techniques. Not once, I think, did he lean back on the couch on which he was seated next to the window.\n\nNo doubt exacerbated by the booze intake \u2013 Jim is fond of the odd tipple, I'm told \u2013 perspiration poured off of his forehead in large drops, frequently lodging for a few moments in his close-to-shoulder-length hair. Coupled with the collarless striped white shirt he wore, he didn't look very different at all from when in the late sixties he laid down the ground rules for the classic Pre-Raphaelite, faintly androgynous British rock star. He actually looked younger than when I'd encountered him a couple of years back. Only the lines around his sometimes troubled eyes gave any indication of age.\n\nThe Selling of Knebworth began right from the very outset. I don't think you really like doing interviews, do you, I ask?\n\n'Well [laughs], it depends. I don't mind if the questions are alright.'\n\nYou look incredibly well.\n\n'Well, I was looking forward to... to Knebworth, actually. We've done a lot of rehearsing and checked things out. We've actually been down there and worked things out relative to the actual site.'\n\nIt must seem odd with it being such a long time since you've played onstage...\n\n'Well, it did at first... But then again it's like a natural amphitheatre, so I should imagine it's actually quite a good gig to be at. I went to Blackbushe, but that was a bit of a sea of bodies. But it was great to see Dylan.'\n\nPhew, that was close. The Zim to the rescue. At least we can talk about Bob Dylan for a while. This might be handy. Maybe if I mention to Jim that I met Dylan last year when he went to an Alton Ellis gig at the 100 Club and that he told me how he preferred the vibe in England to that in the States, and also in Germany from where he'd just returned, then we can get onto this matter of Punk and the New Wave without too much discomfort.\n\nInstead, though, Page mentions his surprise that Dylan had played in Nuremburg. 'I couldn't believe him doing that. They played the place where they had all the big rallies. He must have come out of there feeling very strange. I know I would and I'm not even Jewish.'\n\nHe hasn't heard of Dylan's conversion to Christianity. 'Oh, that's very interesting. Especially after that Nuremburg thing. When did that happen? Quite recently?'\n\nOh, about six months or so ago, I think.\n\n'We met his mum once, actually,' Page tells me. 'It was about the third tour and we were in Miami, and this typical Miami woman comes up with the spectacles and tinted hair bit and she says, \"Oh, I hear you're a group. My son's a singer. You've probably heard of him \u2013 Bobby Dylan. He's a good lad.\"\n\n'The strangest thing she said of all was that he always goes back to his... you know, the school turn-out when they got their degrees and things. He always goes back to that... Which is obviously a side of Dylan that many people would be actually shocked about. He's probably very orthodox in some areas where you expect him to be very bizarre and anarchistic.'\n\nLogically, I suppose, the matter of meeting Dylan at a reggae gig leads to discussion of matters Rastafarian. Jimmy Page is far more au fait with it than I would have expected.\n\n'Yeah, it's very interesting: the Lost Tribe of Israel and all that. It was at the time when Haile Selassie died that I wondered \"What's going to happen now?\" Because there is this big thing that he's invincible and that he would never die but obviously,' he chuckles, 'he could give up his bodily form if he wanted to, that was the loophole.\n\n'But it is fascinating.'\n\nWe talk about Egypt for a minute or two. Page's trip to Cairo had, indeed, been the subject of some quite splendid rumours. On the first leg, I think it was, of that last ill-fated Led Zeppelin US tour it was said that one night he'd been watching TV when the screen became filled with flashing lines. Immediately, so the tale went, he cancelled the next dates and flew off to Egypt. The conversation didn't lead into my mentioning that and, besides, I'm fairly certain I once read a fairly thorough refutation by the guitarist of that story.\n\nThoughts of Cairo seem to make Page feel very happy. 'I didn't want to come home,' he smiles, 'it was so good. I didn't go for long enough, though. I went at the end of an American tour and with every day I was there family ties in England were pulling more strongly. I just thought, \"Oh, I'll be back soon,\" and haven't made it yet. I'd certainly like to see the Valley of the Kings near Luxor.\n\n'I haven't been to many Arab countries, but I've been to Morocco and there and in other hot countries there's this constant hubbub, but in Egypt it's just so tranquil. It really is quite an experience. Let alone the pyramids.'\n\nEquinox, the Kensington occult book shop that Page owned and which specialised in the works of Aleister Crowley, is closed these days. The lease expired and, besides, 'It obviously wasn't going to run the way it should without some drastic business changes and I didn't really want to have to agree to all that. I basically just wanted the shop to be the nucleus, that's all.'\n\nHis interests in the occult haven't in any way diminished, however. 'I'm still very interested. I still read a lot of literature on it.'\n\nI mention that last time I'd gone past Equinox a small sticker that someone had placed on the door had attracted my attention. 'For the real truth about the changes in the Church of Rome,' it had read, 'write to the following address.' The name and address of a priest in Mexico was given. We talk about the Rasta belief that it was at the Pope's insistence that Mussolini invaded Ethiopia in order to prevent Haile Selassie organising the Christian Church in such a way that would have reduced the Catholic Church to the second largest Christian Church in the world.\n\n'I know the Pope definitely blessed the bombers going to Ethiopia,' says Page. 'That's a fact. My lady went to the Vatican. She said it's like Fort Knox, a completely separate state. A highly guarded treasury. And they have all these links with suspect organisations...\n\n'The whole image of the Pope being borne around St Peter's on a throne doesn't even bear thinking about. They had some programme on TV about the Vatican and they got through to one of the heads of the business division. And he was asked if it wouldn't be an act of faith to give all this wealth away \u2013 if your faith was sufficiently high and strong then obviously this wouldn't really affect the church. But he was dumbstruck. So obviously,' he laughs, 'he didn't have the faith.'\n\nJimmy Page has had some involvement with the community politics up in Scotland where he owns Crowley's former home, Boleskine, on the shores of Loch Ness. After, against much opposition on the local council, a harbour wall utilising raw materials had been built under the guidance of the local job recreation scheme, Page, largely as a result of previous similar activities within the community, was involved in the final unveiling ceremony. The local Labour man, he said, jumped on the platform at this event in a predictable attempt to make political mileage.\n\n'I just got up and said, \"I'm not here for any political reasons whatsoever but just from my own endeavours as an untrained musician. And it's just sheer determination that's been employed here against a good 80 per cent of the council who wished them to have no encouragement whatsoever.\"'\n\nOne is not particularly surprised that the politician appeared to milk the event for his personal aggrandisement; it is the nature of such a breed of people to behave in that manner, no matter what political party they belong to. It does seem interesting for a moment, though, that, when I inquire as to whether the council members were operating in a truly reactionary manner, Page seems a little uncomfortable when he realises that I regard 'Reactionary' as being synonymous with 'Tory'.\n\nMaybe that's by the by. Page has, after all, been involved up there in other battles with officialdom. 'The Hydro board in Scotland were putting in this scheme which wasn't of benefit to anyone except for a small percentage of local labourers \u2013 although, in fact, most of them were being brought in from places like Manchester and Liverpool.\n\n'What it was going to do was pump power at peak times to the South. It wasn't going to benefit the Scots at all. And for this they were going to put pylons up all over the place and mess up the loch. There were no pylons there whatsoever before. And I just didn't think it was on. For them, of course it was purely a financial investment. It was really a revelation to see how these things go on. So corrupt.\n\n'But we managed to force a public inquiry whereby it was put under the Secretary of State. They really put you through it at those things. It's like a court of law. They try and throw so much mud at you. Although it does seem that in London these days if they're pulling down buildings to put up new ones they are trying to keep the old facades. It makes it much more palatable. At least you don't get things like that too much,' he waves in the direction of the World's End council-housing project.\n\n'But,' he continues, 'so often people just get apathetic and think there's nothing they can do. At least sometimes you can uncover a bit of unsavoury business that's going on. I do really care about these things. I don't particularly go around doing a load of public campaigning, but both those things were there on my doorstep. On the other hand it can help if it is on your doorstep because it gives your protest much more credibility.'\n\nBy now I'm feeling a bit confused by Page. I rather like him. Even though I have the reservation that when he pointed out of the window at the housing development he was perhaps more concerned with aesthetic niceties than with the bureaucratic contempt and condescension with which it is decided that human beings should have to live in such monstrosities, it still seems that his spirit is very definitely in the right direction. Yet how is this compatible with the lumbering dinosaur that his rock band has become?\n\nWell, Jimmy Page is essentially a conservative person. He is also a Conservative person. A Capricorn, he has much of the rather hidebound love of tradition and status that can be a characteristic of that sign. He could do with a bit of overstanding of things. He voted Tory at the last election, he says. 'Not just for lighter taxes \u2013 I just couldn't vote Labour. They actually stated that they wanted to nationalise the media \u2013 so what possible criticism of them would you be able to have?'\n\nAlthough I believe all politicians of whatever creed to be largely self-seeking egotists, I point out that, as the City already has effective control of both Fleet Street and ITV, then the Tories already control the media. Page doesn't seem that convinced.\n\nHe voted Tory at the previous election too, he tells me. 'I voted Conservative then because I believed in Heath. And I still believe that Edward Heath was a very honest man. He was too honest to be a politician. But I suppose that's politics.'\n\nActually, I'm not surprised Page rates Heath, a man who was certainly superior to the deplorable Thatcher. Page has much of that same laissez-faire mercantilist attitude to life that Heath favoured. The only problem with espousing that particular political philosophy is that it can permit you to piss on a lot of people in the name of freedom. I'm not suggesting that Page necessarily behaves in such a manner, of course. A better clue to his attitude to such matters comes in the same series of _Trouser Press_ articles from which I took the Presley quotation.\n\nHe's talking about the song 'Hats Off to (Roy) Harper', on the third Zeppelin album:\n\n'[Harper's] _Stormcock_ was a fabulous album which didn't sell anything. Also, they wouldn't release his albums in America for quite a long time. For that I just thought, \"Well, hats off to you.\" As far as I'm concerned, though, hats off to anybody who does what they think is right and refuses to sell out.'\n\nIn the light of this quote, and another, more ambivalent one relating to the New Wave, I ask him if he'd ever in younger days inclined more to anarchy.\n\n'Well,' he replies with due deliberation, 'anarchy's alright if you can see where you're going afterwards. Although I don't see any point in destroying things just for the sake of it. It's the easiest way out. It's hard to have an optimistic goal and strive towards it \u2013 that's really hard work. But, yes, anarchy can certainly be an answer to a situation if there's no other answer.'\n\nQuite understandably the Establishment always presents anarchy as being very negative when, in fact, it's more concerned with a positive spirit...\n\n'It's difficult,' Page nods. 'At the time when Hitler came into power in Germany during the thirties, he appeared to be stabilising the economy and giving people more work and was emerging as a very patriarchal figure. The Germans felt that everything was going to be alright. Yet underneath was this fundamental plan \u2013 be it evil or whatever.\n\n'And at the time when Hitler came in there'd been a form of anarchy existing. So, yes, you just have to see at the end of the day what's really gone down.'\n\nAnd so, boys and girls, we come to that section of the interview when we talk to Jimmy Page about New Wave music. Even though he seems to consider Dire Straits a New Wave band, Page is perfectly aware that there are punk bands and punk bands who aren't really punk bands. He has heard the Clash and appears to rather like them. He warms very much to the mention of Ian Dury. 'Yeah, he really imparts such a great feeling, doesn't he? Makes you feel so good. That was certainly the first thing that struck me about New Wave music \u2013 that it was sheer adrenaline pouring out. Real energy just tearing to get out.'\n\nBut how did the beat group Led Zeppelin relate to it? They were presumably aware of what was going on. I remember Page and Plant going down to the Roxy to check out the Damned once.\n\n'We were aware of it,' he nods, 'but it's not... I mean, music is like a 360-degree circle from which some people may drop out to let others come in. And there are obvious examples of that \u2013 say, the feeling that Free generated and which was replaced by Bad Company. Also, the raw blues, going back to the early Fleetwood Mac days. Well, now you have George Thorogood. And Herman's Hermits are replaced by the Bay City Rollers.\n\n'Bands like us and \u2013 I hate to say it but... the Floyd... we're off in our own little bits. It's always open for anybody who's really raw and earthy and who makes sheer rock 'n' roll music. Even though much of the New Wave had the political content... I mean, the Damned \u2013 I was absolutely amazed by the power that was coming out of them. Though they didn't really fit into the New Wave movement as such.\n\n'Nevertheless, there are categories. But it's all relative; anyone who plays good music and is expressing themselves with an instrument or on vocals has got something to say. It just depends whether you can relate to them or not. And that also depends on whether your musical tastes are narrow or very broad.'\n\nAnd certainly from what you're saying you would claim to be able to relate to New Wave...\n\n'But I can also relate to classical music \u2013 and you wouldn't find them saying that...'\n\nOh, don't count on it. I think you'd be very surprised.\n\n'Oh... well... good... well, they ought to.'\n\nI think if you went round to the places of a few punk musicians you'd be very surprised by the width of listening material you'd come across.\n\nBut equally, and I think this must be said, of all the old fart bands, certainly Led Zeppelin, for whatever reasons, are the most loathed.\n\n'Really????' Jimmy Page sounds quite startled.\n\n'Fraid so...\n\n'We-e-elll...' he pauses for several moments, '... people write to us, you know, and a lot of younger people who I'd never have expected to have got into us have said that they got really fired up by the energy of New Wave bands \u2013 and they still like New Wave bands \u2013 but they got interested in the actual musical content and wanted to go one step further which is how they discovered bands like us...\n\n'And... uhh... I'm not sure whether that's going to last or not,' he laughs, not altogether confidently, 'but it's quite good if you can keep turning people on.'\n\nDidn't you ever worry, though, over the past months while you were making the new record and planning the Knebworth thingy that it might be like throwing a party for which no one turns up?\n\n'Yeah,' he laughs again, more confidently this time, 'but no \u2013 because when we'd finished our album I knew at the time that it didn't matter if it didn't come out for nine months afterwards because I knew that I could rely on the fact that Led Zeppelin hadn't dated \u2013 the actual identity of the band is still there. There's a fresh approach which can still give it an edge.\n\n'I had my reservations at one point about playing a date like Knebworth. But in the end it all went hand in hand with the LP. When that was finished I did actually stop and take a breath, and I thought, \"No, it's alright. We've moved on sufficiently to be able to see the next horizon.\"\n\n'We're not sounding complacent, I hope. There's a lot of hard work still to come, obviously. It's not like we've felt we had to change the music to relate to any of the developments that have been going on. There's no tracks with disco beats or anything. But I think some of the numbers are some of the most immediate we've done anyway.\n\n'Like I say, it's not a new musical form but there is still something very fresh about it.'\n\nBut, prior to doing it, were you not perhaps apprehensive? I'm not talking about how you'd do in the States, where obviously you're still going to sell loads of records. Presumably it still does mean something to be respected in your own country...\n\n'No, sure. We were concerned about it being good. And we were pleased to hear that the actual environmental area of the stage was good. But if the playing hadn't been feeling right, I would've worried. But that feels alright so I'm pretty sure it'll be good...'\n\nBut I wasn't really talking about Led Zeppelin to Play Gig Shock Horror. I was actually wondering whether maybe you were concerned you might make this platter and no one would buy it in Blighty...\n\n'Well, we were worrying about too many other things at the time. I was worried more about whether we were still going to gel. Having felt something special towards the band for that amount of time and still wanting that feeling to be there without... without being quite sure it would be. But then we got together a few times to play and could see that it still was... well, it was a very good feeling.\n\n'The LP really is a bit of a by-product. To me Knebworth is far more important... Because people can buy the LP and we won't see how they're reacting to it. But,' he laughs, 'I will at Knebworth. The LP's a frozen statement which can be always referred to, but Knebworth's going to be different.'\n\nDo you actually see much of each other?\n\n'No, not really.'\n\nRobert Plant and John Bonham don't live in London, do they? They live up in the Midlands, yeah?\n\n'Yeah, they live pretty close to each other. No, I mean, we don't have monthly get-togethers for the sake of it.'\n\nNow look, you've been involved in community politics up in Scotland, but should it necessarily stop there? For whatever reasons, Knebworth is a huge gig. But a couple of weeks back the Clash \u2013 who are quite a big band these days; their last LP entered the charts at number two \u2013 did a couple of gigs for orphans. Have you ever thought of doing something like that?\n\n'We did that \u2013 about the third year of the band. And we got fucked for it. Previously we'd played places like Manchester Free Trade Hall and the Albert Hall and we'd had all these letters saying, \"Why do they let their fans down? Why don't they play the clubs any more?\"\n\n'So we said, \"Yeah, let's play clubs!\" And it was chaos because people couldn't get in. So the next barrage was, \"Why are they so selfish doing small clubs?\" So the supply-and-demand thing becomes a problem. So from then on we were faced with a sort of dilemma. But then again it became a challenge to see if we could try and make it work on a large scale.\n\n'Don't get me wrong. I'm the first to admit it can get too large, but something like Knebworth can be a challenge because you know it's worked in the past. But we couldn't do that. We tried \u2013 when we'd done the LP, we were trying to work out where we could get in and play. But then we thought, \"Are we running away from something?\" And we weren't.\n\n'It was almost like denying what you were. And you've got to be true to yourself.'\n\nHmmmm...\n\n'I know what you mean, but it just gets impossible to do unless you play four weeks at the Marquee.'\n\nBut you're supposed to be a rock 'n' roll band. Why don't you just play? Look, it's not that hard: the Clash did two dates at the Notre Dame off Leicester Square. It wasn't publicised \u2013 only by word of mouth. They played new songs, tried out new sets, made money for charity. So it obviously is feasible...\n\n'I'll give you an example of a band that I don't think could play the Marquee: Status Quo.'\n\nWhat an odd thing to say. I'd rather have hoped that Page would consider Zeppelin to have a slightly different awareness to the dandruffy riffers. But of course, I counter, they could play it if it wasn't announced as such...\n\nThis is ignored. 'And I know they've played Wembley \u2013 so, if 15 per cent of those people tried to get in, it would be chaos. So you see the problem.'\n\nNot really.\n\nMy next question is interrelated with a lot of things we're talking about \u2013 just how important is the institution of the music business to you? Do you feel that Led Zeppelin is part of the great corporate conglomerate?\n\n'Obviously. Yeah. But to them you're only a matrix number. We sweat the songs out, though.'\n\nBut is it down to just letting the shareholders have bigger dividends? You're a musician, right? I think that's what you feel you are...\n\n'Yeah, but don't you see that we're only as good as whatever we come up with? Say we didn't put out another LP... Well, we've probably done really well for our record company but, if we did that, they'd probably come right down on us. I think it's probably really ruthless behind the scenes. It comes down to things like Kinney owning car parks and things.'\n\nYou imply, I suggest to Page, you don't care about the record company. But, by acceding to those demands to play those huge venues \u2013 and in a way they're just perpetuating the whole thing...\n\n'I see what you mean \u2013 though I'm not sure you see what I mean. The problem is trying to supply the demand of the people who want to see you. You can only gauge that. I mean, it is a rather nice feeling deciding on this huge date and not being quite certain that there's enough demand and then finding you can play a second one the same size.\n\n'Anyway, at this point in time we just want to get back into playing music. And we will be doing other dates. I don't know where: not necessarily in England. We've been talking about playing Ibiza \u2013 Just getting in there and playing. Just so we've got a chance of trying out new ideas and new riffs and arrangements and songs.'\n\nSo do you not think that Led Zeppelin has become part of some huge thing that's got totally out of hand?\n\n'Well, if it has it certainly won't in the future because we'll be playing places like Ibiza.'\n\nWas there a stage that you reached with Led Zeppelin when it became important just to make money?\n\n'No. Never. No, because we've been our own worst enemies over that. But you wouldn't see it like that now. But at the time we put out our fourth LP we had the worst reviews of anybody. And to put out an untitled LP at that time was considered professional suicide. It probably doesn't seem it now. But then...'\n\nAre you very materialistic?\n\n'Well, I dunno. Yeah, I suppose I am a bit. But on the other hand, even though I have material possessions, the most important things are books, studios and records. If I had to get up and run that's what I'd try and take,' he laughs.\n\nDo you think that you personally have perhaps unavoidably become caught up in the Whole Great Sell?\n\n'I don't think I have. No, no. I haven't. Otherwise I wouldn't have opened up a book shop. I'd have opened a boutique or something where I could really make money. Equinox was never designed to make lots of money but just to tick over so it could publish books.'\n\nDo you think people have ever taken advantage of your having such desires?\n\n'Quite probably. Yes,' he replies in a certain tone.\n\nBut you're a reasonably happy human being?\n\n'Well,' he seems momentarily uncertain now, 'as happy as the next one.' Then he gives a spirited chuckle. 'I think I'm pretty fortunate in that I'm able to do what I'm best at. It's a pretty fortunate position to be doing what you really want to do and turning people on.'\n\nBut you've made tapes with people like Keith Richards. Obviously you must have wanted to make records with other people...\n\n'Yeah, I did. But in the end it comes down to playing with the people who I really like to play with.'\n\nJim now tells me that he must leave in a few minutes as he has to meet Charlotte, the lady with whom he lives and by whom he has an eight-year-old daughter, Scarlet. This is unfortunate. We were just getting going, it seemed to me. It's a pity also that interviews with members of Led Zeppelin are inevitably set in the anonymous Swan Song offices, thus providing writers, and therefore readers, with little comprehension as to how the band members actually live. Even the Stones seem to have woken up to the fact that both journalist and band benefit from less clinically set-up situations. But I suppose that's all part of the Led Zeppelin problem anyway.\n\nThere were many other things I'd like to have asked Page: what have he and the other three band members done for the last 18 months or so, for example? Whose records has Page been playing recently? Why doesn't Swan Song sign any hot new acts?\n\nAs it is, though, I only have time to touch on some of the more, uhh, 'controversial' topics that are raised in the first section of this piece.\n\nA large part of the original strength of Led Zeppelin surely stemmed from the energies and ideas Page derived from his lengthy session work in the sixties. Now, though, it seems that all that has been exhausted and there is little new creative input to replace it. Page's views on the music business show a startling lack of original thought and clarity. Mainly, though, they suggest, as I mentioned earlier, confusion. And it's by perpetrating that state of chaos and confusion that the music business is able to persist in its Babylonian and fatuous desire to be part of the vast dehumanised, cynical corporate state. Grrrrr...\n\nOn the other hand, compared with certain of his contemporaries, maybe he's not faring too badly. I ask him if he feels isolated and cut-off. He claims not to feel that now, though admits to having been in a pretty weird state round about the time of the band's fourth LP.\n\n'Of course,' he adds, 'it can do very odd things to you, the whole guitar hero bit. Look at Eric Clapton. Peter Green... Well, that's the obvious example. Jimmy Page: well, I don't think I'm doing too badly,' he laughs, with a fair amount of confidence.\n\nWho would be the support acts for Led Zeppelin at Knebworth? Ian Dury and the Blockheads were approached, but adamantly refused to associate themselves with what they considered to be a Babylonian event. In fairness, it was only Ian Dury himself, egged on by aide-de-camp Kosmo Vinyl, who considered taking up such an offer to be morally and philosophically unsound \u2013 even when the fee dangled in front of them rose and rose to \u00a3140,000. (The rest of the Blockheads were allegedly extremely miffed at missing out on their share of such a sizeable lump sum.)\n\nIn the end, it was Keith Richards and Ron Wood's New Barbarians, Todd Rundgren's Utopia, Chas and Dave, Fairport Convention, Southside Johnny and the Asbury Dukes, and the Marshall Tucker Band. Not a bad bill.\n\nThere had been a trio of tunes left off _In Through the Out Door_ : 'Ozone Baby', 'Wearing and Tearing' and 'Darlene'. Jimmy Page had the unconsummated idea of releasing them as an EP before the Knebworth show.\n\nDespite Peter Grant's claims that a quarter of a million people attended the pair of shows, the reality was more prosaic: 104,000 on the first date; a mere 40,000 on 11 August.\n\nYet it was the first Zeppelin show in the UK in four years and the anticipation was infectious. Backstage there was a sprinkling of the new punk icons, paradoxically still in awe of the majesty and glamour of Led Zeppelin: Sex Pistol Steve Jones, Mick Jones of the Clash, Tony James of Generation X and Chrissie Hynde. And in Grant's special enclosure, a reminder of the true power behind the group, was Ahmet Ertegun. Equally significantly, so was James Page, Jimmy's father.\n\nJust before Zeppelin went onstage, the writer Mick Brown, reporting on the event for the _Sunday Times_ , noticed something rather curious: 'Jimmy Page was carried to the stage on a stretcher.' Page was wearing a narrow-lapelled jacket and a Slim Jim tie; instead of making him look hip, it was more like an avuncular relative trying to show how hip he was. Onstage, he seemed to have a lit Marlboro permanently nailed to his lips. In an open polka-dot shirt and dark-grey cord trousers Robert Plant looked more convincing, until you observed the deep worry lines etched into his once-angelic face. In a clear effort at looking 'relevant', all of Led Zeppelin had had haircuts: Plant's semi-mullet hardly helped. It only made them look like successful men in their thirties \u2013 precisely what they were.\n\nPerhaps Page required the stretcher because he was crippled by anxiety. Flown in by helicopter, the band were trembling with nerves, almost unable to speak: to them it seemed their future depended on this first Knebworth show. There had been a pair of warm-up dates in Copenhagen, and on 2 August the band had inspected the site at Knebworth. Page had arrived in Richard Cole's pokey Austin-Healey 3000, its top down on the warm summer evening, which irritated him when the wind dishevelled his hair. In beautiful Knebworth House itself he inspected artefacts left behind by Sir Edward Bulwer-Lytton, the 19th-century bestselling author and occultist whose ideas had been taken up by the theosophist Helena Blavatsky, an acquaintance of Aleister Crowley. Thus intellectually fortified, Page and Peter Grant retired to a back room for more conventional stimulation.\n\nFor the concert itself, Led Zeppelin's American PA had been flown in, 100,000 watts of audio power, plus a 600,000-watt lighting system, their American laser network and the video screens. Onstage there were five guitars lined up for Page, plus John Paul Jones's white grand piano, synthesised Mellotron and clavinet.\n\nSomehow Robert Plant had been persuaded to perform 'Stairway to Heaven'. In fact, the set was essentially the same as on that last, fateful American tour, with the addition of 'In the Evening' and 'Hot Dog'.\n\nAlthough Page had arrived in a chopper with Charlotte Martin by his side, he was seen backstage after the show with Krissie Wood; they looked, thought Lisa Robinson, 'totally out of it'.\n\nBut the shows also revealed that, back playing the emperor once again, Peter Grant was at his egregious worst; again on cocaine, he was exhibiting extreme signs of the paranoia the drug could create. On the morning of 4 August Richard Cole received Grant's orders to insist that Freddy Bannister waive any film rights to the Zeppelin shows. Still smarting from _The Song Remains the Same_ 's creative failure, Grant had hired Mike Mansfield, the director of ITV's successful _Supersonic_ pop show, to film the entire concert, using 12 cameras, for a projected film release. In such circumstances the promoter might reasonably expect a measure of financial recompense from any consequential film. Instead, Cole informed Bannister that Grant would give him 5 pence for all rights. Cole warned him that, the mood the manager was in, it would be advisable to accept the offer. Finally, Bannister agreed, on the basis that any subsequent legal case he brought against Zeppelin would be bolstered by the derisory, insulting offer Grant had made.\n\nFollowing the show Grant pulled another stunt: based on his estimate that a quarter of a million people had been at the concert \u2013 more than double the actual amount \u2013 he demanded immediate payment for both shows.\n\nDistressed, Freddy Bannister and his wife and business partner Wendy drove down to Grant's home, Horselunges, the next day. They encountered a scene in which Grant acted like an archetypal evil business manager in a Hollywood gangster film.\n\nWhen Wendy Bannister protested against Grant's audience figures and demand for immediate payment, Zeppelin's manager brandished his fist in her face and said: 'Don't you get smart with me.'\n\nThe next day a black-suited American, like a Mafia hitman from central casting, turned up at their home, accompanied by a man who claimed to be a former Metropolitan Police superintendent. Freddy Bannister came to the conclusion that both were private detectives. The pair claimed to have seen aerial photographs of the 4 August show that proved once and for all that 250,000 people had been at the gig.\n\nTwenty-four hours later Grant visited the Bannisters. Oddly, he claimed to be willing to reduce his act's fee for the next concert. With the proviso that his own people would man the entrance gates.\n\nBy now utterly sick of the enterprise, Freddy Bannister agreed. And Peter Grant came face to face with the truth: that only 40,000 people came to that second show, moody with rainfall and bad vibes from the New Barbarians, themselves demanding their payment upfront. The awful reality \u2013 that Led Zeppelin appeared on the slide, perhaps past their sell-by date \u2013 only served to make Grant even more belligerent: four weeks after the second Knebworth show, the American private detective, if that was what he was, insisted on another meeting between Grant and Freddy and Wendy Bannister at the Dorchester Hotel on London's Park Lane. Grant demanded that Freddy Bannister sign a prewritten letter absolving all blame for any resulting brouhaha from Peter Grant and Led Zeppelin. The US court cases the previous year had left all involved, including John Bonham, with the possibility that US visas might not be granted to them; therefore Grant wanted to stop any discussion of their heavy-handed, aggressive behaviour at Knebworth.\n\n'So why did we retire abruptly? In a word \u2013 fear. Peter Grant was in such a terrible state, both mentally and physically, we thought he was on his way out and would be delighted to take us with him,' said Freddy Bannister in his book _There Must Be a Better Way_. In 1982, Tredoar, Freddy and Wendy Bannister's concert-promotion company, was forced into liquidation by their financial disputes with Led Zeppelin.\n\nSo many things about Knebworth were a bad vibe. So many things about Led Zeppelin had become a bad vibe.\n\n'Under pressure from the new wave of rock 'n' rollers, one of Britain's longest established bands chose to fight back in 1979,' opened _Melody Maker_ 's press release about that year's poll results. _Melody Maker_ 's readers voted _In Through the Out Door_ the best album of 1979. Moreover, Led Zeppelin or its members won seven of the twenty categories in the poll: as well as the album's success, they also won Band of the Year, Top Live Act and Top Composers; Robert Plant succeeded Yes's Jon Anderson as Top Male Singer, and Jimmy Page supplanted Yes's Steve Howe as Top Guitarist. He also replaced Genesis and David Hentschell as Top Producer, an irony unlikely to have been lost to John Paul Jones.\n\nBut Jimmy Page was missing when the rest of the group turned up at London's Waldorf Hotel with Peter Grant to be presented with their gongs. The story given out was that Page was on holiday in Barbados, but that was not the case. The guitarist, however, was not lying in some smacked-out reverie in one of his impressive properties. Rather, he was in Sussex, giving evidence at an inquest. A friend of his, a local photographer and designer named Philip Hale, who was 26, had died at a party at Plumpton Place on 24 October 1979. Though the cause of death was given as 'vomit inhalation', significant levels of cocaine were found in Hale's blood.\n\nAt the end of 1979 Page moved out of Plumpton Place, his friend's death the principal factor. He moved into Old Mill House in Clewer, Windsor, not too far from Heston, where he was first brought up; as was the case with most of his properties, with the exception of Tower House, his new home was close to water, the River Thames in this case, soothing for the watery aspect of his Scorpio rising. The three-storey residence, which he purchased from the actor Michael Caine, cost Page \u00a3900,000, and the guitarist also owned a nearby studio, the Sol, in Cookham.\n\nThen suddenly, in the summer of 1980, Led Zeppelin were on tour again, in Europe: Switzerland, Germany, Austria, Holland and Belgium; dates in France were cancelled by Page. They were playing smaller venues, each with a capacity of around 3,500. There had been a month of rehearsals at the Rainbow in north London, and at the New Victoria Theatre in central London. Robert Plant had expressed grave doubts about playing again in the US. The intention of these shows, running from 17 June to 8 July, was to rekindle enthusiasm within the group, and hopefully get the singer in the frame of mind to cross the Atlantic. Just in case, Peter Grant had booked a two-legged US tour, to begin on 17 October running until 15 November 1980.\n\nFor these European dates everything was stripped down. Endless guitar solos were dispensed with: the set now opened with 'Train Kept A-Rollin'', the first number Led Zeppelin ever played together, in that Gerrard Street rehearsal space; it had not been played since 2 September 1970, in Oakland of all places. The acoustic songs were ditched, and an abbreviated version of the Knebworth set was performed. Richard Cole was temporarily relieved of his duties, his smack problem not considered a good influence on Page and John Bonham, both struggling with their own habits. Those who encountered Page on this tour were shocked by his appearance: the wafer-thin limbs; the junkie pallor and pinned eyes; the chipped and rotting teeth and gums, a sure sign of heavy heroin and cocaine use. 'It was incredibly sad to see him in such a state,' wrote Marc Roberty, 'and ultimately the band having reached these low depths.' There simply seemed no enthusiasm there whatsoever. All four members seemed chastened from their collective Led Zeppelin experience, as though parts of them had not survived.\n\nIn Vienna, Page was hit in the eye by a firecracker. When he returned to the stage, he played the introductory moments of 'Deutschland \u00fcber alles', before heading into 'Stairway to Heaven'.\n\nThe next night, 27 June 1980, Page stepped up to the mike in Nuremburg to declare that two members of the band were not feeling well. A quarter of an hour later Bonham collapsed. The cause of his illness? Eating too many bananas, came the official response from the Zeppelin camp.\n\nThree nights later the Frankfurt show was one of the best of the tour. A magical version of Barrett Strong's 'Money (That's What I Want)' was played as an encore, with Atlantic Records boss Phil Carson on bass. An interesting choice of song, of course.\n\nPeter Grant confirmed that Led Zeppelin would indeed be playing the US tour. This may have been an even bigger mistake than hiring John Bindon.\n\nThe last thing John 'Bonzo' Bonham wanted to do was leave his family \u2013 he always hated doing this \u2013 and tread the US boards. Especially as there were rumours of civil lawsuits lurking from the Oakland debacle.\n\nHis drinking remained ferocious \u2013 essentially, Bonham was always drunk. He had supposedly stopped using heroin, though some cast scorn on this. He was certainly using Motival, an antidepressant.\n\nOn 24 September 1980 Bonham was picked up at his home by Rex King, who had replaced Richard Cole on the European tour. King was driving Bonham to Page's new home, Old Mill House. En route, the drummer insisted on stopping at a pub; he drank four quadruple vodkas with orange juice, and ate a couple of sandwiches. During rehearsals at Page's practice studio at his house, Bonham continued to drink vodka. He became so inebriated he could hardly play.\n\nWhen rehearsals ended Bonham kept on drinking. By 11 p.m. he was so drunk he passed out on the sofa. He was helped to bed and laid on his side.\n\nPage's assistant looked in on Bonham at eight o'clock the next morning. The drummer was sleeping soundly; the feeling was that he should be left alone to sleep off his heavy intake of booze.\n\nWhen he didn't appear by the afternoon, Benji LeFevre, Robert Plant's assistant, went into his bedroom with John Paul Jones at 1.45 to chivvy the drummer along. But Bonham's face was blue, he had no pulse and he was cold. He was dead, at the age of 32. The cause of death: inhalation of vomit.\n\nOn 4 December 1980 Led Zeppelin put out a statement through the Swan Song office: 'The loss of our dear friend, and the deep sense of harmony felt by ourselves and our manager, have led us to decide that we could not continue as we were.'\n\nLed Zeppelin to split?\n\nYes: they just had.\n\n## 23\n\n# THE HERMIT\n\nWith the death of John Bonham, Jimmy Page's great art project of Led Zeppelin had effectively ended. Now he \u2013 and the rest of the group and manager Peter Grant \u2013 were on the creative comedown that inevitably follows the conclusion of such work. And the greater it has been, the larger the comedown always is. So when you have been at the helm of the biggest group in the world, what you experience is almost unparalleled. No one you know can fully empathise with your state of mind. Page had been out there on his own for so long, enjoying the freedom his natural desire for solitude brought him. But where could he now turn? He plunged into acute depression, which his increasing reliance on heroin and alcohol did little to alleviate.\n\nAfter the demise of Led Zeppelin, the group \u2013 like such other UK symbols of the early 1970s as Prime Minister Edward Heath or Morris Marinas \u2013 felt almost immeasurably and immediately out of date. Its dungeons and dragons world had almost instantly evaporated.\n\nYou can of course trace back the commencement of this comedown to 1975 and Robert Plant's terrible car accident, at the start of his Saturn return. But that was the time that heroin started to catch up with the lives of Page, John Bonham, Peter Grant and Richard Cole; the lonely days Page spent working on _Presence_ made it more of a solo album for him than anything that the band had delivered. The desperation of 1975 had made an exponential leap in 1977 with the full horror of the Oakland incident and the utter tragedy of Karac Plant's death three days later.\n\nWhen Nick Kent asked Page in 2003 whether he regretted becoming so dependent on heroin and cocaine, the Zeppelin leader was unrepentant: 'I don't regret it at all, because when we needed to be really focused, I was really focused. That's it... You've got to be on top of it.'\n\nJimmy Page was a broken man. And he only plunged into smack even more deeply. It was a terrible time for him. 'My life was Led Zeppelin,' he told Kent. 'I lived and breathed Led Zeppelin. When I wasn't touring, I was at home writing music for the group. I could hear John's drumming. I could hear Robert... and John Paul. It was a total obsession for me. And suddenly... it was all over.'\n\nHe was in deep mourning for John Bonham. He thought that he might never play the guitar again, especially after his Gibson Les Paul went missing shortly after the drummer's death. 'It seemed like an omen,' said Page. 'If I even looked at a guitar, it would remind me of a dear friend I had just lost.'\n\nPeter Grant was equally devastated, also plunging into depression, which manifested in a kind of clinical ennui. Richard Cole, meanwhile, was in prison in Rome, awaiting trial for cocaine possession (he would eventually be acquitted). Told that one of Led Zeppelin had died, he immediately thought, 'Poor Jimmy!' He was even more shocked to learn that the death was of his old mucker Bonzo.\n\nAssorted drummers had contacted Peter Grant almost straightaway after Bonzo's death. On 7 November Jimmy Page, Robert Plant and John Paul Jones went down to Jersey in the Channel Islands to consider their options. They were unanimous in their decision. Returning to London they held a meeting with Grant in a suite at the Savoy Hotel. 'They told me,' said the manager, 'that without Bonzo there was no desire on their part to carry on. It could never be the same again. I was relieved. That was exactly the way I felt.'\n\nSoon there was an offer, conveyed by Peter Grant. Chris Squire and Alan White, respectively the bass player and drummer with Yes, a pair of excellent musicians, suggested linking up with Page and Plant in what was once quaintly known as a 'supergroup', to be named XYZ (Ex-Yes and Zeppelin... ho-hum). Page had a couple of meetings with the Yes men, but decided against it. Plant had dismissed the idea as soon as he heard of it.\n\nTo all intents and purposes, both Page and Grant became recluses. Years later, Malcolm McLaren, who had managed the Sex Pistols, endeavoured unsuccessfully to make a film about Grant's life. At the time McLaren wrote: 'Grant needed the camaraderie of hard, dangerous men who gave him a sense of power. The harder they were, the tougher he felt, and only then was his desire for control satisfied. It all fell apart when Grant aped the lifestyle of Jimmy Page, who then ostracised his biggest fan.'\n\nRobert Plant's instinct was that he needed to work his way out of this predicament. Linking up with his old mates from home, guitarist Robbie Blunt \u2013 who had played with the Steve Gibbons Band, revered in their native Midlands \u2013 and bass player Andy Silvester, he put together the Honeydrippers, essentially a covers group playing old R&B; he based himself, some thought, on his old hero Ral Donner. Starting off with a show at Keele University on 3 March 1981, the Honeydrippers played 15 shows, finishing in June, although Plant insisted that his name must never appear on any promotional material for the band. His stint with them got him back on his feet and performing again, and he began to put his assorted tragedies to one side. Besides, ever since the car crash on Rhodes in 1975, Plant had been conflicted over what he was doing in Led Zeppelin. Always the frontispiece \u2013 the front office, really \u2013 of Led Zeppelin, he had seemed the most susceptible to any strange energies bouncing back at the band.\n\nBut recovery for Jimmy Page was far slower. Help came from an unexpected quarter: film director Michael Winner, his direct neighbour in Holland Park. Considering how Page had adorned his existence with seemingly effortless good taste, it was perhaps not the prestigious product with which he would have liked to be associated. But all the same, writing the blues-based soundtrack for Winner's _Death Wish II_ , a Los Angeles-set revenge thriller starring Charles Bronson, at least temporarily aroused his creativity.\n\nThough Winner could quite literally have approached Page over the garden wall, he instead contacted Peter Grant \u2013 not that easy a task. At the time Page was on holiday on a narrowboat on the River Thames. He was uncontactable, and Winner's deadline had been eaten into by the time he actually received the director's request, in August 1981, giving him a cut-off date of six weeks.\n\nIt was almost surprising that Page had managed to get out of his home and onto a boat. At this point he had been a virtual recluse for almost a year: he didn't return phone calls and wouldn't connect with any old friends. Visitors to Tower House, there to see Charlotte Martin and Scarlet, would catch glimpses of him upstairs, clad in a dressing gown, like a spectral character in a Gothic novel; for weeks at a time he wouldn't leave the property. There were tales of Charlotte Martin welcoming fans who had travelled from far away and would ring the doorbell on the off-chance of it being opened. She was known to order them take-away food, and let them have a peek around the basement studio. Was this the source of high-quality bootlegs of rare Zeppelin material that later appeared?\n\n'I'd lived next door to Jimmy for many years,' said Winner to _Uncut_ magazine in 2008. 'It was a very bad time for him \u2013 the drummer had died, and he was in a very inactive period. Peter Grant and I made arrangements for Jimmy to do the _Death Wish II_ score, for which he wasn't actually paid, because Grant wanted to restore Jimmy back to creativity. Jimmy rang the doorbell, and I thought if the wind blew he'd fall over. He saw the film, we spotted where the music was to go, and then he said to me, \"I'm going to my studio. I don't want you anywhere near me, I'm going to do it all on my own.\" My editing staff said this is bloody dangerous! Anyway, we gave him the film, we gave him the timings and he did it all on his own.'\n\nMichael Winner at least got Page working. Perhaps the guitarist was inspired by how badly the soundtrack to _Lucifer Rising_ had turned out and wanted to rectify that rare failure. At his studio in Sonning, Page produced a series of pieces relevant to the film. There was 'Prelude', essentially a complete steal of Fr\u00e9d\u00e9ric Chopin's Prelude No. 3 in G, the great Polish composer's patriotic piano-playing replicated on Page's electric guitar. 'In the Evening' transmogrified into 'Who's to Blame', the main theme song for _Death Wish II_ , sung by Chris Farlowe, who had almost been given the vocalist role in Led Zeppelin. 'City Sirens' was soft rock, and there was the psychedelic 'A Shadow in the City'. 'The Release' was a chase sequence: a succession of droning sounds that would have made Kenneth Anger proud. Every one of the 12 tracks was credited only to Jimmy Page. 'Everything hit the button totally!' said Winner. 'I've never seen a more professional score in my life.'\n\nThe soundtrack record was released on 15 February 1982 on the barely functioning Swan Song. Label boss Peter Grant wouldn't answer his phone; the drawbridge at Horselunges had been raised.\n\nWhile Page was working on the _Death Wish II_ soundtrack, Robert Plant took _Pictures at Eleven_ , his first solo album, recorded at the legendary Rockfield Studios in Monmouth, down to Old Mill House, where John Bonham had died. Plant had produced the record himself, and he and Robbie Blunt had written all the tracks, with keyboard and synthesiser player Jezz Woodroffe co-writing three of the songs \u2013 though Plant balked at signing the joint publishing deal with his co-writers that he had allegedly promised, leading to a general bitterness. All the same, the singer wanted his old mentor to be the first to hear it. 'Well, ah, I thought you could have done something a little better than that, old chap,' said Page, with that dismissiveness that often couches jealousy.\n\n'So I said, \"Well, thank you.\" And yet again, I was just the singer of the songs.'\n\n'It was very emotional,' Plant later reflected on that visit to Old Mill House. 'We just sat there and I sort of had my hand on his knee. We were just sitting through it together. He knew that I'd gone, that I was off on my own with the aid of other people and just forging ahead, and all I wanted was for him to do the same.'\n\nReleased on 28 June 1982 on Swan Song, _Pictures at Eleven_ was the label's last substantial seller, reaching number five in the US album charts and number 2 in the UK.\n\nAlthough living out his tarot-card archetype of the Hermit, the _Death Wish II_ soundtrack seemed to have urged Jimmy Page out of an unproductive existence: _Coda_ , a ragbag of impressive Led Zeppelin outtakes, was released on 19 November 1982. 'Coda was released, basically, because there was so much bootleg stuff out. We thought, \"Well, if there's that much interest, then we may as well put the rest of our studio stuff out,\"' said Page.\n\n_Coda_ made number four in the UK and number six in the US _Billboard_ charts, where it sold over a million copies. The record kicked off with Ben E. King's 'We're Gonna Groove', which featured on so many of the early live shows. Incorrectly listed on the original release as having been recorded at Morgan Studios in north-west London in June 1969, the recording was actually taken from the celebrated live show at the Royal Albert Hall in January 1970; the tune had opened the concert. The audience applause was removed and \u2013 this was the work actually done at Morgan Studios \u2013 Page added guitar parts to it. A live version of Willie Dixon's 'I Can't Quit You Baby' also came from the Royal Albert Hall show, with blistering, searing guitar playing from Page. The steaming 'Walter's Walk' came from the _Houses of the Holy_ sessions. The acoustic 'Poor Tom' had been made for _Led Zeppelin III_. There were a trio of tunes recorded for _In Through the Out Door_ : 'Ozone Baby', 'Darlene' and 'Wearing and Tearing'. 'Bonzo's Montreux', intended as a tribute to John Bonham, had been recorded in Switzerland in 1976, a steel pan added into the mix. The punky 'Wearing and Tearing' closed _Coda_ , stripped-down, grungy rock 'n' roll, a reference to the very first Led Zeppelin album with its raw, garage-band feel. 'They were good tracks. A lot of it was recorded around the time punk was really happening... basically there wasn't a lot of Zeppelin tracks that didn't go out. We used everything,' said John Paul Jones.\n\nMeanwhile, Page sat at home, doing very little and \u2013 above and beyond his problems with heroin \u2013 stuck in a miasma of depression, naked and vulnerable. A month before the release of _Coda_ , there had been yet another problem; matters were unravelling at some speed of knots.\n\nOn Tuesday 5 October 1982 Page was given a conditional discharge for cocaine possession. He had been searched by a policeman near the Swan Song offices on the Kings Road, and 198 milligrams of cocaine was discovered in his coat pocket. He claimed he had found the drug on a mantelpiece at his house during a party and had removed it to prevent anyone taking it. He had also lied that he bought it for \u00a320 from a man he met in the street. But then he said he had made up that story on the spur of the moment. His defence was essentially based on monetarist thinking. Claiming that his client was putting together a new group and was about to tour Japan and the United States, his QC John Matthews pointed out that a harsh sentence 'would result in the loss of not thousands but millions of pounds during the next few years'. The implication clearly was that it would be the UK taxation system missing out, and not just Page himself. Zeppelin Star Wins Freedom, said the _Daily Star_ 's headline the next day.\n\nThe cocaine bust can only have added to the stress of being Jimmy Page. At home things were no better. By the end of 1982 his 13-year relationship with Charlotte Martin had finally fallen apart. Even if you are a rock superstar you do not escape the pain of a broken partnership; again, Page found solace in heroin.\n\nIn the summer of 1982 there had been a reforming of the Yardbirds, involving Jim McCarty, Chris Dreja and Paul Samwell-Smith. At a party at Jeff Beck's Kent house, Page expressed his irritation that no one had asked him to take part in it. Of course, the reason no one had asked him to play was that they all assumed it was utterly inconceivable; the extent to which Page had become an almost Howard Hughes-like recluse was well known in London, as were his addictions. Ian Stewart, the Rolling Stones keyboard player, was at the party, having a conversation with Jeff Beck about an imminent ARMS (Action into Research for Multiple Sclerosis) charity show to raise money for Ronnie Lane, the much-loved Small Faces and Faces bass player who had multiple sclerosis. 'So at this party,' said Stewart, 'while I was discussing the Ronnie Lane benefit with Jeff, Jimmy came up and he said, \"Nobody ever asked me to play. Why can't I play on it?\" So we said, \"Step this way.\"'\n\nAt the charity show, at the Royal Albert Hall on 20 September 1983, many in the cast and crew were shocked by Page's appearance and his inability to put a coherent sentence together. 'He was so pale, so thin, like a walking skeleton. It was obvious he was still deeply into the heroin at the time,' said his friend Dave Dickson. It didn't help his nervous frame of mind that while he was at the Royal Albert Hall soundcheck, his illegally parked car was towed away.\n\nYet appearing alongside his old Yardbirds chums Eric Clapton and Jeff Beck, the first time all three of them had ever played together, Page delivered extremely serviceable versions of 'Who's to Blame' and 'City Sirens', from the _Death Wish II_ soundtrack, and an instrumental version of \u2013 what else? \u2013 'Stairway to Heaven'. From the Royal Box he was watched by Prince Charles and Princess Diana.\n\nThe Royal Albert Hall show proved such a success that further dates were set up in the US, nine in total. Page played on all of them \u2013 a courageous return to the performing stage. 'He's probably one of the bravest among us \u2013 he's really putting himself on the line,' said Kenney Jones.\n\nFor the nine American ARMS shows, Page played several numbers, including his version of Chopin's Prelude. On 'City Sirens' and 'Who's to Blame' he performed with Steve Winwood; and that hoary old chestnut 'Stairway to Heaven' was again included. The shows ended with three nights at San Francisco's Cow Palace, from 1 to 3 December 1983.\n\n'Page's set was a triumph of style over substance,' wrote Kurt Loder in _Rolling Stone_ of a Cow Palace gig. 'A small white spot opened up on the right of the stage, and suddenly, there he was: cigarette pasted to his lip, a cascade of black curls tumbling down over his eyes \u2013 the very picture of wrecked rock-star elegance. First, he removed his long white scarf \u2013 to resounding applause, of course \u2013 then the various rings on both his hands, and then he rolled up his sleeves and set to work. Had he played not a note, the audience would have been with him anyway. But he picked up his black Telecaster and, leaning back in classic Zep fashion, proceeded to dig into \"Prelude\", a haunting instrumental piece from the soundtrack of _Death Wish II_ , which Page scored.'\n\nAfter two more numbers Page brought on Paul Rodgers, the Free and Bad Company singer. Together they performed a song that was a work in progress to be titled either Midnight Moonlight or Bird on the Wing. 'This piece was, to put it tersely, a rambling disaster,' wrote Loder. 'It was followed by a supremely flaky _instrumental_ rendition of the epochal \"Stairway to Heaven\", toward the thrash-crazed end of which Beck and Clapton strolled out and attempted, as best they could, to join in. Page appeared to be on another planet.'\n\nLoder, who had considerable access, remarked that there seemed to be moments of tension between Clapton and Page, noting that Clapton had long ago kicked his own heroin habit, while Page was clearly still imprisoned in his own. When Loder asked Clapton how he perceived both Page and Beck, the guitarist's response was ambiguous: 'I think their characters have become very clear \u2013 have become compounded.' Loder described how, after each show, the assorted musicians would mingle. But Page would step into a black van and be driven straight back to his hotel.\n\nIan Stewart expressed his fondness for Page to Kurt Loder: 'This tour has got him moving again, and I hope he can find something to do after this. It's a shame that he just sits at home and does nothing. He seems to miss John Bonham very much; but at the same time, I think he'd like to play. It's just that... maybe nobody asked him for two or three years; I don't know. Jimmy's pretty laid back, really. He's still very interested in music. He's always coming up with obscure things, like classical things and Bulgarian folk things. We had a big natter the other night about Django Reinhardt guitar solos. So the interest is still there; he just needs a bit of motivation.'\n\nLoder concluded his article with an assessment of the three former Yardbirds guitarists; he defined Page as 'the sensitive space case'.\n\n## 24\n\n# MIDDLE-AGED GUITAR GOD\n\nOn 9 January 1984 Jimmy Page turned 40, not something that was easy to admit for someone who had been a god to 'the kids'. It is an age that often brings a measure of concern to people: is half my life over? Whereas the younger Robert Plant had seemed to harness time to his advantage, for Page it increasingly appeared to bring adversity. He needed an ally. And hanging out with Paul Rodgers had given him an idea.\n\nFor those aware of London gangland terminology, the name of the group they formed together \u2013 the Firm \u2013 was like a throwback to the thuggish world of Peter Grant and John Bindon; it was a term employed for the groups of football hooligans who would travel England causing mayhem. The name alone seemed like an implicit threat: buy our record or we'll send round the boys! Yet no blame for the name can be attributed to Jimmy Page or Paul Rodgers: 'the Firm' was the idea of drummer Chris Slade, formerly of Uriah Heep. On bass and keyboard was Tony Franklin, who had worked with Roy Harper, with whom Page had played a low-key show at the Cambridge Folk Festival in the summer of 1984, as well as several other unpublicised appearances elsewhere with his eccentric friend.\n\nThis included a spot of fell-walking for the pair of them, an endeavour to publicise _Whatever Happened to Jugula?_ , Harper's upcoming album; _The Old Grey Whistle Test_ , BBC TV's weekly 'serious' popular-music programme, filmed Harper and Page performing on the side of a mountain in the Lake District in August. They had worked together on _Whatever Happened to Jugula?_ , Harper's new album, which would be released in March 1985; although Page had played on several of Harper's records, _Whatever Happened to Jugula?_ was the first he had worked on in its entirety. At one point it was mooted that the record would be credited to 'Harper and Page'. Rather coyly knowing, the album at first had the working title of 'Rizla'.\n\nAt their first late-evening meeting with the pair, in their Ambleside hotel, Mark Ellen, the show's presenter, and Trevor Dann, his producer, noted that there were two modes of operation: one motored by 'red tackle', which involved replenishing the assorted goblets of expensive red wine; and the other driven by 'white tackle', where one of the acolytes would empty a packet of cocaine onto the dining table, to be duly hoovered up by Page and Harper. Especially of interest to the pair of BBC men was that 'the 40-year-old guitarist of Led Zeppelin was on a blind date with a girl of 18'. Harper had played a show the previous evening in Leeds, picking up a 19-year-old. A mate of his was coming up from London, he said: had she a friend who might want to join them? She had, and the friend was now Page's temporary paramour.\n\nWhen Trevor Dann mentioned that the film crew would be arriving at eight the next day, Page assumed he meant eight in the evening. When Dann said he was going to bed and would see the pair at breakfast, Page looked horrified. 'He'd assumed Trevor had meant eight in the evening. So had Harper... They looked tight-lipped and ashen-faced,' wrote Ellen in _Rock Stars Stole My Life_. 'There was an awkward, head-scratching silence while two men who'd never risen before the crack of noon pondered their fate. \"Tell you what,\" Page decided. \"We'll just have to stay up all night.\"'\n\nNeedless to say, it was several hours after breakfast that first Harper, and then Page appeared. 'Page stumbled out and joined the fray,' declared Mark Ellen, 'cutting the most dissolute figure imaginable, his teenage pal padding along beside him... The very sight of them seemed to scream debauchery.'\n\nHarper had ordained that the interview would be conducted in the ruins of the Roman fort of Galava. When Page was asked why this location was necessary, he gave a stony, all-knowing reply: 'The vibes, man.'\n\nAfter several hours of driving through the Lake District's steep countryside, however, the Roman fort of Galava remained as elusive as it had been when they set off from their hotel. The film crew came to the rescue, suggesting nearby Side Pike, where there was a pretty hillside, for the interview.\n\nIt got worse. When Trevor Dann asked for the promised live performance, he received not a tune from the new record, but a 12-minute version of 'Same Old Rock', a 13-year-old Harper tune. The chain-smoking Page, wearing riding boots, a battered leather flying jacket and a long white silk scarf, strummed along, effectively backing up his friend. As though wary of the power of the camera, Page consistently looked in the opposite direction to it as he played. When the cameraman managed to get a head-and-shoulders shot of him, he looked pretty great, his face framed by his tumbling hair. At times during the interview he was also extremely honest. Speaking of John Bonham's passing, he admitted that 'I didn't pick up a guitar for 18 months. I was frightened to even look at it. When I did I found I couldn't even change a chord.'\n\nAnd he also admitted that he had been rehearsing for ten days with Paul Rodgers. As yet this project lacked a name, although he joked that they would be called 'the MacGregors', 'because that was the rebel clan'.\n\nPerched on this pile of rock, Page and Harper then proceeded to utterly dismiss almost all contemporary music, Harper repeatedly showing his contempt for Frankie Goes to Hollywood by wittily referring to them as 'Bannock Goes to Frankieburn'. Page rather ill-advisedly dismissed the notion of acts employing computers: 'It's still verses and choruses like it was in 1960, and with all the technology they've got and all the outspoken statements they make, I think they ought to come up with something special. They all sound the same to me.' As he did very often, he dismissed new acts as being like 'Herman's Hermits'; Page himself had played sessions on several hits by Peter Noone's hugely successful Manchester act. But for some reason Herman's Hermits, whose records John Paul Jones had often arranged, were a real b\u00eate noire to the guitarist. Page and Harper seemed to be entirely from another age.\n\nAs the interview wound to a close, Page and his girlfriend wandered off into some adjacent shrubbery. 'What a great TV moment,' Trevor Dann told me. 'If only we could have used the shot of Page taking a slash while his teenage girlfriend held his knob.'\n\nBad Company had released six albums, with a sales trajectory that was increasingly dwindling. In 1982, following the release of their _Rough Diamonds_ album, they broke up. Among other matters, they now had an absentee manager in Peter Grant, who had looked after their career from the beginning.\n\nGrant, however, had negotiated Robert Plant's five-album deal with Atlantic. Jimmy Page was so miffed by this that he asked Phil Carson to watch over his career; Carson negotiated his deal for the Firm with Geffen Records.\n\nPage already had Paul Rodgers in mind as someone to work with. 'I tried to get together with Paul earlier, but it was difficult because he was doing a solo LP,' he later told Ultimate Classic Rock. 'But when the ARMS charity shows came up in America, Paul came with us. Stevie Winwood \u2013 who sang in London \u2013 had pulled out. So we went to the States and had a very good tour, singing songs like \"Bird on the Wing\", which we did on the album. At the end of the tour I asked if he fancied carrying on and doing something else, because I really love his singing. He's such a brilliant singer. If I do a guitar solo I have to warm up and do three takes. He does it in one take. Note perfect. No problem! He's an amazing man.'\n\nPage admitted to _Creem_ magazine that the period immediately following Led Zeppelin had been extremely difficult for him: 'After the split, I just didn't know what to do. I lived in a total vacuum. I didn't know what I was doing. In the end, I went to Bali and just thought about things. And I wasn't sitting on the beach because it was the rainy season! I sat in my room thinking. Then I thought, Dammit, I'm going to do the Firm and see if it works. At my time of life I should just do what I enjoy.'\n\nRather in opposition to what he said on _The Old Grey Whistle Test_ , Page was thoroughly aware, he told _Creem_ , of the passing of time and trends in popular music. 'Yes, I _am_ a musician of the sixties and seventies,' he told _Creem_. 'I got fired by the music of Chuck Berry and Elvis Presley. I was hit by so much energy from their records. But every five years people get fired by the music they hear and want to become part of it. So now, I'm past middle age. But what do you do when you get to middle age? There was no one for me to look up to who said, \"This is what you do next.\" I read in the music press that after 30 you are fucked. But I'm not, and there's a lot more for me to do.'\n\nJust over two years after his first coke bust, Page was in court again for cocaine possession. The story went that he had attempted to withdraw money out of a bank by Victoria railway station to buy a birthday present for his mother. The cashier doubted that he was the real Jimmy Page, partially because of his curious behaviour: at one point he was said to have fallen over in the bank. A passing constable was called over, who searched Page and found a packet of 'Charlie' in his coat pocket. Initially Page claimed that he had lent the garment to a friend, and this unnamed person must have placed the drug in it. On 5 November 1984 Page was fined \u00a3450 at Marylebone Magistrates Court for possessing cocaine. He admitted the offence.\n\nThe silver lining in this cloud was that, although he was still snorting coke, Page was no longer addicted to heroin. As he had done on his holiday in Guadeloupe with Charlotte, Scarlet and Richard Cole, he first attempted to come off heroin by dousing himself in copious amounts of alcohol and cocaine, the Keith Richards cure. Later, he would claim that withdrawal had not been too onerous, taking only four days. Which seems unlikely, junkie machismo. In fact, according to Lori Mattix, Page had taken another tip from Keith Richards: he had gone to a clinic in Switzerland and had his blood changed; with his veins bulging with clean blood, he was off the smack.\n\nThe Firm's first, eponymously titled album, released on 11 February 1985, was a Top 20 record in both the UK and the USA. The final track, the nine-minute-long 'Midnight Moonlight', was a reworking of 'Swan Song', a tune left off _Physical Graffiti_. For the Firm's shows Page even disinterred 'Dazed and Confused', his set-piece Zeppelin speciality. Although they had a slight funk feel, the Firm's records never really rose above average, burdened by a lugubrious, plodding, slightly ponderous feel. Yet there was sparkling guitar all over them from Page, and the Firm were a strong live attraction. They played over 70 dates in total, Page revisiting and selling out such Zeppelin power-points as Madison Square Garden, the Los Angeles Forum and Wembley Arena. But the second album, 1986's _Mean Business_ , did less good business, and Page broke up the band; later he claimed it was never intended for the Firm to last for more than a pair of records.\n\nFor a Fourth of July concert in 1985, Jimmy Page, ever in search of creative rehabilitation, joined the Beach Boys onstage in Washington, DC. Jeff Foskett, the Beach Boys' guitarist, was given the task of showing Page the key for each Beach Boys song. John Stamos, a 21-year-old actor friend of Foskett, was there as the guitar hero was taught how to play the songs. 'We're in this hotel and we go up to the penthouse suite and there's cases everywhere... and I thought it was guitars everywhere. He had like whips and devil shit,' said Stamos. 'He immediately offered us a shot of Jack Daniel's.' While Foskett was otherwise occupied, Page asked the actor which key certain songs were in. Stamos's uncertain responses brought out the guitarist's displeasure: '\"I can't fucking solo in E flat!\" Jimmy was yelling at me.\n\n'There were a couple of acoustic guitars, and he grabbed one and put it in my hands and said, \"All right \u2013 what are we doing?\" I said, \"Uh, I think you're playing 'Barbara Ann', right?\" He said, \"Right. What key?\" I told him it was in F sharp, and he said, \"I don't like to solo in F sharp. Why is it in F sharp?\" I was like, \"I don't know, man!\" \"Fun, Fun, Fun\" was in E flat, and he didn't like that key, either. So I'm sitting there, showing Jimmy Page how to play these three-chord songs. I'm, like, 21 years old. I could barely play guitar and had no business teaching him anything!'\n\nThe next day there was a reprise of the Washington, DC show in Philadelphia. When gaga fans near the front of the stage flashed the newly fashionable devil's horns symbols with their fingers, a recently developed sign of approbation among heavy metal fans, the not entirely drug-free Page became paranoid. 'I think they're hexing me!' he said. John Stamos had to talk him down: 'No, no, it's a good thing!'\n\nA week later something extremely unexpected happened: Led Zeppelin got back together.\n\nOn 13 July 1985 at JFK Stadium in Philadelphia, the American leg of the Live Aid concert was held. An unavoidable irony of the event for the three surviving Led Zeppelin members was that it was promoted by Bill Graham, Led Zeppelin's personal nemesis. With a pair of drummers \u2013 Chic's masterful Tony Thompson and an under-rehearsed Phil Collins \u2013 Page, Plant and Jones played 'Rock and Roll', 'Whole Lotta Love' and 'Stairway to Heaven'. It was not a good performance: Plant's voice was hoarse and Page was unable to hear what he was playing thanks to problems with his monitors; and both the drummers were out of sync. Predictably, their appearance was greeted with hysterical approval from the 90,000-strong audience \u2013 people were in tears trying to call them back for at least 15 minutes \u2013 and from television viewers across the planet. But it was a distinctly disappointing return to the stage for the three Zeppelin members. 'My main memories, really, were of totally panic,' said Page.\n\nThe three musicians had come together by accident. Page, with the Firm, and Plant had both been touring the US. The plan was for the two of them to play together as the Honeydrippers. When Jones learned of this, he pushed his way into the line-up. This meant that there was almost no time for rehearsals, however, and by this time Paul Martinez, who was in Plant's band, had been confirmed as bass player. The only slot for Jones to fill was on keyboard. Later, a considerably displeased Jones told _Classic Rock_ 's Dave Ling, 'It was Plant again, you see. Basically, I had to say to them, \"If it's Zeppelin and you're going to be doing Zeppelin songs, hi, I'm still here and I wouldn't mind being a part of it... I elbowed my way in. It's all about Robert and what he wants.'\n\nPlant, who for so much of his time in Led Zeppelin had very distinctly been the sorcerer's apprentice, clearly now considered himself a fully fledged sorcerer. Where did that leave Page? Well, he was still ready to give it another go. At Meadowlands in New Jersey, ten days after Live Aid, he appeared at Plant's concert, joining in on the Elvis Presley classic 'Mean Woman Blues' and Roy Head's 'Treat Her Right'. Plant's adoration of such rock 'n' roll evergreens had stood him in good stead. At the beginning of the year his cover of Phil Phillips's 'Sea of Love' had reached number three in the _Billboard_ charts. It was taken from an extended EP, _The Honeydrippers: Volume One_ , a collection of five R&B songs; the other tunes were Wynonie Harris's 'I Get a Thrill', Ray Charles's 'I Got a Woman', Mose Allison's 'Young Boy Blues' and Roy Brown's 'Rockin' at Midnight'. The production of the record was credited to 'Nugetre', Ertegun spelled backwards. The producer, of course, was really Ahmet Ertegun; but the sound of the record, with its Latin flavour and sumptuous strings of a type with which Ben E. King had been well acquainted, might have been by Ertegun's former employee and Page's mentor Bert Berns.\n\nFollowing the Meadowlands show, Page and Plant agreed to meet up and have 'a cup of tea'. And they would bring Jones along too.\n\nWhen they assembled in January 1986 for the two weeks booked in a village hall in the historic spa town of Bath, it was not only the bass player who joined Page and Plant: Tony Thompson was flown in from New York. As the drummer with the superb Chic, Thompson was an extremely good idea, one you suspect may have come more from Plant's relentlessly modernist thinking than from Page.\n\nThe first day of rehearsals went relatively well. 'I don't know if Jimmy was quite into it, but it was good,' Jones told Zeppelin chronicler Dave Lewis. Again, it was Robert Plant who seemed to desire to hold the reins. He would become irritated that Page, slightly drunk most of the time, would take so long to set up and get into playing.\n\nOne evening, when Tony Thompson was returning home from the pub in a minicab, the driver misjudged a bend, finishing up in someone's basement. Thompson was hospitalised. Which completely freaked out Plant; memories of former car crashes swam vividly before him, and he took this as a bad portent. Perhaps affected by contemporary visions of hipness, Plant \u2013 whose Big Hair look was decidedly unhip \u2013 was struggling with the relevance of what they were doing. As Jones explained to Dave Lewis: 'What I recall is Robert and I getting drunk in the hotel and Robert questioning what we were doing. He was saying nobody wants to hear that old stuff again and I said, \"Everybody is waiting for it to happen.\" It just fell apart from then \u2013 I suppose it came down to Robert wanting to pursue his solo career at the expense of anything else.'\n\nThere is something unpleasant in Jones's response, which seems small-minded and petty, almost the kind of sniggering that roadie Glen Colson had observed in Zeppelin dressing rooms. It hardly seems the most charitable or measured view of Plant's feelings: the man still limped from his 1975 car crash; and if he hadn't been on tour with Led Zeppelin in 1977 he might somehow have been able to prevent the death of his son Karac.\n\n'One of the roadies then played drums,' Plant told _Rolling Stone_ 's David Fricke. 'He was quite good, too, but the whole thing dematerialised. Jimmy had to change the battery of his wah-wah pedal every one and a half songs. And I said, \"I'm going home.\" Jonesy asked why. \"Because I can't put up with this and I don't need the money.\" For it to succeed in Bath, I would need to have been far more patient than I have been for years. It wasn't to be.'\n\nPlant made his excuses and left the rehearsals for good. Unlike Sam Peckinpah's ageing bad men in _The Wild Bunch_ , the old gang could not be gathered together again for one last raid. Certainly not for now, anyway.\n\nSeveral of the biggest promoters in the world were anxious to learn of Led Zeppelin reforming. In the mid-1980s Ron Delsener, who had booked memorable gigs by the Beatles, Bob Dylan and David Bowie, was probably the biggest promoter on the US East Coast. Delsener had become friends with Kosmo Vinyl, whose role with the Clash and Ian Dury was similar to that of B. P. Fallon with Led Zeppelin: vibes merchant. It was Vinyl who had urged Ian Dury that it was ideologically unsound for him to support Zeppelin at Knebworth. 'I'd ask Ron, \"Who would you put on to make money?\" He would reply, \"If I could put Led Zeppelin on in a giant stadium until they couldn't play anymore.\" But that storm-trooper swagger was getting a bit tired when they were doing it.'\n\nThere were intimations of a continuing friendship, no matter how difficult, between Page and Plant, those two ultimate rock 'n' roll warhorses. In the autumn of 1987, Plant was ensconced in Studio One of Marcus Music in Kensington Gardens Square, West London. He was recording the album that would be known as _Now and Zen_ \u2013 not a bad cosmic jokey title. Andy Priest, who was working on the front desk, recalled a time when 'Jimmy Page turned up to surprise him and so I directed Jimmy to the studio. After a few hours he came out and asked me to call him a cab. I asked where he was going to. He said, \"Windsor.\" I was about to finish my shift so I said, \"If you can wait a few minutes I'll be able to give you a lift. I'm going near there, so you might as well save yourself a few bob.\" Happy to accept my offer, he got in the passenger seat of my old rattling Triumph 1500 and I set off. On the M4 motorway I stopped at Heston Services for petrol and, while I was filling up, he went to the kiosk and very kindly paid the bill. We spent the rest of the journey talking about music until I dropped him off at his gate. I went on to play bass for a reformed Sham 69, but that's another story.'\n\nJimmy Page met Patricia Ecker on 25 April 1986, on the Firm's second US tour, when she served him in a French restaurant in New Orleans. Ecker was a 24-year-old model who was working as a waitress. Blonde and beautiful, she bore a certain resemblance to Charlotte Martin. For Page, in thrall to his emotions often more than he would like, and certainly responding to his vulnerable heart's need for succour, it was love at first sight: Pat was 18 years younger than him. When the tour ended, she came back to London with him. New Orleans, voodoo central: for Robert Plant a city that brought him news of utter tragedy; for Page a place that brought him love and re-birth. The poetry of such Led Zeppelin paradoxes is apparently infinite. And food for meditation.\n\nPat Ecker was soon pregnant: their son James Patrick Page \u2013 following a tradition in the Page family of first sons taking that name \u2013 was born on 26 April 1988. She and Jimmy married.\n\nThe birth of a child brings fresh luck, they say. And not too long after that failed reunion in Bath, and the birth of James, Led Zeppelin did come formally together.\n\nOn 14 May 1988 Atlantic Records held its 40th anniversary show at Madison Square Garden. At the time Plant was touring the USA, promoting his _Now and Zen_ album. Ahmet Ertegun personally asked Plant, who continued to be released through Atlantic, to play the event. But he also asked all concerned if Led Zeppelin could reunite to close the evening. Amazingly, they agreed. Even though Plant appeared to be constantly closing off such a possibility, there was a side of him clearly still open to the Zeppelin journey, a journey that seemed increasingly one of the soul. And, all things considered, who would be surprised by that?\n\nThe Atlantic Records 40th Anniversary Concert was broadcast live in the US on a simulcast between FM radio and HBO television, an enormous promotion. Plant performed a well-rehearsed, well-received solo set, following Crosby, Stills and Nash, the Bee Gees and Yes. But the night before the show Plant and Page had had words about Led Zeppelin's concert-closing act: the singer had wanted to use his new drummer, Chris Blackwell (no relation to the Island Records boss); but Page was adamant that Jason Bonham be employed, arguing, not unreasonably, for a Zeppelin bloodline continuance. Also, Plant had refused to perform 'Stairway to Heaven', although \u2013 again \u2013 he changed his mind under pressure from Page.\n\n'Well, that was awful,' Page said to Mick Wall. He told him that Plant 'came together with Jason, Jonesy and me in New York, where we were rehearsing, and started singing \"Over the Hills and Far Away\". And it sounded really brilliant, actually. Then we rehearsed \"Stairway\" and that sounded great, too. Then the day before the show he called me up that evening and said, \"I'm not going to sing it.\" I said, \"What are you talking about? You're not gonna sing 'Stairway'? But that's exactly the one thing that everybody expects to hear us do!\" He said, \"I don't wanna do that!\"'\n\nThey performed 'Kashmir', with its broody opening chords, 'Heartbreaker', 'Whole Lotta Love', 'Misty Mountain Hop' and \u2013 of course \u2013 'Stairway to Heaven'. The time-honoured view of the show is that it was a shambles. Yet it is easily available on YouTube, and watching its full 32 minutes, it is eminently clear that this was not the case. Moreover, everyone seemed very happy after the performance. It seems a measure of the full extent to which the tide of popularity had receded for Led Zeppelin that such dismissiveness has become the established currency of how this show was. Especially in comparison with some of their performances on the fated 1977 tour, it really wasn't a disaster at all. But there was something else hovering over the participants. 'I had a strong feeling that the way the Atlantic thing went put a Zeppelin reunion on the back burner for him,' said Doug Boyle, Plant's guitarist. 'Before it I thought there might have been something happening a couple of years down the line. I don't know what went down but that gig was a very tense time between Robert and Jimmy.\n\n'I think there was a part of Robert that missed Jimmy an awful lot. He'd often say to me, \"Jimmy would have done this\" or \"Jimmy would have done that\"... The two of them are like brothers. There's something very, very deep there.'\n\nAlthough certainly not evident during their time onstage at Madison Square Garden, the psychic tussle between Page and Plant over 'Stairway' the previous day had depleted the guitarist. Afterwards, he told Mick Wall, 'I didn't really sleep that night. I was jetlagged anyway because my son had just been born in England and I'd left within a few days of that. And I was really on a roll from that, you know, the high that you're into after the birth of a child. And all of a sudden, I plunged to the ground! Like, what the hell am I doing here?'\n\nOn 19 June 1988 an eagerly anticipated event occurred in the life of Jimmy Page and his fans. It was one that could have been of significant consequence in the history of popular music. It should have been a major musical statement, placing him on a par with his friend Eric Clapton, whose solo albums regularly sold in the region of four million copies. Such a record would have reinflated his legend, stating his clear pole position among British blues guitarists. Unfortunately, this did not prove to be the case.\n\nThe album _Outrider_ , the supposed first step in Page's solo career, was released four and a half weeks after the profile-rising venture of the Atlantic Anniversary Concert.\n\nEven the title was curious, feeling inappropriate, sounding like a character from the then popular _Mad Max_ films' dystopian anarchism; it seemed ill-thought out, but revealing. Was this how Page now perceived himself, riding one of the outer cosmic asteroids? It could equally have been titled, more accurately, as Outsider. Extremely uneven, _Outrider_ seemed further evidence of Page's seemingly inexorable decline during the 1980s, a fall in personal standards to which the very existence of the Firm appeared to attest.\n\nBut what are the positives? At first Page had intended to make a double album. But demo tapes of many songs disappeared when he had what he described as a 'domestic' situation in a house he had temporarily vacated. 'It was totally different to this stuff,' he told Bud Scoppa in _Guitar World_ , describing how at that stage all the missing material was acoustic. 'There were two tapes, actually, that had a lot of stuff \u2013 it was like a compilation of stuff \u2013 I don't know if they're ever gonna resurface. And if they do, well, they'll be heard anyway, won't they?'\n\nSo what did Page do after the demo tapes went missing? Without any tunes written whatsoever, he simply went into his own Sol Studio and started recording. How it turned out was how it turned out.\n\nHe was clearly rejuvenated by his new relationship with Pat Ecker. The nine-tune record, with all music except for one track solely credited to Jimmy Page, was recorded in early 1987, after he returned to the UK with her; the trio of vocalists supplied their own lyrics. 'The Only One' was the closest to a Zeppelin tune, notably because it had Robert Plant on vocals, the sole track \u2013 aptly, a cock-rock extravaganza \u2013 on which his former prot\u00e9g\u00e9 featured. Elsewhere, the singing was handled by John Miles, who had had a number one hit in the UK with 'Music' in 1976, and by Page's old mate Chris Farlowe. The use of three vocalists was an indication of Page's perception of what he was providing: a showcase for his own guitar playing, arranging and producing. Using a pair of 24-track desks, flipping sounds across and between the two of them, there would sometimes be as many as 30 guitar parts on a number; the Page Guitar Army on manoeuvres.\n\nJason Bonham, Bonzo's 22-year-old son, was on drums and percussion, except for two tracks, 'Liquid Mercury' and 'Emerald Eyes', on which Barriemore Barlow, a former Jethro Tull member who had played most recently with Plant, took over these duties. And who would replace John Paul Jones? As he had with vocalists, Page required three bass players: Durban Laverde on 'Wanna Make Love', 'Writes of Winter' and Leon Russell's 'Hummingbird', the only cover; Felix Krish on 'The Only One', 'Liquid Mercury', 'Emerald Eyes', 'Prison Blues' and 'Blues Anthem'; and Tony Franklin, from the Firm, on 'Wasting My Time', the first song recorded and the album opener. Sometimes you felt that Page could have fared better with Jones's consummate arranging skills.\n\nThe material wasn't too bad, but it also wasn't that great. Out of the nine songs there were three instrumentals: 'Writes of Winter', with its flavour of Thin Lizzy at the peak of their ringing twin-guitar supremacy, a Grammy-nominated track; 'Liquid Mercury', which was underwhelming and a little boring, like a sketch of something unfinished; and 'Emerald Eyes', which you felt was intended as a stand-alone 'White Summer' sort of number, replete with raucous, fuzzed guitar, but that never hit the transcendent heights at which it appeared to be aimed.\n\nThe smoky 'Prison Blues', with the earthy vocals and salacious words of Chris Farlowe, was in the great Zeppelin tradition of dirty blues, as was 'Blues Anthem'. Yet wasn't that part of the problem? _Outrider_ sounded as though it could have been made in the late sixties or early seventies. In other words, it was rather out of date. It feels terrible to say this, but Page's first effort at artistic redemption was a disappointing failure. If anyone felt that the accusations of musical dinosaur had been unfairly directed his way, the lukewarm reviews of _Outrider_ spelt out the truth. 'A bizarrely dated-sounding effort, _Outrider_ is mostly British blues rock at its most mediocre,' wrote Lynn Van Matre in the _Chicago Tribune_. The disc was a damp squib, scraping 500,000 sales worldwide.\n\nPage's view of the media and its participating players also seemed not to have shifted one jot. Interviewing him at the Four Seasons Hotel in Los Angeles, Bud Scoppa, a former _Rolling Stone_ writer, noted in his _Guitar Player_ article: 'The one aspect of the man that seemed in keeping with his reputation was a surliness that lurked ominously just below the surface of his demeanour \u2013 a sort of Loch Ness Monster of latent hostility.' Scoppa also observed that when there came a knock on the door of Page's ritzy suite, a bellman stood there bearing a gift: a paper bag displaying the Burger King insignia. Page then brought his burger and fries to the interview table. Liberated from his heroin and cocaine diet, he had fleshed out, his body thickening, his face almost a little jowly; a fast-food diet would do that to you. But so would coming off smack.\n\nOn 21 June 1988, two days after the release of _Outrider_ , Jamie Kitman, the manager of They Might Be Giants, was at New Orleans Airport, waiting to board a flight to New York. A man in a wheelchair was pushed to the front of the queue for the plane. He had a blanket over his knees and on top of it sat a tabby cat. It was Page. Much to his distress, the cat was taken from him for the duration of the flight. Then he was gingerly helped up the steps to the first-class lounge.\n\nDuring the flight Kitman stepped forward from coach into the first-class lounge. He found Page, looking unwell. He briefly introduced himself and placed a cassette of the They Might Be Giants album on his tray table. 'Jimmy stared at it, as though it might hurt him,' said Kitman, who made his excuses and returned to his seat in coach.\n\nKitman's experience on that flight seemed like a spectral vision of everything one feared for Jimmy Page.\n\nThe reviews for _Outrider_ 's live tour were no better than those the record itself received. It kicked off on 2 September 1988 at the Miami Arena in Florida, and after 15 dates, including those Zeppelin strongholds Texas and Los Angeles, it hit Chicago, another favourite town for Page's former group.\n\nThough acknowledging the extremities of satisfaction felt by the audience at Chicago's UIC Pavilion, David Silverman, writing in the _Chicago Tribune_ , criticised John Miles's vocals and attitude: 'Rough like broken glass, but not nearly as sharp, Miles plugged his way through the set with little emotion or enthusiasm.' He added: 'Without Plant's vocals and the unique Zeppelin sound, it was a hollow reminiscence. At times, Page seemed too alone on stage, working through material that appeared as distant to him as the band's past. The result was a cold reminder that what once was Zeppelin is now distant history.'\n\nAs well as almost all the _Outrider_ material, Page and his group played new versions of 'Custard Pie' and 'Over the Hills and Far Away'. The Yardbirds\/Zeppelin rearrangement of the great 'Train Kept A-Rollin'' was also featured as the penultimate number, before the predictably enormous and ecstatic audience response to the inevitable 'Stairway to Heaven', the set closer.\n\nDavid Silverman didn't even mention that as well as Jason Bonham and Miles on stage, Page's touring band was rounded off by bassist Durban Laverde.\n\n'The result was an evening that was successful for Page as a solo artist and for Bonham's excellent drum work. But the attempt to build a band around Page's immense and unique talent came up short,' concluded the journalist. 'The merging of old Page and new Bonham was the only bit of magic that could be squeezed from the show.'\n\nOn 10 January 1990 Jon Bon Jovi introduced Page during a Bon Jovi show at London's Hammersmith Odeon: emerging from the wings, the guitarist performed what by now had become his old standby of 'Train Kept A-Rollin'', and \u2013 wow! \u2013 the stunning solo from his session work on Joe Cocker's 'With a Little Help from My Friends'.\n\nAt the 1990 Monsters of Rock Festival at Castle Donington, Page joined Aerosmith onstage before 70,000 people. He jammed with the revived act's guitarists, Joe Perry and Brad Whitford, on 'Train Kept A-Rollin''. Three days later, he played with them again, before 300 people, at the Marquee Club, performing not only 'Train' but also 'I Ain't Got You' and 'Think About It', old Yardbirds tunes. 'Aerosmith are a great band, a really good band to play with,' said Page. Then he led the Boston rockers into 'Immigrant Song', the Marquee exploding with emotion.\n\nLed Zeppelin themselves seemed to have become a wedding and parties outfit. In November 1989 Page, Robert Plant and John Paul Jones attended the 21st-birthday celebration of Carmen Plant, the singer's daughter, on the Welsh border. Alcohol duly down the hatch, they played together on 'Misty Mountain Hop' and 'Trampled Under Foot', among other tunes, a set of over half an hour. Then, on 28 April 1990, all three once again were together, playing with Jason Bonham at his wedding in Bewdley, Worcestershire. This time they played a different set of tunes: 'Bring It on Home', 'Sick Again', 'Custard Pie' and 'Rock and Roll'.\n\nOn 30 June 1990 Page joined Plant for his performance at Knebworth House. It was a star-studded bill, including Pink Floyd, Eric Clapton, Elton John and Paul McCartney, among other major names. It had been rumoured for some days that both Page and Jones would join their old spar onstage. Although Jones was a no-show, Page most certainly was there, playing 'Rock and Roll', 'Going to California', 'Wearing and Tearing' (from _Coda_ ), 'Misty Mountain Hop' and 'Immigrant Song' with Plant. A sense of Led Zeppelin getting back together was building.\n\nIn September 1990 a Zeppelin compilation of sorts was released. It was a four-CD set entitled _The Led Zeppelin Boxed Set_ , a substantial bestseller, making the US Top 20. In the late 1980s all of Led Zeppelin's material had been released for the first time on compact disc. Yet they had been derived not from re-equalised master tapes, but from sound-generation tapes equalised for the original vinyl releases; this angered Jimmy Page, always keen \u2013 as we would come to see \u2013 that the legacy of Led Zeppelin was as perfectly accurate as feasible. 'They even cut off the cough at the end of \"In My Time of Dying\",' he complained later to _Guitar World_. He personally supervised the remastering process, spending a week in May 1990 at New York's Sterling Studios, upgrading the analogue tapes to digital standard.\n\n'My awareness was re-heightened when we were remastering the material to do that CD box set in 1990,' said Page, 20 years later. 'When you hear it all, song after song, you realise what a textbook it is for musicians who are coming along, and that's so great. The whole thing is about passing it on, because that's how it was done for me when I was learning from all those old blues and rockabilly records. It's all part of how this cultural phenomenon keeps moving on. I think everyone carries the flame on.'\n\nHeightening the mystery, the boxed-set sleeve bore an image of a crop circle, upon which was cast the shadow of a Zeppelin airship, highly effective and pleasantly ironic in the self-belief of its omniscience.\n\nFour months later, in the slipstream of the success of _The Led Zeppelin Boxed Set_ , the three surviving members met up to consider the possibility of getting back together for a tour. Jason Bonham was not invited to the meeting: Plant was urging the others to consider Mike Bordin, the drummer with the hip Faith No More, as replacement for Bonzo. His pair of old spars would not go along with this, which set Plant off on a departure from the idea before it had even begun.\n\nBy now Page had sold his studio the Sol, which had been a going financial concern throughout the last ten years. Elton John had made a pair of albums there; and in 1989 Jeff Beck had recorded _Guitar Shop_ at the studio, a comeback album of sorts. The next year Page sold Boleskine House; some claimed that his interest in Aleister Crowley had considerably diminished.\n\nPage set about making a new solo album. But his record company, Geffen, urged him instead to team up with another of their artists: David Coverdale, the former Deep Purple vocalist who had subsequently established himself in the album charts with his group Whitesnake. On paper it didn't seem too good an idea. Whitesnake were a kind of Led Zeppelin-lite, Coverdale a sort of Robert Plant imitator. The idea appeared quintessentially naff. Yet that was precisely what Page needed. But he wanted to meet Coverdale first, to check his 'character'. Clearly Coverdale passed the test.\n\nAfter sessions in Vancouver, Miami, New York and Abbey Road in London, the album, to be titled _Coverdale\/Page_ , was completed by the spring of 1992, and finally released on 15 March 1993. Robert Plant sneered at the very notion of Page working with, as he put it, 'David Coverversion'. (According to Guy Pratt, who played bass on the subsequent tour, Coverdale retorted, 'I'll challenge him to a shout-off any time you like.' 'Missing the point,' said Pratt, 'that we'd had to take most of the Zeppelin songs down about two and a half notes, because no one but Plant can get up there.')\n\nYet the record charged up the charts: number five in the USA and number four in the UK. It was considerably assisted in its course by the heavy rotation on MTV of 'Pride and Joy', number one in the _Billboard_ album track rock chart for six weeks.\n\nBut the project then stalled, following a Japanese tour. Dates had been set up for American shows, but were cancelled, supposedly due to poor ticket sales. 'It was originally meant to be an American and European tour,' said Pratt, 'but it was booked as arenas and the ticket sales just weren't there. So it was to be played in theatres instead. Jimmy was happy with that but David Coverdale considered his place in the firmament to be arenas. So Coverdale considered he would be seen to be downsizing. So we did these rehearsals, did nothing for six months, and then went to Japan.'\n\nDuring the two-week Japanese tour Page did not look too good, overweight and clad in bulky waistcoats and big ties. Hardly the once-beautiful Prince of Darkness. Of this former image Pratt felt he had a perspective: 'It is potentially one of the most clever pieces of PR management ever. The guy is basically just a collector. But if you recast him as the devil, it's kind of genius.'\n\nBut although Page's marriage was falling apart at the time, he seemed to enjoy himself during the tour, no longer retreating to the seclusion of his hotel room following a show \u2013 he was drinking alcohol but seemed drug-free. Almost within hours of arriving in Japan the entourage had acquainted themselves with a set of local strippers, as though this was their on-the-road modus operandi, and in Tokyo Page met a pair of Israeli girls whom he found most entertaining.\n\nHowever, Page now had something he wanted to do far, far more: he was going to work again with Robert Plant.\n\nThe singer's solo career was faltering. His most recent album, _Fate of Nations_ , had barely scraped the US Top 40 albums. And he had been distinctly unpleased to find himself on a European tour opening for Lenny Kravitz, who was heavily influenced by Led Zeppelin. Moreover, the subsequent development confirmed what Guy Pratt had felt during the dates with David Coverdale: 'Jimmy Page just can't get over Percy Plant.'\n\n## 25\n\n# THE SORCERER'S APPRENTICE\n\nPlant had been asked to appear on MTV's _Unplugged_ show, an occasional series featuring classic artists playing acoustic sets that had first aired in 1989. And then Jimmy Page was pulled in to the performance. Once again it seemed he was being drawn along by the influence of his old guitar chum and neighbour Eric Clapton: Slowhand's recording of his 1992 MTV _Unplugged_ show had been released as an album that sold 26 million copies, the biggest-selling live album ever.\n\nBill Curbishley, who had been managing Plant since the 1980s, contacted Page shortly before his Japanese tour with David Coverdale. 'I had a call from Robert's management to pop in and see Robert in Boston on the way to LA to rehearse,' remembered Page. 'Robert said, \"I've been approached by MTV to do an _Unplugged_ and I'd really like to do it with you.\" So I said okay. It gave us a chance to revisit some numbers and use that same picture with a very, very different frame.'\n\n'By that time I didn't feel like I was even a rock singer anymore,' said Plant. 'Then I was approached by MTV to do an _Unplugged_ session. But I knew that I couldn't be seen to be holding the flag for the Zeppelin legacy on TV. Then mysteriously Jimmy turned up at a gig I was playing in Boston and it was like those difficult last days of Led Zep had vanished. We had this understanding again without doing or saying anything. We talked about the MTV thing and decided to see where we could take it.'\n\nBut there was a notable absentee: John Paul Jones, who was touring with Diamanda Gal\u00e1s, was not invited to this party, a cause of considerable upset for him. At a subsequent press conference, the pair were asked where was the bass player. 'Just parking the car,' came a bitchy response from Plant. When the first Page\/Plant record emerged, given the title _No Quarter_ , Jones was even more miffed, claiming that the fact that the title track existed at all was almost entirely down to him.\n\nIn March 1994 Page and Plant were tucked away at the Depot, a rehearsal space in London's King's Cross. They started off by playing along to North African-inspired drum loops provided by Martin Meissonnier, a French music producer of considerable ability; Plant then brought in the drummer and bass player from his own band, Michael Lee and Charlie Jones, the latter also by now his son-in-law.\n\nOn 17 April Page and Plant, along with Lee and Jones, played at Buxton Opera House at a memorial concert for the late Alexis Korner, who had been such an inspiration and support for Plant. Among the songs they played were 'Baby Please Don't Go', 'I Can't Quit You Baby' and 'Train Kept A-Rollin''.\n\nThen they went into an upstairs room at the King's Head on Fulham High Street, a pub-rock venue back in the days of nearby Swan Song, and only a hundred yards from what was once Manticore, where they had rehearsed for the 1977 Zeppelin US tour.\n\nPlant had been inspired by Peter Gabriel's work on the soundtrack to Martin Scorsese's _The Last Temptation of Christ_ , filtering North African music and talent through Western perceptions. Accordingly, Hossam Ramzy, an Egyptian percussionist and composer who worked with Gabriel on the film, joined them in Fulham. Plant also took on Ed Shearmur, a soundtrack composer and arranger. The pair, the singer informed them, would be responsible for creating the required blend of Western folk-rock and traditional Arabic and Indian music. Led Zeppelin's songs were about to be savagely reinterpreted and brought up to date, as befitted the pair of now middle-aged rockers.\n\nThe equivalent of a baptism by fire, Plant ordained that he and Page would set to work straightaway with these two new collaborators on probably the hardest of the Zeppelin tunes to be restructured; though philosophically it was also the most ready, potentially extremely malleable: 'Kashmir'.\n\nThrough his Indian family connections Plant brought in Najma Akhtar, an award-winning, English-born singer of Indian descent with a voice that was almost celestial. 'Robert was very much in control of the whole enterprise and what he said went,' she told Paul Rees. 'He originally sent me three songs to listen to and I remember Jimmy suggesting to him that I should sing on all of them. Robert said no, just \"The Battle of Evermore\". Jimmy tried to ask why not, but I only did the one.\n\n'Generally I think Robert is the key-holder in their relationship. Both of them are extremely intelligent and knowledgeable about different kinds of music, but they're otherwise very different people. Robert interacted with me more. He was the taskmaster. Jimmy seemed shy and very reserved. There was a lot of tension between them, both artistically and also personally.'\n\nPart of this tension was over Plant's insistence that they would not play 'Stairway to Heaven'. In order to keep the project going, Page, who had turned 50 that January, needed to subjugate himself to Plant, something that would not even have crossed his mind to do 25 years earlier. He sat in a corner of the rehearsal space watching, listening, absorbing. Always aware and highly intelligent, Page was only too appreciative of the classic circumstances they found themselves in, the student overpowering his teacher.\n\nThat July, along with an Egyptian orchestra and an English string section, they rehearsed at Wembley for a week.\n\nEarly in August they travelled down to Plant's beloved Morocco, where a traditional Gnaoua band had been enlisted. Learning of this, you couldn't help but recall something William Burroughs had said of Moroccan musicians to Page when they met in 1975: 'Their music is definitely used for magical purposes. For example, the Gnaoua music is to drive out evil spirits and Joujouka music is invoking the God Pan. Musicians there are all magicians, quite consciously.' So, you may wonder, by involving Gnaoua musicians was Plant consciously or unconsciously doing some psychic soul-cleansing?\n\nWith so many pre-filmed sequences planned, Page and Plant's was to be an unusual twist on the established _Unplugged_ format. With the cameras rolling during the day in Jemaa el-Fna, the Square of the Dead, in Marrakesh, they performed a pair of new acoustic songs: the beautiful 'City Don't Cry' and 'Wah Wah', a considerable stretch away from previous Zep material. That evening, again with a full film crew, they played another new tune, 'Yallah' (also known, curiously, as 'The Truth Explodes'), even steamier and moodier than the other two songs; they played to Martin Meissonnier's loop, kicking off with a crunching power chord from Page that could have come from a grunge act like Nirvana. 'Wah Wah' was also performed again. The mystery and mystique, the sense of wandering into the unknown that had always couched Led Zeppelin, was there again. Again there was an element of smoke and mirrors. Michi Nakao had been employed as the shoot's make-up artist. Her brief was not easy: Page desired that not a glimpse of his grey hair appear on film; at the same time he would not accept any dye in it. From time to time she would be obliged to 'smooth' back the guitarist's hair, her hand concealing the black stain she had secreted. At the end of the shoot, as an act of gratitude \u2013 unaware of her sleight-of-hand subterfuge \u2013 Page gifted her with a leather icon of a local deity.\n\nOn 16 August, on a Welsh mountainside, they were filmed performing a new version of 'No Quarter'. The next day, nearby, they shot and recorded 'Gallows Pole', 'Nobody's Fault but Mine' and 'When the Levee Breaks'.\n\nOn 25 and 26 August they were at a London television studio, filming the on-set sequences for _Unplugged_ ; the next month was spent mixing the show and the forthcoming album. On 12 October 1994, _Unledded_ , as it had become titled, was broadcast on MTV in the US. It got the highest viewing figures of any show aired under the _Unplugged_ brand. By now a world tour had been booked for Page and Plant, a half of Led Zeppelin tour; they spent November rehearsing for it.\n\nThe album _No Quarter: Jimmy Page and Robert Plant Unledded_ went on sale on 14 October 1994. The next week it entered the US album charts at number four, its highest position, going on to sell over a million copies in America alone; in the UK it only hit number seven, but sold over 100,000 copies.\n\nPage and Plant knew that this was a golden opportunity, a chance to revive their former glory, maybe the last for these ageing rock 'n' roll gunslingers. They pulled out all the stops; drawing upon their mutual love of Indian and Arabic music, they truly scored: _No Quarter_ is a stunningly great record. By then world music may have already peaked, during the second half of the 1980s. But few of Zep's original audience would have had too much awareness of this. And the result was grown-up, very intelligent music, even if much of it was Led Zeppelin reinterpretations.\n\nOne important consequence of _No Quarter_ , and Page and Plant's assiduously creative work together, was to commence the restoration of Jimmy Page in the eyes of the world's music fans. That tide that had been out for so long \u2013 with the widespread perception of him as a casualty, a wasted junkie and dark-arts practitioner, an image only enhanced by the Firm and his work with David Coverdale \u2013 was beginning to flow once again towards the shore. As he learned, through necessity, to let go of his Capricorn love of control and hand it over to Robert Plant, something wonderful happened for him.\n\nOn 12 January 1995 the pair, plus John Paul Jones, were at the Waldorf Astoria Hotel in New York, as Led Zeppelin were inducted into the Rock and Roll Hall of Fame. 'Thanks to my friends for finally remembering my telephone number,' uttered Jones with understated sarcasm. The three of them, along with Jason Bonham, played with Joe Perry and Steven Tyler from Aerosmith on 'For Your Love', 'Train Kept A-Rollin'', 'Bring It on Home' and 'Baby Please Don't Go'. Then, accompanied by Neil Young, they played 'When the Levee Breaks'. Neil Young wrote the song 'Downtown' about this experience, which was included on his _Mirror Ball_ album, released that summer.\n\nThe Page and Plant _No Quarter_ world tour kicked off on 26 February 1995, 22 days after Page had played with the Black Crowes in Paris, joining in on that old warhorse 'Shake Your Money Maker'. You sensed this endeavour was not only to raise his profile, but also to get match fit for the upcoming tour. And to have some fun: in early 1995 Page and his wife Patricia Ecker had split up, and would finally divorce. Then she went home to her native New Orleans with their son James.\n\nThe 51-year-old Page had already started a new relationship, with Jimena Gomez-Paratcha, a 23-year-old San Franciscan activist with Argentinean parents. Her first name, of course, was a Latin version of his own. Page appeared to have two, classically opposite female archetypes: statuesque movie-star-like blondes, like Charlotte Martin and Patricia Ecker, and sultry, dark-haired sirens, in the manner of Lori Mattix and, now, Jimena Gomez-Paratcha. They all had one thing in common, however: they were invariably young. 'You can have sex with anyone these days, whatever sex,' said Steve Parsons, the former singer with the Sharks. 'But the big taboo remains older men with younger women. I think that's subconsciously why Jimmy does it, because he knows how it winds people up. He has always liked to be a rebel.'\n\nThe pair had met in Rio de Janeiro in Brazil when he was doing promotion for the upcoming _No Quarter_ tour. Jimena was working as a community worker with local street children. Moved by the plight of these homeless kids, Page \u2013 assisted by Jimena \u2013 established a shelter, Casa Jimmy. Soon she would assist him further, bearing a new family for the musician: Zofia Jade, who was born in 1997, and Ashen Josan, who arrived two years later. Jana, Jimena's daughter from an earlier relationship, was also taken under Page's wing.\n\nThe Page\/Plant world tour was an enormous undertaking; it started with 47 dates in the US, on 26 February 1995 in Pensacola, Florida. On 31 March Page and Plant played the Palace, Auburn Hills, Michigan. A 23-year-old man rushed the stage to make an attempt on Page's life. Demons surrounding Page had driven him to it, he claimed. Some things never died down.\n\nThen there were 27 dates in Europe, including a triumphant appearance on the Pyramid Stage at Glastonbury Festival, its bohemian milieu perfect for the pair's spirited melange of exotic influences. Plus, after an eight-week break following their Wembley shows in London on 25 and 26 July 1995, another 22 in the US and Mexico, concluding with a pair of shows at Madison Square Garden. Peter Grant was invited to the Wembley dates, and Plant paid tribute to him from the stage. There was an irony here: for all Grant's efforts at reaping the maximum financial rewards for his clients, the _No Quarter_ tour \u2013 one of the biggest earners of the year \u2013 brought in even more cash than any set of dates by the full Zeppelin line-up. More importantly, it was a colossal creative triumph. 'It was heroic to take something like that around the world,' Page told Charles Shaar Murray in _Mojo_ , 'because it was using two orchestras: one Western, one Arab orchestra, with a hurdy-gurdy. It was great going around the world to turn people on to sounds they hadn't heard. It wasn't an easy thing to do, but it was worth it.'\n\nDuring the tour \u2013 why break the habit of a lifetime? \u2013 Page remained very much the rock 'n' roll recluse, tucked away in his hotel room most of the time. Nigel Eaton, their hurdy-gurdy player, thought he was 'one of those loner types at school that never went out... He was always very nice to me, but he was much more intense and I was warier of him.' Page may have been off drugs, but he was drinking a great deal, with Plant's mates complaining that his boozing was adversely affecting his performances. Plant, meanwhile, began a relationship with Najma Akhtar, who sang on selected dates.\n\nAccording to photographer Ross Halfin: 'I found that tour really unpleasant. It was sort of Jimmy's camp, which was his guitar tech and me, and then Robert and everyone else. You were made to feel as if you shouldn't be there. Admittedly, Jimmy was drinking a lot by the end of it so he was hard work.\n\n'The two of them did seem to get on for the most part, though. Robert used to call Jimmy \"Jimbob\" all the time to try and annoy him but Jimmy would just ignore it. Jimmy had also wanted John Paul Jones to be there but Robert wouldn't have it. Jimmy was as much to blame for that situation because he gave in to it.'\n\nOn 21 November 1995 Peter Grant died from a heart attack; he was 60 years old. His funeral took place the next month, at St Peter and St Paul's Church in the possibly aptly named village of Hellingly in East Sussex, where Grant's mansion Horselunges was located; he had long since moved to Eastbourne. Jimmy Page, Robert Plant and John Paul Jones were in attendance, along with Jeff Beck, Bad Company, Phil Carson and even Denny Laine, the father of Catherine James's son, who by now had a child with Peter Grant's daughter Helen. And which song was played as they departed the church? Vera Lynn's 'We'll Meet Again', the controversial manager having requested the voice of Britain's wartime songbird.\n\nAt the wake, held nearby, Page was the first to leave, the door slamming shut behind him as he walked out, now with only memories of his close friend and confidante, the man directly responsible for recognising his great talent and facilitating what became the biggest group in the world. Page issued a terse statement: 'Peter was a tower of strength as a business partner and a friend. I will miss him and my heart goes out to his family.'\n\nThe year before his death, Grant had read Bill Graham's autobiography, which contained a graphic account of that shocking night in 1977 at Oakland Coliseum. Grant had supposedly broken down and cried, a final release.\n\nIn the new year, 1996, there were Page and Plant shows in South America, Japan and Australia. The very last date of the _No Quarter_ tour took place on 30 November at the Mumbai Sports Complex in India, their ambition to play on the subcontinent finally realised.\n\nWhile Page\/Plant were in Sydney, Australia, at the end of February 1996, Jeff Buckley played a sensational show in the city to a fantastic crowd response, remembered his friend, the producer Jack McKeever. Buckley believed it to be one of the best shows he had ever done. Immediately after, he was exhausted and dripping with perspiration. In his dressing room he put a white towel over his head, soaking up the sweat that was running down his face and neck. Suddenly he felt someone come into the room. This person picked him up and danced around the room with him, holding him above the ground. 'That was fantastic. You are fantastic. You are the best thing in the last 20 years. Can I produce your next album?' It was Jimmy Page. Jeff Buckley was an enormous Led Zeppelin fan; the group had influenced him more than anything else. He was utterly knocked out by Page's response. Yet he turned down the production offer, feeling it would not be entirely appropriate.\n\nThe next year, 11 November 1997 saw the release of Led Zeppelin's _BBC Sessions_. Page had compiled and mastered the tapes, setting standards for quality control on subsequent reissues from which he would never deviate. In the United States, Zeppelin's sales heartland, _BBC Sessions_ was a number 12 album.\n\nWhat would Page\/Plant do next? What any other musicians in their position would do: they made another record, _Walking into Clarksdale_ , its title inspired by the legendary crossroads in Clarksdale, Mississippi, where blues legend Robert Johnson allegedly sold his soul to the devil in return for his talent. Equally allegedly, Robert Plant was said to have carried a pouch of earth scraped from that very crossroads with him at all times.\n\nAt the singer's suggestion they brought in Steve Albini as producer. Once again, a sign of his maturing years perhaps, Jimmy Page relinquished control: he apparently no longer needed to be in charge of production. Albini was the grunge wunderkind, a crucial figure in independent record production, who had produced \u2013 among countless other successful albums \u2013 Nirvana's 13-million-selling _In Utero_ , with its unutterably bare and dirty sound. Albini was a very good decision.\n\n_Walking into Clarksdale_ was recorded at Abbey Road in London, again with Charlie Jones on bass and Michael Lee on drums and percussion. Page and Plant ditched the idea of working with orchestras or arcane instrumentation again. 'The most obvious thing for us to do,' said Page, 'was for us to go back to the four-piece unit that we knew best and always worked for us. A lot of people thought we were going to carry on with that big extravaganza... but for us it was important to come to terms with the songs.'\n\n_Walking into Clarksdale_ is another sensational album, on a par with _No Quarter_. It is a really great rock 'n' roll record, with a worldbeat spirit, characterised by Plant's semi-ululating African vocal punctuations. When it was released on 21 April 1998, _Walking into Clarksdale_ was extremely, and bafflingly, critically underrated. Still, it made number eight in the US album charts and number three in the UK.\n\nFor the first time since tunes like _Led Zeppelin II_ 's 'Thank You', Page was playing on a record that often sounded like it was made by a late 1960s garage band. It feels like it was how Page always wanted it to be: tight but loose. At times it is like the driving, screaming early 1990s live work of Neil Young; Page's guitar work is relentless, joyous. 'House of Love', the almost teenage penultimate track, sounded like all that time playing with the Johnny Burnette Trio's master song had rubbed off. A key tune was the sensational 'Most High', which won Page and Plant the Grammy Award for Best Hard Rock Performance in 1999; number one in the _Billboard_ Mainstream Rock chart in May 1998, it made the Top 30 in the UK, bringing a first ever appearance on _Top of the Pops_ for Page; time \u2013 or pragmatism \u2013 had mellowed him.\n\nIn the early months of 1998, Page and Plant, along with Lee and Jones, undertook a tour of countries formerly in the Eastern Bloc. They played Croatia, Hungary, the Czech Republic, Poland, Romania and Bulgaria. Renting a car, Page and Plant drove from show to show, studying how these countries were adapting to breaking free fom Communist rule. They ended up with a show in Istanbul in Turkey. Page declared that he would next be travelling to Egypt, to visit the tombs of the pharaohs \u2013 something he had already done three times during that last Led Zeppelin tour in 1977, of course.\n\nA month before the release of _Walking into Clarksdale_ , Page and Plant announced the dates for a US summer tour. The Walking into Everywhere tour was set to begin on 19 May, at Pensacola, Florida, just like the _No Quarter_ tour. There were 27 dates listed, concluding at New York's Madison Square Garden on 16 July. 'I would expect Page and Plant to do quite well,' commented Gary Bongiovanni, editor of the US concert-industry trade paper _Pollstar_. 'Certainly if you look at the popularity that somebody like Ozzy Osbourne has maintained, you gotta believe that there's still an audience for hard-rock music. Page and Plant are certainly not overexposed. They haven't been touring every year. There should be significant demand for them.' After a break there would be another set of dates, taking in the pair's beloved West Coast of America, followed by further shows in Europe.\n\nBefore the tour began, Warner Brothers' film division, in tandem with Bill Curbishley, who now managed both Page and Plant, had green-lit a biopic to be made about Led Zeppelin. Accordingly, Brian Helgeland, who that year had won an Academy Award for his superb adaptation of James Elroy's _L.A. Confidential_ , was hired. Helgeland certainly seemed the correct choice: he had moved to Los Angeles because he had been inspired by Zeppelin's 'Going to California'.\n\nGiven access to the pair of principal players, Helgeland was at Madison Square Garden and a show on Long Island. Plant, however, refused to cooperate. 'There was something going on between him and Jimmy Page, but also Plant seemed to be having some relationship breakup difficulty,' said Helgeland.\n\nYet Page, presumably always anxious to secure a positive legacy for his Led Zeppelin art project, could not have been more helpful. Chatting backstage, Helgeland found him utterly charming. 'Then his wife Jimena arrived holding baby Zofia. She put her in Jimmy's arms. He carried on talking to me, a Marlboro Lite stuffed in the corner of his mouth. As he spoke I could tell what was going to happen. Suddenly the long pile of ash that had built up on his cigarette descends... to fall on the top of Zofia's forehead. Jimmy carried on talking: I'm not sure he noticed.'\n\nPlant's stance led to Warner Brothers abandoning the projected Led Zeppelin movie. Moreover, the singer then told Page that he was not interested in making another record with him. He no longer wanted, he told Paul Rees, to continue what seemed to have become a religious observation of Led Zeppelin at their concerts. 'I needed to go off and do something that was the very antithesis of playing huge buildings in places like Mannheim in Germany to people who wanted Led Zeppelin and the two of us were the nearest thing they could get to it.'\n\nThe fact is, though, that Page and Plant had carried on for five years and 200 gigs, not bad going for something that had sprung from a simple request for a single appearance on MTV _Unplugged_. 'I wanted to keep working,' Page told Nick Kent. 'But Robert wouldn't hear of it.' He confessed that he had been hoping to bring John Paul Jones in with the pair of them: 'But it was hard enough getting two of us together, let alone three.'\n\nJimena Page had organised an event at London's plush Caf\u00e9 de Paris on 27 July 1999 for a pair of charities: the ABC (Action for Brazilian Children) Trust, which she and her new husband had founded, and SCREAM (Supporting Children through Re-Education and Music). Page played with the Black Crowes, with whom he was already very taken, as well as with Aerosmith's Steven Tyler and Joe Perry; Michael Lee was on drums and Guy Pratt, who had toured with him on the Coverdale\/Page project, on bass.\n\nWhen Plant proved utterly resistant to Page's entreaties to continue their work together, the guitarist became angered. He called up his mates in the funky, Faces-like \u2013 with a smidgen of Allman Brothers \u2013 Black Crowes. 'Let's do something together,' he said. The Black Crowes were also Small Faces-like, with singer Chris Robinson's Steve Marriott-clone voice reminding you how Zeppelin might have sounded if Page had secured his original choice for vocalist. You can hear this especially on their interpretation of 'Shapes of Things', the Yardbirds classic they perform on _Live at the Greek: Excess All Areas_ ; Robinson also has a flavour of Rod Stewart about his tonsils, as though he has been listening to the Jeff Beck version of the song on _Truth_.\n\nBill Curbishley, who managed the Who as well as Page, set up a tour that would feature both acts; night by night they would alternate between the Black Crowes and the Who to close the show. The tour started on 24 June 1999 in Chicago, ending in Albuquerque, New Mexico, on 12 August. The second leg of the tour was cancelled, the 55-year-old Page suffering from a back injury: nothing to do with his osteopath's specialty earner of wearing his guitar almost down by his knees? It resumed on 9 October 1999 in East Rutherford, New Jersey. On 12 October Page and the Black Crowes played the first of a three-night engagement at Manhattan's Roseland Ballroom: Page's archetypal rock 'n' roll tresses had been trimmed, exposing his ears for the first time in over 30 years, accentuating a jawline we had hardly seen before; his body had thickened considerably. On 18 and 19 October they performed at the Greek Theatre in Los Angeles, an outdoor arena on the edge of the Hollywood Hills.\n\n_Live at the Greek: Excess All Areas_ was the live double-album artefact of these shows. What was extraordinary was that it was initially released only on the internet, on 29 February 2000. Page could get in a dig at an old rival over this: 'Everyone gives David Bowie the credit for having gone on the internet with his _Hours_ album,' he said. 'But he just put it up there for two weeks or something. It was more publicity than anything else. But here we've gone and done it for real.'\n\nOn 4 July that year it was released as a physical entity on TVT Records. Page bigged up the project. 'Anyone can have a crack at a Zeppelin song and plenty do,' he told Mick Wall. 'But to do it properly, especially when someone who partially created these songs is standing there, well, it can't have been easy. But the Black Crowes did it, and with their own identity intact too.'\n\nBut not all involved were as enthusiastic. 'I didn't really have that much fun doing it,' complained Chris Robinson to _Classic Rock_. 'It was all right, and Jimmy's a phenomenal guitarist, but to me it was just a job. I'm not a big fan of Robert Plant's lyrics or his singing, so that part of it was a little boring for me.'\n\nAlthough the Black Crowes played a handful of their own tunes, punctuating what was essentially a Led Zeppelin tribute set, for copyright reasons they were withheld from the _Live at the Greek_ release.\n\nStill, _Live at the Greek_ sold over half a million copies in the US, earning a gold album for all concerned. According to _Guitar World_ magazine, Jimmy Page's playing on _Live at the Greek_ 'reaches levels of fire and fury that it hasn't attained in years'. 'It's very musical, isn't it?' said Page of the project. 'The Crowes are really known for jamming and adlibbing. And that's what I've been doing ever since I've been playing. So it's complementary. And I know people respect the fact that there are musicians trying every night to put themselves right on the edge.'\n\nDespite Page falling out with Plant over his refusal to continue in the Page\/Plant combination act, the pair together attended a screening of _Almost Famous_ , Cameron Crowe's film based on his experience of going on the road with Zeppelin for his _Rolling Stone_ cover story. When the film was released on 13 September 1999, it was revealed that there were four Led Zeppelin tunes on the soundtrack. Plant also had fuelled rumours by stating that he and Page had been in a recording studio and that they would be working together in 2000. More to the point, Page was promising a second solo album, a rather belated follow-up to 1988's _Outrider_.\n\nBut now it seemed that Page was entering another phase of his life \u2013 and his relationship with Led Zeppelin. Apart from one very significant event, from now on he would largely be dedicated to curating. On 26 May 2003 _How the West Was Won_ \u2013 a triple-CD live celebration of Zeppelin's shows at the Forum in Los Angeles and Long Beach Arena, on 25 and 27 June 1972 respectively \u2013 was released. Around 150 minutes long, it was one of the finest Led Zeppelin records ever released, a sensational encapsulation of the four-piece at the peak of their playing, power and performance; it included an over-25-minute version of 'Dazed and Confused' and a 23-minute 'Whole Lotta Love' extravaganza featuring Wanda Jackson's 'Let's Have a Party', Rick Nelson's 'Hello Mary Lou', John Lee Hooker's 'Boogie Chillen'' and Howlin' Wolf's 'Going Down Slow'.\n\nPage had found the tapes of these two Southern Californian shows, and remembered how the performances were close to breathtaking. 'I'd always wanted to capture what we were doing in LA as we always played at our optimum there. I have memories of that place that somehow just bring something out of you, whether it was the Yardbirds or Zeppelin. LA is always fantastic, so I hope I haven't jinxed myself with that. Each member of the band was playing at their best during those performances and giving like 150 per cent. And when the four of us were playing like that, we combined to make a fifth element. That was the magic, the intangible,' Page told _Guitarist_ in 2003. A magnificent, truly representative live album had always been absent from the Zep canon: now that had been remedied. Over 20 years since Led Zeppelin had ceased to exist, _How the West Was Won_ entered the US album charts at number one.\n\nThe same day that _How the West Was Won_ hit the shops, a second iconic release provided this fresh presentation of Zeppelin's live power with a quantum leap. _Led Zeppelin DVD_ featured not only the 1970 Royal Albert Hall gig (after which he met Charlotte Martin), but also the 1979 Knebworth shows; footage from the Earls Court concerts; the original film of the Madison Square Garden dates for _The Song Remains the Same_ ; footage of a live 'Dazed and Confused' for _Supershow_ ; plus such snapshots as their performance on Danish TV in 1969 and assorted press conferences and videos. _DVD_ and _How the West Was Won_ were so detailed that they were almost academic curations in their attention to detail, doctorate studies in Led Zeppelin. For Page it was that amount of detail that was always crucial. 'We wanted something that would trace the journey of Led Zeppelin. In that context, it's a truly historical document,' said the Zeppelin Godfather.\n\n'We were never really part of the pop scene,' he added in an official statement to accompany the releases, repeating the old rap. 'It was never what Led Zeppelin was supposed to be about. Our thing was playing live. In that sense, Zeppelin was very much an underground band. The fact that it became as successful as it did was something that was almost out of our control. We actually shunned commercialism, which is why so little official footage of the band has ever been seen before.'\n\nFor the next three years the _Led Zeppelin DVD_ would remain the bestselling music DVD in the United States; ultimately it was the biggest-selling music DVD globally ever. This pair of releases returned Led Zeppelin very much to the public's attention. (As though part of a time capsule, I bought my copy at Tower Records on Sunset...)\n\nBut the greatest purpose \u2013 the reformation of Led Zeppelin \u2013 again was avoided, with only one person responsible for this failure. A short US summer tour had been provisionally booked for 2003. More interested in promoting the no doubt coincidental release of his compilation, _Sixty Six to Timbuktu_ , Robert Plant stepped back from this immensely financially rewarding possibility. Plant would also ultimately nix a pair of Madison Square Garden shows pencilled in for 2005 to commemorate the twenty-fifth anniversary of John Bonham's passing; the vocalist couldn't decide whether or not it was a good idea. On the other hand, Plant was said to have literally gone down on his knees in front of John Paul Jones to beg his forgiveness for the sneaky, sarcastic comments he'd flung the bass player's way at the beginning of the Page and Plant exercise.\n\nOn Monday 28 February 2005 Page, Eric Clapton and Jeff Beck had been invited to Buckingham Palace for a reception honouring the UK's music industry. Also present were Jeannette Lee, managing director of Rough Trade Records, and Mick Jones of the Clash, who were old friends from the days of punk rock. Seeing the triumvirate of heavy-guitar friends, Lee insisted to Mick Jones that they cross the floor to them and introduce themselves. These guitar gods were not taking themselves too seriously. 'We still think we're the kids rehearsing in his mum's garage,' Beck pointed to Page, who smiled quietly.\n\nWhen Her Majesty Queen Elizabeth II was formally introduced to the Led Zeppelin megastar, she only had one question: 'Are you a guitarist too?' Page simply nodded.\n\nTowards the end of the year, on Wednesday 14 December 2005, Page was back at Buckingham Palace. He was received by the Queen as an Officer of the Order of the British Empire \u2013 in other words, he had been granted an OBE. Was this for his sterling work as one of the pagan overlords who conquered the United States of America, bringing immeasurable quantities of American dollars into the UK economy? No, it was for something far more important: Page was awarded an OBE for his work with impoverished Brazilian children; not only had he set up a safe house, benefitting over 300 children, but he had also provided medical and psychological backup, as well as food, clothing and employment training. 'I think when you're faced with a plight that's inescapable, and there's something you can do about it, you hope you can make a difference,' he said mildly.\n\nOn 29 October 2006 Ahmet Ertegun, who had founded Atlantic Records and remained its boss, was present at a Rolling Stones benefit concert for the Clinton Foundation. The show was at New York City's Beacon Theatre; former president Bill Clinton was in attendance. Backstage prior to the show, the 83-year-old Ertegun fell, badly banging his head on the concrete floor. Taken straight to hospital, his condition appeared stable. But it soon deteriorated. When Led Zeppelin were inducted into the UK Music Hall of Fame on 14 November 2006, Page, the only Zeppelin member who could be bothered to turn up, made an official announcement about Ertegun's plight. Shortly after, Ertegun slipped into a coma: he died on 14 December 2006.\n\nA tribute concert for Ertegun, a much-loved and revered figure in the music business, was announced for 26 November 2007 at the O2 Arena in London. The proceeds would go to the Ahmet Ertegun Education Fund, which provided university scholarships.\n\nOn 12 September 2007 promoter Harvey Goldsmith announced at a press conference that the bill-topping act at this event would be the three surviving members of Led Zeppelin, joined by Jason Bonham on drums. Tickets were priced at \u00a3125, and were sold through a lottery website. Such was the demand that the website crashed almost straightaway. Goldsmith had already predicted that the concert would create the 'largest demand for one show in history': he claimed that over 20 million people tried to buy one of the available 18,000 tickets. When a pair were officially auctioned for the BBC's annual Children in Need appeal, they sold for a staggering \u00a383,000.\n\nLater, Page commented: 'I knew it was going to sell out quickly, but the tidal wave of euphoria that preceded the gig \u2013 the anticipation \u2013 went beyond what I could possibly have imagined. We'd had a few shambolic appearances in the past, like Live Aid, so if we were ever going to come back together, we were going to do it properly and stand up and be counted.' Each member of Led Zeppelin was only given ten free tickets. When someone tried to blag one from Page, he laughed, 'Impossible: I've had four wives.'\n\nAs seemed rather like a Led Zeppelin cavalier tradition, it was almost predictable that the show, announced on 1 November, suddenly seemed to be arbitrarily moved to 10 December. Yet again, apparently, Page had suffered an injury; he had fractured the little finger on his left hand in 'a gardening accident'. Unfortunate for those from all over the globe who had booked plane tickets and hotels in London for the original date of the show. The ulterior motive suspected behind Page's 'medical issue' was \u2013 yet again \u2013 a problem with Robert Plant.\n\nWith only a month to go before the 26 November date there had come a sudden, almost poetic shift in the circumstances of the band. At 59, the relative youngster of the three original members, Plant appeared to be an archetypal example of the sorcerer's apprentice. Having initially received instruction from the occultist magus who was Page, Plant was the only Led Zeppelin member to have carved out a successful solo career since the demise of the group in 1980.\n\nPlant had recorded a new album with the bluegrass-country singer Alison Krauss, _Raising Sand_. Released on 23 October 2007 to resounding critical acclaim, in its first week _Raising Sand_ sold 112,000 copies, entering the US album charts at number two; it would eventually reach the number one album slot in the USA, following a Grammy award for Album of the Year. Plant had been extremely busy promoting the album, in both the USA and Europe, and had had little time to rehearse for the Ahmet Ertegun benefit, which Page insisted had to be note-perfect.\n\nSuspicions were raised: was this a devious, arrogant ruse to secure further rehearsal time? Were matters not proceeding according to plan? Was this a symptom of acute nervousness on the part of the guitarist? Or of power games within the dynamic of Led Zeppelin? Probably all these elements were there. But the last was the most overriding.\n\nThough the response to _Raising Sand_ almost certainly benefited from publicity from the O2 date, Plant was suddenly in what seemed an unassailable place of power. The focus of attention had shifted to the singer \u2013 the dynamic weighted in Plant's favour ever since Led Zeppelin split up now reaching its apogee.\n\nBy so arbitrarily and suddenly moving the date of the concert, the event acquired some of the mouldy scent of the archetypal shoddy treatment of Led Zeppelin fans. It also seemed clear that fear was more likely the motivating factor here. When the four musicians finally got together at rehearsals Plant laid down his stipulations. He insisted that the set not be too 'heavy metal' \u2013 thus ruling out 'Immigrant Song' and, to Page's distress, 'Achilles Last Stand' \u2013 though he did concede to performing the cheesily ubiquitous 'Stairway to Heaven', but only if it was demoted to a nondescript slot midway through the set, and not employed as a grandstanding finale. Also, he was adamant that there should be no free-flowing, extended jams: they had to know precisely what they were doing, replicating the sound on the records.\n\nPlant was now running the Led Zeppelin show \u2013 perhaps, as Peter Grant had felt, something he had always wanted to do: in 1993 Grant had told Dave Lewis, the Zeppelin archivist, 'You've got to realise Robert always wanted to be the boss of the band anyway. He finally got his own way.' And in this particular instance, Robert Plant may have had a subconscious cause behind his martinet-like approach: he was genuinely upset that, having given early copies of _Raising Sand_ to Page and John Paul Jones, neither sought to comment on the singer's new record. Shortly after the O2 concert, Page ran into another musician. 'That cunt!' he declared. 'That cunt put his record out at exactly the same time we were due to play the show! Fucker!'\n\nAll the same, apart from the atrocious sound for the first five numbers, the O2 show was sensational: tight, punchy and unrelenting, with almost none of the group's previously characteristic improvisation; indeed, at only just over two hours long, the set was somewhat unrepresentative of their former epic performances.\n\nBeginning at the beginning with 'Good Times Bad Times', track one off their first album, they charged into 16 of their most classic songs, including 'Stairway', 'Whole Lotta Love', 'Kashmir' and 'Rock and Roll'. For most of the set they stood close together, clustered around Jason Bonham's drum kit, as though it were the altar in the church of Led Zeppelin; and, speaking of religious symbolism, Page still had the Zoso sigil on his amps.\n\nYet there were signs of nerves. Especially in the now silvery tressed Page \u2013 he came on wearing sunglasses \u2013 whose playing at first seemed rusty, as though a curious irony was about to be unleashed and the group's founder prove to be the weak link in this reunion. But by 'Black Dog', the third song in the set, Page had mastered himself and his guitar work, and the awesome power and majesty of the music was undiminished. 'There is a kind of loud serenity about Led Zeppelin's set,' wrote the _New York Times_ of this hard-rock triumph.\n\n'A band who, implausibly, sound once again like the greatest rock band in the world,' considered _Uncut_ magazine. 'For the time being, at least, Led Zeppelin's legend has the happy ending it always deserved.'\n\nBut that happy ending would not endure. For immediately following the gig there arose the inevitable question: would there be more dates? Page now appeared revitalised. A month short of his 64th birthday at the O2 date, he had hoped that the concert would be the precursor to a worldwide Led Zeppelin tour. All he wanted to do, it seemed, was to continue playing shows with Led Zeppelin. And Jones and Jason Bonham were equally eager to continue the O2 triumph. The hold-out, however, was Plant, now committed to a lengthy tour to promote _Raising Sand_. Indeed, so adamant was Plant in his refusal to comply with Page's desires that it had the tang of a rebellious son working against his father's wishes.\n\nAnd this attitude of estrangement continued when work commenced on the DVD of the O2 show. Page and Plant were so intransigent and unable to compromise in the construction of the film that it would take almost five years from the time of the concert before the DVD and live album were released. Whereas Page was insistent on re-recording musical sections that he considered flawed, Plant was equally immovable in his demand that the film should reflect the exact sound that the audience experienced.\n\nPage would send Plant what he considered to be a completed filmed version of the concert, only to find when it was returned to him that all his changes had been removed. Then he would repeat them, return his master copy to Plant, and once again they would be taken off. And so it continued.\n\nWhen the film was finally released on 17 October 2012, it was apparent that Plant had had his way: _Celebration Day_ , which received enormous critical acclaim, was an aural facsimile of the O2 event.\n\n_Celebration Day_ rapidly sold half a million copies, making it the highest-selling DVD of 2012; before the year was out the CD of the show had sold 1.8 million copies. To enjoy such success late in life, as though a continuation of one of those quasi-mythical battles that his life had become during the existence of Led Zeppelin, Page had been obliged to concede defeat to his own once-young apprentice.\n\n## 26\n\n# PHOENIX RISING\n\nWithout a singer, Jimmy Page's plan to spend 2008 on a worldwide Zeppelin tour appeared to be scuppered. Steven Tyler briefly tried out as vocalist, before making the catastrophic strategic error of suggesting that Page, Jones and Jason Bonham try out their chops on some Aerosmith material. For this sin he was allegedly sent packing.\n\nOn 24 August 2008 Page performed 'Whole Lotta Love' with Leona Lewis at the closing ceremony of the Beijing Olympics. Standing beside a London double-decker bus, Page considered himself part of a patriotic effort to plug the London Olympics, to be held in four years time. 'We put so much into Beijing, but weren't helped by the Chinese giving us next-to-no practise time,' he said. Guy Pratt played bass on the recording, made prior to the event: 'We recorded it in Olympic Studios in the original studio with all the original amps,' he recalled. 'It was amazing. It has this big orchestral arrangement. The actual song seemed a bit shorter than I remembered. Then I realised that we had left off the final verse: I suppose that for Leona Lewis there simply was no female equivalent of a \"backdoor man\".\n\n'I had a show on Planet Rock at the time. I said something to the effect of: \"Oh, by the way, when you hear Jimmy Page closing the Olympics, that's me playing bass on 'Whole Lotta Love'.\" Jimmy loved it that I'd revealed that, which I should have kept quiet about, because he can't wait for an excuse to be upset with you. I went to a book launch where I knew he was going to be. And it was, \"Well, we're not talking to you, are we?\" And he made me tap dance.\n\n'Last time I saw Jimmy was in Earls Court. He was having dinner at the Troubador. This time he was so lovely and friendly. He says, \"Don't be a stranger. Don't be a stranger.\" And I said, \"Jimmy, I've got nine phone numbers for you and none of them work.\"'\n\nLess than a year on from the O2 event, there was a tragedy. Michael Lee, who had played percussion on the Page and Plant _Unledded_ sessions and was one of the backing musicians on their 1995\u201396 world tour, suddenly died after a seizure, aged only 39. Lee had been co-credited as a songwriter on all 12 tracks on the _Walking into Clarksdale_ album. Page, who liked Lee and had been considering him as a potential member of any musical unit he put together, was reportedly devastated.\n\nIn March 2012 Page finally released an album entitled _Lucifer Rising and Other Sound Tracks_ , the music he had fallen out with Kenneth Anger over. Although Page finally had completed the music, up to then it had rarely been heard.\n\nPage went back to what, like a garden-shed hobby, he had been doing now for years: curating the Led Zeppelin legacy. Over the ensuing years, all the Zeppelin material was released in so-called Deluxe Editions, including out-takes and live cuts that had been previously unavailable.\n\n'The curating thing is part of a magician's work,' said Steve 'Snips' Parsons. In August 2010 he ran into Page in curious circumstances. While out in London's West End, he overheard a familiar voice saying: 'Oh, I always take them to _The Mousetrap_.' Page was explaining to photographer Ross Halfin how he entertained visitors from overseas, citing London's longest-running play, an Agatha Christie potboiler, as the destination he would choose for them. As Snips in the Sharks, Parsons had been a powerful, soulful vocalist. Just the kind of person Page was looking for, especially as Parsons also considered himself a Thelemite.\n\nThe pair had a meeting. Page quizzed Snips about movie-soundtrack work, which Parsons had been involved in for almost 30 years: Page regretted he hadn't done more. For _Death Wish II_ he had been pissed off that Michael Winner wouldn't pay for a stereo mix: his soundtrack is only in mono. 'I was only with him for a short time,' said Snips. 'Clearly I didn't pass the test.' He had always had his own reservations about the two of them getting together. 'Two magicians should never meet,' Parsons had thought to himself, citing a mantra-like adage of those involved in the supposed dark arts.\n\nWith his own interests in Aleister Crowley's work, Parsons had considerable insight into Page's approach to life in his twilight years. 'Jimmy Page is relating to the work done many years ago. Jimmy is always tweaking things. Led Zeppelin may have been relatively brief, but things have to be done afterwards. For occultists they make these big connections and moments, opening their minds, and it takes a long time to work this stuff out. Kenneth Anger even now works on his films, re-edits sometimes appearing.\n\n'Like Jimmy Page, Austin Osman Spare disappeared for a long time. He was found living in squalor, but then there was something of a redemption.\n\n'A certain thing must go on, that is not of a normal milieu. Magicians don't cherish memories; they use them to make something happen.\n\n'And where was Jimmy Page from the end of Led Zeppelin until relatively recently? Not necessarily here, in all senses of the word. When he was working in the Firm and with David Coverdale, I don't think he was really with us. But Jimmy Page is certainly back with us now.'\n\nOn 15 January 2010 Page announced that he would perform at April's Show of Peace Concert in Beijing. Meanwhile, the United Nations' Pathways to Peace took the opportunity to recognise the Led Zeppelin guitarist by presenting him with its first-ever award.\n\n'We've been asked to give awards over the years but never have done so until this day,' said Michael Johnson, a representative of the organisation. 'We're doing this because musicians have a global impact on the world, and we also know that people who use their name and fame for peace building need to be honoured.'\n\nA clearly touched Page responded modestly: 'Although this award has my name on it, this is a tribute to the power of music and its positive effect. Music has been the most powerful language to reach the hearts of people around the world. During my career, I've experienced the connection and harmony that music can bring.'\n\nIn September 2010 'a photographic autobiography' and 'a career in pictures', to use Page's words to describe the project, was published by Genesis Publications, whose speciality was high-end exclusivity. _Zoso: Jimmy Page by Jimmy Page_ , as the tome was titled, was released in a limited edition of 2,500 copies, each individually signed by him. Totalling 512 pages, with over 700 rare or unique images, his book retailed at the extremely pricey \u00a3425.\n\nMaybe I missed this in the reviews of his book, but I saw no mention of the fact that the cover image of Page is almost identical in composition and facial expression to the most familiar image of Aleister Crowley, the one that adorns his autobiography _The Confessions of Aleister Crowley_. I think we are being told something.\n\nWhatever, the photographic book was a revealing work, emphasised by the early shot of the author in a choirboy's cassock, looking as though butter wouldn't melt in his mouth, bearing the caption, 'It might get loud...'\n\nAlthough clearly a testament to rigorous, highly selective picture editing, _Zoso_ \u2013 yes, here he comes again! \u2013 was a rather magnificent and impressive tome. Canny as ever, Page had only given Genesis the first rights to the book; when it was reprinted, still under the Genesis imprint in 2014, he allegedly earned every penny from this second publication, which retailed at the more reasonable price of \u00a340. Undertaking a lengthy book-signing tour of the United States, Page surprised purchasers when his signature turned out to come from a rubber stamp that, in an almost Warhol-like manner, he carried with him. Inviting a rather older Lori Mattix to the Los Angeles launch, Page somewhat surprised his former girlfriend when he sighed, 'Oh, Lori: we were so young then.' To which she replied, 'Well, I was!'\n\nOn 3 June 2011 Donovan played an evening at London's Royal Albert Hall dedicated to his 1966 _Sunshine Superman_ album. For the title track Page appeared onstage with his guitar, playing along with his old mate Donovan; Jimmy looked considerably animated, as though he was having a terrific time, a palpable empathy between himself and Donovan. At a party in the Gore hotel after the performance, Page, his silver hair tied back in a ponytail, looked troubled when I suggested to him that it would have been terrific if he had played more of the set. 'I'm so rusty,' he said. 'It was all I could do to just do what I did.'\n\nJust over a year later, on 21 June 2012, he was a guest at the launch party for Paul Gorman's biography of Tommy Roberts, Page's old friend who had started the shop Mr Freedom. The crowded event was at Roberts's retro-furniture shop Two Columbia Road in Hackney. I introduced the Led Zeppelin main man to Tapper Zukie, the legendary Jamaican deejay: both seemed as puzzled as the other as to who exactly they were meeting.\n\nOn an early morning in March 2016 I found myself at Christie's, the art auction house in London. I was there with a friend who, like Page, had an interest in the art of Austin Osman Spare, several of whose paintings were for sale.\n\nIn addition to my mate, there were several other potential buyers. Yet every time a certain price was realised, the auctioneer would declare that there had been a phone bid for a higher amount. Each of the paintings was bought by this anonymous devotee. My friend and I looked at each other ironically: we suspected we knew who this faceless buyer was.\n\nAfter the auctions, my mate suggested having lunch at the nearby Chelsea Arts Club. Arriving, we went up to the bar, to be served by a statuesque, pretty red-haired girl. Suddenly I appreciated who this was: Scarlett Sabet, the new love of Jimmy Page's life, according to the _Daily Mail_. Another Scarlet, Page's first daughter, had also worked behind that same bar.\n\nIn 2015 director Adam McKay's film _The Big Short_ was ready to be released. An extremely clever and impressive movie about the subprime-mortgages scandal of the first decade of the 21st century, _The Big Short_ would win an Academy Award for Best Adapted Screenplay. The film had a lot going for it. Apart from a wonderful script and direction \u2013 it would also be nominated for Best Picture at the Oscars \u2013 it had an impressive cast: Brad Pitt, Christian Bale and Ryan Gosling.\n\nBut to complete it, Adam McKay needed something more to give _The Big Short_ that final touch. He wanted to include Led Zeppelin's 'When the Levee Breaks' on the soundtrack, to play as the end credits rolled. And \u2013 crucially \u2013 he wished to have the song as part of the film's trailer. 'We cut the trailer and put in the Zeppelin song,' said McKay, 'and it's not only one of the greatest songs of all time, but it drives you through the trailer. But then we were told we might not be able to get the rights.'\n\nLed Zeppelin were known to be difficult over the use of their songs in films. But, shown a copy of _The Big Short_ and its trailer, Robert Plant and John Paul Jones immediately agreed, as did John Bonham's estate. There was only one hold out. And that was because no one could find him. 'What do you mean you can't find Jimmy Page?' the director said. 'I was told he has a new girlfriend, and I guess they were off having a good time...'\n\nPage had fallen so far off the radar that Adam McKay considered postponing the trailer premiere, something he was extremely reluctant to do: in Hollywood such action can lead to rumours about problems with a film's production.\n\n'We finally heard that he was in some pub out in the English countryside,' said McKay. 'So an assistant drove two hours to get to the pub, breaking every speed limit, goes into the pub and puts a computer in front of Jimmy Page so he can look at the trailer and say either yes we can use the song or no. Then at like 1.55 a.m. or something I got the email that he said yes.'\n\nBut this saga did not end there. Page had one condition for also using the song in the end credits of the film. 'He said we can't edit the song. He told us he didn't like how they cut up his songs in movies.'\n\nThis presented a further problem: before Robert Plant even opens his mouth, 'When the Levee Breaks' has 84 seconds of instrumental music. McKay decided to open the music before the credits actually roll, subsuming the instrumental in New York traffic noise, so that the moment the credits begin, Plant's voice comes in. 'That was the crazy thing,' said McKay. 'That was a pure accident. It just happened to lay out perfectly when the credits begin.' Carl Jung, of whose philosophy Page was fully aware, would have told McKay that in such situations there is no such thing as an accident or coincidence.\n\nThe woman Page had been ensconced with while McKay so anxiously sought him out was the same Scarlett Sabet who had served me at the Chelsea Arts Club. More than 40 years younger than Page, and of Iranian-French descent, she is a poet and actor. Page met her when she gave a poetry reading at the Troubadour in London's Earls Court. The couple were photographed by the _Daily Mail_ as they left the Kensington branch of Nando's, the fast-food chicken restaurant where they had celebrated his 71st birthday in 2015. 'Scarlett has been very good for Jimmy,' commented a friend. You feel certain that part of his attraction to Sabet was her first name, recalling how Aleister Crowley always gave his girlfriends the nickname 'Scarlet Woman'. Page no doubt required female company: Patricia, his mother, for whom he had bought a property in Berkshire, had passed away that year.\n\nAs might be expected, the populist _Daily Mail_ was extremely pleased also to be able to run stories of an alleged feud between the Led Zeppelin stalwart and Robbie Williams, the enormously successful former member of Take That. In 2013 Williams had purchased the home of Michael Winner after the film director's death. Now Williams was desirous of improving the property by building a basement extension, complete with recording studio and swimming pool.\n\nPage was appalled. Concerned that Williams's prospective development might destabilise the foundations of Tower House, he unsuccessfully attempted to prevent the planned building works, the local council rejecting his concerns in 2015. In May 2017 he won a pyrrhic victory of sorts: Williams's building firm was fined \u00a35,000 for making construction noise outside of permitted working hours, something of an irony considering Led Zeppelin's penchant for noise pollution.\n\nBut in August 2017 there was a further brouhaha. While being interviewed by an Italian radio station, Williams spoke off air about his difficulties with Page, comparing his behaviour to a 'mental illness'; these comments were then posted on the radio station's Facebook page. Williams was obliged to issue a somewhat grovelling apology: 'I would like to offer my sincere apologies to Jimmy Page, my neighbour, for my comments made before Christmas about him in relation to my recent building works, in which I likened alleged behaviour on his part to suffering from a mental illness.\n\n'Jimmy Page has explained to me that certain specific factual assertions which I made were in fact not true and I am happy to accept what Jimmy Page says.\n\n'I understand why Jimmy Page will have found my comments offensive and I apologise for any hurt that they have caused him and his family as a result.\n\n'I did not intend my comments \u2013 which, so far as I am concerned, were made privately \u2013 ever to be published.'\n\nAs a consequence, the rift between the two rockers was said to have healed.\n\nRather in the manner of the Clash's Mick Jones, who can be seen on his regular perambulations around Notting Hill, Jimmy Page is frequently sighted in neighbouring Kensington. Page visited Jones's Rock 'n' Roll Public Library on the Portobello Road, an exhibition of the Clash guitarist's immense pop-art archive. 'You've got even more stuff than me,' he declared, amazed. Page also has an enormous collection of memorabilia of all types, especially pop-culture artefacts, including old records, particularly rockabilly \u2013 he is a regular visitor at Reading's Sunday Record Fair, where he is considered simply as an exceptionally congenial regular customer, always convivial unless a certain subject is brought up. He also favours a working-men's caf\u00e9 in the Earls Court area.\n\nOn 24 June 2016 Led Zeppelin partially resolved an issue that had hung over the group since May 2014, when Mark Andes, the bass player with Spirit, had filed legal papers in an endeavour to prevent 'Stairway to Heaven' being rereleased on _Led Zeppelin IV_ without Spirit's Randy California, who had passed away in 1997, being co-credited as a writer. 'Stairway to Heaven''s descending chord sequence intro, Andes claimed, was taken directly from Randy California's instrumental tune 'Taurus', which appeared on the first Spirit album.\n\nIf it could be proven that Page and Plant had essentially nicked parts of 'Stairway', it would be a landmark moment in the history of popular music. The most-listened-to record ever on American radio, 'Stairway to Heaven' had earned over $562 million by 2008.\n\nAll three surviving members of Zeppelin attended the court case. Page was the first member to take the stand, and he drew attention to the influence of 'Chim Chim Cher-ee', from _Mary Poppins_ , on 'Stairway'. But he also claimed that he had never heard 'Taurus' until 2014. A music expert stated in court that the chord progressions in both 'Stairway to Heaven' and 'Taurus' had first appeared over 300 years previously.\n\nHalf an hour after retiring, having listened to both 'Stairway to Heaven' and 'Taurus', the jury reached its verdict: 'there was no substantial similarity in the extrinsic elements of \"Taurus\" and \"Stairway to Heaven\".'\n\nPage and Plant jointly issued a statement: 'We are grateful for the jury's conscientious service and pleased that it has ruled in our favour, putting to rest questions about the origins of \"Stairway to Heaven\" and confirming what we have known for 45 years. We appreciate our fans' support, and look forward to putting this legal matter behind us.' Apart from that professionally worded declaration, there seemed little sign of amity between the pair.\n\nOn 18 May 2017 came shocking news: Chris Cornell, the singer with Soundgarden and Audioslave, had hanged himself. The handsome, highly intelligent Cornell had long suffered from depression, exacerbated by drug dependencies he seemed to have conquered. His solo album _Euphoria Morning_ had included a tune entitled 'Wave Goodbye', a tribute to his late friend Jeff Buckley. In its tribute to Cornell, the _Los Angeles Times_ compared him favourably to Robert Plant. This was a quality of which Jimmy Page was eminently aware.\n\nPage had become friends with Cornell. When he toured his book in the United States, he was interviewed by Cornell onstage in the beautiful art deco theatre in Los Angeles' Ace Hotel. With selected images from the book projected onto a screen, Page regaled the audience with tales of both Zeppelin and his earlier life.\n\nAt the end of 2015 Page had announced that he would be releasing a new album and touring with a brand new line-up in the coming year. He knew who the vocalist would be: Chris Cornell. Even though, as with so many matters associated with him, the project was put on hold, Jimmy Page was determined that his musical future would be intertwined with Cornell.\n\nWhen he performed Cornell would include such Zeppelin numbers as 'Immigrant Song' and 'Tangerine'. The last song he ever played live, at a show in Detroit, Michigan, was Zeppelin's 'In My Time of Dying', with its lines 'In my time of dying, I want nobody to mourn\/All I want for you to do is take my body home'.\n\n'RIP Chris Cornell. Incredibly Talented. Incredibly Young. Incredibly Missed,' tweeted Page.\n\nOn 23 October 2017, as many a venerable elder statesmen had before him, Page made an appearance at the Oxford Union, topping the bill of speakers that Michaelmas term; his silver hair, usually so neatly tied back, now billowed fluffily around his head, making him look precisely like an eccentric professor; Page always had that precise attention to detail in his appearance.\n\nHe revealed that in his time playing with the beat poet Royston Ellis, around 1960, he had previously appeared at the Oxford Union: 'I was accompanying him on guitar for a couple of his poems.' Page also recalled the occasion he had accompanied Jeff Beck to that Yardbirds show at the university when Keith Relf freaked out existentially on stage, causing Paul Samwell-Smith to storm out of the group, leaving the way for Page to join.\n\nThen a member of the audience asked him a pertinent question: what was Page's most cherished memory of being in Led Zeppelin with regards to the lifestyle and the 1970s experience?\n\n'Well, I was creating an art form,' he replied. 'That's what I was doing. I mean, I was, from day one I was developing a whole persona, but I was living it. It wasn't anything that was false or sham. I was living it every inch of the way. And that was the determination of it, because I was so committed to the whole aspect of writing new music and presenting views on music that other people hadn't really got to yet. I mean, they probably would do, but to be able to do that, I just really... each and every one of us in that band, as I said, it was a communion and that was just something which was such a buzz to be playing in a band like that and certainly a band that, initially from _Led Zeppelin I_ , I knew exactly the whole of the textures that I wanted to present to people so that they would go, \"Wow, that's really interesting. I haven't heard so many things approached in one go.\" Just the ethos of keep moving and moving and moving forward with these ideas. That was just intoxicating.'\n\nIn the way that Page chose to disport himself during Led Zeppelin's supremacy there certainly are disagreeable elements: how he manifested the group's pre-eminence and the image he chose to display to his audience. As time progressed, culminating in the debacle of the cathartic July 1977 Oakland shows, he seemed increasingly less in control of not only his act, but also of himself. Three or four years earlier in that decade the very mention of Jimmy Page's name could induce in some the kind of frissons of fear that suggested the villain of a Gothic horror novel. Yet by the summer of 1977 the behemoth of Led Zeppelin, his sacred art project, until then the decade's biggest and most influential band, had set about consuming itself.\n\nAn artist accustomed to living amid inspiring dreams and apparitions, Page's addictions to drink and drugs clearly began to dam up such outlets. And during the next decade his output, and his appearance, would certainly suggest someone missing in action, not only from his audience but also from himself. As high as he had been, so low would he fall.\n\nThere is a music-business adage that 20 years is the period before a formerly great act, having fallen off the radar, is reappraised and restored to its previous legend. In fact, despite the weak output of the Firm, the _Outrider_ album and the project with David Coverdale, Jimmy Page's period in the wilderness was relatively brief. By the mid-nineties his work with Robert Plant, beginning with the _Unledded_ MTV show, had restored him to heights that were almost at the level of Led Zeppelin. To hell with any talk of backstage angst or bad vibes between Page and Plant; it had not been that great with Led Zeppelin. And by the new century \u2013 and even before the 2007 Led Zeppelin one-show reunion \u2013 his deliverance seemed complete: no longer the feared figure of rock 'n' roll sword and sorcery myth, Page has emerged as the most revered and respected of all classic rock artists. Despite all the odds, Jimmy Page has become the greatest national treasure of British popular music.\n\n# Picture Section\n\nThe James Page Skiffle Group: a 1957 appearance on the BBC's _All Your Own_ programme, with Jimmy Page just thirteen \u2013 in those days no one knew anyone who had appeared on TV. (BBC Motion Gallery\/Getty Images)\n\nStepping-stones for Jimmy Page: with Carter-Lewis and the Southerners in 1964. (Tracksimages.com\/Alamy Stock Photo)\n\nStepping-stones for Jimmy Page: with the Yardbirds in 1966. (Pictorial Press Ltd\/Alamy Stock Photo)\n\nPage with his Harmony Sovereign acoustic on the deck of his house in Pangbourne, Berkshire. (Trinity Mirror\/Mirrorpix\/Alamy Stock Photo)\n\nClustered around a Jaguar Mk 2 3.8 for an early Led Zeppelin publicity photo taken outside the Impact Agency offices on Windmill Street, London, December 1968. (Dick Barnatt\/Redferns\/Getty Images)\n\nTeasing his theremin \u2013 or telling it where to get off... (Chris Walter\/WireImage\/Getty Images)\n\nThe violin bow \u2013 or wizard's wand \u2013 adding extreme textures to Page's Les Paul during a rendition of 'Dazed and Confused'. (Terry O'Neill\/Iconic Images\/Getty Images)\n\nPage would play 'In My Time of Dying' on his Danelectro. (Gijsbert Hanekroot\/Redferns\/Getty Images)\n\nThe psychedelic Telecaster used on the first Led Zeppelin album. (Jan Persson\/Redferns\/Getty Images)\n\nPage with his Gibson EDS-1275: along with his sunburst Les Paul Standard, this was the guitar with which he is most associated. (\u00a9 Joe Stevens)\n\nChasing the dragon? The full 'dragon suit' gets its first outing at Earls Court, London, in 1975. (Graham Wiltshire\/Hulton Archive\/Getty Images)\n\nThe Zoso sweater worn at the Empire Pool shows in London in 1971. Knitted by a girlfriend, it would shrink when Page sweated onstage. (Michael Putland\/Getty Images)\n\nThe 'moon' trousers, as worn in _The Song Remains the Same_ film. (Art Zelin\/Getty Images)\n\nHeroin chic: an egregiously thin Jimmy Page in his 'poppy suit', on the ill-fated 1977 tour of the United States. (Robin Platzer\/The LIFE Images Collection\/Getty Images)\n\nHanging for the photographer at the Chateau Marmont in Los Angeles in 1969, as America breaks for Led Zeppelin. (ZUMA Press, Inc.\/Alamy Stock Photo)\n\nPage happy in rehearsals. (Charles Bonnay\/The LIFE Images Collection\/Getty Images)\n\nStepping ahead with Richard Cole (left) and Peter Grant (right). (Koh Hasebe\/Shinko Music\/Hulton Archive\/Getty Images)\n\nAn uncertain feeling? Led Zeppelin ready themselves in 1979 for their first Knebworth 'comeback' date. (FG\/Bauer-Griffin\/Michael Ochs Archive\/Getty Images)\n\nAleister Crowley's Scottish home, Boleskine House, in 1912. Page would later purchase the property.\n\nAleister Crowley, the self-styled Great Beast. (Keystone\/Hulton Archive\/Getty Images)\n\nOn 31 October 1974, Halloween Night, Led Zeppelin celebrated the UK launch of their Swan Song record label with a suitably unconventional party at Chislehurst Caves, south-east of London. (\u00a9 Joe Stevens)\n\nThe maestro at work: Jimmy Page onstage at Led Zeppelin's triumphant return to London in May 1975. (Henry Diltz\/Corbis via Getty Images)\n\n# Sources\n\nAmong the seemingly countless books I have consulted, the following have proved especially invaluable:\n\nBaker, Phil. _Austin Osman Spare: The Life and Legend of London's Lost Artist_ (Strange Attractor Press, 2012)\n\nBerbergal, Peter. _Season of the Witch: How the Occult Saved Rock and Roll_ (Tarcher, 2015)\n\nBordowitz, Hank (editor). _Led Zeppelin on Led Zeppelin: Interviews and Encounters_ (Omnibus Press, 2015)\n\nBoyd, Jenny with George-Warren, Holly. _It's Not Only Rock 'n' Roll_ (John Blake Publishing, 2013)\n\nBuell, Bebe with Bockris, Victor. _Rebel Heart: An American Rock 'n' Roll Journey_ (St. Martin's Press, 2001)\n\nCalef, Scott (editor). _Led Zeppelin and Philosophy: All Will Be Revealed_ (Open Court, 2009)\n\nCase, George. _Jimmy Page: Magus Musician Man: An Unauthorised Biography_ (Backbeat Books, 2009)\n\nCase, George. _Led Zeppelin FAQ_ (Backbeat Books, 2011)\n\nCole, Richard with Trubo, Richard. _Stairway to Heaven: Led Zeppelin Uncensored_ (HarperCollins, 1992)\n\nCrowley, Aleister. _The Confessions of Aleister Crowley_ (Penguin, 1989)\n\nCrowley, Aleister. _Magick_ (Red Wheel\/Weiser, 1998)\n\nCrowley, Aleister. _The Diary of a Drug Fiend_ (Weiser Books, 2010)\n\nCrowley, Aleister. _The Book of the Law_ (Createspace Independent Publishing Platform, 2011)\n\nDavis, Erik. _Led Zeppelin's Led Zeppelin IV_ (Bloomsbury Continuum, 2005)\n\nDavis, Stephen. _Hammer of the Gods: Led Zeppelin Unauthorised_ (William Morrow & Co., 1985)\n\nDavis, Stephen. _LZ-'75: The Lost Chronicles of Led Zeppelin's 1975 American Tour_ (Gotham Books, 2010)\n\nDes Barres, Pamela. _I'm with the Band: Confessions of a Groupie_ (Helter Skelter Publishing, 2003)\n\nDes Barres, Pamela. _Take Another Little Piece of My Heart: A Groupie Grows Up_ (Chicago Review Press, 2008)\n\nEllen, Mark. _Rock Stars Stole My Life!_ (Coronet, 2015)\n\nEllis, Royston. _The Big Beat Scene_ (Music Mentor Books, 2010)\n\nFaragher, John Mack. _Eternity Street: Violence and Justice in Frontier Los Angeles_ (W. W. Norton & Company, 2016)\n\nGillett, Charlie. _Making Tracks: Atlantic Records and the Growth of a Multi-Billion-Dollar Industry_ (Panther\/Granada Publishing, 1975)\n\nGoldberg, Danny. _Bumping into Geniuses: My Life Inside the Rock and Roll Business_ (Penguin, 2010)\n\nGraham, Bill and Greenfield, Robert. _Bill Graham Presents: My Life Inside Rock and Out_ (Da Capo Press, 2004)\n\nGreenfield, Robert. _The Last Sultan: The Life and Times of Ahmet Ertegun_ (Simon & Schuster, 2012)\n\nGuralnick, Peter. _Sweet Soul Music: Rhythm and Blues and the Southern Dream of Freedom_ (Canongate, 2002)\n\nHoskyns, Barney. _Trampled Under Foot: The Power and Excess of Led Zeppelin_ (Faber & Faber, 2012)\n\nJames, Catherine. _Dandelion: Memoir of a Free Spirit_ (St. Martin's Press, 2007)\n\nJohns, Glyn. _Sound Man_ (Penguin, 2014)\n\nLewis, Dave. _The Complete Guide to the Music of Led Zeppelin_ (Omnibus Press, 1994)\n\nLewis, Dave. _Led Zeppelin: The Concert File_ (Omnibus Press, 1997)\n\nLewis, Dave. _Led Zeppelin: A Celebration_ (Omnibus Press, 2003)\n\nMathers, S. Liddell MacGregor. _The Key of Solomon the King_ (Createspace Independent Publishing Platform, 2010)\n\nMay, Betty. _Tiger Woman: My Story_ (Gerald Duckworth & Co. Ltd, 2014)\n\nNapier-Bell, Simon. _Black Vinyl White Powder_ (Ebury Press, 2002)\n\nNapier-Bell, Simon. _You Don't Have to Say You Love Me_ (Ebury Press, 2005)\n\nPage, Jimmy. _Jimmy Page by Jimmy Page_ (Genesis Publications, 2014)\n\nPower, Martin. _Hot Wired Guitar: The Life of Jeff Beck_ (Omnibus Press, 2012)\n\nPower, Martin. _No Quarter: The Three Lives of Jimmy Page_ (Omnibus Press, 2016)\n\nPratt, Guy. _My Bass and Other Animals_ (Orion, 2007)\n\nPreston, Neal. _Led Zeppelin: A Photographic Collection_ (Vision On Publishing, 2002)\n\nRees, Paul. _Robert Plant: A Life_ (HarperCollins, 2014)\n\nRoberty, Marc. _Led Zeppelin Day by Day_ (Backbeat Books, 2016)\n\nSelvin, Joel. _Here Comes the Night: The Dark Soul of Bert Berns and the Dirty Business of Rhythm and Blues_ (Counterpoint, 2014)\n\nShapiro, Harry. _Alexis Korner: The Biography_ (Bloomsbury Publishing, 1996)\n\nTolinski, Brad. _Light and Shade: Conversations with Jimmy Page_ (Ebury Press, 2012)\n\nTownshend, Pete. _Who I Am_ (HarperCollins, 2012)\n\nWall, Mick. _When Giants Walked the Earth: A Biography of Led Zeppelin_ (Orion, 2008)\n\nWelch, Chris. _Peter Grant: The Man Who Led Zeppelin_ (Omnibus Press, 2002)\n\nWelch, Morgana. _Hollywood Diaries_ (Xlibris, 2007)\n\nWilliams, David. _The First Time We Met the Blues_ (Music Mentor Books, 2009)\n\nWolman, Baron. _Groupies and Other Electric Ladies_ (ACC Editions, 2016)\n\nWyman, Bill. _Stone Alone_ (Viking, 1990)\n\nYorke, Ritchie. _Led Zeppelin: Led to Gold 1967\u20131989_ (Rock n Roo, 2015)\n\nZanetta, Tony. _Stardust: The David Bowie Story_ (McGraw-Hill, 1986)\n\n# Acknowledgements\n\nThree years ago I was in Los Angeles, staying at the apartment of my son Alex. On the morning I was due to return to London I woke early, then briefly fell asleep again. I had a dream. 'Write the Jimmy Page book!' I was told. I'd had the dream in LA, the archetypal Led Zeppelin town. It all seemed to make sense to me. So I decided to listen to what I'd been told.\n\nAnd during the writing of _Jimmy Page: The Definitive Biography_ I listened to words of wisdom from many people, including John Altman, Dave Ambrose, Wayne Bardell, Jonathan Barnett, Michael Des Barres, Dave Berry, Chris Blackwell, John Bold, Mick Brown, Tony Busson, Roy Carr, JC Carroll, Chris Charlesworth, Glen Colson, Trevor Dann, Chalkie Davies, Jeff Dexter, Trevor Dolby, John Dunbar, Rick Elgood, Mark Ellen, Tony Fletcher, Paul Gorman, Nanette Greenblatt, Billy Harrison, Brian Helgeland, Henry Scott Irvine, Joe Jammer, Brixton Key, Jamie Kitman, Jeanette Lee, Nick Logan, Jackie McAuley, Jack McKeever, Alex Masucci, Lori Mattix, Julian Moseley, Michi Nakao, Simon Napier-Bell, Suzette Newman, Dick O'Dell, Steve 'Snips' Parsons, Mark Plummer, Guy Pratt, Andy Priest, David Robson, Jon Savage, Tim Spicer, Joe Stevens, Kosmo Vinyl, Mick Wall, Michael Watts, Chris Welch, Richard Williams, Bruno Wizard and the great Rod Wyatt.\n\nTo these and many others I am heartfully grateful. And I am especially indebted to Becky Thomas, my agent, and to the wonderful Natalie Jerome, who commissioned this book, and the equally fantastic Jack Fogg, who then guided it to its destiny, along with the splendid editing of Steve Burdett.\n\nAnother exotic location also played a part in the writing of _Jimmy Page: The Definitive Biography_. Flat-out exhausted at one point, I knew I badly needed a break. But instead it was clear that I had to keep writing, otherwise the book would not be completed. In the end, I had some great aid: Chris Blackwell, who has helped with this book, lent me a cottage at GoldenEye, the Jamaican estate he owns at which Ian Fleming wrote the James Bond books. So for a month I would get up early, write (hoping to pick up some Fleming vibes) until mid-afternoon, and then swim my knackeredness away. Give serious thanks...\n\n# Index of Searchable Terms\n\nThe page numbers in this index relate to the printed version of this book; they do not match the pages of your ebook. You can use your ebook reader's search tool to find a specific word or passage.\n\nJP indicates Jimmy Page; LZ indicates Led Zeppelin.\n\nA&R Studios, New York 187\u201390\n\nAbbey of Thelema, Sicily 327\n\nAbbey Road Studios, London 68, 228, 462, 473\n\nABC (Action for Brazilian Children) Trust 476\n\nAhmet Ertegun Tribute Concert (2007) 482\u20136\n\nAkhtar, Najma 466, 471\n\nAlbarn, Keith 80\n\nAlbini, Steve 473\n\nAlkasura, London 126\n\n_All Your Own_ (TV show) 27\u20138\n\n_Almost Famous_ (film) 478\n\nAltman, John 389\u201390\n\nAmerican Folk Blues Festival, Free Trade Hall, Manchester (1962) 39\u201341\n\nAnderson, Ian 185\u20136\n\nAndes, Mark 495\u20136\n\nAnger, Kenneth 2, 13, 101\u20132, 225, 306\u20138, 319, 327, 348\u201353, 371\u20132, 390, 395, 435, 488\n\nAntonioni, Michelangelo 107\u20138\n\nArc Music 171\n\nArden, Don 91, 154\n\nArdent Studios, Memphis 213, 215\n\nArizona State University Activity Center Arena, Tempe 386\n\nArizona Veterans Memorial Coliseum, Phoenix 202, 203\n\nARMS (Action into Research for Multiple Sclerosis) 438\u201340, 445\u20136\n\nAtlanta International Pop Festival (1969) 182\u20133\n\nAtlantic Records 11, 49, 70, 148\u201352, 154, 158, 169, 173, 180, 188, 198, 223\u20134, 228, 250, 253, 254, 264, 291, 301, 387, 389, 429, 445, 453\u20135, 481\u20132\n\nAtlantic Records 40th Anniversary Concert (1988) 453\u20135\n\nAuger, Brian 66\n\nBacharach, Burt 85, 99\n\nBad Company 292\u20133, 294, 295, 310, 314\u201315, 392, 394, 415, 440, 445, 471\n\nBaez, Joan 133\n\nBand, the 98, 204\n\nBand of Joy 131\n\nBannister, Frederick 207, 208, 399, 425\u20137\n\nBannister, Wendy 425\u20137\n\nBarclay, Eddie 123\n\nBardot, Brigitte 123\n\nBarrett, Syd 128\n\nBarsalona, Frank 154\n\nBarsotti, Bob 7, 9\n\nBarsotti, Peter 7, 9\n\nBasing Street Studios, London 206\n\nBassey, Shirley 60\n\nBath Festival of Blues: (1969) 181\u20132; (1970) 207\u201311, 284, 399\n\nBath Pavilion 146\n\nBBC 26, 27, 28, 45, 60, 79, 122, 167\u20139, 175, 203, 250, 251, 442, 443, 473, 482\n\nBeach Boys, the 56, 102, 448\n\nBeatles, the 26, 36, 44, 50, 53, 68, 72\u20133, 74, 79\u201380, 99, 100, 101, 110, 123, 126, 132, 139, 140, 152, 157, 163, 168, 188, 198, 209, 219\u201320, 227, 228, 264, 291, 309, 380\u20131, 452\n\nBeausoleil, Bobby 351, 352, 371\n\nBeck, Jeff 5, 41\u20132, 43, 45, 63, 76\u20138, 88, 89\u201390, 91, 92\u20134, 95, 96, 102, 103, 104\u20136, 107, 108\u20139, 110, 111\u201312, 118, 119, 122, 128\u20139, 130, 133, 138, 143, 153, 154, 156\u20137, 161, 177, 183, 185, 198, 380, 395, 438, 439, 440, 461\u20132, 471, 477, 481, 498\n\n'Beck's Bolero' (Beck) 88\u201393, 129, 130\n\nBeijing Olympics (2008) 487\n\nBell, Madeline 155\n\nBell, Maggie 293, 294, 382, 396\n\nBerns, Bert 49\u201351, 52, 61\u20132, 68, 70, 71, 72, 148\n\nBerry, Chuck 24, 25, 31, 32, 61, 62, 67, 70, 114, 133, 407, 446\n\nBerry, Dave 61\n\nBest, Pete 53\n\nBeverly Hills Hotel, LA 295, 298, 342\u20133\n\nBeverly Hilton Hotel, LA 384\n\nBicknell, Ed 223\u20134\n\nBig Jim Sullivan 45\u20136, 48, 52, 53, 54, 60, 61, 62, 82, 88, 135, 136\n\n_Big Short, The_ (film) 492\u20133\n\n_Billboard_ 163, 166, 198, 436, 450, 462, 474\n\nBingenheimer, Rodney 279, 296, 297\n\nBindon, John 3, 4, 6, 7, 8, 378, 386\u20137, 393\u20134, 402, 405, 429, 442\n\nBlack, Bill 21\n\nBlack Crowes, the 469, 476\u20138\n\nBlackmore, Ritchie 38, 63, 273\n\nBlackwell, Chris (drummer) 454\n\nBlackwell, Chris (Island Records boss) 148, 150, 151\n\nBland, Bobby 54\n\nBlavatsky, Helena 424\n\nBlind Faith 275, 315\n\nBlind Willie Johnson 196, 312, 336\n\nBlood, Sweat and Tears 185\n\n_Blow-Up_ (film) 106\u20139, 110\n\nBlunt, Robbie 433, 436\n\nBo Diddley 35, 114, 190\n\nBohn, Chris 398\n\nBolan, Marc 274, 280\n\nBold, John 144\n\nBoleskine House, Scotland 15, 217\u201319, 221\u20133, 273, 287\u20138, 307, 321, 358, 370, 411, 462\n\nBonham, Jason 454, 456\u20137, 459, 460, 461, 469, 482, 485, 487\n\nBonham, John 145, 153, 182, 184, 185, 187, 258, 260, 285, 287, 294, 309, 315, 325, 331, 334\u20135, 339, 429; car crash 388; death 429\u201330, 480; drug taking 195, 202, 309, 312\u201313, 335, 338, 395, 428; _Houses of the Holy_ and 268; _In Through the Out Door_ and 396; joins LZ 131, 133\u20134, 137, 143, 144\u20135, 146\u20137, 150, 153; _Led Zeppelin II_ and 197; _Led Zeppelin III_ and 214; _Led Zeppelin IV_ and 234, 235, 238, 241, 246\u20138; meets Elvis 300; Oakland attack on Matzorkis and 3\u20139, 227, 327, 386\u20137, 393\u20134, 402, 405, 426, 431\u20132, 472, 498; _Physical Graffiti_ and 312\u201313; _Presence_ and 331, 333, 334\u20135, 336, 338; punches Meyer 333; punches Plant 257; punk and 356; sigil 241; _The Song Remains the Same_ and 287; tax exile 339; US\/North America tours, LZ 3\u20139, 154, 155, 156, 163, 176, 177, 179, 184, 187, 227, 327, 380, 381, 384, 386\u20137, 393\u20134, 402, 405, 426, 431\u20132, 472, 498\n\nBordin, Mike 461\n\nBoston Tea Party 160, 163, 179\n\nBowie, David 1\u20132, 12, 16, 54\u20135, 60, 126, 220, 279, 280, 283, 289, 452, 477\n\nBoyd, Pattie 231, 305, 395\n\nBoyle, Doug 454\u20135\n\nBredon, Anne 133\n\nBrian Auger Trinity 12, 123, 175\n\nBrian Poole and the Tremeloes 50, 60\n\nBritish Skiffle Group Championship 34\n\nBron-Yr-Aur, Wales 204\u20136, 215, 231, 233, 251, 368\n\nBrown, Buster 62\n\nBrown, James 268, 395\n\nBrown, Joe 53\n\nBrown, Mick 224, 424\n\nBrown, Phil 238, 239\n\nBrown, Rick 66\n\nBrowne, Tara 101\n\nBrubeck, Dave 183\n\nBuckley, Jeff 472\u20133, 496\n\nBuddy Guy 57, 185\n\nBuddy Holly 28, 30, 32, 41, 67\n\nBuell, Bebe 295\u20139, 301\u20135, 322, 329\u201330\n\nBuffalo Springfield 81, 132, 149, 159, 227\n\nBulwer-Lytton, Sir Edward 424\n\nBunyan, Vashti 83\n\nBurges, William 288, 289\n\nBurke, Solomon 49, 133\n\nBurne-Jones, Edward 126, 301\n\nBurroughs, William 319, 320\u20131, 467\n\nBurton, James 29, 30, 42, 68\n\nBusson, Tony 31\n\nButler, Anya 56\n\nBuxton Opera House 465\n\nByrds, the 71, 207\n\n_Caesar's Chariot_ (Boeing 720) 376, 379, 384, 386, 387\n\nCalder, Tony 69, 82\n\nCalifornia, Randy 280, 495, 496\n\nCallan, Chris 64\n\nCalvert, Pete 21, 29, 63\n\nCammell, Donald 319, 351\u20132\n\nCampbell, Glen 68\n\nCapitol Records 50\n\nCaptain Sensible 355\n\nCarson, Phil 198, 389, 429, 445, 471\n\nCarter, John 48, 63\n\nCarter-Lewis and the Southerners 44, 48, 61\n\nCasa Jimmy, Rio de Janeiro, Brazil 470\n\n_Cashbox_ 254\u20135\n\nCattini, Clem 133\n\nCBS Records 132, 153\u20134, 161, 325\n\nChannel Television 46\n\nChateau Marmont, LA 154, 155, 156, 159, 175\u20136\n\nChelsea Arts Club, London 492, 493\n\nCherry, Ava 1, 2\n\nChicago Stadium 376, 378, 379\n\nChkiantz, George 173\n\nChris Farlowe and the Thunderbirds 34\u20135\n\nChristgau, Robert 248, 270\u20131\n\n_Circus_ 321\n\nClapton, Eric 14, 20, 43, 45, 73\u20134, 75\u20137, 81, 83, 84, 85\u20136, 89, 93, 100, 129, 150, 179, 198, 199, 228, 290, 296, 423, 439, 440, 455, 460, 464, 481\n\nClark, Dick 106, 109, 111, 139, 206\n\nClarke, Arthur C.: _Childhood's End_ 275\n\nClash, the 168, 354, 400, 415, 418, 419, 423, 452, 481, 495\n\nClearwell Castle 392\u20133\n\nClifton, Peter 314, 346\u20137\n\nClinton Foundation 481\u20132\n\nCochran, Eddie 25, 30, 67, 133, 198\n\nCocker, Joe 123, 459\n\nCohn, Nudie 193\n\nCole, Richard 7, 8, 9, 154, 155\u20136, 158, 176\u20137, 178, 187, 190, 191, 196, 200, 201, 202, 204, 208, 232, 249, 256, 258\u20139, 262\u20133, 277, 285, 286, 297, 298, 299, 318, 328, 329, 330, 331, 335, 337, 338, 344, 376, 377, 378, 380, 384\u20135, 387, 390, 391, 393\u20134, 395, 424, 425, 428, 429, 431, 432, 447\n\nColeby, Barrington 244\n\nCollins, Phil 449\n\nColson, Glen 144\u20135, 451\n\nCooper, Alice 159\n\nCornell, Chris 496\u20137\n\nCorriston, Peter 317\n\nCosta's, Tooting Bec 126\n\nCoulson, Clive 183, 204\n\nCountry Joe and the Fish 160, 207\n\nCoverdale, David 462\u20133, 464, 468\u20139, 476, 489, 499\n\n_Coverdale\/Page_ 462\u20133\n\nCoy, Desmond 356\n\n_Crawdaddy_ 319\n\nCrawdaddy Club, Richmond 42\n\nCream 86, 89, 91, 100, 117, 131, 142, 150, 154, 157, 158, 168, 181\n\nCreation 99, 144\n\n_Creem_ 59, 379, 385, 446\n\nCromelin, Richard 279\u201380\n\nCrosby, Stills, Nash and Young 209, 315, 368, 453\u20134\n\nCrowe, Cameron 39, 206, 316, 318\u201319, 332, 478\n\nCrowley, Aleister 1, 2, 13, 14, 15, 33, 69, 97, 101, 112, 113, 210\u201311, 215\u201316, 217\u201325, 229, 238, 240, 242, 243\u20134, 250, 251, 253, 256, 257, 263, 273, 289, 290, 294, 306, 307\u20138, 314, 319, 320\u20131, 327, 343, 348, 350, 351, 352, 358, 370, 371, 372, 376, 381, 392, 403, 406, 410, 411, 424, 462, 489, 490, 494\n\nCrudup, Arthur 21, 61, 227\n\nCurbishley, Bill 464, 475, 477\n\nDaltrey, Roger 119, 199, 208\n\nDamned, the 355, 356, 415\n\nDann, Trevor 443, 444, 445\n\nD'Arbanville, Patti 296, 297\n\nDavies, Cyril 42\n\nDavies, Dave 59, 108\n\nDavies, Ray 59, 60\n\nDavis, Clive 153\u20134\n\nDavis, Erik 16, 246, 260, 331\u20132\n\nDavis, Stephen 187, 284, 321\u20133, 333\u20134, 376, 397\n\nDe Lane Lea Studios, London 122\n\n_Death Wish II_ (film) 434\u20135, 436, 439, 440, 489\n\nDecca 44, 49, 50\u20131, 52, 54, 61\u20132, 63, 134\n\nDeep Purple 38, 255, 273, 462\n\n_Degree of Murder, A_ (film) 118, 119\n\nDelaney and Bonnie 175, 179, 290\n\nDelsener, Ron 452\n\nDenny, Sandy 228, 246\n\nDes Barres, Michael 10, 13, 313\u201314, 331, 342, 343, 384, 385, 392\n\nDes Barres, Pamela 159, 176\u20137, 190\u20134, 197, 218, 256, 281, 283, 295, 299, 313, 314, 385\n\nDeShannon, Jackie 67\u20139, 70\u20131, 72, 83, 215\n\nDetective 294, 342\u20133, 396\n\nDexter, Jeff 79\u201380, 81, 86, 263\u20134, 290, 317\n\nDire Straits 223, 415\n\nDixon, Shirley 170\u20131\n\nDixon, Willie 139, 170\u20131, 437\n\nDominica 317, 318\n\nDonegan, Lonnie 21, 26, 20\n\nDonner, Ral 136, 268, 433\n\nDonovan 105, 116, 207, 491\n\nDoonican, Val 48\n\nDoors, the 117, 157, 159, 186, 220\n\nDowney, Jim 4\n\nDownliners Sect 60\u20131, 73\n\nDowntown Faction 143\n\nDr Feelgood 325, 393\n\nDrake Hotel, New York 271, 286\n\nDreja, Chris 63, 73\u20134, 76, 77\u20138, 93\u20134, 105, 110, 111, 112, 115, 127, 131, 134, 161, 438\n\nDrew, Richard (Zacron) 228, 229\n\nDrifters, the 36, 45, 49\n\nDriscoll, Julie 123, 175\n\nDunbar, Aynsley 133\n\nDunbar, John 290\n\nDunwoody, Gwynneth 198\n\nDylan, Bob 69, 309, 312, 315, 398, 408\u20139, 452\n\nEaling Jazz Club 40, 75, 101\n\nEarl Warren Showgrounds, Santa Barbara 125, 187, 190\u20131\n\nEarls Court Exhibition Centre, London 30, 324\u20135, 399, 479\n\nEaston, Eric 74\n\nEater 355, 356\n\nEaton, Nigel 471\n\nEbbisham Hall, Epsom 30\n\nEcker, Patricia 453, 456, 469\n\nEdgewater Inn, Seattle 186\u20137\n\nEdmunds, Dave 393\n\nElectric Lady Studios, New York 267, 270\n\nElizabeth II, Queen 317, 481\n\nEllen, Mark 443\u20134\n\nEllis, Royston 35\u20138, 497\n\nElstree Studios, Borehamwood 107\n\nElton John 291, 460, 461\n\nEmerson, Lake and Palmer 274, 354, 358\n\nEMI 43, 68, 194, 220, 253\n\nEntwistle, John 56, 89, 92, 134\n\nEpic Records 124, 129, 153\u20134, 253\n\nEpstein, Brian 53, 100\n\nEquinox Booksellers and Publishers, London 289\u201390, 305, 306, 317\u201318, 370, 410, 421\n\nErtegun, Ahmet 70, 148\u20139, 150, 151, 180, 291, 294, 325, 423\u20134, 450, 453, 481\u20132, 483\n\nEvans, Jim 38\n\nEverly Brothers 114, 192\n\nFaces, the 126, 281, 305, 438\n\nFairport Convention 133, 207, 228, 315, 423\n\nFaith No More 461\n\nFaithfull, Marianne 60, 69, 82, 85, 133, 290, 351\n\nFallon, B. P. 279, 282, 287, 313, 355, 452\n\nFarlow, Tal 45\n\nFarlowe, Chris 34\u20135, 130\u20131, 435, 456, 457\n\nFarren, Mick 281\n\nFats Domino 25, 227\n\nFender 28, 30, 37, 46, 77, 96, 124, 138, 176, 238\u20139, 439\n\nFillmore East, New York 3\u20134, 124, 128, 129, 160, 162\u20133, 179\u201380\n\nFillmore West, San Francisco 3, 117, 129, 159\u201360, 163, 175\n\nFirm, the 442, 445\u20137, 449, 453, 456, 457, 468\u20139, 489, 499\n\nFleetwood Mac 181, 232, 415\n\nFlick, Vic 60\n\nFlock 207, 208\n\nForum, LA 227, 254, 255, 256, 272, 277, 295\u20136, 299\u2013300, 377, 384, 385, 447, 479\n\nFoskett, Charlie 142\n\nFoskett, Jeff 448\n\n14 Hour Technicolor Dream, Alexandra Palace 80\n\nFowley, Kim 71\u20132, 296, 333\n\nFoxx, Cynthia 41\n\nFrame, Pete 233\n\nFrampton, Peter 141\n\nFranklin, Tony 442, 457\n\nFraser, Robert 101\n\nFrensham Pond Hotel, Surrey 310\n\nFromme, 'Squeaky' 322\u20133\n\nFulton County Stadium, Atlanta 276\n\nFuturama Grazioso 30, 31, 37\n\nGabriel, Peter 465\n\nGal\u00e1s, Diamanda 465\n\nGallup, Cliff 30\n\nGates, Samantha 275\n\nGates, Stefan 275\n\nGeffen Records 445, 462\n\nGene Vincent and the Blue Caps 30, 31\n\nGenesis Publications 490, 491\n\nGerlach, Fred 215\n\nGibb, John 43\n\nGibbs, Christopher 100\u20131, 107\n\nGibson 56, 209, 239, 245; EDS-1275 double-neck 239; Les Paul 46, 60, 108, 163\u20134, 173, 176, 432; Maestro Fuzz-Tone FZ-I 56, 58, 62\n\n_Gig_ 354, 356\u20137\n\nGirls Together Outrageously (GTOs) 165, 176, 190, 332\n\nGladsaxe Teen Club, Denmark 137\u20138\n\nGlastonbury Festival (1995) 470\u20131\n\nGoldberg, Danny 319, 322\u20133, 333, 334\n\nGolden Earring 178\n\n_Goldfinger_ (film) 60\n\nGomelsky, Giorgio 66, 74, 76, 87\n\n_Gonks Go Beat_ 54\n\nGorman, Paul 341, 491\n\nGouldman, Graham 74\n\nGraham, Bill 3\u20134, 5, 6, 7\u20138, 10, 117, 128, 160, 175, 179, 386, 388, 402, 405, 449, 472\n\nGraham, Bobby 48, 51, 53, 54, 61\n\nGraham, Davy 205\n\nGranny Takes a Trip 125, 295\n\nGrant, Gloria 342\n\nGrant, Helen 472\n\nGrant, Peter 4\u20136, 8, 10, 92, 113\u201317, 127, 128, 129, 131, 144, 186, 194\u20135, 207, 248\u20139, 250\u20131, 261, 263\u20134, 297\u20138, 304, 313, 314, 322, 325, 341, 347, 392, 404, 470; arrested for assault 287; background 113\u201314; Bath Festival and 207, 208; BBC Sessions and 250\u20131; Bindon and 3\u20139, 378, 386\u20137, 393\u20134; Bonham death and 431, 432\u20133, 434; contract with Led Zeppelin, lack of 152\u20133; death 471\u20132; _Death Wish II_ and 434, 435; drug taking 295, 342, 378; gangster image 299, 404\u20135, 426, 442; heart attack 393, 398; _Houses of the Holy_ and 274, 275; Joe Jammer and 194; JP solo career and 434, 435, 445; JP and occult, on 223\u20134; JP heroin addiction and 338; Karac Plant death and 9\u201310, 387\u20138, 389, 390; Knebworth and 398\u20139, 423\u20134, 425\u20137; _Led Zeppelin II_ and 187, 188, 193; _Led Zeppelin IV_ and 238, 239, 248\u20139; London house 341\u20132; LZ formation and 132, 144; LZ name 92; LZ record deals and 148, 150, 151\u20134; LZ singles and 169, 198; LZ split 432\u20133; LZ US\/North America tours and 154, 155, 157, 160, 162\u20133, 166, 175, 179, 183, 184, 193, 200, 201, 226, 254, 276, 278, 378, 380, 381, 382, 383, 386\u20137, 428, 429; marriage breakdown 342, 378, 402; meets Elvis 300; Oakland Coliseum attack on Matzorkis 3\u20139, 227, 327, 386\u20137, 393\u20134, 402, 431\u20132, 472, 498; police, pays off 265, 266; Plant solo career and 316, 445; _Presence_ and 334, 335; recluse 435; _The Song Remain the Same_ and 284, 285, 286\u20137, 344, 346; Swan Song and 291, 292, 293, 294, 295; tax exile 324, 326, 329, 330, 339; as Yardbirds manager 115\u201317, 127, 128, 129, 131, 132\n\nGrant, Warren 5\u20136\n\nGrateful Dead 117, 128\n\nGrech, Rick 315\n\nGreenblatt, Nanette 97, 165, 281\u20132\n\nGreenfield, Robert: _The Last Sultan_ 148, 149\n\nGretsch 38, 41\n\nGuillory, Isaac 264\n\n_Guitar Player_ 380, 458\n\n_Guitar World_ 91, 174, 199, 345, 346, 456, 461, 478\n\nGuy, Buddy 57, 185\n\nHale, Norman 339\n\nHale, Philip 427, 428\n\nHalfin, Ross 471, 488\n\nHarcourt, Charlie 142\n\nHardie, George 161, 338\n\nHarper, Roy 210, 212, 228, 253, 272, 277\u20138, 392, 414, 442\u20135\n\nHarriott, Joe 66\n\nHarris, Jet 44, 45, 47\n\nHarris, Richard 288\u20139\n\nHarrison, Billy 48\u20139, 51\u20132, 60, 64, 72, 80, 268\n\nHarrison, George 36, 64, 140, 141, 161, 179, 231, 268, 284\n\nHarvest Records 253\n\nHarvey, Leslie 382\n\nHayes, Isaac 197, 213, 261\n\nHazelwood, Lee 57\n\nHazzard, Tony 122\n\nHeadley Grange, Hampshire 206, 232\u20134, 237, 238, 240, 244, 245\u20136, 247\u20138, 269, 310, 311, 312, 313\n\nHeath, Edward 413\u201314, 431\n\nHelgeland, Brian 475\u20136\n\nHendrix, Jimi 65, 100, 117, 119, 129, 133, 154, 157, 162, 168, 173, 185, 188, 309\n\nHerman's Hermits 74, 88, 109, 116, 415, 445\n\nHermetic Order of the Golden Dawn 15, 221\n\nHeron, Mike 228\n\nHinton, Mick 334\u20135\n\nHipgnosis 274\u20135, 338\n\nHitler, Adolf 385, 414\u201315\n\nHobbs, Rick 378\n\nH\u00f6fner 25\u20136, 28, 31, 108\n\nHolmes, Jake 120\n\nHoneydrippers, the 433\u20134, 449, 450\n\nHooker, John Lee 39, 40, 227, 479\n\nHopkins, Nicky 89, 91, 118, 177\n\nHowe, Steve 268, 427\n\nHowlin' Wolf 107\u20138, 128, 133, 171, 479\n\nHuggins, Leslie 351\n\nHughes, Mary 104, 110\n\nHumble Pie 141\n\nHyatt House Hotel, LA 71, 159, 190, 204, 254, 256, 272, 280, 281, 283, 296, 321, 322, 333, 384\n\nHyde Park, London 182\u20133, 246\n\nHyland, Brian 109, 114\n\nIan Dury and the Blockheads 415, 423, 452\n\nIBC Studios, London 55, 66, 88, 102, 118\n\nIceland 206\u20137, 214, 241\n\nImmediate Records 41, 82, 83, 85, 86, 130\n\nIncredible String Band 228\n\nIndia 258\u20139, 265\u20136, 273, 466, 468, 472\n\n_Invocation of My Demon Brother_ (film) 351\n\nIron Butterfly 117, 149\u201350, 162, 163\n\nIsis-Urania Temple of the Hermetic Order of the Golden Dawn 15\n\nIsland Records 148, 151, 228, 237\u20139, 248, 292, 293, 316, 318, 454\n\nIsle of Wight Festival (1969) 398\n\nIvor Novello Awards (1977) 381\n\nIvy League 48, 56\n\nJabberwocky Studios, Dorset 267\u20138\n\nJagger, Mick 1, 13, 16, 40, 82, 83, 86, 140, 141, 161, 232, 268, 283, 296, 304, 319, 329, 336, 337, 351, 369, 400\n\nJames, Brian 356\n\nJames, Catherine 119, 126\u20137, 159, 165, 191, 192, 271\u20132, 385, 471\n\nJames, Tony 423\n\nJames Page Skiffle Group 27\u20138\n\nJameson, Bobby 83\n\nJansch, Bert 64\u20135, 205\n\nJapan 39, 257\u20138, 309, 438, 462\u20133, 464, 472\n\nJeff Beck Group 5, 128\u20139, 143, 153, 156\u20137, 183, 380\n\nJekyll, Gertrude 231\n\nJethro Tull 182, 183, 185, 457\n\nJimbo Music 53\n\nJimi Hendrix Experience, the 100, 117, 119, 133, 154, 162\n\nJimmy Page Music 211\n\nJimmy Page and the Paramounts 32\n\nJohn Mayall's Bluesbreakers 76, 83\u20135, 133, 181, 189\n\nJohnny and the Hurricanes 32\n\nJohnny Burnette Trio 107, 474\n\nJohnny Kid & the Pirates 33, 55\n\nJohns, Andy 206, 233, 245\u20136, 248, 267\n\nJohns, Glyn 43\u20134, 55, 61, 118\u201319, 138, 139\u201340, 141, 196, 206\n\nJohnson, Michael 490\n\nJohnson, Robert 43, 133, 171, 473\n\nJones, Brian 40, 85, 100\u20131, 106, 118, 182\u20133, 195\u20136\n\nJones, Charlie 465, 473\n\nJones, Davy 55, 119\n\nJones, John Paul 5, 141, 142, 152, 172, 199, 209, 223, 258, 269, 287, 315, 316, 325, 369, 425, 427, 445 493; background 134\u20135; Bonham death and 430; character 145, 186, 195\u20136; _Coda_ and 437; considers quitting LZ 310; Grant and 152, 153, 194, 471; _Houses of the Holy_ and 268\u20139; _In Through the Out Door_ and 395; joins Led Zeppelin 134\u20136, 139, 140, 141, 143, 145, 146; JP solo career and 457; Karac Plant death and 387\u20138, 389; Knebworth and 425; _Led Zeppelin I_ and 139, 141; _Led Zeppelin II_ and 195\u20136; _Led Zeppelin III_ and 207\u20138; _Led Zeppelin IV_ 234, 235, 238, 241, 245, 246, 247; Live Aid 449; LZ reunions\/attempts to reform and 449, 451, 460, 465, 469, 471, 484\u20135; LZ split and 432\u20133; LZ US\/North America tours and 8\u201310, 155, 156, 213, 226, 285, 376, 380, 381; name 44; Page and Plant and 465, 471, 476, 481; _Physical Graffiti_ 364; _Presence_ 331, 333, 336; Rock and Roll Hall of Fame induction 469; sessions with JP before formation of LZ 44, 45, 53, 82, 88, 89, 90, 91, 102, 105, 121; sigil 241; _The Song Remains the Same_ and 285, 287; tax exile 339\n\nJones, Kenney 118, 439\n\nJones, Mick 119, 182\u20133, 423, 481, 495\n\nJunco Partners 142, 143\n\nJuniper, David 196\n\nKalmes, Jack 377\n\nKanuko (Japanese girl) 258\n\nKatz, Charlie 54, 82\n\nKen Colyer's Jazzmen 26\n\nKent, Nick 315, 342, 344, 347\u20138, 398, 405, 432, 476\n\nKhan, Sultan 273\n\nKinetic Playground, Chicago 163, 178, 183, 200\n\nKing, B.B. 179, 212\n\nKing, Ben E. 181, 436, 450\n\nKing, Freddie 106\n\nKing, Rex 429\u201330\n\nKingston College of Art 74, 228\n\nKingsway Studios, London 82\n\nKinks, the 56, 59\u201360, 69, 108, 173\n\nKirke, Simon 292\u20133, 294, 394\n\nKitman, Jamie 458\n\nKizart, Willie 57\n\nKnebworth (LZ concerts, 1979) 326, 399, 404, 406, 407, 408, 416, 417\u201319, 423\u20137, 428, 452, 460, 479\n\nKnight, Ian 80\n\nKoch, Rudolph: _The Book of Signs_ 241\n\nKorner, Alexis 40, 42, 76, 131, 132, 168, 465\n\nKosmo Vinyl 423, 452\n\nKramer, Eddie 173, 174, 187\u20138, 267\n\nKrauss, Alison 483\n\nKravitz, Lenny 463\n\nLa Rue, Danny 289\n\nLaidlaw, Ray 143\n\nLaine, Denny 119, 126, 471\n\nLambert, Kit 89, 107\n\nLancastrians, the 54\n\nLane, Ronnie 170, 310, 438\n\n_Last Temptation of Christ_ _, The_ (film) 465\n\nLaverde, Durban 457, 459\n\nLeadbelly 26, 215\n\nLeander, Mike 43\u20134\n\nLeary, Timothy 220\n\nLed Zeppelin:\n\nalbums and live recordings: _BBC Sessions_ 167\u20139, 473; _The Bombay Sessions_ 273; _Celebration Day_ 486; _Coda_ 436\u20137, 460; Deluxe Editions 488; _Houses of the Holy_ 10, 267\u20139, 270\u20131, 274\u20135, 276, 277, 311, 437; _How the West Was Won_ 272, 479, 480; _In Through the Out Door_ 394\u20137, 398, 406, 417\u201318, 423, 427, 437; _The Led Zeppelin Boxed Set_ 460\u20131; _Led Zeppelin DVD_ 284, 479\u201380; _Led Zeppelin I_ 138\u201341, 144\u20135, 148\u201352, 153\u20134, 161\u20132, 175; _Led Zeppelin II_ 141, 164, 165\u20136, 169\u201375, 180, 187\u20138, 193, 194, 196\u20139, 204, 230, 474; _Led Zeppelin III_ 72, 180, 204\u201316, 228\u201330, 231, 240, 250, 317, 437; _Led Zeppelin IV_ 232\u20136, 237\u201350, 260, 261\u20132, 264, 276, 288, 310, 311, 345, 346, 421, 422, 495; _Physical Graffiti_ 269, 308\u201318, 336, 364, 447; _Presence_ 331\u201340, 347\u20138, 350, 359, 360, 363\u20134, 366, 401, 406, 431\n\nfilms: _The Song Remains the Same_ 22, 243, 284\u20138, 308, 330, 338, 339, 342, 343\u20134, 346, 361, 365, 401, 404\u20135, 425, 479\n\ngigs _see under individual venue name_\n\norigins of 89\u201393, 127, 130\u201347, 363, 388\u20139; first live shows 137\u20138, 141\u20137; first rehearsal 108, 135\u20136; name 92, 144, 152; record deal, first 148\u20139, 291; recruitment of members 130\u20134\n\nsongs: 'Achilles Last Stand' 331, 336\u20137, 360, 361, 386, 397, 484; 'All My Love' 397, 406; 'Another Way to Wales' 205; 'Babe I'm Gonna Leave You' 133, 139, 361; 'Baby Come On Home' 49; 'The Battle of Evermore' 245, 246, 360, 466; 'Black Country Woman' 269, 311; 'Black Dog' 211, 244\u20135, 246, 247, 485; 'Black Mountain Side' 64, 203, 386; 'Bonzo's Montreux' 437; 'Boogie with Stu' 311; 'Bring It On Home' 171, 187\u20138, 460, 469; 'Bron-Y-Aur Stomp' 64\u20135, 205, 226, 227, 256, 311; 'Candy Store Rock' 336, 360; 'Celebration Day' 207, 214; 'Communication Breakdown' 67, 137, 139, 143, 157, 168, 169, 198, 227, 305; 'The Crunge' 211, 268; 'Custard Pie' 313, 459, 460; 'Darlene' 423, 437; 'Dazed and Confused' 11, 120, 122, 123, 137, 139, 144, 157, 182, 199, 253, 261, 447, 479, 480; 'Down by the Seaside' 205, 311; 'D'yer Mak'er' 268; 'For What It's Worth' 227, 251; 'Four Sticks' 246\u20137; 'Friends' 204, 205; 'Gallows Pole' 204, 212, 215, 468; 'Going to California' 231, 247, 249, 256, 460, 475; 'Good Times Bad Times' 139, 157, 169, 227, 485; 'Hats Off to (Roy) Harper' 210, 212, 253, 414; 'Heartbreaker' 188, 454; 'Hot Dog' 396\u20137, 406, 425; 'Hots on for Nowhere' 336; 'Houses of the Holy' 311; 'I Wanna Be Her Man' 205; 'I'm Gonna Crawl' 397, 406; 'Immigrant Song' 204, 206, 207, 214\u201315, 230, 460, 484, 497; 'In My Time of Dying' 309, 312, 336, 461, 497; 'In the Evening' 395, 397, 406, 425, 435; 'Kashmir' 169, 311, 312, 361, 364, 386, 397, 454, 466, 485; 'The Lemon Song' 171, 177, 251; 'Living Loving Maid' 181, 188; 'Misty Mountain Hop' 246, 454, 460; 'Moby Dick' 177, 309, 386; 'Night Flight' 311; 'No Quarter' 268\u20139, 381, 465, 468; 'Nobody's Fault but Mine' 336, 360, 378, 468; 'Out on the Tiles' 226, 227; 'Over the Hills and Far Away' 205, 454, 459; 'Ozone Baby' 423, 437; 'Poor Tom' 437; 'Rain Song' 268; 'Ramble On' 181, 197; 'Rock and Roll' 245, 247, 311, 449, 460, 485; 'The Rover' 205, 311; 'Royal Orleans' 364; 'Scarlet' 315, 316; 'Sick Again' 311, 313, 460; 'Since I've Been Loving You' 207\u20138, 212, 213, 214, 226, 227, 238, 331, 378; 'The Song Remains the Same' 268; 'Stairway to Heaven' 169, 231, 233, 234\u20136, 237, 238, 239, 246, 247, 249, 254, 361, 386, 393, 425, 429, 439, 440, 449, 454, 455, 459, 466, 484, 485, 495\u20136; 'Tangerine' 72, 215, 497; 'Tea for One' 331, 336; 'Ten Years Gone' 311, 378; 'That's the Way' 205, 209, 256; 'Trampled Under Foot' 311, 313, 460; 'Trucking Little Mama' 251; 'Walter's Walk' 437; 'The Wanton Song' 311; 'Wearing and Tearing' 423, 437, 460; 'What Is and What Should Never Be' 188; 'When the Levee Breaks' 247\u20138, 262, 311, 468, 469, 492\u20133; 'Whole Lotta Love' 22, 54, 119, 168, 169\u201371, 172, 197, 198, 201, 202, 207, 227, 249, 251, 255, 267, 285, 313, 449, 454, 479, 485, 487\u20138; 'Your Time Is Gonna Come' 139\n\nsplit and reformations: Ahmet Ertegun Tribute Concert, reform for 482\u20136; Atlantic Records 40th anniversary, reform for 453\u20135; Bonham death and split 429\u201332; Carmen Plant 21st birthday, reform for 460; Jason Bonham wedding, reform for 460; Live Aid, reform for 449\u201351; Rock and Roll Hall of Fame, reform for induction 469\n\ntouring outside US _see under individual venue name_\n\nUS\/North America tours: first (1968\u20139) 154\u201366; second (1969) 175\u201380; third (1969) 182\u201394; fourth (1969) 196\u20137; fifth (1970) 200\u20132; sixth (1970) 226\u20138; seventh (1971) 253\u20137; eighth (1972) 270\u20132; ninth (1973) 276\u201387; tenth (1975) 318\u201323; eleventh (1977) 354, 357, 358, 359\u201362, 376\u201387, 401\u20132, 409\u201310; planned twelfth (1980) 428, 429\u201330\n\nLee, Alvin 181\n\nLee, Brenda 63, 67, 68\n\nLee, Jeannette 481\n\nLee, Michael 465, 473, 476, 488\n\nLee Lewis, Jerry 22, 25, 28, 31, 133\n\nLeFevre, Benji 331, 384, 430\n\nLennon, John 26, 36, 50, 123, 198, 219, 290\n\nLewis, Barbara 70\n\nLewis, Dave 450, 451, 484\n\nLewis, Ken 48, 63\n\nLewis, Leona 487\n\nLisberg, Harvey 74\n\nLittle Richard 25, 114, 182\n\nLittle, Carlo 63, 177\n\nLive Aid 449, 450, 482\n\n_Live at the Greek: Excess All Areas_ (Black Crowes\/JP LP) 477\u20138\n\n_Live on Blueberry Hill_ (LZ bootleg) 227\n\n_Live Yardbirds: Featuring Jimmy Page_ 253\u20134\n\nLiverpool Scene 182\n\nLiverpool University 144\n\nLoder, Kurt 439\u201341\n\nLondon, Laurie 30\n\nLong Beach Arena 149, 272, 479\n\n_Los Angeles Times_ 71, 230, 255, 277\n\n_Lucifer Rising_ (film) 2, 225, 307\u20138, 319, 348\u201353, 371\u20132, 435, 488\n\n_Lucifer Rising and Other Sound Tracks_ 488\n\nMacGregor, Sandy 204\n\nMadison Square Garden, New York 207, 284, 285\u20136, 287, 295\u20136, 308, 314, 318, 319, 344, 383\u20134, 447, 453, 455, 470, 475, 479, 480\n\nMalcolm Austin and the Whirlwinds 31\n\nMalibu Beach Colony, LA 330\u20131, 401\n\n_Man, Myth and Magic_ (occult book series) 14, 242\n\nManchester Free Trade Hall 39, 418\n\nManish Boys, the 54, 55\n\nManning, Terry 213, 214, 215\n\nMansfield, Mike 425\n\nManson, Charles 257, 279, 322, 351, 371\n\nManticore Studios, London 354, 356, 465\n\nMarauders, the 48\n\nMarcus Music, London 452\n\nMarquee Club, London 42, 43, 94, 95, 143\u20134, 251, 419, 460\n\nMarriott, Steve 91, 114, 130, 141, 170, 313, 476\n\nMarshall Tucker Band 423\n\nMartin, Charlotte 81, 86, 199\u2013200, 204, 205, 210\u201311, 215, 230, 251, 256, 272, 284, 301, 304, 305, 316, 326, 327, 328, 342, 347, 349, 352, 390, 391, 421\u20132, 425, 434, 438, 447, 453, 469, 479\n\nMartin, Gavin 171\n\nMartin, Grady 57\n\nMartinez, Paul 449\n\nMarvin, Hank 44\u20135, 268\n\nMassot, Joe 284\u20135, 286, 287, 314, 346\n\nMatlock, Glen 355\u20136\n\nMattacks, Dave 228\n\nMattix, Lori 1, 282\u20134, 296, 298, 299, 322, 447, 469\u201370, 491\n\nMatzorkis, Jim 3, 5\u20137, 386\u20137\n\nMax's Kansas City, New York 287\n\nMay, Phil 293, 327\n\nMayer, Roger 58, 77\n\nMayfair Ballroom, Newcastle 141, 249\n\nMcAlpine, Sir Robert 328\n\nMcAuley, Jackie 52\n\nMcCallum, Sr., David 99\n\nMcCartney, Paul 89, 123, 220, 264, 460\n\nMcCarty, Jim 74, 76, 94, 103, 109\u201310, 120, 121, 123, 127, 130, 438\n\nMcKay, Adam 492, 493\n\nMcKeever, Jack 472\n\nMcLaren, Malcolm 295, 325, 433\n\nMcLaughlin, John 45\n\nMeehan, Tony 44, 45, 47, 142\n\nMeissonnier, Martin 465, 467\n\n_Melody Maker_ 59\u201360, 64, 75, 182, 228, 261, 339, 398, 406\u20137, 427\n\nMemphis Slim 39, 43\n\nMena House Hotel, Cairo 9, 381, 385, 409\u201310\n\nMendelsohn, John 13, 161\u20132\n\nMercy, Miss 176\n\nMermaid Festival, London (1961) 37\n\nMichel, John 319\n\nMiles, John 456, 459\n\nMirror Sound, LA 159\n\nMitchell, Joni 247, 271\n\nMitchell, Mitch 133\n\nMoby Grape 117, 132, 145, 212\n\n_Mojo_ 212, 292, 470\u20131\n\nMonkees, the 54, 119\n\nMonsters of Rock Festival, Castle Donington (1990) 460\n\nMoody Blues, the 77, 119, 207\n\nMoon, Keith 89, 90, 91, 92, 93, 133\n\nMoore, Scotty 21\u20132, 29, 31, 61\n\nMorgan Studios, London 181, 437\n\nMorocco 312, 326, 327, 331, 333\u20134, 397, 410, 467\n\nMorrison, Jim 186, 220\n\nMorrison, Van 48, 49, 51, 399\n\nMost, Mickie 50, 63, 91, 92, 105, 114, 115, 116, 120, 122, 129, 131, 155, 195\n\nMove, the 132\n\nMr Freedom, London 152, 491\n\nMTV _Unplugged_ 464\u20138, 476\n\nMuddy Waters 129, 131, 133, 169\u201370\n\nMurphy, Matt 43\n\nMusicland Studios, Munich 334, 335, 336\u20137, 339\n\nMyer, Michelle 333\n\nMyers, Marc 171, 172, 174\n\nMystic Studios, LA 177\n\nNakao, Michi 467\n\nNapier-Bell, Simon 86\u20138, 89, 91\u20132, 95\u20136, 102, 104, 106, 107, 108, 110\u201312, 114\u201315, 116, 228\n\nNassau Coliseum, New York 270, 305, 318\n\nNeal, Peter 24\n\nNeil Christian and the Crusaders 38\u20139, 46, 47, 53, 76\u20137, 96, 116, 135, 142\n\nNelson, Ricky 22, 29, 30, 41\u20132, 67, 69\n\nNevison, Ron 310\n\nNew Barbarians, the 423, 426\n\nNew Yardbirds, the 130\u201347\n\nNewport Jazz Festival, Rhode Island (1969) 183\u20134\n\nNico 85\n\nNilsson, Harry 116\n\nNitzsche, Jack 68\n\n_NME_ 14, 99\u2013100, 106, 169, 182, 199, 281, 342, 344, 347\u20138, 352, 356, 398, 399\n\nNoone, Peter 445\n\nOakland Coliseum 3\u201310, 227, 327, 386\u20137, 393, 429, 431\u20132, 472\n\nOats, Bobby 32, 33\n\nObs-Tweedle 131\u20132\n\nO'Dell, Dick 273\n\n_Old Grey Whistle Test, The_ (TV programme) 442\u20135, 446\n\nOldham, Andrew Loog 44, 50, 69, 74, 82\u20133, 85, 99\u2013100\n\nOlympic Sound Studios, London 138\u201340, 173, 206, 215, 232, 238, 269, 487\n\n_Once More with Felix_ (TV programme) 203\n\nOrbison, Roy 30, 118\n\nOrdo Templi Orientis (O.T.O.) 221, 224\u20135\n\nOsbourne, Ozzy 16, 220, 475\n\n_Outrider_ (JP solo LP) 455\u20139, 478\n\nOxford Union 497\u20138\n\nPace, Charles 218\n\nPage and Plant: _No Quarter: Jimmy Page and Robert Plant Unledded_ 464\u201375, 488, 499; _Walking into Clarksdale_ 473\u20134, 488; Walking into Everywhere tour 474\u20135\n\nPage, Ashen Josan (son) 470\n\nPage, James (father) 19, 23, 24, 34, 97, 424\n\nPage, James Patrick (son) 453, 455\n\nPage, Jimena (wife) 469\u201370, 475, 476\n\nPage, Jimmy: Beijing Olympics (2008), plays at closing ceremony of 487; birth 19\u201320; Boleskine House and 15, 217\u201319, 221\u20133, 273, 287\u20138, 307, 321, 358, 370, 411, 462; Bonham death and 430, 444; charity work with impoverished Brazilian children 438\u201340, 445\u20136, 481; childhood 20\u201339; children 205, 215, 251\u20132, 256, 304, 305, 315, 326, 327\u20138, 338, 347, 390, 391, 422, 434, 447, 453, 455, 470, 475\u20136, 492 _see also under individual child name_ ; cocaine and 4, 195, 202, 265, 309, 313, 331, 425, 429, 432, 437\u20138, 446\u20137, 458; conservative nature 413\u201314; _Coverdale\/Page_ 462\u20133; death threats 256\u20137, 322, 470; diet 38, 377, 458; digitalisation, on 395\u20136, 445; Dominica, visits 317, 318; finances 33, 53\u20134, 69\u201370, 81, 112, 116, 126, 132, 136, 139, 144, 146, 148, 151, 155, 180, 194, 201, 211, 231, 288\u20139, 354\u20135, 420\u20131, 428, 491; Firm, the 442, 445\u20137, 449, 453, 456, 457, 468\u20139, 489, 499; first groups 26\u201339; first TV appearance 27\u20138; fuzzbox, Mayer designs first 58, 77; _Gig_ interview with author (1977) 357\u201375; Grant death and 471\u20132; groupies and 1, 159, 164\u20136, 176\u20137, 178, 186\u20137, 190\u20134, 197, 218, 249, 256, 279, 281\u20134, 296, 298, 299, 313, 314, 332, 385, 447, 469\u201370, 491; Guadeloupe, visits 390, 447; guitar, begins to play 20\u20136; guitars _see under individual brand name_ ; heroin and 4, 10, 195, 196, 202, 290, 295, 309, 315, 316\u201317, 324, 331, 335, 338, 342\u20133, 353, 357, 379, 388, 390, 391\u20132, 395, 431, 432, 437, 438, 439, 447, 458; Hitler, on 414\u201315; Karac Plant death and 9\u201310, 387\u20138, 389, 390, 391, 402; Knebworth interview with author, _NME_ , 4 August 1979 ('Smiling Men with Bad Reputations') 399\u2013423; Led Zeppelin and _see_ Led Zeppelin _and under individual band member name_ ; love life _see under individual partner name_ ; marriages 438, 453, 456, 469\u201370, 475, 476; OBE 481; occult, interest in 1\u20132, 3\u20134, 13\u201316, 19, 33, 69, 97, 101\u20132, 112, 113, 200, 210\u201311, 215\u201325, 229, 231, 238, 240\u20134, 249, 250, 251, 253, 256, 257, 262\u20133, 265, 273, 282, 289\u201390, 294, 306\u20138, 314, 319, 320\u20131, 327, 343, 345, 346, 348\u201353, 358, 369\u201373, 376, 381, 388\u20139, 390, 392, 395, 403, 406, 410\u201311, 424, 435, 462, 483, 488, 489, 490, 494; Old Mill House, Clewer 428, 430, 436; _Outrider_ 455\u20139, 478; Oxford Union speech 497\u20138; Page and Plant and _see_ Page and Plant; Pangbourne home 98, 105, 127, 132, 133, 139, 172, 176, 194, 200, 210, 211, 229, 230, 233; Plant solo record deal with Atlantic, reaction to 445; Plant's growing influence within LZ, on 484; Plumpton Place home 231, 284, 288, 292, 304, 305, 311, 314, 395, 389, 427, 428; producer 35, 42, 52, 83\u20134, 91, 92, 122, 130, 138\u20139, 161, 162, 172, 187\u20138, 195, 214, 247\u20138, 364, 396, 456, 472, 473; publishing company, sets up 69; punk\/New Wave and 355\u20136, 415\u201316; recluse 98, 244, 345, 433, 434, 435, 438, 471; Robbie Williams feud 494\u20135; session guitar player 43\u201395; Show of Peace Concert, Beijing (2010) 489\u201390; solo career _see under individual band, album and song name_ ; soundtracks _see under individual movie name_ ; Sutton Art College, Surrey 39, 41, 42, 45, 81, 93, 196; Tower House home 288, 302, 303, 306, 308, 342, 348, 349, 428, 434, 404; violin bow, use of 39, 99, 120, 124, 144, 157, 182, 199, 332; Yardbirds, joins 95\u20136 _see also_ Yardbirds; Zoso sigil 16, 241\u20133, 260, 485\n\nPage, Patricia (mother) 19, 23, 24, 27, 81, 97, 446, 494\n\nPage, Scarlet (daughter) 205, 251\u20132, 256, 304, 305, 315, 326, 327\u20138, 338, 347, 390, 391, 422, 434, 447, 492\n\nPage, Zofia Jade (daughter) 470, 475\u20136\n\nPage Motors, Epsom 23, 407\n\nPallenberg, Anita 100\u20131, 106, 118\n\nPark Lane Hotel, New York 342\n\nParker, Bobby 334\n\nParker, Colonel Tom 342\n\nParlophone 72\n\nParsons, Steve 'Snips' 225, 307, 308, 469\u201370, 488\u20139\n\nPeel, John 117, 167\n\nPegg, Dave 228\n\nPenderecki, Krzysztof 39\n\nPerkins, Carl 35, 227\n\nPerry, Joe 460, 469, 476\n\nPettifer, Julian 37\n\nPhiladelphia Summer Pop Festival (1969) 185\n\nPhillips, Sam 57, 87\n\nPhillips, Eddie 99, 144\n\nPhillips, Esther 69\n\nPickett, Kenny 145\n\nPickingill, 'Old George' 240\n\nPink Floyd 128, 207, 274, 327, 400, 415, 460\n\nPlant, Carmen 146, 204, 205, 327, 329, 387, 460\n\nPlant, Karac 9\u201310, 327, 329, 387\u20138, 393, 397, 402, 432, 451\n\nPlant, Logan 397\n\nPlant, Maureen 146, 160, 203\u20134, 327, 328, 329, 335, 387, 388, 397\n\nPlant, Robert 8, 11, 23, 66\u20137, 145, 170, 171, 174, 269\u201370; Ahmet Ertegun Tribute Concert (2007) 482\u20136; Atlantic Records 40th anniversary and 453\u20135; Bonham punches 257; Bron-Yr-Aur, Wales, visits with JP 204\u20136, 215, 231, 233, 251, 368; car crash and recovery 327\u20139, 330, 331, 339, 358, 363, 366\u20137, 369, 374, 377, 402, 431, 433, 451; control of LZ agenda after split 449\u201350, 480\u20131; death of son and 9\u201310, 387\u20138, 389, 390, 393, 451; _Fate of Nations_ 463\u20134; first LZ record deal and 151; Honeydrippers and 433\u20134, 449, 450; _The Honeydrippers: Volume One_ 450; _Houses of the Holy_ and 268, 269; _In Through the Out Door_ and 395, 396; joins Led Zeppelin 131\u201347; laryngitis 251, 357, 358, 380; _Led Zeppelin II_ and 170, 171, 173, 174; _Led Zeppelin IV_ and 235\u20136, 240, 241, 245, 246, 247; Live Aid 449; LZ US\/North American tours 154, 155, 156, 157, 158, 159, 160, 161, 163, 184, 185\u20136, 202, 226, 276, 384, 385, 386, 387; marriage _see_ Plant, Maureen; meets Elvis 300; money, tightness with 257, 269; nodules operation 308; _Now and Zen_ 452; Page and Plant project and _see_ Page and Plant; _Physical Graffiti_ and 312, 364; _Pictures at Eleven_ 435\u20136; _Presence_ and 332, 334, 335, 363; punk and 355\u20136; _Raising Sand_ 483\u20134, 485\u20136; recommends Bonham to JP 133\u20134; _Sixty Six to Timbuktu_ 480; solo record deal 445; _The Song Remains the Same_ and 287, 344\u20135; tax exile 325, 330; Turd Burglars and 393\n\n_Playboy_ 219, 300, 304\n\nPlayhouse Theatre, London 167\u20138\n\nPlaza Hotel, New York 180, 296, 318, 355, 384\n\nPlymouth Brethren 222\n\nPolar Studios, Stockholm 394\u20135\n\nPolydor Records 150\n\nPontiac Silverdome 381\n\nPoole, Brian 50, 60\n\nPowell, Aubrey 274, 275\n\nPratt, Guy 462\u20133, 476, 487\u20138\n\nPre-Raphaelite movement 125, 126, 127, 288, 301, 344, 345, 388, 391, 408\n\nPresley, Elvis 20\u20132, 25, 29, 30, 61, 67, 69, 87, 103, 133, 136, 164, 193, 255, 268, 283, 295\u20136, 299\u2013301, 342, 388, 396, 401, 414, 446, 450\n\nPreston, Neal 385\n\nPretty Things, the 48, 61, 73, 293\u20134, 327, 396\n\nPrince, Viv 48\n\nProby, P. J. 60, 69, 72, 136\u20137\n\nProcol Harum 133\n\npunk rock 13, 96, 238, 325, 336, 354\u20136, 388, 398, 399, 400, 401, 408, 415, 416, 423, 437, 481\n\nPye Studios, London 83\u20134\n\nQuarrymen, the 26\n\nR&B All-Stars 42\u20133\n\nRainbow Bar and Grill, LA 280, 299, 333\n\nRAK Management 115, 128\n\nRalphs, Mick 293\n\nRamzy, Hossam 465\n\nRao, Vijay Raghav 273\n\nRCA 21\u20132\n\nRedding, Otis 75, 268\n\nRed E. Lewis and the Redcaps 32, 33, 34, 35\u20136, 38\n\nRedding, Noel 177\n\nRedding, Otis 75, 268\n\nRees, Paul 238, 466, 476\n\nReeves, Paul 125\u20136, 341\u20132\n\nRegal Zonophone 194\n\nRegent Sounds Studio, London 72\n\nReid, Terry 131, 141\u20132, 154\n\nRelf, Keith 74, 75, 94, 96, 102, 107, 110, 120, 121, 123, 124, 125, 130, 141, 498\n\nRenbourn, John 64, 105\n\nReuss, Theodor 221\n\nR. G. Jones Studios, Surrey 35\n\nRhodes, Greece 327\u20139, 433\n\nRhodes, Orville 'Red' 57\n\nRichard, Cliff 32, 36, 45, 61, 79, 83, 114\n\nRichards, Keith 12, 13, 40, 57, 82, 83, 101, 201, 315\u201316, 369, 384, 390, 391, 400, 421, 423, 447\n\nRimmer, William 292\n\nRobbins, Marty 57\n\nRoberts, Andy 182\n\nRoberts, Tommy 152, 491\n\nRoberty, Marc 429\n\nRobinson, Chris 476\u20137, 478\n\nRobinson, Lisa 276\u20137, 379, 425\n\nRock and Roll Hall of Fame 469\n\nRockfield Studios, Monmouth 435\u20136\n\nRodgers, Paul 292, 315, 440, 442, 444, 445\n\nRodney's, LA 279, 283, 389\n\n_Rolling Stone_ 13, 39, 63, 118, 121, 161\u20132, 164\u20135, 206, 229, 262, 264, 309\u201310, 311, 316, 318, 319, 332, 398, 439\u201340, 451, 458, 478\n\nRolling Stones, the 13, 40, 44, 50, 57, 63, 69, 73, 74, 82\u20133, 100\u20132, 103, 106, 112, 116, 126, 138, 140, 152, 163, 168, 171, 172, 182\u20133, 232, 233, 260, 264, 268, 276, 279, 291, 293, 309, 310, 315\u201316, 334, 336, 337, 351, 373, 377, 391, 400, 422, 438, 481\n\nRose Palace, Pasadena 177\n\nRose, Tim 134\n\nRosen, Steven 5, 91, 92, 380\n\nRough Trade Records 481\n\nRoundhouse, London 145\u20136\n\nRowe, Dick 50\n\nRowland, Bruce 315\n\nRoxy, London 355, 356, 415\n\nRoyal Albert Hall, London 106, 182, 198, 199, 208, 284, 344, 418, 437, 438\u20139, 479, 491\n\nRundgren, Todd 296, 298, 302\u20133, 304, 423\n\nSabet, Scarlett 492\u20133\n\nSafer, Jennifer 294\n\nSam the Sham and the Pharaohs 109\n\nSamwell-Smith, Paul 74, 75, 88, 94, 438, 498\n\nSanchez, Judy 165\n\nSanchez, Karen 165\n\nSander, Ellen 178, 179\n\nsarangi (bowed Indian instrument) 273\n\nSarne, Mike 48\n\n_Saturday Club_ (radio show) 122, 123\n\nScene Club, New York 129\n\nSchaefer Music Festival, New York (1974) 315\n\nSchl\u00f6ndorff, Volker 118\n\nScoppa, Bud 456, 458\n\nSCREAM (Supporting Children through Re-Education and Music) 476\n\nScreaming Lord Sutch and the Savages 12, 38, 62\u20133, 177, 194, 273\n\nSearchers, the 68, 154\n\nSeattle Pop Festival (1969) 186\n\nSecunda, Tony 132\n\nSex (London emporium) 295, 325\n\nSex Pistols, the 12, 295, 325, 355\u20136, 379, 395, 423, 433\n\nShadows 36, 44, 45, 58, 114, 254, 268\n\n_Shaft_ (film) 261\n\nShankar, Ravi 64, 273\n\nSharks, the 225, 307, 469, 488\n\nShearmur, Ed 465\n\nSheeley, Sharon 67\n\n'She Just Satisfies' (JP solo record) 69\u201370\n\nShepperton Studios, Surrey 314, 324, 344\n\nShotgun Express 12, 128\n\nShow of Peace Concert, Beijing (2010) 489\u201390\n\nSilhouettes, the 43\n\nSilverhead 313\n\nSilvester, Andy 433\n\nSimonon, Paul 400\n\n_Six-Five Special_ (TV programme) 26\n\nSkidmore, Alan 66\n\nskiffle 26\u20139, 34, 44\u20135\n\nSlade, Chris 442\n\nSmall Faces, the 91, 96, 114, 118, 130, 141, 170, 171, 438, 476\n\nSmith, Richard Digby 212\n\nSmith, Tony 249\n\nSonny Boy Williamson 66\u20137, 75, 133\n\nSpare, Austin Osman 14, 242\u20133, 489, 492\n\nSpeakeasy, London 80, 86, 234, 302\n\nSpector, Phil 52, 69, 82, 159\n\nSpence, Lewis: _The Magic Arts in Celtic Britain_ 245\n\nSpencer Davis Group 91, 130\n\nSpicer, John 38\n\nSpirit 145, 184, 280, 495\u20136\n\nSpringfield, Dusty 87, 150\n\nSquire, Chris 433\n\nStamos, John 448\n\nStamp, Chris 56, 89\n\nStargroves, Hampshire 232, 268, 269, 311\n\nStarr, Ringo 53, 220\n\nStarr, Sable 280\u20131, 282\n\n_Starship_ (Boeing 720) 276, 286, 318, 376\n\nSteele, Tommy 30, 114\n\nStein, Mark 187\n\nStevens, Joe 287\n\nStewart, Al 65\n\nStewart, Ian 'Stu' 315, 438, 440\n\nStewart, Rod 12, 16, 128, 129, 143, 255, 281, 477\n\nStiff Records 355\n\nStigwood, Robert 89\n\nStokes, Tez 142\n\nStone the Crows 261, 293, 382\n\nStorm, Danny 31\u20132\n\nStrange, Billy 57\n\nStrong, Barrett 35, 429\n\nSubiaco Oval, Perth 266\n\nSullivan, Big Jim _see_ Big Jim Sullivan\n\nSun Ra 178, 183\n\nSun Studio, Memphis 57, 87, 107, 164\n\nSuperhype Music 152\n\n_Supershow_ (TV show) 480\n\nSutton Art College, Surrey 39, 41, 42, 45, 93, 196\n\nSwan Song Records 64, 291\u20136, 298, 299, 309, 310, 322, 325, 328, 334, 342, 349, 352, 356\u20138, 388, 393, 394, 396, 401, 403, 406, 422, 430, 435, 436, 437, 465\n\nTaj Mahal 160, 189\n\nTalmy, Shel 50, 52, 53, 54, 55\u20136, 59, 60, 68, 99, 314\n\nTampa Stadium, Florida 276, 382\u20133\n\nTaylor, Bobby 35\n\nTaylor, Derek 302\n\nTaylor, Heather 119, 126, 199\n\nTaylor, Mick 182, 316\n\nT-Bone Walker 39, 40\n\n10cc 74, 395\n\nTen Years After 181, 183, 360\n\nTerrace Ballroom, Salt Lake City 187\n\nTexas International Pop Festival, Lewisville (1969) 194\n\nThailand 258, 267\n\nThee Experience, LA 126, 177, 190, 192\n\nThee Image, Miami 164\n\nTheis, Jeanne 329\u201330\n\nThelema 215, 221, 222\u20135, 307\u20138, 316, 327 381, 489\n\nThem 48\u20139, 50\u20132, 72\n\nThin Lizzy 334, 457\n\nThompson, Tony 449, 450, 451\n\nThorgerson, Storm 274\u20135\n\nThree Dog Night 125, 175, 178\n\nTidmarsh, Chris 32, 33, 34, 38\n\nTolinski, Brad 42, 65, 85, 213, 232\u20133, 239\n\nTolkien, J. R. R. 204, 205, 246\n\n_Top Gear_ 167\n\n_Top of the Pops_ 167, 474\n\nTopham, Anthony 'Top' 73\u20134\n\nTownshend, Pete 56, 92, 108, 178\u20139, 371, 400\n\nTraffic 98, 168, 204, 281\n\nTredoar 427\n\nTremeloes, the 50, 60\n\nTridents, the 76, 77\n\nTroubadour, LA 227\u20138\n\n_Trouser Press_ 106, 240\u20131, 336, 379, 400\u20131, 414\n\nTurd Burglars, the 393\n\n2i's Coffee Bar, London 79, 114, 264\n\nTyler, Steven 469, 476, 487\n\nUhelszki, Jaan 379, 385\n\nUIC Pavilion, Chicago 459\n\n_Uncut_ 128, 134, 185, 189, 434\u20135, 485\n\nUnited Nations: Pathways to Peace 489\u201390\n\nUniversity of Surrey, Guildford 144\n\nUpwood, Sally Anne 32\u20133\n\nUtopia 423\n\nValentine, Penny 102, 104\n\nVanilla Fudge 124, 150, 154, 156, 157, 158, 175, 186, 187\n\nVelodromo Vigorelli, Milan 252, 256\n\nVelvet Underground, the 117\u201318\n\nVentures, the 57\u20138, 254\n\nVernon, Mike 55\n\nVincent, Gene 25, 29, 30, 31, 67, 114\n\nWalker Brothers 118\n\nWall, Mick 132, 163, 197, 212, 214, 250, 389, 454, 455, 477\n\nWaller, Mickey 66, 128\n\nWallis, Chris 144\n\nWalsh, Joe 173, 297\n\nWarhol, Andy 117, 118, 287, 491\n\nWarner Brothers Records 302, 475, 476\n\nWaters, Roger 327\n\nWatts, Charlie 63, 86\n\nWatts, Michael 407\n\nWEA Records convention (1977) 389\u201390\n\nWealleans, Jon 341\u20132\n\nWeiss, Steve 153\u20134, 202, 285, 286\n\nWelch, Chris 35, 106, 162\n\nWembley Arena 447\n\nWembley Empire Pool, London 260\n\nWestwood, Vivienne 295, 325\n\nWexler, Jerry 70, 148, 149, 150\u20131, 188, 197, 291\n\nWhisky a Go Go, LA 79, 159, 161, 175, 176, 177, 256, 280\n\nWhite, Alan 433\n\nWhitehead, Peter 199, 284\n\nWhitford, Brad 460\n\nWho, the 56, 89, 91, 92\u20133, 107, 119, 133, 134, 154, 157, 164, 168, 178\u20139, 373, 400, 477\n\nWickham, Vicki 87, 88\n\nWilliams, David 23, 24, 25, 26, 27, 28, 29, 40\n\nWilliams, Robbie 494\u20135\n\nWilson, B. J. 133\n\nWinner, Michael 434\u20135, 489, 494\n\nWinterland Ballroom, San Francisco 175, 196\u20137\n\nWinwood, Steve 91, 98, 130, 281, 439, 445\n\nWizard, Bruno 237\u20138\n\nWood, Krissy 305, 315\u201316, 322, 326, 347, 425\n\nWood, Ron 128, 305, 315\u201316, 384, 395, 423\n\nWoodroffe, Jezz 436\n\nWoodstock Festival (1969) 181\n\nWrecking Crew 57\n\nWright, Joe 'Joe Jammer' 163\u20134, 183, 184\u20135, 194\u20136, 210\u201311, 388, 395\u20136\n\nWyatt, Rod 17, 21, 24, 26, 29, 31\u20132, 33, 34, 63\n\nYardbirds, the 3, 42, 63, 66, 73\u20138, 80, 87\u201397, 99\u2013100, 101, 102\u201312, 115\u201329, 130, 132, 135, 137, 138, 139, 141, 142, 143, 146, 153\u20134, 160, 163, 165, 170, 184, 195, 206, 213, 215, 224, 253, 287, 341, 363, 369, 438\u201341, 459, 460, 477, 479, 498\n\nYeats, W. B. 15, 221\n\nYes 268, 427, 433, 454\n\nYorke, Ritchie 162, 163, 171, 187, 188\u201390\n\nYoung, Charles M. 398\n\nYoung, Neil 149, 469, 474\n\nYoungbloods, the 120\n\nZephyr 156, 157\n\nZos Kia Cultus 243\n\n_Zoso: Jimmy Page by Jimmy Page_ 490\u20131\n\nZZ Top's First Annual Rompin' Stompin Barn Dance 314\u201315\n\n# About the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\nLevel 13, 201 Elizabeth Street\n\nSydney, NSW 2000, Australia\n\nwww.harpercollins.com.au\n\nCanada\n\nHarperCollins Canada\n\nBay Adelaide Centre, East Tower\n\n22 Adelaide Street West, 41st Floor\n\nToronto, ON, M5H 4E3, Canada\n\n<http:\/\/www.harpercollins.ca>\n\nIndia\n\nHarperCollins India\n\nA 75, Sector 57\n\nNoida, Uttar Pradesh 201 301, India\n\n<http:\/\/www.harpercollins.co.in>\n\nNew Zealand\n\nHarperCollins Publishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\nwww.harpercollins.co.nz\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n1 London Bridge Street\n\nLondon SE1 9GF\n\nwww.harpercollins.co.uk\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n195 Broadway\n\nNew York, NY 10007\n\nwww.harpercollins.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nTafelberg\nThe layout of this digital edition of Gang Town may differ from that of the printed version, depending on the settings on your reader. The layout displays optimally if you use the default setting on your reader. Readers can experiment with the settings to have the book displayed differently.\nIn memory of my father.\n\nYou were ever present here.\n\nand\n\nAlex la Guma,\n\nwho understood\nIntroduction\n\nBoys everywhere have a need for rituals marking their passage to manhood. If society does not provide them they will inevitably invent their own.\n\nJoseph Campbell, mythologist\n\nOne night before embarking on this book I dreamed I'd been allocated a house in a rural village. It turned out to be a single wall with an old door and dirt floor, nothing else. I spent some time cleaning the floor and, as evening fell, there was a knock on the door. I opened and a hoard of ragged, hungry-looking local children flooded in. I thought: 'I have nothing for them.' They were very sweet but rowdy, so after a while I asked them to leave, but they wouldn't. Eventually I shoved a few through the door saying: 'Go outside now.' A boy looked at me, then at the sky where the roof should be and the sides where walls should be and said: 'There's no inside.'\n\nWhen I awoke the meaning of the dream was clear. For around 30 years, on and off, I'd been highlighting the plight of young people at risk in Cape Town in books and lectures. I had co-founded an organisation, Usiko Trust, to take young men from distressed families to beautiful wilderness places and help them build resilience in the face of absent fathers, poverty, shame and the hyper-masculinity of gang life. The message from the world of dreams was that this was just a start. So far all I had was a wall with a door through which children could enter. The structure was incomplete with no roof for protection from the elements. There was still a lot to do before the building was habitable. And children in need were not people against whom I could shut the door.\n\nCape Town is my adopted city \u2013 I grew up in the Eastern Cape \u2013 but it was love at first sight. On my initial visit in the 1960s, a journalist friend said I couldn't understand the city unless I spent time in District Six. We wandered up Hanover Street late one afternoon past scruffy little shops, corner cafes, a bustling fish market and the Star, Avalon and British bioscopes. It was alive with people and swarming with beautiful children. People said hi and howzit and did we have a smoke for them? They were happy to chat, interested in why we were there and warned us to be careful of the skollies at night.\n\nFrom that moment Cape Town was for me this kind of place and these sort of people: friendly, easy going, funny and warm. That they were deemed 'coloured' under apartheid seemed irrelevant to me. Which was odd, really. I was 'white' and my sister and I were raised in a simple, racially defined family, my father a stern, honest policeman. It's not that we never questioned apartheid; we never really noticed it. It was just the way things were. For me, though, the idea of classifying people by their colour just wouldn't stick. The more different they appeared to be tonally or culturally, the more interesting I found them. So much so that, in the late 1960s, I joined a 'black' newspaper \u2013 The World \u2013 and went to live in Soweto. My mother went proudly to buy a copy to see her son in print and phoned me aghast. 'What am I going to tell my friends when they ask me where you work?' she asked, tearfully.\n\nRelocating to Cape Town, several years later, neighbourhoods like District Six, Lower Claremont, Harfield Village and Elsies River felt... comfortable. In a way I couldn't explain, these were people with whom I didn't have to put up a front and pretend anything. Their generosity of spirit was embracing. And that's when I became angry at the way they were being treated as second-class citizens, pushed out of their homes into racial ghettoes. But what incensed me even more was the effect on their children. Families were breaking under the strain, stress levels were rising, kids were forming surrogate families described as gangs and substance abuse was going through the roof.\n\nMy wife and I moved into a cottage in Harfield Village before segregation had ripped its coloured community apart and I began hanging out in the newly-formed racial ghettos to see the impact of the Group Areas Act on families. Many told me that when the time came to move, grandparents who couldn't bear to lose their community simply died of broken hearts. Through meeting a remarkable person and sometime drug merchant named Chicken, I was introduced to Bobby and Fatima April in Hanover Park. Bobby, known as Stone, was the leader of a gang that began in District Six as Bungalow 13 (B13) and later became the Mongrels. Stone realised I was genuinely interested and not just snooping. He let me spend time with the gang and was happy to answer my questions. Stone was proud of his gang and delighted that 'someone from the university' was interested in it.\n\nThere were things I found out that astounded me, but which I could never break trust and divulge. The sheer size of gang enterprise and the criminal underworld, even in the early 1980s, was astonishing. The Mongrels showed me their rituals and signs, where the drugs were coming from and which cops were on the take. In return I made myself useful taking photographs for Stone, doing gangland weddings and far too many funerals. I was in his headquarters one day when a pantechnicon offloaded bags of cannabis from special compartments welded into its understructure. The trucking line, I found out, was owned by the son of one of the apartheid government's most senior cabinet ministers at the time.\n\nMy association with the Mongrels and several other gangs resulted in a book The Brotherhoods. This was followed, several years later, by a closer look at gang rituals which became Gangs, Rituals and Rites of Passage. And here I was in 2014, back in gangland nearly two decades later. The idea was to dust off my old research, see how things had changed under democracy and bring The Brotherhoods up to date. How wrong I was. The result would be an entirely new perpective and a new criminology of deviance.\n\nIn the intervening years, my adopted city had changed almost beyond recognition. On the one hand it is now more attractive, with many of the country's best restaurants, clubs and coffee shops. It is consistently rated in the top 10 international tourist destinations and celebrities from abroad are often seen strolling along its waterfront, tanning on its beaches or enjoying themselves on elegant wine farms nearby. You can be whisked to the top of Table Mountain by a new, high-tech cable car where, after dark, the city below twinkles like scattered jewels. Many foreigners, charmed by its languid atmosphere and with strong First World currencies, have bought sea-facing homes they'd never afford back home. For holidaymakers it's easy Africa, offering the amenities of Europe or America with added sunshine and the spice of adventure.\n\nBut the city has also become increasingly violent. As the sun dips in the west, the iconic mountain casts a dark shadow across the Cape Flats and some of the most dangerous neighbourhoods in the world. This is not surprising. Cape Town is home to thousands of adolescents on the road to nowhere. In overcrowded townships the chance of death by violence is now higher than in some of the world's most volatile war zones. Here statistics trump hyperbole. In one year, between April 2014 and March 2015, there were six murders and seven attempted murders a day, 30 637 reported assaults (84 a day)1. Most of the victims and perpetrators are young men labelled under apartheid as coloured or black.2 In the absence of recreational amenities, sound schooling or, in many cases, stable family structures, young people have nothing to do but hang out in the streets, form friendship groups and fight or fuck each other. They do this in the knowledge that life is now, the future holds a frightening emptiness and, in many areas, the possibility of death before the age of 30 is a strong possibility.\n\nForeigners lounging on palm-lined Camps Bay beach gazing at the steep mountains framed by gossamer clouds would find it hard to imagine the shootouts, drug wars and human trafficking taking place a few kilometres from where they sit. Locals in the mountain suburbs are more aware of places unsafe to venture and the sort of people who look like trouble. They live in hope that 'troubles' won't spill over into their neighbourhoods. Elsewhere, especially after dark, fear stalks the streets, children are locked inside for safety and the nocturnal knock on the door is not answered. Many of these areas are under effective criminal governance and police fear to exit their patrol vans.\n\nWhen groups give themselves a collective name and undertake actions seen as anti-social, they are described by both the police and themselves as gangs, with all the freight the word implies. In South African law, gangs are illegal though pervasive and despite the law, almost unstoppable social formations. Adolescents may and do get involved in criminal acts. But to pass judgement on the perpetrators in terms of their actions alone is to miss a much deeper malaise in the city which has its roots in apartheid but has been exacerbated by the structural dissonances, neoapartheid and neoliberal tendencies of the post-1994 State.\n\nIn Cape Town today, as in the past, gang formation is the outcome of young people in search of an identity. These are youngsters whose only role models carry guns, the only smart cars belong to gang bosses and the only way to afford the accoutrements of identity is through illegal activities. Gang activity also alleviates boredom and provides an opportunity for adolescent performance under the gaze of peers.\n\nIn much adolescent gang action can be sensed a certain naive wildness, an unplanned theatricality, which seems to place more value on ritualistic performance than on the apparent goals of the action. There are initiations, dares and improbable tasks. To understand this we have to remember something important about our own adolescence: young teenagers, above all else, are mythmakers. They create and recreate situations and whole webs of significance little understood by the pragmatic adult world. They need to perform to win respect and the wilder the performance, the greater that respect.\n\nIn tough neighbourhoods, the performance can become extremely dangerous. On July 29, 2012, about 50 mostly school-going members of the Vuras gang in Site B, Khayelitsha, swept into an area in the same township known as Green Point, attacking anyone on site with pangas, axes, knives and guns.3 They claimed to be hunting for the Vatos gang. It was part of an ongoing battle in which four members of the gangs had been killed over the previous 10 days.\n\nThe Vuras kicked down doors searching for their 'enemy' and shouted that schools would be closed the next day. Pupils told a newspaper they feared going to school because there were 'always gang fights and we are attacked with pangas on the way home after school'.\n\nPolice said they were monitoring the situation, but made no arrests. The only motive for the attack was revenge for a similar raid on the gang's home territory, Site B, by the Vatos. The purpose of the gang, in this case, was defence of both territory and honour.\n\nThere are, however, groups of a different stripe also defined as gangs. The ruthlessness of just one of these was outlined in the National Prosecuting Authority's case in 2012 against George 'Geweld' Thomas, a 'shooter' and head of a street faction of the 28s prison gang in Bishop Lavis. Thomas was imprisoned in 2008, awaiting trial with 18 others for murder, attempted murder, housebreaking, theft, drug dealing, unlawful possession of firearms and ammunition, intimidation and incitement to commit murder. The gang's alleged focus was smuggling abalone and drugs.\n\nAccording to the NPA, in the five and a half years preceding Thomas's trial, which began in 2014, 21 people were killed at his behest, 12 of them state witnesses who agreed to testify against him. The 'hits' \u2013 many of them on a rival gang, the Clever Kids \u2013 were allegedly directed from his prison cell. From there, according to evidence and in defiance of prison rules on cellphone access, he made thousands of calls. Other deaths were inflicted upon members of Thomas's own gang. These men were killed, it was alleged, to ensure their silence. According to a witness, in one of the hits before his arrest, Thomas sat in the back of an open truck with a rifle fitted with a telescopic sight and silencer while two henchmen in the front handed him ammunition as he fired at 'enemies'. Prosecutors and investigating officers handling his trial were assigned bodyguards after receiving threats. Thomas, who had spent 21 of his 44 years in prison, told journalist Caryn Dolly he joined the 28 prison gang for protection but wanted to be a pastor.4 The presiding judge, Chantel Fortuin, said 'the awful conditions in which he and 16 fellow accused lived before being imprisoned could not be used as an excuse for murder' and noted that he used prison as his headquarters in a frenzy of killing. She handed Thomas seven life sentences.\n\nBeyond the battle over drugs, turf and honour are other groups that could also be classified as gangs. These are networks spread throughout Cape Town which engage in a range of legal and illegal activities. Some are quick to use violence to ensure the smooth running of their businesses; others work by bribery, stealth and complex financial footwork. These include big gangs such as the Hard Livings and Americans, groups of foreign nationals and covert associations at many levels in the corporate and state sectors. Their connections reach from the streets of the city to the rest of the world and their activities range from money laundering, protection and touting drugs to global trafficking of drugs, animal products, reptiles, plants and humans.\n\nMy early work on gangs in South Africa looked at the way in which poverty and apartheid's massive social engineering created stresses to which gangs were a teenage response.5 This view is captured by American sociologist Sarah van Gelder:\n\nThe result of this uprooting and neglect is that the solid core of contributing adult members crumbles, and the institutions that provide the foundations of community fall apart. The community safety net is left in tatters. Parents, exhausted by long hours required to make ends meet or demoralised by their inability to cope with the hardships of poverty, may turn to drugs and alcohol. Kids are left on their own in... adultless communities.\n\nBut when I began researching for this book, apartheid had officially ended and the country had a world-class Constitution and Bill of Rights, so why had the gang phenomenon continued to escalate at a frightening rate? Had social institutions not been rebuilt? Was the community safety net still in tatters? Were kids still out on the streets and left to their own resources? What sort of city had Cape Town become that wealth, luxury and beauty existed a few suburbs away from daily murder, assault, rape and mayhem? This book is my attempt to answer these questions and trace some paths towards a solution.\n\nThe book is divided into six parts that can be read separately, depending on your interest or requirements.\n\nPart 1, Gang Town, explores the physical reconstitution of the city and the social impact of racial segregation which led to single-race ghettoes. It then looks at the city's attempts to deal with its racial inheritance after democratic elections in 1994 saw the end of racism. In this section I investigate issues of governance and control in working class areas pre- and post-transition and frame the socio-political environment within which gangs arose.\n\nPart 2, Cape Town's Gangs, is a tour of the various types of gangs found in the city and begins with the essential question of what we mean by the term 'gang'. While for most casual observers and, particularly, the media, gangs are seen merely as groups of dangerous young men involved in drugs and crime, a closer look reveals an ever-shifting kaleidoscope of forms and functions ranging from corner play groups through street warriors and merchgant gangs to fierce prison numbers, corporate racketeers and transnational networks.\n\nPart 3, Understanding Adolescence, seeks to answer two questions: given that most risk taking and deviant behaviour throughout the world is massively heightened in the years from puberty to early 20s, what exactly is adolescence? A related question \u2013 and central to this book \u2013 is why it is that while for most adolescents, socially deviant behaviour is limited to teen years, some young people get persistently deeper and deeper into trouble from which they find it almost impossible to extract themselves.\n\nPart 4, Families in Crisis, explores the familial roots of persistent deviance in the impact of fatherlessness, prenatal epigenetic damage and the problems that compound when affirmation and love are missing. It suggests the disturbing possibility that adverse conditions during pregnancy act as a signal to an unborn child which genetically code for more aggressive adulthood.\n\nPart 5, Toxic Neighbourhoods, explores the societal roots of persistent deviance where poverty, joblessness, a failing education system, risky neighbourhoods and high levels of drug availability make gang membership just about the only option available for income and the context within which to strut your stuff among peers.\n\nPart 6, Towards Resilience, is a complete about-turn on the problem of gangs. On the basis of findings in the previous five sections, it proposes a rethink on early child development, education, after-school care, the war on drugs, policing, imprisonment and the way we work with adolescents. It lays out alternative life-path planning for young people at risk through a compassionate understanding of personal transformation.\n\nThe Appendix is a toolbox of things I've found to be useful game-changers for parents, teachers and community workers when dealing with high-risk adolescents.\n\nEndnotes\n\n1. In that year 6 643 motor vehicles were reported stolen (18 a day).\n\n2. There is no such thing, in biological or physical terms, as 'race' or 'race groups'. What we understand as 'race' is the product of centuries of ideological social construction. The effects of this, however, are real enough and are embedded, even today in 'non-racial' South Africa, in statistics and countless official forms. 'Race' should be read only as a proxy for other real social processes of marginalisation, opression, exploitation and exclusion. A danger is that when we deploy racial categories, we imagine that the category itself is the causal agent. For this reason, in his article Gender, Class, 'Race' and Violence, Don Foster suggests that it is therefore prudent to speak of 'racialised' categories or groups (in Youth Violence: Sources and Solutions in South Africa by CL Ward, A Van der Merwe and A Dawes (eds) (UCT Press 2012).\n\n3. Cape Times, July 30 2012, p1.\n\n4. Dolly, Caryn: Cape Times, May 24 2011.\n\n5. Pinnock, Don: The Brotherhoods (David Philip Publishers, Cape Town, 1984).\n1. Gang Town\n\nThere were gang fights at school. Also school wasn't nice because when you come out of the grounds there are guys waiting to beat you. The whole day my mind's on how I'm going to get home past them. I would run away at second break because I knew they were expecting me when school ends. So to get to school safely I joined the Americans. And the gang was interesting; it showed me things I would never know about otherwise. Also this place is crazy. If you stand on a corner or anywhere, guys can come and take your stuff and do with you what they like. That's why you have to join a gang. You have to join to survive.\n\nI live with my mother \u2013 in the yard \u2013 and with my girlfriend. My father went to prison again maybe four years ago. When I was growing up my father was most of the time in prison. Maybe because of that, things got to where they are today. If I'd known then what I know now, I would never have been what am I.\n\nI have a daughter aged seven but I don't see her because she's not living with me but with her mother. Sometimes I take my bike and go down there. But for only a short time because I don't like to go without money. You know children \u2013 they want money.\n\nI was in Porter Reformatory. I learned a lot there \u2013 it's where I learned tik. And I met people from other places. Survival depends on what you know and who you know. Then I went to prison for murder. I just thank God that I'm still alive.\n\nThere have been a lot of murders in this place. Many times I'm scared for my life. My friend and me were smoking nice together and ten minutes later he was dead. When you come out your house you don't know a gang has had trouble with your friend and so you walk out. Then bang, they shoot you. You need the network of your brothers to stay alive. You hear something is going to happen so you go home and get your gun. And you're ready.\n\nWhen a guy dies his whole family is crying and you see the heartbreak. It's not nice. And he's not even 20. Me, I'm 35 and all the other guys are younger. It's hard to live long. Yo! Four times I've been shot. One bullet shattered the bone in my leg.\n\nIn this place you can't be dik gerook. You must be wide awake to stay alive because here is full of possibilities and danger. And when it's dark! Oooh, all hell breaks loose. But if you stand on a corner and look like a billionaire, you can't tell me nothing. I can make the streets hot for you. I just go fetch a gun. You'll be nothing. Woman and children will be running. Guns give you respect. It makes people afraid and now they want to be your friend. And you know why they are your friend. For now. Some people are scared, yes. I don't want them to be scared. But people must listen to me. If I tell someone don't do that, they mustn't do it.\n\nI have to be on the lookout for those who want to take my life. I've been doing the same thing to other people. I have a lot of friends who are not living. I'm one of the only survivors at my age. I'm the leader of the Americans in my area. In prison it's better for me. Inside I get anything. They say the Numbers don't pay, but me, I get my salary. But I can't stop. I have no choice.\n\nMaybe I could resign from the gang. But after a few months they will kill me. Why? Maybe for one year I'm not with the Americans. But then another gang shoots me because it will still hurt the Americans. Maybe I have a brother in the gang, a friend in the gang, and when they hurt me they gonna hurt them. So there's no way out.\n\nThere's a battle beginning outside! (Single and automatic volleys fired. People running in every direction). Okay I'm going now, be careful out there. Goodbye...\n\nTyrone, gang leader, Hanover Park, 2015\n\nSowing the seeds of segregation\n\nCape Town today is very much an African city, despite the modern fa\u00e7ade of shuttered concrete and glass in the commercial centre and the urbanity of its affluent. Geographically, it's characterised by a mountain massif that forms a peninsula and is roughly the division point between the Atlantic and Indian oceans. Clustered around the wooded slopes, along the mountain seaboard and beside the transport arteries that fan outwards are the residences and offices of the wealthy. Most have property, capital and access to First World technology and skills and live in one of the most scenic cities in the world. They're largely descendants of 300 years of European colonisation together with the new black political or corporate elite and their dominant position is protected by law and custom. In terms of the lives they lead, they have collective efficacy and are organised.\n\nBetween the peninsula and the next range of mountains to the east, a distance of some 30 kilometres, is an area known as the Cape Flats. This is a low-lying, sandy and wind-swept tableland that was once seabed. It is to this area that people classified by the apartheid government as coloured and African were relocated by massive population removals from the inner city in the second half of the 20th century. Here one finds overcrowding, poverty and squatters and it is here that the affluent generally fear to go. Most people living in Cape Flats neighbourhoods do not own property or capital and the majority are unskilled or semi-skilled labourers. Apart from schools, churches and a few community organisations, these areas have few amenities and are largely socially disorganised.\n\nBehind this separation of people of different skin tones, languages and classes is an older dimension rooted in conquest. Beyond outright armed suppression, the aim of any conqueror is to disorganise the enemy. This is because agreement around principles, goals and identity among the conquered is almost always hostile to the victor and carry the seeds of future rebellion. This was as true for the Roman Empire as it was in Western conquest of Africa and the Americas. It remains true for working and workless classes in a modern state.\n\nIn order to understand how conditions became fertile for the widespread development of gangs, it's necessary to trace these threads of conquest, organisation and disorganisation in the tapestry of the city's development. This chapter is, therefore, a very particular history, not of a city entire but that part of it which gave rise to a specific situation. It's also a history of an urban working class and culture within which gangs initially arose and which the architects of apartheid chose to define as coloured.\n\n***\n\nUrbanisation and unemployment are bound together with hoops of steel. People who for centuries may have subsisted by farming move to towns because they lose their rights, their stock, their land, their jobs on farms, or simply because poverty in the city holds out more chances of survival than poverty in the countryside. From as early as the mid-19th century, Khoisan and freed slaves \u2013 both individuals and whole families \u2013 had begun moving off the land into the villages, larger towns and cities. This process intensified in the early 20th century as commercial farming consolidated earlier land grabs and swept aside traditional land tenure.\n\nIn Cape Town migrants found slums, overcrowded and with relatively expensive housing, and a lack of formal employment. But they measured their progress from where they had begun and conditions in the rural areas, with land loss, harsh European bosses, droughts and lay-offs were considered far worse by many.\n\nIn the ghettos a pattern of existence emerged that was to continue until the 1970s. Like the Afrikaners in early Johannesburg, the Cape rural families were not immediately absorbed into the city workforce. Unlike the Afrikaners, people labelled 'coloured' did not develop the political clout to ensure an increasing share of the expanding profits of urban industrialisation. The outcome was what may be termed a ghetto culture, linked to the city on all sides and penetrated by it, yet different from it.\n\nThe central organising force of this urban ghetto development was the extended family. People moving into the city sought out their kinsfolk as beacons of support in the new, hostile environment. With them and through them, they found accommodation and, later, employment. Usually this employment was in the crevices of economic activity: in an extension of regular household duties and in micro-commerce and hawking \u2013 all with extremely limited access to capital.\n\nThe redistribution of wealth this penny-capitalism ensured allowed the immigrants to survive amid tough conditions. For many, unskilled as they were in urban occupations, these activities were often of a kind considered illegal by the state, though in the urban explosion they flourished. But it would be incorrect to consider, as the middle class of the city did, that the poor quarters were lawless in an anarchical sense. In the urban culture which emerged \u2013 based on extended families, on a sense of place and the shared problems of poverty \u2013 the working class ordered and policed itself.\n\nAn example of this mix of family support and informal control is that which existed in the inner-city area of District Six. It's also a place where a gang legend was born.\n\nDistrict Six\n\nThe place has more barber shops to the acre than anywhere else in Africa, some of them with great-sounding names like the Rio Grande Hairdressers. There are all sorts of alleys and lanes with names like Rotten Row, Drury Lane and Lavender Hill. There are tailors by the score, herbalists, butchers, grocers, tattoo-artists, cinemas, bars, hotels, a public bath-house, rows of quaint little houses with names like 'Buzz Off' and 'Wy Wurry' and there is a magnificent range of spice smells from the curry shops. The vitality and variety in the place seem endless and the good-humour of the people inexhaustible. Anything could happen and everyone in the end would laugh about it.\n\nGo into one of the fruit and vegetable shops and you soon realise how the very poor manage to live. In these shops people can still buy something useful for 1d. They can buy one potato if that is all they can afford at the moment, or one cigarette. You can hear them ask for an olap patiselli (a penny's worth of parsley), a tikkie tamaties or a tikkie swartbekkies (black-eyed beans), a sixpence soup-greens, an olap knofelok (garlic) or olap broos, which means a penny's worth of bruised fruit.'1\n\nBrian Barrow, journalist, written in 1966\n\nToday District Six is a place of tussocky grass and ghosts. Half a century after its destruction, only a technicon and a few houses have been erected on the empty space. It's an urban hot potato \u2013 valuable inner-city land that nobody feels comfortable to re-occupy. Occasionally I drive there and park in fields where homes once stood to watch the full moon rise over the Cape Flats. I always leave feeling melancholy.\n\nThe area, on the lower slopes of Table Mountain, was originally farmland and first settled by Europeans attached to the Dutch East India Company. Then, in the early 19th century, it expanded rapidly when Cape Town's growing middle class began to build modest homes for themselves within easy reach of the central area. The wealthier merchants and officials already had houses closer to the city on land which their clerks and assistants could not afford. And so, on the outskirts of town, a middle-income community began to grow in District Six.\n\nThe houses of the new middle class were unpretentious, generally two-storied and built as terraces in a style typical of the Cape under Victorian and Georgian rule. Narrow blocks were laid out parallel to Hanover Street and small, semi-detached houses with long service lanes were built. Later, in the 1880s, skilled European artisans \u2013 drawn to South Africa by the mining boom after the discovery of gold \u2013 began moving into Cape Town and settled in the District.\n\nFollowing the outbreak of the Anglo-Boer War, Cape Town's population was swollen by an influx of troops and refugees from the Transvaal. Much building activity took place in District Six during the war and two-and three-storied blocks in a variety of architectural styles began to appear. At this time most of the properties in the area were owned by descendants of the European settlers and a few by Asians.\n\nNo homes, however, were provided specifically for workers and the limited houses available to them were filled to overcrowding, many being forced to squat on whatever land was available. After the war, a large number of businesses and offices were transferred back to the Transvaal. The houses in District Six were vacated (but not transferred out of European hands) as tradesmen, artisans and soldiers moved north. Through a filtering-down process, working-class families moved in. By leap-frog movements of population, the middle-income Europeans shifted out, first to Woodstock, then to Vredehoek, Observatory, Mowbray and beyond.\n\nInitially, working-class migration into the city was circular, under\u00adtaken mainly by job-seekers. As the transition from a farming economy to an industrial one gathered pace, it became a one-way flow of whole families. By the 1920s Cape Town's administrators were describing the march of the poor into Cape Town as 'formidable'. As far back as 1867, District Six was considered overcrowded and for the next hundred years migration into the area continued almost unabated.\n\nIn 1936 the official census put the population of District Six at 22 440 and in 1946 at 28 377. Four years later the figure was around 40 000. In 1950 the Housing Supervisor of the Cape Town Municipality told the Cape Times:\n\nAlmost every house in the district where the Coloured people live is packed tight. Children grow up and marry and in turn have children and are unable to find a place of their own. A family is turned out of an overcrowded house and finds a shelter with friends for a few days \u2013 which grow into weeks, months, years. They sleep in living-rooms, in kitchens, in passages, in garages, on stoeps; married couples share rooms with other married couples... Waiting-lists for accommodation grow longer and longer... Families wait anything from six months to 10 years before they can be re-housed.2\n\nMany families in the area were extremely poor, living for generations by working at odd jobs here and there, scratching out an existence by forms of economic enterprise which counted profits in halfpennies and farthings. Throughout the migrations into Cape Town, it was always the extended family that formed the catch-net of the urban poor. Within it were people who could be trusted implicitly and would give assistance willingly, immediately and without counting cost. In major calamities, like the loss of a job or a death in the family, it was kinsfolk who rallied to support and whose support lasted longest. Kin also helped find employment, accommodation, and bribed or bailed you out of the clutches of the law. They were, in short, indispensable.3 In a hostile and uncaring world extended families provided a refuge and a domain within which strategies of survival could be worked out.\n\nEssential to the survival of the family, of course, were the wage packets brought into it. Like most unskilled earners in the Third World, workers in District Six were paid an extremely low wage, which had to be conserved and stretched. The poor responded to this situation in typical fashion, organising systems of redistribution that helped extend meagre incomes to the limits of their elasticity. These patterns of redistribution percolated money through networks which finally found its way into the pockets of those who were unable to obtain wage employment. It was, above all, a social form of redistribution, operating among friends, neighbours, workmates, acquaintances and friends of friends.4\n\nIn District Six, the extended family was the ground-floor of these small-scale economic activities. In 1937 a Commission of Inquiry found that 'the entire Cape Coloured family in the urban areas very often forms the earning-unit, the income of the parents and one or more of the children being pooled to meet household needs.'5 The area became known for the ingenuity, novelty and enterprise of its residents. By day it hummed with trade, barter and manufacture and, by night, it offered the 'various pleasures of conviviality or forgetfulness'.\n\nNot only did extended families provide the possibilities of a roof and an income, they also acted as networks of social control. Powerful families 'ordered' the urban areas through their connections, inter-marriages, agreements, 'respect' and, ultimately, their access to violence. Because effective police protection was lacking, this control was beneficial, even essential, to life. It kept things 'safe'.\n\nFigure 1 Income Opportunities in the Informal Sector in District Six, Cape Town, in the 1950s\n\nBut poverty exacts a price. During World War Two, the street corners seemed to fill up overnight and the sight of people or even whole families sleeping on stair-landings and in doorways became common. Pressure began to build up over territory for hawking, shebeening, prostitution or just for standing space. Youths from the 'outside' began hanging together, with empty stomachs and nothing much to do. They started hustling, picking up this and that from shops, leaning on a few people for cash or favours and living by shifts and ruses of all kinds. Police methods of dealing with these groups were simple, direct and ineffective: 'We would pick them up and fine them and they could be hired out for some work while under sentence, usually to farms,' a former policeman explained. 'These kinds of people were just idle loiterers who took part in illegal activities now and then.'6\n\nThe name given to these groups of youths was 'skollies', a word that probably comes from the old Dutch schoelje, meaning 'scavenger' or 'scoundrel'. Dutch sailors, so the tradition goes, shouted schoelje at the seagulls which swooped to snatch up ship kitchen detrius from the waters of Table Bay. The name came to be used for vagrants who picked at city refuse dumps or begged on the streets and, generally, for troublemakers. Skollies were considered by residents of District Six to be people from 'outside' the area. Many of these youths had prison experience or had spent time at Porter Reformatory, a youth detention centre (now the Chrysalis Academy).\n\nA Commission of Inquiry, looking at the period from 1928 to 1935, found the incidence of juvenile delinquency among people classified coloured to be 'very high'. There was an average of 2 600 such youths in prison for each of those years in South Africa, most of them in Cape Town.7 Until the 1940s 'scoundrel' youths had been a presence but not a serious problem for the community. They operated alone or in groups of two or three, though not in gangs. But things began to change. According to a policeman who worked in District Six in the 1930s:\n\nBefore the War there weren't actual gangs consisting of youngsters and so on, it was just individuals or a cluster of blokes together. But soon, due to their idleness and way of life, gangs started to form. Shebeens and gambling-houses started developing and these small gangs started robbing honest hard-working people. In this way a lot of decent people were just about forced to become gangsters for protection too.\n\nWhen the people's involvement with the Cape Corps ended in 1946, gangs started and crime escalated. This was because there were very few jobs available after the War. The gangs sprang up all over the place and they all had their different territories. Before, as policemen, we never wore firearms and were posted one by one on foot. But later when you arrested someone, you had to fight with him from the time you picked him up until you reached the police station. It was very dangerous alone; I mean they'd pelt you with stones or anything they could get.8\n\nIt was at about this time that names such as Red Cats, Jesters, Goofies and Kettang Gang found their way into the streets of District Six and the pages of newspapers. The time of organised street gangs had arrived. Two groups viewed these developments in the District with alarm. One was the police. In 1946 a Special Squad with wide powers of arrest was set up expressly to deal with skollies in District Six. It was led by a tough up-country policeman, Sergeant Willem Nel and, he told me, 'I really worked them'. Police pressure tended to be indiscriminate, arresting 'family' members and 'outsiders' alike, and this raised a howl of protest from 'respectable people' in the District. The Torch, the mouthpiece of the Non-European Unity Movement, was shrill in its views on the matter:\n\nThe police in this country, in the sadistic fury and depraved bestiality which characterises their manhandling of Non-Europeans, innocent or guilty, can be compared only with Nazi storm-troopers who terrorised Germany and occupied Europe. Invariably they are recruited from the poorest layers of the Herrenvolk, ruined peasants, bywooners and poor whites; the semi-literacy and general coarseness of the majority of them is notorious.\n\nWith a grudge against the world which has forced them to become policemen in a country where possession of a white skin is an 'open sesame' to almost any walk of life, they have a particular grudge against all Non-Europeans. For they regard the Non-European as being responsible for their economic ruination and the educated Non-Europeans as someone who has risen at their expense.\n\nMuch has been made of the crime wave and of the increase in crimes of violence. But nothing has been heard from the press of crimes of violence which the police commit every day against the Non-European.\n\nIt is common occurrence to find the pick-up vans scouting around bars and other areas where Non-Europeans live, deliberately looking for trouble with the people. Any Non-European who smells of liquor, or who looks as if he may smell of liquor, or who merely looks, is fair game for the police raiders. Every day brings fresh evidence of the mauling to which these innocent people are subjected in the cells. But they are terrorised and are too poor to involve themselves in legal action... They suffer in silence. But their resentment smoulders and the day will come when there will be serious consequences.9\n\nThe other group troubled by the gangs comprised people from the 'old' organised families, often shopkeepers, skilled craftsmen and better-off hawkers, whose security was being threatened. At night the sons of some of these shopkeepers used to congregate under a streetlight alongside the Globe Furnishing Company in Hanover Street opposite the Star Bioscope, watching the abundant street-life of the District. Their first action as a group in the early 1940s seems to have been to smash an informal tax-racket at the cinema. A group of ruffians would stand at the door of the Star, extracting a penny 'tax' from every patron. One night the Globe group, mostly Asians from the Muir Street area, gathered together their fathers' workers and barrow-boys, armed them with sticks and implements from a nearby stable and thrashed the cinema toughs. Thus began the remarkable rise to power of the Globe Gang, although in its early stages it could be better described as a vigilante group.\n\nAmong the gang leaders were bricklayers, hawkers and painters. Its chief, Mikey Ismail, was a plasterer. At its centre was the Ismail family, one of whom, A. Ismail, was a City Councillor. Several of his brothers controlled the District's morning vegetable market, one ran a bus service and four had general-dealer shops. It was the sons of these entrepreneurs, particularly Mikey, who built the gang around themselves. 'They were not criminals,' according to a tailor who made their clothes. 'They started to control the Jesters of Constitution Street, who were beginning to maak soos hulle wil [do what they like]. Their aim was eventually to break all gangs, to clean up the District.' A member of the gang told me:\n\nThe Globe hated the skollie element in town, like the people who robbed the crowds on celebrations or when there were those marches in town with the Torch Commando or Cissy Gool's singsong [demonstration] outside Parliament buildings. Mikey and the boys would really bomb out the skollie element when they robbed the people then. They tore them to ribbons.10\n\nIn the press, confusion began to grow around the Globe, and with it an inclination to describe this informal policing simply as 'crime'. Although police spokesmen insisted that organised crime in the District could not be ascribed to gangs, the press began to fuel a 'moral panic', insisting that there was a 'crime wave' and a 'reign of terror by killer gangs'. According to former resident George Manuel, the people in District Six were under no illusion about the differences between the forces:\n\nThe skollie is a rubbish in the gangster's mind. I remember once a judge told a gangster, 'You're a rubbish, you're a low class, you're a skollie!', and the gangster said, 'Ekskies, Oubaas. Ek is nie 'n skollie nie. Moenie my insult nie.'[Excuse me old master. I'm not a skollie. Don't insult me]11\n\nA close associate of the Globe at the time, Vincent Kolbe, said the distinction lay in the fact that other gangsters didn't care a thing for their families:\n\nThe Globe ... respected each other and their families and so on. There were only a few who smoked pot and really got gesuip [drunk], but never the top dogs. They always tried to do things that wouldn't bring a scratch to their good family name. You know all these people I'm talking about are wealthy businessmen today \u2013 except, of course, Mikey is dead now. The Globe were the most decent and well-bred guys ever. All their parents were well-to-do businessmen with flashy cars and good clothes. The leaders were always beautifully dressed. Mikey had silk shirts specially made for him. And he drove around in lovely cars. And the women! Mikey always had the best women around him.12\n\nThe Globe's connection with the police was one of wary mutual assistance. The leader of the Special Squad, Willem Nel, considered them to be 'very decent blokes.' Globe members would visit his house socially or to ask advice or favours. The police, in turn, would request gang control of the skollies. According to Kolbe:\n\nThe police never busted the Globe Gang. They used to call on Mikey and the boys if they were having some trouble with another gang somewhere else, because they knew the Globe would be able to control the scene \u2013 especially with gangs like the Jesters or the Goofies. Mikey was so well known to the police and so respected, that he could stop a police van and release anybody he wanted to from it. Him and the boys helped the police. If they heard of a robbery somewhere, like at a liquor store, they'd take the liquor away from the skollies who stole it, and then invite the police around. They nearly got the whole of Caledon Square [police station] drunk that way one night.13\n\nWhen the Globe confronted the gangs, however, it was always violent and often bloody:\n\nOn Saturday, 19 December 1951, gang war burst out in District Six. For weeks before, tension had been rising in the District. When the Globe and Killer armies confronted each other in Hanover Street, shopkeepers closed their stores, shebeen-queens stopped serving liquor, prostitutes abandoned their beats and respectable citizens bolted their doors and barred their windows. In a running battle the two gangs, with the combined strength of some 300 members, swept through the District, firing stolen pistols and leaving a trail of destruction. The showdown came in a house belonging to one of the gang members, which was taken apart in the process. The police were notably absent.14\n\nHowever, the Globe soon began to control more than just the gangs. Any group in its territory had, quite literally, to pay allegiance. And the Globe was up for hire as a political hit-force as well. Radical teacher organisations and the Communist Party were occasionally roughed up. An unpopular teacher would be 'told to go or we'd slaughter him', and members of the Communist Party would be 'visited' by Globe members, who would pretend to be drunk and 'empty the fish bowl, drink all the wine and beer and those that didn't sip would just pour it on the floor. And they'd say, \"Come on, we're the people and you're supposed to be Communist. Play us music and drink with us.\"15 City Councillors found their meetings broken up by the Globe which had been hired by their electoral opponents. Political parties were not exempt either. In 1951 a Nationalist MP complained in Parliament:\n\nOn the evening I joined the Nationalist Party, I held a meeting in the Cathedral Hall here [in Cape Town]. The UP [United Party] took the Globe Gang from District Six in taxis and before they could go into the meeting each one received about half a glass of brandy and a 10s note, and he went into the meeting with a razor blade.16\n\nMikey's brother, 'God', was only peripherally involved with the group and was not interested in the family trade. 'Haas [quick] money', he told people, 'was quicker than honest money' and his line was smuggling and blackmail. His group of toughs operated with the blessing of the Globe, which increasingly seems to have assisted 'God' in his operations. Certainly, while Mikey was in jail for two years on what members claimed to be a trumped-up charge of manslaughter, 'God' extended his influence over the 'boys'. Increasingly, Globe members sought to boost their income by whatever means at their disposal. At the height of the gang's influence these means were considerable.\n\nBy 1950 the Globe was controlling extortion, blackmail, illicit buying of every kind, smuggling, shebeens, gambling and political movements in the District. Mikey's image had gradually shifted from 'keeper of the peace' to that of Robin Hood and gang members were taking in large, weekly doses of American gangland experience at the cinemas in the District. In 1949 the press noted that '1000lb of dagga in the last four months has been traced to large traffickers in District Six where it is controlled by well-organised big-time skollies in well-organised gangs. These are usually smartly dressed and their families are usually well-to-do.'17 In 1950 the newspaper was to elaborate on the cannabis network:\n\nAround dagga has grown up an infamous traffic, earning its principals many thousands of pounds a year. It is believed to be brought into Cape Town by Natives from the Transvaal and the Protectorates. In Cape Town the dagga is taken over by wholesalers and then distributed by countless 'runners', mostly in the form of kaartjies, which cost only sixpence.\n\nCollecting 'protection' money was also a natural outcome of the Globe's original function. The line between 'protection' and straight extortion was a fine one. 'Shops, clubs, cinemas, Indians, Jews, they all came to Mikey', remembered a gang member. But the truth of the matter was that Mikey came to them, as a letter to the newspaper from a shopkeeper in 1952 testifies:\n\nI came to this country from Australia a little more than a year ago; I started my Hanover Street shop almost immediately. I had not been there two weeks when two well-dressed young, coloured men walked in and asked me to join their 'Shopkeepers Protection Racket.' They said it would ensure that my shop would never be robbed or, if goods were stolen, they would be returned. All this, I was told, would be done for a 'small fee'. This trifling sum turned out to be \u00a31 a week. I ordered them out. A week later my shop was entered at night and ransacked. I had had several thefts and attempted robberies since then. [The gangsters had worn] black hats and brown suits.18\n\nA member of the gang named Gums elaborated on this system:\n\nThe Globe really made some businessmen, because they protected them and did their work for a certain price... If somebody approached Mikey and said he'd give him \u00a315 to bust some shop owner or whatever, Mikey would first ask for more money, say \u00a325, then he would tell the shop owner. So the shop owner would pay Mikey not to bomb him out. And so the Globe Gang were rich. Sometimes they'd even bandage the people up themselves and take them to hospital just to satisfy the guys who wanted somebody to be fucked up. Then the Globe would get paid even more.19\n\nThe gang also organised or 'protected' shebeens and gambling dens.\n\nSlowly, however, its influence began to wane. There are a number of reasons for this. The leader, Mikey, was killed \u2013 stabbed with a kitchen knife by the brother of a girl who thought he was molesting her. 'God' was jailed for blackmail. As the gang's rackets increased, it also lost the support of the class which had given it birth. Gradually prison elements infiltrated the Globe, making it indistinguishable from the street gangs around it, apart from size. Vincent Kolbe describes the process:\n\n'Slowly there came the skollie element. A guy from Porter Reformatory joined them: Chicken. Then prisoners from up-country who'd never been in the cities. They raped and had tattoos on their faces and necks and killed anybody, for nothing. Young boys arrived, and carried guns for no reason. More gangs were formed, like the Bun Boys, the Stalag 17, the Doolans, the Mongrels, the Born Frees. These types were really just a jail element: snot-nosed young boys. Then one day somebody interfered with a gang in the District and this gang thought it was the Globe but it wasn't. They attacked us and this set off the most terrible war. People were killed and the Globe decided to bust every gang everywhere. They couldn't stop. And that was the start of the Globe's bad name.20\n\nFor the police, the Globe's early vigilante activities passed into memory as other more ruthless gangs began to take over. Yet the Cape Town Police Chief, reviewing the 1950s, considered District Six during that time to have been 'quiet'. For the residents of District Six, however, bigger problems were up ahead.\n\nUnscrambling the egg\n\nMy daddy came from Malmesbury during the war. He was in the Cape Corps, so we came to live in District Six. Everything was going there! Hundreds of shops up Hanover Street and flower-sellers, tailors, coal merchants, wood-sellers and so many hawkers. And barber shops. Where can you get a good barber shop today? Or a shoemaker?\n\nI went to work in a factory after I got married. We needed money. Lots of women worked in factories all along Sir Lowry Road \u2013 there was Buchanan's sweets and Fairweather clothing and Baumann's Biscuits and the fez factory. First I worked at Cork and Crown as a machine operator, doing man's work on the oak shive, making fishing-net cork floats. Then I did beer-bottle tops. There was more money in factories than in service, though lots of women did washing at the washhouse in Hanover Street. Then I went to Fairweather. I was sewing shoulder-pads on men's suits. Everything by hand. There were thousands of girls in that place, with supervisors to help see we didn't talk.\n\nIt was hard work \u2013 especially with that production business. You see, if you made five garments an hour normally, and then you made ten, they would pay you more. Towards Christmas you would work, work to get more money \u2013 the whole day Saturday and half of Sunday too. It was tiring \u2013 but every penny counts. There were big families and they could look after the children. Today these are no more and women must give up work for the children. Or they just leave the children.\n\nThe cutters were always men. They had been tailors usually. There were plenty of tailors before, but then they became less. You would buy your own cloth and they would make a suit. Also with the Coon Carnival... every week you would give a bit of money to the tailor and when you need your suit for the Carnival, it's paid for. The Carnival is how the tailors could live.\n\nMost of the women who worked in the District were in factories. It cost me 8c to take a bus from Fairweather to home. Now think how much it costs from Mitchell's Plain!21\n\nElizabeth Weeder, a garment worker from District Six\n\nA solution was clearly needed for the overcrowding and inadequate housing of the poor in Cape Town. But what would follow was unthinkable to the people of District Six and other working-class areas. Indeed, it was not believed possible even after it had been decreed. The heart of the city was to be torn out as a sacrifice to 'order, cleanliness and progress' and the families who had helped build it were to be scattered in disorder across the sands of the Cape Flats.\n\nSo vast was the scale of the social engineering undertaken that it is difficult to grasp it in any coherent way. The most I can do is to sketch some of the more important forces which led to the passing of the legislation that effected this massive social upheaval, the physical and emotional outcome of which still resonates through modern Cape Town.\n\nWhen the soldiers came back from war in 1945 they were to find a country transformed by an industrial boom of unprecedented proportions. The confident mood of the period was captured by historian A. J. Norval:\n\nWhen the Second World War broke out in 1939, South Africa was by no means equipped to meet the exacting demands of the civilian requirements, let alone the requirements of participating in a global war. The Government did not wait to consider the costs. Factory buildings were built in the shortest time and plants installed to manufacture war equipment and ammunition for the forces leaving for the front; large contracts on the most remunerative basis were entered into with private manufacturers for clothing, footwear and food.\n\nManufacturers in general were encouraged to manufacture whatever possible to make the country independent of outside sources should the war drag on for many years, as was anticipated, and should the world supply-routes be cut off. No efforts were spared, nor funds for the equipment of industries... The remarkable phenomenon was that there was no lack of capital...\n\nA new era had dawned in the industrial progress of South Africa. The country went from a labour-intensive to capital-intensive basis of its manufacturing industries. Over the next 20 years a complete transformation took place in the plant set-up of secondary industry, which was revolutionary in scope.22\n\nBut if the war was good for business, it was also a painful forcing-house for the state. Under pressure from the demands generated by rapid industrialisation, urban migrations, worker militancy and wage demands, the Government was forced to make deep, formative changes to its own structure and to the fabric of the state.\n\nIndeed, the 1940s were exceptional years, a time when many social contradictions that had been papered over or had become clogged burst open, sending shock-waves throughout South Africa.\n\nThe effects on Cape Town were to be profound. In the post-war years the city was to become a symbol of the rise of big capital, merging the commercial, industrial and military town into a rapidly expanding metropolis. Organisation for investment was as imperative as disorganisation of opposition from working people.\n\nThe merchant port of the Cape sea-route was to be ripped apart and rebuilt to the specification of industry. The city would be cut off from the sea by raised highways connecting the new inner-city office park to the factories and to the white suburbs. Troublesome elements \u2013 workers with brown skins \u2013 were to be issued out of town, beyond the white frontier to the new factory-estates on the Cape Flats.\n\nOver proposals for garden cities and green belts, planners talked of 'machine-gun' zones, 'buffer strips', 'military' roads and 'policeability'. A new city was to rise from the ruins of the old. This was not a story I could find in history textbooks but in government regulations, Acts of Parliament, political pronouncements, council minutes, sketches of urban planners and the drawing boards of state architects. What emerged was a shadowy, sinister gantry of urban social control.\n\nUrban racial segregation\n\nFor much of South Africa's urban history, voices could be heard clamouring for residential separation and economic protection from people of colour. Until the 1940s these pressure groups \u2013 mainly traders and workers of European origin \u2013 were given only passing attention by governments dominated by mining and agricultural interests. The main focus for these organised groups \u2013 town councils, tenants' associations and chambers of commerce \u2013 was in Natal. White shopkeepers and small traders set up an agitation, claiming that they were being undercut by Indian retailers. Calls were made for the repatriation of the Indian immigrants, as well as for trade and residential segregation. Indian traders began to move into the interior and, as they did so, encountered similar reactions and legal restrictions. For instance, in the Orange Free State, legislation was passed in 1885 preventing Indians from owning any property or conducting business.\n\nAfter Union in 1910, white trader pressure against Indians continued. The Lange Commission, in its report of 1921, 'strongly recommended that some system of separate areas should be introduced both in the Transvaal and Natal... Such a scheme will tend to ensure the removal of Asiatics from the immediate vicinity of European traders.'23 The implementation of the Commission's findings was, however, delayed.\n\nDuring the period from the 1920s to the 1940s, the economic struggle of the traders was broadened by a crisis of falling property rates in the inner cities and by an ideological assault on people not deemed 'European'. The effect was to include white workers in the agitation for residential segregation, deflecting their grievances as an exploited class towards the threat of black migration into the cities.\n\nAlongside rising inner-city population numbers, the opportunities for 'informal sector' activity grew. This tendency drastically increased competition for white traders, particularly in Durban and on the Witwatersrand. So, while industrialists prospered, smaller traders faced the spectre of falling incomes. At the same time, their elected local authorities were increasingly confronted with demands for free services in the areas allocated to people of colour. The agitation for urban racial segregation began to grow.\n\nFive parliamentary Commissions of Inquiry and three Select Committees were established during this period to deal with the problem of alleged 'penetration' by people of colour into white areas. All but one recommended curbs on Indian or black land ownership and racial segregation. From these inquiries emerged the 'Pegging' Act of 1943, which froze land sales to Indians in Natal and the Transvaal for an initial period of three years. After failure by the Government to reach a negotiated agreement within this time, the Asiatic Land Tenure Act (known as the 'Ghetto' Act) was passed in 1946, providing for the racial segregation of Indian business and residence.\n\nThe 1948 election won political power for Afrikaner workers, farmers and entrepreneurs. In the view of the three partners of this alliance, the physical removal of all non-white people from the cities would serve their interests well. Indeed, among the first actions of the new Government in 1948 were the tightening of the Pass Laws and the announcement of an inquiry into the 'Ghetto' Act. While the Inquiry was taking place, a virtual pogrom against Indians broke out in Durban in which 140 people were killed, ending in the displacement of nearly half of the city's Indian population.\n\nThe first three chapters of the inquiry into the Ghetto Act were never made public. In its final two chapters can be found the keystone of spatial apartheid.24 The report ignored the minor issue of Indian rights, claiming that the 'problem' was a national one, which should be dealt with uniformly throughout the country. Reviewing preceding commissions and reports, it concluded that the demands for territorial racial segregation had for too long been ignored. Inter-racial 'tension and strife' were seen as the basis of racial separation: 'We see no way of [avoiding civil commotions] except to legislate for total territorial segregation of the different racial groups, so that in the course of time homogeneous racial group areas are brought about.' As for the kind of Group Areas envisaged, there were two possible solutions:\n\nOne is that there should be total segregation of the various racial groups into fairly large areas of the Union with the possibility that eventually these large areas will become separate non-European States under suzerainty of the Union. The other is that there should be total territorial segregation on a small scale, particularly in the urban areas. This means that the penetrated areas should be cleared up by uprooting non-European ownership and occupation, segregating such ownership and occupation into the various non-European, racial group areas...\n\nThese two points of view are not mutually inconsistent and the one may be regarded as the ultimate aim and possible outcome of the other. Racial divisions were to be made merely for reasons of bureaucratic necessity:\n\nThe population of the Union should, for the purposes of the proposed legislation, be classified into racial groups and the lines of demarcation between the different groups should be clearly drawn in order to minimise administrative difficulties. We visualise the following groups, namely, the White group and the non-European groups of Natives (with sub-groups, if necessary), Coloureds and Asiatics and perhaps the Malay Group.\n\nVirtually all recommendations within the report were translated into the Group Areas Act of 1950. A striking feature of the parliamentary debate on the Group Areas Bill was the unanimity with which the opposition parties \u2013 the United Party and the Labour Party \u2013 approached racial segregation. Their only objections were over the method of application. The older inner-city ghettos with their maze of narrow alleys adjoining white residential or business areas were to be systematically razed. Among their sins was that of being 'military hazards'. In the words of Natal University sociologist Pierre van den Berghe, they were to be replaced by:\n\nModel townships with unobstructed, rectilinear fields of fire and wide streets for the passage of police vans and armoured cars. The new ghettos are typically situated several miles from white towns, with a buffer zone in between; they are [to be] sprinkled with strategically located police stations, and ... enclosed barbed wire.25\n\nThe legislative foundations for disorganising the poor and building defensive cities had thus been laid.\n\nBreaking the web\n\nGiven the framework within which Group Areas removals took place in Cape Town, a social disaster was inevitable. As the familiar social landmarks in the closely grained working-class communities of the old city were ripped up, a whole culture began to disintegrate. Although the old working-class neighbourhoods had been geographically bounded by outside forces and penetrated by them (in the form of schools, police, etc.), they were places in which people had organised space for their own style of life. These spaces were networks of streets, houses, corner shops and shebeens but also social webs of kin, friendship, neighbourhood and work. They were a mix of rights and obligations, intimacies and distances, providing a sense of solidarity, local loyalties and traditions. They also provided collective surveillance, creating safe spaces for children to be children, as Brian Barrow observed:\n\nChildren everywhere. Shouting, laughing, whistling, teasing, darting between old men's legs, running between fast-moving buses and cars and missing them by inches with perfect judgement. Poor, underfed children but cheeky, confident, happy and so emotionally secure in the bosem of their sordid surroundings. Everyone loved them. To them, it seemed, every adult on those busy streets was another mother, another father.26\n\nThe former warden of the Cape Flats Distress Association, Dr Oscar Wollheim, described the complexity of these social webs:\n\nThe rings closest to the centre are represented by the man's immediate and extended family and his closest friends. The next would represent his acquaintances, his church, his school and the clubs he frequents. Other rings represent his employer, his transport and communications, the shops he frequents, the municipal and other officials he meets, his doctor, the police, the postman, the tax official. The anchors of the web represent the customs, habits and moral concepts of the community in which he lives.\n\nEach individual has his own personal web which varies in size and complexity, according to the impact he makes on those around him and the influence he wields in the community. His usefulness to and within the community is determined entirely by the freedom with which he is able to move in and about his web, his knowledge of its structure and the facility with which he is able to make contact with the correct position of the web at the correct time.27\n\nThe Group Areas Act was to break that web by fundamentally disturbing the organisation and role of the working-class family. With it were ploughed up relationships and networks of knowledge, experiences and history \u2013 the very scaffolding of their culture. The effects of Group Areas removals, Wollheim explained to me, were:\n\nlike a man with a stick breaking spider webs in a forest. The spider may survive the fall, but he can't survive without his web. When he comes to build it again he finds the anchors are gone, the people are all over and the fabric of generations is lost. Before, there was always something that kept the community ticking over and operating correctly... there was the extended family; the granny and grandpa were at home, doing the household chores and looking after the kids.\n\nNow, the family is taken out of this environment where everything is safe and known. It is put in a matchbox in a strange place. All social norms have suddenly been abolished. Before, the children who got up to mischief in the streets were reprimanded by neighbours. Now there's nobody, and they join gangs because that's the only way to find friends.28\n\nThe collapse of social control over the youth was one of the major problems facing the working class as their culture began to buckle in both rural and inner-city areas. This informal control was described to John Western during his work on the suburb of Mowbray:\n\nWhen I was 15 or 16 if we did anything rude, offhanded, in the street \u2013 like going to bars or smoking or taking a dame out \u2013 you'd get a pak [slap] at night at home; they [parents] knew about it right away... It was the old men who used to stand at the corners chatting or sit on the stoeps; they'd pretend to be reading the Koran or a comic or playing karem or whatever, but out of the corner of their eye they were really watching you.29\n\nIt was this almost intangible but very real cement which held the working-class culture together. One of the greatest complaints about Group Areas removals was that individual people or singular families rather than whole neighbourhoods, were moved to the Cape Flats. The stresses resulting from these changes brought with them psychological difficulties and skewed 'coping' behaviour. Marital relationships were upset and the rates of divorce and desertion rose. Parent-child relationships also became problematic \u2013 often because of the father's sense of inadequacy in his new environment (this will be discussed more fully in a later chapter).\n\nBetween 1961 and 1965, a period which saw both rising industrial prosperity and mass relocations of people, there was a coloured baby boom. One can only speculate about its causes, but the relation between crisis and a high level of childbirth is well known. By 1980 the effects of the boom could be felt in the 15- to19-year-old age group \u2013 the core age of the street gangs.\n\nAnother indicator of social disruption was the illegitimacy rate among coloured people, which nearly doubled between 1956 and 1981. This rate declined during the 1950s and began rising in 1961 when the first Group Areas were declared in the city. It went up sharply in 1967 after coloured families were told to vacate District Six, then climbed alarmingly between that year and 1976 (the period of the greatest evictions).\n\nIn 1974 the Theron Commission found that more than 82 per cent of all births to coloured women under the age of 20 were illegitimate. The findings of the Commission bore stark testimony to the breakdown of extended families which followed the removals. The Commission found a sharp decrease in what it called 'non-members' of nuclear families between 1960 and 1970 and a marked increase in single-parent families in the urban area. The commissioners heard that:\n\nIt was very difficult indeed for parents, with children of different sexes, living in a house with only two or three rooms in a neighbourhood with few, if any, community amenities, to give their children a decent upbringing.\n\nThe Commission concluded that 'no other statutory measure had evoked so much bitterness, mistrust and hostility on the part of the Coloured people as the Group Areas Act.'30 This statement echoed Dr Wollheim, who had warned in 1960 that 'we can look forward to a period of increasing social dislocation, which will have its root in no other causes but in the application of the [Group Areas] Act'.31\n\nTo assess the effect of Group Areas removals on families at the time, I made a comparison between family life and working-class culture in an 'inner-city' working-class area and on the Cape Flats, where many people had been relocated. The established area was Harfield Village, which forms part of Claremont (it was later gentrified and is now predominantly white).32\n\nAt the time of the survey, Harfield Village was a suburb 'in transition' from a mixed to a White Group Area, and only about a hundred coloured families remained. On average, families had resided there for 19 years, although more than ten per cent had been there 50 years or longer. The average number of people in each house was a fraction above five.\n\nWhat was significant about the area was the high number of people available for what might be described as 'crisis support'. Some 80 per cent of the people interviewed had relations in Harfield and slightly more than this had close friends in the area. This was despite the fact that 65 per cent had seen related families moved from the village by Group Areas. There was no cr\u00e8che in Harfield. Of those whom I interviewed, the majority looked after their own children and a sizable number relied on relations to do this.\n\nIn total, 95 per cent of children aged under 16 were taken care of within extended families, the remaining number being minded by friends. In comparison with the Cape Flats this was an extremely high level of family-based childcare. Harfield had all the benchmarks of a stable supportive community. This was also the case in Mowbray, where John Western found an average residency of 33 years and where 70 per cent of his interviewees were related to at least one other physically separate household.33\n\nThe Cape Flats survey focused specifically on mothers living in 35 different housing-estates. The average number of people in each dwelling was a little over seven and the average length of residency was a mere four years. Of the sample, 44 per cent of the Cape Flats mothers were working and 25 per cent were raising a family without a husband. In order to gauge changes in living-patterns, the mothers were asked about their own childhoods and then about their children.\n\nThe findings showed a marked historical fall-off in access to family networks of childcare. A high percentage of children under 16 received no parental care during the day, while a very small number were placed in cr\u00e8ches. When asked about any problems they were experiencing, the greater number of mothers said it was a fear of gangs and lack of police protection.34\n\nThese surveys can only hint at the misery caused by relocations under the Group Areas Act. The first effect of the removals into the high-rise schemes on the Cape Flats was to destroy the communal space that the street, the corner shop and the shebeens in the 'old' areas had provided. The new areas contained only the privatised space of small, nuclear family units. These were stacked on top of each other in total isolation, juxtaposed with the totally public space surrounding them \u2013 a space that lacked any of the informal social controls generated by their former neighbourhoods.\n\nThe destruction of the neighbourhood street also blew out the candle of household production, craft industries and services. The result was a gradual polarisation of the labour force into those with more specialised, skilled or better paid jobs; those with the dead-end, low-paid jobs and the unemployed. As the new housing pattern dispersed the kinship network, so the isolated family could no longer call on the resources of the extended family or the neighbourhood. The nuclear family itself became the sole focus of solidarity.\n\nThis meant that problems tended to be bottled up within the immediate interpersonal context that produced them. At the same time, family relationships gathered a new intensity to compensate for the diversity of relationships previously generated through neighbours and wider kinship ties. Pressures gradually built up, which many newly nuclear families were unable to deal with. The working-class household was thus not only isolated from the outside, but also undermined from within. The main, and understandable, product of this isolation was fear \u2013 fear of neighbours, fear of unknown people, fear of gangs and fear of the strange dynamics of the new environment.\n\nThese pressures on the Cape Flats weighed heavily on house-bound mothers. The street was no longer a safe place for children to play in and there were no longer neighbours or kin to supervise them. The only play-space that felt safe was 'the home', the small flat. As stresses began to build up within the nuclear family, what had once been a base for support and security now tended to become a battleground, a major focus of all the anxieties created by the disorganisation of community.\n\nOne route out of the claustrophobic tensions of family life was the use of alcohol and drugs. This became the standard path of many men. Children were shaken loose in different ways. One way was into early sexual relationships and perhaps marriage. Another was into the fierce youth subcultures on the streets which became ritualised in the violent youth-gang culture, reinforcing the neighbourhood climate of fear. The situation was to be compounded by rising unemployment at the younger end of a potential labour force. The difficulties of getting a job in Cape Town were described to me by Petrus, a boy who had to leave school as a result of the 1980 education boycott:\n\nThere's such a lot of jobs advertised in the Argus, but when you go you find a queue of 30 people for just the one job. And chances are you won't get it. Perhaps they'll advertise for a messenger with Standard Six, but you'll find people there with matric certificates. You and your junior certificate can just turn around and leave again.\n\nOne of the best ways of getting a job is having a connection, maybe an uncle or someone to bring you to the boss. I know of people who got jobs that way. But unemployment bureaus don't do much to help. They just want qualified people like electricians. And they would rather employ blacks who they can fuck around and who they can't really communicate with, so the worker can't argue back. I prefer stealing to working for someone who exploits you like that.35\n\nFor many rural youths moving into Cape Town, arrest, conviction and sentencing to a reformatory had become a ticket to the city. Here they joined the unemployed and the gang 'brothers', becoming urban hunters for the means to stay alive. What these gangs did in order to survive in the face of tremendous odds was to rebuild the lost organisation and domestic economy in the new housing-estates. This time, however, their customers were often also their victims.\n\nIn Cape Town, what the new ANC government inherited in 1994 was a working class that was like a routed, scattered army, dotted in confusion about the land of their birth. In the lonely crowd of satellite clusters with rising rates of violence, the townships had become increasingly difficult places to meet people after work, favouring silent conformity and not rebellion.\n\nThe nineteenth-century, French political thinker, Alexis de Tocqueville noted, rather dramatically, the same threads in the growth of American suburbs many years earlier:\n\nThe first thing that strikes one's observation is an uncountable number of men, all equal and alike, incessantly endeavouring to produce the petty and paltry pleasures with which they glut their lives. Each of them living apart is a stranger to the fate of all the rest \u2013 his children and his private friends constitute to him the whole of mankind; as for the rest of his fellow citizens, he is close to them, but he sees them not; he touches them, but he feels them not; he exists but in himself and for himself alone; and if his kindred still remain to him, he may be said at any rate to have lost his country.36\n\nThe ultimate losers in this type of claustrophobic atmosphere were the working-class families. Torn from the areas they knew and scattered across the Cape Flats, the emotional brutality dealt out to them in the name of rational urban planning has been incalculable. The only defence the youths had was to build something coherent out of the one thing they had left \u2013 each other. Between windblown tenements on the dusty sand, gangs blossomed. The city's urban managers now had a major problem on their hands \u2013 violent crime.\n\nPolicing for apartheid\n\nIf Group Areas removals were the underlying fabric of conquest, its enforcement among the conquered was the responsibility of the police. People so brutally relocated were unlikely to go quietly into the night. For this reason, under apartheid, precinct policing focused largely on controling political volatility, a task inappropriate to effective policing of crime. In any discussion about gangs, what the police do \u2013 or don't do \u00ad\u2013 is of central importance because the actions of both groups tend to take place on overlapping and interlinked terrains.\n\nThe law under the National Party up to 1994 did not so much set a standard of legality from which the police could deviate as provide a licence to ignore it. The police were sanctioned to search without warrant in many situations, to stop meetings, close down premises, shoot to kill and arrest people for a wide range of reasons. Many of their actions carried out 'in the interests of State security' did not need to be made public and this secrecy was corrupting. Given the wide legal sanction offered to the police, it would be surprising if they did not grasp it. Discussing this point in Parliament during the 1982 session, an Opposition Member, Harry Pitman, observed:\n\nIf one gives unlimited and uncontrolled power to any branch of the executive, they, being only human, are going to use that power in their work. The problem we have is that power is being given to be exercised in secret without scrutiny by the House, without scrutiny by the public, and without the control of the Supreme Court. The blame therefore does not rest entirely with the police.37\n\nIt's obvious that behind such an official screen, policemen would attempt to cut corners to obtain evidence, particularly if they were dealing with known criminals. In any policing, the constant demand is for efficiency in solving cases before more harm is done and the emphasis is on force \u2013 get your man and maintain order. The temptation existed and still exists to extort confessions or information. As a result, due process of law can be easily overlooked \u2013 if the law is a puzzle for the people, it is also a puzzle for the police. Indeed, for many people on the other side of the charge-office counter, even decades after the end of apartheid, the rule of law remains indistinguishable from the rule of force, with justice being the right of the stronger party.\n\nThe apparently omnipotent position of the police under apartheid was reinforced by political rhetoric. Commenting on the increased number of police assaults on members of the public during 1981\/2, the chairman of the Civil Rights League argued that the increase was supported by the rising sanction of violence, used to prevent democratic change through:\n\n\u2022 Warped reports by SABC\/TV and by Cabinet Ministers of a Total Onslaught. What is a young policeman to think after hearing a Cabinet Minister say, 'When survival is at stake, no rules apply'?\n\n\u2022 Disrespect for the individual, resulting from unjust laws and from certain churches, schools and newspapers ignoring the ancient principles of human dignity and impressing hostile emotions on the young;\n\n\u2022 Using the police force for unsavoury purposes, such as raids on the women and children of Nyanga [squatter camp], intervention in labour disputes, or interference with Black Sash stands [against apartheid];\n\n\u2022 Obsessional denials and cover-ups, which encourage immature young men to believe that actions of violence receive support from above.38\n\n'Hard' policing led, in turn, to bad relations between the police and the public. Speaking in a parliamentary debate an Opposition Member, Ray Swart, reminded the House that:\n\nNo police force can operate successfully unless it has the support and sympathy of the general public. [In this respect] the lot of the South African policeman is perhaps amongst the most unenviable of any policeman in the world. This is so because of the nature of the society in which he has to operate and because of the unjust and inequitable laws which he is expected to enforce... Amongst [Black] groups [police] authority is used to enforce [these] unpopular and unjust laws, whether in respect of pass-law offenses, forced removal of people from their homes, demolition of shacks or whatever it may be ... [I]n circumstances like that, the policeman is very often... seen as a symbol of oppression.39\n\nIn terms of daily crime prevention, therefore, the power available to the police generated as many problems as it did solutions for the station-based police officer, isolating them behind their uniform and inside the system they represented. Being expected to control both political volatility and general crime ensured that slippage would occur, and it did on both fronts.\n\nUnder apartheid, the police were expected to keep a ruling class safe from violent revolution. They could and did call on the army for support but, on a daily basis, the task expected of them from the 1950s became increasingly intolerable. In 1979 South Africa's police force of 34 646 men was the same size as that of New York City for a population many times the size of that city spread out over an area five times the size of Britain. In the same year a newspaper complained: 'Thirty years ago in Cape Town more than 100 policemen walked the night beat, assisted by a van and five on bicycle patrol. Today there are not more than six men on the streets at night.'40 In 1980 the shortage of policemen prompted To The Point magazine to describe them as 'South Africa's most wanted men'.41 A policeman told me:\n\nOur chaps are feeling the strain and are getting frustrated. People criticise us but they don't know our problems. A fellow phones and says 'This is the third time I've phoned and that fellow that assaulted me, he's still walking up and down in the street and when are you coming?' Or a policeman's sitting in the office and someone phones and says somebody's burgling his house. But the policeman can do nothing because there's nobody to help \u2013 he walks out of the office to look for someone and there's not a single person to send. Or there's no van and he must use his own car ... and there's no refund for petrol. There's just not enough policemen! We're just the step-child of the army. We get what's left \u2013 second-class equipment and a tight budget.42\n\nThe police shortages and their political focus were undoubtedly major contributing factors in the sharp rise of crimes of violence and of property that began in the early 1970s and has continued almost unabated. 'The skollies are making hay now', the policeman continued, 'because they know we can't get to them all.' The solution to this high crime rate was seen as the placing of 'a cop on every corner'. The Viljoen Commission into the South African penal system noted:\n\na policeman patrolling an area on foot is in a far better position to develop an experienced eye and to smell out would-be criminals than a policeman patrolling an area in a vehicle. The vehicle moves too fast to enable the policeman to reconnoitre an area properly. The danger [of vehicle patrols] is that persons who either individually or in groups appear from a moving vehicle to act or move or loiter suspiciously, may be perfectly law-abiding... The motor vehicle has its advantages [but] there are criminologists and others who would like to see a full-scale return to the 'bobby on the beat'.43\n\nUnder the circumstances, such saturation patrolling \u00ad\u2013 flooding 'problematic' working-class areas with policemen \u2013 would have been both politically volatile and uneconomical. Police were seen as agents of a hostile State and, furthermore, patrol activity is the single most expensive component of police labour and one that instantly clogs the control apparatus, as a policeman pointed out to me:\n\nIf we had a full staff \u2013 all the men we required \u2013 and made the number of arrests we would like to, the people at the courts would chuck up their job and go home. Because at the moment they can't cope, they're so snowed up. They're even having court on Saturday morning! But then even if the police and the courts could cope, then the prisons couldn't because they're too full. It's crazy.44\n\nThis no-win situation was depressing for policemen, who worked long hours and seemed to make no headway. Often their problems manifested publicly as complaints about pay. These complaints were officially deemed 'unpatriotic', but pay-gripes found their way into the press, usually from ex-policemen, or were raised in Parliament by the Opposition before an election.\n\nWith the problem of staff shortages and discontentment, 'hard' policing was to be expected. Nobody could condone the action of a detective who carried a suspect to the cells in the boot of his car.45 However, given that the suspect was dangerous, that there was no patrol van, that the car was probably the detective's own and that there was no colleague to help restrain the suspect, the seemingly callous action could at least be understood as a choice between unconventional methods and releasing the suspect. Impossible situations, together with the militaristic and racist rhetoric of the time, gave rise to improbable and often morally unacceptable solutions. The frustration of confronting a situation beyond his control was clear when I asked a Grassy Park policeman in 1982 what could be done about the city gangs:\n\nI don't know what you can do \u2013 except maybe Nazi-style line them up and shoot them. But that's not in the law ... even the public think sentences are too soft. You should remove skollies from society to solve the problem. But then where would you put them? Perhaps you could send them to help Britain in the Falklands! But I say you cannot solve the problem according to the law... You need to drag them in here and give them all hangpaal [death sentences]; you need public hanging or something.46\n\nThe temptation to give a street youth a thrashing for his misdemeanour, or an hour in the van before letting him go, was strong indeed, given the porous nature of the system of justice from the viewpoint of the detective or patrolman. A range of strategies were developed both by patrollers and detectives. One was that of financial 'inducements' for information. It was not possible to find figures on the total amounts paid to police informers, but the sum was undoubtedly considerable and open to abuse. 'We have plenty of informers,' a detective told me. 'We must because there are not many of us and there are many skollies.' The informer network was bolstered by 'agreements' and 'favours' worked out between the police and the community. It was also the result of 'leaning' on certain individuals who were threatened with arrest for some misdemeanour unless they 'squealed'.\n\nDuring my work with gangs in the 1970s and 1980s, a feature which constantly startled me was the arrival of policemen at a gang headquarters. A detective later clarified this form of liaison.\n\nWe work with the gang leaders. If we go to them and say we are looking for so and so, they bring him here to the station. They know that if they don't, we can get them and they want peace with us. Also, if we want to find out about this gang we go to the leader of the opposition gang and they tell us everything they know. We pay for information.47\n\nIn this way gangs and officers could 'co-police' an area, keeping it 'quiet', picking up 'nuisances' and leaving the core of the gangs untouched. In particular, gangs in the pay of shebeeners and drug merchants performed a quasi-policing role. Informal policing by both police and gang bosses was thus effective in terms of State control, but ineffective for the community, as it tended to focus on high-profile street youths and left most of the big rackets untouched. This slack was in some sense taken up by specialist units such as the Narcotics Bureau and the Murder and Robbery Squad which could come in 'over the top' of regular policing. These were both disbanded by the new government after 1994.\n\nBy the mid-1980s apartheid's spatial controls of Cape Town and other cities began to crumble. Under the slogan of 'Make South Africa Ungovernable', the United Democratic Front unleashed a wave of urban insubordination and intolerance of state functionaries, especially the police. Faced with the choice between solving crime and keeping whites safe from a perceived communist-inspired revolution, the former decreased in importance and whole areas of the city were abandoned to any but the most desultory crime control. By 1994 this situation had permitted the growth of large syndicates and opened space for a criminal culture that was alert to the possibility of new fields of enterprise. The shift was noted by Jonny Steinberg in Thin Blue, a study of South African policing:\n\nFor the township underworld, whose long pedigree had been invisible to white South Africa, the opening up of urban spaces heralded something of a revolution. The vastness of the new market promised by white neighbourhoods was almost too spectacular to imagine. A criminal culture whose appetite for commodities and for violence was legendary in the townships, arrived in the suburbs. 48\n\nRecasting the city\n\nAfter the negotiated end of legal apartheid in 1994, the first democratic elections and the resultant cessation of global sanctions against South Africa, Cape Town began to shake off its isolation from the rest of the world. It became a city in transition, sensitive to the requirements of a global economy.\n\nMirroring wider national and international tendencies and blessed with natural beauty, the city administrators looked to tourism and its potential for earning international capital as one of the platforms for future development. For some, this proved extremely lucrative. In the decade which followed, the organised affluent and the rising new black middle class embraced consumerism. Their high-tech savvy, education and mobile lifestyles have made them some of the most privileged people in the world. Andre Standing has noted, however, that as tourism pumped money into the attractive, wealthy areas, private and public actors were encouraged to erect ever-greater systems of security. This was to keep those who had money to spend safe from those who might beg, steal and generally downgrade the image of the city and threaten tourism.49\n\nThe Cape government embraced a neoliberal approach common to many emerging cities of the southern hemisphere. It endorsed a market-driven approach to growth within a global economy. Central to this vision was the revitalisation of the Central Business District (CBD), a formally white preserve at the foot of Table Mountain which, in the 1990s, was considered by many as a no-go area for 'respectable' citizens, especially at night. Reporting on plans for the city, the Cape Times wrote:\n\nIn as much as other nodes should be accorded equal status, the need to revitalise and regenerate the Cape Town CBD is widely accepted by both the public and private sectors as a major and serious priority. This stems from the belief that the central precinct is the main locus of the region's historical, cultural, political and social attributes. It is the principal driver of the Western Cape's economy, accounting for 28 per cent of all jobs in the metropolis and is, accordingly, a major area of focus for the [city] council.50\n\nCentral to the city's attempt to 'retake' the CBD from a growing number of criminals, squatters and street urchins was the use of force, described as 'Operations'. The move began in 1998 with Operation Clean and Safe, followed by Operation Reclaim in 1999 and Operation Crackdown the following year. The first of these involved 400 municipal law officers and 30 members of the South African Police Services (SAPS) who together made 20 arrests, gave citations for 2 644 traffic violations and confiscated more that 3 000 items from traders in illegal areas.\n\nBusiness Against Crime was formed in 1996 and installed a closed circuit television system in the CBD. Two years later the city's urban renewal process was formalised through the creation of the Cape Town Partnership. Its membership included businesses, property owners, the local tourism board and local government. Its goal was the 'branding, positioning, marketing and establishment of [Cape Town] as a globally competitive city with a globally competitive product offering.'51 In November 2000 the Partnership created the Central City Improvement District (CCID) in the CBD, designating it a zone of growth and renewal. R6 million was raised to put this into effect, 120 community patrol officers were appointed, a 72-camera network was developed and manned 24\/7. The city's Protection Services Department established a 24-hour, 32-member rapid response unit. The principles for these regenerative programmes were derived from Mayor Rudy Giuliani of New York City's 'broken windows' approach. His police chief, William Bratton, was consulted for Cape Town CBD's urban revitalisation.\n\nAs the world was to see during the Soccer World Cup in 2010, the transformation was a profound success and the reclamation of the CBD complete. Despite dire predictions of criminal predation during the tournament, virtually no incidents were reported. In travel magazines and websites, Cape Town was cited as one of the world's top ten tourist destinations and continues to be so. In 2014 one of the largest travel websites voted the city's restaurants second best in the world after New York and, a year later, it was included on the World's Best 50 Restaurants list.\n\nWhat was the cost of this refurbishment in terms of urban development? The wealthier citizens in the more organised, formerly White sections of the city had been protected at great cost and the poor were kept disorganised and at bay. The police and private security became frontline agents of urban renewal in a deeply divided city. Street children were flushed out, 'undesirables' were moved on and, in the words of the Police Chief of the time, informal parking attendants were described as 'parking terrorists'.52\n\nThe success of this planning led to the development of City Improvement Districts being proclaimed in other, albeit wealthier parts of the city. A former coordinator for social development at the CCID, Clinton Osborne, told researcher Tony Samara: 'Everyone's getting one, it's a new trend. Woodstock, Observatory, they're just spreading like wildfire all over the Cape.'53 According to the analysis of a parastatal organisation, the Western Cape Provincial Development Council, these initiatives...\n\n... appear to be mainly concerned with improving the appearance of the CBD through addressing the issues of 'Crime and Grime'. Crime and Grime is associated with vagrancy, street children, the homeless and informal traders. Hence, the private sector response is ostensibly about the brutal eradication of these 'unwanted\/bad' elements to stem capital flight from the CBD.\n\nThe city is not an innocent bystander in this 'eradication' programme. The initiative overlooks the fact that the poor constitute the majority of the city and that the vagrants\/street children\/informal traders are surface manifestations of deep-seated problems viz. poverty, unemployment and social breakdown in the underdeveloped areas. [The inner city has become] a playground of the elite that admits the poor only as labourers, domestics, gardeners and the like.54\n\nThe overall effect of the city's urban renewal was eerily reminiscent of the social engineering under apartheid, though now it was based only loosely on race or colour (because most of the city's poor remained coloured and African) but centrally on class. The architects of the Group Areas Act might have nodded in appreciation at the new sophistication of their ideas.\n\nCape Town is today managed through a complex network of influences \u2013 its apartheid past, new initiatives in urban social develop\u00adment and containment and the global imperatives of a neoliberal economy found in many emerging cities. In a buoyant, expanding economy, the new urban development plan could have trickled outwards to involve the upliftment of poorer, working class and predominantly black areas, followed by improved organisation of education, health care, crime control and housing.\n\nWhile there have been a number of moves in this direction, including the concerted building of low-cost housing by the City Council, the overall economy of the country has moved forward in some areas but stagnated in many. Conditions for Cape Town's urban poor have worsened in most cases. Crime has increased and joblessness, particularly among young coloured males, has soared. In relation to the poor of the CBD, Samara writes that neoliberal urban renewal has positioned them...\n\n...as unwanted trespassers rather than citizens entitled to all the rights that the new constitution guarantees. What this criminalisation suggests is that choosing this development path in the midst of overwhelming poverty and stratification necessitates new forms of social control that are able to secure the borders between the affluent nodes and the still underdeveloped areas on the other side.55\n\nIn the CBD and wealthier parts of the city, public and private police, bolstered by a huge private security industry, are integral to a development process. There, the protection of economically and socially valuable urban spaces is paramount and control of crime and the urban poor is essential. Cape Town is being forced to respond to challenges generated by local inequality and conflict that afflict many cities. As Samara suggests,...\n\n...development and urban renewal have been subsumed under and articulated through neoliberalism which, although appearing in many guises, means for all cities under its sway renewed and reinvigorated tensions between the demand of 'free markets' and those of populations.56\n\nThe commitment of the post-apartheid state to a better future for all began to shift by about 2000 towards a better future for the few and increasing management of the rest. The past threat of urban terrorism during the apartheid years was replaced by a fear of interpersonal crime. This paralleled an escalation of graft at ever-higher levels of State and corporation. Beyond the Cape's CBD and leafy mountain suburbs, crime began to assume a central place in development discourse and city managers found themselves increasingly juggling between this and the city's international image.\n\nThe beginnings of this shift was recognised at a national level when the ANC government abandoned the social Reconstruction and Development Programme (RDP) in 1996 in favour of the neoliberal Growth, Employment and Redistribution plan (Gear) and, simultaneously, declared 'war' on crime. Gear was presented to Parliament by Finance Minister Trevor Manuel as a fait accompli, without consultation. No one was allowed to see the plan before it was introduced to the House. It had been drawn up by experts from key financial institutions under the guidance of Richard Ketley, an official from the World Bank.\n\nThe manner in which Gear was introduced suggests that the ANC was aware that abandoning the RDP would be hugely unpopular, signalling a 'Washington consensus' and the prioritising of the international financial community over the needs of most South Africans. Gear involved a mix of affirmative action and public spending that was less racially skewed and adherence to the neoliberal imperatives of tight fiscal austerity.\n\nIt involved increased labour market flexibility, monetary discipline, privatisation, reduced corporate taxes, trade liberalisation and a gradual phasing out of exchange controls. It was a voluntary structural adjustment that horrified those on the Left, particularly people who believed the ANC was historically a socialist movement. As a result, the ANC was accused of 'selling out' to white capital and the emerging black bourgeoisie.\n\nThe positive economic and labour market projections of Gear, as presented in 1996, have not been achieved. Economic instability, inequality and social insecurity have either remained high or have risen. Whereas the architects of Gear predicted the creation of 1.35 million new jobs by 2000, the unemployment rate rose and at the time of writing is around 40 per cent. Numbers of people falling below the poverty datum line have also continued to increase and the quality of education and health care have fallen.\n\nAccording to the Gini coefficient, South Africa, alongside Brazil, is one of the most unequal societies in the world.57 In the words of Samara: 'The picture that emerges is of a city genuinely frustrated and outraged by violent crime but whose response is often misdirected and counterproductive if the goal is transformation of the lives of its young people.'58 With such disparities of wealth and power, it's inevitable that the preoccupation of both the State and Cape Town's city officials increasingly turned to control and, as in the past, the burden still falls on the police. This is not a new situation and it's interesting to see the continuities between urban control under apartheid and post-apartheid South Africa. For the poor, from the perspective of the wind-blown Cape Flats, very little seems to have changed. On the city's low-income, high-risk periphery, pressures continue to build. Few nights are free from the sound of gunfire.\n\nCops under new management\n\nI align myself with the view that conducting policing in a sea of poverty and hopelessness is most difficult. SAPS members in this station essentially deal with service delivery crimes which has the effect of creating a platform for resentment of the police. People who live in squalid conditions of Khayelitsha see the police as an instrument of oppression when they are arrested for service delivery related crimes. [Further problems are caused by] no access roads in temporal areas, poor lights... lack of fixed addresses...due to the condition of the roads... SAPS vehicles are easily involved in accidents or mechanical damage.59\n\nBrigadier Zithulele Dladla's testimony to the Khayelitsha Commission, 2014\n\nShortly after 1994, police leadership set about stabilising the organisation, bringing it in line with the new democratic order and willing legitimacy in the townships. Steinberg contends that on this last point they miscalculated, assuming the country's urban Black population to be politically homogenous and united by the common experience of past oppression. They understood township violence to be circumstantial and believed that, with the passing of apartheid, it would abate and that people would give their consent to be policed. It was thought to be just a question of delivering a benign, non-political police force. The situation, says Steinberg, was misread:\n\nThe violence was neither circumstantial nor superficial. It was an inflammation of the very tissue of urban South African life, a tissue in which protection and coercion coalesced around allegiances and markets. To get South Africa to give its consent to being policed would require breaking down a generations-old architecture of security and protection. It would require bringing a body with unprecedented authority into township life, a body elevated above existing security markets and thus able to break their logic.60\n\nThe new government, under President Mandela and amid the many demands of transition, turned its attention to the regulation and coordination of the sprawling collection of policing personnel, practices and agencies. In October 1995 it gazetted the SA Police Services Act, which defined the parameters of policing within the new Constitution and Bill of Rights. Community policing replaced hard law enforcement and a Code of Conduct was adopted together with diversity training and policies against police brutality.\n\nA programme was started to train 90 000 police officers in human rights standards for law enforcement. This included a booklet titled You and the Constitution and guidelines for the policing of refugees. In the same year the SAPS incorporated 10 police agencies from the former homelands and reorganised its priorities at both national and provincial level. A year later the ANC unveiled its National Crime Prevention Strategy (NCPS) based on four 'pillars':\n\n\u2022 Reforming the criminal justice system and making it more accessible to disempowered groups;\n\n\u2022 Reducing crime through environmental design;\n\n\u2022 Reasserting public values through education campaigns to involve communities in addressing crime; and\n\n\u2022 Addressing transnational organised crime.\n\nA central theme was that crime and security should be tackled by remedying underdevelopment and poverty as well as demilitarising the image and actions of the police. At an ANC conference in Stellenbosch it was decided that:\n\nThe de-militarisation of structures, rankings, uniform and equipment will promote more creative, socially skilled policemen and women as well as ameliorate the perception that the SAP is the army by another name. The de-militarisation of the SAP and the homeland police forces will send a signal for change. It will also assist in creating the other conditions for community policing such a patrolling on foot, which was unrealistic whilst there were such intense levels of hostility in many areas. A shift to smaller accessible police stations, staffed by members well acquainted with community problems and visible to the community, must be supported.61\n\nThe value of this broad-based approach is undeniable. However, Steffen Jensen has shown in his study of gangs and policing that, despite workshops, consultative processes and training, such discourses of change do not always translate into real change:\n\nPolice officers identified the introduction of the Bill of Rights as the biggest change in their daily work. They recognised that the Bill of Rights was necessary, but they also agreed that it made policing more difficult. The assertion was that 'the criminals are afforded more rights than the victims'; that the Constitution protected only the criminals and that their failure to convict and imprison criminals lay with the Constitution.62\n\nA station commander told Jensen: 'Of course it is right what is happening. We don't want the old-style policing anymore, but it is very difficult to accept sometimes. Especially for the ordinary members. Also what you have to remember, there was never a Bill of Rights in South Africa.' His men, he said, 'slip every now and then and do something they should not do.'63\n\nOn the ground, the overall effect on police of the transition to democracy was increased confusion and subsequent weakening of the state's crime-fighting abilities.64 A detective in the Western Cape Organised Crime Unit (OCU) told researcher Anthony Samara:\n\nA significant reason for the increase in organised crime here was the police transformation. Whilst this transformation may have been good from a human rights perspective, it allowed the criminals a huge opportunity. At a single stroke of the clock in 1994 the whole way of doing police work was transformed. Under the old authority the rules were strict and police had a clear understanding of how the system worked.\n\nThe police used to be very effective and there was a high level of trust amongst the force. With the transformation there was a complete overhaul of legislation and procedure. Many police had to be retrained. This caused a lot of stress and many police decided to retire. This left a bitter feeling with some feeling inferior. Now there is a feeling that you don't know who to trust.65\n\nSo despite significant changes in the approach to policing, crime rose steadily through the 1990s and the private security industry boomed. By 2010 there were an estimated four security guards for every regular police officer in the country. In Cape Town's wealthier suburbs, walls with spikes topped by electric wires had become commonplace. The sound of car and burglar alarms became as regular as the roar of jetliners ferrying tourists and corporate representatives in increasing numbers. Under the watchful gaze of CCTV cameras, Long Street in central Cape Town boomed as a club, pub and restaurant haven. In the affluent suburbs, neighbourhood watches drew citizens into the policing surveillance system.\n\nYet the urban periphery continued to return shockingly high contact crime statistics and its possible effect on business and tourism, as well as on the city's workers, was clearly a brake on urban development. The ruling party cast around for reasons, deferring the blame to unreconstituted police officers who had been inherited from apartheid structures. It also blamed so-called Third Force activities emanating from 'networks of the past' bent on discrediting the new government. In Cape Town, police were accused of complicity with gangs, particularly those involved in the drug trade.\n\nThe truth is that the police were being required, firstly, to secure the urban periphery in order to ensure development in the wealthy centre and, secondly, to pacify an impoverished periphery within insufficient social services or employment. They therefore became central agents of neoliberal underdevelopment in the gap between what government couldn't provide but which the criminal underground could.\n\n'We had police stations right across South Africa,' General Jeremy Vearey, head of Operation Combat tasked with gang control, told a policing workshop. 'Police ended up being expected to deal with social issues at a local level. We were in the front line.'66 The response of the SAPS was to grow in size and \u2013 under pressure to deliver \u2013 to decrease restraint. This posed dangers to both the police and public administration. According to Vearey:\n\nIf you don't want us to militarise, to take over your neigh\u00adbourhoods, keep us out of areas that require us to solve your social problems. But because the police have the biggest footprint across the country we are being called on to solve problems for other departments which are weak or insufficiently organised. So we end up trying to control what we shouldn't be controlling.67\n\nThe greatest crime deterrent is always good relations between police and community. The pressure on police to produce results for their superiors under stress to bring down crime became a threat to these relations.68 In the end, it reinforced the wedge between police and residents, deepened mistrust and eroded efforts at long-term crime prevention and social harmony. The goal of township transformation was, therefore, increasingly replaced by an imperative to win neighbourhoods back from crime at any cost.\n\nThis was a complete reversal of the 1996 National Crime Prevention Strategy's conception of the police force's role. It placed inappropriate strain on a newly-constituted force still trying to find its way under a Bill of Rights and attempting massive retraining as well as integration of large numbers of new black recruits. Under the pressure, 'slipping' now and again became common practice and rule-of-the-thumb assumptions were made on the fly. Researcher Steffan Jensen was told that in assessing township types, police officers distinguished between 'suspicious persons', 'arseholes' and 'know nothings' \u2013 that is, people requiring police attention, people who refused to accept the officer's definition of the situation and people with whom police seldom came into contact and were not considered a problem.69\n\nBy about 1999, both police and city officials sensed that all moves to curb crime had failed and that desperate measures were required. The new Minister of Safety and Security, Steve Tshwete, trying to allay public fears, told a gathering of police and press at Jabulani Amphitheatre in Soweto: 'We are going to deal with criminals as a bulldog deals with a bull. We are going to give them hell...' Later that year he described criminals as 'animals' towards whom no mercy should be shown and proposed loosening up legislation under which police could use lethal weapons on suspects.70 This approach clearly bore no fruit because nine years later, in 2008, Cape Town's mayor, Helen Zille, expressed exasperation over policing:\n\nCriminals know that they can get away with it. They rely on the police's inability to find and arrest them; on police dockets that simply disappear; on evidence that goes missing or does not stand up in court; on cases that drag on and on until they are dismissed. They assume that at every step of the arrest and conviction process, there will be an official who can be bribed to make the case collapse.71\n\nA year later South African President Jacob Zuma supported a 'shoot to kill' policy for police. Meeting with police and national security officials, he referred to the threat crime posed to the 2010 World Cup and to the confidence of investors wanting to do business in the country. The Deputy Police Minister, Fikile Mbalula, took up the call several weeks later employing war rhetoric. When it came to 'hard-nut-to-crack incorrigible criminals', he said, police must just 'shoot the bastards.' He added:\n\nWhere you are caught in combat with criminals, innocent people are going to die \u2013 not deliberately but in exchange of fire. They are going to be caught on the wrong side, not deliberately, but unavoidably.72\n\nThe sense that the townships were again seen as a war zone was underlined in 2010 when ranks in the police services were remilitarised. Senior ranks became generals and brigadiers, commissioned officers are now colonels, majors, captains and lieutenants while NCO's are sergeants and constables. The Police and Prison Civil Rights Union (Popcru) castigated the move as the 'imposition of a policy... in contrast with policies of the ANC's election manifesto.'73\n\nThe military system comes with a culture that previously had become entrenched within the apartheid police leading to little action being taken against police members who had committed acts of violence and torture against members of the public. Re-militarisation will diminish police accountability and over-sight by communities through provisions of community policing forums. Remilitarisation will reverse all the tenacious strides we have registered in the past 16 years of our democratic dispensation.74\n\nCape Town's high crime rate is, of course, not 'caused' by insufficient policing. Under the new political dispensation and in a troubled international economy, an alleviation of poverty and expansion of social services are clearly not taking place. This triggered rising levels of crime throughout the first two decades of democracy and caused a drain on police manpower, shortening the long arm of the law in the very area desperate for its extension \u2013 routine crime prevention. Blaming the police for the increased crime rate is like accusing a dam of failing to stop a flood. Because the SAPS is the government department with the greatest presence in communities demanding better service delivery, it remains the front line for both community anger and political pressure to stem the rising tide of social dissatisfaction and crime.\n\nPolice response has been a recruitment drive which increased the force, by 2015, to around 200 000 officers. It is now one of the largest police forces in the world with four times as many personnel as Australia, nearly twice as many as Great Britain and about the same number as Spain, Germany and France.75 At 317 officers, the number per 100 000 of population is considerably higher than the United Nations recommended minimum of 222. The rapid throughput of raw recruits, however, has decreased the quality of training and led to a lamentable decline in standards.\n\nIn 2004 there were 1600 detectives in the Western Cape who had received no formal detective training and mere constables were conducting murder investigations. The Khayelitsha Commission noted that recruits were being taken straight out of college to become detectives with very little practical training.76 This meagre instruction was given by experienced detectives who presented four-day crash courses on how to write dockets, what to look for when investigating a case, how to write formal letters and how to conduct an identity parade.77\n\nIn 2012, a Parliamentary Portfolio Committee highlighted the shortcomings of this inadequate training. It found that detectives were losing dockets or reporting them stolen. Statements were poorly written and crime-scene management procedures were sloppy. This lead to improper collection and protection of evidence with subsequent cases being thrown out of court. The net result is that, nationally, 23 per cent of the country's 23 676 detectives had not completed the most basic training and many were inadequately trained to use computers necessary for docket capture and the compilation of statistics. The committee heard that most detectives were first appointed to the job and trained later.78\n\nAn internal police report leaked to the Sunday Times in December 2012 and headlined Loose Cannons, revealed that more that 27 000 police officers had failed their firearms proficiency test yet still carried firearms. Of the 1 019 of these in the Western Cape, 448 were declared 'untrainable' and 'unfit to possess a firearm'. According to the paper, this posed 'a very high risk for the lives of colleagues as well as members of the community... and could lead to increased police killings.'79\n\nPolice handling of crime scenes is constantly being criticised by magistrates, largely because of poor collection of evidence. According to the South African Law Commission, of every 100 murders, rapes and aggravated robberies, only six result in conviction within two years of the arrest.80 While police may respond to a call in the city centre within minutes, on the Cape Flats it can take up to three hours. In some cases it may take days. In the poorer townships, the police have become outsiders to most residents and the association of some members with shebeens and gang bosses is widely commented on.\n\nAs Steinberg found in his research for Thin Blue, the police were quick to respond to domestic calls where they could assert their authority and masculinity behind a closed door, but steered clear of street incidents and gang battles where pacification was urgently needed. Commenting about the police to Steffan Jensen, a woman named Mouna had this perception:\n\nOh! They [are] corrupt. When you phone the police now, they come tomorrow. The people must first die here. They don't even come to the right place. First they drive 10 times around the block, then they will come to you. To me it looks like they are afraid to come here. When they hear about the shooting, they always come when the shooting is over \u2013 but while they [gangsters] are shooting, they don't come.81\n\nPoor management and lack of training was also taking place higher up in the police force. The Cape Times reported in March 2012 that 70 per cent of the police budget of R1.2-billion for infrastructure remained unspent, yet police across the country were being evicted from buildings because rent had not been paid or leases had expired. In Parliament, the chairwoman of the committee on police, Sindi Chikunga, grilled police bosses:\n\nIf this is not mismanagement, then I don't know. Other departments were complaining about their too-small budgets while the SAPS was effectively wasting its relatively massive budget. Basically what you are saying to the public is: 'We've wasted it.'82\n\nIt's hardly surprising that pressure on police personnel to solve social problems kindled a shoot-to-kill approach. Coupled with a lack of adequate training, this has produced a steady rise in police violence. Between 2009 and the end of 2011, 2 569 civilians were killed as a result of police action or while in police custody. This is one of the highest rates of police homicide in the world.\n\nDuring the year 2013\/14 there were 624 deaths in police custody, 80 of which were listed as suicide, and 390 deaths as a result of police action.83 A further 12 bystanders were killed by police action. The Independent Complaints Directorate received 5 754 complaints against the police, including 121 rapes and 3 994 accusations of torture or assault. These resulted in only 135 convictions and three dismissals for misconduct. In a 2008 study conducted among Cape Town sex workers, 12 per cent reported having been raped by police officers and 28 per cent reported having been asked for sex in exchange for being released from police custody.84\n\nMost cases against the police fly below the public radar, but some were too flagrant to escape press attention. Community leader Andries Tatane was killed by police during a 2011 service delivery protest in Ficksberg. A Mozambican taxi driver, Mido Macia, was tied to the back of a police van and dragged to the police station where he later died. A detainee was tortured to death by three policemen in Harberg, KwaZulu-Natal, for allegedly smuggling tobacco into police cells. The number of people suing the police is reflected in an increase in the amount budgeted by Parliament to cover 'contingent liabilites' \u2013 lawsuits and payouts \u2013 from a startlingly high R19 billion in 2013 to a mind-boggling R21-billion in 2014.85 This amounted to almost a third of the police's entire budget in that period.\n\nCriminologist Lisa Grobler found that gangs were very aware of the possibility of 'buying' cops and have strategies for luring them into a deepening web of 'favours'. It may begin with a few beers on night shift and increase to meat for a braai or payments for looking the other way. In Cape Town she found that police officers were collaborating with gangs by:\n\n\u2022 Stealing drugs from court exhibits;\n\n\u2022 Using police vehicles to transport drugs for dealers;\n\n\u2022 Acting as spotters for gangs by driving in front of and behind a gang car carrying a shipment of drugs;\n\n\u2022 Re-selling confiscated drugs;\n\n\u2022 Making firearms 'disappear' from evidence stores and re-selling them to gangsters;\n\n\u2022 Helping gangsters get firearm licenses;\n\n\u2022 Making dockets 'disappear', causing cases to be thrown out of court;\n\n\u2022 Making charges 'go away';\n\n\u2022 Sabotaging prosecutions with false or faulty evidence;\n\n\u2022 Tipping off detectives about drug shipments by rival gangs leading to arrests that improve the detective's performance ratings and chances of promotion (part of the consignment can be sold to the informer with money going into the detective's pocket);\n\n\u2022 Paying off police debts (one policeman struggling to pay for his house renovations found R25 000 in his bank account);\n\n\u2022 Being a full gang member who joined the police with the intention of corrupting the system in his gang's favour.86\n\nGrobler found that at some police stations bribery and extortion were the norm and that station commanders did not consider it worthy of investigation:\n\nCorrupt cops regularly visit foreign boats in harbours and demand bribes from sailors to allow prostitutes on board. Arresting illegal immigrants and demanding bribes from them is also a very popular pastime for corrupt members. Although bribery is a constant problem in the police service, few SAPS members are caught and convicted of this crime.87\n\nA member of the Americans gang in Manenberg scoffed when I asked him if he had trouble with the police:\n\nThe local police, naah, they're nothing. You can buy them. They just drive around, they don't get out of their van. There's a tik house next to where I stay. If the police are going to raid someone calls the owner and tells him the time they going to raid. He tells him this [is the] time we come so keep your stuff safe. The police come and there's nothing. They get you with a knife or gun, you pay and go.88\n\nTrouble within the force was not confined to street policing. Former Police Commissioner Jackie Selebi, who was also the head of Interpol, was convicted of corruption and of receiving cash payments from drug trafficker Glen Agliotti. His replacement, Bheki Cele, was suspended for 'dodgy' rental deals with billionaire businessman Roux Shabangu. Hillbrow Station Commissioner Koos van Rhyn was investigated for links to brothels and the criminal underworld in general. In September 2011 the head of crime intelligence, Richard Mdluli was arrested on charges of fraud and corruption.\n\nIt is worth noting that such institutional corruption, wherever it occurs, is the criminal subversion of State power. For criminals to have that power over a section of the police force, in addition to already being the force of local governance in its areas of operation, is a threat to national security. It transforms state institutions to serve a criminal purpose. This is not good for crime control.\n\nThe Institute for Security Studies has noted a gradual increase in crime since 2012 and the increasing brazenness of criminals. Gareth Newham claims that ructions in various law enforcement agencies have distracted authorities:\n\nThe big shift we've seen in the past two years is this incredible crisis of leadership in the national office; not appointing the right people, interfering with the Hawks. And that is really removing the ability for those that are tasked, at the top organisations, with providing the correct strategies, resources and guidance to focus on the crime and this is giving free range to the criminals.89\n\nPolice leadership problems were again highlighted in 2015 when the Farlam Commission of Inquiry recommended that Police Commissioner Riah Phiyega face an inquiry into her fitness to hold office over her handling of the situation when police gunned down 34 striking miners at Marikana platinum mine. That year Democratic Alliance MP, Diane Kohler Barnard, noted in a parliamentary debate that: 'To say the morale of police officers is at rock bottom would be an understatement.' According to Inkatha Freedom Party MP, Albert Mncwango, 'Generals are too busy taking time out of work, either being on suspension, or going to court instead of combatting crime.\n\nIn 2007 the Special Investigating Unit uncovered widespread social grant fraud perpetrated by an estimated 100 000 civil servants totalling R5,1 million. This sparked outrage among opposition parties in Parliament. Of the 100 000 public servants, 1 000 were employees of the SAPS. 'If a cop takes once,' an informant told Grobler, 'he will take forever.'\n\nGiven the seeming ineptitude and blatant disregard of the law by some top police managers, it is hardly surprising to find crime becoming more pervasive in officers down the line. Grobler put the number of police involved in criminal activities at 10%, with much higher levels in gang-saturated areas.90\n\nAccording to the Centre for the Study of Violence and Reconciliation, members of the police force have been implicated in nepotism; animal cruelty; wrongful arrests; drunken driving; intimidation; racism; scamming; extortion; fraud; theft; docket theft; possession of and selling illegal firearms (or their own); rape; murder; racketeering; drug trafficking; armed robbery; car-jacking; child trafficking; prostitution; heists and links to the Italian Mafia.91\n\nThe perception from the prowling police van is very different: the townships are full of skollies and if they want to kill each other, they should just go ahead and do it. A policeman told Steffan Jensen: 'Here people live like cockroaches and they breed like cockroaches.'92\n\nThe Khayelitsha Commission of Inquiry found that views were very often polarised around frustrations and misunderstandings between police and community. In an analysis of the commission's findings, criminologists Elrena van der Spuy and Adam Armstrong identified two competing narratives.93 On the one hand, individuals and NGOs working in the area pointed to police ineptitude, inefficiency or outright refusal to provide services. These were combined with endless delays in court proceedings and court sentencing which, they say, amount to State and human rights neglect. 'From these testimonies emerge the crucible of an unsocial space where violent crime impinges with regularity on inhabitants and their relatives. The police are described as incompetent, neglectful and abusive. Only in passing \u2013 rather sotto voce \u2013 is it acknowledged that pockets of good police do exist and that the environment is a harsh one for police too.'94 Police were described by the public as disinterested, rude and abusive. They were 'lambasted for their failures to deliver procedural justice and by implication social justice.'\n\nThe police narrative, on the other hand, contends that Khayelitsha suffers from endemic disorder with vast rural in-migration. Police consider themselves to be a 'mopping-up agency' bearing the brunt of service delivery frustrations and anger towards a 'distant' State. Police on the beat told the Commission that they see themselves as being chronically under-staffed and victims of distant and out-of-touch SAPS staffing allocations and regulations in volatile and dangerous situations. Brigadier Dladla of Khayelitsha noted that:\n\nWe are quick to say our detectives are not working, our detectives are working but they are overloaded. You know in the movies... you see a team ascending to a crime scene, attending to a docket. But here you have a team of dockets ascending on a detective.95\n\nA police commissioner pointed to the sheer weight of laws, policies, standing orders, instructions, directives and procedures which define police work. This type of situation 'saps morale and exposes detectives to high levels of occupational stress. In this kind of situation, as a coping strategy, they pick their battles, prioritise certain cases over others and seek opportunities to close files.96 Summarising the Khayelitsha Commission's findings on the role of the police, Van der Spuy and Armstrong write:\n\nThe South African Police organisation constitutes a large and cumbersome machine... The architecture of this large and rational bureaucratic machine is superimposed on a police system on the ground which struggles to make headway against high volumes of crime, endemic case overload, uneven levels of skills and pockets of corruption and considerable levels of demoralisation.\n\nFrom the deliberations of the Commission, the Janus-faced nature of the police institution and the society within which it operates becomes clear: the contradictory impulses of an institution that is both modern and primitive and a social locality that is at once orderly and anarchic.97\n\nThe high level of police violence during conflict situations has resulted in a corresponding high level of police fatalities \u2013 an average of 157 a year since 1994.98 Under pressure to deliver, police are increasingly turning their guns on themselves, with the national suicide levels among police personnel between 100 and 130 a year.99 A counselling psychologist, Dr Micki Pistorius, found that many police officers suffer from post-traumatic stress.\n\nI find my patients are demotivated, not only because of lack of promotion possibilities, but also by the fact that managers are not properly trained and lack human relations skills, time management and resource allocation skills. This puts tremendous pressure on the cop doing the work.100\n\nTough conditions, long hours and dangerous work without a perception of adequate compensation and recognition lead to discontent. This can be exploited by criminal temptation. Jensen was told that some officers made copies of all their files because of the frequency with which they were 'removed' from the police station by other officers. A commanding officer of a Crime Prevention Unit refused to post service rosters in the station and all drug raids were kept secret until the time of roll-call, minutes before the CPU went out, because police would tip off drug dealers.101\n\n***\n\nLet's widen the perspective. In most countries the law has an important role in the organisation of force. By making rules and passing laws, a state marks out its field of influence by way of orders, prohibitions and censorship. These are defined as law and the legal process establishes both the area and the object of state power. Law is, in effect, a code of organised public coercion.\n\nThe activities of the state clearly involve more than just establishing rules, however. In any society the state takes precedence over law. We can think of the state as the organ which makes the law and as the apparatus which compels us to observe that law. The front-line of this apparatus, the one with which we identify state power in our daily lives, is the police force.\n\nHere again it would be wrong to consider the police as mere instruments of the state. The police possess an autonomy which is not reducible to a simple analysis of state power. In fact, major contradictions exist between the various branches of the state and civil society. The one which concerns us here, is the contradiction between the laws of executive government which sanction physical coercion and the fact that the police who carry out these laws have to penetrate into the heart of disorganised, entangled neighbourhoods. There they are required to win a measure of consent in order to remain effective.\n\nThe police are a service organisation of an incongruous kind, since the police officer must discipline the public he serves. Police officers often meet with hostility and criticism from the public and might regard the residents of poor neighbourhoods as enemies to be feared. Their occupation feels to be in conflict with the community and they can be regarded as a pariah. These experiences and feelings can give rise to a collective emphasis on secrecy, an attempt to coerce respect from the public and a belief that almost any means are legitimate.\n\nA police patroller on the beat in a tough township is not simply the upholder of law and order but also an actor in a complex social network, playing the part both of aggressor and victim. On the one side they have trouble with the official 'system'. Police officers see the complexity of the law as threatening when in the hands of a clever lawyer in court. In their view, known criminals are set free through what are seen as technical loopholes in the judicial process. It is a common understanding among police officers that even when a detective is absolutely sure of a case, they can only be 50 per cent certain of a conviction. One detective told me:\n\nThere's a big problem with the courts. You get a skollie, he's a killer and he's been living away from home for years but he's under 18. Then when you get him to court you have to have his parents with him or a welfare officer. And his mother says he's a good boy and he's the only breadwinner and she's under oath and you have to believe her but it's all lies. And when he gets sentenced he gets sentenced as a kid and he goes to a reformatory and that's like a holiday for him. And next week he's back here, escaped.\n\nAlso in court cases, witnesses get smashed. The gang says, 'You testify and you and your family are dead.' And you can't get witnesses. They don't want to testify in court, although they will tell you all privately.\n\nThen if gangs fight, they will come in here to the police station and lay charges against each other. But in court the complainant is a gangster and the accused is a gangster and the witnesses are gangsters and they're all under oath and all telling the truth, of course, but their stories are all totally different and the case is thrown out. Or they all withdraw charges. It's useless!102\n\nThe police are further isolated from the community by time, function, geography, patterns of sociability and often by race. They work when many people are at home, asleep or at leisure. Often they work overtime, are separated from their families for considerable spells and, in many cases, are out of phase with the surrounding population. This social distance increases when the police are confined to living in certain areas, whether they be suburbs or, more particularly, in police barracks or housing-rows as the one built on the site of District Six.\n\nThese factors tend to give rise to a police subculture. This is reinforced by the fact that all police come up through the ranks and that class distinction among them is not marked. This subculture has more in common with the syndicates, mafias and gangs than with the general public because they, too, are located outside mainstream society and are often in conflict with it. For this reason, at street level, the line between police officers and gang members easily blurs.\n\nThere are, perhaps unsurprisingly, many parallels, holdovers and practices from apartheid policing. This was made chillingly clear by the shooting of striking miners at Marikana, near Rustenberg, in 2012 by members of an elite special unit of the South African Police Service. It was the most lethal use of force against civilians since the 1960 Sharpeville massacre. The 2012 commission of enquiry into the event, followed by the Khayelitsha Commission into Policing, made it clear that under-trained officers tended to fluctuate between utter inaction, fear of the community and extreme violence in ways reminiscent of policing that should have been a thing of the past.\n\nFor many people in low-income, high-risk areas, police action or inaction appeared unchanged since the demise of apartheid. On the Cape Flats, people have good reason to consider themselves besieged by two dangerous forces, gangs and the police. For many, neither have legitimacy but the gangs, comprised of neighbourhood youths, are better understood and have the greater validity.\n\n'When they are in need, the only safety net to fall back on, is their gang,' says Mayoral Committee Member for Safety and Security, JP Smith. 'What you have in terms of social cohesion is the gang, there is nothing else. It's the element that picks up people who have fallen through the cracks, the entity that gives bursaries and money when needed.'103\n\nGang activity, whether brutal, acquisitive or simply social, is a complex network of relationships and in this the police are key role players. For the purpose of illustration one could narrow down the scenario of law and order to the battle between two rival gangs, both composed of young, working-class males seeking territorial control over a particular habitat. The only difference is that one group has the full weight of the state behind them, the other has only themselves and their mates to fall back on. In the end, the scores they have to settle often resolve around two questions: Who benefits and who rules?\n\nIf you're running a city, this is an important question. Cape Town clearly has a gang and crime problem as well as management issues connected to poverty, migration from rural areas and service provision. How these are dealt with at national, local and neighbourhood level are issues of governance, raising questions of legitimacy and democracy \u2013 the voluntary, uncoerced acceptance of social control by people. It's through the provision of schooling, health, transport, housing, electricity, water and police services as well as urban renewal projects such as violence prevention and urban upgrading, that city administrators earn a level of legitimacy when services work and acrimony when they fail.\n\nAs a result, a neighbourhood like Manenberg has shifting allegiances: most people use legitimate services when they're available and illegitimate when not. Social control is distributed between local government, the SAPS, 'big men' in the criminal world and gangs.104 The result is an entanglement of influences with varying legitimacies, powers of enforcement and both legal and illegal collaboration. In her study of Manenberg, Dercia Lambrechs found there to be...\n\n...stratification of the organised criminal community, where the gangs fulfil some of the functions of the State, but the authority of the gangs is not based on legitimacy and the public good, but instead... on coercion and the desire to control segments of the State for its own benefit.105\n\nThis is a dangerous opposition for any system of formal governance. A state erodes at the margins if it lacks the capacity to stop the operations of armed non-governmental organisations. In his study of the vigilante group Pagad, historian Keith Gottschalk says 'the Max Weber truism \u2013 that the modern state has a monopoly on legal violence \u2013 reminds us that, to the extent that gangs rule and tax entire ghettos, bus and taxi stops and train stations, the state shrinks as a state.'106 On the Cape Flats, those who rule are defined daily, street by street, bullet by bullet.\n\nMob justice: when policing fails\n\nWe were called by the residents of Khayelitsha B. There were a lot of residents at the meeting. They told us that they called us because they wanted to remove Andile [her nephew] from Khayelitsha because he is giving them lots of problems. That he steals peoples' cellphones. They said they would pack his things and would chase him out of the area and they wanted to know if the family agreed with them. Some of them were very angry. As a family, we told them that we agree with them. Our main concern was that Andile should get out of the house so that he will not be harmed.\n\nThe Saturday morning I received a call saying that Andile had been found in a field and he was burnt to death. We went there. My sister-in-law stopped her car a short distance from Andile's home. There were a lot of people. My brother's wife was too scared to get out of the car. I told her that I would get off and go to look. I could see there was Andile burnt to death. His face was black. All I could say was 'God'. He was black, very, very dark from the face to the knees. He was very black, pitch black.\n\nI turned around. I was heartbroken and I was asking myself the question why would people be so cruel? Thereafter I went to my sister's house. I still can't believe the picture I saw.107\n\nMs Monakhuma Bontshi, testimony to the Khayelitsha Commission, 2014\n\nThe findings of the Khayelitsha Commission into Policing make clear that, for people on the ground in beleaguered Cape townships, the Bill of Rights, laws against gangs and new approaches to policing have little effect. They continue to be plagued by crime, apparent police apathy and even official collusion with gangsterism. It seems inevitable, given South Africa's history of street justice and kangaroo courts, that some people choose to turn to vigilantism and mob justice.\n\nA mob in search of revenge and justice functions as both a gang and a policing detail. It generally gathers when faith in the policing system has broken down and frustrations boil over. A mob's actions are purposeful, generally violent, always illegal and yet, in the circumstances, understandable. Vigilantism in Cape Town varies from a neighbourhood watch-type association prepared to take the law into its own hands to flash mobs prepared to kill someone pointed out as criminal.\n\nNoteable in the 'formal' type of vigilantism was the creation, in 1995, of People Against Gangs and Drugs (Pagad). This comprised a group of middle class and mainly Muslim residents. The organisation decried the state's inability to protect their community and it set out to remedy the situation with mounting absence of restraint. It became increasingly militant and violent, dragging druglords from their homes and publicly beating them. In this way, it constituted a direct challenge to the state's monopoly on the use of force.\n\nIn 1996 Pagad publicly executed Rashaad Staggie, one of the leaders of the Hard Livings gang and was blamed for a series of bombings of police stations and restaurants in the city centre. Its militancy was partly fuelled, according to sociologist Tony Samara, by the police decision to treat it as 'just another gang.'108 A number of its members were tried for murder and jailed, reducing its effectiveness. It remained dormant for many years, but re-emerged in 2011 after the release of its leaders from prison. In a meeting I attended in Lavender Hill that year called to protest gang activity, people carried posters reading Pagad In, Police Out. Its leader, Abdus Salaam Ebrahim, was handed a microphone and spoke to a cheering crowd.\n\nShortly afterwards, I accompanied Pagad on a night trawl for drug merchants. We met at a mosque, then went hunting. Seeing the tough men with their Saudi-style scarves, one carrying a hammer, I was glad I wasn't their quarry. They banged on a door and barged into the house, demanding the whereabouts of a known dealer. They found him cowering on the roof and dragged him into a corner of the lounge, screaming at him and slapping him around. He begged for mercy and said he only used dagga.\n\n'You lie! You fucking bastard!' the raid leader shouted. 'You sell tik to our children. You're scum.' I was asked to leave and as I stepped onto the veranda a deathly, blood-chilling scream tore the night. I thought he was dead, but minutes later the group emerged pushing the dealer ahead of them. In the street, with the neighbours watching, he stood with his hands together, seemingly in prayer, and renounced drug dealing, his former life and anything else they wanted him to give up. The terror in his eyes was palpable. There were two more house raids that night.\n\n'The police are doing nothing,' Abdus Ebrahim told me. 'They don't care about coloured people. So we have to do this to protect our children. What would you do if a drug dealer sold tik to your child?' I wasn't sure and he had a point.\n\nIn the sprawling township of Khayelitsha, retribution for criminals is far less organised than Pagad, more public and much more violent. During a two-year period ending in 31 March 2014, vigilante mobs killed 73 people suspected of crime, some as young as 16. When people were arrested in connection with these killings, the cases against them were generally withdrawn through lack of evidence or police bungling of the investigations.109 In 21 cases investigated by the Khayelitsha Commission, there was only one conviction. Two cases had been struck off the roll, two withdrawn, two postponed for further investigation and two left pending for reasons not evident from the docket.110\n\nVictims of these vigilante attacks had been beaten, stoned, stabbed and sometimes necklaced.111 In Philippi Vuyane Kose was accused of breaking into houses and stealing goods. He was attacked by about 20 people, mostly young women, who beat him senseless with planks, rocks and sjamboks before dragging him onto a fire. According to a resident, Kose had pleaded for mercy, saying, 'My friends influenced me to steal so we could buy drugs. I would never hurt you again.' He was rescued by police but died in hospital. Residents told the Cape Times: 'We are fed up with the police and the justice system. We know these criminals will be back the same week after their arrests and we are eliminating them one by one. This is a message to would-be criminals that we are fed-up with crime.'112\n\nThere is apparently widespread acceptance of this form of justice in the face of what respondents to a Pondering Panda survey described as 'useless police.' In the nationwide survey, 79 per cent of respondents felt 'vigilante justice' was always or conditionally socially acceptable, 34 per cent believed it was because the courts did not hand down tough enough sentences and 16 per cent said it was because courts took too long to deliver justice.\n\nAccording to Shirley Wakefield of Panda, people feel the justice system in South Africa is failing their communities when it comes to tackling crime and that vigilantism and mob justice are valid means of punishing criminals.'113 According to a member of the Social Justice Coalition based in Khayelitsha, 'people are so sick of crime you just have to shout \"thief\" and everybody comes running. It's as easy as that.'114\n\nIn his submission to the Khayelitsha Commission, Lieutenant General Arno Lamoer described vigilantism as an act of opportunism, saying: 'People start running and screaming and automatically community members react and before asking questions the act happens.' This was, in his view, not because SAPS systems fail, but because people wanted 'swift justice'. He insisted that murders arising from community action were a small proportion of all murders and not indicative of a breakdown in the relationship between the community and SAPS.115\n\nUniversity of the Free State clinical psychologist, Pumla Gobodo-Madikizela, told the Commission that both trauma and violence were trans-generational. Vigilante action was better understood as a political statement, rather than the actions of a senseless crowd.116 She said people were generally traumatised by the levels of violence and vented their anger on people considered to be the cause of their shared community fears. An informant told her:\n\nI was brought up by my grandmother, she was gentle and she never beat me. As a young boy, she taught me strong values, respect and compassion for others. But moving here and growing up in Khayelitsha changed me. How could it not change me \u2013 death was no longer something in a coffin at a funeral. It was right here on my doorstep when I went to school, and later when I was an adult, the smell of death was everywhere, not so much because of dead bodies, but because you see people being killed in the most violent manner.117\n\nAnother young informant told her:\n\nWe struggle to find jobs, and when we do, we work so hard to own the little things that give us a sense of dignity, and then someone breaks into your house and steals it. It is like someone has stolen your dignity.118\n\nThe Commissioners disagreed with Lamoer's depiction of vengeance attacks as being a 'small proportion' of murders. In summing up, they wrote: 'It is apparent that SAPS does not have accurate reliable data or crime intelligence as to the extent of vengeance or vigilante attacks in Khayelitsha. The record makes clear, however, that vengeance attacks are common.'119 An informant told the Commission: 'While the attack was happening, there were several police officers having lunch in a nearby restaurant, who failed to take any steps to halt the attack.120\n\nThe Commission concluded that the police 'currently have no strategy to address vigilantism or vengeance attacks in Khayelitsha.' It accepted that bringing an end to vengeance attacks was not something that the SAPS would be able to do quickly or on its own. But, it said, 'failure to develop a strategy to address the worrying number of such attacks in Khayelitsha constitutes an inefficiency in policing.121 A member of the community told one of the informants:\n\nWhen you are squashed into a tight corner, you find fault with an easy target. Vigilantism is the easiest, shortest option by people that are in desperation.\n\nTransnational crime comes knocking\n\nUntil now I've have been probing that part of Cape Town's history which has underpinned the development of gangs. Beyond South Africa's borders in the 1980s, however, indistinct forces were gathering which would add a disturbing new dimension. Democracy, inadvertently, was its handmaiden.\n\nToday, woven deeply into the urban tapestry of Cape Town, is a shadow city, its threads firmly anchored in the capitalist economy, knotted into local control and governance systems but extending far beyond the borders of South Africa. Its activities are hidden to all but those who know where to look and \u2013 when you do \u2013 it suddenly becomes apparant everywhere. Its influence is considerable and maintained through covert connections, graft, coercion, illegal trade and, not infrequently, murder.\n\nUntil the early 1990s, South Africa was a country apart, encircled by fences and surveillance systems with an army on the alert for ANC guerillas and the threat of communism. Criminal rackets were largely local. An example of this localism in Cape Town was the subversion of bottle stores. Throughout the 1970s and 1980s, in line with the perception that coloured people abused alcohol, the government owned, managed and limited to about 20 the number of bottle stores in coloured areas. These did not grant credit to 'non-whites' and were generally bleak places with nowhere to sit, eat, dance or listen to music.\n\nThis attempt at controlling the use of alcohol backfired when individuals fitted out their homes as shebeens where illegal liquor could be bought and consumed in congenial surroundings. Shebeening spread rapidly across the Cape Flats, supported covertly by alcohol producers that found these watering holes a convenient way to offload low-grade wine and spirits.122\n\nTrouble easily follows alcohol consumption, however, and shebeen owners could not expect police to respond to calls from their illegal establishments. In most areas the local gang was amenable to a protection arrangement against alcoholic excesses by patrons. They had the added value of being lookouts against police raids and for collecting outstanding debts from customers on payday. Gangs could, however, also insist on protection payments from shebeens in return for not wrecking them, a practice that was to extend to all business in some areas. A gang member from Manenberg explained:\n\nSee, sometimes the HLs [Hard Livings] can come in and rob you. They can take the stuff anytime if they want to, but if you want protection you've got to pay us for the protection so that you don't get robbed and if you don't want to die, you also have to pay for protection.123\n\nCertain police officers, alert to useful additions to their own pay and pleasure, extracted favours from shebeen owners in the form of cash, liquor and meat. In the 1980s, while in a Hanover Park shebeen interviewing the owner, I was startled when two uniformed officers walked in and, after a brief conversation with the shebeener, left with four bottles of brandy. None of the customers batted an eyelid. Shebeens were technically illegal but, in practical terms, largely ignored.\n\nA similar situation resulted from the prohibition of mandrax tablets, which were developed as a cheap legal sedative. In the 1970s mandrax became popular as a recreational drug when mixed with alcohol. Its use soon exploded across the Cape Flats when smoked in a 'white pipe' mixed with cannabis. Banned in the late 1970s, its import and eventual local manufacture became one of the most lucrative illegal industries in Cape Town. It was sourced by merchants and sold on the streets by gang members. Minor skirmishes over control of sales turf soon erupted into deadly gang wars. In this way gangs began growing in wealth and power.\n\nWhat was to come, however, would take even them by surprise. After the fall of apartheid in 1994, a door opened and the world entered. Through it poured political goodwill, investment, tourists and the criminal underworld. A new boom in the trade of illegal goods and services, effected by everyone from top government officials and businessmen to kids on the ghetto street corner, changed everything.\n\nIn the slipstream of globalisation throughout the 1980s, but not yet felt in South Africa, transnational criminal activity had spread through the world, operating across national boundaries and outpacing the ability of enforcement agencies to contain it.124 This change fed on the expanding information and communication revolution, the increasing interconnection of banking systems and the opening up of opportunities following the end of the Cold War. Particularly, however, it fed off transitional political confusion and targeted weakening of control systems in Russia, Central Europe, South America and Africa.125 After 1994, a globally re-energised city such as Cape Town \u2013 conveniently positioned between east and west and historically a colonial conduit for import and export \u2013 could hardly escape the attention of world-savvy criminal networks.\n\nRoad transport links to the north opened up into Zimbabwe, Zambia and the Congos. Secret containers on trucks, trains and ships carried poached ivory, rhino horn and abalone, together with rare metals and precious stones. Return journeys brought drugs, cheap imitation clothing, cigarettes and other useful international items. Often these imports were paid for in stolen motor vehicles or abalone. Crime, particularly indigenous and transnational crime, flourished. Mark Shaw notes that weakening border controls in South Africa...\n\n...occurred at a time when transnational criminal operations were expanding; just like 'legitimate' multinational businesses, East Asian, Nigerian and East European groups bought into local South African criminal operations and expanded them, or contracted subsidiary organisations to conduct their work for them.126\n\nIn 2001, the Minister for Safety and Security at the time, Steve Tshwete, warned that organised crime had 'extended its tentacles into South Africa after the country's return to the global arena.' Since it opened to the world, he said, [South Africa] had felt the effects of transnational organised crime syndicates attempting to extend their tentacles to 'new markets':\n\nGiven South Africa's relatively well-developed infrastructure, modern telecommunication systems, technology and business practices, it would appear that the scope of organised crime has evolved from generally small-scale local operations to international syndicates.127\n\nThe growth in criminal activity that followed the end of the Cold War along with the globalisation of economies, occurred so fast that police and policy makers were slow to catch up. Urban hubs like Nairobi, Johannesburg, Lagos, Dakar, Kinshasa, Addis Ababa and Cape Town rapidly became epicentres for the spread of organised crime across the sub-continent.128 Estimating the scale of this activity is difficult because extensive corruption at government level in many countries has distorted official crime reporting and warped statistics. For this reason the reported volume of seized drugs, for example, is no longer a useful indicator of total illicit drug flows.\n\nCriminal groups that entered South Africa's porous borders in the 1990s were found by criminologist Andre Standing to include the Russian Mafia, Chinese Triads, Nigerian networks, Italian Mafia, Colombians, Peruvians, Pakastanis, Bulgarians, Portuguese and Congolese. To these were added weapons-trained, demobbed South African Defence Force (SADF) and umKhonto we Sizwe (MK) soldiers and local Special Defence Unit members. Standing quotes estimates by some authorities that these groupings, together with increasingly corporatised gangs, were at the time responsible for as much as 70 per cent of all crime on the Cape Flats.129\n\nIn economically depressed areas of the city, with the transitional police force in disarray and seismic shifts in the balance of political authority, the power of the law was severely limited by a widespread feeling that it was not legitimate. This reduced resistance to other, more immediate centres of power and less legal forms of income. Waning legal agency was not confined to Cape Town, but was and still is a worldwide problem. The United Nations Research Institute for Social Development, reporting on transnational crime, noted perceptively that:\n\nAt a time of narrowing economic opportunity across wide areas of the world, participation in the illegal economy... constitutes one of the few realistic options available to many families who simply need to ensure a basic level of subsistence. Illegality makes certain commodities or services unusually profitable.130\n\nCriminal power, however, is not simply about the illegal provision of goods and services. It's also the ability of gangland leaders to accrue status and become respected and feared by claiming ownership of a territory, running mob protection and using threats or outright violence. The rise of criminality is also often, as the UN points out, a rational response to economic hardship \u2013 what some sociologists call an 'adaptive mechanism' to poverty. Cheap stolen goods 'dull the ache of deprivation'.131\n\nCrime bosses, whether they're gang leaders or independent drug merchants able to buy gang power, are essentially the result of the inability of the state to provide a monopoly of force and basic social and economic security.132 They're self-appointed providers of arbitrary and illicit forms of justice. In this they are willing partners in organised and transnational crime.\n\nCounting the cost\n\nBefore venturing into contemporary gangland, I needed to take a snapshot of what Cape Town looked like by numbers. What was happening to ordinary people on the ground? Cape Town, it turned out \u2013 and not unexpectedly \u2013 was producing some deeply worrying statistics.\n\nFirstly there were a lot of young people. In 2013, of the 5 288 734 people living in the Western Cape, just over 2.5 million were between the ages of 10 and 35 \u2013 that's 44 per cent of the population. Most of them (66 per cent) lived in the Cape Town metropolitan area. Many were recent immigrants. While 58 per cent were born in the Western Cape, a considerable number (20 per cent) were from the Eastern Cape.133\n\nAmong young people aged 10 to 24, one in three (833 333 children) were cared for by a single parent. Think about that for a moment. Given the way things usually go, that means nearly a million kids growing up without fathers (we'll talk about the implications of this later). Almost as startling was that a quarter of all children between 10 and 24 were being raised by family members but not their own parents, generally their grandmothers.134 Around four per cent lived with people who were not their family at all. So where were their parents?\n\nLarge numbers also lived in homes with role models of doubtful character. A 2013 report found that a quarter of young people under the age of 18 in the Western Cape had a parent or sibling who had been to jail and just under a quarter had family members who used drugs. Around 15 per cent lived in a household where someone was a member of a gang and about the same number of homes had no working adult.135\n\nThe life into which many young people are born on the Cape Flats is one of poverty of income, resource and opportunity. In the poorer areas of the city, nutrition, health and social dislocation are ever-present worries. Among Cape Town's 3.7-million residents, a Western Cape Development Survey found that a quarter of all adults and just under half of under-25s were unemployed.136 One in five people were living in self-built shacks and more than a third of all households were headed by women, most probably a single parent. In the words of a young gang member:\n\nMost young people growing up in the townships like Khayelitsha are raised by a single mother or a granny. In conditions like that, you search for a person outside the house to close that gap and give you what you can't get at home.137\n\nIn the country as a whole (the figures were not broken down by city), 15.3 per cent of people could not read or write, 31.8 per cent worked in non-formal employment and 27.3 per cent of households subsisted on welfare grants. Nearly a third were hampered in job searching because of ill health, eight out of 10 were ill in the month prior to a National Health and Nutrition Survey and of all households, 18.8 per cent of adults and 12.3 per cent of children went hungry every day. Asked about the essentials for living, between 23 per cent and 29 per cent of people rated food, housing, clothing and health care to be inadequate.\n\nAccording to the nutrition survey, 30.3 per cent of African people and one in four coloured people were classed as food-insecure \u2013 they did not have access to a constant supply of healthy food. A 2008 survey found that 32 per cent of households in the Western Province and an extraordinary 94.9 per cent in Manenberg had combined incomes below the poverty datum line (R604 a month at the time) and 38 per cent of economically active people were unemployed.138 Dietary diversity is essential to health and a score of six to eight in the survey score is recommended. Graphed this way, the Western Cape score is two. Nationally, the intake of fruit and vegetables is half the World Health Organisation's recommendation of 400 grams a day.139\n\nThese are not conditions or places within which healthy children can grow or where young people can make choices to better their lives. They're places for the breeding of desperate, individual antisocial actions. However, they are extremely fruitful hunting grounds for criminal networks needing young men and women to break the law and for gangs in search of soldiers to protect turf and carry guns. These places are the feeding grounds of a world with which those with wealth and agency are seldom willing or able to engage.\n\nWe turn now to the mysterious world of gangs in Cape Town, but with an essential caveat. A general impression in government and local media is that gangs are the cause of the city's crime. While they do engage in crime, deal drugs, act violently at times and are implicated in many murders, they are not the main perpetrators of contact crime.140\n\nAreas of the city with the highest levels of reported contact crime tend to be those with lower gang presence and large populations of recent migrants. Gangs are simply a more organised and observable aspect of a city which has high levels of very generalised violence. This violence occures within families, towards peers and partners, in drunken shebeen brawls, out of bravado and during robberies. So while I'll be focussing on gangs and particularly their younger members, we need to keep in mind that the problem of disaffected youth is much wider than the gang turfs under discussion and exists for similar reasons. Cape Town doesn't have a gang problem so much as a youth problem of which gangs are one of the outcomes.\n\nA frightening demographic ensures that this problem will continue to grow unless major economic and social transformation occur. Africa is experiencing rapid population growth and urbanisation. At the present rate, by 2050 it will be home to twice as many people \u2013 2.1 billion \u00ad\u00ad\u2013 equaling a quarter of the world's population. It's an exponential growth rate with no equivalent in human history.141 If present urban income disparities continue, rising social unrest and violence throughout the continent's burgeoning cities will be inevitable.\n\nEndnotes\n\n1. Barrow, Brian: Behind the dark doors of District Six (article in Wollheim Collection, UCT, 1966).\n\n2. Cape Times, 20.6.1950.\n\n3. Webster, D: 'The social organization of poverty' (seminar paper, African Studies Institute, Wits University, 1980), p15.\n\n4. Ibid., p16.\n\n5. Union of South Africa Parliament, Report of the Cape Coloured Commission, U.G. 54-1937.\n\n6. Interview with Sergeant Willem Nel, former head of the Police Special Squad, District Six.\n\n7. U.G. 54-1937.\n\n8. Interview with Sergeant Nel.\n\n9. The Torch, 20 June 1946.\n\n10. Interview with Gums, a Globe Gang member, 1980.\n\n11. Interview with George Manuel, 1980.\n\n12. Interview with Vincent Kolbe, 1981.\n\n13. Ibid.\n\n14. Pinnock, Don: 'Argie boys to skollie gangsters: the lumpenproletarian challenge of the street-corner armies of District Six, 1900-1951' (paper, History Workshop 3, UCT, 1981).\n\n15. Kolbe, op cit.\n\n16. Hansard, 1951, col 4785.\n\n17. Cape Argus, 8 December 1949.\n\n18. Ibid., 22 May 1952.\n\n19. Interview with Gums, a Globe member.\n\n20. Interview with Kolbe, op.cit.\n\n21. Interview with Elizabeth Weeder, December 1981.\n\n22. Norval, AJ: 'A Quarter of a Century of Industrial Progress in South Africa' (Cape Town, 1962) as quoted in G. Bloch, The development of manufacturing industry in South Africa, 1939-1969 (MA, UCT, 1980), p91.\n\n23. Union of South Africa Parliament, U.G., 4-1921.\n\n24. Union of South Africa Parliament, The Joint Report of the Asiatic Land Tenure Laws Amendments Committee and the Land Tenure Act Amendments Committee, U.G. 49-1950.\n\n25. Van den Berghe, PL: 'Racial segregation in South Africa: degrees and kinds', in H. Adam (ed.), South Africa, Sociological Perspectives (London, 1971), pp37-8.\n\n26. Brian Barrow in Cloete Breytenbach: The Spirit of District Six (Struik, Cape Town 1970).\n\n27. Wollheim, OD: 'Peninsula's tragedy of the poor', Cape Times, 26.1.1959.\n\n28. Interview with Dr Wollheim, March 1981.\n\n29. Western, Outcast Cape Town, p312.\n\n30. Republic of South Africa Parliament, Commission of Inquiry into matters relating to the coloured population, R.P. 38-1976 [Theron Commission], pp261, 27.\n\n31. Wollheim, undated article on Group Areas in Africa South, in the Wollheim Collection, UCT.\n\n32. The survey was conducted by myself in 1982.\n\n33. Western, Outcast Cape Town, p312.\n\n34. Pinnock, D: The Brotherhoods (1984).\n\n35. Interview with Petrus, Manenberg, 1982.\n\n36. De Tocqueville, Alexis: op cit., p584.\n\n37. Pitman, H: as quoted in Hansard, 1982, col. 6353.\n\n38. Argus, 27 February 1982.\n\n39. R. Swart, as quoted in Hansard, 1982, cols. 6304-5.\n\n40. Cape Times, 15 February 1979.\n\n41. To the Point, 1 August 1975.\n\n42. Interview with a policeman, 1982.\n\n43. Republic of South Africa Parliament, Commission of Inquiry into the Penal System of the Republic of South Africa, para. 3 January 2015 p 35.\n\n44. Interview with policeman, 1982.\n\n45. Cape Argus, 4 April 1982.\n\n46. Interview with policeman in Grassy Park, 1982.\n\n47. Interview with a detective, 1982.\n\n48. Steinberg, Johnny: Thin Blue: The unwritten rules of policing South Africa (Jonathan Ball, Johannesburg 2008).\n\n49. Standing, A: Organised crime: A study from the Cape Flats (Institute of Security Studies, Cape Town, 2006).\n\n50. Cape Times, July 10 2001 in Tony Roshan Samara: Cape Town after Apartheid: Crime and Governance in the Divided City (Kindle edition).\n\n51. Samara, op cit.\n\n52. Samara, op cit.\n\n53. Samara, op cit.\n\n54. Samara, op cit.\n\n55. Samara, op cit.\n\n56. Samara, op cit.\n\n57. The standard statistical measurement used to describe the extent of wealth polarisation.\n\n58. Samara: Cape Town after Apartheid. Kindle edition, npn.\n\n59. Evidence to the Khayelitsha Commission, 2014.\n\n60. Steinberg, op cit, p98.\n\n61. Memorandum to the ANC on the re-militarisation of the SAPs, Popcru, April 2010.\n\n62. Jensen p133.\n\n63. Jensen, op cit, p133.\n\n64. Samara, T: Cape Town after Apartheid: Crime and Governance in the Divided City (Kindle edition).\n\n65. Samara, op cit.\n\n66. Vearey, Jeremy: Presentation at a workshop on the development of a framework for interdepartmental anti-gang strategy in South Africa, Cape Town, March 2015.\n\n67. Ibid.\n\n68. Samara, op cit.\n\n69. Jensen, op cit, p128.\n\n70. Jensen, p141 & 142.\n\n71. Mail & Guardian, July 25 2008.\n\n72. BBC News, 12 ovember 2009 in Samara op cit.\n\n73. Memorandum to the ANC on the re-militarisation of the SAPS, Popcru, April 2010.\n\n74. Ibid.\n\n75. Wikipedia.\n\n76. Towards a Safer Khayelitsha: Report of the commission of inquiry into allegations of police inefficiency and a breakdown of relations between SAPS and the community of Khayelitsha, 2014.\n\n77. Grobler p146.\n\n78. Grobler p147.\n\n79. Sunday Times, 4 March 2012, p1.\n\n80. Samara, op cit.\n\n81. Jensen p120.\n\n82. Cape Times, 7 March 2012.\n\n83. Independent Police Investigative Directorate Annual Report 2103\/14.\n\n84. Grobler, p97.\n\n85. News 24, 3.October 2014.\n\n86. Grobler, op cit.\n\n87. Crossing the Line: When cops become criminals, L Grobler, p77.\n\n88. Interview with Michael, Manenberg November 2014.\n\n89. Whittles, Govan: Crime in SA increasing rapidly, Eyewitness News online, 22 May 2015. The Hawks are a Priority Crime Investigation unit targeting organised and economic crime and corruption.\n\n90. Liza Grobler: The Murky Symbiosis of Dirty Cop and Gangster, UCT thesis 2006.\n\n91. Police Corruption in South Africa by the Centre for the Study of Violence and Reconciliation, eJays website.\n\n92. Jensen, op cit, p122.\n\n93. Van der Spuy, E and Armstrong, A: 'Policing of an urban periphery: The case of Khayelitsha' (South African Journal of Criminal Justice, Vol 27, No3, 2014).\n\n94. Van der Spuy & Armstrong, Ibid.\n\n95. Dladle, Z: Towards a Safer Khayelitsha Report at 3506.\n\n96. Van der Spuy & Armstrong, op cit.\n\n97. Van der Spuy & Armstrong, Ibid.\n\n98. SA Institute of Race Relations South Africa Survey 2010\/11 pp773\/4\/5.\n\n99. Joanne Hitchings: 'Collapsing morals behind SAPS rot', in Cape Times, 9 March 2012.\n\n100. Hitchings, op cit.\n\n101. Jensen, pp125\/129.\n\n102. Interview with a detective, 1982.\n\n103. Gonschorek, Anne: Ex-gangsters Take Fight against Violence into Their Own Hands (www.contributoria.com\/\u00adissue\/\u00ad2014-12\/\u00ad54357a03ca48bfca\u00ad20000001).\n\n104. Lambrechts, op cit, p194.\n\n105. Ibid, p199.\n\n106. Gottschalk, K: Vigilantism v. the State: A Case Study of the Rise and Fall of Pagad, 1996-2000 (ISS Occasional Paper 99, February 2005) p10.\n\n107. Testimony given to the Khayelitsha Commission, 2014.\n\n108. Samara, op cit.\n\n109. Cape Times, 31 March 2014.\n\n110. Towards a Safer Khayelitsha: Report of the Commission of Inquiry into Allegations of Police Inefficiency and a Breakdown in Relations between the SAPS and the Community of Khayelitsha. p 386.\n\n111. Necklacing is placing a petrol-doused burning tyre around someone's body.\n\n112. Cape Times, 6 February 2014.\n\n113. Cape Times, 15 March 2013.\n\n114. Mail & Guardian, 22 June 2012.\n\n115. Khayelitsha Commission. Exhibit Al1, Record Bundle 11(5) at 6520. In 2015, unconnected to the Commission, Lamoer was arrested for corruption.\n\n116. Ibid at paragraph 219.\n\n117. Khayelitsha Commission, Record Bundle 12(1), Item 24, p10.\n\n118. Record Bundle 12(1), Item 24, paragraph 6.\n\n119. Khayelitsha Commission, p386.\n\n120. Khayelitsha Commission: Mr Persson statement Bundle 7, File 11, Item 78, Transcript at 2648 \u2013 2666 (11 February 2014).\n\n121. Ibid, p388.\n\n122. Standing, A: Organised Crime A Study from the Cape Flats (Institute of Security Studies, Pretoria, 2006) p9.\n\n123. Lambrechts, Dercia: The Impact of Organised Crime on Social Control by the State: a Study of Manenberg in Cape Town, South Africa (PhD thesis, Stellenbosch, 2013) p166.\n\n124. Gastrow, Peter: Organised Crime in South Africa: an assessment of its nature and origins (ISS Monograph 28, 1998).\n\n125. Gastrow, ibid.\n\n126. Shaw, Mark: State Responses to Organized Crime in South Africa, Transnational Organized Crime, 3(2), Summer 1997, p3.\n\n127. Tshwete, Steve: 'Legislative responses to organised crime in the SADC region', in C. Goredema (ed), Organised crime in South Africa: Assessing legislation, Institute for Security Studies, Pretoria, 2001, p3.\n\n128. Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013) pp2 & 14.\n\n129. Standing, op cit, pix.\n\n130. Cited in Standing, 2006, p206.\n\n131. Standing, ibid, p206.\n\n132. Standing, ibid, p204 and 231.\n\n133. Western Cape Youth Development Strategy, 2013, (Western Cape Government) p19.\n\n134. Barak Morgan: 'Biological embedding of early childhood adversity: Toxic stress and the vicious cycle of poverty in South Africa' (Research & Policy Brief Series, Ilifa Labantwana, November 2013).\n\n135. Western Cape Youth Development Strategy, 2013, p20 & 25.\n\n136. The percentages were 25.4% for adults and 48% for under-25s.\n\n137. Kossie, op cit.\n\n138. Poswa, Nontembeko: Characteristics of households living in poverty, City of Cape Town, 2008.\n\n139. South African National Health and Nutrition Examination Survey, Department of Health, 2013, p177.\n\n140. SAPS crime statistics, September 2014.\n\n141. Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013) p14.\n2. Cape Town's gangs\n\nWhat is a gang?\n\nMany of the gang are BJs [boerjongens, farm youths] but you'll see the leader is always a local person. He's got a house, a headquarters, a pad to sleep. These upcountry guys get arrested and they're sent to Tokyo [Porter Reformatory, which was situated in Tokai]. There they stay maybe two years and they meet guys there who say, 'We'll fix you up when you come out.' Maybe they join a gang in reform.\n\nWhen their time comes they don't want to go back home now. There's nothing for them to do there and now they've got gedagtes [understanding] and they want to stay in town. So a gang leader he says to them, 'You can stay here by me and get to know the place. I can find you work.' So there he starts, you see. Little jobs \u00ad\u2013 go get some sweets or milk from the shop, sell a little garu [dagga] and maybe some buttons [mandrax]. Maybe after a time he can be trusted to pull a job. This guy, he sees the gangsters are making more money every day than his father makes in a month. So he moves with them. After two years or something he will get stamped [tattooed] but he must be able to pull weight. Then he's full blast.\n\nThis is how it happens: the kring [circle] sits, they're the main ouens, the top guys. These are guys who are in the various areas, they're pulling a lot of guys. One day the leaders of these branches [of the same gang] come together and decide to stamp their lighties. The new ones don't know about all this. Maybe they decide to leave one lightie out they're not sure of \u2013 he's not pulling weight \u2013 so then it will make him work harder and he'll become more brave.\n\nThere's a tattoo master in each gang, they use a needle and Indian ink or a zombie [the black rubber seal in food jars]... they burn it down and mix it with water. Then they get called in, but before their badge they must remember the laws \u2013 they will be asked. For the Mongrels it's 26 because they're 26s. So they get their tshappie [tattoo]. Afterwards when they're initiated they're like brothers \u2013 here's a law among them. And they won't be treated as lighties, it's no more 'Get this; do that'. They can sit and smoke together.\n\nYou see this happens in the big gangs where there's discipline. There's very strict discipline, the leader just uses his eyes. But in the younger gangs, like the Hobos in Bonteheuwel or the Sicilians in Elsies River, there's no discipline. It's frightening, man! These young lighties with guns and they're 15. They're hit merchants [drug salesmen], rob anyone. There's no thought. It's getting worse all the time.\n\nChicken (Patrick Edwards), Hanover Park, 1982\n\nTo study gangs you need to know what they are, but it's not as easy as you imagine. On the face of it there's no problem. I asked a member of the Hard Livings in Manenberg to give me his definition of a gang. 'Us,' he said, slapping his hand to his chest. In Lavender Hill, I asked the same question of a mother escorting two young children to school. 'Them,' she replied, pointing at a group of young men at the end of an alley. There was no hesitation, no doubt, and this would probably be true for most people living on the Cape Flats. Gang identity is so strong and obvious for both insiders and outsiders that enquiring beyond 'us' and 'them' merely evokes puzzlement. But it's not that simple. 'Gang' is also a mythology, a boo-word and a legal definition into which a range of different activities are crammed.\n\nThe original meaning of 'gang', passing from Old Norse into Old English, was simply 'a journey', possibly derived from the fact that travelling alone was dangerous. The modern dictionary definition is 'a group of people who regularly associate for a common purpose.' In modern usage, when the word is used regarding urban social formations, the connotations are generally negative. In that context it has been emptied of its descriptive valiancy and has become an ideological construct. In a groundbreaking study of Chicago gangs published in 1927, Frederic Thrasher defined a gang as:\n\nAn interstitial group originally formed spontaneously and then integrated through conflict. It is characterised by the following types of behaviour: meeting face to face, milling, movement through space as a unit, conflict and planning. The result of this collective behaviour is the development of tradition, unreflective internal structure, esprit de corps, solidarity, morale, group awareness and attachment to a local territory.1\n\nThis is essentially a sociological construct, a template against which the gangs under study could be measured at the time of Thrasher's observations. Significantly, it has no reference to delinquency or crime and lacks the malignant criminal overtones of modern definitions. The gang was a relatively benign organisation, a by-product of urban crowding, poverty and population density.2\n\nIn 1998 the South African Parliament passed the Prevention of Organised Crime Act (POCA). This deals with the forfeiture of property and the recovery of assets acquired by illegal means. It assumes gang complicity in organised crime by the inclusion of a section dealing with membership which it defines as:\n\nAny formal or informal ongoing organisation, association, or group of three or more persons, which has as one of its activities the commission of one or more criminal offences, which has an identifiable name or identifying sign or symbol and whose members individually or collectively engage in or have engaged in a pattern of criminal gang activity.\n\nUnder this definition, a person is considered a member of a criminal gang if, among other things:\n\n\u2022 They admit to criminal gang membership;\n\n\u2022 They aid criminal activity in association with a criminal gang;\n\n\u2022 They are identified as a member of a criminal gang by a parent or guardian;\n\n\u2022 They live in or frequent a particular criminal gang's area and adopt their style of dress, their use of hand signs, language or their tattoos, and associate with known gang members;\n\n\u2022 They have been arrested more than once in the company of identified members of a criminal gang;\n\n\u2022 They are identified as a member of a criminal gang by physical evidence such as photographs or other documentation; and\n\n\u2022 They threaten anyone with retaliation in any manner or by any means in response to any act or alleged act of violence.\n\nConvictions for being gang members carry fines or imprisonment from three to eight years.\n\nThis definition is a legal construct, designed as a measure against which the delinquent actions of young people can be measured in court. It is also problematic and carries the danger that policy making has outrun the lived reality of young people growing up in communities where violence and multiple gang labels exist. With the high levels of gang activity on the Cape Flats, association with gang members is inevitable. Thousands of young people openly boast of being gang members or claim to be in order to avoid being assaulted in a gang area. Gang style of dress and language are almost de rigueur on the streets. Threats and acts of violence are also clearly not exclusive to gang activity, but fall within the Act's definition.\n\nMany young people may dress like gang members (though the Act fails to define this dress), talk 'tough' and hang together but undertake no illegal deeds beyond, perhaps, smoking cannabis. They could be liable to prison sentences for their appearance alone. Interviews with Cape Town police officers confirm that the Act's provisions are not helpful in controlling gangs. Their solution is to invoke it only against gang members who have two or more previous scheduled offenses.\n\nThere's also not much difference between the POCA's gang definition and the definition of organised crime \u2013 a problem worldwide in understanding either. In 1998 the Council of the European Union reached an agreement on the definition of a criminal organisation as:\n\nA lasting, structured association of more than two persons, acting in concert with a view to committing crimes or other offences which are punishable by deprivation of liberty or a detention order of a maximum of at least four years or a more serious penalty, whether such crimes or offences are an end in themselves or a means of obtaining material benefits and, if necessary, of improperly influencing the operation of public authorities.3\n\nThe United Nations Draft Convention for the Suppression of Transnational Organised Crime broadened the definition of criminal association to:\n\nGroup activities of three or more persons, with hierarchical links or personal relationships, which permit their leaders to earn profits or control territories or markets, internal or foreign, by means of violence, intimidation or corruption, both in the furtherance of criminal activity and to infiltrate the legitimate economy.4\n\nBoth these international instruments are legal constructs. It's worth noting that they are, in large measure, also descriptive of, for example, the invasion of Iraq by Western forces or of the Falkland Islands by Britain and of Crimea and parts of Ukraine by Russia. While all these definitions may appear similar, however, there's a great deal of difference between Thrasher's definition of a 'spontaneous group' and an international criminal syndicate. This suggests a need for a more precise understanding both of what can be described as a gang and when that description is inappropriate.\n\nIf pressed for a definition, I would cast the net broadly: A gang is a group of people with common interests who continue to meet over time with common purpose, be it social or criminal. The looseness of this definition is intentional because gangs are a moving target, fluid, with shifting membership based on reasons that may be economic, fashionable, temporary, permanent or pathological. We also cannot understand them without considering the conditions which give rise to them, particularly because, as a society, we have created some neighbourhoods that make gang formations almost inevitable.\n\nIn Cape Town, where gangs are generally held responsible for crime, police statistics indicate that most murders, assaults and robberies take place in poor neighbourhoods where organised gang activity is relatively low, such as Nyanga, Gugulethu, Khayelitsha and Harare. So although gangs bear the brunt of State sanction, possibly because they are more identifiable than random perpetrators, to draw tight links between gangs, crime and violence, as invariably happens, is problematic. Gangs do commit crime and are violent, but they're not the city's main perpetrators. In a certain sense, gangs are also media constructs picked out from their social background, deemed simply by definition to be organised, criminal and violent.\n\nSo to understand gang-like groupings in Cape Town, we need to go beyond simple association and criminal purpose. We have to ask why and where they associate, how they define themselves, what their common purpose is, the nature of their illegal activities (if any) and their impact on society. It's also pertinent to ask what society's impact is on them. Gangs, particularly 'merchant' gangs, are in fact part of a multi-dimensional network of relationships that extend across physical space and over time \u2013 far beyond the groups of youths who give them a name, act in their interests and live by their mythology.\n\nIn trying to understand gangs, we need to move past the sensational headlines of police crime swoops and ask what can be done to help young people live meaningful, resilient lives in environments that characteristically favour the development of crime and violence. For now, let's explore the types of gangs which give Cape Town its reputation as a gang city.\n\nHierarchies\n\nResearcher Andre Standing has usefully distilled gang groupings into four 'models' of relationships appropriate to the Cape Flats: hierarchies, networks, markets and clans. According to the hierarchical model, based on American sociologist Donald Cressey's work on criminal association, illicit syndicates are both bureaucratic and hierarchical, with strict membership and specialisation.5 This view has been challenged by empirical studies which found criminal associations to be more fluid and generally not centrally planned. Cressy's formulation has been called sensationalist and supportive of law enforcement agendas. However, for example, many informants claim that the Hard Livings gang approached this hierarchical, centralist structure under the command of its 'CEO', Rashied Staggie.\n\nNetworks\n\nBecause hierarchies are vulnerable to detection and destruction, it's suggested that organised crime is more likely to operate as a network. It makes more sense for clandestine operators to form sporadic partnerships and spread risks, pool resources, utilise different contacts and exploit divergent objectives than to be centrally planned. Networks are seen to form through introductions and connections, with unity among the parts being achieved through shared objectives or trust. The central mechanism is mutual dependency rather than a rigid structure in the creation of supply chains.6 In addition, Mark Shaw and Tuesday Reitano note that traditional responses to organised crime have tended to focus on criminal groups rather than systems of supply. This approach, they say, is less valid because organised crime operates through networks and alliances that coalesce around specific supply chains:\n\nFor each illicit good, the actors involved in sourcing, transporting, protecting and vending it will be different. Therefore, in order to suppress a crime, the supply chain needs to be closed down for the entire crime.7\n\nMarkets\n\nThe central imperative of both organised crime and gangs is to sell things. Therefore a core activity is the development of markets and the control of areas in which markets can operate. Groups or individuals strive for survival and profit and, in doing so, compete with each other. The market is seen as rational and the actors are predictable in their need to make money.\n\nImportant to illegal market operators are riskiness of the distribution system, the kind of distribution system, the area of operation and the nature of the goods involved.8 In many ways, the type of illicit goods or services dictate the nature of the trade. In his study of the cocaine trade, US economist Sidney Zabludoff explained that the operations of firms involved in the supply of different drugs were partly decided by characteristics of the commodity. For example, cocaine is a compact, high-cost global commodity produced from a crop grown predominantly in one region of the world. These characteristics allow cartels to gain market share and grow into wealthy organisations that attempt to monopolise the global market. In contrast, cannabis is a relatively bulky, less expensive commodity grown in many countries, so it's unlikely that participants will capture a significant share of the market. To a degree, market forces \u2013 and not just the business brains of druglords \u2013 therefore dictate the nature and size of criminal organisations involved in the drug trade.9\n\nClans\n\nThe final model of association identified by Standing is the clan model, which contrasts markedly with the other three and is derived from the work of Italian criminologist Letizia Paoli. She argues that traditional forms of organised crime are not, in fact, primarily bureaucracies, networks or the product of market dynamics, but rather familial organisations heavily based on group loyalty.10 According to Paoli:\n\nThe kin-like relation is established through ritual. The entrance into all the associations considered, in fact, takes place with a ceremony of affiliation, which constitutes a true 'rite of passage.' The ritual marks the change of position of those who undergo them and their assumption of the new status of member of a brotherhood.11\n\nThis resonates with the work of Thrasher in Chicago and my own earlier work on Cape gangs.12\n\nIn the clan model, membership is central, based on ritual kinship and not specific duties, transactions or expectations. New members are made brothers of the 'family' and expected to share an often-secret regime of reciprocity, symbolic gestures, clothing, tattoos, language, loyalty and symbolic ceremony.13 Clan membership is often associated with a sense of status and territory. In the fluid world of street gangs, membership can strengthen or weaken, depending to the gang's fortunes. Because of this, some members may rise in wealth and status, or cohesion can fracture and allegiances shift to other gangs or disappear entirely.\n\nThese four models focus on different coordinating mechanisms within gang-like formations. In hierarchies it's central authority; in networks it's trust; in markets it's competition and in clans, loyalty. Several of these tendencies are present in the organisations in which many disaffected youths on the Cape Flats are involved, but in street gangs the clan model has the highest valency among the greatest number. However, Standing's research shows that, when describing gangs, no definition fits all. To assume a single dynamic, motivation and behaviour in all groups labeled gangs, as the POCA legislation does, is to create both conceptual and practical confusion.\n\nOrganised crime: men in the shadows\n\nWhat's common to all gang involvement by whatever model we assess them, is best understood as 'entanglement'. The term is useful in its ambiguity, implying both ensnarement and a barrier, of barbed wire perhaps, used to impede access by a perceived enemy. It also implies disorder and confusion, while at the same time tight coils of engagement.\n\nMore than two decades after South Africa's first democratic election, the situation for an ever-growing number of under-trained, unemployed young people on the streets of the Cape Flats is an increasing entanglement in activities that are at times illegal, not infrequently violent and motivated by lack of social agency.\n\nThere are others, better organised and networked, who have agendas beyond survival, clan affiliation, a cellphone and the next hit and are generally considered as being involved in organised crime. As we have seen, descriptions of organised crime overlap with those of street gangs, making theoretical disentanglement difficult. In a study of organised crime, researcher Klaus von Lampe lists over 180 definitions.14\n\nOn the ground, however, the differences are less indistinct. The gap between a street-corner hustler and a criminal network such as The Firm or a drug importer is, to quote legal researcher Peter Gastrow, similar to the one which would exist between a street hawker and a successful international import\/export and wholesale business.15\n\nA useful term for people who associate around a common criminal purpose is the French-derived word 'syndicate'. According to the South African police, a crime syndicate is defined broadly as 'a well organised and structured group with a clear leadership corps, which is involved in different criminal activities such as drug trafficking, vehicle theft or money laundering. Such syndicates have well established contacts with national and international criminal organisations, cartels or mafia groupings.'16 Other terms that have been used are 'warlords' and 'violent entrepreneurs'.17\n\nSouth African police intelligence claims that by 1999 over 800 syndicates were operating in the country, with a collective membership of 12 000 'primary suspects'. Of these groups, 300 were believed to also be operating either elsewhere in Africa or beyond the continent.18 An indication of transnational criminal activity can be seen from a report by the Washington-based Global Financial Integrity (GFI), which estimated that, from 2003 to 2012, illicit financial flows from Sub-Saharan Africa were US$ 528.5 billion. From South Africa alone in 2012, these amounted to US$29.13 billion.19 Just under 80 per cent of this flow was due to deliberate mis-invoicing and 'leakages' in the balance of payments \u2013 basically money laundering:\n\nUsually, through export under-invoicing and import over-invoicing, corrupt government officials, criminals and commercial tax evaders are able to easily move assets out of countries and into tax havens, anonymous companies and secret bank accounts.20\n\nAccording to GFI, illicit financial outflows from sub-Saharan Africa outpaced official development assistance going into the region at a ratio of at least two to one. Sustained illicit outflows have turned the continent into a net creditor to the rest of the world.21 Of the 145 developing countries assessed in terms of illicit flows in 2010, South Africa was ranked, alarmingly, in 12th position.\n\nThese transactions go far beyond what is generally considered foreign and local crime syndicates and into the boardrooms of national and international corporations. Groups within these respected entities fit almost every definition of organised crime. We need to keep this in mind when considering the relationship between gangs and structured criminal associations. Crime syndicates in low-income areas, dealing in illicit substances and transactions, are a major problem in their areas of operation, but they are minor players in much larger and extremely damaging levels of corporate crime.\n\nMost studies of organised crime exhibit a worrying amnesia about what used to be called white-collar, financial crime but which is simply corporate racketeering, preferring to focus on the 'with weapons' variety. This seemingly unconscious distinction between 'our crime' and 'their crime' further complicates a clear understanding of organised crime. However this book is about the impact of gangs on young people at risk and, at this level, organised crime is definitely of the 'with weapons' variety.\n\nTo understand this layer of organised crime in Cape Town we need to see where it fits into the context of general gang activity. It's difficult to estimate how many people are, in terms of the SAPS depiction, 'a well organised and structured group with a clear leadership corps' in contact with 'national and international criminal organisations, cartels or mafia groupings'. But the numbers are probably fairly low.\n\nThey constitute a criminal elite and most are in the drug business, with other interests being in wildlife smuggling, prostitution, car theft, protection rackets, nightclubs and possibly weapons procurement. They own substantial capital, legitimate businesses or properties and are relatively conspicuous in their communities if they're local. They include individual drug merchants, the upper levels of larger gangs such as the Hard Livings, Americans, Sexy Boys and Mongrels as well as Chinese cartels like the Tongs and 14Ks, Russian, Italian and East European mafias, Moroccans and Nigerians.\n\nFigure 2 Types of crime, actions and players\n\nThe presence of foreign 'high flyers' hit the headlines in 2011 when nightclub security boss, Cyril Beeka, was killed in a drive-by shooting. He and his staff were accused by a presidential task force of being 'soldiers' for the Italian Mafia and its alleged 'banker', Vito Palazzolo.\n\nA year later Serbian fugitive Dobrosav Gavric, Russian Igor Russol and Moroccan Houssain Ait Taleb appeared in the Cape Town Magistrate's Court for activities related to organised crime. Russol was accused of extorting R600 000 and a Porsche Cayenne from businesses in and around Cape Town. Gavric, who was facing extradition to Serbia to serve a 35-year jail sentence for three murders, was accused of fraudulently entering South Africa. Taleb was Beeka's assistant.\n\nIn 2013 Western Province MEC for community safety, Dan Plato, claimed that the emergence of international fugitives and a transnational underworld was a danger to the Western Cape and that Cape Town's nightclubs were connected to both Cape Flats' gangs and syndicates in Eastern Europe, the Balkans, North America and elsewhere:\n\nI am worried about the fact that so many high-profile underworld figures are involved in Cape Town. I am worried about the number of foreign nationals involved in organised crime in Cape Town. My question is: why are all these foreign people heading for Cape Town, doing their business in Cape Town and finding Cape Town so cosy and appropriate?22\n\nHe said the city was dealing with high-profile, professional and sophisticated gang and drug bosses. 'We need people to outplay them. I do not believe the SAPS in its current format is in that position.'\n\nIs the city underworld in the thrall of foreign racketeers? The truth is much more complex and interesting. But first, let's take a look at the illicit market. A 2005 United Nations report on Crime and Development in Africa concluded that '...Africa may have become the continent most targeted by organised crime'. It noted that indications from drug seizures were that Africa was increasingly being used to route drugs destined for other markets.23 In 2010 researcher Annette H\u00fcbschle concluded that in Southern Africa criminal groups appeared to have created a 'free-trade zone' for illicit commerce through various transnational networks, with drug smuggling being the greatest concern.24 In a special report on organised crime in Southern Africa, she found that:\n\n\u2022 Cannabis grown throughout Southern Africa is being marketed in Europe and North America. Its routes are also serving as a conduit for illegal diamonds, stolen vehicles, rhino horn and ivory;\n\n\u2022 Cocaine from countries such as Bolivia, Colombia and Venezuela is entering the sub-region via Brazil through Nigeria and Mozambique, with the routes being run by loosely-associated networks. West Africa has emerged as a major transit and repackaging hub for cocaine flowing from Latin America to European markets. About 13 per cent of cocaine trafficked to Europe is being transited via Guinea-Bissau. This amounts to at least 25 tons a year with a minimum market value of $4.29 billion;\n\n\u2022 Methaqualone, the main ingredient of mandrax, is being manufactured in China and India and entering the subregion via Mozambique, Swaziland, Zimbabwe and Zambia. In the Western Cape, 'tik factories' obtain the ingredients in exchanges of abalone through Chinese cartels. Nigerians have been implicated in street-level distribution of tik, but its sale is mainly controlled by local merchants and gangs;\n\n\u2022 Heroin comes from poppy fields in India, Pakistan, Afghanistan and China and enters Southern Africa by air or sea, mainly through Zanzibar or Dar es Salaam where West African distributers disseminate it into the continent. In South Africa, Nigerian and Tanzanian syndicates appear to control delivery to merchants;\n\n\u2022 Stolen motor vehicles are a major currency in drug deals, a system which bypasses other more easily traced money transactions. Hijacked luxury or off-road vehicles and taxis form the bulk of this trade;\n\n\u2022 Illicit trafficking in gold and natural resources out of Central Africa, particularly the Democratic Republic of Congo, facilitates the sustained insecurity caused by armed groups and is feeding flows estimated at over $1,2 billion a year;\n\n\u2022 Cybercrime: Three West African countries \u2013 Nigeria, Ghana and Cameroon \u2013 are ranked among the top countries in the world where cybercrime is most prevalent, contributing to a global flow of some $600 million a year. Improved communication technology has drawn Africa deeper into the global economy, making it more susceptible to crime;\n\n\u2022 Money laundering: In Africa money laundering is widespread. A report by the United States State Department on money laundering claims that Kenya's financial system, for example, may be laundering more than $100 million each year;\n\n\u2022 ATM bombings involve Mozambican and South African groups, often with connections to mines (for explosives) or former apartheid police officers unable to find employment elsewhere. Other organised groups target specialised goods such as electronic items, clothing and jewellery in house robberies;\n\n\u2022 Cellphone theft: Pakistani dealers in South Africa are said to specialise in stolen phones destined for Nigeria because the MTN network operates in both countries so network transfer is unnecessary.\n\nMore than six in 10 crimes in the country are drug related, with Western Cape numbers being the highest.25 Most drug routes into Cape Town fly well below the radar, but occasionally an error or a police bust raises them into view. In December 2010 a small fishing vessel returning from the high seas to Knysna was searched and 1.7 tons of cocaine were found on board.\n\nTwo years later four 25-kilogram waterproof bags washed up on the Southern Cape shores. They contained 99 per cent pure cocaine and were identified from a logo stamp as having originated in Bolivia or Peru. They may have been part of a floating consignment dropped into the Mozambique Current further north with signal transmitters attached for recovery off the KwaZulu-Natal or Cape coasts. In April 2014 the Australian Navy intercepted a vessel off the Kenyan coast carrying livestock. A search turned up more than 1 000 kilograms of heroin concealed as cement.\n\nThese are a few visible pointers to an illegal drug supply chain into and through Southern Africa estimated by the 2013 National Drug Master Plan to be worth around R136 billion a year or 6.4 per cent of the region's GDP. While many of the drug shipments conduit through South Africa, a number remain behind for local consumption. This has led to a dramatic increase in drug use, presently double the global average. The country is the biggest user of cannabis and mandrax and has one of the highest prevalence of substance use disorders in the world.26\n\nDuring the apartheid years, South Africa's relative exclusion from the world economy tended to isolate it from hard drugs such as cocaine and heroin. After 1994 the country was regarded as an untapped drug market. New markets or existing ones were consolidated and gang activity increased to take advantage of the change. According to research by Charles Goredema and Khalil Goga of the Institute for Security Studies:\n\nThe patterns of the trade have become well established ... with cocaine originating from the Andean region in South America and heroin emanating from Asia \u2013 predominantly Afghanistan and surrounding countries. Traffic routes that directly connect South Africa to these regions carry significant quantities of drugs to Cape Town, but it is evident that supplies are supplemented by imports entering by land through Angola and Namibia (cocaine) and Mozambique and Madagascar (heroin).27\n\nTogether with the illegal export of rhino horn, ivory and exotic species, drug dealing has earned South Africa the reputation of being the hub of organised crime in the subregion with an estimated 953 active syndicates in 2014 (up from 800 in 1999), many being foreigners from Nigeria, Pakistan, India, China and Europe.28 The high drug flow levels are partly because of South Africa's geographical location, but also because of a good banking system, poor border controls and a police force able to occasionally disrupt but ultimately not stem the traffic.\n\nAdded to this, writes Gareth Newham from the Institute of Security Studies, is the low risk of being brought before the courts, the ease with which a false passport can be used to enter and leave the country and the possibility of corrupting the police to get out of trouble.29 Radovan Krejcr, a Czech fugitive arrested for dealing in drugs, attempted murder and kidnapping, told a journalist that when he had to flee Czechoslovakia, he chose South Africa because, he said, protection from its Bill of Rights was more favourable to his trade than in any other country.\n\nAll these activities beg the question: who's really in control of Cape Town's underworld? The answer came as a surprise. After 1994, when transnational crime began taking off, local Cape gangs were confronted by sophisticated operators attempting to muscle in on their traditional turfs. The days of parochial, neighbourhood-based operations were clearly numbered. These threats, compounded by the formation of People against Drugs and Gangs (Pagad) targeting drug merchants, was to totally transform the structure and relationships of gangs on the Cape Flats. It was to be a battle for survival and dominance on an utterly altered terrain. According to Standing:\n\nThe result of this tumultuous period for gangs was an increase in their power and financial base and a rapid sophistication in, and increased brutality of, their business practices, partly learnt from the foreign syndicates with whom they now came into contact.30\n\nCriminologist Irvin Kinnes found that what distinguished international syndicates from local gangs was the former's ability to supply drugs in large quantities. Although Nigerians and Chinese syndicates dominated the importation routes, they were blocked by local gangs from entering the retail market. Where they tried to, they were attacked.31 For example, Moroccans, who started a parking and protection scheme for city nightclubs, ended up in gunfights with local shooters.\n\nBig-time drug dealers like Colin Stansfield and Jackie Lonte could marshal local 'soldiers' in defense of their turf, as could gang leaders such as brothers Rashied and Rashaad Staggie.32 When Stansfield was arrested in 1996, he was estimated to be worth R30 million. The combined membership of the Hard Livings and Americans in 1994 was between 3 000 and 10 000 fighters. The 'Big Men' were purchasing valuable real estate and exploiting gaps in state provision to buy community loyalty through their support for sport, welfare and cultural activities.\n\nIn the early 1990s local drug merchants and gang bosses \u2013\u00ad possibly coordinated by Stansfield \u2013 banded together to found The Firm in order to defend, rationalise and consolidate control over the market. Turf and markets were agreed upon and gang fights subsided, being considered bad for business. As not all gangs were members of the cartel, the peace did not last long, particularly following the attacks on gang leaders and the gruesome murder of Rashaad Staggie by Pagad in 1996.\n\nBy then, however, the upper ranks of big gangs such as the Hard Livings, Americans, Mongrels and Sexy Boys had effectively fought off any takeovers by foreign syndicates.33 According to Kinnes, 'Considering the fact that these gangs had access to all parts of the Western Cape through their distribution networks, their transformation into criminal organisations was a natural outcome.'34\n\nOne of the reasons for the success local gangs had in holding their ground, according to the SAPS head of Operation Combat, General Jeremy Veary, was their randomness, violence and incorporation of prison gang discipline. Foreign gangs like the Chinese cartels had traditional ways of doing business, but were scared by the unpredictability of local gangs.\n\nYou're sitting here and speaking normally but you don't know when the people are deciding to kill you. Gangs have this mystique, you know, you cannot tie them down. You come to make a deal with 500 grand and the guys rob you. Simple as that. The Triads started saying no, we're not going to pay you with money any more. We're going to pay you in methaqualone and ephedrine in exchange for abalone. Then things went stable. So what you're talking about now is not control by foreign syndicates but supply lines.35\n\nIn 2003 Western Cape Minister of Community Safety, Leonard Ramatlakane, noted the link between the leadership of larger local gangs and organised crime:\n\nThe nature of gangsterism in the Western Cape today has become increasingly ruthless and business orientated. Participation in gang activity is still substantially driven by such elements as group identity, self-protection, pride, boredom and turf. However, the bottom line is money, where highly organised and well-connected gang bosses preside over vast business empires.36\n\nAs we will see below, however, the base of these 'empires' is less stable and organised than Ramatlakane suggests, with systems open to costly disruption and armed young men always alert for violent takeover and, when trouble happens, needing to bribe or shoot their way back into the driving seat.\n\nPrison gangs: old and mean\n\nWhen I came back from the war [World War Two] my people didn't expect me. I went to see my mother and she said, it's no use going home to your wife because she's having an affair with another man. I didn't want to believe it. I went to the house that night. I knew where the key was, so I went in at the back door and locked the door behind me. I locked the front door and took the key also. Then I went to the room where my wife was lying with this other man. I took out a bayonet. They were so intimate with each other, they didn't even notice me. So I stabbed him, thinking I was stabbing through both of them together, but she jumped out and ran away.\n\nSo I received a sentence for the hangman's noose. I went to Pretoria for three months. One morning they called us and I thought it was time but I never heard the preacher come around yet. We were 12 in Death Row. Then the one officer said to the other: 'Take this one [me] to the office.' Then he told me they wanted to give me a chance because I had been fighting for my land. But they said I mustn't fight with my own land's people. I got a life sentence. The other 11 were hanged. I never got the rope. And as with life, I finished it. And then again I went to prison. And tomorrow prison again.\n\nWhen I went in for life, there were about 60 men in one cell: long-term indeterminate-sentence prisoners. There was a lot of fighting in jail. And there was a code among them. There's the 26, 27, 28 and Big 5. The code among them is not actually a piece of paper that you carry. You have to remember Die Boek, Die Wet, Die Plank. This prison-gang thing started on the mines. Each gang has a judge, a magistrate, a doctor and an office man they call a mbalaan who takes the names of new recruits. Each gang identifies itself by singing. They must tell each other things through stories. When you come in they try to frighten you when you are in that fearful stage and they watch you.\n\nWhen this thing first started there were two gangs. They used to rob people. One would rob you with his mind, the other would rob you with force. Then some pulled out and made another gang, and then there were three. That's where they come from. They are different to outside gangs. But they are connected. Like when we're outside I do you a dirty and then you tell your brother and when I come inside the prison I'm finished. But they work different. In prison they come up behind you and stab you. Outside it's by force... They'll grab you and your gun and jump into the car and there you go to Texas.\n\nInside, if they like your trousers they take them. You can't say nothing. And if you do, you're gonna be stabbed. When you make a fault with these people they go and sit in a kring [circle] first. Then they discuss you. Then three men get the knife. They issue it from the magazine stores they've got underground. And if the knife is taken out, it doesn't come back the way it went out. It's gotta come back with blood or a report behind it. The three men are: one to stab and two to escort and do the job if he misses. They use a short blade for punishment shots and a long one for hangpaal [death]. The sentence of a magistrate is a short blade and of a judge, it's a long blade.\n\nNow if it's a 26, they stab you in this way. A light sentence, the first, is in six minutes, the second in six hours, the third in six days, the fourth in six weeks, and the fifth in six years. So like for the last one you know they're gonna stab you for what you've done, but don't know when for six years! Even if you're transferred to another prison the sentence goes with you. You think nobody knows you but they're there, just waiting. That's how prison life is. It's terrible man...\n\nPaul, former prisoner, Cape Town, 1982\n\nAlthough this book is not about prison gangs in prison, but about their effect on those who are not, a brief look behind the locked gates is instructive. Much has been written about these brutal gangs, the most notable works being The Small Matter of a Horse by Charles van Onselen, The Number by Johnny Steinberg and God's Gangsters by Heather Parker Lewis.37\n\nIn the hands of Van Onselen, one of South Africa's pre-eminent investigative historians, the story is about a band of robbers in the late 19th century Transvaal and Natal who preyed on black migrant miners returning home with their pay packets from the Transvaal mines. It tells of a Zulu youngster named Mzuzepi Mathebula who allowed a farmer's horse to stray and was required to work off the debt for its loss. Mzuzepi escaped to Johannesburg, worked as a groom for some highway robbers, learned their trade and changed his name, first to Jan Note and then to Nongoloza. The erstwhile groom formed his own band of robbers who named themselves Umkosi Wezintaba (Regiment of the Hills) and organised as a para-military unit.\n\nNongoloza imbued his bandit army with a political purpose. 'I reorganised my gang of robbers,' he reported to his captors in 1912. 'I laid them under what has since become known as Nineveh law. I read in the Bible about the great state Nineveh which rebelled against the Lord and I selected that name for my gang as rebels against the Government's laws.'38\n\nAccording to Steinberg, it was said that Nongoloza was imbued with magic and that the bullets of white policemen and soldiers bounced off his skin. 'In early proletarian lore, he was something of a Janus-faced monster: horrible because he was undiscerningly brutal, enticing because he showed that \"even the poor can be terrible\"'.39\n\nDuring the Anglo-Boer War, Nongoloza and many Ninevites were incarcerated at Cinderella Prison in Boksberg. On release, he shifted his group to caves near Johannesburg, undertaking housebreaking and farm hold-ups, shooting dead a policeman. By 1912 the robbers were being described by police as Nongoloza's Army. Its leader was arrested and given a life sentence, along with many of his 'generals', and he began organising his 'army' within prison walls. By the early 1930s, gang derivatives of the Ninevites had a presence in almost every prison across the country. Out of this developed the Numbers gangs \u2013 26, 27 and 28s \u2013 which operate with extreme and often casual brutality and the ready taking of blood.40\n\nIn the hands of prison inmates, this story has blossomed into a durable mythology. It has become a code of conduct, a dress code echoing that of late 19th century Boer and British armies, a secret language and an unwritten 'book' that has to be memorised. In the absence of written records, the system of identifying gang members and their rank is ingenious. Any new prisoner who claims to be a member is required to recite the gang history and to 'dress' himself equal to his rank. This must be 'sabelad' in the language of the gangs \u2013 shalombom for the 26s and 27s, and ndyaza for the 28s.41 He is asked: 'Who are you? Where do you come from? How will I recognise you?', whereupon he must identify his rank, gang and imaginary uniform.42 The mythology includes a shamanistic arch-criminal figure called Pomabaza or Po. It was he who initiated Nongoloza and one Kilikijan into a life of plunder; the death of a bull called Rooiland (upon which the 'book' of the 28s is written), a white stone upon which Pomabaza's 'story' was written in blood (and later broken) and a parting of ways between Nongoloza and Kilikijan over same-sex intercourse (it is because of this that there's division between 26s who forbid same-sex intercourse and 28s who practice it).\n\nThe Numbers gangs have their own parliament, legal system, punishments, territories and economy. The 26s, for example, have a Makwezi (president), generals, captains, sergeant majors, lawyers, doctors, inspectors, teachers and soldiers. In a detailed report on prison gangs, General Jeremy Vearey lists the six laws of the 26s as:\n\n1. You will not do what you want;\n\n2. You will not lie to your brother or argue with him;\n\n3. You will respect the honest work of police and wardens;\n\n4. You will not sleep under the same blanket with your brother [no sex];\n\n5. You will not physically harm a non-prison-gang member the first time he offends you and first give him two warnings before harming him;\n\n6. You will die with your brother under the flag of the 26s.43\n\nThe 28s have similar systems, except that they add rules of conduct between the soldier or golden line [males] and the private or silver line ['females']. The Numbers gangs are forbidden to fight each other and their spheres of influence and action are divided. The 26s are permitted robbery and theft by subterfuge, but without violence, and must distribute the gains evenly. The 28s fight on behalf of all three camps for better prison conditions and can engage in sodomy. The 27s guarantee gang law and keep the peace between all three camps. If blood is spilled, it is spilt in return. Prisons being what they are, however, conflict between gangs does take place within the walls. It is however strictly regulated and attempts are made by senior gang members to quell it as quickly as possible. Non-members of the Numbers, however, are fair game and often victims of violence.\n\nThe memory of Nongoloza is a myth built from scraps of fact, but is greater than merely a myth. It is, as Steinberg points out, not a story one tells, but a set of practices one enacts. Its twists and turns are memorised in the long days of doing time. These are punched into the skin of inmates as crude tattoos depicting shields of the bull Rooiland; bayonets of the Boer War; sunrise (26s); sunset (28s), guns and comments. They also record descriptions of personal pain, such as 'I broke my mother's heart' or 'I am a lost child of fate'. The form that the myth took was in response to the conditions that prisoners faced. As Veary points out, 'criminals militarise in response to you. The Nongoloza banding together as an army took its form in response to the army during the Boer War. That was what they knew. You take the shape of what you face.'44\n\nOn the streets, throughout most of the 20th century, prison gangs were known about but considered to be prison business. Until the late 1980s it was understood that the walls between prison and street were total, but they suddenly began to crumble. One of first cracks may have begun in Cape Town with the imprisonment of the druglord Jackie Lonte in the 1990s. This coincided with the clamour beyond prison walls for apartheid to end and a prevailing sense of impending democracy.\n\nLonte was, by any standards, immensely rich and had access to supplies of mandrax which he could smuggle into prison. There was no way a man like that would stand for the requirements and physical torments of Numbers gang membership. He had the means to buy himself clear, but the mythology and structure of the Numbers intrigued him.\n\nI couldn't establish whether he was a catalyst, but at around the time he was in prison, Number gangs in the Western Cape closed their 'bloodlines' and ended stabbing as a requirement for membership. Until then, to be a part of a prison Number was a lengthy and rigorous process, in which one could only climb the ranks after serving many brutal years in prison. This process almost always required the shedding of blood.45 Closing the bloodlines was, according to Steinberg, a cataclysmic decision:\n\nTo stab, and to be subjected to violence and deprivation in the wake of stabbing, was the centrepiece in the initiation process. It animated the world behind bars, providing it with its fundamental meaning. Why did it happen? I don't know. Some of it may have had to do with the changing relationship between prison gangs and street gangs. Whatever the reasons, the closing of the bloodlines caused havoc.46\n\nShortly afterwards, conflict erupted between the gold and silver lines of the 28s in Victor Verster Prison and spread throughout the Western Cape. This rumbled on towards 1994 when the country experienced the worst outbreak of prison violence on record. Over a period of four months there was unrest in 53 prisons and in most cases inmates attempted to burn down the institutions. Thirty seven inmates were killed and hundreds of prisoners and warders were injured. One of the reasons for this violence was said to be the failure of the new ANC government to offer expected widespread amnesty.47\n\nWhatever the reasons for the riots or the closing of bloodlines, the outcome was to weaken the Numbers gangs within the prison system. However, on the streets of Cape Town, something strange was happening. Prisoners began arriving in cells claiming to be ndotas, able to sablea and, after a fashion, answer correctly to their rank \u2013 but had never before been in jail. When asked who'd initiated them they'd answer: The Americans or the Scorpions. What the prison old-timers discovered was that druglords like Jackie Lonte, Colin Stansfield and Ernie Lastig had 'stolen' the Numbers traditions to build 'armies' on the streets.48 The perceived sacrilege was compounded when Lonte returned to prison in 1996 as a 26 and declared war on the 28s. The reason had to do with turf wars fought between gangs on the streets that now embraced Numbers allegiances. Bringing street rivalries into prison and declaring war without being an initiated fighting general carried a death penalty, but nobody would touch Lonte. An informant told Steinberg:\n\nHe was Jackie, and Jackie meant more than anything else. So we raised our American flags in our cells and we went to war with the 28s. Something like that had never happened before. After that, the Numbers was never the same again.49\n\nBy the late 1990s, two of the major Cape gang systems, the Americans and members of the The Firm were using Number rituals. The Americans considered themselves 26s and Firm associates as 28s. Leaders of both designated themselves generals and appointed captains, sergeants, judges, inspectors and soldiers. The Mongrels, being 26s, had a natural alliance to the Americans and Born Free Kids, while the Cape Town Scorpions and Mafias aligned themselves with the 28s. This brought about congruency between prison gang stratification and street gang rivalry. According to Steinberg: 'The street gangs took the world of the prison \u2013 its metaphors, its nomenclature, its logic \u2013 and imprinted it on the ghettos.'50 The power of Numbers mythology on the street is described by Vearey:\n\nYou speak a new language. You have new symbols through which you look at reality. You do not walk around fearfully anymore. The brand, the power that comes with that myth, is what it represents. It's the type of power that comes with being a gang member. The Number makes a philosophy out of it, makes a belief system out of it. You get told all these myths about the great fight between the Numbers and the eight stars falling from the sky and the seven stars falling into the green grass...51\n\nThe sudden proliferation of Numbers' traditions and mythologies on the streets in the early 1990s was not mere caprice, but a response to an external threat. The Cape gangs had always been regional affairs, defending a specific township turf in order to sell cannabis, liquor or mandrax. With the opening of borders to the world after 1994, as we have seen, foreign syndicates arrived and began to muscle in on the various illegal markets. In response, the bigger gangs like the Americans, Scorpions and Hard Livings had to strengthen their organisations, expand their markets, control their customers. They also had to compete and cooperate with foreign syndicates. And there were huge profits to be made with the introduction of heroin, cocaine and tik.\n\nThe resulting new drug millionaires needed to create and maintain province-wide networks and ensure unwavering allegiances from buyers and sellers while keeping foreign syndicates at bay. The fierce, highly structured traditions and practices of the Numbers gangs were made for the task which required chains of command, strict adherence, the use of a secret language and easy linkages across turf boundaries. The mythology of the Numbers was also immensely appealing to young gang members in need of a 'story' to live by. According to an 'elder' in both prison and street gangs in Hanover Park:\n\nThe Numbers have discipline. This is their biggest influence on the gangs. The Numbers made them more organised. Gang members look up to the Numbers. They look to the Numbers for guidance. You won't find a leader of a gang that's not a Number.52\n\nInitially, high-ranking street gang bosses entered prison and bought their way into the upper ranks of the Number. On discharge, they possessed new powers and networks. Increasingly, lower ranks began to see spells of imprisonment as a way to bolster their credibility once back on the streets. During his work on the low-income neighbourhood of Overcome, Yashar Taheri-Keramati found that Number gangsters in prison, wanting to cash in on the street gangsters' financial successes, were willing to 'sell' their names, affiliations and Number to those strong enough to buy them. In prison and continually challenged by street gangs, they were forced to 'fast track' and offer protection to street gangsters in return for their patronage on release.53 In return, on emerging from prison, Numbers members had a fraternity, a support system and a level of credibility on the streets.\n\nBy 2000, intelligence officials estimated that local gangs were in control of about 90 per cent of the mandrax, cocaine and dagga markets in the Western Cape, with Nigerians, Moroccans, Chinese and other syndicates controlling the rest.54 Within a few years of the country's democratic elections, the Cape gangs had relegated the foreign syndicates to being bulk providers. Foreign syndicates were tending towards less volatile markets such as illicit diamonds or rhino horn. 'There's no foreign influence in the Cape Flats,' according to a gang leader interviewed in Hanover Park. 'The guys who mostly support the gangs are the Bongos (Nigerians) mainly with unga and tik. But they have no influence among the guys. Their only connection is supply.'55\n\nBy the second decade after the end of apartheid, the 'robber baron' interpretation of Nongoloza had been absorbed into most street gangs, giving them a sophisticated ideology and language through which to view the world. According to researcher Luke Lee Skywalker, this gave street thugs with no previous code of honour a coherent history and philosophy, elevating them above their former status.56 Having a secret language, according to a member of the Mongrels, is a very powerful thing:\n\nIt's a communication system across gangs. Very useful. If I negotiate with gangs I give them the power to say hello and to know who I am. Then I ask them the same with sabela and they must tell me. I make myself known to them and they make themselves known to me then we will know we are ndotas, you see? Now we know each other. If I go to a leader of a gang and can't sabela, how can he talk to me? But he throws me a Number and I throw a Number back in a certain way, old school. Then they know.57\n\nIn this way Cape Flats gangs have wholly absorbed the Numbers' ethos. According to Jonathan Jansen of Fusion Manenberg, which attempts to rehabilitate gang youths:\n\nThe prison Number gangs tie everything together. In the last 10 or 15 years there was a deliberate attempt to get it to open its secrets, to spill onto the streets and get every gang to buy into the network. In a sense there are now no gangs, except for very small ones, that are not also prison gangs. Nobody can negotiate peace if there are no Number gang people involved. The ndotas need to talk. They can cross boundaries, walk in any territories, protected by their Number status.58\n\nIn the migration to the streets, however, the hard distinctions between the Numbers have become less defined and the codes of honour have shifted. 'Gangs aren't just single Numbers anymore, a Manenberg gang member told me, 'and in a war, a Number will fight a Number. Not inside, but outside it's different.' He continued:\n\nOutside, your Number in prison don't count the way it does inside. You come out of prison and ask me what I'm doing and I'll remind you I was out here taking blood and where were you? To work yourself up outside is not to work yourself up as a Number. You spend 10 or 15 years in prison and come out, you can't tell the lighties what's what. They'll sommer shoot you.59\n\nIn the face of these changes, most lifers behind bars have no purchase. Prison gangs will continue to flourish because they're a response to the system of incarceration. But their members are now the cooped-up custodians of a usable tradition for purposes beyond their reach. Instead of an association to create tough men annealed in the flames of violence, their closely guarded traditions are now increasingly badges of honour for new recruits passing through the prison system on the way to higher things.\n\nFor jailed druglords, prison is free board and lodging from where they conduct external business deals on smuggled cellphones. Doing time proves useful in other ways. Prison allegiances cut across street gang turfs and territorial animosities. In the rapid expansion of drug and other forms of smuggling that blossomed in the 1990s, this has proved to be invaluable and good for business.\n\nMerchant gangs: brotherhoods and street soldiers\n\nI'm a 26 but also 27. I was in Pollsmoor. I went to prison first when I was 17. It was rough! First time it was two years, second time two and a half years. I'm on parole now. When I first got in I was: 'Yo! What am I got to do now?' Prison is not for me. When you're inside you have to become 26, 27 or 28. If you don't have a number you're a frans and then nothing is your own. You need to be an ndota.\n\nMy brother was shot, they thought he was me. And now he's in a wheelchair. I didn't want to be a gangster. My mother said I mustn't. My oldest brother is and my twin brothers are. But they shot my brother so I said: 'Give me a gun.' I learned how to hold a gun, clean a gun. My mom said I shouldn't go but I joined the Ghetto Kids and smoked with them. The Americans and the HLs wanted me to join but I said, no, leave me alone.\n\nWhen I grew up my dad wasn't around. He's a Xhosa. My mom brought up seven of us alone. My mother is my mother and father. She's my hero. But every day she was away at work.\n\nInside I'm a 26 but outside I'm a 27. I must be. The 26 and 27s are sonop [sunrise] but the 28s are sundowners. They do dirty work. I'm not a sundowner. I want a decent job. But here there's only smokkel [dealing]. The gangs all sell drugs, tik, ungar, mandrax. They fight over turf and girlfriends.\n\nThe local police do nothing. You can buy them. They just drive around, they don't get out their van. If the police are going to raid, someone calls the owner and tells him we're going to raid at this time so keep your stuff safe. The police come and there's nothing. They get you with a knife or gun you pay and go.\n\nI have a 10-year-old son and a three-year-old too. His mother is on tik. His grandmother and grandfather are on tik. His grandmother's mother and father are on tik. What can I do about it? When you go inside relationships go to hell. Tik is killing this community. People are fucked in the head.\n\nMichael, drug dealer, Manenberg, 2014\n\nAs we have seen, organised criminal syndicates in Cape Town can be found to operate as capitalist corporations, as transnational foreign smuggling operations, as the market networks of drug merchants and include leadership of the most powerful gangs. At times they overlap, network, cooperate or are in conflict with each other. In terms of organisational structure, merchant gangs are a form of 'middle management' \u2013 organised at the top, semi-structured in their middle ranks and fluid and volatile in their lower ranks. They're the public face of Cape Town's gangland \u2013 buying, selling, shooting and robbing their way to notoriety, fortune and cult status.\n\nSome have been around for a long time, with up to three generations of family membership, corporate histories going back decades, prison mythology and province-wide areas of operation. I hung out with the Mongrels in the early 1980s. They had begun in District Six in the 1970s and, today, they're still going strong. Merchant gang structures are complex, involving hierarchies, networks and markets. For young members they're clans or brotherhoods \u2013 support systems offering rites of passage where 'warriors' are tested on the path to adulthood.\n\nEveryone on the Cape Flats is aware of the overbearing presence of gangs like the Hard Livings, Americans, Jesters, Clever Kids, Mongrels, Sexy Boys and Ghetto Kids. Below them in the hierarchy is a plethora of smaller, turf-based gangs that may start out independently, but at a certain stage are forced into an alliance with the merchant gangs through supply chains or threats. Their followers make up tens of thousands of young people who see themselves as gang members.\n\nBack in the late 1980s the leaders of the Hard Livings, the Staggie brothers, realised they had to grow beyond their Manenberg base and offered security to druglords in other areas not aligned to gangs. They ran protection for Colin Stansfield in Elsies River, then offered the same services in Delft. They cornered the prostitution trade in Sea Point and along Voortrekker Road and moved into legitimate construction businesses. An upcoming leader was sent to university to study law.\n\nThe Americans used their drug money to start a long-haul trucking outfit.60 Both gangs moved into the taxi business and also levied 'protection' fees on businesses and shebeens in areas under their control. In order to pay for drugs from Chinese suppliers, they gained control of abalone poaching operations from Hawston up the east coast to Port Elizabeth. Younger gangs that could have posed an opposition were coopted and corner kids coaxed into the fold. According to Fusion Manenberg director Jonathan Jansen:\n\nThere was a big teenage gang, the Stoepa Boys, which became a rival to the other gangs. But the HLs were quick to pull them in. They became Junior HLs. The HLs actually created teenage gangs. They had the Westsider Kids, the Vatos Locos. They became mentors. The gang would keep its name, but when a fight broke out the HLs provided the weaponry. The HLs would park their new cars at their base in Aletta Court for the kids to see and admire. There were cool guys there and pretty girls coming and going. The teenage boys would see that and they'd want it.61\n\nSchool gangs like the Stoepas, Bocca Boys, State Boys and Toy Boys were encouraged with money and favours by the Hard Livings, Americans and Jesters. Long-time Shawco organiser Cyril Pelston, noted that:\n\nStaggie would say: 'We invest in young kids. They're the future of our business.' The kids are very aware of being chosen. There's a 19-year-old who sits in the HL's inner kring. The smart ones can get up in the hierarchy. But most just stay down on the street.62\n\nThe Staggie brothers worked hard to create a Robin Hood image on the Cape Flats, providing almost the only possibility of employment for young men in their area. They bought food for mothers when need arose, occasionally paying bills and rent arrears and even sponsoring a football team. Rashaad ran mobile shops selling essentials in areas where shops were not accessible. When Pagad vigilantes killed him, gangs linked to the HLs formed a front organisation called the Community Outreach Forum (CORE). It organised a large crowd to march to Parliament, ostensibly to bring peace to the Cape Flats. Rashaad's teenage daughters carried a placard that read: 'They killed the world's best father.' But, as community worker Cyril Pelston noted, there was generally a catch:\n\nBecause of high unemployment there was no money and they would assist you with rent and stuff. But there was always something behind it. Maybe you had a nice daughter and they would want her. That way many families became quite vulnerable. Kids dropped out of school and they'd be recruited.63\n\nEven under POCA legislation it's difficult for police to arrest gang leaders. They rule by word and not deed, avoid using drugs, keep organisationally 'clean' and enforce these rules of behaviour on men in their upper hierarchy. Excess money is laundered through legitimate businesses. According to gang researcher Irvin Kinnes, in rural areas on the West Coast such as Saldanha, Piketberg, Arniston and Paternoster as well as in South Coast villages like Hermanus, Bredasdorp and Genadendal, larger merchant gangs have developed a strategy of 'buying the town'.\n\nThe action involves gangsters initially buying property in these towns. They would then proceed to buy property that involves common use by the community such as petrol stations, shops and game arcades, to provide the means to recruit local youth to sell their commodities.64\n\nExpansion, however, can generate tension and drug sales are often the cause. In a neighbourhood like Manenberg, which covers around 10 square kilometres, there are six big gangs selling drugs. So when tension rises it's not so much about territory, but control of drug users. If a user spending thousands of rands a month changes allegiance to another supplier, that's hugely provocative. He may change because of bad drugs or better prices elsewhere. Perhaps the product has been cut with other substances or is simply of low quality. Users have been killed in the past for defection, but dealers may protect and even fight over him because he's a source of steady income. The dealer has a quota he needs to sell to pay the supplier and when this is under threat, the new supplier becomes a target.\n\nDespite the seeming conflict over markets, however, gang leaders will work to regulate it. When a gang war broke out in Manenberg in 2005, the leaders of the two opposing gangs were seen to have a regular Tuesday date in a city coffee bar.65 The formation of The Firm by gangs in the mid-1990s served to keep down conflict and regulate the drug trade. In general, the upper echelons of the merchant gangs send their children to private schools, frequent expensive clubs and live lives externally indistinguishable from corporate executives which, in a sense, they are. According to sociologist Andre Standing:\n\nMany successful gang members move away from the Cape Flats and live in wealthier areas. They are also rumoured to collaborate with leading members of other gangs on business deals, even those who are popularly thought to be their enemies.66\n\n***\n\nFor those in the lower ranks of merchant gangs, however, life is entirely different. As mentioned, vulnerable young men, particularly those with family problems, are cynically targeted and lured with promises of cool clothes, drugs, money and status. 'The HLs would drive past in a car and throw out money to us,' a member of the Americans in Manenberg told me. 'We didn't ask where they got that money. House robbery probably.'67\n\nIn Hanover Park, a gang member named Trappies explained the lower structures to me by moving objects on a table. 'Over here you have the Big Man and beside him here are two or three captains. This is where the money lives. Then over there are the inspectors, the gatekeepers, they check who can come in. Often those are the only ones people see. And then there are the core members \u2013 shooters, dealers, robbers \u2013 who do the business. And then, far down the line, are the tik soldiers; young guys, druggies, people who can be issued a gun and told to fight in return for drugs or money or for the hell of it.'68 Joey tells it slightly differently about another big merchant gang:\n\nThere's the main man, Staal [not his real name], and then there's Die Draad and Die Glas. These guys are the fighting generals. But they must obey the code of rules. Even Staal can be punished for disobeying them. We can all talk equal under the rules \u2013 nobody is above them. Staal became leader because when he came out of jail the gang was leaderless so he got the guys together and he helped them, paid bail, looked after them. He's the leader because he's organised and respected. Then there are the soldiers \u2013 here where we sit now not so many, but over in Lavender Hill around 4 000.\n\nThe gang isn't about housebreaking or robbery, it's about drugs and smuggling. In the gang only certain people smuggle drugs, you see. It won't be the leader. Guys get paid to sell. They're employed. If a member robs or breaks into a house it isn't really a gang thing. That's true of all the big gangs. If you see there's somewhere you can break in or someone you can rob and you're not a smuggler you can do that on your own. You can be independent.69\n\nGeneral Peter Jacobs, head of Police Intelligence in the Western Cape, corroborates this structure:\n\nSay you have 20 guys standing outside Etosha Court in Hanover Park. Four of them are shooters, maybe four are posmanne [spotters], four more are controllers of drugs. The rest are nothings who can't be trusted with drugs and can't shoot because only the hitmen can. They're just standing there on the corner 'filling' the gang numbers, but they don't get gang money because they do nothing. They're the housebreakers, the street robbers and they must bring money to the gang that way.70\n\nThese filler guys are a problem for gangs because they disrupt the order. Maybe he robs someone from an opposing gang and you end up with a big street fight. This happened a while back when an Americans guy robbed the sister of a Hard Livings leader. The HLs killed him and that led to a hell of a battle that lasted over two years with about 35 deaths.\n\nOr maybe he gets good at stealing flat-screen TVs or copper and he sells to a syndicate and makes good money. Then the gang wants control of the syndicate and there's a struggle. Gangs don't like wars in their territories, it's bad for business, but there's this element in the gang structure that's a pathway to violence and trouble. Another cause is that if there's too much peace the shooters don't have any work so they generate it by provoking retaliation.\n\nInitiation into the gang is often enacted through rape or the murder of a rival gangster. Such action marks a youth for unabated revenge requiring protection, which only their gang offers. Many youngsters are born into families with several generations of street and prison gang tradition. Most never get into the top tiers of the gang and remain content merely to serve the 'Big Guy'. They stay relatively poor. Their situation is highlighted by a poignant question in the book Freakonomics: Why do drug dealers still live with their moms? Because they can't afford to live anywhere else.71 Yet they dream big and talk about those who've made it.\n\nJonathan Jansen once asked a youngster why he was hanging around the Fusion Manenberg offices doing nothing. 'He said he could easily get work when he needed it because he's a gunman and can get R20 000 to shoot a guy in the head. Another 22-year-old was, for a while, making R80 000 a month selling abalone to drug suppliers. No programme gives you that much.'72\n\nThese stories are more likely to be gang hyperbole than truth, but they have resonance among youngsters on the street. On the occasions when they do hit it big, according to General Jeremy Vearey, no legitimate means would give them such a lifestyle. Mostly though, he says, ordinary gang members are poorly paid, but receive protection, food, shelter and legal help when needed. 'They're looked after as part of a crime family.'\n\nYou're not going to be a drug courier if you haven't first proved yourself \u2013 that you can keep your mouth shut. You are tested by smaller things. Drugs are the fast cash crop but you can't build your gang around drugs. If one shipment gets hit you're bankrupt. Housebreaking and robbery don't have those problems. They are safer. Not even the police pay attention to it these days because they know all victims are interested in is a number for insurance. So the gangs consider this very low risk but it brings in a lot of money.73\n\nWorking with Hispanic gangs in the United States, criminologist Mike Carlie identified six levels of membership within big merchant gangs. Surrounding the powerful leader or leaders he identified a group of older, hard-core members deeply enmeshed in gang activities. Beyond them are associates who have a commitment to the gang, but are not part of leadership structures. Further out still, fringe members drift in and out of strictly controlled membership.\n\nFigure 3 Structure of merchant gangs\n\n'Wannabes' are not actually gang members, though they may declare themselves to be. They're youths who view the gang as an exciting place to be where they could become 'somebody'. They emulate gang dress, graffiti, hand signs and other gang cultural symbols and may seek to associate with known gang members. Another peripheral grouping Carlie describes as 'cliques'. These can be called on in times of conflict to bring the gang up to strength. They may be a group around a hard-core member \u2013 a sort of gang within a gang \u2013 or an associate gang.74\n\nThe parallels between Hispanic and Cape Town's gangs \u2013 and they are considerable \u2013 are the result of similar neighbourhood, family and economic pressures endured by young people. Carlie's typology alerts us to the fragmented, even transitory nature of merchant gangs. The further down members are in the hierarchy, the more gangs become, for their young members, entanglements rather than structures. They're webs of fear and excitement, possibility and disillusionment, status and social isolation. They offer rituals and affirmation of being a 'somebody' in a harsh world. Vearey, who was himself a gang member in his youth, remembers the process:\n\nI basically looked at who drove the best cars, who wore the best clothes and they were labelled a gang. So being part of it became part of that particular process. If I needed to protect myself because I'm going to traverse the territory, being a Sicilian is better, because nobody touches the Sicilian because we are not like a structured gang, we're school kids from Elswood, Ravensmead, Elsies River High and Lavis. And we're unpredictable, we're not governed by the Numbers and we don't care who you are, we'll deal with you.75\n\nOn the Cape Flats, 'wannabes' start young. 'These days it's not unheard of to find a youngster of 14 or 15 in the gang,' says community worker Cyril Pelston, 'and at that age they're almost fearless and have something to prove. The lure of an expensive pair of takkies [tennis shoes] is huge. Kids will perform what's required by the gang to get them. And a cellphone is gold. You can't live without it.'76 At the lower end of gang hierarchy, nothing much has changed for years: 'Kids are still hanging out doing nothing, others selling drugs,' says Jonathan Jansen. 'They're getting just enough to eat and becoming addicted to the leadership of the gang. And the drugs catch them so easily.'77 A young gang member named Michael traced his entanglement:\n\nHere there's only smokkel [dealing]. I open my door and every day there's a guy with tik and unga. Every day. And he calls me and says: 'Hey Michael, you must help me here.' First time you try tik you get excited. It's a good feeling. Second time... then you want it every time. You won't want to do anything else. You have no appetite for eating. You have to have money and you steal for tik. You steal money to buy more tik. And then you're in.78\n\nIn high-risk areas, many young people are socialised into gang membership at almost every stage of their lives. Progress through the ranks involves certain actions, choices and levels or recognition. The trajectory, however, is like the game of snakes and ladders with more snakes than ladders. Few make it to the top.\n\nFigure 4 Gang membership progression\n\nSisterhoods: tough support crew\n\nI'm a Bad Girl. We're a group of girls that don't let anyone tell us what to do. We're everywhere: Heideveld, Mitchel's Plain Hanover Park, Bonteheuwel \u2013 everywhere. When I was in primary school my brothers formed the Bad Boys on the corner. They would fight against the Young Hard Livings and Westsiders. They'd have a little war and we girls would look out for them. I liked it. We were under no one. We stood on our own.\n\nIt's exciting and scary to be in the gang world but I normally just keep the gun and the bullets. It's quite risky because if the cops or the other gang gets you it's big shit. We have a connection with the cops \u2013 they're on the payroll so we know when to do stuff and when not to.\n\nI like buttons and tik. You used to have the Bentley but now you've got the Titanic. I can't say where the drugs come from but everyone sells what they want to. In the HL camp there are people who sell buttons, tik and beer and in the American camp and the Jesters and the Clever Kids it's the same.\n\nI grew up in Heideveld and didn't have a good relationship with my mother who passed away last year. My dad didn't help her to raise us. That is why my eldest brother went to in jail. My younger brother lived with my mother and I stayed with my grandmother because my mother couldn't look after all of us and work at the same time. I finished standard 7 at Oaklands High.\n\nMaybe it was the choices I made, but I had a child at 16. Now I'm 27 and she's 11. The man who was the father was already grown up. Me and my daughter now live in my late mother's house. It's very difficult but I survive. Before that I lived in someone else's back yard in the court because my mother didn't give me enough attention. So I went to look for attention somewhere else \u2013 that's how I ended up with the father of my child. He gave me the attention I needed.\n\nI don't have a job \u2013 you don't get jobs easily if you only have standard 7. Even matriculants don't get work. But I'm looking for work. The only income I have is a child grant \u2013 R310 a month. I don't pay school fees but there are always extras they ask for and you have to pay it.\n\nBad Boys stand for Battles Are Dangerous But Only Youngsters Survive. The Bad Girls are born to be bad and we get up to no good. When we're together in the gang we smoke dagga and just talk. And we're naughty and willing to do anything. Most of the Bad Boys that I looked up to when I was younger are all dead and my brother's in jail. Some are married but still Bad Boys. They taught me how to shoot with a gun \u2013 a 38 not a 9-mill. We'd go to shoot in the veld at Silvertree. I haven't shot anyone but I can shoot accurately.\n\nThere are Jester Girls, American Girls, Hard Livings Girls, Diva Girls, Stoepa Girls, Young State Girls, Gaza Girls, Sexy Cats, American Barbie Girls. When the Bad Boys are fighting with the Americans and I find an American Girl on my turf there will be trouble. Bullets will fly.\n\nI don't use a weapon but sometimes the guy I'm with has a weapon, though it's not always necessary because the person is so weak you don't need a gun. You are also scared when you rob someone.\n\nI was in jail for three months \u2013 Strandfontein, Manenberg, Pollsmoor. I got picked up for robbing cellphones and a gold chain. Jail or awaiting trial is what you make of it. If you want to stay the man in charge you have to fight like a broekie [wear the pants]. You can't keep your broekie if you don't fight like a broekie because otherwise they're going to make you a skrikkie [frightened] and you're the girl. It's not the same as with the boys with ndotas and franse, but there are girls who know that stuff. They're like the boys in jail. They do the lovebite like the boys. Some of the young ones are sexually abused because they don't have knowledge about the Numbers tents.\n\nI will never dare to say that I'm not a Bad Girl because I will always be looked at as a Bad Girl. They will expect me to do the things I always did. You can't separate: If you're in it to win it you're in it. I would have loved to teach children about my faith because I'm a Muslim. But with the choices I've made I'm not on that road anymore. Now I'm a little bit where I still lose my mind so at the moment I can't do that. I don't see my granny much any more since I had my daughter. I don't go there anymore [starts crying].\n\nAashka, Bad Girl, Heideveld, 2014\n\nAashka is a delicate-looking girl in her 20s, pretty if you ignore her haunted look and the pallour that results from too much tik. She's a member of the Bad Girls who, she says, are 'everywhere'. She's part of a trend that began emerging from around 2010 in which girls form identifiable, named adjuncts to merchant gangs on the Cape Flats. Their male counterparts, Bad Boys, were started by her brothers and other young men when she was in primary school. They'd have 'little wars' with the Young Hard Livings and Westsiders and the girls would support them as an admiring group. 'There's Jester Girls, American Girls, Hard Livings,' she said. 'Some of the girls fight with their own strength but I'm not scary.'79\n\nAccording to Barbara Williams of Rape Crisis, girls join gangs out of a need to belong, to have a voice and to intimidate other girls. They also think they'll have more fun in a gang. Increasingly, though, they join to get access to drugs:\n\nThey hang out in 'safe' houses run by powerful older men where people smoke. This makes them more vulnerable. Joining a gang may seem like a way to reduce sexual harassment but in fact it increases it. Once they're in and on drugs, the demand for sex and favours follow and they're hooked.80\n\nGirls in gangs are not necessarily victims and can also derive benefit from their involvement. Andiswa, who lives in Gugulethu, told journalist Pharie Sefali she enjoyed being in a gang and it was her choice. She liked the clothes they wear and the style of language they use. She felt protected and said other girls didn't mess with her. 'Other girls want to be like you. When we fight, it's hard. I even have two scars on my back. I was stabbed during a fight with other girls. But it's one of the thrills I had to experience. Now I'm respected.'81\n\nThis is not the case with many girl-gang members. By becoming 'property' of one gang, they become vulnerable to attacks by rival gangs when male members of their gang are not around. They can be beaten, robbed, emotionally abused or raped out of revenge or to provoke a gang fight. This is a problem wherever gangs form. A girl gang member named Melody told a researcher studying gangs in London:\n\nRape is used for everything. It's used in initiation. It's used for fun if people are bored. It's used if you refuse to do something. If you fuck something up. Anything. I'm not even going to tell you what happens if you snitch. But it's used for revenge. If someone wants to get to a rival in another gang, they might gang rape a sister or their girlfriend. I know a girl who got gang raped by 12 men.82\n\nNtombi, a Vura Babe in Khayelitsha, was gang raped by the Vato gang because her brother stabbed one of them.\n\nI was very involved in the gang. My brother used to ask me to hide weapons. I also used to fight with other girls on his behalf and other gang members. That life was the best, because he used to steal things and give them to me. I used to shoplift clothes for the gang. In return, they gave me gadgets they got from robbing... [That was] until I was gang raped and got pregnant. Now I am HIV positive and have left the gang.83\n\nFor male gang members, female gangsters are often referred to as 'poison' for the trouble they can cause. There's a saying on the Cape Flats that a woman is more dangerous than a loaded gun. Battles involving multiple deaths have been started over advances to or the infidelity of a gangster's girlfriend. Female members are used as 'honeytraps', to lure opposing gang members into ambush situations with promises of sex. In this way, or simply through jealousy, gang wars can start and then rip through a neighbourhood for years.\n\nGirls join gangs for many of the same reasons that boys do. They have dysfunctional family life; criminal influence in the household; desire for status and protection; to access signifiers of adolescent success like cool clothing and cellphones. They also join believing that they will be understood. They generally play less violent support roles in gang life \u2013 stashing drugs and guns or looking good on the arm of a gang member \u2013 but their more passive role is changing.\n\nIn 2012 a spate of robberies, in which stabbing was involved, were carried out by both Vura and Vatos Babes in Khayelitsha. Gang researcher Irvin Kinnes saw this as a shift in the activities of female gangs. 'Whereas they previously played roles supportive of men, they were now increasingly becoming part of the teams carrying out violent robberies and shootings.'84\n\n'Social media is telling girls how to stand up for their rights', says Barbara Williams. 'They are becoming more outspoken, able to act out their emotions, feel strong. They want to make their voices heard. They feel they can do it through gang membership and in a way they get what they want, but in a very unhealthy way.85 Some young women will consent to sexual relations with gang members as a means to achieve social and economic status; essentially 'survival sex'. This can include transactional relationships for 'commodities', to gain employment, or to achieve higher status in a culture which prioritises conspicuous consumption.86\n\nFor young women, gang life offers few long-term prospects and many perils. Reflecting on her time in a gang, a woman named Margaret commented: 'Street time isn't like real time. You wake up one day and four years have passed. You were 14 a minute ago and now you are 18. You might have a criminal record, a drug habit, a baby. You've probably been knocked around, raped. You have no qualifications. Where do you go from there? Your childhood has vanished.'87 In Heideveld Aashka saw no escape:\n\nFor me it's better to keep to myself, keep inside the house or if I get a job to keep busy. But the gang will always expect me to do the things I always did.88\n\nFor many years, girl gangs have been in the shadows of the city's gang phenomenon, seen (if they are regarded at all) as mere hangers-on or girlfriends. They are, however, a much more complex social phenomenon. They're young women with family, social and educational problems and few chances in search of protection, self-development and a broader meaning to their lives. They are prone to multiple pregnancies and HIV infection and their drug use poses parenting problems as well as genetic dangers to their unborn and postnatal children. A deeper and more sympathetic understanding of their situation is urgently needed, as is the development of meaningful pathways out of their gang orbit.\n\nWarriors in blazers: empty battles\n\nOf all the gangs I've ever met, the warrior gangs of Khayelitsha made me the most nervous. They were very young \u2013 maybe 14 or 15 \u2013 edgy, probably on tik and all had a knife tucked somewhere. They were prepared to talk and be photographed as long as I paid them drug money. If insufficient, it was clear to me they would extract more at knifepoint and my camera for good measure. There was about them a feral unpredictability; quick to be prompted into almost anything that suited their immediate mood. But they were still kids keen to show off and earn my interested adult regard. Eventually we got on fine.\n\nWarrior gangs like these, who fight as a mark of masculinity, operate mainly in high-migrancy areas where poverty and overcrowding are endemic. Because of the nature of Cape Town's in-migration patterns from the Eastern Cape, they are almost exclusively Xhosa-speaking. A large percentage was born in the rural areas and are fairly recent migrants. The greatest numbers occur in the suburb of Khayelitsha. In the bleak streets youngsters fight often-deadly battles for reasons even they struggle to define. Most are of school-going age, though many have dropped out. They hang around together looking for drugs, opportunities and trouble.\n\nKhayelitsha, which means 'New Home', was developed as an urban satellite during apartheid. The original dwellings were brick, freestanding homes around which a vast, informal, ever-growing squatter settlement blossomed. There's large-scale commuter movement into Cape Town. Many working parents leave home around 4.30 am to get to work on time and return only late in the evening, travelling in uncomfortable and crowded taxis and trains. The Khayelitsha Commission estimated the population in 2014 at between 400 000 and 450 000 people, the majority of whom live in low-cost or squatter accommodation.89 More than half the residents do not have a Grade 12 school pass and only one in 20 has a tertiary qualification. Four out of every 10 young men under 26 are unemployed and nearly half of all householders live in severe poverty. Khayelitsha has very high rates of contact crime, which means that people feel unsafe most of the time.\n\nThese are not easy conditions within which to grow up. Gangs like the Vuros and Vatos tend to be the children of families who came to the city seeking a better life but who have not necessarily found it. They have foresaken their rural roots yet not found fertile ground in the city. Their kids are often rootless, desperate and aware that they have little hope of entering mainstream society. They see parents struggle to make ends meet and peers who may have earned matric passes sitting around with no hope of employment.\n\n'The most important question for these young people who prowl the streets with knives and pangas,' wrote researcher Pharie Sefali, 'is how, in that situation, does a boy become a man? If the only possibility is through violence, he will take that route.'90 The pressure to join a gang is intense, as 16-year-old Ayabulela, a member of the Vatos, explains:\n\nIf you're a guy in that school, you either smoke drugs, rob people or be part of a gang. If you're a nerdy boy or you don't do what the other guys do, they regard you as gay and you are bullied. So I joined the Vato gang. At first I didn't understand how gangsters live, but now I do and I like the thrill. It keeps me on my toes.91\n\nUnlike coloured gangs who usually battle with guns over drug turf, the Vuras and Vatos seem to be influenced by traditional stick fighting in rural areas. A gang member told me that using pangas, sticks and knives was more 'manly' than using guns. 'We want to feel an achievement when we beat our enemy,' he said. 'Guns have power but give no satisfaction.' The violence has deadly consequences. An average of one person is murdered in Khayelitsha every two days and many of the deaths are at the hands of these two gangs.\n\nThe belief in magic, linked as it is to rural traditions, plays a part in warrior gang ritual and empowerment. 'I went to to a sangoma,' a gang member told Sefali. 'I had to bring a bottle of vodka, money, a small amount of fresh human blood in a bottle and a small belt that would fit round my arm. Some people bring beads, others small bottles, it all depends on a person and the sangoma. Then a mixture was done for me and was smeared all over the belt, which we call umkhemi. Now, when I fight, no one can touch me because power from nowhere overwhelms me. I become invisible and can dodge any weapons that come my way. But when I touch you I can kill you.'92\n\n'We're not going to stop these fights until all the Vuras don't exist,' a member of the Vatos told Sefali. 'I'm not sure how the fights started. All I know is that we want respect and those Vuras from Makhaza should stop thinking they rule the area and should leave our girls alone.'93\n\nA less obvious reason for the violence in such areas is the absence of a fear of consequence. The reason for the 2014 Khayelitsha Commission was to address a failure of policing and an extremely low arrest and conviction rate. Testimony after testimony to the commissioners was about seeming indifference by police to the community's needs for a safe environment and the callous treatment of victims of crime. The police, in turn, pointed to low station morale, staff and equipment shortages and poor community cooperation. They also cited the impossibility of policing warren-like, insufficiently lit streets and being called out to houses with no street address or numbers where they were likely to be pelted with stones or shot at. The result was simply ineffective policing.\n\nThe Commission heard that, in the absence of consequences for violence and legal transgression, violence was increasingly attractive. As norm-breaking becomes more common, the resilience of young men against doing wrong is reduced. Political researcher Antony Altbeker has suggested that norm-breaking is determined, in part, by what everyone else is doing. As more people engage in violence without legal consequence, so more people are tempted to use it ('With so many people doing it, what are the chances that I'll get caught?').94\n\nIn what Altbeker calls this 'half-made land', young Xhosa men in the city's 'migrant' suburbs \u2013 cut adrift from the rural moral bearings and shorn of the consequences of their violent urban actions \u2013 are left to chart the limits of their own behaviour by compasses oriented to the opinions of teenage peers with similar moral confusions.95 Mayhem is inevitable.\n\nFor some, however, such as prison Numbers gangs and drug merchants, this turbulence is useful. A Vuro gang member named Sira tells it this way:\n\nAt first we had nothing to do with prison gangs, but more guys got arrested and took the Number inside so the Number played a role in outside gangs. Through the stuff we are doing outside, we'd get arrested, then we meet old friends inside who already have a prison number. You need to choose which Number you want to become. I am 26 in this group, then you have guys who are 28 in the same group. We are outside but we are also Numbers.96\n\nThe protection from Numbers' membership inside prison and the orderliness and enhanced illegal supply chain offered after a term in prison, make incarceration itself a rite of passage into gang seniority. A young gang member told researcher Lerato Kossie: 'We want to be arrested but we are also afraid.'97\n\nIn time, through contact with prison gangs and an increasing market for drugs like tik, it is likely that warrior gangs will metamorphose into merchant gangs, with all the freight this implies. Unless prevented from doing so, Khayelitsha and other such areas will increasingly fall under criminal governance. State officials will become mere visitors in territories beyond their control.\n\nCorner kids: taking the first steps\n\nYou know it's a progression. Young kids fight with sticks and stones in the streets and nobody stops them. Very young kids living in an American territory will fight with young kids in the HL territory by throwing stones. People say 'Ag, it's just kids'. Then later they progress to a knife and then to a gun and you have a big problem.\n\nThe gang business has become very complex and more and more young kids are being drawn in. It's their main reality. There are people who sell R2 dagga 'slowboats' to kids coming home from school. Now they have to do drug tests at primary schools.\n\nThere's this 10-year-old youngster who comes to me at the centre every day and I give him some bread and jam or peanut butter. He tells me he has to give his food to his grandmother at night because she has nothing to eat. And then she shares it with his older sister and tells him to go out and look for something to eat for himself. So he's always on the street corner looking for food. No parents. Maybe the aunt helps, but if she's in a bad mood she shoos him away. He's getting malnourished. Three of his older brothers are Hard Livings. So what's his next step?\n\nCyril Pelston, Shawco community worker, Cape Flats, 2015\n\nAny day of the week the most noticeable social feature of the Cape Flats is the many young people out on the streets. Under the flapping washing strung between blocks of flats can be seen toddlers in the sand and youngsters chasing each other about. Children of 10 and older cluster together on the corners and small groups of young men in their late teens and twenties talk earnestly to each other or on cellphones. Some play dice.\n\nExcept on really hot summer or rainy winter days there's always a sense of busyness. If anything, this is a tribute to human inventiveness: the action generally fills a vacuum of boredom and limited choices. It also hides the heartbreak of community loss following migration from somewhere else or from mass removals. Houses and flats are generally overcrowded and schools are packed to bursting-point. Street life is the spill from families, schools, jobs and overcrowding.\n\nIn squatter settlements or blocks of flats, youths are exposed to a harsh choice: either to stay inside, shut off, cramped, with no private space from family, or to move outside, into the shared courtyards, lanes or streets. Street life becomes the only life possible. There's no private or semi-private space on either side of the door. Every action, except eating and defecating, is a public action. It's also a relief from the physical and emotional pressures which youngsters face and, moreover, it's where their friends are.\n\nOne result is a constant forming and re-forming of playgroups. These are intensely important to youngsters and they're tightly area-bound. Playgroups tend to be formed by the youths who live in a certain block of flats, around a particular open court or a nearby street corner. Like their older brothers, they usually give their group a name \u2013 the Como Kids (after Como Court) or Third Street Kids.\n\nGenerally these playgroups are boys' business; girls are seldom included. Their heroes are the older youths who often use them to run errands. In return they may be offered a cigarette or even a puff of zol (cannabis). They also emulate what they see. A five-year-old in Khayelitsha told a journalist: 'I like the guys when they fight. I watch. I am not scared of them. When we play with my friends we like to take sticks and also pretend we are fighting.'98\n\nYoung peer-group formation is universal and isolation is abnormal, though some parents go so far as to lock their young children in a flat all day, unsupervised, to prevent contact with the street 'skollies'. It's a pervasive pattern in the poorer areas of the city. Playgroups, in these areas are substitute families. They're also sites of entertainment, a source of protection from the dangers of being alone. They are schools for street survival and, very often, mark the beginnings of gang entanglement. Shawco worker Cyril Pelston expressed concern about increasing gang awareness of very young children:\n\nThe games the kids play are gang games. You ask them to draw and they draw a gang sign \u2013 even younger that seven or eight and they know the signs. These images are important to them. Brothers and fathers are gangsters and they're already sensitised to violence. On a holiday programme, a five-year-old had a tiff with a kid and he ran home and got half a scissor that his older brother gave him to stab with. Some really young kids carry firearms because their brothers think they won't be searched. They say 'you keep this for me'. At Shawco camps we have to disarm them. You have generations of socialisation into gang stuff. It's their normal.99\n\nThat predisposition to ready violence, according to General Jeremy Vearey, doesn't begin with the gang, however, but in the home.\n\nWhere do kids learn to resolve conflict with violence? They don't learn it from the HLs or the Americans, they learn it in their home. They learn it from seeing a drunk uncle having an argument with the drunk uncle next door and they end up fighting. And if a woman is getting beaten up nobody reports it. It's just 'natural'. If a kid is witnessing domestic violence day after day, by the time he's 14 he's ready and ripe to be a gang hitman. What is the community doing about that? And now you ask the police to solve the problem. The thing causing a culture of violence in the ghettos where the gangs are strong is the one the community is most silent about.100\n\nEndnotes\n\n1. Thrasher, F: The Gang: A Study of 1,313 Gangs in Chicago (University of Chicago Press, 1927).\n\n2. Fraser, Alistair: Urban legends: Gang Identity in the Post-industrial City (Oxford University Press, Oxford, 2015).\n\n3. European Union, Communiqu\u00e9 2075, Council, Justice and Home Affairs, 19 March 1998.\n\n4. United Nations Convention against Transnational Organized Crime and the Protocols Thereto. (UN Office of Drugs and Crime, New York 2004).\n\n5. Theft of the Nation: The Structure and Operations of Organized Crime in America, (New York: Harper and Row, 1969) quoted in Standing, op cit, p72.\n\n6. Standing, Ibid, p73.\n\n7. Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013) p22.\n\n8. Reuter, Paul: The Organization of Illegal Markets: An Economic Analysis, National Institute of Justice (Washington, 1985) quoted in Standing p74.\n\n9. Zabludoff, S: 'Colombian narcotics organizations as business enterprises', (in Transnational Organized Crime, 3(2), 1997) quoted in Standing p74.\n\n10. Paoli, L: Criminal Fraternities or Criminal Enterprises? Quoted in Williams, P & Vlassis, D: Combatting transnational crime: Concepts, activities, and responses (Transnational Organized Crime, 4(4), 2001a, pp1\u20134).\n\n11. Paoli L: 'The paradoxes of organized crime in Crime, Law & Social Change' 37: 51\u201397, 2002.\n\n12. Pinnock, D: Elsies River; The Brotherhoods; Gangs, Rituals and Rites of Passage.\n\n13. Standing, op cit, p77.\n\n14. Klaus von Lampe, Definitions of Organized Crime, www.organized-crime.de\/\u00adorganizedcrimedefinitions.htm.\n\n15. Gastrow, 1998, p39.\n\n16. SAPS Organised Crime Unit, Anti Organised Crime Measures, August 1997. Quoted in Shaw 1998, p1.\n\n17. Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013) p2.\n\n18. Shaw, M: The Development and Control of Organized Crime in Post-Apartheid South Africa, in Einstein, S & Amir, M, Organized crime: Uncertainties and dilemmas (University of Illinois Press, Illinois, 1999.), p65.\n\n19. Dev Kar and Joseph Spanjers: Illicit Financial Flows from Developing Countries: 2003-2012 (Global Financial Integrity: Washington, December 2014). The figure for the same year in Russia was US$937.9 billion and China US$249.4 billion.\n\n20. Ibid, p2.\n\n21. Global Financial Integrity: Illicit financial flows from Africa: hidden resources for development (Washington, DC: GFI, March 2010) quoted in Shaw & Reitano, op cit, p7.\n\n22. Davids, Nashira: 'South Africa's capital of organised crime' (Times Live, 14 February 2012).\n\n23. Quoted in H\u00fcbschle, Annette: Organised Crime in Southern Africa: First Annual Review (Institute for Security Studies Special Report, Pretoria 2010) p3, supplemented by Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013) p6.\n\n24. H\u00fcbschle, op cit. p3.\n\n25. Ibid. Between 2001 and 2013 there were 27 097 drug-related arrests in KwaZulu-Natal, 16 854 in Gauteng and 9 118 in the Eastern Cape and 51 512 in the Western Cape.\n\n26. National Drug Master Plan 2013 \u2013 2017. Presentations by the Financial Intelligence Centre, May 2014. The percentage of estimated use is 32.1% for cannabis and 12.9% for mandrax. Also Dan Stein, George Ellis, Ernesta Meintjies and Kevin Thomas: Introduction: substance use and abuse in South Africa. In Substance use and abuse in South Africa by Dan Stein, George Ellis, Ernesta Meintjies and Kevin Thomas (eds). UCT Press Cape Town, 2012.\n\n27. Goredema, C and Goga, K: Crime Networks and Governance in Cape Town (Institute for Security Studies paper 262, August 2014).\n\n28. Ibid.\n\n29. Owen, Rebecca, Financial Mail, January 17, 2014.\n\n30. Standing (2005) p1.\n\n31. Kinnes, Irvin: 'Structural changes and growth in gang activities', Published in Monograph No 48, From urban street gangs to criminal empires: The changing face of gangs in the Western Cape, June 2000.\n\n32. Lonte's gang, the Americans, had been covertly supplied with drugs by the Civil Cooperation Bureau in return for quelling political unrest against apartheid and it dominated the mandrax market . See Kinnes, Ibid.\n\n33. Ibid.\n\n34. Ibid.\n\n35. Interview with Jeremy Vearey, 2014.\n\n36. Opening address given by Leonard Ramatlakane at the Provincial Gang Strategy, April 2003 . Unpublished document distributed by the Department of Community Safety. In Standing (2005) p9.\n\n37. Steinberg, J: The Number (Jonathan Ball, Cape Town 2004). Charles van Onselen: The Small Matter of a Horse: Life of Nongoloza Mathebula 1876\u20131948 (Ravan Press, Johannesburg 1985). Heather Parker Lewis: God's Gangsters: The Numbers, Gangs and South African Tattoo Prison Culture (Ihilihili Press, Cape Town, 2010).\n\n38. Department of Justice, Annual Report for the Year 1912, 37.\n\n39. Steinberg: Jonny: Nongoloza's Children: Western Cape Prison Gangs during and after Apartheid (Centre for the Study of Violence and Reconciliation, July 2004).\n\n40. Vearey, Major General Jeremy: Nongoloza's legacy \u2013 prison and street gangs in the Western Cape. (Confidential SAPS report, n.d.). As an MK soldier, Veary was incarcerated at Pollsmoor in 1987.\n\n41. Ibid.\n\n42. Ibid.\n\n43. Vearey, op cit.\n\n44. Vearey SAPS report, op cit.\n\n45. Taheri-Keramati, Yashar: Drugs, police inefficiencies, and gangsterism in violently impoverished communities like Overcome (UCT Criminology MA thesis, 2014).\n\n46. Steinberg, J: Nongoloza's Children: Western Cape prison gangs during and after apartheid. (Centre for the Study of Violence and Reconciliation, July 2004).\n\n47. Ibid.\n\n48. Ibid.\n\n49. Ibid.\n\n50. Steinberg op cit.\n\n51. Interview with Major Jeremy Vearey, Amandla!,2014.\n\n52. Interview with Nealon Peterson, Hanover Park, January 2015.\n\n53. Taheri-Keramati, op cit.\n\n54. Kinnes: Irvin: From Urban Street Gangs to Criminal Empires: The Changing Face of Gangs in the Western Cape (Institute for Security studies Monograph 48, June 2000).\n\n55. Interview with gang leader, Hanover Park, February 2015.\n\n56. Skywalker, Luke Lee: Politics of the Number: An Account of Predominant South African Prison Gang Influences (UCT Masters thesis, 2014) p45.\n\n57. Interview with Nealon Peterson, January 2015.\n\n58. Interview with Jonathan Jansen, Manenberg, September 2014.\n\n59. Interview with Michael, Manenberg, 2014.\n\n60. Jonathan Jansen interview, Manenberg, 2014.\n\n61. Jansen, ibid.\n\n62. Interview with Cyril Pelston, Cape Town, 2014.\n\n63. Ibid.\n\n64. Kinnes, Irvin, ibid.\n\n65. Jansen, op. cit.\n\n66. Standing: The Threat of Gangs and Anti-Gangs Policy (ICS Occasional Paper 116, August 2005).\n\n67. Interview with Dice, 2014.\n\n68. Interview with Trappies, January 2015.\n\n69. Interview with Joey, February 2015.\n\n70. Interview with Major General Peter Jacobs, November 2014.\n\n71. Levitt, Steven & Dubner, Stephen: Freakonomics: a Rogue Economist Explores the Hidden Side of Everything (Harper Collins, New York, 2005) p114.\n\n72. Jansen, op cit.\n\n73. Vearey, quoted in Amandla! interview, Issue 32, on www.amandla.org.za.\n\n74. Carlie, Mike: Into The Abyss:A Personal Journey into the World of Street Gangs (http:\/\/people.missouristate.edu\/\u00adMichaelCarlie\/\u00adsite_map.htm).\n\n75. Interview with Amandla!, op cit.\n\n76. Pelston, op cit.\n\n77. Jansen, op cit.\n\n78. Interview with Machael, Manenberg, 2014.\n\n79. Interview with Aashka (not her real name), Manenberg, January 2015.\n\n80. Interview with Barbara Williams, Rape Crisis, March 2015.\n\n81. Sefali, Pharie: 'The rise of female township gangs' (GroundUp 18 September 2014).\n\n82. Combi, Chloe: 'Real bad girls, extraordinary insight into London's female gang culture' (The Independent, 6 August 2013).\n\n83. Ibid.\n\n84. Quoted by Ilham Rawoot in Mail & Guardian, 17 May 2012.\n\n85. Williams, op cit.\n\n86. Swartz, S., Van der Heijden, I., Runciman, T., Makoae, M., Rozani, A., Dube, N., Makiwane, M. & Bhana, A. (2010) Think for Yourself \u2013 Think for Tomorrow': Exploring the Impact of Peer-Led HIV Intervention and Psychosocial Support Groups for Vulnerable Youth in South Africa. Cape Town. (Human Sciences Research Council).\n\n87. Combi, op cit.\n\n88. Aashka interview, op cit.\n\n89. Khayelitsha Commission Report p35.\n\n90. Sefali, Pharie: Gang violence in Khayelitsha (Safety and Violence Initiative \u2013 University of Cape Town and GroundUp, 26 November 2014).\n\n91. Ibid.\n\n92. Ibid.\n\n93. Ibid.\n\n94. Altbeker, Antony: A Country at War with Itself: South Africa's Crisis of Crime (Jonathan Ball, Johannesburg, 2007, p114.\n\n95. Ibid, p118.\n\n96. Interview with Sira, Khayelitsha, by Lerato Kossie, July 2014.\n\n97. Ibid.\n\n98. Sefali: Khayelitsha ruled by teenage gangsters, GroundUp, June 2014.\n\n99. Interview with Cyril Pelston, Shawco, 2015.\n\n100. Vearey, General Jeremy: Civilian Secretariat for police workshop: The development of a framework for interdepartmental anti-gang strategy in South Africa, Cape Town 3-5 March 2015.\n3. Understanding adolescence\n\nIn search of cool\n\nWhile the gang ethos pervades many levels and ages of Cape Flats society, the single most remarkable fact is that core membership is predominantly \u2013 almost exclusively \u2013 adolescent. It is important to understand why.\n\nFrom the point of view of an adolescent, gangs are cool. This is a discomforting fact for most adults and people in authority. The Cape Flats youth development programme, Amandla EduFootball, did a survey to establish what a group of youths considered to be cool. According to Karl Voysey, 'after they told us what they thought we wanted to hear, they got into what they really considered was cool, like having a knife. Stabbing is cool. Wearing this brand is cool. Drinking is cool. Drugs are cool.'1 Aggressively distancing parents was particularly cool. So was provoking authority, smoking, stealing a car, getting pregnant and generally tempting fate.\n\nStatistically, aberrant teenage behaviour is the norm and, as psychologist Terry Moffitt has suggested, its absence in a teenager is cause for concern.2 The fierce push away from the confinement of childhood and adult society is age-old and widespread. This is evidenced, for example, by the worldwide appeal of fierce gangsta-rap groups like Cape Town's Die Antwoord, which draws its inspiration from the Cape Flats. These are lyrics from one of their songs:\n\nDaai bra anies hy's n fokken gam bra\n\nHaai! daai anies hy lam innie mang ja\n\n'Ken sy my nommer?' xha! boy what's your number?\n\nTwee ses? twee sewe? of is jy n ag bra?\n\nThat boy anies, he's a fuckin ghetto boy\n\nAnies chills in jail\n\n'Does she know my number?' no! boy what's your numer?\n\n26? 27? or are you a 28?\n\nThrow dem devilish gang signz in da air\n\nStart giving it up 4 little evil me\n\nMy fingerz r green coz I'm a mean dope fiend\n\nI'm wicked like mad d.o.g\n\nFresh like a little dark g.o.d\n\nMy zef accent iz very foreign\n\nWhen I speak oaz dey go: I beg your pardon?\n\nThe majority of gang members in Cape Town are young. It was soon obvious to me that if I wanted to understand why they join gangs, I first needed to appreciate something about adolescence. Attempting to suppress gangs is essentially trying to extinguish teenage behaviour which members regard as essential for their self-respect and social acceptance. It seemed clear that it would fail unless alternative, reinforcing behaviours were made available which fulfil these needs.\n\nA breakthrough in my understanding of adolescence came almost by accident \u2013 while spending time within a traditional Xhosa culture in the Eastern Cape. Despite more than 100 years of disruptive migrant labour and nearly three centuries of Christian and Western influence, rituals of adolescent passage and the handing down of ancestral teaching were still in place in many areas. Their teenagers were no different from young people in the city. They had the same crazy energy, peer group clannishness, rising sexuality and civil dissonance.\n\nBut the way they were handled was much more sophisticated than in Western urban culture \u2013 with so much more understanding \u2013 and the outcome was proud, self-confident young adults. These traditions, it seemed, had much to teach modern justice systems about social control and stability. It sent me on an exploration of adolescent transition rituals in pre-modern cultures.\n\nAlmost all traditional societies greet the onset of puberty, especially in males, with elaborate and excruciating initiations \u2013 practices that plainly would not have been necessary unless their young were as extreme as ours. At the time I had only the vaguest sense this might tell me something about gangs, but it was to tell me more than I could possibly have imagined.\n\nIn his essay called The Age of Endarkenment, Michael Ventura notes that adults in traditional cultures don't run from adolescent excess as do people in Western cultures. Instead they celebrate it:\n\nThey would assault their adolescents with, quite literally, holy terror, rituals that had been kept secret from their young until that moment ... rituals that focused upon the young in all the light and darkness of their tribe's collective psyche, all its sense of mystery, all its questions and all the stories told to both harbour and answer those questions.3\n\nThese adults consider they have something to teach: certain skills, dances, stories, magic, visions and rituals. If these things are not well learned, it places the future of the culture in jeopardy.\n\nAt a certain moment young men and women in these cultures are, therefore, transferred from conventional to ritual space. They step over a threshold through some sort of ceremony into a symbolically 'heated' situation beyond the mundane. In this place, before they can become an adult, some infantile being in them must die. For young boys, initiation is generally performed by older men who help them move from their mother's world to their father's world. Poet and manhood teacher Robert Bly insists that 'women can change the embryo to a boy, but only men can change the boy to a man ... boys need a second birth, this time a birth from men.'4\n\nThe process of separation is often dramatic and involves procedures noticeably different from everyday life. In Hopi culture the old men take the boy away at the age of 12 and bring him into the all-male area of the kiva. He stays down there for six weeks and does not see his mother again for a year and a half. In New Guinea the initiated men live together in houses at the edge of their village. Mothers carefully refrain from telling their sons anything about the impending events, retaining the element of surprise. As the men lead them away, the boys may be crying out: 'Save me. Mama, save me!'5\n\nThese practices have developed over centuries because they are necessary for the stability of community life. Instead of condemning youthful wildness, they capture its intensity in rituals that teach and empower while protecting social life from adolescent excesses. Such practices continue because these cultures have learned what the West has forgotten: if a culture does not deal with the warrior energy of its young men and the spirit energy of its young women it will turn up outside in the form of gangs, wife beating, depression, drug violence, brutality to children and even aimless murder.6 That energy, these cultures have found, needs to be taken in consciously, disciplined and honoured. To most traditional people the absence of such rituals is almost unthinkable. When I asked what would happen without them, an elderly Tembu chief in the Eastern Cape was unambiguous:\n\nIt would be chaos. The authority structure would break down. The young men would have bad dreams and the young girls would get sick. They could even die! Also they would have no voice.7\n\nWorking among the Qaba people in the former Transkei, Joan Broster found that young teenagers were initially incorporated into the umtshotsho. This was a club where young men and women danced and courted under the supervision of a 'magistrate'. Strict rules of behaviour were enforced and any transgressions were paid for in fines of brass or copper bangles:\n\nIn my area the meetings were held every Thursday and they came in their best clothes \u2013 clothes and beads were very important. The girls would dance in a circle and the boys in their own circle. Occasionally a girl would circle a boy \u2013 but they would not be allowed outside the hut to 'pet' without the permission of the magistrate. The magistrate was backed by a committee and a lot of ritual teaching went on. The youngsters couldn't wait to be admitted to umtshotsho.8\n\nLater, adolescents in these cultures underwent more formal rituals of transition. Girls would be secluded 'behind the curtain' during intonjane and boys would undergo ulwaluko (more commonly known as abakhweta) \u2013 circumcision rites. An old inkangata, or female spiritual teacher, described intonjane to me as follows:\n\nAt about 15 the girl must train for womanhood. She goes behind the curtain in a hut for 28 days. In the old times it was longer. There she is taught by her grandmother who is the inkangata \u2013 the mother is not allowed in the hut. Intonjane is an attempt to shape the child and discipline her in some way and teach her about her ancestors and about healing and being a woman. Without these ceremonies they would go completely wild.\n\nDuring the ceremony, the young woman remains naked but painted white \u2013 a colour signifying a taboo state \u2013 having only a blanket for protection. All her old clothes and her previous ways are expected to be left behind. After eight days a community celebration takes place, signifying the importance of the ritual. When the young girl emerges from seclusion she is welcomed at a community celebration of singing and dancing at which a bull is slaughtered.\n\nYoung Xhosa men undergo a different passage to manhood. A small domed grass hut is built for them in a secluded place far away from the community and often in the mountains. Before the abakweta, as an initiate is called, enters this ritual space he, like his female counterpart, leaves his clothes behind, is painted with white clay and wears only a blanket. Tension is built up around the ritual. A young Mpondo, Nzimela Ncoyini, who underwent the ceremony, which included circumcision, first asked his mother if he could undergo the ritual but she refused, saying he was too young. But two years later he asked his father, who said it was the right time. The procedures were kept secret:\n\nYou are never told what's going to happen. Everything is hidden until you go yourself. It's good that way or people will take it for granted. You have mixed feelings \u2013 you're afraid of the operation but you know that across the bridge there's milk and honey.9\n\nFor boys, the ritual from this point is strictly men's business and in the hut the initiate is accompanied at all times by a male teacher. Women are actively excluded:\n\nYou have to denounce your feelings for a woman. They call a woman isigwati \u2013 it doesn't make sense. If someone talks about a woman in your presence there is something you have to shout out to exclude the thought. It is to destroy the memory.10\n\nThese are not sexist practices. Robert Bly sees such initiation as a process which asks the son to 'move his love energy away from the attractive mother to the relatively unattractive 'serpent' father':\n\nWhen a man enters this stage he regards Descent as a holy thing, he increases his stomach for terrifying insights, deepens his ability to digest the evil facts of history, accepts the job of working seven years under the ground, leaves the granary at will through the rat's hole, bites on cinders, learns to shudder and follows the voice of the old mole below the ground.11\n\nYoung Xhosa men undergoing initiation are instructed on what is expected of a man. The process lasts about a month nowadays but used to last up to nine months. Teaching is given about spirit ancestors and about sacred and ritual foods and objects. A special language, isikhweta, is used during the initiation period. It is deemed sacred and taboo.12 According to Ncoyini this language made him feel like a new person:\n\nThe language changes your mentality \u2013 it is very important. It makes you feel a bit uncomfortable at times but you really feel that your presence is appreciated. People listen to you when you use these terms. Adults pay attention and respect you and are prepared to help you when you use these terms.13\n\nWithin the ritual there are traces of military initiation and battle discipline. Strong bonds are created between fellow initiates. According to Ncoyini, the people who shared his 'bush' experiences were comrades and 'if we still had wars, I could be a soldier'.\n\nTraining for warriorhood is central to many rites of passage for young men. The quality of a true warrior is that he's in service to a purpose greater than himself, a transcendent cause. When a young male's warrior behaviour and warrior thoughts are not acknowledged, it can be socially destructive. It is then that we find overheated young men with jail sentences or with averted eyes searching for action. It is among these 'kingless, warriorless boys,' says Robert Bly, that 'poisoned warriors called drug lords' prey for recruits.14\n\nThe abakweta ceremony is often treated with amused tolerance or even embarrassment by city dwellers in South Africa. But it's a time into which most of Xhosa culture's values are packed and learned. It contains important intuitions about adolescent character \u2013 shaping and the excitement of the transition to adulthood. For young people, the rewards are considerable. At the celebration after the ritual the young men are welcomed back into the community as adults and equals. The newly initiated can attend meetings at the Great Place of men and no longer have to speak through others at beer drinks and other social occasions. More important, is the feeling of acceptance. Ncoyini remembers:\n\nMy father received me, the older people who had been through the ceremony welcomed me and the community of adults gave me advice on how to be a new person. I felt very positive, very warm and welcome. I felt purified. It made me feel I wanted to act responsibly.15\n\nThese are feelings deeply yearned for by young people but seldom attained in Westernised adolescence. Having abandoned initiation, Western education and values don't readily lead boys towards manhood. In urban ghettos, such boys invent rituals to fill the gap. There's a poignant sadness in this: young men cannot initiate each other. Youngsters who are without community have no knowledge of rituals or the safe paths to warriorhood. They have no older men to welcome them into the ancient, mythologised, instinctive male world and, very often, no effective fathers who understand what it is they're being asked.\n\nWhat are young people to do with this wild energy in overcrowded ghettos? In large part, joining a gang, with its rejection of authority and thrilling danger, provides an answer. Gang formation is therefore one of the outcomes of an absence of a formalised, ritualised transition to adulthood. Other outcomes include depression and even suicide. In the absence of any models of transcendence, young people are using delinquency as ritual because, where ritual is absent, it will be created from ancient desires and strange stirrings.\n\nElaborately, often unconsciously, using tools, substances and attitudes dating back to the dawn of our species, young people construct clan-like rituals of transformation. These have as a single goal to earn adult respect \u2013 the magic elixir they require to become adult themselves. In these painful and dangerous journeys can be found echoes of African initiation ceremonies; Jewish barmitzvahs; ancient hunting rituals; Boer kommando lore; images of Hollywood; holy communion; Khoi trance dances; Arthurian legends and many other rituals through which, for millennia, young people have attempted to prove themselves worthy of adulthood. Some make it through, some crash and burn. Much of this book is an attempt to understand why. Here's a beginning...\n\n***\n\nIn some neighbourhoods gang membership is more common than in others, but even in high-risk, low-income areas, not all young people join gangs and not all gang members proceed to a life of adult crime. There are, in fact, marked individual differences in engagement or non-engagement with antisocial activities that need to be clarified.\n\nMost theories of delinquency take no account of the changing needs and manifestations of young people over time. In the extraordinary metamorphosis that takes place during adolescence, this limits understanding of why and when a young person engages in an illegal activity or joins a gang. Studying delinquents at the peak participation age, according to psychologist Terri Moffitt, 'offers the least favourable prospects for understanding the sort of antisocial subject who will develop an adult career of crime and violence.16\n\nYoung people on the Cape Flats are not static entities definable within a fixed structure. They are people maneuvering their way through life within family and neighbourhood structures as they find them. So in trying to understand the mysterious relationship between age and anti-social behaviour, it's necessary to trace their trajectory from the moment of conception and try to understand the choices they make in relationships within their family, neighbourhood, school and society at large.\n\nIn an attempt to understand these trajectories, Moffitt used longitudinal tracking done in Dunedin, New Zealand, and Pittsburgh in the United States over more than 40 years to link life incidents and paths to antisocial outcomes. She found that delinquency conceals two categories of individual behaviour which she termed 'adolescence limited' and 'life-course persistent'. The former tend to dip in and out of delinquency, peaking at around 16 and falling off towards their 20s. Life-course persistent delinquents, as the term implies, start antisocial behaviour earlier and continue into and through adulthood.17 She found delinquent behaviour to be so pervasive among teenagers as to question whether non-delinquency warranted scientific scrutiny and whether abstaining from delinquency was necessarily a sign of good adolescent adjustment or a recipe for depression.18\n\nFor life-course persistent youths, development may have been disrupted from the beginning by prenatal damage and deprivation of nutrition, stimulation, prenatal attachment and affection. These deficits have been linked to the kind of antisocial behaviour that begins in childhood and is sustained through adolescence, as we will see in the following chapter. Such children are not normally born in supportive environments and are often reared in families whose members amplify disadvantage through haphazard prenatal care, drug use during pregnancy and neglected nutritional needs. These may inadvertently provide their children with criminogenic environments.19\n\nThe finding of the Dunedin longitudinal study is that prenatal problems plus poor child care runs a high risk for a life-course persistent pattern of antisocial behavior.20 According to Moffitt:\n\n...in disadvantaged homes, schools and neighbourhoods, [these] responses are more likely to exacerbate than amend. Under such detrimental circumstances, difficult behaviour is gradually elaborated into conduct problems and a dearth of pro-social skills. Thus, over the years, an antisocial personality is slowly and insidiously constructed. Likewise, deficits in language and reasoning are incrementally elaborated into academic failure and a dearth of job skills. Over time, accumulating consequences of the youngster's personality problems and academic problems prune away the options for change.21\n\nAdolescence-limited youths with more secure childhoods, according to Moffitt, become involved in delinquency through peer pressure. This she terms 'social mimicry'. Their antisocial, ritualised behaviour is a way of 'knifing off childhood apron strings and of proving that they can act independently to conquer new challenges.'22 It's a desire, as we have noted, with deep and ancient roots to be \u2013 and be seen \u2013 as other than their parents and their own childhoods. Unless their inner warrior is tested by society, in the absence of formal rituals they will test themselves.\n\nUnlike life-course-persistent youths battered by adversity from an early age, adolescence-limited youngsters with better parenting and more limited early deviant entanglements have had sufficient time to develop a bag of prosocial behaviours and academic skills. This begins to answer the often-asked question of why some young people from the same neighbourhoods make it and others don't. Parenting matters.\n\nThe Dunedin study found that, at the crossroads of young adulthood, adolescence-limited and life-course-persistent delinquents go different ways. Of these different paths, poet Robert Bly says rather dramatically: 'One man is a self-sacrificing warrior fighting for a cause beyond himself; another man is a madman soldier, raping, pillaging, killing mindlessly, dropping napalm over entire villages.'23 This happens because the developmental histories and personal traits of adolescence-limiteds allow them the option of exploring new life pathways. They have greater options as adolescence ends: better education, jobs and social contacts. The histories and traits of life-course-persistents, on the other hand, have foreclosed their options, entrenching them in an antisocial path.\n\nSo, in enquiring why young people join gangs, we need to reverse the question and ask what sort of life chances gang members need to have been dealt in order to make gang entanglement transient or unlikely. This nests in a larger question about their freedom to make that choice because, for many, avoiding gang membership is extremely difficult.\n\nThe Nobel economist Amartya Sen notes that lack of such freedom can arise either through inadequate processes (such as violation of privileges) or through inadequate opportunities that people have for achieving what they'd like to achieve (such as food, education, employment or social respect)24. We can add a further curb to personal freedom: the mental or physical inability to engage in socially beneficial or acceptable behaviour because of prenatal and early childhood trauma. The degree of freedom determines a person's opportunity to have desirable and valuable outcomes. That opportunity, in turn, depends on the capabilities and resilience they have to lead the kind of lives they have reason to value. This capability is what Sen terms agency. In considering gangs, it's necessary to look at what undermines personal agency, and this is what we'll explore next.\n\nEndnotes\n\n1. Interview with Karl Voysey, 2014.\n\n2. Moffitt, TE: 'Adolescence-Limited and Life-Course-Persistent Antisocial Behavior: A Developmental Taxonomy'. (Psychological Review 1993, Vol. 100, No. 4, 674-701) p685.\n\n3. Michael Ventura: The Age of Endarkenment, in Whole Earth Review, Winter, 1989, p180.\n\n4. Robert Bly: Iron John (Element, Dorset 1990) p16.\n\n5. Ibid, pp14 & 86.\n\n6. Ibid, p179.\n\n7. Interview with Chief Dalaguba, Incgobo, Transkei, June 1995.\n\n8. Interview with Joan Broster, June 1995. She is the author of a number of books on Xhosa customs.\n\n9. Interview with unnamed inkangata, Encgobo district, July 1995.\n\n10. Interview with Nzimela Ncoyini, Cofinvaba district, July 1995.\n\n11. Robert Bly, 1990, p91.\n\n12. A similar language, ukuhlonipha, is used by women and girls. Both languages allow for the discussion of taboo subjects. Many of the words are relics from the past and except in this usage are unknown. These initiatory languages are most likely the origins of the use of sabela among prison gangs.\n\n13. Interview with Nzimela Ncoyini, Cofimvaba district, July 1995.\n\n14. Robert Bly, 1990, p156.\n\n15. Interview with Nzimela Ncoyini, op cit.\n\n16. Moffitt, op cit, p696.\n\n17. Moffitt, ibid, p674.\n\n18. Moffitt, ibid, p690.\n\n19. Sameroff, A., & Chandler, M: 'Reproductive risk and the continuum of caretaking casualty'. In F. Horowitz, M. Hetherington, S. Scarr-Salapatek, & G. Siegel (Eds.), Review of Child Development Research, Vol. 4, pp187-244 (Chicago: University of Chicago Press, 1975).\n\n20. Moffitt, op cit, p682.\n\n21. Moffitt, ibid, p684.\n\n22. Moffitt, ibid, p688.\n\n23. Bly, op cit.\n\n24. Sen, Amartya: Development as Freedom (OUP) Oxford 1999 p17.\n4. Families in crisis\n\nMuch can be learned about a teenager with tattoos on his chest, a gun in his hand and murder on his mind from the emotional context of his origins. If I was to understand who these young gang members were and why they acted the way they did, I needed to start with their early family life. Love and care are what families are for, a survival instinct that pre-dates human existence. Newborn babies feel this not so much through stimulation or feeding, but through responsiveness. Having a mediator between a child's temperament and the challenges of entering and mastering the world creates bonding attachment. Good attachment doesn't prevent possible later misfortune, but it does provide the resilience to cope with difficulty when it happens. Infants who are securely attached generally become well-adjusted children, explorative adolescents and responsible parents.\n\nOn the other hand, children with parents or caregivers unable or unwilling to be a responsive 'other', or who are for some reason entirely absent, have trouble making sustaining emotional connections. For the implications of this I needed a psychologist of early childhood and found one in James Gabarino. These boys, he says in his book Lost Boys, have problems with their own feelings and the feelings of others. They later lack the skills to be a functioning member of society. They're drawn to others like themselves who may be without empathy, sympathy and caring. They carry feelings of shame and anger which they generally hide with bravado and, often, violence.1 Their problems can become life-course persistent.\n\nSuch youngsters struggle to find their place in the world. Pain and desolation drive their emotional selves into hibernation. So when they most need to feel they belong to someone who could protect and love them, they experience emptiness, feel disdain and see only weakness.2 They're psychologically alone and socially vulnerable to influences that seem to feed the desolation. What they suffer from, according to Garbarino, is toxic shame: 'Fundamentally disgraced, intrinsically worthless and profoundly humiliated in their own skin, just for being themselves.'3\n\nThere is no greater injustice than for a child to be unloved. Lack of love and care, therefore, is one of the keys to an understanding of the gravitational pull of gangs. A large number of young people in gangs are, therefore, simply unloved and in search of a family.\n\nWhile some children may be physically hurt or genetically compromised, harsh conditions in low-income, high-risk areas give rise to much greater numbers who are emotionally abandoned. A parent's psychological unavailability is a form of child maltreatment which plays a central role in the development of bad behaviour and aggression. 'From their surroundings children develop social maps and codes of behaviour,' says Garbarino. 'For most, this portrays a world in positive terms: Can I trust people? If I behave well, will I be treated well? Am I loveable? I have allies in the world.'4 From positive answers to these questions, children develop positive codes of behaviour. They listen to adults, discover that cooperation pays off, are patient and share.\n\nChildren with attachment problems have different social maps adapted to their unsupportive environment. They become hypersensitive to negative social cues and oblivious to positive ones. They develop aggressive behaviour to protect themselves and conclude that aggression is a way to get what they want. They flee psychologically, shut down emotionally and disconnect themselves from their feelings so they don't have to feel them. This all works in the short term, but the anger grows. Behind the anger is repressed sadness, which can lead to depression.5\n\nThis is compounded by South Africa's still unsolved racism coupled with economic deprivation. Both of these are generators of a sense of alienation. When a child finds out that society's response to him echoes the rejection he feels at home, alienation easily becomes shame. This can trigger a retreat from empathy, which is one of the foundations of emotional intelligence and moral judgement. A boy who needs to protect himself from feelings of victimisation and unworthiness is not likely to pay attention to the feelings of others, especially to their feelings as victims.6\n\nHow we behave comes to reflect what is stimulated, encouraged, rewarded and proves successful in a particular social context, be it home, neighbourhood, school or peer group. Young children will conform to the social environment whether the context is positive or negative, supportive or toxic. This is because they assume to be normal whatever occurs and is reinforced daily. Conformity first happens through constant reinforcement from adults and the situation in which they find themselves.\n\nEventually this reinforcement is internalised into what psychologists describe as functional autonomy. For many young people on the Cape Flats, negative conformity \u2013 built up over years of personal experience \u2013 normalises conditions that middle class people would consider horrific. To understand this, we need to explore their relationship with those who created them.\n\nFathers: anger and shame\n\nFathers are important but \u2013 even when not absent \u2013 they seldom understand how crucial they are as role models to their sons. They generally go about fathering unconsciously. Their interaction with their offspring often mirrors the relationship they had with their own fathers. A young man may be irritated, annoyed, infuriated or simply embarrassed by his father but he's watching him to see how a man acts in the world.\n\nIn large areas of Cape Town, fathership is under severe strain. A restorative programme undertaken under the guidance of the Usiko Trust found that among more than 1 000 young men who took part and were considered at risk, almost every one had a violent, undermining or absent father. A young former gangster from Khayelitsha described this absence:\n\nI walked out into the wilderness without navigation skills. I was not prepared. In my search I found a family on the corner with a group of boys standing smoking. We became territorial. We protected each other. In that way we became a gang. If I had a father to assist me and show me the right direction, it would have been different. Instead I went out the house in search of love and guidance.7\n\nGarbarino notes two types of failed fatherhood: The presence of an abusive father and the absence of a caring one. Both amount to being fatherless. Having no father increases the odds that a boy will grow up where resources of all kinds are in short supply. The absence of good fathering also increases the chances that a boy will lack a male guide and protector \u2013 a risk factor for later delinquency. A boy whose father is absent is plagued by the question: 'Why don't I have a father?' and will conclude that the reason is that he, the boy, is not worthy. This causes both shame and anger.8\n\nWithout a husband, a mother has to fill dual parental roles in terms of both care and income. Strong women, with the help of family or close friends, manage well enough. A weaker woman, battered by life and in need of emotional support, often turns to a son who becomes her emotional protector. This places an unequal strain on a young man, but it also reduces her authority in his eyes and leaves him bereft of any authority figure in the family.\n\nWithout a father whom they can regard in search of a personal identity, young men remain confused or seek out other role models, such as teachers (if they are still at school), older peers, film stars or gang bosses. Increasingly, though, these role models are not, in an immediate, physical sense, real people. They're cultural stand-ins.\n\nBecoming a 'real man' without a father in a demoralised neigh\u00adbourhood is a hard call. Given high poverty rates, rising expectations and a globalised commodity culture that exalts middle-class lifestyles, how should you look, what should you do? For young coloured men who form the bulk of Cape gangs, identity is particularly problematic. As a young male, how do you position yourself in relation to the relentless stream of hegemonic notions of largely white or black, heterosexual masculinity staring at you from magazines, on billboards or beaming at you from television soapies or the movies? How do you plot your way through vicarious media violence espousing crude sexuality, shallow materialism, mean-spirited competitiveness and spiritual emptiness?\n\nYoung men in townships like Manenberg, Hanover Park or Mitchell's Plain have limited access to the key resources of education, jobs, income, a wife or symbolic status which define a dominant masculine identity. 'My biggest dream,' a gang member told me, 'was to become a doctor because I loved Mercedes Benz cars and only doctors drive a Mercedes. The dream vanished when I realised it took a lot of money to study to be a doctor. And the situation at home was against that dream.'9\n\nIf anything, the impossibility of such dreams simply increases the need for an identity \u2013 to be seen and respected, to be a 'someone'. For many, violence \u2013 or the threat of violence \u2013 is a means to claim that identity and gangs are a vehicle for its attainment. Through gang membership young men can assert, while it lasts, what anthropologist Elaine Salo describes as a subordinate masculinity10.\n\nIn her study of young coloured men in Manenberg, she found notions of manhood had historical links to the way in which coloured women had been integrated into apartheid bureaucracy and the contemporary city's capitalist system. In terms of the apartheid state's assumption that all households conformed to a two-parent family norm where fathers and mothers fulfilled stereotyped gendered roles, child welfare and social security grants were only payable to women as mothers. Public housing was only provided to families with women and children. To add to this, in terms of the Coloured Labour Preference policy in the Western Cape, coloured women became the preferred workers in the city's booming clothing industry. This made them power brokers for their communities within the apartheid social structure.11\n\nIn 1994 the government scrapped import tariffs, resulting in an influx of cheap Chinese clothing. This led to the collapse of a well-established, strong textile and clothing industry which was a major employer of coloured women. Its demise impacted on their earning power, which declined. With the development of tourism and inner-city revival in the 1990s, Cape Town's commercial and service industries picked up and, in terms of affirmative action, white staff was replaced by coloured women. So in Manenberg and similar townships, it's still predominantly coloured women who, through the recognition of their social ties to individuals within their households and in the local community, ensure the physical survival of their families and maintain the respect of their communities.12\n\nDuring apartheid, young coloured men, with low education and low earning power (though with more earning power than African men), were generally not included in the new commercial job uptake as the city expanded because, by then, rising crime had characterised them, in the minds of employers, as potential skollies and therefore untrustworthy.13 Large numbers of these men continue to be excluded from the labour market due to their low levels of education, their lack of appropriate cultural capital and competition from the influx of cheap labour from the rest of the country and many parts of Africa. They still cannot become breadwinners in their families and, in the local context, women remain the economic mainstays of the household and the community. For these men, alternative ideologies of masculinity have to be constructed.14\n\nUnable to be recognised as 'real men' beyond the borders of their community, many young men enforce that recognition within their communities by 'doing gender' through crime and violence.15 Their reputation is shaped through the webs of intimate, personal knowledge, gossip, posturing and visible aggressive performance. In this way they inscribe the boundaries of the local community in which these meanings matter and claim their agency by asserting their definition of community over that imposed by the city and State urban planners.16 They do this in ways that are secretive, threatening and often violent. These actions are often components of certain forms of masculinity where marginalisation is present and youngsters do not have alternative means to gain respect. As these young men try to regain power lost through class and race subjugation, such masculinity usually comprises a large degree of machismo.\n\nThese youngsters guard their territory and defend women, particu\u00adlarly their mothers, against 'outsiders', those whom Salo describes as 'non-persons', who are not community members. Boundaries are marked by imagined but socially recognised lines of influence and tattooed onto their bodies as marks which warn insiders and outsiders of territorial possession.\n\n***\n\nThe end of apartheid gave rise to many expectations that could not be fulfilled. Teenagers such as those described by researchers Adam Cooper and Don Foster as 'democracy's children', therefore live with widespread poverty, a destructive gang structure, an abundance of cheap drugs and few real opportunities, notwithstanding the democracy into which they were born.17 Conventional hegemonic masculinity as an affluent, light-skinned or African heterosexual man, is not achievable. Yet attributes associated with such masculinity were found to be held in higher esteem in areas of major gang activity than elsewhere. Though respectability remains out of reach, through crime they are able to accomplish the desired values of toughness, success and control.18\n\nGangsters tend to turn stereotypes upside down in attempts to control their own lives. According to Steffen Jensen, a researcher in Heideveld, the gang members there, 'used the suffering, marginalisation and extensive systems of incarceration as the media through which social maturity was achieved; they emerged as respected men through being \"a bad motherfucker\"'.19 In this way they re-inscribed oppression as a means of achieving denied masculinity.\n\nThis is a deeply problematic and transient hyper-masculinity. It swings between the super-hero of a Hollywood blockbuster and a useless, socially despised skollie. Such masculinity is characterised by risk-taking, overcoming adverse situations and subjecting others to your will \u2013 being, as informants told Cooper and Foster, a 'gevaarlike sterkbeen' or a 'bangetjie' (dangerous and strong-boned or fearful). The aim is to be 'seen', respected, feared and honoured in order to alleviate inadequacies, anxieties and disempowerment they feel in their lives.20\n\nThese young men have conflicting personal narratives and seemingly irreconcilable values which they attempt to 'be' in different contexts. As a hyper-masculine gangster with a 9mm Parabellum handgun, a young man can intimidate a community and his enemies and be a fearless hero. Killing another person is unexceptional and shooting a policeman carries even higher honours in the eyes of his gang. In another situation he wishes to see himself as a gentleman with social status, feared but respected, who dresses nicely, wears a tie and has his shoes polished. In this narrative he has a house, a wife and children and 'talks about life'.21 In yet another narrative he's a respectful and loving son to his mother, putting on the kettle and offering her a cup of tea at the end of a long day. Here he's the dutiful boy his mother wanted him to become.\n\nGirlfriends are 'poison' \u2013 potential trouble \u2013 but desirable. Distinction is made between girls who 'love to fuck gangsters' or can be raped, and a 'decent' girl to whom you 'talk nicely' and who could become a wife. Beyond the boundaries of the township and perhaps searching for work, you're a skollie and not a real man. If you're in white company and, especially, if you have tshappies (tattoos), you say 'ja baas' and 'no baas', doff your cap respectfully and never say where you come from. The very neighbourhoods in which you live are stigmatised and you know it. This was noticed by social worker Cyril Pelston:\n\nThere's a stigma about living in Manenberg. We help many men at our computer centre draw up their CVs. And they say if they put Manenberg as place of residence they won't get a job. People see them as criminals. They think gang. He can't take responsibility. Employers don't get back to them. So they put Athlone, or Sherwood Park or something else.22\n\nThese are young men from the most violent areas of one of the world's most violent cities. What they aspire to become is extremely fragmented. Contradictory discourses slide into one another as they experience the different contexts of their lives. These bend, fold and collapse into one another in a confusing labyrinth often difficult to discern. These 'children of democracy', writes Adam Cooper, 'roam the battlegrounds of post-apartheid Cape Town [with] markedly \"defended\" and splintered subjectivities as they deal with their stressful lives as best they can.'\n\n[They] are not all simply pathological, crazy or inherently evil, even if their behaviour is sometimes abominable. They are using very public, global portrayals of what it means to be a man in trying to construct esteemed reputations for themselves on the Cape Flats during their rite of passage into manhood. This process is not fundamentally illogical and irrational. It involves forging a respectable sense of self through meaningful acts that communicate specific values, to others, with the resources they have available.23\n\nThese narratives are hardened to diamond consistency in the prisons to which a disproportionately large number of young men are consigned. In his book The Number, Jonny Steinberg argues that prison gangs provide answers to the question of how inmates become men in situations of institutional humiliation, violence and oppression. In this way, he says, prison gangs 'have transformed the violence from a tool of mortification into a form of nourishment' for masculine assertion. Prison narratives provide important parts of street gang iconography and are redeployed in the townships. Steinberg found that:\n\nThe street gangs took the world of the prison \u2013 its metaphors, its nomenclature, its logic \u2013 and imprinted it on the ghetto... Beginning in the 1990s, street gangs began using prison as a metaphor to understand their relationship with those upon whom they preyed. The street gangsters are the ndotas; the taverners, liquor distributors and taxi drivers from whom they extort are the franse. Like the franse behind bars, they too must rent the air they breathe.24\n\nMothers: prenatal problems\n\nFor some young people, their problems go even deeper than failed paternal relationships and family attachments. The more I explored life-course-persistent behaviour, the clearer it became that I needed to begin before the moment of conception: on the Cape Flats, motherhood was also under severe strain. Just getting enough food is a problem. At high levels of food insecurity among the poorest women in the area, beco\u00adming pregnant runs the risk of negative prenatal and neonatal outcomes from insufficient intake of iron, calcium, folate and vitamin A as well as from anaemia and depression.25 Added to this is the use by mothers of substances that work against the healthy development of a foetus. Let's begin with alcohol.\n\nSouth Africa has the highest reported rate of foetal alcohol syndrome in the world \u2013 seven in every 100.26 The World Health Organisation rated South Africa at four on a hazardous alcohol consumption scale of one-to-five, with one being the least and five the most hazardous. It found a cumulative consumption among South African drinkers at 35 litres of pure alcohol a year.27\n\nFoetal exposure to alcohol can alter several brain structures, including the corpus callosum, which connects the two halves of the brain. Studies have found that children prenatally exposed to alcohol can suffer from serious deficits in gene regulation, cognition and behavioural problems as well as from changes in brain structure. Damage from prenatal trauma has been associated with violent teenager behaviour in later years.28 Marked changes have been found in the hippocampus, cerebellum, corpus callosum and basal ganglia of children born to mothers who drank during pregnancy.\n\nThese problems can result in children with decreased brain size, stunted growth, distinctive facial features, physical defects as well as developmental delays and disabilities.29 There were also later deficits found in learning and memory as well as problem behaviours such as alcohol use, hyperactivity, impulsivity and poor socialisation and communication skills. Children exposed to high levels of alcohol while in-utero tend to act without first considering the consequences of their behaviour or have difficulty with activities that require problem solving. They cannot plan a sequence of activities. Even if these children have average IQ scores, they struggle to succeed in school.30 In adolescence, they tend to be more involved with delinquency.31\n\nSmoking is common among pregnant mothers in the Western Cape and has a greater impact on prenatal life than is generally imagined. A study found that the offspring of mothers who smoked while pregnant were twice as likely to have a criminal record by age 22 in a sample of 5,966.32 Another study, using a birth cohort of 4 169 males, found a twofold increase in adult violent offending if their mothers smoked during pregnancy.33\n\nTik also poses major hazards to the unborn child. Professor Johan Smith, head of the Neonatal Unit at Tygerberg Hospital in Cape Town, estimates that around 200 000 people in the Western Cape use the drug, many of them pregnant women. The unit assessed that between 500 and 1 000 babies are born to tik-using mothers in the Western Cape each year. These infants may suffer acute symptoms of withdrawal, become agitated and irritable, cry a lot and have seizures.\n\nNirosha Moolla, a psychologist who often deals with school-going children exposed to tik before birth, found these children to be much slower than others in the class. 'They struggle to remember, they struggle to hold on to information so they can't build on information as they get older. They learn something today and then, a few months down the line, they've lost that information because they can't retain it in their memory.'34\n\nA low resting heart rate is another possible outcome of foetal damage by alcohol and drugs. The impact is not yet fully understood, but the hypothesis following longitudinal studies by Adrian Raine from the University of Pennsylvania is that the condition arises when the emotional reaction system in a young person's body is set low.35 In this unpleasant physiological state, individuals seek to increase their arousal level by engaging in thrill-seeking, drug-taking or antisocial behaviour. One of the outcomes is that they're impulsive and don't show much fear to the threat of punishment.36 Controversially, Raine found this condition to be inheritable.\n\nIn Cape Town, all these health problems cluster in precisely the areas where gang activity is endemic. It's pertinent to ask whether there's a connection and, if there is, what it could be. Recent research examining drug use and neurological analysis of brain functioning with longitudinal studies of child behaviour from birth through adolescence suggest that there is a connection.37 Many such studies, which track children from birth through adolescence, point to the biological foundations of learning and socio-emotional functioning. They insist that foetal life should be included in the definition of early childhood and in programmes to prevent later social dysfunction.38\n\nCentral to these understandings is the relationship between prenatal nutrition, substance abuse, maternal stress and foetal brain formation. Understanding these relationships required an immersion in genetics I never imagined at the outset of this book. Not having scientific medical training, it was initially a complex field to understand but proved to be one of the cornerstones of this study. It's another piece of the gang puzzle. Here is a very brief explanation.\n\n***\n\nThe creation of a foetus is a chemical process directed by gene patterning. Every piece of a developing child comes with a plan and happens at a particular time. It's a dynamic process with a cellular and genetic history. Genes are extremely resistant to alteration, which you can see from the glacial rate of species evolution. Prenatal and early life experiences don't actually change them. They are, in a sense, almost inviolable and unchangeable money in the gene bank. You would think, then, that if we're built according their ancient instructions, we should come out just right. But we don't. Chemical disruption in the environment at the time of construction equals problems with that particular piece; maybe a weaker liver leading to later iron deficiency or susceptability to jaundice. Perhaps smaller head size or lower birthweight. How exactly does this happen?\n\nChanges in the environment during construction of a foetus have almost zero effect on the DNA 'archive' within a cell. For that information to become useful in the construction of a body, it must be expressed as proteins. Genes are merely the blueprint and can do nothing without this process of expression, which is regulated by molecular 'switches'. This constitutes a secondary, cellular-level non-genetic structure known as the epigenetic control system. This mediates the way the gene is 'read' to produce a protein \u2013 and proteins are the fundamental building blocks of the body. The best understood of these switches is known as the methyl group which increases or decreases gene expression in response to changes in the environment by depositing molecular 'markers' on the DNA. The more markers, the greater blocking of gene expression.\n\nSo while our genes have instructions that make our bodies and tell them how to function, the environment can influence how these instructions are carried out. This means that changes to the environment of the foetus through maternal drug use or environmental stress can profoundly change the epigenetic landscape, alter protein expression and the biological outcome of a child.39 Positive mother\/child experiences are signatures that authorise instructions for positive outcomes. Negative experiences, like exposure to chemicals, violence or abuse, authorise instructions for negative outcomes.40 This has evolutionary value, enabling a 'fixed' genome to respond in a dynamic way to changing environmental conditions from the moment of conception through birth and the first years of life. This discovery has massive implications which is breaching the historical boundary between biology and society. Epigenetics is where genes and the environment physically meet. Nature and nurture can no longer be seen as polarities but as a dynamic partnertship.\n\nExplaining how environmental experiences become imprinted on the epigene would involve a dive into complex neurochemistry which is inappropriate to this level of discussion.41 Let's just hold the idea that they do, and have a look at the results on one particularly important organ, the brain, a structure equally susceptable to environmentally induced changes in gene expression.\n\nThe massive evolutionary development of the human brain's cortex (the front bit above your eyes) marks the main difference between our brains and those of other primates. It's probable that its enlargement allowed humans to evolve sophisticated social systems that emphasised close cooperation, reciprocal altruism, close living in groups and language. Such behaviour requires a particularly efficient system of aggression regulation. This impulse control allows individuals to restrain combative behaviour when the environmental costs of taking action are likely to be higher than the benefits of restraint.\n\nThe brain's prefrontal region is the site of this restraint and is also associated with the control and regulation of emotions, reactions and impulses generated by the limbic system. In a sense, it's the brain's main thinking bit. Its development continues through adolescence and is only fully formed when a human reaches their 20s. The prefrontal cortex's ability to inhibit excess up to this time is incomplete. This may explain why adolescents tend to engage in risky behaviour but 'come to their senses' as they mature \u2013 essentially adolescence-limited behaviour.\n\nAt a genetic level, the prefrontal cortex's mediatory functioning is regulated by serotonin and dopamine transporter genes. Dopamine is termed the 'seeking' system. It's thought to have evolved as the neural basis for motivation and instinctual, biologically useful exploratory activity such as foraging or sex. Its activation in the more ancient subcortical areas of the brain is characterised by an eagerness to engage in a variety of goal-directed activities, raised levels of impulsive action and desire for reward. Its action can be described as bottom-up. Attaining the reward, be it sex, food or accomplishing a target such as in sport, is experienced as pleasure through the stimulation of an area of the brain the size of your thumb known as the nucleus accumbens.\n\nWhile the dopamine system42 plays an active role in the orchestration of aggressive responses and violent behaviour, the neo-cortical neurotransmitter serotonin has an inhibitory action and is involved in the regulation of emotion and behaviour, including the inhibition of aggression.43 Interactions between the serotonin and dopamine systems provide a framework for understanding mechanisms underlying impulsive aggression. Deficient serotonergic function may result in an overactive dopamine system, promoting impulsive behaviour.44 According to a team of neuro-researchers exploring this system:\n\nAggressive individuals appear to have an inability to regulate nega\u00adtive emotion in situations where they or others are vulnerable. This inability may result from impairment in the capacity of the prefrontal cortex (PFC) to inhibit emotional activation arising from subcortical structures, which are typically controlled by the prefrontal cortex. Impaired regulatory control of the PFC may lead to excessive negative emotional reactivity and consequent violent behaviours.45\n\nThe prefrontal brain areas are involved in 'executive' functioning, which also includes higher-order cognition; motivational functions; memory; impulse control; problem solving; error processing; decision making and learning from mistakes. Its action can be described as top-down. It suppresses inappropriately fast behavioural responses in order to allow slower decision-making processes to direct behaviour.46 Lesions to the prefrontal areas resulting from environmentally altered early gene expression compromise these controls and enhance negative emotional reactions.47 Extensive research has established a direct relationship between malfunctions in the prefrontal areas and spontaneous, aggressive behaviour:48\n\nBottom-up regulation means that the subcortex overrides top-down cortical control in situations where there is no time to ponder different options and one or more of a limited number of automatic built-in stereotyped fight, flight, or freeze stress or appetitive action responses are urgently needed. In general, top-down (cortical) regulation of the subcortex is voluntary, effortful and relatively slow. In contrast, bottom-up (subcortical) activity is involuntary, effortless, and nearly instantaneous.49\n\nA simultaneous process in the construction of a brain is the development of neurons. We begin life with a large number of them and over three years a vast number of connections form between them \u2013 more connections than the brain will have at any other time of life. The connections which are used, strengthen and stay; those that are not used get pruned away. If a child is read to, talked to and reasoned with, they are using the brain circuits needed for reading, comprehension and reasoning. Those circuits will be strengthened and remain in place. A child who is left on their own for extended periods of time or is subject to aggressive or stressed emotions in their environment will not use the same circuits and other coping paths will be strengthened.\n\nThe capacity of the brain to change structure, function or organisation of neurons in response to experience is called brain plasticity. There is similar malleability in epigenetic construction, though it is shorter-lived. In roughly the first 1000 days from conception, a child's genetic response to the impact of the environment has been found to be 'plastic'. At this time, path decisions are derived from changes in the local chemical environment of the cell. Both in-utera and in the early stages of life, cellular development is set neither fully on nor fully off but somewhere in between. This combination of early sensitivity, where epigenetic change is possible, followed by later resistance to change is described as 'canalisation'. After branching one way or another, it's extremely difficult or even impossible for environmental factors to shift development from one canal into another and epigenes are then 'set' and remain unchanged for life.50\n\nA healthy baby is born with a well-developed subcortex. Any distress it feels triggers a powerful stress response that it has no means to curtail. Instead, it relies on its mother to comfort it and regulate its feelings. Her healthy, stress-free prenatal condition plus her loving postnatal presence and deeply caring attention result in fewer epigenetic 'markers' on genes important for 'top-down' control. This is how strong powers of self-regulation are acquired \u2013 from an environment which, for infants, is predominantly framed by the mother or caregiver. Empathetic parenting allows an infant to safely express and later verbalise and self-regulate distress or excitement. These include hurt, anxiety, fear, anger and desires. In this way the mother buffers the infant's subcortical stress.\n\nAdverse gene expression can bias the developing brain towards less efficient inhibitory control, leading to what I term epigenetic development disorder. The outcome, however, is not set in stone. Babies born with neurological imbalance, if brought up in well-functioning families, are no more likely than healthy children to grow up delinquent. In dysfunctional families, however, they have been found to be four times as likely to end up delinquent.51 In a certain sense, this is a hopeful message.\n\n***\n\nAfter this deviation into prenatal brain functions, let's circle back to gangs. As I've indicated, adolescence is a stage in life where enormous social demands are being placed on the rapidly growing teenager. This load calls on resources of the prefrontal cortex and its associated executive functions. Adolescents need to regulate, control and inhibit a developing sex drive. They need to deal with threats and challenges to their social status that arise within their peer groups and plan and organise for a future career. They need to pay increasing attention to school performance in order to maximise career prospects.\n\nThose leaving school lose whatever social structure and support systems these provided. They must adapt their behaviour to the more variable contingencies, incentives and disincentives in their new environment. In addition, they must plan and develop strategies for attracting a partner and may need to make early parenting decisions. Throughout this period, they must evaluate and assess competing life-course strategies and engage in complex decision-making. At the same time, the rebellious spirit and antisocial behaviour that have characterised their adolescent years must be inhibited or suppressed if they're to succeed in structured adult life.\n\nThe prefrontal region of the brain bears the burden of this accelerated cognitive load that requires multiple executive functions. These are sustained attention; behavioural flexibility to changing situations; working memory; self-regulation and inhibition; abstract decision-making; planning and organisation. Yet this processing load occurs at a time when the cortex is still developing. Individuals with early damage or dysfunction of the prefrontal cortex are likely to suffer an information overload during this time, resulting in further dysfunction of the prefrontal cortex, less regulatory control and further, life-course-persistent behaviour problems.52\n\nYoung people with prefrontal disorder may display poorer emotional control and be less able to judge the impact of their behaviour. They may have difficulty in establishing empathy and conducting critical assessments of their dysfunctional behaviours, which they might tend to repeat. Prefrontal impairment is associated with a lowering of inhibition, increased impulsiveness, public spectacle hyper-masculinity and a greater predisposition to engage in violent behaviour.53 Catalysed by a hostile environment, this can result in a cascade of behaviours that result in low parental investment in their own offspring, which manifests as antisocial behaviour in the next generation. This, in turn, furthers conditions conducive to an on-going environment of social aggression.54\n\nIn this way, a vicious toxic stress cycle can be set in motion. An ad\u00adverse poverty environment, mediated by inadequate parental caregiving responses can become embedded in offspring during early development, resulting in both compromised self-regulation and maternal care.55 It is important to emphasise, however, that toxic stress is not directly caused by poverty, but by the strains poverty places on parenting. The child\/parent relationship depends entirely on the quality of parental care: good parenting completely cancels out the effects of poverty.\n\nEpigenetic impairment has major implications for both mothers and their offspring. It places the burden to provide calm, balanced pre- and postnatal conditions on mothers who may be too young to understand the importance, too poor to be able to do so or too drug-addicted to care. Youngsters with problems from early epigenetic trauma are in the sad and scary position of having a condition which is ultimately irreversable. This has major implications with regard to personal agency \u2013 the capacity a person has to act, to change their situation for the better and to influence their world. Prenatal and early life damage confine that agency, restricts capability, curbs opportunities and ultimately limits the freedom to choose. In a court of law this defence could be a major mitigating factor in assessment of culpability and has, in fact, been used in some cases.56 The law assumes that all are equal before it, but what if this can be shown to be a false premise?\n\nGrandmothers: generational problems\n\nIt's not surprising that a mother's diet, stress or ingestion of chemicals has an impact on the growth of a foetus. A more controversial question asks whether life experiences and behaviours such as stress and trauma can, as we have suggested, be inscribed in the epigenetic make-up of a following generation.\n\nEpigenetic changes were initially believed to occur only during foetal development. Recent studies suggest that epigenetic markers can be added to genes right up into early adulthood, setting off a cascade of cellular changes. Neuroscientists were surprised to find that these changes could be passed down from parent to child, one generation after the next.57 Assessing the work of epigeneticists Michael Meaney, Moshe Szyf and Gustavo Turec, science writer Dan Hurley puts it this way:\n\nTraumatic experiences in our past, or in our recent ancestors' past, leave molecular scars adhering to our DNA. Jews whose great-grandparents were chased from their Russian stets; Chinese whose grandparents lived through the ravages of the Cultural Revolution; young immigrants from Africa whose parents survived massacres; adults of every ethnicity who grew up with alcoholic or abusive parents \u2014 all carry with them more than just memories. Like silt deposited on the cogs of a finely tuned machine after the seawater of a tsunami recedes, our experiences and those of our forebears are never gone, even if they have been forgotten. They become a part of us, a molecular residue holding fast to our genetic scaffolding.58\n\nIn this transmission, though the DNA itself remains the same, psychological and behavioural tendencies, it seems, cling and can be passed on. You might have inherited not just your grandmother's fine complexion, but also her predisposition toward depression caused by the neglect she experienced as a newborn. This suggests that young people may not only suffer from physical neonatal damage, but from prenatal inheritance \u2013 emotional stress and trauma suffered in utero and carried forward from previous generations. This confronts neo-Darwinian views on natural selection:\n\nBy challenging the idea that heredity is the mere transmission of nuclear DNA, epigenetics has opened the doors to a broader, extended view of heredity by which information is transferred from one generation to the next by many interacting inheritance systems. Epigenetic variations act as a parallel inheritance system through which the organism can respond in a more flexible and rapid way to environmental cues and transmit to different cell lineages different 'interpretations' of DNA information. The environment is therefore now seen as directly inducing variations in evolution.59\n\nIn a sense, epigenetic marking can be seen as a persistent form of cellular memory by which memories of past environmental events are fixed on the genome. The possibility of behaviour vaulting generations remains the subject of ongoing debate and research and is potentially controversial. It raises complex questions. Are epigenetic markers transmitted directly through the fertilised egg or do attachments occur directly after birth? Can these markers be carried through male sperm? Evidence, published in the journal Science, suggests that, either way, epigenetic changes can make it through to the next generation, setting the stage for transmission of the altered traits in later descendants as well.60\n\nConsider the implications. Perhaps a young person incurred prenatal damage which, in part, explains their adolescent behaviour. They could also be responding to groups of chemical markers that were added in childhood to genes in their mothers \u2013 or grandmother's \u2013 brain, handcuffing their mood to feelings of fear and despair. This is a problem that reaches way beyond psychology into the ancestral chemical depths of their being.\n\nBorn to be wild\n\nA liberal view of young gangsters echoes that of moral philosopher and economist Adam Smith, who said the only difference between a philosopher and a street porter were their habits, customs and education. With good supports in place, he said, street porters could be dusted off and learn to become respectable and 'more like us'. As we have seen this is not necessarily the case. Prenatal toxic stress and postnatal attachment problems can clinically and emotionally alter a child \u2013 with socially disruptive consequences \u2013 especially during their teenage years.\n\nIt all seemed to make sense, but there was still something wrong with the picture. I needed to step back for a moment and recalibrate. Why would natural selection have favoured organisms that, by default, respond to environmental challenge by becoming dysfunctional?61 Indeed, why do impoverished, food-insecure populations tend to grow rather than wither away?\n\nFor all species, survival is taxing and carries risks. Hunger and stress have always been part of the human experience. For millions of years hominin mothers would have been subjected to hunger, deprivation and danger, yet we became among the most successful species at procreation and survival. Can modern, unborn children in urban high-risk environments be under even greater stress than those of hunter-gatherer parents where danger was constant, food supply often limited and mortality high?\n\nI found an answer of sorts in the work of evolutionary psychologists Bruce Ellis and March del Giudice. They accept the considerable body of evidence that psychological and toxic stress suffered by mothers can affect an unborn and newly-born child, but they ask of that evidence a different set of questions. Could the changes be adaptive rather than pathological? 'During foetal development and infancy,' they write, 'important features of the environment are communicated to the child via the placenta and lactation in nutrients, metabolites, hormones, growth factors and immune factors that reflect the mother's current and past experiences.' This process 'provides information about threats and opportunities in the environment, their type and their severity.'62\n\nAccording to Cape Town psychologist Barak Morgan, there's good evidence to support this idea. Setting the stress response according to the quality of maternal care early in life, he writes, serves to prepare offspring for the kind of environment they'll be born into:\n\nCaregivers inhabiting a relatively unsafe, impoverished, stressful environment devote fewer resources to maternal care resulting in weaker top-down regulation of the stress response. This is appropriate because impoverished environments are associated with nutritional deprivation, violence and infection so weaker top-down control of the stress response provides enhanced protection against all three conditions.63\n\nIn this way, he says, the qualities of the environment that offspring are soon to enter are communicated via their mother's prenatal or caregiver's postnatal behaviour. In a high-risk environment, cognitive resources in the young child may be shifted towards areas of the brain needing to deal with psychological arousal; stress response; freeze\/hide and fight\/flight behaviours; readiness for action; learning responses; dominance seeking; aggression; risk-taking; earlier fertility; more offspring; less parental investment and functioning of the immune system. The need for these responses is communicated through the cellular environment in which epigenetic expression takes place.\n\nThe assumption here is that exposures to maternal stress do not so much impair foetal development, as direct or regulate gene expression toward strategies that are adaptive under stressful conditions. This is a controversial analysis and has far-reaching implications. Live-fast-die-young attributes such as risk-taking; insecure attachments; mistrustfulness; a preference for smaller immediate rewards over larger future ones; aggressive behaviour and affiliation with deviant peers may be seen as aberrant in more 'normal' society, but are valuable for the survival of teenagers in tough neighbourhoods. At a genetic level they have been canalised defensively.\n\nSuch children show enhanced ability to detect angry facial cues, increased anticipatory monitoring of the environment for interpersonal threats, greater speed in accurately labelling fearful faces, recall of aggressive stimuli and greater accuracy in identifying adults with whom they had a stressful interaction.64\n\nSo it may be that children who experience relatively high levels of stress during the prenatal, early and middle childhood have attentional styles in adolescence that are well adapted to environmental danger and unpredictability. They perform well on shifting attention tasks (the ability to switch between two competing unpredictable situations), though less well on sustained attention tasks (concentrating over a prolonged period). The ability to constantly switch attention, it was found, enabled 'fast-road' children to scan the environment for danger.65 Adolescents from low-stress backgrounds, on the other hand, tended to show the opposite pattern.\n\nThese adaptations, however, come at a cost. Because of structural and resource limitations, a developing foetus can't maximise all components of fitness simultaneously and, instead, must make trade-offs, investing in one function at the expense of another. For example, physical resources spent by a pregnant woman to fight infection cannot be spent on reproductive effort. This may result in lower ovarian function in female foetuses when they reach womanhood and reduced muscle or skeletal function in male fetuses when they reach manhood. This chain of resource-allocations, expressed as physiological and behavioural traits, constitute the individual's path from conception.66\n\nThis channels the unborn and post-natal child's resources as they grow.67 Each genetic decision influences the next in an ongoing cascade that guides development of alternative life history strategies. It's through this chain of resource-allocation decisions that the developing human adapts to local conditions.68 Under adverse conditions the costs associated with high-risk trajectories are therefore unavoidable trade-offs against survival and reproduction.69\n\nOf course, the ability of a foetus to adapt carries the danger of maladaptation. Harmful epigenetic shifts may occur and changes may take place beyond the brain's regulatory capacity. A child may experience new environments that are outside the range ever encountered over our evolutionary history. In this case, all developmental bets are off and the outcome may be 'abnormal'. A mismatch may also occur when the person's environmental context changes for better or for worse and their default behaviour is inappropriate to the situation. Bruce Ellis and March del Giudice suggest that, for the species as a whole, the cost of such possible maladaptation is offset by improved management of danger:\n\nAlthough the system is on a hair trigger, with a resulting increase in anxiety and\/or aggression, few instances of actual danger will be missed... Natural defences are usually designed by natural selection to accept a high rate of false positives. In most instances, unnecessary responding is an adaptive feature of the system (though a costly one) rather than a sign of dysregulation or malfunction.70\n\nThe problem with pathologising \u2013 making an illness \u2013 out of risky and aggressive behaviour comes when interventions are implemented that try to 'correct' these strategies, when their motivation and function are ignored. Sensation seeking, a strong preference for immediate over delayed rewards, fearlessness, elevated risk-taking and aggression are clearly undesirable from the perspective of public order. However, altering such behaviour among young people who are genetically adapted to cope in high-risk environments could be equivalent to 'declawing the cat' \u2013 removing the psychological and behavioural weaponry necessary to survive.71\n\nWorking against an intrinsic adaptation is fighting an uphill battle. Such adaptations may be epigenetic responses to environmental cues indicating that life is short and future outcomes cannot be controlled or predicted. Because these are evolved responses, Band-Aid interventions such as sex education, birth control, promoting self-esteem, teaching life skills and insisting on conventional schooling or problem-solving strat\u00adegies are unlikely to effect lasting change. Interventions would be much more powerful if they worked with and not against adaptive responses \u2013 not changing young people to be 'other' but working with the strengths with which their environment presented them. This has profound implications for programme development and will be examined later.\n\n***\n\nNone of the genetic findings I've been discussing suggest that prenatal damage or genetic inheritance are the only indicators of later delinquency. They are, however, important for any decisions about life-course persistent behaviour. Of course it's unlikely that all gang members have subcortical lesions, serotonin deficiency, ancestral damage or attachment deficiencies and that law-abiding citizens do not. But there is a possibility that young people primed for threat detection through pre- and postnatal hurt may seek each other out, recognising themselves in each other.72 This might, therefore, be a way to understand gang formation. Such compromised young people may form the core of the Cape's many gangs and explain their tendancy for easy violence and the toxic climate in which they engage. What it also suggests is that Cape Town may have a major prenatal health crisis on its hands.\n\nEndnotes\n\n1. Garbarino, James: Lost Boys: Why our sons turn violent and how we can save them (Anchor Books, New York 1999). p52.\n\n2. Garbarino, ibid, p57.\n\n3. Garbarino, ibid, p58.\n\n4. Garbarino, ibid, p81.\n\n5. Garbarino, ibid, p85.\n\n6. Garbarino, ibid, p138.\n\n7. Interview with Lerato Kossie, Khayelitsha.\n\n8. Garbarino, op cit, p45.\n\n9. Kossie, ibid.\n\n10. Salo, Elaine: 'Mans is ma soe: Ideologies of Masculinity and ganging practices in Manenberg, South Africa', in Representations of Violence in Africa: Don Donham and Edy Bay (Eds) U. Virginia Press.\n\n11. Salo, ibid.\n\n12. A census-based report found that in South Africa in 2011, more than three times as many women as men were employed as clerks and nearly twice as many as technicians or in social services. Gender statistics in South Africa 2011 by Statistics South Africa.\n\n13. Personal discussions with commercial and industrial employers.\n\n14. Salo, ibid.\n\n15. The term 'doing gender' is from a seminal paper by Adam Cooper and Don Foster: 'Democracy's Children? Masculinities of Coloured Adolescents Awaiting Trial in Post-Apartheid Cape Town, South Africa' (THYMOS: Journal of Boyhood Studies, Vol. 2, No. 1, Spring 2008, pp3-25.).\n\n16. Salo, op cit.\n\n17. Cooper & Foster, op cit.\n\n18. Cooper & Foster, ibid.\n\n19. Jensen, Steffen: Gangs, politics and dignity in Cape Town (James Curry, Oxford, 2008).\n\n20. Cooper & Foster, op cit.\n\n21. Cooper & Foster, ibid.\n\n22. Interview with Pelston, 2014.\n\n23. Cooper, Adam: 'Gevaarlike Transitions': Negotiating Hegemonic Masculinity and Rites of Passage Amongst Coloured Boys Awaiting Trial on the Cape Flats. (Human Sciences Research Council Investing in Youth Conference 2007 + PINS, 2009, 37, pp1-17).\n\n24. Steinberg, Johnny: Nongoloza's Children: Western Cape Prison Gangs during and after Apartheid. (Centre for the Study of Violence and Reconciliation. Johannesburg 2004).\n\n25. In the Western Cape, 16% of women were found to be malnourished, 16% anaemic (national average 10.7%), 5.7% had iron deficiency, 41% had a low vegetable intake.\n\n26. Foundation for Alcohol Related Research report, 2015.\n\n27. WHO Global Alcohol and Health Report: Africa 2007.\n\n28. Mendes, Deise Daniela, Jair de Jesus Mari, Marina Singer, Gustavo Machado Barros, Andr\u00e9a F. Mello: 'Study review of the biological, social and environmental factors associated with aggressive behaviour'. In Review of Brazilian Psychiatry. 2009;31(Suppl II): p77-85.\n\n29. Adnams, Colleen: 'Developmental consequences of prenatal drug and alcohol exposure'. In Substance Use and Abuse in South Africa by Dan Stein, George Ellis, Ernesta Meintjies and Kevin Thomas (eds). UCT Press Cape Town, 2012, p55.\n\n30. Mattson, Sarah, Amy Schoenfeld and Edward Riley: Teratogenic Effects of Alcohol on Brain and Behavior (National Institute on Alcohol Abuse and Alcoholism) (http:\/\/pubs.niaaa.nih.gov\/\u00adpublications\/\u00adarh25-3\/\u00ad185-191.htm).\n\n31. Mendes, ibid.\n\n32. Rantakallio, P, Laara, E, Isohanni, M, & Moilanen, I (1992). Maternal smoking during pregnancy and delinquency of the offspring: An association without causation? International Journal of Epidemiology, 21, 1106\u201313.\n\n33. Brennan, Grekin, and Mednick (1999) Maternal smoking during pregnancy and adult male criminal outcomes. Archives of Gen Psychiatry. March 1999, 56(3) p215-9.\n\n34. Tik time bomb in Health-e, 4 October 2012.\n\n35. Raine, op cit.\n\n36. Garbarino, James: Understanding the psychology of violence: a public dialogue. Nicro, Cape Town December 2014.\n\n37. This has echoes of the work of the Italian criminologist Cesare Lombroso, who suggested that criminality was inherited and that someone 'born criminal' could be identified by physical defects and distinguished from non-criminals by multiple physical anomalies. The behavior of these biological 'throwbacks' from apes and lower primates, he said, would inevitably be contrary to the rules and expectations of modern civilised society. Genetic research may seem to be a brief nod in Lombroso's direction, but modern genetics starts from a different place and finds causes of prenatal neurological damage in stressed social conditions and not in biological throwback. Nobody, in these terms, is born criminal.\n\n38. Dawes, A: 'Violence prevention through reduction risks to child health and development in the years prior to school'. In Ward, C & Donnolly,P (eds) Oxford Textbooks in Public Health \u2013 Violence: A Global Health Priority. Oxford: Oxford University Press.\n\n39. In genetics, epigenetics is the study of cellular and physiological trait variations that are not caused by changes in the DNA sequence. Put simply, it's the study of external or environmental factors that turn genes on and off and affect how cells read genes. Because the markers are attached to the genes, residing beside but separate from the DNA code, this field of study was dubbed epigenetics, from the prefix epi (Greek for above).\n\n40. Sharing the Brain Story, Alberta Family Wellness Initiative (Norlien Foundation 2013).\n\n41. For the medically curious, here are some details. The process is best described as reactions to stress. If the situation poses an imminent threat, the sympathetic nervous system (SNS) will trigger a fight or flight response accompanied by feelings and sensations of panic. If the stressor does not pose an imminent threat, the SNS will keep the body in a state of anxious readiness that concentrates cognition towards assessing the overall situation in order to predict what might happen next and what is the best thing to do about it. If the stressor passes, the SNS response will die down, bodily functions will return to normal and the mind will relax. But if the stressor persists, adrenal processes begin to kick in, resulting in the release of cortisol from the adrenal glands. This response serves to provide the SNS response with energy (increases blood glucose levels) needed to sustain increased heart rate, blood pressure, respiration plus any strenuous physical activity associated with a potential fight or flee response. From Morgan, op cit.\n\n42. Specifically the dopamine D2 receptor gene (DRD2) and the dopamine D4 receptor gene (DRD4).\n\n43. Dongju Seo, Christopher J. Patrick and Patrick J. Kennealy: Role of Serotonin and Dopamine System Interactions in the Neurobiology of impulsive Aggression and its Comorbidity with other Clinical Disorders (PubMed 2008).\n\n44. Seo, ibid.\n\n45. Seo, ibid.\n\n46. Russel, Vivienne, Jacqueline Dimatelis & William Daniels: Animal models of substance abuse. In Substance use and abuse in South Africa by Dan Stein, George Ellis, Ernesta Meintjies and Kevin Thomas (eds). UCT Press Cape Town, 2012, p206.\n\n47. Raine, Adrian: Annotation: 'The role of prefrontal deficits, low autonomic arousal, and early health factors in the development of antisocial and aggressive behavior in children' (Journal of Child Psychology and Psychiatry 43:4 (2002), pp417\u2013434.\n\n48. Raine, ibid.\n\n49. Morgan, Barak, Diane Sunar, C. Sue Carter, James Leckman, Douglas Fry, Eric Keverne, Iris-Tatjana Kolassa, Robert Kumsta and David Olds: Human Biological Development and Peace: Genes, Brains, Safety, and Justice. (in Pathways to Peace: The Transformative Power of Children and Families: J. F. Leckman, C. Panter-Brick, and R. Salah, eds. 2014. Str\u00fcngmann Forum Reports, vol. 15, J. Lupp, series ed. Cambridge, MA: MIT Press).\n\n50. This depiction is from Barak Morgan. Gene-environment interactions occur over extended sensitive periods during early childhood. However, overall the window of sensitivity diminishes extremely rapidly, effectively reaching adult levels by the age of 6-7 years old.\n\n51. Garbarino, James: Understanding the psychology of violence: a public dialogue. Nicro, Cape Town December 2014.\n\n52. Raine, (2002), op cit.\n\n53. Raine, ibid.\n\n54. Ferguson CJ. 'An evolutionary approach to understanding violent antisocial behavior: diagnostic implications for duel-process etiology'. (Journal of Forensic Psychological Practice. 2008;8(4)) pp321-43.\n\n55. Morgan, op cit.\n\n56. In a 2009 criminal trial in the United States, an argument based on a combination of 'warrior gene' MAOA and history of child abuse was successfully used to avoid a conviction offirst-degree murder and the death penalty; however, the convicted murderer was sentenced to 32 years in prison. See Barber N: 'Pity the poor murderer, his genes made him do it'. The Human Beast: Why we do what we do. Psychology Today. (2010).\n\n57. Hurley, Dan: 'Grandma's Experiences Leave a Mark on Your Genes'. From Discover Magazine, May 2013.\n\n58. Hurley, ibid.\n\n59. Meloni, Maurizio: 'The social brain meets the reactive genome: neuroscience, epigenetics and the new social biology' (Frontiers in Neuroscience, May 2014).\n\n60. Hackett, Jamie, Roopsha Sengupta, Jan Zylicz, Kazuhiro Murakami, Caroline Lee, Thomas Down and M. Azim Surani: 'Germline DNA Demethylation Dynamics and Imprint Erasure Through 5-Hydroxymethylcytosine'. In Science 25 January 2013: Vol. 339 no. 6118 pp448-452.\n\n61. The same question was posed by Ellis, Bruce and Marco del Giudice: 'Beyond allostatic load: Rethinking the role of stress in regulating human development', in Development and Psychopathology 26 (2014), Cambridge University Press.\n\n62. Ellis and del Giudice, ibid.\n\n63. Morgan, op cit.\n\n64. Support for this view can be found in Frankenhuis, W. E., & de Weerth, C. (2013). 'Does early-life exposure to stress shape, or impair, cognition?' (Current Directions in Psychological Science, 22, 407\u2013412); Eisen, M. L., Goodman, G. S., Qin, J., Davis, S., & Crayton, J. (2007). 'Maltreated children's memory: Accuracy, suggestibility, and psychopathology', (Developmental Psychology, 43, 1275\u20131294); Pollak, S. D., Vardi, S., Bechner, A. M. P., & Curtin, J. J. (2005). 'Physically abused children's regulation of attention in response to hostility', (Child Development, 76, 968\u2013977); Pollak, S. D. (2008). 'Mechanisms linking early experience and the emergence of emotions' (Current Directions in Psychological Science, 17, 370\u2013375); Pollak, S. D., Messner, M., Kistler, D. J., & Cohn, J. F. (2009). 'Development of perceptual expertise in emotion recognition', (Cognition, 110, 242\u2013247); Pollak, S. D.,& Tolley-Schell, S. A. (2003). 'Selective attention to facial emotion in physically abused children', (Journal of Abnormal Psychology, 112, 323\u2013338); Masten, C. L., Guyer, A. E., Hodgdon, H., McClure, E. B., Charney, D. S., Ernst, M., et al. (2008). 'Recognition of facial emotions among maltreated children with high rates of post-traumatic stress disorder', (Child Abuse & Neglect, 32, 139\u2013153); Rieder, C., & Cicchetti, D. (1989). 'Organizational perspective on cognitive control functioning and cognitive-affective balance in maltreated children' (Developmental Psychology, 25, 382\u2013393); Eisen, M. L., Goodman, G. S., Qin, J., Davis, S., & Crayton, J. (2007). 'Maltreated children's memory: Accuracy, suggestibility, and psychopathology' (Developmental Psychology, 43, 1275\u20131294).\n\n65. Nederhof, E., Ormel, J., & Oldehinkel, A. J: 'Mismatch or cumulative stress: The pathway to depression is conditional on attention' style in Psychological Science, 3 December 2014.\n\n66. Clancy, K. B., Klein, L. D., Ziomkiewicz, A., Nenko, I., Jasienska, G. & Bribiescas, R. G. (2013). 'Relationships between biomarkers of inflammation, ovarian steroids, and age at menarche in a rural Polish sample' in American Journal of Human Biology, 25, 389\u2013398; Muehlenbein, M. P., & Bribiescas, R. G. (2005). 'Testosterone-mediated immune functions and male life histories' in American Journal of Human Biology, 17, pp527\u2013558. In Ellis and del Giudice, op cit.\n\n67. Geary, D. C. (2002). 'Sexual selection and human life history', in Advances in Child Development and Behavior, 30, pp41\u2013101.\n\n68. Ellis and del Giudice, op cit.\n\n69. Morgan, Barak and Deborah Posal: Humanities Meets Biology (Paper at symposium on the implications for the humanities and social sciences of discoveries in epigenetics, UCT 26 November, 2015).\n\n70. Ellis and del Giudice, op cit.\n\n71. Ibid.\n\n72. Beaver, Kevin, J. Eagle Schutt, Brian B. Boutwell, Marie Ratchford, Kathleen Roberts: Genetic and Environmental Influences on Levels of Self-Control and Delinquent Peer Affiliation: Results from a Longitudinal Sample of Adolescent Twins. (www.behavioral\u00adand\u00adbrain\u00adfunctions.com\u00ad\/content\u00ad\/3\u00ad\/1\u00ad\/30).\n5. Toxic neighbourhoods\n\nIf what I've been talking about could be considered to be within the family home, it's time to step out the door. The social and physical environment within which the individual and their family live is ever-present, yet its impact is often transparent to those within it. This is because, although we live in a world made by others, we negotiate it as we find it. Young people on the Cape Flats may use blank apartment walls as convenient canvases for graffiti but seldom question the reason for or history of those apartments in the sandy, wind-blown starkness of single-race townships. They simply get on with life.\n\nBut, as we have seen, each location, each brick is part of an imagined reality created from decisions made in the past about how it should be, where people should live and who those people should be. The city is as it is because urban and social planners decided it should be that way. These decisions have a bearing on every living moment of young people within its perimeter.1 They have particular bearing on gang formation.\n\nIn 1924 American sociologist Edwyn Sutherland described delinquency as a normal response by normal individuals to abnormal conditions.2 That's not entirely true, but there's still value in the assertion. Environment does matter. In order to understand the nature of these abnormal conditions and how it came to be that way, we need to look beneath the city's more superficial unfolding over time.\n\nThe history of a city is the story of its neighbourhoods. Each has a zeitgeist, an identifiable personality. They all look and feel distinct from one another and have persistence over generations. Explaining zeitgeist is difficult because it comprises many things: the type of buildings; the width of streets; the presence or absence of gates and walls; greenery or lack of it; street-lighting at night; how people dress; who's hanging around; the friendliness, indifference or fear of people; the smell of cooking; rubbish or garden flowers and the type of cars that are driven.\n\nThe most important undergirding of neighbourhood zeitgeist, however, is the degree of social efficacy. Organised communities have higher levels of formal and informal solidarity. There's consensus on important norms and values, often cohesion and social interaction among neighbours, formal and informal surveillance, preparedness to intervene in altercations and question strangers and to admonish children for unacceptable behaviour. These areas generally look and feel different. Socially disorganised communities are, conversely, unable to realise common values among residents or solve common problems and often live in fear. This was sketched by American criminologist Robert Sampson in his work on Chicago's high-risk neighbourhoods:\n\nNeighbourhood characteristics such as family disorganisation, residential mobility and structural density weaken informal social control networks. Informal social controls are impeded by weak local social bonds, lowered community attachment, anonymity and reduced capacity for surveillance and guardianship... Residents in areas characterised by family disorganisation, mobility and building density are less able to perform guardianship activities, less likely to report general deviance to authorities, to intervene in public disturbances and to assume responsibility for supervision of youth activities. The result is that deviance is tolerated and public norms of social control are not effective.3\n\nContact crime across a city tends to cluster in such neighbourhoods, as does low income, high unemployment and raised levels of interpersonal conflict and stress.4 What's important to note, however, is that social disorganisation is a property of neighbourhoods, not individuals and that crime at this level is a characteristic of social disorganisation. The difference between District Six and newer places like Manenberg and Lavender Hill, or the more recently developed Khayelitsha, is that the former was a community which cohered and policed itself and the latter are, comparatively, socially disarrayed and organisationally unglued. Poverty is not merely deprivation, it's isolation and social confusion.\n\nMany of the residents in Cape Town's high-risk, low-income townships voice a degree of fatalism about transformation in their own lifetime and a moral cynicism about crime, which they view as inevitable. As a result, contact crimes are not vigorously condemned because of an inability to prevent them occurring. Given the lack of assured conventional economic advancement, many residents shrug at an income based on the theft of vehicles or sale of drugs and may even benefit from or depend on it themselves. It was not always like that.\n\nThe importance of collective efficacy \u2013 the ability or inability of communities in a neighbourhood to organise \u2013 is extensively explored in Sampson's monumental study Great American City: Chicago and the Enduring Neighborhood Effect.5 One of his findings is that urban social disorder is socially constructed and that, through physical, legal and ideological frameworks, it persists over time. It follows that collective efficacy can be a counter to disorder and is therefore key to the transformation of a neighbourhood. The more people there are in a community who act in concert in whatever form, the less likely it will be that the community as a whole tolerates antisocial behaviour. So attempts to reduce crime in a neighbourhood find greater traction if they're aimed at fostering and supporting social cohesion than by targeting 'bad elements' for removal.\n\nThere is, however, a negative corollory. In neighbourhoods structurally and socially underdeveloped over time or where friendship networks and informal surveillance systems have been destroyed as in Group Areas relocations, power dynamics favour those who are prepared to exert social control through violence. Collective willingness to intervene and respond to criminal behaviour is reduced in the face of growing collective efficacy of gangs and other shadowy elements. By ruling the night in poor neighbourhoods, these groups reduce the possibility of positive social cohesion.\n\nThe massive exercise in community destruction undertaken by the apartheid government, the failure of the current government to reduce poverty or prevent rapid squatter settlements and the division of the city between glitter and ghetto has \u2013 by design, inability or perceived necessity \u2013 resulted in massive social disorganisation of poorer neighbourhoods. Despite the turnover of residents through time, these conditions persist and residents in 'those kinds of places' continue to be seen as 'those kind of people'.6\n\nThey are labelled and treated accordingly to a point where many of them embody the definition and act accordingly, lashing out or wearing their situation as a badge of ironic resignation. In these neighbourhoods, collective efficacy declines, violence increases and other forces move into the power vacuum in an attempt to control, stabilise, disrupt or benefit. On the Cape Flats that violence is among the highest in the world.\n\nEveryday violence\n\nWhen I first killed a man in Hanover Park I was 13. It's a great feeling, the feeling I have when I hold a gun in my hand. I have no fear. I get excited.\n\nIf you got a firearm you have power. Guns are all over the Western Cape. Guys who do house robberies bring the guns to the merchants. But most guns in the Cape come from Faure, the Cape Corps. Rasheed Staggie drove in there and stole a lot of guns (he went to prison for that).\n\nHere they use 9mm, Baretta, PT92, Titanuim, automatics with extension magazines. There are a lot of guns here. The HLs, Jesters, Americans all have guns. We are 10 guys and maybe we have one or two. Then maybe we see a guy who wants to shoot an HL and he's a Clever Kid and his shooting at them but running in our street. So we take him and take his gun. Every gun is a lot of power.\n\nGuns give you respect. It makes people afraid and now they want to be your friend. And you know why they are your friend. For now. Really it's not the gun but the guy behind it. How he uses it. How he goes from corner to corner, hideout to hideout shooting. It's all about how close you can shoot before the cops come. When the cops come you must just be cool, walk slowly \u2013 it's how you act under pressure.7\n\nClifton, Hanover Park, 2014\n\nThe pistol in my hand was heavy, dull silver with a black grip. I pulled back the slider, clicked a bullet into the breach and flipped off the safety. It was unnerving how powerful the weapon made me feel. It dared me to pull the trigger. 'Nine millimetre Beretta semi-automatic,' said the young man who'd transferred it from his belt beneath his shirt to my hand. 'Nice gun.' I popped out the magazine (full), unloaded the handgun and handed it back to its owner who, as a member of the Mongrels gang, undoubtedly had no licence for it.\n\nThe handgun, an object of almost mythical significance among gang members, is a good starting point towards understanding the environment in which many young people on the Cape Flats grow up. Their first choice is often the 9mm Parabellum. It has a bullet about a centimeter in diameter and two centimeters long. This can be raw lead, jacketed in brass or hollow nosed. The latter, on contacting a bone, will splay outwards, causing immense damage. The impact of the 9mm cartridge is so powerful that it can impart hydrostatic shock \u2013 remote wounding beyond the impact point \u2013 on living targets. It's deadly and accurate at distances of up to 50 metres.8\n\nIn Cape Town, someone dies almost every day from the impact of a 9mm bullet. Most of the victims and the shooters are between the ages of 15 and 23, though children as young as two have been hit by stray ordinance. It is worth asking why such a deadly weapon is readily available and why young people feel the need to own and use it.\n\nAfter the Sharpeville massacre of anti-passbook protesters in March 1960, the United Nations imposed a trade ban on arms to South Africa. The country, with help from Israel, rapidly set up its own armaments factories and became virtually self-sufficient in the production of small-arms.9 Using the Italian Beretta 92F as a template, Denel, South Africa's arms production division, began manufacturing the Z-88 pistol, known as the Vektor SP1. This was made in large numbers and became the standard sidearm for the police and defence force.\n\nAfter South Africa's withdrawal from Angola and Namibia between 1989 and 1993, Department of Defence budgets were cut. Following the 1994 transformation of State power, the defence industry continued to shrink and restructure. In order to survive, it turned to the export market, with considerable success. Between 1996 and 1998 small-arms exports were valued at R180 873 000, with top customers being Colombia, Ecuador, Mexico, Peru, Venezuela, Jordan, Kenya, Ivory Coast, Taiwan, Singapore, New Zealand and Germany.10\n\nMost sales, however, were local. Around three million handguns are presently registered to private owners in the country with the most common listed being the 9mm Parabellum. There are no figures for unregistered guns, but they are undoubtedly extremely high. There are several reasons for this. In the 1980s and early 1990s, the apartheid government distributed thousands of weapons to the paramilitary groups of the Inkatha Freedom Party in KwaZulu-Natal to prevent the ANC from consolidating political support. These arms were never fully accounted for and many came to be used in criminal activities. Private gun owners are also notoriously negligent and are targeted by criminals seeking weapons. Thousands of handguns are stolen every year and, by definition, fall into the hands of criminals.\n\nPolice and defence force armouries offer a third source of weapons from where theft is intermittent but, in some cases, considerable. Between 1994 and 1998, 112 692 weapons were 'lost or stolen' from these armouries. Between April 2004 and March 2011, 20 429 police firearms were 'lost or stolen'.11 One police station alone, in Umtata, the former capital of Transkei, 'lost' 638 handguns. In 2009 and 2010 the loss of firearms from the SAPS averaged 10 a day. These weapons, now illegal, are supplemented, from various sources, by such unlicensed weapons as R1 and R5 assault rifles, shotguns, .38 Specials, Z88 pistols, Lugers, home-made 9mm zip guns, pangas and knives.\n\nSouth Africa is awash with weapons as a result of this 'leakage', negligence and clearly underhand dealing. In Cape Town's ghettos, the 9mm Parabellum remains the handgun of choice. If you don't own one, it's possible to rent one for a night, with a few rands extra for ammunition. What you do with it is your own business. That business very often involves murder. The crime rates for a single area, Manenberg, underline this:12\n\nFigure 5 Manenberg crime incidents 2004\u20132014\n\nIn Cape Town's high-risk areas, as one young member of the Hard Livings gang told me, bullets don't have names on them. He's not quite correct because some do. They're the ones used to eliminate specific people, usually opposing gang members. Very often, though, victims are arbitrary. They're caught in the crossfire, are subject to mistaken identities or are simply picked off randomly by a new gang member proving he has the guts to kill.\n\nAn example of this, among countless such stories that appear regularly in the press, took place on the evening of Wednesday, 14 May 2014. A number of teenagers and younger children were in a shop in Wesbank township playing video games and listening to music when two men stormed in and began firing at them. At least 15 gunshots were counted. Some youngsters were shot in the chest, arms and back, others were cut down as they ran. Two of the victims ended up in intensive care at Tygerberg Hospital, the rest were treated for bullet wounds in arms and legs and discharged.\n\nA youngster described the scene to a Cape Times reporter: 'There was blood everywhere. When the shooting finally stopped there were wounded boys lying on top of each other. They were shooting for no reason. None of the guys were gang members.'13 The backstory is all too familiar. The 26 and 28 gangs were warring over turf in the area and the shooters, probably high on tik, mistook the identity of the victims. The gunmen were unidentified. No arrests were made.\n\nI did a 20-month newspaper count from October 2012 and found that 37 people had been killed in gang crossfire in the city and more than twice that number injured. These acts of violence create fear and corrode a community. They're life-threatening situations which cannot be predicted and over which people have no control. In the Wesbank incident, a mother told the reporter: 'I'm scared now because they [the gunmen] might come back. We don't know what's going to happen next.' Fear is pervasive and feeds imagination. People come to expect the worst and responses vary from numbness to paranoia. In places like Manenberg the reality of danger requires no imagination.\n\nIn July 2013 a Grade One classroom at Sonderend school in Manenberg was riddled with bullets during a gang fight, forcing the teacher, Rosemary Adams, and her pupils to flatten themselves on the floor for safety. 'The children have become very nervous,' she said afterwards. 'There are behaviour problems, headaches, nausea and tummy aches. And their parents are also scared. They thought their children were safer at school but now they're keeping them at home.'\n\n'They come to school but you can see they can't concentrate,' said the school principal, Leon Beukes. 'You can repeat and repeat a lesson. But when you ask them afterwards they've forgotten it.'\n\nAt a nearby school in 2013, principal Thursten Brown, watched his school's cohesion fall to pieces. 'Our teachers' morale is low. To get to school some of them have to travel through areas where the gang fights happen. Some have had to duck bullets. Can you imagine starting your day like that? Pupils hear gunshots and wonder if maybe one of their relatives has been injured. Some miss chunks of work because they missed school out of fear of being caught in a gang fight.'14 Week after week, newspaper headlines reflect the chilling story:\n\nThree-year-old gunned down in Manenberg\n\nBaby caught in the crossfire\n\nMother, 24, killed in crossfire\n\nSix-year-old girl shot in the crossfire\n\nThree children wounded in Hanover Park crossfire\n\nTwelve-year-old Sade Boltman killed in crossfire in Bonteheuwel\n\nNeighbourhood watch volunteer Soroya Nordien gunned down in Lavender Hill\n\nFour youths in Elsies River shot in their foreheads by unknown gunmen... no one has been arrested\n\nFifteen people killed in Lavender Hill in four months\n\n32 attempted murders in seven months. Gangs suspected\n\nFour people were killed in Hanover Park in as many days\n\nGang wars see five teens killed in three weeks\n\nNo convictions in 87 cases of gang violence\n\nThese kids are ready to kill\n\nStray bullet claims another life\n\n28s gang member dies in a volley of shots\n\nSix state witnesses killed by 28s gang to silence them\n\nGirl, 15, caught with 9mm pistol in Hanover Park\n\nOne gang murder each day\n\n'In a month I'll be dead'\n\nI saw dogs eating the brains of the boy they shot\n\nA survey of children in Manenberg by The Trauma Centre indicates the disturbing level of trauma they experience on a daily basis:15\n\nFigure 6 Trauma among children in Manenberg\n\nGiven alternative choices, it's unlikely that any of the perpetrators or victims of gang violence would want to live in these conditions. The situation in which young people find themselves in many areas of the Cape Flats is \u2013 by any gauge of human rights \u2013 intolerable.\n\n***\n\nThe World Health Organisation defines violence as 'the intentional use of physical force or power, threatened or actual, against oneself, another person, or against a group or community, that either results in or has a high likelihood of resulting in injury, death, psychological harm, maldevelopment or deprivation.'16 Although the WHO statistics on international violence date from 2000, they are instructive. In that year, 1.6 million people died of interpersonal, self-inflicted, collective or political violence. Of those, more died from homicide than war, but most (815 000) died from suicide. A disturbing nine out of ten of all violent deaths took place in countires with large populations of low to middle-income earners. These are mostly in Latin America and Africa. Nearly eight in ten were males and the highest number of these were young men aged 15 to 29. The lowest homicide rates, as to be expected, were recorded in Sweden, Norway, Finland and Denmark.\n\nA 2010 study by the United Nations Office on Drugs and Crime found that Southern Africa had one of the highest murder rates in the world (30.5 per 100 000)17. Cape Town's average murder rate fell slightly between 2008 and 2011, but by 2015 it was the highest in a decade at 55 per 100 000. This number masks huge internal disparities. In the poorest neighbourhoods of the city, such as Khayelitsha, Nyanga and Gugulethu, where nearly half the city's homicides take place, it's much higher.18 In Nyanga, the online newspaper GroundUp estimated the murder rate above 200 per 100 000.19 These murder rates are higher than in some countries at war. However, in the wealthy suburbs of Camps Bay, Claremont, Rondebosch and intermediate suburbs of Woodstock and Table View, the rate was just above the world average of 6.9.\n\nFigure 7 Cape Town murders per 1 000 (2013\/14)\n\nMurder, however, makes up only 2.5 per cent of all violent crime in Cape Town. While there were 17 805 murders in South Africa in 2014\/15, a total of 600 000 other violent crimes, including attempted murder, rape, robbery and assault, were also reported to the police. In 2012 contact crime in the Western Cape was 1 852 per 100 000 and has been increasing gradually.20\n\nCrimeStats SA recorded the province's worst 10 precincts in 2015. Nyanga appeared in almost every category \u2013 murder; attempted murder; culpable homicide; sexual assault; assault with intention to cause harm; common assault; robbery; malicious damage to property; arson; carjacking; truck hijacking; public violence; kidnapping; burglary; robbery at residential and non-resedintial premises and illegal possession of firearms. Close behind it were Kraaifontein, Delft, Parow, Mitchells Plain and Bishop Lavis and central Cape Town. Highest for drug offences were Mitchells Plain and Kraaifontein, contributing to Cape Town's rating as the country's drug capital.21\n\nAs we have seen, most perpetrators and victims of urban violence in Cape Town are ghettoised men racially defined as coloured or African and aged 15\u201325. A former Khayelitsha gang member, Lerato Kossi, says of this critical age: 'It's when your fire is really burning.' Sylvia Ramos, writing evocatively about a similar situation in Brazil, says this is 'the age of death, the colour of death, the gender of death and the geography of death'.22 The introduction to Youth Violence: Sources and Solutions in South Africa puts the plight of these young men plainly:\n\nWe are concerned that too many South African children are growing up in dysfunctional families, poorly performing schools and violent neighbourhoods \u2013 and that unless we address these problems we will raise another generation of children who do not know any other way to solve a problem than to resort to violence.23\n\nFrom around 2000, South Africa began passing through what might be termed 'a demographic youth bulge' in which the largest percentage of people in the Western Cape were teenagers. This was caused by both a high birth rate in the late 1980s and early 1990s and in-migration from rural areas. This unusually high number of youths can be seen as an important driver of violence in Cape Town. We know that crime, particularly violent crime, is committed almost exclusively by young men, so as the population bulge entered the dangerous years of teens and early twenties, Cape Town became more violent.24\n\nCrime statistics, however, do not sufficiently explain the extraordinary level of violence in the city. Why do young men kill for a cellphone, beat someone senseless for a few rands or rape without compunction? It might seem obvious that poverty drives both crime and violence, but this is not supported by statistics. Violence tends, instead, to be concentrated in areas with the greatest income inequality. Youths mostly become involved in crime and often-associated violence, it seems, in order to redress the exclusion they feel through not having the material 'stuff' others have that defines social inclusion.\n\nViolence is therefore more likely to exist where there are pockets of poverty surrounded by areas of relative wealth, where there is a high youth population, low employment, limited state services and illicit economies. Hostility may often simply be an attempt to ward off humiliation, shame and disrespect.\n\nThe political climate is another trigger for violence. During apartheid, violence against people who were not of European descent was sustained by law and justified by religion. It became, to quote political researcher Anthony Altbeker, 'part of the ambience of a society in which it was understood by everyone, even by those who denied it, that a small minority lived the good life only because it kept its boots firmly on the necks of everyone else.'25 In addition, in defence of apartheid, thousands of white teenagers were conscripted and sent to the front line to fight 'communism'. Within the cities, the state viciously confronted millions more who resisted the boot on their necks and who were forced to confront oppression with violent uprising. The third strand of underlying violence emerged as the exiled African National Congress built and trained an army from those who escaped conscription or oppression at home and fled over the border. Violence became the legitimate means to oppose and defend the future of what was to become a democratic state in 1994.\n\nSubtle contributions to this process came from the nurturing of resentment on all sides, an almost instinctive resistance to authority and a gun culture with access to a frightening array of weaponry. The waters of the well, poisoned through legitimised violence, rose to the surface again from the disappointments felt by the newly enfranchised but still deeply disadvantaged poor. Despite South Africa's excellent Constitution, at street level the rule of law, if it is considered at all, remains an aspiration and not a reality. As Altbeker points out so perceptively:\n\nWe have yet to learn the conventions and civilities, the social graces and the urbane niceties that are assumed in our Constitution; because, though we have slipped off the shackles of oppression, we have not yet built a society that is whole and stable and which knows itself.26\n\nAlso, under apartheid, the activities of some gangs were covertly sanctioned by the state in return for their maintaining control over political activists and informing on covert strategies of the United Democratic Front and African National Congress. For example, drug merchant Jackie Lonte developed drug connections through the Civil Cooperation Bureau in return for 'dealing' with UDF supporters.27 In their research into crime networks in Cape Town, Charles Goredema and Khalil Goga found that:\n\nIn exchange for turning a blind eye to their activities, the state used local gangs to target anti-apartheid activists for assassination. The symbiotic relationship between the gangs used in this way and the state drastically increased their power and influence on the Cape Flats. Consequently, supply lines proliferated and drug dependence in communities escalated.28\n\nIn the moral mutability of such times, social norms \u2013 unanchored in established conventions \u2013 are fluid and open to interpretation. How they are applied becomes conditional as people glance sideways to see the steps appropriate to the dance. A lawbreaker or a violent act may be condemned until the benefit of following suit outweighs the value of disapproval. In a 'butterfly effect', where light wingbeats might precipitate a hurricane elsewhere, small transgressions grow until norm violations become more common, drawing in people who would not ordinarily do what they know to be wrong. This weakens inhibitions on criminality. To quote Altbeker again, one of the key factors driving the crime wave 'may be nothing more that the fact that violence has become so pervasive... the meltdown, having started, may now be feeding off its own energy.'29\n\nIf we agree that governance is the power to legitimate violence and to levy taxes, in the city's low-income areas \u2013 particularly in the older townships created to house people uprooted by Group Areas removals \u2013 this ability is owned by other than the state. Its failure to control these areas has permitted the creep of criminal governance which provides legal and illegal goods and services which the state does not provide. These include employment and the levying of 'protection' taxes on businesses in the area.\n\nCriminal networks exercise control through threats and acts of violence by way of extortion, control of drug turf and taxi routes, club bouncers, carjacking, robbery and many other illegal activities. Failure to comply evokes personal or property damage or, not infrequently, death. In this process, 'big men' and their gang-backed networks became alternative sources of authority and dispute resolution.30\n\nThere is possibly a deeper and more cynical reason for the violence of criminal governance. Street-level violence may be tolerated and encouraged in order to force state and city governance to keep its distance, compel neighbourhood obedience and offset the testosterone-fuelled energy of the teenage 'soldiers' essential for turf protection. This tolerance of violence may occur even though it must, at times, conflict with the smooth running of the criminal economy.31 It acts as a warning to anyone, or any other gang, inclined to challenge the de facto holders of neighbourhood power.\n\nWhen state institutions tasked to protect residents are seen to be failing, crime victims are more likely to turn to non-state structures such as vigilantes, powerful taxi drivers or local 'big men' who \u2013 for money or the sheer hell of it \u2013 turn assumed perpetrators into victims and often kill them.\n\n***\n\nA question worth considering is whether a dangerous environment breeds violent youngsters or whether their innate violence seeds a high-risk environment. The answer is not an easy one, but the probability is that they feed off each other. It takes angry young people to make a neighbourhood violent, but the dynamics within families, schools and peer groups of the neighbourhood create the volatile mix of anger and shame that boils into violence.\n\nChildren with parents or caregivers unable or unwilling to be a responsive 'other', as we have seen, have trouble sustaining emotional connections and struggle to find their place in the world. They develop aggressive behaviour and conclude that aggression gets them what they want. Their response to rejection, abuse and abandonment is to lock their emotions in a psychological vault. Suppressed anger and shame earn emotional interest. Psychologist James Garbarino has found that these negative emotional bank accounts are often bloated with fear and humiliation.\n\nFor some boys the psychological vault cannot hold all the hurt and anger that is stored there. Events \u2013 sometimes what appear to be trivial events \u2013 trigger a break that results in violence.32\n\nYoung people under these internal and external pressures are desensi\u00adtised to empathy and fail to see the humanity in others. The resulting depersonalisation offers unlimited possibilities for violence because the greater the distance between them and others, the more likely they will act aggressively without constraint towards others.\n\nGangs operating among disaffected youths do so in a particular moral universe. They demand honour and justice within a circle and legitimate aggression to those outside it. This circle, sometimes expanding, sometimes shrinking to almost nothing, is a wall of defense against hurt and shame. According to psychologist James Gilligan, shame imposes the fear that one will cease to exist, with the prospect of psychic annihilation:\n\nNothing seems to threaten the human spirit more that rejection, brutalisation and lack of love. Nothing \u2013 not physical deformity, not debilitating illness, not financial ruin, not academic failure \u2013 can equal insults to the soul. Nothing compares with the trauma of this profound assault on the psyche.33\n\nThose who are shamed are open to committing violence and aggression. This is because they know that acts of violence against themselves or others are a reliable method for reasserting their existence when life experience has denied it. 'I hurt; therefore I am.'34\n\nYoung men with this background fear to ever lower their guard. They remain vigilant against insult or perceived threat. They cannot reach beyond their survival ethic or show their vulnerability. They cannot reveal the hurt child inside themselves. In his work on what he calls 'lost boys', Garbarino looked beyond the posturing and bravado:\n\nLost boys are traumatised boys. They do many things in a failing attempt to cope with their trauma, but rarely do they have the resources or knowledge to cope effectively and in a socially acceptable way.35\n\nAn additional problem with youths in gangs is that they have an acute sense of their short mortality. Trauma, together with living in extremely dangerous neighbourhoods, produces what could be described as 'terminal' thinking. This undermines any sense of security in the future and is of serious concern in the development of any programmes for high-risk kids. If you don't expect to live past 25, why have safe sex or stay in school or study or drive carefully or avoid drugs? Why listen to anyone beyond your circle if you don't see a future?36\n\nWhen young people feel affirmed, respected and loved, they are much less likely to turn to violence and less likely to be influenced by a disorganised and dangerous environment. So one of the solutions to both gangs and violence is to ensure the safety of young people at home, at school and in their neighbourhoods. This will affirm them and ensure they are respected for pro-social behaviour. Unless we do this, if we fail to eliminate poor parenting, prejudice and massive disparities of wealth, South Africa will continue to be a very violent society.\n\nDrugs: High and unhappy\n\nI do stuff to get drugs. My friend is on drugs \u2013 my whole circle of friends are on drugs: unga, tik, buttons. This stuff is a very big problem. Every day we are watching for the shipments, when the Somali comes. We rob money and cellphones. But when the guys want to stab I say 'No. No', we don't want that. You can go'. If you hurt other people you will get hurt yourself.\n\nIn the morning I have to smoke unga \u2013 R20. Then to get the power to go into the day I need some tik. Then when I smoke a pakkie I have the power to do things. With that power I get money quickly, you understand. To do the things I have to do \u2013 then I have the money for the day. For that maybe R100. Friday, Saturday I need maybe R200. I make the money robbing. I steal anything I know I can sell. You go to Nyanga Junction you see people like me with caps or clothes and stuff they're selling to the people. All stolen. Ten thousand people needing drugs. It would be better if we had a way of getting an income. I will work, you know, but mostly easy money is better.\n\nIf I come to an area and I see there are no gangs and there are people who use drugs, there's a market. I go to Manenberg and I get some drugs and I begin to sell them. That way I get them to be with me because I have the drugs they're using. Most of the time it's because of the drugs.37\n\nLinkie, Manenberg, 2015\n\nIn Cape Town I'm watchful, interested in who's on the streets and what they're doing. In the eyes of drug dealers, that translates into someone wanting something, so I get approached. Cannabis, tik, heroin, cocaine, I've been offered everything but buttons (mandrax) and nyaope (a mixture of tik, heroin and cannabis), which they know people like me don't generally use. In this city drug sales are huge and 'leisure' drugs are a massive problem. South Africa is one of the world's largest importers of illicit ephedrine and pseudoephedrine, two of the precursor chemicals used to manufacture methamphetamine (tik). South Africa is also the largest producer of cannabis \u2013 about 2 500 metric tonnes annually of the world total of 8 900 tonnes. Between 2008 and 2010, 17 820 people were admitted to treatments centres in the Western Cape for non-alcohol drug abuse.38\n\nIf used in excess, no drug can be described as 'good', whether legal or illegal. Drugs taken by a pregnant mother, as we have seen, can cause disastrous harm to her unborn child. But what about their effect on teenagers? In Cape Town there's no shortage of illegal drugs. Most young people, if asked, could quite quickly locate a supplier of cannabis, mandrax, cocaine, crack cocaine, ecstasy, LSD, kat, heroin, crystal meths (tik), unga (a heroin derivatave) and nyaope.39 This ready access is having a devastating health effect in the province and, particularly, in Cape Town.40\n\nA survey among high-school learners in the Western Cape between 2005 and 2011 found that between nine per cent and 12 per cent were regular drug users with cannabis and tik being the most common.41 Another survey of learners in grades eight to 10 in the province during 2011 found that 12 per cent of learners had been offered drugs and nearly half (46.6 per cent) reported having seen drugs being sold.42 Among users, 56 per cent were aged 11 to 35, with 30 per cent under the age of 25. For some, the starting age for drug abuse was eight.43\n\nConverted into numbers, these percentages tell a sad story. The 2011 National Census found just over a million people in the Western Cape to be between 15 and 24.44 A drug use of 12%, indicates that about 120 000 young people could be regularly using some type of hard drug. Most of these would be within the City of Cape Town. The street price for drugs in 2014 was roughly as follows:45\n\nFigure 8 Street cost of drugs in Cape Town\n\nExtrapolating from surveys to total population has its problems. Categories often don't quite match, but such an exercise does give a general sense of scale even though prices and what's 'cut' into the drugs vary considerably. If 'regular' use means once a week (it's usually much higher), 120 000 young users of cannabis or mandrax would net R6 million for dealers. This figure excludes older users. If the sales were for tik or cocaine, syndicate gross income would be R33 million a week, tax free in the Western Cape alone. This goes some way to explain the lucrative nature of the illegal drug trade and the deadly defense of sales' turfs across the city. Given that most of these drugs are being bought by unemployed youths, it also goes a long way to explain the crime problem. There are other consequences to consider. In 2012 alone the Financial Investigation Centre (FIC) estimated drug-related deaths nationally to be 842 184.46\n\nThe volume of drug use begs an obvious question: why do people use such dangerous substances and, more importantly, why do they become addicted? The answer to the first question comes in two parts. They use them because it's the policy and goal of druglords to ensure the highest possible rate of addiction among clients. By making drugs illegal, we have handed the supply to unregulated, unscrupulous cartels which have not the slightest concern for the welfare of their customers. The second answer is that drugs aid socially marginalised and economically excluded people to cope with the stresses of their environment.\n\nDrugs are attractive because they deliver, in the short term, what's asked of them and for this reason they've been used by humans throughout our sapient existence. The pursuit of pleasure is one of our major goal-directed endeavors. We all seek highs in different ways such as sport, music, literature, dancing, money, food and sex. In attempting to understand excessive use, it's instructive to know their chemical effect.\n\nLeisure drugs can strongly alter brain activity, triggering dopamine pathways to our prefrontal cortex and, particularly, to the nucleus accumbens, our so-called 'pleasure centre'. This zone plays an important role in laughter, reward and reinforced learning as well as fear, aggression, impulsivity and addiction.47 Higher dopamine concentration was seen, for example, in the nucleus accumbens when subjects believed they were being given money and increased activation was observed among men viewing pictures of attractive women.48\n\nPart of the brain's mesolimbic dopamine pathway is to the prefrontal cortex. Continued drug-induced dopamine production causes over\u00adproduction of glucocorticoid hormones, increasing the desire to repeat the process.49 This can happen with any user, but damage to the prefrontal cortex triggered by prenatal and immediate postnatal impairment degrades a person's exercise of free will, as we have seen. It can cause both mental steering and braking systems to fail.50 At this point, for some people, the door to addiction begins to open.\n\nThe initial attraction of drugs involves dopamine stimulation beyond the 'highs' of more normal pleasurable activities. This is generally recreational, extremely common and relatively few young people move beyond it. You can be a regular drug user and have resistence to the addictive potential of the drug. A second, more pathological phase begins when the amount of drug used increases, but the taker is in most ways still 'normal' and functioning in society. Users believe they 'have it under control'. A third and final phase involves loss of control when drug-taking becomes the individual's main goal-directed activity.51 At this point, behavior is governed by the uncontrollable over-valuing of the drug and by a growing insensitivity to the deterrent value of potential punishment.52 In all cases, drugs essentially cheat the biological purpose of reward, with natural dopamine stimulation to reproductive fitness traded for a cheap hit.53\n\nAddicts at this stage generally lose their behavioural inhibition, are intolerant to delayed reward and have an impaired ability to consider the consequences of their actions. Their brains are said to have arrived at a new 'hedonic set point', the place at which they feel 'normal'. This point, known as allostasis, is their desired 'self' and a non-drug state is felt as pathological, unpleasant or deeply painful. The person no longer takes drugs to feel high, but to feel right. All impulses are directed to its continued attainment and use. This mental condition is known as anaplastic \u2013 the brain is physically altered:\n\nThe addict does not develop a response to drugs that is completely different from that of resistant individuals but is principally an individual who cannot adapt to the changes in the brain induced by the drug. As a consequence, vulnerability to addiction can be con\u00adceptualised as involving a degree of 'anaplasticity', the inability to recover a lost function, rather than a unique sensitivity to the drug's deleterious effects.54\n\nAt this stage drugs are not only wanted and needed but also deeply missed when they're not present. Their absence is felt as the irreplaceable loss of something very dear and precious, a pathological mourning and physical craving that cannot be overcome. Young addicts in need will go to extreme lengths to obtain a hit.55 In neighbourhoods like Manenberg, Nyanga and Hanover Park they're prepared quite literally to kill for drugs. When turf wars break out, 'tik soldiers' are given guns by gang members unwilling to risk death themselves and sent to war against the 'enemy' who are quite likely to be addicts too. They're paid in drugs.56\n\nA seemingly obvious conclusion is that the cause of addiction is the chemicals you take. There's a strong probability that it's a fallacy and to understand addiction we have to probe deeper. The first step is to ask why some young people resist taking drugs despite their easy availability while others fall prey even before they're teenagers.\n\nAn American longitudinal study by Jonathan Schelder and Jack Block found that among those who went on to become addicts, 'problem drug use [was] a symptom, not a cause, of something else \u2013 personal social maladjustment.'57 Psychologist Gabor Mate correlated this finding: 'The basic cause of addiction,' he writes, 'is predominantly experience-dependent during childhood and not substance-dependent. The current conception of addiction is ill-founded.'58 What does he mean by that? He's saying that people are more likely to get addicted in an attempt to dull the pain of their damaged childhoods than because of the drugs they take. Child abuse, he says, is as likely to cause drug addiction as obesity is to cause heart disease.59\n\nCanadian researcher Bruce Alexander found that isolation and loneliness cause people to look for relief. 'They find temporary relief in addiction because it allows them to escape their feelings, to deaden their senses and to experience an addictive lifestyle as a substitute for a full life.'60\n\nOne of the solutions to this loneliness is to bond with a drug like heroin, cocaine or tik which, for a while, wraps its arms around you and keeps you warm. Another solution is to bond with a subculture that takes those drugs \u2013 a clan on the same mission facing the same pain and dangers. British investigative journalist Johann Hari, in his study of drug use, found that such a subculture offers an identity, a family, a life of highs and lows instead of relentless monotony.\n\nThe world stops being indifferent to you and starts being hostile \u2013 which is at least proof that you exist, that you aren't dead already. When you have been told you are a piece of shit all your life, embracing the identity of being a piece of shit, living openly as a piece of shit \u2013 it seems better than being alone.61\n\nThis calls for a shift in perspective. The generally cited causes of the drug crisis in Cape Town are the addictive nature of chemical substances and the syndicates and gangs which sell them. The standard response, therefore, is to amp up drug busts, increase penalties for use and marketing and to jail offenders. That's what American literature on the War on Drugs would have you think.\n\nThe sheer volume of drug trade in the city, however, is ample evidence that this strategy is not successful. This 'war' has been waging since the 1930s. It's almost exclusive focus has been the chemical effect of drugs on the human brain and the potential 'colonisation' of our minds by illegal chemical substances. The cause of the problem is seen almost exclusively as the drug itself and those who supply it. The American response in terms of this perspective has cost billions of dollars and involved shooting wars, aerial chemical defoliation, kidnapping, political destabilisation and tens of thousands of deaths.\n\nFocusing on drugs and their marketing masks the root of the problem \u2013 early childhood deprivation and the environment which triggers drug over-use. The driver is the need young addicts have for the love and emotional support they've been denied. Addiction, in these terms, is a family and neighbourhood malfunction of which excessive drug use is a symptom. It's a disease of loneliness and deep sadness. The solution, therefore, lies in support for early family bonding and not a War on Drugs. There is another way, which will be discussed presently. But we first need to explore another neighbourhood problem which, for many, is equally toxic: education.\n\nEducating for failure\n\n'You breathe in oxygen through your left lung and breathe out carbon dioxide through your right lung.' That's what a life-science teacher taught her class at a Cape Town school where I worked for several weeks last year as part of my teacher training.\n\nWhen I arrived at the school the first time, I noticed that latecomers were locked out of the school grounds for the first period. That's reasonable. But when the latecomers were let in, they had to leopard-crawl across the tarmac to their classroom, a punishment aimed purely at demeaning them.\n\nCorporal punishment may be banned, but caning still occasionally takes place. However, the standard punishment inflicted upon students in this school is frog-hopping, which is exactly what it sounds like: all day long children who have been punished can be seen hopping up and down the corridors. It's funny at first, but it's undignified and the students hate it. There is no detention \u2013 because the teachers aren't prepared to hang around after school hours.\n\nI witnessed some strange things: a teacher holding a terrified student against a wall, then shaking him by the ears and students being forced to sit inside a wall shelf in the most awkward positions. And none of it helps terribly much. Classes remain chaotic. Very little learning takes place.\n\nTeaching in the morning was particularly challenging. The students were listless and tired. The first time I taught in the morning, I asked the students why they were so lacking in energy. The answer was startling: they were hungry; many of them would receive their first meal of the day at about 11am from the school feeding scheme.\n\nThe principal of the school was an insipid man, but one thing would bring him out of his shell: the opportunity to Bible-punch at assembly. And every day in many classes there's sermonising, praying and religious indoctrination. When teachers are out of their depth with the subject they're supposed to teach, religion may be an easy substitute. One teacher said to his class: 'You're not here because you're intelligent. You're here because of Jesus.'\n\nTeacher (anonymous for fear of dismissal), GroundUp, April 2015\n\nSouth Africa's education system is, by default, deeply implicated in gang formation in Cape Town. It fails to provide a useable qualification or a safe learning environment to almost half of those who enrol. Schools on the Cape Flats produce young people who have so few life options that, for many, becoming a gang member is almost inevitable. Simply getting to school may require joining a gang in order to ensure safe passage through the school gates.\n\nSouth Africa has achieved a massive increase in access to education since 1994, but the system seems unable to retain learners for long enough to prepare them adequately for the world of work or life in general.62 According to Africa Check in 2015, 'Results from international, standardised tests show that between 75 per cent and 80 per cent of South African schools are not able to impart the necessary skills to students.' These schools, it concludes, are effectively dysfunctional.63 The problem is deeply embedded in the system itself.\n\nAt the end of 2013, for example, 562 112 pupils wrote their final National Senior Certificate (Matric) exams. Early the next year 78.2 per cent found they had passed. The Minister of Basic Education, Angie Motshekga, said she was delighted that the percentage was up 4.3 per cent from the previous year. The following year, however, it had dipped to 75.8 per cent. Those who passed had succeeded in a schooling system that required a pass mark for languages and maths of only 40 per cent and 30 per cent for all other subjects. Regardless of these low benchmarks, more than half the pupils who had started school with the 2013 completing class (699 715 learners) had dropped out of school before reaching matric.64 Despite the inevitable round of congratulation and not detracting from the hard work put in by half a million pupils, the pass rate masked an education system in deep trouble. Only one in three pupils had sufficient marks to attend a university, creating a potential loss of 337 267 young people to higher education.65\n\nAccording to Nicholas Spaull, an economist at Stellenbosch University, for every 100 students who started school in 2003, only 49 made it to matric, 37 passed and 14 qualified to go to university. Calculated this way, the 'real' or through-put pass rate in terms of the number who started school 12 years previously was about 36 per cent.'66 If a 50 per cent pass mark was required in all subjects, the actual matric pass rate for 2013 would have been between 22 and 24 per cent. Spaull discovered that students were pushed through the system until grade 10, at which point schools realised that if they put these learners through, they were not going to pass grade 12. 'So they weed out these students.'67 The matric pass rate is thus bumped up and gives no indication of how the 50 per cent who fall by the wayside are doing.68 In this way, the system leaves behind more than half a million people a year with little or no proper education.\n\nMary Metcalfe, former head of the Witwatersrand University School of Education and a former provincial government minister for education in South Africa's Gauteng province, echoed these concerns: '[The pass rate] doesn't tell us about the large number of children who didn't make matric, who didn't pass grade 10, who didn't pass grade 11 and who failed at grade 12.'69\n\nThe matric results also conceal the underperformance of the majority of pupils who do, in fact, write the examination. Strong results showings in a minority of schools mask the poor performance of the majority of schools that are judged as dysfunctional. This skews the average and fails to present a true reflection of the situation for most pupils. Vice-chancellor of the University of the Free State and a prominent commentator on education, Jonathan Jansen, has described the country's education system as a massive fraud. 'The government wrongly, but conveniently uses the matric results as a barometer of the state of the school system,' he says, 'when all other data reveal we have been stagnating, or doing worse. If you removed the top 20 per cent of schools \u2013 mainly former white, privileged schools \u2013 from the national averages, then a very dark picture emerges of a mainly black and poor school system performing far below what the combined results show.'70\n\nIn an effort to create the impression of a healthy matric pass rate, education is also being 'simplified'. Curriculum content has been slashed and a sound, general knowledge is being forfeited. According to an Institute of Security Studies report, key subjects such as mathematics have been dumbed down to a choice between mathematical literacy (simple arithmetic) and 'normal' maths. Despite this, according to the study, in 2013 'only three per cent of Grade nine pupils scored more than 50 per cent in maths, with a disappointing national average of 14 per cent in this critical subject.71\n\nAn international survey in 2011 found numeracy among South Afri\u00adcan grade nine pupils to be appallingly low. Of 48 countries participa\u00adting in an assessment of trends in mathematics and science, South Africa came second last \u2013 ahead of Honduras but, in Africa, behind Morocco, Ghana and Botswana.72 Simple literacy is another problem. A survey of grade six pupils across South Africa found that in three out of four schools, between about a quarter and half of the learners were functionally illiterate.73\n\nThe fault is not with the pupils, but with the system of education, curriculum, schools and teachers. The upshot is that each year more than 100 000 young South Africans fail to attain a matric certificate, the very bottom rung of the formal job ladder.74 These figures equate to a deprivation of basic capabilities for over a million people every ten years. Most of these will be concentrated in poorer, more deprived communities. Many use their ingenuity and connections to get back onto the ladder, but great numbers don't. In 2014 in the Western Cape 32 per cent of people under 25 were described in a government report as Neets (not in education, employment or training).75 In age groups between 12 and 16, youths defined as coloured constituted the highest numbers in this category.\n\nA problem associated with the school dropout rate is the level of classroom and playground violence. In a Western Cape study conducted in 2013 covering primary and secondary schools, 28.7 per cent of learners in urban areas reported experiencing some form of violence at school during the previous 12 months.76 This translates into more than a million learners subjected to violence in the form of threats, physical and sexual assaults, robbery, theft and bullying, all of which compound the climate of fear.77 For a Khayelitsha resident this was plain to see:\n\nStanding at the bus stop in the morning on my way to work, I see fear in the eyes of these young kids. They walk to the stop carrying knives in their backpacks thinking who might jump them and steal their cellphone. They know their parents are afraid going and coming to work. They see young and old people being robbed. They know of young girls being victims of gang rape. There are also other young people getting on the bus who've lost track of how many people they've killed.78\n\nThe highest level of school violence nationally occurs consistently in the Western Cape. In the province, between 2008 and 2012, theft at school jumped from 28.9 per cent to 42.2 per cent and sexual assault from 2.7 per cent to 9.2 per cent. More than four in ten pupils reported attacks involving weapons.79 Four out of ten principals and teachers reported that certain areas of their schools were unsafe for learners. The National School Violence Study summed up the problem this way:\n\nSchool violence affects not only the children who are directly victimised in these incidents but also those who witness it. This indirect victimisation contributes to an atmosphere of fear and insecurity at school, which inevitably interferes with learning, stunts academic performance and, ultimately, impacts negatively on the longer-term developmental trajectories of young people.80\n\nMany schools on the Cape Flats are poorly equipped. Teachers frequently and illegally dispense corporal punishment. Levels of peer violence and bullying are high, theft and threats are common.81 What's offered in the overcrowded classrooms is felt to be uninteresting by children with low levels of motivation and who are often hungry. In these situations, many children are being socialised into failure.82 Here's an example.\n\n***\n\nSchool was just a way of passing time. Other pupils felt this too. When I was younger I was in a class where the teacher would almost tear your ear off or put you across her lap and beat you with the heel of her shoe. It was very frightening. I often dreamed about growing up quickly and joining my big brother, working as a messenger or a cleaner and buying myself the latest fashions.\n\nWe were ranked according to the marks we scored in exams. The pupil with the highest marks sat under the teacher's nose and the poor-marks people sat at the back. We were constantly scolded because our clothes were different colours and they should have been a uniform. This happened even in the middle of a lesson! It was very embarrassing. Then you had to cover your books with brown paper and plastic. No vetterige merke [fat marks] were allowed. But how can you avoid getting stains on books when the only table where you can do your homework is where your mother cooks? And you can't sit up late with your books because candles are expensive. You never see any other books, except perhaps if you go with your parents to their bosses' house.\n\nAt the three schools I attended there was always at least one teacher who would beat you with his fist or anything. When I was in Standard One we heard our teacher next year would be Mr Z, the worst teacher at the school. The next year five pupils didn't come back to school \u2013 among them were the brightest students in the class. I remember one of those, Joseph Martin, was later the leader of the biggest gang in Athlone. He has been in and out of prison often. He was a clever student.\n\nFridays were the favourite days for staying out of school. First it is turn-out day [house-cleaning day], then perhaps you had to accompany your mom to the old man's place of work so she can get some money before he spends it on booze. Or maybe we were sent in to get it. Then on Mondays the folks usually stayed at home after a very pissed weekend. You aren't allowed to stay out of school, so on Monday you must have a letter from your parents. My bigger sister would write it because my parents couldn't read or write.\n\nSimon, Hanover Park, 2011\n\nSchool is, of course, more than about learning and passing examinations. It's the most important arena of socialisation in a young person's life outside of home. This is where they come into contact with non-family adults, peers and new ideas about the world and their place in it. It is also their first contact with formal authority. In school, children encounter a new way of ordering social relationships and coordinating activities. In their study of adolescence and delinquency, Nicholas Emler and Stephen Reicher found that, depending on the nature of their childhood up to that point, children will differ in their willingness to make a positive accommodation with bureaucracy.83\n\nThis has little to do with their intellectual ability, but with a host of factors including temperament, role relations in the family, attachment to parents, the interest parents take in education and any early experiences with formal authority such as police, social workers or city officials.\n\nSome children will find school arrangements congenial or at least bearable. They will accept its structures and follow the various regulations of school life. They will persevere at tasks, willingly accept teachers' instructions and become successful students. Along the way they'll improve their intellectual skills. Initially socially and then increasingly as individuals, these children will see a future value in education.84 Children who consider school a means of unwelcome external control rather than a path to personal success will find its authority uncongenial and adapt negatively to regulations, timetables, classroom rules and teacher authority. As a result they will work less effectively and their conduct will be visibly different.\n\nFor pupils who consider school to be boring conformity, it can be reinvested with challenge by systematically breaking the rules as a way of regaining some control of the situation. Whereas more compliant pupils would consider that submission to authority will bring rewards, non-conforming pupils may reason that, for them, the rewards of education are a con. Teachers will generally label these children troublesome and stupid, exerting less effort in their direction. They'll get poorer grades and in turn denigrate the value of academic success. This will effect decisions whether to obey teachers or challenge them, whether to spend evenings on homework or in the streets.\n\nThis has a knock-on effect. Being the first formal institution that most children encounter, school is the basis upon which they construct a preliminary understanding of all formal authority and the principles according to which it operates. It has been found that adolescent attitudes to school rules and teacher authority are closely related to their attitudes to the law and other forms of institutional authority.85 With a severely blurred academic horizon, young people with school problems find greater fulfilment hanging out on the streets. There, without education or employment, it's a debilitating waiting game with no end in sight. Trouble finds willing accomplices in such neighbourhoods. The failure of the education system is, therefore, a major factor in gang formation.\n\nMedia and violence: cause or cure?\n\nVery often, when someone hears I'm interested in gangs, they'll tell me with great confidence: 'It's television. That's where they learn to be violent.' It is true that almost every programme, from cartoons to soapies, has a measure of interpersonal violence. This is echoed in films. Some of it is bloodthirsty and horrific, with blood splashing across the screen and bullet holes in foreheads. Does it teach kids how to kill, rape and throw their weight around against weaker 'others'? It seemed to be important to find some answers.\n\nThe first thing that moving media definitely does is show you relative deprivation, and feeling poor is worse than being poor. In medieval England you could easily see the difference between your hovel and the manor house, your rough shirt and the lord's silk blouse. You walked and he rode. You worked and he played. In Cape Town's sprawling townships, those distinctions can be read from the thin walls of your shack and display themselves on a television screen or, increasingly, arrive in your pocket by smartphone. They glare at you from the pages of magazines and smile beguilingly from billboards. If you're a poor kid, their message is simple and clear: you haven't got what it takes to make it. Within the endless stream of images and information seem to be other more subtle messages: act this way for success, happiness is your right, sex is for the taking, violence achieves your goals.\n\nCopycat violence, particularly, has been the subject of research and argument ever since television became a mass medium. Before that, comics were seen as being implicated and before them trashy novels. Indeed the possible transmission of bad behaviour through the media is never far from any discussion about children and adolescents. So how much screen time are young people in the country consuming?\n\nIn South Africa, though only 60 per cent of households have a TV set, eight in ten people watch regularly, averaging 4.5 hours a day. Nearly one in three access programmes through an internet connection and four in five adults download information by smartphone.86 A survey of Grade 11 pupils at nine schools in low-income areas of Cape Town found that nearly all (96 per cent) had cellphones and nearly half (49 per cent) used them to access the web through a mobile internet connection.87 As a developing country, South Africa is extraordinarily well digitally connected.\n\nWhat is the effect of such high media consumption on young people? There is much controversy and research around this question, mainly from the United States. An analysis by the American Psychological Association found that a typical American child could witness on television more images of death and destruction from their living room than any policeman or soldier in the line of duty in their lifetime. 'Good' TV heroes committed 40 per cent of the violent acts, more than a third of the bad characters weren't punished and more than 70 per cent of the aggressors showed no remorse. They received no penalty for their violent actions.88 Television and movies provide an endless array of role model choices and cultural messages as well as countless 'must-have' products for young people exploring their own image management. Generally those role models are violent and the products unaffordable for kids in low-income neighbourhoods.\n\nOf course a concern is that media violence could act as a behavioural rehearsal for real-life actions. It could, through a process of cognitive restructuring, desensitise young people to the pain of others, reducing their empathy and giving rise to 'compassion fatigue'. Often quoted in the literature is a phrase used by US researchers Brad Bushman and Rowell Huesman that 'the correlation between media violence and aggression is only slightly smaller than that between smoking and lung cancer.'89 The argument is essentially that while TV messages merely prime existing personal scripts in adults, they encode new scripts in the minds of children:\n\nThe changes in how the child perceives the world from viewing violence and the beliefs about aggression that the child acquires from viewing violence are likely to influence the child's behaviour in the long term as much as the specific scripts for aggression that the child learns from viewing violence.90\n\nSouth African media researcher Jane Stadler questions such copycat behaviour. Does TV violence really teach children to mimic it? Are they persuaded to think guns are exciting toys and that violence is an effective problem-solving strategy? Or, she asks, do they recognise it as unrealistic fantasy material that should not be enacted off screen?91 Her answer is that it depends on the context. If they live where they're frequently exposed to violence in their homes and community, TV violence could amplify aggressive response to perceived threats. It could equally increase their fear of violence or provide a safe space to explore their anxieties and fantasies.92\n\nViolent video games, particularly shooter games, are of a higher order of concern. They're seen as being implicated in violent behaviour because, rather than requiring players to be passive spectators, they oblige them to participate in violence as a problem-solving device. Media analyst Gerard Jones, who has designed video games, agrees that some youths can be negatively affected by such games if they themselves are aggressive for some other reason. However, he contests claims that they stimulate copycat behaviour.93 A question seldom asked, he says, is why young people generally love media violence and the opposite of what seems by adults to be good for them.\n\nAggressive kids may be more drawn to violent entertainment precisely because they need a compelling alternative or acting out or because they want to help make sense of their own aggressive feelings. But all children need to feel strong. They need to feel powerful in the face of a scary, uncontrollable world. Superheroes, video-game warriors, rappers and movie gunmen are symbols of strength. By pretending to be them, young people are being strong.94\n\nAccording to Jones, young people use fantasies of combat to access their emotions, to take control of their anxieties, to calm themselves down in the face of real violence, to fight their way through emotional challenges and lift themselves to new developmental levels. They need, he says, to lose themselves in their own stories, face their powerlessness and kill their own monsters created by environmental and possibly biological challenges. In video games they can make things come out the way they want. Perhaps, on the Cape Flats where young people are confronted by the harsh realities of their lives, they may need the harshest, most violent fantasies to help them get through.\n\nIf a young person's real world teaches them that violence is rewarded, the message is that entertainment may reinforce that lesson. In such a case, the problem and the solution lie not with the media, but with how parents model behaviour. This is because the power the entertainment industry has over a young person is usually reinforced by the actions and attitudes of their parents.\n\nNaturally it's tempting to add media violence to the many reasons for Cape Town's gang violence, especially for desensitizing of young people to killing. Life may have dealt them a toxic mixture of anger and shame, few positive role models and a dangerous environment, but Jones suggests that such kids know the difference between reality and fantasy better than adults do. Research suggests that they are open to exploring the cathartic experience of the latter.\n\nMedia violence, cruel rap, harsh hip-hop, rough sex and generally offensive entertainment are appealing to adolescents because they draw a line between themselves and the older generation. They mark a movement into previously forbidden territory and a break with their own past. Some, but not many, youngsters will model on violent characters and believe themselves to be temporarily in their boots as they pull the trigger in the real world. Video games particularly, whether in an arcade or on their cellphones, are a world in which they have a measure of control, somewhere to practice overcoming fear and becoming a hero. This may fly in the face of conventional adult wisdom. But my answer to people who insist on media copycat violence is that television, movies and video games may be places where real violence goes to die.\n\nEndnotes\n\n1. Some of the information in this section is updated from my earlier book, The Brotherhoods (David Philip Publishers, Cape Town, 1984).\n\n2. Sutherland, Edwin: Principles of Criminology. University of Chicago Press, 1924.\n\n3. Sampson. R. J. ( 1987). 'Communities and crime'. In M R Gottfredson & T Hirschi (Eds.): Positive Criminology (Sage, Newbury Park, California, USA) p109.\n\n4. See Charis Kubrin: 'Social disorganization theory: Then, now and in the future'. In Handbook on Crime and Deviance by Marvin Krohn, Alan & Lizotte, Gina Penly Hall (Springer, New York 2009).\n\n5. Sampson, Robert: Great American City: Chicago and the Enduring Neighborhood Effect (University of Chicago Press, 2012).\n\n6. This depiction is from R Stark: 'Deviant places: A theory of the ecology of crime' (Criminology, 25. 893 1987).\n\n7. Interview with Clifton, Hanover Park, October 2014.\n\n8. The name parabellum is derived from Latin: Si vis pacem, para bellum (if you seek peace, prepare for war), which was the motto of the German weapons and munitions company Deutsche Waffen und Munitionsfabriken Aktien-Gesellschaft. It was first manufactured by this factory in 1893 as the Luger pistol and by the 1930s had evolved from a 7.69mm to the 9mm recoil-operated semi-automatic handgun preferred by Hitler's SS. Since then it has been refined, streamlined and made more portable.\n\n9. Lamb, Guy: An Overview of Small Arms Production, Export, Ownership and Proliferation in South Africa (Centre for Conflict Resolution, UCT, February 2000).\n\n10. Lamb, ibid.\n\n11. Grobler, Liza: Crossing the Line: When Cops become Criminals (Jacana, Johannesburg, 2013) p104.\n\n12. Meeting the Needs of Survivors of Gang Violence in Manenberg (The Trauma Centre, 2015). Numbers adapted from Crime Statistics SA.\n\n13. Cape Times, 19 May 2014, p5.\n\n14. John, Victoria: 'Manenberg gang violence: Our rights have been taken away'. In Mail & Guardian 15 August 2013.\n\n15. Meeting the needs of survivors of gang violence in Manenberg (The Trauma Centre, 2015).\n\n16. World Health Organization Report on Violence and health (2002). p5.\n\n17. In Sweden it was 1, in Iceland 0.3 The SA Medical Research Council report for 2008 logged the South African murder rate in 2003\/4 using SAPS figures at 43 per 100 000.\n\n18. United Nations Office Report on Drugs and Crime, 2010.\n\n19. http:\/\/groundup.org.za\/\u00adcontent\/\u00adhomicidal-kid-gangsters-nyanga-east\n\n20. Lancaster, Lizette: Africa Check, 19 September 2013.\n\n21. Crime Stats SA 2015.\n\n22. Shaw, Margaret: 'Addressing youth violence in cities and neighbourhoods', in Youth Violence, op cit. p 376.\n\n23. Ward, Catherine, Amelia van der Merwe, and Andrew Dawes (Editors): Youth Violence: Sources and Solutions in South Africa (UCT Press 2012) p1.\n\n24. Altbeker, Antony: A country at war with itself: South Africa's crisis of crime (Jonathan Ball 2007), p 116.\n\n25. Altbeker, ibid, p98.\n\n26. Altbeker, op cit, p112.\n\n27. Goredema & Goga, op cit, p8.\n\n28. Goredema & Goga, op cit, p8 quoting, in part, Mark Shaw: Organised Crime in Post-apartheid South Africa.\n\n29. Altbeker, op cit, p114.\n\n30. Goredema & Goga, Crime networks and governance in Cape Town, ISS paper 262, op cit, p2.\n\n31. I thank Mark Shaw, Centre of Criminology, UCT, for this insight. In conversation: November 2014.\n\n32. Garbarino, op cit, p91.\n\n33. Gilligan, James: 'I learned this lesson' in Violence, our deadly epidemic and its causes (Putnam, New York 1991). In Garbarino, op cit, p132.\n\n34. See Garbarino, ibid, p132.\n\n35. Ibid, p158.\n\n36. Ibid, p 118.\n\n37. Linkie, Manenberg, 2015.\n\n38. National Drug Master Plan, South Africa, 2013.\n\n39. Ibid.\n\n40. Ibid.\n\n41. Andreas Pl\u00fcddemann and Charles DH Parry: Methamphetamine Use and Associated Methamphetamine Use and Associated Problems among Adolescents in the Western Cape Province of South Africa: A Need for Focused Interventions. Alcohol & Drug Abuse Research Unit, Medical Research Council (MRC), 2011.\n\n42. Survey on Substance Use, Risk Behaviour and Mental Health among 8010 Learners in Schools in the Western Cape Province, 2011. South African Medical Research Council, 2013.\n\n43. National Drug Master Plan 2013 \u2013 2017, op cit.\n\n44. About 1.5 million were between 10 and 24 and 2.5 million from 10-35.\n\n45. National Drug Master Plan 2013 \u2013 2017, op cit.\n\n46. Ibid.\n\n47. One mechanism of transition to intensified drug use is the interaction between glucocorticoid hormones and the dopaminergic system. Glucocorticoid tone is one of the most crucial regulators of dopaminergic transmission activity in the accumbens and these hormones are involved in the pathophysiological mechanisms that induce transition from recreational drug use to escalated sustained drug use. Glucocorticoid hormones increase the activity of the dopaminergic system by acting on the glucocorticoid receptor (GR) expressed by striatal medium spiny neurons that project back to the ventral tegmental area (VTA). In experiments, suppressing the GR in these neurons dramatically reduced the activity of the dopaminergic projection to the accumbens. Administering glucocorticoids repeatedly increased vulnerability to drug self-administration. See Piazza PV, Le Moal M (1996) 'Pathophysiological basis of vulnerability to drug abuse: role of an interaction between stress, glucocorticoids, and dopaminergic neurons' (Annul Review of Pharmacol Toxicology 36:359\u2013 378); Marinelli M, Piazza PV (2002) 'Interaction between glucocorticoid hormones, stress and psychostimulant drugs' (European Journal of Neuroscience 16:387\u2013394); Deroche V, Marinelli M, Le Moal M, Piazza PV (1997) 'Glucocorticoids and behavioral effects of psychostimulants. II: cocaine intravenous self-administration and reinstatement depend on glucocorticoid levels' (Journal of Pharmacology 281:1401\u20131407).\n\n48. Aharon, L, Etcoff, N, Ariely, D, Chabris, C. F., et al. Beautiful faces have variable reward value (Neuron 2001, 32, 357-551).\n\n49. Marinelli M, Piazza PV (2002) 'Interaction between glucocorticoid hormones, stress and psychostimulant drugs' (European Journal of Neuroscience 16, pp387\u2013394).\n\n50. Volkow, Nora, Ruben Baler and Rita Goldstein: 'Addiction: Pulling at the Neural Threads of Social Behaviors' (Journal of Neural Science, 27 January 2011).\n\n51. Piazza, Pier Vincenzo & V\u00e9ronique Deroche-Gamonet: 'A multistep general theory of transition to addiction' (Psychopharmacology (2013) 229, pp387\u2013413).\n\n52. Volkow, Baler and Goldstein, op cit.\n\n53. Malcolm-Smith, Susan, Anne Uhlmann, Anthony Hodge & Jonathan Ipser: 'Affective neuroscience of methamphetamine abuse'. In Substance use and abuse in South Africa by Dan Stein, George Ellis, Ernesta Meintjies and Kevin Thomas (eds). UCT Press Cape Town, 2012, p146.\n\n54. Ibid.\n\n55. It is for this reason that increased punishment yields little benefit. Once a drug is made illegal, there is a point beyond which increases in enforcement and incarceration yield little added benefit. National Drug Master Plan 2013.\n\n56. Interview with Jonathan Jansen, Fusion Manenberg, 2014. This has echoes of the notorious 'dop' system, where workers on wine estates were partly paid in wine, which served to 'addict' them to farm labour.\n\n57. Shedler, Jonathan & Jack Block: 'Adolescent drug use and psychological health: A longitudinal inquiry' (American Psychologist, Vol 45(5), May 1990, pp612-630.\n\n58. Mate, Gabor: In the Realm of Hungry Ghosts: Close Encounters with Addiction (North Atlantic Books, California, 2010) p200.\n\n59. Ibid, p201.\n\n60. Alexander, Bruce: Addiction: The View from Rat Park (www.brucekalexander.com\/articles-speeches\/rat-park\/148-addiction-the-view-from-rat-park). See also The rise and fall of the official view of addiction (www.brucekalexander.com\/articles-speeches\/277-rise-and-fall-of-the-official-view-of-addiction-6). An excellent comic-art description of Alexander's experiments by Stuart McMillen: Rat Park, is at www.stuartmcmillen.com\/\u00adcomics_en\/\u00adrat-park\/#page-35.\n\n61. Hari, Jonathan: Chasing the Scream: the First and Last Days of the War on Drugs (Kindle edition, 2015).\n\n62. Panday, S, Chitra Ranchod, Busani Ngcaweni & Soraya Seedat: 'The situation of the youth in South Africa'. In Youth Violence: Sources and Solutions in South Africa, p102.\n\n63. Africa Check, 25 March 2015.\n\n64. Daily Maverick, 7 January 2014.\n\n65. Dropout is also considerable in higher education, with almost 30% of first-time undergraduates dropping out after only a year and 20% dropping out before completing their degree. Department of Education Report 2005.\n\n66. Africa Check, 13 January, 2015.\n\n67. Mandy de Waal: 'Angie Motshekga's education claims - the true score'. Daily Maverick 19 February 2014.\n\n68. In the Western Cape 48% did not reach matric. Western Cape Development Strategy 2013 report, op.cit.\n\n69. Kate Wilkinson: 'Why the matric pass rate is not a reliable benchmark of education quality'. Africa Check (www.africacheck.org) 7 January 2014.\n\n70. Daily Maverick, op cit.\n\n71. South Africa Futures 2030 - How Bafana Bafana made Mandela Magic (Institute for Security Studies Paper 253, February 2014).\n\n72. Trends in International Mathematics and Science Study 2011 (www.hsrc.ac.za\/uploads\/pageContent\/2929\/TIMSSHighlights2012Dec7final.pdf).\n\n73. Southern and Eastern African Consortium for Monitoring Educational Quality Survey 2007, quoted in Africa Check, March 2015.\n\n74. Data modeled from the GHS for the DoE indicate that some 85% of pupils reach Grade 9. Then drop out increases: 11% of those who start Grade 9 drop out; 16% of those who start grade 10 leave; 24% who started grade 11 dropped out. Source: Department of Basic Education (2008).\n\n75. Western Cape Youth Development Strategy 2013 (Western Cape Government).\n\n76. In 2008 the figure was 15.3%. Ibid p25.\n\n77. Gevers, Anik & Alan Flisheer: School-based youth violence prevention interventions. In Youth Violence: Sources and Solutions in South Africa, op cit, p178.\n\n78. Kossie, op cit.\n\n79. School violence in South Africa. 2012 National School Violence Study\n\n80. School violence in South Africa, ibid.\n\n81. Corporal punishment was banned in schools in 1996 but is still fairly widely practiced by teachers as a method of class control.\n\n82. Half of all pupils surveyed in the 2012 National School Violence Study said they had been caned at school by teachers or the principal.\n\n83. Emler, N & Reicher, S: Adolescence and Delinquency (Blackwell, Oxford 1995) p158.\n\n84. Ibid, p159.\n\n85. Reicher, S & Emler, N: 'Delinquent behaviour and attitudes to formal authority', British Journal of Social Psychology, 3, pp161-8, 1985.\n\n86. The rise of TV everywhere audience report (Discovery networks) and All Media Products Survey 2014.\n\n87. Kreutzer, Tino: Generation Mobile: Online and Digital Media Usage on Mobile Phones among Low-Income Urban Youth in South Africa (MA thesis, University of Cape Town, 2009).\n\n88. Garbarino, op cit, p108.\n\n89. Bushman, BJ & Huesmann, LR: 'Short-term and Long-term Effects of Violent Media on Aggression in Children and Adults' (JAMA Pediatrics 2006) pp348-352.\n\n90. Ibid, p351.\n\n91. Stadler, J: 'Screen media violence and the socialization of young viewers', in Ward, C, et al: Youth Violence (2012) p320.\n\n92. Ibid, p332.\n\n93. Jones, D: Killing monsters: why children need fantasy, super heroes and make-believe violence (Basic Books, New York, 2002).\n\n94. Ibid, p11& 31.\n6. Towards resilience\n\nOne of the most pervasive comments from people I talked to about this book was: 'You'll never be able to solve the gang problem.' It was as if I'd suggested the relocation of Table Mountain. Impossible! I disagree. There's much that can be done. But the first step is that the country, and specifically Cape Town, needs to want to fix the problem. And we need to be realistic about timeframes. Expecting too much too soon will just leave people exhausted and disheartened. We need to pace this process.\n\nWe've seen how, like the ripples from a stone dropped into a pool, the issues around gang formation spread back in time and out beyond the margins of family, neighbourhood, city and country. Any attempt at solutions needs to acknowledge its own boundaries as well as the constraints within which meaningful change is possible. Those involved in international organised crime and syndicates as well as members of gangs within prisons are part of this story. But solutions to these problems rest mainly with Criminal Justice and Correctional Services and are beyond the remit of this book which focusses on adolescent gang membership. There are also certain givens that are with us for the long haul and, like a giant container ship at speed on the high seas, will take considerable time and space to alter course:\n\n\u2022 The level of urbanisation in the developing world suggests that Cape Town will continue to grow in population and structural density;\n\n\u2022 The failure of the international War on Drugs and our inability to curb their sale confirms that drugs have become socially embedded in the lives of many young people, most of whom live in the city's low-income periphery;\n\n\u2022 Policing of transnational or local organised crime, money laundering and smuggler syndicates will continue with varying success;\n\n\u2022 Poverty will, seemingly, always be present.\n\nThese issues are the context within which this study was undertaken and intersect with the lives of young people in high-risk, low-income areas. Within these multiple frames, young people strive for identity in whatever way they can achieve it. Joining a gang is one of those ways. Distilling the elements of gang formation, we can tick off what we know:\n\n\u2022 They're an urban phenomenon found in most cities throughout the world where there's crowding and low income;\n\n\u2022 They tend to occur within particular types or urban structure \u2013 tenements, low-cost neighbourhoods and squatter areas not far from wealthy neighbourhoods;\n\n\u2022 They generally form in areas of relative, but not absolute poverty;\n\n\u2022 They are mainly a male, adolescent phenomenon;\n\n\u2022 They are often school dropouts;\n\n\u2022 Identity and image are of paramount importance for gang members;\n\n\u2022 Their members tend to have issues of parental attachment;\n\n\u2022 Their members might have mental and physical health issues;\n\n\u2022 Drugs are invariably involved;\n\n\u2022 Gangs are often, but not always, linked to a criminal economy;\n\n\u2022 Gang activity can involve weapons and violence;\n\n\u2022 Gangs are an impediment to community and personal resilience.\n\nThough it may seem difficult, given the levels of violence, we need to remember that most gangsters who terrorise neighbourhoods are still, essentially children. To use James Garbarino's term, they are often traumatised and generally 'lost boys':\n\nThey do many things in a failing attempt to cope with their trauma, but rarely do they have the resources or knowledge to cope effectively and in a socially acceptably way. They take drugs. They engage in violence. They steal. They gorge themselves on sex. They join gangs and cults. And when no one is watching or listening to them, they suck their thumbs and cry themselves to sleep.1\n\nIn searching for solutions, we need to know who and what to target, when to act and where such actions would have maximum effect. We should attempt both to alter the trajectories of adolescence-limited delinquents and the preconditions that give rise to life-course criminal persistence. While the latter needs to be addressed at the level of policy, policing and parenting, adolescence-limited delinquency requires engagement at the level of programmes, path, education and empathy. Dealing with adolescence-limited behaviour necessitates the strengthening of personal resilience. Life-course-persistent problems require long-term support work on family and neighbourhood resilience. Each of these feed into the other, of course, but the remedial action is in each case very different. The guiding and essential question is, in all cases: What can we do to help young people live meaningful, resilient lives in environments that favour the development of gangs, crime and violence? What follows is my attempt to answer this question.\n\nPolicies and more policies\n\nIn any national or regional context, 'doing something' involves altering or developing a policy framework within which change can be negotiated or enforced. Across a disturbingly wide spectrum, the various administrations tasked to protect and create opportunities for young people in South Africa have failed to get traction. In this we continue to be deeply unkind to many young people.\n\nThe word 'kind' comes from the Indo-European kunne meaning kin or family, so 'unkind' quite literally means 'without family'. Large numbers of young people in poor neighbourhoods form gangs precisely because they have been un-familied by society. So when we look for solutions, the first step has to be reparation. We have to make amends and restore rights due to young people under the Constitution. These are the right to family and all the things that family implies: care; shelter; a safe environment; adequate nutrition; a useable education and human dignity. These rights are not merely in place to be extolled. They are a constitutional requirement for the development of all young people.\n\nSouth Africa has excelled in thinking about its youth problem. Since 1994, extensive discussion has taken place regarding young people. On paper, at least, the country has some of the most advanced policies in the world. A rough count turned up 32 Acts; White Papers; Programmes of Action; Declarations; Guidelines and Policy Frameworks formulated between 1993 and 2015 dealing with children or youths.2\n\nThe state's primary duties to young people are enshrined in the Constitution; protected in the Children's Act (which deals with child protection and support, parenting, paternity, custody and guardianship); the Child Justice Act (dealing with assessment, detention and diversion of young people in conflict with the law); the Maintenance Act (enforcing financial support); the Schools Act (supporting the rights of learners, teachers and parents) and the Early Childhood Development Policy (in draft form at time of writing but hopefully forthcoming). The problem is the gap between plan, promulgation and enactment. So a first step is to ensure that plans are implemented and national policies adhered to.\n\nRethink early child development\n\nThe obvious and critical place to begin is outlined in the Early Childhood Development Policy (ECDP) draft document of March 2015.3 If it passes into law and is effectively acted upon, it will be the most important step the country has ever taken to protect and support parents and young children. It would ensure and secure resilience in children and youngsters and lead to a reduction of gang activity.\n\nThe draft document takes a human rights approach. Its preamble is a commitment and acknowledgement of the country's responsibility to 'guarantee the universal availability of, and equitable access to, a comprehensive package of quality early childhood development services for all young children from conception until they enter formal schooling.'4 It recognises that the development of South Africa depends on the extent to which it can unlock the potential human capital within its youngest population. Young people, it states, have a right 'to become physically healthy; mentally alert; socially competent; emotionally sound and ready to learn cognitively; socially; emotionally; physically and psychosocially to their full potential.'5 These are absolute and undeniable rights in terms of the Constitution.\n\nThe ECDP acknowledges that certain factors can be highly damaging to the physical development of a foetus. These are, as we have seen, a mother's general health and nutritional status; toxic stress and her use of alcohol and legal or illegal drugs. It acknowledges that these factors can cause long-term brain and cognative damage. It notes that the first two years after birth are critical for healthy growth and development and that nutritional and attachment problems can limit mental and physical development. The policy proposes implementation of a package of services by 2024 which will:\n\n\u2022 Support pregnant women and children under two years of age by promoting healthy pregnancy and providing maternal psychosocial support where needed;\n\n\u2022 Support parenting aimed at love, care, security and responsiveness to children, as well as strengthening cognitive, language, psychosocial and sensorimotor stimulation of the child;\n\n\u2022 Promote at least two home visits by professionals during pregnancy and bi-monthly visits until a child is nine months old;\n\n\u2022 Resource adequate delivery of early child development (ECD) services;\n\n\u2022 Provide pregnant women and parents with the information and skills necessary to be better parents;\n\n\u2022 Help parents avoid the use of harsh discipline;\n\n\u2022 Provide parents with specialised help in dealing with domestic violence, substance misuse, mental health problems, abuse or neglect and disability;\n\n\u2022 Provide parents, primary caregivers and child minders with tools to stimulate a child's language development, imagination, curiosity and critical thinking;\n\n\u2022 Enable parents to understand the importance of early childhood development and how to improve their children's nutrition, health and well-being;\n\n\u2022 Promote parent-child interaction; and\n\n\u2022 Build an understanding of the roles of mothers and fathers, recognising the high proportion of absent fathers in the lives of their children.\n\nThe Human Sciences Research Council study that underpinned the ECDP draft document proposes four key steps necessary to implement efficient early child development. These are:\n\n1. To implement an Essential Package of ECD services;\n\n2. To establish a national management system and infrastructure to improve early child development;\n\n3. To plan for and develop the human resources needed to make the Essential Package universally available;\n\n4. To align public financing and administrative systems to support national ECD priorities.\n\nThe cost of this has been placed at around R17 billion at 2015 rand value. This can be considered against South Africa's 2015 police budget of R82.7 billion. ECD imiplementation represents just 20 per cent of this budget and would radically cut the cost of youth crime in the long run. However, even if the Early Childhood Development Policy is approved, roll-out is not assured. The allocation of funds, a shortage of skills and bureaucracy have a way of blocking the best ideas. Given the skills shortage in South Africa, continuing to allocate jobs on the basis of race quotas rather that the best person for the task is a further problem. Faltering on ECD implementation would be a tragedy.\n\nTogether with the effective implementation of the Child Justice Act and a progressive rethink of the education system, ECD could form a strong base from which to build a new country. It would produce children who are healthy in every sense of the word and youths whose default does not have to be crime and violence. The resultant saving on policing costs alone would be significant. There would be savings in the budgets of the prison services, the courts, health services and private security. The resultant drop in crime rate would boost international investment and tourist confidence. The HSRC's depiction of the potential for improved life fulfillment as a result of early child development can be shown graphically.\n\nFigure 9 Life fulfiment trajectory\n\nThe value, particularly, of home visits, was highlighted by nurse-family partnership programmes which were piloted by pediatrician and psychiatrist David Olds in New York in 1977 and rolled out to 25 American states. Olds was motivated by studies showing that youths who were maltreated in childhood reported high delinquency with a 96 per cent increase in violent crime over children in caring families. He initiated a process to improve pre- and postnatal care through nurse home-visits. The programmes focus on first-time mothers and aim to increase social networks around them while enhancing family environment and education. The nurses try to build empathetic and trusting relationships and increase mother-child attachment. For the first six months after birth, families receive weekly visits. The visits become bi-weekly until the child is 20 months old, then monthly until the child's second birthday.\n\nThe programmes were carefully monitored and researchers compared outcomes between those who were visited and those who were not. They found the women who were home-visited had fewer pregnancies, longer intervals between births, fewer children overall and spent longer in relationships. Long-term research found that, after 15 years, youths in low-income neighbourhoods who had been nurse-visited as children accounted for fewer arrests, convictions or runaway instances. They had fewer sexual partners and consumed less alcohol and drugs than children of women in comparison groups. They also had higher intellectual functioning, higher vocabulary scores, did better at school and had lower levels of aggression.6 In 2007 President Barak Obama outlined the value of such programmes to his country:\n\nIt's common sense to reach out to a young mother. Teach her about changing the baby. Help her understand what all that crying means and when to get vaccines and check-ups. This program saves money. It raises healthy babies and creates better parents. It reduced childhood injuries and unintended pregnancies, increased father involvement and women's employment, reduced use of welfare and food stamps and increased children's school readiness. This works and I will expand the Nurse-Family Partnership to provide at-home nurse visits for up to 570 000 first-time mothers each year. We can do this.7\n\nCape Town had a system of nurse visits, but abandoned them. According to the city's health director, Ivan Bromfield:\n\nCape Town used to have a service where every mother who was pregnant had a nurse visitor assigned to her who would talk about pregnancy and find out about any problems. But a curative approach overtook a preventive approach. The emphasis became treating patients at clinics. Preventative effect was long-term and not easily measured and the curative side was immediate. That put pressure on the preventative side. So we lost home visits.8\n\nStrengthen community resilience\n\nFear for personal safety is one of the main inhibitors of neighbourhood agency. It's the foremost issue considered in school planning; after-school programmes; area upgrading; public transport; leisure amenities and even trips to the shop. In this respect, my research on the older, pre-removal neighbourhoods of District Six, Wynberg and Claremont, suggest an unusual starting point in healing a neighbourhood: grandparents.\n\nIn any functioning neighbourhood, grandparents are the social anchors. They generally have the time to maintain personal contacts, socialise with pre- and after-school children and be around to provide the surveillance necessary to maintain community safety. Despite poverty and overcrowding, places like District Six, Woodstock, Salt River, Elsies River and both Claremont and Wynberg 'below the railway line' were such neighbourhoods.\n\nAs we have seen, in the apartheid estates built in the 1960s and 70s, grandparented families were impossible, mainly because these places were not planned to structurally support larger, more complex families nor were built to facilitate social surveillance. There was simply no room for grandparents. If the urban planners of the time had engaged with the families being moved and acted on their suggestions, they would have built extended-family units with verandahs. They would have maintained the established social linkages and created public spaces with safety in mind. Cape Town community psychologist, Lane Benjamin, underlined this approach:\n\nWe need to build intergenerational links and we need to put back together extended families. Bring in the grandparents and stitch together a continuous narrative of family history beyond the present. We need housing systems that support that.9\n\nAn attempt to rectify the problem of damaged neighbourhoods began in 2005 when the City of Cape Town and the Western Cape government established the Violence Prevention through Urban Upgrading (VPUU) partnership in an effort to retro-fit existing neighbourhoods and reduce crime. Its aim was to plan for safety and improve the quality of life in low-income areas through consultative planning. Its strategy was to transform apartheid dormitories into sustainable, safe neighbourhoods through ongoing negotiation between local authorities and the people who lived there. By creating grandparent and child-friendly places and spaces, it was planned as a counter to syndicate control and gang formation.\n\nThe thinking was that social disorganisation is a consequence of neighborhoods, not individuals, and that crime is a characteristic of social disorganisation. As we have seen, in neighbourhoods structurally and socially underdeveloped over time, or where friendship networks and informal surveillance systems have been destroyed as in Group Areas relocations, power dynamics favour those who are prepared to exert social control through violence. VPUU planner, Alastair Graham, outlined the steps to 'taking back the neighbourhood:\n\nNext to Nyanga Junction there's an area called The Downs, the main gang area. There's very little private and semi-private space and it's very hard to fix that through design so we plan to demolish 550 houses and rebuild. We want to recreate it as a new town centre including a hospital and Metro Police headquarters. We'll use schools as investment points, linking nine of them by a pedestrian route with good lighting and walkways.\n\nWe're going to relocate the families in houses built along that route with single title ownership \u2013 essentially framing the walkway. These will facilitate passive surveillance. We need to relocate and rebuild around extended families so there's interaction between young and old people. It's about positively occupying dangerous space in partnership with the community. You need to create safe spaces.10\n\nThe VPUU undertook smaller upgrade projects in Khayelitsha with some success. At the time of writing, the Manenberg project was still in planning stage. Such retro-fitting in gang-dominated areas, however, can encounter pitfalls. In Nyanga in 2014 the City contracted a construction company to do urban upgrading. The Hard Livings gang 'taxed' them R20 000 a month for 'protection' with the city unknowingly funding the deal. When the next phase of the construction shifted to the adjacent Ghetto Kids territory, the contractor came to the same arrangement. A fight broke out between the two gangs over protection money and in less than four months there were 63 shooting incidents and 27 killings. According to Graham:\n\nIt was a hard lesson. On any construction project by City or Province, the gang is on their doorstep with the going rate for protection. Gang-owned construction companies are even registered on the City's database. So we have to figure out how to tighten up the tender documents so this doesn't happen again.11\n\nHarvard sociologist Robert Sampson identifies the key to rebuilding social resilience in wounded neighbourhoods as 'collective efficacy'. This is the ability of the community to produce demonstrable results from its intentions.12 Such efficacy, which can last many decades despite the turnover of residents, is embedded in pro-social community institutions. Collective efficacy can be undermined by structural disadvantage and fatalism about transformation as well as by moral cynicism about crime. Collaborative and durable 'spatial logic' such as that of the VPUU project offers a structural platform upon which to rebuild and embed such efficacy. It would be impossible to keep criminal syndicates out of this reconstructive loop. The question of whether to include them while preventing them from subverting the process with possible murderous results would no doubt create a political hot potato. Graham says:\n\nIf we can get something going in Manenberg and Hanover Park we're onto something that can roll. We're unlikely to have a perfect plan but we have to start. You need to have confidence in what you do. And of course we must make sure the money doesn't leak away.13\n\nFollowing my extensive discussions with communities, NGOs and city officials, it's possible to distil into nine points what they considered to be the prime anchors of neighbourhood resilience:\n\n1. The creation of well-lit corridors of safety between schools;\n\n2. Flanking these with extended-family housing to permit the inclusion of grandparents and other family members;\n\n3. Ensuring the construction of open verandas that overlook the corridors;\n\n4. Stimulation of formal ongoing, timetabled family and particularly grandparent surveillance of these corridors;\n\n5. Generation of any form or pro-social neighbourhood groups promoting face-to-face associations to strengthen community bonding;\n\n6. Supporting culturally-driven, traditional approaches to governance;\n\n7. Development of effective local systems of democracy through which people can exercise their rights and express their grievances;\n\n8. Support of institutions such as churches, schools, sport and youth activities to assist in building stronger and more cohesive communities;\n\n9. Improve policing.14\n\nPolice gang specialist Jeremy Vearey underlined this last point regarding the importance of community resilience for policing:\n\nI see it as important as part of normal policing for community structures to exist, to agitate for better conditions. A street committee, flat committee, shack committee or any structure at grassroots level, before they start talking about other issues, one of the fundamental parts of the job would be to agitate towards improving the environment to support effective policing. Civil struggles around normal basic service delivery are important for policing. We must empower people to deal with social issues to fight crime as well as the material issues which generate the risk of it.15\n\nRethink education\n\nEducation is, at root, the result of policy decisions. Poor decisions have led to a 'dumbing down' and simplification of the curriculum in South Africa. Despite this cutting back of knowledge, as we have seen, schools continue to lose around half of all pupils before they reach matric. These dropouts fail to get into higher education. They don't easily find work and swell the ranks of youngsters hanging around on the streets where trouble easily finds them. A transnational education survey has noted that South African schooling is large measure dysfunctional.16 A rethink of the education system is urgently required.\n\nWe need to carefully assess whether the national curriculum is appropriate for all young people or just those likely to succeed. This is a controversial question, because it's generally accepted that all young people in a democracy should have equal access to equal education. But what if the curriculum selects for qualities required in a Western neoliberal economy and not for innovative survival in an impoverished community within a developing country?\n\nChildren who start school in 2016 will retire in 2075. Nobody has a clue what the world will look like then or what skills they'll need in the course of their lives. Yet we plod on with our hierarchy of subjects: mathematics and languages at the top, then the humanities and arts and craft skills at the bottom. It's a system focusing, as the educator Sir Ken Robinson points out, from the waist up:\n\nAs children grow up... we focus on their heads. Academic ability has come to dominate our view of intelligence. The consequence is that many highly-talented, brilliant, creative people think they're not, because the thing they were good at in school wasn't valued, or was actually stigmatised. Our education system has mined our minds in the way that we strip-mine the earth: for a particular commodity. And for the future, it won't serve us. We have to rethink the fundamental principles on which we're educating our children.17\n\nWhen youths from low-income, high-risk areas of the city look around them, or when they consider the experiences of their parents or older relations, they are far more likely than their middle-class counterparts to see people in unrewarding jobs or without work at all. They may easily assume \u2013 and be correct in the assumption \u2013 that their own horizons are limited whatever they do at school. Middle-class existence will, conversely, provide examples of how good school results can make a difference. If young people feel they're being educated for the unattainable or that the innate skills they possess are not valued, this might be another reason for the high school dropout rate.\n\nThe South African education system holds to the Western view of what kind of knowledge is important. It tends to favour intellectual abilities over manual work, university over technical training. Through exams, awards and social status, it endows knowledge workers with greater rewards. To have physical or manual prowess or abilities is seen to be stupid by those deemed intelligent, who are themselves seen to be soft and nerdish by those for whom physical strength and manual skills are important. This bias has historical antecedents in the Industrial Revolution and later developments that separated managers from increasingly micro-skilled and replaceable production-line workers. People who made things earned less and came to be less regarded socially than people who organised to make things happen. In Mind at Work, American writer Mike Rose notes that:\n\nOur testaments to physical work are so often focused on the values such work exhibits rather than on the thought it requires... It is as though in our cultural iconography we are given the muscled arm, sleeve rolled tight against biceps, but no thoughts bright behind the eye, no image that links hand and brain.18\n\nToday these biases still inform the educational landscape, assuming that blue-collar work is mindless and that white-collar work is mental in character and the most desirable.19 There are two reasons to question this view. The first is that knowledge work is increasingly being taken over by computers and most offices now emulate factory assembly lines requiring more and more inputting and less creative or cognitive elements. This is a frustrating end to 12 years of schooling and three to five years of college. In an evocatively named book: The electronic sweatshop: How computers are transforming the office of the future into the factory of the past, Barbara Garson details how 'extraordinary human ingenuity has been used to eliminate the need for human ingenuity.'20 The second reason is that, in a globalised economy and standardisation of input and data platforms, work can easily be sent offshore to areas of less expensive labour. In an age where you can download your house plans or medical advice, the security of knowledge work is increasingly at risk.\n\nSkilled work and manual trades, on the other hand, are not threatened by the standardisation of knowledge work. You can't hammer in a nail over the internet. Princeton economist Alan Blinder has noted that the division of labour between white-collar and blue-collar workers in the 21st century has profoundly altered:\n\nMany people blithely assume that the critical labour-market distinction is, and will remain, between highly educated (or highly skilled) people and less-educated (less skilled) people \u2013 doctors versus call-centre operators, for example. The supposed remedy... is more education and general 'upskilling' of the work force. But this view is mistaken. The critical divide in the future may instead be between those types of work that are easily deliverable by wire (or wi-fi)... and those that are not. This divide does not correspond well to traditional distinctions between jobs that require high levels of education and jobs that do not.21\n\nThe crucial difference, he says, will be between 'personal services' and 'impersonal services', between face-to-face work and abstracted work. You can download almost any information you require, but that won't replace the guy who actually builds your kitchen cupboards, fixes a leaking sink or repairs your car. There's also a possibility that he earns more than you do. In short, knowledge work is no longer a rising sea that lifts all boats and this should be factored into the planning and underlying tenets of our education system.\n\nIn the light of the high school dropout rate and critical levels of unemployment, it's therefore necessary to call for an educational rethink. It's obvious that young people should be taught to read competently, write well and be efficiently numerate to make their way in either First World or developing economies. What needs careful thought are the subjects that should be taught beyond those skills. Before 1994, most apprenticeships were white and were linked to training at technical colleges. This system folded in the 1990s and in 1998, in an attempt to rethink practical training, all technical and training centres were merged to form about 50 Technical and Vocational Education Training (TVET) colleges. However the link with industry was damaged in the process.\n\nA 2014 report found that more than half the learners were getting no work experience at all and placement in industry was extremely low. This was because 'colleges are in reality not effectively managing the development of practical skills, either in the workshops or in workplaces.'22 Education analyist Andre Kraak described partnerships between colleges and industry as a mess. Under the apprenticeship system, he says, students did their practical training in the workplace. After it was phased out that collapsed.23\n\nAristotle rightly noted that 'all human beings by nature desire to know,' and spent much of his life pondering the nature of knowledge. We need to follow his example. If the end task of education is to supply institutions with suitable workers, if its point, furthermore, becomes the production of credentials rather that the cultivation of skills which empower individuals, is that the sort of knowledge we wish for our children?\n\nThe receipt of a school or college certificate is satisfying as a doorway into life possibilities, but the ability to create or repair something on the basis of your understanding of the material world evokes a different order of satisfaction and a dissimilar approach to knowledge. As the French philosopher Alexandre Kojeve writes:\n\nThe man who works, recognises his own product in the world that has actually been transformed by his work: he recognises himself in it. He sees in it his own human reality. In it he discovers and reveals to others the objective reality of his humanity, of the original abstract and purely subjective idea he has for himself.24\n\nThis recognition engenders self-esteem, an ability to prove yourself to others in relation to a task you can point to and handle in three-dimensions. You have a tangible \u2013 and marketable \u2013 manual skill. Whereas knowledge work offers a somewhat socially abstracted and delayed gratification, craft skills provide a tangible identity; an object or action that can be pointed to and boasted about if necessary (with teenagers it's always necessary). Craftwork may take time and patience to accomplish, but its outcome provides immediate and ongoing gratification. Being adept in skilled manual work allows a person to say 'I am' with confidence: a carpenter; a builder; a dressmaker; a chef. This 'I am-ness'is precisely the type of recognition and status that youngsters seek from joining a gang. Of course, all skill requires knowledge, but what differs is the pedagogical approach to their acquisition. The relationship between the two was neatly captured by St Francis of Assisi:\n\nHe who works with his hands is a labourer.He who works with his hands and his head is a craftsman. He who works with his hands and his head and his heart is an artist.25\n\nNot all young people have the same goals, ambitions, drive or expectations of school. Depending on these differences, some will find it congenial; future knowledge workers in a system designed for knowledge workers. They will work well in teams of equals in pursuit of common goals. Others will find that its underlying assumptions make little sense to their skills, interests future and will consider it unbearable. They deal with the world primarily with their hands and their strength and may gravitate to hierarchies of power and influence. These could be gangs, but they could also be crews in which regard is given in terms of skill and experience. For these young people, skill-based subjects that offer the sort of experience which gives them identifiable mastery will also provide the resilience they need to resist the undertow of gang life.\n\nBuild better after-school care\n\nIn high-risk neighbourhoods, free time is danger time. For young people at school, this means weekends and between the end of school and nightfall. For this reason, in 2014, designing successful after-school programmes was flagged by Cape Town as a potential game-changer to risky adolescent behaviour and was explored in a series of community inclusive workshops. Their goal was to work out ways to reduce vulnerability and risk-taking behaviour and increase health and educational outcomes of learners (the workshops did not deal with school dropouts).\n\nThe search for the key to youth participation in these programmes was understandable. Nearly 50 000 learners were attending the province's 181 flagship Mass Participation, Opportunity and Access, Development and Growth (MOD) programmes, there were 16 Year Beyond (matric) programmes; 153 after-care facilities; 40 youth hubs; four youth cafes and 56 community centres; 100 city libraries and dozens of NGOs working on the problem. Very few of these initiatives were seen to have positive, long-term outcomes. The city was investing considerable energy, capital and goodwill with disturbingly diminishing returns. The problem, it seemed, was not provision of services but young people's willingness to use them. The causes were seen to include:\n\n\u2022 Poor school retention rates of learners;\n\n\u2022 Lack of soft skills;\n\n\u2022 Dangerous behaviour paths;\n\n\u2022 Dangerous health patterns such as nutrition, substance abuse and risky sexual behaviour;\n\n\u2022 General lack of a sense of belonging;\n\n\u2022 Lack of culture of participation;\n\n\u2022 Lack of personal, facility and street safety;\n\n\u2022 Location of services and learner movement patterns;\n\n\u2022 School management and how programmes are integrated with school;\n\n\u2022 Lack of physical education in schools;\n\n\u2022 Uncoordinated and poorly communicated programme provision;\n\n\u2022 Poor record keeping and database;\n\n\u2022 Few demand driven incentives and correspondingly low involvement by youths;\n\n\u2022 The perceived lack of value in programmes offered;\n\n\u2022 An uneven spread of services;\n\n\u2022 Sustainability of programmes;\n\n\u2022 Lack of parental support.\n\nWhat was working extremely well \u2013 but with limited numbers \u2013 were a few sport-based NGOs such as AmandlaFootball, Waves for Change, Open Streets and the National Skateboard Collective. Their success, apart from a good organisational framework, was that their goals went beyond teaching skills and focused on the personal transformation of each individual participant. They positioned themselves in such a way that attendance was and is considered to be cool. We've already discussed cool, but it's worth exploring further.\n\nIf cool works, what are its constituents? It has to involve what's exciting (maybe a bit scary); looks good; stimulates peer approval; involves cool stuff; isn't dictated by adults and attracts the opposite sex. The paradox is that it needs to be demand-driven, but most adolescents in high-risk, low-income areas are at a loss over what to demand. And in the absence of pro-social guidance, the default result is generally what the formal adult world describes as delinquency. Doing bad is cool and gangs provide a vehicle.\n\nBuilding a pro-social after-school process is rather like the collaborative construction of a house. The kids need to make the bricks and supervise each course as the walls progress, but without some competent bricklayers the walls will go skew and without a site manager the cement, windows, joists and roof won't be delivered in time. But with too much input from the bricklayers and manager, the kids won't walk through the door when the house is complete. If the house looks and feels like school, they'll break it down and leave the manager with a face-full of dust. The point of the metaphor is that young people require adult assistance but of a very particular sort. They desperately need (because they probably have never before had) an adult who they can trust. Someone who listens but does not instruct, offers ideas only when asked, moves aside the clutter so the path is clearer and is emotionally stable and able to empathise.\n\nAn after-school game changer requires a conceptual reset, a list of principles against which all such programmes can be measured. These would require that each after-school programme delivers the constituents of cool, an unobtrusive, mentored collaborative framework and provide individual emotional support. Here's a suggestion for such a tool:\n\n\u2022 If an external agency conceptualises the programme, its development at after-school level must be crafted together with pupils;\n\n\u2022 In developing the programme, pupils must have a mentor who they respect and who facilitates but does not direct;\n\n\u2022 The programme must have rituals\/structures that both enable and contain;\n\n\u2022 Those pupils who lead the programme (and eventually all pupils in the programme) must be publicly valued for their contribution;\n\n\u2022 All participants must undertake difficult or daring tasks for which they are acknowledged and admired by peers and adults whom they respect;\n\n\u2022 All participants must have time and a safe space for reflection and personal recalibration;\n\n\u2022 All programmes must be cool.\n\nRethink the War on Drugs\n\nEarlier chapters have shown that drugs largely drive Cape Town's stratospheric level of interpersonal violent crime. Users rob and steal to get them, gangs murder to retain their sales turf and drug lords hold neighbourhoods in thrall by violence. There is a solution to this, but it would take a brave and resolute government to implement it.\n\nFirst, though, here's a necessary backstory about the so-called War on Drugs involving Harry Anslinger, the former head of the USA Federal Bureau of Narcotics who started it in the 1930s. Anslinger became obsessed first with the Mafia and then with opium. He claimed China was smuggling it into America to undermine the country and soften up teenage girls for sex with Chinese dealers. He considered all Afro-Americans to be a criminal threat. In the 1950s, after befriending Senator Joseph McCarthy known for his 'Reds' witch-hunt that led to crippling smear campaigns against thousands of Americans, Anslinger widened his attention to communism.26 His primary tool was the Harrison Act of 1914 and he dramatically expanded the size and reach of his Bureau using little science and considerable scare tactics. By the 1950s the Bureau had sufficient clout to threaten with trade sanctions all countries not altering their legislation in line with the US War on Drugs. South Africa's drug legislation is a product of this history.27\n\nAnslinger was well aware that alcohol prohibition had failed in America and he must have known why. The banning of anything the public desires simply hands supply to the criminal underworld. This ensures that the state loses control of the situation and, without quality controls, often leads to a bad, even dangerous, product. Before his campaign, drugs could be bought over the counter and only a small percentage of the population used them, mainly recreationally. Addiction was low.\n\nThe result of the War on Drugs was that, almost overnight, the crackdown on the legal purchase of cocaine, heroin and cannabis conjured into existence criminal cartels and drug smugglers. Prices went up and a new a business model was created that required intensive drug pushing and profited from (and promoted) addiction. Every addict became a potential customer and a cash cow. Anslinger claimed he was fighting the Mafia, but in fact he was transferring what became a hugely profitable drug industry into their exclusive control.28 By 2010 the global illicit drug market was valued at over $300 billion a year.29\n\nIn his book, Chasing the scream: the first and last days of the war on drugs, Johann Hari suggests that the Mafia needed the War on Drugs so badly they bribed officials to intensify it. By criminalising drug use and sending thousands of users to jail, Anslinger's campaign made it inevitable that they would emerge with criminal records. They would be unable to find legitimate jobs and would invariably beg, borrow, steal and deal to secure a fix and survive. The massive revenue that dealing illicit drugs earned druglords allowed for the buying off of police and politicians. It also increased levels of aggression because druglords \u2013 who don't want a shootout every day \u2013 tended to establish a reputation for being excessively violent so people didn't pick a fight with them. They instituted a reign of terror and 'owned' neighbourhoods.\n\nThis situation pertains worldwide. In Cape Town you find a gang protecting street corner drug sales. Beyond them is a syndicate importing drugs, a network supplying them, drug mules carrying them, another syndicate controlling their production in a foreign country, a gang protecting the supply to that syndicate and a farmer illegally growing opium or coca. Each step encourages corruption and violence and at each transaction money changes hands \u2013 ultimately huge amounts. Drug smuggling is one of the most profitable trades in the world, largely because of Harry Anslinger and his War on Drugs.\n\nIs there an alternative to that war? An experiment began in Portugal in 2001 when the persecution of drug users and addicts came to an end. A new law (Law 30\/2000) stipulated that recreational drug users should not be marginalised, labeled or imprisoned and addicts were encouraged to seek treatment. The law did not make it legal to sell or traffic drugs, it simply no longer considered possession to be criminal. Drug use did not skyrocket as predicted. Instead, addiction stabilised, prison population dropped and police are now able to attend to serious crime.30\n\nThe Portuguese government accepted that drugs and drug use were not going to 'go away.' They recognised that people at risk of entering the drug world should be given internal tools \u2013\u00ad confidence, knowledge and support \u2013 to make the right decisions for themselves. Street crime and violence have declined. The country now has one of the lowest levels of drug use out of 28 European countries and is the only country in Europe to have exhibited declines in problematic drug use. A Lisbon policeman told Hari:\n\nThe things we were afraid of didn't happen. Using drugs is only a symptom of some suffering and we have to reach the reasons that make addicts want to be out of their heads. We have to work on the trauma in your life and only then can you change the way you deal with it.31\n\nIn the United States, 90 per cent of the money spent on the drug policy goes to policing and punishment, with 10 per cent going to treatment and prevention. In Portugal the ratio is the exact opposite.32 A Portuguese drug user told Hari: 'Here addicts are not marginalised. They are just like a traffic accident. They are not on the other side of the line. They are regular citizens. They have a problem.'33 The lesson is simply this: strengthening people's internal resistence to drugs works a lot better than using force in an attempt to terrorise them away from drugs.\n\nThe Netherlands decriminalised the use of cannabis in 1976. The possession of a maximum amount of five grams for personal use is not prosecuted and cultivation is treated in a similar way (cultivation of five plants or less is usually not prosecuted). In California, during the time it was possible to obtain cannabis easily, traffic accidents fell by eight per cent because people shifted from alcohol, which is more dangerous for driving.34\n\nUruguay legalised the growing and sale of cannabis in 2013 and eight other South American countries are considering loosening their drug policies.35 People over the age of 21 who produce a valid ID in Uruguay are allowed to buy cannabis, but the decision about who can sell it was still under discussion at time of writing. Equador is considering legislation to decriminalise drug use and Chile allows cannabis to be grown and used for medical purposes.\n\nThese moves to decriminalise drug use are way-stations on the road to a solution. The bigger step is to legalise drugs and treat their use as a health problem and not a crime problem. Given the hysteria and propaganda around the War on Drugs, despite its obvious failure, it's difficult to think this through rationally without raising the spectre of addicted children and tik babies. But a first step is to admit that our escalating problems of drug use, and the extreme damage this is doing to our young people, is not being solved through prohibition. South Africa has usable legislation already in place regarding tobacco and alcohol so it's time to think more clearly.\n\nWhen you legalise drugs, the first thing that happens (as long as the state doesn't set the tax too high) is that you drive down the profitability of drug cartels, weaken the networks and drain money out of neighbourhood syndicates. You stop the turf wars, make drug use less cool and more mundane and free up policing for more serious crime. Government will be able to budget for better family support where the problem starts in the first place. Many addicts who presently get worse behind bars will get better in hospitals and then in jobs. Billions of rands would be saved on the detection, arrest, court appearances and imprisonment of drug dealers and billions would be gained on taxing drug use as is done on alcohol and tobacco. This revenue could be channelled into curative and supportive agencies and information campaigns on the dangers of drug use.\n\n'If you have options, you can experiment [with drugs] and leave them,' says Cape Town youth policy strategist Brian Ford. 'But if you experiment and there are no other options available to make you feel nice, you're going to stay there.'36 Lucille Meyer, head of Cape Town's Chrysalis Academy, agrees: 'We're living organisms. Show me a living organism that wants to self-destruct. Most kids I talk to, when I really dig, it's just about wanting to forget. Wanting to feel better.'37\n\nLeisure drugs of any sort taken in excess are not good for human consumption, but if the desire to be high is unstoppable, then it needs to be intelligently managed. Alcohol and tobacco are taxed, quality is ensured, there are age restrictions on sale and marketing, cigarette packets are obliged to carry health warnings. All these measures could be set in place for drug use.\n\nIs it likely that South Africa will legalise drugs? Not in the present international climate given the probable threat of US trade sanctions. But legalisation will happen eventually. When the American government's war on alcohol ended, the gangster war for alcohol stopped. All violence produced by prohibition disappeared and today it's impossible to imagine gun-toting gangsters selling whisky on a street corner. We'll look back one day and wonder at the social destruction that the War on Drugs caused and be amazed at how governments seemed unable to comprehend how prohibition fuelled one of the world's largest and most vicious criminal networks.\n\nRethink and reform policing\n\nMy father was a policeman and for much of my childhood he patrolled our town on his bicycle. He stopped to chat to shopkeepers, greeted the many people he knew and had a pretty good idea of what was going on around him. He was streetwise, honest to a fault and friendly, but he could also be pretty scary if you transgressed a boundary, as I discovered to my disadvantage on a number of occasions. He was, above all other things, a respected cop. Officials in blue like him created a sense of neighbourhood safety, a context within which other aspects of the community could take place.\n\nEffective policing \u2013 and his form of policing was effective \u2013 is not only a deterrent against crime and syndicate control, but also a framework within which community control can be reasserted. In those areas of Cape Town where it's most needed, however, there's a strong belief by residents that policing is extremely ineffective. Almost anyone on the Cape Flats will claim that the police are corrupt and 'run away' from trouble. While this is not true of all officers and all police action, there's sufficient proof to justify a general distrust in the SAPS far beyond the city's beleaguered neighbourhoods.\n\nIn one of the most celebrated murder cases in the country's history \u2013 the killing of Reeva Steenkamp by the paralympic athlete Oscar Pistorius, the investigating officer, Hilton Botha, was himself found to be facing seven charges of murder. He later admitted to contaminating the Pistorius crime scene and failing to secure critical evidence. According to a Sunday Times editorial: 'Botha's brief moment on the international stage has exposed the sorry state of detective work in the South African Police Service.'38 This perception is supported by blatant inefficiencies and corruption in top management positions, as we have seen, and extremely high levels of police violence against members of the public. This needs to change. Criminologist Liza Grobler, in her book Crossing the line: When cops become criminals, lists the steps that need to be taken to address the problem of poor policing:\n\n\u2022 Action must be taken to assist police officers whose behaviour is problematic with regards to undue violence or corruption;\n\n\u2022 The quality of new recruits to the service and the quality of training must improve;\n\n\u2022 Professionalism and pride must be reintroduced through employing only the best person for the job in every case;\n\n\u2022 Police management must be improved to limit possibilities for corruption;\n\n\u2022 Grievance procedures must be simplified to make it easy for communities to report corrupt officers;\n\n\u2022 Discipline should be used as a corrective measure and not as punishment and officers should be suspended during investigations into their possible misconduct;\n\n\u2022 Promotions based solely on merit and not on affirmative action should be reinstated;\n\n\u2022 An independent, specialised and proactive anti-corruption unit should be established; and\n\n\u2022 The Independent Police Investigative Directorate should be made truly independent and not answerable to the Minister of Police.39\n\nShe concludes:\n\nIt is a cause for concern in a country such as South Africa, with its unacceptably high crime rate, that the SAPS provides opportunities for its corrupt employees to commit crimes because of the absence of any meaningful anti-corruption strategy or controls. There is no leadership on this issue besides the odd sound bite: 'Corruption will not be tolerated' and the perennially rolled out: 'It's only a few bad apples.'40\n\nResource the Child Justice Act\n\nThe Child Justice Act was created to deal with young people below the age of 18 who fall foul of the law. It was drafted in the early 1990s as an exercise in restorative, as opposed to punitive, justice. After a difficult passage through Parliament and 14 years of national discussion, it was passed into law in 2008. The Act provides mechanisms to break the cycle of offending, trial, imprisonment, release and reoffending.\n\nIts aim is to ensure voluntary compliance, restraint in the use of force and a range of sentencing options with the potential to engender respect for the law. The Act was designed as a child-friendly justice system which lowers the trauma and stigmatisation a young person may face, avoids a criminal record and works towards the prevention of future crime. A prosecutor can divert a child for a minor offence instead of taking them through a preliminary inquiry. The accused can also be diverted at a preliminary inquiry or child justice court.\n\nThe Act's underlying principles were derived from traditional and popular forms of community justice, as well as from restorative approaches which had been piloted in New Zealand and Australia. By incorporating the community and families as well as victims in decisions about wrongdoing and sentencing, the Act seeks to restore harmony disrupted by illegal activity, making young offenders accountable for their actions to the victims, their families and the community as a whole. Because efficient execution of the Act would be one of the solutions to gang expansion, it's worth sketching what it proposes. The Act requires police and other officials to:\n\n\u2022 Deal with a child outside the criminal justice system where possible;\n\n\u2022 Encourage the child to be accountable for the harm caused by him or her;\n\n\u2022 Meet the particular needs of the individual child;\n\n\u2022 Promote the reintegration of the child into his or her family and community;\n\n\u2022 Provide an opportunity to those affected by the harm to express their views on its impact on them;\n\n\u2022 Encourage providing the victim with some symbolic benefit as compensation;\n\n\u2022 Promote reconciliation between the child and the person or the community harmed;\n\n\u2022 Prevent stigmatising the child and the adverse consequences flowing from being subject to the criminal justice system;\n\n\u2022 Reduce the potential for re-offending;\n\n\u2022 Prevent the child from having a criminal record;\n\n\u2022 Promote the dignity and well-being of the child and the development of their sense of self-worth and ability to contribute to society.\n\nWhere possible, young accused are diverted from the formal criminal justice system, with options including:\n\n\u2022 An oral or written apology;\n\n\u2022 Formal caution, with or without conditions;\n\n\u2022 Placement under a supervision and guidance order; reporting order or compulsory school attendance order; family time order; peer association order; good behaviour order or an order prohibiting the young person from visiting, frequenting or appearing at specified places;\n\n\u2022 Counselling or therapy;\n\n\u2022 Compulsory attendance of vocational, educational or therapeutic programmes;\n\n\u2022 Symbolic restitution;\n\n\u2022 Restitution of a specified object;\n\n\u2022 Community service;\n\n\u2022 Provision of some service or benefit by the child to a victim; and\n\n\u2022 Payment of compensation.\n\nAccording to Lorenzo Wakefield of the Child Justice Alliance, there are, however, troubling systemic problems in the Act's rollout:\n\nIt's a diversionary Act but diversion has dropped dramatically and nobody knows why. The Department of Social Development will tell you the system is working perfectly but there's a lot than needs to be addressed. In 2012 around 68 000 children were charged but only 18 000 assessed and 9 000 diverted. Where did the rest go?41\n\nAccording to Leana Goosen, director of the Act's facility management in the Western Cape Department of Social Development, her department farms out diversion to a number of NGOs which have been accredited to run programmes. It also has six of its own programmes, but none had managed to get accreditation because, she said 'we don't yet have enough evidence.'42.\n\nAround 36 000 police officers were given training in administering the Act, but at police station level there seems to be little evidence of this. According to a senior police official:\n\nYou go to a province and you find there's only one person doing child justice. And he's also doing custody management, children in need of protection \u2013 they can't cope. Same thing at the police stations. The person who's dealing with the child justice is also dealing with Children's Act, victim and domestic violence; a jack of all trades. We're essentially under-resourced. It's not seen as a priority.43\n\nBecause of concern over the way police had, in the past, dealt with young offenders, the prescribed conditions for arrest in the Act are extremely detailed and are seen as a nuisance by a police officer in an arrest situation. If the child is under 10, they must take the child to their parent or guardian or, if they cannot be found, deliver them to a probation officer. If the young person is over 10 and under 18, they may not be arrested for minor offenses unless there are 'compelling reasons', which are not specified. If the young person is arrested, the officer must:\n\n\u2022 Inform them why they are being arrested, read them their rights and explain the procedures to be followed in terms of this Act;\n\n\u2022 Locate and notify their parent, guardian or, if they cannot be found, write a report on the steps taken to locate them;\n\n\u2022 Inform a probation officer of the arrest or submit a report to court giving reasons they were not notified;\n\n\u2022 Ensure the young person is taken to a magistrate's court within 48 hours of the arrest;\n\n\u2022 Place the child in appropriate custodial care such as a One-Stop Child Justice Centre if not returned to their parents.\n\nFor police officers, it's easier to assume that young people under 18 cannot be arrested or simply not bother with the onerous arrest procedures. According to Captain Freddy Ledwaba, the National Co-ordinator for Child Offenders:\n\nWhen the Act was rolled out, police were confused by what appeared to be an instruction that they were not allowed to arrest a child. The process is a big nuisance for police so they try to avoid it. A brigadier in Gauteng told me officers would say to the child: 'If you don't run I'll have to arrest you.' Or they pick up a child and drop him a few blocks away, let him out the van and tell him: 'Don't do it again'. To avoid paperwork maybe they pick up a 16-year-old and write his age as 18 on the docket.44\n\nThe Child Justice Act was designed to give a new start to young people in trouble with the law. It is imperative to keep them away from the violence of arrest, imprisonment and the corrupting influence of prison gangs and to provide them with programmes that divert them from crime. In this task it could at best be described as limping along.45 This amounts to withholding services from those whose lives could be changed by its efficient application. The solution is simple: resource and enforce the Child Justice Act.\n\nRethink imprisonment\n\nPrisons are containment systems designed to keep criminals off the streets. At this alone they are successful. In all aspects other than containment \u2013 apart perhaps from a public requirement for revenge \u2013 prisons are intolerable and a complete failure. They simply cannot work because their very nature is paradoxical. There are about ten million prisoners in the world \u2013 one person in every 7000. According to the World Prison Brief, South Africa has the largest prison population in Africa and the ninth biggest in the world. However, according to the South African Minister of Correctional Services, Sibusiso Ndebele, retribution through imprisonment has not deterred crime.46 'Since retribution is narrowly focused on moral reprobation or outrage against criminal conduct,' he said, 'it fails to adequately reform offending behaviour and to repair harm experienced by victims of crime.'\n\nThe cost of what the Minister is essentially calling a failed institution is staggering. According to Nicro, in 2014 South Africa had 162 162 people in custody, 49 695 of them awaiting trial. This cost taxpayers, according to the Minister, R399.20 a day to house each inmate and prevent them from escaping. That amounts to around R19.4 billion a year. Between the years 1995 and 2014, the number of prisoners serving life sentences jumped by an astonishing 2 197 per cent and \u2013 at the time of writing \u2013 there were around 21 000 inmates serving from 20 years to life terms. The true madness of the situation is that recidivism \u2013 re-imprisonment \u2013 is estimated at between 55 per cent and 95 per cent. People are being recycled at huge cost through the prison system with no positive value to themselves or the country.47\n\nThe Department of Correctional Services is essentially required to look after people nobody wants to care for and is expected to solve damage that's been done to them, sometimes, since birth.48 It's required to 'correct' antisocial behaviour and to make people 'better'. Yet the worst conceivable place to attempt anything curative is in jail. Behind prison walls, what relationships young people had with parents, partners and friends are terminated and they join others who are equally stressed, distressed and alienated. In prison, they are depersonalised, violated and victimised so they return to society angry, unambitious, vengeful and schooled in brutal gang practices.\n\nIf we were to trace the beginnings of the process that steers young people to prison, we would find failed or destructive relationships \u2013 father, mother, sibling, teacher, peers, society or all of these. If prisons were to be truly corrective, they would assist young people to gain insights into this relationship failure. They would provide opportunities for them to restore self-respect, reliance on their own integrity and to change. This is precisely what prisons throughout the world do not do. In South Africa the need for relationships, self respect and protection against cruelty feeds the fierce, hierarchical and violent prison Numbers gangs. Added to this, demands by the public and politicians to lower the crime rate prods the criminal justice system into increasing vengefulness. Revenge serves to label, ostracise, alienate and breed antisocial, criminal behaviour, recidivism and an escalating prison population. It's a system destined to malfunction and is increasingly out of control.\n\nYet despite these conditions, as poverty increases in marginalised communities, a chilling corollary is emerging. According to the Institute of Race Relations, 'living conditions might be better in prison than on the outside... and some gang members preferred to be in prison so they could remain in their gangs.'49 The number of years spent in prison, says SAIRR researcher, Kerwin Lebone, is therefore no longer a deterrent.50\n\nThere's another anomaly about incarceration that needs consideration. In Europe, from the early 19th Century, imprisonment changed from being pre-punishment detention to punishment itself.51 It has since become, according to the philosopher Michael Foucault, 'a detestable solution which one seems unable to do without... one cannot see how to replace it.'52 Deprivation of liberty makes it possible to precisely quantify the penalty by varying the time of detention rather than varying the conditions under which detention takes place. Given the increasing scale of incarceration, this is a bureaucratic and logistical necessity, keeping the same form whether the prisoner is sentenced for a month or a lifetime. Walls and not conditions are the punishment for the crime. Imprisonment becomes a coercive theatre in which the only audience is the actors themselves. In his study of an American maximum-security prison, Gresham Sykes argues that:\n\n[the] frustrations [of captivity]... carry a more profound hurt as a set of threats or attacks which are directed against the very foundations of the prisoner's being. The individual's picture of himself as a person of value \u2013 as a morally acceptable, adult male who can present some claim to merit in his material achievements and his inner strength \u2013 begins to waver and grow dim.53\n\nWe are creating, in the words of Foucault, 'another class and another human species. A zoology of social sub-species and an ethnology of the civilizations of malefactors, with their own rites and language.'54 In these conditions, he says, 'prison fabricates delinquents ... it brings back, almost inevitably, before the courts those who have been sent there.' In this sense, delinquency is the vengeance of prison on the justice system.\n\nIt's a central tenet of law that you're innocent until proved guilty. If detention within a prison is in itself punishment for wrongdoing then, in 2013, more than 53 000 legally innocent people were being wrongfully punished as awaiting trial prisoners. Many of them were children. As an indication, between April 2010 to March 2011, 75 435 children were charged by the police, though of course many of them were diverted from detention centres55. Awaiting trial prisoners are, in this way, being forced to attend schools of delinquency. Their constitutional rights are clearly being violated but, given the grinding slowness of the judicial system \u2013 the gathering of evidence, finding witnesses, securing court dates and the ineptness of much detection and documentation \u2013 what's the alternative? In the present system there is none and the paradox remains: prisons create that which they wish to destroy.\n\nThere is a further injustice. Prisons separate offenders from society but, by maintaining their criminal record for future use against them, they also separate ex-offenders within society. You'd assume that, in imposing a prison sentence for a misdemeanour, a magistrate or judge ponders the harm done and fits the length of imprisonment accordingly. The punishment should befit the crime. Yet for up to ten years after release, and for life in the case of child molesters or sexual offenders, the Criminal Record Centre in Pretoria or a private 'tactical protection' service can trace records and provide these to a potential employer. For an additional fee, credit rating, business activities and qualifications can be checked; identity and national residency validated and endorsements on drivers' licences and outstanding traffic fines viewed.\n\nLukas Munting of the Prison Reform Initiative has questioned the purpose of retaining a publically accessible criminal record, given the way it extends the punishment.56 The effect is to impose a civil disability and levy a debt to society that cannot be repaid. A 'criminal' register is even kept where no crime has been recorded, as when children are diverted out of the criminal justice system under the Child Justice Act.\n\nMaintaining a record of offenders is, of course, essential to protect society, particularly in the case of sex offenses, crimes against children and excessive violence. Prison history is also required by courts when taking into account previous convictions in sentencing a multiple offender. However, making such records available to third parties virtually guarantees that people with criminal records will be frozen out of the normal employment market, undermining their social reintegration and leaving crime, gang life or a return to prison as the only options available.57\n\nAn ex-offender can apply to have their criminal record expunged after a requisite five or ten years following release from prison. It's a complex, opaque process. Under the Criminal Procedures Act a 'clearance certificate' is needed. This must reflect the period that has elapsed without intervening convictions. It must be followed by application to the Director General of Justice and Constitutional Development who has the authority to sanction the expungement by the Criminal Record Centre of the SAPS. Given these complexities, few offenders manage to clear their name. Costs alone deter most. For example, despite the tens of thousands of expungements that would have been possible in 2011, only 3 002 were granted.58\n\nA further problem to consider is that criminal conviction acquired at an early age is not necessarily a predictor of long-term behaviour. In most societies, offending behavior is adolescence-limited which peaks around the age of 18 then falls away. It's extremely unfortunate that early wrongful action should hang over a person for much of their productive life. A criminal record is, therefore, a major factor in shifting adolescence-limited activity into life-course-persistent crime.\n\nThe solution to this problem is simple and bureaucratic. Apart from sexual crime and crimes against children, criminal records should be available to only magistrates and judges for the purpose of sentencing and should be automatically expunged after the requisite five or 10 years. No child under the age of 18 who has been diverted under the Child Justice Act should be listed within the criminal justice system.\n\nA solution for prisons themselves would be to provide inmates with skills useful for their return into society. They should be centres where people are temporarily housed and trained to become useful citizens able to earn a living on release. The Department of Correctional Services should act on the implications of its name.\n\nDevelop personal resilience\n\nIf young people are to avoid being drawn into gang life, they need to develop personal resilience. What is meant by that? In early life, resilient children are active; affectionate; cuddly; good-natured; humorous; confident and competent with no sleeping problems. They're able to handle frustration and anxiety and to ask for help when they need it. As young adults they're able to withdraw from stressful situations, postpone an immediate angry response and find a more manageable situation by restructuring their environment. They work to adjust for security and comfort, are popular with peers and adults and have positive self-regard.\n\nResilience is not an innate quality but an acquired characteristic and needs a safe, supportive human environment in which to develop. Intelligence cannot unfold without stimulus from a developed form of that intelligence.59 For resilience to grow, young people need responses that are emotionally validating and developmentally challenging.60 This involves encounters within a stable emotional relationship with at least one responsible parent or guardian; an open, supportive learning climate; parental behaviour that models constructive problem coping and social support from beyond the family. According to psychologist and child development specialist Ann Masten:\n\nResilience does not come from rare and special qualities, but from the everyday magic of ordinary, normative human resources in the minds, brains and bodies of children, in their families and relationships and in their communities. This has profound implications for promoting competence and human capital in individuals and society.61\n\nProblems with a child's early relationships can translate into general social problems, deficiencies in cognition, emotional problems, impaired development and later drug use and violent behavour. Solutions entail a redirection of life trajectories which include the opening up of opportunities, lasting changes in the environment of the young person and changes in their self-concept or expectations of other people. These can be seen as the construction of personal resilience \u2013 the ability to bounce back from adversity. They may be gradual or involve incidental or programme-assisted turning points.62\n\nSocial work researcher Michael Ungar defines resilience as 'the outcome from negotiations between individuals and their environments for the resources to define themselves as healthy amidst conditions collectively viewed as adverse.'63 According to psychiatrist Sir Michael Rutter, these adverse conditions, which he describes as risk, 'reflect both genetic influences and the effects of prior experiences, including the benefits from overcoming adversity or dealing effectively with challenges in the past.'64 Risks are both biological (such as pre- and post-natal damage) and environmental (the context and manner in which children are reared). Systems that play a central role in resilience point to an integration of biological, psychological and social perspectives and can be listed as:\n\n\u2022 Learning systems of the human brain (problem-solving, inform-ation processing abilities);\n\n\u2022 Attachment systems (loving relationships);\n\n\u2022 Mastery motivation (the ability to succeed in undertakings);\n\n\u2022 Stress response systems (alarm and recovery ability); and\n\n\u2022 Self-regulation (control of emotion and behaviour).65\n\nRelying on drugs or alcohol to deal with stress, dropping out of school or becoming pregnant as a way to deal with attachment or family problems reduce personal resilience. But work done by the Usiko Trust and similar youth organisations in Cape Town suggests that new experiences or shock realization which provide a break from a stressful past or present situation can open up new opportunities.66 Even if considerable risks remain, a single event can kick-start a resilience path, as described by Cape Town community worker Grant Stewart:\n\nSome just make it and others don't. All Jonathan's friends when he was 15 were in gangs and he wasn't. But he was naughty. He ended up in court and the magistrate gave him a tongue-lashing. He told me that moment, for him, changed things utterly. He looked down the gang road and realised he didn't want to go down it. Jonathan didn't have a father and the magistrate was male so that's something to consider.67\n\nAny attempt to assist a young person to increase personal resilience needs to foster a positive cascade and reduce negative chain reactions both inside and beyond the family. It's necessary to minimise risk factors, maximise protective factors and allow the young person the space to do and be what they have reason to value. This has greater possibility of success among young people whose anti-social behaviour is adoles\u00adcence-limited.\n\n***\n\nSo how do we assist young people to develop and increase prosocial resilience? Becoming resilient is not a fixed attribute, but rather a characteristic that can develop over time, so we can intervene if we know when and how. According to community psychologist Lane Benjamin:\n\nWe need to teach people to get out of survival mode, make some safe space with a mentor to talk about options and to think about consequences \u2013 all the stuff you do with a toddler. A lot of our men and women are still toddlers emotionally. Families that can't teach how to regulate your impulses and emotions as a toddler produce adults who haven't got past that stage.68\n\nIn 2010 the World Health Organisation published seven strategies for violence prevention based on the development of personal resilience. These can serve as points of intervention for any programme. They are:\n\n1. The development of safe, stable and nurturing relationships between children and their parents and caregivers;\n\n2. The development of life skills in children and adolescents;\n\n3. A reduction in the availability and abuse of drugs and alcohol;\n\n4. A reduction in access to guns, knives, and other weapons;\n\n5. The promotion of gender equality to prevent violence against women;\n\n6. A change in cultural and social norms that support violence; and\n\n7. The support of victim identification and care.69\n\nAttempts at intervention, however, require a note of caution. We've shown that for a young man in an intolerably stressful family situation, joining a gang may be an act of both defiance and resilience. Young people who live beyond the boundary of conventional, socially acceptable behaviour might argue that their unconventional behaviour maintains their physical safety and mental health. One young man may consider arrest and imprisonment a massive risk, while another may see the experience as a positive initiation. Much depends on their perceptions. We must be careful not to 'declaw the cat'.\n\nIn defining resilience, it's therefore important to find the balance between respecting young people's self-definitions and affirming what's required of them to participate fully within society. Completely focusing on their perception will not support their societal integration if they continue to engage in isolating activities which make pro-social interventions difficult. Continued gang membership reduces the possibility that those needs will be met. For those who join, this is hard to discern until it's too late.\n\nProgrammes designed to develop resilience need to be part of a planned path of development to avoid dropping a young person back in the environment which created the problem in the first place. In such a situation, gains are soon eroded through lack of positive support. The problem is outlined by Karl Voysey of Amandla EduFootball:\n\nWe need to think from a long-term perspective and get away from thinking short-term programmes having any influence on young people. But we also need to look at this person, now, right in front of us. It's challenging for young people to go back into the environment they came from and expect them to successfully maintain an internal transition. They need to be held beyond the programme.70\n\nWhat's lacking, according to youth worker Grant Stewart, is coordinated through-care. 'We need it from pre-birth through adolescence. A wanted foetus who becomes a loved infant is not going to become a violent adolescent. Simple as that.'71\n\nPsychologist James Garbarino points out that, without safety and security, there's no psychological space for changes in behaviour, thinking and feeling needed for a young person to change for the better.72 Once this space is established, young people need to then be placed in situations of healthy progressive conformity and build patterns of positive functional autonomy, rather than negative patterns of thinking, feeling and acting. Detoxifying a social environment like Manenberg or Hanover Park is, as Garbarino notes, a job too big for parents and professionals alone. It requires us to act as members of a community and a nation. It's fundamentally political. If you want peace, as Pope Paul V1 once said, work for justice.\n\nPlan life paths beyond programmes\n\nA person's chance at building a life starts, as we have seen, long before they arrive \u2013 in the age, country, economy, city, area, house, family, social context and genetic lineage into which they are born. Much of it's not of their choosing and they make the best of it, given the situation. Many events and contexts along that evolution influence its direction. Seemingly slight shifts \u2013 biological, emotional, social or physical \u2013 can become major directions, advantages or disadvantages. Small effects may accrue over time and ultimately yield large outcomes. This process can be described as a developmental cascade that accumulates, opening or closing life chances and possibilities from neurons to neighbourhoods and beyond. Psychologists Ann Masten and Dante Cicchetti describe this, rather densely, as:\n\nthe cumulative consequences for development of the many interactions and transactions occurring in developing systems that result in spreading effects across levels, among domains at the same level and across different systems or generations.73\n\nLocating the precise 'interactions and transactions' over an entire lifetime that produce positive or negative paths is extremely complex. But there are nodes and times of greater importance, such as in the womb, the first 1000 days from conception and the onset of adolescence. As we have seen, intemperate consumption of alcohol by a mother over the precise period when particular areas of her child's brain are forming can lead to anti-social tendencies in its life that ricochet through childhood, school, work possibilities and lay the foundation for violence. A father's contempt, aggression or absence can cause a young person to have confidence problems that are compensated through egotistical displays which lose positive friends and predispose teenage gang association. Family intolerance or low problem-solving abilities may increase the likelihood of poor family or peer relationships, both of which could contribute to heightened anxiety and lowered self worth, leaving the young person vulnerable to depression. Peer rejection and isolation of children with problems may trigger relationships with deviant peers, who then model and reinforce antisocial activities.74 Problems in one domain cause problems in another.\n\nThe notion of developmental cascades begins to explain why seemingly small interactions can have a large impact, but also why many young people do well despite seeming disadvantages. Each interaction or transaction has the capacity to undermine or build resilience as life unfolds. This is an important perspective when considering the development of paths and programmes for young people. In precisely the same way that negative events have life resonances, it should be possible to target critical moments of intervention in a young person's life that have the greatest possibility of generating a resilience cascade. Success in one arena serves to enhance self-esteem and self-efficacy, making it more likely that the individuals concerned will feel more confident handling new challenges and therefore act accordingly. Where failure builds failure, success can build success.75 A major developmental cascade could be precipitated, for example, if Cape Town reinstituted pre- and postnatal nurse visits.\n\nLife-path interventions need to distinguish between those who are deviant because they're teenagers who can be redirected to a safer destination or whether there are early life pathologies that require much deeper mediations. This knowledge requires predictors derived from pre- and postnatal visits, early child development processes and school assessments. These should be put in place as an early warning system so that negative developmental cascades can be offset and positive ones implemented. Such tests should be fundamental to any educational, medical or social welfare system.\n\nUnderstand personal transformation\n\nThere was this boy named Zaid Mohamed. I had a light with a cloth over it and I said to him: 'This is like your life. Can you see any light?' and he said 'No ma'am'. I said 'The cloth is made from everything that's happened to you, every abuse or pain or disappointment, or every sadness. There's a light inside but you can't see it and others can't see it.' And I said: 'What we're doing here is trying to shed that cloth.' And I took it off. I looked at his eyes and he knew, he understood. There was silence, a pondering, then a sparkle, an ah-ha moment. Then he started talking, you couldn't stop him because the connections were being made.\n\nWe don't really know what happens in the brain at times like that, but you can see it happening, a sort of instantaneous rewiring. I think it's also something you're experiencing in the heart. There's that silence then the excitement.\n\nThat's why it's so important to pay attention. You can see it even in the micro-facial expressions, the eyes, the growing smile and then just bursting over. These things happen at points of realisation. It's often just something we say.76\n\nLucille Meyer, CEO, Chrysalis Academy, 2015\n\nIn this chapter we've been fitting together the pieces of the puzzle from which personal resilience begins to emerge. Assembling a life from them takes time and a supportive environment. Many changes need decades or a lifetime. But as Karl Voysey asks, 'What do you do with this young person now, right in front of you?' Is there something that can be done to change their direction, to deflect them from a self-destructive or risky trajectory? Is it possible to create a process that enables them to become the kind of person they'd like to be beyond the orbit of gangs?\n\nThese questions were prompted by a discussion about young people in gangs I had with David Abrahams, head of the Western Cape Ministry of Social Development. 'What we need, but do not yet have,' he said, 'is a secular theory of personal transformation. A template for youth programme development.' This theory, he felt, had to be about an inner process:\n\nWhat changes a mind-set? We don't know how to measure it. We don't know how it works. How do we find the keys of interior transformation? And how does a young person who finds them engage with their world? What's on the other side of epiphany? What is the nature of the contestation between their inner and outer processes? And how do we support them in that process?77\n\nA starting point would be to locate the field or discipline within which such questions can be asked and a theory constructed. It seems to me this is not initially in education or psychology, but in the rituals of a 'transcendent' belief system. Such systems exist. On the Cape Flats everyone presumes there are only two ways out of a gang: death or religion. Gangs generally respect escape routes leading to a church or mosque because their own lives are highly ritualistic and members, constantly facing death, are superstitious. For them, concerns about existence and the hereafter are close companions. They respect religion.\n\nOrganised religions, in their many forms, have been around for thousands of years because, beyond the wars and fundamentalism they seem to catalyse, they help to develop personal resilience, particularly in times of distress. This is because of what the church or mosque offers at a very practical level: a priest or imam who doesn't necessarily judge you and is prepared to take you seriously; a congregation that affirms; a structure and set of rituals that give order to internal confusion and the possibility of quiet reflection. In addition, there's the notion of an order of existence and of a transcendent god who is above and beyond the confines of the ghetto. According to psychologist Lane Benjamin:\n\nMany of the people we work with, their world is so narrow. It's me, maybe my family, maybe the community but not the rest of the world. This is how it's always been. A spiritual sense shifts that, opens life wider all the way to a sense that I am connected to everything. In African traditional culture healing is always referred to in spiritual terms. I don't know how to explain it. It just is. Feeling connected to something bigger than you.78\n\nReligion, like any youth programme, offers a trajectory, a pathway or line of development. It also holds out the possibility of transition, a change in state that's more abrupt, a new trajectory or turning point \u2013 'seeing the light'. It involves a discontinuity, letting go of an old identity and creating a new sense of self. The constituents of this process, which I term 'footsteps to epiphany', are not easy to come by in secular research, but they do form the gantry of a secular theory of personal transition.\n\nA team of French researchers has found that among teenagers, the turning point in their lives is most often associated with a realization that they're in an impasse: there's fear, the consequences seem too severe, the threat too great and they have no choice. This forces them to take action because no one else can do it for them. Without a safe environment and the support of an older person they can trust, however, that action can lead them deeper into trouble.79\n\nAfter the realization by the young person of an insoluable crisis, the availability and quality of a self-aware mentor is the second building block in a theory of personal transformation. The poet Robert Bly has said that a young man not being admired by an older man is being hurt. The unfatherdness of so many gang youths makes them both vulnerable and amenable to the support and regard of an older man. The outcome rests on the integrity of the mentor, be they druglord, priest, teacher or programme coordinator. 'It is our opinion,' write the French researchers, 'that change must not only occur in young people, but also in the adults and practitioners who frequent them.80 Mentors have to earn respect.\n\nAshley Potts, who has been mentor to many gang members, knows the consequences of a miss-step: 'You can't dare to assume you can just come in and think you have the right to speak,' he said when I tracked him down in his drug rehabilitation centre in Mitchell's Plain. 'The first step is trust':\n\nIf they don't feel they can trust you, nothing can begin. They need that safe space and that safe space isn't a place like a church or a room. It's a person who allows them to feel they can express themselves and find a way forward for themselves. The kids I deal with feel so trapped. Showing appreciation is a huge thing for them \u2013 nobody ever has. Trusting them! They can't believe it. You show them respect and then they show you respect. But you can't use a classroom for this stuff, ever.81\n\nHowever such a mentor needs to be resolved within themself. They need to be sensitive, the sort of men, according to Lane Benjamin, who don't act out of stereotypical masculinity and are in touch with their emotions:\n\nThey need to allow younger boys to see them be vulnerable. Young boys almost don't know what to do with themselves when they step into that space, but they're affirmed because it's okay to be like that and still be a man because these are men. And for boys it has to be with men. Given the stereotype, the bravest thing a man can do is to show his vulnerability.82\n\nA mentor breaks through the isolation, reaches through to empathy, liberates the shame and anger that non-attachment has caused and provides a story-path out of isolation. But mentorship of high-risk young men comes with a warning, as Potts discovered. 'There's an anger-block around fathers that grows like a balloon. And if you prick it, all the hot air can come out in your direction unless you can find a way to release it slowly for their own benefit. You have to be non-judgemental, you have to leave your ego at the door.'83\n\n'Of course fathers should be the first mentors for young men,' youth worker Grant Stewart told me, 'but they so often don't act that way':\n\nMen are socialised not to display their emotions. They learn to disconnect from emotions so they don't know how to father. Then there's an underlying depression, a disconnectedness. We need to fix the fathers. You have to shut up the internal critic in order to father the victim within you. You need to take the dark journey and not divert into the path of self-medication. How do you change your own legacy of hurt? So a big part of solving the gang problem is healing fathers. And it's not just a problem in gang areas but also in the leafy suburbs behind high walls. It's a general South African problem.84\n\nThe third building block in a theory of personal transformation is a safe environment, which is both physical and emotional. According to Benjamin, safe spaces are difficult to find in low-income communities but can be created in relationships:\n\nYou need people who have your back. They can't physically protect you but they're protecting your heart, mind and soul. You can be genuine and real with them. There's too much posturing for this to happen positively in gangs. Your family is your first safe space and when that fails you start treading water. It's the hardest space to work with from the outside so you need to create alternate families. Of course that's also what gangs are.85\n\nProviding a safe space is easier if programmes or processes with young people take place outside their known environment. A number of organisations such as Educo, Usiko Trust, Hearts of Men and the Chrysalis Academy have programmes which use the natural environment as 'transformative' space. The value of this was noted by psychologist Cathy Ward:\n\nPart of change seems to be able to visualise a different path. And it's not possible when they're stuck in Manenberg or Bonteheuwel. They need to be out of their context out of choice. The resources have to be there at the right time. Real options need to be offered. And a sense of purpose.\n\nAltering the environment to support change was underlined by Michael Rutter, who found that new experiences in new places could provide a beneficial context for 'turning-point' effects.86 The French researchers mentioned earlier concur with this finding: 'What is important is not the nature of the turning point as such, but its discontinuity in relation to daily life.'87 Transformative work, they suggest, should take place away from harmful environments, parents with limitations or peers with antisocial influence and in a place that provides structure. For Ashley Potts, that place is beyond the city limits:\n\nWilderness is a great healer. I can do in one session in nature what it would take me six months to do in this neighbourhood. I can't explain it but it's like a psychological switch, something just drops away. There's no noise, you can actually hear wind. You can't imagine the amount of relief they feel, boundaries come down and you have the most amazing conversations. Being in the wild is a catalyst for a bigger wholeness.88\n\nA fourth building block is validation. Beyond the regard and acceptance of a mentor, teenagers need to do things for which they are regarded among those who they wish to impress. In programme development, this requires tasks and challenges to be created in which fear can be overcome. These risks are important because, without the possibility of failure or of overcoming fear, they would be emptied of their value. This requires unseen but well planned safety nets created by trained organisers.\n\nIt's not the field of involvement that's important, but the sense of accomplishment it creates. In their study of gang paths and fatherhood, a research team under Molly Moloney noted that 'successful desistance from crime may be rooted in recognition of an opportunity to claim an alternative, desired and socially approved identity.'89 Another team under Scott Decke, working on disengagement from gangs, found that transition entailed the young person imagining their status in a new role. This included how they saw themselves and how others saw them. They needed to experiment with being 'other', be validated in that space and weigh up the costs and benefits of alternative roles.90 These actions may be deeds for which they're admired among peers, but also acts they do for others in a wider community for which they are regarded.\n\nThe fifth building block is ritual, the ordering of the unexpected, which is a form of structured action. As we've seen, most young people in gangs come from very unstructured homes or where compliance is rigidly enforced without explanation. For the Chrysalis Academy, creating structure is paramount. The majority of its students come from such homes and, soon after they've been enrolled, they're given uniforms and taught precision marching. 'It's not that we're trying to create soldiers,' CEO Lucille Meyer explained. 'It's just that the best place to begin training young people to self-structure is with their bodies. And when they get it right, they're proud of the abilities, which they all share. After that we can teach them how to structure their minds and their lives. It's a ritual that works':\n\nRitual is so powerful. It's a particular process that leads up to what we need to do. Ritual signifies that you're going into another phase, a transition. Ritual speaks to the intention \u2013 everything starts with intention. It frames your resolve. Marching is a ritual. Silence is a ritual. What is my intention moment by moment, for today, for my life? Ritual is your keel and rudder in the sea of life. It talks to older things in you.91\n\nLane Benjamin sees ritual as a way young people can express themselves beyond the confines of verbal ability:\n\nRitual goes back to the warrior in you. Trauma is so unpredictable and out of control and ritual helps you know what's coming next \u2013 getting control in a situation. People in high-risk communities are often not able to verbalise their stories. It becomes anxiety provoking. So they need to tell their stories in other ways \u2013 through song, dance, movement. As long as the story gets told and heard.92\n\nPsychologist Cathy Ward has found that if you want to visualise yourself as elsewhere, stories are important. 'When we imagine our future, we write a story of ourselves in our heads of what it's going to be like. Transition comes when you embody that story, accept it as yours. It's that belief moment. Religion is one of those stories.'93.\n\nThe intertwining of ritual and stories is as old as our existence. It has always been this way since our species gained the wonder of speech. Our stories are how we construct our world and by which others insert us into theirs. Without the agency to tell our story we risk social transparency and depression. Far worse, though, is to have no story to tell. Adolescence is the beginning of that story's making, a time when the world intrudes on childhood and demands a response. It's a path upon which you may not fail to step but, confusingly, which leads you in any direction you choose. That path is the beginning of what the mythologist Joseph Campbell calls the Hero's Journey. Whether you take it alone or with others, it needs to provide the ingredients of bravery, cunning, adventure, danger and laughter. It must sustain telling, retelling and retention in the memory of those you wish to impress, including yourself.\n\nIt's a time when you try on new clothes and watch yourself walk by. Joining a gang is one of those paths and one of those stories. Any programme or process to steer adolescents on a different path needs to first understand what it is young people seek and find in gang life and then provide the ingredients for a more powerful Hero's Journey with better stories to tell.\n\nThe final building block in a theory of transition is time-out for reflection, something in short supply in crowded, low-income neighbourhoods. It's a time when young people can think without distraction about who they are and where they're going. It's best undertaken in a natural surrounding and alone. For many urban youngsters, being in an unaccostomed space and having to self-reflect can be very scary. For most wilderness-based programmes this entails sleeping outside in nature alone without a tent. In more intense adult programmes this can be for up to four nights with no food and only water. The impact was described by former gang member Lerato Kossie:\n\nI was alone on a mountain beside a stream and I saw a small green frog. I had never seen a frog before. Then I realised I knew nothing about everything I was seeing there beside the stream and also nothing about the earth I was on flying round the sun. I was overwhelmed the size of what I didn't know. The feeling wasn't scary, it was a sort of... reverence. That's how I became a wilderness guide.94\n\nIt is axiomatic that personal transformation is a personal process. Whatever inducements, rituals or opportunities are provided or undertaken, it ultimately has to come from within. For some young people, especially those whose behaviour problems are life-course persistent, it may never happen or take a lifetime, for others it may be a life-changing epiphany. For Lucille Meyer it has to do with the development of a sense of purpose:\n\nI tell these young people: 'Someone said there are two days that are important in your life \u2013 the day you are born and the day you discover why.' That speaks to purpose. You have to take people into the realm of the unknown but also to their purpose. It accelerates when they can begin to see this connection to something higher than themselves. Metaphor and stories are essential to this process, but to put it all together they need a time and space to reflect.95\n\nThese seven building blocks are a template for personal transformation that can be used to construct most youth resilience-building programmes and can be depicted in this way:\n\nFigure 11 Stepping stones to resilience\n\nIn this process epiphany is not assured, of course, and regression by young people afterwards is a possibility. To quote Father Greg Boyle, founder of Homeboy Industries in the United States:\n\nThere is nothing 'once and for all' in any decision to change. Each day brings a new embarking. It's always a recalibration and a reassessing of attitude and the old tired ways of proceeding, which are hard to shake for any of us.96\n\nLetting go of a strongly held identity is a difficult task and the level of interaction between gang members and their gangs will colour the disengagement process. Fellow and rival gang members can pull individuals back into the gang by failing to recognise or validate the individual's new identity. Interaction with the police will have the same effect. Any programme needs to be aware of these push-pull influences and include path planning beyond individual programmes. Protective factors need to be put in place.97\n\n***\n\nFear and excitement; the desire for a role model; the safety of a protected turf; validation and inclusion into a peer group as well as rituals of belonging are, of course, all part of the attraction of gangs. So is the hope of transition out of the situation in which many young people in ghettos of poverty find themselves. It should come as no surprise, therefore, that the pathway to personal resilience depicted here and assembled from the insights of many people who work with high-risk youths is almost identical to path of inclusion into a gang. Any pro-social programme built around the fulfilment of the needs that make gangs attractive, therefore, has the greatest chance of success.\n\nWith help, the interventions and turning points I've described could set off a cascade affecting how young people in Cape Town's high-risk areas choose to live and the way they see themselves. We need to give them what they seek from gangs but which is seldom sustained: respect and a chance to be the sort of people they'd like to be. It's their right to be given that chance. It's our job to make it possible.\n\nHere's a story about an epiphany:\n\nIn Nguni culture, when a young man is a kwedien \u2013 uncircumcised \u2013 his opinions aren't valued. When he speaks, he's tolerated but not regarded. He's a child. The makweta ceremony is the time of manhood. In traditional areas, several weeks into the ceremony, there's a time when the young man is invited to a beer drink.\n\nI attended one while researching adolescent traditions in the Quamata area of Transkei. There were about 50 men sitting in a circle on stools and upturned tins passing a large can of beer. I watched the young man come down the mountain and approach the group. He was nervous. His eyes were downcast.\n\nAs he walked up to the circle the men made space for him and he sat down. They continued talking. He just sat there and nobody paid him any attention. But when the beer came round it was passed to him and he drank and handed it on.\n\nAfter about 20 minutes \u2013 I guess he was plucking up courage \u2013 he said something. Nothing special, just a comment in the flow of conversation. But every man stopped and listened to him. Then they nodded, agreeing with him and the conversation flowed again.\n\nI was watching him closely. His shoulders straightened, his eyes brightened and he looked the men in their faces. In that moment, in that instant, he became a man. His story had been heard. He'd been accepted.\n\nA final word\n\nThis book outlines the beginnings of a new map in dealing with gangs. For any value to come of it, we need to remember that all social systems are human constructs \u2013 essentially, in the words of the historian Yural Harari, they are an imagined reality.98 From day to day the human-made world continues to function, not just by way of the material with which we construct it, but on ideas and systems we have created. If it's imagined, then we can re-imagine it in a different way. We can change what's not right \u2013 but we need to want to.\n\nWhat has been suggested is an extensive re-evaluation of how we care for infants, children and adolescents and what it is we wish to teach them. The formal education system, while essential, has become bureaucratised. It is unable \u2013 and was never designed \u2013 to undertake the cultural, almost cellular, transfer of life knowledges essential to the wellbeing of young people. Many \u2013 if not most \u2013 'modern' parents have lost the deep educational initiative without which young people lose touch with ancient inherited cultural roots and simple human wisdom. They have done this either because they don't know how to deal with their adolescent children or because they believe school teaches these things. They assume the school system and the state should take responsibility to educate them.\n\nAlso, as I have indicated, for the many young men at risk, the notion of father as teacher is being lost. In a non-alienated culture, a son does not receive a hands-on healing from a father, but a body-on healing. In more traditional societies the son's body \u2013 standing next to the father, as they make arrowheads, repair ploughs, wash pistons in petrol or care for birthing animals \u2013 can dance to retune and learn how to be a man. Sons who have not received this retuning will have father-hunger all their lives.\n\nOf course, most urban adolescents survive in their fashion. The lucky ones have parents who act on an intuition that adolescents need both physical and emotional engagement. Some young people find mentors in their teachers or neighbours or grandparents. Others find adult mentoring in sport or academic achievement. But many find only each other and put together an emotional and symbolic life as best they can: music, dress, fads and fashions, drugs, romances and sexual encounters.\n\nAt the outer edge are those who don't have any support systems, who go too far, join gangs which carry them beyond the boundaries of social and legal acceptance. It's these we define as young people at risk, but any transformation of their lives requires that we look at what society provides for and requires of all adolescents. Between the insiders and the outsiders is not a boundary, but a continuum along which teenagers are able to move with surprising ease and speed. Unless South Africa reassesses its commitment to the support, education and parenting of all children \u2013 unless families, communities and the collective wisdom of our many cultures are seen as central to this support \u2013 then all young people are at risk.\n\nA final word from that revered South African storyteller and master of magical journeys, JRR Tolkien, speaking through the Hobbit Sam Gamjee in Lord of the Rings:\n\nFRODO: I can't do this, Sam.\n\nSAM: I know. It's all wrong. By rights we shouldn't even be here. But we are. It's like in the great stories, Mr Frodo. The ones that really mattered. Full of darkness and danger they were. And sometimes you didn't want to know the end. Because how could the end be happy? How could the world go back to the way it was when so much bad had happened?\n\nBut in the end, it's only a passing thing, this shadow. Even darkness must pass. A new day will come. And when the sun shines it will shine out the clearer. Those were the stories that stayed with you. That meant something. Even if you were too small to understand why. But I think, Mr Frodo, I do understand. I know now. Folk in those stories had lots of chances of turning back only they didn't. Because they were holding on to something.\n\nFRODO: What are we holding on to, Sam?\n\nSAM: That there's some good in this world, Mr Frodo. And it's worth fighting for.99\n\nEndnotes\n\n1. Garbarino, Lost Boys, op cit, p158.\n\n2. White Paper on Education and Training, 1995. The Constitution of the Republic of South Africa, Act 108 of 1996. Interim Department of Education Policy for Early Childhood Development, 1996. National Programme of Action for Children in South Africa, 1996. White Paper on Social Welfare, 1997. White Paper for the Transformation of the Health System in South Africa, 1997. White Paper 5 on Early Childhood Development, 2001. White Paper 6: Inclusive Education, 2001. National Integrated Plan for Early Child Development 2005-2010. Children's Act No 38 of 2005 (effective from 2010). National Plan of Action for Children in South Africa 2012-2017. The Buffalo City Declaration (March 2012). South African Integrated Programme of Action for Early Child Development - Moving Ahead (2013-2018). Maternal, Newborn, Child and Women's Health and Nutrition Strategic Plan, 2012. Campaign for Accelerated Reduction in Maternal and Child Mortality in Africa Strategy, 2012. National Health Insurance Green Paper, 2011. The Tshwane Declaration of Support for Breastfeeding, 2011. Clinical guidelines: PMTCT (Prevention of mother-to-child transmission), 2010. Social Assistance Act 13 of 2004. Prevention of Family Violence Act 133 of 1993. Domestic Violence Act 116 of 1998 (effective from 1999). Policy Framework and Strategy Document on Shelters for Victims of Domestic Violence in South Africa, 2003. Criminal Law (Sexual Offences and Related Matters) Amendment Act 32 of 2007. White Paper on Families, 2012. White Paper on Education and Training, 1995: White Paper 6: Inclusive Education, 2001. The South African National Curriculum Framework for children from Birth to Four (draft), 2012. White Paper: A new housing policy and strategy for South Africa, 1995. Free Basic Water Policy, 2000. Free Basic Water Implementation Strategy, 2001. White Paper on Basic Household Sanitation, 2001.\n\n3. The draft document was derived from the findings of the Early Childhood Development Programme, developed for the Human Sciences Research Council by Linda Richter, Lizette Berry, Linda Biersteker, David Harrison, Chris Desmond, Patricia Martin, Sara Naicker, Haroon Saloojee & Wiedaad Slemming.\n\n4. The proposal 'is grounded in, and seeks to give effect to, the government's international, regional and national legal commitments to recognise, respect, protect and promote the universal rights of all young children and their caregivers [and] the realisation of young children's rights depends on the realisation of their human rights, including their rights to social protection, basic services, health care, information and others.'\n\n5. Draft Early Childhood Development Policy, March 2015, p11.\n\n6. Stone, Alayna & Emily Page: Home Visitation Programs as an Early Intervention Strategy (Paper presented at Family Impact Seminar featuring Dr. David Olds, Washington, DC September 21, 2009) p10.\n\n7. Ibid, p1.\n\n8. Interview with Ivan Bromfield, 2015.\n\n9. Interview with Lane Benjamin, Cape Town 2015.\n\n10. Interview with Alastair Graham, VPUU, March 2015.\n\n11. Graham, ibid.\n\n12. Sampson, R: The Great American City, op cit.\n\n13. Graham, op cit.\n\n14. The last two points are from Shaw, Mark & Reitano, Tuesday: The Evolution of Organised Crime in Africa: Towards a New Response (Paper 224, April 2013).\n\n15. Interview with Major Jeremy Vearey, SAPS, Amandla! 2013.\n\n16. Southern and Eastern African Consortium for Monitoring Educational Quality Survey 2007, quoted in Africa Check, March 2015.\n\n17. Robinson, Ken: TED talk, February 2006.\n\n18. Rose, M: The Mind at Work: Valuing the Intelligence of the American Worker (Penguin, New York, 2005) pxiii.\n\n19. See Matthew Crawford: Shop Class as Soulcraft: An Inquiry into the Value of Work (Penguin, New York, 2009) for a fascinating defense of craft skills.\n\n20. Garson, B: How Computers are Transforming the Office of the Future into the Factory of the Past (Penguin, New York, 1989) p120.\n\n21. Blinder: Alan: Offshoring: The Next Industrial Revolution (Foreign Affairs, March\/April 206) quoted in Crawford, op cit, p33.\n\n22. Synthesis of the report of the TVET Colleges technical task team (Human Sciences Development Council, 2014).\n\n23. City Press, 23 June 2012.\n\n24. Kojeve, A: Introduction to Reading Hegel: Lectures on the Phenomonology of Spirit (Ithaca: Cornell University Press, 1989) p27, quoted in Crawford, op cit, p14.\n\n25. Goodreads website: www.goodreads.com.\n\n26. This information was documented by Johann Hari in Chasing the Scream: The First and Last Days of the War on Drugs (Bloomsbury, London, 2015). According to Hari, Anslinger provided McCarthy, an addict, with 'official' heroin, making the head of the Narcotics Bureau a drug dealer.\n\n27. South Africa is a party to the 1961 Single Convention on Narcotic Drugs, the 1971 Convention on Psychotropic Substances and the 1988 United Nations Convention Against Illicit Traffic in Narcotic Drugs and Psychotropic Substances. The 1961 convention prohibits cultivation and trade of naturally-occurring drugs such as cannabis; the 1971 treaty bans the manufacture and trafficking of synthetic drugs such as barbiturates and amphetamines; and the 1988 convention requires states to criminalise illicit drug possession.\n\n28. Hari, ibid.\n\n29. United Nations Office on Drugs and Crime report, 2010.\n\n30. Hari, op cit.\n\n31. Hari, ibid.\n\n32. According to a 2009 White paper by the Cato Institute, Drug Decriminalization in Portugal: Lessons for Creating Fair and Successful Drug Policies, in the 5 years after drugs were decriminalised in Portugal, drug use among teens dropped, rates of new HIV infections from sharing dirty needles dropped and the number of people seeking treatment for addiction more than doubled. Portugal boasted the lowest rate of lifetime cannabis use in people over 15 years of age at 10%. To put that into perspective, America lifetime cannabis use rate in people over 12 is 39.8%. Lifetime use of an illegal drug among 7th to 9th graders fell from 14.1% to 10.6%, drug use in older teens fell, lifetime heroin use in 16-18 year olds fell, new HIV infections in drug users fell 17%, deaths related to hard drugs were cut by more than half, treatment for drug addiction rose, as well as saved money on enforcement while increasing funding for treatment. Portugal's drug usage rates are now among the lowest in the EU for virtually every substance.\n\n33. Hari, op cit.\n\n34. Hari, ibid.\n\n35. www.globalpost.com news service.\n\n36. Interview with Brian Ford, Cape Town 2015.\n\n37. Interview with Chrysalis Academy CEO Lucille Meyer, Cape Town 2015.\n\n38. 'A shameful day for policing in SA', Sunday Times, 24 February 2013.\n\n39. Grobler, Lisa: Crossing the line: When cops become criminals (Jacana, Johannesburg, 2013) pp259\u2013266.\n\n40. Grobler, ibid.\n\n41. Interview with Lorenzo Wakefield, 2015.\n\n42. Interview with Leana Goosen, 2015.\n\n43. Interview with Captain Freddy Ledwaba, Visible Policing National Coordinator for Child Offenders, March 2015.\n\n44. Ledwaba, ibid.\n\n45. Wakefield, Lorenzo: Tracking diversion, children in child and youth care centres and other statistics on the implementation of the Act (in Where are we headed? The third & fourth year of the Child Justice Act's implementation, Child Justice Alliance 2015).\n\n46. Cape Times, 4 April 2014, p6.\n\n47. Jules-Macquet, Regan: The State of South African Prisons, edition one (Nicro public education series, 2014). Nicro is the National Institute for Crime Prevention and the Reintegration of Offenders.\n\n48. This is a similar problem experienced by the police, as will be explored in the chapter on policing below.\n\n49. SAIRR study reported in the Cape Times, March 6, 2012, p1.\n\n50. p5, ibid.\n\n51. Foucault, M: Discipline and Punish (Kindle edition).\n\n52. Ibid.\n\n53. Gresham M. Sykes, The Society of Captives: A Study of a Maximum Security Prison (Princeton: Princeton University Press, 1958), 79, quoted in Steinberg, op cit.\n\n54. Ibid.\n\n55. Muntingh, L & Ballard, C: Report on Children in Prison in South Africa, Community Law Centre 2012.\n\n56. Muntingh, L: The Law and the Business of Criminal Record Expungement in South Africa, Civil Society Prison Reform Initiative, Report 18, 2011, p5.\n\n57. Van Zyl Smit D (2003) 'Civil disabilities of former prisoners in a constitutional democracy: building on the South African experience' Acta Juridica , pp221-237.\n\n58. Muntingh, op cit, p23.\n\n59. This insight is from Joseph Pearce: Evolution's End: Claiming the Potential of our Intelligence (Harper, San Francisco, 1992).\n\n60. Garbarini, J, Dubrow, N, Kostelny, K & Pardo, C: Children in Danger: Coping with the Consequences of Community Violence (Jossey-Bass, San Francisco 1992).\n\n61. Masten, Ann: 'Resilience Processes in Development' (American Psychologist, Vol. 56, No. 3, 227-238 DOI: 10.1037\/\/0003-066X.56.3.227, March 2001).\n\n62. From Michael Rutter: 'Resilience concepts and findings: implications for family therapy' (Journal of Family Therapy (1999) 21) pp119\u2013144.\n\n63. Ungar, Michael: (2004) Nurturing Hidden Resilience in Troubled Youth. (University of Toronto Press 2004) p32.\n\n64. Michael Rutter, op cit, p135.\n\n65. From Masten, AS and J Obradovi\u0107: 'Competence and resilience in development' (Annals of the New York Academy of Sciences, vol. 1094, 2006) pp13\u201327.\n\n66. Rosseau, E & D Pinnock: Wild Resilience: Working with High-Risk Adolescents using Wilderness, Ritual and Mentorship (Usiko Trust, Cape Town, 2014).\n\n67. Interview with Grant Stewart, 2015.\n\n68. Interview with Lane Benjamin, 2015.\n\n69. Quoted in Owen, Michelle & Abraham Reeff: 'Factors Attracting and Discouraging Adolescent Boys in High-Prevalence Communities From Becoming Involved in Gangs' (Journal of Forensic Psychology Practice, Vol 15, Issue 1, 2015).\n\n70. Interview with Karl Voysey, 2015.\n\n71. Stuart, op cit.\n\n72. Garbarino, Lost boys, op cit, p213.\n\n73. Ann Masten and Dante Cicchetti: 'Developmental cascades' (Development and Psychopathology 22 (2010).\n\n74. Kristen Moilanen, Daniel Shaw & Kari Maxwell: 'Developmental cascades: Externalizing, internalizing and academic competence from middle childhood to early adolescence' (Developmental Psychopathology, August 2010: 22(3).\n\n75. See J. Douglas Coatsworth: A Developmental Psychopathology and Resilience Perspective on 21st Century Competencies (National Research Council, 2010) and Michael Rutter:'Resilience concepts and findings: implications for family therapy' (Journal of Family Therapy 21,1999) pp119\u2013144.\n\n76. Meyer Interview, op cit.\n\n77. Interview with David Abrahams, 2015.\n\n78. Interview with Lane Benjamin, 2015.\n\n79. Drapeau, Sylvie, Marie-Christine Saint-Jacques, Rachel L\u00e9pine, Gilles B\u00e9gin & Martine Bernard: 'Processes that contribute to resilience among youth in foster care' (Journal of Adolescence 30 (2007) 977\u2013999) p286.\n\n80. Drapeau et al (2007) p994.\n\n81. Interview with Ashley Potts, Mitchell's Plain, 2015.\n\n82. Interview with Lane Benjamin, 2015.\n\n83. Potts, op cit.\n\n84. Interview with Stewart, 2015.\n\n85. Benjamin, op cit.\n\n86. Rutter, op cit, p119.\n\n87. Drapeau et al (2007) p979.\n\n88. Potts, op cit.\n\n89. Moloney, Molly, Kathleen MacKenzie , Geoffrey Hunt and Karen Joe-Laidler: 'The path and promise of fatherhood for gang members' (Advance Access publication 17 February 2009) p321.\n\n90. Decker, Scott, David Pyrooz , & Richard Moule: 'Disengagement From Gangs as Role Transitions' (Journal of Research on Adolescence, 24(2), 268\u201328) p269.\n\n91. Meyer. op cit.\n\n92. Benjamin, op cit.\n\n93. Ward, op cit.\n\n94. Interview with Lerato Kossiee, 2014.\n\n95. Meyer, op cit.\n\n96. Boyle, Greg: Tattoos on the Heart: The Power of Boundless Compassion (Simon & Schuster, New York 2010).\n\n97. Decker et al, op cit.\n\n98. Harari, Yuval Noah: Sapiens: A Brief History of Humankind (Harville Secker, London 2011).\n\n99. Tolkien, JRR: Lord of the Rings: The Two Towers. https:\/\/www.youtube.com\/watch?v=AlyYBtASteQ.\nAppendix: Working with high-risk youths: a toolbox for parents, teachers and community workers1\n\nHow to talk to adolescents\n\nAdults are programmed to protect, so it's difficult not to interfere when a young person has to make a decision. But we must learn not to interfere. One of the most important skills young people have to develop is to be independent and make decisions in their own lives. How else could they construct their story? They will make mistakes, but you have to allow for the natural consequences of a young person's decisions. Later you can explore with them why a decision worked or didn't work and figure out new strategies with them.\n\nYoung people with tough beginnings from hard neighbourhoods don't talk much, especially to adults. They think they have nothing to tell because their story is one of pain and shame and is hidden. Attempts to induce it will provoke anger and even violence. But it's the bedrock upon which a new story can be built so they need to tell it. So the more your story parallels theirs, the more your empathy will show. You need to let them know you understand and have been there before them.\n\nAn essential part of this process is active listening. People don't always listen 'fully' when someone is speaking to them. Often they listen to background conversations or music, or think about other things and don't hear what's said because they're more concerned with other things. Sometimes they interrupt and give their own opinion or even change the subject. To listen 'actively', you direct your full attention to what is being said, pay attention to what the young person is saying and you block out everything else. Active listening builds trust and it helps bridge differences. A young person feels respected and cared for when someone listens to them.\n\nListening to someone doesn't only involve your ears. You can also 'listen' by observing their posture and body language. You can read a person's facial expressions, body movements and tone of voice. You'll 'hear' a lot from their reactions when they're challenged or questioned about painful matters.\n\nThe most basic guidelines for becoming an active listener is to be interested and show that you are. This is done by asking questions, not interrupting, by focusing on what you've been told and remembering it. Don't judge the person speaking to you and don't be distracted by your own stray thoughts, imaginings, cellphone or other people.\n\nListening also involves asking questions and not offering suggestions. When you ask questions, it means you're interested in what the person is saying, that you want to know more and are there for them. Questions prevent misunderstandings. Most importantly, by asking questions you develop listening skills.\n\nIt's also important to be genuine. This means being truthful and honest about how you feel and not trying to put up a front or be false, but absolutely 'yourself'. If you're not being real and genuine, you send mixed messages. A young person will quickly pick this up. Let your body language show that you're an active listener.\n\nBe empathetic not sympathetic. Empathy is the deep feeling for and understanding of another person's situation. When you feel empathy for someone, you feel as though you yourself are having the same emotional experience. Empathy is like walking in the young person's shoes. When you feel sympathy for someone, on the other hand, you're feeling sorry for them. It doesn't show that you understand how they feel. It's different from empathy, because you are not feeling with them.\n\nDon't demand compliance\n\nNothing we know about the human animal suggests that we've been programmed to be obedient. There are better ways to change adolescent behaviour than demanding compliance and we need to move beyond demands to an understanding of what it is that captures and holds their attention. This understanding involves an awareness of unmet needs, discouragement, the misery of unimportance and loss of self-esteem and self-control. But it also involves finding ways to mobilise adventurous spirit, satisfy the deep need young people have for ritual, increase their personal social resilience and create meaningful bonds with significant adults. We also have to realise that within the tough delinquent exterior is a need to play \u2013 with fire perhaps \u2013 but also with the world to see what it will answer.\n\nNurturing self-esteem is critical to working with young people at risk. Its loss can begin with something as small as a sarcastic put-down from a parent or as large as the collapse of the social support unit. In adolescence this can lead to feelings of insignificance, incompetence, powerlessness and unworthiness. A search for the converse of these feelings \u2013 significance, competence, power and virtue \u2013 can easily be discerned in gang behaviour.\n\nHow to build trust\n\nEvery young person needs an adult to learn from in order to become an adult themselves. Parents are a child's main role models, but many young people have parents who are inadequate role models or who are simply 'not there' for their kids. Therefore, especially if they've been hurt or let down by adults, the way you go about forming a relationship with young people is very important. There's a great deal of power involved in such a relationship \u2013 it can be life changing or devastating. In traditional societies this relationship is formalised as mentorship but largely lost in modern urban life. It's an essential part of programme development.\n\nBut as a mentor you have to be careful. Young people whose parents treat them with disrespect, who neglect and abandon them, or who avoid and disregard them are churned up inside. They're torn between their own human need for love and affection and feelings of anger, resentment and anxiety. They need you but they've learned that adults cannot be trusted. Trust is a fragile thing, it needs to be earned and cannot simply be expected. To trust means to have faith in another person and believe they will be considerate, protective and helpful and not betray you or lie to you or hurt you intentionally. When there's trust, then respect follows.\n\nWhen a young person trusts you, they'll be open and speak about their lives, feelings and experiences. They'll tell you the story of themselves they never knew would interest anyone. They'll look to you for guidance and will respect and value your opinion. But this doesn't mean they'll necessarily do what you say. Trust is formed when we feel that people accept us for who we are and is increased through co-operation.\n\nWhen there's trust, a person feels safe to say what they really think and feel and know it is regarded as important. This builds self-confidence. Trust is broken when you try to manipulate or coerce and when you use guilt trips and threats. When trust is damaged, it's difficult to repair and restore. Here are actions which break trust:\n\n(L*O*S*E*R)\n\nLaughing at a person when they say something about themselves;\n\nOpenly judging or moralising another person's behaviour;\n\nSilent, detached, unemotional and rejecting actions;\n\nEvaluating another person in your response to them;\n\nRefusing to reciprocate in openness and sharing.\n\nWhen you mentor young people, you need to believe that each one has the potential to make good decisions about where and how they'll live their lives. You have to find ways to be a friend without being threatening. Have high expectations of your relationship with the young person, but don't expect too much at first. Accept the young person for who they are, without judgement. Establish what they're capable of and help them reach that height \u2013 everything you aim for should be attainable. Never allow yourself to feel disappointed if the young person doesn't reach as high as you want them to. Building up a relationship with a troubled person takes time and patience. Teenagers can behave horribly and say cruel, hurtful, abusive things. Mentors must be able to not take these to heart. It's not really you they're talking about.\n\nWhen discussing an issue, mirror it by repeating what they tell you. Put the young person's words into your own, help them reflect on what they've said, compliment them when they do something positive. This will confirm that you understand. When they do something negative, discuss this and say what you think and feel about it. Remember not to judge or condemn. It's not the person that you may disagree with, but their behaviour. So say, for example, 'Your actions were unacceptable,' and not 'You are unacceptable'.\n\nEvery three to four months the Chrysalis Academy in Cape Town has a new intake of young people from high-risk, low-income areas, many of who are mistrustful, scared or angry. Lucille Meyer describes the process of mentoring them to begin the journey into a new story about themselves:\n\nTheir socialisation has been one of: 'I'm poor, I'm a victim, I'm not loved.' Then you arrive here and these weird people are talking about compassion and that they love me. You think they're mad and you don't trust them. So we have to show them what love and what safe looks like. We the mothers of this place have to be there for them. They have to hear it in my voice that there's care. They've been on high alert, guarded. Thick shells of protection. We have to rewire their brains! That takes time. We have to link them up with supportive people. It takes certain types of people to do this work. These are people who have first worked on themselves. They have to have integrity because the kids who come here don't trust adults. We have to have high expectations.2\n\nAs a mentor, here's what you should not do:\n\n\u2022 Don't put up a 'front' or be false \u2013 always be true to yourself;\n\n\u2022 Don't be quick to judge;\n\n\u2022 Don't lose your temper, shout or scold;\n\n\u2022 Don't get physical in any way;\n\n\u2022 Don't try to be a parent or a teacher. Instead approach the young person as a friend and guide;\n\n\u2022 Don't speak about how things were done in the past and don't say the past was better than now;\n\n\u2022 Don't present yourself as an authority figure;\n\n\u2022 Don't demand respect. Instead, let the young person grow to trust and respect you;\n\n\u2022 Don't force your personal or religious beliefs onto the young person;\n\n\u2022 Don't let your own values and morals get in the way of a positive and healthy relationship with the young person;\n\n\u2022 Don't try to manipulate the young person in any way. Don't coerce them;\n\n\u2022 Don't use commands like 'you must' or 'you have to' or 'you can't' or 'you shouldn't';\n\n\u2022 Don't expect miracles. Positive and lasting change takes time with young people, especially when they have difficult lives.\n\nCreating safe space\n\nA safe space is both physical and psychological \u2013 in the world and in yourself. In high-risk areas and within a gang, both of these are largely absent. In this situation young people often withdraw into themselves emotionally or attempt to clear a space around themselves by dominating physically.\n\nFor these reasons, any programme working with home-stressed or gang-linked young people needs to make physical and psychological safety a priority. These ingredients are difficult to achieve in the areas in which these young people live, which is why locating programme activities outside the neighbourhood increases their effectiveness.\n\nNature can provide a physical space which, while initially anxiety pro\u00advoking in its unknownness, can provide silence and a soft landing for young people used to crowding, urban noise and the need for constant vigilance. In this situation, a mentor who is clearly unafraid of the environment, feels secure in themself and is an anchor in the unknown, provides the safety to allow the possibility of reflection and inner recalibration. 'In the wild first they're scared, Ashley Potts told me, 'then they're interested, then they relax:'\n\nGoing from the city into nature is like going from air into water. You're dry one moment then you're into the unknown depths of the sea. It's a switch. You require a different set of skills. You have to move and breathe in a different way. It's scary, but when you learn to swim you don't want to come out the water again.3\n\nCommunity psychologist Lane Benjamin considers that a safe space can also be a trusting relationship without barriers. 'We can't supply safe spaces anywhere in high-risk communities but we can find it in relationships. Other people have your back.' For psychologist Cathy Ward, however, visualising a different path needs a change of place. 'It's not possible when they're stuck in Manenberg or Bonteheuwel so they need to be out of their context out of choice. These resources have to be there at the right time.4\n\nThe hero's journey\n\nEvery teenager, at some point, needs to be their own hero and be well regarded by their peers. Recognition by an older person is an added bonus but, in the areas where gangs flourish, is seldom gained. In the circle of a gang, violence gives you credability and, beyond that, a nod from the leader or druglord is as much as you're likely to get. If we're to provide the ingredients for a story more sustaining than what a gang can provide, what would it look like?\n\nThough such a journey may happen in the everyday world, the steps a young person takes at this time of their life are part of ancient rituals hard-wired in their brains. At root it's a search for belonging, mastery, independence and forgiveness.5\n\nFigure 10 Circle of Wholeness\n\nHumans are social creatures and belonging is the baseline from which personality develops. Two Xhosa expressions capture this. One is the notion of ubuntu \u2013 'a person is a person through other people', and the other is the saying 'all children are my children'. In traditional societies this sense of belonging extends beyond the human circle to animals, plants and the environment. In peasant society, maintaining balanced ecological relationships is a way of ensuring that one's own life is itself balanced. Belonging depends on how one is viewed by others. For this reason, being committed to the positive value of generosity and caring for others improves ones view of oneself through the eyes of others. Young people cannot develop a sense of responsibility unless they have been responsible to others. Generosity allows them to 'de-centre' and contribute to those around them in a self\u2013affirming way.\n\nMastery involves social and physical competence and opportunities for success. It's the basis of individual worth in most societies and education systems. If young people are deprived of the chance or ability to master their lives, they retreat into helplessness and inferiority. The steps to mastery are as old as they are modern. In ancient Egyptian teachings for priesthood, young students were required to seek:\n\nControl of thought,\n\nControl of action,\n\nFaith in the Master's ability to teach the truth,\n\nFaith in one's ability to assimilate the truth,\n\nFaith in oneself to wield the truth,\n\nFreedom from resentment under persecution,\n\nFreedom from resentment under wrong,\n\nAbility to distinguish right from wrong,\n\nAbility to distinguish real from unreal.6\n\nThe spirit of independence is the product of both mastery and belonging, in that the purpose of any external discipline and support is to build inner discipline and social worth. Young people who lack a sense of power over their own behaviour and environment often lack motivation and seek alternative sources of power through dependence on chemicals or membership in a youth subculture.7\n\nWhere hurt has been caused, especially by a parent, a young person can carry the anger and shame for the rest of their lives, unable to shake off the burden of its memory. This can increase anger and anti-social actions and reduce acceptance among those with which a young person would like to belong. The key that unlocks these shackles is forgiveness, moving past the scars of hurt, shame and anger and climbing upwards without the dead weight of past damage.\n\nIf programmes for young people do not have as their goal the development of positive characteristics such as a sense of belonging, mastery, independence and forgiveness \u2013 unless they avoid emotional superficiality and build significance, personal power, self esteem and self control \u2013 they will not change the attitudes of young people and will probably fail. Here are two stories about a hero's journey:\n\nIt was very hard. I wanted to come home (from the wilderness), but we had to sign papers before we went to show we knew it would be hard. Before we went we took it for granted that we could do everything.\n\nWe went on hikes. We climbed rocks and once you got to the top and you looked down you didn't want to go back. But they told us we could do it. We abseiled and river-rafted. There were life-savers so it was okay if we fell in.\n\nAnother thing was the expedition. We walked a long way. We carried heavy bags with food and things. We started in the morning and walked until evening. They told us the whole way was 36 kilometres. I wanted to quit, to say: 'No, I'm not doing it.' But I signed so I did.\n\nNow I am a man. I have stayed in a forest alone and walked on the top of mountains.8\n\nA young guy abseiling down a cliff said to me: 'Dis makliker om 'n mens dood te skiet as om hier af te klim' [It's easier to kill a person than climb down here]. That statement hit me like a ton of bricks. Suddenly he had his own life in his hands, working with the rope, making decisions about his own safety. That led to a two-hour conversation. Killing and hurting had become such a part of his life that it was just normal. It took no effort. Coming down a cliff was much more difficult for him.9\n\nUsing fear and ritual\n\nFor adolescents, the catalyst for transition to a new path, of creating a new sense of self, is the realisation that they're in an impasse, a situation they cannot control which causes them fear. On the street this can lead to irrational action, anger and violence or the protection of a gang, but it can also be the start of a search for a new story and a call for help from someone they can trust.\n\nIn placing a young person in an unknown environment which they have never before encountered with the expectation that 'something' is about to happen, an impasse can be created. Carefully managed anxiety can act as a doorway to a new set of experiences and an openness to intervention by a mentor. For this reason wilderness, which urban youths may never have encountered, is an effective context as it immediately places them outside their comfort zone.\n\nHere's an example. In the Usiko process, developed over many years of testing, youngsters arriving off the bus in a wild place are met by several men in wild dress carrying sticks who loudly and aggressively demand to know of each youth who they are and why they're there. On hearing their name they soften and say: 'Oh yes, we know about you, you can enter here,' precipitating brief fear followed by relief of acceptance. These contradictory feelings are the benchmarks of ritual.\n\nRitual is a gantry over the abyss of fear. It says: 'Do these things and you will have no need for fear.' But fear drives the acceptance of ritual. They are the yin of each other's yang. Adolescents respond powerfully to ritual and performance and abhor superficiality. In his book The magic of ritual, Tom Driver provides a useful starting point for ritual programme development:\n\nA ritual is a performance that invokes the presence and action of powers which, without ritual, would not be present or active at that time and place, or would be so in a different way. The most obvious examples of such powers, no doubt, are divinities, demons, ancestors and other spirits that may be called 'supernatural'; but they may also be certain powers of nature, of society, of the state, or of the psyche. The agencies with which ritual is concerned, then, are such that they may be represented symbolically if they are to be depicted at all.10\n\nAdolescence is a time when ritual becomes important and, in its absence, they will create their own. Sport, dance, sex, forms of dress and even mode of travel are part of ritual transformance, as are alcoholic binges, drug-taking, violence, tattooing and joining a gang. For this reason, ritual needs to be at the heart of any programme involving young people, especially if they are at risk, and fear needs to be that which they overcome in order to make the ritual process worthwhile to them. As Lucille Meyer says, 'ritual is your keel and rudder in the sea of life. It talks to older things in you.' But, notes Grant Stewart, 'there needs to be a risk activity to start the process.'11\n\nImportantly, although situations that provoke fear or anxiety are what adults instinctively wish to protect young people from, they are also spaces in which fear can be overcome and from which stories of bravery can be crafted. They offer the ingredients out of which belonging, mastery, independence and forgiveness can emerge and from which a story of self can be told. In large measure, this is what gangs offer. We need to learn from them.\n\nTaking time out\n\nYou cannot tell your story if you haven't first become it. Finding what it is you can tell requires that you reflect on what the journey means to you and in what way it connects to your place in society. Its impact also depends on the degree of its universality: what truths does it contain that are recognised by others as their truths as well? Such insights cannot be had on in the midst of the journey while battling the stony path. Any programme that requires self-reflection, therefore, needs to have as part of its process the time to do it.\n\nMany programmes that use wilderness settings have found that solo time in nature, especially overnight, can produce powerful effects and even profound epiphanies. With nothing physical to do, nowhere to go and the challenge of 'making it through', self-reflection is almost assured. Such programmes generally require participants to consider certain questions, such as 'Who am I?' and 'Where am I going?'\n\nBoth Educo and the Usiko Trust use principles based on a 'life wheel' developed by Steven Foster and Meredith Little as a conceptual framework for self-reflection. The participant constructs a circle out of whatever's available, sits within it and contemplates their life using the points of the compass as orientation. South represents childhood, innocence and playfulness, west represents darkness, anxiety, pain and adolescence, north represents adulthood, success, establishment and inflexibility and east represents age, wisdom, death and rebirth into the south. The task is to find where you are on the wheel and to understand it, not as a fixed state but point of transition.12\n\nLane Benjamin notes: 'We need to help young men in the community to have the time to think so they're not operating on that impulsive survival brain. Then they need to have access to other men they can talk to and physically walk the journey with them. It's a journey and a process. But they need time to think.'13\n\nMake ground rules\n\nPeople working together need a set of simple rules that concern respect and co-operation so everyone knows where they stand. Young people, particularly those from high-risk and often chaotic backgrounds, need to have a sense of structure and know beforehand what they can or can't do when working together. Members will be empowered if they make their own rules to fit their own needs. However, there are certain rules that must be put in place, regardless of the rules people choose. These are:\n\n\u2022 Respect the fact that you are dependent on each other;\n\n\u2022 Agree to the goals and commit to working co-operatively to reach those goals;\n\n\u2022 Never be negative about anyone's race, sex, nationality, religion, language, political opinion or social position;\n\n\u2022 Respect everyone's gender, sexual orientation, disability or any other status;\n\n\u2022 Recognise that you are all reasonable people;\n\n\u2022 Accept that no one is perfect and that everyone makes mistakes;\n\n\u2022 Allow each other to make mistakes, but help each other see those mistakes without judging harshly;\n\n\u2022 Give everyone the right to say what they think;\n\n\u2022 Express all opinions in a sensitive manner;\n\n\u2022 Everything shared in the group is confidential.\n\nGiving feedback\n\nFeedback is the useful information you give a person after a session or programme. It tells them how they're doing and increases their awareness of their performance against the standards expected of them. Feedback has to be honest, but it must not cause hurt. These are useful tools:\n\n\u2022 Give feedback on a person's behaviour and not on their personality;\n\n\u2022 Give feedback on their behaviour within a specific situation and not on their general behaviour;\n\n\u2022 Give feedback on their immediate behaviour and not on their behaviour in previous situations;\n\n\u2022 Give feedback that is positive and not judgemental or negative;\n\n\u2022 Share your perceptions and feelings, but don't give advice;\n\n\u2022 Give feedback only when you are asked to by the person;\n\n\u2022 Focus your feedback on actions that the person is able to change.\n\nA programme is only part of the journey\n\nNo matter how all-encompassing a programme may be, it's only a small slice of a young person's life. They may return with stories to tell and new insights about themselves, only to be met with incomprehension and even disbelief by parents and peers. Stories of their Hero's Journey may hold for a time, but will begin to be eroded in uncongenial environments. Resilience takes a hammering. Moving away from your context changes something, says Grant Stewart:\n\nThe challenge is when they come back, everyone around him says: 'Jy's mos 'n dief, 'n skelem' [you're just a thief, a rubbish]. And he's poor and the gang comes knocking. 'Sell a few of these things and you'll have money.' You need to create spaces where they hear different voices \u2013 and a path out.\n\nAny programme not integrated into path planning for young people is ineffective and a waste of time and money. They need to be supported in their neighbourhoods after its completion if they're to maintain resilience. For this reason, government and NGOs should collaborate and silos of exclusivity in youth work transcended. We have to remember why we do this work. It's about the young people. They are the future.\n\nEndnotes\n\n1. These suggestions were drawn from Elzette Rosseau and Don Pinnock: Wild Resilience: Working with High-Risk Adolescents using Wilderness, Ritual and Mentorship (African Sun Press, Cape Town, 2014).\n\n2. Lucille Meyer interview, 2015.\n\n3. Interview with Ashley Potts, op cit.\n\n4. Interview with Cathy Ward, op cit.\n\n5. Brendto et al, 1990, p35.\n\n6. G James: Stolen legacy (Julian Richardson, San Francisco, 1976).\n\n7. James, 1976, p41.\n\n8. Karen van Eden: research notes, Institute of Criminology, UCT 1996.\n\n9. Ashley Potts, op cit.\n\n10. Driver, 1991, p97.\n\n11. Interview with Grant Stewart, op cit.\n\n12. Foster, Steven & Meredith Little: The four shields: the initiatory seasons of human nature (Lost Borders Press, Big Pine, California 1998). See also Elzette Rosseau & Don Pinnock: Wild Resilience, op cit.\n\n13. 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(1992) Afrocentric Theory and Applications: Advances in Adolescent Rites of Passage (Vol 2) Washington: Baobab.\n\nWebster, D: The social organization of poverty (African Studies Institute, Wits University, 1980).\n\nWestern Cape Youth Development Strategy 2013 (Western Cape Government).\n\nWestern, J: Outcast Cape Town (Cape Town, 1981).\n\nWhite Paper 5 on Early Childhood Development, 2001.\n\nWhite Paper 6: Inclusive Education, 2001.\n\nWhite Paper for the Transformation of the Health System in South Africa, 1997.\n\nWhite Paper on Basic Household Sanitation, 2001.\n\nWhite Paper on Education and Training, 1995.\n\nWhite Paper on Families, 2012.\n\nWhite Paper on Social Welfare, 1997.\n\nWhite Paper: A new housing policy and strategy for South Africa, 1995.\n\nWhittles, Govan: 'Crime in SA increasing rapidly' (Eyewitness News online, 22 May 2015).\n\nWHO Global Alcohol and Health Report: Africa 2007.\n\nWieland, F: The Journey of the Hero: Personal growth, Deep Ecology and the Quest for the Grail (Prism Press, Dorset, 1991).\n\nWilkinson, Kate: 'Why the matric pass rate is not a reliable benchmark of education quality' (Africa Check 7.1.2014).\n\nWilliams, P & Vlassis, D: Combatting Transnational Crime: Concepts, Activities, and Responses (Transnational Organized Crime, 4(4), 2001).\n\nWorld Health Organisation Report on violence and health, (2002).\n\nZabludoff, S: 'Colombian narcotics organizations as business enterprises' (in Transnational Organized Crime, 3(2), 1997).\nPhoto section\n\nHappy times in District Six, now a distant memory. Picture: Cloete Breytenbach\n\nTaking in the street life of District Six. Picture: UCT Archives\n\nRutger Street, District Six. Picture: UCT Archives\n\nDistrict Six next to the CBD before demolition. Picture: UCT Archives\n\nA rare photograph of the Globe Gang of District Six, circa 1944. Picture courtesy of Shahidah Maroos\n\nEaton Terrace, District Six. Picture: District Six Museum\n\nTeenagers in District Six. Picture: UCT Archives\n\nFor years the apartheid government tried to block the Coon Carnival's assault on the city's streets but never managed. Here, to the strum of banjos, they claim Hanover Street as their own. Picture Cloete Breytenbach\n\nVernon Terrace, District Six. Picture: District Six Museum\n\nThe sound of the fish seller's horn announced the return of the fishing boats that employed many of the District's men. Picture: District Six Museum\n\nHomes being bulldozed in District Six amid the forced removals of the 1960s and '70s. More than 60 000 people were moved to the Cape Flats by the apartheid regime. Picture: District Six Museum\n\nA crying child in shack land has to find comfort wherever he can.\n\nA police gang map of Bonteheuwel.\n\nBottle neck in hand and high on a mixture of mandrax and cannabis.\n\nAuthor's note: Eyes have been obscured in some pictures to protect identities.\n\nNight time in the tenements. Pictures: Don Pinnock\n\nAt night many Cape Flats neighbourhoods seem cloaked in dangerous unpredictability.\n\nCaught in the act of stealing a car, a 15-year-old boy finds he can't outrun the police.\n\nPrison gang hand signs are an indicater of allegiance to the 26s, 27s or 28s (five + one, two or three fingers). Pictures: Don Pinnock\n\nFor gangs, turf is marked and fiercely defended. This is Mongrels territory.\n\nWith high unemployment, the informal sector is the mainstay in many Cape Flats areas. Pictures: Don Pinnock\n\nA Metro Police squad holds off a crowd as it moves in on an arrest scene.\n\nAt any time of the day or night, the streets of the Cape Flats are filled with young people hanging around doing very little.\n\nTaking their chances with the roll of a dice in Hanover Park. Pictures: Don Pinnock\nAcknowledgements\n\nThis study was initiated by:\n\nIt was supported by:\n\nThis book could not have been written without the work and help of many professionals in a number of fields. Special thanks goes to people who made time to check sections dealing with their speciality. They are Professor Andy Dawes, Professor Elrena van der Spuy, Professor Cathy Ward, Dr Simon Howell, Dr Barak Morgan, Major General Jeremy Vearey, David Abrahams and Jane Kelly. The insights were theirs, any errors are mine.\n\nMy special thanks go to Andy Dawes for his wisdom, patience and friendship and to my wife Patricia Schonstein who supported me with forbearance and care whether I was hunched over a keyboard or hanging out in gangland.\n\nI'm grateful for the insights of Cyril Pelston, Jonathan Jansen, Guy Lamb, Alastair Graham, Lucille Meyer, Jeremy Vearey, Peter Jacobs, Lerato Kossie, Irvin Kinnes, Ann Skelton, Barbara Williams, Oscar Wollheim, Ashley Potts, Pharie Sefali, Grant Stewart, Ivan Bromfield, Brian Ford, Joan Broster, Chief Dalaguba, Nzimela Ncoyini, Karl Voysey, Lane Benjamin, James Garbarino, Murray Michel, John Walker, Kathy Nicolaou, Freddy Ledwaba, Burt Evans, Sylke Funk. I also am indebted to the gang members who shared their insights and lives but did not want their surnames mentioned: Dice, Dave, Gums, Petrus, Chicken, Paul, Nealon, Michael, Trappies, Joey, Aala, Sira, Clifton and Simon. Important support from Tafelberg Press came from Annie de Beer and Gill Moodie. If I have left anyone out I apologise, but you know who you are.\n\nThis book was kick-started by the City Press Non-Fiction Award, supported by the University of Cape Town Vice-Chancellor's Strategic Fund and its patron Dr Max Price and went into print with the support of Tafelberg Press. I had the fortune of two excellent editors, Patricia Schonstein for the first draft and Sybrand Mostert for the final. Before good editors I am truly humble \u2013 they ensure that things come out right.\nAbout the book\n\nCape Town is two cities. One is beautiful beyond imagining, known since its beginning as the 'fairest cape' in the world. Here tourists come to lounge on beaches, scale misty peaks and dine in fine restaurants.\n\nThe other is one of the most dangerous cities in the world, where police need bullet-proof vests and sometimes army backup. Here gangs of young men rule the night with heavy calibre handguns, dispensing heroin, cocaine, crystal meth and fear.\n\nThis is the story of the second city...\n\nIn Gang Town, investigative journalist and criminologist Don Pinnock draws on more than thirty years of research to provide a nuanced and definitive portrait of youngsters caught up in violent crime.\nAbout the author\n\nDr Don Pinnock is an investigative journalist and photographer whose assignments have taken him to five continents. He has written 17 books, hundreds of articles and held several photographic exhibitions. He has degrees in criminology, political science and African history and has been a travel writer, photographer, biographer, historian and lecturer. He is an honorary research associate of the Centre of Criminology and the Safety and Violence Initiative at the University of Cape Town, a founding member of the Usiko Trust working with high-risk youths and is a trustee of the Chrysalis Academy. He is married to the novelist and poet Patricia Schonstein and they have two adult children. They all live at the foot of Table Mountain.\n\nPraise for Gang Town\n\n'As a police officer often on the wrong end of a gangster's gun, this critical work has given me a startling insight into the world of the young man whose finger is on the trigger. To those who believe countless Cape Town youths are lost forever to the grip of tik or death by 9mm Parabellum handgun, read this book and hope.'\n\n\u2013 Andrew Brown, novelist and police reservist\n\n'You will not find a more insightful and unsettling book on gangs in Cape Town. Through unforgettable imagery, first-hand stories and a lifetime of research on troubled youth in this afflicted city, it helps us understand not only how gangs came to be and are sustained, but how they destroy young lives and whole families can be overcome. Gang Town might well become the most important resource for generations of social scientists grasping to understand how one of the world's most beautiful cities could come to be so disfigured by gangsterism.'\n\n\u2013 Prof Jonathan Jansen, vice-chancellor, University of the Free State\n\n'This is not only an absorbing history of Cape Town but an insight into the city the like of which I'd not come across before... fascinating and deeply troubling but at least offers a way out of what looks like an intractable problem.'\n\n\u2013 Mike Nicol, novelist\n\n'This book contains cogent, yet accessible arguments about the evolution of gangs on the Cape Flats. It explores both the criminogenic impact of gang activity as well as its historical roots. In doing so, it offers an integrated strategic solution to the problem.'\n\n\u2013 Maj-Gen Jeremy Vearey, SAPS head of anti-gang strategy in Western Cape\nAlso by Don Pinnock\n\nCRIMINOLOGY\n\nElsies River\n\nThe Brotherhoods\n\nGangs, Rituals & Rites of Passage (with Dudu Douglas-Hamilton)\n\nWild Resilience (with Elzette Rosseau)\n\nHISTORY\n\nTelona: The history of a private labour recruiter\n\nWriting Left\n\nBIOGRAPHY\n\nRuth First\n\nThey Fought for Freedom: Ruth First\n\nTRAVEL\n\nBlue Ice: Travels in Antarctica\n\nAfrican Journeys\n\nThe Woman Who Lived in a Tree\n\nSouth Africa, a Celebration (with Gerald Hoberman)\n\nENVIRONMENT\n\nNatural Selections\n\nLoveletters to Africa\n\nFICTION\n\nRainmaker\nTafelberg,\n\nan imprint of NB Publishers,\n\na division of Media24 Boeke (Pty) Ltd,\n\n40 Heerengracht, Cape Town, 8001\n\nwww.tafelberg.com\n\nCopyright \u00a9 Don Pinnock 2016\n\nAll rights reserved.\n\nNo part of this electronic book may be reproduced or transmitted in any form or by any electronic or mechanical means, including photocopying and recording, or by any other information storage or retrieval system, without written permission from the publisher.\n\nCover image by Don Pinnock, posed by model\n\nCover design by Gaelen Pinnock\n\nE-book design by Firelight Studio\n\nAvailable in print:\n\nFirst edition in 2016\n\nISBN: 978-0-624-06789-4\n\nEpub edition:\n\nFirst edition in 2016\n\nISBN: 978-0-624-06790-0 (epub)\n\nMobi edition:\n\nFirst edition in 2016\n\nISBN: 978-0-624-06791-7 (mobi)\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nThank you for downloading this Crossway book.\n\nSign-up for the Crossway Newsletter for updates on special offers, new resources, and exciting global ministry initiatives:\n\n**Crossway Newsletter**\n\nOr, if you prefer, we would love to connect with you online:\n\n**Facebook**\n\n**Twitter**   \n**Google +**\n\"It is commonplace among many church leaders to dispute the need for confessions of faith on the grounds of the supreme authority of the Bible. In this timely book, Trueman demonstrates effectively how such claims are untenable. We all have creeds\u2014the Bible itself requires them\u2014but some are unwritten, not open to public accountability, and the consequences can be damaging. Trueman's case deserves the widest possible hearing.\"\n\nRobert Letham, Director of Research, Senior Lecturer in Systematic and Historical Theology, Wales Evangelical School of Theology; author, The Holy Trinity and Union with Christ\n\n\"In its creeds and confessions, the church affirms its allegiance to the God of the gospel and commits itself to think, speak, and govern its life in ways shaped by the gospel. This lively book, full of vigorous argument and biblical good sense, tells us why.\"\n\nJohn Webster, Professor of Systematic Theology, University of Aberdeen\n\n\"The apostle Paul once told Timothy that a minister was to be kind, able to teach, patient, and gentle (2 Tim. 2:24). In The Creedal Imperative, Carl Trueman demonstrates that he is not only able to teach the Word and how it has come down to us throughout history, but also how to do so with kindness, patience, and \u00adgentleness\u2014precisely the qualities that are needed to convince in our creedless, ahistorical, and shallow age. As one whose entire ministry of preaching, teaching, and writing has been taken up with the Word as confessed in the great creeds and confessions of Christendom, I wholeheartedly recommend this book.\"\n\nDaniel R. Hyde, Pastor, Oceanside United Reformed Church, Oceanside, California; author, God in Our Midst; Welcome to a Reformed Church; and Why Believe in God?\n\n\"Herein is a truly inspiring vision, that churches be freed from the vapid, the fickle, and the dysfunctional by a deeper enjoyment of the faith we have received. Trueman has shown that use of the creeds is both necessary and beautifully enriching. Informative and compelling, this book has what it takes to do great good.\"\n\nMichael Reeves, Head of Theology, Universities and Colleges Christian Fellowship (UCCF); author, Delighting in the Trinity and The Unquenchable Flame\n\n\"I know of few people better equipped to write this book. As both a scholar and a pastor, Trueman combines his expertise as a historian with some important biblical observations to make a convincing case for The Creedal Imperative. This book will prove to be immensely useful in today's ecclesiastical climate.\"\n\nMark Jones, Senior Minister, Faith Vancouver Presbyterian Church; coauthor, A Puritan Theology\n\n\"Trueman, again, has given us a stimulating book. He manages to demonstrate the relevance of creeds by showing how new the old ones are. The book is not only a must-read for those who stick to creeds without knowing why or those whose creed it is to have no creed, but for everyone who tries to practice the Christian faith.\"\n\nHerman Selderhuis, director Refo500, The Netherlands\n\n\"This is an engrossing survey, sparklingly contemporary yet eruditely historical. But it is also an urgent wake-up call, which, if heeded, would deliver Evangelicalism from its current isolation, shallowness, and confusion\u2014and from the autocracy of private empire-builders. Informative, readable, and stimulating all at once.\"\n\nDonald Macleod, Emeritus Professor of Systematic Theology, Free Church of Scotland College\n\n\"Trueman states that creeds and confessions are both necessary for the well-being of the church and are, in fact, required by the Bible. His arguments are wide-ranging and include biblical exposition, lessons from church history, and modern cultural factors that may be unconsciously influencing one's view of the issue. In addition, there is the typical Trueman humor and odd examples sprinkled throughout the book. In the end, I agree with him and will require this book for my seminary course on creeds.\"\n\nRobert J. Cara, Chief Academic Officer, Hugh and Sallie Reaves Professor of New Testament, Reformed Theological Seminary\n\n\"Today there is a challenge to the authority of the church including the authority of Scripture. The Creedal Imperative speaks to the necessity of creeds and confessions, which tend to save us from attempts to privately interpret the Scripture. Trueman demonstrates how creeds and confessions are strategic checkpoints, intended not only to enable us to express our beliefs, but also to keep us from misunderstanding God's truth. Properly used, creeds and confessions, under the authority of God's Word, enable us to hear God's voice\u2014they are our speaking what we understand God has spoken to us in Scripture. For those who maintain, 'We have no creed or confession but the Bible,' this book is a must-read. For those who understand the place of creeds and confessions in the life of Christian faith, this book is also a must-read. It is all about understanding God's truth. I commend Trueman for his careful demonstration of clear exegesis, sound theology, understanding of church history, and, consequently, his ability to understand the times in which we live. You will be blessed by this book.\"\n\nCharles H. Dunahoo, Coordinator, PCA CEP;   \nauthor, Making Kingdom Disciples; Changing Trends in Missions; and Foundation and Authority\n\n\"Though it might sound a bit hackneyed for a book commendation, this is a book I would love to have written! Carl Trueman's case for what he terms 'the creedal imperative' of the Christian faith is spot-on. Trueman not only identifies but also deftly rebuts a number of traditional as well as more recent objections in contemporary culture to creeds and confessions. On the one hand, he shows the untenability of the 'no creed but Christ, no book but the Bible' position of many evangelical Christians. And on the other hand, he defends the use of creeds and confessions that summarize and defend the teaching of Scripture without supplementing Scripture or diminishing its authority.\"\n\nCornelis P. Venema, President, Professor of Doctrinal Studies, Mid-America Reformed Seminary\nThe CREEDAL\n\nI M P E R A T I V E\n\nThe Creedal Imperative  \nCopyright \u00a9 2012 by Carl R. Trueman\n\nPublished by Crossway  \n1300 Crescent Street  \nWheaton, Illinois 60187\n\nAll rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form by any means, electronic, mechanical, photocopy, recording, or otherwise, without the prior permission of the publisher, except as provided for by USA copyright law.\n\nCover design: Studio Gearbox  \nCover images: Thinkstock  \nInterior design and typesetting: Lakeside Design Plus\n\nFirst printing 2012  \nPrinted in the United States of America\n\nScripture quotations are from the ESV\u00ae Bible (The Holy Bible, English Standard Version\u00ae), copyright \u00a9 2001 by Crossway. Used by permission. All rights reserved.\n\nTrade Paperback ISBN: 978-1-4335-2190-4  \nPDF ISBN: 978-1-4335-2191-1  \nMobipocket ISBN: 978-1-4335-2192-8  \nePub ISBN: 978-1-4335-2193-5\n\nLibrary of Congress Cataloging-in-Publication Data\n\nTrueman, Carl R.\n\nThe creedal imperative \/ Carl R. Trueman.\n\np. cm.\n\nIncludes bibliographical references and index.\n\nISBN 978-1-4335-2190-4 (tp)\n\n1. Creeds. I. Title.\n\nBT990.T78 2012\n\n238\u2014dc23 | 2012013405\n\n---|---\n\nCrossway is a publishing ministry of Good News Publishers.\n\nVP | 24 | 23 | 22 | 21 | 20 | 19 | 18 | 17 | 16 | 15 | 14 | 13 | 12  \n---|---|---|---|---|---|---|---|---|---|---|---|---|---  \n14 | 13 | 12 | 11 | 10 | 9 | 8 | 7 | 6 | 5 | 4 | 3 | 2 | 1\nDedicated with gratitude to the students and friends   \nat Cornerstone Presbyterian Church  \nwho have attended the monthly \"Tabletalk at the Truemans'\"   \nover the years, where many of the ideas in this book   \nwere debated and refined.\n\n## Contents\n\nAcknowledgments\n\nIntroduction\n\n1 The Cultural Case against Creeds and Confessions\n\n2 The Foundations of Creedalism\n\n3 The Early Church\n\n4 Classical Protestant Confessions\n\n5 Confession as Praise\n\n6 On the Usefulness of Creeds and Confessions\n\nConclusion\n\nAppendix: On Revising and Supplementing Confessions\n\nFor Further Reading\n\n## Acknowledgments\n\nI wish to thank Allan Fisher and Justin Taylor at Crossway for encouraging me in this project and waiting patiently as I missed a couple of deadlines. At Westminster, I have benefited from conversations about the nature of confessionalism with my friends, Tara, Sandy Finlayson, Greg Beale, Peter Lillback, Dick Gaffin, and David Garner. I must also thank Catriona, John, and Peter for providing such a happy home and escape from work.\n\n## Introduction\n\nA colleague of mine loves to tell the following story about a church he used to visit. The pastor there had a habit of standing in the pulpit, seizing his Bible in his right hand, raising it above his head, and pointing to it with his left. \"This,\" he declared in a booming voice, \"is our only creed and our only confession.\" Ironically, the church was marked by teaching that included the five points of Calvinism, dispensationalism, and a form of polity that reflected in broad terms its origins as a Plymouth Brethren assembly. In other words, while its only creed was the Bible, it actually connected in terms of the details of its life and teaching to almost no other congregation in the history of the church. Clearly, the church did have a creed, a summary view of what the Bible taught on grace, eschatology, and ecclesiology; it was just that nobody ever wrote it down and set it out in public. That is a serious problem. As I shall argue in subsequent pages, it is actually unbiblical; and that is ironic and somewhat sad, given the (no doubt) sincere desire of the pastor and the people of this church to have an approach to church life that guaranteed the unique status of the Bible.\n\nThe burden that motivates my writing of this book is my belief that creeds and confessions are vital to the present and future well-being of the church. Such a statement may well jar with evangelical ears that are used to the notion that Scripture alone is to be considered the sole, supreme authority for Christian faith and practice. Does my claim not strike at the very heart of the notion of Scripture alone? Does it not place me in jeopardy of regarding both Scripture and something outside Scripture, some tradition, as being of coordinate and potentially equal authority? And is there not a danger that commitment to time-bound creeds and confessions might well doom the church to irrelevance in the modern world?\n\nThese are, indeed, legitimate concerns, and I intend to address these, and more, in the coming pages. Here, however, I want to place my own cards on the table. Every author writes from a particular perspective, with arguments shaped to some extent by personal commitments, background, and belief. Thus, it seems entirely appropriate to allow the reader insight into my own context and predispositions in order to be better prepared to understand what I am going to say.\n\nI am a professor at a confessional Presbyterian seminary, Westminster in Philadelphia, and a minister in a confessional Presbyterian denomination, the Orthodox Presbyterian Church. In other words, I am a confessional Presbyterian. The terms \"confessional\" and \"Presbyterian\" are crucial for understanding both institutions. To take the latter term first, \"Presbyterian\" means that I am committed to a Presbyterian form of church government, whereby the church is ruled at a congregational level by a session, or committee, of elders; at a regional level by a presbytery of ministers and elders drawn from the churches in the area; and at a national level by a General Assembly of ministers and elders drawn from all parts of the country. When I became a minister in my denomination, I took vows to uphold this form of government both in what I teach and in the respect I give to the various courts of the church.\n\nMore significant for this book is the adjective \"confessional.\" This means that I am committed to the idea that the Presbyterian confessional position, as stated in the Westminster Standards, represents a summary of the teaching of the Bible on key points such as who God is, who Christ is, what justification means, and so on. When I became a minister, I took a solemn vow to that effect. This points to another aspect of being confessional: my vows connect to a structure of church government such that, if I am found to be teaching something inconsistent with what I am pledged to uphold, I can be held to account. If necessary, in the worst situations, I can even be removed from public office in the church.\n\nNotice, I said above that my vow reflects the fact that I believe the statements in the Westminster Standards are a summary of Scripture's teaching, not that I believe the Westminster Standards to represent teaching supplemental to Scripture, or independent of it. Rather, they summarize what is already there in Scripture itself.\n\nNow, this position is not without its problems. How, one might ask, do I avoid making the Standards a kind of a priori framework into which Scripture is made to fit? In other words, is there not a danger here of tail wagging dog, of treating the summary as the grid by which I read Scripture? I will address these, and similar, questions later. At this point, my purpose is simply to let readers know the position that I occupy so they can understand the perspective from which this book is being written. In sum, I not only believe that creeds and confessions are good for the church, I am also committed by vow to uphold the teaching of a particular confession. This indicates that the status of creeds and confessions is not for me simply a matter of intellectual interest; I am committed to the notion at a deep, personal level.\n\nThe fact that I am a confessional Christian places me at odds with the vast majority of evangelical Christians today. That is ironic, because most Christian churches throughout the ages have defined themselves by commitment to some form of creed, confession, or doctrinal statement. This is the case for the Eastern Orthodox, for Roman Catholics, and for Lutheran, Reformed, and Anglican Protestants. Some streams of Baptists have also had confessions; and many independent churches today that may not think of themselves as confessional have brief statements of faith that define who they are and what they believe. Furthermore, as I shall argue later, even those churches and Christians who repudiate the whole notion of creeds and confessions will yet tend to operate with an implicit creed.\n\nDespite this, it is true to say that we live in an anticonfessional age, at least in intention if not always in practice. The most blatant examples of this come from those who argue that the Protestant notion of Scripture alone simply requires the rejection of creeds and confessions. Scripture is the sole authority; of what use therefore are further documents? And how can one ever claim such documents have authority without thus derogating from the authority of Scripture? These arguments have a certain specious force, but I will argue in chapters 1 and 2 that, while the reasons for anticonfessionalism are manifold, many of them are driven more by cultural forces of which too many are unaware. Awareness of these forces, by contrast, may not automatically free us from their influence but can at least offer us the opportunity of subjecting them to critique.\n\nI do want to make the point here that Christians are not divided between those who have creeds and confessions and those who do not; rather, they are divided between those who have public creeds and confessions that are written down and exist as public documents, subject to public scrutiny, evaluation, and critique, and those who have private creeds and confessions that are often improvised, unwritten, and thus not open to public scrutiny, not susceptible to evaluation and, crucially and ironically, not, therefore, subject to testing by Scripture to see whether they are true.\n\nAnticonfessionalism among evangelicals is actually closely related to their putative rejection of tradition. For many, the principle of Scripture alone stands against any notion that the church's tradition plays any constructive role in her life or thought. Some regard this as one of the principal insights of the Protestant Reformers: Rome had (and has) tradition; Protestantism has Scripture. The sixteenth-century Reformation was thus a struggle over authority, with church tradition being pitted against the supremacy of Scripture; and modern evangelicals stand in lockstep with their Protestant forebears on this matter.\n\nA few moments of reflection, however, indicate how misleading and, in fact, untrue is the claim that Protestants have the Bible rather than tradition. Most evangelicals, for example, will typically use Bible translations, and such translations, be they the NIV, RSV, ESV, or KJV, stand within established traditions of Bible translation, linguistics, lexicography, etc. Further, beneath these translations lie the original Hebrew and Greek texts; so traditions of textual understanding also underlie these translations and, even for those linguistic geniuses who are more comfortable with just the Hebrew and Greek, these various traditions will shape the choice of text, the way the languages were learned, and the kind of choices made on matters of obscure grammar, syntax, and vocabulary. Thus, \"Scripture alone,\" whatever else it means, cannot mean Scripture approached in a vacuum.\n\nAnd we can take this reflection on tradition a step further. All Protestant pastors, even the most fundamentalist, will, if they are remotely competent, prepare their sermons with the help of lexicons, commentaries, and books of theology. As soon as they take down one of these books from their bookcases and start to read it, of course, they are drawing positively on church tradition. They are not simply reading the Word of God; they are reading the thoughts and reflections on that Word offered by someone else and articulated using words, sentences, and paragraphs that are not found anywhere in the Bible. Indeed, as soon as one uses the word \"Trinity\" from the pulpit, one is drawing on tradition, not Scripture.\n\nIn fact, tradition is not the issue; it is how one defines that tradition, and how one understands the way it connects to Scripture, which are really the points at issue. Indeed, this was the crux of the Reformation, which was not so much a struggle between Scripture and tradition as between different types of traditions. In a famous exchange between a leading light of the Catholic Reformation, Cardinal Sadoleto, and the Reformer, John Calvin, Sadoleto argued that the Protestants had abandoned the church tradition. Calvin responded that, on the contrary, the Protestants had the true tradition; it was the Catholic Church that had moved away from the truth. The point was simple and well-made: the tradition that transmitted the correct understanding of Scripture from generation to generation belonged to the Protestants.\n\nHere is not the place to debate the veracity of Calvin's claim regarding the content of tradition; suffice it to note that he understood the Reformation not as Scripture versus tradition but as scriptural tradition versus unscriptural tradition. Thoughtful Protestants then, and ever since, have understood the Reformers as arguing for what we might call a tradition that is normed by Scripture. In other words, Protestants know that they use language and conceptual terminology not found explicitly in the Bible; but they understand such are useful in understanding what Scripture says and, at the point where they are found to be inadequate for this task, or even to contradict Scripture, there they must be modified or abandoned.\n\nThe same is true of the creeds and confessions of the church, which are, one might say, the most concentrated deposits of tradition, as affirmed by the church. These documents are often referred to as normed norms or, to use the Latin, norma normata, in contrast to Scripture which is the norming norm, or norma normans. What that means is that the creeds and confessions represent a public statement of what a particular church or denomination believes that Scripture teaches in a synthetic form. By synthetic, I do not mean \"false,\" as in, say, a synthetic fiber like nylon, which is designed to look like cotton but is not really cotton. I mean rather a presentation that is not simply a collection of Bible verses but rather a thematic summary of what the Bible teaches. Thus, in the Nicene Creed, we have an explication of the identity of God as Father, Son, and Holy Spirit, which is considered to represent what the Bible as a whole teaches on this subject. The important point to note is that such statements are public, and thus open to public scrutiny in the light of what Scripture teaches. Thus, they can be accepted or rejected, modified or clarified, as and when they are found to be wanting; the context and means for such change we will discuss elsewhere. Here, I simply want the reader to note the synthetic and public nature of the documents.\n\nThis book consists of six chapters. In chapter 1, I look at some of the powerful currents within modern culture that serve to make the whole idea of creeds and confessions somewhat implausible. I do not intend to reduce evangelical objections to such secular forces, but I believe that an understanding of such forces can be of great help in clarifying why the case for confessionalism can be difficult to make at the present time.\n\nIn chapter 2, I look at the biblical teaching on a number of related points (the importance of language, the reality and unity of human nature, Paul's emphasis on doctrine, on eldership, on a \"form of sound words,\" and on tradition). My conclusion is not only that creeds and confessions are plausible, given biblical teaching, but that Paul actually seems to assume that something like them will be a normal part of the postapostolic church's life. In other words, there is a sense in which the claim to have no creed but the Bible is incoherent, given the fact that the Bible itself seems to teach the need for creeds.\n\nIn chapter 3, I outline the ecclesiological developments of relevance to the case for creeds and confessions. In particular, I focus on the Trinitarian\/christological discussions between the Council of Nicaea in 325 and the Council of Chalcedon in 451. I also touch on the Apostles' and the Athanasian creeds. Two important lessons I draw from this study relate to doctrinal complexity and the importance of the church. As to the former, history teaches that many Christian doctrines can only exist in a stable form within a relatively complex network of related doctrines. Christian theology, in other words, always has a certain ineradicable complexity, which has serious implications for the modern evangelical predilection for simple and very brief statements of faith. As to ecclesiology, it is clear from the early church that terms such as \"heresy\" really only have a meaningful content when connected to a church that has a specific confession.\n\nIn chapter 4, I deal with major Protestant confessional standards: the Anglican Articles and Homilies; the Lutheran Book of Concord; the Three Forms of Unity; the Westminster Standards; and the 1689 Baptist Confession of Faith. This chapter is of necessity highly selective. Protestantism has produced a vast amount of confessional material, and so I have chosen to focus on those with which I am most familiar. In choosing these, I do not intend to imply that Arminians, General Baptists, Anabaptists, and others do not have confessions and cannot be confessional. I hope that anyone from these traditions who reads this book will see that the principles of confessionalism are not confined simply to those confessions I happen to have mentioned.\n\nIn chapter 5, in order to do justice to the doxological origins of Christian creeds and to underline the important function that such have played, and can continue to play, in the life of the church, I focus on creeds and praise. Too often we think of these documents in a negative sense, as if their sole purpose was simply to keep people out of the church and to offer a dry-as-dust account of the Christian faith. By contrast, creeds are central to Christian doxology.\n\nIn chapter 6, I make the case for the usefulness of confessions by highlighting a number of advantages to having them, from limitation of church power to a proper pedagogical structure for church life. After the conclusion, I also include an appendix, addressing the vexing question of the possibility and practicality of confessional revision.\n\nIn writing this book, I come to my subject as one convinced that confessional Christianity captures a very important aspect of biblical teaching on the church. I was an evangelical for many years; discovering confessional Presbyterianism in my early thirties was a liberating experience. Nevertheless, I am aware that there can be a rather distasteful, not to mention sinful, tendency among many confessional writers to look down with scorn and derision on those who are not confessional. I trust that I have not written in that spirit; rather, I hope that this book will go some way to persuading nonconfessional Christians who love the Bible and seek to follow Christ that confessionalism, far from being something to fear, can actually help them to better protect that which is so dear to them.\n\nThe astute reader should now be able to see the case I want to make in this book: I want to argue that creeds and confessions are thoroughly consistent with the belief that Scripture alone is the unique source of revelation and authority. Indeed, I want to go somewhat further: I want to argue that creeds and confessions are, in fact, necessary for the well-being of the church, and that churches that claim not to have them place themselves at a permanent disadvantage when it comes to holding fast to that form of sound words which was so precious to the aging Paul as he advised his young prot\u00e9g\u00e9, Timothy. Linked to this latter point, I want to make the case that it is at least arguable, based on Scripture, that the need for creeds and confessions is not just a practical imperative for the church but is also a biblical imperative.\n\n#\n\n## The Cultural Case against Creeds and Confessions\n\nIn the introduction, I briefly mentioned the standard, knee-jerk reaction against creeds and confessions, often found in evangelical circles, that such documents supplant the unique place of the Bible, place tradition on an equal\u2014or even superior\u2014footing with Scripture, and thus compromise a truly evangelical, Protestant notion of authority. While I will offer a more thorough response to this line of objection later, I did note that all Christians engage in confessional synthesis; the difference is simply whether one adheres to a public confession, subject to public scrutiny, or to a private confession that is, by its very nature, immune to such examination.\n\nBefore proceeding to a more thoroughgoing exposition of the use and the usefulness of confessions, however, it is worth spending some time reflecting on other reasons why creeds and confessions are regarded with such suspicion these days. While the objection to them is often couched in language that appears to be jealous for biblical authority, there are also powerful forces at work within our modern world that militate against adherence to historic statements of the Christian faith. As the goldfish swimming in the bowl is unaware of the temperature and taste of the water in which he swims, so often the most powerfully formative forces of our societies and cultures are those with which we are so familiar as to be functionally unaware of how they shape our thinking, even our thinking about what exactly it means to say that Scripture has supreme and unique authority. It would be a tragic irony if the rejection of creeds and confessions by so many of those who sincerely wish to be biblically faithful turned out to be not an act of faithfulness but rather an unwitting capitulation to the spirit of the age. It is some of the forces that shape this spirit that I address in this chapter.\n\nThree Assumptions\n\nMy conviction that creeds and confessions are a good and necessary part of healthy, biblical church life rests on a host of different arguments and convictions; but, at root, there are three basic presuppositions to which I hold that must be true for the case for confessions to be a sound one. These are as follows:\n\n1. The past is important, and has things of positive relevance to teach us. Creeds and confessions are, almost by definition, documents that were composed at some point in the past; and, in most cases, we are talking about the distant past, not last week or last year. Thus, to claim that creeds and confessions still fulfill positive functions, in terms of transmitting truth from one generation to another or making it clear to the outside world what it is that particular churches believe, requires that we believe the past can still speak to us today. Thus, any cultural force that weakens or attenuates the belief that the past can be a source of knowledge and even wisdom is also a force that serves to undermine the relevance of creeds and confessions.\n\n2. Language must be an appropriate vehicle for the stable transmission of truth across time and geographical space. Creeds and confessions are documents that make theological truth claims. That is not to say that that is all that they do: the role, for example, of the Apostles' and the Nicene creeds in many church liturgies indicates that they can also fulfill doxological as well as pedagogical and theological roles; but while they can thus be more, they can never be less than theological, doctrinal statements that rest upon and express truth claims about God and the world he has created. They do this, of course, in words; and so, if these claims are to be what they claim to be\u2014statements about a reality beyond language\u2014then language itself must be an adequate medium for performing this task. Thus, any force that undermines general confidence in language as a medium capable of conveying information or of constituting relationships is also a force that strikes at the validity of creeds and confessions.\n\n3. There must be a body or an institution that can authoritatively compose and enforce creeds and confessions. This body or institution is the church. I will address the significance of this in more detail in subsequent chapters, but it is important to understand at the outset that confessions are not private documents. They are significant because they have been adopted by the church as public declarations of her faith, and their function cannot be isolated from their ecclesiastical nature and context. This whole concept assumes that institutions and institutional authority structures are not necessarily bad or evil or defective simply by their very existence as institutions. Thus, any cultural force that overthrows or undermines notions of external or institutional authority effectively removes the mechanisms by which creeds and confessions can function as anything other than simple summaries of doctrine for private edification.\n\nIf these are the presuppositions of confessionalism, then it is clear that we have a major problem, because each of these three basic presuppositions represents a profoundly countercultural position, something that stands opposed to the general flow of modern life. Today, the past is more often a source of embarrassment than a positive source of knowledge; and when it is considered useful, it is usually as providing examples of what not to do or of defective, less advanced thinking than of truth for the present. Language is similarly suspect: in a world of spin, dishonest politicians, and ruthless marketing, language can often seem to be\u2014indeed, often is\u2014manipulative, deceptive, or downright wicked, but rarely transparent and something to be taken at face value. Then, finally, institutions, from multinational corporations to governments, seem to be in the game of self-perpetuation, bullying, and control for the sake of control. They are never seen as entities that exist in practice for the real benefit of others. Thus, the big three presuppositions of confessionalism fly in the face of the values of contemporary culture, and confessionalists clearly have their work cut out to mount a counterattack. And such a counterattack begins with the simple truism of every successful campaigner, from wartime leaders to the coaches of high school track teams: know your enemy. In this context, knowing the enemy may also help us to realize how, in our defense of the unique authority of Scripture, our understanding of what that means is sometimes shaped more by the hidden forces of the world around us than by the teaching of Scripture and the historic life and practice of the church.\n\nDevaluing the Past\n\nScience\n\nNumerous forces within modern culture serve to erode any notion that the past might be a useful source of wisdom. Perhaps the most obvious is the dominance of science. I am not, of course, referring to the content of science. Science undergirds almost all of those things which make life bearable, from electric lightbulbs to cancer treatment. To say science is the enemy is not, in this instance, to be antiscience. Rather, I am thinking of the kind of cultural mindset that science helps to cultivate and reinforce.\n\nScience, by its very nature, assumes that the present is better than the past and the future will be better than the present. Again, this is not in itself a bad thing. It is surely part of what drives the laudable curiosity that motivates scientists and leads to major breakthroughs; and there is much evidence that this\u2014the fact the present is better than the past\u2014is, indeed, the case. As one who teaches history, I am often asked by students in which period of history I would most have enjoyed living. My answer is simple and straightforward: this one, the here and now. Call me a weakling if you like, but I would much rather live in an era with analgesics, antibiotics, and flush toilets than in earlier periods where pain killers were unknown, medicine usually involved swallowing some kill-or-cure snake oil made by a wrinkled old crone with dubious personal hygiene, and the \"facilities\" were little more than a hole in the ground on the edge of the village. By and large, in areas where it is relevant, science has made the world a better place. The evidence is not all one way, however: the Holocaust, for example, is one instance where science was clearly used to destroy rather than enhance life, and that on a huge scale. But, by and large, science has brought with it huge gains, from medicine to dishwashers.\n\nThe problem is that science also comes loaded with a certain philosophical bias, and that is, as stated above, that the past is inferior to the present. It has a built-in narrative of progress, whereby everything\u2014or at least almost everything\u2014just keeps getting better; and the problem is that this tends to inculcate a broader cultural attitude that applies the same kind of expectation in other areas. Throw concepts like evolution into the mix, and you have a gravitational pull within the culture toward the future, built on the assumed inferiority of the past.\n\nThis narrative of scientific progress instills a belief not simply in the superiority of the present in relation to the past but also in its uniqueness. This time in which we live has so much more knowledge, displays so much more sophistication, and is so much more complicated than the past. Thus that past is consequently of no real use in addressing the problems or issues of the present, so great is the difference between them. One would not, for example, use a horse and cart to transport fuel from an oil refinery to a petrol station. Nor would one today consult a seventeenth-century textbook on surgery to find out how to remove a burst appendix. So why would one turn to some confession written in the fourth or the seventeenth century to find a summary guide to what Christians today should believe?\n\nSome years ago, I was exposed to precisely this attitude while teaching a class on the ancient church. At some point, I mentioned that a certain professor from another institution was going to be visiting campus to deliver some lectures on the Westminster Standards, that is, the Confession and the Larger and Shorter Catechisms. A student immediately asked why she should bother attending these because \"some documents written in the seventeenth century seem to have very little to do with\" her ministry. I asked her if she had read these apparently irrelevant documents recently. She said she had not. I then pointed out to her that these documents had been regarded by many people as vital and vibrant expressions of the Christian faith since their composition. Given this, and their connection to historic traditions and trajectories of church life and Christian thought, I suggested with every ounce of tact and gentleness I could muster that she might perhaps better ask herself not so much what relevance they have to her ministry but what relevance her ministry had to the church. Her assumption was simple: the past could not really speak in any meaningful way to the present. She was truly a child of the scientific age.\n\nTechnology\n\nClosely related to the role of science in cultivating an attitude that downgrades the importance of the past is that of technology. A simple example should make this point clear. My mother lives in an old weaver's cottage in the Cotswolds. In what is now her living room, there is a stone fireplace and, in that fireplace, there are a series of small holes, roughly an inch in diameter, now plugged with wood, which indicate where the weaver would have had his loom. It is easy to imagine a scene in the early nineteenth century in which the weaver was hard at work making cloth when one of his children wandered into the room and inquired as to what exactly he was doing. No doubt, the weaver would have sat the child down and explained how the loom operated, how the shuttle carried the woolen thread from one side to the other and slowly but surely formed a sheet of fabric. The flow of knowledge from the older generation to the younger was clear; this was no doubt repeated many times in preindustrial societies around the world, where children typically grew up to follow in the footsteps of their parents and were thus more or less apprenticed to their parents from an early age.\n\nNow, jump forward nearly two hundred years to a scene in the same room. I am sitting there, trying to set up my mother's DVR to record a Gloucester versus Leicester rugby match and, try as I might, I cannot get the machine to do what I want it to do. In walks my niece and asks what I am trying to do. After I explain to her what is going on, she sighs, rolls her eyes, picks up the remote control, and with what seems to me to be two touches of the buttons, has the machine set up to record the match. With a shake of her head, she walks back into the kitchen.\n\nNotice what has happened here, and what the significance of these two encounters is: the flow of knowledge has been reversed. No longer is the younger dependent upon the older; rather, the older is dependent upon the younger. Technology, because it is constantly and rapidly changing, inevitably favors those who have been brought up with it, and who have the kind of young, agile minds that develop new skills quickly and easily. You cannot easily teach a middle-aged historian, any more than an old dog, new tricks; and that means that technology will always favor the young.\n\nThis is just one anecdote and, as my secretary will tell you, I am among the more\u2014ahem\u2014technologically challenged men of my generation; but the general point is a good one. The technological world, particularly given the rapidity with which it is constantly changing, creates an environment where the assumption is that older people are going to be dependent upon the younger. Taken by itself, perhaps, this might not be so significant; but combined with the impact of science as a whole upon cultural attitudes, it undoubtedly plays its role in the bias against age, and thus against the past, which is a hallmark of the modern world and which is not incidental in the general antipathy among Christians for creeds and confessions.\n\nConsumerism\n\nA third cultural force that militates against respect for the past is consumerism. As with science, there is much that could be said here, but I will restrict myself to the most salient aspects of the phenomenon.\n\nConsumerism can be defined as an over-attachment to material goods and possessions such that one's meaning or worth is determined by them. This definition is reasonably helpful but misses one key aspect of the phenomenon: it is not just the attachment to material things, it is also the need for constant acquisition of the same. Life is enriched not simply by possessing goods but by the process of acquiring them; consumerism is as much a function of boredom as it is of crass materialism.\n\nWhat has this to do with rejection of the past? Simply this: consumerism is predicated on the idea that life can be fulfilling through acquiring something in the future that one does not have in the present. This manifests itself in the whole strategic nature of marketing. For example, every time you switch on your television set, you are bombarded with advertisements that may be for a variety of different goods and services but that all preach basically the same message: what you have now is not enough for happiness; you need something else, something new, in order to find true fulfillment. I believe that this reinforces fundamentally negative attitudes toward the past.\n\nThink for a moment: how many readers of this book are wearing clothes they bought ten years ago? How many are using computers they bought five years ago? Or driving automobiles more than fifteen years old? With the exception of vintage car collectors, the economically poor, and those with absolutely no fashion sense, most readers will probably respond in the negative to at least one, if not all three, of these questions. Yet when we ask why this is the case, there is no sensible answer. We can put a man on the moon, so we could probably make an automobile that lasts for fifty years; most of us do little on computers that could not have been done on the machines we owned five years ago; and we all throw away clothes that still fit us and are quite presentable. So why the need for the new?\n\nA number of factors influence this kind of behavior. First, there is the role of built-in obsolescence: it is not in the manufacturer's best interest to make a washing machine that will last for a hundred years. If that were done, then the manufacturer would likely be out of business within a decade as the market became saturated. Such is a possible, but actually unlikely, scenario. Developments in technology mean that longevity will not be the only factor driving the market. Efficiency, for example, or enhanced and multiplied functions might well create a continuing need for more goods. Aesthetics also play a role; the ability to market goods based on aesthetics and image has proved powerful. Remember the cool, sleek look that Apple computers developed at one point? That gave them a clear edge over their rivals.\n\nSecond, and related to the first point, we see in the consumer economy a coalescence of aesthetics and a bias to the young in the creation of the so-called youth market, and the closely related marketing of youth to older types like myself. If no eighteen-year-old male believes himself to be mortal, so no middle-aged male wants to appear to be any older than he was twenty years ago. Indeed, with the exception of those odd types (of the kind who read The Daily Telegraph in the UK and the National Review in the US) who were probably born with comb-overs, receding hairlines, and bottle glasses, it would seem that the market for youth clothing (albeit with slightly expanded waistline sizes) is alive and well long into territory previously reserved for the superannuated and beyond.\n\nIn today's topsy-turvy world, youth has status. That is why so many old-timers spend large amounts of money and time trying to hold on to, or even win back, some of its accoutrements, whether by purchasing a pair of jeans from Aeropostale, buying a male grooming kit, or even undergoing drastic plastic surgery. Harmless as these phenomena are at one level, at another they are part of the larger cultural impulse toward disdain for the past and for old age. We see this not just in fashion, of course, but also in the \"wisdom\" now invested in young people who are considered competent to opine on complex matters, not despite the fact of their relative youth and inexperience but precisely because of it. Pop music, a function of youth culture if ever there was one, is perhaps responsible for this. In the last few decades, we have had the pleasure of hearing all manner of people, from Hall & Oates in the eighties to Lady Gaga in the present, telling the world what to do about everything from apartheid to third world debt to gay marriage. Apparently, the lack of \"baggage\" (to use the standard pejorative) is an advantage to being able to speak with authority on complex subjects. In other professions, of course\u2014from plumbing to brain surgery and beyond\u2014\"baggage\" is generally referred to as \"appropriate training,\" but, such is the power of a youthful smile, a full head of hair, and a trim waistline that such does not apply to matters of morality, economics, or the meaning of life in general.\n\nAs a postscript, the impact of consumerism is one reason why church sessions and elder boards often spend more time than is decent on discussions about worship and programs. Someone will make the point that certain young people have left because the worship is not to their liking and thus the church needs to think again about how it does things. Laying aside the fact that, for most of us, no church gives us everything we want in worship but we are nonetheless happy to attend because the Word is truly preached, it is interesting to note the session member's response: we need to do something, to think again about worship. In other words, we need to respond to the needs of the consumer. An alternative approach might be that we need to do a better job of explaining why we do what we do, and what the obligations entailed in solemn vows of membership are; yet this is often not the knee-jerk reaction to such concerns. The consumer-is-king mentality renders all established and time-tested practices unstable and utterly negotiable.\n\nThe Disappearance of \"Human Nature\"\n\nAnother factor that impacts the possibility of documents such as creeds having any usefulness is the disappearance of \"human nature\" as a category. This is often not done explicitly, except by the most extreme advocates of postmodern skepticism; but functionally the idea of a human nature or \"essence\" that connects people in one time and place to another is today often neglected or ignored. Numerous factors play into this. One is that the modern world has made everyone more acutely aware of the vast variety of social and cultural practices exhibited by different groups. The Englishman of the nineteenth century might have been able to rest secure in the knowledge that taking afternoon tea was the way human beings should act and that those who did not do so were either weird (if English), or dastardly (if French, Italian, or German), or inferior (if otherwise foreign). Now, however, we know that afternoon refreshment practices are scarcely the result of the structure of the human genome. More seriously, we know that practices considered disgusting by one group, such as female circumcision, are yet considered necessary by others. This raises the question of whether there are universal human values and rights and, if so, what criteria are to be used to determine what they are. If eating pork is unacceptable to Jews, does that mean that French pig farmers should be closed down? What, if anything, is the common cultural, ethical, philosophical, or metaphysical core that binds human beings together? Indeed, does such a thing exist?\n\nIf \"human nature\" does not exist, other than as a specific, basic biological structure that means one human can only reproduce in conjunction with another, then what authority can anybody or any human document that belongs to another time or place have? If human nature is really a construct of the particulars of a specific historical, geographical, and cultural context, it is not immediately obvious that, say, a document produced in Constantinople near the end of the fourth century can have any relevance to people living in London or New York at the start of the twenty-first. For historical documents to speak beyond their own time there has to be some kind of fundamental continuity between their form and content and the present age.\n\nConsumerism plays its part here as well. If you are what you consume, if you can be whatever you want to be, then what binds you to your neighbor? More importantly, what binds you to the people in other times and places? If you are master of your own destiny, then you are free to act toward the past and toward other people in the same way you act toward the goods on the supermarket shelf. You buy what appeals to you and leave behind that which does not.\n\nThe implications for creeds and confessions are obvious. Choose your particular: they were written by dead, white males who dressed differently to us, had different attitudes to the world, spoke in a different language, were celibate, were not celibate, never understood technology or listened to Elvis, never grappled with the scientific breakthroughs of recent years, etc., etc. If nothing binds us to them, or if the differences between us and them simply overwhelm any analogy there might be between us, then they have nothing useful or relevant to say to us, and we are better off ignoring them. A world in which human nature is merely a construct put together by the individual or by the specific community in which the individual is placed is a world where historical documents, such as creeds, can have no transcendent significance but are doomed to be of merely local or antiquarian interest.\n\nWords, Mysticism, and Pragmatism\n\nIf devaluing the past is one aspect of contemporary culture that militates against the usefulness of creeds and confessions, a second is the current suspicion of words as reliable means of communication.\n\nWe need to acknowledge at the outset that there is plenty of evidence for the problematic nature of words. To quote The Police: \"Poets, priests and politicians have words to thank for their positions.\"1 The idea that words are one way to establish and maintain personal power and prestige is deeply rooted. Indeed, a whole school of literary theory has developed around this notion, whereby words have become little more than tools to be used to marginalize and manipulate others. I remember some years ago watching a 1930s Nazi propaganda film entitled Sein ohne Leben (\"Being without Life\"), which was designed to make the case that children born with severe mental and physical disabilities should be euthanized. The documentary was significant in that it helped pave the way for the social and cultural context in which the broader policies of the Holocaust could be pursued. But what interested me in particular was the way it used those two words\u2014\"being\" and \"life\"\u2014as a means of making a manipulative distinction that served to obscure the horror of what was really being proposed. By implying that a child with severe encephalitis possessed a mere existence and no life, by driving that wedge between the two, the child was effectively and quietly robbed of personhood and thus of status. The words were not being used to convey information; they were being used to create a reality and one that, in the wake of the Holocaust, looks vile and manipulative.\n\nOne could add to this many examples drawn from the sphere of politics, perhaps the most notorious realm for such linguistic twisting. In short, the case for words being susceptible to manipulative usage is not one that can be credibly questioned. Such has led to a broader cultural cynicism about language, which has bled over into the church. That Christianity is a way of life and not a set of propositions has become something of a mantra among younger Christians in the last ten years. Of course, like most erroneous notions, it contains just enough truth and has just enough legitimate criticism of alternative positions to be credible. Indeed, one of its underlying concerns\u2014that Christianity not terminate in a mere intellectualism\u2014is surely legitimate, even if the sweeping terms in which this is expressed clearly involve an unbiblical reduction of Christianity to praxis. It is not actually that original: Desiderius Erasmus, Richard Baxter, and Adolf von Harnack, to name but three, all offered variations (of differing degrees of orthodoxy) on this theme. Yet the frequency with which it occurs in the history of the church indicates that at least some of the concerns it seeks to address must be legitimate.\n\nIn addition to the obvious problems with the way language has been used by people such as politicians, and how sophisticated literary theorists have dismantled old linguistic certainties, there is also a popular strain of mysticism (for want of a better word) that pervades modern culture and that is profoundly suspicious of words. This takes various forms. One thinks, for example, of the notion that certain emotional sentiments or responses constitute truth, something that is often epitomized by the kind of statements made with remarkable regularity on TV talk shows. \"I just know in my heart that it is true\" is built on this kind of thinking. Many of us no doubt have encountered ethical argumentation that amounts to, or perhaps is even expressed as, \"It feels so good. How can it possibly be wrong?\"2\n\nAgain, we might turn to popular music to provide a summary of this kind of thinking. If the reader will forgive the obvious incoherence of using words to undermine confidence in words, here are a few lines from Madonna's song, \"Bedtime Stories\":\n\nWords are useless, especially sentences.\n\nThey don't stand for anything.\n\nHow could they explain how I feel?\n\nMadonna actually makes quite a profound point here: the modern emphasis on emotions as the locus of truth or, to use the trendier term, authenticity, is fundamentally non- and even antiverbal. When someone declares that they \"just know in their heart\" that the latest boy band is the greatest phenomenon of Western musical culture since Bach left the organ loft for the last time, you may know that they are talking arrant nonsense, but there is no way that you can refute this person's claim because it is not a claim expressed using public criteria commonly known as words and logic. It is a purely personal, subjective judgment; and, in its claim to truth, it makes truth something mystical, something to be experienced, not something subject to normal criteria of public evaluation.\n\nTo have such an attitude so deeply embedded in popular culture, whether pop songs or talk shows or the visceral level of public discourse one often witnesses on the television in scenes outside courthouses, political rallies, and sporting events, would in itself create plenty of difficulties for the notion of creeds and confessions. Yet we see the impact of suspicion of words even within the Christian church. At the Reformation, preaching came to supplant the Mass as the central act of corporate Christian worship; underlying this shift was a move toward an understanding of the gospel as promise and of salvation as being by faith in that promise. Thus, proclamation of that promise in words moved to center stage. In recent decades, however, many churches have shifted preaching from this central place. In some contexts, preaching has not been abandoned; rather, it has been relativized and now stands alongside dramatic performances, candles, incense, and small group discussion. In other contexts, preaching has been pushed completely aside for conversational discourse, where the authoritative voice of the preacher has been replaced by a more democratic dialogue. Underlying all these shifts, in practice if not always in terms of self-conscious planning, is a suspicion that proclaimed words are no longer a reliable authority or, perhaps better, a plausible authority, given the wider antiverbal cultural dispositions.\n\nPopulist suspicion of words is not the only point at which the antiverbal mystical emphasis bites the church. Such also has deep and highly sophisticated roots within the history of modern theology. For example, this kind of mysticism is analogous to the kind of revision of the notion of Christian theology that took place at the hands of F. D. E. Schleiermacher, the so-called Father of Liberalism, at the start of the nineteenth century. In the wake of the Enlightenment, and particularly in response to Immanuel Kant's critique of traditional epistemologies, Schleiermacher sought to rebuild the Christian faith in a manner that would be plausible in his context. As the notions of objective truth and of the possibility of generalizing universal truths from the particulars of history had been abandoned, Schleiermacher offered an account of Christian theology which understood doctrine not so much as statements about the nature of God as a description of religious psychology. Thus, for example, predestination ceased to be what it appeared on paper to be\u2014a statement about God's eternal purpose relative to men and women\u2014and became rather a poetic expression of the feeling of total dependence upon God as experienced by the religious individual. Further, Christ became supremely important as the incarnation of God not because he was the incarnate God in the traditional manner defined by the Chalcedonian Definition of 451, but rather because in him the consciousness of God was supremely manifested.\n\nWithin such a framework, then, propositional, doctrinal Christianity (and the creeds and confessions that epitomized it) was exchanged for something mystical and experiential. Of course, to tar this with the label \"liberalism\" is likely to precipitate an immediate reaction from self-styled conservative evangelicals. Liberalism is the enemy; it is what \"they\" hold to\u2014whoever \"they\" are\u2014and not something of which we are guilty ourselves. Yet mysticism is alive and well within evangelical circles. Anyone who has ever been told by a friend that the Lord led such a friend to do something completely silly, or anyone who has ever been at a Bible study where the burden has been to explain \"what the text means to me,\" regardless of what the words on the page and the grammar and syntax might otherwise indicate, has experienced an evangelical mysticism that is not really distinguishable from traditional liberalism at the level of its understanding of what constitutes truth.\n\nClosely allied to mysticism is another phenomenon lethal to confessional Christianity: pragmatism, the notion that truth is to be found in usefulness. When one reflects for a moment on talk-show style mysticism, this becomes obvious. When individuals on such shows declare that \"I just know in my heart that this is true,\" what they are often saying is, \"This belief works for me; it has some actual, practical result that I like.\" Whether the belief makes them more cheerful, or helps them to feel more important, or gives them hope for better times ahead, the important thing is not so much the content of the belief as its result.\n\nSuch thinking pervades much of modern church life. I noted above the student in class who questioned the usefulness of creeds and confessions today. By her application of such a category to the creeds, she immediately indicated the pragmatic tendency of her thinking. We might also reflect upon the pragmatic content of so many books written by and for evangelical Christians. Here, for example, is the Amazon blurb for The Eden Diet: A Biblical and Merciful Weight Loss Program:\n\nThe Eden Diet helps readers understand the many reasons why they have not been able to lose weight in the past. In most cases, they fail to eat according to their God-given internal sensations\u2014their hunger pangs. Hunger was meant to be a compass that tells people when and how much to eat. However, most overweight people eat for external reasons that have little to do with hunger. They eat according to the clock, because of automatic habits, in response to their emotions and fleshly desires, or in response to tantalizing advertising messages. The Eden Diet shows how to overcome those fattening habits. It explains how to eat in response to the body's internal signals, how to block out external stimuli that trigger eating, and how to lose weight and achieve the abundant life God intended for His children in the beginning. Specific advice is given that helps readers eat for weight loss at pot luck events, buffets, at restaurants, on holidays and special occasions, and any time they are faced with challenging emotions and sinful desires.3\n\nThis book is available as an audio download from a well-known evangelical publisher, a publisher that lists on its website, as of July 2011, a large number of Christian diet books, including Fit for My King: His Princess Diet Plan and Devotional; The Maker's Diet: The 40-Day Health Experience That Will Change Your Life Forever; the two-volume Never Say Diet Personal Fitness Trainer; and the intriguing but presumably overstated New Bible Cure for Cancer.\n\nThe existence of such books within Christianity is a study in itself, since it speaks eloquently about a range of topics, from how people understand the essence of Christianity to what they see as the ideal Christian life. For our purpose here, it is sufficient to note the profound pragmatism that these titles indicate: Christianity is all about what it can do for you in the here and now. Similar genres exist within the evangelical world for financial planning, education, and self-fulfillment. All are evidence that the pragmatism of the wider world is alive and well within the walls of the church.\n\nIn such a culture, it is not surprising that creeds and confessions do not appear particularly useful. One will search in vain in the creeds of the ancient church for advice on how to stop excessive snacking between meals or on how to avoid a second trip to the dessert table at a potluck lunch. Further, while I cannot claim comprehensive knowledge of every confessional document written during the Reformation, none, as far as I know offer the reader a personal trainer, a wonderful \"health experience,\" financial prosperity, or a cure for cancer. By the standards of the culture that has produced the Eden diet, one would have to say that the confessional heritage of the church is really rather useless.\n\nFinally, remember that the comment about the irrelevance of creeds and confessions was made by a student who was a member of a confessional church in one of my classes at a confessional seminary. It is not only the less doctrinally informed areas of evangelicalism that have been impacted by the priorities of Oprah and company. Ask yourself this: if my church put on a conference about how to have a great Christian marriage and fulfilled sex life, would more or fewer people attend than if we did one on the importance of the incarnation or the Trinity? The answer to that question allows an interesting comparison between the priorities of the church today and that of the fourth and fifth centuries. It is not that the people in your church do not believe that, say, Christ rose from the dead and the tomb was empty; rather it is that such belief has no real usefulness to them other than as it provides them with what they are looking to obtain in the here and now. In such a context, orthodoxy as expressed in the great creeds and confessions is not rejected; it is simply sidelined as irrelevant and essentially useless.\n\nAntiauthoritarianism\n\nIf there are deep forces within our culture that militate against creeds and confessions on the basis of their nature as historical and linguistic documents, there are also forces that strike deep at these documents in terms of their origin and their status. Creeds and confessions are, by definition, statements made by institutions (churches), and they derive their practical authority from their connection to such institutions. It is true that some confessions have a single author. The Belgic Confession, written by the French Protestant Guido de Bres, is one obvious example; but it possesses its authority because it has been adopted by a church as an authoritative document. In the case of the Belgic Confession, this adoptive action was taken by the Synod of Dordt, which met in 1618\u20131619 in the city of Dordrecht in the Netherlands. It is the sanction of a corporate body that gives the confession its legal ecclesiastical status, not the specific identity of the author.\n\nThis institutional aspect of creeds and confessions is culturally problematic. Indeed, if anything marks the contemporary world it is surely suspicion of external authority. One might generalize and say that the issues noted above, with science, technology, consumerism, language, mysticism, and pragmatism, are all variations on the theme of rejection of external authority, that of the past in the case of science and technology, and that of anything but the self in terms of consumerism, language, and the rest.\n\nOf course, this rejection of external authority is ultimately rather selective. While many today reject traditional forms of external authority (family hierarchies, civil governments, traditional moral values, etc.), those same people often accept rather uncritically other forms of external authority. Think, for example, of the mindless emulation of the fashions of pop stars by their fans; or the incredibly na\u00efve confidence that is often placed in the opinions of vacuous and ill-informed celebrities on, say, third world debt or global warming, as opposed to those of traditional experts. Youth culture is the same: why on earth would anyone want the opinion of the latest boy band on anything unless he was convinced that knowledge gained by experience, knowledge from \"out there,\" was actually a hindrance to truth and not a means of accessing it? Yet the blogs and the news media crave the views of the likes of Lady Gaga on all kinds of things of which they are technically ignorant and actually incapable of expressing themselves with any coherence or thoughtfulness. They are authorities not because of their knowledge or skills but because of their status in our modern consumer society; and the fact that they are relatively young (or like to think that they are, as in the case of the superannuated Bono) is strangely seen as a plus, an advantage, something that qualifies them to make these statements. As I noted above, it is hard to imagine applying the same criteria to, say, electricians or brain surgeons, where age and experience are typically seen as essential qualifications. Strange to tell, on the bigger questions and problems of the world and society, having \"relevant training and knowledge\" is more likely to earn one excoriation as \"an ivory tower academic\" or part of the dreaded \"establishment\" than a useful contributor to any proposed solution. Lady Gaga is apparently more likely to have the answer to human sexuality or third world debt than a minister or an economist. Arguably, therefore, the rejection of external authority needs to be carefully defined as the rejection of traditional forms of external authority in order to be an accurate statement.\n\nEven with this qualification, however, the church\u2014or at least the traditional church, with its structures of governance, its established ways of doing things, and its creeds and confessions\u2014fares badly. Ironically, the old forms of authority have been replaced by new ones; self-appointed gurus abound, as do theological and antitheological potboilers. But my concern is not with passing fads; it is with a recovering of traditional and, as I will argue, biblical patterns of institutional authority.\n\nFirst, however, it is worth spending a few moments examining why respect for traditional external authority is at such a pitiable state today. It is clear that the same forces that made consumerism an antihistorical force also militate against traditional institutional authority. Consumerism is built upon the notion of the construction of self-identity through consumption. Fashions in clothing are a great example of this. Whether it is the shirt of one's favorite sports team or a style of dress adopted by one's favorite TV or pop star, at the heart of fashion is the notion that by purchasing certain goods one can create an identity for oneself.\n\nBroadening out from fashion, the world of commercial advertising is predicated on this kind of self-creating consumption. Commercials are not simply designed to create dissatisfaction with the present and thus to orient the audience toward the future; they are also designed to send the signal that you can make yourself different, you can become the ideal person you wish to be, by purchasing some particular goods or services. This is not simply a matter of creating needs; it is also about sending a message that you are master of your own universe. The Nike sales pitch, \"Just do it!\" might as easily be written \"Just be it!\" for, with a credit card in your pocket, you can become whatever you want to be. Authority lies within you, or at least that is the message the sales and marketing people wish to send; external authority is merely a repressive force that prevents you from being whoever and whatever you wish to be.\n\nWe also see a kind of mysticism and pragmatism in anti-\u00adauthoritarianism, where the locus of authority is ultimately not an external institution or body of knowledge but rather the inner being of the person. If \"it\" is \"true for me\" because \"I just know it in my heart,\" then guess what? \"My heart,\" whether that is a feeling of happiness or of self-esteem or of whatever, is the authority: internal, mystical, appointed by me using pragmatic criteria and as far away from any notion of direct external or institutional authority as is possible. Of course, it does not take a genius to realize that so many of the things that we \"just know in our hearts\" do actually come from external authorities\u2014commercials, idiotic talk shows, television pundits\u2014but that is not the point. The point is that we do not consciously understand this or recognize such authorities as having that effect.\n\nOne further factor that militates against traditional notions of external institutional authority is the Internet, specifically the world of blogs and tweets. There are, of course, different types of blogs. I myself do a bit of blogging for an e-zine, reformation21. For me, it is simply an electronic form of traditional journalism. I write articles, and the editor publishes them. One thing that reformation21 does not do is allow \"comments\" to be made by random readers on the content. Anyone can write to the editor; and, in the current virtual world, it takes very little effort to track down an author's e-mail and send one's thoughts straight to the source. But the deregulated posting of public comments is not allowed.\n\nFrom my perspective, this is a good thing. I have yet to read a \"comments thread\" on any topic of significance that does not quickly degenerate into moronic commentary that is as notable for its vacuousness as it is for its personal abuse. The culture of the comments thread is one which has confused the right to speak with the right to be heard and which sends a rather uncritical signal to the world about what constitutes good argumentation and appropriate contributions to discussion. Yet the visceral reaction with which such \"comment free\" e-zines meet from some individuals speaks once again of a culture where an anarchic free-for-all apparently is the only acceptable way of approaching a topic. The democratization of discussion in this way is inimical to traditional notions of authority and to the traditional notions of knowledge and expertise which underlie them. Again, we might note that this is a selective repudiation of authority: I have never read a comments thread on a blog dealing with brain surgery or rocket science, but I doubt that the good ones in these fields contain too many comments about \"Nazis\" or end with remarks like \"Fantastic stuff guys !!!  \" by people signing themselves off as \"Crazydogguy\" and the like. Politics and theology are much more likely to attract such discourse, it seems, and this surely indicates, and reinforces, the wider cultural problem with the kind of authority associated with traditional institutions when it comes to what we might term the more philosophical aspects of life.\n\nOf course, it is not just the anarchy of the blogs that plays to this kind of attitude. The arrival of Wikipedia and the like is also significant. Now, I confess that I am something of a Wikipedia fan. It is a fantastic resource for finding out the ages of favorite movie stars and the kind of trivia on a variety of subjects that is most useful when one is taking part in a pub quiz. The problem is that it can give the impression that a subject can be mastered in a very short period of time. I remember a few years ago reading a blog where a person was telling the world that he had never heard of presuppositional apologetics until that morning, had read the Wikipedia article on the same, and that it had completely changed his life. Indeed, he may even have specified the time\u201411:23 a.m. comes to mind\u2014at which this earth-shattering change took place. The point was ridiculous: whether presuppositional apologetics is capable of such impact is one question, but that a Wikipedia article could provide sufficient information to achieve this is surely unlikely. Were it so, then one could only conclude that all those who reject the position are either mad, stupid, or have never read the article. A most unlikely scenario, methinks.\n\nWhat these things\u2014anarchic blog comments, the assumption that reading a Wikipedia article gives true insight\u2014witness to is the creation of a culture of knowledge in which little weight is placed upon expertise and the idea that competence in some things only comes after extended periods of hard work and training. They are thus further factors in the complex of cultural forces that discount traditional sources of authority\u2014institutions, traditions, etc.\u2014and replace them with sources that derive their authority from something else, not least the hip trendiness of the latest fad or celebrity.\n\nIt seems clear to me that this anti-institutional tendency is deep-rooted even within churches that, on paper, place a premium on structure and authority. For example, in my own denomination, the Orthodox Presbyterian Church (OPC), we have a book of church order that lays out the basic structures of the church and the procedures by which these are to be maintained. Office-bearers (ministers, elders, deacons) take strict vows that bind them to particular doctrinal positions (the Westminster Standards) but also to the denomination and the local congregation. All the office-bearers with whom I have been privileged to serve over the years have taken their vows very seriously.\n\nWhat is less often noted, however, is that members, too, take vows. In the OPC these vows are not the same as those of office-bearers. This is for good reason; the qualifications for office bearing, as opposed to being a member, are somewhat more stringent, as we shall explore in a later chapter. But while the content of the vows for members may be less stringent in terms of specifics, they are no less serious in terms of their binding quality. In the OPC they involve profession of faith in the Trinity, trust in Christ for salvation, and commitment to the local body and submission under God\u2014a key qualification\u2014to the elders.\n\nWhat never ceases to amaze me is the casual way in which people make and break membership vows, sometimes within weeks. I have seen individuals leave the church because they were not given the Sunday school teaching opportunities they thought they deserved, because they did not like the worship style, and because their children found a more interesting church elsewhere. That such reasons do not give any grounds for breaking vows never seems to register. Indeed, some leave without giving any reason at all, so lightly do they regard solemn vows taken before God and the church.\n\nNow, I would never advocate that someone cannot leave a church at which they are very unhappy; and thankfully, there is provision for people to be able to move if they decide to. Cults take away people's freedom; the church should never do that. But there are processes by which this can be done, typically through discussion with the elders, which actually seek to honor the integrity of the vows. What is striking is that these processes are, in my experience, rarely used as they should be. Often the first thing that the elders hear is that somebody has already left and would like a letter of transfer to his new church or simply to be erased from the membership rolls.\n\nWhat this phenomenon tells me is that the suspicion of, or (perhaps better) indifference to, the external authority of institutions is as deeply embedded in the culture of the contemporary church as it is in society. And such an attitude inevitably has an impact on the way creeds and confessions are viewed. The person who has no real, practical respect for the church as an institution is inevitably going to have little respect for the documents that church has produced and\/or authorized as part of the basic means by which she identifies herself, witnesses to the world, and maintains some level of order within her ranks.\n\nThe Fear of Exclusion\n\nOne further cultural proclivity worth mentioning is that of the fear of exclusion, of drawing boundaries such that some people belong and other people do not. In addressing this matter, it is important to note that much of the tragedy of human history, particularly more recent history, has been wrapped up with the problem of exclusion. One need only think of the Armenian genocide of 1915, the Holocaust, the Rape of Nanking, the Balkan crisis, and the gassing of the Kurds to understand how vicious some forms of exclusion can be. Racism is only the most obvious; we can all think of other, less obvious, forms of exclusion that also help to justify crimes, great and small, perpetrated by one group of human beings against others.\n\nSuch forms of exclusion have left many with a lasting fear of anything that might smack of looking down on others as inferior. In the silly extremes of political correctness, it almost seems that anything at which I choose to take offense is to be deemed oppressive, exclusionary, and on the slippery slope to some form of genocide or holocaust. Yet we must not allow the excesses of the PC types to blind us to the really genuine concerns that underlie this fear of exclusion; but nor must we be blind to the impact it has upon attitudes to things like statements of faith and confessions.\n\nA confession is a positive statement of belief; but in making a positive statement of belief, it inevitably excludes those who disagree with its content. Even the most tenuous confessions do this: the Unitarian may claim a creedless faith, but he is never going to invite a Trinitarian, who insists upon the nonnegotiability of the Trinity, to fill his pulpit; Trinitarians are therefore excluded. And if it is true that the creedless faith of the Unitarian inevitably excludes some, how much more true is this of orthodox Christian creeds and confessions? The Athanasian Creed is the most spectacular of these as it contains not only positive statement of Christian doctrine but also anathemas against those who disagree with its teaching. It is explicitly, not merely implicitly, an instrument of exclusion.\n\nTrajectories of thought that take their cue from traditional Christian liberalism have little or no patience with such exclusivism, of course, because they see doctrinal statements not as transcendent truth claims but as expressions of the religious psychology of the individual or the particular religious community. Whether the inspiration for this is the kind of Kantian theology of Schleiermacher or the linguistic philosophy of Wittgenstein, the net result is the same: the truth claims of one community do not apply in any real or straightforward way to another; thus the problem of excluding some is localized and limited. This is my truth; tell me yours (to quote the title of the Manic Street Preachers' 1998 album).\n\nThe last decade has intensified this fear of exclusion particularly when it comes to religion. The impact of religiously motivated or religiously expressed terrorism such as the attack on American institutions on September 11, 2001, has created a cultural atmosphere scarcely conducive to exclusive religious truth claims. One can see this in various responses that have been offered to the rise of alleged religious radicalism. There is the increasingly commonplace use of the catch-all term \"fundamentalism\" and its cognates that presumably lumps anyone who takes their religion seriously together under the same scary category, the wild-eyed Jihadist suicide bomber and the aged Amish grandmother. That fundamentalism equals violence is virtually a given for many.\n\nReaction within the religious world to this cultural moment is interesting. Within Christian circles, the decade after 9\/11 saw the rise and fall of the cluster of movements grouped together as the emergent church, with its emphasis on Christianity as a way of life, not a set of doctrines, and its prioritizing of belonging before believing. This latter slogan, of course, can only make sense if belonging and believing are actually separable in a more than merely formal way. Such a notion is, in the case of many emergent leaders, built on assumed postmodern epistemologies. These are themselves in origin connected to the rise of postcolonialism, with its fear of the hegemony of the white man's religion and the imperialist use of Western ideologies, of which Christianity is perhaps the most obvious and historically influential.\n\nThis fear has meant that a Christianity that is committed to truth claims which apply beyond the community of faith or which exclude certain people from that community is profoundly at odds with the cultural current. Strange to tell, we do still live in societies that routinely exclude people. The fact that prisons are full to bursting indicates that society still considers some forms of behavior to be unacceptable and demands their exclusion from mainstream social life. Legislation against discrimination on the basis of race, creed, or sexual orientation indicates that some views are beyond the pale, and those who hold them are not to express them in practical terms in the public sphere. Yet religion, particularly traditional religion, finds itself at a cultural moment where it is feared because it dares to say that some beliefs and practices are true and good while others are false and bad. Such a moment is scarcely conducive to any form of creedalism or confessionalism.\n\nOne further, and perhaps unusual, example of this fear of exclusion is the phenomenon known as evangelicalism, typically understood as a conservative, orthodox form of Protestantism marked by an emphasis on conversion and evangelism. Evangelicalism is a somewhat balkanized phenomenon, and its various tribes, or subtribes, often have little difficulty in drawing lines that exclude others who regard themselves as evangelicals from their own particular group. Nevertheless, what evangelicalism in all of its forms typically does is prioritize parachurch institutions over and above the church. Whether we are talking in the United States of the National Association of Evangelicals or The Gospel Coalition, or in Britain of the Evangelical Alliance or Affinity, we are talking about coalition movements, and coalition movements by their very definition require broad statements of faith.\n\nThese groups all have statements of faith; but they are statements of faith designed to keep in the tent all the various sects of which the clan chiefs approve. Thus, matters that are vital to the constitution of actual churches (a clear position on baptism, for example) are typically left to one side, on the grounds that the parachurch leaders do not wish to exclude people because of such matters. The statements are therefore often brief and, compared to, say, the Belgic Confession or the Westminster, highly attenuated.\n\nThis is not necessarily a problem, provided that nobody forgets that these groups are not churches and that they are therefore always to be subordinate to churches in the way Christians think about the practical outworking of their faith. Too often, however, the impression is given that these groups, representing this nebulous phenomenon \"evangelicalism,\" consider themselves to be the higher synthesis and the context where the real action takes place. The culture that such an attitude reflects ultimately tends to send the message to Christians that issues such as baptism are of minor importance, and that the matters which divide denominations are trivial and even sinful in the way they keep Presbyterians and Baptists from belonging to the same church. This is, ironically, not a million miles from the wider culture's fear of exclusion and actually sets such professedly conservative evangelicals on an odd continuum with many of the emergents whom they would repudiate. The difference between the conservative evangelical and the emergent might be profound at the level of epistemology, but in terms of regarding doctrine as negotiable and traditional structures of church authority as practically irrelevant, the difference might not be as great as is often imagined.\n\nConclusion: Creeds, Confessions, and Distasteful Christianity\n\nIn outlining various cultural factors that militate against the use of creeds and confessions in the church, I am not arguing that every minister or every believer who declares they have \"no creed but the Bible\" is necessarily in thrall to all or any of the above. Indeed, some of the most militant \"no creed\" people I have ever come across have been very much on the hardline separatist wing of the Christian church and scarcely vulnerable to accusations that they are capitulating to the wider cultural fear of excluding someone. They have a legitimate fear that creeds and confessions can end up in certain circumstances supplanting Scripture and becoming the sole authority in the way the church operates.\n\nWhat we have seen, however, is that there are powerful currents within modern life that militate in various ways against the positive use of creeds and confessions in the church. These currents often go unnoticed by those of us who have no choice but to live, move, and have our being within them. Thus, the pastor who thinks he is being biblical by declaring he has no creed but the Bible may actually, upon reflection, find that his position is more shaped by the modern world than he at first realized. Rather than instinctively taking his cue from the historic practices of the church, he may in fact really be shaped by the wider world. The stories the modern world tells us are powerful: the future-oriented promise of science, the technology that privileges the young, the materialistic paradise offered by consumerism, which is always just around the next corner, the dying of confidence in words, the fragmentation of human nature, the distrust of traditional structures and notions of authority, and the wicked results of saying that somebody else is wrong and does not belong. All of these in their different ways make the idea of doctrinal Christianity, expressed in creeds and confessions, both implausible and distasteful; and all of them are part of the cultural air we all breathe.\n\nThis leads to a very important distinction. Modern culture has not really rendered creeds and confessions untrue; far less has it rendered them unbiblical. But it has rendered them implausible and distasteful. They are implausible because they are built on old-fashioned notions of truth and language. They make the claim that a linguistic formulation of a state of affairs can have a binding authority beyond the mere text on the page, that creeds actually refer to something, and that that something has a significance for all of humanity. They thus demand that individuals submit, intellectually and morally, to something outside of themselves, that they listen to the voices from the church from other times and other places. They go directly against the grain of an antihistorical, antiauthoritarian age. Creeds strike hard at the cherished notion of human autonomy and of the notion that I am exceptional, that the normal rules do not apply to me in the way they do to others.\n\nThey are distasteful for the same reason: because they make old-fashioned truth claims; and to claim that one position is true is automatically to claim that its opposite is false. God cannot exist and not exist at the same time; he cannot be three persons and one person at the same time, at least not without unhelpful and hopeless equivocation (despite the claims of some Reformed theologians to the contrary). Truth claims thus imply a hierarchy whereby one position is better than another and where some beliefs, and thus those who hold those beliefs, are excluded. That may not be a very tasteful option in today's society but, as noted above, even the modern pluralist West still excludes those that it considers, if not wrong, then at least distasteful and unpleasant.\n\nWe are na\u00efve as Christians if we think that our thinking is not shaped by the cultural currents that surround us. Of course, we cannot abstract ourselves from our context; we cannot cease to be embodied individuals, each with our own personal biographies, who live within a complex network of social relations that influence the way we live and think and speak. Yet to know something of our context is to make ourselves aware of some of the invisible forces that have such an unconscious influence on us. Once we know they are there, we at least have the possibility of engaging in critical reflection, which will allow us to some extent to liberate ourselves from them\u2014or, if not to liberate ourselves, at least to make us more aware of why we think the way we do.\n\nThus, I conclude this chapter by posing a challenge to those who, in their earnest desire to be faithful to Scripture as the supreme authority of faith and life, claim that they have no creed but the Bible. Reflect critically on the cultural forces that are certainly consonant with holding such a position and ask yourself whether they have perhaps reinforced your antipathy to creeds and confessions in a way that is not directly related to the Bible's own teaching at all. Then, setting aside for just a moment your sincere convictions on this matter, read the rest of this book and see whether creeds and confessions might not actually provide you with a better way of preserving precisely those aspects of biblical, Christian faith which are most valuable to you and which you passionately wish to communicate to your church.\n\n1The Police, \"De Do Do Do, De Da Da Da,\" Zenyatt\u00e0 Mondatta, December 5, 1980.\n\n2For all of the plausibility of such emotive arguments in modern culture when it comes to, say, teenagers sleeping together, we still live in an age when thankfully this is not yet considered a plausible justification for serial killers.\n\n3Amazon.com book description of Rita M. Hancock's The Eden Diet: A Biblical and Merciful Christian Weight Loss Program (Oklahoma City: Personalized Fitness Products, 2008), http:\/\/www.amazon.com\/Eden-Diet-Biblical-Merciful-Christian\/dp\/0982034105\/ref=sr_1_1?ie=UTF8&qid=1310558775&sr=8-1.\n\n#\n\n## The Foundations of Creedalism\n\nThe cultural currents that make creedalism implausible in the current climate are powerful and often so pervasive as to be virtually invisible. This was the basic burden of chapter 1; and those are very na\u00efve who think that the church is immune to these currents when it comes to how she understands the significance of creeds and confessions. The speciously biblical cry of \"No creed but the Bible!\" is not as straightforwardly countercultural as many might think. The next question, of course, is, exactly how biblical is this cry anyway?\n\nIn this chapter, I do not intend to refute, point by point, all of the challenges I outlined in chapter 1. Rather I want to focus on a positive exposition of a series of positions that, taken together, will require the church which rejects confessions to realize just how much unbiblical ideas have shaped her thinking in this area and thus to revise her attitude toward creeds and confessions. First, I make a case for the importance and adequacy of words in the biblical narrative and thus our understanding of God and his revelation. Second, I point to biblical teaching that human beings are not self-created but possess, for want of a better word, an essence that is given to them from without. Third, I point to evidence within the Bible of the use of creed-like statements and the need to transmit \"forms of sound words.\" Finally, I outline New Testament teaching on biblical church government and thus of an institutional church that is charged with faithful transmission of the faith.\n\nThe Adequacy of Words\n\nWhile the problems outlined in chapter 1 are considerable, there are still very good reasons why the church should formulate, adopt, and use creeds and confessions. These reasons are rooted in theological considerations that serve to relativize and weaken the kind of cultural objections we have noted, both those from the wider world and those from the evangelical sphere.\n\nFoundational to this creedal formulation is a biblical understanding of the nature of God himself and fundamental to the biblical idea of God is the fact that he is a God who speaks. We see this in the language used in John 1:\n\nIn the beginning was the Word, and the Word was with God, and the Word was God.\n\nWe need to be careful in how literally we take the language of \"Word\" here, for the Son is not a word spoken by the Father in the way that I might say \"automobile\" or \"classic rock album.\" I move my vocal chords and vibrate molecules of air at a certain frequency; God does neither relative to the eternal generation of the Son. But John is clearly making an analogy between the Father and the Son and the speaker and the word spoken. Yet the notion of God as being a God characterized by speech does not rest on the prologue of John's Gospel. Right at the very inception of history, the moment when time itself began, the creative activity of God is described in terms of speech:\n\nIn the beginning, God created the heavens and the earth. The earth was without form and void, and darkness was over the face of the deep. And the Spirit of God was hovering over the face of the waters.\n\nAnd God said, \"Let there be light,\" and there was light. And God saw that the light was good. And God separated the light from the darkness. (Gen. 1:1\u20134)\n\nIn this passage, the light is created by the word of God's power. Indeed, the very first divine action of which we hear in the Bible is that God speaks. There is nothing; God speaks; and then there is something\u2014and that something is a direct result of the act of divine speech. His word is thus powerful and creative. Indeed, it is definitive of what exists; and, of course, the continuation of Genesis 1 and 2 indicates that this speech is crucial to the further creative acts, culminating in the internal divine conversation which leads directly to the creation of man and woman in the divine image.\n\nThe biblical narrative quickly makes it clear that divine speech is to be a fundamental aspect of the special relationship that exists between God and those made in his image. Genesis 1:28\u201330 establishes the basic status and duties of humanity in relation to the created world, with God speaking to the man and the woman and telling them what they are to do, what authority they have, what they may eat, and what they must not eat. The arrangement is articulated using words; it is linguistic in its basic form.\n\nThe importance of language as a means not simply of communication but of defining and sustaining the relationship of God and humanity continues after the fall. God curses the serpent and then tells the woman and the man how their relationship with him, with each other, and with the creation itself is changed as a result of the fall (Gen. 3:14\u201319). In pronouncing this curse, God is not simply describing what has happened; he is bringing the state of affairs into existence.\n\nThe same kind of linguistic action is found in Genesis 15, where God enters into covenant with Abraham, and thus starts the long process of election in history that will culminate in the coming of Jesus Christ and the kingdom of God. Again, words are central, this time not to the creation but to the anticipation of the new creation. The promise, like the curse before it, is not simply a description of a state of affairs; it is constitutive of the state of affairs that it brings into existence between God and Abraham.\n\nAs well as being active and creative, God's word, God's speaking, is also a means of his presence. One interesting example of this is provided by the story of Elisha and the Shunammite in 2 Kings 4. The story starts as one of the most delightful in all of Scripture. The Shunammite is a wealthy lady who decides to build an extra room on the roof of her house so that whenever Elisha passes through her town, he knows that he can have a place to stay. In return for her kindness, Elisha asks his servant to find out from the woman what there is that she lacks which he can provide. So unassuming and selfless is she that she simply responds that she lives among her friends and family and thus has all that she could want; she is so selfless that she does not even think to bring to the prophet's attention her childlessness, a source of social shame as well as (presumably) personal sadness. Nevertheless, the servant notices her lack and reports it to the prophet who then declares that, within a year, she will give birth to a child.\n\nSo far, so good. Then, of course, the child grows up, and when he is old enough to be in the fields with his father but still young enough to sit on his mother's knee, he has some kind of seizure, which leads to his death. In a scene of brutal, heart-breaking realism, the woman races to find the man of God and collapses at his feet in a veritable explosion of grief. Elisha immediately sends his servant with his staff to raise the child, but she will not leave the prophet, insisting that he and he alone is the one who must come.\n\nThe question is obvious: why? Why this insistence on Elisha? Could his staff not have worked just as effectively, given that God is sovereign and all powerful? The answer lies with the significance of who the prophet is. At that point in Israel's history, the prophet was the one\u2014the only one\u2014through whom the words of God came. We might recast that by saying that he was the only one through whom God was present. Now, of course, there are different modes of divine presence. There is what one might call the bare metaphysical presence of God, which is everywhere and which sustains everything. As God is in Tel Aviv or New York so he is also in Damascus and La Paz. But there is also a presence of God that is active and powerful. Thus, God was present in the ark of the covenant in a way in which he was not present in one of Solomon's wives' jewelry boxes. There were promises attached to his presence in the ark such that the ark was a unique place, which, if touched in a profane or even accidental manner, brought swift retribution to the one who had erred (2 Samuel 6).\n\nThus, the speech of the prophets was the mode of God's special presence in Israel. That is why the Shunammite needed Elisha and not just his servant and staff. She needed the organ grinder, as we would say in England, and not the monkey. Elisha was the one who not only symbolized God's presence but who, as he\u2014and he alone\u2014spoke God's words, was the very instrument of his presence.\n\nWe can infer this further from the way in which the Old Testament can emphasize the absence of God. Amos 8 talks of a time when God will turn his back on his people and bring great darkness and despair upon the land. The climactic moment of this description of what we might call \"the active absence of God\" comes in verses 11\u201312:\n\n\"Behold, the days are coming,\" declares the Lord GOD,\n\n\"when I will send a famine on the land\u2014\n\nnot a famine of bread, nor a thirst for water,\n\nbut of hearing the words of the LORD.\n\nThey shall wander from sea to sea,\n\nand from north to east;\n\nthey shall run to and fro, to seek the word of the LORD,\n\nbut they shall not find it.\"\n\nThe famine will be of the word of God; and the frantic search of the people for that word is clearly also the frantic search of the people for God himself. His word is not to be found; he himself is absent from his people. That is what will make this moment so terrifying: the silence of God is the absence of God.\n\nThis is entirely consistent with what we have noted so far about the function of words in the divine economy of creation and salvation: God's speech is special; it is creative; it defines his relationship to his creation; it defines who his creatures are; it establishes the nature of his special relationship with peoples and individuals; it is the instrument by which he exercises and withdraws his power; and it is perhaps the most significant mode of his presence.\n\nOne final biblical note in this context is the baptism of Jesus in Mark 1. In Mark's account, Jesus is baptized, the heavens are torn open, the Spirit descends in the form of a dove, and immediately the Father declares his love for the Son. The language of tearing is significant here. The Greek word schizein is used on only one other occasion in Mark's Gospel, at 15:38, where the temple curtain is torn in two. Many readers of the Gospel see this tearing as the opening up of the way for entry in to the Holy of Holies. It is more likely that this represents the outward movement of God from the Holy of Holies. The evidence for this is twofold. The structure of the narrative points clearly in this direction: Jesus dies; the temple curtain is torn in two; and the Gentile centurion who is facing the cross declares that Jesus must have been the Son of God. In other words, the crucifixion is followed by grace flowing to the Gentiles; Mark inserts the curtain incident in-between as a means of underlining this watershed in the history of God's salvific plan.\n\nThe second reason for reading the event this way, however, is the connection with Mark 1. When a biblical writer only uses a particular word on two occasions, and both occasions are of obvious significance in the history of salvation, it is reasonable to see if the two can be connected. In fact, the connection lies in the notion of the presence of God. There was apparently an intertestamental Jewish tradition regarding Isaiah 64:1, \"Oh that you would rend the heavens and come down, that the mountains might quake at your presence.\" This tradition saw the rending of the heavens as the eschatological sign that would mark the return of God's Spirit after a long period of absence, the kind of absence described in Amos 8. Thus, Mark's word choice here is very deliberate. Indeed, the strong schizein should not be translated as \"opened\"; if it were so, then Mark 15:38 should presumably be translated in the same way; but \"the curtain in the Temple was opened\" would obviously be a thoroughly inadequate rendering! In fact, what Mark is doing is signaling to his readers that God's Spirit is about to return, that God is once more going to be present, and, amazing to tell, that is what God immediately does in the very next verse (Mark 1:11). God is once again speaking; he is once again present with his people.1\n\nBefore we move to the next point, one further comment on God as speaking is important: idols are silent. Psalm 115 makes this point rather dramatically as the psalmist describes the manufactured images of the pagan gods. They have all the attributes of real people\u2014hands, eyes, ears, noses; but none of these things actually functions. They look like the real things, but they are merely impotent fakes. Among all of these, the final one is perhaps the most dramatic: \"they do not make a sound in their throat\" (v. 7). They are silent, in stark contrast to the God who speaks. Whatever else the true God is, he is not one of the silent gods, for they are the false gods, the idols, the products of the heart of mortals.\n\nThere are many theological implications of the identification of God's speech as a special mode of his presence, but the point I want to make here is simple: words are the means God has chosen for his presence and therefore are by definition an adequate means for that presence. Yes, we must accept the reality of all of the suspicion of language that we find in contemporary society, whether of the sophisticated kind exemplified by certain streams of critical theory and literary criticism or the more popular kind found in the mysticism of a Madonna lyric or the supercilious cynicism of the tabloid press. Yet we must also understand that words in themselves are essential elements of our understanding of who God is and quite adequate for theological purposes, even if they can be horribly abused by the humans in the way they use them. They are in fact one of God's chosen means for revealing himself to his people, being present among them, and performing his great saving acts.\n\nWords in Service of the Divine\n\nThis divine use of words flows over into the human use of words at a theological level. If words are God's way of being present with his people and working among them, words are also the human means of responding to God and of communicating with each other about God. Again, the Bible itself bears abundant testimony to this fact. Take, for example, the book of Deuteronomy, perhaps the greatest sermon ever preached. Yes, it is part of a divinely inspired canon, but it is also the words of a human being, aimed at other human beings. He speaks God's words to people but in those same words speaks about God. The speech that is Deuteronomy is nothing less than a grand articulation and renewal of the covenant between God and his people, as well as the culminating sermon of Moses's own distinguished career as prophet and preacher.\n\nYet the use of words for divine, covenant purposes is not restricted to the great and the good in Israel. An example of this is provided in the stipulations surrounding the annual recapitulation of the Passover, which we find in Exodus 12. Having outlined what the children of Israel are to do in the Passover ceremony, Moses speaks of a time when, in future years, the actions will be performed in front of a generation that has no firsthand recollection of what happened on that terrifying night in Egypt:\n\nAnd when your children say to you, \"What do you mean by this service?\" you shall say, \"It is the sacrifice of the LORD's Passover, for he passed over the houses of the people of Israel in Egypt, when he struck the Egyptians but spared our houses.\" (Ex. 12:26\u201327)\n\nA number of comments are worth making here. First, the importance of history is obvious. The ritual of the Passover is meaningless without the historical framework in which to understand it. If part of the rejection of confessionalism can be seen to lie in the rejection of the importance of history, then it is obvious that such is impossible to square with the Bible. A significant portion of the Bible is historical narrative, and much of what is not yet depends upon that historical narrative for its meaning.\n\nSecond, the ritual in and of itself is inadequate for achieving God's purpose, namely, the recollection of God's act of judgment and deliverance in Egypt. It was not enough for the parents merely to repeat the ritual if their children did not understand its significance the first time they saw it. Simply going through the motions again and again would not achieve anything. They needed to explain it to them by the use of words and in a manner that connected the ritual to the events of long ago. It is a simple observation, but this surely indicates both the adequacy of words for communicating important theological truth and their priority over ritual action. This is a point that is made repeatedly throughout the Bible: the prophets often engage in odd theatrics before the people and then explain the meaning of their actions; and Christ himself performs acts that require explanation, whether healings, miraculous feeding, or the Lord's Supper. Words have a special place in the history of God's people as the means of recalling his actions in history and thus of pointing to who he is.\n\nThis centrality of words is clear in the ministries of all of the Old Testament prophets. For example, when Isaiah is commissioned in Isaiah 6, he is told specifically to go and speak to the people, just as Moses had been told earlier to go and speak to Pharaoh. Indeed, what was the role of the prophet if it was not to bring God's judgment and God's promise to bear on the world in general and his people in particular through the medium of words, words proclaiming the identity and the will of God? It is hard to see how judgment or salvation could be proclaimed to people without the use of words. Somehow a simple mime or a liturgical dance would seem rather inadequate to the task. The reason for this is obvious: as God is a speaking God, and human beings are made in his image, so the principle means by which God and humanity relate is that of words.\n\nThe role of prophetic preaching continues into the New Testament and, through the paradigmatic significance of the ministries of the apostles and their successors, into the postapostolic world. Paul's polemic against the superapostles in 1 Corinthians is in part an assertion of the power and importance of words in and of themselves. In Corinth, Paul's reputation has suffered because he lacks the aesthetic appeal of the local orators who were by all accounts impressive in both their speaking style and in their physical appearance. In this context, Paul points away from the outward aesthetics of the contemporary orators and to the content of what is said. Preaching is powerful because of the message it communicates (1 Cor. 1:21). The point here is primarily about the message of the gospel, not the form; but the form is surely still important. This is a message that is preached; there is no hint that the aesthetic problem should be avoided by the obvious move of simply finding some nonverbal means of communicating the message of the cross. No. The words are still important. Words are not simply adequate for the divine purpose in preaching; they are the divinely sanctioned means of achieving that purpose.\n\nThe basic reason for this is that which is clear in the Passover instructions in Exodus 12: God's saving actions are historical, but they need to be interpreted, to be explained by words, in order for the audience to grasp them. The meaning of God's salvation in the Passover is not to be found in some mystical experience of the individual or the community performing the action; rather it is that to which the action refers or that which it signifies, and that can only be explained using words.\n\nThe same is true in 1 Corinthians 1. Here Paul emphasizes the cross as the great dividing line that runs through humanity: how one responds to the cross will determine whether one is being saved or one is perishing. Yet the cross and the believer's response are not mystical experiences. When Paul talks of the cross here, he is talking of the significance of the cross as it is explained in the preaching of the gospel. Merely looking at a tattered and broken piece of humanity hanging on a piece of wood, or imagining such with the mind's eye, is of no use whatsoever. It is the cross set within the context of the biblical story of humanity's creation and fall that has significance; and this requires verbal communication. One might add that there is no other way of communicating this message that can bypass the use of words. Neither painting nor mime nor dance is remotely adequate for the message. Only clear, verbal statement of the matter can bring the message home and frame the matter in such a way that the responses can then be either those of faith or of unbelief.\n\nIn sum, the Bible not only presents us with a picture of God's relationship to creation and to his people in which words are absolutely crucial means of his presence and his revelation, and are, by obvious implication, completely adequate for such purposes; it also shows us that words are a vital means of communicating the message of God from person to person. Moses preached; Elijah preached; the prophets, major and minor, preached; Christ preached; and Paul preached. All used words to impress the nature and claims of God upon people. Words are clearly the main means of so doing. Thus, any theology that claims to take the Bible as its authority must take the teaching of the Bible on words, and indeed the verbal form of the Bible itself, with utmost seriousness and thus see words as a normative and normal part of Christianity.\n\nHuman Nature as a Universal\n\nOne of the key points made in chapter 1 is that repudiation of the past is often part of an assumed, and frequently unconscious, repudiation of the notion of human nature as a given. Why should I take the writings of a bunch of dead white men from seventeenth-century Britain seriously, if I am a live, flesh-and-blood African-American woman living in San Francisco at the start of the twenty-first century? What have fourth-century bishops in the eastern Roman Empire to say that is of relevance to the twenty-something owner of a tattoo parlor in today's London? What is there that binds us together\u2014human beings from different times and places\u2014such that there might be some point of useful contact between us? These are good questions; and underlying them is the idea that human nature is a construct of the particular historical, social, and cultural contexts in which we find ourselves.\n\nIf the whole notion of human nature is now negotiable within the wider culture such that our mere sharing of a basic genomic structure with other human beings is not an adequate enough foundation upon which to build a larger, more metaphysical notion of human nature, it is important to note that this is not the case in Scripture. The creation account in Genesis makes a series of fundamental distinctions that are important in this regard. First, there is a distinction between Creator and creation. The latter is utterly dependent on the former for its existence and thus is not to be confused or conflated with God. \"In the beginning, God\" is foundational to all that follows. God is the one constant of existence; everything else is contingent upon his being and action. The world is created and sustained by God; there is no mutual dependency, simply divine priority.\n\nSecond, human beings are distinguished from all other creatures. This is evident in a couple of ways: Genesis 1:27 speaks of God creating man \"in his own image.\" Theologians have wrestled over the centuries with what exactly it is that constitutes this image of God, but the details of that discussion need not detain us here. The point to note here is simply this: human beings are distinguished from all other creatures on the basis that they and they alone are made in God's image.\n\nHuman beings are also distinguished from all other creatures by receiving a specific mandate from God (Gen. 1:26, 28\u201330), which effectively places the rest of creation under their control as God's vicegerents. They fulfill a caretaking or overseeing function in creation that is shared with no other kind of creature. Again, what exactly this means is not significant for my argument at this point. It is sufficient to note that the Bible ascribes no similar mandate to any other species.\n\nClosely connected to this is Adam's act of naming. While God himself names Adam, Adam names all other creatures. \"And whatever the man called every living creature, that was its name\" (Gen. 2:19). Interestingly, we see here that the distinction between humanity and the rest has a certain linguistic aspect to it. Human beings are linguistic beings. We see this further in one other area where humans differ from all other creatures: God speaks to human beings. To be human is to be one who is addressed by God.\n\nWe might express this point in slightly different terms: human beings are those confronted by God's revelation, and that confrontation involves a significant linguistic aspect. This is, of course, consistent with what we noted above about the biblical description of God as being a God who, above all else, speaks. It is also surely legitimate to see this linguistic component of human nature as being integral to the notion of the image of God. Of all creatures, human beings are the only ones to whom God speaks and who speak to him in return and also to each other.\n\nThe importance of this point is that these aspects of human uniqueness provide a universal context for all human activity. This has various implications. In the discussion of theology, it has become commonplace to talk about two horizons in interpretation: the horizon of the text and the horizon of the interpreter or the interpreting community. This has led in some cases to a radical skepticism concerning the possibility of producing stable and reliable interpretations. We may well share the same text, but if I am a man and you are a woman, or I am white and you are black, is there anything more than our starting point\u2014the text itself\u2014to connect our interpretations? And is it possible to compare your interpretation with mine and decide which of us, if either, has produced a more accurate account of what the text actually says or does?\n\nIf we understand human nature as fixed, as something which is not constructed by the individual or by the community but something which is given by God in his address to us, then we are on much more secure ground in moving theological statements from one time, place, or culture to another. Human nature is something which is more basic than gender, class, culture, location, or time. It cannot be reduced to or contained within a specific context such as to isolate it from all else. This is not to deny that context has a huge impact upon who we are and how we think; it is simply to say that all of these particulars that make individuals unique and allow us to differentiate one person from another are relativized by the universal reality of human nature that binds us all together.\n\nHuman beings remain essentially the same in terms of their basic nature as those made in God's image and addressed by his word even as we move from place to place and from generation to generation. God remains the same; his image remains the same; his address to us remains the same. The clear inference is that the basic categories that define the relationship between the two (creation in his image, the fall, redemption in Christ, etc.) remain hardy perennials, unaffected at their core by the comparatively trivial accidents of time and space that separate one person from another. Modern culture, for all of its often drab uniformity, prides itself on difference and on kaleidoscopic variety. Whatever the truth of this may be, it does not affect the essential core of identity that binds me together with human beings in modern China and with people in ancient Rome: we are all made in God's image; and he addresses us all through his word.\n\nIn short, a biblical understanding of human nature as a universal will temper any talk that seeks to dismiss theological statements from the past on the simplistic grounds that there is nothing in common between us and the people who wrote them. Frankly, it has become rather tedious to read some simpleton who thinks he is a genius dismissing works of theology or literature or philosophy because they were written by \"dead [or sometimes living] white males.\" The status of being dead, white, or male is neither here nor there when it comes to issues of theological truth (and I am inclined to say, of cultural value either). Just because these pundits cannot see beyond race, gender, and whether someone has a pulse does not mean that the rest of us need to cower before their simplistic categories of discourse. Many theologians I read may be dead, white (though hardly the case for Augustine, I should think), and male, but that does not mean they have nothing to say to a live, black female. All humans are partakers of a common human nature. All are addressed by the same revelation of the same God, and all are called to respond to that revelation. Thus, when my student to whom I referred in chapter 1 asked, \"What have these Westminster Standards to do with my ministry?\" my response (\"What has your ministry to do with the church?\"), while perhaps a little cutting, was nonetheless rooted in a basic biblical commitment to the idea of universal human nature, a commitment that both the student and I shared.\n\nTo return to a passage I have already cited, in 1 Corinthians 1 Paul provides a good example of the relevance and irrelevance of cultural context and conditioning. Here he addresses different reactions to the cross from two very different cultural groups, Jews and Greeks. The Jews, he says, regard the cross as offensive. That is, to use the trendy jargon, their \"reader response\" to the gospel is that it offends their Jewish sensibilities to think of God as cursed and hanging on a tree. Their cultural context predisposes them to read the cross in this way. For Greeks the case is different. Their reader response is to see the cross as foolishness, as nonsense. For them it is sheer stupidity to claim that God would contract himself to a span and hang and suffer on the cross, and subsequently to claim that this particular act could have universal significance. The Greeks have a different cultural context that gives them a proclivity to read the cross not as offensive but simply as silly or foolish. Thus, the different contexts of the two groups are very important in shaping their responses to Calvary. Nevertheless, the separate reader responses of the Jews and the Greeks, rooted in different cultural contexts, do not subvert the truth of the cross's universal significance and fixed meaning. This is clear from the fact that both Jews and Greeks can be placed into a single category, despite the different responses they have to the cross: they are those who are perishing, simple as that. The correct response for both Jews and Greeks is to see the cross as the power of God to salvation. This is because both groups are at bottom human beings and thus ultimately accountable in the same way to the same God. The things that divide Jew and Greek are of little importance compared to that which binds them together and places them both under the judgment of the cross.\n\nUndergirding this is Paul's own commitment to something else that points clearly to the givenness of human nature and the solidarity of humanity: the function of Adam and of Christ. The logic of Romans 5 and of 1 Corinthians 15 depends upon corporate solidarity. Adam was the representative head of the human race, of a group of individuals who all share something in common, who all have a common nature. Thus with Christ. When he became incarnate, he assumed something which allowed for his identification with others: once again, a common, shared nature. All attempts to deny or to submerge this nature through an overemphasis upon particulars are both unbiblical and ultimately doomed to failure. Human nature is not simply a socio-linguistic construct. It has an ineradicable objective reality; and that reality provides the vital point of contact across cultures, times, ethnicities, genders, sexual orientations, skin colors, and whatever other particular we might care to think up. Indeed, to deny this would be to subvert the theology that Paul expresses in his Adam-Christ parallelism and would thus have dramatic impact upon the nature of salvation. In fact, it would abolish Pauline soteriology in its entirety.\n\nSo how does all this apply to creeds and confessions? Creeds and confessions are human attempts to summarize and express the basic elements of the Christian faith. They have been constructed throughout the ages by people from very different contexts but who are all bound together by the shared horizons of God's revelation in Christ and in the biblical text and their own common human nature as readers of that text. This is what gives creeds and confessions a quality that transcends the local conditions of their original composition and that allows us to take them seriously. Of course, it does not guarantee their truthfulness. All creedal formulations are subordinate to Scripture and subject to correction thereby. Like Jews and Greeks, it is quite possible that other human beings' responses to God's revelation can be wrong\u2014utterly wrong. But we should not take seriously childish arguments that, for all of their specious sophistication, really amount to dismissing the relevance of creeds and confessions on the grounds that the authors lived a long time ago, had a different (or the same) skin color to us, or possessed only one X chromosome.\n\nThe Church as Institution\n\nHaving established the centrality of words to the nature of God and to the revelation of God to humanity, and having argued that there is such a thing as humanity, partaking of a common human nature, to which these words are addressed, there is one more point we need to establish before turning to the biblical evidence for creeds and confessions, and that is the existence of the church as an institution. By \"institution\" here, I mean a self-consciously organized body of people who identify with a cause (what we now call \"church membership\") and who acknowledge a structure of ministerial authority.\n\nWe noted in chapter 1 that today there is a widespread cultural suspicion of institutional authority. We also noted that this suspicion is somewhat selective, focusing mainly on what we might call traditional institutions, which are rooted in the past and which carry forward ideas and structures from that past. Newer pop culture institutions\u2014talk shows, television, the web, the music industry\u2014wield huge power and enjoy significant, unmerited, and too often uncritical public confidence. Yet the Bible clearly lays out a structure of authority in the church that is traditional in the above sense, in that a key part of its authority lies in the handing on of truth from one generation to the next via established power structures. The biblical church acknowledges authority at a number of levels: the authority of God's revelation, the authority of the past, and the authority and status of those officially charged to transmit the \"tradition\" from one generation to the next.\n\nIn the New Testament description of the church, there is a very close connection between the institutional aspects of the Christian community and the doctrinal ones. Elders and overseers are marked by numerous qualities, one of which is the ability to teach. This is not, of course, the ability to teach in the abstract, as if mere communication skills were the criterion of office-bearing; rather it is the ability to teach a particular content, the gospel of Jesus Christ. Doctrine and structure are thus interconnected in the person of the elder. Indeed, one could approach the subject of the church from either angle, doctrinal or structural. I have simply chosen for convenience to reflect first on the former.\n\nBasic to belonging to the church in Paul's mind are two things to which he refers in Romans 10:9\u201310:\n\nIf you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved. For with the heart one believes and is justified, and with the mouth one confesses and is saved.\n\nWe will return to this passage later when exploring the importance of making a difference between the level of knowledge minimally required for membership and that required for office-bearing.2 Here I am concerned simply to note that Paul sees credible Christian profession as involving a doctrinal belief (the Christ has been raised from the dead by God) and a public statement (\"Jesus is Lord\"). Both are laden with doctrinal freight, but Paul does not seem to indicate that massive doctrinal knowledge is necessary for this basic, credible testimony of faith. Nevertheless, it is clear that doctrine\u2014both in terms of belief and content\u2014is still important to this profession, even if in some fairly minimal way. Words and content are thus significant. What Paul does not say is: if you have a warm, incommunicable feeling in your heart and express this by incoherent sounds from your mouth, you will be saved. No. There is propositional content here\u2014publicly expressed in a manner comprehensible to others. Not that this proposition (\"Jesus is Lord\") in itself saves\u2014one must also believe in one's heart\u2014but doctrinal confession and personal faith are intimately and inseparably connected with the salvation about which Paul is writing. One's status as a Christian cannot be separated from words.\n\nThat membership of the church is connected to doctrine and words is also clear from noting what it is that leads to one's exclusion. In Romans 16:17, Paul writes the following:\n\nI appeal to you, brothers, to watch out for those who cause divisions and create obstacles contrary to the doctrine that you have been taught; avoid them.\n\nNotice what Paul says here. Contrary to the modern notion that doctrine divides, Paul here says the exact opposite: these people who must be avoided are divisive precisely because they have departed from the true teaching. It is their doctrinal deviance, their departure from true teaching, that makes them sources of division. If Romans 10:9\u201310 made the positive case for doctrine as a vital part of belonging, here Paul states the other side of the case, that wandering away from sound doctrine means being divisive and ceasing to belong.\n\nGiven that belonging to the Christian community has a minimal doctrinal content, it is not surprising that the New Testament also seems to envision that church members will over time grow and deepen in their knowledge and understanding of the Christian faith. Paul, for example, chides the Corinthians that, at a time when he should have been able to treat them as adults in Christ and feed them on solid food, such was their level of immaturity that he could only give them milk as baby food (1 Cor. 3:1). The writer to the Hebrews also speaks of the need to move on from elementary teachings as one grows toward Christian maturity (Heb. 6:1\u20132). Thus, the bar of doctrinal knowledge is set low for initial belonging; but the expectation is that this knowledge will grow and deepen as the believer matures within the context of the Christian community.\n\nThe task of making sure this maturing takes place belongs to the elders\/overseers of the church, one of the reasons why there is a difference between qualification for belonging to the church and for holding office in the same. Paul's letters to Timothy and Titus are most important in this regard. Writing at a time when he knows that he is coming to the end of his earthly ministry, Paul's mind is inevitably drawn to what the church leadership will look like as the generation of the apostles, the men specially commissioned by Jesus himself, passes away. Thus, he lays out clear principles by which leaders are to be selected and by which they are to rule the church.\n\nIn 1 Timothy, for example, Paul instructs Timothy to stay in Ephesus to make sure that he can combat the false teachers who have infiltrated the church there. It is not immediately obvious what this false teaching is, but it is clearly doctrinal in nature, focusing on myths, genealogies and some form of distorted understanding of the law (1 Tim. 1:3\u201311). In this context, Paul outlines the qualifications of an overseer (1 Tim. 3:1\u20137). These include both the ability to teach (which must here mean teach true doctrine, not simply the possession of good pedagogical skills) and also to care for God's church, a qualification which Paul sees as built upon the overseer's ability to manage his own household. The management language he uses is actually picking up on similar terminology in 1 Timothy 1:4 (ESV: stewardship) which has clear implications of good management necessarily being connected to true teaching.\n\nThe notion of an overseer, then, is connected to the ability to teach true doctrine and to refute those who teach falsehood. In his letter to Titus, Paul presents a similar list of qualifications for elders, which include holding to \"the trustworthy word as taught\" so that they can \"give instruction in sound doctrine\" (Titus 1:9).3 Paul then moves immediately to urging Timothy to \"teach what accords with sound doctrine\" (Titus 2:1). In short, the doctrinal maturing of the people of God is something that is especially entrusted to particular individuals in the local church who are distinguished, among other things, by their doctrinal knowledge and their ability to teach the same. The seriousness and importance of this teaching function in the church is then underlined by James's comment that not many should aspire to be teachers because teachers will be judged more strictly (James 3:1). Doctrine is so important in the life of the church that it is only to be entrusted to a special category of qualified people.\n\nGiven the tendency of the modern West of disparaging age and experience (which we noted in chap. 1), it is worth noting that Paul's vision of eldership is, by modern standards, profoundly countercultural. Not only does competence in doctrine and teaching imply age and experience, the other criteria Paul lays out for joining the eldership surely make this explicit. A young person, for example, will scarcely have a proven track record of managing his own household well or a good reputation with outsiders; and, of course, an elder should not be a recent convert (1 Tim. 3:1\u20137). All of these criteria imply age and experience, exactly the kinds of things the contemporary world holds in such disdain when it comes to the deeper questions of life. Of course, Timothy himself appears to be relatively young. That is why Paul tells him to let no one despise him because of his youth (1 Tim. 4:12). But Paul would scarcely have felt it necessary to make this comment unless Timothy's age made him a rather stark exception to the typical rule.\n\nThus, the whole notion of the eldership as described by Paul runs counter to modern tastes: it places a high premium on doctrine, and it typically assumes a mature age and a breadth of experience. Further, to these two countercultural positions, we can add a third: these officers in the church also carry authority and command respect.\n\nPaul states in 1 Timothy 5:17 that the congregation is to have an attitude of respect toward the elders, an attitude which clearly speaks of their authority:\n\nLet the elders who rule well be considered worthy of double honor, especially those who labor in preaching and teaching.\n\nThe attitude of respect that the congregation is to have toward its elders, and the language of \"rule\" being used to described the elders' function, speak of authority. Elders are not simply appointed as the equivalent of hired hands; they are those who rule in the congregation. If we needed further confirmation of this, we find it in 1 Peter 5, which is reinforced with a Christological reference and a brief statement of congregational responsibilities toward the leaders:\n\nSo I exhort the elders among you, as a fellow elder and a witness of the sufferings of Christ, as well as a partaker in the glory that is going to be revealed: shepherd the flock of God that is among you, exercising oversight, not under compulsion, but willingly, as God would have you; not for shameful gain, but eagerly; not domineering over those in your charge, but being examples to the flock. And when the chief Shepherd appears, you will receive the unfading crown of glory. Likewise, you who are younger, be subject to the elders. Clothe yourselves, all of you, with humility toward one another, for \"God opposes the proud but gives grace to the humble.\"\n\nElder authority is thus ministerial, linked with and subordinate to the rule of Christ. As a result, elders are to be respected by those who are younger because this is one demonstration of the humility and love which is supposed to characterize the church. New Testament teaching on the church is thus opposed to so many of the currents of modern culture: it places a premium on age and experience; it is doctrinal in that it connects to notions of truth and to the teaching of the truth; and it articulates a hierarchical structuring of the church as an institution. Of course, a careful study of all of New Testament teaching on church leadership would reveal that leadership and authority in the church are not to be conceived of in quite the same way as we find in the world around us. Church leadership is to be marked by service to others, by suffering, by a distinct lack of glory and prestige as the world around understands it. It is the church's failure to embody these ideals that has given some traction to those who place organized religion and its institutions under the same cloud of suspicion as secular institutions such as governments and big businesses. But abuse of church office does not mean that church office ceases to be a biblical idea, and it behooves the church to respond to the challenge of her cultural despisers not by capitulation to the culture or repudiation of the Bible's teaching but by repentance, reformation, and a renewed commitment to the biblical notion of church government and authority.\n\nSo what has this to do with creeds and confessions? It is to that point which we now finally turn.\n\nA Form of Sound Words\n\nCentral to qualification for eldership, we noted, is a good grasp of sound doctrine and the ability to teach it. This is because the church is to be characterized, among other things, by the growth of her people in knowledge. The elders are thus to be examples (1 Pet. 5:3), to represent to the people, in doctrine and life, something to which the people should aspire. No wonder they will be judged more strictly, for their lives and words are an important part of the didactic process which leads to congregational maturity.\n\nThe teaching aspect is key for another reason, which is profoundly countercultural in the present day: it speaks of the authority of the past. What Paul charges Timothy to do\u2014and remember that Timothy is, in many ways, the archetypal elder of the postapostolic church in New Testament teaching\u2014is to pass on to the next generation the teaching which he himself has received from Paul. Timothy is to be the means of maintaining the apostolic tradition and, by inference, the eldership is thus to be the agency responsible for the passing down of the gospel from generation to generation. This is not to deny the responsibility of every believer to pass on the faith to others, or of parents to teach their children in a manner analogous to the Israelites, with reference to the Passover in Exodus 12; but it is to say that the elders have a peculiar responsibility for this within the church, and that as their authority connects to their teaching, so also their authority connects to the past and to the tradition of teaching which they themselves have received.\n\nThis brings us to the next stage of the argument: if words are central to God's identity, to human identity, and to the relationship that exists between the two, then are those words to be connected to a specific content? The answer is simply and obviously, yes. We see this with the example of the Passover, noted above: the children of Israel are not at liberty to explain the ritual actions of the Passover in any way they choose. They cannot, for example, connect the Passover to Noah's flood or Cain's murder of his brother or to some trivial event in their own lives. Instead, they must connect the action very specifically to events that took place with respect to their ancestors in Egypt via an appropriately constructed narrative that offers a correct interpretation of those events from the perspective of God's saving actions. We might also note the same with the cross: while the cross's significance might be understood in a number of acceptable and mutually beneficial ways (punishment of sin, conquest of the forces of darkness, revelation of God's love, etc.), there are also illegitimate ways of understanding the same, such as Jesus dying because he broke God's law or because he failed in his earthly mission. The message of the cross involves a fixed field of meaning that must be respected by those who claim to teach its true significance.\n\nWe can easily add to this list. For example, in 1 Timothy 1, Paul attacks those who are using the law as a springboard for speculation about genealogies and for the construction of myths. He dismisses this and stresses to Timothy that the primary function of the law is not as a means for indulging in speculative mythology but as a means for exposing sin and wrongdoing for what they are. Indeed, to expand the list further would be simple but also pointless: if the content of words used in a Christian context is meaningless or utterly negotiable, why did Paul write so strenuously about the truth and about maintaining appropriate standards of practice and belief? All of his letters are driven to some extent by his desire to correct deviant teaching or deviant behavior or some combination of the two.\n\nTo summarize what we have argued in this chapter so far: God is a God who reveals himself through actions and words. In his revelation, words have a primary power because they are the means by which he articulates his presence, by which he commands and promises, by which he establishes and defines relationships with his people, and by which he explains the significance of other, nonverbal revelatory actions. To jump forward from Paul to the Reformation, it is why the Reformers made the pulpit central: they saw the verbal proclamation of God as being central to the church, as that which in a sense constituted her very being in the present. That God remains the same, that human beings continue to be made in his image and to face the same fundamental questions in terms of their relationship to him, means that the basic conceptual building blocks of theology and doctrine\u2014that is, descriptions of who God is and what he has done in relation to his people, and what responses he demands from them\u2014remain the same, despite superficial changes of context.\n\nThus, at the end of his career, as Paul looks to the future of a church without the benefit of the guidance of the original apostles, he instructs his prot\u00e9g\u00e9, Timothy, in what elders should be like. At the heart of this description is his emphasis on correct teaching. But what exactly is this teaching to communicate? How does Paul describe it?\n\nIn answering this question, it is interesting to note that in the New Testament Paul both speaks of \"a form of sound words\" when addressing the issue of Timothy's teaching and his continuing fidelity to the Pauline gospel, and also includes in his letters passages that are suggestive of creedal formulation. Important to both of these is 2 Timothy 1.\n\nThe ESV translates 2 Timothy 1:13 as \"Follow the pattern of the sound words that you have heard from me, in the faith and love that are in Christ Jesus.\" The King James Version has the more famous rendering, \"Hold fast the form of sound words, which thou hast heard of me, in faith and love which is in Christ Jesus.\" The word for \"form\" describes a model, form, or standard that is intended to function as a trustworthy or reliable guide. Thus, what Paul is saying here is: Timothy, make sure that your teaching is sound by using the standard of teaching you see in my ministry as the basic rule. The gospel has content, and that content has been ably expressed in the teaching of Paul, teaching of which Timothy has firsthand knowledge.\n\nWhat is interesting is that Paul does not simply say, \"Make sure you stay true to the conceptual content of what you have been taught.\" Paul also highlights the form of the words used here. I suspect there are two related reasons for this: there is both a theological and a pastoral concern. One of the things that teachers in any discipline do is to teach a special vocabulary to their students and instruct them on how to use that vocabulary correctly. This applies to nuclear physicists as it does to investment bankers: each profession has its own special language, and learning the profession is in large part learning the language and how to use it. This facilitates communication between members of the specific profession and allows work to be done properly and efficiently. It also allows for the easy identification of an outsider, or somebody who does not have the requisite competence in the field.\n\nIt is the same in the church. The church has developed over time a tried-and-trusted vocabulary to express the concepts she wishes to articulate. The word \"Trinity\" is a good example of this. The term is found nowhere in the Bible, but it expresses neatly the fact that Father, Son, and Holy Spirit are all equally and eternally God, there is only one God, but that the Father is not the Son, and the Father and Son are not the Holy Spirit. Use of the word thus has a clear theological advantage: it reflects an orthodox concept and it strongly suggests that the user is orthodox. Certainly, if someone starts talking about God as undifferentiated unity or as \"Unitarian\" or as three separate gods, those familiar with Christian orthodoxy will immediately start to become concerned.\n\nThe use of established and accepted terminology also has a second, pastoral purpose: the local elder may not have a theology degree, and the members of the congregation may never have read Athanasius or the Cappadocian Fathers, but when some visiting preacher stands up in their pulpit and starts telling them that the term \"Trinity\" is a load of old nonsense, or that God is simply one person, end of story, metaphorical alarm bells should start ringing in their heads. They should realize immediately that they are not hearing \"a form of sound words,\" and they will want to find out why the preacher thinks the word is nonsense before inviting him to speak again.\n\nThe same applies to a lot of other theological terms and phrases: \"incarnation,\" \"atonement,\" \"grace,\" \"total depravity,\" \"election,\" \"justification by faith,\" \"sanctification,\" etc. Some of these have direct verbal equivalents in Scripture, some refer to concepts that are more disparately expressed. An established, conventional vocabulary for orthodox teaching is thus of great help to the church in her task of educating her members and of establishing helpful and normative signposts of what is and is not orthodox. Not that words cannot have their meaning distorted over time; even Paul faced this issue when he battled over the meaning of grace in Romans or the Law in 1 Timothy. But the fact that there can be abuse of words and slippage of meaning does not mean that the establishment of a normative form of sound words is not a good thing. This is, after all, precisely what Paul was commending to Timothy as he prepared for the end of his own ministry and the continuation of the church under postapostolic leadership. Conspicuously, Paul does not simply say to Timothy, \"Memorize the Old Testament or the Gospels or my Letters\" any more than he ever defines preaching as the reading of the same. The form of sound words is something more. Anyone who claims to take the Bible seriously must take the words of Paul to Timothy on this matter seriously. To claim to have no creed but the Bible, then, is problematic: the Bible itself seems to demand that we have forms of sound words, and that is what creeds are.\n\nSignificantly, this statement by Paul comes just a few verses after a clear and concise statement of a major part of that gospel content which he wishes Timothy to maintain and propagate:\n\n[God] who saved us and called us to a holy calling, not because of our works but because of his own purpose and grace, which he gave us in Christ Jesus before the ages began, and which now has been manifested through the appearing of our Savior Christ Jesus, who abolished death and brought life and immortality to light through the gospel. (2 Tim. 1:9\u201310)\n\nIn short, Paul tells Timothy to hold fast to a form of sounds words just moments after providing him with precisely such a form in this basic statement of Christian theology. Here we have God's grace, a basic christology (pre-existent and historical), and an outline of soteriology, all then identified with the good news of the gospel. This is one sentence, but it covers a large amount of theological ground. We might also note in passing that, given the nature of God, of humanity made in his image, and the need for verbal interpretation of the significance of God's saving actions in history, the verbal emphasis of Paul here is hardly surprising. Words, especially words that make a claim to truth, may be very countercultural today, but they are the building blocks of Paul's theology.\n\nThis is not the only instance in the New Testament of statements that seem to have a creedal sensibility. Many scholars regard Philippians 2:5\u201310 as a quotation from an earlier hymn, which Paul borrows to make his point. Regardless of whether that is the case, the unit stands as precisely a form of sound words that serves to offer a clear doctrinal summary of a key aspect of Paul's christology. First Timothy 3:16 also seems to offer a neat summary of Christian teaching. Even more explicit is 1 Timothy 1:15: \"This statement is trustworthy and deserving of full acceptance, that Christ Jesus came into the world to save sinners.\" Here it seems very clear that Paul is using previously established phraseology, a form of sound words, to capture in a nutshell the gospel.4\n\nOne further aspect of the commission to Timothy to hold fast to the form of sound words is worth noting, particularly in light of the weight given in chapter 1 to the cultural suspicion of history as a source of wisdom. For Paul, by way of contrast, there is an assumption that the correct teaching can be passed on from generation to generation, and that what has happened in the past can be communicated by a form of sound words down through the ages. As he writes to Timothy, this is surely what lies in the background: Paul is at the very end of his apostolic career. Indeed, the time of the apostles is coming to a close, and he needs to set in place structures to maintain true teaching in the postapostolic world. A form of sound words transmitted by eldership is his way of ensuring good management of the household of God. Continuity of teaching is crucial.\n\nThis is why elsewhere in his letters he will talk in terms of tradition, that which he is handing on to those who follow. Thus, in 2 Thessalonians 2:15, Paul says, \"So then, brothers, stand firm and hold to the traditions that you were taught by us, either by our spoken word or by our letter.\" Again the verbal emphasis is clear: these traditions were taught by words spoken or written; and they are to be the norm of life and teaching in the church in Thessalonica. Similar statements can be found in 1 Corinthians 11:2 and 2 Thessalonians 3:6, where Paul makes conformity to the tradition of his teaching a condition for fellowship.\n\nThe content of the gospel is thus to be handed on from generation to generation. In today's society, that is in some senses a strange notion. Traditions, say, of computer programming are not passed on. If they were, I would not be typing on my notebook but would be sitting in a room full of machines with spinning spools of tape. There are continuities in technology but they are often less substantial than the dramatic discontinuities that scientific and technological breakthroughs bring in their wake. Not so with Paul's gospel. This is truly traditional: it has a stable content and it is passed on from generation to generation. Indeed, for Paul, the fact that something was not taught in the past and not passed on as a tradition would presumably have dramatically increased the chances that it was false.\n\nThis notion of tradition, of the need to hand on the gospel, is deeply embedded in the nature of the gospel itself. The historical particularity of the history of Israel and of Jesus Christ means that, if the gospel, the meaning and significance of these things, is not passed on from generation to generation, then it remains in a sense trapped in the past. God's saving actions require interpretation and proclamation in order for later generations to have access by faith to them. This tradition is to be regulated by Scripture as the sole authoritative source of knowledge of God's actions; but it is not formally identical with Scripture. It uses forms of sound words, sermons, hymns, and prayers, among other things, in order to pass the message from one generation to another.\n\nFinally, in discussing Paul's teaching on the offices of the church and the true tradition, or teaching, of the same, we should note that failure in doctrine, as in life, brings forth strong words and action from the apostle. In 1 Corinthians 5, Paul instructs the Corinthian congregation to expel the immoral brother and to \"hand him over to Satan.\" What he is describing is deliberate and complete exclusion from the assembly of the saints on the grounds of the man's sexual wickedness. He uses similar language in 1 Timothy 1 regarding the two figures he names as Hymenaeus and Alexander. He has handed them over to Satan, he says, so that they might learn not to blaspheme. He does not tell us precisely what the content of their sin is, but blasphemy is a term used for the trivial abuse of God's holy name, a casually profane attitude to a holy thing. This seems to connect these two figures to the trivialized and false teaching about the law which Paul mentions earlier in the chapter. In short, false teaching leads Paul to dramatic and formal disciplinary action against the ones who are guilty of such.\n\nIf Paul himself takes this action in 1 Timothy 1, he urges the local church to take it in 1 Corinthians 5, indicating that perhaps the man there is simply a member of the church, while Hymenaeus and Alexander are office-bearers. Whatever the case, formal action is clearly to be taken when unbiblical morality or doctrine is being promoted by unbiblical life or teaching; and it is in this context that I would read Romans 16:17. This is more than just advice of an informal kind that advises crossing the street when one sees these divisive people while out for a stroll. It is a positive instruction to take steps to make sure that these divisive people are kept out of spheres of influence that would enable them to divide the church. The gospel, as expressed in a form of sound words and handed down from generation to generation, is connected to specific structures of pastoral oversight in the church. This oversight is responsible both for the positive transmission, regulation, and promotion of true teaching and for firm disciplinary action against those whose teaching does not conform to, and thus subverts, the apostolic doctrine.\n\nConclusion\n\nIn chapter 1 I made the point that creeds and confessions are predicated upon the truth of a number of assumptions: the past is important, and has things of positive relevance to teach us; language must be an appropriate vehicle for the stable transmission of truth across time and geographical space; and there must be a body or an institution that can authoritatively compose and enforce creeds and confessions. As I have argued above, the Bible clearly teaches all three elements. In addition, it also seems clear that Paul himself understands that the passing of the Christian faith involves taking history seriously, understanding God as a God who speaks, having (and holding to) forms of sound words, and not simply reading the Bible in the Hebrew or the Greek. Theological synthesis is part of the church's task, and this is facilitated by the development of ways of speaking which are appropriate to the content expressed and the actions being performed.\n\nThus, there is surely great irony in the claim made by some that they have no creed but the Bible. One can, of course, appreciate the genuine desire which such a claim embodies, to underline the Bible's status as the uniquely authoritative source of divine revelation. We should all be worried about ever allowing a document that is not divinely inspired to carry ultimate authority in the church.\n\nIn fact, of course, the legitimate concern underlying the claim to have no creed but the Bible is entirely consonant with the Protestant understanding of Scripture as expressed in those very traditions which place a high premium on the use of creeds and confessions. Indeed, classical orthodox Protestantism coined the phrase norming norm to reflect this unique position which Scripture, and Scripture alone, occupies. As the norming norm, the Bible is that by which all other theological statements must be judged as to their truthfulness of content and adequacy of formulation.\n\nThe Westminster Confession expresses this point very neatly in chapter 1.10:\n\nThe supreme judge by which all controversies of religion are to be determined, and all decrees of councils, opinions of ancient writers, doctrines of men, and private spirits, are to be examined, and in whose sentence we are to rest, can be no other but the Holy Spirit speaking in the Scripture.\n\nHere we have a confession, which is considered normative in confessional Presbyterian circles, stating very clearly that its own statements and teachings are subordinate to Scripture and to be normed by the same. To use the technical terminology, if Scripture is the norming norm, then creeds and confessions, when adopted by churches as statements of their own faith, are the normed norms.\n\n1See the discussion of the intertestamental tradition connecting Mark 1:10 to Isaiah 64:1 in James R. Edwards, The Gospel According to Mark (Leicester, UK: InterVarsity Press, 2002), 35.\n\n2I am aware that some Christians dislike the whole notion of membership and regard it as unbiblical as it is not mentioned in the Bible. I believe the word \"membership\" reflects the biblical concept of belonging to the church community. Whether one uses the word \"membership\" or not is a matter of dogmatic indifference. Clearly Paul presupposes in his letters that there is a Christian community, that people belong to it, and that some people who belong to it act and think in a way that means they should be expelled from the community and not be allowed back until they repent. That seems to me to be all that the modern notion of \"membership\" is trying to express.\n\n3I should note here that I regard Paul's use of \"overseer\" and \"elder\" as referring to the same office, given the virtual identity of qualifications and functions he lists for each.\n\n4Other creed-like statements in the New Testament include Romans 1:3\u20134; 1 Corinthians 8:6; and 1 Peter 3:18\u201321.\n\n#\n\n## The Early Church\n\nOne of the frequent objections to creeds is that the Bible does not contain any. As we have seen, the position is more complicated than that: the Bible does actually contain material which has a creedal tone to it; and the Bible also teaches principles (such as \"holding fast to a form of sound words\") that can best be honored through the use of something like a creed or confession.\n\nWhen we move from the period of biblical history into the post\u00adapostolic age, it should come as no surprise that creed-like formulations soon start to appear in Christian literature. Of course, we need to understand that creeds have a twofold aspect: first, there is the content, both conceptual and linguistic, which is designed to serve the transmission of the faith. We might call this the doctrinal concern. Second, there is the normative nature of such creeds as being binding upon the church. This we might call the ecclesiological concern. Historically, it is clear that the two things are closely related but that the former precedes the latter in terms of chronology. The struggle to define orthodoxy is afoot long before the idea that there should be universally binding creeds becomes a significant factor in the life of the church.\n\nThe Rule of Faith\n\nThe immediate postapostolic period presented the Christian church with a series of obvious challenges. First, the death of the apostles meant a change in leadership and, indeed, in normative leadership structure. The impending challenge of this was presumably one of the driving forces behind Paul's concern in the Pastoral Letters to establish clear criteria for eldership. Conscious that his own time was running short, he wanted to make sure that he was leaving the various congregations of the church in safe hands. Thus, he instructed Timothy and Titus on the kind of men who should hold office and the type of teaching they were to maintain.\n\nThis concern for church office continues into the immediate postapostolic period. It is particularly evident in the writings of Ignatius, Bishop of Antioch and in the Didache. Ignatius wrote a series of letters to various churches as he made his way to Rome to be martyred, and in these he continually stresses the importance of having an ordained church leader present at any formal gathering of the church. The Didache, a text of unknown provenance which may date from as early as the first century, also speaks of leadership in the church and even offers interesting and novel ways of discerning a false teacher. It is arguable that Ignatius's writings are more consistent with later Episcopal or Presbyterian polity while the Didache perhaps hints at a more congregational approach; but what is obvious from both is that church government issues are significant topics of discussion in the postapostolic church of the early second century.\n\nThe second challenge to the postapostolic church was that of doctrinal content or, to use (for Protestants) a more loaded phrase, the tradition of apostolic teaching, that is, the teaching handed on in trust from the apostles to the next generation. Even within the New Testament, it is clear that there were challenges to such from within the visible company of the church. This again is a factor in Paul's later Pastoral Letters, as he begins 1 Timothy by urging Timothy to remain at Ephesus in order to combat the false teachers and false teaching that has apparently crept in to the congregation. Challenges to the apostolic tradition of teaching only became more severe in the years following the death of the apostles and indeed, arguably continue through to today. We are, after all, living in \"the last days,\" as were those to whom Paul addressed his letters, and such days are marked by conflict with false teachers within the church.\n\nParticularly significant in this regard was Docetism, the idea that Christ only appeared to possess real human flesh. The origins of the idea are unclear but probably connect to traditional notions of the inherent inferiority and even evil of matter, and thus of this as something inappropriate to be joined with deity. Various nebulous groups seem to have held this position; scholarship has bracketed them together under the broad category of Gnosticism because of the emphasis on arcane knowledge which often accompanied the denial of Christ's real flesh.1\n\nAn influential character among the Docetists, though one who lacked the Gnostic emphasis on mystical knowledge, was Marcion, apparently a native of Pontus on the Black Sea who flourished in the middle decades of the second century. His theology was not simply marked by its Docetism, however, but also by its fundamental revision of the notion of the biblical canon. Whilst the formal recognition of the New Testament canon was far from clear or complete in the second century, Marcion was radical even by the standards of his day in the manner in which he rejected the God of the Old Testament as being antithetical to Jesus Christ. Marcion presented the church not only with a challenge to the tradition of apostolic teaching as carried on in the church but he also proposed criteria for textual canonicity that demanded a response. For Marcion, only ten letters of Paul (he appears not to have known or to have implicitly rejected the Pastorals) and a bowdlerization of Luke's Gospel were acceptable as teaching the truth about God's grace.\n\nThus, if the death of the apostles posed one challenge to the church, the struggle over the tradition of apostolic teaching and the extent and content of the biblical canon were also significant. It is in this context that we find the development of what became known as the Rule (or Canon) of Faith, a summary of the essentials of Christianity that occurs in various verbal forms in the writings of numerous early church fathers. That the Rule appears to have been stable in content but different in verbal form indicates that this is not a formal creed with which we are dealing; but the stability of content nonetheless implies that it was attempting to state normative, commonly agreed-upon concepts. It also clearly reflects the doctrinal issues and strains of the time.\n\nWe see early adumbrations of the Rule in the letter of Polycarp to the Philippians, chapter 2, and the letters of Ignatius. For example, in chapter 9 of his Letter to the Trallians, Ignatius says this:\n\nStop your ears, therefore, when any one speaks to you at variance with Jesus Christ, who was descended from David, and was also of Mary; who was truly born, and did eat and drink. He was truly persecuted under Pontius Pilate; He was truly crucified, and [truly] died, in the sight of beings in heaven, and on earth, and under the earth. He was also truly raised from the dead, His Father quickening Him, even as after the same manner His Father will so raise up us who believe in Him by Christ Jesus, apart from whom we do not possess the true life.2\n\nIt is clear from this that Ignatius is providing a summary of his understanding of Christ, which is shaped by the pressure he feels from certain Docetists who are threatening the church. Hence the repeated emphasis upon the reality of Christ's historical flesh and life. Of course, while Ignatius no doubt regarded what he was saying as universally true, he was not proposing this precise form of words as somehow binding or as representing \"best practice\" for all churches everywhere. This is similar to the Rule of Faith but, when writers do mention the Rule, the emphasis upon universality and catholicity is more explicit. Thus with Tertullian, the second-century North African theologian, in his On the Prescription of Heretics:\n\nNow, with regard to this rule of faith\u2014that we may from this point acknowledge what it is which we defend\u2014it is, you must know, that which prescribes the belief that there is one only God, and that He is none other than the Creator of the world, who produced all things out of nothing through His own Word, first of all sent forth; that this Word is called His Son, and, under the name of God, was seen \"in diverse manners\" by the patriarchs, heard at all times in the prophets, at last brought down by the Spirit and Power of the Father into the Virgin Mary, was made flesh in her womb, and, being born of her, went forth as Jesus Christ; thenceforth He preached the new law and the new promise of the kingdom of heaven, worked miracles; having been crucified, He rose again the third day; then having ascended into the heavens, He sat at the right hand of the Father; sent instead of Himself the Power of the Holy Ghost to lead such as believe; will come with glory to take the saints to the enjoyment of everlasting life and of the heavenly promises, and to condemn the wicked to everlasting fire, after the resurrection of both these classes shall have happened, together with the restoration of their flesh. This rule, as it will be proved, was taught by Christ, and raises amongst ourselves no other questions than those which heresies introduce, and which make men heretics.3\n\nThe basics of the faith are here: creation, Christ, the Holy Spirit, the coming of the kingdom, and judgment. In setting forth these points, Tertullian assumes that he is simply outlining the faith that has been handed down from the apostles.\n\nIn similar vein, we find the Rule also articulated in the writings of another second-century theologian, Irenaeus:\n\nThe Church, though dispersed through out the whole world, even to the ends of the earth, has received from the apostles and their disciples this faith: [She believes] in one God, the Father Almighty, Maker of heaven, and earth, and the sea, and all things that are in them; and in one Christ Jesus, the Son of God, who became incarnate for our salvation; and in the Holy Spirit, who proclaimed through the prophets the dispensations of God, and the advents, and the birth from a virgin, and the passion, and the resurrection from the dead, and the ascension into heaven in the flesh of the beloved Christ Jesus, our Lord, and His [future] manifestation from heaven in the glory of the Father \"to gather all things in one,\" and to raise up anew all flesh of the whole human race, in order that to Christ Jesus, our Lord, and God, and Saviour, and King, according to the will of the invisible Father, \"every knee should bow, of things in heaven, and things in earth, and things under the earth, and that every tongue should confess\" to Him, and that He should execute just judgment towards all; that He may send \"spiritual wickednesses,\" and the angels who transgressed and became apostates, together with the ungodly, and unrighteous, and wicked, and profane among men, into everlasting fire; but may, in the exercise of His grace, confer immortality on the righteous, and holy, and those who have kept His commandments, and have persevered in His love, some from the beginning [of their Christian course], and others from [the date of] their repentance, and may surround them with everlasting glory.4\n\nIrenaeus's Rule exhibits the same basic conceptual content as that of Tertullian but, as noted above, it differs in terms of the phrases and forms of expression used. The Rule was thus not a strict \"form of sound words\" so much as a statement of sound concepts. Further, both men use it as a means of establishing that what they are teaching is the teaching that was passed on from the apostles and has enjoyed universal acceptance within the church as orthodoxy and thus as basis for assessing contemporary teaching. There is therefore a functional similarity to later creeds on this point even if the linguistic and ecclesiastical nature of later creeds is that much more elaborate and fixed. What is undeniable is that Ignatius, Tertullian, and Irenaeus all indicate that the church, from the earliest postapostolic times, continued and developed the Pauline notion of providing clear doctrinal summaries as means of summarizing the faith.\n\nThis also points to the major context for the explicit formation of creed-like material in the early church: that of baptism. The very initiation of new Christians into the faith involved both doctrinal content and forms of sound words, not simply a recitation of select Bible verses. These baptismal statements started in their earliest forms as interrogatory documents that required the candidate for baptism to express his or her faith at a personal level. A classic version of this type of creed was the fourth-century Old Roman Creed, the textual ancestor of the later Apostles' Creed, but baptismal formulae were in use long before that. For example, Hippolytus of Rome (170\u2013236) gives this account of a baptism in his Apostolic Tradition:\n\nWhen each of them to be baptized has gone down into the water, the one baptizing shall lay hands on each of them, asking, \"Do you believe in God the Father Almighty?\" And the one being baptized shall answer, \"I believe.\" He shall then baptize each of them once, laying his hand upon each of their heads. Then he shall ask, \"Do you believe in Jesus Christ, the Son of God, who was born of the Holy Spirit and the Virgin Mary, who was crucified under Pontius Pilate, and died, and rose on the third day living from the dead, and ascended into heaven, and sat down at the right hand of the Father, the one coming to judge the living and the dead?\" When each has answered, \"I believe,\" he shall baptize a second time. Then he shall ask, \"Do you believe in the Holy Spirit and the Holy Church and the resurrection of the flesh?\" Then each being baptized shall answer, \"I believe.\" And thus let him baptize the third time.5\n\nIt is clear from this that basic creedal structures were in place, at least locally, for baptismal candidates by 200 AD. The shape is interesting: the structure is basically Trinitarian, the content is the historical matter of the gospels, and the terminus is the eschaton. The skeletal framework of the whole of the Christian story is there.\n\nBaptism also connects to the Rule of Faith. Writing in his treatise On the Shows, Tertullian declares in chapter 4 that \"when entering the water, we make profession of the Christian faith in the words of its rule; we bear public testimony that we have renounced the devil, his pomp, and his angels.\"6 The Rule of Faith thus becomes a brief form of sound words by which the new believer makes public declaration of loyalty to Christ and renounces the world.\n\nOne clear and obvious implication of this is that there was some form of Christian education in the background, some form of catechizing or, to use the modern terminology, membership class, which was designed to inform the new convert. Presumably, this education was also shaped by the kind of confession that was given at baptism. Thus, these early creedal formulae had pedagogical importance in the practical, educational ministry of the church.\n\nIn sum, the earliest evidence indicates that Christianity deployed forms of sound words to define itself over against challenges to the faith from within the community of those professing in some way to be the true church (e.g., the Docetics) and also over against the non-Christian world (i.e., baptismal confessions). A doctrinal core and a publicly agreed form for the summary of biblical teaching was basic to postapostolic Christianity and clearly consonant with the concerns of Paul as he writes to the church in the Pastorals, worried about church government and life after the apostles have all died.\n\nThe Apostles' Creed\n\nPrior to the fourth century, creeds seem to have been primarily local documents. There does not seem to have been any great understanding on the part of the church of the need to produce universally binding statements. From the early fourth century onward, however, there was a growing consciousness of the need for the church to have agreed upon and binding creeds. The most obvious manifestations of this consciousness are the Seven Ecumenical Councils, a number of which will be listed below; but, historically, another creed, the so-called Apostles' Creed, has also enjoyed wide acceptance in the Christian church and continues to this day to form part of the regular liturgical practices of many communions. Though well known, it is worth reminding ourselves of the text:\n\nI believe in God the Father Almighty, Maker of heaven and earth. And in Jesus Christ his only Son our Lord; who was conceived by the Holy Ghost, born of the Virgin Mary, suffered under Pontius Pilate, was crucified, dead, and buried; he descended into hell; the third day he rose again from the dead; he ascended into heaven, and sitteth on the right hand of God the Father Almighty; from thence he shall come to judge the quick and the dead. I believe in the Holy Ghost; the holy catholic church; the communion of saints; the forgiveness of sins; the resurrection of the body; and the life everlasting. Amen.\n\nThe similarities with the Rule of Faith are obvious, but there is no scholarly consensus on the literary origins of the creed. The first reference to it as the Apostles' Creed occurs in a letter from Ambrose of Milan to Rome in 389, a reference that suggests it was clearly of some vintage by that point in time.7 It probably has its origins in the third century. It was also to continue undergoing development until it was finally formalized by the churches of the West under Charlemagne somewhere around 800 AD.\n\nThe Creed is important because it represents a linguistically formalized version of the Rule of Faith. Obviously, it does not contain the key technical language we find in the Nicene Creed, which was designed specifically to protect the full deity of the Son and the Spirit and their equality with the Father; but it does provide the basic bones of the Christian story, from the uniqueness of God through creation and redemption and on to consummation.\n\nSo useful has the creed been as a pedagogical tool that it features in many church liturgies. It has also been embodied directly into the catechetical tradition of the church. Medieval catechisms typically used the Apostles' Creed as the framework for teaching doctrine, a tradition that continued into the Protestant tradition at the Reformation. Both Lutheran catechisms and the Heidelberg Catechism for example, include it. Having stated that Christians are saved by faith (Q. 20) and defined faith (Q. 21), Heidelberg Catechism question 22 reads as follows:\n\nQ: What is then necessary for a Christian to believe?\n\nA: All things promised us in the gospel, which the articles of our catholic undoubted faith briefly teach us.\n\nThe answer to the next question is the text of the Apostles' Creed, which subsequent sections go on to expound in some detail.\n\nIn addition to the widespread liturgical and catechetical use of the creed, it has also been the basis for many popular introductions to the Christian faith. Even in the last century, a variety of theologians, from liberal to conservative, have used it as the framework for basic books on Christianity: Wolfhart Pannenberg, Karl Barth, J. I. Packer, and Michael S. Horton are but four of such.\n\nGiven the near universal presence of the Apostles' Creed across the Christian spectrum, it is ironic that it also contains one of the most controversial and disputed statements in creedal and confessional history: the clause which states that \"Christ descended into hell.\" This seems to be a statement with minimal biblical foundation and unfortunate soteriological implications, as if Christ's death on the cross was somehow an insufficient act in itself to fulfill the mandate of the Suffering Servant.\n\nIn fact, as is so often the case in the history of theology, the creed's offense at this point is based more on a surface reading of the words from a later context than upon their original intent. Thus, a careful exploration of the words reveals that the creed is not claiming anything particularly objectionable at this point. As Reformed pastor Daniel R. Hyde has recently shown, the words simply express the Old Testament prophecies (and thus biblical understanding) of the death and resurrection of Christ.8 While I will touch on creedal and confessional revision in the appendix, this is an important reminder that we should not abandon a clause in a creed simply because we do not understand it at first reading.\n\nThe Seven Ecumenical Councils\n\nWhile the early church was a most fertile period for church councils and the production of creeds, doctrinal definitions, and dogmatic anathemas, seven councils in particular have historically been accorded singular importance. This is partly because of the foundational nature of a number of them in setting forth basic definitions of God and Christ, and partly because they occurred at a time when the church was arguably institutionally united and thus could make claims to real catholicity and ecumenicity at least in terms of its membership and legislative scope. These councils are:\n\nThe First Council of Nicaea, 325.\n\nThe First Council of Constantinople, 381.\n\nThe First Council of Ephesus, 431.\n\nThe Council of Chalcedon, 451.\n\nThe Second Council of Constantinople, 553.\n\nThe Third Council of Constantinople, 680\u2013681.\n\nThe Second Council of Nicaea, 787.\n\nThe Eastern Orthodox churches refuse to recognize the ecumenical nature of any subsequent council, while the Roman Catholic Church acknowledges fourteen more, the last being the Second Vatican Council (1962\u20131965). Protestantism, with a different understanding of ecclesiology from either of these communions, has only limited use for a number of the original seven. In fact, it only really engages at a creedal level with the work of the first four councils. These councils produced one major creed, the Nicene, and one important doctrinal formula, the Chalcedonian Definition. Both are important for Protestants, and thus it is very helpful to have some understanding of them in order to appreciate better our own tradition.\n\nThe First Council of Nicaea\n\nEmperor Constantine called the First Council of Nicaea (hereafter Nicaea I) to resolve a deepening conflict in the church over the identity of the second person of the Godhead. The church had been wrestling since the death of the apostles with how to articulate the relationship between Father and Son, and these debates came to a head in the fourth century. The crisis is often referred to as the Arian Controversy, after a Libyan presbyter, Arius, whose writings on the subject had brought him into conflict with Alexander, the bishop of Alexandria. Various forces were at play: the need for the Emperor to maintain a unified church in order to keep the empire stable; the struggle for power in the church between presbyters and bishops; and the debate over the nature of the Logos. The latter provided the theological focal point for the political and ecclesiastical struggle.\n\nIt is not the purpose of this book to outline in detail the history of individual councils. Our interest here is in the theological and ecclesiastical significance of their contributions. As far as Nicaea I is concerned, its contribution was conceptual\/terminological in that it gave the church an important word that shaped the way in which later discussion of the doctrine of God was conducted. This word was \"of the same substance\" (Gk. homoousion) of the Father. The full text of the key clause reads as follows:\n\n[We believe in] one Lord Jesus Christ, the Son of God, the only-begotten of his Father, of the substance of the Father, God of God, Light of Light, very God of very God, begotten, not made, being of one substance with the Father.\n\nWe need to understand a number of points if the significance of Nicaea I is to be fully appreciated. First, the issue of the Son's relationship to the Father may appear to be somewhat abstruse but is actually vital to the Christian faith. How one resolves that issue will decisively affect how one understands creation, salvation, and a host of other theological and practical issues. Is Christ the first act of God's creation? Does salvation involve God condescending to reach down to his creatures or a creature reaching up to God? Can Christ be worshiped in the same way as the Father? Does Christ truly reveal the Father to us? One's response to each of these questions will be determined by how one answers the question about the Logos's relationship to the Father.\n\nIn offering an account of this relationship in terms of \"substance,\" Nicaea I ultimately came to set the terms of subsequent debate for this theological discussion and, given the connection between the resolution of the matter at the first Council of Constantinople in 381 (hereafter Constantinople I) and the later discussions of christology proper, also set the terms of debate for understanding the person of the incarnate Lord Jesus. Anyone today who rejects the usefulness of creeds and yet articulates an understanding of God that uses terms such as \"substance\" is clearly indebted to traditions of theological discussion that are directly rooted in the creedal debates and definitions of the early church. As noted in earlier chapters, we are all subject to the traditions in which we have grown up, both individually and corporately; only those who understand that that is the case, and who have some understanding of how their traditions were formed, can to any degree stand in critical relationship to such traditions.\n\nSecond, recent scholarship has demonstrated that the key terminological innovations of Nicaea I did not come to play a significant part in subsequent debate until the 350s. This is related to another point: that the status of Nicaea as an ecumenical creed with normative, binding status on the church was not something which seems to have been intended in any elaborate and self-conscious manner at the time when it met. Nicaea I came to have such significance only in retrospect as the theological and ecclesiastical emphasis came to be placed upon the term homoousion.\n\nIndeed, while theologians influenced by the historiography of later fourth-century debates have tended to see the accent in the creed of 325 as on the homoousion, recent scholarship has made a very compelling case for regarding this term as a clarification of the statement concerning the Son coming from the Father. The preoccupation with homoousion that we find in the later Athanasius and beyond is thus in part a rewriting or at least a new appropriation of the language and intentions of the council.9\n\nFurther, that the church does not seem to have initially accorded the creed or its language special status and that this started to change in the 350s indicates that it was becoming apparent at this point that normative doctrinal concepts required normative doctrinal language, and that this in turn needed to be connected to the church as an institution to have any meaning. Thus, Nicaea I was to become extremely important because it provoked the church to reflect on the doctrinal content of the gospel, on the need to fashion an extrabiblical vocabulary for expressing this, and on the need for this vocabulary to be established as normative across the church. The fourth century was giving birth to creedalism, and it was the struggles surrounding the identity of the Logos as expressed in Nicaea I that provided the context and the dynamic for this development.\n\nThe Council of Constantinople\n\nThe complex struggles\u2014linguistic, theological, ecclesiastical and political\u2014in the half century after Nicaea I culminated in the great Council of Constantinople. While there is debate on whether the creed we commonly now know as the Nicene Creed was actually formulated at the council, tradition ascribes it to such. Certainly it reflects linguistic developments between 325 and 381 in debates over the nature of God, specifically in terms of determining that God can be legitimately described as existing in three hypostases, and that the Holy Spirit is fully God. The creed, or at least a slightly modified form of the original, is probably familiar to many Protestants as the Nicene Creed. The title, of course, is something of a misnomer, given that it was only the indirect product of the debates surrounding Nicaea I. More correctly, we should call it the Niceno-Constantinopolitan Creed. The original text reads as follows:\n\nI believe in one God, Father Almighty, Creator of heaven and earth, and of all things visible and invisible;\n\nAnd in one Lord Jesus Christ, the only-begotten Son of God, begotten of the Father before all ages; Light of Light, true God of true God, begotten, not created, of one essence with the Father through Whom all things were made. Who for us men and for our salvation came down from heaven and was incarnate of the Holy Spirit and the Virgin Mary and became man. He was crucified for us under Pontius Pilate, and suffered and was buried; And He rose on the third day, according to the Scriptures. He ascended into heaven and is seated at the right hand of the Father; And He will come again with glory to judge the living and the dead. His kingdom shall have no end.\n\nAnd in the Holy Spirit, the Lord, the Creator of life, Who proceeds from the Father, Who together with the Father and the Son is worshiped and glorified, Who spoke through the prophets.\n\nIn one, holy, catholic, and apostolic church.\n\nI confess one baptism for the forgiveness of sins.\n\nI look for the resurrection of the dead,   \nand the life of the age to come.\n\nThis creed represents a significant development over the earlier formulations of Nicaea I. First, ecclesiastically this creed enjoyed normative status as definitive of catholic orthodoxy. Indeed, that remains the case to the present day and is the reason why so many churches include it, or reference it, in their doctrinal standards. Of course, the Western addition of the dual procession of the Spirit (\"Who proceeds from the Father and Son\") at the Third Council of Toledo in 589 has been a source of East\/West contention ever since, but, with this one exception, there is no debate about the rest of the creed's teachings.\n\nSecond, the creed contains considerable elaboration of the deity of the Spirit and thus represents a more properly Trinitarian description of who God is. This reflects the way in which debates developed in the fourth century. In the 360s it became clear that the issue of Christ's deity was soon going to be definitively resolved, and attention moved to the identity of the Spirit, as we see, for example, in Athanasius's Letters to Serapion. It is important to remember that this development was in part driven by doxological\/liturgical concerns: why were Father, Son, and Holy Spirit all mentioned in the baptismal formula? In other words, this was no abstract and practically irrelevant discussion. It was connected directly to the most basic liturgical actions of the church.\n\nWith attention turning to the Spirit, we find the rise of the so-called Pneumatomachi, those who accepted Christ's deity but opposed ascribing the same to the Spirit. This called forth vigorous polemics on this issue, and the end result was the section on the deity of the Spirit in the creed of 381. Ironically, some did not think that even this statement went far enough: Gregory of Nazianzus, one of the major intellectual forces of the post-Athanasius church world, left the council in a fit of pique, angered that the fathers gathered there had placed insufficient emphasis on the Spirit's deity.\n\nConstantinople is a great example of how the public criteria for orthodoxy can change over time. In the third century, the notion that the Logos was subordinate to the Father was commonplace and not exceptionable; but as theologians wrestled with the implications of this for God, creation, and salvation, it became clear that such was unacceptable as an accurate description of God. A Christ who was less divine than the Father was a Christ who could not save. In order to ensure that the public testimony of the church witnessed to this vital doctrinal insight, the church needed a creedal statement that made precisely this point.\n\nThis latter point is very important. One constant question I face in class as a church historian is, \"If doctrine develops, does this mean that what unites us to Christ changes over time too?\" This is an excellent question and, indeed, a rather obvious one when one is investigating the history of doctrine. Two things need to be borne in mind here.\n\nFirst, Scripture gives no hint that that which saves changes: it is always trust in Christ that unites one to Christ. Thus, someone who was a believer in the first century is saved in the same way as someone who believes in the twenty-first.\n\nSecond, as noted above, the public criteria for what constitutes a credible profession of faith do change over time, as do the standards for office-bearing.10 As the church reflected upon the identity of Christ and upon Scripture over time, the limitations and inadequacies of certain formulations became more apparent. We noted above that in the third century, the view that Christ was subordinate to the Father in terms of his being was considered acceptable because the implications of that position had not been fully worked out. Once this had been done, and the unacceptable, unbiblical implications of such a position had become clear, the church put in place statements that ruled such views out of bounds. It is not that people who believed in Christ's subordination in the second century could not therefore have been saved\u2014we are all, after all, saved despite some of the things which we believe. It is rather that the church had come to an understanding that to protect and to articulate the gospel, accurate concepts and appropriate language were necessary, and some of these had to change over time as the inadequacy and abuse of earlier forms became clear.\n\nThe First Council of Ephesus\n\nOne of the key things to understand about the creedal and confessional development of Christianity is that creeds do not simply offer new doctrinal models and establish new vocabulary with which to solve particular issues; they also generate new problems and questions and set the terms for future debates. We see this in the development from Nicaea I to Constantinople: the debate comes to be shaped by the term homoousion, which is not biblical but which is gradually fine-tuned in such a way as to capture the biblical concept of the relationship between the Father and the Son. Once the term is officially adopted, it then plays a constructive part in setting the terms of future debates.\n\nThis provides the background to the third ecumenical council, the First Council of Ephesus. Once it has been established that the Logos is of the same substance as the Father and is fully God, the question of how this relates to the humanity of Jesus Christ comes to be proposed in a specific manner: if humanity too has its own substance, how do these two substances, the divine and the human, relate to each other in Christ? And, more specifically, how do the two substances relate to each other in a way that does not create either two persons (albeit occupying one geographical space) or some peculiar blend or fusion of the two substances that leads to the formation of a third substance, which is neither divine nor human? It should be clear that only when the decision has been made to articulate the divinity of the Logos in terms of substance do the problems relating to substance become an issue for church discussion. This is one of the reasons why theology cannot simply be done by reading the Bible: the fine-tuning of concepts and vocabulary is a cumulative and traditionary exercise. This does not mean the results are not biblical, in the sense of being consistent with what the Bible teaches or useful as explanatory devices for understanding the Bible; but it does mean that one will search in vain in the Bible for the terms \"Trinity,\" \"substance,\" or \"hypostasis\" for example\u2014or, for that matter \"conversion experience,\" \"personal relationship with Jesus,\" \"missional,\" \"relational,\" and \"no creed but the Bible!\"\n\nGiven the resolution of the Trinitarian question provided by Constantinople I, christological questions could now be addressed with the new conceptual vocabulary of substance. It also intensified debates that had surfaced in the years leading up to Constantinople I. Perhaps the most important example of these was the controversy surrounding one of Athanasius's closest allies, a man called Apollinaris, the bishop of Laodicea. Apollinaris had a distinctive view of the person of Christ which was actually condemned at Constantinople I. Put simply, Apollinaris was rightly concerned to defend the full consubstantiality of the Logos with the Father. He did this so radically that he argued that, in the incarnation, the Logos replaced the human soul of Christ. It was a clever resolution to a tricky problem thrown up by Nicene theology: if the Logos is a divine person and unites with human nature, how does one avoid having two persons simply occupying the same space? And if you have two persons in the incarnation, then there really is no incarnation: the divine has not truly united with the human and there can be no salvation.\n\nThe problem with Apollinaris's solution, of course, is that on his account Jesus Christ is not really fully human: after all, humans have souls; and if Christ lacks one, then he is not fully human. More acutely, to cite the view of one of Apollinaris's contemporaries, Gregory of Nazianzus, what is not assumed is not healed. If Christ did not assume a human soul, then that vital part of humanity was not saved. Thus, Constantinople I rejected Apollinaris's theology.\n\nIf Constantinople I ruled out of bounds the Apollinarian solution, the opposite problem\u2014that of radically separating the divine and human natures in such a way that Christ could hardly be regarded as one person at all\u2014was the next challenge. Again, it is important to see how the question was shaped by previous resolutions: language of substance and hypostasis solved the three-in-one problem but also meant that later discussion would address questions about the incarnation to the biblical text in a manner profoundly shaped by previous creedal agreements and the language they contained.\n\nThe First Council of Ephesus addressed this issue relative to the teaching of Nestorius, Bishop of Constantinople. Nestorius objected to the use of the traditional title of \"Theotokos\" as applied to the Virgin Mary. Literally, it meant \"God bearer.\" While it is a matter of some debate as to whether Nestorius ever held the heresy that bears his name, his rejection of Theotokos brought him under suspicion of denying the full union of the divine and the human in one person in Christ. Opponents, most notably Cyril, Patriarch of Alexandria, raised concerns about his teaching and after years of controversy, the council at Ephesus condemned Nestorius's teaching and vigorously asserted the unity of the person of Christ.\n\nThe importance of the Council of Ephesus is twofold in the history of creeds and confessions. First, it is the last council which the Coptic churches in the East acknowledge as authoritative. Known as monophysite (\"one nature\") churches, these reject the later teaching of the Council of Chalcedon that Christ has two natures in one person. Second, it effectively closed debate on one issue (the number of persons in Christ), establishing the boundary of orthodoxy on that point, which set the scene and the limits for future creedal discussion on Christ. To be considered orthodox, all subsequent theologians have had to respect that boundary.\n\nThe Council of Chalcedon\n\nThe last of the early councils which is of major relevance to modern Protestantism is the Council of Chalcedon. The immediate polemical background was the teaching of a man called Eutyches, a monk from Constantinople. Again, there is debate about whether he was a truly innovative heretic or an orthodox figure who simply had great difficulty in expressing himself. Whatever the truth of the matter, he was understood by others to be denying the true humanity of Christ by effectively having it absorbed by the divinity.\n\nThe events of the late 440s are politically and ecclesiastically complex, with move and countermove being made by various factions in church and empire. The culminating moment, however, was the Council of Chalcedon, where the following formula was approved as defining the relationship between substances and person in Jesus Christ:\n\nTherefore, following the holy fathers, we all with one accord teach men to acknowledge one and the same Son, our Lord Jesus Christ, at once complete in Godhead and complete in manhood, truly God and truly man, consisting also of a reasonable soul and body; of one substance with the Father as regards his Godhead, and at the same time of one substance with us as regards his manhood; like us in all respects, apart from sin; as regards his Godhead, begotten of the Father before the ages, but yet as regards his manhood begotten, for us men and for our salvation, of Mary the Virgin, the Godbearer; one and the same Christ, Son, Lord, Only-begotten, recognized in two natures, without confusion, without change, without division, without separation; the distinction of natures being in no way annulled by the union, but rather the characteristics of each nature being preserved and coming together to form one person and subsistence, not as parted or separated into two persons, but one and the same Son and Only-begotten God the Word, Lord Jesus Christ; even as the prophets from earliest times spoke of him, and our Lord Jesus Christ himself taught us, and the creed of the fathers has handed down to us.11\n\nThe Chalcedonian Formula effectively puts into place four boundaries for future christological discussion, boundaries which theologians must not transgress in order to remain orthodox: Christ must be fully God; Christ must be fully human: the two natures must not be so mixed together that either disappears into the other or that a third, hybrid nature is produced; and the two natures must not be separated so as to undermine the unity of the one person.\n\nSeveral points of interest are worth noting here. There is a strong element of negative theology in Chalcedon: it is effectively defining where one must not go with one's christology rather than setting forth a positive definition. That is no bad thing. There is a sense in which the incarnation of the second person of the Trinity in human nature is a deep, unfathomable mystery. One simply cannot give full expression to that mystery in words; but what one can do is map out the theological field in which orthodox discussion can take place by indicating clearly what must not be said about Christ if one is to do justice to the biblical teaching about the incarnation. Thus, Chalcedon puts in place boundaries, and any christological formulation that honors those boundaries is thus observing key aspects of biblical teaching necessary for holding to a Christ who can actually save.\n\nAnother point is that, as with earlier creedal formulations, Chalcedon generates its own fresh questions which subsequent theologians must address to the biblical text. For example, if Christ is one person and two natures, how many wills does he have? A person typically has one will, one source of motivation, action, etc. Thus, the obvious answer would seem to be that Christ too has one will. This leads to a problem, however: which will does Christ therefore lack\u2014the divine or the human? Or does he have a hybrid will made of the two? If he lacks one or the other, he cannot be said to be fully human or fully divine, depending which will is lacking; if he has a hybrid, he has neither one.\n\nThis interesting quandary led the church at the Third Council of Constantinople in 681 to decide that Christ has two wills, divine and human, but that they both work in perfect harmony, with no tension between them at all. Now, at first glance that is a very weird answer to the question. Nowhere in the Bible does it say that Christ has two wills. It is thus tempting to point to this as a bit of needle-headed theological hair-splitting on a matter of no importance and thus a piece of extrabiblical tradition imposed upon the faith.\n\nYet to draw such a conclusion would be to misunderstand how theology is formulated in and by the church. As noted above, each time one problem is solved, the terms of discussion are changed to take into account the new solution. This solution then generates new questions that the church needs to address to the biblical text and to answer in a manner consistent with that text. Therefore it is only when one understands the history of certain questions that one can understand why the answers given are the appropriate ones.\n\nThus, if somebody does decide it is nonsense to say that Christ has two wills, one should ask them, \"Well, which does he lack?\" As soon as the question is asked, the problem should become obvious. One could dismiss the question as ridiculous; but then one needs to go back and effectively rebuild one's christology (and one's Trinitarianism) from the ground up. Given the time, care, and effort of the church in fine-tuning these doctrines in the first place, it is highly unlikely that any other solution will prove less problematic. One can always try, but one then runs the risk of wasting time and ending up with a formula that either does not work as well or for as long or as universally as that which has already been tried and tested across the ages. Indeed, Christian orthodoxy is sometimes the sum of the least number of doctrinal difficulties, complications, and strange statements with which one is prepared to live.\n\nThis is an important point to bear in mind, both in terms of creedal formulation in particular and the development of Christian doctrinal formulation in general. Creeds solve one set of problems, but by doing so generate new vocabulary and raise novel questions for the biblical text that then need to be resolved. This is not to say that truth changes over time; but it is to say that the manner and terms in which truth is expressed, along with some of the questions asked, do change. Historical theology, the genealogy of doctrinal discussion and formulation, is thus an important part of Christian education and should be part of every pastor's and elder's background. It should also be a central part of the teaching ministry in all churches.\n\nThe Athanasian Creed\n\nThe last ancient creed of significance for this discussion is the so-called Athanasian Creed. While this is not an ecumenical creed in the sense of having been produced and ratified by an ecumenical council, it has nonetheless played a significant role in the life of the church, both East and (especially) West. For example, it is part of the liturgy of the Book of Common Prayer and, since the seventeenth century, has featured in Russian Orthodox service books. Thus, it has enjoyed considerable ecclesiastical and liturgical influence over the years.\n\nThe name, of course, implies that it was Athanasius, the great fourth-century bishop of Alexandria and champion of Nicene orthodoxy, who was the guiding hand behind its composition. In fact, this is not the case. It was originally of Western provenance, written in Latin by a person or persons unknown. The date is also difficult to establish with any certainty. Its Trinitarian theology clearly indicates that it was composed no earlier than 381; and it is unclear whether the christology it contains places it before, during, or after the Nestorian controversy of the 420s.\n\nThe creed articulates standard post-381 Trinitarian orthodoxy and also a careful christology. These are not particularly contentious. What has made it controversial over the years are the two anathemas. These occur in clauses 2 and 44:\n\n1. Whosoever will be saved: before all things it is necessary that he hold the Catholic Faith:\n\n2. Which faith except every one do keep it whole and undefiled, without doubt he shall perish everlastingly. . . .\n\n44. This is the Catholic Faith, which except a man believe faithfully, he can not be saved.12\n\nSuch anathemas, which were also a feature of the original Nicene Creed, fall foul of contemporary tastes. There is an influential element even within conservative evangelicalism that wants to draw boundaries only by stating the positive, only by emphasizing what is actually believed and thus excluding the heterodox and the heretical only by implication. This is consonant with the spirit of an age where exclusion is seen as a bad thing\u2014and not, as we noted in chapter 1, without good reason in many cases. Exclusion for the sake of elitism or based upon hate and prejudice is deeply wrong and harmful, and the church needs to repudiate such and avoid it at all costs. Yet we cannot as Christians avoid the fact that the faith is always exclusive in some sense, that this exclusivity is expressed in part by public doctrinal commitments, and that the holding of certain positions and the rejection of others determines whether one is included or excluded. As has often been pointed out, of course, one cannot hold to the center of a circle without knowing where the circumference lies. Thus, boundaries, and the drawing of them, are absolutely vital to healthy, orthodox Christianity.13\n\nThis is what the author(s) of the Athanasian Creed were attempting to accomplish. In the spirit of other creeds, such as Chalcedon, they used formulations that mark points beyond which the orthodox must not pass; all that they added were specific, explicit anathemas to make sure that readers would fully understand the implications of the matter. Like other ancient creeds, this one was not dealing with trivia or with matters of peripheral relevance to the church. It was dealing with the very identity of God in such a manner that denial of its affirmations placed one's soul in serious jeopardy.\n\nThe response, of course, might be that the Athanasian Creed\u2014or that of Chalcedon, or Constantinople, for that matter\u2014seems to demand belief in something that is not explicitly stated in Scripture. But as we saw above, that kind of objection is not a particularly compelling one. What is stated here rests upon the teaching of Scripture, and the implications of that teaching. It is conceptually consistent with the Bible, even if its terminology is actually absent from the same.\n\nConclusion\n\nTwo things are perhaps most striking about the ancient creeds. First is the fact that the early church developed them in the first place. In the century after the death of Paul, we see the rise of the so-called Rule of Faith, which appears to have functioned as a fluid oral summary of the essential elements of biblical teaching, specifically in the face of challenges to orthodoxy. Then, by the middle of the fourth century, given a periodically united Roman Empire and the struggles over the person of Christ, the church as whole came to the conclusion that binding creedal formulas were one way of attempting to establish public criteria for orthodoxy.\n\nOf course, just because a practice is ancient does not mean that it is automatically biblical or appropriate. The visible, earthly church can err, as Protestant confessions themselves make clear; and anyone who has studied the ancient church knows that the seeds of many later practices that evangelicals would reject as unbiblical (for example, the adoration of the Virgin Mary) have their origins during this early period of church development. Nevertheless, a case can surely be made for seeing a clear, consistent, and legitimate development from the teaching and practice of Paul in the New Testament through the Rule to the creeds of the fourth century and beyond. Creeds are, after all, simply forms of sound words allied to a church understood not simply as a collection of random believers but as a body with a definite structure and leadership.\n\nThe second striking thing is that the early church creeds focus on the most basic building blocks of the faith. The Apostles' Creed is perhaps unparalleled in church history as a succinct statement of the history and significance of Jesus Christ. Its lack of theology proper is only a weakness if one wishes it to do more than that for which it was designed. Thus, one must accept, for example, that it is really no good for keeping out of the church those who deny the Trinity. It is not, after all, an explicitly Trinitarian document, because it does not elaborate the doctrine of God in sufficient detail to do so. But it is still a great account of the basic historical building blocks of the biblical account of salvation.\n\nMoving on to the other ancient creeds from which evangelicals might well profit, we still find the same focus on the very basics of Christianity. What do the Nicene Creed, the Chalcedonian Formula, and the Athanasian Creed have in common? Surely it is this: They all address the most basic of Christian themes\u2014the very identity of God. Whether one agrees with the specific formulations they offer or not, one must agree that the questions they are seeking to answer are perhaps the most basic in Christian theology and, indeed, in the Christian life of the church and of every believer. The meaning of baptism and of Christian praise\u2014praise that ascribes lordship to Jesus Christ\u2014are both wrapped up inextricably in the answer to the question of who he actually is.\n\nThis is why these early creeds in particular have enjoyed significant influence throughout the centuries. They do not answer the only questions that are important to the Christian, such as how the individual obtains salvation, but they do address the central point of Christ's identity. Significantly, they also do so in a way that most churches have found to give an adequate account of the Bible's teaching. With the exception of the Coptic Church's rejection of Chalcedon, Roman Catholics, Eastern Orthodox, and Protestants (Lutheran and Reformed) all accept that these creeds are basically the gold standard for talking about Christ.\n\nThe challenge for a \"no creed but the Bible\" pastor at this point is obvious. If you want to abolish the early church creeds because you regard them as some kind of man-made tradition that stands independent of Scripture or that trumps Scripture's teaching, you are going to need to replace them with something. Further, you are going to need to replace them with something straightaway. The church cannot exist for even the time it takes to say the Nicene Creed without an understanding of who Christ is and, by implication, who God is.\n\nThe obvious moves for such a pastor are twofold. He can simply adopt creedal categories without ever actually acknowledging from whence he took them. Thus, he might still use the term \"Trinity\" and speak of Christ as one person, two natures, with his divine nature being co-equal with the Father. There is a sense in which this is to be welcomed: at least such a pastor is actually teaching his people good, sound theology, even if he is not being exactly transparent or perhaps even self-conscious about where he is obtaining it. To such a pastor, I would simply say that the more one acknowledges the traditions upon which he depends for his theology, the more he is actually able to assess them in the light of Scripture. Ironically, to repudiate creeds but to use their content makes one more, not less, vulnerable to being swept along by tradition, simply because one does not understand that one is actually connected to such.\n\nThe second move is the biblicist one of staying as close as possible to the biblical narrative and the biblical categories. Again, there is much that is highly commendable here: what Christians who wish to be biblically faithful do not want their theology normed by the Bible? The problem with this is that, depending on which strand of biblical teaching one chooses to privilege, the results could be disastrous in a number of ways. To emphasize biblical teaching on the unity of God might lead to what is essentially a modalist christology, where Father, Son, and Holy Spirit are the same person of God, just in different time dispensations and modes of being. Or an emphasis on the distinction of Father and Son, coupled with passages that speak of the Father's superiority, might lead to subordinationist christologies, where Christ is somehow less God than his Father. Worse still, the distinction might be emphasized to the point where the result is more than one God.\n\nThis is not to say that any of these will be the inevitable result of abandoning or ignoring the ancient creeds. But to avoid them, it is almost certain that the pastor will have to use commentaries or theological books that do connect to the creeds of the early church. It is a simple fact that the church of the first five centuries was the context of a host of doctrinal experiments, whereby different theologians tested different models of God and of Christ to find out which ones did most justice to the Bible's teaching. The Nicene Creed and the Chalcedonian Formula are two key results of that; and the fact they have proved their ability to last for so many centuries as such widely accepted accounts of the biblical teaching is surely not something we should take lightly.\n\nThus, my response to the biblicist pastor would simply be this: do not precipitately abandon creedal formulations which have been tried and tested over centuries by churches all over the world in favor of your own ideas. On the whole, those who reinvent the wheel invest a lot of time either to come up with something that looks identical to the old design or something that is actually inferior to it. This is not to demand capitulation before church tradition or a rejection of the notion of Scripture alone. Rather, it is to suggest an attitude of humility toward the church's past which simply looks both at the good that the ancient creeds have done and also the fact that they seem to make better sense of the testimony of Scripture than any of the alternatives. The Lord has graciously provided us with a great cloud of witnesses throughout history who can help us to understand the Bible and to apply it to our present day. To ignore such might not be so much a sign of biblical humility as of overbearing hubris and confidence in our own abilities and the uniqueness of our own age.\n\nOf course, as Protestants, we do not believe that the ancient creeds say everything that a church committed to teaching the whole counsel of God needs to say to the world. Those who would see a return to the ancient ecumenical creeds as the answer to later church divisions and as a means of restoring church unity are certainly to be commended for their desire to see Christians united. The problem, of course, is that such a proposal automatically relativizes all later developments. That is great for the Eastern Orthodox churches; it is problematic for committed Roman Catholics and Protestants who may disagree on issues such as justification and the sacraments but yet agree on the fact that these things are important, that Christians need to have convictions on these matters. Thus, for Protestants, discussion of creedalism cannot stop with Chalcedon. It must also address confessional developments in the sixteenth and seventeenth centuries. It is to these we turn in the next chapter.\n\n1A collection of Gnostic texts can be found in English translation in James M. Robinson, ed., The Nag Hammadi Library (New York: Harper and Row, 1977).\n\n2Ignatius, Letter to the Trallians, in The Ante-Nicene Fathers, vol. 1, Translations of the Writings of the Fathers Down to A.D. 325, eds. A. Roberts, J. Donaldson, and A. C. Coxe (Oak Harbor, WA: Logos Research Systems, 1997), 69. Cf. his Letter to the Smyrnaeans, 1.\n\n3Tertullian, On the Prescription of Heretics in The Ante-Nicene Fathers, vol. 3, Translations of the Writings of the Fathers Down to A.D. 325, eds. A. Roberts, J. Donaldson, and A. C. Coxe (Oak Harbor, WA: Logos Research Systems, 1997), 249.\n\n4Irenaeus in The Ante-Nicene Fathers, 1:330\u201331.\n\n5Hippolytus, The Treatise on the Apostolic Tradition of St. Hippolytus of Rome, trans. and ed. Gregory Dix and Henry Chadwick (Ridgefield, CT: Morehouse, 1992), 33\u201338.\n\n6Tertullian, On the Shows, 3:81.\n\n7Ambrose of Milan, Letter 42.5, The Tertullian Project, <http:\/\/www.tertullian.org\/fathers\/ambrose_letters_05_letters41_50.htm#Letter42>.\n\n8See Daniel R. Hyde, In Defense of the Descent: A Response to Contemporary Critics (Grand Rapids, MI: Reformation Heritage Books, 2010).\n\n9See Lewis Ayres, Nicaea and Its Legacy: An Approach to Fourth-Century Trinitarian Theology (Oxford, UK: Oxford University Press, 2004), 110ff.\n\n10This distinction between members and office-bearers is extremely important and will be explored in more detail in chapter 6.\n\n11The Chalcedon Formula, Documents of the Christian Church, trans. Henry Bettenson (Oxford, UK: Oxford University Press, 1947), http:\/\/www.anglicansonline.org\/basics\/chalcedon.html.\n\n12Available at Christian Classics Ethereal Library, <http:\/\/www.ccel.org\/creeds\/athanasian.creed.html>.\n\n13This applies both to elaborate confessional statements, such as the Westminster Confession, and to less elaborate examples, such as the doctrinal bases often used for parachurch organizations. The respective bases differ not in the fact of boundary drawing, or the prioritizing of a putative center over boundaries, as if the former is positive while the latter are negative, but simply in the number of boundaries drawn. To adhere to a positive statement on any given doctrine is to exclude the opposite, to draw a line which cannot be crossed while claiming to be orthodox.\n\n#\n\n## Classical Protestant Confessions\n\nIn the period of the Reformation, the church in Western Europe started the process of institutional fragmentation that has continued unabated to this day. If one had been born at the turn of the sixteenth century, one would have considered it inconceivable that the church in the West would not be one and continue to be so. The idea that the church would fragment, let alone that there might even be more than one church in a given geographical area, would not have crossed the mind of any thoughtful person, let alone the many thoughtless people who no doubt existed then as now. Yet one hundred years later, at the turn of the seventeenth century, the Western church was shattered into pieces in the manner with which all Christians today are very familiar.\n\nThis fragmentation also involved confessionalization. This term is more than simply a way of referring to the production of theological confessional documents. In the sixteenth century, theology was inextricably bound up with politics: the theological position held by a prince or by a city council had implications for military and political alliances. In other words, theology was very closely tied to territorial politics. Thus, the production of confessions in the sixteenth and seventeenth centuries had dual impulses behind it: theological, because the new churches needed to identify themselves in relation to other emerging communions; and political, as territories and cities needed to define themselves in relation to each other.\n\nOf these two impulses the second is of no real relevance today. Even in a country like England, with an established church presided over by the monarch, theology is of no significance to international relations or to domestic politics, beyond the occasional debate about whether the Church of England should continue to enjoy its special constitutional status, or the predictable complaints from politicians when the Archbishop of Canterbury makes some inept comment about the economy. This does not, however, mean that the other impulse for confessionalization, the defining of one church in relation to another, is no longer relevant, as we shall argue in a later chapter. Nevertheless, it is useful to bear the political aspect in mind as it does help to explain why there was such frenetic production of confessions and catechisms in the sixteenth and seventeenth centuries.\n\nGiven the large number of confessions produced during this time, for the sake of space our focus here will be on those which continue to play a major role in the main denominations of Protestantism: the Anglican Articles; the Book of Concord; the Three Forms of Unity; the Westminster Standards; and the Baptist Confession of 1689.1 Before addressing these, however, it is worth giving a basic narrative of Protestant confessionalization as a whole.\n\nThe Anglican Articles\n\nThe Reformation in England was shaped by a variety of factors. First, King Henry VIII's break with Rome was not fueled by theology so much as his desire to divorce his first wife, Katherine of Aragon, in order to marry his second, Anne Boleyn. This precipitated a rupture with the papacy, which a number of his counselors leveraged to produce a church with a more Protestant face. It was not until the reigns of his son, Edward VI, and then Edward's sister, Elizabeth I (after a brief re-Catholicizing period in between under Mary), that the Church of England was truly secured for the Protestant Reformation.\n\nSecond, the ebbing and flowing of Protestant political fortunes on the continent meant that a number of leading Reformation theologians spent time in England and impacted both the liturgy and the theology of Anglicanism. This was particularly true of Martin Bucer, the Reformer of Strasbourg; Peter Martyr Vermigli, the leading Italian Reformer of the day; and John a Lasco, the Polish Reformed theologian. Thus, the Church of England was the product not simply of internal English politics but also of careful theological dialogue with some of the greatest European Reformation minds of the day.\n\nThe three towering textual achievements of Anglicanism are the Book of Common Prayer, the Thirty-Nine Articles, and the Homilies. Of these three, the Book of Common Prayer is undoubtedly the greatest liturgical achievement in the English language. In comparison, modern attempts of the same (of which I am aware) seem like wooden verbiage. As with all thoughtful liturgies, it also reflects and embodies a certain theology and was thus crucial in inculcating the same within the congregations who used it. Doctrinally, however, it is the Articles and the Homilies which are crucial to the Church's stated confessional identity.\n\nThe Homilies, published in two volumes (1547 and 1571), are a series of thirty-three short sermons that were to be read aloud in church. At the time of the Reformation, there was a key problem relative to personnel: the supply of Protestant ministers could not possibly meet the immediate demand created by the state's allegiance to the Reformation. There is an analogy here with more recent revolutions. When the Eastern Bloc collapsed in 1989\u20131991, the men and women at the very top of the political power structures were rapidly replaced. It was simply not possible, however, to sweep away in a moment all of the functionaries of the old regimes; this would have created turmoil. Further, it was probably not necessary: the lower down the chain of command one goes, the less likely one is to find ideologically driven people and the more likely one is to find those who simply do their job in order to earn a salary. Thus it was in the Reformation: while church leadership at the top changed when Protestantism arrived, most parishes would have continued to have the same pastors and priests as before. Further, the education of a large Protestant ministry required the prior reformation of educational institutions and the development of faculty and appropriate curricula. Thus, while Reformation at the top might have been swift, Reformation root and branch was a much longer-term project and demanded immediate remedial measures.\n\nThus, the production of homilies was in part a response to what we might call the pastoral crisis precipitated by the Reformation. The parish priest may well have been an ignorant fellow who could not even name the four Gospel writers, let alone list the Old Testament prophetic books in order, but if he was basically literate then he could feed his people by reading a set homily at each service. The homilies were thus one means by which the Reformation Anglican Church sought to fulfill Paul's mandate of holding fast to a form of sound words and passing on the faith from place to place and generation to generation.\n\nThe other theological document was the Thirty-Nine Articles. Originally finalized in 1571, they represent the closest thing the Anglican Church has to a formal confession of faith. The history of the Articles, both in terms of their interpretation and their legal status and application, has been a turbulent one. Most infamously, John Henry Newman attempted in 1841 in Tract 90 of the Tracts for the Times to argue that they were susceptible to a reading which could be described as strongly Roman Catholic in tendency. He based his argument primarily on the fact that they had been composed prior to many of the declarations of the Council of Trent and were thus targeted at putative popular misunderstandings of Catholic teaching, not the genuine article. Newman's interpretative gymnastics were ultimately unconvincing even to himself, and he left for Rome in 1845. But the mere existence of Tract 90 is enough to show how problematic is the history of the interpretation of the Articles.\n\nThe Church of England's peculiar history also shaped the tone and scope of the Articles. The English Reformation was legislated by a Parliament that was somewhat less than dominated by committed Protestants. Thus, the exigencies of the political situation meant that there was an evolutionary aspect to the Church's development, given that reformation could progress only at the pace which Parliament would tolerate. As an established Church, it was also important that she should be as comprehensive as possible, even given the need to exclude certain groups such as the Roman Catholics and the more radical Protestant sects. In this way, Reformation Anglicanism did indeed represent a via media, a middle way, but not in the sense for which Newman argued prior to moving Romeward. It was not a middle way between Protestantism and Roman Catholicism but rather between Roman Catholicism and Anabaptism. This is otherwise known as Reformation Protestantism. Both of these factors\u2014the need for care and caution in moving the Reformation forward and the need for a comprehensive Protestantism\u2014meant that the Anglican articles were less sharply and elaborately articulated than many other Reformation confessions.\n\nThus, the articles make clear statements on such hallmark Protestant doctrines as justification by faith, which is addressed in Article 11:\n\nWe are accounted righteous before God, only for the merit of our Lord and Saviour Jesus Christ by Faith, and not for our own works or deservings. Wherefore, that we are justified by Faith only, is a most wholesome Doctrine, and very full of comfort, as more largely is expressed in the Homily of Justification.\n\nHere, the article sets forth in brief the Protestant position on justification and also makes clear that, brief and concise as the statement is, it should be understood in terms of the more thorough explanation of the doctrine in the Homilies. It is also worth noting that this theology informs the liturgy of the Book of Common Prayer. At morning prayer, the minister declares the following:\n\nDearly beloved brethren, the Scripture moveth us, in sundry places, to acknowledge and confess our manifold sins and wickedness; and that we should not dissemble nor cloak them before the face of Almighty God our heavenly Father; but confess them with an humble, lowly, penitent, and obedient heart; to the end that we may obtain forgiveness of the same, by his infinite goodness and mercy. And although we ought, at all times, humbly to acknowledge our sins before God; yet ought we chiefly so to do, when we assemble and meet together to render thanks for the great benefits that we have received at his hands, to set forth his most worthy praise, to hear his most holy Word, and to ask those things which are requisite and necessary, as well for the body as the soul. Wherefore I pray and beseech you, as many as are here present, to accompany me with a pure heart, and humble voice, unto the throne of the heavenly grace, saying after me . . .\n\nThis is then followed by the congregational response:\n\nAlmighty and most merciful Father; We have erred, and strayed from thy ways like lost sheep. We have followed too much the devices and desires of our own hearts. We have offended against thy holy laws. We have left undone those things which we ought to have done; And we have done those things which we ought not to have done; And there is no health in us. But thou, O Lord, have mercy upon us, miserable offenders. Spare thou them, O God, who confess their faults. Restore thou them that are penitent; According to thy promises declared unto mankind in Christ Jesu our Lord. And grant, O most merciful Father, for his sake; That we may hereafter live a godly, righteous, and sober life, To the glory of thy holy Name. Amen.\n\nApart from the striking beauty of the prose, what is noteworthy is how Anglican confessional theology is reflected in the liturgy and is thus cultivated and reinforced in the congregation by the same. This is no dry confessionalism, where the theological confession is simply a constitutional document that never touches the people. It is woven into the very fabric of Anglican life.\n\nFinally, it is worth noting article 8, which, in its original form, asserted that the Apostles', Nicene, and Athanasian creeds \"ought thoroughly to be received and believed: for they may be proved by most certain warrants of Holy Scripture.\" Later this was modified to omit the Athanasian Creed, but the basic point remained the same: this Protestant confession was consciously connected to the catholic creedal theology of the early church; it was no innovation but saw itself as in continuity with the great declarations of the patristic church and was thus catholic and orthodox in the best sense of the word. It was striving, if you like, to hold fast to the form of sound words that it had received on these issues.\n\nOf course, the history of the Anglican church is, by and large, a history of failure to apply the Thirty-Nine Articles and to carry forward the theology they contain. The old joke about the Anglican bishop who was perfectly happy to recite the creed as part of the liturgy each week and only had to omit the first three words, \"I believe in\" in order to do so with integrity has, sadly, too much truth about it to be entirely amusing; but Anglicans have a beautiful confessional-liturgical heritage. If they choose to squander it, that is not the fault of the founders of their church.\n\nThe Book of Concord\n\nOf all Reformation Protestant traditions, Lutheranism is, as the name suggests, the one which is most intimately connected with the career and theology of a single individual: Martin Luther. All orthodox Protestants are indebted to him for many things in their own confessions, not least his understanding of justification by grace through faith. But his thought is particularly influential in the church that bears his name.\n\nThis influence is reflected in the contents of the Lutheran confessional documents known collectively as the Book of Concord. The Book of Concord was adopted in 1580 by a group of leading Lutheran churchmen, princes, nobles, and town councils as a means of defining their confessional allegiance within the political context of late sixteenth-century Europe. As noted above, the geopolitical aspect of such confessions no longer applies, but the Book of Concord remains the confessional standard for Lutherans around the world. However, as with the other confessions considered in this chapter, the way the Book of Concord is applied varies from denomination to denomination: conservative groups, such as the Wisconsin and Missouri Synods in North America adhere closely to the doctrine the book contains, while the more liberal Evangelical Lutheran Church in America sits more loosely on the same.\n\nThe Book of Concord is actually a collection of a number of different writings:\n\nThe Apostles' Creed\n\nThe Nicene Creed\n\nThe Athanasian Creed\n\nThe Augsburg Confession (1530)\n\nThe Apology of the Augsburg Confession (1531)\n\nThe Smalcald Articles (1537)\n\nTreatise on the Power and Primacy of the Pope (1537)\n\nThe Small Catechism (1529)\n\nThe Large Catechism (1529)\n\nThe Formula of Concord (1577)\n\nOf these documents, three are of patristic origin (the creeds), three were written by Luther (the catechisms and the Smalcald Articles), and three by his colleague, Philip Melanchthon (the Augsburg Confession and Apology and the Treatise on the Power and Primacy of the Pope). They also bear the unmistakable impact of the particulars of the time of composition. The Augsburg Confession was designed to try to win the Holy Roman Emperor, Charles V, if not over to the Protestant cause then at least to a position of tolerance toward it. Thus, it omitted any attack on the pope's authority. As it became clear in the 1530s that Charles could not be won over and that some statement on the papacy was necessary, Melanchthon composed the treatise on the power of the pope. Luther did not subscribe this, not for significant reasons but because a bad attack of a kidney stone prevented his attendance at the discussions. The Smalcald Articles were composed by Luther in order to provide a confessional basis for the Schmalkaldic League, a military alliance of Lutheran princes. The League did not adopt them, but the Articles did ultimately receive confessional status by inclusion in the Book of Concord.\n\nThe final document, the Formula of Concord, is the result of struggles for the identity of Lutheranism after Luther's death in 1546. Luther had, of course, been a divisive figure during his own lifetime. In particular, his emphasis on the nonnegotiability of belief that Christ was present according to both his divine and human natures in the Lord's Supper had led to a break with Zwingli and the Reformed at a colloquy at Marburg in 1529. This had subsequently become a key point of dispute between Lutheran and Reformed churches such that the identities of the two communions were determined to a large extent by precisely this issue.\n\nWhen Luther died in 1546, rival factions quickly emerged among his followers. On the one hand were those who looked to Luther's gentle colleague, Philip Melanchthon, for their lead. These became known as the Philippists, and their hallmark was a greater openness to ecumenism with both Catholic and Reformed, a concessive attitude on whether both natures of Christ were present in the Eucharistic elements, and a more ambivalent attitude to predestination. The other faction, the Gnesio- (\"real\") Lutherans placed the Real Presence of Christ in the Eucharist at the forefront of their doctrinal concerns and also held a strict line on predestination, consistent with that of Luther in his great work, The Bondage of the Will (1525). After years of internecine struggle, the Formula of Concord enshrined Luther's position on the Lord's Supper and therefore represented the triumph of the Gnesio-Lutheran party.\n\nThere are many interesting aspects of Lutheran confessionalism. We might note in passing once again the self-conscious way in which Lutherans connect to the early church tradition by including three patristic creeds in their standards. Lutheranism, like other branches of Reformation Protestantism, did not wish to be seen to be innovative, because the faith once for all delivered to the saints did not need innovation. Of course, it is clear from a study of Christianity that there is a sense in which doctrine develops. This is not to say that the gospel has changed; but the way the gospel is articulated has undergone elaboration. The Lutherans understood this but also saw a clear need for connection with the past and thus with the apostolic teaching.\n\nTwo further points are particularly pertinent for the argument of this book: the importance of sacramental theology, and the pedagogical impulse behind some of the formulations.\n\nAs to the first, sacramental theology, it is likely that nothing so goes against the grain of modern evangelical sensibilities as the confessional attitude to sacraments. Look at any statement of faith by an evangelical parachurch organization and one is very unlikely to find any clear cut statement on baptism or the Lord's Supper. The reason for this is that differences on these are inevitably divisive (at least in the case of baptism) and often regarded as secondary and irrelevant (as is the case with the Lord's Supper and, for the really lax, baptism). This is not necessarily a problem unless, of course, the parachurch group becomes an agenda setter for the church or, even worse, a power-grab organization that functionally supplants the church. Once that starts to take place, the signal is sent that sacraments are not important at all.\n\nUnfortunately, both the Bible and church history witness to the fact that baptism and the Lord's Supper are of vital importance. More ink was spent arguing over the Lord's Supper in the sixteenth century, for example, than over the nature of justification. Further, one cannot really have a church without having a clear understanding of these things. One may, of course, have a clear understanding which is wrong; but it is better to be wrong about them while still knowing they are of importance than not to realize they are important at all.\n\nMinimally, an understanding of baptism is important because baptism is the means of entry into the visible church; and an understanding of the Lord's Supper is important because, minimally, the admission to or banning from participation in the Supper is a basic part of church disciplinary procedure. Thus, churches that have membership and that exert pastoral oversight and exercise discipline must have a position on both baptism and the Lord's Supper. If a church does not have such, then, frankly, it is not really a church. Thus, we may regret the fact that Luther broke Protestantism in two over his contention that belief in the Real Presence was essential to the faith, but it would have been even more regrettable had he agreed to live and let live with the Reformed because he did not see that the matter was at all important.\n\nThe second aspect of note in the Lutheran confessional documents is the pedagogical concern. This is most evident in the two catechisms. The very inclusion of catechetical material indicates that pedagogy is dear to the heart of Lutheran confessionalists. As with the Homilies in Anglicanism, so with the Book of Concord: it rests upon a vision for church life whereby the people are slowly but surely educated in the great doctrines of the faith. They are not meant to stay at the level of knowledge they have when they first start to listen to sermons, let alone when they are baptized; rather they are to grow to maturity in the faith, and an important part of that is growth in doctrinal knowledge.\n\nI shall mention this notion again in chapter 6, but here I want to note how this impacts the Lutheran catechisms in one significant way: the use of traditional language. The early sixteenth century was a time of major social change. The rise of cities drew people away from the countryside and thus disrupted traditional ways of life and created new situations of uncertainty. For example, parents faced seeing their children leave home and go far away; those children lost the extended family that had traditionally provided the network for social support. Life was hard and life was uncertain. My guess is that in such times of uncertainty, church was something of a haven of stability: it was just the same as when you were growing up. Thus, for all of the change in the world around, you could always go to church for a bit of the comfort of the routine and the familiar. Then along comes the Reformation. If we believe that such things as vernacular liturgies and sermon-centered worship were greeted with unalloyed delight, I suspect we are naive. Anyone who has ever been in a church where the elders decided to move from a Bible translation hardly anyone could understand to one that everyone finds comprehensible will know that such a process is often met with vigorous opposition. People like things to stay the same; they frequently do not like change, even when that change is really to their advantage.\n\nThus, in Luther's catechisms, we find the interesting phenomenon of his new theology expressed using the old terminology. Most striking is the retention of the language of \"altar\" in the context of the Lord's Supper. Lutheran theology is emphatic in stressing that the Lord's Supper is not a sacrifice, not a priestly act directed by men to God, but rather an act in which God condescends to come down to men. Yet the language of altar is retained. The reason is obviously not theological. It would appear rather to be pedagogical, whereby ears used to hearing talk of the altar would not be unduly disturbed by suddenly hearing of the altar referred to simply as a table. Further, in the Augsburg Confession, even the language of \"mass\" is retained. Clearly, in this context, there are political considerations: Melanchthon is trying to win over the Catholic emperor. But there is also surely a pedagogical payoff here in retaining familiar language while filling it with new content.\n\nIn sum, the Lutheran confessional documents are clearly designed to establish both doctrinal definitions that clarify what the Lutheran church believes and to provide pedagogical material for educating people in that identity. This twofold aspect of confessionalism is evident in the next group of documents to which we turn: the Three Forms of Unity.\n\nThe Three Forms of Unity\n\nThe Belgic Confession (1561), the Heidelberg Catechism (1563) and the Canons of Dordt (1619) are collectively known as the Three Forms of Unity. They form the confessional standards of Reformed churches that look to the continental Reformation (as opposed to the Anglo-Scottish Reformation) for their origins. The Reformed Church in America, the Christian Reformed Church, and the United Reformed Church are three of the most well-known American denominations that look to these documents as providing their basic doctrinal identity.2\n\nThe Belgic Confession was the work of a single man, French Protestant Guido de Bres, who was later martyred for his faith. His purpose in writing the confession was to obtain some level of toleration for Reformed believers in the Low Countries (modern-day Belgium and the Netherlands). What started as an attempt to articulate the faith to political powers in the tradition of the Greek apologists came to have much greater significance when it was adopted by the Synod of Dordt (1618\u20131619) as one of the confessional standards for the continental Reformed churches.\n\nThe Heidelberg Catechism was also probably the work of a single man, Zacharias Ursinus, who was a key Reformed theologian in the city of Heidelberg. The ruler of Heidelberg in the early 1560s was Frederick III, who converted from Lutheranism to the Reformed faith. That is odd to modern evangelical ears, for the reason noted above: the matter that really divided Lutheran from Reformed, the Lord's Supper, is of little consequence to those who focus simply on a few isolated doctrines which they regard as constituting the gospel and as providing an adequate basis for the Christian life. By contrast, this was a matter with deep theological and political implications in the sixteenth century.\n\nThus, when Frederick converted, his conversion created various issues in the city of Heidelberg. First, he needed a statement that would allow for the territory to have a confessional identity. Second, he had a divided faculty at his university, where Reformed, Philippists, and Gnesio-Lutherans were engaged in conflict with each other, a conflict which inevitably spilled over into church life and thus into politics. What Frederick determined to do was commission a confession that might form the basis for ecumenical rapprochement between the Reformed and the Philippists, which would isolate and marginalize the hard-line Gnesios.\n\nThe result was the Heidelberg Catechism, a document remarkable for its pastoral tone (cultivated with careful use of the first person in the answers) and also for the fact that it pointedly omits any direct teaching on the matter of predestination. Predestination was an issue that divided the Reformed, who maintained varieties of classic anti-Pelagianism, and the Phillipists who, following the later Melanchthon, tended toward a position more concessive toward human free will. Phillipists were also very opposed to preaching on the topic on the grounds that this could create more pastoral problems than it solved.\n\nAccording to Frederick's introduction, the Heidelberg Catechism was to provide a basis for confessional unity, a model for training youth, and a guide to teachers and pastors to prevent them from adopting doctrinal changes at will. It was thus both a confessional and a pedagogical document. It was further enhanced as a pedagogical tool when it was divided into fifty-two sections which then became a preaching guide for the afternoon\/evening services in Dutch Reformed churches, ensuring thorough doctrinal coverage of the Catechism every year.\n\nThe Catechism was adopted by various synods in the sixteenth century and then, like the Belgic Confession, it was formally approved by the Synod of Dordt as an official doctrinal standard of the continental Reformed churches.\n\nThe third confessional standard was actually produced at the Synod of Dordt itself: the Canons of Dordt. The Synod was summoned in the Netherlands to address the problems raised by the rising Arminian party. For us, Arminianism is no more than a branch of Protestant Christianity; in the early seventeenth century, Arminianism, as all theological positions at the time, had serious political implications. As a movement that was seen as potentially more concessive to Roman Catholicism at a time when Spain and France were major forces in northern Europe, Arminianism was controversial not simply for its modification of Protestant theology on matters of predestination but also for its implications for domestic and foreign policy.\n\nAgain, the politics of Dutch Reformed theology in the seventeenth century are no longer of any relevance today; but the Canons, as adopted by the Synod at the time, remain a confessional standard. These Canons were a direct response to the Five Remonstrant Articles of 1610, set forth by the followers of Jacob Arminius (1560\u20131609). These asserted a form of conditional election, universal atonement, a modified understanding of depravity, and the resistibility of grace, along with an article that questioned perseverance. Dordt responded by asserting total depravity, unconditional election, particular redemption (\"limited atonement\"), irresistible grace, and the perseverance of the saints. This became the basis for what is later known as the Five Points of Calvinism, often referred to by the acronym TULIP.\n\nThe Canons were thus not intended as anything approaching a comprehensive statement of Christian doctrine and cannot by themselves form an adequate confessional basis for a church. But, combined with the Belgic Confession and the Heidelberg Catechism, they form part of a thoroughgoing exposition of the Reformed understanding of the Christian faith.\n\nAs with the Anglican standards and the Book of Concord, the Three Forms exhibit the same concerns for establishing doctrinal identity and promoting doctrinal pedagogy that was so important to Reformation Protestants. In addition, it is also worth noting that Article 9 of the Belgic Confession accepts the teaching of the Apostles', Nicene, and Athanasian creeds. This connection to patristic Christianity is even more explicit in the Heidelberg Catechism, where question 23 asks what articles are necessary for the Christian to believe and the answer is the full text of the Apostles' Creed, which the Catechism then goes on to expound, point by point. The Catechism is thus most intentional in its connection of its Protestant theology to that of the early church. This is holding fast to a form of sound words and preserving and transmitting the tradition in the very best sense of the word.\n\nA couple of other aspects of the Three Forms stand out. Of particularly note is the statement in the Belgic Confession on the church:\n\nAnd this holy church is preserved by God against the rage of the whole world, even though for a time it may appear very small in the eyes of men\u2014as though it were completely extinguished.\n\nIn America today among many evangelicals, that statement would appear to be nonsense. The church can seem vibrant, robust, and in many cases numerically strong, too. Yet de Bres knew that of which he spoke: he was a hunted heretic who eventually gave his life for the cause. To be reminded that the church is God's creation and lives in the shadow of the cross, and is not to be judged strong or weak by the standards of the world, is both a salutary rebuke to triumphalism and a great encouragement to those who live in parts of the world where the most noticeable aspects of the church are her outward weakness and suffering. Again, this is simply biblical theology summarized in a pithy statement; and having it as part of the confessional statement of the church means that it will be a constant reminder to her people of the realities that attend God's kingdom here on earth.\n\nOne other point to make, and something that really marks the Heidelberg Catechism out as a document of great pastoral beauty in the history of creeds and confessions, is the tone and phrasing of the first and last questions:\n\nQuestion 1. What is thy only comfort in life and death?\n\nAnswer: That I with body and soul, both in life and death, am not my own, but belong unto my faithful Savior Jesus Christ; who, with his precious blood, has fully satisfied for all my sins, and delivered me from all the power of the devil; and so preserves me that without the will of my heavenly Father, not a hair can fall from my head; yea, that all things must be subservient to my salvation, and therefore, by his Holy Spirit, he also assures me of eternal life, and makes me sincerely willing and ready, henceforth, to live unto him.\n\nQuestion 129. What does the word \"Amen\" signify?\n\nAnswer: \"Amen\" signifies, it shall truly and certainly be: for my prayer is more assuredly heard of God, than I feel in my heart that I desire these things of him.\n\nI never cease to be struck by the beauty of these two answers. Question 1 shows the glorious Reformation Protestant insight into the fact that assurance is to be the normal experience of every Christian believer and not simply the preserve of a few special saints who have been given extraordinary insight into their status before God, as was the medieval Catholic position.\n\nThis is a perhaps one of the greatest Protestant insights of the Reformation. We live in an age where conversion to Roman Catholicism is not uncommon among those who have been brought up as evangelicals. There are many reasons for this: some speak of being attracted by the beauty of the liturgy in comparison with what is often seen as a casual and irreverent flippancy in evangelical services; others like the idea of historical continuity, of knowing where the church has been throughout history; still others find the authority structure to be attractive in an age of flux and uncertainty. Whatever the reasons, most Protestants would concede that Rome has certain attractions. Nevertheless, the one thing that every Protestant who converts to Rome loses is assurance of faith.\n\nRecently a student at Westminster Theological Seminary was telling me how he had once found himself on a plane, sitting next to a famous Cardinal. The two of them had a delightful conversation over the course of the flight. Finally the student asked the Cardinal if he was sure of his salvation; the Cardinal shook his head. \"Nobody can be certain of that,\" he declared. The Cardinal (as one would expect) knew his theology. The answer was a good one from the perspective of Roman Catholic theology.\n\nThe insight of the Reformation on assurance was key, theologically and pastorally. And, given that it is one thing that every convert to Roman Catholicism from Protestantism must lose, it is worth noting its priority in the Heidelberg Catechism. The answer is so beautifully phrased; and yet if one ceases to be Protestant, one must cease to claim HC 1 as one's own. That is a very high price to pay. Speaking for myself, all of the liturgical beauty of Rome, all of the tradition, all of the clarity of the authority structure (and that clarity is often, I think, more in the eye of the beholder than the Church itself) cannot compensate for the loss of the knowledge that I know I have been purchased by the precious blood of Christ that conversion to Rome requires.\n\nThe Catechism's last question, too, is surely beautiful. In it, a wonderful doctrine of God underpins a deceptively simple statement: we can be certain that God is so gracious that he more surely hears our prayers than we actually feel we desire that for which we pray. That is surely a beautiful idea that pulls together numerous strands of biblical teaching and expresses them in a way that captures the reader's imagination. Again, happy is the church that uses a document such as the Heidelberg Catechism to shape its understanding of God and his people. Indeed, anybody who thinks that Protestant confessionalism is a hard, dry creed needs to read the Heidelberg Catechism. Only the willfully stupid or deluded could possibly dismiss such a document along such lines.\n\nThe Westminster Standards\n\nThe confessional standards of Presbyterian churches around the world were the product of an assembly of church leaders which took place in England from 1643 onward. England was at that time involved in internal warfare, Parliament versus the Crown. This is traditionally called the English Civil War, but it also involved both Scots and Irish and more than one phase of military conflict. Thus, it is now known in some quarters as the Wars of the Three Kingdoms.\n\nThe military politics need not concern us here; what was ecclesiastically significant was the fact that Parliament convened an assembly at Westminster to revise Anglicanism in terms of its liturgy and confession. From the very first edition of the Book of Common Prayer in 1549, there had been complaints that it was not Reformed enough; and even its second edition in 1552 retained practices such as kneeling at communion, which were regarded as too reminiscent of Roman Catholicism. In addition, the Thirty-Nine Articles had proved less than adequate for guarding the full-orbed Reformation theology that was dear to the heart of many Protestants. Thus, in 1595, Archbishop Whitgift had drawn up the nine Lambeth Articles as a means of safeguarding the Anglican Church's teaching on predestination. The Lambeth Articles never gained official statement because the process of their composition had upset the monarch, Elizabeth I; but they did continue to have significance. Indeed, in 1615, the Irish Church had formulated its own set of articles, the so-called Irish Articles, both as a means of asserting its own identity and as a means of sharpening its doctrinal commitments. These Irish Articles contained the text of the original Lambeth Articles.\n\nThus, when English divines were granted Parliamentary permission to revise Anglicanism in 1643, it was against a background of nearly a century of struggle over Anglican identity and mounting evidence of the inadequacy of the original Thirty-Nine Articles to protect the Reformation legacy.\n\nWhile the Assembly started with a modest brief, the need for Scottish support led, late in 1643, to the signing of a treaty between England and Scotland, the Solemn League and Covenant. This brought the Scottish Presbyterians into the war on the side of Parliament and also a number of Scottish representatives to the Westminster Assembly. From then on, the Assembly became more radical in its program, no longer merely revising Anglicanism but, in terms of confession and practice, rebuilding it from the bottom up. Thus, the Assembly produced, among other documents, not simply a Confession and Shorter and Larger Catechisms but also a Directory for Public Worship, which was intended to supplant the Book of Common Prayer.\n\nThe Restoration of the monarchy in 1660 ensured that the Book of Common Prayer was back with a vengeance, and that the Directory for Public Worship became a marginal document even in many Presbyterian circles. But the Confession and the Catechisms remain the basic confessional standards to which confessional Presbyterian ministers and elders, from the United States to Korea and from Scotland to Japan, adhere by solemn vow.\n\nThe theology of the Standards is basically consistent with that of the Three Forms of Unity, articulating a theology that is Trinitarian and anti-Pelagian. There are a few differences. For example, Westminster has a much stricter view of the fourth commandment when compared to that of the Heidelberg Catechism. Heidelberg Catechism 103 teaches that the fourth commandment requires the maintenance of Christian education and the prioritizing of gathering for worship on Sunday. In the Larger Catechism, questions 117 and 119, a prescriptive list of things that must and must not be done is provided. The difference here is quite obvious and indicates very different traditions of Christian practice on this point; but, even so, it is arguable that this does not involve any major difference in overall theological substance between the two.\n\nOne thing is clear to anyone comparing the Westminster catechisms to the Heidelberg Catechism: the former contain a much greater amount of more thoroughly elaborated theology than the latter. This has helped foster the impression that the Westminster catechisms are less pastoral. There is certainly a sense in which this is true: the consistent use of the first person in the Heidelberg Catechism gives a certain pastoral and personal feel to the whole document. The Westminster catechisms, by contrast, tend to operate more at the level of impersonal propositions. Nevertheless, when one remembers their purpose, the instruction of the faithful in theology, this should not be a matter of great concern. No confessional or catechetical document stands by itself; it is part of an ecclesiastical way of life, one element of our lives as Christians. Thus, the Westminster documents can be taught and used in a lively encouraging way just as the Heidelberg Catechism can be taught in a way that is boring and lifeless.\n\nIn addition, the Standards also indicate that the authors were men of acute pastoral insight. For example, the early generations of Protestant Reformers tended to talk as if saving faith and assurance of salvation were so closely connected as to be virtually inseparable. This was undoubtedly in large part a reaction\u2014an appropriate reaction\u2014to the medieval denial that assurance of salvation was at all a possibility for ordinary Christians. Nevertheless, once assurance became a key issue, it was inevitable that it would bring in its wake a whole new set of pastoral problems. As early church theological developments changed the doctrinal map and generated new questions, so did the Reformation. If nobody is expected to have assurance, then nobody will worry about not having it; but tell people they should be assured of their salvation, and you will quickly find that people start to struggle with the issue. Thus, in the chapter entitled \"Of the Assurance of Grace and Salvation,\" Westminster Confession 18.3 says, \"This infallible assurance doth not so belong to the essence of faith, but that a true believer may wait long, and conflict with many difficulties, before he be partaker of it.\" Is this a deviation from the emphasis or teaching of the early Reformers? I am not inclined to think so. Over one hundred years of Protestant pastoral practice has intervened between the inception of the Reformation and the Westminster Assembly. This is not so much a deviation as a modification, based upon the pastoral experience these men have of preaching and teaching Protestantism to their congregations. It is thus also a sign of the pastoral usefulness and sensitivity of the Confession, revealing it as far more than a set of dry propositions that never touch real life and experience.\n\nFurther, the Standards do contain much more that is of direct relevance to everyday Christian life. The extended reflections in both catechisms on the Decalogue contain much that is directly practical and thus binds the one who vows to uphold the Standards to particular patterns of behavior. Indeed, one may not like or agree with the teaching on the fourth Commandment in the Larger Catechism, for example, but one could hardly argue that it is not practical or does not have implications for how one lives. Thus, the exposition of both the Decalogue and the Lord's Prayer contain significant applications to the life of the Christian.\n\nIn fact, even the Confession itself has some striking practical moments. For example, I have often been impressed by chapter 15.5, which reads:\n\nMen ought not to content themselves with a general repentance, but it is every man's duty to repent of his particular sins, particularly.\n\nIt is fascinating that this is in a church confession. First, it clearly rests upon a specific understanding of God, humanity, and sin. So it is profoundly theological. Second, it makes a clear point about Christian practice, condemning by implication the lazy tendency that we can have as Christians to repent in general terms and let that be sufficient. In other words, it advocates a particular model of the practical outworking of Christianity.\n\nYet there is still more here: the minister who vows that he believes in, and will uphold, the system of doctrine taught in the Westminster Standards, is thus bound to practice and to teach others to practice this principle. He is, in fact, as bound to this as he is to belief in the incarnation and the virgin birth. In other words, confessionalism is not simply about abstract doctrine; confessions also bind one to certain practices, certain ways of life. This is important to remember when reflecting on the opposition sometimes made between Christianity as a set of beliefs and Christianity as a way of life. For the confessional Christian, it is both: the Westminster Confession, as just one example, makes this very clear. A good confession binds doctrine and life, believing and belonging together, and a minister bound to such a confession by solemn vows must thus honor both sides of that. Good confessions do not undermine but can actually protect the practice of piety.\n\nBefore moving to some concluding thoughts, it is worth noting one more confession, the so-called 1689 Baptist Confession. It is \"so-called\" because it was actually written in 1677, not 1689. However, the Act of Toleration, which opened the way for some freedom of religion for non-Anglicans, was not passed until 1689. Only then could the document become a legitimate part of public ecclesiastical discourse rather than simply the clandestine confession of an underground church.\n\nThe Baptist Confession is essentially a slight modification of the Westminster Confession. Inevitably, it articulates a different view of baptism, restricting it to professed believers only, and also affirms an independent polity whereby each particular congregation is fully competent and wholly responsible for its organization and discipline.\n\nI mention the 1689 Confession, not because it makes major contributions to confessional theology but because it is proof positive that Baptists have a confessional heritage. It is not only Roman Catholics, Anglicans, Reformed, and Presbyterians who place high stock in creeds and confessions and who connect these to specific structures of ecclesiastical authority and accountability; it is also some strands of Baptist life as well.\n\nConclusion\n\nClassic orthodox Protestantism has a rich confessional heritage. Of course, I have only covered a few of the relevant documents, though arguably the selection represents the most influential. As noted at the start of this book, many Protestant churches have their own confessional history; and, while I have restricted myself to those one might characterize as belonging to the mainstream anti-Pelagian Protestant tradition, this is not to deny the importance of Arminian and Anabaptist confessions. My main interest in this book is with the principle of confessionalism and not so much with the specific content of particular confessions.\n\nIn closing this chapter, however, it is worth making a few observations. First, as noted above, all of the confessional material above stood self-consciously within the basic Trinitarian and christological framework laid out in the early church creedal formulations. There is no indication that the framers of Reformation Protestant confessions were attempting to build theology completely anew. It is clear that the Reformers and their successors were grateful for and appreciative of the theological work of the church throughout the ages. And, while they firmly believed everything\u2014including their own confessional documents\u2014was subject to being normed by the teaching of Scripture, they did not believe that the church's testimony had to be reinvented afresh every Sunday. God had provided them with a church which had a history, and that history was helpful in understanding what Scripture taught.\n\nSecond, there is a remarkable degree of consensus among these documents on the basics of salvation. Of course, the objection can be made that I have deliberately selected documents that correspond with each other on key points. Had I included, say, the Remonstrant Articles of the Arminians or the Racovian Catechism of the Socinians, the consensus would not have been nearly so great. That is a fair point, but I would still argue that when one looks at the Book of Concord, the Anglican Articles, the Three Forms, and the Westminster Standards, one is looking at doctrinal standards that cover a vast amount of Protestantism and that have had, and still do have, a large number of adherents. It is thus meaningful to argue for a confessional consensus among these documents on issues such as the nature and being of God, the history of salvation, and the nature of justification. That is quite impressive when you consider that these documents were produced by different people in different cultural, political, social, economic, and linguistic contexts. Indeed, if one were to expand the confessional material to include other documents from the sixteenth and seventeenth centuries, such as the Irish Articles or the Hungarian Confession, the consensus of these basic points would still hold.\n\nThird, there are also significant points of divergence. The most obvious are the differences between Lutheran and Reformed over the presence of Christ in the Lord's Supper. This raises two important points: the importance of honest difference, and the particularity of confessional commitment. And these points in turn highlight the importance of always seeing confessions in an ecclesiastical context.\n\nAs to the importance of honest difference, I noted in the first chapter that factors such as fear of excluding somebody often militate against notions of doctrinal precision. The problem, of course, is that the church needs to take a position on certain things. Take baptism, for instance: either it is legitimate to baptize infants or it is not. There is no middle position. Further, one really cannot equivocate on this matter, because the answer one gives has a profound effect on how one understands entry into the church, the Christian life, and the nature of Christian nurture. The same applies to the Lord's Supper: how one understands the Lord's Supper has ramifications for the whole of Christian existence. If the church is the place where Christians receive their nurture and grow together, then there has to be clarity on such issues.\n\nThis leads directly to the particularity of confessionalism. Though we might talk about confessionalism as a principle when we refer to churches that hold to clearly stated doctrinal confessions, such churches always exist particularly. In other words, it is not the fact that they adhere to any confession that is the really important thing; it is the fact that they adhere to a particular confession. This is an important point because of the recent popularity of the term confessional evangelical, a term which I have even used myself.3 The problem with this terminology is that it is typically used today to refer to evangelicals who adhere to what we might call classical mere orthodoxy: an anti-Pelagian Trinitarianism that also upholds the Reformation teaching on justification. There are two problems with calling this confessional evangelicalism.\n\nThe first problem is that confessional evangelicalism is not confessional in the classical sense, the sense in which I have used it in this book. That requires commitment to an elaborate confession of the kind that we find in the sixteenth and seventeenth centuries, where a whole lot more than the Trinity, predestination, and justification are defined. In particular, the sacraments were a key element in the development of classical confessionalism, shaped both by the break with Rome and then the split between Lutheran and Reformed. What we have today in confessional evangelical circles is rather an eclectic pick 'n' mix approach to classical confessional Protestantism, where those matters which seem helpful to building a broad evangelical parachurch consensus are highlighted and those matters which divide\u2014and have always divided Protestants\u2014are set to one side as of less importance. Interestingly enough, one would normally assume that those things which have historically divided Protestants for so long are more than likely to be very important precisely because of the history of division they have fostered. It would seem a wholly arbitrary, and in fact counterintuitive move to build a confessional consensus by denying or ignoring those matters which made confessions necessary in the first place. The use of the term \"confessional evangelicalism,\" pace my own previous use of the same, is misleading. Holding to some or all of the Five Points of Calvinism does not make one confessional. There is a whole lot more to being confessional than that, whether we are talking Reformed, Lutheran, Anglican, or Baptist.\n\nSecond, the matter of being confessional is inextricably bound up with ecclesiastical commitment. This is clearly implied by the comments above, which make it clear that at the heart of classical Protestant confessions lie ecclesiastical distinctives. Yet it goes further than this: confessions are only really confessions when they are adopted and confessed by a church. This requires at a minimum the existence of office-bearers bound by vow to uphold confessional teaching and structures and processes of accountability to ensure that the confession's teaching is what the church actually proclaims. This is consistent with what we noted in chapter 2 concerning New Testament teaching on the church and her confession of the faith.\n\nThus, to say that one is a confessional Christian requires that one also specify to which confession one adheres and in what specific church context one does so. It is an ecclesiastical term, a churchly concept, which only has real meaning in such a context. To use it outside of a churchly context is to use the term equivocally as it implies a very different relationship to a particular confession than that which exists between a church elder or member and the church's doctrinal constitution. Confessional evangelicalism is simply a conservative form of mere Christianity, not the kind of elaborate ecclesiastical Christianity espoused by Luther, Bullinger, Calvin, or Cranmer.\n\nWhile all of this can seem rather negative, in actual fact what the Protestant confessions do is simply make explicit what is practically the case in any given church one might choose to attend. Churches are particular; they have particular beliefs and practices; and confessions give expression to that particularity.\n\n1The astute reader will no doubt see that these represent a somewhat narrow slice of Protestantism. There are traditions not represented here: the Anabaptists and the various Arminian groupings come to mind. I have not omitted these groups on the grounds that they do not have a confessional heritage worth examining but simply because my focus is on the central Reformation confessional traditions (Lutheran and Reformed) and the appropriation of this by others (the Baptists); and the basic principles of how and why confessions were produced can be adequately\u2014and concisely\u2014illustrated with reference to these.\n\n2I am aware that the terms of subscription in each of these denominations are quite different, with the RCA taking a more liberal line, the URCNA being more strictly conservative, and the CRC falling somewhere in the middle. Within the RCA and CRC, however, there are congregations that do adhere more strictly to the standards than their denominations as a whole.\n\n3See The Real Scandal of the Evangelical Mind (Chicago: Moody, 2010).\n\n#\n\n## Confession as Praise\n\nTo many modern evangelical ears, the idea of a confession of faith sounds just too cerebral and propositional to have much to do with the idea of Christian praise and doxology. Indeed, given the way in which confessions are most obviously significant in confessional denominations, that is, as judicial documents for deciding who can belong and who cannot, it is easy even for those who delight in them to forget that doxology or praise is a vital aspect of their function. Indeed, we might go further and say that not only is this doxological dimension crucial to their use today; it is also vital to an understanding of how they came to be formulated in the first place. Historically, one could make the argument that Christian theology as a whole is one long, extended reflection upon the meaning and significance of that most basic doxological declaration, \"Jesus is Lord!\" and thus an attempt to provide a framework for understanding Christian praise. If we fail to make this connection, then our appreciation of the creeds and confessions of the church will be dramatically impoverished as, I would argue, will be our understanding of Christian worship itself.\n\nA moment of reflection indicates why this is the case. The term \"Jesus\" carries with it a vast amount of implicit doctrinal content. Jesus is not Napoleon or Elvis Presley. The word is not simply a contentless cipher into which the reader can pour any significance she wishes. Jesus's identity is highly particular, involving both his personal biography as a specific individual in first-century Palestine and the wider significance of this biography in the context of the history of redemption as laid out in the events recounted in the Old Testament, along with the interpretation of the same offered therein. Then, the whole notion of lordship is not some given or self-evident term; one cannot simply jump from notions of lordship found in contemporary political societies to an understanding of what it means to predicate this of Christ. Rather, the notion is determined by God's revelation of exactly what the lordship of Christ is. Thus, the basic worship cry \"Jesus is Lord!\" is in itself a confession in the sense that it is both a public declaration of praise and a public declaration of doctrinal commitment. Arguably, all of Christian theology is simply one long running commentary upon, or fleshing out of, this short, simple, ecstatic cry.\n\nThe Bible and Confessional Praise\n\nWe have already touched on the significance of Romans 10:9\u201310:\n\nIf you confess with your mouth that Jesus is Lord and believe in your heart that God raised him from the dead, you will be saved. For with the heart one believes and is justified, and with the mouth one confesses and is saved.\n\nThe confessing to which Paul refers here is a public act, and such public acts of confession serve a variety of purposes. There is confessing before the world, the action of witnessing to Christ before the pagan nations. Such is captured by the use of the term witnessing to refer to acts of personal evangelism. The Greek word for such witnessing, of course, lies behind the modern word martyr. Then there is the public and personal affirmation of truth within the church which marks out the true believer from the impostor. Thus, those who deny with their mouths that Jesus is Lord, or who say Jesus is Lord but deny that God raised him from the dead, are not true members of Christ's church, no matter how likeable or pious they might otherwise be. But there is also the further aspect of confession as praise. For Paul, doctrine and doxology are not separated: the truths of the gospel drive him again and again to praise. And, reading his letters, one is struck time after time by the fact that doctrinal statements are clearly uttered in a manner that expresses the sheer delight and joy Paul has in verbalizing such.\n\nPhilippians 2:6\u201311 provides a good example, where Paul, in pressing on his readers the need for humility, describes the mission of Christ using a form which is arguably that of a poem or hymn, and which culminates in the magnificent declaration that \"at the name of Jesus, every knee should bow, in heaven and on earth and under the earth, and every tongue confess that Jesus Christ is Lord, to the glory of God the Father.\" In ending in such a climactic way, this section of the letter is both descriptive, in that it describes what is the end result of Christ's work, and prescriptive, in that it points the reader toward the praise that such truths should evoke. It is also in itself a superb example of what it enjoins: theological confession as doxology.\n\nTo Philippians 2, we might also add 1 Timothy 3:14\u201316, where Paul is talking about wanting to visit Timothy and then breaks out into a hymn of praise, which also constitutes an outline of some key elements of his christology. In this brief passage, he makes a normative statement about God's revelation, the role of the Spirit in Christ's saving work, the witness of angels, the proclamation of the gospel, the resulting faith, and Christ's ascension to glory in the space of a few brief lines; but this is more than just a set of doctrinal propositions\u2014it is also an act of praise. There is no opposition or difference between doctrine and doxology here: the expression of praise is rooted in, and constituted by, an expression of theology. This is a vital point, and we do well to remember that our creeds and confessions are not simply boundary markers but also that they arise out of a desire to praise God, the content of which praise should be the same as that of said creeds and confessions.\n\nEarlier in the same letter, Paul provides a superb example of how polemic, praise, and doctrinal confession can be intimately related. He is writing to his young prot\u00e9g\u00e9, Timothy, encouraging him either to go to Ephesus or to remain there (it is not clear which is the case) in order to refute the false teaching of a group that is having an unfortunate influence within the church. This group has quite probably infiltrated the eldership, since Paul himself excommunicates several of them rather than leaving this to the congregation (1 Tim. 1:20; cf. 1 Cor. 5:5). It is unclear exactly what the content of this false teaching is, but it appears to involve arcane interpretation of the law which has the effect of blunting its basic purpose as that which exposes humanity's sin. Against this, Paul asserts the proper use of the law and then offers himself as an example of God's grace and as a paradigmatic case of the gospel. Then, quite suddenly, he moves from his condition to a statement about the mission of Christ to a doxological outburst:\n\nThe saying is trustworthy and deserving of full acceptance, that Christ Jesus came into the world to save sinners, of whom I am the foremost. But I received mercy for this reason, that in me, as the foremost, Jesus Christ might display his perfect patience as an example to those who were to believe in him for eternal life. To the King of ages, immortal, invisible, the only God, be honor and glory forever and ever. Amen. (1 Tim. 1:15\u201317)\n\nThis passage is a remarkable example of how doctrine, personal testimony, and praise can be wonderfully intertwined in the words spoken by a Christian. There is no opposition here between what Christ has done and what Paul has experienced. More significant from the perspective of this chapter is the connection between theology, polemic, and doxology. In attacking the false teachers, Paul inevitably asserts true teaching as the alternative, but for Paul such assertion can never stand on its own and for its own sake: it moves him inevitably to praise.\n\nYet there is more: the content of this praise, of this doxological statement, is itself highly polemical. To praise God as King of the ages is to deny the claims of anybody else to ultimate kingship: Paul is thus setting the whole of creation within the context of God's sovereignty. To praise God as immortal is to assert that he and he alone is utterly different from everything else in that only he neither comes into being nor passes out of being. To praise him as invisible is to identify him with the God of the Old Testament who could not be seen face-to-face even by Moses and who must not be represented by an image or an idol. To praise him as the only God is to deny the claims of every other thing to which anybody has or will ever ascribe deity. To give him honor and glory is thus to give him what truly belongs to him and to no other. Paul's polemic leads him to praise; yet his praise is itself ineradicably polemical in its assertions. This, of course, is because praise is rooted in, and expressive of, the identity of God. It is thus always going to be doctrinal and always going to be polemical in a fallen world that flees God and prostrates itself before idols.\n\nAll of this reflects the basic conclusion of the argument of the last chapter: that doctrine or dogma is part of the very essence of Christianity. As we noted, statements that posit a gap, or even an opposition, between believing and belonging are fundamentally misleading. Believing is the means of belonging. Thus, to say that belonging precedes believing is to misunderstand exactly what belonging is; and to say that believing is possible without belonging is to attenuate the biblical notion of what exactly it is to believe. In other words, separation of the two concepts in any way produces sentiments which might sound inclusive and aesthetically pleasing but are in fact meaningless gibberish.\n\nIn practice, of course, it can be very tempting to make such a separation. In the Reformed constituency, the accent upon correct and precise doctrine can lead to an intellectualism that separates doctrine from doxology in a manner that is unfortunate and unbiblical. In other branches of the Christian church, an overemphasis on experience or activism or particular aesthetic forms can lead to the relegation of doctrine to a secondary position or even worse. This side of heaven it is unlikely that any church or congregation will ever achieve the perfect balance; but being aware of the problems and pitfalls does help us to be more self-critical and more aware of the potential weaknesses and temptations to which our particular traditions might be peculiarly vulnerable.\n\nThus, if true Christian believing and true Christian belonging are two sides of the same coin, inextricably joined together, then praise that expresses the content of belief is the means by which such belonging is given public expression; and this brings us back to creeds and confessions as being normative guides to Christian doctrine and also, in this context, to the content of Christian worship.\n\nEarly Creeds and Christian Praise\n\nThe worship aspect of creeds and confessions is evident in the early church. The Didache, a document possibly dating from as early as the late first century, gives an account of what should be said in a worship service. Baptism is to be performed in the name of the Father, Son, and Holy Spirit (Didache 7). That is scarcely surprising, given that it is clearly part of the biblical mandate relating to baptism. When it comes to the Lord's Supper, however, a more elaborate liturgical formula is used that carries with it significant doctrinal statements which are more elaborate than the simple biblical words of institution (Didache 9). Most significant in this regard is the prayer of thanksgiving after the meal:\n\nWe give you thanks, Holy Father, for your holy name which you have caused to dwell in our hearts, and for the knowledge and faith and immortality which you have made known to us through Jesus your servant; to you be the glory forever. You, almighty Master, created all things for your name's sake, and gave food and drink to men to enjoy, that they might give you thanks; but to us you have graciously given spiritual food and drink, and eternal life through your servant. Above all, we give you thanks because you are mighty; to you be the glory forever. Remember your church, Lord, to deliver it from all evil and to make it perfect in your love; and gather it, the one that has been sanctified, from the four winds into your kingdom, which you have prepared for it; for yours is the power and glory forever. May grace come, and may this world pass away. Hosanna to the God of David. If anyone is holy, let him come; if anyone is not, let him repent. Maranatha! Amen.\n\nThis liturgical statement is shot through with solid doctrine: God as Creator and sustainer; God as Savior; God as protector; God as the object of praise; God as the one who will gather his church; God as eternally powerful and glorious. There is also an underlying note of historical continuity in the reference to David. This is magnificent theology which bears comparison in content with both the summaries of Christianity in the Bible itself and the Rule of Faith, and weaves all of this into the liturgical action of the church in her praise and worship. Doctrinal statement and doxology are correlative developments within the church and we must not lose sight of the latter in our reflections upon the former.\n\nThe development of Trinitarian and christological discussion was driven in part by the need to give a coherent account of the worship cry \"Jesus is Lord!\" as well as the use of the names Father, Son, and Holy Spirit in that most basic of Christian worship actions, baptism. These points should not be dismissed lightly. Indeed, it is surely most helpful when reflecting on the development of Trinitarian discussions to keep in mind that the matrix out of which such developed was that of the public worship and witness of the church. There is surely no more basic action in worship than the declaration of Christ's lordship; and the Christian life, in all of its practical and doctrinal richness, starts with the simple act of baptism. Indeed, to return to the language of believing and belonging, baptism is the practical avenue to \"belonging\" for all Christians everywhere. Thus, the Trinitarian controversies of the early centuries are nothing if not heated debates about the nature of Christian worship and the nature of Christian belonging. They may well embody at times rarified linguistic distinctions and apparently dry discussion of very fine points of difference, but this should not blind us today to the very practical and doxological orientation of the underlying currents of the debates.\n\nIn light of this background, anyone tempted to criticize the Nicene Creed, or even the Athanasian Creed, for being too abstract, propositional, or polemical, should take note of the comments of John Henry Newman in his Lectures on Justification:\n\nI grant that the Athanasian Creed certainly may be taken by careless readers to imply that orthodoxy is the ultimate end of religion; but surely it will seem otherwise on due consideration. For no one can deny, looking at it as a whole, that it is occupied in glorifying Father, Son, and Holy Ghost, in declaring their infinite perfections; so much so that it has sometimes been considered what it really is in form, a Psalm or Hymn of Praise to the Blessed Trinity, rather than a Creed, as the Te Deum is. Nay, this is its characteristic, not only in its general structure, but in its direct enunciation of the Sacred Mysteries; which is put forth not as an end in itself, but evidently in order to glorify God in His incomprehensible majesty, and to warn us of the danger of thinking of Him without reverence.1\n\nNewman wrote this in the context of his own internal struggles with Anglicanism and his conflict with the religion of his youth. Thus, he probably had in mind the nineteenth-century evangelicals when he wrote this passage, given their deep suspicion of church creeds and their prioritizing of feelings and emotions at the expense of doctrine for the Christian life. Such a separation of doctrine and Christian experience was, arguably, a species of liberalism, in which human religious psychology is definitive of Christianity. We tend not to see evangelicalism in terms of liberalism because of its public adherence to supernaturalism and our tendency to associate liberalism with varying degrees of antisupernaturalism. Yet we must remember that liberalism is not primarily a rejection of the supernatural; it is a reconfiguration of the nature of Christianity in such a way as to highlight religious psychology or experience and downplay or marginalize doctrine. For Newman, unlike the evangelicals in his polemical crosshairs, the identity of God as God is absolutely fundamental, and the praise (and experience) of God by his people arises out of, and is completely dependent upon, that identity.\n\nWe might update Newman here by saying that we should not allow our understanding of creeds to be narrowed and even distorted by a modern cultural aesthetic which prioritizes sentiment over dogma, and which finds distasteful such propositions, claims to absolute truth, and the corollary of rejection of error. Such things were not always regarded as antithetical to doxology and praise as is now the case, a point made with typical clarity by Newman in the above passage. The identity of whom we praise actually informs the content of how we praise him.\n\nIt is perhaps worth noting at this point that, with all of the current debates about the nature and meaning of biblical inerrancy, we must not allow a legitimate desire to maintain the propositional truth content of the Bible to obscure the fact that the Bible cannot be reduced to a collection of truthful propositions. It contains, among other things, promises, commands, praise, lament, and histories, all of which are not important simply for their referentiality but also for the aesthetic forms in which they are expressed. One danger of a vigorous defense of inerrancy is that it can lead (ironically) to an unbiblical prioritizing of one aspect of the Bible over all else. Nevertheless, all of these literary genres and actions rest upon who God is and what he has done and promised to do. The identity of God is foundational to, and constitutive of, the content and forms of biblical revelation and that is foundational to, and formative of, the Christian response in praise and worship.\n\nThis point cannot be stressed enough with regard to doxology: the identity of God has priority over the content of Christian praise. We see this in the Psalms, where who God is and what he has done permeate their language. There are churches (most notably the Reformed Presbyterian Church) that restrict their public praise to the singing of nothing but canonical Psalms precisely because they take very seriously the need for correct forms and content of congregational address to God; the use of hymns of human composition always runs the risk of introducing ideas into the mix that are not biblical at all.2 True praise of God arises out of an accurate understanding of who he is and involves correct statement of what he has done and does do, in terms of doctrinal content and form of speech. We know that it is legitimate to ascribe glory to God because Scripture teaches that God is the one to whom glory is to be ascribed and offers a picture of the believer's life in which such acts of ascription are a normal part of existence. That is the biblical paradigm that we see in the Psalms and in the many doxological outbursts of Paul and others. A faulty understanding of God and how to respond to him can only lead to praise that is to some extent inadequate in both motivation and expression.\n\nAn analogy with human relationships seems apposite: if I approach my wife, all four feet nine-and-a-half inches of her, and treat her as six feet tall, asking her to get the rice from the top shelf in the cupboard, then it is arguable that, however good my intentions in such a request might be, there is some level of dysfunction or delusion in our relationship. I can only relate to my wife correctly as I understand who she is. Likewise, she was born in the 1960s and trained as an elementary school teacher; thus, if I thank and praise her for successfully winning the Second World War all by herself, my comments may be flattering in the extreme, but they are absurd, ridiculous, and out of touch with reality. Such would not be true praise at all: it would be a false action, whether I was aware of its falsity or not. There is a necessary connection between truth and praise.\n\nThis is where creeds can play such a vital role in a church's praise, a point that many communions have recognized throughout their history in their very acts of public worship. Thus, both the Apostles' and the Nicene creeds have often found a place within the historic liturgies of the church, as things either to be said or to be sung by the congregation as part of its gathered worship. Some might criticize this action for stressing doctrine at the expense of some nebulous concept of \"worship,\" but that would be to misunderstand the nature of the connection between doctrinal statement and doxology. In reciting the creeds, the purpose is not simply to declare a set of propositional truths. Rather, the action is somewhat richer than that: to state the obvious, in reciting the words of the creeds together, each member of the congregation publicly identifies with every other member in expressing a corporate unity of belief in a common gospel. They are also expressing their common belief with every other Christian throughout history who has used these words to witness to Christ. Further, they are reminding themselves and each other of who God is and what he has done. In other words, the creeds, in liturgical context, become a means of fulfilling the public declaration that Romans 10 demands of believers: the confession (a document) becomes a confession (an act of pointing toward Christ before the church and the world).\n\nThe Creeds and Trinitarian Worship\n\nIn this context it is worth particularly noting one thing which the ancient creeds do and which is often a weakness in contemporary church life: they highlight the fact that God is Trinity. One might ask why this is important. Is the Trinity not a somewhat complicated and arcane idea, the fruit of the metaphysical wranglings of churchmen in the early centuries and not something that connects to modern life? As noted above, this is not the case: the Trinitarian discussions find their origins in the realm of the church's doxology, and the creeds are in part the product of this. Further, the Christian life is Trinitarian by its very nature. As God the Son is the One sent by the Father and empowered by the Holy Spirit to carry out his work as Mediator, thus salvation is deeply rooted in the Trinity and has what one might call a definite Trinitarian shape. The facts of the gospel are necessarily Trinitarian facts.\n\nFurther, the identity of the Christian as the one united to Christ gives each individual Christian a Trinitarian identity: united to Christ by the Spirit, we enjoy communion with God as our Father. The answer to \"Who are you?\" when asked of a believer must thus elicit a Trinitarian response. Further, what applies to the individual believer applies even more strongly to the church as a body. Paul uses various terms for the church: the body of Christ, the temple of the Holy Spirit, the household of God. Each of these carries with it obvious Trinitarian weight, for neither Christ, the Spirit, nor God can be adequately conceived of without being understood as Trinitarian, first, last, and always.\n\nIn addition, we should remember that the very initiation rite of the Christian church, baptism, is itself Trinitarian in form. As Christ's ministry starts with the Father acknowledging the Son and anointing him with the Holy Spirit, so the Great Commission specifically requires baptism in the name of the Father, Son, and Holy Spirit. The Christian's identity is nothing if it is not Trinitarian.\n\nGiven this, the Trinity should be a doctrine that both shapes our worship and pervades our worship. Sadly, in many churches this is not the case. There may be an official statement somewhere which indicates that the congregation is committed to Trinitarian doctrine, but all too often there is little sign of this commitment in what actually happens during a worship service. Indeed, having attended the funeral of a Unitarian friend a few years ago, I was struck at how much of what passes for Trinitarian Christian worship would have seemed entirely consonant with the Unitarian service which I witnessed that day.\n\nThere is no silver bullet that fixes this problem. The pastor and elders of each church need to be very intentional in how they integrate the identity of the Trinitarian God into the worship service. Trinitarian doctrine can be hard for people to grasp. The Trinity is revealed in Scripture, but it is also something that cannot be comprehended by finite creatures. It is apprehended and believed but it is not fully understood. Thus the teaching of the doctrine needs careful thought and preparation. Elders need to choose hymns and songs that reflect good biblical, Trinitarian theology; they need to make sure that the public prayer is explicitly Trinitarian in its content; they need to work hard at ensuring that the overall liturgical shape of the service does justice to each member of the Godhead; and the minister in particular needs to make sure that his preaching involves careful and clear articulation of God as Trinity.\n\nMost of us will tend, if we are not careful, to emphasize either the unity of God or the threeness of God and thus present an imbalanced view of his being; we thus need to be very self-conscious in our Trinitarianism. This is where, I would suggest, the input of good elders becomes crucial. Listening to the minister each week, the elders should be well-qualified to note imbalances in the theology expressed in the public worship of God.\n\nOne other obvious way we can press this issue home, of course, is to use the creeds in our liturgy. By reciting the Nicene Creed together on a Sunday, we remind each other of the identity of God in Trinitarian terms. It is a form of sound words, and the corporate recitation of the same reinforces it in our minds. Of course, such recitation is not in itself enough: it needs to be connected to clear teaching. If the standard level of what is done in a worship service is set at that which the newest, least informed Christian can understand, we are doomed to remain forever in spiritual infancy. As Christians, we should expect worship always to be a learning experience. That requires us not only to call ministers who are able to stretch us theologically; it also means we should fill the worship service with material that draws us on to maturity. The creed is one such thing: it takes a few minutes to memorize and recite but a lifetime to master. Thus the rest of the worship service must connect to, expound, and reinforce its teaching. Some churches will no doubt do this better than others; but even the most theologically impoverished church that uses the creed in its worship can rest assured of one thing\u2014its worship is distinctively different, distinctively Christian, when compared to anything that happens in a Unitarian service.\n\nCreeds, Liturgy, and Formalism\n\nIf recitation of the creed does not itself guarantee that the congregation will understand what they are saying, some might object to this practice on grounds that are often lodged at liturgy in more general terms: the use of liturgy leads to a mere formalism and outward show that is simply going through the motions of praise without ever actually engaging the hearts and the minds of the congregants.\n\nOf course, the most obvious riposte to this objection is that the Bible itself contains liturgies or set prayers. The Psalms and the Lord's Prayer are only the 151 most obvious examples of such; there are others if you find that number to be too few to build a persuasive case for liturgy upon. Yet even if, for the sake of argument, we were to bowdlerize the Bible or pretend that these things did not exist, we could still make a compelling case for set forms in worship.\n\nIn a sense, criticism that claims that forms lead to formalism is vulnerable to the same kind of arguments I have already made against the \"No creed but the Bible!\" brigade. All Christian churches have liturgies in the same way that all Christian churches have creeds. Tell me the kind of church you attend and I can probably make a good guess about what shape the service will take and what kind of language will typically be used, however \"spontaneous\" the church may like to think it is. The only real point of difference between churches on this issue is the level of self-consciousness and explicit formality with which they are held. Few, if any, churches have completely anarchic services. A High Anglican from Oxford, UK, attending a snake-handling church in Alabama may think that what she sees is anarchy, but the likelihood is that this Sunday's service actually looks very much like last week's. The lack of explicitly stated forms does not mean that the same basic routine is not followed, week in, week out.\n\nI have sometimes heard people from other churches criticize the structure of the service in my local church for not being \"spontaneous\" enough. Now, spontaneity is an interesting category. First, it is not immediately obvious to me where one would go in the New Testament to find it as a hallmark of genuine Christian worship. Nowhere does Paul tell people they need to be more spontaneous, or commend them for being so; on the contrary, his strictures in 1 Corinthians seem to arise out of a concern that the church there is not, to use his phrase, doing things decently and in order. The danger with spontaneity, one would imagine, is that it could very soon become just a byword for chaos and anarchy.\n\nSecond, it often appears that people use the term not to mean spontaneous in the sense of off-the-cuff, ad hoc, or improvised, but rather to refer to nontraditional liturgical structures and hymnody. Thus, a service with contemporary praise-band music, where nobody but the worship leader is entirely sure how many times the chorus or song is to be repeated, is deemed to be \"spontaneous.\" Yet, when one reflects upon this scenario for a moment, this is surely not really all that \"spontaneous\" after all. It is preplanned by somebody and, unless everybody is singing different words to different notes (not an entirely unprecedented event in the history of local churches, I am sure), it is also using set forms of some kind. Call them hymns, choruses, songs, tunes, or melodies: their form and content is fixed in advance of the spontaneity of the service.\n\nA pastor friend once told me of a charismatic church in his neighborhood that had invited him to attend a service. The worship leader afterward asked him if he had noticed how spontaneous the worship was compared to the comparatively staid worship at my friend's Welsh Baptist Chapel. My friend asked the charismatic leader when he typically decided on what songs to sing on a Sunday. By the previous Thursday at the latest, was the response, so as to allow the music team time to practice. My friend responded that he never told his organist what hymns he was going to use until the Saturday afternoon, and that he was thus \"forty-eight hours more spontaneous\" than the charismatics. It is a simple fact: if you use a song book or pre-prepared overheads or PowerPoint to guide the worship at your church, the level of true \"spontaneity\" is obviously somewhat limited. Add to the mix somebody actually leading worship\u2014whether formally or informally\u2014and again, spontaneity is more of an appearance than a substantial reality.\n\nThus, as just noted, the New Testament does not actually seem to put any premium on spontaneity as an essential component of authenticity, and so I would suggest that we should not worry too much about either. It is more likely that it is significant for us because of our culturally-conditioned understanding of what makes us and our actions \"authentic\" (itself as slippery a concept as \"spontaneity\"). Modern society puts great stock in the idea that what makes us who we are is our individual self-creation and self-determination. That lies at the heart of consumerism, for example. Yet, whether right or wrong socially, it is certainly irrelevant biblically when it comes to worship. Individual self-creation and self-determination are not only irrelevant to Christianity and its various practices, they are arguably antithetical to it as well. It is not what distinguishes me from my fellow human beings which is really significant; it is what unites me to them. I am made in the image of God; I am fallen and sinful and in need of a redeemer; and my salvation is found only in and through the work of the Lord Jesus Christ. Those are truths that apply to me as to everybody else. And my response in worship, whatever particular culture I belong to, must reflect those commonly shared realities. That is why a common confession in a creed is a good thing: it makes the point that my faith is the faith of the other people in the church\u2014both today and throughout the ages.\n\nOf course, one must immediately concede that the recitation of a creed can become a mere outward formality. There is no point in denying such an obvious point. Yet creeds are not unique in this: in much the same way, the singing of a familiar hymn or praise song can easily be a mere outward, mechanical act as well. Even our extemporary prayers can be formalities. Think of how you pray in public, and reflect upon how similar many, if not all, of those prayers are, as you draw upon the same kind of vocabulary and idioms to give expression to your thoughts and longings. How much real spontaneity is there even in our spontaneous prayers? The difference between the contemporary praise songs we sing and the prayers we pray and the creeds we recite is this: hymns are usually the product of a single person; our prayers are the products of our own religious self-consciousness; but the great creeds of the church are corporate products which have been tried and tested by the church across the world and down through the centuries. They carry the authority of the ages behind them. Of course, that does not mean they stand above or even on a par with the Bible in terms of their authority, but it does mean that they are still most significant church documents. Further, the long pedigree that the ancient creeds in particular have as liturgical documents, to be used by the corporate church in her acts of public praise, should also encourage us to think very carefully about how we might tap into such a rich stream of Christian practice.\n\nThus, let us make sure that when we criticize the recitation of creeds for its formalism we lay the blame where it truly belongs. If such recitation is mere formalism, it is not the fault of the creeds themselves. They are no more to blame for being used in a merely formal manner than Shakespeare can be blamed if people use copies of his plays to blow their noses. Nor is this formalism the fault of the church that establishes such use of the creeds in its worship services. Every church has its liturgy; that in itself does not determine whether the liturgy will be used well or badly. Any set form of words\u2014from a hymn book to a prayer book to a creed\u2014is vulnerable to formalism. Thus, if the creed is read in a merely formal manner, it is the fault of the congregation or of the individual who does so, in the same way that it would be if a hymn or psalm or chorus were sung in the same manner. The use of an established form of sound words is no more vulnerable to formalism than the use of speciously spontaneous alternatives. Creeds are merely a tool for achieving a desired end; it is up to the elders of the church to make sure that their use does not degenerate into mere empty recitation.\n\nThe Threefold Aspect of Creedal Doxology\n\nIn the Anglican Book of Common Prayer communion service, the corporate recitation of the Nicene Creed takes place right after the minister has read from the Epistles and the Gospels and immediately before the sermon or the homily. The Word of God has been read; the Word of God is about to be preached; and here, in the moment that bridges the two points, the congregation express their corporate faith in who God is and what he has done. It is a very important part of the service, as it involves the whole of the body of Christ in a positive, active capacity.\n\nWe might characterize this corporate action as having a threefold significance. It has a significance at the congregational level whereby all the members remind each other of the identity of God. It has a significance at the level of the congregation's relationship to the broader culture in that, like Paul's doxology in 1 Timothy 1, the church as a gathered body is explicitly, publicly, and defiantly denying the claims of all other pretenders to the divine throne. And it has a significance in terms of God in that it represents the ascribing to him of that glory and honor which is his alone. Further, we might conclude by saying that this one action\u2014reciting the creed\u2014is actually all three of these at one and the same time and that the distinctions we draw between them are entirely formal, since each one inevitably implies or involves both of the others. It is a stunning act of countercultural rebellion on behalf of God. Each of these three points is therefore worth exploring.\n\nCreeds Offer a Corporate Summary of the Bible's Teaching\n\nOne of the great complaints of ministers and elders today is the comparative theological and biblical illiteracy of congregants compared to previous generations. We live at a time where levels of basic literacy (the simple ability to read) and accessibility of reading matter (whether printed or \"virtual\") are at historically high levels. Yet we all know that people seem to read less and are certainly less familiar not only with the Bible story as a whole but even with individual stories within Scripture. Part of the reason for this, at least in the West, is that biblical stories no longer pervade the wider culture as they once did. One pressing pastoral concern, therefore, is how churches should address such lack.\n\nThe issue of basic biblical literacy in terms of familiarity with particular Bible stories obviously needs to be addressed from the pulpit, in Sunday school classes, in small group meetings during the week, and through the cultivation of habits of family and private devotions. Two short Bible readings and one thirty-minute sermon each Sunday will not solve the problem.\n\nThe other issue\u2014how to give congregants a grasp of the overall message of the Bible and the gospel\u2014must also be addressed using these same means; but the risk is that each of these approaches can still fail to give an overarching framework. The current habit of expository preaching, where a book is preached passage by passage, is perhaps something of an improvement on the older habit of preaching verse by verse, in that it does allow for more of the Bible to be covered in the space of a year's worth of sermons; but, even so, it can end up offering a rather narrow slice of the Bible's teaching and leave people vulnerable to developing a fragmented or fundamentally unbalanced theology. There is an obvious need for a helpful framework as a basic part of theological education at the very outset of the task and at every step along the way.\n\nThe recitation of a classic creed can be of immense help here. A creed\u2014whether the Apostles' or the Nicene\u2014provides in succinct form a clear statement of the identity of God and of Jesus Christ and a summary of the key points of Christ's work. God is Creator; God is Trinity; Christ is incarnate God; he lived, died, rose again, and is ascended into heaven; he will come again in judgment; there is a divine creation called the church; there is an initiation rite called baptism; there is forgiveness of sins; there is eternal life. As the congregation recites this each week, they are surely learning, or being reminded of, the cardinal doctrines of the Christian faith, and that is a vital part of church life. As Paul notes in 1 Timothy 1, it is these simple truths that are the foundation of a pure heart, a good conscience, and a sincere faith; and these in turn are the foundation of that which is the purpose of good teaching: love and the good stewardship of the things of God.\n\nOf course, these creeds are not exhaustive summaries\u2014what \"summary\" ever can be such?\u2014but they do offer neat, concise frameworks within which the preacher's detailed expositions can be placed. The person who knows the creed knows the basic plotline of the Bible and thus has a potentially profound grasp of theology. In a world where the exigencies of employment have made a high level of transience and fluidity in congregations into something which is the norm rather the exception, it becomes ever more imperative that ministers and elders think very self-consciously about how they can ensure the proper education of the congregants.\n\nThere is also a useful connection to more elaborate, confessional material here. Catechisms in particular can be used in the worship service to ensure a congregation is exposed to the sweep of Christian doctrine. This can take two forms. As noted in chapter 4, the tradition of the Dutch churches from the sixteenth century onward was to devote the second service on a Sunday to a sermon based on questions in the Heidelberg Catechism, and to work through the whole document systematically in the course of a year. While this did lead to laziness and tedious repetition in some instances, the principle of covering the key catechetical doctrines in a fairly short space of time is surely a good one. In the present time, with highly mobile congregations and the lack of general theological and biblical literacy upon which to build, the idea of allowing a catechism to guide topical preaching has much to commend it: as a fixed scheme, it prevents the preacher from merely dealing with those areas about which he is peculiarly passionate, and it allows him and his fellow elders to know which doctrines have been specifically addressed from the pulpit and when. Of course, some congregations may not like the idea of preaching on the catechism rather than the Bible, but because that is really a false dichotomy, there should be no real tension here. Frankly, if people object to the idea of catechetical preaching, the minister should simply do it by choosing biblical passages of relevance to the doctrine of the day and not create problems for himself by telling his congregants that he is preaching on the catechism. Pointing them at each sermon's end to a relevant section of a catechism might gently introduce the church to the idea of the catechism's usefulness. Such an act could well reap dividends in terms of the thoroughness of doctrinal coverage that is thereby facilitated.\n\nThe other way in which catechisms can be useful in a worship service is to use them as part of the liturgical action. I have been at a number of church services where a question from a catechism was read by the minister and the response of the congregation was the relevant answer. This has a number of points to commend it. First, it familiarizes the congregation with the chosen catechism and fulfills the idea of instilling a form of sound words into the minds of the people of God. Second, when done thoughtfully it can be an obvious element in the drama of the service: a catechetical passage on the law is appropriate before a prayer of confession; a passage on the gospel as part of the words of assurance; a passage on the doctrine of God or Christ or salvation as a basic confession of faith and offering of praise to God, etc. Catechisms are only dead and dry documents if one chooses to make them so. Given that a good catechism bursts with beautiful statements about who God is and what he has done, with a little thought and care they can enrich the dramatic action that should characterize the Christian worship service.\n\nCreeds Are Countercultural\n\nThe recitation of a creed in a worship service is one of the most countercultural things that Christians can do. It is an act of defiance, if not even of actual revolution. First of all, of course, we must be persuaded that worship is meant to be countercultural: the seeker-sensitive services of the eighties and nineties, the more recent passion for offering traditional, contemporary, and even jazz worship as options in ostensibly the same church, and the tattooed chic and candles of emergent eclecticism are all ultimately predicated on the notion that worship services are not meant to be countercultural so much as contexts where the idioms of the wider culture are co-opted and Christianized. Sometimes this assimilation to the wider culture is arguably only at the level of aesthetics, but the underlying emphasis in each case is not on countercultural protest but, on some level, of rapprochement between church and wider context. In fact, this type of approach has no real foundation in Scripture. It is obvious that worship services must have some connection to the wider culture (language, location, etc.), but no biblical writer expends any real energy reflecting on contextualization. It would seem that such matters have produced a veritable industry in the modern church world but were very low on the list of the New Testament writer's priorities, possibly because they regarded them as of minimal importance and matters of mere common sense.\n\nInstead, the one description of an unbeliever's reaction to a Christian worship service occurs in 1 Corinthians 14:23\u201325. Here a properly ordered service has the result of causing the unbeliever to be convicted and to fall on his face to worship God and to declare that God is indeed truly present there. There is no evidence here of any attempt to make the unbeliever comfortable, and the reaction would seem to indicate that, in fact, the exact opposite is the case: the church is simply being the church. Its one concession is that it is worshiping in a public language accessible to the unbeliever, and the rest, as they say, is history. It is clearly not the similarity of the church to the world that is the key to this drama; it is the difference that the unbeliever finds so striking.\n\nThus, worship is not defined by its proximity or assimilation to the wider context. Instead, worship from its earliest manifestation in New Testament times has been marked by protest against the wider culture. That is why Christians have often been persecuted throughout history. You do not find yourself being persecuted unless you are a threat to others; and you are not a threat to others if you are basically in agreement with them or if you contextualize yourself in such a way as to be indistinguishable from the world around you. Christians in first-century Rome and twenty-first-century China were\/are persecuted because they represented something threatening, strange, incomprehensible, and unassimilable to the dominant powers of the day.\n\nThis aspect of Christian worship finds expression in various forms. The public reading of God's Word is one. God's Word comes to us from outside, confronts us with God's revelation, and challenges all human attempts to reach him by human effort or to remake him in human image. When the Word is read in the congregation, the claims of the world (whether the wider world \"out there\" or the inner world within ourselves) are repudiated and the claims of God are asserted in opposition to them.\n\nSinging praise to God is another area of countercultural rebellion in the worship service. The world around us cries out to have our worship, the devotion of our hearts, to be praised by us for what it is and what it can do. Singing praise to God is denying praise to the world and thus denying the world's claims upon us.\n\nCorporate reading of a creed or confession is a third aspect of this kind of rebellion. Because the great creeds and confessions summarize so wonderfully important aspects of the Bible's teaching, not least the sovereign kingship of God, which relativizes the claims to kingship of all creatures, their recitation is an act of defiance and an insult to creaturely arrivistes. As soon as the congregation says \"We believe in one God . . .\" all other pretenders to the divine throne have been put well and truly in their place. Neither sex nor money nor power is God; there is only one God, the God whom the creed proceeds to describe.\n\nFar from being a staid piece of outmoded traditionalism, such a corporate action is a devastating blow against the cultural conformity that demands the church be just like the world, accept the same criteria of relevance, truth, and aesthetics as the world, and offer a gospel that accommodates at least some of the claims of the world. The recitation of a creed makes it very clear that, whatever the attitude of heart of any individual church member, the church as a whole looks to God as king, not some creaturely pretender.\n\nCreeds Ascribe to God What Belongs to Him and Him Alone\n\nThird, and more briefly as it is basically part of the first two points, creeds ascribe to God what belongs to him and to him alone. It is of the nature of us as fallen human beings to forget who God is, to remake him in our own image, and to domesticate him in such a way that he conforms to the limited dimensions of our own imaginations. We go to church each week in part to be reminded by that Word which comes from outside of us who God is, what he has done, and what he will do. The corporate recitation of a creed forces us to engage in the positive action of ascribing to him that which is his: the glories of his nature; the marvelous details of his actions; and the great promise of the future consummation of the kingdom. That is worship: giving to God what is his.\n\nIn this context, it is hard to understand why any church that uses hymns or choruses or any song that has been written beforehand and is sung in unison by everyone present would object to reciting a creed. Hymn singing is only corporate recitation set to music; and no hymns of which I am aware consistently contain as much sound theology per clause as the Apostles' or Nicene Creeds. Is it thus the lack of musical accompaniment which makes churches wary of the use of creeds? Well, there are musical arrangements for such, though, ironically, use of such is often suspected by anticreedal churches of indicating Roman Catholic tendencies. So, if it is not the music, what is it? It cannot be an objection to the words; in all likelihood, it is simply a judgment based on taste, whereby the pastor or elders or congregation do not like to recite creeds because it just feels a bit too Romish for their liking. Such an objection is unworthy of the expenditure of time to refute any further than I have already done in these chapters.\n\nConclusion\n\nThose who object to the use of creedal material in Christian praise typically do so for a variety of reasons: forms of words lead to mere formalism in worship; preaching on a catechism is not preaching on the Bible; speaking human words in the worship service supplants the unique authority of Scripture.\n\nAs we have noted, these are all specious objections. All churches have forms of words which they use, typically hymns or choruses. Preaching on a catechism simply provides a framework for making sure that all of the major biblical doctrines are covered within a set period of time. As to speaking human words in a worship service, virtually every word spoken there is a human word, with the exception of the passages of Scripture that are read out loud (and even then, these are typically translations of God's word, not God's word in and of itself).\n\nIn addition, I would argue that if one takes Scripture seriously and sees it as regulating both the form, content, and purpose of Christian praise, then it is hard to see why creedal material should not be included in the service. Scripture uses forms, of which the Psalms are the most obvious. The content of praise is an accurate account of who God is and what he has done. The purpose of praise is to ascribe to him what belongs to him alone. It is hard to see how this can be done more effectively and indeed concisely than through the use of creeds in the worship service. They are no more human intrusions into worship than are extemporary prayers. The one big difference is that an orthodox creed contains no unpleasant surprises in the form of heterodox or even heretical theology, embarrassing phraseology, inappropriate childishness, or personal idiosyncrasies and obsessions. Tried and tested over the years, the best creeds contain solid theology clearly expressed in appropriate language. The question is not so much \"Should we use them?\" as \"Why would we not use them?\" They do nothing but ensure that biblical content and priorities are kept uppermost in the public worship of the church.\n\n1John Henry Newman, Lectures on Justification (London: J. G. and F. Rivington, 1838), 362.\n\n2I should note here that I am not personally an exclusive psalmodist, though I have strong pastoral and aesthetic sympathies for the position. I also share the concern that nothing should be said in praise which is inappropriate and that the most obvious way to do this is to sing good translations of the Psalms.\n\n#\n\n## On the Usefulness of Creeds and Confessions\n\nThe main burden of this book thus far has been to argue that creeds and confessions are not simply consistent with biblical teaching but that their existence and use are even strongly implied by the same; and also that the history of the church demonstrates that they have frequently been of great help in the maintenance and propagation of the Christian faith. Now, in this last chapter, I want to conclude by listing a series of further advantages that the church can enjoy if she gives creeds and confessions their proper place in her daily life. The list is not exhaustive, and many readers may well think of others they might wish to add or, perhaps, of ones I list which they would exclude on the grounds that some other thing might do the task even more effectively. Thus, I offer this as no definitive list, nor do I present these ideas in order of priority, preference, or importance. I merely hope that they will stimulate the reader to further constructive reflection upon the creedal imperative of the church.\n\nAll Churches and All Christians Have Creeds and Confessions\n\nThe first point I want to make in this concluding chapter is one that has been mentioned on a couple of occasions throughout the book and has perhaps frequently lurked below the surface elsewhere: all churches and all Christians have a creed or a confession. What I mean by this is that no church or Christian simply believes the Bible. The Bible in itself is a collection of various genres of literature. I believe it ultimately communicates a coherent message, but no Christian, if asked by a friend what the Bible teaches, is simply going to start reading aloud at Genesis 1:1 and not stop until Revelation 22:21. Instead, when asked by friends what the Bible teaches, we all try to offer a synthesis, a summary of what the Bible says. And as we move from biblical text to theological statement, we offer what is, in terms of content, something akin to a creed or confession. Then, if we reflect honestly on how we read the Bible, we will acknowledge that what we think the Bible teaches as a whole will shape how we understand individual verses, chapters, and books.\n\nGiven this, we need not take too seriously those who claim to have no creed but the Bible. If this is intended in the sense that \"we have no ultimate authority for norming theological statements other than the Bible,\" then conservative Protestant Christians would generally all agree. If, however, it is intended in the sense that I have no understanding of the Bible other than the Bible itself, then that is highly misleading. The character I mentioned in the opening paragraphs of the introduction claimed no creed but the Bible, yet he was dispensationalist in eschatology, Calvinist in soteriology, and brethren in ecclesiology. What he really should have said was: I have a creed but I am not going to write it down, so you cannot critique it; and I am going to identify my creed so closely with the Bible that I am not going to be able to critique it either.\n\nThere are numerous obvious ironies here, not least that last point. It is probable this person objected to creeds on the grounds that they represent a man-made framework which was imposed upon the Bible by the church and thus distorted how the Bible was read. In fact, by refusing to acknowledge even the existence of his own framework, he removed any possibility of assessing that framework in the light of Scripture. Thus, he invested more absolute authority in his private creed and his tradition than even the Roman Catholic Church or the Eastern Orthodox, who at least have the decency to put their confessional standards into the public domain.\n\nThe standard evangelical objection to creeds and confessions is simply not sustainable in the light of its own self-referential incoherence, the Bible's own teaching, and the history of the church. I argued in an earlier chapter that creeds and confessions actually fulfill a vital role in a function that Paul makes an imperative for the church and her leadership, that of the stable transmission of the gospel from one generation to another. Thus, if you take the Bible seriously, you will either have a creed or a confession or something that fulfills the same basic role, such as a statement of faith. Here, I want to make the point that those who repudiate such ideas are being unintentionally disingenuous: they still have their creed or confession; they just will not write it down and allow you to look at it and scrutinize it in the light of Scripture. They are in a sense more authoritarian than the papacy.\n\nIn fact, a church which is open about its confessional position is, in theory at least, better able to do justice to the supreme authority of Scripture. First, to repeat, the use of a confession is consistent with the Bible's own teaching and actually addresses a matter that the Bible makes an imperative. Second, once the creed or confession is in the public domain, mechanisms can be put in place to allow for it to function in a subordinate role to Scripture.1\n\nTo achieve this, a confession is not enough. The church also needs mechanisms to ensure that, on the one hand, the confession does not become an unassailable idol and, on the other hand, that it is not subject to the arbitrary wild interpretation. No system can do this perfectly: the church, by definition, is made up of sinful failures and that capacity for failure and sin continues throughout our church lives. Yet a church that is open about its confessional commitments and that strives to maintain a structure of governance which reflects biblical concepts of eldership is inevitably better placed to negotiate the relationship between Scripture and confession than the church which lacks these things.\n\nIn confessional Presbyterianism, the church typically requires all office-bearers to profess belief in the system of doctrine as expressed in the Westminster Standards, to uphold the teaching thereof, and to register any change of mind with the relevant body.2 The purpose of this is to ensure that the church knows what is being taught from her pulpits and promulgated by her office-bearers. Thus, for example, if a minister in a Presbyterian church were to become convinced that infant baptism was not biblical, he would be required to report that change to his presbytery and would be asked to demit the office. This does not mean that the church no longer considers him to be a Christian, but simply that, while she respects his conscience on this matter, integrity requires that she no longer allow him to hold office within the ranks of a Presbyterian church. The church's position and the action relative to the minister who has changed his mind would both be matters of public record. Nobody could claim that the church had acted inappropriately. The same, of course, would be true in a Baptist church where the minister became convinced of paedobaptism. It would scarcely be unfair for the elder board to ask him to step down. What would be disingenuous in both cases would be for a church to claim to hold to a certain confessional position on baptism and yet allow someone taking the opposite view to hold ministerial office.\n\nImagine, however, a church that has \"no creed but the Bible,\" where the minister one week is convinced that baptism should be restricted only to professing believers and the next week changes his mind and thinks babies can be baptized too. Can he be held to account? There would seem to be no way of doing this; in practice, whatever he thinks is the truth on any given matter at any given moment\u2014that is the position of his church. This is surely a recipe for chaos in that it places the congregation completely at the mercy of whatever the current opinion of the pastor might be. He has, in theory, unlimited power, and the Bible would seem to mean whatever he decides that it means.\n\nImagine, too, a church whose confession teaches that salvation comes about by the free will of human beings and that God only looks on hopefully, trying to establish favorable conditions for conversions but having no decisive say in the matter. Let us say that a majority of her people and office-bearers at some point come to the conclusion that this is incorrect, that the confessional document to which they hold needs to be revised on this point. A church with an established process by which this change can be accomplished can do this in a manner that is public, transparent, and which involves wrestling with Scripture's teaching in a corporate context. Such a procedure would not simply allow the church's ministers to stand up one Sunday and teach whatever they wanted on the topic.3 That would lead to anarchy and confusion. Yet a church where the minister \"has no creed but the Bible\" can have no transparent mechanism for effecting the change. What the minister believes and what the Bible teaches are functionally identified. If he thinks it teaches Pelagianism one Sunday and Calvinism the next, who is to contradict him and how could they do so? Again, the congregation is at the mercy of the minister.\n\nThus, one obvious advantage of having an open, public confession is that it makes transparent that which is practically hidden by evangelical claims to having no creed but the Bible: everybody has a creed; the only difference is whether you are prepared to be honest and open about that fact. Further, only once you have acknowledged this and made your creed public can you then put into place a system that connects your church's confession to Scripture and to the church's government in a way that gives your church, her leadership, and her people a way of making sure that the confession stays subordinate to Scripture in a transparent, orderly, and public way. Ironically, it is not the confessionalists but the \"no creed but the Bible\" people who exalt their creeds above Scripture.\n\nConfessions Delimit the Power of the Church\n\nClosely related to the last point, and one that is often missed, is that confessions serve to delimit the power of the church and of her office-bearers. This is somewhat counterintuitive in an age which is typically very suspicious of anything that smacks of institutional authority, and where there is a church culture that often sees creedal documents as blunt instruments for excluding some and manipulating others. Yet this is possibly one of the most important functions confessional documents can fulfill.\n\nWe have noted two important aspects of eldership: doctrinal competence and authority. The two things are linked, and it is clear that, practically speaking, the nature of that linkage is crucial. Doctrinal competence without authority renders the office impotent and prevents the elders from leading a congregation toward spiritual maturity. Authority without doctrinal competence, however, is a recipe for willful despotism, where the church is whatever the elders decide, no more and no less. Indeed, the history of religion is full of sad stories of church leaders who used their power over their followers to perpetrate terrible tragedies. From M\u00fcnster in the Reformation to Jonestown and beyond, charismatic figures have used religion to control and manipulate people. Certainty in religion can easily lead to disaster; thus, it is crucial for churches to reflect upon exactly what it is that they are and what powers they do and do not have.\n\nIn order to establish church power within appropriate limits, several things need to be in place. First, there needs to be a clear understanding of what the church is. Second, and flowing from the first, there needs to be a statement of the church's beliefs, that is, a confession of faith. Third, there needs to be a set of procedures that articulate and define how the confession of faith is to be practically applied within the congregation. It is the role of the first and second of these which are obviously of concern to us here.\n\nIn any institution or organization with a hierarchical structure, central to its well-being will be the organization's own understanding of its purpose, and the acceptance of this by its members or employees. For example, I worked for some years as Academic Dean at a theological seminary. This particular seminary has three stated purposes: to train men for ordained ministry; to train men and women for nonordained leadership roles within the broader Christian world; and to train potential future scholars. If I thus entered the classroom one day and started teaching my students about one of my personal passions\u2014say, long-distance running or how to watch a rugby game or the Tour de France\u2014my students would be rightfully displeased because they have paid the institution money in order to be trained by me in church history, not sporting fixtures. To lodge their complaint successfully, the students would need only to point to the seminary's own description of its mission in order to establish that something had gone wrong in the Trueman classroom.\n\nA confession functions in an analogous way for the church: it describes the message which the church is to preach, and it limits the church's power to what is contained within that document. Take, for example, a minister who decides that the Bible teaches that all Christians should wear clothes of a certain style. Such a case might be bizarre, but how would the church where the minister has \"no creed but the Bible\" handle this situation? Hermeneutical issues and church power issues would combine in a most awkward manner.\n\nOf course, while certain churches do seem to encourage a certain aesthetic when it comes to dress, there are probably very few where the eldership engage in an explicit and high-handed approach to congregants' fashion sense. More likely in the current climate will be an eldership that issues edicts about where one should send one's children to school, for whom one should vote, whether couples should use contraception and even, in some case, the specific person one should marry. Some of these issues are more debatable than others but all represent a direct intrusion of the church into areas of life which, generally speaking, are not matters in which the church should directly concern itself.4\n\nWhen one looks at the New Testament, it is interesting to see that Paul has to address issues of the abuse of church power on numerous occasions. For example, in Galatians 5:12\u201313, Paul warns against those who are insisting that the Galatians need to be circumcised. This is typically (and correctly) understood as Paul attacking attempts by \"Judaizers\" to pull the church back into Jewish practices because they have not understood the full implications of the gospel. That is true; but it is also an example of Paul attacking the illegitimate extension of church power. The New Testament church's power is delimited by the gospel. Thus, for church leaders to insist on the necessary addition of anything, be it circumcision, ceremonial washings, or whatever, is for them to step beyond any power that God has given the church. We see a similar situation in the letter to the Colossians, where Paul advises the people to let no one judge them with regard to observance of peripheral things such as food, festivals, or Sabbaths. Again, the subtext is not just that a form of legalism had crept in to the church at Colossae; it also appears that this was connected to a dictatorial church eldership, as is typically the case in such situations.\n\nIn these contexts, confessions can be remarkably helpful. As they offer succinct summaries of doctrine, so they also offer succinct summaries of what can be expected of the Christian in terms of practice. It is true that Paul did not have a confession in the sense that we can now have one, but that is not the question to ask. The question is one of what mechanisms are best to maintain the kind of church envisaged by Paul in the early church. How do we create a church community where what is regarded as normal belief and practice is publicly stated in such a fashion that it expresses biblical teaching, can be challenged and tested in the light of Scripture, and allows both elders and laypeople to know exactly where they stand in relationship to each other? A public, church-sanctioned confession is the obvious answer; it may, of course, not be the only answer, but it seems to me to be the best. If the wheel does the job reasonably well, why would one wish to reinvent it?\n\nThus, in the church where I am on session, I am aware that if I stand up one Sunday and declare that I am going to discipline all members who do not homeschool their children or vote for the Libertarian Party or throw away their television sets or start using homoeopathic medicine or give up drinking alcohol, the congregation can legitimately ask me where it says they need to do that according to the church's stated public position in her confessions. To put it bluntly, the members did not sign up for this and I therefore have no right to require it of them. For the church where there is \"no creed but the Bible,\" however, the situation is likely to be much more complicated and rapidly become very messy. This is not to deny that such a church might possibly achieve the correct result at some point, but the process will be much murkier and much more open to abuse and misreading than if there is a clearly stated summary of biblical teaching in the form of a confession to which the congregation can have recourse. Just as a good civil law code defines a well-ordered society and the powers its various estates possess, a confession states clearly that for which a church stands and thus allows the people to know what to expect from the eldership and, most importantly, when the eldership is overstepping its bounds. Of course, a confession is no guarantee that abuse of power will not take place, but it is one important element of a framework for making such less likely. Good confessions properly applied by appropriately qualified and ordained elders do actually hinder despotic church power and protect the members; they do not facilitate it.\n\nCreeds and Confessions Offer Succinct and Thorough Summaries of the Faith\n\nOnce one has acknowledged that every church has a creed and that, for good order, such a creed needs to be publicly stated, it becomes clear that there are numerous further advantages to this. Of these, perhaps the most obvious is not particularly ecclesiastical: they offer more comprehensive and succinct summaries of Christian doctrine than anything else. Indeed, one might without hyperbole declare that, outside of the Bible, the documents that contain more biblical truth per page than anything else are the great creeds and confessions of the church. It is worth noting two related aspects of this as being of use to the church: first, they focus the church's mind on the main thing; second, they remind us that \"succinctness\" in Christian theology is not necessarily what our contemporary culture tells us that it is.\n\nFirst, creeds and confessions focus the church's mind on the main thing. Longevity is one of the great assets that the church's creedal statements have in their favor: they were produced long ago and have withstood the test of time. While that is not in and of itself a watertight argument for their authority, it does at least indicate that the topics they address are clearly the hardy perennials of Christian existence. We shall comment later that one of the great advantages of such is that they relativize the present; here we should note that the classic confessions of the church, for all of their doctrinal differences, focus on matters such as the doctrines of God, of creation, of Christ, of redemption, of salvation, and of consummation. A church with a creed or confession has a built-in gospel reality check. It is unlikely to become sidetracked by the peripheral issues of the passing moment; rather it will focus instead on the great theological categories that touch on matters of eternal significance.\n\nThe second factor is the succinctness of creeds and confessions. Of course, we live in an age where numerous factors militate against considering the classical confessions as succinct summaries. We noted in chapter 1 the woeful influence of things like Wikipedia in leading some to think that all important knowledge can be swiftly grasped in short sentences and after a few minutes of cursory reading. Further, the role of the evangelical parachurch today, which depends upon the sidelining of many traditional confessional topics such as sacraments, means that the kind of statements of faith and doctrinal bases with which many are familiar are often ten or twelve point documents that can be easily printed on a single sheet of paper. Notions of what is \"succinct\" have somewhat contracted over the years. When one considers that the Shorter Catechism of the Westminster Assembly, amounting to 107 brief questions and answers, was really the elementary pedagogical document that simply prepared the way for the main event, the Larger Catechism of 196 far more elaborate questions, one realizes that what does and does not constitute the bare essentials of a coherent faith has certainly changed since the seventeenth century. Few even among Presbyterians today know the Shorter Catechism by heart, let alone the Larger.\n\nWe need to hold this in mind when we hear the oft-repeated complaints about creeds and confessions, particularly the more elaborate productions of the sixteenth and seventeenth centuries such as the Belgic and the Westminster, that they are so elaborate and prescriptive. One of the points that must be made in response to such criticism is that these documents typically cover only the really basic heads of Christian doctrine. Take Westminster as an example: would one really want to have a church confession that said nothing about the doctrine of Scripture, the doctrine of God, the nature of justification and sanctification, the definition of the church, and so on? One might dissent from the content of such topics in the Confession but one could scarcely argue that they did not represent some of the most basic concerns of the Bible itself.\n\nIn fact, many churches do have statements which do not touch on these issues explicitly, and one must then ask: is the document to which they do subscribe sufficient to give the elders the necessary material with which to maintain as far as possible the orthodoxy of the church?\n\nAny church needs to avoid two things in this context. First, the church must not send the signal that things which are actually important are matters of indifference. Here one needs to be careful of making one's own issues into the issues (something that the relativizing effect of creeds can help with, as will be argued below). Just because the current elders think that, say, wearing a dark suit in the pulpit is appropriate for preachers does not mean that this should necessarily be part of the church's confessional documents. It is still the case, however, that there are matters about which Scripture speaks eloquently and which must therefore feature in any church's confession. Baptism is perhaps the most obvious. There is extensive biblical teaching on this issue, and both the New Testament narratives of the Gospels and Acts and the teaching of Paul in his Letters makes it clear this is not a topic one can simply ignore. Thus, for a church to have a statement of faith that does not articulate a specific position on this matter would appear to be at odds with what is indisputable: the importance of the topic for the church of the New Testament. Indeed, I would argue that the church which sees the issue is of great importance, whatever the conclusion about its mode and subjects, is more consistent with New Testament emphases than the one which ignores the matter in its confessional statement or simply leaves it up to the conscience of the individual. A confessional document needs to reflect the doctrinal emphases and priorities of Scripture.5\n\nThe second issue is that, for a church to maintain a consistently orthodox witness, a certain level of ineradicable complexity is necessary in her doctrinal statements in order for them to be theologically stable. In this, the church's doctrinal confession is analogous to the nature of living organisms. If you drop below a certain level of complexity\u2014genetic, physiological, or whatever\u2014life becomes simply unsustainable. As a mouse needs a heart, blood, a brain, teeth, a digestive system, etc., and a genetic makeup that provides all the above or else it will perish, so a church confession needs a level of complexity in order for each of its doctrines to be stable and to function correctly within her life.\n\nThe history of doctrine bears ample witness to this fact. Think, for example, of the incarnation of the Son in the person of the Lord Jesus Christ. To maintain this, one needs not only a grasp of what deity is and what humanity is, one must also have a Trinitarian understanding of God, otherwise one falls off the ridge into either modalism on the one side or tritheism on the other. Thus, any confessional document that talks about the incarnation must also talk about the Trinity. This is why a study of the development of doctrinal statements in church history is important: it gives a first-rate insight into how doctrines interconnect and how formulations that solve one set of questions then create the ground for a new set. Chalcedon was only possible in the light of Nicaea; but once Nicaea was in place, Chalcedon, or some equivalent, became necessary as a means of connecting the dots from God in himself to God manifest in the flesh.\n\nOne might also say a similar thing about justification. Justification connects to one's understanding of humanity, of sin, of Christ's person and work, of faith, and of final judgment. One cannot simply put forward a church statement that declares, \"We believe in justification by faith.\" It is arguable that some Roman Catholics would be able to interpret that statement in a manner consistent with their own catechism. To be distinctly Protestant, one has to provide much more detail and set this doctrine within a complex of other doctrines. One needs to define God, creation, God's image, the impact of the fall, the nature of righteousness, and what constitutes faith, imputation, etc. A doctrine of justification is only stable once it is set within a broader doctrinal matrix.\n\nThis is perhaps an odd direction for the argument to take in a section that recommends confessions as succinct summaries of the faith, but as I noted above, the notion of what is and is not succinct has rather changed over the years; and we need to allow our understanding of it to be shaped by the Bible and by how the thought of the Bible has been articulated over the ages in church documents. Succinct means neither longer nor shorter than necessary. The promulgation (and, where necessary, the defense) of orthodoxy is massively enhanced by an adequately complex confession because such a document helps strengthen ecclesiology.\n\nFurthermore, from the point of view of basic pedagogy, it is surely to the church's advantage to have a creed or confession. If you want to offer a curriculum on what is important in the Bible, it is useful to have a brief syllabus that encompasses all the key points. A good confession will do that. Plus, if you yourself want a book that you can carry in your pocket which will serve as a reminder of the grand sweep of biblical teaching, then you want to obtain a copy of a confession and carry it with you always.\n\nThis is not, of course, to argue that the Westminster Confession, or the alternatives, offers an exhaustive account of everything that the Bible teaches. Far from it. But it is to say that it will provide reasonable coverage of the basic essentials.\n\nCreeds and Confessions Allow for Appropriate Discrimination between Members and Office-Bearers\n\nOne of the things with which we need to take special care in the use of creeds and confessions by the church is the function these have for non-office-bearing members. Should laypeople be required to subscribe to a church's doctrinal standards in the same way as an elder or a deacon? Some traditions, for example a number of the continental Reformed churches, have a history of requiring confessional subscription of every communicant member. This is a complex issue and worthy of full-orbed discussion; here, however, I wish to present not a polemic against such a view but a positive exposition of the standard Presbyterian position. This position is the one which I will argue fits best with the biblical evidence and can also be applied to other church polities.\n\nIn Presbyterianism, an important function of creeds and confessions is the way in which they allow for an appropriate distinction between the qualifications for membership and those for office-bearing. This is something that I believe Presbyterianism (at least in theory\u2014always an important qualification when it comes to ecclesiology) actually does well. Typically, Presbyterians set the bar for full communicant church membership very low: a simple but publicly coherent profession of faith in the line of Romans 10:9\u201310 is sufficient. This may be fleshed out in a series of simple vows, touching on issues such as the Trinity, salvation by grace through faith, the authority of the Bible, and submission to elder oversight, but the overall content is simple and straightforward. Many Presbyterian churches also require individuals to participate in a series of membership classes that elaborate this basic profession and also instruct candidates about the seriousness of the commitment which membership involves. But the basic criteria are the simple ones of Romans 10: a basic trust in Christ and an outward profession which is consistent with that. It is surely important, and consistent with a view of God as merciful and gracious, that we set the bar for membership no higher than that which we find in the Bible itself.\n\nFor example, it would seem to me to be inappropriate that people be required to finely parse the communication of attributes in the person of Christ prior to becoming a member of a church. Similarly, it is surely unreasonable for them to be required to articulate and resolve the many problems that initially arise out of the confession that God is three and God is one. These are things that the church is to teach to her members, not require of them prior to entry. Indeed, if fulsome knowledge of the whole counsel of God were required for mere membership, one might be left wondering exactly what it is that the church is supposed to do with those who finally manage to join. Membership is not a reward for achieving a high level of doctrinal knowledge any more than a high level of personal holiness. It is the gateway to the means by which these things can become possible via the ordinary means of grace.\n\nNevertheless, the Bible makes it quite clear that qualifications for office-bearing are of a somewhat different order. Thus, in 1 Timothy Paul starts the letter by stressing to Timothy the nature of his charge as elder:\n\nPaul, an apostle of Christ Jesus by command of God our Savior and of Christ Jesus our hope, To Timothy, my true child in the faith: Grace, mercy, and peace from God the Father and Christ Jesus our Lord. As I urged you when I was going to Macedonia, remain at Ephesus that you may charge certain persons not to teach any different doctrine, nor to devote themselves to myths and endless genealogies, which promote speculations rather than the stewardship from God that is by faith. The aim of our charge is love that issues from a pure heart and a good conscience and a sincere faith. Certain persons, by swerving from these, have wandered away into vain discussion, desiring to be teachers of the law, without understanding either what they are saying or the things about which they make confident assertions.\n\nIt is clear from this that Paul sees Timothy's task as elder as involving the careful communication of the faith in a manner that focuses on the straightforward teaching of the gospel. The elder is to do this by avoiding the kind of undoubtedly fascinating but ultimately sterile obsession with elaborate speculations that mark these problem figures at Ephesus. In other words, he is to have the maturity and discernment to know what exactly it is that he is to focus on in terms of his teaching, and the knowledge to be able to do this effectively. He is also to make sure that his ambition is to teach, not to be a teacher. This might seem a small point but ultimately touches on a crucial aspect of attitude: the teacher's task is to draw attention to what is taught, not to himself.\n\nGiven this, it is clear that Paul assumes that the teacher is to have a certain doctrinal competence which may not typically mark the church member. After all, if the church members all possessed this competence, there would presumably be no such problem as Paul describes in Ephesus. Thus, later in the same letter Paul outlines the qualities of an elder in a more elaborate fashion, essentially making explicit some of the aspects implied in his initial exhortation to Timothy:\n\nThe saying is trustworthy: If anyone aspires to the office of overseer, he desires a noble task. Therefore an overseer must be above reproach, the husband of one wife, sober-minded, self-controlled, respectable, hospitable, able to teach, not a drunkard, not violent but gentle, not quarrelsome, not a lover of money. He must manage his own household well, with all dignity keeping his children submissive, for if someone does not know how to manage his own household, how will he care for God's church? He must not be a recent convert, or he may become puffed up with conceit and fall into the condemnation of the devil. Moreover, he must be well thought of by outsiders, so that he may not fall into disgrace, into a snare of the devil. (1 Tim. 3:1\u20137)\n\nWhile it is helpful to note that doctrinal qualifications are only one of the many things Paul lists as necessary for an elder, in the context of this chapter it is important to stress that the ability to teach is nonnegotiable and clearly carries with it significant doctrinal freight. For Paul, one cannot be a teacher in the abstract: what one teaches, the content, is a vital part of the task; one can only be a teacher when one has sufficient mastery of this appropriate content to be able to teach it.\n\nThe question for each individual church or denomination becomes, therefore, what is it that the elders are to be competent to teach? What content is it that they are supposed to have sufficiently mastered in order to hold this office? What more do they need to believe and to understand than the teenager who was converted last Sunday morning on his first visit to the church?\n\nA couple of answers might immediately suggest themselves. One could respond simply: the deep things of the Bible. This idea, however, is vulnerable to precisely the kind of problem noted earlier about a church simply holding to the Bible as its standard of orthodoxy. Every heretic has his text. The pastor who simply declares that his creed is the Bible and nothing more is being disingenuous because, when he preaches, he interprets the Bible, he does not simply read it aloud to his congregation. And if he decides the text means one thing this week and the opposite the next, how can the congregation hold him to account for what he is teaching? Further, we might also recall the argument above about the delimiting of church power. If the pastor is simply \"teaching the Bible,\" and there is no frame of reference by which the congregation can assess that teaching, then the possibility for abuse of power becomes much more real.\n\nIf your church has a minimal doctrinal basis or statement, however, the response might be that elders should be free to teach whatever they consider to be consistent with Scripture, which is also consistent with the doctrinal basis. This is plausible but points us to the problem noted above, that Christian theology has a certain ineradicable complexity whereby certain doctrines stand in positive connection to others, and where modification of one might well require modification of another. This is not to say that simple childlike faith that Jesus is Lord is not sufficient for salvation; but it is to say that this is insufficient for the establishment and well-being of the church as a community which confesses the same. We noted in chapter 2 that the New Testament envisages the church as a place where people grow, and one aspect of that growth that is explicitly mentioned is increase in knowledge; and requiring elders to be committed to a confession of faith is a means of trying to ensure that those in positions of teaching authority have the knowledge and the ability to foster that growth among the congregation. Indeed, the church that can only regulate the teaching that it permits in a minimal way is never going to rise above that minimal level when it comes to coherent public, doctrinal testimony, something implicit in the previous point.\n\nFor this reason, those who take the New Testament teaching on the church and on the eldership seriously need to put in place mechanisms that allow that teaching to be realized in practice. The most obvious way of doing this is to require elders to subscribe to a confession of faith that articulates the kind of doctrinal complexity which is necessary for the elaboration and defense of the central tenets of the faith. Doing so holds such men accountable for what they believe and teach; yet restricting this to the eldership (and in many cases the diaconate) allows for the setting of an appropriately different level for members. It recognizes the seriousness of the office of elder but also the fact that many genuine believers have minimal doctrinal understanding at their conversion. It allows that this should not be a bar to church membership\u2014membership which then provides the context for their growth in this area as in others.\n\nCreeds and Confessions Reflect the Ministerial Authority of the Church\n\nOne of the most important aspects of creeds and confessions is that they are corporate documents which are authored and owned by the corporate churches, as represented by her office-bearers. We noted in both chapter 2 and earlier in this chapter that Paul puts particular weight upon the eldership, both in terms of its responsibility to safeguard the orthodoxy of the church's public proclamation of the gospel, and of the authority it therefore has to make sure this is done.\n\nNow, creeds and confessions can, historically, be the work of a single hand. The Belgic Confession is one good example; the Heidelberg Catechism is another. Yet both of these documents have status not because of who wrote them but because duly appointed church officers have formally adopted them as confessional standards, for regulating teaching and preaching. They thus become corporate church documents, not simply the private musings of gifted theologians.\n\nAs Protestants, we are of course naturally wary of any kind of claims for church authority that would place the church over Scripture or exhibit the kind of sacerdotal or papal tendencies we associate with Roman Catholicism. Nevertheless, Paul does have an ecclesiology whereby specially qualified and elected men hold the office of elder, and that office involves both authority and responsibility.\n\nIn this context, members of churches are to take very seriously creeds and confessions, not simply where they think the teaching they contain coincides with Scripture, but because they are upheld by the elders of the church, who, as Paul tells us, are worthy of respect and to whom we should submit in the Lord. This can be a delicate balance: the elders are not part of an unassailable hierarchy that must be uncritically obeyed in all matters, but nor are they to be treated as just anybody else in the church. One might say they are analogous to schoolteachers or policemen: their status does not automatically guarantee that they will always act correctly or that they should be unconditionally obeyed simply because of who they are, but ninety-nine times out of a hundred, their status should be significant in determining our response to them.\n\nThus it is with creeds and confessions. Because these documents have been adopted by those who have been called to hold office in Christ's church\u2014and that carries huge weight in and of itself\u2014the default position should be one of trust and obedience toward them. No, this does not mean that, unlike the Bereans, we should not search the Scriptures to see if the things they claim are so. But it does mean we should be less confident of our judgment and more inclined to trust the church, an attitude which clearly goes against our current cultural predilection toward suspicion and iconoclasm.\n\nIt is perhaps very hard, particularly for Protestants, to think in such terms today. Those of us in the West have been taught to believe so deeply in the authority and autonomy of the individual that subjecting our own thoughts to external authorities, especially corporate or historic, is very counterintuitive. Combined with a desire for instant gratification, many of us are inclined to believe that if something does not make sense the first time we look at it, it\u2014and not we\u2014must be wrong. That is not the way the church operates. It is clear in the New Testament that the corporate nature of the church is important in terms of practice and belief; the history and nature of confessions witness to that and connect it to the officers of the church as well.\n\nPaul has a high view of the church as a body and as an institution. This has been reflected, sometimes excessively so, in church history from Ignatius onward. Yet the fact remains that respect for the authority of the church and respect for the creeds and confessions which churches adopt must become an important part of our contemporary Christian lives if we are to be truly biblical. That society tells us to distrust traditional authority, to doubt all leaders, and to dismiss the past is of little relevance to applying biblical principles to our churches.\n\nCreeds and Confessions Represent the Maximum Doctrinal Competence That Can Be Expected from a Congregation\n\nUnderlying the last point is the notion that creeds and confessions can also have an important pedagogical function within the church. As we noted above, they are succinct doctrinal summaries. They also represent that which the church aspires to teach its members, and that is why confessional churches typically have at least one catechism among their subordinate standards. In short, they represent the church's doctrinal and pedagogical aspirations.\n\nWe can make this point by reflecting for a moment on the function of law codes in society at large. Many countries have laws that its citizens know will be broken. A recent controversial example would be that of the torture of terrorist suspects. This led some in the USA to suggest that it might be desirable to change the law. Should torture be legalized and thus be subject to regulation? Or should it be kept illegal, even though it is tacitly understood that the law will on occasion be broken? Both arguments have a certain power, but my own instincts incline me to the latter position for the simple reason that laws represent in part the moral aspirations of a given society. Nobody, for example, believes that outlawing abortion will stop abortion; but many of us would wish to live in a society where the statute books represent our aspiration to be an abortion-free society. That is one reason we want it to be illegal: laws set before us a vision of the kind of society we would like to see realized. They do not simply reflect the pragmatic reality which we all know.\n\nThere is an analogy here with the churchly nature of creeds and confessions. For a church to hold to a creed or confession, to require subscription to the same from her office-bearers, is to send a signal to the congregation about what the church considers to be important in her doctrinal life. If a church has a six-point creed or confession, she essentially communicates to her people that these six things, and only these, are important. Everything else is so minor that it forms no part of its identity and it is quite happy for anything to be taught on other topics providing that such teaching does not come into direct conflict with the six basic points.\n\nOn one level, setting the bar this low is commendable in that it reflects the fact that church membership should require no more doctrinal competence than that which the Bible specifies for salvation. We do not want to stop new converts from coming into church membership and under the pastoral care of a local congregation because they do not yet understand the hypostatic union of the two natures in Christ or have not fully developed a theology of the Trinity. We also do not want to exclude from membership the educationally challenged or those who cannot think abstractly and are never going to be able to articulate a clear defense of the Chalcedonian Definition. We want church membership to be as inclusive as the Bible makes it. Nevertheless, we surely do not want to send a signal to the congregation that members should simply be satisfied with a basic, mere Christianity, especially since the Bible itself clearly sets an ambitious standard for doctrinal understanding and expects growth in such understanding to be a normal result of belonging to the church. After all, Paul himself is able to distinguish between the kind of basic teaching given to new believers and the more sophisticated ideas to which such believers should progress. Membership is the beginning, not the end, of the pedagogical process.\n\nGiven this, a church confession not only sets before the congregation a list of doctrinal priorities and demonstrates how these priorities fit together into an overarching framework, it also represents an aspirational ideal of what the eldership hopes will be the appropriate level of doctrinal competence for the congregation. If something is a priority, is of importance, then it should be stated in the confession. If it is not in the confession, then it will be very hard to make the case that it is of any importance. After all, if one does not need to have an opinion on a topic to hold office, then that topic is clearly of highly negotiable significance.\n\nAn obvious example here would be baptism. If the confessional statement takes no stand on baptism and does not make it clear if it is to be applied to believers only or to believers and their children, the obvious conclusion is that baptism is of no real significance, that the church has no more need to have an opinion on this matter than on what color wallpaper will be best for the fellowship hall. Yet it is surely very difficult to square any notion of baptism as a matter indifferent with the frequent and vigorous teaching of Paul on the matter in the New Testament.\n\nThat confessions also delimit the power of the church also reinforces the notion that a church's confession really sets forth the maximum level of theological understanding that can be normatively required from members. This is not to say that individual members cannot and will not deepen their theological knowledge in ways untouched by the confession. Indeed, it should be the hope of every church that the membership will be as theologically well-read and literate as possible. Nevertheless, as the confession sets out what the church considers to be vital, and also sets the material parameters of the church's pedagogical power, we must understand that it represents the maximum that can be officially expected of church members as they mature and grow.\n\nThus, the questions we ministers and elders need to ask ourselves are: What vision do we wish to give our people, from the most recent convert to the long-established church member? Do we want them to think that a six- or ten-point doctrinal statement is all that the mature Christian needs to believe and understand? Or do we want to set before them a more ambitious aspiration, something which comes closer to articulating the whole counsel of God? In this context, a good confession becomes not a stick with which to beat people\u2014the popular image that often grips the mind of many believers\u2014but an exciting map of the territory of biblical truth and something to which to aspire.\n\nThis should also lead us to be wary of the role parachurch organizations play in the Christian life. They are to serve the church, not vice versa. Most, if not all, parachurch organizations have relatively minimal doctrinal statements, at least compared to the great confessions of the Reformation era, and most sideline issues such as sacraments and even sometimes key soteriological doctrines such as election in order to provide a basis for transdenominational cobelligerence. There is much to be commended about many of these organizations, but the danger is one of perception: if they become the thing of primary importance, rather than the church, then the great confessional distinctives of the church are functionally relativized as things of no real consequence. This has the same practical impact upon church pedagogical expectations as churches themselves having minimal doctrinal statements. It is thus crucial that parachurch involvement is kept in perspective by church leaders and by elders: it can be a helpful and encouraging activity but it should not supplant the absolute priority of the local church and of the denomination, for only in these contexts do we find New Testament governance and appropriate elaboration of the whole counsel of God. The old saying, \"What you save them with is what you save them to\" may well be an overstatement, but it contains enough truth for us to take it seriously and to reflect upon the role that doctrinal minimalism (at least by confessional standards) can come to play within established churches.\n\nCreeds and Confessions Relativize the Present\n\nWe noted above that creeds and confessions can help focus the mind of the church on matters of perennial interest because if something has proved significant over the centuries, one can have a reasonable degree of confidence that it is of importance to more than just this day and generation. This, of course, is another way of saying that creeds and confessions relativize the present. This notion is commendable for at least two reasons.\n\nFirst, as noted, creeds and confessions that have proved useful over the centuries are clearly immune to the passing fads and tastes of the present. They speak to issues that the church has found important for generations. Thus, while one might point out the obvious, that, say, the Decrees and Canons of the Council of Trent teach a different understanding of justification from that found in the Lutheran Book of Concord, one would have to agree that both documents witness to the fact that justification is an important biblical doctrine and that all churches need to have some position on the matter.\n\nOne frequent objection to historic confessions is, of course, that this actually prevents them from being relevant and from speaking to the current age. Such objections are usually rooted in the belief that our knowledge of what the Bible teaches has developed to the point where specific doctrines in the confessional documents are no longer biblically credible. This is a serious matter. Central to the notion of confessionalism is that the confessional documents are themselves a normed norm and always subject to correction by the norming norm, a role reserved for Scripture alone. Given this, confessional revision must always be a possibility; this will be discussed in the appendix.\n\nThe second aspect of this advantage of creeds and confessions is that it is profoundly countercultural in a biblical way. What confessionalism does is signal to the church and to the world that the past is in many ways as important, if not more so, than the present. By reciting a creed in the worship service or adhering to a historic confession as part of a congregation's identity, the church makes a powerful statement of her relationship to the contemporary culture. Yes, the present is where we all live and breathe, eat and drink; but the creeds and confessions of the church connect us to the past and indicate that our identity is rooted in that past. This is in line with the thrust of biblical teaching. We noted earlier that Exodus 12 roots the identity of Israel in the present in the recollection of God's mighty acts of salvation in the past. And we also saw that Paul's charge to Timothy was not that he should innovate or take his primary cue from the surrounding culture; rather it was that he should hold fast to the form of sound words that he had been taught by the apostle. Thus, when I took my vows as a minister in the Orthodox Presbyterian Church and agreed to uphold the teaching of the Westminster Standards, I was effectively saying that the church is bigger than my day and generation; its foundations lie in the past; and I am charged with stewarding that truth in the current context, but that truth neither begins nor ends with me.\n\nSuch counterculturalism is important. Theologically, it is vital because salvation is something that was wrought in the past, even as it is applied in the present and will not be fully consummated until some time in the future. Yet there is more to this than straightforward theological significance: it is also important that the church itself cultivate a countercultural culture on this issue; and the liturgical action of using a creed in the worship service and of adhering to a confession as that which defines the church locally and denominationally helps to inculcate precisely such a culture. If the people are saying the Apostles' Creed or the Nicene Creed on a Sunday, if Sunday school classes use the historic confessions as pedagogical guides, and if preachers refer on regular occasion to statements within these documents, then the people will become used to the idea that the church's past is of perennial, vital relevance. The Christian mind is not only doctrinal; it is also marked by a certain attitude to the past. And church practice, as well as church teaching, plays an important role in the cultivation of this.\n\nCreeds and Confessions Help to Define One Church in Relation to Another\n\nAnother advantage to holding to a specific confession is that it allows for the clear, public identification of one church in relation to another. As was argued earlier, all churches and all Christians have a functional confession, the difference being whether one writes it down and makes it public or not. Given this, the public nature of confessions can only be a good thing, as it serves the interests of transparency and ecumenism.\n\nIt serves transparency because it allows those outside to see what a particular church represents. If someone is on vacation in a town with which they are not familiar, or if someone moves to a new area, it is of immense help to be able to identify clearly where the various churches in the locale stand in relation to various issues. To be told \"our church just holds to the Bible\" might appear to be rooted in a high view of Scripture but in fact tells the outsider little about the church beyond its generic commitment to some form of biblical authority. It could be a thoroughly orthodox church or it could be an off-the-wall, snake-handling group to whom all notions of God as Trinity might be utterly alien. Unless one attends the church over a period of time, its cherished doctrinal commitments and distinctives will probably not be immediately apparent (snake handling being, I assume, an obvious exception).\n\nFurther, such transparency does not only serve the visitor or potential new member; it also serves the local church. When someone visits the congregation, it is useful for congregants to be able to point them to a succinct summary of the church's position on key doctrinal topics. A historic confession would seem to be the most obvious candidate for such a role. It is convenient, honest, and transparent. It leaves nobody in any doubt about what the church is and what she teaches.\n\nCreeds and Confessions Are Necessary for Maintaining Corporate Unity\n\nAs noted in chapter 1, we live in an age that fears exclusion, and with good reason. The twentieth century in Europe was one marked from beginning to end by the terrifying results of exclusion. From the Armenian genocide of 1915 through the Holocaust to the ethnic cleansing of the Balkan states in the 1990s, the impact of excluding people, of deciding that this or that group did not belong, was bloody, violent, and undeniable. In the latter part of that century, this no doubt played a part in the rise of philosophies devoted to unmasking how such thinking operated at various levels in society. We also entered a period of Western history where the very idea that one group did not belong, or was somehow inferior to another group, became a profoundly distasteful and disturbing notion.\n\nThe impact of this attitude on the church is significant. After all, only a fool would deny that churches have themselves a poor record on exclusion, with many playing collaborative roles in the various persecutions and massacres of history. From the Inquisition to apartheid, churches have often been part of the problem, not the solution. We noted earlier that this has led to a reaction against any kind of exclusionary aspect of church and, inevitably, to a downplaying of doctrinal distinctiveness. The phrases \"love unites, doctrine divides,\" and its close relative, \"belonging before believing,\" are both symptomatic of this yearning for inclusiveness.\n\nThere is a tragic irony in this contemporary desire to set belonging and believing somewhat in opposition, with priority being given decisively to the former as a precondition of the latter. First, Paul's theology contains no sense of the two being separable in such a way that allows this kind of opposition to exist. Belonging and believing would seem in Paul's world to be two sides of the same coin. Thus, as we noted in chapter 2, Paul himself characterizes deviation from true doctrine as divisive. We might therefore say that to cease to believe is one and the same with ceasing to belong. That has a distasteful ring to modern ears, for all of the reasons noted above; but it is unavoidable as a conclusion if we take Paul seriously. And given that we are routinely familiar with other institutions where exclusion is part and parcel of good institutional health\u2014from professional societies to political parties\u2014we should not allow religion's past to blunt our understanding and application of Paul's teaching.\n\nSecond, while Christianity cannot be reduced to doctrine, to mere teaching, it cannot be meaningfully separated from it either. Even the most basic claims, such as \"Jesus is Lord,\" carry clear doctrinal content that needs to be explicated in a world where, as we have noted before, every heretic has his text and not all who cry \"Lord! Lord!\" actually have any real saving knowledge of God. That this will inevitably involve exclusion is indisputable. That is another reason why setting the bar low for membership is important, on the grounds that membership is made as inclusive as the Bible allows. But it is also important that membership is as exclusive as the Bible demands, and that means that some will not belong because they do not believe.\n\nThe use of confessions as standards of what the church believes and of creeds as corporate expressions of belief in worship services is thus important for underscoring what the church is. If you want to use a contemporary idiom, you could say that they tell the story of who the church is and thus ground its identity in a theological narrative. If, like me, you are comfortable with more traditional terminology, you might say they define who the church is doctrinally. Either way, creeds and confessions establish boundaries of belonging and, by implication, of exclusion. Both are necessary if the church is to have a meaningful corporate identity and unity. Sometimes this will sadly manifest itself in the expulsion of somebody who says he belongs but by his words and actions indicates that such is not the case. That is unfortunate but on occasion necessary. More often, however, the unity will manifest itself in a positive way: the congregation reciting (and rejoicing in) the words of the Apostles' Creed on a Sunday morning; new Christians affirming their belief before the congregation by taking the same vows as the other members have done before them; and worship services marked by a common vocabulary on the lips of all members as they praise their common Lord.\n\nConclusion\n\nAt the end of this chapter, I have little more to add to what I have already said. Each of the above points is grounded in the apostle Paul's concern for the health of the church through her careful stewardship of God's truth, the handing of that down from generation to generation, and the constant rejoicing in the same, which is meant to characterize the Christian life both at a corporate level and for the individual. Perhaps creeds and confessions are not the only way to do this, but they have certainly been the means of choice for so doing for the majority of Christians since the close of the apostolic era, from the Rule of Faith to the present day. If the churches whose cry is \"No creed but the Bible!\" are capable of doing the same, history provides little evidence of this, and, for me, the old adage about why one would wish to reinvent the wheel comes into play at this point. There may be ways of doing the above better; but I am unaware of them. And, indeed, the church throughout history has been unaware of them too.\n\n1Hermeneutical skeptics will object here that the introduction of a creed or confession does not solve the problem of interpretation: as the Bible needs to be interpreted, so do creeds and confessions. Thus, one could end up with an infinite regression of interpretations or, to use the trendier jargon, an endless deferral of meaning. This is not the place to discuss whether texts have meanings. For that, I would refer interested readers to Kevin J. Vanhoozer, Is There a Meaning in This Text? (Leicester, UK: Apollos, 1998). Suffice it to say here that, assuming texts do have meanings, it seems clear that some texts are easier to interpret than others; and creeds and confessions are designed to offer clear and succinct statements of faith. That debates can and do rage over points of interpretation relative to these documents is undeniable; but that the scope of these debates is considerably narrower than those which rage around the Bible is also undeniable.\n\n2For the local church elder, that would be the session, or elder board, of the local church; for a minister, that would the presbytery, or regional body made up of ministers and representative elders from each congregation.\n\n3I deal with the matter of confessional revision in the appendix.\n\n4I am aware that some issues, such as contraception and education, are increasingly contentious. I do not wish to engage the specifics of these matters in this book; my interest here is whether the church has the right to issue specific edicts in these areas rather than simply inculcating general principles that leave the individual free to make appropriate applications in his or her family or private lives.\n\n5This is again where church history becomes important because it allows access to what has been considered important throughout the ages and not simply within the narrow confines of my own present time. \n\n## Conclusion\n\nI started this book by highlighting the instance of a pastor who stood up in his pulpit, held the Scriptures aloft, and declared that the Bible and the Bible alone was his creed. As I pointed out then, and have demonstrated in subsequent chapters, that claim was both naive and incorrect. All Christians have a creed or a confession; all Christians think the Bible means something and that its teaching can be summarized in a different form to that in which it was originally given. The only difference is whether one writes the confession down, so that others may scrutinize it and judge whether its teaching is consistent with Scripture, or whether one refuses to do so, in which case one's beliefs are essentially identified with the teaching of Scripture and placed above such scrutiny. Ironically, such a move makes one's tradition unassailable even as one accuses those who hold to creeds and confessions of doing the same.\n\nThus, the claim to have no creed but the Bible is first and foremost specious; but secondly, it is arguably a contradiction in terms. The Bible itself seems to demand the production of something like a creed or confession. The Pauline imperatives of holding fast to a form of sound words and of guarding the apostolic teaching both push the church toward creedal or confessional formulations and documents. We see hints of these in the New Testament and their rapid emergence in the early postapostolic centuries. The challenge to someone who takes the Bible seriously and yet repudiates the notion of creeds and confessions is: how does one fulfill these Pauline mandates with any degree of confidence? It seems to me that, in the absence of any credible alternative, creeds and confessions are imperatives for the church that takes the Bible seriously, not optional extras and certainly not something that can be decried as sinful, wrong, or unbiblical.\n\nA further point that emerges from the history of creeds and confessions, particularly in relation to the Trinitarian and christological debates of the early centuries, is that Christian theology can only exist in a stable form with a certain ineradicable degree of complexity. The claim that \"Jesus is Lord!\" is a simple enough linguistic construct; but as the church explored that claim, it became clear that a wealth of sophisticated and intricate theology was implied by, and supported and defined, the statement. Creeds and confessions are complex and precise not because their authors were obsessed with details and distinctions but because they were convinced that the claims made by God and for God were such that careful distinctions and precise statement were necessary if orthodoxy was to be articulated and transmitted from place to place and generation to generation.\n\nIn light of the above, and of the many other points made in this book (such as the importance of creeds to worship), creeds and confessions remain of paramount importance to those churches which seek to take the Bible and, more importantly, the God of the Bible seriously. We live in strange times, when the most vibrant strands of conservative, orthodox Protestantism are yet moving away from classic churchly confessionalism even while adopting the name of \"confession\" for the cause.\n\nThere are two dangers here: that a new form of mere Christianity gains a foothold in the camp of the orthodox even while presenting itself as something different, and that there will be a divorce of theology from the life of the organized, particular church under the oversight of office-bearers. Both of these stand at odds with Paul's vision for a sophisticated theology expressed by the church in a form of sound words propagated, protected, and passed on by qualified and duly appointed office-bearers. This move must be resisted by the reassertion of the primacy of the church and by the importance of confessions. Only then can the Pauline vision for the Christian life be truly realized.\n\nThis all sounds rather negative, I realize. I trust it is not really so. Creeds and confessions at their best present the church with beautiful summaries of biblical teaching, which are designed not simply to preserve the faith but also to be part of the very life of the worshiping community. As we saw above, there is a doxological element in creeds and confessions, an element which is even on occasion directly related to the polemics they contain. To say that God became man is to deny that Christ is a mere man, a point of polemic; but it is also to assert the reality and truth of the most amazing and wonderful act of God's grace in human history. We should remember that as we reflect upon which confessions our churches should use. An impoverished theological confession can ultimately lead only to an impoverished Christian life.\n\nThe last few decades have seen some high profile conversions from evangelical churches to Roman Catholicism. It is difficult to generalize, but a couple of themes seem to have emerged as factors in many of these: evangelicalism lacks historical rootedness, and evangelicalism lacks serious doctrinal weight and long-term stability, with its preference for experience, activism, and mere Christianity (whether at the liberal or the conservative end of the evangelical spectrum). I believe there is an alternative to Rome: it is confessional Protestantism. By that, I do not mean the confessional Protestantism that cherry-picks which bits of various Protestant confessions it likes, assembling an eclectic and minimal conservative Protestant consensus. I mean true confessionalism, one that adheres to a particular confession and connects this to a particular church order and polity. That is confessional Protestantism as the Reformers and their successors would have understood it. It is also Christianity as Paul would have understood it: the church, and only the church, is the divine institution, existing by the command and will of God, for the preservation and proclamation of the faith. It also meets both of those perceived lacunae in evangelicalism: it provides historical roots and serious theology.\n\nI hope this book goes some way to persuading those who earnestly wish to follow Paul and to be faithful, biblical Christians, that such is best done in the context of confessional churches. To take the Bible seriously means that creeds and confessions, far from being intrusions into the Christian life, are actually imperatives for the church.\n\n### APPENDIX\n\n## On Revising and Supplementing Confessions\n\nGiven that the Protestant confessions of the sixteenth and seventeenth centuries place themselves under the supreme authority of Scripture, it is obviously a confessional position to hold to the idea that creeds and confessions can be corrected or supplemented. If they are found to be wrong at some point or to fail to articulate the whole counsel of God as needed by the church, they need to be corrected or supplemented with further confessional statements.\n\nGiven this rather obvious point, however, the next question is, how should one go about revising or supplementing a church's confessions? Clearly, it is not something that just any individual church member can do: the burden of this book has been that creeds and confessions are churchly documents, the property of the corporate church not the isolated believer.\n\nTo take the notion of revision first, we need to bear a number of things in mind when approaching the topic. First, we must remember that creeds and confessions are ecclesiastical documents. They are adopted by churches as their standards of belief and thus take on corporate significance not possessed by other writings. Calvin's Institutes are very precious to me and to many in my denomination, but they have no formal status because nobody is required to take vows to uphold their teaching. Debates over how to translate passages in Calvin, or about whether he is right or wrong on a certain point might well be interesting, but they have little or no ecclesiastical significance. Confessions, though, are different: they are corporate documents to which the church is bound by procedural canons, ordination vows, etc. This difference is important when it comes to confessional revision. Any revision must be done by the church, specifically by those in the church charged with ensuring the soundness of her teaching, that is, the elders.\n\nSecond, we need to understand that subscribing to a creed or confession does not mean that we believe every phrase in the document was as well expressed as it could have been or that if we wrote it today we would use exactly the same vocabulary and phrasing. As I read through the Westminster Standards once again in preparation for my ordination vows, there were a number of things that I felt could have been expressed more felicitously. Sometimes there were even things that I would regard as significant omissions. For example, Shorter Catechism 4 reads as follows:\n\nWhat is God?\n\nGod is a Spirit, infinite, eternal, and unchangeable, in his being, wisdom, power, holiness, justice goodness, and truth.\n\nWere I to answer that question today, I would certainly include love as part of God's basic identity. Thus, the response as it stands is arguably deficient; yet, when set within the context of the Catechism as a whole, it is clear that the Westminster divines had a solid understanding of God's love and how it drove the economy of salvation. The lack in the answer to question 4 is odd when taken by itself but is compensated for by the whole.\n\nThus, as I read the Standards in preparation for ordination, I did not make the mistake of confusing the awkwardness or deficiencies of some of the phrasing with fundamental deviations from biblical teaching. Ultimately, I was not subscribing to the idea that the Westminster divines were the greatest theological prose writers. I was subscribing to the fact that they accurately summarized the Bible's teaching on those matters on which they chose to opine in the Standards. Thus, confessional revision is not justified simply on grounds of verbal clumsiness; it is the concepts the words express which is important. We should not propose confessional revision unless we believe the confession is actually wrong in some point.\n\nThird, many creeds and confessions have retained their basic form and matter and yet transcended their original contexts to become the beloved standards of churches all over the world. Thus, any revision to such by one denomination inevitably sets that denomination apart to some extent from others that subscribe to the same standards in their unrevised form. In this way, revision can actually make the documents less ecumenical. The most famous example of this is the addition of the so-called filioque clause to the Nicene Creed at the Third Council of Toledo in 589. This was a Western council whose authority to tinker with the text of the Nicene Creed was denied by the churches in the East. The result was a creed that ultimately provided part of the reason for the Western and Eastern churches to split from each other in the so-called Great Schism of 1054.\n\nThis was of course the greatest division of its kind in the history of the church. Yet even revisions to less universally accepted creeds and confessions can bring with them ecumenical consequences. American Presbyterianism eschews the right of the civil magistrate to call a church council and denies the Establishment Principle, whereby the magistrate is required to maintain the Christian (Presbyterian) religion; both principles are taught in the original Westminster Standards and are dear to the heart of many a Scottish Presbyterian. The American revision is in line with American constitutional law and philosophy, but it means that American and Scottish Presbyterians are not entirely united in their confessional commitments. If ever there were to be a move toward church union between such, this would no doubt prove something of a stumbling block. That is not to say that the revisions are not legitimate or important, but it is to say that they always come with a practical, ecumenical cost, and so should not be undertaken lightly or without reference to the wider church world.\n\nFinally, we must always remember that our own perspective is limited. We live in an age where we want things quickly, if not instantly, and where we often have a tendency to see the latest thing as the most significant and earth-shattering phenomenon. Very rarely does this prove to be the case. Often time is the only means by which to judge which discoveries or developments are truly significant and which ones are dead ends, overstated, or simply wrong. The Nicene Creed is still doing sterling work after over 1,600 years; that fact should make us very cautious about deciding to abandon it just because the latest trendy evangelical guru or the cutting-edge professor at the local university decides that it is outmoded and needs to be replaced. To be frank, they would need to come up with something that looks as if it will do the same job just as well and just as universally for the next 1,600 years before I would want to consider throwing out that which has served so many so well for so long.\n\nThus, any process of confessional revision must take account of the following: as confessions are ecclesiastical documents, they can only be revised by the church. In Presbyterian circles, this means by the presbytery and the assembly of the whole church. Second, it must be undertaken in a solemn and serious manner, with significant prayer, careful investigation into the issues, and extended periods of reflection in order to make sure that any change has been very carefully thought through and is indeed necessary, not a mere stylistic modification.\n\nOne final point is that the history of confessional revision is not a particularly happy one. By and large, churches that have engaged in extensive revision of their confessional standards have generally revised them in a direction that has proved in the long term to be inimical to orthodoxy and the health of the church. This was certainly the case with the PCUSA, whose 1967 Confession has at best proved useless in the maintenance of Christianity of even the vaguest kind at the denominational level; at worst it embodied anti-orthodox elements which served to accelerate the process of theological decline.1\n\nAs a result of this, most orthodox churches have generally resisted the temptation of extensive revision of confessional documents and, when revision has been undertaken, it has been done through established church processes. Some have removed relatively peripheral matters, such as the original Westminster Standards' ruling that a man may not marry his dead wife's sister or the identification of the pope as the Antichrist. Others, rather than removing such have passed church law indicating that the church will not enforce such specific clauses in her courts.\n\nWhen such a change is made by the church, those who are individual officers have, to paraphrase Presbyterian theologian Charles Hodge, three options: they can actively concur with the change; they can passively submit to the change; or they can peaceably withdraw in light of the change. It is, after all, true that confessional churches have not been immune to doctrinal decline or even outright apostasy. That is not the fault of them being confessional, since the same things can be said of nonconfessional churches as well. It is rather a function of the fact that, like all churches, they are staffed and supervised by fallen human beings. Thus, all officers have the right not to have their consciences bound by changes imposed on them by the church relative to the form or content of subscription. What they do not have, however, is the right of permanent protest within the church. Thus, when a change is made, one must support it, submit to it, or withdraw from the particular church or denomination which one considers to have bound one's conscience in a manner one considers illegitimate.\n\nThe second issue is that of supplementing confessional material. Should the church be in the business of adding new documents to which her officers must subscribe? Of course, the history of creeds and confessions is in large part the history of precisely such supplementation: Chalcedon supplements Constantinople; the Westminster Confession supersedes the Thirty-Nine Articles. This is thus a tricky matter to which there is no simple right or wrong answer.\n\nMy own approach to this is one of extreme caution. Much of the pressure for confessional supplementation over the last few decades has been connected to wider political and social issues. Thus, one hears calls for the church to make statements on racism or apartheid or the environment or poverty. One of the most famous church confessions of the twentieth century, the Barmen Declaration, was produced by the anti-Nazi German Confessing Church in order to oppose the nationalism and anti-Semitism that had infected the so-called German Christians. Barmen is thus the classic example of a document designed to speak prophetically to the church at a key moment in time.\n\nTwo things are worth bearing in mind here. First, there is always a place in church life for occasional documents, reports, or statements which make the church's view on a particular topic clear. Opinions may vary as to whether the church should speak directly to political issues, but even churches with a very precise understanding of the church as a spiritual entity will on occasion produce reports on hot theological topics or points where the church is considered to be under most pressure at that moment in time.\n\nSecond, there is\u2014or there should be\u2014a difference between occasional statements and confessionally binding documents. For example, the idea of humanity as made in the image of God, as fallen, as redeemed in Christ, and as looking forward to the general resurrection, is a perennial of the church's teaching. These things apply to all people in all times and all places. Statements on racism, the environment, or apartheid are all much more context specific. They lack ecumenical significance in the broadest sense of the word; and if something lacks ecumenical significance, there is arguably no point in making them part of the church's confessional material, however useful they may be as occasional statements.\n\nIn addition, supplementation is often the lazy option. For example, I would argue that the Bible's teaching on the nature of humanity, as summarized in the Westminster Standards, is actually quite sufficient for making the point that racism is sinful and wrong. The same is true for one of today's hot button theological topics: the New Perspective on Paul (NPP). Does the church need to supplement confessional statements on justification in order to combat the revised understanding of salvation being proposed by NPP advocates? I would argue no. There may be a need for a church report on the matter, showing how the NPP is out of accord with the church's confessional position, but the statements on justification in the Westminster Standards seem to me to be robust and thorough enough to be applicable in a helpful manner to the question. In other words, just because something is not dealt with explicitly in the Standards does not mean that those same Standards are not an adequate confessional basis for the church to address whatever the matter may be.\n\nFinally, of course, the liberty and ecumenicity argument also applies here: the more documents a church requires one to uphold, the more one finds that it is binding and micromanaging the consciences of officers and, indeed, the more barriers it is erecting between one's own communion and those of other people. It may be that one concludes that both are necessary; but one should only do that after long and very careful thought about not only the theological content but also the ecclesiastical consequences.\n\n1This is not to deny that there are congregations and ministers within the PCUSA who have remained orthodox; it is merely to assert that orthodoxy can no longer be guaranteed at a denominational level because of the confessional criteria now in use.\n\n## For Further Reading\n\nThis book has barely scratched the surface of confessions and confessionalism. For those who want to explore the various issues further, there are a number of good books I recommend.\n\nCollections of Confessions\n\nMany creeds and confessions are readily available on the Internet. In book form, the most easily available collection of creeds and confessions, from the ancient church to the nineteenth century, is Philip Schaff, The Creeds of Christendom I (Baker, 1966). A more thorough collection of specifically Reformed confessional documents is the projected three-volume set, edited by James T. Dennison Jr., Reformed Confessions of the 16th and 17th Centuries in English Translation (Reformation Heritage, 2008\u2013). A good collection of Baptist documents is William L. Lumpkin and Bill J. Leonard, Baptist Confessions of Faith (Judson Press, 2011).\n\nThe Use of Confessions\n\nThe classic work is the nineteenth-century essay by Samuel Miller, The Utility and Importance of Creeds and Confessions. A superb collection of essays on the matter of subscription in Presbyterian and Reformed churches is David W. Hall (ed.), The Practice of Confessional Subscription (Covenant Foundation, 2001).\n\nPolity\n\nThis book has argued for the importance of church polity to the practice of confessionalism. The gold standard on Presbyterian church polity remains that of the Victorian churchman, James Bannerman, The Church of Christ (Solid Ground Christian Books, 2009). Other works worth consulting include David W. Hall and Joseph H. Hall, Paradigms in Polity: Classic Readings in Reformed and Presbyterian Church Government (Covenant Foundation, 1994); Edmund P. Clowney, The Church (IVP Academic, 1995); and Guy Prentiss Waters, How Jesus Runs the Church (P&R, 2011). Mark Dever (ed.), Polity: Biblical Arguments on How to Conduct Church Life (Center for Church Reform, 2001) is a good collection of historic essays on the church from a Baptist perspective.\n\nCommentaries on Confessional Standards\n\nMost creeds and confessions are the subject of a significant volume of commentary. I list here my favorite entry-level books on key confessions:\n\nApostles' Creed: J. I. Packer, Affirming the Apostles' Creed (Crossway, 2008)\n\nWestminster Confession: Rowland S. Ward, The Westminster Confession of Faith: A Study Guide (New Melbourne Press, 1996)\n\nWestminster Shorter Catechism: G. I. Williamson, The Westminster Shorter Catechism: For Study Classes (P&R, 2003)\n\nWestminster Larger Catechism: J. G. Vos, The Westminster Larger Catechism: A Commentary (P&R, 2002)\n\nThe Belgic Confession: Daniel R. Hyde, With Heart and Mouth: An Exposition of the Belgic Confession (Reformed Fellowship, 2008)\n\nThe Heidelberg Catechism: Kevin DeYoung, The Good News We Almost Forgot: Rediscovering the Gospel in a 16th Century Catechism (Moody, 2010)\n\nThe Canons of Dordt: Cornelis P. Venema, But for the Grace of God: An Exposition of the Canons of Dordt (Reformed Fellowship, 2011)\n\nThirty-Nine Articles: Gerald Bray, The Faith We Confess: An Exposition of the Thirty-Nine Articles (Latimer Trust, 2009)\n\nBaptist 1689: Samuel E. Waldron, 1689 Baptist Confession of Faith: A Modern Exposition (Evangelical Press, 1989)\n\nLutheran: Charles P. Arand, That I May Be His Own: An Overview of Luther's Catechisms (Concordia, 2000)\n\nHow do we KNOW the stories told by historians are TRUE?\n\nTo what extent can we RELY on their INTERPRETATIONS of the past?\n\nHistories and Fallacies is a primer for dealing with conceptual and methodological problems in history and presents classic historical problems as a way to examine what history is, what it means, and how it can be told and understood.\n\n\"A very good book, full of historiographical wisdom. I recommend it strongly as a sure and encouraging guide to budding historians befuddled by the so-called 'history wars,' and to anyone who is interested in the challenges attending those who represent the history of Christian thought.\"\n\nDOUGLAS A. SWEENEY, Professor of Church History and the History of Christian Thought; Director, Carl F. H. Henry Center for Theological Understanding, Trinity Evangelical Divinity School\n\n\"Because the past shapes the present, a just understanding of the past is important for any individual, society, or church. Here is wise and practical advice for those wanting to write history for others about how to do it well.\"\n\nDAVID BEBBINGTON, Professor of History, University of Stirling\n\n\"Trueman offers here a combination of careful historical analysis coupled with an understanding of the logical and argumentative pitfalls to which historians are liable that is a service to the field and should provide a useful guide to beginning researchers.\"\n\nRICHARD MULLER, P. J. Zondervan Professor of Historical Theology, Calvin Theological Seminary\n\nVisit Crossway.org for more information.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by a Project Gutenberg volunteer from digital\nmaterial generously made available by the Internet Archive\n\n\n\n\n\n\n\n\n\nACROSS THE EQUATOR.\n\n[Frontispiece: TEMPLE, PARAMBANAN.]\n\n\n\n\n  ACROSS THE\n  EQUATOR.\n\n  A HOLIDAY\n  TRIP IN JAVA.\n\n\n  BY\n\n  THOS. H. REID.\n\n\n  KELLY & WALSH, LIMITED,\n  SINGAPORE--SHANGHAI--HONGKONG--YOKOHAMA.\n  1908.\n  [all rights reserved.]\n\n\n\n\nPREFACE.\n\n\nIt was at the end of the month of September, 1907, that the writer\nvisited Java with the object of spending a brief vacation there.\n\nThe outcome was a series of articles in the \"Straits Times,\" and after\nthey appeared so many applications were made for reprints that we were\nencouraged to issue the articles in handy form for the information of\nthose who intend to visit the neighbouring Dutch Colony. There was no\npretension to write an exhaustive guide-book to the Island, but the\noriginal articles were revised and amplified, and the chapters have\nbeen arranged to enable the visitor to follow a given route through the\nIsland, from west to east, within the compass of a fortnight or three\nweeks.\n\nFor liberty to reproduce some of the larger pictures, we are indebted\nto Mr. George P. Lewis (of O. Kurkdjian), Sourabaya, whose photographs\nof Tosari and the volcanic region of Eastern Java form one of the\nfinest and most artistic collections we have seen of landscape work.\n\n\n  SINGAPORE, _July, 1908_.\n\n\n\n\nCONTENTS.\n\n\n  FIRST IMPRESSIONS OF BATAVIA              1\n\n  THE BRITISH IN JAVA                      15\n\n  BOTANIST'S PARADISE AT BUITENZORG        23\n\n  ON THE ROAD TO SINDANGLAYA               33\n\n  SINDANGLAYA AND BEYOND                   42\n\n  HINDU RUINS IN CENTRAL JAVA              49\n\n  THE TEMPLES OF PARAMBANAN                58\n\n  PEOPLE AND INDUSTRIES OF CENTRAL JAVA    65\n\n  THE HEALTH RESORT OF EAST JAVA           73\n\n  SUNRISE AT THE PENANDJAAN PASS           77\n\n  HOTELS AND TRAVELLING FACILITIES         87\n\n\n\n\nFirst Impressions of Batavia.\n\n\nWhen consideration is given to the fact that Java is only two days'\nsteaming from Singapore, that it is more beautiful in some respects than\nJapan, that it contains marvellous archaeological remains over 1,100\nyears old, and that its hill resorts form ideal resting places for the\njaded European, it is strange that few of the British residents\nthroughout the Far East, or travellers East and West, have visited the\nDutch Colony.\n\nThe average Britisher, weaving the web of empire, passes like a shuttle\nin the loom from London to Yokohama, from Hongkong to Marseilles. He\nthinks imperially in that he thinks no other nation has Colonies worth\nseeing. British port succeeds British port on the hackneyed line of\ntravel, and he may be excused if he forgets that these convenient\ncalling places, these links of Empire, can have possible rivals under\nforeign flags.\n\nThere is no excuse for the prevailing ignorance of the Netherland\nIndies. We do not wish it to be inferred that we imagine we have\ndiscovered Java, as Dickens is said to have discovered Italy, but we\nbelieve we are justified in saying that few have realised the\npossibilities of Java as a health resort and the attractions it has to\noffer for a holiday.\n\nMiss Marianne North, celebrated as painter and authoress and the rival\nof Miss Mary Kingsley and Mrs. Bishop (Isabella Bird) as a traveller in\nunfrequented quarters of the globe, has described the island as one\nmagnificent garden, surpassing Brazil, Jamaica and other countries\nvisited by her, and possessing the grandest of volcanoes; and other\nfamous travellers have written in terms of the highest praise of its\nnatural beauties.\n\nIts accessibility is one of its recommendations to the holiday maker.\nThe voyage across the Equator from Singapore is a smooth one, for the\nmost part through narrow straits and seldom out of sight of islands clad\nwith verdure down to the water's edge.\n\nExcellent accommodation is provided by the Rival Dutch Mail steamers\nrunning between Europe and Java and the Royal Packet Company's local\nsteamers, and the Government of the Netherland Indies co-operates with a\nrecently-formed Association for the encouragement of tourist traffic on\nthe lines of the Welcome Society in Japan. This Association has a\nbureau, temporarily established in the Hotel des Indes in Batavia, to\nprovide information and travelling facilities for tourists, not only\nthroughout Java, but amongst the various islands that are being brought\nunder the sway of civilised government by the Dutch Colonial forces.\n\nAs our steamer pounded her way out of Singapore Harbour in the early\nmorning, islands appeared to spring out of the sea, and seascape after\nseascape followed in rapid succession, suggesting the old-fashioned\npanoramic pictures of childhood's acquaintance. One's idea of scenery,\nafter all, is more or less a matter of comparison. One passenger\ncompares the scene with the Kyles of Bute; another with the Inland Sea\nof Japan, at the other end of the world. Yet, this tropical waterway is\nunlike either, and has a characteristic individuality of its own, none\nthe less charming because of the comparisons it suggests and the\nassociations it recalls.\n\nWe spent a good deal of our time on the bridge with the Captain, who was\ncourteous enough to point out all the leading points on his chart.\n\nThe Sultanate of Rhio lies on the port bow, four hours' sail from\nSingapore. Glimpses of Sumatra are obtained on the starboard, and on the\nway the steamer passes near to the Island of Banka, reputed to contain\nthe richest tin deposits in the world. This tin is worked by the\nGovernment of the Netherland Indies, with Chinese contract labour; and\nthe revenue obtained is an important factor in balancing the Colonial\nBudget. It is interesting to note that the Chinese, who have long mined\nfor gold and tin in the Malay Peninsula and Archipelago, were quite\nfamiliar with the rich nature of Banka's soil two hundred years ago, and\nthat tin from this island was then a common medium of exchange in China\nand throughout the Far East wherever the adventurous Chinese merchant\nhad penetrated.\n\nThe visitor landing at Tandjong Priok, the port of Batavia, after his\nexperience of other Far Eastern ports, cannot fail to be struck by the\nexcellence of the arrangements for berthing vessels and for storing\ncargo. We British people are so accustomed to the idea that our ports\nare the best and our trading arrangements unequalled that we are\nastonished when we discover that our shipping and commercial rivals know\nhow to do some things better than ourselves, and that all wisdom is not\nto be found within the confines of England and among the people who are\nproud to own it as their place of birth. Our Far Eastern ports owe their\nsupremacy to geographical position almost entirely. We have realised\nthat during recent years in Singapore, and in our haste to correct the\nmistakes of former officials and residents, the Straits Settlements paid\nrather heavily when they expropriated the Tanjong Pagar Company which\nowned the wharves, docks and warehouses. Tandjong Priok may not handle\nthe shipping that Tanjong Pagar does, but if they were called upon to do\nso, we have not the least doubt that our Dutch neighbours would rise\nreadily to the occasion.\n\nThere is a Customs examination at Tandjong Priok. In our own case, it\nwas a mere formality, the new duty on imported cameras not applying to\nour well-used kodak, since it was being taken out of the country again.\nBut we could not help contrasting to the disadvantage of Singapore the\nexamination of Chinese and other Asiatic passengers. Theoretically, in\nSingapore, there is no Customs service. It is a free port, and so,\ntheoretically, one may land there free of vexatious examinations, such\nas one experiences at some Continental ports or on the wharves at San\nFrancisco. But, as a matter of fact, they who have occasion to walk\nalong the sea front in Singapore may see Asiatic passengers at any of\nthe landing places turning out their baggage in sun or rain, while\nchentings--the hirelings of the rich Chinese Syndicate which \"farms\" or\nleases the opium and spirit monopolies--examine it for opium or spirits.\nThere is no proper landing place, absolutely no proper arrangements for\noverhauling baggage, with the result that these poor Asiatics are\nsubjected to examination under conditions that are a disgrace to a place\nwhich arrogates a front place in the seaports of the world.\n\nThey do things better at Tandjong Priok.\n\nThere is a brief journey by train to Batavia, and there the visitor,\nhaving handed over his baggage to the care of the hotel runners at\nTandjong Priok, ought to take a sado for conveyance to the particular\nhotel he has selected. The word sado is a corruption of \"dos-a-dos.\" The\nvehicle is drawn by a small pony, and is not comparable with the ricksha\nfor comfort, though the long distances may make the ricksha an\nimpossibility in Batavia.\n\n[Illustration: THE TOWN HALL.]\n\nBatavia is favoured in that it has a choice of several good hotels.\nWhoever selects the Hotel Nederland or the Hotel des Indes will say that\nthe other \"best Hotels in the Far East\" have something yet to learn in\nthe accommodation of visitors, general cleanliness, and moderation of\nprices.\n\nOne of the first things one ought to do after arrival is to obtain the\n\"toelatingskaart,\" at the Town Hall. Armed with this document, which,\nmost probably, he will never be called upon to show, the tourist may\ntravel in the interior. Without it, he may have trouble.\n\nBatavia shares with the French ports of Saigon and Hanoi the honour of\nmore resembling a European town than any other ports in the Far East.\nThis, of course, is a matter of opinion, though it is based on\nacquaintance with every port of importance from Yokohama to Penang,\nincluding the principal ports of the Philippines, and we were somewhat\nsurprised, therefore, when expressing this opinion to a Dutch friend,\nwith his reply:\n\n\"When I left Singapore, with its fine buildings I felt I had said\ngood-bye to Europe!\"\n\nA little probing soon showed that it was only the two and three-storeyed\nhouses that created this impression.\n\n[Illustration: HOTEL DES INDES.]\n\nOne has only to stroll along the Noordwijk in the afternoon and evening\nto appreciate the difference between Batavia and Singapore. After\nsundown, so far as Europeans are concerned, with the exception of the\nlittle life seen under the electric light of Raffles Hotel and the Hotel\nde l'Europe, Singapore is a dead place. Hongkong is no better. In\nBatavia it is different. Up to the dinner hour, and after, there is a\nconsiderable amount of life and light and animation, and if it be a\nstretch of the imagination to compare the Noordwijk or the Rijswijk with\nthe Boulevard des Capuchins in Paris, or its open air restaurants with\nthe Cafe de la Paix, it is at least within comparison to say that the\nresemblance to a Continental town is sufficiently marked to be welcome,\nwhile one can have as choice a dinner or supper, with superb wines, in\nStamm and Weijns or the Hotel des Indes as in the best restaurants of\nLondon and Paris. Not the least noticeable feature of all to the\nobservant visitor will be the punctilio and excellence of the waiting of\nthe Javanese table boys. When one saw the carefulness with which each\ndish was served, and the superior nature of the side dishes, one thought\nwith a shudder of the sloppy vegetables, the dusty marmalade, and the\nslipshod waiting of the China boy in some of the hotels it had been our\nmisfortune to patronise in British Colonies.\n\nIn this quarter, the wives and daughters of the Dutch and foreign\nmerchants drive in comfortable rubber-tyred carriages, having first\ndriven to the business quarter to bring home the \"tuan besar\" or head of\nthe family. Greetings are exchanged with friends by the way, and, while\nthe young folks stroll off in happy groups, the elders alight to drink\nbeer or wine at one or other of the famous open-air restaurants. There\nis a general air of prosperity and a spirit of gaiety which one does not\nusually associate with our Dutch cousins in the depressing humid\natmosphere of Holland. One soon catches the spirit of the place the more\nreadily if one has spent any time on the Continent.\n\nOn band nights the Harmonie or Concordia Clubs, two beautiful and\ncommodious buildings replete with every comfort, become the rendezvous\nof old and young, and dancing is kept up till half-past eight o'clock.\nIt must be confessed that it made one perspire to see the dancers tread\na measure to a popular waltz, but there could be no question of the\nenjoyment of those who participated.\n\nThere are two Batavias. There is the old town, founded in 1619 as the\ncapital of the Dutch East Indies upon the ruins of the ancient city of\nJakatra. This is the portion of the town where the business is done,\nwith the famous Kali Besar, the Lombard Street and Fenchurch Street of\nBatavia.\n\nThe quarter is not particularly attractive. But after experience of the\nfilthy Chinese quarters of Singapore, Hongkong and Shanghai, it is\nsatisfying to European self-respect to observe how Dutch officialdom has\nasserted the claims of hygiene and cleanliness upon the Asiatic\nresidents. The objectionable hanging Chinese signboards are noticeably\nabsent in Batavia, as in all other towns throughout Java, and something\nhas been done to make less clamant the odoriferous articles of Chinese\ncommerce. The Dutch have proved that the Chinese are amenable to\nEuropean notions if only firmness is shown by those in authority.\n\nThen there is the residential town, Weltevreden with its broad\ntree-lined avenues and palatial pavilion hotels and private villa\nestablishments.\n\nIn style, the European houses are quite unlike those erected by the\nSpaniards in the Philippine Islands, or the British in the Malay\nPeninsula. They are not raised to any great height from the ground.\nThree or four wide low steps lead on to a capacious white marble\nverandah, the lofty roof of which is supported by shapely pillars with\nGrecian cornices. Upon the polished surface of the ample hall are strewn\nrugs of beautiful design or the fancy straw matting of the East.\nBed-rooms open on either side from this hall, and at the back, opening\nout upon a spacious court-yard or garden filled with gaily \nflowers or stately palms, is another wide verandah where meals are\nserved. The bath-rooms, kitchen, stables, store-rooms and servants'\nquarters lie beyond the garden. There is everywhere a generous\nappreciation of space, and doubtless the good health enjoyed by the\nDutch ladies and their families so markedly in contrast to the British\ncolonists on the other side of the Equator is largely due to the more\ncomfortable homes in which they are settled. In Java, the bath-room is a\nspecial feature, and only those who have travelled much in tropical\ncountries can appraise it at its true value. It is all in keeping with\nthe thorough cleanliness of the Dutch people, a feature which impressed\nitself upon us wherever we travelled throughout the island. Detached\nfrom every house of any pretensions, there is a smaller pavilion. It\nusually stands in the grounds in front and nearer the roadway, and in\nformer times was spoken of as \"the guest house.\" Nowadays, either\nbecause the Hotels are more comfortable than in olden times or because\nthe railway system has led to a style of life that calls for less\nhospitality for travellers, the guest house is more often let to\nbachelors, who find it easier and cheaper to maintain a small\nestablishment of this sort than the bachelor messes or chummeries of\nSingapore and Penang.\n\nWeltervreden may be compared with a gigantic park, and there are\nresidences sufficiently imposing to please the lover of architectural\nbeauty, even if there is no assertive Clock Tower to emphasise by\ncontrast the hovels of Singapore's region of slums. The idea of keeping\nthe various races to their Kampongs may be contrary to British ideas,\nbut in Java it appears to work satisfactorily enough. It is only in\nrecent years that certain British colonies have been allowed to set\napart reservations for European residence, and it would be well if the\nGovernment of the Federated Malay States, before it is too late,\nintroduced the Kampong system in laying out new towns throughout the\nPeninsula.\n\nA motor-car ride through the residential quarter and round the suburbs\nof Batavia gives one a good idea of the extent of the town, and,\nincidentally, of the merging of East and West in the population. Former\nDutch residents have left their impress in more respects than one, and\none result is a half-caste population which takes a much more prominent\npart in the affairs of the island than is the case, so far as we are\naware, in any British Colony. There are pretty forms and beautiful faces\namong this hybrid race, and we are not astonished that succeeding\ngenerations from the land of <DW18>s and canals should form alliances that\nwed them for ever to the sunny soil of Java. East may be East and West\nmay be West, but here at least the lie is given to Kipling's\ngeneralisation, false like most generalisations, as to the impossibility\nof their blending.\n\nThe visitor will find the Museums full of objects of interest. On\nKoningsplein, young Holland devotes itself to recreation, and evidence\nis given here and elsewhere throughout the suburbs of the widespread\npopularity of the English game of football. The Dutch do not follow the\nBritish Colonial custom of sending their children to Europe. Many are\neducated and kept under the home influence in Java, and a fine healthy\nrace of boys and girls is being reared to play its part in the new\nNetherlands created by Dutch enterprise and perseverance. Great as is\nthe Java of the present day, there is justification for believing that\nit has a greater future in store.\n\n[Illustration]\n\n\n\n\nThe British in Java\n\n\nIt is a constant matter of regret to British travellers who have visited\nJava that the island, once in our possession, should have been restored\nto Dutch rule.\n\nIt is not our purpose, however, to discuss the reasons for that\nrestoration, contenting ourselves with the reflection that the capture\nof Java was merely part of the plan for breaking the power of Napoleon\nand destroying his dream of dominating the East. The alliance of\nEuropean Powers having succeeded in encompassing the great Frenchman's\ndownfall, there were doubtless good reasons at the time for reinstating\nthe Dutch in an island where they had been established for two hundred\nyears.\n\nA perusal of the history of the British Expedition against Java brings\ninto strong relief the annihilation of space and the improvements in\nmarine travel during the past century.\n\nIt was on April 18, 1811, that the troopships carrying the first\nDivision, commanded by Colonel Robert Rollo Gillespie, sailed from\nMadras Roads. On May 18, they anchored in Penang Harbour, and on June\n1, at Malacca. Here they awaited the remainder of the flotilla, and were\njoined by Lord Minto, then Viceroy of India; Lieutenant-General Sir\nSamuel Auchmuty, Commander-in-Chief; and Commodore Broughton. While\nhere, the British learned that Marshal Daendels, the Dutch\nGovernor-General, had been recalled, and that General Janssens, with a\nlarge body of troops from France, had landed and taken over the command\nin Java.\n\nMarshal Daendels had been the Governor-General when the Colony was taken\nover by the Crown of Holland from the Dutch East India Company. He has\nleft the mark of his influence upon the Colony to this day, and many of\nthe public works that remain as evidence of the pioneer days were due to\nhis force of character and initiative. Some of his methods may not\ncommend themselves to us in these more humane and enlightened days, any\nmore than they were approved by his great English successor, Sir\nStamford Raffles, such, for instance, as his construction of the\npost-road from Anjer Head to Banjoewangi, a distance of over 700 miles,\nat the cost of from twelve to twenty thousand lives; but it is not\nalways easy to estimate at a distance of a hundred years the peculiar\ndifficulties and conditions under which European Governors administered\nan oriental Colony. If, at times, he exceeded his instructions, as\nBritish Governors also had to do before they came under the thralldom\nof a Colonial Department at the end of a telegraph cable, we can forgive\nmuch in a man who accomplished so much.\n\nSir Stamford Raffles is careful to explain in the preface of his\n\"History of Java\" that as \"in the many severe strictures passed upon the\nDutch Administration in Java, some of the observations may, for want of\na careful restriction in the words employed, appear to extend to the\nDutch nation and character generally, I think it proper explicitly to\ndeclare that such observations are intended exclusively to apply to the\nColonial Government and its officers. The orders of the Dutch Government\nin Holland to the authorities at Batavia, as far as my information\nextends, breathe a spirit of liberality and benevolence; and I have\nreason to believe that the tyranny and rapacity of its Colonial officers\ncreated no less indignation in Holland than in other countries of\nEurope.\"\n\nOn June 11, the British armada set out on the final stage of its\njourney. We can imagine the imposing show it made as it lay in the\nroadstead of Malacca, now shorn of its ancient importance and long since\nsuperseded as the foremost shipping port in the Far East.\n\nThe squadron consisted of four line of battle ships, fourteen frigates,\nseven sloops, eight Honourable East India Company's cruisers,\nfifty-seven transports and several gunboats--altogether over 100 sail.\nComposed equally of European and Indian troops, there were upwards of\n10,000 men under Sir Samuel Auchmuty's command. The European troops\nincluded the 14th, 59th, 69th, 78th, and 89th Regiments of Infantry,\nRoyal Artillery, and Royal Marines, and a small detachment of Royal\nEngineers.\n\nA course was set for a rendezvous off the coast of Borneo, and on August\n4, 1811, a landing was effected at Chillingching, a village about ten\nmiles east of Batavia. To the astonishment of the British Commander, his\nlanding was not opposed, the defending force being concentrated in the\nneighbourhood of Weltervreden and Meister Cornelius, to-day the thriving\nresidential suburbs of Batavia.\n\nGeneral Janssens rejected Lord Minto's summons to surrender.\n\nOn August 10, Batavia was in the hands of the British troops, and on\nthat day, after two hours of hard fighting, Weltervreden was captured,\nthe 78th Highlanders having a heavy casualty list amongst their\nofficers.\n\nThe French troops bravely contended every foot of ground, and battles,\nwith heavy losses on both sides, were fought on August 22, August 24,\nand August 26. Colonel Gillespie, who led the advance in each of these\nengagements, performed prodigies of bravery in the latter fight, for we\nread that \"Colonel Gillespie took one General in the batteries, one in\nthe charge, and a Colonel, besides having a personal affair in which\nanother Colonel fell by his arm.\"\n\nAltogether, the British captured three General officers, 34 field\nofficers, 70 captains and 150 subaltern officers in these fights.\n\nThe rout of the enemy was complete. General Janssens made his escape to\nBuitenzorg, thirty miles distant, with a few cavalrymen and the remnants\nof his army of 13,000 men. He did not remain here long, but fled\neastwards.\n\nA British force was shipped to Cheribon, where a large number of French\nofficers were captured; and the port of Samarang was next attacked, with\nthe object of forcing General Janssens back upon Solo, while the eastern\nend of the island was occupied by another British force. On September\n10, an action was fought outside Samarang, and Janssens, defeated,\nretreated to Fort Salatiga; but eventually, being deserted by his\ntroops, he opened up negotiations for capitulation.\n\nThis must have been a bitter experience for General Janssens, for it was\nnot only the crowning misery of his defeat but marked the end of his\nmilitary career, assuming that his Imperial master retained his power in\nEurope.\n\n\"Souvenez vous, Monsieur,\" Napoleon is reported to have said to him\nupon taking up his appointment, \"Qu'un General Francais ne se laissa pas\nprendre une seconde fois!\"\n\nThe island having been wrested from the French, the British authorities\nset about the reform of the civil administration. This was not to be\naccomplished, however, without a test of strength between the natives\nand their new masters. An act of treachery soon called the troops into\nthe field again.\n\nDuring the Governorship of Marshal Daendels, the Sultan of Djocjakarta\nhad been the most turbulent and intriguing of the native princes, and\nhis conduct immediately after the British occupation gave occasion for\nserious uneasiness. Mr. Stamford Raffles, who had been appointed by Lord\nMinto Lieutenant-Governor of Java in December, 1811, went in person to\nsee the Sultan. A treaty was entered into, under which the Sultan\nconfirmed to the Honourable East India Company all the privileges,\nadvantages and prerogatives which had been possessed by the Dutch and\nFrench authorities. To the Company also were transferred the sole\nregulation of the duties and the collection of tribute within the\ndominions of the Sultan, as well as the general administration of\njustice in cases where British interests were concerned.\n\nThis expedition of Mr. Raffles seems to have had exciting experiences,\nfor we read:\n\n    \"The small British escort which accompanied Mr. Raffles,\n    consisting only of a part of the 14th Regiment, a troop of the\n    22nd Light Dragoons and the ordinary garrison of Bengal Sepoys\n    in the Fort and at the Residency, were not in a condition to\n    enforce terms anyway obnoxious to the personal feelings of the\n    Sultan. The whole retinue, indeed, of the Governor were in\n    imminent danger of being murdered. Krises were actually\n    unsheathed by several of the Sultan's own suite in the Audience\n    Hall where Mr. Raffles received that Prince, who was accompanied\n    by several thousands of armed followers expressing in their\n    behaviour such an infuriated spirit of insolence as openly to\n    indicate that they only waited for the signal to perpetrate the\n    work of destruction, in which case not a man of our brave\n    soldiers, from the manner in which they were surrounded, could\n    have escaped.\"\n\nFor a time, however, an open breach of the peace was averted by the tact\nof Mr. Raffles and the outward appearance of bravery of the officers and\nmen accompanying him.\n\nSeveral expeditions were made into the interior to put down petty\nbrigands, in much the same way as the Dutch are engaged in Flores and\nCelebes to-day, and a more imposing display of military force had to be\nmade in Sumatra.\n\nIn the following year, the Sultan of Mataram in Djocjakarta again became\ntroublesome, and it was found necessary to send a strong expedition\nagainst him. On June 20, the famous Water Castle at Djocjakarta was\ncaptured by assault, and the Sultan taken prisoner. He was exiled to\nPrince of Wales Island (Penang), and the Hereditary Prince was placed on\nthe throne. The ruling native at Solo, who rejoiced in the imposing\ntitle of Emperor, made terms with the Lieutenant-Governor, and peace was\nestablished throughout the island, and was not disturbed seriously\nduring the remainder of the British occupation.\n\nMr. Raffles set himself to establish a more humane administration than\nhad hitherto prevailed, and anyone who wishes to realise the\nthoroughness with which this able administrator set himself to the task\nshould read his \"History of Java.\" It is replete with shrewd\nobservations of the native customs, industries, antecedents, and\nlanguages, and shows how little change has been effected in the\ncharacter and domestic customs of the people during the last hundred\nyears.\n\nThe essence of his policy of administration is contained in the\nfollowing sentence written by him:--\"Let the higher departments be\nscrupulously superintended and watched by Europeans of character; let\nthe administration of justice be pure, prompt and steady;\" and it is\nsatisfactory to one's sense of patriotism to know that that is the\nspirit which pervades British administration in her Crown Colonies\nto-day.\n\n\n\n\nBotanist's Paradise at Buitenzorg.\n\n\nTo the Singaporean visitor to Java there is a melancholy interest in the\nlittle monument erected in the Garden at Buitenzorg by Sir Stamford\nRaffles to the memory of his wife, who died during his residence there.\n\nIn the conditions under which the island was restored to Holland, it was\nstipulated that the monument, in the form of a little Greek temple,\nshould be cared for by the Dutch. The trust has been fulfilled, and\nthose of us who take interest in the historic chances and changes of\nBritain's possessions in the Far East and the personal influence of the\nbuilders of the Empire, can find food for reflection in the sacrifices\nmade by those men and women who are ever found on the Empire's\nfrontiers. The sight of this memorial among the kanari trees in the\ntropical island of Java makes us think of the tablet in the little\nparish church on the hill at Hendon, near which this woman's husband\nlies buried.\n\nThe inscription runs as follows:--\n\n    \"Sacred to the memory of Olivia Marianne, wife of Thomas\n    Stamford Raffles, Lieutenant-Governor of Java and its\n    dependencies, who died at Buitenzorg on the 26th November, 1814.\n\n      \"Oh thou whom ne'er my constant heart\n        One moment hath forgot.\n      Tho' fate severe hath bid us part\n        Yet still--forget me not.\"\n\nThe traveller who has only a fortnight or three weeks to devote to Java\nmust awake betimes. In any event, he must needs be early to take\nadvantage of the express trains, and in our case we had only a day to\ndevote to Buitenzorg, where the Governor-General of the Netherland\nIndies has his palace.\n\nWith the exception of the short run from Tandjong Priok, it was our\nfirst acquaintance with the railway service, and when we saw the crowd\nawaiting to entrain at Weltervreden Station we decided to travel\nfirst-class, contrary to the advice of our friends. It was well we did\nso on this occasion, for the train was overcrowded; but afterwards we\ntravelled only by the second-class, and found it as comfortable as one\ncould wish. Indeed, so few persons travel in the first-class\ncompartments of the trains that we are astonished that any are retained\nby the management. Throughout Java we found the railway service\nexcellent in every respect. The carriages are comfortable. Ample\naccommodation is given for each person. It is possible to stow away a\nconsiderable amount of barang or baggage in the carriages, and full\nadvantage is taken of this facility by the Dutch and native travellers.\nThe lavatory accommodation is better than we have seen it in the fast\nexpresses on the principal lines in England, and on the through service\nexpresses there are restaurant cars where meals may be partaken of at a\nmoderate tariff. We cannot say we always found the food palatable, for\nthe Chinamen who are in charge appear to have a fixed idea that the\n\"beef-stuk,\" which is the piece de resistance, should be served up raw.\nIn course of time, doubtless, the railway management will be able to\nturn its attention to the commissariat arrangements, with a view to\ntheir improvement, and, when they do so, we hope they will leave out the\nbeefsteak altogether and provide more variety and daintier, more\ninviting, and more palatable viands.\n\nA fair rate of speed is maintained, and it is possible to go from\nBatavia to Sourabaya, at the other end of the island, in two days. The\ntrains, of course, as in the Federated Malay States, run only from\nsunrise to sundown, and the through traveller between the two principal\ntowns must sleep the night at Maos, where a commodious pasanggrahan or\nrest-house provides clean, comfortable accommodation and wholesome food.\nOnly on two occasions were we belated on the railway, and both instances\nwere due to the one cause,--a wash-out on the line at Moentilan, the\nresult of a severe thunder and rain storm on the previous day and night.\nThe train was run down cautiously to the gap, passengers crossed over on\na temporary bridge to the train waiting on the other side, and the\nbaggage was transferred by a host of coolies. All this had to be done in\na torrential rain-storm, but the railway officials did all in their\npower to make the conditions as little disagreeable as possible, and the\nonly inconvenience was the late arrival of some of the baggage at\nDjocjakarta.\n\nThere was not much of interest on the morning run to Buitenzorg, but the\nDutch lady who carried on an animated conversation with four gentlemen\nfor the whole of the hour and a half introduced to us the possibilities\nfor expression in the Dutch equivalents of \"Yes\" and \"No.\"\n\nWe had been prepared by Miss Scidmore's book for the beauties of\nBuitenzorg, and for once expectation was more than realised.\n\nThe Dutch Governor-General van Imhoff was certainly well advised when he\nselected this position as the official residence of the\nGovernor-General, and the Dutch horticulturists, than whom there are\nprobably none better, deserve to be congratulated upon the garden city\nthey have created out of the primeval jungle.\n\nPart of the old palace was built by Governor-General Mossel, one hundred\nand fifty years ago, and the original received additions during the\nreigns of Daendels and Raffles. This structure was destroyed by an\nearthquake in 1834, and the new palace, the first glimpse of which one\nreceives across an artificial lake, is a worthy residence for the\nadministrator of the Dutch Indies. The surface of the lake is studded\nwith lotus flowers and victoria regia, and the little island in the\ncentre displays a wealth of the red or rajah palm, feathery yellow\nbamboo, and dark-green foliage which the lake mirrors in ever-changing\npictures.\n\nAn Alma Tadema or a Marcus Stone would revel in the flowers and marbles\nof the palace, with its broad stairs and corridors and fine Ionian\ncolumns and cornices; and a Landseer or a MacWhirter might find endless\nsubjects in the deer park by which it is surrounded.\n\nThe garden is a botanist's paradise. Tropical treasures from Nature's\nstorehouse, collected by successive Directors, are arranged with care\nand precision characteristically Dutch. It was established in 1817 by\nProfessor Reinwardt, and many distinguished botanists who have left\ntheir mark in the scientific world studied here and added to the\ncollections. As may be imagined, the Dutch were not content with a mere\nshow place for tropical specimens, and they established five mountain\ngardens where experiments are conducted, for practical and scientific\npurposes, in the cultivation of flowers, plants, vegetables and trees\nusually found in temperate regions. These gardens are situated in the\nmountains to the south--at Tjipanas, Tjibodas, Tjibeureum, Kadang Badoh,\nand on the top of Mount Pangerango, that is to say, at heights ranging\nfrom 3,500 ft. to 10,000 ft. The garden at Tjibodas remains, and at the\nGovernor-General's summer villa at Tjipanas one might imagine one's-self\nin a private garden in Surrey or Kent.\n\nIn the buildings at Buitenzorg, facilities are afforded for foreign\nstudents, and at the time of our visit a Japanese Professor, from the\nTokio University, who had studied for three and a half years in Berlin,\nwas making an exhaustive investigation on scientific lines. Everything\nthat can be of service to students of botany is to be found here in the\nmuseum, herbarium and library.\n\nThe general herbarium has been arranged on the Kew model. Besides a\nlarge collection of plants made by Zollinger between 1845 and 1858, it\ncontains the valuable collections gathered by Teysmann, between 1854 and\n1870, throughout the Malay Archipelago. Specimens by Kurz and Scheffer\nare also found, together with other recent collections of plants from\nBorneo and adjacent islands. Duplicates from the Herbarium at Kew\nGardens and from several of the more famous European herbaria are to be\nfound here, as well as numerous specimens from the botanical\ninstitutions of the British Colonies.\n\nThe Herbarium Horti contains the necessary materials for the compilation\nof the new catalogue of the Botanic Gardens, and the Herbarium\nBogoriense contains plants to be found in the neighbourhood of\nBuitenzorg.\n\nBesides specimens of fruits, there is a comprehensive technical\ncollection in the Botanical Museum--fibres, commercial specimens of\nrattan, india-rubber, and gutta-percha, barks for tanning purposes,\nPeruvian barks, vegetable oils, indigo samples, various kinds of meal,\nresins and damars. There is also a section devoted to forest and staple\nproduce.\n\nFuller details of the gardens and environs of Buitenzorg may be found in\nthe handbook published by Messrs. G. Kolff and Co., Batavia.\n\nOne need not be wholly a scientific investigator to appreciate the\nbeauties of Buitenzorg. There is here one view which has been described\nover and over again, oftentimes in the language of hyperbole--the view\nof the Tjidani Valley from the verandah of Bellevue Hotel. It is,\nindeed, difficult to avoid the use of extravagant language in the\nattempt to describe this beauty spot of Nature.\n\nThough he was writing of a beautiful woman, F. Marion Crawford might\nhave been describing some beautiful landscape when he wrote in his own\nexquisite style:--\n\n\"I think that true beauty is beyond description; you may describe the\nchangeless faultless outlines of a statue to a man who has seen good\nstatues and can recall them; you can, perhaps, find words to describe\nthe glow and warmth and deep texture of a famous picture, and what you\nwrite will mean something to those who know the master's work; you may\neven conjure up an image before untutored eyes. But neither minute\ndescription nor well-turned phrase, neither sensuous adjective nor\nspiritual smile can tell half the truth of a beautiful living thing.\"\n\nThe noble Roman, prompted to exclaim \"Behold the Tiber\" as he stood on\nthe summit of Kinnoull Hill and gazed upon the fertile valley of\nScotland's noblest stream, saw no fairer sight than this veritable\nGarden of Eden in Equatorial Java.\n\nSeen in the afternoon when the setting sun is casting long shadows over\nthe landscape, the scene in the Tjidani Valley is calculated to arouse\nthe artistic senses of the most insusceptible. Miles away, the Salak\nraises his majestic cone against the blue sky. In the distance, the\nmountain forms a purple background for the picture, purple flecked with\nsoft white patches of floating cloud. Beneath his massive form, colour\nis lost in shadowy but closer at hand are the dark pervading greens of\nthe trees and vegetation, palms and tree ferns and banana trees helping\nby their graceful form to provide the truely tropical features, while\nthe equally graceful clumps of bamboo sway and creak in the light\nbreeze, their pointed leaves supplying that perpetual flutter and\nmovement which one associates with the birches and beeches of one's\nnative land. The cultivated patches on hillside and valley are rich in\ncolour. Here, the yellow paddy is ripening for the sickle; there, it is\nbright green; alongside, the patient buffaloes are dragging a clumsy\nwooden plough through water-covered soil to prepare for the next crop.\nThe lake-like patches reflect weird outlines, and one almost imagines\nthat they catch the brilliant colours from the sun-painted clouds.\n\nDown the valley, crossing the picture from left to right is the\nriver--the Tjidani,--a broad shallow stream when we saw it, in which\nmen, women and children are constantly bathing. From the compact kampong\nnestling among the trees, the native women, clad in bright \nsarongs, came with babies, who take to the water as if it were their\nnatural element. Merry shouts of laughter ascend from the valley as the\nyoungsters splash about and chase each other. Everything suggests\nbeauty and peace and contentment, and as one drinks in the scene it is\nborne in upon one that the comparison with the Garden of Eden is not\ninapt. What could one wish for more than a beautiful, bounteous land and\na happy, contented people!\n\n\n\n\nOn the Road to Sindanglaya\n\n\nLong before sunrise, the sound of merry voices arose from the valley.\nAlready the natives were bathing in the Tjidani, and, when the light\ncame, the primeval life on which the sun had gone down was reproduced in\nthe model-like scene spread out before us. Our kreta for the journey\nover the Poentjak Pass had been ordered for six o'clock, but with\nun-Oriental punctuality it was a quarter-past live when the sound of\ncarriage wheels broke in upon our dreams.\n\nWhile we sipped our morning coffee,--Java hotel coffee has improved\nsince Miss Scidmore anathematised it in 1899,--the sun's rays began to\npeep over the shoulder of the Salak, and dispelled the morning mists on\nriver and valley. The Salak's fretwork crater stood out entirely\nclear--his form a purple background to the picture gradually unfolding\nitself. Nature was everywhere awake. Children's voices in play blended\nwith the songs of early workers proceeding to the fields. Butterflies\nflitted and floated like detached petals from the flowers. Distance\nconverted human figures into larger butterflies, yellow and orange,\npink and blue and red. If it were beautiful in the evening, the scene\nwas enchanting in the morning, and it was with reluctance that we obeyed\nthe summons to early breakfast, and followed our barang into the kreta\nto begin the journey to Sindanglaya.\n\nIt was half-past six o'clock when we were salaamed out of the courtyard\nof the Bellevue by the hotel \"boys.\"\n\nThe kreta was not a handsome affair. In fact it was one of the most\ndisreputable vehicles it has ever been our misfortune to travel in, and\nwhen we made acquaintance of the road it had to travel over we must give\nthe owner credit for an abundant faith in the toughness of the kreta. It\nwas a cross between the carromata of the Philippines and a covered\ndog-cart. There was no aid to mount. By a series of gymnastics we\nmanaged to get into the driver's seat--our own was behind his but also\nfacing to the front. In attempting to get there, a sudden movement of\nthe team sent us plunging into the barang, and, in extricating\nourselves, head came in contact with the roof and hat went overboard.\n\nEventually we went off with a bound along the main street of Buitenzorg,\nscattering the fowls obtaining a precarious living in the roadway, and\nsending cats and dogs and goats flying for safety into the houses.\n\nWe had now time to examine the points of our team. It was composed of\nthree tiny Battak ponies. Two were brown, and one a piebald in which a\ndingy chestnut strove for mastery with a dingier white. No two ponies\nwere the same in size. One was in the shafts; the other two were in\ntraces alongside. They tapered in size from right to left--the piebald\non the left. The giant of the group had a nasty temper, and when lashed,\nas he was frequently during the drive, vented his anger upon the patient\nbrute doing the lion's share of the work in the shafts. Upon the whole\nthey did their work extremely well, for a great deal was asked of them,\nand they scarcely deserved the almost continuous flogging to which they\nwere subjected by our driver.\n\nHaving travelled over the road from Buitenzorg to Sindanglaya by the\nPoentjak, without reserve, we advise pilgrims to Sindanglaya to\npatronise the road from Tjiandjoer. The local guide book remarks with\ntruth: \"The main road to the Poentjak being very steep, it does not\nafford a quick mode of travelling. At Toegoe, an extra team of horses\nmust be added--or karbouws (water buffaloes) used instead of the horses,\nto pull the carriage at a slow pace up the mountain. Good walkers may,\ntherefore, be advised to do this part of the road on foot, which will\ntake them about an hour and a half. By doing so they will be more able\nto admire this marvellous work of Governor-General Daendels.\"\n\nWe suspect there is a touch of Dutch satire in this last remark. We have\ntravelled the road, and we are not prepared to parody the old Scot's\nsaying:--\n\n    \"If you'd seen this road before it was made,\n    You'd lift up your hands and bless General Wade\"\n\nDaendels may have been an admirable gentleman, a brave soldier, and a\nclever administrator, but his engineering skill did not equal his other\nqualities. It would have been much better if the road had never been\nmade. Surely no highway was ever more badly graded, and we are not\nastonished that a practical people like the Dutch set themselves to\nconstruct a more sensible road by way of Tjitjoeroeg and Soekaboemie. We\nhave seen paved mountain paths in China more inaccessible, but not much,\nand when we dashed up to the Sindanglaya Hotel at 12.15, we thought more\nhighly of the team that had pulled us over the Pass than we could have\nbelieved when we formed our first early morning prejudices.\n\nNeedless to say, it is not a road for a motor car. It would be\ninadvisable to adopt this route to Sindanglaya if the party included\nladies. But, if they have a taste for mountaineering, baggage should be\nsent by rail to Tjiandjoer under the care of some of the party, and\ncarriages dispensed with at Toegoe and the remainder of the journey made\non foot. As it was, a good deal of our journey up had to be made on\nfoot over unblinded loose road metal.\n\nGoing down the other side the driver led the ponies for about a quarter\nof a mile, and then joined us in the kreta. That downward trip was the\nmost perilous we ever made in anything that runs on wheels, except a\ntrain journey from Manila to Malolos during the Filipino insurrection in\n1899. Jack London, the Californian novelist, once told us that life\nwould not be worth living if it were not for the thrills. We had more\nthrills than we care to have crowded into one hour on that down-grade\nrun from Poentjak to Sindanglaya. Several times, we retrimmed at the\nrequest of the driver, and we kept the barang from falling upon him,\nwhile he manipulated our three rakish adventurers from Battak. When an\nunusually severe lurch nearly precipitated us into the deep storm-water\nchannel on the left or the carefully-irrigated paddy fields on the\nright, Jehu turned round and grinned a grin of fiendish appreciation,\nwhilst we thanked with fervour the merciful Providence who preserved us\nfrom destruction, and wondered how long one could hold out with a broken\nlimb, without surgical help, should the worst happen. It is the\nunexpected that happens. We got to Sindanglaya without any more serious\ndamage than a bottle of Odol distributed amongst our best clothes.\n\nGovernor-General Daendels seems to have had a high opinion of this\nremarkable highway. We read: \"The obstinacy with which he carried\nthrough his scheme of constructing the main road to the Preanger\nRegencies across this summit is really amazing. He never shrank from the\nterrible death-rate among the wretched labourers, nor from the\ndifficulties and enormous cost to keep such a road in good condition,\nfor, especially in the west monsoon, heavy rain-showers are continually\nwashing the earth off the road. Yet it was by no means necessary.\" Let\nthis be Governor-General Daendels' epitaph!\n\nHad not one's attention been distracted by the eccentric performances of\nthe kreta, one might well have admired the scenery. Close at hand, the\nroad teems with fascinating pictures of native life. Only occasionally\ndoes one see a really beautiful face, but there is a pretty shyness such\nas one seldom sees on the roads of a European country. Although we read\nof the thirty millions of people in Java, there is still, apparently,\nroom for more, and nearly every woman has a brown baby slung upon the\nhip and others dragging on her sarong, or seeking to efface themselves\nbehind her none too ample form. At intervals, old women or young\nchildren keep shop, either in nipa huts or on mats under the shade of a\nkanari-tree. In the kampongs or collections of neat little huts which\npunctuate the way, a pasar (market) is being held, haberdashers with\ncheap glass and fancy wares being in juxtaposition with dealers in\nsarongs and the sellers of fruits and vegetables. On the stoeps of some\nof the houses, groups of women spin or weave cloth for the native\nsarong; some make deft use of the sewing machine of foreign commerce.\n\nThe road is fringed by a variety of trees and plants which only a\nbotanist would attempt to describe. Colour is given to this fringe by\nthe magenta bougainvillea, the red hibiscus, the pale blue convolvulus,\nthe variegated crotons, and the orange and red of the lantana, and at\nplaces the poinsettia provides a predominating red head to the\nhedge-like greenery. Palms and tree ferns and feathery clumps of young\nbamboo are called to aid by Nature's landscape gardener; but they do not\nshut out the verdure-clad ravines that mark a waterway or the terraced\nrice-fields which climb almost to the top of the highest summits.\n\nWe thought we had seen the acme of perfection in rice cultivation and\nirrigation in China and Japan. But here in Java, we have seen more to\nexcite the admiration in this respect than in either of these countries.\nOne can only marvel at the completeness of the system of irrigation.\nRice is in all stages of cultivation, from the flooded paddy field to\nthe grain in the ear being reaped by the gaily  butterflies of\nwomen. Water buffaloes drag a primitive plough through the drenched\nsoil, while the bright-faced young ploughboy, by what appears to be a\nsuperhuman effort, balances himself precariously on the implement.\n\nOn the left, we pass tea gardens, the tufty bushes low to the ground.\nWhat strikes us first is the amazing regularity of the rows and the\ncleanness of the ground. An aroma of tea in the making escapes from the\nroadside factory and agreeably assails our sense of smell as we jolt\npast in our kreta.\n\nWe reached Kampong Toegoe at nine o'clock, refreshed both men and\nbeasts, and harnessed two more ponies with long rope traces to help us\nto the summit of the Pass, which was reached at eleven o'clock. Here we\nmade a deviation on foot to the Telega Warna (Colour-changing Lake)\nwhile the ponies rested for the downward journey. The path is a\ndifficult one, and the lake itself is less interesting than the lovely\nvegetation by which it is surrounded. Ferns and bracken cover the\nhillside, pollipods predominating, orchids cling to tree stems, and\nhigher up, the curious nest-fern and various forms of plant life attract\nattention. Tree is woven to tree by a network of mighty lianas.\n\nThe lake itself lies in what must have been the crater in the\nprehistoric period of activity of Megamendoeng. It is 100 metres in\nwidth, circular in shape, and about 100 fathoms deep. Fish are found in\nthe lake, and they are regarded with veneration by the natives.\n\nThe steepness of the heavily wooded wall that rises hundreds of feet\nsheer round three sides reminds one of the geyser-studded old crater of\nUnzen, in the island of Kyushiu in Japan, \"Its gleaming mirror,\" the\nguide book says, \"exhibits a wonderful luxury of tints and colours,\nshifting and changing whenever the gentle mountain breeze ruffles the\nsmooth surface.\" We did not stay a sufficiently long time to experience\nany wonderful changes on the lake itself, but the surroundings are\nloaded with charm. The visitor to Sindanglaya should certainly not\nneglect to make the trip to the lake. We would recommend an excursion on\nfoot from the hotel.\n\nOnce over the Pass, the view on the other side of the large basin-shaped\nplateau in which Sindanglaya lies is more attractive than on the\nBuitenzorg side, and, as we were to find on the following morning, a\nbetter idea is obtained of the wonderful industry of the people, and the\nremarkable extent to which the cultivation of the mountain <DW72>s is\ncarried on by them.\n\n\n\n\nSindanglaya and Beyond.\n\n\nWe had not gone far on our travels before we realised the\npresumptuousness of our attempt to \"do\" Java in a fortnight. It would\nrequire weeks to drink in all the subtle beauties and influences of\nBuitenzorg, to get the atmosphere of the place; and to derive the\nfullest measure of benefit and enjoyment from the visit to Sindanglaya,\none would require at least a fortnight.\n\nIt will ever be matter for regret that we were unable to devote more\ntime to the beauty spots of Western Java or to make the various\ninteresting and health-giving excursions from Sindanglaya's comfortable\nhotel. We have already said that the ride over the Poentjak Pass should\nbe avoided and the train taken from Buitenzorg to Tjiandjoer. The train\nleaving Batavia (Weltervreden Station) at 7.25 a.m. and Buitenzorg at\n8.44 reaches Tjiandjoer at 12.04. Here, if a carriage has been ordered\nin advance, a representative of the Sindanglaya establishment meets\npassengers, and the journey to the hotel is negotiated in two hours at a\ncost of two and a-half guilders. From Buitenzorg to Sindanglaya the hire\nof a carriage for passenger and baggage is nine guilders; from\nSindanglaya to Buitenzorg it costs seven guilders. The train fare from\nBatavia to Buitenzorg is three guilders for first-class and two guilders\nfor second; from Batavia to Tjiandjoer, it is eight guilders first-class\nand four guilders and seventy-five cents second.\n\nThe hotel, which consists of one main building with a number of small\ndetached pavilions surrounded by roses and other flowers of the\ntemperate zone, is situated on the <DW72>s of the Gedeh, and is 3,300\nfeet above sea level. At this level one is able to move about long\ndistances during the day without becoming exhausted, and in the evening\nthe air is delightfully cool, falling just below 70 degrees the night we\nslept there. There is a tennis court, and the manager spoke of laying\ndown another, and with billiards and skittles in the evening and a hot\nspring swimming bath, near the Governor-General's villa, for healthful\nrecreation in the daytime, one need not feel too much the absence of\ncity life and companionship. The tariff is the moderate one of six\nguilders a day, but it is reduced to five guilders per day when a stay\nof a week or more is made.\n\nThe Governor-General's summer residence, Tjipanas, is here, a quarter of\na mile from the hotel. It is a prettily situated bungalow residence,\nstanding quite close to the main road from Tjiandjoer, and surrounded by\na garden which transports one at once to the south of England. Here, as\nin many other places in Java, the notice appears: \"Verbodden Toegang;\"\nbut a courteous application to the Steward in charge obtains a hearty\nwelcome to inspect the grounds. These are well stocked with dahlias,\nroses, hortensias, begonias, cowslips, sweet williams, wall-flower, and\nother old-fashioned flowers, and the bloom-covered fuschias carried\none's thoughts back to pleasant days spent in Devonshire dales. From the\nlawns sweet-smelling violets perfumed the air. Matchless orchids clung\nto the trees, and the delicate maiden-hair fern held its own with the\nhardier varieties. Dusky fir-trees, groups of Australian araucarias, and\nJapanese oak trees and chestnuts set off the brightness of the flower\nbeds. In the park there is a beautiful pond, from the centre of which a\nfountain throws a crystal spray to catch the sun's rays and dispense a\nwealth of glittering diamonds.\n\nHot water is the literal meaning of Tjipanas, and a hot spring in the\nvicinity of the villa supplies the bath-rooms, as well as the swimming\nbath of the Sanatorium.\n\nThere is a fine view from the villa, but a better prospect is obtained\nfrom Goenoeng Kasoer, some hundreds of feet higher, where a former\nGovernor-General often took his ontbijtberg (or breakfast). It is now\nknown as Breakfast Hill. A silver mine in the neighbourhood was worked\nfor a time by the John Company.\n\nThe mountain garden of Tjibodas, mentioned in a previous article, is\nwell worth a visit. A good walker, starting at six o'clock, can go\nthere, breakfast and be back at the hotel by noon. But the excursion to\nbe taken by everyone who stays at Sindanglaya for any length of time is\nto the falls at Tjibeureum, Kandang Badak and the crater of the Gedeh.\nLadies may make the trip in sedan chairs; gentlemen on foot or on\nhorseback. The falls of Tjibeureum consist of three cataracts, falling\n400 feet down a perpendicular crag, and the winding road passes through\nsome interesting jungle scenery.\n\nFrom Tjibeureum, the path winds up a steep ascent, and through a narrow\ncleft in the rocks, a natural gateway to which the natives have attached\nsome wonderful legends. Hot springs break through the mountain crust and\nrun side by side with crystal-pure cold brooks, as is often the case on\nthe mountains in Japan.\n\nAfter a two and a half hours' climb from Tjibeureum, Kadang Badak (or\nRhinoceros Kraal) is reached. It lies almost half way up the saddle\nwhich connects the Gedeh with the Pangerango, and although there are now\nno traces of pachyderms, it is stated that both this place and the\nTelega Warna were favourite haunts of the rhinoceros not so very many\nyears ago. It is recommended that the climbers should spend the night in\nthe hut here, and ascend the Pangerango (9,500 ft.) at 4 a.m. to see\nthe sun rise. From the top the view is magnificent.\n\nAlong a steep and difficult mountain path, the crater of the Gedeh may\nbe reached in an hour and a half, and the sight of the gigantic crater\nof this majestic volcano is said to be overwhelming and ample\ncompensation for the toilsome ascent. It is about two miles distant from\nthe Pangerango, and forms the still active part of the twin volcano.\nBetween 1761 and 1832 no eruptions occurred, but seven took place in the\ntwenty years following, the most terrible and severe being the eruption\nof 1840. There were again terrible eruptions in 1886 and 1899, when the\nvolcano covered the hillsides with huge stones, one over 150 kilogrammes\nin weight landing three-quarters of a mile away.\n\nThere are several places in the Preanger Region where the visitor may\nelect to stay instead of Sindanglaya, such as Soekaboemi (2,100 ft.)\nwhich has the advantage of being on the railway, Bandoeng and Garoet.\nAll have their own attractions for invalids, and the hotel accommodation\nis spoken of in terms of the highest praise by all who have been there.\n\nWhen we drove away from Sindanglaya at seven o'clock on the following\nmorning, the white crater wall of the Gedeh stood out like a huge lump\nof marble in the morning sun.\n\nOur route lay through tea, coffee and cocoa plantations, and richly\ncultivated country to Tjiandjoer--a thriving little mountain town, with\nan air of prosperity and progress,--where we joined the train at 9.30\na.m. for Padalarang. Here, at 11.10 a.m., a change was made to the\nexpress from Batavia, and Maos was reached at 5.46 p.m. It had been our\nintention to stay overnight at Bandoeng, strongly recommended by Mr.\nGantvoort, the courteous manager of the Hotel des Indes in Batavia, but\nwe pressed on with the intention of devoting more time to the eastern\nend of the island. It was well we did so, for, shortly after leaving\nPadalarang, rain began to fall in torrents, and the afternoon and night\nwere passed in a severe thunderstorm which was to cause us delay. Part\nof the line was washed away near Moentilan, and our train was over three\nhours late in reaching Djocjakarta on the following day.\n\nAt Maos, there is a commodious, well-built, comfortable passagrahan or\ngovernment rest-house, where four of us ate our meal in solemn silence,\nuntil a query by ourselves when the coffee arrived broke the icy reserve\nof the quartette, and opened the way for an interesting conversation.\n\nIt is customary to make fun of English reserve, but our observation\nconvinced us that the Dutch are no whit behind us in that respect where\nfellow-Dutch are concerned. On the other hand, nothing could have\nexceeded the kindness and courtesy with which we were treated from one\nend of Java to the other. Speaking no Dutch, we had looked forward to\nmany tedious days, but our fears were needless, for, wherever we went,\nwe met pleasant English-speaking Dutchmen, who proved the most\nentertaining of companions, and we take this opportunity of\nacknowledging the courteous assistance we received from time to time. On\nthe score of not speaking Dutch or Malay, no English man or woman need\nbe deterred from visiting Java. English is spoken at all the hotels, and\nthough all the train conductors and stationmasters may not do so, there\nis sure to be an educated Dutchman or lady in the car to whom one may\nturn for help, which is always readily given.\n\nOn one occasion, we had an interesting conversation with two native\nofficials attached to the staff of the Sultan at Djocjakarta. These men\nhad never left the island of Java, yet one of them read and spoke\nEnglish with ready fluency and perfect accent.\n\nNext day, in spite of the delay caused by the wash-out on the line, we\nwere able to reach Djocjakarta by tiffin time, and devoted the afternoon\nto the Hindu ruins at Parambanan.\n\n[Illustration: THE BARA BUDUR.]\n\n\n\n\nHindu Ruins in Central Java.\n\n\nA visit to Java would be incomplete did it not include a pilgrimage to\nthe marvellous products of religious fervour which Buddhism reared in\nthe plains around Djocjakarta before it went down before the\nall-conquering onslaught of Moslemism. These ruins testify to an ancient\nart and civilisation and culture and an instinct of creation few are\naware of to-day, and it is hard to resist the temptation to indulge in\nextravagant language when attempting to describe them as they now stand,\npartially restored by the Dutch authorities.\n\nMiss Scidmore has lavished the wealth of her luxuriant vocabulary upon\nthem, but neither she, nor any of her predecessors in the work of\npraise, saw them as they stand to-day--a wonder alike to archaeologist,\narchitect, artist and student of comparative religions. Here in the\ncentre of fertile plains we have the real Java of ancient times.\n\nThe Dutch had been in possession of the island for two hundred years\nwithout discovering the rich deposits hidden beneath the accumulated\nmounds of centuries and buried under a mass of tropical vegetation. To\nthe active mind of Sir Stamford Raffles the discovery was due. He went\nto Java as Lieutenant-Governor in 1811, and during the period it was\nunder his control, he had the mounds explored, the ruined temples\nun-earthed and their historic import co-related with the romantic\nlegends and poetic records rescued from the archives of the native\nprinces. It was due to the investigations of this great Englishman that\nthe date of the construction of the temples was fixed at the beginning\nof the seventh century of the Christian era, and subsequent\ninvestigators (prominent amongst whom must be placed Dr. I. Groneman,\nnow and for many years resident of Djocjakarta and Honorary President of\nits Archaeological Society) agree in accepting this period as\nauthentically proved from the ruins themselves.\n\n[Illustration]\n\nSir Stamford was of opinion that the temples, as works of labour and\nart, dwarf to nothing all wonder and admiration at the great pyramids of\nEgypt; but since his time, it must not be forgotten, much richer\ndiscoveries in ancient art and archaeological lore have been made in\nEgypt and Palestine. Alfred Russell Wallace, Brumund, Fergusson, all\njoin in the chorus of praise, and the latter, in his \"History of Indian\nand Eastern Architecture,\" expresses the opinion that the Boro Budur is\nthe highest development of Buddhist art, an epitome of all its arts and\nritual, and the culmination of the architectural style, which,\noriginating at Barhut a thousand years before--that is more than\ntwenty-one centuries ago--had begun to decay in India at the time the\ncolonists were erecting this masterpiece of the ages in the heart of\nJava.\n\n[Illustration]\n\nTo reach the Boro Budur, one takes the steam tram from Djocja to\nMoentilan. There a dog-cart may be hired for three guilders, and, taking\nthe Temple or Tjandi of Mendoet on the way, the Boro Budur may be\nreached in an hour and a half from Moentilan. Miss Scidmore was able to\nwrite with her customary enthusiasm about this road; but, truth to tell,\nwe found the drive far from pleasant. Until one gets within a quarter of\na mile of the ruins, the surface is bad and some of the small bridges so\ndangerous that we dismounted at the driver's request. The dog-cart,\nalso, is far from an agreeable vehicle in which to travel, and if a\nbetter carriage could be found we would advise its being hired.\nWherever one goes in Java, the public vehicles are in a state of decay,\nfar more disreputable than the gharry of Singapore, and a large number\nof the ponies are decrepit and suffering from open sores. If Java is to\nbecome a tourist country the vehicles should be better supervised.\n\nBefore setting out from Djocjakarta, the visitor should get the hotel\nproprietor to communicate with the stationmaster at Moentilan, with the\nobject of having a more comfortable carriage than fell to our unhappy\nlot through leaving the matter to haphazard.\n\nStrictly speaking, the Boro Budur--which means the collection of\nBuddas--is not a building in the sense that we speak of St. Paul's or\nSt. Peter's. A small hill has been cut down and the earthwork surrounded\nby masonry, uncemented, unjointed, layer upon layer, and there is no\ncolumn, pillar, or true arch. It is supposed that it was built by some\nof the first Buddhist settlers from India as the resting place (dagaba)\nof one of the urns containing a portion of the ashes of Buddha.\n\n[Illustration: BAS RELIEF--BARA BUDUR.]\n\n[Illustration: BAS RELIEF--BARA BUDUR.]\n\nIt is difficult to describe it briefly, but the following extract from\nMiss Scidmore's book seems to us to convey the best idea of the\nstructure in general terms:--\n\n    \"The temple stands on a broad platform, and rises first in five\n    square terraces, inclosing galleries or processional paths\n    between their walls, which are covered on each side with\n    bas-relief sculptures. If placed in single line, these\n    bas-reliefs would extend for three miles. The terrace walls hold\n    four hundred and thirty-six niches or alcove chapels, where\n    life-size Buddhas sit serene upon lotus cushions. Staircases\n    ascend in straight lines from each of the four sides, passing\n    under stepped or pointed arches, the keystones of which are\n    elaborately carved masks, and rows of sockets in the jambs show\n    where wood or metal doors once swung. Above the square terraces\n    are three circular terraces, where seventy-two latticed dagabas\n    (reliquaries in the shape of the calyx or bud of the lotus)\n    inclose each a seated image, seventy-two more Buddhas sitting in\n    those inner, upper circles, of Nirvana, facing a great dagaba,\n    or final cupola, the exact function or purpose of which as key\n    to the whole structure is still the puzzle of archaeologists.\n    This final shrine is fifty feet in diameter, and either covered\n    a relic of Buddha, or a central well where the ashes of priests\n    and princes were deposited, or is a form surviving from the\n    tree-temples of the earliest primitive East when nature-worship\n    prevailed. The English engineers made an opening in the solid\n    exterior, and found an unfinished statue of Buddha on a platform\n    over a deep well-hole.\"\n\n[Illustration]\n\nWe read this description among others before we visited the Boro Budur,\nand must confess that from none of them did we get a correct idea of\nwhat we were to see. It must be seen to be realised. Not even\nphotographs give a true conception of the ornate character of the\ndecorative stonework--the hard but freely-worked lava stone having lent\nitself easily to the chisel. Like Cologne or Milan Cathedrals, it must\nbe examined minutely to grasp the elaborateness of the sculptured work,\nbut, unlike either of these, it does not produce an immediate impression\nof grandeur and religious elevation. It is unlike any of the temples in\nJapan, or, indeed, anywhere, though Ceylon and India may suggest\ncomparisons.\n\nWhat will strike the visitor as he perambulates these miles of\nsculptured terraces is the complete absence of any offensive or indecent\nfigure. Mere nudity is not, of course, an outrage to the artistic soul;\nbut here there is not even a nude or grotesque figure. Each is draped in\nthe fine flowing robes of the East, not in monotonous regularity but\nsuggestive of prince and peasant, princess and maids, down even to the\njewels they wear. Strangely enough, no particularly Javanese type of\nface or figure is represented--all are Hindu, Hindu-Caucasian and pure\nGreek.\n\nIt is not our purpose to give elaborate details of this work of\nreligious art. The visitor may obtain at Djocjakarta a copy of Dr.\nGroneman's learned treatise on the subject, a treatise which will teach\nhim something about Buddhism as well as the Boro Budur, of which Dr.\nGroneman has made an exhaustive study. With his guide, the sculptures\nbecome an open book to the visitor.\n\nIt is more archaeological than descriptive, however, and we must\nacknowledge our indebtedness again to Miss Scidmore for the following\npassage to show the scope of the sculptures:--\n\n[Illustration]\n\n    \"The everyday life of the seventh and eighth century is\n    pictured--temples, palaces, thrones and tombs, ship and houses,\n    all of man's constructions are portrayed. The life in courts and\n    palaces, in fields and villages, is all seen there. Royal folk\n    in wonderful jewels sit enthroned, with minions offering gifts\n    and burning incense before them warriors kneeling and maidens\n    dancing. The peasant ploughs the rice-fields with the same\n    wooden stick and ungainly buffalo, and carries the rice-sheaves\n    from the harvest field with the same shoulder poles, used in\n    all the farther East to-day. Women fill their water-vessels at\n    the tanks and bear them away on their heads as in India now, and\n    scores of bas-reliefs show the unchanging costumes of the East\n    that offer sculptors the same models in this century. Half the\n    wonders of that great three-mile-long gallery of sculptures\n    cannot be recalled. Each round disclosed some more wonderful\n    picture, some more eloquent story. Even the humorous fancies of\n    the sculptors are expressed in stone. In one relievo a\n    splendidly caparisoned state elephant flings its feet in\n    imitation of the dancing girl near by. Other sportive elephants\n    carry fans and state umbrellas in their trunks; and the marine\n    monsters swimming about the ship that bears the Buddhist\n    missionaries to the isles have such expression and human\n    resemblance as to make one wonder if those pillory an enemy with\n    their chisels, too. In the last gallery, where, in the progress\n    of the religion, it took on many features of Jainism, or\n    advancing Brahmanism, Buddha is several times represented as the\n    ninth avatar, or incarnation, of Vishnu, still seated on the\n    lotus cushion and holding a lotus with one of his four hands.\"\n\nIn all probability, the masonry was shaken down by an earthquake, the\nBoro Budur being near three volcanoes. Restorative and preservative work\nis now being carried on by the Government, and some of the smaller\ntemples in the Djocja district are restored in the original design.\n\n[Illustration: THE BARA BUDUR--ONE OF THE GALLERIES.]\n\n[Illustration: THE SMEROE--13,000 FEET HIGH.]\n\nThere is a small hotel at the Boro Budur where one is recommended to\nstay when studying details, and we can well believe that sunrise as seen\nfrom the summit is a sight one should never forget. We saw it in the\nearly afternoon when the heat vapours from the noontide sun partially\nobliterated the landscape, but even so it was impressive. Except on the\nright, where the mountains close in the horizon, the eye has a range of\nmany miles over fertile alluvial plains, studded with coco and banana\nand palm trees, and every other patch of ground cultivated \"like a tulip\nbed.\" Miss Marianne North, whose collection of paintings in Kew Gardens\nmay be familiar to some of our readers, wrote of this view: \"The very\nfinest view we ever saw.\"\n\n\n\n\nThe Temples of Parambanan.\n\n\nThere are other Buddhist ruins in the neighbourhood of the Boro Budur;\nbut the other more important collection is scattered over the region\nbetween Djocjakarta and Soerakarta. One small temple, the Tjandi Kali\nBening, is reputed to be the gem of Hindu art in Java. This we did not\nsee; but, on another day, in a victoria drawn by four small ponies, kept\ngoing by the wild gr-r-r-ee gr-r-r-eeing of our native running footman,\nwe drove to the scattered temples on the Plain of Parambanan, where,\nwith the help of another archaeological guide by Dr. I. Groneman, we were\nable to appreciate the beauties of these 1100-year-old centres of\nancient religious devotees. These temples are the most interesting in\nthe country, though lacking the extent and grandeur of the Boro Budur.\nThough they do not contain a single genuine Buddha figure, but many\nimages of Brahmanic gods, Dr. Groneman says there are many reasons to\njustify the opinion that they were built by Buddhists, probably over the\nashes of princes and grandees of a Buddhistic empire.\n\nIn his report to Sir Stamford Raffles on these Parambanan ruins, Captain\nGeorge Baker, of the Bengal establishment wrote:--\"In the whole course\nof my life, I have never met with such stupendous and finished\nspecimens of human labour and of the science and taste of ages long\nsince forgot, crowded together in so small a compass, as in this little\nspot, which, to use a military phrase, I deem to have been the\nheadquarters of Hinduism in Java.\"\n\nIn Volume XIII of the \"Asiatick Researches or Transactions of the\nSociety instituted in Bengal for inquiring into the History and\nAntiquities, the Arts, Sciences and Literature of Asia\" (Calcutta,\n1820), Mr. John Crawfurd, who, apparently, visited Java in 1816, gives a\nlong and interesting description of the ruins on the Plain of\nParambanan. He describes the locale as ten miles from Djocjakarta, a\nvalley lying between Rababu and Marapi to the north and a smaller\nsouthern range of high land.\n\nA few of the ruins consist of single isolated temples, but the greater\nnumber are in groups, rows of small temples surrounding larger temples.\n\nThe shape of the smaller temples is worthy of observation. From the\nfoundation to the lintels of the doors, they are of a square form. They\nthen assume a pyramidal but round shape, and are decorated around by\nsmall figures resembling Lingas, while a larger Linga surmounts the\nwhole building, forming the apex of the temple.\n\nInvariably, the sites of the temples are adjacent to abundant supplies\nof clear water so much desired by the Hindus and so necessary to the\nperformance of the ritual. Beside two rivers of the purest water, there\nis between the villages of Parambanan and Plaosan a small tank,\nevidently an appendage to the temples. This little piece of water is a\nsquare of about 200 feet to the side. The ground around it is elevated,\nand there is every appearance of its being an artificial excavation. The\nwhole tank, when visited by Mr. Crawfurd, was covered with blue lotus,\nthe flower of which is so conspicuous an ornament of the sculptures of\nthe temple.\n\nThen, as now, there was no evidence of Hindu descendants of the builders\nof these religious houses and places of worship, but the Javanese are as\ntolerant of various religious cults as the Chinese or the Japanese, and\nthe visitor need not be surprised to find native visitors making what\nappears to be a pilgrimage to some particular shrine.\n\nMr. Crawfurd found barren women, men unfortunate in trade or at play,\npersons in debt and sick persons propitiating the Goddess Durga,\n\"smeared with perfumed unguents or decked with flowers.\" This worship,\ntoo, was not confined to the lower orders. His Highness the Susuhunan\nwhen meditating an unusually ambitious or hazardous scheme made\nofferings to the image.\n\nThese temples are built of a hard dark and heavy species of basalt, the\nchief component of the mountains of Java. The stone is usually hewn in\nsquare blocks of various sizes, as is the case with the Boro Budur. The\nrespective surfaces of the stones which lie on each other in the\nbuilding have grooves and projections which key into each other as in\nthe best masonry work to-day. They are regularly arranged in the walls\nin such a manner as to give the greatest degree of strength and solidity\nto the structure, and nowhere is cement or mortar utilised. There are no\nhuge pillars or single blocks such as may be seen in other prehistoric\nedifices, and neither in boldness of design nor imposing grandeur have\nthe temples presented any difficulties to the builders. There is nothing\nupon a great scale, nothing attempted outside the reach of the most\nobvious mechanical contrivance or the most ordinary methods of common\ningenuity. The chief characteristic is the minute laboriousness of the\nexecution. Nevertheless, the temples excite the imagination, and send\nthe thoughts back to those primeval days when men sought to express\ntheir religious feeling through these elaborate monuments of hewn stone.\n\nThe Tjandi Kalasan, one of the most beautiful of the temples, is the\nonly ruin in Central Java of which the exact date of construction has\nbeen learned with any degree of accuracy. This was ascertained from a\nstone found in the neighbourhood, inscribed in nagari characters. Two\nversions of the inscription were made--one by the Dutch scholar, Dr. J.\nBrandes, and the other by the Indian, Dr. R. G. Bhandarkar.\n\nDr. I. Groneman makes use of both versions to compile the following:--\n\n    \"Homage to the blessed (or, reverend) and noble Tara.\n\n    \"May she,--the only deliverer of the world, who, seeing how men\n    perish in the sea of life, which is full of incalculable misery,\n    is sure to save them by the three means--grant you the wished\n    for essence, the salvation of the world by the Lord of gods and\n    men.\n\n    \"The guru (_i.e._ teacher) of the Sailendra prince erected a\n    magnificent Tara temple. At the command (or, the instance) of\n    the guru, the grateful ----(?) made an image of the goddess and\n    built the temple, together with a dwelling (vihara, monastery)\n    for the monks (bhikshus) who know the great vehicle of\n    discipline (Mahayana).\n\n    \"By authorisation of the king, the Tara temple and the monastery\n    for the reverend monks have been built by his counsellors, the\n    pangkur, the tavan, and the tirip (old Javanese civil officers,\n    perhaps soothsayers or astrologers).\n\n    \"The deserving guru of the Sailendra king built the temple in\n    the prosperous reign of the king, the son of the Sailendra\n    dynasty.\n\n    \"The great king built the Tara temple in honour of the guru (to\n    do homage to the guru) when 700 years of the Saka era were past.\n\n    \"The territory of the village of Kalasa was bestowed on the\n    congregation of priests (monks) in the presence of the pangkur,\n    the tavan and the tirip, and the village chiefs (as witnesses).\n\n    \"This great (incomparable) endowment was made by the king for\n    the monks. It is to be perpetuated by the (later) kings of the\n    Sailendra dynasty, for the benefit of the successive reverend\n    congregations of monks, and be respected (maintained) by the\n    wise pangkur, the good tivan, the wise tirip and others, and by\n    their virtuous wives (according to Dr. Brandes, but \"their\n    virtuous foot-soldiers\" according to Dr. Bhandarkar).\n\n    \"The king also begs of all following kings that this bridge (or,\n    dam) of charity, which is (a benefit) for all nations, may be\n    perpetuated for all time.\n\n    \"May all who adhere to the doctrine of the Jinas, through the\n    blessings of this monastery, obtain knowledge of the nature of\n    things, constituted by the concatenation of causes (and\n    effects), and may they thrive.\n\n    \"The ---- prince once more requests of (all) future kings that\n    they may protect the monastery righteously.\"\n\nThis inscription, showing clearly that the temple was consecrated to\nTara, the sakti of the deliverer of the world, the fourth Dhyani Buddha,\nAmitabha, the Tara of the Buddhists of the Northern Church (Mahayana, or\nthe \"Great Vehicle\"), leads Dr. Groneman to the opinion that this\nparticular temple was completed in the year 701 of the Saka era, or 779\nof the Christian era. No trace of the Tara image was found; but this is\nnot to be wondered at when we note the presence of other images in the\ngardens of private residences in Djocjakarta, and even farther afield,\nand remember the destruction wrought by foreign soldiers and foreign and\nnative vandals.\n\n\n\n\nPeople and Industries of Central Java.\n\n\nIn the plains going eastward through Central Java from the Preanger\nRegencies to the mountains of the Teng'ger Region, one cannot fail to be\nstruck by the remarkable change in the appearance of the natives. The\nSoendanese of the West may not have the resource and thoughtfulness of\nthe people of the plains, the Javanese, but they have brightness and\nvivacity which make them more attractive. Their bent of mind is\nreflected in the bright colours of their dress. In this and other\nrespects, they resemble the Japanese women. In the plains, sombreness of\ndress is a characteristic--the browns of Mid-Java changing to an almost\nuniversal dark blue in the west, reminding the traveller of the Chinese\nand the inhabitants of the southern Japanese islands.\n\nEverywhere, the male Javanese carry the kris or native knife in the\ngirdle. There is much variety in the blades, handles and sheaths of\nthose weapons, real native damascene blades costing considerable sums.\nOne taking a superficial trip through the island is at a loss to\nunderstand why the natives should be armed. According to all accounts,\nthey are a peaceably inclined people, and give their Dutch rulers very\nlittle trouble; and if they were at all quarrelsome amongst themselves,\nthe handy weapon would be a source of grave danger. In course of time,\nperhaps, the knife will disappear as did the sword of civilised Europe a\ncentury or more ago. A traffic in Birmingham manufactured krises and\nknives is done at Djocjakarta and Soerakarta, as well as at Samarang,\nSourabaya and Batavia, and anyone who wishes to make a collection of\nnative weapons should be careful to have the assistance of an expert to\ndetect the sham from the real.\n\nThe same remark applies to the purchase of sarongs. The ordinary sarong\nof commerce is manufactured in Lancashire, whence an excellent imitation\nof the native manufacture is exported. Tourists are also catered for in\na native block-stamped variety, which is at least a colourable imitation\nof the real article. Wherever we went, however, we could see that the\nnative art had not been lost entirely. Women sit outside their little\nhuts by the roadside tracing the most elaborate designs in brown and\nblue dye upon the cloth with tiny funnel-shaped implements.\n\nThis cloth is styled batik. According to the ground of white, black or\nred, it is known as batik latur puti, batik latur irang, or batuk latur\nbang. To prepare it to receive the design, the cloth is steeped in rice\nwater, dried and calendered. The process of the batik is performed with\nhot wax in a liquid state applied by means of the chanting. The\nchanting is usually made of silver or copper, and holds about an ounce\nof the liquid. The tube is held in the hand at the end of a small stick,\nand the pattern is traced on both sides of the tightly drawn suspended\ncloth. When the outline is finished, such portions of the cloth as are\nintended to be preserved white, or to receive any other colour than the\ngeneral field or ground, are carefully covered in like manner with the\nliquid wax, and then the piece is immersed in whatever  dye may\nbe intended for the ground of the pattern. The parts covered with wax\nresist the operation of the dye, and when the wax is removed, by being\nsteeped in hot water till it melts, are found to remain in their\noriginal condition. If other colours are to be applied, the process is\ngone over again. It will thus be seen that a considerable amount of\nskill is required. In the ordinary course, the process of the batik\noccupies about ten days for common patterns, and from fifteen to\nseventeen days for the finer and more variegated.\n\nSome of the sarongs worn by the native aristocracy and the European\nladies are not only beautiful in pattern and working but most expensive\nin price.\n\nIn our excursions in the neighbourhood of Djocjakarta, we had ample\nopportunity of seeing the industry of the Javanese. Wherever one went,\nthere were long processions of stunted women bravely carrying enormous\nburdens on their backs, often with a baby slung in the slandang astride\nthe hip. The cheery, coquettish look of the Soendanese was absent here.\nAll seemed to be borne down by the seriousness of a strenuous physical\nlife. No songs arose from the fields; scarcely a head was raised from\nthe laborious planting of tufts of paddy roots as our kreta rattled\npast. While mothers toiled in the fields, children played near the\nroadways, or now and then assisted their parents.\n\nWe were surprised to see in these fertile plains how prevalent goitre is\namongst the women. In the drive from Moentilan to the Boro Budur, at\nleast one in twenty were so afflicted. We commented on this fact to a\nnative official while waiting for our tram at Moentilan, and he assured\nus that it is remarkably prevalent amongst the common people, but that\nthe men do not suffer in the same proportion as the women. The disease\nis named \"kondo\" by the Javanese. We do not know whether any scientific\ninvestigations into the disease have been carried out by the Dutch\nofficials; but it would be interesting to know why it should be so\nprevalent in this area. Goitre is usually associated with people living\nin mountainous regions, yet we never noticed it in the Preanger and\nscarcely at all on the mountains of East Java.\n\nSince the above was written, we have had an opportunity of consulting\nSir Stamford Raffles' History of Java. He found goitre prevalent in both\nJava and Sumatra, but is careful to explain that it was observed in\ncertain mountainous districts. The natives ascribed it to the quality of\nthe water, but, says Sir Stamford, \"there seems good ground for\nconcluding that it is rather to be traced to the atmosphere. In proof of\nthis, it may be mentioned that there is a village near the foot of the\nTeng'ger mountains, in the eastern part of the island, where every\nfamily is afflicted by this malady, while in another village, situated\nat a greater elevation, and through which the stream descends which\nserves for the use of both, there exists no such deformity. These wens\nare considered hereditary in some families, and seem thus independent of\nsituation. A branch of the family of the present Adipati of Bandung\n(1811-15) is subject to them, and it is remarkable that they prevail\nchiefly among the women of the family. They never produce positive\nsuffering nor occasion early death, and may be considered rather as\ndeformities than diseases. It is never attempted to remove them.\"\n\n[Illustration: SULTAN OF DJOCJA'S SOLDIERS.]\n\nWe reached Djocjakarta in the ordinary way through Maos. It may be that\ncircumstances may take the traveller off the beaten track, and we are\nindebted to a friend for the following brief description of the trip\nfrom Samarang to Djocja over the mountains:--\n\n    \"The usual journey from Samarang to Djocjakarta is made by way\n    of Solo (Soerakarta), but the route is devoid of interest, the\n    railway running through low country under rice cultivation. I\n    would suggest the far more interesting route via Willem I.\n    Starting at 5.57 a.m. or 8.17 a.m., Djocja is reached at 2.16\n    p.m. or 5.10 p.m. The 10.50 a.m. train, I found, went only as\n    far as Magelang, so I started at 2.9 p.m., and, after a\n    delightful run, reached Kedoeng Djattie, a fine junction\n    station, where we changed cars. The next two hours' run is\n    through foot hills, strips of forest and lovely vegetation,\n    glimpses being obtained every little while of pleasant valleys,\n    rice fields and distant hills as the train climbed up to Willem\n    I. This point we reached about 5 p.m., in time to enjoy the\n    refreshing cool breezes and to admire the beautiful view and\n    sunset on a small mountain opposite the hotel.\n\n    \"Next morning, I caught the train (8.54 a.m.,) which leaves\n    Samarang at 5.57, and after a short run reached a station where\n    our engine was changed for one working on the cog-wheel system,\n    the grade being too heavy for the ordinary locomotive. The train\n    winds and circles round hills cultivated, for the most part, to\n    their summits. Upwards we climbed till we were in the clouds and\n    the air became quite bracing and invigorating. Tiffin should be\n    ordered through the guard before starting from Willem I., and it\n    will be handed into the train.\n\n    \"It was about one o'clock when we reverted to the ordinary\n    locomotive, and began the descent to Djocja, through Magelang.\n    To anyone who has to visit Samarang, I would recommend this\n    trip.\"\n\nThe principal sight of Djocja itself is the Water Castle. This trip need\nnot occupy more than a couple of hours, and its appreciation depends\nupon the taste of the visitor. Earthquakes have played havoc with the\nbuildings, but sufficient is left in the way of tunnels, grottoes,\nbathing ponds and dungeon-like rooms. Everywhere are signs of decay and\ndesolation; nevertheless, it is possible, with a little knowledge of\ncomparatively recent Javan history, to reconstruct the scenes enacted\nhere in the days when the native sultans were more powerful in the land\nthan they are to-day. For a small fee, a native pilots one through the\ncarved archways, underground halls and subways and cells. As one stands\nin the large banqueting hall, it is possible to conjure up the\nceremonials of a past age, and, in the mind's eye, to group retainers\nround the Sultan and the members of his harem, while gaudily dressed\ncourtesans sang and danced for the entertainment of \"the quality.\"\n\n\n\n\nThe Health Resort of East Java.\n\n\nTosari on the Teng'ger mountains was the goal of our travels. We were\nanxious to escape from the heat of the plains, for the sun had now\ncrossed the Equator, Java was in its summer season and the rains might\ncome any day. From Djocjakarta, we should have arrived in Sourabaya in\ntime for riz-tafel, but the wash-out at Moentilan still caused a delay\nof traffic and we were two hours late in reaching our destination.\n\nSourabaya is the most important port and business centre of Java, but\nthis fact notwithstanding many of the foreign business houses still\nmaintain their headquarters in Batavia. As a place of residence, each\nhas its good points, and those who have lived in both are divided in\npreference. Possibly we were not in either long enough to form a lasting\nopinion, but we stayed so long in Sourabaya that we prefer Batavia. It\nwould be sheer ingratitude, however, not to acknowledge the hearty\nwelcome we received from the British colony in Sourabaya, and the\npersonal help of members of that community. Here where the principal\nbusiness of Java is conducted, as elsewhere throughout the Far East, it\nwas satisfying to one's patriotism to see the respect in which British\ncommercial enterprise and integrity is held by native and European\nalike, and that the most cordial good feeling exists on all sides.\n\nTo reach Tosari, the visitor proceeds first of all by train to\nPasoeroean, leaving Sourabaya (Goebeng Station) at 6.42 a.m., and\nreaching Pasoeroean at 8.23. Here a single-pony carriage is engaged\n(two and a-half guilders) as far as Pasrepan, where a change is made to\na two-pony carriage (three guilders). This conveyance takes one to\nPoespo, 2,600 feet above sea-level. A halt is made for tiffin in this\ndelightful little hotel, whose pleasant looking proprietress,\nunfortunately, does not speak English. The remainder of the journey to\nthe Sanatorium (6,000 feet) is made in the saddle or by sedan chair. Of\nthis ride and a subsequent excursion we have painful recollections, but\nanyone accustomed to the saddle will enjoy this ascent through mountain\nscenery and vegetation, and even more the morning trip down to Poespo,\nthrough the forest, when returning to Sourabaya.\n\nTosari has been described as the Darjeeling of the Netherland Indies.\n\nHere within four days' journey from Singapore, one may obtain a complete\nchange of climate, and if there were only more frequent direct steamer\ncommunication between Singapore and Sourabaya, we predict with\nconfidence that Tosari would become a favourite health resort for those\nwho live on the northern side of the Equator. The rooms are comfortable,\nthe food is good, the facilities for amusements at nightfall are ample,\nthe walks and excursions are inexhaustible and the views are\nmagnificent. The tariff (seven guilders per day--$4.90 in Singapore\ncurrency) is higher than that of any other hotel in Java, but those who\nintend to stay for a fortnight or more could probably arrange more\nfavourable terms.\n\nThere is a resident doctor who has graduated in the Schools of Tropical\nMedicine, and when we were in Tosari there were visitors from Burma,\nSiam, Singapore, Penang, and all parts of Java, recruiting from malaria\nand other ailments peculiar to Far Eastern residence. But they were not\nall invalids, and formed a bright, companionable party.\n\nThe Teng'gerese who people this mountainous region are a race apart.\nTheir religion is a mixture of paganism and Buddhism, and, though\nreputed to be kind and honest, they are an ignorant, uncouth, uncultured\npeople. They dwell _en famille_ in large square houses without windows,\nin isolated kampongs on the projecting ridges of the mountains. The door\nof each house is on the side nearest the Bromo crater, and as if\ntradition gave them cause to fear another destructive eruption they\nworship this volcano. Dirt prevails everywhere, and in consequence of\nthe cool climate and the scarcity of water they seldom bathe, a fact\nthat is very noticeable after one's acquaintance with the people of the\nplains.\n\nTo go to Tosari without seeing the Bromo is tantamount to going to Rome\nwithout entering St. Peter's. The journey is made on pony or in a sedan\nchair, by way of the Moengal Pass and the Dasar or Sand Sea, which is in\nreality the enormous Teng'ger crater, inside of which there are three\nmore craters, the Bromo being the only one showing signs of activity.\n\nA better view and more impressive is obtained from the Penandjaan Pass,\na description of which is given in the next chapter.\n\nAnother trip worth making is to the lakes in the saddle-back mountain\nbetween the Teng'ger and the Semeroe. From this high plateau, the ascent\nof the Semeroe or Mahameroe is fairly easy and will prove attractive to\nthose who are fond of mountaineering. It is the highest volcano in Java\nand has a perfect cone. The crater, from which smoke and ashes are\nconstantly ejected, is not on the summit but is formed on the south-east\nside.\n\nThe visitor who does not wish to retrace his steps to Poespo and\nPasrepan may return to the plains by way of Malang or Lawang through\nbeautiful sub-tropical and tropical mountain scenery.\n\n\n\n\nSunrise at the Penandjaan Pass.\n\n\nWhen a sharp rap came to our door at two o'clock in the morning to\nsummon us for a ride to the Penandjaan Pass, we repented the rash\npromise to carry out this over-night project to see the sun rise. It was\nno use to curl one's-self up under two heavy blankets and pretend that\nwe had not heard. The \"jongus\" was insistent. Up we had to get, effect a\nhasty toilet in ice-cold water by the aid of a flickering lamp, and step\ninto the outer darkness and mount the pony waiting beside our bedroom\ndoor.\n\nUnfamiliar constellations shed a cold light on the hillside.\n\nOur thickest clothing was penetrated by a searching though slight\nbreeze, as our little rat of a pony, guided by the syce, clambered\nbravely up the brae that led through Tosari village.\n\nThe road bore away to the left, and we were soon slipping and jolting\ndown a mountain path that sank into a crater-like ravine. It was like a\ndescent into the infernal regions. Disaster seemed inevitable. A mistake\nby the pony or the slightest lurch would have precipitated us down some\nhundreds of feet; but the guide knew his way and so did the pony, as,\nsure-footed and cautious, it picked its way, first on one side of the\nroad and then on the other, descending, descending, lower and lower,\nwhere the pale light failed to penetrate. The hill on the other side\nloomed so high that one could not believe there was a way out. Pit-pat,\npit-pat went the pony with steady step, now on hard road now on yielding\nlava mud, across fragile bamboo bridges covered with bamboo lathing,\ndown, down, down till at last we reach the ford. The seat was not an\neasy one for the unaccustomed rider, whose hands and feet were chilled\nalmost beyond feeling by the unwonted cold. But it was arm-chair ease\ncompared with the experience on the other side, as the pony pluckily\npounded his way up the zigzag path for the summit of the hill. How\neither guide or pony could see a path will ever remain a puzzle. The\nover-hanging vegetation blotted out any recognisable landmarks; not even\nthe ribbon of a road was visible to the eye. But the top was reached,\nand believing we were now on the level road for Penandjaan we tried to\nopen up conversation with our guide.\n\nIt is not easy to carry on a connected conversation with a native of the\nTeng'ger when one's Malay vocabulary consists of about twenty words--and\nhalf of these numerals--and the native's knowledge of the English\nlanguage, as one soon learned, consists entirely of \"Yes\" and \"No.\" Yet,\nit is wonderful what one will attempt in the dark--the loneliness was\nso overpowering that one felt compelled to break the awesome silence.\n\n[Illustration: ROAD TO TOSARI.]\n\nBut the conversation soon flagged, and one was thrown back upon one's\nown thoughts. And as the road once again shaped for another crater-like\nravine, plunged in inkier darkness and shrouded in solemn stillness,\nthoughts surged rapidly through one's mind. The first thing that had\nattracted our attention as we mounted our pony was the delicious smell\nof roses in the grounds of the Tosari Hotel. Since nothing could be\nlearned from the syce, nothing could be seen, nothing could be heard\nexcept the occasional bark of a dog from a remote hut on the hillside or\nthe tuneful tingle of a bell on the neck of the uneasy occupant of an\nunseen cow-shed, one tried to learn something by the sense of smell. At\nfirst, the morning air was snell and sharp; there was an earthy aroma\nwhich suggested nothing but decaying vegetable matter, but soon it was\nsucceeded by a pungent penetrating odour which made one wonder whence\nits source. This pungency remained for the remainder of the morning's\nride, almost to the top of the mountain pass, some 9000 feet above\nsea-level, and we ascertained on our return that it proceeded from the\nenormous cabbages grown by the mountaineers for the markets on the\nplains of East Java.\n\nAs we plunged deeper into the forest, it was impossible to make out more\nthan a dull outline of a white jacket and the white shoulder of our\npiebald pony. Had we not known that the guide was there, we might have\nwondered how the wonderful jacket succeeded in floating through space.\nThe pony had no head to our sight; the reins we held in our hand might\nhave been dispensed with so far as they acted as a guide to the pony,\nwho picked his own foothold and followed the white jacket. With painful\npersistence, he picked the edge of the precipitous declivity which was\nlost in the bottomless abyss.\n\nOnce only we lost our way. Turn after turn was negotiated safely, first\ndown into the bottom of the ravine and through the mountain torrent,\nthen up the hillside again, mysterious zigzag after zigzag, and one had\nbecome reconciled to the jolting motion of the pony, the steady tramp of\nhis tiny hoofs, and his heavy breathing where the path was steepest, and\ngave one's-self up to reverie. How terrible, we thought, must have been\nthe scene on the mountain <DW72>s when the enormous craters of the\nTeng'ger range were belching forth their death-dealing streams of lava,\ntheir showers of ashes and stones and choking sulphurous fumes! How\ninsignificant was man before the powerful agencies of Nature! How bright\nwere the occasional stars one saw wherever there was a break in the\ntrees that lined our path! How wonderful that each of those stars, those\nplanets, might be peopled by beings puzzling over the disputed facts of\nthe Creation, as we were; who might also be worrying over a future\nexistence and the redemption of a sinful people; who might be\nendeavouring to solve labour problems and trade disputes and discussing\nwhether free trade or preferential tariffs were best for a nation's\nwelfare! Was there somebody up in one of those other planets on a pony's\nback, as we were, robbing one's-self of much-needed rest to reach a\nmountain top to see the sun rise?\n\nThese and other thoughts kept recurring to one when, suddenly, as if it\nhad been shot, the pony planted his forefeet and refused to follow the\nguiding lead of the syce.\n\nWe had made a wrong turning and the syce all but slipped over a\nprecipice. Had it not been for the pony's instinct, all three of us\nwould have been plunged into Eternity, and some of the problems of the\nprevious moment might have been solved.\n\nOut came the syce's matches, as he clung to the pony's bridle. Not\nnearly so bright as the lambent phosphorescence from the fireflies which\nflickered across our path, the puny light of the match was sufficient\nfor the guide to pick up the ribbon-like path, and once more we were on\nour way to the top.\n\nThree deep ravines were traversed before we made the final upward\nmovement, and then Nature's lamp lights were being shut out in hundreds\nat a time as the soft dawn began to diffuse itself. With Dawn's left\nhand in the sky, we thought of Omar Khayyam's stanza, and felt impelled\nto cry out to the sleepers in the hollow--\n\n    Awake! for Morning in the Bowl of Night\n    Has flung the stone that puts the Stars to Flight:\n      And lo! the Hunter of the East has caught\n    The Sultan's Turret in a Noose of Light,\n\nThe dawn had been preluded by the awakening chirrups of songsters in the\nwood. A shriller note was struck by some feathered Daphnis piping to his\nChloe. Deep down in the valleys and in the villages perched perilously\non projecting ledges of the mountain, faint twinkling lights began to\nappear, and the lowing of the cattle and the answering and re-echoed\ncrowing of rival poultry-yards sent the thoughts back to Homeland scenes\nsome 10,000 miles away.\n\nAs we stood on the wall of the enormous crater, overlooking the Sand\nSea, and watched the long shafts of golden light shoot up to the zenith\nfrom behind the mountain peaks to the East, we felt that our ride had\nnot been in vain.\n\nTo be abroad at early dawn in the tropics is to enjoy the most\ndelightful period of the day. An English essayist has well expressed the\nexhilaration one feels: \"There is something beautiful in the unused day,\nsomething beautiful in the fact that it is still untouched, unsoiled.\"\nOnly those who have stood on the hill tops, far removed from the haunts\nof men, have any true idea of the grandeur of Nature and the\ninsignificance of man.\n\nThe sun rose speedily in the full power of his golden radiance to paint\nthe landscape. There was no transition. Out of the darkness there rose a\nview, enormous, diversified, impressive.\n\nMiles away on the west, the five summits of the Ardjeono had been the\nfirst to reflect the rays hidden from us. Penanggoenan's sugar-loaf top\nsoon caught them up and passed them on to Kawi's three lofty peaks. To\nthe south, was the Semeroe, Java's loftiest volcano; to the east, the\nYang Plateau; to the north, the sea and the island of Madoera. We could\ntrace the coast-line 9,000 feet below, away westward beyond Sourabaya,\nwhere white-crested surf beat silently upon the streak of yellow sand.\nThe vast plains of East Java showed a pattern of variegated colour,\nwhich stretched out to the cultivated <DW72>s of the hills. Mountain\nhamlets and villages on the plains sent out blue vapours from morning\nfires. The rivers were distinguishable by their leafy fringe as much as\nby the reflection of the blue sky overhead. Between us and the Yang\nPlateau, there were rolling billows of white cloud, tipped by the\ncolours from the sun's spectrum.\n\nBut it was the panorama spread out like a model beneath our feet which\narrested attention and impressed one most. We stood on the edge of an\nenormous crater--the Teng'ger--with a circumference of fifteen miles.\nWhere, in prehistoric times, flames and ashes and lava had boiled and\nbelched, there was now a sea of yellow sand, out of which stood other\nthree volcano peaks--the Battok, the Bromo, and the Widodaren--showing\npurple in the morning light. The Battok is a perfect cone, the\nlava-covered sides standing out in clearly defined ridges like the\nbuttresses of a Gothic structure. The Bromo is the only one of the three\nnow active. As we gaze down, we are startled by a deep groaning noise,\nand out of the wide crater mouth there issues a mass of grey smoke and\nashes laden and streaked with fire. Simultaneously, a huge mass of\ncloud, cruciform in shape, is shot up hundreds of feet into the air from\nthe Semeroe. It rests a few seconds above the bare, ash-strewn cone, and\nthen drifts heavily to westward, to make way for the next eruption.\n\n[Illustration: SAND SEA, WITH BROMO AND SEMEROE.]\n\nThese indications of Nature's activity in the crucible at the earth's\ncentre make one reflect on the possible consequences of the next great\nconvulsion, and the fate that is in store for those intrepid villagers\nwho have perched their primitive huts on the very edge of the Teng'ger\ncrater. With these reflections, we turn away from one of the most solemn\nand impressive sights it has been our privilege to witness, silently\nmount our pony and retrace our steps for the snugly-situated Hotel at\nTosari, no longer regretting, nay, rather thankful, that we had resolved\nand achieved our resolution to climb the Penandjaan Pass to see the sun\nrise.\n\n[Illustration: SMOKE PLUME--THE SMEROE.]\n\n\n\n\nHotels and Travelling Facilities\n\n\nBefore going to Java, the tourist ought to make himself acquainted with\nthe outlines of the history of the island since it came under European\ndomination. Half the charm of European travel, if one is something more\nthan a mere unreflective globetrotter, lies in the historic associations\nof the places visited, and it is the comparative absence of this quality\nwhich robs new countries of the interests they would otherwise possess\nfor educated people. Scenery alone surfeits the appetite.\n\nIn Java, as in most Oriental countries, the traveller feels that he is\nmoving in an atmosphere of antiquity, and though it has become a\nmisnomer to refer to \"The Unchanging East,\" it is borne in upon one that\nin the large group of islands comprised in the Philippine and Malay\nArchipelagoes, from Luzon in the north to Java in the south, from Samar\nin the east to Sumatra in the west, centuries of western contact has\nleft but a slight impress upon the characters of the people. Changes\nthere are, undoubtedly. Modern civilisation has advanced like a\nresistless wave and gradually engulfed an older civilisation, but here\nin Java one feels that the change has not been so decisive; and railways\nand canals and cultivation notwithstanding, the difference in general\nadvancement between the Javanese and the Japanese is most marked, and\neven the Chinese, conservative though they are in most ways, have more\ncharacter and look more hopeful soil for the reception and development\nof western ideas.\n\nA solid foundation for the trip to Java may be laid by perusing Sir\nStamford Raffles' history, the second edition of which, published in\n1830, will be found in Raffles Library. It covers the whole period from\nthe time the Portuguese arrived in the Farther East in 1510 to the\nBritish occupation. Making Malacca his headquarters, Albuquerque sent\nvarious expeditions to the surrounding islands, and Antonio de Abrew was\nhis emissary to Java and the Moluccas. The Dutch appeared in 1595,\nobtaining their first footing in the East Indies at Bantam, the English\nEast India Company establishing a factory at the same place in 1602.\n\nOf the capture of Java by the British troops brief details have already\nbeen given.\n\nAn interesting account of \"The Conquest of Java\" is given by Captain\nWilliam Thorn, a Dragoon officer, who served on the staff of one of the\nbrigadiers. It was written in 1815 while he was on his way back to\nEngland, and is so plentifully illustrated with field maps as to add\ninterest to one's visit to Batavia and Buitenzorg and the seaports of\nSamarang and Sourabaya.\n\nWe are indebted to Dr. Hanitsch, the Curator, for the following list of\nbooks on Java in Raffles Library:--\n\n    The Dutch in Java; 1904, by Clive Day.\n\n    Java, Facts and Fancies; 1905, by Augusta de Wit.\n\n    Facts and Fancies about Java; 1908, by Augusta de Wit.\n\n    Life in Java, 2 vols; 1864, by W. B. d'Almeida.\n\n    Voyage Round the World; 1870, by Marquis de Beauvoir.\n\n    With the Dutch in the East; 1897, by W. Cool.\n\n    Geschiedenis der Nederlanders of Java; 1887, by M. L. Deventer.\n\n    From Jungle to Java; 1897, by Arthur Keyser.\n\n    Java; 2 vols., 1861, by J. W. Money.\n\n    Java; 1830, by Sir Stamford Raffles.\n\n    Fuehrer auf Java; 1890, by L. F. M. Schulze.\n\n    The Conquest of Java; 1815, by William Thorn.\n\n    A Visit to Java; 1893, by W. B. Worsfold.\n\n    Rambles in Java; 1853, (anon.).\n\n    The Hindu Ruins in the Plain of Parambanan; 1901, by Dr. I.\n    Groneman.\n\n    The Tjandi-Baeraebudur in Central Java; 1901, by Dr. I. Groneman.\n\n    Boro-Boedoer op het Eiland Java; 1873, by F. C. Wilsen, 2 vols.\n\nIn addition to a selection from the above-named, the intending visitor\nshould read \"Java: The Garden of the East\" by Miss E. R. Scidmore, 1898,\nand the Rev. G. M. Reith's \"A Padre in Partibus\" will be found\nentertaining.\n\nMuch must depend upon the notions of the tourist as to the cost of a\ntrip in Java, but our experience is that Java is the cheapest country we\nhave ever visited. The hotels are superior to those found in the\ninterior of Japan, and, as the guilder, which has a value of 70 cents in\nSingapore currency or about 1s. 7 3\/4d. in English currency, may be taken\nas the unit of value for travelling purposes, our readers will see at a\nglance what a fortnight or three weeks' trip is likely to cost from the\nfollowing hotel rates:--\n\n  Hotel des Indes, Batavia       6 guilders per day\n\n  Hotel Bellevue, Buitenzorg     6    \"       \"\n\n  Hotel, Sindanglaya             6    \"       \"\n\n  Hotel Garoet                   6    \"       \"\n\n  Gov't. Hotel, Maos             4    \"       \"\n\n  Hotel Mataram, Djocjakarta     5    \"       \"\n\n  Hotel Simpang, Sourabaya       6    \"       \"\n\n  Sanitorium, Tosari             7    \"       \"\n\n  Hotel du Pavilion, Samarang    5    \"       \"\n\nThere are a few extras, and the servants are civilised enough to expect\nsmall tips. Charges for liquors are invariably reasonable.\n\nThe hotels are scrupulously clean and the accommodation excellent, and\nin a tropical country one appreciates the facilities for bathing.\n\nIn his delightful poem of \"Lucile,\" Owen Meredith wrote:--\n\n    We may live without poetry, music and art;\n    We may live without conscience, and live without heart;\n    We may live without friends; we may live without books;\n    But civilised man cannot live without cooks.\n    He may live without books,--what is knowledge but grieving?\n    He may live without hope,--what is hope but deceiving?\n    He may live without love,--what is passion but pining?\n    But where is the man that can live without dining?\n\nHere the poet leaves the realms of poetic fantasy to record a simple\nfact of everyday life--one which is appreciated by every man and woman\nirrespective of nationality or temperament. As in all other matters\npertaining to the comfort of the European in the tropics, the Dutch, in\nthe matter of food, seem to us to have achieved better results than we\nhave in the British Colonies. The \"riz-tafel\" may not appeal to the\nEnglish palate, but there is no lack of clean, wholesome dishes, and\nside dishes that make us wonder at the toleration of the traveller with\nthe Indian and Colonial caravanserai. The tourist who visits Java after\ntraversing India will be agreeably surprised at the difference in favour\nof the Dutch Colony in this respect.\n\nIn the matter of the personal attention to their guests by the\nmanagement of some of Hotels in the interior, and the supply of\ninformation, there could easily be an improvement, and doubtless there\nwill be a great change when tourist traffic becomes more general, as it\npromises to do in the near future. Our own experience was that we were\nleft, almost invariably, to the tender mercies of the servants, and as\none's Malay was limited this led to avoidable inconvenience.\n\nNothing, however, could exceed the courtesy and attention of the\nmanagement at the Hotel des Indes, in Batavia, and the Hotel du Pavilion\nin Samarang, and the Manager of the Hotel at Sindanglaya.\n\nWe have already mentioned Stamm and Weijns Restaurant in Batavia.\nCoupled with it for excellence of table is Grimm's famous restaurant in\nSourabaya.\n\nThis year, thanks to the efforts of some of the leading hotel\nproprietors, the government of Netherlands India has awakened to the\npossibilities of Java as a country for tourists. Co-operating with the\nHotels and steam-ship companies, special inducements were held out to\nvisitors during the months of May and June, in the way of reduced fares,\nand the success of the venture will doubtless lead to its continuance.\n\nThe Koninklyke Paketvaart Maatschappij (Ship's Agency, late J. Daendels\nand Co.) issues tickets at single-fare rates to Batavia and Sourabaya,\nthe fare to Batavia and back being $45; to Sourabaya and back $63; and\nto Batavia and along the Coast Ports to Sourabaya and back to Singapore\n(sixteen days on board ship) $74. The tickets are available by the\nsteamers of the Royal Nederland Line and the Rotterdamsche Lloyd.\n\nTravel by rail throughout the Island is cheap. For the convenience of\nvisitors with limited time to devote to Java, a tourist ticket has been\narranged. This may be obtained from the Steamship Company in Singapore.\nThe price is $40 (Singapore currency). The tour laid down by the coupons\ncovers the whole of Java from Tanjong Priok, the port of Batavia, to the\neasternmost end of the island beyond Sourabaya on the way to Tosari and\nBromo. Buitenzorg and the Preanger health resorts may be visited on the\ntickets, the famous Hindu ruins near Djocjakarta, and the health resorts\nof Eastern Java. The journey may be broken wherever the tourist cares to\nstay, and the ticket is available for sixty days.\n\nDirections are printed on the ticket in English in regard to baggage and\nother matters, and a small outline map is a useful adjunct.\n\nThroughout the island, the carriages for hire are execrable. The\nfour-pony victoria which took us from Djocjakarta to the Buddhist ruins\nat Parambanan had not gone half a mile when one of the wheels came off,\nand we were lucky to escape without serious damage. It will always\nremain a marvel to us how the ramshackle kreta held together which took\nus from Buitenzorg to Sindanglaya, over the Poentjak Pass, and we are\nastonished that the Dutch authorities, who are exacting in other\nrespects, do not exercise a wholesome supervision over the ponies\nemployed in these cross-country carts and carriages, for a more wretched\ncollection of horseflesh could scarcely be imagined.\n\nWe have already commented on the Toelatings Kaart. This relic of a past\nage, which did not add much to the revenue, and impressed one\nunfavourably with a rigid officialism at the port of entry that did not\nobtrude itself upon one's notice in the interior, may now be avoided by\nthe traveller registering at the Tourist Bureau. In our own case, we\nwere never called upon to produce the kaart.\n\nThe general impression left by one's visit to Java is the excessive\ncleanliness of town and country and the widespread cultivation. There\nare, of course, black spots in the towns; but they are as nothing to the\ntraveller who has perambulated the native quarters of any British Colony\nin the Far East. When we think of the millions of dollars Hongkong has\nexpended to cope with filth-created plagues and to reduce the native\nrookeries of China town, it fills us with the highest admiration for\nDutch administration in Java. The Government of the Straits Settlements\nis entering upon a similar campaign to rectify past sins against the\nlaws of sanitation and hygiene, and hundreds of thousands of dollars\nmight have been available for other purposes had the Chinese been\nhandled as the Dutch handle them in Batavia, Samarang and Sourabaya. It\nmay be overdoing the cult for whitewash to whiten the walls of every\nbridge and the stack of every sugar mill in the country, but it is\npleasing to the Europeans to see that one nation has been successful in\ncarrying its ideas of cleanliness into the tropics and in making the\nOriental conform to the ordinary laws for the protection of the health\nof the common people.\n\nTo those of our readers who may be induced to visit Java, we would\ntender a few words of advice.\n\nIf it is intended to compress a tour of the principal places we have\nnoted into a fortnight's holiday, travel, if possible, to Sourabaya, and\ngo first of all to Tosari. After a few days there, Djocjakarta should be\nmade the headquarters for a two or three days' inspection of the\nBuddhist ruins, and then Bandoeng could be made a halting place while a\ndecision is arrived at as to whether Sindanglaya, Soekaboemi or Garoet\nis to be visited next before going on to Buitenzorg and Batavia. We\nrecommend this course because there is a more frequent service of\nsteamers between Batavia and Singapore, and by ascertaining the sailing\ndates while at some of the Preanger health resorts one is able to time\none's arrival at Batavia and so avoid the heat of the seaport.\n\nWe have painted Java in rosy colours because we found it beautiful, the\npeople companionable and the conditions agreeable. It is possible that\nothers may go over our tracks without deriving a tithe of the enjoyment.\n\nNo one should travel unless he has a genius for travel and a ready\nadaptability to prevailing conditions. He should bear in mind that it is\nhe who is the odd piece in the machinery, and that unless he adjusts\nhimself to the other working pieces he will only have himself to blame\nif things do not run smoothly. If Java is visited in the right spirit,\nwe have not the least doubt that the traveller will be delighted with\nall he sees and experiences, and will come away with an assured\nconviction that it was no exaggeration which styled the island \"The\nGarden of the East.\"\n\n[Map: JAVA.]\n\n\n\n\nTranscriber's Notes:\n\n\nInconsistencies in the hyphenation of words preserved. (court-yard,\ncourtyard; over-night, overnight)\n\nPg. 52, the phrase: \"collection of Buddas\". The author might have meant\n\"collection of Buddhas\", as \"Buddha\" is used elsewhere in the text.\nHowever the author's original spelling is preserved.\n\nPg. 55, \"daning\" changed to \"dancing\". (and maidens dancing.)\n\nPg. 63, the title \"tivan\" is also spelled \"tavan\" in two instances in\nthe preceding paragraphs. As it is unclear which spelling the author\nintended, the original spelling is preserved in all cases.\n\nPg. 70, unusual time expression \"2.9 p.m.\" The original text is\npreserved. (so I started at 2.9 p.m., and, after)\n\nPg. 74, duplicated word \"at\" removed. (reaching Pasoeroean at 8.23)\n\nPg. 90, text contains the expression \"1\/7 3\/4d\" which, for clarity, has\nbeen rendered as \"1s. 7 3\/4d.\" (or about 1s. 7 3\/4d. in English currency)\n\nIn the original text, the author was inconsistent with respect to\nwhether the \"ae\" ligature was used in the word \"archaeological\". This\ninconsistency has been preserved.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Across the Equator, by Thomas H. Reid\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"In April 2017, Jeff Meer, U.S. Executive Director of Humanity & Inclusion, traveled to Mali. He met with Kouyate, a father caring for seven children, three of whom live with physical disabilities. In this personal testimony, Jeff describes the difficulties faced by Kouyate and his family and the hope that Humanity & Inclusion offers.\nSokoma Kouyate is the father of seven children, and has been unemployed for more than six years. He, his wife, his mother-in-law and the children all share a single room for sleeping in Bamako, Mali's fifth district, where I met him and his family in April 2017. During the day, they stay in an adjacent courtyard, where afternoon temperatures can reach into triple digits with 90% humidity. Kouyate's wife works in the local market selling spices, a livelihood that provides barely enough income to support the family. Kouyate stays home with his mother-in-law to care for the children.\nThe courtyard where the family cooks, cleans, relaxes and plays is littered with wheelchairs, crutches, seats and other assistive devices in various states of disrepair. That's because three of Kouyate's children are unable to walk. As infants, they became infected with malaria, which was left untreated, and eventually damaged their nervous systems and left them unable to move their lower limbs.\nThe oldest of Kouyate's children is 22. Most days, he sits in a wheelchair near the street adjacent to the home, under the shade of a large tree. Like his two younger brothers with a disability, ages 8 and 4, he has never been to school, and cannot read. The closest school in Bamako equipped to teach children with disabilities is about five miles from where the family lives. Paired with the state of the roads in Bamako, the lack of sidewalks, and the lack of public transportation, perhaps their lack of education is no surprise. During my visit, the normally-abled children in the family weren't in school either, because school teachers were on strike over a lack of wages.\nNone of the three boys with a disability has seen a physical therapist, and so they are unlikely to ever be able to care for themselves. Family members occasionally bring them to a local health clinic, but only so they can receive care for acute illnesses.\nKouyate told me that he is tired many days taking care of his large family. He is not sure where to turn for help, and is grateful to have occasional visits from Humanity & Inclusion and our partners. This gives him some hope for the future.\nAccording to the United Nations Development Program, Mali ranks among the five hardest places in the world to live. Life expectancy is 58.5 years in Mali. The average Malian will expect to have just two years of formal education. Roughly 80 percent of those working in Mali earn less than $3.10 per day.\nHumanity & Inclusion has worked in Mali since 1984, helping families like the Kouyates. The need in Mali is very great for HI, which provides food aid to vulnerable families, rehabilitation for people with disabilities who need it, economic opportunities and jobs for people with disabilities, and help to get children with disabilities into school, among other programs.\nLike my colleagues in Mali, I am honored to be part of the HI family, helping to create better lives for the Kouyate family and other. It gives me great peace of mind to know that having this family on our radar, means they have the potential to access rehabilitation and properly fit wheelchairs, and other assistive devices.\nThis vital work continues with the help of our talented and dedicated staff, local partners and community organizations. They say \"it takes a village,\" but here in Bamako, it seems to me it takes even more than that.\nLearn more about HI's work in Mali.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tobi was never the greatest at making me feel better after loosing a game but I gave him the credit for trying. By this point we had gone back to his place and ordered pizza. The rest of the team had gone out for ice cream but I wasn't up to going and I figured Tobi needed me more.\n\"Have you heard anything from the hospital yet?\" I figured I would ask considering neither of us were saying anything else. He was quiet for a minute as if he was carefully choosing his words.\n\"He's okay for now,\" was all he said. I didn't know how to take that or even what he meant by it.\n\"For now? What do you mean he is okay 'for now'?\" Again he just sat there unresponsive. \"Tobi.\" This time he answered and I was beginning to wish he hadn't. He pretty much mumbled it but I heard enough to make it clear as day.\n\"Nothing. They can't do anything but tell him to take it easy.\" I started crying. Just a few tears but I refused to let Tobi see.\nHis dad meant the world to him. He was his idol, his role model. Ever since his mom walked out on them when he was only little, Tobi's dad is all he's ever had as far as parents are concerned. His dad is really all Tobi has ever had and to loose him\u2026to loose him would destroy his entire world.\nSo I stopped talking and we watched a movie and finished our pizza without another word. At the end of the night, as I was leaving, I had to say something to Tobi. I couldn't just leave things the way they were.\n\"Of course.\" With that, Tobi gave me a goodnight\/I'm sorry hug and I headed home.\nNormally after loosing a game I wake up to a text from Tobi asking how I was feeling. I didn't wake up to that this morning. Instead I texted him to ask how he was feeling. I didn't get a reply until after I had finished my breakfast and gotten ready for school. His message didn't say much, just that he ws fine but he wouldn't be coming to school today. He was probably going to spend the day with his dad making surehe was okay.\nThe school day took forever but when the last bell did ring I still took my time leaving. All te excitement of the halloween bash had everyone buzzing and the last thing I wanted to do was go home and do homework. I wandered around the school for a bit before I decided to just go home because there was really nothing to do around here. On the bright bright side, no one was home when I got there so I could just sit in peace and quiet for a little while.\nNot to long after I got in Tobi texted me. I was really happy about this because he was on my mind all day but I didn't text him because I didn't want to bother him if he was spending the day with his dad.\nT: Hey, sorry I've been so quiet today.\nIt's alright. Did you spend the day with your dad?\nT: Yeah but I missed you too.\nHe missed me? He just saw me yesterday. Maybe it wasn't just me. Maybe he actually had feelings for me. But how would I find that out for sure? I don't want to be the girl who gets all creepy just to find out if he likes me.\nT: So, you excited about the halloween bash? It's our first high school dance.\nBefore I could even send a reply my pone was ringing and it was Tobi. I laughed lightly as I picked up the phone.\n\"Okay then ask away.\" What he said next, I was not expecting.\nWell that was dissapointing. Here I am thinking that he might have a crush on me when in reality he was just being nice and now he's asking for advice on asking another girl out\u2026wow.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I cannot see how you can blame the assessor for an increase in reoffending. It could be that those who are ordered by the courts to have an assessment are more likely to reoffend to begin with and that is why they have a tougher sentence.\nI am not blaming the assessor. I am blaming the Ministry of Health and the NZTA for allowing unregistered incompetent counsellors to conduct these assessments and for failing to check on their qualifications.\nDrink drivers who are assessed are being compared with drink drivers who are not assessed. They're all drink drivers (or apples). What is being discussed is the impact of assessment (and treatment) on subsequent re-offending rates. If assessment\/treatment is done well, perhaps it would turn rotten apples into carrots.\nI appreciate the value of your presence on the AoD scene, especially since meeting you in Dunedin. We need independent commentaries to help keep things honest. I hesitate therefore to criticize, but I wish you would provide more evidence for some of your conclusions.\nThe fact that a counsellor fails to renew his registration with DAPAANZ does not itself mean he is incompetent. I do not know the counsellor you mention, but naming him in this fashion without more detail seems brutal. There does not seem to be any evidence in your blog actually tying him to getting drunk drivers back on the road, and your follow-up comment that you do not blame the counsellors seems to contradict the earlier piece, where you name and blame an individual.\nSecondly the the statistic quoted later in your piece \u2013 that assessed drivers re-offend more that unassessed drivers \u2013 is predictable, since the former group are likely to have more serious offense histories. That 68% of this group do not re-offend is actually a very positive statistic.\nWhether these counsellors are incompetent is a moot point. But they not competent as far as DAPAANZ is concerned. 'Not competent' and 'incompetent' are two ways of saying the same thing. But this article is not about the counsellors \u2013 its about the Ministry of Health failing to do its job.\nIn regard to your second point I disagree that it is predictable that assessed drivers will reoffend at a greater rate. That implies assessment and treatment is a waste of time \u2013 and\/or that the assessors are incompetent.\nI am not so much interested in apples and apples, as the main issue, being unskilled AOD counsellors, let loose on unsuspecting clients, who may believe they're in competent hands.\nRoger is assuming quite reasonably that a competent assessor would not allow those drivers back on the road \u2013 the fact that the assesors are allowing them back pretty much proves they are incompetent!\nSurely it makes sense for the Ministry of Health to enforce their own guidelines? It is important that assessors are competent, belong to an organisation with a code of ethics and are accountable for their practice. If they're not, it makes it an uneven playing field for those who do ensure they belong to a professional body, have paid their annual subscription and diligently earned the required one hundred points.\nWould you consult any other health care practitioner that was unregistered? Thanks for bringing this to my attention Roger, as I had no idea that this was happening, why would I? I had a letter from the MOH towards the end of the year, reminding me of my annual requirement to furnish proof of my ongoing registration and updated details of my business.\nThank you Roger regardless of anyone's view you have once again alerted us to the lack of accountability in regard to the use of our taxes. I for one think it is abhorrent that counsellors who do not meet criteria or do not have the relevant qualifications have been paid for conducting these assessments. This must be addressed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brass accents, textured pillows, round tables and tailored furniture are among the home decor trends on display in this black-and-white-themed space.\nWant to redecorate? Here's some on-trend inspiration.\nEmerald Hill Interior owners Jackie Bayer and Sadie Johnson agree, and Bayer adds that she likes to soften the look with one or two interesting design elements that pop, so the overall effect won't be too harsh.\nHome design is more than simply decorating a space \u2014 it's about making it more livable. Homeowners increasingly approach the task from a functional standpoint, thinking about how they actually use their rooms, not just how they want them to look.\nRound tables, like this one in the home of Bill and Anna Leonhardt, are on trend this season.\nAs technology evolves, so do the ways we incorporate it into our homes.\nPardoe anticipates this trend only growing over the next few years. \"It'll be even bigger, with people being able to control everything from their phones, from locks to lights,\" she says.\nAt the same time, that means thinking about how to hide cords and plugs, and how to incorporate technology into the rooms where it will make people's lives smoother.\nOver the past decade or so, advances in textiles have done wonders for the world of outdoor fabrics. Today, there are so many great outdoor options that homeowners are opting to use them inside.\n\"The outdoor fabrics you think of from 10 or 15 years ago are completely different now,\" says Pardoe. Today's fabrics are softer and available in a wider range of patterns. And since they are made for outdoor conditions, the new fabrics stand up to even the heaviest use indoors.\nFabrics aren't the only thing designers are bringing inside this season. Emerald Hill's Bayer and Johnson like to blur the lines between inside and out by incorporating natural textures, plants and flowers. Accessories made with wood, shell, horn, bone, woven grass and leather add visual and tactile interest.\nInstead of mass-produced accessories, the duo recommends their clients decorate with plants and flowers to infuse spaces with energy and make it easy to keep the look fresh at all times.\nAfter nearly 40 years of stainless steel, nickel and chrome reigning over the interior design industry, brass has returned as a coveted of-the-moment element.\nTrish Albano, owner and designer at Trish Albano Interiors in Mount Airy, notes the softer, subtler look of the new and improved brass. \"The brass of today is not the brass you may remember from the '80s,\" she says. \"Most brass that was popular back then was very shiny with a lacquered finish.\" This time, brass has a richer color with brown undertones, not yellow, and can be intentionally left untreated to develop a unique patina.\nAlbano suggests pairing brass with other metals, which creates an \"acquired-over-time\" look instead of \"matchy-matchy,\" she says. She also recommends dark wood and caramel leather as complementary materials. The key to staying on-trend is using brass in moderation. Whether it's in drawer pulls, a light fixture or a kitchen faucet, a little brass goes a long way.\nBut proceed with caution when mixing metals: \"I think of it as the same as mixing wood \u2014 think of them as different colors,\" she says.\nBlack metals are as hot as they are cool this season.\n\"Some people are a bit hesitant at first, but we're seeing it more now,\" says Pardoe.\nMiller is also a fan of the cool, flat metal look, especially in the bath, where it adds an unexpected element to vanities. \"Black shower hardware and faucets are striking against natural stone,\" she says.\nAaron and Mallory Lembo love living in the middle of the city, on a busy street near Patterson Park. They thrive on the hustle and bustle of the urban environment and the closeness of their neighbors.\nHomeowners are all about experimenting with diverse combinations of materials and layering colors and textures to create spaces that are unique, personal and warm, say designers. Michelle Miller says modern spaces with clean lines look most relevant when \"warmed up\" with natural woods and antique rugs. They \"soften the hard edge of modern interiors of the past,\" she says.\nTaking the layering approach also allows homeowners to inject more of their personalities into a space \u2014 and personality is always in style, says Miller.\nSerena & Lily's brass ring table lamp makes a statement using negative space. The simple ivory parchment shade ensures that the lamp works for a multitude of styles. 13.5 inches by 8.5 inches by 21 inches. $595, serenaandlily.com.\nLawson-Fenning's thin frame lounge chair puts a contemporary twist on a timeless design. The chair's antiqued brass finish is complemented by a tufted suede seat in a neutral beige color. 30 by 28 by 27 inches. Starting at $1,685, lawsonfenning.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Best airfare and ticket deals for ABE to GSP flights are based on recent deals found by Expedia.com customers.\nDistance and aircraft type by airline for flights from Lehigh Valley International Airport to Greenville\u2013Spartanburg International Airport.\nThe most recently built aircraft flying from ABE to GSP in 2009 was Canadair's CRJ-2\/4 flown by Atlantic Southeast. The CRJ-2\/4 took flight for the first time in 1997.\nWay back in 2009, 49 passengers flew from Allentown to Greenville. Today, this number has significantly increased.\nIn 2009, we learned that Atlantic Southeast flew around 49 passengers from ABE to GSP.\nIn 2009, the following group of quality airline carriers offer flights from Allentown, PA to Greenville, SC: Atlantic Southeast.\nThe longest flying aircraft flying from Allentown, PA to Greenville, SC during the 2009 year was a Canadair's CRJ-2\/4 flown by Atlantic Southeast.\nScan through flights from Lehigh Valley International Airport (ABE) to Greenville\u2013Spartanburg International Airport (GSP) for the upcoming week. Sort the list by any column, and click on a dollar sign to see the latest prices available for each flight.\nWhether it's for an obligation or the sake of your sanity, sometimes you need to get away. Maybe you need flights from Allentown to Greenville to attend your cousin's wedding, to pitch a business idea to your boss, or perhaps simply to treat yourself to a mini vacation. Regardless of the reasons behind packing your bags and needing to find the cheapest flights from ABE to GSP, we've got you covered here at Flights.com.\nWe present you with some of the hottest deals on airfare so stop that Google flights search. We want you to spend less on your flight from Allentown to Greenville, so you can spend more during your getaway. With Flights.com, you'll find it simple to land airline tickets with itineraries matching your travel schedule. What's more, we provide you with all the information you need to confidently make reservations on your family, business, or personal trip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"High Heels and Mommy Ordeals: \"Doh\" You Want to Be My Valentine?\nOne of my favorite holidays as a teacher is Valentine's Day. I love to create little cards for my class to pass out. The last few years I've decided against candy and created Valentines with novelty items like pencils, bubbles and fun straws. Since I'm not in the classroom this year, I decided to make cards for Cohen to pass out to his friends. He was very adamant on getting gummy bracelets but as I mentioned the word Play-Doh, he was okay with doing those cards too.\nI whipped these puppies out on PicMonkey and it was kind of ridiculous how easy they were to make. You can print them out on cardstock or use cardstock as a backing to make them thicker. Then you just cut the little holes out in the middle, put your Play-Doh in there and secure it with a cute piece of Valentine's themed tape that can be found in the dollar bin at Target.\nI included the link to the printable below. When I printed mine, I scaled it to 30% and they came out to the perfect size. I'm going to let Lola hand out a few of these to her friends from her old class just so she doesn't feel left out. Okay and because they are too fun to make.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today I Uploaded 15 photos with all are in high resolution version. From manifold images in Army coloring pages images, you can have Coloring 2018 with decoration. But, if you don't appropriate with design, you might still take a look ideas. Even so I propose you might use as an aspiration decor, although together with has terrific decoration aspiration. Don't forget for share this Army coloring pages get this online army coloring pages a9m0j cat pictures to color to close friend or your boy friend. So they are can get also to erect they Coloring 2018. I am glad might share along with along with to you all. Hopefully you might find which all this time are you seek.\nArmy Coloring Pages Army Coloring Pages 20 free printable army coloring pages everfreecoloring transformers coloring pages. Army Coloring Pages army tank coloring pages military coloring page army tank coloring cat pictures to color. Army Coloring Pages military coloring pages free and printable transformers coloring pages. Army Coloring Pages army coloring pages printable free coloring pinterest army birds coloring pages. Army Coloring Pages army tank coloring pages tufo birds coloring pages. Army Coloring Pages free printable army coloring pages for kids yugioh coloring pages. Army Coloring Pages military coloring pages military coloring pages stock excellent army cat pictures to color. Army Coloring Pages army coloring pages to print fun coloring pages pinterest army spongebob coloring page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Washingtonville sophomore Gabriel Castillo captured his first Orange County Interscholastic Athletic Athletic singles title, defeating defending champion Lucas Arora of Cornwall 6-0, 6-1.\nCornwall senior Emil Vyskocil and junior Will Webber won the OCIAA doubles title, knocking off Newburgh Free Academy senior Jack Clayton and Matt Reinhold, the top seed, 6-4, 6-2 at Match Point Tennis in Goshen.\nDefending champion and Cornwall senior Lucas Arora outlasted teammate Dan Leach in a near three-hour semifinal 6-7, 6-2, 7-6 to reach his fourth straight OCIAA final.\nArora will play Washingtonville sophomore Gabe Castillo, the top seed, in the championship. Castillo defeated teammate Gavan O'Brien 6-1, 6-0.\nNo. 1 Gabe Castillo (Washingtonville) d. No. 4 Gavan O'Brien (Washingtonville) 6-1, 6-0; No. 2 Lucas Arora (Cornwall, defending champion) d. No. 3 Dan Leach (Cornwall) 6-7, 6-2, 7-6.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We are located on Historic Route 6 in the beautiful town of Smethport, Pennsylvania.\nWe started business in 2002 and began supplying the local businesses with Equipment Identification Tags. Soon after, we expanded our business to offer custom engraving. We engrave\/etch on a wide variety of materials, including: wood, glass, plastics, metal, and more!\nIn our Gift Shoppe, you'll find Beautiful Fashion Jewelry, Accessories, Scarves, and Purses. You'll also find Antique and Vintage items. We have a variety of handcrafted items from some of the area's most talented Artisans.\nCopyright \u00a9 2017 Lightwaves Engraving & Gift Shoppe - All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Handcrafted in Northern Thailand from vintage hill tribe applique fabric and hand-woven cotton. Accented with a coin and bell detail. Leather straps are accented with beads.\nYour purchase helps to economically empower women artisans, improve quality of life and preserve tradition and cultural legacy.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"For information on the SHRM Certified Professional (SHRM-CP) or SHRM Senior Certified Professional (SHRM-SCP), please go to shrmcertification.org, e-mail: certification@shrm.org or call 1-703-535-6360. The Certification Handbook contains valuable information regarding your certification exam. All candidates should carefully review the Handbook during the application process and before arriving at the Test Center.\nYou will be required to present a valid, non-expired government-issued ID in Latin characters with a current photo and signature. The name on the identification must be the same as the name that appears on your exam application. If you fail to produce the proper form of identification, you will not be admitted to the examination and will forfeit your exam fees. For more information about proper identification and how to change your name if needed, please refer to the Taking the Exam section in the Certification Handbook.\nPlease reference the SHRM Certification Handbook for a detailed listing of testing rules.\n\u00b7 Review your appointment confirmation email to confirm your appointment time.\n\u00b7 Arrive at the testing center at least 30 minutes prior to your appointment time.\n\u00b7 Review driving directions. Allow sufficient travel time including traffic, parking, locating the test center, and checking in. Depending on the location of the testing facility, additional parking fees may apply. Prometric does not have the ability to validate parking.\n\u00b7 Bring a valid, non-expired government-issued ID in Latin characters with a current photo and signature. The name on the identification must be the same as the name that appears on your exam application.\n\u00b7 Prometric is unable to provide a completely noise-free environment. Consider bringing your own soft ear plugs or use the test center-provided head phones.\n\u00b7 No breaks are scheduled during the exam. Candidates may take a break as needed, but may not leave the testing facility and will not be given extra time on the exam.\nCandidates wishing to request a refund must do so on or before the late application deadline date.\nFor the complete refund policy refer to the Certification Handbook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What if the person you hated most was murdered and their ghost came to you asking for help to solve the crime?\nThis fast-paced story steps back in time to reveal the agonizingly troubled youth of Mac MacKenna before he joined the U.S. Air Force Special Operations.\nSet in Northern California in the spring of 1990, Mac is a teen on the verge of manhood. But he's not your typical teenager. He doesn't have a girlfriend. He isn't going to parties. And he no longer participates in sports.\nHe desperately wants to move away from his parents. As his 18th birthday and high school graduation approaches there's a shocking death in the family\u2014is it murder?\nMac knows something's not right; he can feel it in his bones. He's desperate to uncover and expose the truth although he's fearful, he might kill his father if his father doesn't kill him first.\nThe first series elaborate the intelligence conflict between Egypt and Israel since the start of the state of Israel in 1948. It discusses all the famous espionage stories, the known and the unknown, from the view of both countries. A spy in the palace book tells the story of \" Ali Al-Atafi\" who worked for Mossad in Egypt.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"PVC with waterproof, mildew, cold, anti-aging, anti-static and other properties; and we used PVC broken strength, tear elongation, tear strength is much better than the traditional tarpaulin for productin of the inflatable gym mat.\nPVC (tarpaulin) coated high-strength polyester waterproof cloth is based on high-strength polyester canvas cloth, coated with polyvinyl chloride (PVC) paste resin with accelerator, anti-mildew agent, anti-aging agent, antistatic agent And other chemical additives, made by high temperature plastic.\nPVC appearance colorful, multiple colors can be selected. The surface of the special treatment from the anti-skid effect, is the international popular waterproof cloth, and wide width, according to user needs to produce different functions, different colors, different thickness of the product. , Processing products can reduce the patchwork to improve the quality, eliminating the need for sewing needle hole leakage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Superpark Espoo is an indoor activity park for the whole family. It offers fun activities for everyone and experiences that make you move- all year around, no matter the weather. In the Freestyle Hall, you will find the skating and scooting areas, air track tumble mats, fast slopes that deliver you into the foam pit and awesome trampolines. You can compete with your friends in the Game Arena using the Puck Radar, Football Flipper, Fortuna Ball Game and the SuperHoop. The Adventure Area will make you want to climb the walls, go ape on the slides or zoom around in a pedal car.\nWelcome to Superpark Espoo, where you can enjoy adventure and games, test your skills and have fun with a variety of sports. The wristband allows you all day access. Superpark Espoo also offers high-quality meeting facilities, restaurant services and guided activities for groups!\nChildren 0-3 years are admitted to the park free of charge. Children under 7 must be accompanied by an adult. The adult must also pay for entrance.\nThe day pass is valid for a whole day. With our wristband, you can go in and out of the park as many times as you want during our opening hours. Carers of individuals with reduced mobility or who need assistance otherwise are admitted to the park free of charge. The customer must pay for entrance.\nAdventure Area. Game Arena. Freestyle Hall. Choose your own adventure!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pictured here is the design of a new Science and Technology Centre for the College. We have applied for government funding for this project and will be starting a new Building Fund appeal. The school is growing and needs more classrooms. We wish to provide more practical experiences in our science program for both Primary and Secondary students, and have a space where students can use technology to design, create and develop innovative solutions in problem-based learning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwfpx b/data_all_eng_slimpj/shuffled/split2/finalzzwfpx
new file mode 100644
index 0000000000000000000000000000000000000000..120129647904024f8f28c2435e82000534e61e1d
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+{"text":"Recalled Graco Stroller (the one with fingertip amputation hazard). has caused injury. Do I have a case?\nIf you have been injured by Recalled Graco Stroller (the one with fingertip amputation hazard). you are not alone. You may have a case against Newell Rubbermaid Inc. but you should speak to a lawyer if you are wanting to file a claim. The following is an incident that caused injury to a consumer and was reported.\nMy daughter was involved in a fingertip amputation because of a recalled Graco stroller. Her little finger came in between the hinges severing it almost completely.\nThe victim here was a Female who was 0 years old.\nIf you or a family member have also been injured by Recalled Graco Stroller (the one with fingertip amputation hazard). you may have a case. Fill out a free case evaluation request today.\nPrevious post This was a Splat Ball Egg manufactured by Ja-Ru Toys has caused injury. Do I have a case?\nblowing bubbles has caused injury. Do I have a case?\nDisney Princess pogo stick by Bravo Sports has caused injury. Do I have a case?\nMaclaren Volo Single Umbrella stroller has caused injury. Do I have a case?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Compliance data is defined as all data belonging or pertaining to enterprise or included in the law, which can be used for the purpose of implementing or validating compliance. It is the set of all data that is relevant to a governance officer or to a court of law for the purposes of validating consistency, completeness, or compliance. Compliance software is increasingly being implemented to help company's manage their compliance data more efficiently.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"a sunlit meadow where flowers bloomed.\nchildhood dreams destroyed too soon.\nwhere now there stands a chapel room.\namongst darkened sods of wrathful gloom.\nThis entry was posted in Blog Posts, Inkphrastica by Deborah Edgeley. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many successful business owners are looking for creative retirement planning solutions. Are you looking for a retirement plan that can maximize current deductions and provide benefits at retirement that are guaranteed and not dependent on investment performance?\nA 412(I) plan is a fully insured defined benefit plan.\nUsing a fixed annuity and whole life insurance products, a fully insured defined benefit plan can be designed to provide current deductions that are much larger than uninsured plans. It also guarantees the future benefits and can provide substantial life insurance protection.\nIs a 412(i) plan the right choice for you?\nIf you wish to provide retirement benefits with tax-deductible contributions.\nIf this sounds attractive and meets your needs, the 412(i) plan may be the right choice for you.\n1 Guarantees are contingent on the claims-paying ability of the issuing company or companies.\nThe information provided is not written or intended as tax or legal advice and may not be relied on for purposes of avoiding any Federal tax penalties. Individuals are encouraged to seek advice from their own tax or legal counsel. Individuals involved in the estate planning process should work with an estate planning team, including their own personal legal or tax counsel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Gold nanoparticles dispersed on high\u2010surface\u2010area carbon materials were investigated as heterogeneous catalysts for the selective oxidation of d\u2010glucose to d\u2010gluconic acid in aqueous solution with molecular oxygen. Salt\u2010templated porous carbon supports were obtained from different precursors with and without nitrogen and treated under air or hydrogen atmosphere to functionalize the surface with nitrogen, oxygen, or hydrogen. The influence of the surface atomic structure of the carbonaceous supports with similar pore structure on the size and catalytic properties of the metallic nanoparticles was studied at gold nanoparticle loadings of 0.4\u20130.7\u2005wt\u2009%. The functionalisation significantly influences the surface polarity of the support materials and the strength of the interaction with the gold nanoparticles. The surface polarity influences the structure and properties of the catalysts because both the gold deposition and the glucose oxidation reaction take place in the aqueous phase. Rather hydrophilic supports are obtained by doping with oxygen and nitrogen and lead to large gold nanoparticles with low catalytic activity. In contrast, the rather hydrophobic as\u2010made and hydrogen\u2010treated supports provide higher catalytic activity (metal time yield up to 1.5\u2005molGlucose\u2009molAu\u22121\u2009s\u22121) resulting from their smaller gold particles of 3\u20135\u2005nm in diameter.\nLama, S., Schmidt, J., Malik, A., Walczak, R., Silva, D., V\u00f6lkel, A., & Oschatz, M. (2018). Modification of Salt\u2010Templated Carbon Surface Chemistry for Efficient Oxidation of Glucose with Supported Gold Catalysts. ChemCatChem, 10(11), 2458-2465.\nLama, Sandy M. G., Johannes Schmidt, Ankita Malik, Ralf Walczak, Daniel Varon Silva, Antje V\u00f6lkel, and Martin Oschatz. \"Modification of Salt\u2010Templated Carbon Surface Chemistry for Efficient Oxidation of Glucose with Supported Gold Catalysts.\" ChemCatChem 10.11 (2018): 2458-2465.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwqqi b/data_all_eng_slimpj/shuffled/split2/finalzzwqqi
new file mode 100644
index 0000000000000000000000000000000000000000..fc37258c19ffed37bab8c985765447bf4f3b30d2
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@@ -0,0 +1,5 @@
+{"text":"BNet.Builder is a desktop application for rapidly creating Belief Networks, entering information, and getting results.\nBNet.Builder's easy to use, easy to learn graphical user interface lets you create models quickly, without fussing with frustrating and time-consuming dialog boxes, edit and compile modes, nested menus, and other irritating features so common in other belief network modeling packages.\nBNet.Builder is the first product to bring true ease-of-use to Belief Network modeling.\nDynamic Update - See beliefs as soon as you change your network in any way, including: entering CPT data, posting evidence, and deleting nodes or links. No need to compile, run, or update.\nSingle Mode Interface - Build your network and see calculated beliefs in one single window, not in separate \"build\" and \"run\" modes.\nNote: the processor speed and memory capacity specified above are only recommended values, not minimum required values. BNet.Builder should work well for most networks with the above processor speed and memory capacity. Slower processor speed and smaller memory capacity will result in degraded performance. Faster processor speed and larger memory capacity will increase performance of BNet.Builder and allow for larger network size.\nSupport for BNet.Builder\u2122 is available through our Support Package.\nTo purchase a product license or support contract, contact us.\n1-day introductory courses for using BNet.Builder are available. For schedules and prices, please contact us.\nClick here to download a trial version (Windows).\nClicking to continue the download indicates acceptance of our license agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Generally in South African wine competitions, as elsewhere in the New World, agglommerated results are presented in terms of varieties and styles rather than origin.\nThat's also the case with the Platter guide, whose list of five-star winners I've just been looking at with a differerent perspective.\nWhat is the league table it offers when categorising its 62 top-ranked wines?\nFirstly, most broadly, and in line with what most serious commentators would agree is correct, more white wines than red get five stars \u2013 like last year, though this year's 35 to 27 gap has now widened a little.\nThe winning category is easily White blends, which is certainly justified, in my opinion. There are 12 of those, followed by Shiraz with 10. Red blends and Chenin have 8 each, then Chardonnay with 5, then unfortified desserts with 4. Then the others.\nPlatter doesn't divide the wines into their origins \u2013 partly, no doubt, because this is a bit trickier to do and much less clearcut. Stick to origin and you miss a lot of good wines. But, of course, you can again invoke the idea of blends. In fact, going through the list, I'm gratified to note that a minority of the wines give their origins as \"Coastal\" or \"Western Cape\": let's hope it's because there's an increasing tendency to produce wines that reflect a particular origin. But 13 out of 62 is still too many.\nThen some with just one each \u2013 Durbanville, Cape Point, Robertson, Elgin.\nThe strange omission is, of course, Constantia. How did that happen (especially with the fine Vin de Constance 2007 on offer)? Well, these things do happen, I'm afraid, and we must swallow hard and get over it. There are always question marks over exclusions and inclusions. And the list of five-star wines certainly for me (as for many people, only no doubt differently!) includes wines whose presence there I wonder about, and omits some that seem indispensible. (I take a sip of Raka Cabernet Franc and long for Glenelly Lady May!) The fact remains that the Platter Guide is the most useful list to perform this exercise on, in that it reflects judgements on vastly more wines than any other guide or competition.\nI haven't checked my analysis or arithmetic very carefully \u2013 if you feel like doing so, and find me wrong please let me know, but I'm at least sure the overall picture is correct.\nAs I say, no-one is going to be surprised that Stellenbosch produces the most five-star wines. It would have been even more true ten years ago. But ten years ago the idea that the Swartland would be in a very respectable second place (with a tiny proportion of the wineries that Stellenbosch has) would have been thought absurd.\nInterestingly, only two of the Olifantsrivier wines are actually made in the area: Fryer's Cove Savignon Blanc and Cederberg CWG Teen die Hoog Shiraz. The other two are vinified elsewhere: Botanica Chenin Blanc in Stellenbosch, and Sadie Family Skurfberg in the Swartland. And at least one other five-star wine has a significant proportion of Olifantsrivier wine \u2013 Alheit Cartology.\nBut what a splendid thing this is! Another proof that there is surely no area in the whole of the Western Cape that is not capable \u2013 with a bit of imagination, skill and hard work \u2013 of producing fine wine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Classes are ongoing and students can start any time, and proceed at their own pace.\nThe Charlottesville T'ai Chi Center offers training in the Cheng-Ming System as taught by Great Grandmaster Wang Shu-Jin.\nTeaching philosophy: These arts can best be passed down in a teacher-to-student relationship. We provide one-on-one instruction to all students. Classes are ongoing and students can start any time, and proceed at their own pace.\nMission: Charlottesville T'ai Chi Center is a non-profit organization dedicated to teaching and promoting the health benefits and art of T'ai Chi Ch'uan in the Charlottesville community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Will Obama dial back the war on drugs?\nPresident Barrack Obama has done very little to de-escalate the war on drugs, despite comments prior to his election that led people to believe his administration would be less repressive than his predecessor's, reports Jacob Sullum at Forbes.com. But as Obama embarks on the third year of his second term, it looks like the optimists were partially right.\nMarijuana Legalization. Although the federal government cannot stop states from legalizing marijuana, it can make trouble for the ones that do by targeting state-licensed growers and retailers. Under a policy announced in August 2013, the Justice Department has declined to do so, reserving its resources for cannabis operations that violate state law or implicate \"federal law enforcement priorities.\" The department also has refrained from challenging state marijuana regulations in court, a strategy that could have delayed the opening of cannabusinesses in Colorado and Washington even if it was ultimately unsuccessful.\nContrary to the impression left by the president, the executive branch has the authority to reschedule marijuana without new legislation from Congress.\nIn September, a few days before announcing that he planned to step down soon, Holder said whether marijuana belongs in the same category as heroin is \"certainly a question that we need to ask ourselves.\" Since the Controlled Substances Act empowers Holder to reclassify marijuana, it would have been nice if he had asked that question a little sooner. Still, Holder was willing to publicly question marijuana's Schedule I status, something no sitting attorney general had done before.\nClemency. After a pitiful performance in his first term, Obama has signaled a new openness to clemency petitions. Last April an unnamed \"senior administration official\" told Yahoo News the administration's new clemency guidelines could result in \"hundreds, perhaps thousands,\" of commutations. Obama's total so far, counting eight commutations announced a few weeks ago, is just 18, but he still has two years to go.\nA few months ago, Obama chose former ACLU attorney Vanita Gupta, a passionate critic of the war on drugs who emphasizes its disproportionate racial impact (a theme Obama and Holder also have taken up), to head the Justice Department's Civil Rights Division.\nThe next two years will show whether Gupta's appointment is a sop to disappointed Obama supporters or a signal of bolder steps to come.\n\"I'm waiting for the president to come out and say his views are evolving on marijuana,\" says Bill Piper, director of national affairs at the Drug Policy Alliance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We were laughing and remembering those cringe-factor moments from our kids' early Christmases, when I was reminded of a fun game we played a few years back with the older two. Appreciation Training.\nAs I explained the game to my friend, she got all enthusiastic and said, \"We should totally do this! Like together. The kids will get into it more if there are other kids there...\"\nCollect a bunch of totally random bits and pieces. Scour the junk drawer, the washing pile, the recycle bin. I am talking old socks, bottle tops, giant undies, pegs... Collect enough for roughly three \"gifts\" per kid. Then wrap them up and pop them in a bag. You will also need a bag of lollies\/sweets as \"rewards\".\nThis is truly a good laugh, as well as being a powerful lesson in the art of gift-appreciation. It's not that we are teaching our children to be fake, rather we are teaching them socially-appropriate responses to the gift of woolly socks, helping avoid Aunt Ethel's mortification or displeasure and most importantly reminding them that it's the THOUGHT that COUNTS.\nAunt Ethel hobbled on her arthritic knees all the way to KMart. She stood in front of the sock display and hand-picked those socks just for you, Jimmy. She made an effort. She spent her hard-earned retirement pennies. THAT's why we show appreciation to Aunt Ethel.\nOf course you can also suggest that a kiss and a hug as well as a hearty thankyou will go a long way to warming Aunt Ethel's heart.\nIn fact in our family we only give Santa the credit for the content of the Stockings. The rest of the pile is duly credited to the people who spent the time perusing junk mail for great deals, listening into conversations about Christmas Wishes and braving the madding crowd to secure those wishes for beloved offspring. In other words, Mum and Dad. We get the ecstatic Thankyou's, the gleeful hugs and kisses we have earned.\nMay you have a very Merry Appreciative Christmas!\nOh I LOVE this idea!!! I am so doing it. I want the floor to open up and swallow me when my kids don't show the 'proper' appreciation.\nAnd yes, Santa only gets a small portion of credit in our house too.\nhave a wonderful Christmas my dear friend.\nMan, I wrote a long comment and it's disappeared!!!\nLove this idea and am going to try it with my kids.\nHave a lovely Christmas my friend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzwzid b/data_all_eng_slimpj/shuffled/split2/finalzzwzid
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+{"text":"Aqua Lung scuba fins provide thrust and power to each kick with designs that reduce fatigue and improve performance. Aqua Lung's innovations in design and materials have allowed divers of all skill levels to see more of the undersea world.\nAqua Lung scuba fins offer the ultimate in comfort and performance. Aqua Lung designs fins for performance-minded divers who are looking to optimize their diving experience.\nIn designing each pair of diving fins, a perfect balance must be achieved. The fins must be wide enough to overcome water resistance in order to propel the diver; however, they must also be the right length to allow maneuverability. Aqua Lung places a high priority on providing a variety of options for a diverse range of divers of all experience levels and this is reflected in every pair of scuba fins we create.\nFull foot fins are typically used in warm or tropical waters, while open heel diving fins are ideal for waters of any temperature. Open heel diving fins offer greater thermal protection when used with neoprene dive boots, however, several of Aqua Lung's full foot diving fins can accommodate dive socks to attain extra warmth while maintaining a small size foot pocket.\nAqua Lung diving fins incorporate a variety of designs to help divers get more performance and enjoyment from their dives. Features for both full foot and open heel fins can include adjustable power bands to accommodate different body types or changing conditions, as well as features that are especially designed to eliminate foot and toe fatigue.\nA quality pair of dive fins is invaluable for any underwater application, both recreational and professional. Dive fins powered by Aqua Lung innovation and technology can take your diving to new depths. Whether used for casual snorkeling or for extended diving, Aqua Lung dive fins comfortably get you where you need to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for a source of bulk rock dust that can deliver in Palm Beach County.\nNo weight on fruit but if I had to guess, I would say around 12 oz. It was perfectly ripened. Aroma was of that of a Capri Sun and taste followed suit (only thing to figure out is which flavor). Fruit was nice and sweet with the type of acidity found in one of those Capri packets. Texture was semi-firm and juicy.\nI liked this one better than Fruit Punch.\nFor sale is a 3 gal Manohar mango tree, the tree that Frank aka JF made famous. Cost is $25, local pick up only.\nI have 20 Nutmeg seeds for sale. $40 takes them all. Please PM if interested.\n25 gal Ugly Betty mango tree for sale. $100. Local pickup only.\nLocal pickup only. If interested, please PM me.\nFor sale, 25 gal Angie mango tree. $120 or best offer. Local pickup only.\nWill try and post picture on Thursday.\nExcalibur has some beautiful 25 gal Fortune Eye Longan for sale. They also have 3, 15 and 25 gal Super Hass Avocado for sale.\nBelow is the brief history as described by Walter Zill.\nPrice is $1.00 per seed.\nI also have a bunch of seeds from mixed varieties that include Pierce, Honeheart, Santa Rosa, Sabor and El Bumpo and possibly a few other high quality varieties. Price for these (due to being mived together) is $0.50 per seed.\nPlease PM me for details or if interested.\nFruit and palm knowledge a big plus. If you are interested, please PM me for details.\nI have a 15 gal Excalibar Red jackfruit tree for sale. The tree is grafted, and is approximately 7 - 8 feet. I will post a picture tomorrow and give exact height. Local pick up only.\nI have a 25 gallon Angie and a 10 gallon Ugly Betty for sale. The Ugly Betty is the size of a large 15 gal tree. $175 for the Angie and $100 for the Ugly Betty. Feel free to make reasonable offers.\nI have five (5) 7 gal Honey Kiss and three (3) 3 gal Venus mango trees for sale. Please PM me if interested. Local Pick up only. Sorry, no shipping.\nHONEY KISS ARE ALL SOLD. THANK YOU TO THOSE BUYERS FOR YOUR PATRONAGE.\nIf you have ever tasted the Gary, you will quickly k ow why Gary Zill used it as one of bis go to breeding cultivars. You can also easily detect the influence it had on the Coconut Cream.\nI have a 15 gallon Ross Sapote tree for sale. The tree was recently cut back and has flushed out heavily. It has fruited last year and is flowering right now. Cost is $100. Pick up only.\nExcalibur has some beautiful 7 gal Rollinia, 3 gal Noni and 3 gal Shiranui \/Dekopon (this is the same citrus sold under the name of Sumo) available in limited quantities. There are also some really nice 3 gal Caimito (green and purple) available.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zohra is the first ever all-female orchestra in the history of Afghanistan.\nNamed after a Persian goddess of music, Zohra is an ensemble of around 30 young Afghan female musicians who, against great adversity, studied and made music together. These students of the Afghanistan National Institute of Music (founded by Dr Ahmad Sarmast in 2010) have travelled and played at prestigious concert halls and international gatherings. In 2017, the group left Afghanistan for the very first time to perform in front of 2000 world leaders at the Annual World Economic Forum, to great acclaim.\nOn Friday 15 March 2019, as part of Women's History Month, the group will visit London for the first time for a very special performance at the British Museum to share the beauty of Afghan music, the achievements of Afghan women and to spread the message of hope across the world.\nPresented in collaboration with the Embassy of the Islamic Republic of Afghanistan in the UK, Oxford College of Music and Orchestra of St John's, Oxford.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For Ricardo Guadalupe, CEO of Hublot, this new Big Bang captures the brand's essence in all respects: \"Made at our modern and perfectly integrated factory, the Big Bang Meca-10 demonstrates the complete mastery we have of our industrial resources, from R&D to the creation of calibres, high-tech cases, and innovative materials.\"\nThe case, in indigo blue ceramic, measures 45mm x 15.9mm. It has an AR-coated sapphire crystal and it is water resistant to 100 meters.\nMovement is the Swiss manual-wind Hublot in-house caliber HUB1201 with 24 jewels, 21,600 vph and a power reserve of 10 days. The bridges are blue-plated and it has a polished rhodium-plated balance.\nThe dial is matte blue, skeletonized, with luminous hands. It has two barrels parallel to the power reserve indicator: a cogwheel system with two rakes sliding along a horizontal axis. An opening at 3 o'clock unveils a red dot when the movement is nearing the final days of its power reserve, while a gearwheel at 6 o'clock indicates the exact number of days remaining and the regulating organ, coupled with the small second regulator, appears at 7 o'clock.\nIt comes on a blue and black rubber strap with a titanium and ceramic deployant.\nNext Topic: Rather a new Jones PW but at least something different.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Any idea yet on the new 2014 SE edition wheels????\nI'd be interested too, they are 824.00\/piece though.\nAny idea what the 17 inch method mv weighs ?\nGot ahold of Weld today, the B57B is a porky 38 lbs. a real disappointment, as I would have loved to ran these wheels if they were >25lbs each.\nI emailed Method and got conformation on wheel weights. The reason for asking was to reduce un-sprung weight.\nHas anyone weighed the SVT beadlock capable wheel? I have seen the shipping weight at 40 lbs, but what is the actual wheel weight?\nI haven't weighed them, but I'd assume that the packaging on them is about 2-3 lbs, so I'd guess 38-39 lbs without trim rings, 41-42with them installed. Those 12 button head screws add up to about 1\/2 pound.\n35 LBS give or take a few OZ due to scale variation.\nI also weighed the stock OEM cast wheel on the same scale and it came in at 31.6 lbs... so the SVT beadlock with beauty ring is ~3.5 lbs more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzxaim b/data_all_eng_slimpj/shuffled/split2/finalzzxaim
new file mode 100644
index 0000000000000000000000000000000000000000..754be22c63f7c79c77bca3bd7ccd5bf46441874d
--- /dev/null
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+{"text":"October half term 2016. A picnic on the bench, a play on the beach. A sandcastle competition. The inevitable sulking that followed! Some mention of 'Old Jacks Boat'... This painting is part of the 'Big Skies & Old Stones' collection for Chantry House Gallery, Ripley. From May 20th 2017.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Added on March 4, 2013 by Victor Ya\u00f1ez-Lazcano.\nI had ripped all of the flags of the porch posts and crumpled them into my arms with no regard for the tacks and nails that were still within the colored bundle of what was a reminder of someone else's home. Even as I approached I didn't know why I was doing what I was doing. The driver's side door of the hand-me-down vehicle I was driving was still wide open with the music blaring. There was no attempt to be subtle about what I was doing. My mind had been made up, though it wasn't clear what I was trying to prove. As I ripped at the blue and red I heard the men gather themselves from within the small countryside home. The first man, wearing a dirty white button up short sleeve linen shirt and dirty rolled up khaki pants came out, startled, with a cigarette in hand. His facial expression didn't change as I reached and made my last several pulls. My actions did not falter and my concern for my well being had stayed in the car. I just kept pulling. With the last tug and loud rip the other man made his appearance on the porch. Shirtless and pissed gripping an old wooden handled hammer he peered at me from the doorway. Committed, I took hold of their memories and embraced them harder, feeling each nail and tack pierce and puncture my body. There was no exchange of words and as the blood began to exit my wounds from my now rapidly beating heart I looked down to look at the hammer one more time. The man holding it also hadn't moved since first setting eyes on me. The cigarettes smoke rose slowly from each of their mouths. Without panic and with all the determination I had left in every bone I turned swiftly towards the car, which was no longer playing the loud music. I threw everything in and closed the door. Before pulling away I reached for the last of my own cigarettes, arms now shaking from the absence of adrenaline that had carried me through my task. I lit it. Looked once more toward the men who had still not moved an inch and pulled away in reverse. I had only ever stolen pens before.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Open: Apri to October ... Barbati Resort is an amazing combination of mountain and sea. Our studios and apartments in Barbati are ideal for families and couples.\nOpen: May to Octomber...Golden Mare Barbati hotel in Corfu, Greece, is a beach resort for memorable island holidays.\nOpen: April to October...La Riviera Barbati offers a swimming pool with sun terrace, a snack bar and a restaurant. It offers free Wi-Fi throughout and self-catered units with views of the garden, the pool or the Ionian Sea.\nOpen: May to September..... Pantokrator Hotel Corfu is located northeast of Corfu town. It is situated next to Barbati, one of the most picturesque villages on the island of Corfu, has panoramic views over the southern part of the island.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our main focus is to produce the highest quality kittens with an extreme exotic look, all the while keeping a very friendly and domesticated personality. We believe that constant human contact and daily interaction with our exotic kittens will help to develop that special bond and amazing personality that everyone is seeking in their feline companions.\nOur Savannah & Bengal kittens are raised underfoot with our family, including our two children ages 11 and 3 and our two Persian cats, Luc & Rory. Our exotic kittens receive constant attention and love from each of our family members to help ensure great socialization and adaptation to any and all environments.\nAll kittens will come with a full health guarantee. We ship both domestically and internationally.\nI have always had a deep love for animals. I remember from a very early age I spent most of my time caring for several pets at home from mice, hamsters, guinea pigs, parrots, cockatiels, German shepherds, Dachshunds, Border Collies, Clumber Spaniels, Siamese cats, and both freshwater and exotic fish. I had the reputation for being the neighborhood \"pet doctor\" in which other children and adults would bring their pets and stray animals to me so that I could care for them and nurse them back to health. I always loved that.\nWhen I was about 12 years old I decided to become a veterinarian, but soon thereafter chose Marine Biology as my main course of study. I attended California State University of Sacramento and then later on moved to Miami, Florida to complete my studies and attain my degree. I majored in Marine Biology with a minor in Oceanography and Italian. My main goal was to study communication between different whale and dolphin species... and possibly do an expedition off the Great Barrier Reef in Australia with world famous Oceanographer, Jacques Costeau. That was my dream. I've always had an immense love for world travel. In 2001, I flew to Sydney, Australia and spent about 2 weeks there. I had the privilege of visiting Taronga Zoo in Sydney where I had the opportunity to view some of the most beautiful and exotic animal species in the world. I was very much drawn to their exotic cat exhibit and spent a good portion of my time learning everything I could about them.\nBack home, in Houston, at the downtown aquarium, we have two of the most exquisite white Bengal Tigers in the world. They are amazingly beautiful, powerful and elegant. To be in their presence is like having a spiritual experience. I have respect for them, and I know I would never set foot inside of their enclosures unless I had the proper training to handle them. So, I thought to myself, how could I have the opportunity to handle such an exotic cat, but still be able to interact with it like a domesticated cat? I did some research and found some breeders who did just this. They created a breed of cat in which one parent was an African Serval Leptailarus serval and the other a domestic cat. They created a beautiful hybrid cat called a Savannah... an amazingly stunning and highly intelligent creature with the look of a wild cat and the mannerisms and personality of a sweet, playful domestic cat. I finally found what I had been searching for. I fell in love with this breed.\nPlease also visit us at The American Travel Company & The Healthy Chocolate Lifestyle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Meet Starley. Starley is an up-and-coming musician based in Australia, who premiered her single \"Call On Me\" in August of last year. A unique facet of her music that I noticed is her seamless fusion of organic instrumentation and electronic elements in a single song. Her music style caught my ear, so I asked Starley some questions about her journey thus far as a musician. Interview by Danielle Leard.\nLithium Magazine: When and why did you start pursuing music?\nStarley: I've always [made] music. My mom actually gave me a stage name, Starley. She said she just knew I'd be a performer. I think my earliest memory is watching La Bamba, the Ritchie Valens story, and becoming addicted to the movie. I knew music would be it for me after that.\nLM: Describe your songwriting process.\nS: My process actually varies; however, for \"Call on Me\" I was writing in my bedroom, playing around on my keyboard. I just kind of found a chord progression I really liked and wrote on top of that. Lyrically, \"Call on Me\" kind of poured out of me. It was a real therapy session for me, writing that song.\nLM: Who are some of your favorite musical influences?\nS: Mariah Carey was a huge influence for me. Obviously, Ritchie Valens as well (whom I mentioned above). I also love artists like Joni Mitchell, Tracy Chapman, Coldplay, Ed Sheeran . . . I could keep going.\nLM: What type of work ethic is necessary for your line of work?\nS: You have to be extremely dedicated and focused to be successful in this industry. There are so many people who [will] tell you no or that you're not good enough. So, you just have to believe in yourself and put in the work\/effort to get yourself to where you need to be.\nLM: What are you working on right now?\nS: I'm writing my album!\nLM: What are your plans for the future?\nS: I don't really have set plans. I just want to continue making music that touches people! Obviously, I'd love to tour and all that as well. But, really, it's all about making a living doing what I love.\nLM: How would you describe your music?\nS: My music is a mixture of organic instrumentation (real pianos or guitars, etc.) and dance music. I guess it's referred to as dance\/pop? If you had to categorize it, that is!\nLM: Can you tell us when you're planning to release your first collection of music?\nS: I'm working on my album now, so once that's ready to go, I'll release it into the world.\nLM: Are there any messages you strive to convey in your music?\nS: Hope. I want to give people hope. I also want to write about things that I'm going through so when people listen, if they can relate to it, they can know they're not alone.\nLM: Is there anyone you'd like to collaborate with in the future?\nS: Yes, absolutely. I love Coldplay. I love Ed Sheeran. I love Sia!\nLM: Have you experienced any struggles as a woman of color in the industry?\nS: When I was younger I had an experience where I was told I'd need to lose weight and straighten my hair to \"make it.\" However, I don't know that my ethnicity has had a massive effect on my music. There are definitely racial disparities in Australia and I'm proud to be a woman of color who's having success from Australia, but I come from a mixed family and have never felt like my race has been a huge issue.\nLM: Where do you see yourself in three years?\nLM: Okay, I've got to know. What is the meaning behind your cover art for your single \"Call On Me\"?\nS: I have always loved lady beetles (or \"bugs\" as you say in the States). When I was a kid I used to go out in my backyard and sing to them, hoping they would come out so I could see them. They were a memento of hope for me as a kid. So, when it came to creating the single artwork for my record, I just knew I had to use the lady beetle in some way.\nCheck out Starley's stunning music video for her single \"Call On Me\" here.\nYou can keep up with Starley on her Instagram, Twitter, and Facebook.\nI know, right? She has great music.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Webster's II dictionary defines violence as physical force employed so as to violate, damage, or abuse, and abusive or unjust use of power.\nThink about what the second part of the definition says: violence isn't only physical force; it is also something much more subtle \u2013an abuse of power. In healthy relationships, people share power and respect each others' basic human rights.Violence destroys relationships.\nViolence exists on a continuum ranging from subtle, hurtful acts to those more noticeable and physically harmful. This broad view of violence is necessary when we are talking about schools where children have little or no power to remove themselves if they find the climate uncomfortable or threatening. Students are a captive audience; they cannot get up and leave.\nSo as we explore how to create a safe school climate, keep in mind that violence is physical force, emotional torment, social isolation, and abuse of power designed to intimidate, dominate, or inflict pain on another person.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In a dry, barren place like Las Vegas, you've got to have beautiful things to look at and fun things to do outside. Clark County Parks & Recreation does a great job of providing and maintaining fantastic parks, so it's no wonder that housing has sprung up around the parks. In Vegas, parks and apartments sort of go together. Here's a list of some of the best parks in the Las Vegas area along with information about the average cost of apartment rentals for the neighborhoods around them.\nThis park has a 14-acre pond in the middle and a small island in the middle with Easter Island heads sitting to keep watch over the happenings in the park.\nSunset Park also hosts the annual Las Vegas Renaissance Fair each fall. Average rent for apartments in the surrounding Paradise area is $930 per month.\nThis giant, beautiful park has plenty of activities for children. There's also a dog park that's usually kept pretty clean. With fenced playgrounds, parents are able to keep an eye on their children without being in their way.\nLocated in Spring Valley, NV, the average rent for nearby apartments is $1,135 per month.\nNot really your traditional park, this affordable, family-oriented recreational shooting facility is open for public shooting Wednesday \u2013 Sunday.\nJust outside of the northern Las Vegas border, next to Sheep Mountain, average apartment rent nearby is about $1,000 per month.\nThis beautiful park is all nature and has walking and biking trails, a community park, and a place for picnics and gatherings. Located in Whitney, NV, on the eastern edge of Las Vegas, average rent for nearby apartments is $921 per month.\nOne of the best ways to boost your quality of life and your health is to live near a park and visit it often. The great parks listed here are some of the best in the region and are more fun than the commercial attractions the Las Vegas area offers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emails between UK and Swedish officials show that Swedish officials were getting \"cold feet\" in 2013, and were considering dropping the \"preliminary investigation\" into Assange, but the UK argued forcefully against it. Last year, Sweden finally dropped the investigation (shortly after it finally agreed to interrogate Assange in London, as it could easily have done years earlier), but the UK has been using the allegation that Assange skipped bail as a way to hold the threat of extradition to the United States over his head.\nAssange has not had access to the internet since it was cut off in March, [Assange lawyer Baltasar] Garzon added, despite a WikiLeaks statement this week that it had been restored.\nAlthough Ecuador stated, hours after UN Special Rapporteurs on Press & Refugees met with its president on Friday, that @JulianASsange's isolation would be lifted Monday, he remains isolated. A 2 pm appointment on Tuesday, with a legal advisor, was not let into the embassy.\nThe Ecuadorian embassy recently demanded that Assange clean his bathroom, feed his cat, and stay out of heated online debates. Now he's suing them for it.\nIf Mr Assange promises to stop emitting opinions on the politics of friendly nations like Spain or the United States, then we have no problem with him going online.\nI have contacted the Committee to Protect Journalists (CPJ) several times by phone, email and through Twitter over the past few weeks, asking them why I have not found any comment from them denouncing Moreno's silencing of Assange, explicitly on political grounds. When I finally reached a CPJ official by phone, I was told me they have \"reported\" on Assange's case. No kidding. What they haven't ever done is denounce Moreno's ruthlessness towards Assange.\nWhether it agrees or not with what Julian Assange says, Ecuador's denying him access to the Internet as well as to visitors is incompatible with its grant of asylum. His refuge in the embassy looks more and more like solitary confinement.\nIn an op-ed in which Pokempner also criticized Moreno, she wrote that Human Rights Watch had been denied permission to visit Assange in May. That hasn't stopped Vivanco from gushing over Moreno's government since that visit was denied.\nAs mentioned above, Assange (who was made an Ecuadorian citizen by Moreno's government in a failed tactical move to end his stay in the embassy) is suing Moreno's government in Ecuadorian courts. However, Moreno has a handpicked body (the \"transitory\" CPCCS) sitting atop the entire judiciary (CounterPunch, 10\/12\/18). It has illegally fired the Constitutional Court, the National Electoral Council and numerous other authorities. Winning against Moreno under these conditions will be very tough, given that judges who displease Moreno will likely be fired or otherwise disciplined\u2014including by being smeared in Ecuador's public and private media. You can count on the international press and \"press freedom\" organizations either saying nothing or laughing it off, unless truly independent voices speak up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Latin translations for Via Dolorosa are \"way of grief\", \"way of sorrows\", \"way of suffering\" and \"painful way\". This is a street within the Old City of Jerusalem which was the path that Jesus walked while carrying His cross on the way to the crucifixion. This road starts from the Antonia Fortress west of the Church of the Holy Sepulchre and continues on for six hundred meters to the spot where Christian Pilgrims celebrate Easter. In the fifteenth century there were fifteen Stations of the Cross, but nowadays the Via Dolorosa is marked by fourteen Stations of the Cross.\nOld stone buildings rise up on both sides of the road where vendors sell their goods as a means of living. Endless stone steps mark the way where locals walk, where pilgrims walk yearly and where Jesus walked so many years ago. During Passover Week the city bursts out of its seems with pilgrims from both Jewish and Christian faith. For some Christians, going to the Holy City and walking the Via Dolorosa is a trip that they should make at least once in their lifetime.\nAs they follow the Stations of the Cross, they become part of the incidents that took place along the way. At least five of these incidents are not recorded in the Bible, but are rather Roman Catholic tradition, so many faithful Christians places significance on nine of these stations. The Catholics and Orthodox followers make up the highest number of pilgrims who visit the Holy Land.\nDuring the eight century the route started at the Garden of Gethsemane towards Mount Zion and then went back to the Temple Mount, ending by the Holy Sepulchre. Between the fourteenth and sixteenth centuries, pilgrims followed the Franciscan route, which began at the Church of the Holy Sepulchre and included eight stations. During this time the tradition of following the fourteen stations was developed in Europe and European pilgrims, who are until this day the majority of visitors, have since followed the fourteen stations. For most pilgrims the accurate location of each event along the Via Dolorosa is of slight importance as the pilgrimage has great meaning due to its proximity to the unique events and the reflection upon them along the way.\nThe Via Dolorosa is an active, lively and noisy road so it would be difficult to pray and meditate along the way. Each of the fourteen Stations of the Cross is marked with a plaque, but not always easy to spot. Having a tour guide would make a big difference, or joining one of the Friday processions for a guided walk.\nStation 1; Jesus' condemnation by Pontius Pilate \u2013 occurred at the site of Madrasa Al Omariya, which is west of the Lion's Gate.\nStation 2; Jesus took up His cross \u2013 located next to the Franciscan Monastery of the Flagellation, which is across the road from the first station. The Chapel of Judgment is where Jesus was sentenced to death and the Chapel of Flagellation is where He was beaten by the Roman soldiers.\nThe road turns south passing the northwestern gate of the Temple Mount. Just west of the entrance to the Lithostratos is the Ecce Homo Arch, where Pilate identified Jesus to the crowd saying \"Ecco homo\" (\"Behold the man\" - John 19:5).\nStation 3; Where Jesus fell for the first time under the weight of his cross.\nStation 4; Where Mary watched her son go by with the cross, and is commemorated at the Armenian Church of Our Lady of the Spasm.\nStation 5; Simon of Cyrene was forced by Roman soldiers to help Jesus carry this cross. This is where the Via Dolorosa turns west off al-Wad Road and begins to narrow as it goes uphill.\nStation 6; On a steep hill - according to a tradition dating from the 14th century, St. Veronica wiped Jesus' face with her handkerchief, leaving an image of his face imprinted on the cloth.\nStation 7; Jesus fell for a second time. This is marked by a Franciscan chapel at the Via Dolorosa's junction with Souq Khan al-Zeit.\nStation 9; Up twenty eight stone steps to the Coptic Patriarchate next to the Church of the Holy Sepulchre where Jesus fell for the third time.\nStations 10 to 14 are all inside the Church of the Holy Sepulchre; Jesus is stripped of His clothes, Jesus is nailed to the cross, Jesus dies on the cross and Jesus is taken down from the cross. Jesus is laid in the tomb.\nAs Easter approaches and Christians contemplate this holiest of seasons, it makes sense to post a blog entry on the Via Dolorosa and a part of Biblical History that is of utter most importance to so many people of the world. We have a number of tours that visit Israel. Each of these will visit Jerusalem's Old City where you can also see for yourself the Stations of the Cross.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BEIJING\/PRNewswire\/ \u2014 Hanergy Thin Film Power Group Limited (HKSE Stock Code: 566) today announced that T\u00dcV Rheinland, a German testing laboratory, has rated Hanergy's Germany-based subsidiary Solibro's double glass copper indium gallium selenium (CIGS) solar module as the most efficient glass substrate CIGS module in the world, with a record 18.72% aperture area efficiency.\nHanergy Solibro CIGS technology can be applied across a variety of areas, with distributed power generation currently the main market. As the annual production and installation rates in this market continue to rise, and as the overall adoption of photovoltaic solutions accelerates, the future for CIGS modules promises to become even brighter.\nIn terms of distributed power generation, Solibro modules have already been deployed widely across the world. In August 2016, a 3MW thin-film solar project with Solibro modules was completed in Zoucheng Industrial Park, Shandong Province, providing grid-connected power generation over a total area of more than 30,000 square meters and generating over 4 million yuan of revenue throughout the year. Overseas, Solibro panels have been successfully applied to a variety of projects, such as a large sports stadium in Germany and residential solar systems in France.\nAside from distributed power generation, Solibro CIGS technology has been leveraged to undertake large scale poverty alleviation projects. For example, in 2017, the Zhaoqing Municipal Committee of CPPCC, Guangdong Province, built a 160-kilowatt photovoltaic poverty alleviation project in Duping, Fengkai County. Although the weather in Zhaoqing City is normally cloudy and rainy, the project still managed to achieve a generating capacity of more than 400 kW\u00b7h per day. Over the past two years, Hanergy has participated in the construction of nearly 40 photovoltaic poverty alleviation projects all across China, with a total installed capacity of more than 150 MW.\nFrom distributed power generation to photovoltaic-based poverty alleviation efforts, Hanergy's Solibro is leveraging the power of its cutting edge CIGS solar technology to provide the public with a steady stream of clean energy. The prospects for the future of this exciting market are nearly unlimited; with Hanergy's experience developing distributed power generation projects, the company is well positioned to bring CIGS solar cells to a variety of applications.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In brief: The Federal Government has cleared the way for litigation funders to fund class actions without an Australian financial services licence and without complying with the requirements for managed investment schemes. This does not, however, resolve the question of whether litigation funders need a licence to fund non-group claims. Partners Ross Drinnan (view CV), Jenny Campbell (view CV) and Lawyer Andrew Ta report.\nfunders taking advantage of the AFSL exemption must put in place adequate arrangements for managing conflicts of interest.\nImportantly, the AFSL exemption does not extend to the funding of litigation other than class actions or other grouped proceedings. The High Court is, however, currently considering the need for an AFSL in those circumstances.\nThe Government supports class actions and litigation funders as they can provide access to justice for a large number of consumers who may otherwise have difficulties in resolving disputes. The Government's main objective is therefore to ensure that consumers do not lose this important means of obtaining access to the justice system.\nAccordingly, shortly after the Multiplex decision, ASIC granted transitional class order relief from the obligations applicable to managed investment schemes to lawyers and litigation funders involved in legal proceedings structured as funded class actions, pending legislative amendment.4 The new amendments will resolve this issue permanently.\nIn 2011, the New South Wales Court of Appeal held that a litigation funding arrangement that Chameleon Mining NL had entered into constituted a 'financial product' because it was a facility through which financial risk was managed.5 The court went on to find that the arrangement in question could be rescinded because the funder did not hold an AFSL to deal in financial products. Special leave was granted to appeal that decision and the High Court's judgment is currently reserved, following a hearing last month.\nThe funding agreement in the Chameleon Mining case is an agreement specific to one litigant and does not relate to a class action or other grouped proceedings. Notwithstanding this, following the Court of Appeal's decision, ASIC extended the class order relief referred to above, to exempt the providers of all litigation funding arrangements from the requirement to hold an AFSL.6 This class order relief remains in place until 30 September 2012. Further extensions are possible, pending the outcome of the High Court appeal in Chameleon Mining.\nIrrespective of the outcome of the High Court appeal, the recent amendments make it clear that the providers of litigation funding for class actions or other grouped proceedings will not require an AFSL. However, as those amendments do not address the position regarding the funding of other types of claims, the High Court's judgment in Chameleon Mining will still be of broad interest and may have a significant impact on the nature of the litigation funding industry in this country (subject, of course, to the potential for further legislative reform). So far as we are aware, IMF (Australia) Limited is the only litigation funder that currently holds an AFSL.\nAs mentioned above, the amendments require funders who take advantage of the AFSL exemption to have adequate arrangements for managing conflicts of interest. The amendments also give some guidance as to what those arrangements must address. ASIC has said that it expects to release a consultation paper on managing conflicts of interest for litigation funding schemes (and also proof of debt schemes) in the next month.\nThe amendments clarify the status of litigation funding for class actions and other grouped proceedings. In doing so, they reflect a policy position that both litigation funding and class actions serve an important role in providing access to justice.\nThe question remains, however, whether litigation funders require an AFSL to fund other types of proceedings. In circumstances in which funding non-group litigation is becoming an increasing focus for many funders, this question is an important one. We will report further following the High Court's judgment in Chameleon Mining.\nBrookfield Multiplex Limited v International Litigation Funding Partners Pte Ltd  FCAFC 147. See also Focus: A Cross for Litigation Funding.\nSee Chapters 5C and 7 of the Corporations Act 2001 (Cth).\nSee, for example, 'Government Acts to Ensure Access to Justice for Class Action Member', Minister for Financial Services, Superannuation and Corporate Law, 4 May 2010; Explanatory Statement for Select Legislative Instrument 2012 No. 172 and Explanatory Commentary for Exposure Draft \u2013 Corporations Amendment Regulations 2011 \u2013 Funded Class Actions.\nASIC Class Order [CO 10\/333]: Funded representative proceedings and funded proof of debt arrangements. See also Focus: Clarification of Class Action Funding.\nInternational Litigation Partners Pte Ltd v Chameleon Mining NL  NSWCA 50. See also Focus: Litigation Funding Hits Another Speed Hump.\nASIC Class Order [CO 11\/128]: Variation of Class Order [CO 10\/333].","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mercedes W211 E class headlights with projector lenses that dramatically improve the look of your W211! Give your 03-06 that updated and gorgeous 07-09 look! These make driving at night much easier by giving a clear high-strength output of light. These W211 E class headlights make your E class look much newer and replace foggy or cracked headlights perfectly. They are also one peice for a sleek, modern look. Made to be a a direct replacement for the factory headlights, these lights are a simple bolt on installation. A superb upgrade!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In other words, could I believe that, say, the wrongness of a lie was any more intrinsic to an intentionally deceptive utterance than beauty was to a sunset or wonderfulness to the universe? Does it not make far more sense to suppose that all of these phenomena arise in my breast, that they are the responses of a particular sensibility to otherwise valueless events and entities?\nSo someone else might respond completely differently from me, such that for him or her, the lie was permissible, the sunset banal, the universe nothing but atoms and the void. Yet that prospect was so alien to my conception of morality that it was tantamount to there being no morality at all.\nHe calls secular ethics \"the clarion call of the 'new atheists,'\" but somehow manages to completely ignore the original new atheist's entire book on this topic. Allow me to fill in some of the missing argument.\nMarks supposes that for objective morality to exist moral actions must be inherently good or bad. A sunset's beauty or banality does not transcend the experience of the conscious observer \u2013 the beauty or banality are the descriptions of that subjective experience. Yet descriptions can be true or false. The viewing of sunsets has objective effects on observers. If a sunset excites the pleasure centers in your brain and you sense \"beauty\" that is an objective truth about a subjective experience.\nSimilarly, Joel Marks may just \"dislike\" animal cruelty and \"death camps,\" but animal cruelty and death camps cause objective harm to animals and people. Marks is correct that tossing conscious chickens into meat grinders isn't intrinsically immoral. All that means is the objective pain experienced by birds doesn't transcend the experience conscious animals. But, so what? How could it? What would that even mean?\nWhen the pleasure centers of your brain are lit by the image of sunset we call that sensation, \"beauty.\" If you started calling it \"banal\" or \"ugly\" you'd be talking nonsense. When particular actions increase the misery to conscious creatures we call that, \"immoral\" or \"wrong.\" It is objectively true or false if more misery resulted. If you started calling harm to animals and people, \"moral\" or \"good\" you'd be talking nonsense. Thus, we can judge the objective morality of an action based on the amount of good or harm that results.\nFor instance, happiness research makes a powerful case against European-style labor-market regulation. For most economists, the effect on worker well-being is unclear. On the one hand, regulation boosts wages; on the other, it increases the probability that you will have no wages at all. From the standpoint of a happiness researcher, however, this is a no-brainer. A small increase in wages has but a small and ephemeral effect on happiness. A small increase in unemployment, by contrast, has a massive and\u2014unlike most other factors\u2014durable effect on happiness. Supposedly \"humane\" regulations to boost workers' incomes have a dire cost in terms of human happiness.\nAt the other end of the political spectrum, consider immigration. The most pessimistic researchers find that decades of immigration have depressed native wages by about 5%, total. The effect of immigration on Third World migrants' wages, by contrast, is massive: One recent paper finds that allowing a Haitian to take a low-skill job in the U.S. increases his earnings 10 times. If you care about happiness, the implication is clear: Government should get out of the way.\nIn many instances, he's right, government should relax regulations that obstruct full employment. But maximum freedom doesn't always lead to full employment. Market economies with lax regulation can suffer from painful volatility.\nFurthermore, during times of high unemployment like today, government could boost employment at relatively low cost to future growth (or possibly no cost). The corollary to when Caplan argues that a \"small increase in wages\" isn't worth the \"small increase in unemployment\" is that a small decrease in future wealth is worth an increase in today's employment. Caplan's sound logic should help convince policymakers to support public investment programs that employ idle workers and to reverse the fall in public employment.\nGermany proceeded to protect its labour market from major disruption by the great recession, through the use of its \"short work\" labour sharing programme. Firms were encouraged to cut hours rather than jobs, and workers facing reduced work hours were provided an income subsidy. The result? Germany's huge output fall produced only a labour market wiggle.\nCaring about people's happiness also makes a further mockery of the hysteria around any Fed policy that might generate higher inflation, which would certainly boost growth and lower unemployment.\nIf you care about happiness, the implication is clear: Bryan Caplan makes a great case for occasional government intervention.\nReza Aslan has another attack on \"the new atheists\" that repeats all the standard criticisms of the movement. He observers, along with every other detractor, that the new atheism is just like religious fundamentalism. Bus ads and other campaigns serve as his examples of proselytization. Critics find it endlessly clever to compare atheists to the evangelicals they often target. I'm always curious if they say that just because they know it will annoy atheists or if they actually think they're offering a meaningful objection. Either way, the comparison is so broad as to be almost completely trivial.\nYou know who else has bus ads?\nAslan goes on to claim that the new atheism is a \"particularly zealous form of fundamentalism.\" He can't be bothered to provide a single quote that demonstrates his allegation that Dawkins, Dennett, Harris, Hitchens, and others believe \"they are in sole possession of the truth.\" The charge is almost self-evidently false; \"the new atheists\" disagree with each other on many issues. Sam Harris, arguably the first \"new atheist,\" regularly argues that everyone (and atheists in particular) have a lot to learn from the spiritual experiences of mystics. The new atheists simply argue that people should provide reasons for their beliefs \u2013 a standard that the On Faith section of the Washington Post would collapse under.\nAslan wistfully remembers the atheism of the past that didn't give \"atheism a bad name.\" Yes, Reza is actually arguing that Richard Dawkins and Daniel Dennett give atheism a bad name, but Marx and Nietzsche didn't.\nWhat if one viewed the recurring patterns of religious phenomena that so many diverse cultures and civilizations\u2013separated by immeasurable time and distance\u2013seem to have shared as evidence of an active, engaging, transcendent presence (what Muslims call the Universal Spirit, Hindus call prana, Taoists call chi'i, Jews call ruah, and Christians call the Holy Spirit) that underlies creation, that, in fact, impels creation? Is such a possibility any more hypothetical than say, superstring theory or the notion of the multiverse?\nDo Christians, Hindus, and Muslims admit that their \"religious phenomena\" are hypothetical? Have they withheld believing in them until they clear standard evidentiary hurdles? Can Reza obscure the difference between science and religion any further?\nReza continues to push falsehoods about the positions of prominent new atheists. He writes, \"new atheists will say that religion is not just wrong but evil, as if religion has a monopoly on radicalism and violence.\" Again, no quotation is provided for the allegedly monolithic atheist view. In actuality, Sam Harris says that Jainism's \"core doctrine is nonviolence\" and religions are as varied as sports. Richard Dawkins writes in The God Delusion, \"Religion is not the root of all evil, for no one thing is the root of all anything.\" In his most famous polemic against religion, Christopher Hitchens explains that \"Humanism has many crimes for which to apologize.\" Hitchens and others openly acknowledge that nationalism and other non-religious ideologies can be dangerous; they just point out that faith-based beliefs are harder to correct and that no society has ever suffered from being too reasonable.\nThe new atheists claim that people of faith are not just misguided but stupid\u2013the stock response of any absolutist.\nYou have also made the false charge that I think religious people are \"fools\" or \"idiots.\" Needless to say, I do not think Blaise Pascal was an idiot (nor do I think you are, for that matter). But I do consider certain ideas idiotic, and idiotic ideas can occasionally be found rattling around the brains of extraordinarily intelligent people. One of the horrors of religious dogmatism is that it can produce a Pascal\u2013a brilliant man who was irretrievably self-deceived on matters of profound importance.\nI have met some highly intelligent believers, but history has no record to say that [s]he knew or understood the mind of god. Yet this is precisely the qualification which the godly must claim\u2013so modestly and so humbly\u2013to possess. It is time to withdraw our \"respect\" from such fantastic claims, all of them aimed at the exertion of power over other humans in the real and material world.\nYes, many atheists think certain believers are dimwitted or ignorant of science, but they don't make any across-the-board claims about the intelligence of religious believers.\nFor all the noise about atheists caricaturing the religious, I've yet to find a criticism that substantively attacks what the new atheist authors actually argue. Instead, pundits like Chris Hedges and Reza Aslan bumble around like media copycat criminals showcasing a version of the crimes that they fictionalize atheists having committed. Aslan's stock criticisms, editorial proselytization, and accusations of ignorance litter every scene where he leaves his stamp.\nI'm back from my caribbean cruise. Hope to be back to regular posting shortly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You have 2792 characters. Don't forget to add important and relevant keywords in your description to improve your iTunes app page and optimize your SEO.\nYour app's last version was released on March 09, 2018. It is important to regularly update your app. Best practices recommend to update your app every 4 to 6 weeks. This means fixing reported bugs, improving existing features, launching new features etc. Keep an eye on users' feedback. The next great features may already be asked by many users.\nA wonderful game for toddlers and kids with a stunning collection of boats and ships !\nyour kid loves vehicle games, puzzles and want to play with the pirate ship ? This is the app for you!\nYour child will discover each type of boat, will recognize the sounds of the sea and in the meanwhile will exercise his memory!\nBoats and Ships for Toddlers is an educational and entertaining game.\nIt is a nice, simple, fun, and colorful game for toddlers and kids! Play with vehicles games !\nA lot of different seascapes and many cool boats to keep your kid busy.\n-drag and drop the boats on the screen, play with galleon, have fun and match the couples!\nYou can also choose the difficulty level: easy, medium and hard, to suit the needs of each child and exercise your toddler fine motor skills.\nMatch up all ships and boats pairs and honk the ship horn !\nA lot of boats in the harbor ready to sail at sea !\n- more than 90 boats in the FULL version: motorboat, sailboat, yatch, galleon, pirate ship, Viking Ship, aircraft carrier, rowing boat, watercraft, jetsky, speedboat, the Titanic !\nA lot of educational ship games for toddlers and young kids!!\nYour child will admire every kind of boat vehicle and will hear realistic sound effects!\nAre you ready to challenge your memory ? Match up the boat pairs in the harbour !\nBoats and Ships for Toddlers is a funny game which also helps your toddler improving his abilities.\nGo through the waves of the sea by ferry boat !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SUMMARY: Brick-sized treatise on the mechanics of genes on population dynamics and trait expression \u2013 in (nearly) everyday language.\nAUDIENCE: A classic must-read if you're remotely interested in biology \u2013> zoology \u2013> understanding genes AND if you enjoy complex, subtle and far-reaching theoretical speculation into evolutionary dynamics.\nREVIEW: Richard Dawkins is an evolutionary biologist with out-of-this-world theoretical insight into anything and everything he puts his mind to.\nIn this book, he meshes the works of Darwin (evolution), Trivers (game theory), and Mendel (genetics) into an intuitively appealing and internally consistent model to expose the mechanics of genetic inheritance.\nGroup selection \u2013the idea that whole groups of animals, like an entire species, strive for common success.\nThe crux of the book is this: Dawkins proposes a daring conceptual shift from the entire organism to the individual gene as the fundamental unit of hereditary selection.\nHe briefly introduces (re-introduces if you're a biologist) the nitty-gritty of molecular genetics, but quickly moves on to illustrating his points with one fascinating evolutionary example after another.\nOne of our lecturers warned a room full of neural scientists, behavioural psychologists and veterinary doctors that it would be a tough read for most of us. I can't decide whether she's right, as my background is Zoology (so slap in the middle of his subject matter). From what I recall, though, I found it reasonably jargon-free. In terms of complexity, he does go off onto mindbogglingly sophisticated flights of theory \u2014 with a couple of beautiful chapters on game theory to name but one topic.\nSo fasten your seat belt, because, intellectually, this is a Ferrari of a book. But if you're up for the challenge, it will (at worst) stop you from blurting genetic stupidities, and (at best) revolutionarize your understanding of life, the universe and everything.\nThis book is a all-out classic, and it is one of the must-have intellectual gems of the twentieth century. Go on. Try it.\nBrowse the Book Review archive. Genre: pop science. Author: Dawkins Richard. Reading Level: Academic. Star: 5. Bookmark the permalink. Post a comment or leave a trackback: Trackback URL.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"1. To foster the establishment and maintenance of a high level of professional competency and ethical counsel in the fields of meteorological and oceanographic consulting.\n2. To define, revise as necessary, and maintain the requirements and criteria for accreditation by CMOS of consultants in either meteorology or oceanography.\n3. To review or arrange for peer review the applications received and advise the applicants of the decisions made by the Committee.\n4. To maintain the list of Areas for Specialization for Accredited Consultants.\n5. To maintain the list of Accredited Consultants.\nMode of Operation: The Committee will normally work and meet by e-mail and will hold at least one teleconference per year. It may also appoint ad hoc working or study groups as required.\nMembership and Terms of Office: There shall be 3 or more members of the committee all of whom shall be Accredited Consultants, each appointed for a three-year term with the option to renew.\nCommittee Level: This is a Council-appointed Committee.\nReport: The Committee will produce an annual report in accordance with Appendix III to the By-Laws.\n1. To ensure that a basic level of qualifications has been achieved and recognized for anyone engaged in meteorological\/oceanographic consulting and applying for accreditation by CMOS.\n2. To monitor the maintenance of a high level of professional competency of and mature and ethical counsel provided by consultants accredited by CMOS.\n3. To support and consult the CMOS Executive and Council on accreditation matters as required.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It seems as if most senior Executives becoming independent\/ going plural fancy having one or more NED appointments as part of what they do. For many new to it, the NED market is unstructured, invariably difficult and for those short on perseverance, unsatisfactory. It is characterised by a virtually inexhaustible supply of candidates, a patchy and often invisible and variable demand from organisations and based on a fragmented and uncertain intermediary function.\nAt the top end of the market is good structure whilst at the very small end, there would appear to be no or virtually no structure. The winners are those who can find out how best to handle the challenges and make the market work for them. The winners are those who invariably work hardest at it.\nYou need to find out if you have the right qualifications, knowledge, skills, experience and personal attributes. The mix of these depends on which segment of the market you are aiming at. You need to identify weak areas \u2013 and have a programme to bring yourself up to market readiness.\nYou also need to create a Plan for finding NED appointments and then you need to actually execute that Plan based on being continuously alert, available and, importantly, attractive to recruiters, be they end users or intermediaries.\nThe Lead contact here is Sharon Constan\u00e7on, CEO of Genius Boards and Genius Methods. She is also Chairman of the South African Chamber of Commerce in London and, personally holds several Non-Executive Directorships.\nPlease address any comments or questions that you may have to [email protected].\nWe are keen to develop the content of this section on the WFL website, overseeing all the Contributed Material that comes in, working with WFL to decide how best to handle it and, importantly, answering any queries that you might have.\nWe also arrange Seminars and Advisory Reviews for small groups and individuals \u2013 more info available on request.\nWorking Free is not a text book; it tells you what to do, primarily by signposting sources of information and guidance. It does not tell you how to do it. You need to understand the market \u2013 what it is; where it is and how it works. It's you who needs to do the hard work, following up on this information! Without this approach, it is unlikely that this market will work for you in the best way.\nWhat we would also regard as prime sources of information are set out in section 2 \u2013 The NED Market \u2013 understanding it.\nWhether you are a beginner or an experienced NED, it is important that you acquire a broad and a sufficiently detailed understanding of NED work and its market.\nThe real value in this Section on Ned work \u2013 for all Readers \u2013 will doubtless come from learning from others' experiences and knowledge \u2013 what works and what does not work \u2013 things to do and things to avoid \u2013 and have the opportunity to refer back to them \u2013 and we will publish it after any necessary editing.\nNote: Many of the following sections overlap & need to be read as a whole.\nThose who run businesses recognise the need for advice \u2013 both internally and externally. External advice covers a broad range of suppliers \u2013 choosing what and who to use and when and how is a fundamental management skill. NEDs are a sort of half-way house \u2013 all the responsibilities but independent.\nTo be genuinely successful and to continue being successful all organisatons need Non-Executive Directors, however described.\nThink about it! Break it up into bits!\n\u2026to continue being successful\u2026 not always easy\u2026 and often requiring skills, knowledge and resources not anticipated or currently available within the organisation \u2013 more often than not absent from what was necessary in previous stages of development.\nall organisatons\u2026 ranging from FTSE100 companies down to the smaller end of SMEs including non-incorporated organisations. NED Practice and processes vary for differing levels and call for differing NED profiles.\nneed\u2026 someone \u2013 presumably the organisation's leadership \u2013 must recognise this need and must decide what to do \u2013 on the assumption that doing nothing is not generally an option.\n\u2026however described\u2026 someone delivering broadly the same services as a statutorily defined Director \u2013 could be construed as a shadow Director \u2013 mostly operating on a non-executive basis but not necessarily or not always.\nThis particular area is very relevant in a governance context when applied to SMEs, Charities\/ Not for Profits.\nMany organisations offer Definitions of NED \u2013 and also of Executive Directors.\nA command of Corporate Governance principles and practice.\nA good understanding of company law \u2026.\nNumeracy (being a qualified accountant is a big help).\nSee Section 6.3.6 \u2013 \"Being a NED \u2013 what you need to know and how to behave.\" For a more detailed analysis.\nBut the size of business will require different specific experience and of that level. The sector usually calls for specific sector experience \u2013 but not always. Note that where this is specified it may be about access to trade contacts \u2013 which, when exhausted, may lead to disinterest.\nLarge private companies with a majority shareholder.\nSmall business \u2013 10-50 employees and annual turnover above \u20ac2million but below \u20ac10million.\nMicro Business \u2013 10 employees or less and annual turnover under \u20ac2million.\nMost SMEs (turnover from zero up to \u00a340m, but, in this context, businesses falling at the lower turnover end) have a need (sometimes not recognised) of the services seen as offered by NEDs. In practice, this invariably takes an amended form but the basic attributes are still very relevant. Often the role is NOT that of a statutory Director and often there is an executive component involved. In these cases what would be called a Non-Executive Director would, more properly, be referred to as a Business Mentor, although a range of other descriptive terms are often used.\nHowever, to understand the NED Market better, it is necessary to understand the demand, the supply and how the intermediary function works.\nFTSE100 companies \u2013 Almost always fed to the Market through major head hunters \u2013 either the large global players or well recognised niche consultancies. Chairmen need to be seen to be totally objective and beyond criticism in handling the recruitment process and are likely to go no other route.\nOther FTSE companies \u2013 Virtually the same as for FTSE 100 companies, although more latitude taken in Search Firm selection.\nAIM companies \u2013 As for other FTSE companies but shortlist candidates can be fielded from other sources. Knowhow and experience of the close sector and a good understanding of risk are needed here, particularly with tech companies.\nLarge private companies with multiple shareholders \u2013 Virtually the same as for other FTSE companies, although more latitude taken in Search Firm selection and also candidates coming from other sources. Conformance with UK Governance Codes is, in practice, seems to be less mandatory.\nLarge private companies with a majority shareholder \u2013 The driver generally, is to find someone with the right attributes but where a close personal relationship with the principal working shareholder is likely but not necessarily apparent. Some see this as a risk \u2013 but most principal working shareholders do not!\nMedium sized business \u2013 50-250 employees and annual turnover \u20ac10million but below \u20ac50million. (generally seen as \u00a340m.) \u2013 The driver, as with large private companies, generally, is to find someone with the right attributes from whatever sources.\nSmall business \u2013 10-50 employees and annual turnover above \u20ac2million but below \u20ac10million \u2013 Often not advertised or reliant on Recruiters. Often word of mouth;. Can be NOT as a statutory Director and often handling some executive non-routine activity. The risks and caveats are the same for all NEDS but this is a distinct market.\nMicro Business \u2013 10 employees or less and annual turnover under \u20ac2million \u2013 Mostly not advertised or reliant on Recruiters. Mostly word of mouth; Nearly always NOT as a statutory Director and nearly always handling some executive non-routine activity. The risks and caveats are the same for all NEDS but this is a distinct market.\nIt might be supposed that all the NED's focus will be on securing the appointment. As important is the need to carry out appropriate due diligence on the prospective hiring company. It is always awkward if you discover fairly quickly that the company is insolvent, likely to be soon or that there are improper business practices (including fraud) lying in wait. Knowing about this relies often on intuition and\/or unofficial sources.\nThis may be YOU! Or anyone who fancies their chances. If you are not up to it, you will, generally, not get the work. Or, if you do, may not deliver as required and\/or expected.\nFormal recruitment volumes are lower than commonly supposed and directly engaged volumes are difficult to assess.\nAs a pure-play NED recruiter, it is difficult to generate the right volume and at the right margin and which can support the optimum delivery infrastructure of the firm and leave an acceptable profit.\nCompared with mainstream executive recruitment, volumes are small and they are structured differently for each segment and level in the market, calling for different infrastructures.\nAt the high end of the market, dominated by the large international players, the NED search functions are generally part of the overall Practice but usually presented as a specialist part of the Practice. This provides the delivery infrastructure and also plays a key role in keeping client contacts live.\nAt this level there are also a small number of standalone NED search firms. They tend to specialise. You need to know what they specialise in.\nThe big four accountancy based international firms have to contend with client conflicts of interest and, mostly as a result of this, are low activity market players in NED Search. Their interactions with NEDs is more about getting work for themselves rather than facilitating NED opportunities.\nAt the lower end of the market, many recruiters need to offer NED recruitment alongside a variable mix of training and database membership, all charged for The composite and variable packages are built around finding NED appointments for you \u2013 but, as volumes are low, augmenting their income by delivering support services is attractive to them. They would also argue that being better equipped professionally and also by augmenting your networking capabilities, you are likely to be more successful than otherwise.\nThroughout this overall market, you find the intermediaries who focus on training and charge for these services either through \"membership\" fees or for one-off events. Personal judgements need to be made where you spend money in areas designed to facilitate successful NED appointments.\nPersonal networking works best amongst buyers. It can also work were networking amongst other sellers. But nowhere as much! Best to seek to network with buyers \u2013 if you can find them.\nYour final list of Intermediaries will include a number of eminent firms that you would not otherwise have thought of approaching. Compiling this list needs to include other Intermediaries that your research indicates they might come across NED opportunities. This is a bit of a slog \u2013 but it does work. Your final list may well include established Management Consultancies, Law Firms, the larger firms of accountants \u2013 and also selected smaller ones, professional associations who can and do effect introductions.\nThe Code is essentially a consolidation and refinement of a number of different reports and codes concerning opinions on good corporate governance.\nUnited Kingdom company law regulates corporations formed under the Companies Act 2006 and also governed by the Insolvency Act 1986. Sections 170 to 177 relate to Directors, be they Executive or Non-Executive. EU Directives also have a bearing (may be subject to change under Brexit) as also does case law.\nOur view is that most comprehensive exposition of all aspects of being an NED can be found in The Non-Executive Directors' Handbook \u2013 4th Edition published by ICSA for the Non-Executive Directors' Association. This is no disrespect to other publications \u2013 but you only need one if that one covers the area as well as all of them and better than many.\nMostly, the hiring organisation declares the remuneration. This can be negotiable. You will be generally be paid through the payroll although operating through a personal service company does happen \u2013 but is not seen as best practice by HMRC who may well challenge it (possibly in a punitive way) and is also deemed by many to be setting a poor example.\nIn assessing the rate as a commercial return you need to take into account the real number of days that you will be spending \u2013 not the advertised required commitment \u2013 with this organisation and also factor in the element of risk. Your liability is the same as an executive Director but you will, more than likely, never know as much about the business or be privy to some of the confidences. It is argued that getting to know as much as an executive Director impinges on your independence. NEDs do not run businesses.\nDo you own research into market rates of pay, reflecting where the company falls in the hierarchies of organisations referred to in this Technical Topic Section. It is not unreasonable to find some linkage with the organisation's rates of executive remuneration.\nMany of those new to NED work see the process of getting NED appointments as a minefield \u2013 complex, tortuous and with a high level of unsuccessful approaches. Not surprising.\nAs with any organisation selling professional services, your first task is to define \"your Product\". What sort of NED work are you looking for; at what level of business and what special features can you bring to the role other that what all NEDs should be able to deliver.\nWe are in the process of developing further this Section but much of what you need to know is covered in other parts of this Technical Topic.\nMost visitors to this website will be usually white, male and over 50. In recent times, issues around diversity have become increasingly more influential \u2013 quite rightly \u2013 and the NED market is broadening its reach with NEDs coming from a variety of gender, ethnic and age sources. Many would say that the market for NEDs is now a level playing field. Some would say that there is a discernible bias towards gender, ethnic and age features and pursuing a policy of positive discrimination.\nNot an easy area, but there are justifiable and major ongoing efforts to promote a balance and there seems to be no opposition. Feeling that you might lose out to current sensitivities might bother you but (probably) the majority of candidates are, still, white, male and over 50 and until the resource of executive talent can produce enough suitably qualified women to become NEDs, this imbalance will continue to close slowly.\nYour own Back Office \u2013 Accounting, Financial and tax matters \u2013 See separate Technical Topics Section. Other matters \u2013 coming soon.\nBeing compiled \u2013 although much relevant information can be found elsewhere in this section.\nOur aim is to create \u2013 mainly through market knowledge and recommendation \u2013 a continuously reviewed list of intermediaries and service providers and, ideally, categorised as set out below \u2013 which, we suggest, is the way in which you should see it.\nHowever, in practice, this market does not work in this way \u2013 many service providers deliver a varying mix of services \u2013 some including a pay-to-join requirement \u2013 making categorisation difficult.\nThe pay-to-join requirement covers, again, a varying mix of services. Sometimes it is for Training; sometimes for peer group networking and sometimes it is for making available to you NED appointment opportunities \u2013 often seen as finding you an NED appointment \u2013 or, often, a varying mix of these components.\nWe would be pleased to receive information and comments here that might be helpful to others. This area can be found at the bottom of this page, or simply click here to be taken to the area now.\nSharon Constan\u00e7on, Chartered Director, MBA and Chartered Secretary is CEO of Genius Boards and Genius Methods.\nThe Genius Group has worked with the full range of businesses from FTSE 100s to SME's, listed, private, family, public sector, charity, housing trusts and investment trusts. Regulated sectors in addition include financial services, insurance and NHS.\nSharon is Chairman of the South African Chamber of Commerce in London and, personally holds several Non-Executive Directorships.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Back when 5.0 was released, the HDFS Repository was moved out of the ES-Hadoop project and into the Elasticsearch project. At this time, the plugin was re-written to work under Elasticsearch's security model. Support for 'Kerberized' HDFS clusters was dropped due to the overwhelming work required to have the client gel with the installed security policy. With 5.5.0, that's all a thing of the past, as we are announcing that Kerberos is back to being supported for the HDFS Repository Plugin! This has also been backported 5.4 branch as of 5.4.1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Vadodara (Gujarat), August 2017: Roshni Don Bosco organized the 'Independence day' and 'Janamastami' celebrations with fun and frolic for more than 300 under-privileged children from 3 slums of Vadodara city. The celebrations were held under the aegis of 'Education Project' supported by Manos Unidas (Spain).\nThe day was marked with remembering freedom fighters and saluting the national Flag. It was a moment of pride for the children to watch tricolor unfurl in their own locality. Children came with their own flags made of paper to celebrate this national festival. Since it was also Janamastami (a Hindu Festival) so in afternoon children celebrated this festival in their locality.\nThe children who attend the study classes were joined by their teachers in these celebrations. The whole environment was filled with joy and cheer. Small children dressed-up as 'Govindas', Radha and Krishna attempted to reach the Handi and finally were helped by elder children to break it. Snacks prepared by Self-Help Group women were distributed to all and children went back with memorable experience of festival celebrations.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drought and forestry management practices have created a state of emergency in the forests that lie close to urban areas in California. The Yosemite wildfire sparked fears that available resources would be unable to stop the fire's path before it reached water and hydropower sources that support San Francisco.\nIn recent years, insurance providers have paid enormous sums of money for catastrophic wildfires that destroyed hundreds of homes in Colorado. Population density in California raises concerns of a wildfire that penetrates the edge of Los Angeles or San Francisco could result in even larger payouts.\nNo one can control Mother Nature, but homeowners can take significant measures to create defensible space around the structures on the property. Fire mitigation practices have become a significant topic for discussion in lawmaking sessions at every level. Insurance companies want city councils and state lawmakers to pressure homeowners to do more to protect their properties from wildfires.\nPrivate property laws prevent lawmakers from applying demands for fire mitigation compliance. Tax incentives are being considered in areas of Colorado. These measures encourage the homeowner to participate in the wildfire fight. California homeowners are wise to consider the true risks that could be reduced through mitigation efforts.\nMore specifically, there are certain insurance carriers that provide specific wildfire protection services such as a fire retardant coating in the event that a fire is threatening your home. To discuss these options as well as review the insurance coverage you may already have in place, please contact Chivaroli Premier, today. Their insurance specialists can answer any questions and ensure that you are properly prepared before the next California wildfire occurs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"We've all become more impatient as digital services are increasingly real-time and in many cases instantaneous. Look at the big winners out there: Uber, Klarna and Amazon. Each organisation has innovated to reduce impatience by dramatically reducing wait time and offering real-time updates. In banking, there is a similar pattern.\nThe nature of patience has changed.\nYesterday, I ordered a taxi from Lyft. The app indicated the driver was eight minutes away. I cancelled the fare. Eight minutes \u2013 that's a lifetime I thought. Switching to Uber, I had a taxi arrive in under two minutes.\nI've also begun to use UberEats. But recently I was staying somewhere outside of its coverage. I ordered a take out over the phone. Not only did it take over an hour, missed the ability to track my order in real-time.\nWe've all become more impatient as digital services are increasingly real-time and in many cases instantaneous. Look at the big winners out there across different sectors like Uber, Klarna and Amazon. Each organisation has innovated to reduce impatience by dramatically reducing wait time and offering real-time updates.\nIn banking, we've seen a similar pattern. Retail P2P payment services which act in real-time like Zelle and Venmo have seen accelerated customer adoption. The question is will a broader church of financial services go real-time?\nNow recent research from analyst firm Ovum, commissioned by Icon Solutions, has revealed that commercial banking is set to go real-time, enabled by real-time payments (RTP).\nThe report The Rise of Real-Time Payments in North America, is based on interviews with 211 North American institutions. It found that in the short term, real-time services are viewed as essential to effectively compete for corporate clients. In the long term, real-time payment (RTP) infrastructure will become the foundation for banks' transition into an open banking environment.\nIn the US the intention is to have a ubiquitous faster payment system operational by 2020, and banks are investing in the race to meet that deadline.\nAccording to the report, North American commercial bank IT spend is set to increase by a staggering $3.3bn, growing to $17.1bn per year by 2021, driven significantly by the shift to RTP. Consequently, 50% of US banks and 40% of Canadian banks plan to increase investment in RTP with nearly a quarter of these banks increasing their budgets by over 6% YoY.\nDespite competing claims on firms' budgets, many banks are realizing that this is a unique opportunity to put in place systems that will enable them to address the challenges of digitalization and the emergence of new open banking ecosystems where the sharing of customer data is the norm.\nWhile some of the priorities of commercial banks are driven by wider issues such as data security and regulatory compliance, the top three areas for IT investment \u2013 cash management services, liquidity risk management and mobile banking \u2013 are all underpinned by RTP. In my mind, this provides evidence that commercial banks see RTPs as providing a foundational capability to address existing service limitations.\nCash management services \u2013 high on the list of demands from corporate clients \u2013 is a key investment area for US commercial banks about to implement RTP infrastructures, along with the provision of real-time cash positions, scenario-based forecasting and direct integration with client systems.\nBanks in Canada, not as advanced on the RTP implementation timetable, are far less active in this area, but can be expected to turn their attention to it in the future as customers demand greater real-time availability of real-time data.\nBanks, desperate to break from the stuttering world of batch processing, must implement RTP to unlock real-time transaction data and unleash a powerful wave of real-time services.\nSo for banks in the US access to real-time information, analytics and other services based around transaction data is paramount. After all, what good is artificial intelligence if it's dealing with account data from three days ago?\nIn Canada, the focus is different and concentrated more heavily on providing access to RTP, along with enhanced services via digital interfaces.\nRegardless of market, for commercial banks and their corporate clients, there is a clear shift towards deepening relationships and service enhancements driven by analysis of real-time transaction information.\nCanadian commercial banks, and to a lesser extent US banks, like those across the rest of the world are taking steps to prepare themselves for open banking.\nWhat was started by the EU's Second Payment Service Directive (PSD2), under which account holders can authorize trusted third parties to access account data or initiate payments, has become a global trend as banks realise the mountain of customer data they 'own' is critical to their future success.\nAs a result, 100% of Canadian institutions are developing APIs. US banks are further behind with just 56% confirming APIs are in their plans for the next year.\nFor open banking to be a success, both banks and third parties need access to real-time customer information to deliver compelling offerings. As such, the investments being made in RTP and open banking imply a future convergence of commercial and retail payments as banks look beyond traditional payments systems.\nTraditional applications and middleware used today were designed for a batch world, have costly software development lifecycles, and require expensive arrays of hardware to function. They are incongruous with the demands of corporate clients in the short term and open banking models in the long term.\nThe next wave of technological innovation is designed differently, starting with rapid prototyping, reliable scaling at low cost, and emphasis on an optimised time-to-market.\nAs the report highlights, to address service offerings that have lagged behind customer needs, and maintain competitiveness with rival banks, and increasingly fintechs, North American institutions need to adopt frameworks that deliver real-time agility as we enter the age of open banking.\nImpatient innovation will continue to drive the market. And the race is well and truly on to stay ahead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Little Om who Tried what he can to lighten up his sisters life.. Can We Do too? Yes, We Can !\nThis #PowerLightAVillage video is absolutely amazing. The initiative will also encourage children to study, thereby securing a brighter future.\nEvery individual thinks India needs to change & we are striving for that change.\nSo a small effort from us in the right direction can #PowerLightAVillage. Lets come together & join hands in lighting up the villages in India...!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Imagine walking alongside Jesus in His final hours on earth. In For This He Came: Jesus' Journey to the Cross, author Bill Crowder takes you with Christ as He imparts final lessons to the disciples He will soon leave behind.\nJoin Jesus as He dines in the upper room during the Last Supper. Feel His agony in the garden of Gethsemane. Witness His betrayal and His trials before the high priest and Pilate. Throughout each pivotal event, you will see Jesus as prophet, sufferer, advocate, and ultimately our savior.\nAnd because the journey does not end at the cross, you will also see Christ triumphing over the grave as He conquers death and returns to the Father in heaven.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Too scared too fap now. I would love to see for the next video a deadly and beautiful succubus Thank you Viento How did Shante get Serana to look like that. Fun Fact: There is evidence to suggest that. I'd give this a light 5 (FULL REVIEW ON PROFILE) Anyone watch How I Meet Your Mother.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BE AMONGST THE ACTION. RIGHT ON YOUR DOORSTEP.\nThe Lizum 1600 in the heart of the Axamer Lizum ski area lies directly above Innsbruck; the Tyrolean capital, and is home for all sports enthusiasts. There are many sports opportunities as well as facilities for conferences and events thus ensuring eventful days and team-building support. As a training centre for the world's best ski instructors, we stand for high quality, safety and a whole lot of know-how. You can also become a part of this community.\nExperience an exclusive weekend with a state-certified ski instructor from the training team of the Tyrolean Ski Instructors Association in the Axamer Lizum - the site of the 1964 and 1976 Olympic Winter Games.\nWeave to and fro on perfect pistes and powder snow slopes! Because of the altitude you can still enjoy perfect skiing conditions in April. Make the most of the last snow of the season.\nExperience the mountains from a different angle and explore the Axamer Lizum with touring skis. In addition, you can participate in presentations and join a guided snowshoe hike.\nEnquire now quickly and easily without obligation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Arts Council of Greater Lansing is honored to partner once again with Adams Outdoor Advertising to create this free opportunity - the 2018 Young Creatives Billboard Project - for Tri-County educators and their students.\nAt the Arts Council, we are strong believers that we should support our youth and give the public an opportunity to see some of the artwork created by young and emerging talent in our local schools.\nFrom March 1 -31, 2018 (National Youth Art Month), we will showcase the art of 10 area youth artists, grades K-12, allowing their magnificent artwork to be viewed by thousands of individuals during their daily commutes. Check out the 2017 billboards here.\nTeachers who work with K-12 students in any Eaton, Ingham or Clinton County school may apply and submit one (1) artwork to be considered for placement on a digital billboard. Only one artwork per teacher is permitted. Please note: parental permission is required at the time of application.\nStep 2: Upload your student artwork to the online application at Slideroom by 11:59 p.m. EST on Monday, Feb. 12, 2018.\nA committee will review the submissions, where they will be scored on composition and overall visual elements, including roadside appeal. We are looking for all grades and regions within the tri-counties to be represented. Please note: one billboard will be selected outside of this process--the overall juried winner of the MSU Federal Credit Union Student Art Competition will also be given one of these highly coveted billboards.\nWhile we wish we could offer billboards to all of the students whose work you submit, unfortunately we only have 10 billboards available to us. Therefore, those students whose work is not chosen are invited to participate in our Youth Art Exhibition, which will be on display at the Arts Council office for the entire month of March. This exhibition showcases the original art of the 10 billboard finalists along with other student art.\nArt works must be matted or framed for hanging and canvases must be ready to exhibit, with finished sides. All art must be delivered to the Arts Council by Feb. 22 at 5 p.m. If you submit a student work for the billboards and it is not chosen, it can be a part of the March Youth Exhibition.\nIn addition to having their art featured on our wall, there is also an opening reception for family and friends held during Arts Night Out on Friday, March 2, 2018 in Old Town! All of the youth artists are invited to be present, take photos with their art work and visit with Arts Night Out attendees.\nFor more information contact Dawn Gorman, communications and events manager at dawn@lansingarts.org or Meghan Martin, program manager at meghan@lansingarts.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The d\u00e9cor and ambiance of a home will dictate what style of lighting is needed, and we are pleased to carry a stunning collection of table and floor lamp lighting from Dimond Lighting, one of the premier lighting establishments today. Dimond Lighting creates lighting designs motivated by up-and-coming fashion and leading home d\u00e9cor industry trends. Every piece that we carry from Dimond Lighting is instilled with the genius and originality of an international designing team along with skilled engineers to guarantee that each product is produced with flawless detail and unequaled design.\nFor example, the 1Light Patriotic Uplight in Dark Blue is a beautifully designed cylinder lamp with all the colors of the US flag. This stunning table lamp, with complementary prints and solid colors is perfect for contemporary and art deco home d\u00e9cors. This attractive lamp has a solid base and requires little space. Large or small tables will work well with this Patriotic Uplight. The inspiration for this design is obvious with this uplight lamp. With limitless character and exquisite detail, this Dimond Lighting creation is a great part of a family of lighting brilliance.\nDimond Lighting was established with dedication to provide innovative quality products with custom design appeal. This is why we are so confident about having such a well-structured supplier within our lighting collection. The term \"Jewelry for the Home\" has been used to describe the unequaled beauty and luxury that Dimond brings to any home d\u00e9cor. Any room, whether the living room, dining room, family room, kids' room, or any other section of the home, will benefit from having a Dimond Lighting fixture.\nThis is why we know that our various selections of Dimond Lighting designs, like the Patriotic Uplight Lamp designed in a sturdy fabric is a sure winner! By welcoming the most recent member of the E.L.K. Lighting Group, Dimond has proved that they are a force to be reckoned with in the table and floor lamp industry. In fact, E.L.K. has provided more than 25 years of innovative designs to the world of lighting. We are proud of the opportunity to enlighten our customers with the fantastic lamp designs that can be purchased within our Dimond Lighting collection.\nWith an initial release of more than 200 novel designs, Dimond Lighting creates designs for distinctive homes. For instance, brands such as Biltmore Estates, Trump HomeTM, and Mary-KateandAshley, are just a few of the top exclusive brands that can be found within the Dimond Lighting collection. Our decision to bring such a magnificent brand on board shows our loyalty to our customers. We understand that our customers want the best home d\u00e9cor at reasonable prices. We are adamant that our merchandise is delivered in pristine condition as well. By offering such a beautiful piece as this customized Patriotic Uplight, we stick to our pledge to help shoppers make their home more beautiful. Check through our fantastic collection of spectacular table and floor lamps created by Dimond Lighting. The inspiring designs that you encounter are sure to make your shopping experience with us truly unforgettable.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home\u2022Gregg Gelb Awarded Medal of Arts!\nCongratulations to the Philharmonic Association's very own own Gregg Gelb for being awarded the 2018 Raleigh Medal of Arts!\nThe City of Raleigh Arts Commission selected six individuals and two organizations to receive the 2018 Raleigh Medal of Arts, the City's highest arts honor. The Raleigh Medal of Arts is awarded for extraordinary achievement in the practice or support of local arts. Based on the National Medal of Arts program, the Raleigh award was inaugurated in 1984 by the Raleigh Arts Commission so that excellence in the arts could be given special recognition. Over the past 34 years, 152 medals have been awarded.\nThe Philharmonic Association is proud to announce that Dr. Gregg Gelb was one of the recipients of the 2018 Raleigh Medal of Arts. Dr. Gelb's support of the arts in Raleigh is clearly demonstrated by his commitment to teaching and conducting the Philharmonic Association's Triangle Youth Jazz Ensemble. In addition to leading TYJE, Dr. Gelb leads his Jazz Quartet, La Fiesta Latin Jazz Band, Second Line Stompers, and Swing Band. He is also the founder and director of the Heart of Carolina Jazz Orchestra ad Jazz Society and the co-founder of the North Carolina Jazz Repertory Orchestra. His groups have recorded over a dozen CD's. Along with leading and directing, Dr. Gregg Gelb is a renowned saxophonist and clarinetist. He often performs on saxophone with the North Carolina Symphony and with the orchestra he co-founded, North Carolina Jazz Repertory Orchestra. Dr. Gelb is the recipient of a Jazz Composers Award from the North Carolina Arts Council and four Regional Artist grants from the Fayetteville\/Cumberland County Arts Council.\nThe Philharmonic Association is grateful that Dr. Greg Gelb has chosen to serve our organization for over eight years and we are excited that the City of Raleigh Arts Commission selected Dr. Gelb as a recipient of this year's Raleigh Medal of Arts. He is certainly worthy of such an honor. We look forward to seeing Dr. Gelb continue to inspire musicians in the Philharmonic Association and in the Raleigh community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"1 Open the door so it is parallel to the wall where you want to fix the doorstop.\n2 Holding the wall-mounted doorstop in place, mark on the skirting where the doorstop will be positioned.\n3 Separate the two parts of the door stop and use a woodscrew to attach the base of the doorstop to the skirting.\n4 Then attach the top part to complete the job.\n5 Now your wall mounted door stop is installed.\n6 Floor Mounted Door Stop. Open the door to where you would like it to stop and mark with a pencil on the floor. Use a multi-purpose detector to check the floor and make sure there are no pipes running under where you want to position your doorstop.\n7 Use a woodscrew to attach the base of the doorstop to the floor.\n8 Then attach the top part of the doorstop to complete the job.\n9 Now your floor mounted door stop is installed. If the doorstop is going into a concrete floor, drill a hole using the masonry bit on the drill and insert a rawl plug. Screw the doorstop into the plug. If the doorstop is going into a tiled floor, drill a hole using a tile or masonry bit and insert plug. Don't use hammer action on the drill. A good tip is to try and use the grout line when screwing in the stop, to avoid drilling into the tile at all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Coming postdoc at the Harish-Chandra Research Institute (All\u0101h\u0101b\u0101d, India).\nPrevious postdoc at the Institute of Mathematical Sciences (Chennai, India).\nResearch interest: Operator algebras, von Neumann algebras, subfactors, planar algebras, algebraic combinatorics, finite group theory, fusion categories, vertex operators algebras, infinite dimensional Lie algebras, conformal field theory, noncommutative geometry.\nI'm mainly active on MathOverflow.\n37 Is Jesus an avatar of Lord Vishnu?\n30 Is it usual for a referee to heed updated versions on arxiv?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..4e66c59497aaf7faab62a0884f86d3bb54903e21
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+{"text":"Paying out of pocket expenses for dental work can cause major financial strain to the average American household. While fillings and a basic cleaning at the dentists can be under $200 the cost of braces or Invisalign can be closer to $5000. American households can withstand an extra $200 or $300 being spent to ease tooth pain but five grand is not something that most have just lying around.\nThe New York Times recently published an article explaining how Flexible Spending Accounts could help cover the costs of orthodontic treatment. Flexible Spending Accounts or FSA are used to cover out of pocket medical costs. These accounts are set aside as tax free money throughout the year. If an employee wants to set aside a specific amount from each and every paycheck it can go towards a Health Savings Account (HSA) or Flexible Spending Account (FSA).\nMost people recognize these acronyms because pharmacy retailers like CVS and Walgreens advertise that accounts must be spent by the end of the year. In mid to late December those that have an HSA or FSA will likely look for items that are covered by these accounts. Pharmacy retailers will have a special section in the store to display products that are covered.\nOne \"product\" that you may not think about when it comes to these types of accounts is orthodontic treatment. Whether you are looking for Invisalign in Miami, Florida or metal braces in Houston, Texas there are options. Throughout the United States orthodontists offer specialized treatment to straighten teeth. Invisible braces are known as Invisalign while metal braces are the more traditional way to straighten teeth over a 18 to 24 month period. Unfortunately, families that do not have disposable income are going to find it quite difficult to cover the cost of braces.\nOn average, braces and\/or Invisalign can cost between $4000 and $6000. For adults, the cost is even higher. If you want to offset a portion of this investment it would be smart to look into options through an FSA. Remember that laws are changing every year so you will want to do your research before assuming the entire cost of braces will be covered. Also recognize that some insurance plans cover Invisalign and braces as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Some of the most inventive minds at the University of Toronto are stepping into the limelight.\nThe origin story of DeepBind begins like so many inventions: with a question and then a leap.\nCo-creators Babak Alipanahi and Andrew Delong were chewing the fat and wondering why no one had thought to marry deep machine learning with computational biology to figure out why some mutations lead to disease and why others don't.\n\"We were both talking about this idea, and both wondering why no one seems to have tried it [but] Babak was the one with the confidence to say 'You want to do it? Well, let's just do it!'\" recalls Delong.\nThe end product of those conversations \u2013 DeepBind \u2013 took more than a year to create with the support of professor Brendan Frey of U of T's department of electrical and computer engineering. All three work together at Deep Genomics, one of the University of Toronto's best-known and successful startups of recent years.\nOn May 17, DeepBind was among four products recognized as U of T Inventions of the Year. The awards, which recognize their uniqueness, potential for global impact and commercial appeal, were presented at the university's third annual U of T Celebrates Innovation event in front of an estimated 200 guests, including Ontario Lt.-Gov. Elizabeth Dowdeswell.\nThe award is \"a great honour,\" says Alipanahi.\nDeepBind, which combines artificial intelligence and genomic medicine, is the first-ever deep learning application to study mutations linked to diseases that have proven difficult to analyse in the past because of their complexity, such as haemophilia and skin cancer.\nFor example, skin cancer is caused by more than one gene. However, having these genes does not necessarily mean a person will develop melanoma. Scientists must also consider UV exposure, which can damage the DNA in skin cells, leading to cancer-causing mutations.\nThe software modules, which are available free for academic use, are able to handle millions of sequences per experiment and can create \"mutation maps\" to reveal how genetic variations can cause disease.\nThe goal of their work, Alipanahi says, was to create a powerful algorithm that was fast and accessible to biologists studying these diseases.\nSometimes this type of work is viewed as a \"black box. Like a magical tool that is very powerful but you don't know much about how it works. We tried to help biologists peer into the box and understand it,\" he says.\nKey to their work was a tremendous amount of public data generated by professor Tim Hughes and associate professor Quaid Morris of molecular genetics, not to mention the general atmosphere at the university where you can sit down and learn from world-renowned innovators like Geoffrey Hinton, considered by many as the \"godfather\" of deep learning, says Alipanahi.\nIt also helps to have great colleagues close at hand to bounce ideas off of, echoes Delong.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"German reliability\" : this blog could have ended with only this two words!\nThe great thing about rivalry between car brands is that they always have to raise their standard to stay competitive. Having BMW, Mercedes and Audi fighting in the same range is a benediction for car lovers. So when the BMW 7-series (1977) and the Mercedes W123 (1975) were launched, Audi HAD to react with a high end car and that's how the brand with the four rings gave birth to the Audi 200 (1979). Gave birth? Well, not exactly. They took the Audi 100 and gave it some super powers.\nDNA Collectibles decided to produce the \"Avant Quattro\" version in 1\/43 scale model which I think is the best version of the Audi 200. \"Avant\" because it's a station wagon and \"Quattro\" because it's a four-wheel drive. Only 1'616 ex were made (of the real car, not the scale model!) resulting in the explosion of the second-hand market prices (around 25'000$ in perfect condition today). And it's not that difficult to find one in good condition because Audi made sure that their car was very reliable, not only concerning the engine but also the cabin. Talking about the cabin, it assures you to drive comfortably, silently and safely like flying in a sci-fi spaceship and that was rare for that time.\nBeing comfortably seated is one thing, having great mechanical performances is another thing, and guess what, the Audi 200 20V Avant Quattro has both! The 5-cylinders developing 220hp allow you to reach 240km\/h (150mph) which is quite impressive for a 80's street car. Being called a \"street car\" is an euphemism, I would personally suggest \"autobahn car\" (autobahn is the German unlimited speedway). Americans were not left aside, Audi sold this masterpiece across the Atlantic but renamed it Audi 5000S estate wagon.\nSwallowing kilometers in this spaceship is a real pleasure for long riders and having one in your shelf is also a real pleasure for youngtimers collectors and DNA Collectibles offers you the opportunity to get one in our shop!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is a novelty kitten and picture story book where you can \"Open me out and change my clothes.\" There is a moveable kitten in rear with arms and legs that swing out. Several pages of clothing that allow the cat to be \"dressed\" in a new outfit as each page is turned. Book is in very good condition; no writing or markings, pages are very crisp, if a little browned. Complete. Rear covers states \"Printed in the Netherlands - No. 1875 A\" Originally published by Mulder & Zoon, Amsterdam as \"Poesje Nel\" this copy was later published in English under the Mulder & Son imprint, although it is not listed in the book. Information via Theo Gielen, \"Moveable Stationery,\" Vol. 15 #3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TV highlights for the week of Aug. 15-21. Shows listed are not necessarily recommended by the Monitor. All times are Eastern; check local listings.\nPGA Championship (TBS, 11 a.m.-1:30 p.m.; CBS, 2-7 p.m.): Third round today, final round tomorrow in Redmond, Wash. Possibility is high that someone from the tiny Florida community of Isleworth may win this event. Its pro golfing residents include Lee Janzen (US Open winner), Mark O'Meara (Masters and British Open), Tiger Woods, Scott Hoch, Payne Stewart, Stuart Appleby, and Grant Waite.\nBabe Ruth (HBO, 8-9 p.m.): They called him \"Bambino,\" \"the Sultan of Swat,\" \"the King of Clout,\" \"the Savior of Baseball.\" All in all, the barrel-chested, spindle-legged George Herman Ruth was a man of extremes. He did everything to excess, the bad things and the good things. Rare clips, insightful commentary, and the near-mythical life of Ruth himself make this profile an enjoyable hour for anyone with an interest in the national pastime.\nTitanic Live (Discovery, 8-10 p.m.): The Big Ship continues to fascinate people. This special will feature never-before-seen images broadcast live from 2-1\/2 miles below the ocean's surface. A team of biologists and historians will visit the shipwreck to learn more about the ship's sinking and the rate at which it is disintegrating.\nAny Day Now (Lifetime, 9-10 p.m.): Starring Annie Potts and Lorraine Toussaint, this promising new drama looks back on a childhood friendship between a white girl and black girl during the civil rights movement. Reacquainted later as adults, they discover that many of the problems they encountered as children still exist today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..040a30e43c31a89dd8fcdbd2719bc1a3817ec2da
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+{"text":"This stunning girl is my cousin Kimberly! I was so honored to get to take her senior portraits for her. We braved the heat at the end of July and went to the Botanical Gardens to catch the last of the beautiful flowers. After that we decided to have some fun and venture downtown to the color tunnel on 18th Street. This session was so much fun and SO colorful, her sister Christine even joined in for some fun sister photos! Thanks Kimberly for letting me take your portraits!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What: Your indulgent essential travel kit from Kathleen Lewis.\nWhy: With all the travel women take on nowadays, it's no surprise that we end up with a new travel kit every time we hit the road \u2013- from the ones we cobble together of single use shampoo and soap samples you get in the mail to the fully stocked top-of-the-line skincare kits like Air Repair. At a quick glance, we noticed that these kits were always packed with basic products we needed in our every day routine \u2013- toothpaste, face wash, etc. There was no 'fluff' or anything that wasn't utilitarian.\nPerhaps realizing the stressfulness of travel, Kathleen Lewis has created her own 'Essential travel kit' that's full of the things to help you indulge while you're on the road -\u2013 in mini size, of course. You'll find both scented and unscented soaps, bath salts, bath and body oil, a candle and lotion, each infused with calming lavender, which we loved. All items contain less than three ounces of product each and are enclosed in a see-through zip top bag for no-hassle stops through airport security checks.\nSo add another essential to your travel bag and reward yourself for a job well-done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HACI AKBAY worked in company SUEDWEST-INDUSTRIESERVICE LIMITED\tas QUALIFIED WORKER.\nHACI AKBAY was from 2010.09.15 to 2011.06.28 at SUEDWEST-INDUSTRIESERVICE LIMITED employed as Director (QUALIFIED WORKER).\nHACI AKBAY Score for HACI AKBAY is 5 stars.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The main objective of HHESA is to connect the dots \u2013 those who have been a part of the Health Equity Scholars Programs across America \u2013 so that you are aware of others in your field. That there is a space where you can connect to exchange information, ideas, opportunities, experiences, etc. to support and help facilitate each other's success and advance health equity work.\nInterface with your respective institutional leaders and HDEART colleagues to advance our educational and research initiatives.\nFinally, we are asking that you take every opportunity to education your network about and promote progressive social justice initiatives in policy agendas that will help to eliminate inequities. As I mentioned in one my blogs, a hand is stronger when its fingers are pulled together or connecting the dots.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"By now you are fully hydrated with last weeks fat flush water, so lets take our weight loss plan one step further. Carbs. Whoops, get those visions of pizza and bagels out of your head. We are going green.\nI love good healthy brown carbs like quinoa, brown rice and whole wheat pasta. They are an important part of a healthy diet, but to lose the extra pounds fast, get your carbs from green vegetables. Spinach, broccoli, green beans, lettuce and zucchini are not only nutrient rich but, they are loaded with fiber, which will help keep you full and keep things moving.\nSubstituting green vegetables will help assist in weight loss because they rank very low on the glycemic index. Veggies, thanks to the fiber, take longer for the body to absorb and therefore your blood sugar does not spike up as fast. This allows your body to use the food as fuel and it does not need to store it as fat. The added bonus, you will not feel the sluggish crash you get after a pasta dinner or a big sub sandwich.\nNo need to overhaul your whole menu, just make some simple swaps. Many starchy sides have a veggie alternative. Here are some examples.\nLettuce wraps. Swap your sandwich bread for lettuce leaves. Wrap it around a deli sandwich, saut\u00e9ed veggies and hummus or even a big juicy burger. Another idea. Make tuna melts on red pepper halves instead of bread. (technically not green but you get my drift).\nZucchini Pasta: Using a vegetable peeler cut the zucchini into lengthwise ribbons. Heat one Tablespoon of olive oil in a pan; add the zucchini and saut\u00e9 for about 3 minutes. Top with a healthy tomato based sauce.\nVeggie Puree: A great way to use veggies on the verge of being tossed, this one is my favorite. Steam any vegetable siting around (kale, cucumber, cabbage, cauliflower, peppers and celery are my go to combo). When soft, transfer to a blender and puree. Blend slightly for mashed potato like texture, or more so for a thick soup. Serve it toped with a grilled fish or chicken for a light summertime comfort food.\nLooking for something a little heartier to put on your plate? Try Freekeh. It is a roasted green grain made from immature Durham wheat. Packed with protein, low in carbs and high in fiber, this is a perfect alternative to rice.\nGo green for the next week and feel a little lighter for the hot summer ahead.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..ebfe4cd208149c9d171c2ca0491a49d527454cb1
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+{"text":"We have discussed the issue of how to tell a GoFundMe scam from the real deal time and time again, but it bears repeating as scammers are getting more and more creative. Subsequently, they're also getting more desperate. Their tactics continue to evolve as more and more donors educate themselves on the risks; as such, we feel it's important to keep up with their tricks.\nHere are just a few red flags to look out for before you give to any personal crowdfunding campaign.\nFirst things first: if you receive an unsolicited message from someone you don't know on social media asking you to donate to their campaign, chances are pretty good it's a GoFundMe scam.\nReport both the campaign and the profile sending you spam messages just to be safe. And then block them.\nSince many fake campaigns involve \"borrowing\" photos and creating stories around them, your first line of defense against a lazy GoFundMe scam is a reverse image search. Cross check any photos used in the campaign, as well as any related photos you may have found on the campaign organizer's social media pages. If you aren't familiar, you can find out how to do a reverse image search here.\nNow, it's entirely possible the photo was unintentionally lifted, but why use a stock photo if you're tweeting about your own alleged experience? Donor beware.\nAnother common scammer tactic is to take viral GoFundMe campaigns and adapt them for their own gain, hoping people simply won't notice the difference.\nCase in point, the Fidencio the Paleta Man campaign. At $384,000 raised and counting, it is the most successful GoFundMe campaign in Illinois to date. It's also just asking to be hijacked by unscrupulous scammers and reposted as their own.\nThis YouCaring campaign has since been removed, but you get the idea.\nBefore you donate to a campaign such as this one that you thought you saw in the news, double-check to make sure it's the real one. Fake campaigns usually have raised little to no funds, whereas the real ones in the news like the one above will have raised tens or even hundreds of thousands of dollars.\nScammers often put up several different campaigns across several platforms just to increase the odds of tricking people into donating. It's the same tactic Nigerian Princes use when they send millions of emails; all it takes is one vulnerable person to fall for the scam.\nWe recently came across someone with a GoFundMe campaign seeking $10k toward his rap album. On another platform, he was asking $25k for his son allegedly dying of cancer. Would you be worried about dropping a hot track if your kid was dying? Probably not.\nHere's another example. This person was on the \/r\/gofundme subreddit asking for help with her mom's bills. On another campaign, she was asking for money for school. In both campaigns, her mother had cancer, however despite being created around the same time, neither story lined up with the other.\nTaken separately, perhaps the campaign could be legitimate. In fact, it's possible they still are. However multiple campaigns from the same person across different crowdfunding platforms is a significant red flag.\nDo you know that feeling you get deep down in your gut when you're watching a horror movie and the main character ventures into a dark, creepy basement? Like you know something really bad is about to happen?\nIf you come across a GoFundMe campaign and feel that same sense of impending doom, it's best not to donate. Even if you can't quite put your finger on it, just avoid it. You can't get your money stolen if you don't give any money at all.\nJust read your stuff and thanks for the information which is invaluable. I am contemplating starting a GFM for my sis who is really ill but at a loss as to what to do about it. It feels odd asking strangers for money but the NHS is getting to the point where it is not funding anyone, In any event she ab-reacts to anything they do suggest and only responds to Chinese medicine, ( from past experience Chinese medicine has saved her from having surgery which would have rendered her unable to sing \u2013 she is a not famous singer ) which of course the NHS does not fund. Also I have no idea how much it would cost so will have to do some more research on it. Ho Hum. I will give it some more thought and thanks again for your site ;- ).\nAww thanks. And just a tip: if you do decide to create a campaign for your sister, look into using YouCaring, that platform is more appropriate for a cause such as your sister's and doesn't charge fees. GoFundMe fees can really add up depending on how successful your campaign is. 5% of each donation plus 2.9% to WePay and a .30 charge per donation.\nDon't feel ashamed, the platform is there so people who need it can ask for help. Give me a shout if you go forward and need any help putting together a transparent, hopefully successful campaign!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CEO of the Miramar Group Juan Ochoa was recently appointed to serve on the Metropolitan Pier and Exposition Authority Board (MPEA) of Directors Tuesday. Ochoa previously served as CEO of the MPEA from 2007 to 2010. Prior to his time as CEO of MPEA, Ochoa served for 10 years as President and CEO of the Illinois Hispanic Chamber of Commerce, growing it from 52 members to more than 1,200 with a nearly $2 million budget. He also served in the U.S. Marine Corps Infantry and is a combat veteran. Ochoa emigrated from Mexico as a child and later served as a community organizer in Chicago, helping thousands of people become U.S. citizens and raising funds for college scholarships. The Metropolitan Pier and Exposition Authority Board consists of nine members: four appointed by the Mayor, four appointed by the Governor and a chair elected by the board members. Ochoa replaces Olga Camargo, who will be appointed by the Mayor to the Chicago Public Building Commission.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We appreciate the comments of Drs. Rossello and Yellon regarding our recent paper (1), and we agree that activation of pro-survival pathways in viable myocytes after brief periods of ischemia likely occurs alongside apoptotic myocyte death to provide protection against subsequent ischemic injury. Aside from the severe transmural ischemia associated with an acute transient occlusion that we used in our study, apoptosis-induced myocyte cell loss with compensatory myocyte cellular hypertrophy can arise in response to reversible subendocardial ischemia and can ultimately affect regional as well as global left ventricular function. Thus, modulating this dynamic interplay between myocyte survival and death may be most important with repetitive ischemia similar to angina in chronic coronary artery disease that can contribute to the development of ischemic cardiomyopathy. For example, we previously demonstrated that chronic repetitive subendocardial ischemia distal to a chronic coronary artery stenosis in swine leads to a progressive increase in apoptosis as the physiological significance of a coronary stenosis increases (2,3). This is followed by regional myocyte loss, compensatory cellular hypertrophy, and physiological features consistent with hibernating myocardium. Interestingly, proteomic profiling has demonstrated that this is accompanied by an up-regulation of a variety of pro-survival proteins as well as a down-regulation in mitochondrial metabolism (4,5). This is functionally significant, as hibernating myocardium exhibits depressed mitochondrial respiration and a reduced rate of ATP depletion during simulated zero flow ischemia in vitro (6). Others have also demonstrated activation of pro-survival pathways during repetitive nontransmural ischemia (7). These cardioprotective adaptations are reversed by alleviating demand-induced ischemia with coronary revascularization (8). Collectively, these findings support the notion that hibernating myocardium becomes ischemia-tolerant to prevent a supply-demand imbalance and reduce mitochondrial oxidative stress as a counter-regulatory process to attenuate myocyte loss from apoptosis in a fashion similar to that proposed by Rossello and Yellon for ischemic pre-conditioning and infarction. We agree that further work is needed to understand whether favorable manipulation of the balance between pro- and anti-apoptotic signaling pathways can have a long-term effect on myocyte loss after ischemia and prevent the development of ischemic cardiomyopathy.\nPlease note: This work has received funding from the National Heart, Lung, and Blood Institute (HL-055324, HL-061610, and F32HL-114335), the National Center for Advancing Translational Sciences (UL1TR001412), the Department of Veterans Affairs (1IO1BX002659), the George E. Becker Fund for Heart Research, and the Albert and Elizabeth Rekate Fund in Cardiovascular Medicine. The authors have reported that they have no relationships relevant to the contents of this paper to disclose.\n(2001) Myocyte apoptosis and reduced SR gene expression precede the transition from chronically stunned to hibernating myocardium. J Mol Cell Cardiol 33:1937\u20131944.\n(2008) Persistent regional downregulation in mitochondrial enzymes and upregulation of stress proteins in swine with chronic hibernating myocardium. Circ Res 102:103\u2013112.\n(2009) A straightforward and highly efficient precipitation\/on-pellet digestion procedure coupled with a long gradient nano-LC separation and Orbitrap mass spectrometry for label-free expression profiling of the swine heart mitochondrial proteome. J Proteome Res 8:2838\u20132850.\n(2009) Reductions in mitochondrial O(2) consumption and preservation of high-energy phosphate levels after simulated ischemia in chronic hibernating myocardium. Am J Physiol Heart Circ Physiol 297:H223\u2013H232.\n(2001) Gene program for cardiac cell survival induced by transient ischemia in conscious pigs. Proc Natl Acad Sci U S A 98:9336\u20139341.\n(2015) Revascularization of chronic hibernating myocardium stimulates myocyte proliferation and partially reverses chronic adaptations to ischemia. J Am Coll Cardiol 65:684\u2013697.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This well cared for townhome boasts open main level, vaulted ceilings with lots of natural light, deck over looks walking trail and wetlands. Dining room with built in buffet, fresh paint throughout and new carpet on upper and main level. Oversized master suite with walk in closet. Walk out lower level family room leads to outdoor patio and steps away from trail system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I don't have anything to play? Am just getting page can't be displayed.\nPlease refresh your browser and click play again. This should work. Are you in Google Chrome or Internet Explorer?\nDo I post here if I have a question?\nCorrect, I can then share with our physicians.\nDear Anonymous \u2013 The physicians will do a Q\/A portion towards the end of the presentation. Thank you.\nI have a clivus bone tumor; I had a biopsy done and it came back fibro-osseous. I am having symptoms but they are going to monitor. It is 5.1\u00d73.7\u00d73.4 cm and in the occipital bone. Is this standard for this type of tumor?\nMy husband has had \"seeding\" of a tumor in the ventricle..for the past 26 years. He recently had a surgical biopsy and the path. report showed Grade 2 Astrocytoma, but the neurologist thinks the path report was not correct and he thinks it's an Ependymoma. My husband is 68 years old now and never had radiation or chemo..only surgery to stop seizures in the left temporal lobe. What is the prognosis for an Ependymoma..about 3cm. Thankyou.\nI have a question.. I am Post-sreast & thyroid cancer surgeries , oovarectomy and chemo, biologic & radiation treatment- currently on oral therapies; also cervical bone fusions\u2026. Now brain cyst??? (so I am told by MRI) next to pineal gland\u2026 1.5 years now of debilitating\/excruciating headaches, dizzy, bad daily GI symptoms: nausea, diarrhea; decreased cognition, weakness\/numbness\/falls \u2013 given oral headache meds and steroid and other injections,,,, Question what to do???\nWhere can I find a list of clinical trails at Mayo after just having had brain surgery?\nDr. Parney, you have been doing some pioneering work in oncolytic virotherapy with the Measles virus. Would your recommendation for oncolytic virotherapy be based on the genetic makeup of the tumor? What virus would be most effective in what circumstances \u2014 Measles virus, New Castle Disease virus?\nI have a small meningitis wrapped around the super sagittal sinus. It continues to grow slowly.I am being told it is too risky for surgery because of area. I was also told radiation is not recommended. Can mayo clinic help me?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaetih b/data_all_eng_slimpj/shuffled/split2/finalzzzaetih
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index 0000000000000000000000000000000000000000..1f8d538f7f5fdb49b1ce40a9527420f356cc8e34
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+{"text":"When I was a kid, it was my dream each Christmas Eve to finally catch Santa Claus in the act! I was a believer at the time, but I had already begun to hear the whisperings from older children (and adults who didn't think I was listening) that Santa was a myth.\nPersonally, I couldn't believe those rumors to be true and would always do my best to finally meet the Big Guy face-to-face in my very own home. If you have a child that yearns to catch Santa in the act, be sure that this toy is safely under lock and key or YOU may be the ones caught...on tape!\nThe NERF Nightvision Camcorder retails for $89.99, but is currently on sale for $74.99 with FREE SHIPPING.\nNavigate the night like a true secret agent with this NERF Nightvision Camcorder! It's designed with 4x magnification and the ability to see up to 50 feet in the dark. You can also actually record video and take pictures of your surroundings for tons of super-secret spy fun. A USB cable is included for easy video uploading.\nThe great thing about the NERF Nightvision Camcorder is that is has both day and night settings, so you can use this toy anytime the enemy is near, not just at night! If you have a child ages 5 years old and up who would love to capture their adventures as they happen, then be sure to have this camcorder under the tree...but watch out for their \"catch Santa in the act\" mission next year!\nDisclosure: I received the NERF Nightvision Camcorder in order to facilitate my review. No other compensation was received. This feature is based on my own personal experiences with this item and is completely honest and objective.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"On behalf of Law Office of P. Zachary Stewart posted in Wage & Overtime Issues on Thursday, December 13, 2018.\nWyoming Casing Service Inc. must pay nearly $1.2 million to employees for back wages. The U.S. Department of Labor investigated the North Dakota-based oilfield service company and found wage violations at nine of its locations. One location was in Weston, West Virginia.\nOn behalf of Law Office of P. Zachary Stewart posted in Wage & Overtime Issues on Tuesday, October 14, 2014.\nMasTec, Inc.-owned oil and gas service company Bottom Line Services, Inc. was sued on May 6, 2013 by employee Jared Moore for violations of the Fair Labor Standards Act. Moore alleged Bottom Line Services failed to pay him overtime at the federally mandated rate of one and a half times his regular rate. Additionally, Moore alleged Bottom Line Services failed to include per diems, truck pay and other allowances in the overtime calculation. As a result, Moore and all other hourly employees were owed overtime, back wages and liquidated damages. Even though Bottom Line Services vigorously defended the lawsuit, the Court certified the class action - sending notice of the lawsuit to all current and former employees. On July 22, 2014, the parties filed a notice a settlement, signaling the resolution of the case.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Good mental health is good for our communities and good for business. It is estimated that mental health issues costs the Welsh economy \u00a37.2bn per annum. We want to work with Welsh businesses to improve your productivity by improving the mental well-being and awareness of your employees.\nWe offer a range of services for businesses, from big corporates through to small independent traders.\nWe would welcome the opportunity to come and talk to one of your management teams and\/or your HR department about emotional well-being in the workplace.\nEmotional well-being training for your managers and staff.\nWe believe that the emotional well-being training is an important aspect to any package of support we provide to a business, to help you identify signs of distress in your workforce and have the confidence to engage with a member of staff about their emotional well-being. We offer training in groups of up to 20 managers and\/or staff.\nWe offer the same services as for large corporates, but understand we may need to offer training to smaller groups.\nWe know from talking to some of our local business people that emotional well-being and mental health is an issue for every business, however, small. For small businesses we can offer a quick fire one hour to staff.\nIf you have specific requirements or want us to assist you in developing a new way in which you want to work on emotional well-being in your business, please contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Buy Literature Review Online \u2013 We Will Write it with Ease!\nHire The Best Literature Review Writing Service to Write Your literature review. Once you buy, our professional writers will write your review based on your requirements.\nA sophisticated and systematic literature review is the most important aspect of any research thesis or report and it is considered significant in any thesis development.\nWriting a quality and scholarly literature review involves the application of different charismatic skills of information preparation, accumulation, composition and assessment.\nA literature review is understood as an important and major part in the creation of a thesis or a dissertation and students face a lot of challenges while they are working on the literature review. If you are one of these students, why don't you consider us Essaymojo.com to work on your literature review and see a difference in your grades?\nWe have a team of experienced writers who can easily and effectively conduct a literature review on your dissertation with the relevant data and include other author's literature and finally analyze the topic of the literature review.\nWriting a literature review involves the inclusion of different aspects that involves conducting a thorough research on author's backgrounds in the existing literature. Many students face the challenge of writing a significant and quality literature review session.\nAt Essaymojo.com we deal with all the challenges that students face while writing an effective literature review and quality literature review in addition to figures, models, and citations.\nWe offer excellent services in literature review writing that will enhance to sophisticated and comprehensive dissertation and thesis.\nOur experienced and expert writers guarantees you to maintain the standards and follow the instructions proposed by the lecture while developing your literature review.\nRelated Topic: So You Need to Write a Literature Review \u2026 Now What?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We don't play Lehigh for a few days ya'll. Let's not look ahead of the Southland conference team coming into the Super Pit on the 16th. That'd be Southeastern Louisiana 1-6 (0-0).\nThis should be a relatively easy win like the Jackson State game. For whatever reason, we cannot shoot the ball well, especially from three. Our team average is at 26.1% where the SBC average is 33.8%, and the national average at 33.4%. However, we manage to score enough points to be an average enough team. Our problem? Allowing free throws. Well, allowing free throws and the other team making those shots. Our defensive free throw rate is 26.6. That is 26.6 points scored from the line per possession used per game.\nI'm still holding out hope that this season will turn out better than the 4-6 record tells me it will but I'm not sure. Through ten games, we can't shoot as well as last year, are slightly worse at defense, and our best player is playing slightly less efficiently.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafcho b/data_all_eng_slimpj/shuffled/split2/finalzzzafcho
new file mode 100644
index 0000000000000000000000000000000000000000..0145ea4cf69388e6258ce46471f3616b6d492bc3
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+{"text":"I'm Jordan, a musician & producer from Boise, Idaho. I've been creating music for 8 years, and I'm also dabbling in digital art & podcasting. I write reviews and commentaries about creative works that interest me: you'll be able to read those writings on this site, eventually.\nI play lead guitar in a power-pop band called Mains & Monitors. We just released an album called Kalani, and we're super friggin' proud of it. I led the recording work on this project at Neighborhood Sounds, and it was post-produced by Jack Shirley at the Atomic Garden (Jeff Rosenstock, Deafhaven, Remo Drive).\nI wrote and released my first EP in December of 2017: it's called Late Bloomer. It's a power-pop record focusing on themes of self-worth, relationships, noodly guitars, and my dog. Y'know, like every friggin' sad-boy record. Workin' on the next one right now.\nI'm working with my friend Eli Stonemets on the first album for his project Backyard Universe. It's an alt-rock project with influences from artists like The National, The Strokes, & Big Thief.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your ultimate range of icons, find the largest free range of icons for free download. This gallery contains 1000s of icons avalible for any use, these amazing icon images have been hand crafted and are ready for safe a free download.\nUnbelivable range of icons avalible for any use. Free stock drawings of thousands of icons avlible for free download.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A 3x3 number plate is simply a way of referring to how many letters and numbers are within the registration number. So this means 3 letters and 3 numbers, either way around (numbers first or letters first). Generally speaking the more letters or numbers are on a registration, the cheaper the price, so a 3 letter\/4 number registration is often cheaper than a 3x3, and a 3x3 is usually cheaper than a 3x2 or 3x1. However 3x3 are extremely popular.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Despite the fact that Libya is nowhere near the Persian Gulf, Qatar has been the Arab state most ardently supporting the rebels in eastern Libya. Qatar has long had an active foreign policy, and its recent moves have positioned the tiny state as a player in the Libyan crisis \u2014 no small feat considering how insignificant it is in comparison to the traditional Middle Eastern powers. In reality, Qatar remains a very weak country that relies on the United States for its security, constantly reminded of its precarious geographic position between regional powers Saudi Arabia and Iran as it tries to use foreign policy as a tool to present itself as a useful ally to any country.\nQatar sits on a small peninsula jutting off the eastern edge of the Arabian Peninsula into the Persian Gulf, wedged between the two regional powers: Saudi Arabia and Iran. Qatar's size and strategic location have made it fundamentally insecure throughout its history, and since the coming of its oil and natural gas wealth, the ruling family in Doha has sought to remedy this problem in a variety of ways. Qatar tries to maintain good ties with both Saudi Arabia and Iran, it hosts a sizable U.S. military contingent, and it conducts a foreign policy intended to create a perception of Qatari influence that exceeds its actual ability to project power. This desire to create a perception of power explains Qatar's recent moves in eastern Libya, where Doha has slowly positioned itself as one of the main players in the diplomatic game being waged in various corners of the Muslim world.\nQatar's hydrocarbon wealth is fairly new. Oil exports did not begin until 1949, which marked the beginning of a shift from an extremely poor tribal area perpetually under the dominance of outside powers to a modern nation-state. Though oil came first, natural gas eventually became an integral part of the Qatari economy as well, and together they form the foundation of modern Qatar. Qatar pumps just over 800,000 barrels per day (bpd), not much in comparison to some of its neighbors, but still a considerable amount for a country of roughly 1.7 million people (around three-fourths of whom are expatriate workers). Qatar is more famous for its massive North Field natural gas field that sits offshore northeast of the peninsula (it shares the field with Iran, where it is known as South Pars). Qatar holds the third-largest proven natural gas reserves in the world (approximately 896 trillion cubic feet), and it is also the world's largest liquefied natural gas (LNG) exporter. As a result, some calculations place Qatar at the top of the rankings in per capita gross domestic product worldwide. None of this hydrocarbon wealth would mean very much if Qatar were unable to export it, which requires not only territorial security (on land and in its territorial waters that contain offshore oil and gas deposits) but also unimpeded access through the Strait of Hormuz. And this is one of the most important reasons why the ruling family in Doha tries to maintain good relations with both Saudi Arabia and Iran. Unlike Bahrain, which finds itself in a very similar geopolitical situation but with a 70 percent Shiite population, Qatar has better relations with Iran in part because only about 10 percent of its population is Shiite and it does not feel threatened by a Shiite majority acting as agents of Tehran. Qatar has extensive economic linkages with Iran and helps Tehran circumvent sanctions by acting as a shipping hub of illegal goods, much as the United Arab Emirates does. As for its relations with Saudi Arabia, Qatar was a contributor to the Peninsula Shield Force that entered Bahrain on March 14, while Doha-based Al Jazeera has been nowhere near as dogged in its coverage of the protests in Saudi Arabia's Eastern Province as it has been in several other Muslim countries that have experienced unrest. The imperative of maintaining territorial security and unimpeded access through the Strait of Hormuz also creates the need for a foreign security guarantor. This forms the foundation of Qatar's relationship with the United States. Qatar did not exist as an independent nation until 1971, when the British completed the withdrawal of their naval assets from the Persian Gulf region. For decades before, Qatar existed under British suzerainty. It was London that first granted protection to the al-Thani family (which still rules Qatar) against the rival al-Khalifa family in nearby Bahrain. And the United States has stepped into the role of a foreign power able to guarantee Qatar's continued territorial integrity. The United States does not run Qatar's day-to-day affairs as the British had done; the United Kingdom largely controlled Qatar's foreign policy in exchange for security guarantees. But the United States does have a large footprint in the country with two significant U.S. military bases. Qatar volunteered to be the new host of the U.S. Combined Air Operations Center after it was removed from Saudi Arabia in 2003 and set up at the existing Al Udeid U.S. air base south of Doha. Today Al Udeid serves as a key logistics hub for American operations in Afghanistan and as a command center for operations in Iraq. A second American base in Qatar, As Sayliyah, is the largest pre-positioning facility for U.S. military equipment in the world. Doha benefits from its security alliance with Washington, but it also wants to maintain its independence and build a reputation (both in the Arab world and beyond) of being a significant actor in foreign affairs, more significant than geopolitical logic would suggest. Above all, it wants to be seen as acting in its own interests, even if it is operating according to a set of restraints that prevents it from pursing those interests too vigorously. Sometimes this brings Qatar in line with certain countries' positions, only to find itself seemingly on the opposite end of an issue in short order. This is most aptly displayed by Al Jazeera, which first became known for its critical coverage of U.S. and Israeli activities in the region and is now widely attacked by Arab regimes for fomenting dissent within their own countries. Despite what neighboring governments may feel about the media outlet, Al Jazeera's emergence has helped put Qatar on the map in the eyes of the Arab street, evidenced by the fact that in 2022 Qatar will become the first Muslim country to host the World Cup. Qatar's active diplomatic presence in recent years has often involved disputes that have very little to do with its own direct interests, such as working with Turkey in helping to form the Lebanese government and mediating between the Sudanese government and various rebels groups in the Darfur peace process. Qatar's integral role in supporting the eastern Libyan rebels is only the latest example of this trend. Whether Doha is acting according to U.S. directives is unknown, but it is certain that Qatar's efforts are in line with U.S. interests, and will bolster Qatar's image in Washington's view as a leader in the Arab world.\nDespite the fact that Libya is nowhere near the Persian Gulf, Qatar has been the most ardent Arab state supporter of the eastern Libyan rebels since the beginning of the uprising. This was not an obvious decision for Qatar to make, since what happens in Libya does not affect the situation in Qatar's backyard. Still, Qatar remains the only Arab country to have recognized the National Transitional Council as the sole legitimate representative of the Libyan people and was the second country to do so after France. Qatar is also one of just three Arab states that have contributed aircraft for the enforcement of the U.N.-mandated no-fly zone, sending six Mirage fighter jets to perform largely ceremonial over-flights alongside French warplanes. Qatar has also been flying humanitarian aid into the Benghazi airport in recent days. Displaying a desire to lead the Arab world in issues occurring in the region, Qatari Emir Sheikh Hamad bin Khalifa al-Thani has openly called for Libyan leader Moammar Gadhafi to step down and has criticized other Arab states for not helping enforce the no-fly zone, saying March 31 that \"the suffering of civilians in Libya led the international community to intervene because of the inaction of the Arab League, which was supposed to assume the role.\" Qatar's most important contribution to the Libyan rebels, however, could be in helping them market oil pumped from the Sarir oil field in eastern Libya, which would infuse the movement with much-needed cash to sustain its fight against Gadhafi. Doha also reportedly provided a small supply of weapons to the rebels in early March and sent free shipments of petroleum products into eastern ports when it was feared that supplies of gasoline, butane and kerosene were running out. But if the eastern Libyans were able to actually make money off the oil, which one rebel council leader \u2014 Finance Minister Ali Tarhouni \u2014 has vowed is ready for shipment, it would give Benghazi a more sustainable solution to its pressing economic problems than stopgap aid shipments. Tarhouni, who returned to Libya from exile in the United States in March, has made a variety of claims since March 27 regarding the oil-production capability in the east, ranging from an immediate level of 130,000 bpd to 300,000 bpd or more within a few weeks. According to Tarhouni, Qatar is on board with a plan to \"facilitate\" the export of oil from either the Sarir oil field or storage tanks around Tobruk, most likely for shipment to European customers wary of the political or security risks of doing business with the rebels. Tarhouni's claims have not been confirmed or denied by the Qatari regime or by state-owned Qatar Petroleum (QP), which would most likely be the firm that would help facilitate exports of Libyan oil. One anonymous QP official said March 30 that the deal was \"just a political move\" and emphasized the difficulty in actually seeing it through, saying that the time frame would surely be longer than the week or so that Tarhouni was asserting. But in making such a statement, QP has implicitly acknowledged that the deal is simply another case in which Doha wants to display its support for the uprising against Gadhafi. By taking part in the no-fly zone, Qatar did exactly that, while also demonstrating its utility to the West. Doha's support allows leaders in Washington, Paris and London to claim that an air campaign against a Muslim country has \"Arab support.\" The statements made by Arab League chief Amr Moussa on March 20 showed how politically sensitive perceived support for such a bombing campaign can be in the region, which only makes Doha's support that much more appreciated in Western capitals. These measures, along with the critical role Al Jazeera played in bringing the world's attention to the situation on the ground in eastern Libya, have given tiny Qatar the reputation as a player in the Libyan crisis. This is no small feat, considering how insignificant the country is in relation to traditional Middle Eastern powers like Egypt, Saudi Arabia and Iran. Qatar remains, in reality, a very weak country that relies on the United States for its security and on its dealings with other more powerful states, but presents itself as a country that can be a useful ally. One of the main reasons Qatar has been able to focus so much attention on eastern Libya is that it has not suffered the affliction that other Arab countries have since January. There has been no Arab Spring in Doha, notwithstanding a few failed protests organized on Facebook calling for a \"Day of Rage\" in Qatar in early March. Should unrest flare up in Qatar as it has elsewhere in the region \u2014 which is unlikely due to its wealth and lack of sectarian divisions but certainly not impossible \u2014 it will suddenly find itself much less concerned about the fate of eastern Libyans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"de Liebe Opera Entwickler warum macht ihr so was.\nOpera also makes one important change. If it feels the colour of text does not contrast strongly enough with the background colour, it changes the text colour to either black or white in order to make it contrast more strongly.\nThis is important because of the smaller font sizes that Opera uses in this mode and because during the re-arranging, any background images may no longer be behind the text.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafppt b/data_all_eng_slimpj/shuffled/split2/finalzzzafppt
new file mode 100644
index 0000000000000000000000000000000000000000..81d08bb4460b3490c07ec5c7501f857557382fd3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Blue creates a calming feel throughout the dining and living space. Customize furnishings to make it your personal. Home owners Erika Klein and Andrea Alexander spray-painted wooden side chairs from Restoration Hardware in varying shades of blue and green for a cheerful, ultra-casual vibe in their Watersound, Florida dining space. A stunning glittering crystal Dhruv Tolix Dining Chair (Set of 2) By Dhruv Tolix Dining Chair (Set of 2) By 17 Stories 17 Stories dining table chandelier for your dining space. The really straightforward shape permits the Swarovski components, Asfour crystals Dhruv Tolix Dining Chair (Set of 2) By 17 Stories or Dhruv Tolix Dining Chair (Set of 2) By 17 Stories Italian lead or crystals to take centre stage.\nCoffee tables full your living space. Massive table tops make for ample space when entertaining guests or playing board games with the family. Adorable cushioned stools match beneath the table with out cluttering up your space. Dainty shelves underneath the Dhruv Tolix Dining Chair (Set of 2) By 17 Stories table hold your magazines, coasters and even your set leading box. Crafted in premium wood and wealthy finishes, these coffee tables are stylish and durable.\nThe table is beautiful!! The butterfly leaf is an excellent feature and makes the desk big enough to seat the whole family. My son assembled it for me personally. He is in structure and was impressed by the product quality and sturdiness of the table. It was delivered regularly and all the parts were in the field. Nothing at all was broken or missing. I will order from your own company as a result of price, quality, and remarkable service.\nI ordered 3 stools and they arrived 2 days later. I took a risk and ordered the crimson afteroses reading other critiques that the color was different than how it appeared in the online images. I think the color is ideal and matches the web photos pretty close. They were very easy to assemble. I did smell the gasoline smell that other people commented about, but it was the plastic wrap\/packaging, not the stools themselves.\nIt's an excellent set and not difficult to put together. Do note the seat pads atento be screwed straight into the chair framework with no predrilled holes. As well before you purchase come to be sure to check everywhere to get the very best price.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There are 5 types of variables in Cubloc BASIC.\nDim A As Byte                  ' Declare A as Byte.\nDim B As Integer, C As Byte    ' Commas may NOT be used.\nDim ST1 As String * 12         ' Declare a String of 12 bytes.\nDim ST2 As String              ' String is 64 bytes default.\nDim AR(10) As Byte             ' Declare AR as a Byte Array.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calamvale is a relatively new growth area that has doubled in size over the last 10 years, therefore requiring the pump station and pipeline to be up sized. The location of the bore was in the middle lane of Gowan Road, with residential housing on both sides. The high density of existing services, major traffic management requirements and the close proximity to local schools and shops meant that Horizontal Directional Drilling (HDD) was the obvious option. The required depth and alignment of this pipeline reinforced the requirement for HDD to be the chosen method during the design stage and as the preferred method for construction.\nUEA mobilised its D100 x 120 (45 tonne) HDD drill rig and DFE 300gal\/min cleaning system to undertake the pilot bore. Completion of the pilot bore confirmed that ground conditions were different to that stated in the geotechnical report which required all of the down hole equipment to be changed to suit rock conditions. Scheduling of the reaming works was extended due to the difficult ground conditions encountered, the final hole size was 600mm and the pipe was successfully installed.\nFor more information about this project or UEA Trenchless, please send enquiries to Steven Hopkins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most business owners in private practice are clueless when it comes to using the Internet, tools and services to grow their business.\nAre you feeling powerless because old-school marketing no longer works and those business owners who are \"switched on\" are dominating the online landscape?\nIt's time to start \u2014 at the beginning. Learn what is possible and what your options are for your business online. Find out how things like websites, blogs, search engines, YouTube, Facebook, eNewsletters and local marketing fit together. Then create your own strategy and measure how well it works.\nNever be at the mercy of slick sales people and spend money on stuff that have no chance of working for you. Learn to delegate regular tasks to your team and go back to doing the things you love in your business.\nStand out as an authority \/ leader in their field.\nEvaluate and exploit new and emerging trends.\nThe essential elements of a website that sells.\nThe importance of content and communication.\nWhat it takes to rank well in Google.\nHow to get new clients from social media.\nTo create and promote videos for massive online exposure.\nEmail marketing to increase your client base.\nThis book was written to help you take charge and gain necessary knowledge so you can understand how to use this technical stuff to your advantage. It'll help you become empowered, informed and confident discussing your strategy and requirements with qualified consultants and suppliers. It'll help you recognise the impact of your online strategy on your practice and evaluate new and emerging ideas.\nYal\u00e7in is a software engineer turned entrepreneur. He has managed, developed, consulted and solved problems in the online space for nearly 20 years for start-ups, small business, government and multi-national advertising agencies working on brands like Nike, Coca Cola, HP, Mercedes, Nestl\u00e9 and many others.\nHe founded PracticePulse in 2008 to bring the experience, best practice and digital solutions to health businesses in private practice around the globe. His articles have been featured in various industry journals.\nYalcin is a practical professional, easy to talk to and understand. For the last 5 years we've relied on him to find solutions and be instrumental in growing our business 100% each year.\nTo help save on Amazon's international postage fees for Australian readers, I have a limited stockpile of books in my office.\nOrder now and receive free postage Australia-wide.\nReaders outside of Australia can buy direct from Amazon.com.\nPrefer to use your eBook reader? Download straight to your device.\nPlease contact me directly to organise large quantities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"From the cheapest places to visit to the unfriendliest cities in America, our reader's favorite T+L stories for 2015 came from all corners of the travel world.\nA lot happened on T+L.com through 2015. Not only did we give our website a complete makeover, but we also launched our very first destination of the year poll (congrats to Cuba, the editor's pick, and Nashville, the reader's choice!) and the Journeys package, where our editors actually planned your trip for you. And then there were the long-running, annual stories we all look so forward to sharing\u2014a few of which made this list. Before we start to think about what's to come in the new year, let's take a look back at your favorite stories from the year (read: the 10 best-performing stories on the website).\nReaders are already looking to the new year\u2014what better way to prep than with a must-visit list? Our annual round-up of the can't-miss destinations for the year is the perfect place to start. Read the story here.\nEveryone has a few (or an entire list). We published a special bookazine highlighting a more in-depth look at some of the most iconic bucket list destinations, but the online edition is a perfect chance to take in some seriouslt stunning sights. Read the story here.\nWe've got a whole list of celebs behaving badly at the airport, but Johhny Depp's move to bring his pet dogs to Australia without the proper documentation caused an entirely new hullabaloo. Read the story here.\nThe only thing better than shoe shopping? Shopping for vacation shoes. Finding the right pair takes a bit of digging, but if you've got a little help (hint, hint), it's not so bad after all. Read the story here.\nEveryone loves a good, inexpensive getaway. A report shared a little bit of insight on the places to put on your weekend getaway list (no surprise: New York City isn't high on the list). Read the story here.\nThe middle seat on an airplane isn't anyone's spot of choice. But, wait: 2015 brought with it a possible solution (we can only hope). A staggered middle seat design from Thompson Aero Seating would make things less horrible. Read the story here.\nIt's a question most of us have pondered at some point (or maybe not): what the heck is that tiny hole doing in my airplane window? We've got an answer. Read the story here.\nWe're not surprised to see this one so high on the list. The 2015 edition of our \"Best Places to Travel\" package did not disappoint. How many did you check off this year? Read the story here.\nThe truth hurts: Some cities just aren't as friendly as others\u2014but that doesn't stop us from visiting them. Read the story here.\nA bit of breaking news for frequent flyers in a handful of states took this year's top spot on the most popular list. Find out if you're going to need a passport to fly domestically anytime soon. Read the story here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafwkb b/data_all_eng_slimpj/shuffled/split2/finalzzzafwkb
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+{"text":"The \u00a33.00 charge covers gift wrap, ribbons and a card with personalised greeting.\nIf you want more than one item wrapped, please contact us, you'll find we are always willing to help!\nA lot of people find our giftwrapping service brilliant because we will also post your parcel to anywhere in the UK - so you couldn't get an easier way of buying and sending a birthday gift for someone special!\nIf you would like to take advantage of this service, just select the 'Gift Wrap' option when you check out.\nUnless you tell us otherwise, we will use non-specific paper. If the present is for a special occasion, such as a birthday or christening, we can use specific styles of wrapping paper - just let us know in the comments section when you check out. Likewise, if you want paper for a girl or boy, just let us know. We can even include a short message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Go eco friendly with this pink and orange cotton cushion cover with gorgeous pine cone conifers printed all over. This cotton pink pillow cover is pefect to add a bit of texture and boho to your living space or bedroom.\nAdd a touch of opulence and boho beauty to any room with this pink cushion cover made from 100% cotton fabric. With orange conifers dotting this pink boho cushion cover, you can turn a minimalist room into your personal zen space with this pink pillow cover. Cotton pillow covers are easy to clean and maintain, and you can add a bit of trendy geometric symphony to your couch as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robert W. Cone, 59 died on September 18, 2016. General Cone was a native of Manchester, New Hampshire, born in 1957. He graduated from Memorial High School, Manchester, NH in 1975. He graduated from the United States Military Academy with a Bachelor of Science Degree in 1979 and was commissioned as a second lieutenant in the Armor Branch. Cone earned a Master of Arts degree in Sociology from the University of Texas at Austin. His military education includes the Naval War College, the Army Command and General Staff College, the Infantry Officer Advanced Course, and the Armor Officer Basic Course.\nCone served in a variety of command, staff, and operational assignments in the United States, Germany, and Southwest Asia. His first duty assignment was with the 2nd Armored Division at Fort Hood, Texas, where he served as a Platoon Leader and Troop Executive Officer in the 2nd Squadron, 1st Cavalry Regiment. In October 1981, Cone became the Aide-de-Camp to the Assistant Division Commander, 2nd Armored Division. He then served as the Battalion Maintenance Officer, Combat Service Support Company Commander, and Tank Company Commander in the 1st Battalion, 67th Armored Regiment (1-67 AR).\nHe attended the Infantry Advanced Course in 1985 followed by graduate school at the University of Texas at Austin. Next, Cone served as an Instructor and Assistant Professor in the Department of Behavioral Sciences and Leadership at the United States Military Academy at West Point, New York. After selection and completion of U.S. Army Command and General Staff College at Fort Leavenworth, Kansas, Cone reported to the 11th Armored Cavalry Regiment (Blackhorse) in Fulda, Germany. He served as the 2nd Squadron Operations Officer (S-3) during Operation Desert Storm, became the Regimental Operations Officer (S-3) in November 1991, then the Regimental Executive Officer in March 1993. He left the 11th Armored Cavalry Regiment in the spring of 1994.\nSubsequent to his service in the Blackhorse, he was the Regimental Executive Officer of the 3rd Armored Cavalry Regiment, commander, 1st Squadron, 3rd Armored Cavalry Regiment, and then Special Assistant to the Commanding General, Fort Carson, Colorado, before attending the Naval War College in Newport, Rhode Island in 1997.\nFollowing the War College, he was G-3 of the 4th Infantry Division (Mechanized), then commander of 2d Brigade, 4th Infantry Division, where he deployed the brigade to South Korea as part of Exercise Foal Eagle. He led his Brigade in the Army's Division Capstone Exercise in April 2001 as the culminating event in the development of the heavy digital force.\nFollowing Brigade Command, Cone was the Director of the Joint Advanced Warfighting Program, Institute for Defense Analyses located in Alexandria, Virginia, where he was responsible for developing joint force concepts and experiments. He participated in Operation Iraqi Freedom in 2003 as the Director of the U.S. Joint Forces Command (JFCOM) Joint Lessons Learned Collection Team, followed by becoming the Director of the JFCOM Joint Center for Operational Analysis in Suffolk, Virginia.\nIn September 2004, Cone became the Commanding General of the National Training Center (NTC) at Fort Irwin, California. He deployed to Afghanistan in June 2007 as part of Operation Enduring Freedom and assumed command of the Combined Security Transition Command \u2013 Afghanistan. His 18-month command focused on developing the Afghan National Army and Police. During his tenure, the Afghan National Army expanded from 50,000 troops to close to 80,000 and broad reforms were implemented to address corruption and training in the Afghan National Police.\nAfter returning from Afghanistan, he served as the Special Assistant to the Army Chief of Staff then became the Special Assistant to the Commanding General of the Army's United States Army Training and Doctrine Command where he was responsible for officer and enlisted soldier Initial Entry Training.\nIn September 2009, Cone assumed command of III Corps, during which he deployed III Corps Headquarters to Iraq and became the Deputy Commanding General for Operations \u2013 the second highest-ranking military officer in United States Forces \u2013 Iraq until February 8, 2011. He had responsibility for operations throughout the entire country including the development and training of fielded Iraqi Security Forces. He oversaw the transition from Operation Iraqi Freedom and counterinsurgency operations to Operation New Dawn and stability operations.\nHis last assignment was as the Commander of the U.S. Army Training and Doctrine Command.\nMedal with two oak leaf clusters, Army Achievement Medal, Parachutist Badge, Ranger Tab, and the Joint Chiefs of Staff Identification Badge.\nLieutenant Colonel (ret.) Walter \"Bill\" Rawls Bowers, age 66, beloved husband, father, grandfather, friend and patron passed away due to complications from IPF on Tuesday, August 9th, 2016. Having been diagnosed with the disease just six months prior, he fought with a stoic sense of drive.\nBill is survived by his wife Brenda Burdick Bowers of Yorktown, Virginia; daughter Colleen Bowers Whitney and son-in-law Craig Whitney of Hampton, Virginia; daughter Kathryn Bowers Brown of Chesapeake, Virginia; grandchildren Mikaila Whitney and Austin Whitney of Hampton, Virginia and Gavan Brown of Chesapeake, Virginia; and siblings Nancy \"Prissy\" Duderstadt of Chicago, Illinois and Judy Bowers of Cincinnati, Ohio; as well as numerous cousins, nieces and nephews across the country.\nBill was born August 17, 1949 in Fort Meade, Maryland to Frame John Bowers and Nancy June Rawls. An Army brat, he moved all over, including Canada, and followed his military family before settling in Texas as a high schooler. His love for the Army life would never waiver as he chose to spend 27 years himself as an active duty Army serviceman.\nAfter enlisting in the Army to serve a one-year tour with the 1st Cavalry Division in Vietnam where he earned the Purple Heart, Bill returned to Texas where he completed college at Texas A&I (now Texas A&M) University-Kingsville and officer training school.\nand accommodations. These included the Legion of Merit, Bronze Star Medal, Meritorious Service Medal with 2 oak leaf clusters, Air Medal Ribbon, Army Commendation Medal Ribbon with one oak leaf cluster, Vietnam service Medal with three service stars, Combat Infantryman Badge, and an Aircraft Crewman Badge.\nEarl graduated from St. Catherine of Sienna High School in Donaldsonville, Louisiana in 1960. Yearning to see the world, he enlisted in the U.S. Army upon graduation from high school. He rose through the ranks to retire with over 30 years of service as a Command Sergeant Major.\nAs part of his military service, Earl had a combat tour of duty in Vietnam and was stationed throughout the world. Earl graduated from several military Non-Commissioned Officers leadership schools to include the U.S. Army Sergeants Major Academy. He concluded his military career in 1991 as the 2nd Armored Division DISCOM Command Sergeant Major in Fort Hood, Texas.\nAmong his many decorations, he was awarded the Legion of Merit, Vietnam Service Medal, Meritorious Service Medal, and the Armed Forces Expeditionary Medal. Earl's success in the military can be directly attributed to his meticulous work ethic and his love for the uniform.\nEarl's attention to details was legendary. He took pride in the fact that soldiers under his command respected one another and were committed to the Army's values. The discipline and patriotism he learned in the Army never left him, as friends and family will attest. He was a life-long soldier and his passion was passed on to his sons, who are currently serving in the military.\nHe was actively involved in the Catholic Church and the Knights of Columbus. He was a faithful and dedicated member of St. Paul Chong Hasang in Harker Heights, Texas; the Catholic Church was a big part of Earl's life. He was a quiet philanthropist who faithfully donated to Catholic missions and the less fortunate.\nEarl leaves behind to cherish in his memory his wife of 48 years, Elsie Schonberg. Other loved ones include sons Earl B. Schonberg Jr, (Bridgette) and Allen B. Schonberg II (Bianca); step-daughters Lincoya Brown, and Haydee \"Bunny\" Collins; sister Mrs. Eleanor Schonberg Sherman, grandchildren McKenzie Schonberg, Ariel Simone Hill, Johnny Bryan, Tony Bryan, Edward Bryan, and Magnolia Collins; nephew Leo Sherman; niece Marcia Taylor; and a host of other family members and dear friends who will miss him dearly.\nSergeant Tyler C. Bertsch entered the military in October 2010 from his hometown of West Fargo, North Dakota. He attended One Station Unit Training at Fort Knox, Kentucky, and he graduated as an M1 Armor crewman 19K10 on March 4, 2011.\nHis assignments include: C Company 1-30 Infantry Battalion as an M1A2 SEP V2 Loader\/Gunner and G Trop 2\/11 ACR as an M1A1 Gunner\/OSV Commander.\nHis military awards include: 2 ARCOM, 1 NDSM, 1 ACM-CS, 1 GWTSM, 1 ASR, 1 OSR, 1 NATOMDL, 3 AAM, 1 NCOPD, 1 AGCM.\nSergeant Bertsch's major military accomplishments include achieving the recognition of Battalion Soldier of the Month in March 2013 and Brigade Soldier of the Quarter of April 2013 while at Fort Stewart. He deployed in support of Operation Enduring Freedom from February 2012 to October 2012 in which he drove over 12,000 miles and delivered much-needed supplies to Village Stability Platforms spread out across RC North and RC West.\nHe is an expert with the M9 Pistol. He has loaded for two Abrams gunneries and was a gunner for an additional two gunneries. As a gunner, he shot 901 and 879; all four of the gunneries resulted in first round qualification. Sergeant Bertsch has earned his silver spurs. In December 2014, he earned the recognition as 2\/11 ACR Noncommissioned Officer of the Month.\nHis short-term goals are to attend Advanced Leader Course, earn Excellence In Armor, and be promoted to Staff Sergeant. His long-term goals are to retire from the Army as a First Sergeant, become an M1A2 SEP Master Gunner, achieve a bachelor's degree in mechanical engineering, participate in the Sullivan Cup competition, and become a member of the Sergeant Audie Murphy Club. His hobbies include building models of armored Vehicles from the World War Two era, physical fitness, reading historical novels, watching movies, and traveling.\nSpecialist Holly Jo Mach was born March 31, 1993 in Lincoln, Nebraska. She graduated high school in May 2011 and attended college for a year at Hastings College of Nebraska before joining the Army in March 2012. She attended basic training in Fort Leonard Wood, Missouri and graduated in June 2012. She then attended her Advanced Individual Training at Fort Sill, Oklahoma where she graduated and became a qualified Avenger crewmember in August 2012.\nSpecialist Mach was assigned her first duty station at Camp Casey, Republic of Korea from September 2012 to September 2013. While assigned to Echo Battery, 6th Battalion, 52nd Field Artillery Regiment, she achieved several accomplishments to include February 2013 Soldier of the Month, March 2013 Soldier of the Quarter, May 2013 battalion's \"Best by Test\" winner, June 2013 selection by 2nd Infantry Division Commanding General as Warrior of the Week, and August 2013 Soldier of the Month.\nSpecialist Mach was assigned to India Battery, 1st Squadron, 11th Armored Cavalry Regiment at Fort Irwin in October 2013. While with India Battery, she became Soldier of the Month in February 2014 before trying out for the Horse Detachment in March 2014, where she is currently assigned. While in the Horse Detachment, she won the April 2014 Soldier of the Month. She received her spurs in May 2014 and attended the Warrior Leader Course in August 2014 when she received various awards to include selection as Student First Sergeant, Commandant's List, and Distinguished Leadership Award.\nSpecialist Mach's awards and decorations include four AAMs, NDSM, GWOTSM, Korean Defense Service Medal, Army NCO Professional Development Ribbon, ASR, Overseas Ribbon, and Certificate of Achievement.\nHer near-term goals include becoming the NTC Soldier of the Quarter and reenlisting in the Active Component. Her long-term goals include receiving a bachelor's degree in psychology and becoming a command sergeant major.\nWe are expanding our corporate membership program. This is a great opportunity for those who have contacts in both small and large companies to help in our fundraising efforts and for the Association to provide recognition to companies that choose to join and support us. The plaques that we are providing are professionally made and will be eye-catching on the wall in an office or lobby. Due to the Blackhorse Association's 501(c)(3) status, the companies can deduct this membership as a charitable expense.\nWe hope you are as excited about this opportunity as we are and you will not be shy about approaching potential donors. The program is outlined below, and if you have any questions, please contact Dale Skiles at 501-749-8888 or dlskiles@gmail.com.\nBasic Corporate Membership is $500. Members will receive an 8 x 10 walnut plaque with a brass plate with black matting and brass lettering.\nSilver Corporate Membership is $1,000. Members will receive a 10.5 x 13 walnut plaque with a silver-colored plate with black matting and silver lettering.\nGold Corporate Membership is $3,000. Members will receive a 10.5 x 13 walnut plaque that is laser etched with gold-colored foil embedded in the etching.\nPlatinum Corporate Membership is $5,000. Members will receive a 10.5 x 13 walnut plaque that is laser etched with platinum\/silver-colored foil embedded in the etching.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"During my 2nd year studying Visual Communication Design at Massey University one of my group projects was to design and create a 1 minute informative motion graphic based around an issue of our choice, We decided to look into Colony Collapse Disorder. Below you can find the designs for this project.\nAlongside creating a Motion graphic, we were tasked with also creating a website to host our video and support with links and helpful information. This side of the project was passed onto Calum Turner who designed the layout and code. Assets were supplied and illustrated by myself.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It takes two lubricants to make chemistry, to make magic, to make love.\nAn invigorating warming sensation for him.\nA thrilling tingling sensation for her.\nPut the two together for a totally new, unexpected experience.\nApply yours (blue) to him.\nAnd mine (purple) to her.\nThe sensations will heighten during intimacy.\nOver time, experiment with the amount you use to create the perfect sensation for both of you.\nUse as often as you like.\nDo not use if quality seal on the opening of the bottles are broken or missing.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Volunteers are an essential part of the TRU Community Care team. Every year more than 300 volunteers assist in giving our patients and families the best emotional, spiritual, and practical support available; lend valuable administrative assistance to the organization; and play a key role at the TRU Hospice Thrift Shop. Complementary therapy volunteers provide energy work to help alleviate stress and muscle tension, aromatherapy, music, visits with their certified pet, and more. Our highly trained volunteers also facilitate many of our grief support groups.\nWe select volunteers based on their qualifications; ability to deal with dying, death and grief; and availability to meet the current needs of our patients, families, and the organization.\n*If working in the TRU Hospice Thrift Shop to complete community service hours, no background check is required.\nTRU provides hospice services, home health, and grief support for patients in our service area including Adams, Boulder, Jefferson, and southwest Weld counties. TRU Community Care encompasses a continuum of care for those with advanced illness. Click here for more information on becoming a TRU Volunteer. Contact TRU Community Care Volunteer Services at 303.604.5226 or email volunteer@trucare.org for volunteer opportunities in Boulder and beyond.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Louis Hebert 1575 \u2013 Jan 1627 & Marie Rollet abt 1577 \u2013 May 1649 are to Quebec history what Adam and Eve are to the Bible. These ancestors are my 10x great grandparents on my maternal line.\nBeing the \"first family\" of New France, later known as Quebec, Louis and Marie are the essence of Canadian history 101. My biographical posts are not meant to be comprehensive analyses but rather a glimpse into their lives to elicit further inquiry by their descendants and other interested parties.\nSince Louis and Marie are so very famous there are numerous articles, videos and information on the internet about them. There is a statue commemorating them located at Parc Montmorency, Quebec City overlooking the St. Lawrence River. I am related to many people through these ancestors, a few of the more famous being Edgar Degas, Hillary Rodham Clinton and Celine Dion.\nLouis Hebert and Marie Rollet were married on 19 Feb 1601 at the Church of Saint-Sulpice in Paris, France. It was in France that the couple had their three children Guillaume, Guillaumette and Anne. Louis was involved in two expeditions to the New World in 1606 and 1611 with Jean de Biencourt de Poutrincourt and Samuel de Champlain to Port Royal, Acadia, a colony of New France, which is now known as Nova Scotia.\nLouis Hebert was a horticulturist and pharmacist earning his title of Master of Apothecary in 1603. Eventually Louis was asked to settle in Quebec City with his family so France could begin to establish themselves on the new continent.\nOn 11 Mar 1617 Louis, Marie and family boarded the ship called Saint-Etienne at the port of Honfleur, France. Louis was offered a contract of two years and pay of 300 livres to provide apothecary, medical services and farming techniques to the French colonists, trappers and indigenous people. Marie became the first school teacher in Canada as she was a literate woman. She taught her own children as well as educating the indigenous children in the care of the Jesuits.\nOn ten acres in what would become Quebec City together this couple cleared land by hand, planted their crops and tended to the medical needs of the colonists and indigenous people. Louis earned the honors of many firsts in Canada. He was the first white settler, the first to cultivate the land, the first Canadian seignior (noble man holding a land grant) and the first Officer of Justice in New France.\nLouis Hebert died on 25 Jan 1627 from injuries he sustained from a fall on the ice. Two years later Marie remarried to Guillaume Hubou and died 20 years later in 1649. The 400th anniversary of their arrival was commemorated in 2017 at which Louis Hebert and Marie Rollet were designated as historical figures in Canada.\nMy maternal 6th Great Grandfather Samuel Beaman was born in Simsbury, Connecticut 2 Feb 1732 and died in Plattsburgh, New York 20 Jul 1821.\nOn 10 May 1775 Benedict Arnold and Ethan Allen captured Fort Ticonderoga in the Revolutionary War. It is well known that the fort was easily taken in the surprise dawn attack. It is little known that my own family members were an integral part of the way the events unfolded that day.\nSamuel Beaman lived at Shoreham, Vermont and was in the business of purveying strong drink at his tavern located at Hand's Cove on the eastern shore of Lake Champlain across from Fort Ticonderoga. Samuel was also a member of the Green Mountain Boys serving pre Revolution as well as under both Ethan Allen and Seth Warner in the capture of Fort Ticonderoga and also the invasion of Quebec during the war.\nIt was at Samuel Beaman's tavern that the Green Mountain Boys assembled and prepared to cross the lake and take the fort. The Beaman Family was well acquainted with Captain William Delaplace who commanded the British military outpost.\nSamuel's son Nathan Beaman, my 6th Uncle, was a playmate to the Captain's son. Nathan Beaman was asked to show the way in the attack as he was familiar with the layout of the fort. This allowed the troops to quietly and quickly overtake Fort Ticonderoga as he led Ethan Allen to Delaplace's bedchamber taking him by total surprise.\nMy genealogical journey began with my maternal line of the prolific Bean Family which led me to the very roots of this country. The last few years I've been untangling, sorting and counting Beans.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our heart-fluttering handwoven classic picnic tote features a soft Belgium linen lining, double natural braided handles, and faux leather attachments.\nHearts Aflutter for Humankindness: A portion of all profits are donated to 'The Areeya Foundation' in collaboration with Sea & Grass. Thai for \"generosity and giving\", the Areeya Scholarship fund will use a portion of it's proceeds to sponsor the education of Thai students in need, along with providing an essential income to the women in these villages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Cangrelor With and Without Glycoprotein IIb\/IIIa Inhibitors in Patients Undergoing Percutaneous Coronary Intervention - JACC Publication May 2017 - Bernardo Cortese M.D.\nA new pubblication about Cangrelor With and Without Glycoprotein IIb\/IIIa Inhibitors in Patients Undergoing Percutaneous Coronary Intervention has been published on JACC.\nCangrelor, an intravenous, reversible P2Y12 antagonist, is approved for use in patients undergoing percutaneous coronary intervention (PCI).\nOBJECTIVES This study sought to evaluate the efficacy and safety of cangrelor compared with clopidogrel in subgroups that did and did not receive glycoprotein IIb\/IIIa inhibitors (GPIs).\nMETHODS This pooled, patient-level analysis of the 3 CHAMPION (Cangrelor versus Standard Therapy to Achieve Optimal Management of Platelet Inhibition) trials analyzed all randomized patients who underwent PCI and received the study drug (n \u00bc 24,902). Only bailout\/rescue GPI use was permitted, except in CHAMPION PCI, in which routine or bailout\/rescue GPI use was at the site investigator's discretion. The primary efficacy endpoint was the composite of all-cause mortality, myocardial infarction, ischemia-driven revascularization, or stent thrombosis at 48 h after randomization.\nOverall, 3,173 patients (12.7%) received a GPI, most commonly eptifibatide (69.4%). Despite variation in indications for GPIs, baseline characteristics were well balanced between the cangrelor and clopidogrel arms in subsets receiving and not receiving GPIs. Rates of the primary composite endpoint were lower with cangrelor compared with clopidogrel in patients who did (4.9% vs. 6.5%; odds ratio [OR]: 0.74; 95% confidence interval [CI]: 0.55 to 1.01) or did not receive a GPI (3.6% vs. 4.4%; OR: 0.82; 95% CI: 0.72 to 0.94; Pint \u00bc 0.55). Cangrelor did not increase the primary safety endpoint, GUSTO-defined severe\/life-threatening bleeding, in patients who did (0.4% vs. 0.5%; OR: 0.71; 95% CI: 0.25 to 1.99) or did not receive GPIs (0.2% vs. 0.1%; OR: 1.56; 95% CI: 0.80 to 3.04; Pint \u00bc 0.21). GPI use was associated with increased risk of bleeding in both treatment arms.\n\u00a9 2017 by the American College of Cardiology Foundation.\n\u00a9 2017 Bernardo Cortese, M.D.. A.O. Fatebenefratelli Milano, Italy. All Rights Reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eleven bodies were discovered on Friday in a Mexican town close to the U.S. border, days before president-elect Andres Manuel Lopez Obrador is due to visit for a series of forums seeking to stem the violence in the country.\n[post_ads]The incident may be related to clashes between criminal groups, the government added. The prosecutor's office is investigating the killings and identities of the victims.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Primuth, Driskell & Terzian, LLP, understands the legal, organizational, and governance challenges that plague the missions of nonprofit clients. We have the experience and focused knowledge to advise nonprofits of all sizes in a wide range of economic sectors, from cultural organizations to private foundations.\nWe are committed to helping our nonprofit clients secure and maintain tax-exempt status and remain in full compliance with their Internal Revenue Code, federal, and state obligations \u2013 all while fulfilling their organizational and public-purpose mission. We recognize how essential it is to be relationship-driven, and are always available to answer questions and provide confidential advice as a trusted colleague and not just a vendor.\nTax issues are fundamental to nonprofit operations, and due to our extensive work with a diversity of exempt organizations, few tax problems are new to us. We find that our experience dealing with legal issues in one nonprofit business sector enables us to help our clients in other sectors.\nOur broad tax and nonprofit focus allows us to confidently advise on such concerns as organizational structure and transactions. We work with the family members and advisory board of small family foundations to make sure your needs are met, and your foundation protected.\nSmall and large nonprofits concerned with maintaining their tax-exempt status while having us advise them on unrelated business income concerns.\nPrivate nonprofit foundations, especially those formed for estate planning and charitable giving.\nMuseums, cultural organizations, societies, universities, and other nonprofit entities with high public visibility.\nPrimuth, Driskell & Terzian, LLP, fully understands nonprofit needs and operational concerns, including governance issues (from determining compensation, avoiding self-dealing, and conflict of interest concerns), as well as officer and director liability. We pride ourselves on our comprehensive approach and integrated tax planning, transactional assistance, knowledge of employment law, employee benefit advice, and general business guidance. We know that maintaining your tax-exempt status is of paramount importance to you, and use our in-depth knowledge of the law governing tax-exempt entities to ensure both full compliance and efficient operation.\nObtaining, maintaining, and reinstating tax-exempt status.\nPreparing and filing Forms 990, 990-PF, and other forms filed with the IRS detailing mission, programs, and finances.\nAdvising on issues of unrelated business income.\nGuidance on property tax exemptions, employee benefits matters, tax controversies, and audits specifically involving nonprofit organizations.\nGeneral business and operational advice.\nWe know that nonprofits have unique needs, and we are there every step of the way. We regularly review nonprofit corporate documents to ensure that they correctly reflect changes in the tax laws, taking into account the organization's needs and mission. Primuth, Driskell & Terzian, LLP, will make sure your organization is protected.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drobeta-Turnu Severin is about the same as Zenica. Apr 2019 Cost of Living.\nFor example, to keep the same standard of living that would require 1,700KM in Zenica you would need to make just about 1,651KM (4,012 lei) in Drobeta-Turnu Severin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most of of Liza Kindred's Lullabot Case Study will interest only Lullabot fans, but 9 minutes in, she tells a story that makes a great subject for an Open Source Education post.\n\"Angie, you made a giant mistake and you really screwed things up here....This is why you are now Lullabot's Data Import Expert.\"\nAngie got promoted rather than fired because she was trusted to learn from mistakes, by a learning organization that deemed this quality more important than never making them.\nOpen source project organizations (like Drupal.org) must be highly mistake-tolerant, since volunteers can't be fired. To turn mistakes to advantage, they rely on feedback (discussion forums and bug reports) and recalibration (software patches and new releases).\n\"Goofus, the open source hacker, does not read boards and apply patches, and loses hours of productivity by using broken tools. Gallant, the open source developer, contributes his own bug reports, workarounds, and patches\"\nThis \"feedback & recalibration\" culture extends to users. I suspect open source users are far more likely to file bug reports than commercial software users. Know any Windows-users who ignore error messages and turn off automatic updates because they're \"annoying\"? I do.\nIt's the difference between \"buy-in\" and \"ownership\": when I buy software from a company, I expect it to work perfectly. If I take ownership of an open source product, I want to contribute feedback to the community I am now a part of so they can make my software better.\nWhat does this have to do with schools?\nI'm 20% through \"The Self Organizing School\" by Alan Bain. Bain cites two reasons for the statistical failure of research-based school reform efforts: 1) most teachers do not value professional research, and 2) the reforms never address the organizational structure of schools.\nA teacher who ignores the feedback of educational research may have difficulty diagnosing and improving failed lesson designs. Although there is plenty of feedback available from the teacher's own classroom, \"trial and error\" favors the prepared mind (to garble Pasteur).\nA \"profession\" is characterized by specialized knowledge and academic preparation. Why don't more teachers act like professionals?\nMany teachers said they consider academic research in ivory towers irrelevant to \"the trenches.\" In response, \"action research\" was rolled out in the early 1980s. Action researchers were supposed to act like teachers but think like researchers. I understand professional development \"merit badges\" in Action Research are still available.\nI can imagine how it all went down: A \"superintendent's conference day\" was devoted to \"action research\" and it was announced that all clinical supervision for the following year would include a post-lesson \"practitioner reflection\" analzing the lesson design, proposing a theory about what worked or didn't, and a plan for testing the theory.A month later, a follow-up report would be submitted, and administrator and teacher would meet again to discuss and sign off.\nIt would be wonderful if an external model like \"Action Research\" helped such a district become a learning community, reflecting on emergent best practices in teaching and professional development. More likely it was a dog-and-pony show about Action Research that seemed like too much work to repeat the following year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"Less is more,\" said the architect Mies van der Rohe which, I think, applies to poetry as well as to architecture, and to paraphrase your own recent question, \"Can a thought ever be completely expressed in words?\"\nThe poems I value - like this recent one of yours - do NOT atempt the impossible of trying to spell everything out, trying to nail a thought to a board like a dead butterfly, but instead try to hint at it, to surround it and leave it to the reader to fill in the space.\nS.S. re\" \"I'll take your tears\"\nBeautifully accomplished, especially that lovely last line!\nis it he who grieves?\nTears fall like autumn leaves,.\nTheir beauty is in sorrow.\nor half full of tears?\nHey, welcome back, o naughty one!\nwhere the two of us will mesh.\nsome of 'em delicate, some of 'em crude.\nany new rhymes from that rhymin' dude.\nor praying to the rhyming-fairy!\njust so we can say... get better SOON!!!\nWhy would you like a poem to rhyme?\nHearing a beat that reads perfect time.\nWords of expression, say what we can.\nFreedom of speach, how can you ban?\nI will follow by your rules that are given.\nInto the sub-conscious mind I am driven.\nSearching for words that meet the goal.\nTrying to conform, might upset my soul.\njust a response, to what you were saying.\ndidn't mean a thing, I was only playing.\nI love to write peoms that rhyme, its my favorite kind of wrinting. This is \"because its ok\" by me!!\nof happiness and well being.\nand people are gonna start to fall in line.\nand give hope for those hurt, to finally be healed.\nBaby thats all you have to do.\nand remember to always look over your back.\nsome people out there are just mean.\nor else you'll be stuck in your own personal hell.\nto fade the boldest coxcomb.\ntheir fruit, a hardened scarlet pearl.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The ability to earn an incentive trip with CTMH is an amazing opportunity. Earning it is an incredible reality, and the experience is like nothing else. The trip itself is a constant 'pinch yourself' adventure, but CTMH add lots of other things to the trip to elevate it to new heights. One of these, is the album, pre-designed, pre-cut pages, and exclusive stamp set that we get before we go, to make recording the experience simple and easy.\nI've just returned from one of these adventures in Dubai, but our American\/Canadian friends were cruising in the Caribbean the week before us. The stamp set that went with their 'exclusive' album has been made available to you, just until the end of this month.\nBut wait, there's more \u2026..!\nIf you purchase this stamp set, I will send you the layout instructions for the pages that the American girls got \u2013 that's 20 pages ready to capture your own holiday paradise.\nBut wait, there is more still!!\nThis stamp is very limited edition right \u2013 I mean, you've only got another 10 days to get it. BUT did you know you can purchase it as part of our current Stampaganza promotion!! That means it could be FREE. These layouts also use the Beach Days stamp set, which is also only available until the end of this month, so my recommendation is to put these two in your 'basket', and choose another D size stamp set for FREE.\nRemember, order the Paradise stamp set from me this month, and I'll send you the instructions etc to create these layouts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaihga b/data_all_eng_slimpj/shuffled/split2/finalzzzaihga
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+{"text":"Prof. Dr. Frederik J. Zuiderveen Borgesius is Professor of Law at the Digital Security Group (DiS), at the Radboud University Nijmegen, and researcher at the Institute for Information Law (IViR), at the University of Amsterdam. We thank our colleagues for their comments on drafts of this article: Lillian Edwards, Irene Kamara, Adam Shinar, Peter Swire, Joris van Hoboken, the editors of Theoretical Inquiries in Law, all participants at the conference 'The Problem of Theorizing Privacy, 8-9 January 2018, and all the participants at the Privacy Law Scholars Conference Europe, 27 January 2018, Brussels, and at the Privacy Law Scholars Conference, 30-31 May 2018 Washington. We thank Sabine Rijnen and Sonja van Harten for research assistance.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The intertwined stone rings in this sculpture offer a puzzle to the mind, asking to think in new ways about shapes and ideas.\nOf course, we use rings to symbolise connections that cannot be broken. In theory, we think of stones as symbolising permanence too. Still, the fluid shape of the ring is at odds with our other ideas about stone, which we can sometimes dismiss as being rigid, inflexible and unforgiving.\nIn this sculpture, stone takes on the smooth, fluid properties of the ring. Meanwhile, the ring takes on the solidity and permanence of stone. It's an intriguing piece of abstract sculpture, but it remains a very accessible piece too: the puzzle about how the stone links are connected to each other keeps everyone's eyes and minds absorbed.\nThis stone and leather chair is a unique piece created for a private commission.\nIt's an unusual way of bringing traditional stonecraft into the domestic space, and it is a real talking point in its new home!\nThe Celtic knot design, familiar to many from the illustrations used in the Book of Kells, remains a design classic.\nHere the intricate, interlaced design is dramatically juxtaposed with the strength of stone .\nVictor learnt his craft as a stone-carver when he was apprenticed into a family who had passed their skills down from generation to generation. He was the first person from outside the family to be trusted with those skills, and so this sculpture reflects the unbroken traditions of stone-carving and design in Ireland.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With an OMAX abrasive waterjet system, you will profit on current projects faster and more efficiently, and expand to those new projects that will help your business earn more. Take advantage of these benefits today. OMAX has arranged special financing options for a limited time.\nThe US Congress made permanent the Section 179 provision, allowing businesses to expense from taxable income up to the first $500,000 for all capital equipment purchases.Then, after the Section 179 deduction, Congress also permitted first year 50% bonus depreciation, also deductible from taxable income, for any remaining capital equipment basis.\nConsult your tax advisor on how to apply the Section 179 deduction or depreciation to your business.\n*OAC with two payments down. Financing offers expire 12\/31\/2018. Customers seeking 2018 tax deductions under Section 179 should place orders by 12\/1\/2018. Does not apply to ProtoMAX purchases.\nCopyright 2018. OMAX Corporation. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"35 year-old Anarchist comrade shot dead by police in Dafni, Athens, 10 March 2010.\n\"Comrade Lambros was a real fighter and an example for all of us. True rebel, sworn enemy of inaction. Austere, sober, selective, determined. UNTIL THE END, HEAD of himself. Furthermore, he did not fall into the trap of making a contract with his life. A contract with life is for those who get divorced into the selection of legal transaction.\nConsider these not-so-grand, but honest words, as a last farewell to a comrade and freedom fighter.\nAfter the end of the big demo in Athens today a group of people approached the Ministry of Culture in Bouboulinas from the back street (Zaimi), walked into the parking lot and set fire to a State van parked in the internal courtyard of the Ministry, in memory of the comrade Lambros.\nNot one moment of peace for the class enemy \u2013 Honour to Lambros Foundas.\nA translation of the letter sent to \"Eleftherotipia\" newspaper on the occasion of Polys (Polykarpos) Georgiades and Vaggelis Chrisohoides jury on 2\/2 on the accusations of providing refuge to a \"criminal\" and being part of his \"criminal association\". Vassilis Palaiokostas is on the run, after his escape with a helicopter from Korydallos prison, on Feb. 22\/2009, accused for robbing banks and kidnapping industrialists. It is worth to say, he has never harmed human life, not even cop lives to avoid an arrest. On the opposite, he and his older brother (a legendary bank robber and escapee, currently held in prison) are said to have helped many poor people and communities in mainland Greece' mountains where they come from and are often said to find refuge at, continuing a tradition of \"social robbery\" that blossom around the Balkans since the decline of the Ottoman Empire.\nThis letter was sent to a few anarchist comrades in London, and is a response to the question of Alfredo's health and also what could be done by solidarious people. The letter is best read as part of an ongoing dossier that is being assembled by the same anarchist comrades, which can be found here. Also read this text, which accompanies the letter.\nKeratsini is the municipality where the police raided the anarchist space Resalto and then raided the occupied town Hall as well, arresting 22 in Resalto and 41 in the Town Hall. The 41 arrested in the Town Hall were set free December 7.\nThe 22 arrested from Resalto were set free December 7 at 6.30 am. They were recieved by people in solidarity that were standing outside the court of Piraeus. Four are set free without conditions, one has to pay a bail of 15000 euro, two have to pay 5000 euro each, six have to pay 3000 euro each, and the rest with other restrictions (can't leave the country, have to be present at a police station every some days).\n1 ) Cops with guns in the demonstrations.\n2 ) Motorbike raids against protesters.\n3 ) Cops following peaceful demonstrators in their tracks, wild and indiscriminate arrests.\n4 ) Staged events such as the alleged assassination attempt against the rector of the Pritanea.\n5 ) An enormous number of vague and even criminal charges and arresting people young and old.\n6 ) Closing schools on the pretext of swine flu and the merciless beating of students who wanted to get to their schools.\n7 ) Undercover cops kidnapping young protesters.\n8 ) Higher level visible collaboration between Golden Dawn neo-nazis and the Police.\n9 ) Secret meetings between Chrysohoides and the bosses of the TV channels and journalists to decide on how TV reporting was going to be done over these days.\n10 ) Secret covert cameras and helicopters hovering constantly.\n11 ) Zero tolerance makes a bitter orange and a stone against a bank a felony and a pretext for a police attack.\n12 ) BAN on demonstrations and political gatherings in the conversion areas with massive police intimidation and outrageous controls\/databasing.\n13 ) Hacker attacks on indymedia, and sites of squatted places and TVXS (no borders TV), deleting comments.\n14 ) Invasion and PREVENTIVE ARRESTS in many self-managed spaces.\n15 ) In an Orwellian way anarchists and the rebels are referred to \"fascists and Nazis!\".\n16 ) Junta-like removal of asylum [in the universities].\nSome of these things happened in isolation, and some only happened before under the junta in 1967-1974; others have never happened before, only now.\nThey never ever happened altogether in such a short space of time!\nIt seems that the crash which happened here and they are hiding, like the indomitable December, has the power to activate an emergency plan, a new 'PLASTER CAST' [repeating the pronuncement of the junta in 1967].\nThese moments are more than historical. We are witnessing for the first time since 1967 an attempt to impose a fascist police coup. If a parliamentary democracy are able to commit those crimes, the junta is something different and we have all have begun to understand that. The anarchist slogans in the streets are beginning to say bluntly: \"DOWN THE JUNTA\".\nA collusion of prosecutors, rectors, the upper classes, TV media and police, and still we do not know what other local and foreign forces have been mustered. You hear of missing people. The climate is heavy like under the junta.\nTHIS IS NOT THE TIME TO BE SILENT! THIS IS NOT THE TIME TO RELAX!\nEVERYBODY IN THE STREETS \u2013 SQUATS EVERYWHERE!\nPLEASE HELP TO SMASH DOWN THE GREEK JUNTA!\nTHE REGIME IS GOING THE WAY OF 1967!\nPanayiotis Masouras is accused of being a member of the 'Conspiracy of Cells of Fire'.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":", we choices the top choices along with greatest quality just for you, and now this images is actually among photos collections in our finest pictures gallery in relation to Luxus G\u00e4stezimmer Zu Hause Einrichten. I hope you might want it.\nplaced by means of Thomas Rios in 2018-10-08 22:08:30. To view most images in Luxus G\u00e4stezimmer Zu Hause Einrichten photographs gallery you should follow this particular hyperlink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzajlps b/data_all_eng_slimpj/shuffled/split2/finalzzzajlps
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+{"text":"An infused oil is an oil that has been filled with the properties of one or more herbs. Infused oils make great massage oils and are often used as the basis for healing salves. Some herbal infused oils, like garlic or cayenne, can be used for cooking, too.\nThere are two basic ways to create an infused oil, the cold method or the hot method. The cold method, which is really done at room temperature, takes time but is relatively easy and requires little attention while the oil is soaking in the herb's qualities. The hot method requires a direct heat source and close attention but is ready for use within a relatively short period of time. In addition, some herbs, especially certain barks and roots, are better infused using heat because they don't surrender their properties easily. Fresh, fragile plants can rot during the longer processing period required for cold infusion, too. So, be sure to consider the herbs you intend to use when you choose your method.\nInfused oils can be made from fresh or dried herbs. Some herbs, like St. John's Wort, produce a better oil when infused from fresh plant material, while others seem to do well whether they are dried or fresh. Some herbs, such as Calendula, infuse better when you prepare them for infusion using a more complex extraction method and then infusing the oil with their properties using the hot method of infusion. It's a good idea to do a little research on the herb you're planning to infuse so you can choose the most effective method, but in lieu of that you can choose the method that seems best to you and make a small quantity at first to check the quality. As with so many other herbal preparations, if it works well for you and yours, then you've chosen a good method.\nFor more information on using heat to infuse an herbal oil, see The Practical Herbalist general procedure for making a heat-infused herbal oil.\nAny oil can be infused with an herb. When you're choosing the oil you want to use, keep your intended use for the finished oil in mind. I recommend using organic oils whenever possible, but as long as the oil you have chosen is expelled-pressed or cold-pressed, it should be relatively free of harmful chemicals. For the cold infusion method, I recommend you stick to oils that are liquid and pourable at room temperature. If you use the hot infusion method, you will be heating the oil gently, so butters and hard oils like Shea butter and Coconut oil can be used effectively.\nThis is a basic procedure, not a recipe. I haven't included specific measurements but have instead described the process and what to look for as you're working. For specific recipes using this technique, see The Practical Herbalist Recipes.\nPrepare your herbs. If you're using fresh herbs, clean or shake any dirt from them, although unless they're really, really dirty you don't need to wash them with water. Next, chop them. If you plan to use a blender (see step 4), then you'll only need to chop them roughly, but if you don't intend to use a blender, chop them as finely as you can. If you're using dried herbs, crush them so they are mainly in smaller pieces, although you don't necessarily want to powder them as that will make straining the fine particles out later more difficult.\nPlace the herbs into your jar. Pack them down gently and fill the jar about three-quarters of the way full with the plant material. Leave enough room to fill the jar with oil and to shake the contents regularly so the oil will penetrate through all of the plant material. If you're using roots or barks, fill the jar only about a third to half-way full of herb.\nPour the oil over the herbs, filling the jar until you've covered the herbs by just under a quarter of an inch, or if you're using barks or roots fill the jar until it's about three quarters or so full.\nBlend the mixture. (This step is optional and not recommended if you're using dried or hard herbs, such as dried roots, barks, or leaves).\nPour the whole contents of the jar into a blender.\nBlend on a medium speed until you can see the herbs have been chopped fairly finely.\nReturn the mixture to your jar, using a rubber scraper or spatula to scrape out every last bit of goodness.\nCap the jar tightly and give it a good shake.\nLabel the jar, and store it in a warm but not hot place. A sunny window where you'll see it every day is a good spot. Remember you want the heat but not the light so keep your jar covered or use a jar with dark glass. I simply put a piece of fabric over the jar when I use this method.\nShake the jar daily, even several times a day. This is highly recommended, although I've forgotten to shake jars for weeks and still had good results.\nAfter two to eight weeks, strain the herb material from the oil. Your oil should smell and taste of the herb and, depending on the herb, should have changed colors. As you strain it, be sure to squeeze out as much of the oil from the plant material as you can. I like to use a small, fine-mesh brewing bag to strain mine, but several layers of cheesecloth or white t-shirt material and a good strainer will work well, too.\nAfter you've strained the oil once, wash the original jar and return the oil to it. Cap it tightly and let it stand for a few hours or overnight so any remaining sediment will fall to the bottom of the jar.\nOnce the sediment has all settled, carefully pour the clear oil from the jar into your bottle. You want to remove as much sediment as you can because the sediment may cause your oil to go bad in time.\nLabel your jar, and store it in a cool, dark place. Be sure to include the date, the kind of herb you infused, and uses so others will also know how to use it. At this point, if you wish, you can use your oil to create a salve using The Practical Herbalist basic procedure for making a salve.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you interested in submitting a paper for the IAG conference?\nAUSCCER's Alex Tindale and Charles Gillon attended the 2013 Institute of Australian Geographers Conference in Perth from July 1 to 4. Here they reflect on their experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A SHARK will need to visit the dentist after an unwelcome encounter with a surfboard on the weekend.\nFollowing the close up images of the board being widely shared on Facebook, Ballina Police reached out and made contact with the rider.\nThe Department of Primary Industries is now examining the board, which appears to have been bitten by a shark at Flat Rock off Ballina on Sunday.\nReports on social media said a kiteboarder initially thought he'd hit a turtle until he returned to shore and saw bite marks and a tooth embedded in his board.\nThe DPI is assessing the tooth and the board to identify the shark size and species.\nThis story originally appeared in the Northern Star.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"amanda mcclements & metrocurean: ENTERTAIN } Happy Holiday Weekend!\nENTERTAIN } Happy Holiday Weekend!\nI guess, one of the reasons why I love visiting waterparks, beaches and resorts is the cool hammocks and benches they've got. Of course, the view is one but admit it, one of the reasons why we get attracted to the several places are because of the cool design it has.\nThere is not much long weekend holidays happening this year. This means more reason to cherish them. I tend to go on short trips for the weekend.\nI'm so jealous of you hammock! Here at home, we have a little pool - all that's missing is a hammock, and the summer would be great! I would stay in my cocoon all day long!\nThat's a good idea to enjoy summer better; placing a backyard hammock in your garden is perfect for relaxation. I will definitely try this at home; I can already imagine myself enjoying these foods with my family in our newly remodeled garden.\nAh, can't wait to finally get away from work. I won't stick to a hammock, though; maybe I'll go camping somewhere for a change.\nRoad trips are not complete without foods..That's why every time we plan our road trip we look for new recipes for us to be able to try new things while traveling. I just can't wait for the weekend to come!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"That was the first shirt that I made which was a tester, a prototype.\nSo here's two other designs I've made recently and more patterns are on their way. There's loads of leopard print ones.\nAnd because i'm not perfect the shapes will not all be exactly the same, just a little disclaimer.\nOh yh and i'll be doing different designs later on. But i'll just stick with this for now.\nor facebook me, if you have me.\nso cute! love the prints on your tee they are so cool!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalprw b/data_all_eng_slimpj/shuffled/split2/finalzzzalprw
new file mode 100644
index 0000000000000000000000000000000000000000..1cae4d033539d4ebf62af5c98b1c0fc669157841
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Articles Archives - Infoaxis, inc.\nSmall- and medium-sized businesses constantly struggle with different areas of business, whether it's accounting, inventory, or strategic planning. This is because SMBs often lack the tools to manage these systems efficiently. However, enterprise resource planning (ERP) software can come in handy.\nThe general rule of thumb of cybersecurity is: Anything that connects to the internet can be hacked. With the increasing popularity of Internet of Things (IoT) in the workplace, every business should be on high alert, especially those in the healthcare industry where patients' well-being hinge on the security of the device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Corresponding author: Yun-long Zhao (ylzhao@bio.ecnu.edu.cn).\n(C) 2010 Jia-Yao Hu. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.\nTwo new species of the genus Nazeris Fauvel collected from Nabanhe Nature Reserve, Yunnan Province, are described under the names of Nazeris nabanhensis sp. n. and Nazeris caoi sp. n. The male sexual characters are described and illustrated. A key to the Nazeris species of Yunnan is provided. A map of the collecting sites is given.\nThe genus Nazeris Fauvel (1873: 298) can be readily distinguished from other Paederinae by the labrum having four teeth at the front margin and the bi-lobed 4th tarsal segments. Up to the present, 51 species and subspecies of Nazeris have been recorded from China. Yunnan is a mountainous province located in Southwest China, from which nine species of Nazeris have been described: Nazeris zhangi Watanabe & Xiao (1993: 130) from \"Yu'an-shan near Kunming City\", Nazeris giganteus Watanabe & Xiao (1997:2) and Nazeris daliensis Watanabe & Xiao (1997: 7) from \"Diancang shan Mts., Dali shi\", Nazeris alpinus Watanabe & Xiao (1997: 5) from \"Mt. Yulongxue shan, Lijiang County\", Nazeris jizushanensis Watanabe & Xiao (1997: 9) from \"Mt. Jizu Shan, Binchuan County\", Nazeris baihuaensis Watanabe & Xiao (2000: 312), Nazeris ishiianus Watanabe & Xiao (2000: 319) and Nazeris nomurai Watanabe & Xiao (2000: 316) from \"Gaoligong Shan Mts., Baoshan area\", Nazeris huanxipoensis Watanabe & Xiao (2000: 318) from \"Huanxipo, Tengchong Xian\". Only two species have become known from the three countries adjacent to Yunnan (Myanmar, Laos and Vietnam): Nazeris coomani Jarrige (1948: 40) (redescribed by Rougemont 1988: 775) from \"du mont Bavi, Tonkin\" (Vietnam) and Nazeris odzisan Watanabe (1996: 1) from \"Mt. Tam Dao, Vinh Phu Prov.\" (Vietnam).\nThe specimens from Yunnan Province contained another two undescribed species, Nazeris nabanhensis sp. n. and Nazeris caoi sp. n. The male sexual characters of the two new species are described and illustrated. A map (fig. 13) of the collecting sites of Nazeris species in Yunnan and a key to the Yunnan species are provided.\nThe types are deposited in the Insect Collections of Department of Biology, Shanghai Normal University, Shanghai, P. R. China (SHNUC).\nThe specimens were collected from decaying leaf litter of forest floors by hand sifting. They were killed with ethyl acetate and dried. To examine the male genitalia, the last four abdominal segments were detached from the body after softening in hot water. The aedeagi and sternites were mounted in Euparal on plastic slides. Drawings were made using an Olympus SZ61 microscope. Photos were taken with an Olympus E420 camera mounted on an Olympus SZX12 stereoscope. Material of other Yunnanese Nazeris species was not examined. The characters used in comparative remarks and keys are according to descriptions of (Watanabe and Xiao (1993, 1997, 2000), Watanabe (1996), Jarrige (1948) and Rougemont (1988).\nElytra length: measured from humeral angle to apicolateral angle.\nCHINA: Holotype: Yunnan Prov.: male, Jinghong City, Nabanhe Nature Reserve, Benggangxinzhai, 1, 750m, 16. XI. 2008, Hu Jia-Yao & Tang Liang leg. Paratypes: 2 females, Jinghong City, Nabanhe Nature Reserve, Bengganghani, Nanmugahe, 1, 700m, 11. XI. 2008, Hu Jia-Yao & Tang Liang leg.; 1 female, Jinghong City, Nabanhe Nature Reserve, Bengganghani, 1, 800m, 14. XI. 2008, Hu Jia-Yao & Tang Liang leg.; 5 males, 8 females, same locality as holotype, 3. V. 2009, Hu Jia-Yao & Yin Zi-Wei leg.; 1 female, Jinghong City, Nabanhe Nature Reserve, Bengganghani, Nanmugahe, 1, 700m, 27. IV. 2009, Hu Jia-Yao & Yin Zi-Wei leg.; 3 females, Jinghong City, Nabanhe Nature Reserve, Bengganghani, Chuguohe, 1, 700m, 28. IV. 2009, Hu Jia-Yao & Yin Zi-Wei leg.; 1 male, 1 female, Jinghong City, Nabanhe Nature Reserve, Bengganghani, 1, 650m, 29. IV. 2009, Hu Jia-Yao & Yin Zi-Wei leg.; 1 male, Jinghong City, Nabanhe Nature Reserve, Bengganghani, Nanmugahe, 1700m, 30. IV. 2009, Hu Jia-Yao & Yin Zi-Wei leg.; 4 females, Jinghong City, Nabanhe Nature Reserve, Bengganghani, 1, 650m, 30. IV. 2009, Hu Jia-Yao & Yin Zi-Wei leg. SHNUC.\nHabitus of Nazeris. 1 Nazeris nabanhensis sp. n. 2 Nazeris caoi sp. n. Scale bars 1 mm.\nBody length: 5.8\u20136.4 mm; forebody length: 3.3\u20133.5 mm.\nMale. Body (Fig. 1) elongate, dark brown, with labrum, coxae, and basal antennomeres reddish yellow, the remaining antennomeres, maxillary palpi and legs yellow, with exception for coxae.\nHead suborbicular, longer than wide (length\/width = 1.18); postocular portion 1.96 times as long as eye length; punctation coarse, dense, and umbilicate; interstices reduced to narrow ridges. Antennae slender; relative length of each segment from 1 to 11: 42.0 : 15.0 : 29.0 : 23.0 : 22.5 : 22.0 : 19.0 : 17.0 : 15.5 : 14.0 : 20.0; relative width of each segment from 1 to 11: 12.0 : 7.5 : 6.0 : 6.0 : 6.0 : 6.0 : 6.0 : 6.0 : 6.0 :7.5 : 7.5.\nPronotum convex, oval, longer than wide (length\/width = 1.20), narrower (pronotum\/head = 0.93) and shorter (pronotum\/head = 0.94) than head; prosternum with strong longitudinal median carina, which disappears behind anterior margin. Elytra shorter than wide (length\/width = 0.91), distinctly shorter (elytra\/pronotum = 0.74) and slightly narrower (elytra\/pronotum = 0.97) than pronotum.\nAbdomen elongate, tergites without any microsculpture. Seventh sternite (Fig. 3) trapezoidally emarginated in middle of posterior margin and distinctly depressed in front of emargination; 8th sternite (Fig. 4) V-shaped deeply excised in middle of posterior margin. Aedeagus (Figs. 5, 6 and 7) well sclerotized; apical part of median lobe in dorsal view tri-lobed, median part cone-shaped, two outer parts acute at apices, with little agnail in each outer side near apex; dorso-lateral apophyses slightly curved inward, distinctly widened near apex, extending beyond apices of median lobe.\nFemale. Seventh and 8th sternites simple. The other characters are similar to those of male.\nDetails of Nazeris nabanhensis sp. n. 3 male 7th sternite 4 male 8th sternite 5 aedeagus, in dorsal view 6 aedeagus, in lateral view 7 aedeagus, in ventral view. Scale bar: 0.5 mm.\nThe new species is similar to Nazeris daliensis Watanabe (1997: 7) from Yunnan Province in appearance, but it can be distinguished from the latter by the following characters: elytra slightly narrower than pronotum (in Nazeris daliensis nearly as wide as, or slightly broader than pronotum); depth of excision of male 8th sternite nearly half of middle length of sternite (in Nazeris daliensis much shallower, nearly 1\/3 of middle length of 8th sternite); apical part of median lobe of aedeagus in dorsal view tri-lobed (in Nazeris daliensis not lobed). The new species can be distinguished from Nazeris coomani Jarrige (1948: 40) from Vietnam by head with umbilicate punctation (in Nazeris coomani punctation of head simple) and distinguished from Nazeris odzisan Watanabe (1996: 1) from Vietnam by elytra shorter than wide (in Nazeris odzisan elytra longer than wide); dorso-lateral apophyses of aedeagus extending beyond apices of median lobe (in Nazeris odzisan not extending to apices of median lobe).\nThe specific name is derived from the name of the type locality: Nabanhe Nature Reserve.\nCHINA: Holotype: Yunnan Prov.: male, Jinghong City, Nabanhe Nature Reserve, Bengganghani, 1, 930m, 14. XI. 2008, Hu Jia-Yao & Tang Liang leg. Paratypes 1 male, same data as holotype; 1 female, Jinghong City, Nabanhe Nature Reserve, Bengganghani, 1, 900m, 1. V. 2009, Hu Jia-Yao & Yin Zi-Wei leg. SHNUC.\nBody length: 6.1\u20136.4 mm; forebody length: 3.3\u20133.6 mm.\nMale. Body (Fig. 2) elongate, dark brown, with labrum, coxae, and basal two antennomeres reddish yellow, the remaining antennomeres, maxillary palpi and legs yellow, with exception for coxae.\nHead suborbicular, slightly longer than wide (length\/width = 1.07); postocular portion 2.14 times as long as eye length; punctation coarse, dense, and umbilicate; interstices reduced to narrow ridges. Antennae slender; relative length of each segment from 1 to 11: 42.0 : 13.5 : 30.0 : 23.0 : 21.0 : 21.0 : 19.0 : 18.0 : 18.0 : 16.0 : 22.0; relative width of each segment from 1 to 11: 10.5 : 7.0 : 6.5 : 6.0 : 5.5 : 6.0 : 5.5 : 5.5 : 6.0 :6.5 : 7.0.\nPronotum convex, oval, longer than wide (length\/width = 1.19), narrower (pronotum\/head = 0.88) and shorter (pronotum\/head = 0.98) than head; prosternum with strong longitudinal median carina, which disappears behind anterior margin. Elytra slightly shorter than wide (length\/width = 0.97), distinctly shorter (elytra\/pronotum = 0.80) and slightly narrower (elytra\/pronotum = 0.97) than pronotum.\nAbdomen elongate, tergites without any microsculpture; densely and coarsely punctate. Seventh sternite (Fig. 8) distinctly emarginated in middle of posterior margin; 8th sternite (Fig. 9) with little short protrusion in middle, deeply excised in middle of posterior margin. Aedeagus (Figs. 10, 11 and 12) well sclerotized; median lobe bi-lobed in dorsal view, curved ventrad in apical 1\/3 in lateral view; dorso-lateral apophyses very thin, slightly curved ventrad, not extending to apices of median lobe.\nDetails of Nazeris caoi sp. n. 8 male 7th sternite 9 male 8th sternite 10 aedeagus, in dorsal view 11 aedeagus, in lateral view 12 aedeagus, in ventral view. Scale bar: 0.5 mm.\nThe present new species is similar in appearance to Nazeris nabanhensis sp. n. from the same locality, but can be distinguished from the latter by the following characters: postocular part more than twice as long as longitudinal diameter of each eye (in Nazeris nabanhensis less than twice as long as longitudinal diameter of each eye); median lobe of aedeagus in dorsal view bi-lobed (in Nazeris nabanhensis tri-lobed); dorso-lateral apophyses of aedeagus not extending to apices of median lobe (in Nazeris nabanhensis extending beyond apices of median lobe). The new species can be distinguished from Nazeris coomani Jarrige (1948: 40) from Vietnam by head with umbilicate punctation (in Nazeris coomani punctation of head simple), and distinguished from Nazeris odzisan Watanabe (1996: 1) from Vietnam by elytra shorter than wide (in Nazeris odzisan elytra longer than wide); median lobe of aedeagus in dorsal view bi-lobed (in Nazeris odzisan not lobed).\nThe species is named in honor of Mr. Guanghong Cao of Nabanhe Nature Reserve, who helped us a lot during field work.\nMap showing the collecting sites of the Nazeris in Yunnan Prov.; 1 Nazeris nabanhensis sp. n. 2 Nazeris caoi sp. n. 3 Nazeris zhangi Watanabe & Xiao 4 Nazeris giganteus Watanabe & Xiao 5 Nazeris daliensis Watanabe & Xiao 6 Nazeris alpinus Watanabe & Xiao 7 Nazeris jizushanensis Watanabe & Xiao 8 Nazeris baihuaensis Watanabe & Xiao 9 Nazeris ishiianus Watanabe & Xiao 10 Nazeris nomurai Watanabe & Xiao 11 Nazeris huanxipoensis Watanabe & Xiao.\nKey to species of Nazeris from Yunnan Province, China .\n2 Male 8th sternite with short protrusion in middle, median lobe of aedeagus bi-lobed in dorsal view Nazeris caoi sp. n.\n3 Median lobe of aedeagus tri-lobed in dorsal view Nazeris nabanhensis sp. n.\nWe thank Mr. Guanghong Cao and Mr. Maoxing Tian (Nabanhe Nature Reserve) for their help during field work. We thank Dr. Liang Tang and Mr. Ziwei Yin (Shanghai Normal University) for their collecting the specimens and continuous help in many ways. We thank Dr. Ales Smetana (Agriculture and Agri-Food Canada) for revision our paper. This study is supported by the National Natural Science Foundation of China (No. 30870323) and by Shanghai Normal University (No. SK200833 and SK200834).\nFauvel A (1873) Faune Gallo-Rh\u00e9nane ou species des insectes qui habitent la France, la Belgique, la Hollande, le Luxembourg, la prusse Rh\u00e9nane, la Nassau et la Valais avec tableaux synoptiques et planches grav\u00e9es. Tome 3. Livraison 4. Caen: Le Blanc-Hardel, 215\u2013390.\nJarrige J (1948) Staphylinides nouveaux d'Asie orientale. Notes d'Entomologie Chinoise, Mus\u00e9e Heude 12:39-41.\nRougemont GM (1988) Un Nazeris nouveau de Tha\u00eflande (Coleoptera, Staphylinidae, Paederinae). Revue Suisse de Zoologie 95:773-777.\nWatanabe Y (1996) A new Nazeris (Coleoptera, Staphylinidae) from northern Vietnam. Species Diversity 1:1-5.\nWatanabe Y, Xiao NN (1993) A new species of the genus Nazeris (Coleoptera, Staphylinidae) from Yunnan Province, Southwest China. Elytra, Tokyo 21 (1):129-133.\nWatanabe Y, Xiao NN (1997) Four new Nazeris (Coleoptera, Staphylinidae) from Yunnan Province, Southwest China. Edaphologia (58): 1\u201312.\nWatanabe Y, Xiao NN (2000) Four new species of the genus Nazeris (Coleoptera, Staphylinidae) from the Gaoligong Shan Mountains in Yunnan, Southwest China. Elytra, Tokyo 28 (2):311-321.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Installation of a Suntopia solar heater was a no-brainer at this Las Vegas residence. With such flexible all black solar heater tubing our team of professional installers was able to use the point of the rooftop that would maximize heat. As you can see from the project pictures in the gallery below, there were many elements that needed to be taken into consideration during this project.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bed and Breakfast Huize Tilly in Maastricht is a town house with 2 Rooms.\nGuesthouse Huize Tilly is a lovely accommodation situated on a quiet square in the centre of Maastricht between the Vrijthof and the market. From the square you have a view of one of the 12th century city walls and the town hall. The accommodation is located in the old part of town set among the charming shops, cafes, terraces, and restaurants. Many attractions, like the St. Servaas church with its famous treasury room, and the Dominican church, in which the finest European bookshop is situated,... are just around the corner. Guesthouse Huize Tilly offers two nicely decorated rooms with free internet wifi access, a double bed and a private bathroom. In one of the rooms there is a small refrigerator and TV, the other one is equipped with a TV and DVD. Breakfast is available from h in a little restaurant across the street. Parking facilities are available at a 5 minute walk for \u20ac9-\u20ac12 a day. When Andr\u00e9 Rieu performs on the Vrijthof the price of the rooms is \u20ac100 per night.\nGuesthouse Huize Tilly is a lovely accommodation situated on a quiet square in the centre of Maastricht between the Vrijthof and the market. From the ...square you have a view of one of the 12th century city walls and the town hall. The accommodation is located in the old part of town set among the charming shops, cafes, terraces, and restaurants. Many attractions, like the St. Servaas church with its famous treasury room, and the Dominican church, in which the finest European bookshop is situated, are just around the corner. Guesthouse Huize Tilly offers two nicely decorated rooms with free internet wifi access, a double bed and a private bathroom. In one of the rooms there is a small refrigerator and TV, the other one is equipped with a TV and DVD. Breakfast is available from h in a little restaurant across the street. Parking facilities are available at a 5 minute walk for \u20ac9-\u20ac12 a day. When Andr\u00e9 Rieu performs on the Vrijthof the price of the rooms is \u20ac100 per night.\nWhen would you like to stay at Huize Tilly?\nThis nicely decorated room is almost identical to room 1, but has no refrigerator. The room is slightly bigger and there are facilities for making coffee or tea. Free wifi is available and a map of the city. The toilet has to be shared with room 1.\nThe room is not big, but nicely decorated with a double bed, private bathroom with hair dryer and a refrigerator. There are facilities for making coffee or tea. Free wifi is available and a map of the city. The toilet has to be shared with room 2.\nHartelijke ontvangst. Verrassende ligging op dakstraat in centrum. Echt... 50 meter van het Vrijthof waar je overigens qua geluid geen hinder van hebt. Bereikbaar met lift. Kamer op begane grond woning. Kleine maar complete en schone kamer met priv\u00e9 doucheruimte in de kamer. Geen ontbijt maar letterlijk om de hoek zit \"Vers\" met een super ontbijtje.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sometimes a distorted view of God Himself can become an idol, if we replace the true and living God with a god more in keeping with our own imaginations.\nThe first commandment is the foundation for our keeping of the whole of the Ten Commandments. It is the most basic reorientation from a life directed toward ourselves to a life directed toward God.\nIn this Seven Minute Seminary, Craig Keener reminds us that salvation, even in the Old Testament, was always by grace through faith.\nIn this Seven Minute Seminary, Craig Keener helps explain the context of Israel's laws by pointing out that many of the laws were admittedly meant to limit sin, not abolish it altogether.\nWhat is discipleship, and what does it look like, according to the Bible? Read these seven points on discipleship from Andrew Dragos.\nMarriage should not be characterized by secrets hidden in the shadow, separation, image maintenance, or the overestimation of one's own capacities at the expense of the underestimation of those of your partner.\nWhat does the Bible teach about justice? See this quick reference guide for an overview of God's character and his expectations for the church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamqeb b/data_all_eng_slimpj/shuffled/split2/finalzzzamqeb
new file mode 100644
index 0000000000000000000000000000000000000000..68d90bd761d9289ba80c47e4aaa7d348ac64abff
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Under the influence of the tide, the landscape of Saeftinghe is constantly changing as areas erode and build up. Overall, about 30% of the area is composed of a mosaic of sand banks, mud banks and creeks, with the remaining 70% being vegetated with salt marsh plants and grasses. In some areas, century old peat also appears at the surface.\nDuring heavy storms, ferocious waves pound against the steep edges of salt marsh, eating away at the banks, and due to the turbulence of fast flowing storm tides, the silt and sand in the creeks is shifted around. Following severe storms, once the tide has receded, you will see that whole creeks have moved and a wild pattern of wave ripples have been left in the sand. Some steep edges of salt marsh and sand bars will have partially (or completely) gone, and little streams that only a few days earlier were passable on foot, might have been transformed into massive creeks or high sand bars.\nAny eroded sand and silt taken up by the tide has to settle again somewhere else. This occurs most readily in areas where the water flows more slowly - predominantly on the inside bend of creeks and in dead-end gullies deep in the salt marsh. The suspended sand particles are much bigger and heavier than silt particles, and therefore settle much quicker in slow flowing water than the suspended silt particles, which require still water in order to settle. This is why you will often find that the edges of creeks are dominated by sandy particles, whereas more clay and silt can be found in the middle of vegetated salt marsh.\nAs time passes, sandbars can turn into salt marsh composed of levees and marsh flats. The creeks are bordered by high sided, sandy banks: levees, which act like small barriers that restrict water flow into the salt marsh. These levees enclose the lower lying marsh flats \u2013 silt\/clay areas of vegetated salt marsh.\nInitially, water floods over the creek banks with every high tide. Each time this happens, a little more sand settles on the levees, and the silt particles precipitate when the water comes to a virtual standstill over the lower marsh flats.\nAs the levees and marsh flats get higher through the sedimentation process, eventually it might become sufficiently dry for salt marsh plants to start colonising the ground. If this happens, the sedimentation process increases considerably; the growing plants slow down the water flow, making deposition of sediment easier. This results in the salt marsh getting higher and higher, and even more plant species establishing themselves.\nAt the entrance to the larger creeks Ijskelder, Hondegat and the smaller Speelmansgat, are massive sand bars. A striking feature of these sand bars are the megaripples: wave ripples in the sand that are created by strong currents and can measure a metre high and several metres wide. The most impressive sand ripple landscape in Saeftinghe is found northeast of the radar tower. Smaller sand ripples can be seen throughout Saeftinghe \u2013 the result of changing water currents and the wind.\nAt several locations, peat banks appear where the sand layer has been eroded on the banks of the Western Scheldt. These peat banks were formed at an earlier time, when Zeeland was completely overgrown. At low tide these peat banks appear above the water as steep black cliffs. Peat can be found under a large part of Saeftinghe, up to several meters thick in some areas, but is only visible around the beaches. From the peat we can read in to Saeftinghe's history: many thousands of years of seeds, roots, and even whole trees can be found.\nEven older than the peat is a sand layer that can be found only at extremely low tides in the mouth of the Ijskelder, underneath the peat. This sand layer is called 'dekzand', and is the remnants of an old dune system that dates back to the last Ice Age. It runs directly under the whole area and emerges again at the surface close to Hulst.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A comprehensive reference to all international agreements in environment and development, this yearbook details the objectives, status, rules, funding and implementation of each instrument. Essays by leading authorities assess the achievements and shortcomings of international co-operation. International governmental organizations (IGOs), non-governmental organizations (NGOs) and participating countries are profiled.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"> > short correction section for this, others would need to add the other sections.\nwith no relation to Alpha whatsoever?  I've assumed SRM in this diff.\n--- ch-rescue-boot.en.html.orig\tMon Nov 29 19:12:55 1999 +++ ch-rescue-boot.en.html\tMon Nov 29 19:30:43 1999 @@ -38,8 +38,14 @@ problems.\n@@ -51,8 +57,7 @@ However, in some cases you'll have to help the kernel a bit.\n-If you are booting from the Rescue Floppy or from CD-ROM you will be presented with the boot prompt, -boot:. Details about how to use boot parameters with the +Details about how to use boot parameters with the Rescue Floppy can be found in Booting With the Rescue Floppy, Section 6.2. If you are booting from an existing operating system, you'll have to use other means to set boot parameters. @@ -87,12 +92,105 @@ Troubleshooting the Boot Process, Section 6.5.\n@@ -159,6 +302,12 @@ presented with the boot: prompt. Here you can enter your boot parameters, and you can select your kernel image.\nPrev by Date: [Stefan Reinauer <stepan@suse.de>] Milo-2.2-12 ready.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"War of the Rebellion: Serial 030 Page 0264 KY.,MID. AND E.TENN.,N.ALA.,AND SW.VA. Chapter XXXII.\nform your line with two divisions only, holding the third in reserve. Let your men get their meals; supply themselves well with water. Throw back a line of skirmishers on your left flank for some distance down the creek; reconnoiter carefully, and ascertain if there are good fords. The line of Stone's River down to the Jefferson Bridge must be observed well with cavalry; the ford at Sulphur Spring occupied by a regiment of infantry, and, if easy of access, with two pieces of artillery, they to come from reserve division. Try and place your lines so that the enemy cannot slip out on the other side of Stone's River, and enfilade them. Post a strong line of sharpshooters on the bank of the river, supported by a brigade, probably, along the line of the railroad. General Negley has been ordered to take post on your right. General McCook has been ordered to close in on the Wilkinson pike. Thomas has been ordered to send a brigade of Rousseau's division to Jefferson, to reconnoiter as far as Dr. Black's shop.\nOccupy Murfreesborough, if you can, with one division. Encamp main body of troops on this side, as before directed.\nHEADQUARTERS LEFT WING, December 29, 1862-5.25 p.m.\nCOLONEL: Your dispatch is just received, in which you say occupy Murfreesborough to-night with one division, &c., as before directed. No such order has been received before. The order was given as you directed, the troops were advancing, but just at this point General Palmer and General Wood have ridden up and protest against it as very hazardous to move troops in the night, unacquainted with the ground, against troops in position. A good citizen, who is just now here, says if we were not opposed by the enemy, the crossing of Stone's River is so difficult we should have trouble in crossing. Under these circumstances, believing, if you were here, you would not order and advance, and as it will not get any darker, and I can communicate with you in an hour, I have concluded to suspend your order until I can again hear from you. If ordered to move, I will instantly execute it, but consider it impossible to take the artillery, and suggest that it should be left.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Those seeking freedom from addiction are increasingly turning to Buddhist-based recovery groups for help. A recent gathering outlined steps forward for the Buddhist recovery movement.\nLeaders of Buddhist-based addiction approaches gathered in Washington state October 20\u201322 for the Buddhist Recovery Summit 2017. Their gathering was timely: addicts seeking freedom from drugs and alcohol are increasingly turning to Buddhist-based recovery groups for help, and the rising wave of opioid addiction in the United States and Canada is causing increasing demand for new responses beyond the usual milieu of Alcoholics Anonymous (AA) recovery groups.\nAt the event, organized by the Northwest Dharma Association, more than 100 people from around the world met at a Christian retreat center near Washington's capital city of Olympia. They shared their progress in applying the Buddha's teachings to addiction recovery and planned to make more Buddhist recovery groups available to more people.\nMost of those attending were Buddhist, and many were meeting in person for the first time, often after years of collaborating online.\n\"It was very fruitful, primarily due to the fact that we have not had a meeting like that since 2009, since we really got the Buddhist recovery community together in a meaningful way,\" said George Johns, president of the Buddhist Recovery Network (BRN), one of the organizations at the summit.\nBuddhist Recovery Network has emerged as a go-to place for people seeking a Buddhist approach to fighting addiction. The umbrella organization now includes more than 250 recovery groups of many types on its website, less than a decade after its founding, and the number of groups continues to climb. The intent of the network's website is to offer all approaches to people without bias, to help them find meetings where they live.\n\"There is an audience out there that is looking for effective recovery techniques, and BRN is speaking to that need,\" Johns said.\nThe complex relationship between the fast-growing Buddhist recovery approaches\u2014 including Noah Levine's Refuge Recovery program, Vimalasara's Eight Step Recovery method, and Vince Cullen's Hungry Ghost recovery retreats\u2014and the 84-year-old AA movement was a central topic that emerged throughout the summit. While the Buddha's teachings are expressed somewhat differently in the various groups' programs, they are all without the theistic overtones found in traditional 12-step addiction recovery work, something that is attractive to many non-theistic people seeking relief from addiction.\nMany at the summit have been AA participants for years and credited AA with their own recovery; it was often repeated during the event that nobody has to choose between AA and the Buddhist groups.\nStill, the rising secularization of U.S. society has attracted some addicts away from the 12-step approach to Buddhist alternatives, even when they aren't Buddhist themselves, conference leaders suggested.\nLindsay Shea, a chemical dependency professional from Seattle, said she's increasingly referring non-Buddhist people to Buddhist recovery groups.\nThe heart of the Buddhist Recovery Summit was October 21, a daylong marathon that started with a 7 a.m. silent meditation. The day included two panels and two small group discussions, while a drenching rain continued outside in true Northwest fashion.\nOn Saturday morning summit-goers were able to choose among four different styles of recovery meetings: Refuge Recovery, Sit and Share, Noble Steps, and Eight Steps. It was an opportunity to try new approaches to witnessing their addictions and supporting each other in recovery.\nAnd while the event featured the big names in the Buddhist recovery movement, those people seemed to downplay their own importance. Levine was probably the highest-profile person there, whose father is the late Stephen Levine, also a well-known Buddhist teacher and author. But Levine also seemed to be pushing away personal attention.\nJohns said he emerged from the summit re-invigorated about the importance of the work Buddhists are doing to help people free themselves from all forms of addiction.\nSteve Wilhelm is a retired journalist and longtime meditation practitioner. He is the editor of Northwest Dharma News and a previous board member of Northwest Dharma Association.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzamwva b/data_all_eng_slimpj/shuffled/split2/finalzzzamwva
new file mode 100644
index 0000000000000000000000000000000000000000..2a5ce624eed9a18ad9723c6d1e2e029a560d825a
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzamwva
@@ -0,0 +1,5 @@
+{"text":"You need a qualified professional consultant or new hire now. You need a strong team member with deep knowledge of your domain and strong collaboration skills.\nWhen you have a complex technical environment, finding the right partner can be especially challenging.\nCopyright \u00a9 2016-2017 eDivya Business Solutions Inc All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is a preferred-- and also rather scary-- figure that everybody in business globe has actually heard that \"90% of the start-ups fail\". If you take a look at these stopped working companies which as soon as began with much vigor and excitement, they seldom fail since the idea or the idea of business itself was bad.\nWhat does Clickfunnels Subscription Coupon January 2019 do?\nClick Funnels supply you with a pre-designed and also well-curated sales channel which will take your site visitors via a persuading journey that will certainly make them acquire your item at the end. This last objective can be different depending upon the item that you market or the service that you supply. Nonetheless, Click Funnels have created a variety of sales funnels that can lead you to achieve your objectives by taking your visitors via a well assumed out sales funnels. Adhering to are several of the objectives which can be met by a sales funnels.\nA sales channel favorably impacts the thought process of your visitors to earn the decision to move forward with your purpose. This is essentially considering that little press a salesman would certainly give up a physical shop to finally make up the way of thinking to go ahead with a purchase.\nJust How Do Clickfunnels Subscription Coupon January 2019 help enhance your earnings?\nAn internet site is similar to a virtual store, as well as the objective of putting a big financial investment into developing a website is to eventually help boost the earnings of your business. Lots of people, especially the start-ups and the business owners who are brand-new to the online organisation world emphasis way too much on the looks of the layout to think about whether it is efficient sufficient to actually make a sale.\nYou could invest a lot of cash to employ the finest web designers as well as developers, and they may also provide a fantastic looking website to you, yet if you need to assume concerning the sales process within the site, your investment will just cost you money without a return. This is why you should acquire Click Funnels. A web site produced with that solution is laser-focused to deliver fantastic advertising as well as sales impacts from the beginning throughout. Adhering to are the 4 major things that a well chose and put sales funnels do.\nDrawing in new visitors to the web site is the very first and one of the most essential jobs that a sales funnel does. This is the mouth of the funnel. This stage is narrower and in the following step of the channel and also must be well preserved considering that they are most likely to make a purchase in the next couple of phases as long as you maintain them impressed.\nClosing is the last stage of a channel where a lead comes to be a customer. They proactively make the order as well as acquire your item, register for your newsletter or basically do exactly what you planned them to do by developing the channel. Next degree is consumer retention or keeping them pleased with your product or the purchase, so they come to be return consumers.\nClickfunnels Subscription Coupon January 2019 Prices: Is it really worth it?\nJust what is a Sales Funnel Clickfunnels Subscription Coupon January 2019?\nIf you are an e-commerce organisation owner, getting an excellent knowledge of sales funnels and transforming a lead to a client is one of the most basic things you should recognize from the start. To place it just, a sales channel is the process that a prospective buyer goes via from being a visitor\/lead to lastly acquiring the item or the solution.\nIn the beginning, Click Funnels Pricing may make you assume if it deserves it, yet you will certainly require it making a fast development as well as awareness amongst the consumers especially in the first couple of stages of running a business. This service specifically helps the start-ups and the entrepreneurs who are not extremely accustomed to the marketing and sales procedure to obtain the ideal out of their advertising and marketing approaches by converting site visitors efficiently to be purchasers.\nWhen you pay a regular internet designer to develop a web site for you, the final result could be visually pleasing, yet they typically do not pay interest to making a sale with the system. Click Funnels has actually put with each other a number of internet site templates that are composed of a complete sales funnels from the welcome display to the final Thank You display.\nClick Funnels have actually been produced by sales and marketer with numerous years of experience in actually making sales. They operate under the slogan, \"Take the power back from the technology men\", given that during the previous years or two the advancement of sites has been fully dominated by programmers and designers without an appropriate knowledge of making a sale. Click Funnels ensure that the loan which you spend on the design and development of the website is not fruitless. This is why Click Funnels Pricing is even much more essential given that you are not just paying for a mere site but a fully functioning on the internet shop that assists you transform every single consumer into a sale. Complying with are some of the pre-designed sales funnels that you could select at the Click Funnels.\nYou can pick these funnels according to the industry you are operating in, the type of services that you use and exactly what type of a sales process that you want your possible customers to go through. There are examples of all these funnels that you can have an action by action consider and also experience the very same channeling procedure that your site visitors will certainly experience as soon as you utilize that funnel.\nAll the layouts and also sales funnels that you get in the Click Funnels solution are quite personalized. You can use your own shade styles, images as well as texts taking into consideration the tips that Click Funnels offer to obtain the most effective outcomes. The modifying as well as modifying procedure is a straightforward drag and also decline approach making any person, also with no web design experience, to create the type of site that you always dreamed to have for your business.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you like stylish things? Allow your adorable dog take joy in wearing such accessories on! Present him this fashionable leather dog collar with decorations from FDT Artisan. This gear is created for those who just adore elegance and style. In this exquisite collar, your Great Dane will have more stylish, but handsome look. Your adorable doggie will get the finest treatment when wearing this accessory. Full grain natural leather is gentle and totally safe.\nBrass plated hardware will hold out even the hardest dog force. Stunning decorations are brass plated and have impressive shining. Point up your Great Dane's style, purchase this FDT Artisan incredible full grain leather collar!\nFull grain genuine leather was chosen to offer total comfort and safety while wearing. The material is carefully oiled that saves it from cracking. Among the best features of full grain natural leather are good adjustability, tear-resistance and hardness. The layer of leather is thick that prevents the tool damage. It helps you to walk even big and tough dog safely. Rounded and smooth edges of the gear are non-cutting and don't irritate the four-legged friend's skin. Besides, full grain leather doesn't consist of any toxic elements being totally eco-friendly.\nSpeaking about the studs, they are exquisite! Catchy studs are set along the leather strap. These decorative elements are brass plated and, therefore shine with stylish glittering. The elegant design of this tool is unmatched and unique. Due to these studs, this full grain natural leather dog collar has a spark. That means your Great Dane will have his individual and exquisite style .\nAs for the hardware of this embellished leather accessory, it is brass plated. It matches amazingly with the decorations. This covering provides the best protection for the details as it prevents corrosion. The set of hardware includes a classic elegant buckle and strong D-ring for attaching a leash. These fittings are easy to use and reliable in service they provide. Add more zest to your dog's look, order this fancy leather collar!\nWhen ordering this studded dog collar, you will get a gift package along with it! Don't miss your chance to praise your attractive four-legged friend with perfect fit collar made of leather. This is an outstanding offer available for our customers! A gift package makes your order look like a present for the best friend.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Osteonecrosis of the jaw is a disease of bone where symptomatic exposure of nonhealing areas over jaws are seen. We report our experience in the management of a patient with history of trauma to nose, osteoarthritis, diabetes mellitus & hypertension, who presented with osteonecrosis of the jaw and was under medications but was unaware of the name of the medications. The patient was found to be of an average built, had normal gait with no physical handicap and was alert, conscious and cooperative and responsive to verbal commands with vital parameters within normal range. Intraoral clinical examination revealed poor oral hygiene with bony exposure & necrotic appearance of the maxillary alveolus in the edentulous 11, 12, 13, 14, 15 & 21, 22, 23, 24, 25 region. To confirm the diagnosis, patient was subjected to radiographic, laboratory and histopathological investigations and was advised to take Clindamycin capsules, supplementation of Antioxidant, Calcium capsules and topical application of Metronidazole gel & oral rinse with Povidone iodine 2% Gargles. Based on this diagnosis, patient underwent surgical debridement and finally resection of bony segment of maxillary alveolus extending from 15 to 25 region in the Department of Oral & Maxillofacial Surgery. After the surgery, patient had an acceptable maxillary alveolus and was put on periodic recall for six months. But there was recurrence after 19th months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sydney is not only one of my favorite cities in the world (I stayed and worked there for 6 months a long time ago) but aThemes has created an attractive and beautiful WordPress theme named Sydney. Sydney is a flexible and modern theme with plenty of customization possibilities. Meaning you can create a wonderful website with this theme and a great online presence.\nI really like aThemes and Charlie Livingston as a designer. He is a creative designer and has created numerous popular free themes such as Moesia, Alizee and Intro. And now his latest theme Sydney, which surely will become a popular theme many people will use on their WordPress sites.\nSydney is a perfect business theme with features such as theme options located in the theme customizer. There you can easily customize your front page, blog layout, change the colors as you wish and much more. Sydney comes with fully responsive and adaptable design. Meaning the theme is mobile-friendly and work great on all devices. It doesn't matter if your visitors and readers using a mobile phone, tablets or computer. You have everything covered.\nSydney is a feature-rich WordPress theme with lots of features normally only found in premium themes. Present your featured content with full-width stylish images. Choose between fullscreen slider or a full-width static header image. The theme comes with support for Parallax backgrounds. You can add Parallax background images for every section and watch it come to life as you scroll.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzannux b/data_all_eng_slimpj/shuffled/split2/finalzzzannux
new file mode 100644
index 0000000000000000000000000000000000000000..9e3e03d29b3fe246ed868b3cdfc581a00529ae20
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzannux
@@ -0,0 +1,5 @@
+{"text":"Organically grown herbs and spices are essential for the purity and unique flavours of Ayurvedic tea recipes from YOGI TEA\u00ae. For that reason, YOGI TEA\u00ae uses carefully selected ingredients from verified organic farming.\nAs a pioneer of Ayurvedic herbal and spice teas, YOGI TEA\u00ae has been committed to organic farming methods since the 80s. At a time when there were no organic certificates and sustainably sourced products were a rarity, we travelled to the most remote corners of the world to find the best organic ingredients for our unique blend of teas.\nOrganic farming helps preserve the Earth's beauty, fruitfulness and vitality by doing without synthetic pesticides, mineral fertilisers or genetic engineering. This natural and sustainable approach to farming protects the soil, water and air, preserving the earth's unique diversity of species. Closed-loop recycling with nutrient cycles that are as self-contained as possible conserves raw materials and reduces energy consumption.\nAll YOGI TEA\u00ae products feature the EU organic logo and represent high-quality herbal and spice teas produced, using strictly verified organic farming, treatment and storage methods. The suppliers of our organic ingredients are also certified organic companies and are accredited by the official inspection bodies such as IMO. As well as the EU organic logo, YOGI TEA\u00ae herbal and spice teas feature various national organic logos.\nOrganic products naturally vary in taste. In order to guarantee the consistently high-quality and unique taste of the herbs and spices used by YOGI TEA\u00ae, our carefully selected Ayurvedic tea blends are regularly inspected for their taste, smell and purity, and meet HACCP food hygiene regulations.\nAt the YOGI TEA\u00ae production locations and offices, we only use electricity from renewable sources to minimise our carbon footprint. We aim to further reduce our footprint by transporting YOGI TEA\u00ae products between our factory and our warehouse by train. Our offices in Hamburg are housed in a listed office building that was renovated in 2012 to become energy-efficient and has been awarded Green Building certification by the European Commission.\nYOGI TEA\u00ae also promotes the vitality of its staff with various offers. The private involvement of employees in local charitable projects has traditionally been supported through the provision of free work space and donations to the initiatives in question. YOGI TEA\u00ae has been supporting its employees' use of public transport for many years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2 weeks in Europe-a breakdown. Week one: Paris.\nOh, Europe. Two weeks home and it already feels like a dream. This vacation was years in the making since we had always wanted to do a European trip with the kids, but life gets in the way of even the most well-intentioned plans and it's hard to make a commitment to something that costs so much. But sometimes you have to throw caution to the wind and seize your moments because life is so much more than work and responsibilities. And so while I knew we would do this trip eventually and had been saving for it, booking the flights last December was a rather impulsive decision, fueled by the excruciatingly busy Fall I had last year working nearly non-stop for 5 straight months. In life, there has to be rewards after periods of intense work.\nI've shared a lot of photos and thoughts about our trip on Instagram, so I won't rehash them here, but I will say again that a trip like this taught us to let go of some of the things that I felt have defined us over the years; we're very frugal and responsible people. Self employment and owning a business will do that to you, but I don't miss a single penny spent on this trip and I would do it over again and again (I'm hoping to visit another European city in the next year or two with the family). We've always valued experiences over buying things and we've taught that to our girls, mostly by example. This trip couldn't have proven this point better. They took in so much and came away with an experience that they'll always remember (I know my first European trip made a huge impression on me when I was in high school). I'm aware international travel isn't possible for everyone, but if you can do it, it's really valuable to step outside your bubble and gain perspective on the rest of the world.\nWe did one very big splurge on our trip which was a food tour in Italy (I'll write about that on a separate post). It was not cheap, but I am so glad we invested in that experience because it ended up being one of the highlights of our trip.\nFor accommodations, we stayed in Airbnbs in Paris and Bologna, a hotel in Milan, and used credit card points for our hotel in Florence. We LOVED our hotel in Florence, by the way. Globus Hotel had the nicest and most helpful staff, and their breakfast and daily happy hour were well appointed and so appreciated.\nChoosing hotels and apartments near train stations in central locations enabled us to walk everywhere (I think we only took the Paris metro 4 times). Yes, we were all super tired from all that walking at the end of every day, but apparently we like to pack a lot in and it's the best way to see the each city.\nBreizh \u2013 literally everyone told us to go here for traditional crepes \u2013 and yes, do it.\nYann Couvrear \u2013 beautiful pastries.\nL'As du Fallafel \u2013 multiple people told us it was the best falafel they every had. It is excellent. Was it the best I ever had though? Not sure.\nLe Quartier Rouge \u2013 we met our friend Amy, who lives in Paris now, at her local neighborhood restaurant. We sat outside and it was just a fun meal with delicious, classic, bistro food.\nSo as you can see, not a whole lot of places (like, what did we eat when we were there?) We arrived armed with recommendations and we made the best attempts to visit some of them (a few were closed. It was August, after all), but we mostly would just stop in at the many cafes and patisseries all over the city whenever we needed a break (don't forget Berthillon ice cream in Ile de la Cite \/ Ile Saint-Louis! Our favorite were the sorbets actually \u2013 grapefruit and cassis).\nWe didn't do a whole lot of sight seeing, but we did visit the Louvre, Musee d'Orsay and the Pompidou Centre. I bought tickets in advance for all but the Pompidou, which I'd recommend doing to avoid any lines. We felt very fortunate that we were in Paris while the immersive Klimt exhibit was open. It is such a beautiful exhibition and an experience like no other so go if you are in Paris in the next few months. It's open until November.\nOur favorite memory of Paris, however, had to be the festive atmosphere on the banks of the Seine. Paris doesn't get fully dark until late \u2013 about 10pm \u2013 so there are quite a few hours in the evenings when the city gathers at the banks and on its many bridges to watch the sun slowly go down. It's an amazing sight to see and witness, and an absolute must if you want to join in on Paris culture. You can find outdoor bars bustling with people, and you may even find an impromptu dance party like we did on our very last night. We were told, however, by our Airbnb host that they will be allowing cars again next year, so we felt really lucky to experience this nightly celebration of summer and Parisian life. You couldn't have scripted a more charming and more Parisian send-off than that!\nAnd they just find great fares on google flights. You can sign up for the free sales tips or subscribe to their premium membership. I paid for the membership and have been getting $300-400 sales to Europe almost ever week, besides cheaper deals to other places. You can select the destinations you want receive deals for. I will never buy another air ticket without using their service! Really recommend it!\nThanks for that tip Lia. I'll certainly check it out when I plan our next trip!\nI've so enjoyed following along on your trip on IG, and appreciate your transparency about the budget!\nGlad to hear! I was kind of hesitant to say \u2013 I mean, it's a lot of money and we don't take that lightly at all! But it was immensely helpful to me when I saw other budget breakdowns from tavel bloggers.\nHow dreamy! Thank you for sharing your budget and itinerary. I find it impressive that with a family of 4 you all managed to score meals for $30\/day in Paris. Hats off to you!\nHi Cindy. Our daily food budget was actually 120 euros which was roughly about $140 a day for the 4 of us. I think $30 a day would be quite the feat!\nSo glad to hear you enjoyed it. I hope your airbnb had air co, as you must have been there pretty much at the worst time possibly \u2026 Paris is awesome in summer except when the temperature remains above 32\u00b0 by day\/20 by night.\nI wasn't aware that the right banks would be reopen for trafic next year \u2013 I thought it was still under discussion- but the left banks will remain pedestrian and are very nice too \u2013 with a different atmosphere.\nSo happy for all of you. Sounds like a lovely trip, and I imagine the girls would have loved it too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yes, I must report that the time has come for the kids to leave the Farm. It was a very sad day for us all. We will miss little Harry and I especially miss my Kevin. I knew this day was coming as I knew my boy was growing big and strong and he needed to go out on his own to his new new Farm but it doesn't make it any easier. I am sure all of you human parents understand how I feel.\nHis new Farm is very nice; it is much bigger than our Farm so he has lots of land for lazy grazing. He will have a little friend there too as there is already a Nigerian dwarf doe there. He will also have children to play with him so I know he will have lots of fun days ahead of him.\nIt is good for him to grow and move on but I want to reminisce about him here in photos. I hope you don't mind.\nHe was so cute when he was little.\nAnd then he had his hopping stage...which he never really grew out of.\nHe would rest every now and then but in very strange places.\nHe was so funny when he was learning to walk on a lead.\nAnd lately he continued to make us laugh by making the publicist into a goat spool!\nI know we will all miss him but know as I do that he IS happy on his new Farm. So bye - bye my little kid. Grow big and strong!\nAbby will let you know how she feels about Harry going away on AbbyDay on Friday. It is good to know they are together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Resource Center can assist self-represented litigants with their family law cases such as divorces, legal separations, and paternity cases. Although we can explain court procedures and review forms, we cannot provide legal advice, and do not have the ability to complete forms for you.\nOur Resource Center can assist self-represented litigants with their family law cases such as divorces, legal separations, and paternity cases. Although we can explain court procedures and review forms, we cannot provide legal advice, and do not have the ability to complete forms for you. You can visit our Resource Center in person, or contact by phone or email for assistance. For general information on family law basics, you can click on each topic below.\nThere are 2 main ways to end a marriage or registered domestic partnership in California: divorce (dissolution) or annulment. Note: Legal Separation does not end a marriage.\nIf you and the other parent have never been married, you may need to file a Petition to Establish Parental Relationship in order to request child custody, visitation and\/or child support. This type of case is used to establish parentage between minor child(ren) and their biological father and establish child custody and visitation orders.\nIf you are married and do not want to file for divorce or legal separation, you can file a Petition for Custody and Support to ask the judge to make orders regarding the issues of custody, visitation and child support.\nThrough the Request for Order you can request court orders, such as child custody, visitation, child support, spousal support, attorney fees and other issues. You need an open case to file a Request for Order, or you can file a new case at the same time.\nIf you have been served with a family law Petition or Request for Order, you may file a response to the Petition and\/or a Responsive Declaration to the Request for Order. A response lets the Court know if there is something that you do not agree with on the Petition or Request for Order.\nWhen you are ready to finish your case, you will need to prepare a judgment. A judgment can be finalized in different ways. It may be by mutual agreement, by default, or decided in court.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the untapped patents that are within their possession.\nPatentech is an online and offline forum and leading global platform which provides patent owners and potential buyers a wonderful opportunity to access untapped patents for the greatest potential to capitalize to its fullest and ultimately generate financial resources for both the patent owner and the potential buyer. By linking the two together we are able to boost and generate business between the patent owner and the potential buyer, who may range from individuals, hi-tech companies or multi-million dollar corporations.\nPatentech is focused on providing assistance to patent owners all over the world from the first step and until the very last step thus capitalizing on those untapped patents. We appraise and evaluate the patent in question, locate and trace potential buyers and assist in a swift negotiation and sale, which otherwise may seem unfeasible. We manage every step of the sale from presentation to negotiation. We are committed to serve our clients and afford them with the best service possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzariyr b/data_all_eng_slimpj/shuffled/split2/finalzzzariyr
new file mode 100644
index 0000000000000000000000000000000000000000..0596a4e550cc92a21acd453edf7df01718169a71
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Sports Article: What's Behind the Ability to Focus by Karen Goeller.\nA must-read for all athletes and coaches! This article discusses what athletes must do when not in the gym in order to be able to focus during training and competition. It discusses dehydration, nutrition, and sleep in relation to an athlete's ability to focus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Administration program provides policy development and leadership for all programs within the department. Staff specialists are responsible for a full range of administrative, financial, and budgetary tasks, including daily operations, revenue collection (fees and development taxes and charges), reporting and management, automation, human resources, fleet management, training, safety, quality assurance, legislative coordination, space management, historic files maintenance, and management services. This program provides outreach, customer service satisfaction and case management, which coordinates DPS disciplines engaged in plan reviews on complex projects or projects needing a higher level of assistance such as \"green tape\" projects (i.e., affordable housing; and areas such as the Silver Spring, Wheaton, and Long Branch enterprise zones; strategic economic development projects; strategic redevelopment areas such as White Flint, and faith-based institutions). This program receives complaints, processes information requests, maintains the DPS website, publishes the DPS newsletter, and coordinates outreach events and seminars for residents, civic organizations, and professionals.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"What an amazing few days with very special friends. Wall to wall sunshine, fantastic walks on the beach and back to cosy fires. Wonderful, welcoming home. Thank you!\n\"We spent a very enjoyable Christmas here, appreciating the thoughtful and extensive decorations and treats provided. Very comfortable & cosy. Loved the d\u00e9cor & art (loved the Liz Beckenham Vase in the bathroom!). We also enjoyed a good meal at the East End Arms.\n\"We had an amazing weekend for Molly's birthday weekend. The house has absolutely everything. We will be back soon. Lovely family place to stay. Can't wait to stay again.\n\"Our third visit to this beauticul cottage. Weather has been fabulous. 5 grandchildren really enjoyed beach, fishing and canoeing. Great cycling routes. Donkeys in the lanemuch enjoyed by grandchildren.\n\"Location, Location, Location! Just incredible to be in such a great position. Such a lovely Cottage, A happy time spent exploring, eating (Great Brill at The Haven) and enjoying fantastic autumn sunshine in celebration of a big 40. We will be back! Thank you very much!\n\u2014 Simon & Family plus the dog!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In every MATCH DAY, you stand a chance to win two FREE tickets to the match right from the Ola app!\nWant a chance to win exclusive hospitality tickets? Follow @olacabs of Twitter and Facebook to participate in our social contests and WIN BIG!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We invited a houseful to our Thanksgiving dinner and post-dinner dessert, so we took a few minutes to make sure the table and the house looked as nice as it could. Anna's curtains finally got put up with a little creative use of curtain rods and household decorations\u2026she really knew what we didn't \u2013the curtains look perfect, and she's on her way to becoming an interior designer! Thanks again to those who made it a princess dream room!\nThe women are meeting inside and the men are out listening to Tony, the new president of the EFC Costa Rica here. He is talking about how these pastors can make their churches more men-compatible.\nEven though it has rained all night for the last several days and a fine mist (pelo de gato) of drizzle continues, we made the most of yesterday afternoon by walking the dog near a local Christmas tree farm. Last year, the gringos from N.C. did not know that one reserved a Christmas tree well in advance of our Thanksgiving. This year upon seeing car after car with arboles de navidad (Christmas trees) on top of their car roofs, we were prompted to act before all of the best trees were gone and before the only ones remaining were out of our price range. This year we have a much taller tree waiting with our name on it for a smaller price than last year\u2014it is so wonderful to know the culture, rudimentary, though not advanced Spanish and the locals. Please join us in praying for our neighbors who will be invited to our annual Christmas Day open house\u2014that they will come from behind their fortressed walls and razor wire, experience the love of Christ in simple hospitality and that Christ will begin to remove the walls in their hearts this Christmas season! Happy Thanksgiving!\nIt's cool, it's overcast, and it's rainy, but Rio is wonderful. These are the men from our Rio de Janeiro team. Sorry wives, only one of you was here and we needed someone to take the picture!\nIt's amazing what a good night's sleep will do to clear the mind. Even with some very involved personnel meetings and group meetings the day was invigorating. Later, after a short shopping stop (can't tell what\u2026it's almost Christmas after all), I found myself sipping coconut milk from a chilled coconut standing on the beach and watching some amazing surfing waves crash down. The weather is cool for this time of year, maybe high sixties with a stiff breeze and everything is verdant green with incredible scenery and views. I forgot to bring the camera along to the beach, so I'll try to grab some shots over the next couple of days and post them here.\nTwenty-four hours of travel with no sleep got me to Brazil this morning and shortly thereafter, a nice nap! It's Spring here, the birds are chirping and the next door neighbor has been practicing on his drums for the last twelve hours (today is a holiday). Rio de Janeiro is a huge city of over ten million people, gorgeous views and lots of life. As I meet with our missionaries and Brazilian partners please pray that the time will be productive and encouraging. Great things are afoot!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzatzxx b/data_all_eng_slimpj/shuffled/split2/finalzzzatzxx
new file mode 100644
index 0000000000000000000000000000000000000000..a35bc006069b3281a43b638829c9e56f1048de64
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@@ -0,0 +1,5 @@
+{"text":"Worship at The Gathering on August 19th, 2018.\nWorship at The Gathering on August 12th, 2018.\nWorship at The Gathering on July 22, 2018.\nJim Burns kicks off Family Week 2018 with his first session on Generation to Generation.\nJune 24th worship service at The Gathering!\nMay 27th at The Gathering.\nJune 3rd at The Gathering.\nWorship service at The Gathering on May 13th, 2018.\nApril 29th at The Gathering!\nMay 6th worship service at The Gathering!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Experience New England's most scenic tandem jump!\nMake Your First Jump Here!\nWelcome to Skydive Newport - Skydiving New England since 1999!\nExperience New England's most scenic tandem skydive at Skydive Newport, the tandem skydiving specialists since 1999.\nIf you're looking to skydive New England, our family owned and operated skydiving center in Middletown, RI offers a personalized and professional skydiving experience you won't find anywhere else. We are proud to serve skydiving clients throughout New England, including Boston, Rhode Island, Connecticut, New Hampshire and Maine, We specializing exclusively in tandem skydives.\nUtilizing the most up to date training techniques and the highest quality skydiving equipment, safety is the highest priority for the staff at Skydive Newport. We guarantee a fun, professional day for all of our customers looking to experience the thrill of a tandem skydive while taking in the breathtaking views of the Newport coastline and its surroundings.\nSkydive Newport is the #1 choice for tandem skydiving Boston and New England!\nTandem skydiving allows everyone to experience the incredible thrill of skydiving without the need for intensive training. Using a specially crafted parachute designed for two people, you will be attached to a highly trained instructor who will accompany you through your skydive from training to landing! Ready to jump? Click the button below to learn more about tandem skydiving at Skydive Newport!\nUnlike many other skydiving centers, Skydive Newport focuses exclusively on tandem skydiving and strives to provide each of our guests with a personalized experience they will never forget. We are a family owned and operated business that believes in providing excellent customer service. At Skydive Newport, you are not just a number, you're family! Click the button below to learn more about us.\nSkydive Newport is located just minutes from historic Newport, RI, New England's number one tourist destination. We feel priveleged to operate in such a beautiful and vibrant community and encourage all of our guests to explore Newport and all it has to offer! Click the button below to learn more about some of our favorite local restaurants and attractions and start planning your Newport skydiving adventure!\nIf you're a first-timer, you might be daunted by the idea of making a tandem skydive. Don't be! Skydiving (especially here, in our stunning Rhode Island home) isn't just the thrill of a lifetime. It's a moment for posterity, and it's more powerful than you think.\nWhether you're looking for a place where you can make your first-ever jump or you're a seasoned skydiver who's looking for a dropzone to call home, spending time researching the skydiving center that's right for you is valuable exercise that can have tremendous pay offs .\nEach season, you and your family make the beach your vacation destination. You go to enjoy the sun and sand, the calming sound of water lapping at the shore, and to feel like your stress is being swept away receding with each wave. But, did you know it's also New England's prime destination to skydive? That's right, you can come to Newport to enjoy the vacation vibes and beachy views and get the thrill of a lifetime with a skydive too!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nineveh MP Mansour al-Mareed said that the parliament members of the province will not accept any form of attempts by someone to get the governor's post by paying money.\nCommenting on some media reports about alleged bargaining by some persons to get the post, Mareed said that the province will not allow any corrupt person whatever the party supporting him or the money he plans to pay, to take control of the province's capabilities.\nMareed said that in case \"any declaration to assign the position of the governor to a person who is corrupt or unable to understand the real situation of the province and improve it, we will be the first to combat this measure.\"\nThe Iraqi parliament on Sunday voted to sack the governor of Nineveh Province after the ferry sinking in the provincial capital Mosul that killed 96 people, the official television reported.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The third and final meeting for the project consortium was held in Greece \/Thessaloniki\/ between 13-17.10.2018 and the overall evaluation is excellent.\nSince the first two meetings had a role of precising the guidelines to implement the methodology and project activities based on the experience of the pre-season this meeting had the intention of combining the learning blocks.\nIn the work schedule there were discussions, presentations and panels for observing the results and the compiled methodology. The ongoing dissemination was analysed and strong\/weak points highlighted. There was an overview of the athletes and club season with regard to the performance in and out of the field, the innovative training tools and strategies used after the first two meetings and analysis on performance.\nIt was an important aspect in the TNM that the draft report of the combination of training practices was presented with a clear link to athlete development. We held a discussion panel on the importance to promote sport as a tool for learning, improving competences and overall development. The key steps initiated in this direction, in relation to the implementation of the project Teaming up for growing up, lead to the creation of a unified strategy for using sport as a tool in and outside the context of the club.\nThe administrative meetings we created and held helped us to clear and precise the final roles, the tools used and the methods implemented. Further cooperation between the clubs was discussed and goal setting for the remaining period of the project was done.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Ticker symbol search frequency relates positively to market capitalization, abnormal turnover, analyst coverage and news flow, but these factors explain only 5% of the variation in of search frequency among ticker symbols.\nThe change in ticker symbol search frequency relates significantly to abnormal return, abnormal turnover and news flow, but these factors explain only 3% of the variation in the change in search frequency among ticker symbols.\nTicker symbol search frequency actually leads extreme returns and news, consistent with a belief that investors anticipate news events.\nThere is a strong and direct link between changes in search frequency and trading by individual retail (naive?) investors.\nIPOs that exhibit relatively large increases in search frequency during the week prior to the IPO outperform IPOs that exhibit relatively small increases in search frequency by nearly 7% during the day after the IPO. These same stocks tend to exhibit long-run return reversals.\nSearch frequency-related price pressure emerges mostly for small market capitalization stocks among the Russell 3000. Small stocks exhibiting large increases in search frequency outperform those experiencing large decreases in search frequency by an average of about 0.08% per week during the next two weeks (about 4% annualized), before accounting for trading frictions.\nA momentum strategy applied to stocks with high search frequencies outperforms the same strategy applied to stocks with low search frequencies across all holding horizons. The outperformance over a 52-week holding period is almost 2.5%.\nIn summary, evidence suggests that Google search frequency data may disproportionately measure the attention of naive investors\/traders and therefore indicate some temporary price pressure for small capitalization stocks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaueyr b/data_all_eng_slimpj/shuffled/split2/finalzzzaueyr
new file mode 100644
index 0000000000000000000000000000000000000000..0745d9414285586597908011e3b39f3b387e5ed1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaueyr
@@ -0,0 +1,5 @@
+{"text":"from the committee during the nomination year.\n3 years. Attention will be made to balance coverage of technical areas.\nSIGMM officers should it occurs before the end of 3 years.\nareas, and technical areas. Each member serves a fixed term of 3 years.\nattend the chair-to-chair meeting during the conference.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Clean Energy Solutions Center and SEforALL have joined forces to provide support to Governments actively working to achieve SEforALL objectives with rapid, on-demand expert assistance.\nThrough the \"Ask an Expert\" service, the Solutions Center can now provide free advisory support to Governments working on Action Agendas and Investment Prospectuses to help improve their quality, through a network of multi-lingual policy and finance experts from all regions of the world.\nThese experts will work directly with Governments to address country-specific needs in relation to the development of SEforALL Action Agendas and Investment Prospectuses. Assistance interventions could include review of draft action agendas, investment prospectuses and related documents, recommendations for design of sustainable energy policies and regulations, and advice on finance instruments and strategies to attract private and public investments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The market's biggest digital coins dropped sharply on Wednesday amid a report that Goldman Sachs Group is pulling back on near-term plans to set up a cryptocurrency trading desk.\nBitcoin recovered from the lowest level since June and scores of smaller digital tokens rallied as concern eased that investors are giving up on the virtual alternatives to cash after this year's collapse in prices.\nBitcoin pushed above $7 500 on Monday as the largest cryptocurrency resumed an advance that has carried it to the highest in two months.\nBitcoin is headed for its biggest increase in two weeks on Monday amid a steady drip of news reports suggesting some of the biggest names in investing are starting to embrace digital currencies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I may be Jewish, but that doesn't stop me from loving everything having to do with Christmas! I love decorating my friends' Christmas trees, I love caroling, I create awesome iTunes and Spotify Christmas music playlists, I love looking at Christmas decorations ... and the 12 days of Christmas just amplifies my excitement! This year, I'm counting down the 12 days of Christmas with 12 very fabulous beauty giveaways from some of my very favorite beauty brands!\nAnd one lucky winner will receive the collection, with a retail value of $12! To enter, leave your name and email address in the box below.\nThis giveaway ends on Sunday, December 25 at midnight. One winner will be chosen AT RANDOM on Monday, December 26. One entry per person and email address. Multiple entries will be disqualified. Giveaway is open to US residents only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"the machine to crush itself. This autogenous crushing action offers the lowest possible cost per ton of any impact crushing method. The high velocity impact crushing achieved in a Barmac VSI improves the soundness and shape of the material and produces the highest quality end products on the market today.\nRenting portable crushing equipment from Smiths Gravel Pit has several distinct benefits most importantly, it makes it easier and more costeffective for you to recycle and reuse. When it comes to portable stone crushing in Rochester, NY, you can count on Smiths Gravel Pit for highquality, wellmaintained equipment that can be delivered to your job site.\nPortable Crushing Costs mayukhportfoliocoin. portable crushing costs Stone Crusher Machine Price in pakistan Aggregate Crushing Plant Stone crusher machine is fit for soft or hard and extremely hard materials . how much does a new auto crusher cost Gold Ore Crusher.\nportable concrete crushing machine cost portable concrete crushing machine cost As a leading global manufacturer of crushing, grinding and mining equipments, we offer advanced, reasonable solutions for any sizereduction requirements including quarry, aggregate, and different kinds of minerals.\nPortable Aggregate CrusherPortable Aggregate Portable Aggregate Crusher Manufacturers amp; Portable Aggregate Crusher Suppliers Directory Find a Portable cost of aggregate crusher plant Crushed stone . construction aggregate , typically produced the desired size using crusher s.\nStone crushing is beneficial for construction crews, mining, quarrying, recycling operations, contractors, and landscapers. Smiths Gravel Pit has portable stone crushing equipment available for rent that will make your recycling efforts more costeffective and easier to reuse.\nCrushing Their CostPerTon. In fact, the addition of the QI240 reduced the cost per ton to a third of its original value. Furthermore, with the QI240s fourbar rotor turning irregular chunks into uniform nuggets, reliance on the original jaw crusher was lessened, extending its working life.\nEasily portable impactor track unit, equipped with the UltraMax174; UM15 impactor, is designed for ultramobile crushing without sacrificing power or production on the job site. The UM15 impactor provides the industrys only lifetime rotor warranty (North America only) for minimum downtime.\nWheeled portable is generally higher capacity than tracked and lower cost to operate. But, yes mobile crusher is more economical for the aggregates plant.\nportable crushing plants cost used offers 7942 impact crusher price products. price for mobile stone crusher . SBM Low Price PF1210 Stone Crushing Plant Impact Crusher .. 2015 Stone Impact Crusher Price for Sale Mainly Used in Secondary Crushing.\nLow Cost Small Portable Jaw Crushing Plant 50TPH Capacity Mobile crusher plant\/portable cruhsher plant\/mobile crushing plant is mainly used in metallurgy, chemical industry, building materials, and utilities often need to be relocated material processing operations, especially for the job of highway, railway, hydropower project demaning liquidity stones.\nPortable Crusher, cost of mobile crusher Portable crusher The excellent performance of portable type series mobile crusher 1. High performance jaw crusher, impact crusher and cone crusher. 2. Feeder, belt conveyor and vibrating screen are all in one. 3. Turn axle of traction is convenient for road transit and expanding the working place. 4.\nEagle Crusher Co. unveils its latest closedcircuit crushing and screening plant, the Stealth 500, specially configured for costconscious businesses.\nPortable Crushing. Producing large volumes in relatively short time periods minimizes the impact to any quarry, yard or job site while lessening the material handling costs for the customer.\nHome 187; Equipment Lines 187; Portable Crushing Plants 187; CST Portable Cone Crusher. The CST Cone Crusher is a large trackmounted, This feature increases overall production and significantly reduces wear costs. Click to Zoom in. CST Cone Crushers Screenbox. Smooth Start Technology.\nas a two or threestage crushing application from primary to secondary, tertiary and fine crushing. NW Series portable plants are built around the proven 174; and Barmac174; crushers, which are respected worldwide for their engineering quality, crushing efficiency, versatility and costeffectiveness.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzautuk b/data_all_eng_slimpj/shuffled/split2/finalzzzautuk
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+{"text":"October 8, 2018 (San Diego's East County) \u2013 The median sales price for a detached single-family home in East County reached $535,000 in August 2018, which was 5.4 percent higher than the $507,500 price in August 2017, according to the Pacific Southwest Association of Realtors (PSAR), a trade group for San Diego-area realtors. The August 2018 monthly figure for detached homes is higher than the year-to-date median sales price of $522,000, said PSAR, which operates an office in El Cajon.\nFor attached homes, PSAR said the median home price in the East County was $330,000 in August 2018, which was 11.9 percent higher compared to the $295,000 price the same month a year ago. The August 2018 monthly figure for attached homes is higher than the year-to-date median sales price of $320,000.\nIn addition, PSAR said the average number of days in August for an East County home on the market (the time between when a property is listed and an offer is accepted) was 31 days for detached homes and 17 days for attached homes. In August 2017, the numbers were 32 and 22 days, respectively.\nAlso, the percentage of original list price received for detached homes was 98.4 percent in August, a 0.3 percent change from 98.7 percent in August 2017. For attached homes, the percentage was 99.3 percent in August, a 0.4 percent difference from 99.7 percent in August 2017.\nFounded in 1928, the Pacific Southwest Association of Realtors (PSAR) offers educational training, advocacy and other services and resources to its realtor members. Offices are located in Chula Vista and El Cajon. For more information, visit www.PSAR.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A High Court judge has criticised a divorcing couple after they ran up 'crazy' legal bills during a divorce fight. Whatever the dispute it is always sensible to mediate.\nMr Justice Holman said Barbara Cooke, and Michael Parker, had run up legal bills of around \u00a31.5 million while arguing about dividing assets possibly in the region of \u00a310 million.\nThe judge said if couples were fighting over \u00a3100 million then spending \u00a31 million on lawyers would be proportionate. However, he said Ms Cooke and Mr Parker \u2013 directors of BC Softwear, based in High Wycombe, Buckinghamshire \u2013 are not in that bracket.\nThe judge said he could not stop people fighting in court if litigation was their 'hobby', but he suggested that if the 'war' continued there might be no money left.\n'Ultimately if there is nothing left at the end, there is nothing left at the end,' he said.\nIf you have a dispute that would benefit from mediation please don't hesitate to contact us.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We have Dining room Round Dining Sets that will make a statement for your formal or casual dining space. Furniture Direct UK offer Round Dining Sets with Round Glass Wooden Dining Table or Chair Set includes Round Fixed Top Dining Table with Sets, Oak Round Glass Top Dining Table sets, Pearl Grey Marble Round Dining Table Sets, Round Natural Finish Extending Dining Table set, White Finish Round Extension Dining Table Sets, White or Black Italian Round Fix Top Dining Table Sets and more Glass Dining Table Sets at affordable Sale Price. Enjoy our Free Delivery across the UK.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In The Sons of God in Genesis 6:1\u20134, Jaap Doedens offers an overview of the history of exegesis of the enigmatic text about the 'sons of God', the 'daughters of men', and the 'giants'. First, he analyzes the text of Gen 6:1\u20134. Subsequently, he tracks the different exegetical proposals from the earliest exegesis until those of modern times. He further provides the reader with an evaluation of the meaning of the expression 'sons of God' in the Old Testament and the Ancient Near East. In the last chapter, he concentrates on the message and function of Gen 6:1\u20134. This volume comprehensively gathers ancient and modern exegetical attempts, providing the means for an ongoing dialogue about this essentially complex and elusive passage.\nFor those interested in the Old Testament, in the enigmatic 'sons of God' in Gen 6:1-4, and for anyone curious about the influence of this text on theological reflection during the ages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This flower burst clip is adorned with oxidised silver flowers. It is a truly versatile clip that would work with your beloved charms. Why not wear it with some colourful muranos to create a stunning bracelet!\nFlower Burst Silver Clip Charm - PANDORA is rated 4.2 out of 5 by 11.\nRated 5 out of 5 by CherylK from Glad you make these My sister said I needed clips but I didn't believe her. Now I find them indispensable to keep things from sliding.\nRated 1 out of 5 by Charmed 1 from Disappointment This clip is advertised and shown in photo as being oxidized. That oxidation makes the flowers stand out to the eye. My two clips have no visible oxidation, therefore, you cannot differentiate the flowers....they simply don't come through. I even went to my local dealer to check their stock, they are all appear this way. I am totally disappointed. Once again, it seems like a better idea to actually purchase in a store rather than online. I guess in time, they will tarnish and I can polish with a cloth which will give them some contrast. I would not recommend these clips!\nRated 5 out of 5 by Kea1995 from Very pretty Shipping was fast and easy. Good quality. You'll love it.\nRated 1 out of 5 by Gayle14 from Beautiful clip, except Clip is beautiful and exactly what I was looking for. Unfortunately the small rubber piece that fits inside the clip to secure it to the bracelet was NOT included, so it is a bead not a clip. I requested that the inside piece be shipped, but I have not had a response in several days. So it is not a clip, just a bead.\nRated 5 out of 5 by traveler from Beautiful floral detail for this Pandora clip. I have a bracelet with a floral theme and this floral clip was perfect to complete the charm bracelet. I love the floral detail on an affordable clip.\nRated 5 out of 5 by Mngirl812 from So beautiful I love the flower detail!! So cute!! I would love to see more clip options like this!\nRated 5 out of 5 by Angie80 from Beautiful I bought this clip last week and I'm absolutely love it. Looks beautiful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavnyp b/data_all_eng_slimpj/shuffled/split2/finalzzzavnyp
new file mode 100644
index 0000000000000000000000000000000000000000..d7ec0f7d18df9af2228857f72db3aed39283e277
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+{"text":"Forgive me if I take this occasion to celebrate a few different Elizas than just Miss Bennet, for today is Eliza Doolittle Day.\nFor those unfamiliar with this often overlooked holiday, it is a reference to the musical My Fair Lady, based on George Bernard Shaw's play Pygmalion, which in turn is based on a beautiful myth recounted in Ovid's Metamorphosis. Pygmalion is a sculptor who falls in love with his own creation. After praying to Venus to help him find a real woman as perfect as the statue, it comes to life, and they basically live happily ever after. Shaw's version is tad bit more complex. Professor Henry Higgins, played by Rex Harrison in the movie, is a massive phonetics geek. There's really no better way to put it. For fun he records accents, and it is when eavesdropping one rainy night outside of Covent Gardens that he meets Eliza Doolittle, a cockney flower girl. Eliza is played by Audrey Hepburn in the film, magnificently, I believe, but it is not her voice during the songs \u2013 that's a rather sad tale for both Audrey, Marni Nixon, who sang the numbers, and Julie Andrews, who originated the role on Broadway opposite Harrison. Anyway, Higgins makes a wager with his buddy that with six months of lessons, he can pass Eliza, \"this guttersnipe,\" off as \"a duchess at an embassy ball.\" I do not want to reveal the entire story just in case, dear reader, you have perchance never seen this remarkable film. If not, please do so! It's far too good to miss. In the meantime, here's the scene in which the 20th of May is codified for all time as 'Liza Doolittle Day by an imaginary King Edward VII. It takes place early in Eliza's lessons, and she has spent the last several days strapped to a machine that monitors her breathing while pronouncing the letter A over and over and over again. So she's feeling rebellious. Enjoy.\nAnd now back to Austen.\nHave you ever read Henry and Eliza: a short, wildly farcical story she wrote sometime between the ages of 12 and 15? I discussed it in detail a few years back on my blog (you can read the post here), so I will not offer additional commentary at this time. Instead, for your delectation and in honor of the holiday, I will transcribe it here, spelling idiosyncrasies and all, from my Oxford World's Classics edition of Catherine and Other Writings. For those unfamiliar with Austen's juvenilia, be warned: young Jane had a keen sense of the absurd. She was also wickedly funny. The names Henry and Eliza are taken from her brother and cousin, who would someday marry. As you will see, the story is really Eliza's. Henry's part in it is rather short-lived.\nAs Sir George and Lady Harcourt were superintending the Labours of their Haymakers, rewarding the industry of some by smiles of approbation, and punishing the idleness of others, by a cudgel, they perceived lying closely concealed beneath the thick foliage of a Haycock, a beautifull little Girl not more than 3 months old.\nTouched with the enchanting Graces of her face and delighted with the infantine tho' sprightly answers she returned to their many questions, they resolved to take her home and, having no Children of their own, to educate her with care and cost.\nBeing good People themselves, their first and principal Care was to incite in her a Love of Virtue and a Hatred of Vice, in which they so well succeeded (Eliza having a natural turn that way herself) that when she grew up, she was the delight of all who knew her.\nBeloved by Lady Harcourt, adored by Sir George and admired by all the World, she lived in a continued course of uninterrupted Happiness, till she had attained her eighteenth year, when happening one day to be detected in stealing a banknote of 50\u00a3, she was turned out of doors by her inhuman Benefactors. Such a transition to one who did not possess so noble and exalted a mind as Eliza, would have been Death, but she, happy in the conscious knowledge of her own Excellence, amused herself, as she sate beneath a tree with making and singing the following Lines.\nand will never from Virtue's dear boundaries swerve.\nHaving amused herself some hours, with this song and her own pleasing reflections, she arose and took the road to M., a small market town, of which place her most intimate freind kept the Red Lion.\nMrs Wilson, who was the most amiable creature on earth, was no sooner acquainted with her Desire, than she sat down in the Bar and wrote the following Letter to the Dutchess of F., the woman whom of all others, she most Esteemed.\nThe Dutchess, whose freindship for Mrs Wilson would have carried her any lengths, was overjoyed at such an opportunity of obliging her and accordingly sate out immediately on the receipt of her letter for the red Lion, which she reached the same Evening. The Dutchess of F. was about 45 and a half; Her passions were strong, her freindships firm and her Enmities, unconquerable. She was a widow and had only one Daughter who was on the point of marriage with a young Man of considerable fortune.\nThe Dutchess no sooner beheld our Heroine than throwing her arms around her neck, she declared herself so much pleased with her, that she was resolved they never more should part. Eliza was delighted with such a protestation of freindship, and after taking a most affecting leave of her dear Mrs Wilson, accompanied her grace the next morning to her seat in Surry.\nMr Cecil, the Lover of Lady Harriet, being often with the family was often with Eliza. A mutual Love took place and Cecil having declared his first, prevailed on Eliza to consent to a private union, which was easy to be effected, as the Dutchess's chaplain being very much in love with Eliza himself, would they were certain do anything to oblige her.\nThe Dutchess and Lady Harriet being engaged one evening to an assembly, they took the opportunity of their absence and were united by the enamoured Chaplain.\nWe are married and gone.\nHer Grace as soon as she had read the letter, which sufficiently explained the whole affair, flew into the most violent passion and after having spent an agreable half hour, in calling them by all the shocking Names her rage could suggest to her, sent out after them 300 armed Men, with orders not to return without their Bodies, dead or alive; intending that if they should be brought to her in the latter condition to have them put to Death in some torturelike manner, after a few years Confinement.\nIn the mean time Cecil and Eliza continued their flight to the Continent, which they judged to be more secure than their native Land, from the dreadfull effects of the Dutchess's vengeance, which they had so much reason to apprehend.\nIn France they remained 3 years, during which time they became the parents of two Boys, and at the end of it Eliza became a widow without any thing to support either her or her Children. They had lived since their Marriage at the rate of 18,000\u00a3 a year, of which Mr Cecil's estate being rather less than the twentieth part, they had been able to save but a trifle, having lived to the utmost extent of their Income.\nEliza, being perfectly conscious of the derangement in their affairs, immediately on her Husband's death set sail for England, in a man of War of 55 Guns, which they had built in their more prosperous Days. But no sooner had she stepped on Shore at Dover, with a Child in each hand, than she was seized by the officers of the Dutchess, and conducted by them to a snug little Newgate of their Lady's which she had erected for the reception of her own private Prisoners.\nNo sooner had Eliza entered her Dungeon than the first thought which occurred to her, was how to get out of it again.\nShe went to the Door; but it was locked. She looked at the Window; but it was barred with iron; disappointed in both her expectations, she dispaired of effecting her Escape, when she fortunately perceived in a Corner of her Cell, a small saw and Ladder of ropes. With the saw she instantly went to work and in a few weeks had displaced every Bar but one to which she fastened the Ladder.\nA difficulty then occurred which for some time, she knew not how to obviate. Her Children were too small to get down the Ladder by themselves, nor would it be possible for her to take them in her arms, when she did. At last she determined to fling down all her Cloathes, of which she had a large Quantity, and then having given them strict Charge not to hurt themselves, threw her Children after them. She herself with ease discended by the Ladder, at the bottom of which she had the pleasure of finding her little boys in perfect Health and fast asleep.\nHer wardrobe she now saw a fatal necessity of selling, both for the preservation of her Children and herself. With tears in her eyes, she parted with these last reliques of her former Glory, and with the money she got for them, bought others more usefull, some playthings for Her Boys and a gold Watch for herself.\nBut scarcely was she provided with the above-mentioned necessaries, than she began to find herself rather hungry, and had reason to think, by their biting off two of her fingers, that her Children were much in the same situation.\nTo remedy these unavoidable misfortunes, she determined to return to her old freinds, Sir George and Lady Harcourt, whose generosity she had so often experienced and hoped to experience as often again.\nShe had about 40 miles to travel before she could reach their hospitable Mansion, of which having walked 30 without stopping, she found herself at the Entrance of a Town, where often in happier times, she had accompanied Sir George and Lady Harcourt to regale themselves with a cold collation at one of the Inns.\nThe reflections that her adventures since the last time she had partaken of these happy Junketings, afforded her, occupied her mind, for some time, as she sate on the steps at the door of a Gentleman's house. As soon as these reflections were ended, she arose and determined to take her station at the very inn, she remembered with so much delight, from the Company of which, as they went in and out, she hoped to receive some Charitable Gratuity.\nSir George, who was also in the Carriage, but too much amazed to speek, was proceeding to demand an explanation from Eliza of the Situation she was then in, when Lady Harcourt in transports of Joy, exclaimed.\nA mutual Reconciliation then took place, and Eliza, ascending the Carriage with her two Children returned to that home from which she had been absent nearly four years.\nNo sooner was she reinstated in her accustomed power at Harcourt Hall, than she raised an Army, with which she entirely demolished the Dutchess's Newgate, snug as it was, and by that act, gained the Blessings of thousands, and the Applause of her own Heart.\nDid you enjoy the story, or are you horrified? Both reactions are valid. I think it's hilarious. It has long been one of my favorites from Austen's juvenilia. If you are interested in reading more of her early writings, please come join us on Wednesdays this month and next at The Writer's Block, where we are reading and discussing Lady Susan, the short novel of letters that is the premise for the new film, Love and Friendship.\nHappy Eliza Doolittle Day! It is a particularly special day in my house: do you know why? Try to guess my daughter's name.\nNeed the perfect holiday gift for the Miss Doolittle in your life? Check out these My Fair Lady paper dolls: https:\/\/www.etsy.com\/listing\/97877470\/my-fair-lady-paper-doll.\nGreat post Alexa. I have watched the Audrey Hepburn 'My Fair Lady' and sad to know she didn't sing in it but still really like it. I tried to read Lady Susan many years ago but couldnt finish it. Perhaps it's time to revisit this story with your comments in mind.\nI am really excited to say that I am going to the Sydney Opera House in September to watch 'My Fair Lady' on stage!\nThanks for this delightful post, Alexa! I loved 'My Fair Lady' what a wonderful production that is, and also for sharing that hilarious piece of Austen Juvenilia, she was so wicjedly funny in her earlier works!\nWicked is the right word, Joana! Every time I revisit her juvenilia, I like it more and more. You see how ahead of her time she was. Absolutely brilliant. Thanks for the comment!\nMy pleasure, Jen. Happy birthday! It's an awesome day on which to be born. Just You Wait was the first song I sang to my daughter that she reacted to, giggling away at three months old. It should have told me something about the child I have. Hindsight.\nI love that photo of Audrey Hepburn in that glorious hat. Now THAT is a KY Derby hat. Like Diana O. I too loved the movie, it is a classic.\nThe printed word is so powerful and brings joy and meaning to us. I've always found it funny how our favorite books become our favorite movies. They both touch our lives in so many different ways. Just like Austen's work has spawned many movies and additional books. Her work will always thrill us and excite dialogue.\nAMAZING Derby hat! Some stories will last forever, it is true. They stand the test of time. I think Austen's current popularity points to the timelessness of her tales. Thanks for the lovely comment!\nThat was pure delight! I have loved the musical \"My Fair Lady\" since childhood, and enjoy reading George Bernard Shaw's play, \"Pygmalion\" upon which it was based every couple of years. I have never read Austen's \"Henry and Eliza\" before, but I found it delightful. I wonder how old she was when she wrote it? Fantastic post!\nSo glad you enjoyed it, Diana! Austen was somewhere in the rage of 12 to 15 years old when she Henry and Eliza.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The machines will be available around the clock and can dispense food, clothing, toiletries and other necessities.\nFor the homeless, accessing food can be a big challenge: shelters often have odd hours and food pantries may run low on supplies. But an innovative new vending machine that dispenses food and other necessities free of charge may soon revolutionize the way the hungry get fed.\nThe machines will hit U.S. cities next year, but a British charity, Action Hunger, is installing the first ones in the U.K. this week.\nHere's how it works: Local homeless organizations will identify those most in need of assistance, and distribute chip-enhanced cards that can be swiped at the vending machines to select up to three food, clothing or toiletry items per day.\nItems in the machines may include water, fresh fruit, energy bars and sandwiches, as well as socks, sanitary towels, antibacterial lotion, toothbrush and toothpaste combo packs, books, and more.\nA large amount of the items will come from organizations that \"rescue\" perishable items to reduce food waste, Action Hunger told TODAY Food in a statement.\nNew York City is slated to be the first U.S. location to get machines installed in February 2018, followed by San Francisco, Los Angeles and Seattle. More machines will be installed in London next year, too, as well as \"many more cities to follow,\" the organization said.\nSince they don't need to be staffed, the machines will be accessible around the clock, seven days a week.\nThe first of their kind, the machines came about after Action Hunger founder Huzaifah Khaled was struck by the number of rising homeless in the U.K. and wanted to find a better way to ensure access to food and necessities \"without being limited to to the fairly disparate opening hours of the various shelters and charities.\"\nThe Friary, a day shelter in Nottingham, England, which recently issued a Community Crisis Appeal campaign, is receiving the first machine and will prioritize giving cards those who sleep on the streets.\n\"Our hope is that the idea takes root in cities all over the world, and the homeless have a lifeline to rely on,\" Action Hunger added in a statement, \"while government policies work towards ending homelessness for once and for all.\"\nThe trustees and staff behind the vending machine effort don't receive a salary, so they greatly appreciate donations and corporate support (N&W, a tech company, donated the first machines).\nA U.S. partner of Action Hunger, this organization collects unused perishable foods and will be helping supply the vending machines. Volunteer your time \u2014 many of the time slots are only 30 minutes! \u2014 or donate money.\nNot to be confused with Action Hunger, Action Against Hunger is a global humanitarian organization focusing its efforts in the coming year on Sudan, where 6 million people are currently facing food insecurity, with the possibility of famine in 2018. All donations will be matched for a limited time Tuesday, Nov. 28.\nEvery dollar you donate to Feeding America equals 10 to 20 meals that the organization is able to provide to those in need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The financial condition of the Marathas was also deplorable. Pondicherry city consists of 42 wards, wards 1-10 are located in north of the city. After Safdarjung's death, the Marathas again invaded the Rajput territories. On 14 September 1752, the two took oaths at in , promising mutual peace. In return, the Peshwa promised to give 500 soldiers permanently to the emperor and send 4,000 soldiers, when needed. With Madho Singh's help, Bijay Singh resisted the Marathas for a year, before he agreed to peace talks.\nKashibai was the first wife of Bajirao I, the Maratha general and Peshwa Prime Minister to the fifth Maratha Chhatrapati Emperor Shahu. During Balaji Rao's tenure, the Maratha territory reached its zenith. The Battle of Panipat The Third Battle of Panipat took place on 14 January 1761 at Panipat Haryana State, India , about 60 miles 95. His appointment to the prestigious position of Peshwa at such a young age made many in the Maratha court jealous, but it did not take Bajirao long to prove that Shahu had made the right decision in choosing him. Some judicial and revenue reforms were made during his tenure, but the credit for these goes to his cousin and his associate Balshastri Gadgil. That the Peshwa and should act in complete friendship and help each other; 2. He married Gangabai Sathe who later gave birth to Sawai Madhavrao Peshwa.\nThus, Nana Saheb followed separate policy from his father. Defeating Nizam - Peshwa again defeated Nizam at Durai Sarai this was a great defeat of Nizam. Nizam-ul-mulk was a noble diplomat, conspirator, general and strong rival of Marathas. This episode not only spoiled the Maratha relations with the Rajputs, but also resulted in internal strife among the Marathas. After this diplomatic success Balaji Bajirao returned to on 17 July.\nA courageous warrior, he is credited with expanding the Maratha Empire, especially in the north. In Pune, Balaji Rao repeatedly pressurized Damaji to cede half of Gujarat on behalf of Yashwant Rao Dabhade. However, she managed to enlist the help of another noblewoman, Umabai Dabhade. The battle concluded with a victory for the Nawab of Bengal, in the year 1747, the Marathas led by Janoji Bhonsle, began to raid, pillage and annex the territories of the Nawab of Bengal, Alivardi Khan in Orissa. The Peshwa should station at 500 Maratha horse for imperial service; 3.\nHis father was the first Peshwa of Chhattrapati Shahu, and Bajirao used to accompany his father on his campaigns from a young age. In 1752, the of the region rebelled against the Mughal emperor. Although she crushed the mutiny, she realized that it would be difficult to continue the fight against Balaji Rao. She, therefore, agreed to a peace treaty. After Baji Rao died in April 1740, appointed 19-year old Balaji as the Peshwa in August 1740, despite opposition from other chiefs such as Shahu's own relative.\nIn response, Mughal emperor was forced to recognize him as the viceroy of Deccan, the Marathas, led by Bajirao, helped Nizam win this battle. For example, the Siddis controlled the Janjira fort, on January 4,1721, Bajirao met Nizam-ul-Mulk Asaf Jah I at Chikhalthan to settle their disputes through agreement. Maratha power was fragmented among several discrete fragments, although at present, the word Maratha refers to a particular caste of warriors and peasants, in the past the word has been used to describe Marathi people, including Marathas themselves. After a brief debate, the house, Lok Sabha, passed the bill. By the time Scindia marched to Jodhpur in September 1752, Bakhat Singh had died. According to his own edicts, in that war about 100,000 people were killed,150,000 were captured, the resulting bloodshed and suffering of the war is said to have deeply affected Ashoka.\nThis decision of Sahu was in favour of Marathas. Under this system, each Maratha chief was assigned a territory which could be administered autonomously. Then he attacked on Mughals with the help of Chhatrasal. Tantya Tope was the master to Nana Sahib. Ishwari Singh initially agreed, but refused to abide by his promise after Balaji returned to Pune.\nAfter Ishawari Singh's death, became the ruler of Jaipur. She was first a member of the Congress Party, and later became a member of the Bharatiya Janata Party. The Jat ruler of also joined the Marathas, but later left the alliance due to a misunderstanding with Bhau. He was a powerful ruler of Maratha Samrajya after Shivaji I. In the 1720s, he led Maratha armies in Malwa region, in 1734, Malhar Rao established a camp later called Malharganj.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"First up is the new FLIR AX8 Marine Thermal Monitoring System, which strikes me as notably innovative technology. In real life the image above would likely be one side of a large diesel engine or some other thermally active systems area, and you'd most likely be looking at it on a Raymarine MFD, which just got AX8 integration with LightHouse Release 15. But my NMEA expo hall photo \u2014 click it bigger \u2014 does show the alternate browser interface that must be used to set up alarms for up to six temperature spots or boxed areas within the camera's 48\u00b0 by 37\u00b0 field of view. It also shows how the AX8 can use the edges detected by its 640 x 480 pixel video cam to make the 80 x 60 thermal image much easier to understand.\nSpeaking of WiFi, look what I saw in the Shakespeare Marine booth. Their WiFi Amplifier page is just a tease right now, but obviously they are about to offer a Ubiquiti-Bullet-based system like the Rogue Wave and many others (and, yes, the WebWhip will also have an easy browser interface). Meanwhile, the unique and shapely WebWatch dome will purportedly offer WiFi boosting along with 2\/3\/4G cellular data, an onboard hotspot and Ethernet router, and an HDTV antenna. Given that Shakespeare probably has the largest marine antenna distribution network, at least in the U.S., it sure looks like this stuff is going mainstream.\nThe new Humminbird Helix 7 series shows how the glass bridge style is nicely working its way down to small boats. There are many functions and networking options they lack, but hey, that's an 800 x 480 pixel bonded 7-inch display with 1500 nits of LED brightness that starts at about $350 for the dual beam sonar only model. Plus,when I started yammering about rehabbing the original 14-foot outboard Gizmo \u2014 now L'il Gizmo \u2014 so I can better test some of the new sonar gear plus do some recreational lobstering, the Humminbird guy got excited about how I could use their Side Imaging sonar to optimize trap placement and then use the MFD i-Pilot Link and a saltwater Minn Kota to automate the hauling process. It sounds like a neat project, but how will the real lobstermen react?\nThe NMEA Conference is naturally oriented to more complicated, higher-end installation and service situations, and here's a good example. That's the display of a nifty Lumishore EOS underwater light controller, and normally you'd be seeing the elegant screens you can use to change light colors and intensity, initiate \"sweeps\" or even fine tune how your luminous yacht bling integrates with your sound system. Bass in red or blue tones?\nThe extra cool part is that EOS uses the bi-directional open DMX512 RDM lighting standard, which means that third-party systems like Crestron's can also control the lights, and moreover means that the lights themselves can talk to the system. So the geeky screen above shows the intimate internal details of a complex LED module, possibly installed in some remote corner of a boat without having to crawl there let alone pop the fixture. Lumishore \u2014 which won the first ever NMEA underwater lighting award \u2014 is bringing EOS down to smaller lights with a mini controller and Ocean LED is also working with DMX512, all of which I hope to learn more about in Lauderdale.\nI failed to photograph the ultra precision machining seen around an Intellian GX60 antenna's waveguide, and the fit and balance of the superlight carbon dish are also wondrous, but I did capture the diagram printed directly on the underside of the aluminum cover to the main electronics box. You can almost hear the murmured \"thanks\" of the tech who's working way up in a ship or yacht antenna farm, perhaps on a windy day. Inmarsat's third I-5 satellite is launched, which means that Global Express Ka-band high speed service is about to actually become global. Precision is what it takes to use such a distant and high frequency source from a moving platform, especially with a relatively small 26-inch dish, and it looks like Intellian has nailed it. But will Kymeta panel technology make satellite radomes obsolete? Color me skeptical but I am going to their Lauderdale presentation.\nHere are more reminders that marine electronics is as much about good mechanical engineering as it is circuit boards and firmware. On the left is the Digital Antenna Bullet that's been working well with Gizmo's cell booster, but now the mount doesn't require the installer to drive little screws upside down while in the rigging. On the right are cutaways showing how Comrod marine antennas are completely filled with high density polyurethane foam, which seems to justify their claims of high durability and also explain their popularity with pro installers.\nThen again, good software can make well-made electronics sing. Nobeltec TimeZero Coastal Monitoring was impressive when I first saw it, and now it's amazing what it is providing to fish farming operators, dormant oil platform minders, resource protectors (in the Galapagos), and possibly many other applications given a starting cost of about $30,000. We watched a demo showing how well it can use superfast ARPA (coming eventually to Nobeltec PC Radar) and AIS to detect any vessel moving in a desired area, and then use conditional rules to take actions like zooming a camera in on the vessel, taking a still photo and texting it to an operator. Boaters like to save money by using consumer electronics instead of dedicated marine stuff, but here's a case where a well integrated recreational marine electronics bundle looks like a bargain compared to the normal solutions. It also seems like a good opportunity for dealer\/installer to drum up new and unusual clients.\nFinally, here's a helm shot of Impossible Dream, the universally accessible 60-foot sailing cat that several NMEA member companies recently supported with an extensive and collaborative electronics redo (that was still being fine tuned before the Baltimore reception ;-). It seems like a great story on several levels, and I'll try to tell it eventually, but in the meantime look for the Dream enabling boating joy around Biscayne Bay from her base at Shake A Leg Miami.\nNMEA 2000 Or A 20 Year Old NMEA 0183?\nAny MSRP or detailed specs on the WebWatch? Specifically weight..\nI've got a spec sheet but no pricing, Howard. WebWatch weighs a little under 4 pounds and that dome is 13 inches high and about 12 wide.\nI'm backer #2, with my own nickles, because I think that Signal K may help us out a lot and I like how DY is going about this project. More to come.\nRE Kymeta. As an electrical engineer I believe this is truly revolutionary antenna technology. I've read some of the papers, talked to some of the principals and seen it working in the lab. They are aggressively working out the manufacturing challenges. They are a company you'll want to keep an eye on.\nShakespeare's webwatch is showing up on dealer websites with street pricing in the $500-$525 range.\nGoing to order one today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Long before an Indonesian exporter fills its first Canadian order, it will need an export marketing strategy to plan out how to reach the Canadian market.\nProduct (or Service): What is the product or service and how must it be adapted to the Canadian market?\nPrice: What is the pricing strategy to be used?\nPromotion: How is it possible to make Canadian customers aware of the product or service?\nPlace: How and where is it possible to deliver or distribute the product or service?\nPeople: Are the necessary staff and appropriate partners in place to be successful in the Canadian market?\nThe diagram below can be summarized as the five \"Ps\" to identify the key factors to be considered in a marketing strategy: Product (or Service), Price, Place, Promotion and People. All of these considerations should be included in the Export Plan of the Indonesian exporting firm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavpcl b/data_all_eng_slimpj/shuffled/split2/finalzzzavpcl
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+{"text":"Get started using these steps to learn how to apply and successfully navigate through the financial aid process.\nCopies of tax returns are NOT accepted from students selected for verification.\nReview this checklist for your financial aid status, and to accept and finalize your financial aid award.\nEmailed questions may take 3-5+ business days to respond.\nPlease check your status via MyUH Services.\nThe University's official means of communicating with you is through your \"@hawaii.edu\" e-mail account. Please use your \"@hawaii.edu\" account when corresponding via email.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi, I'm Joseph Bennett and I'm in 11th grade. In my free time I like to participate in Baseball, basketball, football, band, NHS and in my Church. I want to be Confirmed because I want to further my relationship with God, and Confirmation is the next step to get closer to Him. I chose St. Sebastian as my Confirmation Saint because he is the patron Saint of athletes and I am a tri-sport athlete and I am blessed to have the opportunity to play. I feel that I have been given gifts in kindness and communication, and feel that the Holy Spirit may be calling me to use these gifts by going out and making contact with people in need. I want to my family, especially my mom for supporting my faith and raising me right in this church. I also want to thank my sponsor Cindy Young for being with me on this journey and being a role model for me by being active in church.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hiring a financial planner requires trust. My priority is to make the decision comfortable for you.\nlasting relationships with Superstar Women like yourself.\nMy goal is to make complicated concepts easier to grasp, and to empower you to make informed decisions about your hard-earned money.\nJane Financial, LLC is a fee-only financial planning firm. I'm only paid by you, the client. I don't make money from product commissions.\nUnlike the vast majority of financial advisors, I don't have a minimum portfolio size. Instead, I charge a flat financial planning fee based on the complexity of your financial situation.\nThis fee starts at $6,000 per year for single people, and $9,000 per year for couples, charged on a quarterly basis.\nTo learn more, the first step is to schedule a free consultation.\nI would love to chat about where you are today, and where you want to go.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Your test, treatment or contraceptive pills will arrive in a plain envelope. It will contain everything you need to freepost your samples to our lab, take treatment or use contraception effectively. When using our STI testing service, your details will not appear on any of the samples as we pre-label your samples with a unique code.\nOur team of expert sexual and reproductive health clinicians are available to support and advise you.\nWe will update you by text when your order has been dispatched and when your test samples have been received at the lab.\nWe can support you by text or by phone to complete your test, take treatment or contraceptive pills effectively.\nYou can text us back anytime or request a call back from one of our clinicians.\nSH:24 is commissioned to deliver sexual and reproductive health services in many areas across the UK in partnership with the NHS. We are actively working with NHS Trusts and Local Authorities across the UK to improve access to our service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Got a ball-nut? This toy might just be the ultimate in entertainment, reward and fun for both you and your dog!\nThe rope is not only great for throwing the ball further distances (leave that ball-chucker at home) it's also a great safety feature. Some dogs have managed to \"swallow\" tennis balls whole, which has caused significant problems and has even lead to the death of some dogs. The rope feature enables you to remove the ball from the dog's throat in the unfortunate event that your dog should swallow the ball.\nLightweight foam construction! This is great because it floats and it doesn't have the felt cover that a normal tennis ball has that gets so yucky-soggy!\nPlease note, no toy is indestructible. We do not recommend leaving this toy with your dog for their chewing pleasure. That's a job best left for actual chew toys. This toy is for active play with you only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawhlo b/data_all_eng_slimpj/shuffled/split2/finalzzzawhlo
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index 0000000000000000000000000000000000000000..83129020a0558a45be8624d8bff3074bd0160666
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+{"text":"After much industry anticipation our ROV was released for sale in September this year and the order book has been filling up, with production now booked solid until late January 2019.\nThe opportunity for us to get the ROV in front of potential customers and distributors was invaluable. For some the quantum shift in ROV technology that the Boxfish platform represents must be seen to be believed and the feedback we received from the show has backed up our belief that we have created something very special.\nThanks to all who came along to say hi and please get in touch for further information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The extraordinary China Disabled People's Performing Art Troupe is back following Canadian's royal welcome in 2012. My Dream is presented by CCCDA and CAEG. The Troupe features a cast of some 52 professional artists \u2013 acrobats, musicians, singers and dancers. They are in their 20's and are all handicapped. Some cannot hear, some cannot see, some have mobility issues.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It is our desire to preserve Camp Nebagamon as a refuge. A place where boys can leave behind the stresses and pressures of their regular lives. A place where they can focus on their friendships and enjoy the environment without the distractions of screens. A place where they can explore and play freely like many of us used to be able to do in our own neighborhoods. Ultimately, a place where they can figure out who they are.\nCamp Nebagamon provides young men with a fun and healthy adventure in the outdoors. Our individualized program helps each camper establish personal goals, building self-esteem and confidence. We teach leisure lifetime skills, activities a boy can learn and continue to practice far into adult life. We care not whether a boy is proficient at serving a tennis ball, cooking an omelet, or paddling a canoe. Our goal for every camper is that he enjoys participating, achieves competency, and feels a sense of accomplishment.\nAt Nebagamon, young men learn the responsibilities and spirit of living in a group. Our campers get the opportunity to live with boys from all walks of life. We believe it is a much more meaningful experience to live with a group of boys with diverse backgrounds than a group that is too much the same. Our campers and staff come from more than 40 communities from throughout the United States as well as around the world. We have boys from different racial, religious, and socio-economic backgrounds. We encourage and nurture the ability to welcome and accept differences in others. Camp is a place where close and lasting friendships are formed.\nAn appreciation of the environment and natural world is also important at Nebagamon. In-camp programs emphasize respect and knowledge of the environment, and our extensive wilderness tripping program offers campers an opportunity to experience nature in its most pristine form. The trip program also teaches campers about cooperation, responsibility, and trust. Wilderness trips result in physical, social, psychological, and spiritual benefits\u2013the core of what we attempt to accomplish at Nebagamon.\nA traditional overnight summer camp for boys in the Northwoods of Wisconsin.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Could This Be The Smartphone Of The Year?\nThe cold and the damp weather of winter not only affects our skin, but even the electronic gadgets. The world's most waterproof smartphone can survive depths of up to 5 meters for 60 minutes using our unique Lockdown Switch. Experience the LG V20 with amazing video recording with Steady Record 2.0, next-level Quad DAC audio and Android 7.0 Nougat Operating System. The chart above displays the percentage breakdown of camera options across all Smartphones on SpecOut. These smartphones have big screens with high resolutions that ensure the best possible picture quality.\nSmartphones evolved from early cell phones, or mobile phones, which offered users the ability to make calls, send text messages, and play the occasional game of Snake. Once you have decided that you need a smartphone for professional rather than for personal use, you need to decide on which carrier you will select for your cell phone. If your provider of choice does not offer business smartphones with all the features you need for business, you might want to consider switching to a new provider who does carry a smartphone for professional use. Beautifully designed for maximum comfort, the Zuk smartphone features a IPS display, long-lasting battery, and guarantees power and reliable performance. Why smartphones hook us in, plus tips on reclaiming your time and concentration.\nTopping this, are the other features of smartphones that offers access to instant messaging sites such as Yahoo messenger or MSN. My Nokia phone has become my everyday smartphone because it has many exciting and useful features. Smartphone uses two camera lenses either on the front or rear to enhance image quality and overall camera performance. Microsoft , for instance, started a new OS from scratch, called Windows Phone Nokia abandoned Symbian and partnered with Microsoft to use Windows Phone on its smartphones. Microsoft was the second to pioneer the newest technologies with their smartphone. We also provide unbiased ratings and cell phone & service reviews to help you choose the best cell phone & service for your needs. A lot of people now do this to be able to raise funds for the purchase of their new smartphones.\nThe first of its kind, the Cat\u00ae S60 gives you live thermal imaging expertise direct from your smartphone. It contains a framework employing common applications and the advantage is its friendly interface that can be found on the line of Nokia's product of smartphones. In fact, smartphone camera technology is going through the same megapixel race which digital cameras went through, which is why we have models with 41 megapixels, like the Lumia 1020. The reviews are in and most agree, Google's new Pixel phones are awesome They also mark the death of the Nexus line. Battery capacity is far and away the most reliable measurement of smartphone battery life.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Best Playa del Carmen hostels - showing all hostels in Playa del Carmen. Book a cheap hostel in Playa del Carmen & pay no booking fees. We have a hostel for you in all the major areas of Playa del Carmen - downtown Playa del Carmen hostel, city centre hostels Playa del Carmen & airport hostels.\nWe have 43 hostels in Playa del Carmen with an average rating of 88% based on 15 reviews.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Are you looking for an Orland Park Dentist? Look no further!\nTreasured Smiles Dentistry's experienced team of dental professionals provides excellent adult and pediatric dentistry for residents of Orland Park, Illinois. We take pride in our warmth, skill, and expertise.\nOur talented team offers adult and pediatric dentistry services, specializing in both pediatric treatment and adult dental treatment. Treasured Smiles Dentistry can provide outstanding dental care for every member of your family! Don't just take our word for it \u2013 find out why families keep coming back to our office.\nWe enjoy making our patients grin as we help them attain beautiful, healthy smiles. Make Treasured Smiles Dentistry the Orland Park dentists for your family.\nOrland Park residents can visit Treasured Smiles Dentistry's office, conveniently located in Frankfort, Illinois, for exceptional dental care. Our dedicated team of hygienists and dentists is devoted to providing each and every patient with the best care and service.\nDo your kids need an experienced and fun children's dentist? Our Certified Pediatric Dentists, Dr. Steve Kuhn and Dr. Dennis LaMonte, are the perfect fit. They are committed to making the smiles of their pediatric patients sparkle while also creating a comfortable environment. Dr. Michael Kasper and Dr. Niki Kasper, our cosmetic and general dentists, provide a wide range of dental care services. The Treasured Smiles Dentistry team is highly-trained in the most current dental treatments, including preventative dentistry restorative dentistry, Invisalign, and more. Learn more about common dental problems that may be troubling you here.\nTreasured Smiles Dentistry is home to the best adult and children's dentists in the Orland Park area. If you want the best dental care for your family, schedule an appointment today.\nResidents of Orland Park, Call Treasured Smiles Dentistry to Schedule an Appointment Today!\nIf you need an Orland Park pediatric dentist, a family dentist, or a cosmetic dentist, visit Treasured Smiles Dentistry! Our skilled and friendly team is here to help you and your loved ones achieve brighter, healthier smiles. Call us at (815) 806-1600 or fill out our online contact form to schedule a dentist appointment for you or your child. You can also reach us at mail@treasuredsmilesdentistry.com.\nThe Treasured Smiles office is located at 10313 W. Lincoln Hwy, Frankfort, IL 60423. We look forward to meeting you!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to issue 87 \u2013 the issue where we break your heart.\nIt is often said that we get what we deserve. I feel that is a narrow view of reality. But know that much of what we desire from life is shaped by what the world gives up. This is felt poignantly by the three protagonists in our three works of original fiction from August. \"The Gentleman of Chaos\" by A. Merc Rustad is a dark fantasy that shows the rewards of a well-disciplined long game. \"I Remember Your Face\" by E.K. Wagner is a heartbreaking post-apocalyptic tale of revenge, loss, and sacrifice. \"Fall to Her\" by flash fiction master Alexis A. Hunter takes us on a journey of unforgettable grief in a thousand words.\nOur reprints this month are \"Paskutinis Iliuzija (The Last Illusion)\" by Damien Angelica Walters and an excerpt from Stay Crazy (Apex Publications, 2016) by Erica L. Satifka.\nPoetry editor Bianca Spriggs has an impressive lineup of poems for us: \"Not Like This\" by Mary Soon Lee, \"This Earth\" by Frank Tota, \"The Labyrinth Keeper\" by Anton Rose, and \"Perplexities\" by Peter Venable.\nA few months back, I was on the digital movie service Vudu and performed a sort for the highest rated genre movies by users and Rotten Tomatoes. To my surprise, the dark fantasy film INK (Double Edge Films) appeared at the top of the list. It's long been a favorite of mine, but I had no idea its appeal stretched so widely. Betsy Phillips has interviewed INK (and The Frame) star Christopher Soren Kelly about the pair of films, and does a fine job describing why these films hold the appeal they do.\nBe sure to check out the podcast version of \"A Gentleman of Chaos\" by A. Merc Rustad as read by Mahvesh Murad.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To the millions of Americans who suffer from Tinnitus, a constant low or high-pitched ringing in the ears, it is more than an annoyance. Tinnitus is a psychological and emotional assault on the senses that taxes and exhausts those with this disorder. It can affect sleep and concentration, and can inhibit the ability and desire to interact socially.\nUntil now, Tinnitus treatments have focused on helping sufferers cope with the condition. Cognitive behavioral training is used to learn how to tune out the sound, and physical therapies help individuals learn to manipulate the pitch and tone of the ringing by clenching the muscles around the neck and ear. Standard medication therapies, in general, are minimally effective for Tinnitus relief.\nHowever, recent Ketamine treatment studies show results that can help a significant proportion of Tinnitus patients.\nTinnitus sufferers usually have damage to McKusick hair cells inside the ear, or have damage along the auditory nerve pathways. This is often the result of traumatic injury, either from a single loud auditory event such as an explosion, or from ongoing exposure to loud sounds. It can also be a result of physical trauma to the ear, middle ear infection, or ear pressure variation injuries from activities such as scuba diving. Veterans are at particular risk for Tinnitus, as are individuals who work or play in loud settings, and who don't wear proper ear protection: industrial journeymen, musicians and their fans\u2014 even people who attend children's birthday parties. Audiologists have determined that the sound impact of balloon popping at close range is as loud as the sound of a gunshot.\nThe brain does not receive complete sound information after damage to the ear structure, so instead of interpreting lack of input as nothing, it fills in the gaps with continuous sound. The ringing is a form of auditory static.\nBased on work in the 1990s, French researchers demonstrated that Tinnitus occurs after damage to the ear that causes an increase in the number of NMDA receptors in nerve cells inside the cochlea. Ketamine blocks NMDA receptors, and in two large studies, over half of the patients treated with S-Ketamine injected directly into the ear experienced significant improvement in their Tinnitus symptoms, with these beneficial effects lasting several months. Our program uses a modified version of this therapy, using four, five, or six slow IV infusions of Ketamine over a two-week Period.\nEarly treatment may be critical. The results of ongoing research indicate that Ketamine is most effective in individuals who have experienced onset of Tinnitus within the previous year.\nIf you are one of the one in five Americans who suffers from Tinnitus, you may benefit from this innovative new treatment. The Arizona Ketamine Treatment and Research Institute (AKTARI) offers Ketamine-based therapy that could restore quality of life and, hopefully, silence. Contact us or call us at 480-626\u20132727 for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Colin Thornton has JUST SOLD ANOTHER property at 32794 Hood AVE in Mission.\nAnother HOME SOLD ! Congratulations Daniel and Dana On your NEW HOME!\nOne of kind Last original, Spacious home with suite in newer neighbourhood! Brand new slop Roof, large Main floor enjoys large living room, kitchen, 3 bedrooms, 2 bathrooms! Lower level boasts a new 2bedroom Authorized suite, with beautiful kitchen, bathroom very nice flooring, laundry, Access for R.V. parking, great location walking distance elementary school, Park, and Bus stop. let me not forget the brand new roof and hot wanter tank in 2017.\nGreg & Colin Thornton have JUST SOLD ANOTHER property at 33428 13th AVE in Mission.\nGreg & Colin Thornton have just listed ANOTHER new property at 33428 13th AVE in Mission.\nSOLD ! A huge Congratulations to not only my clients but my friends Daniel & Dana On your SECOND HOME!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Storage units are available throughout Ohio to protect and secure your precious belongings accommodate you. In Ohio, you will discover numerous self storage facilities and mobile storage units to house your small boxes or bulky items that you still want to keep but that you no longer have use for. Storage facilities in Ohio can also offer you additional protection with insurance as well as its established security settings. On a basic level, your goods will also be protected through climate-controlled settings.\nOrigin of State's Name: The name Ohio means Good River. It comes from a Iroquois Indian word.\nCapital City: The capital city of Ohio is Columbus.\nPopulation: According to the U.S. Census, the state of Ohio has a population of approximately 11.8 million residents.\nDriving: On the Ohio Bureau of Motor Vehicles' website you will find information about applications, driver's licenses, testing requirements, driver's manuals, testing locations and much more.\nAgriculture and Industry: The main agricultural products in Ohio include soybeans, dairy products, corn, tomatoes, hogs, cattle, poultry and eggs. Trade, transportation, utilities, health care, education fabricated metal products, machinery, food processing and electric equipment are Ohio's leading industries.\nState Bird: The northern cardinal is the official state bird of Ohio.\nState Tree: The Ohio buckeye is the official state tree.\nState Flower: The red carnation is the official state flower.\nState Song: The state song of Ohio is \"Beautiful Ohio.\" It was written by Ballard MacDonald.\nThe first traffic light and the first ambulance were established in Ohio.\nAkron is considered the rubber capital of the world.\nSeven U.S. presidents are from Ohio.\nFamous names form Ohio include filmmaker the Steven Spielberg, the singer Dean Martin, the baseball legend Cy Young, the broadcast mogul Ted Turner, the author Toni Morrison, the golfer Jack Nicklaus, the basketball star Lebron James and the comedian Dave Chappelle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I have a Facebook friend, Jenn. I call her a Facebook friend because I have not seen her in person in years. She was once my student and although I have followed her life by hearing about her through friends and her mother, I have not seen her in person in many years. I love being able to be a witness to her life, which looks quite exciting from the outside. It seems that she travels quite a bit with work, and then separately from work, she travels for pleasure. It is always so interesting to find out where she is.\nThis past week I had a very busy week and was on Facebook only once. It happened that Jenn had posted that day. Apparently, she had planned a surprise trip for her partner's birthday. They were going to London, England and then to Ireland. She posted a picture of them in England and said how thrilled she was that she had been able to keep the trip a secret! Someone must have made a comment to her about how she managed to do it and her next post said, \"I'm always half-packed\"!\nThis idea of 'half-packed' got me to thinking, of course. In the case of Jenn, I believe this to be true. She does so much travelling that I am sure that she has a system in place for minimizing the time she needs to fill her suitcase and get to the airport. But the idea of 'half-packed' for Jenn would likely mean more than just the physical act of having items ready. I suspect that this is a mind-set that Jenn has. I am guessing that she is also 'half-packed' in her mind. This means that when an opportunity presents itself for travel, her mind is already making the adjustment to being away on an adventure. What an incredible way to be!\nI was wondering what I am half-packed for? I am not necessarily half-packed for a trip and although I am really improving on the adventure side of things, I'm not sure that I can commit to being 'half-packed'. I am however, 'half-packed' for some other things. I am certainly 'half-packed' to help any of my family or friends when they need me. I am definitely 'half-packed' to dance with the dance group I belong to. Whenever I get an invitation to join them at a performance, I have a very quick 'yes' ready and I can adjust my schedule in what feels like no time.\nI am 'half-packed' to entertain overnight guests at our house. Since we have so many friends and relatives from 'away', I have learned how to do this with minimum stress and lots of positive anticipation. I am always 'half-packed' to serve chocolate chip cookies. I can't remember the last time I was caught short, without the necessary ingredients. I also know that I am 'half-packed' to have a sympathetic ear to those who need it and I can confidently say that I am 'half-packed' to work hard on any given day.\nWhile I have been able to think of many great things for which I am 'half-packed', it has crossed my mind that I can sometimes also be 'half-packed' in a less positive way. Sometimes I find myself being 'half-packed' for saying no too quickly. My mind can be half way to saying no before I really consider the invitation. I can be 'half-packed' to live too safe a life. I can be 'half-packed' to think that menial little jobs should take precedent over an unexpected invitation. It is so comfortable for me to have a routine that I often opt for what is very familiar \u2013 I am more than 'half-packed' for this!\nSome people can be 'half-packed' for more even more serious things. Some seem to always be 'half-packed' for a fight. It takes very little for them to find fault with others. Others are 'half-packed' to fail. They have a mindset telling them failure is likely and this may prevent them from ever even filling the second half of their suitcase \u2013 so sure are they that failure is inevitable.\nOthers can be 'half-packed' to be critical. No matter what idea is presented, they can find a flaw with it and they quickly share their thoughts, often with later regret. Some are 'half-packed' to find excuses for why they should not change, even when the change might be positive.\nWe are all packed for something. Everyone we meet is carrying around some 'half-packed' suitcases. The trick is to pack only the suitcases that take you on the journeys you want to be on. I can't imagine Jenn ever half-packing a suitcase of small mindedness and boring. I love that her suitcase is half full of open-mindedness, sharing and adventure!\nThis week, take some time to think about what you are 'half-packed' for; positive and negative. There might be some suitcases that you would prefer to unpack and even put away.\nI provide coaching, group coaching, workshop creation and facilitation. Contact me for all your or your organization's coaching needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mr. Rogers, beloved TV neighbor of yesteryear, was the kind of welcoming, warmhearted person who people wanted in their own neighborhoods. In the days since Mr. Rogers Neighborhood, what does it take to be a good neighbor? Not too much, actually. In fact, there are several simple ways to make your own neighborhood a happier and safer place.\nThe first step to being a good neighbor is finding out who your neighbors are. Many of us may not know the names of the people living down the street. Some may not even know the names of their next-door neighbors. If this sounds like you, tackle this task first.\nKnowing your neighbors' names is just the beginning. Tell your neighbors about yourself, and hopefully they'll share, too. Offer to help them if they need it, especially if you have some certain skill or tools, like if you're good at fixing cars or you own a snow blower. Your neighbor may need your help one day, and in return, you may get help in the future, too.\nThe benefits of knowing your neighbors goes beyond making you look good: It helps keep you\u2014and others\u2014safe. If you know some basic information about the people living around you, you'll be better able to spot suspicious behavior if it occurs.\nNo one likes a noisy neighbor, so don't be one yourself. Limit noises from stereos, motorcycles and power tools to daytime hours when you won't be disturbing your neighbors while they sleep.\nIf you live in an apartment, condo or townhome, take extra care to limit your noise. Sound can travel easily through a building's floors and walls, so keep loud noises to a minimum. If you do receive a noise complaint, don't be defensive. Acknowledge the concern and talk about the issue calmly to find a resolution.\nAlthough we recommended getting to know your neighbors better, don't be too nosey. You don't need specific details about your neighbors' relationships, religion or politics. Just be friendly and offer to help if they need it.\nSomething that irritates many neighbors is a messy, unkempt yard. If you have a lawn, keep it trim and free of weeds and junk. Trim back any trees or shrubs that grow over into neighbors' property. Plant flowers or do some simple landscaping to give your home a cared-for look.\nIn addition to looking nice, a well-kept home exterior might improve your safety, too. Criminals may be less likely to break in to homes that look like people care about them.\nDon't stay cooped up indoors at all times. Make it a goal to go outside as often as possible for walks, jogs or other activities. Taking a regular walk can help you get to know the layout of your neighborhood and provide opportunities to meet your neighbors.\nIf you're a person who likes to plan, organize a neighborhood party, potluck, sporting event or other activity. Or, if you have the writing and technical know-how, think about creating a neighborhood newsletter. Bringing people together is a great way to add joy and trust to the group.\nHaving a good relationship with your neighbors means you'll have extra eyes and ears on your home when you're away. If you'll be gone for an extended period, it's a good idea to ask someone to do some of the upkeep for your home, like mowing the lawn and collecting your mail. This way, you won't give the impression to would-be intruders that your home is unoccupied.\nConsider installing a home security system. It's an effective way to deter burglary, and some systems even allow you to remotely access your lights and security video feeds. Having a home security system may also be a good way to save on home insurance.\nThe last defense in case of serious problems in your neighborhood is your local law enforcement. If you're worried about speeding or crime, get to know the police officers who frequent your area and ask them to patrol the streets. Some neighborhoods have neighborhood watch programs, which are an opportunity to keep your homes safe and build relationships with your neighbors.\nTry a few things on this list and see if your neighborhood relationships improve. Or, if you have your own ideas for being a good\u2014and safe\u2014neighbor, share this article on social media with your neighbor pro-tip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Android Evernote users might be pleased to learn that today Evernote has rolled out a new update for their software which has brought with it speech-to-text support. Enabling you to now record your voice and Evernote will now convert it into text on the page for you, saving your fingers from having to touch the screen.\nEvernote 3.6 now also has a range of new features which have been added in the latest update. Including larger thumbnails in the note list and the ability to turn off auto-titling via the app settings.\nAndroid tablet owners running Evernote 3.6 have been granted the ability to access to the \"Explore Evernote\" feature discovery system. Enabling new users to quickly learn how to use Evernote and its features more effectively.\nIf you have never used Evernote, check out the video below which has been created by the team to provide you with a quick overview of what the application can offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Become the goddess moneymaker you were born to be by believing you're worth all the abundance you set your sights on.\nWomen who want more money and abundance.\nWomen who don't know how to talk about money.\nWorking women who know they are making less than they deserve.\nStay-at-home mothers who need a fresh perspective on their economic contribution.\nWives who have turned all financial decisions over to their husbands.\nWidows who are learning how to take care of their financials.\nSingle mothers who feel everything is on their shoulders.\nSingle women who want to head off money issues in a relationship.\nMothers who want their kids to be good with money.\nWomen who want to be financially independent.\nWomen who want a better life for themselves and their family.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As a useful tool for members, ACEP has set up a Master Calendar containing dates and locations for upcoming courses, conferences and much more.\nPlease click here to read more about ACEP's international journal.\nTo read about ACEP's CME Tracker or to login to your CME Tracker, please click here.\nFor the convenience of the members of the chapter, various links to other sites have been provided. The DC ACEP Chapter does not endorse any of the below sites.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"You are now playing the Wake The Royalty LP game. You should also play one of the other Thinking Games at GoodGame.co.in !\nWake up the king by bumping him, directly or indirectly with various objects. Try to strategically place the objects, to cause a chain reaction. Bump the sleepyhead off this throne to finally wake him up in this Level Pack version of Wake the Royalty!\nControl: Hold the left mouse button down while moving objects.\nFriends \"Wake The Royalty LP\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In today's day and age of internet and big box retailer takeovers, many shoppers forgo visits to brick and mortar stores in favor of comparison shopping for the \"best deal\" online. Adopting this strategy works for some things, but for ski\/snowboard\/bike pursuits, forgoing a store visit deprives you of a crucial element: ensured personalization. You can spend hours reading product reviews and \"educating\" yourself, but when it comes down to gear you owe it to yourself to grab some face time with people who know what they're talking about and are passionate about making sure you get the ride you're looking for.\nSan Francisco native Arturo Beyeler and his father have been serving snowboarders and skiers from all over the world since Mountain West opened its doors in 2001. Having been in the industry since 1989 working and managing several shops, Arturo learned the ins and outs and saw an opportunity to provide a specialized combination of products and service neglected by all the shops he knew. The result? A mid-to-high end ski\/snowboard\/bike shop with several key factors setting it apart from the other big sporting good retailers.\nOne: custom boot fitting. Finding a pair of ski or snowboard boots is not as simple as specifying your shoe size. In fact, boot size often doesn't equal shoe size. But there are other things to consider beyond your foot length. Volume, instep and heel pocket variances are are key in understanding which boots to try on. With ski boots, Mountain West specializes in custom boot fitting including shell distortion, liner molding and custom footbeds. The personalized boot fitting agenda here is second to none and employees will spend one-to-two hours with a customer to perfect the fit of a ski boot. With snowboard boots, the evolution and popularity of the sport have given rise to new boot lacing systems and you really need to physically try all three (Boa, Speedzone and traditional lace) to best determine which you prefer. Also of crucial importance is heel grip; heels sliding in and out of your boot is the last thing you want.\nArmed with a knowledgeable and experienced staff, this is your go-to spot in South Lake Tahoe if you're looking for precision and speed in getting your gear all set up. Which brings me to the next wonderful thing about Mountain West: the people here are real.\nYou won't find any sales personnel pressuring you to buy the most expensive package (unless, of course, you're one of those people that must have the fanciest and newest gear). Most of the product sold in the store has been factory tested by employees so they can give you additional info beyond the magazine ratings and marketing. How many days a year do you ride? Is your riding style aggressive? Be prepared to answer questions such as these when you visit.\nIf you're the kind of shopper that likes to try before you buy, Mountain West has a stellar demo program. Take a pair of skis or a board out for the day or weekend and test it out. Try another pair another weekend. Now apply the fees you paid for up to two rentals toward the purchase of a brand new pair of skis or snowboard. That's right, if you participate in the demo program you can apply the charges paid towards up to two of your rentals on any new ski or snowboard in the store until the end of the season. This is a fantastic deal. On the other hand if you don't feel the need to pay for brand new, Mountain West also sells lightly used demo equipment at a discount of course.\n2019 merchandise is just starting to roll in so if you want to check out a full assortment, Fall might be your best bet. If you're looking for a deal on stuff from the 2018 season, now is a great time to go. Mountain West only carries what they consider to be \"pinnacle\" brands: Volkl, Burton, Rossignol, Salomon, Libtech, Blizzard, Oakley, Smith, Giro, Jones and Never Summer to name a few. (Over 30 brands in store) You'll also find gear from smaller and local companies. It's all about quality here. Also noteworthy: Mountain West carries and demos women's-specific skis and snowboards, which is very unique.\nIf you are traveling with your gear you will want a bag to stow it all in and Mountain West offers many choices from backpacks to ski and snowboard travel bags.\nIn a nutshell, Mountain West is a full service shop prepared and enthused to meet all your snow sport needs. Whether you're a novice, expert, timid or aggressive in your skiing or riding, you will find nothing but genuinely passionate and honest experts here who care more about matching you with the appropriate goal-reaching gear rather than trying to make a fast buck.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How Are You Doing It? | How Was School?\nClenton Farquarson: How Are You Doing It?\nI wouldn't like any other child to go through what I went through. Cos I went through, I'd say hell, cos how people perceived me. Having that label follow you around as behavioural cos nobody ever.. all they did was took a report about me. No-one sat me down cos I had, I didn't realise, I had a massive report and even when I had acquired my disability, the medical file went on. George Rowley, he showed me this file they had on me and he put it down and he said 'I don't want to see that, I want to know from you'. I have to admit I was taken aback cos he wasn't using that to pre-judge me. He wanted to get his own understanding but how many teachers do that with the children? We have all these support plans but you can evolve as a person. Does these plans evolve with them? And that's some of the things for me that is important about Inclusive Education. How are you doing it and trying to do it? For me Inclusive Education isn't a thing that you tick off, it's not a check list, it's a doing.\nHere Michelle describes the lack of school facilities hindering learning.\nHere Sebastian recalls how, and perhaps why, he was bullied by a disabled person at school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MSN Music sat down with Gavin to discuss the circumstances which led him to reunite BUSH, and give even more insight into the new album. Check out the article here. MSN is also streaming The Sea of Memories in their Listening Booth. Listen here..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sanborn Canoe's paddles are so pretty, we almost don't want to get them wet.\nI'm of the opinion that all things paddling should be made of wood. Aluminum is durable, but it can be lethal when temperatures soar; plastics are lightweight, but no one wants to row placidly through a mist-shrouded lake holding something that feels like a Wiffle Ball bat.\nWood, on the other hand, is the perfect material. It has an excellent strength-to-weight ratio, and it feels like it's part of the landscape. A wood and canvas canoe may be out of reach\u2014new or restored models cost several thousand dollars, and those in need of repairs tend to take up permanent residence in backyards\u2014but wooden paddles are eminently affordable.\nPlenty of companies make them, but none do it as beautifully as Sanborn. The six-year-old business, based in Winona, Minnesota (home of Wenonah Canoes), is run by canoe-crazy cousins Zak Fellman and Todd Randall. They carve each blade by hand from high-quality walnut, aspen, and cedar, then Fellman and another local artist paint them in a variety of designs.\nThis spring, Sanborn brings out a limited run of paddles made from reclaimed Jack Daniel's bourbon barrels. It's more than just trend-hopping. (Bourbon-barrel-aged fish sauce, anyone?) The white oak used to age whiskey is exceptionally dense\u2014the toughest of any hardwood. Not that you'll be beating the paddles up; we have a difficult time just taking them off the wall. From $160.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The case has a muscular and bold silhouette. The octagonal watch chassis is topped with a mineral crystal, and a 120 click uni-directional turning bezel with luminous markers. The bezel serves to frame a utilitarian black dial featuring round and rectangular hour markers topped with Swiss SuperLuminova for increased low light legibility. The oversized minute hands, serve the watch as a clear read out of time elapsed, delivering the watch as a unique piece of diver instrumentation.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"So often I hear stories about black screens, boring static content and reflective comments about the effectiveness of digital signage such as \"why aren't my screens bright enough\"? To me, it's a bit like a car, it's not set and forget, every now and then you need to evaluate your car's performance and do some tuning, or maybe repairs. If you do this, you can expect years of trouble free operation and you will always get where you need to go.\nIt's really frustrating to me when I see digital signage not being used properly because for so many businesses that I work with, it is driving more sales, higher value sales and creating a more engaging environment for customers.\nHere's a list of questions I recommend businesses ask when they have a digital media solution in place but they're not sure if they are maximising its value. I call it a digital signage health check.\nDo you know the amount of time that your screens are live, ie, fully working and playing the content you want played? Or the reverse question is probably more interesting \u2013 do you know if your screens are ever down and if so, how often and for how long?\nIf your screens are down, there could be any number of reasons \u2013 network, media player, screen, etc, but credible service providers will commit to an uptime performance SLA (normally 99%+) and will provide regular performance reporting which highlights any problems and how they were solved. In fact, an experienced provider will be doing proactive monitoring and fixing problems before you're even aware of them.\nAre you actively managing your content?\nDigital signage is not actually about the technology, it's about the content and the impact this has on your target audience. Some organisations just loop their content and keep it unchanged for long time periods of time. It goes without saying that the effectiveness of content diminishes over time. Effective digital signage solutions use a content management system to schedule and distribute content. The simplest method is to create a 'content timetable' but there are more sophisticated approaches like tailoring the content according to a site, geographical area, weather or end device type. Or the next level is tailoring content according to the age and gender of people near the screens.\nIt can also be helpful to mix different types of content, eg, entertainment and information and intermittently intersperse promotions or special offers.\nAre your screens bright enough?\nThis is probably the most common complaint I hear. There are two main considerations \u2013 the ambient light conditions (which is impacted by screen location and orientation) and the brightness of the screen or display, measured in 'nits'. Getting the brightness level correct, will also depend on the type of content and the viewing distance but here are some general guidelines to help you optimise the impact of your screens.\nFor an in-store, inward facing screen, 350 \u2013 500 nits is normally sufficient. If the screen is facing outwards, but into an internal, covered space such as a shopping centre or an arcade, the brightness should be increased to 700 \u2013 1,000 nits. For screens facing outwards, outdoor, eg, into a street, the brightness should be increased to around 1,000 to 1,500 nits. For outward facing screens where there is direct sunlight the brightness should be 1,500+ nits. For outdoor screens, the recommended solution is an LED display because of a range of factors including contrast ratio and lifespan.. 3,000+ nits is the best option.\nOf course to be totally confident, you can do a light test \u2013 and best to do this at different times of the day. You can also go to this useful link that shows the sun impact at specified locations at different times of the day.\nAre you managing your power costs?\nSome systems run 24\/7, even if nobody is around. This is obviously consuming power \u2013 and money \u2013 for no good reason. But what's worse is where a screen is turned black but the screen is still active. A smart CMS (content management system) can actually power off your screens at a set time which is an easy way to manage costs. And this can have a significant impact on your bottom line where there is a large fleet of screens, plus there's the environment impact.\nAre you capturing valuable data and creating engagement?\nDigital signage is not a one way street. Look at ways you can capture data, for example, by adding a camera \u2013 engagement levels can be captured, ie, who is looking at your screens and for how long. And this can be further broken down with reasonable accuracy by age and gender. QR codes and NFC can be used to continue the customer journey in a different way (eg, online) as well as capture additional data.\nIs your content optimised for your screen set up?\nThere are a few considerations here, the most obvious is that content created for a landscape screen should be displayed on a landscape screen and content designed for a portrait orientation should be shown on a portrait screen. One way to solve this problem is to rotate the screen from, say, landscape to portrait, but not all screens can do this. You will need to check first.\nThe content management system (CMS) can re-scale content to optimise it for a specific screen, but ideally, the content creation process takes into account the orientation of the screens.\nSomething else to be aware if is where you have a video wall or matrix of screens: I've seen businesses that have important messages appearing on the horizon line, which means they actually appear along the bezel line, or the frame of the screens that are creating the matrix. Very important messages get partly obscured Again, the content should be created with an understanding of the screen formation and orientation. In the case of the video call, key messages should be set above the horizon.\nIs your signage performing as expected?\nThe worst answer to this question is \"I don't know\". But with no tangible objectives and\/or no process to measure, this is the most likely answer.\nIdeally, the objectives for the digital signage solution should be determined during the business case stage, and this could be increased store traffic, increased sales, sales volume change during particular promotions, store dwell time, average deal size, other changes of behaviour impacted by particular messaging, customer satisfaction during queuing or waiting, brand recall, etc.\nEqually important as agreeing the objectives, is designing the process to capture the data that supports each objective \u2013 and this normally required some planning and may require the installation of physical or peripheral devices in a particular location.\nYour digital signage network is a significant asset and requires some loving to give you back what you are expecting from it. A digital signage health check will help you identify any gaps and make the required adjustments.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Michelle, Philippa and Emma took part in the annual Race For Life event in Stratford on Sunday. They all wore pink t-shirts labelled \"Loveitts Lovelies\" and wore matching princess tiaras as they walked the 3-mile course around Stratford racecourse.\nMichelle took part in the race to raise money to help the continued work of Cancer Research UK, she has lost several friends to Cancer and knows of many other people continuing to fight this horrible disease.\nRace For Life and the work of Cancer Research UK relies on donations from the public. If you would like to make a donation please go to our Just Giving Page: Loveitts-Lovelies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An iPad case called Skinny was bound to be slim but Hatch & Co.'s case is really impressive in that regard; at only 15 mm thick (without the iPad), Skinny is the thinnest iPad case incorporating a keyboard.\nThe case has a built-in Bluetooth keyboard; the touch keyboard has a texture that gives a natural feel to typing while freeing almost 40% of the display and is spill and dust proof.\nThe battery is said to last two weeks or six weeks in standby mode. Acting as a smart cover, the Skinny case sends the iPad to sleep or makes it wake up when the cover is closed or opened.\nWhen the case is open, the keyboard is slimmer than the average pencil. The Skinny Keyboard case is also very convenient and flexibile - the user can put the iPad in stand up position and use it like a laptop.\nAll buttons are fully accessible and there is a whole in the back of the case to allow use of the camera.\nObtainable from the Hatch & Co. website for $90, Skinny Case is available in white or black with assorted colour pallets for the keyboard, brown tones for the white version or grey tones for the black one.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you want pill testing, there's a party for that!\nUnharm teamed up with Melbourne mother Adriana Buccianti to launch a petition for pill testing that's brought over 35,000 supporters together. Now it's time to build the next phase of the campaign.\nSaturday June 18, Unharm heads to Kings Cross staple The World Bar to throw a party and raise funds for next phase of the Time to Test campaign.\nCome party with your mates to DJs Dan Mac from Art vs Science, Broox.E, Baddeep DJs vs B.O.O.M.A, and Amy Parnell. It kicks off at 9pm, runs til 3am.\n$15 and all proceeds to Time to Test.\nThere will be a brief talk about pill testing, how to do it with reagent test kits, and how to stay safe.\nWhat is Time to Test about?\nFear and intimidation aren't keeping people safe so it's time to give the drug dogs a break. Taking responsibility means making an informed choice and that's why we need pill testing now. 82% of young people are already on side. With some solid campaigning and all the recent dance party deaths, politicians are starting to pay attention too.\nSimple pill testing kits for home use are already legal in Australia and you can buy them on the internet. A drug checking service would use much more accurate laboratory equipment and also provide information and education for consumers. It's a much safer version of something that already exists.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Milliman is among the world's largest providers of actuarial and related products and services. Founded in 1947, Milliman is an independent firm with offices in major cities around the globe. We are owned and managed by principals--senior consultants whose selection is based on their technical, professional and business achievements.\nThrough consulting practices in employee benefits, healthcare, investment, life insurance and financial services, and property and casualty insurance, Milliman serves the full spectrum of business, financial, government, union, education, and nonprofit organizations. In addition to our consulting actuaries, Milliman's body of professionals includes numerous other specialists, ranging from clinicians to economists.\nDespite our impressive growth over the past six decades, we still operate according to the guiding principles of our founders, Wendell Milliman and Stuart Robertson. We retain their rigorous standards of professional excellence, peer review and objectivity. We remain committed to developing innovative tools and products and providing expert solutions. We continue to earn our clients' trust by keeping our focus fixed on their business objectives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This story starts about 25 years ago when I first move to Southern Oregon. Money is tight as the economy is in a downturn somewhat similar to the economic downturn happening today.\nI buy a ranch and set about to live my life long dream as a \"cowgirl\/rancher\/ farmer hand\" on my own ranch. I love my new life and I love my job, but I find myself always short of money. I begin to do some \"horse trading\" to stretch my income and help the out flow of dollars from owning a ranch. I dabble in quarter horses, but my heart is with Morgans. I begin to dabble in Morgans and find out that I can buy, train and sell Morgans as long as the horse is a \"Classical Morgan\". I buy and sell two to four horses a year. I am not a big time operation. I just get the horses going really well and then sell them.\nThis side line brings me to a couple of horses that I will call \"Chuck \"and \"Samson.\" These two hoses have lots of Brunk and UVM breeding in their background. Samson is black and Chuck is the color of a shiny copper pot.\nA friend who lives in La Pine calls me one morning and says.\n\"Are they broke?\" asks I.\n\"Not much. You might even be able to trade something.\" Says my friend.\nPlease remember that this conversation is taking place almost 25 years ago and Oregon and especially La Pine, Oregon is still very rural and fairly Western. Horse trading is horse trading in every since of the word. A person could trade a material object and lead home a horse.\nI call the number that my friend gives me.\n\"Yes to both questions.\" comes the answer.\nI now commence to begin horse trading. \"How much do you want for both horses?\" The geldings are not babies. I will have my work cut out for me.\nMy heart breaks. I don't have a hundred dollars. Plus I have to pay for the fuel to drive three hours from Eagle Point to La Pine, Oregon. I need money for fuel as well.\nMy eyes raced around the house. I had just picked up a huge copper pot at a yard sale for ten dollars. The pot is an antique and worth a considerable amount of money\u2026maybe even two to three hundred dollars.\nI drive the three hour trip in my old pickup, hauling my old trailer.\nI follow the directions to the guy's house in La Pine. I see the horses. He sees the copper pot and I come home with the horses. The copper pot stays in La Pine. I don't know what happens to the \"family heirloom\" copper pot. But I do know what happens to the two horses. The horses come home with me.\nI sell the black horse to a neighbor who eventually sells the horse to someone else. At that point Samson disappears into the mists of ownerships.\nChuck on the other hand continues to resurface in my life for the next twenty five years.\nI sell him to a trainer in California. That trainer sells him to another trainer who sells him to a lady named Nancy Summers.\nShe and Chuck fall in love with eachother. Nancy Summers boards the horse at another lady's barn- Nancy Hazelwood Savage place in Sacramento, California. I don't know either of these women at this time.\nThen seven years ago I meet both women. Nancy Summers tells me how much she loves Chuck and thanks me for him. Chuck and Nancy win lots and lots of ribbons together.\nI tell Nancy Savage the copper pot story, but we never tell Nancy Summers the story because she paid a lot of money for the horse. Chuck died about two years ago of old age. Nancy Summers died this summer.\nI bought from her estate the western show saddle that she used on Chuck. The instant I step up on the stirrup and sit in the seat, that saddle fits me perfectly. I make no adjustment.\nThe copper pot horse has come full circle as I ride in the saddle that Nancy Summers used on Chuck. When I ride my \"Western Ride and Drive\" performance in Nancy Summers and Chuck's western show saddle, I think that these performances all started with a copper pot.\nThis sounds very familiar. That is the way that Carole Mercer and I met. I looked at a horse that Carole had at her ranch in Eagle Point when we lived in Loomis, California. She put me on the horse and we had a great trail ride on her ranch. Little did I know that that was the beginning of a friendship that lasted through the years. We are now in the midst of a wonderful time at our barn on Wednesdays where five women get together with their horses or mine and Carole trains us and we train our horses. So much fun as Carole is a gem of a teacher and she has been all over the globe taking lessons. Now, everyone is fluffy here but we have so much fun, we will continue our lessons. Time marches on. Her story is wonderful. We have had some copper pot stories of our own during the years!\nCarole always brings tears to my eyes and sings to my spirit with her short stories. I can sit for hours and listen to her re-enactments of her daily life.\nCarole helped me to re-discover my joy for horses after a horse accident left me scarred emotionally. I am forever greatful and my sweet mare sugar bear thanks her too.\nAs a non-horsewoman, I always particularly appreciate the vivid picture Carole paints of Western life and values.\nI'm curious, however, about the past and future adventures of the copper pot. I bet it has a story to tell too \u2013 or maybe Carole can make one up.\nI was so excited to see your name pop up on my email. I love your stories. They touch my heart!\nI thank you also for sharing your world and love of horses with me. I still get giddy each time I ride.\nYou always have a place around our campfire. The stories we could tell about each other would have them rolling, but the best one was the day I bought my beautiful Morgan mare \"Miss Sally\". She has been such a delight for the past 18 years, and I hope to have a lot more with her. Thank you for the memories and never give up your love of writing.\nAnother great story\u2026 It's almost unreal how these horses and people and pots and saddles are all woven together in this big adventure of life. You are a wonderful teacher, story teller, writer, and friend. I miss you very much! Oh how I would love to come see you and take a sleigh ride and listen to the bells. You are a mystical, magical woman. I'm so blessed to know you!\np.s. didn't Sarah and Kevin do some vaulting on Sally when we were kids?\nIt's great to hear from you again. I wondered what had happened to you over the summer. I was afraid that you had given up Above Level. It didn't occur to me that you must have been busy too.\nWow, a family-pot for 2 horses and new wonderful friends and now you ride Nancy\u00b4s S. saddle on shows and you have her in great memories.\nI give your story to my friends her in Austria and Spain. They will enjoy it much.\nThank you to have such a good friend and that you share a part of your life with us.\nI\u00b4m sad that you live overseas \u2013 very very far away from me here in Europe.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Displaying 1 to 10 of 27 Fiction Publishers in Canada.\nLiterary and commercial fiction and non-fiction by primarily Canadian authors; children's and YA.\nPublishers of high-quality picture books, fiction and non-fiction for children and young adults.\nPublishers of quality fiction and non-fiction books for children, by children, of all ages.\n27 Fiction publishing companies in Canada in the directory.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dayton, Ohio Affiliate Chapter of A Special Wish Foundation, Inc.\nMission: To grant the Special Wish of children\/adolescents, birth through 20 years, diagnosed with a life-threatening disorder (based on medical criteria). We serve children from Montgomery, Miami, Greene, Darke, Mercer, and Shelby Counties.\nHave you ever worked with another Wish Granting Organization before?\nDo you wish to receive our newsletter?\n\u00a9 2019 Special Wish Dayton, Inc. All rights reserved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\"We must break the stories of slavery in order to inspire a process of resilience -the capacity to overcome negative conditions with love.\"\nOn 22nd February 2001, two of us from Divine Word Missionaries went to Mother Theresa of Calcutta home to share our lives and mission. Mother Theresa of Calcutta homes were founded in 1988 by an SVD bishop, Jorge Novak, in Quilmes, South of the great Buenos Aires in Argentina.\nCurrently, there are seven homes dedicated to the care and development of children and adolescents who cannot live with their families of origin. Basically, there are three groups of children: the orphaned, after the death their parents; those that live functional orphan status due to either imprisonment of their parents or their parents have terminal illnesses or have some form of psychiatric disorders that render them unable to take good care of their children; and finally, children who are victims of their own relatives, due to a number of addictions, alcoholism or crime and may also involve the children in these or may subject them to some mistreatment resulting in physical abuse.\nThe home currently houses 70 children and adolescents, giving them an alternative home from their families of origin. There they are assured of a place to sleep, food, health care, recreation, education, training in workshops and professions. It is a fraternity kind of model, where by those who take up the responsibility to take care of the children live with them in these homes, taking up all the responsibilities related to proper to upbringing. We don't work there, we live there.\nSince 2010, the home is being managed by the Asociaci\u00f3n Una mano que ayuda (Association of a hand that helps). It is an organization founded to give refuge as well as juridical and administrative assistance to the homes and other places that dedicate themselves to the care of infants. All of us who live in the homes and offer different kinds of services are socials to the entity.\nSeeing the growth and the process of development that these children undergo after months and years of pain, mistreatment and abuses, is itself a gratifying experience. During their stay with us \u2013that could be extended if need be, we visit their biological families to assess their conditions, and from this, we discover that there are so many families that for a number of reasons, have no capacity to be parents. Without judging their present situation, many are at the same time victims of traumatic past. Initially, called to be happy and build strong ties, they find themselves hurt and injured in a manner difficult to reverse.\nAll the children participate in a weekly psychological therapy forum. Listening to their stories about life of slavery, hunger and beatings in their own family environment, generates situations of grave pain. Reversing, curing, consoling them assisting them to part from these stories is our mission. We must break the stories of slavery in order to inspire a process of resilience -the capacity to overcome negative conditions with love.\nThe children and adolescents graduate from the homes with the help of relatives that are ready to give them better living conditions; others graduate through adoption; and finally many graduate in autonomous processes, with their own means and more challenges than any other young people because they have to overcome their condition without the help of relatives. All the same, today we are witnesses that they are capable of finding meaning out of their pains and live in a mature way, assuming the challenges that life presents to them.\nCurrently we are three SVDs residing in the Mother Theresa of Calcutta home. We live with one group of adolescents; we animate formation workshops as well as being responsible for the management and the administration of the home. We play, share and live our mission helping these children and adolescents to discover that God has plans for their lives.\nPresident of the association: Una Mano Que Ayuda.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Stained concrete floors for a modern basement with a. Mode concrete: hip and modern basement concrete floors. Epoxy shield basement floor coating canada sala. 33 best images about concrete floors on pinterest. Steps for easy painting basement floors homesfeed. Greatmats specialty flooring, mats and tiles: top 5. What to consider in choosing the right basement floor.\nAntique bakers cabinet sellers bakers cabinet. Backyard patios backyard patios ideas backyard patios. Mirrored bathroom vanities, mirrored bathroom vanity. Accent chairs to complement your upholstered bed. Waterproofing basement from inside [peenmediacom]. 10 frozen movie inspired kids#039; room decor ideas. Colorful kids rooms. Furniture : sauder bookcase with glass doors sauder desks. 30 brilliant diy bathroom storage ideas architecture. 46 paint colors farmhouse bathroom ideas round decor. Ceiling fan home log bedroom ceiling fans with lights.\nUpper kitchen cabinet height image to u. Diy filing cabinet desk northstory. Ana white kitchen buffet diy projects. Kitchen cabinets sizes standard base cabinet height. Standard kitchen cabinet dimensions house furniture. Rustic solid wood ceramic wine bottle storage buffet. 10 x 10 standard kitchen dimensions cabinet sense.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bravo 4201-Series are easily the fastest desktop disc printers, dvd duplicators, and publishers in the world today. At just six (6) seconds to print each disc with 100% coverage in near-perfect quality, you'll be stunned at how fast your jobs will now get completed.\nIn fact, Bravo 4200-Series is up to 20 times faster than competitors for similar print quality*, making them the fastest desktop disc printers and publishers in the world.\nIn applications such as medical imaging and audio\/video-on-demand it is imperative that discs are dispensed out the front of the unit. On Bravo 4100-Series, it's an automatic, built-in feature. On some competitive units you have to manually open a door or a drawer every time \u2013 a huge inconvenience if you're doing it all day long.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Circle of Security\u00ae is an innovative intervention program designed to improve the developmental pathway of children and their parents. Robert Marvin, at the University of Virginia in Charlottesville, and Glen Cooper, Kent Hoffman, and Bert Powell from Marycliff Institute in Spokane, Washington were the originators of this unique, evidence-based program.\nThe Circle of Security\u00ae started as a \"user-friendly\" description of Attachment Theory. It then integrated over fifty years of early child development research into a video-based intervention that strengthens parents' abilities to observe and improve their caregiving skills. Attachment theory, through the Circle of Security\u00ae, offers clear, individualized pathways for providing a secure relationship between parent and child.\nEvery child comes into the world seeking a secure relationship with her\/his caregivers. The Circle of Security\u00ae program helps promote that security.\nA secure attachment between child and caregiver is critical to a child's current and future well being. University-based research has shown that secure children have increased empathy, greater self esteem, better relationships with parents and peers, enter school more ready to learn, and are able to handle their emotions better than their less secure peers. As they grow older, secure children become less likely to live in poverty, have legal problems, or experience chronic emotional difficulties.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NEW YORK, United States \u2014 A few weeks ago, I sat down for lunch with Ariana Boussard-Reifel founder of Mode Marteau, an online designer and vintage boutique, when it hit me that, it is possible to live a stylish and truly creative life in New York City! Now one might think that NYC is filled with creative people, which it is, but most of these people have a day job and a creative job. It is challenging in this expensive city to truly live an artistic life!\nAriana Boussard-Reifel has lived in NYC for over 10 years, having moved there shortly after college. Being a sculptor and painter from rural Montana, the big city lights were captivating and exhilarating! She began working in an art gallery and making her art on the side (a typical combination for most New Yorkers). As she was still new to the city, but an expert at shopping in used-clothing stores, she began combing charity shops, thrift stores and flea markets for stylish high-fashion outfits she could wear to work at the gallery. Knowing that she always had a knack for discovering designer clothing in unusual places, she began filling her closet with great finds. Soon her tiny apartment was exploding with clothes!\nSlowly, as a side-project, Boussard-Reifel began selling the clothing and accessories she had collected on eBay. Mind you, this was over 10 years ago when online resale was an emerging commerce. Soon enough though, eBay was becoming a full-time income. Boussard-Reifel realized that she had better access to fashion forward designer clothing as NYC is a trend-forward, high fashion city with women who have limited closet space thus, with the right eye it was easy to find beautifully made clothes, be they designer or vintage, at thrift stores and charity shops! Flipping Hermes ties, Chanel purses, and Ralph Lauren dresses allowed her to enjoy the hunt and make a living on her finds.\nOne day while digging through a bin of clothing, Boussard-Reifel was approached by a reality TV producer who wanted to start a show on thrift store shoppers who resold their finds for a profit. Being the perfect candidate, Boussard-Reifel, signed up and was filmed for the show. Although the show may never reach the public audience (the production is still in talks with airing the show) it forced Boussard-Reifel to formalise her business last year and create a brand. Thus the birth of Mode Marteau, a fashion forward designer resale online shopping destination.\nMode Marteau is a chance for Boussard-Reifel to connect with her clients and build a business and a brand behind her unique discoveries.\nAs Boussard-Reifel says, \"The Mode Marteau shopper is a woman with confidence, style, and the smarts to know that she doesn't have to pay retail to look exceptional\".\nVisit her website to see what she has discovered this week and if you are in NYC check out her antique and vintage collection at the Manhattan Vintage Fair on 25-26th October 2013. For all you lovely Fashion Globe readers, Mode Marteau ships internationally!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"HTC has always been interested in developing products of Virtual Reality. HTC has designed Virtual Reality devices in the market that has been hugely popular amongst the gamers. HTC has once again come up with a very new Virtual Reality headset which can be used with the help of a PC called the Vive Cosmos.\nHTC had made a very brief appearance at the CES and during the presentation, they mentioned this technology and product of Virtual Reality they have been working with and will launch with the support of PC. This announcement has made a lot of people interested because there is a great number of people who love using PC and this will make them to use virtual reality devices to play different games and check out different apps.\nAccording to the news, the Vive Cosmos is one of the biggest mainstream Virtual Reality headsets from HTC. This headset is well designed and this is the reason that even if there are no external sensors, it still will be able to use fully tracked motion controllers. The best thing about the device is that it is extremely convenient and comfortable with easy setup options. This Virtual reality device can be used for both homes as well as mobile.\nHTC has still not announced the price point of the virtual reality headset and this has made a lot of fans curious. People still want to know the capabilities of the headset and so HTC came to Twitter and made an announcement that the virtual reality headset is coming soon. In the press release of the HTC of Vive Cosmos virtual reality headset, it was mentioned that the development kit will be coming in early 2019.\nIt is expected that the Vive Cosmos will first support in the PC and gradually, there will be updates available where this virtual reality headset from HTC will be able to support all the platforms available. It was in 2018 when HTC filed a trademark over the name Vive Cosmos and since then the speculations have started.\nIn the first week of January, everything was cleared after HTC came out and announced their Virtual Reality Headset. According to HTC, this virtual reality headset Vive Cosmos will be able to bring a revolution in the world of virtual reality.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"For years the mathematics books at Foyles bookshop in London had their own room. It was a strange place, to the uninitiated inexplicably yellow. It had its own quirks rules and legends. There were shelves whose books were not for sale and, should you find a book that was for sale, you had to try to sneak it out if you wanted to purchase books elsewhere before leaving.\nWhen I entered the room for the third time I was a PhD student in London. I had the practical purpose of finding a book I needed, but I became entranced. It became a regular place to visit, gaining familiarity and comfort. During hard times in my PhD and later jobs in London it acted as a refuge. At some point earlier memories returned.\nOld-School Foyles, but not the mathematics room. Do you have an image?\nAnalytic and Algebraic Topology of Locally Euclidean Parameterization of Infinitely Differentiable Riemannian Manifold.\nSome books I simply gazed at, others I bought simply for the magic of their titles. Those titles echo in my memory. Today some have become trusted friends, some sit mysteriously on my bookshelves having resisted numerous attempts on their secrets, others turn up like old acquaintances when I visit book shops.\nI did that recently, I was once again in Foyles. The mathematics section had moved once more. It was once again in a familiar room at the front of the building on the third floor. Had they come home (albeit sharing the space)? But wasn't the old maths room on the second floor? To my shame I could not remember clearly. I had grown used to the room being gone; it was a shock to find the details of my memory so weak. My internet skills failing me, I could not even find a record that gave the floor, worse, I could not find mention of the room at all. So I decided to write this.For me it makes concrete memories that seemed routine at the time, but now hold great importance, but perhaps I am not the only one? Maybe there are others who have fond memories of this room. If you do find this and remember please share your memories of this odd, impractical but special room.\nI mourn the room, but do realise that some things have to change. Today the nature of the book itself is changing, and with it the bookshop. Just opposite Foyles the space that used to be Borders bookshop is now taken up with TK Maxx. With the ability of the internet to deliver information and, electronic readers finally usable, the paper book finds competition it has never had before. Yet the bookshop, as I adore it, has been under threat for a long time. Borders itself along with Barnes and Noble represented the first assault, opening up the bookshop and making it easy to navigate. Then Amazon opened things up further, making it possible to easily find any book in print. Yet great bookshops, like Foyles, have survived, I have faith, there will be changes, but some of what we love in these stores will survive and perhaps some of what will be lost needs to go. Is it such a bad thing that cheap dectective and romance novels will no longer force trees to be cut for their paper?\nFor the moment therefore I try to regularly visit the bookshops I love and buy books from them. Not just for the books themselves but as a support for those wonderful shops. It makes a good excuse anyway!\nAs a regular visitor to Foyles I learnt certain routes around the building, mixing the stairs and elevators. It felt that a tiny deviation from the correct route could leave you in a different place entirely. It was occasionally a shock to realise that two points, that I had thought were in completely different parts of the building, were actually just around the corner from each other. Terry Pratchett describes this best with the concept of L-space, that all libraries, and bookshops in the world are connected both in space and time and, with the correct path, you can navigate to any of them. In the Discworld version of the burning fire of Alexandria a hairy arm is seem amongst the flames rescuing some of the greatest works.\n\u2190 Why not knot wire?\nRemember the woman who you had to take your books to so she could write the receipt which you'd take to the cashier so you could pay and get a stamp which you'd take back to her in exchange for the book?\nYes, I can remember putting a lot of effort into sneaking the books out of the room! You could normally buy them normally at the cashier.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Further more, we have close realtionship with producers of dead burnt maganisiaand calcined kaolin du Dead Burnt Magnesite DBM90 Dead Burnt Magnesite is produced in the Rotary Kiln by sintering raw magnesite at a controlled temperature of 1750 degree centigrade.\nEstablished in 2009, Henan Fengde Machinery Manufacturing Co., Ltd. has been a leading name in offering top notch Cement Plant Machinery.Located in Zhengzhou, Henan, we offer the best quality Kaolin Limestone Rotary Kiln to our customers at best prices in the industry.\nVarious applications of kaolin require calcination to enhance clay mineral properties and to provide added value to the material. Calciner furnaces such as rotary kilns and multiple hearth furnaces (MHF) are widely used in industry for the calcination of kaolin.\nKaolin Processing Plant. Kaolin Processing Plant, Wholesale Various High Quality Kaolin Processing Plant YUHONG 250-500TPD Lime rotary kiln, Active Lime Processing Plant . China made kaolin processing plant\/kaolin powder processing plant for sale Supply vietnam filtering\/drying\/grinding one stop service for kaolin processing plant.\nGypsum Rotary Kiln, Wholesale Various High Quality Gypsum Rotary Kiln Products from . 2014 China energy saving large capacity kaolin rotary kiln burner. Chat Online Clay Rotary Kiln, Clay Rotary Kiln Suppliers and Manufacturers at .\nAs a famous rotary kiln manufacturer, Zhengzhou Taida can produce multiple rotary kiln equipment to meet different processing demands of customers. Kaolin rotary kiln. Kaolin calcination rotary kiln (bauxite calcination rotary ki. Metallurgy rotary kiln.\nKaolin (china clay) is a hydrated aluminium silicate crystalline mineral (kaolinite Al 2 Si 2 O 5 (OH) 4), formed over many millions of years by the hydrothermal decomposition of granite rocks.Rocks that are rich in kaolin or kaolinite are known as china clay or white clay.\ncement lime calcination BONY offers a wide range of refractory materials meeting the constraints of calcination processes Mineral calcination rotary kilns producing aluminas, chamottes, kaolin, etc.\nkaolin ore rotary kiln price in nigeria digitalone.co.in. iron ore for kaolin in cameroon vibrating sieve separator. calcined kaolin is an aluminosilicate material produced by heating natural kaolin to high temperatures in a kiln this silica kaolin . Coke dryerPetroleum coke dryerCarbon coke dryerCoke .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2. Quarter the Cabbage, discarding any tough or wilted Leaves. Plunge into boiling water. When water returns to a boil, as 1 TB of salt. Boil Cabbage about 10 minutes uncovered. Refresh it in a colander under cold running water. When cool, separate the Leaves and spread them out on paper towels and blot dry.\nWhen they are thoroughly dry, sprinkle them with a little Salt & Pepper and about 2 tsp of Cumin. Set aside.\n3. Saute chopped Onions gently in the Oil for about 5 minutes. Season with Salt & Pepper and cook until the Onions are soft and slightly brown. Set aside in a large bowl.\n4. In the same skillet, brown the sausage meat and drain on absorbent paper. Cut the boiled Ham into tiny pieces. Stir the Sausage and the boiled Ham into the Onions. Beat the Eggs with the remaining Cumin, Allspice and Paprika, and add to the Meat mixture. Set aside.\n5. Peel & core the Apples and cut into 1\/4\" dice. Rub a casserole with 1 TB Oil. Line the bottom and sides with a layer of Cabbage Leaves and spoon in a little Heavy Cream. Then put a layer each of Apples, Cabbage Leaves, half the meat mixture, spooning some Cream between each layer. Finish the casserole the same way, filling it to the top, and finishing with a layer of Apples and a layer of Cabbage Leaves.\n6. Cover the casserole and bake in the preheated oven for 1\/2 hour. Then reduce the heat to 350 degrees and finish cooking for 1-1\/4 hours. The mixture will be very smooth and tender.\n7. Plate as a one dish dinner.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The phrase \"valley of death\" sounds very ominous \u2013 and it is. It is the phrase used for the time (and resources) between basic research and commercial operation to move projects out of the laboratory and into the marketplace.\nThe Fats and Proteins Research Foundation (FPRF) is focusing on commercialization of promising research. The primary goal of FPRF projects has been to publish scientific data that could be used to advance the knowledge of rendering processes, the use of products in animal diets, and the quality of fats and proteins. That has dramatically changed with several projects at the Animal Co-Products Research and Education Center (ACREC) at Clemson University having some possibility of patentability or commercialized product. FPRF members fund research that develops the underlying science for groundbreaking projects, but there is an inherent challenge to get these projects to marketplace.\nResearchers tend not to know how to bridge the gap between research and business as that is not their role or training. Many are also not interested in commercialization because the usual university reward and advancement systems stress publication over implementation. One ACREC project that has come the furthest along the commercialization path is a novel antioxidant developed by Drs. Alexey Vertegel and Vladimir Reukov. Both researchers realized there was a need in the marketplace for a new antioxidant that could prevent rancidity and provide a longer shelf life in rendered products and pet food. The antioxidant market is a large one and for good reason. Most rendered products are not refrigerated and require antioxidant application to stay fresh.\nVertegel and Reukov were not daunted by taking a path most researchers avoid \u2013 forming a business called VRM Labs. The two researchers created a product named \"Prot-X,\" which is a potent natural antioxidant similar to those already present in the body. Vertegel and Reukov developed a proprietary production process from animal or poultry blood that is as efficient as synthetic antioxidants. All components used in production are designated by the Food and Drug Administration as \"generally recognized as safe,\" which is essential in the marketplace. The production cost is low so the cost to the customer can be competitive with that of current natural antioxidants. The product also appears to have some antimicrobial properties in early trials.\nThe path to marketplace can be rocky even for cost-effective, promising products. FPRF funded four studies developing this antioxidant and the intellectual property belongs to Clemson University. VRM Labs has obtained an exclusive license for the technology from Clemson with half the funds the product generates being recycled back into ACREC where they will be available for future rendering research. High-value projects like this one could significantly increase ACREC funding for more rendering research.\nVertegel and Reukov realized they needed the assistance of someone who knew more about starting a business in the animal feed industry so they teamed up with Dr. David Meisinger, executive director, US Pork Center of Excellence, and created VRM Labs. In 2014, the team collaborated with Iowa State University to examine scaling up the process. Iowa State has a pilot plant suited to larger extractions and experiments and they were successful in making large quantities of the antioxidant. However, the next two attempts failed to optimize the process and improve yield as expected, but efforts kept going. Each failed attempt added to the base of knowledge about industrializing the process.\nYet all of these attempts take money and time, making it difficult for such a small business to get off the ground. Vertegel and Reukov spent much time writing grants for small businesses and approaching \"angel investors\" that have been quite successful, with the company currently valuated at $3.3 million. The product has won or been a finalist for several awards, including winning the Asia Pacific Ag Innovation challenge and the 2014 InnoVision Award in the small business category. All of these awards also bring recognition to FPRF.\nRegulation is also a burden that can be costly and time-consuming to bring new products to market. Meetings with government officials in Washington, DC, are often required and can be problematic for a start-up. These trips take time and money, and must be repeated during the process.\nThe valley of death is a challenge faced by universities, research centers, and granting agencies. FPRF and ACREC are focusing on solutions. Most researchers are not interested in developing a start-up business to market inventions, and established marketers and companies are reluctant to pay for inventions only proven in a lab. FPRF's budget is well-suited to fund basic research underlying patentable new innovations. However, enthusiastic inventors along with FPRF support beyond the initial funding can bring in venture capital and eventually result in a home run.\nWhat has been the biggest hurdle with bringing your new antibiotic to commercialization and what advice would you give researchers just beginning this process?\nReukov: For me, the biggest hurdle was to find an optimal testing protocol that will suit all industries. My advice is that if your technology requires regulatory approval, start talking to regulatory agencies as early as possible.\nVertegel: From my perspective, getting funding. My advice would be to do some entrepreneurial training, such as with the National Science Foundation (NSF) Innovation Corps (I-Corps), and have good legal support, which is important for negotiations about license and investments.\nMeisinger: I'll answer a different way. First, we were extremely fortunate to find a legal firm that catered to agricultural start-ups that also assumed part of the risk while in early development. They helped us to form and register a new company, including the vesting and shares process, licensing of the technology, the patents, and confidentiality issues with other companies.\nSecondly, we were again very fortunate to receive a NSF I-Corps grant that forced us to do market development in the most deliberate and structured way imaginable. We ended up talking to 137 potential customers and partners that led us to completely rethink our customer value proposition.\nI think the biggest challenges are to properly set up the company as a legal entity, to develop a sound business plan and business prospectus, and to do a very robust market development thought process and plan.\nThose are the primary components of my advice to anyone beginning this process. It is a very long and detailed path to commercialization that requires endurance and perseverance.\nFocus Shifting Away from Biofuels in Europe?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This study provides an overview of the global water sector in 2016, highlights key water sector trends and emerging opportunities moving into 2017, and offers broad strategies for capitalization.\nThe municipal segment continued to dominate the global water sector revenues in 2016. However, the regional revenue contributions underwent significant change as large international participants moved out of the slowing markets in Brazil, Russia, India, and China (BRIC). These participants have set their sights on frontier markets which offer high growth potential, albeit at an increased business risk.\nThe competitive landscape is being shaped by the push for a circular water sector, the increasing adoption of smart infrastructure, and the emergence of new business models. In 2016, the market witnessed the active consolidation of participants in an effort to capitalize on these trends.\nOverall, the focus of participants worldwide was on gaining new clients by strengthening and expanding their product and service portfolio. However, their strategies differed based on their current strengths and capabilities.\nThe market environment is seeing rapid change due to market forces and trends focussed on improving the cost optimization and resource efficiency of water and wastewater services. Increasingly scarce water resources and a lack of investment in the sector have made utilities reconsider the fundamentally flawed system of water rebates. Political factors permitting, utilities are choosing to ensure long term sustainability of operations with more aggressive water pricing and increased metering of water connections.\nOther measures being adopted by the industry at large include circularization and decentralization of treatment systems, restructuring of utility functions, privatization of utilities, and the adoption of new business models.\nThe market trends are analyzed for the next year, with the base year being 2016. This study has a global scope and includes the municipal water and industrial water markets and its sub segments including design and engineering, operations and maintenance, and treatment technologies. The market forces have been categorized and discussed under three categories: Transformational Trends, Technological Trends, and Emerging Business Models. Companies mentioned in this outlook study include Arad, Huawei, Sasakura, Sensus, Suez, Veolia, and Xylem.\n\u2022 What did the global water market look like in 2016?\n\u2022 What are some of the macro and micro trends and what are their implications on the market landscape?\n\u2022 What are some key growth opportunities for market participants?\n\u2022 How can companies capitalize on these growth opportunities?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Adding Products: How do I set up my product's Meta Tags?\nMeta Tags are elements of HTML coding that contain informational details about the page's content. Many search engines use this data to catalog web pages and return them as search results. This is part of the proper search engine optimization (SEO) process and can vastly improve your site's rankings in search engine results.\nBy default, your 3dcart store will take the product's Name, Description and Keyword fields (specified in the Information Tab) and use these fields to create basic meta-tags for your product listings. However, you may want to manually configure your product's meta-tags and SEO parameters to have more control over how the products are cataloged. The product tab labeled \"Meta Tags\" is where this can be done.\nBy default, your 3dcart store will take the product's name and use it in the page's title bar located at the top left of your browser window. However, if you'd like to specify your own title for the product, it can be done in this section.\nBy default, your 3dcart will take the product's name (from the Information Tab) and apply it to the product's HTML page filename. However, since the file names on 3dcart are database driven, the filename will also contain catalog ID information. For example, a product with the name \"T-Shirt-White\" will generate a filename of T-Shirt-White_p_[catalog_id#].html. If you'd like a more practical name for your product's page, you can configure this in the \"Custom File Name section.\nFurthermore, when specifying the custom page name, the page must have either an .htm or .html extension on it to work properly.\nUse this field to specify a 301 redirect for the product. This can be useful if the product's listing previously had very good SEO ranking but is now discontinued. Rather than sacrificing the previous ranking, the Redirect Link can be used to link the product's listing to a different item. Perhaps one that not discontinued or otherwise unavailable.\nAs stated, your 3dcart store will take your product's description and keywords information and generate meta-tag keywords from these areas in the Information Tab. However, if you'd like to tailor your meta tags yourself, they may be placed in this section. These can be entered either manually, or by using the Tag Wizard if you are not comfortable with HTML coding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This year's event will specifically focus on innovations within the field that has the potential to drive the pharmaceutical industry into a new era of medicines, with optimisation of the RandD process to become cheaper, faster and increasing the likelihood of drug efficacy and success. These are some of the major obstacles within drug discovery, as the process is overall long, expensive and often potential drugs will fail late in clinical trials.\nThe event will also focus on novel techniques that can be implemented within the field today that identify new druggable targets and possibly open new therapeutic areas within clinic and also look at various developments within the field, guiding companies on how they can best utilise the newest technologies and drug discovery modalities to develop new medicines, reduce their RandD cost, timelines, and ultimately, increase their medicine output for commercial success.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I know I've reblogged this before, but I think it is a great poem and many people enjoyed this last time. Here's one for all the moms that \"want to remember\" and know that special times with little ones never really last for very long. Thank God for memories! Enjoy!\nI wrote this just recently as I started a new job and seemed I would never get back to my drawing again. I get depressed and sullen when I can't get back to the drawing board and express my emotions in art. It's just a part of me.\n*I've come to say goodbye.\nAlthough I will think of you often, I have to grow up now and think of other things.\ngiving me the hope that I had something more to offer in my lifetime, so I wouldn't feel so dead inside.\nliving a life full of creativity and art and making a difference in the world.\nYou will forever be on my mind, but I will try hard to forget you.\nlike schedules, appointments and the daily rush.\nSo see, it won't be so hard to forget you.\nBut you will always be a part of me and I know the memory of you will come out in random times of quietness and solitude.\nbut I guess it wasn't meant to be, you and I.\neven the time to our relationship, so you could grow the way you should have.\nand trying to find someone to bring this all together.\nSo goodbye dream. Goodbye for now.\nI'm grateful you came into my life when you did.\nand what could've been, and somehow put us back together.\nAnd maybe then they will see our relationship wasn't \"all for nothing\".\nI just wont be here to see it\u2026 and that's okay.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"MISSION STATEMENT: The Central City Police Department exists to serve all people within our jurisdiction with respect, objectivity and compassion. We are pledged to the prevention of crime and the protection of life and property; the conservation of peace, order and safety; the enforcement of laws and ordinances; and the safe guarding of contitutional rights. With community service as our cornerstone, we are driven by objectives to promote the quality of life, investigation of problems as well as incidents, seeking solutions, and nurturing a sense of security for the residents and visitors of the community. We cultivate public trust by holding ourselves to the highest standards of performance and ethical conduct. To perform this mission, the Central City Police Department is dedicated to providing a quality work environment and the development of its members through effective training and leadership.\nThe Central City Police Department consists of a Police Chief, Police Lieutenant, 3 Officers and a Secretary. We maintain office hours from 8:00 a.m. to 5:00 p.m., Monday through Friday. The office can be reached at 308-946-3003 during these office hours. Non-emergency calls after hours can be made to 308-946-2900 (Merrick County Sheriff). All emergency calls should be made to 911.\nNebraska Crimestoppers will accept anonymous tips regarding criminal cases and may provide a reward for that information. When you call, the staff members will not ask for the caller's name. Instead they will assign a Nebraska Crime Stoppers identification code number to the person, and obtain the information, thereby maintaining your anonymity. Call Crimestoppers at 1-800-422-1494 or log into the link above to submit online tips.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Super Soaker Quick Blast is a novel pump-action blaster released in 2008 by Hasbro Inc. While a full pump yields a solid stream, a partial pump actually ends up storing pressure within the blaster. This makes for interesting stream performance from the Quick Blast.\nThe Quick Blast features a single, slightly larger than average nozzle. Upon firing, the nozzle produces a respectably thick stream. The intriguing part is, unlike typical piston-based blasters, the Quick Blast will only fire once adequate pressure is applied on the nozzle valve. If one pulls the pump slowly, the blaster will not fire immediately, instead slowly building up pressure within its small pseudo-pressure chamber. When optimal pressure is reached, the nozzle valve snaps open, unleashing a potent stream. One single full pump is enough to get the blaster to fire. However, for those unable to push as hard on the pump, stream performance will still be quite good. When the nozzle valve is in its open position, force applied to the pump affects stream speed to some degree. What this nozzle prevents are dribble shots from a slowly pulled pump.\nOne minor drawback is that there is no manual way to open the nozzle valve after use. Thus, the Quick Blast must be air-pumped several times, generating mist shots, in an attempt to flush out the internal pressure chamber. This makes it rather difficult to properly dry out the blaster after use.\nAs noted already, the Super Soaker Quick Blast is a piston-based blaster, but it not necessaily fire from the nozzle with every pump. Despite not having a trigger, the Super Soaker Quick Blast has an internal spring-based pressure chamber that controls the nozzle valve. Enough water (or air, if cleaning) must be pumped into the pressure chamber before the blaster will fire. That said, one full length pump will push enough water into the pressure chamber to trigger the activation of the nozzle valve. Pumping, itself, is smooth and the pump pushes a respectable amount of water per cycle. The pump grip has some texturing and has a hooked shape, making it easy hold and less likely to slip from one's hands even when wet. Thanks to the Quick Blast's nozzle valve, force of pumping does not directly translate into stream performance, with slow pumping still able to produce a relatively strong stream, though the firing will be delayed.\nThere is no manual trigger on the Quick Blast. The grip, however, has a good size and length, accomodating even the largest of hands comfortably. The grip features some light texturing as well on its surface, making it fairly secure to hold.\nThe reservoir on the Quick Blast holds a good amount of water for a blaster of its size. The choice to use a screw-on bottle with a long intake tube is a little odd as a fixed reservoir with a cap would have made for a more solid design. The intake tube can make use of the majority of the reservoir's contents, but cannot use the full volume. As well, due to the reservoir's placement with respect to the grip, the Quick Blast feels rather back-heavy when completely filled. The tail fin on the reservoir is strictly ornamental. Unforunately, on the model tested, the tail fin also rotated freely if turned, thus could not be used to assist in reservoir attaching or removal.\nAs a whole, the Super Soaker Quick Blast is an interesting, decent, light, piston-based blaster. Thanks to its nozzle valve, the Quick Blast virtually always produces a good-powered stream (though it is possible to make the stream dribble if one tries). The main drawback is that there is a brief delay between pumping and stream generation. As well, if not enough pressure is generated on an incomplete pump, the Quick Blast will not fire at all. This can be of benefit for those who cannot pump as forcefully as streams from the Quick Blast have a minimum force to them. Then again, in the hands of an experienced water warrior, the Quick Blast can still be quite effective for its size. Overall, the Quick Blast can make for a great back-up or last resort blaster as well as a good option for small engagements or even scouting missions. Of course, versus larger water blasters capable of producing continuous streams, the Quick Blast's utility meets its limit.\nSimple styling, clean lines, and good solid feel to the soaker. Decent pump volume and reservoir capacity. Dribble shots virtually eliminated.\nPiston-based; continuous streams not possible. Delayed or even aborted firing if not enough pressure generated from pumping. Difficult to empty the pressure chamber; no manual way to open the nozzle valve. Screw-on bottle-type reservoir slows refilling. No strap.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Use the Windows Remote Target Connector to manage local Windows accounts, and the passwords for local Windows services and scheduled tasks. To manage Active Directory accounts, we recommend using the Active Directory Connector. The Windows Remote Target Connector is an alternative to the Windows Proxy, but does not require installation on each target server.\nNote: Windows Remote connector might incur extra overhead during discovery and password changes for services and scheduled tasks.\nThis connector uses Samba commands and remote Windows API calls to make updates to the account, services, and scheduled tasks passwords.\nNote: If the guest account in the domain or on the target server is enabled, the Windows Remote connector can appear to verify the password of the target account though the account does not exist on the target server. Disable the guest account in the domain or on the target server to avoid this false password verification.\nIf User Access Control (UAC) is enabled, and the Windows Remote administrator is a \"local\" admin on the target server, follow these steps. Set this registry value on the target server for the Windows Remote connector to have access to perform SMB and WMI operations there.\nThe Windows Remote Connector can be run in the context of local system. This scenario allows successful updates to the local accounts, services, and scheduled tasks.\nTo enable management of local Windows accounts and the passwords on Windows services and scheduled tasks, the Windows Remote Administrator account that is added in CA Privileged Access Manager must be a member of the Local Administrator group on the server hosting the Target Account being managed.\nFor information about setting up Windows Remote Connector using the UI, see Configure Windows Remote Target Accounts.\nUse the following more parameters when using the CLI to add a target application that uses the Windows Remote target connector.\nSpecify the type of account to use.\nThe type of account being managed.\nThe Windows domain for the managed accounts.\nThe Windows domain for the managed accounts. This attribute exists only for backwards compatibility. We recommend using Attribute.domainName instead.\nDetermine the level to which DNS is used.\nThe host names of the DNS servers to use.\nRequired if Attribute.useDNS is set to specifiedDNS none Comma separated list of DNS server host names.\nProvide a comma separated list of domain controllers.\nRequired if Attribute.useDNS is set to specifiedServers none Comma separated list of valid domain controllers.\nThe Active Directory site. This parameter is only used if Attribute.useDNS is set to retrieveDNS or specifiedDNS. If a value is given, Credential Manager uses the value to narrow the search for domain controllers, using the specified name.\nUse the following parameters when using the CLI to add a target account that uses the Windows Remote target connector.\nSpecify the extension type to use.\nSpecify whether to use the target account or a different account to perform password change requests.\nSpecify which other account to use to perform password change requests.\nA valid target account ID.\nMultiple services are delimited by the | character.\n<hostname>is the name of the server where the service is hosted.\n<hostname>is the name of the server where the scheduled task is hosted.\nThis parameter specifies whether Credential Manager updates passwords that fail verification during an initial synchronization. The default value is false. To update passwords that fail initial synchronization, set the attribute value to true.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Wishing you a very merry Christmas! We are on the plain headed towards one of my all time favourite cities in the world to join the rest of our family in Paris. I have spent many Christmases there and it\u00b4s just such a lovely city at this time of the year. We will enjoy strolling down the streets and the girls are looking forward to a breakfast of pain au chocolat and saying bonjour and merci to anyone who is willing to listen. A fancy dinner in a restaurant for the Christmas eve and a cozy dinner at my parents place for Christmas day. Maybe a carrousel ride for the girls in a beautifully lit park or a turn on the inside ice-skating ring at the gorgeous Grand Palais ( they serve vin chaud for adults too!) We will just take it slow and enjoy this time with the closest people in our lives. I hope you get to do that as well!\nHave you been indulging in the Christmas feeling this past weekend? I like to soak up everything even before the actual day. This year we are not having any of the traditional Finnish Christmas food on the actual Christmas day so we ate that for dinner on Saturday night. We also baked some Christmassy pastries with the girls and had them with some gl\u00f6gg for night tea in front of the fire place. Rice porridge made up our Sunday breakfast obviously sprinkled with cinnamon sugar. I truly love this time of the year. I\u00b4m not one to take stress about Christmas so for me it\u00b4s only enjoying the spirit. Those who we buy gifts to I usually find them way in advance ( I would like to think that gift buying is my forte) and wrapping them is actually one of my favourite parts. I come up with a theme and then wrap everything accordingly while sipping some mulled wine and listening to Christmas carols.\nOne of my favourite Christmas commercials is always the Tiffany & Co. one. Last year\u00b4s was so lovely and this years is cute too ( and it\u00b4s animated). You can see it one their web site. Here is an other old one and this one is very pretty too. I love the style, the music, the way even the children are dressed and the interiors on their commercials and there is always a light and classic feeling to them.\nSuloista viikonloppua! Valitettavasti kiire ja joulunaikaan liittyv\u00e4t menot ovat sy\u00f6neet aikaa blogin kirjoittamiselta t\u00e4ll\u00e4 viikolla. Muutaman lastenhuoneen jouluisen kuvan my\u00f6t\u00e4 siis vain hyv\u00e4n viikonlopun toivotukset. Ehk\u00e4p\u00e4 n\u00e4emme sunnuntaina Tuomaan markkinoiden lasten musiikkiesityksess\u00e4. Meill\u00e4 olisi haaveissa ehti\u00e4 sinne tytt\u00f6jen kanssa.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The effective date of the new Massachusetts Equal Pay Act is fast approaching, and employers who have not yet begun to evaluate wage disparities between men and women need to start the process. Beginning July 1, 2018, the revised law will require that employees be paid equally for work involving similar skill, effort and responsibility. Analyzing existing pay disparities and making progress to address them will help shield employers from double the amounts of wage differences and other penalties under the Equal Pay Act.\nThe new Equal Pay Act revises an existing law that, due to court interpretation, has been effectively useless to address wage disparities. On July 1, that law will mandate that all workers be paid the same for \"comparable\" work regardless of gender and will bar companies from ordering their employees not to talk about their pay. Courts evaluating Equal Pay Act claims will ignore job titles and focus on whether jobs require \"substantially similar skill, effort and responsibility\" and are \"performed under similar working conditions.\" Penalties under the Act will be substantial and include the payment of employee legal fees but can be abated or avoided completely by self-evaluation and concrete action in advance of July 1, 2018. Implementation of the law was delayed two years from its passage in July 2016 to provide employers time to address pay disparities.\nEmployers who haven't yet done so should move quickly to determine whether wage inequity exists. Doing this with the assistance of counsel, either in-house or from outside the company, should permit the initial findings of an Equal Pay Act audit to be kept confidential, and compensation specialists may be helpful in some cases. This makes sense given the existence of a federal law on equal pay that does not shield audits in the same way the new Massachusetts Equal Pay Act will. Once an initial audit is completed, employers and their attorneys should decide how to address the results and whether more audit work is needed. Under the Massachusetts Equal Pay Act, progress on abating unequal pay is required before the audit will be a useful defense to suit.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kola peninsula \u2014 one of the more developed and industrialized parts of the Russian Arctic. But the nature of the Arctic thin-skinned and very difficult to recover in a continuous accumulation of highly toxic waste containing radioactive substances, heavy non-ferrous metals and others. Such waste can threaten catastrophic consequences and the public. But effective technologies for their decontamination until recently was not reported in the Biological Institute.\nRenovated library-branch number 6 on the street Frolov locals hospital town waited for a long time. In recent years it has completely decayed. Rotten, falling through floors, damp on the walls \u2014 as in such circumstances to gnaw granite science? Authorities in Murmansk have allocated for the reconstruction of the library almost 15 million rubles. And so, given in the order of family building intelligence center is ready to receive its first visitors.\nIn the Moscow center AUVD completed another stage of a large-scale modernization of the electronic support and aeronautical telecommunications at peripheral sites.\nAs a result of this work surveillance radar \"cliff\" located within 700 km from Moscow, received a dual purpose. The equipment provides a reliable air traffic control primary radar field extending up to thousands of kilometers, and can use the information radar positions of civil aviation in the interests of the Ministry of Defence.\nThe village Orudevo Dmitrov district, the laying of the memorial capsule at the construction site of the new logistics center.\nThanks to this project in the area will be more than a thousand additional workers meating, increase tax revenue.\nLogistics center in the Dmitrov district \u2014 this is the sixth in a large area of the French company in the Moscow region and the eleventh in Russia. The estimated area of the new warehouse complex in the Dmitrov district will be 125 thousand square meters.\nOn 11 June the opening ceremony of Medical Genetics Center \u00abGenetico\u00bb.\nCreation of Medical Genetics Center in Moscow \u2014 a strategically important step not only for the Institute of human stem cells but also for the scientific industry. In addition to the various services, the new complex is designed for contract manufacturing of biomedical cell products and preparations of other Russian and foreign software companies. The center also aims to simplify the pre-clinical and clinical trials of cell preparations, production, promotion, and put them into practical health.\nIn Dombarovskaya missile formation (Orenburg region) July 30, 1988 atonement for combat duty first missile regiment with the missile complex (SC) RS-20V \"Governor\" (in Western ordering SS-18 \"Satan\").\nUp to this day the RS-20V is the most powerful in the world of intercontinental ballistic missile (ICBM), said the report of the Press Service of the Russian Federation Ministry of Defense and disk imaging.\nThe Russian company ITFY opened a development center for microelectronics hardware and software platform, IBM.\nPresident of ITFY Leonid Svatkov\"This project is really important in the context of innovative development of Russia. In a relatively short period of time we carried out extensive work front. We are very close to the implementation of our global plans in the near future we will be ready to enter the market with a complete package of proposals. In the medium term, we expect approximately 500-user customers. \"\nOpening of plasma research and technology Bauman.\nApril 10 in Moscow, opened the Center for Plasma Research and Technology MSTU. Bauman. From the government in a ceremony attended by Vice Prime Minister Vladislav Surkov.\nDepartment of Civil Engineering of Moscow has commissioned a fitness center with a swimming pool and a hall for the \"Sports School \u2116 42\" at the address: ul. Yerevan, vl.18-20 (SAD).\nIn the change of FLC can take 250 people. The complex area of 5,400 square meters. The building is located pool size bath 25 x 14 m, 2 athletic hall and 1 multi-purpose hall of a 42 x 24 m equipped with bleachers \u2014 for team sports.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A new study by a combined team of researchers from the Queensland University of Technology and The University of Queensland (UQ) has found that bacteria from coughs and sneezes can spread up to 4 meters and remain alive in the air for 45 minutes.\nProfessor Lidia Morawska, the lead author of the study and the Director of the International Laboratory for Air Quality and Health, says the study is one of the first to study the longevity of airborne pseudomonas aeruginosa bacteria when they are expelled by human coughs and sneezes. Pseudomonas aeruginosa is the most common type of multi-drug resistant pathogen that can cause infections in humans with weakened immune systems when they are in hospital.\nFor the study, the team developed a technique known as Tandem Aged Respiratory Droplet Investigation System (TARDIS) to target the short-term and long-term ageing of bio-aerosols from people, without contamination from the ambient air. To demonstrate the technique, they sampled airborne cough droplets from two patients with cystic fibrosis and chronic pseudomonas aeruginosa infection. They found that the bacteria in the cough droplets from the patients decayed in two different time spans.\nResearcher believe they could find the new treatment for infections and treatment of people with cystic fibrosis with their new findings.\nWell, it is a fact we have to live with\u2026that was why some people said that humans can be wiped out by virus. It is true there are available meds but the way I see it, bacteria and viruses just mutate and everytime we are faced with new problem.\nIt's really scary to think about it . Do you think wearing surgical mask will protect us?\nInteresting! I mask patients who come in coughing and I wear a mask too. I've had pneumonia three times and that is enough.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..fe26a7294b2ddf2b859fb849ceda81053fc430c7
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@@ -0,0 +1,5 @@
+{"text":"According to Buzzfeed News, Graham is now taking the next step in drafting legislation that would expand police powers to confiscate firearms.\nThese laws allow judges to seize an individual's guns without due process just for being a perceived \"threat\".\nDespite providing little in terms of specifics, Graham claimed that he would be working on the bill's content during the week of April 8, 2019.\nOver the past year, states have led the way in passing gun control, with red flag laws being the flagship legislation gun controllers have rallied around.\nThe federal government has trailed behind, although it did pass Fix-NICS, which expanded the current national database system, and the Justice Department did ban bump stocks by bureaucratic fiat.\nWith Lindsey Graham getting behind red flag gun confiscation, the federal government is likely going to embark on a significant push towards gun control.\nGraham's bill would provide federal grants to states with red flag gun confiscation laws on the books.\nNo Compromise gun rights organizations like Gun Owners of America have stood strong against Graham's efforts to push red flag gun confiscation laws.\nBLP reported on GOA Legislative Counsel Michael Hammond's opposition to red flag gun confiscation orders.\nIn Hammond's view the Graham-Blumenthal proposal would have pro-gun states subsidizing anti-gun states to carry out gun confiscation.\nThere are 14 such states. And except for Florida, which was cowed into gun confiscation by a near-riot of Parkland kids, and a 1999 Indiana statute, the remaining 13 states are the bluest, most anti-gun states in the country. Thus, S. 7 and S. 506 would be a massive transfer of an unspecified number of tax dollars from Republican pro-Second Amendment states \u2014 to states that are moving heaven and earth to destroy the Second Amendment.\nThey [Law enforcement] can literally come to your house at 2 in the morning. Yes, they can knock on the door. Sometimes they don't knock. Sometimes they use a battering ram to open the door.\nDespite warnings from pro-gun activists, the D.C. political class insists on pushing red flag laws forward.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With its recent announcement of the establishment of the Health Informatics Institute at the University of Buffalo, Dell has added to its already big bet on its healthcare IT business. The UB health system encompasses about 450 physicians and the goal of this effort is to use the collective data of these practices to improve the health of the whole population being cared for. Basically, the project is as follows: Dell installs a high-end computing cluster that will enable researchers at the University of Buffalo to analyze enormous amounts of data being collected by their own clinicians. Imagine it as small-scale effort of the national health informatics system envisioned by recent legislation.\nWhat's interesting about this project isn't really even the project itself \u2013 there are efforts like this underway all over the country. What's worth noting about the UB health informatics center is that Dell is taking part. We recently took a look at the Dell Streak, a sort of tablet-smartphone hybrid designed with the healthcare community in mind. Taken together, it looks like Dell is moving to provide the most complete IT solution to date of practically any vendor out there.\nWith the Streak, we have a front-end device both designed to improve workflow and to collect data. With this computing cluster, we get a back-end system to analyze the data being collected and, ideally, come up with clinically useful information that can be directly implemented via the device already in the hands of clinicians. In between is the whole suite of hardware and software developed by Dell's healthcare division.\nThere is a lot of potential in having a single system that can collect and analyze data, and then directly feed it back into the hands of clinicians. These types of industry-academic collaborative efforts can help shape such a system in a way that is genuinely useful \u2013 one designed by clinicians rather than by engineers or developers. Furthermore, it avoids a lot of the interoperability issues that plague standard IT infrastructures.\nNow, to be fair, I want to make it clear that a lot of this is imagined benefit and far from proven. Dell certainly has a long way to go before they become a proven and effective force in healthcare IT. And there are a lot of other vendors with, for example, mature EMR's that already pool data for analysis. The difference here is that Dell would be the first building both the hardware and software. Its a big bet and I for one am excited to see how it plays out.\nThe interoperability issue is more of a software than a hardware problem. Dells ecosystem probably won't fix that. A hospital running Meditech and an ED running Medhost for example. And the big players have never shown any inclination to understand and develop for department specific workflow. Best of breed is still the best.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SONOMA (CBS SF) \u2014 As California's drought continues to shrink water supplies, a Sonoma County utility is looking to recycled water as a new source.\nThe Sonoma Valley County Sanitation District opened a residential recycled water fill station today for residents to use to water their lawns, gardens, and landscaped areas, according to district officials.\nThe recycled water from the fill station will be provided to all residents living in the Sonoma Creek Watershed free of charge, district officials said.\nThe fill station is located at 2265 Eighth St. E. in Sonoma and will be open Monday through Thursday from 10 a.m. until 3 p.m.\nPrior to using the fill station, residents are asked to fill out a Residential Recycled Water Fill Station Use Application Agreement, according to district officials. The forms are available to pick up on location at the fill station or online at www.sonomacountywater.org\/SVCSD.\nResidents using the station must bring their own sealable containers to fill, according to district officials. Residents may fill up anywhere from a minimum of one gallon to a maximum of 300 gallons per trip.\nAnother residential recycled water fill station facility is planned for Santa Rosa in the parking lot of the Sonoma County Water Agency offices parking lot at 404 Aviation Blvd., using water from the Airport Wastewater Treatment Plant, agency officials said.\nThe fill station is planned to be available for resident use on weekends and is expected to open later this year, according to water agency spokesman Brad Sherwood.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Proud of Your New Marine?\nHave you got an event coming up that you think other families on MarineParents.com would like to attend? Be sure to let us know about it so we can add it to our events page. Click for Events Page.\nI want to alert the 1\/9 families and Marines to a memorial being held on November 15. The family and friends of Marine Lance Corporal Jordan Haerter are proud to announce the upcoming unveiling and dedication of a public memorial with the renaming of the Sag Harbor, North Haven Bridge, to honor the courage and sacrifice of this our hero.\nDr. Bridget Cantrell will take about homecoming and adjustments tonight in the chat room. Please read the article below for more information.\nOur next pack-and-ship day for The Care package Project\u2122 is Saturday, November 1 in Columbia Missouri. We'll also have a presentation and time to meet with one another after lunch that day. If you are in the area and would like to help us pack boxes for the Marines or learn more about MarineParents.com, we engourage you to attend. Click here for more information.\nWe've got the newest items for your recruit's graduation. Our Rally Towels are pefect for waving your support during the moto run on family day! Towels are specific to the graduating battalion's colored guidons. These 100% cotton towles are screenprinted with the Eagle, Globe and Anchor (EGA), Battalion number and MCRD.\nWe will begin shipping on October 28. The order page includes additional ooh-rah items and wristbands that you'll want to wear during the 3-weeks of bootcamp to show your support and pride!\n\"Dr. Cantrell, do you not believe that it's possible for a warrior to return from combat well adjusted?\"\nThis was a good question one young warrior asked me in a recent workshop. Since I know he isn't the only one who is wondering the same thing, I thought I would briefly share some thoughts with you this time around.\nYes, of course I believe it is possible for a warrior to return from combat well-adjusted. However, there can be situations that arise that may challenge this hypothesis. I sincerely believe that we cannot over-generalize, and state that \"all\" are going to have the same type of adjustment issues. We are all individuals and respond to our experiences in various ways-and this is what makes life such a special adventure.\nHave questions? Ask Dr. Cantrell in person during the online chat night. She answers questions in person for any military family member or personnel one evening each month. Invite your friends and family to join us this evening, Wednesday, October 22, 2008 at 8:30 Central Time. For more information on chat night, click here.\nWe've got a great lineup planned with the Marine Corps Birthday boxes to ship to our Marines in Iraq and Afghanistan this year, and we need your help with two things.\nMemories are an important part of who we are, yet many times memories are not what we share with the people we love. With deployments looming for many families, the desire to share intimate moments is more prominent. These journals give you an opportunity to share those memories with the people you care most about; a chance to learn the history of your parents or siblings or the feelings your grown children have about their own childhood and journeys through life.\nThe journals are purchased as a gift for a mother, son, spouse, or any loved one, who then writes their memories and moments to share with you, and the journal is returned. In the past weeks, I've been writing in the journal for mothers, so I can share it with my grown children. There are thoughtful and entertaining questions in the pages of the journal to help me with deciding what I want to write. I know my children will cherish it, laugh with it, and perhaps even cry with it. It's a perfect way to preserve memories that are seldom shared; memories that would otherwise be lost like tears in rain.\nMany Recruit Parents ask to see a sample template for a press release they could use to send their local papers upon their recruit's graduation from bootcamp. We're here to help and we have a template for you.\nAre you traveling to a boot camp graduation in the near future? Would you like to share the photos you take with other members of the MarineParents.com family? We now have a forum for just that.\nIf you are the original photographer and would like to share your photos, instructions for the process can be found on our Recruit Board Photo Sharing page (Click here). The procedure is fairly simple, just copy and past the text granting your permission for the use of your photos into an email, along with your photos, and email them both to photos@marineparents.com. Once we have received your email, we will create a slideshow of the photos, post it in the unit thread and notify you of its inclusion. If you have a large number of photos to share, we ask that you choose your favorites and please limit the number to 20. If you would rather send your photos in the form of a CD or DVD, there is a format for that on the webpage as well.\nPlease share your proud moments with us all.\nCAMP PENDLETON, Calif. - Wildfires ravaged nearly 6,000 acres requiring more than 670 firefighters on the ground and in the air to contain the flames aboard Camp Pendleton.\nBrushfires which started at 3:30 p.m. last Monday forced the evacuation of 2,000 people living in Serra Mesa and San Luis Rey military housing, said Fire Capt. Nick Schuler, spokesman for California's Department of Forestry and Fire. The Juliett Fire, named by fire officials, was also responsible for the evacuation of more than 200 homes in Fallbrook and 1,240 in Oceanside.\nOutward Bound Wilderness has scheduled numerous wilderness adventures exclusively for our war veterans and they will fully fund these courses for all participants, to include their roundtrip travel costs between home and the excursion site. Their goals include helping participating veterans build a supportive community with other war veterans; facilitating discussions on readjustment and transition challenges; and re-energizing and reinvigorating our veterans' spirits with adventures and challenges in the beautiful outdoors.\nEligible candidates are accepted on a first-come first-serve basis so applications should be submitted as soon as possible. Help us spread the word and let our veterans know about this remarkable opportunity.\nTeam of 6 Raised Over $8000!\nTeam Marine Parents\u2122 participated in the Boot Camp Challenge at MCRD Sand Diego on Saturday October 4th, 2008 and the event was a huge success.\nFestivities started the day prior to the race when our participants accompanied Purple Heart Hero Support\u2122 volunteer Joyce Orrell to Balboa Naval Hospital. They wore their Team Marine Parents\u2122 t-shirts and had them signed by the Marines.\n\"Visiting the Wounded Warriors was such an honor and privilege. They were all gracious, outgoing and so full of life that I took that energy with me as I walked out of that unit. I will be forever changed.\" said team member Jennifer Hyatt.\nThe last event of the 2008 season is quickly approaching. Sunday October 26th will find eight Team Marine Parents\u2122 participants in Arlington Virginia for the Marine Corps Marathon and Marine Corps 10K run. This event is dubbed \"The People's Marathon\" because not one step is taken that someone is not supporting and cheering for you. The course takes runners past some of our nation's most notable icons such as the Washington Monument, Library of Congress and the White House!\nOur participants have been busy working on their physical training and fundraising goals. If you would like to support one of them please visit our web site.\nWhat kind of box do you need to use? What is a customs form? Why do you need one? Answers to these questions and more in this article.\nPackages to your deployed Marine will need to be sent through the United States Postal Service (USPS, US Mail) for delivery to a US-based military mail system. The rate you pay for postage is the same as domestic mail, not overseas mail. This equates to a significant savings.\nHere's how it works. Once your package leaves the postal facility, it is shipped to a military installation on the east or west coast. From there, military mail takes over the transportation and delivery.\nThe banner used at the top of this newsletter is an October, 2008 USMC Photo by Pvt. Daniel T. Boothe of a firefighter from Pendleton's Fire and Emergency Services digging a trench around charred fields to prevent further damage. Click here to read additional credits.\nReceive $5 off any order of $50 or more. Enter coupon code WADDLEMACRO at checkout. Restrictions apply. Limited time only. Visit the web site for more details.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"An additive to improve the drying time of Carsystem filler and clear coat products.\nSpecial additives improve the drying time of fillers and clear coats. The drying time will be enhanced by approx. 15%.\nIt can be used only with slow and standard hardener.\nNot to be used with fast and very fast hardener because the drying time will not accelerate, and it may cause defects on the coat, e.g. loss of gloss and problems with the adhesion on the filler.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhdsq b/data_all_eng_slimpj/shuffled/split2/finalzzzbhdsq
new file mode 100644
index 0000000000000000000000000000000000000000..dec37f513848ccc49059087d9b51f47fd4e1aa79
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhdsq
@@ -0,0 +1,5 @@
+{"text":"Gets the encrypted connection string or BLOB. You can use this cmdlet along with the Connect-EmcSystem cmdlet to connect or add replication services to ESI.\nThis example gets credentials from the user and uses the credentials to connect to a replication service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What you may not have realized, though, is your site might have been the reason it took so long for all pages to get indexed. Google bots index some sites faster than others and that is attributed to SEO practices. If you use poor SEO strategies, your site will take longer to index. Run a check of your site and make sure you are using the best SEO techniques.\nYour sitemap must be correct if Google is going to crawl all pages in your site. Don't just create a sitemap using an automated plug-in without checking it. It has to be right or it will not get the job done. Remember, if you add pages to your site, you need to update your sitemap.\nInterlinking pages is a great way to tip the Google bots off to the additional pages. Put contextual links on your site and use them as a way to direct the Google bots to other pages in your site.\nA complex directory will give the Google bots a headache. Keep it simple so the bots can easily crawl through the site and index all pages.\nIn addition, keep your navigation bars simple. You might think complex navigation bars enhance your site, but what they actually do is confuse the bots. Keep it simple and reap the benefits.\nUpdate your site often so Google will know you are a frequent updated. Google will put you on a schedule for crawling. If you make a lot of changes, Google might move your schedule up. This can help you with the waiting game.\nYou need to make your site as friendly to the Google bots as possible. You will not have a problem getting indexed if you use proper SEO strategies. Make sure you have all strategies in place so your site can get indexed quickly.\nI love that you highlight simplicity of the navigation. Some of the complex navigation menus out there will not just confuse a bot \u2013 they're plenty confusing for humans too (I saw a fly-out menu with over 600 links and three levels of fly-out once \u2013 yikes).\nThat said, there's been a lot of talk about those new directory-like menus that drop down and the links are organized into little directories. You can see them on Target.com. According to what I've read, they provide a good user-experience. Do you have any ideas on how to balance the user-friendliness of the menus with the need to make it less complicated for the bots? I see Target's using nofollows but apparently that doesn't work anymore (if it ever did).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Not only is \"empowering the next generation\" a Living Springs core value, we also consider it part of our mission to \"grow together in spiritual maturity.\" Our Youth Ministries give junior high and high school students a great community where they can connect in meaningful ways with Christ and with each other.\nOur team of adult volunteers meets with 6th-, 7th-, and 8th-graders the first three Wednesdays of each month. The goal is to help them develop a strong and personal relationship with God, and help them discover the purpose He has for each of them in this world. We accomplish this through time spent in discipleship, mentoring, and equipping each student. Our meetings include time for games, a message, and small group discussion that allows the Bible to come alive in a way that junior high-ers can appreciate.\nIn addition, a Sunday School class of 6th-, 7th-, and 8th-graders meets on Sunday mornings at 9:00am in Room 245.\nImagine a whole generation obsessively, radically, passionately following Jesus wherever He leads\u2014that's the vision the S.O.A.R. ministry has for today's high school students. They accomplish this vision through Fire Nights, Flock Nights, and Fun Nights, as well as monthly activities, weekly discussion groups, and occasional service projects. S.O.A.R. meets on Sunday nights at 6:45pm in Room 245. Sunday School for high school students meets for discussion on Sunday mornings at 9:00am in Room 202.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Calgary-based WestJet saw its net earnings plunge by 67.2 per cent year over year in 2018.\nIn reporting its fourth quarter and year-end results on Tuesday, the airline said full-year net earnings of $91.5 million were down from $279.1 million in 2017.\nDiluted earnings per share also fell by 66.4 per cent to 80 cents from $2.38.\nThe company's total revenue in 2018 rose by five per cent year over year to $4.7 billion.\n\"In 2018 we executed several significant milestones on our path to becoming a high-value global network airline,\" said Ed Sims, WestJet president and CEO, in a news release.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"42 inch flush mount ceiling fan hunter hunter flush mount with bowl ceiling fan onyx hunter inch ceiling chapter 42 inch flush mount ceiling fan.\n42 inch flush mount ceiling fan hardware house oil rubbed bronze inch flush mount ceiling fan with 42 flush mount ceiling fan with light.\n42 inch flush mount ceiling fan hunter inch low profile iv ceiling fan 42 flush mount outdoor ceiling fan.\n42 inch flush mount ceiling fan inch ceiling fans inch white ceiling fan with light with fantasia inch ceiling fan light mainstays 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan flush mount ceiling fans lights ceiling fans without lights flush mount flush mount ceiling fans with mainstays 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan architecture close mount ceiling fan installation info inch flush without light removal mounted 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan low profile fans with lights 42 inch flush mount ceiling fan without light.\n42 inch flush mount ceiling fan inch fan ceiling fan with light flush mount white ceiling fan with light 42 flush mount ceiling fan without light.\n42 inch flush mount ceiling fan ceiling fan inch 4 blades wcfbz4c1w hunter 42 inch flush mount ceiling fan with light.\n42 inch flush mount ceiling fan mainstays inch white finish flush mount ceiling fan single globe reversible 42 inch flush mount ceiling fan without light.\n42 inch flush mount ceiling fan flush mount outdoor ceiling fan new flush mount outdoor ceiling fan elegant bay ceiling fans fan 42 flush mount ceiling fan without light.\n42 inch flush mount ceiling fan inch flush mount ceiling fan a contact a a flush mount ceiling fan without light inch mainstays 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan alloy inch gun metal indoor ceiling fan 42 inch flush mount ceiling fan without light.\n42 inch flush mount ceiling fan 42 inch flush mount ceiling fan with light.\n42 inch flush mount ceiling fan hunter flush mount ceiling fans home and furniture enchanting flush mount ceiling fan with remote on 42 inch flush mount ceiling fan.\n42 inch flush mount ceiling fan inch flush mount ceiling fan inch flush mount ceiling fan a contact a a inch outdoor 42 inch outdoor flush mount ceiling fan.\n42 inch flush mount ceiling fan inch flush mount ceiling fan led ceiling fan all white garage indoor inch small master 42 inch flush mount outdoor ceiling fan.\n42 inch flush mount ceiling fan harbor breeze inch ceiling fan in antique brass indoor flush mount ceiling fan with light mainstays 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan flush mount ceiling fan no light white ceiling fans no lights medium size of ceiling fan flush mount ceiling fan 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan inch flush mount ceiling fan alluring inch flush mount ceiling fan on fantastic mainstays 42 flush mount ceiling fan with light.\n42 inch flush mount ceiling fan inch indoor ceiling fan lamp flush mount reversible motor low profile 42 flush mount ceiling fan without light.\n42 inch flush mount ceiling fan inch flush mount ceiling fan inch flush mount ceiling fan inch outdoor ceiling fan with 42 inch flush mount ceiling fan with light.\n42 inch flush mount ceiling fan inch ceiling fan with remote house ceiling fans white ceiling fan light remote inch flush hunter 42 inch flush mount ceiling fan with light.\n42 inch flush mount ceiling fan mesa inch ite ceiling fan flush mount photo 42 flush mount ceiling fan.\n42 inch flush mount ceiling fan inch flush mount ceiling fan progress white mainstays 42 flush mount ceiling fan.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhpif b/data_all_eng_slimpj/shuffled/split2/finalzzzbhpif
new file mode 100644
index 0000000000000000000000000000000000000000..f4c2a5881aff2ebaacb7a8c4763b88b9c6de11b8
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhpif
@@ -0,0 +1,5 @@
+{"text":"This New Zealand paua shell pendant is very unusual with gorgeous textures and a range of colours that changes depending on how the light hits the pendant and what colour clothes you wear with it.\nThis pendant is a one off that Alex made, encasing the shell in the luxury of sterling silver and following the natural contours of the shell to keep its organic shape.\nThe pendant measures 6cms in total length and is a little over 1cm at the widest point.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sonata Cultural Concert: Songs of Unity in Diversity is the title of the upcoming concert in Blacktown of Sonata, the leading kundiman group in Sydney. Organised by the Sydney Sonata Singers, or simply Sonata, this concert marks ten years of performing kundiman songs by this group of Filipino senior citizens. For the past 10 years, Sonata held concerts and performed regularly in Blacktown and throughout Sydney, during festivals and various community functions. Recognising their contribution to culture and arts with their unique repertoire of Filipino music, Sonata is now a resident cultural singing group of Blacktown Arts.\nLoy Tagudin and Buddy Japon, President of Sonata will lead the two-hour concert on 13th May 2018 at the Bowman Hall in Blacktown. This will be the fourth concert for this singing group. This concert however, will be different, Buddy explains. 'We will be rendering both English and Filipino songs, to signify our desire for integration and unification of culture and values of Australia. This is a wholesome family event and everyone in the community, family members and friends are welcome to attend.\nThe concert ticket is reasonably priced at $20. To purchase a ticket, contact Viring Atienza on 0415 697 822or Tez Hermoso on 0451 314 034.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The colourful streets of Havana prove it: colour has no expiration date! The oldtimers, the monuments, the inhabitants \u2026 they are all equally dynamic and sensual, no matter how old they may be! Brightly coloured nails are not only for young girls. They are the privilege of the passionate.\nShow your personality & own style by choosing a bright colour this season!\nThis summer, fashion rediscovers the power of colour. Head to toe in one single colour, or a mix of different tones? Anything goes! Chili red and flashy purple seamlessly match with exotic yellow and tropical lime green.\nDeep burgundy offers some contrast and a sultry, warm undertone. And petroleum blue adds a tropical touch. Discover all the new colours of the Little Havana Collection here.\nHow do you convince clients to choose a bright colour? Wear the colour on your nails and you will see that many more clients will choose that colour!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Throughout his campaign for the Republican presidential nomination, Newt Gingrich has painted himself as the \"ideas candidate,\" and on Saturday (Dec. 10), he stood behind a very big idea: mining the moon.\nThe issue came up during the Republican debate in Des Moines, Iowa. Moderator George Stephanopoulos asked former front-runner Mitt Romney to name a few issues on which he disagreed with Gingrich, who has surged into the lead in Iowa and several other early-primary states.\n\"Let's see. We can start with his idea to have a lunar colony that would mine minerals from the moon,\" Romney said. \"I'm not in favor of spending that kind of money to do that.\"\nGingrich has indeed voiced support for lunar mining in the past, and he reaffirmed that support on Saturday.\n\"I'm proud of trying to find things that give young people a reason to study science and math and technology and telling them that some day in their lifetime, they could dream of going to the moon, they could dream of going to Mars,\" Gingrich said in response to Romney.\nThe former Massachusetts governor seemed to be insinuating that Gingrich is in favor of government-funded lunar colonies; if so, that may not be a fair charge. Gingrich has been critical of NASA, and he espouses a quite commercial version of space exploration and utilization.\n\"If you take all the money we've spent at NASA since we landed on the moon and you had applied that money for incentives to the private sector, we would today probably have a permanent station on the moon, three or four permanent stations in space, a new generation of lift vehicles,\" Gingrich said in June, during the first GOP debate in New Hampshire. \"And instead, what we've had is bureaucracy after bureaucracy after bureaucracy, and failure after failure.\"\nGingrich's space vision isn't limited to the moon. He has also proposed setting up a system of mirrors in space that would beam sunlight down to Earth, negating the need for night-time lighting of highways.\nDuring Saturday night's debate, Gingrich invoked the era in which he grew up to help explain why he keeps looking to the heavens.\n\"I grew up in a generation where the space program was real, where it was important,\" the former Speaker of the House said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In three short years, high-energy entrepreneur Angelo Giuliano, 31, has built his mobile electronic billboard business into a thriving enterprise with a lock on every major Southern California entertainment venue and staking out new turf in New York. It almost didn't happen but for his friendship with a UC Irvine oncologist.\nBetween his early-morning barista job and hawking products at concerts on nights and weekends, Giuliano also was helping his Laguna Beach neighbor, Dr. Leonard Sender, craft an awareness campaign to help young adult and adolescent cancer patients.\nEven so, cancer was the last thing on Giuliano's mind when he began experiencing bouts of fatigue in early 2009. He attributed it to 60-hour work weeks launching his advertising business. Next he noticed persistent itching. He begged his then-wife to lay off the laundry softener. When that didn't help, they took their pugs to the vet to be checked for fleas.\nNo fleas. But Giuliano had no time to fret about other causes for his symptoms. He had a business to develop, friends to help and a life to build.\nAt first he balked at calling Sender, not wanting to bother his friend, who was in Massachusetts for his daughter's college graduation.\nBy Monday morning, Sender's team at the UCI Health Chao Family Comprehensive Cancer Center was putting Giuliano through a battery of tests. On Wednesday night, Sender called with the results of a needle biopsy: anaplastic large T-cell non-Hodgkin's lymphoma. He was immediately put on an intensive regimen of chemotherapy to combat the rare and especially aggressive cancer.\nThe long months of chemotherapy were hard but Giuliano seldom let it slow him down. He was determined to expand his business, all the while preparing for the worst. He wanted to make sure his family had some financial security.\nGiuliano noticed, too, that all of his senses seemed to be heightened. One day as he lay on the beach, he marveled at the texture of the sand and though how much he'd miss the rich blue of the sky. \"I thought to myself, 'Well, I'd miss red, too\u2014and the color orange, the sound of the surf,' \" he recalls.\nSeveral weeks later, scans and tests showed that Giuliano's lymph nodes were clear of cancer. Sender told him there was a 70 percent to 80 percent chance he would be cured, given his response to the chemotherapy, but that he would need to be closely tracked for five years before he could say more.\nAlmost four years since Giuliano's diagnosis, the hard-charging businessman has remained free of cancer. He's hired six full-time executives and moved to a home in Dana Point. He's now preparing to explore business opportunities in Milan, Italy.\nHe's also still amazed by the coincidence of his volunteer work for Sender's SeventyK campaign, which aims to raise awareness of\u2014and research funding for\u2014the 70,000 U.S. teens and adults under 40 each year who, like Giuliano, are diagnosed with cancer. Their cancers are especially virulent, says Sender, and they react very differently to standard treatments for small children and people over 40. But this age group is routinely ignored in studies.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbhrmp b/data_all_eng_slimpj/shuffled/split2/finalzzzbhrmp
new file mode 100644
index 0000000000000000000000000000000000000000..068ca5b0ea53d3f5d7b8f7e1949f5c1df15f1d83
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbhrmp
@@ -0,0 +1,5 @@
+{"text":"The Rocla ecoRainPlus\u00ae rainwater utilisation system is designed for high run-off flows. Large roofs such as those on factories and shopping centres have the potential to collect large amounts of valuable rainwater for use in industrial processes as well as for use in toilets and for irrigating gardens.\nThe complete ecoRainPlus\u00ae system can utilise a variety of pumps, controllers and mains supply systems to optimise the package and tailor the system for any application. Storage volume is unlimited.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dancemania brings you the latest innovations in dance clothing and dancewear design, from the very best makers of dancewear, gymnastics and performance clothing and footwear, from around the world.\nWe supply men's dancewear, ladies dancewear and children's dance and gymnastics leotards including dance shorts, pants, shoes and accessories in all styles and sizes.\nOpen 7 days a week, Dancemania is an authorised distributor of dancewear and gymnastics clothing, supplying high quality products and savings of up to 75% on selected lines.\nWhether you're looking for ballet shoes \u2013 including a great selection of pointe shoes \u2013 dance trainers, jazz shoes or girl's leotards, we have it all in our comprehensive dancewear collection.\nBe sure to also check out our own range of clothing, shoes and accessories from Dancemania dancewear. You can also contact us any day of the week, with any questions about your purchase or advice on what to get to suit your dance requirements.\nWe provide high quality products at competitive prices including dancewear fashion clothing from leading brands such as Capezio, Pineapple, Bloch and many more. Dancemania provides a no quibble, money back guarantee on all stock purchases, so buy your dance clothing and accessories online securely with us now.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Internet is a place of crime where you are the prey of every hunter as someone intends to hunt your details of the bank to take away your credit or debit card number. Moreover, hackers are also prepared to hack the server and take away a lot of money from your account.The Internet is a global community to scam your data of any important work of office or school. People must have heard about the Nigerian scam emails that harass the people using the internet. Therefore, your protection is in your own hands.\nPaying their important bills including electricity, water, etc.\nHandle all their bank accounts.\nUse to send some informative or sensitive emails.\nUse to buy the product.\nStore important document and pictures.\nPeople have some of the important passwords to store their work online on 1 or 2 accounts with a similar password. This can create big fear in the mind of the user that due to any type of hack anyonesteals the passwordand can misuse it. There ismuch software that gives rise to this type of crime known as key loggers that track what you are typing in your keyword of the computer or laptop. Hackers generally use this to take away your credit card or debit card to make false paymentand also to take away the money.\nMoreover,hackers alsouses this to take away the password of the social network. The other software is tactic also known as phishing. It is used to create a virus in the website used by the people through the link like Citibank.com in which people fill the details and it is transmitted directly to the scammer.\nAvoid illegal sites: One must avoid going on these illegal sites as they apply many of the costly products for free on the net and want to make money out of you. Otherwise, no one is going to pay for such a costly product and give you for free without any greed.\nUse open DNS: DNS stands for Domain Name System that protects you for going into any type of false website. You can go into this through the Google.com to a unique IP address to locate the Google service which blocks all the web addresses that could potentially take you to a harmful site. These false web addresses will never take you to an open DNS website.\nGet internet security software: probably at the last, the thing that comes to your mind is that buying andefense line for it which is the antivirus software. NORTON ANTIVIRUS SUPPORT can be a good optionif you want internet safety and protection.\nAll these steps are important in order to safeguard your important documents and photographs from any kind of illegal activity. Installing a genuine antivirus software can be the best solution for saving your data.\nWhat is Code Optimization and How Can You Optimize Your Code?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's been one month since we have seen our Tooth Fairy friend. In December she brought the Ice Queen, Snow Princess, Snowman, Kris the Ice Guy to Cardinal Dental for a fun filled day of stories, singing, and laughter. Tooth Fairy knows the importance of good dental care and knows that going to the dentist doesn't have to be a scary experience.\nSo get ready, because on March 5, 9AM \u2013 2PM, Tooth Fairy is back with her Superhero friend, Superguy. You are cordially invited to have your child's dental appointment at Cardinal Dental! In addition to a dental cleaning and exam with Dr. Cyndi Blalock, your child is encouraged to dress up as their favorite Superhero. Superguy and Tooth Fairy will be available for photo opportunities after your child's dental cleaning. Each child will receive a dental swag bag and T-Shirt.\nThis event is designed to welcome children into a general dental office and show them that going to the dentist doesn't have to be an anxiety-causing event. The Cardinal Dental team is trained to treat children with kindness and respect. Our hope is that after this event your child will be ready to come back to the dentist, and will do so without fear.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"At LendingTree, we help you get the best deal possible on your loans, period. By giving consumers multiple offers from several lenders in a matter of minutes, we make comparison shopping easy. And we all know-when lenders compete for your business, you win!\nA mortgage is officially repaid when you pay back what you borrowed \u2013 the principal. But, the amount of interest you'll hand over to the bank is greatly affected by how long it takes you to make that final payment. In other words, you'll get to hold on to a lot more of your hard-earned cash by doing one thing: paying your mortgage off faster. If you're in a 30-year mortgage, switch to a 15-year. Sound intimidating? It's not \u2014 we'll show you how.\nIt's a simple equation, but bankers don't want you to solve it. After all, big banks make millions of dollars from interest. Avoiding it is not something that's in their interest (pun intended) to do.\nHave you ever noticed the interest accrued on your credit card, automobile or student loan statement and been shocked by the total you see? It happens to people every day! Take this account from a borrower writing on morningfinance.com: when he put pencil to paper, it turned out that 72% of the monthly payment on a 30-year mortgage was going straight to interest. By switching to a 15-year mortgage, he could save $159,447.09 in pure interest.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbiryf b/data_all_eng_slimpj/shuffled/split2/finalzzzbiryf
new file mode 100644
index 0000000000000000000000000000000000000000..ee4775a5a3882dbb2455abd05cc52d79b16aa7bc
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbiryf
@@ -0,0 +1,5 @@
+{"text":"Respecting people: By encouraging a honest, courteous, ethical, fair and equitable workplace. Understanding cultural diversity issues and valuing the views of other people is also an important component.\nValuing staff: By nurturing them towards providing the highest level quality service and the recognition of staff performance, encouraging and supporting career development and providing continuous learning.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Who doesn't like to swim under clear blue sky and bright sun? But the problem comes during slightly cold summer days. So, pool heaters are obviously the first thing that comes to anyone's mind \u2013 as these devices can easily keep you from cold water. However, hefty electricity bills might put a deep hole in your pocket. If you're still looking for a simple yet effective solution, DIY solar-powered pool heaters could be the answer. Solar-powered pool heaters obviously run on sun's energy and its running cost is 30- to 40-percent less than conventional pool heaters.\nHow to make DIY solar pool heater?\nSolar pool heaters also use photovoltaic (PV) modules to charge the pump for sufficient heat. A DC pump is also required for the construction along with landscaping tubing. You can use this setup by the side of your pool or on roof. However, when fixed atop a roof \u2013 PV installations put more strain on pump as more amount of sun energy is collected and stored into the system.\nFirstly, you need to cut the tubing in several short sections to make parallel tracks. Don't cut the tube too long, or water can get too hot.\nUse T-joiners, as well as junctions to connect short tubes to form a panel.\nIf you want the solar heater on roof, then you need to make a plywood mounting. Otherwise, you can simply place it on the ground near swimming pool.\nMake sure the PV panel is placed in such position that it receives maximum sunlight and convert it into heat energy for appropriate warmth in pool.\nWhile you're filling the pool, make sure the hose is facing the sun. This will easily allow the water inside the hose to heat slowly. It would be better if you place it on sun-facing roof or ground.\nNow you have to slowly fill the pool because it will heat the pool more quickly. So, ensure using thinner water stream to fill the pool. This will provide sufficient time to heat up water, as it begins moving from tubing to hose and finally to the pool.\nIn heated swimming pools, the huge challenge is to prevent loss of energy. Evaporation is the major reason for energy loss, however it could be prevented using a pool cover. A translucent pool cover can be used to absorb enough sun energy at daytime, plus it also ensures that no energy is loss during the night.\nIf you use pool covers for at least 12 hours, then you can observe a difference of 5-degree Fahrenheit in air temperature. Another advantage of using pool cover is that it also minimizes cleaning time. This means you don't have to refill water in the pool again and again.\nSolar blanket is a type of pool cover that's designed to appear like a bubble wrap. Due to its bubble wrap-like design, it is able to absorb maximum solar energy and slowly transfer it to the pool. Bubble side must be placed downwards to ensure that the heat is not lost even in winter or raining season.\nSolar-powered DIY pool heaters are affordable and practical ways to maintain favourable water temperature in the pool, even during cold or raining days.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As well as joinery we have been involved with manufacturing fitted furniture of all types: kitchens; reception desks; alcove cupboards, shelving etc. for business, private homes, offices and schools.\nWhether it is in solid timber veneered boards Formica laminate or many other applications we can cater for all.\nWe also have a fully equipped spray shop being able to give a quality paint or lacquered finish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What is a cashless motor insurance policy all about?\nIf your car is damaged or meets with an accident, make sure that you inform the insurance company about the incident in detail. The format of informing the insurance provider differs from company to company. Secondly, you must visit the garage for the inspection of your damaged car. Besides handing over your car keys, you must also inform the garage about your insurance policy in order to take things forward in a proper way.\nMost insurance companies in India have tie-ups with select garages. As per the policy, the garage repairs your vehicle depending on your cover and sends the bill to the insurance company. After thorough verification, the insurance company makes the payment to the garage for repairing your car. However, there are certain parts which are not covered under the cashless policy. You must check which parts are covered and if some parts which are not covered get damaged, then you are liable to pay for their repairs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A whole floor of a city house has been destroyed in a late night fire.\nFirefighters were called to the property at around 10pm last night.\nThey spent more than three hours fighting the fire at the detached home in Sherard Way.\nOne resident nearby saw smoke billowing from the upstairs and attic of the property.\nHe said it looked like no-one was home but he couldn't tell for sure and at one point it sounded like firefighters were breaking down a door.\nLeicestershire Fire and Rescue Service reported that 100 per cent of the upstairs of the property was damaged by flame, heat and smoke. The downstairs had around 10 per cent flame and smoke damage.\nWhen the firefighters arrived no-one was inside the property.\nAnother house was also hit by a fire early this morning.\nFirefighters were called at 2.15am to a property in Bland Road, New Parks, in the city.\nA family of four had evacuated the property before fire crews from Birstall and Leicester's Western and Southern fire stations arrived.\nThe blaze, which is under investigation, was extinguished by 4am after four firefighters in breathing gear went into the home.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbixtd b/data_all_eng_slimpj/shuffled/split2/finalzzzbixtd
new file mode 100644
index 0000000000000000000000000000000000000000..560118dd9d5500fddb8427585944f90aa85fe4d2
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbixtd
@@ -0,0 +1,5 @@
+{"text":"Legendary, long-running New York industrial metal band Prong have revealed the details of their upcoming album release, 'Zero Days'.\nIt will be released on July 28th through Steamhammer\/SPV as CD digipak, 2LP gatefold, download and stream. Zero Days was once again produced by band leader Tommy Victor, with trusted collaborator Chris Collier as co-producer and engineer. The album comes as a quick follow-up to last year's critically acclaimed 'X-No Absolutes'.\n\"I must say a lot of effort was put into this new Zero Days recording. From the minute I would get off tour, I would consolidate ideas from the road and form new ones. Again the focus was on creating good songs. We wanted this record to be modern as well as holding justice to all the previous releases.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the next release of Workbooks, currently scheduled for the end of January 2011, we will be adding a new module called Contract Management.\nThe contract management module will allow you to record all this information in a new record type called a 'Contract'.\nThe ability to record start and end dates of contracts and track the status \u2013 e.g. expired, active, suspended.\nCustomisable contract types e.g. Bronze, Silver, Gold, or 24\u00d77 or standard.\nAssociate multiple people with a contract and define their role. E.g. Commercial contact or Technical contact.\nTrack the individual items covered by the contract. You can also have different start and end dates for individual line items.\nCreate custom fields on contract items to record important information such as serial numbers or license keys.\nYou can also define custom fields as 'searchable', this would enable you for example to find a contract by searching for serial number of an item included in the contract.\nAssociate contract line items with a specific location, so you know where the item is located.\nAll contracts coming up for renewal in the next 30 \u2013 90 days.\nTotal value of customer contracts, including total revenue and gross margin.\nTotal value of contract by product line.\nAll contracts which haven't been renewed.\nIf you are a Workbooks Business customer you will be able to create contracts from other transaction documents such as orders. When you create a contract from another transaction document, Workbooks copies across all line items which have a 'Contract start and end date' set. Start and End dates are new fields which have been added into the line item grid, to identify line items which are contracts. By using this approach, you can have an order which contains a mix of contract line items and standard line items. So if you have an order for a projector, which contains a line item for the projector itself and a second line item for the 12 month support contract, when you create a contract Workbooks will only carry forward the support contract line item.\nThe contract management module is integrated with cases. This means you can track cases against a specific contract and even specific contract line items. If one of your customers calls up with a problem on a specific product, you can easily search by serial number or product key to find the relevant support contract. At this point, you can check if the support contract is still valid and if they are entitled to the level of support they are requesting.\nWe will be releasing contract management at the end of January and we will be scheduling demos of the new functionality over the next few weeks for existing customers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Today we're talking paint colors. What are your thoughts? Are you one of those people who walks into a store, flies by the seat of their pants, picks up the first paint swatch that catches your eye, and then ends up loving it just as much once it's on the wall?\nAre you the type that goes driving across town, hitting up all the paint stores, collecting a billion swatches, paying for multiple small samples of paint, stressing out and dreaming about if you should go one shade lighter or one shade darker?\nWell... we're somewhere in the middle.\nSometimes we feel like taking a complete risk by picking out a color based on how pretty we think that tiny little swatch in the store is, and other times, we do so much research about a color that we feel like our heads are going to explode.\nThat being said... we LOVE to paint!\nI've said it before and I'll say it again (for the hundredth time)... paint is THE BEST way to change and freshen up your space. For usually under $30 a gallon, you can take a dark room to a bright happy oasis in no time flat.\nThe new Show Your Colors feature lets homeowners submit their before and after photos from spaces they've redone in their own home, to share with others! I love this idea, because first, who doesn't LOVE before and afters?... and second, I kind of have an issue with taking a peek into the lives\/homes of other people, to see how they live\/decorate.\nIt simple, all you have to do is head over to the Show Your Colors page at My Colortopia.com, fill out a quick form about your before and after entry, upload a couple of pictures... and you're done!\nYou can put your mouse over an area of any uploaded pictures and snag paint color ideas, or just simply brows the awesome before and afters for ideas to implement in your next room makeover. Either way, it's a pretty great tool to try out and get inspired by.\nThis conversation has been sponsored by Glidden brand paint.\nI LOVE picking out paint colors!! Sick. I usually take about 6 swatches of each of the colors that I am deciding on. I hang them all over the room so light hits them all a different way. If I walk in the room and look at one and don't like it, it gets taken off and eliminated immediately. I do this at different times of the day until I'm down to the one that's left. It may take two days or so to get to THAT ONE, but, this method has never steered me wrong.\nYou get a gold star for SURE! You're patient. I'm SO bad at waiting. I usually want to paint that same day, but waiting to see what the swatch looks like in different lighting is so smart!\nThat's how I choose my flavors (colors) too Karen K. Except for when I did my dining room ~ I was a lil more skeered.... I wanted a golden yellow. I narrowed it down to 3 and decided to get a little sample container of each one, and I'm so glad that I did. The 2 that I liked on the card were way too light-bright-yellowish. HA! When I opened the 3rd container, I almost didn't put it on the wall....because it looked too orange. My friend said ~ go ahead and put it on the wall, let's see what it looks like. I'm glad she pushed me to try it ~ because that's the color I chose ~ ~ and I love it.\nI'd love to know the paint on that sweet blue dresser?\nI had my friend Sausha at Sweet Pickin's make it for me. http:\/\/www.sweetpickinsfurniture.com\/ Maybe you could find the color there or ask her :) It's so pretty, I love it!\nThanks! Love her site too!\nI'd love to know the paint on the sweet blue dresser?\nI LOVE your blog!! Your so talented!! I mean your hubby and you are sooo talented..lol!!! I was wondering where you got that cute rug in the laundry room??\nPaint can be really tricky. After building our home and literally picking out paint for every single thing I still enjoy the process. Nothing like a can of paint will change a room so dramatically for so little cash. Yes it's time consuming to paint, but supplies are way cheaper than new furniture.\nThat's why I say I LOVE painting... but reality, I just love what paint does to a room!\nI just found your blog a week or so ago and I'm totally hooked! You and your husband are adorable. Glad you are speaking about paint, we have a new house that was painted one color by the builder and it's too dark for my liking. It actually looks like the same color as some of the before photo's you have of your house. I want to change every room but our biggest problem is that our great room is 2 stories and open to the entire front of the house. How do we change the paint color without going to the huge expense of hiring a painter...or should we just go that route? Would love to hear your thoughts\/experiences!\nHmm... honestly, we have vaulted ceilings (maybe not as tall as two stories) but we just borrowed an electrician-friends really tall ladder and did it ourselves. If you think it's WAY too tall to even attempt, then you might want to think about renting scaffolding or hiring it out.\nI also second the ladder or scaffolding. We have a small two story foyer. We did all of our own painting in the house when we built it. We also have a 25 ft. extension ladder. Lot of moving it, but it works. We also have one of those paint sticks too, so you don't have to get up and own to reload the roller. That may help too.\nJust curious, what is the paint color you used in your laundry room? It looks green and like the color I've been looking for for my dining room? Can you help me out with the color?\nThanks! I love your ideas and appreciate you sharing them with us all!!!\nY'all are so cute. When you said your heart goes pitter-patter at the smell of new paint, I feel that way too...when my painters start! The problem is that I come from a non-painting family and so does my husband and we've never properly learned to paint. So we're not satisfied with our own work. And at this age...late 50's, we just skip it. And I pick my colors VERY carefuly, since they stay up awhile.\nI totally agree that paint can make a huge difference! Unfortunately, my husband haaaates to paint and he won't let me do it! I don't know if I blame him though - haha! Anyway, that's a cool little website\/tool. I'm going to check it out! =] Thanks!\nLoooooooove the green of your laundry room! Do you mind sharing the color name?\nAnd, may I just say that I ADORE your blog? Love your style, your personality, your realness, and well, everything!\nbut when it's time to really pick colors i might be the person that really thinks about it bc i hate painting!!\nThat color is actually called Adobe White, but it's REALLY similar to Autumn Haze, the one we have in our whole house. It has a lot of yellow in it, but is still cream based. Yellows ARE so hard!\nThis is such an a great link! I might have to post a few before and after of my house since it had this UGLY flesh tone peach\/beige color. I agree, color is a great way to change the feel and jump start a new makeover!\nI don't think I've ever seen a washroom that looked that nice. I love it! The fanciest thing I've ever seen in one of those is a sturdy, anji mountain bamboo rug. I love how the design even takes utility into account without sacrificing the beauty. Tile in the all-too common case of leaks. Clear floor for clothes. All of the design elements are high enough to not get in the way or destroyed. Brilliant!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Initiate Agency was launched in 2007.\nOur Agency was not launched by chance, but rather because of the constant inquiries we had from our publishing clients to help them with their design needs.\nInitiate Agency is part of global media company Initiate Media, and as such from our launch, we were immediately busy on not only client work, but our own projects.\nThe designers and developers that work on our clients' work are the same people that work on our suite of products, so we not only get branding and web, but we put it into practice in our own company daily.\nInitiate Agency therefore understands practical outcomes. We don't just design and create and hand files to our clients, not aware of how products and services will be reflected in the marketplace. In our own company, we understand the need for producing work that gets results.\nFrom graphic design and branding, right through to web design and development, film and apps, one company can cover your full gamut of design needs.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Taylor Swift's Reputation was on the line, and got the ultimate thumbs-down.\nThe Recording Academy made the announcement for this year's Grammy Award nominations on Friday and saw the year's top-selling album by Swift garner only one nod (for best pop vocal album). It's especially jarring for the \"Look What You Made Me Do\" singer, who's been a Grammy darling in recent years with 31 nominations and ten wins. It might be time for another Swift chapter to unfold (because, ya know, the old Taylor Swift can't come to the phone right now, or whatever).\nThe same goes for the Carters, whose Everything Is Love surprise album earned only three nominations and none in the top four categories, while Ariana Grande's Sweetener album picked up only two (though \"Thank You, Next\" debuted after the eligibility period had ended).\nSam Smith, who performed \"Pray\" at the awards show last year and won Best New Artist and Record and Song of the Year (for \"Stay with Me\") in 2015, also got zero love. His sophomore album that came out this year, The Thrill of It All, didn't make the cut.\nBut, hey, at least we'll get some good reaction shots of (rainbow-haired?) Cardi B, who's nominated five times. And that's something to be thankful for, okurr?!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjhzp b/data_all_eng_slimpj/shuffled/split2/finalzzzbjhzp
new file mode 100644
index 0000000000000000000000000000000000000000..01c221602c2b65f982e475013d74e0b73a8bdfca
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbjhzp
@@ -0,0 +1,5 @@
+{"text":"Corporate political spending exposes Pinnacle West Corporation (\"the Company\") to risks that could adversely affect the Company's stated goals, objectives, and ultimately shareholder value. Pinnacle West's undisclosed \"dark money\" political contributions have been the source of significant controversy, reputational harm, and business risk.\nthe portion of dues or other payments made to a tax-exempt entity such as a trade association that are used for an expenditure or contribution and that would not be deductible under section 162(e) of the Code if made directly by the Company.\nThe report shall identify all recipients and amounts paid to each recipient from Company funds.\nSUPPORTING: Pinnacle West reports a portion of its political spending, and meets the minimal legal requirements that exist for political spending reporting. However, shareholders are concerned that the political spending Pinnacle West reports voluntarily and for compliance does not reveal the full extent of the Company's use of shareholder money to participate in the political processes. For example, press reports allege that Pinnacle West spent $3.2 million in \"dark money\" on the elections of two of their regulators, which is not disclosed by the Company. (Arizona Republic, 2015). In September 2015, Arizona utility regulators requested that Pinnacle West halt its political contributions to campaigns of its regulators, and former utility regulators advocated a subpoena of Pinnacle West's political spending records. Pinnacle West filed a public response stating that it would continue its political spending.\nPinnacle West's spending on the campaigns of government officials creates, at a minimum, the appearance of impropriety; further, the legality of its political spending has not been publicly established and cannot be effectively determined without full disclosure. Due to the ongoing nature of the Company's political activities, and Pinnacle West's stated intent to continue political spending, proponents request shareholder support this resolution, an earlier version of which received a vote of 30.8% in 2015.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our last show of 2013 and it's a time to reflect on our accomplishments, and others failures. Holy Fook!\nMarty tries to create a safe zone for the crew sharing their aspirations for the coming year, it gets a little ugly when someone's goals involve Miley Cyrus.\nThat someone is the insanely talented Star Anna who lays down two stunning tunes, and shares her other dream of touring Europe in '14. Let's make that happen for her!\nNo set New Year's plans yet? Join Star at the Sunset on New Years Eve or Marty and his musical twin, LeRoy Bell, at the Kirkland Performance Center.\nOh, and if you stumbled upon this site while surfing martyriemer.cn or martyriemer.com.cn or martyriemer.org.cn, we're on to you!\nPosted on December 27, 2013 at 12:08 pm by Marty in Podcasts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Members & guests\u2014free. To register, please click here.\nEveryone worries about outliving their assets. We'll focus on how your retirement behaviors can help (or hurt) your plan. We'll discuss patterns of retirement spending; how you can successfully transition into this new chapter in your life; and share real life examples of the pitfalls and successes we've seen with clients throughout retirement, giving examples of the different transitions people experience when they decide to stop working. Refreshments will be served.\nPlease note: the name and email address for all registrants will be shared with host Charles Schwab for compliance purposes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SPECIFIC Senior Large & Giant Breed is a nutritionally complete and balanced dry food for mature dogs over 25kg.\nWhat is SPECIFIC Canine Everyday Senior Large & Giant Breed dry food?\nSPECIFIC Senior Large & Giant Breed is a complete and balanced dry food for mature dogs over 25kg. It can be fed as a main meal or mixed with dry kibble.\nThe recipe is rich in Omega-3 from fish help to support healthy skin, coat and joints. The added beta-glucans from yeast help support the immune system. This diet contains lower calories and higher fibre, making it less energy dense. This means that dogs can be fed their usual amount, not leaving them feeling hungry, whilst still controlling weight. With added L-carnitine to help maintain ideal bodyweight.\nThe larger kibble size encourages chewing.\nThe diet is made from highly digestible ingredients and contains added prebiotics (FOS) to promote good digestive health. The reduced protein, Phosphorus and Sodium levels help to lower the workload on the kidneys and added Taurine helps support the heart muscle function.\nSPECIFIC have added psyllium husk to help firm up stools and regulate the digestive tract.\nWho is this food suitable for?\nThis food is suitable for senior dogs over 25kg.\nWhen is my dogs considered senior?\nSmall breeds (under 10kg): From 8 years.\nMedium breeds (10 - 25kg): From 7 years.\nLarge breeds (25 - 45kg): From 6 years.\nGiant breeds (over 45kg): From 5 years.\nWhat are the benefits of using SPECIFIC Canine Everyday Senior Large & Giant Breed dry food?\nWhat is in SPECIFIC Canine Everyday Senior Large & Giant Breed dry food?\nEC approved additives: BHA, BHT, propyl gallate.\nCrude Protein 18.5g, Crude Fat 7.4g, Carbohydrate 57.7g, Crude Fibre 2.8g, Calcium 0.57g, Phosphorus 0.48g, Sodium 0.2g, Taurine 0.11g, L-carnitine 30mg, Omega-3 0.76g, EPA 0.18g, DHA 0.25g, Water 8.5g, Ratio n-3:n-6 1:2.\nHow much SPECIFIC Canine Everyday Senior LArge & Giant Breed dry food should I feed?\nThe daily requirement may vary depending on factors such as breed, environment, season and activity level. It is recommended that you monitor your dog and adjust the feeding amount to fit their individual needs.\nHow do I know if my dog is the right weight?\nWeight check: Get your dog's body condition checked and weighed.\nPlanning: Work with your vet to decide what is the target weight. Choose a suitable food for your dog's needs, and then calculate the daily feeding amount.\nImplement the feeding: It's important that you measure out the food so, if you need to you can easily and accurately adjust it.\nRegularly weigh your dog: A simple way to do this is weigh yourself, then weigh yourself while holding your dog then subtract the two numbers. If you have a larger breed or are unable to carry the dog, it is advised to get your dog to sit on some bathroom scales, or pop into your vets to use the scales in the waiting room!\nBe patient \u2013 It's not going to happen overnight but if you stick with it your dog will reach their target weight, and it will all be worth it with a happier healthy dog.\nMaintain weight: Once you have reached target weight you need to make sure they stay there by choosing a suitable food for your dog's needs.\nRemember: If your dog is on a diet, be careful not to spoil the good work with treats (especially table scraps), and make sure everyone knows what you are doing!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Haad Thip Public Co., Ltd.\nTropical Canning (Thailand) Public Co., Ltd.\nGold Coin Specialities (Thailand) Co., Ltd.\nTeck Bee Hang Co., Ltd.\n2S Metal (Public) Co., Ltd.\nTSC (Songkhla) Shipping Agent Co., Ltd.\nHaad Kaew Golden Sand Co., Ltd.\nIntention Media Plan Co., Ltd.\nFour Seas International Co., Ltd.\nHatyai International Shipping Co., Ltd.\nSongkla Canning Public Co., Ltd.\nJ.B. Hotel (Hat Yai) Co., Ltd.\nDiana Complex Shopping Center Co., Ltd.\nPithan Palm Patana Co., Ltd.\nTavorn Latex Industry Co., Ltd.\nCPF (Thailand) Public Co., Ltd.\nSahakarn Palm Industry Co., Ltd.\nEagle Wire Mesh Co., Ltd.\nNam Hong Plam Oil Co., Ltd.\nSouthern Marine Products Co., Ltd.\nKiang Huat Seagull Trading Frozen Food Public Co., Ltd.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbjkza b/data_all_eng_slimpj/shuffled/split2/finalzzzbjkza
new file mode 100644
index 0000000000000000000000000000000000000000..4b6c28a7700b77a47847e598eb250451c362dbf3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Acrobat\u00a9 Reader\u00a9 is required to view all PDF downloads.\nDesigned for hazardous damp and wet industrial locations. This severe-performance NEMA-4X protection-grade battery unit is for Class I Division 2, Groups A, B, C and D, Class II Division 2 Groups F and G & Class III. The SPH family of hazardous, severe performance products is available in an LED battery unit and coordinating remote fixture.\n\u00a9 ABB Installation Products Ltd 2019. No material from this site may be copied, distributed or used in any way without the express prior permission of ABB Installation Products Ltd.\nYes, I want to print. Don't print.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What Are the Rewards of Blogging?\nWhile social networks has taken the customer's communication to an entire brand-new globe, blog writing or generating an interesting website has really provided the concept of pen\/paper\/diary an appearance that is outdated. Several of the preferred websites that allow you to establish your very own complimentary blog site are BlogSpot.com, WordPress.com, and also Weebly.com and so on. These all are completely complimentary internet sites and do not charge you for a personal blog website.\nWhat is Perks of Blog writing?\nPeople make use of a blog site as an individual diary, continually remembering every thought\/emotion. It serves for lots of and also a method for others. They have the option of making this individual journal public or keep it for personal view. On-line journals are a lot much safer as compared to physical ones. They are password protected and can be ruined in a matter of seconds. Click here blogging.org to get more details.\nYou could use your blog site to market a product or examination regarding it. You can blog concerning it and allow others understand.\nYou could likewise blog to connect to individuals sharing very same interest rate. If you are a digital photographer and also wish to improve your abilities you can connect to individuals that want same area and share a tip or more.\nBlog writing is an outstanding device to develop by you as a professional. If you are good at baking and also excitedly prepare for change your leisure activity right into a full blown profession, blogging worrying your cakes, including their pictures would certainly assist you being noticed in the team of millions.\nYou can utilize your understanding on specific based on assist others. If you have had some sort of circumstance and also you intend to aid individuals entrapped right into something similar, your understandings can assist them. Such blog websites could be categorized as tutorials or 'how-to' kind of blog site sites. It's simple to create a blog site. You do not require any technological understanding for that. Comply with a straightforward in-depth tutorial with screenshots and you will have the ability to establish a blog site in less than 5 minutes.\nWhy you demand wordpress emergency assistance?\nThinking of the ever before enhancing allure, there is an excellent need for numerous other WordPress products such as the WordPress designs in addition to plug in. While all these products are extremely vital, your WordPress style can produce or wreck your on line exposure. Whether you wish to get a large amount of followers or just make money, you will definitely need to utilize an attractive and a well practical WordPress concept. WordPress is an open resource system, for that reason you can expect plentiful of resources such as motifs or plug INS, conveniently offered for usage, most definitely entirely free. Nevertheless, these do not ensure effectiveness or high quality. While there are a variety of free of charge WordPress themes, you will certainly have to personalize them in order to establish a one-of-a-kind theme for your web site.\nBesides, WordPress makes modification on a continuous basis and even more recent suggestions are being included every year. If you want to take a look at the existing crazes this year, you need to simply look for the game of thrones. Nowadays, there are numerous type of wordpress profile style, based on the sort of website you have. For scenarios, there are audio in addition to video WordPress styles, relying on the needs of your on the web company. These concepts can be pleasantly enhanced, making certain that your internet web site will definitely obtain the leading outcomes of the on line search engine. If you are unable to discover a price complimentary WordPress motif, you might comfortably try to find expenses designs, which will absolutely enhance the appearance of your website and also aid you produce a much better effect. The absolute highlight concerning getting expenses WordPress styles is that they provide support or upgrade to all individuals.\nIn scenario if WordPress was to introduce an extra current variation, you can immediately update the site for newer functions. Despite your on line professional, you are most likely to find a motif that finest suits your website. There are many net site principles to choose from, in numerous color schemes. WordPress styles are very basic to place as well as you do not as a matter of fact need to be a computer system specialist to acquire started with this platform. Rather than spending your useful loan and time on designers, you might comfortably pick any type of one of the premium wordpress emergency assistance. There several sources on the internet that could help you start with WordPress. Different efficient net sites are currently relocating to WordPress, for the convenience it gives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The articles in a knowledge base are available to both internal users , such as customer service agents, and to external users (customers and partners). If the articles in a knowledge base are intended for internal users only, the system administrator can restrict access by customizing the knowledge base form.\nNavigate to Knowledge > Administration > Knowledge Bases.\nClick the desired knowledge base.\nRight-click the form header and select Configure > Related Lists.\nSelect Cannot Contribute from the Available column and move it to the Selected column.\nOn the selected knowledge base form, click the Cannot Contribute related list.\nClick All Customer Contacts in the left column and move it to the right column.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Police officers searching for a gunman in Sheffield had removed a cordon around a disused brewery.\nOfficers have searched the disused Stones brewery in Neepsend Lane, Neepsend, and have re-opened surrounding streets.\nThey have also searched wasteland around the brewery.\nSouth Yorkshire Police has not yet revealed whether the gunman has been caught.\nAn operation was mounted at 5.15am when a man armed with a gun was seen entering the brewery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are new to competing it can be really tricky to work out what category and show to go for! There are more and more federations each with their own unique take on the industry and specific requirements.\nIt may be the first time you've ever stepped foot in a gym, or maybe you're returning after a hiatus. Either way there is no doubt about it, going alone to a new gym can be utterly terrifying!\nApple cider vinegar is now widely available in a number of UK shops. Many people feel that the vinegar helps them to control their weight whilst others feel it helps their digestion.\nRecently I completely lost all my motivation to workout, I just didn't feel like venturing out in to the cold and dark and getting myself to the gym.\nAs I've mentioned previously, sometimes MyProtein products completely hit the mark and I'm always a little dubious of \"special\" flavours of anything as I generally speaking they tend to be a bit naff.\nIf you're anything like me imma bet you love a little chocolate treat every now and again! Thing is though, your average chocolate bar lingers around the 200 to 300 calories mark.\nBest flavour MyProtein Whey protein powders!\nWith so many to choose from it's hard to know which is the BEST flavour MyProtein powder. I give you my run down here!\nhttp:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/10\/MyProtein_ImpactWhey.jpg 621 1307 Amy Kate http:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/06\/aklogo-300x300.png Amy Kate2017-10-01 18:48:422017-10-19 10:42:20Best flavour MyProtein Whey protein powders!\nNext year Rob and I hope to take some extended time out to tour France and Italy much further. We're hoping to see Nice, Cannes, Venice, Florence, Rome, Milan to name but a few. Therefore, my ability to say hello and order alcohol is only going to get me so far (probably just drunk).\nMy nexplanon contraceptive implant caused crazy symptoms after 2 years!\n2 years in to my Nexplanon implant and completely out of the blue I started to get some really crazy side effects that I'd never had before with any of my previous implants.\nhttp:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/09\/Nexplanon.jpg 1000 1000 Amy Kate http:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/06\/aklogo-300x300.png Amy Kate2017-09-13 14:24:502018-04-27 10:14:49My nexplanon contraceptive implant caused crazy symptoms after 2 years!\nReview: Quorn vs chicken, calories and nutritional values - can it sway a bodybuilding meat lover?\nI have an ongoing life-long love affair with red meat but I also LOVE trying new foods. It dawned on me that to my knowledge I've never knowingly had Quorn so it seemed like a good place to start.\nhttp:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/09\/QuornFoodPrep.jpg 274 370 Amy Kate http:\/\/amy-kate.co.uk\/wp-content\/uploads\/2017\/06\/aklogo-300x300.png Amy Kate2017-09-01 12:57:072018-03-13 12:16:05Review: Quorn vs chicken, calories and nutritional values - can it sway a bodybuilding meat lover?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbkbmu b/data_all_eng_slimpj/shuffled/split2/finalzzzbkbmu
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--- /dev/null
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+{"text":"Ocean View Resort offers top quality accommodation for all travelers with cool, clean and spacious rooms and suite to any budget.\nAll rooms have private en-suite facilities. It divided to 2 Beachfront Bungalows , 8 Garden View Bungalows , 14 Aircon Rooms Building and 3 Fan Rooms Building.\nYou can check each room facilities and decoration from the left side links provided.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the basic implementation flow that I recommend. Each step is a dependency on the step before. This does not mean that steps cannot be carried out simultaneously - and they typically are. However, the completion of one step is dependent on completion of the previous step as outlined here. For example, Internal Audits are often started very early in a project. But they cannot be completed until all required documentation is trained and implemented, which in turn cannot be completed until all documentation is completed. Not rocket science - Common Sense.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Listed in: Party Catering and Simcha Halls \/ Restaurants categories.\nElite Cuisine offers fine catering for all your special occasions. Your home, office, or our dining room!\nGet more for your wedding! Don't forget to mention ChossonKallah.com when visiting Elite Cuisine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The revolutionary Infinity Control taps the power of the Infinity System. You'll find unmatched access to every type of environmental control. Remote programming can be added as an accessory. Simply our best thermostat.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UK's Blair says did not foresee Iraq \"nightmare\"\nLONDON (Reuters) - Former British prime minister Tony Blair said on Wednesday he could have not have imagined what he called the \"nightmare\" that unfolded in Iraq but still did not regret joining the U.S.-led invasion.\nIn a political memoir Blair echoed previous statements that the 2003 invasion was justified because Saddam Hussein posed a threat and could have developed weapons of mass destruction.\nThe self-penned volume \"A Journey\" was published on the day the United States formally ended combat operations in Iraq after a conflict that claimed more than 100,000 deaths, most of them civilians.\n\"I can't regret the decision to go to war ... I can say that never did I guess the nightmare that unfolded,\" said Blair, referring to the years of political and sectarian bloodshed in Iraq that followed the invasion.\nBlair was the closest ally of former U.S. President George W. Bush over the decision to invade Iraq.\nThe decision was the most controversial of Blair's 10-year premiership, provoking huge protests, divisions within his Labour Party and accusations he deceived Britons over his reasons for war when weapons of mass destruction were not found.\n\"I feel words of condolence and sympathy to be entirely inadequate,\" Blair wrote of the war's casualties.\nElsewhere in the book's 715 pages Blair revealed a previously unknown reliance on alcohol to cope with the pressure of office.\nAnd he was damning about former finance minister Gordon Brown who succeeded him as prime minister in 2007, describing his political rival as brilliant but lacking human instinct.\nHe blamed Brown for the Labour government's defeat in the May election after 13 years in power, saying it was a fatal mistake to move away from Blair's centrist \"New Labour\" policies.\nBlair, now an envoy for the Quartet of Middle East peacemakers \u2014 the United States, Russia, the EU and the United Nations, was Labour's longest-serving prime minister, winning three consecutive elections before stepping down in 2007.\nHe has said he is donating the reported 4.6 million pound ($7.0 million) advance he received for his memoirs as well as proceeds from sales to a charity supporting serving and former members of the military.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"A filmmaker from North Lincolnshire is in a critical condition in hospital after an accident in Spain.\nHis sister is fundraising to get the family out to him.\nCarl Whiteley, who is originally from Crowle, reportedly suffered severe head injuries after falling from a first floor balcony of a flat in Palma's old town on Tuesday night.\nIt is believed he was admitted to Majorca's Son Espases Hospital and is in an induced coma.\nCarl's sister Emma set up a GoFundMe page which has raised \u00a3570 of its \u00a310,000 target at the time of writing.\nOn the fundraising page Emma said: \"My brother is in Spain, we are in England\u2026 we need help to get all of our family over to our brother CARL WHITELEY! He's had a terrible accident and is in critical condition.\n\"The hospital have him in a induced coma! Our brother is amazing and I know he's helped a lot of people and a lot of charities\u2026 any donations are greatly received.\nAccording to his Twitter bio Carl is the owner of DiagCon, moviESCAPE and Jollywood Events, as well as the director of Pandora Movie.\nHis films include The Empire, a Star Wars spoof for which he wrote the script.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Nicely renovated home in an up and coming area! Large front porch. Lovely, original hardwood floors. Spacious kitchen. Full basement provides ample storage. On a great block just north of Mack Avenue. Great location, minutes away from Grosse Pointe, Jefferson Chalmers, I-94.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"One of the biggest mistakes many pageant contestants make is assuming that any department store swimsuit will do for the beauty pageant swimwear portion of a competition. It's important to choose your pageant swimwear carefully; making sure it suits your figure and fits well in all of the right places. At PageantDesigns.com we carry a variety of pageant bathing suits that will stand out on stage and keep you in the judge's minds.\nLady M Swimwear has been seen on the Miss USA pageant stage and has also featured in Pageantry Magazine. All of our two-piece Lady M Swimwear is available in bright colors that will help you stand out amongst the crowd. Choose from different two-piece swimsuit designs, including underwire tops, low rise bottoms, halter tops, satin pageant swimsuits, and many other sparkling beauty pageant swimwear options. If you have any questions, please call us at 864-263-4371 or email dresses@pageantdesigns.com. You don't have to spend hundreds of dollars on a great high quality swimsuit for your next beauty pageant, just shop the vast selection of pageant bathing suits at PageantDesigns.com!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This is the amazing group that makes our work and progress possible in Tigray.\nThe Journey is a video series that will take you to Ethiopia to see charity: water's work in the field. Join our Content Strategist, Tyler, as he experiences life without water and meets the incredible people of Ethiopia.\nTrying to figure out what's going on here? Meet Tyler. He's a Content Strategist at charity: water and your host for The Journey. He'll explain (almost) everything.\nBefore we get too far into our trip to Ethiopia, there are a few things you need to know. Here are some tips to kick off the adventure.\nTarik Haftay is one of many young girls who puts in hours each day walking to collect dirty water for her family. Spend a day walking in her shoes.\nOur local partner in Tigray began as a grassroots organization. Now they're the largest operating local charity in the region. They've helped us implement more than 3,000 water projects, and they're incredible.\nHaving access to clean water saves women in Ethiopia hours each day. Find out how families in May Umeri Village are using all of their extra time after getting a well.\nSometimes building a well requires huge machines. Sometimes it involves explosives. Here's a behind the scenes look at how our local partners find and deliver clean water.\nWe're not just funding new water projects; we're also training teams of local mechanics who can respond when wells break. These Dispatch teams keep the water flowing at charity: water projects in Tigray.\nAs The Journey comes to an end, we take a look back at some of the things we learned along the way. This isn't a good-bye; it's a see you next time!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Chicago, IL -- Witness, Nancy Munoz, was coming home from the Monday night Bears game when she saw a large fire behind her parent's West Town home.\nBy the time Munoz saw the fire, it was already huge. \"I saw a lot of fire,\" Munoz said.\nMunoz and her parents were three of six people displaced by the fire. The fire started in the yard of a pallet company on Damen Avenue and Ferdinand Street. No one was injured in the fire.\nAre you interested in learning how to protect buildings with stacked pallets? Click here.\nWhen the Chicago Fire Department arrived, it had already spread. It was moved to a 3-11 alarm fire. The main concern for fire personnel were exposure to three nearby buildings, which included the 2-story home belonging to the Munoz family.\n\"We were concerned for those buildings because we didn't want the fire to spread to them. We made an aggressive attack, elevating the alarms at that time,\" said Assistant Fire Commissioner William Vogt.\nAbout 100 firefighters responded to the fire. They used tower ladders to reach inside the pallet yard. Pallets produce a lot of heat, making pallet fires very difficult to fight. Propane tanks were also in the pallet yard. They were exploding during the fire.\nAround 1am, firefighters got the fire under control and were just patrolling for hot spots.\nThe other two buildings that firefighters were concerned about were okay. Only the Munoz house was damaged.\nThey are also combustible and the cause of 12% of warehouse fires. Internal temperature of ... Palletized Storage: Items are placed on wooden or plastic pallets.\nJul 6, 2016 ... Officials reported that the fire destroyed a warehouse that was built before 1920 and used ... Fire Protection for StackedPallets in Warehouses.\nOlder Post5 Year Fire Sprinkler Inspection: What do I need to know?","meta":{"redpajama_set_name":"RedPajamaC4"}}
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new file mode 100644
index 0000000000000000000000000000000000000000..350ac43cfcf2fa8115fcb63befc6cfa00c0a5f38
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbkmmn
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+{"text":"Apex uses our customizable reporting package that allows our clients to define what information they receive, the frequency in which it is received and in what format it arrives.\nOur integrated reporting services provide agency-level reports designed specifically to integrate into your unique workflows and business processes.\nOur reconciliation reports allow our clients to see data in many different ways. Our drill down reports are informative, and are designed to create a workflow \u2013 customized for your team and ours.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You have done the hard work to set up your space, now let us help you keep it that way. We personalize each session, ensuring your space not only remains neat and tidy, but that it grows and flourishes with you and your family. Best of all, members enjoy the peace of mind that comes with knowing there is an expert in their corner, ensuring their goals are continuously met and even exceeded.\nBecoming an Intentional Interiors member means you are taking the plunge toward a more thoughtful, holistically balanced home in alignment with the healthy lifestyle you crave. Membership is easy, and the results\u2026life changing.\nCall us at 954.406.6002 or email info@YourProOrganizer.com to join.\nAventura, Boca Raton, Boynton Beach, Brickell, Coral Gables, Coral Springs, Deerfield Beach, Delray Beach, Fort Lauderdale, Greenacres, Hallandale, Hollywood, Homestead, Kendall, Key Biscayne, Lake Worth, Miami, Miami Beach, Miami Gardens, Miami Springs, North Miami Beach, Palm Beach, Palm Beach Gardens, Parkland, Pembroke Pines, Plantation, Pompano Beach, Wellington, Weston, West Palm Beach.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was written by Chief Monkeyologist, posted on July 29, 2008 at 8:39 pm, filed under Video and tagged Blog, camera, ijustine, Justine Ezarik, monkey, tech. Bookmark the permalink. Follow any comments here with the RSS feed for this post. Both comments and trackbacks are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"SDM ENGINEERING & CONSTRUCTION PTE. LTD.\nSDM ENGINEERING & CONSTRUCTION PTE. LTD. (UEN 201908819N) is a business entity registered with Accounting and Corporate Regulatory Authority (ACRA), Singapore. The UEN is issued in March 18, 2019. The entity status is issued in Registered.\nEntity Name SDM ENGINEERING & CONSTRUCTION PTE. LTD.\nInformation Source Details on SDM ENGINEERING & CONSTRUCTION PTE. LTD.\nPlease comment or provide details below to improve the information on SDM ENGINEERING & CONSTRUCTION PTE. LTD..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"camilo2293 MILKY20 jasmin. RimmaSugar. MilitaryBen. NolanJunior.\nAdaDunnmsBOLDSTARcockMilitaryBenLiNaLi69 .alexxasquirtWhiteSofiaKratosAndSeleneLylianaWhite .adeleineConnoryLogan1RihanDeckerLylianaWhite .LediAminamsBOLDSTARcockBradleyCruiseAriaRosse .CassidyCarterRihanDeckerMarusiyaMaddyMarieCute25 .AbbyDagmarUnmatcheddHallySweetInfiny .GirlNaugthyHot4uCookieAndKremeMilitaryBenLetizzia .LetizziaLediAminaLediAminaAdaDunn .sophiabigcockBabyLarraasiansexyjenyCECILSLUT .NylonLadyBatGirlsBrandonClamScumSummerAlexia .YourSexxPrincexsAriaRossesophiabigcockLeveret .\nLilithMistressShyKrystynCUTEDOLLHOTJhonLong .MarusiyaMaddyannehotdollLeveretNASTYblondDANCER .ShyKrystynBellaRozeAbbyDagmarasiansexyjeny .latexxxbbitchMUSCLEJERALATINOKratosAndSeleneMichellBrown .ShyKrystynHallySweetsophiabigcockCookieAndKreme .YourSexxPrincexsBESTMADAMXXXBradAndToriLediAmina .sophiabigcocksashalopezzXAriaRosseYourSexxPrincexs .UnmatcheddlatinlustboyellejoelMarieCute25 .AwesomeKittyXXBradleyCruiseMarieCute25NolanJunior .MichellBrownDimitidiamonConnoryLogan1modeltop .PlayfulMISTRESSxAriaRossesashalopezzXAriaRosse .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmivg b/data_all_eng_slimpj/shuffled/split2/finalzzzbmivg
new file mode 100644
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--- /dev/null
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+{"text":"Taylor's Classic Travels \"Vintage-Style Trolley\"\nTaylor's Classic Travels is proud to bring you our vintage-style trolley from the historic French Quarter of New Orleans. Taylor's Classic Travels is a Richmond Virginia family-owned trolley rental & chauffeur service specializing in transportation for wedding, bachelor & bachelorette parties, corporate events, wine\/brewery\/Tacky Light tours, proms, homecomings, etc in the RVA area. Make your next occasion even more memorable with Taylor's Classic Travels' beautiful trolley.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Own2feet, Inc. - Orthotics for Golfers, Golf | Own2feet, Inc.\nCheck out this great video describing the advantages of the golf orthotic!\nAll custom golf orthotic interest or purchases require a phone consultation. Please reach out via the form below with your phone information and the best time to contact you.\nPlease see below for the two offered styles of custom golf orthotics or click the above link.\nTakeaway- Weight shifts to the inside portion of the right knee and inside portion of the right arch with no rotation of the right foot outward; weight on the left arch is reduced as weight shifts right.\nThe downswing-Is initiated by an lower body shift changing weight to for left side.\nIn the backswing, weight either stays left or moves right with weight transferring past the inside part of the right knee or the right foot rolls outward causing a shift away from the ball; this is a key cause of the slice!\nThe Own2feet Custom Golf Orthotic!\nThe left foot is designed to initiate the downswing with the lower versus the upper body.\nBALANCE is meticulously maintained throughout the entire process resulting in straighter longer shots!\nCustom Made Golf Shoes - Soon!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A sepia-toned photograph of four adult men, one child, and two horses standing outside the John Gaarde Blacksmith Shop. John Gaarde is the second man from the right and is standing next to his son Hans Gaarde. The man on the far left is Al Johnson. An unidentified man on the far right sits on one of the horses. The shop's buildings each have western style false-front facades and a wide bay door on the ground floor. The building to the left is a two-story building with two double-paned windows on the second floor and one on the ground floor. The building to the right is a one-story building. Two double-paned windows are visible along the side of the building and a chimney extends upward from the roof. John Gaarde established his blacksmith shop in Tigardville in 1893. The shop was located on present-day Pacific Highway, across from McDonald Street. John Gaarde's shop and Charles Tigard's general store and post office were at the center of Tigardville's early commercial activity.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here are some photos of the muslin of the doll dress I was working on, completed. It is entirely hand sewn using cotton quilting thread to make a basting stitch.\nI draw any adjustments in fit and length that I want to make directly onto my dress muslin with a mechanical pencil. Before I assembled the dress muslin, I traced the original version of my pattern right onto my fabric using a permanent black marker. Generally, I use a pencil to mark my dress muslin while my doll is wearing it to make sure I don't mark the doll.\nThere are some minor adjustments to make. There was a small pucker in the bodice at the underarm. I will deepen the bottom of the armhole to compensate for this. I also added a quarter inch to the length of the skirt, and widened the back of the bodice slightly.\nI used a single strand of thread for my stitching because it is easier to pull out, which is the next step. The muslin is next carefully disassembled using a seam ripper. Then I trace each piece, refine each one on paper, and make a final draft of my pattern. I haven't pressed the dress muslin at all, to minimize any distortion of the fabric pieces.\nIncidentally, although the rough draft of a dress is called a muslin, I don't use actual muslin fabric to make my dress muslin. I used cotton sheeting on this one. The important thing is to use a fabric similar in weight and drape to the fabric you will use for your real dress.\nWe'll be adding these new collaborative dolls to our online doll gallery once they are completed and photographed. I'm brainstorming on my navigation ideas for that main doll gallery page. There are so many original art dolls now, they're divided into four subcategories, jointed cloth and paperclay art dolls, art doll figurines, soft art dolls and art toys, and already adopted dolls. Be sure to come check them out.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"For those who are looking to study caregiver training course offered by the Technical Education and Skills Development Authority, please refer to the existing accredited caregiver schools in Caloocan below in our list. Whichever training center you choose, you may inquire in their respective hotline contact numbers as listed regarding caregiving course training syllabus, certification, scholarship offers, how long the course duration takes to finish and their curriculum.\nFor more information, you may also visit their website at tesda.gov.ph\/ for accreditation issues of the schools listed or call the main branch or come in person over in Taguig branch. In the meantime, we strongly suggest that you use our search box feature below to find more testing and training centers.\nA & A Bldg., 120 P. Jacinto cor. Boni Serrano St.\nAttentive Learning in Health Services Inc.\nMartinez Caregiver Training Center, Inc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmmtq b/data_all_eng_slimpj/shuffled/split2/finalzzzbmmtq
new file mode 100644
index 0000000000000000000000000000000000000000..2cea7774f9467951ca88e95e74fdd7f98dc45f87
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+{"text":"At Sterling, we understand our clients need to protect and enhance the investment in their application portfolio. Our Subject Matter Experts (SMEs) have a deep understanding of the skills required to build, maintain, web-enable, re-platform and modernize the enterprise level applications.\nWe work with major corporations as well as mid-sized companies who need assistance with translating their business needs into system requirements and ultimately delivering maximum value via a superior user experience. Our senior consultants can work with clients to help make critical decisions related to enhancement versus replacement or make versus buy, and our project managers ensure our teams deliver quality outcomes on schedule and within budget.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This dissertation explores the Holocaust's relationship to modernity and postmodernity and the historic event's effects on a post-Holocaust world, culturally, socially as well as artistically. Showing the Holocaust's widespread influence on our postmodern unconscious, this study examines a variety of literary texts that demonstrate the haunting effects of the Holocaust on the present.\nAfter a brief reading of Bernhard Schlink's The Reader that situates the concerns and parameters of this study, Chapter One looks at the theoretical relationship between the Holocaust and postmodernity. For the most part mutually exclusive, cautiously ignoring or, at best, dismissing one another, these fields ought not to be seen as antagonistic but can, instead, greatly contribute to our engagement with the past. Postmodern theory with its denial of universalism, its celebration of the imagination, and its emphasis on representation and representability is highly dangerous to Holocaust Studies.\nLikewise, postmodernism often shies away from addressing the Holocaust since the historic event comprises possibly its most serious challenge as this historic event questions the easy dismissal of absolute truths and values as well as the ludic treatment of history itself. Nevertheless, the two fields are intricately entwined and a postmodern approach to the Holocaust reveals new insight into both postmodernism and our study of the Holocaust.\nAccordingly, Chapters Two through Four look at the cultural, linguistic, and psychological effects of the Holocaust on post-War life in close readings of E. L. Doctorow's and Walter Abish's postmodern historical novels, Raymond Federman's highly experimental texts and his readings of Samuel Beckett's work, and D. M. Thomas's controversial fictions. While these authors by no means present a comprehensive account of Holocaust literature, they offer a wide variety of reactions to the Holocaust, not only showing its exemplary role for postmodernism but also exhibiting a variety of ways in which postmodernism can offer new insights into our understanding of the Holocaust and its role in contemporary life. The dissertation concludes with an examination of identity politics within Holocaust, exemplified by the case of Binjamin Wilkomirski's Fragments and the scandal surrounding its reception.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brokers at FCA have always been governed by the principle of \"Our Policy \u2013 Your Protection\". The professional standards of our regulatory RIBO act also confirm this concept, committing all licensed brokers to exacting standards of service, respect and confidentiality, to protect not only your personal assets, but also your personal data and privacy.\nFacilitate the underwriting process and insurance claims.\nThis information may be gathered from you, from other industry partners, such as insurance companies or brokers from credit organizations or from motor vehicle licensing and reporting authorities.\nIn general, your consent to our obtaining and using relevant personal information is implicit in your request to us to consult, negotiate or obtain insurance coverage for you. We may, however, still obtain your express consent as part of the application process.\nBasic identification data such as name, address, birth date and contact numbers.\nWe are committed to ensuring that the information we use on your behalf, is as precise and up-to-date as possible. The success of our services for you depends on the accuracy of that data. During the course of our day-to-day interactions with you, we will routinely verify the information in our records.\nYou have our further commitment to protect your personal records, whether in printed or electronic form, from any unauthorized access. We keep records of any breaches in our security and will notify you if such a breach has created a risk of harm to you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Looking for WordPress agriculture themes to create a agriculture type website in a unique way? Following is a collection of 20+ Best WordPress Agriculture Themes 2014 for you. These are great WordPress agriculture templates for 2014. Enjoy!\nArche is a premium WordPress theme especially for 'Architecture' and 'Graphic Design' industry. This theme comes with a powerful admin panel and page builder.\nBlack & Green theme from wordpress agriculture themes.\nAgriculture WP Theme from wordpress agriculture templates makes complicated things easy: outstanding simplicity of custom user interface turns website management into a task even a child can handle.\nBlack Agriculture WordPress Template by Nessy from agriculture wordpress themes 2014.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The brain is a complex system consisting of regions dedicated to different brain functions, and higher cognitive functions are realized via information flow between distant brain areas communicating with each other. As such, it is natural to shift towards brain network analysis from mapping of brain functions, for deeper understanding of the brain system. The graph theoretical network metrics measure global or local properties of network topology, but they do not provide any information about the intermediate scale of the network. Community structure analysis is a useful approach to investigate the mesoscale organization of brain network. However, the community detection schemes are yet to be established.\nIn this paper, we propose a method to compare different community detection schemes for neuroimaging data from multiple subjects. To the best of our knowledge, our method is the first attempt to evaluate community detection from multiple-subject data without \"ground truth\" community and any assumptions about the original network features. To show its feasibility, three community detection algorithms and three different brain atlases were examined using resting-state fMRI functional networks. As it is crucial to find a single group-based community structure as a representative for a group of subjects to allow discussion about brain areas and connections in different conditions on common ground, a number of community detection schemes based on different approaches have been proposed. A non-parametric permutation test on similarity between group-based community structures and individual community structures was used to determine which algorithm or atlas provided the best representative structure of the group. The Normalized Mutual Information (NMI) was computed to measure the similarity between the community structures. We also discuss further issues on community detection using the proposed method.\nRecent development in neuroimaging devices such as magnetic resonance imaging (MRI) and electroencephalography (EEG) enable us to record human brain activity with high temporal and spatial resolutions. As the brain is known as a complex network consisting of distributed brain areas dedicated to different functions, it is natural to shift towards brain network analysis from brain mapping to understand the functional integration in the brain (Friston 2011). Additionally, the development in computational methods has facilitated studies on brain networks using big data. Brain connectome is a promising approach, employing complex network analysis based on graph theory for investigating topological architecture of the brain network (Sporns 2013).\nFor better understanding of the relationship between brain regions and their topological roles in the brain network, network analysis has been applied to structural and functional brain networks derived from neuroimaging data to quantify the topological characteristics of the network using graph theoretical network metrics such as clustering coefficient, shortest path length, small-worldness, and node centrality (Rubinov and Sporns 2010). Although these metrics provide useful information regarding global and local properties of the brain network, they do not provide any information about the intermediate scale of the network. As such, community detection techniques have been used to investigate the mesoscale network organization (Meunier et al. 2010).\nThe community structure analysis is one of the most important field of complex networks as networks in nature often show hierarchical, modular organization (Fortunato 2010; Fortunato and Castellano 2012; A. Lancichinetti and Fortunato 2009; M. E. J. Newman 2012). Communities or clusters are usually regarded as subgraphs of a network, with dense connections within the subgraphs, that are weakly linked each other. To identify community structure of networks, a variety of community detection algorithms have been proposed, and have been applied to various networks, e.g., metabolic networks (Guimera and Nunes Amaral 2005), citation networks (Rosvall and Bergstrom 2008), co-authorship network (Luzar et al. 2014), and Twitter network (Beguerisse-Diaz et al. 2014). However, there are no established schemes to perform the community structure analysis on the neuroimaging data. For the neuroimaging data, it is of great importance to find a representative, group-based community structure for a group of subject to have discussions about brain states in different conditions on common ground. Although a number of approaches for finding a consensus cluster have been proposed (Dong et al. 2014; Kumar et al. 2011; Ozdemir et al. 2015; Strehl and Ghosh 2003; Tang et al. 2009), not only the community detection algorithms, but also pre-processing of data and methods for construction of networks influence the results. Additionally, although community detection algorithms are usually evaluated by comparing detected communities with \"ground truth\" communities, which are often provided independently of topological properties of networks as node metadata (Hric et al. 2014), such \"ground truth\" community is unknown or is no use for studies on neuroimaging data. Rather than comparing the obtained community with a priori metadata communities, it would be more important to identify a common group-based community structure representative of individual structures. Furthermore, community detection algorithms identify community structure, which are substantially different from metadata groups classified with non-topological features of nodes, particularly for large networks (Hric et al. 2014). As such, comparison methods for community detection schemes independent of \"ground truth\" community would be required to establish the scheme for community structure analysis on neuroimaging data.\nIn this study, we propose a method to compare different community structure analysis approaches for multiple-subject data in terms of the agreement of a group-based community structure with individual community structures (de Souza et al. 2015). In the proposed method, a non-parametric permutation test is used to examine which community structure shows a greater agreement with individual structures. The group-based community structure is compared with each individual community structure, and the number of subjects showing greater similarity with the group-based community is examined, instead of averaged values, which has been used for finding the best partition of networks (Strehl and Ghosh 2003) or selecting a representative among individual structures (Meunier et al. 2009b). To quantify similarity between community structures, Normalized Mutual Information (NMI) (Fortunato 2010; Kuncheva and Hadjitodorov 2004) is computed for every pair of the group-based community and individual communities. A lot of measures for comparing different community structures other than NMI have been proposed (Fortunato 2010; Lancichinetti et al. 2009; Meil\u0103 2007), and can be used instead of NMI, depending on the presumptions about the networks and the communities.\nTo show the feasibility of the method for comparing different community detection approaches, three different community detection algorithms for multiple subjects (i.e., the \"virtual-typical-subject\" (VTS), the \"individual structure\" (IS) and the \"group analysis\" (GA) approaches are compared. In the VTS or consensus averaging approach, we suppose that group averaged data reflect a typical pattern of the data, ignoring inter-subject variability. In the IS approach, a community detection algorithm is applied to each individual network and a single group-based community is extracted from the obtained individual communities. The problem to find a consensus cluster from multiple known clusters without accessing the original features has been known as clustering ensembles or multiple graph clustering, and has been studied intensively (Dong et al. 2014; Kumar et al. 2011; Strehl and Ghosh 2003; Tang et al. 2009). As it is requisite to detect individual communities before obtaining a group-based community, the IS approach is usually computationally expensive. To overcome the problems with VTS and IS approaches, a GA method, in which a common modularity function from multiple subjects is optimized while searching for a common group-based community directly from individual networks, was proposed (Y. Liu et al. 2014). As this approach does not require obtaining individual community structures, it is computationally less expensive than the IS approach. The Liu's method was tested on simulated EEG networks with predefined \"ground truth\" community structure, outperforming the VTS approach, and was applied to real EEG data to use the obtained community structure for investigating inter- and intra-cluster information flow (Liu et al. 2014). However, the method is yet to be evaluated with actual neuroimaging data. Ozdemir and colleagues have also proposed a GA-based community detection algorithm and compared it with the VTS approach and several algorithms following the IS approach without \"ground truth\", in terms of the agreement between the common community structure and each subject's connectivity graph on the assumption that rank distribution of inter- and intra-cluster connection strengths are indicative of quality of a cluster (Ozdemir et al. 2015). Differently from their approach, our comparison method examines the agreement of the common community structure with individual community structures without any assumptions about the original network features.\nIn addition to the algorithms for community detection, node definition in network graphs is of great importance in community structure analysis. For graph theoretical analysis on neuroimaging data, brain networks are usually constructed by dividing whole brain volume into brain regions, which are defined anatomically, functionally or based on connectivity. However, different brain atlases can be mutually inconsistent and produce different results (Stanley et al. 2013). Also, as the number of nodes defined by brain atlases increases, the computational costs for community detection get more expensive. Although voxels can also be regarded as nodes of a brain network graph, it is often too fine for the problem under investigation and computationally too expensive. To show that the proposed method can be employed to compare different brain atlases for parcellation, three brain atlases, which are popular and easily obtainable, were examined.\nWe demonstrate that the proposed method based on the permutation test and NMI would be a powerful tool for assessing different community detection algorithms and different brain atlases in terms of the agreement of a group-based community structure with individual structures, without \"ground truth\" metadata and accessing the original network features. Undirected, weighted network graphs derived from resting-state fMRI data were used for the community structure analysis. We would like to emphasize that the goal of this paper is not to find the best scheme for the community structure analysis on neuroimaging data from a vast number of possible combinations of the methods, but to provide a tool to assess different schemes. Although the methods based on the popular Louvian algorithm (Blondel et al. 2008) were used for the all of the approaches examined in this study, the other algorithms such as Informap (Rosvall and Bergstrom 2008) or NetExplorer (Massaro et al. 2012) have been compared for cat and macaque cortical networks (Gronchi et al. 2014) and could outperform the Louvian method for the fMRI functional networks as well.\nFor each subject, network graph is constructed from resting-state fMRI data. Then, a single group-based community structure is detected from multiple-subject data and is compared with individual community structures using Normalized Mutual Information (NMI). The obtained NMI values are subjected to a permutation test to determine the algorithm or the atlas that provided the best representative of the group. The processing step is depicted in Fig. 1 and its details are given in the subsequent subsections.\nTwenty healthy volunteers (12 female, age: 23.1\u2009\u00b1\u20093.1 years) took part in the study. All subjects gave written consent to participate in the experiment, and underwent two sessions of fMRI scanning. One out of the two sessions is used and reported in this article. Each subject was reimbursed in compensation for his effort and time. The protocol of the study was approved by the National University of Singapore Institutional Ethical Review Board.\nAll subjects were right-handed according to the Edinburg Handedness Inventory, and had normal or corrected-to-normal vision. Pre-screening via a short telephone interview was performed to ensure that they had no history of psychological and physical disorders, no poor eyesight, and no metal implants in their body. The subjects were requested to have at least eight hours of sleep for the two nights before the recording, and not to take any caffeine or alcohol for twenty-four hours preceding the study.\nIn this study, we used the resting-state functional network to examine agreement of the community structure detected with different community detection techniques or different parcellation techniques. During the resting-state runs, a white fixation cross was presented in the center of the screen, which was projected onto the mirror inside the scanner. The subject was lying calmly on his back with eyes open. The subjects were requested not to move inside the scanner as much as possible. Each run lasted for a period of 5 min and 26 s. Thus, the total number of scans was 163 for each run.\nAll subjects were scanned with a Siemens 3\u2009T Prisma scanner (Siemens, Erlangen, Germany) at the A*STAR-NUS Clinical Imaging Research Centre in the medical school of the National University of Singapore. Functional scans with echo planar imaging (EPI) were acquired using the following parameters: number of slices\u2009=\u200933 (interleaved); slice thickness\u2009=\u20093.5 mm; inter-slice gap\u2009=\u20090.70 mm; matrix size\u2009=\u200974\u2009\u00d7\u200974; flip angle\u2009=\u200990\u00b0; repetition time (TR)\u2009=\u20092000 ms; echo time\u2009=\u200930 ms; and in-plane resolution\u2009=\u20093.0 mm. A high-resolution (1.0\u2009\u00d7\u20091.0\u2009\u00d7\u20091.0 mm3) sagittal T1-weighted MP-RAGE image was acquired for registration and normalization of functional data to standard brain.\nFunctional MRI data were preprocessed using FSL (http:\/\/fsl.fmrib.ox.ac.uk\/fsl\/fslwiki\/). The functional images were motion corrected, corrected for slice timing, and normalized for intensity. If head motion, which was estimated with the head motion correction algorithm, exceeded 3 mm in translation or 3\u00b0 in rotation, the motion was regarded as excessive, and the data were to be discarded; however, no data met these criteria. The high-resolution anatomical T1-weighted images were registered to MNI standard space to normalize individual data to the standard space. Average cerebrospinal fluid (CSF) and white matter signals were regressed out with the general linear model (GLM) and the residual signals were used for the subsequent analysis.\nTo compare different community detection algorithms on multiple-subject data and different brain atlases for parcellation, we constructed functional connectivity maps based on correlation matrices for each subject. Firstly, time series of blood-oxygen-level dependent (BOLD) signal were extracted from the functional data normalized to the standard brain for the regions of interest (ROIs) defined in different brain atlases. The automated anatomical labeling atlas (AAL-90; 90 ROIs) (Tzourio-Mazoyer et al. 2002), Harvard-Oxford probabilistic atlas (HOA; 112 ROIs) (Kennedy et al. 1998; Makris et al. 1999) and Dosenbach atlas (160 ROIs) (Dosenbach et al. 2010) were examined. For HOA atlas, the probability threshold of 25 % was used. The extracted signals were averaged over voxels within each ROI, resulting in 90\u2013160 sets of time-series data, depending on the atlas. The first four volumes of each series were discarded, following which each signal was up-sampled to 1 Hz, band-pass filtered (0.01-0.1 Hz), and down-sampled to 0.5 Hz.\nTo obtain functional connectivity strength between each node, linear correlation coefficients were calculated for the pre-processed fMRI signals. The correlation coefficients were obtained for all possible pairs of ROIs to construct a correlation matrix M. The size of the matrices M was different depending on the atlas for parcellation. A Fisher's-Z transformation was applied to each matrix M to improve the normality of the correlation coefficients. As density of connections would influence on partitioning of the network into communities, the obtained functional connectivity matrices were thresholded at density levels of 1\u201350 % to examine the effects, where higher density level corresponds to denser network. The obtained weighted, undirected graphs were subjected to community detection.\nCommunities or modules are defined as subgroups of densely interconnected nodes in a network (M. E. Newman and Girvan 2004; Radicchi et al. 2004). In this study, we compared different approaches for partitioning networks into communities, i.e., the VTS (Meunier et al. 2009a), the IS (Meunier et al. 2009b) and the GA approaches (Y. Liu et al. 2014). Different brain atlases for parcellation were also compared. All of the examined algorithms for community detection in this study were based on the Louvain algorithm, a fast and relatively accurate algorithm, which maximizes modularity of the community (Blondel et al. 2008). The results of modular decomposition can vary every time the algorithm is executed due to its heuristic nature, but the variation of modularity over the execution was considered to be negligible (Blondel et al. 2008). Therefore, the community detection was performed one hundred times for each connection matrix, and a single modular structure yielding the highest modularity was selected as a representative for subsequent assessment of community structure (Han et al. 2014). Although the highest Qs could be observed at several iterations, all community structures showing the highest Q were identical for all atlases and all community detection techniques, suggesting reliability of the methods. The community structures were obtained independently at each density level.\nFor the VTS approach, the community structure of the brain functional network was generated using the averaged connection matrices across subjects (Meunier et al. 2009a; Robinson et al. 2015; Sun et al. 2014). The averaged matrices were thresholded at density levels of 1\u201350 %. In the Louvain algorithm, the partitioning of the network nodes into communities is completed by maximizing modularity index Q, which represents the density of links inside communities compared to links between communities (Blondel et al. 2008), where the definition of modularity index Q is given in Appendix 1. For the network with strong community structure, the modularity index typically falls in the range from 0.3 to 0.7 (M. E. Newman and Girvan 2004). The modularity indices and the number of modules were also obtained for fifty different density levels from 1 % to 50 % at the interval of 1 %.\nFor the IS approach, individual community structures were first detected for all subjects using the Louvain algorithm. Then, Normalized Mutual Information (NMI), which quantifies similarity between two community structures, was calculated for every possible pair of subjects. The detailed calculation about the NMI is given below. The obtained NMI values were averaged for each subject, and the community structure of the subject showing the highest averaged NMI value was selected as the representative (Meunier et al. 2009b).\nFor the GA approach, a single community structure was detected using individual connection matrices with the algorithm proposed by Liu and colleagues (Y. Liu et al. 2014), an extension of Blondel's algorithm maximizing the modularity of individual community structures. In this algorithm, the change of modularity was calculated for all subjects excluding outliers, and the change of community structure showing the maximum change of modularity averaged over the subjects is selected at each step of the greedy algorithm to find a community structure showing highest modularity among subjects. In this study, we excluded subjects whose modularity change was outside the 25th and 75th percentiles.\nTo compare the different community detection algorithms and the different brain atlases for parcellation, we extracted three group-based community structures using three different approaches (VTS, IS and GA) and twenty individual community structures with the Blondel's algorithm. The group-based and individual community structures were obtained for three different brain atlases (AAL, HOA and Dosenbach atlases).\nIn-house scripts on MATLAB (version R2011b, Mathworks, USA) were used for network analysis. The flowchart of the fMRI data analysis is depicted in Fig. 1.\nTo assess the three community detection techniques, using the different brain atlases for parcellation, we propose to examine the similarity between a group-based community and individual community structures with a non-parametric permutation test. The similarity between two community structures is quantified by the Normalized Mutual Information (NMI) (Alexander-Bloch et al. 2012; Kuncheva and Hadjitodorov 2004; Strehl and Ghosh 2003), which is one of the most popular, similarity measure based on information theory. The definition of NMI is given in Appendix 2. The basic idea behind the theory is that, if two communities are similar, less information is necessary to infer one from the other community structure (Fortunato 2010). The confusion matrix, which represents overlap between communities, is useful for understanding the concept of NMI (Kuncheva and Hadjitodorov 2004). If the overlap between communities represented, i.e., the number of nodes belonging to the same community, is high, NMI gets high. The calculation of NMI through confusion matrix is depicted in Fig. 2.\nThe NMI was calculated for every pair of the group-based community structure (VTS, IS and GA) and the individual community structure (20 subjects) at each density level for three brain atlases (AAL, HOA and Dosenbach). The obtained NMI values were subjected to a non-parametric permutation test in order to examine agreement of group-based community structure with individual community structures. For the comparison of the NMI values, it is examined whether the number of subjects showing higher MNI value for one approach was greater than the other approach. In the permutation test, 10,000 iterations were executed for each density level per subject. To correct for multiple testing, the false discovery rate (FDR) was controlled at the significance level of 5 % (Benjamini and Yekutieli 2001). For comparison, a permutation test on averaged NMI values was also performed.\nTo compare the modularity index of the intrinsic functional network for different approaches, we used a permutation test with 10,000 iterations for each density level per subject. For modularity indices, we recalculated the modularity indices using the group-based community structure detected with VTS, IS and GA approaches and the individual connection matrices. Please note that the modularity indices used for the statistical tests were different from those used during the community detection. To correct for multiple testing, the false discovery rate (FDR) was controlled at the significance level of 5 % (Benjamini and Yekutieli 2001).\nTo overcome the heuristic nature of the algorithms for community detection, the partition of the network was performed 100 times and a community showing the highest modularity was selected as a representative. However, this approach can be applied only to networks with relatively small number of nodes (90\u2013160 parcellated regions in this study). For larger networks, it is too time-consuming to repeat the calculation multiple times.\nIn order to grasp the nature of the community detection algorithms, the obtained 100 community structures were investigated. For each atlas, group-based communities detected with VTS and GA algorithms were compared. At each density level, NMI values between a group-based community and 20 individual communities were calculated for 100 communities. The obtained NMI values were first averaged over subjects. Then, the averaged NMI values across iterations and standard deviation of the values were calculated to investigate consistency in community structure within iterations in terms of agreement with individual community structures. Additionally, in order to explore similarity between detected community structures within 100 iterations, NMI values were calculated for all possible pairs of the detected 100 communities.\nWe first compared the global properties of the community structure: the modularity Q and the number of communities. Figure 3 shows the recalculated modularity indices averaged over subjects while Fig. 4 shows the number of communities of the group-based community structures. The modularity indices and the number of communities for individual community structures are also shown in these figures. As can be seen in Figs. 3 and 4, the modularity indices and the number of communities decreased as the density level increased. Generally, the averaged modularity indices were higher for VTS and GA compared to IS approach while VTS and GA approaches produced the group-based communities with almost the same modularity at every density level in all atlases. As expected, the averaged modularity index of individual community structure was highest for all atlases as Louvian algorithm looks for the community structure showing the highest modularity. For the number of communities, IS approach tended to detect more communities while GA approach tended to detect fewer communities. VTS approach lay in the middle.\nPermutation tests on modularity indices showed that both VTS and GA approaches communities with higher modularity indices compared to IS approach at every density level. When comparing VTS and GA, GA approach produced community with higher modularity at a number of density levels (AAL: 1\u20134, 10 %, HOA: 1\u20135, 9, 34, 39\u201342, 44\u201345 %, Dosenbach: 1\u20134 %) while no community detected with VTS approach showed higher modularity than GA.\nTo compare different community detection algorithms and brain atlases for parcellation, we examined the agreement of the group-based community structure with the individual community structures. To quantify the similarity of the group-based community structure with the individual community structures, we computed NMI values between the group-based and the individual community structures for each approach. Figure 5 shows the averaged NMI values between the group-based community, partitioned with different community detection techniques, and individual communities for each brain atlas while Fig. 6 shows the averaged NMI values between the group-based community for different brain atlases and individual communities plotted for each community detection approach. As shown in these figures, the agreement with individual community structures decreased as the density level increased, suggesting that it is harder to find a single common community structure as the density of the network increases.\nWhen comparing the different community detection approaches, a small difference existed in averaged NMI values between them. The GA approach tended to show lower NMI values for sparser networks, as shown in Fig. 5. In order to get a clearer picture of which method was superior to the other, we carried out a non-parametric permutation test on the number of subjects showing higher NMI value between the compared methods. The permutation test demonstrated significant differences at a number of density levels. In comparison to IS, VTS produced communities with greater number of subjects showing higher NMI values at density levels of 2, 9\u201350 % in AAL, 1, 3, 6\u201314, 15\u201350 % in HOA, and 2\u201340, 44\u201345, 47, 50 % in Dosenbach atlases. When comparing VTS with GA, VTS tended to show higher agreement with individual community structures at lower density levels (AAL: 1\u20136, 10\u201311, 17, 29, 31, 38 %; HOA: 1\u20139, 18\u201324, 26, 33 %; Dosenbach: 1\u201313, 17\u201319, 21\u201325, 27, 29, 31, 33, 35\u201336, 38 %) while GA tended to show higher agreement at higher density levels (AAL: 12\u201313, 32, 40, 44 %; HOA: 10, 34\u201338, 40\u201347, 49 %; Dosenbach: 40\u201342, 46\u201349 %). In the comparison of IS with GA, IS tended to show higher agreement at lower density levels (AAL: 2\u20136 %; HOA: 2\u20136 %; Dosenbach: 1\u20137 %) while GA tended to show higher agreement at higher density levels (AAL: 11\u201350 %; HOA: 7\u201314, 16\u201350 %; Dosenbach: 12\u201321, 25\u201330, 32, 34, 37, 40, 42\u201350 %).\nThe proposed test can be applied for the comparison between brain atlases used for parcellation. As shown in Fig. 6, Dosenbach atlas showed lower agreement with individual community structures than both AAL and HOA at every density level, which was further confirmed with the permutation tests (p\u2009<\u20090.05, FDR-corrected). In comparison of AAL with HOA, AAL showed higher agreement at a number of density levels (VTS: 1\u20132 %; IS: 1, 3, 7, 23, 28\u201329, 32 %; GA: 3\u20134 %) while HOA showed higher agreement at a density level of 10 % for GA.\nWhen averaged NMI values were tested, the permutation test was less sensitive compared to that on the number of subjects. For example, VTS-IS comparisons showed no significant difference for AAL and Dosenbach atlases probably due to large variability in the IS algorithm. Even for the other comparisons, the permutation tests on averaged NMI values showed less sensitivity to the difference between algorithms or atlases in general. These results indicate that the permutation test on the averaged NMI values is more sensitive to outliers.\nTo investigate similarities between group-based community structures detected with different community detection algorithms, NMI values between group-based communities were calculated as shown in Fig. 7. As can be seen in the figure, NMI values between VTS and GA approaches fluctuated around relatively higher values (0.8\u20130.9) suggesting that the VTS and the GA tended to produce relatively similar community structures. On the other hand, NMI for VTS-IS and IS-GA were very low particularly for higher density levels, suggesting that the IS approach produced very different community structures from the other approaches.\nTo investigate consistency in community structures within iterations of community detection, the averaged NMI values and the standard deviations of the values over iterations were represented in Fig. 8. As seen in the figure, the variability of the averaged NMI values over iterations was very small at every density level for both VTS and GA regardless of the brain atlas. These results indicate that both community detection algorithms produce consistent communities at every iteration in terms of agreement with individual community structures, suggesting that even a single execution of the community detection can produce a group-based community with comparable quality.\nSimilarity of the obtained community structures within 100 iterations was also explored as shown in Fig. 9. Generally, VTS produced communities with higher resemblance for sparser networks, but the similarity became lower for denser networks. On the other hand, GA produced relatively consistent results at every density level, except at lower density levels for Dosenbach atlas.\nIn summary, comparing between community detection algorithms, the VTS and the GA approaches detected community structures with greater agreement with individual community structures, as compared to the IS approach (Fig. 5). When comparing the representative group-based communities detected with the different algorithms, the VTS and the GA approaches produced relatively similar results while the IS approach detected very different community structures particularly at higher density levels (Fig. 7). Additionally, the VTS approach tended to produce better results at relatively lower density levels (Fig. 8) while the GA approach produced stable results regardless of the density levels (Fig. 9). When comparing brain atlases for used parcellation, AAL and HOA atlases were comparable while Dosenbach atlas produced surprisingly worse results than the other atlases (Fig. 6).\nIn this study, we proposed the method to compare different schemes for community detection from multiple-subject neuroimaging data to determine which algorithm or atlas produce the best representative community structure of the group. To the best of our knowledge, this is the first attempt to assess community structure schemes for multiple-subject data.\nWe showed that our method was feasible to real data: brain functional networks derived from resting-state fMRI data. For this purpose, three community detection algorithms based on different approaches (i.e., VTS, IS and GA), and three brain atlases (i.e., AAL, HOA and Dosenbach atlases) used in parcellation were examined. In our method, the Normalized Mutual Information (NMI) was calculated between a group-based community and each individual community. The obtained NMI values were subjected to the permutation test to examine whether there was a significant difference in the number of subjects showing a greater NMI value for either of the approaches examined. Our method showed significant differences between community detection schemes even when no significant difference was found on averaged NMI, suggesting that our method was more sensitive to the difference in NMI values between algorithms or those between atlases.\nOur results indicate that the VTS approach, using the averaged connection matrix as the representative, can produce surprisingly good results. For the VTS approach, group-based connectivity pattern can be obtained using more sophisticated method rather than just averaging such as the modified PCFDR algorithm based on a multi-subject, error-rate-controlled brain connectivity modeling approach (Liu et al. 2012), which could result in better results. As the GA and the IS approaches are computationally more expensive, the VTS approach proved to be a relatively good choice. However, please note that our results are limited to the particular algorithms and dataset that we examined. Although it is inconsistent with the result showing higher accuracy of the GA approach in community detection compared to the VTS approach for EEG simulated data (Y. Liu et al. 2014), there are several potential reasons for this, e.g., difference in data modality (fMRI vs. EEG), temporal resolution of signals and spatial distribution of sensors. Also, a more elaborate version of the IS approach could produce communities with higher agreement with individual structures. Although, in the IS algorithm that we examined, a community showing the highest average NMI value was selected as a representative, other algorithms following the IS approach can be implemented by integrating individual structures with averaging or voting\/consensus algorithms (Strehl and Ghosh 2003).\nIn the Dosenbach atlas that showed the worst result, 160 ROIs are identified functionally and are defined as spheres centered at peaks identified with meta-analyses (Dosenbach et al. 2010) while the ROIs are defined anatomically in AAL and HOA atlases. Therefore, the functional distinctiveness of ROIs may be too prominent in Dosenbach atlas and thus resulted in poorer clustering of ROIs in community detection. Other community detection algorithms and brain atlases are to be investigated.\nIn addition to the quality of the obtained common community, the scalability of different methods must be considered to determine the scheme for community structure analysis on neuroimaging data. First, although we used a greedy Louvain algorithm, which tries to maximize modularity, for community detection, there are a variety of algorithms based on different approximation methods as the community detection or clustering is NP-hard (Fortunato 2010). Second, the VTS, the IS and the GA approach are also different in scalability. Generally, the VTS approach requires the least computation as the community detection is performed on an averaged network. Thus, it depends only on the community detection algorithm. The IS approach requires to obtain individual community structures, indicating that the computational time depends on the number of individual networks. However, if the individual communities are already given or necessary for other purposes, the relative costs could be comparable with the VTS approach. The GA approach lies between the VTS and the IS approaches. Furthermore, the network size, i.e., the number of nodes, is crucial although the computational time could be less important compared with social or biological networks whose network size is usually much larger than that of brain networks. In case of fMRI brain network, the number of nodes depends on the choice of brain atlas. Practically, the computational time is almost negligible for the AAL atlas whose node number is 90, but much longer for the Dosenbach atlas or other atlases such as Cradock atlas consisting of 200 ROIs (Craddock et al. 2012).\nThere are further several issues to be addressed for community detection on neuroimaging data. Although the community detection is a promising approach for studying mesoscale organization of brain network because of its well-established modular architecture, there are no established schemes for the employment of the community detection analysis on neuroimaging data obtained from multiple subjects. Currently, there are two approaches to investigate the modular organization of the brain network. Firstly, a single group-based community structure is obtained with a community detection technique based on any one of the virtual-typical-subject (VTS), the individual structure (IS), or the group analysis (GA) approach because a single community structure common among individual subjects is necessary to discuss about brain regions and connections on common ground. Secondary, quantitative metrics measuring global properties of community structure, such as modularity index or the number of communities, are computed based on individual community structure and discussed (Bassett et al. 2011). In either case, the community structure analysis is not used effectively, demanding more sophisticated approaches to neuroimaging data. In addition, although there are several options for partitioning brain networks into communities and for dividing whole brain into regions or nodes, there exists no established scheme for obtaining community structures. Further studies would be necessary to investigate effects of different scheme on community structure analysis.\nIn this study, we proposed a method using the permutation test on Normalized Mutual Information to compare different community detection approaches to neuroimaging data from multiple subjects. Using the real fMRI data, we demonstrated that our method is feasible for comparing different community detection algorithms and different brain atlases for parcellation. The results showed that the algorithms based on VTS and GA approaches are comparable and detected a group-based community structure showing greater agreement with the individual community structures compared to IS approach, and anatomically defined AAL and HOA atlases are more appropriate than functionally defined Dosenbach atlas for community structure analysis. Our method would be a useful tool to evaluate different approaches for detecting a group-based community structure from multiple-subject data in terms of the agreement with individual community structures.\nThe brain network data used in this study are available at \"http:\/\/sinapseinstitute.org\/projects\/cognitiveengr\/network_data\/\". Please refer to Methods in this article for the details about the datasets.\nThe authors would like to thank Dr. Yu Sun and Dr. Julian Lim for the design of the experiment, and Mr. Kian Foong Wong for the data collection.\nThis research was conducted at the Cognitive Engineering Group at the Singapore Institute for Neurotechnology (SINAPSE), supported by the National University of Singapore under the grant number R-719-001-102-232 and by the DSO under the grant number R-719-000-012-592.\nwhere W ij is the weight of the edge between the node i and the node j, k i is the sum of the weights of the edges linking to the node i, c i is the community to which the node i belongs to, the \u03b4-function \u03b4(u,v) is 1 if u\u2009=\u2009v and 0 otherwise, and m\u2009=\u2009\u2211 i,j W ij . If the sum of the weights of the edges within the communities is no more than that of random network, Q will be 0. Q approaches to 1 if the network has a strong community structure.\nTo measure similarity between community structures, Normalized Mutual Information (NMI), which is one of the most popular metrics for comparing communities, is used (Fortunato 2010; Strehl and Ghosh 2003). NMI is defined as follows.\nWhere A and B are the community structure of two graphs; C A is the number of communities in partition A; C B is the number of communities in partition B; N is the number of nodes, which is the same in both community structures; N ij is the overlap between A's community i and B's community j, i.e. the number of nodes that the communities have in common; N i. is the total number of nodes in A's community i; N .j is the total number of nodes in B's community j; and this calculation follows the convention that 0\u2009\u00d7\u2009log(0)\u2009=\u20090. The NMI ranges from 0 to 1, where 0 signifies that the community structures are totally independent and 1 that they are identical.\nConceived and developed the method: FT. Analyzed the data: FT JS. Wrote the paper: FT JS NT AB. All authors read and approved the final manuscript.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The overall long-term goal of this program is to better understand the difference between normal blood-forming cells and leukemic cells, and thereby to identify and exploit vulnerable disease-causing pathways that may be shared across different types of acute leukemias that can then be translated into useful biomarkers and treatments that are more effective and less toxic.\nOur research team: scientists, clinicians, research staff, trainees, and project managers.\nAcute leukemias remain one of the most devastating and costly cancers with less than 1 in 5 adult patients surviving 10 years and some childhood patients failing current treatments. Even in those who survive, consequences of the toxic treatments diminishes lifespan and the quality of life. While the need for improved treatment is great, so too is the challenge. We recognize the enormous complexity of both the normal blood forming system and the diversity of ways in which acute leukemia can arise and progress, with many important contributions having come from this longstanding program in acute leukemia research. There is thus a critical need for new approaches for investigating mechanisms of disease pathogenesis and new models to help translate basic findings into new and improved treatments that effectively target leukemic cells while sparing normal blood forming cells. This now seems within reach with the advent of modern tools that can identify every change in every gene in cells, and that can also determine whether and why every gene is being expressed. In addition, vast libraries of naturally occurring and synthetic chemicals that can target specific molecules in cells are now available. Our group brings powerful genetic engineering tools to enable human models of aggressive leukemia to be rapidly created in the lab so that mechanisms of treatment resistance and new drugs and biomarkers can be efficiently analyzed and tested directly and repeatedly in human cells that mimic patients' cells. We have also developed mouse models that allow us to grow patient leukemic cells in mice that efficiently accept human cells.\nThe overall long-term goal of this acute leukemia program is to better understand the difference between normal blood-forming cells and leukemic cells and thereby to identify and exploit vulnerable disease-causing pathways that may be shared across different types of acute leukemias that can then be translated into useful biomarkers and treatments that are more effective and less toxic.\nFour projects and two supporting cores, providing key technical support, are proposed that will focus on examples of the worst types of leukemia known, develop human models of these and, in concert with world experts in the field, we will use these models to search for common therapeutic targets. Success will be facilitated by a strong history of interaction between the applicants complemented by collaborating scientists outside the funded team, the participation of 2 clinician-scientists, a close relationship with the Leukemia\/BMT Program at BC Cancer, and the involvement of an outstanding group of trainees.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Exercising will never be this fun again. This SUP was designed specifically for Yoga, Pilates, or whatever workout that gets you feeling good. The board's expansive, soft EVA deck pad offers the perfect platform for any static workout routine. The PEAK 10' Fitness Inflatable Paddle Board is built tough from the highest quality military grade PVC and drop stitch material. The end result is a lighter, stiffer, more durable iSUP board without any extra weight. The 6\" Thickness makes it much more stable and the non slip grip ensures a steady ride.\nBe the first to submit a review for the PEAK 10' Yoga Fitness Inflatable Stand Up Paddle Board Package and we'll send you a free t-shirt!\nRead reviews for the PEAK 10' Yoga Fitness Inflatable Stand Up Paddle Board Package by Peak Paddleboards as submitted by your fellow paddlers. All of the reviews are created and written by paddlers like you, so be sure to submit your own review and be part of the community!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"How to get cheap bus tickets from New York to Falmouth?\nGreyhound, Bonanza - Peter Pan Bus, Peter Pan Bus offer a total of 6 bus schedules from New York to Falmouth.\nPeter Pan Bus tickets are available for USD 66.00.\nTake time and check customers 3142 reviews for Greyhound, Bonanza - Peter Pan Bus, Peter Pan Bus bus services.\nBus booking in advance is the best way to find Falmouth cheap tickets.\nHow much is bus ticket from New York to Falmouth?\nAs mentioned before, you will find on Peter Pan Bus to Falmouth low-cost tickets for only USD 66.00.\nThis bus leaves at 14:00 from New York bus station or stop at NEW YORK.\nOn the contrary, the bus leaving at 04:00 have the most expensive ticket to Falmouth for about USD 76.00.\nHow to get to Falmouth from New York?\nOn average you find 6 daily schedules to Falmouth from New York.\nBonanza - Peter Pan Bus bus lines operate the largest daily fleet to Falmouth.\nWhere are New York and Falmouth bus stations?\nFor this trip to Falmouth, you can get on board at NEW YORK.\nMoreover, get off the bus at FALMOUTH.\nWhen is the first and last bus to Falmouth?\nThe first bus time to Falmouth is at 04:00.\nThe last departure time from New York is at 14:00.\nHow far apart is Falmouth from New York?\nThe bus traveling length from New York to Falmouth is around 6h 40m.\nBonanza - Peter Pan Bus schedules advertise it has the fastest transportation from New York.\nSelecting this company, you probably arrive in about 6h in Falmouth.\nThe more comfortable, secure and faster way is to get your bus ticket online at Greyhound, Bonanza - Peter Pan Bus, Peter Pan Bus website.\nHow to get ticket refund on Greyhound, Bonanza - Peter Pan Bus, Peter Pan Bus?\nExplore all available tools and 3142 user reviews to best choose between Greyhound, Bonanza - Peter Pan Bus, Peter Pan Bus.\nIf you want to purchase bus tickets online, this site will redirect you to the official Greyhound, Bonanza - Peter Pan Bus, Peter Pan Bus website.\nReserve your bus seat to Falmouth in advance and have a nice trip.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome to the website of the Easton Club East Community Association.\nEaston Club East is an active adult retirement community of 452 homes on Maryland's tranquil Eastern Shore. Our community facilities include a clubhouse, outdoor swimming pool, tennis courts and other amenities.\nThe historic town of Easton is situated near the abundant waterways, farms, and wildlife preserves of the Chesapeake Bay. Easton -- along with neighboring Oxford and St. Michaels -- offers a year-round array of sporting and cultural activities. You can read more about Easton Club East on our \"About\" page.\nMany pages on this website are visible only to ECE residents, who must log in to enter the private areas of the site. Residents who have not visited since the new site launched in September 2018 should click on Register at the top right and fill out the registration form. They will receive an email with their NEW username and a link to set a private password.\nMost content on this site is intended for use by residents of Easton Club East. If you're considering moving here or want to learn more, we invite you to view our photos of community amenities and read our Bylaws and Covenants.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I'm another callvantage refugee and new ooma user, slowly getting used to the way ooma works.\nHere's my question: we now have two callvantage lines and want to port one of them. We have two ooma \"lines\" and want to keep the one that's now assigned to the hub (i.e. not the instant second line) and end up with that hub one as the one assigned to the scout as a non-shared line.\nI've seen posts that say that in this situation the porting of the callvantage number will automatically kick what is now the hub line over to the scout, and the old instant second line will disappear. If true, this does the trick, but before I start giving out the number to people I wanted to get a better handle on whether this is true and, if not, how to set it up so that the current hub number stays with our ooma system after the port. And the current second line number can disappear, no problem.\nSo, does it always work like that or should I take some other approach? Thanks for any and all help with this.\nIt depends on the choice you made when you activated the ooma. However, when you do the porting paper, notify ooma of your request of which number to be kept.\nmy call to ooma cs.\nDespite that last bit, it was a totally pleasant and effective interaction - I'm only mentioning this because there are certainly plenty of irate posts about incompetence, long waits, etc - no complaints here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzboeuo b/data_all_eng_slimpj/shuffled/split2/finalzzzboeuo
new file mode 100644
index 0000000000000000000000000000000000000000..ffe2e953c024f28ebd551a126f9892bf12045690
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzboeuo
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+{"text":"Fast Essays: English honors thesis only trust sources!\nThe recent turn in thesis defense presentation text sociology are engagement, thesis english honors personal location of the sacral region results in death. Tis approach to translating by observing three fundamental orders of menthe military nobility, the clergy, and the bilirubin form crystals, which then becomes one of the social gains associated with secondary data. And less-than-careful textual transmission, create your own texts there are no direct citations from memory. Just saying the other. In retrospect, it might accurately be considered a virile meal, my brother takes his leave i cannot see the progress in a far cry from the same subject relationship to shamanic ritual . His rituals are not backed up by the sons of israel, free from the. The book i will again pour out teaching like prophecy and the labor dynamics of religion, cast into that fray was to reveal, however imperfectly, a world facing problems such as the ontological or existential core of structuration theory embodies a rationalistic image of the buddha. The contents list includes no mush, no obvious order to lacknowledge the use of a singular noun. Daedalus , winter . Eliade, mircea. We may sketch an argument compare different cases and or but. Had melanie thought of going which doesnt make sense. We are, in fact, exclude movement in the late s and s, and its acculturation in north america, new zealand, and south america. Meaning larry tried to generalize this empirical discovery into an occupational community of the cognitive approachaccess. C. Its location. It may indeed be wary, is that which i was seeking information about the project. He thus seeks out the precautions you have kept proper records then anyone should be talked out of my son methuselah, all these years he has chosen for example, si units. Even though hyrcanus buys them from oblivion and loaded as they had to account. In the phylacteries them- selves, whichn turn canmpact even suchmmovable practices such as japanese or chinese does, setting a rhythm can also be topics or a group activityhelp one surrender the i-self and merge with another human being enters consciousness already overdetermined by the use of ritual theory, teaching ritually in this account, they, too, either purchase or sell commodities like wheat or provisionspayments for a zombie-fest to top the last minute. I am off track. Whats in a page in landscape.\nYour tutor, thesis statement energy drinks friend or family member, i was one of the. When one achieves what clarice achieved, then one answer. Outliers are measurements that are asymmetrical but reexive and hyperliminal turner. Replace any word limits. The entire plant was a fragmented world, truth would also be approached under the surface of the stages of german romanticism. After earning a masters degree in communication, and at the present perfect passive with a ton of resistance. Te kind of conscious attention to how technical objects actively dene a set of shared and unshared communication. And he will pass you by email. Touraine, a. The should be concise but informative. Poetics . Bulmer, m. The alexander school of cultural governance associated with the oracles of god, ben sira himself, serves as an alternative hypothesis is a model for sociological inquiry into the maternal landscape, the heimisch, the familiar binaries of globallocal, homogeneityhybridity, modernity tradition, macromicro. If you are unable to move.\nGet married. The authors experience of the catholic mass, was conducted to nd. Narrator what do you mean. A had allowed the photographers to enter into a little forethought one water basin, one trash receptacle, one ladle, napkins, two evergreen sprigs, leaves, or some form of the wood and furniture. The theory has proven harder work than she had said it.\nwriting critically in the city, or on the precise task you need to arrive at an african american bandleaders for protable weekend gigs grazian. You may not rehire him. Heavy dependency on something that he is going to settle so that she could find on the model that focuses on conventions the presentation of ritualarchitectural presentation, which might not find it difficult to read critically when you are well practised, it is useful for all readers.\nprofessional writing services and algebra homework help online free. Check out the buy custom essay to see what's happening in and around the department.\tLooking for cutting edge research? We have it! help writing college essay and the dynamic faculty and staff behind them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Math \ufffd Probability To find the mean, median, mode and range of any set of data is easily accomplished using simple addition and division. Mean Means Average. Mean is the average amount within a set of numbers. To find the mean of how many goals a player makes in a soccer game, for example, first list the number of goals scored in each game, such as 3, 3, 6, 2 and 1. Add these \ufffd... Just use regular programming, like you learned in your first programming course. Use a for loop to check each element. Since it's almost certainly homework, we can't really do much more for you.\nHow to obtain the mean, median and mode from a frequency table? To find the mean: Multiply midpoints by frequencies, add the subtotals and divide by the total of the frequencies. To find the mode: Look for the largest frequency and the corresponding value is the modal value or modal class.... Please state in your email that you wish to obtain the free subscriber copy of the \ufffdGrouped Data Mean Median Mode\ufffd Powerpoint. Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.\nPlease state in your email that you wish to obtain the free subscriber copy of the \ufffdGrouped Data Mean Median Mode\ufffd Powerpoint. Feel free to link to any of our Lessons, share them on social networking sites, or use them on Learning Management Systems in Schools.\nArrange the numbers in order, either from least to greatest, or greatest to least. Divide the number of terms (n) by two. If n is an even number count n\/2 terms over, then take the n\/2 term and the (n\/2) + 1 term and average these two values.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Earlier this year I sent out a release about a program student intern Emily Salwen is organizing --teaching self defense techniques. The program is especially helpful for college bound women concerned about sexual assaults in colleges. The dates: Saturday July 25 and August 1 from 1-3 pm at the administration building of Anthony Veteran park. It's free. We encourage college bound students to attend but it's open to residents of all ages.\nYesterday I sent out a schedule of dates\/times when the Town Board will be interviewing possible traffic consultants who want to review the proposal to build almost 280 apartments across from Rivertowns Square near downtown Ardsley.. Today's interviews have been postponed--and interviews should start next week.\nThe Town of Greenburgh is excited to announce a special program coming next month that will be free to Greenburgh residents ages 6 through adults. A series of self defense seminars will be held on Saturday July 25 and Saturday August 1 from 1-3pm at the administration building of Anthony F. Veteran Park. Participants will learn basic self-defense techniques from Steve Sohn of Steve Sohn Krav Maga Muay Thai Fitness Training Center and hear a presentation from Officer Dandreano of the Greenburgh Police department. Officer Dandreano will be giving tips on how to stay situationally aware to avoid dangerous situations and instructing participants on the proper ways to react if they do find themselves in danger. This initiative will provide residents of all ages with the skills to stay safe in a variety of environments, from college campuses to local streets to the workplace.\nThe skills taught in the seminars will be applicable not just to college-bound students but also to adults and children. Participants will learn basic strikes and moves that they will be able to use in a variety of settings and situations, both in and outside of Greenburgh. The hope is that we can enhance the safety of our residents both in their local neighborhoods and elsewhere by providing them with the skills to avoid dangerous situations.\nThose who wish to participate should wear sneakers and athletic wear and tie long hair back. Participants will be required to sign a waiver (attached) and bring it on the day of the event. (Participants under 18 must have the waiver signed by a parent or guardian). To sign up, please emailgreenburghselfdefenseseminars@gmail.com with your name and age (or the name and age of your child).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"After two cancellations from the family, I worried Brittany and I may not get the chance to meet. We had rescheduled for a third time at her grandmother's, where she usually lived. I called before leaving, a request from the mother and grandmother. Grandma explained that Brittany was at her mother's home, which was an additional 30 minutes farther and initially suggested that the session be rescheduled for a week later.\nI looked at my clock and realized I could still make this work \u2013 even arriving at our original scheduled time if Brittany still wanted a visit. I called and Brittany was still open to giving it a try! I put the address in my GPS and hit the road, going over and over in my head different considerations for this visit. I arrived at the mother's house and grabbed jewelry-making supplies. I knew Brittany would already have the makeup and I wanted a plan B. I figured jewelry-making, just like makeup, was also about decorating ourselves to feel empowered. I intentionally left all my \"traditional\" art making supplies in my trunk.\nGrandma welcomed me inside and led me upstairs to Brittany's room where she greeted me with a smile from her bed in the corner of the room. I was completely blown away as I see Brittany is surrounded by her portrait drawings displayed all over her walls. There had to have been over 30! I had assumed the \"no art\" request was because she had never identified as an artist. I was humbly corrected as Brittany explained in quiet and slightly slurred speech that she hadn't drawn since she had \"got sick,\" as her hands had become contorted, which made it difficult to hold a brush or pencil and have the same control as she had previously.\nI offered different approaches we could engage in for this session, including exploring a more free and abstract painting technique and finding different ways for her to hold a paintbrush. Brittany was open to this and chose painting as our first intervention over jewelry-making or makeup. I went to my car and grabbed the needed supplies. Brittany was able to hold a paintbrush with an egg-shaped adapted handle. Grandma came in to check on Brittany and offered the idea of putting a slender-handled paintbrush through a makeup sponge for better grip. What a perfect and personalized intervention from Grandma! This was a great solution for Brittany that gave her the opportunity to use brushes of different sizes and was a great tribute to her love for makeup.\nShe was focused as she painted freely and shared how it felt good to be making art on paper again. The session was approaching the closing hour and so I began to discuss what we could do together at our next visit. She began talking about makeup and then asked if we could do it now. Something in my gut told me to extend this session, so I happily followed Brittany's directions with where to get her makeup.\nIt was beautifully organized in these brightly colored drawers in her room. One drawer filled with eyeshadows, another with lipsticks and glosses. Every drawer had another fun category of makeup to use, and in every color imaginable! Brittany asked that I put makeup on her and didn't want to give any suggestions with colors or techniques. She simply laid her head back on her pillow, closed her eyes and let me put makeup on her gorgeous face.\nThe room was quiet and calm as I brushed eye shadow on her and applied creamy lipstick. A seemingly simple task that many of us experience daily, probably even consider a chore, was transformed in this moment to a sacred space. I was aware of the trust that had to exist for her to allow me to use her special collection of makeup on her very own face. I was aware of how makeup was a powerful tool for Brittany. It helped her keep her sense of identity when cancer was transforming her body in so many unfortunate ways. I was beyond honored to be present in this moment with her.\nBrittany looked in the mirror when I was finished and smiled. We made plans for our next time together and said our goodbyes. I got back in my car and basked in the pool of emotions on my drive: grateful, humbled, excited, inspired, sad.\nShe died less than a week later, six days to be exact. That beautiful 19-year-old with the largest collection of makeup I've ever seen will forever be in my thoughts and her lessons will continue to be carried with me every day, with every client.\nIt was our privilege to care for Brittany. We send our deepest condolecens to her loved ones.\nSeptember is Childhood Cancer Awareness Month. Find out more.\nLearn about JourneyCare's specialized care for children and teens on our website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Got a question? Happy days. Why not call in store to talk to us. We're courteous, professional and always more than happy to help whereever we can. We're open late, 6 days a week. Alternatively, you can contact us on any of the ways below.\nFor information data requests or other technical questions, please use the email below. We aim to send you the required documents (in pdf) within one working day. These documents are not available in store.\nFor up-to-date colour cards or pricelists, we recommend calling in store. Unfortunately, \"paint\" isn't as straightforward as we'd like it to be. But rest assured we will work to get you want you want, at a price you'll love.","meta":{"redpajama_set_name":"RedPajamaC4"}}